text stringlengths 4 2.78M |
|---|
---
abstract: 'We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.'
address:
- '$^1$Ufa Institute of Mathematics, Russian Academy of Science, 112, Chernyshevsky Street, Ufa 450008, Russian Federation'
- '$^2$Bashkir State University, 32 Validy Street, Ufa 450076 , Russian Federation'
author:
- 'I T Habibullin$^{1,2}$, A R Khakimova$^{2}$ and M N Poptsova$^1$'
title: On a method for constructing the Lax pairs for nonlinear integrable equations
---
Introduction
============
In the present article we suggest an algorithm for constructing the Lax pairs for nonlinear integrable models. Our scheme is based on the symmetry approach. Let us explain the algorithm with the evolutionary type PDE of the form $$\begin{aligned}
u_t =f(x,t,u,u_1,u_2,...,u_k), \qquad u_j=\frac{\partial^{j} u}{\partial x^j}. \label{I1}\end{aligned}$$ Recall that equation $$\begin{aligned}
u_{\tau} =g(x,t,u,u_1,u_2,...,u_m) \label{I2}\end{aligned}$$ is called a symmetry for the equation (\[I1\]) if function $ g $ satisfies the following differential equation $$\begin{aligned}
\left(D_t - \frac{\partial f}{\partial u} - \frac{\partial f}{\partial u_1} D_x - ... -\frac{\partial f}{\partial u_k} D_x^k\right) g = 0 \label{I3}\end{aligned}$$ where $D_t$ and $D_x$ are the operators of the total derivatives with respect to $t$ and $x$ correspondingly. Note that equation (\[I3\]) is overdetermined and for any fixed value of $ m $ one can effectively find all of its solutions of the form $g=g(x,t,u,u_1,u_2,...,u_m)$ (see [@IbragimovShabat]).
An ordinary differential equation $$\begin{aligned}
u_m =G(x,t,u,u_1,u_2,...,u_{m-1}) \label{I4}\end{aligned}$$ defines an invariant manifold for (\[I1\]) if the following condition is satisfied $$\begin{aligned}
D_t G - D^m_x f|_{(1.1),\,(1.4)} = 0. \label{I5}\end{aligned}$$ Here $ D_t G $ is evaluated by means of the equation (\[I1\]) and all the $x$-derivatives of the order greater than $m-1$ are expressed due to the equation (\[I4\]) and its differential consequences. Equation (\[I5\]) generates a PDE with unknown $ G $ admitting a large class of solutions. However it is a hard problem to find these solutions since the equation is not overdetermined. Some of the invariant manifolds to (\[I1\]) can be found from the stationary part of the symmetry (\[I2\]) by taking $ g(x,t,u,u_1,...,u_m) = 0. $
We now concentrate on the linearization of the equation (\[I1\]) around its arbitrary solution $ u = u(x,t): $ $$\begin{aligned}
v_t - \frac{\partial f}{\partial u} v - \frac{\partial f}{\partial u_1} v_1 - \dots-\frac{\partial f}{\partial u_k} v_k = 0 . \label{I6}\end{aligned}$$ Actually (\[I6\]) defines a family of differential equations, depending on $u$. The important fact that $u$ ranges the whole set of the solutions to (\[I1\]) is formalized in the following way. We assume that in addition to the natural independent dynamical variables $ v, v_1, v_2, ... $ the variables $u, u_1, u_2, ... $ are also considered as independent ones.
Let us find the invariant manifold of the form $$\begin{aligned}
v_m = \sum^{m-1}_{j=0} a(j) v_j , \quad v= v_0, \label{I7}\end{aligned}$$ to the equation (\[I6\]). This means that the condition $$\begin{aligned}
D_t \left(\sum^{m-1}_{j=0} a(j) v_j\right) - D^m_x \left(\sum^{k}_{j=0} \frac{\partial f}{\partial u} v_j\right)|_{(1.1),\, (1.6),\, (1.7)} = 0 \label{I8}\end{aligned}$$ holds identically for all values of $ u, u_1, u_2, \dots . $ Here we assume that $ \forall j$ function $a(j) $ depends on $ x, t $ and a finite number of the dynamical variables $ u, u_1, u_2, ... $. In (\[I8\]) the variables $ u_t, u_{xt}, u_{xxt}, ... $ are expressed by means of the equation (\[I1\]) and the variables $v_t$, $v_{1,t}$ ,..., $v_{m-1,t}$ are exresssed due to (\[I6\]). Equation (\[I8\]) splits down into a system of $m$ partial differential equations with the unknown functions $a(0),a(1),\dots, a(m-1)$, coefficients of the decomposition (\[I7\]). Due to the presence of the additional dynamical variables $u,u_1,u_2,...$ the system is overdetermined and hence can effectively be investigated. Actually, equation (\[I7\]) defines a bundle of the manifolds depending on infinite set of the variables $u, u_1, u_2, ... $. Suppose that such an invariant manifold is found. Then we can interpret a pair of the equations (\[I6\]), (\[I7\]) as the Lax pair to the equation (\[I1\]). In the examples the order $k$ of the equation (\[I1\]) and the order $m$ of the equation (\[I7\]) coincide.
As an illustrative example we consider the well known KdV equation $$\label{s1kdv}
u_{t}=u_3+uu_1.$$ Linearized equation $$\label{s1KdV:lin}
v_t = v_{3} + u v_1 + u_1 v$$ obtained by virtue of the rule (\[I6\]) admits the third order invariant manifold defined by the equation $$\label{s1KdV:L}
v_{3} = \frac{u_{2}}{u_1} v_{2} - \left( \frac{2}{3} u + \lambda\right) v_1 + \left( \left( \frac{2}{3} u + \lambda \right)\frac{u_{2}}{u_1} - u_1 \right)v.$$ It is easily checked that equations (\[s1KdV:lin\]) and (\[s1KdV:L\]) are consistent if and only if the function $u=u(x,t)$ satisfies the equation (\[s1kdv\]). Therefore these equations constitute the Lax pair to the KdV equation. It differs from the usual one found earlier in [@Lax68]. Stress that there is no any second order invariant manifold of the form $v_{2}=a(u,u_1,u_2)v_{1}+b(u,u_1,u_2)v$ for the equation (\[s1KdV:lin\]). But there is a first order one $v_1=\frac{u_2}{u_1}v$ which however does not contain $\lambda$ and therefore does not generate any true Lax pair.
Recursion operator is an important attribute of the integrability theory. It gives a compact description for the hierarchies of both symmetries and conservation laws. Various methods for studying the recursion operators can be found in literature (see, for instance [@GursesKarasu]-[@mikhailov] and the references therein).
We observe that invariant manifold for the linearized equation is closely connected with the recursion operator for the original equation. Indeed examples in §6 show that equation defining the invariant manifold, which provides the second operator of the Lax pair, can be rewritten as a formal eigenvalue problem of the form $$Rv=\lambda v$$ for the recursion operator $R$. For instance, equation (\[s1KdV:L\]) is easily rewritten as (see §6) $$(D_x^2+\frac{2}{3}u+\frac{1}{3}u_xD_x^{-1})v=\lambda v$$ where the operator at the l.h.s. is nothing else but the recursion operator for the KdV hierarchy.
Therefore our scheme of constructing the Lax pairs provides an alternative tool for searching the recursion operator. On the other hand side when the recursion operator is known the invariant manifold and hence the Lax pair can be found by simple manipulations (see §6). At this point we have an intersection with the pioneering articles [@IbragimovShabat], where nonstandard Lax pairs are given for some of the KdV type equations in terms of the recursion operators. However these Lax pairs differ from those found within our scheme since they are nonlocal and do not contain any spectral parameter.
There is a great variety of approaches for searching the Lax pairs from the Zakharov-Shabat dressing [@ZakharovShabat74; @ZakharovShabat79] and prolongation structures by Wahlquist and Estabrook [@WahlquistEstabrook] to 3D consistency approach developed in [@NijhoffWalker]-[@Nijhoff]. We mention also approaches proposed in [@Yamilov82; @Xenitidis; @IbragimovShabat]. An advantage of our scheme is that it can be applied to any integrable model (at least in 1+1 -dimensional case), the first of the operators is easily found and the second is effectively computed. They allow finding the conservation laws, higher symmetries and invariant surfaces for the corresponding nonlinear equation. The found Lax pairs are more complicated since they are based on the differential operators of the orders greater than usual ones. This is their disadvantage. The question remains open whether the Lax pairs of this kind allow to find any new solution for the well studied models.
In the case of the hyperbolic type integrable equations the algorithm should be slightly modified. Let us explain it with the example of the sine-Gordon equation $$\label{s1.sin}
u_{xy}=\sin u.$$ Let us find the simplest but nontrivial, i.e. depending on a parameter, invariant manifold for the linearized equation $$\label{s1.lin-sin}
v_{xy}=(\cos u) v.$$ It is the three dimensional surface in the space of the dynamical variables $v,v_x,v_y,v_{xx},v_{yy},...$ defined by the following two linear equations, with the coefficients, depending on the field variable $u(x,t)$ and its derivatives $$\label{s1.sin:inv1}
v_{yy} - u_y(\cot u) v_y + \lambda \frac{u_y}{\sin u} v_x - \lambda v = 0,$$ $$\label{s1.sin:inv2}
v_{xx} - u_x(\cot u) v_x + \lambda^{-1} \frac{u_x}{\sin u} v_y - \lambda^{-1} v = 0.$$ Here $\lambda$ is a complex parameter. Note that equations (\[s1.sin:inv1\]), (\[s1.sin:inv2\]) are not independent. One of them is immediately found from the other by differentiation by means of the equations (\[s1.sin\]), (\[s1.lin-sin\]). A triple of the equations (\[s1.lin-sin\])-(\[s1.sin:inv2\]) can be rewritten (see the end of §3) as a pair of the systems of ordinary differential equations providing the Lax pair realized in $3\times3$ matrices. The method for deriving the Lax pair from the triple (\[s1.lin-sin\])-(\[s1.sin:inv2\]) is based on constructing the infinite Laplace cascade for the linearized equation (\[s1.lin-sin\]) and obtaining the finite reduction of the cascade.
Let us give a brief comment on the structure of the article. In §§2,3 we discuss the well-known Laplace cascade for linear and nonlinear hyperbolic type equations. In §3 also the problem of finding finite reductions of the infinite Laplace sequence is studied. The Lax pair to the sine-Gordon equation is derived from the Laplace cascade. In §4 the definition of the invariant manifold for the hyperbolic type equations is recalled. The Lax pair is construted via invariant manifolds for hyperbolic equation (\[eq12:main\]) found in [@SokolovMeshkov]. In §5 the Lax pairs are constructed by evaluating invariant manifolds for the evolutionary type integrable equations. Beside the explanatory examples here we consider two equations (\[eq12:sym1\]) and (\[eq12:sym2\]) found in [@SvinolupovSokolov] as equations possessing infinite hierarchies of conserved quantities. To the best of our knowledge the Lax pairs for the equations (\[eq12:main\]), (\[eq12:sym1\]) and (\[eq12:sym2\]) have never been found before. In §6 we illustrate applications of the newly found Lax pairs. The Lax pair obtained in the previous sections is used to construct conservation laws for a Volterra type chain. We also show that the second operators of our Lax pairs are closely connected with the recursion operators for the associated nonlinear equations. In Appendix we give all of the computational details appeared when we evaluated the invariant manifold for the linearization of the sine-Gordon equation.
Laplace cascade for the linear hyperbolic type equations
========================================================
Let us recall the main steps of the Laplace cascade method (see [@Darboux], [@Goursat]). Consider a linear second order hyperbolic type PDE of the form $$\label{s2eq1}
v_{xy}+a(x,y)v_x+b(x,y)v_y+c(x,y)v=0.$$ It can easily be checked that functions
$$\label{s2eq2}
h_{[0]}=a_x+ab-c,\quad k_{[0]}=b_y+ab-c$$
do not change under the linear transformation $ v\rightarrow \lambda(x,y)v $ with arbitrary smooth factors $ \lambda(x,y) $ applied to equation (\[s2eq1\]). They are called the Laplace invariants for (\[s2eq1\]).
We rewrite equation (\[s2eq1\]) as a system of two equations: $$\label{s2eq3}
\left( \frac{\partial}{\partial y}+a \right)v=v_{[1]}, \quad \left(\frac{\partial}{\partial x}+b\right)v_{[1]}=h_{[0]}v.$$
When the invariant $ h_{[0]} $ does not vanish then one can exclude $ v $ from (\[s2eq3\]) and obtain a linear PDE for $ v_{[1]} $ $$\label{s2eq4}
v_{[1]xy}+a_{[1]}v_{[1]x}+b_{[1]}v_{[1]y}+c_{[1]}v_{[1]}=0$$ where the coefficients are evaluated as follows $$\label{s2eq5}
a_{[1]}=a-\frac{\partial}{\partial y}\log(h_{[0]}),\quad b_{[1]}=b,\quad c_{[1]}=a_{[1]}b_{[1]}+b_{[1]y}-h_{[0]}.$$ Thus we define a transformation of the equation (\[s2eq1\]) into the equation (\[s2eq4\]). This transformation is called the Laplace $y$-transformation. Iterations of the transformation generate a sequence of the equations $$\label{s2eq6}
v_{[i]xy}+a_{[i]}v_{[i]x}+b_{[i]}v_{[i]y}+c_{[i]}v_{[i]}=0$$ for $ i\geq1 $ where the coefficients are given by $$\label{s2eq7}
a_{[i]}=a_{[i-1]}-\frac{\partial}{\partial y}\log(h_{[i-1]}), \quad b_{[i]}=b_{[i-1]},\quad c_{[i]}=a_{[i]}b_{[i]}+b_{[i]y}-h_{[i-1]}.$$ Here we assume that $ a_{[0]}=a$, $b_{[0]}=b$, $c_{[0]}=c$. Eigenfunctions $ v_{[i]} $ are related by the equations $$\label{s2eq8}
\left(\frac{\partial}{\partial y}+a_{[i]}\right)v_{[i]}=v_{[i+1]}, \quad \left(\frac{\partial}{\partial x}+b_{[i]}\right)v_{[i+1]}=h_{[i]}v_{[i]}.$$ Due to the relation $ c_{[i]}=\frac{\partial}{\partial x}a_{[i]}+a_{[i]}b_{[i]}-h_{[i]} $ system (\[s2eq7\]) is rewritten as $$\label{s2eq9}
a_{[i]}=a_{[i-1]}-\frac{\partial}{\partial y}\log(h_{[i-1]}),\quad h_{[i]}=h_{[i-1]}+a_{[i]x}-b_{[i]y}, \quad b_{[i]}=b.$$ Reasonings above define the functions $ a_{[i]}$, $b_{[i]}$, $h_{[i]}$ only for $ i\geq 1 $. However they can be prolonged for $ i\leq 0$ by virtue of the same formulas rewritten as follows $$\label{s2eq10}
h_{[i-1]}=h_{[i]}-a_{[i]x}-b_y,\quad a_{[i-1]}=a_{[i]}+\frac{\partial}{\partial y}\log(h_{[i-1]}),\quad b_{[i-1]}=b_{[i]}.$$ Summarising the computations above we get a dynamical system of the form $$\label{s2eq11}
\frac{\partial}{\partial y}\log(h_{[i]})=a_{[i]}-a_{[i+1]},\quad a_{[i]x}=h_{[i]}-h_{[i-1]}+b_y,\quad b_{[i]}=b,$$ which is reduced to the well-known Toda lattice $$\label{s2eq12}
\frac{\partial}{\partial y}\log(h_{[i]})=p_{[i]}-p_{[i+1]},\quad p_{[i]x}=h_{[i]}-h_{[i-1]},$$ where $ p_{[i]}=a_{[i]}-\tilde{b} $ and $ \tilde{b}_x=b_y. $
Define two linear operators $$\label{s2eq13}
L_{i}=D_y+a_{[i]}-D_{[i]}, \quad M_{i}=D_x+b_{[i-1]}-h_{[i-1]}D_{i}^{-1},$$ where $ D_x, D_y $ are the operators of differentiation with respect to $ x, y $ correspondingly and $ D_{i} $ is the shift operator acting as follows $ D_{i}a_{[i]}=a_{[i+1]}, D_{i}h_{[i]}=h_{[i+1]} $, etc. We summarize all the reasonings above as a statement.
[**Proposition 1**]{}. The operators $ L_{i}, M_{i} $ commute for all $ i $ iff their coefficients satisfy the system (\[s2eq9\]).
[**Corollary**]{}. Equations (\[s2eq8\]) constitute the Lax pair for the system (\[s2eq11\]).
The Laplace $x$-transformation can be interpreted in a similar way.
Laplace cascade for the nonlinear hyperbolic type equations.\
Formal Lax pairs
=============================================================
Let us explain how the Laplace cascade is adopted to nonlinear case [@Goursat] (see also [@SokolovZhiber]). Consider a second order nonlinear hyperbolic type PDE $$\label{s3eq1}
u_{xy}=F(x,y,u,u_x,u_y).$$ Its linearization around a solution $ u(x,y) $ derived by substituting $ u=u(x,y,\varepsilon)=u(x,y,0)+\varepsilon v(x,y)+\ldots $ with $\displaystyle{v(x,y)=\frac{\partial u(x,y,\varepsilon)}{\partial\varepsilon}|_{\varepsilon=0}}$ into (\[s3eq1\]) is an equation of the form $$\label{s3eq2}
v_{xy}+av_x+bv_y+cv=0$$ where the coefficients $ a=-\partial F/\partial u_x$, $b=-\partial F/\partial u_y,$ $c=-\partial F/\partial u $ depend explicitly on the independent variables $ x$, $y $ and the dynamical variables $u$, $u_x$, $u_y$. Let us assign the Laplace sequence (\[s2eq8\])-(\[s2eq11\]) to the linearized equation (\[s3eq2\]). However now instead of the operators $\partial/\partial x$, $\partial/\partial y $ in (\[s2eq8\])-(\[s2eq11\]) we use the operators $ D_x$, $D_y $ of the total differentiation with respect to $x$, and $y$. Denote through $u_i$, $\bar u_i$, $i=0,1,\ldots $ the $i$-th order derivatives of the variable $ u $ with respect to $ x $ and $ y $ correspondingly $$\label{s3eq3}
D_x^iu=u_i,\quad D_y^iu=\bar u_i.$$ Evidently we have explicit expressions for the operators $ D_x$, $D_y $ acting on the class of smooth functions of $ x, y $ and a finite number of the dynamical variables $ u_i, \bar u_i $ $$\label{s3eq4}
D_x= \frac{\partial}{\partial x} + \sum^{\infty}_{i=0}u_{i+1}\frac{\partial}{\partial u_i}+ \sum^{\infty}_{i=1}D_y^{i-1}(F)\frac{\partial}{\partial \bar u_i},$$ $$\label{s3eq5}
D_y= \frac{\partial}{\partial y} + \sum^{\infty}_{i=0}\bar u_{i+1}\frac{\partial}{\partial \bar u_i}+ \sum^{\infty}_{i=1}D_x^{i-1}(F)\frac{\partial}{\partial u_i}.$$ The Laplace invariants corresponding to the equation (\[s3eq2\]) are evaluated as $$\label{s3eq6}
h_{[0]}=D_x(a)+ab-c, \quad k_{[0]}=D_y(b)+ab-c.$$ The linear system (\[s2eq8\]) in this case converts into $$\label{s3eq7}
(D_y+a_{[i]})v_{[i]}=v_{[i+1]}, \quad (D_x+b_{[i]})v_{[i+1]}=h_{[i]}v_{[i]}.$$ The coefficients $ a_{[i]}, b_{[i]}, h_{[i]} $ are evaluated due to the equations $$\label{s3eq8}
\begin{array} {l}
D_y(\log(h_{[i]}))=a_{[i]}-a_{[i+1]}, \quad D_x(a_{[i]})=h_{[i]}-h_{[i-1]}+D_y(b),\\
b_{[i]}=b,\quad a_{[0]}=a.
\end{array}$$ All of the mixed derivatives $ u_{xy}, u_{xxy}, \ldots $ are replaced by means of the equation (\[s3eq1\]) and its differential consequences.
Infinite-dimensional system (\[s3eq7\]) defines a sequence of the linear operators $$\label{s3eq9}
L_{i}=D_y+a_{[i]}-D_{i}, \quad M_{i}=D_x+b_{[i-1]}-h_{[i-1]}D_{i}^{-1},$$ satisfying the commutativity conditions $$\label{s3eq10}
\forall i \quad [L_{i},M_{i}]=0.$$ Thus one can define a pair of commuting operators $L_{i}$, $M_{i}$ depending on an integer parameter $i$ for an arbitrarily chosen equation (\[s3eq1\]). Roughly speaking the sequence of the commuting operators (\[s3eq9\]), (\[s3eq10\]) recovers equation (\[s3eq1\]). Hence system (\[s3eq7\]) defines a (formal) Lax pair for the arbitrary (generally non-integrable) equation (\[s3eq1\]). It is not very surprising since for non-integrable case system (\[s3eq7\]) is of infinite dimension. However as it is approved below by several examples for the integrable case the system is either finite (Liouville type equations) or admits a finite dimensional reduction (sine-Gordon type equations).
[**Example 1**]{}. As an illustrative example of the non-integrable equation with the infinite dimensional Lax pair consider the equation $$\label{s3eq11}
u_{xy}=u^2.$$ For its linearization $$\label{s3eq12}
v_{xy}=2uv$$ we have $a_{[0]}=b_{[0]}=0$, $h_{[0]}=h_{[-1]}=2u$, $a_{[1]}=-\frac{u_y}{u}$, $h_{[1]}=u+\frac{u_xu_y}{u^2}$. One can find all of the coefficients $a_{[j]}$, $h_{[j]}$ due to the equations (\[s3eq8\]) as functions of the dynamical variables and therefore define completely the system (\[s3eq7\]). Now go back, suppose that the system evaluated above is consistent and show that its consistency defines uniquely equation (\[s3eq11\]). Indeed the consistency of (\[s3eq7\]) implies $D_x(a_{[1]})=h_{[1]}-h_{[0]}$ equivalent to $(-\frac{u_y}{u})_x=\frac{u_xu_y}{u^2}-u$ which gives (\[s3eq11\]).
[**Example 2**]{}. As an example with the finite system (\[s3eq7\]) we take the Liouville equation $$\label{s3eq13}
u_{xy}=e^{u}$$ for which $ h_{[0]}=h_{[-1]}=e^{u} $ and $ h_{[1]}=h_{[-2]}=0. $ For $ i>1 $ and $ i<-2 $ the Laplace invariants $ h_{[i]} $ are not defined. The coefficient $ a_{[i]} $ is defined only for the following three values of $ i: a_{[1]}=-u_y, a_{[0]}=0, a_{[-1]}=u_y. $ Evidently $ b_{[0]}=0. $ System (\[s3eq7\]) for the equation (\[s3eq13\]) contains only seven equations $$\begin{aligned}
(D_y-u_y)v_{[1]}=v_{[2]}, \quad D_xv_{[2]}=h_{[1]}v_{[1]} \\
D_yv_{[0]}=v_{[1]}, \quad D_xv_{[1]}=h_{[0]}v_{[0]} \\
(D_y+u_y)v_{[-1]}=v_{[0]}, \quad D_xv_{[0]}=h_{[-1]}v_{[-1]} \\
D_xv_{[-1]}=h_{[-2]}v_{[-2]}\end{aligned}$$
There is a freedom in choosing of $ v_{[2]}$, $v_{[-2]}.$ Put them equal to zero. Then the obtained system gives the Lax pair for (\[s3eq13\]) $$\Psi_x=A\Psi, \quad \Psi_y=B\Psi,$$ where $ \Psi=(v_{[1]}, v_{[0]}, v_{[-1]})^{T} $ and
$$\begin{aligned}
A=\left( \begin {array}{ccc} 0&e^{u}&0\\ 0&0&e^{u} \\ 0&0&0 \end{array}\right),
B=\left( \begin {array}{ccc} u_y&0&0\\ 1&0&0 \\ 0&1&-u_y \end{array}\right).\end{aligned}$$
Remark that there are some degenerate cases where the commutativity condition of the operators (\[s3eq9\]) defines not exactly the initial equation (\[s3eq1\]) but some other equation connected with (\[s3eq1\]) by a Miura type transformation. Illustrate it with the following example.
[**Example 3**]{}. Consider the equation $$\label{s3eq14}
u_{xy}=e^{u+u_x}.$$ The coefficients of its linearization $$\label{s3eq15}
v_{xy}=e^{r}(v+v_x)$$ depend on $r=u+u_x$. Thus the commutativity condition of the operators (\[s3eq9\]) assigned to (\[s3eq14\]) implies the equation $r_{xy}=e^r+r^{r_x}$, connected with (\[s3eq14\]) by a very simple Miura type transformation $u_x+u=r$.
A more symmetrical form of the Laplace sequence
-----------------------------------------------
Let us change the dependent variables in the system (\[s3eq7\]) to make formulas more symmetrical. Introduce new dependent variables $w_{[i]}$, $i\in(-\infty,\infty)$ in such a way that $ w_{[i]}=v_{[i]} $ for $ i\geq0 $ and $ w_{[i]}=h_{[-1]}h_{[-2]}...h_{[i]}v_{[i]} $ for $ i\leq-1. $
Then the set of equations (\[s3eq7\]), (\[s3eq8\]) is changed to the form below, where $ i\geq0: $ $$\begin{aligned}
\label{s3.1eq3}
\begin{array}{l}
(D_y+a_{[i]})w_{[i]}=w_{[i+1]}, \quad (D_x+b_{[0]})w_{[i+1]}=h_{[i]}w_{[i]}, \\
D_y(\log(h_{[i]}))=a_{[i]}-a_{[i+1]}, \quad D_x(a_{[i]})=h_{[i]}-h_{[i-1]}+D_y(b_{[0]}), \\
(D_x+\hat{b}_{[-i]})w_{[-i]}=w_{[-i-1]}, \quad (D_y+a_{[0]})w_{[-i-1]}=h_{[-i-1]}w_{[-i]}, \\
\hat{b}_{[-i-1]}=\hat{b}_{[-i]}-D_x(\log(h_{[-i-1]})), \quad \hat{b}_{[0]}=b_{[0]}
\end{array}\end{aligned}$$
Sine-Gordon equation
--------------------
More than two decades ago an important property of the Laplace invariants of the Liouville type integrable equations has been observed [@SokolovZhiber], [@AndersonKamran]. It was proved that the hyperbolic equation (\[s3eq1\]) is an integrable equation of the Liouville type if and only if the set of its Laplace invariants is terminated on both sides. Mention also recent results obtained in [@Smirnov2015]. The problem of describing the properties of the Laplace invariants characterizing the sine-Gordon type integrable PDE is discussed in [@SokolovZhiber2001]. Our investigation convinces that a connection between the sine-Gordon type equations and the Laplace cascade is clearly formulated in terms of the cascade eigenfunctions. Let us explain our observation with an example.
We consider the sine-Gordon equation $$\label{s3.1eq1}
u_{xy}=\sin u.$$ It can be shown that the Laplace invariants $ h_{[i]} $ for the linearized equation $$\label{s3.1eq2}
v_{xy}=(\cos u) v$$ do not vanish identically for any integer $ i. $ Thus system (\[s3eq7\]) provides an infinite-dimensional Lax pair for (\[s3.1eq1\]). Below we show that for this case (\[s3eq7\]) admits a finite-dimensional reduction. Bring (\[s3eq7\]) to the symmetric form (\[s3.1eq3\]).
[**Proposition 2**]{}. The system (\[s3.1eq3\]) corresponding to the sine-Gordon equation (\[s3.1eq1\]) with $w_{[0]}=v$, $a_{[0]}=0$, $b_{[0]}=0$, $c_{[0]}=-\cos u$ is consistent with the following cutting off boundary conditions $$\label{s3.1eq4}
\begin{array}{l}
w_{[2]}=\alpha(-1)w_{[-1]}+\alpha(0)w_{[0]}+\alpha(1)w_{[1]}, \\
w_{[-2]}=\beta(-1)w_{[-1]}+\beta(0)w_{[0]}+\beta(1)w_{[1]}
\end{array}$$ where $$\label{s3.1eq5}
\begin{array}{l}
\alpha(-1)=-\lambda\frac{u_y}{\sin u}, \quad \alpha(0)=\lambda, \quad \alpha(1)=\frac{u_y}{\cos u \sin u}, \\[10pt]
\beta(-1)=\frac{u_x}{\cos u \sin u}, \quad \beta(0)=\lambda^{-1}, \quad \beta(1)=-\lambda^{-1}\frac{u_x}{\sin u},
\end{array}$$ $\lambda$ is a complex parameter.
Sketch of proof. Look for the functions (\[s3.1eq5\]) providing the consistency of the following overdetermined system of equations obtained from (\[s3.1eq3\]) by imposing (\[s3.1eq4\]) $$\label{s3.1eq6}
\left \{
\begin{array}{l}
w_{[1]y}=(\alpha(1)-a_{[1]})w_{[1]}+\alpha(0)w_{[0]}+\alpha(-1)w_{[-1]}, \\
w_{[0]y}=w_{[1]},\\
w_{[-1]y}=h_{[-1]}w_{[0]},
\end{array}
\right.$$ and $$\label{s3.1eq7}
\left \{
\begin{array}{l}
w_{[1]x}=h_{[0]}w_{[0]}, \\
w_{[0]x}=w_{[-1]},\\
w_{[-1]x}=\beta(1)w_{[1]}+\beta(0)w_{[0]}+(\beta(-1)-\hat{b}_{[-1]})w_{[-1]}.
\end{array}
\right.$$ The compatibility conditions $ (w_{[i]x})_y=(w_{[i]y})_x $ generate a system of nonlinear equations for the functions $ \alpha(j), \beta(j) $ searched: $$\label{s3.1eq8}
\begin{array}{l}
D_y(\beta(-1))+\beta(1)\alpha(-1)=h_{[-2]},\\
D_y(\beta(0))+\beta(1)\alpha(0)+\beta(-1)h_{[-1]}=0,\\
D_y(\beta(1))+\beta(1)(\alpha(1)-a_{[1]})+\beta(0)=0,\\
D_x(\alpha(-1))+\alpha(-1)(\beta(-1)-\hat{b}_{[-1]})+\alpha(0)=0,\\
D_x(\alpha(0))+\alpha(-1)\beta(0)+\alpha(1)h_{[0]}=0,\\
D_x(\alpha(1))+\alpha(-1)\beta(1)=h_{[1]}.
\end{array}$$ Here all the given coefficients $ a_{[1]}$, $\hat{b}_{[-1]}$, $h_{[-2]}$, $h_{[-1]}$, $h_{[0]}$, $h_{[1]} $ are linear functions of the derivatives $u_x$, $u_y$: $$\label{s3.1eq9}
\begin{array}{l}
h_{[0]}=h_{[-1]}=\cos u, \quad h_{[1]}=h_{[-2]}=\frac{1}{\cos u}+\frac{u_x u_y}{\cos^2 u},\\
a_{[1]}=u_y \tan u, \quad \hat{b}_{[-1]}=u_x\tan u,
\end{array}$$ therefore we can assume that $ \alpha(j), \beta(j) $ also linearly depend on the first derivatives of $u$ $$\label{s3.1eq10}
\begin{array}{l}
\alpha(j)=\alpha(j,u,u_y)=p(j,u)u_y+q(j,u), \\
\beta(j)=\beta(j,u,u_x)=r(j,u)u_x+s(j,u), \quad j=1,0,-1.
\end{array}$$ Substitute expressions (\[s3.1eq9\]), (\[s3.1eq10\]) into equations (\[s3.1eq8\]) and then compare the coefficients before the independent combinations of the dynamical variables $ u_xu_y, u_x, u_y $. As a result one gets twenty four equations for twelve functions $p(j,u)$, $q(j,u)$, $r(j,u)$, $s(j,u)$, $j=1,0,-1$, depending on $u$ only. We write down explicitly only a part of the equations since the others are obtained from these by applying the replacement $p(j)\leftrightarrow r(-j)$, $q(j)\leftrightarrow s(-j)$.
$$\label{s3.1eq11}
\begin{array}{l}
\frac{dr(-1)}{du}+r(1)p(-1)=\frac{1}{\cos^2 u}, \quad \frac{ds(-1)}{du}+s(1)p(-1)=0, \\
r(1)q(-1)=0, \quad r(-1)\sin u+s(1)q(-1)=\frac{1}{\cos u}, \\
\frac{dr(0)}{du}+r(1)p(0)=0, \quad \frac{ds(0)}{du}+s(1)p(0)=0, \\
r(-1)\cos u+r(1)q(0)=0, \quad r(0)\sin u+s(-1)\cos u+s(1)q(0)=0,\\
\frac{dr(1)}{du}-r(1)\tan u+r(1)p(1)=0, \quad \frac{ds(1)}{du}-s(1)\tan u+s(1)p(1)=0,\\
r(0)+r(1)q(1)=0, \quad r(1)\sin u+s(0)+s(1)q(1)=0.
\end{array}$$
By solving the overdetermined system of equations (\[s3.1eq11\]) we find explicit expressions (\[s3.1eq5\]) for the coefficients of the constraint (\[s3.1eq4\]). Now the systems (\[s3.1eq6\]), (\[s3.1eq7\]) can be rewritten as follows
$$\label{s3.1eq12}
\Psi_x=A\Psi, \quad \Psi_y=B\Psi$$
where $ \Psi=(w_{[1]},w_{[0]}, w_{[-1]})^T $ and $$\label{s3.1eq13}
A=\left( \begin {array}{ccc} 0&\cos u&0\\ 0&0&1 \\ \frac{-u_x}{\lambda\sin u}&\frac{1}{\lambda}&u_x \cot u \end{array}\right), \,
B=\left( \begin {array}{ccc} u_y\cot u& \lambda & \frac{-\lambda u_y}{\sin u}\\ 1&0&0 \\ 0& \cos u &0 \end{array}\right).$$ It is easily checked that (\[s3.1eq12\]), (\[s3.1eq13\]) defines the Lax pair for the sine-Gordon equation (\[s3.1eq1\]). We failed to reduce it to the well-known usual one found in [@Ablowitz73].
Invariant manifolds of the hyperbolic type PDE
==============================================
Let us recall the definition of the invariant manifold of the hyperbolic type equation (\[s3eq1\]). Consider an equation of the form $$\label{s4eq1}
G(x,y,u_k,u_{k-1}, \ldots u, \bar{u}_1, \bar{u}_2, \ldots \bar{u}_m)=0.$$ Note that $ G $ depends on $ x, y $ and a set of the dynamical variables $ u, u_1, \bar{u}_1, \ldots$, where $u_j=\frac{\partial^j u}{\partial x^j}$, $\bar u_j=\frac{\partial^j u}{\partial y^j}$. Take the differential consequences of (\[s4eq1\]) $$\label{s4eq2}
G_1(x,y,u_{k+1}, \ldots u, \bar{u}_1, \bar{u}_2, \ldots \bar{u}_m)=0,$$ $$\label{s4eq3}
G_2(x,y,u_k, \ldots u, \bar{u}_1, \bar{u}_2, \ldots \bar{u}_{m+1})=0,$$ where $ G_1, G_2 $ are evaluated by applying the operators $ D_x,D_y: G_1=D_xG, G_2=D_yG $ and subsequent replacement of the mixed derivatives by means of the equation (\[s3eq1\]) and its differential consequences. Equation (\[s4eq1\]) defines an invariant manifold for (\[s3eq1\]) if the following equation is satisfied $$\label{s4eq4}
D_xD_yG|_{(3.1), (4.1) - (4.3)}=0.$$
[**Example 4**]{}. Show that equation $$\label{s4eq5}
u_{xx}+\frac{1}{2} u^2_x \tan u=0$$ defines an invariant manifold for the sine-Gordon equation (\[s3.1eq1\]). Here\
$G=u_{xx}+\frac{1}{2} u^2_x \tan u$, $G_1=D_xG=u_{xxx}+\frac{1}{2} u^3_x $ and $ G_2=D_yG=\frac{u_x}{\cos u}+\frac{u^2_xu_y}{2(\cos u)^2}. $ It is easily verified that $$D_xD_yG=D_x\left( \frac{u_x}{\cos u}+\frac{u^2_xu_y}{2(\cos u)^2}\right)=0 \quad \mbox{mod} ((4.5),G=0,G_1=0,G_2=0).$$ Therefore equation (\[s4eq4\]) holds and thus (\[s4eq5\]) defines an invariant manifold for (\[s3.1eq1\]).
From the Laplace cascade to invariant manifolds
-----------------------------------------------
Show that reduced system (\[s3.1eq3\]), (\[s3.1eq4\]) is closely connected with the invariant manifolds of the linearized equation (\[s3.1eq2\]). Indeed, equations (\[s3.1eq3\]) imply that $w_{[2]}=(D_y+a_{[1]})(D_y+a_{[0]})w_{[0]}$, $w_{[1]}=(D_y+a_{[0]})w_{[0]}$, $w_{[-1]}=(D_x+b_{[0]})w_{[0]}$, $ w_{[-2]}=(D_x+\hat b_{[-1]})(D_x+b_{[0]})w_{[0]}$. Therefore since $a_{[0]}=b_{[0]}=0$, the boundary conditions (\[s3.1eq4\]) turn into the equations $$(D_y+a_{[1]})D_y w_{[0]}=\alpha(-1)D_x w_{[0]}+\alpha(0) w_{[0]}+\alpha(1) D_y w_{[0]},$$ $$(D_x+\hat b_{[-1]})D_x w_{[0]}=\beta(-1)D_x w_{[0]}+\beta(0) w_{[0]}+\beta(1) D_y w_{[0]}.$$ Simplify the equations obtained by using explicit expressions (\[s3.1eq5\]), (\[s3.1eq9\]) and find equations $$\label{s4eq6}
\fl L_yw_{[0]}:=(D^2_y-u_y \cot u D_y+\lambda \frac{u_y}{\sin u} D_x-\lambda)w_{[0]}|_{(3.17), w_{[0]xy}=(\cos u)w_{[0]}}=0,$$ $$\label{s4eq7}
\fl L_xw_{[0]}:=(D^2_x-u_x \cot u D_x+\lambda^{-1} \frac{u_x}{\sin u} D_y-\lambda^{-1})w_{[0]}|_{(3.17), w_{[0]xy}=(\cos u)w_{[0]}}=0$$ which define the invariant manifold for (\[s3.1eq2\]) discussed in Introduction (see (\[s1.sin:inv1\]), (\[s1.sin:inv2\]) above).
This observation leads to an alternative algorithm to look for the Lax pair. Instead of the cutting off boundary conditions to the lattice (\[s3.1eq3\]) one searches an invariant manifold for the linearized equation (\[s3.1eq2\]).
By construction we have $$\label{sin:inv3}
Mw_{[0]}:= (D_xD_y - \cos u)w_{[0]}.$$ Commutators of the operators $L_x$, $L_y$, $M$ satisfy the following relations $$\begin{aligned}
&&[L_x,L_y]=2A_{xy}(\lambda^{-1}L_y-\lambda L_x),\nonumber\\
&&[M,L_x]=B_{xx}L_y-B_{xy}L_x+(A_{xx}-\lambda^{-1} A_{xy})M,\, \label{commutativity}\\
&&[M,L_y]=B_{yy}L_x-B_{xy}L_y+(A_{yy}-\lambda A_{xy})M \nonumber\end{aligned}$$ where $ {A=\log \cot \frac{u}{2}}$, $B=\log\sin u$, $A_x=D_x(A)$, $A_y=D_y(A)$, $A_{xx}=D_x^2(A)$ and so on. Consequently any element of the Lie ring generated by the operators $L_x$, $L_y$, $M$ is represented as a linear combination of the same three operators.
Linear equations (\[s4eq6\]), (\[s4eq7\]) define a manifold parametrized by $w_{[0]}$, $w_{[0]x}$, $w_{[0]y}$ and the dynamical variables $u$, $u_1$, $\bar u_1, ...\,$. By applying $D_x$ to the equations (\[s4eq6\]), (\[s4eq7\]) and then simplifying due to the equations (\[s3.1eq1\]), (\[s3.1eq2\]) one gets another parametrization of the manifold $$\label{sin:inv4}
w_{[0]xxx} = \frac{u_{xx}}{u_x} w_{[0]xx} + (\lambda^{-1} - u^2_x)w_{[0]x} - \lambda^{-1} \frac{u_{xx}}{u_x}w_{[0]},$$ $$\label{sin:inv5}
w_{[0]y} = -\lambda \frac{\sin u}{u_x} w_{[0]xx} + \lambda (\cos u) w_{[0]x} + \frac{\sin u}{u_x} w_{[0]},$$ where the parameters $w_{[0]}$, $w_{[0]x}$, $w_{[0]xx}$ are taken as independent ones. It is shown below that this parametrization is closely connected with the Lax pair for the potential KdV equation being a symmetry of the sine-Gordon equation.
Evaluation of the invariant manifolds and the Lax pair for the equation $u_{xy} = f(u) \sqrt{1+u^2_x}$, $f''=\gamma f$
----------------------------------------------------------------------------------------------------------------------
In this section we construct a Lax pair to the equation $$\label{eq12:main}
u_{xy} = f(u) \sqrt{1+u^2_x}, \quad f'' = \gamma f$$ found in [@SokolovMeshkov]. It is known that S-integrable equation of the form (\[eq12:main\]) by an appropriate point transformation can be reduced either to the case $f(u)=u$ or $f(u)=\sin u$ (see [@SokolovMeshkov]). By analogy with the sine-Gordon equation considered in the previous section we look for the invariant manifold of the form $$\label{eq12:inv}
v_{yy} + a v_y + b v_x + c v = 0$$ for the linearized equation $$\label{eq12:lin}
v_{xy} = f'(u)\sqrt{1+u^2_x}v + \frac{f(u) u_x}{\sqrt{1+u^2_x}} v_x.$$ Apply the operator $D_x$ to (\[eq12:inv\]) and rewrite the result as $$\begin{aligned}
\fl v_{xx} = -\frac{1}{b}\left( 2 v u_x f(u)f'(u)+ v_xf^2(u)+D_x(b) v_x+c v_x+D_x(c) v+D_x(a) v_y\right) \nonumber\\
-\frac{\bigl(u_y u_x v_x+(v_y +a v)(1+u_x^2)\bigr) f'(u)+\bigl(p u_y v(1+ u_x^2)+a u_x v_x\bigr) f(u)}{b \sqrt{1+u^2_x}}. \label{eq12:eq1}\end{aligned}$$
Now apply $D_y$ to (\[eq12:eq1\]), simplify the result due to the equations above and get an equation of the form $$\label{eq12:eq2}
v_{yy} + \tilde{a} v_y + \tilde{b} v_x + \tilde{c} v = 0,$$ with the coefficients $\tilde{a}$, $\tilde{b}$, $\tilde{c}$ depending on a finite number of the dynamical variables. According to the definition of the invariant manifold equations (\[eq12:inv\]) and (\[eq12:eq2\]) should coincide. This fact implies a system of three equations on the sought functions $a$, $b$, $c$ $$\begin{aligned}
\fl \Bigl(2 f(u) u_x b f'(u)+D_x(c) b-D_y(b) D_x(a)+D_yD_x(a) b-a b D_x(a)\Bigr)\sqrt{1+u_x^2}\nonumber \\
+(2 p u_y b u_x^2-D_x(a) u_x b+2 p u_y b)f(u)-(u_x^2+1)D_y(b) f'(u) = 0,\label{eq12:eq1:coeff:vy}
\end{aligned}$$ $$\begin{aligned}
\fl \Bigl((-D_y(b) u_x^2+a b-D_y(b)) f(u)^2+u_y b (3+2 u_x^2) f'(u) f(u)\Bigr)\sqrt{1+u_x^2}\nonumber\\
-\bigl( b^2 D_x(a)-D_y(c) b-D_yD_x(b) b+D_y(b) c+D_y(b) D_x(b)\bigr)(1+u_x^2)^{3/2} \nonumber\\
+(u_{xx} b^2+(p u_y^2 u_x b+D_y(a) u_x b-D_y(b) a u_x)(1+u^2_x)) f(u)\nonumber\\
+u_x (1+u_x^2) (b^2 u_x+u_{yy} b+a u_y b-D_y(b) u_y) f'(u)=0,\label{eq12:eq1:coeff:vx}\end{aligned}$$ $$\begin{aligned}
\fl \Bigl(2 p f(u)^2 u_y u_x b+u_x (a b-2 D_y(b)) f'(u) f(u)+3 f'(u)^2 u_x u_y b\Bigr.\nonumber\\
\Bigl.-D_y(b) D_x(c)-c b D_x(a)+D_yD_x(c) b\Bigr) \sqrt{1+u_x^2}+b (u_x^2+3) f'(u) f(u)^2\nonumber\\
+\left((-D_y(b) a+D_y(a) b+D_x(b) b+p u_y^2 b)(1+u^2_x)+u_x u_{xx} b^2\right) f'(u)\nonumber\\
+\left(p( b u_{yy} +a u_y b+u_x b^2-D_y(b) u_y)(1+ u_x^2)-D_x(c) u_x b\right) f(u)= 0.\label{eq12:coeff:v}\end{aligned}$$
Assuming that the searched functions depend only on $u$,$u_x$, $u_y$ i.e. $a = a(u,u_x,u_y)$, $b=b(u,u_x,u_y)$ and $c = c(u,u_x,u_y)$ substitute these functions into (\[eq12:eq1:coeff:vy\]), (\[eq12:eq1:coeff:vx\]) and (\[eq12:coeff:v\]) and eliminate mixed derivatives of $u$ using (\[eq12:main\]) from the resulting equations. Thus we obtain three equations of the following form $$\alpha_{i}(u,u_x,u_y)u_{xx}u_{yy} + \beta_{i}(u,u_x,u_y)u_{xx} + \gamma_{i}(u,u_x,u_y)u_{yy} + \delta_{i}(u,u_x,u_y) = 0,$$ $i=1,2,3$. These relations are satisfied only if the following conditions: $$\label{eq12:mainsys}
\fl \alpha_{i}(u,u_x,u_y) = 0, \quad \beta_{i}(u,u_x,u_y) = 0, \quad \gamma_{i}(u,u_x,u_y) = 0,\quad \delta_{i}(u,u_x,u_y) = 0$$ hold identically for all values $u$, $u_x$ and $u_y$, $i=1,2,3$. Here $$\begin{aligned}
\alpha_{1} & = & (b a_{u_xu_y}-a_{u_x} b_{u_y})\sqrt{1+u^2_x},\\
\alpha_{2} &= &( b b_{u_xu_y}-b_{u_x} b_{u_y})(1+u^2_x)^{3/2}, \\
\alpha_{3} &= & (b c_{u_xu_y}-c_{u_x} b_{u_y})\sqrt{1+u^2_x}, \\
\beta_{1} &= & \left(b c_{u_x}+b u_y a_{u u_x}-b_{u} u_y a_{u_x}-a b a_{u_x}\right) \sqrt{1+u_x^2}\nonumber\\
& & +f(u) (1+u_x^2) (b a_{u_x u_x}-b_{u_x} a_{u_x}),\\
\beta_{2} &=& -(1+u_x^2) (b_{u} u_y b_{u_x}-b u_y b_{u u_x}+b^2 a_{u_x}) \sqrt{1+u_x^2}\\
&&+f(u) \left((b b_{u_xu_x}-b^2_{u_x})(1+u^2_x)^2 + b b_{u_x}u_x(1+u^2_x)+b^2\right),\\
\beta_{3} &=& (b u_y c_{u u_x}-c b a_{u_x}-b_{u} u_y c_{u_x}) \sqrt{1+u_x^2}\\
&&+(1+u_x^2) (b c_{u_x u_x}-b_{u_x} c_{u_x}) f(u)+b (b_{u_x}(1+u^2_x)+b u_x) f'(u),\\
\gamma_{1} &=& u_x (-a_{u} b_{u_y}+b a_{u u_y}) \sqrt{1+u_x^2}\\
&&+(1+u_x^2) \bigl((b a_{u_y u_y}-b_{u_y} a_{u_y}) f(u)-f'(u) b_{u_y}\bigr),\\
\gamma_{2} &=& (1+u_x^2) \Bigl[(-b_{u_y} b_{u} u_x+b b_{u u_y} u_x-f(u)^2 b_{u_y}-c b_{u_y}+b c_{u_y}) \sqrt{1+u_x^2}\Bigr.\\
&&+\bigl((b b_{u_y u_y}-b_{u_y}^2)(1+u^2_x)-a u_x b_{u_y}+u_x b a_{u_y}\bigr) f(u)\\
&&+u_x (b-b_{u_y} u_y) f'(u)\Bigl.\Bigr],\\
\gamma_{3} &=& u_x (-c_{u} b_{u_y}-2 b_{u_y} f(u) f'(u)+b c_{u u_y}) \sqrt{1+u_x^2}\\
&&+(1+u_x^2) \bigl((p b-b_{u_y} c_{u_y}+b c_{u_y u_y}-p u_y b_{u_y}) f(u)\nonumber\\
&&+(-a b_{u_y}+b a_{u_y}) f'(u)\bigr),\\
\delta_{1} &=& \Bigl[(1+u_x^2) (-b_{u_x} a_{u_y}+b a_{u_x u_y}) f(u)^2+(-u_x^2 b_{u_x}+2 u_x b-b_{u_x}) f'(u) f(u)\Bigr.\nonumber\\
&&\Bigl.-u_x (b_{u} u_y a_{u}+a b a_{u}-b c_{u}-b u_y a_{u u})\Bigr] \sqrt{1+u_x^2}\\
&&+b (u_y a_{u u_y}+ c_{u_y}+2 p u_y -a a_{u_y}+ a_{u u_x} u_x)(1+u^2_x)f(u)\\
&&\Bigl(b a_{u}-(b_{u_x} a_{u} u_x+b_{u} u_y a_{u_y})(1+u^2_x)\Bigr) f(u)\\
&&+(1+u_x^2) (b a_{u_x} u_x-b_{u} u_y+b a_{u_y} u_y) f'(u),\\
\delta_{2} &=& \Bigl[ (bb_{u_xu_y}-b_{u_x}b_{u_y})(1+u^2_x)^2f^2(u)\\
&& + (1+u^2_x)\bigl( b a_{u_x}u_x - a b_{u_x}u_x + b b_{u_y}u_x - b_u u_y\bigr)f^2(u)\Bigr. \\
&& + ab f^2(u)+u_y (2 u_x^2 b+3 b-u_x^3 b_{u_x}-u_x b_{u_x}) f'(u) f(u)\\
&&\Bigl.-(1+u_x^2) (-b u_y b_{u u} u_x+b^2 a_{u} u_x+b_{u}^2 u_y u_x+c b_{u} u_y-b c_{u} u_y)\Bigr] \sqrt{1+u_x^2}\\
&&+(1+u_x^2) \Bigl[ (b c_{u_x}+b b_{u}-b^2 a_{u_y} +b b_{u u_y}u_y -b_{u} b_{u_y}u_y +b b_{u u_x} u_x \\
&&-c b_{u_x}-b_{u_x} b_{u} u_x -b_{u_x}f^3(u))(1+u^2_x) +u_xu_y(ba_u-ab_u+pbu_y)\\
&&\Bigl.+((b b_{u_y} u_y+b b_{u_x} u_x)(1+u^2_x)+b^2 u_x^2+u_x a u_y b-u_x b_{u} u_y^2) f'(u)\Bigl],\\
\delta_{3} &=& \Bigl[(1+u^2_x)\bigl((b c_{u_x u_y}-p u_y b_{u_x}-b_{u_x} c_{u_y})+2 p u_y u_x b\bigr)f^2(u)\Bigr.\\
&&+\bigl( (b a_{u_x}+b b_{u_y}-a b_{u_x})(1+u^2_x) -2 u_x b_{u} u_y+u_x a b\bigr) f'(u) f(u)\\
&&\Bigl.+3 f'^2(u)b u_x u_y -u_x (c b a_{u}+b_{u}c_{u} u_y -b c_{u u}u_y)\Bigr] \sqrt{1+u_x^2}\\
&&+(3 b+ b u_x^2-2b_{u_x} u_x^3 -2 b_{u_x} u_x) f'(u) f^2(u)\\
&&+\bigl( (1+u^2_x)(p u_x b^2+b u_y c_{u u_y}-c b a_{u_y}-b_{u} u_y c_{u_y}-b_{u_x} c_{u} u_x-p u_y^2 b_{u} \bigr.\\
&&\bigl.+b c_{u u_x} u_x+a p u_y b) +b c_{u}\bigr)f(u)\\
&&-(1+u_x^2) (a b_{u} u_y-b c_{u_x} u_x-b b_{u} u_x-b a_{u} u_y-b c_{u_y} u_y-p bu_y^2 ) f'(u).\end{aligned}$$ Thus the problem is reduced to a system of equations (\[eq12:mainsys\]).
We look for the functions $a$, $b$ and $c$ depending on the variable $u_y$ linearly: $$\begin{aligned}
a = a_1(u,u_x)u_y + a_2(u,u_x), \\
b = b_1(u,u_x)u_y + b_2(u,u_x), \\
c = c_1(u,u_x)u_y + c_2(u,u_x).\end{aligned}$$ Then equation $\alpha_2 = 0$ is essentially simplified $b_2 (b_1)_{u_x} - b_1 (b_2)_{u_x} = 0$. Assume that $b_2 \equiv 0$, then $$\label{eq12:b}
b = b_1(u,u_x)u_y.$$ From equations $\alpha_1 = 0$ and $\alpha_3 = 0$ we obtain $$u_y a_{u_x u_x} - a_{u_x} = 0, \quad u_y c_{u_x u_x} - c_{u_x} = 0.$$ Assume that $a_{u_x} = c_{u_x} \equiv 0$ then $$\label{eq12:ac}
a = a_1(u) u_y + a_2(u), \quad c = c_1(u) u_y + c_2(u).$$
Substituting the functions (\[eq12:b\]), (\[eq12:ac\]) into $\gamma_1 = 0$ (see (\[eq12:mainsys\] above)) we obtain the following equation $$\fl b_1(u,u_x)\left[(f'(u)+a_1(u)f(u))(1+u^2_x)+\sqrt{1+u^2_x}u_xa'_2(u)\right]) = 0.$$ Since functions $f$, $a_1$ and $a_2$ depend only on $u$, we obtain $$f'(u)+a_1(u)f(u)=0, \quad a'_2(u) = 0.$$ Consequently $$a_1(u) = -\frac{f'(u)}{f(u)}, \quad a_2(u) = a_3,$$ where $a_3$ is an arbitrary constant. Then from the equality $\gamma_3 = 0$ we get $$c_1(u) = -a_3\frac{f'(u)}{f(u)}, \quad c_2(u) = -f^2(u) + c_3,$$ where $c_3$ is an arbitrary constant. Analyzing the equation $\delta_1 = 0$ we define that $$b_1 = \frac{b_3}{f(u) \sqrt{1+u^2_x}},$$ where $b_3 \neq 0$ is an arbitrary constant. From the equation $\delta_2 = 0$ we obtain that $a_3 =0$ and $c_3 = -b_3$. It is easily checked that equalities $\beta_{i}=0$, $i=1,2,3$, $\gamma_2=0$, $\delta_3 = 0$ are automatically satisfied.
Thus summarizing the reasonings above we can claim that equations $$\label{eq12:vyy1}
v_{yy} - \frac{f'(u)}{f(u)} u_y v_y + \frac{\lambda u_y}{f(u) \sqrt{1+u^2_x}} v_x - (f^2(u) + \lambda) v = 0,$$ $$\label{eq12:vxx1}
v_{xx} -\left(\frac{f'(u)}{f(u)}+\frac{u_{xx}}{u_x^2+1}\right) u_x v_x + \frac{u_x \sqrt{u_x^2+1}}{\lambda f(u)} v_y - \left( \frac{u_x^2+1}{\lambda}\right) v = 0$$ define an invariant manifold for the linearized equation (\[eq12:lin\]). Here $\lambda$ is the spectral parameter.
Derive the Lax pair for the equation (\[eq12:main\]) from the invariant manifold (\[eq12:vyy1\]), (\[eq12:vxx1\]). To this end evaluate the Laplace sequence of the form (\[s3.1eq3\]) for the linearized equation (\[eq12:lin\]). In what follows we will need in explicit expressions for several first coefficients of the system (\[s3.1eq3\]) $$\begin{aligned}
\label{coeff}
a_{[0]}=a_{[-1]}=-\frac{u_x f(u)}{\sqrt{u_x^2+1}}, \quad b_{[0]}=b_{[1]}=b_{[-1]}=0, \nonumber \\
k_{[0]}=f'(u) \sqrt{u_x^2+1}, \quad h_{[0]}=-\frac{u_xx f(u)}{(u_x^2+1)^\frac{3}{2}}+\frac{f'(u)}{\sqrt{u_x^2+1}}, \\
\fl a_{[1]}=\frac{(u_x^2+1)f(u) (\gamma u_y \sqrt{u_x^2+1}-f'(u)) + u_x u_{xx} f^2(u)-f'(u) u_y u_{xx} \sqrt{u_x^2+1} }{\sqrt{u_x^2+1} (f'(u)(u_x^2+1) - f(u)u_{xx} )}. \nonumber \end{aligned}$$ The constraints (\[eq12:vyy1\]), (\[eq12:vxx1\]) generate the cutting off boundary conditions of the form $$\label{eq12:bc}
\left \{
\begin {array}{l}
v_{[2]}=\alpha(1) v_{[1]}+\alpha(0) v_{[0]}+\alpha(-1) v_{[-1]}, \\
v_{[-2]}=\beta(1) v_{[1]}+\beta(0) v_{[0]}+\beta(-1) v_{[-1]}
\end{array}
\right.$$ imposed on the infinite system (\[s3.1eq3\]). Find the coefficients $\alpha(i)$ and $\beta(i)$ in the relation (\[eq12:bc\]). Evidently equations (\[s3.1eq3\]) imply $$\label{eq12:bc-diff}
(D_y+a_{[1]})(D_y+a_{[0]})v_{[0]}=v_{[2]}, \\
(D_x+b_{[-1]})(D_x+b_{[0]})v_{[0]}=v_{[-2]}.$$ Set $v:=v_{[0]}$ and simplify (\[eq12:bc-diff\]) by virtue of (\[eq12:bc\]) and the relations $v_{[1]}=(D_y+a_{[0]})v$ and $v_{[-1]}=(D_x+b_{[0]})v$. As a result we obtain $$\label{eq12:vyy2}
\fl \left.
\begin {array}{l}
v_{yy}+(a_{[0]}+a_{[1]}-\alpha(1))v_y-\alpha(-1)v_x+(a_{[0]y}+a_{[0]}a_{[1]}-a_{[0]}\alpha(1)-\alpha(0)-\alpha(-1)b_{[0]})v=0, \\
v_{xx}+(b_{[-1]}-\beta(-1))v_x-\beta(1)v_y+(b_{[0]x}+b_{[0]}b_{[-1]}-a_{[0]}\beta(1)-\beta(0)-\beta(-1)b_{[0]})v=0,
\end{array}
\right.$$ The last equations should coincide with (\[eq12:vyy1\]), (\[eq12:vxx1\]). Comparison of the corresponding coefficients allows one to derive explicit formulas for the sought functions $\alpha(i)$ and $\beta(i)$ $$\left \{
\begin {array}{l}
\alpha(1)=a_{[0]}+a_{[1]}+\frac{u_y f'(u)}{u}, \\
\alpha(-1)=-\frac{\lambda u_y}{f(u) \sqrt{u_x^2+1}}, \\
\alpha(0)=f(u)^2+\lambda+a_y-a_{[0]}^2-a_{[0]}\frac{u_y f'(u)}{f(u)}+\frac{\lambda u_y b_{[0]}}{f(u) \sqrt{u_x^2+1}},\\
\end{array}
\right.$$ $$\left \{
\begin {array}{l}
\beta(1)=-\frac{u_x \sqrt{u_x^2+1}}{\lambda f(u)},\\
\beta(-1)=b_{[-1]}+\frac{u_x f'(u)}{f(u)} + \frac{u_x u_{xx}}{u_x^2+1}, \\
\beta(0)=b_{[0]x}-a_{[0]}(-\frac{u_x \sqrt{u_x^2+1}}{\lambda f(u)})-(\frac{u_x f'(u)}{f(u)} + \frac{u_x u_{xx}}{u_x^2+1})b_{[0]}+\frac{u_x^2+1}{\lambda}.
\end{array}
\right.$$ Now the infinite system of equations (\[s3.1eq3\]) is reduced to a pair of the third order systems of ordinary differential equations $$\left \{
\begin {array}{l}
v_{[0]y}=v_{[1]}-a_{[0]}v_{[0]}, \\
v_{[1]y}=(\alpha(1) -a_{[1]}) v_{[1]}+\alpha(0) v_{[0]}+\alpha(-1) v_{[-1]}, \\
v_{[-1]y}=k_{[0]}v_{[0]}-a_{[0]}v_{[-1]},\\
\end{array}
\right.$$ $$\left \{
\begin {array}{l}
v_{[0]x}=v_{[-1]}-b_{[1]}v_{[1]},\\
v_{[1]x}=h_{[0]}v_{[0]}-b_{[0]}v_{[1]},\\
v_{[-1]x}=\beta(1) v_{[1]}+\beta(0) v_{[0]}+(\beta(-1) - b_{[-1]})v_{[-1]}
\end{array}
\right.$$ which can be specified as follows $$\label{eq12:dy}
\fl \left( \begin {array}{l} v_{[0]} \\ v_{[1]} \\ v_{[-1]} \end{array}\right)_y=
\left( \begin {array}{ccc} \frac{u_x f(u)}{\sqrt{u_x^2+1}} &1&0\\ \lambda & {\frac{u_y f'(u)}{f(u)} - \frac{u_x f(u)}{\sqrt{u_x^2+1}}} & -\frac{u_y \lambda }{ f(u) \sqrt{u_x^2+1}} \\ f'(u)\sqrt{u_x^2+1} &0& \frac{u_x f(u)}{\sqrt{u_x^2+1}} \end{array}\right)
\left( \begin {array}{l} v_{[0]} \\ v_{[1]} \\ v_{[-1]} \end{array}\right),$$
$$\label{eq12:dx}
\fl \left( \begin {array}{l} v_{[0]} \\ v_{[1]} \\ v_{[-1]} \end{array}\right)_x=\left( \begin {array}{ccc} 0&0&1\\ {\frac{f'(u)}{\sqrt{u_x^2+1}}-\frac{u_{xx}f(u)}{(u_x^2+1)^{\frac{3}{2}}}}&0&0 \\ \frac{1}{\lambda}&-\frac{u_x \sqrt{u_x^2+1}}{\lambda f(u)}& \frac{u_x f'(u)}{f(u)} + \frac{u_x u_{xx}}{u_x^2+1} \end{array}\right)\left( \begin {array}{l} v_{[0]} \\ v_{[1]} \\ v_{[-1]} \end{array}\right).$$
Systems (\[eq12:dy\]), (\[eq12:dx\]) define the Lax pair for the equation (\[eq12:main\]).
Searching the Lax pairs for the evolutionary type integrable equations
======================================================================
Let us consider the evolutionary type integrable equations, for which the Laplace cascade is not defined. Here we use the scheme set out in the Introduction.
Korteweg-de Vries equation.
---------------------------
As an illustrative example we consider the Korteweg-de Vries equation $$\label{KdV:main}
u_t = u_{xxx} + u u_x.$$ Its linearization evidently has the form $$\label{KdV:lin}
v_t = v_{xxx} + u v_x + u_x v.$$ Direct computations show that equation (\[KdV:lin\]) does not admit any invariant manifold of the form $v_{xx}=a(u,u_x,u_{xx})v_{x}+b(u,u_x,u_{xx})v$ fit for arbitrary solution $u(x,t)$ of (\[KdV:main\]).
Let us look for the invariant manifold of order three $$\label{KdV:inv}
v_{xxx} = a v_{xx} + b v_x + c v,$$ where the coefficients $a$, $b$, $c$ depend on a finite number of the dynamical variables $u$, $u_1$, $u_2, ...\,$. According to the definition the following condition $$\label{KdV:compcond}
(v_{xxx})_t = (v_{t})_{xxx}$$ should be valid.
Replacing in (\[KdV:compcond\]) $v_{xxx}$ and $v_t$ due to (\[KdV:inv\]) and (\[KdV:lin\]) respectively and then comparing the coefficients before the independent variables $v_{xx}$, $v_x$ and $v$ we obtain the following equations $$\begin{aligned}
\fl 3 a D_x(b)+6 u_{xx}+D^3_x(a)+3 D_x(a) b+u_x a+u D_x(a)+3 D_x(a)^2\nonumber\\
+3 a D^2_x(a)+3 a^2 D_x(a)+3 D_x(c)-D_t(a)+3 D^2_x(b) = 0,\label{KdV:coeff:vxx}\\
\fl D^3_x(b)+3 D^2_x(c)+3 b D_x(b)+3 D_x(a) D_x(b)+u D_x(b)+4 u_{xxx}+3 a b D_x(a)\nonumber\\
+3 D^2_x(a) b-3 a u_{xx}-D_t(b)+3 c D_x(a)+2 u_x b=0,\label{KdV:coeff:vx}\\
\fl 3 a c D_x(a)+u_{xxxx}+u D_x(c)+D^3_x(c)+3 D^2_x(a) c-a u_{xxx}+3 D_x(b) c\nonumber\\
-D_t(c)+3 u_x c+3 D_x(a) D_x(c)-b u_{xx}=0.\label{KdV:coeff:v}\end{aligned}$$ It is reasonable to assume that $a = a(u,u_x,u_{xx})$, $b=b(u,u_x,u_{xx})$ and $c = c(u,u_x,u_{xx})$. We substitute these expressions into (\[KdV:coeff:vxx\]), (\[KdV:coeff:vx\]) and (\[KdV:coeff:v\]) and then exclude all the mixed derivatives of $u$ due to the equation (\[KdV:main\]). As a result we obtain three equations of the following form $$\begin{aligned}
\fl \alpha_{i}(u,u_x,u_{xx},u_{xxx})u_{xxxx} &-\beta_{i}(u,u_x,u_{xx}) u^3_{xxx} - \gamma_{i}(u,u_x,u_{xx}) u^2_{xxx}\\
&-\delta_{i}(u,u_x,u_{xx}) u_{xxx}
- \epsilon_{i}(u,u_x,u_{xx}) = 0, \quad i=1,2,3.\end{aligned}$$ Since $u_{xxxx}$, $u^3_{xxx}$, $u^2_{xxx}$, $u_{xxx}$ are independent variables then these equations split down into fifteen equations as follows $$\label{KdV:mainsys}
\begin{array}{c}
\alpha_{i}(u,u_x,u_{xx},u_{xxx}) = 0, \quad \beta_{i}(u,u_x,u_{xx}) = 0, \quad \gamma_{i}(u,u_x,u_{xx}) = 0,\\
\delta_{i}(u,u_x,u_{xx}) = 0, \quad \epsilon_{i}(u,u_x,u_{xx})=0
\end{array}$$ hold for all values of $u$, $u_x$ and $u_{xx}$, $i=1,2,3$. Thus the searched coefficients $a$, $b$, $c$ satisfy a highly overdetermined system of differential equations (\[KdV:mainsys\]). Specify and analyse the system. It can be verified that the three equations $\beta_i = 0$, $i=1,2,3$ immediately imply $a_{u_{xx} u_{xx} u_{xx}} = 0$, $\beta_{u_{xx} u_{xx} u_{xx}} = 0$ and $c_{u_{xx}u_{xx}u_{xx}} = 0$ therefore $$\begin{aligned}
a &= a_1(u,u_x) u^2_{xx} + a_2(u,u_x)u_{xx} + a_3(u,u_x),\\
b &= b_1(u,u_x) u^2_{xx} + b_2(u,u_x)u_{xx} + b_3(u,u_x),\\
c &= c_1(u,u_x) u^2_{xx} + c_2(u,u_x)u_{xx} + c_3(u,u_x).\end{aligned}$$ Equations $\alpha_i = 0$, $i=1,2,3$ are of the form $$\begin{aligned}
a_{u_{xx} u_{xx}} u_{xxx} +a a_{u_{xx}} + a_{u u_{xx}} u_x + b_{u_{xx}} + a_{u_x u_{xx}}u_{xx} &= 0,
\label{KdV:uxxxx:1}\\
b_{u_{xx} u_{xx}} u_{xxx} + c_{u_{xx}} + b_{u u_{xx}} u_x + b_{u_x u_{xx}}u_{xx} + b a_{u_{xx}} &= 0 , \label{KdV:uxxxx:2}\\
3 c_{u_{xx} u_{xx}} u_{xxx} + 3 c_{u u_{xx}}u_x + 3 c_{u_x u_{xx}}u_{xx} + 3 c a_{u_{xx}} + 1 & = 0 . \label{KdV:uxxxx:3}\end{aligned}$$ Then since functions $a$, $b$ and $c$ depend only on the variables $u$, $u_x$ and $u_{xx}$ the coefficients at $u_{xxx}$ vanish, i.e. we get $$\begin{aligned}
a = a_1(u,u_x)u_{xx} + a_2(u,u_x), \\
b = b_1(u,u_x)u_{xx} + b_2(u,u_x), \\
c = c_1(u,u_x)u_{xx} + c_2(u,u_x).\end{aligned}$$ Substituting $a$, $b$ and $c$ into the equations (\[KdV:uxxxx:1\]), (\[KdV:uxxxx:2\]) and (\[KdV:uxxxx:3\]) we obtain $$\begin{aligned}
\bigl(a^2_1+ (a_1)_{u_x}\bigr)u_{xx} + a_1 a_2 + u_x (a_1)_u + b_1 = 0,\\
\bigl( (b_1)_{u_x} + a_1 b_1 \bigr)u_{xx} + c_1 + u_x (b_1)_u + a_1 b_2 = 0,\\
3 \bigl( (c_1)_{u_x} + a_1 c_1 \bigr)u_{xx} + 3 u_{x}(c_1)_u + 3 a_1 c_2 + 1 = 0.\end{aligned}$$ Since functions $a_{i}$, $b_{i}$ and $c_{i}$, $i=1,2$ depend only on $u$ and $u_x$ the coefficients at $u_{xx}$ vanish and we get the following system of equations $$\eqalign{a^2_1+ (a_1)_{u_x} = 0, \\
(b_1)_{u_x} + a_1 b_1 = 0, \\
(c_1)_{u_x} + a_1 c_1 = 0,\\
a_1 a_2 + u_x (a_1)_u + b_1 = 0, \\
c_1 + u_x (b_1)_u + a_1 b_2 = 0 = 0, \\
3 u_{x}(c_1)_u + 3 a_1 c_2 + 1 = 0.}\label{KdV:eq13}$$ Concentrate on the last system. It is easy to check that $a_1 \neq 0$. Assume that $b_1 \neq 0$ and $c_1 \neq 0$. From system (\[KdV:eq13\]) we find $$\begin{array}{lll}
\displaystyle a_1 = \frac{1}{u_x + a_4(u)}, &\displaystyle b_1 = \frac{b_4(u)}{u_x + a_4(u)}, & \displaystyle c_1 = \frac{c_4(u)}{u_x + a_4(u)},\\
\displaystyle a_2 = -\frac{u_x (a_1)_u + b_1}{a_1}, & \displaystyle b_2 = -\frac{c_1 + u_x (b_1)_u}{a_1}, &\displaystyle c_2 = -\frac{1+3u_x (c_1)_u}{3 a_1}.
\end{array}$$ Now equations $\gamma_{i} = 0$, $i=1,2,3$ are satisfied automatically. From equation $\delta_1 = 0$ we obtain that $b_4 = b_5$ and $a_4 = a_5 u + a_6$, where $b_5$, $a_5$ and $a_6$ are arbitrary constants. From equation $\delta_2 = 0$ we get $c_4 = \frac{2}{3} u + c_5$, where $c_5$ is an arbitrary constant. Equation $\delta_3 = 0$ implies $a_5 = b_5$. Then from equation $\epsilon_1 = 0$ we obtain $a_5=a_6=b_5 = 0$. It is easy to check that equations $\epsilon_2 = 0$ and $\epsilon_3 = 0$ are identically satisfied.
Thus equation (\[KdV:inv\]) is of the form $$\label{KdV:L}
v_{xxx} = \frac{u_{xx}}{u_x} v_{xx} - \left( \frac{2}{3} u + \lambda\right) v_x + \left( \left( \frac{2}{3} u + \lambda \right)\frac{u_{xx}}{u_x} - u_x \right)v.$$ Here $\lambda=c_5$ is an arbitrary parameter.
[**Proposition 3**]{}. [*Pair of equations (\[KdV:lin\]), (\[KdV:L\]) defines the Lax pair for the KdV equation.*]{}
Potential and modified KdV equations
------------------------------------
Concentrate on the potential KdV equation $$\label{symsin:main}
u_t = u_{xxx} + \frac{1}{2} u^3_x.$$ Its linearization $$\label{symsin:lin}
v_t = v_{xxx} + \frac{3}{2}w^2 v_x,\, \mbox{where}\, w=u_x$$ does not admit any second order invariant manifold of the necessary form $v_{xx}=a(u,u_x,u_{xx},...)v_x+b(u,u_x,u_{xx},...)v$. However it admits a third order invariant manifold given by $$\label{symsin:inv1}
v_{xxx} = \frac{w_{x}}{w} v_{xx} - \left( w^2 + \lambda\right) v_x + \lambda \frac{w_{x}}{w}v.$$ Here $\lambda$ is an arbitrary parameter. The consistency condition of the equations (\[symsin:lin\]), (\[symsin:inv1\]) is equivalent to the mKdV equation $$\label{mKdV}
w_t = w_{xxx} + \frac{1}{2}w^2 w_x$$ connected with (\[symsin:main\]) by a very simple substitution $w=u_x$. In other words (\[symsin:lin\]), (\[symsin:inv1\]) define the Lax pair for the equation (\[mKdV\]) as well.
Now let us return to the sine-Gordon equation (\[s3.1eq1\]). Recall that equation (\[symsin:main\]) is a symmetry of the sine-Gordon equation. It is easily seen that equation (\[symsin:inv1\]) coincides with (\[sin:inv4\]) up to the notations.
Lax pairs for the KdV type equations from Svinolupov-Sokolov list
-----------------------------------------------------------------
Consider the following two third order differential equations $$\begin{aligned}
u_t = u_{yyy} - \frac{\gamma}{2} u^3_y - \frac{3}{2} f^2(u) u_y, \label{eq12:sym1}\\
u_{\tau} = u_{xxx} - \frac{3u_x u^2_{xx}}{2(1+u^2_{x})} - \frac{\gamma}{2} u^3_x \label{eq12:sym2}\end{aligned}$$ possessing infinite hierarchies of conservation laws [@SvinolupovSokolov]. As it is established in [@SokolovMeshkov] these equations are symmetries of the equation (\[eq12:main\]). We have proved above in §4 that equations $$\label{eq12:inv:y}
v_{yy} - \frac{f'(u)}{f(u)} u_y v_y + \frac{\lambda u_y}{f(u) \sqrt{1+u^2_x}} v_x - (f^2(u) + \lambda) v = 0,$$ $$\label{eq12:inv:x}
v_{xx} -\left(\frac{f'(u)}{f(u)}+\frac{u_xx}{u_x^2+1}\right) u_x v_x + \frac{u_x \sqrt{u_x^2+1}}{\lambda f(u)} v_y - \left( \frac{u_x^2+1}{\lambda}\right) v = 0$$ define an invariant manifold for the linearized equation (\[eq12:lin\]). It is reasonable to expect that invariant manifolds for the linearizations $$\label{eq12:sym1:lin}
v_t = v_{yyy} - \frac{3}{2} (\gamma u^2_y+ f^2(u))v_y - 3 f(u) f'(u) u_y v$$ and $$\label{eq12:sym2:lin}
v_{\tau} = v_{xxx} - \frac{3 u_x u_{xx}}{1+u^2_x} v_{xx} - \frac{3}{2} \left( \frac{(1-u^2_x)u^2_{xx}}{(1+u^2_x)^2} + \gamma u^2_x \right) v_x$$ of the symmetries (\[eq12:sym1\]) and (\[eq12:sym2\]) are closely connected with the same manifold. Indeed by applying the operators $D_y$ and $D_x$ to the equations (\[eq12:inv:y\]) and respectively to (\[eq12:inv:x\]) one can deduce the following two third order ordinary differential equations $$\begin{aligned}
\fl v_{yyy} - \frac{u_{yy}}{u_{y}} v_{yy} - \bigl( \gamma u^2_y + f^2(u) + \lambda \bigr) v_y \nonumber\\
+\left( \frac{(f^2(u)+\lambda)u_{yy}}{u_y} - 3 f(u)f'(u)u_y \right) v = 0, \label{eq12:vyyy}\end{aligned}$$ $$\begin{aligned}
\fl v_{xxx} - \frac{(1+3 u^2_x)u_{xx}}{(1+u^2_x)u_x} v_{xx} \nonumber\\
-\left( \lambda^{-1} + \gamma u^2_x + \frac{u_x u_{xxx}}{1+u^2_x} - \frac{3 u^2_x u^2_{xx}}{(1+u^2_x)^2} \right) v_x +\frac{u_{xx}}{u_x} \lambda^{-1} v=0. \label{eq12:vxxx}\end{aligned}$$ It is easily checked by a direct computation that equations (\[eq12:vyyy\]) and (\[eq12:vxxx\]) define invariant manifolds for (\[eq12:sym1:lin\]) and (\[eq12:sym2:lin\]) correspondingly. Conclude the reasonings with the following statement.
[**Proposition 4**]{}.
*1) Linear equations (\[eq12:sym1:lin\]), (\[eq12:vyyy\]) define the Lax pair for the equation (\[eq12:sym1\]);*
2\) linear equations (\[eq12:sym2:lin\]), (\[eq12:vxxx\]) define the Lax pair for the equation (\[eq12:sym2\]).
Volterra type integrable chains
-------------------------------
In this section we discuss the semi-discrete equations of the form $\frac{\partial}{\partial t}u_n=f(u_{n+1},u_n,u_{n-1})$ with the sought function $u=u_n(t)$, depending on the discrete $n$ and continuous $t$. The direct method for constructing the Lax pairs through linearization can be applied to the discrete models as well. As illustrative examples we consider the modified Volterra chain $$\label{sd2}
\frac{d p_{n}}{dt} = -p^2_{n}(p_{n+1}-p_{n-1})$$ and the equation $$\label{sd0}
\frac{d u_{n}}{dt} = \frac{1}{u_{n+1}-u_{n-1}}$$ found in [@Yamilov83]. These two equations are related to each other by a very simple Miura type transformation $$\label{sd1}
\frac{d u_{n}}{dt} = p_{n}, \qquad p_{n}=\frac{1}{u_{n+1}-u_{n-1}}.$$
Note that the coefficients of the linearization $$\label{sd3}
\frac{d v_{n}}{dt} = -p^2_{n} (v_{n+1}-v_{n-1})$$ of the equation (\[sd0\]) depend on the variable $p_n$. This explains why we study these two equations together. Look for the invariant manifold of the third order: $$\label{sd4}
v_{n+2} = a v_{n+1} + b v_{n} + c v_{n-1}$$ to the equation (\[sd3\]) with the coefficients $a$, $b$, $c$ depending on a finite set of the dynamical variables $p_n,p_{n\pm 1},...\,$. Actually we suppose that (\[sd4\]) defines an invariant manifold for any choice of the solution $p=p_n(t)$ to the equation (\[sd2\]).
The coefficients $a$, $b$, $c$ are found from the equation $$\label{sd5}
\frac{d}{dt} ( a v_{n+1} + b v_{n} + c v_{n-1}) = D_{n}^2 \left( -p^2_{n} (v_{n+1}-v_{n-1}) \right).$$ Studying the equation (\[sd5\]) we assume that the variables $\left\{p_k\right\}_{k=-\infty}^{\infty}$, $v_n$, $v_{n+1}$, $v_{n-1}$ are independent dynamical variables. Omitting the simple but tediously long computations we give only the answer: $$a=-\frac{p_{n}}{p_{n+1}} + \frac{\lambda}{p^2_{n+1}},\quad b=1 - \frac{\lambda}{p_{n}p_{n+1}},\quad c=\frac{p_{n}}{p_{n+1}}.$$ Therefore the invariant manifold searched is of the form: $$\begin{aligned}
\fl v_{n+2} = \left( -\frac{p_{n}}{p_{n+1}} \right. & + \left.\frac{\lambda}{p^2_{n+1}}\right) v_{n+1}\nonumber\\
&+ \left( 1 - \frac{\lambda}{p_{n}p_{n+1}} \right) v_{n}
+ \frac{p_{n}}{p_{n+1}} v_{n-1}. \label{invMv}\end{aligned}$$
[**Proposition 5**]{}. [*The consistency condition of the equations (\[sd3\]) and (\[invMv\]) coincides with the equation (\[sd2\]).*]{}
Construction of the recursion operators and conservation laws via newly found Lax pairs
=======================================================================================
It was observed that the Lax pairs for the integrable equations found above essentially differ from their classical counterparts. In this section we discuss some useful properties of the newly found Lax pairs. We show, for instance, that they provide a very convenient tool for searching the recursion operators and conservation laws for integrable models. As illustrative examples we take the KdV and pKdV equations, the Volterra type chain (\[sd0\]) etc. We show that the equation of the invariant manifold to the linearized equation is easily transformed into the recursion operator.
Evaluation of the recursion operators for the KdV type equations
----------------------------------------------------------------
Let us start with the KdV equation (\[s1kdv\]). We can rewrite equation (\[s1KdV:L\]) of the invariant manifold in the following form $$\begin{aligned}
\left( D^3_x - \frac{u_{xx}}{u_x} D^2_x + \frac{2 u}{3} D_x + u_x - \frac{2 u u_{xx}}{3 u_x} \right) v = \lambda u_x D_x \frac{1}{u_x} v. \label{6.1}\end{aligned}$$ We now multiply (\[6.1\]) from the left by the operator $ u_x D^{-1}_x \frac{1}{u_x} $ and obtain a formal eigenvalue problem of the form $$\begin{aligned}
R v = \lambda v \label{6.2}\end{aligned}$$ for the operator $$\begin{aligned}
R = u_x D^{-1}_x \frac{1}{u_x} \left( D^3_x - \frac{u_{xx}}{u_x} D^2_x + \frac{2 u}{3} D_x + u_x - \frac{2 u u_{xx}}{3 u_x}\right) \label{6.3}\end{aligned}$$ An amazing fact is that $ R $ coincides with the recursion operator for the KdV equation. Indeed, we have $$\begin{aligned}
R v = u_x D^{-1}_x \left( \frac{v_{xxx} u_x - u_{xx} v_{xx}}{u^2_x} + \frac{2}{3} \frac{u v_x}{u_x} + v - \frac{2}{3} \frac{u u_{xx} v}{u^2_x} \right) = \\ v_{xx} + u_x D^{-1}_x \left( \frac{2}{3} \frac{u v_x}{u_x} + v - \frac{2}{3} \frac{u u_{xx} v}{u^2_x} \right). \end{aligned}$$ It can be simplified due to the relation $$\begin{aligned}
\frac{u v_x}{u_x} - \frac{u u_{xx} v}{u^2_x} =\left( \frac{u v}{u_x}\right)_{x} - v\end{aligned}$$ and reduced to the form $$\begin{aligned}
R v = \left( D^2_x + \frac{2}{3} u + \frac{1}{3} u_x D^{-1}_x\right) v.\end{aligned}$$ In a similar way we derive from (\[symsin:inv1\]) the recursion operator to the potential KdV equation (\[symsin:main\]). Indeed rewrite (\[symsin:inv1\]) as follows $$\begin{aligned}
\left( D^3_x - \frac{u_{xx}}{u_x} D^2_x + u^2_x D_x \right) v = - \lambda u_x D_x \frac{1}{u_x} v. \label{6.4}\end{aligned}$$ Equation (\[6.4\]) implies $$\begin{aligned}
R v = - \lambda v\end{aligned}$$ where $$\begin{aligned}
R = u_x D^{-1}_x \frac{1}{u_x} \left( D^3_x - \frac{u_{xx}}{u_x} D^2_x + u^2_x D_x \right).\end{aligned}$$ Simplify the expression for $Rv$: $$\begin{aligned}
R v= u_x D^{-1}_x \left( \frac{v_{xxx} u_x - u_{xx} v_{xx}}{u^{2}_x} + u_x v_x \right) = v_{xx} + u_x D^{-1}_x u_x D_x v.\end{aligned}$$ Apparently operator $ R = D^2_x + u_x D^{-1}_x u_x D_x $ coincides with the recursion operator for (\[symsin:main\]).
Invariant manifolds (\[eq12:vyyy\]) and (\[eq12:vxxx\]) for the linearized equations (\[eq12:sym1:lin\]) and (\[eq12:sym2:lin\]) allow constracting of the recursion operators $$\begin{aligned}
R = D^2_y-\gamma u^2_y - \frac{f^2(u)}{u_y} + u_y D^{-1}_y \left( \gamma u_{yy} - f f'\right)\end{aligned}$$ and $$\begin{aligned}
R = D^2_x - \frac{2 u_x u_{xx}}{1 + u^2_x} + u_x D^{-1}_x \left( \frac{u_{xxx}}{1 + u^2_x} + \frac{u_x u^2_{xx}}{(1 + u^2_x)^2} - \gamma u_x \right) D_x\end{aligned}$$ for the Svinolupov-Sokolov equations (\[eq12:sym1\]) and (\[eq12:sym2\]) respectively.
Recursion operators via the Lax pair for a Volterra type chain
--------------------------------------------------------------
We proceed with an example of the discrete model (\[sd0\]). Let us write equation (\[invMv\]) defining the invariant manifold for (\[sd3\]) as follows $$\begin{aligned}
\left( D^2_n + \frac{p_n}{p_{n+1}} D_n - 1 -\frac{p_n}{p_{n+1}} D^{-1}_n \right) v_n = \frac{\lambda}{p_{n+1}} \left( D_n - 1 \right) \frac{v_n}{p_n}, \label{6.2.1}\end{aligned}$$ where $ D_n $ is the shift operator acting due to the rule $ D_n a(n) = a(n+1) $. We multiply (\[6.2.1\]) from the left by the factor $ p_n \left(D_n - 1\right)^{-1} p_{n+1} $ and then get $$\begin{aligned}
R v = \lambda v \label{6.2.2}\end{aligned}$$ where $ R = p_n \left(D_n - 1\right)^{-1} p_{n+1} \left( D^2_n + \frac{p_n}{p_{n+1}} D_n - 1 -\frac{p_n}{p_{n+1}} D^{-1}_n \right). $ After some elementary transformations the operator $ R $ is reduced to the form $$\begin{aligned}
R = p^2_n \left(D_n +D^{-1}_n\right) + 2 p_n p_{n-1} + 2 p_n \left(D_n - 1\right)^{-1} \left(p_{n-1} - p_{n+1}\right) \label{6.2.3}\end{aligned}$$ Operator $ R $ given by (\[6.2.3\]) defines the recursion operator for the chain (\[sd0\]). For instance, by applying the operator $ R $ to $ \frac{1}{u_{n+1} - u_{n-1}} $ we obtain the r.h.s. of the equation constructed in [@Tongas] $$\begin{aligned}
\frac{\partial}{\partial\tau}u_{n} = \frac{1}{\left(u_{n+1} - u_{n-1}\right)^2} \left(\frac{1}{u_{n+2} - u_{n}} + \frac{1}{u_{n} - u_{n-2}}\right) \label{6.2.4}\end{aligned}$$ as a fifth-point symmetry of the equation (\[sd0\]). Note that the whole hierarchy of the symmetries for the chain (\[sd0\]) is described in [@svinin].
Conservation laws via the Lax pair for a Volterra type chain
------------------------------------------------------------
Let us consider an equation of the form $$\frac{d u_{n}}{dt} = \frac{1}{u_{n+1}-u_{n-1}}. \label{maindiscrEq}$$ Due to the relations (\[sd1\]) between $u$ and $p$ a pair of the equations (\[sd3\]), (\[invMv\]) define a Lax pair to the chain (\[maindiscrEq\]) as well.
Rewrite the scalar Lax pair (\[sd3\]), (\[invMv\]) in the matrix form $$\label{VLax}
y_{n+1} = f y_n, \qquad \frac{{\rm d} y_{n}}{\rm{dt}} = g y_n,$$ where $$\begin{aligned}
\label{VLgf}
f = \left( \begin{array}{ccc}
-\frac{p_n}{p_{n+1}} + \frac{\lambda}{p^2_{n+1}} & 1- \frac{\lambda}{p_n p_{n+1}} & \frac{p_n}{p_{n+1}}\\
1 & 0 & 0\\
0 & 1 & 0
\end{array} \right),\\
g= \left( \begin{array}{ccc}
p_n p_{n+1} - \lambda & \frac{p_{n+1}}{p_n} \lambda & -p_n p_{n+1}\\
-p^2_n & 0 & p^2_n\\
p_{n-1} p_n & -\frac{p_{n-1}}{p_n} \lambda & -p_{n-1}p_n + \lambda
\end{array} \right).\end{aligned}$$ To construct the conservation laws, we apply the method of the formal diagonalization suggested in [@HabLOMI; @HabYang] and developed in [@HabPoptsova].
The first equation in (\[VLax\]) has singular point $\lambda = \infty$ ($f$ has a pole at $\lambda = \infty$). It can be checked that the potential $f$ is represented as follows $$f = \alpha Z \beta,$$ where $$\begin{aligned}
\alpha = \left(\begin{array}{ccc}
\frac{1}{p^2_{n+1}} - \frac{p_n}{p_{n+1}}\lambda^{-1} & 0 & 0\\
\lambda^{-1} & \frac{p_{n+1}}{p_n} & 0\\
0 & 1 & 1
\end{array} \right), \\
\beta = \left( \begin{array}{ccc}
1 & - \frac{p_{n+1}}{p_n} & \frac{p_n p_{n+1}}{\lambda - p_n p_{n+1}}\\
0 & 1 & -\frac{p^2_n}{\lambda - p_n p_{n+1}}\\
0 & 0 & \frac{p^2_n \lambda}{\lambda - p_n p_{n+1}}
\end{array} \right)\end{aligned}$$ are analytic and non-degenerate around $\lambda = \infty$, $Z$ is a diagonal matrix of the form $$Z = \left( \begin{array}{ccc}
\lambda & 0 & 0\\
0 & 1 & 0\\
0 & 0 & \lambda^{-1}
\end{array} \right).$$ The change of variables $\psi = \beta y$ is reducing the first system in (\[VLax\]) to the special form $$\label{Vsf}
\psi_{n+1} = PZ \psi_n,$$ where $$\begin{aligned}
\fl P = D_n(\beta) \alpha =
\left( \begin{array}{ccc}
\frac{1}{p^2_{n+1}} - \frac{p_n + p_{n+2}}{p_{n+1}} \lambda^{-1} & -\frac{p_{n+2}}{p_n} \left( 1 - \frac{p_n p_{n+1}}{\lambda - p_{n+1} p_{n+2}}\right) & \frac{p_{n+1} p_{n+2}}{\lambda - p_{n+1} p_{n+2}}\\
\lambda^{-1} & \frac{p_{n+1}}{p_n} \left( 1 - \frac{p_n p_{n+1}}{\lambda - p_{n+1} p_{n+2}} \right) & -\frac{p^2_{n+1}}{\lambda - p_{n+1} p_{n+2}}\\
0 & \frac{p^2_{n+1} \lambda}{\lambda - p_{n+1} p_{n+2}} & \frac{p^2_{n+1} \lambda}{\lambda - p_{n+1} p_{n+2}}
\end{array} \right).\end{aligned}$$ The function $P(\lambda)$ together with $P^{-1}(\lambda)$ are analytic around $\lambda = \infty$: $$\label{VP}
P(\lambda) = \sum_{i \geq 0}^{\infty} P^{(i)} \lambda^{-i},$$ where $$\begin{aligned}
\fl P^{(0)} = \left( \begin{array}{ccc}
\frac{1}{p^2_{n+1}} & -\frac{p_{n+2}}{p_n} & 0\\
0 & \frac{p_{n+1}}{p_n} & 0\\
0 & p^2_{n+1} & p^2_{n+1}
\end{array} \right), \quad
P^{(1)} = \left( \begin{array}{ccc}
-\frac{p_{n+1}(p_n + p_{n+2})}{p^2_{n+1}} & p_{n+1} p_{n+2} & p_{n+1} p_{n+2}\\
1 & -p^2_{n+1} & - p^2_{n+1}\\
0 & p^3_{n+1}p_{n+2} & p^3_{n+1}p_{n+2}
\end{array}
\right),\\
\fl P^{(2)} = \left( \begin{array}{ccc}
0 & p^2_{n+1} p^2_{n+2} & p^2_{n+1} p^2_{n+2} \\
0 & -p^3_{n+1} p_{n+2} & -p^3_{n+1} p_{n+2}\\
0 & p^4_{n+1}p^2_{n+2} & p^4_{n+1}p^2_{n+2}
\end{array} \right), \quad
P^{(3)} = \left( \begin{array}{ccc}
0 & p^3_{n+1} p^3_{n+2} & p^3_{n+1} p^3_{n+2} \\
0 & -p^4_{n+1} p^2_{n+2} & -p^4_{n+1} p^2_{n+2}\\
0 & p^5_{n+1}p^3_{n+2} & p^5_{n+1}p^3_{n+2}
\end{array} \right) \quad \mbox{etc.}\end{aligned}$$ The leading principal minors of $P(\lambda)$ $$\begin{aligned}
\det_1 P (\lambda = \infty) = \frac{1}{p^2_{n+1}}, \qquad \det_2 P (\lambda = \infty) = \frac{1}{p_n p_{n+1}} ,\\
\det_3 P(\lambda = \infty) = \det P(\lambda = \infty) = \frac{p_{n+1}}{p_n} \end{aligned}$$ do not vanish if the variable $p_n$ satisfy the inequality $p_n \neq 0$ for all $n$. According to the Proposition 1 in [@HabYang] the first system in (\[VLax\]) can be diagonalized, i.e. there exist formal series $$\begin{aligned}
T = T^{(0)} + T^{(1)} \lambda^{-1} + T^{(2)} \lambda^{-2} + \cdots, \label{VT}\\
h = h^{(0)} + h^{(1)} \lambda^{-1} + h^{(2)} \lambda^{-2} + \cdots \label{Vh}\end{aligned}$$ such that the formal change of the variables $\psi = T \varphi $ converts the system (\[Vsf\]) to the system of the diagonal form $$\label{diagformV}
\varphi_{n+1} = h Z \varphi_n.$$ Thus we see that the formal change of variables $y=R \varphi = \beta^{-1} T \varphi $, reduces the first system in investigated Lax pair (\[VLax\]) to the form (\[diagformV\]). By construction $R = \beta^{-1} T$ is a formal series of the form $$\label{VR}
R = R^{(0)} + R^{(1)} \lambda^{-1} + R^{(2)} \lambda^{-2} + \cdots$$ It follows from (\[Vh\])–(\[VR\]) that diagonalizable system (\[diagformV\]) admits an asymptotic representation of the solution to the direct scattering problem $$\label{Vformalsolution}
y_n(\lambda) = R(n,\lambda)e^{\sum_{s=n_0}^{n-1} \log h(s, \lambda)} Z^n$$ with the “amplitude” $A = R(n,\lambda)$ and “phase” $\phi = n \log Z + \sum_{s=n_0}^{n-1} \log h(s,\lambda)$.
Turn back to the problem of diagonalization. By solving the following equation $$\label{VThPeq}
D_n(T) h = P (\lambda) \bar{T}, \qquad \bar{T} = ZTZ^{-1}$$ we find the formal series $T$ and $h$ $$\begin{aligned}
\fl T = \left(\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & p_{n-1} p_ n & 1
\end{array} \right) +
\left( \begin{array}{ccc}
0 & \frac{p^2_{n+1} p_{n+2}}{p_{n}} & 0\\
p^2_{n} & 0 & 0\\
0 & p_n p^2_{n-1}(p_n - p_{n-2}) & 0
\end{array}
\right) \lambda^{-1} \\
+ \left( \begin{array}{ccc}
0 & \frac{p^2_{n+1} p^2_{n+2} (p_{n+1}+p_{n+3})}{p_n} & 0\\
p^3_n (2 p_{n-1} + p_{n+1}) & 0 & p_n p_{n+1}\\
p^4_n p^2_{n-1} & T^{(2)}_{32} & 0
\end{array} \right) \lambda^{-2} + \cdots,\end{aligned}$$ $$\begin{aligned}
\fl h = \left( \begin{array}{ccc}
\frac{1}{p^2_{n+1}} & 0 & 0\\
0 & \frac{p_{n+1}}{p_n} & 0\\
0 & 0 & p^2_{n+1}
\end{array} \right)
+ \left( \begin{array}{ccc}
-\frac{p_{n+1} (p_n + p_{n+2})}{p^2_{n+1}} & 0 & 0\\
0 & \frac{p^2_{n+1} ( p_{n+2} - p_n)}{p_n} & 0\\
0 & 0 & p^3_{n+1} (p_n + p_{n+2})
\end{array} \right) \lambda^{-1} \\
+ \left( \begin{array}{ccc}
-p_n p_{n+2} & 0 & 0\\
0 & h^{(2)}_{22} & 0\\
0 & 0 & p^4_{n+1}(3 p_n p_{n+2} + p^2_n + p^2_{n+2})
\end{array} \right) \lambda^{-2} + \cdots\end{aligned}$$ where $$\begin{aligned}
\fl T^{(2)}_{32} = p_n p_{n-1} (p^2_{n-2} p^2_{n-1} + p^2_{n-2}p_{n-1}p_{n-3} + p^2_{n-1}p^2_n + 2 p^2_{n-1} p_n p_{n-2} - p_{n+2} p^2_{n+1} p_n),\\
h^{(2)}_{22} = -\frac{p^2_{n+1} (-p^2_{n+2} p_{n+3} - p^2_{n+2}p_{n+1} +p^2_n p_{n-1} + p_n p_{n+1} p_{n+2})}{p_n}.
\end{aligned}$$ According to the general scheme the second system of the Lax pair (\[VLax\]) is diagonalized by the same linear change of the variables $y = R \varphi$, where $$\begin{aligned}
\fl R= \beta^{-1} T = \left( \begin{array}{ccc}
1 & -\frac{p_{n+1}}{p_n} & 0\\
0 & 1 & 0\\
0 & -p_{n-1} p_n & p^2_n
\end{array} \right) \\
\fl + \left( \begin{array}{ccc}
0 & - \frac{p^2_{n+1} p_{n+2}}{p_n} & p_n p_{n+1}\\
-p^2_n & p_n p_{n+1} & -p^2_n\\
p_{n-1} p^3_n & -p_{n-1} p^2_n p_{n+1} - p^2_{n-1} p^2_n - p_{n-2} p^2_{n-1} p_n & p^3_n(p_{n+1} + p_{n-1})
\end{array} \right) \lambda^{-1} \\
+ \left( \begin{array}{ccc}
R^{(2)}_{11} & R^{(2)}_{12} & R^{(2)}_{13}\\
R^{(2)}_{21} & R^{(2)}_{22} & R^{(2)}_{23}\\
R^{(2)}_{31} & R^{(2)}_{32} & R^{(2)}_{33}
\end{array} \right) \lambda^{-2} + \cdots,\end{aligned}$$ $$\begin{aligned}
R^{(2)}_{11} = p_n p^2_{n+1} p_{n+2}, \\
R^{(2)}_{12} = -\frac{p^2_{n+1} p_{n+2} (p_{n+2} p_{n+3} + p_{n+1} p_{n+2} + p_n p_{n+1})}{p_n},\\
R^{(2)}_{13} = p_n p^2_{n+1} (p_n + p_{n+2}),\\
R^{(2)}_{21} = -p^3_n (2 p_{n-1} + p_{n+1}),\\
R^{(2)}_{22} = p_n p^2_{n+1} p_{n+2} + 3 p_{n-1} p^2_n p_{n+1} + p^2_n p^2_{n+1},\\
R^{(2)}_{23} = -3 p^3_n p_{n+1},\\
R^{(2)}_{31} = 2 p^2_{n-1} p^4_{n} + p_{n-1} p^4_n p_{n+1} + p_{n-2} p^2_{n-1} p^3_{n},\\
R^{(2)}_{32} = -p_{n-1} p_n (3 p_{n-1} p^2_n p_{n+1} + p^2_n p^2_{n+1} + p_{n-2} p_{n-1} p_n p_{n+1} + p^2_{n-2} p^2_{n-1} \\
+ p_{n-3} p^2_{n-2} p_{n-1} + p^2_{n-1} p^2_n + 2 p_{n-2} p^2_{n-1} p_n).\end{aligned}$$ This change of the variables reduces the second system in (\[VLax\]) to the form $\frac{{\rm d} y_n}{{\rm d} t} = S y_n$ with $$\begin{aligned}
\fl S = -R^{-1} R_t + R^{-1} g R
= \left( \begin{array}{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -1
\end{array} \right) \lambda\\
+
\left( \begin{array}{ccc}
2 p_n p_{n+1} & 0 & 0\\
0 & p_n (p_{n-1} - p_{n+1}) & 0\\
0 & 0 & -2p_n p_{n+1}
\end{array} \right)
+\left(\begin{array}{ccc}
S^{(2)}_{11} & 0 & 0\\
0 & S^{(2)}_{22} & 0\\
0 & 0 & S^{(2)}_{33}
\end{array} \right) \lambda^{-1} + \cdots,\end{aligned}$$ where $$\begin{aligned}
S^{(2)}_{11} = p_n p_{n+1} (p_{n+1} p_{n+2} + p_{n-1} p_n),\\
S^{(2)}_{22} = p_n p_{n+1} (p_{n-1} p_n - p_{n+1} p_{n+2}),\\
S^{(2)}_{33} = -p_n p_{n+1} (p_{n+1} p_{n+2} + p_{n-1} p_n).\end{aligned}$$ According the paper [@HabYang] the equation $$D_t \ln h = (D_n - 1) S$$ generates the infinite series of conservation laws for the equations (\[sd2\]). We write down in an explicit form three conservation laws from the infinite sequence obtained by the diagonalization procedure $$\begin{aligned}
\fl D_t \left(\ln \frac{1}{p_{n+1}}\right) = (D_n - 1) p_n p_{n+1},\\
\fl D_t (-p_{n+1} (p_n + p_{n+2})) = (D_n - 1) p_n p_{n+1} (p_{n-1}p_n + p_{n+1} p_{n+2} ),\\
\fl D_t \left( -2 p_n p^2_{n+1} p_{n+2} - \frac{1}{2} p^2_n p^2_{n+1} - \frac{1}{2} p^2_{n+1} p^2_{n+2} \right) \\
= (D_n - 1) p^2_n p^2_{n+1} (2 p_{n-1} p_{n+2} + p_{n+1} p_{n+2} + p_{n-1} p_n ).\end{aligned}$$ Being rewritten in terms of $u_n$ these relations give the conservation laws for the equation (\[maindiscrEq\]): $$\begin{aligned}
\fl D_t \ln (u_{n+2} - u_n) = (D_n-1)\frac{1}{(u_{n+1}-u_{n-1})(u_{n+2} - u_n)},\\
\fl D_t \frac{u_{n-1}-u_{n+3}}{(u_{n+1}-u_{n-1})(u_{n+2} - u_n)(u_{n+3} - u_{n+1})} \\
= (D_n-1) \left( \frac{1}{(u_n - u_{n-2})(u_{n+1}-u_{n-1})^2 (u_{n+2}-u_n)} \right.\\
\left.+ \frac{1}{(u_{n+1}-u_{n-1})(u_{n+2}-u_n)^2(u_{n+3} - u_{n+1})} \right),\\
\fl D_t \left( -\frac{2}{(u_{n+1}-u_{n-1})(u_{n+2}-u_n)^2(u_{n+3}-u_{n+1})} \right.\\
-\frac{1}{2(u_{n+1}-u_{n-1})^2(u_{n+2} - u_n)^2}\\
\left.- \frac{1}{2 (u_{n+2}-u_n)^2 (u_{n+3}-u_{n+1})^2}\right) \\
= (D_n-1) \left( \frac{2}{(u_n - u_{n-2})(u_{n+1}-u_{n-1})^2 (u_{n+2} - u_n)^2 (u_{n+3} - u_{n+1})} \right. \\
+ \frac{1}{(u_{n+1} - u_{n-1})^2 (u_{n+2} - u_n)^3 (u_{n+3} - u_{n+1})} \\
\left.+ \frac{1}{(u_n - u_{n-2})(u_{n+1} - u_{n-1})^3 (u_{n+2} - u_n)^2} \right).\end{aligned}$$ These conservation laws coincide with those found earlier (see, for instance, [@Yamilov82], [@Yamilov83]).
Conclusions {#conclusions .unnumbered}
===========
There is a large set of classification methods allowing classes of integrable nonlinear PDEs and their discrete analogues to be described. For studying the analytical properties of these equations one needs the Lax pairs. Therefore the problem of creating convenient algorithms for constructing the Lax pairs is relevant. In the present article such a method is suggested. For the evolution type integrable equation the Lax pair consists of the linearized equation and the equation of its invariant manifold. In the case of the hyperbolic equations to obtain the Lax pair we use the Laplace cascade in addition to the invariant manifold. The method is applied to equations (\[eq12:main\]), (\[eq12:sym1\]) and (\[eq12:sym2\]) known to be integrable for which the Lax pairs have not been constructed before.
An interesting observation is connected with the Laplace cascade of the sine-Gordon equation. It is proved that in this case the cascade admits a finite-dimensional reduction which generates the Lax pair to the sine-Gordon model. We conjecture that the Laplace cascade corresponding to any hyperbolic type integrable equation admits a finite-dimensional reduction.
We considered examples showing that our method leads to true Lax pairs having useful applications. For the Lax pair of the Volterra type chain we found an asymptotic eigenfunction which allowed an infinite set of conservation laws to be constructed. It is also shown that these Lax pairs allow the recursion operators to be constructed describing infinite hierarchies of the higher symmetries and invariant manifolds for the given nonlinear equation.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors gratefully acknowledge financial support from a Russian Science Foundation grant (project 15-11-20007).
Appendix. How to look for the invariant manifolds for the linearization of a nonlinear hyperbolic type equation
===============================================================================================================
It is well known that the linear hyperbolic type equation $v_{xy}=p(x,y)v_x+q(x,y)v_y+r(x,y)v$ might admit linear invariant manifolds (see [@Kaptsov]). Here we consider equation (\[s3eq2\]) obtained by linearizing an essentially nonlinear equation (\[s3eq1\]). We request that at least one of the coefficients $a$, $b$, $c$ in (\[s3eq2\]) depends on at least one of the dynamical variables $u$, $u_1$, $\bar u_1$. We look for a linear invariant manifold of the form $$\label{s4eq77}
(p(j)D_y^j+p(j-1)D_y^{j-1}+ ... +p(1)D_y+p(0)+q(1)D_x +...+q(k)D_x^{k})v=0.$$ It is assumed that all the coefficients $p(i)$ and $q(m)$ can depend on $x$, $y$ and on a finite set of the dynamical variables $u,u_1,\bar u_1,u_{2}, \bar u_{2},...$. More precisely we study a family of equations (\[s3eq2\]), (\[s4eq77\]) depending on a functional parameter $u(x,y)$ such that when $u=u(x,y)$ ranges a set of all solutions to the equation (\[s3eq1\]) then (\[s4eq77\]) ranges a set of invariant manifolds for the corresponding equation (\[s3eq2\]). In this case equation (\[s4eq77\]) generates an overdetermined system of differential equations for defining the searched coefficients $p(i)$ and $q(m)$.
As an illustrative example we consider equation (\[s3.1eq2\]) obtained by linearizing the sine-Gordon equation. We search an invariant manifold for (\[s3.1eq2\]) in the following form $$\label{sin:inv}
v_{yy} + a v_y + b v_x + c v = 0.$$ Here $a$, $b$ and $c$ are functions depending on a finite number of the dynamical variables $u,u_1,\bar u_1,u_{2}, \bar u_{2},...$. Apply the operator $D_x$ to (\[sin:inv\]) and rewrite the obtained result as follows $$\begin{aligned}
\label{sin:eq5}
\fl v_{xx} =
\frac{1}{b} \left( \sin(u) u_y v-\cos(u) (v_y+a v)-D_x(a) v_y-D_x(b) v_x-D_x(c) v-c v_x \right).\end{aligned}$$
Now apply the operator $D_y$ to this equation and simplify the result by means of the equation (\[s3.1eq2\]). We arrive at the equation $$v_{yy} + \tilde{a} v_y + \tilde{b} v_x + \tilde{c} v = 0,$$ which should coincide according to the definition above with equation (\[sin:inv\]), i.e. $$\tilde{a} = a, \quad \tilde{b} = b, \quad \tilde{c} = c.$$ These conditions give rise to the following equalities $$\begin{aligned}
\fl D_yD_x(a) b-D_y(b) \cos(u)-2 \sin(u) u_y b+D_x(c) b- D_x(a)(a b+D_y(b)) = 0,\label{sin:coeff:vy}\\
\fl D_yD_x(b) b-D_y(b) D_x(b)+D_y(c) b-D_y(b) c-b^2 D_x(a) = 0,\label{sin:coeff:vx}\\
\fl D_yD_x(c) b-D_y(b) D_x(c)-c b D_x(a)
+ \cos(u)\left(- u_y^2 b+D_y(a) b+D_x(b) b-D_y(b) a \right)\nonumber\\
+\sin(u)\left(- u_{yy} b-b^2 u_x-a u_y b+D_y(b) u_y\right) = 0.\label{sin:coeff:v}\end{aligned}$$
Assume that $a = a(u,u_x,u_y)$, $b=b(u,u_x,u_y)$ and $c = c(u,u_x,u_y)$ and substitute these functions into (\[sin:coeff:vy\]), (\[sin:coeff:vx\]) and (\[sin:coeff:v\]). Eliminating the mixed derivatives of $u$ due to equation (\[s3.1eq1\]) we obtain three equations of the following form $$\alpha_{i}(u,u_x,u_y)u_{xx}u_{yy} + \beta_{i}(u,u_x,u_y)u_{xx} + \gamma_{i}(u,u_x,u_y)u_{yy} + \delta_{i}(u,u_x,u_y) = 0,$$ $i=1,2,3$. Since the functions $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$ and $\delta_{i}$, $i=1,2,3$ depend only on $u$, $u_x$ and $u_y$, we should have the coefficients at $u_{xx} u_{yy}$, $u_{xx}$, $u_{yy}$ and remaining terms equal to zero, i.e. $$\begin{aligned}
\fl \alpha_{i}(u,u_x,u_y) &= 0, \qquad \beta_{i}(u,u_x,u_y) = 0, \qquad
\gamma_{i}(u,u_x,u_y) &= 0,\qquad \delta_{i}(u,u_x,u_y) = 0\label{sin:mainsys}\end{aligned}$$ for all $u$, $u_x$ and $u_y$, $i=1,2,3$. Here $$\begin{aligned}
\alpha_1 & = & b a_{u_xu_y}-b_{u_y} a_{u_x},\label{sin:alpha1}\\
\alpha_2 &=& b b_{u_xu_y}-b_{u_y} b_{u_x}, \label{sin:alpha2}\\
\alpha_3 & =& b c_{u_xu_y}-c_{u_x} b_{u_y},\label{sin:alpha3} \\
\beta_1 &= & b c_{u_x}-a b a_{u_x}-b_u u_y a_{u_x}-b_{u_x} \sin(u) a_{u_x}\nonumber\\
&&+b a_{u_x u_x} \sin(u)+b a_{u u_x} u_y,\label{sin:beta1}\\
\beta_2 &= & b b_{u u_x} u_y+b b_{u_x u_x} \sin(u)-b_u u_y b_{u_x}-b^2 a_{u_x}-b_{u_x}^2 \sin(u),\label{sin:beta2}\\
\beta_3 & = & b c_{u_x u_x} \sin(u)+b c_{u u_x} u_y+b b_{u_x} \cos(u)-c b a_{u_x}\nonumber\\
&& -b_{u_x} \sin(u) c_{u_x}-b_u u_y c_{u_x},\label{sin:beta3}\\
\gamma_1 &=& -\cos(u) b_{u_y}+b \sin(u) a_{u_y u_y}+b u_x a_{u u_y}-b_{u_y} a_u u_x-b_{u_y} a_{u_y} \sin(u),\nonumber\\
\gamma_2 &=& b u_x b_{u u_y}-b_{u_y} b_u u_x+b \sin(u) b_{u_y u_y}-b_{u_y}^2 \sin(u)+b c_{u_y}-c b_{u_y},\nonumber\\
\gamma_3 &=& -b_{u_y} c_u u_x-b \sin(u)-b_{u_y} c_{u_y} \sin(u)+b a_{u_y} \cos(u)+\sin(u) u_y b_{u_y}\nonumber\\
&&+b u_x c_{u u_y}-a \cos(u) b_{u_y}+b \sin(u) c_{u_y u_y},\nonumber\\
\delta_1 &= &(b_{u_x} a_{u_y}-b a_{u_xu_y}) \cos^2(u)- b_{u_x}\cos(u) \sin(u)\nonumber\\
& &+(b a_{u_y} u_y+b a_{u_x} u_x-b_{u} u_y) \cos(u)\nonumber\\
& &+(b a_{u}-2 u_y b+b a_{uu_y} u_y-b_{u} u_y a_{u_y})\sin(u)\nonumber\\
& & + (-a b a_{u_y}-b_{u_x} a_{u} u_x+b u_x a_{uu_x}+b c_{u_y}) \sin(u)\nonumber\\
& &+b u_x a_{uu} u_y-b_{u} u_y a_{u} u_x-a b a_{u} u_x+b a_{u_xu_y}-b_{u_x} a_{u_y}+b c_{u} u_x,\nonumber\\
\delta_2&=& (b_{u_x} b_{u_y}-b b_{u_xu_y}) \cos^2(u)+b (b_{u_x} u_x+b_{u_y} u_y) \cos(u)\nonumber\\
&&+(-b_{u} u_y b_{u_y}-b_{u_x} b_{u} u_x+b u_x b_{uu_x})\sin(u)\nonumber\\
&&+(b b_{uu_y} u_y-b^2 a_{u_y}-c b_{u_x}+b b_{u}+b c_{u_x}) \sin(u)\nonumber\\
&&+b b_{u_xu_y}+b c_{u} u_y-b_{u}^2 u_y u_x-c b_{u} u_y-b^2 a_{u} u_x+b u_x b_{uu} u_y-b_{u_x} b_{u_y},\nonumber\\
\delta_3 &=& (b_{u_x} c_{u_y}-b c_{u_xu_y}-u_y b_{u_x}) \cos^2(u)+(b a_{u_x}-a b_{u_x}+b b_{u_y}) \cos(u) \sin(u)\nonumber\\
&& +(b b_{u} u_x-a b_{u} u_y+b a_{u} u_y+b c_{u_x} u_x+b c_{u_y} u_y-u_y^2 b) \cos(u)\nonumber\\
&& +(b u_x c_{uu_x}-a u_y b+b c_{uu_y} u_y-b_{u_x} c_{u} u_x)\sin(u)\nonumber\\
&&+(-c b a_{u_y}-b_{u} u_y c_{u_y}-b^2 u_x+b c_{u}+u_y^2 b_{u}) \sin(u)\nonumber\\
&&+b u_x c_{uu} u_y-b_{u} u_y c_{u} u_x-c b a_{u} u_x+u_y b_{u_x}-b_{u_x} c_{u_y}+b c_{u_xu_y}.\nonumber\end{aligned}$$ Hence the problem of searching the equation (\[sin:inv\]) is reduced to the system of equations (\[sin:mainsys\]).
We now look for the functions $a$, $b$ and $c$ depending linearly on the variable $u_y$: $$\begin{aligned}
a = a_1(u,u_x)u_y + a_2(u,u_x),\label{sin:abc:a}\\
b = b_1(u,u_x)u_y + b_2(u,u_x), \label{sin:abc:b}\\
c = c_1(u,u_x)u_y + c_2(u,u_x).\label{sin:abc:c}\end{aligned}$$ Then equation $\alpha_2 = 0$ (see (\[sin:alpha2\])) takes the form $b_2 (b_1)_{u_x} - b_1 (b_2)_{u_x} = 0$. This equation is satisfied only if at least one of the following conditions holds $$b_1 = 0, \qquad b_2 = 0, \qquad \frac{(b_1)_{u_x}}{b_1} = \frac{(b_2)_{u_x}}{b_2}.$$ Let us consider the case $b_2 = 0$. Then function $b$ defined by the formula (\[sin:abc:b\]) takes the form $$\label{sin:b}
b = b_1(u,u_x)u_y.$$ Furthermore equations $\alpha_1 = 0$ and $\alpha_3 = 0$ become $$u_y a_{u_x u_x} - a_{u_x} = 0, \quad u_y c_{u_x u_x} - c_{u_x} = 0.$$ These equations hold if at least one of the following four conditions is satisfied: $$\begin{aligned}
1) a_{u_x} = 0,\, c_{u_x}=0;& &\\
2) a_{u_x} = 0,\, c_{u_x} \neq 0,& \quad & \frac{c_{u_x u_x}}{c_{u_x}} = \frac{1}{u_y};\\
3) c_{u_x} = 0, \, a_{u_x} \neq 0,& &\frac{a_{u_x u_x}}{a_{u_x}} = \frac{1}{u_y};\\
4) a_{u_x} c_{u_x} \neq 0,& &\frac{a_{u_x u_x}}{a_{u_x}} = \frac{c_{u_x u_x}}{c_{u_x}}.\end{aligned}$$
It can be proved that cases 2)-4) lead to contradiction. Concentrate on the case 1) which implies $$\label{sin:ac}
a = a_1(u) u_y + a_2(u), \quad c = c_1(u) u_y + c_2(u).$$ By substituting the functions (\[sin:b\]), (\[sin:ac\]) into equalities $\beta_i = 0$, $i=1,2,3$ (see (\[sin:beta1\]),(\[sin:beta2\]) and (\[sin:beta3\])) one can verify that equation $\beta_1 = 0$ is satisfied. The equations $\beta_2 = 0$ and $\beta_3 = 0$ lead to $$b_1 (b_1)_{u u_x} - (b_1)_u (b_1)_{u_x} = 0 \quad \mbox{and}\quad(\cos u) u^2_y b_1 (b_1)_{u_x} = 0$$ correspondingly. Thus $b_1(u,u_x) = F_1(u)$ for some function $F_1(u)$. The equality $\gamma_1 = 0$ becomes $$F_1(u)(\cos u+u_x a'_2(u)+a_1(u)\sin u)=0.$$ Clearly, the functions $a_1$ and $a_2$ are defined by the formulas $$a_1(u) = -\mathrm{\cot}u, \quad a_2(u) = C_1,$$ where $C_1$ is an arbitrary constant. According to that we rewrite the equality $\gamma_3 = 0$ as $$F1(u)(u_xc'_2(u)+c_1(u)\sin u+\cos uC_1) = 0.$$ Since $u$ and $u_x$ are regarded as independent variables the last equation leads to $$c_1(u) =-C_1\mathrm{\cot}u, \quad c_2(u) \equiv C_2,$$ where $C_2$ is an arbitrary constant. Let us turn to the equation $\delta_1 = 0$ which gives: $$u^3_y u_x(F_1(u)\cos u+F'_1(u)\sin u) = 0.$$ Integration of the equation leads to $$F_1 = \frac{C_3}{\sin u}.$$ Now the equation $\delta_2 = 0$ takes the form $$C_3u^2_y\bigl(C_1 u_y \sin u+(C_3 +C_2)\cos u\bigr) = 0$$ We get $C_2 + C_3 = 0$, $C_1 = 0$. It is easy to check that equalities $\delta_3 = 0$ and $\gamma_2 = 0$ are identically satisfied. Thus we have $$a = -u_y \mathrm{\cot}u, \quad b = \lambda\frac{u_y}{\sin u}, \quad c = - \lambda.$$ Therefore the invariant manifold (\[sin:inv\]) is of the form $$v_{yy}-u_y\cot u \,v_y+\lambda\frac{u_y}{\sin u} v_x-\lambda v=0$$ and coincides with (\[s4eq6\]) while (\[sin:eq5\]) gives (\[s4eq7\]).
References {#references .unnumbered}
==========
[20]{}
Ibragimov N H and Shabat A B 1979 The Korteweg-de Vries equation from the point of view of transformation groups [*Sov. Phys. Dokl.*]{} [**24**]{}(1) 15–17. Ibragimov N H, Shabat A B (1980). Evolution equations admitting a nontrivial Lie-Backlund group. Funct. Anal. Appl, 14(1), 25-36.
Lax P D 1968 Integrals of nonlinear equations of evolution and solitary waves [*Commun. Pure Appl. Math.*]{} [**21**]{}:5, 467–90
Gurses M, Karasu A and Sokolov V V 1999 [*J. Math. Phys.*]{} [**40**]{} 6473–-90
Zhang D and Chen D 2002 [*J. Phys. [A]{}: Math. Gen.*]{} [**35**]{} 7225–41
Demskoi D K and Sokolov V V 2008 [*Nonlinearity*]{} [**21**]{} 1253–64 arXiv:nlin/0607071v1 (2006).
Wang J P 2009 [*J. Math. Phys.*]{} [**50**]{} 023506 arXiv:0809.3899v1 \[nlin.SI\] (2008).
Gerdjikov V S, Grahovski G G, Mikhailov A V and Valchev T I 2011 [*SIGMA*]{} [**7**]{} arXiv:1108.3990v2 \[nlin.SI\].
Malykh A A, Nutku Y and Sheftel M B 2004 Partner symmetries and non-invariant solutions of four-dimensional heavenly equations [*J. Phys. [A]{}: Math. Gen.*]{} [**37**]{} 7527–-45, arXiv:mathph/0403020
Marvan M and Sergyeyev A 2012 Recursion operators for dispersionless integrable systems in any dimension, [*Inverse Problems*]{} [**28**]{} 025011, 12 p., arXiv:1107.0784
Morozov O I 2014 The four-dimensional Martínez Alonso-Shabat equation: differential coverings and recursion operators [*J. Geom. Phys.*]{} [**85**]{} 75-–80, arXiv:1309.4993
Khanizadeh F, Mikhailov A V and Wang J P 2013 Darboux transformations and recursion operators for differential-difference equations [*Theor. Math. Phys.*]{} [**177**]{} 1606–54
Zakharov V E and Shabat A B 1974 A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem I. [*Funct. Anal. Appl.*]{} [**8**]{}:3 226–35
Zakharov V E and Shabat A B 1979 Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II [*Funct. Anal. Appl.*]{} [**13**]{} 166–74
Wahlquist H D and Estabrook F B 1975 Prolongation structures of nonlinear evolution equations :1 1–7
Nijhoff F W and Walker A J 2001 The discrete and continuous Painlevé VI hierarchy and the Garnier system [*Glasgow Math. J.*]{} [bf 43A]{} 109–-23
Bobenko A I and Suris Yu B 2002 Integrable systems on quad-graphs [*Int. Math. Res. Notes*]{} [**11**]{} 573-–611.
Nijhoff F W 2002 Lax pair for the Adler (lattice Krichever-Novikov) system :1–2 49–58
Yamilov R I 1982 On the classification of discrete equations Integrable Systems ed A B Shabat (Ufa: Soviet Academy of Sciences) 95-–114 (in Russian)
Xenitidis P 2009 Integrability and symmetries of difference equations: the Adler-Bobenko-Suris case [*Proc. 4th Workshop “Group Analysis of Differential Equations and Integrable Systems”*]{} 226–42, arXiv:0902.3954.
[*Theor. Math. Phys.*]{} [**146**]{}:2 170–82
Sokolov V V and Meshkov A G 2011 Hyperbolic equations with third-order symmetries [*Theor. Math. Phys.*]{} [**166**]{}:1 43–57
Svinolupov S I and Sokolov V V 1982 Evolution equations with nontrivial conservative laws [*Func. Anal. Appl.*]{} [**16**]{}:4 317–19
Darboux G 1896 [*Lecons sur la théorie générale des surfaces et les applications geometriques du calcul infinitesimal*]{} (Paris:Gauthier-Villars) vol 1 p 513 vol 2 p 579 vol 3 p 512 vol 4 p 547
Goursat E 1896, 1898 [*Lecons sur l’int$\acute{e}$gration des $\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles du second ordre a deux variables indep$\acute{e}$ndantes*]{} (Paris: Hermann) vol 1 p 226 vol 2 p 345
Sokolov V V and Zhiber A V 1995 On the Darboux integrable hyperbolic equations 303–8
Anderson I M and Kamran N 1997 The variational bicomplex for hyperbolic second order scalar partial differential equations in the plane [*Duke Math. J.*]{} [**87**]{}:2 265–319
Smirnov S V 2015 Darboux integrability of discrete two-dimensional Toda lattices [*Theor. Math. Phys.*]{} [**182**]{}:2 189–210
Zhiber A V and Sokolov V V 2001 Exactly integrable hyperbolic equations of Liouville type [*Russian Mathematical Surveys*]{} [**56(1)**]{}:61 http://dx.doi.org/10.1070/RM2001v056n01ABEH000357
Ablowitz M J, Kaup D J, Newell A C and Segur H 1973 Method for solving the sine-Gordon equation :25 1262
Yamilov R I 1983 Classification of discrete evolution equations [*Uspekhi Mat. Nauk*]{} [**38**]{}:6 155–56
Tongas A, Tsoubelis D and Papageorgiou V 2005 Symmetries and group invariant reductions of integrable partial difference equations. In Proceedings of 10th International Conference in MOdern GRoup ANalysis [**222**]{} 230.
Svinin A K 2011 On some integrable lattice related by the Miura-type transformation to the Itoh-Narita-Bogoyavlenskii lattice [*J. Phys. [A]{}: Math. Theor.*]{} [**44**]{} 465210.
Khabibullin I T 1988 [*J. Soviet Math.*]{} [**40**]{} 108–15
Habibullin I T and Yangubaeva M V 2013 [*Theoret. and Math. Phys.*]{} [**177**]{} 1655–79
Habibullin I T and Poptsova M N 2015 [*J. Phys. [A]{}: Math. Theor.*]{} [**48**]{} 115203
Kaptsov O V 1992 Invariant sets of evolution equations [*Nonlin. Anal.*]{} [**19**]{}:8 753–61
|
---
abstract: 'Under the historical definition of entanglement, namely that encountered in Einstein–Podolski–Rosen (EPR)–type experiments, it is shown that a particular Slater determinant is entangled. Thus a definition which holds that any Slater determinant is unentangled, found in the literature, is inconsistent with the historical definition. A generalization of the historical definition that embodies its meaning as quantum correlation of observables, in the spirit of the work of many others, but is much simpler and more physically transparent, is presented.'
title: On Fermion Entanglement
---
The concept of entanglement is important to the conceptual structure of quantum computation and information transfer, as seen by the many recent papers dealing with it in this context, e.g., [@bennett; @zanardi01; @zanardi02; @zanardi02'; @zanardi04; @shi; @schliemann; @eckert; @barnum0; @barnum; @ortiz]. The term entanglement was coined by Schrödinger [@schrodinger] regarding the essentially non-classical behavior discussed in the famous Gedanken experiment of Einstein, Podolski and Rosen (EPR). [@einstein] The original concept, as simplified by Bohm [@bohm], is as follows. Consider two spin-1/2 particles, e.g. two atoms in a molecule or (I add) the two electrons in a He atom, such that the system is initially in its ground state with total spin $S=0$, i.e. the singlet. Then the force that binds them together is removed, the particles moving apart so that their interaction becomes negligible. Despite the latter, a measurement of the spin of one of the particles (undeterminable before the measurement!) will immediately predict what a spin measurement on the other one will yield. These correlations differ markedly from those expected in classical mechanics, where each spin can be known at all times. The essential difference is expressed technically by the fact that quantum-mechanically the two spins can be correlated in the usual sense, $<s_1^\zeta
s_2^\zeta>\ne<s_1^\zeta><s_2^\zeta>$ (as in the singlet state). Whereas *classically* $<s_1^\zeta
s_2^\zeta>=<s_1^\zeta><s_2^\zeta>,$ the variables, accordingly, being said to be uncorrelated. Here $<\cdot>$ denotes a pure state average. This is at one instant of time, so, each spin classically having a definite value, averages of functions of the spins are simply those functions. I.e., at a given time there are no fluctuations in the variables, whereas quantum mechanically there are such fluctuations, even in a pure state.) [@barnum0; @bohm2] The general idea of identifying entanglement with these quantum correlations is widespread, e.g. see [@zanardi01; @zanardi02; @zanardi02'; @zanardi04; @barnum0; @barnum; @ortiz; @peres].
The essential idea of this experiment can be simplified a bit further: There is no need to consider the time evolution of the prepared state; one can consider measurement immediately after the state is prepared. \[15, p.148\]. A further comment is in order here, namely I have used $s_1^\zeta$ and $ s_2^\zeta$ as observables; the fact that they are not permutation symmetric, which will probably raise objections for the electron example, will be justified below.
Until fairly recently, emphasis has been placed on the spin state, without regard to the rest of the wave function. But the question of how the fermionic (or bosonic) nature of particles tangles with the concept of entanglement has been discussed more recently. Here I consider three modifications of the concept. Two of them, due to Zanardi and coworkers [@zanardi01; @zanardi02; @zanardi02'; @zanardi04], and to Barnum et al [@barnum0; @barnum; @ortiz], have in common that they apply both to spin systems and to indistinguishable particles, plus the fact that an essential idea behind their definitions is entanglement depends on what observables are being considered. The other, due to Schliemann *et al.* and Shi [@schliemann; @eckert; @shi], is specifically for fermions, and considers a state to be entangled or not, regardless of the observables being measured. The latter definitions [@schliemann; @eckert; @shi] have entanglement depending only on whether the state is a single Slater determinant or not (see below).
An important point of this paper is to note that definitions of entanglement that are based on the determinantal criterion are inconsistent with the original concept. Also we give and use in examples a definition that embodies the essential physics of the historical concept, and is far simpler than the recent works cited.
A basic idea of the modifications [@zanardi01; @zanardi02; @zanardi02'; @zanardi04; @barnum0; @barnum; @ortiz], as I understand them, is that whether or not a given pure quantum state is entangled depends on what observables are being considered as correlated or not. More precisely, *a pure state $\psi$ is entangled with respect to observables $A$ and $B$ if they are correlated in that state, i.e. if* $$C_{AB}^\psi\equiv <\psi|AB|\psi>-<\psi|A|\psi><\psi|B|\psi>\ne 0;\label{1}$$ *otherwise $\psi$ is unentangled with respect to $A$ and $B$*. I’ll call this definition $\mathcal{D}$. [^1] Note: (i) $C_{AB}^\psi={C_{BA}^\psi}^*$, so that this definition is independent of the order of $A$ and $B$ insofar as its being zero or not. (ii) If $\psi$ is an eigenstate of $A$, then it is unentangled with respect to $A$ and *any other* observable $B$; but $\psi$ being an eigenstate of $A$ (or $B$) is not necessary for it to be unentangled with respect to $A$ and $B$. (iii) $A$ and $B$ need not commute. Under this definition, *any* pure state is unentangled with respect to *some* observables. We will see that this mathematically very simple definition corresponds quite directly to the essential measurements characterizing EPR-type experiments.
A familiar example from the spin-only class: $$\chi=[\alpha_\uparrow(s_1)\alpha_\downarrow(s_2)-\alpha_\downarrow(s_1)\alpha_\uparrow(s_2)]/\sqrt{2};\label{2}$$ $\alpha_\uparrow(s)$ and $\alpha_\downarrow(s)$ are the familiar “spin-up" and “spin-down" states for a spin-1/2 particle. \[To establish my notation: With $\hat{z}$ a unit vector in the up direction, $s_i^z\alpha_\sigma(s)=\sigma\alpha_\sigma(s)$ where $\sigma$ takes on numerical values $\pm1$ corresponding to pictorial values $\uparrow,\downarrow$; thus we take $s_i^z$ as having eigenvalues $\pm 1$, so our $s_i^z$ is what is commonly called $\sigma_i^z$. In the usual representation, the argument $s$ takes on those eigenvalues.\] We see that this state $\chi$ is entangled with respect to $s_1^z$ and $s_2^z$ (and with respect to $s_1^\zeta$ and $s_2^\zeta$ for any direction $\zeta$), but is unentangled with respect to $\mathbf{s}^2$ and $B$, where $\mathbf{s}=\mathbf{s}_1+\mathbf{s}_2$ is the total spin and $B$ is an arbitrary observable. The historical definition of entanglement is the same as this definition, with the restriction that the observables be “local", i.e. $A$ refers to one particle and $B$ to the other; $\mathbf{s}$ would be called a global observable. The definition with respect to such global variables, while possibly of interest for some purposes, removes the charm or “mystery" of the historical view, which involves the idea of “spooky action-at-a- distance". [@mermin]
Shi [@shi], Schliemann *et al.* [@schliemann], and Eckert *et al.* [@eckert] have defined entanglement of fermions as follows: A state that is not a single Slater determinant (i.e. not reducible to such a determinant), is entangled; any single Slater determinant is unentangled. I will show that this definition is not a generalization of the earlier concept, but rather is inconsistent with that concept. Note that any normalized 2-electron Slater determinant $D$ is, by definition of the creation operators, $a^\dagger$ and $
b^\dagger$ $$D=a^\dagger b^\dagger|0>=\frac{1}{\sqrt{2}}\mathcal{A}a(\xi_1)b(\xi_2).\label{3}$$ $\mathcal{A}$ is the antisymmetrizer ($\mathcal{A}f(\xi_1,\xi_2) \equiv f(\xi_1,\xi_2)-f(\xi_2,\xi_1)$), we can take $\xi_i=\mathbf{r}_i,s_i$ in the Schrödinger representation ($\mathbf{r}_i$ is the position of the $i^{th}$ particle), and $a(\cdot), b(\cdot)$ are orthonormal 1-electron states.
Consider the single determinant where a single spatial function $u(\bf{r})$ (orbital) is occupied by two electrons: $$D_0 = a_\uparrow^\dagger a_\downarrow^\dagger|0>,\label{4}$$ where $a_\sigma^\dagger$ creates a particle in the state $u(\mathbf{r})\alpha_\sigma(s)$. I consider the Schrödinger representation, and will not distinguish between the state vector and this representation. It is easy to see that $$D_0=u(\mathbf{r}_1)u(\mathbf{r}_2)\chi(s_1,s_2),\label{5}$$ where $\chi$ is the singlet (\[2\]). In form, $D_0$ is seen to be the familiar Hartree–Fock ground state of the He atom or the H$_2$ molecule in the molecular orbital approximation. Clearly $$C_{s_1^z,s_2^z}^{D_0}=-1,\label{6}$$ so that according to $\mathcal{D}$, the single Slater determinant $D_0$ is entangled (with respect to $s_1^z$ and $s_2^z$). Thus we’ve proved what we said we would, namely, the definition in references [@shi; @schliemann; @eckert] is inconsistent with $\mathcal{D}$, which includes the historical definition of entanglement.
But some questions need be asked. One concerns our use of $s_1^z$ and $s_2^z$ as observables, a usage proscribed by at least two authors: in \[18, p. 388\], and \[15, p. 127\], it is stated that any observable for a system of identical particles must be invariant under permutations of the particles. This has some force in view of the agreed-upon fact that there is no way in principle to distinguish between such particles in quantum mechanics. Following a suggestion of Birge [@birge], I considered the *z*-component of the spin density $$s(\mathbf{R})=\sum_i s_i^z\delta(\mathbf{R}-\mathbf{r}_i),\label{7}$$ which clearly satisfies the required symmetry. Of course (\[7\]) is related to the field operators $\hat{\Psi}_\sigma(\mathbf{R})$ by $s(\mathbf{R})=\hat{\Psi}_\uparrow(\mathbf{R})^\dagger
\hat{\Psi}_\uparrow(\mathbf{R})-\hat{\Psi}_\downarrow(\mathbf{R})^\dagger \hat{\Psi}_\downarrow(\mathbf{R})$. Taking the observables as $A=s(\mathbf{R}), B=s(\bf{R'})$ with $\mathbf{R}\ne\mathbf{R'}$, yields the same conclusion: the correlation $C_{A,B}^{D_0}\ne 0$, so that the determinant $D_0$ is entangled with respect to the spin density at two different spatial points. Actually, this gives a way to closely mimic the EPR experiment: there the measurement apparatus consists of two detectors, at positions $\bf{R}$ and $\bf{R}'$. And one accepts only coincidences where there’s a particle at $\bf{R}$ and a particle at $\bf{R}'$. Thus one measures the average $<s(\bf{R})s(\bf{R}')>$ *conditional* on a particle being at $\bf{R}$ the other one being at $\bf{R}'$. In the state $D_0$, one can see that this average is just that $<s_1^zs_2^z>$ calculated in (\[6\]) (which is independent of $\bf{R}$ and $\bf{R}'$). I add that the “mystery" is present in the sense that the distance $|\bf{R}-\bf{R}'|$ can be large enough to render any interactions negligible. [^2]
The fact that working with the properly symmetric operator (\[7\]) yields the same result as treating the non-symmetric $\mathbf{s}_1$ and $\mathbf{s}_2$ as observables suggests that the proscription against considering, in general, an operator that is not permutation symmetric as an observable might be too strong. If one interprets $\mathbf{r}_1, s_1^z$, not as the position and spin of particle \#1, but rather as the position and spin of *a* particle, it seems that this might be acceptable, and consistent with other approaches. It would be useful if so, since, as one can see, it is formally simpler than working through the (symmetric) spin density.
Zanardi’s discussion of the 2-site Hubbard model [@zanardi02'] is consistent with definition $\mathcal{D}$. This is seen, e.g., when the interaction $U=0$. The ground state is just (\[5\]), with $u(\mathbf{r})=[w_1(\mathbf{r})+w_2(\mathbf{r})]/\sqrt{2},$ the bonding orbital; $a^\dagger_\sigma$ in (\[4\]) is $2^{-1/2}(c^\dagger_{1\sigma}+ c^\dagger_{2\sigma})|0>$, where $c_{i\sigma}^\dagger|0>
=w_i(\mathbf{r})\alpha_\sigma$, $w_i(\mathbf{r})$ being the Wannier function at site $i$. Thus, according to $\mathcal{D}$, $D_0$ is unentangled with respect to e.g. the $U=0$ Hamiltonian $H_{Hubb}^0$ and any other variable (which corresponds to Zanardi’s entropy $S_0=0$ at $U/4t=0$ in his FIG. 1, $\Lambda^*$), but is *entangled* with respect to the two spins $s_1^\zeta,s_2^\zeta$ (corresponding to FIG. 1, $\Lambda$). Actually, for the purpose of the present paper, the main point is that there are variables with respect to which *this single determinant is entangled*, rather than the degree of entanglement, so that the particular choice of such variables is beside the point. However, some preliminary results that show the possibility of basing a definition of degree of entanglement on $\mathcal{D}$ are given in the Appendix. Incidentally, Eq. (11) of [@zanardi02'] is incorrect; it is probably just a misprint, Eq. (12) being correct for $U=0$.
I have discussed only a particular Slater determinant (\[4\]), which is entangled with respect to the two spins, unentangled with respect to the positions of the particles. Instead, the determinant $c_{1\uparrow}^\dagger c_{2\uparrow}^\dagger|0>$ is entangled with respect to the positions, unentangled wrt the spins. More general determinants will be entangled with respect to both spins and positions. It is clear that a Slater determinant will always be entangled with respect to to some “local" observables (i.e. $A(\mathbf{p}_1,\mathbf{r}_1, \mathbf{s}_1)$ and $B(\mathbf{p}_2,\mathbf{r}_2 ,\mathbf{s}_2)$), because the antisymmetry prohibits the determinant from being a product of single-particle states. ($\mathbf{p}$ is linear momentum.) A question then is, is it possible in any sense to talk about fermions entangled or not with respect to such local observables? The answer is yes, under special conditions, by considering mapping of the true fermion wave functions to non-fermion states. One example is the familiar case of a collection of hydrogen atoms, or, simpler, Hubbard atoms, when the overlap of the Wannier functions (or the hopping parameter $t$) is small enough (*but nonzero)*; then low-lying states are governed to a good approximation by the Heisenberg Hamiltonian, and are in $1-1$ correspondence with site-spin states, and the latter have no permutation symmetry restrictions. Thus for some observables (e.g. low-temperature thermodynamic properties), one can ascribe the entanglement or lack thereof in spin states to the corresponding fermion states (see [@kaplan]). An example where this correspondence is exact occurs if one is interested only in “perfect half-swaps" (see [@kaplan2]).
In summary, the definition of fermion entanglement that includes the statement, any Slater determinant is unentangled, has been shown to be inconsistent with the historical definition. This was done by pointing out that experimental investigation of a particular Slater determinant (\[4\]) or (\[5\]) will lead to the EPR type of phenomena wherein the famous peculiarly quantum correlations between the spin components of the two “entangled" fermions occur. The discussion involved the spin density operators at two widely separated spatial points, which formally served as observables that showed the correlation effects. It allowed a close correspondence to the actual experiment, which measures the conditional average of the product of the spin densities at the points of the measuring detectors, this conditional average being identical to the average of the product of the spins of the two particles. Examples given in the text and in the Appendix show that the question of whether a wave function is a single determinant or not is irrelevant to the question of its entanglement. It was noted that the definition $\mathcal{D}$ of entanglement adopted here is consistent with the idea that a true generalization of the historical concept can be made such that entanglement depends on the observables being considered. Furthermore, $\mathcal{D}$ directly embodies the idea that entanglement means correlation of observables, and is far simpler than other definitions which also embody this dependence on the observables being measured.\
**Acknowledgements**\
I thank C. Piermarocchi, S. D. Mahanti, N. Birge and M. Dykman for encouragement and useful discussions. Thanks to A. Mizel for important correspondence, and to R. Merlin for making me aware of the fermion entanglement problem.\
**Appendix: On the Degree of Entanglement.**\
A straightforward definition of the *degree* of entanglement $E_{AB}^\psi$, based on the previous discussion and definition $\mathcal{D}$ of whether or not a state is entangled, is simply $|C_{AB}^\psi|$; preferable would be to normalize it by the maximum of $|C_{AB}^\phi|$ over all states $\phi$ of interest, which is what we suggest: $$E_{AB}^\psi\equiv |C_{AB}^\psi|/\max_\phi |C_{AB}^\phi|.\label{8}$$ Then $0\le E_{AB}^\psi\le 1$, putting this quantity on the same scale as the common definition via von Neumann entropy. [@bennett]
To get a feeling for the physical significance of this (tentative) definition, we apply it to the ground state $\Psi$ of the 2-site 2-electron Hubbard model (following Zanardi [@zanardi02']), for various choices of $A,B$. We put $U/(4t) = x$, where $U,t$, both $\ge0$, are the Coulomb repulsion and hopping parameter. The maximization in the denominator of (\[8\]) is over all $\phi$ in the (6-dimensional) space of 2-particle states appropriate to the model. The calculation of all averages in $\Psi$ is straightforward; I found the calculation of the denominator in (\[8\]) not so straightforward.\
(i) $\{A,B\}=\{s_1^z,s_2^z\}$ (electron spins)
Since the ground state in this model is a product of a (symmetric) spatial state and the singlet, it is obvious that $<\Psi|s_1^zs_2^z|\Psi>=<\chi|s_1^zs_2^z|\chi> = -1$ for all $x$. Similarly $<\Psi|s_i^z|\Psi>=0$, so that $$C_{s_1^z,s_2^z}^\Psi=-1, \ \mbox{independent of}\ x.\label{9}$$ One can see [@kaplan3] that this gives the maximum of $|C_{s_1^z,s_2^z}^\phi|$, so $E_{s_1^z,s_2^z}^\Psi=1$ ($\Psi$ maximally entangled) for all $x$. (This includes $x=0$ where $\Psi$ is a single determinant, and, as discussed above, is clearly appropriate to an EPR type experiment.)\
(ii) $\{A,B\}=\{S_1^z,S_2^z\}$ (site spins)
The site spins are rather standardly defined as $$S_i^z=N_{i\uparrow}-N_{i\downarrow};\label{10}$$ $N_{i\sigma}=c_{i\sigma}^\dagger c_{i\sigma}$ are the site-spin occupation numbers. (Similar definitions are made for the other components, of course.) One finds straightforwardly $$\begin{aligned}
C^\Psi_{S_1^z,S_2^z}&=&-\frac{1}{1+f(x)^2},\nonumber\\
f(x)&=&\sqrt{1+x^2}-x,\label{11}
\end{aligned}$$ and it can be shown [@kaplan3] that $max_\phi|C^\phi_{S_1^z,S_2^z}|=1.$ (\[11\]) is the same as (\[9\]) for $x\rightarrow\infty$. Interestingly, it differs for finite $x$, the largest difference, a factor of two, occurring for $x=0$. This raises the question, under what circumstances are the site spins observed in experiment. In the usual derivation of the Heisenberg Hamiltonian (for large $x$), the spins that appear are the site spins. (It should be realized that this is a physics question, beyond any question of entanglement definition.)\
(iii) $\{A,B\}=\{n_{\uparrow},n_{\downarrow}\}$ (bonding orbital occupation numbers)
With $a_\sigma\equiv(c_{1\sigma}+c_{2\sigma})/\sqrt{2}$, the bonding orbital occupation numbers are defined as $ n_\sigma=a_\sigma^\dagger a_\sigma$. One finds $$E_{n_\uparrow,n_\downarrow}^\Psi =4g(x)[1-g(x)],\label{12}$$ where $$g(x)=\frac{[1+f(x)]^2}{2[1+f(x)^2]}.$$ The normalizing maximum is 1/4. [@kaplan3] We see that $E_{n_{\uparrow},n_{\downarrow}}^\Psi = 0$ at $x=0$, as is understandable since in this limit, the wave function, a Slater determinant, is an eigenstate of both these observables. As in the discussion above of Zanardi’s example, this noninteracting ground state is unentangled with respect to one set of observables, but (maximally) entangled for the other set ($s_1^z,s_2^z$) considered. At the other extreme, $E_{n_\uparrow,n_\downarrow}^\Psi\rightarrow 1$ as $x\rightarrow\infty$. Thus, according to the definition (\[8\]), the ground state is maximally entangled with respect to these “non-interacting occupation numbers" in the strongly-interacting limit–quite reasonable.\
(iv) $\{A,B\} = \{N_1, N_2\}$ (site occupation numbers)
These are defined as $N_i=N_{i\uparrow} + N_{i\downarrow}$. One finds easily $$C_{N_1,N_2}^\Psi = -\frac{f(x)^2}{1+f(x)^2}.\label{13}$$ This $\rightarrow0$ as $x\rightarrow\infty$, so the ground state in the strongly-interacting limit is unentangled with respect to to these site variables. Again this is because this state is an eigenstate of $N_1$ and $N_2$. Note that this state is *not a single determinant, nor can it be reduced to one*. Thus we have another example of *un*entanglement of a state that is *not* a single Slater determinant (the other obvious ones are $\Psi$ for all $x > 0$, with respect to $\mathbf{s}^2$ and $B$, arbitrary $B$). In the non-interacting limit the magnitude of (\[13\]) is 1/2, showing that this state, a single determinant, is entangled with respect to these observables, but not maximally. (The normalizing maximum can be shown to be unity [@kaplan3].)
[99]{}
C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wooters, *Mixed-state entanglement and quantum error correction*, *Phys. Rev. A* [**54**]{} (1996), 3824.
P. Zanardi, *Virtual quantum subsystems*, *Phys. Rev. Lett.* [**87**]{} (2001), 077901. P. Zanardi and X. Wang, *Fermionic entanglement in itinerant systems*, *J. Phys. A:Mathematical and General;* [**35**]{} (37) (2002), 7947. P. Zanardi, *Quantum entanglement in fermionic lattices*, *Phys. Rev. A* [**65**]{} (2002), 042101.
P. Zanardi, D. A. Lidar and S. Lloyd, *Quantum tensor product structures are observable induced*, *Phys. Rev. Lett.* [**92**]{} (2004), 060402. Y. Shi, *Quantum entanglement of identical particles*, *Phys. Rev. A* **67** (2003), 024301.
J. Schliemann, D. Loss and A. H. MacDonald, *Double-occupancy errors, adiabaticity, and entanglement of spin qubits in quantum dots.* *Phys. Rev. B* [**63**]{} (2001), 085311. K. Eckert, J. Schliemann, D. Bruss, and M. Lewenstein, *Quantum correlations in systems of indistinguishable particles*, *Annals of Physics* [**299**]{} (2002), 88. H. Barnum, E. Knill, G. Ortiz, , R. Somma and L. Viola, *A subsytem-independent generalization of entanglement*, *Phys. Rev. Lett.* [**92**]{}, 107902 (2004). H. Barnum, E. Knill, G. Ortiz and L. Viola, *Generalizations of entanglement based on coherent states and convex sets*, *Phys. Rev. A* [**68**]{} (2004), 032308. G. Ortiz, R. Somma, H. Barnum, E. Knill and L. Viola, *Entanglement as an observer-dependent concept: an application to quantum phase transitions*, quant-ph/0403043 v1 4 Mar 2004. E. Schrödinger, *Die gegenwärtige Situation in der quantenmechanik*, *Naturwissenschaften* [**23**]{} (1935), 823, 844. A. Einstein, B. Podolsky and N. Rosen, *Can quantum-mechanical description of physical reality Be considered complete?*, *Phys. Rev.*[**47**]{} (1935), 477. D. Bohm, *Quantum Mechanics*, Prentice-Hall (1951) sec. 22.16. A. Peres, *Quantum Theory:Concepts and Methods, Kluwer Academic Publishers*, Dordrecht, Boston (1993). Bohm [@bohm] uses the term correlated in a different sense for the classical case. This phrase is attributed to Einstein. See *The Born-Einstein Letters*, with comments by M. Born, Walker, New York (1971), referred to in N. D. Mermin, *Is the moon really there when nobody looks? Reality and the quantum theory*, Physics Today/April, pg.38 (1985). G. Baym, *Lectures in Physics*, W. A. Benjamin, New York (1969). N. Birge, private comm. (2004).
T. A. Kaplan and C. Piermarocchi, *Spin Swap and $\sqrt{SWAP}$ vs. Double Occupancy in Quantum Computation*, in Proc. of INDO-US Workshop, “Nanoscale Materials: From Science to Technology", Puri, India, April 5-8, 2004, to appear. T. A. Kaplan and C. Piermarocchi, *Spin swap vs. double occupancy in quantum gates*, *Phys. Rev. B* **70** (2004), 161311(R) . T. A. Kaplan, unpublished work.
[^1]: This is consistent with examples in [@zanardi02'; @zanardi04], although there and in [@barnum0; @barnum] the entanglement criterion is expressed in different, mathematically quite complicated, terms, which I frankly do not follow in detail. I will use the much simpler definition $\mathcal{D}$, but will make contact with some results of [@zanardi02'] (see the 2-site Hubbard model discussion below).
[^2]: The short range interaction of neutrons might make them preferable for the experiment.
|
[\
***River scaling analysis and the BHP universality hypothesis***\
]{}
[**Rui Gonçalves (email: rjasg@fe.up.pt)**]{}
[[*Faculdade de Engenharia do Porto*]{}]{}
[**Nico Stollenwerk (email: nks22@cus.cam.ac.uk)**]{}
[[*Instituto Gulbenkian de Ciência*]{}]{}
[**Alberto Pinto (email: aapinto@fc.up.pt)**]{}
[[*Centro de Matemática da Universidade do Porto*]{}]{}
[[**Abstract:** The XY-model shows in two dimensions in the strong coupling regime a universal distribution, named BHP, which in turn also describes other models of criticality and self-organized criticality and even describes natural data as river level and flow. We start by analysing the two dimensional XY-model and calculate the BHP probability density function. The results obtained for several dissimilar phenomenons which includes the deseasonalised Danube height data raised the universality hypothesis for rivers. This hypothesis is tested for the Iberian river Douro. Deviations from the BHP are found especially for medium and small runoffs. For regimes closer to the natural flow the fluctuations tend to follow the universal curve again.\
]{}]{}
[[**PACS numbers:** 05.40.–a, 05.65+b, 64.70.Nd\
]{}]{}
Introduction
============
In their pioneer paper the authors Bak, Tang and Wiesenfeld, [@BTW88], proposed the hypothesis that under very general conditions nonequilibrium systems consisting of many interacting constituents may exhibit universal behaviour. This behaviour among other features is characterised by the formation of correlations and should occur naturally without any parameter tuning from the outside. For these systems the dynamical response is complex but the statistical properties are governed by simple power laws. Since only dimensionless numbers can be raised to arbitrary powers in a meaningful way, this self-organisation of the system implies independence of any particular scale of length or time. Hence the behaviour of the systems on these self-organized states is close to those of equilibrium systems at the critical point. This phenomenon is known as self-organized criticality (SOC).\
In equilibrium thermodynamics criticality is linked with continuous phase transitions, [@Binneyetal]. The phase transition is continuous when the rate of change of some order parameter, and not the order parameter itself, is discontinuous at the critical point. In these systems when temperature is equal to the transition temperature the spin correlation function instead of an exponential decay reveals a power law behaviour and the corresponding exponent is called the critical exponent, in this case for the correlation function. As a consequence any local perturbation can be propagated through out the entire system and hence any member of the system affects all the others. At the critical point the contribution of the interaction between widely separated points for the large scale fluctuations of the order parameter are not exponentially rare. In this frame the central limit theorem is not applicable and Gaussianity is not present at all.\
The existence of power laws statistics and critical exponents raises the question of classification of these systems. The phenomenon, whereby dissimilar systems exhibit the same critical exponents, is called universality. Two systems are assigned to the same universality class if they share the same dimensionality $d$ of the underlying lattice (number of spatial dimensions) and the same dimensionality of the order parameter, $D$. All the systems in the same universality class have the same critical exponents.\
In this work we follow Bramwell, Holdsworth, Pinton, [@BHP1998] in calculating the probability density function (PDF) for the fluctuations of the magnetic order parameter of a two-dimensional model (2dXY) for spins in the strong coupling (low temperature) regime using the spin wave approximation where it shows a universal distribution, i.e., independent of system size and critical exponent $\nu$. This universal PDF describes other models of criticality and SOC [@bramwell2000] hence we show the data collapse for the deseasonalized Danube river height data at Nagymaros (80 years) following the steps of [@DahlstedtJensen2005] and compare it with the daily mean runoff data of river Douro at Régua (26 years). According to those authors no important differences occur when height or runoff measures are used.\
This article is organised as follows the second section describes the two-dimensional Ising model (2dIsing) and its frame set as the preliminary for the 2dXY calculations presenting, order parameter and PDF’s at different regimes of temperature; subcritical critical and supercritical. In the third section we present the analytics for the spin-wave (SW) approximation of the 2dXY model and the magnetic order parameter PDF for the critical (low temperature) regime named BHP, after Bramwell, Holtsworth and Pinton [@bramwell2000]. In the fourth section we show the data collapse of the deseasonalized Danube river height data and we compare it with the Douro river in two cases, the entire year and the winter regime. Apparently the deviations of the Douro river from the BHP are due to some river flow regulations in order to have fluctuations closer to the Gaussian PDF. For larger values of the streamflow the regulation is no longer possible due to storage limitations and the river dams are set open. For larger values the river fluctuations are much closer to the BHP form. The main differences between the Danube and Douro are related to the amount of water on the river basin and its distribution in time. The Douro is a southern European river where the summer is usually very dry contrasting with the Danube basin where there is lots of water the year around and regulation seems not to have any major effect on the natural flow.
The Ising model
===============
Ernst Ising suggested a very simple thermodynamic model to understand spontaneous magnetization (Ising, 1925, [@Ising]). Statistical mechanics as used by Ising considers equilibrium distributions $$p^*(\sigma _1, ... , \sigma _N)=\frac{1}{Z}
\cdots e^{\cal{ H} (\sigma _1, ... , \sigma _N)}
\label{pequilibrium}$$ and quantities derived from them, like the magnetization per spin $\langle M \rangle := \langle \frac{1}{N}
\sum _{i=1}^{N} \sigma _i \rangle $ and magnetic susceptibility $\chi := \langle M^2 \rangle - \langle M \rangle ^2$. Here the $N$ spins $\{ \sigma _i \}_{i=1}^{N}$, $\sigma _i \in \{-1,+1\} $, are distributed in $p^*(\sigma _1, ... , \sigma _N)$ according to Boltzmann weights $e^{\cal H} $ (originally introduced to represent well known distributions in ideal gas theory, the Maxwell distribution of velocities) with a function $${\cal H} (\sigma _1, ... , \sigma _N) =
\sum_{i=1 }^{N} \sum_{j=1 }^{N} V_{ij} \sigma _i \sigma _j +
\sum_{i=1 }^{N} h_i \sigma _i
+\sum_{i=1 }^{N} c_i
\label{hamiltonian}$$ and a normalization, the partition function $Z$. In order to obtain reasonably sized numbers we use $C=\sum_{i=1 }^{N} c_i =-N \mbox{ln} 2$ for the $2^N$ possible configurations of spins up and down, hence $$Z=
\sum_{\sigma _1=\pm 1 } ... \sum_{\sigma _N=\pm 1 }
e^{
\sum_{i=1 }^{N} \sum_{j=1 }^{N} V_{ij} \sigma _i \sigma _j +
\sum_{i=1 }^{N} h_i \sigma _i
}
\cdot \frac{1}{2^N}
\quad.
\label{partitionfunction}$$ It is $V_{ij}:=V \cdot J_{ij}$ with coupling strength $V$ and adjacency matrix $J_{ij} \in \{0,1 \}$ and $h:=h_i$ the external magnetic field. Then the distribution of the magnetization per spin is given by $$p(M)=
\sum_{\sigma _1=\pm 1 } ... \sum_{\sigma _N=\pm 1 }
\delta \left( M - \frac{1}{N} \sum _{i=1}^{N} \sigma _i \right)
p^*(\sigma _1, ... , \sigma _N)
\label{magdistribution}$$ from simply applying Bayes’ rule $p(M)= \sum_{\sigma _1=\pm 1 } ...
\sum_{\sigma _N=\pm 1 } p(M|\sigma _1, ... , \sigma _N)
\cdot p(\sigma _1, ... , \sigma _N)$ for joint and conditional probabilities.
The mean magnetization, $\langle M \rangle=\sum_{i=0}^{N} M \cdot p(M)$, with $M=-1+i\cdot \Delta M$, $\Delta M =2/N$ is given by $$\langle M \rangle
=\sum_{\sigma _1=\pm 1 } ... \sum_{\sigma _N=\pm 1 }
\left(
\sum_{i=0}^{N} M \cdot
\delta \left( M - \frac{1}{N}
\sum _{i=1}^{N} \sigma _i \right)
\right)
p^*(\sigma _1, ... , \sigma _N)
\quad.
\label{mag}$$ Using the $\delta $-function the mean magnetization is then $$\langle M \rangle
=
\sum_{\sigma _1=\pm 1 } ... \sum_{\sigma _N=\pm 1 }
\left(
\frac{1}{N} \sum _{i=1}^{N} \sigma _i
\right)
\frac{1}{Z} e^{\cal H}
\quad.
\label{mag2}$$ The mean magnetization $\langle M \rangle $ is an easily accessible quantity in physical systems, whereas the distribution of the magnetization $p(M)$ is easily accessible in small systems. Also the mean $\langle M \rangle $ is dominated by the maximum $M_{max}$ of the distribution $p(M)$ for large systems. The Ising model was proposed by Lenz[@Lenz] and solved by Ising for one dimension. The two dimensional case shows already a phase transition. In 1944 Onsager[@Onsager], solved the Ising system for $d=2$ in the absence of an external magnetic field. Here we present a numerical simulation for the magnetization distribution for the 2d Ising model for a $N=4\times4$ lattice with different coupling strength, $V$. The external magnetic field is absent, $h=0$.
![[]{data-label="fig:vertroh0145"}](vertroh0130.eps){width="\linewidth"}
![[]{data-label="fig:vertroh0145"}](vertroh0140.eps){width="\linewidth"}
![[]{data-label="fig:vertroh0145"}](vertroh0145.eps){width="\linewidth"}
For low temperature (strong coupling), fig. \[fig:vertroh0130\] the spins are strongly correlated and tend to align with their neighbours so the distibution mode is for zero magnetization. Near critical temperature, fig. \[fig:vertroh0140\] small variations of spins can be propagated through the lattice and will affect spins at large distances. That is the point of phase transition. Fluctuations can easily happen for values between $M\approx -0.5$ and $M\approx +0.5$, for which all magnetizations in this range are about equally likely. For temperatures just above criticality, fig. \[fig:vertroh0145\], two maxima start appearing symmetrically around the origin.
The XY-model in two dimensions
------------------------------
In the XY model spins are able to rotate freely in the XY-plane, i.e. the spin variables are $\underline S_i$ with $$\underline S = \left(
\begin{array}{c}
cos(\theta _i) \\
sin(\theta _i) \\
\end{array}
\right)
\quad
\label{xyspins}$$ where the angle $\theta _i $ changes between $-\pi$ and $+\pi$.\
The stationary distribution is again given by $$p^*(\underline S _1, ... ,\underline S _N)=\frac{1}{Z}
\cdot e^{{\cal H} (\underline S _1, ... ,\underline S _N)}
\label{pxyequilibrium}$$ and $${\cal H} (\underline S _1, ... ,\underline S _N) =
\sum_{i=1 }^{N} \sum_{j=1 }^{N} V_{ij}
\underline S_i \underline S _j +
\sum_{i=1 }^{N} \underline H \, \underline S _i
+C
\label{xyhamiltonian}$$ where later the constant $C$ will be fixed to obtain reasonably small values for the partition function $Z$ (irrelevant for all physically relevant quantities, e.g. the stationary distribution $p^* $).
Since the spins are unit vectors rotating we can replace them just by angles between two interacting spins $$\underline S_i \underline S _j
= |S_i| \cdot |S_j| \cdot cos (\theta _i -\theta _j)
= cos (\theta _i -\theta _j)
\label{xythetainteraction}$$ or the angle between spin and outer magnetic field (see [@Binneyetal; @ZinnJustin]) $$\underline H \; \underline S _i
= |H| \cdot |S_i| \cdot cos (\theta _i )
= h \cdot cos (\theta _i )\quad.
\label{xythetamagneticfield}$$ Hence $$p^*(\theta _1, ... ,\theta _N)=\frac{1}{Z}
\cdot e^{{\cal H} (\theta _1, ... ,\theta _N)}
\label{pxyequilibriumtheta}$$ and $${\cal H} (\theta _1, ... \theta _N) =
V \sum_{i=1 }^{N} \sum_{j=1 }^{N} J_{ij}
cos( \theta _i - \theta _j) +
\sum_{i=1 }^{N} h \cdot cos(\theta _i)
+C
\label{xyhamiltoniantheta}$$ with partition function as normalization constant $$Z=\int_{-\pi }^{\pi} \, d\theta _1 ... \int_{-\pi }^{\pi} \, d\theta _N
\; \; e^{{\cal H} (\theta _1, ... ,\theta _N)}\quad.
\label{xypartitionfct}$$ The absolute value of the magnetization as order parameter for the phase transition [@KosterlitzThouless] is given by
$$m:= \frac{1}{N} \parallel \sum_{i=1}^{N} \underline S_i
\parallel = \frac{1}{N} \sqrt{\left(
\sum_{i=1}^{N} cos(\theta _i)
\right)^2
+
\left(
\sum_{i=1}^{N} sin(\theta _i)
\right)^2
}
=: M(\theta _1, ... ,\theta _N)
\label{xymagnetization}$$
and the distribution of this quantity $$p(m)=\int_{-\pi }^{\pi} \, d\theta _1 ... \int_{-\pi }^{\pi} \, d\theta _N
\; \; \delta (m-M(\theta _1, ... ,\theta _N) )
\; \;\frac{1}{Z} e^{{\cal H} (\theta _1, ... ,\theta _N)}\quad.
\label{xydistribmagnetization}$$ This defines completely the XY-model in any dimension (where the dimension is fixed by specifying the adjacency matrix $J$ in ${\cal H} (\theta _1, ... ,\theta _N) $). This will be referred to the non-quadratic model, since later for analytical treatability of the strong coupling regime the $cos$-interaction will be replaced by the quadratic Taylor expansion (valid for small angles between the spins only or spins and external magnetic field). The BHP-distribution will turn up by evaluating the Eq. (\[xydistribmagnetization\]) in the case of quadratic approximation of the above model. Using the second order Taylor series expansion for the cosine, $$cos(x)=1-\frac{1}{2} x^2 + {\cal O} (x^4)
\label{cosTaylor}$$ hence we can replace in ${\cal H} (\theta _1, ... ,\theta _N) $ in the strong coupling case ($V>>1$) $$cos(\theta _i - \theta _j)
\approx 1-\frac{1}{2} (\theta _i - \theta _j)^2
\label{cosTaylortheta}$$ i.e. approximating all relevant quantities in quadratic form in the exponential function and then use Gaussian integration. This approximation is valid for strong coupling $V>>1$, where in the physical model spin waves are the only relevant dynamical phenomena.\
The partition function as normalization constant $$Z=\int_{-\pi }^{\pi} \, d\theta _1 ... \int_{-\pi }^{\pi} \, d\theta _N
\; \; e^{{\cal H} (\theta _1, ... ,\theta _N)}
\label{xypartitionfct2}$$ becomes in quadratic approximation $$Z=\int_{-\infty }^{\infty } \, d\theta _1 ...
\int_{-\infty }^{\infty } \, d\theta _{N-1} ...
\int_{-\pi }^{\pi} \, d\theta _N
\; \; e^{J_{tot}\cdot V}
\; \; e^{-V \cdot \underline{\theta} ^{tr} A \underline{\theta} }
\label{xypartitionfctquadratic}$$ It is $A:= Q\cdot \mathbbm{1} -J$ with $J$ the adjacency matrix, $Q:= \sum_{j=1 }^{N} J_{ij}$ the number of neighbours, and $J_{tot}:= \sum_{i=1 }^{N} \sum_{j=1 }^{N} J_{ij}$ the total number of connections in the adjacency matrix. Changing variables $\underline y = T \underline{\theta}$ with the transformation matrix $$T := ( u_{nk}) = \left(
\frac{1}{\sqrt{N}}
e^{-2 \pi i \frac{1}{L} \underline k \cdot \underline n}
\right)
\label{fouriertransform}$$ diagonalizes matrix $A$ such that $T^{\dagger} AT = \Lambda $ with $ \Lambda $ the diagonal matrix of eigenvalues of $A$ and $\underline u _k$ the eigenvectors of $A$ due to the structure of the adjacency matrix $J$. $T^{\dagger} $ is the Hermitean matrix of the complex matrix $T$, i.e. the transposed and complex conjugate. In 2 dimensions $$\underline k \cdot \underline n
=
\left(
\begin{array}{c}
k_x \\
k_y \\
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
n_x \\
n_y \\
\end{array}
\right)
\label{fouriertransformvectors}$$ with $k_x$ etc. having values 1,2,..,L, and with $N=L\times L$, the variable $$k:=k_x + (k_y-1) \cdot L
\label{fouriertransformbetraege}$$ has values 1,2,..., N. Likewise for $\underline n$. Also in 2d the eigenvalues are $$\lambda_k =
4-2 \cdot \cos \left( 2 \pi \frac{1}{L} k_x \right)
-2 \cdot \cos \left( 2 \pi \frac{1}{L} k_y \right)\quad.
\label{eigenwertevonA}$$ In the general case the partition function is then given in changed coordinates, the eigencoordinates of $A$, $$Z= e^{J_{tot}\cdot V}
\int_{-\infty }^{\infty } \, dy _1 ...
\int_{-\infty }^{\infty } \, dy _{N-1 } ...
\int_{-c }^{c} \, dy _N
\; \; e^{-V \cdot \underline{y} ^{tr} \Lambda \underline{y} }
\cdot |det(T)|
\label{xypartitionfcteigencoordinates}$$ with $|det(T)|=1$ and transformed integration boundaries for the zero eigenvalue coordinates $c$. It turns out that the $N^{th}$ coordinate is the zero eigenvalue coordinate, in which case $c= (u^{\dagger }_{NN})^{-1} \pi = \sqrt{N} \cdot \pi$.
Performing the Gaussian integration[^1] of Eq. \[xypartitionfcteigencoordinates\] gives $$Z= e^{J_{tot}\cdot V}
\prod _{k=1}^{N-1 }
\sqrt{\frac{\pi}{V\cdot \lambda _k}}
\cdot (2\sqrt{N}\pi)
\label{xypartitionfctgaussintegration}$$ where the last factor comes from the eigenvalue zero. Using the Fourier representation of the delta function in \[xydistribmagnetization\], we obtain
$$p(m)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\int_{-\pi }^{\pi} \, d\theta _1\cdots d\theta _N
\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{i(m-M(\underline{\theta})x}dx\frac{1}{Z} e^{{\cal H} (\theta _1, \cdots ,\theta _N)}\quad.
\label{xydistribmagnetization3}$$
For simplicity purposes it’s more convenient to write the magnetization in terms of the spin variables instead of vector spins,
$$M=\frac{1}{N}\sum_{r=1}^N(\theta_r-\overline{\theta})^2
\label{magnetizationabsolute}$$
this is just the average, $\langle\theta_{r}\rangle$ for unconfined spins. For the case of periodic spins, following [@Archambault1997], the instantaneous magnetization direction, $\overline{\theta}$, is defined by,
$$\overline{\theta} ={\it arctan} \left( {\frac {\sum _{r=1}^{N}\sin{{\theta_r }}}{\sum _{r=1}^{N}\cos{{\theta_r }}}}
\right)$$
which is the same as the average for large $N$. Then we may write $M({\underline{\theta}})= 1-\frac{1}{2N}\underline{\theta}^{tr}\underline{\theta}$. The magnetization PDF can be written using the quadratic approximation,
$$p(m)=\frac{e^{J_{tot}}}{Z}\int_{-\infty}^\infty \frac{dx}{2\pi}\int_{-\infty}^\infty \cdots\int_{-\infty}^\infty\int_{-\pi}^{\pi} d\underline{\theta}e^{ix\left(m-1+\frac{1}{2N}\underline{\theta}^{tr}\underline{\theta}\right)-V\underline{\theta}^{tr}A\underline{\theta}}
\label{xydistribmagnetization4}$$
where $d\underline{\theta}=\prod_{k=1}^N d\theta_k$. Simplifying for Gaussian integration,
$$p(m)=\frac{e^{J_{tot}}}{Z}\frac{1}{2\pi}\int_{-\infty}^\infty \frac{dx}{2\pi}\int_{-\infty}^\infty \cdots\int_{-\infty}^\infty\int_{-\pi}^{\pi} d\underline{\theta}e^{\underline{\theta}^{tr}\left(\frac{ix}{2N}\mathbbm{1}-VA\right)\underline{\theta}}\quad.
\label{xydistribmagnetization5}$$
Using the linear transformation (\[fouriertransform\]) and writing $B=VA-\frac{ix}{2N}\mathbbm{1}$ there is a diagonal matrix $\tilde{\Lambda}$ such that $B=T\tilde{\Lambda} T^{\dagger}$. Using the multivariate Gaussian integral, we obtain
$$\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}d\underline{\theta}e^{-\underline{\theta}^T B\underline{\theta}}=\sqrt{\frac{\pi^N}{\det{B}}}\quad.
\label{multivariategaussianintegral}$$
The matrix $B$ is equal to $q(A)$ where $q()$ is a polynomial of $A$. Then we may write,
$$\det(B)=\prod_{k=1}^N \ell_k=\prod_{k=1}^N q\left(\lambda_k\right)
\label{determinantofB}$$
where $\left\{\lambda_k\right\}_{k=1}^N$ are the eigenvalues of matrix $A$ and $\left\{\ell_k\right\}_{k=1}^N$ are the eigenvalues of $B$. The magnetization PDF is,
$$p(m)=\frac{(2c)e^{J_{tot}}}{Z}\int_{-\infty}^{\infty}\frac{dx}{2\pi}e^{ix(m-1)}\sqrt{\frac{\pi^{N-1}}{\prod_{k=1}^{N-1} \ell_k}}
\label{xydistribmagnetization6}$$
where $c=\sqrt{N}\pi$. Simplifying we have,
$$p(m)=\int_{-\infty}^\infty\frac{dx}{2\pi}e^{ix(m-1)}\left(\prod_{k=1}^{N-1}\frac{ \ell_k}{V\lambda_k}\right)^{-1/2}\quad.
\label{xydistribmagnetization7}$$
The mean and standard deviation magnetization can be calculated using the distribution of the magnetization absolute value, \[xydistribmagnetization6\]. Under the same assumption for the $A$ matrix we have
$$\begin{aligned}
\left\lbrace \begin{array}{c}
\langle m\rangle=1-\frac{1}{2N}\sum_{k=1}^{N-1}\frac{1}{2V\lambda_k}\\
\\
\sigma_m^2=\frac{1}{2N}\sum_{k=1}^{N-1}\frac{1}{2}\left(\frac{1}{N\lambda_k}\right)^2\quad.
\end{array}
\right.
\label{BHPmoments}\end{aligned}$$
At this point one has to do the appropriate normalisation in order to get the universal critical PDF. The distribution function for the magnetization has been discussed by several authors (see references in [@Archambault1997]). Binder, 1992[@Binder], argued that from the Ising model with distribution $f(m)$ there is scaling of the form
$$p(m)\sim L^{\beta/\nu}p(mL^{\beta/\nu},L/\zeta)
\label{BHPscaling1}$$
where $\zeta$ is the correlation length. We know that very close to the critical point the correlation length is going to be much bigger than the system size. In this case $L$ is the only important scale so $f$ should depend only on the variable $mL^{\beta/\nu}$,
$$p(m)\sim L^{\beta/\nu}p(mL^{\beta/\nu})\quad.
\label{BHPscaling2}$$
This form is not surprising since that for this system the mean $\langle m \rangle$ scales with $L^{-\beta/\nu}$ and being critical the standard deviation, $\sigma$ is also scales in the same way as the mean value. These facts explain the universal form of $p(mL^{\beta/\nu})$. Hence we can rewrite equation (\[BHPscaling2\]) as
$$p(m)\sim \frac{1}{\sigma_m}f(\frac{m}{\sigma_m})\quad.
\label{BHPscaling3}$$
The standard deviation is the correct normalization for the order parameter. Defining $\mu$ as
$$\mu=\frac{m-\langle m\rangle}{\sigma_m}
\label{normalizedm}$$
it is straightforward to see that the PDF $p(\mu)=\sigma_mp(m)$ and this PDF is normalized not depending on system size
$$\sigma _m \cdot p(m) \sim L^0 \cdot
f\left( \frac{m-\langle m \rangle}{\sigma _m} \cdot L^0 \right)\quad.
\label{BHPscaling4}$$
The form of the normalized and consequently universal BHP is,
$$\begin{aligned}
p(\mu)=\int_{-\infty}^{\infty}\frac{dx}{2\pi}
\sqrt{\frac{1}{2N^2}\sum_{k=1}^{N-1}\frac{1}{\lambda_k^2}}
&\!&e^{ix\mu\sqrt{\frac{1}{2N^2}\sum_{k=1}^{N-1}\frac{1}{\lambda_k^2}}
-\sum_{k=1}^{N-1}\left[\frac{ix}{2N}\frac{1}{\lambda_k}-\frac{i}{2}
\mbox{arctan}\left(\frac{x}{N\lambda_k}\right)\right]}\nonumber\\\!
&\!&.e^{-\sum_{k=1}^{N-1}\left[\frac{1}{4}\mbox{ln}{\left(1+\frac{x^2}{N^2\lambda_k^2}\right)}\right]}
\quad.
\label{BHPfinal}\end{aligned}$$
In figure \[fig:vertpm12\] we show a BHP PDF simulation for $L=10$, $N=L^2$, using eq. (\[BHPfinal\]).
![[]{data-label="fig:vertpm12"}](vertpm12.eps){width="8cm"}
At this point we should mention that the magnetic order parameter density of the 2dXY model approaches this universal PDF when $T\rightarrow0$. Surprisingly the quadratic approximation remains close to the true PDF for almost all the critical regime. It has been shown recently [@MackPalmaVergara2005; @BanksBramwell] that the weak temperature dependence gets noticeable near the KT phase transition.
BHP and the Danube and Douro data
=================================
Jánosi and Gallas [@ImreJason99] analysed statistics of the daily water level fluctuations of the Danube collect over the period 1901-97 at Nagymaros, Hungary. The authors found in the Danube data similar characteristics to those of company growth[@Stanleyetal]. This suggested that a universal description of the statistics should exist for both systems. Those authors noticed a data collapse for the conditional PDF of the one day logarithmic rate of changes for the mean adjusted river height.\
Following [@ImreJason99], let $h(t)$ be the 87$\times$365 data points[^2] the time series of height measurements and define a season mean and, following [@bramwellfennelleurophys2002], a season standard deviation, $\overline{h}(t)$ and $\sigma_h$, respectively,
$$\overline{h}(t)=\frac{1}{87}\sum_{j=0}^{86} h(t,j)
\label{seasonadjustment1}$$
where $h(t,j)=\left[\left\lbrace h(t+j*365)\right\rbrace_{j=0}^{86}\right] _{t=1}^{365}$,
$$\sigma_h(t)=\sqrt{\frac{\sum_{j=0}^{86} h(t,j)^2}{87}-{\overline{h}(t)}^2}\quad.
\label{seasonadjustment2}$$
The seasonality adjustment consists in normalizing each value of the original time series by subtracting the respective component of the season mean and dividing by the respective component of the season standard deviation, $\left(\frac{h-\overline{h}}{\sigma_h}\right) $. The water plays for the river the same role as the coupling strength in the 2dXY model.
![[]{data-label="fig:donaubinlogdata"}](donaubin3.eps){width="\linewidth"}
![[]{data-label="fig:donaubinlogdata"}](donaubinlogdata3.eps){width="\linewidth"}
In figures \[fig:donaubin3\],\[fig:donaubinlogdata\] we show the Histogram and Log histogram of $\sigma_hP(h)$ against $\left(\frac{h-\overline{h}}{\sigma_h}\right)$.
![[]{data-label="fig:donaubinlog3"}](donaubinlog3.eps){width="8cm"}
As it may seen in figure \[fig:donaubinlog3\] the reverse Log-BHP falls on top of the Log-histogram of the deseasonalised Danube data.\
The analogue histograms for the Douro river show a higher concentration of values below the origin (mean for the original data). This is not surprising because for the Douro basin there is no melting ice to feed the river in the spring and in the long dry summer where the natural flow is much less than that of winter. The mean daily flow of Douro is close to the percentile 70. For the Danube data the average height is about the same as the median.
![[]{data-label="fig:dourobinlogdata"}](dourobinnew.eps){width="\linewidth"}
![[]{data-label="fig:dourobinlogdata"}](dourobinlogdata.eps){width="\linewidth"}
The predominance of small runoffs for the Douro may be seen in the deseasonalised histogram and Log histogram, figures \[fig:dourobinnew\],\[fig:dourobinlogdata\]. In this last graphic the differences for the Danube are made very clear because of the spike at the origin.
![[]{data-label="fig:dourobinloggausswinter"}](dourobinloggauss.eps){width="\linewidth"}
![[]{data-label="fig:dourobinloggausswinter"}](dourobinloggausswinter.eps){width="\linewidth"}
Finally, in figure \[fig:dourobinloggauss\] is shown the Log of the BHP and the Gaussian fit[^3] on top of the Log histogram of the deseasonalised daily runoff data for the full year and only for the winter regime. The BHP PDF comes from a system with no outside tuning so it should model data that is very close to the natural flow of the river. The river flow is more close to its natural form when regulation is not applicable. This is the case when the river carries so much water that the dams have to be set open, see figures \[fig:dourobinloggauss\] and \[fig:dourobinloggausswinter\]. The differences in the left branch of figures \[fig:dourobinloggauss\] and \[fig:dourobinloggausswinter\] appears to be caused by a regulation for small Winter runoffs which seems not to be possible or desirable for the summer resulting in a bad fit for the full year. The PDF of the deseasonalised daily Douro runoff data in Winter results is a mixture of a Gaussian PDF for small and medium runoffs and BHP for runoffs above a certain threshold. Future work passes by searching for stronger evidence in favour or against the allegations concerning the differences between the so called *universal rivers* and those which seem not to be universal and also the differences between the winter regimes.\
The BHP PDF has been found to be an explanatory model for fluctuations of several phenomenon such as, width power in steady state systems, variations in river heights and flow, numerous self-organised critical systems among others, (see references in [@BanksBramwell]). Still it is not known yet why BHP captures the essential behaviour of systems of different universality classes. Finally there could be a relation between the Fisher-Tippett-Gumbel distribution of extremal statistics and the BHP but recent works [@Bramwelletal2001; @DahlstedtJensen2001], showed asymptotic differences between the two so in this direction there is also work to be done.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Imre Jánosi for providing the Danube data and him, Jason Gallas and Henrik Jensen for discussions and literature.
[99]{}
Gonçalves, R., Stollenwerk, N. and Pinto, A. Análise de scaling em rios e a hipótese de universalidade da distribuição BHP, [*Actas do XIII Congresso Anual da Sociedade Portuguesa de Estatistica.*]{}, 389–400.
Jánosi, I.M., & Gallas, J.A.C. (1999) Growth of companies and water-level fluctuations of the River Danube, [*Physica* ]{}[**A 271**]{}, 448–457.
Dahlstedt, K., & Jensen, H.J. (2005) Fluctuation spectrum and size scaling of river flow and level, [*Physica* ]{}[**A 348**]{}, 596–610.
Dahlstedt, K., & Jensen, H.J. (2001) Universal fluctuations and extreme-value statistics, [*J. Phys. A: Math. Gen.* ]{}[**34**]{}, 11193–11200.
Bramwell, S.T., Fortin, J.Y., Holdsworth, P.C.W., Peysson, S., Pinton, J.F., Portelli, B. & Sellitto,M. (2001) Magnetic Fluctuations in the classical XY model: the origin of an exponential tail in a complex system, [*Phys. Rev* ]{}[**E 63**]{}, 041106. Also as [*cond-mat/0008093v2, 12. Dec. 2000*]{}.
Bramwell, S.T., Fennell, T., Holdsworth, P.C.W., & Portelli, B. (2002) Universal Fluctuations of the Danube Water Level: a Link with Turbulence, Criticality and Company Growth, [*Europhysics Letters* ]{}[**57**]{}, 310. Also as [*cond-mat/0109117v2, 8. Sep. 2001*]{}.
Bramwell, S.T., Christensen, K., Fortin, J.Y., Holdsworth, P.C.W., Jensen, H.J., Lise, S., López, J.M., Nicodemi, M. & Sellitto,M. (2000) Universal Fluctuations in Correlated Systems, [*Phys. Rev. Lett.* ]{}[**84**]{}, 3744–3747.
Bramwell, S.T., Holdsworth, P.C.W., & Pinton, J.F. (1998) Universality of rare fluctuations in turbulence and critical phenomena [*Nature* ]{}[**396**]{}, 552–554.
Archambault, P., Bramwell, S.T., & Holdsworth, P.C.W. (1998) Magnetic fluctuations in a finite two-dimensional XY model, [*J. Phys. A: Math. Gen.* ]{} [**30**]{}, 8363–8378.
Bak, P., Tang, C., & Wiesenfeld, K. (1988) Self-organized criticality. [*Phys. Rev.* ]{}[**A 38**]{}, 364–374.
Banks, S. T., & Bramwell, S. T. (2005) Temperature dependent fluctuations in the two-dimensional *XY* model , [*cond-mat/0507424v1,18 July 2005*]{}.
Sinha-Ray, P., Borda Água, L., Jensen, H. J. (2001) Threshold dynamics, multifractality and universal fluctuations in the SOC forest-fire: Facets of an auto-ignition model, [*Physica D*]{} [**157 3**]{}, 186 - 196.
Stanley, M.H.R., Amaral, L. A. N., Buildrev, S. V., Havlin, S., Leschhorn, H. Maass, P., Salinger, M.A. & Stanley, E. (1996) Scaling behaviour in the growth of companies. [*Letters to Nature*]{} [**379**]{}, 29, 804-806.
Mack, G., Palma, G., & Vergara, L. (2005) Corrections to Universal Fluctuations in Correlated Systems: the 2D XY-model, [*cond-mat/0506713v1,28 June 2005*]{}.
Lenz, W. (1920) Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern. [*Phys. Zeitschr.*]{} [**21**]{}, 613-615.
Ising, E. (1925) Beitrag zur Theorie des Ferromagnetismus. [*Zeitschrift für Physik* ]{}[**31**]{}, 253-258.
Onsager, L. (1944) Crystal Statistics. I. A Two-Dimensional Model with a Order-Disorder Transition. [*Phys. Rev.*]{} [**65**]{}, 117-149.
Kosterlitz, J M. & Thouless, J. (1973) Ordering, metastability and phase transitions in two-dimensional systems. [*J. Phys. C.*]{}[**6**]{}, 1181-1203.
Binney, J.J., Dowrick, N.J., Fisher, A.J., & Newman, M.E.J. (1992). [*The Theory of Critical Phenomena, An Introduction to the Renormalization Group*]{} (Oxford University Press, Oxford).
Binder, K. (1992). [*Computational Methods in Field Theory*]{} (Springer, Berlin).
Yeomans, J.M. (1992). [*Statistical Mechanics of Phase Transitions*]{} (Oxford University Press, Oxford).
Zinn-Justin, J. (1989). [*Quantum Field Theory and critical phenomena*]{} (Oxford University Press, Oxford).
[^1]: Let $B$ be a complex symmetric matrix with a non-negative real part and non zero eigenvalues, then, $ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}d\underline{\theta}\,e^{-\underline{\theta}^T B\underline{\theta}}=\sqrt{\frac{\pi^N}{\det{B}}}$.
[^2]: all the $29^{\mbox{th}}$ of February were discarded
[^3]: only a subset of data was used for the fit.
|
---
abstract: 'Using state-of-the-art dynamical simulations of globular clusters, including radiation reaction during black hole encounters and a cosmological model of star cluster formation, we create a realistic population of dynamically-formed binary black hole mergers across cosmic space and time. We show that in the local universe, 10% of these binaries form as the result of gravitational-wave emission between unbound black holes during chaotic resonant encounters, with roughly half of those events having eccentricities detectable by current ground-based gravitational-wave detectors. The mergers that occur inside clusters typically have lower masses than binaries that were ejected from the cluster many Gyrs ago. Gravitational-wave captures from globular clusters contribute 1-2 $ \rm{Gpc}^{-3}\rm{yr}^{-1}$ to the binary merger rate in the local universe, increasing to $\gtrsim 10$ $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z\sim 3$. Finally, we discuss some of the technical difficulties associated with post-Newtonian scattering encounters, and how care must be taken when measuring the binary parameters during a dynamical capture.'
author:
- 'Carl L. Rodriguez'
- 'Pau Amaro-Seoane'
- Sourav Chatterjee
- Kyle Kremer
- 'Frederic A. Rasio'
- Johan Samsing
- 'Claire S. Ye'
- Michael Zevin
title: 'Post-Newtonian Dynamics in Dense Star Clusters: Formation, Masses, and Merger Rates of Highly-Eccentric Black Hole Binaries'
---
Introduction {#sec:intro}
============
In early 2016, Advanced LIGO reported the first detection of gravitational waves (GWs) from a binary black hole (BBH) merger [@Abbott2016a], demonstrating for the first time the feasibility of GW astronomy. With the subsequent detection of four more BBH mergers [@Abbott2017e; @Abbott2017c; @Abbott2017; @Abbott2016] and one binary neutron star inspiral [@Abbott2017d], we are rapidly approaching an era where catalogues of GWs will supplant single detections. While individual events can provide significant physical insight (particularly when coupled to electromagnetic observations), the power of GW astrophysics will soon come from comparing entire populations of compact-object mergers to detailed astrophysical models.
A significant amount of work has been done to try and understand the various possible formation pathways for merging BBHs. Broadly, most formation channels fall into one of two categories: isolated binary stellar evolution, where the BBH is formed as the remnant of a massive stellar binary [@Belczynski2002; @Voss2003; @Podsiadlowski2003; @Sadowski2007a; @Belczynski2010; @Dominik2012; @Dominik2014; @Dominik2013; @Belczynski2016], and dynamical formation, where the BBH is created and hardened through dynamical interactions in a dense stellar environment [@PortegiesZwart2000; @Banerjee2010; @Ziosi2014; @Banerjee2017; @OLeary2006; @OLeary2007; @Moody2009; @Downing2010; @Downing2011; @Tanikawa2013; @Bae2014; @Rodriguez2015a; @Rodriguez2016a; @Rodriguez2016b; @Askar2016; @Giesler2017; @Rodriguez2018; @Banerjee2017; @Hong2018; @Choksi2018; @Fragione2018]. Both of these broad categories can produce BBHs with masses, spins, and merger rates consistent with the current LIGO/Virgo constraints [e.g., @Amaro-Seoane2016], and there is no reason to suspect that multiple channels do not operate simultaneously, making the question of their origins particularly challenging.
However, one key observable that is unique to the dynamical formation channels is the potentially high orbital eccentricity of BBHs as they enter the sensitivity band of the GW detectors. Historically, eccentricity has not been included in many of the studies of compact-object mergers, because the emission of GWs can efficiently circularize a binary long before it reaches the frequency band of ground-based detectors. While this is certainly true for BBHs formed from isolated stellar binaries, many dynamical channels can produce BBHs mergers that retain significant eccentricities even up to a GW frequency of $10\,$Hz or greater. This can either arise from the influence of a third bound object, which effectively “pumps up” the binary eccentricity via the Lidov-Kozai mechanism [@Antonini2012a; @Antonini2016; @VanLandingham2016; @Leigh2017; @Silsbee2017; @Petrovich2017; @Hoang2018; @Antonini2017; @Rodriguez2018a; @Arca-Sedda2018], or by directly forming the BBHs with very high eccentricities near the detection threshold for terrestrial detectors [@Samsing2014; @OLeary2009; @Samsing2017a; @Samsing2017; @Amaro-Seoane2016; @Samsing2018; @Samsing2017b; @Rodriguez2018; @Zevin2018]. While the latter is nearly impossible in the purely Newtonian case, the inclusion of post-Newtonian (pN) corrections to the equations of motion offers a new pathway for BBH formation. In particular, the 2.5pN term describing non-conservative radiation reaction allows for the formation of bound BBHs by the emission of GWs [*during*]{} a close BH encounter [e.g., @Hansen1972; @Quinlan1989]. These encounters are most likely to occur during three- and four-body scatterings, where the long-lived chaotic states offer many opportunities for unbound BHs to pass sufficiently close that they emit a pulse of GWs and form a highly-eccentric binary [e.g., @Gultekin2006; @Samsing2014]. These encounters occur frequently in the cores of dense stellar environments such as globular clusters (GCs), where a binary will continue to encounter other stars and binaries until it is either disrupted, ejected from the cluster, or merges due to GW emission [e.g., @Benacquista2013],
Recently, semi-analytic [@Samsing2018] and fully-numerical [@Rodriguez2018] models of GCs have shown that these GW-driven captures can contribute significantly to the BBH merger rate, with roughly 5% of sources from dense star clusters entering the LIGO/Virgo detection band with an eccentricity of 0.1 or higher. These sources emit GWs with a unique spectral signature, providing a telltale sign of dynamical formation. In parallel, much work has been done to develop waveforms that can model these unique signatures, placing the measurement of orbital eccentricity from BBH mergers within the grasp of Advanced LIGO/Virgo [@Mikoczi2015; @Huerta2018; @Hinder2010; @Tanay2016; @Gondan2017; @Moore2016; @Huerta2017; @Yunes2009; @Pierro2001; @Sperhake2008; @Huerta2014]. The latest generation of waveforms, combining pN theory with numerical relativity and BH perturbation theory, are approaching sufficient fidelity that they can enable both the detection and characterization of eccentric mergers [@Huerta2018]. To avail ourselves of this new, dynamically-rich parameter space, we must better understand the formation, evolution, and mergers of eccentric BBHs across cosmic space and time.
In this paper, we expand upon our previous work [@Rodriguez2018], to fully explore the formation, masses, and merger rates, of eccentric BBH mergers from GCs. Using state-of-the-art dynamical models of GCs, along with a recently developed cosmological model for cluster formation [@El-Badry2018; @Rodriguez2018b], we perform a complete population survey of eccentric BBH mergers from the cores of dense star clusters. In Section 2, we describe the changes to our method from [@Rodriguez2018], and how we combine the cosmological model for GC formation from [@El-Badry2018] with our $N$-body models of GC evolution to produce a realistic population of BBH mergers across cosmic time. In Section 3, we explore the various mechanisms by which BBH mergers are produced in GCs, and describe the expected eccentricities and masses for each sub-channel, both globally and in the local universe ($z<1$), as well as the formation of BBH mergers with GW frequencies *inside* the LIGO/Virgo band. Finally, in Section 4, we explore the cosmological rate of eccentric BBH mergers from GCs. Throughout this paper, we assume a flat $\Lambda \rm{CDM}$ cosmology with $h= 0.679$ and $\Omega_M = 0.3065$ [@PlanckCollaboration2015], and that all BHs from stars are born with no spin (though they can attain spin through mergers with other BHs), which allows for the retention of BH merger products [see e.g., @Rodriguez2018] in the cluster.
Methods
=======
Monte Carlo Models {#sec:mccosmo}
------------------
We generate our GC models using the Cluster Monte Carlo (`CMC)` code, a Hénon-stlye [@Henon1971; @Henon1971a] Monte Carlo code for modeling the evolution of massive, spherical star clusters. `CMC` has been developed over many years [@Joshi1999; @Pattabiraman2013], and can model all of the relevant physical processes which drive the evolution of dense star clusters. This includes orbit-averaged two-body relaxation, single and binary stellar evolution [@Hurley2000; @Hurley2002; @Chatterjee2010], three-body binary formation from single BHs [@Morscher2012; @Morscher2015], three- and four-body scattering encounters between stars and black holes [@Fregeau2004; @Fregeau2007], physical collisions, and galactic tides. This approach has been shown to produce GC models similar to those generated with direct $N$-body calculations [e.g. @Wang2016] in a fraction of the computational time [@Rodriguez2016]. `CMC` uses prescriptions for stellar winds, supernovae, pulsational-pair instabilities, and BH formation [@Rodriguez2016a; @Rodriguez2018] which are identical to the single stellar evolution prescriptions used in the most recent population synthesis approaches to modeling BBHs formed from massive stellar binaries [@Belczynski2010; @Fryer2012; @Dominik2013; @Belczynski2016; @Belczynski2016a]. `CMC` also includes pN corrections to encounters between BHs and BBHs using the technique developed in [@Antognini2014; @Amaro-Seoane2016; @Rodriguez2018], allowing us to self-consistently model GW emission and BBH mergers during strong resonant encounters.
We consider a grid of 48 GC models, covering a wide range of initial conditions. As in [@Rodriguez2018], we created 24 GC models spread across a 4x3x2 grid in initial mass, metallicity, and virial radius. Our models span four initial particle numbers ($2\times10^6$, $10^6$, $5\times10^5$, and $2\times10^5$) corresponding roughly to initial masses of $1.2\times10^6M_{\odot}$, $6\times10^5M_{\odot}$, $3\times10^5M_{\odot}$, and $10^5M_{\odot}$, three separate stellar metallicities ($0.01Z_{\odot}$, $0.05Z_{\odot}$, and $0.25Z_{\odot}$) at different galactocentric radii (20 kpc, 8 kpc, and 2 kpc respectively, roughly following the correlation in the Milky Way), and two initial virial radii (1 pc and 2 pc). The masses of single stars and the primary masses of binaries are chosen from a Kroupa initial mass function [@Kroupa2003] in a range between $0.08 M_{\odot}$ to $150M_{\odot}$. For each of these models, we assume 10% of our particles are initially in binaries, with semi-major axes drawn from a distribution flat in the log from stellar contact to the local hard/soft boundary, eccentricities drawn from a thermal distribution ($p(e)de= 2ede$), and secondary masses drawn from a uniform distribution between 0 and the primary mass. We assume that the binary fraction is independent of the stellar mass. In addition to the 4x3x2 grid of models, we generate 4 additional, statistically-independent realizations for each of the models with $2\times10^5$ initial particles. This was done to increase the statistics for mergers from smaller clusters (some of which may disrupt before the present cosmological era), where any individual cluster may only produce a few (if any) BBH mergers over its lifetime. This yields a total number of 48 GC models.
In previous works employing realistic GC models [e.g., @Downing2011; @Rodriguez2016a; @Askar2016], it was assumed that all GCs formed at the same epoch, i.e. that all GCs formed exactly 12 Gyr ago at redshift 3.5. However, this well-known approximation ignores the measured spread in GC ages and the correlation between age and cluster metallicity [e.g., @Forbes2010]. Furthermore, it ignores the continued formation of dense star clusters up to the present day (e.g. super-star clusters), and the correlation between formation times and galaxy halo mass [though this was done correctly in @Chatterjee2017]. Here, we extend our GC models to redshift 10, corresponding to an age of 13.3 Gyr, and convolve their time evolution with a cosmological model for GC formation which we now describe.
Comparing collisional models of GC evolution to detailed cosmological simulations is well-beyond both the scope of this paper and the resolution of most cosmological simulations. However, recent models have been developed that can predict the formation and evolution of GCs across cosmic times and galaxy type, either using halo merger trees and detailed prescriptions for gas-cloud collapse [e.g. @Boylan-Kolchin2017; @Choksi2018; @El-Badry2018], or semi-analytic treatments of cluster evolution and disruption in the galaxy [e.g. @Fragione2018a]. These models have been used to estimate the rates of various transients from dense star clusters [e.g. @Hong2018; @Fragione2017; @Fragione2018; @Rodriguez2018b; @Choksi2018a] while properly accounting for the formation, destruction, and evolution of said clusters across cosmic time.
In [@Rodriguez2018b], we used the GC formation model from [@El-Badry2018] to estimate the BBH merger rate from GCs formed in different halo masses across cosmic times. [This was done using the GC models presented in here.]{} We extend this formalism, using it both to draw our initial distribution of cluster ages and to determine how to weigh the BBH mergers from the 48 cluster models to best represent a realistic population of GCs. To that end, we use the fitting formula from [@Rodriguez2018b], to Figure 5 of [@El-Badry2018] which captures the formation rate of GCs per galaxy halo mass at a given redshift (accounting for the halo mass function). See Appendix A of [@Rodriguez2018b]. Integrating this rate directly over all halo masses (from $10^8M_{\odot}$ to $10^{14}M_{\odot}$) would give us the distribution of formation times for GCs. However, we can improve upon this simple estimate by binning the integral according to the median metallicity of star formation occurring in halos of a certain mass at a given redshift. To do this, we place our GC models in the center of three logarithmically spaced bins in metallicity space; i.e., we assume our $0.01Z_{\odot}$ clusters represent all clusters formed with $Z < 0.022 Z_{\odot}$, the $0.25Z_{\odot}$ clusters cover the bin where $Z > 0.11 Z_{\odot}$, and the $0.05Z_{\odot}$ clusters represent clusters in the middle bin. To determine the formation rate for a specific metallicity, we only integrate over halo masses where the median gas metallicity at a given redshift for that halo mass lies within the metallicity bin of interest:
$$\dot{M}_{\rm GC}(\tau) = \int_{\left<M_{\rm Halo}\right>(Z_{\rm low})}^{\left<M_{\rm Halo}\right>(Z_{\rm high})} \left. \frac{ \dot{M}_{\rm GC}} {d\log_{10}M_{\rm
Halo}} \right|_{z(\tau)} dM_{\rm Halo}
\label{eqn:elbad}$$
The median star formation metallicity for a given halo mass at a given redshift is taken from [@Behroozi2013], and the relation between stellar metallicity and gas metallicity is taken from [@Ma2016]. This is identical to the stellar enrichment prescription used in [@El-Badry2018], and we find it to produce similar results.
We show the distribution of formation times for clusters with specific metallicities in Figure \[fig:formation\]. As would be expected from any reasonable cosmological model, the higher metallicity clusters form at later time, and all cluster formation in the local universe occurs in high-metallicity clusters. In the bottom panel of Figure \[fig:formation\], we show the distribution of merger rates from our cluster models, convolved with the cluster formation times calculated above. Even though they represent a smaller total fraction of clusters in the local universe, the continuous formation of clusters leads to the high-metallicity GCs playing the largest contribution to the BBH mergers in the local universe. To generate the population of BBH mergers that we will use for the remainder of this paper, we assign to each BBH merger 100 unique formation times from Figure \[fig:formation\], allowing us to convolve the BBH merger rate with the cosmological model from [@El-Badry2018].
Weighting the GC Models
-----------------------
To get a representative sample of BBH mergers from GCs, we must apply a weighting scheme to our grid of models, in order to better represent the observed and theoretically-predicted properties of GCs. In [@Rodriguez2016a], this was accomplished by binning the population of GCs in mass/metallicity space, then drawing with replacement from our sample of BBH mergers according to the weight of each model (i.e. models with higher weights contributed more BBHs to our effective population in the local universe). Here, we can take a more cosmologically-motivated approach. We assign to each model a weight based on the cluster initial mass function (CIMF) and metallicity. We assume that GCs form following a $1/M^2$ distribution, then assign to each cluster a weight based on its initial mass, assuming that the cluster occupies the center of a linearly-spaced bin in the CIMF. The least massive clusters ($\sim10^5M_{\odot}$) get a weight of $\sim 50\%$, while the most massive clusters ($\sim1.2\times10^6M_{\odot}$) contribute only $\sim 10\%$ of their binaries. Of course, many of our smallest clusters disrupt before the present day, meaning that the contribution from these clusters to the merger rate in the local universe will either arise from clusters that were formed at late times or from BBHs which were ejected from their parent clusters prior to disruption. Additionally, to each model we assign a metallicity weight. This is simply the fraction of GCs that formed in each metallicity bin from Figure \[fig:formation\]. The weight assigned to each BBH merger is simply the product of the mass and metallicity weights assigned to each cluster.
Throughout the rest of this paper, any quantities we quote (averages, histograms, cumulative fractions, etc.) will be weighted according to this scheme, except where otherwise noted. We also note that there still remains uncertainty in the CIMF, especially given observational evidence of an exponential-like truncation at higher masses in young massive star clusters [e.g., @PortegiesZwart2010]. However, to test the sensitivity of our results on the CIMF, we also considered a weighting scheme based on the present-day GC mass function (GCMF), taken by shifting the peak the observed luminosity function of GCs [@Harris2014] upwards by a factor of 4 to account for a mass-to-light ratio of 2 [@Bell2003] and the (roughly factor of 2) mass loss experienced by GCs over their $\sim12$Gyr lifetimes [e.g. @Morscher2015]. Because the most massive clusters contribute the majority of sources, and because the number of BBH mergers within a Hubble time from a single cluster scales super-linearly with the cluster mass [e.g. @Rodriguez2016a], the most massive clusters from the current grid still dominate the BBH mergers in the local universe in either case; however, we find only minimal changes in the properties of BBH mergers when using the CIMF versus the observed GCMF. Therefore, we only report the results using the CIMF for the remainder of this work.
pN Integrations of BH Encounters {#sec:pnfix}
--------------------------------
We have made several changes to the pN implementation described in [@Rodriguez2018], which we will now detail. First, for the runs described in this work, we only integrate BH encounters using purely Newtonian forces plus the 2.5pN term, which is responsible for the emission of GWs. This represents a departure from [@Amaro-Seoane2016; @Rodriguez2018], where both the 1pN and 2pN terms were included as well[, but is consistent with the models used for the cosmological rate estimate in [@Rodriguez2018b]]{}. While including all terms up to and including the 2.5pN terms obviously includes more physics, this can introduce significant errors in the classification of bound systems in the `Fewbody` integrator.
By default `Fewbody` reclassifies the entire hierarchical structure of the system every 500 timesteps, to determine when to terminate any encounter and how to report the outcome.However, this classification scheme depends on the energy and angular momentum of bound systems in the encounter, which are converted into orbital elements (semi-major axis and eccentricity). With the inclusion of pN terms, the Newtonian energy and angular momentum are no longer formal constants of the motion (even without the dissipative 2.5pN terms), and can vary significantly over a single orbital period. Furthermore, because the pN expansion is only valid up to the next highest order in $(v/c)$, there do not exist well-defined constants of the motion that can be used to identify particles and bound or unbound. Even the pN definitions of the energy and angular momentum are only conserved in the orbit-averaged approximation. For a binary whose component velocities are not constant over the orbit (such as the highly-eccentric systems studied here), the pN energy and angular momentum can vary significantly over a single orbit.
In the top panel of Figure \[fig:Fewbody\], we show the fractional change in the energy over a single orbit of a moderately eccentric binary ($m_1 = m_2 =
20M_{\odot}$, $e_0 = 0.85$ and $a_0 = 500M$, where $M$ is the total mass of the binary in geometric units, i.e. $(m_1 + m_2) G/c^2$). As the velocity changes from apocenter to pericenter by a factor of $\sim 12$, the energy increases by nearly 2.5% when only 1pN corrections are considered. When both 1 and 2pN terms are included, the change in energy decreases by nearly two orders of magnitude. Note that we are using the pN definitions of the energy [e.g. @Mora2004] corresponding to each pN order. We find that the fractional change in energy from apocenter to pericenter is directly proportional to the next-to-next-to leading order pN correction. In other words, the change in the 1pN energy is proportional to $(v/c)^6$, while the 2pN energy difference scales as $(v/c)^8$. While the increase in the example in Figure \[fig:Fewbody\] is relatively minor (especially when 2pN terms are included), this change can be extreme for the GW captures we are interested in, which frequently occur at the $e \rightarrow 1$ boundary between unbound and bound systems. In these cases, the variation in the energies can be much more extreme, and for binaries where $(v/c) \sim 0.1$ at pericenter, the variation over a single orbit can be as much as 1000% (10%) for the 1pN (2pN) equations of motion. Because of these issues, we choose to remove the 1 and 2pN terms from our integration in this work. This decreases the number of mergers with eccentricities of $\sim 10^{-3}$ at a GW frequency of 10Hz that were reported in [@Rodriguez2018], which arose from the classification of unbound systems as highly-eccentric (and bound) binaries.
[When considering the long-term dynamics of triple systems, particularly hierarchical triples that undergo Kozai-Lidov oscillations, the contributions from the conservative pN terms can be substantial, since the relativistic precession of the binary pericenter can suppress (or in rare cases enhance) the eccentricity growth of the inner binary [@Naoz2013]. This can be particularly important when considering the formation of long-lived triples in dynamical environments, which may contribute at the $\sim 1\%$ level in GCs [@Antonini2016], or even higher in open clusters [@Banerjee2018a]. CMC does not currently track the secular evolution of hierarchical systems, and so we do not consider these effects here.]{} But for the scattering encounters that dominate the mergers presented here, the conservative 1 and 2pN terms do not appear to have a significant impact. Recent work [@Zevin2018] has shown that the inclusion of conservative pN terms during scattering encounters such as these do not have any noticeable influence on the statistical outcomes of these scattering experiments. This, combined with the difficulties in correctly measuring the energies of these systems, was the primary motivation for our exclusion of the conservative pN terms. [However, because the inclusion of higher-multiplicity BH systems would only increase the number of highly-eccentric BBH mergers, we consider our results here to be a conservative lower limit.]{}
[Finally, we note that an error was discovered in the pN physics from our previous work [@Rodriguez2018] that changed the strength of the relativistic terms for scattering encounters with non-equal mass components. This error did not significantly alter our previous results (the fraction of mergers occurring in the cluster decreased by $\sim 10\%$ in the local universe) but we mention it here for consistency. This error was discovered in [@Rodriguez2018b], and does not effect the rate estimates quoted there.]{}
![A pN integration of an isolated binary with close separation and high eccentricity, typical of GW capture BBH mergers with high eccentricities. In the top panel, we show the energy of the binary over two orbits when only the conservative terms are included. Because the pN expansion explicitly does not conserve energy beyond the highest order in $v/c$, the binary introduces a significant energy increase during the high-velocity pericenter passage. This error does not occur in the purely Newtonian case. In the bottom panel, we show the variation in the Newtonian eccentricity near merger. The integration of Equations and from initial conditions extracted at instantaneous separations of $10M$ and $100M$ are shown in dashed orange and dotted red, respectively.[]{data-label="fig:Fewbody"}](fewbody.pdf)
Measuring Eccentricity during GW Captures
-----------------------------------------
To better measure the eccentricities of any BBHs that merge during resonant encounters, we also sample the orbital elements of merging BHs at multiple discrete separations. We record the semi-major axes and eccentricities for all pairs of BHs (bound or unbound) when they approach within 500M, 100M, 50M, and 10M, where M is the total mass of the BH pair. To determine when a binary crosses into the LIGO/Virgo band, we must know the eccentricity and semi-major axis when that binary’s GW frequency passes 10 Hz. For a circular binary, the GWs are all emitted at the lowest-order harmonic ($n=2$) of the orbital frequency, such that the GW frequency is simply twice the orbital frequency. However, an eccentric binary emits GWs across a range of harmonics of the orbital frequency. The dominant frequency at which an eccentric binary emits GWs can be approximated as [@Wen2003]
$$f_{\rm GW} = \frac{\sqrt{G M}}{\pi} \frac{(1 + e)^{1.1954}}{\left[
a(1-e^2)\right]^{3/2}}~,
\label{eqn:wen}$$
where $a$ and $e$ are the semi-major axis and eccentricity, respectively. To determine when a binary will enter the LIGO/Virgo band, we can simply integrate the orbit-averaged equations for the evolution $a$ and $e$ from [@Peters1964]:
$$\begin{aligned}
\left<\frac{da}{dt}\right> &= -\frac{64}{5} \frac{G^3 m_1 m_2 (m_1+m_2)}{c^5 a^3 (1-e^2)^{7/2}} \left( 1+\frac{73}{24}e^2 + \frac{37}{96}e^4 \right)~,\label{eqn:dadt}\\
\left<\frac{de}{dt}\right> &= -\frac{304}{15} e \frac{G^3 m_1 m_2 (m_1+m_2)}{c^5 a^4 (1-e^2)^{5/2}} \left( 1 + \frac{121}{304} e^2 \right)~,\label{eqn:dedt}\end{aligned}$$
until Equation equals 10Hz. Here $m_1$ and $m_2$ are the component masses of the binary.
To ensure that we are measuring the correct $a$ and $e$ to use as initial conditions for the Peters equations, we pick the largest $(a,e)$ pair sampled from $10M$ to $500M$ where the system is bound, in order to minimize the error introduced by sampling the binary in the strong-field regime. These Newtonian orbital elements can vary over an orbital timescale, particularly in the strong-field regime near merger (see the bottom panel of Figure \[fig:Fewbody\]). Note that there are two cases where this procedure may not provide a solution: first, it is possible that sources can form *inside* the LIGO/Virgo band, at such a close separation that their dominant frequency is already greater than 10Hz when the binary forms. This typically occurs when the eccentricity is extremely large, so we simply report these systems as having eccentricities of 0.99 at 10Hz. Secondly, we find that a handful of BBHs are technically unbound when they merge in a `Fewbody` encounter. However, because we treat two BHs as having merged whenever they pass within 10M of one another, we cannot resolve whether these systems were the result of true collisions between unbound particles, or whether they were binaries that formed with pericenters less than 10M. We record these systems as having eccentricities $e\gtrsim 1$, and will discuss both cases in Section \[sec:pericenter\].
Eccentricities of Merging Binaries
==================================
For the purposes of this study, we will divide our BBH mergers into four categories [see also @Zevin2018]:
- **Primordial Binaries** - mergers that occur though isolated binary stellar evolution in the GC. Of course, BBHs from binary stars can undergo dynamical encounters in the cluster before merging. Here we label a binary as primordial only if it never participated in a strong dynamical encounter.
- **Ejected Mergers** - BBHs undergoing many hardening encounters before being ejected from the cluster, only to merge later in the field [e.g. @Downing2011; @Rodriguez2016a]
- **In-Cluster Mergers** - binaries merging inside the GC after a dynamical encounter, but [*not*]{} due to significant GW emission [*during*]{} the encounter.
- **GW Captures** - BHs merging [*during*]{} a resonant encounter due to strong GW emission at a very close passage.
For the remainder of this paper, we will focus on the ejected, in-cluster, and GW capture channels, since these are unique to the dynamical environments in which they are formed.
In Figure \[fig:ecc\], we show the total distribution of eccentricities at a GW frequency of 10Hz from our GC models at all redshifts and in the local universe ($z<1$). The final eccentricities span a wide range, from $e = 10^{-8}$ all the way to $e\sim 1$. However, each of the four types of BBH mergers inhabit a distinct range of the eccentricity space. Both the primordial binaries (which mostly circularize during mass transfer) and the ejected binaries peak near $10^{-6}$. This is consistent with previous results that have shown that both ejected cluster binaries and BBHs from the field should have relatively similar final eccentricities [@Breivik2016; @Amaro-Seoane2016; @Nishizawa2016], with the cluster binaries having a tail extending to higher eccentricities.
The total fraction of mergers in each sub-population changes as a function of redshift. For the total population (out to redshift 10) ejected BBHs are the dominant contribution, comprising 55% of all mergers from GCs. The remaining 45% of mergers occur in the cluster, with 5% resulting from primordial binaries, 28% merging as isolated binaries in the cluster, and 12% forming as GW captures. In the local universe ($z < 1$), the contribution from ejected binaries increases, since many of the BBHs that were ejected early in the cluster lifetime may not merge for many Myr or Gyr (see Section \[sec:rates\]). At these later times, the ejected BBHs comprise 64% of all mergers, with primordial binaries, in-cluster mergers, and GW captures contributing 1%, 25%, and 10%, of the total BBH mergers, respectively.
The red histogram in Figure \[fig:ecc\] shows the distribution of eccentricities from GW captures. Unlike [@Rodriguez2018], the distribution ranges from as low as $10^{-3}$ at a peak GW frequency of 10Hz, all the way to $e\sim1$. This is largely due to our more conservative physics for the pN scatterings described in Section \[sec:pnfix\]. While the total number of GW captures has increased significantly, the fraction of mergers with measurable eccentricities is identical to our previous results, with roughly $4\%$ of all mergers from GCs entering the LIGO/Virgo band with eccentricities greater than 0.1. This fraction increases to $\sim6\%$ if we consider the lower threshold ($e \gtrsim 0.05$) for measurably eccentric BBHs recently proposed by [@Lower2018], although we note that study focused on BBH mergers with masses similar to GW150914 ($30M_{\odot}+30M_{\odot}$), which is more massive than the typical highly-eccentric merger identified here (see Section \[sec:masses\]).
Why BBHs Merge Where They Do
----------------------------
Of these dynamical channels, what primarily differentiates the three? As binaries interact with other BHs and stars in the cluster, hard binaries (those whose binding energy is greater than the typical kinetic energy of surrounding stars and BHs) will preferentially harden after each encounter, shrinking their semi-major axes and leaving the encounter with some fraction of that binding energy converted to kinetic energy. This statistical inevitability, known as Heggie’s law [@Heggie1975], will continue, producing harder and harder binaries until the binary either merges (due to GW emission) or is ejected from the cluster by the recoil of the encounter. Which of these two available pathways a binary takes is largely a matter of timescales. After an encounter, the survival of a binary is dictated by the competition of two timescales: the timescale for GW emission to drive a binary to merge, given by
$$T_{\rm GW} \propto a^4(1-e^2)^{7/2}~,$$
where $a$ is the semi-major axis of the binary, and $e$ is the eccentricity, and the average time between successive binary encounters, which scales as
$$T_{\rm bs} \propto n a^2 \sigma \left(1+\frac{G M}{2a\sigma^2}\right)~,$$
where $n$ is the number density of single particles and $\sigma$ is the typical velocity dispersion. As shown by Heggie, each resonant encounter (between objects of near-equal masses) will produce binaries with eccentricities drawn from a thermal distribution, $p(e)de = 2ede$. Because of the extreme dependence of $T_{\rm GW}$ on the orbital eccentricity, if a binary leaves any scattering encounter with a sufficiently high eccentricity, it can easily merge before being disrupted or disturbed by a third body.
In Figure \[fig:eccevol\], we show the post-encounter eccentricities and semi-major axes for each BBH after its last encounter in the cluster. For ejected BBHs, this is also the encounter responsible for its ejection. Here, it is immediately obvious that the in-cluster mergers are preferentially selected from BBHs which leave their last encounters with a very high eccentricity, while the ejected BBHs leave the cluster with a distribution of eccentricities very close to thermal. Because of the steep dependence of the inspiral time on eccentricity, the semi-major axes of the in-cluster mergers after their last encounter can be significantly larger than the ejected binaries, with a few percent of in-cluster mergers having semi-major axes in excess of 10 AU. As an example, a $30M_{\odot}+30M_{\odot}$ binary with a separation of $a=10\rm{AU}$ and an eccentricity of $e=0.9999$ will merge in $10^4\rm{yr}$, well before another binary encounter (though this will depend on the concentration of the cluster). This is obvious in the bottom panel of Figure \[fig:eccevol\]. Typically, most in-cluster mergers occur in less than $\sim 10$Myr after their last encounter in the cluster, while the ejected BBHs can may have a significant delay between their ejection and merger (up to many Hubble times).
### Predicting Where BBHs Merge
[ Each of the three dynamical channels operates on a unique timescale, $\tau$, which allows us to make significant analytic progress. For ejected mergers, that timescale is roughly a Hubble time, $\tau_{\rm ej} \sim T_{H}$, since after ejection the BBH need only merge within the lifetime of the Universe to be of interest to LIGO/Virgo. For in-cluster mergers, a BBH needs to merge before its next encounter in the cluster, making its maximum lifetime roughly $\tau_{\rm in} \sim T_{\rm bs}$, while for GW captures, the timescale is roughly the orbital timescale of the binary at the beginning of an encounter ($\tau_{\rm GW} \sim T_{\rm orb}$) [see e.g., @2018MNRAS.481.5445S; @2018MNRAS.481.4775D]. These timescales can be used to make specific predictions for the fraction of mergers that occur in each dynamical channel, as has been done in [@2018ApJ...853..140S; @2018PhRvD..97j3014S; @2018MNRAS.481.5445S; @2018MNRAS.481.4775D]. We can compare this formalism, developed by comparing the difference in timescales at each stage in a binary’s life in the cluster to our numerical results, demonstrating that this simple consideration of timescales can predict the relative fraction of in-cluster mergers and GW captures from GCs. ]{}
[ We start by deriving the probability, $P(t < T)$, for an assembled BBH to merge due to GWs within time $T$. Assuming the eccentricity distribution of the dynamically-assembled BBHs follows a thermal distribution $P(e)de =2ede$, one can show that [@2018ApJ...853..140S; @2018PhRvD..97j3014S], ]{} $$\begin{aligned}
P(t < T) &\approx \left(\frac{T}{T_{\rm GW}^{e=0}}\right)^{2/7} \nonumber \\
&\propto T^{2/7} a^{-8/7}M^{6/7},\end{aligned}$$
where $T_{\rm GW}^{e=0}$ is the GW inspiral time of a given binary (assuming a circular orbit) [@Peters1964]. [ To derive the relative number of events of each outcome, we assume that BBHs form in their clusters at the hard-soft boundary, where the binding energy of the binary is equal to the average kinetic energy of nearby particles, since binaries with lower energies are typically destroyed during strong encounters [@Heggie1975]. We then assume that these binaries are hardened during binary-single encounters with other BHs, with each encounter reducing the binary’s semi-major axis by a fixed fraction ($\delta = 7/9$). This continues until the binary reaches a separation, $a_{\rm ej}$, where the liberation of $\sim 20\%$ of the binding energy is sufficient to eject the BBH from the cluster. Of course, during each encounter, there exists some probability that the binary will either merge promptly due to GW emission, or will leave with such a large eccentricity that it merges before its next BH encounter. As shown in [@2018PhRvD..97j3014S], the probability for a BBH to undergo a merger of outcome type $i$ can be written as, ]{} $$P_{i} \approx F_{i} \times \left(\frac{\tau_{i}(a_{\rm ej})}{t_{\rm GW}^{e=0}(a_{\rm ej})}\right)^{2/7},$$
[ where $P_{i}$ is the probability for that a BBH initially formed at the hard-soft boundary and is hardened until ejection results in outcome $i$, and $F_{i}$ is a pre-factor based on the typical number of encounters a BBH undergoes in the cluster, weighted according to the GW merger probability of each encounter. As shown in [@2018PhRvD..97j3014S], for in-cluster mergers $F_{\rm in} \approx (7/10)/(1-\delta) \approx 3$, while for GW captures, $F_{\rm GW} \approx (7/5)/(1-\delta) \times N_{\rm MS} \approx 120$, where $N_{\rm MS}$ is the number of intermediate, meta-stable BBH states a binary forms *during* a resonant three-body encounter. Evaluating $P_{i}$ for each outcome, assuming $a_{\rm ej} \sim 0.5$ AU, $M \sim 30M_{\odot}$, $\tau_{\rm in} \sim 10^7$ years, and $\tau_{\rm GW} \sim 0.1$ year, we find that $P_{\rm in} \approx 0.15$ and $P_{\rm GW} \approx 0.03$. This implies that approximately 82% of dynamically-formed BBHs will be ejected from the cluster. Of this ejected population a fraction $P(t_{\rm GW} (a_{\rm ej}) < T_{\rm H})
\approx 0.35$ will merge within $T_{\rm H}$, meaning that the probability for a BBH formed at the hard-soft boundary to get ejected and merge within a Hubble time is approximately $0.82 \times 0.35 \approx 0.3$. ]{}[ Of course, the escape speed and central concentration of a GC can change by factors of a few over the evolution of the cluster system, changing both the rate at which BBHs are produced and their typical semi-major axes at ejection [e.g., @Rodriguez2016a]. However, as a simplifying assumption we assume that BBHs are dynamically formed at the hard-soft boundary at a steady rate. The total contribution to the BBH merger rate from each outcome is simply the probabilities from the previous paragraph normalized to the sum of the three outcomes, i.e. $P_{i}/\sum_{i} P_{i}$. Applying this, we find that ejected mergers, in-cluster mergers, and GW captures contribute $62\%$, $31\%$, and $7\%$ of the total cluster merger rate, respectively. Note that for this estimate we have not included binary-binary interactions, which have been shown to contribute about $30\%$ of all GW captures [@2018arXiv181000901Z]. If this were accounted for, we would expect that $\sim 10\%$ of BBH mergers would come from GW captures. These numbers are in good agreement with the numerical results presented here, indicating that the analytic approach developed in [@2018PhRvD..97j3014S; @2018MNRAS.481.5445S; @2018MNRAS.481.4775D] can provide a reasonable and effective estimate of where and how BBHs merge from dense star clusters. ]{}
### Distinguishing In-cluster Mergers from GW Captures
It is natural to worry that our distinction between in-cluster mergers and GW captures is entirely a result of our termination criterion for scattering encounters. By default, `Fewbody` will end any encounter that has resolved into bound, hierarchical systems that are moving away from one another, while the pN corrections from [@Antognini2014] introduce an additional criteria that the pericenter velocity of any binary be less than 5% the speed of light. One could easily imagine, however, that binaries could be formed by the emission of GWs during an encounter and be classified by `Fewbody` as stable systems, causing the merger to be classified as an isolated, in-cluster merger.
To test the robustness of our classification scheme, we track the GW emission during each scattering encounter by directly evaluating $\frac{dE}{dt}$ between each pair of particles in `Fewbody` [e.g., @Iyer1995 c.f. Equation 2.10], then summing the contributions and integrating over the total time of the encounter. In Figure \[fig:egw\], we show the distributions of total energies lost due to GWs during the last encounter before each merger, normalized to the total energy (kinetic + potential) at the beginning of each encounter. Clearly, the three populations show distinct distributions in terms of energy emitted, with the GW captures showing significantly more energy loss than either the ejected or in-cluster mergers. The ejected and in-cluster mergers, on the other hand, typically show fractional energy losses several orders-of-magnitude below unity. We note that the in-cluster mergers show a suggestive bimodalitiy, with peaks at $E_{\rm GW}/E_0 \sim 10^{-4}$ and $\sim10^{-2}$, which does appear to weakly correlate with the eccentricity of the binary after its last encounter before merger (which we do not show). However, our current implementation of `Fewbody` cannot discriminate between high-eccentricity systems that were created by GW emission, or binaries that were created and emitted GWs as an isolated system before the termination of the `Fewbody` integration.
Masses of Eccentric BBH Mergers {#sec:masses}
-------------------------------
During the evolution of a GC, the most massive BHs are preferentially ejected first, since they are able to most efficiently segregate into the center of the cluster and participate in dynamical encounters [e.g. @Morscher2015]. By the present day, most GCs are expected to be entirely depleted of their heaviest BHs, leaving behind a population of lower mass BHs to participate in dynamical encounters. Because of this, it would be natural to expect that the masses of BBHs that merge inside clusters in the local universe are typically lower than those that may have been ejected many Gyr ago, when the cluster still contained significantly more massive BHs. This was in fact the result in [@Rodriguez2018], which did not include any cosmological treatment of GC formation, and assumed that all GCs formed precisely 12 Gyr ago. Here, by considering a more realistic distribution of GC formation times (Section \[sec:mccosmo\]), we find that the difference in masses for ejected mergers, in-cluster mergers, and GW captures are not nearly as clean cut.
In the top panel of Figure \[fig:masses\], we show the weighted distribution of BBH masses which merge in the local ($z<1$) universe. As with previous results [@Rodriguez2016a; @Rodriguez2016b], the peak of the distribution occurs at a total source-frame binary mass of $M_{\rm tot}\sim 40M_{\odot}$, with a reasonable range from $25M_{\odot}$ to $60M_{\odot}$, and a long tail that extends to higher masses. This is consistent with the picture from [@Rodriguez2018], where all GCs formed at $z\sim3.5$, and in-cluster mergers were limited to $\lesssim 60M_{\odot}$, since BHs more massive that $30M_{\odot}$ were ejected from the cluster in the early universe. However, the young clusters in our cosmological model have not ejected all of their heavier BHs by the present-day, allowing these massive BHs to still participate in mergers in the local universe. Of course, this effect is limited by the correlation between stellar metallicity and the maximum BH mass. Clusters forming in the local universe have preferentially higher metallicities, and correspondingly lower maximum BH masses [e.g. @Belczynski2010]. In the bottom panel of Figure \[fig:masses\], we show the fraction of sources from each formation pathway as a function of the total mass. As the mass of the BHs increases, the fractional contribution from ejected BBHs increases, since there exist very few low-metallicity young clusters that have retained their heavy BBHs up to the present day. For the lower-mass BBHs, the contribution of ejected BBHs to the merger rate increases from roughly 50% at $25M_{\odot}$ to nearly 100% at $100M_{\odot}$. The fraction of in-cluster mergers decreases over the same interval from 30% to $\sim 10\%$, while the fraction of all mergers that occur from GW captures decreases from $\sim10\%$ to $\sim 5\%$.
This trend is more obvious in Figure \[fig:masses\_1st\], where we focus only on the “first-generation” of BBH mergers. In this case, we have excluded the 15% of BBH mergers which have components formed from the previous mergers of BBHs in the cluster. The cutoff at $\sim 81M_{\odot}$ arises from the pair-production instability, where stellar core masses above $45M_{\odot}$ lose mass until they no longer undergo stellar pulsations, leaving behind BHs of at most $40.5M_{\odot}$ [@Belczynski2016a]. Here, the fraction of ejected mergers increases smoothly from 50% to 100% from $25M_{\odot}$ to $100M_{\odot}$, while the in-cluster mergers decrease smoothly from 30% to 20% at $70M_{\odot}$, before dropping precipitously to $\sim 5\%$ at high masses. Somewhat surprisingly, the fraction of GW capture events is relatively constant between $30M_{\odot}$ and $60M_{\odot}$, before decreasing to roughly 5%. For the heavier BBHs that merge in the local universe, the fraction that occur as isolated binaries versus GW captures are nearly equal.
Mergers at $e\sim 1$ {#sec:pericenter}
--------------------
The most striking feature in the eccentricity distribution of GW captures (Figure \[fig:ecc\]) is the sharp peak at $e\sim1$. This unique feature is consistent with previous scattering experiments [@Samsing2017; @Samsing2018; @Zevin2018], and arises from the small fraction of BBH captures that form *inside* the LIGO/Virgo detection band, with a peak GW frequency (equation \[eqn:wen\]) greater than 10Hz. We will now examine these systems in detail, which were set to $e\equiv0.99$ in the previous sections.
In the top panel of Figure \[fig:pericenter\], we zoom in on the eccentricity distribution of the GW captures. At an instantaneous separation of $500M$ (the largest separations reported by `Fewbody`), the distribution of eccentricities is biased towards very high values, with no GW captures passing through $500M$ with an eccentricity less that 0.1. By the time these systems reach a peak GW frequency of $10{\rm Hz}$, the majority have been driven to $e<0.1$, with a tail extending to $e\sim0.6$. However, there are three peaks at higher values that bear mentioning. First, at $e\sim0.8$, we see the beginning of the very-highly-eccentric mergers, where the systems form at such close pericenter distances that they cannot radiate away significant eccentricity before entering the LIGO/Virgo band. The second peak, at $e=0.99$, represents BBHs which formed with peak GW frequencies greater than 10Hz. Each of these sources form with very high eccentricities, which push the peak of GW emission to correspondingly high frequencies. As stated earlier, these sources do not formally have a peak frequency at 10Hz, as they formed with such high eccentricities that their peak frequencies are already greater than 10Hz. However, all of these sources have peak frequencies of between 10Hz and 40Hz (when evaluated at $500M$). If we instead ask what eccentricity each binary would have when it passes a peak GW frequency of 40Hz (the red dotted line in Figure \[fig:pericenter\]), we find that all GW capture BBHs have well-defined eccentricities.
The third peak at $e \gtrsim 1$ are systems which were not bound at the time of merger. Because we treat any BHs which pass within $10M$ of one another as merged, and because the pN approximation becomes unreliable at these separations, we cannot reliably track GW captures that form with pericenter separations of less than $10M$. However, scattering experiments using the Monte Carlo models presented here as initial conditions [@Zevin2018] have found almost no direct collisions occur between BHs when the Schwarzschild radius is used as the collision criterion. These systems, which form with pericenter separations and velocities where the pN approximation breaks down, may retain significant eccentricities up to the point of merger. However, since the energy emitted in GWs decreases as the eccentricities approach very large values [@Sperhake2008], these systems are unlikely to produce observable GWs.
Merger Rates {#sec:rates}
============
In [@Rodriguez2018b], we found that the total merger rate of BBHs from GCs was around 14 $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z<0.1$. This was calculated [by combining the models presented here]{} with the cosmological model for GC formation from [@El-Badry2018], and is consistent with other recent estimates in the literature [e.g., @Fragione2018; @Hong2018; @Choksi2018a]. Because of the distinct delay times between the in-cluster and ejected BBH mergers, we used separate phenomenological fits to the two populations, and found that the in-cluster mergers peaked earlier in redshift than BBHs that were ejected prior to merger. For that work, the GW captures were classified as in-cluster mergers. Of course, if the in-cluster mergers and GW captures follow different delay time distributions, it would be necessarily to separately fit all three populations.
In Figure \[fig:delay\], we show the cumulative distribution of merger times for the ejected, in-cluster, and GW capture populations. As was pointed out in [@Rodriguez2018b], the ejected and in-cluster BBHs merge at preferentially different times; this is largely due to the additional delay time incurred by the ejected BBHs after their last encounter in the cluster, but before their eventual merger, which can be anywhere from many Myrs to many Gyrs later, depending on the escape speed of the cluster [e.g., @Rodriguez2016a].
However, the distribution of merger times for the in-cluster and GW capture systems are nearly identical. This is hardly surprising, given that both follow the evolution of the retained BH subsystem in the cluster, and depend strongly on the instantaneous encounter rate between BBHs and other stars/BHs. Because of the nearly identical distributions, we calculate the rate of GW captures by multiplying the in-cluster merger rates from [@Rodriguez2018b] by the relative fraction of GW captures to all in-cluster mergers.
In Figure \[fig:rates\], we show the comoving merger rates for the GW captures as a function of redshift. The solid red line shows the fraction of our standard model from [@Rodriguez2018b] that occur as GW captures, while the dashed and dotted lines show the merger rate if it were assumed that all clusters are born with initial virial radii of 1 or 2pc. The total rate of GW captures varies from roughly 1 to 2 $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z < 0.1$, with a peak anywhere from about 7 to 18 $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at redshift 2.9 (2.7) for clusters with $r_v=1\rm{pc}$ ($r_v=2\rm{pc}$). This decreases to 0.7 (0.5) $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at $z<0.1$ when we restrict ourselves to the fraction of GW captures with measurable eccentricities of $e > 0.05$ ($e > 0.1$). Note that these rates assumed the “standard” merger rate estimate of [@Rodriguez2018b], which assumed a $1/M^2$ CIMF for clusters (as we have done here). However, this rate is sensitive to the contribution from high mass clusters, and can decrease by a factor of 3 as the CIMF maximum mass is decreased from $10^7M_{\odot}$ to $10^6M_{\odot}$, which would further decrease the rates presented here. At the same time, the fraction of GW captures does depend strongly on the contribution from binary-binary BBH encounters, which become more important at lower cluster concentrations and masses [@Zevin2018].
Given that the LIGO/Virgo merger rates for BBH mergers in the local universe are anywhere from $32^{+33}_{-22}~\rm{Gpc}^{-3}\rm{yr}^{-1}$ (if a log-uniform BH mass function is assumed) to $103^{+110}_{-63}~\rm{Gpc}^{-3}\rm{yr}^{-1}$ (if a BH mass function following a $m^{-2.35}$ power law is assumed), the rate of highly-eccentric mergers is obviously not a dominant component of the total BBH merger rate. Taking the upper and lower 90% credible regions from the LIGO/Virgo rate as a bound, measurably eccentric GW captures may contribute anywhere from 0.25% to 5% of the total BBH merger rate. While this represents only a small fraction of the total merger rate, the distinct eccentricities demonstrated here would provide a key discriminant between the many formation channels for BBH mergers.
Conclusions
===========
In this paper, we have explored the formation and eccentricities of merging BBHs formed dynamically in GCs, with a particular focus on binaries that form during resonant encounters between BHs and BBHs, and taking into account the gravitational radiation reaction. For the first time, we are able the study these systems in a fully cosmological context [@El-Badry2018] with a realistic population of GC models, whereas previous studies have been limited to isolated scattering experiments [e.g., @Gultekin2006; @Samsing2018; @Samsing2014; @Samsing2017] or isolated models that were not representative of the observed population of clusters [e.g., @Rodriguez2018]. We find that, when considering clusters with realistic initial conditions, GW captures contribute $12\%$ ($10\%$) of all BBH mergers from GCs at all redshifts (for redshifts $z<1$). These mergers tend to have lower total masses than the BBHs that merge after ejection from the cluster, since the most massive BHs in the cluster were ejected many Gyr before the present day. However, this trend is less pronounced when considering a realistic formation history of GCs across cosmic time, and becomes even weaker when multiple generations of BHs are allowed to form.
Combining the results of these simulations with the cosmological merger rate estimate from [@Rodriguez2018b], we find that GW captures from GCs occur at a rate of 1-2 $\rm{Gpc}^{-3}\rm{yr}^{-1}$, in the local universe, increasing to a rate of 6 to 18 $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at redshift 2.7 to 2.9, depending on what assumptions are made about the initial virial radius of the cluster. When restricting ourselves to mergers that enter the LIGO/Virgo band with measurable eccentricities, we find local merger rates of 0.7 and 0.5 $\rm{Gpc}^{-3}\rm{yr}^{-1}$ for binaries that have eccentricities greater than 0.05 and 0.1, respectively. While this represents a small fraction of the total LIGO/Virgo merger rate (thought to be between 10 and 213 $\rm{Gpc}^{-3}\rm{yr}^{-1}$ at 90% confidence), the anticipated detection rate of one BBH merger per week for the upcoming LIGO/Virgo observing run suggests that the detection of a BBH with measurable eccentricity may be imminent. Furthermore, proposed third-generation GW observatories, such as Cosmic Explorer [@Abbott2017f] and the Einstein Telescope [@Hild2011], can potentially measure BBH mergers beyond redshift 10. Coupled with an anticipated increase in the sensitivity to lower-frequency GWs, we see that the merger rates presented in Figure \[fig:rates\] may be directly measurable across cosmic time. However, this will depend on how well the proposed GW detectors can measure orbital eccentricity: while some studies [@Lower2018] have suggested measurement accuracies as low as $e\sim 10^{-4}$, it has been noted that the parameter-estimation degeneracies between the BH spins and orbital eccentricities may complicate efforts to measure eccentricities below 0.1 [@Huerta2018].
While eccentricity is a clear indicator of dynamical processes in the formation of BBH mergers, there is significant work to be done to ascertain the eccentricity distributions associated with different dynamical channels. There are several dynamical formation channels, including mergers from isolated field triples [@Antonini2017], GW captures around super-massive BHs [@OLeary2009], triples and captures formed in young open clusters [@Banerjee2017; @Banerjee2018; @Banerjee2018a], and triples around SMBHs [@Antonini2014] that can produce BBH mergers with eccentricities detectable by LIGO/Virgo. Further work will be needed to compare the global properties of these populations (such as the masses, spins, redshifts, and eccentricity distributions) to distinguish between these various dynamical scenarios. Both the work presented here and recent work on GW captures in galactic centers [@Gondan2017a] have begun to probe the connection between the BH masses and eccentricities for some of these formation channels.
We thank Emanuele Berti, Carl-Johan Haster, Scott Hughes, Cliff Will, and Nico Yunes for useful discussions. CR is supported by a Pappalardo Postdoctoral Fellowship at MIT. This work was supported by NASA Grant NNX14AP92G and NSF Grant AST-1716762 at Northwestern University. PAS acknowledges support from the Ram[ó]{}n y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain and the COST Action GWverse CA16104. CR and MZ thank the Niels Bohr Institute for its hospitality while part of this work was completed, and the Kavli Foundation and the DNRF for supporting the 2017 Kavli Summer Program. CR and FR also acknowledge support from NSF Grant PHY-1607611 to the Aspen Center for Physics, where this work was started.
[113]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , , , , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , **** ().
, , , , , , , , , ****, ().
, , , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, , , , ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, , , , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , , , , **** ().
, , , ****, ().
, pp. ().
, ****, ().
().
, , , , , , , ****, ().
, , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, ****, ().
, , ().
, , , ****, ().
, , , ****, ().
().
, ****, ().
, **** ().
, , , ****, ().
, , , , ().
, ****, ().
, ****, ().
, , , ****, (), .
, **** ().
, , , ****, ().
, , , , , , , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , , , ****, ().
, , , , , ****, ().
, , , , ().
, ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, ****, ().
, , , , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, **** ().
, , , **** ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , , ().
, , , ****, ().
, , , , , , , ****, ().
, , , ****, ().
, , , , , , , , , , , ****, ().
, , , , ****, ().
, **** ().
Naoz S., Kocsis B., Loeb A., Yunes N., 2013, ApJ, 773, 187
, ****, ().
, ****, ().
, , , , pp. ().
, , , , , ****, ().
, , , , ****, ().
, ****, ().
J. [Samsing]{}, M. [MacLeod]{}, and E. [Ramirez-Ruiz]{}, ApJ [**784**]{}, 71 (2014), 1308.2964.
J. [Samsing]{}, M. [MacLeod]{}, and E. [Ramirez-Ruiz]{}, ApJ [**853**]{}, 140 (2018), 1706.03776.
J. [Samsing]{}, PRD [**97**]{}, 103014 (2018), 1711.07452.
J. [Samsing]{} and D. J. [D’Orazio]{}, MNRAS [**481**]{}, 5445 (2018), 1804.06519.
D. J. [D’Orazio]{} and J. [Samsing]{}, MNRAS [**481**]{}, 4775 (2018), 1805.06194.
M. [Zevin]{}, J. [Samsing]{}, C. [Rodriguez]{}, C.-J. [Haster]{}, and E. [Ramirez-Ruiz]{}, ArXiv e-prints (2018), 1810.00901.
J. [Samsing]{}, A. [Askar]{}, and M. [Giersz]{}, ApJ [**855**]{}, 124 (2018), 1712.06186.
, ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , , ****, ().
|
---
abstract: 'We present results for the mass of the flavor singlet meson calculated on two-flavor full QCD configurations generated by the CP-PACS full QCD project. We also investigate topological charge fluctuations and their dependence on the sea quark mass.'
address:
- 'Center for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan'
- 'Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan'
- 'Institute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo 188-8502, Japan'
- 'High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan'
- 'DAMTP, 21 Silver St., University of Cambridge, Cambridge, CB3 9EW, England, U.K.'
author:
- '[CP-PACS Collaboration]{}: A. Ali Khan, S. Aoki, R. Burkhalter$^{\rm a,b}$, S. Ejiri$^{\rm a}$, M. Fukugita, S. Hashimoto, N. Ishizuka$^{\rm a,b}$, Y. Iwasaki$^{\rm a,b}$, K. Kanaya$^{\rm a,b}$, T. Kaneko$^{\rm a}$, Y. Kuramashi$^{\rm d}$, T. Manke$^{\rm a}$, K. Nagai$^{\rm a}$, M. Okawa$^{\rm d}$, H.P. Shanahan, A. Ukawa$^{\rm a,b}$ and T. Yoshié$^{\rm a,b}$'
title: |
\
\
Eta meson mass and topology in QCD with two light flavors[^1]
---
INTRODUCTION
============
Recently considerable progress has been made in the simulation of full QCD [@ref:mawhinney]. In particular, sea quark effects have been found to lead to a closer agreement of the light meson spectrum with experiment [@ref:review98; @ref:kaneko].
Missing from the calculated spectrum, however, has been the flavor singlet meson $\eta'$. Due to the difficulty of the determination of the disconnected contribution, only preliminary lattice results have been available [@ref:Itoh; @ref:Kuramashi; @ref:Venkataraman].
In the first half of this article, we present new results on this problem. Since these are obtained with two flavors of dynamical quark, we call the flavor singlet meson as $\eta$ and reserve the name $\eta'$ for the case of $N_f\!=\!3$.
The $\eta'$ meson is expected to obtain a large mass through the connection to instantons. This leads us to an investigation of topology in full QCD, presented in the latter half of this article.
‘?=?
$N_s^3\!\times\!N_t$ $\beta$ $N_s a$ \[\] $a^{-1}$ \[\] $m_{\rm PS}/m_{\rm V}$
---------------------- --------- -------------- --------------- ------------------------
$12^3\!\times\!24$ 1.8 2.58(3) 0.92(1) 0.81–0.55
$16^3\!\times\!32$ 1.95 2.45(3) 1.29(2) 0.80–0.59
$24^3\!\times\!48$ 2.1 2.60(5) 1.82(3) 0.80–0.58
$24^3\!\times\!48$ 2.2 2.06(6) 2.30(7) 0.80–0.63
: Parameters of lattices used in this calculation.[]{data-label="tab:runs"}
Calculations have been performed on configurations of the CP-PACS full QCD project [@ref:review98; @ref:kaneko]. These have been generated using an RG-improved gauge action and a tadpole-improved SW clover quark action at four different lattice spacings and four values of the sea quark mass corresponding to $m_{\rm PS}/m_{\rm V}\!\sim\!0.8$–0.6. An overview of the simulation parameters is given in Table \[tab:runs\]. More details can be found in [@ref:review98; @ref:kaneko].
FLAVOR SINGLET MESON
====================
The mass difference $\Delta m$ between the flavor singlet ($\eta$) and non-singlet ($\pi$) meson can be extracted from the ratio $$R(t) = \frac{\langle\eta(t)\eta(0)\rangle_{\rm disc}}
{\langle\eta(t)\eta(0)\rangle_{\rm conn}}
\rightarrow 1 - B\exp(-\Delta mt),
\label{eq:ratio}$$ where the right hand side indicates the expected behavior at large time separation $t$.
In this work, the connected propagator was calculated with the standard method. For the disconnected propagator we used two methods. In the first instance, it was calculated using a volume source without gauge fixing (the Kuramashi method)[@ref:Kuramashi]. This measurement was made after every trajectory in the course of configuration generation for all runs listed in Table \[tab:runs\]. As for the second method, we employed a U(1) volume noise source with 10 random noise ensembles for each color and spin combination. This was performed only at $\beta=1.95$ on stored configurations separated by 10 HMC trajectories.
Figure \[fig:ratio\] compares the ratio $R(t)$ for the two methods. We observe that they are consistent with each other but that the error is smaller for the first method. This might be due to the fact that there are 10 times more measurements with it, although binning is made over 50 HMC trajectories for both cases to take into account auto-correlations. In the following we only use data obtained with the first method.
In Fig. \[fig:ratio\] we also see that the error of $R(t)$ increases exponentially, which make the determination of $\Delta m$ via Eq. \[eq:ratio\] impossible at large time separations. The data, however, shows the expected behavior already beginning from small $t$ and a fit with Eq. \[eq:ratio\] is possible from $t_{min}\!=\!2$. Increasing $t_{min}$ leads to stable results, as can be seen in Fig. \[fig:chireta\].
The chiral extrapolation of $m_{\eta}^2$ linear in the quark mass is shown in Fig. \[fig:chireta\]. Contrary to the pion mass, the flavor singlet meson remains massive in the chiral limit.
Figure \[fig:eta\] shows the $\eta$ meson mass at all measured lattice spacings after the chiral extrapolation. The scale is set using the $\rho$ meson mass. A linear extrapolation to the continuum limit gives $m_\eta=863(86)$ MeV. This value lies between the experimental $\eta$(547) and $\eta'$(958) masses. We emphasize that a proper comparison with experiment requires the introduction of a third (strange) quark and a mixing analysis.
TOPOLOGY
========
Studies of topology on the lattice have encountered several difficulties. In addition to the ambiguity of defining a lattice topological charge, it was found that topological modes have a very long auto-correlation time in the case of full QCD with the Kogut-Sussind quark action [@ref:topo-full].
We employ the field theoretic definition of the topological charge together with cooling. For the charge we use a tree-level improved definition which includes a $1\!\times\!2$-plaquette, hence the $O(a^2)$ terms are removed for instanton configurations. For the cooling we compare two choices of improved actions, both including a $1\!\times\!2$-plaquette term: 1) a tree-level Symanzik improved (LW) action and 2) the RG improved Iwasaki action.
Using different actions for cooling can lead to different values of the topological charge. This ambiguity is only expected to vanish when the lattice is fine enough. We have tested this explicitly by simulating the pure SU(3) gauge theory at three lattice spacings in the range $a \sim
0.2$–0.1 fm and at a constant size of 1.5 fm. As Fig. \[fig:suscCont\] shows, the topological susceptibilities $\chi_t\!=\!\langle Q^2\rangle/V$ for the two cooling actions converge to a common value towards the continuum limit. Using $\sqrt{\sigma}\!=\!440$ MeV we obtain $\chi_t\!=\!(178(9)\,{\rm MeV})^4$, in agreement with previous studies [@ref:topo-pure].
=5.55cm
In full QCD we have so far measured the topological charge at $\beta\!=\!1.95$. Figure \[fig:history\] shows the time history for two quark masses. Auto-correlation times are visibly small even for the smallest quark mass. For the Wilson quark action rather short auto-correlation times have been reported in Ref. [@ref:SESAM]. The fact that we find even shorter auto-correlations might be explained by the coarseness of our lattice.
Based on the anomalous flavor-singlet axial vector current Ward identity, one expects the topological susceptibility to vanish in the chiral limit. Indeed, Fig. \[fig:history\] shows the width to be shrinking with the quark mass. The decrease, however, is not sufficient; as we find in Fig. \[fig:suscept\], the dimensionless ratio $\chi_t/\sigma^2$, with $\sigma$ calculated for each sea quark mass, does not vary much with the quark mass, and takes a value similar to that for pure gauge theory. To understand the origin of this behavior, more investigations at different lattice spacings will be needed.
This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 09304029, 10640246, 10640248, 10740107, 11640250, 11640294, 11740162). SE and KN are JSPS Research Fellows. AAK, TM and HPS are supported by JSPS Research for the Future Program. HPS is supported by the Leverhulme foundation.
[9]{}
R. Mawhinney, review in these proceedings.
R. Burkhalter, Nucl. Phys. [**B**]{} (Proc.Suppl) [**73**]{} (1999) 3.
T. Kaneko [*et al.*]{}, CP-PACS collaboration, these proceedings.
S. Itoh [*et al.*]{}, Phys. Rev. [**D36**]{} (1987) 527.
Y. Kuramashi [*et al.*]{}, Phys. Rev. Lett. [**72**]{} (1994) 3448.
L. Venkataraman and G. Kilcup, hep-lat/9711006.
Y. Kuramashi [*et al.*]{}, Phys. Lett. [**B313**]{} (1993) 425; B. Allés [*et al.*]{}, Phys. Lett. [**B389**]{} (1996) 107.
For a recent review see J. Negele, Nucl. Phys. [**B**]{} (Proc.Suppl) [**73**]{} (1999) 92.
B. Allés [*et al.*]{}, Phys. Rev. [**D58**]{} (1998) 071503.
[^1]: talk presented by R. Burkhalter at Lattice ’99
|
---
abstract: |
The main purpose of this paper is twofold. We first want to analyze in details the meaningful geometric aspect of the method introduced in the previous paper [@F2], concerning regularity of families of irreducible, nodal “curves” on a smooth, projective threefold $X$. This analysis highlights several fascinating connections with families of other singular geometric “objects” related to $X$ and to other varieties.
Then, we generalize this method to study similar problems for families of singular divisors on ruled fourfolds suitably related to $X$.
author:
- Flaminio Flamini
title: Equivalence of families of singular schemes on threefolds and on ruled fourfolds
---
[^1] [^2]
Introduction {#S:intro .unnumbered}
============
The theory of families of singular curves with fixed invariants (e.g. geometric genus, singularity type, number of irreducible components, etc.) and which are contained in a projective variety $X$ has been extensively studied from the beginning of Algebraic Geometry and it actually receives a lot of attention, partially due to its connections with several fields in Geometry and Physics.
Nodal curves play a central role in the subject of singular curves. Families of irreducible and $\delta$-nodal curves on a given projective variety $X$ are usually called [*Severi varieties*]{} of irreducible, $\delta$-nodal curves in $X$. The terminology “Severi variety” is due to the classical case of families of nodal curves on $X= {\mathbb{P}^2}$, which was first studied by Severi (see [@Sev]).
The case in which $X$ is a smooth projective surface has recently given rise to a huge amount of literature (see, for example, [@CH], [@CC], [@CL], [@CS], [@F1], [@GLS], [@Harris], [@Ran], [@S] just to mention a few. For a chronological overview, the reader is referred for example to Section 2.3 in [@F] and to its bibliography). This depends not only on the great interest in the subject, but also because for a Severi variety $V$ on an arbitrary projective variety $X$ there are several problems concerning $V$ like non-emptyness, smoothness, irreducibility, dimensional computation as well as enumerative and moduli properties of the family of curves it parametrizes.
On the contrary, in higher dimension only few results are known. Therefore, in [@F2] we focused on what is the next relevant case, from the point of view of Algebraic Geometry: families of nodal curves on smooth, projective threefolds.
The aim of this paper is twofold: first, we want to study in details the meaningful geometric aspect of the method introduced in [@F2]. As a result of this analysis, we discover several intriguing and fascinating connections with families of other singular geometric “objects” related to $X$. Then, we generalize this method to study similar problems for families of singular divisors on ruled fourfolds suitably related to $X$.
To be more precise, let $X $ be a smooth projective threefold and let ${\mathcal F}$ be a rank-two vector bundle on $X$, which is assumed to be globally generated with general global section $s$ having its zero-locus $V(s)$ a smooth, irreducible curve $D = D_s$ in $X$. The geometric genus of $D$ is given by$$2g(D) -2 = 2p_a(D) -2= deg({\mathcal L}\otimes \omega_X \otimes
{{\hbox{\goth O}}}_D),$$where ${\mathcal L}:= c_1({\mathcal F}) \in Pic(X)$ and $\omega_X$ is the canonical sheaf of $X$.
Take now ${{\mathbb P}}(H^0(X, {{\mathcal F}}))$; from our assumptions on ${\mathcal F}$, its general point parametrizes a global section whose zero-locus is a smooth, irreducible curve. This projective space somehow gives a scheme dominating a subvariety in which the curves move.
Given a positive integer $\delta \leq p_a(D)$, it makes sense to consider the locally closed subscheme: $$\begin{aligned}
{{\mathcal V}}_{\delta}({{\mathcal F}}) := & \{[s] \in {{\mathbb P}}(H^0(X,{{\mathcal F}})) \; | \; C_s := V(s) \subset X \;
{\rm is \; irreducible} \\
& {\rm with \; only} \; \delta \; {\rm nodes \; as \; singularities} \};
\end{aligned}$$(cf. ). These are usually called [*Severi varieties*]{} of global sections of ${\mathcal F}$ whose zero-loci are irreducible, $\delta$-nodal curves in $X$, of arithmetic genus $p_a(D)$ and geometric genus $g= p_a(D) - \delta$ (cf. [@BC], for $X={\mathbb{P}^3}$, and [@F2] in general). This is because such schemes are the natural generalization of the (classical) Severi varieties on smooth, projective surfaces recalled before.
When ${{\mathcal V}}_{\delta} ({{\mathcal F}})$ is not empty then its expected codimension in ${{\mathbb P}}(H^0(X,{{\mathcal F}}))$ is $\delta$ (see Proposition \[prop:0\]). Thus, one says that a point $[s] \in {{\mathcal V}}_{\delta} ({{\mathcal F}})$ is a [*regular point*]{} if it is smooth and such that $dim_{[s]}({{\mathcal V}}_{\delta} ({{\mathcal F}}))$ equals the expected one (cf. Definition \[def:0\]).
It is clear from the definition of regularity that it is fundamental to determine the tangent space to a Severi variety at a given point. In [@F2] we introduced the following cohomological description of the tangent space $T_{[s]}({{\mathcal V}}_{\delta} ({{\mathcal F}}))$.
[**1**]{} (cf. Theorem \[prop:3.fundamental\]) Let $X $ be a smooth projective threefold. Let ${\mathcal F}$ be a globally generated rank-two vector bundle on $X$ and let $\delta$ be a positive integer. Fix $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ and let $C=V(s) \subset X$. Denote by $\Sigma$ the set of nodes of $C$.
Let$${\mathcal P} := {{\mathbb P}}_{X}({{\mathcal F}}) \stackrel{\pi}{\longrightarrow} X$$be the projective space bundle together with its natural projection $\pi$ on $X$ and denote by ${{\hbox{\goth O}}}_{\mathcal P}(1)$ its tautological line bundle.
Then, there exists a zero-dimensional subscheme $\Sigma^1 \subset
{\mathcal P}$ of length $\delta$, which is a set of $\delta$ rational double points for the divisor $G_s \in |{{\hbox{\goth O}}}_{\mathcal P}(1)|$ corresponding to the given section $s \in H^0(X, {\mathcal F})$.
In particular, each element $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) $ corresponds to a divisor $G_s \in |{{\hbox{\goth O}}}_{\mathcal P}(1)|$, which contains the $\delta$ fibres $L_{p_i} = \pi^{-1}(p_i) \subset {\mathcal P}$, for $p_i \in \Sigma$, $1 \leq i \leq \delta$, and which has $\delta$ rational double points each of which are on exactly one of the $\delta$ fibres $L_{p_i}$.
By using the above result, one can translate the regularity property of $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ into the surjectivity of some maps among spaces of sections of suitable sheaves on $X$ (cf. Proposition \[prop:corr1\] and Remark \[rem:localdescr\]). This allows us to find several equivalent and sufficient conditions for the regularity of Severi varieties ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ (cf. [@F2] and also Corollaries \[cor:propcorr1\], \[rem:reg\], \[cor:explanetion\], \[cor:tangentspace\] and Theorem \[thm:9bis\] in this paper).
On the other hand, Theorem 1 also introduces a meaningful and fascinating connection between elements in $ {{\mathcal V}}_{\delta}({{\mathcal F}}) $ and other singular schemes in $X$ and in ${\mathcal P}$. The aim of this paper is to investigate in details the deep geometric meaning of $T_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}}))$ and its several connections with these families of singular geometric objects related to $X$ and to ${\mathcal P}$.
More precisely, we show that there exists a natural connection among:
- a nodal section $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$,
- the corresponding singular divisor $G_s$ in $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$ on ${\mathcal P}$ having $\delta$ rational double points over the nodes of $C = V(s)$,
- for any $s + \epsilon \; s' \in T_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}}))$, the surface $V(s \wedge s') \subset X$ which is singular along $\Sigma$ and which belongs to the linear system $| {\hbox{\goth I}}_{C/X} \otimes {\mathcal L}|$ on $X$,
- given $G_s$ and $G_{s'}$ in $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$ corresponding to $s$ and $s'$ respectively, the “complete intersection” surface in ${\mathcal P}$, ${{\mathcal S}}_{s,s'} := G_s \cap G_{s'}$, which is singular along $\Sigma^1$ and which dominates $V(s \wedge s')$,
(cf. Propositions \[prop:corr1\], \[prop:corr2\] and Remarks \[rem:propcorr2\], \[rem:2propcorr2\]).
In particular, we prove:
[**2**]{} (cf. Theorem \[thm:fundamental\] and Corollary \[cor:explanetion\]) The following conditions are equivalent:
- $s + \epsilon s' \in T_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}}))$, where $\epsilon^2 = 0$;
- $V (s \wedge s') \subset X$ is a surface which contains $C$ and which is singular along $\Sigma$;
- the divisor $G_{s'}$ passes through $\Sigma^1$
- the surface ${\mathcal S}_{s,s'}:= G_s \cap G_{s'} \subset {\mathcal P}$ is singular along $\Sigma^1$;
By using Theorem 2 we can also give several further equivalent conditions for the regularity of $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ (cf. Remark \[rem:explanetion\]).
Furthermore, thanks to the correspondence introduced in Theorems 1 and 2, we also consider a generalization of Severi varieties of nodal curves. Indeed, we denote by $${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))
:= \{G_s \in |{\hbox{\goth O}}_{{\mathcal P}}(1) | \; {\rm s.t.} \; [s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) \}$$the schemes parametrizing families of expected codimension $\delta$ in $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$, whose elements correspond to divisors which are irreducible and with only $\delta$ rational double points as singularities. For brevity sake, these are called ${\mathcal P}_{\delta}$-[*Severi varieties*]{} (cf. Definition \[def:rdelta\] and ).
One can obviously give a similar definition of [*regularity*]{} for ${\mathcal P}_{\delta}$-[*Severi varieties*]{} (cf. Definition \[def:expdim\]). We prove:
[**3**]{} (cf. Theorem \[thm:tangentspace\] and Corollary \[cor:tangentspace\]) Let $[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$ and let $\Sigma^1 $ be the zero-dimensional scheme of the $\delta$-rational double points of $G_s \subset {\mathcal P}$. Then $$T_{[G_s]} ({\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))) \cong \frac{H^0({\hbox{\goth I}}_{\Sigma^1/{\mathcal P}}
\otimes {\hbox{\goth O}}_{{\mathcal P}}(1))}{<G_s>}.$$
In particular, $$[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) \; {\rm is \; a \; regular \; point} \Leftrightarrow
[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) \; {\rm is \; a \; regular \; point}.$$
Finally, we first improve some regularity results of [@F2] for Severi varieties ${{\mathcal V}}_{\delta}({{\mathcal F}})$ of irreducible, $\delta$-nodal sections on $X$; then, we use Theorem 3 to deduce regularity results also for ${\mathcal P}_{\delta}$-Severi varieties ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) $ on ${\mathcal P}$. Precisely, we have:
[**4**]{} (cf. Theorems \[thm:9bis\] and \[thm:regrdelta\]) Let $X$ be a smooth projective threefold, ${\mathcal E}$ be a globally generated rank-two vector bundle on $X$, $M$ be a very ample line bundle on $X$ and $k \geq 0$ and $\delta >0$ be integers. Let ${\mathcal P}: = {\mathbb P}_X ({\mathcal E}\otimes M^{\otimes k})$ and ${\hbox{\goth O}}_{{\mathcal P}}(1)$ be its tautological line bundle. If$$(*)\;\;\;\;\;\;\delta \leq k+1,$$then both ${{\mathcal V}}_{\delta}({{\mathcal E}} \otimes M^{\otimes k})$ on $X$ and ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$ are regular at each point.
The upper-bounds in $(*)$ are also shown to be almost-sharp (cf. Remark \[rem:10\]).
What we want to stress is the following fact: the regularity condition for the schemes ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) $ on ${\mathcal P}$ is equivalent to the separation of suitable zero-dimensional schemes by the linear system $|{\hbox{\goth O}}_{{\mathcal P}}(1)|$ on the fourfold ${\mathcal P}$ (cf. Corollary \[cor:tangentspace\]). In general, it is well-known how difficult is to enstablish separation of points in projective varieties of dimension greater than or equal to three (cf. e.g. [@AS], [@EL] and [@K]). In some cases, some separation results can be found by using technical tools like [*multiplier ideals*]{} as well as the Nadel and the Kawamata-Viehweg vanishing theorems (see, e.g. [@E], for an overview). In our situation, thanks to the correspondence between ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ and ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$, we deduce regularity conditions for ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ from those already obtained for ${{\mathcal V}}_{\delta}({{\mathcal F}})$.
The paper consists of seven sections. Section \[S:1\] contains some terminology and notation. In Section \[S:a\] we briefly recall some fundamental definitions in [@F2], which are frequently used in the whole paper. Section \[S:b\] briefly recall one of the main result in [@F2] concerning the correspondence beteween elements in ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ and those in ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$ (cf. Theorem \[prop:3.fundamental\]).
In Section \[S:c\] we describe how to associate elements of $T_{[s]}{{\mathcal V}}_{\delta}({{\mathcal F}})$ to singular divisors in $X$. Section \[S:d\] contains one of the main result of the paper (cf. Theorem \[thm:fundamental\]) which proves the equivalence of several singular geometric “objects” related to $X$ and to ${\mathcal P}$. In Section \[S:ef\] we focus on ${\mathcal P}$-Severi varieties on ${\mathcal P}$; we give a description of tangent spaces at points of such schemes as well as we find conditions for their regularity (cf. Theorem \[thm:tangentspace\] and Corollary \[cor:tangentspace\]). Section \[S:ghil\] is devoted to the determination of almost-sharp upper-bounds on $\delta$ implying the regularity of ${{\mathcal V}}_{\delta} ({{\mathcal F}})$ on $X$ as well as of ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$.
[*Acknowledgments:*]{} Part of this paper is related to the previous paper [@F2], which was prepared during my permanence at the Department of Mathematics of the University of Illinois at Chicago (February - May 2001). Therefore, my deepest gratitude goes to L. Ein, not only for the organization of my visit, but mainly for all I have learnt from him as well as for having suggested me to approach this pioneering area. My very special thanks go also to GNSAGA-INdAM and to V. Barucci, A. F. Lopez and E. Sernesi for their confidence and their support during my period in U.S.A.
I am greatful to L. Caporaso and to the organizers of the Workshop [*“Global geometry of algebraic varieties”*]{} - Madrid, December 16-19, 2002. In fact, their invitations to give talks at the Dept. of Mathematics of “Roma Tre” and at the Workshop, respectively, have given “life” to these new results, because of the preparation of the talks on the subjects in [@F2]. I am indebted to L. Chiantini and to C. Ciliberto for having always stimulated me to investigate more in this research area.
Notation and Preliminaries {#S:1}
==========================
We work in the category of algebraic ${\mathbb{C}}$-schemes. $Y$ is a *$m$-fold* if it is a reduced, irreducible and non-singular scheme of finite type and of dimension $m$. If $m=1$, then $Y$ is a (smooth) [*curve*]{}; $m=2$ and $3$ are the cases of a (non-singular) [*surface*]{} and [*threefold*]{}, respectively. If $Z$ is a closed subscheme of a scheme $Y$, ${\hbox{\goth I}}_{Z/Y}$ denotes the *ideal sheaf* of $Z$ in $Y$, ${{\mathcal N}}_{Z/Y}$ the [*normal sheaf*]{} of $Z$ in $Y$ whereas ${{\mathcal N}}_{Z/Y}^{\vee} \cong {{\hbox{\goth I}}_{Z/Y}}/{{\hbox{\goth I}}_{Z/Y}^2}$ is the [ *conormal sheaf*]{} of $Z$ in $Y$. As usual, $h^i(Y, \; -):=\text{dim} \; H^i(Y, \; -)$.
Given $Y$ a projective scheme, $\omega_Y$ denotes its dualizing sheaf. When $Y$ is a smooth variety, then $\omega_Y$ coincides with its canonical bundle and $K_Y$ denotes a canonical divisor s.t. $\omega_Y \cong {\hbox{\goth O}}_Y(K_Y)$. Furthermore, ${{\mathcal T}}_Y$ denotes its tangent bundle whereas $\Omega^1_Y$ denotes its cotangent bundle.
If $D$ is a reduced curve, $p_a(D)=h^1({\hbox{\goth O}}_D)$ denotes its [*arithmetic genus*]{}, whereas $g(D)= p_g(D)$ denotes its *geometric genus*, the arithmetic genus of its normalization.
Let $Y$ be a projective $m$-fold and ${\mathcal E}$ be a rank-$r$ vector bundle on $Y$; $c_i({\mathcal E})$ denotes the *$i^{th}$-Chern class* of ${\mathcal E}$, $1 \leq i \leq r$. As in [@Ha] - Sect. II.7 - ${\mathbb P}_{Y} ({{\mathcal E}})$ denotes the [*projective space bundle*]{} on $Y$, defined as $Proj(Sym({{\mathcal E}}))$.
There is a surjection $\pi^*({\mathcal E}) \to {\hbox{\goth O}}_{{\mathbb P}_{Y} ({{\mathcal E}})}(1)$, where ${\hbox{\goth O}}_{{\mathbb P}_{Y} ({{\mathcal E}})}(1)$ is the [*tautological line bundle*]{} on ${\mathbb P}_{Y} ({{\mathcal E}})$ and where $\pi : {\mathbb P}_{Y} ({{\mathcal E}}) \to Y$ is the natural projection morphism. Recall that ${\mathcal E}$ is said to be an [*ample*]{} (resp. [*nef*]{}) vector bundle on $Y$ if ${\hbox{\goth O}}_{{\mathbb P}_{Y} ({{\mathcal E}})}(1)$ is an ample (resp. nef) line bundle on ${\mathbb P}_{Y} ({{\mathcal E}})$ (see, e.g. [@H1]).
For non reminded terminology, the reader is referred to [@BPV], [@Fr] and [@Ha].
Families of nodal “curves” on smooth, projective threefolds {#S:a}
===========================================================
In this section we briefly recall some definitions and results from [@F2] which will be frequently used in the sequel.
Let $X $ be a smooth projective threefold and let ${\mathcal F}$ be a rank-two vector bundle on $X$. If ${\mathcal F}$ is globally generated on $X$, it is not restrictive if from now on we assume that the zero-locus $V(s)$ of its general global section $s$ is a smooth, irreducible curve $D = D_s$ in $X$ (for details, see [@F2]; for general motivations and backgrounds, the reader is referred to e.g. [@OSS] and to [@Sz], Chapter IV).
From now on, denote by ${\mathcal L}\in Pic(X)$ the line bundle on $X$ given by $c_1({\mathcal F})$. Thus, by the [*Koszul sequence*]{} of $({{\mathcal F}}, s)$: $$\label{eq:koszul}
0 \to {{\hbox{\goth O}}}_X \to {{\mathcal F}} \to {{\hbox{\goth I}}}_{V(s)} \otimes {\mathcal L}\to 0,$$we compute the geometric genus of $D$ in terms of the invariants of ${\mathcal F}$ and of $X$. Precisely $$\label{eq:numeriX}
2g(D) -2 = 2p_a(D) -2= deg({\mathcal L}\otimes \omega_X \otimes
{{\hbox{\goth O}}}_D).$$
This integer is easily computable when, for example, $X$ is a general complete intersection threefold. In particular, when $Pic(X) \cong {\mathbb{Z}}$ (e.g $X= {\mathbb{P}^3}$ or $X$ either a prime Fano or a complete intersection Calabi-Yau threefold) one can use this isomorphism to identify line bundles on $X$ with integers. Therefore, if $A$ denotes the ample generator class of $Pic(X)$ over ${\mathbb{Z}}$ and if ${\mathcal F}$ is a rank-two vector bundle on $X$ such that $c_1({{\mathcal F}}) = n A$, we can also write $c_1({{\mathcal F}}) = n$ with no ambiguity.
Thus, if e.g. $X = {\mathbb{P}^3}$ and if we put $c_i = c_i({{\mathcal F}}) \in {\mathbb{Z}}$, we have $$\label{eq:numeriP3}
deg(D) = c_2 \; {\rm and} \; g(D)= p_a(D) = \frac{1}{2} (c_2 (c_1 -4))+ 1,$$i.e. $D$ is subcanonical of level $(c_1 -4)$.
Take now ${{\mathbb P}}(H^0(X, {{\mathcal F}}))$; from our assumptions on ${\mathcal F}$, the general point of this projective space parametrizes a global section whose zero-locus is a smooth, irreducible curve in $X$. Given a positive integer $\delta \leq p_a(D)$, it makes sense to consider the subset $$\label{eq:severi var}
\begin{aligned}
{{\mathcal V}}_{\delta}({{\mathcal F}}) := & \{[s] \in {{\mathbb P}}(H^0(X,{{\mathcal F}})) \; | \; C_s := V(s) \subset X \;
{\rm is \; irreducible} \\
& {\rm with \; only} \; \delta \; {\rm nodes \; as \; singularities} \};
\end{aligned}$$therefore, any element of ${{\mathcal V}}_{\delta}({{\mathcal F}})$ determines a curve in $X$ whose arithmetic genus $p_a(C_s)$ is given by (\[eq:numeriX\]) and whose geometric genus is $g= p_a(C_s) - \delta$. We recall that ${{\mathcal V}}_{\delta}({{\mathcal F}})$ is a locally closed subscheme of the projective space ${{\mathbb P}}(H^0(X, {{\mathcal F}}))$; it is usually called the [*Severi variety*]{} of global sections of ${\mathcal F}$ whose zero-loci are irreducible, $\delta$-nodal curves in $X$ (cf. [@BC], for $X={\mathbb{P}^3}$, and [@F2] in general). This is because such schemes are the natural generalization of the (classical) Severi varieties of irreducible and $\delta$-nodal curves in linear systems on smooth, projective surfaces (see [@CC], [@CH], [@CS], [@F1], [@GLS], [@Harris], [@Ran], [@S] and [@Sev], just to mention a few).
For brevity sake, we shall usually refer to ${{\mathcal V}}_{\delta}({{\mathcal F}})$ as the [*Severi variety of irreducible, $\delta$-nodal sections*]{} of ${\mathcal F}$ on $X$.
First possible questions on such Severi varieties are about their dimensions as well as their smoothness properties.
A preliminary estimate is given by the following standard result:
\[prop:0\] Let $X $ be a smooth projective threefold, ${\mathcal F}$ be a globally generated rank-two vector bundle on $X$ and $\delta$ be a positive integer. Then $$expdim ({{\mathcal V}}_{\delta}({{\mathcal F}})) = \begin{cases}
h^0(X, {{\mathcal F}}) - 1 - \delta, \; \; {\rm if} \; \delta \leq
h^0(X, {{\mathcal F}}) - 1= dim({{\mathbb P}}(H^0({{\mathcal F}}))),\\
-1, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm if}
\; \delta \geq
h^0(X, {{\mathcal F}}).\\
\end{cases}$$
See Proposition 2.10 in [@F2].
\[ass:main\]
By Proposition \[prop:0\], it is natural to state the following:
\[def:0\] Let $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$, with $\delta \leq {\rm min} \{ h^0(X, {{\mathcal F}}) - 1, \; p_a(C) \}$. Then $[s]$ is said to be a [*regular point*]{} of ${{\mathcal V}}_{\delta}({{\mathcal F}})$ if:
- $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ is a smooth point, and
- $dim_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}})) = expdim ({{\mathcal V}}_{\delta}({{\mathcal F}}))=
dim({{\mathbb P}}(H^0(X,{{\mathcal F}}))) - \delta$.
${{\mathcal V}}_{\delta}({{\mathcal F}})$ is said to be [*regular*]{} if it is regular at each point.
One of the main result in [@F2] has been to present a cohomological description of the tangent space $T_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}}))$ which translates the regularity property of a given point $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ into the surjectivity of some maps among spaces of sections of suitable sheaves on the threefold $X$. This description allowed us to find also several sufficient conditions for the regularity of Severi varieties ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$.
One of the aim of this paper is to study in more details the deep geometric meaning of the cohomological description of the tangent space $T_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}}))$ and its several connections with families of other singular geometric objects related to $X$ and to ${\mathcal F}$.
To do this, we have first to recall some results contained in [@F2], since these are the starting point of our analysis.
Association of elements of ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ to singular divisors in ${{\mathbb P}}_{X}({{\mathcal F}}) $ {#S:b}
========================================================================================================================================
In this section we want to briefly recall the correspondence given in [@F2] between elements of ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ and suitable singular divisors in the tautological linear system $|{{\hbox{\goth O}}}_{\mathcal P}(1)|$ on the projective space bundle ${\mathcal P}:={{\mathbb P}}_{X}({{\mathcal F}})$, which is a fourfolds ruled over $X$. Instead of referring the reader to [@F2], we prefer to briefly recall here the proofs of some results contained in there, not only because we give here more precise statements and proofs, but mainly because the strategy of the proofs as well as their technical details will be fundamental for the analysis in the whole paper.
From now on, with conditions as in Assumption \[ass:main\], let $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$. Then:
\[prop:3.fundamental\] (cf. Theorem 3.4 (i) in [@F2]) Let $X $ be a smooth projective threefold. Let ${\mathcal F}$ be a globally generated rank-two vector bundle on $X$ and let $\delta$ be a positive integer. Fix $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ and let $C=V(s) \subset X$. Denote by $\Sigma$ the set of nodes of $C$.
Let$${\mathcal P} := {{\mathbb P}}_{X}({{\mathcal F}}) \stackrel{\pi}{\longrightarrow} X$$be the projective space bundle together with its natural projection $\pi$ on $X$ and denote by ${{\hbox{\goth O}}}_{\mathcal P}(1)$ its tautological line bundle.
Then, there exists a zero-dimensional subscheme $\Sigma^1 \subset
{\mathcal P}$ of length $\delta$, which is a set of $\delta$ rational double points for the divisor $G_s \in |{{\hbox{\goth O}}}_{\mathcal P}(1)|$ corresponding to the given section $s \in H^0(X, {\mathcal F})$.
In particular, each element $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) $ corresponds to a divisor $G_s \in |{{\hbox{\goth O}}}_{\mathcal P}(1)|$, which contains the $\delta$ fibres $L_{p_i} = \pi^{-1}(p_i) \subset {\mathcal P}$, for $p_i \in \Sigma$, $1 \leq i \leq \delta$, and which has $\delta$ rational double points each of which are on exactly one of the $\delta$ fibres $L_{p_i}$.
Consider the smooth, projective, ruled fourfold $${\mathcal P} := {{\mathbb P}}_{X}({{\mathcal F}}) \stackrel{\pi}{\longrightarrow} X,$$together with its tautological line bundle ${{\hbox{\goth O}}}_{\mathcal P}(1)$ such that $\pi_*({{\hbox{\goth O}}}_{\mathcal P}(1)) \cong {{\mathcal F}}$. Since $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) $, in particular $s \in H^0 (X, {\mathcal F})$; then, one also has $$0 \to {{\hbox{\goth O}}}_{{\mathcal P}} \stackrel{\cdot s}{\to} {{\hbox{\goth O}}}_{{\mathcal P}}(1).$$Therefore, the nodal curve $C \subset X$ corresponds to a divisor - say $G_s $ - on the fourfold ${\mathcal P}$ which belongs to the tautological linear system $|{{\hbox{\goth O}}}_{{\mathcal P}}(1) |$.
It is clear that $G_s$ contains all the $\pi$-fibres over the zero-locus $C=V(s)$; precisely, it contains the surface $\mathbb{F}:= {{\mathbb P}}^1_C = Proj(Sym({\mathcal F}|_C))$ which is ruled over $C$. Therefore, in particular $G_s$ contains the locus $\Lambda := \bigcup_{i=1}^{\delta} L_{p_i}$, where $\Sigma = \{p_1, \ldots, p_{\delta} \}$.
The geometry of $G_s$ is strictly related to the one of $C$. Indeed, if $p \in \Sigma = Sing(C)$, take $U_p \subset X$ an affine open set containing $p$, where the vector bundle ${{\mathcal F}}$ trivializes. Since $p$ is a [*planar singularity*]{}, one can choose local coordinates $\underline{x} = (x_1, x_2, x_3)$ on $U_p \cong {\mathbb A}^3$ such that $\underline{x}(p) = (0,0,0)$ and such that the global section $s$ is $$s|_{U_p} = (x_1x_2, \; x_3).$$For what concerns the divisor $G_s \in |{{\hbox{\goth O}}}_{\mathcal P}(1)|$ corresponding to $s \in H^0 (X, {\mathcal F})$, since $U_p$ trivializes ${\mathcal F}$, then ${{\mathcal P}} \; |_{U_p} \cong U_p \times {{\mathbb P}}^1.$ Taking homogeneous coordinates $[u,v] \in {{\mathbb P}}^1$,we have ${{\hbox{\goth O}}}_{{\mathcal P}}(\pi^{-1}(U_p)) \cong
{{\mathbb{C}}}[x_1, x_2, x_3, u, v]$. Thus, $${{\hbox{\goth O}}}_{G_s}(\pi^{-1}(U_p)) \cong {{\mathbb{C}}}[x_1, x_2, x_3, u, v]/(u x_1 x_2 + v x_3).$$This implies that the local equation of $G_s$ in $\pi^{-1}(U_p)$ is given by $$\label{eq:loccomp0}
u x_1 x_2 + v x_3 = 0.$$
In the open chart where $v \neq 0$, $G_s$ is smooth, whereas where $u \neq 0$, we see that is the equation of a quadric cone in $\mathbb{A}^4$ having vertex at the origin. This means that $G_s$ has a rational double point along the $\pi$-fibre $L_p:= \pi^{-1} (p) \subset {\mathcal P}$.
Globally speaking, one can state that there exist $\delta$ distinguished points on ${\mathcal P}$. Such distinguished points determine a $0$-dimensional subscheme $$\Sigma^1 \subset {{\mathcal P}}$$along which the divisor $G_s \subset {\mathcal P}$ is singular. Thus $\Sigma^1$ is a set of $\delta$ rational double points for $G_s$, each line of ${\Lambda} = \pi^{-1}(\Sigma)= \bigcup_{i=1}^{\delta} L_{p_i} $ containing only one of such $\delta$ points; more precisely, each point $p^1_i\in \Sigma^1$ is a rational double point for $G_s$ and it belongs to the fibre $L_{p_i} \subset {\mathcal P}$, where $p_i \in \Sigma$ is a node of $C = V(s)$. Therefore, the isomorphism$$\Sigma^1 \cong \Sigma$$is directly given by the natural projection $\pi : {\mathcal P}\to X$.
The above result introduces a correspondence between elements of ${{\mathcal V}}_{\delta}({{\mathcal F}})$ and suitable divisors in $|{{\hbox{\goth O}}}_{\mathcal P}(1)|$; this correspondence has been used in [@F2] to give a cohomological description of the tangent space $T_{[s]}({{\mathcal V}}_{\delta}({{\mathcal F}}))$, which has been a fundamental point in order to determine several sufficient conditions for the regularity of Severi varieties ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ (cf. Theorems 4.5, 5.9, 5.25, 5.28 and 5.36 in [@F2]).
The main ideas are as follows: $C$ is local complete intersection in $X$, whose normal sheaf is the rank-two vector-bundle ${{\mathcal N}}_{C/X} \cong {{\mathcal F}}|_C$. Let $T^1_C$ be the [*first cotangent sheaf*]{} of $C$, i.e. $T^1_C \cong {\mathcal Ext}^1(\Omega^1_C, {{\hbox{\goth O}}}_C)$, where $\Omega_C^1$ is the sheaf of K$\ddot{a}$hler differentials of the nodal curve $C$ (for details, see [@LS]). Since $C$ is nodal, $T^1_C$ is a sky-scraper sheaf supported on $\Sigma$, such that $T^1_C \cong \bigoplus_{i=1}^{\delta} {{\mathbb{C}}}_{(i)}$. Furthermore, one has the exact sequence: $$\label{eq:T1}
0 \to {{\mathcal N}}'_C \to {{\mathcal N}}_{C/X} \stackrel{\gamma}{\to} T^1_C \to 0,$$where ${{\mathcal N}}'_C$ is defined as the kernel of the natural surjection $\gamma$ (see, for example, [@S]).
\[prop:corr1\] (cf. Theorem 3.4 (ii) in [@F2]) With assumptions and notation as in Theorem \[prop:3.fundamental\], denote by ${\hbox{\goth I}}_{\Sigma^1/{{\mathcal P}}}$ the ideal sheaf of $\Sigma^1$ in ${\mathcal P}$. Then the subsheaf of ${\mathcal F}$, defined by $$\label{eq:svolta}
{{\mathcal F}}^{\Sigma} := \pi_* ({{\hbox{\goth I}}}_{\Sigma^1/{{\mathcal P}}} \otimes {{\hbox{\goth O}}}_{\mathcal P}(1)),$$is such that its global sections (modulo the one dimensional subspace $<s>$) parametrize first-order deformations of $s \in H^0(X, {\mathcal F})$ which are equisingular.
Precisely, we have $$\label{eq:reg}
\frac{H^0(X, {{\mathcal F}}^{\Sigma})}{< s >} \cong T_{[s]} ({{\mathcal V}}_{\delta}({{\mathcal F}})) \subset T_{[s]}
({{\mathbb P}}(H^0({{\mathcal F}}))) \cong \frac{H^0(X, {{\mathcal F}})}{< s >}.$$
By the correspondence given in Theorem \[prop:3.fundamental\], we can consider the closed immersion $ \Sigma^1 \subset {\mathcal P}$ and so the natural exact sequence $$\label{eq:idealP}
0 \to {{\hbox{\goth I}}}_{\Sigma^1/{{\mathcal P}}} \otimes {{\hbox{\goth O}}}_{\mathcal P}(1) \to {{\hbox{\goth O}}}_{\mathcal P}(1)
\to {{\hbox{\goth O}}}_{\Sigma^1} \to 0,$$which is defined by restricting ${{\hbox{\goth O}}}_{\mathcal P}(1) $ to $\Sigma^1$.
Since $\pi_*({{\hbox{\goth O}}}_{\mathcal P}(1)) \cong {{\mathcal F}}$, $\pi_*({{\hbox{\goth O}}}_{\Sigma^1}) = \pi_*({{\hbox{\goth O}}}_{\pi^{-1}(\Sigma^1)}) = \pi_*(\pi^*({{\hbox{\goth O}}}_{\Sigma}))
\cong {{\hbox{\goth O}}}_{\Sigma}$ and since we have ${{\mathcal F}} \to \!\!\! \to {{\hbox{\goth O}}}_{\Sigma}$, by applying $\pi_*$ to the exact sequence (\[eq:idealP\]), we get ${\mathcal R}^1 \pi_*({{\hbox{\goth I}}}_{\Sigma^1/{{\mathcal P}}}
\otimes {{\hbox{\goth O}}}_{\mathcal P}(1)) = 0$. Thus, we define ${{\mathcal F}}^{\Sigma}$ as in , so that $$\label{eq:idealX}
0 \to {{\mathcal F}}^{\Sigma} \to {{\mathcal F}} \to {{\hbox{\goth O}}}_{\Sigma} \to 0,$$holds.
Observe that ${{\mathcal F}}^{\Sigma} $ fits in the following exact diagram: $$\label{eq:(*1)}
\begin{aligned}
\begin{array}{rcccclr}
&0 & & 0 & & & \\
& \downarrow & & \downarrow & & & \\
0 \to &{{\hbox{\goth I}}}_{C/X} \otimes {{\mathcal F}} & \stackrel{\cong}{\to} & {{\hbox{\goth I}}}_{C/X}\otimes {{\mathcal F}} &\to & 0
& \\
& \downarrow & & \downarrow & & \downarrow & \\
0 \to & {{\mathcal F}}^{\Sigma} & \to & {{\mathcal F}} & \to & {\hbox{\goth O}}_{\Sigma} & \to 0 \\
& \downarrow & & \downarrow & & \downarrow^{\cong}& \\
0 \to & {{\mathcal N}}'_C & \to & {{\mathcal F}} |_{C} & \to & T^1_C & \to 0 \\
&\downarrow & & \downarrow & &\downarrow & \\
& 0 & & 0 & & 0 & .
\end{array}
\end{aligned}$$
From the commutativity of diagram , the vector space $$\frac{H^0(X, {{\mathcal F}}^{\Sigma})}{< s >}$$parametrizes the first-order deformations of $[s]$ in ${\mathbb P}(H^0(X, {{\mathcal F}})) $ which are [*equisingular*]{}; indeed, these are exactly the global sections of ${{\mathcal F}}$ which go to zero at $\Sigma$ in the composition $$\label{eq:compo}
{{\mathcal F}} \to \!\! \to {{\mathcal F}}|_C \to \!\! \to T^1_C \cong {{\hbox{\goth O}}}_{\Sigma}.$$
Notice that gives a completely general characterization of the tangent space $T_{[s]} ({{\mathcal V}}_{\delta}({{\mathcal F}}))$ on $X$. In particular, with assumptions and notation as in Theorem \[prop:3.fundamental\] and in Proposition \[prop:corr1\], we get the following results:
\[cor:propcorr1\] Each global section $s' \in H^0(X, {{\mathcal F}}^{\Sigma})$ corresponds to a divisor $G_{s'} \in | {\hbox{\goth I}}_{\Sigma^1/ {\mathcal P}}
\otimes {\hbox{\goth O}}_{{\mathcal P}}(1) |$ on ${\mathcal P}$. In particular, for $\epsilon \in {\mathbb{C}}[T]/(T^2)$, we have:
$s + \epsilon s' \in T_{[s]} ({{\mathcal V}}_{\delta}({{\mathcal F}}))$ on $X$ $\Leftrightarrow$ $s' \in
H^0(X, {{\mathcal F}}^{\Sigma})$ $\Leftrightarrow$ $G_{s'} \in |{\hbox{\goth O}}_{{\mathcal P}}(1) |$ and $\Sigma^1 \subset
G_{s'}$.
It directly follows from the definition of ${\mathcal F}^{\Sigma}$ and from the correspondence in Theorem \[prop:3.fundamental\].
\[rem:reg\] (cf. Corollary 3.9 in [@F2]) From , it follows that $$\label{eq:regbis}
\begin{array}{rcl}
[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) \; {\rm is \; regular} & \Leftrightarrow &
H^0(X , {{\mathcal F}}) \stackrel{\mu_X}{\to \!\!\! \to} H^0(X, {{\hbox{\goth O}}}_{\Sigma})\\
& \Leftrightarrow &
H^0({{\mathcal P}} , {{\hbox{\goth O}}}_{\mathcal P}(1)) \stackrel{\rho_{{\mathcal P}}}{\to \!\!\! \to} H^0({{\mathcal P}},{{\hbox{\goth O}}}_{\Sigma^1}).
\end{array}$$
It follows from Proposition \[prop:0\], from Theorem \[prop:3.fundamental\] and from Proposition \[prop:corr1\].
\[rem:localdescr\]
Association of elements in $T_{[s]}{{\mathcal V}}_{\delta}({{\mathcal F}})$ to singular surfaces in $| {\hbox{\goth I}}_{C/X} \otimes c_1({\mathcal F}) |$ on $X$ {#S:c}
=================================================================================================================================================================
The connection between singular schemes defined in Sections \[S:a\] and \[S:b\] can be further analyzed in order to determine interesting geometric interpretations of first-order deformations given by sections in $H^0(X, {{\mathcal F}}^{\Sigma})$.
With notation as in §\[S:a\] and in Assumption \[ass:main\], let $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$, $C = V(s)$, $\Sigma = Sing(C)$. Denote by ${\mathcal L}:= c_1({\mathcal F}) \in Pic(X)$.
By , one has: $$\label{eq:tspaces}
T_{[s]} ({{\mathcal V}}_{\delta}({{\mathcal F}})) \subset T_{[s]} ({{\mathbb P}}(H^0(X, {{\mathcal F}})) ) \cong \frac{H^0(X, {{\mathcal F}})}{H^0(X, {\hbox{\goth O}}_X)}
\hookrightarrow H^0 (X, {\hbox{\goth I}}_{C/X} \otimes {\mathcal L}).$$Therefore, first-order deformations of $[s]$ in ${{\mathbb P}}(H^0(X, {{\mathcal F}}))$, as well as in the Severi variety ${{\mathcal V}}_{\delta}({{\mathcal F}})$, can be related to suitable divisors moving in the linear system $|{\mathcal L}|$ on $X$ and containing the nodal curve $C$.
Indeed, we have:
\[prop:corr2\] Let $X $ be a smooth projective threefold. Let ${\mathcal F}$ be a globally generated rank-two vector bundle on $X$ and let ${\mathcal L}= c_1({\mathcal F})$. Let $\delta$ be a positive integer, $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ and $C=V(s)$ be the corresponding irreducible, nodal curve in $X$. Denote by $\Sigma$ the set of nodes of $C$. Let ${\mathcal F}^{\Sigma}$ be the sheaf on $X$ defined in Proposition \[prop:corr1\].
Then:
- Each global section $s' \in H^0(X, {{\mathcal F}}^{\Sigma}) \setminus < s >$ determines a divisor $ V( s \wedge s') \in \; | {\hbox{\goth I}}_{C/X} \otimes {\mathcal L}| $ in X which is singular along $\Sigma$.
Precisely, $$\label{eq:propcorr2}
s' \in H^0(X, {{\mathcal F}}^{\Sigma}) \Leftrightarrow V (s \wedge s') \; {\rm contains} \; C \; {\rm
and \; it \; is \; singular \; along} \; \Sigma.$$
- The singularities of $V(s \wedge s')$ are along $\Sigma$ and along the (possibly empty) intersection scheme $C \cap V(s')$.
\(i) Consider $s' \in H^0(X, {{\mathcal F}}^{\Sigma})$. By Proposition \[prop:corr1\] and Remark \[rem:localdescr\], this corresponds to a global section of ${\mathcal F}$ which is in the kernel of the map $\mu_X$ - i.e. a global section which goes to zero in the composition ${\mathcal F}\to\!\!\to {\mathcal F}|_C \to\!\!\to T^1_C$ in diagram .
Since the situation is local, we may work locally around each node, in some open subset where ${\mathcal F}$ trivializes. Thus, fix a node $p \in \Sigma$ and a suitable neighborhood $U = U_p$ of $p$, whose local coordinates are denoted by $(x_1, x_2, x_3)$. We assume that $C$ is defined in $U$ by two equations $ f_1 = f_2 = 0$, where $s|_U = (f_1, f_2)$, $f_1, f_2 \in {\hbox{\goth O}}_X(U)$.
Thus, the kernel of $\mu_X$ is given by those sections which are, at each node $p$, in the image of the Jacobian map $$\label{eq:jacobian}
{{\mathcal T}}_{{{\mathbb{C}}}^3}|_C \stackrel{J(s)}{\longrightarrow} {{\mathcal N}}_{C/{{\mathbb{C}}}^3} \to T^1_C$$given by $$\label{eq:jacobian2}
J(s) := \left( \begin{array}{ccc}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} &
\frac{\partial f_1}{\partial x_3}\\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} &
\frac{\partial f_2}{\partial x_3}
\end{array}
\right)$$in $U$.
Assume $s' \in H^0(X, {{\mathcal F}}^{\Sigma})$ and suppose $s'|_U = (g_1, g_2)$. Therefore, for each $p \in \Sigma$, $s' \in H^0(X, {{\mathcal F}}^{\Sigma})$ if and only if $s'(p) = (g_1(p), g_2(p)) \in {\mathcal F}_p$ is linear dependent on each pair $(\frac{\partial f_1}{\partial x_i}(p), \frac{\partial f_2}{\partial x_i}(p))$, $1 \leq i \leq 3$. This is equivalent to the following conditions: $$det \left( \begin{array}{ll}
g_1(p) & g_2(p) \\
\frac{\partial f_1}{\partial x_i}(p) & \frac{\partial f_2}{\partial x_i}(p)
\end{array}
\right)= 0, \;{\rm for \; each}\; 1 \leq i \leq 3,$$i.e. $$\label{eq:fico}
g_1 (p) \frac{\partial f_2}{\partial x_i}(p) - g_2 (p) \frac{\partial f_1}{\partial x_i}(p) = 0, \;{\rm for \; each}\;
1 \leq i \leq 3.$$
On the other hand observe that, since in particular $s' \in H^0(X, {\mathcal F})$, then it defines a divisor in $| {\mathcal L}|$ on $X$ containing $C$. Indeed, consider $$\tau := (s, s') : {\hbox{\goth O}}_X \oplus {\hbox{\goth O}}_X \to {\mathcal F},$$where $$\tau :=\left( \begin{array}{ll}
f_1 & g_1 \\
f_2 & g_2
\end{array}
\right)$$in the given open subset $U$. The degeneration locus of the map $\tau$ is given by$$V (det (\tau)) = V ( s \wedge s'),$$whose local equation in $U$ is given by $$\label{eq:loceq}
f_1g_2 - f_2 g_1 = 0.$$Thus, since ${\mathcal L}= c_1({\mathcal F})$, $V(s \wedge s')$ corresponds to a divisor in $|{\mathcal L}|$ containing $C = V(s)$, i.e. it belongs to the linear system $| {\hbox{\goth I}}_{C/X} \otimes {\mathcal L}|$ on $X$.
By the local equation of $V(s \wedge s')$ in $U$ and by the fact that $p \in C = V(s)$, we have that $$\label{eq:contiloc}
\frac{\partial}{\partial x_i}(f_1 g_2 - g_1 f_2) (p) =
g_1 (p) \frac{\partial f_2}{\partial x_i}(p) - g_2 (p) \frac{\partial f_1}{\partial x_i}(p), \;
{\rm for \; each} \; 1 \leq i \leq 3.$$Therefore, $s' \in H^0(X, {{\mathcal F}}^{\Sigma})$ if and only if the associated divisor in $|{\hbox{\goth I}}_{C/X} \otimes {\mathcal L}|$ is singular at the nodes of $C$ (cf. [@BC], for the case $X = {\mathbb{P}^3}$).
\(ii) Let $q \in C$ be a point and let $s = (f_1, f_2)$, $s' = (g_1, g_2)$ be the local expressions of $s$ and $s'$ around $q$. Fix $(x_1, x_2, x_3)$ local coordinates of $X$ around $q$. Since $C = V(s)$ and $q \in C$, we have that holds, for each $1 \leq i \leq \delta$.
Assume that $q \in C \setminus \Sigma$ and that $s' (q) \neq (0,0)$; in this case, if is equal to $0$ at $q$, for each $1 \leq i \leq 3$, we would have that $(g_1(q), g_2(q))$ is linear dependent on each pair $$\label{eq:boh3}
(\frac{\partial f_1}{\partial x_i}(q), \frac{\partial f_2}{\partial x_i}(q)), \; 1 \leq i \leq 3.$$In particular, the three pairs in would be linearly dependent. This is a contradiction; indeed, since $q \in C \setminus \Sigma$, the Jacobian map in and is surjective at such a point $q$, i.e. $T^1_{C,q} = 0$. On the other hand, since ${{\mathcal N}}_{C/X, q} \cong {\hbox{\goth O}}_{C,q}^{\oplus 2}$, then we must have that two of the three pairs in are linearly independent.
This implies that $V(s\wedge s')$ cannot be singular outside $\Sigma \cup (C \cap V(s'))$. Indeed, in the other cases - i.e. either $q \in \Sigma$ or $q \in (C \cap V(s'))$ or both - it is easy to observe that always vanishes at $q$, so that $V(s \wedge s')$ is singular at each such a point.
By using the “divisorial” approach introduced in Theorem \[prop:3.fundamental\], we shall give in Theorem \[thm:fundamental\] other interpretations of the equivalence in Proposition \[prop:corr2\], which highlights the deep connection between these apparently distinct approaches.
\[rem:propcorr2\]
Connection among various singular subschemes of $X$ and of ${\mathcal P}$ {#S:d}
=========================================================================
The aim of this section is to study in details the deep connection among:
- a nodal section $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$,
- the corresponding singular divisor $G_s$ in $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$ on ${\mathcal P}$ having $\delta$ rational double points over the nodes of $C = V(s)$,
- for any $s' \in H^0 (X, {{\mathcal F}}^{\Sigma}) \setminus <s>$, the singular surface $V(s \wedge s') \subset X$ belonging to the linear system $| {\hbox{\goth I}}_{C/X} \otimes {\mathcal L}|$ on $X$,
- the singular “complete intersection” surface in ${\mathcal P}$, ${{\mathcal S}}_{s,s'} := G_s \cap G_{s'}$, dominating $V(s \wedge s')$ (cf. Remark \[rem:2propcorr2\]).
As a consequence of the analisys given in Sections \[S:b\], \[S:c\], we have the following:
\[thm:fundamental\] Let $X $ be a smooth projective threefold. Let ${\mathcal F}$ be a globally generated rank-two vector bundle on $X$ and let $\delta$ be a positive integer. Fix $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ and let $C=V(s)$ be the corresponding nodal curve on $X$. Denote by $\Sigma$ the set of nodes of $C$.
Let ${\mathcal P} := {{\mathbb P}}_{X}({{\mathcal F}})$ be the projective space bundle, ${{\hbox{\goth O}}}_{\mathcal P}(1)$ be its tautological line bundle and $\Sigma^1 \subset {\mathcal P}$ be the zero-dimensional scheme of $\delta$-rational double points of the divisor $G_s \in |{{\hbox{\goth O}}}_{\mathcal P}(1)|$ which corresponds to $s \in H^0(X, {\mathcal F})$ (cf. Theorem \[prop:3.fundamental\]). Let ${\mathcal F}^{\Sigma}$ be the subsheaf of ${\mathcal F}$ defined as in Proposition \[prop:corr1\].
Then, the following conditions are equivalent:
- $s' \in H^0(X, {{\mathcal F}}^{\Sigma})\setminus <s>$;
- $V (s \wedge s') \subset X$ is a surface which contains $C$ and which is singular along $\Sigma$;
- the divisor $G_{s'}$ passes through $\Sigma^1$
- the surface ${\mathcal S}_{s,s'}:= G_s \cap G_{s'} \subset {\mathcal P}$ is singular along $\Sigma^1$;
Some of the implications are already proved in the previous results. Indeed:
5 pt
$(i) \Leftrightarrow (ii)$: this has already been proved in Proposition \[prop:corr2\].
5 pt
$(ii) \Leftrightarrow (iii)$: By the very definition of ${\mathcal F}^{\Sigma}$, it is obvious that $(iii)$ is equivalent to $(i)$; therefore, from the step above we would have finished. Anyhow, we give a direct proof of the equivalence of this two conditions, since it highlights the strict connection between the two apparently different approaches and it allows to give another interpretation of the equivalence in .
To this aim, let $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ and let $G_s$ be the corresponding divisor in ${\mathcal P}$ which is singular along $\Sigma^1$. As usual, take $p \in \Sigma$, $U = U_p$ an open subset such that $U \cap (\Sigma \setminus \{ p \}) = \emptyset$, where ${\mathcal F}$ trivializes and whose local coordinates are $\underline{x} = (x_1, x_2, x_3)$, such that $\underline{x}(p) = \underline{0} \in U \cong
\mathbb{A}^3$. If $s|_U = (f_1,f_2)$, we recall that the local equation of $G_s$ in $\pi^{-1} (U)$ is $$F(x_1, x_2, x_3, u ,v):= u f_1 + v f_2$$ Therefore, we have a singular point on $G_s$ in $\pi^{-1}(U)$ if, and only if, there exists a solution of $$\label{eq:system1}
F = \frac{\partial F}{\partial x_1} = \frac{\partial F}{\partial x_2} =
\frac{\partial F}{\partial x_3} =
\frac{\partial F}{\partial u} = \frac{\partial F}{\partial v} = 0.$$Observe that is equivalent to $$\label{eq:system2}
\begin{array}{l}
uf_1+vf_2 = \frac{\partial f_1}{\partial x_1} u + \frac{\partial f_2}{\partial x_1} v =
\frac{\partial f_1}{\partial x_2} u + \frac{\partial f_2}{\partial x_2} v = \\
= \frac{\partial f_1}{\partial x_3} u + \frac{\partial f_2}{\partial x_3} v =
f_1 = f_2 = 0.
\end{array}$$By the last two equations, we find as in Theorem \[prop:3.fundamental\] that the singular point of $G_s$ must be on the $\pi$-fibre over a point of $C$. Let $q \in U \cap C$ be such a point and let $L = L_q$ be this fibre.
We can restrict the system to $L$. We thus get: $$\label{eq:system3}
\frac{\partial f_1}{\partial x_1}(q) u + \frac{\partial f_2}{\partial x_1}(q) v =
\frac{\partial f_1}{\partial x_2}(q) u + \frac{\partial f_2}{\partial x_2}(q) v =
\frac{\partial f_1}{\partial x_3}(q) u + \frac{\partial f_2}{\partial x_3}(q)v = 0.$$Therefore, there exists a solution $[u,v] \in L \cong {\mathbb P}^1$ if, and only if, has rank less than or equal to one. This is equivalent to saying that $$\label{eq:propvectors}
(\frac{\partial f_1}{\partial x_1}(q) , \frac{\partial f_1}{\partial x_2}(q),
\frac{\partial f_1}{\partial x_3}(q)) = \lambda (\frac{\partial f_2}{\partial x_1}(q) ,
\frac{\partial f_2}{\partial x_2}(q),
\frac{\partial f_2}{\partial x_3}(q)),$$for some $\lambda \in {\mathbb{C}}^*$; this is equivalent to: $$\label{eq:system4}
\begin{array}{l}
\frac{\partial f_1}{\partial x_1}(q) \frac{\partial f_2}{\partial x_2}(q) -
\frac{\partial f_2}{\partial x_1}(q) \frac{\partial f_1}{\partial x_2}(q) = \\
\frac{\partial f_1}{\partial x_1}(q) \frac{\partial f_2}{\partial x_3}(q) -
\frac{\partial f_2}{\partial x_2}(q) \frac{\partial f_1}{\partial x_3}(q) = \\
\frac{\partial f_1}{\partial x_2}(q) \frac{\partial f_2}{\partial x_3}(q) -
\frac{\partial f_1}{\partial x_3}(q) \frac{\partial f_1}{\partial x_2}(q) = 0.
\end{array}$$In such a case, we have: $$\label{eq:solution}
\begin{array}{rcl}
[u,v] & = & [-\frac{\partial f_2}{\partial x_1}(q), \frac{\partial f_2}{\partial x_1}(q)]
= [-\frac{\partial f_2}{\partial x_2}(q), \frac{\partial f_1}{\partial x_2}(q)] \\
& = & [-\frac{\partial f_2}{\partial x_3}(q), \frac{\partial f_2}{\partial x_3}(q)]
= [- \lambda, 1].
\end{array}$$Therefore, $G_s$ is singular at $[- \lambda, 1] \in L$ if, and only if, $C = V(s) $ has a node at $q \in U$. This implies that $q=p$, since $p$ was the only node in $U$ by assumption; thus $L = L_p$ and - once again - the singularities of $G_s$ are on the $\pi$-fibres of the nodes of $C$.
Let now $s' \in H^0(X, {\mathcal F}^{\Sigma})$. Assume that $s'|_U = (g_1, g_2)$, for some $g_1, g_2 \in {\hbox{\goth O}}_{X}(U)$. Then, $G_{s'}$ passes through the singular point of $G_s$ along $L_p$, i.e. $[-\lambda, 1]$ if, and only if, $$\label{eq:propvectors2}
[-g_2(p), g_1(p)] = [-\lambda, 1].$$This means that $[-g_2(p), g_1(p)]$ is a solution of (where $q=p$, as proved above). This is exactly equivalent to and so to the fact that the surface $V( s \wedge s')$ is singular at $p$.
5 pt
$(iii) \Leftrightarrow (iv)$: trivial consequence of the fact that $G_s$ is always singular at $\Sigma^1$.
\[rem:2propcorr2\]
Observe that by Theorem \[thm:fundamental\], we can improve the statement of Corollary \[cor:propcorr1\].
\[cor:explanetion\] $s + \epsilon s' \in T_{[s]} ({{\mathcal V}}_{\delta}({{\mathcal F}}))$, s.t. $\epsilon^2 = 0$ $\Leftrightarrow$ $V(s \wedge s')$ is singular along $\Sigma$ $\Leftrightarrow $ ${\mathcal G}_{s'} \in |{\hbox{\goth I}}_{\Sigma^1/{\mathcal P}} \otimes {\hbox{\goth O}}_{{\mathcal P}}(1) | $ $\Leftrightarrow $ ${\mathcal S}_{s,s'}:={\mathcal G}_s \cap {\mathcal G}_{s'} \subset {\mathcal P}$ is singular along $\Sigma^1$.
\[rem:explanetion\]
To sum up, the regularity condition for points of the scheme ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ not only translates into the independence of conditions imposed by a $0$-dimensional scheme on the tautological linear system $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$ on ${\mathcal P}$, but also to the independence of [*singularity conditions*]{} imposed on suitable families of singular surfaces in $X$ as well as in ${\mathcal P}$.
${\mathcal P}_{\delta}$-Severi varieties of singular divisors on ${\mathcal P}$ {#S:ef}
===============================================================================
The aim of this section is to study families of singular divisors in ${\mathcal P}$, which are related to Severi varieties ${{\mathcal V}}_{\delta}({{\mathcal F}}) $ of nodal sections of ${\mathcal F}$ on $X$.
\[def:rdelta\] Given ${\mathcal P}$ and $\delta$ as in Assumption \[ass:main\], Theorem \[prop:3.fundamental\] and Proposition \[prop:corr1\], consider the scheme $$\label{eq:rdelta}
{\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))
:= \{G_s \in |{\hbox{\goth O}}_{{\mathcal P}}(1) | \; {\rm s.t.} \; [s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) \}.$$These schemes parametrize families of divisors in the tautological linear system of ${\mathcal P}$, $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$, which are irreducible and have only $\delta$ rational double points as the only singularities. For brevity sake, these will be called ${\mathcal P}_{\delta}$-[*Severi varieties*]{}.
By the above definition and by Assumption \[ass:main\], from now on we consider $\delta \leq \; min\{ h^0({\mathcal F}) -1, \frac{1}{2}deg ({\mathcal L}\otimes \omega_X \otimes {\hbox{\goth O}}_D) + 1\}$ and ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) \neq \emptyset$.
It is clear that: $$\label{eq:expdim}
expdim ({\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))) = dim (|{\hbox{\goth O}}_{{\mathcal P}}(1)|) - \delta;$$indeed, imposing a rational double point gives at most $5$ conditions on $|{\hbox{\goth O}}_{{\mathcal P}}(1)|$; each such point varies on any of the $\pi$-fibre over $X$.
As in Definition \[def:0\], from it is natural to give the following:
\[def:expdim\] Let $[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$. Then $[G_s]$ is said to be a [*regular point*]{} of ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ if:
- $[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ is a smooth point, and
- $dim_{[G_s]}({\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))) = expdim ({\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)))=
dim (|{\hbox{\goth O}}_{{\mathcal P}}(1)|) - \delta$.
The ${\mathcal P}_{\delta}$-Severi variety ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ is said to be [*regular*]{} if it is regular at each point.
As in Theorem \[prop:3.fundamental\], in order to find regularity conditions for a given point $[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$, it is crucial to give a description of the tangent space at $[G_s]$ to the given ${\mathcal P}_{\delta}$-Severi variety.
\[thm:tangentspace\] Let $[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$ and let $\Sigma^1 $ be the zero-dimensional scheme of the $\delta$-rational double points of $G_s \subset {\mathcal P}$. Then, we have: $$\label{eq:tangentspace}
T_{[G_s]} ({\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))) \cong \frac{H^0({\hbox{\goth I}}_{\Sigma^1/{\mathcal P}} \otimes {\hbox{\goth O}}_{{\mathcal P}}(1))}{<G_s>}.$$
In particular, if $\epsilon \in {\mathbb{C}}[T]/(T^2)$, then: $$G_s + \epsilon \; G_{r} \in T_{[G_s]} ({\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) \Leftrightarrow
G_r \in |{\hbox{\goth O}}_{{\mathcal P}}(1)) | \; {\rm and} \; \Sigma^1 \subset G_r.$$
The divisor $G_s \subset {\mathcal P}$, related to the point $[G_s] \in {\mathcal R}_{\delta}({\hbox{\goth O}}_{{\mathcal P}}(1))$, corresponds to a section $[s] \in {{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$.
Therefore, as in the proof of Theorem \[prop:3.fundamental\], we may locally work around a node of $C = V(s) \subset X$. Let $p \in \Sigma = Sing(C)$ be such a node and let $U = U_p \subset X$ be an affine open set containing $p$, where the vector bundle ${{\mathcal F}}$ trivializes. We thus can choose local coordinates $\underline{x} = (x_1, x_2, x_3)$ on $U \cong {\mathbb A}^3$ and homogeneous coordinates $[u,v] \in {{\mathbb P}}^1$, such that $\underline{x}(p) = (0,0,0)$, $s|_{U} = (x_1x_2, \; x_3)$ and the local equation of $G_s$ in $\pi^{-1}(U) \cong U_p \times {{\mathbb P}}^1$ is given by $u x_1 x_2 + v x_3 = 0$ (cf. ).
Recall that in the open chart where $v \neq 0$, $G_s$ is smooth, whereas, in the open chart where $u \neq 0$, we see that the local equation of $G_s $ in ${\mathbb A}^3 \times {\mathbb A}^1 \cong \mathbb{A}^4$ is $$\label{eq:palle}
G_s = V( x_1 x_2 + x_3 t), \; {\rm where} \; t= \frac{v}{u}.$$This is the equation of a quadric cone in $\mathbb{A}^4$ having vertex at the origin of the $\mathbb{A}^4$ having coordinates $(x_1, x_2, x_3, t)$.
We can consider the Jacobian map of $G_s$ in this $\mathbb{A}^4$. This is given by: $$\begin{array}{rcl}
{\mathcal T}_{\mathbb{A}^4 |_{G_s}} & \stackrel{J_{G_s}}{\longrightarrow} & {{\mathcal N}}_{G_s/\mathbb{A}^4} \\
\partial /\partial x_1 & \longrightarrow& x_2 \\
\partial / \partial x_2 & \longrightarrow & x_1 \\
\partial / \partial x_3 & \longrightarrow & t \\
\partial / \partial t & \longrightarrow & x_3,
\end{array}$$where ${{\mathcal N}}_{G_s/\mathbb{A}^4}$ is locally free of rank one on $G_s$. It is then clear that $J_{G_s}$ is surjective except at the origin $\underline{0} = (0,0,0,0)$.
By the local computations we analytically get: $$coker (J_{G_s}) \cong \frac{{\mathbb{C}}[[x_1, x_2, x_3, t]]/(x_1 x_2 + x_3 t)}{(x_1, x_2, x_3, t)} \cong {\mathbb{C}};$$
Globally speaking, given $G_s \subset {\mathcal P}$, whose singular scheme is $\Sigma^1$, we have the exact sequence of sheaves on $G_s$: $$\label{eq:palle1}
{\mathcal T}_{{\mathcal P}|_{G_s}} \stackrel{J_{G_s}}{\to} {{\mathcal N}}_{G_s/{\mathcal P}} \to T^1_{G_s} \to 0,$$where $T^1_{G_s} $ is a sky-scraper sheaf supported on $\Sigma^1$ and of rank one at each point.
As in for nodal curves, denote by ${\mathcal N}'_{G_s}$ the kernel of $J_{G_s}$ in . This is the so-called [*equisingular sheaf*]{}, whose global sections give equisingular first-order deformations of $G_s$ in ${\mathcal P}$.
By standard exact sequences, one sees that there is an injection $$\frac{H^0({\mathcal P}, {\hbox{\goth I}}_{\Sigma^1/{\mathcal P}} \otimes {\hbox{\goth O}}_{{\mathcal P}}(1))}{H^0({\mathcal P}, {\hbox{\goth O}}_{{\mathcal P}})} \hookrightarrow
H^0(G_s, {\mathcal N}'_{G_s}),$$which is an isomorphism when ${\mathcal P}$ - equivalently $X$ - is regular, i.e. $h^1({\mathcal P}, {\hbox{\goth O}}_{{\mathcal P}}) = 0$. Therefore, the vector space on the left-hand-side of the injection actually parametrizes equisingular first-order deformations of $G_s$ in $|{\hbox{\goth O}}_{{\mathcal P}}(1)|$.
\[rem:nfi\]
To conclude the section denote, as in , by$$\rho_{{\mathcal P}} : H^0({\mathcal P}, {\hbox{\goth O}}_{{\mathcal P}}(1))
\to H^0({\mathcal P}, {\hbox{\goth O}}_{\Sigma^1})$$the natural restriction map. Then, from Theorem \[thm:tangentspace\], it immediately follows:
\[cor:tangentspace\] With assumptions and notation as in Theorem \[thm:tangentspace\], we have: $$\begin{array}{lcl}
[G_s] \in {\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) \; {\rm is \; a \; regular \; point} & \Leftrightarrow &
\rho_{{\mathcal P}} \; {\rm is \; surjective} \\
& \Leftrightarrow & [s] \in {{\mathcal V}}_{\delta}({{\mathcal F}}) \; {\rm is \; a \; regular \; point}\\
& & {\rm (in \; the \; sense \; of \; Definition \; \ref{def:0}).}
\end{array}$$
The first equivalence is a direct consequence of and Theorem \[thm:tangentspace\]. The other follows from Corollary \[rem:reg\].
Some uniform regularity results for ${{\mathcal V}}_{\delta}({{\mathcal F}})$ and ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) $ {#S:ghil}
===============================================================================================================================================
In this section we first improve some regularity results of [@F2] for Severi varieties ${{\mathcal V}}_{\delta}({{\mathcal F}})$ of irreducible, $\delta$-nodal sections of $X$, then we use Corollary \[cor:tangentspace\] to deduce regularity results also for ${\mathcal P}_{\delta}$-Severi varieties ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) $ of irreducible divisors in $|{\hbox{\goth O}}_{{\mathcal P}}(1) |$ on ${\mathcal P}$. We find upper-bounds on the number $\delta$ of singular points which ensure the regularity of ${{\mathcal V}}_{\delta}({{\mathcal F}})$ as well as of ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) $; these upper-bounds are shown to be almost sharp (cf. Remark \[rem:10\]).
What we want to stress is the following fact: even if the regularity of the schemes ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1)) $ on ${\mathcal P}$ is defined by means of separation of suitable zero-dimensional schemes by the linear system $|{\hbox{\goth O}}_{{\mathcal P}}(1)|$ on the fourfold ${\mathcal P}$, one can avoid to consider this intricate situation. Indeed, it is well-known how difficult is to enstablish separation of points in projective varieties of dimension greater than or equal to three (cf. e.g. [@AS], [@EL] and [@K]). In general some results can be found by using the technical tools of [*multiplier ideals*]{} together the Nadel and the Kawamata-Viehweg vanishing theorems (see, e.g. [@E], for an overview). Anyhow, sometimes no answers are given by using these techniques.
In our situation, thanks to the correspondence between ${{\mathcal V}}_{\delta}({{\mathcal F}})$ on $X$ and ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ on ${\mathcal P}$, we deduce regularity conditions for the scheme ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ from those of the scheme ${{\mathcal V}}_{\delta}({{\mathcal F}})$.
From now on, let $X$ be a smooth projective threefold, ${\mathcal E}$ be a globally generated rank-two vector bundle on $X$, $M$ be a very ample line bundle on $X$ and $k \geq 0$, $\delta >0 $ be integers. With notation and assumptions as in Section \[S:a\], we shall always take$${{\mathcal F}} = {{\mathcal E}} \otimes M^{\otimes k}$$and consider the scheme ${{\mathcal V}}_{\delta}({{\mathcal E}} \otimes M^{\otimes k})$ on $X$.
By using Theorem \[prop:3.fundamental\] and Corollary \[rem:reg\], here we determine conditions on the vector bundle ${\mathcal E}$ and on the integer $k$ and uniform upper-bounds on the number of nodes $\delta$ implying that each point of $ {{\mathcal V}}_{\delta}({{\mathcal E}} \otimes M^{\otimes k})$ is regular. Indeed, the next result is a generalization of Theorem 4.5 in [@F2].
\[thm:9bis\] Let $X$ be a smooth projective threefold, ${\mathcal E}$ be a globally generated rank-two vector bundle on $X$, $M$ be a very ample line bundle on $X$ and $k \geq 0$ and $\delta >0$ be integers. If $$\label{eq:uniform}
\delta \leq k+1,$$then ${{\mathcal V}}_{\delta}({{\mathcal E}} \otimes M^{\otimes k})$ is regular.
If $k =0$, then we consider $\delta = 1$; therefore, by the hypothesis on ${\mathcal E}$, it follows that $$H^0 ({\mathcal E}) \to H^0({\hbox{\goth O}}_p^{\oplus 2})$$is surjective, for each $p \in X$. By the description of the map $\mu_X$ in Remark , this implies that ${{\mathcal V}}_1({{\mathcal E}})$ is regular at each point.
When $k > 0$, first of all observe that, since $M$ is very ample, then $M^{\otimes k}$ separates any set $\Sigma$ of $\delta$ distinct point of $X$ with $\delta \leq k+1$. This is equivalent to saying that the restriction map $$\label{eq:prop7}
\rho_k: \; H^0(X, M^{\otimes k}) \to H^0({{\hbox{\goth O}}}_{\Sigma})$$is surjective, for each such $\Sigma \subset X$. Thus, implies there exist global sections $\sigma_1, \ldots, \sigma_{\delta} \in H^0(X, M^{\otimes k})$ s. t. $${\sigma}_i(p_j) = \underline{0} \in {{\mathbb{C}}}^{\delta}, \; {\rm if \; i \neq j }, \;{\rm and} \;
\sigma_i(p_i) = (0, \ldots, \stackrel{i-th}{1}, \ldots, 0 ), \; 1 \leq i \leq
\delta.$$On the other hand, since ${\mathcal E}$ is globally generated on $X$, the evaluation morphism $$H^0(X, {{\mathcal E}}) \otimes {{\hbox{\goth O}}}_X \stackrel{ev}{\to} {\mathcal E}$$is surjective. This means that, for each $p \in X$, there exist global sections $s_1^{(p)}, \; s_2^{(p)} \in
H^0 (X, {{\mathcal E}})$ such that $$s_1^{(p)}(p) = (1,0), \; s_2^{(p)}(p) = (0,1) \in {{\hbox{\goth O}}}_{X,p}^{\oplus 2}.$$Therefore, it immediately follows that $$H^0(X, {{\mathcal E}} \otimes M^{\otimes k}) \to \!\!\! \to H^0({{\hbox{\goth O}}}_{\Sigma}^{\oplus 2}) \cong
{{\mathbb{C}}}^{2 \delta}.$$If we compose with diagram , we get: $$\begin{array}{ccl}
H^0( {{\mathcal E}} \otimes M^{\otimes k}) & \to\!\!\! \to & H^0({{\hbox{\goth O}}}_{\Sigma}^{\oplus 2}) \cong {{\mathbb{C}}}^{2 \delta} \\
\downarrow^{\mu_X} & & \downarrow \\
H^0({{\hbox{\goth O}}}_{\Sigma}) & \stackrel{\cong}{\to} & H^0({{\hbox{\goth O}}}_{\Sigma}) \cong {{\mathbb{C}}}^{\delta}.\\
& & \downarrow \\
& & 0
\end{array}$$thus $\mu_X$ is surjective (cf. Remark \[rem:localdescr\]). By , one can conclude.
\[rem:10\]
For what concerns ${\mathcal P}_{\delta}$-Severi varieties on ${\mathcal P}$, we get:
\[thm:regrdelta\] Let $X$ be a smooth projective threefold, ${\mathcal E}$ be a globally generated rank-two vector bundle on $X$, $M$ be a very ample line bundle on $X$ and $k \geq 0$ and $\delta >0$ be integers.
Let ${\mathcal P}: = {\mathbb P}_X ({\mathcal E}\otimes M^{\otimes k})$ and let ${\hbox{\goth O}}_{{\mathcal P}}(1)$ be its tautological line bundle. Let ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ be the ${\mathcal P}_{\delta}$-Severi variety of irreducible divisors on ${\mathcal P}$ having $\delta$-rational double points on ${\mathcal P}$. Then, if: $$\label{eq:uniform2}
\delta \leq k+1,$$ ${\mathcal R}_{\delta} ({\hbox{\goth O}}_{{\mathcal P}}(1))$ is regular.
From Theorem \[thm:9bis\], we know that is a sufficient condition for the regularity of ${{\mathcal V}}_{\delta}({{\mathcal E}} \otimes M^{\otimes k})$ on $X$. One can conclude by using Corollary \[cor:tangentspace\].
[ADSE]{} Anghern U., Siu Y.-T., Effective freeness and point separation for adjoint bundles, [*Invent. Math.*]{}, [**122**]{} (1995), 291-308. Ballico E., Chiantini L., A look into the Severi varieties of curves in higher codimension. Dedicated to the memory of Fernando Serrano, [*Collect. Math.*]{}, [**49**]{} (1998), no. 2-3, 191-201. Barth W., Peters C., Van de Ven A., [*Compact Complex Surfaces*]{}, Springer, Berlin, 1984. Caporaso L., Harris J., Counting plane curves of any genus, [*Invent. Math.*]{}, [**131**]{} (1998), 345-392. Chiantini L., Ciliberto C., On the Severi varieties of surfaces in ${\mathbb{P}^3}$, [*J. Algebraic Geom.*]{}, [**8**]{} (1999), 67-83. Chiantini L., Lopez A. F., Focal loci of families and the genus of curves on surfaces, [*Proc. Amer. Math. Soc.*]{}, [**127**]{} (1999), no.12, 3451-3459. Chiantini L., Sernesi E., Nodal curves on surfaces of general type, [*Math. Ann.*]{}, [**307**]{} (1997), 41-56. Ein L., Multiplier ideals, Vanishing Theorems and Applications, [*Proc. Symposia Pure Math.*]{}, [**62.1**]{} (1997), 203-219. Ein L., Lazarsfeld R., Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, [*J. A.M.S.*]{}, [**6**]{} (1993), 875-903. Flamini F., [*Families of nodal curves on projective surfaces*]{}, Ph.D thesis, Consortium Universities of Rome “La Sapienza” and “Roma Tre”, (2000). Available on http://www.mat.uniroma3.it/dipartimento/esterni/flamin${\rm i}_{-}$ricerca.html Flamini F., Moduli of nodal curves on smooth surfaces of general type, [*J. Algebraic Geom.*]{}, [**11**]{} (2002), no. 4, 725-760. Flamini F., Families of nodal curves on projective threefolds and their regularity via postulation of nodes, [*to appear on Trans. Amer. Math. Soc.*]{}, (2003), [*preprint: math.AG/0202022, 4 Feb. 2002*]{}. Friedman R., [*Algebraic surfaces and holomorphic vector bundles*]{} (UTX), Springer-Verlag, New York, 1998. Greuel G.M., Lossen C., Shustin E., Geometry of families of nodal curves on the blown-up projective plane, [*Trans. Amer. Math. Soc.*]{}, [**350**]{} (1998), no.1, 251-274. Harris J., On the Severi problem, [*Invent. Math.*]{}, [**84**]{} (1986), 445-461. Hartshorne R., [*Ample Subvarieties of Algebraic Varieties*]{}, Springer LNM, [**156**]{}, Springer-Verlag, Berlin, 1970. Hartshorne R., [*Algebraic Geometry*]{} (GTM No. 52), Springer-Verlag, New York - Heidelberg, 1977. Kawamata Y., On Fujita’s conjecture for $3$-folds and $4$-folds, [*Math. Ann.*]{}, [**308**]{} (1997), 491-505. Okonek C., Schneider M., Spindler H., [*Vector bundles on complex projective spaces*]{}, Progress in Mathematics, [**3**]{}, Boston-Basel-Stuttgart, Birkhäuser, 1980. Lichtenbaum S. Schlessinger M., The cotangent complex of a morphism, [*Trans. Amer. Math. Soc.*]{}, [**128**]{} (1967), 41-70. Ran Z., On nodal plane curves, [*Invent. Math.*]{}, [**86**]{} (1986), 529-534. Sernesi E., On the existence of certain families of curves, [*Invent. Math.*]{}, [**75**]{} (1984), 25-57. Severi F., [*Vorlesungen über algebraische Geometrie*]{}, Teubner, Leipzig, 1921. Szpiro L., [*Lectures on equations defining space curves*]{}, TATA Institute of fundamental research, Bombay, Springer, 1979.
[^1]: 2000 [*Mathematics Subject Classification*]{}. 14H10, 14J30, 14J60, 14J35, 14C20.
[^2]: The author is a member of Cofin GVA, EAGER and GNSAGA-INdAM
|
---
abstract: |
Some graphs admit drawings in the Euclidean $k$-space in such a (natural) way, that edges are represented as line segments of unit length. Such drawings will be called $k$ dimensional unit distance representations. When two non-adjacent vertices are drawn in the same point, we say that the representation is degenerate. The dimension (the Euclidean dimension) of a graph is defined to be the minimum integer $k$ needed that a given graph has non-degenerate $k$ dimensional unit distance representation (with the property that non-adjacent vertices are mapped to points, that are not distance one appart).
It is proved that deciding if an input graph is homomorphic to a graph with dimension $k \geq 2$ (with the Euclidean dimension $k \geq 2$) are NP-hard problems.\
[**Keywords**]{}: unit distance graph; the dimension of a graph; the Euclidean dimension of a graph; degenerate representation; complexity;\
[**Mathematics Subject Classification**]{}: 05C62, 05C12, 05C10,
author:
- |
Jan Kratochvíl[^1]\
Charles University, Prague,\
Czech Republic\
Boris Horvat[^2]\
IMFM, University of Ljubljana,\
Slovenia\
Tomaž Pisanski[^3]\
IMFM, University of Ljubljana, and University of Primorska,\
Slovenia
title: 'On the computational complexity of degenerate unit-distance representations of graphs'
---
Introduction and background
===========================
Define a [*representation*]{} $\rho := (\rho_{V},\rho_{E})$ of a graph $G$ in a set $M$ as a mapping $\rho_{V}: V(G) \ \rightarrow M$ and a mapping $\rho_{E}: E(G) \ \rightarrow 2^{M}$, such that if $v$ is an end-vertex of an edge $e=u \sim v$, then $\rho_{V}(v) \in \rho_{E}(e)$ (and $\rho_{V}(u) \in \rho_{E}(e)$) [@pearlsgt]. If the converse is also true, that is, that for $v \in V(G)$ and $e \in E(G)$, if $\rho_{V}(v) \in \rho_{E}(e)$ is true, then $v$ is an end-vertex of $e=u \sim v$, the representation is called a [*realization*]{}. If there is no danger of confusion we drop the subscripts and denote both mappings $\rho_{V}$ and $\rho_{E}$ by $\rho$. Observe, that the intersection $\rho(e) \cap \rho(\ell)$ of realizations of non-adjacent edges $e, \ell \in E(G)$ can be non-empty, but cannot contain a realization of any vertex of $G$. Thus, representations of edges of $G$ may cross, which is different from the usual graph embedding case. If the mapping $\rho$ is not injective on $V(G)$ it is called [*degenerate*]{}.
In this paper we consider only representations in the Euclidean $k$-space, i.e. ([*$k$ dimensional Euclidean*]{}) representations that have $M = {\mathbb{R}}^k$, and where an edge is always represented by the line segment between the representations of its end-vertices. Each Euclidean realization of a connected graph on at least three vertices is non-degenerate, see [@HorvatPisanski2]. An Euclidean representation is called a [*unit distance representation*]{} if each edge is represented as a line segment of length one. A graph is called a $k$ dimensional [*unit distance graph*]{} if it has a $k$ dimensional unit distance realization. For example, the well known Petersen graph $G(5,2)$ is a unit distance graph with exactly 18 different degenerate unit distance representations, see [@HorvatPisanski].
Erdös [*et al.*]{}, see [@ErdosHararyTutte], defined the [*dimension*]{} of a graph $G$, denoted as $\dim(G)$, as the minimum integer $k$ needed that $G$ is a $k$ dimensional unit distance graph. Other authors (e.g. [@BuckleyHarary; @UDGMAEHARA4; @UDGMAEHARA2]) defined an [*unit distance embedding*]{} to be a unit distance realization that maps non-adjacent vertices to points with distances other than one. The smallest integer $k$ needed that a given graph $G$ has an unit distance embedding in ${\mathbb{R}}^k$ is called the [*Euclidean dimension*]{} of $G$, see for example [@UDGMAEHARA3], and is, as usual, denoted as $\e(G)$. A graph with $\e(G) = k$ is called a $k$ dimensional [*strict unit distance graph*]{} (or, following Boben [*et al.*]{} [@BobenPisanski], $k$ dimensional [*unit distance coordinatization*]{}). As any $k$ dimensional strict unit distance graph is a $k$ dimensional unit distance graph, it follows that for a given graph $G$, $\dim(G) \leq \e(G)$. Wheh $\dim(G)=2$ (or $\e(G)=2$) we will be talking about [*planar*]{} unit distance graphs (or planar strict unit distance graphs). A planar graph with a planar unit distance realization in the plane is called a [*matchstick*]{} graph.
If $G$ is composed of the connected components $G_{1}, G_{2}, \ldots, G_{s}$, then $G$ admits a unit distance representation in ${\mathbb{R}}^{k}$ if and only if each component $G_{i}$, $1 \leq i \leq s$, admits a unit distance representation in ${\mathbb{R}}^{k}$. By definition, loops are not valid unit distance Euclidean edge representations, and there exists only one line segment of length one between two points in the Euclidean space. Therefore, we may restrict our attention to connected simple graphs.
A mapping $f: V(G) \rightarrow V(H)$ from a graph $G$ into a graph $H$ is called a [*graph homorphism*]{}, if $f$ maps vertices of $G$ into vertices of $H$, such that $u \sim v \in E(G)$ implies $f(u) \sim f(v) \in E(H)$ ($f$ preserves adjacencies). Each graph morphism $f: V(G) \rightarrow V(H)$ induces a unique mapping (denoted by the same letter) $f: E(G) \rightarrow E(H)$, such that for $e=u \sim v$, $f(e)=f(u)\sim f(v)$ (see e.g. [@GAHHELL]). Let $G$ and $H$ be graphs, let $f:G \rightarrow H$ be a graph homomorphism, and let $k \in {\mathbb{N}}$. Any unit distance representation of $H$ in ${\mathbb{R}}^{k}$ (if it exists) can be extended to a (possibly degenerate) unit distance representation of $G$ in ${\mathbb{R}}^{k}$, see [@HorvatPisanski2].
The union $G_{1} \union G_{2}$ of graphs $G_{1}$ and $G_{2}$ is the graph with vertex set $V(G_{1}) \cup V(G_{2})$ and edge set $E(G_{1}) \cup E(G_{2})$. The disjoint union of two graphs $G_{1}$ and $G_{2}$, denoted $G_{1} \disjunion G_{2}$, is the graph obtained by taking the union of $G_{1}$ and $G_{2}$ on disjoint vertex sets, $V(G_{1})$ and $V(G_{2})$. The join of two simple graphs $G_{1}$ and $G_{2}$, written $G_{1} \join G_{2}$ is the graph obtained by taking the disjoint union of $G_{1}$ and $G_{2}$ and adding all edges joining $V(G_{1})$ and $V(G_{2})$. A wheel graph $W_{n}$ on $n \geq 4$ vertices is defined as $W_{n} = W_{1, n-1} := K_{1} \join C_{n-1}$, where $K_{1}$ is the one vertex graf and $C_{n-1}$ the cycle graph on $n-1$ vertices. It is well known [@BuckleyHarary] that $\dim(W_{n}) = 3$ for $n \neq 7$ and $\dim (W_{7}) = 2$. Even though $\e(W_{5}) = \dim(W_{5}) = 3$, there exists degenerate planar unit distance representations of the graph $W_{5}$; see Figure \[fig:degudgrepr\].
It is therefore interesting to answer a question “Does, for a given graph $G$ and a natural number $k$, there exists a degenerate $k$ dimensional unit distance representation of $G$?”. One can instead, answer a question “For a given natural number $k$ and a graph $G$, is $G$ homomorphic to a graph with dimension $k$?”. Of course, similar questions concerning the Euclidean dimension can be asked.
The only graph with zero dimension is $K_{1}$, and the same holds for the Euclidean dimension. It can be easily shown that the only graphs with dimension one are path graphs on at least two vertices. Again, the same is true for the Euclidean dimension. Thus, we can check in polynomial time whether a given graph is homomorphic to a linear strict unit distance graph. In Section 2 we prove that deciding if an input graph is homomorphic to a planar unit distance graph (planar strict unit distance graph) are NP-hard problems. In Section 3 we consider the same problems in higher dimensions.
Planar unit distance representations
====================================
Deciding if an input graph $G$ is homomorphic to a graph $H$ with $\dim(H) = 2$ (with $\e(H) = 2$) are NP-hard problems.
The proof is based on the Moser graph (see Fig. \[fig:Moser\]) which is a well known planar strict unit distance graph. It has the property that every homomorphism to a unit distance graph $H$ is injective and the image $H$ is isomorphic to the Moser graph itself. In particular, adding any edge to the Moser graph results in a graph with dimension at least 3. It is not hard to see, that the dimension and the Euclidean dimension of the Moser graph are both equal to two. Using Laman’s Theorem, see e.g. [@Hendrickson:1992p3225], it is not hard to show that the unit distance coordinatization of Moser graph is rigid.
The NP-hardness reduction goes from . Given a formula $\Phi$ with a set of variables $X$ and a set of clauses $C$ (each clause containing exactly 3 literals), we construct a graph $G_{\Phi}$, such that $G_{\Phi}$ is homomomorphic to a strict unit distance graph (namely to the Moser graph) if $\Phi$ is satisfiable, while $G_{\Phi}$ is not homomorphic to any unit distance graph if $\Phi$ is unsatisfiable. This will prove both claims.
The cornerstone of the construction is a copy of the Moser graph with vertices $\{u,T,F,u',u'',v,w\}$ and edges as drawn in the shaded part of the Fig. \[fig:variable\]. For every variable $x \in X$, we add an edge $x' \sim x''$, where $x'$ represents the positive literal and $x''$ the negation of $x$. Also each $x'$ and $x''$ are made adjacent to both $u$ and $u'$. The part of the construction of $G_{\Phi}$ that corresponds to a variable $x \in X$ can be seen in the bolded part of the Fig. \[fig:variable\].
For every clause $c \in C$, say $c = (\lambda_{c,1} \vee \lambda_{c,2} \vee \lambda_{c,3})$, the clause gadget is depicted in Fig. \[fig:clausegadget\]. It contains six vertices, three of them being connected to the literals appearing in $c$ (one to each) and one adjacent to the special vertex $F$ of the Moser graph.
Formally, $$V(G_{\Phi}) = \{u, v, w, u', u'', T, F\} \cup \bigcup_{x \in X} \{x', x''\} \cup \bigcup_{c \in C} \{c_{1}, c_{2}, c_{3}, c_{4}, c_{12}, c_{34}\}$$ and $$\begin{aligned}
E(G_{\Phi}) & = & \{u \sim T, u \sim F, T \sim F, T \sim u', F \sim u', u \sim v, u \sim w, v \sim w, \\
& & \indent v \sim u'', w \sim u'', u' \sim u'' \} \cup \\
& & \bigcup_{x \in X} \{x' \sim x'', x' \sim u, x'' \sim u, x' \sim u', x'' \sim u'\} \cup \\
& & \bigcup_{c \in C} \{c_{1} \sim c_{2}, c_{2} \sim c_{3}, c_{3} \sim c_{4}, c_{4} \sim c_{1}, c_{1} \sim c_{12}, c_{2} \sim c_{12}, c_{3} \sim c_{34}, \\
& & \indent c_{4} \sim c_{34}, c_{12} \sim F, c_{2} \sim \lambda_{c,1}, c_{3} \sim \lambda_{c,2}, c_{34} \sim \lambda_{c,3}\}.\end{aligned}$$
Suppose first that $G_{\Phi}$ is homomorphic to a graph $H$ with $\dim(H) = 2$ (to a planar unit distance graph $H$). Consider a planar unit distance realization of $H$. There are at most two paths of length two between two vertices of a planar unit distance graph [@UDGMAEHARA4]. Since there are four paths of length two between $u$ and $u'$ in $G_{\Phi}$, the homomorphism either places $x'$ in $T$ and $x''$ in $F$, or vice versa. This placement defines a truth assignment on the variables of $\Phi$ - we say that $x$ is [true]{} if $x'$ is placed in $T$, and that it is [false]{} otherwise. We claim that $\Phi$ is satisfied by this assignment. Suppose there is an unsatisfied clause, say $c = (\lambda_{c,1} \vee \lambda_{c,2} \vee \lambda_{c,3})$. Then all three vertices $\lambda_{c,1}, \lambda_{c,2}, \lambda_{c,3}$ must be placed in $F$, and the clause gadget must map onto another copy of the Moser graph (with $F$ being its degree four vertex). But then, in the plane, the edge $c_2 \sim c_3$ cannot have a unit length. We have a contradiction.
Suppose now that $\Phi$ is satisfied by a truth assignment $\phi: X\longrightarrow \{{\sf true, false}\}$. We fix a strict planar unit distance realization of the Moser graph, place $x'$ in $T$ and $x''$ in $F$ whenever $\phi(x)={\sf true}$ (and vice versa if $\phi(x)={\sf false}$). Following the case analysis in Fig. \[fig:placement\], the vertices of the clause gadgets can be placed in vertices $u,T$ and $F$. Hence, we have constructed a well defined surjective homomorphism of $G_{\Phi}$ onto the Moser graph. Therefore, $G_{\Phi}$ is homomorphic to a graph with the Euclidean dimension 2.
$k$ dimensional unit distance representations
=============================================
Let $\text{\pS}_{k}(\vec{s}, r)$ denote the $k$ dimensional (hiper)sphere in ${\mathbb{R}}^{k}$ with center in $\vec{s}$ and radius $r$. When the center and the radius of a sphere are not important, the abbrevation $\text{\pS}_{k}$ will be used.
\[le:presekksferjekm1\] Let $k>1$ be a natural number. A non-empty non-degenerated intersection of two $k$ dimensional spheres with distinct centers is $k-1$ dimensional sphere.
Let $c, r, R > 0$ be positive real numbers. We can, without loss of generality, assume, that the first sphere is centered at the origin $\vec{0}$ and the second one is centered on the $x_{1}$ axes in $\vec{c} = (c, 0, \ldots, 0)$. Spheres $\text{\pS}_{k}(\vec{0},r)$ and $\text{\pS}_{k}(\vec{c},R)$ are determined by equations $$x_1^2+x_2^2+ \ldots +x_k^2 = r^2$$ and $$(x_1-c)^2+x_2^2+ \ldots +x_k^2 = R^2.$$ We subtract the above equations to obtain a linear equation for $x_1$ with a solution $a := {r^2+c^2-R^2 \over 2c}$. When $a$ is inserted into the first equation, the equality $x_2^2+ \ldots +x_k^2=r^2-a^2$ is obtained. When $r^2-a^2 < 0$, the system does not have any solution and this case corresponds to a situation, where spheres do not intersect. A situation $|a| = r$ corresponds to spheres that are tangent to each other and their intersection is a point - a degenerated sphere. Assume now, that $r^2-a^2 > 0$. The equation $x_2^2+ \ldots +x_k^2 = r^2 - a^2$ determines the $k-1$ dimensional sphere in the hyperplane $x_1 = a$ in ${\mathbb{R}}^{k}$.
\[lm:chileq3\] Let $\text{\pS}_{2}$ denote the circle in the Euclidean plane, which is circumscribed to a unit distance realization of the complete graph $K_{3}$ on three vertices. Let $G$ be a connected graph with (possibly degenerate) planar unit distance representation, which places all vertices of $G$ into points that lay on $\text{\pS}_{2}$. Then $\chi(G) \leq 3$.
Graph $K_{3}$ has unique (unique up to isometries) unit distance realization in the plane. It can be represented as an equilateral triangle with all sides of unit length, which has unique circumscribed circle. Denote the circumscribed circle by $\text{\pS}_{2}$. Every point on $\text{\pS}_{2}$ has exactly two points that are unit distance appart from it and belong to $\text{\pS}_{2}$. Graph $G$ has a unit distance representation on $\text{\pS}_{2}$ and can hence occupy only three distinct points on $\text{\pS}_{2}$, which are exactly the vertices of (possibly another) equilateral triangle with all sides of length one. Thus, there exists a proper 3 coloring of $G$ - colors being the vertices of the equlateral triangle.
Lemma \[lm:chileq3\] can be generalized. Let $G_{k, r, \alpha}$ denote the graph with vertices being points of $k$ dimensional sphere $S_{k}(\vec{0}, r)$ with radius $r$ in ${\mathbb{R}}^{k}$, where two vertices are connected if and only if they are at distance $\alpha$. L. Lovász [@Lovasz:1983p585] proved the following inequalities.
\[th:chileqlovasz\] Let $k \geq 3$ be a natural number. Then, for $0 \leq \alpha \leq 2$ holds $$k \leq \chi(G_{k,1,\alpha})$$ and $$\chi\left(G_{k,1,\sqrt{2(k+1) \over k}}\right) \leq k+1.$$
\[cor:chileqk\] Let $k \geq 3$ be a natural number and let $\text{\pS}_{k}$ denote the $k$ dimensional sphere in ${\mathbb{R}}^{k}$, which is circumscribed to the unit distance realization of the complete graph $K_{k+1}$ on $k+1$ vertices. Let $G$ be a connected graph with (possibly degenerate) $k$ dimensional unit distance representation, which places all vertices of $G$ into points that lay on $\text{\pS}_{k}$. Then $\chi(G) \leq k+1$.
It is known, that the circumradius of the (hiper)sphere that is circumscribed to the regular simplex with $k+1$ vertices and all sides of length $\ell$, equals to $\ell \sqrt{k \over 2 (k+1)}$. Thus, the circumradius of sphere $\text{\pS}_{k}$ equals to $\sqrt{k \over 2 (k+1)}$. The representation of graph $G_{k, \sqrt{k \over 2 (k+1)}, 1}$ on sphere $\text{\pS}_{k}$ can be (down)scaled to obtain the representation on $k$ dimensional unit sphere (with radius one), that is circumscribed to the regular simplex with all sides of length $\sqrt{2(k+1) \over k}$. Following Theorem \[th:chileqlovasz\], $\chi\left(G_{k,1,\sqrt{2(k+1) \over k}}\right) \leq k+1$. The proper $k+1$ coloring of $G_{k,1,\sqrt{2(k+1) \over k}}$ gives rise to the proper $k+1$ coloring of $G_{k, \sqrt{k \over 2 (k+1)}, 1}$. Since $G$ is a subgraph of $G_{k, \sqrt{k \over 2 (k+1)}, 1}$, $\chi(G) \leq k+1$.
Let $k \geq 3$ be a natural number. Deciding if an input graph $G$ is homomorphic to a graph $H$ with $\dim(H) = k$ (with $\e(H) = k$) are NP-hard problems.
Let $K'_{k}$ and $K''_{k}$ be two copies of the complete graph with $k \geq 3$ vertices. Let $v,w',w''$ be additional vertices, such that $v,w',w'' \notin V(K'_{k}) \cup V(K''_{k})$. Denote $M'_{k} = K'_{k} \join \{v, w'\}$, $M''_{k} = K''_{k} \join \{v, w''\}$ and $M_{k} = M'_{k} \union M''_{k} \union \{w' \sim w''\}$, e.g. Figure \[fig:kcoldeg1\]. Graph $M_{k}$ is well known Moser-Raiskii spindle.
Note that $\e(K'_{k}) = \dim(K'_{k}) = k-1$. Since $K'_{k} \join v$ is the complete graph on $k+1$ vertices, $\dim(K'_{k} \join v) = \e(K'_{k} \join v) = k$. Applying another graph join does not change any of dimensions. Thus, $e(M'_{k}) = e(M''_{k}) = \dim(M'_{k}) = \dim(M''_{k}) = k$. Both $M'_{k}$ and $M''_{k}$ are rigid in ${\mathbb{R}}^{k}$. Consider a unit distance realization of $M'_{k} \union M''_{k}$ in ${\mathbb{R}}^{k}$. We can rotate $M''_{k}$ around the representation of the vertex $v$, such that $w'$ and $w''$ results distance one appart. Hence $\dim(M_{k}) = k$. Additionally, we can rotate the subgraph $K''_{k}$ in such a way, that $\e(M_{k})=k$. Note that $H_{k}$ has the property that every $k$ dimensional unit distance realization in ${\mathbb{R}}^{k}$, places vertices $v$ and $w'$ in distinct points (this is guaranteed by the edge $w' \sim w''$).
Let us construct the graph $H_{k}$, such that $V(H_{k}) := V(G) \cup V(M_{k})$ and $$E(H_{k}) := E(G) \union E(M_{k}) \union \bigcup_{u \in V(G)} \{u \sim v, u \sim w'\},$$ where vertices $v, w'$ are vertices from $M_{k}$.
We reduce from graph . We will prove that, when $H_{k}$ is homomorphic to a graph with the dimension $k$, then $\chi(G) \leq k$, while from $\chi(G) \leq k$ follows, that $H_{k}$ is homomorphic to a graph with the Euclidean dimension $k$. Suppose $H_k$ is homomorphic to a graph with the dimension $k$. Then $H_k$ has (possibly degenerate) unit distance representation $\rho$ in ${\mathbb{R}}^{k}$. All vertices of $G$ are one appart from $\rho(v)$ and from $\rho(w')$, and they all belong to the intersection of two unit $k$ dimensional spheres with centers in $\rho(v)$ and $\rho(w')$. Using Lemma \[le:presekksferjekm1\], a non-empty non-degenerated intersection of two $k$ dimensional spheres with distinct centers is $k-1$ dimensional sphere. All vertices of the complete graph $K'_{k}$ belong to the same intersection and hence to the same $k-1$ dimensional sphere. If $G$ is connected, we can use Corollary \[cor:chileqk\] and $\chi(G) \leq k$. If $G$ is not connected, we can observe its components in similar matter.
Assume now, that $\chi(G) \leq k$. Using proper $k$-coloring of $G$ we can map vertices of $G$ into vertices of $K'_{k}$, see [@GAHHELL]. Hence a homomorphic image of $H_{k}$ is $M_{k}$ and $\e(M_{k}) = k$.
As any $k$ dimensional strict unit distance graph is a $k$ dimensional unit distance graph, it follows that for a given graph $G$, $\dim(G) \leq \e(G)$. Hence we proved both claims of this Theorem.
Concluding remarks
==================
There are several results in connection to computational complexity and unit distance graphs. Firstly, it is known, that a graph reconstruction problem is strongly NP-complete in $1$ dimension and strongly NP-hard in higher dimensions, see [@saxe]. For general edge-lenghts, graph realization problems and problems concerning rigidity of structures in connection to computational complexity, were observed in [@Hendrickson:1992p3225] and [@Yemini:1979p3248].
P. Eades and S. Whitesides proved in [@Eades:1996p3245] that the decision problem: “Is a planar graph $G$ a matchstick graph?” is NP-hard. They used a reduction from the well known NP-complete problem 3SAT, a special instance of the problem, in which, each term has to have at least one true and at least one false literal. Their proof was constructive, they used a matchstick graph to construct a mechanical device, called [*logic engine*]{}, that was used to simulate the 3SAT problem instance. Cabello [*et al.*]{} in [@Cabello:2004p560] used similar approach to prove that the decision problem: “Is a planar $3$-connected infinitesimally rigid graph $G$ a matchstick graph?” is NP-hard.
Additionally, P. Eades and N. C. Wormald proved in [@Eades:1990p3289] that the decision problem: “Is a $2$-connected graph $G$ a matchstick graph?” is NP-hard, too.
[99]{}
M. Boben, T. Pisanski. Polycyclic configurations. *European J. Combin.* **24** (2003), 4:431–457.
F. Buckley, F. Harary. On the Euclidean dimension of a Wheel. *Graphs Combin.* **4** (1988), 23–30.
S. Cabello, E. Demaine, G. Rote. Planar embeddings of graphs with specified edge lengths. *Lect. Notes Comput. Sc.* **2912** (2004), 283–294.
P. Eades, S. Whitesides. The logic engine and the realization problem for nearest neighbor graphs. *Theor. Comput. Sc.* **169** (1996), 1:23–37.
P. Eades, N. C. Wormald. Fixed edge-length graph drawing is NP-hard. *Discrete Appl. Math.* **28** (1990), 2:111–134.
P. Erdös, F. Harary, W. T. Tutte. On the dimension of a graph. *Mathematika* **12** (1965), 118–122.
N. Hartsfield, G. Ringel. *Pearls in Graph Theory: A Comprehensive Introduction*. Revised and augmented. Academic Press, San Diego, 1994.
P. Hell, J. Ne[š]{}et[ř]{}il. *Graphs and Homomorphisms*. Oxford University Press, 2004.
B. Hendrickson. Conditions For Unique Graph Realizations. *SIAM J. Comput.* **21** (1992), 1:65–84.
B. Horvat, T. Pisanski. Unit distance representations of the Petersen graph in the plane. *Ars Combin.* (to appear).
B. Horvat, T. Pisanski. Products of unit distance graphs. *Discrete Math.* (to appear).
L. Lovász. Self-dual polytopes and the chromatic number of distance graphs on the sphere. *Acta Sci. Math. (Szeged)* **45** (1983), 1-4:317–323.
H. Maehara. Note on Induced Subgraphs of the Unit Distance Graph. *Discrete Comput. Geom.* **4** (1989), 15–18.
H. Maehara. On the Euclidean dimension of a complete multipartite graph. *Discrete Math.* **72** (1988), 285–289.
H. Maehara, V. Rödl. On the Dimension to Represent a Graph by a Unit Distance Graph. *Graphs Combin.* **6** (1990), 365–367.
J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In *Proc. 17th Al lerton Conf. Commun. Control Comput.*, 480–489, 1979.
Y. Yemini. Some theoretical aspects of position-location problems. In *Proc. 20th Annu. IEEE Sympos. Found. Comput. Sci.*, 1–8, 1979.
[^1]: `honza@kam.mff.cuni.cz`
[^2]: `Boris.Horvat@fmf.uni-lj.si`
[^3]: `Tomaz.Pisanski@fmf.uni-lj.si`
|
---
abstract: 'The advantages of weak measurements, and especially measurements of imaginary weak values, for precision enhancement, are discussed. A situation is considered in which the initial state of the measurement device varies randomly on each run, and is shown to be in fact beneficial when imaginary weak values are used. The result is supported by numerical calculation and also provides an explanation for the reduction of technical noise in some recent experimental results. A connection to quantum metrology formalism is made.'
author:
- 'Y. Kedem'
title: 'Using technical noise to increase the signal to noise ratio in weak measurements. '
---
In 1988 Aharonov, Albert and Vaidman (AAV) [@AAV88] discovered that the measured value of an observable can be 100 times bigger than its biggest eigenvalue, provided the measurement interaction is weak and a postselection is employed. They showed that a system which is coupled weakly to another, pre- and postselected system, described by the [*two-state vector*]{} $\langle \Phi |~|\Psi\rangle $, via an observable $C$, is effectively coupled to the weak value of the observable [@AV90] $$\label{wv}
C_w \equiv { \langle{\Phi} \vert C \vert\Psi\rangle \over
\langle{\Phi}\vert{\Psi}\rangle } .$$ The replacement of an operator with its weak value, which is a complex number [@Joz], is known as the AAV effect and the procedure in which the weak value is measured is referred to as a weak measurement. The promise that this phenomenon holds for improving precision measurements had recently started to materialize in observation of the spin hall effect of light [@HK] and ultra sensitive measurement of beam deflection [@How]. Other areas where the use of weak measurements was investigated include measuring small longitudinal phase shifts [@Brunner; @phase], charge sensing [@charge], frequency measurements [@freq], and Kerr nonlinearities [@stein].
The general use of quantum effects for precision enhancements, known as quantum metrology [@metro], is showing significant results [@inter] and lately, much attention is drawn to practical issues such as the effects of an environment [@env; @env2], noise [@esch], and technical limitations [@real]. According to [@HK; @How] the use of imaginary weak values in the measurement process, allows a reduction in technical noise. In this letter we will analyze the process of weak measurement as a method for precision measurements. Furthermore, we will consider a specific model for technical noise, affecting the preparation of the measurement device (meter), and show that in the presence of such a noise the precision is enhanced.
Consider a physical interaction: $$\label{h}
H = g(t) PC ,$$ where $C$ is an observable on a system, $P$ is an operator on a meter and $g(t)$ is a coupling function satisfying $ \int g(t) dt =k$. Our concern is estimating the size of $k$, or in some cases simply observing the interaction. A straight forward approach is to put the system in an eigenstate of $C$ having some eigenvalue $c$, and the meter in a Gaussian state: $$\label{Pgauss}
\Psi_{M} (Q) =(\Delta ^2 \pi )^{-1/4} e^{ -{Q ^2 \over 2 \Delta ^2}},$$ where $Q$ is a variable conjugate to $P$, and $\Delta$ is its quantum uncertainty. An estimate of $k$ can be obtained from the shift in $Q$ due to the interaction, $\langle Q \rangle = kc$ , and its precision is determined by the standard deviation ${1 \over \sqrt{2}} \Delta.$ In the case $kc \ll \Delta$, little information is acquired from a single measurement, but by repeating the procedure $N$ times and averaging the results, the precision is enhanced. Strictly speaking, the amount of information gathered, regarding $k$, is measured by the Fisher information [@esch], but for our purposes we can use the more intuitive concept of [*signal to noise ratio*]{} ($S/N$) [@snr], which in this case is $$\label{snr}
S/N = \sqrt{N }{kc \over \Delta }.$$
Since our interest is in the regime where $kc \ll \Delta$, which is a condition for the AAV effect [@wu], we will, for now, assume that the AAV effect occurs and later examine its validity in more detail. Thus, we will consider the system to be initially in a state $|\Psi\rangle$ and take into account the meter results only when the system was found in a state $|\Phi\rangle$, after the interaction. The shift in $Q$ is given by $\langle Q \rangle_{\Phi} = k ReC_w$ [@AAV88], and $$\label{snr2}
S/N = \sqrt{N_{\Phi} }{k {\text Re}C_w \over \Delta },$$ where $N_{\Phi} \sim N \left| \langle \Phi |\Psi\rangle \right|^2$ is the number of times the system was found in a state $|\Phi\rangle$. In order for $C_w$ to be larger than any eigenvalue of $C$, the scalar product, $\langle \Phi |\Psi\rangle,$ has to be small, so we can see that we cannot improve (\[snr2\]) significantly, relative to (\[snr\]). It is, however, a remarkable fact that by using only a small portion of our potential data, we get the same quality of information. In practice, there are many set ups where a rare postselection is beneficial, especially when there is a detection constraint, such as saturation limits or dead time.
Another option is to measure the meter in the $P$ basis. Assuming the meter initial state is (\[Pgauss\]), which we can write in the $P$ basis as: $$\label{Pgaussp}
\Psi_{M} (P) = (\Delta ^{-2} \pi )^{-1/4} e^{ -{\Delta ^2 P ^2 \over 2 }},$$ the final shift in $P$ is given by $\langle P \rangle_{\Phi} = k \Delta^{-2}{\text Im} C_w$ [@Joz] and the standard deviation is ${1 \over \sqrt{2}} \Delta^{-1}$, giving us $$\label{snr3}
S/N = \sqrt{N_{\Phi} }{k {\text Im} C_w \over \Delta }.$$ Surprisingly, for ${\text Im} C_w = {\text Re} C_w$, the S/N for this case is the same as (\[snr2\]) and it seems measuring an imaginary weak value is ineffective. However, as we will now show, this is not the case.
In calculating the $S/N$ (\[snr\]), (\[snr2\]) and (\[snr3\]) we considered only the quantum uncertainty, sometimes called shot noise, and not any technical issues. Since the set ups used in advance experiments are highly intricate, there is an enormous range of possible technical issues and conceiving a general model for their effect is beyond the scope of this letter. Instead, we will restrict our discussion to faults in the preparation of the meter, causing its initial state to be shifted with respect to (\[Pgauss\]) or (\[Pgaussp\]).
Let us start by considering a shift, $Q_0$, in the $Q$ basis only, making the initial state of the meter $$\label{gaussq}
\Psi_{M} (Q) =(\Delta ^2 \pi )^{-1/4} e^{ -{(Q-Q_0) ^2 \over 2 \Delta ^2}}.$$ A measurement of $Q$, after an interaction (\[h\]) with a pre and postselected system $ \langle \Phi |~~|\Psi\rangle $, will yield $$\begin{aligned}
\label{Q1}
\langle Q \rangle_{\Phi} &=& Q_0 + k {\text Re}C_w, \nonumber \\
\langle Q^2 \rangle_{\Phi} &=& {\Delta ^2 \over 2}+ (Q_0 + k{\text Re}C_w)^2. \end{aligned}$$ Since the shift $Q_0$ can be different for every run, some distribution should be used when averaging over the results. We assume an uncorrelated distribution with vanishing average $ \overline{Q_0}=0$, which can be seen as white noise. A finite average would describe a systematic error while correlations can appear, for example, if $Q_0$ has some time dependency which is relevant to the frequency in which the runs occur or to their total time. In order to treat such disturbances, an analysis using Allan variance [@allan] is needed which we will not discuss here. In [@stein], weak measurements was shown to be beneficial for noise with long correlation time, however, their results about its ineffectiveness for white noise was based on the measurements of real weak values.
For simplicity, we assume the distribution of the shift is Gaussian as well, i.e. the probability for the shift $Q_0$ to occur is $$\label{prob}
\Pr(Q_0)= (\Delta_Q \sqrt {\pi} )^{-1} e^{ -{Q_0 ^2 \over \Delta_Q ^2}},$$ where $ \Delta_Q$ is the width of the distribution of the shift. An average over $Q_0$ will result in $$\begin{aligned}
\label{Q2}
\overline{\langle Q \rangle_{\Phi} } &=& k{\text Re}C_w, \nonumber \\
\overline{\langle Q^2 \rangle_{\Phi} } &=& {\Delta ^2 \over 2}+ {\Delta_Q ^2 \over 2} + (k{\text Re}C_w)^2,\end{aligned}$$ meaning the same shift as it was for (\[Pgauss\]) but a larger standard deviation, making $$\label{snr4}
S/N = \sqrt{N_{\Phi} }{k {\text Re} C_w \over\sqrt{ \Delta ^2+ \Delta_Q ^2} }$$ smaller than (\[snr2\]). Similarly we can get $S/N = \sqrt{N} k c / \sqrt{ \Delta ^2+ \Delta_Q ^2}$ if the system is in an eigenstate of $C$ with eigenvalue $c$.
By writing the meter state, in the $P$ basis, after the interaction and postselection: $$\label{gaussfp}
\Psi_{M} (P) = \mathcal{N} (\Delta ^{-2} \pi )^{-1/4} e^{ -{\Delta ^2 P ^2 \over 2 } + i (Q_0 - k C_w) P},$$ where $ \mathcal{N} = Exp \left[ - k^2 \Delta ^{-2} ( {\text Im} C_w) ^2 / 2 \right]$ is the re-normalization factor due to the postselection, one can see that a measurement of $P$ will yield $$\begin{aligned}
\label{P1}
\langle P \rangle_{\Phi} &=& k \Delta ^{-2} {\text Im} C_w, \nonumber \\
\langle P^2 \rangle_{\Phi} &=& {\Delta ^{-2} \over 2}+ ( k \Delta ^{-2} {\text Im} C_w )^2. \end{aligned}$$ This means that the $S/N$ for this case, $$\label{snr5}
S/N = \sqrt{N_{\Phi} }{k {\text Im} C_w \over \Delta },$$ is the same as (\[snr3\]), the $S/N$ for the case of an ideal initial state.
This is the first result of our letter: when one has a dominant technical issue in the preparation of a variable conjugate to the interaction operator, measurements of an imaginary weak value can eliminate its effect.
Let us now consider a shift, $P_0$, in the $P$ basis, making the initial state of the meter $$\label{gaussp2}
\Psi_{M} (P) =(\Delta ^{-2} \pi )^{-1/4} e^{ -{ \Delta ^2 (P-P_0) ^2 \over 2}},$$ with probability $$\label{prob1}
\Pr(P_0)= (\Delta_P \sqrt {\pi} )^{-1} e^{ -{P_0 ^2 \over \Delta_P ^2}}.$$ After an interaction (\[h\]) with a pre and postselected system $ \langle \Phi |~~|\Psi\rangle $ the meter is in a state $$\label{gaussfp2}
\Psi_{M} (P) = \mathcal{N}_{P_0} (\Delta ^{-2} \pi )^{-1/4} e^{ -{\Delta ^2 (P-P_0) ^2 \over 2 } - i k C_w P},$$ where $\mathcal{N}_{P_0}$ is the re-normalization factor due to the postselection. A final measurement of $P$, will yield $$\begin{aligned}
\label{P2a}
\langle P \rangle_{\Phi} &=& P_0 + k \Delta ^{-2} {\text Im} C_w, \nonumber \\
\langle P^2 \rangle_{\Phi} &=& {\Delta ^{-2} \over 2}+ (P_0 + k \Delta ^{-2} {\text Im} C_w)^2. \end{aligned}$$ In order to calculate the average over $P_0$ we have to consider the probability of postselection $$\begin{aligned}
\label{post}
\Pr(|\Phi\rangle \mid P_0) &=& \left|\langle\Phi\vert\Psi\rangle\right|^{2} e^{ k {\text Im} C_w \left(2 P_0 + k {\text Im} C_w \Delta^{-2}\right)} \nonumber \\
&=& \left|\langle\Phi\vert\Psi\rangle\right|^{2} \mathcal{N}_{P_0}^{-2},\end{aligned}$$ which was of no importance for a shift in $Q$, since it did not depend on $Q_0$. This means that if we prepare an ensemble of $N$ meters, with states (\[gaussp2\]) according to the distribution (\[prob1\]), and then, after an interaction (\[h\]), we postselect to $| \Phi \rangle$, the postselected ensemble of meters will have a different distribution: $$\begin{aligned}
\label{prob3}
\Pr(P_0 \mid |\Phi\rangle ) &=& {\Pr(P_0) \Pr(|\Phi\rangle \mid P_0) \over \Pr(|\Phi\rangle )} \nonumber \\
&=& (\Delta_P \sqrt {2 \pi} )^{-1} e^{ -{(P_0 - k {\text Im} C_w \Delta_P ^2)^2 \over \Delta_P ^2}}.\end{aligned}$$ Calculating the averages using (\[prob3\]) we get $$\begin{aligned}
\label{P2}
\overline{ \langle P \rangle_{\Phi}} &=& k (\Delta ^{-2}+ \Delta_P ^2) {\text Im} C_w, \nonumber \\
\overline{ \langle P^2 \rangle_{\Phi}} &=&{\Delta ^{-2} \over 2}+ {\Delta_p ^2 \over 2} + \left(k (\Delta ^{-2}+ \Delta_P ^2) ImC_w \right)^2,\end{aligned}$$ yielding: $$\label{snr6}
S/N = \sqrt{N_{\Phi} } k {\text Im} C_w \sqrt{\Delta ^{-2} + \Delta_p ^2 }.$$ While for $ \Delta_p = 0$ this $S/N$ equals (\[snr5\]), for $ \Delta_p > 0$ it is bigger.
This is the main result of our letter: In the regime where the AAV effect occurs, a non coherent spread in the variable appearing in the interaction improves the precision of the measurement.
In order to examine the conditions for the AAV effect, in the context of an imperfect meter preparation, we can look on the evolution (up to normalization): $$\begin{aligned}
\label{rem}
\langle\Phi\vert e^{-i k P C}\vert\Psi\rangle &=& \langle\Phi\vert\Psi\rangle e^{-i k C_{w}P} \\
&+&\langle\Phi\vert\Psi\rangle\sum_{n=2}^{\infty}\frac{(i k P)^{n}}{n!}[(C^{n})_{w}-(C_{w})^{n}] . \nonumber\end{aligned}$$ The AAV effect means that the final state of the meter is determined by the first term, so we need to require that its initial state is such that the second term is negligible. When examining the expectation value of this expression, we will consider only the first term in the sum, i.e. $n=2$, assuming the rest are smaller. An additional simplification can be done assuming $ |(C^{n})_w | < | C_w |^n $ for $n>1$, which is the case, in general, for a weak value that is larger than any eigenvalue, limiting our concern to verifying the condition: $ | k C_w | ^2 \langle P^2 \rangle \ll 1$. For the state (\[gaussq\]), it amounts to $ |k C_w |^2 \Delta^{-2} \ll 1 $, implying that there is no dependency on the distribution of $Q_0$ and also that for a purely real (imaginary) weak value, the $S/N$ (\[snr2\]) ( (\[snr3\]) ) have to be small for $N_{\Phi} =1$. Thus, a small $S/N$ per measurement is a necessary condition for the AAV effect. For the state (\[gaussp2\]), with a distribution (\[prob1\]), we have $$\label{cond}
|k C_w |^2 (\Delta ^{-2} + \Delta_p ^2 ) \ll 1,$$ implying that in order to make the S/N (\[snr6\]) large, for any value of $ \Delta_p$, one has to perform many measurements.
We support our results with a numerical calculation of a simple example, in which the pre and postselected system is a two level system (qubit) described by $(4+4w^2)^{-1/2}\left( \left\langle \uparrow \right| + \left\langle \downarrow \right|\right)~~\left((1+ iw) \left|\uparrow\right\rangle + (1- iw) \left| \downarrow \right\rangle\right)$, where $ \left|\uparrow\right\rangle$ ($\left| \downarrow \right\rangle$) is an eigenstate of $C$ with eigenvalue 1 (-1). The weak value is given by $C_w = i w$, but our calculation is not based on the AAV effect. For an initial state of the meter that is described by (\[gaussp2\]) and (\[prob1\]), we find that the distribution of a final measurement of $P$ is given by $$\label{dist}
\rho(P)= {2e^{k^2 \Delta_T ^2 - {P^2 \over \Delta_T ^2 }} \left|\cos(k P) + w \sin(k P)\right|^2 \over \left(1 - w^2 + \left(1+w^2 \right) e^{k^2 \Delta_T ^2}\right)\sqrt{\pi} \Delta_T },$$ where $ \Delta_T = \sqrt{\Delta ^{-2} + \Delta_p ^2}$. This distribution and the $S/N$ per measurement, $\overline{P} / \sqrt{\overline{P^2} - \overline{P}^2}$, for it, are plotted in Fig. \[dist1\].
![The expected results of a final measurement of $P$, based on equation (\[dist\]): The probability distribution for $w = 8$ (TOP) and the $S/N$ per measurement (Bottom). The center of the distribution is shifted in proportion to $\Delta_T$, however, when $w k \Delta_T \sim 1$ part of the distribution start appearing on the wrong side. For small $ \Delta_T$, the $S/N$ is increasing linearly, in agreement with (\[snr6\]). The maximum is around $w k \Delta_T \sim 1$. From (\[cond\]) it is clear that for larger values of $ \Delta_T $ the AAV effect is not valid, and thus the $S/N$ in getting smaller, as expected for standard measurements. In an experiment aimed at measuring a tiny effect, such as [@HK] and [@How], the interaction strength, $k$, would be very small making $ k \Delta_T \ll 1$ the relevant regime and only with the factor $ \sqrt{N_{\Phi} } \sim \sqrt{N}/w$, due to $N$ repetitions, can the $S/N$ be larger than unity. []{data-label="dist1"}](fig1b.jpg "fig:"){width="48.00000%"} ![The expected results of a final measurement of $P$, based on equation (\[dist\]): The probability distribution for $w = 8$ (TOP) and the $S/N$ per measurement (Bottom). The center of the distribution is shifted in proportion to $\Delta_T$, however, when $w k \Delta_T \sim 1$ part of the distribution start appearing on the wrong side. For small $ \Delta_T$, the $S/N$ is increasing linearly, in agreement with (\[snr6\]). The maximum is around $w k \Delta_T \sim 1$. From (\[cond\]) it is clear that for larger values of $ \Delta_T $ the AAV effect is not valid, and thus the $S/N$ in getting smaller, as expected for standard measurements. In an experiment aimed at measuring a tiny effect, such as [@HK] and [@How], the interaction strength, $k$, would be very small making $ k \Delta_T \ll 1$ the relevant regime and only with the factor $ \sqrt{N_{\Phi} } \sim \sqrt{N}/w$, due to $N$ repetitions, can the $S/N$ be larger than unity. []{data-label="dist1"}](fig2a.jpg "fig:"){width="48.00000%"}
In order to put our results in an experimental context, we analyze two experiments, [@HK] and [@How], where weak measurements were used to detect tiny modifications in a paraxial light beam. In [@HK] the beam was displaced by the spin hall effect of light, creating a polarization dependent change in its transverse spatial distribution. They considered an effective Hamiltonian of the form of (\[h\]), with $C$ being a polarization variable, $P$ the transverse momentum and $k$ was a small coefficient that needed to be estimated. Polarizers were used for the pre and postselection, making $C_w$ purely imaginary, and a position sensor was located in a distance such that the center of the spatial distribution was determined by the transverse momentum immediately after the interaction.
In [@How], a Sagnac interferometer was used where the angle of one of the mirrors changed the beams direction, depending on which path it took in the interferometer. The analogue interaction of the type (\[h\]) is having $C$ as the which-path variable, $P$ as transverse position and $k$ as the angle of the mirror times the light wave number. The interferometer was set up to make the weak value purely imaginary, and lenses were used to make the transverse position of the beam at detection proportional to the transverse position immediately after the interaction, up to a geometrical optical factor.
Thus, even though the interactions were of different nature, both results should agree with (\[P2\]). The manifestation of $ \Delta ^{-2} + \Delta_p ^2$ in an experiment would be the square of the width of the final measurement, and indeed, in both experiments the final result was proportional to this quantity. It was also mentioned in [@How], [@HK] and [@Brunner] that this method was especially beneficial for technical noise. Distinguishing between the coherent width $\Delta ^{-1}$ , and the one caused by technical issues, $ \Delta_p $, can be rather difficult, but it is unnecessary in our formalism.
Unlike the common practice in quantum metrology [@metro; @esch], our results do not require the meters to be entangled. The correlations created by the postselection can be viewed as classical ones and thus the precision scales as $\sqrt{N}$. The way this method improves the Cramér-Rao bound is simply by increasing $ \Delta H$, a task that can be done in a non-coherent way, and thus might be much simpler, experimentally, than the creation of entanglement.
An important aspect of our result is the effect of the meter on the system, usually called backaction. One can also consider the measurement interaction (\[h\]) with reversed roles, i.e. regard $P$ as acting on a system and $C$ on a meter. $k$ could be measured, for example, by putting the meter in an initial state $|\Psi\rangle$, the system in a state $P=P_0$ and measuring the probability to find it in a state $|\Phi\rangle$: $ \left| \langle{\Phi} \vert e^{-i k P_0 C} \vert\Psi\rangle \right|^2 = \left| \langle{\Phi} \vert \Psi\rangle \right|^2(1+2 k {\text Im} C_w P_0 )+ O(k P_0)^2$. The binomial distribution gives an $S/N$ of $$\label{snrPhi}
S/N = 2 k {\text Im} C_w P_0 \sqrt{N \left| \langle{\Phi} \vert\Psi\rangle \right|^2 \over (1 - \left| \langle{\Phi} \vert\Psi\rangle \right|^2) },$$ which is comparable to (\[snr6\]) with the replacement $P_0 \leftrightarrow \sqrt{\Delta ^{-2} + \Delta_p ^2 }$. This highlights some of the differences in the experimental challenges each method presents, with regard to the preparation and measurement of $P$. Naturally, experimental issues concerning $\langle \Phi |~|\Psi\rangle $ can also have different effects depending on the chosen method.
The AAV effect is valid for qubit meter as well [@KV] and, when it is used, technical issues in its preparation can naturally lead to reduced $S/N$. Since the uncertainty of any qubit variable is bounded, measurements of imaginary weak values cannot yield an improved $S/N$ relative to other methods, according to our results. However, such measurements can yield the same $S/N$ regardless if the meter uncertainty is coherent or not.
We have shown that in the scenario of measurement of imaginary weak values, a shortcoming in the ability to prepare the meter in an exact known state, does not diminish the precision and the result of some flawed preparation can in fact increase the precision. This phenomenon explains some remarkable recent results where technical noise was overcome and it has the potential to improve many quantum metrology schemes in a novel way.
This work has been supported in part by grant 32/08 of the Binational Science Foundation, and grant 1125/10 of the Israel Science Foundationt.
[9]{} Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. [**60**]{}, 1351 (1988).
Y. Aharonov and L. Vaidman, Phys. Rev. A **41**, 11 (1990).
R. Jozsa, Phys. Rev. A **76**, 044103 (2007).
O. Hosten and P. Kwiat, Science, [**319**]{}, 787 (2008).
P. B. Dixon, [*et al*]{}., Phys. Rev. Lett. [**102**]{}, 173601 (2009).
N. Brunner and C. Simon, Phys. Rev. Lett. [**105**]{}, 010405 (2010).
C. F. Li, [*et al*]{}., Phys. Rev. A **83**, 044102 (2011).
O. Zilberberg, A. Romito, and Y. Gefen, Phys. Rev. Lett. [**106**]{}, 080405 (2011).
D. J. Starling, [*et al*]{}., Phys. Rev. A **82**, 063822 (2010)
A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. [**107**]{}, 133603 (2011)
V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. [**96**]{}, 010401 (2006)
M. Napolitano, [*et al*]{}., Nature [**471**]{}, 486 (2011)
G. Goldstein, [*et al*]{}., Phys. Rev. Lett. [**106**]{}, 140502 (2011)
D. Braun and J. Martin, Nat. Commun. [**2**]{}, 223 (2011).
B. M. Escher, R. L. de MatosFilho, and L. Davidovich, Nature Phys. [**7**]{}, 406 (2011).
N. Thomas-Peter, [*et al*]{}., Phys. Rev. Lett. [**107**]{}, 113603 (2011).
D. J. Starling, [*et al*]{}., Phys. Rev. A **80**, 041803 (2009).
S. Wu and Y. Li, Phys. Rev. A **83**, 052106 (2011).
D.W. Allan, IEEE Trans. Ultrason. Ferroelectr. Freq. Control [**34**]{}, 647 (1987).
Y. Kedem and L. Vaidman, Phys. Rev. Lett. [**105**]{}, 230401 (2010).
|
---
author:
- ZEUS Collaboration
title: |
Measurement of high-${Q^2}$ charged current\
cross sections in $\mathbf{e^-p}$ deep inelastic\
scattering at HERA
---
|
---
abstract: 'We investigate the transmission of electrons through a quantum point contact by using a quasi-one-dimensional model with a local bound state below the band bottom. While the complete transmission in lower channels gives rise to plateaus of conductance at multiples of $2e^{2}/h$, the electrons in the lowest channel are scattered by the local bound state when it is singly occupied. This scattering produces a wide zero-transmittance (anti-resonance) for a singlet formed by tunneling and local electrons, and has no effect on triplets, leading to an exact $0.75(2e^{2}/h)$ shoulder prior to the first $2e^{2}/h$ plateau. Formation of a Kondo singlet from electrons in the Fermi sea screens the local moment and reduces the effects of anti-resonance, complementing the shoulder from 0.75 to 1 at low temperatures.'
author:
- Ye Xiong
- 'X. C. Xie'
- 'Shi-Jie Xiong'
title: 'Anti-Resonance and the 0.7 Anomaly in Conductance through a Quantum Point Contact '
---
The quantization of conductance in mesoscopic systems has been observed in quantum point contacts (QPC) where a series of plateaus at multiples of $%
2e^{2}/h$ appears in curves of conductance $G$ versus gate voltage $V_{g}$ [@1; @2]. Prior to the first integer plateau, a shoulder of 0.7($2e^{2}/h$) has often been observed and is called the 0.7 anomaly" [@3; @4; @5; @6]. Considerable experimental and theoretical efforts have been devoted to understanding this anomaly [@7; @8; @9; @a1; @a2]. In Ref. it is suggested that the anomaly originates from the 3:1 triplet-singlet statistical weight ratio if two electrons form bound states by some attractive interaction and the triplet has a lower energy. From calculations for two electrons in the Hubbard model or the Anderson Hamiltonian, two resonance peaks in transmission spectrum are shown, corresponding to singlet and triplet states with weights 0.25 and 0.75 [a2]{}. Some attention has been focused on the Kondo effect in such systems [@10; @11]. In a recent experiment the temperature and magnetic-field dependence of the differential conductance was investigated in detail to highlight the connection to the Kondo problem[@12], and subsequently a theoretical study was carried out in further support of Kondo physics in QPC [@13].
Since relevant states undergo empty, single and double occupations even within the width of one plateau or shoulder, including global charge variations and fluctuations in calculations is important. The “0.7 anomaly” appears only prior to the first integer plateau, and only at relatively higher temperatures, for which the intensity of the Kondo effect is negligible. The shape of the shoulder is much different from the two resonance peaks of the singlet and triplet states [@a2], and there is no evidence of the attractive interaction from which two-electron bound states can be formed [@a1]. To address the above mentioned questions, in this Letter we theoretically study the transport through a quantum point contact(QPC) by using a model which includes the Coulomb interaction, the charge fluctuations, and the multi-channel structure on an equal footing. By using the singlet-triplet representation to label spin states of the tunneling electron and the local electron, we show a wide anti-resonance for the singlet channel near the band bottom, giving rise to the 0.75 shoulder. The shoulder is complemented to $1$ by the formation of a Kondo singlet at low temperatures, and manifests itself at higher temperatures when the Kondo singlet collapses. Thus, the role of the Kondo singlet is to suppress the anti-resonance, quite different from the Kondo physics discussed in Ref. [@13]. A simple scaling curve for conductance is obtained and is found to compare well with the experimental one. The results provide consistent explanations for a wide range of characteristics observed in experiments.
A QPC can be described with a narrow and short bar connected to the left and right leads which serve as reservoirs. Thus, one obtains several continuous 1D subbands. Some local levels, may be virtual bound states[@13], can be created by the specific QPC geometry. These states are isolated from the leads and should not be included in the band continuum. The potential of the bar area, including both the band continuum and the local levels, is tuned by the gate voltage. The single-electron energies of subbands and levels are shown in Fig. 1(a). We include on-level Coulomb repulsion $U$ of electrons confined in one local state. In equilibrium the Fermi energy of reservoirs is fixed and we set it as the energy zero. The occupations of levels is controlled by tuning the gate voltage. In Figs. 1(b), 1(c) and 1(d) we show the empty, single and double occupations of the level nearest the band bottom. When the Fermi level crosses the band bottom, the first subband contributes to the conductance and the first plateau appears. At this moment most of the local levels are fully occupied and have no effect on the transport, except the one closest to the band bottom which may be singly occupied due to the on-level interaction. It is sufficient to include this level in our model. By tuning $V_{g}$ further this level is also fully occupied and the plateau structure is determined only by the subbands.
![(a) Sketchy illustration of dispersion relation of 1D subbands and local levels. $\protect\delta_0$ is energy spacing between subbands, and $\protect\epsilon_0$ is the position of the local level. (b) Empty level. $%
E_F$ denotes the Fermi level. (c) Singly occupied level. (d) Doubly occupied level. []{data-label="FIG.1"}](yekondofig1.eps){width="6.8cm"}
The Hamiltonian of the system can be written as $$H=\sum_{m,i,\sigma }t_{0}(a_{m,i,\sigma }^{\dag }a_{m,i+1,\sigma }+\text{H.c.%
})$$$$+\sum_{m,i,\sigma }[(m-1)\delta _{0}-eV_{g}]a_{m,i,\sigma }^{\dag
}a_{m,i,\sigma }$$$$+\sum_{\sigma }t^{\prime }(a_{1,0,\sigma }^{\dag }d_{\sigma }+\text{H.c.}%
)+\sum_{\sigma }(\epsilon _{0}-eV_{g})d_{\sigma }^{\dag }d_{\sigma }$$$$+Ud_{\uparrow }^{\dag }d_{\uparrow }d_{\downarrow }^{\dag }d_{\downarrow },$$where $a_{m,i,\sigma }$ and $d_{\sigma }$ are the annihilation operators of electrons on the $i$th site of the $m$th continuum channel and at the local state, respectively, with $\sigma $ being the spin index. $\delta _{0}$ is the energy spacing between subband bottoms. $\epsilon _{0}$ stands for the position of the local level. $t_{0}$ is the hopping integral of the channels, and $t^{\prime }$ is the coupling of the lowest channel and the bound state at a site ($i=0$). Only the coupling of the local level to the lowest channel ($m=1$) is considered, because local level coupling has no effect on the transmission in higher channels due to the double occupation.
For $V_{g}\ll 0$ and at the zero temperature, the potential is too high and the states in the bar, including the local state and the continuum chains, are empty. With increasing $V_{g}$, the local state becomes singly occupied in the range of $\epsilon _{0}<eV_{g}<\epsilon _{0}+U$. For $eV_{g}>\epsilon
_{0}+U$, it is doubly occupied and has no effect on the conductance. On the other hand, the $m$th continuum channel gives $2e^{2}/h$ contribution to the conductance if $eV_{g}>(m-1)\delta _{0}-2t_{0}$, because in this case the Fermi level is higher than the bottom of this subband. This provides the integer plateaus in the $G$-$V_{g}$ curves with plateau width $\delta _{0}/e$.
Since the local level is below the bottom of the whole band continuum, the tunneling in the first channel is affected by its charge and spin states. There are 4 states of the local level: the empty, $|\phi _{1}\rangle
=|0\rangle $, the spin up and down single occupation, $|\phi _{2}\rangle
=|\uparrow \rangle $ and $|\phi _{3}\rangle =|\downarrow \rangle $, and the double occupation, $|\phi _{4}\rangle =|\uparrow \downarrow \rangle $. If an electron is injected into the channel, it will be scattered by the state on the local level. At first we consider only the tunneling and local electrons and ignore the other electrons in the Fermi sea. These considered electrons form a many-body state $$|\psi \rangle =\sum_{n=1}^{4}\sum_{i,\sigma }p_{n;1,i,\sigma }|\phi
_{n}\rangle \bigotimes |1,i,\sigma \rangle ,$$where $|1,i,\sigma \rangle $ is the orbital at site $i$ with spin $\sigma $ in the first channel, and $p_{n;1,i,\sigma }$ is the corresponding coefficient. By applying the Hamiltonian on $|\psi \rangle $ one obtains the Schrödinger equations for the coefficients. These equations can be expressed with an equivalent single-particle network in which every site represents a combination of indices $(n;1,i,\sigma )$ of the coefficients [@14]. In the present case the network is an 8-channel one where every channel stands for a combination of $n$ and $\sigma $, and a site in a channel corresponds to a coordinate $i$ in the chain. In the network the channels with $(n=2,\sigma =\uparrow )$ and $(n=3,\sigma =\downarrow )$, and the channels with $n=1,4$, corresponding to the empty and doubly occupied local states, are independent. The other 2 channels are connected at site $%
i=0$ but can be easily decoupled with the transformation $|S,i\rangle =\frac{%
1}{\sqrt{2}}(|\phi _{2}\rangle \bigotimes |1,i,\downarrow \rangle -|\phi
_{3}\rangle \bigotimes |1,i,\uparrow \rangle )$ and $|T,i\rangle =\frac{1}{%
\sqrt{2}}(|\phi _{2}\rangle \bigotimes |1,i,\downarrow \rangle +|\phi
_{3}\rangle \bigotimes |1,i,\uparrow \rangle )$. The final network with 8 independent channels is shown in Fig. 2. For single occupation there is one singlet channel with one scatterer (6) and three pure triplet channels (3, 4, and 5) without scattering by the local level. Different from Ref. [a1]{}, the singlet-triplet notation used here is merely to label spin states of the tunneling and local electrons that are not bound together. For $m>1$ and for the non-scattering pure channels of $m=1$, the transmission coefficient is $$\tau _{m}(\epsilon )=\left\{
\begin{array}{l}
1,\text{ for }|\epsilon -(m-1)\delta _{0}+eV_{g}|<2t_{0}, \\
0\text{ otherwise. }%
\end{array}%
\right. , \label{trans}$$where $\epsilon $ is energy of the injected electron. For the singlet (empty) channel of $m=1$, $$\tau _{\text{s(e)}}=\frac{4[u_{\text{s(e)}}-2\cos (k)]^{2}\sin ^{2}(k)}{v_{%
\text{s(e)}}^{4}+4[u_{\text{s(e)}}-2\cos (k)]^{2}\sin ^{2}(k)} \label{tran}$$for $|\epsilon +eV_{g}|<2t_{0}$ and $\tau _{\text{s(e)}}=0$ otherwise, where $k$ is the momentum of the tunneling electron determined by $\epsilon
=2t_{0}\cos (k)-eV_{g}$, and $u_{\text{s}}=(\epsilon _{0}+U)/t_{0}$, $u_{%
\text{e}}=\epsilon _{0}/t_{0}$, $v_{\text{s}}=\sqrt{2}t^{\prime }/t_{0}$, $%
v_{\text{e}}=t^{\prime }/t_{0}$.
![Eight independent channels for a tunneling electron. The state of the local level is illustrated by circles, and the spin of the tunneling electron is represented by an arrow outside the circle. []{data-label="FIG.2"}](yekondofig2.eps){width="6.8cm"}
For the singlet channel the scatterer produces an anti-resonance at $%
\epsilon =\epsilon _{0}-eV_{g}+U$ with a zero transmission coefficient at the minimum and with semi-width ${t^{\prime }{}^{2}}/t_{0}|\sin (k)|$. Near the subband bottom where we focus, $\sin (k)\sim 0$, so the width of the dip is extremely large even though $t^{\prime }$ may be much smaller than $t_{0}$. As a result, the dip of $\tau _{\text{s}}$ develops to a flat plateau with nearly zero height. This is different from the results of Ref. [@a2] where the singlet and triplet states give two separate resonance peaks with different weights. Without the magnetic field and ignoring the effects of other electrons, in the range of single occupation comma the conductance is governed by $e^{2}/2h$ times the sum of the transmission coefficients of the singlet and triplet channels which are non-zero only for the tunneling electron with energy higher than the subband bottom. Here the prefactor $1/2$ stands for the weight of one spin state of the local electron. As mentioned above, $\epsilon _{0}\lesssim -2t_{0}$, so $\epsilon _{0}+U$ may be above the subband bottom. Thus, in the range of $-2t_{0}<eV_{g}<\epsilon _{0}+U$ conductance plateaus at height $0.75(2e^{2}/h)$. For $eV_{g}>\epsilon _{0}+U$, the conductance is dominated by the double-occupation channels (7 and 8 in Fig. 2), leading to a jump from 0.75 to 1 at $eV_{g}=\epsilon _{0}+U$. This is the origin of the 0.7 anomaly within our model.
![Linear conductance as a function of the gate voltage. The parameters are: $\protect\delta_0=0.02t_0$, $U=0.016t_0$, $\protect\epsilon%
_0=-2.008t_0$, $T_K = 0.0005t_0 $ and $t^{\prime}=0.05t_0$. Upper inset: Differential conductance as a function of bias voltage for $eV_g = -1.993t_0$. Lower inset: Linear conductance as a function of scaled temperature $T/T_K$. Solid line: The calculated conductance. Dashed line: The experimentally fitted conductance.[]{data-label="FIG.3"}](yekondofig3.eps){width="8.5cm" height="7cm"}
Now we consider the effect of the electrons in the Fermi sea. It is known that at low temperatures the electrons from Fermi sea are coupled with the local electron to form a Kondo singlet. In this case the local state is screened by the Fermi electrons and is no longer available for the injected electron in the singlet channel, resulting in the disappearance of the side-coupled scatterer for this channel. This removes the anti-resonance and enhances the transmission coefficient from nearly zero to one. If the probability of annihilating the Kondo singlet is $p_k$, the average transmission coefficient of the singlet channel is $\bar{\tau_{\text{s}}} =
(1-p_k) \tau_1 +p_k \tau_{\text{s}} $. We choose $p_k=tanh(-T/T_{K})^{2} $. This simple form reflects the physics that at low temperature the Kondo singlet peak in the spectral density goes as $1-\alpha T^2$ [@Costi]. The formation of the Kondo singlet has no effect for the triplet channels since they are without scattering.
Combining all the contributions, the linear conductance at temperature $T$ can be calculated as $$\label{cond}
G(T) = -\frac{2e^2}{h} \int d \epsilon \frac{\partial f (\epsilon, T) }{%
\partial \epsilon} \tau(\epsilon),$$ $$\tau (\epsilon) = P_0(T) \tau_{\text{e}}(\epsilon) +P_2(T) \tau_1(\epsilon)$$ $$+\frac{P_1(T)}{4} [\bar{\tau}_{\text{s}}(\epsilon)+3\tau_1(\epsilon)]+
\sum_{ m \geq 2 } \tau_m(\epsilon) ,$$ where $f(\epsilon,T)$ is the Fermi-Dirac distribution, $P_l(T)$ is the thermal probability of finding the local level occupied by $l$ electrons, $$P_{1}(T) = \frac{2}{Z} e ^{-\frac{\epsilon_0-eV_g}{k_B T} },\, P_{2}(T) =
\frac{1}{Z} e^{-\frac{2\epsilon_0-2eV_g+U}{k_BT}},$$ $$P_{0}(T)= \frac{1}{Z}, \, Z= 1 +2 e^{-\frac{\epsilon_0-eV_g}{k_BT} } + e^{ -%
\frac{2\epsilon_0-2eV_g+U}{k_BT}} .$$
In Fig. 3 we plot the conductance as a function of the gate voltage for several values of $T$. The local level $\epsilon _{0}$ is below and close to the band bottom, $\epsilon _{0}+U$ is above it, and $t^{\prime }\ll t_{0}$, reflecting the localized nature of the level. The value of $t_{0}/\delta _{0}
$ corresponds to the number of channels. A $0.75$ shoulder is clearly seen at high temperatures $T>T_{K}$, meanwhile it is complemented to 1 at low temperatures $T<T_{K}$. This accounts for the basic features observed in the recent experiment [@12].
At low temperatures $P_{l}$ and $f(\epsilon )$ can be approximated with step functions and for $-2t_{0}<eV_{g}<\epsilon _{0}+U$, Eq. (\[cond\]) becomes $$G(T)=\frac{2e^{2}}{h}\left( \frac{3}{4}+\frac{1}{4}F(\widetilde{T})\right) ,
\label{scal}$$where $F(\widetilde{T})$ is a universal function of rescaled temperature $%
\widetilde{T}=T/T_{K}$, describing the influence of the Kondo effect. Here, the range of $F$ is $[0,1]$, the prefactor of $F$ is $1/4$ and the constant is $3/4$, reflecting the fact that the formation of the Kondo singlet influences only the contribution from the singlet channel. In Ref. [@12], the same functional form as Eq.(7) is used for $G(T)$, however, the corresponding prefactor and constant are both $1/2$. But the range in which the measured points can be fitted well by a universal function is only from $%
1/2$ to $1$, as can be seen from Fig. 2(b) in Ref. [@12]. This implies that the experimental data only confirm the $1/4$ of the conductance that is influenced by the Kondo effect. The other $1/4$ may also vary with temperature but perhaps due to other physical influences. In Ref. [@17], from the data of quantum dots the prefactor is $1$ and the constant is $0$, implying the full Kondo effect for the tunneling. This may indicate the consequence of different structures between a QPC and a quantum dot: the Kondo impurity is an extra local level in QPC as studied in this work while it may be embedded in the tunneling path in quantum dots. It is known that there is no anti-resonance associated with the latter case [@XRWang]. It is easy to see that at low $T$ $F(\widetilde{T})\sim
1-p_{k}(T)=1-tanh(-T/T_{K})^{2}$. In the lower inset of Fig. 3 we show (solid line) the scaling conductance of Eq.(\[scal\]). For a comparison, the experimentally obtained scaling conductance (dashed line) is also shown. The experimental conductance is obtained from Eq.(1) and Eq.(2) of Ref. [12]{}. The experimental $T_{K}$ is twice the value of the theoretical one.
By applying a bias voltage $V_b$ between two leads the current is $I=\frac{2e%
}{h}\int d\epsilon [f(\epsilon-eV_b/2)-f(\epsilon+eV_b/2)]\tau (\epsilon)$, function $F(\widetilde{T})$ in Eq.(\[scal\]) becomes $F(\widetilde{T}) \exp%
[-(eV_b/w)^2]$, reflecting the width $w$ ($\sim k_BT_K$) of the Kondo peak in the spectral density [@Meir2]. We calculate the differential conductance $g=\frac{dI}{dV_b}$, shown in the upper inset of Fig. 3. The zero-bias peak, originated from the Fermi-level dependence of the Kondo effect, exists only at low temperatures for which the 0.7 shoulder is complemented to 1. By applying a magnetic field, the degeneracy of spins is lifted, and the spin of the local level is along the field direction. As a result, the singlet-triplet representation is no longer suitable and all the plateaus become multiples of $0.5 (2e^2/h)$ due to the splitting of spin subbands.
In summary, we demonstrate that a combination of anti-resonance of the singlet channel and Kondo physics provides a satisfying account of basic features associated with the “0.7 anomaly” in quantum point contacts.
We thank Junren Shi for helpful conversations. This work is supported by DOE Grant No. DE/FG02-04ER46124, NSF-EPSCoR, and NSF-China.
B.J. van Wees [*et al.*]{}, Phys. Rev. Lett. [**60**]{}, 848 (1988).
D.A. Wharam [*et al.*]{}, J. Phys. C [**21**]{}, L209 (1988).
K.J. Thomas [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 135 (1996).
K.J. Thomas [*et al.*]{}, Phys. Rev. B [**58**]{}, 4846 (1998).
A. Kristensen [*et al.*]{}, Physica (Amsterdam) [**249B**]{} - [**251B**]{}, 180 (1998).
K.S. Pyshkin [*et al.*]{}, Phys. Rev. B [**62**]{}, 15842 (2000).
D.J. Reilly [*et al.*]{}, Phys. Rev. B [**63**]{}, 121 311 (2001).
K. Hashimoto [*et al.*]{}, Jpn. J. Appl. Phys. [**40**]{}, 3000 (2001).
B. Spivak and F. Zhou, Phys. Rev. B [**61**]{}, 16730 (2000).
V.V. Flambaum and M. Yu. Kuchiev, Phys. Rev. B [**61**]{}, R7869 (2000).
T. Rejec, A. Ramšak and J.H. Jefferson, Phys. Rev. B [**62**]{}, 12985 (2000); [*ibid.*]{} [**67**]{}, 075311 (2003).
C.K. Wang and K.F. Berggren, Phys. Rev. B [**57**]{}, 4552 (1998).
H. Bruus, V.V. Cheianov, and K. Flensberg, Physica (Amsterdam) [**10E**]{}, 97 (2001).
S.M. Cronenwett [*et al.*]{}, Phys. Rev. Lett. [**88**]{}, 226805 (2002).
Y. Meir, K. Hirose, and N.S. Wingreen, Phys. Rev. Lett. [**89**]{}, 196802 (2002); [*ibid.*]{} [**90**]{}, 026804 (2003).
Shi-Jie Xiong and Ye Xiong, Phys. Rev. Lett. [**83**]{}, 1407 (1999).
D. Goldhaber-Gordon [*et al.*]{}, Phys. Rev. Lett. [**81**]{}, 5225 (1998).
T.A. Costi, Phys. Rev. Lett. [**85**]{}, 1504 (2000).
X.R. Wang, Yupeng Wang, and Z.Z. Sun, Phys. Rev. [**B65**]{}, 193402 (2002).
T.K. Ng and P.A. Lee, Phys. Rev. Lett. [**61**]{}, 1768 (1988); L.I. Glazman and M.E. Raikh, JETP Lett. [**47**]{}, 452 (1988); Y. Meir, N.S. Wingreen, P.A. Lee, Phys. Rev. Lett. [**70**]{}, 2601 (1993).
|
---
abstract: 'We have studied the ground state of the Gross-Pitaevskii equation (nonlinear Schrodinger equation) for a Morse potential via a variational approach. It is seen that the ground state ceases to be bound when the coupling constant of the nonlinear term reaches a critical value. The disappearence of the ground state resembles a saddle node bifurcation.'
address:
- 'Department of Theoretical Physics, S.N.Bose National Centre For Basic Sciences, JD-Block, Sector-III, Salt Lake City, Kolkata-700098, India'
- 'Harish-Chandra Research Institute, Chhatnag road, Jhunsi, Allahabad-211019, India'
author:
- Sukla Pal
- 'J. K. Bhattacharjee'
title: Ground state of the one dimensional Gross Pitaevskii equation with a Morse potential
---
Gross-Pitaevskii Equation (GPE),Trapped Bose Einstein Condensate,Morse Potential,Localisation,Bifurcation
Introduction {#1}
============
The nonlinear Schrodinger equation (NLSE) has been a particularly active field of research in condensed matter physics in the last two decades because of its close relationship to the Gross Pitaevskii equation (GPE)[@a; @b] which describes the dynamics of the condensate in the process of Bose Einstein Condensation (BEC)[@7; @11]. In the quasi–1D regime, GPE reduces to the one-dimensional NLSE with an external potential. This regime holds when the transverse dimensions of the condensate are of the order of its healing length and the longitudinal dimension is much longer than its transverse dimensions. In this regime the BEC remains phase coherent and the governing equations are one dimensional. This is in contrast to a truly 1D mean-field theory which requires transverse dimensions of the order of or less than the atomic interaction length. In this paper we will deal with 1D GPE and hence with NLSE with Morse type of external potential in x-direction.
Traditionally the evolution equation of a complex variable $\psi(x,t)$ has been called NLSE when it has the structure $i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi+g|\psi|^2\psi$. It becomes the GPE when it has a potential (trapping potential in the case of BEC) V(x) on the right hand side. The most common trapping potential is the simple harmonic oscillator $V(x)=\frac{1}{2}m\omega^2x^2$ which gives a localised ground state. For $V(x)=0$, one possible set of solutions of the NLSE is $\psi(x,t)=\sqrt{\frac{N}{L}}e^{-\frac{iE_kt}{\hbar}}e^{ikx}$, which is the usual free particle plane wave solution and gives the energies $E_k=\frac{\hbar^2k^2}{2m}+\frac{gN}{L}$. This energy spectrum has a gap which violates the Hugenholtz Pines theorem for the interacting Bose gas (repulsive interaction). This motivated us to look for a potential $V(x)$ which can be tuned from a harmonic oscillator potential to a free particle one and study the evolution of the ground state with the evolution of the potential. The potential concerned is the Morse potential $V(x)=D(e^{-2ax}-2e^{-ax})$; where the constant $D$ carries the dimension of energy and ‘$a$’ carries the dimension of inverse length. If $a\rightarrow 0$, then the potential tends to the constant value $-D$ which is effectively a free particle and for $a\rightarrow\infty$, we have a strongly confining potential. So we can tune the potential by tunning ‘$a$’ and accordingly we will see a transition in the nature of the ground state of the particle.
Many applications of the NLSE to BEC’s or GPE have dealt with the ground-state properties as well as various intriguing features of BEC in harmonic oscillator potential [@12], double well potential [@c; @c1], anisotropic potential [@15], optical lattice [@16; @c2], etc and has been successfully extended to trapped dipolar BEC [@13; @14]. There is also growing interest in the possibility of generating topological excitations of a condensate, which may well be described by excited-state solutions of the NLSE [@d]. However, in this paper we will concentrate on the ground state properties of GPE in well known Morse potential. We are already familiar with the existence of localized ground state of GPE in presence of harmonic trap and after switching off the trapping potential the rapid expansion of BEC leading no longer to a localized state [@9].
We present the results of a variational calculation in sec 2 and end with some observation in sec 3.
Variational Calculation
=======================
The one dimensional GPE describing the dilute bosons with repulsive interaction under a trapping potential $V_{ext}$ is given by $$\begin{aligned}
\label{eq1}
i\hbar\partial_t\psi =-\frac{\hbar^2}{2m}\nabla^2\psi+g|\psi|^2\psi+V_{ext}(x)\psi\end{aligned}$$ with $$\begin{aligned}
\label{eq2}
\int_{-\infty}^{\infty}|\psi(x)|^2 dx=N\end{aligned}$$ N is the total number of particles trapped. Eq. (\[eq1\]) can be written as $i\hbar\frac{\partial\psi}{\partial t}=\frac{\delta E[\psi]}{\delta \psi}$. Where $E[\psi]$ is the energy functional: $$\begin{aligned}
\label{eq3}
E[\psi(x)]=\int^{\infty}_{-\infty}[\frac{\hbar^2}{2m}|\nabla\psi|^2+V_{ext}|\psi|^2+\frac{g}{2}|\psi|^4]dx\end{aligned}$$ We will write Morse potential of sec I in the form:
$$\begin{aligned}
\label{eq4}
V(bx)=D(e^{\frac{-2bx}{k}}-2e^{\frac{-bx}{k}})\end{aligned}$$
Where, ‘D’ is the well depth having the dimension of energy and ‘b’ is an inverse confining length $b=\sqrt{\frac{8mD}{\hbar^2}}$, related to ‘$a$’ by $b=ak$, where, $k$ is a dimensionless number. Hereafter we will keep $D$ fixed and vary the confining length by varying the dimensionless number $k$. We have considered a variable transfer of the form $y=ke^{-ax}$ where $k^2=\frac{8mD}{\hbar^2 a^2}$. We fix $D$ at 1.
![Figure above shows the plot of $V(bx)$ vs $bx$ for different values of $k$. This clearly shows that with the change of $k$, the nature of the potential changes.[]{data-label="fig1"}](morse-pot-x.eps)
For large values of $k$ system becomes essentially free though strictly bounded in the left ($x<0$) at a large distance. As the value of $k$ increases the change in the shape of the potential given in Fig. \[fig1\] suggests that the system falls under more weaker trap.\
We try a variational form for the ground state as: $$\begin{aligned}
\label{eq5}
\psi(y)=N_cy^{\alpha}e^{-\beta y}\end{aligned}$$ After substituting Eq. (\[eq5\]) into the energy functional $E[\psi]$, Eq. (\[eq3\]) reduces to $$\begin{aligned}
\label{eq6}
E[\psi(y)]&=&Na^2\Big(\frac{\alpha}{2}+\frac{\alpha^2}{4\beta^2}+\frac{\alpha}{8\beta^2}-\frac{\alpha k}{2\beta}\Big)\nonumber\\&~&+\frac{mgaN^2}{2^{4\alpha}\hbar^2}\frac{\Gamma(4\alpha)}{\Gamma(2\alpha)^2}\end{aligned}$$ We consider separately two cases.
$g=0$
-----
Bosons are non interacting in this case and hence Eq. (\[eq6\]) takes the simple form given below: $$\begin{aligned}
\label{eq7}
E[\psi(y)]_{g=0}= Na^2\Big(\frac{\alpha}{2}+\frac{\alpha^2}{4\beta^2}+\frac{\alpha}{8\beta^2}-\frac{\alpha k}{2\beta}\Big)\end{aligned}$$ differentiation with respect to the parameters gives following two constraints $$\begin{aligned}
1+\frac{\alpha}{\beta^2}+\frac{1}{4\beta^2}=\frac{k}{\beta}\label{eq8}\\
\alpha+\frac{1}{2}=\beta k\label{eq9}\end{aligned}$$ With the help of Eq. (\[eq8\]) and Eq. (\[eq9\]), $\beta$ comes out to be equal to 0.5 and hence independent of $k$. If we substitute this value of $\beta$ in Eq. (\[eq8\]), we will readily find $\alpha=\frac{k-1}{2}$. Hence the ground state wave function becomes $$\begin{aligned}
\label{eq10}
\psi_{g=0}(y)|_{\tiny{variation}}=[\frac{aN}{\Gamma (k-1)}]^{1/2}y^{\frac{k-1}{2}}e^{-y/2}\end{aligned}$$ with ground state energy $$\begin{aligned}
\label{eq11}
E_{g=0}|_{variation}=-\frac{N\hbar^2 a^2}{2m}\big(\frac{1}{2}-\frac{\sqrt{2mD}}{a\hbar})^2\end{aligned}$$ Eq. (\[eq10\]) and Eq. (\[eq11\]) with $N=1$ gives the exact ground state energy as obtained by solving (by applying Laplace and inverse Laplace transformation) Schrodinger Equation for a single particle in Morse potential [@5].
$g\ne0$
-------
In this case bosons have repulsive interaction. Differentiation with respect to parameter gives the following two constraint conditions in this case. $$\begin{aligned}
\label{eq12}
\alpha+\frac{1}{2}&=&\beta k\end{aligned}$$ $$\begin{aligned}
\begin{aligned}\label{eq13}
1-\frac{1}{4\beta^2}&=& \frac{kg'}{\sqrt{2}}\frac{\Gamma(4\alpha)}{2^{4\alpha}[\Gamma(2\alpha)]^2}\Big[2\Psi(2\alpha)+4ln2\\ &~&-\Psi(4\alpha)\Big]
\end{aligned}\end{aligned}$$ Where, $g'=\frac{\sqrt{m}gN}{\hbar\sqrt{D}}$ and $g=\frac{2\hbar^2}{ma_s}$ [@6], $a_s$ being the s wave scattering length and $g'$ being the dimensionless quantity. $\Psi$ is the digamma function defined as $\Psi(x)=\frac{d}{dx}ln[\Gamma(x)]$. To solve Eq. (\[eq12\]) and (\[eq13\]) we proceed in the following way. At first the expression of $\beta$ from Eq. (\[eq12\]) is substituted into Eq. (\[eq13\]) $$\begin{aligned}
1-\frac{k^2}{(2\alpha+1)^2}&=&\frac{kg'}{\sqrt{2}}C(\alpha)\Big[2\Psi(2\alpha)+4ln2-\Psi(4\alpha)\Big]\nonumber\\
\Rightarrow f_1(\alpha)&=&g'f_2(\alpha)\label{eq14}\end{aligned}$$ Where, $C(\alpha)=\frac{\Gamma(4\alpha)}{2^{4\alpha}[\Gamma(2\alpha)]^2}$, $f_1(\alpha)$=$1-\frac{k^2}{(2\alpha+1)^2}$ and $f_2(\alpha)=\frac{k}{\sqrt{2}}C(\alpha)\Big[2\Psi(2\alpha)+4ln2-\Psi(4\alpha)\Big]$ with energy functional given below: $$\begin{aligned}
\begin{aligned}\label{eq15}
\frac{E[\psi(y)]}{ND}&=&\frac{4}{k^2}\Big(\frac{\alpha}{2}+\frac{\alpha^2}{4\beta^2}+\frac{\alpha}{8\beta^2}-\frac{\alpha k}{2\beta}\Big)\\&~&+C(\alpha)\frac{kg'}{2\sqrt{2}}
\end{aligned}\end{aligned}$$
All the energy values obtained from Eq. (\[eq15\]) will be in scale of $ND$. Now we will plot both $f_1$ and $f_2$ as a function of $\alpha$ for constant values of $k$ and $g'$ and we will search for the point of intersection.
Observations
============
Here we consider fixed value of $k$ at first and then we analyse the case with different values of $k$, i.e., by tunning the shape of the trapping potential.
$k$ is fixed at a constant value: $(k=3.0)$
--------------------------------------------
For $g'$ close to 0, there is a pair of solutions for $\alpha$. Keeping $k$ fixed at 3.0 as $g'$ is increased, there is a critical value of $g'$ $(= g'_c)$ at which there is only one solution of $\alpha$. Beyond $g'_c$ there is no solution for any value of $g'$. Hence the solutions bifurcate at $g=g_c$.
![In this figure we have plotted $f_1(\alpha)$ and $f_2(\alpha)$ vs. $\alpha$ for $k=3.0$ and $g'=0.1 $. It shows that we have two possible values of $\alpha$ ($\alpha_1=1.2 $and $\alpha_2=6.2 $). This qualitative nature (i.e, two sets of solution for $\alpha$ and $\beta$) of the above figure remains same up to $g'<g'_c$[]{data-label="fig2"}](g-k31-new.eps)
From this two values of $\alpha$ ($\alpha_1= 1.2$, $\alpha_2= 6.2$) as shown in Fig. \[fig2\], $\beta_1$= 0.56 and $\beta_2= 2.23$ are obtained from Eq. (\[eq12\]). Then with the help of Eq. (\[eq15\]), the energy of the corresponding ground states are calculated as $E_1|_{g'=0.1}=-0.418$ corresponding to $\alpha_1$ and $\beta_1$ and $E_2|_{g'=0.1}=0.463$ corresponding to $\alpha_2$ and $\beta_2$ i.e, there is one positive energy solution and one negative energy solution. We proceed to analyze the energy profile at ($\alpha_1$,$\beta_1$) and ($\alpha_2$,$\beta_2$) in the parameter space of $\alpha$ and $\beta$. We find that at ($\alpha_1$,$\beta_1$): $$\begin{aligned}
\label{eq16}
\begin{aligned}
\frac{\partial^2 E}{\partial\beta^2}=10.61;{~~}
\frac{\partial^2 E}{\partial\alpha^2}=1.66;{~~}
\frac{\partial^2 E}{\partial\alpha\partial\beta}=1.1
\end{aligned}\end{aligned}$$ and at ($\alpha_2$,$\beta_2$) $$\begin{aligned}
\label{eq17}
\begin{aligned}
\frac{\partial^2 E}{\partial\beta^2}=0.84;{~~}
\frac{\partial^2 E}{\partial\alpha^2}=-437.75;{~~}
\frac{\partial^2 E}{\partial\alpha\partial\beta}=-0.22
\end{aligned}\end{aligned}$$ Eq. (\[eq16\]) and Eq. (\[eq17\]) suggest that ($\alpha_1$,$\beta_1$) is the point of stable minima and ($\alpha_2$,$\beta_2$) is a saddle point in parameter space. This leads to the conclusion that the negative energy solution corresponding to ($\alpha_1$,$\beta_1$) is the only bound state energy. In Fig. \[fig3\] we have plotted the energy profile in $\alpha$,$\beta$ parameter space.
![In this figure we have plotted the energy profile ($E$ in the Z-axis is in the scale of $ND$) given in Eq. (\[eq15\]) for $k=3$ and $g'=0.1$ in the $\alpha$,$\beta$ parameter space. This clearly signifies that ($\alpha_1$,$\beta_1$) is a global minima.[]{data-label="fig3"}](energy.eps)
![This figure shows the plot of $f_1(\alpha)$ and $f_2(\alpha)$ vs. $\alpha$ for $k=3.0$ and $g'=0.17 =g'_c $. The single point of intersection at $\alpha= 2$ indicates the merge of two sets of solutions at $g'_c$ and appearence of 1 set of solution only for $\alpha$ and $\beta$. []{data-label="fig4"}](g-k32-new.eps)
The solution of $\alpha$ corresponds to the ground state energy $E|_{g'=g_c}=-0.31$ where $E|_{g_c}<E_1|_{g_{c-}}$ considering the magnitude only. This clearly signifies that with the increase of the value of $g'$ the localized nature of the system diminishes and the free particle character comes into play.
If we increase $g'$ keeping $k$ fixed at 3.0, then after a certain value of $g'=g'_{c-}$, both $E_1$ and $E_2$ become negative. Further increase of $g'$, decreases the magnitude of $E_1$ and increases that of $E_2$ as illustrated in Table 1 and finally when $g'=g_c$, $f_2(\alpha)$ becomes tangent to $f_1(\alpha)$ as depicted in Fig. \[fig4\]. Further increasing $g'$, there will be no solution for $\alpha>0$ (only $\alpha>0$ is physically relevant) which will be clear from Fig. \[fig5\].\
---------------- -------- --------
$g'$ $E_1$ $E_2$
\[0.5ex\] 0.10 -0.418 0.463
0.12 -0.407 0.168
0.14 -0.395 -0.029
0.155 -0.37 -0.177
0.17 -0.31 -0.31
\[1ex\]
---------------- -------- --------
: Results showing how the energy values changes with $g'$. $k$ is keeping fixed at 3.0. After $g'=0.12=g'_{c-}$ both $E_1$ and $E_2$ becomes negative and at $g'=g'_c=0.17$ only one bound state exist, $f_1$ and $f_2$ being tangent to each other and at $g'>g'_c$ no bound state exists at all.
![Above figure shows the plot of $f_1(\alpha)$ and $f_2(\alpha)$ vs. $\alpha$ for $k=3.0$ and $g'= 1.0$. It shows that we have no solution for $\alpha$ at $g'>g'_c$. []{data-label="fig5"}](g-k33-new.eps)
Different values of $k$ is considered
-------------------------------------
So far we have explained the case keeping $k$ fixed at 3.0 and now we will consider different values of $k$. The qualitative nature of the solutions remain the same as $k=3.0$ but the values of $g_c'$ and the ground state energy at $E|_{g_c'}$ changes with $k$ as is obvious from Table 2.
[c c c ]{} $k$ & $g'_c$ & $E_{g'_c}$\
\[0.5ex\] 2 & 0.445 & -0.048\
3 & 0.170 & -0.310\
4 & 0.095 & -0.459\
5 & 0.061 & -0.546\
\[1ex\]
From table 2, it can readily be said that with the increase of the value of $k$, critical value of $g$ decreases but $|E_{g_c'}|$ increases. $g_c'$, $|E_{g_c'}|$ approaches to the following limit: $g_c'\rightarrow 0$ and $E_{g_c'}\rightarrow-1$ as $k\rightarrow\infty$. Hence free particle nature of the bosons predominates more rapidly. The decrease of the values of ${g_c'}$ with the increase of $k$ implies that very small repulsive interatomic interactions at this point is sufficient to destroy the localized state.\
From Eq. (\[eq8\]) and Eq. (\[eq9\]), it can readily be shown that for $k=1$, $\alpha$ is equal to 0 when there is no interaction present in the system. But we want the bound state of the form of Eq. (\[eq5\]), hence our treatment will only be valid for $k\ge2$.
![Above figure shows the density profile of the localized state for different values of interaction strength $g'$ for $k=3$. In Y-axis $|\psi(y)|^2$ is in the scale and dimension of $\sqrt{\frac{NmD}{\hbar^2}}$. With the increase of $g'$, peaks are shifted towards right.[]{data-label="fig6"}](psi-k3.eps)
![This figure shows the density profile of the localized state for different values of $g'$ for $k=5$. The scale of Y axis remains the same as Fig. (\[fig6\]).[]{data-label="fig7"}](psi-k5.eps)
![This figure shows the density profile of the localized state at critical value of $g'$ i.e., at $g_c'$ for different values of $k$. The scale of Y axis remains the same as Fig. \[fig5\]. The width of the curves increases with the increase of the value of $k$. []{data-label="fig8"}](diff-k-gc.eps)
In Fig. \[fig6\], we have plotted $|\psi(y)|^2$ vs y for a particular value of $k$ $(k=3)$ which shows that with the decrease of the value of $g'$, the curve starts broadening which seems apparently counter-intuitive . But the careful observation reveals that though the peaks of the curves are around $y=2$, the peaks are going far apart from the trap centre as $g'$ is increased indicating higher repulsive interaction between the particles causing poor localization inside the trap and the particles are more prone to get rid of the confining potential.
In Fig. \[fig7\] also we have plotted the same for $k=5$ and we observe that the peaks are now located around $y=4$ and herealso the peaks are going far away from the trap centre with the increase of $g'$. Smaller magnitudes of $g'$ in this case than those of $k=3$ indicates that the small repulsive interaction are now sufficient to cause delocalisation.
In Fig. \[fig8\] we have shown the density profile at different values of $g_c'$ corresponding to different $k$’s. With the increase of the value of $k$, the increasing width of the curves indicates that as the system falls under weak confinement, the localised nature of the system at $g_c'$ diminishes.
Conclusion
==========
As a summary, in this work starting from Gross-Pitaevskii energy functional we have explored analytically the ground state behavior of GPE which develops an interesting feature when the system is trapped under Morse potential. As the potential approaches to simple harmonic one, the system being highly trapped, the ground state solution becomes more and more localized, whereas in the opposite limit we have explained the free nature of the system dominates which is very relevant. This interesting limiting feature of the Morse potential is very unique. We have shown here for the first time that a bifurcation in the ground state solution appears when GPE is under the influence of Morse potential. Since GPE is nothing but NLSE, our work has indeed revealed an important characteristics of NLSE in Morse potential.\
This treatment can be extended to coupled Gross Pitaevskii equation also and in the limiting cases this should agree with the results which are already known in literature.
Acknowledgements {#acknowledgements .unnumbered}
================
One of the authors, Sukla Pal would like to thank S. N. Bose National Centre for Basic Sciences for the financial support during the work. Sukla Pal acknowledges Harish-Chandra Research Institute for hospitality and kind support during visit.
[0]{} E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20 (1961) 454–457\
\]L. P. Pitaevskii, Soviet Phys. JETP, 13 (1961) 451–454.\
M. H. Anderson et al. Science 269 (1995) 198\
K. B. Davis et al. Phys Rev Lett 75 (1995) 3969\
A. Leggett Rev Mod Phys 73 (2001) 307–356\
C. J. Pethick and H. Smith. Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge, 2002\
M. J. Holland, D. S. Jin, M. L. Chiofalo and J. Cooper Phys Rev Lett 78 (1997) 3801\
P. Capuzzi and E. S. Hernandez, Phys Rev A 59 (1999) 1488.\
G-S Paraoanu, S. Kohler, F. Sols and A. J. Leggett J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 4689–4696\
A. B. Tacla, C. M. Caves New J. Phys. 15 (2013) 023008\
C. Tozzo, M. Kramer, F. Dalfovo Phys Rev A 72 (2005) 23613\
Gh.-S. Paraoanu Phys. Rev. A [67]{} (2003) 023607\
Y. Cai, M. Rosenkranz, Z. Lei and W. Bao Phys Rev A 82 (2010) 043623\
N. G. Parker, C. Ticknor, A. M. Martin, D. H. J. O’Dell1 Phys Rev A 79 (2009) 013617\
L. D. Carr, Charles W. Clark, and W. P. Reinhardt PRA 62 (2000) 063610.\
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari Rev Mod Phys 71 (1999) 463\
G. Chen, Phys Letters A 326 (2004) 55-57\
M. Olshanii Phys Rev Lett 81 (1998) 938\
|
---
abstract: 'We analyze the excitation spectra of a spin-phonon coupled chain in the presence of a soliton. This is taken as a microscopic model of a Spin-Peierls material placed in a high magnetic field. We show, by using a semiclassical approximation in the bosonized representation of the spins that a trapped magnetic state obtained in the adiabatic approximation is destroyed by dynamical phonons. Low energy states are phonons trapped by the soliton. When the magnetic gap is smaller than the phonon frequencies the only low energy state is a mixed magneto-phonon state with the energy of the gap. We emphasize that our results are relevant for the Raman spectra of the inorganic Spin-Peierls material CuGeO$_3$.'
address:
- |
Departamento de Física, Universidad Nacional de Rosario,\
and Instituto de Física Rosario, Avenida Pellegrini 250,\
2000 Rosario, Argentina
- |
Instituto de Astronomía y Física del Espacio,\
Casilla de Correos 67, Sucursal 28,\
1428 Buenos Aires, Argentina.
author:
- Ariel Dobry
- David Ibaceta
title: 'Mixing of magnetic and phononic excitations in incommensurate Spin-Peierls systems'
---
The discovery in 1993 by Hase et al [@Hase] of the first inorganic Spin-Peierls compound CuGeO$_3$ has opened the possibility of study the physics of this collective phenomena in a deep way. Several experimental proves have given an exhaustive information about the excitation spectra of this system and its evolution with an applied magnetic field. The effect of non-magnetic impurities has been investigated also.
Theoretical studies have focused on simplified magnetic model. The excitation spectra in the low temperature phase have been analyzed using a dimerized and frustrated Heisenberg chain as a minimal model for this material[@UhrigS; @RieraK; @Bouzerar]. The logic underlying these studies are: the competition between magnetic and elastic energies resolves in the low temperature phase in the dimerization of the lattice. Once this process takes place, phononic and magnetic excitation completely decouple and the magnetic excitations are the same as the chemical dimerized system. This point of view is based on an adiabatic approximation supposing that the energy scale of the magnetic process are high enough respect to the phononic ones. As it has been recently emphasized [@Uhrig], this relation is not fulfilled for CuGeO$_3$ where the phonons relevant for the dimerization process are about one order of magnitude more energetic than the magnetic gap. The adiabatic approximation is questionable for this system. An antiadiabatic approach has been developed. The frustrated interaction arises, in this context, from the integration of the in-chain phonons and the explicit dimerization from the interchain interaction treated in a mean field approximation[@Uhrig2]. The same frustrated-dimerized Hamiltonian is therefore obtained but with a reinterpretation of the parameters. What is clearly missed in these studies is a general understanding on how spin and phonons mix as elementary excitation and how the spectra of Spin-Peierls systems is built as a result of this mixing. Some recent numerical results have partially addressed this question [@Augier].
In this paper, we analyze the excitation spectra of one-dimensional spin-phonon system by semiclassical techniques on the bosonized representation of the spins subsystem. We focus on the properties of this system in a high magnetic field. In the dimerized phase the system is in a singlet ground state. Coupling with a magnetic field is not effective up to a critical field value where some dimer break and their spins are freely from their singlet. The lattice also relaxes in this process forming the so called soliton lattice. This relaxation is a first indication that spin and phonons do not act as independent excitation. It has been experimentally shown by RX measurements that the lattice becomes incommensurate following a soliton pattern[@Kiry]. The magnetic profile of the soliton has been analyzed by NMR measurement [@fagot].
More recently, optical proves have shown the spectral signature of the incommensurate phase. In the uniform dimerized phase a low energy resonance appears at 30 cm$^{-1}$ which is a smaller energy than two times the magnon gap[@Lemmens]. This peak has been adjudicated to a bound state of two triplet [@Bouzerar]. When the incommensurate phase appears the spectral weight is transfered from this state to a lower energy peak at the position of the magnon gap[@Loa]. Direct magnon process could not be seen in the optical response due to spin conservation. Soliton assisted one magnon excitation was recently been proposed as the origin of this peak[@Loa] in similarity with the situation in the presence of non-magnetic impurities[@Els].
This approach is based on the image of the soliton as an isolated spin in the externally dimerized chain so that an adiabatic approximation is supposed. We will show that this state in fact disappears when the dynamics of the phonons is included . We will also show that a trapped magneto-phonon state with the energy of the gap appears in the antiadabatic regime so explaining the optical data.
Let us proceed more formally. Our starting Hamiltonian for an Spin-Peierls compound is:
$$\begin{aligned}
H=H_{ph}+H_{mg} \label{H}\end{aligned}$$
$$\begin{aligned}
H_{ph}=\sum_i\frac{P_i^2}{2M}+\frac K2(u_{i+1}-u_i)^2\end{aligned}$$
$$\begin{aligned}
H_{mg}=\sum_iJ(1+\frac \alpha 2(u_{i+1}-u_i)){\bf S}_i\cdot {\bf S}_{i+1}\end{aligned}$$
${\bf S}_i$ are spin-1/2 operators of the $i$ ion and $\alpha $ the magneto-elastic coupling . $H_{ph}$ represents our simplified model for the phonons. It contains a scalar coordinate $u_i$ and its conjugate momentum $%
P_i$ which are supposed to be the relevant ion coordinates for the dimerization process. As we will only retain the phonons relevant for SP transition, the specific dispersion of the phonons is not important in our approach. The tridimensional character of the phonons field is essential to account for the finite Spin-Peierls temperature and the excitations in the low temperature dimerized phase[@domain]. We will discuss later its effect on the incommensurate phase.
The low energy spectrum could be studied by bosonization. The spin variables are approximately represented by the bosonic field $\phi (x)$ and its conjugated momentum $\Pi (x)$[@Affleck]. In addition, we retain only the phonon modes producing a smooth deviation of the dimerized pattern. So, we make the replacement $(-1)^iu_i\rightarrow u(x)$. The low energy Hamiltonian becomes: $$\begin{aligned}
H_{ph} &=&\int dx\{\frac{aP(x)^2}{2M}+\frac{2K}au(x)^2\} \nonumber \\
H_{bos} &=&\int dx\{\frac 1{2\pi }[\frac{v_s}\eta (\partial _x\phi (x))^2
\nonumber \\
&&+v_s\eta (\pi \Pi (x))^2]+\frac{J\alpha u(x)}{\pi a_0}\sin \phi (x)\}
\label{Hbos}\end{aligned}$$ where $a$ is the lattice constant of the original chain and $a_0$ a short range cutoff introduced in the bosonization procedure. $v_s$ is the spin wave velocity and $\eta $ is related with the exponent of the correlation functions. For the isotropic Heisenberg model with nn interaction we have $%
v_s=\frac{Ja\pi }2$ and $\eta =2$.
We use a semiclassical approach in the form of a self consistent harmonic approximation(SHA) as it has been originally proposed by Nakano and Fukuyama (NF) [@NF1]. In this approach the boson field $\phi $ is split in a classical component $\phi _{cl}$ and the quantum fluctuation $\hat{\phi}$. The last term of (\[Hbos\]) is developed up to second order around $\phi _{cl}$ and then treated self-consistently by the following replacement:
$$\begin{aligned}
\sin\phi \rightarrow \nonumber \\
e^{-\langle \hat{\phi}^2\rangle /2}[\sin(\phi_{cl})(1-\frac{(\hat{\phi}^2-
\langle \hat{\phi}^2\rangle)}2)+\cos(\phi_{cl})\hat{\phi}] \label{SHA}\end{aligned}$$
$\langle \hat{\phi}^2\rangle $ is the ground state expectation value.
In their original work NF have fixed the displacement field $u$ to its equilibrium classical value ($u_{cl}$), so that an adiabatic approximation was assumed. Let us summarize the main results of this study and its consequence for the spectra:
- The classical equations for $\phi $ and $u(x)$ have a soliton solution of the form: $\sin (\phi _{cl})=\tanh (x/\xi )\nonumber\\ %
u_{cl}(x)=u_0\tanh (x/\xi )$ . $\xi $, the soliton width, is $v_s/\Delta $ and $\Delta $ is the gap over the homogeneous dimerized state (the ’magnon’ gap) and $u_0$ are the displacements in this homogeneous dimerized state. The soliton carries $S_z=1/2$ spin. In the presence of an external magnetic field these solitons will condense in the ground state and accommodate in a soliton lattice structure.
- The creation energy of the soliton as well as the excitations over this solitonic vacuum could be evaluated once the eigenvalues of the fluctuation operator are known. This eigenvalue problem corresponds to a one of a Schroedinger-like equation which reads as:
$$\begin{aligned}
\{-\nabla _{\hat{x}}^2+\tanh ^2(\hat{x})\}\psi _\lambda =\frac{\Omega
(\lambda )^2}{\Delta ^2}\psi _\lambda (\hat{x}) \label{sch} \\
\hat{x}\equiv \frac x\xi \nonumber\end{aligned}$$
$\Omega (\lambda )$ are the frequency oscillations in presence of the soliton. Eq. (\[sch\]) has one bound state at $\frac{\Omega _b}{\Delta}
=\sqrt{\frac{\sqrt{5}-1}2}\sim 0.786$ and a continuum started at $\Omega =\Delta $. The excitation spectra of the theory is spanned by the following state:
<!-- -->
- [The ground state of the quantum soliton builded around $\phi _{cl}$. Their creation energy ($E_s$) is given by its classical energy plus the difference between the sum of the zero point energies of the oscillators $%
(\Omega (\lambda ))$ and the ones in absence of the soliton. This creation energy measures the critical field for the commensurate- incommensurate transition[@NF1].]{}
- [The excited state of the soliton with energy $E_s^{*}=E_s+\Omega _b$. It is related with the bound state of Eq. (\[sch\]).]{}
- [Labeling by q the continuum of level of Eq. \[sch\], they are $%
\Omega (q)=v_s\sqrt{q^2+1/\xi ^2}$ This is just the kinetic energy of a particle in the soliton free sector. In terms of the original spin chain this is the low energy dispersion of the $S_z=0$ component of a magnon. Therefore this state corresponds to the scattering of a magnon in presence of the soliton. Moreover, when one of the continuum modes is excited once, we get a two particle magnon-soliton state.]{}
The two last adjudication of states have not been done in the original analysis of the NF work. As our formalism breaks SU(2) symmetry at intermediate stage, the total spin of each state is not directly accessible and should be carefully reanalyzed. We have previously stated that the states below the continuum correspond to the emission of a magnon in the presence of the soliton. Therefore they have total spin-3/2. The bound state has the same total spin as the soliton. Then, it is an optical active mode and it is the analogue in our formalism of the so called soliton assisted magnon process founded in the strong dimerization limit[@Els].
Moreover, the previous predicted spectra could be tested by numerical exact diagonalization on finite chain. To this end, we solve iteratively the adiabatic equations for $u_i$ arising from Hamiltonian Eq. \[H\]. The ground state of the spin system was recalculated at each iteration step by Lanczos diagonalization. We considered chains of odd number of site up to $L=23$ sites with periodic boundary conditions and look for the equilibrium positions in the subspace $S_z=1/2$. The numerical details of the method was given in [@Feiguin; @Sch].
Once the ionic coordinate converge we diagonalize the spin Hamiltonian, with fixed ${u_i}$, to obtain the low lying excitation in the $S_z=1/2$ and $S_z=3/2$ subspaces. The results are shown in Fig.\[fig1\] as a function of the $1/L^2$. Parameter $\frac{K}{(\alpha J)^2}=\frac14$ has been chosen in order to have a thin soliton and to reduce finite size effect. The bound state predicted by the bosonized theory is clearly seen. The higher energy states collapse in a continuum of total spin-3/2 in the thermodynamical limit. By linear extrapolation, we found for $L\rightarrow \infty$ the ratio $0.84$ between the bound state and the border of the continuum. This value compares well with our previous prediction $0.786$. We conclude that the method gives at least a qualitative features of the low energy spectra. The states appearing in the border of the continuum in Fig. (\[fig1\]) are an artifact of the strongly thin soliton we are considering. We have check that for bigger values of $\frac{K}{(\alpha J)^2}=\frac14$ these states in fact disappear.
.5 truecm
We discuss in the following two effects that could change our estimated value of the bound state from the SHA method:
1. [ In going from \[H\] to \[Hbos\], we have neglected as NF in their original work a term proportional to $cos(2\phi)$ in the bosonized theory. Even though this operator plays an important role in the case of Ising anisotropy, it is a marginally irrelevant operator in the Heisenberg case and therefore it should not change the qualitative features previously discussed. Moreover, NF have studied in a further contribution [@NF2] the effect of this term. Unfortunately, the estimation of the bound state value is strongly dependent on the value of the cuttoff in the bosonization procedure ($a_0$) in this situation. If we assume as NF that $\pi a_0=a$, the bound state becomes ${\Omega _b}{\Delta}=\sqrt{\frac{\sqrt{3}}2}\sim 0.93$. We can conclude as a general fact that the presence of this term tend to increase the bound state value.]{}
2. [It has been recently noted [@Fabrizio; @Uhlig3] that a full self-consistent treatment (where $ <\phi^2>$ in Eq. \[SHA\] is not fixed to its value in the ground state) changes the form of the soliton. Moreover, it has been shown in Ref.([@Uhlig3]) that this treatment produces a smoother soliton than the one considered here. Therefore, we conclude that this treatment will reduce the bound state value.]{}
In the following we analyze the evolution of the spectra when non-adiabatic effects are included. For simplicity we do not introduce the two additional effects previously discussed. These effects will not affect the prediction of the trapped magneto-phonon state we will find.
[*Inclusion of non-adiabatic effects.*]{} To go beyond this static adiabatic approximation in this semi-classical calculation it is necessary to include fluctuations in the displacement field $u(x)$. Therefore, we split $u(x)$ as $%
u_{cl}+\hat{u}$ and replace it in Eq.(\[Hbos\]). The classical equations are the same as before. The fluctuation operator is now a 2 x 2 differential operator with component over $u$ and $\phi$. The eigenvalue problem is now given by:
$$\begin{array}{lllll}
\Delta^2[-\nabla^2_{\hat{x}} + \tanh^2(\hat{x})]\psi_1 & + & \frac{u_0}{a}%
8\pi K v_s {\rm sech}(\hat{x})\psi_2 & = & \Omega^2 \psi_1 \\
\omega^2 u_0{\rm sech}(\hat{x}) \psi_1 & + & \omega^2 \psi_2 & = & \Omega^2
\psi_2
\end{array}
\label{fluc2}$$
$\psi_1$ and $\psi_2$ are the component of the fluctuation eigenvector over $%
\phi$ and $u$ respectively. $\omega=\sqrt{\frac{4K}{M}}$ is the phonon frequency at $q=\pi$ The eigenfrequency are as previously given by $\Omega$. We obtain from the second equation: $$\begin{aligned}
\psi_2=\frac{u_0 \omega^2{\rm {sech}(\hat{x})}} {\Omega^2-\omega^2}\psi_1
\label{psi12}\end{aligned}$$ Replacing in the first equation we have a kind of Schroedinger equation where the potential depends on the energy. This equation reads:
$$\begin{aligned}
\{-\nabla^2_{\hat{x}} + \tanh^2(\hat{x})\}\psi_1+ (\frac{\omega}{\Delta})^2
\frac{{\rm {sech}}^2(\hat{x})}{(\frac{\Omega}{\Delta})^2- (\frac{\omega}{\Delta})^2}%
\psi_1=(\frac{\Omega}{\Delta})^2\psi_1 \label{schef}\end{aligned}$$
It is worthy to note that, the only relevant parameter is $\frac{\omega}{\Delta}$.Then, we can follow the evolution of the spectra from the adiabatic to the antiadiabatic limit by moving this parameter.
We have numerically solve Eq. (\[schef\]). In Fig. \[fig2\], we show the evolution of the excitation spectra as a function of $\frac \omega \Delta $. The soliton creation energy has been taken as the zero of the energy.
.5 truecm
The main results are:
- [There is a zero energy solution which was not present in the adiabatic calculation. This mode arises in the complete translation invariance of the theory. It is nothing but the translational mode of the soliton. In a realistic situation solitons are not isolated, they form a soliton lattice. This zero mode will acquire some dispersion giving rise to the so called phason modes of the soliton lattice. If the magnetic field is not much higher than the critical one the solitons will be rather separate and we can neglect their interference. ]{}
In the direction perpendicular to the magnetic chains, solitons will accommodate in a domain wall, i.e. they form an array of parallel solitons. Their coherency is assured by the interchain elastic coupling. The zero mode as well as all the states of the spectra will have a transversal dispersion, they will depend on the momentum perpendicular to the chains. The spectrum shown in Fig. (\[fig2\]) gives the q=0 excitations. We will analyze their dispersive character in a forthcoming work.
- [The upper branch corresponds to the excited state of the soliton previously found in the adiabatic approximation. Its energy increases with $%
\omega $ up to a critical value $\frac{\omega ^2}{\Delta ^2}=\frac 12$ where this state disapear. For this $\omega $ and $\Omega =\Delta $, the energy dependent potential vanishes. The lattice soliton produces an harmonic potential over the magnetic soliton where it oscillates. In the adiabatic approximation the lattice soliton is frozen and the magnetic one oscillates. For finite $%
\omega $, the lattice soliton can also move and vibrate with respect to the magnetic deformation. Finally, at the critical value of $\omega $ there is no more a localized vibration and the soliton becomes an unique entity both for the lattice and for the spins. This explains the disparition of the trapped state at a critical phonon frequency.]{}
- [The straight line of slope one in Fig.\[fig2\] corresponds to $%
\Omega =\omega $. The eigenstate with this energy corresponds to the excitation of a phonon in the presence of the soliton. The eigenvalues near above are satellite phonons corresponding to phonons trapped by the soliton (i.e. particle vibrations near the soliton). They have only spectral weight on the phonons as it can be seen putting in Eq.(\[psi12\]) $\Omega \sim \omega $. For increasing phonon frequency, most of these eigenvalues lose but the lowest survives acquiring the energy of the magnetic gap. This state is a mixing of a magnetic and a phononic excitation as we show in the following. ]{}
- [Four Peierls active phonons have been identified for CuGeO$_3$[@wernergross]. They have frequencies of 3.12, 6.53, 11.1 and 24.6 THz i.e. the smaller one is about 150K. The spin gap is rather small for this system $%
\Delta =24K$. As we previously stated the real parameter regime where this material lives is $\omega >\Delta $ and the only survival bound state has the energy of the gap. This is a singlet state which we associate with the low energy peak seen in Raman scattering in the incommensurate phase [@Loa]. The $\psi _1$ and $\psi _2$ components of the corresponding eingenfunction are different from zero for this state.]{}
Their mixed character could be advocated by analyzing separately the magnetic and phononic spectral response. The magnetic Raman operator of a dimerized chain is given by ${\cal R}_{mg}=\sum_i(1+\gamma _{mg}(-1)^i{\bf S}%
_i\cdot {\bf S}_{i+1})$[@Muthukumar] $\gamma _{mg}$ is a microscopic parameter. In our bosonized formalism the most relevant contribution comes from the staggered part and it is given by $\frac{\gamma _{mg}}{\pi a_0}%
sin(\phi )$. By using the SHA given by Eq. (\[SHA\]), retaining the term linear in $\hat{\phi}$ and developing in the basis of the eigenvalues of the fluctuation operator the magnetic spectral weight ($M_n$) of an state of energy $\Omega _n$ is given by:
$$\begin{aligned}
M_n = \frac{\gamma_{mg}^2}{2 \Omega_n}\left\{\frac{4 K}{J \alpha a} \frac{%
\Omega_n- \omega^2}{\omega^2} \int \psi_2 dx \right\}^2\end{aligned}$$
where we have used Eq. (\[psi12\]) to write $\psi_1$ as a function of $\psi_2$. Phonons contribution to Raman scattering is proportional to the square of the transition elements of the normal coordinates ${\cal R}_{ph}=\gamma
_{ph}\int \frac{dx}au(x)$, the phonons spectral weight of the n-state is given by:
$$\begin{aligned}
P_n = \frac{\gamma_{ph}^2}{2 \Omega_n}\left\{ \int \psi_2 \frac{dx}a
\right\}^2\end{aligned}$$
The relation between the magnetic and phononic contribution to the n-th peak of the spectra is:
$$\begin{aligned}
\frac{M_n}{P_n} = \frac{\gamma_{mg}^2}{\gamma_{ph}^2} \left\{ \frac{4 K}{%
J\alpha}\right\}^2 \left\{ \frac{\Omega_n-\omega^2}{\omega^2} \right\}^2
\label{MP}\end{aligned}$$
We take the first two factors as a measure of the spin-phonon coupling , we fix this parameters when moving $\omega $. Eq. (\[MP\]) shows that the magnetic component of a given Raman peak increases when its position shifts from a phonon frequency. The lower energy eigenvalue of Fig([\[fig2\]) starts being a trapped phonons for $\frac \omega \Delta <1$ and transmutes in a magneto-phonon in the opposite limit where the real material lives. Its existence is a consequence of the quasi-onedimensional character of our system because a one-dimensional well has always one bound state. ]{}
In summary, we have shown that nonadiabatic effects are relevant for incommensurate Spin-Peierls system. In the antiadiadatic limit applicable to CuGeO$_3$ we show the apparition of a trapped mixed state with the energy of the magnon gap. It gives an alternative explanation of the Raman induced peak found in the incommensurate phase of CuGeO$_3$. A.D. acknowledges G. Uhrig and A. Greco for useful discussion. We are grateful to J. Riera for useful discussions and computational help. We are grateful to P. Lemmens for discussions and for pointing out Ref. [@Loa].
M. Hase, I. Terasaki and K. Uchinokura, Phys. Rev. Lett. [**70**]{}, 3651 (1993).
I. Loa [*et al.*]{}, Solid St. Commun. [**111**]{}, 181 (1999) , P.H.M. van Loosdrecht et al Phys. Rev. B [**54**]{}, R3730 (1996)
G. S. Uhrig and H. J. Schulz, Phys. Rev. B [**54**]{}, R9624 (1996).
J. Riera and S. Koval, Phys. Rev. B [**53**]{}, 770 (1996)
G. Bouzerar, A. P. Kampf and G. I. Japaridze, Phys. Rev. B [**59**]{}, 6266 (1999)
G. S. Uhrig, Phys. Rev. B [**57**]{}, R14004 (1998).
G. S. Uhrig, cond-mat/9908225 (1999).
D. Augier [*et al.*]{}, Phys. Rev. B [**58**]{}, 9110 (1998). , R. Bursill, R. McKenzie and C. Hamer, Phys. Rev. Lett. [**83**]{}, 408 (1999)
V. Kiryukhin [*et al.*]{}, Phys. Rev. Lett. [**76**]{}, 4608 (1996)
Fagot-Revurat et al, Phys. Rev. Lett. [**77**]{}, 1871 (1996)
P. Lemmens [et al.]{} Physica B [**535**]{} (1996)
G. Els [et al.]{} Europhys. Lett. [**43**]{}, 462 (1998)
A. Dobry and D. Ibaceta Phys. Rev. B [**58**]{}, 3124 (1998)
I. Affleck, Fields Strings and Critical Phenomena, edited by E. Brézin and J.Zinn-Justin (North-Holland, Amsterdam 1990), pg. 563.
T. Nakano and H. Fukuyama, J. Phys. Jpn [**49**]{}, 1679 (1980).
T. Nakano and H. Fukuyama, J. Phys. Jpn [**50**]{}, 2489 (1981).
A. Feiguin, J. Riera, A. Dobry and H. Ceccatto, Phys. Rev. B [**56**]{}, 14607 (1997).
F. Schöenfeld, G. Bouzerar, G. Uhrig and Müller-Hartmann, Eur. Phys. J. B. [**5**]{}, 521 (1998)
M. Fabrizio, R. Mélin and J. Souletie, Eur. Phys. J. B. [**10**]{}, 607 (1999).
G. Uhlig, Physica B, 280, 308 (2000)
R. Werner, C. Gross and M. Braden, Phys. Rev. B, (1999)
V. N. Muthukumar et. at Phys. Rev. B [**54**]{}, R9635 (1996).
|
---
author:
- Stathis Antoniou and Sofia Lambropoulou
title: Topological Surgery in Nature
---
Introduction {#Intro}
============
Topological surgery is more than a mathematical definition allowing the creation of new manifolds out of known ones. As we point out in [@SSN1], it is a process triggered by forces that appear in both micro and macro scales. For example, in dimension 1 topological surgery can be seen in DNA recombination and during the reconnection of cosmic magnetic lines, while in dimension 2 it happens when genes are transferred in bacteria and during the formation of black holes. Inspired by these natural processes, in [@SSN1] we enhance the formal definition of topological surgery with the observed [*forces*]{} and [*dynamics*]{}, thus turning it into a continuous process.
Furthermore, in [@SSN1] we observe that phenomena like tension on soap films or the merging of oil slicks are undergoing $1$-dimensional surgery but they happen on surfaces instead of $1$-manifolds. Similarly, moving up one dimension, during the biological process of mitosis and during tornado formation, $2$-dimensional surgery is taking place on $3$-dimensional manifolds instead of surfaces. Thus, in order to fit natural phenomena where the interior of the initial manifold is filled in, in [@SSN1] we extend the formal definition by introducing the notion of [*solid topological surgery*]{} in both dimensions 1 and 2.
Finally, in [@SSN1] we notice that in some phenomena exhibiting topological surgery, the ambient space is also involved. For example in dimension 1, during DNA recombination the initial DNA molecule which is recombined can also be knotted. In other words, the initial $1$-manifold can be a knot (an embedding of the circle) instead of an abstract circle. Similarly in dimension 2, the processes of tornado and black hole formation are not confined to the initial manifold and topological surgery is causing (or is caused by) a change in the whole space. We therefore define the notion of [*embedded topological surgery*]{} which allows us to model these kind of phenomena but also to view all natural phenomena exhibiting topological surgery as happening in $3$-space instead of abstractly.
The notions and ideas presented in this paper can be found in [@SSN1; @SSN2]. However, [@SSN1; @SSN2] contain a much deeper analysis of the dynamical system modeling 2-dimensional surgery, which is discussed here very briefly.
The paper is organized as follows: it starts by recalling the formal definitions of surgery in Section \[definitions\]. Dynamic topological surgery is then presented in natural processes and introduced as a schematic model in Section \[Dynamics\]. Next, solid surgery is defined in Section \[DSolid\] where the dynamical system modeling 2-dimensional surgery is also presented. Finally, embedded surgery and its differences in dimensions 1 and 2 are discussed in Section \[Embedded\].
We hope that the presented definitions and phenomena will broaden our understanding of both the topological changes exhibited in nature and topological surgery itself.
The formal definitions of surgery {#definitions}
=================================
We recall the following well-known definition:
\[surgery\] An *m-dimensional n-surgery* is the topological procedure of creating a new $m$-manifold $M'$ out of a given $m$-manifold $M$ by removing a framed $n$-embedding $h:S^n\times D^{m-n}\hookrightarrow M$ and replacing it with $D^{n+1}\times S^{m-n-1}$, using the ‘gluing’ homeomorphism $h$ along the common boundary $S^n\times S^{m-n-1}$. Namely, and denoting surgery by $\chi$: $$M' = \chi(M) = \overline{M\setminus h(S^n\times D^{m-n})} \cup_{h|_{S^n\times S^{m-n-1}}} (D^{n+1}\times S^{m-n-1}).$$
Further, the *dual m-dimensional $(m-n-1)$-surgery* on $M'$ removes a dual framed $(m-n-1)$-embedding $g:D^{n+1}\times S^{m-n-1}\hookrightarrow M'$ such that $g|_{S^n\times S^{m-n-1}}=h^{-1}|_{S^n\times S^{m-n-1}}$, and replaces it with $S^n\times D^{m-n}$, using the ‘gluing’ homeomorphism $g$ (or $h^{-1}$) along the common boundary $S^n\times S^{m-n-1}$. That is: $$M = \chi^{-1}(M') = \overline{M'\setminus g(D^{n+1}\times S^{m-n-1})} \cup_{h^{-1}|_{S^n\times S^{m-n-1}}} (S^n\times D^{m-n}).$$
From the above definition it follows that $M = \chi^{-1}(\chi(M))$ and $n+1 \leq m$. For further reading see [@Ra; @PS; @Ro]. We shall now apply the above definition to dimensions 1 and 2.
1-dimensional 0-surgery {#1D_FormalE}
-----------------------
We only have one kind of surgery on a 1-manifold $M$, the *1-dimensional 0-surgery*: $$M' = \chi(M) = \overline{M\setminus h(S^0\times D^{1})} \cup_{h|_{S^0\times S^{0}}} (D^{1}\times S^{0}).$$ The above definition means that two segments $S^0\times D^1$ are removed from $M$ and they are replaced by two different segments $D^1 \times S^0$ by reconnecting the four boundary points $S^0\times S^0$ in a different way. In Fig. \[1\_2D\_Formal\] (a) and \[1\_2D\_Formal\_twisted\] (a), $S^0\times S^0 =\{1, 2, 3, 4\}$. As one possibility, if we start with $M=S^1$ and use as $h$ the standard (identity) embedding denoted with $h_s$, we obtain two circles $S^1 \times S^0$. Namely, denoting by $1$ the identity homeomorphism, we have $h_s:S^0\times D^{1}=D^{1} \amalg D^{1} \xrightarrow{1 \amalg 1} S^0\times D^{1} \hookrightarrow M$, see Fig. \[1\_2D\_Formal\] (a). However, we can also obtain one circle $S^1$ if $h$ is an embedding $h_t$ that reverses the orientation of one of the two arcs of $S^0\times D^1$. Then in the substitution, joining endpoints 1 to 3 and 2 to 4, the two new arcs undergo a half-twist, see Fig. \[1\_2D\_Formal\_twisted\] (a). More specifically, if we take ${D^1}=[-1, +1]$ and define the homeomorphism $\omega:D^{1}\to D^{1} ; t \to -t$, the embedding used in Fig. \[1\_2D\_Formal\_twisted\] (a) is $h_t:S^0\times D^{1}=D^{1} \amalg D^{1} \xrightarrow{1 \amalg \omega} S^0\times D^{1} \hookrightarrow M $ which rotates one $D^{1}$ by 180. The difference between the embeddings $h_s$ and $h_t$ of $S^0\times D^1$ can be clearly seen by comparing the four boundary points $1, 2, 3$ and $4$ in Fig. \[1\_2D\_Formal\] (a) and Fig. \[1\_2D\_Formal\_twisted\] (a).
![Formal **(a)** 1-dimensional 0-surgery **(b~1~)** 2-dimensional 0-surgery and **(b~2~)** 2-dimensional 1-surgery using the standard embedding $h_s$.[]{data-label="1_2D_Formal"}](1_2D_Formal.jpg){width="11cm"}
![Formal **(a)** 1-dimensional 0-surgery **(b~1~)** 2-dimensional 0-surgery and **(b~2~)** 2-dimensional 1-surgery using a twisting embedding $h_t$.[]{data-label="1_2D_Formal_twisted"}](1_2D_Formal_twisted.jpg){width="11cm"}
Note that in dimension one, the dual case is also an 1-dimensional 0-surgery. For example, looking at the reverse process of Fig. \[1\_2D\_Formal\] (a), we start with two circles $M'=S^1\amalg S^1$ and, if each segment of $D^1 \times S^0$ is embedded in a different circle, the result of the (dual) 1-dimensional 0-surgery is one circle: $ \chi^{-1}(M')=M=S^1$.
2-dimensional 0-surgery {#2D_FormalE0}
-----------------------
Starting with a 2-manifold $M$, there are two types of surgery. One type is the *2-dimensional 0-surgery*, whereby two discs $S^0\times D^2$ are removed from $M$ and are replaced in the closure of the remaining manifold by a cylinder $D^1\times S^1$, which gets attached via a homeomorphism along the common boundary $S^0\times S^1$ comprising two copies of $S^1$. The gluing homeomorphism of the common boundary may twist one or both copies of $S^1$. For $M=S^2$ the above operation changes its homeomorphism type from the 2-sphere to that of the torus. View Fig. \[1\_2D\_Formal\] ([b~1~]{}) for the standard embedding $h_s$ and Fig. \[1\_2D\_Formal\_twisted\] ([b~1~]{}) for a twisting embedding $h_t$. For example, the homeomorphism $\mu:D^{2}\to D^{2} ; (t_1, t_2) \to (-t_1, -t_2)$ induces the 2-dimensional analogue $h_t$ of the embedding defined in the previous example, namely: $h_t:S^0\times D^{2}=D^{2} \amalg D^{2} \xrightarrow{1 \amalg \mu} S^0\times D^{2} \hookrightarrow M $ which rotates one $D^{2}$ by 180. When, now, the cylinder $D^1\times S^1$ is glued along the common boundary $S^0\times S^1$, the twisting of this boundary induces the twisting of the cylinder, see Fig. \[1\_2D\_Formal\_twisted\] ([b~1~]{}).
2-dimensional 1-surgery {#2D_FormalE1}
-----------------------
The other possibility of 2-dimensional surgery on $M$ is the *2-dimensional 1-surgery*: here a cylinder (or annulus) $S^1 \times D^1$ is removed from $M$ and is replaced in the closure of the remaining manifold by two discs $D^2 \times S^0$ attached along the common boundary $S^1 \times S^0$. For $M=S^2$ the result is two copies of $S^2$, see Fig. \[1\_2D\_Formal\] ([b~2~]{}) for the standard embedding $h_s$. Unlike Fig. \[1\_2D\_Formal\] ([b~1~]{}) where the cylinder is illustrated vertically, in Fig. \[1\_2D\_Formal\] ([b~2~]{}), the cylinder is illustrated horizontally. This choice was made so that the instances of 1-dimensional surgery can be obtained by crossections of the instances of both types of 2-dimensional surgeries, see further Remark \[Rot\]. Fig. \[1\_2D\_Formal\_twisted\] ([b~2~]{}) illustrates a twisting embedding $h_t$, where a twisted cylinder is being removed. In that case, taking $D^{1}=\{h:h \in [-1, 1]\}$ and homeomorphism $\zeta$:
$\zeta:S^1\times D^{1}\to S^1\times D^{1};$\
$\zeta: (t_1, t_2, h) \to (t_1\cos{\frac{(1-h)\pi}{2}}-t_2\sin{\frac{(1-h)\pi}{2}}, t_1\sin{\frac{(1-h)\pi}{2}}+t_2\cos{\frac{(1-h)\pi}{2}}, h)$
the embedding $h_t$ is defined as: $h_t:S^1\times D^{1} \xrightarrow{\zeta} S^1\times D^{1} \hookrightarrow M $. This operation corresponds to fixing the circle $S^1$ bounding the right side of the cylinder $S^1 \times D^1$, rotating the circle $S^1$ bounding the left side of the cylinder by 180 and letting the rotation propagate from left to right. This twisting of the cylinder can be seen by comparing the second instance of Fig. \[1\_2D\_Formal\] ([b~2~]{}) with the second instance of Fig. \[1\_2D\_Formal\_twisted\] ([b~2~]{}), but also by comparing the third instance of Fig. \[1\_2D\_Formal\] ([b~1~]{}) with the third instance of Fig. \[1\_2D\_Formal\_twisted\] ([b~1~]{}).
It follows from Definition \[surgery\] that a dual 2-dimensional 0-surgery is a 2-dimensional 1-surgery and vice versa. Hence, Fig. \[1\_2D\_Formal\] ([b~1~]{}) shows that a 2-dimensional 0-surgery on a sphere is the reverse process of a 2-dimensional 1-surgery on a torus. Similarly, as illustrated in Fig. \[1\_2D\_Formal\] ([b~2~]{}), a 2-dimensional 1-surgery on a sphere is the reverse process of a 2-dimensional 0-surgery on two spheres. In the figure the symbol $\longleftrightarrow $ depicts surgeries from left to right and their corresponding dual surgeries from right to left.
\[Rot\] The stages of the process of 2-dimensional 0-surgery on $S^2$ can be obtained by rotating the stages of 1-dimensional 0-surgeries on $S^1$ by 180 around a vertical axis, see Fig. \[1\_2D\_Formal\] ([b~1~]{}). Similarly, the stages of 2-dimensional 1-surgery on $S^2$ can be obtained by rotating the stages of 1-dimensional 0-surgeries on $S^1$ by 180 around a horizontal axis, see Fig. \[1\_2D\_Formal\] ([b~2~]{}). It follows from the above that 1-dimensional 0-surgery can be obtained as a cross-section of either type of 2-dimensional surgery.
Dynamic topological surgery in natural processes {#Dynamics}
================================================
In this section we present natural processes exhibiting topological surgery in dimensions 1 and 2 and we incorporate the observed dynamics to a schematic model showing the intermediate steps that are missing from the formal definition. This model extends surgery to a continuous process caused by local forces. Note that these intermediate steps can also be explained through Morse theory, see Remark \[Morse\] for details.
Dynamic 1-dimensional topological surgery {#1D}
-----------------------------------------
We find that $1$-dimensional 0-surgery is present in phenomena where 1-dimensional splicing and reconnection occurs. It can be seen for example during meiosis when new combinations of genes are produced, see Fig. \[1D\_Nature\] (3), and during magnetic reconnection, the phenomena whereby cosmic magnetic field lines from different magnetic domains are spliced to one another, changing their pattern of conductivity with respect to the sources, see Fig. \[1D\_Nature\] (2) from [@DaAn]. It is worth mentioning that $1$-dimensional 0-surgery is also present during the reconnection of vortex tubes in a viscous fluid and quantized vortex tubes in superfluid helium. As mentioned in [@LaRiSu], these cases have some common qualitative features with the magnetic reconnection shown in Fig. \[1D\_Nature\] (2).
In fact, all the above phenomena have similar dynamics. Namely, $1$-dimensional 0-surgery is a continuous process for all of them. Furthermore, as detailed in [@SSN1], in most cases, surgery is the result of local forces. These common features are captured by our model in Fig. \[1D\_Nature\] (1) which describes the process of dynamic $1$-dimensional 0-surgery locally. The process starts with the two points (in red) specified on any 1-dimensional manifold, on which attracting forces are applied (in blue). We assume that these forces are caused by an attracting center (also in blue). Then, the two segments $S^0\times D^1$, which are neighborhoods of the two points, get close to one another. When the specified points (or centers) of the two segments reach the attracting center they touch and recoupling takes place, giving rise to the two final segments $D^1 \times S^0$, which split apart. In Fig. \[1D\_Nature\] (1), case (s) corresponds to the identity embedding $h_s$ described in Section \[1D\_FormalE\], while (t) corresponds to the twisting embedding $h_t$ described in Section \[1D\_FormalE\].
![**(1)** Dynamic 1-dimensional surgery locally **(2)** The reconnection of cosmic magnetic lines **(3)** Crossing over of chromosomes during meiosis.[]{data-label="1D_Nature"}](1D_Nature.jpg){width="12cm"}
As also mentioned in Section \[1D\_FormalE\], the dual case is also a 1-dimensional 0-surgery, as it removes segments $D^1 \times S^0$ and replaces them by segments $S^0\times D^1$. This is the reverse process which starts from the end and is illustrated in Fig. \[1D\_Nature\] (1) as a result of the orange forces and attracting center which are applied on the ‘complementary’ points.
Finally, for details about the described dynamics and attracting forces in the aforementioned phenomena, the reader is referred to the analysis done in [@SSN1].
\[Morse\] It is worth mentioning that the intermediate steps of surgery presented in Fig. \[1D\_Nature\] (1) can also be viewed in the context of Morse theory [@Mil]. By using the local form of a Morse function, we can visualize the process of surgery by varying parameter $t$ of equation $x^2-y^2=t$. For $t=-1$ it is the hyperbola shown in the second stage of Fig. \[1D\_Nature\] (1) where the two segments get close to one another. For $t=0$ it is the two straight lines where the reconnection takes place as shown in the third stage of Fig. \[1D\_Nature\] (1) while for $t=1$ it represents the hyperbola of the two final segments shown in case (s) of the fourth stage of Fig. \[1D\_Nature\] (1). This sequence can be generalized for higher dimensional surgeries as well, however, in this paper we will not use this approach as we are focusing on the introduction of forces and of the attracting center.
Dynamic 2-dimensional topological surgery {#2D}
-----------------------------------------
We find that $2$-dimensional surgery is present in phenomena where 2-dimensional merging and recoupling occurs. For example, *2-dimensional 0-surgery* can be seen during the formation of tornadoes, see Fig. \[2D\_Nature\] (2) (this phenomenon will be detailed in Section \[E2D0\_D3S0\]). Further, it can be seen in the formation of Falaco solitons, see Fig. \[2D\_Nature\] (3) (note that each Falaco soliton consists of a pair of locally unstable but globally stabilized contra-rotating identations in the water-air discontinuity surface of a swimming pool, see [@Ki] for details). It can also be seen in gene transfer in bacteria where the donor cell produces a connecting tube called a ‘pilus’ which attaches to the recipient cell, see Fig. \[2D\_Nature\] (4); in drop coalescence, the phenomenon where two dispersed drops merge into one, and in the formation of black holes (this phenomena will be discussed in Section \[E2D0\]), see Fig. \[ES2D0\_B3\] (2) (Source: www.black-holes.org).
On the other hand, *2-dimensional 1-surgery* can be seen during soap bubble splitting, where a soap bubble splits into two smaller bubbles, see Fig. \[2D\_Nature\] (5) (Source: soapbubble.dk); when the tension applied on metal specimens by tensile forces results in the phenomena of necking and then fracture, see Fig. \[2D\_Nature\] (6); also in the biological process of mitosis, where a cell splits into two new cells, see Fig. \[2D\_Nature\] (7).
![**(1)** Dynamic 2-dimensional surgery locally **(2)** Tornadoes **(3)** Falaco solitons **(4)** Gene transfer in bacteria **(5)** Soap bubble splitting **(6)** Fracture **(7)** Mitosis[]{data-label="2D_Nature"}](2D_Nature.jpg){width="12cm"}
Phenomena exhibiting 2-dimensional 0-surgery are the results of *two colinear attracting forces which ‘create’ a cylinder*. These phenomena have similar dynamics and are characterized by their continuity and the attracting forces causing them, see [@SSN1] for details. These common features are captured by our model in Fig. \[2D\_Nature\] (1) which describes both cases of dynamic $2$-dimensional surgeries locally. Note that we have used non-trivial embeddings (recall Section \[2D\_FormalE0\]) which are more appropriate for natural processes involving twisting, such as tornadoes and Falaco solitons. In Fig. \[2D\_Nature\] (1), both initial discs have been rotated in opposite directions by an angle of $3\pi/4$ and we can see how this rotation induces the twisting of angle $3\pi/2$ of the final cylinder.
The process of *dynamic $2$-dimensional 0-surgery* starts with two points, or poles, specified on the manifold (in red) on which attracting forces caused by an attracting center are applied (in blue). Then, the two discs $S^0\times D^2$, neighbourhoods of the two poles, approach each other. When the centers of the two discs touch, recoupling takes place and the discs get transformed into the final cylinder $D^1\times S^1$, see Fig. \[2D\_Nature\] (1). The cylinder created during 2-dimensional 0-surgery can take various forms. For example, it is a tubular vortex of air in the case of tornadoes, a transverse torsional wave in the case of Falaco solitons and a pilus joining the genes in gene transfer in bacteria.
On the other hand, phenomena exhibiting 2-dimensional 1-surgery are the result of *an infinitum of coplanar attracting forces which ‘collapse’ a cylinder*, see Fig. \[2D\_Nature\] (1) from the end. As mentioned in Section \[2D\_FormalE1\], the dual case of 2-dimensional 0-surgery is the 2-dimensional 1-surgery and vice versa. This is illustrated in Fig. \[2D\_Nature\] (1) where the reverse process is the *2-dimensional 1-surgery* which starts with the cylinder and a specified circular region (in red) on which attracting forces caused by an attracting center are applied (in orange). A ‘necking’ occurs in the middle which degenerates into a point and finally tears apart creating two discs $S^0\times D^2$. This cylinder can be embedded, for example, in the region of the bubble’s surface where splitting occurs, on the region of metal specimens where necking and fracture occurs or on the equator of the cell which is about to undergo a mitotic process.
\[Rot2\] From Definition \[surgery\] we know that surgery always starts by removing a thickened sphere $S^n\times D^{m-n}$ from the initial manifold. As seen in Fig. \[2D\_Nature\] (1), in the case of 2-dimensional 0-surgery, forces (in blue) are applied on two points, or $S^0$, whose thickening comprises the two discs, while in the case of the 2-dimensional 1-surgery, forces (in orange) are applied on a circle $S^1$, whose thickening is the cylinder. In other words the forces that model a $2$-dimensional $n$-surgery are always applied to the core $n$-embedding $e=h_{|}: {S^n}={S^n} \times \{0\} \hookrightarrow M$ of the framed $n$-embedding $h:S^n\times D^{2-n}\hookrightarrow M$. Also, note that Remark \[Rot\] is also true here. One can obtain Fig. \[2D\_Nature\] (1) by rotating Fig. \[1D\_Nature\] (1) and this extends also to the dynamics and forces. For instance, by rotating the two points, or $S^0$, on which the pair of forces of 1-dimensional 0-surgery acts (shown in red in the last instance of Fig. \[1D\_Nature\] (1)) by 180 around a vertical axis we get the circle, or $S^1$, on which the infinitum of coplanar attracting forces of 2-dimensional 1-surgery acts (shown in red in the last instance of Fig. \[2D\_Nature\] (1)).
Finally, it is worth pointing out that these local dynamics produce different manifolds depending on the initial manifold where they act. For example, 2-dimensional 0-surgery transforms an $S^0\times S^2$ to an $S^2$ by adding a cylinder during gene transfer in bacteria (see Fig. \[2D\_Nature\] (4)) but can also transform an $S^2$ to a torus by ‘drilling out’ a cylinder during the formation of Falaco solitons (see Fig. \[2D\_Nature\] (3)) in which case $S^2$ is the pool of water and the cylinder is the boundary of the tubular neighborhood around the thread joining the two poles.
Defining solid topological surgery {#DSolid}
==================================
Looking closer at the phenomena exhibiting 2-dimensional surgery shown in Fig. \[2D\_Nature\], one can see that, with the exception of soap bubble splitting that involves surfaces, all others involve 3-dimensional manifolds. For instance, what really happens during a mitotic process is that a solid cylindrical region located in the center of the cell collapses and a $D^3$ is transformed into an $S^0\times D^3$. Similarly, during tornado formation, the created cylinder is not just a cylindrical surface $D^1\times S^1$ but a solid cylinder $D^2\times S^1$ containing many layers of air (this phenomena will be detailed in Section \[E2D0\_D3S0\] ). Of course we can say that, for phenomena involving 3-dimensional manifolds, the outer layer of the initial manifold is undergoing 2-dimensional surgery. In this section we will define topologically what happens to the whole manifold.
The need of such a definition is also present in dimension 1 for modeling phenomena such as the merging of oil slicks and tension on membranes (or soap films). These phenomena undergo the process of 1-dimensional 0-surgery but happen on surfaces instead of 1-manifolds.
We will now introduce the notion of solid surgery (in both dimensions 1 and 2) where the interior of the initial manifold is filled in. There is one key difference compared to the dynamic surgeries discussed in the previous section. While the local dynamics described in Fig. \[1D\_Nature\] and \[2D\_Nature\] can be embedded in any manifold, here we also have to fix the initial manifold in order to define solid surgery. For example, as we will see next, we define separately the processes of *solid 1-dimensional 0-surgery on $D^2$* and *solid 1-dimensional 0-surgery on $D^2 \times S^0$*. However, the underlying features are common in both.
Solid 1-dimensional topological surgery {#Solid1D}
---------------------------------------
The process of solid 1-dimensional 0-surgery on $D^2$ is equivalent to performing 1-dimensional 0-surgeries on the whole continuum of concentric circles included in $D^2$, see Fig. \[Solid\] (a). More precisely, and introducing at the same time dynamics, we define:
![**(a)** Solid 1-dimensional 0-surgery on $D^2$ **(b~1~)** Solid 2-dimensional 0-surgery on on $D^3$ **(b~2~)** Solid 2-dimensional 1-surgery on $D^3$[]{data-label="Solid"}](Solid.jpg){width="12cm"}
*Solid 1-dimensional 0-surgery on $D^2$* is the following process. We start with the $2$-disc of radius 1 with polar layering: $$D^2 = \cup_{0<r\leq 1} S^1_r \cup \{P\},$$ where $r$ the radius of a circle and $P$ the limit point of the circles, which is the center of the disc and also the circle of radius zero. We specify colinear pairs of antipodal points, all on the same diameter, with neighbourhoods of analogous lengths, on which the same colinear attracting forces act. See Fig. \[Solid\] (a) where these forces and the attracting center are shown in blue. Then the antipodal segments get closer to one another or, equivalently, closer to the attracting center. Note that here, the attracting center coincides with the limit point of all concentric circles, which is shown in green from the second instance and on. Then, we perform 1-dimensional 0-surgery on the whole continuum of concentric circles. We define 1-dimensional 0-surgery on the limit point $P$ to be the two limit points of the resulting surgeries. That is, the effect of *solid 1-dimensional 0-surgery on a point is the creation of two new points*. Next, the segments reconnect until we have two copies of $D^2$.
The above process is the same as first removing the center $P$ from $D^2$, doing the 1-dimensional 0-surgeries and then taking the closure of the resulting space. The resulting manifold is $$\chi(D^2) := \cup_{0<r\leq 1}\chi(S^1_r) \cup \chi(P),$$ which comprises two copies of $D^2$.
We also have the reverse process of the above, namely, *solid 1-dimensional 0-surgery on two discs $D^2 \times S^0$*. This process is the result of the orange forces and attracting center which are applied on the ‘complementary’ points, see Fig. \[Solid\] (a) in reverse order. This operation is equivalent to performing 1-dimensional 0-surgery on the whole continuum of pairs of concentric circles in $D^2\amalg D^2$. We only need to define solid 1-dimensional 0-surgery on two limit points to be the limit point $P$ of the resulting surgeries. That is, the effect of *solid 1-dimensional 0-surgery on two points is their merging into one point*. The above process is the same as first removing the centers from the $D^2 \times S^0$, doing the 1-dimensional 0-surgeries and then taking the closure of the resulting space. The resulting manifold is $$\chi^{-1}(D^2 \times S^0) := \cup_{0<r\leq 1}\chi^{-1}(S^1_r \times S^0) \cup \chi^{-1}(P \times S^0),$$ which comprises one copy of $D^2$.
Solid 2-dimensional topological surgery {#Solid2D}
---------------------------------------
Moving up one dimension, there are two types of solid 2-dimensional surgery on the $3$-ball, $D^3$, analogous to the two types of 2-dimensional surgery. More precisely we have:
\[cont2D\] We start with the $3$-ball of radius 1 with polar layering: $$D^3 = \cup_{0<r\leq 1} S^2_r \cup \{P\},$$ where $r$ the radius of the 2-sphere $S^2_r$ and $P$ the limit point of the spheres, that is, their common center and the center of the ball. *Solid 2-dimensional 0-surgery on $D^3$* is the topological procedure whereby 2-dimensional 0-surgery takes place on each spherical layer that $D^3$ is made of. More precisely, as illustrated in Fig. \[Solid\] ([b~1~]{}), on all spheres $S^2_r$ colinear pairs of antipodal points are specified, all on the same diameter, on which the same colinear attracting forces act. The poles have disc neighborhoods of analogous areas. Then, 2-dimensional 0-surgeries are performed on the whole continuum of the concentric spheres using the same embedding $h$ (recall Section \[2D\_FormalE0\]). Moreover, 2-dimensional 0-surgery on the limit point $P$ is defined to be the limit circle of the nested tori resulting from the continuum of 2-dimensional 0-surgeries. That is, the effect of *2-dimensional 0-surgery on a point is defined to be the creation of a circle*.
The process is characterized on one hand by the 1-dimensional core $L$ of the solid cylinder which joins the two selected antipodal points of the outer shell and intersects each spherical layer at its two corresponding antipodal points, and on the other hand by the embedding $h$. The process results in a continuum of layered tori and can be viewed as drilling out a tunnel along $L$ according to $h$. Note that in Fig. \[Solid\], the identity embedding has been used. However, a twisting embedding, which is the case shown in Fig. \[2D\_Nature\] (1), agrees with our intuition that, for opening a hole, *drilling with twisting* seems to be the easiest way. Examples of these two embeddings can be found in Section \[2D\_FormalE0\].
Furthermore, *solid 2-dimensional 1-surgery on $D^3$* is the topological procedure where on all spheres $S^2_r$ nested cylindrical peels of the solid cylinder of analogous areas are specified and the same coplanar attracting forces act on all spheres, see Fig. \[Solid\] ([b~2~]{}). Then, 2-dimensional 1-surgeries are performed on the whole continuum of the concentric spheres using the same embedding $h$. Moreover, 2-dimensional 1-surgery on the limit point $P$ is defined to be the two limit points of the nested pairs of 2-spheres resulting from the continuum of 2-dimensional surgeries. That is, the effect of *2-dimensional 1-surgery on a point is the creation of two new points*. The process is characterized by the 2-dimensional central disc of the solid cylinder and the embedding $h$, and it can be viewed as squeezing the central disc $C$ or, equivalently, as pulling apart the left and right hemispheres with possible twists, if $h$ is a twisting embedding. This agrees with our intuition that for cutting a solid object apart, *pulling with twisting* seems to be the easiest way. Examples of the identity and the twisting embedding can be found in Section \[2D\_FormalE1\].
For both types of solid 2-dimensional surgery, the above process is the same as: first removing the center $P$ from $D^3$, performing the 2-dimensional surgeries and then taking the closure of the resulting space. Namely we obtain: $$\chi(D^3) := \cup_{0<r\leq 1}\chi(S^2_r) \cup \chi(P),$$ which is a solid torus in the case of solid 2-dimensional 0-surgery and two copies of $D^3$ in the case of solid 2-dimensional 1-surgery.
As seen in Fig. \[Solid\], we also have the two dual solid 2-dimensional surgeries, which represent the reverse processes. As already mentioned in Section \[2D\_FormalE1\], the dual case of 2-dimensional 0-surgery is the 2-dimensional 1-surgery and vice versa. More precisely:
\[solto2D\] The *dual case of solid 2-dimensional 0-surgery on $D^3$* is the solid 2-dimensional 1-surgery on a solid torus $D^2 \times S^1$. This is the reverse process shown in Fig. \[Solid\] ([b~1~]{}) which results from the orange forces and attracting center. Given that the solid torus can be written as a union of nested tori together with the core circle: $D^2 \times S^1=(\cup_{0<r\leq 1}S^1_r \cup \{0\}) \times S^1$, solid 2-dimensional 1-surgeries are performed on each toroidal layer starting from specified annular peels of analogous sizes where the same coplanar forces act on the central rings of the annuli. These forces are caused by the same attracting center lying outside the torus. It only remain to define the solid 2-dimensional 1-surgery on the limit circle to be the limit point $P$ of the resulting surgeries. That is, the effect of *solid 2-dimensional 1-surgery on the core circle is that it collapses into one point, the attracting center*. The above process is the same as first removing the core circle from $D^2 \times S^1$, doing the 2-dimensional 1-surgeries on the layered tori, with the same coplanar acting forces, and then taking the closure of the resulting space. Hence, the resulting manifold is
$$\chi^{-1}(D^2 \times S^1) := \cup_{0<r\leq 1}\chi^{-1}(S^1_r \times S^1) \cup \chi^{-1}(\{0\} \times S^1),$$ which comprises one copy of $D^3$.
Further, *the dual case of solid 2-dimensional 1-surgery on $D^3$* is the solid 2-dimensional 0-surgery on two $3$-balls $D^3$. This is the reverse process shown in Fig. \[Solid\] ([b~2~]{}) which results from the blue forces and attracting center. We only need to define the solid 2-dimensional 0-surgery on two limit points to be the limit point $P$ of the resulting surgeries. That is, as in solid 1-dimensional surgery (see Fig. \[Solid\] (a)), the effect of *solid 2-dimensional 0-surgery on two points is their merging into one point*. The above process is the same as first removing the centers from the $D^3 \times S^0$, doing the 2-dimensional 0-surgeries on the nested spheres with the same colinear forces and then taking the closure of the resulting space. The resulting manifold is $$\chi^{-1}(D^3 \times S^0) := \cup_{0<r\leq 1}\chi^{-1}(S^2_r \times S^0) \cup \chi^{-1}(P \times S^0),$$ which comprises one copy of $D^3$.
Note that remarks \[Rot\] and \[Rot2\] are also true here. One can obtain Fig. \[Solid\] (b~1~) and Fig. \[Solid\] (b~2~) by rotating Fig. \[Solid\] (a) respectively by 180 around a vertical axis and by 180 around a horizontal axis.
The notions of 2-dimensional (resp. solid 2-dimensional) surgery, can be generalized from $S^2$ (resp. $D^3$) to a surface (resp. a handlebody) of genus $g$ creating a surface (resp. a handlebody) of genus $g \pm 1$ or a disconnected surface (resp. handlebody).
A dynamical system modeling solid 2-dimensional 0-surgery {#DS}
---------------------------------------------------------
In [@SSN1; @SSN2] we present and analyze a dynamical system for modeling 2-dimensional 0-surgery. This system was introduced in [@SaGr1] and is a generalization of the classical Lotka–Volterra system in three dimensions where the classic predator-prey population model is generalized to a two-predator and one-prey system. The parameters of the system affect the dynamics of the populations and they are analyzed in order to determine the bifurcation properties of the system.
![Solid 2-dimensional 0-surgery by changing parameter space from (a) to (b).[]{data-label="LV_Surgery"}](LV_Surgery.jpg){width="12cm"}
In Fig. \[LV\_Surgery\] we reproduce the numerical simulations done in [@SaGr1; @SaGr2], illustrating how by changing the parameters space, solid 2-dimensional 0-surgery is performed on the trajectories of the system. We have used the same colors as in Fig. \[Solid\] ([b~1~]{}) to make comparison easier. What is even more striking is that the new topological definition stating that the effect of 2-dimensional 0-surgery on a point is defined to be the creation of a circle (recall Section \[Solid2D\]) also has a meaning in the language of dynamical systems. More precisely, the limit point in the spherical nesting of trajectories shown in green in Fig. \[LV\_Surgery\] (a) is a steady state point and the core of the toroidal nesting of trajectories shown in green in Fig. \[LV\_Surgery\] (b) is a limit cycle. This connection is detailed in [@SSN1], see also [@SSN2].
Defining embedded topological surgery {#Embedded}
=====================================
As mentioned in the Introduction, we noticed that the ambient space is also involved in some natural processes exhibiting surgery. As we will see in this section, depending on the dimension of the manifold, the ambient space either leaves ‘room’ for the initial manifold to assume a more complicated configuration or it participates more actively in the process. Independently of dimensions, embedding surgery has the advantage that it allows us to view surgery as a process happening inside a space instead of abstractly. We define it as follows:
*An embedded $m$-dimensional $n$-surgery* is a $m$-dimensional $n$-surgery where the initial manifold is an $m$-embedding $e:M \hookrightarrow S^d$, $d\geq m$ of some $m$-manifold $M$, and the result is also viewed as embedded in $S^d$. Namely, according to Definition \[surgery\]: $$M' = \chi(e(M)) = \overline{e(M)\setminus h(S^n\times D^{m-n})} \cup_{h|_{S^n\times S^{m-n-1}}} D^{n+1}\times S^{m-n-1} \hookrightarrow S^d.$$
Since in this analysis we focus on phenomena exhibiting embedded 1- and 2-dimensional surgery in 3-space, from now on we fix $d=3$ and, for our purposes, we consider $S^3$ or ${\mathbb R}^3$ as our standard 3-space.
Embedded 1-dimensional surgery {#Embedded1D}
------------------------------
In dimension 1, the notion of embedded surgery allows the topological modeling of phenomena with more complicated initial 1-manifolds. Let us demonstrate this with the example of site-specific [**DNA recombination**]{}. In this process, the initial manifold is a (circular or linear) DNA molecule. With the help of certain enzymes, site-specific recombination performs a 1-dimensional 0-surgery on the molecule, causing possible knotting or linking of the molecule.
![DNA Recombination as an example of embedded 1-dimensional 0-surgery.[]{data-label="1D_Embedded"}](1D_Embedded.jpg){width="7cm"}
The first electron microscope picture of knotted DNA was presented in [@WaDuCo]. In this experimental study, we see how genetically engineered circular DNA molecules can form DNA knots and links through the action of a certain recombination enzyme. A similar picture is presented in Fig. \[1D\_Embedded\], where site-specific recombination of a DNA molecule produces the Hopf link. It is worth mentioning that there are infinitely many knot types and that 1-dimensional 0-surgery on a knot may change the knot type or even result in a two-component link (as shown in Fig. \[1D\_Embedded\]). Since a knot is by definition an embedding of $M=S^1$ in $S^3$ or ${\mathbb R}^3$, in this case embedded 1-dimensional surgery is the so-called *knot surgery*. A good introductory book on knot theory is [@Ad] among many others.
We can summarize the above by stating that for $M=S^1$, embedding in $S^3$ allows the initial manifold to become any type of knot. More generally, in dimension 1 the ambient space which is of codimension 2 gives enough ‘room’ for the initial 1-manifold to assume a more complicated homeomorphic configuration.
\[EmbededSolid1D\] Of course we also have, in theory, the notion of embedded solid 1-dimensional 0-surgery whereby the initial manifold is an embedding of a disc in 3-space.
Embedded 2-dimensional surgery {#Embedded2D}
------------------------------
Passing now to 2-dimensional surgeries, let us first note that an embedding of a sphere $M=S^2$ in $S^3$ presents no knotting because knotting requires embeddings of codimension 2. However, in this case the ambient space plays a different role. Namely, embedding 2-dimensional surgeries [*allows the complementary space of the initial manifold to participate actively in the process*]{}. Indeed, while some natural phenomena undergoing surgery can be viewed as ‘local’, in the sense that they can be considered independently from the surrounding space, some others are intrinsically related to the surrounding space. This relation can be both *causal*, in the sense that the ambient space is involved in the triggering of the forces causing surgery, and *consequential*, in the sense that the forces causing surgery, can have an impact on the ambient space in which they take place.
Let us recall here that the ambient space $S^3$ can be viewed as ${\mathbb R}^3$ with all points at infinity compactified to one single point: $S^3 = {\mathbb R}^3 \cup \{\infty\}$. Further, it can be viewed as the union of two $3$-balls along the common boundary: $S^3 = B^3 \cup D^3$ where a neighbourhood of the point at infinity can stand for one of the two $3$-balls. Finally, $S^3$ can be viewed as the union of two solid tori along their common boundary: $S^3 = V_1\cup V_2$.
As mentioned in the introduction of Section \[DSolid\], in most natural phenomena that exhibit 2-dimensional surgery, the initial manifold is a *solid* 3-dimensional object. Hence, in the next subsections, we describe natural phenomena undergoing solid 2-dimensional surgeries which exhibit the causal or consequential relation to the ambient space mentioned above and are therefore better described by considering them as embedded in $S^3$ or in ${\mathbb R}^3$. In parallel, we describe how these processes are altering the whole space $S^3$ or ${\mathbb R}^3$.
### A topological model for black hole formation {#E2D0}
Let us start by considering the formation of [**black holes**]{}. Most black holes are formed from the remnants of a large star that dies in a supernova explosion. Their gravitational field is so strong that not even light can escape. In the simulation of a black hole formation in [@Ott], the density distribution at the core of a collapsing massive star is shown. Fig. \[ES2D0\_B3\] (2) shows three instants of this simulation, which indicate that matter performs solid 2-dimensional 0-surgery as it collapses into a black hole. In fact, matter collapses at the center of attraction of the initial manifold $M=D^3$ creating the singularity, that is, the center of the black hole (shown as a black dot in instance (c) of Fig. \[ES2D0\_B3\] (2)), which is surrounded by the toroidal accretion disc (shown in white in instance (c) of Fig. \[ES2D0\_B3\] (2)). Let us be reminded here that an accretion disc is a rotating disc of matter formed by accretion.
![**(1)** Embedded solid 2-dimensional 0-surgery on $M=D^3$ (in ${\mathbb R}^3$) **(2)** Black hole formation []{data-label="ES2D0_B3"}](ES2D0_B3.jpg){width="11cm"}
![Embedded solid 2-dimensional 0-surgery on $M=D^3$ (in $S^3$)[]{data-label="ES2D0_B3_Com"}](ES2D0_B3_Com.jpg){width="12cm"}
Note now that the strong gravitational forces have altered the space surrounding the initial star and that the singularity is created outside the final solid torus. This means that the process of surgery in this phenomenon has moreover altered matter outside the manifold in which it occurs. In other words, *the effect of the forces causing surgery propagates to the complement space, thus causing a more global change in 3-space*. This fact makes black hole formation a phenomenon that topologically undergoes embedded solid 2-dimensional 0-surgery.
In Fig. \[ES2D0\_B3\] (1), we present a schematic model of embedded solid 2-dimensional 0-surgery on $M=D^3$. From the descriptions of $S^3$ in Section \[Embedded2D\], it becomes apparent that embedded solid 2-dimensional 0-surgery on one 3-ball describes the passage from the two-ball description to the two-solid tori description of $S^3$. This can be seen in ${\mathbb R}^3$ in instances (a) to (c) of Fig. \[ES2D0\_B3\] (1) but is more obvious by looking at instances (a) to (c) of Fig. \[ES2D0\_B3\_Com\] which show the corresponding view in $S^3$.
We will now detail the instances of the process of embedded solid 2-dimensional 0-surgery on $M=D^3$ by referring to both the view in $S^3$ and the corresponding decompacified view in ${\mathbb R}^3$. Let $M=D^3$ be the solid ball having arc $L$ as a diameter and the complement space be the other solid ball $B^3$ containing the point at infinity; see instances (a) of Fig. \[ES2D0\_B3\_Com\] and (a) of Fig. \[ES2D0\_B3\]. Note that, in both cases $B^3$ represents the hole space outside $D^3$ which means that the spherical nesting of $B^3$ in instance Fig. \[ES2D0\_B3\] (a) extends to infinity, even though only a subset of $B^3$ is shown. This joining arc $L$ is seen as part of a simple closed curve $l$ passing by the point at infinity. In instances (b) of Fig. \[ES2D0\_B3\_Com\] and (b) of Fig. \[ES2D0\_B3\], we see the ‘drilling’ along $L$ as a result of the attracting forces. This is exactly the same process as in Fig. \[Solid\] ([b~1~]{}) if we restrict it to $D^3$. But since we have embedded the process in $S^3$ or ${\mathbb R}^3$, the complement space $B^3$ participates in the process and, in fact, it is also undergoing solid 2-dimensional 0-surgery. Indeed, the ‘matter’ that is being drilled out from the interior of $D^3$ can be viewed as ‘matter’ of the outer sphere $B^3$ invading $D^3$. In instances (c) of Fig. \[ES2D0\_B3\_Com\] and (c) of Fig. \[ES2D0\_B3\], we can see that, as surgery transforms the solid ball $D^3$ into the solid torus $V_1$, $B^3$ is transformed into $V_2$. That is, the nesting of concentric spheres of $D^3$ (respectively of $B^3$) is transformed into the nesting of concentric tori in the interior of $V_1$ (respectively of $V_2$). The point at the origin (in green), which is also the attracting center, turns into the core curve $c$ of $V_1$ (in green) which, by Definition \[cont2D\] is 2-dimensional 0-surgery on a point. As seen in instance (c) of Fig. \[ES2D0\_B3\_Com\] and (c) of Fig. \[ES2D0\_B3\] (1), the result of surgery is the two solid tori $V_1$ and $V_2$ forming $S^3$.
The described process can be viewed as a double surgery resulting from a single attracting center which is inside the first 3-ball $D^3$ and outside the second 3-ball $B^3$. This attracting center is illustrated (in blue) in instance (a) of Fig. \[ES2D0\_B3\] but also in (a) of Fig. \[ES2D0\_B3\_Com\], where it is shown that the colinear attracting forces causing the double surgery can be viewed as acting on $D^3$ (the two blue arrows) and also as acting on the complement space $B^3$ (the two dotted blue arrows), since they are applied on the common boundary of the two 3-balls. Note that in both cases, the attracting center coincides with the limit point of the spherical layers that $D^3$ is made of, that is, their common center and the center of $D^3$ (shown in green in (a) of Fig. \[ES2D0\_B3\] and (a) of Fig. \[ES2D0\_B3\_Com\]). For more details on the descriptions of $S^3$ and their relation to surgery, the reader is referred to [@SSN1].
The reverse process of embedded solid 2-dimensional 0-surgery on $D^3$ is an embedded solid 2-dimensional 1-surgery on the solid torus $V_2$, see instances of Fig. \[ES2D0\_B3\_Com\] in reverse order. This process is the embedded analog of the solid 2-dimensional 1-surgery on a solid torus $D^2 \times S^1$ defined in Definition \[solto2D\] and shown in Fig. \[Solid\] ([b~1~]{}) in reverse order. Here too, the process can be viewed as a double surgery resulting from one attracting center which is outside the first solid torus $V_1$ and inside the second solid torus $V_2$. This attracting center is illustrated (in orange) in instance (c) of Fig. \[ES2D0\_B3\_Com\] where it is shown that the coplanar forces causing surgery are applied on the common boundary of $V_1$ and $V_2$ and can be viewed as attracting forces along a longitude when acting on $V_1$ and as attracting forces along a meridian when acting on the complement space $V_2$.
One can now directly appreciate the correspondence of the physical phenomena (instances (a), (b), (c) of Fig. \[ES2D0\_B3\] (2)) with our schematic model (instances (a), (b), (c) of Fig. \[ES2D0\_B3\] (1)) . Indeed, if one looks at the formation of black holes and examines it as an isolated event in space, this process shows a decompactified view of the passage from a two 3-ball description of $S^3$, that is, the core of the star and the surrounding space, to a two solid tori description, namely the toroidal accretion disc surrounding the black hole (shown in white in instance (c) of Fig. \[ES2D0\_B3\] (2)) and the surrounding space.
\[Duality2d0\] It is worth pinning down the following spatial duality of embedded solid 2-dimensional 0-surgery for $M=D^3$: the attraction of two points lying on the boundary of segment $L$ by the center of $D^3$ can be equivalently viewed in the complement space as the repulsion of these points by the center of $B^3$ (that is, the point at infinity) on the boundary of the segment $l-L$ (or the segments, if viewed in ${\mathbb R}^3$). Hence, the aforementioned duality tells us that the attracting forces from the attracting center that are collapsing the core of the star can be equivalently viewed as repelling forces from the point at infinity lying in the surrounding space.
### A topological model for the formation of tornadoes {#E2D0_D3S0}
Another example of global phenomenon is the formation of [**tornadoes**]{}, recall Fig. \[2D\_Nature\] (2). As mentioned in Section \[DSolid\] this phenomenon can be modelled by solid 2-dimensional 0-surgery, recall Fig. \[Solid\] (b~2~) (from right to left). However, here, the initial manifold is different than $D^3$. Indeed, if we consider a 3-ball around a point of the cloud and another 3-ball around a point on the ground, then the initial manifold is $M=D^3 \times S^0$. If certain meteorological conditions are met, an attracting force between the cloud and the earth beneath is created. This force is shown in blue in see Fig. \[TornadoSequence\] (1). Then, funnel-shaped clouds start descending toward the ground, see Fig. \[TornadoSequence\] (2). Once they reach it, they become tornadoes, see Fig. \[TornadoSequence\] (3). The only difference compared to our model is that here the attracting center is on the ground, see Fig. \[TornadoSequence\] (1), and only one of the two 3-balls (the 3-ball of cloud) is deformed by the attraction. This lack of symmetry in the process can be obviously explained by the big difference in the density of the materials.
![**(1)** Attracting force between the cloud and the earth **(2)** Funnel-shaped clouds **(3)** Tornado[]{data-label="TornadoSequence"}](TornadoSequence.jpg){width="11.7cm"}
During this process, a solid cylinder $D^2\times S^1$ containing many layers of air is created. Each layer of air revolves in a helicoidal motion which is modeled using a twisting embedding as shown in Fig. \[2D\_Nature\] (1) (for an example of a twisting embedding, the reader is referred Section \[2D\_FormalE0\]). Although all these layers undergo local dynamic 2-dimensional 0-surgeries which are triggered by local forces (shown in blue in Fig. \[TornadoSequence\] (1)), these local forces are not enough to explain the dynamics of the phenomenon. Indeed, the process is triggered by the difference in the conditions of the lower and upper atmosphere which create an air cycle. *This air cycle lies in the complement space of the initial manifold* $M=D^3 \times S^0$ *and of the solid cylinder* $D^2\times S^1$ *, but is also involved in the creation of the funnel-shaped clouds that will join the two initial 3-balls*. Therefore in this phenomenon, surgery is the outcome of global changes and this fact makes tornado formation an example of embedded solid 2-dimensional 0-surgery on $M=D^3 \times S^0$.
It is worth mentioning that the complement space containing the aforementioned air cycle is also undergoing solid 2-dimensional 0-surgery. The process can be seen in ${\mathbb R}^3$ in instances (a) to (d) of Fig. \[ES2D0\_ES2D1\] while the corresponding view in $S^3$ is shown in instances (a$'$) to (d$'$) of Fig. \[ES2D0\_ES2D1\].
![Embedded solid 2-dimensional 0-surgery on $D^{3}_{1} \amalg D^{3}_{2}$ (from left to right) and embedded solid 2-dimensional 1-surgery on $D'^3$ (from right to left)[]{data-label="ES2D0_ES2D1"}](ES2D0_ES2D1.jpg){width="11cm"}
More precisely, let us name the two initial 3-balls ${D^{3}_{1}}$ and ${D^{3}_{2}}$, hence $M=D^3 \times S^0 ={{D^{3}_{1}}} \amalg {{D^{3}_{2}}}$. Further, let $B^3$ be the complement of ${D^{3}_{1}}$ in $S^3$. This setup is shown in (a$'$) of Fig. \[ES2D0\_ES2D1\] where $S^3$ is viewed as the union of the two 3-balls ${D^{3}_{1}} \cup B^3$, and here too, $B^3$ represents everything outside ${D^{3}_{1}}$. The complement space of the initial manifold, $S^3 \setminus M=B^3\setminus D^{3}_{2}$, is the 3-ball $B^3$ where ${D^{3}_{2}}$ has been removed from its interior and its boundary consists in two spheres $S^2 \times S^0$, one bounding $B^3$ or, equivalently, ${D^{3}_{1}}$ (the outside sphere) and one bounding ${D^{3}_{2}}$ (the inside sphere). Next, ${D^{3}_{1}}$ and ${D^{3}_{2}}$ approach each other, see Fig. \[ES2D0\_ES2D1\] (b$'$). In (c$'$) of Fig. \[ES2D0\_ES2D1\], ${D^{3}_{1}}$ and ${D^{3}_{2}}$ merge and become the new 3-ball $D'^3$, see Fig. \[ES2D0\_ES2D1\] (c$'$) or Fig. \[ES2D0\_ES2D1\] (d$'$) for a homeomorphic representation.
At the moment of merging, the spherical boundary of ${D^{3}_{2}}$ punctures the boundary of $B^3$; see the passage from (b$'$) to (c$'$) of Fig. \[ES2D0\_ES2D1\]. As a result, the complement space is transformed from $B^3\setminus D^{3}_{2}$ to the new deformed 3-ball $B'^3$, see Fig. \[ES2D0\_ES2D1\] (c$'$) or Fig. \[ES2D0\_ES2D1\] (d$'$) for a homeomorphic representation. Note that, although the complement space undergoes a type of surgery that is different from the ones defined in Section \[DSolid\] and shown in Fig. \[Solid\], it can still be defined analogously. In short, we have a double solid 2-dimensional 0-surgery which turns $M={D^{3}_{1}} \amalg {D^{3}_{2}}$ into ${D'^3}$ and the complement space $S^3 \setminus ({D^{3}_{1}} \amalg {D^{3}_{2}})$ into $B'^3$. This process is initiated by the attracting center shown (in blue) in Fig. \[ES2D0\_ES2D1\] (a$'$). The created colinear forces can be viewed as acting on ${D^{3}_{1}} \amalg {D^{3}_{2}}$ or, equivalently, as acting on the complement space $S^3 \setminus ({D^{3}_{1}} \amalg {D^{3}_{2}})$ (see the two blue arrows for both cases).
Going back to the formation of tornadoes, the above process describes what happens to the complement space and provides a topological description of the behavior of the air cycle during the formation of tornadoes. The complement space $B^3\setminus D^{3}_{2}$ in ${\mathbb R}^3$ is shown in red in Fig. \[ES2D0\_ES2D1\] (a) and its behavior during the process can be seen in instances (b) to (d) of Fig. \[ES2D0\_ES2D1\]. Note that in Fig. \[ES2D0\_ES2D1\] (a), $B^3\setminus D^{3}_{2}$ represents the hole space outside $D^{3}_{1}$, which means that the red layers of Fig. \[ES2D0\_ES2D1\] (a) extend to infinity and only a subset is shown.
### Embedded solid 2-dimensional 1-surgery on $M=D^3$ {#E2D1}
We will now discuss the process of *embedded solid 2-dimensional 1-surgery* in $S^3$. Taking $M=D'^3$ as the initial manifold, embedded solid 2-dimensional 1-surgery is the reverse process of embedded solid 2-dimensional 0-surgery on $D^3 \times S^0$ and is illustrated in Fig. \[ES2D0\_ES2D1\] from right to left. The process is initiated by the attracting center shown (in orange) in (d$'$) of Fig. \[ES2D0\_ES2D1\]. The created coplanar attracting forces are applied on the circle which is the common boundary of the meridian of $D'^3$ and the meridian of $B'^3$ and they can be viewed as acting on the meridional disc $D$ of the 3-ball $D'^3$ (see orange arrows) or, equivalently, in the complement space, on the meridional disc $d$ of $B'^3$ (see dotted orange arrows). As a result of these forces, in Fig. \[ES2D0\_ES2D1\] (c$'$), we see that while disc $D$ of $D'^3$ is getting squeezed, disc $d$ of $B'^3$ is enlarged. In Fig. \[ES2D0\_ES2D1\] (b$'$), the central disc $d$ of $B'^3$ engulfs disc $D$ and becomes $d \cup D$, which is a separating plane in ${\mathbb R}^3$, see Fig. \[ES2D0\_ES2D1\] (b). At this point the initial 3-ball $D'^3$ is split in two new 3-balls $D_1^3$ and $D_2^3$; see Fig. \[ES2D0\_ES2D1\] (b$'$) or Fig. \[ES2D0\_ES2D1\] (a$'$) for a homeomorphic representation. The center point of $D'^3$ (which coincides with the orange attracting center) evolves into the two centers of $D_1^3$ and $D_2^3$ (in green) which by Definition \[cont2D\], is 2-dimensional 1-surgery on a point. This is exactly the same process as in Fig. \[Solid\] ([b~2~]{}) if we restrict it to $D'^3$, but since we are in $S^3$, the complement space $B'^3$ is also undergoing, by symmetry, solid 2-dimensional 1-surgery.
All natural phenomena undergoing embedded solid 2-dimensional 1-surgery take place in the ambient 3-space. The converse, however, is not true. For example, the phenomena exhibiting 2-dimensional 1-surgery discussed in Section \[2D\] are all embedded in 3-space, but they do not exhibit the intrinsic properties of embedded 2-dimensional surgery, since they do not demonstrate the causal or consequential effects discussed in Section \[Embedded2D\] involving the ambient space. Yet one could, for example, imagine taking a solid material specimen, stress it until necking occurs and then immerse it in some liquid until its pressure causes fracture to the specimen. In this case the complement space is the liquid and it triggers the process of surgery. Therefore, this is an example of embedded solid 2-dimensional 1-surgery where surgery is the outcome of global changes.
\[Duality2d1\] Note that the spatial duality described in embedded solid 2-dimensional 0-surgery, in Remark \[Duality2d0\], is also present in embedded solid 2-dimensional 1-surgery. Namely, the attracting forces from the circular boundary of the central disc $D$ to the center of $D'^3$ shown in (d$'$) of Fig. \[ES2D0\_ES2D1\], can be equivalently viewed in the complement space as repelling forces from the center of $B'^3$ (that is, the point at infinity) to the boundary of the central disc $d$, which coincides with the boundary of $D$.
\[SumsUp\] One can sum up the processes described in this section as follows. The process of embedded solid 2-dimensional 0-surgery on $D^3$ consists in taking a solid cylinder such that the part $S^0 \times D^2$ of its boundary lies in the boundary of $D^3$, removing it from $D^3$ and adding it to $B^3$. Similarly, the reverse process of embedded solid 2-dimensional 1-surgery on $V_2$ consists of taking a solid cylinder such that the part $S^1 \times D^1$ of its boundary lies in the boundary of $V_2$, removing it from $V_2$ and adding it to $V_1$. Following the same pattern, embedded solid 2-dimensional 1-surgery on $M=D^3$ consists of taking a solid cylinder in $D'^3$ such that the part $S^1 \times D^1$ of its boundary lies in the boundary of $D'^3$, removing it from $D'^3$ and adding it to $B'^3$. Similarly, the reverse process of embedded solid 2-dimensional 0-surgery on $S^3 \setminus ({D^{3}_{1}} \amalg {D^{3}_{2}})$ consists of taking a solid cylinder such that the two parts $S^0 \times D^2={D^{2}_{1}} \amalg {D^{2}_{2}}$ of its boundary lie in the corresponding two parts of the boundary of $S^3 \setminus ({D^{3}_{1}} \amalg {D^{3}_{2}})$, removing it from $S^3 \setminus ({D^{3}_{1}} \amalg {D^{3}_{2}})$ and adding it to ${D^{3}_{1}} \amalg {D^{3}_{2}}$. Note that, for clarity, in the above descriptions the attracting centers causing surgery are always inside the initial manifold. Of course a similar description starting with the complement space as an initial manifold and the attracting center outside of it would also have been correct.
Conclusions
===========
In this paper, inspired by natural phenomena exhibiting topological surgery, we introduced the notions of dynamic, solid and embedded surgery in dimensions 1, 2 and 3. There are many more natural phenomena exhibiting surgery and we believe that the understanding of their underpinning topology will lead to the better understanding of the phenomena themselves, as well as to new mathematical notions, which will in turn lead to new physical implications.
We wish to express our gratitude to Louis H.Kauffman and Cameron McA.Gordon for many fruitful conversations on topological surgery. We would also like to thank the Referee for his/her positive comments and for helping us clarify some key notions.\
We further wish to acknowledge that this research has been co-financed by the European Union (European Social Fund - ESF) and the Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.
[00]{}
Adams, C.: The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society (2004)
Antoniou, S., Lambropoulou, S.: Extending Topological Surgery to Natural Processes and Dynamical Systems. ArXiv:1603.03644 \[math.GT\]. Submitted To PLOS ONE: minor revision (2016)
Dahlburg, R.B., Antiochos, S.K.: Reconnection of antiparallel magnetic flux tubes. J Geophys Res 100, No. A9: 16991–16998 (1995) doi: 10.1029/95JA01613
Kiehn, R.M.: Non-equilibrium systems and irreversible processes - Adventures in applied topology vol. 1 - Non equilibrium thermodynamics, pp. 147, 150. University of Houston Copyright CSDC Inc (2004)
Laing, C.E., Ricca, R.L., Sumners, D.: Conservation of writhe helicity under anti-parallel reconnection. Scientific Reports 5, No 9224 (2014) doi: 10.1038/srep09224
Lambropoulou, S., Antoniou, S.: Topological Surgery, Dynamics and Applications to Natural Processes. To appear in Journal of Knot Theory and its Ramifications (2016)
Milnor, J.: Morse Theory. Princeton University Press (1963) Ott, C.D., et al: Dynamics and Gravitational Wave Signature of Collapsar Formation. Phys Rev Lett 106, 161103 (2011) doi: 10.1103/PhysRevLett.106.161103
Prasolov, V.V., Sossinsky, A.B.: Knots, links, braids and 3-manifolds. AMS Translations of Mathematical Monographs 154 (1997)
Ranicki, A.: Algebraic and Geometric Surgery. Oxford Mathematical Monographs, Clarendon Press (2002)
Rolfsen, D.: Knots and links. Publish or Perish Inc. AMS Chelsea Publishing (2003)
Samardzija, N., Greller, L.: Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model. Bull Math Biol 50, No. 5: 465–491 (1988) doi: 10.1007/BF02458847
Samardzija, N., Greller, L.: Nested tori in a 3-variable mass action model. Proc R Soc Lond A Math Phys Sci 439, No. 1907: 637–647 (1992) doi: 10.1098/rspa.1992.0173
Wasserman, S.A, Dungan, J.M., Cozzarelli, N.R.: Discovery of a predicted DNA knot substantiates a model for site-specific recombination. Science 229, 171–174 (1985)
|
---
address:
- 'Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstra[ß]{}e 11, 30157 Hannover, Germany'
- 'Department of Geotechnical Engineering, Tongji University, Siping Road 1239, 200092 Shanghai, P.R.China'
- 'Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, 200092 Shanghai, P.R.China'
author:
- Yiming Zhang
- Xiaoying Zhuang
bibliography:
- 'Reference.bib'
title: 'A softening-healing law for self-healing quasi-brittle materials: analyzing with Strong Discontinuity embedded Approach'
---
Traction separation law ,Self-healing ,Quasi-brittle materials ,Strong Discontinuity embedded Approach (SDA)
|
---
abstract: |
We determine the critical exponent and the recurrence function of complementary symmetric Rote sequences. The formulae are expressed in terms of the continued fraction expansions associated with the S-adic representations of the corresponding standard Sturmian sequences. The results are based on a thorough study of return words to bispecial factors of Sturmian sequences. Using the formula for the critical exponent, we describe all complementary symmetric Rote sequences with the critical exponent less than or equal to 3, and we show that there are uncountably many complementary symmetric Rote sequences with the critical exponent less than the critical exponent of the Fibonacci sequence. Our study is motivated by a conjecture on sequences rich in palindromes formulated by Baranwal and Shallit. Its recent solution by Curie, Mol, and Rampersad uses two particular complementary symmetric Rote sequences.
*Keywords: Critical Exponent, Recurrence Function, Rote Sequence, Sturmian Sequence, Return Word, Bispecial Factor*
*2000MSC:* 68R15
address: 'Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19, Prague 1, Czech Republic'
author:
- Ľubomíra Dvořáková
- Kateřina Medková
- Edita Pelantová
title: 'Complementary symmetric Rote sequences: the critical exponent and the recurrence function'
---
§[S]{}
\[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Observation]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Notation]{} \[thm\][Remark]{} \[thm\][Example]{}
Introduction {#introduction .unnumbered}
============
We study the relation between the critical exponents of two binary sequences $\vv = v_0v_1v_2 \cdots$ and $\uu = u_0u_1u_2 \cdots$ over the alphabet $\{0,1\}$, where $u_i = v_i+v_{i+1} \mod 2$ for each $i \in \N$. We write $\uu = \S(\vv)$. Our study is motivated by a conjecture formulated by Baranwal and Shallit in [@BaSh]. They searched for binary sequences rich in palindromes with a minimum critical exponent. They showed that the value of this critical exponent is greater than $2.707$. Moreover, they found two sequences $\vv^{(1)}$ and $\vv^{(2)}$ having the critical exponent equal to $2+ \tfrac{1}{\sqrt{2}}$ and they conjectured that this is the minimum value. Both of these sequences belong to the class of complementary symmetric Rote sequences. Their conjecture has been recently proved by Curie, Mol, and Rampersad in [@CuMoRa].
A Rote sequence is a binary sequence $\vv$ containing $2n$ factors of length $n$ for every $n \in \N, n \geq 1$. If the language of $\vv$ is invariant under the exchange of letters $0\leftrightarrow 1$, the sequence $\vv$ is called a complementary symmetric (CS) Rote sequence. Already in his original paper [@Ro94], Rote proved that these sequences are essentially connected with Sturmian sequences. He deduced that a binary sequence $\vv$ is CS Rote sequence if and only if the sequence $\uu =\S(\vv)$ is Sturmian. Both CS Rote sequences and Sturmian sequences are rich in palindromes, see [@BlBrLaVu], [@DrJuPi].
The formula for the critical exponent of Sturmian sequences was provided by Damanik and Lenz in [@DaLe]. The relation between the critical exponent of a CS Rote sequence $\vv$ and the associated Sturmian sequence $\S(\vv)$ is not straightforward: While the minimum exponent among all Sturmian sequences is reached by the Fibonacci sequence and it is $3+ \frac{2}{1+\sqrt{5}}$ (see [@MiPi]), the two CS Rote sequences $\vv^{(1)}$, $\vv^{(2)}$ whose critical exponent equals $2+ \frac{1}{\sqrt{2}}$, i.e., it is minimum among all binary rich sequences, are associated with the Sturmian sequences $\S(\vv^{(1)})$ and $\S(\vv^{(2)})$ whose critical exponent is $3+\sqrt{2}$.
In this paper, we will first derive the relation between the critical exponents of the sequences $\vv$ and $\S(\vv)$, where $\vv$ is a uniformly recurrent binary sequence whose language is closed under the exchange of letters, see Theorem \[Lem\_PrevodKritExpoRoteNaSturm\]. Using this relation, we will determine the formula for the critical exponent of any CS Rote sequence, see Theorem \[T\_KritExpo\].
One of the consequences of this theorem is for instance the fact that the languages of the sequences $\vv^{(1)}$ and $\vv^{(2)}$ are the only languages of CS Rote sequences with the critical exponent less than $3$, see Proposition \[Prop\_MalyExponent\]. In this context, let us mention that in [@CuMoRa] the authors showed that there are exactly two languages of rich binary sequences with the critical exponent less than $\tfrac{14}{5}$ and they are the languages of the sequences $\vv^{(1)}$ and $\vv^{(2)}$. Furthermore, we show that there are uncountably many CS Rote sequences with the critical exponent strictly less than the critical exponent of the Fibonacci sequence, see Theorem \[Thm\_Exponent72\].
Our main technical tool is the description of return words to bispecial factors of Sturmian sequences in terms of the continued fraction expansions related to the S-adic representations of Sturmian sequences. As a by-product, we obtain an explicit formula for the recurrence function of CS Rote sequences, see Theorem \[Thm\_RecFunctionRoteDetailed\]. When formulating our results, we use the convergents $\bigl(\tfrac{p_N}{q_N}\bigr)$ of an irrational number $\theta=[0,a_1,a_2,a_3,\ldots]$, where the coefficients $a_i$’s in the continued fraction expansion of $\theta$ correspond to the S-adic representation of the standard Sturmian sequence associated to a given CS Rote sequence.
There are many generalizations of Sturmian sequences to multiliteral alphabets, see [@BaPeSt]. The critical exponent and the recurrence function were studied for two of these generalizations. Justin and Pirillo described in [@JuPi] the critical exponent of substitutive Arnoux-Rauzy sequences. Recently, Rampersad, Shallit, and Vandomme in [@RaShVa], and Baranwal and Shallit in [@BaShBalanced] determined the minimal threshold for the critical exponent of balanced sequences over alphabets of cardinality $3, 4$, and $5$, respectively. The recurrence function of Sturmian sequences was found by Morse and Hedlund in [@MoHe], and their result was generalized by Cassaigne and Chekhova in [@CaCh] for Arnoux-Rauzy sequences.
The paper is organized as follows. We first introduce basic notions from combinatorics on words in Section \[S\_Preliminaries\]. In Section \[S\_CriticalExponent\], we recall how to simplify the formula for the critical exponent using return words to bispecial factors. The definitions of the already mentioned mapping $\S$ and complementary symmetric Rote sequences and their basic properties are provided in Section \[S\_mapS\_CSRote\]. The relation between the critical exponents of the sequences $\vv$ and $\S(\vv)$ is described in Section \[S\_v\_andSv\]. The main tool for further results – a thorough study of return words to bispecial factors of Sturmian sequences using the S-adic representation – is carried out in Section \[S\_retwordsBS\_Sturmian\]. An explicit formula for the critical exponent of CS Rote sequences is given in Section \[S\_CriticalExponentRote\]. CS Rote sequences with a small critical exponent are studied in Section \[S\_smallexponent\]. And finally, in Section \[S\_recurrencefunction\], an explicit formula for the recurrence function of CS Rote sequences is derived.
Preliminaries {#S_Preliminaries}
=============
An *alphabet* $\A$ is a finite set of symbols called *letters*. A *word* over $\A$ of *length* $n$ is a string $u = u_0u_1 \cdots u_{n-1}$, where $u_i \in \A$ for all $i \in \{0,1, \ldots, n-1\}$. The length of $u$ is denoted by $|u|$. The set of all finite words over $\A$ together with the operation of concatenation form a monoid $\A^*$. Its neutral element is the *empty word* $\varepsilon$ and we denote $\A^+ = \A^* \setminus \{\varepsilon\}$.
If $u = xyz$ for some $x,y,z \in \A^*$, then $x$ is a *prefix* of $u$, $z$ is a *suffix* of $u$ and $y$ is a *factor* of $u$. We sometimes use the notation $yz = x^{-1}u$.
To any word $u$ over $\A$ with the cardinality $\#\A = d$, we assign its *Parikh vector* $\vec{V}(u) \in \N^{d}$ defined as $(\vec{V}(u))_a = |u|_a$ for all $a \in \A$, where $|u|_a$ is the number of letters $a$ occurring in $u$.
A *sequence* over $\A$ is an infinite string $\uu = u_0u_1u_2 \cdots$, where $u_i \in \A$ for all $i \in \N = \{0,1,2, \ldots\}$. We always denote sequences by bold letters. A sequence $\uu$ is *eventually periodic* if $\uu = vwww \cdots = v(w)^\omega$ for some $v \in \A^*$ and $w \in \A^+$. Otherwise $\uu$ is *aperiodic*.
A *factor* of $\uu$ is a word $y$ such that $y = u_iu_{i+1}u_{i+2} \cdots u_{j-1}$ for some $i, j \in \N$, $i \leq j$. The number $i$ is called an *occurrence* of the factor $y$ in $\uu$. In particular, if $i = j$, the factor $y$ is the empty word $\varepsilon$ and any index $i$ is its occurrence. If $i=0$, the factor $y$ is a *prefix* of $\uu$. If each factor of $\uu$ has infinitely many occurrences in $\uu$, the sequence $\uu$ is *recurrent*. Moreover, if for each factor the distances between its consecutive occurrences are bounded, $\uu$ is *uniformly recurrent*.
The *language* $\mathcal{L}(\uu)$ of the sequence $\uu$ is the set of all factors of $\uu$. A factor $w$ of $\uu$ is *right special* if both words $wa$ and $wb$ are factors of $\uu$ for at least two distinct letters $a,b \in \A$. Analogously we define a *left special* factor. A factor is *bispecial* if it is both left and right special. Note that the empty word $\varepsilon$ is a bispecial factor if at least two distinct letters occur in $\uu$.
The *factor complexity* of a sequence $\uu$ is a mapping $\mathcal{C}_\uu: \N \to \N$ defined by $$\mathcal{C}_\uu(n) = \# \{w \in \mathcal{L}(\uu) : |w| = n \}\, .$$ The aperiodic sequences with the lowest possible factor complexity are called *Sturmian sequences*. In other words, it means that a sequence $\uu$ is *Sturmian* if it has the factor complexity $\mathcal{C}_\uu(n) = n+1$ for all $n \in \N$. Clearly, all Sturmian sequences are defined over a binary alphabet, e.g., $\{0, 1\}$. There are many equivalent definitions of Sturmian sequences, see a survey in [@BaPeSt].
A *morphism* over $\A$ is a mapping $\psi: \A^* \to \A^*$ such that $\psi(uv) = \psi(u)\psi(v)$ for all $u, v \in \A^*$. The morphism $\psi$ can be naturally extended to sequences by $$\psi(\uu)=\psi(u_0u_1u_2 \cdots) = \psi(u_0)\psi(u_1)\psi(u_2) \cdots\,.$$ A *fixed point* of a morphism $\psi$ is a sequence $\uu$ such that $\psi(\uu) = \uu$. The *matrix* of a morphism $\psi$ over $\A$ with the cardinality $\#\A = d$ is the matrix $M_\psi \in \N^{d\times d}$ defined as $(M_\psi)_{ab} = |\psi(a)|_b$ for all $a,b \in \A$. The Parikh vector of the $\psi$-image of a word $w\in \A^*$ can be obtained via multiplication by the matrix $M_\psi$, i.e., $$\label{multi}\vec{V}({\psi(w)}) = M_\psi \vec{V}(w)\,.$$
Consider a prefix $w$ of a recurrent sequence $\uu$. Let $i < j$ be two consecutive occurrences of $w$ in $\uu$. Then the word $u_iu_{i+1} \cdots u_{j-1}$ is a *return word* to $w$ in $\uu$. The set of all return words to $w$ in $\uu$ is denoted $\mathcal{R}_\uu(w)$. If the sequence $\uu$ is uniformly recurrent, the set $\mathcal{R}_\uu(w)$ is finite for each prefix $w$, i.e., $\mathcal{R}_\uu(w) = \{r_0, r_1, \ldots, r_{k-1}\}$. Then the sequence $\uu$ can be written as a concatenation of these return words: $$\uu = r_{d_0}r_{d_1}r_{d_2} \cdots$$ and the *derived sequence* of $\uu$ to the prefix $w$ is the sequence $\dd_\uu(w) = d_0d_1d_2 \cdots$ over the alphabet of cardinality $\# \mathcal{R}_\uu(w) = k$. The concept of derived sequences was introduced by Durand in [@Dur98].
The critical exponent and its relation to return words {#S_CriticalExponent}
======================================================
Let $z \in \mathcal{A}^+$ be a prefix of a periodic sequence $u^\omega$ with $ u \in \mathcal{A}^+$. We say that $z$ has the *fractional root* $u$ and the *exponent* $e = |z|/|u|$. We usually write $z =u^e$. Let us emphasize that a word $z$ can have multiple exponents and fractional roots. A word $z$ is *primitive* if its only integer exponent is $1$.
Let $\uu$ be a sequence and $u$ its non-empty factor. The supremum of $e \in \Q$ such that $u^e$ is a factor of $\uu$ is the *index* of $u$ in $\uu$: $$\begin{aligned}
\ind_\uu(u) = \sup\{e \in \Q: u^e \in \mathcal{L}(\uu)\}\,.\end{aligned}$$ If the sequence $\uu$ is clear from the context, we will write $\ind(u)$ instead of $\ind_\uu(u)$.
The *critical exponent* of a sequence $\uu$ is $$\begin{aligned}
\emph{cr}(\uu) &= \sup \, \{e \in \Q : \text{ there is a non-empty factor of } \uu \text{ with the exponent } e\} \\
&= \sup \, \{ \emph{ind}_\uu(u) : u \text{ is a non-empty factor of } \uu\}\,.\end{aligned}$$
\[remark\] Let us comment the above definition.
1. If a non-empty factor $u \in \mathcal{L}(\uu)$ is non-primitive, i.e., $u=x^k$ for some $x \in \A^+$ and $k \in \mathbb{N}, k\geq 2$, then $\ind_\uu(x) = k\, \ind_\uu(u) > \ind_\uu(u)$. Therefore, only primitive factors play a role for finding $\CR(\uu)$.
2. If some non-empty factor occurs at least twice in $\uu$, then $\ind(x)> 1$ for some non-empty factor $x$ and so $ \CR(\uu) >1$. Consequently, $ \CR(\uu) >1$ for each sequence $\uu$.
3. We say that $u$ is an *overlapping* factor in $\uu$, if there exist $x, y \in \mathcal{A}^*$ such that $xu=uy\in \mathcal{L}(\uu)$ and $ 0< |x| <|u|$. If $\uu$ has an overlapping factor, then $\CR(\uu) > 2$. Indeed, by [@Lo83] the equality $xu=uy$ implies that there exist $a,b \in \mathcal{A}^*$ and $k \in \mathbb{N}$ such that $u=(ab)^ka$, $x=ab$, and $y=ba$. If $a$ is empty then the assumption $|u| >|x|>0$ forces $k\geq 2$, otherwise $k \geq 1$. In both cases $ \ind_\uu(ab)>2$.
4. If $\uu$ is eventually periodic, then $\CR(\uu)$ is infinite.
5. If $\uu$ is aperiodic and uniformly recurrent, then each factor of $\uu$ has a finite index. Nevertheless, $\CR(\uu)$ may be infinite. As an example of such a sequence may serve a Sturmian sequence, for which the coefficients in the continued fraction expansion of its slope are not bounded, see [@DaLe].
6. If $\uu$ is a binary sequence, then either $11$, $00$, or $0101$ occur in $\uu$. It means that the critical exponent of a binary sequence is at least $2$. This value is attained by the famous Thue-Morse sequence, which is, of course, overlap-free, see [@Thue] or [@Berstel].
\[Lem\_CriticalExponentByReturnWords\] Let $\uu$ be a uniformly recurrent aperiodic sequence. Then $\emph{cr}(\uu) = \sup \, \{ \emph{ind}_\uu(u) : u \in \mathcal{M}\}$, where $$\begin{aligned}
\mathcal{M} = \{ u : u \text{ is a return word to a bispecial factor of } \uu \}\,.\end{aligned}$$
Let $u \in \mathcal{L}(\uu)$ be a non-empty factor with the index $\ind(u) >1$. Denote $|u| = n$. When searching for supremum, we may assume without loss of generality that $u$ is a factor having the largest index among all factors of $\uu$ of length $n$, i.e., $\ind(u) \geq \ind(v)$ for all $v \in \mathcal{L}(\uu)$ of length $n$. Since $\uu$ is uniformly recurrent, $\ind(u)$ is finite. We denote $$\begin{aligned}
z = u^{\ind(u)} = u'u'' \cdots u'u''u' \quad \text{ and } \quad b = u^{\ind(u)-1}\,,\end{aligned}$$ where $u = u'u''$ and $u'' \neq \varepsilon$. Clearly, $z = bu''u' = u'u''b$.
Let us show that the word $b$ is a bispecial factor of $\uu$. The word $z$ is a factor of $\uu$ and so $z$ occurs in $\uu$ at some position $j$, i.e., $$z = u_ju_{j+1} \cdots u_{j + |z| -1}\,.$$ Then the letter $u_{j + |z|}$ which follows the word $z$ is distinct from the first letter of $u''$. Otherwise, we could prolong $z$ to the right, which contradicts the definition of the index of $u = u_ju_{j+1} \cdots u_{j+n-1}$. Similarly, the letter $u_{j-1}$ which precedes $z$ is distinct from the last letter of $u''$. Indeed, if those letters are the same, then the factor $u_{j-1}u_{j} \cdots u_{j + n-2}$ of length $n$ has the index at least $\ind(u) + \frac{1}{n}$, which contradicts the choice of $u$. We can conclude that the factor $b$ is a bispecial factor of $\uu$.
Moreover, since $z = bu''u' = u'u''b$, the word $u = u'u''$ is a concatenation of the return words to the bispecial factor $b$. It suffices to prove that only the cases when $u$ is a return word to the bispecial factor $b$ have to be inspected. Let us assume that $u$ is a concatenation of at least two return words to $b$. It means that $$\label{Eq_Rovnice}
z = ub = sbt \quad \text{ for some } s, t \text{ such that $s$ is a prefix of } u \text{ and } 0 < |s| < |u|\,.$$ We will find another factor of $\uu$ with the index strictly larger than $\ind(u)$, which means that such a factor $u$ can be omitted. We distinguish three cases:
- If $\ind(u) \geq 2$, then $|b| \geq |u|$ and both words $u$ and $s$ are prefixes of $b$. Therefore, the relation implies $us = su$ and we can easily conclude that there is a word $x$ and an integer $k > 1$ such that $u = x^k$. As mentioned in Remark \[remark\], $\ind(x)> \ind(u)$.
- If $1 < \ind(u) <2 $ and $\CR(\uu) \leq 2$, then by Item (3) of Remark \[remark\], $\uu$ has no overlapping factor. Clearly, $z=u'u''u'$ and $b=u'$ for $u', u'' \neq \varepsilon$. Then the relation implies $u'v = su'$ for some $v$ and $|u| > |s| \geq |u'|$. Indeed, if $0 < |s| < |u'|$, then $u'$ is an overlapping factor, which is not possible. Therefore, $u'$ is a prefix of $s$ and we can easily deduce that $$\ind(s) \geq \frac{|s|+ |u'| }{|s|} > \frac{|u|+ |u'| }{|u|} = \ind(u) \,.$$
- If $1 < \ind(u) < 2$ and $\CR(\uu) > 2$, then there is a factor $x \in \mathcal{L}(\uu)$ with $\ind(x) > 2 > \ind(u)$.
\[rem\_CriticalExponentByReturnWords\] In fact, we proved that it suffices to consider the set $$\begin{aligned}
\mathcal{M'} = \{ u : u \text{ is a return word to a bispecial factor of } \uu \text{ with the fractional root } u \}\,\end{aligned}$$ or, even more specifically, the set $$\begin{aligned}
\mathcal{M''} = \{ u : u \text{ is a return word to the bispecial factor } b = u^{\ind(u)-1} \text{ of } \uu \}\,\end{aligned}$$ instead of $\mathcal{M}$. Clearly, $\mathcal{M''} \subset \mathcal{M'} \subset \mathcal{M}$.
In Remark \[remark\], we emphasize that only primitive factors are relevant for finding the critical exponent. Let us verify that all return words from the set $\mathcal{M'}$ (and so $\mathcal{M}''$, too) are primitive. We prove it by contradiction. Let us suppose that $u \in \mathcal{M'}$ is non-primitive, i.e., $u = x^k$ for some non-empty $x$ and $k \in \N$, $k > 1$. Since $b$ has the fractional root $u$, $b = u^\ell = x^{k \ell}$ for some $\ell \in \Q$. Therefore, $ub = x^{k(\ell +1)} = xx^{k\ell}x^{k-1} = xbx^{k-1}$, which contradicts that $u$ is a return word to $b$ in $\uu$.
The mapping $\mathcal{S}$ on binary words and complementary symmetric Rote sequences {#S_mapS_CSRote}
====================================================================================
In this section, we introduce a mapping $S$ which enables us to describe the properties of CS Rote sequences using Sturmian sequences. Nevertheless, this mapping $S$ can be applied to any binary sequence.
\[S\] By $\S$ we denote the mapping $\S: \{0,1\}^+ \mapsto \{0,1\}^*$ such that for every $v_0\in \{0,1\}$ we put $\S(v_0) = \varepsilon$ and for every $v = v_0v_1 \cdots v_{n} \in \{0,1\}^+$ of length at least $2$ we put $\S(v_0v_1 \cdots v_{n}) = u_0u_1 \cdots u_{n-1}$, where $$u_{i} = v_i + v_{i+1} \mod 2 \ \ \text{ for all } \ i \in \{0,1, \ldots, n-1\}\,.$$ Moreover, we extend the domain of $\S$ naturally to $\{0,1\}^\N$: for every $\vv \in \{0,1\}^\N$ we put $\S(\vv) = \uu$, where $$u_{i} = v_i + v_{i+1} \mod 2 \ \ \text{ for all } \ i \in \N \,.$$
By $E:\{0,1\}^*\mapsto \{0,1\}^*$ we denote the morphism which exchanges the letters, i.e., $E(0)=1$, $E(1)=0$.
We have $E(001110) = 110001$ and $\S(001110) = \S(110001) = 01001$.
Clearly, the images of $v$ and $E(v)$ under $\S$ coincide for each $v \in \{0,1\}^*$. Moreover, $\S(x) = \S(y)$ if and only if $x = y$ or $x = E(y)$. The following rule follows directly from the definition of $\S$: $$\label{jasne}
\S(v_0v_1 \cdots v_{n}) = \S(v_0 v_1\cdots v_k) \S(v_k v_{k+1} \cdots v_{n})\quad \text{ for any $k = 0, \ldots, n$}.$$ These observations hold also for infinite sequences.
\[lem\_Bispecial\] Let $\vv$ be a binary sequence whose language $\mathcal{L}(\vv)$ is closed under $E$. Then $w\neq \varepsilon$ is a right (left) special factor in $\vv$ if and only if $\S(w)$ is a right (left) special factor in $\S(\vv)$.
We will prove the statement for right special factors. The proof for left special factors is analogous. Let $c$ be the last letter of $w$.
$(\Longrightarrow)$: Let $w0$ and $w1$ belong to $\mathcal{L}(\vv)$. Then $\S(w0)$ and $\S(w1)$ belong to $\mathcal{L}(\S(\vv))$. By the rule , $\S(w0) =\S(w)\S(c0)$ and $\S(w1) =\S(w)\S(c1)$. As $\S(c0) \neq \S(c1)$, the factor $\S(w)$ is right special in $\S(\vv)$.
$(\Longleftarrow)$: Let $\S(w)0$ and $\S(w)1$ be in $\mathcal{L}(\S(\vv))$. Then $\S(w)0 = \S(w)\S(cc) = \S(wc)$ and $\S(w)1 = \S(w)\S(cE(c)) = \S(wE(c))$. Since $\mathcal{L}(\vv)$ is closed under the exchange of letters, all factors $wc$, $E(wc)$, $wE{(c)}$ and $E(w)c$ are in $\mathcal{L}(\vv)$. It means that $w$ and $E(w)$ are right special factors in $\vv$.
A *Rote sequence* is a sequence $\vv$ with the factor complexity $\mathcal{C}_\vv(n) = 2n$ for all $n \in \N, n \geq 1$. Clearly, all Rote sequences are defined over a binary alphabet, e.g., $\{0,1\}$. If the language of a Rote sequence $\vv$ is closed under the exchange of letters, i.e., $E(v)\in \mathcal{L}(\vv)$ for each $v \in \mathcal{L}(\vv)$, the Rote sequence $\vv$ is called *complementary symmetric* (shortly CS).
Rote in [@Ro94] proved that these sequences are essentially connected with Sturmian sequences.
\[Prop\_SturmRote\] Let $\uu$ and $\vv$ be two sequences over $\{0,1\}$ such that $\uu=\S(\vv)$. Then $\vv$ is a complementary symmetric Rote sequence if and only if $\uu$ is a Sturmian sequence.
Let us emphasize that to a given CS Rote sequence $\vv$ there is the unique *associated* Sturmian sequence $\uu$ such that $\uu = \S(\vv)$. On the other hand, for any Sturmian sequence $\uu$ there exist two *associated* CS Rote sequences $\vv$ and $E(\vv)$ such that $\uu = \S(\vv) = \S(E(\vv))$. However, $\mathcal{L}(\vv) = \mathcal{L}(E(\vv))$.
Analogously, to a given factor $v \in \mathcal{L}(\vv)$ there is a unique associated word $u$ such that $u = \S(v)$ and this word $u$ is a factor of $\uu$. In addition, to a given factor $u \in \mathcal{L}(\uu)$ there are exactly two associated words $v, E(v)$ such that $S(v) = \S(E(v)) = u$ and both these words $v, E(v)$ are factors of $\vv$.
\[Fib\] Let us underline that for Sturmian sequences $\uu$ and $E(\uu)$ the languages of their associated CS Rote sequences may essentially differ. Consider the Fibonacci sequence $${\mathbf f}=abaababaaba\cdots\,,$$ which is the fixed point of the Fibonacci morphism $F: a \to ab$, $b \to a$.
- If $a=0$ and $b=1$, then the associated CS Rote sequence starting with $0$ is $\vv=001110011100\cdots$. The prefix $w$ of $\vv$ of length 7 is $w=0011100=(00111)^{\frac{7}{5}}$, i.e., $w$ has the fractional root $00111$ and the exponent $\frac{7}{5}$.
- If $a=1$ and $b=0$, then the associated CS Rote sequence starting with $0$ is $\vv'=011011001001\cdots$. The prefix $w'$ of $\vv'$ of length 7 is $w'=0110110=(011)^{\frac{7}{3}}$, i.e., $w'$ has the fractional root $011$ and the exponent $\frac{7}{3}$.
We will show later in Example \[Ex\_Rozdil\_cr\_u\_Eu\] that even the critical exponent of CS Rote sequences associated with $\uu$ and $E(\uu)$ may be different.
In the next section, we will explain that the relation between the shortest fractional root of a factor $v$ and the shortest fractional root of $\S(v)$ is influenced by the number of letters $1$ occurring in the shortest fractional root of $\S(v)$. This is the reason for the following definition and lemma.
A word $u = u_0u_1 \cdots u_{n-1} \in \{0, 1\}^*$ is called *stable* if $|u|_1 = 0 \mod 2$. Otherwise, $u$ is *unstable*.
\[pomocny\] Let $\S: \{0,1\}^+\mapsto \{0,1\}^*$.
- If $0$ is a prefix of $v \in \{0,1\}^*$, then $\S(v0)$ is stable.
- For every $u \in \{0,1\}^*$ there exists a unique $w \in \{0,1\}^+$ with a prefix $0$ such that $u = \S(w)$. Moreover, $w$ has a suffix $0$ if and only if $u$ is stable.
- If $0$ is a prefix of $w$, then $\S(vw) = \S(v0)\S(w)$.
- Let $0$ be a prefix of $v$ and $v'$. Then $\S(v')$ is a prefix of $\S(v)$ if and only if $v'$ is a prefix of $v$.
<!-- -->
- Let $v = v_0v_1\cdots v_{n-1}$, $n =|v|$, and $v_0=0$. Put $v_n =0$. Then $\S(v0) = u_0u_1\cdots u_{n-1}$, where $u_i =v_{i} + v_{i+1} \mod 2 $ for every $ i\in \{0,1, \ldots, n-1\}$. It implies $$|\S(v0)|_1 = \sum_{i=0}^{n-1}u_i = v_0 + v_n = 0 \mod 2\,.$$
- Let $u= u_0u_1\cdots u_{m-1}$ and $m =|u|$. We look for $w =w_0 w_1\cdots w_m$ such that $u_i = w_i+w_{i+1} \mod 2$ for every $i\in \{0,1,\ldots, m-1\}$. Clearly, these equations can be equivalently rewritten as $$\label{Eq_Kongruence}
u_i = w_{i+1} - w_{i} \mod 2 \quad \text{ for every } i\in \{0,1,\ldots, m-1\}\,.$$ Then starting with $w_0=0$ and summing up the equations for $i=\{0, 1, \ldots, j-1\}$, we determine the letter $w_j$ of $w$ as $w_{j} = \sum_{i=0}^{j-1} u_i \mod 2$. In particular, $w_m = |u|_1\mod 2$.
- It is a particular case of the equation .
- It follows directly from the definition of $\S$.
The relation between the indices of factors in $\vv$ and $\S(\vv)$ {#S_v_andSv}
==================================================================
In this section, we provide a tool for determining the critical exponent of a binary sequence $\vv$ whose language is closed under the exchange of letters. For any factor $v$ of such a sequence, $\ind_\vv(v) =\ind_\vv(E(v))$ and we can consider only factors of $\vv$ starting with $0$ without loss of generality.
\[prevod\] Let $\vv$ be a binary aperiodic uniformly recurrent sequence whose language is closed under $E$. Denote $\uu = \S(\vv)$. For a non-empty factor $v \in \mathcal{L}(\vv)$ with the prefix $0$ and $\ind_\vv(v) > 1$, there exists a stable factor $u \in \mathcal{L}(\uu)$ such that $$\label{vztah}
\ind_\uu(u) + \tfrac{1}{|u|} = \ind_\vv(v) \quad \text{and} \quad u= \S(v0)\,.$$ And vice versa, for a non-empty stable factor $u \in \mathcal{L}(\uu)$, there exists a factor $v \in \mathcal{L}(\vv)$ with the prefix $0$ satisfying .
For a given $n \in \mathbb{N}, n\geq 1$, consider the set $\mathcal{K}_n$ of factors $v \in \mathcal{L}(\vv)$ of length $n$ with the prefix $0$ and $\ind_\vv(v) > 1$. First, we show that the mapping $v \mapsto \S(v0)$ is a bijection between $\mathcal{K}_n$ and the set of all stable factors of $\uu$ of length $n$.
Indeed, if $v \in \mathcal{K}_n$, then $v0 \in \mathcal{L}(\vv)$. The factor $u: = \S(v0)$ belongs to $ \mathcal{L}(\uu)$, $|u| = |v|$, and by Item (i) of Lemma \[pomocny\], $u$ is stable. On the other hand, if $u \in \mathcal{L}(\uu)$ is stable and of length $n$, then by Item (ii) of Lemma \[pomocny\], there exists a unique $w$ such that $0$ is a prefix and a suffix of $w$ and $\S(w) = u$. As $\mathcal{L}(\vv)$ is closed under $E$, necessarily $w \in \mathcal{L}(\vv)$ and $w =v0$ for some $v$ with the prefix $0$. In particular, $\ind_\vv(v) > 1$. As $u =\S(w) =\S(v0)$, the lengths of $u$ and $v$ coincide.
Now we show that any $v \in \mathcal{K}_n$ and its image $u =\S(v0)$ satisfy . Find $k \in \mathbb{N}, k\geq 1$, and $\theta \in (0, 1]$ such that $\ind_\vv(v) =k + \theta$. Denote $v' = v^\theta$. Obviously, $v' \neq \varepsilon$, $v'$ is a prefix of $v$ and $0$ is a prefix of $v'$. Applying Item (iii) of Lemma \[pomocny\], we get $\S(v^kv')= (\S(v0))^k \S(v')$. Clearly, $|\S(v')| = |v'| -1$, $|u| = |v|$, and by Item (iv) of Lemma \[pomocny\], $\S(v')$ is a prefix of $\S(v0)$. For $u = \S(v0)$ it means that $$\ind_\uu(u) \geq k + \tfrac{|S(v')|}{|u|} = k + \tfrac{|v'|}{|v|} - \tfrac{1}{|u|} = k + \theta - \tfrac{1}{|u|} = \ind_\vv(v) - \tfrac{1}{|u|}\,.$$
To show the opposite inequality, we find $\ell \in \mathbb{N}$ and $\eta \in [0,1)$ such that $\ind_\uu(u) =\ell + \eta$. Denote $u' = u^\eta$. Using Item (ii) of Lemma \[pomocny\], we find $v'$ with the prefix $0$ such that $u' = \S(v')$. By Item (iv), $v'$ is a prefix of $v$, and by Item (iii), $u^\ell u' = (\S(v0))^\ell \S(v') = \S(v^\ell v')$. Therefore, $v^\ell v' \in \mathcal{L}(\vv)$ and $$\ind_\vv(v) \geq \ell+\eta + \tfrac{1}{|u|} = \ind_\uu(u)+ \tfrac{1}{|u|}\,.$$
As explained in Lemma \[Lem\_CriticalExponentByReturnWords\], only return words to bispecial factors play a role for the determination of the critical exponent of a sequence. More specifically, we can restrict ourselves to factors from the set $\mathcal{M}'$ (or $\mathcal{M}''$) introduced in Remark \[rem\_CriticalExponentByReturnWords\].
\[CoJeToProU\] Let $\vv$ be a binary sequence whose language is closed under $E$. Assume that $v$ with the prefix $0$ is a return word in $\vv$ to a bispecial factor $b = v^{e-1}$, where $e> 2$. Denote $u =\S(v0)$. Then
– either $u$ is a stable return word in $\S(\vv)$ to a bispecial factor with the fractional root $u$;
– or $u = x^2$, where $x$ is an unstable return word in $\S(\vv)$ to a bispecial factor with the fractional root $u$.
The factor $\S(b)$ is bispecial in $\S(\vv)$ by Lemma \[lem\_Bispecial\]. Moreover, by the rule , we can write $\S(b) = \S(v^{e-1}) = \S(v0)^{f}$ for $f = e-1-\frac{1}{|v|} \geq 1$. Thus $\S(b)$ has the fractional root $u = \S(v0)$.
The word $vb$ is a complete return word to $b$ in $\vv$ and thus $vb =bw$ for some $w$. Note that $0$ is the first letter of $b$ and denote $z$ the last letter of $b$. By the rule , we get $\S(v0)\S(b) = \S(b)\S(zw)$. It means that $\S(b)$ is a prefix and a suffix of the word $\S(v0)\S(b)$. We discuss two cases:
– $\S(b)$ has exactly two occurrences in $\S(v0)\S(b)$, one as a prefix and one as a suffix. In this case $u= \S(v0)$ is a return word to $\S(b)$ in $S(\vv)$ and by Item (i) of Lemma \[pomocny\], $u$ is stable.
– $\S(b)$ occurs in $\S(v0)\S(b)$ as an inner factor. In this case, there exists a return word $u'\neq \varepsilon$ to $\S(b )$ such that $|u'| <|u| $ and $u'\S(b)$ is a proper prefix of $\S(v0)\S(b)$. We take the word $b'$ such that $b= v b'$. Clearly, $b'$ has the prefix $0$ and so $\S(b) = u\S(b')$. Then $u'\S(b) = u'u\S(b')$ is a proper prefix of $\S(v0)\S(b) = u\S(b) = uu\S(b')$. In other words, $u'uu'' = uu$ for some non-empty $u''$ and consequently, $u =u'u'' = u''u'$. This implies the existence of $x \in \{0,1\}^+$ and $k', k'' \in \mathbb{N}, k', k''\geq 1$ such that $u' =x^{k'}$ and $u'' = x^{k''}$. If we denote $k =k'+k'' \geq 2$, we can write $u=x^k$.
We show that $x$ is unstable and $k=2$. Indeed, assume $x$ is stable, then by Item (ii) of Lemma \[pomocny\], we find a unique $y$ with the prefix $0$ such that $ x =\S(y0)$. Applying Item (iii), we obtain $\S(v0)=u= x^k = (\S(y0))^k = \S(y^k0)$ and thus $v =y^k$. Nevertheless, the factor $v$ is primitive as explained in Remark \[rem\_CriticalExponentByReturnWords\]. Thus this is a contradiction.
Since $x$ is unstable and $u=x^k$ is stable, necessarily $k = 2p$ for some integer $p\geq 1$. Now we deduce that $k=2$. Indeed, if $p\geq 2$, then $u$ is a $p$-power of the stable factor $x^2$ which yields a contradiction with the primitivity of $v$ as above. Finally, $k=2$ implies $k' = 1$ and $u' = x$ is an unstable return word to the bispecial factor $\S(b)$ in $S(\vv)$.
\[Lem\_PrevodKritExpoRoteNaSturm\] Let $\vv$ be a binary aperiodic uniformly recurrent sequence whose language is closed under $E$. Denote $\uu = \S(\vv)$,
$A_1 = \left\{ \emph{ind}_\uu(u) + \tfrac{1}{|u|} : u \text{ is a stable return word to a bispecial factor of } \uu \right\} \ $ and
$A_2 = \left\{\tfrac12 \bigl(\emph{ind}_\uu(u) + \tfrac{1}{|u|} \bigr) : u \text{ is an unstable return word to a bispecial factor of } \uu \right\} \,.$
Then $$\emph{cr}(\vv) = \sup \bigl(A_1\cup A_2\bigr)\,.$$
First we show that $$\label{mensi}
\ind_{\vv}(v)\leq \sup \bigl( A_1\cup A_2\bigr) \quad \text{ for any non-empty} \ v \in \mathcal{L}(\vv)\,.$$
If $\ind_{\vv}(v)\leq 2$, then the inequality is trivially satisfied as $A_1$ contains the number $\ind_{\uu}(0) + \tfrac{1}{|0|} \geq 2$ (note that $0$ is a stable return word in $\uu$ to the bispecial factor $\varepsilon$). Now we assume that $\ind_{\vv}(v) = e> 2$. By Lemma \[Lem\_CriticalExponentByReturnWords\] and Remark \[rem\_CriticalExponentByReturnWords\], we may focus only on $v$ which is a return word to the bispecial factor $b =v^{e-1}$ and $v$ has the prefix $0$. By Lemma \[prevod\], $\ind_{\vv}(v) = \ind_{\uu}(u) + \tfrac{1}{|u|}$, where $u= \S(v0)$. By Lemma \[CoJeToProU\], the factor $u$ is either a stable return word to a bispecial factor in $\uu$, or $u = x^2$, where $x$ is an unstable return word to a bispecial factor in $\uu$. In the first case we have $\ind_{\vv}(v) \leq \sup A_1$, while in the second case we have $\ind_{\vv}(v) = \ind_{\uu}(u) + \tfrac{1}{|u|} = \tfrac{1}{2}\ind_{\uu}(x) + \tfrac{1}{2|x|} \leq \sup A_2$. We may conclude that $ \CR(\vv) \leq \sup \bigl(A_1\cup A_2\bigr)\,.$
To prove the opposite inequality, we show $$A_1\cup A_2 \setminus \left[0,1\right] \subset \{\ind_\vv(v) : v \in \mathcal{L}(\vv), v\neq \varepsilon\}\,.$$ If $H \in A_1$, then there exists a stable factor $u$ in $\uu$ such that $H= \ind_\uu(u) + \frac{1}{|u|}$, and by Lemma \[prevod\], we find $v$ in $\vv$ such that $H=\ind_\vv(v)$. Analogously, if $H \in A_2$ and $H >1$, then $\ind_\uu(u) = 2H - \frac{1}{|u|} \geq 2$ for some unstable factor $u \in \mathcal{L}(\uu)$. Thus the word $y=uu \in \mathcal{L}(\uu)$, it is a stable factor of $\uu$ and its index in $\uu$ is $\tfrac12\ind_\uu(u)$. By Lemma \[prevod\], there exists $v$ in $\vv$ such that $\ind_\vv(v) = \ind_\uu(y) + \tfrac{1}{|y|} = \tfrac12\ind_\uu(u) + \tfrac{1}{2|u|} = H.$
Theorem \[Lem\_PrevodKritExpoRoteNaSturm\] will be used in the next sections to determine the critical exponent of a complementary symmetric Rote sequence $\vv$ by exploiting the indices of factors in the Sturmian sequence $\S(\vv)$. The following example shows an opposite application of Theorem \[Lem\_PrevodKritExpoRoteNaSturm\]. But before that, let us state a simple auxiliary statement reflecting the behaviour of fractional roots under the application of a morphism.
\[pridano\] Let $\phi :\mathcal{A}^* \mapsto \mathcal{A}^* $ be a morphism and let $w \in \mathcal{A}^*$ be a prefix of $\phi(a)$ for each $a\in \mathcal{A}$. If $u$ is a fractional root of $z$, then $\phi(u)$ is a fractional root of $\phi(z)w$.
Let us consider the Thue–Morse sequence $\tt = 01101001\cdots$, which is fixed by the morphism $\psi: 0\mapsto 01$ and $1\mapsto 10$. As $\psi$ is primitive, the sequence $\tt$ is uniformly recurrent. It is well-known that its language is closed under the exchange of letters and $\CR(\tt) = 2$. The corresponding sequence $\uu = \S(\tt) = 1011101 \cdots$ is called the period doubling sequence and it is fixed by the morphism $\phi : 0\mapsto 11$ and $1\mapsto 10$, see [@rigo].
We determine the critical exponent of $\uu$. Theorem \[Lem\_PrevodKritExpoRoteNaSturm\] implies $\CR(\uu)\leq 4$, as otherwise $\CR(\tt) >2$, which is a contradiction. Now we show that the value $4$ is attained.
By Observation \[pridano\] and the fact that both $\phi(0)$ and $\phi(1)$ have the prefix $1$, the morphism $\phi$ has the following two properties:
1. If $w \in \mathcal{L}(\uu)$, then $\phi(w)1 \in \mathcal{L}(\uu)$.
2. If $u$ is a fractional root of $w$, then $\phi(u)$ is a fractional root of $\phi(w)1$.
We will construct two sequences $\bigl(u^{(n)}\bigr)$ and $\bigl(w^{(n)}\bigr)$ of words belonging to $\mathcal{L}(\uu)$. We start with $u^{(0)}=1$ and $w^{(0)}=111\in \mathcal{L}(\uu)$ and for each $n\in \mathbb{N}$ we define $$w^{(n+1)} = \phi(w^{(n)})1 \qquad \text{and} \qquad u^{(n+1)} = \phi(u^{(n)})\,.$$ Note that $u^{(0)}=1$ is a fractional root of $w^{(0)}=111$. Because of the property (2), the word $u^{(n)}$ is a fractional root of $w^{(n)}$ for each $n\in \N$. Moreover, the specific form of the morphism $\phi$ implies $| u^{(n+1)}| = 2| u^{(n)}|$ and $| w^{(n+1)}| =2 | w^{(n)}| +1$. It gives $| u^{(n)}| = 2^n$ and $ | w^{(n)}| = 2^{n+2} -1$. Therefore, $\ind_\uu(u^{(n)}) \geq \frac{2^{n+2} -1}{2^n} \rightarrow 4$. We may conclude that $\CR(\uu) = 4$.
Return words to bispecial factors of Sturmian sequences {#S_retwordsBS_Sturmian}
=======================================================
The main goal of this article is to describe the critical exponent and the recurrence function of CS Rote sequences. Proposition \[Prop\_SturmRote\] and Theorem \[Lem\_PrevodKritExpoRoteNaSturm\] transform the first task to the computation of the indices of return words to bispecial factors in the associated Sturmian sequences.
This is a preparatory section for this computation. We introduce the directive sequence of a standard Sturmian sequence and recall some known results on bispecial factors, their return words, and derived sequences. It allows us to describe the longest factor of $\uu$ with the fractional root $u$, where $u$ is any return word to a bispecial factor of a Sturmian sequence $\uu$ (Lemma \[mocniny\]). Further on, we explain how to express the lengths of these factors explicitly (Proposition \[ParikhRSB\]), and eventually in Section \[S\_CriticalExponentRote\], we determine the indices. First, we recall that a binary sequence $\uu \in \{0,1\}^\N$ is Sturmian if it has the factor complexity $\mathcal{C}_\uu(n) = n+1$ for all $n \in \N$. If both sequences $0\uu$ and $1\uu$ are Sturmian, then $\uu$ is called a *standard Sturmian sequence*. It is well-known that for any Sturmian sequence there exists a unique standard Sturmian sequence with the same language. Since all properties which we are interested in (indices of factors, critical exponent, special factors, return words, recurrence function) depend only on the language of the sequence, we restrict ourselves to standard Sturmian sequences without loss of generality.
In the sequel, we use the characterization of standard Sturmian sequences by their directive sequences. To introduce them, we define two morphisms $$G = \
\left\{ \, \begin{aligned}
0 &\to 10 \\
1 &\to 1\,
\end{aligned}\right.
\ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ \ \ \
D = \
\left\{ \, \begin{aligned}
0 &\to 0 \\
1 &\to 01\,
\end{aligned} \right.$$ with the corresponding matrices $$M_G = \left(\begin{array}{cc}
1 & 0\\
1 & 1\\
\end{array} \right)\quad \quad \ \text{and} \quad \quad \quad
M_D = \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array} \right)\,.$$
Let us note that $G = E \circ F$ and $D = F \circ E$, where $E$ is the morphism which exchanges letters, i.e., $E: 0 \to 1$, $1 \to 0$, and $F$ is the Fibonacci morphism, i.e., $F: 0 \to 01$, $1 \to 0$.
\[Lem\_Standard\] For every standard Sturmian sequence $\uu$ there is a uniquely given sequence ${\bf \Delta} = \Delta_0\Delta_1\Delta_2 \cdots \in \{G, D\}^\N$ of morphisms and a sequence $(\uu^{(n)})_{n \geq 0}$ of standard Sturmian sequences such that $$\uu = {\Delta_0\Delta_1 \ldots \Delta_{n-1}}(\uu^{(n)})\, \ \text{for every } \ n \in \N\,.$$ Moreover, the sequence ${\bf \Delta}$ contains infinitely many letters $G$ and infinitely many letters $D$, i.e., $${\bf \Delta} = G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots \ \text{ or } \ {\bf \Delta} = D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots \quad \text{for some sequence } (a_i)_{i \geq 1} \text{ of positive integers} \,.$$ The sequence ${\bf \Delta}$ is called the *directive sequence* of $\uu$.
\[Rem\_Zamena\] Let us note that $\uu$ has the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots$ if and only if $E(\uu)$ has the directive sequence $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots$. Obviously, both sequences $\uu$ and $E(\uu)$ have the same structure up to the exchange of letters $0 \leftrightarrow 1$. In particular, any Sturmian sequence with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots$ can be written as a concatenation of the blocks $1^{a_1}0$ and $1^{a_1 + 1}0$, while any Sturmian sequence with the directive sequence $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots$ can be written as a concatenation of the blocks $0^{a_1}1$ and $0^{a_1 + 1}1$.
By Vuillon’s result [@Vui01], every factor of any Sturmian sequence has exactly two return words. Thus for a given bispecial factor $b$ of $\uu$, we usually denote by $r$ the more frequent and by $s$ the less frequent return word to $b$ in $\uu$. In this notation, the sequence $\uu$ can be decomposed into the blocks $r^ks$ and $r^{k+1}s$ for some $k \in \N, k \geq 1$.
We need to know how bispecial factors and their return words change under the application of morphisms $G$ and $D$. The following description can be found in [@MePeVu], where several partial statements from [@KlMePeSt18] are accumulated.
\[Lem\_ImageG\] Let $\uu', \uu$ be standard Sturmian sequences such that $\uu = G(\uu')$.
- For every bispecial factor $b'$ of $\uu'$, the factor $b = G(b')1$ is a bispecial factor of $\uu$.
- Every bispecial factor $b$ of $\uu$ which is not empty can be written as $b = G(b')1$ for a uniquely given bispecial factor $b' \in \mathcal{L}(\uu')$.
- The words $r', s'$ are return words to a bispecial prefix $b'$ of $\uu'$ if and only if $r = G(r'), s =G(s')$ are return words to a bispecial prefix $b = G(b')1$ of $\uu$. Moreover, the derived sequences satisfy $\dd_\uu(b) = \dd_{\uu'}(b')$.
\[Lem\_ImageD\] Let $\uu', \uu$ be standard Sturmian sequences such that $\uu = D(\uu')$.
- For every bispecial factor $b'$ of $\uu'$, the factor $b = D(b')0$ is a bispecial factor of $\uu$.
- Every bispecial factor $b$ of $\uu$ which is not empty can be written as $b = D(b')0$ for a uniquely given bispecial factor $b' \in \mathcal{L}(\uu')$.
- The words $r', s'$ are return words to a bispecial prefix $b'$ of $\uu'$ if and only if $r =D(r'), s = D(s')$ are return words to a bispecial prefix $b = D(b')0$ of $\uu$. Moreover, the derived sequences satisfy $\dd_\uu(b) = \dd_{\uu'}(b')$.
Any prefix of a standard Sturmian sequence is a left special factor. Moreover, a factor of a standard Sturmian sequence $\uu$ is bispecial if and only if it is a palindromic prefix of $\uu$. Therefore, we can order the bispecial factors of a given standard Sturmian sequence $\uu$ by their lengths: we start with the empty word $\varepsilon$, which is the $0^{th}$ bispecial factor, then the first letter of $\uu$ is the $1^{st}$ bispecial factor of $\uu$ etc.
\[Rem\_DerivedSeq\] If $\uu$ has the directive sequence $\Delta_0\Delta_1\Delta_2 \cdots \in \{G, D\}^\N$, the derived sequence $\dd_{\uu}(b)$ to the $n^{th}$ bispecial factor $b$ of $\uu$ has the directive sequence $\Delta_{n}\Delta_{n+1}\Delta_{n+2} \cdots$. Indeed, we denote $\uu'$ the sequence with the directive sequence $\Delta_{n}\Delta_{n+1} \Delta_{n+2}\cdots$. It has the bispecial factor $\varepsilon$ and by the definition $\dd_{\uu'}(\varepsilon) = \uu'$. If we apply $n$ times Lemmas \[Lem\_ImageG\] or \[Lem\_ImageD\], we get $\dd_{\uu}(b) = \dd_{\uu'}(\varepsilon) = \uu'$.
Let us formulate a direct consequence of the relation and Lemmas \[Lem\_ImageG\] and \[Lem\_ImageD\].
\[obrazyBS\] Let $k,h\in \N$. Let $b'$ be the $k^{th}$ bispecial factor of a standard Sturmian sequence $\uu'$ and $u'$ be a return word to $b'$ in $\uu'$. Let ${\bf \Delta} = \Delta_0\Delta_1 \Delta_2 \cdots \in\{G, D\}^\N$ be the directive sequence of $
\uu'$.
1. If $\uu = G^h(\uu')$, then the $(k+h)^{th}$ bispecial factor $b$ of $\uu$ and a return word $u$ to $b$ satisfy $$\vec{V}(b) = \left(\!\!\begin{array}{cc}1&0\\ h&1 \end{array}\!\!\right)\vec{V}(b') + h \left(\!\!\begin{array}{c}0\\ 1 \end{array}\!\!\right) \quad \text{and}\quad \vec{V}(u) = \left(\!\!\begin{array}{cc}1&0\\ h&1 \end{array}\!\!\right)\vec{V}(u')\,.$$ The directive sequence of $\uu$ is $G^h\Delta_0\Delta_1 \Delta_2 \cdots$.
2. If $\uu =D^h(\uu')$, then the $(k+h)^{th}$ bispecial factor $b$ of $\uu$ and a return word $u$ to $b$ satisfy $$\vec{V}(b) = \left(\!\!\begin{array}{cc}1&h\\ 0&1 \end{array}\!\!\right)\vec{V}(b') + h \left(\!\!\begin{array}{c}1\\ 0 \end{array}\!\!\right) \quad \text{and}\quad \vec{V}(u) = \left(\!\!\begin{array}{cc}1&h\\ 0&1 \end{array}\!\!\right)\vec{V}(u') \,.$$ The directive sequence of $\uu$ is $D^h\Delta_0\Delta_1 \Delta_2 \cdots$.
As we have seen in Remark \[rem\_CriticalExponentByReturnWords\], when determining the critical exponent it suffices to take into account only bispecial factors whose fractional roots are equal to its return words. Lemma \[mocniny\] says that all bispecial factors of a Sturmian sequence are of this type, and moreover, it enables to determine the indices of their return words. The first auxiliary statement is a slightly strengthened variant of Observation \[pridano\] for the morphisms $G$ and $D$.
\[Lem\_FractionalRoot\] Let $\uu$ be a binary sequence and let $u \in \mathcal{L}(\uu)$. If $z$ is the longest factor in $\mathcal{L}(\uu)$ with the fractional root $u$, then $G(z)1$ is the longest factor in $\mathcal{L}(G(\uu))$ with the fractional root $G(u)$ and, analogously, $D(z)0$ is the longest factor in $\mathcal{L}(D(\uu))$ with the fractional root $D(u)$.
\[mocniny\] Let $b$ be a bispecial factor of a standard Sturmian sequence $\uu$. Let $r$ and $s$ be the return words to $b$ in $\uu$ and let $k \in \N$, $k \geq 1$, be such that $\uu$ is concatenated from the blocks $r^ks$ and $r^{k+1}s$. Then $r^{k+1}b$ is the longest factor of $\uu$ with the fractional root $r$ and $sb$ is the longest factor of $\uu$ with the fractional root $s$.
We proceed by induction on the length of $b$. Without loss of generality, we assume that $\uu$ has the directive sequence ${\bf \Delta} = G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots$.
The bispecial factor $b = \varepsilon$ has the return words $r= 1, s = 0$, and by Remark \[Rem\_Zamena\], $\uu$ is concatenated from the blocks $1^{a_1}0 = r^{a_1}s$ and $1^{a_1 +1}0 = r^{a_1 + 1}s$. Clearly, $r^{a_1+1}b = 1^{a_1+1}$ is the longest factor of $\uu$ with the fractional root $1$. Similarly, $sb = 0$ is the longest factor of $\uu$ with the fractional root $0$.
Let $b$ be a bispecial factor of $\uu$ with $|b| \geq 1$ and let $\uu$ be concatenated from the blocks $r^{k}s$ and $r^{k+1}s$ for the return words $r,s$ to $b$ in $\uu$ and some $k \in \N$, $k \geq 1$. By Proposition \[Lem\_Standard\], there is a unique standard Sturmian sequence $\uu'$ such that $\uu = G(\uu')$. By Lemmas \[Lem\_ImageG\] and \[Lem\_ImageD\], there is a unique bispecial factor $b'$ of $\uu'$ with the return words $r'$ and $s'$ such that $b = G(b')1$, $r = G(r')$, and $s = G(s')$. Moreover, $\uu'$ is concatenated from the blocks $(r')^{k}s'$ and $(r')^{k+1}s'$. Clearly, $|b'| < |b|$ and so by the induction hypothesis, the words $(r')^{k+1}b'$ and $s'b'$ are the longest factors of $\uu'$ with the fractional root $r'$ and $s'$, respectively. But then by Observation \[Lem\_FractionalRoot\], the words $r^{k+1}b = G((r')^{k+1}b')1$ and $sb = G(s'b')1$ are the longest factors of $\uu$ with the fractional root $r = G(r')$ and $s = G(s')$, respectively.
Having in mind our goal to describe the critical exponent of any CS Rote sequence and Theorem \[Lem\_PrevodKritExpoRoteNaSturm\], we need to determine the indices of return words to bispecial factors in standard Sturmian sequences, i.e., the lengths of factors from Lemma \[mocniny\]. We also want to distinguish, which of these return words are (un)stable. Both of these tasks can be solved using the Parikh vectors of the relevant bispecial factors and their return words. We deduce the explicit formulae for the needed Parikh vectors in Proposition \[ParikhRSB\]. For this purpose, we adopt the following notation.
\[Theta\] To a standard Sturmian sequence $\uu$ with the directive sequence ${\bf \Delta} = G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $ or ${\bf \Delta} = D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $ we assign an irrational number $\theta \in (0,1)$ with the continued fraction expansion $$\theta = [0, a_1,a_2, a_3, \ldots ]\,.$$ For every $N\in \mathbb{N}$, we denote by $\tfrac{p_N}{q_N}$ the $N^{th}$ *convergent* to the number $\theta$ and by $\frac{p'_N}{q'_N}$ the $N^{th}$ convergent to the number $\frac{\theta}{1+\theta}$.
\[SlizeneZlomky\] Let us recall some basic properties of convergents. They can be found in any number theory textbook, e.g., [@Hensley].
1. The sequences $({p_N})$, $({q_N})$, and $({q'_N})$ fulfil the same recurrence relation for all $N \in \N, N \geq 1$, namely $$X_{N} = a_{N} X_{N-1} +X_{N-2}\,,$$ but they differ in their initial values: $p_{-1} = 1, p_0 = 0$; $q_{-1} = 0, q_0 = 1$; $q'_{-1} =q'_0 = 1$. It implies for all $N \in \N$ $$p_N +q_N = q'_N\,.$$
2. For all $N \in \N, N \geq 1$, we have
$
\left(\!\!\begin{array}{cc}1&0\\ a_1&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_2\\ 0&1 \end{array}\!\!\right) \cdots
\left(\!\!\begin{array}{cc}1&0\\ a_{2N-1}&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_{2N}\\ 0&1 \end{array}\!\!\right) =
\left(\!\!\begin{array}{cc}p_{2N-1}&p_{2N}\\ q_{2N-1}&q_{2N} \end{array}\!\!\right)\,;
$
$
\left(\!\!\begin{array}{cc}1&0\\ a_1&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_2\\ 0&1 \end{array}\!\!\right) \cdots \left(\!\!\begin{array}{cc}1&a_{2N-2}\\ 0&1 \end{array}\!\!\right)
\left(\!\!\begin{array}{cc}1&0\\ a_{2N-1}&1 \end{array}\!\!\right)=
\left(\!\!\begin{array}{cc}p_{2N-1}&p_{2N-2}\\ q_{2N-1}&q_{2N-2} \end{array}\!\!\right)\,.
$
\[density\] For the description of a standard Sturmian sequence $\uu$ we use the number $\theta$. Usually, a standard Sturmian sequence is characterized by the so-called *slope*, which is equal to the density of the letter 1 in the sequence $\uu$. In our notation, the slope of $\uu$ is $\frac{\theta}{1+\theta} = [0, 1+a_1, a_2, a_3, \ldots ]$ if the directive sequence ${\bf \Delta}$ starts with $D$, otherwise the slope is $\frac{1}{1+\theta} = [0, 1, a_1, a_2, a_3, \ldots ]$.
In the sequel, we will need two auxiliary statements on convergents $\tfrac{p_N}{q_N}$ to $\theta$.
\[Lem\_Convergents\] For all $N \in \N$ we have $$\left(\!\!\begin{array}{c}p_N\\ q_N\end{array}\!\!\right)\not =\left(\!\!\begin{array}{c}0\\ 0\end{array}\!\!\right) \mod 2 \qquad \text{and} \qquad \left(\!\!\begin{array}{c}p_N\\ q_N\end{array}\!\!\right)\not =\left(\!\!\begin{array}{c}p_{N-1}\\ q_{N-1}\end{array}\!\!\right) \mod 2 \,.$$
The first statement is a consequence of the fact that $p_N$ and $q_N$ are coprime. We show the second statement by contradiction. Assume that there exists $K \in \N$ such that $\left(\!\!\begin{array}{c}p_K\\ q_K\end{array}\!\!\right) =\left(\!\!\begin{array}{c}p_{K-1}\\ q_{K-1}\end{array}\!\!\right) \mod 2$. Let $K$ denote the smallest integer with this property. As $q_{-1}= 0$ and $q_{0} =1$, necessarily, $K>0$. Using the recurrence relation satisfied by the convergents, we can write $$\left(\!\!\begin{array}{c}p_K\\ q_K\end{array}\!\!\right) = a_{K}\left(\!\!\begin{array}{c}p_{K-1}\\ q_{K-1}\end{array}\!\!\right) + \left(\!\!\begin{array}{c}p_{K-2}\\ q_{K-2}\end{array}\!\!\right) =\left(\!\!\begin{array}{c}p_{K-1}\\ q_{K-1}\end{array}\!\!\right) \mod 2\,.$$
If $a_K$ is even, then the previous equation gives $ \left(\!\!\begin{array}{c}p_{K-2}\\ q_{K-2}\end{array}\!\!\right) =\left(\!\!\begin{array}{c}p_{K-1}\\ q_{K-1}\end{array}\!\!\right) \mod 2$, which is a contradiction with the minimality of $K$.
If $a_K$ is odd, then the previous equation gives $ \left(\!\!\begin{array}{c}p_{K-2}\\ q_{K-2}\end{array}\!\!\right) =\left(\!\!\begin{array}{c}0\\ 0\end{array}\!\!\right) \mod 2$, which is a contradiction with the first statement.
\[matice\_konvergenty\] For all $N \in \N$, $N \geq 1$, we have $$a_N \left(\!\!\begin{array}{c}p_{N-1}\\ q_{N-1} \end{array}\!\!\right) + a_{N-1} \left(\!\!\begin{array}{c}p_{N-2}\\ q_{N-2} \end{array}\!\!\right) + \cdots + a_2\left(\!\!\begin{array}{c}p_{1}\\ q_{1} \end{array}\!\!\right) + a_1\left(\!\!\begin{array}{c}p_{0}\\ q_{0} \end{array}\!\!\right) =
\left(\!\!\begin{array}{c}p_{N}\\ q_{N} \end{array}\!\!\right) + \left(\!\!\begin{array}{c}p_{N-1}\\ q_{N-1} \end{array}\!\!\right) - \left(\!\!\begin{array}{c}1\\1 \end{array}\!\!\right)\,.$$
It can be easily proved by induction on $N$.
The Parikh vectors of the bispecial factors of $\uu$ and the corresponding return words can be easily expressed using the convergents $\frac{p_N}{q_N}$ to $\theta$. We will use these expressions essentially in the next sections.
\[ParikhRSB\] Let $b$ be the $n^{th}$ bispecial factor of $\uu$ with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4}\cdots $. We denote $r$ and $s$ the more and the less frequent return word to $b$ in $\uu$, respectively. Put $a_0=0$ and write $n$ in the form $n =m+ a_0+ a_1+a_2+\cdots +a_N $ for a unique $N \in \N$ and $0\leq m<a_{N+1}$. Then
1. $\vec{V}(r) = \left(\!\!\begin{array}{c}p_N\\ q_N\end{array}\!\!\right)$;
2. $ \vec{V}(s) = \left(\!\!\begin{array}{c}m\,p_{N}+p_{N-1}\\ m\,q_{N}+q_{N-1}\end{array}\!\!\right)$;
3. $\vec{V}(b)= (m+1)\left(\!\!\begin{array}{c}p_{N}\\ q_{N} \end{array}\!\!\right) + \left(\!\!\begin{array}{c}p_{N-1}\\ q_{N-1} \end{array}\!\!\right) - \left(\!\!\begin{array}{c}1\\1 \end{array}\!\!\right) $.
First we suppose that $N$ is even and denote $\uu'$ the standard Sturmian sequence with the directive sequence $G^{a_{N+1}}D^{a_{N+2}}G^{a_{N+3}}\cdots $. By Remark \[Rem\_Zamena\], $\uu'$ is concatenated from the blocks $1^{a_{N+1}}0$ and $1^{a_{N+1}+1}0$. Thus its $m^{th}$ bispecial factor is $b'=1^m$ and the return words to $b'$ in $\uu'$ are $r' =1$ and $s' = 1^m0$.
By Lemmas \[Lem\_ImageG\] and \[Lem\_ImageD\] and Remark \[Rem\_DerivedSeq\], $$r = G^{a_1}D^{a_2}\cdots G^{a_{N-1}}D^{a_{N}}( r')\qquad \text{and}\qquad s= G^{a_1}D^{a_2}\cdots G^{a_{N-1}}D^{a_{N}}( s')\,.$$ By Corollary \[obrazyBS\] and Lemma \[matice\_konvergenty\], the Parikh vectors of $r$ and $s$ satisfy $$\begin{aligned}
\vec{V}(r) &= \left(\!\!\begin{array}{cc}1&0\\ a_1&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_2\\ 0&1 \end{array}\!\!\right) \cdots
\left(\!\!\begin{array}{cc}1&0\\ a_{N-1}&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_{N}\\ 0&1 \end{array}\!\!\right) \left(\!\!\begin{array}{c}0\\ 1\end{array}\!\!\right) =
\left(\!\!\begin{array}{c}p_{N}\\ q_{N} \end{array}\!\!\right) \,; \\
\vec{V}(s) &= \left(\!\!\begin{array}{cc}1&0\\ a_1&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_2\\ 0&1 \end{array}\!\!\right) \cdots
\left(\!\!\begin{array}{cc}1&0\\ a_{N-1}&1 \end{array}\!\!\right) \left(\!\!\begin{array}{cc}1&a_{N}\\ 0&1 \end{array}\!\!\right) \left(\!\!\begin{array}{c}1\\ m\end{array}\!\!\right) =
\left(\!\!\begin{array}{c}m\,p_{N}+p_{N-1}\\ m\,q_{N}+q_{N-1} \end{array}\!\!\right) \,.\end{aligned}$$ To find the Parikh vector of $b$ we start with the bispecial factor $b' = 1^m$ and $N$ times apply Corollary \[obrazyBS\]. Eventually, we rewrite the arising products of matrices by Lemma \[matice\_konvergenty\] and we get $$\begin{aligned}
\vec{V}(b) &= m\, \left(\!\!\begin{array}{c}p_{N}\\ q_{N} \end{array}\!\!\right) + a_N \left(\!\!\begin{array}{c}p_{N-1}\\ q_{N-1} \end{array}\!\!\right) + a_{N-1} \left(\!\!\begin{array}{c}p_{N-2}\\ q_{N-2} \end{array}\!\!\right) + \cdots + a_2\left(\!\!\begin{array}{c}p_{1}\\ q_{1} \end{array}\!\!\right) + a_1\left(\!\!\begin{array}{c}p_{0}\\ q_{0} \end{array}\!\!\right) \,.\end{aligned}$$ This together with Lemma \[matice\_konvergenty\] implies the statement of Item (3) for $N$ even. The proof for $N$ odd is analogous.
\[Rem\_GandDexchanged\] If we assume in Proposition \[ParikhRSB\] that $\uu$ has the directive sequence ${\bf \Delta} = D^{a_1}G^{a_2}D^{a_3}G^{a_4}\cdots $, then by Remark \[Rem\_Zamena\], the coordinates of the Parikh vectors will be exchanged, i.e., $$\vec{V}(r) = \left(\!\!\begin{array}{c}q_N\\ p_N\end{array}\!\!\right), \ \vec{V}(s) = \left(\!\!\begin{array}{c} m\,q_{N}+q_{N-1} \\ m\,p_{N}+p_{N-1}\end{array}\!\!\right), \ \text{and} \ \vec{V}(b)= (m+1)\left(\!\!\begin{array}{c}q_{N}\\ p_{N} \end{array}\!\!\right) + \left(\!\!\begin{array}{c}q_{N-1}\\ p_{N-1} \end{array}\!\!\right) - \left(\!\!\begin{array}{c}1\\1 \end{array}\!\!\right).$$
The critical exponent of CS Rote sequences {#S_CriticalExponentRote}
==========================================
We are going to give an explicit formula for the critical exponent of a CS Rote sequence $\vv$. We will use Theorem \[Lem\_PrevodKritExpoRoteNaSturm\] which requires the knowledge of the indices of return words to bispecial factors in the Sturmian sequence $\S(\vv)$. It is well-known that there is a unique standard Sturmian sequence $\uu$ such that both $\S(\vv)$ and $\uu$ have the same language. Since the critical exponent depends only on the language, we can work with the standard Sturmian sequence $\uu$ instead of $\S(\vv)$.
In the following proposition and theorem, we use Notation \[Theta\].
\[Prop\_KritExpo\] Let $\uu$ be a standard Sturmian sequence with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4}\cdots$ or $D^{a_1} G^{a_2} D^{a_3}G^{a_4} \cdots $ and let $n \in \N$ be given. We put $a_0 = 0$ and we denote $r$ and $s$ the more and the less frequent return words to the $n^{th}$ bispecial factor of $\uu$, respectively.
1. If $n =m+ a_0+a_1+a_2+\cdots +a_N $, where $0\leq m<a_{N+1}$, then $\emph{ind}(r) = a_{N+1} +2 + \frac{q'_{N-1}-2}{q'_N}$.
2. If $n =a_0+ a_1+a_2+\cdots +a_N$, then $ \emph{ind}(s) = a_{N} +2 + \frac{q'_{N-2}-2}{q'_{N-1}}$.
3. If $n =m+a_0+ a_1+a_2+\cdots +a_N $, where $0<m<a_{N+1}$, then $\emph{ind}(s) = 2 + \frac{q'_{N}-2}{ m\,q'_{N} +q'_{N-1}}$.
Let us comment what is meant by $q'_{-2}$ in the case $N=0$ in Item (2): we define $q'_{-2}$ to satisfy the recurrent relation $1=q'_{0} = a_0q'_{-1} + q'_{-2} = q'_{-2 }$.
We assume that $N$ is even (the case of $N$ odd is analogous). Moreover, we assume that $\uu$ has the directive sequence $G^{a_1} D^{a_2} G^{a_3} D^{a_4} \cdots $.
Since $b$ is the $n^{th}$ bispecial factor in $\uu$, the derived sequence $\dd_{\uu}(b)$ is standard Sturmian with the directive sequence $G^{k}D^{a_{N+2}}G^{a_{N+3}}\cdots $, where $k=a_{N+1}-m$, see Corollary \[obrazyBS\]. By Remark \[Rem\_Zamena\], $\dd_{\uu}(b)$ is a concatenation of the blocks $1^k0$ and $1^{k+1}0$. Therefore, the sequence $\uu$ is concatenated from the blocks $r^ks$ and $r^{k+1}s$, where $r$ and $s$ are the return words to $b$. By Lemma \[mocniny\], the factor $r^{k+1}b$ is the longest factor of $\uu$ with the fractional root $r$ and $sb$ is the longest factor of $\uu$ with the fractional root $s$. In other words, $r^{\ind(r)}=r^{k+1}b$ and $s^{\ind(s)}= sb$. By Proposition \[ParikhRSB\] and Remark \[SlizeneZlomky\], we have $$|r| = p_N +q_N = q'_N, \quad \ |s| = mq_N' + q'_{N-1} \,, \quad \text{and} \ \quad |b| = (m+1)q'_N + q'_{N-1} - 2\,.$$ As $|r^{k+1}b| = (k+1)|r| + |b| = (a_{N+1}-m+1) |r|+|b| = (a_{N+1}+2) q'_N + q'_{N-1} -2$, we get $$\ind(r)=\tfrac{|r^{k+1}b|}{|r|} = a_{N+1} +2 + \tfrac{q'_{N-1}-2}{q'_N}\,.$$ As $|sb| = (2m+1)q_N' + 2q'_{N-1} - 2$, we get for $m=0$ $$\ind(s)=\tfrac{|sb|}{|s|} = \tfrac{q_N' + 2q'_{N-1} - 2}{ q'_{N-1} } = \tfrac{a_Nq'_{N-1}+q'_{N-2}+ 2q'_{N-1} - 2}{ q'_{N-1} } = a_{N} +2 + \tfrac{q'_{N-2}-2}{q'_{N-1}}\,$$ and for $m>0$ $$\ind(s)= \tfrac{|sb|}{|s|} = \tfrac{(2m+1)q_N' + 2q'_{N-1} - 2}{m\,q'_{N} +q'_{N-1}} = 2 + \tfrac{q'_{N}-2}{ m\,q'_{N} +q'_{N-1}}\,.$$
If the directive sequence equals $D^{a_1} G^{a_2} D^{a_3} G^{a_4} \cdots$, only the coordinates of the Parikh vectors of $r$, $s$, $b$ are exchanged (see Remark \[Rem\_GandDexchanged\]).
\[T\_KritExpo\] Let $\vv$ be a CS Rote sequence and let $\uu$ be the standard Sturmian sequence such that $\mathcal{L}(\S(\vv)) = \mathcal{L}(\uu)$. Then $
\emph{cr}(\vv) = \sup (M_1 \cup M_2 \cup M_3)\,, \text{ where }
$ $$\begin{aligned}
\ \ M_1 &= \left\{a_{N+1} + 2 + \frac{q'_{N-1}-1}{q'_N} : \ q_N \text{ is even, } N\in \N\right\}\,; \\
M_2 &= \left\{\frac{a_{N+1} + 2}{2} + \frac{q'_{N-1}-1}{2 q'_N} : \ q_N \text{ is odd, } N \in \N\right\}\,; \\
M_3 &= \left\{2 + \frac{q'_{N}-1}{q'_{N-1} + q'_N} : \ q_{N-1} , q_{N} \text{ are odd and } a_{N+1}>1, N \geq 1\right\} \,\end{aligned}$$ if the directive sequence of $\uu$ is $G^{a_1}D^{a_2}G^{a_3}D^{a_4}\cdots$, and $$\begin{aligned}
M_1 &= \left\{a_{N+1} + 2 + \frac{q'_{N-1}-1}{q'_N} : \ p_N \text{ is even, } N\in \N\right\}\,; \\
M_2 &= \left\{\frac{a_{N+1} + 2}{2} + \frac{q'_{N-1}-1}{2 q'_N} : \ p_N \text{ is odd, } N \in \N\right\}\,; \\
M_3 &= \left\{2 + \frac{q'_{N}-1}{q'_{N-1} + q'_N} : \ p_{N-1} , p_{N} \text{ are odd and } a_{N+1}>1, N \geq 1\right\} \,\end{aligned}$$ if the directive sequence of $\uu$ is $D^{a_1}G^{a_2}D^{a_3}G^{a_4}\cdots$.
Let us recall that every CS Rote sequence is uniformly recurrent and aperiodic. In addition, to a CS Rote sequence $\vv$ we can always find a unique standard Sturmian sequence $\uu$ such that $\uu$ has the same language as $\S(\vv)$. It is important to realize that Theorem \[Lem\_PrevodKritExpoRoteNaSturm\] holds for the pair $\vv$ and $\uu$, too.
First, we assume that $\uu$ has the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4}\cdots$. We compute the suprema of the sets $A_1$ and $A_2$ defined in Theorem \[Lem\_PrevodKritExpoRoteNaSturm\], since by this theorem, $\CR(\vv) = \sup(A_1\cup A_2)$.
Let us decompose $A_ 1$ into $A_1 = \bigcup_{N =0}^{\infty} A_1^{(N)}$, where $$A_1^{(N)} = \Bigl\{\ind_\uu(u) + \tfrac{1}{|u|} :\ u \text{ is a stable return word to the $n^{th}$ bispecial f. of } \uu, \ \sum_{k=0}^N a_k \leq n < \sum_{k=0}^{N+1} a_k \Bigr\}\,.$$ By definition, a word $u$ is stable if the number of ones occurring in $u$ is even, i.e., the second component of its Parikh vector $\vec{V}(u)$ is even. Combining Propositions \[ParikhRSB\] and \[Prop\_KritExpo\], we obtain that $A_1^{(N)} $ contains
- $a_{N+1} +2 + \frac{q'_{N-1}-1}{q'_N} $ if $q_N$ is even,
- $ a_{N} +2 + \frac{q'_{N-2}-1}{q'_{N-1}}$ if $q_{N-1}$ is even,
- the subset $ B_1^{(N)} =\{2 + \frac{q'_{N}-1}{ m\,q'_{N} +q'_{N-1}} : m\,q_{N} +q_{N-1} \text{ even}, \ 0< m < a_{N+1}\}$.
First we look at $A_1^{(0)} $. Since $q_0 = 1$ is odd, $q_{-1}=0$ is even, $a_0 =0$, $q'_{0} = q'_{-1}= q'_{-2}= 1$, we get $A_1^{(0)} = \{2\}$. Since we know that $\CR(\vv) > 2$, we can consider only $N\geq 1$. Let us note that all elements in $B_1^{(N)}$ are strictly less than $3$. If $q_N$ or $q_{N-1}$ is even, the set $A_1^{(N)} $ contains an element $\geq 3$ and the set $B_1^{(N)}$ does not play any role for $\sup
A_1$. If both $q_N$ and $q_{N-1}$ are odd, there is an element in $B_1^{(N)}$ only for odd $m < a_{N+1}$, and obviously, $\sup B_1^{(N)}$ is attained for $m=1$ (if $a_{N+1} = 1$ the set is empty). Together it gives $\sup A_1 =\sup( M_1\cup M_3)$.
Analogously we define the sets $A_2^{(N)}$ for unstable return words. Then $A_2^{(N)}$ consists of
- $\tfrac12\big(a_{N+1} +2 + \frac{q'_{N-1}-1}{q'_N}\bigr)$ if $q_N$ is odd,
- $\tfrac12\big( a_{N} +2 + \frac{q'_{N-2}-1}{q'_{N-1}}\bigr) $ if $q_{N-1}$ is odd,
- the subset $ B_2^{(N)} =\{ \tfrac12\big(2 + \frac{q'_{N}-1}{ m\,q'_{N} +q'_{N-1}} \bigr) : m\,q_{N} +q_{N-1} \text{ odd}, \ 0< m < a_{N+1}\}$.
We easily compute that $\sup A_2^{(0)} = \tfrac12(a_1+2)$. All elements in $B_2^{(N)}$ are strictly less than $\tfrac32$. Thus if $q_N$ or $q_{N-1}$ is odd, the set $B_2^{(N)}$ does not play any role for $\sup A_2$. If $q_N$ and $q_{N-1}$ are even, then the set $B_2^{(N)}$ is empty. It means that $\sup A_2 = \sup M_2$. We can conclude that $\CR(\vv) = \sup(A_1 \cup A_2) =\sup(M_1\cup M_2
\cup M_3)$.
If the directive sequence equals $D^{a_1} G^{a_2} D^{a_3} G^{a_4} \cdots$, only the coordinates of the Parikh vectors of $r$ and $s$ are exchanged (see Remark \[Rem\_GandDexchanged\]).
CS Rote sequences with a small critical exponent {#S_smallexponent}
================================================
In this section, we present some corollaries of Theorem \[T\_KritExpo\]. As we have mentioned, Currie, Mol, and Rampersad proved in [@CuMoRa] that there are exactly two languages of rich binary sequences with the critical exponent less than $\tfrac{14}{5}$. Both of them are languages of CS Rote sequences.
Let us remind that the critical exponent depends only on the language of a sequence and not on the sequence itself. Therefore, there are infinitely many CS Rote sequences with the critical exponent less than $\tfrac{14}{5}$, but all of them have one of two languages. We show that among all languages of CS Rote sequences only these two languages have the critical exponent less than $3$. We also describe all languages of CS Rote sequences with the critical exponent equal to $3$.
\[Prop\_MalyExponent\] Let $\vv$ be a CS Rote sequence associated with the standard Sturmian sequence $\uu = \S(\vv)$. If $\emph{cr}(\vv) \leq 3$, then the directive sequence of $\uu$ is of one of the following forms:
1. $G^{a_1} (D^{2}G^{2})^\omega$, where $a_1 = 1$ or $a_1 = 3$; in this case $\emph{cr}(\vv) = 2+ \tfrac{1}{\sqrt{2}}$;
2. $G^{a_1} D^{4}(G^{2}D^2)^\omega$, where $a_1 = 1$ or $a_1 = 3$; in this case $\emph{cr}(\vv) = 3$;
3. $G^{a_1}D^{1} G^{a_3}(D^{2}G^{2})^\omega$, where $a_1= 2$ or $a_1 = 4$ and $a_3 = 1$ or $a_3 = 3$; in this case $\emph{cr}(\vv) = 3$;
4. $D^{1}G^{a_2} (D^{2}G^{2})^\omega$, where $a_2= 1$ or $a_2 = 3$; in this case $\emph{cr}(\vv) = 3$.
For each $N \in \N$ we denote $ \beta_N =a_{N+1} + 2 + \frac{q'_{N-1}-1}{q'_N}$ the number which is a candidate to join the set $M_1$. We can easily compute that $\beta_0 = a_1 + 2$, $\beta_1 = a_2 + 2$ and $\beta_N > 3$ for every $N \geq 2$. Indeed, it suffices to realize that $q'_{-1} = q'_{0} = 1$, $q'_{1} = a_1 + 1 > 1$ and $(q'_N)_{N \geq 1}$ is an increasing sequence of integers, so $\frac{q'_{N-1}-1}{q'_{N}} \in (0,1)$ for all $N \geq 2$. Since we look for a sequence $\vv$ with $\CR(\vv) \leq 3$, we have to ensure that $\beta_N \notin M_1$ for all $N \geq 2$ by the parity conditions. It is also important to notice that $\sup M_3 \leq 3$. Indeed, since $\frac{q'_{N}-1}{q'_{N} + q'_{N-1}} \in \left[0,1\right)$ for all $N \in \N$, all elements of $M_3$ are less than $3$.
First we assume that the directive sequence of $\uu$ is $G^{a_1} D^{a_2}G^{a_3}D^{a_4}\cdots$. By Theorem \[T\_KritExpo\], if $q_{N}$ is even, then $\beta_N \in M_1$, otherwise $\frac{1}{2}\beta_N \in M_2$. To ensure $\sup M_1 \leq 3$, $q_N$ has to be odd for all $N \geq 2$. Moreover, to ensure $\sup M_2 \leq 3$, $\beta_N \leq 6$ and so $a_{N+1} \leq 3$ for all $N \geq 2$. Since $q_0=1$ is odd, $\frac{1}{2}\beta_0 = \frac{a_1}{2} + 1 \in M_2$ and so $a_1 \leq 4$. We distinguish two cases.
\(i) If $q_1 = a_1$ is odd, then $M_1$ is empty and $\frac{1}{2}\beta_1 = \frac{a_2}{2} +1 \in M_2$. Thus $a_2 \leq 4$. The recurrent relation $q_N = a_Nq_{N-1}+q_{N-2}$ with the odd initial conditions $q_0$ and $q_1$ produces $q_N$ odd for all $N \geq 2$ if and only if $a_N$ is even for all $N \geq 2$. We can summarize that $a_1 \in \{1, 3\}$, $a_2 \in \{2, 4\}$ and $a_N = 2$ for all $N \geq 3$.
Let us observe that if $a_2 = 4$, then $\sup M_2 = 3$. Since $\sup M_3 \leq 3$ and $M_1$ is empty, we conclude that $\CR (\vv) = \sup M_2 = 3$. It gives us Item (2) of our proposition.
If $a_2 = 2$, then it is easy to check that all elements of $M_2$ are smaller than $\tfrac52$ and thus $\sup M_2 \leq \tfrac52$. Since $M_1$ is empty, to prove Item (1), it remains to compute $$\sup M_3= \sup \left\{2 + \frac{q'_{N}-1}{q'_{N-1} + q'_N} : N\in \N\right\} = 2 + \frac{1}{\sqrt{2}} > \frac{5}{2}\,.$$ The sequence $(q'_N)_{N \geq 2}$ fulfils the recurrence relation $q'_N = 2q'_{N-1}+ q'_{N-2}$ with the initial conditions $q'_0 =1$ and $q'_1 = a_1+ 1$. This linear recurrence has the solution $$\label{reseni}
q'_{N} = \tfrac{1}{2\sqrt{2}} \Bigl( ( a_1+\sqrt{2})(1+\sqrt{2})^N - ( a_1-\sqrt{2})(1-\sqrt{2})^N \Bigr)\,.$$ Now it suffices to use the expression to verify that $$\lim_{N\to \infty} \ \ \frac{q'_{N}-1}{q'_{N-1} + q'_N} = \frac{1}{\sqrt{2}} \quad \text{ and } \quad \frac{q'_{N}-1}{q'_{N-1} + q'_N} \leq \frac{1}{\sqrt{2}} \ \text{ for all } N \in \N\,.$$
\(ii) If $q_1 = a_1$ is even, then $M_1 = \{\beta_1 = a_2 + 2\}$ since all the others $q_N$ are odd. Thus $a_2 = 1$ and $\CR(\vv) \geq 3$. The recurrent relation $q_N = a_Nq_{N-1} + q_{N-2}$ with the initial conditions $q_0 = 1$, $q_1 = a_1 \in \{2,4\}$ produces odd $q_N$ for all $N \geq 2$ if and only if $a_3$ is odd and $a_N$ is even for all $N \geq 4$. Moreover, to ensure $\sup M_2 \leq 3$, we have to take $a_1 \leq 4$ and $a_{N} \leq 3$ for every $N \geq 3$. Together with the parity conditions we get $a_1 \in \{2,4\}$, $a_2 = 1$, $a_3 \in \{1,3\}$, and $a_N = 2$ for all $N \geq 4$. Since $\sup M_3 \leq 3$, the set $M_3$ can be omitted. Together it means that in this case $\CR(\vv) = 3$ and it corresponds to Item $(3)$ of our statement.
Now we assume that the directive sequence of $\uu$ is $D^{a_1}G^{a_2} D^{a_3}G^{a_4}\cdots$. By Theorem \[T\_KritExpo\], if $p_{N}$ is even, then $\beta_N \in M_1$, otherwise $\frac{\beta_N}{2} \in M_2$. Since $p_0=0$ is even, $\beta_0 = a_1+2 \in M_1$. Therefore, $a_1 = 1$ and $ \CR(\vv) \geq 3$. Since $p_1 = a_1p_0 + p_{-1} = 1$ is odd, $\frac{\beta_1}{2} = \frac{a_2}{2} + 1 \in M_2$ and so $a_2 \leq 4$. To guarantee $\CR(\vv) \leq 3$, $p_N$ has to be odd and $a_{N+1} \leq 3$ for all $N \geq 2$. The recurrence relation $p_N = a_Np_{N-1} + p_{N-2}$ with the initial conditions $p_0=0$ and $p_1 = 1$ produces $p_N$ odd for all $N \geq 2$ if and only if $a_2$ is odd and $a_N$ is even for all $N\geq 2$. Clearly, the set $M_3$ can be omitted. We may conclude that $\CR(\vv) = 3$ and $a_1 = 1$, $a_2 \in \{1,3\}$, and $a_N = 2$ for all $N \geq 3$, which corresponds to Item (4).
Let us emphasize that all standard Sturmian sequences from Proposition \[Prop\_MalyExponent\], i.e., which are associated with CS Rote sequences with the critical exponent $\leq 3$, are morphic images of the fixed point of the morphism $D^2G^2$. If follows directly from the fact that their directive sequences have the periodic suffix $(D^2G^2)^\omega$.
\[Ex\_Rozdil\_cr\_u\_Eu\] In Proposition \[Prop\_MalyExponent\], we have shown that the CS Rote sequence $\vv$ such that $S(\vv)$ has the directive sequence $G(D^2 G^2)^{\omega}$ has the critical exponent $\CR(\vv)=2+\frac{1}{\sqrt{2}}$. Let us determine the critical exponent $\CR(\vv')$ of the CS Rote sequence $\vv'$ associated to the standard Sturmian sequence $S(\vv')$ obtained by the exchange of letters from $S(\vv)$, i.e., $S(\vv')=E(S(\vv))$.
By Remark \[Rem\_Zamena\], the directive sequence of $S(\vv')$ equals $D(G^2D^2)^{\omega}$. Thus we have $\theta=[0,1,2,2,2,\dots]$ and it is readily seen that $p_N$ is even if and only if $N$ is even. Let us calculate $\CR(\vv')$ by Theorem \[T\_KritExpo\]. We have $$M_1=\{a_{2N+1} + 2 + \frac{q'_{2N-1}-1}{q'_{2N}}: N \in \N\}=\{3\}\cup \{4 + \frac{q'_{2N-1}-1}{q'_{2N}}: N \in \N, N \geq 1\}\,.$$ Using equation , we can check that the sequence $\bigl(\frac{q'_{2N-1}-1}{q'_{2N}}\bigr)$ is increasing and has the limit $\frac{1}{1 + \sqrt{2}}$, and therefore $\sup M_1=4+\frac{1}{1+\sqrt{2}}$. Since the elements of $M_2$ and $M_3$ are $\leq 3$, we can conclude that $\CR(\vv')=4+\frac{1}{1+\sqrt{2}}$.
It is well-known that among Sturmian sequences the sequence with the lowest possible critical exponent is the Fibonacci sequence ${\bf f}$, which has $\CR({\bf f}) = 3+ \tfrac{2}{1+\sqrt{5}} \sim 3. 602$. The following theorem implies that there are uncountably many CS Rote sequences with the critical exponent smaller than $\CR({\bf f})$.
\[Thm\_Exponent72\] Let $G^{a_1} D^{a_2}G^{a_3}D^{a_4}\cdots $ be the directive sequence of a standard Sturmian sequence $\uu$ and let $\vv$ be the CS Rote sequence associated with $\uu$. Then $\emph{cr}(\vv) < \tfrac72$ if and only if the sequence $a_1a_2 a_3 \cdots$ is a concatenation of the blocks from the following list:
$L_0$: $111$;
$L_1$: $s1$, where $s \in\{2,4\}$;
$L_2$: $cs31$, where $ c \in \{1,3\}$ and $s \in\{2,4\}^*$;
$L_3$: $c{\bf s}$, where $ c \in \{1,3\}$ and ${\bf s} \in \{2,4\}^\mathbb{N}$,
and if the block $L_0$ appears in $a_1a_2a_3\cdots$, then it is a prefix of $a_1a_2a_3 \cdots$.
As in the proof of Proposition \[Prop\_MalyExponent\], we again use the sets $M_1$ and $M_2$ from Theorem \[T\_KritExpo\]. The set $M_3$ can be omitted since $\sup M_3\leq 3$.
Let us recall that $q_0 =1$, $q_1 = a_1$, and if $q_N$ is even, then $\beta_N = a_{N+1} + 2 + \frac{q'_{N-1}-1}{q'_N} \in M_1$, otherwise $\frac{1}{2}\beta_N \in M_2$. First we suppose that $\CR(\vv) < \tfrac72$ and we deduce several auxiliary observations for each $N \in \N$:
1. If $q_N$ even, then $a_{N+1} = 1$.
Proof: It follows from the inequality $ \beta_N < \tfrac72$.
2. If $q_N$ odd, then $a_{N+1} \in \{1,2,3,4\}$.
Proof: It follows from the inequality $ \tfrac12\beta_N < \tfrac72$.
3. If $q_{N-1}$ odd and $q_{N}$ even, then $q_{N+1}$ odd.
Proof: It follows from Item (1) and the relation $q_{N+1} = a_{N+1}q_{N}+q_{N-1} = q_N + q_{N-1}$.
4. If $q_{N}$ even, $q_{N+1}$ odd, and $q_{N+2}$ even, then $a_{N+2} \in \{2,4\}$.
Proof: It follows from Item (2) and the relation $q_{N+2} = a_{N+2}q_{N+1}+q_{N}$.
5. If $q_{N}$ even, $q_{N+1}$ odd, and $q_{N+2}$ odd, then $a_{N+2} \in \{1,3\}$.
Proof: It follows from Item (2) and the relation $q_{N+2} = a_{N+2}q_{N+1}+q_{N}$.
6. If $q_{N}$ odd, $q_{N+1}$ odd, and $q_{N+2}$ odd, then $a_{N+2} \in \{2,4\}$.
Proof: It follows from Item (2) and the relation $q_{N+2} = a_{N+2}q_{N+1}+q_{N}$.
7. If $q_{N}$ odd, $q_{N+1}$ odd, and $q_{N+2}$ even, then $a_{N+2}=\{1,3\}$. Moreover, if $a_{N +2} = 1$, then $N = 0$ and $a_1 = 1$.
Proof: Item (2) and the relation $q_{N+2} = a_{N+2}q_{N+1}+q_{N}$ imply $a_{N+2} \in \{1,3\}$. Assume that $a_{N+2} = 1$. By Item (1), $a_{N+3} = 1$. As $q_{N+2}$ is even, $\beta_{N+2} \in M_1$ and so $\beta_{N+2} = 3 + \frac{q'_{N+1}-1}{q'_{N+2}} < \tfrac{7}{2}$. By some simple rearrangements and applications of the recurrence relation, we can rewrite this inequality equivalently as $(a_{N+1} -1)q'_{N} + q'_{N-1} < 2$. It is easy to verify that this inequality holds only for $N = 0$ and $a_1 = 1$ or $N = 1$ and $a_2 = 1$. Nevertheless, the second case leads to a contradiction with the assumption that both $q_1, q_2 = a_2q_1 +1$ are odd.
8. Let $M> N+2$. If $q_{N}$ even, $q_{M}$ even, and $q_{K}$ odd for all $K, N<K <M$, then $a_{N+2} \in \{1,3\}$, $a_M = 3$ and $a_{K} \in \{2, 4\}$ for all $K, N+2 <K < M$.
Proof: Item (6) implies $a_{K} \in \{2, 4\}$ for all $K$, $N+2 <K < M$. By Item (5), $a_{N+2} \in \{1,3\}$ and by Item (7), $a_M =3$.
Using the previous claims we show that for each $J$ for which $q_J$ is even, at the position $J+1$ ends one of the blocks $L_0$, $L_1$, or $L_2$. Moreover, the block $L_0$ can only occur as a prefix of the sequence $a_1a_2 a_3 \cdots$, while each of the blocks $L_1$ and $L_2$ is either a prefix of $a_1a_2 a_3 \cdots$ or it starts at the position $I+2$, where $I$ is the greatest integer smaller than $J$ for which $q_I$ is even. We discuss three cases:
- Let $q_1$ be even. Then Item (1) implies $a_2 = 1$. And since $a_1 = q_1$ is even and $q_0 = 1$ is odd, by Item (2), we get $a_1\in \{ 2, 4\}$. Thus the prefix $a_1a_2$ of the sequence $a_1a_2a_3 \cdots$ is of the form $L_1$ from our list.
- Let $J>1$ be the first index such that $q_J$ is even. As $q_1 = a_1$, $a_1$ is odd, and by Item (2), we get $a_1 \in \{1,3\}$. Item (6) implies $a_K \in \{ 2,4\}$ for all $K, 1<K<J$. By Item (7), $a_J = 3$ or $a_J = 1$. But if $a_J = 1$, then $J = 2$ and $a_1 = 1$. Finally, Item (1) implies $a_{J+1} = 1$. Thus the prefix $a_1a_2\cdots a_Ja_{J+1}$ of the sequence $a_1a_2a_3 \cdots$ is of the form $L_0$ or $L_2$ from our list.
- Let $I$ be an index such that $q_I$ is even and let $J$ be the smallest index greater than $I$ for which $q_J$ is even. By Item (1), $a_{I+1} = 1$. The word $a_{I+2} \cdots a_Ja_{J+1}$ is by Item (4) or Item (8) either of the form $L_1$ or $L_2$.
If there are infinitely many even denominators $q_N$, then we have shown that the sequence $a_1a_2a_3 \cdots$ is concatenated from the blocks $L_0$, $L_1$, and $L_2$ ($L_0$ can only be a prefix). It remains to consider the case when only finitely many denominators $q_N$ are even.
- Let $q_N$ be odd for every $N \in \N$. Then $q_1 = a_1$ is odd. Especially, since both $q_0$ and $q_1 = a_1$ are odd, Item (2) implies $a_ 1 \in\{1, 3\}$ and it follows from Item (6) that $a_2a_3a_4\cdots \in \{2,4\}^\mathbb{N}$. Therefore, the sequence $a_1a_2a_3 \cdots$ is equal to the block $L_3$ from our list.
- Let $L \geq 1$ be the last index such that $q_L$ is even. In particular, it means that $q_N$ is even only for a finite number of indices. Item (1) implies that $a_{L+1} = 1$. By Items (5) and (6), the suffix $a_{L+2}a_{L+3}a_{L+4}\cdots$ of the sequence $a_1a_2a_3 \cdots$ equals to $L_3$.
Now we have to show that any directive sequence $G^{a_1} D^{a_2}G^{a_3}D^{a_4}\cdots$ such that $a_1a_2a_3 \cdots$ is concatenated of the blocks from the list gives a standard Sturmian sequence $\uu$ such that the CS Rote sequence $\vv$ associated to $\uu$ has the critical exponent less than $\tfrac72$.
If $q_N$ is odd, then $\tfrac12 \beta_N = \tfrac12 a_{N+1} + 1+\tfrac{q'_{N-1} -1}{2q'_{N}} < 3 + \tfrac{q'_{N-1}}{2q'_{N-1} + q'_{N-2}}< \tfrac72$, as each $a_{N+1}$ is $\leq 4$.
If $q_N$ is even, it is easy to prove by induction on $N$ that there is a block of the form $L_0$, $L_1$, or $L_2$ ending at the position $N+1$ (and $L_0$ only for $N = 2$). In particular, it means that $a_{N+1}=1$ and $a_N \geq 2$ or $a_1 = a_2 = a_3 = 1$. In the first case, we get $ \beta_N \leq 3+ \tfrac{q'_{N-1}}{2q'_{N-1} + q'_{N-2}}< \tfrac72$, while in the second case, we get $\beta_2 = 3 + \frac{a_1 + 1 - 1}{a_2(a_1 + 1) + 1} = 3 + \frac{1}{3} < \tfrac72$.
To show that $\sup (M_1\cup M_2)< \tfrac72$, we need to show that $\sup \tfrac{q'_{N-1}}{2q'_{N-1} + q'_{N-2}} < \tfrac12$. As $a_{N} \leq 4$ for all $N$, we can estimate $\tfrac{q'_{N-2}}{q'_{N-1}} \geq \tfrac{q'_{N-2}}{4q'_{N-2} +q'_{N-2} } = \tfrac15$ and thus $ \tfrac{q'_{N-1}}{2q'_{N-1} + q'_{N-2}} = \tfrac{1}{2 +\tfrac{q'_{N-2}}{q'_{N-1}} } \leq \tfrac5{11}$.
It would be interesting to reveal some topological properties of the set $$\CR_{Rote} := \{\CR(\vv) : \vv \text { is a CS Rote sequence}\},$$ for instance, to find its accumulation points in the interval $(3, \tfrac72)$. The proof of the previous theorem implies that there is no CS Rote sequence with the critical exponent between $3+\tfrac5{11}$ and $ 3+\tfrac12$. We even believe that for any CS Rote sequence $\vv$, the following implication holds: If $\CR(\vv)<3+\frac{1}{2}$, then $\CR(\vv)<3+\frac{1}{1+\sqrt{3}}$.
The recurrence function of CS Rote sequences {#S_recurrencefunction}
============================================
The main result of this section is Theorem \[Thm\_RecFunctionRoteDetailed\], where we describe the recurrence function of any CS Rote sequence in terms of the convergents related to the associated Sturmian sequence. To obtain this result, we proceed similarly as in the previous parts concerning the critical exponent, i.e., we transform our task of finding the recurrence function of a CS Rote sequence into studying some properties of its associated Sturmian sequence. Let us emphasize that we may still restrict our consideration to CS Rote sequences associated with standard Sturmian sequences without loss of generality.
Let $\uu$ be a uniformly recurrent sequence. The mapping $R_\uu: \N \to \N$ defined by $$R_\uu(n)=\min \{N \in \N : \text{each factor of $\uu$ of length $N$ contains all factors of $\uu$ of length $n$}\}$$ is called the *recurrence function* of $\uu$.
The definition of the recurrence function may be reformulated in terms of return words [@CaCh].
\[Obs\_RnRetWords\] Let $\uu$ be a uniformly recurrent sequence. Then $$R_\uu(n)=\max \{|r|\in \N : \text{$r$ is a return word to a factor of $\uu$ of length $n$}\}+n-1.$$
Moreover, to determine $R_\uu(n)$ we can restrict our consideration to return words to bispecial factors of $\uu$.
\[Lem\_BN\] Let $\uu$ be a uniformly recurrent aperiodic sequence. For $n \in \N$, we denote $$\mathcal{B}_{\uu}(n)= \{ b\in \mathcal{L}(\uu) : \exists w \in \mathcal{L}(\uu), |w| = n, \ \text{such that $b$ is the shortest bispecial factor containing $w$} \}\,.$$ Then $$R_\uu(n)=\max \{|r|: \text{$r$ is a return word to $b \in \mathcal{B}_{\uu}(n)$}\}+n-1.$$
For evaluation of $R_\uu(n)$ we use Observation \[Obs\_RnRetWords\]. Let $w\in \mathcal{L}(\uu)$ and $ |w| = n $.
If $w$ is not right special, then there exists a unique letter $x$ such that $wx \in \mathcal{L}(\uu)$. Obviously, the occurrences of $w$ and $wx$ in $\uu$ coincide. Therefore, return words to $w$ and $wx$ coincide as well.
If $y$ is not left special, then there is a unique letter $y$ such that $yw\in \mathcal{L}(\uu)$. If $r$ is a return word to $w$, then the word $yry^{-1}$ is a return word to $yw$ and the return words $r$ and $yry^{-1}$ are of the same length.
These two facts imply that the lengths of return words to $w$ equal the lengths of return words to the shortest bispecial factor containing $w$.
The following lemma shows that for a CS Rote sequence $\vv$ associated with the Sturmian sequence $\uu$ the sets $\mathcal{B}_\vv(n+1)$ and $\mathcal{B}_\uu(n)$ correspond naturally for every $n \in \N$. Thus to determine the set $\mathcal{B}_\vv(n+1)$, we first describe the set $\mathcal{B}_\uu(n)$ for a standard Sturmian sequence $\uu$.
\[Lem\_ShortestBSinRoteSturm\] Let $w$ be a factor of length $n+1$ in a CS Rote sequence $\vv$ and let $v$ be the shortest bispecial factor of $\vv$ containing $w$. Then the factor $\S(v)$ is the shortest bispecial factor of the associated Sturmian sequence $\S(\vv)$ such that $\S(v)$ contains $\S(w)$.
The statement is a consequence of the simple fact that $\S(v)$ is a bispecial factor of $\S(\vv)$ if and only if $v$ and $E(v)$ are bispecial factors of $\vv$. (See Lemma \[lem\_Bispecial\].)
In the sequel, we will essentially use a characterization of Sturmian sequences by palindromes from [@DrPi]. Let us first remind some basic notions. Consider an alphabet $\A$. The assignment $w=w_0w_1\cdots w_{n-1} \to \overline{w} =
w_{n-1}w_{n-2}\cdots w_{0}$ is called a *mirror mapping*, and the word $\overline{w}$ is called the *reversal* or the *mirror image* of $w \in \A^*$. A word $w$ which coincides with its mirror image $\overline{w}$ is a *palindrome*. If $p$ is a palindrome of odd length, then the *center* of $p$ is a letter $a$ such that $p = sa \overline{s}$ for some $s \in \A^*$. The center of a palindrome $p$ of even length is the empty word $\varepsilon$.
\[Thm\_DrPi\] A sequence $\uu$ is Sturmian if and only if $\uu$ contains one palindrome of every even length and two palindromes of every odd length.
Moreover, when studying in detail the proof of Droubay and Pirillo \[Thm\_DrPi\], we deduce that any two palindromes of the same odd length have distinct centers, one has the center $0$ and the other one has the center $1$. In fact, we get the following corollary.
\[Coro\_OnePalExt\] A binary sequence $\uu$ is Sturmian if and only if every palindrome in ${\mathcal L}(\uu)$ has a unique palindromic extension, i.e., for any palindrome $p \in \mathcal{L}(\uu)$ there exists a unique letter $a\in \{0,1\}$ such that $apa \in
\mathcal{L}(\uu)$.
We believe that Theorem \[Thm\_shortestBS\] is already known. However, since we have not found it in the literature, we add its proof. For this purpose, we need an auxiliary lemma. Let us remind that the language $\mathcal{L}(\mathbf{u})$ of a Sturmian sequence $\uu$ is *closed under reversal*, i.e., $\mathcal{L}(\mathbf{u})$ contains with every factor $w$ also its reversal $\overline{w}$, and all bispecial factors of $\uu$ are palindromes.
\[Lem\_CenterOfBS\] Let $\uu$ be a Sturmian sequence. Let $p \in \mathcal{L}(\uu)$ be a palindrome and let $v$ be the shortest bispecial factor containing $p$.
1. Then $p$ is a central factor of $v$, i.e., $v = sp\overline{s}$ for some word $s$.
2. If $v'$ is the shortest bispecial factor with the same center as $p$ and $|v'|\geq |p|$, then $v'=v$.
<!-- -->
1. Let $v = sp$ be the shortest left special factor containing $p$, in particular $0sp$ and $1sp$ belong to ${\mathcal L}(\uu)$. Since the language $\mathcal{L}(\uu)$ is closed under reversal, $p\overline{s}$ is right special, i.e., $p\overline{s}0$ and $p\overline{s}1$ belong to ${\mathcal L}(\uu)$. As $s$ is the only possible extension of $p$ to the left by a factor of length $|s|$, both $sp\overline{s}0$ and $sp\overline{s}1$ belong to ${\mathcal L}(\uu)$. By the same argument, $\overline{s}$ is the only possible extension of $p$ to the right by a factor of length $|s|$. Therefore, $0sp\overline{s}$ and $1sp\overline{s}$ belong to ${\mathcal L}(\uu)$. Thus $sp\overline{s}$ is the shortest bispecial factor containing $p$.
2. Assume for contradiction that $v\not =v'$. Since $v$ and $v'$ are palindromes with the same centers, there exists a palindrome $q$ such that $v'=s'q\overline{s'}$ and $v=sq\overline{s}$. Let $q$ be the longest palindrome with this property. If $|q|=|v'|$, then necessarily $v'=v$. If $|v'|>|q|$, then the last letters of $s'$ and $s$ are distinct and $q$ is a palindrome with two distinct palindromic extensions. This contradicts Corollary \[Coro\_OnePalExt\].
\[Thm\_shortestBS\] Let $\uu$ be a Sturmian sequence and $n \in \N, n \geq 1$. Find the shortest bispecial factors $P_\varepsilon$, resp. $P_0$, resp. $P_1$ of length greater than or equal to $n$ with the center $\varepsilon$, resp. $0$, resp. $1$. Then the following statements hold:
1. Let $w$ be a factor of $\uu$ of length $n$ and let $v$ be the shortest bispecial factor containing $w$. Then $v \in \{P_\varepsilon, P_0, P_1\}$.
2. If $\uu$ contains no bispecial factor of length $n-1$, then for each $i\in \{\varepsilon, 0,1\}$ there exists a factor $w$ of $\uu$ of length $n$ such that the shortest bispecial factor containing $w$ is $P_i$.
3. If there exists a bispecial factor $v$ of $\uu$ of length $n-1$ and let $i \in \{\varepsilon, 0,1\}$ be the center of the palindrome $v$, then for each $j\in \{\varepsilon, 0,1\}, \ j\not =i,$ there exists a factor $w$ of $\uu$ of length $n$ such that the shortest bispecial factor containing $w$ is $P_j$, while $P_i$ is not the shortest bispecial factor containing $w$ for any factor $w$ of $\uu$ of length $n$.
Consider the [*Rauzy graph*]{} of $\uu$ of order $n-1$, i.e., a directed graph $\Gamma_{n-1}$ whose vertices are factors of $\uu$ of length $n-1$ and edges are factors of $\uu$ of length $n$. An edge $e$ starts in the vertex $x$ and ends in the vertex $y$ if $x$ is a prefix and $y$ is a suffix of $e$. Denote $\ell$, resp. $r$ the vertex corresponding to the unique left special, resp. right special factor of length $n-1$. Further on, denote $p_A$ the shortest path from $\ell$ to $r$, and $p_B$ and $p_C$ the shortest paths of non-zero length starting in $r$ and ending in $\ell$. If $\uu$ has no bispecial factor of length $n-1$, then $p_A$ has a positive length, see Figure \[Fig\_Rauzy\](a). If $\uu$ has a bispecial factor of length $n-1$, then the path $p_A$ consists of a unique vertex – the bispecial factor $b$, see Figure \[Fig\_Rauzy\](b).
Observing these Rauzy graphs, it is obvious that for each edge $e$ from the path $p_x$, where $x \in \{A,B,C\}$, the shortest bispecial factor containing $e$ is the same as the shortest bispecial factor containing $p_x$.
Since the language ${\mathcal L}(\uu)$ is closed under reversal, the mirror mapping restricted to the factors of length $n-1$ and $n$ is an automorphism of the graph $\Gamma_{n-1}$. Let us suppose that $p_x$ contains a palindrome $q$ of length $n-1$ or $n$. As any palindrome is mapped onto itself, $\overline{r}=\ell$, and $\overline{\ell}=r$, this path $p_x$ is mapped onto itself, i.e., $\overline{p_x}=p_x$, and the palindrome $q$ is a central factor of $p_x$. On one hand, it means that $p_x$ is a palindrome with the same center as $q$, on the other hand, it also means that $p_x$ cannot contain any other palindrome of length $n-1$ and $n$.
By Theorem \[Thm\_DrPi\] and the comment after it, there are exactly three palindromes among all vertices and edges of $\Gamma_{n-1}$ (all factors of length $n-1$ or $n$), and moreover, they have distinct centers. We may conclude that each path $p_A$, $p_B$, $p_C$ contains exactly one palindrome of length $n-1$ or $n$. Therefore, all these paths are palindromes and their centers are distinct. The rest of the proof follows from Lemma \[Lem\_CenterOfBS\].
\[Obs\_ParikhPal\] Let $P$ be a palindrome. Then its Parikh vector satisfies:
1. $\vec{V}(P)= \binom{0}{0} \mod 2$ if and only if $P$ has the center $\varepsilon$;
2. $\vec{V}(P)=\binom{1}{0} \mod 2$ if and only if $P$ has the center $0$;
3. $\vec{V}(P)=\binom{0}{1} \mod 2$ if and only if $P$ has the center $1$.
Let us recall that a factor of a standard Sturmian sequence $\uu$ is bispecial if and only if it is a palindromic prefix of $\uu$. Therefore we can order the bispecial factors of a given standard Sturmian sequence $\uu$ according to their lengths and denote $\emph{BS}(k)$ the $k$-th bispecial factor of $\uu$. Thus $\emph{BS}(0)=\varepsilon$, $\emph{BS}(1)=a$, where $a$ is the first (and the more frequent) letter of $\uu$ etc.
The sequences $(p_N)$, $(q_N)$, and $(q'_N)$ we use in the remaining part of the paper were introduced in Notation \[Theta\], the notation $\mathcal{B}_{\uu}(n)$ comes from Lemma \[Lem\_BN\].
\[Thm\_P01E\] Let $\uu$ be a standard Sturmian sequence with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $ or $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $, and $n \in [q_N', q_{N+1}')$ for some $N \in \N$. Put $M = a_0+a_1+a_2+ \cdots +a_{N}$, where $a_0=0$.
- If $n \in [q_N', q_{N+1}'-1)$ and $n-1$ is not the length of a bispecial factor, then $$\mathcal{B}_{\uu}(n) = \{BS(M+m), BS(M+m+1), BS(M+a_{N+1}+1)\} \ \text{for some $m \in \{0,1,\dots,a_{N+1}-1\}$}.$$
- If $n \in [q_N', q_{N+1}'-1]$ and $n-1$ is the length of a bispecial factor, then $$\mathcal{B}_{\uu}(n) = \{BS(M+m), BS(M+a_{N+1}+1)\}\ \text{ for some $m \in \{0,1,\dots,a_{N+1}\}$}.$$
Assume that $\uu$ has the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $. By Proposition \[ParikhRSB\], the Parikh vectors of bispecial factors satisfy $$\label{alternace}
\vec{V}\bigl(BS(M+i+1)\bigr) = \vec{V}\bigl(BS(M+i)\bigr) + \left(\!\!\begin{array}{c}p_N\\ q_N\end{array}\!\!\right) \quad \text{for } \ i=0,1,\ldots, a_{N+1} -1$$ $$\label{jiny}
\text{and} \qquad \vec{V}\bigl(BS(M+a_{N+1}+1)\bigr) = \vec{V}\bigl(BS(M+a_{N+1})\bigr) + \left(\!\!\begin{array}{c}p_{N+1}\\ q_{N+1}\end{array}\!\!\right).$$ Using Observation \[Obs\_ParikhPal\] and the relation $\left(\!\!\begin{array}{c}p_N\\ q_N\end{array}\!\!\right)\not =\left(\!\!\begin{array}{c}0\\ 0\end{array}\!\!\right) \mod 2$ from Lemma \[Lem\_Convergents\], we deduce that the centers of palindromes $BS(M+i)$, where $i=0,1, \ldots, a_{N+1}$, alternate between two distinct elements of $\{\varepsilon, 0,1\}$. The third element of $\{\varepsilon, 0,1\}$ is the center of the palindrome $BS(M+a_{N+1}+1)$, as $\left(\!\!\begin{array}{c}p_N\\ q_N\end{array}\!\!\right)\not =\left(\!\!\begin{array}{c}p_{N+1}\\ q_{N+1}\end{array}\!\!\right) \mod 2$, see Lemma \[Lem\_Convergents\]. By Proposition \[ParikhRSB\], the length of $BS(M-1)$ equals $q_N'-2$ and the length of $BS(M+a_{N+1}-1)$ equals $q_{N+1}'-2$.
- Let us discuss the case $n = q_{N+1}'-1$. The palindromes $BS(M+a_{N+1}-1)$, $BS(M+a_{N+1})$ and $BS(M+a_{N+1}+1)$ have distinct centers, and $n-1$ is the length of the palindrome $BS(M+a_{N+1}-1)$. Item (3) of Theorem \[Thm\_shortestBS\] implies that all factors of length $n$ occurs in $BS(M+a_{N+1})$ and $BS(M+a_{N+1}+1)$. Therefore, the set $\mathcal{B}_\uu(n)$ consists of these two bispecial palindromes.
- Now we assume that $n \in [q_{N}', q_{N+1}'-2]$. Clearly, the length of $BS(M-1)$ is strictly smaller then $n-1$ and $n$ does not exceed the length of $BS(M+a_{N+1}-1)$. We choose the smallest $m \in \{0,1,\dots,a_{N+1}-1\}$ such that $n\leq |BS(M+m)|$. The bispecial factors $ BS(M+m)$, $ BS(M+m+1)$, and $BS(M+a_{N+1}+1)$ have distinct centers.
If $n-1$ is not the length of any bispecial factor, then Item (2) of Theorem \[Thm\_shortestBS\] implies that $ \mathcal{B}_\uu(n) = \{BS(M+m), BS(M+m+1), BS(M+a_{N+1}+1)\}$.
If $n-1$ is the length of a bispecial factor, then Item (3) of Theorem \[Thm\_shortestBS\] together with the fact that the centers of $BS(M+i)$ alternate for $i=0,1,\ldots, a_{N+1} -1$ implies that $ \mathcal{B}_\uu(n) = \{BS(M+m), BS(M+a_{N+1}+1)\}$.
If $\uu$ has the directive sequence $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $, then the proof will be analogous, only the coordinates of the Parikh vectors will be exchanged, see Remark \[Rem\_GandDexchanged\].
The recurrence function of a standard Sturmian sequence $\uu$ with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $ or $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $ is known to satisfy $R_\uu(n)=q_{N+1}'+q_N'+n-1$ for every $n \in [q_N', q_{N+1}')$. Let us show that this formula is a consequence of the previous statements. Indeed, by Lemma \[Lem\_BN\], we have to find the longest return word to the bispecial factor from the set $\mathcal{B}_{\uu}(n)$ described in Theorem \[Thm\_P01E\]. Using Proposition \[ParikhRSB\], we find that the longest one is the return word $s$ corresponding to the bispecial factor $BS(M+a_{N+1}+1)$. Its length is $|s| = q_{N+1}'+q_N'$. And thus Lemma \[Lem\_BN\] implies the above mentioned formula, which was obtained by Hedlund and Morse already in 1940, see [@MoHe].
We have prepared everything we need to derive the formula for the recurrence function of CS Rote sequences. For this purpose, we recall Theorem 3.10 from [@MePeVu]:
\[Thm\_RetWordsRote\] Let $\vv$ be a CS Rote sequence associated with the standard Sturmian sequence $\uu=S(\vv)$. Let $v$ be a non-empty prefix of $\vv$ and $u=S(v)$. Let $r$, resp. $s$ be the more frequent, resp. the less frequent return word to $u$ in $\uu$ and let $\ell$ be a positive integer such that $\uu$ is a concatenation of the blocks $r^{\ell }s$ and $r^{\ell+1}s$. Then the prefix $v$ of $\vv$ has three return words $A, B, C$ satisfying:
1. If $r$ is stable and $s$ unstable, then $S(A0)=r, \quad S(B0)=sr^{\ell}s, \quad S(C0)=sr^{\ell+1}s$.
2. If $r$ is unstable and $s$ stable, then $S(A0)=s, \quad S(B0)=rr, \quad S(C0)=rsr$.
3. If both $r$ and $s$ are unstable, then $S(A0)=rr, \quad S(B0)=rs, \quad S(C0)=sr$.
We will use the previous theorem for the determination of return words to bispecial factors of CS Rote sequences (which are by Lemma \[Lem\_ShortestBSinRoteSturm\] associated to bispecial factors of Sturmian sequences). In particular, we focus on $v$ such that $\S(v)$ is a bispecial factor of the Sturmian sequence $\uu = \S({\vv})$ and $\S(v)$ belongs to the set $\mathcal{B}_\uu(n)$ described in Theorem \[Thm\_P01E\].
\[KratkeBS1\] Let $\vv$ be a CS Rote sequence and $\uu = \S(\vv)$ be the associated standard Sturmian sequence with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $ or $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $. Put $M= a_0+a_1+a_2 +\cdots + a_N$, where $a_0=0$. Let $x$ and $y$ be the bispecial factors in $\vv$ such that $\S(x) = BS(M+a_{N+1}+1)$ and $\S(y) = BS(M+m)$, where $m \in \{0,1,\ldots, a_{N+1} - 1\}$. Then at least one return word to $x$ in $\vv$ is longer than every return word to $y$ in $\vv$.
On one hand, by Lemmas \[Lem\_ImageG\], \[Lem\_ImageD\] and Remark \[Rem\_DerivedSeq\], the derived sequence $\dd_{\uu}(\S(y))$ is a standard Sturmian sequence with the directive sequence $G^{a_{N+1}-m}D^{a_{N+2}}G^{a_{N+3}} \cdots $ or $D^{a_{N+1}-m}G^{a_{N+2}}D^{a_{N+3}} \cdots$. It implies that $\uu$ is a concatenation of the blocks ${r'}^\ell s'$ and ${r'}^{\ell+1} s'$, where $\ell = a_{N+1}-m$. The return words to $\S(y)$ are by Proposition \[ParikhRSB\] of length $|r'| =p_N+ q_N= q_N'$ and $|s'| = m\, q_{N}'+q_{N-1}'$. Regardless of (un)stability of the return words to $\S(y)$, the longest return word to $y$ in $\vv$ is by Theorem \[Thm\_RetWordsRote\] of length at most $$(\ell +1)|r'| + 2|s'| =
(a_{N+1}-m +1) q_{N}' + 2(m\, q_{N}'+q_{N-1}') = q_{N+1}' +(m+1)q_{N}'
+ q_{N-1}'\leq 2q_{N+1}'.$$ On the other hand, the return words $r, s$ to $\S(x)$ in $\uu$ have by Proposition \[ParikhRSB\] either lengths $ |r| = q_{N+1}'$ and $|s| = q_{N+1}' + q_{N}'$ (if $a_{N+2}>1 $), or lengths $ |r| =q_{N+1}' + q_{N}' $ and $|s| = q_{N+1}'$ (if $a_{N+2}=1 $). Regardless of (un)stability of the return words to $\S(x)$, one of the return words to $x$ in $\vv$ is of length at least $$|r|+|s| = 2 q_{N+1}' + q_{N}'\,.$$
\[KratkeBS2\] Let $\vv$ be a CS Rote sequence and $\uu = \S(\vv)$ be the associated standard Sturmian sequence with the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $ or $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $. Put $M=a_0+ a_1+a_2 +\cdots + a_N$, where $a_0=0$. Let $x$ and $y$ be the bispecial factors in $\vv$ such that $\S(x) = BS(M+a_{N+1}+1)$ and $\S(y) = BS(M+a_{N+1})$. Then at least one return word to $x$ in $\vv$ is longer than every return word to $y$ in $\vv$.
Let us denote the return words to $\S(y)$ by $r'$ and $s'$, and the return words to $\S(x)$ by $r$ and $s$. By Proposition \[ParikhRSB\], $|r'| = q_{N+1}'$ and $|s'| = q_{N}'$.
First, we assume that $r'$ is unstable. Then by Theorem \[Thm\_RetWordsRote\], the return words to $y$ in $\vv$ are of length at most $2|r'| +|s'| = 2q_{N+1}'+ q_{N}'$. Regardless of (un)stability of the return words to $\S(x)$, one of the return words to $x$ in $\vv$ is of length at least $|r|+|s| = 2 q_{N+1}' + q_{N}'\,. $
It remains to discuss the case when $r'$ is stable. Let us assume that $\uu$ has the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots$. By Proposition \[ParikhRSB\], it means that $|r'|_1 = q_{N+1}$ is even. We use Theorem \[Thm\_RetWordsRote\] to find the longest return word to $y$. Similarly as in the proof of Lemma \[KratkeBS1\], we determine that $\ell = a_{N+2}$ and thus the longest return word to $y$ in $\vv$ is of length $$L'=2|s'| + (\ell +1) |r'|=2\, q_{N}' + (a_{N+2} +1)q_{N+1}' = q_{N+2}' + q_{N+1}' + q_{N}'\,.$$
Let us compare $L'$ with the length of the return words to $x$ in $\vv$. If $a_{N+2}>1$, then by Proposition \[ParikhRSB\], $|r|_1 = |r'|_1 = q_{N+1}$ and $r$ is stable as well. The longest return word to $x$ is by Theorem \[Thm\_RetWordsRote\] the return word $sr^{\ell+1}s$, where $\ell = a_{N+2}-1$. Its length is $$L= 2|s| + a_{N+2}|r| = 2(q_{N+1}'+ q_N') + a_{N+2}q_{N+1}'=q_{N+2}'+2q_{N+1}'+ q_N' > L'\,.$$
If $a_{N+2}=1$, then by Proposition \[ParikhRSB\], $|r|_1 = q_{N+2}=q_{N+1} + q_{N}$ and $|s|_1 = q_{N+1}$. Since $q_{N+1}$ is even, necessarily $q_{N}$ is odd (as follows from the well-known relation $p_Nq_{N+1} -p_{N+1}q_{N} = (-1)^{N+1}$ for all $N$). It means that $r$ is unstable and $s$ is stable. Thus the longest return word to $x$ in $\vv$ has the length $$L =2|r| + |s| = 2 (q_{N+1}'+ q_N') +q_{N+1}' = 2q_{N+2}' + q_{N+1}' > L'.$$
If $\uu$ has the directive sequence $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $, the proof is analogous, we only have to take into account that the coordinates of the Parikh vectors are exchanged (see Remark \[Rem\_GandDexchanged\]). In particular, instead of considering $q_N$ when determining the number of ones, we consider $p_N$.
\[Coro\_RecFunctionRote\] Let $\vv$ be a CS Rote sequence. Let $\uu$ be the standard Sturmian sequence such that $\mathcal{L(\S(\vv))} = \mathcal{L}(\uu)$ and let $\uu$ have the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $ or $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $. Put $M= a_0+a_1+a_2 +\cdots + a_N$, where $a_0=0$. Let $L$ be the length of the longest return word to the bispecial factor $v$ of $\vv$ such that $\S(v)$ is the bispecial factor $ BS(M+a_{N+1}+1)$ in $\uu$. Then the recurrence function of $\vv$ satisfies $R_\vv(n+1)=L+n$ for any $n \in [q_N', q_{N+1}')$, where $N \in \mathbb N$.
Let $\vv'$ be a CS Rote sequence associated with the standard Sturmian sequence $\uu$ such that $\mathcal{L}(\S(\vv)) = \mathcal{L}(\uu)$. Clearly, $\mathcal{L}(\vv) = \mathcal{L}(\vv')$. Since the recurrence function depends only on the language of the sequence and not on the sequence itself, we can work with the CS Rote sequence $\vv'$ instead of $\vv$.
It follows from Lemma \[Lem\_BN\] that $R_\vv(n+1) = R_{\vv'}(n+1)=L+n$, where $L$ is the length of the longest return word to a bispecial factor from the set $\mathcal{B}_{\vv'}(n+1)$. Lemma \[Lem\_ShortestBSinRoteSturm\] shows the correspondence between the bispecial factors from the set $\mathcal{B}_{\vv'}(n+1)$ and the bispecial factors from the set $\mathcal{B}_\uu(n)$. In particular, if $v \in \mathcal{B}_{\vv'}(n+1)$, then $\S(v) \in \mathcal{B}_{\uu}(n)$. For every $n \in [q_N', q_{N+1}')$, where $N \in \N$, the set $\mathcal{B}_\uu(n)$ is described in Theorem \[Thm\_P01E\]. Together with Lemmas \[KratkeBS1\] and \[KratkeBS2\], it implies that the bispecial factor $v$ such that $S(v)$ equals $BS(M+a_{N+1}+1)$ has the longest return word among all bispecial factors from the set $\mathcal{B}_{\vv'}(n+1)$.
\[Thm\_RecFunctionRoteDetailed\] Let $\vv$ be a CS Rote sequence and let $\uu$ be the standard Sturmian sequence such that $\mathcal{L}(\S(\vv)) = \mathcal{L}(\uu)$. If $\uu$ has the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $, then the value of the recurrence function $R_\vv$ for $n \in [q_N', q_{N+1}')$, $N \in \mathbb N$, is given by
Case $q_N$ even
: $ R_\vv(n+1) = \left\{\begin{array}{ll} 2q_{N+1}'+q_N'+n & \quad \text{if \ \ } a_{N+2}>1,\\ 2q_{N+2}'+n & \quad \text{if \ \ } a_{N+2}=1. \end{array}\right. $
Case $q_{N+1}$ even
: $ R_\vv(n+1) = \left\{\begin{array}{ll} q_{N+2}'+2q_{N+1}'+q_N'+n & \quad \text{if \ \ } a_{N+2}>1\\ 2q_{N+2}'+q_{N+1}'+n & \quad \text{if \ \ } a_{N+2}=1. \end{array}\right.$
Case $q_N, q_{N+1}$ odd
: $ R_\vv(n+1) = \left\{\begin{array}{ll} 3q_{N+1}'+q_{N}'+n & \quad \text{if \ \ } a_{N+2}>1\\ q_{N+3}'+q_{N+2}'+q_{N+1}'+n & \quad \text{if \ \ } a_{N+2}=1. \end{array}\right.$
If $\uu$ has the directive sequence $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $, then the value of the recurrence function $R_\vv$ for $n \in [q_N', q_{N+1}')$, $N \in \mathbb N$, is given by
Case $p_N$ even
: $ R_\vv(n+1) = \left\{\begin{array}{ll} 2q_{N+1}'+q_N'+n & \quad \text{if \ \ } a_{N+2}>1,\\ 2q_{N+2}'+n & \quad \text{if \ \ } a_{N+2}=1. \end{array}\right. $
Case $p_{N+1}$ even
: $ R_\vv(n+1) = \left\{\begin{array}{ll} q_{N+2}'+2q_{N+1}'+q_N'+n & \quad \text{if \ \ } a_{N+2}>1\\ 2q_{N+2}'+q_{N+1}'+n & \quad \text{if \ \ } a_{N+2}=1. \end{array}\right.$
Case $p_N, p_{N+1}$ odd
: $ R_\vv(n+1) = \left\{\begin{array}{ll} 3q_{N+1}'+q_{N}'+n & \quad \text{if \ \ } a_{N+2}>1\\ q_{N+3}'+q_{N+2}'+q_{N+1}'+n & \quad \text{if \ \ } a_{N+2}=1. \end{array}\right.$
By Proposition \[Coro\_RecFunctionRote\], $R_\vv(n+1) = L+n$, where $L$ is the length of the longest return word to the bispecial factor $v$ in $\vv$ such that $\S(v) = BS(M+a_{N+1}+1)$. Consider first that $\uu$ has the directive sequence $G^{a_1}D^{a_2}G^{a_3}D^{a_4} \cdots $. By Proposition \[ParikhRSB\], the Parikh vectors of the return words $r$ and $s$ to the bispecial factor $BS(M+a_{N+1}+1)$ are
1. $\vec{V}(r) = \left(\!\!\begin{array}{c}p_{N+1}\\ q_{N+1}\end{array}\!\!\right) , \ \ \vec{V}(s) = \left(\!\!\begin{array}{c} p_{N+1}+p_{N}\\ q_{N+1}+q_{N}\end{array}\!\!\right) \ \text{if } \ a_{N+2} >1$;
2. $\vec{V}(r) = \left(\!\!\begin{array}{c} p_{N+1}+p_{N}\\ q_{N+1}+q_{N}\end{array}\!\!\right), \ \ \vec{V}(s) = \left(\!\!\begin{array}{c}p_{N+1}\\ q_{N+1}\end{array}\!\!\right) \ \text{if } \ a_{N+2} =1$.
Let us emphasize that at most one of the numbers $q_{N}$ and $q_{N+1}$ is even. It follows from the well-known relation $p_Nq_{N+1} -p_{N+1}q_{N} = (-1)^{N+1}$ for all $N$. Moreover, let us recall that $p_N+q_N = q'_N$.
First we discuss the case $a_{N+2}>1$.
- If $q_N$ is even, then $q_{N+1}$ is odd, i.e., $r$ and $s$ are unstable. Since $|r|< |s|$, Item (3) of Theorem \[Thm\_RetWordsRote\] gives $L = |rs| = 2q_{N+1}' + q_{N}'$.
- If $q_{N+1}$ is even, then $q_{N}$ is odd, i.e., $r$ is stable and $s$ is unstable. We use Item (1) of Theorem \[Thm\_RetWordsRote\] with $\ell =a_{N+2} - 1$. Clearly, $L = 2|s| + (\ell +1)|r| = 2(q_{N+1}' + q_{N}') + a_{N+2}q_{N+1}' = q_{N+2}' + 2q_{N+1}' + q_{N}'$.
- If both $q_{N}$, $q_{N+1}$ are odd, then $r$ is unstable and $s$ is stable. Item (2) of Theorem \[Thm\_RetWordsRote\] implies $L = |rsr| = 3q_{N+1}' + q_{N}'$.
It remains to discuss the case $a_{N+2}=1$.
- If $q_N$ is even, then $q_{N+1}$ is odd, i.e., $r$ and $s$ are unstable. Since $|r|> |s|$, Item (3) of Theorem \[Thm\_RetWordsRote\] gives $L = |rr| = 2q_{N+1}' + 2q_{N}' = 2q_{N+2}'$.
- If $q_{N+1}$ is even, then $q_{N}$ is odd, i.e., $r$ is unstable and $s$ is stable. Item (2) of Theorem \[Thm\_RetWordsRote\] implies $L = |rsr| = 3q_{N+1}' + 2q_{N}'= 2q_{N+2}' + q_{N+1}'$.
- If both $q_{N}$, $q_{N+1}$ are odd, then $r$ is stable and $s$ is unstable. We use Item (1) of Theorem \[Thm\_RetWordsRote\] with $\ell =a_{N+3}$. Thus $L = 2|s| + (\ell +1)|r| = 2q_{N+1}' + (a_{N+3}+1)(q_{N+1}' + q_{N}') = q_{N+3}' + q_{N+2}' + q_{N+1}' $.
If $\uu$ has the directive sequence $D^{a_1}G^{a_2}D^{a_3}G^{a_4} \cdots $, then the statement of Theorem \[Thm\_RecFunctionRoteDetailed\] will stay the same, only $q_N$ and $q_{N+1}$ will be replaced by $p_N$ and $p_{N+1}$ because the Parikh vectors of $r$ and $s$ have the coordinates exchanged, see Remark \[Rem\_GandDexchanged\].
By Proposition \[Prop\_MalyExponent\], the critical exponent of the CS Rote sequence $\vv$ such that $S(\vv)$ has the directive sequence $G(D^2 G^2)^{\omega}$ is $\CR(\vv)=2+\frac{1}{\sqrt{2}}$. In Example \[Ex\_Rozdil\_cr\_u\_Eu\], we have shown that the CS Rote sequence $\vv'$ associated to the Sturmian sequence $S(\vv')=E(S(\vv))$ has the critical exponent $\CR(\vv')=4+\frac{1}{1+\sqrt{2}}$.
Let us find an explicit formula for the recurrence function $R_{\vv}$, resp. $R_{\vv'}$ of the CS Rote sequence $\vv$, resp. $\vv'$. We will see that these recurrence functions differ essentially, too. In the proof of Proposition \[Prop\_MalyExponent\], we have shown that all $q_{N}$ are odd and we have found an explicit formula for $q_{N}'$, see . Applying Theorem \[Thm\_RecFunctionRoteDetailed\], we obtain for every $n \in [q_N', q_{N+1}')$ $$R_{\vv}(n+1)=3q_{N+1}'+q_{N}'+n= n+ \frac{1}{2\sqrt{2}} \Bigl( (4+3\sqrt{2})(1+\sqrt{2})^{N+1} - (4-3\sqrt{2})(1-\sqrt{2})^{N+1}\Bigr)\,.$$ Further on, $p_N$ is even if and only if $N$ is even. Therefore, we obtain for every $n \in [q_{2N}', q_{2N+1}')$ $$R_{\vv'}(n+1)=2q_{2N+1}'+q_{2N}'+n=n+\frac{1}{2\sqrt{2}}\Bigl( (1+\sqrt{2})^{2N+3} - (1-\sqrt{2})^{2N+3}\Bigr)\,;$$ and for every $n \in [q_{2N-1}', q_{2N}')$ $$R_{\vv'}(n+1)=q_{2N+1}'+2q_{2N}'+q_{2N-1}'+n=n+\frac{1}{\sqrt{2}}\Bigl( (1+\sqrt{2})^{2N+2} - (1-\sqrt{2})^{2N+2}\Bigr)\,.$$
Acknowledgement {#acknowledgement .unnumbered}
===============
The research received funding from the Ministry of Education, Youth and Sports of the Czech Republic through the project no. CZ.02.1.01/0.0/0.0/16\_019/0000778 and from the Grant Agency of the Czech Technical University in Prague through the grant no. SGS20/183/OHK4/3T/14.
[10]{} , [*Sturmian jungle (or garden?) on multiliteral alphabets*]{}, RAIRO Theor. Inf. Appl. 44 (2010), 443–470.
, [*Critical exponent of infinite balanced words via the Pell number system*]{}, in: R. Mercas and D. Reidenbach (eds.), Proceedings WORDS 2019, Lecture Notes in Computer Science, vol. 11682, Springer (2019), 80–92.
, [*Repetitions in infinite palindrome-rich words*]{}, in: R. Mercas and D. Reidenbach (eds.), Proceedings WORDS 2019, Lecture Notes in Computer Science, vol. 11682, Springer (2019), 93–105.
, [*Axel Thue’s papers on repetitions in words: a translation,*]{} Publications du LaCIM 20, Université du Québec à Montréal (1995).
, [*Palindromic complexity of codings of rotations*]{}, Theoret. Comput. Sci. 412(46) (2011), 6455–6463.
, [*Fonctions de récurrence des suites d’Arnoux-Rauzy et réponse à une question de Morse et Hedlund*]{}, Ann. Inst. Fourier (Grenoble) 56(7) (2006), 2249–2270.
, [*The repetition threshold for binary rich words*]{}, Discrete Math. Theoret. Comput. Sci. 22(1) (2020), no. 6.
, [*The index of Sturmian sequences*]{}, European J. Combin. 23 (2002), 23–29.
, [*Episturmian words and some construction of de Luca and Rauzy*]{}, Theor. Comput. Sci. [255]{} (2001), 539–553.
, [*Palindromes and Sturmian words*]{}, Theoret. Comput. Sci. [223]{} (1999), 73–85.
, [*A characterization of substitutive sequences using return words*]{}, Discrete Math. 179 (1998), 89–101.
, [*Continued Fractions*]{}, World Scientific Publishing (2006).
, [*Episturmian words and episturmian morphisms*]{}, Theoret. Comput. Sci. 276 (2002), 281–313.
, [*Fixed points of Sturmian morphisms and their derivated words*]{}, Theoret. Comput. Sci. 743 (2018), 23–37.
, [*Combinatorics on Words*]{}, Encyclopaedia of Mathematics and its Applications, vol. 17, Addison-Wesley, Reading, Mass. (1983). Reprinted in the Cambridge Mathematical Library, Cambridge University Press (1997).
, [*Derived sequences of complementary symmetric Rote sequences*]{}, RAIRO-Theor. Inf. Appl. 53 (2019), 125–151.
, [*Repetitions in the Fibonacci words*]{}, RAIRO-Theor. Inf. Appl. 26 (1992), 199–204.
, [*Symbolic dynamics II. Sturmian trajectories*]{}, Amer. J. Math. 62 (1940), 1–42.
, [*Languages invariant under more symmetries: overlapping factors versus palindromic richness*]{}, Discrete Math. 313 (2013), 2432–2445.
, [*Critical exponents of infinite balanced words*]{}, Theor. Comp. Sci. [777]{} (2019), 454–463.
, [*Formal Languages, Automata and Numeration Systems*]{}, 2 vols., Wiley (2014).
, [*Sequences with subword complexity $2n$*]{}, J. Number Theory 46 (1994), 196–213.
, [*Probleme über Veränderungen von Zeichenreihen nach gegebene Regeln*]{}, Christiana Videnskabs-Selskabs Skrifter, I. Math.-naturv. Klasse 10 (1914).
, [*A characterization of [S]{}turmian words by return words*]{}, Eur. J. Combin. 22 (2001), 263–275.
|
---
abstract: 'The impact of control sequences on the environmental coupling of a quantum system can be described in terms of a filter. Here we analyze how the coherent evolution of two interacting spins subject to periodic control pulses, at the example of a nitrogen vacancy center coupled to a nuclear spin, can be described in the filter framework in both the weak and the strong coupling limit. A universal functional dependence around the filter resonances then allows for tuning the coupling type and strength. Originally limited to small rotation angles, we show how the validity range of the filter description can be extended to the long time limit by time-sliced evolution sequences. Based on that insight, the construction of tunable, noise decoupled, conditional gates composed of alternating pulse sequences is proposed. In particular such an approach can lead to a significant improvement in fidelity as compared to a strictly periodic control sequence. Moreover we analyze the decoherence impact, the relation to the filter for classical noise known from dynamical decoupling sequences, and we outline how an alternating sequence can improve spin sensing protocols.'
author:
- Andreas Albrecht
- 'Martin B. Plenio'
bibliography:
- 'filt\_design.bib'
title: Filter design for hybrid spin gates
---
The interaction of a quantum systems with its environment is due both to coherent control fields and environmental noise. Suppressing the environmental noise is crucial for extending the coherence time and for colored noise is frequently performed by dynamical decoupling techniques[@du09; @biercuk09; @delange10]. Such approaches aim at refocusing undesired couplings by means of a control field or control pulse sequence, without the need of additional ancillas or measurements. Whereas these concepts are well-understood for protecting single isolated qubits, their extension to multiple and interacting quantum systems turns out to be much more challenging[@xu12; @vandersar12; @gordon07]. In particular the question arises how to decouple a qubit from a noise background while at the same time retaining very specific desired interactions. Such a situation occurs in the construction of decoupled quantum gates: Even though their feasibility has been theoretically proven[@khodjasteh08; @khodjasteh09; @west10], an actual implementation requires concepts that are tailored to the specifics of the system. Experimental realizations range from decoupled single[@xu12; @zhang14] and multiqubit[@vandersar12; @dolde13; @taminiau13] spin gates to single[@timoney11] and multiqubit gates[@piltz13] in trapped ions. Another example is given by dipolar recoupling techniques in NMR experiments[@lin09; @paravastu06], which aim at the controlled and selective reintroduction of specific interactions for structural information gain.
More generally, control sequences can be used to construct and design filters[@cywinski08; @kofman01; @biercuk11; @sousa09; @gordon07] transmissive exclusively for very specific interactions and frequencies. This insight has lead to the construction of quantum ‘lock-in’ setups[@kotler11; @cai13] or the design of Hamiltonians[@ajoy13] by choosing appropriate control parameters. Moreover decoupling sequences can be intuitively interpreted in the filter framework[@cywinski08; @gordon07; @uhrig07], with decoherence arising as the overlap of a filter with the corresponding noise spectrum. Importantly, scanning a narrowband filter in frequency allows for the tomography of the noise spectrum[@alvarez11; @bylander11], the detection of ac-fields[@delange11; @pham12] and even the sensing of (single) individual spins[@zhao12; @taminiau12; @kolkowitz12; @london13; @mueller14]. Furthermore, the insight following from filter descriptions has been used for the optimization of quantum memories[@khodjasteh13] and for the construction of sensitive mass[@zhao13] and mechanical motion[@kolkowitz12mech] spectrometers.
Here we analyze the filter description for the coherent coupling of hybrid spin systems. As an explicit example, though not limited to, we consider the nitrogen vacancy center (NV$^-$-center) in diamond, characterized by an electronic spin-1 ground state, and its (hyperfine) coupling to individual (nuclear) spins. Both the detection[@zhao12; @taminiau12; @kolkowitz12; @london13; @mueller14] and manipulation[@taminiau12; @taminiau13] of single nuclear spins, e.g. ${}^{13}$C, ${}^{29}$Si or ${}^{14}$N nuclear spins, via the center spin have been demonstrated. Two relevant regimes emerge and are well-described in such a filter formalism: the weak and strong coupling limit which correspond to close[@vandersar12] and distant nuclear spins[@zhao12; @taminiau12; @kolkowitz12; @taminiau13] from the center, respectively.
Section\[sect1\] is devoted to the filter description in the weak and the strong coupling regime, which represents a good description for short interaction times and hence small rotation angles. This description reveals universal properties around the resonance points of the filter, which allow for tuning of the interaction type and strength. Section\[sect2\] then shows how to extend the applicability of these concepts beyond the short time limit and reveals the ability for tunable gate interactions based on alternating control pulse sequences. An application of this insight is provided in section\[sect3\], linking first the gate formalism to the decoherence description, followed by providing concepts for the use of the alternating sequences in spin sensing protocols with improved frequency resolution. Appendix\[append\_limit\] analyzes in detail the limitations of the filter formalism. Appendix\[append\_deriv\] provides the derivation of the filter formulas and Appendix\[append\_strong1h2\] an analysis of the strong coupling limit for a spin 1/2 control spin. Last, Appendix\[append\_grating\] reviews the filter-grating description which simplifies the analysis and filter calculation for alternating pulse sequences.
Filter formalism for coherent spin interactions {#sect1}
===============================================
We will consider the generic Hamiltonian $$\label{ham1} H=\frac{\omega}{2}\,\sigma_z+\frac{A}{2}\,\sigma_x\otimes S_z(t)$$ describing the interaction of a nuclear (‘target’) spin with energy separation $\omega$, hyperfine coupled with frequency $A$ to an electronic (‘control’) spin. Here $\sigma_k$ ($k$$\in$$\{x,y,z\}$) denotes the Pauli matrices acting on the target qubit, and for clarity the analogs on the control spin are denoted by $s_k$ in what follows. $S_z(t)$ describes a $z$-type coupling on the control spin, projected on the (pseudo)spin-1/2 submanifold as selected by a sequence of population inverting $\pi$-pulses; the latter reflected in the time dependence. Such a form holds for any type of spin couplings with significantly different energy scales which lead to the exclusion of population exchange with the control spin system. In particular we assume the control spin to be of spin-1/2 or spin-1 as in the nitrogen vacancy (NV) center in diamond and the target (nuclear) spin to be a spin-1/2 which applies to a wide variety of potential nuclear spin targets. It is important to note that the target spin is not restricted to spin-1/2 and higher spins can be introduced straightforwardly into the formalism by replacing the spin matrices by their higher order analogs.
For the control spin two specific configurations are considered, dependent on the two-level (sub)-system involved in the control sequence: A spin-1/2 control qubit or the spin-1 sub-manifold $\{m_s$=$\pm$1$\}$, both denoted by ‘*spin-1/2 control spin*’ in the following and describable by $S_z(t)=s_z s(t)$. A spin-1 sub-system composed of $m_s$=0 and one of the states $m_s=\pm1$, referred to as ‘*spin-1 control*’, leading to $S_z(t)=(1/2) (s_z\,s(t)\pm\mathds{1})$ with $s_z$ and $\mathds{1}$ the Pauli z-matrix and identity defined on the coupled sub-manifold, respectively. Herein the stepfunction $s(t_0)=1$ at the initial time with a sign change at each $\pi$-pulse time $t_k$, i.e. $s(t)=\sum_{k=0}^N (-1)^k \theta(t_{k+1}-t)\,\theta(t-t_k)$ for $N$-pulses with the final time $t_f\equiv t_{N+1}$ and the Heaviside stepfunction $\theta$.
The effective energy separation of an environmental spin with Larmor frequency $\boldsymbol{\omega_0}=\gamma\,\boldsymbol{B}$ ($H_0=(1/2)\boldsymbol{\omega_0}\,\boldsymbol{\sigma}$) coupled to a control spin-1 by the hyperfine interaction $\boldsymbol{A}$ ($H_{\rm hyp}=\boldsymbol{A}\boldsymbol{\sigma}\otimes 1/2(s_z s(t)\pm \mathds{1}$) is characterized by the dressed states of the unconditional (‘$s_z$-independent’) contributions. Thus, $\omega=|\boldsymbol{\omega_0}\pm\boldsymbol{A}|$[@zhao14; @taminiau12], which allows for distinguishing even target spins of the same species by their position dependent coupling properties. The orthogonal hyperfine component in (\[ham1\]) is then just the orthogonal component with respect to the dressed energy levels; parallel components can be neglected in the perturbative limit evolution considered below. Adding a continuous resonant Rabi driving $\Omega$ on the target spin as has been performed in [@vandersar12], can be described by the Hamiltonian $H=(\Omega/2)\,\sigma_x+(A/2)\,\sigma_z\otimes S_z(t)$, and thus corresponds to a global coordinate rotation of the generic form (\[ham1\]) by replacing $\omega\sigma_z \to \Omega\sigma_x$, $A\sigma_x \to A\sigma_z$ and $\sigma_y\to-\sigma_y$.
In the following the hyperfine coupling induced interaction is analyzed both in the strong and in the weak coupling limit under the influence of a periodic CPMG $N$-pulse control sequence[@meiboom58; @vandersar12] of pulse separation $2\tau$ and total time $t=2N\tau$, i.e. $t_k=-\tau+k\,(2\tau)$ ($k$$\in$\[1,N\]). The effective coupling strength can then be described by a control pulse dependent filter $\mathcal{F}$ with resonances characterized by widths $1/t$ in frequency. Such a description is based on a first order Magnus expansion and holds true in the limit of small total evolution angles as shown in Appendix\[append\_limit\]. The extension to much longer timescales is discussed in the subsequent section. Importantly, besides allowing for specific frequency selective gate interactions, the CPMG control sequence leads to a decoherence decoupling[@delange10; @naydenov11], e.g. extends the coherence time $T_2$ for a quasi-static noise bath to[@sousa09] $T_2^{[N]}=N^{2/3}T_2$.
Weak coupling limit $A\ll \omega$
---------------------------------
The weak coupling limit is essentially insensitive to the control spin being either of the ‘spin-1/2’ or ‘spin-1’ type, the latter requiring simply a substitution $A\to A/2$ in the subsequent expressions as a result of its effectively weaker hyperfine coupling. The total *conditional* evolution in a rotating frame with respect to $H_0=(\omega/2)\sigma_z$ follows as $$\label{uint_wc} U_{\rm int}=\exp\left( -i\frac{A}{2}\,t\,\mathcal{F}_{w}(\omega\tau,N) \,\sigma_\phi\otimes s_z+\mathcal{O}([At]^2)\right)$$ with the frequency selective filter given by (see Appendix\[append\_deriv\]) $$\label{filt_weak} \mathcal{F}_w(\omega\tau,N)= \frac{2}{N(\omega\tau)} \left| \frac{\sin^2(\omega\tau/2)}{\cos(\omega\tau)}\,\sin(N\omega\tau+\{\pi/2 \}) \right|$$ and the argument in curly brackets being absent/present for $N$ even and odd, respectively. Such a filter defines the effective coupling frequency as $(A\,\mathcal{F}_w)$ and is illustrated in figure\[b\_01filter\](a). It reveals sharp resonances around $\omega\tau=(2k+1)\,\pi/2$ ($k\in \mathds{N}_0$) with a frequency width inversely to the total time, and a total peak height decreasing as $\propto 1/(\omega\tau)$. The rotation axis $\sigma_\phi$ in the resonance region $$\label{rescond_weak} \omega\tau=(2k+1)\,\pi/2+c\,\pi/(2N)$$ follows a simple linear behavior, that is $$\label{rot_wc} \sigma_\phi=(-1)^k\,[\cos\phi \,\sigma_x-\sin\phi\, \sigma_y ] \quad \text{with }\phi=c\cdot (\pi/2)\,,$$ where $c\in (-2,2)$, except for $N$=1 & $k$=0 in which case the restriction $c\in (-1,1)$ holds. This angle dependence is valid in any limit of $N$. Moreover in the limit of large $N$ ($N\gtrsim 4$), or for any $N$ at $ c$=0, the filter (\[filt\_weak\]) takes on a universal behavior around the resonances $$\label{filt_weak_approx} \mathcal{F}_w(c,k)\simeq\frac{4}{\pi^2}\,\frac{|\sin(c\,\pi/2)|}{|c|\,(2k+1)}$$ resulting in the filter amplitudes $F_w(0,k)=(2/\pi) 1/(2k+1)$ for a $c$=0 conditional $\sigma_x$, and $\mathcal{F}_w(1,k)\simeq(4/\pi^2) 1/(2k+1)$ for a $c$=1 $\sigma_y$-type rotation. Importantly, the $N$ independence in (\[filt\_weak\_approx\]) ensures a linear accumulation of the rotation in time. On the other hand, the $k$ dependence reflects a reduced coupling for higher order resonances. In the validity range of (\[filt\_weak\_approx\]), the total rotation angle for a given $c$ and $N$ turns out to be independent of $k$, however associated with a longer total time for increasing $k$. Moreover, the filter is symmetric in $c$ exclusively in the limit (\[filt\_weak\_approx\]) of large $N$.
Strong coupling limit $A\gg\omega$
----------------------------------
In the strong coupling limit the achievable interactions crucially depend on the spin properties of the control qubit. Only for the ‘spin-1 configuration’ a conditional interaction that is scaling linearly in time can be achieved and will be focused on in the following; the ‘spin-1/2 configuration’ leads at most to a scalable (less relevant) unconditional evolution and is briefly outlined in Appendix\[append\_strong1h2\].\
For a ‘spin-1’ control qubit the evolution in a rotating frame with respect to the hyperfine Hamiltonian can be described by $$\label{uint_sc}\begin{split} U_{\rm int}(t)=&\exp\left(-i\,\frac{\omega}{2}t\,\left[ \mathcal{F}_{\rm c}(A\tau, N)\,\sigma_{\phi}^c\otimes s_z\right.\right.\\
&\left.\left.+\mathcal{F}_{\rm u}(A\tau, N)\,\sigma_{\phi}^u\otimes \mathds{1} \right]+\mathcal{O}([\omega t]^2)\right) \end{split}$$ connected to the the original (\[ham1\]) frame evolution by $U(t_n)=\exp(\mp i(A/4)t_n\,\sigma_x)\,U_{\rm int}(t_n)$ at the end of a basic decoupling sequence $t_n=t_0+2n\tau$. Upper and lower signs here and in the following are in accordance with $S_z(t)=(1/2) (s_z\,s(t)\pm\mathds{1})$, dependent on the spin-1 subsystem connected by the control pulses as previously defined. Thus the evolution is characterized by both conditional and unconditional contributions, with the corresponding filters of width $\Delta A \propto 1/t\propto 1/N$ defined as (see Appendix\[append\_deriv\]) $$\begin{aligned}
\label{filter_sc} \mathcal{F}_{\rm c}(A\tau, N) &=&\frac{1}{2N}\Bigl| \sin\left(N \frac{A\tau}{2}+\Bigl\{\frac{\pi}{2}\Bigr\}\right) \,\tan\left(\frac{ A\tau }{2}\right) \Bigr| \\
\mathcal{F}_{\rm u}(A\tau, N) &=&\frac{1}{2N}\,\left| \left[\frac{2}{A\tau}+\cot\left( \frac{A\tau}{2} \right)\right]\,\sin\left(\frac{N A\tau}{2}\right) \right| \nonumber\end{aligned}$$ and illustrated in figure\[b\_01filter\](b). Again the contribution in wavy brackets is present exclusively for $N$ being odd.
![\[b\_01filter\] Gate filter for periodic control pulses. Filter function $\mathcal{F}$ describing the coherent spin interaction in the limit of small total evolution angles and for $N=6$ periodic $\pi$-CPMG-pulses with interpulse delay $2\tau$ on the control spin. **(a)** Weak coupling limit $A\ll\omega$, which leads to a purely conditional interaction as follows out of (\[uint\_wc\]). The corresponding rotation axis $\sigma_\phi$ around the resonance peaks as given by (\[rot\_wc\]) is illustrated in the inset. Amplitudes for higher order resonances decay as $1/(\omega\,\tau)$ (black dash-dotted), whereas the resonance peak width is inversely proportional to the total time. **(b)** Strong coupling limit $A\gg\omega$ with both conditional (blue) and unconditional (green) resonances and filters. For both (a) and (b) red dashed lines denote the universal amplitude behavior (\[filt\_weak\_approx\]) and (\[univ\_sc\]) valid in the limit of large $N$’s. ](01_filter_illustr.pdf)
The resonance (region) condition, in a notation analogous to (\[rescond\_weak\]) is given by $$\begin{split} (A\,\tau)_c&=(2k+1)\,\pi+c\,\pi/N,\\ (A\,\tau)_u&=k\,(2\pi)+c\,\pi/N \end{split}$$ for the conditional ($c$) and unconditional ($u$) filter, respectively. The corresponding rotation axis follows as $$\label{rota_sc}\begin{split} \sigma_\phi^c&=\mp\,(\cos(\phi)\sigma_z\pm\sin(\phi)\,\sigma_y) \\
\sigma_\phi^u& = \cos(\phi)\,\sigma_z\pm\sin\phi\,\sigma_y
\end{split}$$ with $\phi=c\,\pi/2$ for $c\in (-2,2)$, except $c\in(-1,1)$ for $\sigma_\phi^c$ and $N$=1, and restricted $k$ dependent c-ranges for $\sigma_\phi^u$ and $N=1,2$.
For large $N$ ($N\gtrsim 4$) both the conditional and unconditional filter are well-approximated by the universal form $$\label{univ_sc} \mathcal{F}_{c/u}\simeq \dfrac{\sin\left( c\pi/2\right)}{c\,\pi}\,,$$ which exactly holds for any $N$ on the resonance $c$=0. Thus, $c$=0 leads to a $\sigma_z$-type rotation with $\mathcal{F}_{c/u}=1/2$, whereas $|c|$=1 corresponds to a $\sigma_y$ rotation with $\mathcal{F}_{c/u}\simeq1/\pi$. The $N$ independence of (\[univ\_sc\]) indicates the additivity in time; the additional $k$ independence reveals that the coupling amplitude, despite a modification of the ability for noise decoupling, is preserved for higher order resonances. Out of (\[filter\_sc\]) and (\[univ\_sc\]) it directly follows that the conditional filter is symmetric in $c$ in any limit of $N$, whereas this holds true for the unconditional filter exclusively for sufficiently large $N$. The (un)conditional contribution at the unconditional (conditional) resonance peak does not scale in $N$ and is thus negligible for a sufficiently large number of pulses; moreover it is zero for $c$=0 and $|c|$=1 for an even and odd number of pulses $N$, respectively, in any limit except for $N$=1[^1].
Sliced gate evolutions and alternating pulse sequences {#sect2}
======================================================
The filter description and its associated properties discussed previously hold true as long as the interaction time and thus the evolution angle is small $\theta\ll 1$ (see Appendix\[append\_limit\]). For gate interactions this forms a severe limitation, which can be overcome by alternating pulse sequences as shown in the following.
We will assume that the total evolution can be split into $s$ slices $[t_j,t_{j+1})$ ($j\in\,$\[0,$s$-1\]), with slice $j$ consisting of $n_j$-periodic CPMG-control pulses with characteristic timescale $\tau_j$ as shown in figure\[b\_sliced\](d). Moreover $t_0$=0, the total time $t$=$t_s$ and the slice time $\Delta t_j$=$t_{j+1}$-$t_j$. Then the total evolution can be cast into the form $$\label{slice_evol1}U=\prod_{j=0}^{s-1} {\rm e}^{-i\,H\,\Delta t_j}={\rm e^{-i\,H_0\,t}}\,\prod_{j=0}^{s-1}{\rm e}^{i\,H_0\,t_j}U^{(j)}_{\rm int}(\Delta t_j) {\rm e}^{-i\,H_0\,t_j}$$ where $H_0=(\omega/2)\,\sigma_z$ and $H_0=\pm (A/4)\,\sigma_x$ for the weak and strong coupling configuration, respectively. $U_{\rm int}^{(j)}$ denotes the rotating frame evolution operator as defined in(\[uint\_wc\]) and (\[uint\_sc\]) with the index $(j)$ indicating that the replacement $s_z$$\to$$-s_z$ has to be made in these expressions for an odd number of preceding pulses. Assuming the rotation angle per slice to be small, its evolution is well-described in the filter description; e.g. for $A$$\ll$$\omega$ the rotation angle $\theta_j$=$2(A/\omega)$$(n_j\omega\tau_j)\,\mathcal{F}_w(\omega\tau_j, n_j)$, which is small as long as $n_j$ is. Following (\[slice\_evol1\]) the only modification consists of an effective coordinate $\exp(iH_0 \Delta t_j)$ rotation between subsequent slices $j$ and $j+1$.
The rotation of the coordinate frame is closely linked to the rotation axis angle of the evolution. This property is analyzed for the weak coupling limit in the following, however the same conclusions hold true for the strong coupling counterpart. After a single slice $j$ around a rotation axis $\phi=c\,\pi/2$ following out of conditions (\[rescond\_weak\]) and (\[rot\_wc\]), the local coordinate system can be interpreted as being rotated by $$\vartheta_j=2n_j\,(\omega\tau_j)=(2k+1)n_j\pi+c\,\pi$$ for the subsequent slice. This rotation, for $n_j$ being even, is just twice the rotation axis angle $\vartheta_j=2\phi$ (see figure\[b\_sliced\](a)). An additional $\pi$-inversion occurs for $n_j$ being odd, i.e. $\vartheta_j=2\phi+\pi$. However in that latter case also $s_z$$\to$$-s_z$ as outlined previously by the index $(j)$ to the evolution operator, and thus the rotation axis is $\pi$-inverted simultaneously and the subsequent conclusions remain unchanged[^2].
Thus, to globally maintain a fixed rotation axis $\phi$, the subsequent slice has to be tuned to $-c$. An alternating evolution sequence $c\to -c\to c\to -c\dots$ as depicted in figure\[b\_sliced\](d) therefore allows for significant rotation angles beyond the perturbative limit around an arbitrary but fixed rotation axis (\[rot\_wc\]) as characterized by $\phi=c\,\pi/2$. This is illustrated for a $\pi/2$ evolution in figure\[b\_sliced\](c), comparing a non-alternating $N$=$20$ pulse sequence for $c$=1 to a $c$=$\pm 1$ alternating sequence of $s$=10 slices with $n_j$=$2$ pulses each. Whereas the former simulation shows a clear deviation from the filter based expectation as a result of the small-angle approximation break-down, the latter sliced evolution leads to an (almost) perfect $\sigma_y$-type conditional $\pi/2$-rotation with fidelity $F=99.9\%$.
For an even number of alternating $\pm c$ slices (or for $c$=0), $\exp(iH_0\,t)\propto \mathds{1}$ such that the rotating frame evolution is equal to the original one. Moreover in this case and for $n_j$$\equiv$$n$, the effective ‘filter’ for the $c$-axis rotation in analogy to(\[uint\_wc\]) is given by $$\mathcal{F}_{\rm eff}=1/(2n)\,((c+n)\,\mathcal{F}_{\rm +c}+(-c+n)\mathcal{F}_{\rm -c})\,,$$ or simply $\mathcal{F}_{\rm eff}=\mathcal{F}_{\pm c}$ whenever $\mathcal{F}$ is symmetric in $c$.
Remarkably the $c$=0 resonance peak is purely additive even in the long time limit without any adjustments, a fact that can already be seen in the $N$-independence of the corresponding resonance condition(\[rescond\_weak\]). On the contrary, two subsequent $c$=+1 slices would be subtractive to first order as the coordinate frame rotation inverts the global rotation axis for the second slice.
The above insight can also be applied for the interpretation of the long time (large $N$) evolution of equidistant CPMG sequences ($\tau_j\equiv\tau$). For that purpose, an $N$-pulse sequence characterized by a resonance parameter $c$ according to (\[rescond\_weak\]) can be formally sliced into $s$ segments of $n'=N/s$ pulses each. The resonance and rotation properties for each slice are then characterized by $c'=(n'/N)c$, as follows by comparing the resonance condition per slice $\omega \tau=(2k+1)\pi/2+c' \pi/(2n')$ to (\[rescond\_weak\]). Choosing sufficiently many slices such that the rotation per slice is small allows for the interpretation within the filter formalism. Namely, as illustrated in figure\[b\_sliced\](b), the rotation of each slice with $c'$ is followed by a coordinate system rotation $\vartheta'=c'\pi$; up to a total rotation $\vartheta=c\pi$ for the total sequence. Essentially this equals a rotation of the rotation axis in time. Therefore, based on the slicing approach, the filter description retains its usefulness beyond the perturbative limit.
Spin sensing and frequency filter for alternating pulse sequences {#sect3}
=================================================================
The conditional coherent interaction induced by Hamiltonian (\[ham1\]) leads to a mutual, time-periodic entanglement of the control and target spin. This effect can be observed as a coherence decay on the control spin subsystem, which along with the frequency selectivity of the control pulse sequence is widely used for sensing individual environmental spins[@zhao12; @taminiau12; @kolkowitz12]. More precise, the coherence follows as (with $s_+$ the ladder operator for the control spin) ${\langle s_+(t) \rangle}={\langle s_+(0) \rangle}\,\text{tr}[ {U_{\rm int}^{(s_z=+1)\dagger}} U_{\rm int}^{(s_z=-1)} \rho_{\rm c} ]$, which in the $A\ll\omega$ limit and assuming the target initial state $\rho_{\rm c}=(1/2)\mathds{1}$ to be completely mixed, leads to $$\label{spin_sens1} {\langle s_+(t) \rangle}\simeq {\langle s_+(0) \rangle}\,\cos\left(At\,\mathcal{F}_w(\omega\tau,N)\right)\,.$$
For sufficiently short times, the coherence evolution follows as ${\langle s_+(t) \rangle}\simeq {\langle s_+(0) \rangle}\,\exp(-(1/2)\,A^2\,t^2 \mathcal{F}_w(\omega\tau,N)^2)$. This latter expression is formally equivalent to the decay under classical noise[@zhao12], with the corresponding filter $\mathcal{F}_{\rm class}=\mathcal{F}_w^{\,2}$[@cywinski08; @sousa09] and assuming a noise bath with discrete spectrum $S(\omega')=\delta(\omega-\omega')$.
The validity range of (\[spin\_sens1\]) is limited by the applicability range of the filter formalism. In practice, as simulated for a CPMG sequence in figure\[b\_sensing\](a), the coherence is well-described by (\[spin\_sens1\]) within the initial sign inverting regime ${\langle s_+(0) \rangle}$$\to$$-{\langle s_+(0) \rangle}$, i.e. up to a rotation angle $\pi$ of the conditional evolution. It forms the typical regime used for spin-sensing. Going beyond this timescale reduces the dip depth of the main resonance again, whereas the vicinity regions still increase in depth. This then leads to an oscillatory broadened regime associated with a loss of frequency resolution. More precise, such a regime makes it hard to distinguish different spins of similar frequency as a result of their overlapping signal envelopes; in addition to the drawbacks associated with the (potentially) reduced coherence dip amplitude and the more intricate signal pattern. Moreover the filter formula description (\[spin\_sens1\]) breaks down at these timescales; except for $c$=0 ($\omega\tau_0$=$\pi/2$ in figure\[b\_sensing\](a)) as is clear out of the ‘slicing’ analysis of the preceding section. Thus there exists a trade-off between an anticipated increasing sensing resolution in time out of the filter description ($\Delta\omega \propto 1/t$) and an upper limitation by the onset of the oscillatory broadening at longer times; besides the limitation on the coherence time set by the classical noise background.
Extending the time until the onset of the oscillatory regime can be achieved by lowering the effective coupling amplitude. That way the filter description and its time inverse frequency width scaling retains its validity on a longer timescale, which, provided the (decoupled) coherence time exceeds that timescale, enables to further increase the sensing resolution[@zhao14]. Thus, for a fixed total time, appropriately tuning the coupling strength allows to turn a signal with previously oscillatory broadened envelope and reduced dip depth, into a clearly distinct peak structure of larger or maximal peak depth, each peak width scaling with the inverse total time.
Such an amplitude damping effect is realized by higher order resonances ($k$$>$0)[@taminiau12], by superimposing a continuous Rabi driving on the control spin[@mkhitaryan15], or by lowering the effective coupling amplitude by gradually reducing the periodicity of the control sequence[@zhao14]. This latter concept can be achieved by an $N$-pulse sequence composed of $s$ alternating $\pm c$slices with $n$ pulses each (figure\[b\_sliced\](d)), choosing the inter-pulse timescales such that $$\tau_{\pm c}=\tau_0\pm c\,\frac{\tau_0}{(2k+1)\,n}\,.$$ Here $\tau_0=(1/2)\,(\tau_{+c}+\tau_{-c})$ or simply $\tau_0=t/(2N)$ for an even number of slices $s$. That way, at the resonance condition $\omega\tau_0=(2k+1)\,\pi/2$, the system is alternatingly driven at the $\pm c$ resonance, i.e. $\omega\,\tau_{\pm c}=(2k+1)\pi/2\pm c\,\pi/(2n)$. This corresponds to an (approximate) single axis rotation with a well-controlled and reduced coupling amplitude as discussed in the preceding section.
However, a reduced periodicity comes at a price, namely additional peaks appearing in the sensing spectra. This is best understood by noting that the filter for an $M$ times periodic repetition of an $m$-pulse block can be described as the overlap of a grating $\mathcal{G}$ with a block filter $\mathcal{F}_{\rm block}$[@ajoy13; @zhao14] (see Appendix\[append\_grating\]) $$\label{filtgrating} \mathcal{F}_{\pm c}(\omega)^2=\frac{1}{t^2}\,\mathcal{G}(\omega\,t,M)\,\mathcal{F}_{\rm block}(\omega\tau_0,m,c)^2\,.$$ We will assume the number of slices $s$ to be even, in which case $M=s/2$ and $m=2n$ with a ‘block’ consisting of a single ‘$c\to-c$’ sequence. The grating is then given by ($m$ even) $\mathcal{G}=\sin^2(\omega t/2)/\sin^2(\omega t/(2M))$ and the block filter is calculated analogue to the CPMG filter (see Appendix\[append\_deriv\]) by just adapting $s(t)$ to the modified pulse sequence. Thus the total filter can be interpreted as the overlap of a grating with peak width inversely to the total time $\Delta \omega\propto1/t$ modulated by a broader block filter of width $\Delta\omega\propto (t/M)^{-1}$. Compared to a perfectly ‘symmetric’ grating ($M$=$N$, $m$=1), in which the grating consists of peaks $\omega\tau_0=(2k+1)\pi/2$, ($m$-$1$) additional peaks per resonance order $k$ and separated by $\Delta(\omega\tau_0)=\pi/m$ emerge for $M=N/m$, gradually introduced into the filter by reducing the symmetry of the $m$-pulse blocks[@zhao14].
Figure\[b\_sensing\](b) illustrates the filters for a sequence of $\pm c$ alternating slices, a fixed total pulse number $N$=24 and different pulse numbers $n$ per slice. Previously located in the oscillatory regime for a fully periodic $N$=24 sequence (figure\[b\_sensing\](a)), the alternation leads to well-defined peaks of frequency width given by the inverse total time and well-described in the filter framework; though at the expense of the emergence of additional peaks for a given frequency.
Conclusion
==========
We have shown how the evolution operator of interacting spins subject to a control pulse sequence can be described in a filter description. This holds true for (hyperfine) coupled spins without population exchange in both the limiting cases of strong and weak coupling. A universal behavior can be ascribed to the resonance region, allowing for a straightforward identification of both the interaction type and amplitude. Importantly, it has been shown that the interpretation in the filter framework can be extended to the previously inaccessible limit of long times and large rotation angles. Thus its insight can be used for the construction of tunable decoupled quantum gates. It should be noted that obtaining the exact solution of the conditional spin evolution is a straightforward task[@vandersar12; @taminiau12; @zhao12; @kolkowitz12], though its formulation is more involved, and quickly becomes a numerical task involving the inversion of trigonometric functions. In contrast, the filter formulation provides an intuitive approach and combining both methods can lead to further optimizations whenever the limiting cases of strong and weak coupling are not strictly fulfilled. Moreover, we have analyzed the impact on the (control spin) subsystem coherence. Remarkably, the sensing relevant regime is well-described in the filter formulation and is closely related to the filter for classical noise. Furthermore it has been shown that sensing resolution can be improved by alternating sequences with well defined coupling amplitudes as follow directly out of the filter description.
This work was supported by an Alexander von Humboldt Professorship, the ERC Synergy grant BioQ and the EU projects DIADEMS, SIQS and EQUAM.
Limitations to the filter description {#append_limit}
=====================================
Hamiltonian (\[ham1\]) in a rotating frame with respect to $H_0=(\omega/2)\, \sigma_z$, well suited for the filter analysis in the weak coupling regime $A\ll\omega$, follows as $H_{\rm int}=(A/2)\,[\sigma_+ \exp(i\omega t)+\text{h.c.}]\otimes S_z(t)$. The evolution operator in the Magnus expansion can then be written as $$\label{mag1} U_{\rm int}=\exp\left(\sum_{i=1}^\infty \Omega_i(t) \right)$$ with the first two expansion contributions $$\label{l2} \begin{split} \Omega_1(t)&=-i\,A/2\,\int_0^t \tilde{H}_{\rm int}(t')\,\mathrm{d}t' \\
\Omega_2(t)&= -\frac{1}{2}\,\left(\frac{A}{2}\right)^2\int_0^t\,\mathrm{d}t_1\int_0^{t_1}\mathrm{d}t_2 \left[ \tilde{H}_{\rm int}(t_1),\tilde{H}_{\rm int}(t_2) \right]\,
\end{split}$$ and the definition $H_{\rm int}=(A/2)\,\tilde{H}_{\rm int}$. A completely analogue treatment can be carried out for the strong coupling limit in a rotating frame with respect to the hyperfine contribution.\
A sufficient condition ensuring absolute convergence of the sum appearing in (\[mag1\]) is given by [@casas07] $\int_0^t \lVert H_{\rm int}(t') \rVert\mathrm{d}t'<\pi$, with the matrix norm here and in the following defined as the operator norm.
As the filter description is based on the first order contribution alone, its validity is restricted to timescales on which higher order contributions are sufficiently small. With $\lVert \tilde{H}_{\rm int}(t) \rVert = J$, a time independent quantity as a result of the unitarily invariant matrix norm, it follows that $\lVert \Omega_1(t) \rVert\leq (A/2) J\,t$ and $\lVert \Omega_2(t) \rVert\leq 1/2\,(A/2)^2\,t^2\,J^2$ and under the assumption of absolute convergence[@khodjasteh08] $$\left\lVert\sum_{i=k}^\infty \Omega_i(t)\right\rVert\leq C_k \left(\frac{A}{2} \right)^k(J\,t)^k$$ with a constant $C_k=\mathcal{O}(1)$.\
Recalling the filter definition (see Appendix\[append\_deriv\]) $\mathcal{F}=(1/t)|\int_0^t\exp(i\omega t')s(t')\mathrm{d}t'|$, the first order contribution takes the form $||\Omega_1(t)||=|A|/2 J\,\mathcal{F}\,t$ and the above estimations allow for an upper bound on time for the filter description validity.
More approximate, for sufficiently many pulses $N$ and close to resonance $$\begin{split} \left\lVert\Omega_2(t) \right\rVert &\leq \left(\frac{A}{2}\right)^2 \int_0^t \mathrm{d}t_1 \left\lVert \tilde{H}_{\rm int}(t_1) \right\rVert \left\lVert \int_0^{t_1} \mathrm{d}t_2 \tilde{H}_{\rm int}(t_2) \right\rVert \\
&\lesssim \left( \frac{A}{2} \right)^2\,J^2\,\mathcal{F}\,\int_0^t\,\mathrm{d}t_1\,t_1=\left(\frac{A}{2}J t \right)^2\frac{1}{2}\mathcal{F}
\end{split}$$ where the ‘coarse grain’ estimation $\lVert \int_0^{t_1}\mathrm{d}t_2\,\tilde{H}_{\rm int}(t_2) \rVert\lesssim \mathcal{F}J t_1$, approximately valid in the quasi-resonant regime for sufficiently long times (large $N$), has been used. Thus, $\lVert\Omega_2(t)\rVert\lesssim ((A/2)\,F\,t\,J)^2$, which compared to the first order contribution and its associated rotation angle $\theta$ $\lVert \Omega_1(t) \rvert =(A/2 \mathcal{F}tJ)=\theta/2\,J$, reveals that the second order contribution is of $\mathcal{O}([\theta/2]^2)$. Thus, the general validity of the filter description is limited to small rotation angles $\theta$; except for the special cases discussed in the main text (e.g. $c$=0 or alternating sequences).
In the weak coupling limit, the evolution operator (\[uint\_wc\]) up to second order takes the form $U_{\rm int}\simeq \exp(-i[\theta/2\,\sigma_\phi\otimes s_z+ (\theta/2)^2\,(\mathcal{F}^{(2)}/\mathcal{F}_w^2)\,\sigma_z\otimes \mathds{1}])$ with $\theta=A\mathcal{F}_w t$ the first order rotation angle, and $\mathcal{F}_w$ and $\mathcal{F}^{(2)}$ the first and second order filter, respectively. These filter contributions are compared in figure\[b\_2ndorder\], verifying that in the resonance region $\mathcal{F}^{(2)}/\mathcal{F}_w\lesssim1$ and thus indeed the second order contribution $\lesssim (\theta/2)^2$ as expected out of the previous discussion.
![\[b\_2ndorder\] Comparison of the first and second order filter contribution in the weak-coupling limit for a number $N$=6 of periodic pulses. ](04_2ndorder.pdf)
Filter function derivation {#append_deriv}
==========================
Weak coupling limit
-------------------
The rotating frame evolution, retaining only the first order Magnus expansion contribution, takes the form $$U_{\rm int}\simeq \exp\biggl(-i\frac{A}{2}\,\bigl[\sigma_+ \int_0^t\,e^{i\omega\,t'}\,s(t')\mathrm{d}t'+\text{h.c.} \bigr]\otimes s_z\biggr)$$ with $s(t)$$\in$$\{-1,1 \}$ the control pulse jump function as defined in the main text. Thus it essentially remains to calculate $\chi(t)=\int_0^t \exp(i\omega t')s(t')\mathrm{d}t'$ and the filter(\[filt\_weak\]) and rotation angle(\[rot\_wc\]) subsequently follow as $\mathcal{F}_w=(1/t)\,|\chi(t)|$ and $\tan\phi=\text{Im}(\chi(t))/\text{Re}(\chi(t))$, respectively.
Using that the pulse times for an equidistant $N$-pulse sequence are given by $t_k=-\tau+k\,(2\tau)$, with start and end time $t_0=0$ and $t_{N+1}=t\equiv 2N\tau$ $$\label{wc_chi1}\begin{split} &\chi(t)=\sum_{k=0}^N(-1)^k\int_{t_k}^{t_{k+1}} e^{i\omega t'}\mathrm{d}t'\\
&=\frac{1}{i\omega}\left((-1)+(-1)^N e^{i\omega (2N\tau)}-2e^{-i\omega\tau}\sum_{k=1}^N\left[-e^{i\omega\,2\tau}\right]^k \right) \end{split}$$ and upon evaluation of the sum using the geometric series property $$\label{chi_wc} \chi(t)=\frac{i}{\omega}\,\left( 1-(-1)^N e^{i\omega\,t}-2\,\frac{e^{i\omega N\tau}}{\cos(\omega\tau)}\,{\genfrac{\lbrace}{\rbrace}{0pt}{}{-i\sin(\omega N\tau)}{\cos(\omega N\tau)}} \right)$$ with the upper and lower case corresponding to $N$ being even and odd, respectively.
Strong coupling limit {#deriv_sc}
---------------------
Hamiltonian (\[ham1\]) in the rotating frame with respect to the hyperfine contribution and for the ‘spin-1’ control spin configuration $S_z(t)=1/2(s_z(t)\pm \mathds{1})$ (corresponding to upper and lower signs in what follows) takes the form $$\label{hint_sc} H_{\rm int}=\frac{\omega}{2}\,\bigl( \varsigma_+ e^{i\,(A/2)\,[s_z\,s_f(t)\pm t ]}+\text{h.c.} \bigr)$$ with $\varsigma_+$ the ladder operator in the $\sigma_x$ eigenbasis, i.e. $\varsigma_++\text{h.c.}=\sigma_z$ and $i\,\varsigma_++\text{h.c.}=\sigma_y$, and $s_F(t)=\int_0^t s(t')\mathrm{d}t'$. For a periodic (CPMG) $N$-pulse sequence $s_F(t)$ can be expressed as $$\label{as_1} s_F(t)=\sum_{k=0}^N (-1)^k\,\left(t-k\,2\tau \right)\,\theta(t_{k+1}-t)\,\theta(t-t_k)$$ with the pulse times $t_k=-\tau+k\,(2\tau)$ and the initial and final time $t_0$=$0$ and $t_{N+1}$=$t$, respectively.\
A separation of (\[hint\_sc\]) into a conditional and unconditional part is obtained by noting that $$\begin{split} \cos\bigl([A/2] s_F(t) s_z\bigr)&=\cos\bigl([A/2] s_F(t)\bigr) \mathds{1} \\
\sin\bigl([A/2] s_F(t) s_z\bigr)&=\sin\bigl([A/2] s_F(t)\bigr)\, s_z \,. \end{split}$$ The total evolution in the rotating frame and approximated by the first order Magnus expansion then follows as $$\footnotesize U_{\rm int}\simeq\exp\left(-i\frac{\omega}{2}\,\biggl[(\varsigma_+\,\chi^{\pm}_c+\text{h.c.}) \otimes s_z+(\varsigma_+\chi^{\pm}_u+\text{h.c.})\otimes\mathds{1}\biggr] \right)$$ with $$\label{chi_sc}
\begin{split}
\chi^{\pm}_c&=i\,\int_0^t\,\mathrm{d}t' \sin\biggl(\frac{A}{2}\,s_F(t')\biggr)\,e^{\pm i\,(A/2)\,t'} \\
\chi^{\pm}_u&=\,\,\,\int_0^t\,\mathrm{d}t' \cos\biggl(\frac{A}{2}\,s_F(t')\biggr)\,e^{\pm i\,(A/2)\,t'}\,.
\end{split}$$ The corresponding conditional and unconditional filter as defined in(\[uint\_sc\]) and (\[filter\_sc\]) then follow as $\mathcal{F}_{c/u}=(1/t)|\chi_{c/u}^+|=(1/t)|\chi_{c/u}^-|$, and the rotation angle as defined in (\[rota\_sc\]) is given by $\tan\phi=\text{Im}(\chi_{c/u}^+)/\text{Re}(\chi_{c/u}^+)$.
Thus it remains to evaluate the expressions (\[chi\_sc\]). Inserting the expression (\[as\_1\]) for $s_F(t)$ $$\small \chi_c^{\pm}=\frac{1}{2}\sum_{k=0}^N(-1)^k \int_{t_k}^{t_{k+1}}\mathrm{d}t'\bigl(e^{i(A/2)(t'-k (2\tau))}-\text{h.c.} \bigr) e^{\pm i(A/2) t'}$$ and analogue for $\chi_u^{\pm}$. Performing the integration, inserting the expressions for $t_k$=$-\tau+k(2\tau)$, separating the ‘special times’ $t_0$ & $t_{N+1}=t$, and finally using the geometric series property in close similarity to the weak coupling treatment (\[wc\_chi1\]), results in the expressions $$\begin{split}
\chi_c^{\pm}&=\left[1-(-1)^N e^{\pm i A N \tau} \right]\,\left(\frac{i}{2A}\mp\frac{\tau}{2} \right)+e^{\pm i (A/2) N \tau}\\ &\left[\frac{i}{A}\mp\frac{\tau}{\cos\bigl((A/2)\tau\bigr)}e^{\pm i(A/2)\tau} \right] {\genfrac{\lbrace}{\rbrace}{0pt}{}{\pm i\sin\bigl((A/2) N\tau\bigr)}{-\cos\bigl((A/2)N\tau\bigr)}}
\end{split}$$ and $$\begin{split}
\chi_u^{\pm}&=\left[ 1- e^{\pm i A N \tau} \right] \left( \frac{\pm i}{2A}+\frac{\tau}{2}\right)+e^{\pm i(A/2)N\tau}\\ & \left[\frac{1}{A}+\frac{\tau}{\sin\bigl((A/2)\tau\bigr)}e^{\pm i(A/2)\tau} \right] \sin\left(\frac{A}{2}N\tau\right)\,.
\end{split}$$ Upper and lower cases in wavy brackets correspond to the total pulse number $N$ being even and odd, respectively.
Spin-1/2 control spin in the strong coupling limit {#append_strong1h2}
==================================================
For the strong coupling limit $A\gg \omega$ and assuming the control qubit to be of the ‘spin-1/2 type’, the rotating frame Hamiltonian is given by $$H_{\rm int}=\frac{\omega}{2}\,\left(\varsigma_+\,e^{i\,A s_F(t)\,\,s_z}+\text{h.c.} \right)$$ with $\varsigma_+$ and $s_F(t)=\int_0^t s(t')\mathrm{d}t'$ as defined in Appendix\[deriv\_sc\].
Thus, the rotating frame evolution follows as $$U_{\rm int}=\exp\left(-i\frac{\omega}{2}\,t\,\left[ \mathcal{F}_c\,\sigma_y\otimes s_z +\mathcal{F}_u\,\sigma_z\otimes\mathds{1} \right] + \mathcal{O}\bigl((\omega t)^2\bigr) \right)$$ with $\mathcal{F}_c=(1/t)\text{Im}(\zeta)$, $\mathcal{F}_u=(1/t)\text{Re}(\zeta)$ and the definition $\zeta=\int_0^t\mathrm{d}t' \exp(i A s_F(t'))$. With (\[as\_1\]) and using the same procedure as in Appendix\[append\_deriv\] the filters are readily evaluated to $$\begin{split} \mathcal{F}_c&={\genfrac{\lbrace}{\rbrace}{0pt}{}{0}{1/(N A \tau) \bigl(1-\cos(A\tau)\bigr)}} \\ \mathcal{F}_u&=1/(A\tau)\,\sin(A\tau)\,. \end{split}$$ Upper and lower cases in curly brackets correspond to an even and odd number of total pulses $N$, respectively. Remarkably, the conditional filter $\mathcal{F}_c$ does not lead to a scalable evolution, i.e. it scales inversely with the pulse number or is even zero for an even number of pulses. More precise, at most a single pulse unit contributes to the evolution and thus such a configuration is not suitable for conditional gate interactions. In contrast, the unconditional contribution is scalable and does reveal a discrete peak structure.
Grating block-filter construction in the weak coupling limit {#append_grating}
============================================================
The filter in accordance with the definition (\[uint\_wc\]) as follows from a first order Magnus expansion in the weak coupling limit is given by $$\label{grat_proof1} \mathcal{F}(t)=\left|\frac{1}{t}\,\int_0^t\,e^{i\omega\,t'}\,s(t')\,\mathrm{d}t'\right|$$ with the step function $s(t)\in\{\pm 1\}$ as previously defined describing a pulse sequence consisting of $M$ periodic repetitions of $m$-pulse blocks. Taking into account that the time integration involved can be split into $M$ periodic time intervals $[t_k, t_{k+1}]$ with $t_1=0$, $t_{M+1}=t$ and $\Delta t=t_{k+1}-t_{k}=t/M$, allows to rewrite (\[grat\_proof1\]) as $$\label{grat_proof2}\begin{split} &\mathcal{F}(t)=\left|\frac{1}{t}\sum_{k=1}^{M}\int_{t_k}^{t_{k+1}} e^{i\omega t'}\,s(t')\,\mathrm{d}t'\right|\\&=\frac{1}{t}\left|\sum_{k=1}^M\,\left[ (-1)^m \right]^{k+1} e^{i\omega(k-1)\,(t/M)} \int_0^{t/M}\,e^{i\omega t'}\,s(t')\,\mathrm{d}t'\right| \end{split}$$ where in the last step the periodicity property $s(t_k+t')=[(-1)^m]^{k+1} s(t')$ has been used. Out of (\[grat\_proof2\]) and (\[filtgrating\]) it is straightforward to identify the grating as $$\mathcal{G}(\omega t,M,m)=\left| \sum_{k=1}^M\left[(-1)^m \right]^{k+1}\,e^{i\omega(k-1)\,(t/M)} \right|^2$$ that upon evaluation of the sum using the geometric series property results in $$\label{grating_def} \mathcal{G}=\begin{cases} \frac{\sin^2(\omega\, t/2)}{\sin^2(\omega t/(2M))}\qquad\qquad\qquad\,\,\,\, m\text{ even} \\ \frac{1}{\cos^2(\omega t/(2M))}\,\begin{cases} \sin^2(\omega t/2)\quad m\text{ odd}, M\text{ even}\\ \cos^2(\omega t/2) \quad m\text{ odd}, M \text{ odd.} \end{cases} \end{cases}$$ Correspondingly the ‘block filter’ is identified as $\mathcal{F}_{\rm block}=\bigl|\int_{0}^{t/M}\exp(i\omega t')\,s(t')\mathrm{d}t'\bigr|$. For a $\pm c$ block with $n$ pulses per slice, i.e. $m=2n$, the block filter follows as $\mathcal{F}_{\rm block}^{\pm c}(2n)=|\chi_n(\tau_{+c},t=2n\tau_{+c})+(-1)^n\,\exp(i2n\omega\tau_{+c})\,\chi_n(\tau_{-c},t=2n\tau_{-c})|$ with $\chi$ as defined in(\[chi\_wc\]).
[^1]: The conditional filter at the unconditional resonance is both zero for $c$=0 and $|c|$=1 for an odd number of pulses.
[^2]: Such a coordinate frame $\pi$-inversion does not occur at the unconditional resonance in the strong coupling limit in accordance with the absence of $s_z$ in the relevant evolution operator.
|
---
abstract: 'For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan–Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.'
title: Irreducible projective representations of the symmetric group which remain irreducible in characteristic $2$
---
=3mu plus 1mu minus 1mu
0
Introduction
============
For any finite group $G$ and prime number $p$, it is an interesting question to ask which ordinary irreducible representations of $G$ remain irreducible in characteristic $p$. This question was answered some time ago for the symmetric groups [@jm1; @jmp2; @slred; @mfred; @mfirred], and more recently [@mfaltred] for the alternating groups. In this paper we address double covers of the symmetric groups; since (in all but finitely many cases) these are Schur covers, we implicitly also address the question of which ordinary irreducible projective representations of the symmetric groups remain irreducible in characteristic $p$.
We denote a chosen double cover of $\sss n$ by ${\tilde{\mathfrak{S}}_}n$ (though in fact our results apply equally well to the other double cover); thus ${\tilde{\mathfrak{S}}_}n$ has a central involution $z$ such that ${\tilde{\mathfrak{S}}_}n/\lan z\ran\cong\sss n$. In any characteristic one can separate the irreducible representations of ${\tilde{\mathfrak{S}}_}n$ into “linear” representations (on which $z$ acts trivially) and “spin” representations (on which $z$ acts as $-1$). The irreducible linear representations are also irreducible representations of $\sss n$, and so the answer to our main question for linear representations of ${\tilde{\mathfrak{S}}_}n$ follows at once from the answer for $\sss n$. So we need only consider spin representations. In fact in this paper we prefer to work in the language of characters: the ordinary irreducible spin characters of ${\tilde{\mathfrak{S}}_}n$ are labelled by $2$-regular partitions of $n$: there is a single character ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ for every $2$-regular partition $\la$ of $n$ with an even number of even parts, and a pair of associate characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+,{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$ for every $\la$ with an odd number of even parts. In the latter case, we write ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ to mean “either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$”, so our main question becomes “for which $\la$ is the $p$-modular reduction of ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ irreducible?”.
There has been a recent resurgence of interest in the representation theory of ${\tilde{\mathfrak{S}}_}n$ in odd characteristic. We hope to address this case in a future paper, but in the present paper we work instead in characteristic $2$. Apart from the fundamental paper [@bo] of Bessenrodt and Olsson in 1997 from which we take some important results, this situation has been relatively neglected, but we hope that the present paper will spark a revival. Note that in characteristic $2$ the central involution $z$ must act as $1$ on an irreducible representation, so there are no irreducible spin representations. So the irreducible Brauer characters in characteristic $2$ are precisely those coming from $\sss n$, namely the characters ${\varphi(\mu)}$ of the “James modules” ${\mathrm D^{\mu}}$. Thus the $2$-modular reduction of any ordinary irreducible spin character decomposes as a sum of characters ${\varphi(\mu)}$. This means that the decomposition matrix of ${\tilde{\mathfrak{S}}_}n$ in characteristic $2$ consists of the decomposition matrix of $\sss n$ extended downwards by a “spin matrix” with rows corresponding to ordinary irreducible spin characters. Although this spin matrix can be computed algorithmically given the decomposition matrix of $\sss n$, it remains fairly mysterious in general. A key contribution in the present paper is to describe part of this matrix (specifically, the part with rows labelled by $2$-regular partitions having no even parts greater than $2$) for Rouquier blocks of ${\tilde{\mathfrak{S}}_}n$.
We now indicate the structure of the paper and of the proof of our main theorem. In \[backsec\] we set out the background information we shall need; we try to provide enough information to make the paper reasonably self-contained, though we do make some effort at concision. In \[mainthmsec\] we recall the notion of a *$2$-Carter partition* which enables us to state our main theorem. In \[degreesec\] we examine character degrees, which are an important ingredient in our proof. The key observation here is that if ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ are ordinary characters having a $2$-modular constituent in common, and if the degree of ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ is greater than the degree of ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$, then the $2$-modular reduction of ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ cannot be irreducible. It turns out that (in contrast to the more famous hook-length formula for the degrees of linear characters) the bar-length formula for spin characters allows us to compare degrees quite easily. In \[rouqandsep\] we consider spin characters lying in a “Rouquier” block of ${\tilde{\mathfrak{S}}_}n$. While Rouquier blocks of symmetric groups have been studied extensively in the last few years, this appears to be the first time they have been considered for the double covers in characteristic $2$. By constructing various projective characters and exploiting connections with symmetric functions, we describe part of the decomposition matrix for a Rouquier block. We then extend this beyond Rouquier blocks to spin characters labelled by what we call *[separated]{}* partitions, and give a proof of our main theorem for [separated]{}partitions. In \[mainproofsec\] we complete the proof of our main theorem, by considering non-[separated]{}partitions. Here we use an inductive proof (similar in spirit to that used for the alternating groups in [@mfaltred]), based on Kleshchev’s modular branching rules. In \[futuresec\] we consider possibilities for future work, and in \[indexnotnsec\] we provide an index of notation.
The author would like to express his gratitude to the referee for a careful and thorough reading of the paper, and some very helpful comments.
The research presented in this paper would not have been possible without extensive computations using GAP [@gap].
Background {#backsec}
==========
*Throughout this section $\bbf$ is a fixed field of characteristic $2$.*
Symmetric groups, Hecke algebras and Schur algebras {#snsec}
---------------------------------------------------
We let $\sss n$ denote the symmetric group of degree $n$, and $\bbf\sss n$ its group algebra over $\bbf$. The representation theory of $\bbf\sss n$ is well-studied, and in the last twenty years or so has become increasingly intertwined with the representation theory of a certain (Iwahori–)Hecke algebra. Specifically, we let ${\calh_}n$ denote the Iwahori–Hecke algebra of $\sss n$ over $\bbc$ with quantum parameter $-1$; this has generators $T_1,\dots,T_{n-1}$, with defining relations $(T_i+1)^2=0$ for each $i$, together with the braid relations $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$ for $1\ls i\ls n-2$ and $T_iT_j=T_jT_i$ for $1\ls i<j-1\ls n-2$. The representation theory of ${\calh_}n$ is closely related to the representation theory of $\bbf\sss n$: the canonical labelling sets for the Specht modules and simple modules for these algebras are the same, and (as we shall see) there is a close relationship between their decomposition matrices. The main advantage of introducing ${\calh_}n$ into the subject is that its decomposition numbers are better understood than those of $\bbf\sss n$ (indeed, they can be computed using the *LLT algorithm* [@llt]). We shall exploit this later, although the relationship between the symmetric group and the Hecke algebra does not extend well to double covers of $\sss n$.
Another important family of algebras in this area is the family of *Schur algebras* introduced by Green [@gbook], and more generally the *$q$-Schur algebras* defined by Dipper and James [@dj Definition 2.9]. We just concentrate on two of these algebras, for each $n$: let ${\cals(n)}$ denote the classical Schur algebra over $\bbf$ (which is $\cals_\bbf(1,n)$ in the notation of Dipper–James), and let ${\cals^\circ(n)}$ denote the $q$-Schur algebra over $\bbc$ with quantum parameter $q=-1$, i.e. $\cals_\bbc(-1,n)$ in the Dipper–James notation. ${\cals(n)}$ is related to $\bbf\sss n$ via the *Schur functor*, which shows in particular that the decomposition matrix of $\bbf\sss n$ can be obtained from that of ${\cals(n)}$ by deleting some of the columns. ${\cals^\circ(n)}$ is related to ${\calh_}n$ is a similar way. As we shall see, the decomposition matrix of ${\cals(n)}$ also appears in the adjustment matrices of Rouquier blocks.
Double covers of $\sss n$ and projective representations {#doublecoversec}
--------------------------------------------------------
Now we consider double covers. In this paper we work with the group ${\tilde{\mathfrak{S}}_}n$ with generators $s_1,\dots,s_{n-1},z$, subject to relations $$\begin{aligned}
z^2&=1,\\
zs_i&=s_iz\tag*{for $1\ls i\ls n-1$,}\\
s_i^2&=1\tag*{for $1\ls i\ls n-1$,}\\
s_is_{i+1}s_i&=s_{i+1}s_is_{i+1}\tag*{for $1\ls i\ls n-2$,}\\
s_is_j&=zs_js_i\tag*{for $1\ls i\ls j-2\ls n-3$.}\end{aligned}$$ Then ${\tilde{\mathfrak{S}}_}n$ is a double cover of the symmetric group $\sss n$: the subgroup $\lan z\ran$ is central, and setting $z$ to $1$ in the above presentation yields the Coxeter presentation of $\sss n$.
In fact, ${\tilde{\mathfrak{S}}_}n$ is a *Schur cover* of $\sss n$ for $n\gs4$. This means that linear representations of ${\tilde{\mathfrak{S}}_}n$ are equivalent to projective representations of $\sss n$, i.e. homomorphisms from $\sss n$ to $\mathrm{PGL}(V)$ for a vector space $V$. Thus our main theorem can be phrased in terms of projective representations. For $n\ls3$, $\sss n$ is its own Schur cover (note that the final relation in the presentation above never occurs when $n\ls3$, so that ${\tilde{\mathfrak{S}}_}n$ is just the direct product of $\sss n$ and $\lan z\ran$). Nevertheless, it makes sense to work with the group ${\tilde{\mathfrak{S}}_}n$ for all $n\gs0$, and we do so in this paper to help with our induction proofs.
We remark that for $n\gs4$ there is an alternative Schur cover obtained by replacing the relation $s_i^2=1$ with $s_i^2=z$; this is isomorphic to ${\tilde{\mathfrak{S}}_}n$ only when $n=6$, but in general is isoclinic to ${\tilde{\mathfrak{S}}_}n$, and so from our point of view its representation theory is the same.
Now we consider the different types of irreducible representations of ${\tilde{\mathfrak{S}}_}n$. In an irreducible representation of ${\tilde{\mathfrak{S}}_}n$ over any field, the central element $z$ must act as $1$ or $-1$. Irreducible representations in which $z$ acts as $1$ correspond exactly to irreducible representations of $\sss n$, and we refer to these as *linear* representations. Studying linear representations of ${\tilde{\mathfrak{S}}_}n$ is equivalent to studying representations of $\sss n$, so in this paper we shall be chiefly concerned with the representations where $z$ acts as $-1$, which we call *spin* representations.
For an introduction to the representation theory of ${\tilde{\mathfrak{S}}_}n$, we refer the reader to the book by Hoffman and Humphreys [@hh].
Partitions {#partnsec}
----------
The representations of the symmetric groups and their double covers are conventionally labelled by partitions. In this paper, a *partition* is an infinite weakly decreasing sequence $\la=(\la_1,\la_2,\dots)$ of non-negative integers with finite sum, which we denote $|\la|$. If $|\la|=n$, then we say that $\la$ is a partition of $n$. The integers $\la_1,\la_2,\dots$ are called the *parts* of $\la$. When writing partitions, we conventionally omit the trailing zeroes and group together equal parts with a superscript (so that we write the partition $(4,4,3,1,1,1,0,0,0,\dots)$ as $(4^2,3,1^3)$), and we write the unique partition of $0$ as $\varnothing$. A partition is *$2$-regular* if it does not have two equal non-zero parts, and *$2$-singular* otherwise. We write $\calp(n)$ for the set of partitions of $n$, and $\cald(n)$ for the set of [$2$-regular partitions]{}of $n$, and set $\calp=\bigcup_{n\gs0}\calp(n)$ and $\cald=\bigcup_{n\gs0}\cald(n)$.
If $\la\in\calp$, the *conjugate partition* $\la'$ is defined by setting $$\la'_r=\left|\lset{k\gs1}{\la_k\gs r}\right|$$ for all $r$. The *dominance order* on $\calp(n)$ is defined by saying that $\la$ *dominates* $\mu$ (and writing $\la\dom\mu$) if $$\la_1+\dots+\la_r\gs\mu_1+\dots+\mu_r$$ for all $r\gs1$.
The *Young diagram* of a partition $\la$ is the set $$\lset{(r,c)}{r\gs1,\ 1\ls c\ls\la_r}\subset\bbn^2.$$ We often identify a partition with its Young diagram; for example, we may write $\la\subseteq\mu$ to mean that $\la_i\ls\mu_i$ for all $i$. The elements of the Young diagram of $\la$ are called the *nodes* of $\la$. More generally, we use the term *node* for any element of $\bbn^2$. We draw Young diagrams as arrays of boxes using the English convention, in which $r$ increases down the page and $c$ increases from left to right.
A node of $\la$ is *removable* if it can be removed from $\la$ to leave the Young diagram of a smaller partition, while a node not in $\la$ is an *addable node* of $\la$ if it can be added to $\la$ to leave the Young diagram of a larger partition. The *residue* of a node $(r,c)$ is the residue of $c-r$ modulo $2$. We refer to a node of residue $i$ as an $i$-node.
Now we introduce some notions relating to spin representations. The *[spin residue]{}* of a node $(r,c)$ is the residue of $\inp{c/2}$ modulo $2$; that is, $0$ if $c\equiv0$ or $1\ppmod4$, and $1$ otherwise. If $\la$ is a [$2$-regular partition]{}and $i\in\{0,1\}$, we write ${\la{\downarrow}_{i}}$ for the smallest [$2$-regular partition]{}which can be obtained from $\la$ by removing nodes of [spin residue]{}$i$; the nodes removed are called *$i$-[spin-removable nodes]{}* of $\la$.
For example, let $\la=(9,6,4,3,1)$. Then ${\la{\downarrow}_{0}}=(7,6,4,3)$ and ${\la{\downarrow}_{1}}=(9,5,4,2,1)$. The following diagram shows the [spin residues]{}of the nodes of $\la$, with the [spin-removable nodes]{}highlighted.
$$\young(0110011!{\Ylinethick{1.3pt}}00!{\Ylinethick{.3pt}},01100!{\Ylinethick{1.3pt}}1!{\Ylinethick{.3pt}},0110,01!{\Ylinethick{1.3pt}}1,0!{\Ylinethick{.3pt}})$$
**Some notation for partitions**
In this paper we shall use some notation for partitions which, although quite natural, is not particularly standard.
Suppose $\la$ and $\mu$ are partitions and $a\in\bbq$ such that $a\la_i\in\bbz_{\gs0}$ for all $i$. Then we write
- $a\la$ for the partition $(a\la_1,a\la_2,\dots)$;
- $\la+\mu$ for the partition $(\la_1+\mu_1,\la_2+\mu_2,\dots)$;
- $\la\sqcup\mu$ for the partition obtained by combining all the parts of $\la$ and $\mu$ and arranging them in decreasing order.
Furthermore, we may use these operations simultaneously, and they take precedence in the order they appear above, so that $\la\sqcup a\mu+\nu$ means $\la\sqcup((a\mu)+\nu)$.
For example, if $\la=(11,7,3)$, $\mu=(3,1^2)$ and $\nu=(10,2)$, then we can define the partition $$\la+4\mu\sqcup\nu=(23,11,10,7,2).$$
$2$-cores and $2$-quotients {#2coresec}
---------------------------
We now set out some of the combinatorics of partitions relevant to the $2$-modular representation theory of the symmetric group. Everything we shall say has analogues for all primes $p$, but we stick to $p=2$ in the interest of brevity.
Suppose $\la\in\calp$. A *rim $2$-hook* of $\la$ is a pair of (horizontally or vertically) adjacent nodes of $\la$ which may be removed to leave a smaller Young diagram; we call the rim hook *horizontal* or *vertical* accordingly. The *$2$-core* of $\la$ is the partition obtained by repeatedly removing rim $2$-hooks until none remain. The $2$-core is well defined, and has the form $(c,c-1,\dots,1)$ for some $c\gs0$. The *$2$-weight* of $\la$ is the number of rim $2$-hooks removed to reach the $2$-core.
This may be visualised using the abacus. Take an abacus with two infinite vertical runners, and label these $0$ and $1$ (with $0$ being the runner on the left). On runner $a$ mark positions $\dots,a-4,a-2,a,a+2,a+4,\dots$, so that position $2i+1$ is directly to the right of position $2i$ for all $i$. Now given a partition $\la$, place a bead on the abacus at position $\la_r-r$ for each $r$. The resulting configuration is called the *abacus display* for $\la$. The key observation motivating the abacus is that removing a rim $2$-hook corresponds to sliding a bead into an empty position immediately above. Hence an abacus display for the $2$-core of $\la$ may be obtained by repeatedly sliding beads up until every bead has a bead immediately above it.
The abacus also allows us to define the $2$-quotient of a partition. Given the abacus display for $\la$ and given $a\in\{0,1\}$, examine runner $a$ in isolation, and let $\la^{(a)}_i$ be the number of empty positions above the $i$th lowest position, for each $i$. Then $\la^{(a)}=(\la^{(a)}_1,\la^{(a)}_2,\dots)$ is a partition, and the pair $(\la^{(0)},\la^{(1)})$ is the *$2$-quotient* of $\la$.
The partition $\la=(6,4^2,3)$ has $2$-core $(2,1)$, as we see from the Young diagram (in which we indicate removed rim $2$-hooks) and the abacus displays of these two partitions. $$\tikz[baseline=-2cm]{\tyng(0cm,0cm,6,4^2,3)\Yfillopacity{0}{\Ylinethick{1.3pt}}\tgyoung(0cm,0cm,::_2_2,:|2|2|2:,|2:,:_2)}\qquad\qquad
\abacus(vv,bb,nn,nb,nb,bn,nb,nn,vv)\qquad\qquad\abacus(vv,bb,bb,nb,nb,nn,nn,nn,vv)$$ Examining the runners in the display for $\la$, we see that the $2$-quotient of $\la$ is $((3),(2,1^2))$.
Later we shall need the following simple lemma (a version of this for arbitrary $p$ appears in [@mfirred §1.1.1], among other places).
\[quoconj\] Suppose $\mu\in\calp$, and let $(\mu^{(0)},\mu^{(1)})$ be the $2$-quotient of $\mu$. Then the $2$-quotient of $\mu'$ is $((\mu^{(1)})',(\mu^{(0)})')$.
The final thing we need relating to cores and quotients is the *$2$-sign* of a partition. This was originally introduced by Littlewood [@litt p. 338], but we use the alternative definition from [@j10 p. 229]. Suppose $\mu$ is a partition, and construct the $2$-core of $\mu$ by repeatedly removing rim $2$-hooks. Let $a$ be the number of vertical hooks removed. $a$ is not well-defined, but its parity is, and so we may safely define the $2$-sign $\epsilon(\mu)=(-1)^a$ of $\mu$. We will need the following very simple observation.
\[parityconj\] Suppose $\mu$ is a partition of $2$-weight $w$. Then $\epsilon(\mu')=(-1)^w\epsilon(\mu)$.
Irreducible modules, irreducible characters and decomposition numbers {#irrdecsec}
---------------------------------------------------------------------
Having established the necessary background relating to the combinatorics of partitions, we can discuss modules and decomposition numbers for the groups and algebras studied in this paper.
For each $\la\in\calp(n)$, we write ${\mathrm S^{\la}}$ for the *Specht module* for $\sss n$, as defined (over an arbitrary field) by James [@jbook]; in particular, ${\mathrm S^{(n)}}$ is the trivial module. Working over our field $\bbf$ of characteristic $2$, suppose $\la\in\cald(n)$; then ${\mathrm S^{\la}}$ has a unique irreducible quotient ${\mathrm D^{\la}}$, which we call the *James module*. The modules ${\mathrm D^{\la}}$ give all the irreducible $\bbf\sss n$-modules as $\la$ ranges over $\cald(n)$.
A similar situation applies for the Hecke algebra ${\calh_}n$: for each $\la\in\calp(n)$ there is a Specht module for ${\calh_}n$ (which we shall also denote ${\mathrm S^{\la}}$), which has a unique irreducible quotient ${\mathrm D^{\la}}$ when $\la$ is $2$-regular, and the modules ${\mathrm D^{\la}}$ are the only irreducible ${\calh_}n$-modules up to isomorphism.
Now we consider the Schur algebras. For every $\la\in\calp(n)$ there is a *Weyl module* ${\Delta^{\la}}$ for ${\cals(n)}$, which has a unique irreducible quotient ${\mathrm L^{\la}}$. Note that the labelling we use for these modules is that used in James’s paper [@j10] (and is the opposite of that in most other places), so that the Specht module ${\mathrm S^{\la}}$ is the image of ${\Delta^{\la}}$ (and not ${\Delta^{\la'}}$) under the Schur functor. Similarly for the $(-1)$-Schur algebra ${\cals^\circ(n)}$ we have a Weyl module ${\Delta^{\la}}$ and an irreducible module ${\mathrm L^{\la}}$ for every $\la\in\calp(n)$.
With these conventions established, we can define decomposition numbers. For any $\la,\mu\in\calp(n)$, we define $D_{\la\mu}$ to be the composition multiplicity ${[{\Delta^{\la}}:{\mathrm L^{\mu}}]}$ for the Schur algebra ${\cals(n)}$. The fact that ${\cals(n)}$ is a quasi-hereditary algebra gives the following.
\[unitriang\] Suppose $\la,\mu\in\calp(n)$. Then $D_{\la\la}=1$, and $D_{\la\mu}=0$ unless $\mu\dom\la$.
We emphasise that we use the convention from [@j10] for labelling Weyl modules; with the more usual convention the symbol $\dom$ would become $\domby$ in the above . This convention also means that when $\mu\in\cald(n)$, $D_{\la\mu}$ is also the composition multiplicity ${[{\mathrm S^{\la}}:{\mathrm D^{\mu}}]}$ for $\sss n$ in characteristic $2$. The matrix with entries $(D_{\la\mu})_{\la,\mu\in\calp(n)}$ is the *decomposition matrix* for ${\cals(n)}$, and the matrix with entries $(D_{\la\mu})_{\la\in\calp(n),\mu\in\cald(n)}$ is the decomposition matrix for $\sss n$ in characteristic $2$. We may write either of these matrices simply as $D$, if the context is clear.
A similar situation applies for Hecke algebras and $(-1)$-Schur algebras. For $\la,\mu\in\calp(n)$ we write ${\mathring D}_{\la\mu}$ for the composition multiplicity ${[{\Delta^{\la}}:{\mathrm L^{\mu}}]}$ for the $(-1)$-Schur algebra ${\cals^\circ(n)}$; if $\mu\in\cald(n)$, then this also equals the multiplicity ${[{\mathrm S^{\la}}:{\mathrm D^{\mu}}]}$ for the Hecke algebra ${\calh_}n$. In addition, \[unitriang\] holds with $D$ replaced by ${\mathring D}$. Furthermore, the decomposition numbers $D_{\la\mu}$ and ${\mathring D}_{\la\mu}$ are related by the theory of adjustment matrices: if we fix $n$ and let $D$ denote the decomposition matrix of ${\cals(n)}$ and ${\mathring D}$ the decomposition matrix of ${\cals^\circ(n)}$, then the square matrix $A$ defined by $D={\mathring D}A$ is called the *adjustment matrix* for ${\cals(n)}$. $A$ is automatically lower unitriangular (i.e. \[unitriang\] holds with $D$ replaced by $A$), and remarkably $A$ has non-negative integer entries; this statement appears in [@mathbook Theorem 6.35], and derives from the theory of decomposition maps [@geck2]. By taking only the rows and columns of $A$ labelled by [$2$-regular partitions]{}we obtain the adjustment matrix for $\sss n$ in characteristic $2$, which we also call $A$; then if we let $D,{\mathring D}$ denote the decomposition matrices for $\sss n$ and ${\calh_}n$, we again have $D={\mathring D}A$.
Now we come to double covers. Here (and for the rest of the paper) we work with characters, rather than modules. This is because we do not have particularly nice constructions of the modules affording the irreducible characters of double covers, and because statements about the decomposition of projective modules are more readily expressed in terms of projective characters. Since we are only concerned with composition factors and not with module structures, there is no loss in working with characters.
With this in mind, we introduce notation for the characters of the $\sss n$-modules described above: let ${\llbracket\la\rrbracket}$ denote the character of the Specht module ${\mathrm S^{\la}}$ over $\bbc$, so that $\lset{{\llbracket\la\rrbracket}}{\la\in\calp(n)}$ is the set of ordinary irreducible characters of $\sss n$. For $\la\in\cald(n)$, let ${\varphi(\la)}$ denote the $2$-modular Brauer character of the James module ${\mathrm D^{\la}}$. Since $\sss n$ is a quotient of ${\tilde{\mathfrak{S}}_}n$, the characters ${\llbracket\la\rrbracket}$ and ${\varphi(\la)}$ naturally become characters of ${\tilde{\mathfrak{S}}_}n$, and we shall always view them in this way. Note that in characteristic $2$, the central element $z$ lies in the kernel of any irreducible representation of ${\tilde{\mathfrak{S}}_}n$, so the irreducible representations of ${\tilde{\mathfrak{S}}_}n$ are precisely those arising from the irreducible representations of $\sss n$. Thus $\lset{{\varphi(\la)}}{\la\in\cald(n)}$ is the set of irreducible $2$-modular Brauer characters of ${\tilde{\mathfrak{S}}_}n$.
Now we look at spin characters. For each $\la\in\cald(n)$, let ${\operatorname{ev}(\la)}$ denote the number of positive even parts of $\la$, and write $$\cald^+(n)=\lset{\la\in\cald(n)}{{\operatorname{ev}(\la)}\text{ is even}},\qquad\cald^-(n)=\lset{\la\in\cald(n)}{{\operatorname{ev}(\la)}\text{ is odd}}.$$ Then for each $\la\in\cald^+(n)$, there is an irreducible spin character ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ of ${\tilde{\mathfrak{S}}_}n$, while for each $\la\in\cald^-(n)$, there are two irreducible spin characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$ of ${\tilde{\mathfrak{S}}_}n$. The definition of these characters goes back to Schur, who showed that they are all the irreducible spin characters of ${\tilde{\mathfrak{S}}_}n$. A uniform construction of irreducible representations affording these characters was given much later, by Nazarov [@naz]. We do not give the representations or their characters explicitly in this paper; the properties we need will be summarised in later sections.
In fact, for $\la\in\cald^-(n)$ the characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$ behave very similarly: their values differ only up to sign, and in particular their degrees are the same and their $2$-modular reductions are the same. So to make things simpler in this paper we adopt the convention that we write ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ to mean “either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$” when $\la\in\cald^-(n)$.
We also follow the usual convention of omitting brackets from a partition when using the notation ${\llbracket\ \rrbracket}$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\ {\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$; thus we may write ${\llbracket4,2^2,1\rrbracket}$ instead of ${\llbracket(4,2^2,1)\rrbracket}$. We remark that historically the spin characters have been denoted $\lan\la\ran$ or $\lan\la\ran_\pm$; we use slightly different notation here because the brackets $\lan\ \ran$ are already used for two other purposes in this paper. We trust that there will be no confusion with the use of ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ in [@bmo §4] to denote a projective character of ${\tilde{\mathfrak{S}}_}n$ in characteristic $3$.
We write ${\left(\ \ \!:\!\ \ \right)}$ for the usual inner product on ordinary characters, with respect to which the irreducible characters are orthonormal. For any character $\chi$ of ${\tilde{\mathfrak{S}}_}n$, we write ${\widebar{\chi}}$ for the $2$-modular reduction of $\chi$, i.e. the $2$-modular Brauer character obtained by restricting $\chi$ to the $2$-regular conjugacy classes of ${\tilde{\mathfrak{S}}_}n$. Our main question is then: for which ordinary irreducible characters $\chi$ is ${\widebar{\chi}}$ irreducible?
We write ${[\ \ :\ \ ]}$ for the inner product on $2$-modular Brauer characters with respect to which the irreducible characters are orthonormal. Hence the decomposition number $D_{\la\mu}$ defined above equals ${[{{\widebar{{\llbracket\la\rrbracket}}}}:{\varphi(\mu)}]}$, for $\la\in\calp(n)$ and $\mu\in\cald(n)$. Given $\la,\mu\in\cald(n)$, we write ${D^{\mathsf{spn}}_{\la\mu}}={[{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}:{\varphi(\mu)}]}$. The matrices of integers $D_{\la\mu}$ and ${D^{\mathsf{spn}}_{\la\mu}}$ together constitute the decomposition matrix of ${\tilde{\mathfrak{S}}_}n$ in characteristic $2$.
The decomposition matrix of ${\tilde{\mathfrak{S}}_}5$ in characteristic $2$ is given below. The top part of the matrix is the decomposition matrix of $\sss5$, with the $(\la,\mu)$-entry being $D_{\la\mu}$. The lower part gives the decomposition numbers for spin characters, with entries ${D^{\mathsf{spn}}_{\la\mu}}$; note that for $\la\in\cald^-(5)$ there are identical rows corresponding to ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$. (In all matrices given explicitly in this paper, we use a dot to mean $0$.) $$\begin{array}{r|ccc|}
&{\rotatebox{90}{$(5)$}}&{\rotatebox{90}{$(4,1)$}}&{\rotatebox{90}{$(3,2)$}}\\\hline
(5)\phantom{_-}&1&\cdot&\cdot\\
(4,1)\phantom{_-}&\cdot&1&\cdot\\
(3,2)\phantom{_-}&1&\cdot&1\\
(3,1^2)\phantom{_-}&2&\cdot&1\\
(2^2,1)\phantom{_-}&1&\cdot&1\\
(2,1^3)\phantom{_-}&\cdot&1&\cdot\\
(1^5)\phantom{_-}&1&\cdot&\cdot\\
\hline
(5)\phantom{_-}&\cdot&\cdot&1\\
(4,1)_+&2&\cdot&1\\
(4,1)_-&2&\cdot&1\\
(3,2)_+&\cdot&1&\cdot\\
(3,2)_-&\cdot&1&\cdot\\\hline
\end{array}$$
This example shows an interesting feature of the answer to our main question: We can re-cast our main question and ask “which irreducible Brauer characters arise as the $2$-modular reductions of ordinary irreducible characters?”. In the case $n=5$ we see that the Brauer character ${\varphi(3,2)}$ does not arise as ${\widebar{{\llbracket\la\rrbracket}}}$ for any $\la$, but does arise as ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$. So introducing spin characters can provide constructions of irreducible Brauer characters which might otherwise be hard to obtain.
A central result in modular representation theory (which we shall use without comment) is Brauer reciprocity, which gives a connection between decomposition numbers and indecomposable projective characters. Given $\mu\in\cald(n)$, the projective cover of the James module ${\mathrm D^{\mu}}$ may be lifted to an ordinary representation of ${\tilde{\mathfrak{S}}_}n$, and we write ${\operatorname{prj}(\mu)}$ for the character of this representation; this is called an *indecomposable* projective character, and the characters ${\operatorname{prj}(\mu)}$ for $\mu\in\cald(n)$ give a basis for the space spanned by all projective characters (i.e. characters which vanish on $p$-singular elements of ${\tilde{\mathfrak{S}}_}n$).
Brauer reciprocity says that ${\operatorname{prj}(\mu)}$ is given in terms of irreducible characters by the entries in the column of the decomposition matrix corresponding to $\mu$, i.e. $${\operatorname{prj}(\mu)}=\sum_{\la\in\calp(n)}D_{\la\mu}{\llbracket\la\rrbracket}\ \ {+}\sum_{\la\in\cald^+(n)}{D^{\mathsf{spn}}_{\la\mu}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\ \ {+}\sum_{\la\in\cald^-(n)}{D^{\mathsf{spn}}_{\la\mu}}({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-).$$ We will use this extensively in \[rouqandsep\] to derive information on decomposition numbers.
The inverse of the decomposition matrix of the Schur algebra {#inversesec}
------------------------------------------------------------
Now we state some results which we shall need later concerning the inverses of the matrices $D$, ${\mathring D}$ and $A$. These are taken from James’s seminal paper [@j10], and derive ultimately from Steinberg’s tensor product theorem.
Given partitions $\alpha,\mu$ with $|\mu|=2|\alpha|$, let $(\mu^{(0)},\mu^{(1)})$ be the $2$-quotient of $\mu$, and define $$\kappa(\alpha,\mu)=
\begin{cases}
a^\alpha_{\mu^{(0)}\mu^{(1)}}&(\text{if the $2$-core of $\mu$ is $\varnothing$})\\
0&(\text{otherwise}).
\end{cases}$$ (This is a special case of [@j10 Definition 2.13].) Here (and henceforth) $a^\alpha_{\beta\gamma}$ denotes the Littlewood–Richardson coefficient corresponding to partitions $\alpha,\beta,\gamma$ with $|\alpha|=|\beta|+|\gamma|$.
Now we can state two results on the inverse of the decomposition matrix. We begin with a result for even $n$; this is a special case of [@j10 Corollary 6.9]. Recall that $\epsilon(\mu)$ denotes the $2$-sign of a partition $\mu$, defined in \[2coresec\].
\[steinberg\] Suppose $\la,\mu\in\calp(n)$ and all the columns of $\la$ are of even length, and write $\la={\alpha\sqcup\alpha}$. Then $$\begin{aligned}
{\mathring D}_{\la\mu}{^{-1}}&=(-1)^{n/2}\epsilon(\mu)\kappa(\alpha,\mu)\\
\intertext{and}
D_{\la\mu}{^{-1}}&=(-1)^{n/2}\epsilon(\mu)\sum_{\beta\in\calp(n/2)}D_{\alpha\beta}{^{-1}}\kappa(\beta,\mu).\end{aligned}$$
The reader following the reference to [@j10] may find Corollary 6.9 hard to decipher. Lemma 2.21(iii) in the same paper is also needed to express the term $\upsilon(\beta,\sigma,\mu)$ from Corollary 6.9 in terms of $\kappa(\sigma,\mu)$.
Note also that the first part of \[steinberg\] follows from the second part of [@j10 Corollary 6.9]; the condition $ep>n$ given there is regarded as automatically true when working over a field of characteristic zero.
As a consequence of \[steinberg\] we can derive some information on the adjustment matrix $A$.
\[adj\] Suppose $\la,\mu\in\calp(n)$ and all the columns of $\la$ are of even length, and write $\la={\alpha\sqcup\alpha}$. Then $$A{^{-1}}_{\la\mu}=
\begin{cases}
D_{\alpha\beta}{^{-1}}&(\text{if $\mu={\beta\sqcup\beta}$ for $\beta\in\calp(n/2)$})\\
0&(\text{if $\mu$ has a column of odd length}),
\end{cases}$$ and hence $$A_{\la\mu}=
\begin{cases}
D_{\alpha\beta}&(\text{if $\mu={\beta\sqcup\beta}$ for $\beta\in\calp(n/2)$})\\
0&(\text{if $\mu$ has a column of odd length}).
\end{cases}$$
The two parts of \[steinberg\] combine to give $$\begin{aligned}
D_{\la\mu}{^{-1}}&=\sum_{\beta\in\calp(n/2)}D_{\alpha\beta}{^{-1}}{\mathring D}_{({\beta\sqcup\beta})\mu}{^{-1}}\\
\intertext{for all $\mu$. But the definition of the adjustment matrix also gives}
D_{\la\mu}{^{-1}}&=\sum_{\nu\in\calp(n)}A{^{-1}}_{\la\nu}{\mathring D}_{\nu\mu}{^{-1}}\end{aligned}$$ for all $\mu$. Since the rows of ${\mathring D}{^{-1}}$ are linearly independent, this enables us to deduce the given expression for $A{^{-1}}_{\la\mu}$.
To get the result for $A_{\la\mu}$, consider the row vector $a$ with entries $$a_\mu=
\begin{cases}
D_{\alpha\beta}&(\text{if $\mu={\beta\sqcup\beta}$ for $\beta\in\calp(n/2)$})\\
0&(\text{if $\mu$ has a column of odd length}).
\end{cases}$$ Using the result already proved for $A{^{-1}}$, we have $$\begin{aligned}
(aA{^{-1}})_\mu&=\sum_{\nu\in\calp(n)} a_\nu A{^{-1}}_{\nu\mu}\\
&=\sum_{\gamma\in\calp(n/2)}D_{\alpha\gamma}A{^{-1}}_{({\gamma\sqcup\gamma})\mu}\\
&=
\begin{cases}
\sum_{\gamma\in\calp(n/2)}D_{\alpha\gamma}D{^{-1}}_{\gamma\beta}&(\text{if $\mu={\beta\sqcup\beta}$ for $\beta\in\calp(n/2)$})\\
0&(\text{if $\mu$ has a column of odd length})
\end{cases}\\
&=\delta_{\la\mu}.\end{aligned}$$ Hence $a$ is the $\la$-row of the adjustment matrix $A$, as required.
Now we give a corresponding result for odd $n$; this is also a special case of [@j10 Corollary 6.9].
\[steinbergod\] Suppose $\la,\mu\in\calp(n)$ and all the columns of $\la$ are of even length except the first column. Write $\la={\alpha\sqcup\alpha}\sqcup(1)$, and let $M$ be the set of partitions obtained which can be obtained by removing one node from $\mu$. Then $$\begin{aligned}
{\mathring D}_{\la\mu}{^{-1}}&=(-1)^{(n-1)/2}\sum_{\nu\in M}\epsilon(\nu)\kappa(\alpha,\nu),\\
\intertext{and}
D_{\la\mu}{^{-1}}&=(-1)^{(n-1)/2}\sum_{\beta\in\calp((n-1)/2)}D_{\alpha\beta}{^{-1}}\sum_{\nu\in M}\epsilon(\nu)\kappa(\beta,\nu).\end{aligned}$$
Again, we deduce information about the adjustment matrix $A$; this is proved in exactly the same way as \[adj\].
\[adjod\] Suppose $\la,\mu\in\calp(n)$ and all the columns of $\la$ are of even length except the first, and write $\la={\alpha\sqcup\alpha}\sqcup(1)$. Then $$A{^{-1}}_{\la\mu}=
\begin{cases}
D_{\alpha\beta}{^{-1}}&(\text{if $\mu={\beta\sqcup\beta}\sqcup(1)$ for $\beta\in\calp((n-1)/2)$})\\
0&(\text{if $\mu$ has a column (other than the first column) of odd length}),
\end{cases}$$ and hence $$A_{\la\mu}=
\begin{cases}
D_{\alpha\beta}&(\text{if $\mu={\beta\sqcup\beta}\sqcup(1)$ for $\beta\in\calp((n-1)/2)$})\\
0&(\text{if $\mu$ has a column (other than the first column) of odd length}).
\end{cases}$$
The degree of an irreducible spin character {#degsec}
-------------------------------------------
Let ${\operatorname{deg}}(\chi)$ denote the degree of an irreducible character, i.e. the value $\chi(1)$. In this section we recall the “bar-length formula” which gives the degrees of the irreducible spin characters. This goes back to Schur [@schu]. Recall that for $\la\in\cald$ we write ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ to mean either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$.
\[barlength\] Suppose $\la\in\cald(n)$ has length $m$. Then $${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}=2^{\lfloor\frac12(n-m)\rfloor}\frac{n!}{\prod_{1\ls i\ls m}\la_i!}\prod_{1\ls i<j\ls m}\frac{\la_i-\la_j}{\la_i+\la_j}.$$
We shall use this formula extensively in \[degreesec\].
Regularisation and doubling {#regnsec}
---------------------------
One of the most useful results in the modular representation theory of $\sss n$ is James’s regularisation theorem, which gives an explicit composition factor (occurring with multiplicity $1$) in the $p$-modular reduction of ${\llbracket\la\rrbracket}$, for each $\la$. To state this result for $p=2$, we need to introduce the $2$-regularisation of a partition. For $l\gs0$, define the $l$th *ladder* in $\bbn^2$ to be the set of nodes $(r,c)$ for which $r+c=l+2$. The intersection of this ladder with the Young diagram of a partition $\la$ is referred to as the $l$th ladder of $\la$. Now define the *$2$-regularisation* $\la{^{\operatorname{reg}}}$ of $\la$ to be the [$2$-regular partition]{}whose Young diagram is obtained by moving the nodes of $\la$ as far up their ladders as possible. For example, if $\la=(4^2,3^4,1)$, then $\la{^{\operatorname{reg}}}=(8,6,5,2)$, as we see from the following diagrams, in which we label nodes according to the ladders in which they lie. $$\young(0123,1234,234,345,456,567,6)\qquad\qquad\young(01234567,123456,23456,34)$$ Now we can state a simple form of James’s theorem.
[j1]{}[Theorem A]{}\[jreg\] Suppose $\la\in\calp$. Then $D_{\la(\la{^{\operatorname{reg}}})}=1$.
Of course, this theorem is extremely useful in determining whether ${{\widebar{{\llbracket\la\rrbracket}}}}$ is irreducible, since it tells us that if ${{\widebar{{\llbracket\la\rrbracket}}}}$ is irreducible, then ${{\widebar{{\llbracket\la\rrbracket}}}}={\varphi(\la{^{\operatorname{reg}}})}$. For the main result in the present paper it will be useful to have an analogue of \[jreg\] for spin characters. Fortunately there is such a result; this is due to Bessenrodt and Olsson, though a special case was proved earlier by Benson [@ben Theorem 1.2]. To state this, we need another definition. Given $\la\in\cald$, define the *double* of $\la$ to be the partition $$\la{^{\operatorname{dbl}}}=\big(\lceil\la_1/2\rceil,\lfloor\la_1/2\rfloor,\lceil\la_2/2\rceil,\lfloor\la_2/2\rfloor,\lceil\la_3/2\rceil,\lfloor\la_3/2\rfloor,\dots\big).$$ In other words, $\la{^{\operatorname{dbl}}}$ is obtained from $\la$ by replacing each part $\la_i$ with two parts which are equal (if $\la_i$ is even) or differ by $1$ (if $\la_i$ is odd). The fact that $\la$ is $2$-regular guarantees that $\la{^{\operatorname{dbl}}}$ is a partition. We write $\la{^{\operatorname{dblreg}}}$ to mean $(\la{^{\operatorname{dbl}}}){^{\operatorname{reg}}}$.
Now we can give the “spin-regularisation theorem”; recall that ${\operatorname{ev}(\la)}$ denotes the number of positive even parts of a partition $\la$.
[bo]{}[Theorem 5.2]{}\[spinreg\] Suppose $\la\in\cald$. Then ${D^{\mathsf{spn}}_{\la(\la{^{\operatorname{dblreg}}})}}=2^{\lfloor{\operatorname{ev}(\la)}/2\rfloor}$.
This result is extremely useful for our main problem: it tells us that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ can only be irreducible if $\la$ has at most one non-zero even part, and that if ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={{\varphi(\la{^{\operatorname{dblreg}}})}}$.
In order to exploit \[spinreg\], we will often want to show that $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$ for certain $\la,\mu\in\cald(n)$ without actually calculating $\la{^{\operatorname{dblreg}}}$ or $\mu{^{\operatorname{dblreg}}}$. We do this using the combinatorics of *slopes*. For $l\gs0$, define the $l$th slope in $\bbn^2$ to be the set of all nodes $(r,c)$ satisfying $2r+\inp{c/2}=l+2$. Define the $l$th slope of a partition $\la$ to be the intersection of this slope with the Young diagram of $\la$. (n.b. the slopes are essentially the “ladders in the $\bar4$-residue diagram” considered in [@bo §3], but we prefer not to over-tax the word “ladder”. The reader should also note that unlike in [@bo] we do not consider shifted Young diagrams.) Now we have the following result, which is implicit in [@bo].
\[slopelad\] Suppose $\la$ is a [$2$-regular partition]{}and $l\gs0$. Then the number of nodes in the $l$th slope of $\la$ equals the number of nodes in the $l$th ladder of $\la{^{\operatorname{dbl}}}$. Hence if $\mu$ is another [$2$-regular partition]{}, then $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$ $\la$ and $\mu$ have the same number of nodes in slope $l$ for each $l$.
For example, in the following diagram we label the nodes of $(14,8,7,1)$ with the numbers of the slopes containing them, and the nodes of $(14,8,7,1){^{\operatorname{dbl}}}=(7^2,4^3,3,1)$ with the numbers of the ladders containing them. $$\young(01122334455667,23344556,4556677,6)\qquad\young(0123456,1234567,2345,3456,4567,567,6)$$
At the level of Young diagrams, replacing $\la$ with $\la{^{\operatorname{dbl}}}$ involves replacing the nodes $(i,1),\dots,(i,\la_i)$ with $$(2i-1,1),\dots,(2i-1,\lceil\la_i/2\rceil),\quad(2i,1),\dots,(2i,\inp{\la_i/2})$$ for each $i$. The former nodes lie in slopes $$2i-2,2i-1,2i-1,2i,2i,\dots,2i-3+\lceil\la_i/2\rceil,2i-2+\inp{\la_i/2}$$ respectively, while the latter nodes lie in ladders $$2i-2,2i-1,\dots,2i-3+\lceil\la_i/2\rceil,\quad2i-1,2i,\dots,2i-2+\inp{\la_i/2}$$ respectively. The result follows.
Blocks
------
We now recall the $2$-block classification for ${\tilde{\mathfrak{S}}_}n$, which will be very useful in this paper. We begin by looking at the $2$-blocks of $\sss n$. The following result is a special case of the Brauer–Robinson Theorem [@brauer; @gdbr], first conjectured by Nakayama.
\[brarob\] Suppose $\la,\mu\in\calp(n)$. Then ${\llbracket\la\rrbracket}$ and ${\llbracket\mu\rrbracket}$ lie in the same $2$-block of $\sss n$ $\la$ and $\mu$ have the same $2$-core.
The same statement applies for $2$-blocks of ${\tilde{\mathfrak{S}}_}n$, and a corresponding statement [@jm1 Theorem 4.29] holds for blocks of ${\calh_}n$. Since we only consider characteristic $2$ in this paper, we will henceforth say “block” to mean “$2$-block”. Given \[brarob\], we may speak of the *core* of a $2$-block $B$ of $\sss n$, meaning the common $2$-core of the partitions labelling the linear irreducible characters in $B$. These partitions necessarily have the same $2$-weight as well, and we call this the *weight* of the block. We can immediately deduce the distribution of irreducible Brauer characters into blocks: if $\mu\in\cald(n)$, then since ${\varphi(\mu)}$ occurs as a composition factor of ${{\widebar{{\llbracket\mu\rrbracket}}}}$, it lies in the same block as ${\llbracket\mu\rrbracket}$.
The block classification may alternatively be expressed in terms of residues. Recall that the *residue* of a node $(r,c)$ is the residue of $c-r$ modulo $2$. Define the *$2$-content* of a partition $\la$ to be the multiset of $0$s and $1$s comprising the residues of all the nodes of $\la$. We write a multiset of $0$s and $1$s in the form ${\{0^{a},1^{b}\}}$. For example, the $2$-content of the partition $(7,4,1^3)$ is ${\{0^{8},1^{6}\}}$, as we see from the following “$2$-residue diagram”. $$\yngres(2,7,4,1^3)$$ It is an easy combinatorial exercise to show that two partitions have the same $2$-content they have the same $2$-core and $2$-weight, so the block classification may alternatively be stated by saying that ${\llbracket\la\rrbracket}$ and ${\llbracket\mu\rrbracket}$ lie in the same block $\la$ and $\mu$ have the same $2$-content. Accordingly, we can define the *content* of a block $B$ to be the common $2$-content of the partitions labelling the linear irreducible characters in $B$.
Now we consider spin characters. Since the only irreducible $2$-modular characters are the characters ${\varphi(\la)}$, the spin characters fit into the $2$-blocks described above, i.e. those containing linear characters. The distribution of the spin characters among these blocks is given by the following theorem, which was originally conjectured by Knörr and Olsson [@ol p. 246].
[bo]{}[Theorem 4.1]{}\[spinblockclass\] Suppose $\la\in\cald(n)$, and let $\sigma$ be the $2$-core of $\la{^{\operatorname{dbl}}}$. Then ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in the $2$-block of ${\tilde{\mathfrak{S}}_}n$ with $2$-core $\sigma$.
It will be helpful to have alternative descriptions of this block classification, for which we need different notions of core and content. If $\la\in\cald$, we define the *[$4$-bar-core]{}* of $\la$ to be the [$2$-regular partition]{}obtained by repeatedly applying the following operations to $\la$:
- removing all even parts;
- removing any two parts whose sum is a multiple of $4$;
- replacing any odd part $\la_i\gs5$ with $\la_i-4$, if $\la_i-4$ is not already a part of $\la$.
The [$4$-bar-core]{}of $\la$ is easily seen to be well defined, and equals either $(4l-1,4l-5,\dots,3)$ or $(4l-3,4l-7,\dots,1)$ for some $l\gs0$. Note that these are precisely the partitions whose double is a $2$-core. Moreover, the double of the [$4$-bar-core]{}of $\la\in\cald$ coincides with the $2$-core of $\la{^{\operatorname{dbl}}}$ [@bo Lemma 3.6]. So \[spinblockclass\] may alternatively be stated by saying that if $\tau$ is the [$4$-bar-core]{}of $\la\in\cald(n)$, then ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in the block of ${\tilde{\mathfrak{S}}_}n$ with $2$-core $\tau{^{\operatorname{dbl}}}$. In particular, two spin characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lie in the same block $\la$ and $\mu$ have the same [$4$-bar-core]{}. So we can define the *[$4$-bar-core]{}* of a block $B$ to be the common [$4$-bar-core]{}of the [$2$-regular partitions]{}labelling spin characters in $B$.
The [$4$-bar-core]{}of $\la$ is a partition of $n-2w$ for some $w$, which we call the *[$4$-bar-weight]{}* of $\la$. By the comments above, the [$4$-bar-weight]{}of $\la$ equals the weight of the block containing ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
Consider the block $B$ of ${\tilde{\mathfrak{S}}_}{10}$ with $2$-core $(3,2,1)$. The partitions of $10$ with $2$-core $(3,2,1)$ are $(7,2,1)$, $(5,4,1)$, $(5,2,1^3)$, $(3,2^3,1)$ and $(3,2,1^5)$. $(3,2,1)$ is the double of the [$4$-bar-core]{}$(5,1)$, so $(5,1)$ is the [$4$-bar-core]{}of $B$. The partitions in $\cald(10)$ with [$4$-bar-core]{}$(5,1)$ are $(9,1)$ and $(5,4,1)$. So the ordinary irreducible characters in $B$ are $${\llbracket7,2,1\rrbracket},{\llbracket5,4,1\rrbracket},{\llbracket5,2,1^3\rrbracket},{\llbracket3,2^3,1\rrbracket},{\llbracket3,2,1^5\rrbracket},$$ $${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}9,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}},{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5,4,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+,{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5,4,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-,$$ and the irreducible $2$-modular characters in $B$ are ${\varphi(7,2,1)}$ and ${\varphi(5,4,1)}$.
Branching rules {#brnchsec}
---------------
If $n>0$, then ${\tilde{\mathfrak{S}}_}{n-1}$ is naturally embedded in ${\tilde{\mathfrak{S}}_}n$, and a lot of information can be obtained by inducing and restricting characters between these two groups. In the modular representation theory of the symmetric groups, more delicate information can be gleaned by using the so-called *$i$-induction* and *$i$-restriction* functors. We summarise the key points here, specialising to characteristic $2$ (where the results automatically extend to ${\tilde{\mathfrak{S}}_}n$). The definition of the $i$-induction and $i$-restriction functors goes back to Robinson [@gdbrbook], though our main reference will be the survey by Brundan and Kleshchev [@bk].
Given a character $\chi$ of ${\tilde{\mathfrak{S}}_}n$, we write $\chi{{\downarrow}}_{{\tilde{\mathfrak{S}}_}{n-1}}$ for its restriction to ${\tilde{\mathfrak{S}}_}{n-1}$, and $\chi{{\uparrow}}^{{\tilde{\mathfrak{S}}_}{n+1}}$ for the corresponding induced character for ${\tilde{\mathfrak{S}}_}{n+1}$. Now suppose $\chi$ lies in a single block $B$, with content ${\{0^{a},1^{b}\}}$. Then we write ${\mathrm{e}_{0}}\chi$ for the component of $\chi{{\downarrow}}_{{\tilde{\mathfrak{S}}_}{n-1}}$ lying in the block with content ${\{0^{a-1},1^{b}\}}$ if there is such a block, and set ${\mathrm{e}_{0}}\chi=0$ otherwise. Similarly, we write ${\mathrm{e}_{1}}\chi$ for the component of $\chi{{\downarrow}}_{{\tilde{\mathfrak{S}}_}{n-1}}$ lying in the block with content ${\{0^{a},1^{b-1}\}}$ if there is such a block, and set ${\mathrm{e}_{1}}\chi=0$ otherwise. We extend the functions ${\mathrm{e}_{0}},{\mathrm{e}_{1}}$ linearly. These functions ${\mathrm{e}_{0}},{\mathrm{e}_{1}}$ can be applied to either ordinary characters or $2$-modular Brauer characters. In any case, it turns out that for any character $\chi$ we have $\chi{{\downarrow}}_{{\tilde{\mathfrak{S}}_}{n-1}}={\mathrm{e}_{0}}\chi+{\mathrm{e}_{1}}\chi$.
The effect of these functions on ordinary irreducible characters is well understood, via the following “branching theorems”. The first of these dates back to Young, but the second is much more recent, due to Dehuai and Wybourne [@dw §8]. To state these, we introduce some notation. Suppose $\la$ and $\mu$ are partitions, and $i\in\{0,1\}$. We write $\mu{\ifthenelse{\equal{r}1}{\stackrel{i}\longrightarrow}{\stackrel{i^r}\longrightarrow}}\la$ to mean that $\la$ is obtained from $\mu$ by adding $r$ addable $i$-nodes (omitting the superscript $r$ when it equals $1$). Similarly, we write $\mu{\ifthenelse{\equal{r}1}{\stackrel{i}\Longrightarrow}{\stackrel{i^r}\Longrightarrow}}\la$ to mean that $\la$ is obtained from $\mu$ by adding $r$ $i$-[spin-addable nodes]{}.
\[classbrnch\] Suppose $\la\in\calp(n)$ and $i\in\{0,1\}$, and let $$\La=\lset{\mu\in\calp(n-1)}{\mu{\ifthenelse{\equal{1}1}{\stackrel{i}\longrightarrow}{\stackrel{i^1}\longrightarrow}}\la}.$$ Then $${\mathrm{e}_{i}}{\llbracket\la\rrbracket}=\sum_{\mu\in\La}{\llbracket\mu\rrbracket}.$$
\[spinbrnch\] Suppose $\la\in\cald(n)$ and $i\in\{0,1\}$, and let $$\La=\lset{\mu\in\cald(n-1)}{\mu{\ifthenelse{\equal{1}1}{\stackrel{i}\Longrightarrow}{\stackrel{i^1}\Longrightarrow}}\la}.$$
1. If $\la\in\cald^+(n)$, then $$\begin{aligned}
{\mathrm{e}_{i}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}&=\sum_{\mu\in\La\cap\cald^+(n-1)}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\ +\ \sum_{\mu\in\La\cap\cald^-(n-1)}({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-).
\\
\intertext{\item
If $\la\in\cald^-(n)$, then}
{\mathrm{e}_{i}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm&=\sum_{\mu\in\La\cap\cald^+(n-1)}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\ +\ \sum_{\mu\in\La\cap\cald^-(n-1)}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm.\end{aligned}$$
To state further results, we need to consider powers: ${\mathrm{e}_{0}}$ and ${\mathrm{e}_{1}}$ are defined for any $n$, so we can define powers ${\mathrm{e}_{i}}^r$ and (if we allow $\bbq$-linear combinations of characters) divided powers ${{\mathrm{e}_{i}}^{(r)}}={\mathrm{e}_{i}}^r/r!$. For any non-zero character $\chi$ and $i\in\{0,1\}$, we can then define $$\epsilon_i\chi=\max\lset{r\gs0}{{\mathrm{e}_{i}}^r\chi\neq0},\qquad{{{\mathrm{e}_{i}}^{(\operatorname{max})}}}\chi={{\mathrm{e}_{i}}^{(\epsilon_i\chi)}}\chi.$$
The following result comes immediately from the classical branching rule.
\[ordmaxbranch\] Suppose $\la\in\calp$ and $i\in\{0,1\}$, and let $\la^-$ be the partition obtained by removing all the removable $i$-nodes from $\la$. Then ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{\llbracket\la\rrbracket}={\llbracket\la^-\rrbracket}$.
For the spin branching rule, a result of the same form holds, but it is more difficult to keep track of multiplicities. We begin with the following result, which will also be useful in later sections.
\[spinbranchpower\] Suppose $\la\in\cald(n)$, $\mu\in\cald(n-r)$, $i\in\{0,1\}$, and $\mu{\ifthenelse{\equal{r}1}{\stackrel{i}\Longrightarrow}{\stackrel{i^r}\Longrightarrow}}\la$. Let $$c=\big|\lset{x\gs1}{\text{there are nodes of $\la\setminus\mu$ in columns $x$ and $x+1$}}\big|.$$
- If $\la\in\cald^+(n)$ and $\mu\in\cald^+(n-r)$, then $$\begin{aligned}
{\left({{\mathrm{e}_{i}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&=2^{\lfloor r/2\rfloor-c}.
\\
\intertext{\item
If $\la\in\cald^+(n)$ and $\mu\in\cald^-(n-r)$, then}
{\left({{\mathrm{e}_{i}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\right)}&=2^{\lfloor(r+1)/2\rfloor-c}.
\\
\intertext{\item
If $\la\in\cald^-(n)$ and $\mu\in\cald^+(n-r)$, then}
{\left({{\mathrm{e}_{i}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&=2^{\lfloor(r-1)/2\rfloor-c}.
\\
\intertext{\item
If $\la\in\cald^-(n)$ and $\mu\in\cald^-(n-r)$, then}
{\left({{\mathrm{e}_{i}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\right)}&=2^{\lfloor r/2\rfloor-c}.\end{aligned}$$
Recall that we write ${\operatorname{ev}(\nu)}$ for the number of positive even parts of $\nu\in\cald(n)$, and that by definition $\nu\in\cald^+(n)$ ${\operatorname{ev}(\nu)}$ is even. When we remove a [spin-removable node]{}from $\nu$, we change the parity of ${\operatorname{ev}(\nu)}$, unless the removed node is the only node in its row. Applying this observation $r$ times, we find the following, when $\mu{\ifthenelse{\equal{r}1}{\stackrel{i}\Longrightarrow}{\stackrel{i^r}\Longrightarrow}}\la$:
- if ${\operatorname{ev}(\la)}+{\operatorname{ev}(\mu)}+r$ is even, then none of the nodes of $\la\setminus\mu$ lies in the first column (i.e. $\la$ and $\mu$ have the same length);
- if ${\operatorname{ev}(\la)}+{\operatorname{ev}(\mu)}+r$ is odd, then one of the nodes of $\la\setminus\mu$ does lie in column $1$ (and so in particular $i=0$).
Now we use induction on $r$, with the case $r=0$ being trivial. For the inductive step, we consider only the case where $\la\in\cald^+(n)$, $\mu\in\cald^+(n-r)$ and $r$ is odd; the other cases are similar (and often simpler). Let $C$ be the set of column labels of the nodes of $\la\setminus\mu$.
By the above remarks, we must have $i=0$ in this case, and $1\in C$. Apply the spin branching rule, and consider the partitions $\nu\in\cald(n-1)$ such that $\mu\subseteq\nu\subset\la$. These come in three types.
1. The partition $\nu$ obtained from $\la$ by removing the node at the bottom of column $1$. Then $\nu\in\cald^+(n-1)$, ${\left({\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=1$ by the branching rule, and by induction ${\left({{\mathrm{e}_{0}}^{(r-1)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=2^{(r-1)/2-c}$.
2. There are $r-1-2c$ different partitions $\nu$ obtained by removing a node in column $x>1$, where neither $x-1$ nor $x+1$ lies in $C$. In these cases $\nu\in\cald^-(n-1)$, ${\left({\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+\right)}={\left({\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\right)}=1$ by the spin branching rule, and by induction ${\left({{\mathrm{e}_{0}}^{(r-1)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=2^{(r-3)/2-c}$.
3. For each pair $(x,x+1)$ of consecutive integers in $C$, there is one [$2$-regular partition]{}$\nu$ obtained by removing a node in column $x$ or $x+1$: if the nodes of $\la\setminus\mu$ lying in columns $x$ and $x+1$ are of the form $(m,x),(m,x+1)$, then $\nu$ is obtained by removing the node $(m,x+1)$, while if they have the form $(m,x),(m-1,x+1)$, then $\nu$ is obtained by removing $(m,x)$. In any case $\nu\in\cald^-(n-1)$, ${\left({\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+\right)}={\left({\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\right)}=1$ by the spin branching rule, and by induction ${\left({{\mathrm{e}_{0}}^{(r-1)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\nu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=2^{(r-3)/2-(c-1)}$.
So in total we find that $$\begin{aligned}
{\left({{\mathrm{e}_{0}}^{(r-1)}}{\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&=2^{(r-1)/2-c}+(r-2c-1)\times2\times2^{(r-3)/2-c}+c\times2\times2^{(r-3)/2-(c-1)}\\
&=r\times2^{(r-1)/2-c},\end{aligned}$$ which is what we need.
We give an example which illustrates a different case of \[spinbranchpower\]. Take $\la=(11,9,7,5,4,1)$ and $r=2$. The $0$-[spin-removable nodes]{}of $\la$ are highlighted in the following diagram. $$\young(01100110011,01100110!{\Ylinethick{1.3pt}}0!{\Ylinethick{.3pt}},0110011,0110!{\Ylinethick{1.3pt}}0!{\Ylinethick{.3pt}},011!{\Ylinethick{1.3pt}}0,0)$$ The spin branching rule gives $$\begin{aligned}
{\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm=\ &{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,4{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,3,1){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,4,1){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}.
\\
\intertext{The case $r=1$ of the \lcnamecref{spinbranchpower} gives}
{\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,4{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm&={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,3{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,4{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}},\\
{\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}&={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,3{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,4,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,4,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\\
&\phantom=\ +{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-,\\
{\mathrm{e}_{0}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,4,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}&={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,4{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-,
\\
\intertext{and we obtain}
{{\mathrm{e}_{0}}^{(2)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm&={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,5,3{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,4{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+\tfrac12({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,4,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,9,7,4,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-)\\
&\phantom=\ +{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}11,8,7,5,3,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\end{aligned}$$ as predicted by \[spinbranchpower\].
Now we can deduce the following about ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ for $\la\in\cald$; this will be central to the proof of our main theorem.
\[spinmaxbranch\] Suppose $\la\in\cald$ and $i\in\{0,1\}$. Suppose $\la$ has $r$ $i$-[spin-removable nodes]{}, and let $\la^-$ be the partition obtained by removing all these nodes. Then:
1. if $\la^-\in\cald^+(n-r)$, then ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}=a{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^-{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ for some $a\in\bbn$;
2. if $\la^-\in\cald^-(n-r)$, then ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}=a{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^-{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++b{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^-{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$ with $a+b\in\bbn$.
Analogous results apply to induced modules. For a character $\chi$ of ${\tilde{\mathfrak{S}}_}n$, we have $\chi{{\uparrow}}^{{\tilde{\mathfrak{S}}_}{n+1}}={\mathrm{f}_{0}}\chi+{\mathrm{f}_{1}}\chi$, where (if $\chi$ lies in the block with content ${\{0^{a},1^{b}\}}$) ${\mathrm{f}_{0}}\chi$ is the component of $\chi{{\uparrow}}^{{\tilde{\mathfrak{S}}_}{n+1}}$ lying in the block with content ${\{0^{a+1},1^{b}\}}$, and ${\mathrm{f}_{1}}\chi$ is defined similarly. For a non-zero character $\chi$ we define $\varphi_i\chi=\max\lset{r\gs0}{{\mathrm{f}_{i}}^r\chi\neq0}$, and we set ${{{\mathrm{f}_{i}}^{(\operatorname{max})}}}\chi={{\mathrm{f}_{i}}^{(\varphi_i\chi)}}\chi$. Then we have analogues of \[classbrnch,spinbrnch,ordmaxbranch,spinbranchpower,spinmaxbranch\] (which follow from these results by Frobenius reciprocity), in which we add nodes rather than removing nodes. We will not state these results explicitly, but instead refer to the “induction versions” of the restriction results above.
Next we recall some of Kleshchev’s “modular branching rules”, to describe what happens to the irreducible Brauer characters ${\varphi(\mu)}$ under the $i$-restriction operators. This involves the combinatorics of *normal nodes*.
Suppose $\mu\in\cald$ and $i\in\{0,1\}$, and construct a sequence of $+$ and $-$ signs by reading along the edge of the Young diagram of $\mu$ from top to bottom and writing a $+$ for each addable $i$-node and a $-$ for each removable $i$-node. This sequence is called the *$i$-signature* of $\mu$. Now successively delete all adjacent pairs $+-$ from this sequence until none remain. The resulting sequence is called the *reduced $i$-signature* of $\mu$. The removable nodes corresponding to the $-$ signs in the reduced $i$-signature are called the *normal $i$-nodes* of $\mu$, and the addable nodes corresponding to the $+$ signs are the *conormal $i$-nodes* of $\mu$. Now we have the following result. See [@bk] (in particular, the discussion following Lemma 2.12) for this and more general modular branching results.
\[modbranch\] Suppose $\mu\in\cald$ and $i\in\{0,1\}$. Let $\mu^-$ be the [$2$-regular partition]{}obtained by removing all the normal $i$-nodes from $\mu$, and let $\mu^+$ be the partition obtained by adding all the conormal $i$-nodes to $\mu$. Then $\mu^-,\mu^+$ are $2$-regular, and $${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{\varphi(\mu)}={\varphi(\mu^-)},\qquad{{{\mathrm{f}_{i}}^{(\operatorname{max})}}}{\varphi(\mu)}={\varphi(\mu^+)}.$$ In particular, if $\chi$ is an irreducible Brauer character of ${\tilde{\mathfrak{S}}_}n$, then ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}\chi$ and ${{{\mathrm{f}_{i}}^{(\operatorname{max})}}}\chi$ are irreducible Brauer characters.
The last statement in this theorem is extremely useful for us, and will form the basis of our induction proof of our main theorem.
Let $\mu=(15,11,8,6,5,2)$ and $i=0$. The Young diagram of $\mu$ with the residues of addable and removable nodes indicated, is as follows. 01 $$\yngres(2,15,11,8,6,5,2)$$ So the $0$-signature of $\mu$ is $-++---+$, and hence the reduced $0$-signature is $--+$, with the normal nodes being $(1,15)$ and $(6,2)$, and the conormal node $(7,1)$. Hence $${{{\mathrm{e}_{0}}^{(\operatorname{max})}}}{\varphi(\mu)}={\varphi(14,11,8,6,5,1)},\qquad{{{\mathrm{f}_{0}}^{(\operatorname{max})}}}{\varphi(\mu)}={\varphi(15,11,8,6,5,2,1)}.$$
Symmetric functions {#symfnsec}
-------------------
The combinatorics of partitions involved in representation theory is closely connected with the combinatorics involved in symmetric functions, and in this paper we shall exploit this connection. In this section we set out the notation and background results we shall need. We give just the bare essentials, since symmetric function theory is covered in detail elsewhere.
We consider the space ${\mathrm{Sym}}$ of symmetric functions over $\bbc$ in infinitely many variables. For $\la\in\calp$, let $s_\la$ denote the corresponding *Schur function*. Let ${\lan\,,\,\ran}$ be the inner product on ${\mathrm{Sym}}$ for which the Schur functions are orthonormal.
For each $r$, we let $h_r$ denote the complete homogeneous symmetric function of degree $r$ (which coincides with the Schur function $s_{(r)}$). For any partition $\mu$, we write $h_\mu$ for the product $h_{\mu_1}h_{\mu_2}\dots$. The *Pieri rule* says that for any $r$ and for $\la\in\calp$ we have $h_rs_\la=\sum s_\nu$, summing over all partitions $\nu$ which can be obtained from $\la$ by adding $r$ nodes in distinct columns. Inductively, this enables us to express any $h_\mu$ as a sum of Schur functions; the coefficients obtained are the *Kostka numbers*. Two immediate consequences of this are that ${\lanh_\mu,s_\mu\ran}=1$ for each $\mu$, and that ${\lanh_\mu,s_\la\ran}$ is non-zero only if $\la\dom\mu$.
We write $e_r$ for the $r$th elementary symmetric function (which coincides with $s_{(1^r)}$), and for any $\mu\in\calp$ we write $e_\mu=e_{\mu_1}e_{\mu_2}\dots$. The *dual Pieri rule* says that for any $r$ and any $\la$ we have $e_rs_\la=\sum_\nu s_\nu$, summing over all partitions $\nu$ which can be obtained from $\la$ by adding $r$ nodes in distinct rows. Inductively, this enables us to express any $e_\mu$ as a sum of Schur functions (again, in terms of Kostka numbers). This has the consequences that ${\lane_\mu,s_{\mu'}\ran}=1$ for any $\mu$ and that ${\lane_\mu,s_\la\ran}$ is non-zero only if $\mu'\dom\la$. The combination of the Pieri and dual Pieri rules yields the identity ${\lane_\mu,s_\la\ran}={\lanh_\mu,s_{\la'}\ran}$.
The sets $\lset{s_\la}{\la\in\calp}$, $\lset{h_\la}{\la\in\calp}$ and $\lset{e_\la}{\la\in\calp}$ are all bases for ${\mathrm{Sym}}$. We shall need to consider the transition coefficients which allow us to express a complete homogeneous function $h_\la$ in terms of the elementary functions $e_\mu$. So define ${\spadesuit_{\la,\mu}}$ for all $\la,\mu\in\calp$ by $h_\la=\sum_\mu{\spadesuit_{\la,\mu}} e_\mu$. In the case where $\la=(k)$, these coefficients are given by a special case of the *second Jacobi–Trudi formula*, which says that $h_k$ equals the determinant of the $k\times k$ matrix $$\left\lvert\begin{matrix}e_1&e_2&e_3&\cdots&e_k\\1&e_1&e_2&\ddots&\vdots\\0&1&e_1&\ddots&e_3\\\vdots&\ddots&\ddots&\ddots&e_2\\0&\cdots&0&1&e_1\end{matrix}\right\rvert.$$
We now derive some consequences of this for partitions with only even parts.
\[ekdouble\] If $\mu\in\calp(k)$, then ${\spadesuit_{(2k),2\mu}}=(-1)^k{\spadesuit_{(k),\mu}}$ for any partition $\mu$.
Using the second Jacobi–Trudi formula and expanding the determinant in terms of permutations, we find that ${\spadesuit_{(k),\mu}}$ is sum of the signs of all the permutations $\pi\in\sss k$ with the property that $\pi(i)\gs i-1$ for each $i$ and the integers $\pi(1),\pi(2)-1,\dots,\pi(k)-k+1$ equal $\mu_1,\dots,\mu_k$ in some order. Let $\Pi_\mu$ be the set of such permutations. Then it suffices to show that there is a bijection from $\Pi_\mu$ to $\Pi_{2\mu}$ which preserves the sign of every permutation if $k$ is even, or changes it if $k$ is odd.
To construct the required map from $\Pi_\mu$ to $\Pi_{2\mu}$, we take $\pi\in\Pi_\mu$, and define $\hat\pi\in\Pi_{2\mu}$ by $$\hat\pi(i)=
\begin{cases}
2\pi((i+1)/2)&(i\text{ odd})\\
i-1&(i\text{ even}).
\end{cases}$$ To construct the inverse map, we take $\sigma\in\Pi_{2\mu}$, and observe that because $\sigma(i)-i$ is odd and $\sigma(i)\gs i-1$ for every $i$, we must have $\sigma(i)=i-1$ for all even $i$. Now we can define $\check\sigma\in\Pi_\mu$ by $\check\sigma(i)=\sigma(2i-1)/2$ for each $i$. It is clear that these two maps are mutually inverse, so it remains to show that ${\operatorname{sgn}(\hat\pi)}=(-1)^k{\operatorname{sgn}(\pi)}$ for every $\pi\in\Pi_\mu$.
Regarding $\pi$ as an element of $\sss{2k}$ by the usual embedding of $\sss k$ in $\sss{2k}$, we can write $\hat\pi=\kappa\circ\pi\circ\iota$, where $\iota,\kappa\in\sss{2k}$ are given by $$\iota(i)=
\begin{cases}
(i+1)/2&(i\text{ odd})\\
i/2+k&(i\text{ even}),
\end{cases}
\qquad
\kappa(i)=
\begin{cases}
2i&(i\ls k)\\
2(i-k)-1&(i>k).
\end{cases}$$ It is easy to see that the signs of $\iota$ and $\kappa$ are $(-1)^{\binom k2}$ and $(-1)^{\binom{k+1}2}$ respectively, which gives the result.
By writing $h_\la$ as the product $h_{\la_1}h_{\la_2}\dots$ and applying \[ekdouble\], we obtain the following result. Note that for partitions $\nu^1,\nu^2,\dots$ we have $e_{\nu^1}e_{\nu^2}\dots=e_{2\mu}$ each $\nu^i$ has only even parts and $e_{\frac12\nu^1}e_{\frac12\nu^2}\dots=e_\mu$, i.e. $\frac12\nu^1\sqcup\frac12\nu^2\sqcup\dots=\mu$.
\[ekdoublep\] Suppose $\la,\mu\in\calp(k)$. Then ${\spadesuit_{2\la,2\mu}}=(-1)^k{\spadesuit_{\la,\mu}}$.
Finally we note the following simple result.
\[simpleep\] Suppose $\mu\in\calp(k)$ and $\la\in\calp(2k)$ with $\la_i$ odd for at least one value of $i$. Then ${\spadesuit_{\la,2\mu}}=0$.
$h_\la$ includes a factor $h_{\la_i}$; when this is expressed in terms of elementary symmetric polynomials $e_\nu$, each $\nu$ satisfies $|\nu|=\la_i$, and so must have at least one odd part. Hence each $e_\nu$ appearing when $h_\la$ is expressed in terms of elementary symmetric polynomials must contain a factor $e_l$ for some odd $l$.
$2$-Carter partitions and the main theorem {#mainthmsec}
==========================================
Having set out the background results we need, we are now able to state our main theorem. This relies on another combinatorial definition: Say that $\la\in\calp$ is *$2$-Carter* if for every $r\gs1$, $\la_r-\la_{r+1}+1$ is divisible by a power of $2$ greater than $\la_{r+1}-\la_{r+2}$.
The $2$-Carter partitions $\la$ with $\la_1\ls5$ are $$\varnothing,(1),(2),(2,1),(3),(3,2,1),(4),(4,1),(4,3,2,1),(5),(5,2),(5,2,1),(5,4,3,2,1).$$
The notion of a $2$-Carter partition was introduced for the classification of irreducible Specht and Weyl modules. We shall need the following result later (recall that ${\cals(n)}$ is the Schur algebra over a field $\bbf$ of characteristic $2$).
[jm1]{}[Theorem 4.5]{}\[irredweyl\] Suppose $\la\in\calp(n)$. Then the Weyl module ${\Delta^{\la}}$ for ${\cals(n)}$ is irreducible $\la$ is a $2$-Carter partition.
(We remark that [@jm1 Theorem 4.5] is phrased in terms of hook lengths in the Young diagram of $\la$. The fact that this formulation is equivalent to the one we have given is explained by James in [@j0 Lemma 3.14].)
Now we come to the classification of ordinary irreducible characters for ${\tilde{\mathfrak{S}}_}n$ that remain irreducible in characteristic $2$. For the characters ${\llbracket\la\rrbracket}$ of the Specht modules, the answer is the same as for $\sss n$, and is given by the following theorem.
[jmp2]{}[Main Theorem]{}\[jmp2main\] Suppose $\la\in\calp$. Then ${\widebar{{\llbracket\la\rrbracket}}}$ is irreducible either $\la$ or $\la'$ is a $2$-Carter partition, or $\la=(2^2)$.
So it remains to consider spin characters. Recall that when $\la\in\cald^-(n)$ we write ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ to mean either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$, and ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is then unambiguously defined, since ${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}={\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-}}$. So we can phrase the main question for spin characters simply as “for which $\la\in\cald$ is ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ irreducible?”. Now we can state our main theorem.
\[main\] Suppose $\la\in\cald$. Then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible one of the following occurs.
1. $\la=\tau+4\alpha$, where $\tau=(4l-1,4l-5,\dots,3)$ for some $l\gs0$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l$.
2. $\la=\tau+4\alpha$, where $\tau=(4l-3,4l-7,\dots,1)$ for some $l\gs1$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l$.
3. $\la=\tau+4\alpha\sqcup(2)$, where $\tau=(4l-1,4l-5,\dots,3)$ for some $l\gs0$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l$.
4. $\la=\tau+4\alpha\sqcup(2)$, where $\tau=(4l-3,4l-7,\dots,1)$ for some $l\gs1$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l-1$.
5. $\la$ equals $(2b)$ or $(4b-2,1)$ for some $b\gs2$.
6. $\la=(3,2,1)$.
\[degreesec,rouqandsep,mainproofsec\] are devoted to the proof of this theorem.
Degrees of irreducible characters {#degreesec}
=================================
In this section we use the bar-length formula to compare the degrees of certain characters. The motivating observation here is the following.
\[samerdoub\] Suppose $\la,\mu\in\cald(n)$, with $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$ and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$. Then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is reducible.
By \[spinreg\], ${{\varphi(\la{^{\operatorname{dblreg}}})}}$ occurs as a constituent of ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$, so if ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ were irreducible, we would have ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={{\varphi(\la{^{\operatorname{dblreg}}})}}$, and in particular ${\operatorname{deg}}({{\varphi(\la{^{\operatorname{dblreg}}})}})={\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$. But ${{\varphi(\la{^{\operatorname{dblreg}}})}}={{\varphi(\mu{^{\operatorname{dblreg}}})}}$ also occurs as a constituent of ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$, so ${\operatorname{deg}}({{\varphi(\la{^{\operatorname{dblreg}}})}})\ls{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}<{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
We now construct various families of pairs of partitions $\la,\mu$ satisfying the hypotheses of \[samerdoub\]. Our first result is as follows.
\[dimen1\] Suppose $m\gs2$, and let $$\begin{aligned}
\la^m&=(4m,4m-3,4m-7,\dots,9,5),\\
\mu^m&=(4m+1,4m-3,4m-7,\dots,9,4).\end{aligned}$$ Then $(\la^m){^{\operatorname{dblreg}}}=(\mu^m){^{\operatorname{dblreg}}}$, and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^m{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^m{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
$\mu^m$ is obtained from $\la^m$ by replacing the node $(m,5)$ with the node $(1,4m+1)$. Since these nodes both lie in the same slope, we have $(\la^m){^{\operatorname{dblreg}}}=(\mu^m){^{\operatorname{dblreg}}}$ by \[slopelad\].
Now we consider the degrees. We use induction on $m$, with the case $m=2$ an easy check. For the inductive step, it suffices to prove that when $m\gs3$ $$\frac{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^m{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{m-1}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^m{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{m-1}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}>1.$$ Directly from the bar-length formula, this ratio equals $$\frac{(4m-5)(4m-1)(4m+1)^2(4m+5)}{(2m+1)(2m+3)(4m-3)(8m-7)(8m-3)}.$$ To show that this is always greater than $1$, we show that the numerator exceeds the denominator for all $m\gs2$. The difference between the numerator and the denominator is $f(m)=256m^4+368m^3-1220m^2-368m+214$. We have $f(2)$ and $f(0)$ positive, while $f(1)$ and $f(-1)$ are negative, so that all the roots of $f$ are less than $2$.
Now we give a similar dimension argument which will help when we consider Rouquier blocks.
\[firstrdim\] Given $a>0$ and $m\gs0$, define $$\begin{aligned}
\la^{a,m}&=(4a+4m-3,4a+4m-7,\dots,1)\sqcup(4a),\\
\mu^{a,m}&=(4a+4m+1,4a+4m-3,\dots,4m+5,4m-3,4m-7,\dots,1).\end{aligned}$$ Then $(\la^{a,m}){^{\operatorname{dblreg}}}=(\mu^{a,m}){^{\operatorname{dblreg}}}$, and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
To see that $(\la^{a,m}){^{\operatorname{dblreg}}}=(\mu^{a,m}){^{\operatorname{dblreg}}}$ we use \[slopelad\], showing that $\mu^{a,m}$ can be obtained from $\la^{a,m}$ by moving some nodes, but keeping each node in the same slope. There are two cases.
1. Suppose $a>m$. Let $A$ be the set of nodes of $\la^{a,m}$ comprising
- the last four nodes in each of rows $a+1,\dots,a+m$, and
- the unique node in row $a+m+1$.
Now observe that $\mu^{a,m}$ can be obtained from $\la^{a,m}$ by replacing each node $(r,c)\in A$ with $(r-a,c+4a)$. Since $(r,c)$ and $(r-a,c+4a)$ lie in the same slope, the result follows.
2. Now suppose $a\ls m$. Let $A$ be the set of nodes of $\la^{a,m}$ comprising
- the last three nodes in row $m+1$, and
- the last four nodes in each of rows $m+2,\dots,a+m$.
Now $\mu^{a,m}$ can be obtained from $\la^{a,m}$ by replacing each node $(r,c)\in A$ with $(r-m,c+4m)$, and also replacing the node $(a+m+1,1)$ with $(1,4a+4m+1)$. Again, each moved node remains in the same slope, so the result follows.
Now we consider degrees. Direct from the bar-length formula we get $$\begin{aligned}
\frac{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}=&\frac{(4a+4m+1)!\big((8a+8m-2)(8a+8m-6)\dots(4a+8m+6)\big)\big(4m(4m-4)\dots4\big)}
{(4a)!(4m)!\big((4a+4m-2)(4a+4m-6)\dots(4m+2)\big)}\\
&\times\frac{\big((4m-3)(4m-7)\dots1\big)\big((4a-1)(4a-5)\dots3\big)}{\big((8a+4m-3)(8a+4m-7)\dots(4a+1)\big)\big((4a+4m)(4a+4m-4)\dots(4a+4)\big)}.\tag*{(\textasteriskcentered)}\end{aligned}$$ Now we proceed by induction on $a$. In the case $a=1$, () becomes $(4m+3)/2$, which is greater than $1$. For the inductive step, it suffices to show that $$\frac{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a+1,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a+1,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}\frac{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}>1.$$ From (), this ratio equals $$\frac{(4a+4m+5)(4a+4m+3)^2(4a+4m+1)}{(8a+4m+5)(8a+4m+1)(2a+4m+3)(2a+1)}.$$ The difference between the numerator and denominator in this fraction is $$\begin{aligned}
&256m^4+896am^3+960a^2m^2+256a^3m+704m^3+1728am^2+1056a^2m+64a^3\\
&+656m^2+984am+204a^2+244m+152a+30\end{aligned}$$ which is obviously positive.
The next three results are proved in exactly the same way.
\[secondrdim\] Given $a>0$ and $m\gs0$, define $$\begin{aligned}
\la^{a,m}&=(4a+4m+1,4a+4m-3,\dots,1)\sqcup(4a+2),\\
\mu^{a,m}&=(4a+4m+5,4a+4m+1,\dots,4m+9,4m+1,4m-3,\dots,5,2,1).\end{aligned}$$ Then $(\la^{a,m}){^{\operatorname{dblreg}}}=(\mu^{a,m}){^{\operatorname{dblreg}}}$, and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
\[thirdrdim\] Given $a>0$ and $m\gs0$, define $$\begin{aligned}
\la^{a,m}&=(4a+4m-1,4a+4m-5,\dots,3)\sqcup(4a),\\
\mu^{a,m}&=(4a+4m+3,4a+4m-1,\dots,4m+7,4m-1,4m-5,\dots,3).\end{aligned}$$ Then $(\la^{a,m}){^{\operatorname{dblreg}}}=(\mu^{a,m}){^{\operatorname{dblreg}}}$, and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
\[fourthrdim\] Given $a>0$ and $m\gs0$, define $$\begin{aligned}
\la^{a,m}&=(4a+4m-1,4a+4m-5,\dots,3)\sqcup(4a+2),\\
\mu^{a,m}&=(4a+4m+3,4a+4m-1,\dots,4m+7,4m-1,4m-5,\dots,3,2).\end{aligned}$$ Then $(\la^{a,m}){^{\operatorname{dblreg}}}=(\mu^{a,m}){^{\operatorname{dblreg}}}$, and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^{a,m}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
To make these results more general, we now consider row removal. We start with the following lemma.
\[domdim\] Suppose $\la,\mu\in\calp(n)$ with $\mu\doms\la$, and that $l$ is an integer greater than $\mu_1$. Then $$\prod_{i\gs1}\frac{l+\mu_i}{l-\mu_i}>\prod_{i\gs1}\frac{l+\la_i}{l-\la_i}.$$
We may assume that $\mu$ covers $\la$ in the dominance order, in which case $\la$ is obtained from $\mu$ by moving a single node down to a lower row. So suppose that for some $j<k$ we have $\mu_j=a+1$, $\la_j=a$, $\mu_k=b$, $\la_k=b+1$, and that $\la_i=\mu_i$ for all $i\neq j,k$. Then the ratio of the left-hand expression to the right-hand expression is $$\frac{(l-a)(l-b-1)(l+a+1)(l+b)}{(l+a)(l+b+1)(l-a-1)(l-b)}.$$ The difference between the numerator and denominator here is $2l(a-b)(a+b+1)$, which is positive, so the ratio is greater than $1$.
\[dimrowrem\] Suppose $\la,\mu\in\cald(n)$ with $\mu\doms\la$, and $l$ is an integer with $l>\mu_1$. Define $$\begin{aligned}
\la^+&=(l,\la_1,\la_2,\dots),\\
\mu^+&=(l,\mu_1,\mu_2,\dots).\end{aligned}$$ Then $$\frac{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la^+{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu^+{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}>\frac{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}.$$ Furthermore, if $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$, then $(\la^+){^{\operatorname{dblreg}}}=(\mu^+){^{\operatorname{dblreg}}}$.
From the bar-length formula, the ratio of the left-hand side to the right-hand side is $$\prod_{i\gs1}\frac{l+\mu_i}{l-\mu_i}\frac{l-\la_i}{l+\la_i},$$ and by \[domdim\] this is greater than $1$. The second statement follows by considering the slopes containing the nodes of $\la$ and the corresponding nodes of $\la^+$.
Rouquier blocks and [separated]{}partitions {#rouqandsep}
===========================================
Rouquier blocks are a certain class of particularly well-behaved blocks of symmetric groups (and more generally of Iwahori–Hecke algebras and $q$-Schur algebras). In this section we summarise some of the important properties of Rouquier blocks of symmetric groups and their double covers in characteristic $2$, and then examine the decomposition numbers for Rouquier blocks of ${\tilde{\mathfrak{S}}_}n$. We compute the rows of the spin part of the decomposition matrix labelled by [$2$-regular partitions]{}with only odd parts, showing that some of the corresponding spin characters are irreducible in characteristic $2$. We extend these results to what we call “[separated]{}” [$2$-regular partitions]{}, and prove our main theorem for spin characters labelled by [separated]{}partitions.
Rouquier blocks
---------------
Suppose $B$ is a block of ${\tilde{\mathfrak{S}}_}n$, with $2$-core $\sigma=(c,c-1,\dots,1)$ and weight $w$. We say that $B$ is *Rouquier* if $w\ls c+1$.
The first thing that makes Rouquier blocks easy to understand is a simple description of the [$2$-regular partitions]{}labelling characters in a Rouquier block.
\[trpsinrouq\] Suppose $\sigma=(c,c-1,\dots,1)$ and that $\la$ is a partition with $2$-core $\sigma$ and $2$-weight $w\ls c+1$. .
1. $\la$ is $2$-regular.
2. The length of $\la$ is at most $c+1$.
3. $\la$ has the form $\sigma+2\alpha$, where $\alpha\in\calp(w)$.
[(1$\Rightarrow$2)]{}
: Suppose $\la$ is $2$-regular, and suppose for a contradiction that the length of $\la$ is greater than $c+1$. Then $(c+2,1)$ is a node of $\la$, so (since $\la$ is $2$-regular) $(c+1,2),(c,3),(c-1,4),\dots,(1,c+2)$ are all nodes of $\la$. But then $|\la|\gs\frac12(c+2)(c+3)>|\sigma|+2w$, a contradiction.
[(2$\Rightarrow$3)]{}
: We use induction on $w$, with the case $w=0$ being trivial. Assuming $w>0$, let $r\ls c+1$ be maximal such that $\la_r\gs c+2-r$ (there must be such an $r$, since $\la\supset\sigma$). Then we claim that $\la_r\gs\la_{r+1}+2$. If $r\ls c$ this is immediate from the choice of $r$, so suppose $r=c+1$. The node $(c+1,1)$ is a node of $\la$ but not of $\sigma$, so must be part of one of the rim $2$-hooks added to obtain $\la$ from $\sigma$. The other node in this rim $2$-hook must be $(c+1,2)$, since $(c,1)$ is a node of $\sigma$, and (by assumption) $(c+2,1)$ is not a node of $\la$. Hence $\la_{c+1}\gs2=\la_{c+2}+2$ as claimed.
Hence we can remove the rim $2$-hook $\{(r,\la_r-1),(r,\la_r)\}$ from $\la$ and leave a partition $\mu$ which satisfies the hypotheses of part (2). By induction $\mu=\sigma+2\beta$ for some $\beta\in\calp(w-1)$, and hence $\la=\sigma+2\alpha$, where $\alpha$ is the partition obtained from $\beta$ by adding a node at the end of row $r$.
[(3$\Rightarrow$1)]{}
: Suppose $\la=\sigma+2\alpha$ with $\alpha\in\calp(w)$. Then the length of $\alpha$ is at most $w\ls c+1$. So $\la_r=0$ for $r>c+1$, and $\la_r-\la_{r+1}=1+2(\alpha_r-\alpha_{r+1})>0$ for $r\ls c$, and hence $\la$ is $2$-regular.
We can also describe the [$2$-regular partitions]{}labelling spin characters in Rouquier blocks. This requires some preliminary work.
\[rouqlem3\] Suppose $\tau=(4l-1,4l-5,\dots,3)$ for $l\gs0$, and $\la$ is a [$2$-regular partition]{}with [$4$-bar-core]{}$\tau$ and [$4$-bar-weight]{}$w\ls2l+1$. Then $\la$ has no parts congruent to $1$ modulo $4$, and $\la$ includes all the integers $3,7,11,\dots,4t-1$, where $t=\lceil l-\frac12w\rceil$.
We use induction on $w$. In the case $w=0$ we have $\la=\tau$ and the result holds. Now take $w>0$ and suppose the result holds for smaller $w$. By the definition of [$4$-bar-core]{}, there is a [$2$-regular partition]{}$\mu$ satisfying one of the following.
- $\mu$ is obtained from $\la$ by reducing some $\la_i$ by $4$ (and re-ordering). By the inductive hypothesis $\mu$ has no parts congruent to $1$ modulo $4$, and hence neither does $\la$. $\mu$ has [$4$-bar-weight]{}$w-2$, so by hypothesis $\mu$ includes all the integers $3,7,\dots,4t+3$. Hence $\la$ includes the integers $3,7,\dots,4t-1$.
- $\la=\mu\sqcup(2)$. In this case, the assumption that $\mu$ satisfies the given conditions means that $\la$ does too.
- $\la=\mu\sqcup(3,1)$. This means that $3\notin\mu$; but since $\mu$ has [$4$-bar-weight]{}$w-2$ the inductive hypothesis says that $3,7,\dots,4t+3\in\mu$, so that $t<0$, i.e. $w>2l+1$, contrary to assumption.
\[rouqspinform3\] Suppose $B$ is a Rouquier block of ${\tilde{\mathfrak{S}}_}n$ with [$4$-bar-core]{}$\tau=(4l-1,4l-5,\dots,3)$ and [$4$-bar-weight]{}$w$, and that $\la\in\cald(n)$ such that ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in $B$. Then $\la$ can be written in the form $\tau+4\alpha\sqcup2\beta$, where $\alpha\in\calp$, $\beta\in\cald$ and $2|\alpha|+|\beta|=w$. Furthermore, if $\la_r$ is any even part of $\la$, then $\la$ includes all positive integers less than $\la_r$ which are congruent to $3$ modulo $4$.
By sorting the parts of $\la$ according to parity, we can write $\la=\nu\sqcup2\beta$ where all the positive parts of $\nu$ are odd. Then the [$4$-bar-core]{}of $\nu$ is also $\tau$, and the [$4$-bar-weight]{}of $\nu$ is at most $w$. Since $B$ is a Rouquier block we have $w\ls 2l+1$, so by \[rouqlem3\] all the positive parts of $\nu$ are congruent to $3$ modulo $4$. So if we let $m={l(\nu)}$ and let $\upsilon$ denote the [$4$-bar-core]{}$(4m-1,4m-5,\dots,3)$, then we have $\nu=\upsilon+4\alpha$ for some partition $\alpha$. But then the [$4$-bar-core]{}of $\nu$ is clearly $\upsilon$, and so we have $\upsilon=\tau$. So we can write $\la=\tau+4\alpha\sqcup2\beta$, and it follows that $2|\alpha|+|\beta|=w$.
The [$4$-bar-weight]{}of $\nu$ is $2|\alpha|$, so applying the second statement of \[rouqlem3\] with $\nu$ in place of $\la$, we find that $\nu$ (and hence $\la$) contains all the integers $3,7,\dots,4(l-|\alpha|)-1$. Now any even part of $\la$ equals $2\beta_s$ for some $s$, and $$2\beta_s\ls2|\beta|=2w-4|\alpha|\ls4l-4|\alpha|+2.$$ Hence $\la$ contains all positive integers less than $2\beta_s$ which are congruent to $3$ modulo $4$.
In a very similar way, we obtain the following.
\[rouqlem1\] Suppose $\tau=(4l-3,4l-7,\dots,1)$ for $l\gs1$, and $\la$ is a [$2$-regular partition]{}with [$4$-bar-core]{}$\tau$ and [$4$-bar-weight]{}$w\ls2l$. Then $\la$ has no parts congruent to $3$ modulo $4$, and $\la$ includes all the integers $1,5,9,\dots,4t-3$, where $t=\lceil l-\frac12w\rceil$.
\[rouqspinform1\] Suppose $B$ is a Rouquier block of ${\tilde{\mathfrak{S}}_}n$ with [$4$-bar-core]{}$\tau=(4l-1,4l-5,\dots,3)$ and [$4$-bar-weight]{}$w$, and that $\la\in\cald(n)$ such that ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in $B$. Then $\la$ can be written in the form $\tau+4\alpha\sqcup2\beta$, where $\alpha\in\calp$, $\beta\in\cald$ and $2|\alpha|+|\beta|=w$. Furthermore, if $\la_r$ is any even part of $\la$, then $\la$ includes all positive integers less than $\la_r$ which are congruent to $1$ modulo $4$.
Note in particular that by \[rouqspinform3,rouqspinform1\], if $\la\in\cald$ has no even parts and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in a Rouquier block, then the [$4$-bar-weight]{}of $\la$ must be even.
Decomposition numbers for Rouquier blocks
-----------------------------------------
An advantage of working with Rouquier blocks is that their decomposition numbers are relatively well understood. We summarise this situation, beginning with Hecke algebras, where the decomposition numbers for Rouquier blocks are known explicitly in terms of Littlewood–Richardson coefficients. This result is due to James and Mathas [@jmq-1 Corollary 2.6]; here we only give two special cases.
\[heckerouqdec\] Suppose $\sigma=(c,c-1,\dots,1)$ and $w\ls c+1$.
1. If $\la,\mu\in\calp(w)$, then $$\begin{aligned}
{\mathring D}_{(\sigma+2\la)(\sigma+2\mu)}&=\delta_{\la\mu}.
\\
\intertext{
\item
If $\la\in\calp(w-1)$ and $\mu\in\calp(w)$, then}
{\mathring D}_{(\sigma+2\la\sqcup(1^2))(\sigma+2\mu)}&=
\begin{cases}
1&\text{if $\la\subset\mu$}\\
0&\text{if $\la\nsubset\mu$}.
\end{cases}\end{aligned}$$
Now we consider Rouquier blocks of symmetric groups. The following result is due to Turner.
[turn]{}[Theorem 132]{}\[snrouqdec\] Suppose $B$ is a Rouquier block of $\sss n$ with $2$-core $\sigma=(c,c-1,\dots,1)$ and weight $w\ls c+1$. Then $$D_{(\sigma+2\la)(\sigma+2\mu)}=D_{\la\mu}$$ for all $\la,\mu$ in $\calp(w)$.
Hence the adjustment matrix $A$ for the block $B$ is just the decomposition matrix of ${\cals(w)}$, i.e. $$A_{(\sigma+2\la)(\sigma+2\mu)}=D_{\la\mu}$$ for $\la,\mu$ in $\calp(w)$.
Some virtual projective characters in a Rouquier block {#projcharsec}
------------------------------------------------------
We now examine the decomposition numbers in a Rouquier block of ${\tilde{\mathfrak{S}}_}n$ by considering projective characters. We fix some notation.
**Assumptions and notation in force for \[projcharsec,somedecevensec\]:** $B$ is a Rouquier block of ${\tilde{\mathfrak{S}}_}n$ with $2$-core $\sigma$ and weight $w$. $\tau$ is the [$4$-bar-core]{}of $B$, i.e. the common [$4$-bar-core]{}of the [$2$-regular partitions]{}labelling spin characters in $B$.
Recall that a projective character is one which vanishes on $2$-singular elements; a *virtual* projective character is a $\bbz$-linear combination of projective characters. It is well known that induction and restriction send (virtual) projective characters to (virtual) projective characters, and hence so do the functors ${\mathrm{e}_{i}}$ and ${\mathrm{f}_{i}}$.
Recall that ${\left(\ \ \!:\!\ \ \right)}$ is the usual inner product on ordinary characters, and that ${\lan\,,\,\ran}$ is the standard inner product on the space of symmetric functions.
\[induceproj\] Suppose $\mu\in\calp(w)$. There is a projective character $\psi^\mu$ of ${\tilde{\mathfrak{S}}_}n$ with the following properties.
1. ${\left(\psi^\mu\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}={\lane_{\mu'},s_\nu\ran}$ for any $\nu\in\calp(w)$. In particular, ${\left(\psi^\mu\!:\!{\llbracket\sigma+2\mu\rrbracket}\right)}=1$, while ${\left(\psi^\mu\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}=0$ if $\mu\ndom\nu$.
2. If the column lengths of $\mu$ are all even, say $\mu={\alpha\sqcup\alpha}$, then ${\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}={\lane_{\alpha'},s_\beta\ran}$ for every $\beta\in\calp(w/2)$. In particular, ${\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=1$ while ${\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$ if $\alpha\ndom\beta$.
3. If $w$ is even but $\mu$ has at least one column of odd length, then ${\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$ for every $\beta\in\calp(w/2)$.
We prove \[induceproj\] by applying induction functors. Note that all the addable nodes of $\sigma$ have the same residue; we assume for the rest of this section that they have residue $0$. (The proof in the opposite case is identical, but with $0$ and $1$ swapped throughout.)
Given $r\in\bbn$, consider the induction functor ${{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}$. The classical branching rule gives the following.
\[induceord\] Suppose $r\ls v\ls w$, and $\xi$ is a partition with $2$-core $\sigma$ and $2$-weight $v-r$. Suppose $\nu\in\calp(v)$.
1. If $\xi$ is $2$-regular, say $\xi=\sigma+2\kappa$, then ${\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{\llbracket\xi\rrbracket}\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}={\lane_rs_\kappa,s_\nu\ran}$.
2. If $\xi$ is $2$-singular, then ${\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{\llbracket\xi\rrbracket}\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}=0$.
This result is effectively a special case of [@ct Lemma 3.1], but exploiting this requires a lot of translation of notation and transfer of results from one context to another, so we give a full proof here.
First note that ${{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{\llbracket\xi\rrbracket}$ lies in the block with $2$-core $\sigma$ and weight $v$, and the condition $v\ls w$ guarantees that this is a Rouquier block. Now the branching rule says that ${{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{\llbracket\xi\rrbracket}$ is the sum of ${\llbracket\rho\rrbracket}$ over all pairs of partitions $(\pi,\rho)$ such that $\xi{\ifthenelse{\equal{r}1}{\stackrel{0}\longrightarrow}{\stackrel{0^r}\longrightarrow}}\pi{\ifthenelse{\equal{r}1}{\stackrel{1}\longrightarrow}{\stackrel{1^r}\longrightarrow}}\rho$. This gives (2) straight away, because if $\xi$ is $2$-singular then by \[trpsinrouq\] the length of $\xi$ is at least ${l(\sigma)}+2$, so $\sigma+2\nu\nsupseteq\xi$.
So we are left with (1). Suppose $\xi$ is $2$-regular and write $\xi=\sigma+2\kappa$. If $\nu$ is obtained from $\kappa$ by adding $r$ nodes in distinct rows, then let $\pi$ be the partition obtained from $\xi$ by adding one node at the end of each of these rows. Then (since all the addable nodes of $\xi$ have residue $0$) we have $\xi{\ifthenelse{\equal{r}1}{\stackrel{0}\longrightarrow}{\stackrel{0^r}\longrightarrow}}\pi{\ifthenelse{\equal{r}1}{\stackrel{1}\longrightarrow}{\stackrel{1^r}\longrightarrow}}\sigma+2\nu$, so ${\llbracket\sigma+2\nu\rrbracket}$ occurs in ${{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{\llbracket\xi\rrbracket}$, and clearly occurs once only. Conversely, if we have $\xi{\ifthenelse{\equal{r}1}{\stackrel{0}\longrightarrow}{\stackrel{0^r}\longrightarrow}}\pi{\ifthenelse{\equal{r}1}{\stackrel{1}\longrightarrow}{\stackrel{1^r}\longrightarrow}}{\sigma+2\nu}$, then $\xi\subset\sigma+2\nu$, so $\kappa\subset\nu$. Moreover, the nodes added to $\xi$ to obtain $\pi$ all have the same residue, so lie in different rows, and so $\sigma+2\nu$ differs from $\xi$ in at least $r$ rows. So $\nu$ differs from $\kappa$ in at least $r$ rows, so $\nu$ must be obtained from $\kappa$ by adding $r$ nodes in distinct rows.
Now we prove a corresponding result for spin characters. Recall that we write ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ to mean “either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$” when $\xi\in\cald^-(n)$.
\[inducespin\] Suppose $r\ls v\ls w$ with $v$ even, and $\xi$ is a [$2$-regular partition]{}with [$4$-bar-core]{}$\tau$ and [$4$-bar-weight]{}$v-r$. Suppose $\beta\in\calp(v/2)$.
1. Suppose all the parts of $\xi$ are odd, say $\xi=\tau+4\alpha$ for $\alpha\in\calp((v-r)/2)$. Then ${\left({{\mathrm{f}_{0}}^{(r)}}{{\mathrm{f}_{1}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}={\lane_{r/2}s_\alpha,s_\beta\ran}$.
2. If $\xi$ has at least one even part (and in particular if $r$ is odd), then ${\left({{\mathrm{f}_{0}}^{(r)}}{{\mathrm{f}_{1}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$.
Let $\rho=\tau+4\beta$. By the spin branching rule, ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\rho{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ occurs in ${{\mathrm{f}_{0}}^{(r)}}{{\mathrm{f}_{1}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ if and only if there is $\pi$ such that $\xi{\ifthenelse{\equal{r}1}{\stackrel{0}\Longrightarrow}{\stackrel{0^r}\Longrightarrow}}\pi{\ifthenelse{\equal{r}1}{\stackrel{1}\Longrightarrow}{\stackrel{1^r}\Longrightarrow}}\rho$, i.e. $\xi$ can be obtained from $\rho$ by removing $r$ nodes of [spin residue]{}$1$ followed by $r$ nodes of [spin residue]{}$0$. The form of $\rho$ means that there are two nodes of [spin residue]{}$1$ at the end of each non-empty row of $\rho$; removing both of these nodes from a row then exposes two nodes of [spin residue]{}$0$ at the end of that row (or one, if the row initially only has length $3$). So the only way $\pi$ can have $r$ nodes of [spin residue]{}$0$ that can be removed is if $r$ is even, and the nodes removed from $\rho$ to obtain $\pi$ occur in pairs at the end of $r/2$ different rows; then the only possible way to remove $r$ nodes of [spin residue]{}$0$ from $\pi$ is to remove them in pairs from these same $r/2$ rows. Thus if $\xi{\ifthenelse{\equal{r}1}{\stackrel{0}\Longrightarrow}{\stackrel{0^r}\Longrightarrow}}\pi{\ifthenelse{\equal{r}1}{\stackrel{1}\Longrightarrow}{\stackrel{1^r}\Longrightarrow}}\rho$, then we must have $\rho_i=\xi_i+4$ for $r/2$ different values of $i$, with $\rho_i=\xi_i$ for all other values of $i$. Hence $\xi=\tau+4\alpha$, where $\alpha$ is obtained from $\beta$ by removing $r/2$ nodes from distinct rows.
This is sufficient to prove the , except that in the case where $\beta$ is obtained from $\alpha$ by adding nodes in distinct rows we must show that the coefficient of ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\rho{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ in ${{\mathrm{f}_{0}}^{(r)}}{{\mathrm{f}_{1}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ is $1$. But this follows from the induction version of \[spinbranchpower\].
Now we can prove \[induceproj\].
We use induction on $w$. In the case $w=0$, we can define $\psi^\varnothing$ to be the unique indecomposable projective character in the block with $2$-core $\sigma$ and weight $0$, i.e. ${\operatorname{prj}(\sigma)}$. By \[jreg,spinreg\], this equals ${\llbracket\sigma\rrbracket}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
Now suppose $w\gs1$, and let $\rho$ be the partition obtained by removing the last non-empty column (of length $r$, say) from $\mu$, so that $e_{\mu'}=e_re_{\rho'}$. By the inductive hypothesis the holds with $\rho$ in place of $\mu$, and we define $\psi^\mu={{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}\psi^\rho$. We must check that the conditions given in the hold for $\psi^\mu$, given that they hold for $\psi^\rho$.
Suppose $\nu\in\calp(w)$. By \[induceord\] we have $$\begin{aligned}
{\left(\psi^\mu\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}&={\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}\psi^{\rho}\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}\\
&=\sum_{\kappa\in\calp(w-r)}{\left(\psi^{\rho}\!:\!{\llbracket\sigma+2\kappa\rrbracket}\right)}{\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{\llbracket\sigma+2\kappa\rrbracket}\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}\\
&=\sum_{\kappa\in\calp(w-r)}{\lane_{\rho'},s_\kappa\ran}{\lane_rs_\kappa,s_\nu\ran}\\
&={\lane_re_{\rho'},s_\nu\ran}\\
&={\lane_{\mu'},s_\nu\ran}.\end{aligned}$$
Now suppose $w$ is even and $\beta\in\calp(w/2)$. If $r$ is odd, then by \[inducespin\](2) we have $${\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}={\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}\psi^{\rho}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0.$$ So suppose instead that $r$ is even. If any of the column lengths of $\rho$ are odd, then by the inductive hypothesis ${\left(\psi^\rho\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\gamma{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$ for every $\gamma\in\calp((w-r)/2)$, so by \[inducespin\](2) ${\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$. On the other hand, if the column lengths of $\mu$ are all even, then write $\mu={\alpha\sqcup\alpha}$, so that $\rho={\delta\sqcup\delta}$ where $\delta$ is obtained by removing the last column from $\alpha$. Then $$\begin{aligned}
{\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&={\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}\psi^{\rho}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\\
&=\sum_{\gamma\in\calp((w-r)/2)}{\left(\psi^{\rho}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\gamma{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}{\left({{\mathrm{f}_{1}}^{(r)}}{{\mathrm{f}_{0}}^{(r)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\gamma{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\\
&=\sum_{\gamma\in\calp((w-r)/2)}{\lane_{\delta'},s_\gamma\ran}{\lane_{r/2}s_\gamma,s_\beta\ran}\\
&={\lane_{\alpha'},s_\beta\ran}.\qedhere\end{aligned}$$
We take $w=4$ and $\sigma=(3,2,1)$, so that $\tau=(5,1)$. We will show how to construct the characters $\psi^{(2^2)}$ and $\psi^{(2,1^2)}$. In both cases we start from $\psi^\varnothing={\llbracket3,2,1\rrbracket}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
Applying ${{\mathrm{f}_{0}}^{(2)}}{{\mathrm{f}_{1}}^{(2)}}$ to $\psi^\varnothing$, we obtain $$\begin{aligned}
\psi^{(1^2)}&={\llbracket5,4,1\rrbracket}+{\llbracket5,1^4\rrbracket}+{\llbracket3,2^3,1\rrbracket}\\
&\phantom=\ +{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}9,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5,4,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5,4,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-.\\
\intertext{
Applying ${{\mathrm{f}_{0}}^{(2)}}{{\mathrm{f}_{1}}^{(2)}}$ again, we obtain
}
\psi^{(2^2)}&={\llbracket7,6,1\rrbracket}+{\llbracket7,4,3\rrbracket}+{\llbracket5,4,3,2\rrbracket}+\chi_1\\
&\phantom=\ +{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}13,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}9,5{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}+\chi_2,\end{aligned}$$ where $\chi_1$ is a sum of characters of the form ${\llbracket\nu\rrbracket}$ with $\nu$ $2$-singular, and $\chi_2$ is a sum of characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ for $\xi$ not of the form $\tau+4\beta$ for any $\beta$. Note that the characters ${\llbracket5,1^4\rrbracket}$ and ${\llbracket3,2^3,1\rrbracket}$ do not contribute any terms ${\llbracket\nu\rrbracket}$ with $\nu$ $2$-regular, and the characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}5,4,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm$ do not contribute any terms ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
Now we look at $\mu=(2,1^2)$. Starting with $\psi^\varnothing$ and applying ${{\mathrm{f}_{0}}^{(3)}}{{\mathrm{f}_{1}}^{(3)}}$, we obtain $$\begin{aligned}
\psi^{(1^3)}&={\llbracket5,4,3\rrbracket}+{\llbracket5,4,1^3\rrbracket}+{\llbracket5,2^3,1\rrbracket}+{\llbracket3^3,2,1\rrbracket}\\
&\phantom=\ +{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}9,2,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}9,2,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-+{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}6,5,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}6,5,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-.\\
\intertext{
Applying ${\mathrm{f}_{0}}{\mathrm{f}_{1}}$, we obtain
}
\psi^{(2,1^2)}&={\llbracket7,4,3\rrbracket}+{\llbracket5,4,3,2\rrbracket}+\chi,\end{aligned}$$ where $\chi$ is a sum of characters of the form ${\llbracket\nu\rrbracket}$ with $\nu$ $2$-singular, or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ with $\xi$ not of the form $\tau+4\beta$.
Next we want to prove a dual result to \[induceproj\], using virtual projective characters.
\[induceprojdual\] Suppose $\la\in\calp(w)$. There is a virtual projective character $\upsilon^\la$ with the following properties.
1. ${\left(\upsilon^\la\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}={\lanh_\la,s_\nu\ran}$ for any $\nu\in\calp(w)$.
2. If $\la_i$ is even for every $i$, say $\la=2\gamma$, then ${\left(\upsilon^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=(-1)^{w/2}{\lanh_\gamma,s_\beta\ran}$ for every $\beta\in\calp(w/2)$.
3. If $w$ is even but $\la_i$ is odd for some $i$, then ${\left(\upsilon^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$ for every $\beta\in\calp(w/2)$.
Recall the coefficients ${\spadesuit_{\la,\mu}}$ defined by $h_\la=\sum_\mu{\spadesuit_{\la,\mu}} e_\mu$, and define $$\upsilon^\la=\sum_{\mu\in\calp(w)}{\spadesuit_{\la,\mu}}\psi^{\mu'}.$$ Now for $\nu\in\calp(w)$ we have $$\begin{aligned}
{\left(\upsilon^\la\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}&={\lan\textstyle\sum_\mu{\spadesuit_{\la,\mu}} e_\mu,s_\nu\ran}\\
&={\lanh_\la,s_\nu\ran}.\end{aligned}$$
Now suppose $w$ is even. Then for $\beta\in\calp(w/2)$ we have $${\left(\upsilon^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=\sum_{\alpha\in\calp(w/2)}{\spadesuit_{\la,2\alpha}}{\lane_\alpha,s_\beta\ran}$$ by \[induceproj\]. If $\la_i$ is odd for some $i$, then by \[simpleep\] ${\spadesuit_{\la,2\alpha}}=0$ for every $\alpha$, so that ${\left(\upsilon^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$. On the other hand, if $\la=2\gamma$, then ${\spadesuit_{\la,2\alpha}}=(-1)^{w/2}{\spadesuit_{\gamma,\alpha}}$ by \[ekdoublep\], so that $$\begin{aligned}
{\left(\upsilon^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&=(-1)^{w/2}{\lan\textstyle\sum_\alpha{\spadesuit_{\gamma,\alpha}} e_\alpha,s_\beta\ran}\\
&=(-1)^{w/2}{\lanh_\gamma,s_\beta\ran}.\qedhere\end{aligned}$$
We complete this subsection by examining a third set of virtual projective characters with a nice symmetry property.
By \[trpsinrouq\], the indecomposable projective characters are the characters ${{\operatorname{prj}(\sigma+2\mu)}}$ for $\mu\in\calp(w)$; by Brauer reciprocity these satisfy ${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}=D_{(\sigma+2\la)(\sigma+2\mu)}$ for all $\la$, and hence by the first statement in \[snrouqdec\] ${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}=D_{\la\mu}$ for all $\la,\mu\in\calp(w)$. The invertibility of the decomposition matrix of the Schur algebra then implies that for each $\mu$ there is a unique virtual projective character $\omega^\mu$ in $B$ such that ${\left(\omega^\mu\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}=\delta_{\la\mu}$ for each $\la$. In fact, we can write $$\omega^\mu=\sum_{\nu\in\calp(w)}D_{\nu\mu}{^{-1}}{{\operatorname{prj}(\sigma+2\nu)}},$$ where (as set out in \[irrdecsec\]) $D$ is the decomposition matrix of the Schur algebra in characteristic $2$, i.e. $D_{\la\mu}={[\Delta(\la):L(\mu)]}$. Our aim is to study the multiplicities of the spin characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ in the $\omega^\mu$ when $w$ is even, which will enable us to deduce information about the decomposition numbers.
The triangularity of the projective characters $\psi^\mu$ (i.e. the second statement in \[induceproj\](1)) means that the set $\lset{\psi^\mu}{\mu\in\calp(w)}$ is a basis for the space of virtual projective characters in $B$. Hence each $\omega^\la$ can be written uniquely as a linear combination of the $\psi^\mu$. So define coefficients $a_{\la\mu}$ by $\omega^\la=\sum_{\mu\in\calp(w)} a_{\la\mu}\psi^\mu$. The next result shows that these coefficients can also be used to write $\omega^\la$ in terms of the virtual characters $\upsilon^\mu$.
\[omegaups\] For any $\la\in\calp(w)$, $$\omega^\la=\sum_{\mu\in\calp(w)}a_{\la'\mu'}\upsilon^\mu.$$
For any $\nu\in\calp(w)$ we have $$\begin{aligned}
{\left(\sum_{\mu\in\calp(w)}a_{\la'\mu'}\upsilon^\mu\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}&=\sum_{\mu\in\calp(w)}a_{\la'\mu'}{\lanh_\mu,s_\nu\ran}\\
&=\sum_{\mu\in\calp(w)}a_{\la'\mu'}{\lane_\mu,s_{\nu'}\ran}\\
&=\sum_{\mu\in\calp(w)}a_{\la'\mu}{\lane_{\mu'},s_{\nu'}\ran}\\
&=\sum_{\mu\in\calp(w)}a_{\la'\mu}{\left(\psi^\mu\!:\!{\llbracket\sigma+2\nu'\rrbracket}\right)}\\
&={\left(\omega^{\la'}\!:\!{\llbracket\sigma+2\nu'\rrbracket}\right)}\\
&=\delta_{\la'\nu'}\\
&=\delta_{\la\nu}\\
&={\left(\omega^\la\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}.\end{aligned}$$ Now the uniqueness property defining $\omega^\la$ gives the result.
Now we can deduce the following symmetry property of the coefficients ${\left(\omega^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$.
\[virtualsym\] Suppose $w$ is even, $\la\in\calp(w)$ and $\beta\in\calp(w/2)$. Then $${\left(\omega^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=(-1)^{w/2}{\left(\omega^{\la'}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta'{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}.$$
On the one hand, we have $$\begin{aligned}
{\left(\omega^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&=\sum_{\mu\in\calp(w)}a_{\la\mu}{\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\\
&=\sum_{\alpha\in\calp(w/2)}a_{\la({\alpha\sqcup\alpha})}{\lane_{\alpha'},s_\beta\ran}
\\
\intertext{by \cref{induceproj}(2,3). On the other hand,}
{\left(\omega^{\la'}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta'{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}&=\sum_{\mu\in\calp(w)}a_{\la\mu'}{\left(\upsilon^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta'{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\tag*{by \cref{omegaups}}\\
&=\sum_{\gamma\in\calp(w/2)}a_{\la(2\gamma)'}(-1)^{w/2}{\lanh_\gamma,s_{\beta'}\ran}\tag*{by \cref{induceprojdual}}\\
&=(-1)^{w/2}\sum_{\gamma\in\calp(w/2)}a_{\la({\gamma'\sqcup\gamma'})}{\lane_\gamma,s_\beta\ran}.\end{aligned}$$ Replacing $\gamma$ with $\alpha'$ gives the result.
Some decomposition numbers for Rouquier blocks of even weight {#somedecevensec}
-------------------------------------------------------------
We now use the results of the previous subsection to find some explicit decomposition numbers for spin characters in Rouquier blocks. We retain the assumptions set out at the start of \[projcharsec\], and *we assume throughout \[somedecevensec\] that $w$ is even*. Our aim is to study the decomposition numbers ${D^{\mathsf{spn}}_{(\tau+4\alpha)(\sigma+2\mu)}}$, for $\alpha\in\calp(w/2)$ and $\mu\in\calp(w)$, and to express them in terms of the adjustment matrix of the Schur algebra.
We start by defining several matrices. Let $E$ be the matrix with rows indexed by $\calp(w/2)$ and columns by $\calp(w)$, with $$\begin{aligned}
E_{\alpha\mu}&={D^{\mathsf{spn}}_{(\tau+4\alpha)(\sigma+2\mu)}}.
\\
\intertext{Now define a matrix $J$ with the same indexing as $E$, with}
J_{\alpha\mu}&=
\begin{cases}
1&(\mu={\alpha\sqcup\alpha})\\
0&(\text{otherwise}).
\end{cases}\end{aligned}$$ Let $D$ be the decomposition matrix of the Schur algebra ${\cals(w)}$ in characteristic $2$, i.e. $D_{\la\mu}=[\Delta(\la):L(\mu)]$. Then (as explained in \[irrdecsec\]) $D={\mathring D}A$, where ${\mathring D}$ is the decomposition matrix of the $(-1)$-Schur algebra ${\cals^\circ(w)}$ and $A$ is its adjustment matrix. Now we can state the main result of this section.
\[allodd\] Suppose $B$ is a Rouquier block of even weight $w$, and define $E,J,A$ as above. Then $E=JA$.
For example, if $w=4$ then we have the following matrices, and we see that \[allodd\] is true in this case. $$\begin{aligned}
2
E&=
\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(2,1^2)}&{\rotatebox{90}{$(4)$}}&{\rotatebox{90}{$(3,1)$}}&{\rotatebox{90}{$(2^2)$}}&{\rotatebox{90}{$(2,1^2)$}}&{\rotatebox{90}{$(1^4)$}}\\\hline
(2)&\cdot&\cdot&1&\cdot&\cdot\\
(1^2)&\cdot&\cdot&1&\cdot&1
\end{array}
&\qquad
J&=
\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(2,1^2)}&{\rotatebox{90}{$(4)$}}&{\rotatebox{90}{$(3,1)$}}&{\rotatebox{90}{$(2^2)$}}&{\rotatebox{90}{$(2,1^2)$}}&{\rotatebox{90}{$(1^4)$}}\\\hline
(2)&\cdot&\cdot&1&\cdot&\cdot\\
(1^2)&\cdot&\cdot&\cdot&\cdot&1
\end{array}
\\[6pt]
D&=\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
&{\rotatebox{90}{$(4)$}}&{\rotatebox{90}{$(3,1)$}}&{\rotatebox{90}{$(2^2)$}}&{\rotatebox{90}{$(2,1^2)$}}&{\rotatebox{90}{$(1^4)$}}\\\hline
(4)&1&\cdot&\cdot&\cdot&\cdot\\
(3,1)&1&1&\cdot&\cdot&\cdot\\
(2^2)&\cdot&1&1&\cdot&\cdot\\
(2,1^2)&1&1&1&1&\cdot\\
(1^4)&1&\cdot&1&1&1
\end{array}&\qquad
{\mathring D}&=\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
&{\rotatebox{90}{$(4)$}}&{\rotatebox{90}{$(3,1)$}}&{\rotatebox{90}{$(2^2)$}}&{\rotatebox{90}{$(2,1^2)$}}&{\rotatebox{90}{$(1^4)$}}\\\hline
(4)&1&\cdot&\cdot&\cdot&\cdot\\
(3,1)&1&1&\cdot&\cdot&\cdot\\
(2^2)&\cdot&1&1&\cdot&\cdot\\
(2,1^2)&1&1&1&1&\cdot\\
(1^4)&1&\cdot&\cdot&1&1
\end{array}
\\[6pt]
A&=\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
&{\rotatebox{90}{$(4)$}}&{\rotatebox{90}{$(3,1)$}}&{\rotatebox{90}{$(2^2)$}}&{\rotatebox{90}{$(2,1^2)$}}&{\rotatebox{90}{$(1^4)$}}\\\hline
(4)&1&\cdot&\cdot&\cdot&\cdot\\
(3,1)&\cdot&1&\cdot&\cdot&\cdot\\
(2^2)&\cdot&\cdot&1&\cdot&\cdot\\
(2,1^2)&\cdot&\cdot&\cdot&1&\cdot\\
(1^4)&\cdot&\cdot&1&\cdot&1
\end{array}&&\end{aligned}$$
\[induceproj\] gives us the following information about the matrix $E$, which shows that \[allodd\] is true up to a triangular adjustment.
\[triang\] Suppose $\beta\in\calp(w/2)$ and $\mu\in\calp(w)$.
1. If $\mu$ has at least one column of odd length, then $E_{\beta\mu}=0$.
2. If the columns of $\mu$ all have even length, say $\mu={\alpha\sqcup\alpha}$, then $E_{\alpha\mu}=1$ while $E_{\beta\mu}=0$ unless $\alpha\dom\beta$.
$E_{\beta\mu}$ is the coefficient ${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$. So we examine the projective characters ${{\operatorname{prj}(\sigma+2\la)}}$, for $\la\in\calp(w)$.
By the triangularity of the decomposition matrix of the symmetric group we have $${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{\llbracket\sigma+2\mu\rrbracket}\right)}=1,\qquad{\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}=0\text{ for }\mu\ndom\nu.$$ By \[induceproj\] the same property holds if we replace ${{\operatorname{prj}(\sigma+2\mu)}}$ with the projective character $\psi^\mu$ introduced in that . Since $\psi^\mu$ is a linear combination of the characters ${{\operatorname{prj}(\sigma+2\la)}}$ with non-negative coefficients, we can obtain ${{\operatorname{prj}(\sigma+2\mu)}}$ from $\psi^\mu$ by subtracting multiples of the characters ${{\operatorname{prj}(\sigma+2\nu)}}$ for $\nu\domsby\mu$. That is, $${{\operatorname{prj}(\sigma+2\mu)}}=\psi^\mu-\sum_{\nu\domsby\mu}x_\nu{{\operatorname{prj}(\sigma+2\nu)}}$$ with each $x_\nu$ a non-negative integer. In particular, for any $\beta\in\calp(w/2)$ $${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\ls{\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)},$$ so (by \[induceproj\]) ${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\beta{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=0$ if $\mu$ has at least one column of odd length or $\mu={\alpha\sqcup\alpha}$ with $\alpha\ndom\beta$. Hence if $\mu={\alpha\sqcup\alpha}$, the coefficient ${\left({{\operatorname{prj}(\sigma+2\nu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$ is zero for any $\nu\domsby\mu$, and so $${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}={\left(\psi^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=1.\qedhere$$
Now we note that the same property holds for the matrix $EA{^{-1}}$.
\[ehv\] Suppose $\beta\in\calp(w/2)$ and $\mu\in\calp(w)$.
1. If $\mu$ has at least one column of odd length, then $(EA{^{-1}})_{\beta\mu}=0$.
2. If the columns of $\mu$ all have even length, say $\mu={\alpha\sqcup\alpha}$, then $(EA{^{-1}})_{\alpha\mu}=1$ while $(EA{^{-1}})_{\beta\mu}=0$ unless $\alpha\dom\beta$.
We have $(EA{^{-1}})_{\beta\mu}=\sum_{\la\in\calp(w)}E_{\beta\la}A{^{-1}}_{\la\mu}$, and by \[triang\](1) we may restrict the range of summation to those $\la$ with all columns of even length; that is, $$(EA{^{-1}})_{\beta\mu}=\sum_{\rho\in\calp(w/2)}E_{\beta({\rho\sqcup\rho})}A{^{-1}}_{({\rho\sqcup\rho})\mu}.$$ If $\mu$ has a column of odd length, then by \[adj\] $A{^{-1}}_{({\rho\sqcup\rho})\mu}=0$ for every $\rho$, which gives (1). If $\mu={\alpha\sqcup\alpha}$, then by \[adj\] $$(EA{^{-1}})_{\beta\mu}=\sum_{\rho\in\calp(w/2)}E_{\beta({\rho\sqcup\rho})}A{^{-1}}_{({\rho\sqcup\rho})({\alpha\sqcup\alpha})}=\sum_{\rho\in\calp(w/2)}E_{\beta({\rho\sqcup\rho})}K{^{-1}}_{\rho\alpha},$$ where $K$ is the decomposition matrix of the Schur algebra ${\cals(w/2)}$. The term $E_{\beta({\rho\sqcup\rho})}$ is non-zero only if $\rho\dom\beta$ (and equals $1$ when $\rho=\beta$), while the term $K{^{-1}}_{\rho\alpha}$ is non-zero only if $\alpha\dom\rho$ (and is $1$ if $\alpha=\rho$). The result follows.
\[ehv\] may be alternatively phrased as follows: there is a square matrix $T$ with rows and columns indexed by $\calp(w/2)$, such that $EA{^{-1}}=TJ$. Furthermore, $T$ is lower unitriangular in the sense that $T_{\alpha\alpha}=1$ for each $\alpha$, while $T_{\alpha\beta}=0$ for $\beta\ndom\alpha$.
Our aim is to prove that $T$ is the identity matrix. To do this, we consider the matrices $ED{^{-1}}$ and $J{\mathring D}{^{-1}}$, and show that they both satisfy a certain symmetry property. Given a matrix $B$ with rows indexed by $\calp(w/2)$ and columns by $\calp(w)$, we say that $B$ is *[conjugate-symmetric]{}* if $$B_{\alpha'\la'}=(-1)^{w/2}B_{\alpha\la}$$ for every $\alpha\in\calp(w/2)$ and $\la\in\calp(w)$.
\[jfunny\] The matrix $J{\mathring D}{^{-1}}$ is [conjugate-symmetric]{}.
The definition of $J$ means that $(J{\mathring D}{^{-1}})_{\alpha\mu}={\mathring D}_{({\alpha\sqcup\alpha})\mu}{^{-1}}$, and this is given explicitly by \[steinberg\] as $(-1)^{w/2}\epsilon(\mu)\kappa(\alpha,\mu)$, where $\kappa(\alpha,\mu)$ is defined in \[inversesec\]. Now $\mu$ and $\mu'$ have the same $2$-core, and if this $2$-core if not $\varnothing$ then $\kappa(\alpha,\mu)=\kappa(\alpha',\mu')=0$, so that $(J{\mathring D}{^{-1}})_{\alpha\mu}=(J{\mathring D}{^{-1}})_{\alpha'\mu'}=0$. So assume that $\mu$ and $\mu'$ have empty $2$-core, and therefore have $2$-weight $w/2$. Then by \[parityconj\] we have $\epsilon(\mu)\epsilon(\mu')=(-1)^{w/2}$, so we just need to show that $\kappa(\alpha,\mu)=\kappa(\alpha',\mu')$ for any $\alpha$. It is a standard property of Littlewood–Richardson coefficients that $a^\alpha_{\beta\gamma}=a^{\alpha'}_{\gamma'\beta'}$ for any $\alpha,\beta,\gamma$, so the result follows from \[quoconj\].
\[efunny\] The matrix $ED{^{-1}}$ is [conjugate-symmetric]{}.
Recall that $E_{\alpha\mu}$ is the multiplicity ${\left({{\operatorname{prj}(\sigma+2\mu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$. Recall also that the virtual projective characters $\omega^\mu$ from §\[projcharsec\] are given by $$\omega^\mu=\sum_{\nu\in\calp(w)}D_{\nu\mu}{^{-1}}{{\operatorname{prj}(\sigma+2\nu)}}.$$ This means that $(ED{^{-1}})_{\alpha\mu}={\left(\omega^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$. Now \[virtualsym\] gives the required result.
For example, in the case $w=4$ the matrix $J{\mathring D}{^{-1}}=ED{^{-1}}$ is the following. $$\begin{array}{r|ccccc}
&{\rotatebox{90}{$(4)$}}&{\rotatebox{90}{$(3,1)$}}&{\rotatebox{90}{$(2^2)$}}&{\rotatebox{90}{$(2,1^2)$}}&{\rotatebox{90}{$(1^4)$}}\\\hline
(2)&1&-1&1&\cdot&\cdot\\
(1^2)&\cdot&\cdot&1&-1&1
\end{array}$$
Now we can complete the proof of \[allodd\].
We have seen that there is a matrix $T$ such that $EA{^{-1}}=TJ$, and that $T$ is unitriangular in the sense that $T_{\alpha\alpha}=1$ and $T_{\alpha\beta}=0$ if $\alpha\ndomby\beta$. So to prove that $T$ is the identity matrix, we just need to prove that $T_{\alpha\beta}=0$ for all $\alpha\domsby\beta$, and we do this by induction on $\beta$ using the dominance order. So assume that we have proved that $T_{\alpha\gamma}=0$ whenever $\gamma\doms\alpha,\beta$.
Multiplying the equation $EA{^{-1}}=TJ$ by ${\mathring D}{^{-1}}$, we obtain $ED{^{-1}}=TJ{\mathring D}{^{-1}}$. Consider the column of $J{\mathring D}{^{-1}}$ labelled by the partition ${\beta'\sqcup\beta'}$. For any $\gamma$ we have $(J{\mathring D}{^{-1}})_{\gamma({\beta'\sqcup\beta'})}={\mathring D}_{({\gamma\sqcup\gamma})({\beta'\sqcup\beta'})}{^{-1}}$, and the triangularity of ${\mathring D}$ means that $(J{\mathring D}{^{-1}})_{\gamma({\beta'\sqcup\beta'})}=1$ if $\gamma=\beta'$, and $0$ if $\gamma\ndomby\beta'$. The triangularity of $T$ then means that the same is true with $J{\mathring D}{^{-1}}$ replaced by $TJ{\mathring D}{^{-1}}=ED{^{-1}}$.
Using the fact that both $J{\mathring D}{^{-1}}$ and $ED{^{-1}}$ are [conjugate-symmetric]{}(and the fact that the dominance order is reversed by conjugating partitions) we get $$(J{\mathring D}{^{-1}})_{\alpha(2\beta)}=
\begin{cases}
(-1)^{w/2}&\text{if }\alpha=\beta\\
0&\text{if }\alpha\ndom\beta
\end{cases}$$ and the same with $ED{^{-1}}$ in place of $J{\mathring D}{^{-1}}$. Hence for $\alpha\domsby\beta$ we have $$\begin{aligned}
0&=(ED{^{-1}})_{\alpha(2\beta)}\\
&=(TJ{\mathring D}{^{-1}})_{\alpha(2\beta)}\\
&=\sum_\gamma T_{\alpha\gamma}(J{\mathring D}{^{-1}})_{\gamma(2\beta)}.\end{aligned}$$ The term $(J{\mathring D}{^{-1}})_{\gamma(2\beta)}$ is zero unless $\gamma\dom\beta$. If $\gamma\doms\beta$ then by our induction hypothesis $T_{\alpha\gamma}=0$, so we only need to consider the term with $\gamma=\beta$, and we obtain $0=(-1)^{w/2}T_{\alpha\beta}$.
Hence $T$ is the identity matrix, and therefore $E=JA$.
Some decomposition numbers for Rouquier blocks of odd weight {#somedecoddsec}
------------------------------------------------------------
We now prove a similar result to \[allodd\] for the case where $w$ is odd. We retain the assumptions set out at the start of \[projcharsec\], and *we assume throughout \[somedecoddsec\] that $w$ is odd*.
As before, we let $D$ be the decomposition matrix of the Schur algebra ${\cals(w)}$, and factorise $D$ as ${\mathring D}A$, where ${\mathring D}$ is the decomposition matrix of the corresponding $(-1)$-Schur algebra and $A$ is the adjustment matrix. We now define $E$ to be the matrix with rows indexed by $\calp((w-1)/2)$ and columns by $\calp(w)$, and $$\begin{aligned}
E_{\alpha\mu}&={D^{\mathsf{spn}}_{(\tau+4\alpha\sqcup(2))(\sigma+2\mu)}}.
\\
\intertext{$J$ is defined to have the same indexing as $E$, with}
J_{\alpha\mu}&=
\begin{cases}
1&(\mu={\alpha\sqcup\alpha}\sqcup(1))\\
0&(\text{otherwise}).
\end{cases}\end{aligned}$$
Now we have the following.
\[allodd2\] Suppose $B$ is a Rouquier block of odd weight $w$, and define $E,J,A$ as above. Then $E=JA$.
For example, taking $w=5$, we have the following matrices.
[ $$\begin{aligned}
2
E&=
\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(3,1^2)}&{\rotatebox{90}{$(5)$}}&{\rotatebox{90}{$(4,1)$}}&{\rotatebox{90}{$(3,2)$}}&{\rotatebox{90}{$(3,1^2)$}}&{\rotatebox{90}{$(2^2,1)$}}&{\rotatebox{90}{$(2,1^3)$}}&{\rotatebox{90}{$(1^5)$}}\\\hline
(2)&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot\\
(1^2)&\cdot&\cdot&\cdot&\cdot&1&\cdot&1
\end{array}
&\qquad
J&=
\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(3,1^2)}&{\rotatebox{90}{$(5)$}}&{\rotatebox{90}{$(4,1)$}}&{\rotatebox{90}{$(3,2)$}}&{\rotatebox{90}{$(3,1^2)$}}&{\rotatebox{90}{$(2^2,1)$}}&{\rotatebox{90}{$(2,1^3)$}}&{\rotatebox{90}{$(1^5)$}}\\\hline
(2)&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot\\
(1^2)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1
\end{array}
\\[6pt]
D&=\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(3,1^2)}&{\rotatebox{90}{$(5)$}}&{\rotatebox{90}{$(4,1)$}}&{\rotatebox{90}{$(3,2)$}}&{\rotatebox{90}{$(3,1^2)$}}&{\rotatebox{90}{$(2^2,1)$}}&{\rotatebox{90}{$(2,1^3)$}}&{\rotatebox{90}{$(1^5)$}}\\\hline
(5)&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
(4,1)&\cdot&1&\cdot&\cdot&\cdot&\cdot&\cdot\\
(3,2)&1&\cdot&1&\cdot&\cdot&\cdot&\cdot\\
(3,1^2)&2&\cdot&1&1&\cdot&\cdot&\cdot\\
(2^2,1)&1&\cdot&1&1&1&\cdot&\cdot\\
(2,1^3)&\cdot&1&\cdot&\cdot&\cdot&1&\cdot\\
(1^5)&1&\cdot&\cdot&1&1&\cdot&1
\end{array}&\qquad
{\mathring D}&=\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(3,1^2)}&{\rotatebox{90}{$(5)$}}&{\rotatebox{90}{$(4,1)$}}&{\rotatebox{90}{$(3,2)$}}&{\rotatebox{90}{$(3,1^2)$}}&{\rotatebox{90}{$(2^2,1)$}}&{\rotatebox{90}{$(2,1^3)$}}&{\rotatebox{90}{$(1^5)$}}\\\hline
(5)&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
(4,1)&\cdot&1&\cdot&\cdot&\cdot&\cdot&\cdot\\
(3,2)&\cdot&\cdot&1&\cdot&\cdot&\cdot&\cdot\\
(3,1^2)&1&\cdot&1&1&\cdot&\cdot&\cdot\\
(2^2,1)&\cdot&\cdot&1&1&1&\cdot&\cdot\\
(2,1^3)&\cdot&1&\cdot&\cdot&\cdot&1&\cdot\\
(1^5)&1&\cdot&\cdot&1&\cdot&\cdot&1
\end{array}\\[6pt]
A&=\begin{array}{@{}r@{\,}|@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
\phantom{(3,1^2)}&{\rotatebox{90}{$(5)$}}&{\rotatebox{90}{$(4,1)$}}&{\rotatebox{90}{$(3,2)$}}&{\rotatebox{90}{$(3,1^2)$}}&{\rotatebox{90}{$(2^2,1)$}}&{\rotatebox{90}{$(2,1^3)$}}&{\rotatebox{90}{$(1^5)$}}\\\hline
(5)&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
(4,1)&\cdot&1&\cdot&\cdot&\cdot&\cdot&\cdot\\
(3,2)&1&\cdot&1&\cdot&\cdot&\cdot&\cdot\\
(3,1^2)&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot\\
(2^2,1)&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot\\
(2,1^3)&\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot\\
(1^5)&\cdot&\cdot&\cdot&\cdot&1&\cdot&1
\end{array}&&\end{aligned}$$ ]{}
To prove \[allodd2\], we use restriction, comparing $B$ with the block $B_-$ of weight $w-1$ and $2$-core $\sigma$ and exploiting \[allodd\]. We define ${\mathrm{e}_{\bullet}}$ to be:
- ${\mathrm{e}_{0}}{\mathrm{e}_{1}}$ if $\sigma$ is one of $\varnothing,(2,1),(4,3,2,1),\dots$ (and hence $\tau$ is one of $\varnothing,(3),(7,3),\dots$);
- ${\mathrm{e}_{1}}{\mathrm{e}_{0}}$ if $\sigma$ is one of $(1),(3,2,1),(5,4,3,2,1),\dots$ (and hence $\tau$ is one of $(1),(5,1),(9,5,1),\dots$).
As in the even-weight case, for each $\la\in\calp(w)$ we define $\omega^\la$ to be the unique virtual projective character in $B$ for which ${\left(\omega^\la\!:\!{\llbracket\sigma+2\mu\rrbracket}\right)}=\delta_{\la\mu}$.
Now consider applying ${\mathrm{e}_{\bullet}}$ to a character in $B$, and examining the coefficients of ${\llbracket\sigma+2\la\rrbracket}$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ in this restriction.
\[ebullord\] Suppose $\la\in\calp(w-1)$ and $\xi\in\calp(n)$ with ${\llbracket\xi\rrbracket}$ in $B$. Then ${\left({\mathrm{e}_{\bullet}}{\llbracket\xi\rrbracket}\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}$ equals: $$\begin{cases}
1&\text{if $\xi=\sigma+2\mu$ where $\mu$ is obtained by adding a node to $\la$;}\\
1&\text{if $\xi=\sigma+2\la\sqcup(1^2)$;}\\
0&\text{otherwise.}
\end{cases}$$
This is a simple application of the branching rule. Assume that ${\mathrm{e}_{\bullet}}={\mathrm{e}_{0}}{\mathrm{e}_{1}}$ (the other case is very similar). Then by the branching rule ${\left({\mathrm{e}_{\bullet}}{\llbracket\xi\rrbracket}\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}$ equals $1$ if there is a partition $\pi$ with $\sigma+2\la{\ifthenelse{\equal{1}1}{\stackrel{0}\longrightarrow}{\stackrel{0^1}\longrightarrow}}\pi{\ifthenelse{\equal{1}1}{\stackrel{1}\longrightarrow}{\stackrel{1^1}\longrightarrow}}\xi$, and $0$ otherwise. Since ${l(\sigma)}\gs w-1\gs{l(\la)}$, all the addable nodes of $\sigma+2\la$ have residue $0$, so if $\pi$ exists then the two nodes added to $\sigma+2\la$ to obtain $\xi$ must be adjacent. If these nodes are both in the same row, say row $i$, then $\xi=\sigma+2\mu$, where $\mu$ is obtained from $\la$ by adding a node in row $i$. On the other hand, if the added nodes are both in the same column, then (since $\sigma+2\la$ is $2$-regular) this must be column $1$, and hence $\xi=\sigma+2\la\sqcup(1^2)$.
\[ebullomega\] Suppose $\mu\in\calp(w)$ and $\la\in\calp(w-1)$. Then ${\left({\mathrm{e}_{\bullet}}\omega^\mu\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}$ equals $2$ if $\la\subset\mu$, and $0$ otherwise. Hence ${\mathrm{e}_{\bullet}}\omega^\mu=2\sum_\la\omega^\la$, summing over those partitions $\la$ obtained by removing a node from $\mu$.
To apply \[ebullord\], we need to know the multiplicities ${\left(\omega^\mu\!:\!{\llbracket\sigma+2\la\sqcup(1^2)\rrbracket}\right)}$. We have $$\begin{aligned}
{\left(\omega^\mu\!:\!{\llbracket\sigma+2\la\sqcup(1^2)\rrbracket}\right)}&=\sum_{\nu\in\calp(w)}D{^{-1}}_{\nu\mu}{\left({{\operatorname{prj}(\sigma+2\nu)}}\!:\!{\llbracket\sigma+2\la\sqcup(1^2)\rrbracket}\right)}\\
&=\sum_{\nu\in\calp(w)}A{^{-1}}_{(\sigma+2\nu)(\sigma+2\mu)}D_{(\sigma+2\la\sqcup(1^2))(\sigma+2\nu)}\tag*{by \cref{snrouqdec}}\\
&=(DA{^{-1}})_{(\sigma+2\la\sqcup(1^2))(\sigma+2\mu)}\\
&={\mathring D}_{(\sigma+2\la\sqcup(1^2))(\sigma+2\mu)},\end{aligned}$$ and by \[heckerouqdec\](2) this equals $1$ if $\la\subset\mu$ and $0$ otherwise.
Now we can apply \[ebullord\]: $$\begin{aligned}
{\left({\mathrm{e}_{\bullet}}\omega^\mu\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}&=\sum_\xi{\left(\omega^\mu\!:\!{\llbracket\xi\rrbracket}\right)}{\left({\mathrm{e}_{\bullet}}{\llbracket\xi\rrbracket}\!:\!{\llbracket\sigma+2\la\rrbracket}\right)}\\
&={\left(\omega^\mu\!:\!{\llbracket\sigma+2\la\sqcup(1^2)\rrbracket}\right)}+\sum_{\substack{\nu\in\calp(w)\\\nu\supset\la}}{\left(\omega^\mu\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}\tag*{by \cref{ebullord}.}\end{aligned}$$ The first term is $1$ if $\la\subset\mu$ and $0$ otherwise from above, while the second term is $1$ if $\la\subset\mu$ and $0$ otherwise by the definition of $\omega^\mu$.
Now the final statement follows from the fact that the set $\lset{\omega^\la}{\la\in\calp(w-1)}$ is a basis for the space of virtual projective characters in $B_-$, with ${\left(\omega^\la\!:\!{\llbracket\sigma+2\nu\rrbracket}\right)}=\delta_{\la\nu}$ for all $\la,\nu\in\calp(w-1)$.
Now we consider applying ${\mathrm{e}_{\bullet}}$ to spin characters.
\[ebullspin\] Suppose $\alpha\in\calp((w-1)/2)$ and $\pi\in\cald(n)$ with ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\pi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ in $B$. Then ${\left({\mathrm{e}_{\bullet}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\pi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$ equals $1$ if $\pi=\tau+4\alpha\sqcup(2)$, and $0$ otherwise. Hence if $\omega$ is a virtual projective character in $B$, then ${\left(\omega\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}=\frac12{\left(e_\bullet\omega\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$.
We apply the spin branching rule. Assume first that ${\mathrm{e}_{\bullet}}={\mathrm{e}_{0}}{\mathrm{e}_{1}}$. In order to have ${\left({\mathrm{e}_{\bullet}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\pi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}>0$, we need $\tau+4\alpha{\ifthenelse{\equal{1}1}{\stackrel{0}\Longrightarrow}{\stackrel{0^1}\Longrightarrow}}\rho{\ifthenelse{\equal{1}1}{\stackrel{1}\Longrightarrow}{\stackrel{1^1}\Longrightarrow}}\pi$ for some $\rho\in\cald$. We have ${l(\tau)}\gs(w-1)/2\gs{l(\alpha)}$, so every addable node of $\tau+4\alpha$ has [spin residue]{}$0$, and every addable node also has a node of [spin residue]{}$0$ to its immediate right, with the exception of the addable node in the first column. So the only possible way of adding two nodes of [spin residues]{}$0$ and then $1$ is to add the addable node at the bottom of the first column, and then the addable node at the bottom of the second column, which gives the partition $\pi=\tau+4\alpha\sqcup(2)$. The spin branching rule then gives the coefficient ${\left({\mathrm{e}_{\bullet}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\pi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$ as $1$.
The case where ${\mathrm{e}_{\bullet}}={\mathrm{e}_{1}}{\mathrm{e}_{0}}$ is very similar: in this case ${l(\tau)}>{l(\alpha)}$, so $\tau+4\alpha$ has addable nodes in columns $1$ and $2$, and the only way to add a node of [spin residue]{}$1$ followed by a node of [spin residue]{}$0$ and end up with a [$2$-regular partition]{}is to add the node in column $2$ followed by the node in column $1$.
The second statement now follows easily, except that we must take the two characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm$ into account. Because $\omega$ is a virtual projective character, we have ${\left(\omega\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+\right)}={\left(\omega\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-\right)}$ (so it makes sense to write ${\left(\omega\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}$ for either of these multiplicities), and the characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm$ both contribute terms ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ when we apply $e_\bullet$. Hence we have the factor $\frac12$ in the final expression.
Take $\mu\in\calp(w)$ and $\alpha\in\calp((w-1)/2)$, and let $M=\lset{\la\in\calp(w-1)}{\la\subset\mu}$. Let $E_-$, $J_-$, $D_-$, ${\mathring D}_-$ be the matrices defined in \[somedecevensec\] for the block with $2$-core $\sigma$ and weight $w-1$ (so $D_-$ is the decomposition matrix of ${\cals(w-1)}$, and so on). Then $$\begin{aligned}
(ED{^{-1}})_{\alpha\mu}&=\sum_{\nu\in\calp(w)}D_{\nu\mu}{^{-1}}{\left({{\operatorname{prj}(\sigma+2\nu)}}\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\\
&={\left(\omega^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha\sqcup(2){\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\\
&=\tfrac12{\left({\mathrm{e}_{\bullet}}\omega^\mu\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\tag*{by \cref{ebullspin}}\\
&=\sum_{\la\in M}{\left(\omega^\la\!:\!{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\tau+4\alpha{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}\right)}\tag*{by \cref{ebullomega}}\\
&=\sum_{\la\in M}(E_-D_-{^{-1}})_{\alpha\la}\\
&=\sum_{\la\in M}(J_-{\mathring D}_-{^{-1}})_{\alpha\la}\tag*{by \cref{allodd}}\\
&=\sum_{\la\in M}({\mathring D}_-{^{-1}})_{({\alpha\sqcup\alpha})\la}.\end{aligned}$$ Now comparing \[steinberg,steinbergod\] we see that this equals ${\mathring D}{^{-1}}_{({\alpha\sqcup\alpha}\sqcup(1))\mu}=(J{\mathring D}{^{-1}})_{\alpha\mu}$. Hence $ED{^{-1}}=J{\mathring D}{^{-1}}$, and so $E=JA$.
Irreducibility of spin characters in Rouquier blocks {#irrspinrouq}
----------------------------------------------------
From the results of the last two subsections we can finally deduce some information about irreducibility of spin characters modulo $2$.
We continue to assume that $B$ is a Rouquier block with weight $w$, $2$-core $\sigma$ and [$4$-bar-core]{}$\tau$. We no longer make any assumption on the parity of $w$. In this section we will classify the spin characters in $B$ that remain irreducible in characteristic $2$.
\[rouqirred\] Suppose $\alpha\in\calp(\lfloor w/2\rfloor)$, and let $$\la=
\begin{cases}
\tau+4\alpha&(\text{if $w$ is even})\\
\tau+4\alpha\sqcup(2)&(\text{if $w$ is odd}).
\end{cases}$$ Then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible $\alpha$ is a $2$-Carter partition.
By \[allodd\] or \[allodd2\], the row of the decomposition matrix corresponding to $\la$ equals the row of the adjustment matrix of ${\cals(w)}$ corresponding to ${\alpha\sqcup\alpha}$ or ${\alpha\sqcup\alpha}\sqcup(1)$. By \[adj\] or \[adjod\], the sum of the entries of this row equals the sum of the entries of the row of the decomposition matrix of ${\cals(\lfloor w/2\rfloor)}$ labelled by $\alpha$. Thus ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible the Weyl module $\Delta(\alpha)$ is irreducible, and by \[irredweyl\] this happens $\alpha$ is a $2$-Carter partition.
We now want to show that if $\la$ is not of the form $\tau+4\alpha$ or $\tau+4\alpha\sqcup(2)$ then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is reducible.
\[sameregdoub\] Suppose ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in $B$, and that $\la$ has the form $\tau+4\alpha\sqcup(4a)$ for some $a\gs1$. Then there is $\mu\in\cald(n)$ such that $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$ and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
We assume first that $\tau=(4l-1,4l-5,\dots,3)$ for $l\gs0$. Let $k$ be the length of $\alpha$. By \[rouqspinform3\], $\la$ includes all the integers $3,7,\dots,4a-1$, which is the same as saying that $k\ls l-a$. So if we let $m=l-a-k$, then the partition $(\la_{k+1},\la_{k+2},\dots)$ equals $(4a+4m-1,4a+4m-3,\dots,3)\sqcup(4a)$, i.e. the partition $\la^{a,m}$ from \[thirdrdim\]. Furthermore, if $k>0$ then $\la_k>4a+4m+3$, so we can define a partition $$\begin{aligned}
\mu=(&4l-1+4\alpha_1,\dots,4l-4k+3+4\alpha_k,\\
&4a+4m+3,4a+4m-1,\dots,4m+7,\\
&4m-1,4m-5,\dots,3),\end{aligned}$$ and we then have $\la_i=\mu_i$ for $i=1,\dots,k$, while the partition $(\mu_{k+1},\mu_{k+2},\dots)$ coincides with the partition $\mu^{a,m}$ from \[thirdrdim\]. Now by combining \[thirdrdim,dimrowrem\] we obtain $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$ and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
The case where $\tau=(4l-3,4l-7,\dots,1)$ is the same, using \[firstrdim,rouqspinform1\] instead of \[thirdrdim,rouqspinform3\].
In the same way (using \[secondrdim,fourthrdim\] rather than \[firstrdim,thirdrdim\]) we obtain the following.
\[sameregdoub2\] Suppose ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in $B$, and $\la$ has the form $\tau+4\alpha\sqcup(4a+2)$ for some $a\gs1$. Then there is $\mu\in\cald(n)$ such that $\la{^{\operatorname{dblreg}}}=\mu{^{\operatorname{dblreg}}}$ and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
Now we can give the main result for Rouquier blocks.
\[mainirrrouq\] Suppose $\la\in\cald$ with [$4$-bar-core]{}$\tau$, and that ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in a Rouquier block. Then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible $\la$ has the form $\tau+4\alpha$ or $\tau+4\alpha\sqcup(2)$ for $\alpha$ a $2$-Carter partition.
By \[spinreg\], we may assume $\la$ has at most one non-zero even part. Hence by \[rouqspinform3\] or \[rouqspinform1\] we can write $\la=\tau+4\alpha$ or $\tau+4\alpha\sqcup(2b)$ for some natural number $b$. In the case where $\la=\tau+4\alpha$ or $\tau+4\alpha\sqcup(2)$, the result follows from \[rouqirred\]. If $\la=\tau+4\alpha\sqcup(2b)$ with $b\gs2$, then by \[sameregdoub\] or \[sameregdoub2\] we can find another [$2$-regular partition]{}$\mu$ with the same regularised double as $\la$ and such that ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ has smaller degree than ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$. Hence by \[samerdoub\] ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is reducible.
Separated partitions
--------------------
Our aim in the remainder of this section is to extend the results of \[irrspinrouq\] beyond Rouquier blocks. So now we drop the assumption that we are working in a Rouquier block, but we make a definition motivated by \[rouqspinform3,rouqspinform1\]: given $\la\in\cald$ and $i=1$ or $3$, we say that $\la$ is *$i$-[separated]{}* if
- $\la$ has at most one non-zero even part,
- all the odd parts of $\la$ are congruent to $i$ modulo $4$, and
- if $\la$ includes an even part $2b$, then $\la$ includes all positive integers less than $2b$ which are congruent to $i$ modulo $4$.
Say that $\la$ is *[separated]{}* if it is either $1$- or $3$-[separated]{}. If we let $\tau$ be the [$4$-bar-core]{}of $\la$, then $\la$ is [separated]{}if $\la$ has one of the following two forms:
- $\tau+4\alpha$, where $\alpha\in\calp$ with ${l(\alpha)}\ls{l(\tau)}$;
- $\tau+4\alpha\sqcup(2b)$, where $b>0$, $\alpha\in\calp$ and ${l(\alpha)}$ is at most the number of parts of $\tau$ which are greater than $2b$.
So let’s write ${\operatorname{sep}(\tau,\alpha,0)}=\tau+4\alpha$ and ${\operatorname{sep}(\tau,\alpha,b)}=\tau+4\alpha\sqcup(2b)$ when $\tau$ is a [$4$-bar-core]{}and $\alpha$ a partition.
The idea of the definition of [separated]{}[$2$-regular partitions]{}is that the corresponding characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ are well-behaved, inheriting properties of corresponding spin characters in Rouquier blocks. The terminology “[separated]{}” is intended to reference the term “$2$-quotient separated” introduced by James and Mathas [@jmq-1] for a class of partitions labelling characters which behave in a similar way for symmetric groups and Hecke algebras.
Our aim is to prove the following, which generalises \[rouqirred\].
\[voddirr\] Suppose $\la$ is a [separated]{}[$2$-regular partition]{}, and write $\la={\operatorname{sep}(\tau,\alpha,b)}$ with $b\gs0$. Then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible $b\ls1$ and $\alpha$ is a $2$-Carter partition.
To prove \[voddirr\], we use a downwards induction. The inductive step is achieved via the following s.
\[voindstep\] Given $l\gs1$, let $\tau=(4l-3,4l-7,\dots,1)$ and let $\upsilon=(4l-1,4l-5,\dots,3)$. Suppose $\la={\operatorname{sep}(\tau,\alpha,b)}$ is [separated]{}, and let $\mu={\operatorname{sep}(\upsilon,\alpha,b)}$.
1. If $b=0$, then $${{{\mathrm{f}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}},\qquad{{{\mathrm{e}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}.$$
2. If $b\gs1$, then $${{{\mathrm{f}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm=\tfrac12({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-),\qquad{{{\mathrm{e}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm=\tfrac12({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-).$$
Hence ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is.
1. $\la$ has two $1$-[spin-addable nodes]{}in each of rows $1,\dots,l$. So by the induction version of \[spinbranchpower\], ${{{\mathrm{f}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\mathrm{f}_{1}}^{(2l)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$. Similarly, since $\mu$ has two $1$-[spin-removable nodes]{}in each row, ${{{\mathrm{e}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\mathrm{e}_{1}}^{(2l)}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$.
2. We let $k$ be such that $\la_k=2b$, and consider two cases according to the parity of $b$. If $b$ is even, then $\la$ has two $1$-[spin-addable nodes]{}in each of rows $1,\dots,l+1$ except row $k$. Adding these nodes yields $\mu$. $\mu$ has no other $1$-[spin-removable nodes]{}, so removing all $1$-[spin-removable nodes]{}from $\mu$ leaves $\la$. Now the expressions for ${{{\mathrm{f}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ and ${{{\mathrm{e}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ follow from \[spinbranchpower\] and its induction inversion.
If $b$ is odd, then each of rows $1,\dots,l+1$ contains two $1$-[spin-addable nodes]{}of $\la$ except rows $k$ and $k+1$, which each contain one (note that the [separated]{}condition means that $\la_{k+1}=2b-1$, so that $(k+1,\la_{k+1}+2)$ is not a [spin-addable node]{}of $\la$). Adding all these nodes yields $\mu$, and $\mu$ has no other $1$-[spin-removable nodes]{}. Again, the expressions for ${{{\mathrm{f}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ and ${{{\mathrm{e}_{1}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ follow from \[spinbranchpower\] and its induction inversion.
In either case, we obtain $${{{\mathrm{f}_{1}}^{(\operatorname{max})}}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}},\qquad{{{\mathrm{e}_{1}}^{(\operatorname{max})}}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}.$$ so the final statement follows from \[modbranch\].
In a similar way we obtain the following.
\[voindstep2\] Given $l\gs0$, let $\tau=(4l-1,4l-5,\dots,3)$ and $\upsilon=(4l+1,4l-3,\dots,1)$. Suppose $\la={\operatorname{sep}(\tau,\alpha,b)}$ is [separated]{}, and let $\mu={\operatorname{sep}(\upsilon,\alpha,b)}$.
1. If $b=0$, then $${{{\mathrm{f}_{0}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}},\qquad{{{\mathrm{e}_{0}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}.$$
2. If $b\gs1$, then $${{{\mathrm{f}_{0}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm=\tfrac12({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-),\qquad{{{\mathrm{e}_{0}}^{(\operatorname{max})}}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_\pm=\tfrac12({{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_++{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-).$$
Hence ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is.
We use downwards induction on the size of the [$4$-bar-core]{}of $\la$. Note that by \[mainirrrouq\] the result holds if ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in a Rouquier block. In general if we assume that $\la$ is [separated]{}and write $\la={\operatorname{sep}(\tau,\alpha,b)}$, then for any [$4$-bar-core]{}$\upsilon$ larger than $\tau$, the partition ${\operatorname{sep}(\upsilon,\alpha,b)}$ is also [separated]{}and has the same [$4$-bar-weight]{}as $\la$, i.e. $2|\alpha|+b$. So if we fix $\alpha$ and $b$ and allow $\tau$ to vary, then for sufficiently large $\tau$ the character ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}{\operatorname{sep}(\tau,\alpha,b)}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ lies in a Rouquier block. So to prove \[voddirr\] for a specific $\la={\operatorname{sep}(\tau,\alpha,b)}$, it suffices to show that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}{\operatorname{sep}(\upsilon,\alpha,b)}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, for some [$4$-bar-core]{}$\upsilon$ larger than $\tau$. If $\la$ is $1$-[separated]{}this follows from \[voindstep\], while if $\la$ is $3$-[separated]{}then we use \[voindstep2\] instead.
Now we come to the conclusion of this section, which is that \[main\] is true for spin characters labelled by [separated]{}partitions.
Looking at the six cases listed in \[main\], we see that $\la$ is [separated]{}in (and only in) cases 1–4: in cases 1 and 2 we have $\la={\operatorname{sep}(\tau,\alpha,0)}$ with $\alpha$ a $2$-Carter partition, and cases 3 and 4 we have $\la={\operatorname{sep}(\tau,\alpha,1)}$ with $\alpha$ a $2$-Carter partition. So when $\la$ is [separated]{}, \[main\] is equivalent to \[voddirr\].
Proof of \[main\] for non-[separated]{}partitions {#mainproofsec}
=================================================
Now we prove \[main\] in the case where $\la$ is not [separated]{}. To do this, we introduce some notation for the various classes of partitions appearing in \[main\]:
- let ${\scrx_3}$ denote the set of [$2$-regular partitions]{}$\tau+4\alpha$, where $\tau=(4l-1,4l-5,\dots,3)$ with $l\gs0$ and $\alpha$ a $2$-Carter partition with ${l(\alpha)}\ls l$;
- let ${\scrx_1}$ denote the set of [$2$-regular partitions]{}$\tau+4\alpha$, where $\tau=(4l-3,4l-7,\dots,1)$ with $l\gs1$ and $\alpha$ a $2$-Carter partition with ${l(\alpha)}\ls l$;
- let ${\scry_3}$ denote the set of [$2$-regular partitions]{}$\tau+4\alpha\sqcup(2)$, where $\tau=(4l-1,4l-5,\dots,3)$ with $l\gs0$ and $\alpha$ a $2$-Carter partition with ${l(\alpha)}\ls l$;
- let ${\scry_1}$ denote the set of [$2$-regular partitions]{}$\tau+4\alpha\sqcup(2)$, where $\tau=(4l-3,4l-7,\dots,1)$ with $l\gs1$ and $\alpha$ a $2$-Carter partition with ${l(\alpha)}\ls l-1$;
- let ${\scrz}$ denote the set of [$2$-regular partitions]{}$(2b)$ or $(4b-2,1)$ for $b\gs2$.
Now let ${\scri}={\scrx_3}\cup{\scrx_1}\cup{\scry_3}\cup{\scry_1}\cup{\scrz}\cup\{(3,2,1)\}$.
To prove the “if” part of \[main\] in the case where $\la$ is not [separated]{}, we need the following theorem of Wales.
[wal]{}[Theorem 7.7]{}\[wales\] Suppose $p$ is a prime. Then the $p$-modular reduction of ${\llanglen{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ is irreducible unless $n$ is odd and $p$ divides $n$.
In particular, we have ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}2b{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ irreducible for every $b$. Next consider the case $\la=(4b-2,1)$ and let $\mu=(4b-2)$. Then (as we have just noted) ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, and by examining [spin residues]{}we see that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={{{\mathrm{f}_{0}}^{(\operatorname{max})}}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\mu{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$, so by \[modbranch\] ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible too. The final case to consider to complete the “if” part of the proof is $\la=(3,2,1)$, for which one can just check the readily-available decomposition matrix (for example, at the Modular Atlas homepage www.math.rwth-aachen.de/homes/MOC/).
Now we proceed with the “only if” part of \[main\], which is more complex. We will prove by induction on $|\la|$ that if $\la$ is a [$2$-regular partition]{}not in ${\scri}$ then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is a reducible $2$-modular character. The key result which underlies the combinatorics of the proof is the following (recall that we write ${\la{\downarrow}_{i}}$ for the partition obtained by removing all the $i$-[spin-removable nodes]{}from $\la\in\cald$).
\[rsprop\] Suppose $\la\in\cald$ and $i\in\{0,1\}$. If ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, then so is ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}{\la{\downarrow}_{i}}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$.
If ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, then by \[modbranch\] so is ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$. But by \[spinmaxbranch\], ${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ equals $a{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}{\la{\downarrow}_{i}}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ for some $a\in\bbn$, so ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}{\la{\downarrow}_{i}}{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ must be irreducible (and in fact $a$ must equal $1$).
So using our inductive hypothesis we may assume that for $i=0,1$ either ${\la{\downarrow}_{i}}\in{\scri}$ or ${\la{\downarrow}_{i}}=\la$. We also assume that $\la$ is not [separated]{}, and by \[spinreg\] we can assume that $\la$ has at most one non-zero even part.
We summarise our assumptions for ease of reference.
**Assumptions and notation in force for the rest of \[mainproofsec\]:**
$\la$ is a [$2$-regular partition]{}which has at most one non-zero even part, is not [separated]{}and does not lie in ${\scri}$. For $i=0,1$, either ${\la{\downarrow}_{i}}\in{\scri}$ or ${\la{\downarrow}_{i}}=\la$. We let $\mu=\la{^{\operatorname{dblreg}}}$.
Recall that by \[spinreg\], if ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, then ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={\varphi(\mu)}$. We will often exploit this by finding an induction or restriction functor (or a composition of such functors) which kills ${\varphi(\mu)}$ but not ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$, so that ${\varphi(\mu)}$ and ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ cannot be equal.
The assumptions above have several immediate consequences.
- ${\la{\downarrow}_{1}}$ cannot lie in ${\scrx_3}$ or ${\scry_3}$, since partitions in ${\scrx_3}\cup{\scry_3}$ all have $1$-[spin-removable nodes]{}(with the exception of $\varnothing$, but in this case $\la$ would also equal $\varnothing$, contrary to assumption).
- Similarly, ${\la{\downarrow}_{0}}$ cannot lie in ${\scrx_1}$ or ${\scry_1}$.
- Neither ${\la{\downarrow}_{0}}$ nor ${\la{\downarrow}_{1}}$ can lie in ${\scrz}$, since then $\la$ would lie in ${\scrz}$ as well.
- Neither ${\la{\downarrow}_{0}}$ nor ${\la{\downarrow}_{1}}$ can equal $(3,2,1)$.
So our assumptions give ${\la{\downarrow}_{0}}\in{\scrx_3}\cup{\scry_3}\cup\{\la\}$ and ${\la{\downarrow}_{1}}\in{\scrx_1}\cup{\scry_1}\cup\{\la\}$. We now consider several cases.
0
1
Case : ${\la{\downarrow}_{0}}=\la$, ${\la{\downarrow}_{1}}\in{\scrx_1}$ {#case-ladownarrow_0la-ladownarrow_1inscrx_1 .unnumbered}
-----------------------------------------------------------------------
In this case, write $${\la{\downarrow}_{1}}=(4l-3,4l-7,\dots,1)+4\nu$$ with $\nu$ a $2$-Carter partition of length $m\ls l$. We can obtain $\la$ from ${\la{\downarrow}_{1}}$ by adding $1$-[spin-addable nodes]{}, and there are two of these nodes which can be added in each row from $1$ to $l$. We must add at least one node in each of these rows, since otherwise $\la$ would have a $0$-[spin-removable node]{}, so that ${\la{\downarrow}_{0}}\neq\la$. On the other hand, $\la$ has at most one even part, so there can be at most one row where we add exactly one node. If we add two nodes in each row, then $$\la=(4l-1,4l-5,\dots,3)+4\nu\in{\scrx_3},$$ contrary to assumption, so instead there must be some $k$ such that we add one node in row $k$ and two nodes in each other row. That is, $$\la=(4l-1,4l-5,\dots,4l-4k+7,\quad 4l-4k+2,\quad 4l-4k-1,\dots,7,3)+4\nu.$$ Hence $$\la{^{\operatorname{dbl}}}=(2l,2l-1,\dots,2l-2k+3,\quad (2l-2k+1)^2,\quad 2l-2k,2l-2k-1,\dots,1)+2({\nu\sqcup\nu}).$$ We now consider several subcases; recall that $m$ is the length of $\nu$.
1. If $k>m$, then $\la$ is [separated]{}, contrary to assumption.
2. If $k=m=l=1$, then $\la\in{\scrz}$, again contrary to assumption.
3. \[subef\] If $k=m=l>1$, then $$\mu=(2l,2l-1,\dots,2)+2({\nu\sqcup\nu}).$$ We apply the modular branching rules to compute ${{\mathrm{f}_{0}}^{(2l-1)}}{\varphi(\mu)}$. Note that $\mu$ has an addable $0$-node at the end of each row from $1$ to $2l-1$, and (since $\nu_l>0$) a removable $0$-node at the end of row $2l$ and an addable $0$-node at the start of row $2l+1$. Hence all the addable $0$-nodes except the one in row $2l-1$ are conormal, so ${{\mathrm{f}_{0}}^{(2l-1)}}{\varphi(\mu)}={\varphi(\xi)}$, where $$\xi=(2l+1,2l,\dots,5,4,2,0,1)+2({\nu\sqcup\nu}).$$ Applying the modular branching rules again we obtain ${\mathrm{e}_{1}}{{\mathrm{f}_{0}}^{(2l-1)}}{\varphi(\mu)}=0$. On the other hand, the spin branching rules give ${\mathrm{e}_{1}}{{\mathrm{f}_{0}}^{(2l-1)}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$, so we cannot possibly have ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={\varphi(\mu)}$.
4. If $l>k\ls m$, then $\nu_k>\nu_{k+1}$ (since $\nu$ is a $2$-Carter partition and hence $2$-regular), which means that $$\mu=(2l,2l-1,\dots,2l-2k+3,\quad 2l-2k+2,(2l-2k)^2,\quad 2l-2k-1,2l-2k-2,\dots,1)+2({\nu\sqcup\nu}).$$ This partition has an addable $1$-node in row $2k$ and removable $1$-nodes in all other rows from $1$ to $2l$. But the fact that $k<l$ means that one of these removable nodes (namely, the one in row $2k+1$) is not normal, so ${\mathrm{e}_{1}}^{2l-1}{\varphi(\mu)}=0$. But the spin branching rules give ${\mathrm{e}_{1}}^{2l-1}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$, so ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq{\varphi(\mu)}$.
Take $\la=(31,15,6)$. Then ${\la{\downarrow}_{0}}=\la$ and $${\la{\downarrow}_{1}}=(29,13,5)=(9,5,1)+4(5,2,1),$$ so that $k=l=m=3$. We have $$\la{^{\operatorname{dbl}}}=(16,15,8,7,3,3),$$ giving $$\mu=(16,15,8,7,4,2)=(6,5,4,3,2)+2(5,5,2,2,1,1).$$ Now we examine the residues of the nodes of $\mu$ and the spin-residues of the nodes of $\la$: $$\begin{aligned}
\la&=\young(0110011001100110011001100110011,011001100110011,011001)
\\
\mu&=\yngres(2,16,15,8,7,4,2).\end{aligned}$$ The branching rules give ${{\mathrm{f}_{0}}^{(5)}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}33,17,6,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ and ${{\mathrm{f}_{0}}^{(5)}}{\varphi(\mu)}={\varphi(17,16,9,8,4,2,1)}$. And since ${\mathrm{e}_{1}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}33,17,6,1{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0={\mathrm{e}_{1}}{\varphi(17,16,9,8,4,2,1)}$ we deduce that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq{\varphi(\mu)}$, so that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is reducible.
1
Case : ${\la{\downarrow}_{0}}=\la$, ${\la{\downarrow}_{1}}\in{\scry_1}$ {#case-ladownarrow_0la-ladownarrow_1inscry_1 .unnumbered}
-----------------------------------------------------------------------
In this case, write $${\la{\downarrow}_{1}}=(4l+1,4l-3,\dots,5,2,1)+4\nu$$ with $\nu$ a $2$-Carter partition of length $m\ls l$. $\la$ is obtained from ${\la{\downarrow}_{1}}$ by adding $1$-[spin-addable nodes]{}, and sufficiently many such nodes must be added that $\la$ has no $0$-[spin-removable nodes]{}. In particular, at least one node must be added in each of rows $1,\dots,l$, and exactly one node must be added in each of rows $l+1$ and $l+2$. But this means in particular that $\la_{l+2}=2$. Since by assumption $\la$ has at most one even part, all of $\la_1,\dots,\la_l$ must be odd, and therefore we must have $$\la=(4l+3,4l-1,\dots,7,3,2)+4\nu\in{\scry_3},$$ contrary to assumption.
1
Case : ${\la{\downarrow}_{0}}\in{\scrx_3}$, ${\la{\downarrow}_{1}}=\la$ {#case-ladownarrow_0inscrx_3-ladownarrow_1la .unnumbered}
-----------------------------------------------------------------------
Now write $${\la{\downarrow}_{0}}=(4l-1,4l-5,\dots,3)+4\nu$$ with $\nu$ a $2$-Carter partition of length $m\ls l$. $\la$ is obtained from ${\la{\downarrow}_{0}}$ by adding nodes of [spin residue]{}$0$, and at least one such node must be added in each of rows $1,\dots,l$ to ensure that ${\la{\downarrow}_{1}}=\la$. $\la$ can have at most one even part, so there can be at most one of these rows where we add a single $0$. We then also have the possibility of adding a $0$ in row $l+1$. If we add two nodes to each of rows $1,\dots,l$, then (regardless of whether we also add a node in row $l+1$) $\la$ is [separated]{}, contrary to assumption. So there is $1\ls k\ls l$ such that we add a single node in row $k$ and two nodes in all other rows from $1$ to $l$.
Again we consider several subcases. First suppose $\la$ does have a node in row $l+1$; that is, $$\la=(4l+1,4l-3,\dots,4l-4k+9,\quad4l-4k+4,\quad4l-4k+1,4l-4k-3,\dots,1)+4\nu.$$ Then $$\la{^{\operatorname{dbl}}}=(2l+1,2l,\dots,2l-2k+4,\quad(2l-2k+2)^2,\quad2l-2k+1,2l-2k,\dots,1)+2({\nu\sqcup\nu}).$$
1. \[subc\] If $k\ls m$, then $$\mu=(2l+1,2l,\dots,2l-2k+4,\quad2l-2k+3,2l-2k+1,\quad2l-2k+1,2l-2k,\dots,1)+2({\nu\sqcup\nu}).$$ and the modular branching rules give ${\mathrm{e}_{0}}^{2l}{\varphi(\mu)}=0$. But ${\mathrm{e}_{0}}^{2l}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$, so ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq{\varphi(\mu)}$.
2. If $k>m$, then $\la$ is [separated]{}, contrary to assumption.
Now suppose $\la$ does not have a node in row $l+1$, i.e. $$\la=(4l+1,4l-3,\dots,4l-4k+9,\quad4l-4k+4,\quad4l-4k+1,4l-4k-3,\dots,5)+4\nu.$$
1. If $m\gs k<l$, then (similarly to subcase \[subc\]) ${\mathrm{e}_{0}}^{2l-1}{\varphi(\mu)}=0$, while ${\mathrm{e}_{0}}^{2l-1}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$, so ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq{\varphi(\mu)}$.
2. If $l=1$, then $\la\in{\scrz}$, contrary to assumption.
3. If $1<k=l$, then (similarly to subcase \[subef\]) ${\mathrm{e}_{0}}{\mathrm{f}_{1}}^{2l-2}$ kills ${\varphi(\mu)}$ but not ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ so we cannot have ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={\varphi(\mu)}$.
4. Finally suppose $m<k<l$, and define $$\xi=(4l+1,4l-3,\dots,9,4)+4\nu.$$ Then by applying \[dimrowrem\] $k-1$ times followed by \[dimen1\], we obtain $\la{^{\operatorname{dblreg}}}=\xi{^{\operatorname{dblreg}}}$ and ${\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}>{\operatorname{deg}}{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\xi{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$, so that by \[samerdoub\] ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is reducible.
1
Case : ${\la{\downarrow}_{0}}\in{\scry_3}$, ${\la{\downarrow}_{1}}=\la$ {#case-ladownarrow_0inscry_3-ladownarrow_1la .unnumbered}
-----------------------------------------------------------------------
Now write $${\la{\downarrow}_{0}}=(4l-1,4l-5,\dots,3,2)+4\nu$$ with $\nu$ a $2$-Carter partition. We obtain $\la$ from ${\la{\downarrow}_{0}}$ by adding sufficiently many $0$-[spin-addable nodes]{}to eliminate any $1$-[spin-removable nodes]{}. We certainly have to add the node $(l+2,1)$ (to eliminate the [spin-removable node]{}$(l+1,2)$). We also have to add at least one node in each of rows $1,\dots,l-1$; since $\la$ has at most one even part, we must add two nodes in each of these rows. If in addition we add two nodes in row $l$, then $\la=(4l+1,4l-3,\dots,5,2,1)+4\nu\in{\scry_1}$, contrary to assumption. So we must add no nodes in row $l$, and this means that $\nu_l=0$, since otherwise $\la$ has a $1$-[spin-removable node]{}.
So we have $$\la=(4l+1,4l-3,\dots,9,3,2,1)+4\nu.$$ In particular, this implies $l>1$, since by assumption $\la\neq(3,2,1)$. So $$\begin{aligned}
\la{^{\operatorname{dbl}}}&=(2l+1,2l,\dots,5,4,2,1^4)+2({\nu\sqcup\nu}),\\
\intertext{which implies that}
\mu&=(2l+1,2l,\dots,5,4,3,1)+2({\nu\sqcup\nu}\sqcup(1)).\end{aligned}$$ Now the modular branching rules (together with the fact that $l>1$) give ${\mathrm{e}_{0}}{\mathrm{f}_{1}}^{2l-2}{\varphi(\mu)}=0$, while ${\mathrm{e}_{0}}{\mathrm{f}_{1}}^{2l-2}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$.
1
Case : ${\la{\downarrow}_{0}}\in{\scrx_3}\cup{\scry_3}$, ${\la{\downarrow}_{1}}\in{\scrx_1}\cup{\scry_1}$ {#case-ladownarrow_0inscrx_3cupscry_3-ladownarrow_1inscrx_1cupscry_1 .unnumbered}
---------------------------------------------------------------------------------------------------------
In this case, let $b$ denote the total number of [spin-removable nodes]{}of $\la$. We claim first that the length of $\mu$ is at most $b$. This follows by considering the various possibilities for ${\la{\downarrow}_{0}}$ and ${\la{\downarrow}_{1}}$. Note that for each $i$ we have $\la_i=\max\left\{({\la{\downarrow}_{0}})_i,({\la{\downarrow}_{1}})_i\right\}$; since $\la$ can have at most two [spin-removable nodes]{}in row $i$, this means that $({\la{\downarrow}_{0}})_i$ and ${\la{\downarrow}_{1}})_i$ differ by at most $2$. Moreover, ${l({\la{\downarrow}_{1}})}-{l({\la{\downarrow}_{0}})}$ is either $0$ or $1$. This leaves six distinct possibilities. We tabulate these below, giving the number $b$ for each, together with the length of $\la{^{\operatorname{dbl}}}$. $$\begin{array}{ccccc}\hline
{\la{\downarrow}_{0}}&{\la{\downarrow}_{1}}&b&{l(\la{^{\operatorname{dbl}}})}\\\hline
(4l-1,4l-5,\dots,3)+4\nu&(4l-3,4l-7,\dots,1)+4\xi&2l&2l\\
(4l-1,4l-5,\dots,3)+4\nu&(4l+1,4l-3,\dots,1)+4\xi&2l+1&2l+1\\
(4l-1,4l-5,\dots,3)+4\nu&(4l-3,4l-7,\dots,5,2,1)+4\xi&2l&2l+1\\
(4l-1,4l-5,\dots,3,2)+4\nu&(4l+1,4l-3,\dots,1)+4\xi&2l+1&2l+2\\
(4l-1,4l-5,\dots,3,2)+4\nu&(4l-3,4l-7,\dots,5,2,1)+4\xi&2l&2l+2\\
(4l-1,4l-5,\dots,3,2)+4\nu&(4l+1,4l-3,\dots,5,2,1)+4\xi&2l+1&2l+3\\\hline
\end{array}$$ Now note that in the third and fourth cases the last two parts of $\la{^{\operatorname{dbl}}}$ are both equal to $1$, so that the length of $\mu$ is less than the length of $\la{^{\operatorname{dbl}}}$, and hence at most $b$. In the fifth and sixth cases, the last three parts of $\la{^{\operatorname{dbl}}}$ are all equal to $1$, so the length of $\mu$ is at most ${l(\la{^{\operatorname{dbl}}})}-2$, and hence less than or equal to $b$.
So our claim is proved. Now if $\mu$ has fewer than $b$ normal nodes, then for some $i\in\{0,1\}$ and some $r$ we have ${\mathrm{e}_{i}}^r{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$ while ${\mathrm{e}_{i}}^r{\varphi(\mu)}=0$, so that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq{\varphi(\mu)}$. So we can assume that $\mu$ has $b$ normal nodes, and since $\mu$ has length at most $b$ this means that $\mu$ must have a normal node at the end of every non-empty row. But the only way this can happen is if the nodes at the ends of the non-empty rows of $\mu$ all have the same residue (since if the nodes at the ends of rows $k$ and $k+1$ have different residues, then the one at the end of row $k+1$ is not normal). But if the removable nodes of $\mu$ all have residue $i$, then ${\mathrm{e}_{1-i}}{\varphi(\mu)}=0$, while ${\mathrm{e}_{1-i}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0$ (since by assumption ${\la{\downarrow}_{1-i}}\neq\la$).
Take $\la=(21,11,5,2,1)$. Then $$\begin{aligned}
{\la{\downarrow}_{0}}&=(19,11,3,2)=(11,7,3,2)+4(2,1)\in{\scry_3},\\
{\la{\downarrow}_{1}}&=(21,9,5,2,1)=(13,9,5,2,1)+4(1)\in{\scry_1}.\end{aligned}$$ So we are in the final case in the table above, with $l=3$. $\la$ has seven [spin-removable nodes]{}. We have $$\la{^{\operatorname{dbl}}}=(11,10,6,5,3,2,1^3),\qquad\mu=(11,10,7,5,4,2,1)$$ so that $\mu$ can have at most seven normal nodes. But in fact $\mu$ has only three normal nodes (all of residue $0$). Hence ${\mathrm{e}_{1}}{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}\neq0={\mathrm{e}_{1}}{\varphi(\mu)}$, so that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is reducible.
This completes the proof of \[main\].
Future directions {#futuresec}
=================
The alternating group {#ansec}
---------------------
Let ${\mathfrak{A}_}n$ denote the alternating group, and define define ${\tilde{\mathfrak{A}}_}n$ to be the inverse image of ${\mathfrak{A}_}n$ under the natural homomorphism ${\tilde{\mathfrak{S}}_}n\to\sss n$. Then ${\tilde{\mathfrak{A}}_}n$ is a double cover of ${\mathfrak{A}_}n$, and (except when $n\ls3$ or $n=6,7$) is the unique Schur cover of ${\mathfrak{A}_}n$. We can ask our main question for characters of ${\tilde{\mathfrak{A}}_}n$ as well, and the answer is closely related to that for ${\tilde{\mathfrak{S}}_}n$. In this section we make a few observations about this problem, which we hope to solve in a future paper. Throughout this section we work over fields large enough to be splitting fields for ${\mathfrak{A}_}n$.
As with ${\tilde{\mathfrak{S}}_}n$, the case of linear characters follows from the corresponding result for ${\mathfrak{A}_}n$, which is given in [@mfonaltred Theorem 3.1]. So here we need only be concerned with spin characters of ${\tilde{\mathfrak{A}}_}n$. We say that two characters of ${\tilde{\mathfrak{A}}_}n$ are *conjugate* if we can transform one into the other by conjugating all elements of ${\tilde{\mathfrak{A}}_}n$ by an odd element of ${\tilde{\mathfrak{S}}_}n$. The ordinary irreducible spin characters of ${\tilde{\mathfrak{A}}_}n$ are easily constructed from those for ${\tilde{\mathfrak{S}}_}n$: for every $\la\in\cald^+(n)$ the restriction ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}{{\downarrow}}_{{\tilde{\mathfrak{A}}_}n}$ is a sum of two conjugate irreducible characters ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+,{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$, while for $\la\in\cald^-(n)$ the restriction ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+{{\downarrow}}_{{\tilde{\mathfrak{A}}_}n}={{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-{{\downarrow}}_{{\tilde{\mathfrak{A}}_}n}$ is a self-conjugate irreducible character which we denote ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$. The set $$\lset{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+,{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-}{\la\in\cald^+(n)}\cup\lset{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}{\la\in\cald^-(n)}$$ is a complete set (without repeats) of ordinary irreducible spin characters of ${\tilde{\mathfrak{A}}_}n$. To help us to determine which of these characters remain irreducible in characteristic $2$, we consider restricting from ${\tilde{\mathfrak{S}}_}n$, using the fact that restriction to subgroups commutes with modular reduction. *We now no longer allow ourselves to write ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ to mean “either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$” when $\la\in\cald^-(n)$.* So ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ for $\la\in\cald^-(n)$ will unambiguously mean the spin character for ${\tilde{\mathfrak{A}}_}n$ labelled by $\la$.
As with ${\tilde{\mathfrak{S}}_}n$, there are no irreducible spin characters of ${\tilde{\mathfrak{A}}_}n$ in characteristic $2$, so the irreducible $2$-modular characters of ${\tilde{\mathfrak{A}}_}n$ are the same as those for ${\mathfrak{A}_}n$. These are also constructed by restriction from $\sss n$, but to explain the situation here we need a definition: say that $\la\in\cald$ is an *S-partition* if for every odd $l$ the $l$th ladder of $\la$ contains an even number of nodes. Then ${\varphi(\la)}{{\downarrow}}_{{\mathfrak{A}_}n}$ is a sum of two conjugate irreducible characters ${\varphi(\la)}_+,{\varphi(\la)}_-$ if $\la$ is an S-partition, and otherwise ${\varphi(\la)}{{\downarrow}}_{{\mathfrak{A}_}n}$ is a self-conjugate irreducible character. The irreducible characters arising in this way are (without repeats) all the irreducible $2$-modular characters of ${\mathfrak{A}_}n$. This result is due to Benson [@ben Theorem 1.1], though as far as we are aware the characterisation of S-partitions we have given is new.
Now we consider what happens when we take an ordinary irreducible spin character of ${\tilde{\mathfrak{S}}_}n$, restrict to ${\tilde{\mathfrak{A}}_}n$ and reduce modulo $2$.
Take $\la\in\cald^-(n)$. Then the fact that restriction commutes with modular reduction gives $${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}={\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}{{\downarrow}}_{{\tilde{\mathfrak{A}}_}n}.$$ So in order for ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ to be irreducible, ${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}$ must be irreducible, and must remain irreducible on restriction to ${\tilde{\mathfrak{A}}_}n$. By \[spinreg\], this means that ${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}={\varphi(\la{^{\operatorname{dblreg}}})}$, and that $\la{^{\operatorname{dblreg}}}$ is not an S-partition.
In fact, it is easy to check that the second condition: an easy exercise shows that an S-partition must have even $2$-weight, whereas all the partitions in \[main\] that lie in $\cald^-$ have odd [$4$-bar-weight]{}, with the exception of the partitions $(4b)$ for $b\gs1$. If $\la=(4b)$ then $\la{^{\operatorname{dblreg}}}=(2b+1,2b-1)$ is an S-partition. So for $\la\in\cald^-$ we conclude that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible ${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}$ is irreducible and $\la\neq(4b)$ for $b\gs1$.
Now consider $\la\in\cald^+(n)$. In this case restricting and reducing modulo $2$ gives $${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}+{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-}}={{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}{{\downarrow}}_{{\tilde{\mathfrak{A}}_}n}.$$ Since ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ and ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$ are conjugate characters, ${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+}}$ is irreducible ${\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-}}$ is, and this happens ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}{{\downarrow}}_{{\tilde{\mathfrak{A}}_}n}$ has exactly two irreducible constituents. Hence either
- ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ is irreducible, or
- ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ has exactly two irreducible constituents, which restrict to conjugate irreducible $2$-modular characters of ${\mathfrak{A}_}n$.
The main theorem of the present paper tells us exactly when the first situation occurs. For the second situation to occur, the two irreducible constituents of ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}$ would have to be equal in order for their restrictions to ${\tilde{\mathfrak{A}}_}n$ to be conjugate and irreducible, so by \[spinreg\] we would have to have ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}=2{\varphi(\la{^{\operatorname{dblreg}}})}$. In particular, this implies that ${\operatorname{ev}(\la)}$ would have to be $2$.
So to solve the main problem for spin characters of ${\tilde{\mathfrak{A}}_}n$, it remains to classify partitions $\la\in\cald$ with exactly two non-zero even parts, such that ${{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}=2{\varphi(\la{^{\operatorname{dblreg}}})}$. Such partitions do occur; the first example is $(4,2,1)$, and we will see a family of examples below.
Decomposition numbers for Rouquier blocks
-----------------------------------------
$$\begin{array}{r|c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c}
&{\rotatebox{90}{$(7)$}}
&{\rotatebox{90}{$(6,1)$}}
&{\rotatebox{90}{$(5,2)$}}
&{\rotatebox{90}{$(5,1^2)$}}
&{\rotatebox{90}{$(4,3)$}}
&{\rotatebox{90}{$(4,2,1)$}}
&{\rotatebox{90}{$(4,1^3)$}}
&{\rotatebox{90}{$(3^2,1)$}}
&{\rotatebox{90}{$(3,2^2)$}}
&{\rotatebox{90}{$(3,2,1^2)$}}
&{\rotatebox{90}{$(3,1^4)$}}
&{\rotatebox{90}{$(2^3,1)$}}
&{\rotatebox{90}{$(2^2,1^3)$}}
&{\rotatebox{90}{$(2,1^5)$}}
&{\rotatebox{90}{$(1^7)$}}
\\\hline
(3),(1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot
&\cdot&\cdot&\cdot&\cdot&\cdot\\
(2,1),(1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&\cdot&1&\cdot&\cdot\\
(1^3),(1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot
&\cdot&\cdot&\cdot&\cdot&1\\\hline
(2),(3)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&2&\cdot
&\cdot&1&1&\cdot&\cdot\\
(1^2),(3)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&2&\cdot
&2&1&1&1&1\\
(2),(2,1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&2&\cdot&\cdot&\cdot\\
(1^2),(2,1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&2&\cdot&2&\cdot\\\hline
(1),(5)&\cdot&\cdot&\cdot&4&\cdot&\cdot&2&1&4&1
&2&2&1&1&1\\
(1),(4,1)&\cdot&\cdot&\cdot&8&\cdot&\cdot&4&2&8&6
&4&4&2&2&\cdot\\
(1),(3,2)&\cdot&\cdot&\cdot&8&\cdot&\cdot&\cdot&2&8&6
&4&\cdot&2&\cdot&\cdot\\\hline
\varnothing,(7)&4&4&2&4&1&\cdot&2&1&2&1
&2&1&\cdot&1&1\\
\varnothing,(6,1)&8&8&12&16&2&4&4&2&8&6
&4&2&2&2&\cdot\\
\varnothing,(5,2)&16&8&16&24&6&12&4&8&12&8
&4&2&2&\cdot&\cdot\\
\varnothing,(4,3)&8&8&4&8&6&8&4&6&4&2
&\cdot&2&\cdot&\cdot&\cdot\\
\varnothing,(4,2,1)&8&\cdot&4&8&\cdot&8&\cdot&6&4&2
&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}$$
$$\begin{array}{r|c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c}
&{\rotatebox{90}{$(7)$}}
&{\rotatebox{90}{$(6,1)$}}
&{\rotatebox{90}{$(5,2)$}}
&{\rotatebox{90}{$(5,1^2)$}}
&{\rotatebox{90}{$(4,3)$}}
&{\rotatebox{90}{$(4,2,1)$}}
&{\rotatebox{90}{$(4,1^3)$}}
&{\rotatebox{90}{$(3^2,1)$}}
&{\rotatebox{90}{$(3,2^2)$}}
&{\rotatebox{90}{$(3,2,1^2)$}}
&{\rotatebox{90}{$(3,1^4)$}}
&{\rotatebox{90}{$(2^3,1)$}}
&{\rotatebox{90}{$(2^2,1^3)$}}
&{\rotatebox{90}{$(2,1^5)$}}
&{\rotatebox{90}{$(1^7)$}}
\\\hline
(3),(1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot
&\cdot&\cdot&\cdot&\cdot&\cdot\\
(2,1),(1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&\cdot&1&\cdot&\cdot\\
(1^3),(1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&\cdot&\cdot&\cdot&1\\\hline
(2),(3)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&2&\cdot
&\cdot&1&1&\cdot&\cdot\\
(1^2),(3)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&2&\cdot&1&1&1\\
(2),(2,1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&2&\cdot&\cdot&\cdot\\
(1^2),(2,1)&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
&\cdot&\cdot&\cdot&2&\cdot\\\hline
(1),(5)&\cdot&\cdot&\cdot&3&\cdot&\cdot&2&\cdot&2&1
&2&1&1&1&1\\
(1),(4,1)&\cdot&\cdot&\cdot&2&\cdot&\cdot&4&2&4&6
&4&2&2&2&\cdot\\
(1),(3,2)&\cdot&\cdot&\cdot&2&\cdot&\cdot&\cdot&2&4&6
&4&\cdot&2&\cdot&\cdot\\\hline
\varnothing,(7)&4&3&2&3&1&\cdot&2&\cdot&\cdot&1
&2&\cdot&\cdot&1&1\\
\varnothing,(6,1)&4&6&12&10&2&4&4&2&4&6
&4&\cdot&2&2&\cdot\\
\varnothing,(5,2)&4&2&16&16&6&12&4&8&8&8
&4&2&2&\cdot&\cdot\\
\varnothing,(4,3)&\cdot&2&4&6&6&8&4&6&4&2
&\cdot&2&\cdot&\cdot&\cdot\\
\varnothing,(4,2,1)&\cdot&\cdot&4&6&\cdot&8&\cdot&6&4&2
&\cdot&\cdot&\cdot&\cdot&\cdot\\
\end{array}$$
One of the main results in this paper (\[allodd,allodd2\]) is the calculation of some of the rows of the spin part of the decomposition matrix for a Rouquier block of ${\tilde{\mathfrak{S}}_}n$; specifically, the rows corresponding to [$2$-regular partitions]{}with no even parts greater than $2$. It would be interesting to extend this to determine the whole of the decomposition matrix for a Rouquier block. We expect this would be amenable to the same techniques, once suitable combinatorial expressions for the entries are found.
By way of example, we consider the spin part of the decomposition matrix for a Rouquier block of weight $7$. Let $\sigma$ be a $2$-core of length at least $6$ and $\tau$ the corresponding [$4$-bar-core]{}, and let $B$ be the block with $2$-core $\sigma$ and weight $7$. Then we write $E$ for the spin part of the decomposition matrix of $B$; we label the columns of $E$ by partitions of $7$, and the rows by pairs $(\alpha,\beta)$ of partitions with $\beta$ $2$-regular and $2|\alpha|+|\beta|=7$, and set $$E_{(\alpha,\beta)\mu}={D^{\mathsf{spn}}_{(\tau+4\alpha\sqcup2\beta)(\sigma+2\mu)}}.$$ We also define $A$ to be the adjustment matrix for ${\cals(7)}$. The matrices $E$ and $EA{^{-1}}$ are given in \[eav\].
Note that the submatrix consisting of the first three rows of $EA{^{-1}}$ is the matrix $J$ from \[somedecoddsec\], in agreement with \[allodd2\]. We leave the reader to try to work out the pattern in the rest of this matrix. We cannot even see why the entries of $EA{^{-1}}$ are necessarily non-negative!
Another point to observe in the matrix $E$ is the sixth row, which shows a spin character whose $2$-modular reduction just has two equal composition factors. This is relevant to the discussion of the double cover of the alternating group in \[ansec\]: it shows that if $\la$ is one of $$(19,7,4,3,2),(21,9,5,4,2,1),(23,11,7,4,3,2),(25,13,9,5,4,2,1),\dots$$ then the ordinary spin characters of ${\tilde{\mathfrak{A}}_}n$ labelled by $\la$ remain irreducible in characteristic $2$.
Index of notation {#indexnotnsec}
=================
For the reader’s convenience we conclude with an index of the notation we use in this paper. We provide references to the relevant subsections.
### Basic objects {#basic-objects .unnumbered}
---------------------------- ---------------------------------------------------------------- --------------------
$\bbf$ a field of characteristic $2$
$\sss n$ the symmetric group of degree $n$ \[snsec\]
${\mathfrak{A}_}n$ the alternating group of degree $n$ \[ansec\]
${\calh_}n$ the Iwahori–Hecke algebra of $\sss n$ over $\bbc$, with $q=-1$ \[snsec\]
${\cals(n)}$ the Schur algebra of degree $n$ over $\bbf$ \[snsec\]
${\cals^\circ(n)}$ the $q$-Schur algebra of degree $n$ over $\bbc$, with $q=-1$ \[snsec\]
${\tilde{\mathfrak{S}}_}n$ a double cover of $\sss n$ \[doublecoversec\]
---------------------------- ---------------------------------------------------------------- --------------------
### Partitions {#partitions .unnumbered}
----------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------
$\calp$ the set of all partitions \[partnsec\]
$\calp(n)$ the set of all partitions of $n$ \[partnsec\]
$\cald$ the set of all $2$-regular partitions \[partnsec\]
$\cald(n)$ the set of all $2$-regular partitions of $n$ \[partnsec\]
$\dom$ the dominance order on $\calp(n)$ \[partnsec\]
$a\la$ the partition $(a\la_1,a\la_2,\dots,)$ \[partnsec\]
$\la+\mu$ the partition $(\la_1+\mu_1,\la_2+\mu_2,\dots)$ \[partnsec\]
$\la\sqcup\mu$ the partition obtained by arranging all the parts of $\la$ and $\mu$ together in decreasing order \[partnsec\]
${\la{\downarrow}_{i}}$ the partition obtained by removing all the $i$-[spin-removable nodes]{}of $\la$ \[partnsec\]
$(\la^{(0)},\la^{(1)})$ the $2$-quotient of $\la$ \[2coresec\]
$\epsilon(\mu)$ the $2$-sign of $\mu$ \[2coresec\]
${\operatorname{ev}(\la)}$ the number of positive even parts of $\la$ \[irrdecsec\]
$\cald^+(n)$ $\lset{\la\in\cald(n)}{{\operatorname{ev}(\la)}\text{ is even}}$ \[irrdecsec\]
$\cald^-(n)$ $\lset{\la\in\cald(n)}{{\operatorname{ev}(\la)}\text{ is odd}}$ \[irrdecsec\]
$a^\alpha_{\beta\gamma}$ the Littlewood–Richardson coefficient corresponding to $\alpha,\beta,\gamma\in\calp$ with $|\alpha|=|\beta|+|\gamma|$ \[inversesec\]
$\kappa(\alpha,\mu)$ $a^\alpha_{\mu^{(0)}\mu^{(1)}}$ if $\mu$ has empty $2$-core, $0$ otherwise \[inversesec\]
$\la{^{\operatorname{reg}}}$ the $2$-regularisation of $\la\in\calp(n)$ \[regnsec\]
$\la{^{\operatorname{dbl}}}$ the double of $\la\in\cald(n)$ \[regnsec\]
$\la{^{\operatorname{dblreg}}}$ $(\la{^{\operatorname{dbl}}}){^{\operatorname{reg}}}$ \[regnsec\]
$\mu{\ifthenelse{\equal{r}1}{\stackrel{i}\longrightarrow}{\stackrel{i^r}\longrightarrow}}\la$ $\la$ is obtained from $\mu$ by adding $r$ nodes of residue $i$ \[brnchsec\]
$\mu{\ifthenelse{\equal{r}1}{\stackrel{i}\Longrightarrow}{\stackrel{i^r}\Longrightarrow}}\la$ $\la$ is obtained from $\mu$ by adding $r$ $i$-[spin-addable nodes]{} \[brnchsec\]
${\scrx_3}$ the set of [$2$-regular partitions]{}$\tau+4\alpha$, where $\tau=(4l-1,4l-5,\dots,3)$ for $l\gs0$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l$ \[mainproofsec\]
${\scrx_1}$ the set of [$2$-regular partitions]{}$\tau+4\alpha$, where $\tau=(4l-3,4l-7,\dots,1)$ for $l\gs1$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l$ \[mainproofsec\]
${\scry_3}$ the set of [$2$-regular partitions]{}$\tau+4\alpha\sqcup(2)$, where $\tau=(4l-1,4l-5,\dots,3)$ for $l\gs0$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l$ \[mainproofsec\]
${\scry_1}$ the set of [$2$-regular partitions]{}$\tau+4\alpha\sqcup(2)$, where $\tau=(4l-3,4l-7,\dots,1)$ for $l\gs1$ and $\alpha$ is a $2$-Carter partition with ${l(\alpha)}\ls l-1$ \[mainproofsec\]
${\scrz}$ the set of [$2$-regular partitions]{}$(2b)$ or $(4b-2,1)$ for some $b\gs2$ \[mainproofsec\]
${\scri}$ ${\scrx_3}\cup{\scrx_1}\cup{\scry_3}\cup{\scry_1}\cup{\scrz}\cup\{(3,2,1)\}$ \[mainproofsec\]
----------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------
### Modules and characters {#modules-and-characters .unnumbered}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------
${\mathrm S^{\la}}$ the Specht module for $\sss n$ or ${\calh_}n$ corresponding to $\la\in\calp(n)$ \[irrdecsec\]
${\mathrm D^{\la}}$ the James module for $\sss n$ or ${\calh_}n$ corresponding to $\la\in\cald(n)$ \[irrdecsec\]
${\Delta^{\la}}$ the Weyl module for ${\cals(n)}$ or ${\cals^\circ(n)}$ corresponding to $\la\in\calp(n)$ \[irrdecsec\]
${\mathrm L^{\la}}$ the irreducible module for ${\cals(n)}$ or ${\cals^\circ(n)}$ corresponding to $\la\in\calp(n)$ \[irrdecsec\]
$D_{\la\mu}$ the decomposition number ${[{\Delta^{\la}}:{\mathrm L^{\mu}}]}$ for ${\cals(n)}$ (equals the decomposition number ${[{\mathrm S^{\la}}:{\mathrm D^{\mu}}]}$ for $\bbf\sss n$ if $\mu\in\cald(n)$) \[irrdecsec\]
${\mathring D}_{\la\mu}$ the decomposition number ${[{\Delta^{\la}}:{\mathrm L^{\mu}}]}$ for ${\cals^\circ(n)}$ (equals the decomposition number ${[{\mathrm S^{\la}}:{\mathrm D^{\mu}}]}$ for ${\calh_}n$ if $\mu\in\cald(n)$) \[irrdecsec\]
$A_{\la\mu}$ the $(\la,\mu)$ entry of the adjustment matrix for ${\cals(n)}$ \[irrdecsec\]
${\llbracket\la\rrbracket}$ the character of ${\mathrm S^{\la}}$ over $\bbc$ \[irrdecsec\]
${\varphi(\la)}$ the Brauer character of ${\mathrm D^{\la}}$ \[irrdecsec\]
${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ the irreducible spin character labelled by $\la\in\cald^+(n)$ \[irrdecsec\]
${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+,{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$ the irreducible spin characters labelled by $\la\in\cald^-(n)$ \[irrdecsec\]
${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}$ either ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_+$ or ${{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}_-$, when $\la\in\cald^-(n)$ \[irrdecsec\]
${\left(\ \ \!:\!\ \ \right)}$ the standard inner product on characters \[irrdecsec\]
${\widebar{\chi}}$ the $2$-modular reduction of a character $\chi$ \[irrdecsec\]
${D^{\mathsf{spn}}_{\la\mu}}$ the decomposition number ${[{{\widebar{{{\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}\la{\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.7\wd\@brx\usebox{\@brx}}}}}}}:{\varphi(\mu)}]}$ \[irrdecsec\]
${\operatorname{prj}(\mu)}$ the character of the projective cover of ${\mathrm D^{\mu}}$ \[irrdecsec\]
${\operatorname{deg}}(\chi)$ the degree of a character $\chi$ \[degsec\]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------
### Branching rules {#branching-rules .unnumbered}
----------------------------------------------------- ----------------------------------------------------------------------------- --------------
$\chi{{\downarrow}}_{{\tilde{\mathfrak{S}}_}{n-1}}$ the restriction of $\chi$ to ${\tilde{\mathfrak{S}}_}{n-1}$ \[brnchsec\]
$\chi{{\uparrow}}^{{\tilde{\mathfrak{S}}_}{n+1}}$ the character obtained by inducing $\chi$ to ${\tilde{\mathfrak{S}}_}{n+1}$ \[brnchsec\]
${\mathrm{e}_{i}}$ Robinson’s $i$-restriction functor \[brnchsec\]
${\mathrm{f}_{i}}$ Robinson’s $i$-induction functor \[brnchsec\]
${{\mathrm{e}_{i}}^{(r)}}$ ${\mathrm{e}_{i}}^r/r!$ \[brnchsec\]
${{\mathrm{f}_{i}}^{(r)}}$ ${\mathrm{f}_{i}}^r/r!$ \[brnchsec\]
$\epsilon_i\chi$ $\max\lset{r\gs0}{{\mathrm{e}_{i}}^r\chi\neq0}$ \[brnchsec\]
$\varphi_i\chi$ $\max\lset{r\gs0}{{\mathrm{f}_{i}}^r\chi\neq0}$ \[brnchsec\]
${{{\mathrm{e}_{i}}^{(\operatorname{max})}}}\chi$ ${{\mathrm{e}_{i}}^{(\epsilon_i\chi)}}\chi$ \[brnchsec\]
${{{\mathrm{f}_{i}}^{(\operatorname{max})}}}\chi$ ${{\mathrm{f}_{i}}^{(\varphi_i\chi)}}\chi$ \[brnchsec\]
----------------------------------------------------- ----------------------------------------------------------------------------- --------------
### Symmetric functions {#symmetric-functions .unnumbered}
-------------------------- --------------------------------------------------- --------------
${\mathrm{Sym}}$ the space of symmetric functions \[symfnsec\]
$s_\la$ the Schur function corresponding to $\la\in\calp$ \[symfnsec\]
$h_r$ the $r$th complete homogeneous function \[symfnsec\]
$h_\la$ $h_{\la_1}h_{\la_2}\dots$ \[symfnsec\]
$e_r$ the $r$th elementary symmetric function \[symfnsec\]
$e_\la$ $e_{\la_1}e_{\la_2}\dots$ \[symfnsec\]
${\lan\ \ ,\ \ \ran}$ standard inner product on ${\mathrm{Sym}}$ \[symfnsec\]
${\spadesuit_{\la,\mu}}$ the coefficient of $e_\mu$ in $h_\la$ \[symfnsec\]
-------------------------- --------------------------------------------------- --------------
[99]{}
|
---
abstract: 'An extension of the classic Enneper–Weierstrass representation for conformally parametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional $CP^1$ and $CP^2$ sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3- and 8-dimensional spaces, respectively. Relations of Weierstrass type systems to the equations of these sigma models are established. In particular, it is demonstrated that the generalised Weierstrass representation can admit different CMC-surfaces in ${\R}^3$ which have globally the same Gauss map. A new procedure for constructing CMC-surfaces in ${\R}^n$ is presented and illustrated in some explicit examples.'
---
\[grundland-firstpage\]
Introduction
============
In this paper we study two-dimensional surfaces conformally immersed into multi dimensional spaces with Euclidean metric. We present explicit formulae for the position vector $X:
{\mathcal D} \in {\mathbb C} \rightarrow {\R}^3$ of a surface for which $X$ satisfies the Gauss–Weingarten and Gauss–Codacci equations. Such formulae describing minimal surfaces (i.e. zero mean curvature $H=0$) imbedded in three-dimensional space were first formulated by Enneper [@Enneper] and Weierstrass [@Weierstrass] about one and half century ago. These authors considered two holomorphic functions $\psi(z)$ and $\phi(z)$ of a complex variable $z \in {\mathbb C}$ and introduced a three component complex vector valued function $w=(w_1,w_2,w_3): {\mathcal D}
\rightarrow {\mathbb C}^3$ which is required to satisfy the following linear differential equations $$\begin{gathered}
\partial w_1 = \frac{1}{2} \left(\psi^{2} - \phi^{2}\right),
\qquad
\partial w_2 = \frac{i}{2}\left(\psi^{2} + \phi^{2}\right),
\qquad
\partial w_3= - \psi \phi, \nonumber\\
\bar{\partial} \psi = 0, \qquad \bar{\partial} \phi = 0,\label{1.1}
\end{gathered}$$ where the derivatives are abbreviated $\partial = \partial/ \partial z$ and the bar denotes the complex congugation. Then they showed that the real vector valued functions $$\label{1.2}
X = \left( {\rm Re} \int_{0}^{z} \frac{1}{2}\left(\psi^{2} - \phi^{2}\right)\,dz',
{\rm Re} \int_{0}^{z} \frac{i}{2}\left(\psi^{2} + \phi^{2}\right)\,dz',
- {\rm Re} \int_{0}^{z} \psi \phi\,dz'\right)$$ can be considered as components of a position vector of a minimal surface, immersed into ${\R}^3$, with the conformal metric $$\label{1.3}
ds^2 = \left(|\psi|^{2} + |\phi|^{2}\right)\sp2\,dz\,d \bar{z},$$ where $z$ and $\bar{z}$ are local coordinates on ${\cal D}$ and the coordinate lines $z={\rm const}$, and $\bar z={\rm const}$ describe geodesics on this surface. Since then this idea has been developed by many authors, for a review of the subject see e.g. [@book1; @K; @book2] and references therein. The theory of minimal, or in general, constant mean curvature (CMC)-surfaces plays an important role in several applications to problems arising both in mathematics and in physics. In particular, many interesting applications can be found in such diverse areas of physics as: the fields of two-dimensional gravity [@book3; @Carroll], string theory [@Land], quantum field theory [@Vis; @book3], statistical physics [@book5; @book6], fluid dynamics [@book7], theory of fluid membranes [@Z; @book5]. One interesting application involves the Canham–Helfrich membrane model [@Can; @Hel]. This model can explain some basic features and equilibrium shapes both for biological membranes and liquid interfaces [@book8].
Our approach involves modifying the original Enneper–Weierstrass representation (\[1.2\]) by adding to it extra terms. For this purpose it is convenient to exploit the connection between Weierstrass systems, $CP^1$ and $CP^2$ sigma model equations, and their Lax representations. We demonstrate that, through these links, conformal immersion of CMC-surfaces into 3- and 8-dimensional spaces can be formulated. We show that a large classes of solutions of the Weierstrass system can be obtained and, consequently, can provide new classes of conformally parametrised CMC-surfaces in multi-dimensional spaces.
The paper is organized as follows. In Section 2, we rederive the classical Enneper–Weierstrass representation for minimal surfaces immersed into ${\R}^3$. In the next section we describe, in detail, the generalised Weierstrass formulae for CMC-surfaces into ${\R}^3$ in the context of the $CP^1$ sigma model and discuss some geometric aspects of $CP^1$ maps. The following section deals with $CP^2$ maps and the corresponding Weierstrass representation for conformally parametrised surfaces immersed into ${\R}^8$. It also presents some geometric characteristics of CMC-surfaces. In Section 5 our theoretical considerations are illustrated by explicit examples and new interesting CMC-surfaces are found. The last section presents final remarks and mentions possible future developments.
The Enneper–Weierstrass formulae\
for minimal surfaces in $\boldsymbol{{\R}^{3}}$
===============================================
Let ${\mathbb M}^2$ be a smooth orientable surface in $3$-dimensional Euclidean space ${\R}^3$. The surface ${\mathbb M}^2$ is described by a real vector-valued function $$\label{2.1}
X = (X_{1}, X_{2}, X_{3}) : D \rightarrow {\R}^3,$$ where $D$ is a region in the complex plane ${\mathbb C}$. The metric is assumed to be conformally flat $$\label{2.2}
ds^2 = e^{2u} \, dz \, d \bar{z}$$ for any real valued function $u$ of $z$ and $\bar{z}$. The conformal parametrisation of the surface ${\mathbb M}^2$ implies the following normalization of the position vector $X(z, \bar{z})$ $$\label{2.3}
( \partial X, \partial X) = 0,\qquad
( \partial X, \bar{\partial} X) = \frac{1}{2} e^{2u},$$ where the brackets $(\cdot ,\cdot )$ denote the standard scalar product in ${\R}^3$. The tangent vectors $\partial X$ and $\bar{\partial} X$ and the real unit normal vector $N$ on the surface ${\mathbb M}^2$ satisfy the obvious relations $$\label{2.4}
(\partial X, N) = 0, \qquad
(N, N) = 1.$$
Equations of a moving complex frame $\xi = ( \partial X, \bar{\partial}
X, N)^{T}$ satisfy the following Gauss–Weingarten equations (see e.g. \[5\]) $$\label{2.6}
\partial \xi = U \xi, \qquad
\bar{\partial} \xi = V \xi,$$ where $3\times 3$ matrices $U$ and $V$ have the form $$\label{2.7}
U = \left(
\begin{array}{ccc}
2 \partial u & 0 & J \\
0 & 0 & \frac{1}{2} H e^{2u} \\
-H & -2 e^{-2u} J & 0 \\
\end{array} \right), \qquad
V= \left(
\begin{array}{ccc}
0 & 0 & \frac{1}{2} H e^{2u} \\
0 & 2 \bar{\partial} u & \bar{J} \\
-2 e^{-2u} \bar{J} & - H & 0 \\
\end{array} \right),$$ and the following notation has been introduced $$\label{2.8}
J = (\partial^2 X, N), \qquad H = 2 e^{-2u}
(\partial \bar{\partial} X, N).$$ Formulae (\[2.6\]) are compatible with the scalar products (\[2.3\]) and (\[2.4\]). From (\[2.6\]) and (\[2.7\]) we can derive the equation for the unit normal vector $N$ $$\label{2.9}
\partial \bar{\partial} N + (\partial N,
\bar{\partial} N ) N + \bar{\partial} H
\partial X + \partial H \bar{\partial} X= 0.$$ The corresponding Gauss–Codazzi equations of the conformally parametrised surface ${\mathbb M}^2\!\subset {\R}^3$ are the compatibility conditions of equations (\[2.6\]) and have the following form $$\begin{gathered}
\partial \bar{\partial} u + \frac{1}{4} H^2 e^{2u}
- 2 |J|^2 e^{-2u} = 0, \label{2.10}\\
\bar{\partial} J = \frac{1}{2} \partial H e^{2u},
\qquad \partial \bar{J} = \frac{1}{2} \bar{\partial} H e^{2u}.
\label{2.11}
\end{gathered}$$
The aim of this section is to rederive the original Enneper–Weierstrass formulae [@Enneper; @Weierstrass] for inducing minimal surfaces in ${\R}^3$. For surfaces with $H=0$ the formulae given above simplify considerably. We focus our attention on the construction of the explicit formula for the position vector $X(z, \bar{z})$ of conformally parametrised surfaces into ${\R}^3$ for which equations (\[2.3\]), (\[2.10\]) and (\[2.11\]) are fulfilled.
For computational purposes, it is useful to examine equations (\[2.3\]), (\[2.10\]) and (\[2.11\]) in terms of a two-component object which, in fact, is a spinor, but its spinorial nature is not relevant to our discussion: $\phi = (\psi_{1}, \psi_{2})^{T} \in {\mathbb C}^2$. We show that, by quadratures, we can determine the coordinates of the position vector $X(z, \bar{z})$ in terms of the components of $\phi$ satisfying equations (\[2.3\]) and (\[2.10\])–(\[2.11\]).
Let us consider the complex vector $\vec{w}$ in ${\mathbb C}^3$ equal to one of the tangent vectors, say, $\partial X$ $$\label{2.12}
\vec{w} = ( w_{1}, w_{2}, w_{3}) = \partial X,
\qquad w_{i} \in {\mathbb C}, \quad i=1,2,3,$$ the $2$ by $2$ traceless matrix $$\label{2.13}
w = \left(
\begin{array}{cc}
w_{3} & w_{1} - i w_{2} \\
w_{1} + i w_{2} & - w_{3} \\
\end{array} \right),
\qquad \mbox{tr}\, w = 0,$$ and the map $$\label{2.14}
\vec{w} : {\mathbb C} \rightarrow w = w_{i} \sigma_{i}\in sl(2,{\mathbb C}),$$ where $\sigma_{i}$ are the Pauli matrices $$\label{2.15}
\sigma_{1} = \left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array} \right), \qquad
\sigma_{2} = \left(
\begin{array}{cc}
0 & -i \\
i & 0 \\
\end{array} \right), \qquad \sigma_{3} = \left(
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array} \right).$$ The map (\[2.14\]) satisfies $$\label{2.16}
\vec{w}^2 = - \det w.$$ From (\[2.16\]), the determinant of the matrix $w$ vanishes if and only if the vector $\vec{w}$ is null, which coincides with the first condition in (\[2.3\]). Hence, using (\[2.13\]), we can express uniquely the null vector $\vec{w}$ in terms of the complex two-component vector $\phi$ as follows $$\label{2.17}
w_{1} = \frac{1}{2} \left( \psi_{1}^{2} - \psi_{2}^{2}\right),
\qquad
w_{2} = \frac{i}{2} \left( \psi_{1}^{2} + \psi_{2}^{2}\right),
\qquad
w_{3} = - \psi_{1} \psi_{2}.$$ From the assumption (\[2.12\]) that the null vector $\vec{w}$ is equal to the tangent vector $\partial X$, we can express $\partial X$ in terms of $\psi_{1}$ and $\psi_{2}$ as follows: $$\label{2.18}
\partial X_{1} = \frac{1}{2} \left( \psi_{1}^2 - \psi_{2}^2\right),
\qquad
\partial X_{2} = \frac{i}{2} ( \psi_{1}^2 + \psi_{2}^2),
\qquad
\partial X_{3} = - \psi_{1} \psi_{2},$$ which coincide with expression (\[1.1\]). The Enneper–Weierstrass representation for surfaces in ${\R}^3$ are obtained under the additional assumption that $\psi_{1}$ and $\psi_{2}$ are arbitrary holomorphic functions of the complex variable $ z \in {\mathbb C}$. Then, integrating equations (\[2.18\]) and taking into account the reality condition of the position vector $$\label{2.19}
X(z, \bar{z}) = \bar{X} ( z, \bar{z}),$$ we obtain the following representation [@Weierstrass] $$\begin{gathered}
X_{1} = \frac{1}{2} \int_{0}^{z}
\left( \psi_{1}^{2} - \psi_{2}^2\right) dz'
+ \frac{1}{2} \int_{0}^{\bar{z}}
\left( \bar{\psi}_{1}^2 - \bar{\psi}_{2}^2\right) dz',
\nonumber\\
X_{2} = \frac{i}{2} \int_{0}^{z}
\left(\psi_{1}^2 + \psi_{2}^{2}\right) dz'
- \frac{i}{2} \int_{0}^{\bar{z}}
\left( \bar{\psi}_{1}^2 + \bar{\psi}_{2}^2\right) d \bar{z}',
\nonumber\\
X_{3} = - \int_{0}^{z} \psi_{1} \psi_{2} \,dz'
- \int_{0}^{\bar{z}} \bar{\psi}_{1} \bar{\psi}_{2} \, d \bar{z},
\label{2.20}
\end{gathered}$$ which, in fact, is equivalent to (1.2). Next, from (\[2.20\]) and invoking the second condition (\[2.3\]) we find that $$\label{2.21}
u = \ln \left(|\psi_{1}|^2 + |\psi_{2}|^2\right).$$ Substituting (\[2.21\]) into the Gauss–Codazzi equations (\[2.10\])–(\[2.11\]), we obtain $$\label{2.23}
\partial \bar{\partial} \ln p^2
-2 |J|^2 p^{-2} = 0,$$ where $$\label{2.22}
p= |\psi_{1}|^2 + |\psi_{2}|^2.$$ By virtue of (\[2.20\]), we find that $J$ and $H$ defined by (2.7), when expressed in terms of $\psi_1$ and $\psi_2$ become $$\label{2.24}
J = \psi_{1} \partial \psi_{2} - \psi_{2} \partial \psi_{1},\qquad H=0$$ and $J$ is analytic, i.e. $
\bar{\partial} J = 0$.
Note that the direction of $\phi = ( \psi_{1}, \psi_{2})^{T}$ is arbitrary, but its length is fixed by (\[2.23\]). Note also that after the change of variable $\varphi = 2 \ln p$ equation (\[2.23\]) becomes $$\label{2.26}
\partial \bar{\partial} \varphi
= 2 |J|^{2} e^{- \varphi},
\qquad
\bar{\partial} J = 0.$$
The generalised Weierstrass formulae\
for CMC-surfaces in $\boldsymbol{{\R}^3}$
=========================================
The Wierstrass–Enneper formulae for inducing minimal surfaces, and their generalisations, has been studied for a long time by many authors (e.g. [@book1; @book9; @book12] and references therein). This topic has most recently been treated by B Konopelchenko and I Taimanov [@Tai]. In this paper, Konopelchenko and Taimanov, established a direct connection between certain classes of CMC-surfaces and an integrable finite-dimensional Hamiltonian system. For a summary of their results, see [@Land2]. There it is shown that to any solution $\phi = (\psi_{1}, \psi_{2})^{T}$ of the first order equations (which resemble Dirac equations) $$\label{3.1}
\partial \psi_{1} = p \psi_{2},\qquad \bar{\partial} \psi_{2} = - p
\psi_{1}, \qquad p = |\psi_{1}|^2 + | \psi_{2}|^2,$$ one can associate a CMC-surface immersed into ${\R}^3$ with radius vector $X(z, \bar{z})$ of the form (\[2.20\]) $$\begin{gathered}
X_{1} = \int_{\gamma} \left(\psi_{1}^2 - \psi_{2}^2 \right) dz'
+ \left( \bar{\psi}_{1}^{2} - \bar{\psi}_{2}^2\right) d \bar{z}',
\nonumber\\
X_{2} = \int_{\gamma} \left(\psi_{1}^2 + \psi_{2}^2\right) dz'
- \left(\bar{\psi}_{1}^2 + \bar{\psi}_{2}^2\right) d \bar{z}',
\nonumber\\
X_{3} = - \int_{\gamma} \psi_{1} \psi_{2} dz'
+ \bar{\psi}_{1} \bar{\psi}_{2} d\bar{z}',
\label{3.2}
\end{gathered}$$ where $\gamma$ is an arbitrary curve, which does not depend on the trajectory but only on its endpoints $z$ in ${\mathbb C}$. Note that these equations are really written for surfaces with $H=1$. To see how to reduce more general cases down to this case – see [@Tai]. Note further that $\psi_1$ and $\psi_2$ are now functions of both $z$ and $\bar z$. The formulae (\[3.1\]) are the starting point of our analysis of CMC-surfaces in this paper, and according to [@K], we will refer to system (\[3.1\]) as the generalised Weierstrass (GW) system.
In this paper, we examine certain aspects of CMC-surfaces in ${\R}^n$ in the context of relating them to solutions of low dimensional sigma models. In particular, we focus our attention on constructing a Weierstrass representation for generic two-dimensional surfaces immersed in ${\R}^8$, whose explicit form has not been known up to now. For the sake of convenience our investigation starts with a derivation of the position vector $X (z, \bar{z})$ of a surface in ${\R}^3$ from the Lax pair for a GW system (\[3.1\]). As it was shown in [@Bra3] the GW system (\[3.1\]) is in a one-to-one correspondence with the solutions of the equations of the completely integrable two-dimensional Euclidean $CP^1$ sigma model $$\label{3.3}
[ \partial \bar{\partial} P, P ] = 0,$$ where $P$ is a projector $$\label{3.4}
P = \frac{1}{A} \left(
\begin{array}{cc}
1 & \bar{w} \\
w & |w|^2 \\
\end{array}\right)
,\qquad
A = 1 + |w|^2,$$ or equivalently, the solutions of $$\label{3.5}
\partial \bar{\partial} w - \frac{2 \bar{w}}{1 + |w|^{2}} \partial w
\bar{\partial} w = 0.$$ In [@Ken] it was shown that if $\psi_{1}$ and $\psi_{2}$ are solutions of the GW system (\[3.1\]), then function $w$ defined by $$\label{3.6}
w= \frac{\psi_{1}}{\bar{\psi}_{2}},$$ is a solution of the equations of the $CP^1$ sigma model, namely, (\[3.5\]). The converse is also true [@Bra3]. Thus, if $w$ is a solution of (\[3.5\]), then $\psi_{1}$ and $\psi_{2}$ of the GW system (\[3.1\]) have the form $$\label{3.7}
\psi_{1} = \epsilon w \frac{(\bar{\partial} \bar{w})^{1/2}}{1 + |w|^2},
\qquad
\psi_{2} = \epsilon \frac{(\partial w)^{1/2}}{1 + |w|^2},
\qquad
p = \frac{|\partial w|}{1 + |w|^2},
\qquad \epsilon = \pm 1.$$ Note that equation (\[2.9\]) with $H=1$ for the unit normal vector $N= (n_{1}, n_{2}, n_{3})$ to a CMC-surface adopts the well known form of the equation of the $SO(3)$ sigma model $$\label{3.8}
\partial \bar{\partial} N + (\partial N, \bar{\partial} N) N = 0,
\qquad
(N,N)=1.$$ Combining the map of the unit vector $N$ onto the unit sphere $S^2$ with the stereographic projection, we obtain the Gauss map $$\label{3.9}
w = \frac{n_{1} + in_{2}}{1 + n_{3}} = \frac{\psi_{1}}{\bar{\psi}_{2}}.$$ which satisfies the $CP^1$ model equation (\[3.5\]).
However, as shown by Zakharov and Mikhailov [@book10] the equation (\[3.3\]) can be considered as a compatibility condition for two linear spectral problems $$\label{3.10}
\partial \Psi = \frac{2}{1 + \lambda} [ \partial P, P] \Psi,
\qquad
\bar{\partial} \Psi = \frac{2}{1- \lambda} [ \bar{\partial} P, P ],$$ where $\lambda \in {\mathbb C}$ is a spectral parameter. The compatibility condition for (\[3.10\]) can also be written the form of a conservation law $$\label{3.11}
\partial K - \bar{\partial} K^{\dagger} = 0,$$ where the traceless 2 by 2 matrices $K$ and $K^{\dagger}$ expressed in terms of $w$ have the form $$\begin{gathered}
K = [ \bar{\partial} P, P] = \frac{1}{A^2} \left(
\begin{array}{cc}
\bar{w} \bar{\partial} w - w \bar{\partial} \bar{w} &
\bar{\partial} \bar{w} + \bar{w}^2 \partial w \vspace{1mm}\\
- \bar{\partial} w - w^2 \bar{\partial} \bar{w} &
w \bar{\partial} \bar{w} - \bar{w} \bar{\partial} w \\
\end{array} \right)
\nonumber\\
- K^{\dagger} = [ \partial P, P] = \frac{1}{A^2} \left(
\begin{array}{cc}
\bar{w} \partial w - w \partial \bar{w} &
\partial \bar{w} + \bar{w}^2 \partial w \vspace{1mm}\\
- \partial w - w^2 \partial \bar{w} &
w \partial \bar{w} - \bar{w} \partial w \\
\end{array} \right),
\label{3.12}
\end{gathered}$$ and the Hermitian conjugate is denoted by $\dagger$.
Next we derive the explicit form of matrices $K$ and $K^{\dagger}$ in terms of $\psi_{1}$ and $\psi_{2}$ in order to find the corresponding conservation laws for the GW system (\[3.5\]). For computational purposes, it is useful to express the first derivatives of $w$ in terms of $\psi_{1}$ and $\psi_{2}$. Note that in the $CP^1$ case the quantity $J$ defined in (\[2.8\]) is given by $$J =\frac{\partial W \partial \bar{W}}{A\sp2}.$$ Using (3.2) we find that it is given by $$\label{3.13}
J =\bar{\psi}_{1} \partial
\psi_{2} - \psi_{2} \partial \bar{\psi}_{1},$$ and so is a holomorphic function, i.e. which satisfies, $$\label{3.14}
\bar{\partial} J = \bar{\partial} (\bar{\psi}_{1}
\partial \psi_{2} - \psi_{2} \partial \bar{\psi}_{1})
= - p \partial p + p \partial p = 0,$$ whenever (\[3.1\]) holds. Actually, in the $CP^1$ case $J$ is a component of the energy-momentum tensor. Note also that (3.13) is different from (2.23). It becomes, formally, equal to it under the substitution $\bar \psi_1 \rightarrow \psi_1$. Using equations (\[3.1\]), (\[3.6\]) and (\[3.13\]) we can express the first derivatives of $w$ in terms of $\psi_{1}$, $\psi_{2}$ and $J$ $$\label{3.15}
\partial w = A^{2} \psi_{2}\sp2 ,\qquad
\bar{\partial} w = - \bar{J} \bar{\psi}_{2}^{-2},$$ where $$\label{3.16}
A= 1 + \frac{|\psi_{1}|^2}{|\psi_{2}|^2}.$$ As a consequence of (\[3.11\]) and (\[3.12\]) we find that the GW system possesses at least three further conservation laws $$\begin{gathered}
\partial (\psi_{1} \bar{\psi}_{2} + \bar{R} \bar{\psi}_{1} \psi_{2})
- \bar{\partial} ( \bar{\psi}_{1} \psi_{2} + R \psi_{1} \bar{\psi}_{2}) =
0, \nonumber\\
\partial \left(\psi^2_{1} - \bar{R} \psi_{2}^{2}\right)
+ \bar{\partial} \left( \psi_{2}^{2} - R \psi_{1}^{2}\right) = 0,
\nonumber\\
\partial \left( \bar{\psi}_{2}^{2} - \bar{R} \bar{\psi}_{1}^{2}\right)
+ \bar{\partial} \left( \bar{\psi}_{1}^{2} - R \bar{\psi}_{2}^{2}\right) = 0,
\label{3.17}
\end{gathered}$$ where we have introduced $$\label{3.18}
R = \frac{J}{p^2}.$$ Note that formulae (\[3.17\]) differ from the conservation laws derived in [@K] as they contain additional terms involving $R$. If we put $R=0$ in equations (\[3.17\]) then we recover the expressions given in [@K] $$\label{3.19}
\partial (\psi_{1} \bar{\psi}_{2})- \bar{\partial}
( \bar{\psi}_{1} \psi_{2}) = 0, \qquad
\partial \left(\psi_{1}^{2}\right) + \bar{\partial} \left(\psi_{2}^{2}\right) =0,
\qquad
\partial \left(\bar{\psi}_{2}^{2}\right) + \bar{\partial} \left( \bar{\psi}_{1}^{2}\right) = 0.$$ As a result of the conservation laws (\[3.17\]), we can introduce three real-valued functions $X_{i} (z, \bar{z})$ given by $$\begin{gathered}
X_{1} = \frac{i}{2} \int_{\gamma}
\left[ \bar{\psi}_{1}^{2} + \psi_{2}^{2} - R \left( \psi_{1}^{2}
+ \bar{\psi}_{2}^{2}\right)\right] dz'
-\left [ \psi_{1}^{2} + \bar{\psi}_{2}^{2} - R \left(\bar{\psi}_{1}^{2}
+ \psi_{2}^{2}\right) \right] d \bar{z}', \nonumber\\
X_{2} = \frac{1}{2} \int_{\gamma}
\left[\bar{\psi}_{1}^{2} - \psi_{2}^{2} + R \left(\psi_{1}^{2} -
\bar{\psi}_{2}^{2}\right)\right] dz'
+ \left[ \psi_{1}^{2} - \bar{\psi}_{2}^{2} + R \left( \bar{\psi}_{1}^{2} -
\psi_{2}^{2}\right)\right] d \bar{z}',
\nonumber\\
X_{3} = - \int_{\gamma} \left[ \bar{\psi}_{1} \psi_{2} + R \psi_{1}
\bar{\psi}_{2}\right] dz'
+ \left[ \psi_{1} \bar{\psi}_{2} + \bar{R} \bar{\psi}_{1} \psi_{2}\right] d
\bar{z}',
\label{3.20}
\end{gathered}$$ where $\gamma$ is any curve from a fixed point $z$ in ${\mathbb C}$. The functions $X_{i}$, $i=1,2,3$ can be considered as components of a position vector of a surface locally parametrised by $z$ and $\bar{z}$ and immersed in ${\R}^{3}$ $$\label{3.21}
X(z, \bar{z}) = ( X_{1} (z, \bar{z}), X_{2} (z, \bar{z}), X_{3} (z,
\bar{z})).$$ Using conformal changes of coordinates on the surface ${\mathbb M}^2$ we can, without loss of generality, put $J=1$ (when, of course $J\ne0$). As a consequence it is easy to show that representation (\[3.20\]) with $R=1/p^2$ cannot be reduced to the Weierstrass formulae (\[3.2\]). This means, as we will see latter, that the additional terms involving $R$ play an important role in the construction of surfaces in ${\R}^3$.
The tangents and the normal unit vector to the surface ${\mathbb M}^2$ are given by $$\begin{gathered}
\partial X = \left(i \left[ \bar{\psi}_{1}^{2} + \psi_{2}^{2} - R\left( \psi_{1}^{2} +
\bar{\psi}_{2}^{2}\right)\right], \left[ \bar{\psi}_{1}^{2} - \psi_{2}^{2} + R
\left(\psi_{1}^{2} -
\bar{\psi}_{2}^{2}\right)\right],\right.\nonumber\\
\qquad\qquad \left.{} -2 \left(\bar{\psi}_{1} \psi_{2} + R \psi_{1}
\bar{\psi}_{2}\right)\right),
\nonumber\\
\bar{\partial} X = \left(-i\left[\psi_{1}^{2} + \bar{\psi}_{2}^{2} - \bar{R}
\left(\bar{\psi}_{1}^{2} + \psi_{2}^{2}\right)\right], \left[ \psi_{1}^{2} - \bar{\psi}_{2}^{2}
+ \bar{R} \left(\bar{\psi}_{1}^{2} - \psi_{2}^{2}\right)\right],\right.\nonumber\\
\qquad\qquad \left.{} -2 \left(\psi_{1}
\bar{\psi}_{2} + \bar{R} \bar{\psi}_{1} \psi_{2}\right)\right),
\label{3.22}
\end{gathered}$$ and $$\label{3.23}
N = \frac{1}{p} \left( i \left(\bar{\psi}_{1} \bar{\psi}_{2} - \psi_{1} \psi_{2}\right),
\bar{\psi}_{1} \bar{\psi}_{2} + \psi_{1} \psi_{2},
|\psi_{1}|^{2} - |\psi_{2}|^{2}\right),$$ respectively. The first and second fundamental forms of the surface ${\mathbb M}^2$ are given by $$\begin{gathered}
I = (d X, dX) = 4 \left(J dz^2 + p^2\left(1 + |R|^2\right) dz \, d \bar{z} + \bar{J} d
\bar{z}^{2}\right), \label{3.24}
\\
II = \left(d^2 X, N\right) = \left(4 J + R + \bar{R}\right) dz^2 + \left(2 p + i (R - \bar{R})\right) dz
d \bar{z}+ \left(4 \bar{J} - R - \bar{R}\right) d \bar{z}^2.
\nonumber
\end{gathered}$$ These quadratic forms contain the Hopf differential $Jdz^2$ and are invariant under any conformal changes of coordinates. The Gauss and mean curvatures are $$\label{3.25}
K = - p^{-2} \partial \bar{\partial} \ln p,
\qquad H = 1,$$ respectively.
Note that if $J=0$ then $R=0$ and so the components of the fundamental forms (3.24) become $$\label{3.26}
g_{12} = 2 p^2, \qquad
g_{11} = g_{22} = 0, \qquad
b_{12}=2 p, \qquad
b_{11}= b_{22}=0.$$ In this case the solutions of GW system (\[3.1\]) expressed in terms of $w$ are represented by (\[3.7\]), where $w(z)$ is any holomorphic function. According to [@book11], the energy $$\label{3.27}
E = \iint_{\!\!D} \frac{\partial w \bar{\partial} w}{1 + |w|^{2}}\,
dz \wedge d \bar{z},$$ is finite when the function $w(z)$ is a ratio of polynomials in $z$. Geometrically, such functions $\psi_{i}$ parametrise an immersed sphere $S^2 \subset
{\R}^3$, since $J=0$ implies the proportionality of fundamental forms $I$ and $II$.
Note also that if in equations (\[3.24\]) we put $J \neq 0$ then there is no conformal immersion of surfaces in ${\R}^3$. Hence, equations (\[3.22\]) and (\[3.23\]) imply that the representation (\[3.20\]) can admit different CMC-surfaces which globally have the same Gauss map (\[3.9\]). This is due to the fact that the tangent vectors ${\partial X}$ and ${\bar\partial} X$ depend on $J$ while the unit normal vector $N$ is independent of $J$. In [@Abe; @Hof], using the isometric immersions, formulae similar to (\[3.22\]) and (\[3.23\]) for particular cases of isothermic surfaces have been discussed.
Let us now discuss the meaning of conservation laws (\[3.17\]). As $J$ is a holomorphic function so, according to (\[2.11\]), we are dealing with CMC-surfaces. If the $CP^1$ model is defined over $S^2$ then solutions $w$ of (\[3.5\]) are either holomorphic or antiholomorphic functions and so $J=0$. However, if the $CP^1$ model is defined on ${\R}^2$ then the function $w$ is not necessarily holomorphic or antiholomorphic and $J \neq 0$. Note that when $J\ne0$ the solutions are defined on ${\R}^2/\{a\}$, where $\{a\}$ is a small set of points of ${\R}^2$. Subtracting (\[3.19\]) from (\[3.17\]) and introducing new independent variables $\eta$ and $\bar{\eta}$ according to $$\label{3.28}
d \eta = J^{1/2} \, dz, \qquad
d \bar{\eta} = \bar{J}^{1/2} d \bar{z},
\qquad
\bar{\partial} J = 0,$$ we obtain the following set of expressions: $$\begin{gathered}
|J|^{2} \left[ \partial_{\eta} \left(\frac{\bar{\psi}_{1} \psi_{2}}{p^2}\right)
- \bar{\partial}_{\bar{\eta}} \left(\frac{\psi_{1} \bar{\psi}_{2}}{p^2}\right)\right] =
0,
\qquad
|J|^{2} \left[ \partial_{\eta} \left(\frac{\psi_{2}^{2}}{p^2}\right)
+ \bar{\partial}_{\bar{\eta}} \left( \frac{\psi_{1}}{p^2}\right)\right] = 0,
\nonumber\\
|J|^{2} \left[ \partial_{\eta} \left( \frac{\bar{\psi}_{1}^{2}}{p^2}\right)
+ \bar{\partial}_{\bar{\eta}} \left( \frac{\bar{\psi}_{2}^{2}}{p^2}\right)\right] = 0,
\label{3.29}
\end{gathered}$$ where the derivatives are abbreviated $\partial_{\eta}= \partial/
\partial \eta$ and $\bar{\partial} \bar{\eta} = \partial / \partial
\bar{\eta}$. Equations (\[3.29\]) suggest that we should consider two separate cases, namely $J=0$ which has been already treated in [@K] and $J \neq 0$. In the latter case, under the change of variables (\[3.28\]) the GW system (\[3.1\]) adopts the form $$\label{3.30}
\partial_{\eta} \psi_{1} = \frac{p}{J}\, \psi_{2},
\qquad
\bar{\partial}_{\bar{\eta}} \psi_{2} = - \frac{p}{\bar{J}}\, \psi_{1},$$ and the expression for $J$, given in (\[3.13\]), provides the following differential constraint (DC) on $\psi_{1}$ and $\psi_{2}$ $$\label{3.31}
\bar{\psi}_{1} \partial_{\eta} \psi_{2} - \psi_{2}
\partial_{\eta} \bar{\psi}_{1} = 1.$$
Keeping in mind that the complex coordinates $\eta$ and $\bar{\eta}$ are is defined up to a conformal transformation, we can without loss of generality put $J=1$. If $\psi_{1} \neq 0$ then the system (\[3.30\]), subject to DC (\[3.31\]), can be written in an equivalent form $$\label{3.32}
\partial \psi_{1} = p \psi_{2}, \qquad
\partial \psi_{2} = \bar{\psi}^{-1} ( 1 + \psi_{2} \partial
\bar{\psi}_{1}), \qquad
\bar{\partial} \psi_{2} = - p \psi_{1}.$$ The compatibility condition for (\[3.32\]) does not imply any new DC on first order derivatives of $\psi_{1}$. Hence, the system (\[3.32\]) is integrable and the derivatives $\bar{\partial} \psi_{1}$ and $\partial \psi_{1}$ are undetermined.
The Gaussian curvature and mean curvature are $$\label{3.33}
K = \left(|\psi_{2}|^2 - |\psi_{1}|^{2}\right)
\left [ |\bar{\partial} \psi_{1}|^{2} + |\psi_{2}|^{-2}
\left(1 + \psi_{2} \partial \bar{\psi}_{1} +\bar{\psi}_{2} \bar{\partial}
\psi_{1}\right)\right], \qquad H=1,$$ respectively.
Note that the equations of the complex frame (\[2.6\]) are specified by DC (\[3.31\]) and are compatible with the scalar products (\[2.3\]) and (\[2.4\]). After the change of dependent variables $$\label{3.34}
p = e^{\varphi/2},$$ the corresponding Gauss–Codazzi equations (\[2.10\])–(\[2.11\]) take the form of the elliptic Sh-Gordon equation $$\label{3.35}
\partial \bar{\partial} \varphi + 4 \sinh \varphi = 0.$$ Hence the CMC-surfaces are determined by formulae (\[3.20\]), where $\psi_{1}$ and $\psi_{2}$ have to obey equations (\[3.32\]) with $p$ determined by (\[3.34\]) and (\[3.35\]). In terms of arbitrary conformal coordinates, we have proved that $(\psi_{1}, \psi_{2}, p )$ can be viewed as the Weierstrass data of the CMC-surface ${\mathbb M}^2$ in ${\R}^3$.
To summarize: the generalised Weierstrass representation for the immersion of a CMC-surface into ${\R}^3$ is described by formulae (\[3.20\]), where $\psi_{1}$ and $\psi_{2}$ obey the GW system of equations (\[3.1\]).
Let us add also, as shown in [@Fer], that under the changes of independent variables (\[3.28\]) and dependent variables $S= 2 p^2 |J|^{-1}$, the GW system (\[3.1\]) is decoupled into a direct sum of equations $$\label{3.36}
\partial_{\eta} \bar{\partial}_{\bar{\eta}} \ln S = S^{-1} - S,
\qquad
\bar{\partial}_{\bar{\eta}} J = 0,
\qquad
\partial_{\eta} \bar{J} = 0.$$ Hence, the GW system (\[3.1\]) is completely integrable. Examples of $N$ solitons solutions can be found in [@Bra3].
The $\boldsymbol{CP^2}$ maps and the Weierstrass representation\
for surfaces in eight dimensional Euclidean spaces
================================================================
The aim of this section is to demonstrate a connection between the recently proposed generalised Weierstrass (GW) system [@G] $$\begin{gathered}
\partial \psi_{1} = \left(1+ \frac{|\psi_{2}|^2}{|\varphi_{2}|^2}\right)
\bar{\varphi}_{1} \bar{Q} - \frac{1}{2} \left[
\frac{\psi_{1} \bar{\psi}_{2}}{\varphi_{2}}
+ \frac{|\psi_{1}|^2}{|\varphi_{1}|^2}
\frac{\bar{\varphi}_{2}^2}{\bar{\varphi}_{1}}\right] \bar{P},
\label{4.1}\\
\partial \psi_{2} = \left(1 + \frac{|\psi _{1}|^2}{|\varphi_{1}|^2}\right)
\bar{\varphi}_{2} \bar{P} - \frac{1}{2} \left[
\frac{\bar{\psi}_{1} \psi_{2}}{\varphi_{1}}
+ \frac{|\psi_{2}|^2}{|\varphi_{2}|^2}
\frac{\bar{\varphi}_{1}^2}{\bar{\varphi}_{2}}\right] \bar{Q},
\label{4.2}\\
\bar{\partial} \varphi_{1} =
- \frac{1}{2} \left[\left( \frac{\psi_{2}}{\bar{\varphi}_{2}} P
+ 2 \frac{\psi_{1}}{\bar{\varphi}_{1}} Q\right) \varphi_{1} + P \psi_{1}
\frac{\varphi_{2}^2}{|\varphi_{1}|^2}\right],
\label{4.3}\\
\bar{\partial} \varphi_{2} = - \frac{1}{2} \left[\left(
\frac{\psi_{1}}{\bar{\varphi}_{1}} Q
+ 2 \frac{\psi_{2}}{\bar{\varphi}_{2}} P\right) \varphi_{2} + Q \psi_{2}
\frac{\varphi_{1}^2}{|\varphi_{2}|^2}\right],
\label{4.4}
\end{gathered}$$ where $$P = \frac{\psi_{1} \bar{\psi}_{2} \bar{\varphi}_{1}}{\varphi_{2}}
+ \left( |\varphi_{2}|^2 + |\psi_{2}|^2\right) \frac{\bar{\varphi}_{2}}{\varphi_{2}},
\qquad
Q = \frac{\bar{\psi}_{1} \psi_{2} \bar{\varphi}_{2}}{\varphi_{1}}
+ \left(|\varphi_{1}|^2 + |\psi_{1}|^2 \right) \frac{\bar{\varphi}_{1}}{\varphi_{1}}
\label{4.5}$$ and the equations of the $CP^2$ sigma model [@book11] $$\begin{gathered}
\partial \bar{\partial} w_{1} - \frac{2 \bar{w_{1}}}{A}
\partial w_{1}
\bar{\partial} w_{1} - \frac{\bar{w}_{2}}{A}
(\partial w_{1} \bar{\partial} w_{2} + \bar{\partial} w_{1} \partial
w_{2}) = 0,
\label{4.6}\\
\partial \bar{\partial} w_{2} - \frac{2 \bar{w}_{2}}{A} \partial w_{2}
\bar{\partial} w_{2} - \frac{\bar{w}_{1}}{A} (\partial w_{1}
\bar{\partial} w_{2}
+ \bar{\partial} w_{1} \partial w_{2}) = 0,
\label{4.7}\\
A = 1 + |w_{1}|^2 + |w_{2}|^2.
\label{4.8}
\end{gathered}$$
Next, we exploit this connection and use the conservation laws for the GW system (\[4.1\])–(\[4.4\]) to define real valued functions $X^{i} (z, \bar{z})$, $i=1, \ldots, 8$ in terms of functions $\varphi_{\alpha}$, $\psi_{\alpha}$, $\alpha=1,2$ which are identified as the coordinates in $8$-dim Euclidean space ${\R}^8$. The formulae (\[4.1\])–(\[4.4\]) and (\[4.6\])–(\[4.8\]) are the starting point for our analysis. In this paper, when we refer to system (\[4.1\])–(\[4.4\]), we will describe it as the modified version of the original Weierstrass system (\[3.1\]).
Note that in equations (\[4.1\])–(\[4.4\]) only four out of eight derivatives of functions $\psi_{i}$ and $\varphi_{i}$ are known in terms of complex functions $\psi_{i}$ and $\varphi_{i}$ and their complex conjugates while the others are unspecified. Note also that if the functions $\psi_{\alpha}$ tend to $\psi/\sqrt{2}$ and $\varphi_{\alpha}$ tend to $\varphi$, i.e. then the system (\[4.1\])–(\[4.4\]) reduces to the Weierstrass formulae (\[3.1\]) for the CMC-surfaces immersed in ${\R}^3$ $$\begin{gathered}
\partial \psi = \left(|\psi|^2 + |\varphi|^2\right) \varphi,
\qquad
\bar{\partial} \varphi = - \left(|\psi|^2 + |\varphi|^2\right) \psi.
\end{gathered}$$ In terms of $w_{i}$, $i=1,2$ the above limit takes the form $$\label{4.9}
w_{i} \rightarrow \frac{1}{\sqrt{2}}\, w,
\qquad i=1,2,$$ and then the $CP^2$ sigma model (\[4.6\])–(\[4.8\]) reduces to the $CP^1$ sigma model (\[3.5\]). These limits characterise the properties of the solutions of systems (\[4.1\])–(\[4.4\]) and (\[4.6\])–(\[4.8\]).
First we show that there exists a one to one correspondence between $GW$ system (\[4.1\])–(\[4.4\]) and the equations of the $CP^2$ sigma model (\[4.6\])–(\[4.8\]). For this purpose, we define two new complex valued functions $$\label{4.10}
w_{1} = \frac{\psi_{1}}{\bar{\varphi}_{1}},
\qquad
w_{2} = \frac{\psi_{2}}{\bar{\varphi}_{2}}$$ and using the GW system (\[4.1\])–(\[4.4\]), we obtain $$\begin{gathered}
\partial w_{1} = A \left[ w_{1} \bar{w}_{2} \varphi_{2}^2
+ \left( 1 + |w_{1}|^2\right) \varphi_{1}^2\right],
\nonumber\\
\partial w_{2} = A \left[ \bar{w}_{1} w_{2} \varphi_{1}^2
+ \left(1 + |w_{2}|^2\right) \varphi_{2}^2\right].
\label{4.11}
\end{gathered}$$ These relations generate the following transformation from the variables $(w_{1}, w_{2})$ and their derivatives to the variables $( \varphi_{1}, \varphi_{2}, \psi_{1}, \psi_{2})$ $$\begin{gathered}
\varphi_{1} = \epsilon A^{-1}
\left[ \left( 1 + |w_{2}|^2 \right) \partial w_{1} - w_{1} \bar{w}_{2}
\partial w_{2}\right]^{1/2},
\label{4.12}\\
\varphi_{2} = \epsilon A^{-1} \left[- \bar{w}_{1} w_{2} \partial w_{1}
+ \left( 1 + |w_{1}|^2\right) \partial w_{2} \right] ^{1/2},
\label{4.13}\\
\psi_{1} = \epsilon w_{1} A^{-1}
\left[\left(1 + |w_{2}|^2\right) \bar{\partial} \bar{w}_{1} - \bar{w}_{1} w_{2}
\bar{\partial} \bar{w}_{2} \right]^{1/2},
\label{4.14}\\
\psi_{2} = \epsilon w_{2} A^{-1} \left[- w_{1} \bar{w}_{2}
\bar{\partial} \bar{w}_{1} + \left(1 + |w_{1}|^2\right)
\bar{\partial} \bar{w}_{2}\right]^{1/2}.
\label{4.15}
\end{gathered}$$
We can now state the following: if the complex valued functions $(\varphi_{1}, \varphi_{2}, \psi_{1}, \psi_{2})$ are solutions of GW system (\[4.1\])–(\[4.4\]), then the functions $(w_{1}, w_{2})$, defined by (\[4.10\]), solve the equations of the $CP^2$ sigma model (\[4.6\])–(\[4.8\]).
Conversely, if the complex valued functions $(w_{1}, w_{2})$ are solutions of the $CP^2$ sigma model equations (\[4.6\])–(\[4.8\]), then the complex valued functions $(\varphi_{1}, \varphi_{2}, \psi_{1}, \psi_{2})$ defined by (\[4.12\])–(\[4.15\]) in terms of functions $(w_{1}, w_{2})$ and their 1st derivatives satisfy the GW system (\[4.1\])–(\[4.4\]).
The proof of our statement is straightforward. The differentiation of equations (\[4.11\]) with respect to $z$ and $\bar{z}$, respectively, yields $$\begin{gathered}
\partial \bar{\partial} w_{1} = A \left[ \bar{w}_{2} \varphi_{2}^2
\bar{\partial}
w_{1} + w_{1} \varphi_{2}^2 \bar{\partial}
\bar{w}_{2} + 2 w_{1} \bar{w}_{2} \varphi_{2}
\bar{\partial} \varphi_{2}
+2 \left( 1 + |w_{1}|^2\right) \varphi_{1} \bar{\partial} \varphi_{1}\right.
\\
\left. \phantom{\partial \bar{\partial} w_{1} =}{} +
\left( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial}
\bar{w}_{1}\right)
\varphi_{1}^2 \right]
+ \left[ w_{1} \bar{w}_{2} \varphi_{2}^2 + \left(1 + |w_{1}|^2\right) \varphi_{1}^2\right]\\
\phantom{\partial \bar{\partial} w_{1} =}{}\times
\left( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1}
+ \bar{w}_{2} \bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{2}\right),
\end{gathered}$$ and $$\begin{gathered}
\partial \bar{\partial} w_{2} = A \left[ w_{2} \varphi_{1}^2 \bar{\partial}
\bar{w}_{1}
+ \bar{w}_{1} \varphi_{1}^2 \bar{\partial} w_{2} + 2 \bar{w}_{1} w_{2}
\varphi_{1}
\bar{\partial} \varphi_{1}
+ 2 \left(1 + |w_{2}|^2\right) \varphi_{2} \bar{\partial} \varphi_{2}\right.
\\
\left.\phantom{\partial \bar{\partial} w_{2} =}{}+
\left( \bar{w}_{2} \bar{\partial} w_{2} + w_{2} \bar{\partial}
\bar{w}_{2}\right)
\varphi_{2}^2\right]
+ \left[ \bar{w}_{1} w_{2} \varphi_{1}^{2} + \left(1 + |w_{2}|^2 \right) \varphi_{2}^2\right]\\
\phantom{\partial \bar{\partial} w_{2} =}{}\times
\left( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1} +
\bar{w}_{2}
\bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{2}\right)
\end{gathered}$$ and their respective complex conjugate equations.
Substituting (\[4.11\]) and into the left-hand side of the first equation (\[4.6\]), we obtain $$\begin{gathered}
\partial \bar{\partial} w_{1} - \frac{2 \bar{w}_{1}}{A} \partial w_{1}
\bar{\partial} w_{1} - \frac{\bar{w}_{2}}{A} (\partial w_{1}
\bar{\partial} w_{2}
+ \bar{\partial} w_{1} \partial w_{2})
\nonumber\\
\qquad {}= \left[ A w_{1} \varphi_{2}^2 + w_{1} |w_{2}|^2 \varphi_{2}^2 + \left(1 +
|w_{1}|^2\right) w_{2}
\varphi_{1}^2\right]
\left( w_{1} \bar{w}_{2} \bar{\varphi}_{1}^2 +\left(1 + |w_{2}|^2 \right)
\bar{\varphi}_{2}^2\right)
\nonumber\\
\qquad{}+ \left[ A w_{1} \varphi_{1}^2 + w_{1}^2 \bar{w}_{2}^2 \varphi_{2}^2 +
\left( 1 + |w_{1}|^2\right) w_{1} \varphi_{1}^2\right] \left( \bar{w}_{1} w_{2}
\bar{\varphi}_{2}^2
+ \left( 1 + |w_{1}|^2 \right) \bar{\varphi}_{1}^2\right)
\nonumber\\
\qquad {} + 2 \left[ w_{1} \bar{w}_{2} \varphi_{2} \bar{\partial} \varphi_{2}
+ \left( 1 + |w_{1}|^2 \right) \varphi_{1} \bar{\partial} \varphi_{1}\right].
\label{4.16}
\end{gathered}$$
Making use of the equations (\[4.1\])–(\[4.4\]) we find that the equation (\[4.16\]) is satisfied identically. An analogous result takes place for the second equation (\[4.7\]), since the $CP^2$ sigma model equations (\[4.6\])–(\[4.8\]) are invariant under $$\label{4.17}
w_{1} \leftrightarrow w_{2}$$ This observation implies that the left-hand side of (\[4.7\]) vanishes as well whenever (\[4.1\])–(\[4.4\]) holds.
Conversely, differentiating (\[4.12\]) with respect to $\bar{z}$ and using (\[4.12\]), we get $$\begin{gathered}
\bar{\partial} \varphi_{1} = \frac{1}{2 A^2 \varphi_{1}}
\left[ \left( \bar{w}_{2} \bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{2}\right)
\partial w_{1} + \left(1 + |w_{2}|^2\right) \partial \bar{\partial} w_{1}\right.\nonumber\\
\left.\phantom{\bar{\partial} \varphi_{1} =}{}
-\left(\bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial}
\bar{w}_{2}\right) \partial w_{2}
- w_{1} \bar{w}_{2} \partial \bar{\partial} w_{2}\right]
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}- \frac{\varphi_{1}}{A}
\left[\bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1}
+ \bar{w}_{2} \bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{2}\right].
\label{4.18}
\end{gathered}$$ Using equations (\[4.11\]) and (\[4.6\])–(\[4.8\]), we can eliminate first and second derivatives of $w_{1}$ and $w_{2}$ in expression (\[4.18\]) and obtain $$\begin{gathered}
\bar{\partial} \varphi_{1} = \frac{1}{2 \varphi_{1}} \left\{ w_{1}^2
\bar{w}_{2}
|w_{2}|^2 \bar{\varphi}_{1}^2 \varphi_{2}^2 + \left(1 + |w_{2}|^2\right) w_{1}
|w_{2}|^2
|\varphi_{2}|^4 + \left(1 + |w_{1}|^2\right) w_{1} |w_{2}|^2 |\varphi_{1}|^4\right.
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}
+\left(1 + |w_{1}|^2 \right)\left( 1 + |w_{2}|^2\right) w_{2} \varphi_{1}^2
\bar{\varphi}_{2}^2
- w_{1} | w_{1}|^2 | w_{2}|^2 |\varphi_{1}|^4
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}
- \left(1 + |w_{2}|^2\right) |w_{1}|^2
w_{2} \varphi_{1}^2 \bar{\varphi}_{2}^2
- \left(1 + |w_{2}|^2\right) w_{1}^2 \bar{w}_{2} \bar{\varphi}_{1}^2 \varphi_{2}
- \left( 1 + |w_{2}|^2\right)^2 w_{1} |\varphi_{2}|^4
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}
- 2 A \varphi_{1}^2
\left(w_{2} \bar{\varphi}_{2}^2 + w_{1} \bar{\varphi}_{1}^2\right)
+ \frac{1}{A^2} \left(A \left[ w_{1} \bar{w}_{2}^2 \varphi_{2}^2 + \bar{w}_{2}
\left(1 + |w_{1}|^2 \right) \varphi_{1}^2\right] \right.
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}
+ \left(1 + |w_{2}|^2\right)
\left[ w_{1} \bar{w}_{2}^2 \varphi_{2}^2 + \bar{w}_{2} \left(1 + |w_{1}|^2\right)
\varphi_{1}^2\right]
\nonumber\\
\left.\phantom{\bar{\partial} \varphi_{1} =}{}
- \left(A + |w_{2}|^2\right) |w_{1}|^2 \bar{w}_{2} \varphi_{1}^2
- \left(A +1 + |w_{2}|^2\right) w_{1} \bar{w}_{2}^2 \varphi_{2}^2 -2 A \bar{w}_{2}
\varphi_{1}^2\right) \bar{\partial} w_{2}
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}
+ \frac{1}{A^2} \left(\left(1 + |w_{2}|^2 \right) \left[ \left(A + |w_{1}|^2\right) \bar{w}_{2}
\varphi_{2}^2
+ \left(A + 1 + |w_{1}|^2\right) \bar{w}_{1} \varphi_{1}^2\right]\right.
\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{}
- A \left[ \bar{w}_{1} | w_{2}|^2 \varphi_{1}^2 + \left(1 + |w_{2}|^2\right)
\bar{w}_{2}
\varphi_{2}^2\right]
\nonumber\\
\left.\left.\phantom{\bar{\partial} \varphi_{1} =}{}
-\left[ \bar{w}_{1} | w_{1}|^2 |w_{2}|^2
\varphi_{1}^2
+ \left( 1 + |w_{2}|^2\right) |w_{1}|^2 \bar{w}_{2} \varphi_{2}^2\right]
-2 A \bar{w}_{1} \varphi_{1}^2 \right) \bar{\partial} w_{1} \right\}.
\label{4.19}
\end{gathered}$$ Collecting all the coefficients of the derivatives $\bar{\partial} w_{1}$ and $\bar{\partial} w_{2}$ in expression (\[4.19\]) we find that these coefficients vanish identically. In fact, we have $$\begin{gathered}
\bar{\partial} w_{1} : \ \left(A + 1 + |w_{1}|^2 + \left(A + 1 + |w_{1}|^2\right)
|w_{2}|^2 - A |w_{2}|^2 - |w_{1}|^2 |w_{2}|^2 - 2 A\right) \bar{w}_{1}
\varphi_{1}^2\\
\qquad {}+ \left(A + |w_{1}|^2 + \left(A + |w_{1}|^2\right) |w_{2}|^2 - A \left(1 + |w_{2}|^2 \right)
- \left(1 + |w_{2}|^2\right) |w_{1}|^2 \right) \bar{w}_{2} \varphi_{2}^{2} \equiv 0.\!
\end{gathered}$$ and $$\begin{gathered}
\bar{\partial} w_{2} : \ \left( A \left(1 + |w_{1}|^2 \right) + A + |w_{1}|^2 |w_{2}|^2
- \left(A + |w_{2}|^2\right) |w_{1}|^2 - 2 A\right) \bar{w}_{2} \varphi_{1}^2
\nonumber\\
\qquad {}+\left( A + 1 + |w_{2}|^2 - A -1 - |w_{2}|^2\right) w_{1} \bar{w}_{2} \varphi_{2}^2
\equiv 0.
\label{4.20}
\end{gathered}$$ Hence, (\[4.20\]) becomes $$\begin{gathered}
\bar{\partial} \varphi_{1} = - \frac{1}{2} \left\{ \bar{w}_{2} w_{1}^2
\frac{\bar{\varphi}_{1}^2 \varphi_{2}^2}{\varphi_{1}} +
\left(1 + |w_{2}|^2\right) w_{1} \frac{|\varphi_{2}|^4}{\varphi_{1}}
+ 2 A w_{2} \varphi_{1} \bar{\varphi}_{2}^2\right.\nonumber\\
\left.\phantom{\bar{\partial} \varphi_{1} =}{}
+ 2 A w_{1} | \varphi_{1}|^2 \varphi_{1}
- w_{1} | w_{2}|^2 \frac{|\varphi_{1}|^4}{\varphi_{1}}
-\left( 1 + |w_{2}|^2\right) w_{2} \varphi_{1} \bar{\varphi}_{2} \right\}
\end{gathered}$$ Performing the transformation (\[4.10\]) we obtain the first equation of (\[4.1\]), i.e. $$\begin{gathered}
\bar{\partial} \varphi_{1} = - \frac{1}{2} \Big\{
\frac{\varphi_{2}}{\varphi_{1}}
\bar{\psi}_{2} \psi_{1}^2 + \left( 1 + |w_{2}|^2\right)
\frac{|\varphi_{2}|^4}{|\varphi_{1}|^2}\nonumber\\
\phantom{\bar{\partial} \varphi_{1} =}{} +
\left( A +1 + |w_{1}|^2\right) | \varphi_{1}|^2 \psi_{1} + \left(A + |w_{1}|^2\right)
\varphi_{1}
\bar{\varphi}_{2} \psi_{2} \Big\}.\label{4.21}
\end{gathered}$$ Since the equations (\[4.1\])–(\[4.4\]) are invariant under $$\label{4.22}
\varphi_{1} \leftrightarrow \varphi_{2},$$ an analogous result holds for (\[4.2\]). Differentiation of (\[4.10\]) with respect to $z$ gives $$\label{4.23}
\partial \psi_{1} = \bar{\varphi}_{1} \partial w_{1} + w_{1}
\partial \bar{\varphi}_{1}.$$ Substituting (\[4.11\]) and the complex conjugate equation of (\[4.1\]) into (\[4.23\]) we get (\[4.3\]). Making use of the discrete symmetry (\[4.22\]) in (\[4.3\]), we obtain equation (\[4.4\]), which completes the proof.
An interesting property of the GW system (\[4.1\])–(\[4.4\]) in the context of the $CP^2$ sigma model (\[4.6\])–(\[4.8\]) is the existence of a gauge freedom in the definition of the variables $w_{1}$ and $w_{2}$ given by formula (\[4.10\]). This is due to the fact that the numerator and the denominator of (\[4.10\]) can be multiplied by any complex functions $f_{i} : {\mathbb C} \rightarrow {\mathbb C}$, $i=1,2$. This means that if we introduce a new set of complex valued functions $(\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2})$ which are related to functions $(\varphi_{1}, \varphi_{2}, \psi_{1}, \psi_{2})$ in the following way $$\label{4.24}
\varphi_{i} = f_{i} (z, \bar{z}) \alpha_{i}, \qquad
\psi_{i} = \bar{f} ( z, \bar{z}) \beta_{i}, \qquad i=1,2,$$ then the transformation (\[4.24\]) leaves the functions $w_{1}$, $w_{2}$ invariant $$\label{4.25}
w_{1} = \frac{\beta_{1}}{\bar{\alpha}_{1}},
\qquad
w_{2} = \frac{\beta_{2}}{\bar{\alpha}_{2}}.$$
We show now that if the complex valued functions $w_{1}$, $w_{2}$ are solutions of the $CP^2$ sigma model equations (\[4.6\])–(\[4.8\]), then for any two holomorphic functions $f_{i}$, $i=1,2$ the complex functions $(\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2})$ defined by $$\begin{gathered}
\alpha_{1} = \epsilon f_{1}^{-1} A^{-1}
\left[\left( 1 + |w_{2}|^2\right) \partial w_{1} - w_{1} \bar{w}_{2} \partial
w_{2}\right]^{1/2},
\nonumber\\
\alpha_{2} = \epsilon f_{2}^{-1} A^{-1}
\left[ - \bar{w}_{1} w_{2} \partial w_{1} + \left( 1 + |w_{1}|^2\right)
\partial w_{2}\right]^{1/2},
\nonumber\\
\beta_{1} = \epsilon w_{1} \bar{f}_{1}^{-1} A^{-1}
\left[ \left( 1 + |w_{2}|^2\right) \bar{\partial} \bar{w}_{1} - \bar{w}_{1} w_{2}
\bar{\partial} \bar{w}_{2}\right]^{1/2},
\nonumber\\
\beta_{2} = \epsilon w_{2} \bar{f}_{2}^{-1} A^{-1}
\left [-w_{1} \bar{w}_{2} \bar{\partial} \bar{w}_{1}
+ \left(1 + |w_{1}|^2\right) \bar{\partial} \bar{w}_{2}\right]^{1/2},
\qquad \bar{\partial} f_{i} = 0,
\label{4.26}
\end{gathered}$$ satisfy the GW system (\[4.1\])–(\[4.4\]).
Indeed, the result is obtained directly by substituting (\[4.24\]) and (\[4.25\]) into $CP^2$ sigma model equations (\[4.6\])–(\[4.8\]). This leads to differential constraints for the functions $f_{i}$ and their first derivatives $$\label{4.27}
\left(f_{i}^2 - 1\right) \bar{\partial} f_{j} = 0, \qquad
\left(\bar{f}_{i}^2 - 1\right) \partial \bar{f}_{j} = 0, \qquad i,j = 1,2.$$ Hence, the general solutions of this system are given by any holomorphic functions $f_{i}$ i.e. $$\label{4.28}
\bar{\partial} f_{i} = 0, \qquad i=1,2$$ Then invoking the main result of the last section we note that the transformation (\[4.12\])–(\[4.15\]) becomes the one given by (\[4.26\]).
Another interesting property in the context of the $CP^2$ sigma model (\[4.6\])–(\[4.8\]) and the GW system (\[4.1\])–(\[4.4\]) is the existence of the quantity $J$ [@book11] (which is a generalization of (3.17)). $$\label{4.29}
J = A^{-2} \left\{ \partial w_{1} \partial \bar{w}_{1} + \partial w_{2}
\partial \bar{w}_{2} + (\bar{w}_{1} \partial \bar{w}_{2}
- \bar{w}_{2} \partial \bar{w}_{1}) ( w_{1} \partial w_{2}
- w_{2} \partial w_{1}) \right\},$$ whose derivative with respect to $z$ vanishes identically whenever equations (\[4.6\])–(\[4.8\]) are satisfied $$\label{4.30}
\bar{\partial} J = 0.$$ This means that $J$, given by (\[4.29\]), is holomorphic.
Note that if the functions $(\varphi_{1}, \varphi_{2}, \psi_{1},
\psi_{2})$ are solutions of GW system (\[4.1\])–(\[4.4\]), then $J$, when written in terms of functions $(\varphi_{1}, \varphi_{2}, \psi_{1}, \psi_{2})$, takes the form $$\label{4.31}
J = \varphi_{1} \partial \bar{\psi}_{1} - \bar{\psi}_{1} \partial
\varphi_{1}
+ \varphi_{2} \partial \bar{\psi}_{2} - \bar{\psi}_{2} \partial
\varphi_{2}$$ and it satisfies $$\label{4.32}
\bar{\partial} J = 0,$$ whenever equations (\[4.1\])–(\[4.4\]) hold.
Next we exploit the observation [@book10] that the equations of the $CP^2$ sigma model (\[4.6\])–(\[4.8\]) can be written as the compatibility condition for two linear spectral problems $$\label{4.36}
\partial \Phi = \frac{2}{1 + \lambda} [ \partial P, P] \Phi,
\qquad
\bar{\partial} \Phi = \frac{2}{1 - \lambda} [ \bar{\partial} P, P] \Phi,
\qquad
\lambda \in {\mathbb C},$$ where the $3 \times 3$ $P$ is given by $$\label{4.37}
P = A^{-1} M, \qquad M = \left(
\begin{array}{ccc}
1 & w_{1} & w_{2} \\
\bar{w}_{1} & |w_{1}|^2 & \bar{w}_{1} w_{2} \\
\bar{w}_{2} & w_{1} \bar{w}_{2} & |w_{2}|^2 \\
\end{array} \right),$$ and $\lambda$ represents the spectral parameter. Using matrix $P$, the compatibility conditions of equations (\[4.36\]) imply $$\label{4.38}
[\partial \bar{\partial} P, P] = 0,$$ which are satisfied whenever equations (\[4.6\])–(\[4.8\]) hold. Equivalently, formula (\[4.38\]) can be rewritten, in a divergent form, as $$\label{4.39}
\partial [ \bar{\partial} P, P] + \bar{\partial}
[\partial P, P ] = 0.$$ Hence, from equations (\[4.37\]) and (\[4.39\]) we obtain the explicit form of the local conservation laws for the $CP^2$ sigma model $$\label{4.40}
\partial K + \bar{\partial} L = 0,$$ where we have introduced the following notation for the traceless matrices $K$ and $L$: $$\label{4.41}
K = \frac{1}{A^2}\, [ \bar{\partial} M, M], \qquad
L = -K^\dagger = \frac{1}{A^2}\, [ \partial M, M],
\qquad {\rm tr}\; K = {\rm tr}\; L =0.$$ Explicitly, the matrix elements of $K$ and $L$ are of the form $$\begin{gathered}
k_{11} = A^{-2} \left\{ ( \bar{w}_{1} \bar{\partial} w_{1}
+ \bar{w}_{2} \bar{\partial} w_{2})
- ( w_{1} \bar{\partial} \bar{w}_{1} + w_{2} \bar{\partial} \bar{w}_{2})\right \},\nonumber\\
k_{12} = A^{-2} \left\{ |w_{1}|^2 \bar{\partial} w_{1} + w_{1} \bar{w}_{2}
\bar{\partial} w_{2} - (\bar{\partial} w_{1} + w_{1}
( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1})\right.\nonumber\\
\left.\phantom{k_{12}=}{} + w_{2} ( \bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial}
\bar{w}_{2})) \right\},
\nonumber\\
k_{13} = A^{-2} \left\{ \bar{w}_{1} w_{2} \bar{\partial} w_{1}
+ |w_{2}|^2 \bar{\partial} w_{2} - (\bar{\partial} w_{2} + w_{1}
( \bar{w}_{1} \bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{1})\right.\nonumber\\
\left.\phantom{k_{13}=}{}
+ w_{2} ( \bar{w}_{2} \bar{\partial} w_{2} + w_{2} \bar{\partial}
\bar{w}_{2})) \right\},
\nonumber\\
k_{21} = A^{-2}\left \{ \bar{\partial} \bar{w}_{1} + \bar{w}_{1}
( \bar{w}_{1} ( \bar{w}_{1} \bar{\partial} w_{1} + w_{1}
\bar{\partial} \bar{w}_{1}) + \bar{w}_{2}
( \bar{w}_{1} \bar{\partial} w_{2} + w_{2}
\bar{\partial} \bar{w}_{1})\right.
\nonumber\\
\left.\phantom{k_{12}=}{}- (|w_{1}|^2 \bar{\partial} \bar{w}_{1}
+ \bar{w}_{1} w_{2} \bar{\partial} \bar{w}_{2}) \right\},
\nonumber\\
k_{22} = A^{-2}\left\{ w_{1} \bar{\partial} \bar{w}_{1} \!+ w_{1} \bar{w}_{2}
(\bar{w}_{1} \bar{\partial} w_{2}\! + w_{2} \bar{\partial} \bar{w}_{1})\!
- (\bar{w}_{1} \bar{\partial} w_{1}\! + \bar{w}_{1} w_{2}
(\bar{w}_{2} \bar{\partial} w_{1} \!+ w_{1} \bar{\partial} \bar{w}_{2}))\right \},
\nonumber\\
k_{23} = A^{-2} \left\{ w_{2} \bar{\partial} \bar{w}_{1} + \bar{w}_{1} w_{2}
( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1} )
+ |w_{2}|^2 (\bar{w}_{1} \bar{\partial} w_{2} + w_{2} \bar{\partial}
\bar{w}_{1})\right.
\nonumber\\
\left.\phantom{k_{23}=}{}- \left[ \bar{w}_{1} \bar{\partial} w_{2} + |w_{1}|^2 (\bar{w}_{1}
\bar{\partial} w_{2}
+ w_{2} \bar{\partial} \bar{w}_{1}) + \bar{w}_{1} w_{2} (\bar{w}_{2}
\bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{2}) \right] \right\},
\nonumber\\
k_{31} = A^{-2} \left\{ \bar{\partial} \bar{w}_{2} + \bar{w}_{1}
( \bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{2})
+ \bar{w}_{2} ( \bar{w}_{2} \bar{\partial} w_{2} + w_{2}
\bar{\partial} \bar{w}_{2})\right.
\nonumber\\
\left.\phantom{k_{31}=}{}- \left( w_{1} \bar{w}_{2} \bar{\partial} \bar{w}_{1} +
| w_{2}|^2 \bar{\partial} \bar{w}_{2}\right) \right\},
\nonumber\\
k_{32} = A^{-2} \left\{ w_{1} \bar{\partial} \bar{w}_{2} + |w_{1}|^2
(\bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{2})
+ w_{1} \bar{w}_{2} ( \bar{w}_{2} \bar{\partial} w_{2} + w_{2}
\bar{\partial} \bar{w}_{2})\right.
\nonumber\\
\left.\phantom{k_{32}=}{}- \left[ \bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{w}_{2}
( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1} )
+ |w_{2}|^2 |\bar{w}_{2}|^2 (\bar{w}_{2} \bar{\partial} w_{1} + w_{1}
\bar{\partial} \bar{w}_{2}) \right] \right\},
\nonumber\\
k_{33} = A^{-2}\left\{ w_{2} \bar{\partial} \bar{w}_{2} + \bar{w}_{1} w_{2}
( \bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{2})\right.
\nonumber\\
\left.\phantom{k_{33}=}{} - ( \bar{w}_{2} \bar{\partial} w_{2} + w_{1} \bar{w}_{2} ( \bar{w}_{1}
\bar{\partial}w_{2} + w_{2} \bar{\partial} \bar{w}_{1})) \right\},
\label{4.42}
\end{gathered}$$ and $$\begin{gathered}
l_{11} = A^{-2} \left\{ \bar{w}_{1} \partial w_{1} + \bar{w}_{2} \partial
w_{2}
- ( w_{1} \partial \bar{w}_{1} + w_{2} \partial \bar{w}_{2}) \right\},
\nonumber\\
l_{12} = A^{-2} \left\{ |w_{1}|^2 \partial w_{1} + w_{1} \bar{w}_{2}
\partial
w_{2}
- [ \partial w_{1} + w_{1} ( \bar{w}_{1} \partial w_{1}+ w_{1} \partial
\bar{w}_{1})\right.
\nonumber\\
\left.\phantom{l_{12} =}{} + w_{2} (\bar{w}_{2} \partial w_{1} + w_{1} \partial \bar{w}_{2}) ] \right\},
\nonumber\\
l_{13} = A^{-2} \left\{ \bar{w}_{1} w_{2} \partial w_{1} + |w_{2}|^2
\partial w_{2} - [ \partial w_{2} + w_{1} (\bar{w}_{1} \partial w_{2} +
w_{2} \partial
\bar{w}_{1}) \right.\nonumber\\
\left.\phantom{l_{13}=}{}+ w_{2} ( \bar{w}_{2} \partial w_{2} + w_{2} \partial
\bar{w}_{2})] \right\},
\nonumber\\
l_{21} = A^{-2} \left\{ \partial \bar{w}_{1} + \bar{w}_{1} (\bar{w}_{1}
\partial w_{1} +
w_{1} \partial \bar{w}_{1}) + \bar{w}_{2} ( \bar{w}_{1} \partial w_{2} +
w_{2}
\partial \bar{w}_{1}) \right.\nonumber\\
\left.\phantom{l_{21}=}{}- ( |w_{1}|^2 \partial \bar{w}_{1} + \bar{w}_{1}
w_{2}
\partial \bar{w}_{2}) \right\},
\nonumber\\
l_{22} = A^{-2} \left\{ w_{1} \partial \bar{w}_{1} + w_{1} \bar{w}_{2}
( \bar{w}_{1} \partial w_{2} + w_{2} \partial \bar{w}_{1})
- [ \bar{w}_{1} \partial w_{1} + \bar{w}_{1} w_{2} (\bar{w}_{2}
\partial w_{1} + w_{1} \partial \bar{w}_{2}) ] \right\},
\nonumber\\
l_{23} = A^{-2} \left\{ w_{2} \partial \bar{w}_{1} + \bar{w}_{1} w_{2}
( \bar{w}_{1} \partial w_{1} + w_{1} \partial \bar{w}_{1}) + |w_{2}|^2
(\bar{w}_{1} \partial w_{2} + w_{2} \partial \bar{w}_{1})\right.
\nonumber\\
\left.\phantom{l_{23}=}{}- [ \bar{w}_{1} \partial w_{2} + |w_{1}|^2 (\bar{w}_{1} \partial w_{2}
+ w_{2} \partial \bar{w}_{1}) + \bar{w}_{1} w_{2} (\bar{w}_{2}
\partial w_{2} + w_{2} \partial \bar{w}_{2})] \right\},\end{gathered}$$ $$\begin{gathered}
l_{31} = A^{-2} \left\{ \partial \bar{w}_{2} + \bar{w}_{1}
( \bar{w}_{2} \partial w_{1} + w_{1} \partial \bar{w}_{2})
+ \bar{w}_{2} ( \bar{w}_{2} \partial w_{2} + w_{2} \partial \bar{w}_{2})\right.
\nonumber\\
\left.\phantom{l_{31}=}{} - ( w_{1} \bar{w}_{2} \partial \bar{w}_{1} + |w_{2}|^2 \partial
\bar{w}_{2} ) \right\},
\nonumber\\
l_{32} = A^{-2} \left\{ w_{1} \partial \bar{w}_{2} + |w_{1}|^2
( \bar{w}_{2} \partial w_{1} + w_{1} \partial \bar{w}_{2})
+ w_{1} \bar{w}_{2} ( \bar{w}_{2} \partial w_{2} + w_{2} \partial
\bar{w}_{2})\right.
\nonumber\\
\left.\phantom{l_{32}=}{}- [ \bar{w}_{2} \partial w_{1} + w_{1} \bar{w}_{2} ( \bar{w}_{1}
\partial w_{1} + w_{1} \partial \bar{w}_{1}) + |w_{2}|^2
(\bar{w}_{2} \partial w_{1} + w_{1} \partial \bar{w}_{2})] \right\},
\nonumber\\
l_{33} = A^{-2} \left\{ w_{2} \partial \bar{w}_{2} + \bar{w}_{1} w_{2}
( \bar{w}_{2} \partial w_{1} + w_{1} \partial \bar{w}_{2}) \right.
\nonumber\\
\left.\phantom{l_{33}=}{}-
(\bar{w}_{2} \partial w_{2} + w_{1} \bar{w}_{2} ( \bar{w}_{1}
\partial w_{2} + w_{2} \partial \bar{w}_{1})) \right\},
\label{4.43}
\end{gathered}$$ respectively. Finally from equations (\[4.40\]), (\[4.42\]) and (\[4.43\]) we see that there exists only five independent conservation laws for $CP^2$ sigma model (\[4.6\])–(\[4.8\]). Namely we have the following independent conserved quantities $$\begin{gathered}
\partial \left\{ A^{-2} \left[ \bar{w}_{1} \bar{\partial} w_{1} + \bar{w}_{2}
\bar{\partial} w_{2} - ( w_{1} \bar{\partial} \bar{w}_{1} + w_{2}
\bar{\partial} w_{2} ) \right] \right\}\nonumber\\
\qquad {} +\bar{\partial} \left\{ A^{-2} \left[ \bar{w}_{1} \partial w_{1} + \bar{w}_{2}
\partial w_{2}- ( w_{1} \partial \bar{w}_{1} + w_{2}
\partial \bar{w}_{2}) \right] \right\} = 0, \nonumber\\
\partial \left\{ A^{-2} \left[ w_{1} \bar{\partial} \bar{w}_{1} + w_{1}
\bar{w}_{2}
( \bar{w}_{1} \bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{1} )
- \bar{w}_{1} \bar{\partial} w_{1} - \bar{w}_{1} w_{2}
( \bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{2}) \right]
\right\}
\nonumber\\
\qquad {}+\bar{\partial} \left\{ A^{-2} \left[ w_{1} \partial \bar{w}_{1} + w_{1}
\bar{w}_{2}
( \bar{w}_{1} \partial w_{2} + w_{2} \partial \bar{w}_{1}) - \bar{w}_{1}
\partial w_{1} \right.\right.
\nonumber\\
\qquad\left.\left.{}- \bar{w}_{1} w_{2} ( \bar{w}_{2} \partial w_{1} + w_{1}
\partial \bar{w}_{2})\right] \right\}=0,\nonumber\\
\partial \left\{ A^{-2} \left[ |w_{1}|^2 \bar{\partial} w_{1} + w_{1} \bar{w}_{2}
\bar{\partial} w_{2} - \bar{\partial} w_{1} - w_{1}
( \bar{w}_{1} \bar{\partial} w_{1} + w_{1} \bar{\partial} \bar{w}_{1})
\right.\right.
\nonumber\\
\qquad \left.\left.{}
- w_{2} (\bar{w}_{2} \bar{\partial} w_{1} + w_{1} \bar{\partial}
\bar{w}_{2}) \right] \right\}
+ \bar{\partial}\left\{ A^{-2} \left[ |w_{1}|^2 \partial w_{1} + w_{1}
\bar{w}_{2}
\partial w_{2} - \partial w_{1}
\right.\right.
\nonumber\\
\qquad\left.\left.{} - w_{1} ( \bar{w}_{1} \partial w_{1} +
w_{1}
\partial \bar{w}_{1}) - w_{2} ( \bar{w}_{2} \partial w_{1} + w_{1}
\partial \bar{w}_{2}) \right] \right\} \nonumber = 0,
\nonumber\\
\partial \left\{ A^{-2}\left[\bar{w}_{1} w_{2} \bar{\partial} w_{1} -
\bar{\partial} w_{2} - w_{1} ( \bar{w}_{1}
\bar{\partial} w_{2} + w_{2} \bar{\partial} \bar{w}_{1}) - w_{2}^2
\bar{\partial} \bar{w}_{2}\right]\right\}
\nonumber\\
\qquad{}+ \bar{\partial}\left\{ A^{-2} \left[ \bar{w}_{1} w_{2} \partial w_{1} -
\partial
w_{2}
- w_{1} ( \bar{w}_{1} \partial w_{2} + w_{2} \partial \bar{w}_{1}) -
w_{2}^2
\partial \bar{w}_{2} \right] \right\} = 0,
\nonumber\\
\partial \left\{ A^{-2} \left[ w_{2} \bar{\partial} \bar{w}_{1} + \bar{w}_{1}^2
w_{2}
\bar{\partial} w_{1} + |w_{2}|^2 w_{2} \bar{\partial} \bar{w}_{1}\right] \right.
\nonumber\\
\qquad\left.{}- A^{-2}
\left[ \bar{w}_{1} \bar{\partial} w_{2} + |w_{1}|^2 \bar{w}_{1} \bar{\partial}
w_{2}
+ \bar{w}_{1} w_{2}^2 \bar{\partial} \bar{w}_{2} \right]\right\}
\nonumber\\
\qquad{}+ \bar{\partial} \left\{ A^{-2} \left[ w_{2} \partial \bar{w}_{1} + \bar{w}_{1}^2
w_{2}
\partial w_{1} + |w_{2}|^2 w_{2} \partial \bar{w}_{1} \right]\right.\nonumber\\
\qquad \left.{}
- A^{-2} \left[ \bar{w}_{1} \partial w_{2} + |w_{1}|^2 \bar{w}_{1} \partial
w_{2}
+\bar{w}_{1} w_{2}^2 \partial \bar{w}_{2} \right] \right\} = 0.
\label{4.44}\end{gathered}$$
Consequently, as a result of the conservation laws (\[4.44\]) there exist eight real-valued functions $X^{i} (z, \bar{z})$, $i=1, \ldots, 8$ expressed in terms of functions $(w_{1},w_{2})$, i.e.$$\begin{gathered}
X^1 = \int_{C} A^{-2} \left\{ -\left[ \bar{w}_{1} \partial w_{1} +
\bar{w}_{2}
\partial w_{2} - (w_{1} \partial \bar{w}_{1} + w_{2} \partial
\bar{w}_{2})\right] dz\right.\\
\left.\phantom{X^1=}{} + \left[ \bar{w}_{1} \bar{\partial} w_{1} + \bar{w}_{2} \bar{\partial} w_{2}
-(w_{1} \bar{\partial} \bar{w}_{1} + w_{2}
\bar{\partial} \bar{w}_{2}) \right] d \bar{z} \right\},\\
X^2 = \int_{C} A^{-2} \left\{ \left[ \left(1 + |w_{2}|^2 \right) (w_{1} \partial
\bar{w}_{1} -\bar{w}_{1}
\partial w_{1}) + |w_{1}|^2 (\bar{w}_{2} \partial w_{2} - w_{2} \partial
\bar{w}_{2}) \right] dz\right.\\
\left.\phantom{X^2=}{}+ \left[ \left( 1 + |w_{2}|^2\right)(w_{1} \bar{\partial} \bar{w}_{1} - \bar{w}_{1}
\bar{\partial} w_{1})
+ |w_{1}|^2 (\bar{w}_{2} \bar{\partial} w_{2} - w_{2} \bar{\partial}
\bar{w}_{2})\right] d \bar{z} \right\},\\
X^3 = i \int_{C} -A^{-2} \left\{ \left[ - \left(1 + \bar{w}_{1}^2 + |w_{2}|^2\right)
\partial w_{1}\right.\right.\\
\left.\phantom{X^3=}- \left(1 + w_{1}^2 + |w_{2}|^2\right) \partial \bar{w}_{1}
+ \bar{w}_{2} (w_{1} - \bar{w}_{1})
\partial w_{2} + w_{2} ( \bar{w}_{1} - w_{1}) \partial \bar{w}_{2} \right] dz
\\
\phantom{X^3=}{}+ \left[ - \left(1 + \bar{w}_{1}^2 + |w_{2}|^2 \right) \bar{\partial} w_{1} -
\left( 1 + w_{1}^2 + |w_{2}|^2\right) \bar{\partial} \bar{w}_{1} + \bar{w}_{2}
(w_{1} - \bar{w}_{1}) \bar{\partial} w_{2} \right.\\
\left.\left.\phantom{X^3=}{}+ w_{2} (\bar{w}_{1} - w_{1} )
\bar{\partial} \bar{w}_{2} \right] d \bar{z} \right\},
\end{gathered}$$ $$\begin{gathered}
X^4 = \int_{C} A^{-2} \left\{ \left[ \left(1 - \bar{w}_{1}^2 + |w_{2}|^2\right) \partial
w_{1}\right.\right.
\nonumber\\
\left.\phantom{X^4=}{}+ \left(-1 + w_{1}^2 - |w_{2}|^2\right) \partial \bar{w}_{1} - \bar{w}_{2} (w_{1}
+ \bar{w}_{1})
\partial w_{2} + w_{2} ( w_{1} + \bar{w}_{1}) \partial \bar{w}_{2} \right] dz
\nonumber\\
\phantom{X^4=}{}+ \left[ \left(-1 + \bar{w}_{1}^2 - |w_{2}|^2\right) \bar{\partial} w_{1}
+ \left(1 - w_{1}^2 + |w_{2}|^2\right) \bar{\partial} \bar{w}_{1} + \bar{w}_{2}
(w_{1} + \bar{w}_{1}) \bar{\partial} w_{2} \right.\nonumber \\
\left.\left.\phantom{X^4=}{}- w_{2} (w_{1} + \bar{w}_{1})
\bar{\partial} \bar{w}_{2} \right] d \bar{z} \right\}, \nonumber\\
X^5= i \int_{C} - A^{-2} \left\{ \left[ \bar{w}_{1} ( w_{2} - \bar{w}_{2})
\partial w_{1}
+ w_{1} (\bar{w}_{2}- w_{2}) \partial \bar{w}_{1}\right.\right.
\nonumber\\
\left.\phantom{X^5=}{}- \left(1 + |w_{1}|^2
+ \bar{w}_{2}^2\right) \partial w_{2} - \left(1 + |w_{1}|^2 + w_{2}^2\right) \partial
\bar{w}_{2}\right] dz
\nonumber\\
\phantom{X^5=}{}+ \left[ \bar{w}_{1} ( w_{2} - \bar{w}_{2}) \bar{\partial} w_{1} + w_{1}
(\bar{w}_{2} - w_{2})
\bar{\partial} \bar{w}_{1} - \left(1 + |w_{1}|^2 + \bar{w}_{2}^2\right)
\bar{\partial} w_{2} \right.\nonumber \\
\left.\left.\phantom{X^5=}{}- \left(1 + |w_{1}|^2 + w_{2}^2\right)
\bar{\partial} \bar{w}_{2}\right] d \bar{z} \right\},
\nonumber\\
X^6 = \int_{C} A^{-2} \left\{ \left[ - \bar{w}_{1} (w_{2} + \bar{w}_{2}) \partial
w_{1} + w_{1} (w_{2} + \bar{w}_{2}) \partial \bar{w}_{1}\right.\right.
\nonumber\\
\left.\phantom{X^6=}{}+ \left(1 + |w_{1}|^2 - \bar{w}_{2}^2\right)
\partial w_{2} - \left(1 + |w_{1}|^2 -w_{2}^2\right) \partial \bar{w}_{2} \right] dz
\nonumber\\
\phantom{X^6=}{}+ \left[ \bar{w}_{1} (w_{2} + \bar{w}_{2}) \bar{\partial} w_{1} - w_{1}
(w_{2} + \bar{w}_{2})
\bar{\partial} \bar{w}_{2} - \left(1+ |w_{1}|^2 -\bar{w}_{2}^2\right) \bar{\partial}
w_{2} \right.\nonumber \\
\left.\left.\phantom{X^6=}{}+\left(1+|w_1|^2-w_{2}^2\right) \bar{\partial} \bar{w}_{2}\right] d \bar{z}
\right\},
\nonumber\\
X^7 = i \int_{C} - A^{-2} \left\{ \left[ \bar{w}_{2} \left(1 + |w_{2}|^2\right) +
\bar{w}_{1}^{2} w_{2} \right]
\partial w_{1} + \left[ w_{2} (1 + |w_{2}|^2) + w_{1}^2 \bar{w}_{2} \right] \partial
\bar{w}_{1}\right.
\nonumber\\
\left.\phantom{X^7=}{}-\left[ \bar{w}_{1} \left( 1 + |w_{1}|^2 \right) + w_{1} \bar{w}_{2}^2\right]
\partial w_{2}
- \left[ w_{1} \left(1 + |w_{1}|^2\right) + \bar{w}_{1} w_{2}^2 \right] \partial \bar{w}_{2} \right\}
dz
\nonumber\\
\phantom{X^7=}{}+ A^{-2} \left\{ \left[ \bar{w}_{2} \left(1 + |w_{2}|^2 \right) + \bar{w}_{1}^2 w_{2} \right]
\bar{\partial} w_{1}
+ \left[ w_{2} \left(1 + |w_{2}|^2\right) + w_{1}^2 \bar{w}_{2}\right] \bar{\partial}
\bar{w}_{1}\right.
\nonumber\\
\left.\phantom{X^7=}{}- \left[ \bar{w}_{1}
\left(1 + |w_{1}|^2\right) + w_{1} \bar{w}_{2}^2 \right] \bar{\partial}
w_{2}
- \left[ w_{1} \left( 1 + |w_{1}|^2\right) + \bar{w}_{1} w_{2}^2\right] \bar{\partial}
\bar{w}_{2} \right\} d \bar{z},
\nonumber\\
X^8 = \int_{C} A^{-2}\left\{\left[ \bar{w}_{2} \left(1 + |w_{2}|^2\right) - \bar{w}_{1}^2
w_{2}\right] \partial w_{1}
+ \left[ - w_{2} \left(1 + |w_{2}|^2\right) + w_{1}^2 \bar{w}_{2}\right] \partial \bar{w}_{1}\right.
\nonumber\\
\left.\phantom{X^8=}{}+
\left[ \bar{w}_{1}\left( 1 + |w_{1}|^2\right) - w_{1} \bar{w}_{2}^2\right] \partial w_{2} +
\left[-w_{1}
\left(1 + |w_{1}|^2\right) + \bar{w}_{1} w_{2}^2 \right] \partial \bar{w}_{2} \right\} dz
\nonumber\\
\phantom{X^8=}{}+ A^{-2} \left\{ \left[ - \bar{w}_{2} \left(1 + |w_{2}|^2\right) + \bar{w}_{1}^2 w_{2}\right]
\bar{\partial} w_{1}
+\left[ w_{2} \left(1 + |w_{2}|^2\right) - w_{1}^2 \bar{w}_{2}^2\right] \bar{\partial}
\bar{w}_{1}\right. \nonumber\\
\left.\phantom{X^8=}{}+\left[ - \bar{w}_{1} \left( 1 + |w_{1}|^2\right) + w_{1} \bar{w}_{2}^2\right]
\bar{\partial} w_{2}
+ \left[ w_{1} \left(1 + |w_{1}|^2\right) - \bar{w}_{1} w_{2}^2 \right] \bar{\partial}
\bar{w}_{2} \right\} d \bar{z}.
\label{4.45}
\end{gathered}$$ Note that by virtue of the conservation laws (\[4.44\]) for the $CP^2$ sigma model (\[4.6\])–(\[4.8\]) the r.h.s. in expression (\[4.45\]) do not depend on the choice of the contour $C$ but only on its endpoints. This is due to the fact that (\[4.42\]) are integrals of exact differentials of real valued functions. We identify the functions $X^{i} (z, \bar{z})$, $i=1, \ldots, 8$ with the coordinates of the radius vector $$\boldsymbol{X} (z, \bar{z}) =
\left( X^1 (z, \bar{z}), \ldots, X^8 (z, \bar{z}) \right),$$ of a two-dimensional surface immersed into eight-dimensional Euclidean space ${\R}^8$. Substituting (\[4.10\]) into (\[4.45\]) we express the radius vector $\boldsymbol{X} (z, \bar{z})$ in terms of functions $(\varphi_{1}, \varphi_{2}, \psi_{1}, \psi_{2})$ and obtain $$\begin{gathered}
X^1 = 2 \int_{C} (\bar{\psi}_{1} \varphi_{1}+ \bar{\psi}_{2}
\varphi_{2}) dz
+ ( \psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2}) d \bar{z},
\\
X^2 = 2 \int_{C} \left\{ \frac{\bar{\psi}_{1}}{\varphi_{1}} \left[
\varphi_{1}^2
- \frac{\psi_{1}}{\bar{\varphi}_{1}} ( \bar{\psi}_{1} \varphi_{1}
+ \bar{\psi}_{2} \varphi_{2}) - \frac{\psi_{1}}{\bar{\varphi}_{1}}
\partial \left(A^{-1}\right)\right]\right.\end{gathered}$$ $$\begin{gathered}
\left.\phantom{X^2=}{}- \frac{A \bar{\varphi}_{2} \psi_{1}}{\Omega}
\left[ \frac{J}{A^2} \frac{\psi_{2}}{\bar{\varphi}_{2}} + \left(\partial \left(A^{-1}\right)
+ \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}\right) \varphi_{2}^2\right]
\right\} dz
\nonumber\\
\phantom{X^2=}{}+ \left\{ \frac{\psi_{1}}{\bar{\varphi}_{1}} \left[ \bar{\varphi}_{1}^2
- \frac{\bar{\psi}_{1}}{\varphi_{1}} ( \psi_{1} \bar{\varphi}_{1} +
\psi_{2} \bar{\varphi}_{2}) - \frac{\bar{\psi}_{1}}{\varphi_{1}}
\bar{\partial} \left(A^{-1}\right)\right]\right.
\nonumber\\
\left.\phantom{X^2=}{}- \frac{A \varphi_{2} \bar{\psi}_{1}}
{\bar{\Omega}} \left[ \frac{\bar{J}}{A^2} \frac{\bar{\psi}_{2}}{\varphi_{2}}
+ \left( \bar{\partial}\left(A^{-1}\right) + \psi_{1} \bar{\varphi}_{1}
+ \psi_{2} \bar{\varphi}_{2}\right) \bar{\varphi}_{2}^2 \right] \right\} d \bar{z},
\nonumber\\
X^3 = i \int_{C} - \left\{ - \frac{\bar{\psi}_{1}}{\varphi_{1}}
\left[ \partial \left(A^{-1}\right) + 2 (\bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2}
\varphi_{2})\right]\right.
\nonumber\\
\phantom{X^3=}{}- \frac{A \bar{\varphi}_{1} \bar{\varphi}_{2}}{\Omega} \left[\frac{J}{A^2}
\frac{\psi_{2}}{\bar{\varphi}_{2}} + \left( \partial\left(A^{-1}\right)
+ \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}\right) \varphi_{2}^2\right]
\nonumber\\
\left.\phantom{X^3=}{}+ \left[ - \varphi_{1}^2 + \frac{\psi_{1}}{\bar{\varphi}_{1}}
(\bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}) +
\frac{\psi_{1}}{\bar{\varphi}_{1}}
\partial \left(A^{-1}\right)\right] \right\} dz
\nonumber\\
\phantom{X^3=}{}+ \left\{ - \frac{\psi_{1}}{\bar{\varphi}_{1}} \left[ \bar{\partial} \left( A^{-1}\right) +
2 (\psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})\right]\right.
\nonumber\\
\phantom{X^3=}{}- \frac{A \varphi_{1} \varphi_{2}}{\bar{\Omega}} \left[ \frac{\bar{J}}{A^2}
\frac{\bar{\psi}_{2}}{\varphi_{2}} + \left( \bar{\partial} \left(A^{-1}\right) + \psi_{1}
\bar{\varphi}_{1} + \psi_{2} \varphi_{2}\right) \bar{\varphi}_{2}^2 \right]
\nonumber\\
\left.\phantom{X^3=}{}+ \left[ - \bar{\varphi}_{1}^2 + \frac{\bar{\psi}_{1}}{\varphi_{1}}
(\psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})
+ \frac{\bar{\psi}_{1}}{\varphi_{1}} \bar{\partial} \left(A^{-1}\right)\right] \right\} d
\bar{z},
\nonumber\\
X^4 = \int_{C} \left\{ - \frac{\bar{\psi}_{1}}{\varphi_{1}}
\left[ \partial \left(A^{-1}\right) + 2 ( \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2}
\varphi_{2})\right]\right.
\nonumber\\
\phantom{X^4=}{}- \frac{A \bar{\varphi}_{1} \bar{\varphi}_{2}}{\Omega} \left[ \frac{J}{A^2}
\frac{\psi_{2}}{\bar{\varphi}_{2}} + \left( \partial\left(A^{-1}\right) + \bar{\psi}_{1}
\varphi_{1} + \bar{\psi}_{2} \varphi_{2}\right) \varphi_{2}^2\right]
\nonumber\\
\left.\phantom{X^4=}{}- \left[ - \varphi_{1}^2 + \frac{\psi_{1}}{\bar{\varphi}_{1}} (
\bar{\psi}_{1}
\varphi_{1} + \bar{\psi}_{2} \varphi_{2}) +
\frac{\psi_{1}}{\bar{\varphi}_{1}}
\partial \left(A^{-1}\right) \right] \right\} dz
\nonumber\\
\phantom{X^4=}{}+\left\{ - \frac{\psi_{1}}{\bar{\varphi}_{1}} \left[ \bar{\partial} \left(A^{-1}\right) + 2
(\psi_{1}
\bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})\right]\right.
\nonumber\\
\phantom{X^4=}{}- \frac{A \varphi_{1} \varphi_{2}}
{\bar{\Omega}} \left[\frac{\bar{J}}{A^2} \frac{\bar{\psi}_{2}}{\varphi_{2}} +
\left( \bar{\partial} \left(A^{-1}\right) + \psi_{1} \bar{\varphi}_{1} + \psi_{2}
\bar{\varphi}_{2}\right)
\bar{\varphi}_{2}^2\right]
\nonumber\\
\left.\phantom{X^4=}{}- \left[ - \bar{\varphi}_{1}^2 + \frac{\bar{\psi}_{1}}{\varphi_{1}}
(\psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})
+ \frac{\bar{\psi}_{1}}{\varphi_{1}} \bar{\partial} \left(A^{-1}\right)\right]\right\} d
\bar{z},
\nonumber\\
X^5 = i \int_{C} - \left\{ - \frac{\bar{\psi}_{2}}{\varphi_{2}} \left[ \partial
\left(A^{-1}\right)
+ 2 ( \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2})\right]\right.
\nonumber\\
\phantom{X^5=}{}+ \frac{A \bar{\varphi}_{1} \bar{\varphi}_{2}}{\Omega}
\left[ \left( \partial \left(A^{-1}\right) + \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2}
\varphi_{2}\right)
\varphi_{1}^2 + \frac{J}{A^2} \frac{\psi_{1}}{\bar{\varphi}_{1}} \right]
\nonumber\\
\left.\phantom{X^5=}{}+ \left[ - \varphi_{2}^2 + \frac{\psi_{2}}{\bar{\varphi}_{2}}
(\bar{\psi}_{1}
\varphi_{1}
+ \bar{\psi}_{2} \varphi_{2}) + \frac{\psi_{2}}{\bar{\varphi}_{2}}
\partial
\left(A^{-1}\right) \right] \right\} dz
\nonumber\\
\phantom{X^5=}{}+ \left\{ - \frac{\psi_{2}}{\bar{\varphi}_{2}} \left[ \bar{\partial} \left(A^{-1}\right)
+ 2 ( \psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})\right]\right.
\nonumber\\
\phantom{X^5=}{}+ \frac{A \varphi_{1} \varphi_{2}}{\bar{\Omega}} \left[ \left( \bar{\partial}
\left(A^{-1}\right) + \psi_{1} \bar{\varphi}_{1} + \psi_{2}
\bar{\varphi}_{2}\right) \bar{\varphi}_{1}^2 + \frac{\bar{J}}{A^2}
\frac{\bar{\psi}_{1}}{\varphi_{1}}\right]\nonumber
\\
\left.\phantom{X^5=}{}+ \left[ - \bar{\varphi}_{2}^2 + \frac{\bar{\psi}_{2}}{\varphi_{2}}
(\psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})
+ \frac{\bar{\psi}_{2}}{\varphi_{2}} \bar{\partial} \left(A^{-1}\right) \right] \right \} d \bar{z},\end{gathered}$$ $$\begin{gathered}
X^6 = \int_{C} \left\{ - \frac{\bar{\psi}_{2}}{\varphi_{2}}
\left[ \partial \left(A^{-1}\right) + 2 (\bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2}
\varphi_{2}) \right]\right.
\nonumber\\
\phantom{X^6=}{}+ \frac{A \bar{\varphi}_{1} \bar{\varphi}_{2}}{\Omega} \left[ \left( \partial \left(
A^{-1}\right) +
\bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}\right) \varphi_{1}^2
+ \frac{J}{A^2} \frac{\psi_{1}}{\bar{\varphi}_{1}} \right]
\nonumber\\
\left.\phantom{X^6=}{}- \left[ - \varphi_{2}^2 + \frac{\psi_{2}}{\bar{\varphi}_{2}}
(\bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}) +
\frac{\psi_{2}}{\bar{\varphi}_{2}}
\partial \left(A^{-1}\right) \right] \right\} dz
\nonumber\\
\phantom{X^6=}{}+ \left\{ - \frac{\psi_{2}}{\bar{\varphi}_{2}} \left[ \bar{\partial} \left( A^{-1}\right)
+ 2 ( \psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2})\right]\right.
\nonumber\\
\phantom{X^6=}{}+ \frac{A \varphi_{1} \varphi_{2}}{\bar{\Omega}} \left[ \left( \bar{\partial}
\left(A^{-1}\right)
+ \psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2}\right)
\bar{\varphi}_{1}^2
+ \frac{\bar{J}}{A^2} \frac{\bar{\psi}_{1}}{\varphi_{1}}\right]
\nonumber\\
\left.\phantom{X^6=}{}- \left[ - \bar{\varphi}_{2}^2 + \frac{\bar{\psi}_{2}}{\varphi_{2}} (
\psi_{1}
\bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2}) +
\frac{\bar{\psi}_{2}}{\varphi_{2}}
\bar{\partial} \left(A^{-1}\right) \right] \right\} d \bar{z},
\nonumber\\
X^7 = i \int_{C} - \left\{ \frac{\bar{\psi}_{2}}{\varphi_{2}} \left[
\varphi_{1}^2
- \frac{\psi_{1}}{\bar{\varphi}_{1}} ( \bar{\psi}_{1} \varphi_{1} +
\bar{\psi}_{2} \varphi_{2})
- \frac{\psi_{1}}{\bar{\varphi}_{1}} \partial \left(A^{-1}\right)\right]\right.
\nonumber\\
\phantom{X^7=}{}+ \frac{A \bar{\varphi}_{2} \psi_{1}}{\Omega} \left[ \left(\partial \left(A^{-1}\right) +
\bar{\psi}_{1}
\varphi_{1} + \bar{\psi}_{2} \varphi_{2}\right) \varphi_{1}^2 + \frac{J}{A^2}
\frac{\psi_{1}}{\bar{\varphi}_{1}}\right]
\nonumber\\
\phantom{X^7=}{}+ \frac{A \bar{\varphi}_{1} \psi_{2}}{\Omega}\left[ \frac{J}{A^2}
\frac{\psi_{2}}{\bar{\varphi}_{2}}
+\left( \partial \left(A^{-1}\right) + \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2}
\varphi_{2}\right) \varphi_{2}^2\right]
\nonumber\\
\left.\phantom{X^7=}{} + \frac{\bar{\psi}_{1}}{\varphi_{1}} \left[ - \varphi_{2}^2 +
\frac{\psi_{2}}{\bar{\varphi}_{2}}
( \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}) +
\frac{\psi_{2}}{\bar{\varphi}_{2}}
\partial \left(A^{-1}\right)\right] \right\} dz
\nonumber\\
\phantom{X^7=}{}+ \left\{ \frac{\psi_{2}}{\bar{\varphi}_{2}} \left[ \bar{\varphi}_{1}^2 -
\frac{\bar{\psi}_{1}}{\varphi_{1}}
(\psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2}) -
\frac{\bar{\psi}_{1}}{\varphi_{1}}
\bar{\partial} \left(A^{-1}\right)\right]\right.
\nonumber\\
\phantom{X^7=}{}+ \frac{A \varphi_{2} \bar{\psi}_{1}}{\Omega} \left[ \left(\bar{\partial}
\left(A^{-1}\right)
+ \psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2}\right)
\bar{\varphi}_{1}^2 + \frac{\bar{J}}{A^2}
\frac{\bar{\psi}_{1}}{\varphi_{1}}\right]
\nonumber\\
\phantom{X^7=}{}+ \frac{A \varphi_{1} \bar{\psi}_{2}}{\bar{\Omega}}\left[
\frac{\bar{J}}{A^2}
\frac{\bar{\psi}_{2}}
{\varphi_{2}} + \left( \bar{\partial} \left(A^{-1}\right) + \psi_{1} \bar{\varphi}_{1} +
\psi_{2}
\bar{\varphi}_{2}\right) \bar{\varphi}_{2}^2\right]
\nonumber\\
\left.\phantom{X^7=}{}+ \frac{\psi_{1}}{\bar{\varphi}_{1}}\left[ - \bar{\varphi}_{2}^2
+ \frac{\bar{\psi}_{2}}{\varphi_{2}} ( \psi_{1} \bar{\varphi}_{1} +
\psi_{2} \bar{\varphi}_{2})
+ \frac{\bar{\psi}_{2}}{\varphi_{2}} \bar{\partial} \left(A^{-1}\right) \right] \right\} d
\bar{z},
\nonumber\\
X^8 = \int_{C} \left\{ \frac{\bar{\psi}_{2}}{\varphi_{2}} \left[ \varphi_{1}^2 -
\frac{\psi_{1}}{\bar{\varphi}_{1}}
( \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}) -
\frac{\psi_{1}}{\bar{\varphi}_{1}}
\partial \left(A^{-1}\right)\right ]\right.
\nonumber\\
\phantom{X^8=}{}+ \frac{A \bar{\varphi}_{2} \psi_{1}}{\Omega} \left[ \left( \partial \left(A^{-1}\right)
+ \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2} \varphi_{2}\right) \varphi_{1}^2
+ \frac{J}{A^2}
\frac{\psi_{1}}{\bar{\varphi}_{1}}\right]
\nonumber\\
\phantom{X^8=}{}- \frac{A \bar{\varphi}_{1} \psi_{2}}{\Omega} \left[ \frac{J}{A^2}
\frac{\psi_{2}}{\bar{\varphi}_{2}}
+ \left( \partial \left(A^{-1}\right) + \bar{\psi}_{1} \varphi_{1} + \bar{\psi}_{2}
\varphi_{2}\right)
\varphi_{2}^2\right]
\nonumber\\
\left.\phantom{X^8=}{}- \frac{\bar{\psi}_{1}}{\varphi_{1}} \left[ - \varphi_{2}^2 +
\frac{\psi_{2}}{\bar{\varphi}_{2}} ( \bar{\psi}_{1} \varphi_{1} +
\bar{\psi}_{2}
\varphi_{2}) + \frac{\psi_{2}}{\bar{\varphi}_{2}} \partial \left(A^{-1}\right) \right] \right\}
dz
\nonumber\\
\phantom{X^8=}{}+ \left\{ \frac{\psi_{2}}{\bar{\varphi}_{2}} \left[ \bar{\varphi}_{1}^2 -
\frac{\bar{\psi}_{1}}{\varphi_{1}}
(\psi_{1} \bar{\varphi}_{1} + \psi_{2} \bar{\varphi}_{2} ) -
\frac{\bar{\psi}_{1}}{\varphi_{1}}
\bar{\partial} \left( A^{-1}\right)\right]\right.
\nonumber\\
\phantom{X^8=}{}+ \frac{A \varphi_{2} \bar{\psi}_{1}}{\bar{\Omega}}
\left[ \left( \bar{\partial} \left(A^{-1}\right) + \psi_{1} \bar{\varphi}_{1} + \psi_{2}
\bar{\varphi}_{2}\right)
\bar{\varphi}_{1}^2 + \frac{\bar{J}}{A^2}
\frac{\bar{\psi}_{1}}{\varphi_{1}}\right]\nonumber
\\
\phantom{X^8=}{}- \frac{A \varphi_{1} \bar{\psi}_{2}}{\bar{\Omega}} \left[
\frac{\bar{J}}{A^2} \frac{\bar{\psi}_{2}}{\varphi_{2}}
+ \left( \bar{\partial} \left(A^{-1}\right) + \psi_{1} \bar{\varphi}_{1} + \psi_{2}
\bar{\varphi}_{2}\right)
\bar{\varphi}_{2}^2\right]
\nonumber\end{gathered}$$ $$\begin{gathered}
\left.\phantom{X^8=}{}- \frac{\psi_{1}}{\bar{\varphi}_{1}} \left[ - \bar{\varphi}_{2}^2
+ \frac{\bar{\psi}_{2}}{\varphi_{2}} (\psi_{1} \bar{\varphi}_{1}
+ \psi_{2} \bar{\varphi}_{2})
+ \frac{\bar{\psi}_{2}}{\varphi_{2}} \bar{\partial} \left(A^{-1}\right) \right] \right\} d
\bar{z},
\label{4.47}
\end{gathered}$$ where we have introduced the following notation $$\label{4.48}
\Omega = \varphi_{1} \psi_{2} | \varphi_{1}|^2 - \varphi_{2} \psi_{1} |
\varphi_{2}|^2,
\qquad
\bar{\Omega} = \bar{\varphi}_{1} \bar{\psi}_{2} | \varphi_{1}|^2
- \bar{\varphi}_{2} \bar{\psi}_{1} | \varphi_{2}|^2.$$
Note that when $J$ vanishes, as can be checked, the position vector $X$ given by (\[4.47\]) obeys the following relations $$\label{4.49}
( \partial X, \partial X) = 0,$$ and $$\label{4.50}
( \partial \bar{\partial} X, \partial \bar{\partial} X) = ( \partial X,
\bar{\partial} X)^{2} \neq 0,$$ whenever the equations of the $CP^2$ sigma model (\[4.6\])–(\[4.8\]) are satisfied. The explicit form of (\[4.50\]), when written in terms of $w$’s or $\psi$’s and $\phi$’s, is very complicated and so we shall not reproduce it here. As a consequence of (\[4.49\]) and (\[4.50\]) the components of induced metric are $$\label{4.51}
g_{zz} = g_{\bar{z}\bar{z}} = 0,\qquad g_{z\bar{z}} \neq 0$$ and the norm of the mean curvature vector $\bar{H} = (g_{z\bar{z}})^{-1}
\partial
\bar{\partial} X$, as expected, is equal to one, i.e. $|\bar{H}|^{2} =
1$. We would like to note that when $w_{i}$ are holomorphic functions then $J
= 0$ and formulae (\[4.45\]) define a surface on $SU(3)$. Then using expressions in [@Fok] we can calculate, in a closed form, all geometric characteristics of a given surface.
Thus we have proved that the conformal immersions of CMC-surfaces into ${\R}^8$ are determined by formulae (\[4.45\]) or (\[4.47\]), where the complex functions $w_i$ obey the equations of $CP^2$ sigma model (\[4.6\])–(\[4.8\]), (or complex functions $\psi_i$ and $\phi_i$ obey the first order system (\[4.1\])–(\[4.4\])) and $J$ given by (\[4.29\]) (or (\[4.31\])) vanishes.
Examples and applications
=========================
In this section, based on our results of previous sections we construct certain classes of two-dimensional CMC-surfaces immersed into ${\R}^8$. For this purpose we use the $CP^2$ sigma model defined over $S^2$. Note that for such a model all solutions of the Euler–Lagrange equations (\[4.6\])–(\[4.8\]) are well known [@book11]. Under the requirement of the finiteness of the action they split into three separate classes, i.e. analytic (i.e. $w_i = w_i(z)$), antianalytic (i.e. $w_i =
w_i(\bar{z})$) and mixed ones. The latter ones can be determined from either the holomorphic or antiholomorphic functions by the following procedure.
Consider three arbitrary holomorphic functions $f_i = f_i(z)$ and define for each pair the Wronskian $$\label{5.1}
F_{ij} = f_i\partial f_j -
f_j \partial f_i,
\qquad
\bar {\partial}f_i = 0,
\qquad
i,j = 1,2,3$$ Next determine three complex valued functions $$\label{5.2}
g_i = \sum^3_{k\neq i} \bar{f_k}
(f_k\partial f_i -
f_i \partial f_k),\qquad
i = 1,2,3$$ Then the mixed solutions $w_i$ of $CP^2$ sigma model (\[4.6\])–(\[4.8\]) can be determined as ratios of the components of $g_i$, i.e. $$\label{5.3}
w_1 = \frac{g_1}{g_3},
\qquad
w_2 = \frac{g_2}{g_3},
\qquad
g_3\neq 0.$$ Alternatively, similar class of solutions can be obtained when we consider three arbitrary antiholomorphic functions $f_i = f_i(\bar{z})$ and construct $g_i$ in the same way as above, but using $\bar{\partial}$ instead of $\partial$ in the equations (\[5.2\]).
Now, let us discuss some classes of CMC-surfaces in ${\R}^n$ which can be obtained directly by applying the Weierstrass representation (\[4.45\]) and (\[5.3\]).
[**1**]{}. One of the simplest solutions which corresponds to the analytic choice of functions is $$\label{5.4}
w_1 = z,
\qquad
w_2 = 1,
\qquad
A = 2 + |z|^2,
\qquad
J = 0.$$
Using the $CP^2$ representation (\[4.45\]) we can find that the associated CMC-surface is immersed in ${\R}^3$ and is given in a polar coordinates $\left(r = \left(x^2+y^2\right)^{1/2}, \varphi\right)$ by $$\begin{gathered}
X^1 = -2 X^2 = 2 X^6 = 2 \sqrt2 \left(2+r^2\right)^{-1},
\qquad
X^3 = - X^7 = -2 r \left(2+r^2\right)^{-1} \sin\varphi,
\nonumber\\
X^5 = 0, \qquad
X^4 = X^8 = 2 r \left(2+r^2\right)^{-1} \cos\varphi.
\label{5.5}
\end{gathered}$$ The metric is conformally flat $$\label{5.6}
I = \frac{2}{\left(2+r^2\right)^2}\,\left(dr^2 + r^2 d\varphi^2\right)$$ This case corresponds to the immersion of the $CP^2$ model into the $CP^1$ model.
[**2**]{}. Another class of two-soliton solutions of the $CP^2$ model (\[4.6\])–(\[4.8\]) is determined, for example, by two analytic functions; i.e. we can take, for example: $$\label{5.7}
w_1 = z^2,
\qquad
w_2 = \sqrt2 z,
\qquad
A = \left(1+|z|^2\right)^2,
\qquad
J = 0.$$
Integrating formulae (\[4.45\]) we obtain the associated CMC-surface which can be written in a polar coordinates as follows $$\begin{gathered}
X^1 = 2 \left(1+r^2\right)^{-2},
\qquad
X^2 = -2 \left(1+2r^2\right)\left(1+r^2\right)^{-2},
\nonumber\\
X^3 = -2 r^2 \left(1+r^2\right)^{-2} \sin2\varphi,
\qquad
X^4 = 2 r^2 \left(1+r^2\right)^{-2} \cos2\varphi,
\nonumber\\
X^5= -2\sqrt2 r \left(1+r^2\right)^{-2} \sin\varphi,
\qquad
X^6 = 2\sqrt2 r \left(1+r^2\right)^{-2} \cos\varphi,
\nonumber\\
X^7 = \sqrt2 \left(2r^2 -3\right)r\left(1+r^2\right)^{-2}\sin\varphi,
\nonumber\\
X^8 = \sqrt2 \left(2r^2 -3\right)r\left(1+r^2\right)^{-2}\cos\varphi,
\label{5.8}
\end{gathered}$$ The corresponding first fundamental form is conformal $$\label{5.9}
I = \frac{2}{\left(1+r^2\right)^2}\,\left(dr^2 + r^2 d\varphi^2\right).$$
[**3**]{}. A class of mixed solutions of the $CP^2$ model (\[4.2\]) is represented by $$\begin{gathered}
w_1 = \frac{\bar{z} + z}{1 - |z|^2},
\qquad
w_2 = \frac{\bar{z} - z}{1 - |z|^2},
\qquad
A = \left(\frac{1 + |z|^2}{1 - |z|^2}\right)^2,
\qquad
J = 0.
\label{5.10}
\end{gathered}$$ From (\[4.32\]) and using (\[5.2\]) we obtain the expression for the associated surface which can be written in polar coordinates as follows $$\begin{gathered}
X^1 = X^2 = X^4 = X^5 = X^7 = 0,
\nonumber\\
X^3 = \frac{-4 r}{1 + r^2}\, \sin\varphi,
\qquad
X^6 = \frac{-4 r}{1 +r^2}\, \cos\varphi,
\qquad
X^8 = \frac{4}{1 +r^2}\,
\label{5.11}
\end{gathered}$$ Hence this CMC-surface is really immersed in ${\R}^3$. The metric is conformal $$\label{5.12}
I = \frac{16}{\left(1 +r^2\right)^2}\,\left(dr^2 + r^2 d\varphi^2\right).$$ This case corresponds to the immersion of the $CP^2$ model into the $CP^1$ model.
If we take a more complicated example of this class, say, $$\begin{gathered}
w_1 = \frac{-\bar{z}\left(3+2|z|^2\right)}{3 - |z|^4},
\qquad
w_2 = \frac{\sqrt{3}z\left(2+|z|^2\right)}{3 - |z|^4},
\nonumber\\
A = \frac{\left(1 + |z|^2\right)\left(|z|^6+6|z|^4+12|z|^2+9\right)}
{\left(3 - |z|^4\right)^2},\qquad
J = 0.
\label{5.113}
\end{gathered}$$ then the expressions become very complicated but, in this case, we do have a genuine $CP^2$ solution.
[**4**]{}. An interesting class of meron-like solutions of the $CP^1$ model (\[3.5\]) is given by $$\label{5.13}
w = \left(\frac{z}{\bar{z}}\right)^{\beta},
\qquad
A = 2,
\qquad
J = \frac{-\beta^2}{2 z^2}.$$ Here $\beta$ can be an integer of a half-integer, with $2\beta$ being the meron number. Note also that all merons are located at $z=0$ and so this solution is defined on ${\R}^2\backslash\{0\}$.
Then using the transformation (\[3.7\]) we find that the solution of the GW system (\[3.1\]) is given by $$\label{5.14}
\psi_1 = \frac{\sqrt{\beta}}{2\sqrt{\bar z}}\left(\frac{z}{\bar z}\right)
\sp{\frac{1}{2}},
\qquad
\psi_2 = \frac{\sqrt{\beta}}{2\sqrt{ z}}\left(\frac{z}{\bar z}\right)
\sp{\frac{1}{2}},
\qquad
p = \frac{\beta}{2|z|}.$$ The associated surface is obtained by integrating formulae (\[3.20\]) and we find $$\label{5.15}
X_1 = \frac{-3}{2} \sin 2 \beta\varphi,
\qquad
X_2 = \frac{-3}{2} \cos 2 \beta\varphi,
\qquad
X_3 = \frac{\beta}{2} \ln r.$$ Thus the CMC-surface is a cylinder which is covered $2\beta$ times. Of course, the Gauss and mean curvatures are $$\label{5.16}
K = 0,
\qquad
H = 1.$$
We note that our procedure of using the ‘multimeron’ solution of the $CP^1$ model to construct our CMC-surface has effectively involved mapping ${\R}^2\backslash\{0\}$ onto the equator of the unit sphere (merons) and then turning this circle into an infinite cylinder (based on this circle). This was done by effectively ‘undoing’ the radial projection of the previous map. Hence the two singular points of the multi-meron configuration (\[5.13\]) i.e. ($z=0$ and the point at $\infty$) have got mapped at the ‘ends’ of the cylinder.
Final remarks and future developments
=====================================
In this paper we have demonstrated links between the $CP^1$ and $CP^2$ sigma models and Weierstrass representations for two-dimensional surfaces immersed into Euclidean spaces ${\R}^3$ and ${\R}^8$, respectively. These links enabled us to present an algorithm for the construction of CMC-surfaces immersed into ${\R}^n$. This new approach has been tested in Section 5. It has proved to be effective, as we were able to reproduce easily the known results which, before, were obtained by much more complicated procedures. Its potential for providing new meaningful results has been exhibited in the case of mixed solutions of the $CP^2$ sigma model leading to new interesting surfaces in ${\R}^8$.
The analytic method of the construction of the CMC-surfaces immersed into ${\R}^n$ presented here is limited by several assumptions. For example we have studied only $CP^1$ and $CP^2$ models defined over $S^2$ (i.e. to consider $J\ne 0$) The question the arises as to whether our approach can be extended to higher $CP^N$ models and to Weierstrass systems describing surfaces immersed in multi-dimensional Euclidean and pseudo-Riemannian spaces. If this is the case our approach may provide new classes of solutions and consequently new classes of surfaces in these multi-dimensional spaces. Other requirement of the proposed method, worth investigating further is the $CP^N$ models involving maps from ${\R}^2$ (not necessarly $S^2$). We can expect that taking $J\ne 0$ can broaden the applicability of our approach.
Finally, it is worth noting that the CMC-surfaces can be used “in reverse” to address certain physical problems. Namely, we sometimes know the analytical description of CMC-surfaces in a physical system for which analytic models are not fully developed. Using our approach we can, perhaps, select an appropriate sigma model corresponding to the given Weierstrass representation and characterise the class of equations describing the physical phenomena in question. This was attempted successfully for Weierstrass representation for CMC-surfaces in 3-dimensional Euclidean space [@book13], but not to our knowledge for multi-dimensional spaces. These and other questions will be addressed in future work.
Acknowledgements {#acknowledgements .unnumbered}
----------------
AMG thanks A Strasburger (University of Warsaw) for helpful and interesting discussions on the topic of this paper. WJZ would like to thank the CRM, Universite de Montreal and Center for Theoretical physics, MIT for the support of his stay and their hospitality. AMG would like to thank the University of Durham for the award of Alan Richards fellowship to him that allowed him to spend two terms in Durham during the academic years 2001 and 2002. Partial support for AMG’s work was provided also by a grant from NSERC of Canada and the Fonds FCAR du Quebec.
[88]{}
Enneper A, [*Nachr. Konigl. Gesell. Wissensch. Georg - Augustus Univ. Gottingen*]{} [**12**]{} (1868), 258–277.
Weierstrass K, Fortsetzung der Untersuchung uber die Minimalflachen, Mathematische Werke, Vol. 3, Veragsbuch-handlung, Hillesheim, 1866, 219–248.
Oserman R, A Survey of Minimal Surfaces, Dover, New York, 1996.
Konopelchenko B, Induced Surfaces and Their Integrable Dynamics, [*Stud. Appl. Maths.*]{} [**96**]{} (1996), 9–51.
Bobenko A I, Surfaces in Terms of 2 by 2 Matrices.Old and New Integrable Cases, in Harmonic Maps and Integrable System, Editors: Fordy A P and Wood J C, Vieweg, Wiesbaden, 1994, 193–202.
Gross D G, Pope C N and Weinberg S (Editors), Two-Dimensional Quantum Gravity and Random Surfaces, World Scientific, Singapore, 1992.
Carroll R and Konopelchenko B, Generalised Weierstrass–Enneper Inducing Conformal Immersions and Gravity, [*Int. J. Mod. Phys.*]{} [**A11**]{}, Nr. 7 (1996), 1183–1216.
Konopelchenko B and Landolfi G, On Classical String Configurations, [*Modern Phys. Letts.*]{} [**12**]{} (1997), 3161–3179.
Viswanathan K and Parthasarathy R, [*Phys. Rev. D*]{} [**D51**]{} (1995), 5830–5879.
Nelson D, Piran T and Weinberg S, Statistical Mechanics of Membranes and Surfaces, World Scientific, Singapore, 1989.
Amit D, Field Theory, the Renormalization Group and Critical Phenomena, McGraw-Hill, New York, 1978.
Rozdestvenski B I and Yanenko N N, Systems of Quasilinear Equations and their Applications to Gas Dynamics, AMS. Transl., Vol. 55, Providence, RI, 1983.
Ou-Yang Z O, Liu J and Xie Y Z, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, World Scientific, Singapore, 1999.
Canham P B, [*J.Theor. Biol.*]{} [**26**]{} (1970), 61–81.
Helfrich W, [*Z. Naturforsch.*]{} [**28**]{} (1973), 693–709.
Safran S A, Statistical Thermodynamics of Surfaces. Interfaces and Membranes, Addison Wesley, Reading, 1994.
Eisenhart L P, Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1909.
Willmore T J, Total Curvature in Riemannian Geometry, Ellis Horwood, New York, 1982.
Konopelchenko B and Taimanov I, Constant Mean Curvature Surfaces via an Integrable Dynamical System, [*J. Phys.*]{} [**A29**]{} (1996), 1261–1265.
Konopelchenko B and Landolfi G, Generalised Weierstrass Representation for Surfaces in Multi-Dimensional Riemann Spaces, [*J. Geom. Phys.*]{} [**29**]{} (1999), 314–333.
Bracken P and Grundland A M, Symmetry Properties and Explicit Solutions of the Generalised Weierstrass System, [*J. Math. Phys.*]{} [**42**]{}, Nr. 3 (2001), 1250–1282.
Kenmotsu K, Weierstrass Formula for Surfaces of Prescribed Mean Curvature, [*Math. Ann.*]{} [**245**]{} (1979), 89–99.
Zakharov V E and Mikhailov A V, Relativistically invariant two dimensional models in field theory which are integrable by means of the inverse scatering problem method, [*Sov. Phys.-JETP.*]{} [**47**]{} (1979), 1017–1059.
Zakrzewski W J, Low Dimensional Sigma Models, Adam Hilger, Bristol, 1989.
Abe K and Erbacher J, Isometric Immersions with the Same Gauss Map, [*Math. Ann.*]{} [**215**]{} (1975), 197–201.
Hoffman D and Osserman R, The Gauss Map of Surfaces in ${\R}^n$, [*J. Diff. Geom.*]{} [**18**]{} (1983), 733–754.
Ferapontov E V and Grundland A M, Links Between Different Analytic Descriptions of Constant Mean Curvature Surfaces, [*J. Nonlinear Math. Phys.*]{} [**7**]{} (2000), 14–21.
Grundland A M and Zakrzewski W J, The Weierstrass Representation for Surfaces Immersed into ${\R}^8$ and $CP^2$ Maps, [*J. Math. Phys.*]{} [**43**]{} (2002), 3352–3362.
Fokas A S and Gelfand I M, Surfaces on Lie Groups, on Lie Algebras and Their Integrability, [*Comm. Math. Phys.*]{} [**177**]{} (1996), 203–220.
Grosse-Brauckman K and Polthier K, Constant Mean Curvature Surfaces Derived from Delauney’s and Wente’s Examples, in Visualization and Mathematics: Experiments, Simulations and Environments, Editors: Hege H C and Polthier K, Springer-Verlag, Berlin, 1997, 386–419.
\[grundland-lastpage\]
|
---
abstract: 'This paper introduces a formally second-order direction-splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spherical coordinate transforms. This approach allows to avoid the singularities at the poles and keeps the grid size relatively uniform. The downside is that the spherical shell is subdivided into two equally sized, overlapping subdomains that requires the use of Schwarz-type iterations. The temporal second order accuracy is achieved via an Artificial Compressibility (AC) scheme with bootstrapping (see [@doi:10.1137/140975231; @Guermond201792],). The spatial discretization is based on second order finite differences on the Marker-And-Cell (MAC) stencil. The entire scheme is implemented in parallel using a domain decomposition iteration, and a direction splitting approach for the local solves. The stability, accuracy and weak scalability of the method is verified on a manufactured solution of the Navier-Stokes-Boussinesq system and on the Landau solution of the Navier-Stokes equations on the sphere.'
address: 'Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada'
author:
- Aziz Takhirov
- Roman Frolov
- Peter Minev
bibliography:
- 'references.bib'
title: 'A direction splitting scheme for Navier-Stokes-Boussinesq system in spherical shell geometries'
---
Splitting methods, Navier-Stokes equations on the sphere, Parallel algorithm.
Introduction
============
This article presents a new direction-splitting scheme for solving the incompressible Navier-Stokes-Boussinesq system: $$\label{eq:eq_ns}
\begin{aligned}
\frac{\partial \bu}{\partial t}+(\bu \cdot \GRAD) \bu + \GRAD p - \Pr \LAP \bu & = \bg \Pr \Ra T \text{ in }~\Omega \times (0, T_{f}]\\
\nabla \cdot \bu&=0 \text{ in }~\Omega \times (0, T_{f}] \\
\bu & = \mathbf{0}\text{ on }~ \partial \Omega \times (0, T_{f}]
\end{aligned}$$ $$\label{eq:eq_heat_convec}
\begin{aligned}
\frac{\partial T}{\partial t}+(\bu \cdot \GRAD) T - \LAP T &
= 0 \text{ in }~\Omega \times (0, T_{f}] \\
T & = 0 \text{ on }~ \partial \Omega \times (0, T_{f}]
\end{aligned}$$ in a spherical shell domain that can be defined in terms of a spherical coordinate triple $\left(r,\theta,\phi \right)$ as: $$\Omega = \left\lbrace \left(r,\theta,\phi \right) \in [R_1, R_2] \times \left[ 0, \pi \right] \times \left[ 0, 2 \pi \right) \right\rbrace.$$ In the above, $\bg$ is the unit vector in the direction of gravity, and $\Pr, \Ra$ are the Prandtl and Rayleigh numbers, respectively. The system - models the flow of a heat conducting fluid, under the assumption that the temperature-induced density variation influences significantly only the buoyancy force and the fluid remains incompressible. It is widely applied to model the flow in the atmospheric boundary layer ([@MKMMKVGHJ2016]), oceanic flows ([@SH2006]), as well as, if combined with an equation for the magnetic field, the flow in the Earth’s dynamo ([@SJNF2017]). Even though for the most part of the discussion, we assume homogeneous Dirichlet boundary conditions on the two spherical surfaces $r=R_1, r=R_2$, the approach is applicable to Neumann and Robin boundary conditions as well.
One widely used approach for numerical approximations of differential equations in spherical shell geometries is based on the use of a spherical transformation that transforms the domain into a parallelepiped. The obvious advantage of this approach is the simple computational domain, which allows for the use of structured grids and the efficient schemes developed for them. Moreover, the grid can naturally follow the geometry of the domain, without requiring too many cells, as would possibly be in the case of a Cartesian formulation. However, the singularity of the transformation and the grid convergence near the poles have for many years been a difficulty in the development of accurate finite difference and pseudo-spectral schemes. Several different treatments have been proposed for dealing with these problems. For example, in [@HUANG1993254], the pole singularity issue is avoided by replacing the equations at the poles with equations, analogous to boundary conditions, while in [@MOHSENI2000787], a redefinition of the singular coordinates is proposed. Other suggested approaches include applying L’Hospital’s rule [@GRIFFIN1979352] to singular terms and switching to Cartesian formulations around the poles [@doi:10.2514/6.1997-760]. On the other hand, the grid convergence has been a more serious problem. In particular, it produces a solution with uneven resolution, requires very small time steps for explicit or IMEX schemes since the time step size is limited by the minimum grid size, and causes convergence problems for iterative solvers. Therefore, different grid systems have been suggested in the literature that give quasi-uniform resolution and avoid the grid convergence problems. One such approach is the “cubed sphere” of [@RONCHI199693], which is a grid that covers a spherical surface with six components corresponding to six faces of a cube. Even though, the resulting grid is quasi-uniform, it still has singularities at the corner points of the faces and it is non-orthogonal. Some of the other suggested unstructured grids include the isocahedral grid of [@Baumgardner1985] and non-orthogonal rhombahedral grid of [@doi:10.1029/2000JB900003].
In this study we adopt an alternative approach, proposed by [@doi:10.1029/2004GC000734], employing the so-called Yin-Yang grids. It starts with a decomposition of the domain into two overlapping subdomains, combined with two different spherical transforms whose axes are perpendicular to each other, cf. Fig. \[fig:YYgrid\]. As a result, both subdomains are transformed into identical parallelepipeds that can be gridded with the same uniform grids. This approach automatically removes the transforms singularities at the poles, at the expense of the introduction of two subdomains, so that the two local solutions must be coupled by means of Schwarz-type iterations. It has been used for simulations of mantle convection [@TACKLEY20087], core collapse supernovae [@refId0], atmospherical general circulation model [@doi:10.1175/2010MWR3375.1] and visualization in spherical regions [@OHNO20091534]. Some advantages of Yin-Yang approach are that the metric tensors are simple, the resolution is quasi-uniform, and it requires modest programming effort for extending the code from a single latitude-longitude grid. The main novelty of this paper is that the Yin-Yang domain decomposition is combined with a direction splitting time discretization that, in case of linear parabolic equations, is unconditionally stable on grids on the spherically transformed domains. The advection can be included either in an IMEX fashion, or by including the linearized advection operator into the entire operator that is further split direction-wise. The resulting splitting scheme is conditionally stable, since the direction-wise operators are not positive, but our numerical experience demonstrated that the second approach yields an algorithm that has better stability performance. This is why the rest of the paper concerns only this type of schemes. To our knowledge, the stability of the direction splitting approach has not been rigorously studied in the context a spherical coordinate system. Therefore, we prove below that it is unconditionally stable in case of a scalar heat equation, in a simply shaped domain (in terms of spherical coordinates). The case of the full Navier-Stokes-Boussinesq system is more involved and we do not provide a rigorous proof here. However, our numerical experience shows that the a direction splitting is still unconditionally stable if the advection terms are omitted and if the velocity-pressure decoupling is done via the AC method proposed in [@doi:10.1137/140975231].
The rest of the paper is organized as follows. In the next section, we briefly recall the definition of the Yin-Yang domain decompostion. In Section $3$ we present the numerical scheme for the advection-diffusion and Navier-Stokes equations on each of the subdomains. In Section $4$, we discuss the implementation details, and in Section $5$ we present the numerical experiments.
Spatial discretization and the Yin-Yang grid
============================================
In this section, we briefly recall the definition of the composite Yin-Yang grid following [@doi:10.1029/2004GC000734]. The grid consists of two identical overlapping latitude-longitude grids whose axes are perpendicular to each other. The Yin grid is based on a spherical transformation $$\begin{aligned}
\begin{cases}
x &= r \sin \theta \cos \phi \\
y &= r \sin \theta \sin \phi \\
z &= r \cos \theta ,
\end{cases}\end{aligned}$$ and covers the region $$\Omega_1: = \left\lbrace \left(r,\theta,\phi \right) \in [R_1, R_2] \times \left[ \frac{\pi}{4}-\varepsilon, \frac{7\pi}{4}+\varepsilon \right] \times
\left[ \frac{\pi}{4}-\varepsilon, \frac{3\pi}{4}+\varepsilon \right] \right\rbrace,$$ where $\varepsilon \ll 1$ is a parameter determining the overlap. The Yang grid is obtained via another spherical transformation: $$\begin{aligned}
\begin{cases}
x &= - r \sin \ttheta \cos \tphi \\
y &= r \cos \ttheta \\
z &= r \sin \ttheta \sin \tphi,
\end{cases}\end{aligned}$$ such that its axes is perpendicular to the axes of the Yin transform, and covers the region $$\Omega_2: = \left\lbrace \left(r,\ttheta,\tphi \right) \in [R_1, R_2] \times \left[ \frac{\pi}{4}-\varepsilon, \frac{7\pi}{4}+\varepsilon \right] \times
\left[ \frac{\pi}{4}-\varepsilon, \frac{3\pi}{4}+\varepsilon \right] \right\rbrace.$$ The choice of the second axes should be such that the Yang grid fully covers the gap of the Yin one, and the overlapping subregions are of the same size (see Fig. \[fig:YYgrid\],). Otherwise, it is identical to the Yin grid modulo two rotations. The resulting Yin-Yang grids are quasiuniform, the coordinate transformations from $(r, \theta, \phi)$ to $(r, \ttheta, \tphi)$ and its inverse, as well as the metric tensors on both grids are identical. As a consequence, the methods and codes developed for the standard latitude-longitude grid can be applied to both grids.
![Yang (left) and Yin-Yang (right) grids. Each subgrid is further decomposed into blocks for a parallel implementation corresponding to a CPU distribution $1\times3\times2$.[]{data-label="fig:YYgrid"}](./Pics/Yang){width="\textwidth"}
![Yang (left) and Yin-Yang (right) grids. Each subgrid is further decomposed into blocks for a parallel implementation corresponding to a CPU distribution $1\times3\times2$.[]{data-label="fig:YYgrid"}](./Pics/YinYang){width="\textwidth"}
Direction-splitting time discretization
=======================================
Preliminaries
-------------
In the sequel of the paper we will frequently make use of the following notations. For a time sequence $w^k, k=1, 2,\dots$ we denote the average between two time levels as $w^{k+1/2}=(w^{k+1}+w^k)/2$, and the explicit extrapolation to level $k+1/2$ by $w^{*,k+1/2}=3w^{k}/2-w^{k-1}/2$. For two regular enough functions $u, v$ defined in the spherical shell we denote their weighted $L^2$ inner product as: $(u,v)_{\omega}: = \int \limits_{R_1}^{R_2} \int \limits_{0}^{\pi} \int \limits_{0}^{2 \pi} u v \omega d r d \theta d \phi, $ where $\omega$ denotes a non-negative weight. In most cases the weight is given by the weight of the spherical transform $\omega = r^2 \sin \theta$, however, in some of the estimates given below, the weight will be appropriately modified. The corresponding norm is given by $\| u\|_{\omega}^2 = (u,u)_{\omega}$.
Direction splitting of the advection-diffusion equation
-------------------------------------------------------
Since the PDEs are identical in both domains, it is sufficient to develop the numerical scheme for the Yin domain. Then the Schwarz domain decomposition method can be used to iterate between the subdomains.
We first present Douglas [@Douglas1962] type direction splitting scheme for the heat equation. Consider $$\label{eq:eq_temp}
\begin{aligned}
\partial_t T - \kappa \LAP T &= 0 \text{ in }~\Omega_1 \times (0, T_{f}],\\
T &=0 \text{ on }~ \partial \Omega_1 \times (0, T_{f}],
\end{aligned}$$ where the Laplacian in spherical coordinates is given by $$\Delta = \Drr + \Dthetatheta + \Dphiphi, \Drr: = \frac{1}{r^2}\dr \left(r^2 \dr \right),
\Dthetatheta: = \frac{1}{r^2 \sin \theta} \partial_\theta\left( \sin \theta \partial_\theta \right),
\text{ and } \Dphiphi: = \frac{\dphiphi}{r^2 \sin^2 \theta}.$$ The Douglas direction splitting scheme for this equation can be summarized in the following factorized form: $$\begin{aligned}
\left[ \mathrm{I} - \frac{\dt}{2} \Drr \right] \left[ \mathrm{I} - \frac{\dt}{2} \Dthetatheta \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Dphiphi \right] \frac{\delta T^{n+1}}{\dt} =
\Delta T^{n} ,
\label{eq:HeatDouglas0}\end{aligned}$$ where $ \delta T^{n+1} : = T^{n+1} - T^n$ denotes the first time difference of the time sequence $T^k$, $\dt$ is the time step, and $ \mathrm{I}$ is the identity operator. We first notice that this splitting can be considered as an Euler explicit scheme whose time difference operator is multiplied by $\left[ \mathrm{I} - \frac{\dt}{2} \Drr \right] \left[ \mathrm{I} - \frac{\dt}{2} \Dthetatheta \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Dphiphi \right] $, that is a consistent perturbation of $ \mathrm{I}$ and stabilizes the scheme. If the spatial derivative operators are positive and commute with respect to some inner product, the stability of this scheme is not hard to establish.
Unfortunately, $\Drr, \Dthetatheta,$ and $\Dphiphi$ do not commute with respect to the weighted product $(.,.)_\omega$, and their positivity is far from being clear. The main obstacle to commutativity of the one-dimensional operators comes from the non-constant terms in the denominators of $\Dthetatheta$ and $\Dphiphi$. Therefore, the scheme should be modified as follows. We first introduce the modified spatial operators: $$\DHthetatheta : = \frac{1}{R_1^2 \sin \theta} \partial_\theta\left( \sin \theta \partial_\theta \right), \DHphiphi := \frac{\dphiphi}{R_1^2 \sin^2 \theta_1},
\ \text{ and } \hat{\LAP} := \Drr + \DHthetatheta + \DHphiphi,$$ where $\displaystyle \theta_1 = \frac{\pi}{4}-\varepsilon $. Then it is easy to check that $\Drr, \DHthetatheta$ and $\DHphiphi$, supplied with zero Dirichlet boundary conditions, commute. Moreover, $$-\left( \DHthetatheta T,T \right)_{\omega} \ge 0, \quad
- \left( \DHphiphi T,T \right)_{\omega} \ge 0 \label{eq:HeatStab2}$$ and $$- \left( \left[\DHthetatheta - \Dthetatheta \right]T,T \right)_{\omega} \ge 0 \text{ and }
- \left( \left[\DHphiphi - \Dphiphi \right]T,T \right)_{\omega} \ge 0.
\label{eq:HeatStab1}$$ These inequalities immediately yield that: $$\left( -\hat{\LAP} T, T \right)_{\omega} \ge \left( - \LAP T, T \right)_{\omega} .$$ In order to obtain an unconditionally stable second order scheme, we start from the second order Adams-Bashforth scheme: $$\frac{\delta T^{n+1}}{\dt} =
\Delta T^{*,n+1/2},
\label{eq:AB2}$$ and stabilize it by multiplying the time difference in the left hand side by $\left[ \mathrm{I} - \frac{\dt}{2} \Drr \right] \left[ \mathrm{I} - \frac{\dt}{2} \DHthetatheta \right] \left[ \mathrm{I} - \frac{\dt}{2} \DHphiphi \right] .$ Since this perturbation is only first order consistent with the identity operator, we subtract from the right hand side the first order perturbation term, taken at the previous time level. The resulting splitting scheme reads: $$\begin{aligned}
\left[ \mathrm{I} - \frac{\dt}{2} \Drr \right] \left[ \mathrm{I} - \frac{\dt}{2} \DHthetatheta \right]
\left[ \mathrm{I} - \frac{\dt}{2} \DHphiphi \right] \frac{\delta T^{n+1}}{\dt} =
\Delta T^{*,n+1/2} - \frac{1}{2} \hat{\Delta} \delta T^n .
\label{eq:HeatDouglas}\end{aligned}$$ Note that, assuming enough regularity of the exact solution in space and time, this is a second-order perturbation of the second-order explicit Adams-Bashforth scheme , the perturbation being given by: $$\frac{\dt^2}{2} \hat{\Delta} \frac{\delta^2 T^{n+1}}{\dt^2} + \left[ \frac{\dt^2}{4} (\Drr \DHthetatheta + \Drr \DHphiphi + \DHthetatheta \DHphiphi)
- \frac{\dt^3}{8} \Drr \DHthetatheta \DHphiphi \right] \frac{\delta T^{n+1}}{\dt} .$$ We have the following stability result for the scheme .
Assuming enough regularity of the exact solution $T$ of the semi-discrete scheme , it is unconditionally stable; more precisely, it satisfies the following estimate: $$\begin{aligned}
\tau \sum \limits_{n=1}^{N-1} \frac{\| T^{n+1} - T^n \|^2_{\omega}}{\tau^2} &+
\frac{1}{2} \|\nabla T^{N} \|^2_{\omega} + \frac{1}{4} \left( \|\partial_\theta \left( T^N - T^{N-1} \right) \|^2_{\omega_1} +
\|\partial_\phi \left( T^N - T^{N-1} \right) \|^2_{\omega_2} \right) \label{eq:Stability} \\
& \le \frac{1}{2} \|\nabla T^{1} \|^2_{\omega} + \frac{1}{4} \left( \|\partial_\theta \left( T^{1} - T^{0} \right) \|^2_{\omega_1} + \|\partial_\phi \left( T^{1} - T^{0} \right) \|^2_{\omega_2} \right), \notag\end{aligned}$$ where $\omega_1 = \left( 1- \frac{r^2}{R^2_1} \right) \sin \theta \ge 0$ and $\omega_2 = \left( \frac{r^2}{R^2_1} - 1 \right) \frac{\sin \theta}{\sin^2 \theta_1} \ge 0$.
Expanding the left hand side of we get: $$\begin{aligned}
\left[ \mathrm{I} - \frac{\dt}{2} \hat{\Delta} \right. & + \left. \frac{\dt^2}{4} \left(\Drr \DHthetatheta +
\Drr \DHphiphi + \DHthetatheta \DHphiphi \right)
- \frac{\dt^3}{8}\Drr \DHthetatheta \DHphiphi \right] \frac{\delta T^{n+1}}{\dt}
= \Delta T^{*,n+1/2} - \frac{1}{2} \hat{\Delta} \delta T^n.
\label{eq:StabProof1}\end{aligned}$$ Rearranging all the $\Delta$ and $\hat{\Delta}$ terms, we obtain $$\begin{aligned}
\frac{\delta T^{n+1}}{\dt} & - \frac{1}{2} \left[\Delta - \hat{\Delta} \right] \left(T^{n+1} - 2T^n + T^{n-1} \right)
- \LAP T^{n+1/2} \notag \\
& + \left[ \frac{\dt^2 }{4} \left(\Drr \DHthetatheta +
\Drr \DHphiphi + \DHthetatheta \DHphiphi \right)
- \frac{\dt^3 }{8}\Drr \DHthetatheta \DHphiphi \right]\frac{\delta T^{n+1}}{\dt} = 0.
\label{eq:StabProof2}\end{aligned}$$ Next we multiply by $v = \delta T^{n+1}$ and integrate by parts. Then the second term gives $$\begin{aligned}
& - \frac{1}{2} \left( \left[\Delta - \hat{\Delta} \right] \left(T^{n+1} - 2T^n + T^{n-1} \right), T^{n+1}-T^{n} \right)_{\omega} \notag \\
& = \frac{1}{4} \left[ \| \partial_\theta \left( T^{n+1} - T^n \right)\|^2_{\omega_1} -
\| \partial_\theta \left( T^{n} - T^{n-1} \right) \|^2_{\omega_1} + \| \partial_\theta \left( T^{n+1} - 2T^n + T^{n-1} \right)\|^2_{\omega_1} \right]
\label{eq:StabProof3} \\
& + \frac{1}{4} \left[ \| \dphi \left( T^{n+1} - T^n \right)\|^2_{\omega_2} -
\| \dphi \left( T^{n} - T^{n-1} \right) \|^2_{\omega_2} + \| \dphi \left( T^{n+1} - 2T^n + T^{n-1} \right)\|^2_{\omega_2} \right]. \notag\end{aligned}$$ The third term is $$\begin{aligned}
- \left( \Delta T^{n+1/2}, T^{n+1}-T^{n} \right)_{\omega}
= \frac{1}{2} \left( \| \nabla T^{n+1} \|_{\omega}^2 - \|\nabla T^n \|_{\omega}^2 \right).
\label{eq:StabProof4}\end{aligned}$$ The remaining terms are all dissipative: $$\begin{aligned}
\left( \Drr \DHthetatheta \delta T^{n+1}, \delta T^{n+1} \right)_{\omega}^2
=\int \limits_{\Omega} \frac{r^2 \sin \theta}{R^2_1} | \partial_{r \theta} \delta T^{n+1} |^2,
\label{eq:StabProof5}\end{aligned}$$ $$\begin{aligned}
\left( \Drr \DHphiphi \delta T^{n+1}, \delta T^{n+1} \right)_{\omega}
= \int \limits_{\Omega} \frac{r^2 \sin \theta}{R^2_1 \sin^2 \theta_1} | \partial_{r \phi} \delta T^{n+1} |^2,
\label{eq:StabProof6}\end{aligned}$$ $$\begin{aligned}
\left( \DHthetatheta \DHphiphi \delta T^{n+1}, \delta T^{n+1} \right)_{\omega}
= \int \limits_{\Omega} \frac{r^2 \sin \theta}{R^4_1 \sin^2 \theta_1} | \partial_{\theta \phi} \delta T^{n+1} |^2,
\label{eq:StabProof7}\end{aligned}$$ and $$\begin{aligned}
- \left( \Drr \DHthetatheta \DHphiphi \delta T^{n+1}, \delta T^{n+1} \right)_{\omega}
= \int \limits_{\Omega} \frac{r^2 \sin \theta}{R^4_1 \sin^2 \theta_1} | \partial_{r\theta \phi} \delta T^{n+1} |^2.
\label{eq:StabProof8}\end{aligned}$$ Substituting - into , and summing for $n=1, \dots,N-1$ completes the proof.
The factorized scheme for the advection-diffusion equation is obtained in a similar fashion and takes the following form: $$\begin{aligned}
\left[ \mathrm{I} - \frac{\dt}{2} \left( \Drr - u_r^{n+1/2} \dr \right) \right]
\left[ \mathrm{I} \right. & - \left. \frac{\dt}{2} \left( \DHthetatheta - u_\theta^{n+1/2} \frac{\dtheta}{r} \right) \right]
\left[ \mathrm{I} - \frac{\dt}{2} \left( \DHphiphi - u_\phi^{n+1/2} \frac{\dphi}{r \sin \theta} \right) \right] \frac{\delta T^{n+1}}{\dt} \notag \\
& = \Delta T^{*,n+1/2} - \frac{1}{2} \hat{\Delta} \delta T^n + \bu^{n+1/2} \cdot \nabla T^n.
\label{eq:Convect_Douglas}\end{aligned}$$
Direction-splitting discretization of the Navier-Stokes system
--------------------------------------------------------------
Now we present the direction splitting scheme for the Navier-Stokes equations . Our numerical scheme is based on the AC regularization: $$\begin{aligned}
\partial_t \bu_1 +(\bu_1 \cdot \GRAD) \bu_1 + \GRAD p_1 - \frac{1}{\Reynolds} \LAP \bu_1 &
= \mathbf{0} \\
\chi \dt \partial_t p_1 + \nabla \cdot \bu_1 &=0,
\end{aligned}
\label{eq:ac1}$$ where $\chi = \mathcal{O}\left(1 \right)$ is an artificial compressibility regularization parameter, and $\Reynolds$ is the Reynolds number. It is well-known that the resulting approximation $\left( \bu_1, p_1 \right)$ is first-order accurate in time (see [@Shen_1995]). A second order scheme can be constructed using the bootstrapping approach of [@doi:10.1137/140975231; @Guermond201792], which requires additionally to solve the system: $$\begin{aligned}
\partial_t \bu_2 + (\bu_2 \cdot \GRAD) \bu_2 + \GRAD p_2 - \frac{1}{\Reynolds}\LAP \bu_2 &
= \mathbf{0} \\
\chi \dt \partial_t \left( p_2 - p_1 \right) + \nabla \cdot \bu_2&=0,
\end{aligned}
\label{eq:ac2}$$ $p_1$ being given by . In the following, for the sake of brevity, we will only discuss the direction splitting implementation of the first order approximation . The higher order correction for $\bu_2, p_2$ is solved identically. First, consider the standard semi-implicit Crank-Nicholson approximation of the system for $\left( \bu_1, p_1 \right)$: $$\begin{aligned}
\frac{\bu^{n+1}_1-\bu^{n}_1}{\dt} + \bu^{*,n+1/2}_2 \cdot \GRAD \bu^{n+1/2}_1 + \GRAD p_1^{n+1/2} - \frac{1}{\Reynolds} \LAP \bu_1^{n+1/2}&
= \mathbf{0} \\
\chi \left( p_1^{n+1}-p^n_1 \right) + \nabla \cdot \bu_1^{n+1/2} &=0\end{aligned}$$ Note that we use the second order velocity $\bu_2$ as advecting velocity, which allows us to assemble a single linear system for both systems. We can rewrite the momentum equation by eliminating $p^{n+1}_1$ from the first equation: $$\begin{aligned}
\frac{\bu^{n+1}_1-\bu^{n}_1}{\dt} + \bu^{*,n+1/2}_2 \cdot \GRAD \bu^{n+1/2}_1 + \GRAD p_1^n - \frac{1}{\Reynolds} \LAP \bu_1^{n+1/2}& - \frac{1}{2 \chi}\GRAD \DIV \bu_1^{n+1/2}
= \mathbf{0} \\
p_1^{n+1} = p^n_1 & - \frac{1}{\chi}\nabla \cdot \bu_1^{n+1/2}.\end{aligned}$$ In order to produce a factorized scheme for each velocity component, the $\GRAD \DIV$ operator must be also split somehow, and we use the Gauss-Seidel type splitting of the $\GRAD \DIV$ operator, which was originally proposed in [@Guermond201792] in the Cartesian case: $$\GRAD \DIV \bu^{n+1/2} \simeq
\begin{pmatrix}
\partial_r \biggl( \frac{\dr \left(r^2 u^{n+1/2}_r \right)}{r^2} + \frac{\dtheta \left( \sin \theta u_\theta^{*,n+1/2} \right)}{r \sin \theta}
+ \frac{\dphi u_\phi^{*,n+1/2}}{r \sin \theta} \biggr) \\
\\
\frac{\partial_\theta}{r} \biggl( \frac{\dr \left(r^2 u_r^{n+1/2} \right)}{r^2} + \frac{\dtheta \left( \sin \theta u^{n+1/2}_\theta \right)}{r \sin \theta}
+ \frac{\dphi u_\phi^{*,n+1/2}}{r \sin \theta} \biggr) \\
\\
\frac{\partial_\phi}{r \sin \theta} \biggl( \frac{\dr \left(r^2 u_r^{n+1/2} \right)}{r^2} + \frac{\dtheta \left( \sin \theta u_\theta^{n+1/2} \right)}{r \sin \theta}
+ \frac{\dphi u^{n+1/2}_\phi}{r \sin \theta} \biggr)
\end{pmatrix} :=
\begin{pmatrix}
D_{11} + D_{12} + D_{13} \\
D_{21} + D_{22} + D_{23} \\
D_{31} + D_{32} + D_{33}
\end{pmatrix} \bu^{n+1/2}$$
### Equation for the $r$-component of the velocity
Using the mass conservation equation $\nabla \cdot \bu = 0$, it is possible to write the first component of the system as follows: $$\partial_t u_r + \bu \cdot \nabla u_r - \frac{\LAP u_r}{\Reynolds} + \dr p
+ \frac{1}{\Reynolds}\frac{2 u_r}{ r^2} - \frac{1}{\Reynolds} \frac{2}{r^3}\partial_r \left(u_r r^2 \right) - \frac{u^2_\theta + u^2_\phi}{r} = 0,$$ where $\bu \cdot \GRAD v = u_r \dr v + u_\theta \frac{\dtheta v}{r} + u_\phi \frac{\dphi v}{r \sin \theta}$ is the advection operator. Let $\Lrr, \Lrtheta$ and $\Lrphi$ be the differential operators that act in each space direction: $$\Lrr u = \frac{1}{\Reynolds} \left( \Drr u - \frac{2 u}{r^2} + \frac{2 \partial_r \left(r^2 u\right)}{r^3} \right) + D_{11}u -
u^{*,n+1/2}_{2,r} \cdot \dr u_r, \Lrtheta u = \left(\frac{\DHthetatheta }{\Reynolds} - u^{*,n+1/2}_{2,\theta} \cdot \frac{\dtheta }{r} \right)u,$$ $$\Lrphi u = \left( \frac{\DHphiphi }{\Reynolds} - u^{*,n+1/2}_{2,\phi} \cdot \frac{\dphi }{r \sin \theta} \right)u \text{ and }
L_r = \Lrr + \Lrtheta + \Lrphi$$ The factorized scheme for the $r$-component takes the following form: $$\begin{aligned}
\left[ \mathrm{I} - \frac{\dt}{2} \Lrtheta \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Lrphi \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Lrr \right] \frac{u_{1,r}^{n+1}-u_{1,r}^{n}}{\dt}
& = L_r u_{1,r}^{*,n+1/2} + \hat{\LAP} u^{n-1/2}_{1,r} - \dr p_1^{n}
+ \frac{D_{12} u_{1,\theta}^{*,n+1/2} + D_{13}u_{1,\phi}^{*,n+1/2}}{2 \chi} \notag \\
& + \frac{ \left( u^{*,n+1/2}_\theta \right)^2 + \left( u^{*,n+1/2}_\phi \right)^2}{r}. \label{eq:r_comp}\end{aligned}$$
### Equation for the $\theta$–component of the velocity
Again using $\nabla \cdot \bu = 0$, the $\theta$-component of the momentum equation can be expressed as: $$\begin{aligned}
\partial_t u_\theta + \bu \cdot \nabla u_\theta - \frac{\LAP u_\theta}{\mathrm{Re}} + \frac{\dtheta p}{r}
+ \frac{1}{\Reynolds}\frac{u_\theta}{r^2 \sin^2 \theta} - \frac{2 \cos \theta}{\Reynolds}\frac{\dtheta \left(u_\theta \sin \theta \right)}{r^2 \sin^2 \theta}
- \frac{2}{\Reynolds} \frac{\dtheta u_r}{r^2} & - \frac{2 \cos \theta}{\Reynolds} \frac{\dr \left(u_r r^2 \right)}{r^3 \sin \theta} \notag \\
& + \frac{u_r u_\theta - u_\phi ^2 \cot \theta}{r}= 0. \end{aligned}$$ Let $\Lthetar, \Lthetatheta$ and $\Lthetaphi$ be defined as follows: $$\Lthetar u = \left(\frac{\Drr }{\mathrm{Re}} - u^{*,n+1/2}_{2,r} \cdot \dr \right) u,
\Lthetaphi u = \left( \frac{\DHphiphi }{\mathrm{Re}} - u^{*,n+1/2}_{2,\phi} \cdot \frac{\dphi }{r \sin \theta} \right) u,$$ $$\Lthetatheta u = \frac{1}{\Reynolds} \left( \DHthetatheta u - \frac{u}{r^2 \sin^2 \theta} +
\frac{2 \cos \theta}{\sin \theta}\partial_\theta \left(u \sin \theta \right) \right) + \frac{u \cdot u^{*,n+1/2}_{2,\phi} \cot \theta}{r}
+ u^{*,n+1/2}_{2,\theta} \cdot \frac{\dtheta u}{r} + \frac{D_{22} u}{2 \chi},$$ $$\text{ and } L_\theta = \Lthetar + \Lthetatheta + \Lthetaphi$$ The factorized scheme for the $\theta$-component takes the following form: $$\begin{aligned}
\left[ \mathrm{I} - \frac{\dt}{2} \Lthetaphi \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Lthetar \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Lthetatheta \right] \frac{u_{1,\theta}^{n+1}-u_{1,\theta}^{n}}{\dt}
& = L_\theta u_{1,\theta}^{*,n+1/2} + \hat{\LAP} u^{n-1/2}_{1,\theta} - \frac{\dtheta p_1^{n}}{r} + \frac{D_{21} u_{1,r}^{n+1/2} + D_{23} u_{1,\phi}^{*,n+1/2}}{2 \chi} \notag \\
& + \frac{1}{\Reynolds} \left( \frac{2 }{r^2} \dtheta u_{1,r}^{n+1/2} + \frac{2 \cos \theta}{r^3 \sin \theta} \dr \left(u_{1,r}^{n+1/2} r^2 \right) \right) \label{eq:theta_comp} \\
& - \frac{u_r^{*,n+1/2} \cdot u^{*,n+1/2}_\phi}{r}. \notag\end{aligned}$$
### Equation for the $\phi$–component of the velocity
The $\phi$-component of the momentum equation is given by: $$\begin{aligned}
\partial_t u_\phi + \bu \cdot \nabla u_\phi + \frac{u_r u_\phi + u_\theta u_\phi \cot \theta}{r} - \frac{\LAP u_\phi}{\Reynolds} +
\frac{\dphi p}{r \sin \theta}
+ \frac{1}{\mathrm{Re}} \left( \frac{u_\phi}{r^2 \sin^2 \theta} - \frac{2 \cos \theta}{r^2 \sin^2 \theta}\partial_\phi u_\theta
- \frac{2}{r^2 \sin \theta}\partial_\phi u_r \right) = 0
\end{aligned}$$ Let $\Lphir, \Lphitheta$ and $\Lphiphi$ be defined as follows: $$\Lphir u = \left( \frac{\Drr}{\mathrm{Re}} - u^{*,n+1/2}_{2,r} \cdot \dr \right) u \text{ and }
\Lphitheta u = \left(\frac{\DHthetatheta }{\Reynolds} - u^{*,n+1/2}_{2,\theta} \cdot \frac{\dtheta }{r} \right)u$$ $$\Lphiphi u = \frac{1}{\Reynolds} \left(\DHphiphi - \frac{1}{r^2 \sin^2 \theta} \right)u - \frac{u_\phi^{*,n+1/2} \cdot u}{r \sin \theta}
- \frac{u_{2,r}^{*,n+1/2} + u^{*,n+1/2}_{2,\theta} \cot \theta}{r} u \text{ and } L_\phi = \Lphir + \Lphitheta + \Lphiphi$$ The factorized scheme for the $\phi$-component is then: $$\begin{aligned}
& \left[ \mathrm{I} - \frac{\dt}{2} \Lphir \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Lphitheta \right]
\left[ \mathrm{I} - \frac{\dt}{2} \Lphiphi \right]
\frac{ u_{1,\phi}^{n+1}-u_{1,\phi}^{n}}{\dt} = L_\phi u_{1,\phi}^{*,n+1/2} + \hat{\LAP} u^{n-1/2}_{1,\phi}
\notag \\
& - \frac{\dphi p_1^{n}}{r \sin \theta} + \frac{1}{\mathrm{Re}} \left(\frac{2}{r^2 \sin \theta} \dphi u_{1,r}^{n+1/2} + \frac{2 \cos \theta}{r^2 \sin^2 \theta} \dphi u_{1,\theta}^{n+1/2} \right)
+ D_{31} u_{1,r}^{n+1/2} + D_{32} u_{1,\theta}^{n+1/2}. \label{eq:phi_comp}\end{aligned}$$
### Pressure update
$$p_1^{n+1} = p_1^{n} - \frac{1}{\chi}\DIV \bu_1^{n+1/2}. \label{eq:nst_Mass1}$$
Implementation and parallelization
==================================
The equations , - are solved as a sequence of 1D equations in each space direction. For example, solving consists of the following steps: $$\begin{aligned}
\frac{\xi^{n+1}}{\tau} & := \frac{1}{2} \Delta T^{*,n+1/2} - \frac{1}{2} \hat{\Delta} \delta T^n + \bu^{n+1/2} \cdot \nabla T^n \\
\frac{\eta^{n+1}}{\tau} & := \left[ \mathrm{I} - \frac{\tau}{2} \DHthetatheta \right]
\left[ \mathrm{I} - \frac{\tau}{2} \DHphiphi \right] \frac{T^{n+1}-T^n}{\tau} \Rightarrow
\left[ \mathrm{I} - \frac{\tau}{2} \Drr \right]\eta^{n+1} = \xi^{n+1} \\
\frac{\zeta^{n+1}}{\tau} & := \left[ \mathrm{I} - \frac{\tau}{2} \DHphiphi \right] \frac{T^{n+1}-T^n}{\tau} \Rightarrow
\left[ \mathrm{I} - \frac{\tau}{2} \DHthetatheta \right]\zeta^{n+1} = \eta^{n+1} \\
\left[ \mathrm{I} - \frac{\tau}{2} \DHphiphi \right] \left( T^{n+1} - T^n \right) & = \zeta^{n+1} \Rightarrow T^{n+1} = \left( T^{n+1}- T^n \right) + T^n.\end{aligned}$$ Similar strategy is applied for the Navier-Stokes approximation. Each 1D system is spatially approximated using second-order centered finite differences on a non-uniform grid. In order to ensure the inf-sup stability, the unknowns are approximated on a MAC grid, where the velocity components are stored at the face centers of the cells, while the scalar variables are stored at the cell centers. To solve the system on each domain in parallel we use the approach developed in [@doi:10.1002/fld.2583], where we first perform Cartesian domain decomposition of both computational grids using MPI, and then solve the resulting set of tridiagonal linear systems using domain-decomposition-induced Schur complement technique. Note, that the Schur complement can be computed explicitly (see [@doi:10.1002/fld.2583] for details) and so the system in each direction can be solved directly by the Thomas algorithm, avoiding the need of iterations on each of the two subdomain. Then, in order to obtain the approximation on the entire spherical shell, we iterate between the Yin and Yang grids using either additive or multiplicative overlapping Schwarz methods. The solution on each grid is computed using only boundary data that is interpolated from the currently available solution on the other grid, using Lagrange interpolation.
In the additive Schwarz implementation, we use an even total number of CPUs. Then we split the global communicator into two equal parts, and assign to each grid one of the communicators. In the multiplicative Schwarz implementation, we use the global communicator to solve the problem on each grid sequentially.
The overall solution procedure in case of the multiplicative Schwarz iteration can be summarized as follows:
Repeat until convergence:
- For $i = 1,2$
- Obtain interpolated boundary values $T_{bd}$ for $\partial \Omega_i$ from $\Omega_{3-i}$.
- Solve the temperature equation in $\Omega_i$ with using extrapolated velocity values $\bu_2^{*,n+1/2}$.
- Obtain interpolated boundary values $\bu_{bd}$ for $\partial \Omega_i$ from $\Omega_{3-i}$.
- If $\left| \bigintsss\limits_{\partial \Omega_i} \bu_{bd} \cdot \mathbf{n} \right| \ge \mathrm{tol}$, then minimize the functional ($\varepsilon \ll 1$): $$\mathrm{J}\left( \bv \right): = \frac{1}{2} \left|\bv - \bu_{bd} \right|^2_{\ell^2}
+ \frac{1}{2 \varepsilon |\partial \Omega_i|^2}
\left| \bigintssss\limits_{\partial \Omega_i \cap \{\theta, \phi \text{ bdry }\}} \bv \cdot \mathbf{n} +
\bigintssss\limits_{\partial \Omega_i \cap \{r \text{ bdry }\}} \bu_{bd} \cdot \mathbf{n} \right|^2,$$ using the Conjugate Gradient Algorithm until $\mathrm{J}\left( \cdot \right) \le \mathrm{tol}$.
- Update $\bu_{bd}: = \bv$ and solve the momentum equation in $\Omega_i$ with the interpolated Dirichlet boundary conditions in $\theta, \phi$ directions and with the original boundary conditions in the $r$ direction.
- Compute the pressure in $\Omega_i$ using the second equation in .
- Interpolate the pressure values at the boundary of $\partial \Omega_i$ using the available pressure on $\Omega_{3-i}$.
- End for.
Step 4 is meant to ensure that there is no spurious mass flux generated through the internal (artificial) boundaries due to the interpolation. It is optional, and as our numerical experience shows, it rarely changes significantly the results. Therefore, it is skipped while producing the numerical results presented in the next section. Skipping Step 7, however, can seriously reduce the rate of convergence of the Schwarz iteration, as observed in the numerical simulations. Clearly, the AC method for the Navier-Stokes equations does not require boundary conditions on the pressure. Nevertheless, the exchange of the pressure values does influence the pressure gradient that appears in -, and thus it seems to influence significantly the convergence of the overall iteration. This effect is not well understood and while some other authors (see for example [@TANG2003567]) also interpolate the pressure values near the internal boundaries, others (e.g. [@MERRILL201660] ) interpolate only the velocity on the internal boundaries.
Another interesting feature of the domain decomposition iteration described above is that it allows to use the previously computed iterates in order to reduce the splitting error of the direction splitting approximation. For example, if the factorized form of the direction-split approximation for a given quantity $\psi$ is given by: $$(I-L_{\psi,r})(I-L_{\psi,\theta})(I-L_{\psi,\phi})(\psi^{n,k} - \psi^{n-1}) = G$$ where the superscript $n$ denotes the time level of the solution and $k$ denotes the domain decomposition iteration level, then the splitting error can be reduced by using the modified equation: $$(I-L_{\psi,r})(I-L_{\psi,\theta})(I-L_{\psi,\phi})(\psi^{n,k} - \psi^{n,k-1}) = G+(\psi^{n,k-1}-\psi^{n-1})-L_{\psi} (\psi^{n,k-1}-\psi^{n-1}),
\label{eq:er}$$ where $L_{\psi}= L_{\psi,r}+L_{\psi,\theta}+L_{\psi,\phi}$. Indeed, in , the splitting error term $(L_{\psi,r}L_{\psi,\theta}+L_{\psi,r}L_{\psi,\phi}+L_{\psi,\theta}L_{\psi,\phi}-L_{\psi,r}L_{\psi,\theta}L_{\psi,\phi})(\psi^{n,k} - \psi^{n-1}) $ at iteration level $k$ has been reduced by the same term at the previous iteration level $(L_{\psi,r}L_{\psi,\theta}+L_{\psi,r}L_{\psi,\phi}+L_{\psi,\theta}L_{\psi,\phi}-L_{\psi,r}L_{\psi,\theta}L_{\psi,\phi})(\psi^{n,k-1} - \psi^{n-1}) $. If this error reduction is employed, then the iteration becomes a block-preconditioned overlapping domain decomposition iteration, the preconditioner being the factorized operator $(I-L_{\psi,r})(I-L_{\psi,\theta})(I-L_{\psi,\phi})$. We must also remark here that this iteration needs to converge to an accuracy of the order of $\dt^2$ for the solution of equation and $\dt^3$ for the solution of equation , in order to preserve the second order accuracy of the overall algorithm.
Numerical tests
===============
Time and space convergence
--------------------------
We verify the convergence rates in space and time using the following manufactured solution, given in a Cartesian form: $$\bu = \cos(t) \left(2 x^2 y z, - x y^2 z, -x y z^2 \right)^\mathrm{T} , p = \cos(t) x y z, T = 2 \cos(t) x^2 y z.
\label{eq:Exact}$$ The parameters used in this test are $R_1=1, R_2=2, Ra=1, Pr=1$, and the grids used in the tests are uniform in each direction. The convergence of the approximation is tested using both, the additive and multiplicative versions of the scheme. The grid used for the time convergence tests consists of $20\times 92 \times 192 $ MAC cells on each of the two subdomains. The solution error is computed at the final time $T_f=10$. For the space convergence tests, the time step is chosen small enough to not influence the overall error, $\dt=0.0001$, and the final time is $T_f=1$. The grid diameter is computed as the maximum diameter of the MAC cells in Cartesian coordinates. In both cases the domain decomposition iterations are converged so that the $l^2$ norm of the difference between two subsequent iterates, for any of the computed quantities is less than $10^{-6}$ ($l^2$ norm denotes the standard mid-point approximation to the $L^2$ norm). Also, the splitting error reduction, as outlined by equation , is employed at each iteration.
The graphs of the $l^2$ norm of the errors in both cases are presented in Figure \[fig:Conv\_Additive\_Schwarz\]. They clearly demonstrate the second order accuracy of the scheme in space and time. Next we verify the accuracy of the proposed algorithm on a physically more relevant analytic solution of the Navier-Stokes equations in a spherical setting, due to Landau (see [@L44] and [@LYY18] for a recent review). The source term of the equations is equal to zero in this case, and the solution is steady and axisymmetric. In all cases presented in figure \[fig:Conv\_Landau\] the multiplicative Schwarz version of the algorithm is used with its convergence tolerance being set to $10^{-6}$, the time step is equal to $10^{-3}$, and $R_1=1, R_2=2$. We first present in the top left graph of figure \[fig:Conv\_Landau\] the $l^2$ error for the velocity and pressure as a function of the grid diameter, at $Re=1$ and the overlap is $\epsilon=0.1$ The scheme clearly exhibits again a second order convergence rate in space. In the top right graph we demonstrate the influence of the overlap size on the error at $Re=1$, the grid size in the $r,\phi, \theta$ directions being $2.7778 \times 10^{-2}, 1.7027 \times 10^{-2}, 3.6121 \times 10^{-2}$ correspondingly. The effect of the overlap on the error is insignificant, however, it seriously impacts the stability of the algorithm i.e. the increase of the overlap improves the stability, particularly at large Reynolds numbers. Finally, the bottom graph demonstrates the effect of the Reynolds number on the error. Again, the overlap is $0.1$ and the grid sizes are equal to $2.7778 \times 10^{-2}, 1.7027 \times 10^{-2}, 3.6121 \times 10^{-2}$ . We should note that the exact solution for the velocity scales like $Re^{-1}$ and therefore the errors in the graph are multiplied by the corresponding Reynolds number. Clearly, the oscillations in the error decrease slower with the increase of the Reynolds number. These oscillations are due to the artificial compressibility algorithm, since the initial data for the pressure corresponds to a divergence-free velocity, while the pressure evolution is determined by a perturbed continuity equation (see [@OA10] and [@DLM17] for a detailed discussion on this issue).
\
Weak parallel efficiency
------------------------
Next we test the parallel efficiency of the code based on the scheme introduced in the previous section. Since we are interested in solving large size problems, we only measure the weak scalability of our code. The problem size is $100\times 100 \times 100$ grid cells per each of the Yin and Yang grids on each CPU, and the maximum number of CPUs used is $960$. Besides, since in the possible applications of this technique (atmospheric boundary layer, Earth’s dynamo) the thickness of the spherical shell is much smaller than the diameter of the shell, we use a two-dimensional grid of processors for the grid partitioning. It must be noted though, that making the grid partitioning three dimensional does not change much the parallel efficiency results presented in this section. The scaling efficiency is computed as the ratio of the CPU time on 32 cores divided by the CPU time on $n\geq32$ cores. The reason for this definition of efficiency is that the particular cluster used in the scaling tests has processors containing 32 cores each, and the efficiency drops very significantly between 1 and 32 cores (to about 75%). After this, when the number of cores is a multiple of 32 the efficiency remains very close to the one at 32 cores. One possible explanation of this phenomenon is that in case when the number of cores is significantly less than 32 cores, they need to share the memory bandwidth and cache with a smaller number of cores, since presumably the rest of the available cores on the given processor are idle (see e.g. [@Keating], p. 152). Again, we are interested in very large computations, and therefore, using a minimum of 32 cores is very reasonable.
The scaling results are performed using the Compute Canada (see https://www.computecanada.ca/) Graham cluster of $2.1$GHz Intel $E5-2683$ v4 CPU cores, $32$ cores per node, and each node connected via a $100$ Gb/s network. The results were calculated using the wall clock time taken to simulate $10$ time steps. We ran two tests, using a fixed number of $1$ and $10$ domain decomposition iterations, and we present the scaling results in Fig. \[fig:Scaling\]. The parallel efficiency is very slightly dependent on the number of domain decomposition iterations, and remains above 90% for the number of cores ranging between 32 and 960 (the maximum allocatable without a special permission on the particular cluster). In our opinion, this is an excellent scaling result for an implicit scheme for the incompressible Boussinesq equations.
![Parallel scalability using up to $960$ CPU cores[]{data-label="fig:Scaling"}](./Pics/Scaling.pdf){width="100mm"}
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to acknowledge the support, under a Discovery Grant, of the National Science and Engineering Research Council of Canada (NSERC).
This research was enabled in part by support provided by Compute Canada (www.computecanada.ca).
References {#references .unnumbered}
==========
|
---
abstract: 'A new approach to the quantization of Chern-Simons theory has been developed in recent papers [@FR; @BR1; @BR2; @AGS]. It uses a “simulation” of the moduli space of flat connections modulo the gauge group which reveals to be related to a lattice gauge theory based on a quantum group. After a generalization of the formalism of q-deformed gauge theory to the case of root of unity, we compute explicitely the correlation functions associated to Wilson loops (and more generally to graphs) on a surface with punctures, which are the interesting quantity in the study of moduli space. We then give a new description of Chern-Simons three manifolds invariants based on a description in terms of the mapping class group of a surface. At last we introduce a three dimensional lattice gauge theory based on a quantum group which is a lattice regularization of Chern-Simons theory.'
author:
- 'E.Buffenoir[^1], Centre de Physique Theorique Ecole Polytechnique91128 Palaiseau CedexFrance'
title: 'Chern Simons theory on a lattice and a new description of 3-manifolds invariants.'
---
\#1[\#1V]{} Ł\#1[\#1L]{} ¶\#1[\#1P]{} §\#1[\#1S]{} \#1[\#1]{}
0[0u]{} u\#1[\#1u]{}
\#1[\#1 g]{}
\#1[\#1 ]{}
\#1[\#1 W]{}
\#1[\#1z]{}
\#1[[R]{}\^[(\#1)]{}]{} \#1\#2[[\#1 ]{}\^[(\#2)]{}]{} \#1[[U]{}\^[(\#1)]{}]{} \#1\#2[[\#1 ]{}\^[(\#2)]{}]{} \#1[\#1]{}
\#1\#2[\#1 \#2 R]{} \#1\#2[\#1 \#2 ]{} \#1\#2[\#1 \#2 ]{} \#1\#2[\#1 \#2 ]{} \#1\#2[\#1 \#2 ]{} \#1\#2[\#1 \#2 \^[-1]{}]{}
\#1[[ U]{}]{}
å[a]{} \#1[\_[\#1]{}]{} \#1[\_[\#1]{}]{} \#1[[U]{}(\#1)]{} \#1[[U]{}(\#1)]{}
\#1\#2\#3\#4\#5\#6[{}\_q]{}
\#1\#2\#3\#4\#5\#6\#7[(.)\_[\#7]{}\^]{}
\#1\#2\#3\#4\#5\#6\#7[(.)\_[\#7]{}\^]{}
\#1\#2\#3\#4\#5\#6\#7\#8\#9[ [I]{}(\#1 [\#2 \#3]{} \#4 [\#5 \#6]{} \#7 [\#8 \#9]{})]{} \#1\#2\#3\#4\#5\#6\#7\#8\#9[ [O]{}(\#1 [\#2 \#3]{} \#4 [\#5 \#6]{} \#7 [\#8 \#9]{})]{}
\#1\#2\#3\#4\#5\#6[ [I]{}(\#1 [\#2 \#3]{} \#4 [\#5 \#6]{})]{} \#1\#2\#3\#4\#5\#6[ [O]{}(\#1 [\#2 \#3]{} \#4 [\#5 \#6]{})]{}
\#1\#2\#3\#4\#5\#6\#7\#8[ [I]{}(\#1 \#4 [\#5 \#6]{} )]{} \#1\#2\#3\#4\#5\#6\#7\#8[ [O]{}(\#1 \#4 [\#5 \#6]{} )]{}
Introduction
============
This paper is the third part of a study of combinatorial quantization of Chern-Simons theory[@BR1; @BR2]. Several ideas developed here are in fact products of those introduced by V.V.Fock and A.A.Rosly in their study of Poisson structures on the moduli space of flat connections[@FR]. To understand the motivation of these papers we must recall some general facts about 3D Chern-Simons theory. Chern-Simons theory is a gauge theory in 3 dimensions defined by the action principle \_[CS]{}= tr( \_[[M]{}]{} [A]{}d[A]{}+[2 3]{}[A]{}\^3 ) where ${\cal M}$ is a 3-manifold, $k$ a positive integer and ${\cal A}$ a connection associated to a semisimple Lie algebra ${\cal G}.$ If we first suppose that the manifold locally looks like a cylinder $\Sigma \times R$, $R$ considered as the the time direction, we can consider the Chern-Simons theory in an Hamiltonian point of view. We will denote by $A$ the two space-components of the gauge field taken to be the dynamical variables of the theory, and the time component $A_0$ will become a Lagrange multiplier. With these notations the action can be written: \_[CS]{}= tr( \_[[M]{}]{} (- A \_0 A + 2 A\_0 F )dt). The first term gives the Poisson structure: { A\^a\_i(x,y), A\^b\_j(x’,y’) } = \^[ab]{}\_[ij]{}\^[(2)]{}((x,y),(x’,y’)). \[Poisson\] The hamiltonian is a combination of constraints and the second term imposes as a constraint that the curvature of the connection $A$ is zero: F=dA+A\^2=0.\[constraint\] Computing the Poisson brackets of the constraints we obtain that they are first class: { F\^a(x,y), F\^b(x’,y’) } = f\^[ab]{}\_[c]{}F\^c(x,y)\^[(2)]{}((x,y)(x’,y’)). where the $f^{ab}_{c}$ are the structure constants of the Lie algebra ${\cal
G}.$ The constraints (\[constraint\]) generate the infinitesimal gauge transformations of the gauge field then the phase space of the hamiltonian Chern-Simons theory is the space of flat connections modulo gauge transformations.\
Quantizing the latter Poisson structure of gauge fields in the usual way, = \^[ab]{}\_[ij]{}\^[(2)]{}((x,y)(x’,y’)) we are led to work with an infinite dimensional algebra, observables becoming functionals over the elements of this algebra. The idea of V.V.Fock and A.A.Rosly is completely different.Let us briefly describe it.\
The space of flat connections modulo the gauge group having only a finite number of freedom degrees, we reduce the connection to live on a graph encoding the topology of the surface and equip this “graph connection” with a Poisson structure such that the Poisson structure induced on the gauge invariant observables is compatible with that induced from the usual symplectic structure. We obtain a “simulation” of hamiltonian Chern-Simons theory in the sense that the operator algebra derived from both descriptions are the same.\
We will consider a graph dividing the surface into contractile plaquettes. This graph will be equiped with an additionnal structure of “ciliated fat graph”, i.e. a graph with a linear order between adjacent links at each vertex. Let $x,y$ be neighbour vertices of the graph, we will denote by $U_{[x,y]}$ the parallel transport operator associated to the link and the connection. The gauge group acts in the usual way on this object: U\_[\[x,y\]]{} g\_[x]{}U\_[\[x,y\]]{}g\^[-1]{}\_[y]{} We can define a Lie Poisson structure on such objects [@FR] owning the following fascinating property:\
the space of flat graph connections modulo graph gauge group is Lie Poisson isomorphic to the space of flat connections modulo gauge group on the surface.\
The problem has been reduced to a finite dimensional problem by limiting the gauge group to act at a finite number of sites.\
The quantization of such a Lie Poisson structure leads us to an exchange algebra on which acts a quantum group. The object of [@BR1; @AGS] was to define this algebraic structure.\
In a second time we found a projector in this algebra imposing “a posteriori” the flatness condition, the result was then a two dimensional lattice gauge theory based on a quantum group. The correlation functions associated to gauge invariant objects in this theory being related to expectation values in Chern-Simons theory.\
In our second paper [@BR2] we further investigated the algebra of gauge invariant elements, particularily the algebra associated to loops. Our aim was to describe a new approach to knots invariants, showing in a well defined framework the relation between Reshetikhin-Turaev invariants and Chern-Simons theory.\
The aim of the following paper is double:\
- to investigate further the computation of the correlation functions of our theory on any surface and construct the derived invariants associated to the mapping class group, and establish a new description of three manifold invariants using a generalization of our previous construction to the case of roots of unity\
- to build up a three dimensional lattice q-gauge theory associated to triangulations of any 3-manifolds which will describe a well defined finite path integral formula for Chern-Simons theory and the way to compute any correlation functions of this theory.\
This work is a tentativ to revisit the work of E.Witten [@W1] with a well defined formalism allowing a lot of new computations and specially computations of invariants associated to intersecting loops in Chern-Simons theory.
Lattice gauge theory based on a quantum group
=============================================
Quantum groups and exchange algebras associated to fat graphs
-------------------------------------------------------------
In this chapter, after a brief summary of the results of [@AC] [@MS], we will further develop the notion of quantum group at root of unity in the dual version and then, using this construction, generalize the results on gauge fields algebra developed in [@BR1; @BR2] in a way quite different from that described in [@AGS]. We will consider a Hopf algebra $({\cal A},m,1,\Delta,S,\epsilon).$ To simplify we will take ${\cal A}={\cal U}_q ( sl_2 )$ with $q$ being a complex number different from $\pm 1.$ As usual we will refer to ${\bf R}$ as the universal $R-$matrix associated to ${\cal A}$ ( we will often write ${\bf R}=\sum_i a_i \otimes b_i$ ), $u$ the element defined by $u= \sum_i S(a_i)b_i$ verifying the usual properties and $v$ the ribbon element defined by $v^2=uS(u)$ (for details see [@Dr; @RT1; @RT2]).\
Depending on whether $q$ is a root of unity or not, the representation theory of ${\cal A}$ is completely different. We will denote by $Irr({\cal A})$ the set of equivalence classes of finite dimensional irreducible representations of ${\cal A}.$ In each class $\dot{\alpha}$ we will pick out a representativ $\alpha$ and, like often in physics, denote equivalently by $\alpha$ or $V_{\alpha}$the representation space associated to $\alpha.$ We will denote by $\bar{\alpha}$ (resp. $\tilde{\alpha}$) the right (resp. the left ) contragredient representation build up from the antipode (resp. the inverse of the antipode) by $\bar{\alpha}={}^{t}\alpha
\circ S$ and $0$ the one dimensional representation associated to $\epsilon.$ The tensor product of two representations is defined by the coproduct $\Delta.$ If q is a root of unity the decomposition of the tensor product of two irreducible representations can involve indecomposable representations (i.e representations which are not irreducible but cannot ye! t be decomposed in a direct sum of stable ${\cal A}-$modules). We are able to introduce $\Psi_{\alpha \beta}^{\gamma,m}$ and $\Phi_{\gamma,m}^{\alpha \beta}$ respectively projection of the tensor product of $\alpha$ and $\beta$ on the m-th isotypic component $\gamma$ and the inclusion of $\gamma$ in the tensor product $\alpha \otimes \beta.$ Using this notation we will make one more restriction between “physical” representations verifying $\Psi_{\alpha
\bar{\alpha}}^{0}\Phi_{0}^{\alpha \bar{\alpha}} \not= 0$ and the other representations, we will denote by $Phys({\cal A})$ this subset of $Irr({\cal A})$. We will introduce a new tensor product between elements of $Phys({\cal A})$ simply realizing a truncation of the previous one, defined by: = \_[Phys([A]{})]{}N\^\_ . $N$ is the fusion matrix of ${\cal A}$ and we will also use the notation $\delta(\alpha \beta
\gamma)$ to be equal to 1 or 0 depending on whether $\gamma$ occurs or not in the decomposition of the tensor product $\alpha \otimes \beta.$ If q is generic the tensor product of two elements of $Irr(A)$ can be decomposed in a direct sum of elements of $Irr(A).$ Moreover all irreducible representations are “physical”, so we do not have to change the tensor product in this case. We will associate new projection and inclusion operators $\psi_{\alpha \beta}^{\gamma,m}$ and $\phi_{\gamma,m}^{\alpha \beta}$ build up from the truncated tensor product. We will also use the $6-j$ notation $\sixj{\alpha}{\beta}{\gamma}{\delta}{\mu}{\nu}$ defined in the usual way from projection and inclusion operators (see [@KR] for definitions and properties).\
We will use the following notation replacing the coproduct by a truncated coproduct for any element $\xi$ of the algebra ${\cal A}$: = \_[Phys([A]{})]{} \^\_[,m]{} \^[,m]{}\_. The antipode and counity maps do not change through truncation and we will again have: = \^[t]{} [| ]{} The first trivial properties of the projection and inclusion operators are: &&\_\^[[’]{},m’]{}\_[,m]{}\^=\_[m,m’]{}\_[,[’]{}]{}()id\_\
&&\_[Phys([A]{}),m]{} \^\_[,m]{}\^[,m]{}\_=[ ]{}\
&&\^[0]{}\_=\^[0 ]{}\_=\_[0]{}\^=\_[0 ]{}\^=\_ id\_. The essential fact is that when the truncation is not trivial (i.e in the root of unity case) the representations $((\alpha \otimes \beta)\otimes \gamma)$ and $(\alpha \otimes (\beta \otimes \gamma))$ are no more equal but are equivalent, the intertwiner map between them being ${\buildrel {\alpha \beta \gamma} \over
\Theta}$ defined by: =\_[,,Phys([A]{})]{} \_\^\_\^\^\_\^\_ ( where we have omited the multiplicities to simplify the notation, it will often be the case in the following). We will denote by ${\buildrel {\alpha \beta \gamma} \over
{\Theta^{-1}}}$ its quasi-inverse.
We will often use the notation ${\buildrel {\alpha \beta \gamma} \over
\Theta}_{123}=\sum_{i} {\buildrel \alpha \over {\theta_i^{(1)}}} \otimes
{\buildrel \beta \over
{\theta_i^{(2)}}} \otimes {\buildrel \gamma \over {\theta_i^{(3)}}},$ and the coproduct notations ${\buildrel {(\alpha \otimes \beta) \gamma \delta} \over
\Theta}_{1234}=\sum_{i} {\buildrel \alpha
\over {\theta_i^{(11)}}} \otimes {\buildrel \beta \over {\theta_i^{(12)}}}
\otimes {\buildrel \gamma
\over {\theta_i^{(2)}}} \otimes {\buildrel \delta \over {\theta_i^{(3)}}}...$\
In the case where $q$ is generic ${\buildrel {\alpha \beta \gamma} \over
\Theta}$ is simply the identity but more generally it is possible to collect some interesting properties in the root of unity case. Using the pentagonal identity and other trivial identities on $6-j$ symbols we can verify: &=&()[ ]{}([ ]{})\[pentagon\]\
[ ]{}=[ ]{}&=& [ ]{}= [ ]{}\[theta0\] and other similar identities for $\Theta^{-1}.$ Moreover we have the quasi-inverse properties: =[ ]{} =[ ]{} recalling that, here, ${\buildrel {\alpha \otimes \beta} \over {\bf 1}} $ is simply a projector. Let us now define intertwiners between $\alpha\otimes\beta$ and $\beta\otimes\alpha$ using our basic objects $\psi,\phi$: &&P\_[12]{} = \_[Phys([A]{}),m]{} \_\^[1 2]{} \^\_[,m]{}\_[ ]{}\^[,m]{}\[Rdef\]\
&& P\_[12]{} = \_[Phys([A]{}),m]{} \_\^[-[1 2]{}]{} \^\_[,m]{}\_[ ]{}\^[,m]{}\
where $R'=\sigma(R)$ and $\lambda_{\alpha \beta
\gamma}=(\frac{v_{\alpha}v_{\beta}}{v_{\gamma}})$ where $v_{\alpha}$ is the Drinfeld casimir, equal to $q^{C^{(2)}_{\alpha}}$, where $C^{(2)}_{\alpha}$ is the quadratic Casimir. We will denote in the following $\Rab=\sum_i {\buildrel
\alpha \over a_i} \otimes {\buildrel \beta \over b_i}$ and $\Rab^{-1}=\sum_i
{\buildrel \alpha \over
c_i} \otimes {\buildrel \beta \over d_i}$ and use sometimes the notation $R^{(+)}=R$ and $R^{(-)}=R'^{-1}$ . Using the hexagonal identities on the $6-j$ symbols it can be shown that: &&=[ ]{}\
&&=[ ]{}\[quasitriang\] which is simply the analog of the quasitriangularity property of $R-$matrices.\
This matrix is no more inversible but we have: = ([ ]{}) = [ ]{}.\[Rinv\] Let us now study the properties of the antipodal map and develop the analog of the ribbon properties [@AC]. We will denote by ${\buildrel \alpha \over A}$ and ${\buildrel \alpha \over B}$ the matrices defined by $\psi^{0}_{ \bar{\alpha}
\alpha}=
< . , {\buildrel \alpha \over A} .>$ and $\phi_{0}^{\alpha \bar{\alpha}}= (\lambda \rightarrow \lambda \sum_i
{\buildrel \alpha \over B}
{\buildrel \alpha \over{e_i}} \otimes {\buildrel {\bar{\alpha}} \over {e^i}}$) where ${\buildrel
\alpha \over {e_i}}$ (resp. ${\buildrel {\bar{\alpha}} \over {e^i}}$) is a basis of the representation space of $\alpha$ (resp. $\bar{\alpha}$) and $<.,.>$ is the duality bracket. To choose the normalisation of $\phi$ and $\psi$s we will impose the ambiant isotopy conditions: \_i \^[(1)]{}\_i B S(\^[(2)]{}\_i) A \^[(3)]{}\_i = 1 \_i S([\^[-1]{}]{}\^[(1)]{}\_i) A [\^[-1]{}]{}\^[(2)]{}\_i B S([\^[-1]{}]{}\^[(3)]{}\_i) = 1 In order to generalize the known properties relative to the antipode, we will also introduce some notations which will be useful in the following: &&[ G ]{} = \_[i,j]{} ([ ]{}) ([ )]{})([A]{} )([ ]{})([ ]{}\
&&[ D ]{} = \_[i,j]{} ([ ]{}) ([ ]{})( [B]{} ) ( ) ([ ]{})\
&&[ f ]{}= \_i ([ )]{} ) [ G ]{} it can be shown that the latter matrices verify: &&[ ]{}= [ A ]{} = [ B ]{}\[deltaA\]\
&&\^\_=[ ]{}\^[t]{}\^[|]{}\_[||]{} \_\^= \^[t]{}\_[|]{}\^[||]{} We endly introduce the element $u$ associated to the square of the antipode, defined by: u=\_[i,j]{} S(\^[-1 (2)]{}\_i B S(\^[-1 (3)]{}\_i))S(b\_j)A a\_j\^[-1 (1)]{}\_i, u is invertible and 1&=&u\_[i,j]{} S\^[-1]{}(\^[-1 (1)]{}\_i) S\^[-1]{}(A d\_j)c\_j\^[-1 (2)]{}\_i B \^[-1 (3)]{}\_i=\
&=&S\^[2]{}(\_[i,j]{} S\^[-1]{}(\^[-1 (1)]{}\_i) S\^[-1]{}(A d\_j)c\_j\^[-1 (2)]{}\_i B \^[-1 (3)]{}\_i)u moreover we have as usual the essential property ,S\^2()=u u\^[-1]{} and the usual corollaries &&S\^[2]{}(u)=u\
&&uS(u)=S(u)u\
&&\_i S(b\_i) A a\_i=S(A)u=S(u)u\_i S(c\_i)Ad\_i\[contract\]\
&&(u)=1 We will denote by $v$ the element satisfying: &&v\^2=uS(u)\
&& S(v)=v (v)=1. and by $\mu$ the element $uv^{-1}.$ Then it can be shown that: = [ ]{} [ ]{}([ ]{}) using the latter notations it can be shown that $\phi_{0}^{\alpha
\bar{\alpha}}=
<S(A)\mu\;.\;,\;.>$ and $\psi^{0}_{\bar{\alpha} \alpha}=(\lambda \rightarrow
\lambda \sum_i
{\buildrel {\bar{\alpha}} \over {e^i}} \otimes \mu^{-1}S(B){\buildrel \alpha
\over {e_i}}).$ In the following the q-dimension of the representation $\alpha$ will be defined by $[d_{\alpha}]=tr_{\alpha}(S(A) \mu B).$
Now, using the latter framework we can give a well defined construction of the quantum group in the dual version for any value of $q$. As a vector space this algebra, called $\Gamma$, is generated by $\{ {\buildrel \alpha \over {g^i_j}}
\mbox{ with $\alpha \in Phys({\cal A})$ and $i,j=1 \cdots dim(\alpha)$} \}$ and the product is simply defined by: \_1 \_2= \_[Phys([A]{})]{}\_\^\^\_. This product is not associative but verify: \_[123]{}((\_1 \_2)\_3)= (\_1(\_2 \_3)) [ ]{}\_[123]{}. \[quasi-associativity\] Moreover we have the exchange relation: \_[12]{} \_1 \_2= \_2 \_1 \_[12]{}. \[exchange\] This algebra can be equiped with a coproduct and a counity: ()=\_i ()=\^i\_j. Moreover it can be shown that the antipodal map $S$ defined to be the linear map verifying $S({\buildrel \alpha \over {g^i_j}})={\buildrel {\bar{\alpha}}
\over {g^j_i}}$ owns the properties: && S(g)\^i\_k A\^k\_l g\^l\_j=A\^i\_j \[antipode1\]\
&&(S(A))\^k\_l g\^l\_j S(g)\^i\_k=(S(A))\^i\_j \[antipode2\]\
&&S([ ]{})S([ ]{})=[ ]{}\_[12]{}S([ ]{} [ ]{})\_[12]{}\
&&S\^[2]{}([ ]{}) = [ ]{}[ ]{}[ ]{}
Our aim is now to define as in our first paper the gauge theory associated to this gauge symmetry algebra.
Let $\Sigma$ be a compact connected oriented surface with boundary $\partial\Sigma$ and let ${\cal T}$ be a triangulation of $\Sigma.$ Let us denote by ${\cal F}$ the oriented faces of ${\cal T},$ by $\Li$ the set of edges counted with their orientation. If $l$ is an interior link, $-l$ will denote the opposite link. We have $\Li={\Li}^{int}\cup{\Li}^{\partial \Sigma},$ where ${\Li}^{int}, {\Li}^{\partial \Sigma}$ are respectively the set of interior edges and boundary edges.
Finally let us also define $\Ve$ to be the set of points (vertices) of this triangulation, $\Ve={\Ve}^{int}\cup {\Ve}^{\partial \Sigma}$, where ${\Ve}^{int},
{\Ve}^{\partial \Sigma}$ are respectively the set of interior vertices and boundary vertices.
If $l$ is an oriented link it will be convenient to write $l=xy$ where $y$ is the departure point of $l$ and $x$ the end point of $l.$ We will write $y=d(l)$ and $x=e(l).$
Let us define for $z\in \Ve,$ the Hopf algebra $\Gamma_z=\Gamma\times \{z\}$ and $\hat
\Gamma=\bigotimes_{z\in \Ve} \Gamma_{z}.$ This Hopf algebra was called in [@BR1] “the gauge symmetry algebra.” If $x$ is a vertex we shall write $\ga_x$ to denote the embedding of the element $\ga$ in $\Gamma_x.$
In order to define the non commutative analogue of algebra of gauge fields we have to endow the triangulation with an additional structure[@FR], an order between links incident to a same vertex, the [*cilium order*]{}.
A ciliation of the triangulation is an assignment of a cilium $c_z$ to each vertex $z$ which consists in a non zero tangent vector at z. The orientation of the Riemann surface defines a canonical cyclic order of the links admitting $z$ as departure or end point. Let $l_1, l_2$ be links incident to a common vertex $z,$ the strict partial cilium order $<_{c} $ is defined by:
$l_1<_{c}l_2$ if $l_1\not=l_2, -l_2$ and the unoriented links $c_z,l_1,l_2$ appear in the cyclic order defined by the orientation. If $l_1, l_2$ are incident to a same vertex $z$ we define:
$$\epsilon(l_1,l_2)=\left\{ \begin{array}{ll}
+1 &\mbox{if}\,l_1<_{c} l_2\\
-1& \mbox{if}\,l_2<_c l_1
\end{array}
\right.$$
The algebra of [**]{} gauge fields [@AGS][@BR1] $\Lambda$ is the non associative algebra generated by the formal variables $\ua(l)^i_j$ with $l\in \Li,\alpha\in
Phys({\cal A}),i,j=1\cdots dim(\alpha)$ and satisfying the following relations:
[**Commutation rules**]{} $$\begin{aligned}
& &\Rab_{12} \ua (yx)_1 \ub (yz)_2 = \ub (yz)_2 \ua (yx)_1\label{SS}\\
& &\ua (xy)_1 {}(S \otimes id)(\Rab_{12}) \ub (yz)_2 =\ub (yz)_2 \ua
(xy)_1\label{ES}\\
& &\ua (xy)_1 \ub (zy)_2 (S \otimes S)(\Rab_{12}) =
\ub (zy)_2 \ua (xy)_1 \label{EE}\\
& &\, \forall \,\,(yx), (yz) \in \Li\, x\not= z \,\,\,{\rm and}\,\,\,
xy<_{c}yz\nonumber\\
& &\ua(l){\buildrel \alpha \over A}\ua(-l)={\buildrel \alpha \over
B}\label{ES=1}\\
& &\forall \,\,l \in \Li^i, \nonumber\\
& &\ua (xy)_1 \ub (zt)_2 = \ub (zt)_2 \ua (xy)_1 \label{Udisjoint}\\
& &\forall\,\, x, y, z, t \mbox{ pairwise distinct in}\, \Ve\nonumber\end{aligned}$$
[**Decomposition rule**]{} (l)\_1 (l)\_2&=&\_[,m]{}\_[,m]{}\^[,]{} (l)\_[,]{}\^[,m]{}[ ]{} P\_[12]{}, \[UCG\]\
0(l)&=&1,l.
[**Quasi-associativity**]{} Let ${\cal M}_P$ be a monomial of gauge fields algebra elements with a certain parenthesing $P.$ For each vertex $x$ of the triangulation we construct a tensor product of representations of ${\cal A}$ by replacing each ${\buildrel \alpha \over {u_l}}$ in the monomial by the vector space $\alpha$ (resp. $\bar{\alpha}$ resp. $0$) depending on whether $x$ is the endpoint (resp. departure point resp. not element) of the edge $l,$ and keeping the previous parenthesing. Let us consider two different parenthesing $P_1$ and $P_2$ of the same monomial. We can construct for both, as described before, the corresponding vector spaces for each $x$ and deduce the intertwiner $\Theta_x$ relating them. The relation of quasi-associativity is then simply: (\_[x ]{} \_x )[M]{}\_[P\_1]{}=[M]{}\_[P\_2]{}\[assocalg\]
$\Lambda$ is a right $\hat \Gamma$ comodule defined by the morphism of algebra $\Omega:\Lambda\rightarrow \Lambda\otimes\hat\Gamma$ : ((xy))=\_x (xy) S(\_y). The definition relations of the gauge fields algebra are compatible to the coaction of the gauge symmetry algebra.
The subalgebra of gauge invariant elements of $\Lambda$ is denoted $\Lambda^{inv}.$
Invariant measure, holonomies, zero-curvature projector
-------------------------------------------------------
It was shown (provided some assumption on the existence of a basis of $\Lambda$ of a special type) [@AGS][@BR1] that there exists a unique non zero linear form $h\in \Lambda^{\star}$ satisfying:
1. (invariance) $(h\otimes id)\Omega(A)= h(A)\otimes 1 \,\,\forall A\in
\Lambda$
2. (factorisation) $h((A)(B))=h(A)h(B)\\
\forall A\in\Lambda_X, \forall B\in\Lambda_Y,\forall X, Y\subset L,\,\,
(X\cup -X) \cap (Y\cup - Y)=\emptyset$
(we have used the notation $\Lambda_{X}$ for $X\subset {\cal L}$ to denote the subalgebra of $\Lambda$ generated as an algebra by $\ua_{l}$ with $l\in X$).
It can be evaluated on any element using the formula: h((x,y)\^i\_j)=\_[,0]{} where $0$ denotes the trivial representation of dimension $1,$ i.e $0$ is the counit.
It was convenient to use the notation $\int d h$ instead of $h.$ We obtained the important formula: h((y,x)\_1 (S id)(\_[12]{}) v\_\^[-1]{}(x,y)\_2)= [1]{}P\_[12]{}[B]{}\_1. \[ortho\]
A path $P$ (resp. a loop $P$) is a path (resp.a loop) in the graph attached to the triangulation of $\Sigma$, given by the collection of its vertices, it will also denote equivalently the continuous curve (resp. loop) in $\Sigma$ defined by the links of $P.$ In this article we will denote by $P=[x_n,x_{n-1},\cdots,x_1,x_0]$ a link with departure point $x_0$ and end point $x_n$. Following the definition for links, the departure point of $P$ is denoted $d(P)$ and its endpoint $e(P).$ The set of vertices (resp. edges) of the path $P$ is denoted by ${\cal V}(P)$ (resp. ${\cal L}(P)$ ), the cardinal of this set is called the “length” of $P$ and will be denoted by $Length(P).$
Properties of path and loops such as self intersections, transverse intersections will always be understood as properties satisfied by the corresponding curves on $\Sigma.$
Let $P=[x_n,...,x_0]$ be a path, we defined the sign $\epsilon(x_i, P)$ to be $-1$ (resp. $1$) if $x_{i-1}x_i<_{c}x_ix_{i+1}$ (resp. $x_ix_{i+1}<_{c}x_{i-1}x_i$).
If $P$ is a simple path $P=[x_n,\cdots,x_0]$ with $x_0\not= x_n$, we define the holonomy along $P$ by \_P=v\_\^[[12]{}\_[i=1]{}\^[n-1]{}(x\_i, P)]{}((x\_n x\_[n-1]{})[A]{} (x\_[n-1]{}x\_[n-2]{})[A]{} (x\_[1]{}x\_[0]{})) . When $C$ is a simple loop $C=[x_{n+1}=x_0, x_n,\cdots, x_0],$ we define the holonomy along $C$ by \_C=v\_\^[[12]{}(\_[i=1]{}\^n (x\_i, C)- (x\_0,C))]{}((x\_0 x\_[n]{})[A]{} (x\_[n]{}x\_[n-1]{})[A]{} (x\_[1]{}x\_[0]{})). We define an element of $\Lambda$, called [*Wilson loop*]{} attached to $C:$ \_C= tr\_(S([A]{})\_C).\[Wilsonloop1\] the interior parenthesing being irrelevant because the loop is simple ( the vector space attached to a vertex occurs, first with $\alpha$, second with $\bar{\alpha}$ and the other times in the trivial representation) and moreover we have the relations (\[theta0\]). We will also use the notation $\Wa_C=\Wa_{[x_0,x_n\cdots,x_1]}.$
The properties shown in our first paper are easily generalized:
The element $\Wa_C$ is gauge invariant and moreover it does not depend on the departure point of the loop $C.$ Moreover it verifies the fusion equation: \_C \_C=\_[Phys(A)]{} N\_\^ \_C
The gauge invariance is quite obvious because of relations (\[antipode1\]), (\[antipode2\]). To show the cyclicity property we must put our Wilson loop in another form called “expanded form” in our first paper. Using relations (\[contract\]),(\[Rinv\]) we easily obtain: &&(C)=v\_\^[-[12]{}(\_[xC]{}(x,C))]{} tr\_[\^[n]{}]{}( (S([A]{}))\^[n]{}\_[i=n]{}\^1 P\_[ii-1]{}\
&&(\_[i=n]{}\^[1]{}(x\_[j+1]{} x\_[j]{})\_j (S id )(R\_[jj-1]{}\^[((x\_[j]{},C))]{})) (x\_1 x\_0)\_[1]{})\[Wilsonloop2\].In this form the cyclicity invariance is obvious using the commutation relations (\[ES\]).\
The fusion relation is less trivial to show. We first show a lemma describing the decomposition rules for holonomies: ([u]{}\_P)\_1 ([u]{}\_P)\_2 = \^\_\_P \^\_ P\_[12]{} [ ]{}\_[21]{}. Indeed, using (\[assocalg\])(\[ES\]), we easily obtain for a path $P=[x,y,z]$: &&(v\_\^[[12]{}(y, P)]{}((x y)[A]{} (yz))\_1(v\_\^[[12]{}(y, P)]{}((x y)[A]{} (yz))\_2=\
&&=\_[i,j,k,l]{}v\_\^[[12]{}(y, P)]{}v\_\^[[12]{}(y, P)]{}((x y)\_1(x y)\_2)(S(\^[(1)]{}\_l)A \^[(1)]{}\_i b\_j \^[-1 (2)]{}\_k \^[(31)]{}\_l)\_1\
&&(S(\^[(2)]{}\_l) S(\^[-1(1)]{}\_k)S(a\_j)S(\^[(2)]{}\_i) A \^[(3)]{}\_i \^[-1 (3)]{}\_k \^[(32)]{}\_l)\_2((y z)\_1(y z)\_2)=\
&&=v\_\^[[12]{}(y, P)]{}v\_\^[[12]{}(y, P)]{}((x y)\_1(x y)\_2)\_[21]{} ((y z)\_1(y z)\_2)\
the last equality is obtained by using successively (\[quasitriang\]) and (\[pentagon\]). Now, using (\[deltaA\])(\[Rdef\]), we obtain the announced result for a two links path. Proceeding by induction we can prove it for any simple open path.\
Let us now consider the loop $C$ as formed by two pieces $[xy]$ and $[yx]$, we have, using the same properties as before: &&(v\_\^[[12]{}((y, P)-(x,C))]{}tr\_(S([A]{})(x y) [A]{} (y x))) (v\_\^[[12]{}((y, P)-(x,C))]{}tr\_(S([A]{})(x y) [A]{} (y x)))=\
&&=\_[[ ]{}]{}(v\_v\_)\^[[12]{}((y, P)-(x,C))]{}tr\_((S(\^[-1 (2)]{}\_p) S(\^[(3)]{}\_m) S(c\_l) S( \^[-1 (2)]{}\_i) S(A) \^[-1 (1)]{}\_i \^[(1)]{}\_m \^[-1 (11)]{}\_p)\_1\
&&(S(\^[-1 (3)]{}\_p) S(A) \^[-1 (3)]{}\_i d\_l \^[(2)]{}\_m \^[-1 (12)]{}\_p)\_2 ((x y)\_1 (x y)\_2)(S(\^[-1 (11)]{}\_q) S(\^[(1)]{}\_n) S( \^[-1 (1)]{}\_j)\
&&A \^[-1 (2)]{}\_j b\_k \^[(3)]{}\_n \^[-1 (2)]{}\_q)\_1 (S(\^[-1 (12)]{}\_q) S(\^[(2)]{}\_n) S(a\_k) S(\^[-1 (3)]{}\_j) A \^[-1 (3)]{}\_q)\_2 ((y x)\_1 (y x)\_2))=\
&&= (v\_v\_)\^[[12]{}((y, P)-(x,C))]{}tr\_((S S)(G\_[21]{}R\_[12]{})() ((x y)\_1 (x y)\_2) (G\_[21]{}R\_[21]{}\^[-1]{})\
&&((y x)\_1 (y x)\_2) )=\
&&= \_N\^\_v\_\^[[12]{}((y, P)-(x,C))]{}tr\_(S([A]{})(x y) [A]{} [u]{}(y x)) which ends the proof of the theorem.
It can also be shown that $[\Wa_C, \Wb_{C'}]=0 $ for all simple loops $C, C'$ without transverse intersections.
Although the structure of the algebra $\Lambda$ depends on the ciliation, it has been shown in [@AGS] that the algebra $\Lambda^{inv}$ does not depend on it up to isomorphism. This is completely consistent with the approach of V.V.Fock and A.A.Rosly: in their work the graph needs to be endowed with a structure of ciliated fat graph in order to put on the space of graph connections ${\cal A}^l$ a structure of Poisson algebra compatible with the action of the gauge group $G^{l}.$ However, as a Poisson algebra ${\cal A}^l/G^{l}$ is canonically isomorphic to the space ${\cal M}^G$ of flat connections modulo the gauge group, the Poisson structure of the latter being independent of any choice of r-matrix [@FR].
We introduced a Boltzmann weight attached to any simple loop $C$ and defined by: \_[C]{}=\_[Phys(A)]{}\[d\_\] \_[C]{}. This element satisfies the flatness relation : $$\begin{aligned}
\delta_{C}\ua_{C}{}^i_j&=&{\buildrel \alpha \over
B}^i_j\delta_{C}.\label{flatness}\end{aligned}$$ moreover we have $$\begin{aligned}
(\frac{(\delta_{C})}{\sum_{\alpha \in Phys({\cal A})}
[d_{\alpha}]^{2}})^{2}&=&(\frac{(\delta_{C})}{\sum_{\alpha \in Phys({\cal A})}
[d_{\alpha}]^{2}}).\label{flatness}\end{aligned}$$
Using the same properties as in the computation of fusion relations, we obtain : &&\_[Phys([A]{})]{}\[d\_\](v\_\^[[12]{}((y, P)-(x,C))]{}tr\_(S([A]{})(x y) [A]{} (y x)))\
&&(v\_\^[[12]{}((y, P)-(x,C))]{}tr\_( S([A]{})(x y) [A]{} (y x)))=\
&&=\_[[ ]{}]{} \[d\_\](v\_v\_)\^[[12]{}((y, P)-(x,C))]{}tr\_((S(\^[-1 (2)]{}\_p) S(\^[(3)]{}\_m) S(c\_l) S( \^[-1 (2)]{}\_i) S(A) \^[-1 (1)]{}\_i \^[(1)]{}\_m\
&&\^[-1 (11)]{}\_p)\_1 (\^[-1 (3)]{}\_i d\_l \^[(2)]{}\_m \^[-1 (12)]{}\_p)\_2 ((x y)\_1 (x y)\_2) (S(\^[-1 (11)]{}\_q) S(\^[(1)]{}\_n) S( \^[-1 (1)]{}\_j) A \^[-1 (2)]{}\_j b\_k \^[(3)]{}\_n \^[-1 (2)]{}\_q)\_1\
&&(S(\^[-1 (12)]{}\_q) S(\^[(2)]{}\_n) S(a\_k) S(\^[-1 (3)]{}\_j) A \^[-1 (3)]{}\_q)\_2 ((y x)\_1 (y x)\_2)S(\^[-1 (3)]{}\_p)\_2)=\
&&=\_[,Phys([A]{})]{}\[d\_\] \_[p,m]{} tr\_(( S(\^[(3)]{}\_m) S(A) \^[(2)]{}\_m \^[-1 (12)]{}\_p)\_1\
&&( \^[(1)]{}\_m \^[-1 (11)]{}\_p)\_2 \^\_ (x y) A (y x) v\_\^[[12]{}((y, P)-(x,C))]{} \^\_ f\^[-1]{}\_[21]{} S(\^[-1 (2)]{}\_p)\_1 S(\^[-1 (3)]{}\_p)\_2) the last equality uses again the quasitriangularity properties. Then we obtain, for any matrix $V$ in $End(\alpha)$:
&&\_C tr\_( S(A) V \_C )=\
&&=\_[Phys([A]{})]{}\_[p,m]{} tr\_( (\_[Phys([A]{})]{}\[d\_\] \^\_ f\^[-1]{}\_[21]{} (S(\^[-1 (2)]{}\_p) S(\^[(3)]{}\_m) S(A) \^[(2)]{}\_m \^[-1 (12)]{}\_p)\_1\
&&( S(\^[-1 (3)]{}\_p) S(A) V \^[(1)]{}\_m \^[-1 (11)]{}\_p)\_2 \^\_ ) ((x y) A (y x) v\_\^[[12]{}((y, P)-(x,C))]{}) ) Due to the intertwining properties and the normalizations mentioned before of the $\phi,\psi$s we can conclude (see [@AGS]) that it does exist some complex coefficients $A(\alpha \beta
\gamma)$ such that: &&id\_K \^[0]{}\_[|]{} = \_A() \^\_[ ]{}\^\_[| ]{}\
&&\
&&\^\_=A()(\^\_[|]{}[ ]{}id\_) (\^[|]{}\_[0]{}id\_) we obtain &&\_[p,m]{} \_[Phys([A]{})]{}\[d\_\] \^\_ f\^[-1]{}\_[21]{} (S(\^[-1 (2)]{}\_p) S(\^[(3)]{}\_m) S(A) \^[(2)]{}\_m \^[-1 (12)]{}\_p)\_1\
&&( S(\^[-1 (3)]{}\_p) S(A) V \^[(1)]{}\_m \^[-1 (11)]{}\_p)\_2 \^\_ =\[d\_\]id\_ tr\_(S(A) V B) then for any matrix $V$ we have $\delta_C \; tr_{\alpha}({\buildrel \alpha
\over S(A)}
\mua {\buildrel \alpha \over V }\ua_C )=\delta_C \; tr_{\alpha}( {\buildrel
\alpha \over S(A)} \mua
{\buildrel \alpha \over V }{\buildrel \alpha \over B})$, and the linear independance of the generators of our algebra ensures the final result.\
The last formula of the proposition is a trivial consequence of the last result.
We were led to define an element that we called $a_{YM}=\prod_{f\in {\cal
F}}\delta_{\partial
f}.$ This element is the non commutative analogue of the projector on the space of flat connections.
In [@AGS][@BR1] it was proved that $\delta_{\partial f}$ is a central element of $\Lambda^{inv}$ and the algebra $\Lambda_{CS}=\Lambda^{inv}a_{YM}$ was shown to be independant, up to isomorphism, of the triangulation. The proof is based on the lemma of decomposition rules of the holonomies shown before and on the quite obvious property: let $C_1$ and $C_2$ be two simple contractile loops which interiors are disjoint and with a segment $[x y]$ of their boundary in common: dh(u(xy)) \_[C\_1]{} \_[C\_2]{} = \_[, ]{}(C\_1 \# C\_2)
As a result it was advocated that $\Lambda_{CS}$ is the algebra of observables of the Chern Simons theory on the manifold $\Sigma\times [0,1].$ This is supported by the topological invariance of $\Lambda_{CS}$ (i.e this algebra depends only on the topological structure of the surface $\Sigma$) and the flatness of the connection.
Our aim is now to construct in the algebra $\Lambda_{CS}$ the observables associated to any link in $\Sigma\times [0,1].$
Links, chord diagrams and quantum observables
---------------------------------------------
In the following subsection and in the chapter $3$ the computations will be made in the case of $q$ generic to simplify the notations but the generalization to $q$ root of unity can be made exactly in the same way.\
We will consider a compact connected surface $\Sigma$ with boundary $\partial
\Sigma$. The boundary is a set of disjoint simple closed curves which are designed to be “In” or “Out”. Let us draw some oriented curves on the surface $\Sigma$ defining a link $L$ , assuming that their boundary is contained in $\partial \Sigma$ and with simple, transverse intersections, with the specification of over- or undercrossing at each intersection. We will also consider that representations of the quantum group are attached to connected components of the link.\
The data ( surface with boundary + colored link ) will be called “striped surface”, the data of “In” (resp. “Out”) boundary of $\Sigma$ and $L$ with corresponding colors will be called the “In state” (resp. the “Out state”) of the striped surface.\
To describe such objects we will choose a Morse function which gives a time direction and the set of “equitime planes” $({\cal P}_t )_{t_i \leq t \leq t_f }$ cutting the surface. An equitime plane $P_t$ divides the surface in two parts called respectively “future” and “past”. On any simple curve drawn on an equitime plane the time direction give us an orientation of the curve, if we impose moreover a departure point $x$ on this curve we are able to decide if a point $z$ is on the left (resp. on the right) of another point $z',$ if $x, z, z'$ (resp. $x, z', z$) appear in the order given by this orientation. We will consider the surface in a canonical position defined by the following conditions. The intersection between the surface and the plane for $t<t_i$ or $t>t_f$ is empty. The “In” (resp. “out”) boundary is contained in the $t=t_i$ (resp. $t=t_f$) plane. The intersection between the $({\cal P}_t )
_{t_i \leq t \leq t_f }$ and the surface is a set of disjoint simple closed curves $(C^t_i)_{i=1,..., n(t)} $ (not necessary disjoint at the singular times), where $n(t)$ is the number of connected components of $\Sigma \cap {\cal P}_t.$ We will call “$\varphi^3-$diagram” of the surface a graph drawn on it which intersections with $({\cal P}_t )
_{t_i \leq t \leq t_f }$ determine departure points on each closed curve in these sets. We impose that the $\varphi^3-$diagram never turn around any handle of the surface.\
Our aim is now to define a ciliated fat graph which will encode the topology of the striped surface,i.e. this decomposition involves only contractile plaquettes, it is sufficiently fine to allow us to put the link on the graph in a generic position and allow us to distinguish two situations related by a Dehn twist of the surface. We then decompose the surface in blocks, their number being chosen with respect to the singularities of the Morse function ( considered as a function over the points of the surface and of the link). The information contained in the Morse function is not sufficient to deal with the problem of possible non trivial cycles of $L$ around handles of $\Sigma$. We will rule out this problem by adding fictively two disjoint $\varphi^3-$diagrams of $\Sigma$ to the link $L,$ the intersections between the link and the $\varphi^3-$diagrams will detect the rotation of the link around an handle of $\Sigma$, we will then refine the decomposition with respect to these datas. We will assume that the singularities of the Morse function $f$, considered now as a function of the points of the surface, points of the link and points of the $\varphi^3-$diagrams, correspond to different times. We will denote by $t_0=t_i,t_1,\cdots,t_{n-1},t_n=t_f$ the different instants corresponding to the singularities of the Morse function. We will consider a decomposition of the surface and of the link in “elementary blocks” ${\cal B}_0,\cdots,{\cal B}_n$ corresponding to the subdivision $[t_i,t_f] = [\tau_0,\tau_1] \cup [\tau_1,\tau_2] \cup
\cdots \cup [\tau_n,\tau_{n+1}]$ where $\tau_0=t_0, \tau_1={1\over2}(t_0+t_1), \cdots,
\tau_n={1\over2}(t_{n-1}+t_{n}), \tau_{n+1}=t_n$ are called “cutting times”. An example of a striped surface with the block decomposition described before is shown in the following figure:
We will consider a triangulation ${\cal T}$ and a ciliation induced by this block decomposition. The set ${\cal V}$ of vertices of ${\cal T}$ contains all elements of the sets $L \cap {\cal P}_{\tau_k}$ and the singularities of the Morse function considered as a function of the link and of the surface. The edges of the triangulation are either a segment of the link or of the $\varphi^3-$diagram between two consecutiv vertices, or a segment drawn on the surface at the same time $\tau_i$ between two emerging strings. The plaquettes of the triangulation are the connected regions of $\Sigma$ surrounded by the edges described before. Let us denote by $(x^k_l)$ the intersections of the link with the cutting planes $({\cal P}_{\tau_k}).$ The ciliation is chosen to be: at each vertex $x^k_l$, directed to the past and to the left, just “before” the equitime line, and at each crossing of the oriented link, between the two outgoing strands. We will choose a practical indexation satisfying the following properties: $x^k_l \in C^{\tau_k}_i$ for $l \in \{ 1+\sum_{j<i}Length(C^{\tau_k}_j),
\cdots$ $\cdots, \sum_{j
\leq i}Length(C^{\tau_k}_j)\}$ and the $x^k_l$ belonging to the same $C^{\tau_k}_i$ are ordered from the departure point and from left to right. The set of all $(x^k_l)$ for a given k is denoted by ${\cal
V}_{\tau_k}$. We will say that, at a vertex of $L$, the orientation of the link is “in the sense of time” or “against the sense of time” according to the position of the link with respect to the cutting plane, the two corresponding subset of ${\cal V}$ will be respectively denoted by ${\cal V}_{-}$ and ${\cal V}_{+}.$ The set of all edges of ${\cal T}$ is again denoted by ${\cal L}$ and we will denote by ${\cal L}_{\tau_k}$ (resp. ${\cal L}_{\tau_k < t < \tau_{k+1}}$ ) the set of edges contained in ${\cal P}_{\tau_k}$ (resp. between ${\cal
P}_{\tau_k}$ and ${\cal P}_{\tau_k+1}$ ). The elements of ${\cal L}_{\tau_k < t < \tau_{k+1}}$ will be currently denoted as $l^k_j$. We index the elements of this set with the following rule: begining from the departure point fixed by the $\varphi^3-$diagram we order the links in an obvious way from left to right if they do not cross themselves and, at a crossing, the first one is that which overcross the other. At last we will denote by ${\cal F}_{\tau_k \le t \le \tau_{k+1}}$ the set of faces of the triangu lation corresponding to the block ${\cal B}_k$. To each element $l$ of ${\cal L}$ we will associate a subset of ${\cal F}$ called $Present(l)$ defined by the following rule: if $l$ belongs to a crossing and is the overcrossing (resp.undercrossing) strand, $Present(l)$ is the set of the four plaquettes surrounding the crossing (resp. the set is empty), if $l$ is an annihilation or a creation, $Present(l)$ is the set of the two surrounding plaquettes, elsewhere $Present(l)$ contains only the plaquette just at the left of $l$. We then define $Past(l)$ to be the subset of ${\cal F}\setminus Present(l)$ such that $P \in Past(l)$ if it is on the left of $l$ in the same block or anywhere in a past block. A link in $\Sigma\times [0,1]$ is an embedding of $(S^1)^{\times
p'}\times([0,1])^{\times p''}$ into $\Sigma\times [0,1],$ with $\partial L \subset \partial \Sigma \times
[0,1]$. On the set of links we can define a composition law, denoted $*$ defined as follows:let $L$ and $L'$ be to links in $\Sigma\times[0,1]$ considered up to ambiant isotopy. We define $L*L'$ to be the link obtained by putting $L$ in $\Sigma \times [{1 \over 2},1]$ and $L'$ in $\Sigma \times [0,{1\over2}].$ This composition is associative and admit the empty link as unit element. This composition law is commutative if and only if $\Sigma$ is homeomorphic to the sphere. Let us denote by $(\L{i})_{i=1\cdots p}$ the connected components of the link $L,$ $\alpha_i\in Phys({\cal A})$ the colour of this component, and by $\P{i}$ the colored loop obtained by projecting $\L{i}$ on $\Sigma.$ It is very convenient to associate to the link $L$ a coloured chord diagram [C]{} [@Va] which will encode intersections of the loops. This chord diagram is constructed as follows: the projection of the link on $\Sigma$ defines $p'$ colored loops and $p''$ coloured open paths (with boundary on the boundary of $\Sigma$) on $\Sigma$ with transverse intersections, this configuration of paths defines uniquely a coloured chord diagram by the standard construction. Let us denote by $(\S{i})_{i=1\cdots p=p'+p''}$ the coloured circles and arcs of the chord diagram corresponding to loops and open paths belonging to the link ( we will call abusively “circles” the circles or the arcs of the chord diagrams). The family of coloured circles (resp. arcs) $(\S{i})_{i=1\cdots p'}$ (resp. $(\S{i})_{i=p'+1\cdots p}$) will be denoted ${\cal C}_1$ (resp. ${\cal C}_2$). Each circle $\S{i}$ is oriented, we will denote by $(\yi_j)_{j=1\cdots n_i}$ the intersection points of the circle $\S{i}$ with the chords. We will assume that they are labelled with respect to the cyclic order defined by the orientation of the circles. Let $Y=\cup_{i=1}^p \{\yi_j,j=1\cdots n_i\}$, we define a relation $\sim$ on the set $Y$ by : $y\sim y' \,\,\mbox{if and only if} \,y\,\mbox{and}\, y' \,\mbox{are
connected by a chord.}$ We will denote by $\varphi$ the immersion of the chord diagram in $\Sigma$, in particular we have $\P{i}=\varphi(\S{i}).$ Every intersection point of the projection of $L$ on $\Sigma$ have exactly two inverse images by $\varphi$ in the chord diagram and these points are linked by a unique chord. We will denote by $\zi{}^k_j\in
\S{i}$ the points such that $\zi{}^k_j\in ]\yi_{j}\yi_{j-1}[$ and $\varphi(\zi{}^k_j)$ is a vertex of the triangulation corresponding to the cutting time $\tau_k$. We will denote by $Z_i$ the set of all points of type $z$ in the $i-$th component, Z the union of these sets, $Z^{\partial \Sigma}$ the subset of $Z$ formed by the points of $Z$ which belong to the boundary of $\Sigma.$ Let us denote by $\Se_i$ the family of segments forming the corresponding circle and $\Se$ the union of these families for all components. To each segment $s=[pq]$ we will associate two vector spaces $V_{q^{-}}$ and $V_{p^{+}}$ such that $V_{q^{-}}=
V_{p^{+}}=\V{\alpha_i}.$
$\Se$ being a finite set, let us choose on it a total ordering. This ordering allows us to define two vector spaces $V_-$ and $V_+:$ $V_-=\bigotimes_{x\in Y\cup Z}V_{x^-}$ and $V_+=\bigotimes_{x\in Y\cup Z}V_{x^+}$ where the order in the tensor product is taken relativ to it.
Let $a, b\in Y\cup Z$ and $\xi, \eta\in \{+, -\},$ and assume that $\phi(a)=\phi(b),$ we will use as a shortcut the notation: $\epsilon(a^{\xi}b^{\eta})=\epsilon(l(a^{\xi}), l(b^{\eta})).$
We define the space $\Lambda_{\Se}$ by : $\Lambda_{\Se}=\Lambda\otimes \bigotimes_{s\in\Se}
\End(V_{d(s)^{-}},V_{e(s)^{+}}).$
If $s$ is an element of $\Se_i$ we denote by $j_s$ the canonical injection $j_s:\Lambda\otimes\End(V_{d(s)^{-}},V_{e(s)^{+}})\hookrightarrow
\Lambda_{\Se}.$
Let us define two types of holonomy along $s$: $u_{s}\in\Lambda\otimes\End(V_{d(s)^{-}},V_{e(s)^{+}})$ is defined by $u_{
s}=u_{\varphi(s)},$ (the right handside has already been defined so that there is no risk of confusion) and $U_{s}\in\Lambda_{\Se}$ is defined by $U_{s}=j_{s}(u_{\varphi(s)}).$
Let $a, b$ two points of $Y\cup Z$ such that $\phi(a)=\phi(b)$ and define the endomorphism:
$R^{(a^{\xi}b^{\eta})}\in \End(V_{a^{\xi}}\otimes V_{b^{\eta}})$ (resp $
\End(V_{b^{\eta}}\otimes V_{a^{\xi}})$) if $s(a^{\xi})\triangleleft
s(b^{\eta})$ (resp if $s(b^{\eta})\triangleleft s(a^{\xi}))$ by:
$$R^{(a^{\xi}b^{\eta})}=\left\{ \begin{array}{ll}
(\alpha_1\otimes\alpha_2)(R^{(\epsilon(a^{\xi}b^{\eta}))}) &\mbox{if
$s(a^{\xi})\triangleleft s(b^{\eta})$}\\
P_{a^{\xi}b^{\eta}}(\alpha_1\otimes
\alpha_2)(R^{(\epsilon(a^{\xi}b^{\eta}))})P_{a^{\xi}b^{\eta}}& \mbox{if
$s(b^{\eta})\triangleleft s(a^{\xi})$}
\end{array}
\right.$$ Let $<$ be any fixed strict total order on $Y,$ we define a family $\{\cR{y}\}_{y\in Y}$ of elements of $\bigotimes_{s\in\Se} End(V_{d(s)^-},V_{e(s)^{+}})$ as follows: let $\{y, y'\}$ be any pair of points of $Y$ such that $y\sim y',$ we can always assume (otherwise we just exchange $y$ and $y'$) that $y< y',$
={
[ll]{} R\^[(y\^[-]{}y\^[+]{})-1]{}\
R\^[(y\^[+]{}[y’]{}\^[+]{})]{} R\^[(y\^[-]{}y\^[+]{})-1]{} R\^[(y\^[-]{}[y’]{}\^[+]{})-1]{}
. ={
[ll]{} R\^[([y’]{}\^[-]{}[y’]{}\^[+]{})-1]{}\
R\^[([y’]{}\^[-]{}[y’]{}\^[+]{})-1]{}R\^[([y]{}\^[-]{}[y’]{}\^[-]{})]{}R\^[([y’]{}\^[-]{}[y]{}\^[+]{})]{}\_[[y’]{}\^[-]{}[y]{}\^[-]{}]{}
. This definition defines completely the elements $\cR{y}$ for $y\in Y.$ Similarly if $z$ is an element of $Z$ we will define $\cR{z}=R^{(z^{-}z^{+})-1}.$
We have now defined the framework necessary to associate to $L$ an element of $\Lambda$ denoted $W_{L}$ which generalizes the construction of Wilson loops. We denote by $<_l$ be the strict lexicographic order induced on $Y$ by the enumeration of the connected components of $L$ and a choice of departure point for each of these components, i.e $\yi_{p}<_l \,\yj_{q}$ if and only if $i<j$ or ($i=j$ and $p>q.$)
Let $P$ be a connected piece of one of the $\S{i}$. Let us choose for simplicity $P=[\zi_{n+1},
\yi_n, \zi_n, \cdots, \yi_1, \zi_{1}]$, we will denote the holonomy associated to it, by \_P=(P)U\_[\[\_[n+1]{}\_[n]{}\]]{}(<\_l)U\_[\[\_n \_n\]]{} (<\_l) (<\_l) U\_[\[\_1 \_1\]]{}, where $\omega(P)=v_{\alpha_i}^{-{1\over 2}
\sum_{x\in P\setminus\{\zi_{n+1}, \zi_1\}}\epsilon(\phi(x),\S{i})}.$ We will denote by $\cUS{i}$ the holonomy associated to the entire circle $\S{i}$. We will also define the permutation operator: $\sigma_P = \prod_{x=\yi_n}^{\yi_1} P_{\zi_{n+1}, x}$ (where the order is given by the order of vertices along $P$) and $\sig{i}$ will denote $\sigma_{\S{i}}.$
To each link in $\Sigma\times[0,1]$ we associate an element $W_L$ by the following procedure: let us denote by ${\cal W}_{L}$ the element \_[L]{}= \_\_[i=1]{}\^p \_[i=1]{}\^[p]{} ; where $\mu_{\Se}=\bigotimes_{x \in Z \setminus Z^{\partial \Sigma}}\mu_{x^+}$ :
The element associated to the link $L$ is defined by W\_[L]{}=tr\_[\_[x Z Z\^]{}V\_[x\^+]{}]{}[W]{}\_[L]{} where $tr_{V_{+}}$ means the partial trace over the space $V_{+}$ after the natural identification $V_{+}=V_{-}.$
This element satisfies important properties described by the following theorem [@BR2]:
Let $L$ be a link satisfying the set of assumptions, then $W_L$ does not depend on the labelling of the components nor does it depend on the choice of departure points of the components. As a result W is a function on the space of links with values in $\Lambda\otimes
\bigotimes_{P \in {\cal C}_2}End(V_{d(P)^-},V_{e(P)^+}).$ Moreover this mapping is invariant under the coaction of the gauge group at a vertex interior to the surface. If $L$ and $L'$ are two links , we have the morphism property $W_{L*L'}=W_{L}W_{L'}.$
Our principal aim is the computation of the correlation function defined in an obvious way:
The correlation function of the link $L$ considered as immersed in $\Sigma\times [0,1]$ is simply defined by: \_[q-YM()]{}= \_[l \^[int]{}]{}dh(U\_l) W\_L \_[F]{} \_[F]{}
The observable associated to $L$ will be denoted by ${\widehat W}_L=W_L\prod_{F\in {\cal F}}\delta_{\partial F}.$ This element of $\Lambda_{CS}$ depends only on the regular isotopy class of the link $L,$ i.e it satisfies the Reidemeister moves of type 0,2,3. This fact was established in [@BR2].\
Moreover, let $L$ be as usual a link in $\Sigma\times [0, 1]$ and $P$ the set of projected curves on $\Sigma$ and let $L^{\propto\pm}$ be another link whose projection $P^{\propto\pm}$ differs from $P$ by a move of type I
applied to a curve colored by $\alpha$, we have the following relation: \_[L\^]{}=v\_\^[1]{} [W]{}\_[L]{} The expectation value of a Wilson loop on a Riemann surface can be considered as an invariant associated to a ribbon glued on the surface with the blackboard framing.
Computation of the correlation functions
========================================
Our first aim is the computation of the invariants associated to links drawn on a closed Riemann surface embedded in $S^3.$ To realize this program we want to decompose the computation by introducing surfaces with boundaries and links drawn on them, already called “striped surfaces”, and by describing the gluing operation of the latter.
This decomposition allows us to reduce the striped surface to the gluing of the following objects, called “elementary blocks” :
1. [**the cups**]{}
2. [**the caps**]{}
3. [**the (n,m)(n+m) trinions**]{}
4. [**the (n+m)(n,m) trinions**]{}
5. [**the free propagation of n strands**]{}
6. [**the propagation of n strands with one overcrossing**]{}
7. [**the propagation of n strands with one undercrossing**]{}
8. [**the (n-2)(n) creation**]{}
9. [**the (n)(n-2) annihilation**]{}
These objects are described in the following figure\
The correlation functions can be put in a more convenient form to reduce the computation to the gluing of elements associated to elementary blocks.
Let us consider an element $l$ of ${\cal L}$ and an element $P$ of ${\cal F}$ then P Past(l) \_P = \_P.
this result is a trivial consequence of the choice of ciliation and of the commutation properties developed in [@BR1]
From now the order induced by the orientation of the link will not be convenient anymore, prefering time ordering we will introduce the vector spaces $V_{x^a}$ and $V_{x^b}$ (“after” and “before”) rather than $V_{x^+}$ and $V_{x^-}.$ Let $\alpha$ denote the representation associated to the circle where x is taken, then $V_{x^+}=V_{x^-}=\V{\alpha}.$ If $x$ is in ${\cal
V}_+$, then $V_{x^a}=V_{x^b}=\V{\alpha}$ and we will introduce the canonical identification maps $id_{(V_{x^a},V_{x^+})}$ and $id_{(V_{x^b},V_{x^-})}.$ If $x$ is in ${\cal
V}_-$, then $V_{x^a}=V_{x^b}=\V{\bar{\alpha}}$ and we will introduce the canonical maps ${\phi^{\bar{\alpha}\alpha}_{0}}_{(V_{x^b},V_{x^+})}$ and ${\psi^{0}_{\bar{\alpha}\alpha}}_{(V_{x^a},V_{x^-})}.$ We can define a new holonomy by ${U}^{\#}_{[ x, y]}$ : $$\begin{aligned}
{U}^{\#}_{[x, y]}&=& id_{(V_{x^a},V_{x^+})} \gua_{[ x, y ]}
id_{(V_{y^b},V_{y^-})}, \mbox{ if } x \in {\cal V}_+, y \in {\cal V}_+\\ &=&
{\psi_{\bar{\alpha}\alpha}^{0}}_{(V_{x^b},V_{x^+})} {\gua}_{[ x, y ]}
{\phi_{0}^{\bar{\alpha}\alpha}}_{(V_{y^a},V_{y^-})}= {\guabar}_{[ y, x ]},
\mbox{ if } x \in {\cal
V}_-, y \in {\cal V}_-\\ &=&
{\psi_{\bar{\alpha}\alpha}^{0}}_{(V_{x^b},V_{x^+})} {\gua}_{[ x, y ]}
id_{(V_{y^b},V_{y^-})}, \mbox{ if } x \in {\cal V}_-, y \in {\cal V}_+\\ &=&
id_{(V_{x^a},V_{x^+})}
{\gua}_{[ x, y ]} {\phi_{0}^{\bar{\alpha}\alpha}}_{(V_{y^a},V_{y^-})}, \mbox{
if } x \in {\cal
V}_+, y \in {\cal V}_- \end{aligned}$$ Despite its apparent complexity, this definition has a very simple meaning. It describes the usual fact that a strand in the direction of the past coloured by a representation $\alpha$ can be described by a strand in the direction of the future coloured by a representation $\bar{\alpha}.$
We will denote in the following: A\_k = \_[j=1]{}\^[Card([L]{}\_[\_k t \_[k+1]{}]{})]{} ( (\_[P Present(l\^k\_j)]{} \_[P]{}) [U]{}\^[\#]{}\_[l\^k\_j]{} ),
the elements ( $(\prod_{P \in Present(l)}\delta_P U^{\#}_l)$ if $l$ does not belong to a crossing and $(\prod_{P \in Present(l)}\delta_P U^{\#}_l U^{\#}_{l'})$ if $l$ and $l'$ cross themselves ) will be called “square plaquettes” elements in the following.\
We will also use the following permutation operator: \^[\_k]{} = \_[j=1]{}\^[Card([V]{}\_[\_k]{})]{} (\_[i=Card([V]{}\_[\_[k+1]{}]{})]{}\^[1]{} P\_[(x\^k\_j)\^a,(x\^[k+1]{}\_i)\^a]{} )
Using these definitions, the element associated to the striped surface can be put in a form which respects the ordering induced by the time order: \_L=v\_\^[[12]{}\_[x \^[int]{}]{} (x\^b,x\^a)]{} tr\_[\_[x \^[int]{}]{}V\_[x\^a]{}]{}((\_[k=1]{}\^[n]{}\^[\_k]{})(\_[k=0]{}\^[n]{}A\_k)) We begin with the ordering of the holonomies attached to the link. Using the commutation relations and the properties of the $R$ matrix we obtain: $$\begin{aligned}
W_L&=& tr_{\bigotimes_{x \in {\cal
V}^{int}}V_{x^+}}((\prod_{k=1}^{n}\prod_{j=1}^{Card({\cal V}_{\tau_k})}
\prod_{i=Card({\cal V}_{\tau_{k+1}})}^{1} P_{(x^k_j)^+,(x^{k+1}_i)^+} )\times\\
&\times&(\otimes_{x\in {\cal V}^{int}}\mu_{x^+})
(\prod_{k=0}^{n}\prod_{j=1}^{Card({\cal L}_{\tau_k \le t \le \tau_{k+1}})}
{U}_{l^k_j}))
v_{\alpha}^{{1\over2}(\sum_{x \in {\cal V}^{int}_+} \epsilon(x^-,x^+) +
\sum_{x \in {\cal V}^{int}_-} \epsilon(x^+,x^-))}\end{aligned}$$ now the commutation lemma gives easily: (\_[P ]{}\_P)(\_[k=0]{}\^[n]{} \_[j=1]{}\^[Card([L]{}\_[\_k t \_[k+1]{}]{})]{} [U]{}\_[l\^k\_j]{})=\_[k=0]{}\^[n]{}\_[j=1]{}\^[Card([L]{}\_[\_k t \_[k+1]{}]{})]{} ( (\_[P Present(l\^k\_j)]{} \_P) [U]{}\_[l\^k\_j]{} ), and with the definition of ${U}^{\#}_{l^k_j}$ we then obtain: $$\begin{aligned}
{\widehat W}_L=v_{\alpha}^{{1\over2}\sum_{x \in {\cal V}^{int}}
\epsilon(x^b,x^a)}
tr_{\bigotimes_{x \in {\cal V}^{int}}V_{x^a}}((\prod_{k=1}^{n}
\prod_{j=1}^{Card({\cal
V}_{\tau_k})} \prod_{i=Card({\cal V}_{\tau_{k+1}})}^{1}
P_{(x^k_j)^a,(x^{k+1}_i)^a} )
(\prod_{k=0}^{n}A_k))\end{aligned}$$
This lemma leads us to a new definition of elements associated to “striped surfaces” which is based on gluing chronologically ordered elementary blocks.
Let us consider a “striped surface” $\Sigma+L$. Let us define an element corresponding to $\Sigma+L$ ( which will be denoted by ${\cal A}_{\Sigma+L}$) by the following rules:\
- if $\Sigma+L$ is an elementary block ${\cal B}_1$, the element of the gauge algebra associated to it is: \_[[B]{}\_1]{}=\_[l \_[\] t,t’ \[ ]{}]{} dh(U\_l) \_[j=1]{}\^[Card([L]{}\_1)]{} ( (\_[P Present(l\^1\_j)]{} \_P) [U]{}\^[\#]{}\_[l\^1\_j]{} ) v\_\^[[12]{}\_[x \^[t’]{}]{} (x\^b,x\^a)]{}
- if $\Sigma+L$ is a disjoint union of $N$ elementary blocks placed between $t$ and $t'$ then the element of the algebra associated to $\Sigma+L$ is obviously the product of the elements associated to each elementary block, the order between them being irrelevant because they are commuting.\
- if there exists a time $t''$ between $t$ and $t'$ such that $\Sigma+L$ is obtained by gluing two “striped surfaces” $\Sigma_1+L_1$ and $\Sigma_2+L_2$ placed respectively between $t$ and $t''$, and between $t''$ and $t'.$ The element associated to $\Sigma+L$ will be defined by: \_[+L]{}&=&[A]{}\_[\_1+L\_1]{} \_[\_2+L\_2]{}\
&=& \_[l \_[t”]{}]{} dh(U\_l) tr\_[\_[x \^[t”]{}]{}V\_[x\^a]{}]{}(\^[t”]{}[A]{}\_[\_1+L\_1]{} [A]{}\_[\_2+L\_2]{})( the canonical choice of ciliation defined for any striped surface is obviously compatible with the gluing operation )\
these properties give us a new way to compute the invariants associated to links on a closed surface: \_[q-YM()]{}= [A]{}\_[+L]{}.
From now the computation of the correlation functions is reduced to the computation of elements associated to elementary blocks. After some definitions we will give the result of the explicit computation of these elements.
Let us consider a striped surface $\Sigma+L$. A connected component of its “In state” is a simple loop ${\cal
C}=[x_{n+1}=x_1, x_n, \cdots, x_1]$ oriented in the inverse clockwise sense with $n+1$ strands going through it at each $x_i$ in the direction of the past with a representation $\alpha_i.$\
( a strand in the direction of the future with a representation $\alpha$ is reversed to the past by changing its representation in $\bar{\alpha}$). The ciliation at each $x_i$ is chosen as in the general construction of striped surfaces. Then, choosing $n$ other representations $(\beta_i)_{i=1,...,n}$ we define ${\cal O} \in \Lambda \otimes End(\otimes_{x \in {\cal C}} V_{x^b},{\bf C})$ and ${\cal I} \in \Lambda \otimes End({\bf C},\otimes_{x \in {\cal C}} V_{x^a})$:\
if there is at least one emerging strand through ${\cal C}$, $$\begin{aligned}
\Outstate{\beta_n
}{\alpha_n}{x_n}{\beta_{n-1}}{\alpha_{n-1}}{x_{n-1}}{\beta_1}{\alpha_1}{x_1}&=&
v^{-1}_{\beta_n}tr_{\V{\beta_n}}
(\psi^{\beta_n}_{\beta_1
\alpha_1}\guf{\beta_1}_{[x_1,x_2]}\psi^{\beta_1}_{\beta_2 \alpha_2}
\cdots \psi^{\beta_{n-1}}_{\beta_n\alpha_{n}}\guf{\beta_n}_{[x_{n},x_1]}
\Rmff{\beta_n}{\alpha_1}\muf{\beta_n})\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Instate{\beta_n}{\alpha_n}{x_n}{\beta_{n-1}}{\alpha_{n-1}}{x_{n-1}}{\beta_1}{\alpha_1}{x_1}&=&
v_{\beta_n}tr_{\V{\beta_n}}(\muf{\beta_n}\Rff{\beta_n}{\alpha_1}
\guf{\beta_n}_{[x_1,x_{n}]}\phi^{\beta_n \alpha_{n}}_{\beta_{n-1}} \cdots
\phi^{\beta_2
\alpha_2}_{\beta_1}\guf{\beta_1}_{[x_2,x_{1}]}\phi^{\beta_1
\alpha_1}_{\beta_n}) \end{aligned}$$ Here and in the following we will often forget the multiplicities $m_i$ for readibility.\
If the connected component of this “In state” has no emerging strand we will define ${\cal O}$ and ${\cal
I}$ to be: (\_0)=W\^[\_0]{}\_[C\^[-1]{}]{}(\_0)=W\^[\_0]{}\_[C]{}
The properties of the latter objects are described in the following lemma. The properties of the In and Out states are generalizations of those of Wilson loops.\
Cyclicity &=&\
&=&\
Gauge transformation ()&=& \_[i=1]{}\^[n]{} S(\_[x\_i]{})\
()&=& \_[i=1]{}\^[n]{} \_[x\_i]{}\
Scalar product \_[l ]{}dh(U\_l)&&tr\_[\_[i]{}V\_[x\_i\^a]{}]{}(\_[[C]{}]{}\
&&)= \_[i=1]{}\^[n]{}\_[\_i,[’]{}\_i]{}
The cyclicity property is not completely obvious. We give here a detailed proof of this fact: &&=\
&&=v\^[-1]{}\_[\_n]{}tr\_[V\_[\_n]{}]{} (\^[\_n]{}\_[\_1 \_1]{}\_[\[x\_1,x\_2\]]{}\^[\_1]{}\_[\_2 \_2]{}\_[\[x\_2,x\_3\]]{} \^[\_[n-1]{}]{}\_[\_n\_[n]{}]{}\_[\[x\_[n]{},x\_1\]]{} )\
&&=v\^[-1]{}\_[\_n]{}tr\_[V\_[\_1]{}]{} (\_[\[x\_1,x\_2\]]{}\^[\_1]{}\_[\_2 \_2]{} \_[\[x\_2,x\_3\]]{} \^[\_[n-1]{}]{}\_[\_n\_[n]{}]{}\_[\[x\_[n]{},x\_1\]]{} \^[\_n]{}\_[\_1 \_1]{} v\_[\_2]{})\
&&=v\^[-1]{}\_[\_n]{}tr\_[V\_[\_1]{}\_[\_1]{}]{} (P\_[V\_[\_1]{},[V’]{}\_[\_1]{}]{}\_[\[x\_1,x\_2\]]{}\^[[’]{}\_1]{}\_[\_2 \_2]{} \_[\[x\_2,x\_3\]]{} \^[\_[n-1]{}]{}\_[\_n\_[n]{}]{}\_[\[x\_[n]{},x\_1\]]{} \^[\_n]{}\_[[’]{}\_1 \_1]{} v\_[\_2]{})\
&&=\_[(i),(j)]{}v\^[-1]{}\_[\_n]{}tr\_[V\_[\_1]{}]{} (\^[\_1]{}\_[\_2 \_2]{} b\_[(i)]{}\^[\_2]{} \_[\[x\_2,x\_3\]]{} \^[\_[n-1]{}]{}\_[\_n\_[n]{}]{}\_[\[x\_[n]{},x\_1\]]{} S(a\_[(j)]{}\^[\_n]{}) \^[\_n]{}\_[\_1 \_1]{} v\_[\_2]{} b\_[(j)]{}\^[\_1]{}\_[\[x\_1,x\_2\]]{} S\^[2]{}(a\_[(i)]{}\^[\_1]{}))\
&&=\_[(i)]{}v\^[-1]{}\_[\_n]{}tr\_[V\_[\_1]{}]{} (S\^[2]{}(a\_[(i)]{}\^[\_1]{})b\_[(i)]{}\^[\_1]{} \^[\_1]{}\_[\_2 \_2]{} \_[\[x\_2,x\_3\]]{} \^[\_[n-1]{}]{}\_[\_n\_[n]{}]{}\_[\[x\_[n]{},x\_1\]]{}\^[\_n]{}\_[\_1 \_1]{} \_[\[x\_1,x\_2\]]{} v\_[\_n]{})\
&&=\_[(i)]{}v\_[\_1]{}\^[-1]{}tr\_[V\_[\_1]{}]{} (\^[\_1]{}\_[\_2 \_2]{} \_[\[x\_2,x\_3\]]{} \^[\_[n-1]{}]{}\_[\_n\_[n]{}]{}\_[\[x\_[n]{},x\_1\]]{}\^[\_n]{}\_[\_1 \_1]{} \_[\[x\_1,x\_2\]]{} \_[\_1]{})\
&&= The gauge transformation is very simple to derive using the decomposition rules of the elements of the group and we can proove the scalar product property using simply the integration formula and the unitarity relations of Clebsch-Gordan maps. &&\_[l ]{}dh(U\_l)tr\_[\_[i]{}V\_[x\_i\^a]{}]{}(\_[[C]{}]{}\
&&)=\
&&=\_[l ]{}dh(U\_l) v\^[-1]{}\_[\_n]{}v\_[[’]{}\_n]{}tr\_[V\_[\_n]{} V\_[[’]{}\_n]{}]{} (\^[\_n m\_n]{}\_[\_1 \_1]{}\_[\[x\_1,x\_2\]]{}\^[\_1 m\_1]{}\_[\_2 \_2]{} \^[\_[n-1]{} m\_[n-1]{}]{}\_[\_n\_[n]{}]{} \_[\[x\_[n]{},x\_1\]]{}\
&&\_[\[x\_1,x\_[n]{}\]]{} \^[-1]{} \^[[’]{}\_n \_[n]{}]{}\_[[’]{}\_[n-1]{} [m’]{}\_[n-1]{}]{} \^[[’]{}\_2 \_2]{}\_[[’]{}\_1 [m’]{}\_1]{}\_[\[x\_2,x\_[1]{}\]]{}\^[[’]{}\_1 \_1]{}\_[[’]{}\_n [m’]{}\_n]{})=\
&&=\_[l ]{}dh(U\_l) tr\_[V\_[\_n]{}]{} (\^[\_n m\_n]{}\_[\_1 \_1]{}\_[\[x\_1,x\_2\]]{}\^[\_1 m\_1]{}\_[\_2 \_2]{} \^[\_[n-1]{} m\_[n-1]{}]{}\_[\_n\_[n]{}]{} \^[\_n \_[n]{}]{}\_[[’]{}\_[n-1]{} [m’]{}\_[n-1]{}]{}\
&&\^[[’]{}\_2 \_2]{}\_[[’]{}\_1 [m’]{}\_1]{}\_[\[x\_2,x\_[1]{}\]]{}\^[[’]{}\_1 \_1]{}\_[[’]{}\_n [m’]{}\_n]{})=\
&&=tr\_[V\_[\_n]{}]{} ()\_[i=1]{}\^[n-1]{}(\_[\_i,[’]{}\_i]{}\_[m\_i,[m’]{}\_i]{})=\_[i=1]{}\^[n]{}(\_[\_i,[’]{}\_i]{}\_[m\_i,[m’]{}\_i]{}).\
This ends the proof of the lemma.\
All elements associated to elementary blocks can be computed in terms of “In” and “Out” states of the latter form. Let us give here the expression of the elements associated to the elementary blocks enumerated before:\
$$\begin{aligned}
&&{\bf {\cal A}^{elem}_{cup}}= \sum_{\beta_0} [d_{\beta_0}] {\cal O}(\beta_0)\\
&&{\bf {\cal A}^{elem}_{cap}}= \sum_{\beta_0} [d_{\beta_0}] {\cal I}(\beta_0)\\
&&{\bf {\cal A}^{elem}_{(n,m)(n+m) tri.}}= \sum_{\beta_1,\cdots, \beta_{n+m}}
[d_{\beta_{n+m}}]^{-1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Intri{{\beta'}_n}{{\alpha'}_n}{{x'}_n}{{\beta'}_1}{{\alpha'}_1}{{x'}_1}\Intri{{\beta''}_n}{{\alpha''}_n}{{x''}_n}{{\beta''}_1}{{\alpha''}_1}{{x''}_1}
\times\\
&&\times\Outtri{{\beta}_n}{{\alpha}_n}{{x}_n}{{\beta}_1}{{\alpha}_1}{{x}_1}
\delta_{{\beta'}_n, {\beta''}_m, {\beta}_{n+m}, \beta_m}
\prod_{k=1}^{n}\delta_{\alpha_{m+k}, {\alpha'}_{k}}
\prod_{k=1}^{m}\delta_{\alpha_{k}, {\alpha''}_{k}}
\prod_{k=1}^{n-1}\delta_{\alpha_{m+k}, {\alpha'}_{k}}
\prod_{k=1}^{m-1}\delta_{\alpha_{k}, {\alpha''}_{k}}\nonumber\\
&&{\bf {\cal A}^{elem}_{(n+m)(n,m) tri.}}= \sum_{\beta_1,\cdots, \beta_{n+m}}
[d_{\beta_{n+m}}]^{-1}
\Intri{{\beta}_n}{{\alpha}_n}{{x}_n}{{\beta}_1}{{\alpha}_1}{{x}_1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Outtri{{\beta'}_n}{{\alpha'}_n}{{x'}_n}{{\beta'}_1}{{\alpha'}_1}{{x'}_1}\times\\
&&\times
\Outtri{{\beta''}_n}{{\alpha''}_n}{{x''}_n}{{\beta''}_1}{{\alpha''}_1}{{x''}_1}
\delta_{{\beta'}_n, {\beta''}_m, {\beta}_{n+m}, \beta_m}
\prod_{k=1}^{n}\delta_{\alpha_{m+k}, {\alpha'}_{k}}
\prod_{k=1}^{m}\delta_{\alpha_{k}, {\alpha''}_{k}}
\prod_{k=1}^{n-1}\delta_{\alpha_{m+k}, {\alpha'}_{k}}
\prod_{k=1}^{m-1}\delta_{\alpha_{k}, {\alpha''}_{k}}\nonumber\\
&&{\bf {\cal A}^{elem}_{free}}=\sum_{\beta_1,\cdots,\beta_n}
\Intri{\beta_n}{\alpha_{n}}{x_{n}}{\beta_1}{\alpha_1}{x_1}
\Outtri{{\beta'}_n}{{\alpha'}_{n}}{{x'}_{n}}{{\beta'}_1}{{\alpha'}_1}{{x'}_1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\prod_{i=1}^{n}\delta_{\beta_i,{\beta'}_i}\prod_{i=1}^{n}\delta_{\alpha_i,{\alpha'}_i}\\
&&{\bf {\cal A}^{elem}_{creation}}=\sum_{{\beta'}_1,\cdots,{\beta'}_n}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Incross{\beta_n}{\alpha_{k+2}}{x_{k+2}}{\beta_k}{\alpha_{k-1}}{x_{k-1}}{\alpha_1}{x_1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Outtri{{\beta'}_n}{{\alpha'}_{n}}{{x'}_{n}}{{\beta'}_1}{{\alpha'}_1}{{x'}_1}\times\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&&\times\prod_{i=k+2}^{n}(\delta_{\beta_i,{\beta'}_i}\delta_{\alpha_i,{\alpha'}_i})
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\delta_{{\beta'}_{k+1},\beta_k,{\beta'}_{k-1}}\delta_{{\alpha'}_{k+1},{\bar{\alpha'}}_{k}}
(v_{{\beta'}_k} v_{{\beta}_k})^{{1\over 2}}
(\frac{[d_{{\beta'}_k}]}{[d_{{\beta}_k}]})^{{1\over 2}}
N^{{\beta'}_k,{m'}_k}_{{\beta'}_{k-1}\alpha_k} \prod_{i=1}^{k-1}
\delta_{\alpha_i,{\alpha'}_i}
\prod_{i=1}^{k-2} \delta_{\beta_i,{\beta'}_i}\nonumber\\ &&{\bf {\cal
A}^{elem}_{annihil.}}
=\sum_{{\beta'}_1,\cdots,{\beta'}_n}
\Intri{{\beta'}_n}{{\alpha'}_{n}}{{x'}_{n}}{{\beta'}_1}{{\alpha'}_1}{{x'}_1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Outcross{\beta_n}{\alpha_{k+2}}{x_{k+2}}{\beta_k}{\alpha_{k-1}}{x_{k-1}}{\alpha_1}{x_1} \times\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&&\times\prod_{i=k+2}^{n}(\delta_{\beta_i,{\beta'}_i}\delta_{\alpha_i,{\alpha'}_i})
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\delta_{{\beta'}_{k+1},\beta_k,{\beta'}_{k-1}}\delta_{{\alpha'}_{k+1},{\bar{\alpha'}}_{k}}
(v_{{\beta'}_k} v_{{\beta}_k})^{-{1\over 2}}
(\frac{[d_{{\beta'}_k}]}{[d_{{\beta}_k}]})^{{1\over 2}}
N^{{\beta'}_k,{m'}_k}_{{\beta'}_{k-1}\alpha_k} \prod_{i=1}^{k-1}
\delta_{\alpha_i,{\alpha'}_i} \prod_{i=1}^{k-2}
\delta_{\beta_i,{\beta'}_i}\nonumber\\ &&{\bf {\cal
A}^{elem}_{overcross.}}=\sum_{\beta_1,\cdots,\beta_n,{\beta'}_k}
\Incross{\beta_n}{\alpha_{k+1}}{x_{k+1}}{\beta_k}{\alpha_k}{x_k}{\alpha_1}{x_1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Outcross{{\beta'}_n}{{\alpha'}_{k+1}}{{x'}_{k+1}}{{\beta'}_k}{{\alpha'}_k}{{x'}_k}{{\alpha'}_1}{{x'}_1}\times\nonumber\\
&&\times\prod_{i\not=
k}\delta_{\beta_i,{\beta'}_i}\prod_{i\not=k,k+1}\delta_{\alpha_i,{\alpha'}_i}
\frac
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
{tr_q(\psi_{\beta_{k-1}\alpha_{k-1}}^{\beta_{k-2}}\psi_{\beta_{k}\alpha_{k}}^{\beta_{k-1}}\Rtff{\alpha_k}{\alpha_{k-1}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\phi^{\beta_{k}\alpha_{k-1}}_{{\beta'}_{k-1}}\phi^{{\beta'}_{k-1}\alpha_{k}}_{\beta_{k-2}})}{[d_{\beta_{k-2}}]v_{{\beta}_k}^{{1\over
2}}v_{{\beta'}_k}^{-{1\over 2}}}
\delta_{\alpha_{k+1},{\alpha'}_k}\delta_{\alpha_{k},{\alpha'}_{k+1}}\\ &&{\bf
{\cal
A}^{elem}_{undercross.}}=\sum_{\beta_1,\cdots,\beta_n,{\beta'}_k}
\Incross{\beta_n}{\alpha_{k+1}}{x_{k+1}}{\beta_k}{\alpha_k}{x_k}{\alpha_1}{x_1}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\Outcross{{\beta'}_n}{{\alpha'}_{k+1}}{{x'}_{k+1}}{{\beta'}_k}{{\alpha'}_k}{{x'}_k}{{\alpha'}_1}{{x'}_1}\times\nonumber\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&&\times\prod_{i\not=k}\delta_{\beta_i,{\beta'}_i}\prod_{i\not=k,k+1}\delta_{\alpha_i,{\alpha'}_i}
\frac
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
{tr_q(\psi_{\beta_{k-1}\alpha_{k-1}}^{\beta_{k-2}}\psi_{\beta_{k}\alpha_{k}}^{\beta_{k-1}}\Rtmff{\alpha_k}{\alpha_{k-1}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\phi^{\beta_{k}\alpha_{k-1}}_{{\beta'}_{k-1}}\phi^{{\beta'}_{k-1}\alpha_{k}}_{\beta_{k-2}})}{[d_{\beta_{k-2}}]v_{{\beta}_k}^{{1\over
2}}v_{{\beta'}_k}^{-{1\over 2}}}
\delta_{\alpha_{k+1},{\alpha'}_k}\delta_{\alpha_{k},{\alpha'}_{k+1}}\end{aligned}$$
\
The result for ${\cal A}^{elem}_{cup}$ and ${\cal A}^{elem}_{cap}$ is clearly given by the Boltzmann weight . The computation of the other elements need a careful description. The idea is very simple. We first absorb each link segment in the attached boltzmann weight to put each elements associated to “square plaquettes” in a same practical form where all edges of the boundary appear one and only one time and always in the same order. This form allows us to reduce the gluing of “plaquettes” to one commutation plus one integration only.\
In the case of an empty square plaquette the corresponding Boltzmann weight is already in the reduced form.\
For example the square plaquette element involved in the computation of a free propagation is given by: &&\_[\[[x’]{}\_n,[x]{}\_[n]{},[x]{}\_[n-1]{},[x’]{}\_[n-1]{}\]]{}\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}= \_[\_[n-1]{}[’]{}\_[n-2]{}]{}\[d\_[\_[n-1]{}]{}\]\_[\_[n-1]{} \_[n-1]{}\_[n-2]{}]{}\^[-1]{}\
&&tr\_[V\_[\_[n-1]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{} \_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\^[\_[n-1]{} \_[n-1]{}]{}\_[[’]{}\_[n-2]{}]{}\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}\_[\_[n-1]{} \_[n-1]{}]{}\^[[’]{}\_[n-2]{}]{} \_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})where the notations are summarized on the following figure:
In the case of a creation element for example, with a similar computation and the same notations, we have : &&\_[\[[x’]{}\_n,[x]{}\_[n]{},[x]{}\_[n-1]{},[x’]{}\_[n-1]{}\]]{}\_[\[[x’]{}\_[n]{},[x’]{}\_[n-1]{}\]]{}\^[\#]{}=\
&&=\_[\_n,[’]{}\_n]{} (\[d\_[\_n]{}\]\[d\_[[’]{}\_n]{}\])\^[1 2]{} N\^[[’]{}\_n \_n]{}\_[\_n]{}tr\_[V\_[\_n]{}]{}( \_[\[[x’]{}\_n,x\_n\]]{}\_[\[x\_n,x\_[n-1]{}\]]{}\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}\
&&\_[[’]{}\_n \_n]{}\^[\_n]{} \_[\[[x’]{}\_n,[x’]{}\_[n-1]{}\]]{}\_[\_n |[\_n]{}]{}\^[[’]{}\_n]{}\_[\_n |[\_n]{} [’]{}\_n]{})
When the square contains a crossing the reduction is less obvious and is given by the following lemma: With the notations of the figure:\
it was shown in our last work [@BR2] that: &&dh(U\_[\[x\_[n-1]{},y\]]{})dh(U\_[\[y,[x’]{}\_n\]]{})dh(U\_[\[x\_[n]{},y\]]{})dh(U\_[\[y,[x’]{}\_[n-1]{}\]]{})\
&&\_[\[y x\_n x\_[n-1]{}\]]{}\_[\[y x\_[n-1]{} [x’]{}\_[n-1]{}\]]{}\_[\[y [x’]{}\_[n-1]{} [x’]{}\_[n]{}\]]{} \_[\[y [x’]{}\_n x\_[n]{}\]]{}\_[\[x\_n y [x’]{}\_[n-1]{}\]]{} \_[\[x\_[n-1]{} y [x’]{}\_n\]]{}=\
&&=\_[\_n]{}\[d\_[\_n]{}\]()\^[1 2]{}[tr\_q(\_[\_[k-1]{}\_[k-1]{}]{}\^[\_[k-2]{}]{}\_[\_[k]{}\_[k]{}]{}\^[\_[k-1]{}]{} \^[\_[k]{}\_[k-1]{}]{}\_[[’]{}\_[k-1]{}]{} \^[[’]{}\_[k-1]{}\_[k]{}]{}\_[\_[k-2]{}]{})]{}[\[d\_[\_[k-2]{}]{}\]]{} tr\_[V\_[\_n]{}]{}( \_[\[[x’]{}\_[n]{} [x]{}\_[n]{}\]]{}\
&&\^[\_n \_n]{}\_[\_[n-1]{}]{}\_[\[[x]{}\_[n]{} [x]{}\_[n-1]{}\]]{} \^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\_[\[[x]{}\_[n-1]{} [x’]{}\_[n-1]{}\]]{} \^[\_[n-2]{}]{}\_[[’]{}\_[n-1]{} \_n]{}\_[\[[x’]{}\_[n-1]{} [x’]{}\_n\]]{}\_[\_[n-1]{}\_n ]{}\^[[’]{}\_[n-1]{}]{}) and the analog relation for the undercrossing.
Now all square plaquettes elements are in the reduced form. Then the problem of computing the whole elementary block element is reduced to the gluing of elements associated to each of the square plaquettes given in the reduced form, i.e. a commutation $+$ an integration. Now, let us give a careful computation in the case of ${\cal A}_{free}^{elem}$ and the other ones, very similar to this one, will be led to the reader.
The notations are summarized on the figure:
([**Remark:**]{} In the following computation we forget again the multiplicities of representations in all decompositions, but we must take care of them...) Let us first describe the gluing of two square plaquettes:
&&dh(U\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}) \_[\[[x’]{}\_n,[x]{}\_[n]{},[x]{}\_[n-1]{},[x’]{}\_[n-1]{}\]]{}\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{} \_[\[[x’]{}\_[n-1]{},[x]{}\_[n-1]{},[x]{}\_[n-2]{},[x’]{}\_[n-2]{}\]]{}\_[\[x\_[n-2]{},[x’]{}\_[n-2]{}\]]{}=\
&&=dh(U\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}) \_[\_[n-1]{},\_[n-2]{},[’]{}\_[n-2]{},[’]{}\_[n-3]{}]{}\[d\_[\_[n-1]{}]{}\]\[d\_[\_[n-2]{}]{}\]\_[\_[n-1]{} \_[n-1]{}\_[n-2]{}]{}\^[-1]{}\_[\_[n-2]{} \_[n-2]{}\_[n-3]{}]{}\^[-1]{}\
&&tr\_[V\_[\_[n-1]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\^[\_[n-1]{} \_[n-1]{}]{}\_[[’]{}\_[n-2]{}]{}\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}\_[\_[n-1]{} \_[n-1]{}]{}\^[[’]{}\_[n-2]{}]{} \_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})\
&&tr\_[V\_[\_[n-2]{}]{}]{}(\_[\[[x’]{}\_[n-1]{},[x]{}\_[n-1]{}\]]{}\_[\[x\_[n-1]{},[x]{}\_[n-2]{}\]]{}\^[\_[n-2]{} \_[n-2]{}]{}\_[[’]{}\_[n-3]{}]{}\_[\[x\_[n-2]{},[x’]{}\_[n-2]{}\]]{}\_[\_[n-2]{} \_[n-2]{}]{}\^[[’]{}\_[n-3]{}]{}\_[\[[x’]{}\_[n-2]{},[x’]{}\_[n-1]{}\]]{})=\
&&=\_[(i),(j)]{}dh(U\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{}) \_[\_[n-1]{},\_[n-2]{},[’]{}\_[n-2]{},[’]{}\_[n-3]{}]{}\[d\_[\_[n-1]{}]{}\]\[d\_[\_[n-2]{}]{}\]\_[\_[n-1]{} \_[n-1]{}\_[n-2]{}]{}\^[-1]{}\_[\_[n-2]{} \_[n-2]{}[’]{}\_[n-3]{}]{}\^[-1]{}\
&&tr\_[V\_[\_[n-1]{}]{}V\_[\_[n-2]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\^[\_[n-1]{} \_[n-1]{}]{}\_[[’]{}\_[n-2]{}]{}a\_[(i)]{}\^\^[-1]{}\
&&\_[\[x\_[n-1]{},[x’]{}\_[n-1]{}\]]{} \_[\[[x’]{}\_[n-1]{},[x]{}\_[n-1]{}\]]{}\_[\[x\_[n-1]{},[x]{}\_[n-2]{}\]]{} \^[\_[n-2]{} \_[n-2]{}]{}\_[[’]{}\_[n-3]{}]{}\_[\[x\_[n-2]{},[x’]{}\_[n-2]{}\]]{}\_[\_[n-2]{} \_[n-2]{}]{}\^[[’]{}\_[n-3]{}]{}\
&&\_[\[[x’]{}\_[n-2]{},[x’]{}\_[n-1]{}\]]{}S(a\_[(j)]{}\^[\_[n-2]{}]{}) \_[\_[n-1]{} \_[n-1]{}]{}\^[[’]{}\_[n-2]{}]{} b\_[(i)]{}\^[\_[n-1]{}]{}b\_[(j)]{}\^[\_[n-1]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})=\
&&=\_[(i),(j)]{}\_[\_[n-1]{},\_[n-2]{},[’]{}\_[n-2]{},[’]{}\_[n-3]{}]{}\[d\_[\_[n-1]{}]{}\]\[d\_[\_[n-2]{}]{}\]\_[\_[n-1]{} \_[n-1]{}\_[n-2]{}]{}\^[-1]{}\_[\_[n-2]{} \_[n-2]{}[’]{}\_[n-3]{}]{}\^[-1]{}\
&&tr\_[V\_[\_[n-1]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\_[\[x\_[n-1]{},[x]{}\_[n-2]{}\]]{} \^[\_[n-2]{} \_[n-2]{}]{}\_[[’]{}\_[n-3]{}]{}\_[\[x\_[n-2]{},[x’]{}\_[n-2]{}\]]{}\
&&\_[\_[n-2]{} \_[n-2]{}]{}\^[[’]{}\_[n-3]{}]{}\_[\[[x’]{}\_[n-2]{},[x’]{}\_[n-1]{}\]]{}S(a\_[(j)]{}\^[\_[n-2]{}]{}) S\^2(a\_[(i)]{}\^) \_[\_[n-1]{} \_[n-1]{}]{}\^[\_[n-2]{}]{} b\_[(i)]{}\^[\_[n-1]{}]{}b\_[(j)]{}\^[\_[n-1]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})=\
&&=\_[\_[n-1]{},\_[n-2]{},[’]{}\_[n-3]{}]{}\[d\_[\_[n-1]{}]{}\]\_[\_[n -1]{}\_[n-1]{}\_[n-2]{}]{}\^[-1]{}\_[\_[n-2]{} \_[n-2]{}[’]{}\_[n-3]{}]{}\^[-1]{} tr\_[V\_[\_[n-1]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\
&&\^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\_[\[x\_[n-1]{},[x]{}\_[n-2]{}\]]{} \^[\_[n-2]{} \_[n-2]{}]{}\_[[’]{}\_[n-3]{}]{}\_[\[x\_[n-2]{},[x’]{}\_[n-2]{}\]]{}\_[\_[n-2]{} \_[n-2]{}]{}\^[[’]{}\_[n-3]{}]{}\_[\[[x’]{}\_[n-2]{},[x’]{}\_[n-1]{}\]]{}\
&&\_[\_[n-1]{} \_[n-1]{}]{}\^[\_[n-2]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})\
In the same way we can glue $n-1$ Boltzmann weights and links. As a result we have obviously: &&\_[i=1]{}\^[n-1]{} dh(U\_[\[x\_[i]{},[x’]{}\_[i]{}\]]{}) \_[i=1]{}\^[n-1]{}(\_[\[[x’]{}\_[i+1]{},[x]{}\_[i+1]{},[x]{}\_[i]{},[x’]{}\_[i]{}\]]{}\_[\[x\_[i]{},[x’]{}\_[i]{}\]]{})=\
&&=\_[\_[n-1]{},,\_[2]{},\_[1]{}]{}\[d\_[\_[n-1]{}]{}\]\_[i=2]{}\^[n-1]{}\_[\_i \_[i]{}\_[i-1]{}]{}\^[-1]{}\_[\_[1]{} \_[1]{}[’]{}\_[n]{}]{}\^[-1]{} tr\_[V\_[\_[n-1]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\
&&\^[\_[1]{} \_[1]{}]{}\_[[’]{}\_[n]{}]{}\_[\[x\_[1]{},[x’]{}\_[1]{}\]]{}\_[\_[1]{} \_[1]{}]{}\^[[’]{}\_[n]{}]{}\_[\_[n-1]{} \_[n-1]{}]{}\^[\_[n-2]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})\
The computation of ${\cal A}^{elem}_{free}$ can be achieved by the gluing of the last Boltzmann weight to obtain the cylinder with $n$ strands.This Boltzmann weight must be changed to the square plaquette element corresponding to crossings,... in the computations of the other blocks elements. &&\_[i=1]{}\^[n]{} dh(U\_[\[x\_[i]{},[x’]{}\_[i]{}\]]{}) \_[i=1]{}\^[n]{}(\_[\[[x’]{}\_[i+1]{},[x]{}\_[i+1]{},[x]{}\_[i]{},[x’]{}\_[i]{}\]]{}\_[\[x\_[i]{},[x’]{}\_[i]{}\]]{})=\
&&=dh(U\_[\[x\_[n]{},[x’]{}\_[n]{}\]]{})dh(U\_[\[x\_[1]{},[x’]{}\_[1]{}\]]{})\
&&(\_[\_n,[’]{}\_[n-1]{}]{}\[d\_[\_n]{}\]\_[\_n \_n [’]{}\_[n-1]{}]{}\^[-1]{}tr\_[V\_[\_n]{}]{}(\_[\[x\_1,x\_n\]]{} \^[\_n \_n]{}\_[[’]{}\_[n-1]{}]{}\_[\[x\_n,[x’]{}\_n\]]{}\_[\_n \_n]{}\^[[’]{}\_[n-1]{}]{}\_[\[[x’]{}\_n,[x’]{}\_1\]]{}\_[\[[x’]{}\_1,[x]{}\_1\]]{}))\
&&(\_[\_[n-1]{},,\_[2]{},\_[1]{}]{}\[d\_[\_[n-1]{}]{}\]\_[i=2]{}\^[n-1]{}\_[\_i \_[i]{}\_[i-1]{}]{}\^[-1]{}\_[\_[1]{} \_[1]{}[’]{}\_[n]{}]{}\^[-1]{} tr\_[V\_[\_[n-1]{}]{}]{}(\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{} \^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\
&&\^[\_[1]{} \_[1]{}]{}\_[[’]{}\_[n]{}]{}\_[\[x\_[1]{},[x’]{}\_[1]{}\]]{}\_[\_[1]{} \_[1]{}]{}\^[[’]{}\_[n]{}]{}\_[\_[n-1]{} \_[n-1]{}]{}\^[\_[n-2]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{}))=\
&&=dh(U\_[\[x\_[n]{},[x’]{}\_[n]{}\]]{})dh(U\_[\[x\_[1]{},[x’]{}\_[1]{}\]]{}) \_[[’]{}\_[n-1]{},\_n,,\_[2]{},[’]{}\_[1]{}]{}\[d\_[\_[n-1]{}]{}\]\[d\_[\_n]{}\] \_[i=2]{}\^[n-1]{}\_[\_i \_[i]{}\_[i-1]{}]{}\^[-1]{}\_[\_[1]{} \_[1]{}[’]{}\_[n]{}]{}\^[-1]{} \_[\_n \_n [’]{}\_[n-1]{}]{}\^[-1]{}\
&&tr\_[V\_[\_n]{} V\_[\_[n-1]{}]{}]{}(\_[\[x\_1,x\_n\]]{} \^[\_n \_n]{}\_[[’]{}\_[n-1]{}]{} a\_[(j)]{}\^[\_[n-1]{}]{} \^[-1]{} \_[\[x\_n,[x’]{}\_n\]]{}\_[\[[x’]{}\_[n]{},[x]{}\_[n]{}\]]{} \_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\
&&\^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\_[\[x\_[1]{},[x]{}\_[n]{}\]]{}b\_[(i)]{}\^[\_1]{}\^[\_[1]{} \_[1]{}]{}\_[[’]{}\_[n]{}]{} \_[\_n \_n]{}\^[[’]{}\_[n-1]{}]{}b\_[(j)]{}\^[\_n]{}\_[\[[x’]{}\_n,[x’]{}\_1\]]{} \_[\[[x’]{}\_1,[x]{}\_1\]]{}\_[\[x\_[1]{},[x’]{}\_[1]{}\]]{}S(a\_[(i)]{}\^[[’]{}\_[n]{}]{}) \_[\_[1]{} \_[1]{}]{}\^[[’]{}\_[n]{}]{}\
&&\_[\_[n-1]{} \_[n-1]{}]{}\^[\_[n-2]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{}))=\
&&=\_[\_n,,\_[1]{}]{}\[d\_[\_n]{}\] \_[i=1]{}\^[n]{}v\_[\_[i]{}]{}\^[-[1 2]{}]{}tr\_[V\_[\_n]{}]{}(\_[\[x\_1,x\_n\]]{} \^[\_n \_n]{}\_[\_[n-1]{}]{}\_[\[x\_[n]{},[x]{}\_[n-1]{}\]]{}\^[\_[n-1]{} \_[n-1]{}]{}\_[\_[n-2]{}]{}\_[\[x\_[1]{},[x]{}\_[n]{}\]]{}b\_[(i)]{}\^[\_1]{}\^[\_[1]{} \_[1]{}]{}\_[\_[n]{}]{}\
&&\^[-1]{}S(a\_[(i)]{}\^[\_[n]{}]{})) tr\_[V\_[\_[n-1]{}]{}]{}(S\^2(a\_[(j)]{}\^[\_[n-1]{}]{})\_[\_n \_n]{}\^[\_[n-1]{}]{} b\_[(j)]{}\^[\_n]{} \_[\[[x’]{}\_n,[x’]{}\_1\]]{} \_[\_[1]{} \_[1]{}]{}\^[[’]{}\_[n]{}]{}\_[\_[n-1]{} \_[n-1]{}]{}\^[\_[n-2]{}]{}\_[\[[x’]{}\_[n-1]{},[x’]{}\_[n]{}\]]{})=\
&&=\_[\_1,m\_1,,\_n,m\_n]{}\
This concludes the computation of ${\cal
A}^{elem}_{overcross}$, ${\cal A}^{elem}_{undercross}$, ${\cal A}^{elem}_{creation}$, ${\cal
A}^{elem}_{annihil}$, and ${\cal A}^{elem}_{free}.$\
The computations of ${\cal A}^{elem}_{(n,m)(n+m)
tri.}$ and ${\cal
A}^{elem}_{(n,m)(n+m) tri.}$ need one more step. It uses naturally the expression of ${\cal
A}^{elem}_{free}$ as a basic object. Indeed we compute the element associated to the trinion by gluing one more plaquette to the cylinder with $n$ strands as it is shown in the following figure.
the computation is realized by the usual techniques:\
&&[A]{}\_[(n,m)(n+m) tri.]{}=dh(U\_[\[x\_1,x\_[m+n]{}\]]{})dh(U\_[\[x\_[m+1]{},x\_[m]{}\]]{}) \_[\[x\_1,x\_m,x\_[m+1]{},x\_[m+n]{}\]]{}\
&&\_[\_1,,\_[m+n]{}]{} =\
&&dh(U\_[\[x\_1,x\_[m+n]{}\]]{})dh(U\_[\[x\_[m+1]{},x\_[m]{}\]]{}) \_[[’]{}\_[m+n]{}]{}\[d\_[[’]{}\_[m+n]{}]{}\]v\_[[’]{}\_[m+n]{}]{}\^[-1]{} tr\_[V\_[[’]{}\_[m+n]{}]{}]{}(\_[\[x\_[m+n]{},x\_1\]]{} \_[\[x\_[1]{},x\_m\]]{}\
&&\_[\[x\_[m]{},x\_[m+1]{}\]]{} \_[\[x\_[m+1]{},x\_[m+n]{}\]]{}) \_[\_[m+n]{},,\_[1]{}]{} tr\_[V\_[\_[n+m]{}]{}]{}( \_[\[x\_1,x\_[n+m]{}\]]{}\^[\_[m+n]{}\_[m+n]{}]{}\_[\_[m+n-1]{}]{}\
&&\_[\[x\_[2]{},[x]{}\_[1]{}\]]{}\^[\_[1]{} \_[1]{}]{}\_[\_[m+n]{}]{}) =\
&&=dh(U\_[\[x\_1,x\_[m+n]{}\]]{})dh(U\_[\[x\_[m+1]{},x\_[m]{}\]]{}) \_[[’]{}\_[m+n]{},\_[m+n]{},,\_[1]{}]{} \[d\_[[’]{}\_[m+n]{}]{}\]v\_[[’]{}\_[m+n]{}]{}\^[-1]{}\
&&tr\_[V\_[[’]{}\_[m+n]{}]{}V\_[\_[n+m]{}]{}]{} ( S\^[-1]{}(b\^[\_[m+n]{}]{}\_[(j)]{})\^[-1]{} (\_[\[x\_[m+n]{},x\_1\]]{})\_[\[x\_1,x\_[n+m]{}\]]{})\
&&a\_[(j)]{}\^[[’]{}\_[n+m]{}]{}a\_[(i)]{}\^[\_[n+m]{}]{} \_[\[x\_[1]{},x\_m\]]{} \^[\_[m+n]{}\_[m+1]{}]{}\_[\_[m+n-1]{}]{}a\_[(l)]{}\^[\_[n+m-1]{}]{}\_[\_[m]{}]{}\^[\_[m+1]{}\_[m+1]{}]{} S\^[-1]{}(b\^[\_[m]{}]{}\_[(k)]{})\^[-1]{}\
&&(\_[\[x\_[m]{},x\_[m+1]{}\]]{}\_[\[x\_[m+1]{},x\_[m]{}\]]{}) a\_[(k)]{}\^[[’]{}\_[n+m]{}]{}\_[\[x\_[m+1]{},x\_[m+n]{}\]]{}b\_[(i)]{}\^[[’]{}\_[m+n]{}]{} S(b\^[[’]{}\_[m+n]{}]{}\_[(l)]{}) \_[\_[m-1]{}]{}\^[\_m \_m]{}\
&&\_[\[x\_[2]{},[x]{}\_[1]{}\]]{}\^[\_[1]{} \_[1]{}]{}\_[\_[m+n]{}]{}) =\
&&=\_[[’]{}\_[m+n]{},\_[m+n]{},,\_[1]{}]{}v\_[\_m]{} \[d\_[\_[m+n]{}]{}\]\^[-1]{} \_[\_[m+n]{},\_m,[’]{}\_[m+n]{}]{}\
&&tr\_[V\_[\_[m]{}]{}]{} ( \_[\[x\_[1]{},x\_m\]]{} \^[\_[m]{}\_[m]{}]{}\_[\_[m-1]{}]{}\_[\[x\_[2]{},x\_1\]]{} \_[\_[m]{}]{}\^[\_[1]{}\_[1]{}]{})\
&&tr\_[V\_[\_[m+n]{}]{}]{}(a\_[(i)]{}\^[\_[n+m]{}]{} \_[\_[m+n-1]{}]{}\^[\_[m+n]{} \_[m+n]{}]{}a\_[(l)]{}\^[\_[n+m-1]{}]{}\^[\_[m+1]{} \_[m+1]{}]{}\_[\_[m+n]{}]{}\_[\[x\_[m+1]{},x\_[m+n]{}\]]{} S(b\^[\_[m+n]{}]{}\_[(l)]{})b\_[(i)]{}\^[\_[m+n]{}]{})\
&&=\
&&= \_[\_1,, \_[n+m]{}]{} \[d\_[\_[n+m]{}]{}\]\^[-1]{}\
&& \_[[’]{}\_n, [”]{}\_m, \_[n+m]{}, \_m]{} \_[k=1]{}\^[n]{}\_[\_[m+k]{}, [’]{}\_[k]{}]{} \_[k=1]{}\^[m]{} \_[\_[k]{}, [”]{}\_[k]{}]{} \_[k=1]{}\^[n-1]{}\_[\_[m+k]{}, [’]{}\_[k]{}]{} \_[k=1]{}\^[m-1]{} \_[\_[k]{}, [”]{}\_[k]{}]{}\
&&=\_[\_1,, \_[n+m]{}]{} \[d\_[\_[n+m]{}]{}\]\^[-1]{}\
&& \_[[’]{}\_n, [”]{}\_m, \_[n+m]{}, \_m]{} \_[k=1]{}\^[n]{}\_[\_[m+k]{}, [’]{}\_[k]{}]{} \_[k=1]{}\^[m]{} \_[\_[k]{}, [”]{}\_[k]{}]{} \_[k=1]{}\^[n-1]{}\_[\_[m+k]{}, [’]{}\_[k]{}]{} \_[k=1]{}\^[m-1]{}\_[\_[k]{}, [”]{}\_[k]{}]{}\
This ends the proof of the Theorem.\
These results can be easily generalized to the case where $q$ is a root of unity. It suffices to realize the following replacements:\
The expression for the In state becomes $$\begin{aligned}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
&&\Instate{\beta_n}{\alpha_n}{x_n}{\beta_{n-1}}{\alpha_{n-1}}{x_{n-1}}{\beta_1}{\alpha_1}{x_1}=\\
&&=v_{\beta_n}tr_{V_{\beta_n}}(S(A)\muf{\beta_n}
{\buildrel {\alpha_1} \over {\theta^{(1)}_l}}
{\buildrel {\beta_n} \over {\theta^{(2)}_l}}
\Rff{\beta_n}{\alpha_1}
{\buildrel {\beta_n} \over {\theta^{-1 (1)}_k}}
{\buildrel {\alpha_1} \over {\theta^{-1 (2)}_k}}
\guf{\beta_n}_{[x_1,x_{n}]}
S({\buildrel {\beta_n} \over {\theta^{(1)}_m}}){\buildrel {\beta_n} \over {A}}
{\buildrel {\beta_n} \over {\theta^{(2)}_m}}\otimes {\buildrel {\alpha_n} \over
{\theta^{(3)}_m}}
\phi^{\beta_n \alpha_{n}}_{\beta_{n-1}} \cdots\\ &&\cdots
\phi^{\beta_2 \alpha_2}_{\beta_1}\guf{\beta_1}_{[x_2,x_{1}]} S({\buildrel
{\beta_1} \over
{\theta^{(1)}_i}}) S({\buildrel {\beta_1} \over {\theta^{(1)}_j}}){\buildrel
{\beta_1} \over
{A}}{\buildrel {\beta_1} \over {\theta^{(2)}_j}}\otimes {\buildrel {\alpha_1}
\over
{\theta^{(3)}_j}} \phi^{\beta_1 \alpha_1}_{\beta_n} {\buildrel {\beta_n} \over
{\theta^{(2)}_i}}
{\buildrel {\beta_n} \over {B}}
S({\buildrel {\beta_n} \over {\theta^{(3)}_i}})
S({\buildrel {\beta_n} \over {\theta^{-1 (3)}_k}})
S({\buildrel {\beta_n} \over {\theta^{(3)}_l}}))\end{aligned}$$ and the analog formula for the Out state. The $6-j$ associated to the crossing is changed to obtain an intertwiner.\
The other objects remain unchanged and the summations are restricted to physical representations only.\
We have the following corollary: If $L$ is a link without boundaries in $D\times[0,1]$ we have for any value of $q$: = RT\_[[U]{}\_q([G]{})]{}(L) where $RT$ is the Reshetikhin-Turaev’s quantum invariant of coloured links. This result has already been shown in [@BR2]. and generally if $\Sigma$ is a closed surface, the invariant associated to $\Sigma+L$ is simply a generalization to the case of a surface of the Reshetikhin-Kirillov invariant in the shadow world [@KR]. This theorem can be considered as a proof of the equivalence of Invariants arising from Chern-Simons theory and Reshetikhin-Turaev quantum invariants.
We can also generalize our invariant to admit other objects called “coupons” defined to be respectively represented on the following figure:
and which expressions are given by: \_[(n+k)(n)coupon]{}=\_Shadow(coupon)\
[A]{}\_[(n)(n+k)coupon]{}=\_Shadow(coupon) with $Shadow(coupon)$ being defined as usual by the following rule:
A new description of invariants of three manifolds
==================================================
In this chapter $q$ will be a root of unity.
Heegaard splitting and surgery of 3-manifolds
---------------------------------------------
In the following ${\cal M}$ is a compact orientable 3-manifold given by a simplicial complex $K$.Let us recall standard definitions that can be found in [@Si]. A [**canonical region**]{} ${\cal R}$ of ${\cal M}$ is a region within which there are p non intersecting 2-cells $(E_i)_{i=1 \cdots p}$(the [**canonical cells**]{}) with boundaries $e_i$ (the [**canonical curves**]{}) on the boundary ${\cal L}$ of ${\cal R}$ such that we obtain a 3-cell by cutting ${\cal R}$ at each $(E_i).$ A surface ${\cal L}$ is said to be a [**canonical surface**]{} of a 3-manifold ${\cal M}$ if it satisfies these conditions:
- ${\cal L}$ is a subcomplex of ${\cal M}$ and is a compact, connected 2-dimensional manifold
- ${\cal M}= {\cal R}_1 + {\cal L} + {\cal R}_2$ with ${\cal R}_1,{\cal
R}_2$ canonical regions and ${\cal L}=\partial {\cal R}_1 = \partial {\cal R}_2$
such a decomposition is called [**canonical decomposition**]{}. It is important to recall that, if $g$ is the genus of ${\cal L}$, in this case, ${\cal R}_1$ and ${\cal R}_2$ are homeomorphic to a genus $g$ handlebody.
[**Remark:** ]{} It is easy to give, for each ${\cal M}$, at least one canonical decomposition. To this aim, let us consider $\{A_0^i\}$, $\{A_1^j\}$, $\{A_2^k\}$, $\{A_3^l\}$ the sets of $0-,1-,2-,3-$cells of $K$. Let $\{B_1^j\}$, $\{B_2^k\}$, $\{B_3^l\}$ be respectively the middle of $\{A_1^j\}$, $\{A_2^k\}$, $\{A_3^l\}$. The complex $K'$ obtained by adding the $B_{1}s$, the $B_{2}s$ and the $B_{3}s$ to the vertices of $K$ is called the [**first derived complex of K**]{}, its 3-simplexes are of the form $(A_{0}B_{1}B_{2}B_{3})$. The [**second derived complex of K**]{}, denoted $K"$ is the complex complex generated from $K'$ by adding the vertices $C_{1}s,C_{2}s,C_{3}s$ middle of the $1-,2-,3-$simplexes of $K'$. Let us denote by ${\cal R}_1$ the set of all $3-$simplexes of $K''$ of the type $(A_{0}C_{1}C_{2}C_{3})$ or $(B_{1}C_{1}C_{2}C_{3})$, by ${\cal R}_2$ the set of all $3-$simplexes of $K"$ of the type $(B_{2}C_{1}C_{2}C_{3})$ or $(B_{3}C_{1}C_{2}C_{3})$ and by ${\cal L}$ the common frontier of ${\cal R}_1$ and ${\cal R}_2$. If we call $G$ (resp.$G^{\star}$) the linear graph generated by the 1-simplexes of $K$ (resp. its dual) we can see that ${\cal
R}_1$ and ${\cal R}_2$ are respectively $K''-$neighbourhood of $G$ and $G^{\star}$. Then we have that ${\cal M}= {\cal R}_1 + {\cal L} + {\cal R}_2$ is a canonical decomposition, it is called the [**canonical decomposition derived from the triangulation**]{}. It is easy to check that for each canonical decomposition, there exits a triangulation of ${\cal M}$ such that the decomposition is in fact the canonical decomposition derived from the triangulation.\
A Heegaard splitting of a 3-manifold ${\cal M}$ is a set $(g,f)$ where $g$ is a non negative integer and $f$ is a diffeomorphism of a genus $g$ surface ${\cal L}_g$ such that ${\cal M}$ is the manifold obtained by gluing two copies of the handlebody ${\cal T}_g$ (the interior of ${\cal L}_g$) along their boundaries after having acted on one of them by f: $${\cal M} = {\cal T}_g \#_f {\cal T}_g$$ A Heegaard diagram is a set $({\cal L},(e_i)_{i=1\cdots g},(f_j)_{j=1 \cdots
g})$ where ${\cal L}$ is a compact connected 2-dimensional manifold of genus $g$ and $(e_i)_{i=1\cdots g}$ (resp.$(f_j)_{j=1 \cdots g})$) are canonical curves of ${\cal R}_1$, the region interior to ${\cal L}$ (resp. canonical curves of ${\cal R}_2$ the exterior of ${\cal L}$).This data is sufficient to reconstruct an element $f$ of $Diff({\cal L})$ such that $f(e_j)=f_j.$ Two Heegaard diagrams are said to be equivalent if they describe homeomorphic 3-manifolds. Let $({f'}_i)_{i=1...g}$ be $g$ other canonical curves in ${\cal L}$ such that $({\cal L},(e_i)_{i=1\cdots g},({f'}_j)_{j=1 \cdots g})$ is a Heegaard diagram of the sphere $S^3$ then $({\cal L},(e_i)_{i=1\cdots g},(f_j)_{j=1 \cdots
g},({f'}_j)_{j=1 \cdots g})$ is said to be an augmented Heegaard diagram. We must recall that any element of the moduli space of a surface can be written as the composition of Dehn twists. A Dehn twist can be described by the following replacement of a regular neighbourhood of the corresponding curve:
There is an important theorem due to Singer [@Si] describing the relation between equivalent Heegaard diagrams.
Let us describe a set of elementary moves on the Heegaard diagrams:
- replace a curve by another curve isotopic to it, or to its inverse, or reembedded the canonical surface ${\cal L}$ in a different way in $S^3$.\
[**type 1: solid handlebodies diffeomorphisms**]{}
- replace one canonical curve of the set $(e_i)_{i=1\cdots g}$ (resp. $(f_j)_{j=1\cdots g}$ ) by the composition of this curve with another one in this set.
- making a Dehn twist along one of the $e_{i}s$.\
[**type 2: $g \rightarrow g+1$ moves**]{}
- add a handle to ${\cal L}$ and define $e_{p+1}$ (resp. $f_{p+1}$) to be the $a-$cycle (resp. the $b-$cycle) of this handle, or erase a handle with cycles for which $e_{p+1}$ is the $a-$cycle (resp. $f_{p+1}$ is the $b-$cycle).
Then we have the following classification theorem [@Si]:
If the diagrams $D$ and $D'$, related by a finite number of Singer’s moves, give rise to the manifolds ${\cal M}$ and ${\cal M}'$ then ${\cal M}$ and ${\cal M}'$ are homeomorphic. Conversely, if $D$ and $D'$ are any two Heegaard diagrams whatsoever arising from a manifold ${\cal M}$ then $D$ and $D'$ are related by a finite number of Singer’s moves. A more generally used description of three manifolds is “the surgery presentation”. Let us recall some facts about this description [@Li]. Let $(R,r)=\cup_{i=1}^{n}(R_i,r_i)$ be a framed link in the oriented sphere $S^3.$ We can define a manifold ${\cal M}$ by “surgery” from $(R,r)$ using the following procedure:\
remove from $S^3$ pairwise disjoint tubular neighbourhoods $V_i$ of the curves $R_i$ and resew them identifying a meridian $z_i$ in $\partial V_i$ with a curve $y_i \in \partial(S^3\setminus V_i^{int})$ which links $R_i$ exactly $r_i$ times.\
Moreover, every 3-manifold ${\cal M}$ can be obtained from a certain framed link by this procedure [@Li]. It is relatively easy to relate the Heegaard and Surgery points of view [@NL]. Let us consider a Heegaard diagram based on a gluing diffeomorphism $f$ described in terms of Dehn twists of the surface. We first remark that splitting $S^3$ along ${\cal L}$ then doing a Dehn twist along a certain curve and resewing the handlebody is equivalent to do a surgery along the ribbon glued on the surface along this curve as it can be seen on the figure:
Let $(R_i)_{i=1..n}$ be a set of ribbons trivially embedded on the surface ${\cal L}$, $f_i$ the corresponding Dehn twists. We want also define the framed link $L$ defined to be the set of ribbons $(R_i \times \epsilon_i)_{i=1 \cdots
n}$ for $0 \le \epsilon_1 \le \cdots \le \epsilon_n \le 1.$ We consider a partition of $S^3$ in three pieces: ${\cal L}\times [0,1]$, the handlebody ${\cal H}_g$ interior to ${\cal L}\times \{0\}$ and the handlebody ${\cal H}^{'}_g$ exterior to ${\cal L}\times \{1\}.$ Let us consider the manifold ${\cal M}(f_1,f_2,\cdots,f_n)$ obtained by gluing the manifolds ${\cal
H}_g$, ${\cal H}^{'}_g$, $({\cal L}\times [\epsilon_{i-1},\epsilon_i])_{i=1
\cdots n}$ with the gluing diffeomorphisms $id, f_1, \cdots, f_n.$ Obviously the manifold ${\cal M}(f_1,f_2,\cdots,f_n)$ is the manifold defined by the Heegaard data $({\cal L}, f_n \circ f_{n-1} \circ \cdots \circ f_1)$, but it is also obvious that this manifold is that defined by the surgery data $R.$ We will say that these surgery and Heegaard presentation of the same manifold are “related” description of ${\cal M}.$
Invariants associated to Heegaard diagrams and Lattice q-gauge theory
---------------------------------------------------------------------
Our principal aim in this section is to prove the following theorem:
Let $({\cal T}_g, (x_j)_{j=1,...,g}, (y_j)_{j=1,...,g}, (z_j)_{j=1,...,g})$ be an augmented Heegaard diagram associated to a manifold ${\cal M}$ then the expectation value : \_[[M]{}]{}= is an invariant of the manifold ${\cal M}.$ Moreover this value is equal to the Reshetikhin-Turaev invariant associated to the manifold ${\cal M}.$ The normalization by the expectation value associated to the sphere is chosen to obtain an $3-$manifold invariant equal to $1$ for the sphere. [**Remark 1:**]{} The latter definition of the correlation function is in fact very natural from the general construction of q-gauge theory. Indeed putting a delta function associated to a plaquette $P$ corresponds to imposing that any ribbon, i.e. holonomy defined in terms of the gauge fields algebra, can be displaced through $P$ without torsion and without changing the expectation value. So, adding to the projector associated to the surface some delta functions corresponding to the $x_i$s, i.e. canonical curves of the interior handlebody, and, at a future time, the delta functions of the $y_i$s, i.e. canonical curves of the exterior handlebody, allows us to displace any curve through a handle of any of the two Heegaard components.This is exactly what we want to do in the framework of Chern-Simons theory.\
[**Remark 2:**]{} Using the properties &&()\^2=()\
&&()()=() ()we can replace easily the correlation function by one where we put all Lickorish generators rather than the canonical curves only. This fact will be useful in the next section.\
We are going to proove the last theorem through two lemmas describing some properties of this invariant.
The expectation value $<\prod_{i=1}^{g}\delta_{y_i}\prod_{i=1}^{g}\delta_{x_i}>_{q-YM({\cal T}_g)}$ associated to a Heegaard diagram $({\cal T}_g, (x_j)_{j=1,...,g},
(y_j)_{j=1,...,g})$ of a manifold ${\cal M}$ is invariant under any Singer’s move applied to the diagram.\
[**Trivial moves:**]{}\
it is a fact already established that the expectation value is invariant under any isotopic deformation of any curve in the surface, simply because of the flatness condition.\
We have the formula $\delta_C=\delta_{C^{-1}}$, which is exactly the second trivial move.\
[**Handlebodies Diffeomorphisms:**]{}\
The flatness condition implies trivially the following property for any curves $C_1,C_2$: \_[C\_1]{}\_[C\_2]{}=\_[C\_1]{}\_[C\_1 \# C\_2]{} moreover the $\delta_{e_i}$s (resp.$\delta_{f_i}$s ) commute one with the others. We then obtain the invariance under the first type 1 move.\
It is easy to see, with the expression of the block element ${\cal A}_{free}$, that we can do the following replacement along any curve $x_i$.:
which implies the invariance under the second type 1 move.\
[**$g \rightarrow g+1$ moves:**]{}\
Let us consider a Heegaard diagram with one handle with its $a-$ and $b-$cycles. We cut the surface along a certain $2-$cell to obtain a torus with a puncture on which are drawn the two cycles as in the following figure. We choose the minimal fat graph describing this object to describe the partial integration of the expectation value over the edges of the latter object.
we obtain easily: &&dh(U\_[\[y,t\]]{})dh(U\_[\[x,z\]]{}) \_[\[y,t,x,z,t,y,z,x\]]{}\_[\[x,y,z,x\]]{} \_[\[x,y,t,x\]]{} =\
&&= dh(U\_[\[y,t\]]{})dh(U\_[\[x,z\]]{}) \_[\[y,t,x,z,t\]]{}\_[\[x,y,z,x\]]{} \_[\[x,y,t,x\]]{}\
&&= dh(U\_[\[y,t\]]{})dh(U\_[\[x,z\]]{}) \_[\[t,x,z\]]{}\_[\[x,y,z,x\]]{} \_[\[x,y,t,x\]]{}\
&&= dh(U\_[\[y,t\]]{})dh(U\_[\[x,z\]]{}) \_[\[t,x,y,z\]]{}\_[\[x,y,z,x\]]{} \_[\[x,y,t,x\]]{}\
&&=\_[\[t,x,y,z\]]{}\
the last line is easily obtained by using the property that the integration just “pick” the zero component associated to a link. The latter result establishes the invariance under the type 2 Singer move. This ends the proof of the lemma and shows that the expectation value is an invariant of the manifold ${\cal M}.$
For any augmented Heegaard diagram $({\cal L},(x_i)_{i=1 \cdots g}, (y_i)_{i=1
\cdots g}, (z_i)_{i=1 \cdots g})$ describing a manifold ${\cal M}$ there exists a framed link $L$ which is a surgery data describing the same manifold ${\cal
M}$ and verifying: = RT([M]{}) where RT is the Reshetikhin Turaev invariant of the manifold computed from $L.$
The trick already used in the proof of the invariance under the second Singer move can be used also here. We first use a natural property of delta functions that can be described by the following figure :
\_[[C]{}\_2]{}\_[[C]{}\_1]{}=\_[[C’]{}\_2]{}\_[[C]{}\_1]{} to transform the correlation function in a new one\
$<(\sum_{\alpha_1,\cdots, \alpha_g}(\prod_{i}
[d_{\alpha_i}])W((R_i,\alpha_i)_{i=1\cdots
g})\prod_{i=1}^{g}\delta_{x_i}>_{q-YM({\cal L})}$ where the $R_i$s are ribbons glued on the surface with the same framings and knotted in $S^3$ in the same way as the $y_i$s but with a support now included in the area described in the following figure:
Using now the usual flatness property (\[flatness\]) we can deform again the latter knot to put its crossings in the “discs”, the rest of the knot being composed of parallel strands along handles zones.\
Then we are able to do the same calculus as in the verification of the invariance under type 2 Singer’s move. The integration “picks” again the zero component on each segment of the skeleton. We then obtain the equality: \_[q-YM([A]{})]{}=\_[\_1,, \_g]{}(\_[i]{} \[d\_[\_i]{}\])(\_[j]{} I\_[disc\_j]{}) with $I_{disc_j}$ being the invariant associated, by our construction, to the knot contained in the j-th disc, placed on the sphere $S^2$ and with four coupons picking the zero component on the boundary of the disc.\
It is then easy to see, using the equivalence already established in section (3) between our invariant on the sphere and the Reshetikhin invariant of link , that the quantity $I_{disc_j}$ is exactly the Reshetikhin invariant associated to this framed link with coupons.\
Now the proof can be achieved by establishing, using the Reshetikhin-Turaev framework, that the latter data is a surgery data of the manifold ${\cal M}.$ Let us first recall that, using the “related” surgery and Heegaard descriptions, we can replace the set of curves $y_i$ describing the manifold ${\cal M}$ by a link composed of the curves $z_i$ associated to the Heegaard description of $S^3$ placed at a time $t$ and the curves $R_i \times {t_i}$ ( with $t_i \le t$ ) associated to the composition of Dehn twists describing the Heegaard gluing diffeomorphism encoded in the $y_i$s. If we compute, with the notations of Reshetikhin and Turaev in [@RT2], the invariant associated to the framed link described before, with an insertion of two “coupons” for each handle picking the zero component, we obtain easily that this invariant is equal to the invariant associated to the link $L=\cup_i R_i \times {t_i}$ only. This property uses trivially the fact that : \_[,[’]{}\_1,[’]{}\_n]{}tr\_[V\_]{}(\^[\_n \_[n-1]{}]{}\_[[’]{}\_[n-1]{}]{} \^[\_2 \_1]{}\_[[’]{}\_1]{}\^[[’]{}\_1 ]{}\_[0]{} \_[[’]{}\_1 ]{}\^[0]{}\_[\_2 \_1]{}\^[[’]{}\_1]{}\_[\_n \_[n-1]{}]{}\^[[’]{}\_[n-1]{}]{})= id\_[V\_[\_1]{}]{} id\_[V\_[\_n]{}]{} We then obtain: =\_[()]{}(\_[i]{}\[d\_[\_i]{}\]) RT((R\_i,\_i)\_[i=1 n]{}) Using now the celebrated result of [@RT2] this object is a non trivial invariant of the 3-manifold ${\cal M},$ it is the Reshetikhin-Turaev’s invariant of three manifolds.\
Chern-Simons theory on a lattice and Three dimensional Lattice q-gauge theory
------------------------------------------------------------------------------
We will define here a three dimensional gauge theory which extends in some sense the previous construction on a surface. The definition of this theory is based on a choice of a simplicial presentation of the manifold which exhibits naturally a canonical decomposition of the manifold. Let us consider a 3-manifold ${\cal M}$ given by a complex $K$. We impose here that all vertices of $K$ are tetravalent. We will denote $K^{\star}$ the dual complex of $K$. We will denote as before $A_0^i,A_1^j,A_2^k,A_3^l$ the $0-,1-,2-,3-$simplexes of $K$ , $A_0^{\star i},A_1^{\star j},A_2^{\star k},A_3^{\star l}$ the $0-,1-,2-,3-$simplexes of $K^{\star}$ and $B_1^j$ (resp.$B_1^{\star j}$) the middle of the $A_1^j$ (resp. $A_1^{\star j}$).
Let us define another tetravalent complex $ K^{\#}$ build up from the previous one as follows: A couple of $B_1^j$ and $B_1^{\star j}$ are said to be a couple of neighbours if $B_1^j$ is the middle of an edge of a certain $A_2^k$ and $B_1^{\star j}$ is in the middle of this $A_2^k.$ We denote by $J$ the set of $1-$simplexes defined by the set of couples of neighbour points. We now define the $0-simplexes$ of $K^{\#}$ to be the middles of the elements of $J$. The $1-simplexes$ of $K^{\#}$ are then given by the set of couples of $0-simplexes$ corresponding to elements of $J$ having one vertex in common, if this vertex is a $B_1^j$ (resp. a $B_1^{\star j}$) then this $1-$simplex is said “of type $K$” (resp. “of type $K^{\star}$”). Now the $2-$simplexes are defined to be of three types: one $2-$simplex is associated to each closed curve formed by type $K$ $1-$simplexes only, one to each closed curve formed by type $K^{\star}$ $1-$simplexes only, and one to each closed curve formed alternatively by type $K$ and type $K^{\star}$ $1-$simplexes.We will refer us to “the $e_{i}$s” , “the $f_{j}$s”, and “the $P$s” to denote respectively these three types of $2-$simplexes. Finally the $3-$simplexes are defined in an obvious way by considering each connected region around the vertices of $K$ and $K^{\star}$. We will denote by $K^{\#}_0,K^{\#}_1,K^{\#}_2,K^{\#}_3$ the sets of $0-,1-,2-,3-$simplexes respectively. A piece of this new complex is shown in the following figure:
The graph $K^{\#}$ build up from any triangulation $K$ describing a manifold ${\cal M}$ owns the following properties:
If we denote by ${\cal R}_1$ (resp. ${\cal R}_2$) the region defined by the set of $3-$cells associated to vertices of $K$ (resp.$K^{\star}$) and by ${\cal L}$ the surface defined by the set of Ps.The decomposition: ${\cal M}={\cal R}_1 + {\cal L} + {\cal R}_2$ is a canonical decomposition and $K^{\#}$ is homeomorphic to $K$. The set formed by the elements of $K^{\#}_0$,the elements of $ K^{\#}_1$ and all $P$s forms the complex L associated to the triangulation of the canonical surface [L]{}(for this reason these sets of $0-,1-$ and $2-$simplexes will be also denoted respectively by $L_0,L_1,L_2$)
The set of canonical $2-$cells of ${\cal R}_1$ (resp ${\cal R}_2$) is a subset of the $e_i$s (resp.the $f_j$s).
This decomposition is equivalent to the Heegaard decomposition “derived” from the complex $K.$
As a consequence of the property that $K^{\#}_0=L_0$ and $K^{\#}_1=L_1$, we can define as before the exchange algebra associated to the elements of $K^{\#}_1$ by imposing the coaction of the gauge symmetry algebra at each element of $K^{\#}_0$ and by choosing a cilium order on the surface.We can define as before the Wilson loops attached to each closed path formed by elements of $K^{\#}_1$ (i.e. drawn on L) and delta functions associated to each $2-$cell. In fact we define the Yang-Mills weight associated to a 2-cell $P$ of area $A_P$ to be: \^\_P=\_[Phys(A)]{}\[d\_\] e\^[- ]{}\_[P]{} where $C_{\alpha}$ is the quadratic casimir of the representation $\alpha$ and $\beta$ is a coupling constant of the Yang-Mills theory. We define the expectation value associated to any element ${\cal A}$ of $\Lambda^{inv}$ in the 3 dimensional q-Yang Mills theory to be: \_[[M]{}]{}:= \_[lK\^[\#]{}\_1]{} dh(U\_l) (\_[j]{} \^\_[f\_j]{})(\_[PL\_2]{}\^\_P) ( \_[i]{} \^\_[e\_i]{}) in the limit $q \rightarrow 1$ this theory becomes the well known Yang-Mills theory on a lattice associated to a manifold ${\cal M}.$
Let $L$ be a link drawn on the 1-skeleton $K^{\#}_1$ of ${\cal M}$. Using again the properties of the complex $K$, $L$ is in fact drawn on the canonical surface and we can define $W_L$ in the framework defined in this article. The correlation function associated to $L$ in the limit $\beta \rightarrow 0$ is then \_ = this formula can be considered as a description of Reshetikhin-Turaev invariants, i.e. of Chern-Simons invariants in term of a well defined lattice gauge theory and a definition of the Witten’s path integral formulas.
\
The expectation value is simply the same as that introduced in the last subsection but with a very special Heegaard decomposition where the gluing diffeomorphism is simply the identity.\
[**Acknowledgements:**]{} It is a pleasure to thank my friend P.Roche for his constant support. I want also aknowledge illuminating discussions with N.Reshetikhin and C.Mercat.
[10]{}
V.V.Fock, A.A.Rosly, , (1992).
E.Buffenoir, Ph.Roche, and [Comm.Math.Phys. 170 669-698]{} (1995).
E.Buffenoir, Ph.Roche, and [CPTH Preprint ]{} (1995).
A.Y.Alekseev, H.Grosse, V.Schomerus, , (1994).
E.Witten, , 351-399 (1989).
D.Altschuler and A.Coste, ,(1992).
G.Mack and V.Schomerus, , (1992) 185-230.
V.G.Drinfeld, , 321 (1990).
N.Yu.Reshetikhin, V.G.Turaev, , (1990).
N.Yu.Reshetikhin, V.G.Turaev, , 547-597 (1991).
A.N.Kirillov, N.Reshetikhin, , (1988).
V.A.Vassiliev, , 9 (1990).
J.Singer, , 88-111 (1933).
W.Lickorish, , (1962) 531-540.
Ning Lu, , (1992) 1.
[^1]: e-mail: buffenoi@orphee.polytechnique.fr
|
---
author:
- 'R. Paladino'
- 'M. Murgia'
- 'E. Orr[ù]{}'
bibliography:
- 'paper\_LOW.bib'
date: 'Received; Accepted'
title: 'Radio spectral index images of the spiral galaxies NGC 0628, NGC 3627, and NGC 7331'
---
i14[I$_{\rm 1.4}$]{} ¶[$\bar{\rm P}$]{} 0[N$_{0}$]{}
[**A.**]{} Sources parameters\
[ccccccc]{} Name & $\alpha$(J2000) & $\delta$(J2000)& d & i& S$_{\rm RC}$&S$_{\rm FIR}$\
&($h\ m\ s\ $) & ( )& (Mpc)&()&(mJy)& (Jy)\
NGC0628& 01 36 41.7 &+15 46 59&7.3&24&180&20.86\
NGC3627& 11 20 15.0& +12 59 30& 11.1&63 & 458 & 67.8\
NGC7331&22 37 04.1& +34 24 56&15.1 &62&372 &35.29\
[**B.**]{} Observations details\
[ccccccccc]{} Name & IF & Frequency & Bandwidth& Configuration&Visibilities& HPBW& rms&Date\
& Number& MHz& MHz&&Number&$\arcsec \times \arcsec$&mJy/beam &\
NGC0628&2&321.5-327.5 &3.125 &C&524446& 65 $\times$ 57 &2.3&01 sep 05\
NGC3627&2&329.1-323.1 &6.250 &B&662708& 20 $\times$ 16.6 &2 & 30 oct 07\
NGC7331&2&329.1-323.1 &6.250 &B&349475&17.5 $\times$ 15.5&1.1 & 03 nov 07\
[**A.**]{} Coordinates, distances, inclinations from [@helfer03]. Inclination is defined assuming i=0 for face-on galaxies. The S$_{\rm RC}$ of NGC0628, and NGC3627 are 1.4 GHz global flux from @condon90, the S$_{\rm RC}$ of NGC7331 is from [@condon87]. S$_{\rm FIR}$ is 60 $\mu$m flux, taken from @rice90 for NGC0628, and for NGC7331, while from the IRAS Bright Galaxy Sample [@soifer89] for NGC3627. [**B.**]{} VLA observational properties.
\[tab1\]
Introduction
============
It is well known that the radio continuum of spiral galaxies consists of a mixture of non-thermal and thermal emission [e.g. @condon92]. The non-thermal component is due to the synchrotron emission of cosmic-ray (CRe) electrons with GeV energies spiralling in the galaxy’s magnetic field. The thermal one is due to the free-free emission from ionized gas at $T\sim 10^4$ K. The two emissions are characterized by different spectra. The synchrotron component generally has a steep spectral index $\alpha\sim 0.8$ ($S_{\nu}\propto \nu^{-\alpha}$) and dominates at low-frequencies while the free-free emission has a much flatter spectrum, $\alpha\simeq 0.1$, and may become relevant at frequencies greater than about 10 GHz [@niklas97]. Synchrotron emission is thus best studied at frequencies below a few GHz. At these frequencies close correlations are observed between the radio continuum and several star formation tracers like the $H_{\alpha}$ emission from ionized gas, the infrared radiation (IR) from thermal dust [e.g. @yun01], and the CO emission from molecular clouds (@rickard77 [-@rickard77], @israel97 [-@israel97], @adler91 [-@adler91], @murgia02 [-@murgia02]). It has been established that these correlations hold not only on global but also on local scales, down to hundreds of parsecs, within the disks of spiral galaxies (@matteo05 [-@matteo05], @io06 [-@io06]). Although it is commonly accepted that all these correlations ultimately originate from the process of massive stars formation itself, the details of the physical mechanisms on how this occurs are not completely understood.
One main difficulty in the study of the non-thermal radio continuum is due to the fact that the synchrotron radiation essentially traces the product of CRe and magnetic field energy densities. Disentangling these two contributions is not directly possible unless making hardly verifiable assumptions (like e.g. energy equipartition between field and particles). A second source of uncertainty is related to the diffusion length of CRe electrons which is poorly known. The CRe diffuse from their injection (or re-acceleration) sites and, as a consequence, their density and energy spectrum at a given location in the disk is the result of the balance between the efficiencies of different processes such as the injection (or re-acceleration) rate, the confinement time, and the energy losses.
Since the mean free path of dust-heating photons ($\sim$100 pc) is far shorter than the presumed diffusion length of CR electrons [@bicay89], the first attempts to model the propagation of CRe have been made by comparing radio and IR images of spiral galaxies. [@bicay90] smeared IRAS images of spiral galaxies by using parametrized kernels in order to match the observed morphology seen in the radio. They found that the diffusion length of CRe electrons is of the order of $\sim$ 1-2 kpc. Recently [@murphy06L] applied this phenomenological image-smearing model to new [*Spitzer*]{} images. They found that the galaxies examinated exhibit a wide range of ages and spreading scale lengths of their CR electrons, going from the youngest populations, recently injected within star-forming regions (l$\sim$500 pc), to the oldest CR electrons, making up a galaxy’s underlying synchrotron disk (l $\sim$ 3 kpc). The way we will use to understand the cosmic ray diffusion mechanism, and thus its correlation with the sites of intense star formation, is the comparison of spatially resolved radio spectral index images of galaxies and their IR distribution. Such a study is only now possible due to the unprecedented high spatial resolution and sensitivity of the [*Spitzer Space Telescope*]{} at IR wavelengths. We recently studied the point-by-point relation between the radio spectral index and the IR emission in the galaxy M51 (Paladino et al., 2006). By comparing the [*Spitzer*]{} 70 $\mu$m image and the spectral index image obtained between 1.4 and 4.9 GHz, we found an anticorrelation between the IR emission and the spectral index. Regions in which the IR is higher than the average tend to have a flatter radio spectrum with respect to the surroundings. This result suggests that the CRe electrons confinement time in the star forming regions is shorter than their radiative lifetime and that they age as they diffuse in the disk (Murgia et al 2005).
{width="56.00000%"}
{width="58.00000%"}
{width="58.00000%"}
This work would be a starting point for the program of extending this kind of analysis to a large sample of galaxies. Here we report on the spectral index analysis of three spiral galaxies: NGC0628, NGC3627, and NGC7331. We observed these galaxies at 327 MHz with the Very Large Array (VLA). In order to obtain detailed spectral index maps we complemented our data set with sensitive archival observations at 1.4 GHz. Since at these frequencies the contribution from thermal emission is likely to be minimized the spectral index can be used to study the spectrum of the non-thermal emission.
The paper is organized as follows: in Sect. \[obs\] we describe the observations and the data reduction. The spectral index analysis and the corresponding results are reported in Sect. \[res\]. Finally, we summarize our results in Sect. \[summ\].
Observations and data reduction {#obs}
===============================
VLA 327MHz observations {#images}
-----------------------
Our aim is to investigate the variations of the non-thermal spectral index in the sample of 22 nearby spiral galaxies presented in Murgia et al. (2005) and Paladino et al. (2006), by mean of low-frequency radio observations. In this work we report on the first results for three galaxies taken from the aforementioned sample.
We present new radio observations of the spiral galaxies NGC0628, NGC3627, and NGC7331 at 327 MHz with the VLA. The basic parameters of these galaxies are listed in Table \[tab1\].A. The galaxy NGC0628 is a clear grand-design spiral galaxy seen almost face-on. In the UV bright knots embedded in diffuse emission trace the spiral pattern and many of these knots are also bright in $\rm H_\alpha$ [@marcum01]. Futhermore these authors have suggested that the entire disk has undergone active star formation within the past 500 Myr and that the inner regions have experienced more rapidly declining star formation than the outer regions. NGC3627, the brightest member of the Leo Triplet, is an interacting spiral galaxy with a bar and two asymmetric spiral arms. The western arm is accompanied by weak traces of star formation, while the eastern arm contains a straight actively star-forming segment in its inner part [@soida01]. NGC7331 is a relatively highly inclined spiral galaxy with a ring, visible in CO [e.g. @regan01], radio continnum [e.g. @cowan94], submillimeter [@bianchi98] and [*Spitzer*]{}-infrared (@regan04 [-@regan04], @regan06 [-@regan06]) images.
We observed NGC3627 and NGC7331 with the VLA in B configuration while NGC0628 was observed with the VLA in C configuration. A summary of the observations, including the VLA configuration, frequency, bandwidth, visibilities number, date of observations, is reported in Table \[tab1\].B.
The observations were conducted in spectral line mode (see Tab. \[tab1\].B for details on frequencies and bandwidths), in order to reduce the effects of bandwidth smearing and to allow for more accurate radio-frequency interference (RFI) excision.
The data were calibrated and imaged with the NRAO package AIPS (Astronomical Image Processing System). Both flux density and bandpass calibration were based directly on observations of the strong calibrator sources 3C48, 3C123, 3C147, and 3C286. The amplitude gains were tied to the flux scale of [@baars77]. For the initial phase calibration we used, when available, further calibrators closest to the observed sources. A careful data editing operation has been made in order to remove RFI present in individual channels. At the end of this procedure $\sim$ 10 $\%$ of data were flagged in the observations of NGC0628. A higher fraction of data, about 35 $\%$ and 28$\%$ for NGC3627 and NGC7331, respectively, was flagged in the 2007 runs, when some EVLA antennas started to be included in the VLA.
The visibility data have been spectrally averaged, retaining 5 channels with a resolution of 488 KHz for NGC0628, and 6 channels with a resolution of 781 KHz for NGC3627 and NGC7331.
The entire field was imaged using the AIPS program IMAGR in 3D mode. Owing to its large field of view at 327 MHz, the VLA is not coplanar at this frequency. Therefore, the primary beam was divided in small overlapping “facets” over which the VLA can be safely assumed to be coplanar [@cornwelper92]. Confusion lobes of surrounding sources far from the center of the field are often so strong that their side-lobes contribute measurably to the noise level in the central region. Thus, by using the NRAO VLA Sky Survey (NVSS) catalog as a reference, we also cleaned the strongest sources over an area of about $\sim~60\degr$ in radius by placing small facets around each of them (task SETFC).
Final images were created by an iterative process of CLEANing and self-calibration including all these “facets”. In the self-calibration process, we used only the strongest components inside the primary beam in order to increase the accuracy of the phase calibration model. The use of smaller clean boxes decreased the possibility of “over-cleaning” and minimized the CLEAN bias [see @condon98 for details].
Figures \[N0628\_90cm\]-\[N7331\_90cm\] show the 327 MHz images of the three observed sources, with typical noise level reported in Table \[tab1\].B.
NGC0628, which has the largest angular extent ($\sim 6\arcmin$) among the three galaxies, has been observed with the VLA in C configuration with an angular resolution of 65$\times$57. The observations of NGC3627 and NGC7331 ($\sim$ 5) were taken with the VLA in B configuration at a resolution of 20$\times$16.6 and 17.5$\times$15.5, respectively.
Archival VLA observations at 1.4 GHz {#20cm}
------------------------------------
For the purposes of the spectral index imaging, we complemented our data set with archival observations at 1.4 GHz with the VLA in D and C configurations. The archival data we recovered is described in Table \[20cmtab\]. The data were calibrated and imaged with the package AIPS following the standard procedures. The coverage of the spatial frequencies of the 1.4 GHz observations is very similar to that at 327 MHz. However, in order to ensure a perfect match in angular resolution, we convolved the final images at both frequencies to the largest beam (see section \[local\_spix\]).
--------- -------- -------------------------- -------------- ------------
Source VLA HPBW rms Program ID
config $\arcsec \times \arcsec$ $\mu$Jy/beam
NGC0628 D 42.72 $\times$ 40.06 54 AA161
NGC3627 C 16.52 $\times$ 14.02 150 AS541
NGC7331 C 17.49 $\times$ 15.48 40 AB345
--------- -------- -------------------------- -------------- ------------
: VLA archival observations at 1.4 GHz.
\[20cmtab\]
IR [*Spitzer*]{} images
-----------------------
The galaxies studied in this work were observed as part of the [*Spitzer*]{} Infrared Nearby Galaxies Survey (SINGS; [@sings03]) and IR images have been released. For details on observations and data processing see delivery document for Fifth (and last) SINGS Data Delivery, April 2007[^1].
The observations obtained with the Multiband Imaging Photometer for [*Spitzer*]{} (MIPS; [@rieke04]) at 70 $\mu$m have resolution of 18. We convolved these images to the beam of our 327 MHz radio images in order to compare the spectral index distribution with the IR one. As reported in the Data Release, due to special processing, the quality of the NGC7331 70 $\mu$m data is somewhat worse than the quality of the other 70 $\mu$m images released. However, we use this image to compare the spectral index and IR distribution in NGC7331, handling with care the results obtained from this analysis.
Analysis and results {#res}
====================
Integrated radio spectra {#intspec}
------------------------
--------- ----------- --------------------- -----------
Source Frequency Flux density Reference
MHz Jy
NGC0628 57.5 2$\pm$1 1
324.4 0.49 $\pm$0.03 This work
1400 0.15 $\pm$0.01 2
1515 0.16 $\pm$ 0.01 This work
4850 0.060 $\pm$ 0.005 3
NGC3627 57.5 2.3$\pm$ 0.7 1
160 1.6 $\pm$ 0.3$^a$ 4
325.5 0.97 $\pm$ 0.03 This work
408 0.93 $\pm$ 0.05$^a$ 5
611 0.87 $\pm$ 0.1$^a$ 4
1370 0.5 $\pm$ 0.01 6
1425 0.38 $\pm$ 0.01 This work
2695 0.2 $\pm$ 0.04 4
4850 0.14$\pm$ 0.02 7
NGC7331 74 1.12$\pm$ 0.18 8
325 0.935$\pm$ 0.004 9
325.9 0.94$\pm$ 0.02 This work
1489 0.32$\pm$ 0.01 This work
1400 0.33 $\pm$ 0.01 2
1369 0.35 $\pm$ 0.01 6
2380 0.18$\pm$ 0.01 10
4850 0.096$\pm$ 0.013 7
--------- ----------- --------------------- -----------
: Integrated flux densities on the observed sources.
\[sed\]
$^a$ Flux density scale from [@wyllie69].
References and telescopes:
1. [@israel90]: Clark Lake radio Telescope
2. [@NVSS]: NVSS – NRAO VLA D array
3. [@GB6]: NRAO Green Bank Telescope
4. [@CSIRO75]: CSIRO telescope
5. [@large81]: Molonglo Radio Telescope
6. [@braun07]: Westerbork Telescope WSRT
7. [@gregory91]: NRAO Green Bank
8. [@cohen07]: NRAO VLA B array
9. [@rengelink97]: WENSS – WSRT
10. [@dressel78]: Arecibo Telescope.
We measured total flux densities at 327 MHz from the new images of the observed galaxies. The values obtained are reported in Table \[sed\].
------------------------------------ ------------------------------------ ------------------------------------
{height="5"} {height="5"} {height="5"}
------------------------------------ ------------------------------------ ------------------------------------
--------- --------- ---------- ---------------- ------------------ ------------------
$\alpha$ $\nu_{\rm br}$ $\nu_{\rm peak}$ $\chi^2/\rm ndf$
MHz MHz
NGC0628 Pow 0.78 – – 0.76
NGC3627 Pow+SSA 0.66 – 54 4.3
CI 0.4 937 – 2.9
NGC7331 Pow+SSA 0.79 – 112 2.5
CI+SSA 0.5 1025 92 0.98
--------- --------- ---------- ---------------- ------------------ ------------------
: Results of the fit with models in detail in Sect. \[intspec\]: Pow = Powerlaw SSA = Synchrotron Self Absorption CI = Continuous injection
\[fittab\]
In order to investigate if some absorption process affects the emission from these galaxies, in particular in the range of frequencies between 327 MHz and 1.4 GHz, we have analyzed their integrated radio spectra. From a careful assessment of published values between 57 and 5000 MHz we derived the integrated radio spectra of the three observed galaxies. The values of the integrated flux density are reported in Table \[sed\]. The telescopes from which the observations have been taken are reported in correspondence of each reference.
Most of the measurements were tied to the flux scale of [@baars77]. Some values (reported with an indication on Table \[sed\]) are on the scale of [@wyllie69], which [@baars77] concluded is very close to their own.
Figure \[Spectr\_NED\] shows the integrated radio spectra of NGC0628, NGC3627, and NGC7331, in which our new 327 MHz flux density measurements are indicated by filled triangles. The integrated spectrum of NGC0628 is shown on the left panel of Figure \[Spectr\_NED\]. The data are very well fitted by a single power law with a spectral index of $\alpha$=0.78. This value is not unusual for a spiral galaxy; the average spectral index of spiral galaxies is 0.74 $\pm$ 0.03 [@gioia82].
------------------------------------ ------------------------------------ ------------------------------------
{height="5"} {height="5"} {height="5"}
------------------------------------ ------------------------------------ ------------------------------------
{width="96.00000%"}
{width="96.00000%"}
{width="96.00000%"}
{width="96.00000%"}
{width="96.00000%"}
{width="96.00000%"}
The integrated spectra of galaxies NGC3627 and NGC7331 present deviations from the classical power law. Most spiral galaxies do indeed show a change in spectral index with frequency and different possible reasons have been proposed (e.g., @israel90 [-@israel90]; @hummel91 [-@hummel91]; @pohl92 [-@pohl92]). We fitted the integrated spectra of these galaxies by introducing an absorption at low frequency. This seems to be necessary in particular to fit the low frequency point in NGC7331. The integrated spectra have been fitted with a power law, S$_{\rm Pow}$(${\nu}$) $\propto \nu^{-\alpha}$, modified by low-frequency absorption as in [@basicsync]: $$S(\nu)\propto(\nu/\nu_{\rm peak})^{(\alpha+2.5)}
(1- e^{-(\nu/\nu_{\rm peak})^{-(\alpha+2.5)}})\cdot S_{\rm Pow}(\nu)
\label{pach}$$
where $\nu_{\rm peak}$ is the frequency at which the optical depth is equal to 1, while $\alpha$ is the spectral index in the optically thin power law regime. For the optically thick regime we assume a spectral index 2.5, as expected in the case of an homogeneous synchrotron self-absorbed source. The fitted parameters, along with the reduced $\chi^{2}$ values ($\chi^{2}$/ndf, where ndf is the number of degrees of freedom), are reported in table \[fittab\]. The frequency peaks for both galaxies resulting from the fit are at $\nu_{\rm peak} \lesssim$ 100 MHz, thus the region between 327 MHz and 1.4 GHz should be relatively free from absorption.
A hint of steepening is observed in the spectra above a frequency of $\sim$ 1 GHz. We tried to fit the data with the continuous injection model (CI), described in [@matteo99]. In this model it is assumed that the radio sources are continuously replenished by constant flow of fresh relativistic particles with a power law energy distribution ( N(E)$\propto E^{-\delta}$) and that the magnetic field is constant, the radio spectrum has a standard shape [@kardashev62], with injection spectral index $\alpha_{\rm inj}$=$(\delta - 1)/2 $ below a critical frequency $\nu_{br}$, and $\alpha=\alpha_{inj}+0.5$ above $\nu_{br}$. The steepening of the spectral index at high frequencies is due to particle energy losses.
By comparing the $\chi^2/\rm ndf$ values, we can notice that the observed high frequency spectra are better fitted by a CI model, rather than a power law, for both galaxies. In NGC3627 the CI model fit do not requires synchrotron self-absorption. Nevertheless, the interpretation of the high frequency break should be considered carefully being the assumptions of the CI model too simple to describe the complex structures and physics of spiral galaxies.
In closing, the analysis of integrated spectra of the three galaxies considered in this work shows that the region between 327 MHz and 1.4 GHz should be relatively free from absorptions.
Local variations in the radio spectral index {#local_spix}
--------------------------------------------
We investigate the variations in the non-thermal spectral index as a function of position within the galaxies. The integrated spectra show that the region between 327 MHz and 1.4 GHz is relatively free from (self-)absorption, even in the case of NGC7331 where a low frequency turnover seems to be present. Moreover, at these frequencies the contribution from thermal emission is likely to be minimized, and thus spectral index can be used to study the spectrum of the non-thermal emission.
The spectral index images between 327 MHz and 1.4 GHz, have been obtained by considering only those pixels where the brightness was above 3$\sigma$ at both frequencies. To accurately determine the spectral index distribution identical beam size and shape at different frequencies is required. We convolved NGC0628 images to a 66 round beam while for NGC3627 and NGC7331 we used a beam of 20. In addition, to avoid any positional offsets, the images were aligned and interpolated to identical projections and field center. The primary beam correction has been applied to the images at both frequencies.
The spectral index images, calculated between 327 MHz and 1.4 GHz, of the three sources are shown in left panels of Figures \[N0628\_spix\], \[N3627\_spix\] and \[N7331\_spix\]. The dynamic range of images at 327 MHz is typically lower than the one at 1.4 GHz, therefore the cut in these spectral index maps in most of the points is driven by the 327 MHz images. Right panel of the same Figures show the spectral index uncertainty. The comparison of the $\alpha$ image with the error image show that most of the observed spectral index features are statistically significant.
The 66 beam of NGC0628 partly hidden the spiral structure of the galaxy, bright clumps in spiral arms in radio images correspond to flatter regions in the $\alpha$ image (Fig. \[N0628\_spix\]). The spectral index image of NGC3627 (Fig. \[N3627\_spix\]) clearly shows that the bar and its brights ending regions have a spectral index flatter than the underlying disk. The radio spectrum of NGC7331 (Fig. \[N7331\_spix\]) is almost uniformly flat in the galaxy’s center and steepens towards the peripheral regions of the disk.
For a quantitative analysis we measured the point-to-point brightness on the radio images at 327 MHz and 1.4 GHz as follows. We overlaid regular grids of rectangular beam-sized boxes on the radio images, and we averaged the brightness in each box. We then calculate the spectral index between 327 MHz and 1.4 GHz.
Figures \[alfaRC\], \[N0628\_spix\_rad\], \[N3627\_spix\_rad\] and \[N7331\_spix\_rad\] show the results of this point-to-point comparison. Only points exceeding 5 times the rms of the computed spectral index are shown. This severe clipping increases our confidence in spectral index variations.
Figure \[alfaRC\] shows the variations of spectral index with the radio brightness in the three studied galaxies. A common feature in these galaxies is that the spectral index is anticorrelated with the radio brightness. Bright regions, like the bar in NGC3627 or the circumnuclear region in NGC7331, are characterized by a flatter spectrum with respect to the underlying disk. In all the three galaxies, the brighter regions have a spectral index of about $\alpha\simeq 0.5-0.6$. In the faintest regions, the radio spectrum steepen to $\alpha\simeq 1.0-1.2$.
Left panels of Figures \[N0628\_spix\_rad\], \[N3627\_spix\_rad\], and \[N7331\_spix\_rad\] show the spectral index versus the distance from the center of the sources. The anticorrelation between the brightness and the spectral index causes a systematic steepening of the radio spectrum with the increasing distance from the center of these galaxies, where typically brightest regions are located.
Comparison with IR emission
---------------------------
The far infrared luminosity from galaxies is emitted by interstellar dust heated by the general interstellar radiation field and the more intense radiation in star formation regions. The 70 $\mu$m emission can be used as star formation diagnostic. To investigate the connection between star formation sites and cosmic rays propagation mechanism, we compare 70 $\mu$m distribution to spectral index in galaxies.
We used 70 $\mu$m [*Spitzer*]{} images [@sings03] convolved to the same beam of our spectral images. We measured the IR brightness using the same grid described in section \[local\_spix\]. Right panels in Figures \[N0628\_spix\_rad\],\[N3627\_spix\_rad\] and \[N7331\_spix\_rad\] show the radio spectral index versus the 70 $\mu$m brightness. Despite the very different morphologies, the radio spectral index shows a common behaviour in all the three galaxies: an anticorrelation exists between the radio spectral index and the infrared brightness. Regions in which the IR is higher than the average tend to have a flatter radio spectrum with respect their surroundings. The radio spectral index shows the same anticorrelation with the radio brightness. The observed trend is expected on the basis of the local correlation between the radio continuum and the infrared emission. This result is consistent with the idea that the CRe electrons confinement time in the star forming regions is shorter than their radiative lifetime.
Summary {#summ}
=======
We have presented new 327 MHz VLA radio images of the three spiral galaxies NGC0628, NGC3627, and NGC7331. Based on these images, we measured the integrated flux densities and we performed a spatially resolved study of the synchrotron spectral index distribution.
We have calculated the integrated radio spectra of these galaxies, using the measured integrated fluxes, together with flux densities measurements taken from the literature. In the considered frequency range, the radio spectrum of NGC0628 is well fitted by a single power law with an index $\alpha\simeq 0.79$. The integrated spectra of galaxies NGC3627 and NGC7331 present deviations from the classical power law. We fitted these integrated spectra and we noticed that the observed high frequency spectra are better reproduced by a continuous injection model. In NGC7331 a synchrotron self-absorption process well reproduce the low frequency spectrum. This is one of the possible mechanisms producing the low-frequency turnover and further low-frequency observations, are needed to confirm it.
We complemented our data set with sensitive archival observations at 1.4 GHz and we studied the variation of the radio spectral index to within the disks of these spiral galaxies. We found that the spectral index is anticorrelated with the radio brightness. Bright regions, like the bar in NGC 3627 or the circumnuclear region in NGC 7331, are characterized by a flatter spectrum with respect to the underlying disk. This causes a systematic steepening of the spectral index with the increasing distance from the center of these galaxies. Furthermore, by comparing the radio images with the 70 $\mu$m images of the Spitzer satellite we found that a similar anticorrelation exists between the radio spectral index and the infrared brightness, as expected on the basis of the local correlation between the radio continuum and the infrared emission. Our results support the idea that in regions of intense star formation, where the injection rate of electrons is presumably higher, the CRe electrons confinement time is shorter than their radiative lifetime [@matteo05], then the electron diffusion must be efficient. We also observed the anticorrelation between RC brightness and spectral index, this may imply that the cosmic ray density and the magnetic field strength are significantly higher in these regions than in their surroundings.
Finally, we note that this study adds to a growing list of examples confirming the importance of high-resolution, low-frequency radio observations in providing important tool to understand the physical mechanism of production and diffusion of cosmic rays. This also points to the great potential of the new generation of much more sensitive low frequency instruments, as LOFAR. Future work will involve a detailed comparison of the radio spectral index distribution and the star-formation tracers, as IR and molecular emission, in a larger sample of spiral galaxies, in order to analyse the connection of cosmic ray production and propagation with the star formation processes.
We would like to thank the anonymous referee for insightful comments which improved this manuscript. E.O.acknowledges financial support of Austrian Science Foundation (FWF) through grant number P18523-N16. The National Radio Astronomy Observatory is operated by Associated Universities, Inc., under contract to the National Science Fundation. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and of CATS database Astrophysical CATalogs support System.
[^1]: http:$//$data.spitzer.caltech.edu$/$popular$/$sings$/$20070410\_enhanced\
\_v1$/$Documents$/$sings\_fifth\_delivery\_v2.pdf
|
---
abstract: 'Let $F(s)$ be a function belonging to Selberg class. Chandrasekharan and Narasiman proved the approximate functional equation for $F(s)$. In this paper, we shall generalize this formula for the derivatives of $F(s)$.'
author:
- Yoshikatsu Yashiro
title: '**Approximate functional equation for the derivatives of functions in Selberg class**'
---
Introduction
============
Selberg [@SEL] introduced a class of zeta functions satisfying the following properties:
(a) The function $F(s)$ is written as an absolutely convergent Dirichlet series $F(s)=\sum_{n=1}^\infty a_F(n)n^{-s}$ for ${\rm Re\;}s>1$.
(b) There exists $m\in\mathbb{Z}_{\geq0}$ such that $(s-1)^mF(s)$ is an entire function of finite order.
(c) The function $\Phi(s)=Q^s\prod_{j=1}^q\Gamma(\lambda_js+\mu_j)F(s)$ satisfies $\Phi(s)=\omega\overline{\Phi}(1-s)$ where $q\in\mathbb{Z}_{\geq1}$, $Q\in\mathbb{R}_{>0}$, $\lambda_j\in\mathbb{R}_{>0}$, $\mu_j\in\mathbb{C}: {\rm Re\;}\mu_j\geq0$, $\omega\in\mathbb{C}: |\omega|=1$, and $\overline{X}$ denotes $\overline{X}(s)=\overline{X(\overline{s})}$.
(d) The Dirichlet coefficients $a_F(n)$ satisfy $a_F(n)=O(n^\varepsilon)$ for any $\varepsilon\in\mathbb{R}_{>0}$.
(e) The function $\log F(s)$ is written as $\log{F(s)}=\sum_{n=1}^\infty{b_F(n)n^{-s}}$ where $b_F(n)$ satisfy $b_F(n)=0$ for $n\ne p^r\;(r\in\mathbb{Z}_{\geq1})$ and $b_F(n)=O(n^\theta)$ for some $\theta\in\mathbb{R}_{<1/2}$,
which is called the Selberg class $\mathcal{S}$. For example, we see that the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty n^{-s}$ and the $L$-function attatched to cusp forms $L_f(s)=\sum_{n=1}^\infty\lambda_f(n)n^{-s}$ belong to $\mathcal{S}$, where $f$ is a cusp form of weight $k$ given by $f(z)=\sum_{n=1}^\infty\lambda_f(n)n^{\frac{k-1}{2}}e^{2\pi inz}$. By Landau’s [@LAN] result (see (15) of p.214), the conditions (a)–(d) of $\mathcal{S}$ imply that the average of $a_F(s)$ is approximated as $$\begin{aligned}
\sum_{n\leq x}a_F(n)
=\begin{cases} x\sum_{r=0}^{p_F-1}c_r(\log x)^r+O(x^{\frac{d_F-1}{d_F+1}+\varepsilon}), & p_F\in\mathbb{Z}_{\geq1}, \\ O(x^{\frac{d_F-1}{d_F+1}+\varepsilon}), & p_F\in\mathbb{Z}_{\leq0}, \end{cases} \label{LANA}\end{aligned}$$ where $c_r\in\mathbb{C}$, $p_F$ is the order of pole at $s=1$ for $F(s)$, and $d_F$ is given by $d_F=2\sum_{j=1}^q\lambda_j$ called the degree of $F$. The conditon [(c)]{} implies that the $m$-th derivatives of $F(s)$ holds $$\begin{aligned}
F^{(m)}(s)=\sum_{r=0}^m(-1)^r\binom{n}{r}\chi_F^{(m-r)}(s)\overline{F}^{(r)}(1-s) \label{FDFE}\end{aligned}$$ for $s\in\mathbb{C}$ where $\chi_F(s)$ is given by $$\begin{aligned}
\chi_F(s)=\omega Q^{1-2s}\prod_{j=1}^q\frac{\Gamma(\lambda_j(1-s)+\overline{\mu_j})}{\Gamma(\lambda_js+\mu_j)}. \label{FCHI}\end{aligned}$$
Chandrasekharan and Narasiman proved the approximate functional equation for a class of zeta functions (see Theorem 2 of p.53). In the Selberg class, this equation is written as $$\begin{aligned}
F(s)=\sum_{n\leq y}\frac{a_F(n)}{n^s}+\chi_F(s)\sum_{n\leq y}\frac{a_F(n)}{n^{1-s}}+O(|t|^{\frac{d_F}{2}(1-\sigma)-\frac{1}{d_F}}(\log y)^{p_F}) \label{AFES}\end{aligned}$$ under the condition $a_n\geq0$ where $F\in\mathcal{S}$, $s=\sigma+it: 0\leq\sigma\leq 1, |t|\geq1$ and $y=(Q\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})|t|^{{d_F}/{2}}$. On the other hand, the author [@YY3] showed the approximate functional equation for the derivatives of $L$-function attached for cusp form as $$\begin{aligned}
L_f^{(m)}(s)=\sum_{n\leq\frac{|t|}{2\pi}}\frac{\lambda_f(n)(-\log n)^m}{n^s}+\chi_{L_f}(s)\sum_{n\leq\frac{|t|}{2\pi}}\frac{\lambda_f(n)(-\log n)^m}{n^{1-s}}+O(|t|^{1/2-\sigma+\varepsilon}),\end{aligned}$$ and introduced the mean value formula for $L_f^{(m)}(s)$ as $$\begin{aligned}
&\int_0^T|L_f^{(m)}(\sigma+it)|^2dt\\&=
\begin{cases}
A_fT(\log T)^{2m+1}+O(T(\log T)^{2m}), & \sigma=1/2,\\ \displaystyle T\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}+O(T^{2(1-\sigma)}(\log T)^{2m}), & \sigma\in(1/2,1), \\ \displaystyle T\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}+O((\log T)^{2m+2}), & \sigma=1.
\end{cases}\end{aligned}$$ where $\chi_{L_f}(s)=(-1)^{k/2}(2\pi)^{1-2s}\Gamma(1-s+\tfrac{k-1}{2})/\Gamma(s+\tfrac{k-1}{2})$ and $A_f$ is a positive constant depending on $f$. These results are generalized for those results for $L_f(s)$ proved by Good [@GD1]. For $F\in\mathcal{S}, \sigma>1/2$ and $T>0$, let $N_{F^{(m)}}(\sigma,T)$ be a number of zeros for $F^{(m)}(s)$ in the region ${\rm Re\;}s\geq\sigma$, $0<{\rm Im\;}s\leq T$. As the application of those results, the author [@YY4] obtained a zero-density estimate for $L_f^{(m)}(s)$, which is $$\begin{aligned}
N_{L_f^{(m)}}(\sigma,T)
=O\left(\frac{T}{\sigma-1/2}\log\frac{1}{\sigma-1/2}\right) \quad (T\to\infty)\end{aligned}$$ for $\sigma>1/2$. This result corresponds the estimate of zero-density estimate for $\zeta^{(m)}(s)$ $$\begin{aligned}
N_{\zeta^{(m)}}(\sigma,T)=O\left(\frac{T}{\sigma-1/2}\log\frac{1}{\sigma-1/2}\right) \quad (T\to\infty)\end{aligned}$$ for $\sigma>1/2$ which is shown by Aoki–Minamide . However, zero-density estimates for derivatives for zeta-function belonging to Selberg class is not known.
In this paper we study the approximate functional equation for $F^{(m)}(s)$ in order to establish tools for estimating $N_{F^{(m)}}(\sigma,T)$ in next paper. To attain the above object we shall use Good’s [@GD1] method and the result . Let $\mathcal{R}$ be a class of $C^\infty$-class functions $\varphi:[0,\infty)\to\mathbb{R}$ satisfying $\varphi(\rho)=1$ for $\rho\in[0,1/2]$ and $\varphi(\rho)=0$ for $\rho\in[2,\infty)$, which is called characteristic functions. Put $\varphi_0(\rho):=1-\varphi(1/\rho)$ and $\|\varphi^{(j)}\|_1=\int_0^\infty|\varphi^{(j)}(\rho)|d\rho$ where $\varphi^{(j)}$ is the $j$-th derivative function of $\varphi\in\mathcal{R}$. Then we see that $\varphi_0\in\mathcal{R}$ and $\|\varphi^{(j)}\|_1<\infty$. For $j\in\mathbb{Z}_{\geq0}$, $r\in\{0,\dots,m\}$, $\rho\in\mathbb{R}_{>0}$ and $s\in\mathbb{C}: |t|\gg1$, we define $$\begin{aligned}
{{\gamma}_{j}^{(r)}}(s;\rho):=&\frac{1}{2\pi i}\int_{\mathcal{F}}\frac{{g_{F}}(s+w)}{{g_{F}}(s)}\frac{1}{w\cdots(w+j)}{\frac{\chi_F^{(r)}}{\chi_F}}(1-(s+w))\times\notag\\
&\times\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+{\mu_j})}{\Gamma(\lambda_js+{\mu_j})}(\rho e^{-\frac{\pi}{2}i{\rm sgn\;}(t)})^{\frac{d_F}{2}w}dw, \label{DEFGM}
\\{{\delta}_{j}^{(r)}}(s;\rho):=&\frac{1}{2\pi i}\int_{\mathcal{F}}\frac{\overline{g_{F}}(s+w)}{\overline{g_{F}}(s)}\frac{1}{w\cdots(w+j)}{\frac{\chi_F^{(r)}}{\chi_F}}(1-(s+w))\times\notag\\&\times\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+\overline{\mu_j})}{\Gamma(\lambda_js+\overline{\mu_j})}(\rho e^{-\frac{\pi}{2}i{\rm sgn\;}(t)})^{\frac{d_F}{2}w}dw \label{DEFDM}\end{aligned}$$ where ${\rm sgn}(t)=\pm1$ if $t\gtrless 0$, $\mathcal{F}=\{(1/2\pm 1)-\sigma+\sqrt{|t|}e^{i\pi(\mp1/2+\theta)}\mid \theta\in[0,1]\}\cup\{u-\sigma\pm i\sqrt{|t|}\mid u\in[-1/2,3/2]\}$ (double-sign corresponds), and $g_F(s)$ is given by $$\begin{aligned}
g_{F}(s)=&\left(s(1-s)\right)^{p_F+m}\textstyle\prod_{j=1}^q(A_j(s)\overline{A_j}(1-s))^{m+1}.
\label{DEFG}\end{aligned}$$ Here $A_{j}(s)$ is given by $$\begin{aligned}
A_j(s)&=\begin{cases} 1, & {\rm Re\;}\mu_j>\lambda_j/2, \\[-0.2em] \prod_{n=0}^{\nu_j}(\lambda_js+\mu_j-n), & {\rm Re\;}\mu_j\leq\lambda_j/2 \end{cases} \label{DEFA} \end{aligned}$$ where $\nu_j=0$ when ${\rm Re\;}\mu_j>\lambda_j/2$ and $\nu_j=[\lambda_j/2-{\rm Re\;}\mu_j]$ when ${\rm Re\;}\mu_j\leq\lambda_j/2$.
Then we obtain the approximate functional equation for $F^{(m)}(s)$ containing characteristic functions:
\[THM1\] For any $F\in\mathcal{S}$, $\varphi\in\mathcal{R}$, $m\in\mathbb{Z}_{\geq0}$, $l\in\mathbb{Z}_{>M_F}$, $s\in\mathbb{C}: \sigma\in[0,1], |t|\gg1$ and $y_1,y_2\in\mathbb{R}_{>0}: y_1y_2=(Q\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^2|t|^{d_F}$, we have $$\begin{aligned}
F^{(m)}(s)=&\sum_{n=1}^\infty\frac{a_F(n)(-\log n)^m}{n^s}\varphi\left(\frac{n}{y_1}\right)+\\
&+\sum_{r=0}^m(-1)^r\binom{m}{r}\chi_f^{(m-r)}(s)\sum_{n=1}^\infty\frac{\overline{a_F(n)}(-\log n)^m}{n^{1-s}}\varphi_0\left(\frac{n}{y_2}\right)+R_\varphi(s)\end{aligned}$$ where $M_F$ is some positive constant and $R_\varphi(s)$ is given by $$\begin{aligned}
&\hspace{-1.2em}R_\varphi(s)\\
:=&\sum_{\frac{y_1}{2}\leq n\leq 2y_1}\frac{a_F(n)(-\log n)^m}{n^s}\sum_{j=1}^l\varphi^{(j)}\left(\frac{n}{y_1}\right)\left(-\frac{n}{y_1}\right)^r\gamma_{j}^{(m)}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)\times\\
&+\chi_F(s)\sum_{r=0}^m(-1)^r\binom{m}{r}\sum_{\frac{y_2}{2}\leq n\leq 2y_2}\frac{\overline{a_F(n)}(-\log n)^r}{n^{1-s}}\times\\
&\times\sum_{j=1}^l\varphi^{(j)}\left(\frac{n}{y_2}\right)\left(-\frac{n}{y_2}\right)^j{\delta_{j}^{(r)}}\left(1-s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)+\\
&+O\left(y_1^{1-\sigma}(\log y_1)^{m+\max\{p_F-1,0\}}|t|^{-\frac{l}{2}}\|\varphi^{(l+1)}\|_1\right)+\\
&+O\left(y_2^\sigma|t|^{d_F(\frac{1}{2}-\sigma)-\frac{l}{2}}\|\varphi_0^{(l+1)}\|_1\sum_{r=0}^m(\log y_2)^{r+\max\{p_F-1,0\}}(\log|t|)^{m-r}\right).\end{aligned}$$
Next introducing new functions $\varphi_\alpha\in\mathcal{R}$ and $\xi\not\in\mathcal{R}$, replacing $\varphi$ to $\varphi_\alpha$ in the above theorem, and choosing $\alpha\in\mathbb{R}_{>0}$ in order to minimize the error term, we obtain the approximate functional equation for $F(s)$:
\[THM2\] For any $F\in\mathcal{S}$, $m\in\mathbb{Z}_{\geq0}$ and $s=\sigma+it: 0\leq\sigma\leq1, |t|\gg1$ we have $$\begin{aligned}
F^{(m)}(s)=&\sum_{n\leq y_1}\frac{a_F(n)(-\log n)^m}{n^s}+\sum_{r=0}^m(-1)^r\binom{m}{r}\chi_F^{(m-r)}(s)\sum_{n\leq y_2}\frac{\overline{a_F(n)}(-\log n)^r}{n^{1-s}}+\notag\\
&+O(y_1^{1-\sigma+\varepsilon}|t|^{-\frac{1}{2}})+O(y_2^{\sigma+\varepsilon}|t|^{d_F(\frac{1}{2}-\sigma)-\frac{1}{2}}) \label{NCAFE}\end{aligned}$$ where $y_1, y_2\in\mathbb{R}_{>0} : y_1y_2=(Q\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^2|t|^{d_F}$. By choosing\
$y_1=y_2=(Q\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})|t|^{d_F/2}$, the error terms of are $O(|t|^{\frac{d_F}{2}(1-\sigma)-\frac{1}{2}+\varepsilon})$.
As an example of Theorem \[THM2\], we shall give the approximate functional equation for derivatives of Rankin-Selberg $L$-function. Let $f$ and $g$ be holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$ given by $f(z)=\sum_{n=1}^\infty\lambda_f(n)n^{\frac{k-1}{2}}e^{2\pi inz}$ and $g(z)=\sum_{n=1}^\infty\lambda_g(n)n^{\frac{k-1}{2}}e^{2\pi inz}$, the Rankin-Selberg $L$-function is defined by $$L_{f\times g}(s)=\sum_{n=1}^\infty\frac{\lambda_f(n)\overline{\lambda_g(n)}}{n^{-s}}.$$ By results of Rankin [@RAN], Selberg [@SEL2] and Delinge [@DEL], we see that $L_{f\times g}(s)$ belongs to $\mathcal{S}$. Hence we obtain $$\begin{aligned}
L_{f\times g}^{(m)}(s)=&\sum_{n\leq\frac{|t|^2}{4\pi^2}}\frac{\lambda_{f\times g}(n)(-\log n)^m}{n^s}+\\
&+\sum_{r=0}^m(-1)^r\binom{m}{r}\chi_{L_{f\times g}}^{(m-r)}(s)\sum_{n\leq\frac{|t|^2}{4\pi^2}}\frac{{\overline{\lambda_{f\times g}(n)}}(-\log n)^r}{n^{1-s}}+O(|t|^{\frac{3}{2}-2\sigma+\varepsilon})\end{aligned}$$ where $m\in\mathbb{Z}_{\geq0}$, $s=\sigma+it: \sigma\in[0,1], |t|\gg1$, and $\chi_{L_{f\times g}}(s)$ is given by $$\chi_{L_{f\times g}}(s)=(2\pi)^{4s-2}\frac{\Gamma({1-s})\Gamma(1-s+k-1)}{\Gamma(s)\Gamma(s+k-1)}.$$
In next section we shall show preliminary lemmas to prove Theorems \[THM1\] and \[THM2\]. Using these lemmas we shall give Theorems \[THM1\] and \[THM2\] in Section 3 and 4 respectively.
Some Lemmas {#LEMP}
===========
First for $\varphi\in\mathcal{R}$ we define $$\begin{aligned}
K_\varphi(w):=&w\int_0^\infty\varphi(\rho)\rho^{w-1}d\rho\quad({\rm Re\;}w>0). \end{aligned}$$ Then $K_\varphi(w)$ has the following properties:
\[KPL\] The function $K_\varphi(w)$ is analytically continued to the whole $w$-plane. Furthermore $K_\varphi(w)$ has the functional equation $$\begin{aligned}
K_\varphi(w)=K_{\varphi_0}(-w). \label{KPW1}\end{aligned}$$ and the integral representation $$\begin{aligned}
K_\varphi(w)=\frac{(-1)^{l+1}}{(w+1)\cdots(w+l)}\int_0^\infty\varphi^{(l+1)}(\rho)\rho^{w+l}d\rho \label{KPW2}\end{aligned}$$ for any $l\in\mathbb{Z}_{\geq0}$. Especially $K_\varphi(0)=1$.
Next to approximate $(\chi_F^{(r)}/\chi_F)(s)$, we shall use the following lemma:
\[LIB\] Let $F$ and $G$ be holomorphic functions in the region in $D$ satisfying $\log F(s)=G(s)$ and $F(s)\ne0$ for $s\in D$. Then for any $r\in\mathbb{Z}_{\geq1}$ there exist $\ell_1,\dots,\ell_r\in\mathbb{Z}_{\geq0}$ and $C_{\ell_1,\dots,\ell_r}\in\mathbb{Z}_{\geq0}$ such that $$\begin{aligned}
\frac{F^{(r)}}{F}(s)
=\sum_{1\ell_1+\cdots+r\ell_r=r}C_{\ell_1,\dots,\ell_r}(G^{(1)}(s))^{\ell_1}\cdots(G^{(r)}(s))^{\ell_r}.\end{aligned}$$ for $s\in D$. Especially $C_{r,0\cdots,0}=1$.
Before approximating $(\chi_F^{(r)}/\chi_F)(s)$ we shall show the following formulas:
\[HAFF\] Let $D=\{z\in\mathbb{C} \mid {\rm Re\:}z<\delta, |{\rm Im\:}z|<1\}$ where $\delta\in\mathbb{R}_{\geq0}$. For any $s\in\mathbb{C}\setminus D$ we satisfy the following formulas:
(i) $\displaystyle \sum_{n=1}^\infty\frac{1}{(s+n)^l}=\begin{cases} O(|t|^{-(l-1)}), & |t|\gg1, \\ O(1), & |t|\ll1, \end{cases}$ where $l\in\mathbb{Z}_{\geq2}$.
(ii) $\displaystyle \sum_{n=1}^\infty\left(\frac{1}{s+n}-\frac{1}{n}\right)=\begin{cases} -\log|t|+i{\pi}{\rm sgn}(t)/2-\gamma+O(|t|^{-1}),& |t|\gg1, \\ O(1), & |t|\ll1, \end{cases}$\
where $\gamma$ is the Euler’s constant given by $\gamma=1-\int_1^\infty\{u\}u^{-2}du$.
Using partial summation we have $$\begin{aligned}
\sum_{n=1}^\infty\frac{1}{(s+n)^l}=
l\int_1^\infty\frac{u-\{u\}}{(s+u)^{l+1}}du
\ll \int_1^\infty\left(\frac{1}{|s+u|^l}+\frac{|s|}{|s+u|^{l+1}}\right)du. \label{HP11}\end{aligned}$$ The estimates $$\begin{aligned}
|u+s|\geq \begin{cases}|{\rm Im}(u+s)|\geq |s\sin\varepsilon|, & u\leq|s| {\rm\:and\:}\arg s\in[-\pi+\varepsilon,\pi-\varepsilon], \\ |{\rm Re}(u+s)|\geq |s|\cos\varepsilon, & u\leq|s| {\rm\:and\:}\arg s\in[-\varepsilon,\varepsilon],\\ |{\rm Re}(u+s)|\geq u, & u\geq|s|, \end{cases} \label{IFTE}\end{aligned}$$ give that the right-hand side of is $$\begin{aligned}
&\ll\int_1^{|s|}\left(\frac{1}{|s\sin\varepsilon|^l}+\frac{|s|}{|s\sin\varepsilon|^{l+1}}\right)du+\int_{|s|}^\infty\left(\frac{1}{u^l}+\frac{|s|}{u^{l+1}}\right)du
\ll\frac{1}{|s|^{l-1}} \label{HP12}\end{aligned}$$ for any $s\in\mathbb{C}\setminus D$ and $l\in\mathbb{Z}_{\geq2}$. Combining and , and estimating $|s|^{-(l-1)}$ we obtain the formula (i). Similally to (i), we calculate $$\begin{aligned}
&\sum_{n=1}^\infty \left(\frac{1}{s+n}-\frac{1}{n}\right)\notag\\
&=\int_1^\infty\left(\frac{1}{(s+u)^2}-\frac{1}{u^2}\right)(u-\{u\})du\notag\\
&=\int_1^\infty\left(\frac{1}{s+u}-\frac{1}{u}\right)du+\int_1^\infty\frac{(-s)}{(s+u)^2}du-\int_1^\infty\frac{\{u\}}{(s+u)^2}du+1-\gamma. \label{HP21}\end{aligned}$$ Then for $s\in\mathbb{C}\setminus D$ the first and second term of right-hand side of are $$\begin{aligned}
&=-\log(s+1)\notag\\
&=-\log s+O(|s|^{-1})=\begin{cases} -\log|t|+i\pi{\rm sgn}(t)/2+O(|t|^{-1}), & |t|\gg1, \\ O(1), & |t|\ll1 \end{cases} \label{HP22}\end{aligned}$$ and $$\begin{aligned}
=-1+\frac{1}{1+s}=\begin{cases} -1+O(|t|^{-1}), & |t|\gg1, \\ O(1), & |t|\ll1. \end{cases} \label{HP23}\end{aligned}$$ respectively, where and $\log s=\log|t|+i\pi{\rm sgn}(t)/2+O(|t|^{-1})$ (see [@GD1 p.335]) were used. By combining – the formula (ii) is obtained.
Using the infinite product of $\Gamma(s)$ and applying the above lemma with $F=\chi_F$, we obtain the approximate formula for $(\chi_F^{(r)}/\chi_F)(s)$ as follows:
\[CHIDF\] For any $F\in\mathcal{S}$ and $r\in\mathbb{Z}_{\geq0}$ the function $(\chi_F^{(r)}/\chi_F)(s)$ has pole of order $r$ at $s=-(\mu_j+n)/\lambda_j,\:1+(\overline{\mu_j}+n)/\lambda_j$ where $n\in\mathbb{Z}_{\geq0}$ and $j\in\left\{1,\dots,q\right\}$. Put $$\begin{aligned}
D&=\{z\in\mathbb{C}\mid a\leq{\rm Re\;}s\leq b\}, \\
E_1&=\left\{z\in\mathbb{C}\:\left|\:{\rm Re\;}z<\max_{j}\tfrac{-{\rm Re\;}\mu_j+\delta}{\lambda_j},\:\:\min_{j}\tfrac{-{\rm Im\;}\mu_j-1}{\lambda_j}<{\rm Im\;}z<\max_{j}\tfrac{-{\rm Im\;}\mu_j+1}{\lambda_j}\right.\right\},\\
E_2&=\left\{z\in\mathbb{C}\:\left|\:{\rm Re\;}z>\max_{j}\tfrac{{\rm Re\;}\mu_j-\delta}{\lambda_j}-1,\:\:\min_{j}\tfrac{{\rm Im\;}\mu_j-1}{\lambda_j}+1<{\rm Im\;}z<\max_{j}\tfrac{{\rm Im\;}\mu_j+1}{\lambda_j}+1\right.\right\} \end{aligned}$$ where $a,b\in\mathbb{R}$ and $\delta\in\mathbb{R}_{>0}$. Then for any $s\in\mathbb{C}\setminus(E_1\cup E_2)$ we have $$\begin{aligned}
\frac{\chi_F^{(r)}}{\chi_F}(s)=\begin{cases} \left(-\log(C_F|t|^{d_F})\right)^m+O\left(\dfrac{(\log|t|)^{m-1}}{|t|}\right), & |t|\gg1, \\ O(1), & |t|\ll1 \end{cases}\end{aligned}$$ where $C_F=(Q\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{2}$.
First we check the location and order of pole for $(\chi_F^{(m)}/\chi_F)(s)$. Taking logarithmic differentiation in the both-hand side of $(1/\Gamma)(s)=se^{\gamma s}\prod_{n=1}^\infty(1+s/n)e^{-s/n}$ and we have $$\begin{aligned}
-\frac{\Gamma'}{\Gamma}(s)&=\frac{1}{s}+\gamma+\sum_{n=1}^\infty\left(\frac{1}{s+n}-\frac{1}{n}\right), \label{DFCH2}\\
\frac{\chi_F'}{\chi_F}(s)&=-2\log Q-\sum_{j=1}^q\lambda_j\left(\frac{\Gamma'}{\Gamma}(\lambda_js+\mu_j)+\frac{\Gamma'}{\Gamma}(\lambda_j(1-s)+\overline{\mu_j})\right) \label{DFCH1}\end{aligned}$$ respectively. Put $G^{(l)}(s)=(d^{l-1}/ds^{l-1})G^{(1)}(s)$ and let $G^{(1)}(s)$ be the right-hand side of . Applying to , we get $$\begin{aligned}
G^{(l)}(s)=\begin{cases}
\displaystyle-2\log Q+d_F\gamma+\sum_{j=1}^q\lambda_j\Biggl(\frac{1}{\lambda_js+\mu_j}+\frac{1}{\lambda_j(1-s)+\overline{\mu_j}}+ & \\
\displaystyle+\sum_{n=1}^\infty\left(\frac{1}{\lambda_js+\mu_j+n}-\frac{1}{n}+\frac{1}{\lambda_j(1-s)+\overline{\mu_j}+n}-\frac{1}{n}\right)\Biggr), & l=1, \\
\displaystyle \sum_{j=1}^q\lambda_j^l\sum_{n=0}^\infty\left(\frac{(-1)^{l-1}(l-1)!}{(\lambda_js+\mu_j+n)^l}+\frac{(l-1)!}{(\lambda_j(1-s)+\overline{\mu_j}+n)^l}\right), & l\in\mathbb{Z}_{\geq2}.
\end{cases}\label{GLSAF}\end{aligned}$$ Hence $G^{(l)}(s)$ has pole of order $l$ at $s=-(\mu_j+n)/\lambda_j,\;1+(\overline{\mu_j}+n)/\lambda_j\;(n\in\mathbb{Z}_{\geq0})$. By using Lemma \[LIB\] the first statement of Lemma \[CHIDF\] is showed.
Lastly we shall approximate $G^{(l)}(s)$. Since ${\rm Re}(\lambda_js+\mu_j)<\delta$, $|{\rm Im}(\lambda_js+\mu_j)|<1$ for $s\in E_1$ and ${\rm Re}(\lambda_j(1-s)+\mu_j)<\delta$, $|{\rm Im}(\lambda_j(1-s)+\overline{\mu_j})|<1$ for $s\in E_2$, Lemma \[HAFF\] gives that $$\begin{aligned}
G^{(j)}(s)=\begin{cases}
\displaystyle O(|t|^{-(j-1)}), & j\in\mathbb{Z}_{\geq2}, |t|\gg1, \\ O(1), & j\in\mathbb{Z}_{\geq1}, |t|\ll1. \end{cases}\end{aligned}$$ for $s\in\mathbb{C}\setminus(E_1\cup E_2)$. Especially in the case of $j=1$ and $|t|\gg1$, since ${\rm sgn}({\rm Im}(\lambda_js+\mu_j))+{\rm sgn}({\rm Im}(\lambda_j(1-s)+\overline{\mu_j}))=0$, $G^{(1)}(s)$ is approximated as $$\begin{aligned}
G^{(1)}(s)&=-2\log Q+d_F\gamma+\sum_{j=1}^q\lambda_j(-2\log|\lambda_jt+{\rm Im\;}\mu_j|-2\gamma)+O(|t|^{-1})\\
&=-\log\bigl((Q\lambda_1^{\lambda_1}\cdots \lambda_q^{\lambda_q})^2|t|^{d_F}\bigr)+O(|t|^{-1})\end{aligned}$$ for $s\in\mathbb{C}\setminus(E_1\cup E_2)$ and $|t|\gg1$. By Lemma \[LIB\] a desired approximation is obtained: $$\begin{aligned}
\frac{\chi_F^{(m)}}{\chi_F}(s)=&(G^{(1)}(s))^m+O(|G^{(1)}(s)|^{m-2}|G^{(2)}(s)|)\\
=&\begin{cases} (-d_F\log(C_F|t|))^m+O(|t|^{-1}{(\log t)^{m-1}}), & |t|\gg1, \\ O(1), & |t|\ll1. \end{cases}\end{aligned}$$
Next in order to estimate the gamma-factors of and , we shall use the following estimates:
For $a,b\in\mathbb{R}$ put $D:=\{z\in\mathbb{C}\mid a\leq{\rm Re\:}z\leq b\}$ and $E_{-}:=\{z\in\mathbb{C}\mid {\rm Re\:}z<1/2, |{\rm Im\:}z|<1\}$. Then for any fixed $s\in D$ and $C_0\in\mathbb{R}_{>0}$ we have $$\begin{aligned}
\left|\frac{\Gamma(s+w)}{\Gamma(s)}
(e^{-i\frac{\pi}{2}{\rm sgn}(t)})^{w}\right|
\leq\begin{cases}\displaystyle C_1\frac{(1+|t+v|)^{\sigma+u-1/2}}{|t|^{\sigma-1/2}}, & \text{if\;} s+w\in D\setminus E_{-}, \\ C_2|t|^{u}, & \text{if\;} |w|\leq C_0\sqrt{|t|}. \end{cases}\end{aligned}$$ where $C_1$ and $C_2$ are constants.
Replace $s\mapsto\lambda_js+\mu_j$ and $w\mapsto\lambda_jw$ in the above lemma. Using the trivial estimate $(1+|\lambda_j(t+v)+{\rm Im\:}\mu_j|)^{\lambda_j(\sigma+u)+{\rm Re\:}\mu_j-1/2}\asymp \lambda_j^{\lambda_ju}(1+|t+v|)^{\lambda_j(\sigma+u)+{\rm Re\:}\mu_j-1/2}$ for $s\in D$ and $s+w\in D\setminus E_{1}$, and multiplying the above formula for $j\in\{1,\dots,q\}$, a desired estimate is obtained:
\[GFEST\] Let $D$ and $E_1$ be those of Lemma \[CHIDF\] respectively. Then for any fixed $s\in D$ and $c_0\in\mathbb{R}_{>0}$ we have $$\begin{aligned}
&\left|\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+\mu_j)}{\Gamma(\lambda_js+\mu_j)}
(e^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}\right|\\
&\leq\begin{cases}\displaystyle c_1(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{u}\frac{(1+|t+v|)^{\frac{d_F(\sigma+u)+e_F-q}{2}}}{|t|^{\frac{d_F\sigma+e_F-q}{2}}}, & \text{if\;}s+w\in D\setminus E_1,\\
c_2(\lambda_1^{\lambda_1}\cdots\lambda_r^{\lambda_r}|t|^{\frac{d_F}{2}})^u, & \text{if\;} |w|\leq c_0\sqrt{|t|}, \end{cases}\end{aligned}$$ where $e_F:=2\sum_{j=1}^q{\rm Re\;}\mu_j$ and $c_1$, $c_2$ are constants depending on $\lambda_1, \mu_1, \dots, \lambda_q, \mu_q$.
By using Stirling’s formula and residue theorem the functions $\gamma_{j}^{(r)}(s;\rho)$ and\
$\delta_{j}^{(r)}(s;\rho)$ are approximate as follows:
\[RCDE\] For any $j, r\in\mathbb{Z}_{\geq0}$ and $s\in\mathbb{C}: |t|\gg1$ we have $$\begin{aligned}
{\gamma_{j}^{(r)}}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)=&\begin{cases} \displaystyle (\chi_F^{(r)}/\chi_F)(1-s), & j=0, \\
\displaystyle O({|t|^{-1}}(\log|t|)^r), & j=1, \\ \displaystyle O({|t|^{-\frac{j}{2}}}(\log|t|)^r), & j\in\mathbb{Z}_{\geq2}. \end{cases} \label{GJEE}\end{aligned}$$ The function ${\delta}_{j}^{(r)}(s;(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{-\frac{2}{d_F}}|t|^{-1})$ equals the right hand side of .
Since $|w|\ll\sqrt{|t|}$ for $w\in\mathcal{F}$, from Lemma \[GFEST\] we can obtain a desired formula in the case of $j\in\mathbb{Z}_{\geq2}$: $$\begin{aligned}
\gamma_{j}^{(r)}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)\ll\int_{\mathcal{F}}\frac{(\log|t|)^{r}}{|t|^{\frac{j+1}{2}}}|dw|\ll\frac{(\log|t|)^{r}}{|t|^{\frac{j}{2}}}. $$ Using Cauchy’s residue theorem we have $\gamma_0^{(r)}(s,\rho)=(\chi_F^{(r)}/\chi_F)(1-s)$ and $$\begin{aligned}
&\gamma_{1}^{(r)}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)\notag\\
&={\frac{\chi_F^{(r)}}{\chi_F}}(1-s)-\frac{{g_F}(s-1)}{{g_F}(s)}\frac{\chi_F^{(r)}}{\chi_F}(2-s)\prod_{j=1}^q\frac{\Gamma(\lambda_j(s-1)+{\mu_j})}{\Gamma(\lambda_js+{\mu_j})}(\lambda_jit)^{\lambda_j} \label{G1A1}\end{aligned}$$ where we used $|t|e^{\frac{\pi}{2}i{\rm sgn}(t)}=it$. It is clear that $$\begin{aligned}
\frac{{g_F}(s-1)}{{g_F}(s)}=\frac{1+O(|s|^{-1})}{1+O(|s|^{-1})}=1+O\left(\frac{1}{|t|}\right). \label{G1A2} \end{aligned}$$ Stirling’s formula $\Gamma(s)=\sqrt{2\pi}s^{s-1/2}e^{-s}(1+O(|s|^{-1}))$ and the trivial approximation $\lambda_js+\mu_j=i\lambda_jt(1+O(|t|^{-1}))$ give that $$\begin{aligned}
\frac{\Gamma(\lambda_j(s-1)+\mu_j)}{\Gamma(\lambda_js+{\mu_j})}
&=\frac{(i\lambda_jt)^{\lambda_j(s-1)+{\mu_j}-1/2}e^{-i\lambda_jt}(1+O(|t|^{-1}))}{(i\lambda_jt)^{\lambda_js+{\mu_j}-1/2}e^{-i\lambda_jt}(1+O(|t|^{-1}))}\notag\\
&=(\lambda_jit)^{-\lambda_j}(1+O(|t|^{-1})), \label{G1A3}\end{aligned}$$ Combining – and using Lemma \[CHIDF\] we obtain $$\begin{aligned}
\gamma_{1}^{(r)}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)
=&{\frac{\chi_F^{(r)}}{\chi_F}}(1-s)-\frac{\chi_F^{(r)}}{\chi_F}(2-s)+O\left(\frac{1}{|t|}\left|\frac{\chi_F^{(r)}}{\chi_F}(2-s)\right|\right)\notag\\
=&O(|t|^{-1}{(\log|t|)^{r}}).$$ By the same discussion, the approximate formula of ${\delta}_{j}^{(r)}(s;1/((\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|))$ is also obtained.
Finally, in order to prove Theorem \[THM2\] from Theorem \[THM1\], we introduce new functions. For $\varphi\in\mathcal{R}$, $\alpha\in\mathbb{R}_{\geq0}$ and $|t|\gg1$ we set $$\begin{aligned}
\xi(\rho)&:=\begin{cases} 1, & \rho\in[0,1], \\ 0, & \rho\in[1,\infty). \end{cases} \\
\varphi_\alpha(\rho)&:=\begin{cases} 1, & \rho\in[0,1-(2|t|^\alpha)^{-1}], \\ \varphi(1+(\rho-1)|t|^\alpha), & \rho\in[1-(2|t|^\alpha)^{-1},1+|t|^{-\alpha}], \\ 0, & \rho\in[1+|t|^{-\alpha},\infty), \end{cases}\\
\varphi_{0\alpha}(\rho)&:=1-\varphi_\alpha(1/\rho).\end{aligned}$$ Then these function have the following properties:
\[CFF\] For any $\alpha\in\mathbb{R}_{\geq0}$ and $\varphi\in\mathcal{R}$ we satisfy the following statements (i)–(iv):
(i) $\varphi_\alpha,\varphi_{0\alpha}\in\mathcal{R}$.
(ii) $(\varphi_\alpha-\xi)(\rho)=0, \ (\varphi_{0\alpha}-\xi)(\rho)=0$ for $\rho\in[0,1-(2|t|^\alpha)^{-1}]\cup[1+|t|^{-\alpha},\infty)$.
(iii) $ \varphi_\alpha^{(j)}(\rho)=0, \:\: \varphi_{0\alpha}^{(j)}(\rho)=0$ for $j\in\mathbb{Z}_{\geq1}$ and $\rho\in[0,1-(2|t|^\alpha)^{-1}]\cup[1+|t|^{-\alpha},\infty)$.
(iv) $\varphi_\alpha^{(j)}(\rho)\ll|t|^{\alpha j}, \ \varphi_{0\alpha}^{(j)}(\rho)\ll|t|^{\alpha j}, \ \|\varphi_\alpha^{(j)}\|_1\ll|t|^{\alpha(j-1)}, \ \|\varphi_{0\alpha}^{(j)}\|_1\ll|t|^{\alpha(j-1)}$ for $\rho\in[0,\infty)$ and $j\in\mathbb{Z}_{\geq0}$.
Proof of Theorem \[THM1\] {#THM1P}
=========================
First for $s=\sigma+it: \sigma\in[0,1], |t|\gg1$, we shall use Cauchy’s integral theorem in the region $$D_\sigma=\{w\in\mathbb{C}\mid w=u+iv,\;-1/2-\sigma\leq u\leq 3/2-\sigma,\;v\in\mathbb{R}\}.$$ Then from and Lemma \[LIB\] the following lemma is obtained:
\[PRO1\] For any $m\in\mathbb{Z}_{\geq0}$, $F\in\mathcal{S}$, $s=\sigma+it: \sigma\in[0,1], |t|\gg1$, $\varphi\in\mathcal{R}$ and $x\in\mathbb{R}_{>0}$ we have $$\begin{aligned}
F^{(m)}(s)=G_m(s;x,\varphi)+\chi_F(s)\sum_{r=0}^m(-1)^r\binom{m}{r}H_r(1-s;1/x,\varphi_0)\end{aligned}$$ where $G_r(s;x,\varphi)$ and $H_r(s;x,\varphi)$ are given by $$\begin{aligned}
&\hspace{-1em}{G_r}(s;x,\varphi)\notag\\
=&\frac{1}{2\pi i}\int_{(\frac{3}{2}-\sigma)}\frac{{g_{F}}(s+w)}{{g_{F}}(s)}\frac{K_{\varphi}(w)}{w}{\frac{\chi_F^{(m-r)}}{\chi_F}}(1-(s+w)){F^{(r)}}(s+w)\times\notag\\
&\times\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+{\mu_j})}{\Gamma(\lambda_js+{\mu_j})}
(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}dw, \label{GSDEF}\\
&\hspace{-1em}H_r(s;x,\varphi)\notag\\
=&\frac{1}{2\pi i}\int_{(\frac{3}{2}-\sigma)}\frac{\overline{g_{F}}(s+w)}{\overline{g_{F}}(s)}\frac{K_{\varphi}(w)}{w}{\frac{\chi_F^{(m-r)}}{\chi_F}}(1-(s+w))\overline{F^{(r)}}(s+w)\times\notag\\
&\times\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+\overline{\mu_j})}{\Gamma(\lambda_js+\overline{\mu_j})}(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}dw \label{HSDEF}\end{aligned}$$ respectively. Here $(3/2-\sigma)$ denotes $\left\{3/2-\sigma+iv\mid v\in\mathbb{R}\right\}$.
For $r\in\{0,1,\dots,m\}$ and $|v|\gg|t|$ let $$\begin{aligned}
I_r(v)=\:&\frac{1}{2\pi i}\int_{-1/2-\sigma+iv}^{3/2-\sigma+iv}\frac{g_{F}(s+w)}{g_{F}(s)}\frac{K_\varphi(w)}{w}\frac{\chi_F^{(m-r)}}{\chi_F}(s+w)F^{(r)}(s+w)\times\notag\\
&\times\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+\mu_j)}{\Gamma(\lambda_js+\mu_j)}
(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}dw. \label{DFIV}\end{aligned}$$
First we shall show that the integrand of is holomorphic in $D_\sigma\setminus\{0\}$. From Lemma \[KPL\], $K_\varphi(w)(Q^{{2}/{d_F}}xe^{-i\pi{\rm sgn}(t)/2})^{{d_Fw}/{2}}$ is holomorphic in $D_\sigma$. Since $F^{(m)}(w+s)$ has pole of order $p_F+m$ at most at $w=1-s$, we see that $$\begin{aligned}
(s+w-1)^{p_{F}+m}F^{(m)}(s+w) \label{GARR}\end{aligned}$$ is holomorphic in $w\in D_\sigma$.
Here we shall consider the holomorphicity of gamma-factor of . In the case of ${\rm Re\:}\mu_j>\lambda_j/2$, since ${\rm Re}(\lambda_j(s+w)+\mu_j)\geq -\lambda_j/2+{\rm Re\:}\mu_j>0$ for $w\in D_\sigma$ we see that $A_j(s+w)\Gamma(\lambda_j(s+w)+\mu_j)=\Gamma(\lambda_j(s+w)+\mu_j)$ is holomorphic in $D_\sigma$. On the other hand, in the case of ${\rm Re\;}\mu_j\leq\lambda_j/2$, we have ${\rm Re}(\lambda_j(s+w)+\mu_j+[\lambda_j/2-{\rm Re\;}\mu_j]+1)=1-\{\lambda_j/2-{\rm Re\;}\mu_j\}>0$ for $w\in D_\sigma$. Hence the functional equation for $\Gamma(s)$ implies that $$\begin{aligned}
A_j(s)\Gamma(\lambda_j(s+w)+\mu_j)=\Gamma(\lambda_j(s+w)+\mu_j+[\lambda_j/2-{\rm Re\:}\mu_j]+1) \label{GBRR}\end{aligned}$$ is holomorphic in $D_\sigma$.
Next we consider the existence of pole for $(\chi_F^{(m-r)}/\chi_F)(s+w)$ in $D_{\sigma}$. In the case of ${\rm Re\;}\mu_j> \lambda_j/2$, since $$\begin{aligned}
&{\rm Re}(-s-(\mu_j+n)/\lambda_j)\leq-\sigma-({\rm Re\;}\mu_j)/\lambda_j<-1/2-\sigma,\\
&{\rm Re}(1-s+(\overline{\mu_j}+n)/\lambda_j)\geq1-\sigma+{\rm Re\;}\mu_j/\lambda_j> 3/2-\sigma\end{aligned}$$ for $n\in\mathbb{Z}_{\geq0}$, we see that $(\chi_F^{(m-r)}/\chi_F)(s+w)$ does not have pole in $D_{\sigma}$. On the other hand, in the case of ${\rm Re\;}\mu_j\leq\lambda_j/2$, since $$\begin{aligned}
&{\rm Re}(-s-(\mu_j+n)/\lambda_j)\in\begin{cases}[-\sigma-1/2,-\sigma], & n\in\mathbb{Z}_{\geq0}\cap\mathbb{Z}_{\leq[\lambda_j/2-{\rm Re\;}\mu_j]}, \\ (-\infty,-1/2-\sigma), & n\in\mathbb{Z}_{>[\lambda_j/2-{\rm Re\;}\mu_j]}, \end{cases} \\ &{\rm Re}(1-s+(\overline{\mu_j}+n)/\lambda_j)\in\begin{cases} [1-\sigma,3/2-\sigma], & n\in\mathbb{Z}_{\geq0}\cap\mathbb{Z}_{\leq[\lambda_j/2-{\rm Re\;}\mu_j]}, \\ (3/2-\sigma), & n\in\mathbb{Z}_{>[\lambda_j/2-{\rm Re\;}\mu_j]}, \end{cases} \end{aligned}$$ from Lemma \[CHIDF\] the points $w=-s-(\mu_j+n)/\lambda_j,\;1-s+(\overline{\mu_j}+n)/\lambda_j$ for $n\in\mathbb{Z}_{\geq0}\cap\mathbb{Z}_{\leq[\lambda_j/2-{\rm Re\;}\mu_j]}$ are pole of order $(m-r)$ for $(\chi_F^{(m-r)}/\chi_F)(s+w)$ in $D_\sigma$. Therefore $$\begin{aligned}
\prod_{j=1}^q(A_j(s+w)\overline{A_j}(1-(s+w)))^{m-r}\frac{\chi_F^{(m-r)}}{\chi_F}(s+w) \label{GCRR}\end{aligned}$$ is holomorphic in $D_{\sigma}$. Combining – we find that the integrand of is holomorphic $w\in D_\sigma\setminus\{0\}$.
Next we shall show $I_r(v)\to0$ when $v\to\infty$. Trivial estimate gives $$\begin{aligned}
\frac{g_{F}(s+w)}{g_{F}(s)}\ll|v|^{2(p_{F}+m)+(m+1)f_F} \label{GGEE}\end{aligned}$$ where $f_F=2\sum_{j=1}^q\max\{0,[\lambda_j/2-{\rm Re\:}\mu_j]\}$. To obtain an estimate of $F^{(r)}(s+w)$ in $w\in D_\sigma$, we shall consider estimates of $\chi_F(s)$ and $(\chi_F^{(r)}/\chi_F)(s)$. By the same method of and $it=|t|e^{i\frac{\pi}{2}{\rm sgn}(t)}$, we have $$\begin{aligned}
\frac{\Gamma(\lambda_j(1-s)+\overline{\mu_j})}{\Gamma(\lambda_js+\mu_j)}
&=\frac{(-i\lambda_jt)^{\lambda_j(1-s)+\overline{\mu_j}-1/2}e^{i\lambda_jt}}{(i\lambda_j t)^{\lambda_js+\mu_j-1/2}e^{-i\lambda_jt}}(1+O(|t|^{-1}))\notag\\
&=(\lambda_j|t|)^{\lambda_j(1-2s)-2i{\rm Im\:}\mu_j}e^{i(2\lambda_j+\frac{\pi}{2}(-\lambda_j-2{\rm Re\:}\mu_j+1){\rm sgn}(t))}(1+O(|t|^{-1}))\notag\\
&=(\lambda_j|t|)^{\lambda_j(1-2\sigma)}e^{i\theta_j(t)}(1+O(|t|^{-1}))\label{GFAF0}\end{aligned}$$ where $\theta_j(t)=2\lambda_j+{\rm sgn}(t)\cdot(-\lambda_j-2{\rm Re\:}\mu_j+1)\pi/2-\log(\lambda_j|t|)^{2(\lambda_j+{\rm Im\:}\mu_j)}$. Hence we obtain $$\begin{aligned}
\chi_F(s)=&\omega {C_F}^{\frac{1}{2}-\sigma}e^{i\theta_F(t)}|t|^{d_F(\frac{1}{2}-\sigma)}(1+O(|t|^{-1}))\label{CHI0E}\end{aligned}$$ where $\theta_F(t)=d_F+{\rm sgn}(t)\cdot(-d_F/2-e_F+q)\pi/2-\log(C_F'|t|^{d_F+e_F})$ and $C_F'=(Q\prod_{j=1}^q\lambda_j^{\lambda_j+{\rm Im\:}\mu_j})^2$. Combining , and Lemma \[CHIDF\] we have $$\begin{aligned}
F^{(r)}(s)\ll |t|^{d_F(\frac{1}{2}-\sigma)}\sum_{j=0}^r(\log|t|)^{r-j}|F^{(j)}(1-\sigma+it)|\ll |t|^{d_F(\frac{1}{2}-\sigma)}(\log|t|)^r \end{aligned}$$ for $s\in\mathbb{C}: \sigma\in\mathbb{R}_{<0}, |t|\gg1$. Phragmén-Lindelöf theorem implies $$\begin{aligned}
F^{(r)}(s+w)\ll|v|^{\frac{d_F}{2}(\frac{3}{2}-(\sigma+u))}(\log|v|)^{r} \label{DFEE}\end{aligned}$$ uniformly for $u\in[-1/2-\sigma,3/2-\sigma]$. By , , Lemmas \[CHIDF\], \[GFEST\] we get $$\begin{aligned}
I_r(v)
&\ll\int_{-1/2-\sigma}^{3/2-\sigma}|v|^{2(p_F+m)+(m+1)f_F}\frac{\|\varphi^{(l+1)}\|_1}{|v|^{l+1}}(\log|v|)^r\times\\
&\quad\;\times |v|^{\frac{d_F}{2}(\frac{3}{2}-(\sigma+u))}(\log|v|)^{m}|v|^{\frac{d_F(\sigma+u)+e_F-q}{2}}du\\
&\ll|v|^{\frac{3d_F}{4}+\frac{e_F-q}{2}+2(p_F+m)+(m+1)f_F-l-1}(\log|v|)^m\|\varphi^{(l+1)}\|_1\end{aligned}$$ when $|v|\gg|t|$. Choosing $l\in\mathbb{Z}_{>M_F}$ we find that $I_r(v)\to0$ when $v\to\infty$, where $M_F=3d_F/4+(e_F-q)/2+2(p_F+m)+(m+1)f_F$. Cauchy’s integral theorem gives $$\begin{aligned}
F^{(m)}(s)=&\frac{1}{2\pi i}\left(\int_{(3/2-\sigma)}-\int_{(-1/2-\sigma)}\right)\frac{g_{F}(s+w)}{g_{F}(s)}\frac{K_\varphi(w)}{w}F^{(m)}(s+w)\times\notag\\
&\times\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+\mu_j)}{\Gamma(\lambda_js+\mu_j)}(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}dw. \label{FMRES}\end{aligned}$$ Here the first term of right-hand side of is $$\begin{aligned}
=G_m(s;x,\varphi). \label{FMRES1}\end{aligned}$$ Since implies $$\begin{aligned}
\frac{\chi_F(s+w)}{\chi_F(s)}=\frac{1}{Q^{2w}}\prod_{j=1}^q\frac{\Gamma(\lambda_j(1-s-w)+\overline{\mu_j})}{\Gamma(\lambda_j(1-s)+\overline{\mu_j})}\times \prod_{j=1}^q\frac{\Gamma(\lambda_js+\mu_j)}{\Gamma(\lambda_j(s+w)+\mu_j)},\end{aligned}$$ combining this formula and we obtain $$\begin{aligned}
&\hspace{-1em}F^{(m)}(s+w)\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+\mu_j)}{\Gamma(\lambda_js+\mu_j)}\\
=&\frac{\chi_F(s)}{Q^{2w}}\prod_{j=1}^q\frac{\Gamma(\lambda_1(1-s-w)+\overline{\mu_1})}{\Gamma(\lambda_1(1-s)+\overline{\mu_1})}\sum_{r=0}^m(-1)^{r}\binom{m}{r}\frac{\chi_F^{(m-r)}}{\chi_F}(s+w)\overline{F^{(r)}}(1-s-w).\end{aligned}$$ By replacing $w\mapsto-w$ and using this formula, and $\overline{g_F}(1-s)=g_F(s)$, the second term of right-hand side of is $$\begin{aligned}
=&\chi_F(s)\sum_{r=0}^m(-1)^{r}\binom{m}{r}\times\frac{1}{2\pi i}\int_{(3/2-(1-\sigma))}\frac{\overline{g_{F}}(1-s+w)}{\overline{g_{F}}(1-s)}\frac{K_{\varphi_0}(w)}{w}\times\notag\\
&\times\overline{F^{(r)}}(1-s+w)\frac{\chi_F^{(m-r)}}{\chi_F}(1-(1-s+w))\prod_{j=1}^q\frac{\Gamma(\lambda_j(1-s+w)+\overline{\mu_j})}{\Gamma(\lambda_j(1-s)+\overline{\mu_j})}\times\notag\\
&\times(Q^{\frac{2}{d_F}}x^{-1}e^{-i\frac{\pi}{2}{\rm sgn}(-t)})^{\frac{d_F}{2}w}dw\notag\\
=&\chi_F(s)\sum_{r=0}^m(-1)^{r}\binom{m}{r}H_r(1-s;1/x,\varphi_0). \label{FMICL}\end{aligned}$$ Therefore, from and Proposition \[PRO1\] is obtained.
Next applying residue theorem to the functions ${G}_r(s;x,\varphi)$ and ${H}_r(s;x,\varphi)$, then these functions are approximated as follows:
\[PRO2\] For any $F\in\mathcal{S}$, $m\in\mathbb{Z}_{\geq0}$, $r\in\left\{0,\dots,m\right\}$, $s=\sigma+it: \sigma\in[0,1], |t|\gg1$, $\varphi\in\mathcal{R}$, $l\in\mathbb{Z}_{>M_F}$ and $x, y\in\mathbb{R}_{>0}: \sqrt{C_F}(x|t|)^{\frac{d_F}{2}}=y$, we have $$\begin{aligned}
&\hspace{-1em}{G_r}(s;x,\varphi)\notag\\
=&\sum_{n\leq 2y}\frac{{a_F(n)}(-\log n)^r}{n^s}\sum_{j=0}^l\varphi^{(j)}\left(\frac{n}{y}\right)\left(-\frac{n}{y}\right)^j{\gamma_{j}^{(m-r)}}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots{\lambda_r}^{\lambda_r})^{\frac{2}{d_F}}|t|}\right)+\notag\\
&+O(y^{1-\sigma}(\log y)^{r+\max\{p_F-1,0\}}|t|^{-\frac{l}{2}}(\log|t|)^{m-r}\|\varphi^{(l+1)}\|) \label{AGRB} \\
&\hspace{-1em}{H_r}(s;x,\varphi)\notag\\
=&\sum_{n\leq 2y}\frac{\overline{a_F(n)}(-\log n)^r}{n^s}\sum_{j=0}^l\varphi^{(j)}\left(\frac{n}{y}\right)\left(-\frac{n}{y}\right)^j{\delta_{j}^{(m-r)}}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots{\lambda_r}^{\lambda_r})^{\frac{2}{d_F}}|t|}\right)+\notag\\
&+O(y^{1-\sigma}(\log y)^{r+\max\{p_F-1,0\}}|t|^{-\frac{l}{2}}(\log|t|)^{m-r}\|\varphi^{(l+1)}\|), \label{AHRB}\end{aligned}$$ where $M_F$ is some positive constant, and ${{\gamma}_{j,r}}(s;\rho)$, ${{\delta}_{j,r}}(s;\rho)$ are given by , respectively.
In order to show , we use and write ${F^{(r)}}(s)=(\sum_{n\leq\rho y}+\sum_{n>\rho y})\times\times{a_F(n)}(-\log n)^rn^{-s}$ for ${\rm Re\;}s>1$ and $\rho\in\mathbb{R}_{>0}$. Then $$\begin{aligned}
{G_r}(s;x,\varphi)=I_1+I_2 \label{GHDIV}\end{aligned}$$ where $I_1, I_2$ are given by $$\begin{aligned}
I_1=&\frac{1}{2\pi i}\int_{(\frac{3}{2}-\sigma)}\frac{{g_{F}}(s+w)}{{g_{F}}(s)}
\left(\int_0^\infty\varphi^{(l+1)}(\rho)\rho^{w+l}\sum_{n\leq\rho y}\frac{{a_F(n)}(-\log n)^r}{n^{s+w}}d\rho\right)
\times\notag\\
&\times\frac{(-1)^{l+1}}{w\cdots(w+l)}{\frac{\chi_F^{(m-r)}}{\chi_F}}(s+w)\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+{\mu_j})}{\Gamma(\lambda_1s+{\mu_1})}(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}dw, \label{J1DEF}\\
I_2=&\frac{1}{2\pi i}\int_{(\frac{3}{2}-\sigma)}\frac{{g_{F}}(s+w)}{{g_{F}}(s)}\left(\int_0^\infty\varphi^{(l+1)}(\rho)\rho^{w+l}\sum_{n>\rho y}\frac{{a_F(n)}(-\log n)^r}{n^{s+w}}d\rho\right)\times\notag\\
&\times\frac{(-1)^{l+1}}{w\cdots(w+l)}{\frac{\chi_F^{(m-r)}}{\chi_F}}(s+w)\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+{\mu_j})}{\Gamma(\lambda_js+{\mu_j})}
(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w}dw. \label{J2DEF}\end{aligned}$$ To approximate $I_1$ and $I_2$, we define $L_{\pm j}$, $C_j$ ($j=1,2$) as $$\begin{aligned}
L_{\pm j}=\{\sigma_j\pm iv\mid v\in[\sqrt{t},\infty)\}, \quad
C_j=\{\sigma_j+\sqrt{|t|}e^{-i\pi(\pm1/2+\theta)}\mid \theta\in[0,1]\},\end{aligned}$$ respectively, where $\sigma_1=-1/2-\sigma$ and $\sigma_2=3/2-\sigma$. The residue theorem gives $$\begin{aligned}
I_1=I_1'+{\rm Res\;}(I_1,\mathcal{F}), \quad I_2=I_2' \label{IJDEF}\end{aligned}$$ where $I_1', I_2'$ are replaced by $L_{-1}+C_1+L_{+1}$, $L_{-2}+C_2+L_{+2}$ respectively, and ${\rm Res\;}(I_1,\mathcal{F})$ is the sum of residue for the integrand of in $\mathcal{F}$. Here by using the result $$\begin{aligned}
\int_\mu^\infty\varphi^{(l+1)}(\rho)\rho^{w+l}d\rho=&\sum_{j=0}^l\varphi^{(l-j)}(\mu)\mu^{w+l-j}(-1)^{l-j}(w+l-j+1)\cdots(w+l)+\\
&+(-1)^{l+1}w\cdots(w+l)\int_\mu^\infty\varphi(\rho)\rho^{w-1}d\rho\end{aligned}$$ for $\mu\in\mathbb{R}_{\geq0}$ (see p.337 of [@GD1]) and the residue theorem, ${\rm Res\;}(I_1,\mathcal{F})$ is written as $$\begin{aligned}
&\hspace{-1em}{\rm Res\;}(I_1,\mathcal{F})\notag\\
=&\sum_{n\leq2y}\frac{{a_F(n)}(-\log n)^r}{n^s}\times\frac{1}{2\pi i}\int_{\mathcal{F}}\frac{{g_F}(s+w)}{{g_F}(s)}\frac{(-1)^l}{w\cdots(w+l)}{\frac{\chi_F^{(m-r)}}{\chi_F}}(s+w)\times\notag\\
&\times\left(\int_{\frac{n}{y}}^\infty\varphi^{(l+1)}(\rho)\rho^{w+l}d\rho\right)\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+{\mu_j})}{\Gamma(\lambda_js+{\mu_j})}\left(Q^{\frac{2}{d_F}}\frac{x}{n^{\frac{2}{d_F}}}e^{-i\frac{\pi}{2}{\rm sgn}(t)}\right)^{\frac{d_F}{2}w}dw\notag\\
=&\sum_{n\leq2y}\frac{{a_F(n)}(-\log n)^r}{n^s}\sum_{j=0}^l\varphi^{(j)}\left(\frac{n}{y}\right)\left(-\frac{n}{y}\right)^j{\gamma_{j}^{(m-r)}}\left(s;Q^{\frac{2}{d_F}}\frac{x}{y^{\frac{2}{d_F}}}\right). \label{REI1}\end{aligned}$$
Next we shall estimate $I_1'$ and $I_2'$ as the error term of approximate functional equation. Partial summation and the assumption give $$\begin{aligned}
&\int_{0}^{\infty}\varphi^{(l+1)}(\rho)\rho^{w+l}\sum_{n\leq\rho y}\frac{a_F(n)(-\log n)^r}{n^{s+w}}d\rho\notag\\
&\ll\int_{1/2}^{2}|\varphi^{(l+1)}(\rho)|\rho^{u+l}\left( \frac{(\log\rho{y})^{r+\max\{p_F-1,0\}}}{(\rho y)^{\sigma+u-1}}+\int_1^{\rho{y}}\frac{(\log u)^{r+\max\{p_F-1,0\}}}{u^{\sigma+u}}du \right)d\rho\notag\\
&\ll y^{1-(\sigma+u)}(\log y)^{r+\max\{p_F-1,0\}}\|\varphi^{(l+1)}\|_1\label{IJGEA}\end{aligned}$$ for ${\rm Re}(s+w)\leq-1/2$ and $$\begin{aligned}
&\int_{0}^{\infty}\varphi^{(l+1)}(\rho)\rho^{w+l}\sum_{n>\rho y}\frac{a_F(n)(-\log n)^r}{n^{s+w}}d\rho\notag\\
&\ll \int_{1/2}^2|\varphi^{(l+1)}(\rho)|\rho^{u+l}\left(\int_{\rho{y}}^{\infty}\frac{(\log u)^{r+\max\{p_F-1,0\}}}{u^{\sigma+u}}du\right)d\rho\notag\\
&\ll y^{1-(\sigma+u)}(\log y)^{r+\max\{p_F-1,0\}}\|\varphi^{(l+1)}\|_1. \label{IJGEB}\end{aligned}$$ for ${\rm Re}(s+w)\geq3/2$. From Lemma \[CHIDF\] and \[GFEST\] we get the following estimate: $$\begin{aligned}
&\frac{(-1)^l}{w\cdots(w+l)}{\frac{\chi_F^{(m-r)}}{\chi_F}}(1-(s+w))\prod_{j=1}^q\frac{\Gamma(\lambda_j(s+w)+{\mu_j})}{\Gamma(\lambda_js+{\mu_j})}(Q^{\frac{2}{d_F}}xe^{-i\frac{\pi}{2}{\rm sgn}(t)})^{\frac{d_F}{2}w} \notag\\
&\ll \begin{cases} \displaystyle \frac{(\log|t|)^{m-r}}{|t|^{\frac{l+1}{2}}}(\sqrt{C_F}(x|t|)^{\frac{d_F}{2}})^u, & w\in C_1\cup C_2\cup\mathcal{F}, \\ \displaystyle \frac{(\log|v|)^{m-r}}{|v|^{l+1}}\dfrac{(1+|t+v|)^{\frac{d_F(\sigma+u)+e_F-q}{2}}}{|t|^{\frac{d_F\sigma+e_F-q}{2}}}(\sqrt{C_F}x^{\frac{d_F}{2}})^u, & w\in L_{\pm1}\cup L_{\pm2}. \end{cases} \label{IJGE1}\end{aligned}$$ Trivial estimate gives $$\begin{aligned}
\frac{{g_F}(s+w)}{{g_F}(s)}\ll\begin{cases} 1, & w\in C_1\cup C_2\cup \mathcal{F}, \\ \dfrac{(1+|t+v|)^{2(m+p_{F})+(m+1)f_F}}{|t|^{2(m+p_{F})+(m+1)f_F}}, & w\in L_{\pm1}\cup L_{\pm2}. \end{cases} \label{IJGE3}\end{aligned}$$ Hence by combining –, $I_1'$ is estimated as $$\begin{aligned}
I_1'&\ll\int_{C_{1}}\frac{(\log|t|)^{m-r}}{|t|^{\frac{l+1}{2}}}(\sqrt{C_F}(x|t|)^{\frac{d_F}{2}})^uy^{1-(\sigma+u)}(\log y)^{r+\max\{p_F-1,0\}}\|\varphi^{(l+1)}\|_1|dw|+\notag\\
&\quad\;+\int_{L_{\pm1}}\frac{(\log|v|)^{m-r}}{|v|^{l+1}}\frac{(1+|t+v|)^{\frac{d_F(\sigma+u)+e_F-q}{2}+2(m+p_{F})+(m+1)f_F}}{|t|^{\frac{d_F(\sigma+u)+e_F-q}{2}+2(m+p_{F})+(m+1)f_F}}\times \notag\\
&\quad\;\times(\sqrt{C_F}(x|t|)^{\frac{d_F}{2}})^{u} y^{1-(\sigma+u)}(\log y)^{r+\max\{p_F-1,0\}}\|\varphi^{(l+1)}\|_1dv\notag\\
&\ll y^{1-\sigma}(\log y)^{r+\max\{p_F-1,0\}}|t|^{-\frac{l}{2}}(\log|t|)^{m-r}\|\varphi^{(l+1)}\|_1 \label{JDE2}\end{aligned}$$ under the condition $\sqrt{C_F}(x|t|)^{\frac{d_F}{2}}=y$, where the following estimate was used: $$\begin{aligned}
&\left(\int_{\pm\sqrt{|t|}}^{\pm\frac{|t|}{2}}+\int_{\pm\frac{|t|}{2}}^{\pm2|t|}+\int_{\pm2|t|}^{\pm\infty}\right)
\frac{(1+|t+v|)^{M_F-d_F}}{|t|^{M_F-d_F}}\frac{(\log|v|)^{m-r}}{|v|^{l+1}}dv
=:J_1+J_2+J_3.\end{aligned}$$ Since $1\ll1+|t+v|\ll|t|$ when $v\in[-2|t|,-|t|/2]\cup[|t|/2,2|t|]$ and $$\begin{aligned}
1+|t+v|\asymp\begin{cases} |t| & \text{when\;} v\in[-|t|/2,-\sqrt{|t|}]\cup[\sqrt{|t|},|t|/2], \\ |v| & \text{when\;} v\in(-\infty,-2|t|]\cup[2|t|,\infty), \end{cases}\end{aligned}$$ $J_j$ ($j=1,2,3$) were estimated as $$\begin{aligned}
J_1&\ll(\log|t|)^{m-r}\int_{\pm\sqrt{|t|}}^{\pm\frac{|t|}{2}}\frac{dv}{|v|^{l+1}}\ll|t|^{-\frac{l}{2}}(\log|t|)^{m-r},\\
J_2&\ll|t|^{\max\{0,d_F-M_F\}-(l+1)}(\log|t|)^{m-r}\int_{\pm\frac{|t|}{2}}^{\pm2|t|}\frac{dv}{(1+|t+v|)^{\max\{0,d_F-M_F\}}}\\
&\ll|t|^{-l+\max\{0,d_F-M_F\}}(\log|t|)^{m-r}\ll|t|^{-\frac{l}{2}}(\log|t|)^{m-r},\\
J_3&\ll\frac{1}{|t|^{M_F-d_F}}\int_{\pm2|t|}^{\pm\infty}
\frac{(\log|v|)^{m-r}}{|v|^{l+1-M_F+d_F}}dv\ll|t|^{-l}(\log|t|)^{m-r},\end{aligned}$$ where $l$ was chosen as $l\in\mathbb{Z}_{\geq 2\max\{0,d_F-M_F\}}$. By the same discussion of estimate of $I_1'$, the estimate of $I_2'$ is obtained: $$\begin{aligned}
I_2'\ll& y^{1-\sigma}(\log y)^{r+A}|t|^{-\frac{l}{2}}(\log|t|)^{m-r}\|\varphi^{(l+1)}\|_1. \label{JDE1} \end{aligned}$$ Therefore combining –, – we obtain . Since we can obtain from the same discussion in the above, Proposition \[PRO2\] is showed.
Finally, for any $x\in\mathbb{R}_{>0}$ we choose parameters $y_1, y_2\in\mathbb{R}_{>0}$ such that $\sqrt{C_F}(x|t|)^{\frac{d_F}{2}}\\=y_1$ and $\sqrt{C_F}(x^{-1}|t|)^{\frac{d_F}{2}}=y_2$, that is, $y_1y_2=C_F|t|^{d_F}$. By combining Propositions \[PRO1\], \[PRO2\] and using , $F^{(m)}(s)$ is approximated as $$\begin{aligned}
&\hspace{-1em}
F^{(m)}(s)\notag\\
=&\sum_{n\leq 2y_1}\frac{a_F(n)(-\log n)^m}{n^s}\sum_{j=0}^l\varphi^{(j)}\left(\frac{n}{y_1}\right)\left(-\frac{n}{y_1}\right)^j\gamma_{j}^{(0)}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)+\notag\\
&+\chi_F(s)\sum_{r=0}^m(-1)^r\binom{m}{r}\sum_{n\leq 2y_2}\frac{\overline{a_F(n)}(-\log n)^r}{n^{1-s}}\sum_{j=0}^r\varphi_0^{(j)}\left(\frac{n}{y_2}\right)\left(-\frac{n}{y_2}\right)^j\times\notag\\
&\times{\delta}_{j}^{(m-r)}\left(1-s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_q^{\lambda_q})^{\frac{2}{d_F}}|t|}\right)+O(y_1^{1-\sigma}(\log y_1)^{m+A}|t|^{-\frac{l}{2}}\|\varphi^{(l+1)}\|_1)+\notag\\
&+O\left(y_2^\sigma|t|^{d_F(\frac{1}{2}-\sigma)-\frac{l}{2}}\|\varphi_0^{(l+1)}\|_1\sum_{r=0}^m(\log y_2)^{r+A}(\log|t|)^{m-r}\right). $$ Using Lemma \[RCDE\] and dividing sums of $j\in\mathbb{Z}_{\geq0}$ to term of $j=0$ and sum of $j\in\mathbb{Z}_{\geq1}$, we complete proof of Theorem \[THM1\].
Proof of Theorem \[THM2\] {#THM2P}
=========================
In order to prove Theorem \[THM2\] from the approximate functional equation containing characteristic functions, we use Lemma \[CFF\]. For any functions $X:[0,\infty)\to\mathbb{R}$, we define $M_{X}(s)$ to $$\begin{aligned}
M_{X}(s):=&\sum_{n=1}^\infty\frac{a_F(n)(-\log n)^m}{n^s}X\left(\frac{n}{y_1}\right)+\\ &+\sum_{r=0}^m(-1)^r\binom{m}{r}\chi_F^{(m-r)}(s)\sum_{n=1}^\infty\frac{\overline{a_F(n)}(-\log n)^r}{n^{1-s}}X_0\left(\frac{n}{y_2}\right). \end{aligned}$$ Replacing $\varphi\mapsto\varphi_{\alpha} \;(\alpha\in\mathbb{R}_{\geq0})$ in Theorem \[THM1\] we can write $$\begin{aligned}
F^{(m)}(s)=M_{\varphi_{\alpha}}(s)+R_{\varphi_\alpha}(s)=M_\xi(s)+O(M_{\varphi_\alpha-\xi}(s)+R_{\varphi_\alpha}(s)) \label{NAF1}\end{aligned}$$ Here $M_\xi(s)$ and $M_{\varphi_\alpha-\xi}(s)+R_{\varphi_\alpha}(s)$ are written as $$\begin{aligned}
M_{\xi}(s)=\sum_{n\leq y}\frac{a_F(n)(-\log n)^m}{n^s}+\sum_{r=0}^m(-1)^r\binom{m}{r}\chi_F^{(m-r)}(s)\sum_{n\leq y}\frac{\overline{a_F(n)}(-\log n)^m}{n^{1-s}} \label{NAF2}\end{aligned}$$ and $$\begin{aligned}
M_{\varphi_\alpha-\xi}(s)+R_{\varphi_\alpha}(s)
=&\:E(s)+\sum_{n=1}^\infty\frac{a_F(n)(-\log n)^m}{n^s}S_{0}\left(\frac{n}{y_1}\right)+\\
&+\chi_F(s)\sum_{r=0}^m(-1)^r\binom{m}{r}\sum_{n=1}^\infty\frac{\overline{a_F(n)}(-\log n)^r}{n^{1-s}}T_{m-r}\left(\frac{n}{y_2}\right)\end{aligned}$$ respectively, where $E(s)$, $S_{0}(\rho)$ and $T_{r}(\rho)$ are given by $$\begin{aligned}
E(s)&=O\left(y_1^{1-\sigma}(\log y_1)^{m+\max\{p_F-1,0\}}|t|^{-\frac{l}{2}}\|\varphi_\alpha^{(l+1)}\|_1\right)+\notag\\
&\quad+O\left(y_2^\sigma|t|^{d_F(\frac{1}{2}-\sigma)-\frac{l}{2}}\|\varphi_{0\alpha}^{(l+1)}\|_1{\textstyle\sum_{r=0}^m}(\log y_2)^{r+\max\{p_F-1,0\}}(\log|t|)^{m-r}\right).\\
S_0(\rho)&=(\varphi_\alpha-\xi)(\rho)+\sum_{j=1}^l\varphi_\alpha^{(j)}(\rho)(-\rho)^j\gamma_{j}^{(0)}\left(s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_r^{\lambda_r})^{\frac{2}{d_F}}|t|}\right),\\
T_{m-r}(\rho)&=(\varphi_{0\alpha}-\xi)(\rho)\frac{\chi_F^{(m-r)}}{\chi_F}(s)+
\\&\quad+
\sum_{j=1}^l\varphi_{0\alpha}^{(j)}(\rho)(-\rho)^j\delta_{j}^{(m-r)}\left(1-s;\frac{1}{(\lambda_1^{\lambda_1}\cdots\lambda_r^{\lambda_r})^{\frac{2}{d_F}}|t|}\right),\end{aligned}$$ Since $S_{0}(\rho)=0$ and $T_{m-r}(\rho)=0$ for $\rho\in[0,(1+|t|^{-\alpha})^{-1}]\cup[1+|t|^{-\alpha},\infty)$ by Lemmas \[RCDE\] and \[CFF\], we have $$\begin{aligned}
E(s)&\ll y_1^{1-\sigma}(\log y_1)^{m+\max\{p_F-1,0\}}|t|^{(\alpha-\frac{l}{2})l}+\notag \\
&\quad+y_2^\sigma|t|^{d_F(\frac{1}{2}-\sigma)+(\alpha-\frac{1}{2})l}\textstyle\sum_{r=0}^m(\log y_2)^{r+\max\{p_F-1,0\}}(\log|t|)^{m-r}\label{MUUE}\end{aligned}$$ for any $\alpha\in\mathbb{R}_{\geq0}$, and $$\begin{aligned}
S_0(\rho)&\ll1+\sum_{j=1}^l|t|^{\alpha j}|t|^{-\frac{j}{2}}\ll 1,\label{MS0E}\\
T_{m-r}(\rho)&\ll(\log|t|)^{m-r}+\sum_{j=1}^l|t|^{\alpha j}|t|^{-\frac{j}{2}}(\log|t|)^{m-r}\ll(\log|t|)^{m-r}
\label{MTRE}\end{aligned}$$ for $\rho\in[(1+|t|^{-\alpha})^{-1}, 1+|t|^{-\alpha}]$ under the condition $\alpha\in[0,1/2]$. Now we choose $\alpha=1/2-\varepsilon$ and $l\in\mathbb{Z}_{\geq1/(2\varepsilon)}$. By the condition (d) of Selberg class and the estimates , – and $(1+|t|^{-\alpha})y_1-(1+|t|^{-\alpha})^{-1}y_1\leq 2|t|^{-\alpha}y_1$, $M_{\varphi_\alpha-\xi}(s)+R_{\varphi_\alpha}(s)$ is estimated as $$\begin{aligned}
&\hspace{-1em}M_{\varphi_\alpha-\xi}(s)+R_{\varphi_\alpha}(s)\notag\\
\ll&\;y_1^{1-\sigma}(\log y_1)^{m+\max\{p_F-1,0\}}|t|^{(\alpha-\frac{1}{2})l}+\notag \\
&+y_2^\sigma|t|^{d_F(\frac{1}{2}-\sigma)+(\alpha-\frac{1}{2})l}\sum_{r=0}^m(\log y_2)^{r+\max\{p_F-1,0\}}(\log|t|)^{m-r}+\notag\\
&+\sum_{(1+|t|^{-\alpha})^{-1}y_1\leq n\leq(1+|t|^{-\alpha})y_1}\frac{|a_F(n)|(\log n)^m}{n^\sigma}+\notag\\
&+|t|^{d_F(\frac{1}{2}-\sigma)}\sum_{r=0}^m(\log|t|)^{m-r}\sum_{(1+|t|^{-\alpha})^{-1}y_2\leq n\leq(1+|t|^{-\alpha})y_2}\frac{|\overline{a_F(n)}|(\log n)^r}{n^{1-\sigma}}+\notag\\
\ll &\; y_1^{1-\sigma+\varepsilon}|t|^{-\frac{1}{2}}+y_2^{\sigma+\varepsilon}|t|^{d_F(\frac{1}{2}-\sigma)-\frac{1}{2}}.
\label{NAF3}\end{aligned}$$ Hence combining , and we complete the proof of Theorem \[THM2\].
[20]{} M. Aoki and M. Minamide, *A zero density estimate for the derivatives of the Riemann zeta function*, Journal for Algebra and Number Theory Academia **2** (2012), 361–365. K. Chandrasekharan and R. Narasimhan, *The approximate functional equation for a class of zeta functions*, Math. Ann. **152** (1963), 30–64. P. Deligne, *La conjecture de Weil. I*, Publ. Math. Inst. Hautes Études Sci. **43** (1974), 273–307. A. Good, *Approximative Funktionalgleichungen und Mittelwertsätze für Dirichletreihen, die Spitzenformen assoziiert sind*, Comment. Math. Helv. **50** (1975), 327–361. E. Landau, *Über die Anzahl der Gitterpunkte in Gewissen Bereichen. II*, Nachr. Ge Wiss Göttingen (1915), 209–243. R. A. Rankin, *Contributions to the theory of Ramanujan’s function $\tau(n)$ and similar functions. II. The order of the Fourier coefficients of integral modular forms*, Proc. Cambridge Phil. Soc. **35** (1939), 357–373. A. Selberg, *Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist*, Arch. Math. Naturvid. **43** (1940), 47–50. A. Selberg, *Old and new conjectures and results about a class of Dirichlet series*, In Proc. Amalfi Conf. Analytic Number Theory, E. Bombieri et al. eds., 367–385, Universitàdi Salerno 1992; Collected Papers, Vol. II, 47–63, Berlin-Heidelberg-New York 1991. Y. Yashiro, *Approximate functional equation and mean value formula for the derivative of $L$-functions attached to cusp forms*, Funct. Approx. Comment. Math. (1) **53** (2015), 97–122. Y. Yashiro, *Zeros of the derivates of L-functions attached to cusp forms*, Bulletin Polish Acad. Sci. Math. **64** (2016), 147–164.
|
---
author:
- 'Jeffrey Hawke, Alex Bewley, Ingmar Posner[^1]'
title: 'What Makes a Place? Building Bespoke Place Dependent Object Detectors for Robotics'
---
=1
[^1]: The authors are with the Oxford Robotics Institute, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom. Email: [@robots.ox.ac.uk]{}
|
---
abstract: 'This paper considers linear quadratic team decision problems where the players in the team affect each other’s information structure through their decisions. Whereas the stochastic version of the problem is well known to be complex with nonlinear optimal solutions that are hard to find, the deterministic counterpart is shown to be tractable. We show that under a mild assumption, where the weighting matrix on the controller is chosen large enough, linear decisions are optimal and can be found efficiently by solving a semi-definite program.'
author:
- 'Ather Gattami[^1]'
bibliography:
- '../../ref/mybib.bib'
title: Deterministic Team Problems with Signaling Incentive
---
Team Decision Theory, Game Theory, Convex Optimization.
Notation {#notation .unnumbered}
========
--------------------- -------------------------------------------------------------------------------------------
$\mathbb{S}^n$ The set of $n\times n$ symmetric matrices.
$\mathbb{S}^n_{+}$ The set of $n\times n$ symmetric positive
semidefinite matrices.
$\mathbb{S}^n_{++}$ The set of $n\times n$ symmetric positive
definite matrices.
$\mathcal{C}$ The set of functions $\mu:\mathbb{R}^p\rightarrow \mathbb{R}^m$ with
$\mu(y)=(\mu_1(y_1), \mu_2(y_2), ..., \mu_N(y_N))$,
$\mu_i:\mathbb{R}^{p_i}\rightarrow \mathbb{R}^{m_i}$, $\sum_{i} m_i=m$, $\sum_{i} p_i=p$.
$\mathbb{K}$ $\{K\in \mathbb{R}^{m\times p}| K=\oplus \sum K_i, K_i\in \mathbb{R}^{m_i\times p_i}\}$
$A^{\dagger}$ Denotes the pseudo-inverse of the
square matrix $A$.
$A_{\perp}$ Denotes the matrix with minimal number
of columns spanning the nullspace of $A$.
$A_i$ The $i$th block row of the matrix $A$.
$A_{ij}$ The block element of $A$ in position$(i, j)$.
$\succeq$ $A\succeq B$ $\Longleftrightarrow$ $A-B\in \mathbb{S}^n_{+}$.
$\succ$ $A\succ B$ $\Longleftrightarrow$ $A-B\in \mathbb{S}^n_{++}$.
$\mathbf{Tr}$ $\mathbf{Tr}[A]$ is the trace of the matrix $A$.
$\mathcal{N}(m,X)$ The set of Gaussian variables with
mean $m$ and covariance $X$.
--------------------- -------------------------------------------------------------------------------------------
Introduction
============
The team problem is an optimization problem, where a number of decision makers (or players) make up a team, optimizing a common cost function with respect to some uncertainty representing *nature*. Each member of the team has limited information about the global state of nature. Furthermore, the team members could have different pieces of information, which makes the problem different from the one considered in classical optimization, where there is only one decision function that has access to the entire information available about the state of nature.
Team problems seemed to possess certain properties that were considerably different from standard optimization, even for specific problem structures such as the optimization of a quadratic cost in the state of nature and the decisions of the team members. In stochastic linear quadratic decision theory, it was believed for a while that certainty-equvalence holds between estimation and optimal decision with complete information, even for team problems. The certainty-equivalence principle can be briefly explained as follows. First assume that every team member has access to the information about the entire state of nature, and find the corresponding optimal decision for each member. Then, each member makes an estimate of the state of nature, which is in turn combined with the optimal decision obtained from the full information assumption. It turns out that this strategy does *not* yield an optimal solution (see [@radner]).
A general solution to static stochastic quadratic team problems was presented by Radner [@radner]. Radner’s result gave hope that some related problems of dynamic nature could be solved using similar arguments. But in 1968, Witsenhausen [@witsenhausen:1968] showed in his well known paper that finding the optimal decision can be complex if the decision makers affect each other’s information. Witsenhausen considered a dynamic decision problem over two time steps to illustrate that difficulty. The dynamic problem can actually be written as a static team problem: $$\begin{aligned}
\text{minimize } & \mathbf{E}\hspace{1mm}\left\{k_0u_0^2+(x+u_0-u_1)^2\right\}\\
\text{subject to } & u_0=\mu_0(x),\hspace{2mm}
u_1=\mu_1(x+u_0+w),
\end{aligned}$$ where $x$ and $w$ are Gaussian with zero mean and variance $X$ and $W$, respectively. Here, we have two decision makers, one corresponding to $u_0$, and the other to $u_1$. Witsenhausen showed that the optimal decisions $\mu_0$ and $\mu_1$ are not linear because of the *signaling*/*coding incentive* of $u_0$. Decision maker $u_1$ measures $x+u_0+w$, and hence, its measurement is affected by $u_0$. Decision maker $u_0$ tries to *encode* information about $x$ in its decision, which makes the optimal strategy complex.
The problem above is actually an information theoretic problem. To see this, consider the slightly modified problem $$\begin{aligned}
\text{minimize } & \mathbf{E}\hspace{1mm}(x-u_1)^2\\
\text{subject to } & u_0=\mu_0(x), \hspace{2mm} \mathbf{E}\hspace{1mm}u_0^2 \leq 1, \hspace{2mm}
u_1=\mu_1(u_0+w)
\end{aligned}$$ The modification made is that we removed $u_0$ from the objective function, and instead added a constraint $ \mathbf{E}\hspace{1mm}u_0^2 \leq 1$ to make sure that it has a limited variance (of course we could set an arbitrary power limitation on the variance). The modified problem is exactly the Gaussian channel coding/decoding problem (see Figure \[teamchannel\])! The optimal solution to Witsenhausens counterexample is still unknown. Even if we would restrict the optimization problem to the set of linear decisions, there is still no known polynomial-time algorithm to find optimal solutions. Another interesting counterexample was recently given in [@lipsa:2008].
In this paper, we consider the problem of distributed decision making with information constraints under linear quadratic settings. For instance, information constraints appear naturally when making decisions over networks. These problems can be formulated as team problems. Early results considered static team theory in stochastic settings [@marschak:1955], [@radner], [@ho:chu]. In [@didinsky:basar:1992], the team problem with two team members was solved. The solution cannot be easily extended to more than two players since it uses the fact that the two members have common information; a property that doesn’t necessarily hold for more than two players. [@didinsky:basar:1992] uses the result to consider the two-player problem with one-step delayed measurement sharing with the neighbors, which is a special case of the partially nested information structure, where there is no signaling incentive. Also, a nonlinear team problem with two team members was considered in [@bernhard:99], where one of the team members is assumed to have full information whereas the other member has only access to partial information about the state of the world. Related team problems with exponential cost criterion were considered in [@krainak:82]. Optimizing team problems with respect to *affine* decisions in a minimax quadratic cost was shown to be equivalent to stochastic team problems with exponential cost, see [@fan:1994]. The connection is not clear when the optimization is carried out over nonlinear decision functions. In [@gattami:bob:rantzer], a general solution was given for an arbitrary number of team members, where linear decision were shown to be optimal and can be found by solving a linear matrix inequality. In the deterministic version of Witsenhausen’s counterexample, that is minimizing the quadratic cost with respect to the worst case scenario of the state $x$ (instead of the assumption that $x$ is Gaussian), the linear decisions where shown to be optimal in [@rotkowitz:2006].
We will show that for static linear quadratic minimax team problems, where the players in the team affect each others information structure through their decisions, linear decisions are optimal in general, and can be found by solving a linear matrix inequality.
Main Results {#minimaxteam}
============
The deterministic problem considered is a quadratic game between a team of players and nature. Each player has limited information that could be different from the other players in the team. This game is formulated as a minimax problem, where the team is the minimizer and nature is the maximizer. We show that if there is a solution to the static minimax team problem, then linear decisions are optimal, and we show how to find a linear optimal solution by solving a linear matrix inequality.
Deterministic Team Problems with Signaling Incentive
====================================================
Consider the following team decision problem
$$\label{minimax_signaling}
\begin{aligned}
\inf_{\mu} \sup_{v\in \mathbb{R}^p, 0\neq w\in \mathbb{R}^q}\hspace{1mm} &
\frac{L(w,u)}{\|w\|^2+\|v\|^2}\\
\text{subject to }\hspace{1mm} & y_i = \sum_{j=1}^ND_{ij}u_j+E_iw + v_i\\
& u_i = \mu_i(y_i)\\
& \text{for } i=1,..., N,
\end{aligned}$$
where $u_i\in \mathbb{R}^{m_i}$ and $E_i\in \mathbb{R}^{p_i\times q}$, for $i=1, ..., N$,
$L(w,u)$ is a quadratic cost given by $$L(w,u)=
\left[
\begin{matrix}
w\\
u
\end{matrix}
\right] ^T
\left[
\begin{matrix}
Q_{ww} & Q_{wu}\\
Q_{uw} & Q_{uu}
\end{matrix}
\right]
\left[
\begin{matrix}
w\\
u
\end{matrix}
\right] ,$$ $Q_{uu}\in \mathbb{S}^m_{++}$, $m=m_1+\cdots + m_N$, and $$\left[
\begin{matrix}
Q_{ww} &Q_{wu}\\
Q_{uw} & Q_{uu}
\end{matrix}
\right] \in \mathbb{S}^{m+n}_+.$$
The players $u_1$,..., $u_N$ make up a *team*, which plays against *nature* represented by the vector $w$, using $\mu\in \mathcal{C}$. This problem is more complicated than the static team decision problem studied in [@gattami:bob:rantzer], since it has the same flavour as that of the Witsenhausen counterexample that was presented in the introduction. We see that the measurement $y_i$ of decision maker $i$ could be affected by the other decision makers through the terms $D_{ij}u_j$, $j=1, ..., N$.
Note that we have the equality $y = Du+Ew+v$ which is equivalent to $v = Du+Ew-y$. Using this substitution of variable, the team problem (\[minimax\_signaling\]) is equivalent to $$\label{minimax_signaling2}
\begin{aligned}
\inf_{\mu\in\mathcal{C}} \sup_{y\in \mathbb{R}^p, 0\neq w\in \mathbb{R}^q}\hspace{1mm} &
\frac{L(w,\mu(y))}{||D\mu(y)+Ew-y||^2+\|w\|^2}\\
\end{aligned}$$
\[ass2\] $$\gamma^\star \leq \bar{\gamma} :=
\inf_{Du \neq 0}\hspace{1mm}
\frac{u^TQ_{uu}u}{u^TD^TDu} .$$
\[signaling\_team\] Let $\gamma^\star$ be the value of the game (\[minimax\_signaling\]) and suppose that Assumption \[ass2\] holds. Then the following statements hold:
- There exist linear decisions $\mu_i(y_i)=K_iy_i$, $i=1,
..., N$, where the value $\gamma^\star $ is achieved.
- If $\gamma^\star<\bar{\gamma}$, then for any $\gamma\in [\gamma^\star~,~\bar{\gamma})$, a linear decision $Ky$ with $K\in \mathbb{K}$ that achieves $\gamma$ is obtained by solving the linear matrix inequality
$$\label{team_computation}
\begin{aligned}
\text{find }\hspace {2mm} & K\\
\text{subject to }\hspace {2mm} & K=\text{diag}(K_1, ..., K_N) \\
&
C = \left[
\begin{matrix}
I & 0
\end{matrix}
\right] \in \mathbb{R}^{p\times(p+q)}, \hspace{3mm} Q_{uu}(\gamma )\in \mathbb{S}^{m\times m}
\end{aligned}$$
$$\begin{aligned}
\left[
\begin{matrix}
Q_{xx}(\gamma) & Q_{xu}(\gamma)\\
Q_{ux}(\gamma) & Q_{uu}(\gamma)
\end{matrix}
\right] &=
\left[
\begin{matrix}
Q_{ww} & 0 &Q_{wu}\\
0 & 0 & 0\\
Q_{uw} & 0 & Q_{uu}
\end{matrix}
\right] - \gamma
\left[
\begin{matrix}
E^TE & - E^T & - E^TD\\
- E & I & - D\\
- D^TE & - D^T & D^TD
\end{matrix}
\right]
\end{aligned}$$
$$\left[\begin{matrix}
Q_{xx}(\gamma) + Q_{xu}(\gamma)KC+C^TK^TQ_{ux}(\gamma) & C^TK^T\\
KC & -Q_{uu}^{-1}(\gamma)
\end{matrix}
\right] \preceq 0 ,$$
($i$) Note that $$y = Du + Ew + v \Longleftrightarrow v = y - Du - Ew\Rightarrow$$ $$\Rightarrow \frac{L(w,u)}{||v||^2+\|w\|^2} = \frac{L(w,u)}{||y - Du - Ew||^2+\|w\|^2}.$$ Now introduce $x\in \mathbb{R}^{n}$, $n=p+q$, such that $$x = \left[
\begin{matrix}
w\\
y
\end{matrix}
\right] ,$$ and $$\label{QR}
\begin{aligned}
Q &=
\left[
\begin{matrix}
Q_{ww} & 0 & Q_{wu}\\
0 & 0 & 0\\
Q_{uw} & 0 & Q_{uu}
\end{matrix}
\right] , \\
R &=
\left[
\begin{matrix}
E^TE & - E^T & - E^TD\\
- E & I & - D\\
- D^TE & - D^T & D^TD
\end{matrix}
\right].
\end{aligned}$$ Then, $$\begin{aligned}
J(x,u) &:= \left[
\begin{matrix}
x\\
u
\end{matrix}
\right]^T
Q
\left[
\begin{matrix}
x\\
u
\end{matrix}
\right] = L(w,u),
\\
F(x,u) &:= \left[
\begin{matrix}
x\\
u
\end{matrix}
\right]^T
R
\left[
\begin{matrix}
x\\
u
\end{matrix}
\right] = ||y - Du - Ew||^2+\|w\|^2,
\end{aligned}$$ and $
y = Cx.
$ Hence, we have that $$\frac{L(w,u)}{||v||^2+\|w\|^2} = \frac{L(w,u)}{||y - Du - Ew||^2+\|w\|^2} = \frac{J(x,u)}{F(x,u)}.$$ Then, for any $\gamma\in (\gamma^\star~,~\bar{\gamma})$, there exists a decision function $\mu\in\mathcal{C}$ such that $$J(x,\mu(Cx)) - \gamma F(x,\mu(Cx)) =
\left[
\begin{matrix}
x\\
\mu(Cx)
\end{matrix}
\right] ^T
\left[
\begin{matrix}
Q_{xx}(\gamma) & Q_{xu}(\gamma)\\
Q_{ux}(\gamma) & Q_{uu}(\gamma)
\end{matrix}
\right]
\left[
\begin{matrix}
x\\
\mu(Cx)
\end{matrix}
\right]
\leq 0$$ for all $x$. Under Assumption \[ass2\], we have that $$Q_{uu}(\gamma) = Q_{uu} -\gamma D^TD\succ 0$$ for any $\gamma\in (\gamma^\star~,~\bar{\gamma}]$. Thus, we can apply Theorem 1 in [@gattami:bob:rantzer], which implies that there must exist linear decisions that can achieve any $\gamma\in (\gamma^\star~,~\bar{\gamma}]$. By compactness, there must exist linear decisions that achieve $\gamma^\star$.\
($ii$) Let $\mu(Cx) = KCx$ for $K\in \mathbb{K}$. Then
$$\left[
\begin{matrix}
x\\
KCx
\end{matrix}
\right] ^T
\left[
\begin{matrix}
Q_{xx}(\gamma) & Q_{xu}(\gamma)\\
Q_{ux}(\gamma) & Q_{uu}(\gamma)
\end{matrix}
\right]
\left[
\begin{matrix}
x\\
KCx
\end{matrix}
\right]
\leq 0, ~ ~ \forall x$$ $$\Updownarrow$$ $$\left[
\begin{matrix}
I\\
KC
\end{matrix}
\right] ^T
\left[
\begin{matrix}
Q_{xx}(\gamma) & Q_{xu}(\gamma)\\
Q_{ux}(\gamma) & Q_{uu}(\gamma)
\end{matrix}
\right]
\left[
\begin{matrix}
I\\
KC
\end{matrix}
\right]
\preceq 0$$ $$\Updownarrow$$
$$Q_{xx}(\gamma)+ Q_{xu}(\gamma)KC+C^TK^TQ_{ux}(\gamma) + C^TK^T Q_{uu}(\gamma) KC\preceq 0$$
$$\Updownarrow$$ $$\left[\begin{matrix}
Q_{xx}(\gamma) + Q_{xu}(\gamma)KC+C^TK^TQ_{ux}(\gamma) & C^TK^T\\
KC & -Q_{uu}^{-1}(\gamma)
\end{matrix}
\right] \preceq 0,$$ and the proof is complete.
Linear Quadratic Control with Arbitrary Information Constraints
===============================================================
Consider the dynamic team decision problem $$\label{lq}
\begin{aligned}
\inf_{\mu} \sup_{w, v\neq 0}\hspace{2mm} & \frac{\sum_{k=1}^{M}\left[
\begin{matrix}
x(k)\\
u(k)
\end{matrix}
\right]^T
\left[
\begin{matrix}
Q_{xx} & Q_{xu}\\
Q_{ux} & Q_{uu}
\end{matrix}
\right]
\left[
\begin{matrix}
x(k)\\
u(k)
\end{matrix}
\right]}{\sum_{k=1}^M \|w(k)\|^2+\|v(k)\|^2}\\
\text{subject to } & x(k+1) = Ax(k)+Bu(k)+w(k)\\
& y_i(k) = C_ix(k)+v_i(k)\\
& u_i(k) = [\mu_k]_{i}(y_i(k)), i=1,..., N.
\end{aligned}$$ Now write $x(t)$ and $y(t)$ as [$$\label{expansion}
\begin{aligned}
x(t) &= \sum_{k=1}^{t}A^{k}Bu(M-k)+\sum_{k=1}^{t}A^{k}w(M-k),\\
y_i(t) &= \sum_{k=1}^{t}C_iA^{k}Bu(M-k)+\sum_{k=1}^{t}C_iA^{k}w(M-k)+v_i(k).
\end{aligned}$$]{} It is easy to see that the optimal control problem above is equivalent to a static team problem of the form (\[minimax\_signaling\]). Thus, linear controllers are optimal under Assumption \[ass2\].
Consider the deterministic version of the Witsenhausen counterexample presented in the introduction: $$\begin{aligned}
\inf_{\mu_1,\mu_2} \hspace{1mm} & \gamma \\
\text{s. t. } &\frac{ {k^2}\mu_1^2(y_1) +({x_1}-\mu_2(y_2))^2}{x_0^2+w^2} \leq \gamma\\
&{y_1 = x_0}\\
&{x_1} = {x_0}+\mu_1(y_1)\\
&y_2 = x_1+w = {x_0}+\mu_1(y_1)+w
\end{aligned}$$ Substitue $x_0=y_1$, $x_1 = y_1 + \mu_1(y_1)$ and $
w^2 = ({x_0}+\mu_1(y_1)-y_2)^2
$ in the inequality $$\ {k^2}\mu_1^2(y_1) +({x_1}-\mu_2(y_2))^2 \leq \gamma ({x_0^2}+w^2).$$ Then, we get the equivalent problem $$\begin{aligned}
\inf_{\mu_1, \mu_2} \hspace{1mm} & \gamma \\
\text{s. t. } & {k^2}\mu_1^2(y_1)+(y_1+\mu_1(y_1)-\mu_2(y_2))^2
\leq \gamma (y_1^2+(y_1+\mu_1(y_1)-y_2)^2)
\end{aligned}$$ Completing the squares gives the following equivalent inequality $$\left[
\begin{matrix}
y_1\\
y_2\\
\mu_1(y_1)\\
\mu_2(y_2)
\end{matrix}
\right]^T
\left[
\begin{matrix}
1-2\gamma & \gamma & 1-\gamma & -1\\
\gamma & -\gamma & \gamma & 0\\
1-\gamma & \gamma & {1+k^2 - \gamma } & {-1}\\
-1 & 0 & {-1} & {1}
\end{matrix}
\right]
\left[
\begin{matrix}
y_1\\
y_2\\
\mu_1(y_1)\\
\mu_2(y_2)
\end{matrix}
\right] \leq 0$$ For $k^2=0.1$, we can search over $\gamma <\bar{\gamma} = k^2 = 0.1$, and we can use Theorem \[signaling\_team\] to deduce that linear decisions are optimal, and can be computed by iteratively solving a linear matrix inequality, where the iterations are done with respect to $\gamma$. We find that $${\gamma^\star \approx 0.0901},$$ $${\mu_1(y_1)=-0.9001 y_1},$$ $${\mu_2(y_2)=-0.0896 y_2}.$$
For $k^2 = 1$, we iterate with respect to $\gamma < 1$, and we find optimal linear decisions given by $$\begin{aligned}
{\mu_1(y_1)} & {=} {-0.3856 y_1}\\
{\mu_2(y_2)} & {=} {0.3840 y_2}
\end{aligned}$$ $$\Downarrow$$ $$\gamma^\star = 0.3820$$
Consider the deterministic counterpart of the multi-stage finite-horizon stochastic control problem that was considered in [@lipsa:2008]: $$\inf_{\mu_k:\mathbb{R}\rightarrow \mathbb{R}} \sup_{x_0, v_0, ..., v_{m-1}\in \mathbb{R}} \frac{(x_m-x_0)^2+ \sum_{k=0}^{m-2} \mu^2_k(y_k)}{x_0^2+v_0^2+\cdots+v_{m-1}^2}$$ subject to the dynamics $$\begin{aligned}
x_{k+1} &= \mu_k(y_k) \nonumber\\
y_k &= x_k+v_k \nonumber .\end{aligned}$$ It is easy to check that $\bar{\gamma}=1$ and $Q_{uu} -\gamma D^T D\succ 0$ for $\gamma<\bar{\gamma}$ (compare with Assumption \[ass2\]) . Thus, linear decisions are optimal. This is compared to the stochastic version, where linear decisions where not optimal for $m>2$.
Conclusions
===========
We have considered the static team problem in deterministic linear quadratic settings where the team members may affect each others information. We have shown that decisions that are linear in the observations are optimal and can be found by solving a linear matrix inequality.
For future work, it would be interesting to consider the case where the measurements are given by $y = Du +Ew + Fv$, for an arbitrary matrix $F$.
Acknowledgements
================
The author is grateful to Professor Anders Rantzer, Professor Bo Bernhardsson, and the reviewers for valuable comments and suggestions.
This work is supported by the Swedish Research Council.
[^1]: Ather Gattami is with the Automatic Control Laboratory, Electrical Engineering School, KTH-Royal Institute of Technology, 100 44, Stockholm, Sweden. E-mail: gattami@kth.se
|
---
abstract: 'For $N\geq 3$, we show Tate’s conjecture for the elliptic modular surface $E(N)$ of level $N$ over $\mathbb{F}_p$ for a prime $p$ satisfying $p\equiv 1\mod N$ outside of a set of primes of density zero. We also prove a strong form of Tate’s conjecture for $E(N)$ over any finite field of characteristic $p$ prime to $N$ under the assumption that the formal Brauer group of $E(N)$ is of finite height.'
author:
- Rémi Lodh
bibliography:
- 'lmodsurfbib.bib'
title: 'On Tate’s conjecture for elliptic modular surfaces over finite fields'
---
Introduction {#introduction .unnumbered}
============
In this paper we study cohomology classes of divisors on the elliptic modular surface $E(N)$ of level $N$, where $N\geq 3$. By definition, $E(N)$ is the universal object over the moduli space $X(N)$ of generalized elliptic curves with level $N$ structure. Fix a prime $p$ which does not divide $N$. Our first result is the following theorem, which goes back to Shioda [@shioda1 Appendix] for $N=4$.
\[main\] Assume the partial semi-simplicity conjecture (\[SS\]) is true for $E(N)_{\mathbb{F}_p}$. If $p\equiv 1\mod N$, then Tate’s conjecture holds for each connected component of $E(N)_{\mathbb{F}_p}$. Moreover, the Mordell-Weil group of a generic fibre of $E(N)_{\mathbb{F}_p}\to X(N)_{\mathbb{F}_p}$ is isomorphic to $(\mathbb{Z}/N)^2$.
The partial semi-simplicity conjecture for a smooth projective surface $S$ over a finite field $k$ is the following statement: if $q=\#k$, then $$\label{SS}\tag{PS}
H^2_{\operatorname{cris}}(S)^{(F_q-q^2)=0}=H^2_{\operatorname{cris}}(S)^{F_q=q}$$ where $H^2_{\operatorname{cris}}(S)=H^2_{\operatorname{cris}}(S/W(k))\otimes\mathbb{Q}$ is the second crystalline cohomology group of $S$ and $F_q$ is the $q$-power Frobenius endomorphism. It is a consequence of Tate’s conjecture. For $E(N)_{\mathbb{F}_p}$, it is related to estimates for the absolute value of the Fourier coefficients of normalized newforms of weight 3, see [@colemanedixhoven]. Such estimates can be shown to hold if the field of coefficients of the newform is totally real, or for CM forms, but the general case is unknown. In any case, we can show that it holds for all primes $p$ outside of a set of density zero, viz.
The conclusion of Theorem \[main\] holds for all $p\equiv 1\mod N$ outside of a set of primes of density zero.
For arbitrary $p$, we can also show Tate’s conjecture under a completely different assumption, namely that the formal Brauer group of $E(N)_{\mathbb{F}_p}$ is a formal Lie group of finite height. In fact, writing $T_p\,A=\varprojlim_nA[p^n]$ where $A[m]\subset A$ is the subgroup of $m$-torsion ($m\in\mathbb{Z}$) for any abelian group $A$, we have
\[fh\] If the formal Brauer group of a connected component of $E(N)_{\bar{\mathbb{F}}_p}$ has finite height, then $T_p\operatorname{Br}(E(N)_{\bar{\mathbb{F}}_p})=0$.
However, it seems to be a difficult arithmetic problem to determine for which $p$ the reduction of $E(N)$ has finite height formal Brauer group. Shioda [@shioda1; @shioda2] has shown that $E(4)_{\bar{\mathbb{F}}_p}$ is a K3 surface whose formal Brauer group is of finite height if and only if $p\equiv 1\mod 4$, but we do not know what happens for $N>4$.
The starting point for the proofs of both Theorems \[main\] and \[fh\] is the following fact: $$\label{hodgetateK}\tag{HT}
V_p\operatorname{Br}(E(N)_{\bar{\mathbb{Q}}})\;\text{is a Hodge-Tate representation with weights}\pm 1.$$ Here $\operatorname{Br}(-):=H^2(-,\mathbb{G}_m)$ denotes the cohomological Brauer group and for any abelian group $A$ we write $V_pA:=T_pA\otimes\mathbb{Q}$.
We can summarize our method of proof of Theorem \[main\] as follows. Let $I_p$ be the extension to $E(N)$ of the morphism of $X(N)$ obtained by multiplying the level structure by $p\in(\mathbb{Z}/N)^*$ and let $U\subset V_p\operatorname{Br}(E(N)_{\bar{\mathbb{Q}}})$ be the subset on which $I_p$ acts trivially. Then (modulo (\[SS\])) (\[hodgetateK\]) and the Eichler-Shimura congruence relation imply $D_{\operatorname{cris}}(U)^{\varphi=1}=0$, where $\varphi$ is the Frobenius. For $p\equiv 1\mod N$, $I_p$ is the identity and the theorem follows. In the case $p\not\equiv 1\mod N$ we only know Shioda’s result [@shioda2] for $N=4$.
The proof of Theorem \[fh\] is much more straightforward: it makes no use of Hecke operators and applies more generally to any proper smooth surface over a finite extension of $\mathbb{Z}_p$ satisfying (\[hodgetateK\]) and whose formal Brauer group has finite height (cf. Corollary \[subquot\]), thereby establishing a relationship between (\[hodgetateK\]) and Tate’s conjecture in this case.
Notation {#notation .unnumbered}
--------
All sheaves and cohomology will be for the étale topology, except where otherwise stated.
We denote by $k$ a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K_0=W[1/p]$, $\bar{k}$ an algebraic closure of $k$, $\bar{K}$ be an algebraic closure of $K_0$, $G_{K_0}=\operatorname{Gal}(\bar{K}/K_0)$, $\hat{\bar{K}}$ the completion of $\bar{K}$ for the $p$-adic norm.
A general result
================
We assume familiarity with Fontaine’s basic theory [@font; @font2; @colfont].
Autodual crystalline representations
------------------------------------
Let $V$ be a $p$-adic representation of $G_{K_0}$. We say that $V$ is *autodual* if it is isomorphic to its dual, i.e. it has a non-degenerate bilinear form $$V\otimes_{\mathbb{Q}_p}V\to\mathbb{Q}_p$$ which is a homomorphism of $G_{K_0}$-modules.
\[key\] Let $V$ be an autodual crystalline representation of $G_{K_0}$ and let $D:=D_{\operatorname{cris}}(V)$ be the associated filtered $\varphi$-module. Suppose the endomorphism $T:=\varphi+\varphi^{-1}$ of $D$ satisfies $T(F^1D)\subset F^1D$. If $D^{(\varphi-1)^2=0}=D^{\varphi=1}$ and $V^{G_{K_0}}=0$, then $D^{\varphi=1}=0$.
The bilinear form on $V$ induces a non-degenerate bilinear form $\cdot$ on $D$. Endow $D^{\varphi=1}$ with the induced filtration from $D$. Since $V$ is crystalline we have $F^0D^{\varphi=1}=V^{G_{K_0}}=0$. If $D^{\varphi=1}=F^0D^{\varphi=1}$, then we are done. If not, then there is $i<0$ and $x\in F^iD^{\varphi=1}\setminus F^{i+1}D^{\varphi=1}$. Since $V$ is autodual, the map $c:D\to D^*:=\operatorname{Hom}_{K_0}(D,K_0)$ induced by $\cdot$ is an isomorphism of filtered $\varphi$-modules, so we have $x^*:=c(x)\in F^iD^*\setminus F^{i+1}D^*$. Note that $x^*$ is the map $D\ni y\mapsto x\cdot y\in K_0$. Since by definition $$F^iD^*=\{f\in D^*:f(F^jD)\subset F^{j+i}K_0\;\forall j\in\mathbb{Z}\}$$ the condition $x^*\notin F^{i+1}D^*$ means that there is $j$ such that $x^*(F^jD)\not\subset F^{j+i+1}K_0$, where $K_0$ has the trivial filtration, i.e. $$F^kK_0=
\begin{cases}
K_0 & k\leq 0 \\
0 & k>0.
\end{cases}$$ If $x^*(F^jD)\not\subset F^{j+i+1}K_0$, then we must have $x^*(F^jD)\neq 0$, hence $x^*(F^jD)=K_0$. So to say that $x^*(F^jD)\not\subset F^{j+i+1}K_0$ but $x^*(F^jD)\subset F^{i+j}K_0$ is equivalent to the condition $j+i=0$. Hence $j=-i>0$, and there is an element $y\in F^1D$ such that $x\cdot y\neq 0$.
Now, up to dividing $y$ by $x\cdot y$ we may assume that $x\cdot y\in\mathbb{Q}_p$. Let $P(t)\in W[t]$ be such that $P(T)y=0$. Since $\varphi(x)=x$ we have $x\cdot T(d)=T(x\cdot d)$ for all $d\in D$, hence $$0=x\cdot P(T)y=P(T)(x\cdot y)=(x\cdot y)P(2).$$ So $P(2)=0$ and we deduce that $P(t)=(t-2)^eQ(t)$ for some $e\in\mathbb{N}$ and some polynomial $Q(t)$ not divisible by $t-2$. Let $z:=Q(T)y$. Note that $x\cdot z=(x\cdot y)Q(2)\neq 0$. Multiplying the equation $(T-2)^ez=0$ by $\varphi^e$ we find $(\varphi-1)^{2e}z=0$, hence $\varphi(z)=z$ since $D^{(\varphi-1)^2=0}=D^{\varphi=1}$. As $F^1D$ is stable under $T$ by assumption, we have $z\in F^1D$. Thus, $z\in F^1D^{\varphi=1}\subset V^{G_{K_0}}=0$, hence $x=0$, and this completes the proof.
The above argument no longer works if one replaces $\varphi$ by a power $\varphi^r$. The problem is related to the fact that, unlike the case $r=1$, for $r>1$ we may have $F^1B_{\operatorname{cris}}^{\varphi^r=1}\neq 0$.
Application to surfaces {#mainsection}
-----------------------
Let $E\to\operatorname{Spec}(W)$ be a smooth projective morphism with geometrically connected fibres of dimension 2. Let $K_0^{\operatorname{ur}}$ be the maximal unramified extension of $K_0$ in $\bar{K}$. The Kummer sequence gives an exact sequence of $G_{K_0}$-representations $$0\to NS(E_{\bar{K}})\otimes\mathbb{Q}_p\to H^2_{\operatorname{\acute{e}t}}(E_{\bar{K}},\mathbb{Q}_p)(1)\to V_p\operatorname{Br}(E_{\bar{K}})\to 0$$ where $NS:=\operatorname{Pic}/\operatorname{Pic}^0$ is the Néron-Severi group. By $p$-adic Hodge theory, applying the functor $D_{\operatorname{cris}}$ we get an exact sequence $$0\to D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p})\to H^2_{\operatorname{cris}}(E_k/K_0)[1]\to D_{\operatorname{cris}}(V_p\operatorname{Br}(E_{\bar{K}}))\to 0$$ where for a filtered $\varphi$-module $D$ we denote $D[1]$ the filtered $\varphi$-module whose underlying $K_0$-module is $D$ with $\varphi_{D[1]}:=p^{-1}\varphi_D$ and $F^iD[1]:=F^{i+1}D$. On the other hand, there is the specialization map $$\operatorname{sp}:NS(E_{\bar{K}})\to NS(E_{\bar{k}})$$ ([@sga6 Exp. E, appendix, 7.12]) which is functorial in $E$ (hence $G_{K_0}$-equivariant) and injective up to torsion (since the Néron-Severi group is finitely generated this can be checked using $l$-adic cohomology). So $NS(E_{\bar{K}})\otimes\mathbb{Q}_p$ is an unramified discrete representation of $G_{K_0}$ hence is $K_0^{\operatorname{ur}}$-admissible. Thus, $D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p})=\left(NS(E_{\bar{K}})\otimes K_0^{\operatorname{ur}}\right)^{\operatorname{Gal}(K_0^{\operatorname{ur}}/K_0)}$, and $D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p})=\left(NS(E_{\bar{k}})\otimes K_0^{\operatorname{ur}}\right)^{\operatorname{Gal}(K_0^{\operatorname{ur}}/K_0)}$ and we have a commutative diagram $$\xymatrix{
D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p}) \ar[r]^{\operatorname{sp}} \ar[d] & D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p}) \ar[d]^{c_1} \\
H^2_{\operatorname{cris}}(E_k/K_0)[1] \ar@{=}[r] & H^2_{\operatorname{cris}}(E_k/K_0)[1]
}$$ where $c_1$ is the first Chern class. In fact, $c_1$ is injective since $$NS(E_{\bar{k}})_{\mathbb{Q}_p}\subset \left(H^2_{\operatorname{cris}}(E_k/K_0)[1]\otimes_{K_0}K^{\operatorname{ur}}_0\right)^{\varphi=1}.$$ Therefore, defining $C:=H^2_{\operatorname{cris}}(E_k/K_0)[1]/D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p})$, we have a commutative diagram with exact rows $$\xymatrix{
0 \ar[r] & D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p}) \ar[r] \ar[d] & H^2_{\operatorname{cris}}(E_k/K_0)[1] \ar[r] \ar@{=}[d] & D_{\operatorname{cris}}(V_p\operatorname{Br}(E_{\bar{K}})) \ar[r] \ar[d] & 0 \\
0 \ar[r] & D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p}) \ar[r] & H^2_{\operatorname{cris}}(E_k/K_0)[1]\ \ar[r] & C \ar[r] & 0
}$$ and so setting $M:=D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p})/D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p})$ we deduce an exact sequence of $\varphi$-modules $$0\to M\to D_{\operatorname{cris}}(V_p\operatorname{Br}(E_{\bar{K}}))\to C\to 0.$$
\[mainth\] Let $D:=D_{\operatorname{cris}}(V_p\operatorname{Br}(E_{\bar{K}}))$ and $T:=\varphi+\varphi^{-1}$. If $D^{(\varphi-1)^2=0}=D^{\varphi=1}$, $T(F^1D)\subset F^1D$ and $V_p\operatorname{Br}(E_{\bar{K}})^{G_{K_0}}=0$, then $M^{\varphi=1}=0=C^{\varphi=1}$.
By Poincaré duality, cup product is non-degenerate on $H^2_{\operatorname{\acute{e}t}}(E_{\bar{K}},\mathbb{Q}_p)(1)$, and since numerical and algebraic equivalence coincide up to torsion for divisors ([@kleiman 9.6.17]), it is also non-degenerate on $NS(E_{\bar{K}})_{\mathbb{Q}_p}$. It follows that $V_p\operatorname{Br}(E_{\bar{K}})\cong\left(NS(E_{\bar{K}})_{\mathbb{Q}_p}\right)^{\bot}$ has a non-degenerate symmetric bilinear form we may apply Proposition \[key\] to obtain $D^{\varphi=1}=0$. Moreover, the restriction of this form to $M$ is non-degenerate since cup product is non-degenerate on both $D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p})$ and $D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p})$. Thus, $C\cong M^{\bot}$ and hence $C^{\varphi=1}=0$.
\[maincor\] Under the assumptions of Theorem \[mainth\], Tate’s conjecture holds for $E_k$ and we have $$NS(E_{\bar{K}})^{G_{K_0}}\otimes\mathbb{Q}=NS(E_{\bar{k}})^{G_{K_0}}\otimes\mathbb{Q}.$$
Tate’s conjecture is well-known to be equivalent to the statement $C^{\varphi=1}=0$ ([@milneAT]). For the second statement, note that we have an exact sequence $$0\to D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p})^{\varphi=1}\to D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p})^{\varphi=1}\to M^{\varphi=1}$$ so since $M^{\varphi=1}=0$ we have $$\left(NS(E_{\bar{K}})\otimes\mathbb{Q}_p\right)^{G_{K_0}}=D_{\operatorname{cris}}(NS(E_{\bar{K}})_{\mathbb{Q}_p})^{\varphi=1}=D_{\operatorname{cris}}(NS(E_{\bar{k}})_{\mathbb{Q}_p})^{\varphi=1}=\left(NS(E_{\bar{k}})\otimes\mathbb{Q}_p\right)^{G_{K_0}}$$ as claimed.
Elliptic modular surfaces
=========================
We fix throughout an integer $N$ and a prime number $p$ which does not divide $N$.
Definition
----------
For $N\geq 3$, let $Y(N)$ to be moduli $\mathbb{Z}[1/N]$-scheme of elliptic curves with level $N$ structure and let $X(N)$ be its modular compatification. $X(N)$ classifies generalized elliptic curves with level $N$ structure whose singular fibres are Néron $N$-gons. $X(N)$ is smooth over $\mathbb{Z}[1/N]$ and the normalization of $\mathbb{Z}[1/N]$ in $X(N)$ is $\mathbb{Z}[\zeta_N,1/N]$, where $\zeta_N$ is a primitive $N$th root of unity. See [@delrap] for details. We denote the universal generalized elliptic curve by $$g:E(N)\to X(N).$$ $E(N)$ is the *elliptic modular surface of level $N$* studied in [@shioda1]. That it is smooth over $\mathbb{Z}[1/N]$ follows from the results of [@delrap VII].
Application of Hodge theory
---------------------------
Assume $\zeta_N\in W$ (note that this is always possible if $p\equiv 1\mod N$, for then $\zeta_N^p=\zeta_N$, so $\zeta_N\in\mathbb{Z}_p$). To simplify the notation write $$E:=E(N)\otimes_{\mathbb{Z}[\zeta_N]}W,\,X:=X(N)\otimes_{\mathbb{Z}[\zeta_N]}W,\,Y:=Y(N)\otimes_{\mathbb{Z}[\zeta_N]}W,\,\Sigma:=X\setminus Y.$$ Let $L$ be the conormal sheaf of the zero section of $g:E\to X$, and let $\omega=\Omega^1_{X}(\log\Sigma)$ denote the line bundle of differential forms on $X$ with logarithmic poles along $\Sigma$.
\[faltings\] There are $G_{K_0}$-equivariant isomorphisms $$H^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))\otimes_{\mathbb{Q}_p}\hat{\bar{K}}=H^1(X,L^{\otimes -1})\otimes_{W}\hat{\bar{K}}(1)\oplus H^0(X,L\otimes\omega)\otimes_{W}\hat{\bar{K}}(-1)$$ $$\tilde{H}^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))\otimes_{\mathbb{Q}_p}\hat{\bar{K}}=H^1(X,L^{\otimes -1})\otimes_{W}\hat{\bar{K}}(1)\oplus H^0(X,L\otimes\Omega^1_X)\otimes_{W}\hat{\bar{K}}(-1)$$ where $\tilde{H}^1:=\operatorname{im}(H^1_c\to H^1)$ is the parabolic cohomology.
This is a special case of the $p$-adic Eichler-Shimura isomorphism of [@famodular].
Let $I\subset G_{K_0}$ be the inertia group.
If $X_H$ is representable, then $H^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))$ is a Hodge-Tate representation with weights $\pm 1$. In particular, $H^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))^{I}=0$.
\[ker\] Let $E'=E\times_XY$. Then
(i) $H^2(E'_{\bar{K}},\mathbb{Q}_p(1))=H^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))\oplus\mathbb{Q}_pe$, where $e$ denotes the characteristic class of the zero section of $g$
(ii) $H^2(E_{\bar{K}},\mathbb{Q}_p(1))^{I}$ is generated as a $\mathbb{Q}_p$-vector space by the characteristic classes of the irreducible components of singular fibres of $g$ together with $e$.
Since $Y_{\bar{K}}$ is an affine curve, the Leray spectral sequence $$H^i(Y_{\bar{K}},R^jg_*\mathbb{Q}_p(1))\Rightarrow H^{i+j}(E'_{\bar{K}},\mathbb{Q}_p(1))$$ gives an exact sequence $$0\to H^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))\to H^2(E'_{\bar{K}},\mathbb{Q}_p(1))\to H^0(Y_{\bar{K}},R^2g_*\mathbb{Q}_p(1))\to 0$$ so $H^2(E'_{\bar{K}},\mathbb{Q}_p(1))^{I}\subset H^0(Y_{\bar{K}},R^2g_*\mathbb{Q}_p(1))=\mathbb{Q}_p$. In fact we must have equality since the class $e$ of the zero section of $g$ cannot be trivial. So $e$ gives a splitting of the sequence, proving (i). For (ii) it suffices to note that the kernel of the map $H^2(E_{\bar{K}},\mathbb{Q}_p(1))\to H^2(E'_{\bar{K}},\mathbb{Q}_p(1))$ is generated by the classes of the components of the fibres over the cusps.
Note that combined with the Shioda-Tate formula ([@shioda1 1.5]) this implies that the rank of the Mordell-Weil group of the generic fibre of $g$ is zero, a result of Shioda [@shioda1 5.1].
\[HT\] $V_p\operatorname{Br}(E_{\bar{K}})\otimes_{\mathbb{Q}_p}\hat{\bar{K}}=H^2(E,\mathcal{O}_E)\otimes\hat{\bar{K}}(1)\oplus H^0(E,\Omega^2_E)\otimes\hat{\bar{K}}(-1)$. In particular, $V_p\operatorname{Br}(E_{\bar{K}})^{I}=0$.
We have $V_p\operatorname{Br}(E_{\bar{K}})\subset V_p\operatorname{Br}(E'_{\bar{K}})$ ([@brauer II, 1.10]) and the latter is a quotient of $H^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))$ by the last corollary, hence $V_p\operatorname{Br}(E_{\bar{K}})$ is a Hodge-Tate representation with weights contained in $\{\pm 1\}$. In particular, $V_p\operatorname{Br}(E_{\bar{K}})\otimes_{\mathbb{Q}_p}\hat{\bar{K}}\cap H^1(E,\Omega^1_E)\otimes\hat{\bar{K}}=0$, and so $V_p\operatorname{Br}(E_{\bar{K}})\otimes_{\mathbb{Q}_p}\hat{\bar{K}}\subset H^2(E,\mathcal{O}_E)\otimes\hat{\bar{K}}(1)\oplus H^0(E,\Omega^2_E)\otimes\hat{\bar{K}}(-1)$. Since $$\begin{aligned}
\dim_{\mathbb{Q}_p}V_p\operatorname{Br}(E_{\bar{K}}) &=& \dim_{\mathbb{Q}_p} H^2(E_{\bar{K}},\mathbb{Q}_p(1))-\dim_{\mathbb{Q}_p} NS(X_{\bar{K}})\otimes\mathbb{Q}_p \\
&\geq &\dim_{\mathbb{Q}_p} H^2(E_{\bar{K}},\mathbb{Q}_p(1))-\dim_{\hat{\bar{K}}}H^1(E,\Omega^1_E)\otimes\hat{\bar{K}} \\
&=&\dim_{\hat{\bar{K}}}H^2(E,\mathcal{O}_E)\otimes\hat{\bar{K}}(1)+\dim_{\hat{\bar{K}}}H^0(E,\Omega^2_E)\otimes\hat{\bar{K}}(-1)\end{aligned}$$ this implies the result.
\[parabolic\] There is a canonical isomorphism $V_p\operatorname{Br}(E_{\bar{K}})=\tilde{H}^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))$.
Let $E':=E\times_XY$ and write $V:=\tilde{H}^1(Y(\mathbb{C}),R^1g_*\mathbb{Z}(1))$. By the classical Eichler-Shimura isomorphism (cf. Theorem \[faltings\]), $V$ is a weight 0 Hodge structure of type $\{(1,-1),(-1,1)\}$. We have $V\subset H^1(Y(\mathbb{C}),R^1g_*\mathbb{Z}(1))\subset H^2(E'(\mathbb{C}),\mathbb{Z}(1))$ and since $NS(E'_{\mathbb{C}}):=\operatorname{im}(NS(E_{\mathbb{C}})\to H^2(E'(\mathbb{C}),\mathbb{Z}(1))$ is a Hodge structure of type $(0,0)$ we have $(V\cap NS(E'_{\mathbb{C}}))\otimes\mathbb{Q}=0$, hence $V\otimes\mathbb{Q}\subset H^2(E'(\mathbb{C}),\mathbb{Q}(1))/NS(E_{\mathbb{C}}')\otimes\mathbb{Q}$.
Now $E\setminus E'=\coprod_{x\in\Sigma}g^{-1}(x)$ is the disjoint union of the singular fibres of $g$, and we have an exact sequence in singular cohomology $$H^2(E(\mathbb{C}),\mathbb{Z}(1))\to H^2(E'(\mathbb{C}),\mathbb{Z}(1))\to \oplus_{x\in\Sigma(\bar{K})}H^3_{g^{-1}(x)}(E(\mathbb{C}),\mathbb{Z}(1)).$$ We also have a commutative diagram $$\begin{CD}
0 @>>> NS(E_{\mathbb{C}}) @>>> H^2(E(\mathbb{C}),\mathbb{Z}(1)) @>>> H^2(E(\mathbb{C}),\mathbb{Z}(1))/NS(E_{\mathbb{C}}) @>>> 0 \\
@. @VVV @VVV @VVV @. \\
0 @>>> NS(E_{\mathbb{C}}') @>>> H^2(E'(\mathbb{C}),\mathbb{Z}(1)) @>>> H^2(E'(\mathbb{C}),\mathbb{Z}(1))/NS(E_{\mathbb{C}}') @>>> 0
\end{CD}$$ in which the left vertical map is surjective, hence the right vertical map is injective and we deduce an exact sequence $$0\to H^2(E(\mathbb{C}),\mathbb{Z}(1))/NS(E_{\mathbb{C}})\to H^2(E'(\mathbb{C}),\mathbb{Z}(1))/NS(E_{\mathbb{C}}')\to \oplus_{x\in\Sigma(\bar{K})}H^3_{g^{-1}(x)}(E(\mathbb{C}),\mathbb{Z}(1)).$$ By Poincaré duality $H^3_{g^{-1}(x)}(E(\mathbb{C}),\mathbb{Q}(1))^*=H^1(g^{-1}(x)(\mathbb{C}),\mathbb{Q}(1))=\mathbb{Q}(1)$ (since $g^{-1}(x)$ is a Néron polygon), hence $\oplus_{x\in\Sigma(\bar{K})}H^3_{g^{-1}(x)}(E(\mathbb{C}),\mathbb{Z}(1))$ is a Hodge structure of weight 2 and therefore the map $$V\to \oplus_{x\in\Sigma(\bar{K})}H^3_{g^{-1}(x)}(E(\mathbb{C}),\mathbb{Q}(1))$$ is zero. Thus, $$V\otimes\mathbb{Q}\subset H^2(E(\mathbb{C}),\mathbb{Q}(1))/NS(E_{\mathbb{C}})\otimes\mathbb{Q}.$$ Now by the Eichler-Shimura isomorphism we have $\dim V\otimes\mathbb{Q}=2\dim H^0(X,L\otimes\Omega^1_X)$, and since $H^0(X,L\otimes\Omega^1_X)=H^0(E,\Omega^2_E)$ ([@schuettshioda th. 6.8]) from Corollary \[HT\] (and Serre duality) we get $\dim V=\dim V_p\operatorname{Br}(E_{\bar{K}})$. Since $\left(H^2(E(\mathbb{C}),\mathbb{Z}(1))/NS(E_{\mathbb{C}})\right)\otimes\mathbb{Q}_p=V_p\operatorname{Br}(E_{\bar{K}})$ we have $V\otimes\mathbb{Q}_p=V_p\operatorname{Br}(E_{\bar{K}})$.
Shioda [@shioda1] shows that $H^1(E,\Omega^1_E)\otimes\hat{\bar{K}}$ is generated by the classes of divisors, which together with the Hodge-Tate decomposition gives another proof of Corollary \[HT\]. Combining this with Corollary \[parabolic\], this gives another proof that $\tilde{H}^1(Y_{\bar{K}},R^1g_*\mathbb{Q}_p(1))$ is a Hodge-Tate representation with weights $\pm 1$.
Application of Hecke operators
------------------------------
The Eichler-Shimura congruence relation relates the $p$th Hecke operator $T_p$ to the Frobenius morphism at $p$. We exploit this relationship to obtain the following
\[eichlershimura\] If $p\equiv 1\mod N$ and $K_0=\mathbb{Q}_p$, then $T:=\varphi+\varphi^{-1}$ is an endomorphism of $D:=D_{\operatorname{cris}}(V_p\operatorname{Br}(E_{\bar{K}}))$ which satisfies $T(F^1D)\subset F^1D$.
Recall ([@delrap V, 1.14]) that there is a regular proper $\mathbb{Z}[1/N]$-scheme $X(N,p)$ (denoted $\mathcal{M}_{\Gamma(N)\cap\Gamma_0(p)}$ in loc. cit.; in [@deligne] one only considers the dense open $M_{N,p}=\mathcal{M}_{\Gamma(N)\cap\Gamma_0(p)}^0$) classifying isomorphism classes of $p$-isogenies $\phi:(E\to S,\alpha)\to (E'\to S,\alpha')$. It is smooth away from $p$ and has semi-stable reduction at $p$. Consider the two canonical morphisms $$q_1:X(N,p)\to X(N):\phi\mapsto (E\to S,\alpha)$$ and $$q_2:X(N,p)\to X(N):\phi\mapsto (E'\to S,\alpha').$$ One can show that each $q_i$ is finite and flat. The universal object over $X(N,p)$ is a $p$-isogeny $$\phi:q_1^*E\to q_2^*E$$ where $E\to X(N)$ is the universal curve. By definition (cf. [@deligne 3.18]), the Hecke correspondence $T_p$ on $E$ is the correspondence $$\xymatrix{
& q_1^*E \ar[dl]_{q_2\circ\phi} \ar[dr]^{q_1} & \\
E & & E
}$$ that is, $T_p=q_{1,*}\phi^*q_2^*$. Now the Eichler-Shimura relation states that $$T_p|_{E_k}=F+I_p{\,^{\operatorname{t}}{F}}$$ where $F$ is the Frobenius, ${\,^{\operatorname{t}}{F}}$ is its transpose as a correspondence, and $I_p$ is the morphism of $X(N)$ defined $I_p(\mathcal{E},\alpha):=(\mathcal{E},p\alpha)$. This can be proven in the same way as [@deligne 4.8], and we sketch the proof.
We first check the equality over a dense open of $X(N,p)$. As shown in [@deligne 4.3], there is a dense open of $X(N,p)_{\mathbb{F}_p}$ isomorphic to the disjoint union of two copies of the dense open $Y(N)_{\mathbb{F}_p}^h\subset Y(N)_{\mathbb{F}_p}$, complement of the supersingular points. Over one of these copies we have $q_1=\text{id}$, and $q_2=F_{Y(N)}$ the Frobenius of $Y(N)^h_{\mathbb{F}_p}$, over the other $q_1=F_{Y(N)}$ and $q_2=I_p$ (cf. [@delrap V, 1.17]). In the first case the universal isogeny is the relative Frobenius $F_{E/Y(N)}:(E,\alpha)\to (F^*_{Y(N)}E,F^*_{Y(N)}\alpha)$, in the second the universal isogeny is its transpose, the Verschiebung ${\,^{\operatorname{t}}{F_{E/Y(N)}}}:(F^*_{Y(N)}E,F^*_{Y(N)}\alpha)\to (E,p^{-1}\alpha)=I_p^*(E,\alpha)$. So in the first case the Hecke correspondence is $$\xymatrix{
& & E \ar[dl]_{F_{E/Y(N)}} \ar@{=}[ddrr] & & \\
& F_{Y(N)}^*E \ar[dl]_{F_{Y(N)}} & & & \\
E & & & & E
}$$ which is obviously $F$. In the second case it is $$\xymatrix{
& & F^*_{Y(N)}E \ar[dl]_{{\,^{\operatorname{t}}{F_{E/Y(N)}}}} \ar[ddrr]^{F_{Y(N)}} & & \\
& I_p^*E \ar[dl]_{I_p} & & & \\
E & & & & E
}$$ i.e. $I_p{\,^{\operatorname{t}}{F}}$.
For the general case, first recall that the open immersion $Y(N)^h_{\mathbb{F}_p}\coprod Y(N)^h_{\mathbb{F}_p}\to X(N,p)_{\mathbb{F}_p}$ extends to a morphism $X(N)_{\mathbb{F}_p}\coprod X(N)_{\mathbb{F}_p}\to X(N,p)_{\mathbb{F}_p}$ which identifies the domain with the disjoint union of the irreducible components of the target [@delrap V, §1], hence is surjective. By the above this gives a surjective morphism $E_k\coprod F_{X(N)}^*E_k\to q_1^*E_k$. Therefore, $\operatorname{im}(q_1^*E_k\to E_k\times_k E_k)=\operatorname{im}(E_k\coprod F_{X(N)}^*E_k\to E_k\times_kE_k)$ has at most two irreducible components. Its irreducible components are therefore the transpose of the graph of $F$ and the graph of $FI_p^{-1}$ (note that these are irreducible since $E_k$ is). This proves the Eichler-Shimura relation.
Now since $p\equiv 1\mod N$, $I_p$ is the identity map and we obtain the relation $$\label{ES} T_p|_{E_{k}}=F+{\,^{\operatorname{t}}{F}}.$$ Since we have $p\varphi=F$ and ${\,^{\operatorname{t}}{F}}F=p^2$ ($\dim E_k=2$), it follows that $pT=T_p|_{E_{k}}$.
Finally, let $Z:=\operatorname{im}(q_1^*E\to E\times_WE)$. Note that since $q_1^*E$ is flat over $W$, so is $Z$. The correspondence $T_p$ acts as $\operatorname{pr}_{2,*}\circ c_Z\circ\operatorname{pr}_1^*$, where $c_Z$ is the cup product with the characteristic class of $Z$, and $\operatorname{pr}_i:E\times_WE\to E$ are the projections. Therefore, to complete the proof it remains only to notice that the canonical isomorphism $H^*_{\operatorname{dR}}(E_{K_0}\times_{K_0}E_{K_0})=H^*_{\operatorname{cris}}(E_k\times_kE_k/K_0)$ is compatible with taking characteristic classes of closed subschemes of $E\times_WE$ which are flat over $W$. Indeed, the isomorphism is compatible with Chern classes of line bundles, hence also (by the splitting principle) with Chern classes of vector bundles, and the structure sheaf of a closed subscheme of $E\times_WE$ has a finite resolution by vector bundles which, if flat over $W$, reduces to give a resolution on $E_k\times_kE_k$.
As a corollary we obtain Theorem \[main\].
\[tate\] If $k=\mathbb{F}_p$ and $p\equiv 1\mod N$, then under hypothesis (\[SS\]) we have
(i) Tate’s conjecture holds for $E_k$
(ii) $NS(E_{\bar{K}})^{G_{K_0}}\otimes\mathbb{Q}=NS(E_{\bar{k}})^{G_{K_0}}\otimes\mathbb{Q}$
(iii) the Mordell-Weil group of the generic fibre of $E_k\to X_k$ is isomorphic to $(\mathbb{Z}/N)^2$.
Note that (\[SS\]) implies that $D_{\operatorname{cris}}(V_p\operatorname{Br}(E(N)_{\bar{K}})^{(\varphi-1)^2=0}=D_{\operatorname{cris}}(V_p\operatorname{Br}(E(N)_{\bar{K}}))^{\varphi=1}$. So by Theorem \[eichlershimura\], (i) and (ii) follow from Corollary \[maincor\]. It follows from (ii) that the Mordell-Weil group is finite, hence isomorphic to $(\mathbb{Z}/N)^2$ by [@shioda1 Appendix, Cor. 3].
Validity of (\[SS\])
--------------------
We can show that (\[SS\]) holds for all primes outside of a set of density zero. For the sake of brevity we only sketch the argument. Let $Y_1(N)$ denote the Deligne-Mumford moduli stack of pairs $(\mathcal{E}\to S,P,P')$ where $\mathcal{E}\to S$ is an elliptic curve over a $\mathbb{Z}[1/N]$-scheme $S$, $P\in\mathcal{E}[N](S)$ a point of exact order $N$ and $P'\in(\tfrac{\mathcal{E}[N]}{<P>})(S)\cong\mu_N(S)$ a point of exact order $N$. For $N\geq 5$ it is known to be a $\mathbb{Z}[1/N,\zeta_N]$-scheme with geometrically connected fibres. Let $g:E_1(N)\to Y_1(N)$ be the universal elliptic curve and consider $V_N:=\tilde{H}^1(Y_1(N)_{\bar{K}},R^1g_*\mathbb{Q}_p)(1)$. We will first explain why (\[SS\]) holds for $D_{\operatorname{cris}}(V_N)$ outside a set of primes of density zero.
For every proper divisor $d$ of $N$ there are pairs of maps $\pi_i:V_{N/d}\to V_N$ ($i=1,2$) defined just like for modular forms. (One map is just the canonical “inclusion” and the other arises from the inclusion $\bigl(\begin{smallmatrix}
d&0\\ 0&1
\end{smallmatrix} \bigr)\Gamma_1(N)\bigl(\begin{smallmatrix}
d&0\\ 0&1
\end{smallmatrix} \bigr)^{-1}\subset\Gamma_1(N/d)$.) The image of these maps is a subspace $V_N^{\operatorname{old}}$ of $V_N$. Its orthogonal complement under the cup product is a subspace $V_N^{\operatorname{new}}$ which corresponds to newforms. The latter splits under the action of the Hecke algebra as a direct sum $$V_N^{\operatorname{new}}=\oplus_{i=1}^mV(f_i)$$ where $f_1,...,f_m$ are a choice of representatives of the $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-conjugacy classes of weight 3 normalized newforms for $\Gamma_1(N)$, and $V(f_i)(-1)$ is the Galois representation associated to $f_i$ by Deligne. Now, by autoduality we have $$V_N^{\operatorname{old}}\subset\prod_{d|N,d>1}V_{N/d}^{\oplus 2}$$ so by induction on $N$ we may assume (\[SS\]) to hold for $D_{\operatorname{cris}}(V_N^{\operatorname{old}})$. The space $V(f_i)$ is free of rank 2 over $K_{f_i}\otimes\mathbb{Q}_p$, where $K_{f_i}$ is the field of coefficients of $f_i$. If $\varphi$ is the Frobenius, we want to show that $D_{\operatorname{cris}}(V(f_i))^{(\varphi^n-1)^2=0}=D_{\operatorname{cris}}(V(f_i))^{\varphi^n=1}$ for some fixed $n$. This is immediate if $f_i$ has CM: in this case $\varphi^2$ acts as a diagonal matrix [@ribet]. If $f_i$ does not have CM and (\[SS\]) does not hold, then the minimal polynomial of $\varphi^n$ is $(t-1)^2$, so both eigenvalues of $\varphi^n$ are equal to $1$. Since the trace of $\varphi$ is equal to $\tfrac{a_p}{p}$, where $a_p$ is the $p$th coefficient of $f_i$, this implies that $a_p=(\zeta+\zeta')p$ for some $n$th roots of unity $\zeta,\zeta'$. By [@serrecebotarev], this can only happen for a set of primes of density zero (dependent on the choice of $n$). For given $N$ and $K_0$ we can take $n$ such that $p^n=\#k$.
Finally, if $\zeta_{N^2}\in K_0$, then there is a dominant morphism $Y_1(N^2)_{K_0}\to Y(N)_{K_0}$ arising from the inclusion $\bigl(\begin{smallmatrix}
N&0\\ 0&1
\end{smallmatrix} \bigr)\Gamma_1(N^2)\bigl(\begin{smallmatrix}
N&0\\ 0&1
\end{smallmatrix}\bigr)^{-1}\subset\Gamma(N)$. From this one can deduce (\[SS\]) for $D_{\operatorname{cris}}(\tilde{H}^1(Y(N)_{\bar{K}},R^1g_*\mathbb{Q}_p(1)))=D_{\operatorname{cris}}(V_p\operatorname{Br}(E(N)_{\bar{K}}))$.
\[tatecor\] For $k=\mathbb{F}_p$ and $p\equiv 1\mod N$, the conclusion of Corollary \[tate\] holds outside of a set of primes of density zero.
The finite height case
======================
Let $W\subset V$ be a finite extension of discrete valuation rings, $k_V$ (resp. $K$) the residue (resp. fraction) field of $V$, and set $T:=\operatorname{Spec}(V)$.
The formal Brauer group
-----------------------
Let $f:X\to T$ be a proper smooth morphism. Artin and Mazur considered the sheaves $$S\rightsquigarrow\widehat{\mathbb{G}}_m(S):=\ker(\mathbb{G}_m(S)\to\mathbb{G}_m(S_{\operatorname{red}}))$$ on the big étale site of $X$ and $$\Phi^i:=R^if_*\widehat{\mathbb{G}}_m$$ on the big étale site of $T$, and proved the following ([@artinmazur II, 2.12]).
If $\Phi^{i-1}$ is a formally smooth functor and $H^{i+1}(X_{k_V},\mathcal{O}_{X_{k_V}})=0$, then $\Phi^i$ is pro-representable by a formal Lie group over $V$.
Assume $\dim X_{k_V}=2$. If $\Phi^1$ is formally smooth, then $\Phi^2$ is pro-representable by a formal Lie group over $V$, called the formal Brauer group.
Now if we assume $\Phi^2$ is a formal Lie group of *finite height* (i.e. a connected $p$-divisible group), then we obtain the following.
\[repcor\] Assume $k$ is finite and $\dim X_{k_V}=2$. Let $\bar{V}$ be the normalization of $V$ in $\bar{K}$. If $\Phi^1$ is formally smooth and $\Phi^2$ is of finite height, then the canonical map $$\rho:\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V'/p^nV'},\mathbb{G}_m)\to\operatorname{Br}(X_{\bar{k}})$$ is surjective on $p$-primary torsion components, where the direct limit is taken over all finite extensions of discrete valuation rings $V\subset V'$ contained in $\bar{V}$.
It suffices to show that $\rho$ is surjective and $\ker(\rho)$ is a $p$-divisible abelian group. To this end, consider the exact sequence of étale sheaves on $X_{V'/p^nV'}$ $$\label{gmdevissage} 1\to\widehat{\mathbb{G}}_m\to\mathbb{G}_{m,X_{V'/p^nV'}}\to\mathbb{G}_{m,X_{k_{V'}}}\to 1.$$ Note that since $V'$ is a henselian discrete valuation ring with finite residue field for any torsion étale sheaf $\mathcal{F}$ on $V'$ we have $H^i(V',\mathcal{F})=0$ for $i>1$ and $H^1(V',\mathcal{F})=(\mathcal{F}(\bar{k}))_{G_{k_{V'}}}$ (coinvariants), where $G_{k_{V'}}:=\operatorname{Gal}(\bar{k}/k_{V'})$. Let $V'\subset\mathcal{V}$ be the maximal unramified extension. From the Leray spectral sequence for the Galois covering $V'\subset\mathcal{V}$ we get an exact sequence $$0\to H^1(X_{\mathcal{V}/p^n\mathcal{V}},\widehat{\mathbb{G}}_m)_{G_{k_{V'}}}\to H^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)\to H^2(X_{\mathcal{V}/p^n\mathcal{V}},\widehat{\mathbb{G}}_m)^{G_{k_{V'}}}\to 0$$ i.e. $$0\to\Phi^1(\mathcal{V}/p^n\mathcal{V})_{G_{k_{V'}}}\to H^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)\to \Phi^2(V'/p^nV')\to 0.$$ Now since the ideal $\ker(\mathcal{O}_{X_{V'/p^nV'}}\to\mathcal{O}_{X_{k_{V'}}})$ is nilpotent, the sheaf $\widehat{\mathbb{G}}_m$ has a finite filtration with quotients isomorphic to $\mathcal{O}_{X_{k_{V'}}}$, and this implies that $H^i(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)$ is finite for all $i$. So taking the inverse limit in the exact sequence we obtain an exact sequence $$0\to\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})_{G_{k_{V'}}}\to\varprojlim_nH^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)\to\Phi^2(V')\to 0.$$ Now assume the following: $$\label{assump}\tag{D} \varinjlim_{V'\subset\mathcal{V}}\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})_{G_{k_{V'}}}\;\text{is a $p$-divisible abelian group.}$$ This implies that $$\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)$$ is a $p$-divisible abelian group: by the above we have a surjective map $$\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)\to\varinjlim_{V'\subset\bar{V}}\Phi^2(V')=\Phi^2(\bar{V})$$ whose kernel is $p$-divisible by (\[assump\]), and by [@tate Cor. to Prop. 4] $\Phi^2(\bar{V})$ is a $p$-divisible abelian group. Taking cohomology in the exact sequence (\[gmdevissage\]) we get an exact sequence $$H^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)\to H^2(X_{V'/p^nV'},\mathbb{G}_m)\to\operatorname{Br}(X_{k_{V'}})\to 0$$ (the exactness on the right follows from $\dim X_{k_{V'}}=2$). Since $H^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)$ is finite taking limits we get an exact sequence $$\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V'/p^nV'},\widehat{\mathbb{G}}_m)\to\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V'/p^nV'},\mathbb{G}_m)\overset{\rho}{\to}\operatorname{Br}(X_{\bar{k}})\to 0$$ and this implies the result.
So it suffices to show (\[assump\]). We claim that $M:=\left(\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})\right)\otimes\mathbb{Z}/p$ is a discrete $G_{k_{V'}}$-module (i.e. the stabilizer of every element is open in $G_{k_{V'}}$). Indeed, taking cohomology in (\[gmdevissage\]) we get an exact sequence $$0\to \Phi^1(\mathcal{V}/p^n\mathcal{V})\to\operatorname{Pic}(X_{\mathcal{V}/p^n\mathcal{V}})\to\operatorname{Pic}(X_{k_{\mathcal{V}}})$$ whence an exact sequence $$0\to\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})\to\varprojlim_n\operatorname{Pic}(X_{\mathcal{V}/p^n\mathcal{V}})\to\operatorname{Pic}(X_{k_{\mathcal{V}}}).$$ Now, denoting by $\widehat{X_{\mathcal{V}}}$ the formal completion of $X_{\mathcal{V}}$ along its special fibre over $\operatorname{Spec}(\mathcal{V})$, we have $$\varprojlim_n\operatorname{Pic}(X_{\mathcal{V}/p^n\mathcal{V}})=\operatorname{Pic}(\widehat{X_{\mathcal{V}}})$$ so it clearly suffices to show that $\operatorname{Pic}(\widehat{X_{\mathcal{V}}})\otimes\mathbb{Z}/p$ is a discrete $G_{k_{V'}}$-module. But taking cohomology in the exact sequence on $\widehat{X_{\mathcal{V}}}$ $$1\to 1+p^2\mathcal{O}_{\widehat{X_{\mathcal{V}}}}\to\mathbb{G}_{m,\widehat{X_{\mathcal{V}}}}\to\mathbb{G}_{m,X_{\mathcal{V}/p^2\mathcal{V}}}\to 1$$ we get an exact sequence $$H^1(\widehat{X_{\mathcal{V}}},1+p^2\mathcal{O}_{\widehat{X_{\mathcal{V}}}})\to\operatorname{Pic}(\widehat{X_{\mathcal{V}}})\to\operatorname{Pic}(X_{\mathcal{V}/p^2\mathcal{V}})$$ and one sees easily (cf. §\[comparisonsection\]) that the (usual) logarithm induces an isomorphism $$\log:H^1(\widehat{X_{\mathcal{V}}},1+p^2\mathcal{O}_{\widehat{X_{\mathcal{V}}}})\cong H^1(\widehat{X_{\mathcal{V}}},p^2\mathcal{O}_{\widehat{X_{\mathcal{V}}}})$$ and the claim follows from this. Finally, note that since $\Phi^1$ is formally smooth by assumption, the maps $\Phi^1(\mathcal{V}/p^{n+1}\mathcal{V})\to\Phi^1(\mathcal{V}/p^n\mathcal{V})$ are surjective for all $n$, so in particular $\varprojlim_n^{(1)}\Phi^1(\mathcal{V}/p^n\mathcal{V})=0$. If $\gamma\in G_{k_{V'}}$ is a topological generator, then we have an exact sequence $$0\to \Phi^1(\mathcal{V}/p^n\mathcal{V})^{G_{k_{V'}}}\to \Phi^1(\mathcal{V}/p^n\mathcal{V})\overset{\gamma-1}{\to}\Phi^1(\mathcal{V}/p^n\mathcal{V})\to \Phi^1(\mathcal{V}/p^n\mathcal{V})_{G_{k_{V'}}}\to 0.$$ Noting that $\Phi^1(\mathcal{V}/p^n\mathcal{V})^{G_{k_{V'}}}=\Phi^1(V'/p^nV')$, hence $\varprojlim_n^{(1)}\Phi^1(\mathcal{V}/p^n\mathcal{V})^{G_{k_{V'}}}=0$ (again by the formal smoothness of $\Phi^1$), and taking limits in this sequence we find $$\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})_{G_{k_{V'}}}=\left(\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})\right)_{G_{k_{V'}}}.$$ Since direct limits commute with tensor product we get $$\left(\varinjlim_{V'\subset\mathcal{V}}\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V})_{G_{k_{V'}}}\right)\otimes\mathbb{Z}/p=\varinjlim_{V'\subset\mathcal{V}}\left((\varprojlim_n\Phi^1(\mathcal{V}/p^n\mathcal{V}))\otimes\mathbb{Z}/p\right)_{G_{k_{V'}}}=\varinjlim_{V'\subset\mathcal{V}}M_{G_{k_{V'}}}.$$ Now as $M$ is a torsion discrete $G_{k_{V'}}$-module we have $$\varinjlim_{V'\subset\mathcal{V}}M_{G_{k_{V'}}}=\varinjlim_{V'\subset\mathcal{V}}H^1(k_{V'},M)=0$$ which proves (\[assump\]).
### Example: elliptic surfaces {#elliptic}
Suppose that $f:X\to T$ is a projective flat morphism with geometrically integral fibres. Then $R^1f_*\mathbb{G}_m$ is representable by a scheme $\operatorname{Pic}_{X/T}$ which is locally of finite type over $T$ ([@kleiman 9.4.8]). Moreover, for all $t\in T$, $H^1(X_t,\mathcal{O}_{X_t})$ is the tangent space of $(\operatorname{Pic}_{X/T})_t=\operatorname{Pic}_{X_t/t}$ at the origin ([@kleiman 9.5.11]), hence $\operatorname{Pic}_{X_t/t}$ is smooth over $t$ if and only if $$\dim\operatorname{Pic}_{X_t/t}\geq\dim H^1(X_t,\mathcal{O}_{X_t})$$ in which case we have equality since the reverse inequality always holds.
Now let $Y\to T$ be a smooth projective curve, and let $g:X\to Y$ be an elliptic fibration, i.e. $g$ is a projective morphism with a section whose geometric fibres are connected of dimension 1 and whose generic fibre in each fibre over $T$ is smooth of genus 1. Assume further that $X_t$ is a non-isotrivial smooth elliptic surface for all $t\in T$ (in particular $X$ is smooth over $T$). Then the map $$\operatorname{Pic}^0_{Y_t/t}\to\operatorname{Pic}^0_{X_t/t}$$ is an isomorphism for all $t\in T$ ([@schuettshioda th. 6.12]). In particular, the map $H^1(Y_t,\mathcal{O}_{Y_t})\to H^1(X_t,\mathcal{O}_{X_t})$ is an isomorphism for all $t\in T$, hence $\dim H^1(X_t,\mathcal{O}_{X_t})$ is independent of $t$, since this is true for $Y$. $\operatorname{Pic}_{X_K/K}$ being smooth ($\operatorname{char}(K)=0$), we get $\dim\operatorname{Pic}_{X_K/K}=\dim H^1(X_k,\mathcal{O}_{X_k})$. Since the connected component $\operatorname{Pic}^0_{X/T}$ of $0$ in $\operatorname{Pic}_{X/T}$ is proper over $T$ ([@kleiman 9.6.18]), we have $$\dim\operatorname{Pic}_{X_{k_V}/k_V}=\dim\operatorname{Pic}^0_{X_{k_V}/k_V}\geq\dim\operatorname{Pic}^0_{X_K/K}=\dim H^1(X_{k_V},\mathcal{O}_{X_{k_V}})$$ so $\operatorname{Pic}_{X_{k_V}/k_V}$ is smooth. By [@kleiman 9.5.20], this implies that $\operatorname{Pic}^0_{X/T}$ is smooth over $T$. (Note however that this does not imply that $\operatorname{Pic}_{X/T}$ is smooth over $T$, cf. [@kleiman 9.5.22].) Since $\Phi^1$ is the completion of $\operatorname{Pic}^0_{X/T}$ along its zero section, it is formally smooth in this case.
Relationship between $\operatorname{Br}(X_{\bar{V}})$ and $\operatorname{Br}(X_{\bar{k}})$ {#comparisonsection}
------------------------------------------------------------------------------------------
For a scheme $S$ write $S_n:=S\otimes\mathbb{Z}/p^n$. Let $f:X\to T$ be a proper smooth morphism with geometrically connected fibres, and let $\hat{X}:=\varinjlim_nX_n$ be the $p$-adic completion of $X$. In the sequel let $j:X_K\subset X$ and $i:X_1\subset X$ be the inclusions. Then we have exact sequences of étale sheaves on $X$ $$0\to j_!\mathbb{G}_m\to\mathbb{G}_m\to i_*i^*\mathbb{G}_m\to 0$$ and $$0\to 1+p^2i_*i^*\mathcal{O}_{X}\to i_*i^*\mathbb{G}_m\to i_*\mathbb{G}_{m,X_2}\to 0$$ where, for a morphism $g$ of schemes, $g^*$ always denotes the inverse image functor and not module pullback.
\[torsion\] We have $i^*\mathbb{G}_m[p^n]=i^*f^*(\mathcal{O}^*_T[p^n])=\mathbb{G}_{m,\hat{X}}[p^n]$ for all $n$, where $\mathbb{G}_{m,\hat{X}}=\varprojlim_n\mathbb{G}_{m,X_n}$.
Since the generic fibre of $f$ is geometrically connected, $K$ is algebraically closed in the fraction field of $X$. Let $x\to X_{k_V}$ be a geometric point and let $V'$ be the normalization of $V$ in $\mathcal{O}_{X,x}$. If $\pi\in V$ is a uniformizer, by flatness we have $V'/\pi V'\subset\mathcal{O}_{X,x}/\pi\mathcal{O}_{X,x}=\mathcal{O}_{X_{k_V},x}$, hence $\pi\in V'$ is also a uniformizer and $V'$ is unramified over $V$. In particular any $p$-power root of unity in $\mathcal{O}_{X,x}$ already lies in $V$, i.e. $i^*\mathbb{G}_m[p^n]=i^*f^*(\mathcal{O}^*_T[p^n])$. For the second equality, note that since $X$ is excellent, the formal fibres of $\mathcal{O}_{X,x}$ are geometrically integral ([@ega IV, 18.9.1]), hence $\mathcal{O}_{X,x}$ is integrally closed in its completion (since $X$ is normal). In particular, $\mathbb{G}_{m,\hat{X}}[p^n]=i^*\mathbb{G}_m[p^n]$.
\[logexp\] Let $A$ be a $p$-adically complete ring. For all $n\geq 1$ the logarithm induces a natural isomorphism $1+p^{n+1}A\cong p^{n+1}A$. In particular, $(1+p^{n+1}A)^p=1+p^{n+2}A$.
The logarithm induces a map $$\begin{aligned}
\log:1+p^{n+1}A &\to& p^{n+1}A \\
1+p^{n+1}x &\mapsto& -p^{n+1}\sum_{r>0}\dfrac{p^{(n+1)(r-1)}(-x)^{r}}{r}\end{aligned}$$ and the exponential $$\begin{aligned}
\exp:p^{n+1}A &\to& 1+p^{n+1}A \\
p^{n+1}x &\mapsto& 1+p^{n+1}\sum_{r>0}\dfrac{p^{(n+1)(r-1)}x^{r}}{r!}\end{aligned}$$ is its inverse. Recall that $v_p(r!)\leq\dfrac{r-1}{p-1}$, so in particular $v_p(r!)<(n+1)(r-1)$ and $\lim_{r\to+\infty}(n+1)(r-1)-v_p(r!)=+\infty$ for all $n\geq 1$, hence these homomorphisms are well-defined.
\[logexp2\] The inclusion $(1+p^{n+1}i_*i^*\mathcal{O}_{X})^p\subset 1+p^{n+2}i_*i^*\mathcal{O}_{X}$ is bijective and $\left(1+p^{n+1}i_*i^*\mathcal{O}_{X}\right)\otimes\mathbb{Z}/p^m\cong p^{n+1}\mathcal{O}_X/p^{n+1+m}\mathcal{O}_X$ for all $n,m\geq 1$.
This is local so we may assume that $X=\operatorname{Spec}(R)$ is the spectrum of a henselian regular local ring with $p$ in its maximal ideal. If $\hat{R}$ denotes the $p$-adic completion we have a commutative diagram with exact rows $$\xymatrix{
1 \ar [r] & (1+p^{n+1}R)^p \ar[d] \ar[r] & 1+p^{n+1}R \ar[d] \ar[r] & C \ar[d] \ar[r] & 0 \\
1 \ar[r] & (1+p^{n+1}\hat{R})^p \ar[r] & 1+p^{n+1}\hat{R} \ar[r] & p^{n+1}R/p^{n+2}R \ar[r] & 0
}$$ where the middle vertical map is injective and the bottom right term follows from Lemma \[logexp\]. The statement amounts to showing that the right vertical map is injective since it is already clearly surjective. For this it suffices to show that the cokernel $D$ of the left vertical map is $p$-torsion free. This follows immediately from the fact, already noted in the proof of Lemma \[torsion\], that $R$ is integrally closed in $\hat{R}$.
\[logexp3\] For all $n$ the canonical map $$\mathbb{G}_{m,X}/p^{n}\to i_*\mathbb{G}_{m,\hat{X}}/p^{n}$$ is bijective.
Since $j_!\mathbb{G}_m$ is $p$-divisible ($j_!$ being exact), we have $\mathbb{G}_{m,X}/p^n=i_*i^*\mathbb{G}_{m,X}/p^n$. Now we have a commutative diagram $$\begin{CD}
0 @>>> 1+p^{2}i^*\mathcal{O}_X @>>> i^*\mathbb{G}_{m,X} @>>> \mathbb{G}_{m,X_2} @>>> 0 \\
@. @VVV @VVV @| @. \\
0 @>>> 1+p^{2}\mathcal{O}_{\hat{X}} @>>> \mathbb{G}_{m,\hat{X}} @>>> \mathbb{G}_{m,X_2} @>>> 0 \\
\end{CD}$$ so reducing modulo $p^n$ we see that the result follows from Proposition \[logexp2\] and the five lemma.
\[surj1\] The canonical map $\operatorname{Br}(X)\to H^2(X_1,\mathbb{G}_{m,\hat{X}})$ is surjective on $p$-primary torsion components.
Let $\mathcal{T}\subset\mathbb{G}_{m,X}$ and $\mathcal{F}\subset\mathbb{G}_{m,\hat{X}}$ be the $p$-power torsion subsheaves, and define $\mathbb{G}'_{m,X}:=\mathbb{G}_m/\mathcal{T}$ and $\mathbb{G}'_{m,\hat{X}}:=\mathbb{G}_m/\mathcal{F}$. By Lemma \[torsion\] $\mathcal{F}$ is a finite constant sheaf and we have an exact sequence $$\label{exseqtorsion} 0\to j_!\mathbb{Q}_p/\mathbb{Z}_p(1)\to\mathcal{T}\to\mathcal{F}\to 0.$$ By the proper base change theorem we have $H^n(X,\mathcal{T})=H^n(X_1,\mathcal{F})=H^n(X,\mathcal{F})$ for all $n$, hence we deduce a commutative diagram with exact rows $$\begin{CD}
H^2(X,\mathcal{F}) @>>> H^2(X,\mathbb{G}_m) @>>> H^2(X,\mathbb{G}'_{m,X}) @>>> H^3(X,\mathcal{F}) \\
@| @VVV @VVV @| \\
H^2(X_1,\mathcal{F}) @>>> H^2(X_1,\mathbb{G}_{m,\hat{X}}) @>>> H^2(X_1,\mathbb{G}'_{m,\hat{X}}) @>>> H^3(X_1,\mathcal{F})
\end{CD}$$ and since $\operatorname{Br}(X)=H^2(X,\mathbb{G}_m)$ is torsion ([@brauer II, 1.4]) an easy diagram chase shows that it suffices to prove that the map $$H^2(X,\mathbb{G}'_{m,X})\to H^2(X_1,\mathbb{G}'_{m,\hat{X}})$$ is surjective on $p$-primary torsion. Now since the functor $j_!$ is exact, $j_!\mathbb{Q}_p/\mathbb{Z}_p(1)$ is divisible, so from the sequence (\[exseqtorsion\]) for all $n$ we get an exact sequence $$0\to\mathcal{F}/p^n\to\mathbb{G}_{m,X}/p^n\to\left(\mathbb{G}'_{m,X}\right)/p^n\to 0$$ and therefore by Corollary \[logexp3\] we deduce isomorphisms $$\mathbb{G}_{m,X}'/p^n=\mathbb{G}_{m,\hat{X}}'/p^n$$ for all $n$. Hence for all $n$ we have a commutative diagram with exact rows $$\begin{CD}
H^1(X,\mathbb{G}'_{m,X}/p^n) @>>> H^2(X,\mathbb{G}'_{m,X})[p^n] @>>> 0 \\
@| @VVV @. \\
H^1(X_1,\mathbb{G}_{m,\hat{X}}'/p^n) @>>> H^2(X_1,\mathbb{G}_{m,\hat{X}}')[p^n] @>>> 0
\end{CD}$$ and the result follows.
Denote by $\varprojlim_n^{(m)}$ the right derived functors of $\varprojlim_n$.
\[limexact\]
(i) Suppose $\{\mathcal{F}_n\}_{n\in\mathbb{N}}$ is an inverse system of sheaves of abelian groups on $X_1$ such that for any étale map $h:U\to X_1$ with $U$ affine we have
(a) the transition maps $\mathcal{F}_{n+1}(U)\to\mathcal{F}_n(U)$ are surjective for all $n$
(b) $H^i(U,\mathcal{F}_n)=0$ for all $i>0$ and any $n$.
Then $\varprojlim_n^{(m)}\mathcal{F}_n=0$ for all $m\geq 1$.
(ii) $\varprojlim_n^{(m)}\mathbb{G}_{m,X_n}=0$ for $m\geq 1$.
(iii) There are canonical isomorphisms $H^i(X_1,\mathbb{G}_{m,\hat{X}})=H^i_{\operatorname{cont}}(X_1,\{\mathbb{G}_{m,X_n}\})$ for all $i$, where $H^*_{\operatorname{cont}}$ is the continuous étale cohomology, i.e. the derived functor of $\varprojlim_nH^0(X_1,-)$ ([@jannsen]).
First note that since $\operatorname{Hom}$ commutes with products in the right argument, $h^*$ commutes with products. If $d_n:\mathcal{F}_{n+1}\to\mathcal{F}_n$ are the transition maps, then we have a left-exact sequence $$0\to\varprojlim_n\mathcal{F}_n\to\prod_n\mathcal{F}_n\overset{\text{id}-d_n}{\longrightarrow}\prod_n\mathcal{F}_n\to 0$$ which is also right-exact by virtue of property (a) and the well-known exact sequence ([@jannsen 1.4]) $$0\to\varprojlim_n\mathcal{F}_n(U)\to\prod_n\mathcal{F}_n(U)\overset{\text{id}-d_n}{\longrightarrow}\prod_n\mathcal{F}_n(U)\to\varprojlim_n{}^{(1)}\left(\mathcal{F}_n(U)\right)\to 0.$$ An object $\{\mathcal{I}_n\}_n$ of the category of inverse systems of sheaves of abelian groups is injective if and only if each $\mathcal{I}_n$ is injective and the transition maps $d_n:\mathcal{I}_{n+1}\to \mathcal{I}_{n}$ are split surjective ([@jannsen 1.1]). In particular, the sequence $$0\to\varprojlim_n\mathcal{I}_n\to\prod_n\mathcal{I}_n\overset{\text{id}-d_n}{\longrightarrow}\prod_n\mathcal{I}_n\to 0$$ is exact. Now let $\{\mathcal{F}_n\}\subset\{\mathcal{I}_n\}$ with $\{\mathcal{I}_n\}$ an injective inverse system of sheaves of abelian groups, and consider the exact sequence $$0\to \mathcal{F}_n\to \mathcal{I}_n\to \mathcal{G}_n\to 0$$ where $\mathcal{G}_n=\mathcal{I}_n/\mathcal{F}_n$. Since for any étale $U\to X_1$ with $U$ affine we have $H^1(U,\mathcal{F}_n)=0$, it follows that $\mathcal{G}_n(U)=\mathcal{I}_n(U)/\mathcal{F}_n(U)$. In particular, the sequence $$0\to\prod_n\mathcal{F}_n\to\prod_n\mathcal{I}_n\to\prod_n\mathcal{G}_n\to 0$$ is exact. From this and the above we deduce an exact sequence $$0\to\varprojlim_n\mathcal{F}_n\to\varprojlim_n\mathcal{I}_n\to\varprojlim_n\mathcal{G}_n\to 0$$ hence $\varprojlim_n^{(1)}\mathcal{F}_n=0$ and $\varprojlim_n^{(m)}\mathcal{F}_n=\varprojlim_n^{(m-1)}\mathcal{G}_n$ for $m>1$. Since $\{\mathcal{G}_n\}$ also satisfies (a) and (b), part (i) follows by induction on $m$.
For (ii), consider the exact sequence ($n\geq 2$) $$1\to 1+p^2\mathcal{O}_{X_n}\to\mathbb{G}_{m,X_n}\to\mathbb{G}_{m,X_2}\to 1.$$ Since $X$ is flat over $T$, the isomorphism of Lemma \[logexp\] induces canonical isomorphisms $1+p^2\mathcal{O}_{X_n}\cong p^2\mathcal{O}_{X_n}$, compatible for varying $n$. So taking limits in the exact sequence we get surjective maps $$\varprojlim_n{}^{(m)}p^2\mathcal{O}_{X_n}\twoheadrightarrow\varprojlim_n{}^{(m)}\mathbb{G}_{m,X_n}$$ for all $m\geq 1$. Thus it suffices to show that $\varprojlim_n^{(m)}p^2\mathcal{O}_{X_n}=0$ for $m\geq 1$, which follows from (i). Since $\varprojlim_n$ commutes with taking global sections and preserves injectives ([@jannsen 1.17]), (iii) follows from (ii).
If $k$ is finite, then the canonical map $H^2(X_1,\mathbb{G}_{m,\hat{X}})\to\varprojlim_nH^2(X_1,\mathbb{G}_{m,X_n})=\varprojlim_nH^2(X_n,\mathbb{G}_m)$ is surjective on $p$-primary torsion components.
By Proposition \[limexact\] (iii) and [@jannsen 3.1] we have an exact sequence $$0\to\varprojlim_n{}^{(1)}H^1(X_n,\mathbb{G}_{m,X_n})\to H^2(X_1,\mathbb{G}_{m,\hat{X}})\to\varprojlim_nH^2(X_n,\mathbb{G}_{m,X_n}) \to 0$$ so it suffices to show that $\varprojlim_n^{(1)}H^1(X_n,\mathbb{G}_{m,X_n})=\varprojlim_n^{(1)}\operatorname{Pic}(X_n)$ is $p$-divisible. Now let $P:=\operatorname{Pic}(X_1)$, $P_n:=\text{im}(\operatorname{Pic}(X_n)\to P)$ and $Q_n:=\ker(\operatorname{Pic}(X_n)\to P)$. We have an exact sequence $$H^1(X_{k_V},1+p\mathcal{O}_{X_n})\to\operatorname{Pic}(X_n)\to P\to H^2(X_{k_V},1+p\mathcal{O}_{X_n}).$$ Consider the finite filtration of $1+p\mathcal{O}_{X_n}$ $$1=1+p^n\mathcal{O}_{X_n}\subset 1+p^{n-1}\mathcal{O}_{X_n}\subset\cdots\subset 1+p\mathcal{O}_{X_n}.$$ It has length $n-2$ and its graded is annihilated by $p$. It follows that for all $i$ the group $H^i(X_{k_V},1+p\mathcal{O}_{X_n})$ is annihilated by $p^{n-1}$. Since $k$ is finite, this group is finite and we have $\varprojlim_n^{(1)}Q_n=0$, $\varprojlim_n^{(1)}\operatorname{Pic}(X_n)=\varprojlim_n^{(1)}P_n$, and for all $n$ we have $p^{n-1}P\subset P_n$, so we have surjective maps $P/p^{n-1}P\to P/P_n$. Since $P$ is a finitely generated abelian group (again since $k$ is finite), $P/p^{n-1}P$ is finite for all $n$ and we get a surjective map $$P\otimes\mathbb{Z}_p=\varprojlim_nP/p^{n-1}P\twoheadrightarrow\varprojlim_nP/P_n.$$ So $\varprojlim_n^{(1)}P_n=\left(\varprojlim_nP/P_n\right)/P$ is a quotient of the $p$-divisible abelian group $(P\otimes\mathbb{Z}_p)/P$, hence is also $p$-divisible.
Together with Proposition \[surj1\] this implies
\[surj2\] If $k$ is finite, then the canonical map $\operatorname{Br}(X)\to\varprojlim_nH^2(X_n,\mathbb{G}_m)$ is surjective on $p$-primary torsion components.
From Corollary \[repcor\] we can now deduce the following (cf. [@artinmazur IV, 2.1]).
\[finiteheight\] Assume $k$ is finite. If $\Phi^1$ is formally smooth and $\Phi^2$ is of finite height, then the canonical map $V_p\operatorname{Br}(X_{\bar{V}})\to V_p\operatorname{Br}(X_{\bar{k}})$ is surjective.
By Corollary \[surj2\], the canonical map $$\operatorname{Br}(X_{V'})\to\varprojlim_nH^2(X_{V',n},\mathbb{G}_m)$$ is surjective on $p$-primary torsion, hence so is the map $$\operatorname{Br}(X_{\bar{V}})\to\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V',n},\mathbb{G}_m).$$ So by Corollary \[repcor\] the composition $$\operatorname{Br}(X_{\bar{V}})\to\varinjlim_{V'\subset\bar{V}}\varprojlim_nH^2(X_{V',n},\mathbb{G}_m)\to\operatorname{Br}(X_{\bar{k}})$$ is surjective on $p$-primary torsion. Finally we have $\operatorname{Br}(X_{V'})\subset \operatorname{Br}(X_{V'}\otimes_VK)$ ([@brauer II, 1.10]), hence $\operatorname{Br}(X_{\bar{V}})\otimes\mathbb{Z}_p\subset\operatorname{Br}(X_{\bar{K}})\otimes\mathbb{Z}_p$ is a finite direct sum of subgroups of $\mathbb{Q}_p/\mathbb{Z}_p$, and this implies that the map $$V_p\operatorname{Br}(X_{\bar{V}})\to V_p\operatorname{Br}(X_{\bar{k}})$$ is surjective.
\[subquot\] Assume $k$ is finite. If $\Phi^1$ is formally smooth and $\Phi^2$ is of finite height, then $V_p\operatorname{Br}(X_{\bar{k}})$ is a subquotient of $V_p\operatorname{Br}(X_{\bar{K}})$.
Applied to our elliptic modular surface $E=E(N)\otimes_{\mathbb{Z}[\zeta_N]}W$ with $N\geq 3$, this implies
\[fhcor\] If $k$ is finite and the formal Brauer group of $E_k$ has finite height, then $T_p\operatorname{Br}(E_{\bar{k}})=0$.
As noted in \[elliptic\], $\Phi^1$ is formally smooth over $W$, so $\Phi^2$ is representable by a formal Lie group over $W$. Moreover, it is easy to see that $\Phi^2$ has finite height since $\Phi^2\otimes_Wk$ does by assumption. So by the last corollary $V_p\operatorname{Br}(E_{\bar{k}})$ is a subquotient of $V_p\operatorname{Br}(E_{\bar{K}})$. It follows from Corollary \[HT\] that $V_p\operatorname{Br}(E_{\bar{k}})$ is a Hodge-Tate representation whose weights are contained in $\{\pm 1\}$. Since it is an unramified representation, this is only possible if it is zero.
|
Introduction
============
Let me start by formulating the problem I am going to discuss: How can one calculate thermal expectation values like $$\begin{aligned}
C(t_1 - t_2) = \langle {\cal O}(t_1) {\cal O}(t_2)\rangle
{\label{c}}\end{aligned}$$ in a non-Abelian gauge theory, when the leading order contribution is due to spatial momenta of order $g^2 T$? The operator ${\cal
O}(t)$ is a gauge invariant function of the gauge fields $A_\mu(t,{{{\bf x}}})$ at time $t$. When I said the leading order contribution is due to momenta of order $g^2 T$, I referred to the Fourier components of the gauge fields entering ${\cal O}(t)$. Finally, $$\begin{aligned}
\langle \cdots \rangle =
\frac{{{\rm tr}}\{(\cdots)e^{- H/T}\}}{{{\rm tr}}\{e^{-H/T}\}}\end{aligned}$$ denotes the thermal average, where $H$ is the Hamiltonian. As one may anticipate, it is not possible to compute such a correlation function in perturbation theory.
This problem arises in the context of electroweak baryon number violation at temperatures, well above 100GeV, when the electroweak symmetry is unbroken. Then the SU(2) gauge theory is very similar to hot QCD. Fortunately, it is a bit simpler because the gauge coupling is small.
In my talk I will try to explain how such correlation functions can be computed at leading order in the gauge coupling [@letter]. We will see that this requires the use of an effective theory which results from integrating out field modes with hard ($p\sim T$) and semi-hard ($p\sim gT$) spatial momenta [^1]. Due to the smallness of the gauge coupling this can be done in perturbation theory. This effective theory turns out to have a relatively simple structure and it can be easily used for numerical calculations on a lattice.
Most of the discussion will be restricted to a pure gauge theory, i.e., without matter fields. I will comment on the role of matter fields when necessary.
Hot electroweak baryon number violation
=======================================
[\[sec:baryon\]]{}
One of the most intriguing aspects of the electroweak theory is its non-trivial vacuum structure which is related to baryon number violating processes. These processes play an important role in understanding the observed baryon-asymmetry of the universe [@rubakov].
Baryon number is not conserved in the electroweak theory due to the chiral anomaly. The baryon number current satisfies [^2] $$\begin{aligned}
\partial_\mu j^\mu_B = n_f \frac{ g^2}{32\pi^2} {{\rm tr}}{\left(}F\tilde{F}{\right)}{\label{anomaly}}\end{aligned}$$ where $F$ is the SU(2) field strength tensor. The rhs of [Eq. [(\[anomaly\])]{}]{} is a total derivative, $$\begin{aligned}
{{\rm tr}}{\left(}F\tilde{F}{\right)}= \partial_\mu K^\mu.\end{aligned}$$ Thus, integrating [Eq. [(\[anomaly\])]{}]{} over 3-space and over time from $t_i$ to $t_f$ we can write the change of baryon number $B$ as $$\begin{aligned}
B(t_f) - B(t_i) = n_f ( N_{\rm CS}(t_f) - N_{\rm CS}(t_i))
{\label{deltaB}}\end{aligned}$$ where $$\begin{aligned}
N_{\rm CS} = \frac {g^2}{32\pi^2} \epsilon_{ijk} \int d^3x
\left( F^a_{ij} A^a_k - \frac{g}{3} \epsilon^{abc}
A^a_i A^b_j A^c_k \right)
{\label{chern}}\end{aligned}$$ is the so called Chern-Simons winding number of the gauge fields. Unlike $N_{\rm CS}$ itself, the difference $\Delta N_{\rm CS}=N_{\rm
CS}(t_f) - N_{\rm CS}(t_i)$ is gauge invariant since it can be written as a space-time integral of $F\tilde{F}$.
The Chern-Simons number is intimately related to the non-trivial topology of the gauge-Higgs field configuration space. There are paths in this space leading from the vacuum to the vacuum which cannot be continuously deformed to a single point. The minimal field energy along such a path cannot be made arbitrarily small by continuous deformation. There is a minimal energy barrier which has to be crossed. The corresponding field configuration is the so called sphaleron. Following such a non-contractible path, the Chern-Simons number changes by an integer.
The simplest mechanical analogue of this situation I can think of is a rigid pendulum in a gravitational field which can swing all the way around in a circle. The Chern-Simons number corresponds to $\phi/(2\pi)$ where $\phi$ is the rotation angle. The vacuum corresponds to the pendulum at rest and a non-contractible loop corresponds to net rotation of the pendulum by an integer times $2\pi$.
(50,50)(40,0) (85,165)[energy]{} (250,40)[$N_{\rm CS}$]{}
[\[fig:chern\]]{}
The gauge-Higgs field configuration space is of course much more complicated, it is infinite dimensional rather than one dimensional as in the case of the pendulum. [Fig. \[fig:chern\]]{} displays a one dimensional slice of this space. It corresponds to a path on which the field configuration for a given value of $N_{\rm CS} $ has a minimal energy.
If there were only gauge and Higgs fields, nothing spectacular would happen in a vacuum to vacuum transition which changes $N_{\rm CS} $. However, in the presence of fermions, the change of $N_{\rm CS} $ is accompanied by a change of baryon number due to [Eq. [(\[deltaB\])]{}]{}.
At zero temperature, $N_{\rm CS} $-changing transitions are possible only by tunnelling. The tunnelling probability is so small that it has no measurable physical effect.
This situation changes at high temperature. Then it becomes possible to make such transitions by thermal activation across the energy barrier. The suppression factor is then given by a Boltzmann factor rather than the tunnelling amplitude. For $T$ of the order of the electroweak phase transition or crossover temperature $T_{\rm c}\sim 100$GeV, the transition probability becomes unsuppressed.
When the temperature is sufficiently below $T_{\rm c}$, most transitions are passing the energy barrier close to the sphaleron and the transition rate $\Gamma$ can be calculated in a saddle point approximation.
For $T\sim T_{\rm c}$ this approximation breaks down and for $T>
T_{\rm c}$ there is no sphaleron solution at all. Therefore one has to employ non-perturbative methods to calculate the rate for this case. It has become popular to call the rate for $T> T_{\rm c}$ the [*hot sphaleron rate*]{}. The idea how to compute it is the following: The Chern-Simons number is expected to perform a random walk. Then, for large $t$, thermal average of ${\left[}N_{\rm CS} (t) -
N_{\rm CS} (0) {\right]}^2 $ grows linearly with time, $$\begin{aligned}
\left\langle {\left[}N_{\rm CS} (t) - N_{\rm CS} (0) {\right]}^2 \right\rangle
\to V t \Gamma
{\label{csdiffusion}}\end{aligned}$$ where $V$ is the space volume. The coefficient $\Gamma$ is the probability for a $N_{\rm CS}$-changing transition per unit time and unit volume. The rate for baryon number violation $\Gamma_{\Delta B}$ is proportional to $\Gamma$. Thus $\Gamma_{\Delta B}$ can be obtained by computing the real time correlation function [(\[csdiffusion\])]{}. At very high temperature $T\gg T_{\rm c}$ the thermal mass of the Higgs field becomes large so that it decouples. Then it is sufficient to evaluate [Eq. [(\[csdiffusion\])]{}]{} in the pure gauge theory.
For the following discussion it is important to identify both the relevant length scale $R$ of the problem and the relevant size of the gauge field fluctuations $\Delta{{{\bf A}}}$ [@khlebnikov]. First of all, we have to consider fluctuations which are large enough to make a change $\Delta N_{\rm CS}$ of order unity. To estimate $\Delta N_{\rm
CS}$, recall that $F\tilde{F} = {{{\bf E}}}{\!\cdot\!}{{{\bf B}}}$. For the electric fields we have ${{{\bf E}}}= - \dot{{{{\bf A}}}}$ in $A_0 = 0 $ gauge. Thus $\int dt {{{\bf E}}}\sim \Delta{{{\bf A}}}$. Note that the relevant time scale drops out in this estimate. Furthermore, we estimate ${{{\bf B}}}\sim \Delta{{{\bf A}}}/R$ and $\int
d^3 x\sim R^3$. Thus we need $$\begin{aligned}
\Delta N_{\rm CS} \sim
g^2 \int d^4 x \,
{{\bf E}}\cdot{{\bf B}}
\sim g^2 R^2 (\Delta{{\bf A}})^2
\stackrel{!}{\sim} 1
{\label{deltachern}}.\end{aligned}$$ In addition, we have to require that the energy of the relevant fluctuations does not exceed the temperature. Otherwise the transition rate would be Boltzmann suppressed. Proceeding as above we require $$\begin{aligned}
{\rm energy} \sim \int d^3 x \,
{{\bf B}}^2
\sim R \, (\Delta{{\bf A}})^2
\stackrel{!}{{\mathop{{\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}}}} T
{\label{energy}}.\end{aligned}$$ Combining Eqs. [(\[deltachern\])]{} and [(\[energy\])]{} we find $$\begin{aligned}
R {\mathop{{\raise0.3ex\hbox{$>$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}}}(g^2 T)^{-1} \qquad
\Delta{{\bf A}} \sim (g R)^{-1}.\end{aligned}$$ On the other hand, we know that the correlation length of magnetic fields in a hot non-Abelian plasma is of order $(g ^2 T)^{-1}$. Thus we can have only $R{\mathop{{\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}}}(g^2 T)^{-1}$ which finally gives $$\begin{aligned}
R \sim (g^2 T)^{-1} \qquad
\Delta{{\bf A}} \sim g T
{\label{scale}}.\end{aligned}$$
As I stated above, the relevant time scale $t$, which we need to know in order to estimate the hot sphaleron rate, has dropped out in these considerations. It has to be determined from the dynamics and it cannot be obtained from purely thermodynamic considerations. Once we know $t$ we can estimate the hot sphaleron rate as $$\begin{aligned}
\Gamma \sim t^{-1} R^{-3}
{\label{rate}}.\end{aligned}$$ While [Eq. [(\[scale\])]{}]{} has been well established for a quite a while, the characteristic time scale has been understood only recently [@letter]. I hope that, by the end of my talk, you will be convinced that the correct answer is given by [Eq. [(\[timescale\])]{}]{}.
Non-perturbative physics at high temperature
============================================
After the discussion of electroweak baryon number violation we now return to our original problem, which is to calculate correlation functions like [(\[c\])]{}.
In the previous section we have seen that the relevant length scale for electroweak baryon number violation is $(g^2 T)^{-1}$ corresponding to momenta $p\sim g^2 T$. It is well known that finite temperature perturbation theory in non-Abelian gauge theories breaks down for momenta as small as $g^2 T$ [@linde80; @gross]. One can see this by considering the typical size of the transverse gauge field fluctuations which can be estimated from the equal time correlation function. For the transverse gauge fields we have $$\begin{aligned}
A_{{\rm t}}({{{\bf x}}}) \sim g T
{\label{sizeax}}.\end{aligned}$$ Thus the two terms in the covariant derivative $\partial_i - i g A_i$ are of the same size which makes perturbation theory impossible. Note that the thermal fluctuations are of the same size as the change $\Delta {{{\bf A}}}$ which we have estimated in [Eq. [(\[scale\])]{}]{}.
When I said that perturbation theory does not work for soft momenta, one may ask: What about the plasmon at rest which has ${{{\bf p}}}= 0$ and the celebrated calculation of it’s damping rate [@damping]? The point is that the plasmon oscillation has an amplitude $\sim g^2 T$, i.e., it is a small oscillation. What we are interested in is the time evolution which makes the field change by an amount equal to it’s typical size $\sim g T$.
For the case of equal time correlation functions the non-perturbative physics associated with the scale $g^2 T$ is relatively well understood. It is determined by an effective 3-dimensional pure gauge theory. This effective theory is the result of integrating out the field modes with non-zero Matsubara frequency (“dimensional reduction”) and the 0-component of the gauge fields [@farakos; @nieto]. For sufficiently small coupling this can be done in perturbation theory. The 3-dimensional theory can be used for 3-dimensional Euclidean lattice simulations which allow for a much higher precision than 4-dimensional ones [@lattice]. It should be noted that the use of dimensional reduction is more a matter of practical convenience rather than a matter of principle. The full 4-dimensional can also be evaluated directly in a 4-dimensional Euclidean lattice simulation.
The situation is different for unequal time correlation functions. Dimensional reduction is not possible because we have to consider time dependent quantities. 4-dimensional Euclidean lattice simulations are of no use either: Formally, real time correlation functions can be obtained by an analytic continuation of their imaginary time counterparts. However, it is not possible to do an analytic continuation of a function which can be evaluated only numerically.
One simplification of the problem is that the soft field modes behave classically. This is due to the fact that the momentum scale of interest is small compared to the temperature. Then the number of field quanta in one mode with wave vector ${{{\bf p}}}$, given by the Bose-Einstein distribution function $$\begin{aligned}
n(p)=
\frac{1}{e^{p/T} - 1} \simeq \frac{T}{p}
,\end{aligned}$$ is large. In this case we are close to the classical field limit. Thus the dynamics of the soft field modes should be governed by classical equations of motion. [^3]
The first estimate of the hot sphaleron rate, due to Khlebnikov and Shaposhnikov [@khlebnikov], was based on the assumption that only the soft fields play a role while the high momentum modes decouple at leading order. Then the only scale in the problem would be $g^2 T$ and, on dimensional grounds, the hot sphaleron rate would have to be of the form $$\begin{aligned}
\Gamma_{\rm KS} =\kappa (g^2 T)^4
{\label{ks}}\end{aligned}$$ with a non-perturbative numerical coefficient $\kappa$. Subsequently attempts were made to compute $\kappa$ by solving the classical Hamiltonian equations of motion for the gauge fields and measure $\Gamma$.
However, it turns out that in the case of unequal time correlation functions is much more complex than in the equal time case. The main difficulty is to understand the role of the different momentum scales which are present in the problem. We will see that both momenta of order $T$ and momenta of order $gT$ are relevant to the leading order dynamics of the soft fields.
Therefore one has to integrate out the high momentum modes to obtain an effective classical theory. Schematically, the equations of motion will be of the form $$\begin{aligned}
\frac{\delta S_{\rm eff}[A]}{\delta A_\nu(x)} = 0
{\nonumber}\end{aligned}$$ where the effective action $S_{\rm eff}[A]$ consists of the tree level action and the terms which are generated by integrating out the field modes with $p\sim T$ and $p\sim gT$. Deriving this equation of motion will be the main subject of my talk.
The reason why the hard modes have to be integrated out to obtain an effective classical theory for the soft modes is pretty obvious: For $p\sim T$ the Bose-Einstein distribution is of order 1 so that these modes certainly do not behave classically. For semi-hard momenta, however, we have $n(p)\ll 1$. The main reason why they have to be integrated out as well is that it does not seem to be possible to construct a lattice formulation of the effective theory for momenta of order $gT$ and $g^2 T$ which allows to take the continuum limit (cf. Ref. [@bms]). Another reason is that after integrating out the scale $gT$ we will obtain an effective theory for the soft field modes which contains only one length scale and one time scale. Then it will be trivial to estimate the parametric form of the hot sphaleron rate.
Integrating out the hard modes
==============================
The hard modes constitute the bulk of degrees of freedom in the hot plasma. Their physics is that of almost free massless particles moving on straight lines. Nevertheless, they have a significant influence on the soft dynamics because they are so numerous.
Integrating out the hard modes means that we have to calculate loop diagrams with external momenta $\ll T$ and internal momenta of order $T$. This generates effective propagators and vertices for the field modes with $p\ll T$. To leading order, we can restrict ourselves to one loop diagrams and we can make a high energy or eikonal approximation of the integrand. The result is nothing but the so called hard thermal loops [@bellac]. The generating functional of the hard thermal loops has the remarkable property to be gauge invariant.
The main difference between QED and non-Abelian theories is that for the former the only hard thermal loop is the 2-point function, while for the latter there are also hard thermal loop $n$-point functions for all $n$. As we will see later, this has a significant effect on the soft dynamics, it is in fact qualitatively different in Abelian and non-Abelian theories.
For the moment, however, we restrict the discussion to the hard thermal loop 2-point function. This will make clear that the hard modes have a significant influence on the soft dynamics which was pointed out by Arnold, Son, and Yaffe [@asy].
We have to consider the transverse (or magnetic) components of the gauge fields. The transverse polarisation function is given by $$\begin{aligned}
\delta \Pi_{{\rm t}}(P) = \frac12 {m^2_{\rm D}}{\left[}\frac{p_0^2}{p^2}
+ p_0{\left(}1 - \frac{p_0^2}{p^2}{\right)}{\int\!\frac{d\Omega_{{\rm v}_{}}}{4\pi}}\frac{\! 1}{v{\!\cdot\!}P}{\right]}{\label{deltapi}}\end{aligned}$$ where ${m^2_{\rm D}}$ is the leading order Debye mass squared. In a pure SU($N$) gauge theory, it is given by $$\begin{aligned}
{m^2_{\rm D}}= \frac13 N g^2 T^2. \end{aligned}$$ In the hot electroweak theory it receives additional contributions due to the Higgs and fermion fields. Furthermore, $v^\mu \equiv
(1,{{\bf v}})$, and the integral $\int d\Omega_{{{{\bf v}}}}$ is over the directions of the unit vector ${{{\bf v}}}$, $|{{{\bf v}}}| =1$ .
For the naive “natural” scale of the problem, i.e., $p_0$, $p\sim g^2 T$ we have $$\begin{aligned}
\delta \Pi_{{\rm t}}(P) \sim g^2 T^2\qquad
(p_0\sim g^2 T, \,\,p \sim g^2 T),\end{aligned}$$ i.e., $\delta \Pi_{{\rm t}}$ is much bigger than the tree level kinetic term. Therefore $\delta \Pi_{{\rm t}}$ must be resummed. The resulting transverse propagator $$\begin{aligned}
{{}^\ast\!}\Delta_{ {\rm t}}(P) =
\frac{1}{-P^2 + \delta \Pi_{ {\rm t}}(P)}
{\label{propagator}} \end{aligned}$$ is smaller than the tree level propagator by a factor $g^2$ when $p_0\sim g^2 T, \,\,p \sim g^2 T$. That means that for the naive “natural” frequency there are only small perturbative fluctuations which are of the same amplitude as the plasmon oscillation. It also means that the estimate [(\[ks\])]{} cannot be correct.
In order to obtain large non-perturbative fluctuations we obviously need to make $\delta \Pi_{{\rm t}}$ smaller. We know that $\delta \Pi_{{\rm t}}$ vanishes [^4] for $p_0 \to 0$. Thus we have to consider small values of $p_0$. In the limit $p_0\ll p$ we have $$\begin{aligned}
\delta \Pi_{{\rm t}}(P) \simeq - \frac{i}{4} {m^2_{\rm D}}\frac{p_0}{p}
{\label{deltapilimit}}\end{aligned}$$ which is purely imaginary. The imaginary part of $\delta \Pi_{{\rm t}}(P)$ reflects the fact that the dynamics of the soft modes is Landau-damped by the hard modes. In fact, it is over-damping because the damping term is much larger than the tree level kinetic term $p_0^2$. In the same limit the transverse propagator [(\[propagator\])]{} becomes $$\begin{aligned}
{{}^\ast\!}\Delta_{{\rm t}}(P) \simeq \frac{1}{p^2}
\frac{i\gamma_p}{p_0 + i \gamma_p}
{\label{propagator.5.4}}\end{aligned}$$ with $$\begin{aligned}
\gamma_p = \frac{4p^3}{\pi{m^2_{\rm D}}}.\end{aligned}$$ Now we see how small $p_0$ must be in order to obtain the desired large fluctuations, namely $$\begin{aligned}
p_0 \sim \gamma_p \sim g^4 T,\end{aligned}$$ corresponding to the time scale $$\begin{aligned}
t\sim \gamma_p^{-1} \sim {\left(}g^4 T{\right)}^{-1}.\end{aligned}$$ Based on these considerations Arnold, Son and Yaffe estimated the hot sphaleron rate as $$\begin{aligned}
\Gamma_{\rm ASY} = \kappa g^{10} T^4.\end{aligned}$$ Subsequently, attempts were made by Huet, Son [@huet; @son] and Arnold [@arnold] to construct a numerical algorithm to compute the coefficient $\kappa$ on a lattice.
In this discussion it is assumed that interactions do not change these order of magnitude estimates. As we will see in the next section, this is not the case: The field modes with momenta of order $gT$ lead to interactions of the hard modes which cannot be neglected relative to the hard thermal loops.
Beyond hard thermal loops
=========================
[\[sec:beyond\]]{}
I will now argue that the hard thermal loop approximation for the high momentum modes ($p\gg g^2 T$) is not sufficient to obtain the correct effective theory for the soft modes.
Consider a hard thermal loop and imagine adding a self energy insertion on an internal line. This gives the diagram in Fig.\[fig:selfenergy\]a in which the hard loop momentum $Q$ is on shell, $Q^2 = 0$. Since the external momentum $P$ is soft, the momentum $Q+P$ is almost on shell. The order of magnitude of the self energy insertion is well known from the calculation of the lifetime of hard particles [@bellac]. From loop momenta $K$ of order $gT$ one gets a contribution $\sim g^2 T^2$. Compared to the hard thermal loop, diagram \[fig:selfenergy\]a also contains an additional propagator which can be approximated as $$\begin{aligned}
\frac{1}{(Q+P)^2} \simeq \frac{1}{2 q}\, \frac{\! 1}{v{\!\cdot\!}P}
\sim \frac{1}{T}\, \frac{\! 1}{v{\!\cdot\!}P}
{\label{lee}}.\end{aligned}$$ Thus we can estimate the diagram in Fig. \[fig:selfenergy\]a as $$\begin{aligned}
\Pi^{\rm (a)}(P) \sim
\delta \Pi_{{\rm t}}(P) \times
\frac{g^2 T}{v{\!\cdot\!}P}
{\label{pia}}\end{aligned}$$ and, by power counting, this diagram is not suppressed relative to the hard thermal loop [(\[deltapi\])]{} when $P\sim g^2 T$. Note that in this estimate it did not play any role that we are considering a non-Abelian theory. The same type of diagram is also present in QED and the estimate would be exactly the same. However, there is an essential difference between the Abelian and the non-Abelian case which will become clear in a moment.
[\[fig:selfenergy\]]{}
Similarly, one can see that the ladder-type diagram \[fig:selfenergy\]b is as large as the hard thermal loop as well. Again, this estimates are the same in QED and QCD. However, in QED the large contributions from the two diagrams turn out to be the same with opposite signs and cancel.
I will now argue that this cancellation does not occur in a non-Abelian theory. Let me first give a formal explanation in terms of diagrams. At the end of Sect. \[sec:logarithmic\], I will come back to this point and I will give a more intuitive argument.
Here is the diagrammatic argument: We have seen that the two diagrams both contain a hard and a semi-hard loop momentum. Imagine now that we calculate them in two steps: First do the integral over the hard momentum with $K$ kept fixed. This can be interpreted as follows: The first step generates an effective 4-point vertex for momenta of order $gT$ and $g^2 T$. The result of such a calculation is well known: At leading order this is nothing but the hard thermal loop four point function. And we also know that there is no such vertex in QED, even though individual diagrams would give such a contribution. Summing over the permutations of external lines the large contributions cancel. In a non-Abelian theory each of these diagrams comes with a different colour factor and the sum over permutations gives a non-vanishing result. In the second step we integrate over momenta of order $gT$ which can be done using the well known expression for the hard thermal loop 4-point function.
Integrating out the semi-hard modes
===================================
[\[sec:semihard\]]{}
We have seen in the previous section that, in order to obtain an effective theory for the soft momentum fields, we cannot restrict ourselves to the hard thermal loops. Not only the hard modes but also the semi-hard ones affect the leading order dynamics of the soft fields. At first sight this appears like an additional complication. Somewhat surprisingly, things become much simpler instead.
We have also seen that it is convenient to proceed in two steps: In the first step we integrate out momenta of order $T$ which yields the hard thermal loop effective theory. In the second step this effective theory is used to integrate out the scale $gT$. In this way the calculation will be simplified because we have a partial cancellation, corresponding to the QED-type contributions, already “built in”.
One loop diagrams
-----------------
[\[sec:one.loop\]]{}
[\[fig:htlvertex\]]{}
We start by considering the one loop diagram in the hard thermal thermal loop effective theory depicted in Fig. \[fig:htlvertex\]a. It corresponds to the sum of the diagrams of the original theory in Fig. \[fig:selfenergy\]. We neglect the external soft momentum relative to the semi-hard one whenever it’s possible. In this way we obtain an expression in which the loop integration is logarithmically infrared divergent. This should not bother us since we do not want to compute the diagram exactly. All we want to do is to integrate out momenta of order $g T$. Therefore we need a scale $\mu$ which separates semi-hard momenta from soft ones, $$\begin{aligned}
g^2 T\ll \mu \ll gT.\end{aligned}$$ This scale appears as a lower limit in our integral so that it is no longer divergent. The integral is difficult evaluate. What it not so difficult, is to compute the terms which are singular for $\mu \to 0$. One finds [@eff] $$\begin{aligned}
\Pi_{\mu\nu}^{(4)} (P) &=& -\frac{i}{4\pi}
{m^2_{\rm D}}N g^2 T \log{\left(}\frac{gT}{\mu}{\right)}{\nonumber}\\ &&
\times p_0 {\int\!\frac{d\Omega_{{\rm v}_{}}}{4\pi}} \frac{v_\mu v_\nu
}{(v{\!\cdot\!}P)^2}\,\,
{\label{pi4}}.\end{aligned}$$ This result is gauge fixing independent in a general covariant gauge.
Obviously, [Eq. [(\[pia\])]{}]{} is not suppressed relative to the hard thermal loop selfenergy [(\[deltapi\])]{}: The prefactor $g^2 T$ is compensated by one additional factor $v{\!\cdot\!}P \sim g^2 T$ in the denominator. [Eq. [(\[pia\])]{}]{} is even larger than [(\[deltapi\])]{} by a factor of $\log(gT/\mu)$.
Note that [(\[pi4\])]{} is not transverse, which would be necessary for the effective theory, which we are going to derive, to be gauge invariant. There is, however, another one loop diagram in the hard thermal loop effective theory which is of the same size as [Eq. [(\[pia\])]{}]{} and which is depicted in Fig. \[fig:htlvertex\]b. With the same approximations as above one obtains [@eff] $$\begin{aligned}
\Pi_{\mu\nu}^{(3)} (P) &=&
\frac{i}{\pi^2} {m_{\rm D}}^2 N g^2 T \log{\left(}\frac{gT}{\mu} {\right)}{\nonumber}\\ && {}
\hspace{-2cm}
\times p_0
{\int\!\frac{d\Omega_{{\rm v}_{1}}}{4\pi}}\frac{v_{1\mu}}{v_1{\!\cdot\!}P} {\int\!\frac{d\Omega_{{\rm v}_{2}}}{4\pi}} \frac{v_{2\nu}}{v_2{\!\cdot\!}P}
\frac{({{{\bf v}}}_1{\!\cdot\!}{{{\bf v}}}_2)^2}{\sqrt{1 - ({{{\bf v}}}_1{\!\cdot\!}{{{\bf v}}}_2)^2}}
{\label{pi3}}.\end{aligned}$$ One easily verifies that the sum of Eqs. [(\[pi4\])]{} and [(\[pi3\])]{} is transverse, $$\begin{aligned}
P^\mu{\left(}\Pi_{\mu\nu}^{(4)} (P) + \Pi_{\mu\nu}^{(3)} (P){\right)}= 0.\end{aligned}$$
Higher loops
------------
There are higher loop diagrams in the hard thermal loop effective theory which are not suppressed relative hard thermal loops either. Obviously these terms need to be resummed. As I already said, the use of the hard thermal loop effective theory has simplified the calculation of [(\[pi4\])]{} and [(\[pi3\])]{} quite a bit. Nevertheless, higher loop diagrams are still difficult to calculate because the number of terms in the hard thermal loop $n$-point functions grows rapidly with increasing $n$.
It is much more efficient to use another formulation of the hard thermal loop effective theory, which is in terms of kinetic equations. The degrees of freedom in this description are the gauge fields $A_\mu(x)$ and, in addition, the fields $W(x,v)$ which transform under the adjoint representation of the gauge group. The fields $W(x,v)$ depend both on the space-time coordinates and on the unit vector ${{{\bf v}}}$. They describe the deviation of the distribution of hard particles with 3-velocity ${{{\bf v}}}$ from thermal equilibrium.
The equations of motion for these fields are the non-Abelian generalisation of the Vlasov equations for a QED plasma [@blaizot93], $$\begin{aligned}
{\label{maxwell}}
[D_\mu, F^{\mu\nu}(x)]= {m^2_{\rm D}}\int\frac{d\Omega_{{\bf v}}}{4\pi}
v^\nu W(x,v),\end{aligned}$$ $$\begin{aligned}
[v \cdot D, W(x,v)] = {{\bf v}}\cdot{{\bf E}} (x)
{\label{vlasov}},\end{aligned}$$ where ${{\bf E}}$ is the non-Abelian electric field, and the 4-vector $v$ is defined as in [Eq. [(\[deltapi\])]{}]{}. The rhs of [Eq. [(\[maxwell\])]{}]{} is the current due to the hard particles. The conserved Hamiltonian corresponding to Eqs. [(\[maxwell\])]{} and [(\[vlasov\])]{} is [@nair; @blaizot94] $$\begin{aligned}
H&=&\int d^3 x {{\rm tr}}\Bigg\{
{{\bf E}}(x)\cdot{{\bf E}}(x) +{{\bf B}}(x)\cdot{{\bf B}}(x) {\nonumber}\\ &&
\hspace{1cm}
{}+
{m^2_{\rm D}}\int\frac{d\Omega_{{\bf v}}}{4\pi}
W(x,v) W(x,v)\Bigg\}
{\label{hamiltonian}}.\end{aligned}$$ One advantage of this formulations that this theory is local. What is even more important, however, is that the algebraic complexity is significantly reduced.
What does it mean to integrate out the semi-hard field modes in this formulation? First we have to split the fields into two pieces. The fields $A$, ${{\bf E}}$ and $W$ are decomposed into soft and a semi-hard modes [^5], $$\begin{aligned}
A &\to& A + a{\nonumber}\\
{{\bf E}} &\to& {{\bf E}} + {{\bf e}}{\nonumber}\\
W &\to& W + w
{\label{separation}}.\end{aligned}$$ The soft modes $A$, ${{\bf E}}$ and $W$ contain the spatial Fourier components with $p<\mu$ while the semi-hard modes $a$, ${{{\bf e}}}$ and $w$ consist of those with $k>\mu$.
The interpretation of $W$ and $w$ is the following: Both describe deviation of the distribution of the hard particles from thermal equilibrium. $W$ ($w$) is the slowly (rapidly) varying piece of this distribution varying on length scale greater (less) than $1/\mu$.
After this split we have two sets of equations of motion, one for the soft fields and one for the semi-hard fields. Due to the non-linear terms these sets are coupled.
The equations for the soft fields can be written as $$\begin{aligned}
[D_\mu, F^{\mu\nu}(x)] &=& {m^2_{\rm D}}\int\frac{d\Omega_{{\bf v}}}{4\pi}
v^\nu W(x,v)
{\label{maxwellsoft}},\end{aligned}$$ $$\begin{aligned}
{\left[}v \cdot D, W(x,v) {\right]}&=& {{\bf v}}\cdot{{\bf E}} (x)
+ \xi(x,v) ,
{\label{vlasovsoft}}\end{aligned}$$ where [^6] $$\begin{aligned}
\xi^a (x,v) = -g f^{abc} {\left(}v\cdot a^b (x)
w^c(x,v) {\right)}_{\rm soft} .
{\label{xi}}\end{aligned}$$ The subscript “soft” indicates that only spatial Fourier components with $p<\mu$ are included. The field strength tensor and the covariant derivatives in Eqs. [(\[maxwellsoft\])]{} and [(\[vlasovsoft\])]{} contain only the soft gauge fields. Note that terms which couple soft and semi-hard fields have been neglected in [Eq. [(\[maxwellsoft\])]{}]{}.
Since the semi-hard modes can be treated perturbatively, one can neglect their self-interaction. One can also neglect interactions in the first Vlasov equation, so that we have $$\begin{aligned}
\partial_\mu f^{\mu\nu}(x) &=&
{m^2_{\rm D}}\int\frac{d\Omega_{{\bf v}}}{4\pi}
v^\nu w(x,v)
{\label{maxwellhard}},\end{aligned}$$ $$\begin{aligned}
v{\!\cdot\!}\partial w(x,v) &=&
{{{\bf v}}}{\!\cdot\!}{{{\bf e}}}(x) + h (x,v),
{\label{vlasovhard}}\end{aligned}$$ where $f^{\mu\nu} = \partial^\mu a^\nu - \partial^\nu a^\mu$. The only interaction which affects the time evolution of the semi-hard fields at leading order appears in [Eq. [(\[vlasovhard\])]{}]{}, it is due to the term $$\begin{aligned}
h^a (x,v) &=&
-g f^{abc}\Big[ v{\!\cdot\!}A^b(x) w^c(x,v) {\nonumber}\\ &&
\hspace{2cm} {} +v{\!\cdot\!}a^b(x) W^c(x,v)\Big].\end{aligned}$$
Integrating out the scale $gT$ means that we solve the classical equations of motion [(\[maxwellhard\])]{} and [(\[vlasovhard\])]{} for the semi-hard fields on the soft background. Then we have to perform the thermal average over their initial conditions using the Hamiltonian [(\[hamiltonian\])]{}. In this way we eliminate the semi-hard fields and we end up with equations of motion for the soft fields only. This can be done in perturbation theory, where one expands in powers of $h(x,v)$. We obtain an expansion $$\begin{aligned}
a(x) &=& a_0(x) + a_1(x) + a_2(x) + \cdots{\nonumber}\\
w(x,v) &=& w_0(x,v) + w_1(x,v) + w_2(x,v) + \cdots
{\label{solution}}\end{aligned}$$ in which the $n$-th order term contains $n$ powers of the soft fields $A$, $W$. The terms $a_0$, $ w_0$ are independent of the soft background and they only depend on the initial conditions for the semi-hard fields. Note that the solution [(\[solution\])]{} is still formal since it contains the yet unknown non-perturbative soft fields.
Inserting [(\[solution\])]{} into [Eq. [(\[xi\])]{}]{} we obtain a series $$\begin{aligned}
\xi(x,v) = \xi_0(x,v) + \xi_1(x,v) + \xi_2(x,v) + \cdots
{\label{xiexpansion}}.\end{aligned}$$ Each term in [(\[xiexpansion\])]{} is bilinear in free fields $a_0$ and $w_0$. The term $\xi_n$ is of $n$-th order in the fields $A$ and $W$. This expression has to be plugged into the second Vlasov equation [(\[vlasovsoft\])]{} for the soft fields, which then becomes $$\begin{aligned}
[v \cdot D, W(x,v)] &=& {{\bf v}}\cdot{{\bf E}} (x){\nonumber}\\ &&
\hspace{-2cm}
{}+ \xi_0(x,v) + \xi_1(x,v) + \xi_2(x,v) + \cdots
{\label{vlasovsoft.2}}.\end{aligned}$$
As it stands, [Eq. [(\[vlasovsoft.2\])]{}]{} is not very useful yet. On the rhs we have an infinite series of terms, each depending on the yet unknown soft fields and on the initial conditions for the semi-hard fields. In general, the average over these initial conditions can be performed only after the equations for the soft fields have been solved.
Fortunately, we can simplify [Eq. [(\[vlasovsoft.2\])]{}]{} significantly since we are only interested in the leading order dynamics of the soft field modes. Recall that each term $\xi_n$ is bilinear in the fields $a_0$ and $w_0$. Furthermore, $a_0$ and $w_0$ are linear in the initial values for their time evolution. After solving the equations of motion for the soft fields [(\[maxwellsoft\])]{} and [(\[vlasovsoft.2\])]{}, one has to average over initial conditions. Due to the scale separation the average of the bilinear terms of semi-hard fields can be approximated by disconnected parts. But this just means that we can perform the average over initial conditions for the semi-hard fields already in the equation of motion [(\[vlasovsoft.2\])]{}! The only exception is the term $\xi_0$ because the thermal average of this term vanishes due to colour conservation: Thermal averages like $\langle a_0^a a_0^b\rangle$ are proportional to $\delta^{a b}$. Inserted into [Eq. [(\[xi\])]{}]{} this gives zero due to the antisymmetry of the structure constants $f^{abc}$. The highest correlation function of $\xi_0$ we have to take into account is the two point function $\langle \xi_0(x_1,v_1)
\xi_0(x_2,v_2)\rangle$. Put differently, the term $\xi_0$ acts like a Gaussian random force. It is random because it is independent of the soft fields.
Since the expectation value of $\xi_0$ vanishes, it is not sufficient to keep only this term on the rhs of [Eq. [(\[vlasovsoft.2\])]{}]{}. We also have to take into account $\xi_1$ which we can approximate by $$\begin{aligned}
\xi_1(x,v) \simeq {\langle\!\langle}\xi_1(x,v) {\rangle\!\rangle}{\label{simplify.xi1}}\end{aligned}$$ where ${\langle\!\langle}\cdots {\rangle\!\rangle}$ denotes the average over initial conditions for the semi-hard fields. The higher order terms $\xi_n$ in [Eq. [(\[vlasovsoft.2\])]{}]{} can be neglected. Then we have $$\begin{aligned}
[v \cdot D, W(x,v)] &\simeq& {{\bf v}}\cdot{{\bf E}} (x){\nonumber}\\ &&
\hspace{-2cm}
{}+ \xi_0(x,v) + {\langle\!\langle}\xi_1(x,v){\rangle\!\rangle}{\label{vlasovsoft.3}}\end{aligned}$$ in which the correlation functions of $\xi_0$ can be treated as Gaussian.
The logarithmic approximation
=============================
[\[sec:logarithmic\]]{}
The next step towards the effective equations of motion for the soft fields is to evaluate the terms in the second line of [Eq. [(\[vlasovsoft.3\])]{}]{}. As in the diagram calculation in Sec.\[sec:one.loop\] one encounters terms which are logarithmically sensitive to the separation scale $\mu$. Only these logarithmic terms will be kept. The reason why this is sufficient will become clear in a moment.
We have seen that all we need to know about the noise term $\xi_0$ is its 2-point function. The leading logarithmic result reads $$\begin{aligned}
\Big\langle
\xi_0^{a}(x_1,v_1)
\xi_0^{b}(x_{2},v_{2})
\Big\rangle
&=&
-2 N \frac{g^2 T^2}{{m^2_{\rm D}}}
\log{\left(}\frac{g T}{\mu}{\right)}I(v_1,v_2){\nonumber}\\ && \times
\delta^{ab} \delta^{(4)}(x_1 - x_2),
{\label{xi0correlator4}}\end{aligned}$$ with $$\begin{aligned}
{\label{k}}
I(v,v_1) \equiv -{\delta^{(S^2)}}({{{\bf v}}}- {{{\bf v}}}_1)
+ \frac{1}{\pi^2}
\frac{({{{\bf v}}}\cdot{{{\bf v}}}_1)^2}{\sqrt{1 - ({{{\bf v}}}\cdot{{{\bf v}}}_1)^2}} ,\end{aligned}$$ where ${\delta^{(S^2)}}$ is the delta function on the two dimensional unit sphere: $$\begin{aligned}
\int d\Omega_{{{{\bf v}}}_1} f({{{\bf v}}}_1) {\delta^{(S^2)}}({{{\bf v}}}- {{{\bf v}}}_1) = f({{{\bf v}}}) .\end{aligned}$$ With the same approximations the result for the last term in [Eq. [(\[vlasovsoft.3\])]{}]{} is $$\begin{aligned}
{\langle\!\langle}\xi_1(x,v){\rangle\!\rangle}&=&
N g^2 T \log{\left(}\frac{g T}{\mu}{\right)}{\nonumber}\\&&\times
{\int\!\frac{d\Omega_{{\rm v}_{1}}}{4\pi}} I(v,v_1)
W(x,v_1). \end{aligned}$$ Inserting this into [Eq. [(\[vlasovsoft.3\])]{}]{} we obtain $$\begin{aligned}
[v \cdot D, W(x,v)] &=& {{\bf v}}\cdot{{\bf E}} (x)+ \xi_0(x,v)
{\nonumber}\\ &&
\hspace{-2cm} {} + N g^2 T \log{\left(}\frac{g T}{\mu}{\right)}{\int\!\frac{d\Omega_{{\rm v}_{1}}}{4\pi}} I(v,v_1)
W(x,v_1) .
{\label{boltzmann}}\end{aligned}$$
I will now argue that, for the leading order dynamics of the soft fields, the lhs of [Eq. [(\[boltzmann\])]{}]{} can be neglected. The only spatial momentum scales which are left in the problem are $\mu$ and $g^2
T$. The field modes we are ultimately interested in, are the ones which have only momenta of order $g^2T$. The cutoff dependence on the rhs must drop out after solving the equations of motion for the fields with spatial momenta smaller than $\mu$. Thus, after the $\mu$-dependence has cancelled, the logarithm must turn into $\log(gT/(g^2 T)) = \log(1/g)$.
We will now simplify [Eq. [(\[boltzmann\])]{}]{} by neglecting terms which are suppressed by inverse powers of $\log(1/g)$. We introduce moments of $W(x,v)$ and $\xi_0(x,v)$, $$\begin{aligned}
W^{i_1\cdots i_n} (x) &\equiv& {\int\!\frac{d\Omega_{{\rm v}_{}}}{4\pi}} v^{i_1} \cdots v^{i_n} W(x,v)
,{\nonumber}\\
\xi_0^{i_1\cdots i_n} (x) &\equiv& {\int\!\frac{d\Omega_{{\rm v}_{}}}{4\pi}} v^{i_1} \cdots v^{i_n}
\xi_0(x,v) .\end{aligned}$$ Taking the first moment of Eq. [(\[boltzmann\])]{} gives $$\begin{aligned}
[D_0,W^i(x)] - [D^j,W^{ij}(x)] &=& \frac13 E^i(x) {\nonumber}\\
&& \hspace{-3.5cm} {}+ \xi_0^i(x)
- \frac{N g^2 T}{4\pi} \log(1/g) W^i(x)
{\label{wi.2}}.\end{aligned}$$ Assuming that (cf. Ref. [@eff]) $W^{ij}$ is of the same size as $W^i(x)$ and $D_0{\raise0.3ex\hbox{$<$\kern-0.75em\raise-1.1ex\hbox{$\sim$}}}g^2 T$, the lhs of Eq. [(\[wi.2\])]{} is logarithmically suppressed relative to the term $\propto W^i(x)$ on the rhs and can be neglected. Then we can trivially solve [(\[wi.2\])]{} for $W^i(x)$, $$\begin{aligned}
W^i(x) = \frac{4\pi}{N g^2 T \log(1/g)} {\left(}\frac13 E^i(x) + \xi_0^i(x) {\right)}.\end{aligned}$$ Inserting the current $j^i(x) = {m^2_{\rm D}}W^i(x)$ into the rhs of [Eq. [(\[maxwellsoft\])]{}]{} gives $$\begin{aligned}
[D_\mu,F^{\mu i}(x)] =
\gamma E^i(x) + \zeta^i(x)
{\label{langevin1}}\end{aligned}$$ where we have introduced $$\begin{aligned}
\zeta^i(x) \equiv \frac{4\pi{m^2_{\rm D}}}{Ng^2T\log(1/g)}
\xi_0^i(x)\end{aligned}$$ and $$\begin{aligned}
\gamma = \frac{4\pi{m^2_{\rm D}}}{3 Ng^2 T\log(1/g)}.\end{aligned}$$ The term $\zeta$ is a Gaussian white noise. Its correlator is easily obtained from [Eq. [(\[xi0correlator4\])]{}]{}, $$\begin{aligned}
\Big\langle
\zeta^{ia}(x_1)
\zeta^{jb}(x_{2})
\Big\rangle
=
2 T \gamma
\delta^{ij } \delta^{ab} \delta^{(4)}(x_1 - x_2)
{\label{langevin2}}.\end{aligned}$$
[Eq. [(\[langevin1\])]{}]{} is the main result of our calculation. It is remarkable that, at the order we are considering, the effect of the high momentum modes can simply be described by a local damping term, together with a Gaussian white noise which keeps the soft modes in thermal equilibrium. This surprising result is somewhat counter-intuitive since Landau damping is generally known as a non-local effect. Qualitatively, it can be understood as follows: Recall that the hard particles have a mean free path of order $(g^2
T\log(1/g))^{-1}$ [@bellac]. To understand the physical picture behind the logarithmic approximation, we have to imagine that $\log
(1/g)$ is a large number. Then the mean free path of the hard particles is small compared to the length scale $(g^2 T)^{-1}$ on which the soft non-perturbative dynamics occurs. That is, the soft field modes are Landau damped only over a small length scale.
There is an essential difference compared to Abelian gauge theories. There the mean free path is of order $(g^2 T\log(1/g))^{-1}$ as well. However, the mean free path is determined by the total cross section of the hard particles which is dominated by small angle scattering. Thus, in a typical scattering event, the momentum of a charged particle in a QED plasma is hardly affected. The Landau damping can continue as a coherent process after such a scattering occurs. In a non-Abelian theory even a scattering under arbitrarily small angle can change the colour charge of a hard particle. It is precisely this colour exchange which leads to a loss of coherence in the process of Landau damping.
The hot sphaleron rate
======================
With [Eq. [(\[langevin1\])]{}]{} we are now able to estimate the characteristic time scale $t$ of non-perturbative gauge field fluctuations and thus of the rate for hot electroweak baryon number violation. Neglecting the term $[D_0,F^{0 i}(x)]$ (see below) the lhs can be estimated as $$\begin{aligned}
[D_j,F^{j i}(x)] \sim R^{-2} \Delta {{{\bf A}}}\sim g^4 T^2 \Delta {{{\bf A}}}{\label{lhs}}.\end{aligned}$$ On the rhs we have in $A_0=0$ gauge $$\begin{aligned}
\gamma {{{\bf E}}}\sim \frac{T}{\log(1/g)} \frac{\Delta {{{\bf A}}}}{t}
{\label{rhs}}.\end{aligned}$$ Putting [(\[lhs\])]{} and [(\[rhs\])]{} together we find $$\begin{aligned}
t\sim {\left(}g^4 \log(1/g)T {\right)}^{-1}
{\label{timescale}}.\end{aligned}$$ Therefore the hot sphaleron rate, at leading order, has the form $$\begin{aligned}
\Gamma = \kappa g^{10}\log(1/g)T^{4}
{\label{rate.2}}\end{aligned}$$ where $\kappa$ is a non-perturbative coefficient which does not depend on the gauge coupling.
An effective theory for the soft modes
======================================
[\[sec:effective\]]{}
I will now discuss how the non-perturbative coefficient $\kappa$ in [Eq. [(\[rate.2\])]{}]{}, and more generally, correlation functions like [(\[c\])]{} can be calculated on a lattice for which it is convenient to work in the $A_0 = 0$ gauge.
The time scale for non-perturbative dynamics is much larger than the corresponding length scale. Therefore time derivatives on the lhs of [Eq. [(\[langevin1\])]{}]{} are negligible and the non-perturbative dynamics of the soft gauge fields at leading order is correctly described by the equation $$\begin{aligned}
[D_j,F^{j i}(x)] =
-\gamma \dot{A}^i(x) + \zeta^i(x)
{\label{langevintag}},\end{aligned}$$ which should be easy to implement in a lattice calculation.
After solving [Eq. [(\[langevintag\])]{}]{}, the result has to be plugged into the operator ${\cal O}(t)$ of interest. The corresponding correlation function is then given by the average over the random force $\zeta$ using [Eq. [(\[langevin2\])]{}]{}.
Summary and discussion
======================
[\[sec:summary\]]{}
We have obtained an effective theory for the non-perturbative dynamics of the soft field modes by integrating out the hard ($p\sim T$) and semi-hard modes ($p\sim gT$) in perturbation theory. This effective theory is described by the Langevin equation [(\[langevintag\])]{}.
Furthermore, we have determined the parametric form of the hot electroweak baryon number violation rate at leading order. It contains a non-perturbative numerical coefficient which can be evaluated using [Eq. [(\[langevintag\])]{}]{}.
One would expect corrections to [Eq. [(\[rate.2\])]{}]{} to be suppressed by a factor $(\log(1/g))^{-1}$, which, in the electroweak theory, is not small [^7]. It would therefore be interesting, both from the theoretical and from the practical point of view, to see how sub-leading terms can be computed.
This work was supported in part by the TMR network “Finite temperature phase transitions in particle physics”, EU contract no. ERBFMRXCT97-0122.
[**Note added:**]{} While this paper was being completed, I received a preprint by Arnold, Son and Yaffe [@asy98], in which they derive Eq. [(\[langevintag\])]{} using the concept of colour conductivity. They also show that the effective theory described by this equation is insensitive to the ultraviolet which means that its lattice implementation does not require any renormalization.
D. Bödeker, Phys. Lett. B426 (1998) 351.
For a review, see V.A. Rubakov, M. E.Shaposhnikov, Usp. Fis. Nauk 166 (1996) 493, hep-ph/9603208.
S. Yu. Khlebnikov, M. E. Shaposhnikov, Nucl. Phys. B 308 (1988) 885.
A. D. Linde, Phys. Lett. 96B (1980) 289.
D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. 53 (1981) 43.
E. Braaten and R. Pisarski, Phys. Rev. D 42 (1990) 2156.
K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 425 (1994) 67; Nucl. Phys. B 442 (1995) 317;\
K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 458 (1996) 90;\
A. Jakovac, K. Kajantie and A. Patkos, Phys. Rev. D 49 (1994) 6810.
E. Braaten, A. Nieto, Phys. Rev. D 51 (1995) 6990.
For a review, see K. Rummukainen, Nucl. Phys. Proc. Suppl. 53 (1997) 30.
D. Bödeker, Nucl. Phys. B 486 (1997) 500.
D. Bödeker, M. Laine and O. Philipsen, Nucl. Phys.B 513 (1998) 445.
W. Buchmüller, A. Jakovac, Nucl. Phys. B521 (1998) 219.
D. Bödeker, L. McLerran and A.V. Smilga, Phys. Rev. D 52 (1995) 4675.
For a review, see M. Le Bellac, [*Thermal field theory*]{}, Cambridge University Press (1997).
P. Arnold, D. Son and L.G. Yaffe, Phys. Rev. D 55 (1997) 6264.
P. Huet and D.T. Son, Phys. Lett. B 393 (1997) 94.
D.T. Son, UW/PT-97-19, hep-ph/9707351.
P. Arnold, Phys. Rev. D 55 (1997) 7781.
D. Bödeker, in preparation.
J. P. Blaizot, E. Iancu, Nucl.Phys. B 417 (1994) 608.
V.P.Nair, Phys. Rev. D 50 (1994) 4201.
J. P. Blaizot, E. Iancu, Nucl.Phys. B 421 (1994) 565.
P. Arnold, D. Son and L.G. Yaffe, UW/PT 98-10, MIT CTP-2779, hep-ph/9810216.
[^1]: For spatial vectors I use the notation $k=|{{{\bf k}}}|$. Four-vectors are denoted by $K^\mu =
(k^0,{{{\bf k}}})$ and I use the metric $K^2 = k_0^2 - k^2$.
[^2]: Baryon plus lepton number is violated in the electroweak theory while baryon minus lepton number is conserved.
[^3]: It should be noted that this argument is somewhat handwaving and it is not easy to justify quantitatively [@hbar; @anharmonic; @wb].
[^4]: The longitudinal hard thermal loop polarisation tensor stays large even for $p_0\to 0$. Thus only the transverse fields can produce large, non-perturbative fluctuations.
[^5]: For notational simplicity I do not introduce new symbols for the soft modes. From now on $A$, ${{\bf E}}$ and $W$ will always refer to the soft fields only.
[^6]: The structure constants $f^{abc}$ are defined such that $[T^a,T^b] = i f^{abc} T^c$.
[^7]: We have $g\simeq 0.66$ for $T\sim 100$GeV which gives $\log (1/g)\simeq 0.4$.
|
---
abstract: 'We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur–Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur–Ulam property.'
address: 'Departamento de An[á]{}lisis Matem[á]{}tico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.'
author:
- 'Antonio M. Peralta'
title: On the extension of surjective isometries whose domain is the unit sphere of a space of compact operators
---
Introduction
============
The problem of extending surjective isometries between the unit spheres of two Banach spaces has experienced a substantial turn with the introduction, by M. Mori and N. Ozawa, of the so-called strong Mankiewicz property. The celebrated Mazur–Ulam theorem has been a source of inspiration for many subsequent research. A key piece among the different generalizations that appeared later is due to P. Mankiewicz [@Mank1972]. Let us recall that a convex subset $K$ of a normed space $X$ is called a *convex body* if it has non-empty interior in $X$. P. Mankiewicz proved in [@Mank1972] that every surjective isometry between convex bodies in two arbitrary normed spaces can be uniquely extended to an affine function between the spaces. M. Mori and N. Ozawa introduced the strong Mankiewicz property in [@MoriOza2018]. According to the just quoted paper, a convex subset $K$ of a normed space $X$ satisfies the *strong Mankiewicz property* if every surjective isometry $\Delta$ from $K$ onto an arbitrary convex subset $L$ in a normed space $Y$ is affine. It is established by Mori and Ozawa that for a Banach space $X$ satisfying that the closed convex hull of the extreme points, $\partial_e (\mathcal{B}_X),$ of the closed unit ball, $\mathcal{B}_X$, of $X$ has non-empty interior in $X$, every convex body $K\subset X$ satisfies the strong Mankiewicz property (see [@MoriOza2018 Theorem 2]). By the Russo-Dye theorem every convex body of a unital C$^*$-algebra satisfies the strong Mankiewicz property, and the same property holds for convex bodies in real von Neumann algebras (see [@MoriOza2018 Corollary 3]) and JBW$^*$-triples (cf. [@BeCuFerPe2018 Corollary 2.2]).
Based on the strong Mankiewicz property, M. Mori and N. Ozawa proved that unital C$^*$-algebras and real von Neumann algebras are among the spaces satisfying the Mazur–Ulam property, that is, every surjective isometry from the unit sphere, $S(A),$ of a unital C$^*$-algebra, or of a real von Neumann algebra $A$, onto the unit sphere, $S(Y)$, of an arbitrary real Banach space $Y$, admits a unique extension to a surjective real linear isometry between the spaces (see [@MoriOza2018]). J. Becerra Guerrero, M. Cueto-Avellaneda, F.J. Fern[á]{}ndez-Polo and the author of this note showed that every JBW$^*$-triple $M$ which is not a Cartan factor of rank two always satisfies the Mazur–Ulam property (cf. [@BeCuFerPe2018]). In a recent collaboration with O.F.K. Kalenda we prove that every rank-2 Cartan factor satisfies the Mazur–Ulam property [@KalPe2019 Theorem 1.1], and consequently, every JBW$^*$-triple enjoys the Mazur–Ulam property [@KalPe2019 Corollary 1.2]. These results simply are the most recent advances of a long list of paper studying the extension of isometries between the unit spheres of two Banach spaces (see, for example, [@CabSan19; @CuePer18; @CuePer19; @FerJorPer2018; @FerPe17c; @FerPe17b; @FerPe17d; @FerPe18Adv; @JVMorPeRa2017; @Mori2017; @MoriOza2018; @Pe2019; @PeTan16; @Ting1987; @WH19] and the surveys [@YangZhao2014; @Pe2018]).
So the natural question is: what can we say in the case of Banach spaces or C$^*$-algebras whose closed unit ball contains no extreme points? This is the case of the space $K(H)$, of all compact operators on an infinite dimensional complex Hilbert space $H$, where $\partial_e(\mathcal{B}_{K(H)}) = \emptyset.$ The lacking of extreme points of the closed unit ball makes impossible a straight application of the arguments based on the strong Mankiewicz property.
In [@PeTan16], R. Tanaka and the author of this note proved that every surjective isometry between the unit spheres of two compact C$^*$-algebras (in particular between the unit spheres of two spaces of compact linear operators on a complex Hilbert space) extends (uniquely) to a surjective real linear isometry between the two C$^*$-algebras. In collaboration with F.J. Fern[á]{}ndez-Polo we showed that the same conclusion remains valid for a surjective isometry between the unit spheres of two complex Banach spaces in the strictly wider class of weakly compact JB$^*$-triples (cf. [@FerPe18Adv]). The reader can get access to the concrete definitions of these objects in the subsequent subsection \[subsec:background\]. In this paper we prove that weakly compact JB$^*$-triples satisfy a much stronger property, namely, every weakly compact JB$^*$-triple $E$ satisfies the Mazur–Ulam property, in equivalent words, every surjective isometry from the unit sphere of $E$ onto the unit sphere of an arbitrary Banach space $Y$ extends to a surjective real linear isometry from $E$ onto $Y$ (cf. Theorem \[t weakly compact JB\*-triples satisfy MUP\]).
Our strategy to solve the problem determines the structure of this note. In Section \[sec:new facial properties\] we gather some new results derived from our knowledge on the facial structure of elementary JB$^*$-triples. New technical properties of elementary JB$^*$-triples, established in Propositions \[p new geometric facial property\] and \[p first consequence from the new geometric facial property\], are applied to deduce that every surjective isometry $\Delta: S(K) \to S(Y),$ where $K$ is an elementary JB$^*$-triple and $Y$ is a real Banach space, is affine on every non-empty convex subset $\mathcal{C}\subset S(K)$ (cf. Corollary \[c affine on convex subsets of the sphere in reflexive\]).
Section \[sec: elementary JB\* triples satisfy MUP\] is aimed to prove that every elementary JB$^*$-triple satisfies the Mazur–Ulam property (see Theorem \[t elementary JBstar triples satisfy the MUP\]). In particular, for any complex Hilbert space $H$, the space $K(H)$ satisfies the Mazur–Ulam property (cf. Corollary \[c KH satisfies the MUP\]). This closes a natural question which remained open until now. Let us observe that in case that $H$ is infinite dimensional the closed unit ball of $K(H)$ lacks of extreme points. As shown in [@JVMorPeRa2017], $c_0$ satisfies the Mazur–Ulam property. Probably, the results in this note show the first example of a non-commutative non-unital C$^*$-algebra satisfying this property.
As we know from many other mathematical problems, certain questions are easier to answer from a more general point of view. This is the case of the Mazur–Ulam property for the space of compact operators; the arguments in the wider class of elementary and weakly compact JB$^*$-triples are probably more abstract but offer an accesible proof.
Our goal in the second part of the paper is to prove that every weakly compact JB$^*$-triple satisfies the Mazur–Ulam property (see Theorem \[t weakly compact JB\*-triples satisfy MUP\]). For this purpose, we shall first show that the closed unit ball of any weakly compact JB$^*$-triple enjoys the strong Mankiewicz property (cf. Corollary \[c weakly compact JB\* triples satisfy the SMP\]). A consequence of this second main result, which is worth to be stated by its own importance, shows that every compact C$^*$-algebra (that is, a C$^*$-algebra which coincides with a $c_0$-sum of spaces of compact operators) also has the Mazur–Ulam property (cf. Corollary \[c compact C\*-algebras satisfy MUP final\]).
Basic background and references {#subsec:background}
===============================
The Riemann mapping theorem is one of the best known results in the theory of holomorphic functions of one variable. As it was already observed by H. Poincar[é]{} in 1907, for higher dimensional Banach spaces the conclusion of the Riemann mapping theorem is no longer valid, there are lots of simply connected domains which are not biholomorphic to the open unit ball. *Bounded symmetric domains* in finite dimensions were introduced and completely classified by E. Cartan. L. Harris proved in 1974 that the open unit ball of every C$^*$-algebra is a bounded symmetric domain [@Harris74], a conclusion which remains true for JB$^*$-algebras (see [@BraKaUp78]). The most conclusive study was obtained by W. Kaup who proved that every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB$^*$-triple (cf. [@Ka83]).
A complex Banach space $E$ is called a *JB$^*$-triple* if it can be equipped with a continuous triple product $\J ... : E\times E\times E \to E,$ which is conjugate linear in the middle variable and symmetric and bilinear in the outer variables satisfying the following axioms:
1. (Jordan Identity) $L(a,b) L(x,y) - L(x,y) L(a,b)= L(L(a,b)x,y) - L(x, L(b,a) y),$ for all $a,b,x,y,$ in $E$, where $L(x,y)$ is the linear operator defined by $L(a,b) (z) =\{a,b,z\}$ ($\forall z\in E$);
2. The operator $L(a,a): E\to E$ is hermitian and has non-negative spectrum;
3. $\|\J aaa\| = \|a\|^3$, for every $a\in E$.
It can be found in the previously mentioned references that every C$^*$-algebra $A$ is a JB$^*$-triple with respect to the triple product $$\label{eq triple product on Cstaralg} (a,b,c)\mapsto \{a,b,c\} =1/2 (a b^* c + c b^* a), \ \ (a,b,c\in A).$$ This triple product also induces a structure of JB$^*$-triple for the space $B(H_1,H_2)$ of all bounded linear operators between two complex Hilbert spaces $H_1$ and $H_2$, and for every closed subspace of $B(H_1,H_2)$ which is closed for this triple product. In particular every complex Hilbert space and the space $K(H_1,H_2),$ of all compact linear operators from $H_1$ to $H_2,$ are JB$^*$-triples. The class of JB$^*$-triples is also widen with all JB$^*$-algebras when they are equipped with the triple product given by $$\label{eq triple product JB*-algebras} \J abc = (a \circ b^*) \circ c + (c\circ b^*) \circ a - (a\circ c) \circ b^*.$$
A subspace $B$ of a JB$^*$-triple $E$ is a JB$^*$-subtriple of $E$ if $\{B,B,B\}\subseteq B$. A JB$^*$-subtriple $I$ of $E$ is called an *inner ideal of $E$* if $\{I,E,I\}\subseteq I$. A subspace $I$ of a C$^*$-algebra $A$ is called an *inner ideal* if $I A I \subseteq I$. For example, if $p$ and $q$ are projections in a C$^*$-algebra $A$, the subspace $I= p A q$ is an inner ideal of $A$. Inner ideals of JB$^*$-triples are studied and characterized in [@EdRutt92].
A *JBW$^*$-triple* is a [JB$^*$-triple]{} which is also a dual Banach space. Every von Neumann algebra is a JBW$^*$-triple. The theory of JB$^*$-triple runs in parallel to the theory of C$^*$-algebras. For example, the second dual of a JB$^*$-triple $E$ is a JBW$^*$-triple under a triple product extending the product of $E$ [@Di86]. It is also known that every JBW$^*$-triple admits a unique isometric predual, and its triple product is separately weak$^*$ continuous [@BarTi86].
Let us recall that an element $e$ in a C$^*$-algebra $A$ is called a *partial isometry* if $ee^*$ (equivalently, $e^*e$) is a projection. It is known that $e$ is a partial isometry if and only if $ee^* e =e$. It is easy to see that, in terms of the triple product given in , an element $e\in A$ is a partial isometry if and only if $\{e,e,e\}=e$. In the wider framework of JB$^*$-triples, elements satisfying $\{e,e,e\}$ are called *tripotents*. For each tripotent $e\in E$ the eigenvalues of the operator $L(e,e)$ are precisely $0,1/2$ and $1$. For $j\in \{0,1,2\}$, by denoting by $E_j (e)$ the $\frac{j}{2}$-eigenspace of $L(e,e)$, the JB$^*$-triple $E$ decomposes as the direct sum $$E= E_{2} (e) \oplus E_{1} (e) \oplus E_0 (e).$$ This decomposition is called the *Peirce decomposition* of $E$ with respect to the tripotent $e$, and the projection of $E$ onto $E_j(e)$, which is denoted by $P_j(e)$, is called the Peirce $j$-projection (see [@Loos2]). It is further known that Peirce projections are contractive (cf. [@FriRu85]) and satisfy the following identities $P_{2}(e) = Q(e)^2,$ $P_{1}(e) =2(L(e,e)-Q(e)^2),$ and $P_{0}(e) =Id_E - 2 L(e,e) + Q(e)^2,$ where for each $a\in E$, $Q(a):E\to E$ is the conjugate linear map given by $Q(a) (x) =\{a,x,a\}$. Consequently, in a JBW$^*$-triple Peirce projections are weak$^*$ continuous and Peirce subspaces are weak$^*$ closed.
Triple products among Peirce subspaces satisfy certain rules, known as *Peirce rules* or *Peirce arithmetic*, asserting that, for $k,j,l\in \{0,1,2\}$ we have $$\begin{aligned} \{ E_k(e) , E_j (e), E_l (e)\} &\subseteq E_{k-j+l} (e), \hbox{ if $k-j+l \in\{0,1,2\}$, and } \\
\{ E_k(e) , E_j (e), E_l (e)\} &= \{0\}\hbox{ otherwise.}
\end{aligned}$$ A tripotent $e$ in $E$ is called *complete* (respectively, *unitary* or *minimal*) if $E_0(E)=\{0\}$ (respectively, $E_2(e)=E$ or $E_2(e)=\CC e \neq \{0\}$).
Orthogonality in JB$^*$-triples is another notion required in this note. Elements $a,b$ in a JB$^*$-triple are said to be *orthogonal* (written $a\perp b$) if $L(a,b) =0$. It is known that $a\perp b$ if, and only if, $L(b,a)=0$ if, and only if, $\{a,a,b\} =0$ if, and only if, $\{b,b,a\}=0$ (see [@BurFerGarMarPe Lemma 1] for more equivalent reformulations). This notion is consistent with the usual concept of orthogonality in C$^*$-algebras. Let $a,b$ be elements in $B(H)$ (or in a general C$^*$-algebra). We say that $a$ and $b$ are *orthogonal* if $ a b^* = b^* a$. It is well known that $\|a\pm b\|= \max\{\|a\|,\|b\|\}$ whenever $a\perp b$ in (see [@FriRu85 Lemma 1.3$(a)$]). For each non-zero finite rank partial isometry $e\in K(H)$ there exists a finite family of mutually orthogonal minimal partial isometries $\{e_1,\ldots, e_m\}$ in $K(H)$ such that $ e = e_1+ \ldots + e_m$. We say that a tripotent $e$ in a JB$^*$-triple $E$ has *finite rank* if it can be written a sum of finitely many mutually orthogonal minimal tripotents.
To make easier our subsequent arguments we remark the following property: let $u$ and $e$ be two orthogonal tripotents in a JB$^*$-triple $E$. Clearly $e+u$ is a tripotent in $E$ and we can easily deduce from Peirce arithmetic that $$\label{eq Peirce 2 of an orthogonal sum} E_{2} (e+u) = E_2(e) \oplus E_2(u) \oplus E_1(e) \cap E_1 (u).$$
A subset $S$ of a JB$^*$-triple $E$ is called *orthogonal* if $0 \notin S$ and $x \perp y$ for every $x\neq y$ in $S$. The minimal cardinal number $r$ satisfying $card(S) \leq r$ for every orthogonal subset $S \subseteq E$ is called the *rank* of $E$ (cf. [@Ka97] and [@BeLoPeRo] for basic results on the rank of a Cartan factor and a JB$^*$-triple).
Let $B$ be a subset of a JB$^*$-triple $E.$ We shall denote by $B^\perp$ the *(orthogonal) annihilator of $B$* defined by $
B^\perp= B^{\perp}_{_E}:=\{ z \in E : z \perp x , \forall x \in B \}.$ Given a tripotent $e$ in $E$ the inclusions $$E_2(e) \oplus E_1(e)\supseteq \{e\}_{_E}^{\perp \perp}= E_0(e)^{\perp} \supseteq E_2(e)$$ always hold (see [@BurGarPe11 Proposition 3.3]). It is also known that the equality $\{e\}_{_E}^{\perp \perp}= E_2(e)$ is not always true. The following counterexample\[eq counterexample biorthog complement\] can be found in [@BurGarPe11 Remark 3.4]: suppose $H_1$ and $H_2$ are two infinite dimensional complex Hilbert spaces and $p$ is a minimal projection in $B(H_1)$. If $E$ denotes the orthogonal sum $p B(H_1) \oplus^{\infty} B(H_2)$ and we consider $e=p$ as an element in $E$, it can be checked that $p$ is a non-complete tripotent in $E$, $\{e\}_{_E}^{\perp} = B(H_2)$ and $\{e\}_{_E}^{\perp\perp} = E_2(e) \oplus E_1(e) =
p B(H_1)\neq \mathbb{C} p = E_2 (e).$
Despite of the previous counterexample, if we assume that $E$ is a Cartan factor and $e$ is a non-complete tripotent in $E,$ then the equality $\{e\}^{\perp \perp}=E_0(e)^\perp= E_2(e)$ is always true (see [@Ka97 Lemma 5.6]).
This seems an appropriate moment to refresh the definition of Cartan factors. A JB$^*$-triple is called a Cartan factor of type 1 if it is a JB$^*$-triple of the form $B(H_1, H_2)$, where $H_1$ and $H_2$ are two complex Hilbert spaces. Let $j$ be a conjugation on a complex Hilbert space $H$, and consider the linear involution $x\mapsto x^t:=jx^*j$ on $B(H).$ A Cartan factor of type 2 (respectively, type 3) is a JB$^*$-triple which coincides with the subtriple of $B(H)$ formed by the $t$-skew-symmetric (respectively, $t$-symmetric) operators. All we need to know in this note about types 4, 5 and 6 Cartan factors is that the first one is reflexive while the last two are finite dimensional (see [@Ka97] for more details).
According to [@BuChu], given a Cartan factor of type $j\in \{1,\ldots, 6\},$ the elementary JB$^*$-triple $K_j$ of type $j$ is defined in the following terms: $K_1 = K (H_1, H_2)$; $K_i = C \cap K(H)$ when $C$ is of type $i = 2 , 3$, and $K_i = C$ if the latter is of type $ 4, 5,$ or $6$. Obviously, if $K$ is an elementary JB$^*$-triple of type $j$, its bidual is precisely a Cartan factor of $j$.
We establish next a version of [@Ka97 Lemma 5.6] for elementary JB$^*$-triples.
\[l biorthogonal of a non-complete tripotent in an elementary\] Let $K$ be an elementary JB$^*$-triple. Suppose $e$ is a non-complete tripotent in $K.$ Then we have $\{e\}^{\perp \perp}=K_0(e)^\perp= K_2(e).$
If $K$ is an elementary JB$^*$-triple of type $j\in \{4,5,6\}$ we know that $K$ is a Cartan factor of the same type and hence the conclusion follows from [@Ka97 Lemma 5.6].
Suppose $K$ is an elementary JB$^*$-triple of type $j\in \{1,2,3\}$. The corresponding bidual $K^{**}= C$ is a Cartan factor of type $j$. By Goldstine’s theorem $K$ is weak$^*$ dense in $C$. Since $e\in K$, and $P_0(e)$ is weak$^*$ continuous in $C$, we deduce that $K_0(e) =\{e\}_{_K}^{\perp}$ and $K_2(e)$ are weak$^*$ dense in $C_0(e) =\{e\}_{_C}^{\perp}$ and $C_2(e)$, respectively. Thus the desired conclusion also follows from [@Ka97 Lemma 5.6]. Namely, we know that $\{e\}_{_K}^{\perp \perp}= K_0(e)^{\perp} \supseteq K_2(e)$ [@BurGarPe11 Proposition 3.3]. Let us take $x\in \{e\}_{_K}^{\perp\perp}$. In this case, $\{x,x,z\} =0$ for all $z\in \{e\}_{_K}^{\perp}= K_0(e)$. It follows from the weak$^*$ density of $K_0(e)$ in $C_0(E)$ and the separate weak$^*$ continuity of the triple product of $C$ that $\{x,x,a\}=0$ for all $a\in \{e\}_{_C}^{\perp}= C_0(e)$. By [@Ka97 Lemma 5.6] we have $x\in \{e\}_{_C}^{\perp \perp}=C_0(e)^\perp= C_2(e),$ and consequently $x\in K_2(e) = C_2(e) \cap K$.
According to [@BuChu], an element $x$ in a JB$^*$-triple $E$ is called *weakly compact* (respectively, *compact*) if the operator $Q(x):E\rightarrow E$ is weakly compact (respectively, compact). We say that $E$ is *weakly compact* (respectively, *compact*) if every element in $E$ is weakly compact (respectively, compact). If we denote by $K(E)$ the Banach subspace of $E$ generated by its minimal tripotents, then $K(E)$ is a (norm closed) triple ideal of $E$ and it coincides with the set of weakly compact elements of $E$ (see Proposition 4.7 in [@BuChu]). It follows from [@BuChu Lemma 3.3 and Theorem 3.4] that a JB$^*$ triple, $E,$ is weakly compact if and only if one of the following statement holds:
1. $K(E^{**})=K(E)$.
2. $K(E)=E.$
3. $E$ is a $c_0$-sum of elementary JB$^*$-triples.
Obviously each non-zero tripotent in a weakly compact JB$^*$-triple is of finite rank. It was observed in [@PeTan16 Corollary 2.5] that the results in [@BuChu] can be applied to deduce that a JB$^*$-triple $E$ is weakly compact if and only if it contains every tripotent of $E^{**}$ which is compact relative to $E$ in the sense of [@EdRu96; @FerPe06].
It should be remarked here that weakly compact JB$^*$-triples are the JB$^*$-triple analogue of compact C$^*$-algebras in the sense employed in [@Alex; @Yli], with the exception that a C$^*$-algebra is compact if, and only if, it is weakly compact (see [@Yli]).
\[c biorthogonal in an elementary through the orthogonal minimal tripotents\] Let $K$ be an elementary JB$^*$-triple. Suppose $e$ is a non-complete tripotent in $K.$ Let $x$ be an element in $K$ satisfying $x\perp v$ for every minimal tripotent $v\in K$ with $v\perp e$. Then $x\in K_2(e)$.
Let us observe that from Peirce arithmetic the Peirce 0-subspace $K_0(e) \neq \{0\}$ is an inner ideal of $K$. Corollary 3.5 in [@BuChu] together with the fact that $K$ is a factor show that $K_0(e)$ must be a weakly compact JB$^*$-triple. By Remark 4.6 in [@BuChu] every element in the weakly compact JB$^*$-triple $K_0(e)$ can be approximated in norm by finite positive linear combinations of mutually orthogonal minimal tripotents in $K_0(e)$. Since every minimal tripotent in $K_0(e)$ is a minimal tripotent in $K$ which is orthogonal to $e$, it follows from the hypothesis that $x\perp K_0(e)^{\perp}= \{e\}^{\perp\perp}$. Lemma \[l biorthogonal of a non-complete tripotent in an elementary\] implies that $x\in K_2(e)$.
Let us observe that Corollary \[c biorthogonal in an elementary through the orthogonal minimal tripotents\] can be also proved by a direct argument in the case in which $K=K(H)$ where $H$ is a complex Hilbert space.
We are actually interested in finding conditions on a tripotent $e$ in a weakly compact JB$^*$-triple $E$ to guarantee that the equality $\{e\}^{\perp \perp}=K_0(e)^\perp= K_2(e)$ holds (compare Lemma \[l biorthogonal of a non-complete tripotent in an elementary\] and the counterexample in page ).
\[l biorthogonal of a non-complete tripotent in a wk JBstar\] Let $\displaystyle E = \bigoplus_{i\in I}^{c_0} K_i$ be a weakly compact JB$^*$-triple, where each $K_i$ is an elementary JB$^*$-triple. For each $i\in I$, let $\pi_i$ denote the projection of $E$ onto $K_i$. Suppose $e$ is a tripotent in $E$ such that $\pi_i (e)$ is a non-complete tripotent in $K_i$ for every $i\in I$. Then the identity $\{e\}^{\perp \perp}_{_E}=E_0(e)^\perp= E_2(e)$ holds.
For each $x\in E$, we shall write $x= (x_i)_{i\in I}$ with $x_i = \pi_i(x)$. Since $e$ is a tripotent in $E$ the set $I_1:=\{ i \in I \ : \ \pi_i(e)\neq 0\}$ must be finite. We observe that $$\{e\}_{_E}^{\perp} = \left\{x\in E \ : \ x_i \in \{e_i\}_{_{K_i}}^{\perp} \hbox{ for all } i \in I_1\right\},$$ and hence $$\{e\}_{_E}^{\perp\perp} = \left\{x\in E \ : \ x_i\in \{e_i\}_{_{K_i}}^{\perp\perp} \hbox{ for all } i \in I_1, \ x_i =0 \hbox{ for all } i \in I\backslash I_1 \right\}.$$ Since for $i\in I_1$, $e_i$ is a non-complete tripotent in $K_i$ it follows from Lemma \[l biorthogonal of a non-complete tripotent in an elementary\] that $\{e_i\}_{_{K_i}}^{\perp\perp} = (K_i)_2(e_i)$, and consequently $\{e\}_{_E}^{\perp\perp} =K_2(e)$.
Let us observe that the conclusion in the previous Lemma \[l biorthogonal of a non-complete tripotent in a wk JBstar\] remains true when $E$ is replaced with an $\ell_{\infty}$-sum of Cartan factors.
New properties derived from the facial structure of an elementary JB$^*$-triple {#sec:new facial properties}
===============================================================================
Along this paper, given a Banach space $X$, the symbols $\mathcal{B}_{X}$ and $S(X)$ will stand for the closed unit ball and the unit sphere of $X$, respectively.
The main goal of this paper is to prove that every weakly compact JB$^*$-triple satisfies the Mazur-Ulam property. In a first step we shall study this property in the case of an elementary JB$^*$-triple $K$. If $K$ is reflexive it follows that $K=K^{**}$ is a Cartan factor, and hence the desired property follows from [@BeCuFerPe2018 Theorem 4.14, Proposition 4.15 and Remark 4.16] when $K$ has rank one or rank bigger than or equal to three and from [@KalPe2019 Theorem 1.1.] in the remaining cases. We shall therefore restrict our study to the case in which $K$ is a non-reflexive elementary JB$^*$-triple (equivalently, an infinite dimensional elementary JB$^*$-triple of type 1, 2 or 3).\[label restriction to non-reflexive elementary\]
As in many previous studies, the facial structure of the closed unit ball of a Banach space $X$ is a key tool to determine if $X$ satisfies the Mazur–Ulam property. The main reason being the fact that a surjective isometry $\Delta$ between the unit spheres of two Banach spaces $X$ and $Y$ maps maximal proper faces of $\mathcal{B}_{X}$ to maximal proper faces of $\mathcal{B}_{Y}$ (cf. [@ChenDong2011 Lemma 5.1], [@Tan2014 Lemma 3.5] and [@Tan2016 Lemma 3.3]).
We recall that a convex subset $F$ of a convex set $\mathcal{C}$ is called a *face* of $\mathcal{C}$ if for every $x\in F$ and every $y,z\in \mathcal{C}$ such that $x= t y + (1-t) z$ for some $t\in [0,1]$, we have $y,z\in F$. Let us observe that every proper (i.e., non-empty and non-total) face of the closed unit ball of a Banach space $X$ is contained in $S(X)$. Following the notation in [@MoriOza2018], a closed face $F \subseteq S(X)$ is called an *intersection face* if $$F = \bigcap \Big\{ E \ : \ E \subseteq S(X) \hbox{ a maximal face containing } F \Big\}.$$ If $X$ is a complex C$^*$-algebra or the the predual of a von Neumann algebra, or more generally, a JB$^*$-triple or the predual of a JBW$^*$-triple, every proper norm closed face of $\mathcal{B}_{X}$ is an intersection face (see [@Tan2017b Corollary 3.4] and [@FerGarPeVill17 Proof of Proposition 2.4 and comments after and before Corollary 2.5]). It should be remarked that this conclusion can be also derived from the main results in [@EdFerHosPe2010]. These facts together with [@MoriOza2018 Lemma 8] are employed in the next result which was already stated in [@KalPe2019 Lemma 2.2].
\[l intersection faces MoriOzawa L8\][([@KalPe2019 Lemma 2.2], [@MoriOza2018 Lemma 8], [@FerGarPeVill17 Proposition 2.4], [@EdFerHosPe2010 Corollary 3.11])]{} Let $\Delta: S(E)\to S(Y)$ be a surjective isometry where $E$ is a JB$^*$-triple and $Y$ is a real Banach space. Then $\Delta$ maps proper norm closed faces of $\mathcal{B}_{E}$ to intersection faces in $S(Y)$. Furthermore, if $F$ is a proper norm closed face of $\mathcal{B}_{E}$ then $\Delta(-F) = -\Delta(F)$.
The structure of all norm closed faces of the closed unit ball of a JB$^*$-triple $E$ was completely determined in [@EdFerHosPe2010], where it is shown that each norm closed face of $\mathcal{B}_{E}$ is univocally given by a tripotent in $E^{**}$ which is compact relative to $E$ (see [@EdFerHosPe2010 Theorem 3.10 and Corollary 3.12] and the concrete definitions therein). As we observed before, each weakly compact JB$^*$-triple $E$ contains all tripotents in $E^{**}$ which are compact relative to $E$ (see [@PeTan16 Corollary 2.5]). Therefore, the norm closed faces of the closed unit ball of a weakly compact JB$^*$-triple $E$ are completely determined by the tripotents in $E$, consequently Theorem 3.10 in [@EdFerHosPe2010] in this concrete setting assures that for each proper norm closed face $F$ of $\mathcal{B}_{E}$ there exists a non-zero finite rank tripotent $e\in E$ such that $$\label{eq structure of norm closed proper faces in the closed unit ball of weakly compact JB*triples} F= F_e = e + \mathcal{B}_{E_0(e)} = (e+E_0(e))\cap \mathcal{B}_{E}.$$
Let us comment some of the difficulties we can find when applying our current knowledge. Suppose $\Delta: S(K(H)) \to S(Y)$ is a surjective isometry, where $H$ is an infinite dimensional complex Hilbert space and $Y$ is a real Banach space. For each non-zero partial isometry $e\in K(H)$, Lemma \[l intersection faces MoriOzawa L8\] shows that $\Delta (F_e)$ is an intersection face in $S(Y)$ and the restriction $\Delta|_{F_{e}} : F_e \to \Delta (F_e)$ is a surjective isometry too. Henceforth, given an element $x_0$ in a Banach space $X,$ we shall write $\mathcal{T}_{x_0}: X\to X$ for the translation mapping defined by $\mathcal{T}_{x_0} (x) = x+x_0$ ($x\in X$). By considering the commutative diagram $$\begin{tikzcd}
F_e \arrow{r}{\Delta|_{F_e}} \arrow[swap]{d}{\mathcal{T}_{-e}}
& \Delta({F}_e) \\
(1-ee^*) \mathcal{B}_{ K(H)} (1-e^*e) \arrow[dashrightarrow, "\Delta_{e}"]{ru} &
\end{tikzcd}$$ we realize that if we could prove that $\mathcal{B}_{(1-ee^*) K(H) (1-e^*e)}$ satisfied the strong Mankiewicz property, we could get some progress to determine the behavior of $\Delta$ on $F_e$. However, $(1-ee^*) K(H) (1-e^*e)$ is a JB$^*$-triple whose closed unit ball contains no extreme points (let us observe that $H$ is infinite dimensional with $ee^*$ and $e^* e$ finite rank projections). So, our current technology is not enough to attack the problem from this perspective. We shall develop a new facial argument not contained in the available literature.
The following result is implicit in [@FerMarPe2012 Remark 20] and a detailed explanation can be found in [@PeTan16 Lemma 3.3] from where it has been taken.
\[l two finite rank tripotents at distance 1 are orthogonal\][[@PeTan16 Lemma 3.3]]{} Let $e$ be a tripotent in a JB$^*$-triple $E$. Suppose $x$ is an element in $\mathcal{B}_E$ satisfying $\|e\pm x\| = 1$. Then $x\perp e$.
We shall need the next consequence.
\[l two finite rank tripotents at distance leq 1 are orthogonal\] Let $e$ be a tripotent in a JB$^*$-triple $E$. Suppose $x$ is an element in $S(E)$ satisfying $\|e\pm x\| \leq 1$. Then $x\perp e$.
Since $S(E)\ni x = \frac12 (x+e )+ \frac12 (x-e)$, we deduce from $\|e\pm x\| \leq 1$ that $\|e\pm x\| = 1$. Lemma \[l two finite rank tripotents at distance 1 are orthogonal\] implies that $ x\perp e$ as desired.
The next result has been borrowed from [@KalPe2019].
\[c Tingley antipodal thm for tripotents\][@KalPe2019 Corollary 2.4] Let $\Delta: S(E)\to S(Y)$ be a surjective isometry where $E$ is a JB$^*$-triple and $Y$ is a real Banach space. Suppose $e$ is a non-zero tripotent in $E$, then $\Delta(-e) = -\Delta(e)$.
We are now in position to establish a new geometric result, based on the facial structure of $\mathcal{B}_{K(H)}$, which provides a new tool to prove the Mazur–Ulam property in the case of elementary JB$^*$-triples.
\[p new geometric facial property\] Let $\Delta: S(K) \to S(Y)$ be a surjective isometry, where $K$ is an elementary JB$^*$-triple and $Y$ is a real Banach space. Then for each tripotent $e\in K$ and each minimal tripotent $u\in K$ with $e\perp u$ the set $\Delta(u+ \mathcal{B}_{K_2(e)})$ is convex and the restriction $\Delta|_{u+ \mathcal{B}_{K_2(e)}}: u+ \mathcal{B}_{K_2(e)} \to \Delta(u + \mathcal{B}_{K_2(e)})$ is an affine mapping. Consequently, there exists a real linear isometry $T^{u}_e$ from $K_2(e)$ onto a norm closed subspace of $Y$ satisfying $\Delta( \mathcal{T}_{u} (x) )=\Delta( u +x) = T^{u}_e (x) + \Delta(u)$ for all $x\in \mathcal{B}_{K_2(e)}.$
As we commented in page , by [@BeCuFerPe2018 Theorem 4.14, Proposition 4.15 and Remark 4.16] and [@KalPe2019 Theorem 1.1.], we can assume that $K$ is non-reflexive (i.e. an infinite dimensional elementary JB$^*$-triple of type 1, 2 or 3), and hence every tripotent in $K$ is non-complete and of finite rank and $K$ has infinite rank.
Let $e$ and $u$ be non-zero tripotents in $K$ such that $u$ is minimal and $u\perp e$. The conclusion clearly holds for $e=0$, we can therefore assume that $e$ is a non-zero finite rank tripotent. Set $w= e + u$. Keeping the notation in for each non-zero tripotent $v\in K$ we write $F_v = v + \mathcal{B}_{K_0 (v)}$ for the proper norm closed face of $\mathcal{B}_{K}$ associated with $v$.
Let $\mathcal{D}_0 =\Delta(F_{u})$. Lemma \[l intersection faces MoriOzawa L8\] implies that $\mathcal{D}_0$ is an intersection face in $S(Y)$. Let us fix an arbitrary minimal tripotent $v\in K$ such that $v\perp w$. We set $$\mathcal{D}_1^v := \left\{ y\in \mathcal{D}_0 \ : \ \|y\pm \Delta(v) \|\leq 1 \right\}.$$ We claim that $\mathcal{D}_1^v$ is norm closed and convex. Namely, given $y_1,y_2\in \mathcal{D}_1^v$ and $t\in [0,1]$ the convex combination $t y_1 + (1-t) y_2 \in \mathcal{D}_0$ because $\mathcal{D}_0$ is convex. Furthermore $$\| t y_1 + (1-t) y_2 \pm \Delta (v) \| \leq t\ \| y_1 \pm \Delta (v) \| + (1-t)\ \| y_2 \pm \Delta (v) \| \leq 1,$$ witnessing that $t y_1 + (1-t) y_2\in \mathcal{D}_1^v$. Clearly $\mathcal{D}_1^v$ is norm closed.
Let us consider the inner ideal $K_2(e)$. Clearly $K_2(e)$ is a finite dimensional JBW$^*$-triple. We shall next prove that $$\label{eq image of BM + pm} \Delta\left(\mathcal{B}_{K_2(e)} + u \right)\! =\! \bigcap \Big\{ \mathcal{D}_1^v : v \hbox{ a minimal tripotent in } K \hbox{ with } v\perp w \Big\}.$$
$(\subseteq)$ Let $v\in K$ be a minimal tripotent with $v\perp w$. We take an element $u+z \in u + \mathcal{B}_{K_2(e)}$. Since $u+ z \perp v,$ $\Delta(-v) = -\Delta(v)$ (see Lemma \[c Tingley antipodal thm for tripotents\]), and $\Delta$ is an isometry we have $$\left\| \Delta(u +z) \pm \Delta (v) \right\| = \left\| z+u \pm v \right\| =\max\{\left\|z+u\right\|, \|v\| \} =1.$$
$(\supseteq)$ Suppose next that $y\in \mathcal{D}_1^v $ for every minimal tripotent $v$ in $K$ with $v\perp w$. It follows from the definition that $y \in \mathcal{D}_0 =\Delta(F_{u})$ and hence there exists $x\in F_{u}$ satisfying $\Delta(x) =y$. Thus, by Lemma \[c Tingley antipodal thm for tripotents\] and the assumptions, we have $1\geq \|y\pm \Delta(v) \| = \| x\pm v \|,$ for every $v$ as above. Lemma \[l two finite rank tripotents at distance leq 1 are orthogonal\] implies that $x\perp v$ for every minimal tripotent $v\in K$ with $v\perp w$. It follows from Corollary \[c biorthogonal in an elementary through the orthogonal minimal tripotents\] that $x\in K_2 (w)$, and since $x\in F_{u}= u + \mathcal{B}_{K_0 (u)}$ we can easily see, for example from , that $x\in u+ \mathcal{B}_{K_2(e)}$, as desired.
We consider next the following commutative diagram $$\label{eq diagram Delta u e} \begin{tikzcd} u + \mathcal{B}_{K_2(e)} \arrow[swap]{d}{\tau_{-u}} \arrow{r}{\Delta} & \Delta\left( u + \mathcal{B}_{K_2(e)} \right) \\
\mathcal{B}_{K_2(e)} \arrow[dashrightarrow]{r}{\Delta^{u}_e} & \Delta\left( u + \mathcal{B}_{K_2(e)} \right)-\Delta(u) \arrow[u, "\tau_{\Delta(u)}"]
\end{tikzcd}$$ It follows from that $\Delta\left( u + \mathcal{B}_{K_2(e)} \right),$ and hence $\Delta\left( u + \mathcal{B}_{K_2(e)} \right)-\Delta(u),$ is a norm closed convex subset of $S(Y)$. Since $K_2(e)$ is a finite dimensional JBW$^*$-triple, it follows from [@BeCuFerPe2018 Corollary 2.2] that $\mathcal{B}_{K_2(e)}$ satisfies the strong Mankiewicz property. Therefore, by the strong Mankiewicz property there exists a surjective real linear isometry $T_e^u $ from $K_2(e)$ onto a norm closed subspace of $Y$ whose restriction to $\mathcal{B}_{K_2(e)}$ is $\Delta^{u}_e$, that is $$\Delta (u +x) =\Delta(u) + \Delta^{u}_e (x) = \Delta(u) + T^{u}_e (x),$$ for all $x\in \mathcal{B}_{K_2(e)}$.
Let us remark a consequence of the previous Lemma \[l two finite rank tripotents at distance leq 1 are orthogonal\] and Corollary \[c biorthogonal in an elementary through the orthogonal minimal tripotents\]. The statement has been actually outlined in the proof of the previous proposition.
\[l sphere M minimal tripotents orhtogonal to e\] Let $e$ be a non-complete tripotent in an elementary JB$^*$-triple $K$. Let $\{e\}^{\perp}_{min}=\{v \hbox{ minimal tripotent in } K \hbox{ with } v\perp e\}.$ Then $$\mathcal{B}_{K_2(e)} = \{ x\in \mathcal{B}_{K} : \|x-v\|\leq 1 \hbox{ for all } v\in \{e\}^{\perp}_{min} \}.$$
$(\subseteq)$ Take $x\in \mathcal{B}_{K_2(e)}$. It follows from Peirce arithmetic that $x\perp v$ for all $v\in \{e\}^{\perp}_{min}$, and thus $\|x- v\| = \max\{\|x\|,\|v\|\} \leq 1$.
$(\supseteq)$ Assume now that $x\in \mathcal{B}_{K}$ with $\|x-v\|\leq 1$ for all $v\in \{e\}^{\perp}_{min}$. We observe that $-v\in \{e\}^{\perp}_{min}$ for all $v\in \{e\}^{\perp}_{min}$. Therefore $\|x\pm v\|\leq 1$ for all $v\in \{e\}^{\perp}_{min}$. Lemma \[l two finite rank tripotents at distance leq 1 are orthogonal\] implies that $x\perp v$ for all $v\in \{e\}^{\perp}_{min}$. Corollary \[c biorthogonal in an elementary through the orthogonal minimal tripotents\] proves that $x\in K_2(e)$ as desired.
In the following proposition we explore the properties of the real linear isometries $T^{u}_e$ given by Proposition \[p new geometric facial property\].
\[p first consequence from the new geometric facial property\] Let $\Delta: S(K) \to S(Y)$ be a surjective isometry, where $K$ is an elementary JB$^*$-triple and $Y$ is a real Banach space. Suppose $e$ and $v$ are tripotents in $K$ with $v$ minimal and $v\perp e$. Let $T^{v}_e : K_2(e) \to Y$ be the real linear isometry given by Proposition \[p new geometric facial property\]. Then the following statements hold:
1. $T^{v}_e = T^{-v}_{-e} = T^{-v}_e = T^{v}_{-e}$;
2. Suppose $u$ is a minimal tripotent with $u\perp v$. Then $\Delta(u) = T_{u}^{v} (u)$;
3. Suppose $u$ is a minimal tripotent in $K_2(e)$. Then $\Delta(u) = T_{e}^{v} (u)$;
4. For each $x\in S(K_2(e))$ we have $ T^{v}_e (x) = \Delta(x);$
5. If $w$ is another minimal tripotent with $w\perp e$, the real linear isometries $T^{v}_e$ and $T^{w}_e$ coincide;
If $K$ is reflexive, the desired conclusion is an easy consequence of [@BeCuFerPe2018 Theorem 4.14, Proposition 4.15 and Remark 4.16] and [@KalPe2019 Theorem 1.1.] because the latter results show that $\Delta$ admits an extension to a surjective real linear isometry. We shall therefore assume that $K$ is non-reflexive. As we commented before, under our hypotheses, $K_2(e)$ is a (weakly compact) finite dimensional JBW$^*$-triple, and thus every element in $K_2(e)$ can be written as a finite positive combination of mutually orthogonal minimal projections in $K$.
$(a)$ It follows from the arguments in the previous paragraph that it suffices to show that $T^{v}_e (u)= T^{-v}_{-e} (u) = T^{-u}_e (u) = T^{u}_{-e} (u)$ for every minimal tripotent $u$ in $K_2(e)= K_2(-e)$. Fix a minimal tripotent $u\in K_2(e)$. By Lemma \[c Tingley antipodal thm for tripotents\] and Proposition \[p new geometric facial property\] we have $$\Delta(v+u) = \Delta(v) + T^{v}_e (u) = - \Delta(-v-u) = \Delta(v) - T^{-v}_e (-u) = \Delta(v) + T^{-v}_e (u),$$ $$\Delta(v-u) = \Delta(v) - T^{v}_e (u) = - \Delta(-v+u) = - \Delta(-v) - T^{-v}_{-e} (u)= \Delta(v) - T^{-v}_{-e} (u),$$ witnessing the desired equalities.
$(b)$ The elements $u\pm v $ belong to the convex set $u + \mathcal{B}_{K_2(v)}$. By Proposition \[p new geometric facial property\] $\Delta$ is affine on $u + \mathcal{B}_{K_2(v)}$. Therefore $\Delta(u) = \frac12 \Delta(u +v ) +\frac12 \Delta(u-v).$ By applying Proposition \[p new geometric facial property\], Lemma \[c Tingley antipodal thm for tripotents\] and $(a)$ we deduce that $$\Delta(u) = \frac12 \Delta(u +v ) +\frac12 \Delta(u-v) = \frac12 \left(\Delta(v) + T_{u}^v (u) + \Delta(-v) + T_{u}^{-v} (u) \right) = T_{u}^v (u).$$
$(c)$ By Proposition \[p new geometric facial property\] and statement $(b)$ we get $$\Delta(v) + T_e^v(u) =\Delta (v+u ) = \Delta(v) + T_u^v(u) = \Delta(v) + \Delta (u),$$ which proves the desired identity.
$(d)$ Let us take $x \in S(K_2(e))$. Having in mind the arguments at the beginning of this proof, we can find mutually orthogonal minimal tripotents $e_1,\ldots, e_m$ such that $\displaystyle x = e_1 +\sum_{k=2}^m t_k e_k$, where $t_2,\ldots, t_m\in (0,1]$. Set $w= e_2+\ldots+e_m$. Proposition \[p new geometric facial property\] applied to $w$ and $e_1$ guarantees that $$\Delta(x) = \Delta (e_1) + \sum_{k=2}^m t_k T_w^{e_1} (e_k) = \Delta (e_1) + \sum_{k=2}^m t_k \Delta (e_k)$$ $$= T_e^{v} (e_1) + \sum_{k=2}^m t_k T_e^{v} (e_k)=T_e^{v}\left( e_1 +\sum_{k=2}^m t_k e_k \right) = T_e^{v} (x),$$ where in the second and third equalities we applied $(c)$.
$(e)$ This is a trivial consequence of $(d)$ because by this statement the real linear isometries $T^{v}_e$ and $T^{w}_e$ coincide with $\Delta$ on $S(K_2(e))$.
We can establish next a version of [@KalPe2019 Corollary 2.7] within the framework of elementary JB$^*$-triples.
\[c affine on convex subsets of the sphere in reflexive\] Let $\Delta: S(K) \to S(Y)$ be a surjective isometry, where $K$ is an elementary JB$^*$-triple and $Y$ is a real Banach space. Let $\mathcal{C}\subset S(K)$ be a non-empty convex subset. Then $\Delta|_{\mathcal
{C}}$ is an affine mapping.
If $K$ is reflexive, then the conclusion follows from [@KalPe2019 Corollary 2.7]. Let us assume that $K$ is non-reflexive. Let $\mathcal{C}\subset S(K)$ be a non-empty convex subset, $x,y\in \mathcal{C}$ and $t\in [0,1]$. By the structure of elementary JB$^*$-triples, for each $0<\varepsilon<\frac12$, there exists a (finite rank) tripotent $e$ in $K$ and elements $x_1,y_1\in S(K_2(e))$ satisfying $\|x-x_1\|, \|y-y_1\|<\varepsilon$ (cf. [@BuChu Remark 4.6] and the fact that $K$ is an infinite dimensional elementary JB$^*$-triple). An standard argument shows that $$\left\| t x + (1-t) y - (t x_1 + (1-t) y_1) \right\| < \varepsilon,$$ and $$\left\| \frac{t x_1 + (1-t) y_1}{\|t x_1 + (1-t) y_1\|} - (t x_1 + (1-t) y_1) \right\| < \varepsilon.$$
Since $K$ has infinite rank, we can find a minimal tripotent $v$ in $K$ which is orthogonal to $e$. Let $T_e^v: K_2(e) \to Y$ be the real linear isometry given by Proposition \[p new geometric facial property\]. Having in mind that $x_1,y_1\in K_2(e)$, it follows from the just quoted proposition, the hypothesis on $\Delta$ and Proposition \[p first consequence from the new geometric facial property\]$(d)$ that $$\begin{aligned} & \left\| t\Delta(x) + (1-t) \Delta(y) - \Delta (t x + (1-t) y) \right\| \\
&\leq \left\| t\Delta(x) + (1-t) \Delta(y) - t\Delta(x_1) - (1-t) \Delta(y_1) \right\| \\
& + \left\| tT_e^v (x_1) + (1-t) T_e^v (y_1) - T_e^v (t x_1 + (1-t) y_1) \right\| \\
& + \left\|T_e^v (t x_1 + (1-t) y_1) - T_e^v \left( \frac{t x_1 + (1-t) y_1}{\|t x_1 + (1-t) y_1\|} \right) \right\| \\
& + \left\|\Delta \left( \frac{t x_1 + (1-t) y_1}{\|t x_1 + (1-t) y_1\|} \right) - \Delta (t x + (1-t) y) \right\| <4 \varepsilon.
\end{aligned}$$ The arbitrariness of $0<\varepsilon < \frac12$ implies that $t\Delta(x) + (1-t) \Delta(y) = \Delta (t x + (1-t) y)$ as desired.
Elementary JB$^*$-triples satisfy the Mazur–Ulam property {#sec: elementary JB* triples satisfy MUP}
=========================================================
Our goal in this section is to prove that every weakly compact JB$^*$-triple satisfies the Mazur–Ulam property. By this result we shall exhibit an example of a non-unital and non-commutative C$^*$-algebra containing no unitaries but satisfying the Mazur–Ulam property. The new geometric properties related to the facial structure of elementary JB$^*$-triples developed in the previous section will be the germ to prove our result. Compared with the techniques in [@MoriOza2018; @BeCuFerPe2018; @CuePer19] and [@KalPe2019], we shall not base our proof on [@MoriOza2018 Lemma 6] nor on [@FangWang06 Lemma 2.1].
\[t elementary JBstar triples satisfy the MUP\] Every elementary JB$^*$-triple $K$ satisfy the Mazur–Ulam property, that is, for every real Banach space $Y$, every surjective isometry $\Delta: S(K)\to S(Y)$ admits a (unique) extension to a surjective real linear isometry from $K$ onto $Y$.
As we already commented in previous sections, if $K$ is reflexive the conclusion follows from [@BeCuFerPe2018 Theorem 4.14, Proposition 4.15 and Remark 4.16] and [@KalPe2019 Theorem 1.1 and Corollary 1.2]. As in the proof of Proposition \[p new geometric facial property\] we shall assume that $K$ is non-reflexive (i.e. an infinite dimensional elementary JB$^*$-triple of type 1, 2 or 3), and hence every tripotent in $K$ is non-complete and of finite rank and $K$ has infinite rank.
Let $\mathcal{F}=\mathcal{F}(K)$ denote the linear subspace of $K$ generated by all minimal tripotents in $K$. In our case $\mathcal{F}$ is a normed subspace of $K$ which is norm dense but non-closed. We shall define a mapping $T: \mathcal{F}\to Y.$ By definition every element $x$ in $\mathcal{F}$ can be written as a finite positive combination of mutually orthogonal minimal tripotents in $K$. We can therefore find a tripotent $e$ in $K$ such that $x\in K_2(e)$. Since $K$ has infinite rank we can always find a minimal tripotent $v\in K$ with $v\perp e$. We set $T(x) := T_{e}^v(x)$. Clearly $T(0) = 0 $.
We claim that $T$ is well defined. Pick $0\neq x\in \mathcal{F}$. Suppose $e_1,e_2$ are tripotents in $K$ with $x\in K_2(e_j)$ for $j=1,2$ and $v_1,v_2$ are two minimal tripotents in $K$ with $v_j \perp e_j$ for $j=1,2$. Proposition \[p first consequence from the new geometric facial property\]$(d)$ implies that $$T_{e_1}^{v_1} \left(\frac{x}{\|x\|}\right)= \Delta\left(\frac{x}{\|x\|}\right) = T_{e_2}^{v_2} \left(\frac{x}{\|x\|}\right),$$ and hence $T$ is well defined.
We shall next show that $T$ is linear. For each $x\in \mathcal{F}$, each pair of tripotents $e,v$ in $K$ with $v$ minimal, $v\perp e$ and $x\in K_2(e)$, and each real number $\alpha$, we have $\alpha x\in K_2(e)$ and $T(\alpha x) = T_e^v (\alpha x) = \alpha T_e^v(x) = \alpha T(x)$. Let us now take $x,y \in \mathcal{F}$. We can find a tripotent $e\in K$ such that $x,y, x+y\in K_2(e)$. Let $v\in K$ be a minimal tripotent with $v\perp e$. By definition $$T(x+y ) = T_e^v (x+y) = T_e^v (x) + T_e^v (y) = T(x) + T(y).$$
Since $T: \mathcal{F}\to Y$ is a real linear isometry and $\mathcal{F}$ is norm dense in $K$, we can find a unique extension of $T$ to a surjective real linear isometry from $K$ to $Y$ which will be denoted by the same symbol $T$.
We shall finally show that $T$ coincides with $\Delta$ on $S(K)$. By continuity and norm density of $\mathcal{F}$, it suffices to prove that $T$ coincides with $\Delta$ on $S(\mathcal{F})$. By definition, given $x\in S(\mathcal{F})$, tripotents $e,v$ in $K$ with $v\perp e$ and $x\in K_2(e)$, Proposition \[p first consequence from the new geometric facial property\]$(d)$ assures that $T(x) = T_e^v (x) = \Delta(x)$, which concludes the proof.
The main results in [@FerPe18Adv] prove that every surjective isometry between the unit spheres of two elementary JB$^*$-triples $K_1$ and $K_2$ can be extended to a surjective real linear isometry between $K_1$ and $K_2$ (see [@FerPe18Adv Theorems 4.4-4.7 and 4.9] and some precedents in [@PeTan16]). Our previous Theorem \[t elementary JBstar triples satisfy the MUP\] offers an strengthened conclusion by showing that every surjective isometry from the unit sphere of an elementary JB$^*$-triple onto the unit sphere of any real Banach space extends to a surjective real linear isometry.
The following straightforward consequence is interesting by itself.
\[c KH satisfies the MUP\] For each complex Hilbert space $H$, the space $K(H)$ of all compact linear operators on $H$ satisfies the Mazur–Ulam property.
More on the strong Mankiewicz property
======================================
In this section we pursue a result showing that every weakly compact JB$^*$-triple satisfies the strong Mankiewicz property. We begin by establishing a technical result which is valid for an abstract class of Banach spaces including all weakly compact JB$^*$-triples.
\[p strong Mankiewicz property for c0 sums\] Let $(X_{i})_{i\in I}$ be a family of Banach spaces, and let $X= \displaystyle \bigoplus_{i\in I}^{c_{0}} X_i$. Suppose that the following hypotheses hold: for each $x,y\in X$ and each $\varepsilon>0$ there exist a finite subset $F\subseteq I$ [(]{}depending on $x,y$ and $\varepsilon>0$[)]{}, closed subspaces $Z_i\subseteq X_i$ and elements $a_i,b_i\in Z_i$ [(]{}$i\in F$[)]{} such that
1. For each $i\in F$ there exists a subset $M_i\subseteq \mathcal{B}_{X_i}$ satisfying $$\mathcal{B}_{Z_i} = \left\{ x_i \in \mathcal{B}_{X_i} : \|x_i -m_i\|\leq 1 \hbox{ for all } m_i \in M_i \right\};$$
2. $\|x - (a_i)_{i\in F}\|, \|y - (b_i)_{i\in F}\| <\varepsilon$ [(]{}we obviously regard $\displaystyle \bigoplus_{i\in F}^{\ell_{\infty}} Z_i$ as a closed subspace of $X$[)]{};
3. The closed unit ball of the space $\displaystyle \bigoplus_{i\in F}^{\ell_{\infty}} Z_i$ satisfies the strong Mankiewicz property.
Then every convex body $K\subset X$ satisfies the strong Mankiewicz property.
By [@MoriOza2018 Lemma 4] it suffices to prove that the closed unit ball of $X$ satisfies the strong Mankiewicz property. To this end let $\Delta :\mathcal{B}_{X}\to L$ be a surjective isometry, where $L$ is a convex subset in a normed space $Y$. We shall show that $\Delta$ is affine.
Let us take $x,y\in \mathcal{B}_{X}$ and $t\in (0,1)$. Fix an arbitrary $\varepsilon >0.$ It follows from our hypotheses that there exist a finite set $F\subseteq I$, closed subspaces $Z_i\subseteq X_i$ and elements $a_i,b_i\in \mathcal{B}_{Z_i}$ ($i\in F$) such that the closed unit ball of the space $\displaystyle Z= \bigoplus_{i\in F}^{\ell_{\infty}} Z_i$ satisfies the strong Mankiewicz property, $\|x - (a_i)_{i\in F}\| <\varepsilon$ and $\|y - (b_i)_{i\in F}\| <\varepsilon$.
We also know the existence of subsets $M_i\subseteq \mathcal{B}_{X_i}$ ($i\in I$) satisfying $(1)$. Let $M\subseteq X$ denote the set given by $$M:= \left\{ x=(x_i)_i\in \mathcal{B}_X : x_i\in M_i \hbox{ for all } i \in F,\ x_j\in \mathcal{B}_{X_j} \hbox{ for } j\in I\backslash F\right\}.$$ We also set $$L_1 :=\left\{ y\in L : \|y - \Delta(m)\|\leq 1 \hbox{ for all } m\in M \right\}.$$ Having in mind that $L$ is convex, it is easy to see that $L_1$ also is convex.
We claim that $$\label{eq Delta BZ equals L1} \Delta(\mathcal{B}_{Z}) = L_1.$$
Indeed, by the assumptions each $z\in \mathcal{B}_{Z}$ satisfies that $\|z-m\|\leq 1$ for all $m\in M$, and hence $\|\Delta(z) -\Delta (m)\|\leq 1$ for all $m\in M$. This proves that $\Delta(\mathcal{B}_{Z}) \subseteq L_1$.
Take now, $y\in L_1$. Since $\Delta$ is surjective there exists (a unique) $z=(z_i)_i\in \mathcal{B}_{X}$ with $\Delta(z) = y.$ Since $\Delta$ is an isometry and $y\in L_1$ we deduce that $$\|z-m\|=\|\Delta(z) -\Delta (m)\|\leq 1,\hbox{ for all } m\in M,$$ in particular, $$\hbox{$\|z_i -m_i\|\leq 1,$ for all $m_i\in M_i$ and for all $i\in F$}$$ and $$\hbox{$\|z_i - w_i\|\leq 1,$ for all $w_i\in \mathcal{B}_{X_i}$ and all $i\in I\backslash F$.}$$ It follows from the hypothesis $(1)$ that $z_i\in \mathcal{B}_{Z_i}$ for all $i\in F,$ and clearly $z_i =0$ for all $i\in I\backslash F$. Therefore $z\in \mathcal{B}_{Z}$ which finishes the proof of .
We deduce from and the preceding comments that $\Delta|_{\mathcal{B}_{Z}}:\mathcal{B}_{Z}\to L_1$ is a surjective isometry and $L_1$ is a convex subset of $Y$. We apply now that the closed unit ball of the space $Z$ satisfies the strong Mankiewicz property to deduce that $\Delta|_{\mathcal{B}_{Z}}$ is affine, and thus $$\Delta \left(t (a_i)_{i\in F}+ (1-t) (b_i)_{i\in F} \right) = t \Delta \left( (a_i)_{i\in F} \right) + (1-t) \Delta \left((b_i)_{i\in F} \right),$$ and consequently $$\begin{aligned}\|& \Delta (t x + (1-t) y) - (t \Delta (x) + (1-t) \Delta(y)) \| \\ & \leq \|\Delta (t x + (1-t) y) - \Delta \left(t (a_i)_{i\in F}+ (1-t) (b_i)_{i\in F} \right) \| \\
&+ t \| \Delta \left( (a_i)_{i\in F}\right)- \Delta (x) \| + (1-t) \| \Delta \left((b_i)_{i\in F} \right) - \Delta(y) \| < 2\varepsilon.
\end{aligned}$$
The arbitrariness of $\varepsilon>0$ implies that $\Delta (t x + (1-t) y) = t \Delta (x) + (1-t) \Delta(y)$, which concludes the proof.
An appropriate framework to apply the previous proposition is provided by weakly compact JB$^*$-triples.
\[c weakly compact JB\* triples satisfy the SMP\] Every weakly compact JB$^*$-triple $E$ satisfies the hypotheses of the previous Proposition \[p strong Mankiewicz property for c0 sums\]. Consequently every convex body $K\subset E$ satisfies the strong Mankiewicz property.
As we commented in subsection \[subsec:background\] every weakly compact JB$^*$-triple coincides with a $c_0$-sum of a family $\{K_i: i\in I\}$ of elementary JB$^*$-triples (cf. [@BuChu]). Given $x,y\in E$ and $\varepsilon>0$ there exist a finite subset $F\subseteq I$ satisfying $\| x - (x_i)_{i\in F}\|, \| y - (y_i)_{i\in F}\|< \frac{\varepsilon}{2}$.
Fix an arbitrary $i\in F$. If $K_i$ is reflexive (i.e. it is finite dimensional or an elementary JB$^*$-triple of type 4), we know that $K_i$ is a JBW$^*$-triple. In this case we take $Z_i = K_i$, $a_i = x_i,$ $b_i = y_i,$ and $M_i = \{0\}$. Clearly, $$\mathcal{B}_{K_i} = \{ x_i \in K_i : \| x_i -0 \|\leq 1\}.$$
Suppose next that $K_i$ is not reflexive (i.e. an infinite dimensional elementary JB$^*$-triple of type 1, 2 or 3). By [@BuChu Remark 4.6] and/or basic theory of compact operators we can find a finite rank (and hence non-complete) tripotent $e_i$ (i.e. a tripotent which is a finite sum of mutually orthogonal minimal tripotents in $K_i$) such that $\|x_i -P_2(e_i) (x_i)\|,\|y_i -P_2(e_i) (y_i)\| <\frac{\varepsilon}{2}$. The subtriple $Z_i = (K_i)_2 (e_i)$ is finite dimensional and thus a JBW$^*$-triple. Take $a_i = P_2(e_i) (x_i),$ $b_i = P_2(e_i) (y_i)\in Z_i$. Set $M_i:=\{e_i\}_{min}^{\perp} \cap K_i =\{ v \hbox{ minimal tripotent in } K_i : v\perp e_i\}\subseteq \mathcal{B}_{K_i}$. Lemma \[l sphere M minimal tripotents orhtogonal to e\] assures that $$\mathcal{B}_{Z_i}= \mathcal{B}_{ (K_i)_2 (e_i)} = \left\{ x_i \in \mathcal{B}_{K_i} : \|x_i -m_i\|\leq 1 \hbox{ for all } m_i \in M_i= \{e_i\}_{min}^{\perp} \cap K_i \right\}.$$ By construction, $$\|x-(a_i)_{i\in F}\|\leq \| x - (x_i)_{i\in F}\| + \| (x_i)_{i\in F}- (a_i)_{i\in F}\|< \varepsilon.$$
Finally, for each $i\in F$, $Z_i$ is finite dimensional or reflexive, therefore $\displaystyle \bigoplus_{i\in F}^{\ell_{\infty}} Z_i$ is a JBW$^*$-triple, and thus its closed unit ball satisfies the strong Mankiewicz property by [@KalPe2019 Corollary 1.2].
\[remark compact C\* algebras satisfy strong Mankiewicz property\] It is worth to note that every compact C$^*$-algebra of the form $\displaystyle A= \bigoplus_{i\in I}^{c_0} K(H_i),$ where the $H_i$’s are complex Hilbert spaces [(]{}cf. [@Alex Theorem 8.2][)]{}. Obviously, $A$ is a weakly compact JB$^*$-triple, and hence every convex body $K\subset A$ satisfies the strong Mankiewicz property.
We have already developed enough tools to address the question whether every weakly compact JB$^*$-triple satisfies the Mazur–Ulam property.
\[t weakly compact JB\*-triples satisfy MUP\] Every weakly compact JB$^*$-triple $E$ satisfies the Mazur–Ulam property, that is, for every real Banach space $Y$, every surjective isometry from $ S(E)$ onto $S(Y)$ admits a (unique) extension to a surjective real linear isometry from $E$ onto $Y$.
We begin with an observation. For each non-zero tripotent $u\in E,$ we consider the proper face $F_u = u + \mathcal{B}_{E_0(u)}$. Lemma \[l intersection faces MoriOzawa L8\] assures that $\Delta(F_u)$ is an intersection face of $S(Y)$, in particular a convex subset. Let us observe that $E_0(u)$ is a weakly compact JB$^*$-triple. Corollary \[c weakly compact JB\* triples satisfy the SMP\] implies that $\mathcal{B}_{E_0(u)}$ satisfies the strong Mankiewicz property. We consider the next commutative diagram $$\begin{tikzcd} F_u =u + \mathcal{B}_{E_0(u)} \arrow[swap]{d}{\tau_{-u}} \arrow{r}{\Delta} & \Delta\left( u + \mathcal{B}_{E_0(u)} \right) \\
\mathcal{B}_{E_0(u)} \arrow[dashrightarrow]{r}{\Delta_{u}} & \Delta\left( u + \mathcal{B}_{E_0(u)} \right)-\Delta(u) \arrow[u, "\tau_{\Delta(u)}"]
\end{tikzcd}$$ Since $E_0(u)$ satisfies the strong Mankiewicz property and $\Delta_{u}$ is a surjective isometry from $\mathcal{B}_{E_0(u)} $ onto a convex set, we can guarantee the existence of a linear isometry $T_u: E_0(u) \to Y$ satisfying $$\label{eq definition of Tu i nlast theorem} \Delta (u + x) = T_u (x) + \Delta (u), \hbox{ for all } x\in \mathcal{B}_{E_0(u)},$$ (cf. [@MoriOza2018]). In particular $\Delta|_{F_u}$ is affine.
We claim that $$\label{eq main property of Tu in last theorem} \Delta (w) = T_u (w), \hbox{ for every non-zero tripotents } u,w \in E \hbox{ with } w\in {E_0(u)} .$$ Namely, for each non-zero tripotent $w\in {E_0(u)}$ ($u\perp w$) the element $u \pm w$ is a tripotent in $F_u$. Therefore $$\Delta (u) = \Delta\left(\frac12 (u+w) + \frac12 (u-w) \right) = \frac12 \Delta\left( u+w\right) + \frac12 \Delta\left(u-w \right).$$ By Lemma \[c Tingley antipodal thm for tripotents\] (se also [@KalPe2019 Corollary 2.4]) we have $- \Delta\left(u-w \right) = \Delta\left(- u+w) \right)$, and by similar arguments to those given above, but now with respect to the face $F_w$, we have $$\Delta (w) = \Delta\left(\frac12 (u+w) + \frac12 (-u+w) \right) = \frac12 \Delta\left( u+w\right) + \frac12 \Delta\left(-u+w \right)$$ $$= \frac12 \Delta\left( u+w\right) - \frac12 \Delta\left(u-w \right).$$ It then follows that $$\label{eq Delta on the sum of orthogonal tripotents last theorem} \Delta (u) + \Delta (w) = \Delta (u + w).$$ The claim in is a straight consequence of and .
We continue with our argument. If $E$ is an elementary JB$^*$-triple the result follows from Theorem \[t elementary JBstar triples satisfy the MUP\]. We can therefore assume that $E$ decomposes as the orthogonal sum of two non-zero weakly compact JB$^*$-triples $A$ and $B$. Let us pick two non-zero tripotents $u_1\in A$ and $u_2\in B$ and the corresponding linear isometries $T_{u_j}: E_0(u_j) \to Y$ given by .
Let us observe that $A\subseteq E_0(u_2),$ $B\subseteq E_0(e_1)$ and $E= A\oplus^{\ell_{\infty}} B$. Therefore the mapping $T: E\to Y$, $T(a+b) = T_{u_2} (a) + T_{u_1} (b)$ is a well-defined linear operator. We shall next show that $$\label{eq T coincides with Delta final thm}
T(x) = \Delta(x), \hbox{ for all } x\in S(E).$$
Let us fix $x\in S(E)$. By Remark 4.6 in [@BuChu] there exists a possible finite at most countable family $\{e_n\}$ of mutually orthogonal minimal tripotents in $E$ and $(\lambda_n)\subseteq \mathbb{R}^+$ such that $\lambda_1 =1$ and $\displaystyle x = \sum_{n\geq 1} \lambda_n e_n$. Each $e_n$ lies in $A$ or in $B$, and hence it follows from the definition of $T$ that $T(e_n) = T_{u_2} (e_n)$ if $e_n\in A$ and $T(e_n) = T_{u_1} (e_n)$ if $e_n\in B.$ We deduce from that in any case we have $\Delta (e_n) = T (e_n)$ for all $n\geq 1$. Now we regard $x$ as an element in the face $F_{e_1}$ and we apply the properties of $T_{e_1}$ and to deduce that $$\Delta (x) = \Delta \left(e_1 + \sum_{n\geq 2} \lambda_n e_n\right) = \Delta(e_1) + \sum_{n\geq 2} \lambda_n T_{e_1} (e_n) = \Delta(e_1) + \sum_{n\geq 2} \lambda_n \Delta (e_n)$$ $$= T(e_1) + \sum_{n\geq 2} \lambda_n T (e_n) = T \left(e_1 + \sum_{n\geq 2} \lambda_n e_n\right) = T(x),$$ witnessing the validity of .
Finally, since $T$ is linear we derive from and the hypothesis on $\Delta$ that $T$ is a surjective real linear isometry.
We have previously mentioned that compact C$^*$-algebras are examples of weakly compact JB$^*$-triples. The next corollary perhaps deserves its own place.
\[c compact C\*-algebras satisfy MUP final\] Suppose $\displaystyle A = \bigoplus_{i\in I}^{c_0} K(H_i)$ is a compact C$^*$-algebra, where each $H_i$ is a complex Hilbert space. Then every surjective isometry from the unit sphere of $A$ onto the unit sphere of any other real Banach space $Y$ admits a (unique) extension to a surjective real linear isometry from $A$ onto $Y$.
**Acknowledgements** Author partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, Programa Operativo FEDER 2014-2020 and Consejer[í]{}a de Econom[í]{}a y Conocimiento de la Junta de Andaluc[í]{}a grant number A-FQM-242-UGR18, and Junta de Andalucía grant FQM375.
[0]{}
J. C. Alexander, Compact Banach algebras, *Proc. London Math. Soc.* (3) **18**, 1–18 (1968).
J.T. Barton, R.M. Timoney, Weak$^*$-continuity of Jordan triple products and applications, *Math. Scand.* **59**, 177–191 (1986).
J. Becerra Guerrero, G. L[ó]{}pez P[é]{}rez, A. M. Peralta, A. Rodr[í]{}guez-Palacios, A. Relatively weakly open sets in closed balls of Banach spaces, and real ${\rm JB}^*$-triples of finite rank, *Math. Ann.* **330**, no. 1, 45–58 (2004).
J. Becerra Guerrero, M. Cueto-Avellaneda, F.J. Fern[á]{}ndez-Polo, A.M. Peralta, On the extension of isometries between the unit spheres of a JBW$^*$-triple and a Banach space, to appear in *J. Inst. Math. Jussieu*. doi:10.1017/S1474748019000173 arXiv: 1808.01460
R.B. Braun, W. Kaup, H. Upmeier, A holomorphic characterization of Jordan-C$^*$-algebras, *Math. Z.* **161**, 277–290 (1978).
L.J. Bunce, C.-H. Chu, Compact operations, multipliers and Radon-Nikodym property in $JB^*$-triples, [*Pacific J. Math.*]{} **153**, 249–265 (1992).
M. Burgos, F.J. Fern[á]{}ndez-Polo, J. Garc[é]{}s, J. Mart[í]{}nez, A.M. Peralta, Orthogonality preservers in C$^*$-algebras, JB$^*$-algebras and JB$^*$-triples, *J. Math. Anal. Appl.* **348**, 220–233 (2008).
M. Burgos, J. Garc[é]{}s, A.M. Peralta, Automatic continuity of biorthogonality preservers between weakly compact JB$^*$-triples and atomic JBW$^*$-triples, *Studia Math.* **204**, no. 2, 97–121 (2011).
J. Cabello-S[á]{}nchez, A reflection on Tingley’s problem and some applications, *J. Math. Anal. Appl.* **476** (2), 319–336 (2019).
L. Cheng, Y. Dong, On a generalized Mazur–Ulam question: extension of isometries between unit spheres of Banach spaces, *J. Math. Anal. Appl.* **377**, 464–470 (2011).
M. Cueto-Avellaneda, A.M. Peralta, The Mazur–Ulam property for commutative von Neumann algebras, *Linear and Multilinear Algebra* **68**, No. 2, 337-362 (2020). M. Cueto-Avellaneda, A.M. Peralta, On the Mazur–Ulam property for the space of Hilbert-space-valued continuous functions, *J. Math. Anal. Appl.* **479**, no. 1, 875–902 (2019). S. Dineen, The second dual of a JB$^*$-triple system, In: Complex analysis, functional analysis and approximation theory (ed. by J. Múgica), 67-69, (North-Holland Math. Stud. 125), North-Holland, Amsterdam-New York, (1986).
C.M. Edwards, F.J. Fern[á]{}ndez-Polo, C.S. Hoskin, A.M. Peralta, On the facial structure of the unit ball in a JB$^*$-triple, *J. Reine Angew. Math.* **641**, 123–144 (2010).
C.M. Edwards, G.T. Rüttimann, A characterization of inner ideals in JB$^*$-triples, *Proc. Amer. Math. Soc.* **116**, no. 4, 1049–1057 (1992).
C.M. Edwards, G.T. Rüttimann, Compact tripotents in bi-dual JB$^*$-triples, *Math. Proc. Camb. Phil. Soc.* **120**, 155–173 (1996).
X.N. Fang, J.H. Wang, Extension of isometries between the unit spheres of normed space $E$ and $C(\Omega)$, *Acta Math. Sinica* (Engl. Ser.), **22**, 1819–1824 (2006).
F.J. Fernández-Polo, J.J. Garc[é]{}s, A.M. Peralta, I. Villanueva, Tingley’s problem for spaces of trace class operators, *Linear Algebra Appl.* **529**, 294–323 (2017).
F.J. Fern[á]{}ndez-Polo, E. Jord[á]{}, A.M. Peralta, Tingley’s problem for $p$-Schatten von Neumann classes, to appear in *J. Spectr. Theory*. arXiv:1803.00763
F.J. Fern[á]{}ndez-Polo, J. Mart[í]{}nez, A.M. Peralta, Contractive perturbations in JB$^*$-triples, *J. London Math. Soc.* (2) **85** 349–364 (2012).
F.J. Fernández-Polo, A.M. Peralta, Closed tripotents and weak compactness in the dual space of a JB$^*$-triple, *J. London Math. Soc.* **74**, 75–92 (2006).
F.J. Fernández-Polo, A.M. Peralta, Tingley’s problem through the facial structure of an atomic JBW$^*$-triple, *J. Math. Anal. Appl.* **455**, 750–760 (2017).
F.J. Fernández-Polo, A.M. Peralta, On the extension of isometries between the unit spheres of a C$^*$-algebra and $B(H)$, *Trans. Amer. Math. Soc.* **5**, 63–80 (2018). F.J. Fernández-Polo, A.M. Peralta, On the extension of isometries between the unit spheres of von Neumann algebras, *J. Math. Anal. Appl.* **466**, 127-143 (2018). F.J. Fernández-Polo, A.M. Peralta, Low rank compact operators and Tingley’s problem, *Adv. Math.* **338**, 1–40 (2018).
Y. [Friedman]{}, B. [Russo]{}. , **356**, 67–89, (1985).
L.A. Harris, [Bounded symmetric homogeneous domains in infinite dimensional spaces]{}. In: *Proceedings on infinite dimensional Holomorphy (Kentucky 1973)*; pp. 13–40. Lecture Notes in Math. 364. Berlin-Heidelberg-New York: Springer 1974.
A. Jim[é]{}nez-Vargas, A. Morales-Campoy, A.M. Peralta, M.I. Ram[í]{}rez, The Mazur–Ulam property for the space of complex null sequences, *Linear and Multilinear Algebra* **67**, no. 4, 799–816 (2019).
O.F.K. Kalenda, A.M. Peralta, Extension of isometries from the unit sphere of a rank-2 Cartan factor, preprint 2019. arXiv:1907.00575
W. Kaup, A Riemann Mapping Theorem for bounded symmentric domains in complex Banach spaces, *Math. Z.* **183**, 503–529 (1983).
W. Kaup, On real Cartan factors, *Manuscripta Math.* **92**, 191–222 (1997).
O. Loos, *Bounded symmetric domains and Jordan pairs*, Math. Lectures, University of California, Irvine 1977.
P. Mankiewicz, On extension of isometries in normed linear spaces, *Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys.* **20**, 367–371 (1972).
M. Mori, Tingley’s problem through the facial structure of operator algebras, *J. Math. Anal. Appl.* **466**, no. 2, 1281–1298 (2018).
M. Mori, N. Ozawa, Mankiewicz’s theorem and the Mazur–Ulam property for C$^*$-algebras, *Studia Math.* **250**, no. 3, 265–281 (2020).
A.M. Peralta, A survey on Tingley’s problem for operator algebras, *Acta Sci. Math. (Szeged)* **84**, 81–123 (2018).
A.M. Peralta, Extending surjective isometries defined on the unit sphere of $\ell_\infty(\Gamma)$, *Rev. Mat. Complut.* **32**, no. 1, 99–114 (2019). A.M. Peralta, R. Tanaka, A solution to Tingley’s problem for isometries between the unit spheres of compact C$^*$-algebras and JB$^*$-triples, *Sci. China Math.* **62**, no. 3, 553–568 (2019).
R. Tanaka, A further property of spherical isometries, *Bull. Aust. Math. Soc.* **90**, 304–310 (2014).
R. Tanaka, The solution of Tingley’s problem for the operator norm unit sphere of complex $n \times n$ matrices, *Linear Algebra Appl.* **494**, 274-285 (2016).
R. Tanaka, Tingley’s problem on finite von Neumann algebras, *J. Math. Anal. Appl.*, **451**, 319–326 (2017).
D. Tingley, Isometries of the unit sphere, *Geom. Dedicata* **22**, 371–378 (1987).
R. Wang, X. Huang, The Mazur–Ulam property for two-dimensional somewhere-flat spaces, *Linear Algebra Appl.* **562**, 55–62 (2019).
X. Yang, X. Zhao, On the extension problems of isometric and nonexpansive mappings. In: *Mathematics without boundaries*. Edited by Themistocles M. Rassias and Panos M. Pardalos. 725-748, Springer, New York, 2014.
K. Ylinen, *Compact and finite-dimensional elements of normed algebras*, Ann. Acad. Sci. Fenn. Ser. A I, No. **428**, 1968.
|
---
author:
- 'V. B. Bezerra'
- 'G. L. Klimchitskaya'
- 'V. M. Mostepanenko'
- 'C. Romero'
date: 'Received: date / Revised version: date'
title: 'Constraining axion-nucleon coupling constants from measurements of effective Casimir pressure by means of micromachined oscillator'
---
Introduction
============
Starting from the prediction of axions in 1978 [@1; @2], axion physics has become a wide subject stimulating development of elementary particle theory, gravitation and cosmology (see [@3; @4] for a review). Axions are pseudoscalar particles which appear as a consequence of breaking the Peccei and Quinn symmetry [@5]. They provide an elegant solution for the problem of strong [*CP*]{} violation and large electric dipole moment for the neutron in QCD. Since the proper QCD axions were constrained to a narrow band in parameter space [@5a], a lot of invisible axion-like particles have been proposed in different unification schemes. Among others, the models of the [*hadronic*]{} (KSVZ) [@6; @7] and the GUT (DFSZ) [@8; @9] axions, which can be used to solve the problem of strong [*CP*]{} violation in QCD, have attracted particular attention (see, for instanse, a number of variants of the model of hadronic axion containing the relationship between the axion-nucleon coupling constant and the Peccei-Quinn symmetry breaking scale [@9a; @9b]). At the moment axion-like particles with masses $m_a$ from approximately $10^{-5}\,$eV to $10^{-2}\,$eV and of about 1MeV are not excluded by astrophysical constraints [@4; @10]. Keeping in mind that the latter may be more model-dependent than the laboratory constraints [@11; @12], it seems warranted to look for some alternative phenomena which could be used for constraining axion-like particles of any mass. Additional interest in this subject is due to the role of axions as possible constituents of dark matter [@12a; @12b].
Previously constraints on axion-nucleon coupling constants have been obtained [@13; @14; @15] from the laboratory experiments of Eötvos [@15; @16] and Cavendish [@17; @21a] type. At first, this analysis was performed for massless axions but later it was generalized [@18] for the case of massive ones. The resulting constraints were found in the range of axion masses from approximately $10^{-9}\,$eV to $10^{-5}\,$eV. In [@19] constraints on the axion-nucleon coupling constants were obtained from measurements of the thermal Casimir-Polder force between a Bose-Einstein condensate of ${}^{87}$Rb atoms and a SiO${}_2$ plate [@48]. These constraints refer to larger axion masses from $10^{-4}\,$eV to 0.3eV. In fact, the effective potential arising between two fermions from the exchange of a pseudoscalar axion-like particle is spin-dependent [@18]. Taking into account that the test bodies in the experiments [@15; @16; @17; @21a; @48] are unpolarized, the aditional force constrained in [@18; @19] comes from the two-axion exchange.
Using the same approach, in [@20] stronger constraints on axion-nucleon coupling constants over the wide range of axion masses from $3\times 10^{-5}\,$eV to 1eV were obtained from measurements of the Casimir force gradient between a sphere and a plate coated with nonmagnetic and magnetic metals performed by means of dynamic atomic force microscope [@21; @22; @23; @24; @25]. The strengthening up to a factor of 170, as compared to the constraints of [@19], was achieved. This demonstrates that various experiments on measuring the Casimir interaction [@26] are promising for further constraining the parameters of an axion. In the past, these experiments were successfully used to obtain stronger constraints on the Yukawa-type corrections to Newtonian gravity due to exchange of light scalar particles [@27] and from extra-dimensional physics with low-energy compactification scale [@28] (see review [@29] and the most recent results [@30; @31; @32; @33; @34; @35]).
In this paper, we obtain stronger constraints on the pseudoscalar coupling constants of axion-like particles to a proton and a neutron from measurements of the effective Casimir pressure by means of micromechanical torsional oscillator [@36; @37]. For this purpose, we calculate the additional effective pressure in the configuration of two parallel plates arising due to two-axion exchange between a sphere and a plate (note that the experimental configuration [@36; @37] involves a sphere oscillating in the perpendicular direction to the plate, so that the effective pressure arises in the proximity force approximation [@26; @29]). The stronger limits on axion-nucleon coupling constants are obtained over the range of axion masses from $10^{-3}\,$eV to 15eV. The strengthening by factors from 2.2 to 3.2 in comparison with the limits of [@20] is achieved over the range of axion masses from $10^{-3}\,$eV to 1eV, respectively. As compared to the limits of [@19], the obtained constraints are stronger up to a factor of 380. Our model-independent constraints are applicable on equal terms to axions and axion-like particles. Because of this, below both terms are used synonymously. All equations are written in the system of units with $\hbar=c=1$.
Pressure between two metallic plates due to two-axion exchange
==============================================================
In the experiment [@36; @37], the effective Casimir pressure between two Au plates was determined from dynamic measurements using a micromechanical torsional oscillator. The oscillator consisted of a heavily doped polysilicon plate of area $500\times 500\,\mu\mbox{m}^2$ and thickness $D=5\,\mu$m suspended at two opposite points above the platform at the height of about $2\,\mu$m. Two independent electrodes located on the platform under the plate were used to measure the capacitance between the electrodes and the plate. They were also used to induce oscillation in the plate at the resonance frequency of the micromachined oscillator. A large sapphire sphere coated with layers of Cr and Au was attached to the optical fiber above the oscillator. The sphere radius was measured to be $R=151.3\,\mu$m. A silicon plate below the sphere was also coated with layers of Cr and Au.
In the dynamic measurements, the vertical separation between the sphere and the plate was varied harmonically with the resonance frequency of oscillator, $\omega_r$, in the presence of the sphere. The Casimir force between the sphere and the plate caused the difference between $\omega_r$ and the natural frequency of the oscillator $\omega_0$. This difference has been measured and recalculated into the gradient of the Casimir force acting between the sphere and the plate, $F_{sp}^{\prime}(a)$, using the solution for the linear oscillator motion ($a$ is the absolute sphere-plate separation). According to the proximity force approximation (PFA) [@26; @29], $$F_{sp}(a)=2\pi RE(a),
\label{eq1}$$ where $E(a)$ is the Casimir energy per unit area of two parallel plates (semispaces). Calculating the negative derivative of both sides of (\[eq1\]), one obtains the effective Casimir pressure between two parallel plates $$P(a)=-\frac{1}{2\pi R} F_{sp}^{\prime}(a),
\label{eq2}$$ which is the physical quantity indirectly measured in [@36; @37]. Note that under the condition $a\ll R$ the relative error in the gradient of the Casimir force computed using (\[eq1\]) does not exceed $(0.3-0.4)a/R$ [@38; @38a; @39; @40; @41]. Taking into account that below we consider separations $a<300\,$nm, this is of less than 0.1% error.
Now we calculate the additional pressure between two parallel semispaces separated with a gap $a$ due to two-axion exchange between nucleons. In this section, we consider homogeneous semispaces and postpone the account of finite thickness of the plate and layer structure of both test bodies to Secs. 3 and 4. First we perform a direct derivation of the additional pressure $P_{\rm add}(a)$ by summing up the energies of pair nucleon-nucleon interactions over the two semispaces and calculating the negative derivative of the obtained result. This pressure can be considered as an addition to the indirectly measured Casimir pressure (\[eq2\]) if the additional force between a sphere and a plate due to two-axion exchange is related to the additional energy per unit area of two parallel plates by the PFA, so that $$\begin{aligned}
&&
F_{sp,{\rm add}}(a)=2\pi RE_{\rm add}(a),
\label{eq3}\\
&&
P_{\rm add}(a)=-\frac{1}{2\pi R} F_{sp,{\rm add}}^{\prime}(a).
\nonumber\end{aligned}$$ Then we determine the application region of (\[eq3\]) from the comparison with the exact result for $F_{sp,{\rm add}}^{\prime}(a)$.
Let the coordinate plane $x,\,y$ coincide with the boundary plane of the lower semispace and let the $z$ axis be perpendicular to it. The effective potential due to two-axion exchange between two nucleons (protons or neutrons) situated at the points $\mbox{\boldmath$r$}_1$ and $\mbox{\boldmath$r$}_2$ of the upper and lower semispaces, respectively, is given by [@18; @42; @43] $$V_{kl}(|\mbox{\boldmath$r$}_1-\mbox{\boldmath$r$}_2|)=
-\frac{g_{ak}^2g_{al}^2\,m_a}{32\pi^3m^2}\,
\frac{K_1(2m_a|\mbox{\boldmath$r$}_1-
\mbox{\boldmath$r$}_2|)}{(\mbox{\boldmath$r$}_1-\mbox{\boldmath$r$}_2)^2}
.
\label{eq4}$$ Here, $g_{ak}$ and $g_{al}$ are the coupling constants of an axion to a proton ($k,\,l=p$) or a neutron ($k,\,l=n$) interaction, $m=(m_n+m_p)/2$ is the mean of the neutron and proton masses, and $K_1(z)$ is the modified Bessel function of the second kind. Equation (\[eq4\]) was derived under the condition $|\mbox{\boldmath$r$}_1-\mbox{\boldmath$r$}_2|\gg 1/m$. Taking into acount that in the experiment [@36; @37] we have $a>160\,$nm, this condition is satisfied with large safety margin.
The additional energy per unit area of the two semispaces due to two-axion exchange can be written as $$\begin{aligned}
&&
E_{\rm add}(a)=2\pi\sum_{k,l}n_{k,1}n_{l,2}
\int_{a}^{\infty}\!\!\!dz_1
\int_{-\infty}^{0}\!\!\!dz_2
\int_{0}^{\infty}\!\!\!\rho d\rho
\nonumber \\
&&~~~~~~~~~~~~~~~~~~~~
\times
V_{kl}(\sqrt{\rho^2+(z_1-z_2)^2}),
\label{eq5}\end{aligned}$$ where $V_{kl}$ is defined in (\[eq4\]) and $$n_{p,i}=\frac{\rho_i}{m_{\rm H}}\,\frac{Z_i}{\mu_i},
\qquad
n_{n,i}=\frac{\rho_i}{m_{\rm H}}\,\frac{N_i}{\mu_i}.
\label{eq5a}$$ Here $i=1,\,2$ numerates semispaces, $\rho_{1,2}$ are the respective densities, $Z_{1,2}$ and $N_{1,2}$ are the numbers of protons and the mean number of neutrons in the atoms (molecules) of respective semispaces. The quantities $\mu_{1,2}$ are given by $\mu_{1,2}=m_{1,2}/m_{\rm H}$, where $m_{1,2}$ and $m_{\rm H}$ are the mean masses of the atoms (molecules) of the semispaces and the mass of the atomic hydrogen, respectively. The values of $Z/\mu$ and $N/\mu$ for the first 92 elements of the Periodic Table with account of their isotopic composition can be found in [@27].
Calculating the negative derivative of (\[eq5\]) with respect to $a$, one obtains the additional pressure between two semispaces $$P_{\rm add}(a)=-\frac{m_a}{m^2m_{\rm H}^2}C_1C_2
\frac{\partial}{\partial a}
\int_{a}^{\infty}\!\!\!dz_1\,I(z_1),
\label{eq6}$$ where $$I(z_1)\equiv\int_{-\infty}^{0}\!\!\!dz_2
\int_{0}^{\infty}\!\!\!\rho d\rho
\frac{{ K}_1(2m_a\sqrt{\rho^2+(z_1-z_2)^2})}{\rho^2+(z_1-z_2)^2}.
\label{eq7}$$ Here, the coefficients $C_{1,2}$ for the materials of the semispaces are defined as $$C_{1,2}=\rho_{1,2}\left(\frac{g_{ap}^2}{4\pi}\,
\frac{Z_{1,2}}{\mu_{1,2}}+\frac{g_{an}^2}{4\pi}\,
\frac{N_{1,2}}{\mu_{1,2}}\right).
\label{eq8}$$
Using the integral representation [@44] $$\frac{K_1(z)}{z}=\int_{1}^{\infty}\!\!\!du\sqrt{u^2-1}e^{-zu}
\label{eq9}$$ and introducing the new variable $v=\sqrt{\rho^2+(z_1-z_2)^2}$, one can rearrange (\[eq7\]) into the form $$I(z_1)=2m_a\int_{-\infty}^{0}\!\!\!dz_2
\int_{z_1-z_2}^{\infty}\!\!\!dv
\int_{1}^{\infty}\!\!\!du\sqrt{u^2-1}e^{-2m_auv}.
\label{eq10}$$ By integrating here with respect to $v$ and $z_2$, we arrive at $$I(z_1)=\frac{1}{2m_a}
\int_{1}^{\infty}\!\!\!du\frac{\sqrt{u^2-1}}{u^2}e^{-2m_az_1u}.
\label{eq11}$$ Substituting this in (\[eq6\]) and differentiating with respect to $a$, we finally obtain $$\begin{aligned}
P_{\rm add}(a)&=&
-\frac{1}{2\pi R}F_{sp,{\rm add}}^{\prime}(a)
\nonumber \\
&=&-\frac{C_1C_2}{2m^2m_{\rm H}^2}
\int_{1}^{\infty}\!\!\!du\frac{\sqrt{u^2-1}}{u^2}e^{-2m_aau}.
\label{eq12}\end{aligned}$$
Now we determine the application region of (\[eq12\]) in the experimental configuration of [@36; @37] which involves not the two parallel plates, but a sphere above a plate. By summing the potential (\[eq4\]) over the volumes of a sphere and a semispace it was shown [@20] that $$\begin{aligned}
&&
-\frac{1}{2\pi R}F_{sp,{\rm add}}^{\prime}(a)=-\frac{C_sC_p}{2Rm^2m_{\rm H}^2}
\int_{1}^{\infty}\!\!\!du\frac{\sqrt{u^2-1}}{u^2}
\nonumber \\
&&~~~~~~~~~~~~~~~~~
\times
e^{-2m_aau}
\Phi(R,m_au),
\label{eq13}\end{aligned}$$ where the function $\Phi(r,z)$ is defined as $$\Phi(r,z)=r-\frac{1}{2z}+e^{-2rz}\left(
r+\frac{1}{2z}\right)
\label{eq14}$$ and $C_s$ and $C_p$ are the constants for the sphere and plate materials as defined in (\[eq8\]). From (\[eq14\]) we can see that $$\begin{aligned}
&&
~~~~~~~~\frac{\Phi(R,m_au)}{R}=1-\frac{1}{2Rm_au}
\nonumber \\
&&~~~~~~~~~~~~~~~~~~
+e^{-2Rm_au}\left(
1+\frac{1}{2Rm_au}\right).
\label{eq15}\end{aligned}$$ Thus, (\[eq13\]) leads to approximately the same results as (\[eq12\]) under the condition $Rm_a\gg 1$. Numerical computations show that (\[eq12\]) and (\[eq13\]) deviate less than approximately 1% under the condition $Rm_a> 10$. Because of this, for the experimental parameters of [@36; @37], (\[eq12\]) can be used for axion masses $m_a>10^{-2}\,$eV. For smaller masses, calculations of forces due to two-axion exchange using the PFA become not sufficiently exact. In this case one should compute the additional effective pressure using (\[eq13\]). Note that similar results concerning the application region of the PFA to Yukawa-type forces are obtained in [@45; @46].
Estimation of boundary effects
==============================
Here, we consider the sphere above the plate of finite thicknes and finite area and estimate errors in the additional force gradient arising from treating this plate as infinitely large. For convenience in calculations, we replace the square of the area $500\times 500\,\mu\mbox{m}^2$ by the disc of radius $L=250\,\mu$m. The replacement of a square by a disc of smaller area may only increase the boundary effects which, as we show below, are sufficiently small.
By summing the potential (\[eq4\]) over the volumes of a sphere and a plate (disc) of thickness $D$ and radius $L$, an additional contribution due to the two-axion exchange to the quantity measured in [@36; @37] can be presented in the form [@20] $$\begin{aligned}
&&
-\frac{1}{2\pi R}F_{\rm add}^{\prime}(a)=
-\frac{m_aC_sC_p}{2Rm^2m_{\rm H}^2}
\label{eq16} \\
&&~~
\times
\frac{\partial}{\partial a}
\int_{a}^{2R+a}\!\!\!dz_1\left[R^2-(R+a-z_1)^2\right]
G(z_1,m_a),
\nonumber\end{aligned}$$ where $$\begin{aligned}
&&
G(z_1,m_a)\equiv\frac{\partial}{\partial z_1}
\int_{-D}^{0}\!\!\!\!dz_2
\int_{0}^{L}\!\!\!\!\rho d\rho
\nonumber \\
&&~~~~~~~~~~~~~~~~
\times\frac{{K}_1(2m_a\sqrt{\rho^2+(z_1-z_2)^2})}{\rho^2+(z_1-z_2)^2}.
\label{eq17}\end{aligned}$$ Using (\[eq9\]), introducing the variable $v$ defined above and integrating with respect to it, we obtain $$\begin{aligned}
&&
G(z_1,m_a)=\int_{1}^{\infty}\!\!du\frac{\sqrt{u^2-1}}{u}
\frac{\partial}{\partial z_1}\int_{-D}^{0}\!\!\!dz_2
\nonumber \\
&&~~
\times\left[e^{-2m_au(z_1-z_2)}\right.
\left.-e^{-2m_au\sqrt{L^2+(z_1-z_2)^2}}\,\right].
\label{eq18}\end{aligned}$$ After integrating and differentiating in (\[eq18\]) over $z_2$ and $z_1$, respectively, we get $$\begin{aligned}
&&
G(z_1,m_a)=\int_{1}^{\infty}\!\!du\frac{\sqrt{u^2-1}}{u}
\left[e^{-2m_aau}\left(1-e^{-2m_aDu}\right)\right.
\nonumber \\
&&~~~~~~~~
\left.-e^{-2m_au\sqrt{L^2+z_1^2}}+
e^{-2m_au\sqrt{L^2+(z_1+D)^2}}\,\right].
\label{eq19}\end{aligned}$$
Now we substitute (\[eq19\]) in (\[eq16\]). In doing so, we integrate only the first term on the right-hand side of (\[eq19\]) with respect to $z_1$ and perform the differentiation with respect to $a$. The result is $$\begin{aligned}
&&
-\frac{1}{2\pi R}F_{\rm add}^{\prime}(a)=
-\frac{C_sC_p}{2Rm^2m_{\rm H}^2}
\int_{1}^{\infty}\!\!du\frac{\sqrt{u^2-1}}{u^2}
\nonumber \\
&&~
\times\left[e^{-2m_aau}
\left(1-e^{-2m_aDu}\right)\Phi(R,m_au)
\right.
\nonumber \\
&&~~~~~~~~~~~~~~~~~~
\left.\vphantom{\left(e^{-2m_aDu}\right)}
-
Y(m_au,L,D)\right],
\label{eq20}\end{aligned}$$ where the function $\Phi(r,z)$ is defined in (\[eq15\]) and the following notation is introduced $$\begin{aligned}
&&
Y(m_au,L,D)\equiv 2m_au\int_{a}^{2R+a}\!\!dz_1
(R+a-z_1)
\nonumber \\
&&~~
\times
\left[e^{-2m_au\sqrt{L^2+z_1^2}}-
e^{-2m_au\sqrt{L^2+(z_1+D)^2}}\,
\right].
\label{eq21}\end{aligned}$$
From the comparison of the right-hand sides of (\[eq20\]) and (\[eq13\]), it is seen that the first term of (\[eq20\]) generalizes (\[eq13\]) for the case of a plate of finite thickness $D$. In the limiting case $D\to\infty$ the first term of (\[eq20\]) coincides with (\[eq13\]). The second term on the right-hand side of (\[eq20\]) takes into account the boundary effects.
Now we estimate the relative role of boundary effects in the calculation of the additional force gradient due to two-axion exchange using the experimental parameters of [@36; @37]. Taking into account that the quantity in square brackets on the right-hand side of (\[eq21\]) is positive, one can only increase the integral by omitting the part of the integration domain where the quantity in the round brackets is negative. This results in the inequality $$\begin{aligned}
&&
Y(m_au,L,D)< 2m_au\int_{a}^{R+a}\!\!dz_1
(R+a-z_1)
\nonumber \\
&&~~
\times
\left[e^{-2m_au\sqrt{L^2+z_1^2}}-
e^{-2m_au\sqrt{L^2+(z_1+D)^2}}\,
\right].
\label{eq22}\end{aligned}$$ The second exponent on the right-hand side of this equation under the condition $D\ll L$ can be approximated as $$\begin{aligned}
&&
e^{-2m_au\sqrt{L^2+(z_1+D)^2}}
\approx
e^{-2m_au\sqrt{L^2+z_1^2}}\,
e^{-m_au\frac{2z_1D+D^2}{\sqrt{L^2+z_1^2}}}
\nonumber \\
&&~~~~~~
\approx
e^{-2m_au\sqrt{L^2+z_1^2}}\left(1-
m_au\frac{2z_1D+D^2}{\sqrt{L^2+z_1^2}}\right),
\label{eq23}\end{aligned}$$ where the last transformation is performed for small axion masses $m_a\sim 1/R$ leading to the largest boundary effects \[the dominant contribution to the integral (\[eq20\]) is given by $u\sim 1$\]. Substituting (\[eq23\]) in (\[eq22\]), one obtains $$\begin{aligned}
&&
Y(m_au,L,D)< 2(m_au)^2\int_{a}^{R+a}\!\!dz_1
(R+a-z_1)
\nonumber \\
&&~~~~~
\times
e^{-2m_au\sqrt{L^2+z_1^2}}
\frac{2z_1D+D^2}{\sqrt{L^2+z_1^2}}
\nonumber \\
&&~~
<2(m_au)^2e^{-2m_auL}\frac{D}{L}
\nonumber \\
&&~~~~~~~~~~~~~~~
\times
\int_{a}^{R+a}\!\!\!dz_1
(R+a-z_1)(2z_1+D)
\nonumber \\
&&~~
\approx\frac{2RD}{3L}(m_aRu)^2e^{-2m_auL}.
\label{eq24}\end{aligned}$$
From (\[eq24\]) it is seen that under the conditions $m_aR\approx 1$ and $u\sim 1$ it follows: $$Y(m_au,L,D)<3\times 10^{-4}R.
\label{eq25}$$
On the same conditions, the contribution of the remaining terms in the square brackets of (\[eq20\]) is equal to $3\times 10^{-2}R$. Thus, the boundary effects contribute less than 1% under the integral (\[eq20\]). We have checked by means of numerical computations that the contribution of the boundary effects to the normalized gradient of the additional force also does not exceed 1%. Because of this, the role of additional forces due to two-axion exchange in the experiment [@36; @37] can be calculated under the assumption of the infinitely large area of the oscillator plate.
Account of layer structure of test bodies
=========================================
As was mentioned in Sec. 2, the test bodies in the experiment [@36; @37] were not homogeneous. The Si plate of an oscillator of finite thickness $D$ was coated with a Cr layer of thickness $\Delta_p^{\!\rm Cr}=10\,$nm and with an outer Au layer of thickness $\Delta_p^{\!\rm Au}=210\,$nm. The sapphire (Al${}_2$O${}_3$) sphere was coated with a Cr layer of thickness $\Delta_s^{\!\rm Cr}=10\,$nm and then with an Au layer of thickness $\Delta_s^{\!\rm Au}=180\,$nm. The densities of all these materials are presented in the second column of Table 1.
[cccc]{} Material&$\rho\,(\mbox{g/cm}^3)$&$\frac{Z}{\mu}$& $\frac{N}{\mu}$\
Au &19.28 & 0.40422 & 0.60378\
Cr& 7.15 & 0.46518 & 0.54379\
Si & 2.33 & 0.50238 & 0.50628\
Al${}_2$O${}_3$ & 4.1 & 0.49422 & 0.51412\
Now we adapt the results of Sec. 2 for the additional effective pressure due to two-axion exchange for the case of experimental layer structure of both bodies and finite thickness of the oscillator plate. We begin with (\[eq12\]), which can be used in the experimental configuration of [@36; @37] within the application region of the PFA. The layers are taken into account one by one. For instance, to account for the Au layer on the plate, we subtract from (\[eq12\]), written for two Au semispaces, the effective pressure between the same semispaces, but separated by the gap $a+\Delta_p^{\!\rm Au}$. Then we add the effective pressure for Au-Cr semispaces separated by the same gap and subtract the pressure for these semispaces separated by the gap $a+\Delta_p^{\!\rm Au}+\Delta_p^{\!\rm Cr}$ etc. Similar procedure is used to account for the layer structure of the upper plate. Finally, for the experimental configuration one obtains $$\begin{aligned}
&&
P_{\rm add}(a)=
-\frac{1}{2\pi R}F_{sp,{\rm add}}^{\prime}
=
-\frac{1}{2m^2m_{\rm H}^2}\int_{1}^{\infty}\!\!\!du
\frac{\sqrt{u^2-1}}{u^2}
\nonumber \\
&&~~~~~~~~~~
\times
e^{-2m_aau}X_p(m_au)X_s(m_au),
\label{eq26}\end{aligned}$$ where $$\begin{aligned}
&&
X_p(m_au)\equiv C_{\rm Au}\left(1-e^{-2m_au\Delta_p^{\!\rm Au}}
\right)
\nonumber \\
&&~~~
+C_{\rm Cr}e^{-2m_au\Delta_p^{\!\rm Au}}
\left(1-e^{-2m_au\Delta_p^{\!\rm Cr}}
\right)
\nonumber \\
&&~~~
+C_{\rm Si}e^{-2m_au(\Delta_p^{\!\rm Au}+\Delta_p^{\!\rm Cr})}
\left(1-e^{-2m_auD}
\right),
\nonumber \\[1mm]
&&
X_s(m_au)\equiv C_{\rm Au}\left(1-e^{-2m_au\Delta_s^{\!\rm Au}}
\right)
\nonumber \\
&&~~~
+C_{\rm Cr}e^{-2m_au\Delta_s^{\!\rm Au}}
\left(1-e^{-2m_au\Delta_s^{\!\rm Cr}}
\right)
\nonumber \\
&&~~~
+C_{\rm Al_2O_3}
e^{-2m_au(\Delta_s^{\!\rm Au}+\Delta_s^{\!\rm Cr})}.
\label{eq27}\end{aligned}$$ Here, the coefficients $C_{\rm Au}$, $C_{\rm Cr}$ and $C_{\rm Si}$ are defined in (\[eq8\]). They are calculated using the respective values for $Z/\mu$ and $N/\mu$ presented in the third and fourth columns of Table 1 [@27]. The quantities $Z/\mu$ and $N/\mu$ for Al${}_2$O${}_3$ are also given in Table 1 [@20].
As was found in Sec. 2, in the experimental configuration [@36; @37], the PFA is applicable to calculate additional forces due to two-axion exchange under the condition $m_a>10^{-2}\,$eV. For axions of smaller masses a more exact expression (\[eq13\]) should be used. It can be adapted for the experimental layer structure using the procedure described above. The result is $$\begin{aligned}
&&
-\frac{1}{2\pi R}F_{sp,{\rm add}}^{\prime}=
-\frac{1}{2m^2m_{\rm H}^2R}\int_{1}^{\infty}\!\!\!du
\frac{\sqrt{u^2-1}}{u^2}
\nonumber \\
&&~~~~~~~~~
\times
e^{-2m_aau}X_p(m_au)\tilde{X}_s(m_au),
\label{eq28}\end{aligned}$$ where the function $\tilde{X}_s$ is defined as $$\begin{aligned}
&&
\tilde{X}_s(m_au)\equiv C_{\rm Au}\left[
\vphantom{e^{-2m_au\Delta_s^{\!\rm Au}}}
\Phi(R,m_au)
\right.
\label{eq29}\\
&&~~~~~~~~~~~~
\left.
-e^{-2m_au\Delta_s^{\!\rm Au}}
\Phi(R-\Delta_s^{\!\rm Au},m_au)\right]
\nonumber \\
&&~~~
+C_{\rm Cr}e^{-2m_au\Delta_s^{\!\rm Au}}
\left[
\vphantom{e^{-2m_au\Delta_s^{\!\rm Au}}}
\Phi(R-\Delta_s^{\!\rm Au},m_au)
\right.
\nonumber \\
&&~~~~~~~~~~~~
\left.
-e^{-2m_au\Delta_s^{\!\rm Cr}}
\Phi(R-\Delta_s^{\!\rm Au}-\Delta_s^{\!\rm Cr},m_au)
\right]
\nonumber \\
&&~~~
+C_{\rm Al_2O_3}
e^{-2m_au(\Delta_s^{\!\rm Au}+\Delta_s^{\!\rm Cr})}
\Phi(R-\Delta_s^{\!\rm Au}-\Delta_s^{\!\rm Cr},m_au).
\nonumber\end{aligned}$$ Here, the functions $X_p$ and $\Phi$ are given in (\[eq27\]) and (\[eq14\]), respectively.
Constraints on axion-nucleon coupling constants
===============================================
The experimental data of [@36; @37] for the effective Casimir pressure were obtained at separations $a>160\,$nm and found to be in good agreement with the Lifshitz theory [@47] under the condition that the low-frequency behavior of the dielectric permittivity of Au is described by the plasma model (the Casimir force is entirely determined by the outer Au layers on both test bodies and, as opposed to the additional force due to two-axion exchange, is not influenced by the layers situated below). No signature of any additional interaction was observed in the limits of the total experimental error, $\Delta P(a)$, in the pressure measurements.
This means that the effective additional pressure should satisfy the following inequality: $$\left|-\frac{1}{2\pi R}F_{sp,{\rm add}}^{\prime}\right|
\leq\Delta P(a).
\label{eq30}$$ The left-hand side of this inequality is given by the magnitudes of either (\[eq26\]) (for axion masses allowing the use of the PFA) or (\[eq28\]) (for axion of smaller masses). The total experimental error in the indirectly measured pressures, $\Delta P(a)$, recalculated with the 67% confidence level for convenience in comparison with the previously obtained constraints, is equal to 0.55, 0.38, and 0.22mPa at separations $a=162$, 200, and 300nm, respectively.
We have found numerically (see Fig. 1) the values of the axion to nucleon coupling constants $g_{ap},\,g_{an}$ and masses $m_a$ satisfying the inequality (\[eq30\]). For this purpose, the expressions (\[eq26\]) and (\[eq28\]) were substituted in (\[eq30\]) over the mass intervals $10^{-2}\,\mbox{eV}<m_a<15\,$eV and $10^{-3}\,\mbox{eV}<m_a<10^{-2}\,$eV, respectively. We do not consider the axion masses $m_a<10^{-3}\,$eV because in this case the respective Compton wavelengths become too large and one cannot neglect the role of boundary effects (see Sec. 3). For $m_a>15\,$eV the constraints on $g_{ap}$ and $g_{an}$ following from this experiment become much weaker. In different intervals of $m_a$, the strongest constraints follow from the inequality (\[eq30\]) considered at different separation distances. Thus, for $m_a<0.1\,$eV the strongest constraints result at $a=300\,$nm and for $0.1\,\mbox{eV}\leq m_a<0.5\,$eV and $0.5\,\mbox{eV}\leq m_a<15\,$eV at $a=200\,$nm and 162nm, respectively.
In Fig. 1, we present the obtained strongest constraints on the constants $g_{ap(n)}^2/(4\pi)$ as functions of the axion mass $m_a$. The lines correspond to the equality sign in (\[eq30\]). In Fig. 1 the three lines from bottom to top are plotted under the conditions $g_{ap}^2=g_{an}^2$, $g_{an}^2\gg g_{ap}^2$, and $g_{ap}^2\gg g_{an}^2$, respectively, for axion masses below 2eV. The regions of the $(m_a,g_{ap(n)}^2)$ plane above each line are prohibited by the results of experiment [@36; @37], because the coordinates of their points violate inequality (\[eq30\]). The regions below each line are allowed by the results of this experiment. As can be seen in Fig. 1, for axions with masses $m_a<10^{-2}\,$eV the obtained constraints are almost independent of $m_a$. In an inset to Fig. 1 we plot the obtained constraints over a wider range of $m_a$ (up to 15eV) under the condition $g_{ap}^2=g_{an}^2$. As is seen in this figure, with increasing $m_a$ the strength of constraints quickly decreases. In Table 2, we present the maximum allowed values of the axion-nucleon coupling constants over the most interesting region of masses from $m_a=10^{-3}\,$eV to 2eV (column 1) partially overlapping with an axion window. The values in column 2 are obtained under the conditions $g_{ap}^2=g_{an}^2$, and columns 3 and 4 contain the maximum values of $g_{an}^2/(4\pi)$ and $g_{ap}^2/(4\pi)$ found under the conditions $g_{an}^2\gg g_{ap}^2$ and $g_{ap}^2\gg g_{an}^2$, respectively.
[cccc]{} $m_a\,$(eV) & $\frac{g_{ap}^2}{4\pi}=\frac{g_{an}^2}{4\pi}$ & $\frac{g_{an}^2}{4\pi}\gg\frac{g_{ap}^2}{4\pi}$ & $\frac{g_{ap}^2}{4\pi}\gg\frac{g_{an}^2}{4\pi}$\
$0.001$& $8.51\times 10^{-5}$&$1.49\times 10^{-4}$& $1.98\times 10^{-4}$\
0.01 &$8.74\times 10^{-5}$&$1.52\times 10^{-4}$& $2.04\times 10^{-4}$\
0.05 &$9.66\times 10^{-5}$&$1.67\times 10^{-4}$& $2.30\times 10^{-4}$\
0.1 &$1.08\times 10^{-4}$&$1.84\times 10^{-4}$& $2.59\times 10^{-4}$\
0.2&$1.28\times 10^{-4}$&$2.16\times 10^{-4}$& $3.11\times 10^{-4}$\
0.3&$1.48\times 10^{-4}$&$2.49\times 10^{-4}$& $3.62\times 10^{-4}$\
0.4&$1.71\times 10^{-4}$&$2.88\times 10^{-4}$& $4.21\times 10^{-4} $\
0.5&$1.96\times 10^{-4}$&$3.29\times 10^{-4}$& $4.85\times 10^{-4} $\
0.6&$2.22\times 10^{-4}$&$3.73\times 10^{-4}$& $5.50\times 10^{-4} $\
0.7&$2.51\times 10^{-4}$&$4.21\times 10^{-4}$& $6.24\times 10^{-4} $\
0.8&$2.84\times 10^{-4}$&$4.76\times 10^{-4}$& $7.06\times 10^{-4} $\
0.9&$3.21\times 10^{-4}$&$5.37\times 10^{-4}$& $7.97\times 10^{-4} $\
1.0&$3.62\times 10^{-4}$&$6.05\times 10^{-4}$& $8.99\times 10^{-4}$\
1.5&$6.48\times 10^{-4}$&$1.08\times 10^{-3}$& $1.61\times 10^{-3}$\
2.0&$1.13\times 10^{-3}$&$1.88\times 10^{-3}$& $2.81\times 10^{-3}$\
In Fig. 2(a) the constraints derived in this paper are compared with those found previously [@19; @20] from measurements of the thermal Casimir-Polder force [@48] and from experiments on measuring the gradient of the Casimir force between Au surfaces [@21; @22]. For the sake of definiteness, the comparison is made under the most reasonable condition $g_{ap}^2=g_{an}^2$. The solid line in Fig. 2(a) reproduces the lower line in Fig. 1 obtained here. The dashed lines 1 and 2 reproduce the constraints obtained [@19; @20] from measurements of the Casimir-Polder force and the gradient of the Casimir force over the regions of axion masses $m_a\leq 0.3\,$eV and $m_a\leq 1\,$eV, respectively. The regions of the plane above each line are prohibited and below each line are allowed by the results of the respective experiment.
As can be seen in Fig. 2(a), at $m_a=10^{-3}\,$eV and 1eV our present constraints are stronger by the factors of 2.2 and 3.2, respectively, than those obtained from measurements of the gradient of the Casimir force (the dashed line 2). In comparison with the constraints from measurements of the Casimir-Polder force (the dashed line 1), the present constraints are stronger up to a factor of 380. This strengthening is achieved for the axion mass $m_a=0.3\,$eV.
Now we compare the obtained here strongest model-independent constraints on the coupling constant $g_{an}$ (the lower line in Fig. 1) with other model-independent constraints obtained to the present day. The line 1 in Fig. 2(b) shows the constraints found [@53a] with the help of a magnetometer using spin-polarized K and ${}^3$He atoms. These constraints are obtained in the region of axion masses from $10^{-10}$ to $6\times 10^{-6}\,$eV. The line 2 shows the constraints found in [@18] from the Cavendish-type experiment [@17; @21a] for $m_a$ from $10^{-9}$ to $10^{-5}\,$eV. The results of a more modern Cavendish-type experiment [@53b] were used to constrain $g_{an}$ in the region from $m_a=10^{-6}\,$eV to $m_a=10^{-2}\,$eV [@53c]. These results are shown by the line 3 in Fig. 2(b). Our constraints obtained here are shown by the line 4. As is seen in Fig. 2(b), the model-independent constraints become weaker with increasing $m_a$ (the same takes place for the constraints on Yukawa-type corrections to Newtonian gravity arising from the exchange of scalar particles [@29; @30; @31; @32; @33; @34; @35]). It can be seen, however, that in the range of axion masses from $2\times 10^{-3}$ to 0.3eV our constraints following from the Casimir effect are the strongest model-independent constraints.
A lot of constraints on an axion were obtained using some model approaches. Thus, the planar Si(Li) detector placed inside the low-background setup was used to detect the $\gamma$-quanta appearing in the deexcitation of the nuclear level excited by a solar axion [@53d]. In the framework of the model of hadronic axions, where the coupling constant is a function of the mass, the upper limits for the axion mass $m_a\leq 159\,$eV [@53d] and $m_a\leq 145\,$eV [@53e] were obtained. From the neutrino data of supernova SN 1987A it was found [@53f] that for the model of hadronic axions $g_{ap(n)}<10^{-10}$ or $g_{ap(n)}>10^{-3}$ with a narrow allowed region in the vicinity of $g_{ap(n)}=10^{-6}$. From astrophysical arguments connected with stellar cooling by the emission of hadronic axions a similar bound $g_{ap(n)}<3\times 10^{-10}$ was obtained [@53g; @53h]. It should be noted, however, that the emission rate suffers from significant uncertainties related to dense nuclear matter effects [@53h]. In addition to a pseudoscalar coupling of axions to nucleons, it is possible also to introduce the scalar one [@53i] and consider respective coupling constants $g_{ap(n)}^{(s)}$. Several constraints on the product of constants $|g_{an}g_{an}^{(s)}|$ were obtained from experiments on neutron diffraction [@53j]. Thus, it was shown [@53j] that $|g_{an}g_{an}^{(s)}|<10^{-11}$ within the range of axion masses $2\times 10^{-5}\,\mbox{eV}<m_a<2\times 10^{3}\,$eV. When the axion mass increases up to $2\times 10^6\,$eV, the respective constraint becomes less stringent: $|g_{an}g_{an}^{(s)}|<10^{-7}$.
At the end of this section, we note that subsequent independent measurements of the gradient of the Casimir force in [@21; @22; @23; @24; @25] confirmed both the experimental results of [@36; @37] and their agreement with the Lifshitz theory under the condition that the low-frequency behavior of the dielectric permittivity of Au is described by the plasma model (the conclusion of [@49], claiming an agreement with the Drude model low-frequency behavior over the same range of separations was shown [@50] to be based on an unaccounted systematic error).
Conclusions and discussion
==========================
In this paper we have derived stronger constraints on the pseudoscalar coupling constants of an axion to a proton and a neutron from measurements of the effective Casimir pressure by means of a micromachined oscillator. For this purpose, we have calculated the additional pressure between two parallel plates due to two-axion exchange and determined the validity region of the PFA when it is applied to the forces of axion origin. The role of boundary effects due to a finite area of the oscillator plate was determined.
The obtained constraints are applicable over a wide region of axion masses from $10^{-3}\,$eV to 15eV, partially overlapping with an axion window. Under the assumption that $g_{ap}=g_{an}$, they are stronger up to a factor of 380 than the previously known laboratory constraints in this mass range derived from measurements of the thermal Casimir-Polder force and up to a factor of 3.15 than those found from measurements of the gradient of the Casimir force by means of AFM.
The obtained results demonstrate that measurements of the Casimir interaction using different laboratory techniques are useful in searching axion-like particles and constraining their coupling constants to nucleons. In future, it seems promising to consider the potentialities of more complicated experimental configurations, specifically, with corrugated boundary surfaces, for obtaining stronger constraints on the parameters of axion-like particles.
The authors of this work acknowledge CNPq (Brazil) for partial financial support. G.L.K. and V.M.M. are grateful to M. Yu. Khlopov for useful discussions. They also acknowledge the Department of Physics of the Federal University of Paraíba (João Pessoa, Brazil) for hospitality.
[99]{} S. Weinberg, Phys. Rev. Lett. [**40**]{}, 223 (1978). F. Wilczek, Phys. Rev. Lett. [**40**]{}, 279 (1978). J. E. Kim, G. Carosi, Rev. Mod. Phys. [**82**]{}, 557 (2010). J. Beringer [*et al.*]{} (Particle Data Group), Phys. Rev. D [**86**]{}, 010001 (2012). R. D. Peccei, H. R. Quinn, Phys. Rev. Lett. [**38**]{}, 1440 (1977). K. Baker [*et al.*]{} Ann. Phys. (Berlin) [**525**]{}, A93 (2013). J. E. Kim, Phys. Rev. Lett. [**43**]{}, 103 (1979). M. A. Shifman, A. I. Vainstein, V. I. Zakharov, Nucl. Phys. B [**166**]{}, 493 (1980). A. P. Zhitnitskii, Sov. J. Nucl. Phys. [**31**]{}, 260 (1980). M. Dine, F. Fischler, M. Srednicki, Phys. Lett. B [**104**]{}, 199 (1981). Z. G. Berezhiani, M. Yu. Khlopov, Z.Phys. C — Particles and Fields [**49**]{}, 73 (1991). M. Khlopov, [*Fundamentals of Cosmic Particle Physics*]{} (CISP-Springer, Cambridge, 2012). A. V. Derbin, S. V. Bakhlanov, I. S. Dratchnev, A. S. Kayunov, V. N. Muratova, Eur. Phys. J. C [**73**]{}, 2490 (2013). J. Jaeckel, E. Massó, J. Redondo, A. Ringwald, F. Takahashi, Phys. Rev. D [**75**]{}, 013004 (2007). P. Brax, C. van de Bruck, A.-C. Davis, Phys. Rev. Lett. [**99**]{}, 121103 (2007). J. E. Kim, Phys. Rep. [**150**]{}, 1 (1987). Yu. N. Gnedin, Int. J. Mod. Phys. A [**17**]{}, 4251 (2002). E. Fischbach, D. E. Krause, [Phys. Rev. Lett.]{} [**82**]{}, 4753 (1999). E. Fischbach, D. E. Krause, [Phys. Rev. Lett.]{} [**83**]{}, 3593 (1999). G. L. Smith, C. D. Hoyle, J. H. Gundlach, E. G. Adelberger, B. R. Heckel, H. E. Swanson, Phys. Rev. D [**61**]{}, 022001 (1999). J. H. Gundlach, G. L. Smith, E. G. Adelberger, B. R. Heckel, H. E. Swanson, Phys. Rev. Lett. [**78**]{}, 2523 (1997). R. Spero, J. K. Hoskins, R. Newman, J. Pellam, J. Schultz, Phys. Rev. Lett. [**44**]{}, 1645 (1980). J. K. Hoskins, R. D. Newman, R. Spero, J. Schulz, Phys. Rev. D [**32**]{}, 3084 (1985). E. G. Adelberger, E. Fischbach, D. E. Krause, R. D. Newman, [Phys. Rev. D]{} [**68**]{}, 062002 (2003). V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko, C. Romero, Phys. Rev. D [**89**]{}, 035010 (2014). J. M. Obrecht, R. J. Wild, M. Antezza, L. P. Pitaevskii, S. Stringari, E. A. Cornell, Phys. Rev. Lett. [**98**]{}, 063201 (2007). V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko, C. Romero, arXiv:1402.2528; Phys. Rev. D, to appear. C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, U. Mohideen, Phys. Rev. B [**85**]{}, 165443 (2012). A. A. Banishev, C.-C. Chang, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, U. Mohideen, Int. J. Mod. Phys. A [**27**]{}, 1260001 (2012). A. A. Banishev, C.-C. Chang, G. L. Klimchitskaya, V. M. Mostepanenko, U. Mohideen, Phys. Rev. B [**85**]{}, 195422 (2012). A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, U. Mohideen, Phys. Rev. Lett. [**110**]{}, 137401 (2013). A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, U. Mohideen, Phys. Rev. B [**88**]{}, 155410 (2013). G. L. Klimchitskaya, U. Mohideen, V. M. Mostepanenko, Rev. Mod. Phys. [**81**]{}, 1827 (2009). E. Fischbach, C. L. Talmadge, [*The Search for Non-Newtonian Gravity*]{} (Springer, New York, 1999). I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B [**436**]{}, 257 (1998). M. Bordag, G. L. Klimchitskaya, U. Mohideen, V. M. Mostepanenko, [*Advances in the Casimir Effect*]{} (Oxford University Press, Oxford, 2009). V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko, C. Romero, Phys. Rev. D [**81**]{}, 055003 (2010). V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko, C. Romero, Phys. Rev. D [**83**]{}, 075004 (2011). G. L. Klimchitskaya, U. Mohideen, V. M. Mostepanenko, Phys. Rev. D [**86**]{}, 065025 (2012). V. M. Mostepanenko, V. B. Bezerra, G. L. Klimchitskaya, C. Romero, Int. J. Mod. Phys. A [**27**]{}, 1260015 (2012). G. L. Klimchitskaya, U. Mohideen, V. M. Mostepanenko, Phys. Rev. D [**87**]{}, 125031 (2013). G. L. Klimchitskaya, V. M. Mostepanenko, Grav. Cosmol. [**20**]{}, 3 (2014). R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, V. M. Mostepanenko, Eur. Phys. J. C [**51**]{}, 963 (2007). R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, V. M. Mostepanenko, Phys. Rev. D [**75**]{}, 077101 (2007). C. D. Fosco, F. C. Lombardo, F. D. Mazzitelli, Phys. Rev. D [**84**]{}, 105031 (2011). L. P. Teo, M. Bordag, V. Nikolaev, Phys. Rev. D [**84**]{}, 125037 (2011). G. Bimonte, T. Emig, R. L. Jaffe, M. Kardar, Europhys. Lett. [**97**]{}, 50001 (2012). G. Bimonte, T. Emig, M. Kardar, Appl. Phys. Lett. [**100**]{}, 074110 (2012). L. P. Teo, Phys. Rev. D [**88**]{}, 045019 (2013). S. D. Drell, K. Huang, Phys. Rev. [**91**]{}, 1527 (1953). F. Ferrer, M. Nowakowski, Phys. Rev. D [**59**]{}, 075009 (1999). I. S. Gradshtein, I. M. Ryzhik, [*Table of Integrals, Series and Products*]{} (Academic Press, New York, 1980). R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. López, V. M. Mostepanenko, Phys. Rev. D [**79**]{}, 124021 (2009). E. Fischbach, G. L. Klimchitskaya, D. E. Krause, V. M. Mostepanenko, Eur. Phys. J. C [**68**]{}, 223 (2010). E. M. Lifshitz, L. P. Pitaevskii, [*Statistical Physics*]{}, Part II (Pergamon, Oxford, 1980). G. Vasilakis, J. M. Brown, T. R. Kornack, M. V. Romalis, Phys. Rev. Lett. [**103**]{}, 261801 (2009). D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle, H. E. Swanson, Phys. Rev. Lett. [**98**]{}, 021101 (2007). E. G. Adelberger, B. R. Heckel, S. Hoedl, C. D. Hoyle, D. J. Kapner, A. Upadhye, Phys. Rev. Lett. [**98**]{}, 131104 (2007). A. V. Derbin, A. L. Frolov, L. A. Mitropol’sky, V. N. Muratova, D. A. Semenov, E. V. Unzhakov, Eur. Phys. J. C [**62**]{}, 755 (2009). A. V. Derbin, V. N. Muratova, D. A. Semenov, E. V. Unzhakov, Phys. Atom. Nucl. [**74**]{}, 596 (2011). J. Engel, D. Seckel, A. C. Hayes, Phys. Rev. Lett. [**65**]{}, 960 (1990). W. C. Haxton, K. Y. Lee, Phys. Rev. Lett. [**66**]{}, 2557 (1991). G. Raffelt, Phys. Rev. D [**86**]{}, 015001 (2012). J. E. Moody, F. Wilczek, Phys. Rev. D [**30**]{}, 130 (1984). V. V. Voronin, V. V. Fedorov, I. A. Kuznetsov, JETP Lett. [**90**]{}, 5 (2009). D. Garcia-Sanches, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, H. X. Tang, Phys. Rev. Lett. [**109**]{}, 027202 (2012). M. Bordag, G. L. Klimchitskaya, V. M. Mostepanenko, Phys. Rev. Lett. [**109**]{}, 199701 (2012).
|
---
abstract: 'We probe the effective field theory extending the Standard Model with a sterile neutrino in B meson decays at B factories and lepton colliders, using angular and polarization observables. We put bounds on different effective operators characterized by their distinct Dirac-Lorentz structure, and probe the $N$-mediated B decays sensitivity to these interactions. We define a Forward-Backward asymmetry $A_{FB}^{\ell \gamma}$ between the muon and photon directions for the $B \to \mu \nu \gamma$ decay, which allows us to separate the SM contribution from the effective lepton number conserving and violating processes, mediated by a near on-shell $N$. Using the most stringent constraints on the effective parameter space from Belle and BaBar we find that a measurement of the final polarization $P_{\tau}$ in the rare $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ decays can help us infer the scalar or vector interaction content in the $N$ production or decay vertices. We find that the B meson decays are more sensitive to scalar operators.'
author:
- Lucía Duarte
- Gabriel Zapata
- 'Oscar A. Sampayo'
bibliography:
- 'Bib\_N\_5\_2020.bib'
title: 'Angular and polarization observables for Majorana-mediated B decays with effective interactions.'
---
Introduction.
=============
[\[intro\]]{}
Besides the remarkable performance of the standard model (SM) of particle physics in describing nature, neutrino oscillations are currently the most compelling experimental evidence of the need to extend the SM in order to include mechanisms for neutrino mass generation. In the recent years, the LHC experiments have put stringent constraints on the existence of new physics involving colored states, but still the possible extensions of the electroweak (EW) sector are far less restricted. A variety of new physics scenarios may be hidden at energies well above the EW scale, and their study is being consistently tackled by the use of the standard model effective field theory (SMEFT) [@Buchmuller:1985jz; @Grzadkowski:2010es].
But also new physics might involve weakly coupled fields at the EW scale. Many models leading to neutrino masses predict the existence of sterile right-handed neutrinos with Majorana masses, as the Type I [@Minkowski:1977sc; @Mohapatra:1979ia; @Yanagida:1980xy; @GellMann:1980vs; @Schechter:1980gr] and also the linear and inverse seesaw mechanisms. This possibly not-that-heavy degrees of freedom can be described by an EFT including them (SMNEFT) [@delAguila:2008ir; @Aparici:2009fh; @Liao:2016qyd; @Bhattacharya:2015vja], and here we concentrate on a simplified scenario with only one right-handed neutrino added [@delAguila:2008ir]. The phenomenology of models extending the Type I seesaw renormalizable Lagrangian with effective interactions of higher dimension for the right-handed neutrinos are being studied [@Caputo:2017pit; @Jones-Perez:2019plk], and recently complemented with the implications of the Minimal Flavor Violation ansatz on the new heavy neutrino interactions [@Graesser:2007yj; @Barducci:2020ncz].
The simplified scenario considering only one sterile state $N$ has started to get attention in connection to the novel dimension five Higgs-neutrino interactions [@Butterworth:2019iff] and the heavy neutrino magnetic moment dipole portal [@Magill:2018jla]. In particular, some studies have set constraints on the effective operators in this simplified scenario using existing LHC searches [@Alcaide:2019pnf] and recent work explores the matching between the off-shell EW-scale operator basis and a low-energy on-shell basis [@Chala:2020vqp; @Dekens:2020ttz].
On the other hand, works on approaches as the so called neutrino non standard interactions (NSI) and general neutrino interactions (GNI) are incorporating right-handed neutrinos to the SMEFT [@Bischer:2019ttk; @Han:2020pff] considering their Majorana and/or Dirac nature. Also effective neutrino long-range interactions are being studied [@Bolton:2020xsm] in an EFT approach.
The presence of a Majorana mass term for the right-handed neutrinos is the source of lepton number violation (LNV) in these models. While the phenomenology of LNV has been thoroughly studied in the past in the context of seesaw scenarios, for recent reviews see [@Drewes:2019byd; @Cai:2017mow], the study of lepton number violating and conserving (LNC) interference effects in final states with light neutrinos is usually discarded. Here we assess the chances of disentangling possible contributions from effective operators with distinct Dirac-Lorentz structure to B leptonic decays where the final neutrinos do not allow for the identification of LNV or LNC interactions.
In this article we focus on the simplified scenario with only one heavy Majorana neutrino $N$, and neglect the effect of the renormalizable Yukawa term $N L \phi$ giving the heavy-active neutrino mixings $U_{l N}$, given that it is strongly constrained not only by the naive seesaw relation $U_{lN}^2 \sim m_{\nu}/M_N \sim 10^{-14}-10^{-10}$ required to account for the light $\nu$ masses [@Cai:2017mow; @Atre:2009rg], but also by the experimental constraints on a toy-like model in which the SM is extended with a massive Majorana neutral fermion, assumed to have non-negligible mixings with the active states, without making any hypothesis on the neutrino mass generation mechanism [@Abada:2017jjx; @Pascoli:2018heg]. Such a minimal SM extension leads to contributions to LNV observables which are already close, or even in conflict, with current data from meson and tau decays, for masses $M_N$ below $10$ GeV (see [@Abada:2017jjx; @Abada:2018nio] and the references therein).
Our group has studied the Majorana $N$ decays [@Duarte:2015iba; @Duarte:2016miz] and phenomenology mostly in colliders for masses $m_N$ above the EW-scale [@Peressutti:2011kx; @Peressutti:2014lka; @Duarte:2014zea; @Duarte:2018xst; @Duarte:2018kiv] focusing on LNV processes, and in the order $1-10 ~GeV$ scale, where its decay is dominated by the $N\to \nu \gamma$ channel [@Duarte:2016caz], which was also considered in [@Yue:2017mmi].
The pure leptonic and radiative decays of pseudoscalar mesons mediated by a sterile $N$ in this effective scenario were initially studied in [@Yue:2018hci], where the authors only took into account vectorial operators contributions. The study of $N$-mediated lepton number violation in rare B meson decays has been pursued, for example, in [@Chun:2019nwi; @Abada:2017jjx; @Asaka:2016rwd; @Cvetic:2016fbv; @Cvetic:2015naa; @Wang:2014lda; @Cvetic:2010rw; @Zhang:2010um; @Helo:2010cw; @Atre:2009rg; @Ali:2001gsa] and the references therein. Also, the tension with the SM values of the ratios of branching fractions $R(D)$ and $R(D^*)$ in semileptonic B decays [@Bifani:2018zmi] measured at Belle, BaBar and LHCb, has led to proposals involving sterile neutrinos as solutions, in seesaw scenarios [@Cvetic:2017gkt] and also including EFTs with right-handed neutrinos [@Mandal:2020htr], with sterile $N$ interactions mediated by leptoquarks [@Azatov:2018kzb] or a $W^{\prime}$ $SU(2)_L$ singlet vector boson [@Greljo:2018ogz; @Robinson:2018gza].
In this work we go on studying the effect on B meson decays of the presence of a Majorana $N$ with effective interactions. In a previous paper [@Duarte:2019rzs] we studied the constraints imposed by the bound on the $B^{-}\to \mu^{\pm}\mu^{\mp}\pi^{+}$ decay by LHCb [@Aaij:2014aba] and in the radiative leptonic $B\to \mu \nu \gamma$ decay by Belle [@Gelb:2018end] on the effective operators. Here we propose to use final tau polarization and angular observables to disentangle the possible contributions of a Majorana neutrino $N$ with effective interactions to the $B^{-} \to \tau^{-} \nu$, the radiative $B\to \mu \nu \gamma$ and the LNV $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ decays in future experiments.
The paper is organized as follows. In Sec. \[sec:eff\_form\] we introduce the effective Lagrangian formalism for the Majorana $N$, discuss the existing bounds on the couplings weighting the different effective operators contributions \[sec:bounds\_alpha\], and review the experimental prospects for B meson leptonic decays \[sec:B\_lept\_dec\_exp\]. In Sec.\[sec:BtotauN\] we study the B tauonic decay. The radiative $B\to \ell \nu \gamma$ decay is assessed in Sec.\[sec:bdec\_rad\] and the LNV $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ in Sec.\[sec:NmedBdec\_taus\]. We summarize our results in Sec.\[sec:summary\].
Effective interactions formalism. {#sec:eff_form}
---------------------------------
We extend the SM Lagrangian including only *one* right-handed neutrino $N_R$ with a Majorana mass term, giving a relatively light massive state $N$ as an observable degree of freedom. The new physics effects are parameterized by a set of effective operators $\mathcal{O}_\mathcal{J}$ constructed with the SM and the Majorana neutrino fields and satisfying the $SU(2)_L \otimes U(1)_Y$ gauge symmetry [@delAguila:2008ir; @Liao:2016qyd; @Wudka:1999ax].
The effect of these operators is suppressed by inverse powers of the new physics scale $\Lambda$. The total Lagrangian is organized as follows: $$\begin{aligned}
\label{eq:lagrangian}
\mathcal{L}=\mathcal{L}_{SM}+\sum_{n=5}^{\infty}\frac1{\Lambda^{n-4}}\sum_{\mathcal{J}} \alpha_{\mathcal{J}} \mathcal{O}_{\mathcal{J}}^{(n)}\end{aligned}$$ where $n$ is the mass dimension of the operator $\mathcal{O}_{\mathcal{J}}^{(n)}$ [^1].
[c l c r c l c r]{} Operator & Notation & Type & Coupling & Operator & Notation & Type & Coupling\
$\;\; (\phi^{\dag}\phi)(\bar L_i N \tilde{\phi}) \;\;$ & $\;\; \mathcal{O}^{(i)}_{LN\phi} \;\;$ & S & $\alpha^{(i)}_{\phi}$ & & &\
$\;\; i(\phi^T \epsilon D_{\mu}\phi)(\bar N \gamma^{\mu} l_i)\;\;$ & $ \;\;\mathcal{O}^{(i)}_{Nl\phi}\;\;$& V & $\alpha^{(i)}_W$ & $\;\; i(\phi^{\dag}\overleftrightarrow{D_{\mu}}\phi)(\bar N \gamma^{\mu} N) \;\;$& $\;\; \mathcal{O}_{NN\phi} \;\;$& V & $\alpha_Z$\
$\;\;(\bar N \gamma_{\mu} l_i) (\bar d'_i \gamma^{\mu} u'_i)\;\;$& $\;\; \mathcal{O}^{(i)}_{duNl} \;\;$& V & $\alpha^{(i)}_{V_0}$ & $\;\;(\bar N \gamma_{\mu}N) (\bar f_i \gamma^{\mu}f_i) \;\;$& $\;\; \mathcal{O}^{(i)}_{fNN} \;\;$ & V & $\alpha^{(i)}_{V_{f}}$\
$\;\; (\bar Q'_i u'_i)(\bar N L_i) \;\;$& $\;\;\mathcal{O}^{(i)}_{QuNL}\;\;$ & S & $\alpha^{(i)}_{S_1}$ & $ \;\;(\bar L_i N)\epsilon (\bar L_i l_i)\;\; $ & $\;\;\mathcal{O}^{(i)}_{LNLl}\;\;$& S & $\alpha^{(i)}_{S_0}$\
$\;\;(\bar L_i N) \epsilon (\bar Q'_i d'_i)\;\;$& $\;\;\mathcal{O}^{(i)}_{LNQd}\;\;$& S & $\alpha^{(i)}_{S_2}$ & $\;\;|\bar N L_i|^2\;\; $& $\;\;\mathcal{O}^{(i)}_{LN}\;\;$& S & $\alpha^{(i)}_{S_4}$\
$\;\;(\bar Q'_i N)\epsilon (\bar L_i d'_i)\;\;$ & $\;\;\mathcal{O}^{(i)}_{QNLd}\;\;$ & S & $\alpha^{(i)}_{S_3}$ & & &\
$\;\; (\bar L_i \sigma^{\mu\nu} \tau^I N) \tilde \phi W_{\mu\nu}^I \;\;$& $\;\;\mathcal{O}^{(i)}_{NW}\;\;$ & T & $\alpha^{(i)}_{NW}$ & $\;\;(\bar L_i \sigma^{\mu\nu} N) \tilde \phi B_{\mu\nu}\;\;$& $\;\;\mathcal{O}^{(i)}_{NB}\;\;$ & T &$\alpha^{(i)}_{NB}$\
The dimension 5 operators were studied in detail in [@Aparici:2009fh]. These include the Weinberg operator $\mathcal{O}_{W}\sim (\bar{L}\tilde{\phi})(\phi^{\dagger}L^{c})$ [@Weinberg:1979sa] which contributes to the light neutrino masses, $\mathcal{O}_{N\phi}\sim (\bar{N}N^{c})(\phi^{\dagger} \phi)$ which gives Majorana masses and couplings of the heavy neutrinos to the Higgs (its LHC phenomenology has been studied in [@Caputo:2017pit; @Graesser:2007yj; @Jones-Perez:2019plk]), and the operator $\mathcal{O}^{(5)}_{NB}\sim (\bar{N}\sigma_{\mu \nu}N^{c}) B^{\mu \nu}$ inducing magnetic moments for the heavy neutrinos, which is identically zero if we include just one sterile neutrino $N$ in the theory.
In the following, as the dimension 5 operators do not contribute to the studied processes -discarding the heavy-light neutrino mixings- we will only consider the contributions of the dimension 6 operators, following the treatment presented in [@delAguila:2008ir; @Liao:2016qyd], and shown in Tab.\[tab:Operators\].
The effective operators above can be classified by their Dirac-Lorentz structure into *scalar*, *vectorial* and *tensorial*. The couplings of the tensorial operators are naturally suppressed by a loop factor $1/(16\pi^2)$, as they are generated at one-loop level in the UV complete theory [@delAguila:2008ir; @Arzt:1994gp].
In this paper we will consider the B decays $B\to \tau N$ in Sec.\[sec:BtotauN\], $B^- \to \mu^- \nu \gamma$ in Sec.\[sec:bdec\_rad\] and $B^-\to \ell^-_{1}\ell^-_{2} \pi^+$ in Sec.\[sec:NmedBdec\_taus\], mediated by a Majorana neutrino $N$. We can thus take into account the following effective Lagrangian terms involved in those processes: $$\begin{aligned}
\label{eq:lag_tree}
\mathcal{L}^{tree}& = &\mathcal{L}_{SM} + \frac{1}{\Lambda^2} \sum_{i,j}\Big\{ -\alpha^{(i)}_W \frac{ ~v ~m_W}{\sqrt{2}}\, \overline{l_i} \gamma^{\nu} P_R N \, W^{-}_{\mu}
+ \alpha^{(i)}_{V_0} \, \overline{u'_j} \gamma^{\nu} P_R d'_j \, \, \overline{l_i} \gamma_{\nu} P_R N \nonumber \\
&& + \alpha^{(i)}_{S_1}\, \, \overline{u'_j} P_L d'_j \, \, \overline{l_i} P_R N - \alpha^{(i)}_{S_2}\, \overline{u'_j} P_R d'_j \, \, \overline{l_i} P_R N + \alpha^{(i)}_{S_3}\, \overline{u'_j} P_R N \, \, \overline{l_i} P_R d'_j + \mbox{h.c.} \Big\} \end{aligned}$$ and $$\begin{aligned}
\label{eq:lag_1loop}
\mathcal{L}^{1-loop}= -i \frac{\sqrt{2} v}{\Lambda^2} {(\alpha_{NB}^{(i)}c_W + \alpha_{NW}^{(i)}s_W) (P^{(A)}_{\mu} ~\bar \nu_{L,i} \sigma^{\mu\nu}N_R~ A_{\nu})}.\end{aligned}$$ The one-loop generated effective Lagrangian contributes to the $N\to \nu \gamma$ decay. Here $-P^{(A)}$ is the 4-momentum of the outgoing photon and $s_W$ and $c_W$ are the sine and cosine of the weak mixing angle.
In the effective four-fermion terms in the quark fields are flavor eigenstates with family $j=1,2,3$. In order to find the contribution of the effective Lagrangian to the $B^{-}$ decays we are studying, we must write it in terms of the massive quark fields. Thus, we consider that the contributions of the dimension 6 effective operators to the Yukawa Lagrangian are suppressed by the new physics scale with a factor $\frac{1}{\Lambda^2}$, and neglect them, so that the matrices that diagonalize the quark mass matrices are the same as in the pure SM.
Writing with a prime symbol the flavor fields, we take the matrices $U_{R}, ~U_{L}, D_{R}$ and $D_{L}$ to diagonalize the SM quark mass matrix in the Yukawa Lagrangian. Thus the left- and right- handed quark flavor fields (subscript $j$) are written in terms of the massive fields (subscript $\beta$) as: $$\begin{aligned}
\label{eq:mass_basis}
&& u^{'}_{(R,L) j} = U_{(R,L)}^{j,\beta} u_{(R,L) \beta} \;, \; \; \; \; \; \overline{u^{'}_{(R,L)j}}=\overline{u_{(R,L)\beta}} (U_{(R,L)}^{j, \beta})^{\dagger} \nonumber \\
&& d^{'}_{(R,L) j} = D_{(R,L)}^{j,\beta} d_{(R,L) \beta} \;, \; \; \; \; \; \overline{d^{'}_{(R,L)j}}=\overline{d_{(R,L)\beta}} (D_{(R,L)}^{j, \beta})^{\dagger} .\end{aligned}$$ With this notation, the SM $V_{CKM}$ mixing matrix corresponds to the term $V^{\beta \beta'} = \sum_{j=1}^{3} (U_{L}^{j \beta})^{\dagger}D_L^{j \beta'} $ appearing in the charged SM current $J^{\mu}_{CC}= \overline{u_{\beta}} ~V^{\beta \beta'} \gamma^{\mu} P_L d_{\beta'}$.
Thus in the mass basis the tree-level generated Lagrangian in can be written in terms of the massive quarks as $$\begin{aligned}
\label{eq:lag_massiveq}
\mathcal{L}^{tree}&&= \mathcal{L}_{SM} + \frac{1}{\Lambda^2} \Big\{ -\sum_{i} \alpha^{(i)}_W \frac{ ~v ~m_W}{\sqrt{2}}\, \overline{l_i} \gamma^{\nu} P_R N \, W^{-}_{\nu} \nonumber \\
&& + \sum_{i,j} \left( \alpha^{(i)}_{V_0}~ U_{R}^{\beta j~*} ~D_{R}^{j \beta'} \, \overline{u}_{\beta} \gamma^{\nu} P_R d_{\beta'} \, \, \overline{l_i} \gamma_{\nu} P_R N + \alpha^{(i)}_{S_1}~ U_{R}^{\beta j~*} ~D_{L}^{j \beta'} \, \, \overline{u}_{\beta} P_L d_{\beta'} \, \, \overline{l_i} P_R N \right. \nonumber \\
&& \left. - \alpha^{(i)}_{S_2}~ U_{L}^{\beta j~*} D_{R}^{j \beta'} \, \overline{u}_{\beta} P_R d_{\beta'} \, \, \overline{l_i} P_R N + \alpha^{(i)}_{S_3}~ U_{L}^{\beta j~*} D_{R}^{j \beta'} \,\, \overline{u}_{\beta} P_R N \, \, \overline{l_i} P_R d_{\beta'} \right) +\mbox{h.c.} \Big\}.\end{aligned}$$ For the sake of simplicity, we will rename the new quark mixing matrix element products as $$\begin{aligned}
\label{eq:mix_beta_betap}
Y^{\beta \beta'}_{RR} \equiv \sum_{j} U_{R}^{\beta j*} ~D_{R}^{j \beta'}, \qquad
Y^{\beta \beta'}_{RL} \equiv \sum_{j} U_{R}^{\beta j*} ~D_{L}^{j \beta'}, \qquad
Y^{\beta \beta'}_{LR} \equiv \sum_{j} U_{L}^{\beta j*} D_{R}^{j \beta'} \qquad \end{aligned}$$ These new quark flavor-mixing matrices combinations $Y^{\beta \beta'}$ are unknown, and in principle their entries may be found by independent measurements, as is done in the case of the SM $V_{CKM}$ matrix. In this occasion we will make an ansatz and consider that all the $Y^{\beta \beta'}$ values in shall be of the order of the SM $V^{\beta \beta'}_{CKM}$ value, taking it as a measure of the strength of the coupling between the respective quarks: $Y^{\beta \beta'}_{RR} = Y^{\beta \beta'}_{RL} = Y^{\beta \beta'}_{LR} \approx V_{CKM}^{\beta \beta'}$. This will be done in the numerical calculations, while we leave the explicit expressions in our analytical equations[^2].
Numerical bounds on the effective couplings {#sec:bounds_alpha}
-------------------------------------------
Besides depending on the quark mixing matrices $Y^{\beta \beta'}$ values, all the quantities we calculate of course depend on the numerical values of the effective couplings $\alpha_{\mathcal{J}}^{(i)}$ which weight the contribution of each operator $\mathcal{O}_{\mathcal{J}}^{(i)}$ in Tab. \[tab:Operators\] to the Lagrangian . Their numerical value can be constrained considering the current experimental bounds on the active-heavy neutrino mixing parameters $U_{\ell N}$ in low scale minimal seesaw models appearing in the charged $(V-A)$ interactions $N L W$ when neutrino mixing is taken into account [@Atre:2009rg]. Inspired in this interaction we consider the combination $$\label{eq:u2}
U^2=(\alpha v^2 /(2 \Lambda^2))^2$$ which is derived from the Lagrangian term coming from the operator $\mathcal{O}^{(i)}_{LN\phi}$ in Tab.\[tab:Operators\] and allows a direct comparison with the mixing angles in the Type I seesaw scenarios [@Atre:2009rg]. Updated reviews of the existing bounds on the Type I seesaw mixings $U_{\ell N}$ can be found in refs. [@Bolton:2019pcu; @Chun:2019nwi].
Some of the operators involving the first fermion family (with indices $i=1$) are strongly constrained by the neutrinoless double beta decay bounds, currently obtained by the KamLAND-Zen collaboration [@KamLAND-Zen:2016pfg]. Following the treatment already made in [@Duarte:2016miz], the values of the $0\nu \beta \beta$-decay constrained couplings $\alpha^{(1)}_W,\,\alpha^{(1)}_{V_0},\,\alpha^{(1)}_{S_{1,2,3}}$ are taken as equal to the bound $\alpha^b_{0\nu\beta\beta}=3.2\times 10^{-2}\left(\frac{m_N}{100 GeV} \right)^{1/2}$ for $\Lambda=1~ TeV$. These operators, although not directly involved in the processes we consider in this work, appear as contributions to the $\Gamma_N$ total width. The numerical treatment of the second and third fermion family effective couplings $\alpha_{\mathcal{J}}^{(i)}$ for $i=2,3$ (involved in the processes studied in this work) is explained below.
As we would like to disentangle the kind of new physics contributing to the Majorana neutrino interactions, for the numerical analysis we will consider different benchmark scenarios for the effective couplings, where we switch on/off the operators with distinct Dirac-Lorentz structure: vectorial, scalar and the tensorial one-loop generated operators. If we call ($V, S, T$) the factors multiplying the vectorial, scalar and tensorial one-loop generated operators respectively, we can define six sets, presented in the table \[tab:alpha-sets\].
----------- ----------- ---------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- --
Operators Couplings $\;$ Type $\;$ **[Set1]{} & **[Set2]{} & **[Set3]{} & **[Set4]{} & **[Set5]{} & **[ Set6]{}\
$\mathcal{O}_{L N \phi}$, $\mathcal{O}_{duNL}$ & $\alpha^{(i)}_{W}$ $\alpha^{(i)}_{V_0}$ & V & 1 & 1 &0 &1 & 1& 0\
$\mathcal{O}_{QuNL}$, $\mathcal{O}_{LNQd}$, $\mathcal{O}_{QNLd}\qquad$ & $\alpha^{(i)}_{S_1} $ $\alpha^{(i)}_{S_2}$ $\alpha^{(i)}_{S_3}$ & S & 1 & 0 & 1& 1 & 0 & 1\
$\mathcal{O}_{NB}$, $\mathcal{O}_{NW}$ & $\alpha^{(i)}_{NB}$ $\alpha^{(i)}_{NW}$ & T & 1 & 1 & 1& 0 & 0 & 0\
************
----------- ----------- ---------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- --
: Effective operators benchmark sets. []{data-label="tab:alpha-sets"}
Throughout the paper we will take all the vectorial couplings numerical values equal to a generic $\alpha_{V}^{(i)}$, the scalar couplings as $\alpha_{S}^{(i)}$ and the tensorial ones as $\alpha_{T}^{(i)}$ for $i=2,3$. In the different sections of the paper, we will take complementary approaches to assign these values, that will be explained in a timely manner. However, we will use the sets defined in Tab. \[tab:alpha-sets\] to see which operators are taken into account in each case.
For the calculation of the total decay width of the Majorana neutrino $\Gamma_{N}$ we include all the kinematically allowed channels for $m_N$ in the range $1 ~GeV< m_N < 5 ~GeV$. The details of the calculation are described in our ref. [@Duarte:2016miz]. The sets 1, 2 and 3 in Tab. \[tab:alpha-sets\] take into account the contributions of the tensorial one-loop-generated effective couplings in to the $N$ decay width. In particular these sets allow for the existence of the $N\to \nu \gamma$ decay channel. As we found in [@Duarte:2016miz], this channel gives the dominant contribution to the $N$ decay width for the low mass $m_N$ range considered in this work. The sets 4, 5 and 6 discard this contribution. The total $\Gamma_N$ width is around three orders of magnitude higher in the sets 1, 2 and 3 than in the sets 4, 5 and 6 [@Duarte:2019rzs].
B meson leptonic and semileptonic decays {#sec:B_lept_dec_exp}
----------------------------------------
Many ongoing and future experiments will copiously produce B mesons, enabling the study of its leptonic and semileptonic decays. The number of B mesons is expected to be over $10^{10}$ at Belle II, over $10^{13}$ at the $300 fb^{-1}$ luminosity LHCb upgrade, and a similar number is expected at SHiP. The estimated number of B mesons to decay in the MATHUSLA detector volume is over $10^{14}$, and the expected number of B meson pairs to be produced at the Z peak at the FCC-ee is above $10^{11}$ [@Chun:2019nwi].
Measuring pure leptonic B decays is challenging. The $B \to \tau \nu$ predicted branching ratio is of order $10^{-5}$ in the SM. For muonic and electronic B decays it is much smaller due to the $m^2_{\ell}$ factor coming from helicity suppression. The spinless B meson -like the pion- prefers to decay to the heaviest possible charged lepton because balancing the spins of the outgoing leptons requires them to have the same handedness, and the neutrino forces its charged partner into the unfavored helicity. The SM decay $B^{-} \to \tau^{-} \bar{\nu}_{\tau}$ (or its conjugate) has not yet been measured by one single experiment with $5\sigma$ significance, and in Sec. \[sec:BtotauN\] we use the value of the combined measurement [@Tanabashi:2018oca] to put bounds on the effective couplings $\alpha_{\mathcal{J}}$ involved in the $B \to \tau N$ decay. However, the prospects are for Belle II to achieve a measurement with approximately $2 ~ab^{-1}$ (assuming the SM branching ratio)[@Kou:2018nap]. This would be a first step towards disentangling the interference with possible new physics, as $B \to \tau N$ decays, for instance with the aid of the final tau lepton polarization information.
Final taus are the only fermions whose polarization is accessible by means of the energy and angular distribution of its decay products. These measurements rely on the dependence of kinematic distributions of the observed tau decay products on the helicity of the parent tau. Tau lepton polarization measurements at Belle have focused on the $R_{(D^*)}$ anomaly [@Hirose:2016wfn; @Hirose:2017dxl]. In Sec. \[sec:NmedBdec\_taus\] we find that measuring the final tau polarization in eventual LNV $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ decays with $\ell^-_{1 ~\text{and/or}~ 2}=\tau^{-}$ could help to discern between the contributions of different types of effective operators. The Belle II prospects for measuring SM rare semi-tauonic modes as $B^{+}\to \tau^{+} \tau^{-} \pi^{+}$ are discouraging, because it is very difficult to reconstruct both final taus, and the lepton flavor violating $B^{+}\to \tau^{+} \mu^{-} \pi^{+}$ decays with final taus and muons are expected to be easier to measure [@Kou:2018nap]. One can hope that the LNV semi-tauonic decays could be discovered in experiments with more B meson events, as MATHUSLA and SHiP [@Chun:2019nwi]. However, these would not allow for the reconstruction of the taus polarization, leaving us with the eventual measurements at the FCC-ee [@Kamenik:2017ghi; @Abada:2019zxq] to test our proposal.
B factories as Belle II produce B mesons at the center of mass (CM) energy of the $\Upsilon(4S)$ resonance, which decays in 50% to $B^+ B^-$ pairs. The decay of the $\Upsilon(4S)$ produces B mesons in pairs: if one (the tag $B_{tag}$) is reconstructed, the rest of the event must be a B meson (the signal $B_{sig}$). In hadronic B tags all the charged and neutral particles are identified, and used to reconstruct the tag-side B meson. The efficiency of this method is low -of order $10^{-3}$- but a very pure sample of B mesons is obtained. In semileptonic B tags a charmed meson is reconstructed together with a high momentum lepton. The efficiency of this method is higher, of order $10^{-2}$, though the obtained sample is not so pure. Inclusive tagging combines the four-momenta of all particles in the rest of the event of the signal-side B candidate. The achieved tagging efficiency is usually one order of magnitude above the hadronic and semileptonic tagging approaches [@Kou:2018nap].
The SM radiative leptonic B decays have been extensively studied in the literature [@Beneke:2011nf; @Wang:2016qii; @DescotesGenon:2002mw; @Korchemsky:1999qb; @Wang:2018wfj; @Beneke:2018wjp], as they are a means of probing the strong and weak SM interactions in a heavy meson system. While the measurement of pure leptonic B decays is very difficult due to helicity suppression and the fact of having only one detected final state particle, the radiative modes can be larger as they escape helicity suppression, and are easier to reconstruct because of the extra real final photon. In Sec. \[sec:NmedBdec\_taus\] we use the bounds imposed on the effective couplings by the last Belle measurements on the $B \to \mu \nu \gamma$ partial branching fraction [@Gelb:2018end]. Now, as the signal yields for the $B \to \mu \nu \gamma$ decay are expected to be three times higher in Belle II compared to Belle [@Kou:2018nap], in Sec. \[sec:bdec\_rad\] we explore the possibility to, if the signal is found, use angular information from the hard photon and the muon flight directions to study the interference of $N$ mediation in this decay. We are particularly interested in the inclusive tagging technique mentioned above, which allows for the reconstruction of the $B_{sig}$ meson flight direction, and the use of its rest frame, without a high loss in efficiency, as has been recently done in [@Prim:2019gtj]. This technique could be used for implementing the angular Forward-Backward asymmetry $A_{FB}^{\ell \gamma}$ between the final charged lepton and photon flight directions we propose in Sec. \[sec:subsec\_AFB\_lg\].
B tauonic decay: final tau polarization {#sec:BtotauN}
=======================================
In this section we aim to study the possibility to measure the effects of the presence of a Majorana neutrino $N$, considering the final tau polarization in the SM decay $B^{-} \to \tau^{-} \bar{\nu}_{\tau}$ and the effective $B \to \tau N$ decay. The latter mode can leave the same final tau plus missing energy $\slashed{E_{T}}$ signal as the SM process when the $N$ escapes the detector before decaying into observable particles.
The current value of the branching fraction $Br(B \to \tau \nu)= (1.09 \pm 0.24)\times 10^{-4}$ [@Tanabashi:2018oca], which averages the latest measurements by Belle [@Adachi:2012mm; @Kronenbitter:2015kls] and BaBar [@Lees:2012ju; @Aubert:2009wt] allows us to impose bounds on the effective couplings $\alpha_{\mathcal{J}}$ involved in the $B \to \tau N$ decay.
The expression for the effective decay width $\Gamma_{B \to \tau N}$ is given below. The details of the calculation[^3] can be found in our ref. [@Duarte:2019rzs]. The result is $$\begin{aligned}
\label{eq:BtotauN_width}
\Gamma_{B \to \tau N}&=&\frac{1}{16\pi m_B}\left(\frac{f_B m_B^2}{2 \Lambda^2} \right)^2 \left\{ |A_V|^2 \left[
(1+x_{\tau}^{2}-x_N^{2})(1-x_{\tau}^{2}+x_N^{2})-(1-x_{\tau}^{2}-x_N^{2})\right]
\right.
\nonumber \\
&+& \left.
|A_S|^2\frac{(1-x_{\tau}^{2}-x_N^{2})}{(x_u+x_b)^2} + (A^*_S A_V+A^*_V A_S) \frac{x_{\tau}}{(x_u+x_b)}(1-x_{\tau}^{2}+x_N^{2}) \right\}
\nonumber \\
& \times & ((1-x_{\tau}^{2}+x_N^{2})^2-4 x_N^{2})^{1/2},\end{aligned}$$ where $x_{\tau}= m_{\tau}/m_B , \;\; x_N=m_{N}/m_B, \;\; x_{u} = m_{u}/m_B, \;\; x_{b}=m_{b}/m_B$, $$A_V = \left( \alpha^{(3)}_{V_0} Y^{ub}_{RR} + \alpha_W^{(3)} V^{ub} \right) \;\text{and}\;\;
A_S = \left( \alpha^{(3)}_{S_1} Y^{ub}_{RL} + (\alpha^{(3)}_{S_2} + \frac12 \alpha^{(3)}_{S_3}) Y^{ub}_{LR} \right).$$ Here $f_B$ is the B meson decay constant.
The effective couplings in $A_{V,S}$ -as the subscript indicates- correspond to *vectorial* and *scalar* interactions. We find that, due to the pseudo-scalar nature of the B meson the scalar operators contribution to the $B\to \tau N$ decay width is enhanced with respect to the vectorial contribution. This leads to the presence of the quark masses $m_u$ and $m_b$ in the denominator of the scalar term in . In turn, this will enable us to put more stringent bounds on the scalar operators contributions to this process [@Duarte:2019rzs]. In order to find the allowed values for the effective couplings in this context, we compare the experimental value $Br^{exp}(B \to \tau \nu)= (1.09 \pm 0.24)\times 10^{-4}$ to the theoretical prediction $Br^{th}= (\Gamma^{SM}_{B \to \tau \nu}+ \Gamma_{B \to \tau N})/\Gamma^{total}_{B} $ subject to the constraint $\tau_{N}> 10^{3} ~ps$ on the $N$ lifetime, enforcing it to escape undetected [@Aaij:2014aba].
The SM result for the branching ratio is $$Br^{SM}_{B \to \tau \bar{\nu}_{\tau}} = \frac{\Gamma^{SM}}{\Gamma_B}= \frac{G^2_F ~ m_B ~ m^{2}_{\tau} }{8 \pi \Gamma_B} \left(1- \frac{m^{2}_{\tau}}{m^{2}_{B}}\right)^{2} f^2_B ~|V^{ub}|^{2}.$$ The uncertainty in this calculation comes from the B meson decay constant $f_B$, the CKM mixing $V^{ub}$, the total width $\Gamma_B$ and the values of the masses. Summing in quadrature the uncertainties in the SM ($\Delta Br^{SM}$) and experimental ($\Delta Br^{exp}=0.24\times 10^{-4}$) values, we find the region in the $(m_N, \alpha)$ plane consistent with the conditions $$Br^{exp}-\sqrt{(\Delta Br^{SM})^2+ (\Delta Br^{exp})^2} \leq Br^{th} \leq Br^{exp}+ \sqrt{(\Delta Br^{SM})^2+ (\Delta Br^{exp})^2}$$ and $\tau_{N}> 10^{3} ~ps$.
In the numerical calculation we consider all the values of the effective couplings to be equal: $\alpha_V^{(i)}=\alpha_S^{(i)}=\alpha_T^{(i)}= \alpha$, and let them be on/off depending on the benchmark set defined in Tab.\[tab:alpha-sets\] considered. As we need the Majorana $N$ to decay outside the detector, the sets 1, 2 and 3 do not give a contribution, as the $N$ decays very fast if the radiative $N\to \nu \gamma$ channel is available. Thus we are left with sets 4, 5 and 6.
The allowed regions are shown in Fig.\[fig:Bnutaupol\], for sets 4 and 6. The left panel shows the results for set 4, which includes all the scalar and vector operators, while the right panel shows the result for set 6, where only scalar interactions are active. As we mentioned above, the constraints for the vectorial operators in set 5 are loose as compared to the scalar operators, and we do not show them here. The dashed border (light gray) is obtained when we consider that only the third lepton family (tau) can have effective interactions: this is implemented numerically taking $\alpha^{(1,2)}_{S}=\alpha^{(1,2)}_{V}=0$, with only $\alpha^{(3)}_{S}=\alpha^{(3)}_{V}= \alpha \neq 0$. The dotted border (dark gray), includes the three fermion families in both sets. As can be seen in the plot, this condition reduces the allowed region, because the $N$ lifetime is shortened when it can decay to the light charged leptons, which is not allowed if only the third family interactions are included.
We now turn to calculate the polarized $B \to \tau \nu$ and $B \to \tau N$ decays. We will consider the final tau polarization observable $P_{\tau}$ in terms of the decay widths $\Gamma(h)$, with $h=\pm 1$ for the positive and negative tau helicity states. Thus we write $$P_{\tau}=\frac{\Gamma(+1)-\Gamma(-1)}{\Gamma(+1)+\Gamma(-1)}.$$ We will also assume that tau decays preserve $CP$ invariance and therefore the distributions for $h=\pm$ anti-taus $\tau^{+}$ follow those of $h=\mp$ taus $\tau^{-}$.
The SM calculation result, summarized in the Appendix \[sec:apx\_polarSM\], is $P_{\tau}^{SM}=1$. The expression for the polarized decay width $\Gamma(h)_{B\to \tau N}$ is $$\begin{aligned}
\label{eq:BtotauN_h}
&&\Gamma(h)_{B\to \tau N}=\frac{1}{32\pi m_B}\left(\frac{f_B m_B^2}{2 \Lambda^2} \right)^2 \left\{ |A_V|^2 \left[
x_{\tau}^{2}+x_N^{2}-(x_{\tau}^{2}-x_N^{2})(x_{\tau}^{2}-x_N^{2}+ h R)\right]
\right.
\nonumber \\
&+& \left.
|A_S|^2\frac{1}{(x_u+x_b)^2} (1-x_{\tau}^{2}-x_N^{2}-h R) + (A^*_S A_V+A^*_V A_S) \frac{x_{\tau}}{(x_u+x_b)}(1-x_{\tau}^{2}+x_N^{2}-h R) \right\}
\nonumber \\
& \times & ((1-x_{\tau}^{2}+x_N^{2})^2-4 x_N^{2})^{1/2},\end{aligned}$$ where $R=[(1-x_N^2)^2-2(1+x_N^2)x_{\tau}^2+x_{\tau}^4]^{1/2}$ and we use the notation in . We find that the pure scalar contribution in the second term is divided by the squared sum of the quark mass quotients $x_u$ and $x_b$. This enhances its contribution (and also the contribution of the second mixed term) compared to the pure vectorial contribution of the first term.
The contour curves in the $(m_N, \alpha)$ plane that give values of $P_{\tau}=0.8, ~0.85, ~0.9, ~0.95$ are shown in Fig.\[fig:Bnutaupol\] for the sets 4 and 6 in solid black lines. While set 4 takes into account both the vectorial and scalar operators contribution to the to the polarized $B\to \tau N$ decay, the set 5 only considers the vectorial contributions, and set 6 the scalars. As we saw from eq. , the departure from the SM value of the polarization ($P^{SM}_{\tau}=1$) is expected to be dominated by the scalar contribution present in sets 4 and 6.
B radiative decay: lepton-photon angular asymmetry {#sec:bdec_rad}
==================================================
In this section we investigate the sensitivity of a Forward-Backward asymmetry between the flight directions of the final lepton and photon in $B\to \ell \nu \gamma$ to the different kinds of -vectorial or scalar- effective interactions.
For the case of the B leptonic radiative decay, we consider the lepton number conserving (LNC) and violating (LNV) contributions to $B^{-}\to \ell^{-} \bar{\nu} \gamma $ and $ B^{-}\to \ell^{-} \nu \gamma$, as in the experiment we cannot distinguish between neutrinos and anti-neutrinos in the final state.
{width="80.00000%"}
In the case of a specific final lepton flavor, take $\ell=\mu$, we consider the amplitude in the SM, which conserves lepton number: $B^{-} \to \mu^{-} \bar{\nu}_{\mu} \gamma$ ($\mathcal{M}^{SM}_{\bar{\nu}_{\mu}}$) and the one mediated by the Majorana neutrino represented in Fig.\[fig:Blnugamma\]. This effective process gives a LNC contribution to the same final state ($\mathcal{M}^{N}_{\bar{\nu}_{\mu}}$) together with LNC contributions to final states with the other two anti-neutrino flavors ($\mathcal{M}^{N}_{\bar{\nu}_{(e,\tau)}}$) and the LNV $B^{-} \to \mu^{-} \nu_{j} \gamma$ ($\mathcal{M}^{N}_{LNV}$) with final neutrinos. The first two must be added coherently[^4] in order to calculate the squared matrix element, while the last must be added incoherently. Thus we write: $$|\mathcal{M}_{B^{-}\to \ell^{-} \nu(\bar{\nu}) \gamma}|^2= | \mathcal{M}^{SM}_{\bar{\nu}_{\mu}} + \mathcal{M}^{N}_{\bar{\nu}_{\mu}}|^2 + | \mathcal{M}^{N}_{\bar{\nu}_{(e,\tau)}}|^2 + | \mathcal{M}^{N}_{{LNV}}|^2.$$ The SM amplitude is written following the treatment in [@Beneke:2011nf] $$\label{eq:Mlnug_SM}
\mathcal{M}^{SM}= e G_F V_{ub} \sqrt{2} \left(\overline{u_{\ell}}(p_{\ell}\right) \gamma^{\alpha} v_{\nu}(p_{\nu}))~ \varepsilon^{* \delta}(k) \left[f_V ~\epsilon_{\alpha \delta \rho \omega}~ v^{\rho} k^{\omega} -i f_A ~(g_{\alpha \delta}~ v.k - v_{\alpha} k_{\delta})\right],$$ with the B meson momentum $q^{\rho}= m_B v^{\rho}$, $p_{\ell}$ and $p_{\nu}$ the final lepton and anti-neutrino momenta, and $k$ the photon momentum. The constants $f_V$ and $f_A$ are the vector and axial-vector form factors in [@Beneke:2011nf; @Wang:2018wfj], $e$ is the electron charge and $G_F$ is the Fermi constant.
The LNC Majorana contribution to the amplitude is $$\label{eq:munugamma_LNC}
\mathcal{M}^{N}_{{LNC}}= \frac{i}{2} \mathbf{C} f_B P_{N}(p_{N}^{2}) k^{\alpha} \varepsilon^{* \delta}(k) \left( \overline{u_{\ell}}(p_{\ell}) (C_{V} \slashed{q} + C^B_{S}) \slashed{p}_{N} \sigma^{\alpha \delta} P_{L} v_{\nu}(p_{\nu})\right)$$ and the LNV part is $$\label{eq:munugamma_LNV}
\mathcal{M}^{N}_{LNV}= -\frac{i}{2} m_N \mathbf{C} f_B P_{N}(p_{N}^{2}) k^{\alpha} \varepsilon^{* \delta}(k) \left( \overline{u_{\ell}}(p_{\ell}) (C_{V} \slashed{q} + C^B_{S}) \sigma^{\alpha \delta} P_{R} v_{\nu}(p_{\nu})\right).$$ Here $p_{N}$ is the intermediate Majorana momentum and $P_{N}(p_{N}^{2})= 1/(p_{N}^{2}-m_N^2)$. The coefficients are $~\mathbf{C}=\frac{v \sqrt{2}}{\Lambda^4}(\alpha_{NB}^{(j)} c_W + \alpha_{NW}^{(j)} s_W)$, $$C_{V}= \left(\alpha_{V_0}^{(i)} Y^{ub}_{RR} + \alpha_{W}^{(i)} V^{ub}\right) \;\text{and}\;\; C^{B}_{S}=\left(\alpha_{S_1}^{(i)} Y^{ub}_{RL}+ (\alpha_{S_2}^{(i)}+ \frac12 \alpha_{S_3}^{(i)}) Y^{ub}_{LR} \right)\frac{m_B^2}{m_{u}+m_{b}}.$$ The indices $(j)$ in $\mathbf{C}$ correspond to the final (anti-)neutrino flavor, and $(i)$ in $C_V$ and $C^{B}_{S}$ to the final charged lepton flavor. A sum over the corresponding indices $j=1,2,3$ in is understood. The details of the calculation can be found in the Appendix C in our ref. [@Duarte:2019rzs].
With the amplitudes above, we perform a phase space Monte Carlo integration, and use it to calculate a Forward-Backward asymmetry between the final charged lepton and photon flight directions in the B rest frame, which can be used as an observable to disentangle the effective Majorana from the SM contribution to the leptonic radiative B decay.
### Forward-Backward lepton-photon asymmetry {#sec:subsec_AFB_lg}
The Forward-Backward asymmetry $A_{FB}^{\ell \gamma}$ between the final charged lepton and photon flight directions is defined as $$\label{eq:AFB_lg}
A_{FB}^{\ell \gamma}= \frac{N_{+}-N_{-}}{N_{+}+N_{-}},$$ where $N_{\pm}$ is the number of events with a positive (negative) value of $\cos(\theta)$, the angle between the final lepton and the photon flight directions in the B meson rest frame.
In Fig.\[fig:AFB\_E\] we show our results for the value of the $A_{FB}^{\ell \gamma}$ asymmetry for the SM process $B^{-}\to \mu^{-} \bar{\nu}_{\mu} \gamma$ considering a final muon, for each photon energy bin $E_{\gamma}$ (black dots with error bars). The SM asymmetry value is negative in all the allowed photon energy range. The photon has maximum energy when the muon and neutrino have parallel momenta, opposed to $\vec{p_{\gamma}}$. In this case, the asymmetry value is strictly $A_{FB}^{\ell \gamma}=-1$, as can be seen in the plot. In the massless muon limit, the energy allowed for the photon in the B rest frame is $$\label{eq:E_gamma_max}
E_{\gamma}= \frac{m_B}{2}+ \frac{E_{\mu}E_{\nu}}{m_B}\left(\cos(\varphi)-1\right)$$ where $\varphi$ is the angle between the muon and neutrino momenta $\vec{p_{\mu}}$ and $\vec{p_{\nu}}$. Thus the photon has maximum energy when the muon and neutrino have the same flight directions in the B rest frame, meaning the photon and muon momenta are back-to-back.
The error bars $\delta_i$ for the SM calculation in each photon energy bin are obtained by adding in quadrature the statistical and theoretical uncertainties in the numbers of events $N_{\pm}$. The leading theoretical error comes from the uncertainty in the values of the form factors $f_A$ and $f_V$ in , which are taken from [@Wang:2018wfj].
### Sensitivity to effective couplings
The numerical value of the effective contribution to the Forward-Backward asymmetry $A_{FB}^{\ell \gamma}$ depends on the Majorana neutrino mass $m_N$ and the effective couplings $\alpha_{\mathcal{J}}$.
If the number of events in Belle II allows us to detect the $B^{-}\to \mu^{-} \nu (\bar\nu) \gamma$ process, and measure the Forward-Backward lepton-photon asymmetry in , we would like to see which values of the effective couplings $\alpha_{\mathcal{J}}$ would be compatible (or not) with the SM prediction.
As we saw previously, the effective radiative decay can only occur when the tensorial couplings $\alpha_{T}$ are included in the calculation. However, the vector and scalar contributions to the numerical value of the asymmetry can change, predicting different results. For the sake of concreteness, we will consider a scenario where the tensorial couplings take a numerical value as $\alpha_{T}= \frac{|\alpha_{V}|+|\alpha_S|}{2}$ to avoid unnatural cancellations, and let the numerical values of $\alpha_V$ and $\alpha_S$ change freely. This means that when both $\alpha_V=\alpha_S=0$ there is no effective contribution to $A_{FB}^{\ell \gamma}$.
In Fig.\[fig:AFBs1v0\] we show the result for the sum of the effective and SM contributions $A^{Tot}_{FB}$ to the asymmetry for four different $m_N$ values taking into account only scalar contributions, fixing $\alpha_S=1$ and $\alpha_V=0$. Fig.\[fig:AFBs0v1\] shows the result for pure vectorial contributions, fixing $\alpha_S=0$ and $\alpha_V=1$.
The different curves in both panels in Fig.\[fig:AFB\_E\] show that the effective contribution to the asymmetry is moderate, except for an interval in the photon energy $E_{\gamma}$ range, where it steps towards less negative values. This means that for this energy interval, some of the effective events contribute with a positive value of $\cos(\theta)$ to the asymmetry. This interval is different for different $m_N$ values.
The effect can be explained by kinematical reasons. In the effective $B \to \mu N \to \mu \nu \gamma$ decay, for each intermediate $m_N$ mass, the photon energy $E_{\gamma}$ in the B meson rest frame can take the values shown in Fig.\[fig:reg\_ene\]. This is due in part to the cut value $E_{cut}=1~GeV$ introduced to ensure the validity of the QCD treatment in the calculation [@Wang:2018wfj], and to the minimal allowed energy $E_{\gamma}^{min}=E_N (1+ \beta_N)/2$ the photon can have in the B rest frame, according to the energy $E_N$ and boost velocity $\beta_N$ of the intermediate Majorana neutrino. The maximal energy is given by the formula corresponding to for $m_{\mu} \neq 0$. Thus, the effective contribution to the asymmetry only exists in the region between the curves in Fig.\[fig:reg\_ene\]. The minimum value of $E_{\gamma}$ in Fig.\[fig:reg\_ene\] for each $m_N$ is the value where the effective contribution starts [@Duarte:2019rzs]. This contribution stops to rise the asymmetry value when the scalar product $ \vec{p_{\mu}}. \vec{p_{\gamma}}= |\vec{p_{\mu}}| |\vec{p_{\gamma}}| \cos(\theta)$ starts to be negative again: $$\label{eq:costheta}
\cos(\theta)= \frac{1}{\beta_{N}}\left(\frac{m_N}{2 \gamma_N E_{\gamma}}-1\right).$$ In Fig.\[fig:cos\_theta\], we plot eq. for the $m_N$ values in Fig.\[fig:AFB\_E\] as an aid to visualize the $E_{\gamma}$ value where the sign of $\cos(\theta)$ starts to be negative.
Let us consider the curves for $m_N=4~GeV$ in Fig.\[fig:AFB\_E\]. The step in both panels starts at $E_{\gamma}= 1.5 ~GeV$, which is the minimum value we obtain from the curve in Fig.\[fig:reg\_ene\]. At this energy $\cos(\theta)$ is positive (it is entering the allowed range $|\cos(\theta)|\leq 1$), and changes sign when $E_{\gamma}=1.95 ~GeV$, as can be seen in Fig.\[fig:cos\_theta\]. Here the step in the curves in Fig.\[fig:AFB\_E\] comes to an end. We also note that when we consider only vectorial operators, in Fig.\[fig:AFBs0v1\], the curve for $m_N=4 ~GeV$ stands between the SM value error bars, while the scalar contribution in Fig.\[fig:AFBs1v0\] increases above them. This can be studied by considering a $\Delta \chi^2$ distribution to see which values of the effective couplings $\alpha_{\mathcal{J}}$ would be compatible (or not) with the SM prediction. We construct a $\Delta \chi^2$ function as $$\Delta \chi^2= \sum_{E_i}\frac{ \left(A^{SM}_{FB}(E_i)- A^{Tot}_{FB}(E_i, m_N,\alpha_V, \alpha_S)\right)^2}{\delta_i^2},$$ where $E_i$ are the photon energy bins shown in Fig.\[fig:AFB\_E\], and $\delta_i$ are the errors for the SM prediction, and $A^{Tot}_{FB}$ is the sum of the SM and the effective value for $A_{FB}^{\ell \gamma}$ in each photon energy bin. In Fig.\[fig:chi2\] we show the contours in the $(\alpha_S,\alpha_V)$ plane corresponding to o a coverage probability $(1-\alpha)$ of 68.27% ($1 \sigma$), 90% and 95% from the SM prediction for $A_{FB}^{\ell \gamma}$, for different $m_N$ values.
Let us consider the panel for $m_N=4~GeV$ in Fig.\[fig:chi\_m4\]. The point $\alpha_S=0, \alpha_V=1$ lies in the region inside the $1\sigma$ contour, and thus the effective contribution to $A^{Tot}_{FB}$ lies between the error bars in Fig.\[fig:AFBs0v1\], while the point $\alpha_S=1, \alpha_V=0$ is far outside the outer contour, and the effective contribution can be well distinguished from the pure SM curve in Fig.\[fig:AFBs1v0\]. The same effect occurs for other $m_N$ values: the asymmetry is more sensitive to the scalar interactions contribution. This can be seen easily by inspection of eqs. and . The scalar contribution term $C^B_S$ is weighted by a factor of order $m_B/(m_u+m_b)$ with respect to the vectorial contribution, and this is again due to the pseudoscalar nature of the B meson.
N mediated LNV B decays with final taus {#sec:NmedBdec_taus}
=======================================
In this section we study the LNV decay $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$, mediated by a Majorana neutrino, as depicted in Fig.\[fig:Bllpi\]. In the case where one or two of the final leptons are tau leptons, we would like to see if a measurement of their polarization could give a hint on the kind of new physics involved in the production or decay of the intermediate $N$. As we will see, the polarization of the final taus in these decays could be used to distinguish the vectorial and scalar operators contributions to the $N$ production and decay vertices, complementing those proposed for a higher $m_N$ scale in [@Duarte:2018kiv].
As we want to keep track of the final taus polarization, we perform a calculation of the decay amplitude considering the intermediate $N$ to be near the mass shell, but without recurring to the narrow width approximation. The details of the calculation are summarized in the Appendix \[sec:apx\_polar\_LNV\].
The amplitude for the LNV decay $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ can be written as $$\label{eq:M_B-llpi}
\mathcal{M}^{N}_{B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+}= -i \frac{f_{\pi} f_{B}}{4 \Lambda^4} m_N P_{N}(p_{N}^2)~
\overline{u}(p_2) \left(C_V^{(II)} \slashed{k} - C_S^{\pi} \right) P_R \left(C_V^{(I)} \slashed{q} - C_S^{B} \right) v(p_1),$$ with $$C_{V}^{(I)}= \left(\alpha_{V_0}^{(i)} Y^{ub}_{RR} + \alpha_{W}^{(i)} V^{ub}\right) \;\;\;\; C^{B}_{S}=\left(\alpha_{S_1}^{(i)} Y^{ub}_{RL}+ (\alpha_{S_2}^{(i)}+ \frac12 \alpha_{S_3}^{(i)}) Y^{ub}_{LR} \right)\frac{m_B^2}{m_{u}+m_{b}}$$ and $$C_{V}^{(II)}= \left(\alpha_{V_0}^{(j)} Y^{ud}_{RR} + \alpha_{W}^{(j)} V^{ud}\right) \;\;\;\; C^{\pi}_{S}=\left(\alpha_{S_1}^{(j)} Y^{ud}_{RL}+ (\alpha_{S_2}^{(j)}+ \frac12 \alpha_{S_3}^{(j)}) Y^{ud}_{LR} \right)\frac{m_{\pi}^2}{m_{u}+m_{d}}.$$ The indices $(i)$ and $(j)$ for the effective couplings correspond to the $\ell_1$ and $\ell_2$ flavors, respectively. In order to keep track of the final leptons polarizations, we use the relations in , which allow us to perform a Monte Carlo simulation and obtain the number of decay events with positively or negatively polarized taus in the final state.
{width="80.00000%"}
We study the polarization of final taus in the processes depicted in Fig.\[fig:Bllpi\]. The different possible processes interchanging the final leptons flavor ($\ell_1=\tau, \mu$ and $\ell_2=\tau, \mu$) depend on the value of the mediator mass $m_N$. The possibilities for the different regions are summarized in Tab. \[tab:kinematic\_regions\].
---------------------------------------------------------------------------------------------------------------------------------------- -- --
**[I]{} & $\;$ **[II]{} $\;$ & **[III]{}\
$m_{\pi}+m_{\mu} < m_{N} < m_{\pi}+m_{\tau}$ & $m_{\pi}+m_{\tau} < m_{N} < m_{B}-m_{\tau}$ & $m_{B}-m_{\tau} < m_{N} < m_{B}-m_{\mu}$\
$\ell_{1}= \tau, $ $ \ell_{2}= \mu $ & $\ell_{1}= \tau, \mu $ $ \ell_{2}= \mu, \tau $ & $\ell_{1}= \mu, $ $ \ell_{2}= \tau $\
******
---------------------------------------------------------------------------------------------------------------------------------------- -- --
: Kinematic regions[]{data-label="tab:kinematic_regions"}
In the case of only one final tau, we define the final-state polarization as $$\label{eq:pol_tau_}
P_{\tau}=\frac{N_{+}-N_{-}}{N_{+}+N_{-}}=\frac{N_{+}-N_{-}}{N},$$ and the two-taus final state polarization in $B^{-} \to \tau^{-} \tau^{-} \pi ^{+}$ as $$\label{eq:pol_taus}
P_{\tau}=\frac{N_{++}+N_{+-}-N_{-+}-N_{--}}{N_{++}+N_{+-}+N_{-+}+N_{--}}.$$ Here the number of events ($N_{\pm}$, $N_{{\pm}{\pm} }$) with subscripts $+$ and $-$ correspond respectively to the number of $h=+$ and $h=-$ polarization states of each final tau $\ell_1$ or $\ell_2$ in Fig.\[fig:Bllpi\] measured in the experiment, as considered in . The defined final state polarizations, being a quotient, are independent of the total number of B decay events considered.
In our previous paper [@Duarte:2019rzs], we studied the bounds that can be set on the different couplings $\alpha_{\mathcal{J}}$ in the effective dimension 6 Lagrangian involved in $N$ mediated B decays by exploiting the experimental results existing on the $B^{-} \to \mu^{-} \mu^{-} \pi^{+}$ [@Aaij:2014aba] and $B^{-} \to \mu^{-} \nu \gamma$ [@Gelb:2018end] processes. The Fig.6 in [@Duarte:2019rzs] shows different $U^2 (m_N)$ curves for the upper bounds obtained for the $U^2$ combination varying with the $m_N$ mass. The most restrictive one corresponds to a benchmark scenario (set 1) where all the types of operators involved in the effective $B^{-} \to \mu^{-} \mu^{-} \pi^{+}$ decay process (vectorial, scalar and tensorial) are considered to be present in the Lagrangian. In this numerical calculation we use this curve $U^2 (m_N)$ to fix the value of the remaining couplings (generically $\alpha_{\mathcal{J}}$, not involved in $0\nu \beta \beta$-decay) inverting for the $\alpha(m_N)$ value in . While this constraint is obtained from interactions involving the $\ell= \mu, ~i=2$ family, we will use it here to fix also the $\alpha^{(3)}_{\mathcal{J}}$ couplings in , for the sake of simplicity[^5].
To appreciate the ability of the final taus polarization to determine the kind of effective operators involved in the studied interaction, we define a parameter $\lambda \in [0,1]$ to measure the proportion of vectorial and scalar operators contributing to the processes. Thus we multiply the vector operators by $\lambda$ and the scalars by $(1-\lambda)$, so that $\lambda=1$ means pure vectorial and $\lambda=0$ means pure scalar interactions.
In our numerical code, given a value of the scalar - vector interaction content $\lambda$, we let the value of the intermediate $m_N$ mass vary in the allowed interval for each region in Tab. \[tab:kinematic\_regions\]. After fixing the numerical values of the effective couplings ($\alpha$) and mixing matrices ($Y$) in , this gives us as output an interval of values for the final-state polarization $P_{\tau}$ as defined in eqs. and for one or two tau processes, consisting of a band in the ($\lambda$, $P_{\tau}$) plane.
In Fig.\[fig:Pol\_maxmin\_lam\] we show the curves of maximum and minimum values giving the allowed values for the final tau polarization $P_{\tau}$ in depending on the $\lambda$ parameter for Majorana neutrino masses $m_N$ in the kinematic regions [**[I]{}**]{} (Fig.\[fig:polmutauI\]), [**[II]{}**]{} (Fig.\[fig:polmutauII\]) and [**[III]{}**]{} (Fig.\[fig:polmutauIII\]) defined in Table \[tab:kinematic\_regions\]. For the case of two final taus , the curves are shown in Fig.\[fig:poltautau\].
In the experiment, such as FCC-ee as we discussed in Sec.\[sec:B\_lept\_dec\_exp\], the first and second leptons can in principle be distinguished, as $\ell_1$ should have the higher momentum. Then, the $m_N$ mass range can be reconstructed with a measurement of the invariant mass $M(\ell_2, \pi)$. This would allow us to distinguish the three regions [**[I]{}**]{}, [**[II]{}**]{} and [**[III]{}**]{}.
If only one tau lepton is found in the final state, we can distinguish between regions [**[I]{}**]{} and [**[III]{}**]{} by comparing its momentum with the muon momentum. If $p_{\tau} > p_{\mu}$, we should be in region [**[I]{}**]{}, and the invariant mass $M(\mu, \pi)$ value should confirm this. In this case, the final tau is produced together with the $N$ in the primary vertex (see Fig.\[fig:Bllpi\]). This means in this case we are probing the $C_V^{(I)}$ and $C_S^{B}$ coefficients in . As we find in Fig.\[fig:polmutauI\], this tau leptons are expected to be mostly negatively-polarized if we only consider scalar interactions ($\lambda=0$), and an unpolarized final tau state could be reached in the case of purely vector interactions ($\lambda=1$). If $p_{\mu} > p_{\tau}$, we should be in region [**[III]{}**]{}, which can be checked with the invariant mass $M(\tau, \pi)$. Here, by measuring the tau polarization we access the $N$ decay vertex, and test the $C_V^{(II)}$ and $C_S^{\pi}$ coefficients in . Fig.\[fig:polmutauIII\] shows that finding a negative $P_{\tau}$ would point towards a dominant scalar interaction, while a positive value would signal a vector-dominated $N$ decay.
The case of region [**[II]{}**]{} shows that we expect mostly a negative final state tau polarization, indicating the dominance of the scalar interactions in both the $N$ production and decay vertices. This is consistent with the fact that the scalar interactions are weighted in both cases by the quark masses in the denominators in , which as we mentioned in the discussion of , enhances the scalar operators contribution, due to the pseudoscalar nature of both the B and $\pi$ mesons.
Summary {#sec:summary}
=======
The effective field theory extending the standard model with right-handed neutrinos (SMNEFT) parameterizes new high-scale weakly coupled physics in a model independent manner. We consider massive Majorana neutrinos coupled to ordinary matter by dimension 6 effective operators, focusing on a simplified scenario with only one right-handed neutrino added, which provides us with a manageable parameter space to probe. These massive neutrinos would be produced in leptonic and semileptonic decays of B mesons, a theoretically clean system to probe new physics effects in lepton colliders and B factories. We exploit the known bounds on B meson decays to constrain the effective parameter space, and propose to use final taus polarization and angular observables to disentangle vectorial or scalar type effective contributions.
The B meson couples preferably to the scalar operators in the left column of Tab.\[tab:Operators\] in comparison to the vectorial ones. Thus, as we already found in [@Duarte:2019rzs], probing its decay to a Majorana neutrino $N$ we can either put more stringent bounds or give more precise predictions on the scalar operators effective couplings.
In Sec.\[sec:BtotauN\] we considered the bounds on the tauonic B decay [@Adachi:2012mm; @Kronenbitter:2015kls; @Lees:2012ju; @Aubert:2009wt] to constrain the allowed regions in the $(m_N, \alpha)$ plane (Fig.\[fig:Bnutaupol\]), and find that a measure of the final tau polarization could help to separate the pure SM decay, which gives $P_{\tau}=1$, from the effective $B\to \tau N$, where the $N$ escapes undetected. This contribution can lower the $P_{\tau}$ value to $0.8$ in the allowed region (with $m_N< 3.5 ~GeV$) for heavy neutrinos coupling only to the third lepton family. In Tab.\[tab:alpha-sets\] we define six different numerical coupling sets which turn on/off the scalar, vector or tensorial interactions. We find that the sets 4 and 6, which involve scalar $N$ interactions, are the ones giving the more interesting results.
In Sec.\[sec:bdec\_rad\] we define a Forward-Backward asymmetry $A_{FB}^{\ell \gamma}$ between the muon and photon directions for the $B \to \mu \nu \gamma$ decay, which allows us to separate the SM lepton number conserving contribution from the effective lepton number conserving and violating processes, mediated by a near on-shell $N$. We find that the effective contributions tend to deviate $A_{FB}^{\ell \gamma}$ from the SM value in photon energy intervals depending on the mass $m_N$, according to the boost of the $N$ in the B meson rest frame (Fig.\[fig:AFB\_E\]). This measure could be done in Belle II with the aid of inclusive tagging techniques, which allow for the reconstruction of the signal B frame [@Prim:2019gtj]. Again, the asymmetry is more sensitive to the scalar interactions, as is found in Fig.\[fig:chi2\].
In Sec.\[sec:NmedBdec\_taus\] we study the final taus polarization in the lepton number violating $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ decay when $\ell^-_{1 ~\text{and/or}~ 2}=\tau^{-}$. Using the bounds on the effective couplings obtained in [@Duarte:2019rzs] we probe the chances to disentangle different effective operators contributions to these decays. Weighting the vectorial and scalar operators by a factor $\lambda \in [0,1]$ with $\lambda = 1$ (purely vectorial) and $\lambda = 0$ (purely scalar) contributions. In Fig.\[fig:Pol\_maxmin\_lam\] we find that a measurement of the final polarization $P_{\tau}$ can help us infer the scalar or vector content in the $N$ production or decay vertices, provided that the invariant mass $M(\ell_2, \pi)$ can be reconstructed.
The B meson provides a rich environment to search for and constrain possible high-energy physics effects in the case a GeV-mass heavy $N$ is the only new accessible state. This study complements probes of lepton number violating same-sign dilepton signals, Higgs-neutrino interactions and other collider searches to characterize the Standard Model Neutrino effective field theory parameter space.
Polarized final taus in B decays {#sec:apx_polar}
================================
Tau polarization in the SM B tauonic decay {#sec:apx_polarSM}
------------------------------------------
In section \[sec:BtotauN\] we showed the $B \to \tau \bar\nu_{\tau}$ polarized decay result in the SM. For reference purposes, we summarize the calculation here. The SM amplitude can be written as $$\mathcal{M}_{B\to \tau \bar\nu_{\tau}}= i\frac{g^2}{2}\frac{V^{ub}}{m_W^2} \langle 0|\bar{u}\gamma^{\beta} P_L b|B^-\rangle \langle \tau^- \bar\nu_{\tau} |\bar{\tau} \gamma_{\beta} P_L \nu|0 \rangle = i\frac{g^2}{2}\frac{V^{ub}}{m_W^2} i f_B q^{\beta} (\bar{u}_{\tau}(p) \gamma_{\beta} P_L v_{\nu}(k)),$$ with $q\ll m_W$ the B meson momentum and $q= p+k$, the tau and anti-neutrino momenta, respectively.
We take the squared amplitude $|\mathcal{M}|^{2}$, considering the spin projectors for final leptons with mass $m$, spin $s$ and momentum $p$, with $p.s=0$: $$\begin{aligned}
\label{eq:ApX_polar}
u(p,s)_{\rho}~ \overline{u}(p,s)_{\omega} &=&\left[(\slashed{p}+m)\frac{(1+ \gamma^{5} \slashed{s})}{2} \right]_{\rho \omega} \nonumber \\
v(p,s)_{\rho}~ \overline{v}(p,s)_{\omega} &=& \left[(\slashed{p}-m)\frac{(1+ \gamma^{5} \slashed{s})}{2} \right]_{\rho \omega}.\end{aligned}$$ The leptons spin vectors can be written in terms of their helicity $h=\pm$ as $$\label{eq:Apx_h}
s^{\beta}=h \left( \frac{|\vec{p}|}{m},\frac{E}{m} \frac{\vec{p}}{|\vec{p}|} \right).$$ Thus we have, in the B meson rest frame $$|\mathcal{M}_{B\to \tau \bar\nu_{\tau}}|^2= G_F^2 f_B^2 {V^{ub}}^2 m_{\tau}^2 (m_B^2-m_{\tau}^2 +2 m_{\tau} ~q.s),$$ which gives us a polarized decay width $$\Gamma_{B\to \tau \bar\nu_{\tau}}(h)= G_F^2 f_B^2 ~|V^{ub}|^2 m_B^3 x_{\tau}(1-x_{\tau}^2)(1+h),$$ with $x_{\tau}=m_{\tau}/m_B$, which shows that we only have taus with $h=1$ in the final state, giving a polarization $P_{\tau}=1$ in the SM.
Final taus polarization in $B\to \ell \ell \pi$ decays {#sec:apx_polar_LNV}
------------------------------------------------------
In section \[sec:NmedBdec\_taus\] we presented the decay $B^{-} \to \ell_1^{-} \ell_2^{-} \pi^{+}$ mediated by the Majorana neutrino $N$, where we considered final polarized taus $\ell^-_{1 ~\text{and/or}~ 2}=\tau^{-}$.
In the calculation, the amplitude $\mathcal{M}^{N}({B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+})$ is initially written as $\mathcal{M}=\langle \ell_1^{-} \ell_2^{-} \pi^{+}| \hat{\mathcal{O}}|B^{-} \rangle$. The interaction $\hat{\mathcal{O}}= \hat{\mathcal{O}}_{I} \cdot \hat{\mathcal{O}}_{II}$ includes the terms coming from the $N$ production vertex $I$ in Fig.\[fig:Bllpi\], considering the Lagrangian terms in with the up-type quark $\bar{u}_\beta=u$, the down-type quark $d_\beta'=b$, the charged lepton $l_i=\ell_1$ and the $N$ decay vertex $II$, with $\bar{u}_\beta=u$, $d_\beta'=d$ and $l_i=\ell_2$.
The leptonic and hadronic parts of the final state can be separated and one considers the intermediate $N$ in $\langle 0|T N \overline{N^{c}}|0 \rangle= i \frac{\slashed{p}_N+ m_N}{(p_N^{2}-m_N^{2})}$, taking the charge conjugate of the $N$ decay terms. The only non-zero hadronic currents for a pseudoscalar meson M (this includes both the B and the pion) are the the pseudo-vector $\langle 0| \bar{u}\gamma^{\mu}\gamma^{5} d| M \rangle = i f_M q^{\mu}$ and the pseudo-scalar $\langle 0| \bar{u}\gamma^{5} d| M \rangle= -i f_M \frac{m_{M}^2}{m_u+m_d}$ currents, where $u$ and $d$ are up-type and down-type quarks, $f_M$ is the meson constant, and $q$ its momentum.
The amplitude is finally written as in $$\label{eq:M_B-llpi_apx}
\mathcal{M}^{N}_{B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+}= -i \frac{f_{\pi} f_{B}}{4 \Lambda^4} m_N P_{N}(p_{N}^2)~
\overline{u}(p_2) \left(C_V^{(II)} \slashed{k} - C_S^{\pi} \right) P_R \left(C_V^{(I)} \slashed{q} - C_S^{B} \right) v(p_1).$$ Here $p_{1,2}$ are the $\ell_{1,2}$ momenta, $f_B$ and $f_{\pi}$ are the $B$ and $\pi$ mesons constants and $$C_{V}^{(I)}= \left(\alpha_{V_0}^{(i)} Y^{ub}_{RR} + \alpha_{W}^{(i)} V^{ub}\right) \;\;\;\; C^{B}_{S}=\left(\alpha_{S_1}^{(i)} Y^{ub}_{RL}+ (\alpha_{S_2}^{(i)}+ \frac12 \alpha_{S_3}^{(i)}) Y^{ub}_{LR} \right)\frac{m_B^2}{m_{u}+m_{b}}$$ are the terms corresponding to the hadronic $B$ current in the $N$ production vertex. On the $N$ decay side we have the terms corresponding to the hadronic $\pi$ current, $$C_{V}^{(II)}= \left(\alpha_{V_0}^{(j)} Y^{ud}_{RR} + \alpha_{W}^{(j)} V^{ud}\right) \;\;\;\; C^{\pi}_{S}=\left(\alpha_{S_1}^{(j)} Y^{ud}_{RL}+ (\alpha_{S_2}^{(j)}+ \frac12 \alpha_{S_3}^{(j)}) Y^{ud}_{LR} \right)\frac{m_{\pi}^2}{m_{u}+m_{d}}.$$ We then take the squared amplitude, and as we did in \[sec:apx\_polarSM\], we use the expressions in eq. and to calculate the polarized decay width.
[^1]: Note that we do not include the Type-I seesaw Lagrangian terms giving the Majorana and Yukawa terms for the sterile neutrinos.
[^2]: The matrices in could also be reabsorbed into the definition of the effective couplings $\alpha_{\mathcal{J}}$, as is done for instance in [@Bolton:2020xsm].
[^3]: Changing $\mu \to \tau$ in the given formulas.
[^4]: In the Majorana case only the $ \mu \overline{\nu}_{\mu} \gamma $ final state adds coherently to the SM amplitude.
[^5]: This treatment yields more stringent bounds than those obtained from previous Belle bounds [@Liventsev:2013zz] using the relation in .
|
---
author:
-
bibliography:
- 'bibliography.bib'
title: 'MEC-enhanced Information Freshness for Safety-critical C-V2X Communications'
---
|
---
author:
- 'Pont, F.'
- 'Hébrard, G.'
- 'Irwin, J. M.'
- 'Bouchy, F.'
- 'Moutou, C.'
- 'Ehrenreich, D.'
- 'Guillot, T.'
- 'Aigrain, S.'
- 'Bonfils, X.'
- 'Berta, Z.'
- 'Boisse, I.'
- 'Burke, C.'
- 'Charbonneau, D.'
- 'Delfosse, X.'
- 'Desort, M.'
- 'Eggenberger, A.'
- 'Forveille, T.'
- 'Lagrange, A.-M.'
- 'Lovis, C.'
- 'Nutzman, P.'
- 'Pepe, F.'
- 'Perrier, C.'
- 'Queloz, D.'
- 'Santos, N.C.'
- 'Ségransan, D.'
- 'Udry, S.'
- 'Vidal-Madjar, A.'
bibliography:
- 'art606\_arxiv.bib'
date: 'Received ; accepted '
title: 'Spin-orbit misalignment in the planetary system[^1]'
---
Introduction {#intro}
============
HD 80606 is a solar-type star with a gas giant planetary companion on a highly eccentric 111-day orbit [@nae01]. With $e=0.93$, the planet receives about a thousand times more star light at periastron than at apastron, which makes it a key system to study the atmospheric and thermal properties of hot gas giant planets. By a lucky coincidence (about 1 percent probability for a randomly oriented orbit), the orbital plane is aligned with the line-of-sight, so that both the secondary eclipse and primary transit were detected. The secondary eclipse was measured during a long photometric run with the Spitzer space telescope [@lau09]. In @mou09 [hereafter M09], we presented our detection of the primary transit, simultaneously measured in photometry and spectroscopy with the 1.2-m and 1.93-m telescopes at Observatoire de Haute-Provence, France. The spectroscopic transit data seemed to indicate that the orbital plane of the planet was not aligned with the stellar rotation axis. But since the transit ingress was not observed, a large degree of uncertainty remained in this parameter, as well as in the latitude of the transit and radius of the host star.
Photometric data of the same event are available from two other teams and locations. @fos09 [F09] presented data obtained at the Mill Hill London Observatory, England, with two telescopes (a 35-cm Celestron and a 25-cm Meade), on a time span that almost matches that of our OHP observations the same night: data were obtained during the main portion of the flat part of the transit, the whole egress, and a few hours after its end. @gar09 [G09] presented data obtained with a 60-cm telescope at the Esteve Duran Observatory, Spain, on a shorter time span: the observations started just before the egress. No detection of the transit ingress has been reported at the time of writing.
In this paper, we present photometric data of taken from Mt Hopkins, Arizona on the same night with the MEarth network [@irw09; @nut08]. These data show no flux variation, but they do provide a powerful constraint on the system parameters by imposing a strict lower limit to the beginning of the transit. We apply a Bayesian analysis to the whole data set, together with previous radial-velocity monitoring and the [*Spitzer*]{} observations near secondary eclipse, to calculate accurate values of the system parameters, including the radii of the host star and planet, and the spin-orbit angle.
Observations {#obs}
============
MEarth photometry
-----------------
A single field containing HD 80606 and HD 80607 was monitored continuously using one telescope of the MEarth observatory located at the Fred Lawrence Whipple Observatory on Mount Hopkins, Arizona, for the night of 2009 February 13. Additional observations were taken on 2009 February 14th and 15th but these are not used in the present work. MEarth uses a non-standard $715\ {\rm nm}$ long-pass filter, with the response limited at the red end by the long-wavelength tail of the CCD quantum efficiency curve.
Observations were started at the end of nautical twilight (solar elevation $12^\circ$ below the horizon), and continued until the start of nautical twilight in the morning. The airmass of the field varied from $1.9$ at the start of observations, to a minimum of $1.1$ at meridian transit, which occurred at UT 07:11, and at the end of observations was $2.5$. A total of $1695$ $4.8\ {\rm s}$ exposures were obtained at an average cadence of $\sim 23\ {\rm s}$ including overheads (CCD readout time, and re-centering the field after each exposure).
Due to the extremely short exposures necessary to avoid saturation, and the small telescope aperture of $0.4\ {\rm m}$, errors on individual measurements are dominated by atmospheric scintillation. It was not possible to defocus the telescope due to the small ($\sim 21''$) separation between HD 80606 and HD 80607. We use the formula of @you67 to estimate the contribution to our observational error bars arising from scintillation. As noted by @rya98, the typical coherence length for this effect is $\sim 12''$, so the standard formula should remain a reasonable approximation.
Data were reduced using the standard MEarth reduction pipeline, which is at present largely identical to the Monitor project pipeline described in @irw07. An aperture radius of $10\ {\rm
pixels}$ (corresponding to $7\farcs6$ on-sky) was used to extract differential photometry. In order to estimate the contamination in the aperture for one star from the other given this large aperture size, we use measurements in multiple concentric apertures from single-stars in the same field as our target to derive a simple curve-of-growth. From this we estimate that $< 0.3\%$ of the flux from HD 80607 falls inside the aperture centered on HD 80606, and vice versa. This is negligible for our purposes, since it would lead to a $< 1\%$ underestimation in transit depths, which is smaller than the observational error in this quantity.
We used HD 80607 as a comparison star to derive differential light curves of HD 80606 from this photometry. Since these stars have similar positions on the detector and almost identical colors, effects such as color-dependent atmospheric scintillation and flat fielding error are minimized by doing this, and we find no advantage to attempting to use other stars of comparable brightness on the field as additional comparison stars. MEarth uses German Equatorial Mounts, so the entire telescope and detector system must be rotated through $180^\circ$ relative to the sky upon crossing the meridian. Using HD 80607 as a comparison star, we see little evidence for flat fielding errors in the data for HD 80606, which normally manifest as different base-line levels in the light curve for positive and negative hour angle. We therefore apply no correction for this effect in the present analysis.
The MEarth data is given in Table \[mearth\_tab\].
Date \[HJD\] F$_{715}$ \[mag\] $\sigma_F$
---------------- ------------------- ------------
2454875.589851 8.116900 0.005537
2454875.590117 8.133620 0.005522
2454875.590395 8.142982 0.005507
2454875.590684 8.160600 0.005493
2454875.590950 8.126864 0.005480
2454875.591217 8.142918 0.005471
2454875.591494 8.129089 0.005450
2454875.591749 8.153709 0.005435
2454875.592015 8.074484 0.005425
... .. ...
: Photometric times series for HD 80606 from MEarth (full table available electronically)[]{data-label="mearth_tab"}
OHP 120-cm photometry
---------------------
The photometric observations of and HD80607 performed at the 120-cm telescope at OHP during the nights 2009 February 12 and 13 were presented by M09. The transit was detected during the second night, which is the only one of the two that we use in the present work. 326 frames were secured during the transit night, with 20 to 30-second exposure times. The negative slope after egress tends to suggest that a correction for airmass variations should be taken into account in deriving the flux. The airmass is 1.5 at the beginning of the night, then reaches 1.0 at the middle of the egress, and increases up to 1.66 at the end of the night. Assuming that the slope seen out of transit is due to airmass changes, and that the effect of airmass on the photometry is linear, we find a correction of $(2-{\rm airmass})\times 0.004$. The impact of the correction is to create a small slope at the beginning of the transit sequence, i.e. during the flat section of the transit. The significance of this new slope is low. Since this correction dominates the error budget on the transit parameters, it is important to take correlated noise into account in the analysis.
Doppler spectroscopy
---------------------
Radial velocities of measured with the spectrograph at the 1.93-m telescope of OHP were presented by M09. A continuous sequence of 39 measurements was acquired during the transit night (2009, $13-14$ February), as well as nine extra measurements between $8\mathrm{th}$ and $18\mathrm{th}$ February, out of the transit. The radial velocities were obtained from a weighted cross-correlation of the spectra with a G2-type numerical template. All the spectra present a similar signal-to-noise ratio, S/N $\simeq 47$ per pixel at 550 nm. Together with the parameters of the cross-correlation function of the spectra (full width at half maximum of $7.33 \pm 0.02$ , and a contrast representing $48.2 \pm 0.3$ % of the continuum), this corresponds to a $\sim2.4$ photon-noise uncertainty on the radial velocity measurements. We quadratically added 3.5 due to telescope guiding errors and 1 due to wavelength calibration, resulting in radial velocities with a typical accuracy of $\sim4.5$ .
Published Doppler and photometric data
--------------------------------------
We complemented our measurements with published radial velocities from three other instruments: 74 measurements from *ELODIE* at OHP [@nae01; @mou09], 46 *HIRES* measurements from Keck [@but06], and 23 *HRS* measurements from the Hobby-Eberly Telescope [@wit09]. The typical accuracies of these three extra datasets are 12.3, 5.0, and 8.3 respectively, and their time spans are Nov. 1999–Nov. 2003, Apr. 2001–Feb. 2005, and Dec. 2004–Mar. 2007, respectively. None of those measurements were obtained during a primary transit. These measurements are used to refined the Keplerian orbit of the planet. Systematic radial velocity shifts between the different datasets are unknown, and left as free parameters. The residuals around the model orbit show that the *ELODIE* uncertainties are underestimated, and we scaled these uncertainties upwards by a factor 1.8 in order to obtain coherent normalized residuals. The egress of the transit of 14th February 2009 was observed in photometry from three locations covering part of the transit center and the transit egress (M09, F09, G09; see Section \[intro\]).
was also monitored for nearly 24 hours around the time of the secondary eclipse at 8 microns with the Spitzer space telescope [@lau09]. These data provide strong additional constraints on the system parameters by measuring the time and duration of the secondary eclipse.
We include these data sets in our combined solution (we use only the Celestron time series from F09).
Analysis
========
Figures \[vr\_fig\], \[rm\_fig\] and \[tr\_fig\] display the photometric and spectroscopic data for the radial velocity curve, the spectroscopic transit and the photometric transit. The secondary eclipse is plotted in @lau09. Figure \[geom\] shows the configuration of the system according to the best-fit solution in Section \[results\].
The procedures to infer physical parameters from observations of transiting planetary systems have now been firmly established, and numerous descriptions can be found in the recent literature [see e.g. contributions to @iau253 for reviews and examples]. The case of HD 80606b requires all the toolbox of the trade for several reasons: the extreme eccentricity makes the relation between physical parameters and observable quantities even more non-linear than usual, the incomplete coverage of the transit in photometry and spectroscopy requires sound Bayesian statistics, most data sets are dominated by correlated noise rather than random errors, and different types of information (transit and secondary eclipse photometry, radial velocity orbit, stellar evolution models) provide partial constraints of comparable importance. For these reasons, we need to use a fully Bayesian approach, with physically meaningful priors on all parameters. We also need proper accounting for correlated noise.
Although the data is incomplete in some respects (absence of detection of the transit ingress), the abundant datasets leave little leeway for most parameters. In particular, the combination of the secondary eclipse duration, the egress duration and the orbital parameters tightly constrain the mass and size of the host star and the planet.
Method
------
We use a Bayesian approach to determine the probability distribution of the physical parameters given all the available observations and prior information such as stellar evolution models. We perform the integration with a Markov Chain Monte Carlo (MCMC) algorithm, using Metropolis-Hastings sampling. This is similar to the procedure described for instance in @cam07.
This technique is particularly sensitive to the assumptions used on the distribution of measurement uncertainties. MCMC integration samples the merit function used to measure the agreement between model and observations according to the Bayesian prior probability, and as such it cannot be better than the merit function used. If a traditional “observed vs. measured” sum-of-squares is chosen, the error underestimation due to correlated noise that plague straightforward fitting methods will also be observed. To avoid biased results and unrealistically narrow posterior probability distributions for the output parameters, it is essential to use realistic error estimates for all data, including systematic errors. In the photometric and transit spectroscopic time series of the dominant source of systematics are fluctuations correlated in time (e.g. airmass and seeing effects). @pon06 discuss the effects of this “red noise” and a possible way to integrate it in the merit function.
Correlated noise
----------------
For most datasets pertaining to HD 80606, red (i.e. correlated) noise is by far the dominant source of uncertainties: (1) the coverage of the spectroscopic transit was acquired during a single night. It is known that in such conditions, data are sensitive to weather-related systematics at the level of a few meters-per-second; (2) the photometric times series also cover only the transit egress. When only a partial transit is observed with ground-based photometry, the correction of slow trends is difficult and dominates the error budget; (3) the out-of-transit baseline flux of the [*Spitzer*]{} observations is made variable at the $10^{-4}$ level by instrumental effects and by the light reflected on the day side of the planet, a factor which, from the point of view of parameter determination, is equivalent to red noise.
We model the red noise as described in Pont et al. (2006), with a single $\sigma_r$ parameters for each data set, describing the amplitude of correlated noise over the relevant timescale. For the photometric time series we use the “$V(n)$” method described in that paper, which estimates $\sigma_r$ from the departure of binned versions of the data compared to purely white noise. We find $\sigma_r=7\times 10^{-4}$ for the OHP photometry, $8\times 10^{-4}$ for the MEarth photometry, $3.6\times10^{-4}$ for the F09 photometry , $8\times 10^{-4}$ for the G09 photometry, 4 $m\, s^{-1}$ for the Sophie transit spectroscopy and $2\times 10^{-4}$ for the [*Spitzer*]{} time series. The first three values are coherent with our experience of ground-based transit spectroscopy, with $\sigma_r$ between 4 and $10 \times 10^{-4}$ being typical for single-object rapid cadence time series. The value of the correlation parameter for the time series is also compatible with previous experience [e.g. @heb09]. Finally, there is enough out-of-transit data in the [*Spitzer*]{} time series to measure $\sigma_r$ precisely. Our red noise analysis thus indicates that the F09 Celestron data is subject to lower systematics than the M09 and G09 photometric measurements (as visible as well in the behaviour of the residuals compared to model lightcurves). This implies that the first data set will get about twice as much weight as each of the other two in the combined analysis. To account for the clearly much lower quality of the photometric data at the beginning and end of the night, we increase $\sigma_r$ by a factor three for these measurements ($JD<2454876.34$).
Models
------
We use the @man02 algorithm to build model transit and secondary eclipse lightcurves, with a linear limb-darkening law with $u=0.66$ (suitable for a solar-type star; at this point the accuracy of the data does not require higher-order modelling of the limb darkening). We use the @oht05 analytical description of the RM effect, and a Keplerian radial-velocity orbit. We compute the 3-D position of the planet relative to the star at each date. Two additional parameters are introduced to describe the photometric signal of the secondary eclipse: a planet-to-star surface-brightness ratio in the Spitzer band, and an out-of-transit continuum flux. The brightness variation of the planet around the time of secondary eclipse is neglected.
MCMC
----
We use a chain with 200000 steps, starting with the parameters in M09. The size of the steps is set to 0.05 times the uncertainties quoted by that study. The free parameters are the following: $P, T_0, i, e, \varpi, V_0^{1..4}, M_{\rm pl}, R_{\rm pl}, M_{\rm s}, R_{\rm s}, dT, \lambda, V_{\rm rot}$, respectively the period, epoch of periastron, orbital inclination, orbital eccentricity, argument of periastron, center-of-mass radial velocity (for each spectrograph independently), planet mass and radius, star mass and radius, planet-to-star surface brightness ratio in the Spitzer band, spin-orbit angle and projected stellar spin. We use Gibbs sampling for the steps in the chain (next step accepted with a probability equal to the ratio of likelihoods). We include the constraints from stellar evolution models directly in the prior and in the merit function used in the MCMC: the MCMC chain moves in an interpolation of the @gir02 stellar evolution models with uniform steps in mass, age and metallicity (corresponding to a flat prior in these quantities). Each evolution model produces values for the stellar mass, radius and temperature, $M_s$, $R_s$ and $T_{\rm eff}$, that are used to estimate the merit function. How this method solves Bayes’ theorem is studied in more details in the context of stellar ages in @pon04.
Merit functions
---------------
The merit function in the MCMC is taken as the likelihood of the observations given the model, assuming gaussian error distributions and purely “white+red” noise [@pon06], using all the photometric and radial velocity data, as well as the spectroscopically determined stellar parameters:
$$\begin{aligned}
\chi^2 = \sum_{\rm phot} \frac{(F_{\rm obs}-F_{\rm mod})^2}{\sigma_{Fw}^2+N\sigma_{Fr}^2} + \sum_{\rm rv} \frac{(V_{\rm obs}-V_{\rm mod})^2}{\sigma_{Vw}^2+N \sigma_{Vr}^2} \\
+ \sum_{\rm star} \frac{(S_{\rm obs}-S_{\rm mod})^2}{\sigma_S^2}\end{aligned}$$
where $F$ are the photometric data, $V$ the radial velocity data, $S$ the stellar parameters, and the [*obs*]{} and [*mod*]{} subscripts denote observed and model values respectively. The $w$ and $r$ subscript indicate white and red noise parameters, and $N$ is the number of data points during the correlation timescale. $F_{\rm mod}$ is the model light curve for the transit and secondary eclipse, $V_{\rm mod}$ the model radial velocity curve including the RM anomaly during transit, and $S_{\rm mod}$ the set of observable parameters of the host star according to the @gir02 models. The sums are made respectively on the individual photometric observations, spectroscopic observations, and input stellar parameters of the evolution models (age, mass and metallicity). We take the duration of the egress as the relevant correlation timescale to evaluate $N$ ($\sim$ 25 points for photometric time series, 5 points for the radial velocity sequence).
Prior distributions
-------------------
We define the steps in the MCMC corresponding to flat prior distributions in the following quantities: $P$, $T_0$, $\cos i$, $V_i$, $e \cos \varpi$, $e \sin \varpi$, $K\sqrt{1-e^2}$, $R_{p}/R_{\rm s}$, $M_s$, $\tau_{\rm s}$, \[Fe/H\], $dT$, $\cos \lambda$. The combinations were chosen either for physical reason (isotropic orientation of spin and orbit in space, random epoch), to avoid strong covariance in the MCMC (radius ratio instead of radius, eccentricity dependence of $K$ integrated in the prior) or linear when the prior is not significant compared to the observational constraints ($P, V_i$).
The prior distribution in the parameters of the parent star ($M_s, R_s, T_{\rm eff}$) corresponds to a flat distribution in age, mass and metallicity, as described above. We therefore do not reduce the information from stellar evolution models to a single term in the merit function, but make use of the full information provided by the stellar evolution models, thus taking into account the correlation of stellar mass, radius and temperature, and the respective probability of different models for a field star in the solar-neighbourhood.
We set the stellar rotational velocity to $V_{rot}=1.8$ km$\,$s$^{-1}$ [@fis05]. This parameter influences the shape of the spectroscopic transit radial velocity curve, but is poorly constrained by the current data. The results do not vary significantly if we repeat the procedure with a moderate uncertainty on this value (up to about 1 km$\,$s$^{-1}$). If this parameter is left completely free, the Markov chain does not converge, because different combinations of the rotation velocity, spin-orbit angle and impact parameter produce similar predictions for the observed portion of the spectroscopic transit.
Altogether, three prior distributions have a significant effect on the solution: the orbital inclination angle (penalizing grazing transits compared to a non-Bayesian fit), the stellar age prior (penalizing rare or inexistant combinations of stellar parameters), and the stellar rotation velocity (lifting the main degeneracy in the spectroscopic transit).
Results and discussion {#results}
======================
Table \[param\] gives the median values and central 68% confidence intervals for the parameters of the HD80606 system given by the MCMC integration. Note that because the median values correspond to different individual solutions for each parameter, the orbital parameters in the table do not correspond to the best-fit orbit. The best-fit orbit is $P=111.43605$ days, $K=476.48$ m$\,$s$^{-1}$, e=0.9332, $T_0=2454424.8529$ BJD, $\varpi=300.71^0$.
Figures \[lam\] and \[Ttr\] show the probability distribution functions for the spin-orbit angle and transit duration (first to fourth contact).
---------------------------------------- -----------------------------------
Center-of-mass velocity $V_0$ (Elodie) 3.787 $\pm$ 0.004 km$\,$s$^{-1}$
$V_0$(SOPHIE) 3.910 $\pm$ 0.004 km$\,$s$^{-1}$
$\Delta V_0$(Keck) $-0.001 \pm 0.002$ km$\,$s$^{-1}$
$\Delta V_0$(HET) $-0.019 \pm 0.003$ km$\,$s$^{-1}$
Orbital period $P$ 111.4357 $\pm$ 0.0008 days
Orbital eccentricity $e$ 0.9332 $\pm$ 0.0008
Velocity semi-amplitude $K$ 474 $\pm$ 4 m$\,$s$^{-1}$
Epoch of periastron $T_0$ 2 454 424.852 $\pm$ 0.008 BJD
Argument of periastron $\varpi$ 300.80 $\pm$ 0.22 $^o$
Orbital inclination $i$ 89.32 $\pm$ 0.06 $^o$
Semi-major axis $a$ 0.449 $\pm$ 0.006 AU
Epoch of transit $T_{\rm tr}$ 2454876.316 $\pm$ 0.023 BJD
Epoch of eclipse $T_{\rm e}$ 2454424.719 $\pm$ 0.009 BJD
Transit duration $T_{1-4} $ 11.9 $\pm$ 1.3 hours
Transit duration $T_{2-3} $ $9.6^{-1.3}_{+0.8}$ hours
Radius ratio $R_p /R_s$ 0.103 $\pm$ 0.003
Spin-orbit alignment $\lambda$ $ 50 ^o$ \[ 14 – 111 \] $^o$
Impact parameter $b$ 0.75 $\pm$ 0.06
Star Mass $M_s$ 0.97 $\pm$ 0.04 M$_\odot$
Star Radius $R_s$ 0.978 $\pm$ 0.015 R$_\odot$
Planet mass $M_p$ 3.94 $\pm$ 0.11 $M_J$
Planet radius $R_p$ 0.98 $\pm$ 0.03 $R_J$
---------------------------------------- -----------------------------------
: Parameters for the HD 80606 system. Uncertainties from the 68% central probability interval of the posterior distribution traced by the Markov chain.[]{data-label="param"}
The strongest covariance between the output parameters is that of the impact parameter with the transit duration. In general, the transit duration depends mainly on the radius of the host star and impact parameter. When the whole transit is measured, the main covariance is between impact parameter and host star radius. However, in our case, the star radius is relatively well constrained by the duration of the secondary eclipse, and the transit duration is weakly constrained because the transit ingress was not observed, so that the main covariance is between the transit duration and impact parameter (higher impact parameters corresponding to shorter transit). Fig. \[mc\_fig\] shows the posterior probability density in the $D$ vs. $b$ plane, and in the $D$ vs $R_p$ plane. The core of the probability distribution in each parameter is well described by Normal distributions, with a tail of low-probability solutions at high impact parameter, low transit duration and larger planet.
Figure \[MR\_fig\] shows the probability density in stellar mass and radius. The main correlation is $M\sim R^{3}$, because the transit parameters are degenerate in these quantities. The sharper limit towards larger masses is due to stellar evolution models.
Uncertainty intervals
---------------------
F09 quote very small uncertainties for the transit duration, orbital inclinations and planetary radius (1.029 $\pm$ 0.017 $R_J$, 12.1 $\pm$ 0.4 hours, $89.285 \pm 0.023$ degrees). The uncertainties are small because F09 keep several important parameters fixed (the host star radius and orbital parameters), and purely uncorrelated noise is assumed in the photometry. These simplifications are useful for a rapid initial analysis, but will yield underestimated uncertainties. As extensively discussed in the context of previous observations of transiting planets with high-cadence photometry, the uncertainty on the density ($M\, R^{-1/3}$) of the primary and the systematics in the photometric noise are actually the dominant sources of uncertainties on most system parameters, including the planetary radius, inclination angle and transit duration. This explains why our uncertainty intervals are larger than those quoted in F09, in spite of being based on a larger ensemble of data.
The larger uncertainties are also due to the fact that the M09, F09 and G09 light curves indicate slightly different shapes for the transit egress (M09 and G09 favour a longer egress, therefore a higher-latitude transit across a larger star). The shape difference between the three lightcurves clearly illustrates that correlated noise dominates the error budget. The presence of correlated noise is also apparent in the sequence, since no parameters for the RM effect can reproduce the sudden jump shortly after the beginning of the data sequence.
For these reasons, as well as the arguments and examples presented in @pon06, the probability distributions described by the MCMC integration including correlated noise, uncertainties on all parameters, and host star properties constrained by stellar evolution models, arguably provide a better description of the actual implications of the data in terms of physical parameters and confidence intervals.
Spin-orbit angle
----------------
The posterior probability distribution for the projected spin-orbit angle $\lambda$ corresponds to configurations in which the planet crosses mainly the receding limb of the star. An aligned orbit ($\lambda=0$) is excluded to a $>95$% level of confidence. The MCMC calculation shows that a spin-orbit alignment in the system is made unlikely by the combination of the shape of the SOPHIE radial velocity time series during transit and the fact that the MEarth photometry excludes a low-latitude transit by placing a strict upper limit on the transit duration.
The result is sensitive to the level of correlated noise assumed in the radial velocity data. This is easily understood: since the amplitude of the RM anomaly is of the order of 15 m$\,$s$^{-1}$, assuming systematics of similar amplitude can reconcile the observed shape with that expected in case of spin-orbit alignment by attributing most of the observed variation to systematics. The presence of some systematics in the SOPHIE data is apparent from the mismatch between the first few measurements during transit, to the level of a few m$\,$s$^{-1}$, but the excellent correspondence between the moment of photometric and spectroscopic egress indicates that the systematics do not dominate the SOPHIE sequence, and therefore that such a high value of $\sigma_r$ is unlikely (note that in our standard solution we already increase the correlated noise for the first measurements in the night to 12 m$\,$s$^{-1}$, which is conservative).
Therefore, our conclusion is that the combined data indicate a tilted system, with the planet crossing the receding side of the star during the transit. is the second highly tilted planetary orbit known, after XO-3 [@heb09; @win09b], out of the twelve measured systems [@fab09]. More strikingly, four of these planets have eccentric orbits, and two of those have tilted orbit. We may therefore be seeing early indications that in addition to having a broad eccentricity distribution, extrasolar gas giants have a bimodal distribution of spin-orbit angles [@fab09]. A possible caveat is that both XO-3 and are high-mass gas giants, which may have formed differently than Jupiter-mass planets [@rib07].
Planetary radius
----------------
The size of the planet is well constrained by the data, in spite of the fact that only the transit egress was observed. This may seem surprising. It is due to the observation of the secondary eclipse and the peculiarity of the orbit of HD80606b: the planet is much nearer to the star during the secondary eclipse than during the transit, because of the large eccentricity and the position of the periastron. As a result, the mere fact that the planet is transiting implies that the secondary eclipse occurs at low latitude behind the star (see Fig. \[geom\]). This lifts the usual degeneracy between stellar radius and impact parameter, and means that the former can be derived reliably from the duration of the secondary eclipse, even in the absence of the constraint from the fractional duration of the ingress/egress. The probability distribution of the stellar radius is further narrowed by the Bayesian prior on the inclination angle, which penalizes high-latitude transits, by the shape of the transit egress, and by stellar evolution models.
The main remaining uncertainty on the size of the planet is due to global systematics in the photometry. The slightly longer egress duration favoured by the M09 and G09 data would indicate a higher impact parameter, therefore a larger star and a bigger planet. However, this is partly compensated by the fact that these datasets also favour a slightly shallower transit, therefore a smaller planet. Altogether, the uncertainty on the planetary radius is small enough for useful comparison with models. We confirm that is not a “bloated” hot Jupiter, as discussed in M09, and has a size compatible with current models for irradiated gas giants. Given its large mass, a planet like HD 80606b would require more additional energy that Jupiter-mass planets to be inflated to a higher radius. However, two other exoplanets with similar masses, CoRoT-Exo-2b ($3.3 M_J$) and OGLE2-TR-L9b ($4.5 M_J$) have radii around 1.5 $R_J$, showing that such inflated sizes are indeed possible for these heavy planets. The two other known transiting planets in the 3-5 M$_{\rm J}$ mass range, HD 17156b and WASP-10b, have radii of $1.0$ and $1.1 R_J$ respectively.
The size of these planets is conspicuously correlated with the amount of flux received from the parent star. CoRoT-Exo-2b and OGLE2-TR-L9b follow close orbits (1.7 and 2.4-day periods), while HD 17156b and HD 80606b have the widest orbits of known transiting planets, with WASP-10b being an intermediate case. The observed correlation between size and incident flux for transiting gas giants may therefore extend to several Jupiter masses, which provides another benchmark that any successful explanation of anomalous exoplanet radius must meet.
Figure \[model\] shows the evolution of the radius of HD 80606b according to models described in @gui06, with the position of our reference solution indicated. The different panels show the radius evolution of a planet with a solar-composition envelope (containing about 25M$_\oplus $ in heavy elements), but either without core (left panel), with a 100M$_\oplus$ core (middle panel) or with a 200M$_\oplus$ core (right panel). Clearly, although uncertainties in the distribution of heavy elements and equations of state have to be taken into account [e.g. @bar08], our models point towards the existence of a large mass in heavy elements (at least 60M$_\oplus$) present in the planet. All possibilities above 60 and 200$_\oplus$ are consistent with the measured radius. Solutions with even larger cores are possible, but we regard them as unlikely given that realistic critical core masses are smaller than 100M$_\oplus$ [@iko06] and planets with masses larger than Jupiter tend to scatter planetesimals much more efficiently than they accrete them [e.g. @gui00].
Given that HD80606 is metal-rich, this large inferred core mass is consistent with a correlation between heavy elements in the star and in the planet [@gui06; @bur07; @gui08]. It should be noted that if the planet had the same composition as the star, its total mass of heavy elements should be of order 50M$_\oplus$, clearly smaller than the estimates that are derived here.
Interestingly, using the concept of a maximum mass in heavy elements in the planet yields an upper limit on the rate of tidal dissipation that it may undergo. As shown by the dotted lines in Fig. \[model\], a maximum rate of dissipation compatible with the observations and models with core of at most 200M$_\oplus$ is of order $10^{27}\rm\,erg\,s^{-1}$. While this may seem large, it is several times smaller that the dissipation due to tides for this planet if it has been brought in by a Kozai mechanism [@wu03], estimated by $GM_* M_{\rm p}/(a_{\rm p} \tau_{\rm migration})\approx 5\times 10^{27}\,\rm erg\,s^{-1}$ (where we used $a_{\rm p}$ as the present semi-major axis of the planet, and $\tau_{\rm migration}\approx 1\,$Ga from WM03). Several possibilities exist: (i) The planet may have a tidal Q larger than the $3\times 10^{5}$ assumed by WM03 so that it would have migrated on a longer timescale; (ii) Heat dissipation may occur in a shallow region close to the atmosphere of the planet and be reradiated quickly after the approach to the star; (iii) The planet may have a core that is much larger than envisioned in the present study. Except in the last case, our calculations indicate that the planet’s intrinsic effective temperature should be of order 200 to 300K, much smaller than the 600 to 700K envisioned by @wu03 and @lau09.
Future observations
-------------------
Given the brightness of the host star and its peculiar orbit, HD 80606 is likely to remain an important target for further studies. Table \[ephem\] shows the predicted times of ingress and egress for the next transit events according to our reference solution and 68% central confidence intervals.
Date ingress \[BJD\] egress \[BJD\]
-------------------- ----------------------- -------------------------
5 June 2009 $2454987.53 \pm 0.05$ $2454987.974 \pm 0.005$
24 September 2009 $2455098.96 \pm 0.05$ $2455099.409 \pm 0.005$
13-14 January 2010 $2455210.40 \pm 0.05$ $2455210.844 \pm 0.006$
5 May 2010 $2455321.84 \pm 0.05$ $2455322.281 \pm 0.006$
24-25 August 2010 $2455433.27 \pm 0.05$ $2455433.717 \pm 0.006$
14 December 2010 $2455544.71 \pm 0.05$ $2455545.152 \pm 0.007$
4-5 April 2011 $2455656.14 \pm 0.05$ $2455656.588 \pm 0.008$
: Predicted ephemerides for the next transit events[]{data-label="ephem"}
When the transit duration will be measured by observation of future transit events, the corresponding value of the most probable impact parameter and planetary radius can be read directly off Fig. \[mc\_fig\].
Conclusion
==========
Our analysis of combined photometric and spectroscopic observations of HD 80606 have led to new estimates of the stellar and planetary parameters of the system. These values allow relatively tight constraints on the planet’s composition, with a likely mass in heavy elements between 60 and 200M$_\oplus$, which imply a rather efficient accretion of planetesimals most probably during the planet’s formation and early evolution.
We have also shown that the planet’s orbit is probably not aligned with the star’s spin: this strengthens the argument that the planet may have migrated inwards through a Kozai mechanism and thus acquired its large eccentricity (Wu & Murray 2003).
Our analysis can be readily put to the test with future observations of planetary transits, as we predict a total duration of the eclipse of $T_{1-4} = 11.9\pm 1.3$ hours. Furthermore, the analysis of the planet’s primary and secondary transits, and of the planet’s irradiation in the infrared when far from the star will also be extremely fruitful to understand how tidal energy is dissipated: we predict that the intrinsic effective temperature of the planet is between 200 and 300K, smaller than the zero-albedo mean equilibrium temperature due to stellar irradiation, 400K. In contrast, larger effective temperatures ($\sim$700K) have been hypothetized, due to efficient dissipation of heat in the planet (Laughlin et al. 2009).
Financial support for the Consortium from the “Programme national de planétologie” (PNP) of CNRS/INSU, France, and from the Swiss National Science Foundation (FNSRS) are gratefully acknowledged. We also acknowledge support from the UK Science and Technology Facilities Council, the French CNRS and Fundação para a Ciência e a Tecnologia, Portugal.
[^1]: Based on observations made with the 1.20-m and 1.93-m telescopes at Observatoire de Haute-Provence (CNRS), France, by the consortium (program 07A.PNP.CONS), and with a 16-inch telescope at Mt. Hopkins, Arizona, USA, by the MEarth team.
|
Introduction {#sec:introduction}
============
The discovery of carbon nanotubes by Iijima in 1991 [@iijima] has marked the starting point of a scientific revolution [@dresselhauss_book; @ebbesen_book; @ajayan:ebbesen_rpp; @ajayan_cr; @mauro_rev]. This discovery has opened a whole new perspective in the nanoscopic regime of Materials Science and Engineering. Their mechanical [@treacy; @zettl; @krishnan; @lieber; @hernandez; @lum; @yakobson; @buongiorno], electrical [@hamada; @wildoer] and magnetic [@lu; @knechtel; @deHeer] properties provide ample opportunity for the fabrication of nano-scale devices. Indeed some such devices have already been reported in the literature [@stm; @field; @devices1; @devices2]. Since Iijima’s, discovery nanotubes of other chemical composition have also been synthesized, such as $\mbox{B}_x\mbox{C}_y\mbox{N}_z$ composite nanotubes [@bcn1; @bcn2; @bcn3; @bcn4; @bcn5], the so-called inorganic nanotubes consisting of layers of $\mbox{MoS}_2$ or $\mbox{WS}_2$ [@tenne1; @tenne2; @tenne3], or $\mbox{NiCl}_2$ nanotubes[@sloan]. All these compounds have phases which consist of layered structures, and this has lead to the prediction that other materials also capable of crystallizing in layered structures can in principle produce nanotubes. Indeed, theoretical arguments have been presented in the literature for the viability of BN [@corkill_bn; @blase_bn], $\mbox{BC}_3$ [@miyamoto_bc3], $\mbox{BC}_2\mbox{N}$ [@miyamoto_bc2n], GaN [@lee:lee], B [@boustani], GaSe [@cote] and P [@seifert:hernandez] nanotubes. Interestingly, nanotubular structures can also be constructed from biochemical compounts, such as peptides, as demonstrated experimentally by Ghadiri [*et al.*]{} [@ghadiri] and theoretically by Carloni [*et al.*]{} [@carloni].
The tubes first detected by Iijima [@iijima] were multi-wall nanotubes (MWCNT’s), [*i.e.*]{} concentric shells of cylindrical shape, in which each shell is separated from the next by approximately the same distance as the interlayer spacing in graphite. Each shell can be characterized by a pair of indices, (n,m), which determine how the folding of the graphene sheet must be carried out in order to obtain the shell. The shells are usually classified into three different types: (n,0) or [*zig-zag*]{} shells, (n,n) or [*arm-chair*]{} shells, and [*chiral*]{} shells, of indices (n,m) where $\mbox{n} > \mbox{m} > \mbox{0}$. Zig-zag and arm-chair shells are said to be achiral (they can be superimposed onto their mirror images). The ordering of the shells in a MWCNT is usually [*turbostratic*]{}, [*i.e.*]{} the pattern of atomic arrangement may vary (and in general does vary) from one shell to the next, or in other words, different shells usually have different chiralities [@ajayan:ebbesen_rpp].
After the discovery of MWCNT’s, a procedure for synthesizing single-wall nanotubes (SWCNT’s) was found [@thess]. These tubes are found to aggregate into bundles or ropes, and their diameter distribution peaks at around 1.4 nm, although more recently this distribution has been found to vary according to the synthesis conditions [@bandow]. The production of SWNT’s has allowed the experimental corroboration [@wildoer] of a theoretical prediction made by Hamada and coworkers [@hamada] soon after the discovery of MWCNT’s, that the electrical conductivity properties of SWCNT’s are dependent on the (n,m) indices. This prediction stated that SWCNT’s can be either metallic or semi-conducting; if the indices (n,m) of a nanotube obey the relation $\mbox{n}-\mbox{m} = 3 \mbox{q}$ (q = 0, 1, 2, 3, …), then the tube is metallic, otherwise the tube is semi-conducting. This dependence of the electric characteristics of SWCNT’s upon their structure has raised an interest in the possibility of devising new synthetic methods that allowed a structural selection of nanotubes, not only according to their diameter, but also to their chirality. A first step in this direction was achieved by Redlich [*et al.*]{} [@redlich], Carroll [*et al.*]{} [@carroll], and by Terrones [*et al.*]{} [@mauro_apl], who, by adding a certain amount of boron during the synthesis, obtained boron doped MWCNT’s which, interestingly, have an improved crystallinity and a larger aspect ratio (quotient of length to diameter) with respect to tubes obtained in the absence of boron. It was shown that boron appears mostly in the form of clusters associated with the tips of the nanotubes. But most importantly, a recent combined experimental and theoretical study by Blase and coworkers [@blase:terrones] has demonstrated that the MWCNT’s thus obtained consist mostly of zig-zag shells. This study also sheds some light on the role played by boron in favoring the zig-zag structure over others, and tries to explain the larger aspect ratio observed in boron assisted synthesis. This latter issue, though, was addressed by First-Principles (FP) Density Functional Theory (DFT) Molecular Dynamics simulations, and the large computational costs of this technique prevented Blase and coworkers from pursuing a detailed enough study which clarified completely this question.
In this paper we address the problem of boron assisted nanotube growth using a Tight Binding model [@tb_review]. Tight Binding (TB) is an approximate method which nevertheless is capable of providing extremely accurate results in favorable systems. Its main advantage with respect to FP methodologies is its comparatively low computational cost, which often allows a more extensive study than is practical or even possible with higher levels of theory. In this work we have used the Density-Functional Tight Binding (DFTB) model due to Porezag [*et al*]{} [@porezag], about which we give more details in Section \[sec:computational\]. This model has proved to be very accurate for carbon based systems. We have used DFTB to perform a series of static and dynamical simulations of SWCNT’s, with and without boron present, with the aim of understanding the effects of the presence of boron on the structural properties of the resulting NT’s. The structure of this paper is as follows: in Section \[sec:computational\] we describe briefly the DFTB model, and provide previous examples of its successes in order to justify its use here. We also describe the calculations which are reported in the remaining of the paper. Section \[sec:results\] is devoted to a discussion of our simulation results, and we summarize our conclusions in Section \[sec:concs\].
Computational Details {#sec:computational}
=====================
Model {#sub:model}
-----
DFTB is is a non-orthogonal Tight Binding scheme in which a parametrisation is constructed directly from DFT calculations using atomic-like orbitals in the basis set, and adopting a two-center approximation for the Hamiltonian matrix elements. For more details on the parametrisation used here the reader should consult references [@porezag; @widany]. The DFTB scheme has proved to be extremely successful in the modeling of carbon-based systems, in particular carbon clusters and nanostructures. Fowler [*et al.*]{} [@fowler] have used it to analyze the energetic ordering of all 426 cage structures containing 5, 6 and 7-membered rings in $\mbox{C}_{40}$. Ayuela [*et al.*]{} [@ayuela] have found, using DFTB and other five semi-empirical methods, a heptagon-containing isomer of $\mbox{C}_{62}$ which was predicted to be more stable than any of the other 2385 classical fullerene isomers ([*i.e.*]{} isomers containing only pentagons and hexagons). DFTB has also been used to determine the mechanical properties of single-wall C, BN and some $\mbox{B}_x\mbox{C}_y\mbox{N}_z$ nanotubes [@hernandez], providing results which are in excellent agreement with the available experimental data. More recently DFTB has also been used to study the structural, mechanical and electronic properties of a novel family of laminar carbon structures known as [*Haeckelites*]{}, as well as those of their tubular counterparts [@haeckelites]. Haeckelites consist of pentagons and heptagons in equal number, with an arbitrary number of hexagons. The suitability of DFTB for performing Molecular Dynamics simulations in carbon-based systems has been most recently demonstrated by Fugaciu [*et al.*]{} [@fugaciu], who have used it to study the conversion of diamond nanoparticles to concentric shell fullerenes. The many examples of the use of DFTB in the context of carbon based systems and the accuracy of the results reported give us confidence in the reliability of this theoretical model.
Like in other Tight Binding models [@tb_review], in DFTB the total energy is calculated as the sum of two contributions: the [*band structure*]{} contribution, which is mostly attractive, and the repulsive [*pair-potential*]{} contribution, which accounts for the core-core repulsion and the double-counting of the electron-electron interaction which is implicit in the band structure term. The band structure energy is calculated by straight forward diagonalisation of the Hamiltonian, summing the eigen-values of the occupied states weighted according to their occupation numbers, [*i.e*]{} $$\begin{aligned}
E_{bs} = 2 \sum_n f_n \epsilon_n,
\label{eq:energy}\end{aligned}$$ where $f_n$ is the population of state [*n*]{} (which in our case is equal to 1 for occupied states, and to zero for unoccupied, [*i.e.*]{} we have assumed 0 K electronic temperature), and $\epsilon_n$ is the eigen-value of state [*n*]{}. The factor of two accounts for the degeneracy of spin. The band structure contribution to the atomic force $\mbox{\bf F}_i$ is then calculated from the Hellmann-Feynman theorem: $$\begin{aligned}
\nabla_i\, \epsilon_n = 2 f_n \mbox{\bf C}_n^\dag\left( \nabla_i\, \mbox{\bf H}
- \epsilon_n \nabla_i\, \mbox{\bf S} \right) \mbox{\bf C}_n,
\label{eq:force}\end{aligned}$$ where $\mbox{C}_n$ is the vector representation of eigen-state [*n*]{}, [**H**]{} is the Hamiltonian matrix, and [**S**]{} is the overlap. The repulsive pair potential energy and force contributions are trivially added to equations \[eq:energy\] and \[eq:force\], respectively.
Calculations {#sub:calculations}
------------
We have performed two types of calculations: a) structural relaxation calculations, in which, using the Conjugate Gradients (CG) technique [@numericalrecipes], the positions of the atoms in a system are displaced until a minimum in the potential energy hyper-surface is found; and b) molecular dynamics (MD) simulations, which we have employed to study the time evolution of the systems considered here at a range of temperatures. More specifically we have performed canonical ensemble molecular dynamics calculations, in which the conserved quantities are the number of atoms, the volume and the average temperature (NVT-MD). We have implemented the NVT-MD algorithm of Nosé as modified by Hoover [@nose; @hoover]. In this algorithm the physical system of interest is extended by the addition of an extra degree of freedom, playing the role of a thermostat, which interacts with the physical system in such a way as to fix the average temperature. In this algorithm the mass associated with the thermostat is somewhat arbitrary, as it does not affect the value of the average temperature, only the size of the instantaneous temperature fluctuations. We have chosen this mass to be equal to the total mass of the physical system. We have used the implementation of the Nosé-Hoover algorithm described by Frenkel and Smit [@frenkel:smit]. Although the total energy is not conserved in NVT-MD, there is a conserved quantity which can be monitored to ensure the correctness of the implementation. This magnitude plays the role of total energy of the extended system. In our simulations we have used a time-step of 1 fs, which has proved to be sufficiently small to maintain the conserved quantity oscillations smaller than 1 part in 10000 during the length of the simulations, with no appreciable drift.
Our studies have covered a range of temperatures, namely 1000, 2000, 2500 and 3000 K. At each temperature the dynamics of the system were monitored during at least 10 ps. The simulations were carried out sequentially, using the coordinates and velocities at the end of a run at a given temperature as a seed for the simulation at the next higher temperature. The transition from one temperature to another was made by gradually increasing the temperature of the thermostat at a rate of 0.5 K/fs. To use a lower heating rate would be prohibitive, in view of the computational costs involved. Nevertheless, once the thermostat reached the desired temperature, the system was allowed to reach equilibrium at the new temperature during 1 ps, in order to minimize as much as possible the effects of such a high heating rate, before performing the simulation at the new temperature.
Simulation Results {#sec:results}
==================
Carbon nanotubes in the absence of boron {#sub:undoped}
----------------------------------------
Before addressing the effect of boron on the structure and growth mechanism of nanotubes, it is necessary to study the dynamics of carbon nanotubes in the absence of boron. In so doing we can also assess the quality and appropriateness of the model. In order to study the dynamics of the open ended tubes, we first considered the systems illustrated in Figure \[fig:starting\]. These systems are a (10,0) nanotube, with one end saturated by ten H atoms, and a (5,5) nanotube, also with one end saturated by ten H atoms. Both tubes consist of 120 C atoms, plus the 10 H atoms. Figure \[fig:starting\] shows the structures obtained after a CG relaxation, which were then used as starting configurations for the NVT-MD simulations. To facilitate the analysis of the MD simulations, the 10 H atoms were kept fixed throughout the dynamical simulations. While this is an approximation, we have chosen a tube section as large as was practical in order to minimise the possible effects of having the H atoms frozen on the dynamics at the other end of the tube.
[**(a)**]{}
[**(b)**]{}
We first discuss the case of the (10,0) nanotube. At the lowest temperature considered (1000 K) the structure of the nanotube remains unaltered except for the normal thermal oscillations around the equilibrium atomic positions. All the hexagonal rings present in the initial structure (Figure \[fig:starting\]) retain their identity, as can be seen in Figure \[fig:1000-2000K\](a), which illustrates the configuration attained after 10 ps of MD run at this temperature. In view of the fact that the simulation of the open nanotube end at 1000 K failed to produce any structural reordering within this time scale, we proceeded to heat the system up to 2000 K. At this temperature, structural changes begin to appear in the open edges of the nanotube, as can be seen in Figure \[fig:1000-2000K\](b). Indeed, some hexagonal rings at the edge have been broken, resulting in chains of carbon atoms, while others have fused into pentagon-heptagon (5/7) pairs. After 10 ps at 2000 K, two pentagonal rings can be seen, as well as one heptagon. These non-hexagonal rings are produced by the fusing of two hexagons, to give a pentagon and a heptagon. However, we find that heptagons appear to be less stable, and break more easily into carbon chains that remain attached to the nanotube edge, while the pentagons remain stable. The presence of these pentagons induces curvature in the structure, so the edge of the nanotube bends inwards.
[(a)]{}
[(b)]{}
Clearly, a tendency to form a fullerene-like cap is observed, a situation that is after all energetically more stable than the open edge. Therefore, we continued the simulation at a temperature of 2500 K. Like before, the temperature of the thermostat was increased at a rate of 0.5 K/fs, and once the desired thermostat temperature was achieved the combined system of nanotube and thermostat was allowed to equilibrate during 1 ps. The dynamics of the system were then monitored during a further 20 ps. Figure \[fig:dynamics\] illustrates six representative configurations of the system at this temperature. As can be seen, the process of tube closure initiated at 2000 K is further facilitated at 2500 K. The chains of carbon atoms which were seen already at 2000 K can now form links bridging across the open edge. This, combined with the presence of the pentagonal rings at or close to the edge, has the overall effect of bending inwards the nanotube walls. As the simulation proceeds, the number of freely oscillating chains is reduced, due to their tendency to bond into the dome-like structure being formed. Eventually the tube closure is completed, and all carbon atoms are three-coordinated in an $\mbox{\em sp}^2$ hybridisation fashion. Although the resulting closed structure is highly strained, due to the presence of some small rings (a tetragon can be seen) and to the adjacency of pentagons, this is certainly a deep local minimum of the potential energy hyper-surface, as the structure, once closed, remains unaltered (except for thermal oscillations), even after heating up the system to 3000 K and following the dynamics at this temperature for a further 10 ps. Obviously, the time-scale for structural re-orderings once the nanotube has closed is much larger than the time-scale for the closure itself. This is because structural changes now must happen via the Stone-Wales [@stone:wales] mechanism, which has a high activation barrier, and therefore occurs very infrequently [@tiffany].
[(a)]{}
[(b)]{}
[(c)]{}
[(d)]{}
[(e)]{}
[(f)]{}
We now discuss the results obtained from the simulation of the (5,5) nanotube. During the first 10 ps of dynamics, in which the average temperature was fixed at 1000 K, one of the edges of a hexagon at the end of the nanotube is broken, and as a result a carbon chain is formed which oscillates under the influence of the thermal motion, but no (5/7) complexes are observed at this temperature. When the temperature is increased to 2000 K two more hexagons break, but still no (5/7) pairs are formed. At 2500 K nearly all edge hexagons have broken, and consequently there is an abundance of carbon chains. Among these the first two pentagons can be observed, but there are still no signs of heptagons. This is probably because the mechanism of pentagon formation in this case is somewhat different than in the (10,0) nanotube. In the latter, pentagons and heptagons occur in pairs, resulting from the fusion of two edge hexagons. Here the pentagons appear from the re-bonding of chains resulting from the breaking of edge hexagons. The presence of these pentagons induces curvature in the structure, which will eventually cause the tube closure. At 3000 K a total of three pentagons are seen, and the structure is rapidly evolving towards complete closure, but this only happens beyond 10 ps of dynamics at this temperature.
Our results show unambiguously that the open edges of nanotubes are unstable, and at sufficiently high temperatures close spontaneously by forming half-fullerene caps. The time scales we find for the nanotube closure are only orientative, as they may vary slightly for different initial conditions, and certainly they are sensitive to the temperature, but we can say that above 2000 K, closure occurs within a few tens of ps. This finding is in agreement with previous first-principles results [@charlier]. At temperatures between 1000 and 2000 K the time scale for closure (which remains beyond the scope of the simulation times covered here) could be in the order of hundreds of ps or even in the ns range, but even so this seems to us to be too short a time scale for permitting nanotube growth, and this is presumably why single-wall nanotubes cannot be obtained without using transition metal nano-particles which act as catalysts. However, the microscopic details of this catalytic growth process are as yet far from being understood [@kanzow].
In order to gain some insight into the experimental observation that boron doping can result in the production of longer and more crystalline multi-wall nanotubes, and in particular how it favors the zig-zag type tubes over the arm-chair or chiral ones, we have performed a number of static and dynamical simulations in the (10,0) nanotube system with different degrees of boron substitution.
Boron doped nanotubes {#sub:doped}
---------------------
We first address the question as to why boron tends to accumulate at the ends of the nanotubes, as has been experimentally observed [@blase:terrones]. To investigate this site preference, we have performed a series of relaxation calculations in which we compare the energy of nanotube and flat edge structures with a boron atom substituting a carbon atom at different sites. We have considered two different nanotube structures, one a zig-zag nanotube with indices (10,0) (the same as illustrated in Figure \[fig:starting\]), and a (6,6) arm-chair nanotube. For each of the two structures we have taken into account five possible boron substitutional sites, namely: the boron atom substitutes at an edge carbon site (site $\alpha$), the boron is located at a site one bond away from the edge site (site $\beta$), two bonds away (site $\gamma$), three (site $\delta$) and six bonds away from the edge (site $\epsilon$). These positions are illustrated schematically in Figure \[fig:sites\].
=6.5cm
=6.5cm
Our results, given in Table \[table:energies\], show that in both types of nanotubes the most favorable position for boron to substitute at is in the vicinity of the edge . However, there are marked differences between both types of edges. In the zig-zag case, the energy of the structure with boron substituted in site $\beta$ is 1.4 eV higher than the structure with boron substituted at the $\alpha$ site, almost the same as for substitution of boron far (site $\epsilon$) from the edge (1.41 eV). So, in the case of the zig-zag edge it is much more favorable for boron to substitute at the edge than elsewhere in the nanotube. In the arm-chair edge, however, it turns out that the most stable structure is that in which the boron atom is placed at a site one bond away from the edge, i.e. site $\beta$ in Figure \[fig:sites\](b), which turns out to be 0.58 eV more stable than the structure obtained by placing the boron atom directly at the edge (site $\alpha$). Nevertheless, boron will still preferentially locate itself close to the edge rather than far away from it, as the energy difference between the most stable configuration and that with boron placed in site $\epsilon$ is 0.88 eV. These findings can be easily rationalized: in the zig-zag case it is more favorable for boron to be at the edge than elsewhere in the tube because in this way the number of carbon dangling bonds at the nanotube edge is reduced, given that boron has one electron less than carbon. In the arm-chair case, however, the same argument does not apply, because the carbon atoms at the edge are able to pair up forming a stable double bond. By placing a boron atom at the edge a double bond cannot form, and the configuration that results is thus less stable than configuration $\beta$. The same trends are evident from the calculations involving the flat graphene edges, indicating that the curvature of the nanotubes does not play any significant part in determining the energetics of the boron-substituted structures.
----------- ---------- --------- ---------- -----------
bonds (10,0) zig-zag (6,6) arm-chair
from edge nanotube edge nanotube edge
0 0.00 0.00 0.58 0.48
1 1.39 0.99 0.00 0.00
2 0.57 0.67 0.60 0.66
3 1.30 1.42 0.83 0.75
6 1.41 1.35 0.88 0.88
----------- ---------- --------- ---------- -----------
: Energies of boron-substituted structures for zig-zag and arm-chair nanotube and graphene edges. The energies are given in eV relative to the most stable structure in each case. \[table:energies\]
These static calculations illustrate how boron could play a role in favoring zig-zag structures over arm-chair ones, as has been found experimentally. Our findings are in very good agreement with the DFT plane-wave pseudo-potential results reported by Blase and coworkers [@blase:terrones]. However, these results do not give any explanation for the increased length of the nanotubes synthesized in the presence of boron. Thus, in order to gain some insight into how boron can assist the production of longer nanotubes, we have performed a series of MD simulations of a (10,0) nanotube with varying degrees of boron substitution at the edge. We have employed the same structure as illustrated in Figure \[fig:starting\], but placing four, six, eight and ten boron atoms at the tube edge, substituting as many carbon atoms in each case. Starting from such configurations, we have performed MD simulations following the same pattern described above for the undoped carbon nanotubes. Let us comment briefly on the details of the dynamics in each case. In all instances, at the lowest temperature considered (1000 K), the structures remain unaltered during the time spanned at this temperature (10 ps); all hexagonal rings present initially maintain their identity during this time, and it is not until the system is run at a temperature of 2000 K that structural changes begin to appear.
When only four boron atoms are present, after 10 ps of dynamics at 2000 K two (5/7) pairs are formed at the tube edge. The structure of these (5/7) pairs is such that the common edge between both rings always incorporates a boron atom \[see Figure \[fig:5/7\](a)\]. This is a structural feature that repeats itself frequently in the other cases considered, as will be seen below. At this temperature, a transient $\mbox{C}_4$ ring, and a $\mbox{B}_2\mbox{C}_6$ octagonal ring are also seen. When the temperature is raised further to 2500 K, the boron containing pentagons still survive, but the heptagonal rings break, and carbon chains linked to the pentagons are formed. These chains, of length equal to two or three bonds, oscillate widely, eventually linking with similar chains or unsaturated bonds along the open edge, thus initiating the tube closure. The tube is closed after approximately 8.5 ps at this temperature. Given the low concentration of boron at the tip, the dynamics observed here is rather similar to that already described for the pure carbon case.
[**(a)**]{}
=6.5cm
[**(b)**]{}
=6.5cm
[**(c)**]{} =6.5cm
When the number of boron atoms is increased to 6, at the temperature of 2000 K two (5/7) complexes like those observed in the four-boron atom case are formed, but the structure is otherwise unaltered. These boron-containing (5/7) pairs are remarkably stable; they remain intact even after 10 ps at 2500 K. In this particular case, it is necessary to rise the temperature further, to 3000 K, for the tube closure to occur. This happens only after 8 ps of dynamics at this temperature, which is indicative of a higher stability of the open edge when compared to the tube with only four boron atoms. In the case of the nanotube doped with eight boron atoms, the structure develops two (5/7) pairs within 10 ps at 2000 K, much like in the previous cases. However, once the temperature is elevated to 2500 K, the structure closes completely within 7.5 ps. This result appears to contradict the thesis that increasing the amount of boron at the tube edge delays the closure process, but in fact this is not necessarily so, as will be argued below.
Finally, let us consider the case of the tube with ten boron atoms. Like in all the cases analyzed earlier, structural changes at the open edge begin to appear at a temperature of 2000 K, at which a boron-containing (5/7) pair is formed. Three such complexes can be seen after 10 ps at 2500 K, but the structure is otherwise unchanged. It is necessary to reach a temperature of 3000 K and monitor the dynamics of the system for 11 ps at this temperature before the tube can be said to be completely closed. We note that Blase and coworkers [@blase:terrones], who have also performed MD simulations of this particular system using DFT with a basis set of plane-waves and the pseudo-potential approximation up to temperatures of 2500 K for 10 ps, failed to observe the closure. The results reported here indicate that, at such a temperature, it is most likely that the closure would not occur below 10 ps, as it required 11 ps at 3000 K to observe the closure in the simulations that we have performed.
[**(a)**]{}
=3.5cm
[**(b)**]{}
=3.5cm
[**(c)**]{}
=3.5cm
[**(d)**]{}
=3.5cm
Figure \[fig:boron\_dynamics\] illustrates the structures obtained at the end of the MD simulations described above. As can be seen, regardless of the amount of boron present initially at the tube edge, the final structures are invariably closed, but as seen in the discussion, there are notable differences observed during the dynamics in each case. That the nanotubes evolve in such a way as to achieve a closed structure is, after all, not so surprising, given that the closed structures, with all atoms connected in an $\mbox{\em sp\/}^2$ network are always more stable than the corresponding open structures. For example, the structure illustrated in Figure \[fig:boron\_dynamics\](d), once relaxed at 0 K, is 13.7 eV more stable than the corresponding open structure (a figure to be compared with an energy difference of 17 eV, obtained for the pure carbon case using the same two structures). Nevertheless, as can be seen from the discussion above, the details of the closure process vary according to the amount of boron present at the edge.
Our MD results seem to indicate that a delay of the tube closure occurs when boron is present in the system. However, it is difficult to establish this beyond doubt purely on the basis of a small number of MD trajectories, because at each temperature and boron composition there will be a distribution of possible closure times (depending on the initial conditions) for a given type of nanotube. To get a clear picture of how the closure time distribution changes as the amount of boron is increased would require a very large sample of MD simulations, much larger than would be practical to carry out given the computational costs involved. Our simulations are clearly insufficient in number to reveal unambiguously the changing trends in closure times with increasing boron content, which could explain the apparently contradicting result obtained for the tube with eight boron atoms, which closes faster and at a lower temperature than the tube with only six boron atoms. Nevertheless, these simulations are sufficient to clarify the role played by boron when present at the open edge of a nanotube, and are thus extremely revealing. They indicate that before the tube closure can begin to occur, (5/7) complexes form at the edge. This is true of both the pure carbon and the boron doped tubes. Under the effect of the rapid thermal motion the heptagons seem to be more prone to breaking, and the chains of atoms thus formed initiate or can participate in the tube closure. Interestingly, when boron is present, it seems to be always associated with these structures once they are formed, in the manner depicted in Figure \[fig:5/7\](a).
These observations have lead us to perform a new set of static structural relaxation calculations, in which we investigate the stability of these edge (5/7) complexes in both (n,0) and (n,n) nanotubes. We have compared the energy of a normal zig-zag edge structure, such as that shown in Figure \[fig:starting\] with that of a zig-zag edge containing a (5/7) pair, both in the pure carbon case and in the case in which the (5/7) pair contains a boron atom situated between the pentagon and heptagon at the edge \[Figure \[fig:5/7\](a)\]. It turns out that a (5/7) pair placed at the edge of a pure (10,0) tube lowers the total energy by about 0.56 eV. When boron is present between the pentagon and heptagon, the energy gain is slightly larger, 0.67 eV, according to our calculations. This seems to indicate that the formation of such boron containing edge complexes stabilizes the open edge, and consequently delays its closure.
A (5/7) complex is also possible at an arm-chair nanotube edge, where it is also possible to construct a more complicated edge structure, involving a pentagon and two heptagons, which we call a (7/5/7) edge complex; both structures are illustrated schematically in Figure \[fig:5/7\](b) and \[fig:5/7\](c). Structure \[fig:5/7\](b) is reminiscent of the ring-defect structure discussed by Crespi [*et al.*]{} [@crespi]. We have investigated the stability of these, both in the pure carbon case and when boron is present in the different possible substitutional sites. The results from these calculations are listed in Table \[table:arm\_chair\_edge\_results\]. It turns out that the only reconstructed arm-chair edge which is more stable than the structure $\beta$ shown in Figure \[fig:sites\](b) is that illustrated in Figure \[fig:5/7\](b), with the boron atom placed between the two heptagonal rings, but not forming part of the pentagon ([*i.e.*]{} site $\beta$). Even so, the energy difference is only small (0.33 eV), certainly smaller than the energy gain obtained by the reconstruction in the zig-zag edge.
Position Structure (b) Structure (c)
---------------- --------------- ---------------
$\alpha$ 1.10 3.40
$\beta$ -0.33 3.17
$\gamma$ 0.46 2.67
$\alpha\prime$ – 2.64
: Energies of boron-substituted reconstructed edges for the (6,6) nanotube. The labeling of the edges corresponds to that of Figure \[fig:5/7\] (b) and (c). The energies are given in eV relative to the most stable unreconstructed substituted arm-chair edge structure.[]{data-label="table:arm_chair_edge_results"}
The results obtained from these structural relaxation calculations provide a possible mechanism for explaining both the observed preference for the zig-zag structure and the improved aspect ratio of nanotubes synthesized in the presence of boron. The reconstruction of a zig-zag edge in which boron is present from the saw-tooth structure illustrated in Figure \[fig:starting\] to one containing a (5/7) pair is exothermic, resulting in a significant stabilization of the nanotube. In the case of an undoped nanotube edge, this reconstruction also lowers the energy, but by a smaller amount. Note also that, while similar reconstructions are possible in the case of a boron doped arm-chair nanotube, only one such reconstructions lowers the energy with respect to the unreconstructed edge, and only by a small amount. Boron, thus, is seen to significantly stabilize zig-zag edges compared to arm-chair ones and we expect the former tubes to grow better; furthermore the edge (5/7) relaxation mechanism we have found is only effective in zig-zag edges, a fact that contributes to favor this type of tube even further.
Conclusions {#sec:concs}
===========
We have performed an extensive theoretical study of the effects of the presence of varying amounts of boron in the edge of growing nanotubes using TB MD and structural relaxation calculations. In spite of the approximate nature of the TB model used here, it gives results that compare quite well with the available FP data from the work of Blase [*et al.*]{} [@blase:terrones].
We have shown that both carbon and boron-doped nanotubes spontaneously close in a time scale of a few ($\leq 20$) ps in a temperature range of 2500-3000 K. In the case of boron-doped tubes the MD simulations reveal a pattern of longer closure times as the amount of boron at the tip of the nanotube is increased. The presence of boron in the proximity of the tube edge has a stabilizing effect, but this effect is larger in the case of zig-zag nanotubes than in the case of arm-chair ones. Furthermore, we have shown that in (n,0) edges a reconstruction of the edge involving (5/7) pairs stabilizes further the open edge. Although similar reconstructions are topologically possible in the case of (n,n) edges they do not result in any significant stabilization of this type of edge.
These results contribute to a better understanding of the experimental observations [@redlich; @mauro_apl; @blase:terrones] that boron accumulates at the nanotube tips, that it favors zig-zag nanotubes over other structures, and that the nanotubes synthesized in the presence of boron have a larger aspect ratio than other nanotubes.
\* to whom correspondence should be addressed; emai: ehe@icmab.es
E.H. wishes to thank X. Blase, M. Terrones, N. Grobert, W.K. Hsu, H. Terrones and M.J. López for enlightening discussions. E.H. and P.O. thank the EU for financial support under project SATURN (IST-1999-10593). A.R and J.A.A ackowledge support by the DGES (Grant: PB98-0345), JCyL (Grant: VA28/99), and EU TMR NAMITECH project (ERBFMRX-CT96-0067 (DG12-MITH)). I.B. acknowledges support by the DGES (SAB 1995-0670P) of Spain during the sabbatical stay at the University of Valladolid, by the DFG (Deutsche Forschungsgemeinschaft Project SPP-Polyeder), and finally by the Fonds der Chemischen Industrie. We acknowledge the C$^4$ (Centre de Computació i Comunicacions de Catalunya) and CEPBA (European Centre for Parallelism of Barcelona) for the use of their computer facilities.
[99]{}
S. Iijima, Nature (London) [**354**]{}, 56 (1991).
M.S. Dresselhaus, G. Dresselhaus and P.C. Eklund, [*Science of Fullerenes and Carbon Nanotubes*]{} (Academic Press, New York 1996).
T.W. Ebbesen (Ed.), [*Carbon Nanotubes, Preparation and Properties*]{} (CRC Press, Boca Raton 1997).
P.M. Ajayan and T.W. Ebbesen, Rep. Prog. Phys. [**60**]{}, 1025 (1997).
P.M. Ajayan, Chem. Rev. [**99**]{}, 1787 (1999).
M. Terrones, W.K. Hsu, H.W. Kroto and D.R.M. Walton, Topics in Current Chemistry [**199**]{}, 189 (1999).
M.M.J. Treacy, T.W. Ebbesen and J.M. Gibson, Nature (London) [**381**]{}, 678 (1996).
N.G. Chopra and A. Zettl, Solid State Comm. [**105**]{}, 297 (1998).
A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yanilos and M.M.J. Treacy, Phys. Rev. B [**58**]{}, 14013 (1998).
E.W. Wong, P.E. Sheehan and C.M. Lieber, Science [**277**]{}, 1971 (1997).
E. Hernández, C. Goze, P. Bernier and A. Rubio, Phys. Rev. Lett. [**80**]{}, 4502 (1998).
J.P. Lu, Phys. Rev. Lett. [**79**]{}, 1297 (1997).
B.I. Yakobson, C.J. Brabec and J. Bernholc, Phys. Rev. Lett. [**76**]{}, 2411 (1996).
M. Buongiorno Nardelli, B.I. Yakobson and J. Bernholc, Phys. Rev. B [**57**]{}, R4277 (1998).
N. Hamada, S. Sawada and A. Oshiyama, Phys. Rev. Lett. [**68**]{}, 1579 (1992).
J.W.G. Wildöer, L.C. Venema, A.G. Rinzler, R.E. Smalley and C. Dekker, Nature (London) [**391**]{}, 59 (1998).
J.P. Lu, Phys. Rev. Lett. [**74**]{}, 1123 (1995).
W.H. Knechtel, G.S. Düsberg, W.J. Blau, E. Hernández and A. Rubio, Appl. Phys. Lett. [**73**]{}, 1961 (1998).
P. Poncharal, Z.L. Wang, D. Ugarte and W.A. de Heer, Science [**283**]{}, 1513 (1999).
H. Dai, J.H. Hafner, A.G. Rinzler, D.T. Colbert and R.E. Smalley, Nature (London) [**384**]{}, 147 (1996).
W.A. de Heer, A. Chatelain and D. Ugarte, Science [**270**]{}, 1179 (1995); A.G. Rinzler, J.H. Hafner, P. Nikolaev, L. Lou, S.G. Kim, D. Tománek, P. Norlander, D.T. Colbert and R.E. Smalley, Science [**269**]{}, 1550 (1995).
P.G. Collins, A. Zettl, H. Bando, A. Thess and R.E. Smalley, Science [**278**]{}, 100 (1997).
S.J. Tans, A.R.M. Verschueren and C. Dekker, Nature (London), [**393**]{}, 49 (1998)
N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie and A.Zettl, Science [**269**]{}, 966 (1995).
A. Loiseau, F. Willaime, N. Demoncy, G. Hug and H. Pascard, Phys. Rev. Lett. [**76**]{}, 4737 (1996).
M. Terrones, W.K. Hsu, H. Terrones, J.P. Zhang, S. Ramos, J.P. Hare, R. Castillo, K. Prasides, A.K. Cheetham, H.W. Kroto and D.R.M. Walton, Chem. Phys. Lett. [**259**]{}, 568 (1996).
M. Terrones, A.M. Benito, C. Manteca-Diego, W.K. Hsu, O.I. Osman, J.P. Hare, D.G. Reid, H. Terrones, A.K. Cheetham, K. Prasides, H.W. Kroto and D.R.M. Walton, Chem. Phys. Lett. [**257**]{}, 576 (1996).
K. Suenaga, C. Colliex, N. Demoncy, A. Loiseau, H. Pascard and F. Willaime, Science [**278**]{}, 653 (1997).
L. Rapoport, Y. Bilik, Y. Feldman, M. Homyonfer, S.R. Cohen and R. Tenne, Nature (London) [**387**]{}, 791 (1997).
Y. Feldman, E. Wasserman, D.J. Srolovitz and R. Tenne, Science [**267**]{}, 222 (1995).
R. Tenne, L. Margulis, M. Genut and G. Hodes, Nature (London) [**360**]{}, 444 (1992).
Y.R. Hacohen, E. Grunbaum, R. Tenne, J. Sloan and J.L. Hutchinson, Nature (London) [**395**]{}, 336 (1998).
A. Rubio, J.L. Corkill and M.L. Cohen, Phys. Rev. B [**49**]{}, 5081 (1994).
X. Blase, A. Rubio, S.G. Louie and M.L. Cohen, Europhys. Lett. [**28**]{}, 335 (1994).
Y. Miyamoto, A. Rubio, S.G. Louie, and M.L. Cohen, Phys. Rev. B [**50**]{} 18360 (1994).
Y. Miyamoto, A. Rubio, S.G. Louie, and M.L. Cohen, Phys. Rev. B [**50**]{}, 4976 (1994).
S.M. Lee, Y.H. Lee, Y.G. Hwang, J. Elsner, D. Porezag and Th. Frauenheim, Phys. Rev. B [**60**]{}, 7788 (1999).
I. Boustani, A. Quandt, E. Hernández and A. Rubio, J. Chem. Phys. [**110**]{}, 3176 (1999).
M. Cote, M.L. Cohen and D.J. Chadi, Phys. Rev. B [**58**]{}, 4277 (1998).
G. Seifert and E. Hernández, Chem. Phys. Lett. [**318**]{}, 355 (2000).
M.R. Ghadiri, J.R. Granja, R.A. Milligan, D.E. McRee and N. Khazanovich, Nature (London) [**366**]{}, 324 (1993).
P. Carloni, W. Andreoni and M. Parrinello, Phys. Rev. Lett. [**79**]{}, 761 (1997).
A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y.H. Lee, S.G. Kim, A.G. Rinzler, D.T. Colbert, G.E. Scuseria, D. Tománek, J.E. Fischer and R.E. Smalley, Science [**273**]{}, 483 (1996).
S. Bandow, S. Asaka, Y. Saito, A.M. Rao, L. Grigorian, E. Richter and P.C. Eklund, Phys. Rev. Lett. [**80**]{}, 3779 (1998).
P. Redlich, J. Loeffer, P.M. Ajayan, J. Bill, F. Aldinger and M. Ruhle, Chem. Phys. Lett. [**260**]{}, 465 (1996).
D.L. Carroll, P. Redlich, P.M. Ajayan, S. Curran, S. Roth and M. Ruhle, Carbon [**36**]{}, 753 (1998).
M. Terrones, W.K. Hsu, A. Schilder, H. Terrones, N. Grobert, J.P. Hare, Y.Q. Zhu, M. Schwoerer, K. Prassides, H.W. Kroto and D.R.M. Walton, Appl. Phys. A [**66**]{}, 307 (1998).
X. Blase, J.-C. Charlier, A. de Vita, R. Car, P. Redlich, M. Terrones, W.K. Hsu, H. Terrones, D.L. Carroll and P.M. Ajayan, Phys. Rev. Lett. [**83**]{}, 5078 (1999).
For a review on tight-binding see C.M. Goringe, D.R. Bowler and E. Hernández, Rep. Prog. Phys. [**60**]{}, 1447 (1997).
D. Porezag, T. Frauenheim, T. Köhler, G. Seifert and R. Kashner, Phys. Rev. B [**51**]{}, 12947 (1995).
J. Widany, T. Frauenheim, T. Köhler, M. Sternberg, D. Porezag, G. Jungnickel and G. Seifert, Phys. Rev. B [**53**]{}, 4443 (1996).
P.W. Fowler, T. Heine, D. Mitchell, G. Orlandi, R. Schmidt, G. Seifert and F. Zerbetto, J. Chem. Soc., Faraday Trans. [**92**]{}, 2203 (1996).
A. Ayuela, P.W. Fowler, D. Mitchell, R. Schmidt, G. Seifert and F. Zerbetto, J. Phys. Chem. [**100**]{}, 15634 (1996).
H. Terrones, M. Terrones, E. Hernández, N. Grobert, J.-C. Charlier and P.M. Ajayan, Phys. Rev. Lett [**84**]{}, 1716 (2000).
F. Fugaciu, H. Hermann and G. Seifert, Phys. Rev. B [**60**]{}, 10711 (1999).
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, [*Numerical Recipes, the Art of Scientific Computing*]{} 2nd Edition (Cambridge University Press 1992).
S. Nosé, J. Chem. Phys. [**81**]{}, 511 (1984).
W.G. Hoover, Phys. Rev. A [**31**]{}, 1695 (1985).
D. Frenkel and B. Smit, [*Understanding Molecular Simulation*]{}, Academic Press, San Diego (1996).
A.J. Stone and D.J. Wales, Chem. Phys. Lett. [**128**]{}, 501 (1986).
T.R. Welsh and D.J. Wales, J. Chem. Phys. [**109**]{}, 6691 (1998).
J.-C. Charlier, A. De Vita, X. Blase, and R. Car, Science [**275**]{}, 646 (1997).
H. Kanzow and A. Ding, Phys. Rev. B [**60**]{}, 11180 (1999).
V.H. Crespi, M.L. Cohen and A. Rubio, Phys. Rev. Lett. [**79**]{}, 2093 (1997).
|
---
abstract: 'We use exact diagonalization combined with mean-field theory to investigate the phase diagram of the spin-orbital model for cubic vanadates. The spin-orbit coupling competes with Hund’s exchange and triggers a novel phase, with the ordering of $t_{2g}$ orbital magnetic moments stabilized by the tilting of VO$_6$ octahedra. It explains qualitatively spin canting and reduction of magnetization observed in YVO$_3$. At finite temperature an orbital Peierls instability in the $C$-type antiferromagnetic phase induces modulation of magnetic exchange constants even in absence of lattice distortions. The calculated spin structure factor shows a magnon splitting due to the orbital Peierls dimerization.'
author:
- Peter Horsch
- Giniyat Khaliullin
- 'Andrzej M. Oleś'
date: 'April 11, 2003'
title: ' Dimerization versus Orbital Moment Ordering in the Mott insulator YVO$_3$ '
---
Many transition metal oxides are Mott-Hubbard insulators, in which local Coulomb interaction $\propto U$ suppresses charge fluctuations and leads to strongly correlated $3d$ electrons at transition metal ions [@Ima98]. When degenerate $d$ orbitals are partly filled, the [*orbital degrees of freedom*]{} have to be considered on equal footing with electron spins and the magnetic properties of undoped compounds are described by spin-orbital superexchange (SE) models [@Kug82; @Tok00]. Such SE interactions are typically strongly frustrated on a perovskite lattice, leading to enhanced quantum effects [@Fei97]. In systems with $e_g$ orbital degeneracy (manganites, cuprates) this frustration is usually removed by a structural transition that occurs well above the magnetic ordering temperature and lifts the orbital degeneracy via the cooperative Jahn-Teller (JT) effect.
A different situation arises when $t_{2g}$ orbitals are partly filled like in titanium and vanadium oxides. As the JT coupling is much weaker, the intrinsic frustration between spin and orbital degrees of freedom may show up in this case. Unusual magnetic properties of titanates have recently been discussed in terms of coupled spin-orbital SE dynamics [@Kha00]. In addition to SE, spin and orbital occupancies of $t_{2g}$ levels are coupled also via atomic spin-orbit interaction, $H_{so}\propto\Lambda (\vec S\cdot\vec l)$, which is particularly relevant for vanadates with a triplet $^3T_2$ ground state of V$^{3+}$ ions. As a result of [*intersite*]{} SE and [*on-site*]{} $\Lambda$ interactions spin and orbital orderings/fluctuations are strongly coupled, as observed in the canonical spin-orbital system V$_2$O$_3$ [@Bao97], as well as in cubic LaVO$_3$ [@Miy00].
The magnetic properties of YVO$_3$ are particularly puzzling [@Ren00], and indicate dimerization along the FM direction within the $C$-AF phase [@Ulr02]. In this Letter we argue that such an exotic $C$-AF phase follows from the realistic spin-orbital model for vanadates that emphasizes the competition between SE bond physics and intraatomic spin-orbit coupling $\propto\Lambda$. We investigate the phase diagram of this model and show that [*orbital moments*]{} are induced in the $C$-phase by finite $\Lambda$, and form at larger $\Lambda$ a novel orbital moment (OM) phase.
The superexchange in cubic vanadates originates from virtual charge excitations, $d^2_id^2_j\rightarrow d^3_id^1_j$, by the hopping $t$ which couples pairs of identical orbitals. When such processes are analyzed on individual bonds $\langle ij\rangle\parallel\gamma$ along each cubic axis $\gamma=a,b,c$, one finds the spin-orbital Hamiltonian with $S=1$ spins ($J=4t^2/U$) [@Kha01], $$\label{som}
{\cal H}=J\sum_{\gamma}\sum_{\langle ij\rangle\parallel\gamma}\left[
\frac{1}{2}({\vec S}_i\cdot {\vec S}_j+1)
{\hat J}_{ij}^{(\gamma)} + {\hat K}_{ij}^{(\gamma)}\right]
+H_{so},$$ where the orbital operators ${\hat J}_{ij}^{(\gamma)}$ and ${\hat K}_{ij}^{(\gamma)}$ depend on the pseudospin $\tau=1/2$ operators ${\vec\tau}_i=\{\tau_i^x,\tau_i^y,\tau_i^z\}$, given by two orbital flavors active along a given direction $\gamma$. For instance, $yz$ and $zx$ orbitals are active along $c$ axis, and we label them as $a$ and $b$, as they lie in the planes orthogonal to these axes. The general form of the superexchange (\[som\]) was discussed before, and we have shown that strong quantum fluctuations in the orbital sector provide a mechanism for the $C$-AF phase [@Kha01]. When $c$ ($xy$) orbitals are occupied ($n_{ic}=1$), as suggested by the electronic structure [@Saw96] and by the lattice distortions in YVO$_3$ [@Ren00; @Bla01], the electron densities in $a$ and $b$ orbitals satisfy the local constraint $n_{ia}+n_{ib}=1$. The interactions along the $c$ axis simplify then to: $$\begin{aligned}
\label{jc}
{\hat J}_{ij}^{(c)}\!&=&\!
(1+2R)\Big({\vec\tau}_i\cdot {\vec\tau}_j\!+\!\frac{1}{4}\Big)\!
-\! r\Big({\vec\tau}_i\otimes{\vec\tau}_j\!+\!\frac{1}{4}\Big) \!-\! R,
\\
%
\label{kc}
{\hat K}_{ij}^{(c)}\!&=&\!
R\Big({\vec\tau}_i\cdot {\vec\tau}_j\!+\!\frac{1}{4}\Big)
+ r\Big({\vec\tau}_i\otimes{\vec\tau}_j\!+\!\frac{1}{4}\Big); \end{aligned}$$ they involve the fluctuations of $a$ and $b$ orbitals $\propto{\vec\tau}_i\cdot {\vec\tau}_j$, and ${\vec\tau}_i\otimes{\vec\tau}_j=
\tau_i^x\tau_j^x -\tau_i^y\tau_j^y +\tau_i^z\tau_j^z$, while the interactions along the $\gamma=a(b)$ axis depend on the static correlations $\propto n_{ib}n_{jb}$ ($n_{ia}n_{ja}$) only; for instance: $$\begin{aligned}
\label{ja}
{\hat J}_{ij}^{(a)}\!&=&
\frac{1}{2}\Big[(1-r)(1+n_{ib}n_{jb})-R(n_{ib}-n_{jb})^2\Big], \\
\label{ka}
{\hat K}_{ij}^{(a)}\!&=&\frac{1}{2}(R+r)(1+n_{ib}n_{jb}).
\end{aligned}$$ The Hund’s exchange $\eta=J_H/U$ determines the multiplet structure of $d^3$ excited states which enters via the coefficients: $R=\eta/(1-3\eta)$ and $r=\eta/(1+2\eta)$. The pseudospin operators in Eqs. (\[jc\])–(\[kc\]) may be represented by Schwinger bosons: $\tau_i^x= (a_i^{\dagger}b_i^{}+b_i^{\dagger}a_i^{})/2$, $\tau_i^y=i(a_i^{\dagger}b_i^{}-b_i^{\dagger}a_i^{})/2$, $\tau_i^z=(n_{ia}^{}-n_{ib}^{})/2$.
The individual VO$_6$ octahedra are tilted by angle $\phi_i=\pm\phi$, which alternate along the $c$ axis [@Bla01]. As the $xy$ orbital is inactive, two components of the orbital moment $\vec l_i$ are quenched, while the third one (2$\tau_i^y$), parallel to the local axis of a VO$_6$ octahedron, couples to the spin projection. Because of AF correlations of $\tau_i^y$ moments, spin-orbit coupling induces a staggered spin component. As the spin interactions are FM, weak spin-orbit coupling would give no energy gain, if the spins were oriented along the $c$ axis. Thus, finite $\Lambda$ breaks the SU(2) symmetry and favors easy magnetization axis within the $(a,b)$ plane. As quantization axis for $\vec l_i$ ($\vec S_i$) we use the octahedral axis (and its projection on the $(a,b)$ plane), respectively. The spin-orbit coupling in Eq. (\[som\]) is then given by: $$\label{hso}
H_{so}=2\Lambda\sum_i\left(S_i^x\cos\phi_i
+S_i^z\sin\phi_i\right)\tau_i^y,$$ and we use $\lambda=\Lambda/J$ as a free parameter. In order to understand the important consequences of the tilting for the interplay between spin and orbital degrees of freedom, we consider first the idealized structure with $\phi=0$. The coherent spin-and-orbital fluctuations lower then the energy due to on-site correlations $\langle S_i^x\tau_i^y\rangle<0$. Since these fluctuations do not couple to the spin order, no orbital moments can be induced at small $\lambda$ as long as $\phi=0$.
Even in the absence of the spin-orbit term ($\lambda=0$), the vanadate spin-orbital model (\[som\]) poses a highly nontrivial quantum problem. We obtained first qualitative insight into the possible types of magnetic and orbital ordering by investigating the stability of different phases within the mean-field approximation (MFA), but including the leading orbital fluctuations on FM bonds along the $c$ axis. In the absence of Hund’s exchange ($\eta=0$), two orbital flavors experience an antiferro-orbital (AO) coupling on these bonds due to ${\hat J}_{ij}^{(c)}$ Eq. (\[jc\]), but are decoupled within the $(a,b)$ planes (${\hat K}_{ij}^{(a,b)}=0$ [@noteja]). This one-dimensional (1D) system is unstable towards [*dimerization*]{} with orbital singlets and FM interactions at every second bond along the $c$ axis [@She02], stabilizing the orbital valence bond (OVB) phase. The spin interactions ${\hat J}_{ij}^{(a,b)}$ Eq. (\[ja\]) and the intersinglet interactions along $c$ axis are AF. In contrast, for large $\eta$ more energy is gained when the orbital singlets resonate along the $c$ direction, giving uniform FM interactions in the $C$-AF phase [@Kha01]. We determined the quantum energy due to orbital fluctuations using the orbital waves found in the Schulz approximation, known to be accurate for weakly coupled AF chains [@Sch96]. Using this approach, the transition from the OVB to $C$-AF phase \[$\langle S_i^z\rangle=S^z e^{i{\vec R}_i{\vec Q}_C}$ with ${\vec Q}_C=(\pi,\pi,0)$\] takes place at $\eta_c\simeq 0.09$ (Fig. \[fig:phd\]), and the orbital ordering, $\langle\tau_i^z\rangle=\tau^z e^{i{\vec R}_i{\vec Q}_G}$ with ${\vec Q}_G=(\pi,\pi,\pi)$, sets in, promoted by the AO interactions ${\hat K}_{ij}^{(a,b)}$ Eq. (\[ka\]).
The ground state changes qualitatively at finite $\lambda$ and $\phi>0$. The magnetic moments $\langle S_i^z\rangle$ induce then the [*orbital moments*]{}, $2\langle\tau_i^y\rangle=l^z e^{i{\vec R}_i{\vec Q}_A}$, which stagger along the $c$ axis with ${\vec Q}_A=(0,0,\pi)$. This novel type of ordering with $l_i^z\neq 0$ can be described as a staggered ordering of complex orbitals $a \pm ib$ (corresponding to $l_i^z=\mp 1$ eigenstates) that competes at $\eta>\eta_c$ with the staggered ($a/b$) ordering of real orbitals with $\tau^z\neq 0$ in the $C$-AF phase. Already at small $\lambda$ the orbital moments induce in turn opposite to them weak $\langle S_i^x\rangle\neq 0$ moments, lowering the energy by $\langle S_i^x\tau_i^y\rangle<0$. In the OM phase favored at large $\lambda$ (Fig. \[fig:phd\]), the spin order has therefore two components: $\langle S_i^z\rangle=S^z e^{i{\vec R}_i{\vec Q}_C}$, and $\langle S_i^x\rangle=S^x e^{i{\vec R}_i{\vec Q}_G}$.
![ Phase diagram of the spin-orbital model in the $(\eta,\lambda)$ plane at $T=0$, reflecting the competition between orbital valence bond (OVB), staggered orbitals (C-AF) and orbital moment ordering (OM), as obtained by the ED of a four-site embedded chain for $\phi=11^{\circ}$ (circles), and in the MFA (dashed lines). Orbital moments in the OM phase (violet arrows) induce spin canting (blue arrows) with angle $\psi$. (YVO$_3$: $\eta\simeq 0.12$, $\lambda\sim 0.3 - 0.4$ [@notept]). []{data-label="fig:phd"}](omo1n.eps "fig:"){width="8.7cm"} -.1cm
An unbiased information about the spin and orbital degrees of freedom was obtained by the accurate treatment of quantum effects within the exact diagonalization (ED) method. Thereby the coupled spin-orbital excitations, terms $\propto S_i^{\alpha}S_j^{\alpha}\tau_i^{\beta}\tau_j^{\beta}$ in Eq. (\[som\]), are now fully included. We performed ED of four-site chains along $c$ axis, both for free and periodic boundary conditions (PBC).
We were surprised to see that the exact ground state of a [*free chain*]{} at $\eta=\lambda=0$ consists indeed in a very good approximation of two orbital singlets on the external (12) and (34) FM bonds ($\langle\vec\tau_i\cdot \vec\tau_{i+1}\rangle=-0.729$), connected by an AF bond (23) with decoupled orbitals ($\langle\vec\tau_i\cdot \vec\tau_{i+1}\rangle=-0.038$). The spin correlations are FM/AF on the external/central bond, $\langle\vec S_i\cdot\vec S_{i+1}\rangle=0.95$ and $-1.56$. With increasing $\eta$ the AF interaction weakens, the sequence of spin multiplets labelled by the total spin $S_t$ is inverted at $\eta_c\simeq 0.12$, and the ground state changes from a singlet ($S_t=0$) to a high-spin ($S_t=4$) state. At finite $\lambda$ no level crossing occurs, but the nondegenerate ground state describes a smooth crossover in the spin and orbital correlations with increasing $\eta$. We verified that several excited states lie within $0.1J$ away from the ground state — all of them would contribute to thermal fluctuations already at temperatures $T\simeq 30$ K.
![ Pair correlations along $c$ axis: (a) orbital $\langle{\vec\tau}_i\cdot{\vec\tau}_{i+1}\rangle$, (b) spin $\langle {\vec S}_i\cdot{\vec S}_{i+1}\rangle$, and (c) spin components $\langle S_i^z\rangle$ (full lines) and $\langle S_i^x\rangle$ (dashed, long-dashed lines), as functions of $\lambda$ for the OVB ($\lambda<\lambda_c\simeq 0.37$) and the OM ($\lambda>\lambda_c$) phase, found by the ED at $\eta=0.07$. FM/AF bonds $(ij)$ in the OVB phase are shown by solid/dashed lines in (a) and (b).[]{data-label="fig:cor"}](omo2n.eps "fig:"){width="6.8cm"} -.1cm
We simulate a [*cubic system*]{} by including infinitesimal symmetry-breaking dimerization field which favors the orbital singlets on the bonds (12) and (34) in a cluster with the PBC, embedded within one of three phases stable in the MFA (Fig. \[fig:phd\]), with mean-fields determined self-consistently in each phase. All phases are characterized by finite magnetic moments $\langle S_i^z\rangle$, either staggered pairwise (OVB phase), or aligned ($C$-AF phase) along $c$ axis, and weak $\langle S_i^x\rangle$ moments. In addition, the orbital ordering ($l^z\neq 0$) appears in the OM phase, while the orbitals stagger ($\tau^z\neq 0$) in the $C$-AF phase.
By computing the energies of different phases we obtained the phase diagram that confirms the qualitative picture extracted from the MFA (Fig. \[fig:phd\]). All transitions are accompanied by reorientation of spins (Figs. \[fig:cor\] and \[fig:mom\]). The orbital and spin fluctuations change only weakly at small values of $\lambda$, weak $\langle S_i^x\rangle$ moments are ordered pairwise on the bonds (12) and (34), and the spin correlations on the intersinglet bonds are almost classical (Fig. \[fig:cor\]). These features show that the spins and orbitals are almost decoupled and the OVB phase is robust. At $\lambda>\lambda_c$ the on-site correlations $\langle S_i^x\tau_i^y\rangle<0$ dominate, while the orbital fluctuations are suppressed, and the correlation functions $\langle {\vec\tau}_i\cdot {\vec\tau}_{i+1}\rangle$ approach the classical value $-\frac{1}{4}$. As the staggered spin moments $\langle S_i^x\rangle$ are induced, the spin correlations $\langle {\vec S}_i\cdot{\vec S}_{i+1}\rangle$ become soon AF within the OM phase. In this regime the spins follow the spin-orbit coupling $\lambda$, and the FM interaction $J_c$ is frustrated.
![ Crossover from the $C$-AF to the OM phase for increasing $\lambda$ at $\eta=0.12$: (a) orbital order parameters: $\tau^z$ (dashed line), and $l^z$ (solid lines) at $\phi=11^{\circ}$; (b) spin canting angle $\psi=\arctan(S^x/S^z)$ (see Fig. \[fig:phd\]), as obtained for the tilting: $\phi=5^{\circ}$, $11^{\circ}$ and $20^{\circ}$ (dashed, solid and dotted line). []{data-label="fig:mom"}](omo3n.eps "fig:"){width="6.8cm"} -.1cm
At the transition from the $C$-AF to the OM phase the orbital ordering changes from staggered [*real orbitals*]{} ($\tau^z\neq 0$) to staggered [*orbital moments*]{} ($l^z\neq 0$), as shown in Fig. \[fig:mom\](a). As a precursor effect of the forthcoming OM phase, the orbital moments $l_i^z$ are induced already in the $C$-AF phase by increasing $\lambda$. The transition to the OM phase is accelerated by the increasing tilting angle $\phi$ \[Fig. \[fig:mom\](b)\]. Also the spin correlations change here discontinuously at the transition (not shown), similar to the OVB/OM transition \[Fig. \[fig:cor\](b)\]. The staggered spin components in the OM phase $\langle S_i^x\rangle$ are similarly large to those shown in Fig. \[fig:cor\](c) for smaller $\eta$, and the spin canting angle $\psi$ approaches $\frac{\pi}{2}-\phi$ in the regime $\lambda\gg 1$. For realistic parameters for YVO$_3$: $J\sim 30$ meV [@Kha01], $\eta\sim 0.12$ (estimated with $J_H=0.64$ eV and intraorbital element $U=5.5$ eV for V$^{2+}$ ions [@Zaa90]), and $\lambda\sim 0.3 - 0.4$ (considering $\Lambda\simeq 13$ meV for free V$^{3+}$ ions [@Abrag]), one finds a competition between the staggered $a/b$ orbital order parameter ($\tau^z\sim 0.25$) and the orbital magnetic moments ($l^z\sim 0.30-0.35$). This reflects the interplay between intersite SE and on-site spin-orbit couplings, and hence orbital and spin moments are not collinear (except for large $\lambda$ values), — in contrast to the conventional picture where orbital moments induced by $\lambda$ coupling are antiparallel to spin, as suggested e.g. for V$_2$O$_3$ [@Tan02].
Finally, we turn to finite temperatures. While the $C$-phase cannot dimerize at $T=0$, it dimerizes at finite $T$ due to the intrinsic instability towards alternating orbital singlets [@Sir02]. The spin correlations $\langle {\vec S}_i\cdot{\vec S}_{i+1}\rangle$ \[Fig. \[fig:dim\](a)\], found using open boundary conditions, alternate between strong and weak FM bonds due to the orbital Peierls dimerization, $2\delta_{\tau}=
|\langle {\vec\tau}_i\cdot {\vec\tau}_{i+1}\rangle
-\langle {\vec\tau}_i\cdot {\vec\tau}_{i-1}\rangle|$, which has a distinct maximum at $T\simeq 0.24 J$ for $\eta=0.12$. Consistent with our discussion above, the modulation of $\langle{\vec S}_i\cdot{\vec S}_{i+1}\rangle$ vanishes in the C-phase at low $T$.
![ Dimerization in the $C$-AF phase at $T>0$: (a) spin-spin correlations $\langle{\vec S}_i\cdot{\vec S}_{i+1}\rangle$ on strong and weak FM bonds (solid and dashed line); (b) spin response $S({\bf q},\omega)$ in the dimerized $C$-AF phase for $q=0, \pi/2, \pi$. Inset: filled (open) circles indicate strong (weak) features in $S({\bf q},\omega)$; lines show the fitted spin-wave dispersion. Parameters: $\eta=0.12$, $\lambda=0.4$. []{data-label="fig:dim"}](omo4a.eps "fig:"){width="7.1cm"} ![ Dimerization in the $C$-AF phase at $T>0$: (a) spin-spin correlations $\langle{\vec S}_i\cdot{\vec S}_{i+1}\rangle$ on strong and weak FM bonds (solid and dashed line); (b) spin response $S({\bf q},\omega)$ in the dimerized $C$-AF phase for $q=0, \pi/2, \pi$. Inset: filled (open) circles indicate strong (weak) features in $S({\bf q},\omega)$; lines show the fitted spin-wave dispersion. Parameters: $\eta=0.12$, $\lambda=0.4$. []{data-label="fig:dim"}](omo4b.eps "fig:"){width="6.8cm"} -.1cm
Up to now, the sole experimental evidence for dimerization of the $C$-phase is the splitting of FM spin waves in the neutron scattering study of Ulrich [*et al.*]{} [@Ulr02]. Fig. 4(b) shows the dynamical spin structure factor $S(q,\omega)$ [@Aue94] obtained by exact diagonalization of a 4-site cluster with PBC at $T=0$, assuming the same orbital dimerization $\langle{\vec\tau}_i\cdot{\vec\tau}_{i+1}\rangle$ as found above for $T/J=0.25$. At $q=\pi/2$ we observe a splitting of the spin wave similar to experiment. The finite energy of the $q=0$ mode results from the $\lambda$-term and the mean-field coupling to neighbor chains. Additional features seen in $S(q,\omega)$, e.g. for $q=\pi$ at $\omega \sim 1.25 J$, we attribute to the coupling to orbital excitations. The spin-wave energies can be fitted by a simple Heisenberg model with two FM coupling constants: $J_{c1}=5.7$, $J_{c2}=3.3$ meV, and a small anisotropy term, as shown in the inset (solid lines). Although these values are strongly reduced by spin-orbit coupling Eq. (\[hso\]), they are still larger than those extracted from the spin waves in YVO$_3$: $J_{c1}^{exp}=4.0$ and $J_{c2}^{exp}=2.2$ meV at $T=85$ K [@Ulr02]. We attribute this overestimate of exchange interactions to quantum fluctuations involving the occupancy of $xy$-orbitals; this will be treated elsewhere.
Summarizing, we have shown that the spin-orbit coupling $\Lambda$ competes with Hund’s exchange in the spin-orbital model for cubic vanadates. It leads to a new orbital moment ordered phase at large $\Lambda$ and can explain qualitatively the spin canting and large reduction of magnetization in the $C$-phase at smaller $\Lambda$. We argue that the 1D orbital Peierls instability observed recently in the $C$-AF phase of YVO$_3$ [@Ulr02] (along the $c$ axis) emerges from a combination of quantum effects due to orbital moments at $\Lambda>0$, and thermal fluctuations which favor dimerized orbital and spin correlations.
We thank B. Keimer and C. Ulrich for stimulating discussions. This work was supported by the Committee of Scientific Research (KBN), Project No. 5 P03B 055 20.
M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. [**70**]{}, 1039 (1998).
K.I. Kugel and D.I. Khomskii, Sov. Phys. Usp. [**25**]{}, 231 (1982).
Y. Tokura and N. Nagaosa, Science [**288**]{}, 462 (2000).
L.F. Feiner, A.M. Oleś, and J. Zaanen, , 2799 (1997).
B. Keimer [*et al.*]{}, , 3946 (2000); G. Khaliullin and S. Maekawa, [*ibid.*]{} [**85**]{}, 3950 (2000); G. Khaliullin and S. Okamoto, [*ibid.*]{} [**89**]{}, 167201 (2002).
W. Bao [*et al.*]{}, , 507 (1997); , 12727 (1998); L. Paolasini [*et al.*]{}, , 4719 (1999).
S. Miyasaka, T. Okuda, and Y. Tokura, , 5388 (2000).
Y. Ren [*et al.*]{}, , 6577 (2000); M. Noguchi [*et al.*]{}, [*ibid.*]{} [**62**]{}, R9271 (2000).
C. Ulrich [*et al.*]{}, arXiv:cond-mat/0211589, unpublished (2002).
G. Khaliullin, P. Horsch, and A. M. Oleś, , 3879 (2001).
H. Sawada [*et al.*]{}, , 12 742 (1996); H. Sawada and K. Terakura, [*ibid.*]{} [**58**]{}, 6831 (1998).
G.R. Blake [*et al.*]{}, , 245501 (2001).
Spins are AF, so ${\hat J}_{ij}^{(a,b)}$ do not contribute either.
S.Q. Shen, X.C. Xie, and F.C. Zhang, , 027201 (2002).
H. Schulz, , 2790 (1996).
Below $T_{N1}=$77 K, the $C$-AF phase of YVO$_3$ changes to the $G$-type AF spin structure [@Ren00]. As we have argued [@Kha01], the $G$-AF phase is stabilized at low temperature by the JT effect, and the phase transition at $T_{N1}$ is promoted by the larger (spin and orbital) entropy of the $C$-AF phase.
J. Zaanen and G.A. Sawatzky, J. Solid State Chem. [**88**]{}, 8 (1990).
A. Abragam and B. Bleaney, [*Electron Paramagnetic Resonance of Transition Ions*]{} (Oxford University Press, New York, 1970).
A. Tanaka, J. Phys. Soc. Jpn. [**71**]{}, 1091 (2002).
See, e.g.: A. Auerbach, [*Interacting Electrons and Quantum Magnetism*]{} (Springer, New York, 1994).
Similar conclusion is also reached by J. Sirker and G. Khaliullin, , 100408(R) (2003).
|
---
abstract: 'We consider a three-node decode-and-forward (DF) half-duplex relaying system, where the source first harvests RF energy from the relay, and then uses this energy to transmit information to the destination via the relay. We assume that the information transfer and wireless power transfer phases alternate over time in the same frequency band, and their [*time fraction*]{} (TF) may change or be fixed from one transmission epoch (fading state) to the next. For this system, we maximize the achievable average data rate. Thereby, we propose two schemes: (1) jointly optimal power and TF allocation, and (2) optimal power allocation with fixed TF. Due to the small amounts of harvested power at the source, the two schemes achieve similar information rates, but yield significant performance gains compared to a benchmark system with fixed power and fixed TF allocation.'
author:
- 'Zoran Hadzi-Velkov, Nikola Zlatanov, Trung Q. Duong, and Robert Schober [^1] [^2] [^3] [^4]'
title: 'Rate Maximization of Decode-and-Forward Relaying Systems with RF Energy Harvesting'
---
Energy harvesting, wireless power transfer, decode-and-forward relays.
Introduction
============
Energy harvesting (EH) technology may provide a perpetual power supply to energy-constrained wireless systems, such as sensor networks. In order to maintain reliable EH-based communication, dedicated far-field radio frequency (RF) radiation may be used as energy supply for the EH transmitters, an approach known as wireless power transfer (WPT) \[\[lit1\]\], \[\[lit2\]\]. If the signal used for information transfer is also used for simultaneous energy transfer, then a fundamental tradeoff exists between energy and information transfer, as has been shown, e.g. for the noisy channel in \[\[lit1\]\], the multi-user channel in \[\[lit3\]\], and the multiple-input multiple-output (MIMO) broadcast channel in \[\[lit4\]\]. In \[\[lit1\]\] and \[\[lit3\]\], the receiver is able to decode information and extract power simultaneously, which, as argued in \[\[lit4\]\], may not be possible in practice and the information and energy signals have to be separated either by time-switching or power-splitting architectures at the receiver.
EH may also be beneficial in relay networks \[\[lit5\]\]-\[\[lit8\]\]. In \[\[lit6\]\], the authors studied the average data rate of an amplify-and-forward (AF) relaying system for both time-switching and power-splitting protocols. The average data rate and optimal power allocation for decode-and-forward (DF) relaying systems were considered in \[\[lit7\]\]. DF relaying with multiple source-destination pairs transmitting at fixed rates was studied in \[\[lit8\]\]. In \[\[lit5\]\]-\[\[lit8\]\], the relay is an EH node that harvests RF energy from the source, whereas the source sends both information and energy to the EH relay.
In this paper, we study the performance of a DF relaying system, where the source first harvests RF energy from the relay, and then transmits its information to the destination via the relay. This scenario may arise, for example, in an EH sensor network where the sensor is an EH device that transmits some measured data to a base station via a fixed relay, which is used for coverage extension and for powering the sensor by WPT. To the best of the authors’ knowledge, \[\[lit9\]\] is the only other work that considers the case of a relay sending RF energy to an EH source. However, the system model in our paper is somewhat different from the model in \[\[lit9\]\], as it assumes: (a) random channel fading, and (b) adaptive time division between the EH and information transmission (IT) phases.
System Model
============
We consider a relaying system that consists of an EH source (EHS), a relay, and a destination $D$, where the direct source-destination link is not available. The EHS does not have a permanent power supply, but instead has a rechargeable battery that harvests RF energy broadcasted by the relay. The relay is used both for DF relaying of information from the EHS to $D$, and for broadcasting RF energy to the EHS.
We assume that the source-relay and relay-destination links experience independent fading and independent additive white Gaussian noise (AWGN) with power $N_0$. The fading in both channels follows the quasi-static (block) fading model, i.e., the channels are constant during each fading block, but change from one block to the next. Let the duration of one fading block be denoted by $T$. In block $i$, the fading power gains of the EHS-relay and relay-$D$ channels are denoted by $x_i'$ and $y_i'$, respectively. For convenience, these gains are normalized by the AWGN power, yielding $x_i = x_i'/N_0$ and $y_i = y_i'/N_0$ with average values $\Omega_X = E[x_i']/N_0$ and $\Omega_Y = E[y_i']/N_0$, respectively, where $E[\cdot]$ denotes expectation. The reverse relay-EHS and $D$-relay channels are assumed to be reciprocal to the respective EHS-relay and relay-$D$ channels.
The information transmission from the EHS to $D$ and the WPT from the relay to the EHS are realized as alternating half-duplex signal transmissions over the same frequency band, organized in transmission epochs. The transmission epochs have a duration of $T$, and therefore, one epoch coincides with the duration of a single fading block. Each epoch consists of three phases: an EH phase, an IT phase 1 (IT1), and an IT phase 2 (IT2). During the EH phase, the relay broadcasts RF energy, which is harvested by the EHS. During IT1, the EHS spends the total energy harvested in the EH phase to transmit information to the relay, which decodes it, and then forwards this information to $D$ in IT2.
Let us consider $M \to \infty$ transmission epochs. In epoch $i$, the duration of the EH phase is set to $\tau_i T$, and the durations of IT1 and IT2 are both set to $(1-\tau_i)T/2$, where $\tau_i$ is referred to as the [*time-fraction*]{} (TF) parameter ($0 < \tau_i < 1$). The relay’s output power in epoch $i$ is denoted by $p_i$, and assumed constant for the entire epoch duration. In fact, since the relay is primarily a communication device (i.e., its circuitry is not specialized for WPT), the transmit power constraints for the EH and IT phases are similar.
The RF energy received during the EH phase at the source is denoted by $E_{Si}$, and given by $E_{Si} = N_0 p_i x_i \tau_i T$. In IT1, the EHS spends the total harvested energy $E_{Si}$ for transmitting a complex-valued Gaussian codeword of duration $(1-\tau_i) T/2$ with an output power $$\label{rav1}
P_S(i) = \frac{E_{Si}}{(1 - \tau_i) T/2} = \frac{2N_0 p_i x_i \tau_i}{1 - \tau_i}.$$ In IT2, the relay forwards all information received from the EHS in IT1 to the destination. Thereby, the relay transmits a complex-valued Gaussian codeword of length $T(1-\tau_i)/2$ with output power $p_i$. Thereby, the maximum amount of data that can be transmitted successfully from the EHS to the destination via the relay in epoch $i$ is given by $R(i) = \min \{ C_1(\tau_i, p_i), C_2(\tau_i, p_i) \}$, where $$\label{c1def}
C_1(\tau_i, p_i) = \frac{1 - \tau_i}{2} \log \left( 1 + \frac{p_i a_i \tau_i}{1 - \tau_i} \right),$$ $$\label{c2def}
C_2(\tau_i, p_i) = \frac{1 - \tau_i}{2} \log \left( 1 + p _i y_i \right), \qquad$$ with $a_i = 2N_0 x_i^2$. Note that (\[c1def\]) is obtained by inserting (\[rav1\]) into $\log(1+P_S(i)x_i)$. As $M\to\infty$, the achieved average data rate with the proposed scheme is given by $$\begin{aligned}
\label{eq_1}
\bar R = \lim\limits_{M\to\infty}\frac{1}{M} \sum_{i=1}^M \min \{ C_1(\tau_i, p_i), C_2(\tau_i, p_i) \}.\end{aligned}$$
Maximization of the Average Data Rate
=====================================
We aim at maximizing $\bar R$, given by (\[eq\_1\]), subject to the relay’s average power constraint $\bar P$, by proposing two schemes for allocation of the relay’s output power $p_i$ and the TF parameter $\tau_i$: (1) the jointly optimal power and time allocation (JOPTA) scheme, and (2) the optimal power allocation (OPA) scheme. In the JOPTA scheme, $p_i$ and $\tau_i$ are jointly optimized, whereas, in the OPA scheme, $p_i$ is optimized for some fixed TF parameter. The data rates achieved with the proposed schemes are feasible if, in each epoch $i$, the three nodes have perfect channel state information (CSI) of both fading links, $x_i$ and $y_i$. These data rates can serve as upper bounds for the imperfect CSI case.
Jointly Optimal Power and Time Allocation
-----------------------------------------
We define the following optimization problem for the proposed JOPTA scheme: $$\label{op1}
\underset{p_i\geq 0, \, 0<\tau_i<1, \, \forall i} {\text{max}} \ \frac{1}{M} \sum_{i=1}^M \min \{ C_1(\tau_i, p_i), C_2(\tau_i, p_i) \} \notag$$ $$\begin{aligned}
\label{rav4}
\text{s.t.} &\text{C1}:& \frac{1}{M} \sum_{i=1}^M p_i \frac{1+\tau_i}{2} \leq \bar P.\end{aligned}$$ In (\[rav4\]), $\text{C1}$ represents the relay’s average power constraint, which incorporates the relay’s power expenditure during all $M$ epochs. The solution of (\[rav4\]) is given in the following theorem.
The optimal value of the TF parameter $\tau_i$ is given by $$\label{Sol2}
\tau_i^* = \min \{\tau_{1i}, \tau_{2i} \},$$ where $$\label{Sol3}
\tau_{1i} =\left\{
\begin{array}{rl}
\tau_{0i}, & a_i/\lambda > 2
\\
1, & \text{otherwise}
\end{array}
\right. \textrm{\quad and \quad}
\tau_{2i} = \frac{y_i}{a_i + y_i}.$$ In (\[Sol3\]), $\tau_{0i}$ is the root of $$\label{Sol4}
1 + \frac{a_i \, \tau_{0i}^2}{\lambda \, (1 + \tau_{0i}^2)} \, \log \left[\frac{a_i \, \tau_{0i}}{\lambda \, (1+\tau_{0i})} \right] = \frac{a_i \, \tau_{0i}}{\lambda \, (1+\tau_{0i})},$$ which satisfies $(a_i/\lambda - 1)^{-1} < \tau_{0i} < 1$. The optimal power allocation at the relay is given by $$\label{Sol1}
p_i^* =\left\{
\begin{array}{rl}
p_{1i}, & \tau_{1i} \leq \tau_{2i}
\\
p_{2i}, & \text{otherwise},
\end{array}
\right.$$ where $$\label{Sol6}
p_{1i} = \frac{1-\tau_{1i}}{1+\tau_{1i}} \left(\frac{1}{\lambda} - \frac{1 + \tau_{1i}}{a_i \, \, \tau_{1i}} \right)^+,$$ $$\label{Sol7}
p_{2i} = \frac{1-\tau_{2i}}{1+\tau_{2i}} \left(\frac{1}{\lambda} - \frac{2}{a_i} - \frac{1}{y_i} \right)^+,$$ with $(\cdot)^{+} = \max\{0, \cdot \}$. The constant $\lambda$ is found such that constraint $\text{C1}$ in (\[op1\]) holds with equality.
Please refer to Appendix A.
For $a_i/\lambda > 2$, (\[Sol4\]) has two roots, denoted by $\tau_{0i}'$ and $\tau_{0i}''$, which satisfy $0 < \tau_{0i}' < \tau_{0i}'' < 1$. The root with the smaller value, $\tau_{0i}'$, is calculated as $\tau_{0i}' = (a_i/\lambda -1)^{-1}$, but leads to a trivial value for the relay’s output power, $p_{i}^* = 0$. The root with the larger value, $\tau_{0i}''$, is the relevant root that leads to $p_{i}^* > 0$.
Optimal Power Allocation
------------------------
In the OPA scheme, the TF parameter is assumed to have a fixed value, i.e., $\tau_i=\tau_0$, $\forall i$. Assuming $M\to\infty$, the relay’s power allocation that maximizes the average data rate is obtained as the solution of the following optimization problem, $$\label{op2}
\underset{p_i \geq 0, \forall i} {\text{max}} \ \frac{1}{M} \sum_{i=1}^M \min \left\{ C_1(\tau_0, p_i), C_2(\tau_0, p_i) \right\} \notag$$ $$\begin{aligned}
\text{s.t.} &\text{C1}:& \frac{1+\tau_0}{2M} \sum_{i=1}^M p_i \leq \bar P.\end{aligned}$$ The solution of (\[op2\]) is given in the following theorem.
The relay optimal power allocation is given by $$\label{op2Sol}
p_i^* =\left\{
\begin{array}{rl}
\left(\frac{1}{\lambda} - \frac{1-\tau_0}{a_i \, \tau_0} \right)^+, & \tau_0 \leq \frac{y_i}{a_i + y_i}
\\
\left(\frac{1}{\lambda} - \frac{1}{y_i} \right)^+, & \text{otherwise},
\end{array}
\right.$$ where constant $\lambda$ is found such that $\text{C1}$ in (\[op2\]) holds with equality.
Please refer to Appendix B.
Introducing (\[op2Sol\]) into the objective function of (\[op2\]), the average data rate $\bar R$ is maximized for given $\bar P$ and $\tau_0$. Let us denote this data rate by $\bar R (\tau_0)$. Clearly, $\tau_0$ should be selected so as to maximize $\bar R (\tau_0)$ for given $\bar P$, as $$\label{op2cmaxmax}
\tau_0^* = \underset{0 < \tau_0 < 1} {\text{arg} \max} \ \bar R(\tau_0).$$ Since $\bar R(0) = \bar R(1) = 0$ and $\bar R(\tau_0) > 0$ for $0 < \tau_0 < 1$, according to the Rolle’s theorem, there must be a point $\tau_0^*$ between $0$ and $1$ where $\bar R(\tau_0)$ attains a maximum.
The Lagrange multiplier $\lambda$ can be determined by using the following lemma.
Given $\bar P$ and $\tau_0$, the constant $\lambda$ in (\[op2Sol\]) is determined numerically as the solution of the following expression $$\frac{2\bar P}{1 + \tau_0} = \int_{\frac{\lambda (1 - \tau_0)}{\tau_0}}^{\infty} \int_{\frac{a \tau_0}{1-\tau_0}}^{\infty} \left(\frac{1}{\lambda} - \frac{1-\tau_0}{a \tau_0} \right) f_Y(y) f_A(a) dy da \notag \\$$ $$\label{lambdafind}
+ \int_{\lambda}^{\infty} \int_{\frac{y (1-\tau_0)}{\tau_0}}^{\infty} \left( \frac{1}{\lambda} - \frac{1}{y}\right) f_A(a) f_Y(y) da dy,$$ where $f_A(a)$ and $f_Y(y)$ are the probability density functions (PDFs) of $a_i = 2N_0 x_i^2$ and $y_i$, respectively. Note that $f_A(a)$ can be expressed in terms of the PDF of $x_i$, $f_X(x)$, as $f_A(a) = f_X \big(\sqrt a/(2N_0) \big)/\big(2 \sqrt {2 N_0 a}\big)$.
Once $\lambda$ is determined from (\[lambdafind\]), the maximum average data rate is calculated as $$\begin{aligned}
\label{op2cmax}
\bar R(\tau_0) = \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \notag \\
= \frac{1-\tau_0}{2} \int_{\frac{\lambda (1 - \tau_0)}{\tau_0}}^{\infty} \int_{\frac{a \tau_0}{1-\tau_0}}^{\infty} \log \left(\frac{a\tau_0}{(1-\tau_0) \lambda} \right) f_Y(y) f_A(a) dy da \notag\end{aligned}$$ $$+ \frac{1-\tau_0}{2} \int_{\lambda}^{\infty} \int_{\frac{y (1-\tau_0)}{\tau_0}}^{\infty} \log \left( \frac{y}{\lambda} \right) f_A(a) f_Y(y) da dy, \quad \,\,$$
Since $M \to \infty$, the time averages in both $\text{C1}$ and the objective function of (\[op2\]) are calculated from their corresponding statistical averages, which yields (\[lambdafind\]) and (\[op2cmax\]).
Benchmark: Fixed Time and Fixed Power Allocation
------------------------------------------------
The improvement offered by the two schemes is compared to a fixed power and fixed time allocation (FPTA) scheme. The FPTA scheme assumes fixed relay output power, $P_0$, and a fixed TF parameter, $\tau_0$. For a fair comparison, given the power budget, $\bar P$, for the JOPTA and OPA schemes, the relay’s power for the FTPA scheme is fixed to $P_0 = 2 \bar P/(1 + \tau_0) $, whereas $\tau_0$ is set to the same value as the TF parameter of the OPA scheme, calculated based on (\[op2cmaxmax\]). In this case, in epoch $i$, the EHS transmits with power $2N_0 P_0 x_i \tau_0/(1 - \tau_0)$. As $M \to \infty$, the achievable rate of the FPTA scheme is determined as $$\label{refrate}
R_0 = \frac{1 - \tau_0}{2 M} \sum_{i=1}^M \log \left(1 + P_0 \cdot \min \left\{y_i, \frac{a_i \tau_0}{1 - \tau_0} \right\} \right).$$ In order to achieve $R_0$, both the EHS and relay should transmit with this fixed rate in all epochs by using infinitely-long Gaussian codewords that span infinitely many fading states.
Numerical Results
=================
Fig. 1 illustrates the average achievable data rates of the proposed schemes in block Rayleigh fading. The deterministic path loss is calculated as $E[x_i'] = E[y_i'] = 10^{-4}\, D_i^{-\alpha}$, with the distance between all the nodes set to $D_i = 10$m, the pathloss exponent set to $\alpha = 3$, and the pathloss at a reference distance of $1$m set to 40dB. We assume an AWGN power spectral density of $-150$dBm/Hz and a bandwidth of $1$MHz, yielding $N_0 = 10^{-12}$ Watts.
Both proposed schemes outperform the FPTA benchmark scheme. The rates achieved by OPA are close to those for JOPTA. However, compared to JOPTA, OPA is simpler and therefore more practical. Namely, the TF parameter of the OPA scheme is fixed for given $\bar P$ (for example, $\tau_0^* = 0.34$ for $\bar P = 1$ Watts, and $\tau_0^* = 0.26$ for $\bar P = 10$ Watts). Additionally, the nodes utilizing OPA can use a single code-book and fixed length codewords, whereas the nodes utilizing JOPTA must change codebooks and the codeword lengths in each transmission epoch.
{width="3.7in"}
\[fig3\]
Proof of Theorem 1
==================
Following \[\[litNewNew1\], Eqs. (18), (19)\], after introducing the change of variables $e_i = p_i(1+\tau_i)/2$, (\[op1\]) is rewritten as $$\label{op1alt0}
\underset{e_i\geq 0,\; 0<\tau_i<1, \forall i} {\text{max}} \ \frac{1}{M} \sum_{i=1}^M \min \left\{ \bar C_{1}\left(\tau_i, e_i\right), \bar C_{2}\left(\tau_i, e_i\right) \right\} \notag$$ $$\begin{aligned}
\text{s.t.} &C1:& \frac{1}{M} \sum_{i=1}^M e_i \leq \bar P\end{aligned}$$ where $\bar C_{1}\left(\tau_{i}, e_i \right) = C_1(\tau_i, 2e_i/(1+\tau_i))$ and $\bar C_{2}\left(\tau_{i}, e_i \right) = C_2(\tau_i, 2e_i/(1+\tau_i))$, with $C_1(\cdot)$ and $C_2(\cdot)$ given by (\[c1def\]) and (\[c2def\]), respectively.
Optimization problem (\[op1alt0\]) is non-convex. Nevertheless, based on \[\[litNew1\], Theorem 1\], we can still apply the Lagrange duality method to solve (\[op1alt0\]) because of a zero duality gap. In particular, (\[op1alt0\]) is in the form of \[\[litNew1\], Eq. (4)\]. For any fixed $\tau_i$, $\bar C_{1}(\tau_i,e_i)$ and $\bar C_{2}(\tau_i,e_i)$ are concave in $e_i$. Therefore, for any fixed set of $\tau_i, \forall i$, the objective function of (\[op1alt0\]) is concave in $(e_1, e_2, ..., e_M)$ for that set of $\tau_i$s, and constraint $C1$ is affine (i.e. convex) in $(e_1, e_2, ..., e_M)$. According to \[\[litNew1\], Definition 1\], the [*time-sharing*]{} condition is thus satisfied, implying the zero duality gap.
The Lagrangian of (\[op1alt0\]) is defined as $$\label{LD1}
L\left(e_i, \tau_i, \lambda \right) = \sum_{i=1}^M \min \left\{ \bar C_{1}\left(\tau_i, e_i\right), \bar C_{2}\left(\tau_i, e_i\right) \right\}
- \lambda e_i,$$ where $\lambda$ is the Lagrange multiplier associated with C1 of (\[op1alt0\]). Then, the Lagrangian dual function of (\[op1alt0\]) is expressed as $$\begin{aligned}
\label{LD2}
L\left(\lambda \right) = \underset{e_i\geq 0,\; 0<\tau_i<1, \forall i} {\text{max}} L\left(e_i, \tau_i, \lambda \right).\end{aligned}$$ Applying Lagrangian dual decomposition, (\[LD2\]) can be decoupled into $M$ subproblems (one for each fading state), $$\begin{aligned}
\label{LD3}
\underset{e_i\geq 0,\; 0<\tau_i<1} {\text{max}} \left[ \min \left\{ \bar C_{1}\left(\tau_i, e_i\right), \bar C_{2}\left(\tau_i, e_i\right) \right\} - \lambda \, e_i \right] \notag \\
= \underset{e_i\geq 0} {\text{max}} \left[H_i(e_i) - \lambda \, e_i \right], \qquad \qquad \qquad\end{aligned}$$ where $$\begin{aligned}
\label{Hdef}
H_i(e_i) = \max_{0<\tau_i<1} \, \left\{\min \left[ \bar C_{1}\left(\tau_i, e_i\right), \bar C_{2}\left(\tau_i, e_i\right) \right] \right\}.\end{aligned}$$
### Solution of (\[Hdef\])
For any fixed $e_i$, we consider two critical points for $\tau_i$: (1) the maximum of $\bar C_{1}(\tau_i, e_i)$, and (2) the intersection between $\bar C_{1}(\tau_i, e_i)$ and $\bar C_{2}(\tau_i, e_i)$. These critical points are denoted by $\bar \tau_{1i}$ and $\bar \tau_{2i}$, respectively.
The [*critical point (1)*]{} is found from $\partial \bar C_{1}(\tau_i, e_i)/\partial \tau_i = 0$, yielding $$\label{rav5}
\frac{2 e_i a_i (1+\bar \tau_{1i}^2)}{(1 + \bar \tau_{1i}) [1 + (2 e_i a_i - \bar \tau_{1i}) \bar \tau_{1i}] } = \log \left(1 + \frac{2 e_i a_i \bar \tau_{1i}}{1 - \bar \tau_{1i}^2} \right).$$ For any given value of the product $e_ia_i$, $\bar C_{1}(\tau_i, e_i)$ has a single maximum at $\tau_{i} = \bar \tau_{1i}$, which is calculated as the root of (\[rav5\]). The maximizer $\bar \tau_{1i}$ is a monotonically decreasing function of $e_i$, which satisfies $\bar \tau_{1i} \to 1$ as $e_i \to 0$, and $\bar \tau_{1i} \to 0$ as $e_i \to \infty$.
The [*critical point (2)*]{} is found from the equality $\bar C_{1}(\bar \tau_{2i}, e_i) = \bar C_{2}(\bar \tau_{2i}, e_i)$, as $$\label{tau2}
\bar \tau_{2i} = \frac{y_i}{a_i + y_i}.$$
If $\bar \tau_{1i} \leq \bar \tau_{2i}$, $\min\{$$\bar C_{1}(\bar \tau_{1i}, e_i), \bar C_{2}(\bar \tau_{1i}, e_i)\}$ $=$ $\bar C_{1}(\bar \tau_{1i}, e_i)$ and $H_i(e_i) = \max_{0<\tau_i<1} \{\bar C_{1}(\bar \tau_{1i}, e_i),$ $\bar C_{1}(\bar \tau_{2i}, e_i)\} = \bar C_{1}(\bar \tau_{1i}, e_i)$. If $\bar \tau_{1i} > \bar \tau_{2i}$, $\min\{\bar C_{1}(\bar \tau_{1i}, e_i),$ $\bar C_{2}(\bar \tau_{1i}, e_i)\}$ $=$ $\bar C_{2}(\bar \tau_{1i}, e_i)$ and $H_i(e_i) = \max_{0<\tau_i<1} \{\bar C_{2}(\bar \tau_{1i}, e_i), \bar C_{2}(\bar \tau_{2i}, e_i)\} = \bar C_{2}(\bar \tau_{2i}, e_i)$. Thus, $$\label{Hdef1}
H_i (e_i) =\left\{
\begin{array}{rl}
U_i(e_i), & \bar \tau_{1i} \leq \bar \tau_{2i}
\\
V_i(e_i), & \bar \tau_{1i} > \bar \tau_{2i}.
\end{array}
\right.$$ where $U_i(e_i) \triangleq \bar C_{1}(\bar \tau_{1i}, e_i)$ and $V_i(e_i) \triangleq \bar C_{1}(\bar \tau_{2i}, e_i)$ $=$ $\bar C_{2}(\bar \tau_{2i}, e_i)$.
### Solution of (\[LD3\])
Considering (\[Hdef1\]), we can now solve (\[LD3\]) by setting the first derivative of $H_i(e_i) - \lambda \, e_i$ w.r.t. $e_i$ to zero. The found critical points are maxima because both $U_i(e_i)$ and $V_i(e_i)$, and therefore $H_i(e_i)$, are concave and monotonically increasing in $e_i$. The concavity of $U_i(e_i)$ and $V_i(e_i)$ is proven from their respective second derivatives w.r.t. $e_i$, which satisfy $U_i''(e_i) < 0$ and $V_i''(e_i) < 0$ for any $e_i > 0$.
Critical point (1): If $\bar \tau_{1i} \leq \bar \tau_{2i} $, (\[LD3\]) reduces to $$\label{dual1subproblem}
\underset{e_i \geq 0} {\text{max}} \ U_{i}(e_i) - \lambda e_i.$$ The solution of (\[dual1subproblem\]) is found from $U_{i}'(e_i) = \lambda $, where $U_{i}'(e_i)$ denotes the first derivative of $U_i(e_i)$ w.r.t. $e_i$. Since $U_i(e_i)= C_{1}(\bar \tau_{1i}, e_i)$, $U_{i}'(e_i)$ is determined as the total derivative of $C_{1}(\bar \tau_{1i}, e_i)$ w.r.t. $e_i$. At the critical point (1), $\left. \partial\bar C_{1}(\tau_i, e_i) /\partial \tau_i \right|_{\tau_i = \bar \tau_{1i}} = 0$, and, therefore, $$\label{rav11}
U_{i}'(e_i) = \frac{\partial \bar C_{1}\left(\bar \tau_{1i}, e_i\right)}{\partial e_i} = \frac{a_i \, \bar \tau_{1i} (1 - \bar \tau_{1i})}{1 + 2 e_i a_i \bar \tau_{1i} - \bar \tau_{1i}^2}.$$ Thus, (\[dual1subproblem\]) is solved as $$\label{dual1sol}
e_{1i} = \frac{1 - \bar \tau_{1i}}{2} \left(\frac{1}{\lambda} - \frac{1+\bar \tau_{1i}}{a_i \, \bar \tau_{1i}} \right)^+,$$ where the $(\cdot)^+$ operator is due to the constraint $e_{i} \geq 0$. Eqs. (\[rav5\]) and (\[dual1sol\]) constitute a set of two equations with two unknowns, $e_{1i}$ and $\bar \tau_{1i}$, yielding (\[Sol4\]) and (\[Sol6\]).
Critical point (2): If $\bar \tau_{1i} > \bar \tau_{2i} $, (\[LD3\]) reduces to $$\label{dual2subproblem}
\underset{e_i \geq 0} {\text{max}} \ V_{i}(e_i) - \lambda e_i.$$ The solution of (\[dual2subproblem\]) is the root of $V_{i}'(e_i) = \lambda $ w.r.t $e_i$, yielding $$\label{dual2sol}
e_{2i} = \frac{1 - \bar \tau_{2i}}{2} \left(\frac{1}{\lambda} - \frac{1+\bar \tau_{2i}}{1-\bar \tau_{2i}} \ \frac{1}{y_i} \right)^+,$$ where the $(\cdot)^+$ operator is again due to the constraint $e_{i} \geq 0$. Combining (\[tau2\]) and (\[dual2sol\]), we obtain (\[Sol7\]).
Proof of Theorem 2
==================
The convex optimization problem (\[op2\]) is rewritten as $$\underset{p_i \geq 0} {\text{max}} \ \frac{1}{M} \sum_{i=1}^M \, H_i(p_i), \quad \text{s. t.} \quad \frac{1}{M} \sum_{i=1}^M p_i \leq 2\bar P/(1+\tau_0),$$ where $$\begin{aligned}
\label{op2dual}
H_i(p_i) &=& \min \left\{ C_{1}(\tau_0, p_i), C_{2}(\tau_0, p_i) \right\} \notag \\
&=&\left\{
\begin{array}{rl}
C_{1}(\tau_0, p_i), & y_i \geq a_i \tau_0/(1-\tau_0)
\\
C_{2}(\tau_0, p_i), & \text{otherwise}.
\end{array}
\right.\end{aligned}$$ Decomposing Lagrangian dual function into $M$ subproblems (one for each fading state), we obtain $\max_{p_i \geq 0} H_i(p_i) - \lambda p_i$, which is solved as the root of $H'(p_i) = \lambda$ w.r.t. $p_i$.
[99]{}
L. R. Varshney, “Transporting information and energy simultaneousely,” *Proc. IEEE ISIT*, Toronto, Canada, July 2008, pp. 1612-1616 \[lit1\]
P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and power transfer,” *Proc. IEEE ISIT*, Austin, June 2010, pp. 2363-2367 \[lit2\]
A. Fouladgar and O. Simeone, “On the transfer of information and energy in multi-user systems,” *IEEE Commun. Letters*, vol. 16, pp. 1733-1736, Nov. 2012 \[lit3\]
R. Zhang and C. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” *IEEE Trans. on Commun.*, vol. 12, pp. 1989-2001, May 2013 \[lit4\]
B. Medepally and N. B. Mehta, “Voluntary energy harvesting relays and selection in cooperative wireless networks,” *IEEE Trans. Wireless Commun.*, vol. 9, no. 11, pp. 3543-3553, Nov. 2010. \[lit5\]
A. A. Nasir, X. Zhou, S. Durrani, and R. Kennedy, “Relaying protocols for wireless energy harvesting and information processing,” *IEEE Trans. Wireless Commun.*, vol. 12, no. 7, pp. 3622-3636, Jul. 2013. \[lit6\]
A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Throughput and ergodic capacity of wireless energy harvesting based DF relaying network,” *Proc. IEEE ICC*, Sydney, Australia, June, 2014, pp. 4066-4071. \[lit7\]
Z. Ding, S. M. Perlaza, I. Esnaola, and H. V. Poor, “Power allocation strategies in energy harvesting wireless cooperative networks,” *IEEE Trans. Wireless Commun.*, vol. 13, no. 2, pp. 846-860, Feb. 2014. \[lit8\]
X. Huang and N. Ansari, “Optimal cooperative power allocation for energy harvesting enabled relay networks”, submitted for publication, arXiv: 1405.5764 \[lit9\]
H. Ju and R. Zhang, “Optimal resource allocation in full-duplex wireless-powered communication network”, *IEEE Trans. Commun.*, vol. 62, no. 10, pp. 3528-3540, Oct. 2014 \[litNewNew1\]
W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems”, *IEEE Trans. Commun.*, vol. 54, no. 7, pp. 1310-1322, July 2006 \[litNew1\]
[^1]: Z. Hadzi-Velkov is with the Faculty of Electrical Engineering and Information Technologies, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia (email: zoranhv@feit.ukim.edu.mk).
[^2]: N. Zlatanov is with the Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC 3800, Australia (e-mail: nikola.zlatanov@monash.edu).
[^3]: T. Q. Duong is with the School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K. (e-mail: trung.q.duong@qub.ac.uk).
[^4]: R. Schober is with the Institute of Digital Communications, Friedrich-Alexander University, Erlangen, Germany (email: robert.schober@fau.de).
|
---
abstract: 'The magnetization process of the orthogonal-dimer antiferromagnet SrCu$_2$(BO$_3$)$_2$ is investigated in high magnetic fields of up to 118 T. A 1/2 plateau is clearly observed in the field range 84 to 108 T in addition to 1/8, 1/4 and 1/3 plateaux at lower fields. Using a combination of state-of-the-art numerical simulations, the main features of the high-field magnetization, a 1/2 plateau of width 24 T, a 1/3 plateau of width 34 T, and no 2/5 plateau, are shown to agree quantitatively with the Shastry-Sutherland model if the ratio of inter- to intra-dimer exchange interactions $J^{\prime}/ J =0.63$. It is further predicted that the intermediate phase between the 1/3 and 1/2 plateau is not uniform but consists of a 1/3 supersolid followed by a 2/5 supersolid and possibly a domain-wall phase, with a reentrance into the 1/3 supersolid above the 1/2 plateau.'
author:
- 'Y. H. Matsuda'
- 'N. Abe'
- 'S. Takeyama'
- 'H. Kageyama'
- 'P. Corboz'
- 'A. Honecker'
- 'S. R. Manmana'
- 'G. R. Foltin'
- 'K. P. Schmidt'
- 'F. Mila'
bibliography:
- 'SCBO.bib'
date: 'August 19, 2013'
title: 'Magnetization of SrCu$_2$(BO$_3$)$_2$ in ultrahigh magnetic fields up to 118 T'
---
Geometrical frustration can induce very interesting phases in quantum magnets [@HFM]. For instance, the orthogonal dimer antiferromagnet SrCu$_2$(BO$_3$)$_2$ exhibits fascinating phenomena due to frustration. The nearest neighbor (NN) $S$=1/2 spins of Cu ions are antiferromagnetically coupled and form singlet dimers through the exchange interaction $J$. Since the inter-dimer exchange interaction $J^{\prime}$ between the next nearest-neighbor (NNN) Cu ions is antiferromagnetic as well, the orthogonal configuration induces geometrical frustration [@Kage99]. Quite remarkably, the crystal lattice is topologically equivalent to the Shastry-Sutherland lattice that was initially investigated out of pure theoretical interest [@SS]. Since its discovery, SrCu$_2$(BO$_3$)$_2$ has thus logically been the subject of a vast number of experimental and theoretical studies [@Miyahara03JPCM; @Takigawa10; @TM].
Quantum phase transitions have been theoretically predicted to take place when the ratio $J^{\prime} /J$ is tuned. It is clear that the ground state is a product of dimer singlets if $J^{\prime} /J = 0$, and that it supports antiferromagnetic Néel order when $J^{\prime} /J \rightarrow +\infty$. An intermediate gapped plaquette phase has been predicted to appear [@Koga2000; @Takushima01; @Chung01; @Laeuchli02] when $0.675 \lesssim
J^{\prime} /J \lesssim 0.77$ [@lou12; @corboz2013]. SrCu$_2$(BO$_3$)$_2$ is believed to be located at $J^{\prime} /J \simeq 0.63$, thus to have an exact dimer singlet ground state [@Miyahara03JPCM; @Takigawa10].
In addition to the interest raised by the exotic ground state of the Shastry-Sutherland model, the presence of several magnetization plateaux in SrCu$_2$(BO$_3$)$_2$ has attracted significant attention. Distinct 1/8, 1/4, and 1/3 plateaux have been reported early on in the magnetization process [@Kage99; @Onizuka00]. More recently, additional plateaux between 1/8 and 1/4 have been observed [@Sebastian08; @Takigawa12], and evidence in favor of the presence of the long predicted 1/2 plateau has been provided by magnetostriction measurements [@Jaime12]. However, the entire 1/2 plateau phase has not been unveiled in Ref. because of the technical upper limit of the magnetic field at 100.75 T.
The 1/2 plateau has been predicted to be less stable than the 1/3 plateau and to disappear for large $J'/J$ [@MT00PRB]. In fact, according to Ref. , the length of the 1/2 plateau is less than half that of the 1/3 plateau, although the 1/2 plateau can be expected to be quite stable considering the checkerboard pattern of the triplet excitation suggested by the boson picture. Hence, the experimental determination of the stability range of the 1/2 plateau is of particular interest in itself, and also important for checking the validity of the theoretical model. Moreover, in addition to the 1/2 plateau, exotic high-field spin states have been predicted such as supersolid phases between the 1/3 and 1/2 plateaux and above the 1/2 plateau [@MT00PRB; @lou12]. The quantum spin state realized when the density of triplets becomes very high has not been uncovered yet.
In the present work, we have investigated the spin states of SrCu$_2$(BO$_3$)$_2$ by magnetization measurements in high magnetic fields up to 118 T. A clear 1/2 magnetization plateau phase has been observed in the field range from 84 to 108 T, and at the upper critical field, a sharp magnetization increase suggests a first-order phase transition. Theoretical calculations based on the infinite projected entangled-pair state (iPEPS) tensor network algorithm [@verstraete2004; @jordan2008; @corboz2010; @corboz2011; @bauer2011], exact diagonalizations, density-matrix renormalization group (DMRG) simulations and series expansions have shown that the 1/2 and 1/3 plateaux can be quantitatively reproduced by the Shastry-Sutherland model with a ratio $J'/J\simeq 0.63$, and they predict a variety of exotic phases between the 1/3 and 1/2 plateaux and above the 1/2 plateau, including several types of supersolid phases, in particular a first-order transition to a 1/3 supersolid above the upper critical field of the 1/2 plateau.
 Pickup coil signal proportional to the time derivative of the magnetization ($\text{d}M/\text{d}t$) plotted as a function of time. The magnetic field waveform $H(t)$ is also shown as a solid red line. ](fig1){width="1\columnwidth"}
#### Experimental procedure.—
A single crystal of SrCu$_2$(BO$_3$)$_2$ was used for the experiment. Pulsed magnetic fields of up to 118 T were generated by a destructive method; the vertical-type single-turn coil technique [@TakeMag12JPSJ] was used. The field was applied parallel to the $c$-axis of the crystal. The magnetization ($M$) was measured using a pickup coil that consists of two small coils (1 mm diameter, 1.4 mm length for each). The two coils have different polarizations and are connected in series. The sample is inserted into one of the coils. An induction voltage proportional to the time derivative of $M$ ($\text{d}M/\text{d}t$) is obtained when the sample gets magnetized by a pulsed magnetic field $H(t)$, where $t$ is the time. The induction voltage due to $\text{d}H/\text{d}t$ is almost canceled out between the opposite polarization coils. The detailed experimental setup for the magnetization measurement using this vertical type single-turn coil method has been described elsewhere [@TakeMag12JPSJ]. A liquid helium bath cryostat with the tail part made of plastic has been used; the sample was immersed in liquid helium and a measurement temperature of about 2 K has been reached by reducing the vapor pressure.
 The magnetization curve at 2.1 K up to 109 T (Shot-A). Applied fields $H$ are parallel to the $c$-axis of the crystal. The magnetic field derivative of the magnetization ($\text{d}M/\text{d}H$) curve is displayed as a function of magnetic field $H$. The dotted curve is the magnetization curve reported previously [@Onizuka00]. $\text{d}M/\text{d}H$ curve of another measurement up to 118 T (Shot-B) is plotted in the inset. ](fig2_crop){width="1\columnwidth"}
#### Experimental results.—
The pickup coil signal proportional to $\text{d}M/\text{d}t$ is shown as a function of time in Fig. \[fig:dmdt\] together with the magnetic field waveform. The obtained maximum field is 109 T and we name this experiment Shot-A in this paper. Distinct peak structures denoted by labels $a$, $b$, $c$, $c^{\prime}$, $b^{\prime}$ and $a^{\prime}$ are present in $\text{d}M/\text{d}t$. They correspond to magnetization jumps at the phase boundaries of different spin states. Indeed, a stepwise magnetization increase gives rise to a peak in $\text{d}M/\text{d}t$ curve, and the peak is positive (negative) for increasing (decreasing) field. The one to one correspondence between $a$ and $a^{\prime}$, $b$ and $b^{\prime}$, and $c$ and $c^{\prime}$ indicates that stepwise transitions take place at these magnetic fields for both field-increasing and decreasing processes without significant hysteresis.
The magnetization curve is obtained by a numerical integration of $\text{d}M/\text{d}t$; the resulting magnetization $M$ is normalized by the expected saturation magnetization $M_S$. The magnetic field derivative of the magnetization $\text{d}M/\text{d}H$ is obtained from the ratio $\text{d}M/\text{d}t \times 1/(\text{d}H/\text{d}t)$.
Figure \[fig:mcurve\] shows the magnetization process and the magnetic field dependence of $\text{d}M/\text{d}H$ at 2.1 K (Shot-A). We also show for comparison the magnetization $M/M_S$ up to 55 T previously reported in Ref. , and the agreement is good. In the present work, we only analyze the result of the field-increasing process because the magnetic field is less homogeneous for the field decreasing process due to the mechanical deformation of the single-turn coil and to the background non-linear offset of the signal which disturbs the precise measurement [@TakeMag12JPSJ]. The $\text{d}M/\text{d}H$ curve shows clear peaks labeled $H_{cn}$ ($n=1-6$) : $H_{c1,c2, c3}$ are attributed to structure $a$ in Fig. \[fig:dmdt\], $H_{c4, c5}$ to structure $b$, and $H_{c6}$ to structure $c$.
We show the $\text{d}M/\text{d}H$ curve obtained from another experiment up to 118 T (Shot-B) in the inset of Fig. \[fig:mcurve\]. The upward behavior at high fields over 100 T is due to the increase of the background noise: the noise becomes relatively larger near the top of the magnetic field curve because the $\text{d}M/\text{d}t$ signal becomes small when $\text{d}H/\text{d}t$ is small. Although the background noise makes it difficult to obtain a very precise magnetization curve by a numerical integration for Shot-B, peaks in $\text{d}M/\text{d}H$ are clearly observed at nearly identical values as for Shot-A. The obtained peak fields are shown in Table \[table:peak\].
$H_{c1} (T)$ $H_{c2} (T)$ $H_{c3} (T)$ $H_{c4} (T)$ $H_{c5} (T)$ $H_{c6} (T)$
-------- -------------- -------------- -------------- -------------- -------------- --------------
Shot-A 26 33 39 73 84 108
Shot-B 27 33 40 75 83 108
: Transition magnetic fields $H_{cn}$ obtained from the $\text{d}M/\text{d}H$ peaks for Shot-A and -B. The precision of the magnetic field value is likely to be about $\pm1$ T.
\[table:peak\]
The $\text{d}M/\text{d}H$ peaks at $H_{c1}$, $H_{c2}$, and $H_{c3}$ are attributed to the magnetization jumps at fields where the spin state enters 1/8, 1/4, and 1/3 plateau phases, respectively. Additional features probably related to extra plateaux [@Takigawa12] are also present between $H_{c1}$ and $H_{c2}$, but steady field measurements are more accurate in that field range, and we will not attempt to discuss them. While the measurement temperature 2.1 K seems to be too high to observe the 1/8 plateau [@kodama02] the adiabatic cooling owing to the first sweep speed of the magnetic field leads to an actual temperature lower than 0.5 K [@Levy08]. In the magnetization curve, the 1/3 plateau is observed in the field range from 39 to 73 T for Shot-A. Here note that we calibrate the absolute value of $M/M_S$ using the magnetization at the 1/3 plateau phase. The field region for the 1/3 plateau is in good agreement with the previous reports [@Onizuka00; @Sebastian08].
After the 1/3 plateau, there is a change of slope around 74 T. Above that critical field $H_{c4}$, there is an almost smooth increase of the magnetization, followed by the appearance of the 1/2 plateau at around $H_{c5} \simeq 84~T$. Note however that a trapezoid or broad flap-top peak is expected if the slope increase was monotonous and had no anomaly. Since a peak structure is clearly observed both in up and down sweeps between the 1/3 and 1/2 plateau (see in particular feature $b'$ in down sweep), some kind of transition probably takes place between the 1/3 and 1/2 plateaux.
The 1/2 plateau starts at 84 T and continues up to 108 T. The starting magnetic field seems to be slightly higher compared to the previously reported value around 82 T detected by magnetostriction [@Jaime12]. This might be partly due to the different ways of detection (magnetostriction versus magnetization), and also to the experimental uncertainty in the present work (the error of the absolute value of the magnetic field is within 3%). The magnetic field absolute value of the single-turn coil method contains a few percent experimental error owing to the technical limit of the precision [@TakeMag12JPSJ]. However, even if there is an error bar on the absolute value of the magnetic field, the relative change in the field value has a smaller error bar. Hence it is safe to conclude that the plateau length of the 1/2 plateau $\Delta H \simeq 24$ T is considerably shorter than that of 1/3 plateau $\Delta H \simeq 34$ T. At higher fields, considering the appearance of a sharp peak $H_{c6}$, a first-order magnetic phase transition is expected to occur after the 1/2 plateau at a field of 108 T.
 Phase diagram of the Shastry-Sutherland model in a magnetic field obtained with iPEPS. ](res_ssm_pd6){width="1\columnwidth"}
#### Theory.—
A good starting point to describe the magnetization process of SrCu$_2$(BO$_3$)$_2$ is provided by the spin-1/2 Heisenberg model on the Shastry-Sutherland lattice defined by: $$H=J'\sum_{<i,j>}\bm S_{i}\cdot \bm S_{j}+J\sum_{\ll i,j\gg}\bm
S_{i}\cdot \bm S_{j}-h\sum_{i}S_i^z$$ where the $\ll i,j\gg$ bonds with coupling $J$ build an array of orthogonal dimers while the $<i,j>$ bonds with coupling $J'$ denote inter-dimer couplings. While a lot of effort has been devoted in the past to the magnetization curve up to $1/3$ [@dorier08; @abendschein08; @nemec12], in the range where a sequence of plateaux has been reported, comparatively little attention has been paid so far to the magnetization curve above $1/3$. Shortly after the discovery of plateaux in SrCu$_2$(BO$_3$)$_2$, Momoi and Totsuka [@MT00PRB] have predicted the presence of $1/3$ and $1/2$ plateaux separated by supersolid phases. This prediction has been left unchallenged until the recent investigation of magnetostriction in very high field [@Jaime12]. These measurements have revealed the presence of an anomaly above the $1/3$ plateau that has been interpreted as a $2/5$ plateau, an interpretation backed by a DMRG (density matrix renormalization group) calculation at $J'/J=0.62$. However, a recent tensor-network calculation based on MERA (multi-scale entanglement renormalization ansatz) has just confirmed the presence of $1/3$ and $1/2$ plateaux without any evidence of a $2/5$ plateau [@lou12].
In view of the importance of this issue for the interpretation of the present results, we have decided to reinvestigate the high-field magnetization process of the Shastry-Sutherland model with a variety of state-of-the-art numerical approaches: exact diagonalizations of finite-size clusters up to 40 spins, DMRG on clusters of size up to $12 \times 10$ spins, high-order series expansions, and iPEPS – a tensor network method for two-dimensional systems in the thermodynamic limit. The various methods yield a rather consistent picture (see supplemental material for a detailed comparison). The most complete phase diagram, shown in Fig. \[fig:ipeps\], has been obtained with iPEPS. Above the $1/3$ plateau, it consists of two additional plateaux at $2/5$ and $1/2$, three supersolid phases with the symmetries of the $1/3$, $2/5$ and $1/2$ plateaux, and a phase with domain walls separating regions of $1/2$ plateau structures. Note that we confirm the presence of a 2/5 plateau for $J'/J=0.62$, in agreement with the DMRG results of Ref. .
 Comparison between the experimental magnetization curve and the iPEPS simulation results for $J'/J=0.63$. The extent of the 1/3 and 1/2 plateaux predicted by the other methods is shown on top of the plateaux. ](res_ssm63paper){width="1\columnwidth"}
For our present purpose, the most important messages of this phase diagram are: i) The 1/2 plateau does not extend beyond a critical value of the order of , in qualitative agreement with Momoi and Totsuka [@MT00PRB]; ii) The $2/5$ plateau does not extend beyond $J'/J\simeq 0.625$. Since the present experimental data do not reveal any evidence of a $2/5$ plateau but show a rather broad $1/2$ plateau, $J'/J$ can neither be too large nor too small, and a comparison of the critical fields of the $1/2$ and $1/3$ plateaux with the experimental ones point to a ratio $J'/J\simeq0.63$.
A detailed comparison of the experimental magnetization curve with the theoretical predictions of the various methods at $J'/J\simeq0.63$ above the 1/4 plateau is shown in Fig. \[fig:expt\_theory\]. First of all, the critical fields $H_{c3}$ to $H_{c6}$ are accurately reproduced by iPEPS. The predictions of the other methods are scattered around the iPEPS values, but altogether they support the main features of the iPEPS results (for a detailed comparison as a function of $J'/J$, see supplemental material). Secondly, the magnetization jumps at $H_{c3}$ and $H_{c6}$, which point to first-order transitions, are well accounted for by the theoretical results: at $H_{c3}$, there is a first-order transition between the 1/4 and 1/3 plateau, while at $H_{c6}$, there is one between the 1/2 plateau and the 1/3 supersolid. The smoother transitions at $H_{c4}$ and $H_{c5}$ also correspond to much weaker anomalies in the theoretical results. For the upper boundary of the 1/3 plateau, series expansions point to a gap closing when increasing $H$, hence to a second order phase transition, around $65$ T, significantly below $H_{c4}$. This is not incompatible with the broad onset of magnetization around $H_{c4}$, with a slope that takes off around $65$ T in shot-A and $70$ T in shot-B. Below the lower boundary of the 1/2 plateau at $H_{c5}$, iPEPS predicts a series of first order phase transitions from a 1/3 supersolid to a 2/5 supersolid, then to a phase with domain walls, and then finally to the 1/2 plateau. In the magnetization curve, these transitions translate into small jumps. This is presumably related to the peak observed in both shots around $80$ T, i.e., between the 1/3 and 1/2 plateaux, consistent with the prediction that the intermediate field range between these plateaux is not a single phase.
Finally, let us comment on the experimental slope of the 1/2 plateau between $H_{c5}$ and $H_{c6}$, which is anomalously large as compared, e.g., to that of the 1/3 plateau. This slope is definitely too large to be due to Dzyaloshinskii-Moriya interactions, but it might be simply explained as a temperature effect. Indeed, the difference in energy per spin between the 1/2 plateau and the competing 1/3 supersolid state obtained with iPEPS is very small ( $<0.004 J$), whereas the competing phases are definitely higher in the middle of the 1/3 plateau.
#### Conclusion.—
To summarize, we have performed ultra-high field measurements of the magnetization of SrCu$_2$(BO$_3$)$_2$, revealing for the first time the extent of the 1/2 plateau. The length of the 1/2 plateau has been found to be around 70% of that of the 1/3 plateau. We have not found any indication of the 2/5 plateau that was previously suggested on the basis of magnetostriction measurements. As revealed by large-scale numerical simulations, these results are consistent with the Shastry-Sutherland model provided the ratio of inter to intra-dimer coupling is neither too small, in agreement with recent results on Zn doped samples [@Yoshida], nor too large, the best agreement being reached for a ratio of about $0.63$. These numerical simulations further predict that the magnetization between the 1/3 and 1/2 plateau and above the 1/2 plateau is not uniform, but that the system is always in a phase that breaks the translational symmetry, either to form a supersolid, or because of the spontaneous appearance of domain walls in the 1/2 plateau phase. It would be very interesting to test this prediction with measurements that can detect a change of lattice symmetry such as X-rays or neutrons, or with a local probe such as NMR. Given the field range of interest, this is however a huge experimental challenge.
#### Acknowledgement.—
Y. H. M. thanks M. Takigawa for fruitful discussions. A. H. and P. C. acknowledge support through FOR1807 (DFG / SNSF). We acknowledge allocation of CPU time at the HLRN Hannover. The iPEPS simulations have been performed on the Brutus cluster at ETH Zurich.
|
---
abstract: 'Stabilized FeSe thin films in ambient pressure with tunable superconductivity would be a promising candidate for superconducting electronic devices yet its superconducting transition temperature ($T_c$) is below 10 K in bulk materials. By carefully controlling the depositions on twelve kinds of substrates using pulsed laser deposition technique, high quality single crystalline FeSe samples were fabricated with full width of half maximum $0.515^{\circ}$ in the rocking curve and clear four-fold symmetry in $\varphi$-scan from x-ray diffractions. The films have a maximum $T_c$ $\sim$ 15 K on the CaF$_2$ substrate and do not show obvious decay in the air for more than half a year. Slightly tuning the stoichiometry of the FeSe targets, the $T_c$ becomes adjustable from 15 to $<$ 2 K with quite narrow transition widths less than 2 K, and shows a positive relation with the out-of-plane (*c*-axis) lattice parameter of the films. However, there is no clear relation between the $T_c$ and the surface atomic distance of the substrates. By reducing the thickness of the films, the $T_c$ decreases and fades away in samples of less than 10 nm, suggesting that the strain effect is not responsible for the enhancement of $T_c$ in our experiments.'
author:
- Zhongpei Feng
- Jie Yuan
- Ge He
- Zefeng Lin
- Dong Li
- Xingyu Jiang
- Yulong Huang
- Shunli Ni
- Jun Li
- Beiyi Zhu
- Xiaoli Dong
- Fang Zhou
- Huabing Wang
- Zhongxian Zhao
- Kui Jin
title: Promoting superconductivity in FeSe films via fine manipulation of crystal lattice
---
Before sending further, pass this check list: \[ \] find and resolve all “???” \[ \] Spell-check \[ \] order of using/defining abbreviations \[ \] citation order (use RefTest) \[ \] figure order & reference order \[ \] Check LaTeX output files (\*.log) for warnings \[ \] Check BibTeX output (screen) for warnings \[ \] update PACS \[ \] decide on color figures \[ \] Re-read paper in the morning
After completing this list, if you made at least one correction, re-do this check-list from the beginning, until no corrections will be done.
Introduction
============
The Fe-based superconductors have attracted blooming attention for its superconducting nature and promising applications.[@Kamihara; @Haindl] Among various Fe-based superconductors, $\beta$-FeSe possesses the simplest structure for an anti-PbO-type structure (space group of $P4/nmm$) with stacks of Fe$_2$Se$_2$ layers, but displays the most multifarious physical properties.[@Hsu; @Paglione] The FeSe bulk crystals exhibit a superconducting transition temperature ($T_c$) of 9 K.[@Chen; @Kasahara] Under external pressure, the $T_c$ can be enhanced up to 38 K for FeSe, which is attributed to a decrease of anion height from the Fe-square planes, highlighting the impact of crystal lattice on superconductivity.[@Chen; @Sun; @Medvedev] Unexpectedly, the $T_c$ can be raised further for one unit cell (UC) of FeSe on SrTiO$_3$ substrate.[@Wang; @Liu; @Peng; @Ge; @Lee] Because of the extreme sensitivity to oxygen, such ultra-thin films of one or a few UC can only be achieved in high vacuum for in-situ fabrications and characterizations, which limits the researches and applications. Therefore, more stabilized FeSe films comparable to or even better than the bulk crystals are highly desired for the next generation of superconducting electronic devices.
{width="100.00000%"}
The fabrication of FeSe thin films has been widely studied, and among these researches, pulsed laser deposition (PLD) and molecular beam epitaxy (MBE) techniques are most commonly used.[@Haindl; @Wang2; @Agatsuma; @Maeda] From the application point of view, PLD is much more efficient for the growth of films with moderate thickness (above 100 nm).[@Agatsuma; @Maeda; @Han; @Chen2; @Jung; @Nie] Basically, the FeSe can be grown onto various substrates, such as LaAlO$_3$, SrTiO$_3$ and MgO, but their $T_c$ values are generally equal to or lower than that of bulk crystals. A recent report showed that another substrate, CaF$_2$, could enhance the $T_c$ up to 11.4 K with thickness of 150 nm, considerably higher than that of bulk FeSe, which was attributed to the in-plane compressive strain from the substrate.[@Nabeshima] The strain by mismatch between the substrate and the film may play a role in promoting the $T_c$ but usually takes effect within limited film thickness, plausibly for the ultra-thin FeSe films where the $T_c$ decreases quickly from 1 UC to 3 UC. [@Tan] Therefore, it is still an open question why the $T_c$ is enhanced in thick films like 160 nm. Besides the strain induced by the lattice mismatch,[@Nie; @Nabeshima; @Song] other effects, such as modification of the out-of-plane lattice parameter,[@Medvedev] sample inhomogeneity by Fe vacancies,[@McQueen; @Fang] as well as the growth conditions,[@Agatsuma] could also influence the superconductivity. To uncover the substrate effects on the superconductivity, it is important to systematically study the crystal lattice and superconducting properties of FeSe films on various substrates.
{width="0.9\columnwidth"}
In this letter, we report the success in synthesizing a series of high-quality single crystalline superconducting FeSe films on various substrates by PLD technique, with maximum $T_{c}$ up to 15 K. Besides, different growth parameters such as the ratio of Fe to Se in the targets and the thickness of the films are also elaborately tuned to arrive at a broad range of zero-resistance transition temperature $T_{c0}$. Based on abundant high quality and stabilized samples, the relation between the crystal lattice and the tunable $T_{c0}$ has been carefully studied.
Experimental
============
FeSe polycrystalline targets were fabricated by the solid-state reaction method. The original materials of Fe (4N, Alfa Aesar Inc.) and Se (5N, Alfa Aesar Inc.) powders were mixed with designed ratio of stoichiometry, then heat-treated at 420 $^{\circ}$C for 24 hours in evacuated quartz tubes. The as-prepared material was grinded and sintered at 450 $^{\circ}$C for 48 hours, and such process was repeated more than three times for final targets. FeSe thin films were prepared by PLD technique with a KrF laser. The background vacuum of the deposition chamber is better than 10$^{-7}$ Torr. The FeSe thin films were grown in vacuum with the target-substrate distance of $\sim$ 50 mm, the laser repetition of 2 Hz, and the substrate temperature of 350 $^{\circ}$C.
X-ray diffraction (XRD) measurements of the thin films were performed on a Rigaku SmartLab (9 kW) X-ray diffractometer with Ge(220) $\times$ 2 crystal monochromator. Figure 1(a) and (b) show the XRD data for the FeSe thin films grown on various substrates. In order to avoid the possible epitaxial strain from the substrates, the thicknesses of all films are above 160 nm. All XRD patterns show a fine (00*l*) orientated growth. The corresponding (002) peaks demonstrate slightly leftward shift with $T_c$ increasing as shown in Fig. 1(c). The *c*-axis lattice parameters were calculated from $\theta-2\theta$ XRD patterns by Bragg’s law, and the results will be discussed in the latter part. The full width at half maximum (FWHM) of the x-ray rocking curve is $0.515^{\circ}$, showing high crystalline quality. In addition, the fine epitaxy of films is demonstrated by a clear four-fold symmetry in $\varphi$ scan pattern, as shown in Fig. 1(e).
Results and discussions
=======================
First, all the films on various substrates were grown with the same thickness for a better comparison. The transport properties of films were measured in the Physical Property Measurement System (PPMS-9 T). Figure 2 shows the $R-T$ curves for these films. Since the fabrication process is identical, the superconductivity seems to strongly depend on the substrates. In most samples, a zero-resistance transition can be observed except the films on the substrates of MAO, LSAO, and LSAT. Instead, a low-$T$ upturn occurs in the $R-T$ curves of films on MAO and LSAO substrates. While, the films on CF, LF and STO are worthy of more attention, where high onset $T_c$ of 15 K, 13 K and 11.5 K are respectively reached. These values are higher than the previous reports on the same substrates.[@Wang2; @Agatsuma; @Maeda; @Chen; @Jung; @Song]
Since the thickness is considered as a key factor for the superconductivity of thin films, especially for the ultra-thin FeSe system,[@Wang; @Liu; @Song] we study the thickness dependence effect of the FeSe/CaF$_2$ films with the highest $T_c$. In Fig. 3(a), we show the temperature dependence of the resistance for the FeSe films with various thickness adjusted only by the counts of laser pulse. The superconductivity still remains as the thickness is reduced to 10 nm (about 18 UC), where $T_{c0}$ = 2 K. In previous work, strong disorder was usually expected to induce a quantum transition in an ultra-thin system, which could completely suppress the superconductivity.[@Nabeshima; @Schneider] The existence of superconductivity in the present 10 nm film exhibits the high quality with less disorder. Increasing film thickness, the $T_{c0}$ also ascends gradually and the films display a bulk-like behavior when the thickness exceeds 160 nm.
Considering the composition off-stoichiometry for the superconductivity of FeSe films, namely the Fe or Se vacancy,[@Chen2] we prepared the targets with subtle adjustment in the nominal ratio of Fe:Se, including 1:1.10, 1:1.05, 1:1.03, 1:1.00, 1.00:0.99, 1.00:0.97, 1:0.95, 1:0.90, and so on. For comparison, the films were grown with the same thickness of $\sim$160 nm and on the same CaF$_2$ substrate. Figure 3(b) shows the $R-T$ curves for films grown with different targets. $T_{c}$ of the films can be well tuned from below 2 K to 15 K with narrow superconducting transitions ($\Delta T <$ 2 K). It should be noted that the best sample deposited by Fe:Se = 1:0.97 target shows $\Delta T =$ 1.2 K, $R-T$ curve RRR $\sim$ 5, the $T_{c} =$ 15 K, and stable superconductivity for more than half a year, which upgrades the record of the bulk-like FeSe films grown by traditional methods.[@Nabeshima]By precise adjustment of target composition, it seems that the ratio of Fe to Se directly determines the superconductivity of the final films. Both energy dispersive x-ray spectra (EDX) and inductively coupled plasma atomic emission spectroscopy (ICP-AES) have been used to check the chemical composition. However, the composition between different superconducting films cannot be clearly defined by these two methods.
{width="0.9\columnwidth"}
Figure 4(a) gives the zero resistance $T_{c0}$ with respect of the corresponding lattice constant *c* of FeSe films deposited on various substrates. There is an obvious positive correlation between $T_{c0}$ and lattice constant *c* of FeSe films as guided by the dash line, but yet between $T_{c0}$ and the surface atomic distance (*d*) of the substrates (see Fig. 4(b)), indicating the strain from substrate has been relieved in the present bulk-like films. We emphasize that depositing FeSe on the TiO$_2$(100) substrate with rectangular lattice (*b* = 4.593 Å, *c* = 2.958 Å) will introduce an anisotropic epitaxial pressure. However, the FeSe/TO film still display superconductivity which is comparable with that of certain films with isotropic strain. Therefore, in comparison with composition for targets, film thickness and substrates, the *c*-axis lattice constant is more closely related to the superconductivity of FeSe. Comparably, the superconductivity of the multi-layered FeSe-based superconductors, i.e., (Li$_{1-x}$Fe$_x$)OHFeSe, are observed as a similar *c*-axis constant dependence behavior,[@Dong] which reinforces our understanding on the profile of lattice parameters on the superconductivity.
{width="1\columnwidth"}
Conclusions
===========
In conclusion, we have successfully prepared high-quality superconducting FeSe films on such as CaF$_2$, LiF, SrTiO$_3$, LaAlO$_3$, TiO$_2$(100), MgO, BaF$_2$, MgF$_2$, Nb-doped SrTiO$_3$, La$_{0.3}$Sr$_{0.7}$Al$_{0.65}$Ta$_{0.35}$O$_3$, (Sr,La)AlO$_4$ and MgAl$_2$O$_4$. Among them, the films on the CaF$_2$ substrate possess a maximum $T_{c}$ of 15 K, which is the highest value reported so far. By slightly adjusting the ratio of Fe to Se in the targets, a series of FeSe/CaF$_2$ films with tunable $T_{c}$ from $<$ 2 K to 15 K are obtained. The superconductivity of the films on various substrates is found to be mainly dependent on the *c*-axis lattice parameter of FeSe films, in a positive correlation. However, there is no direct correlation between the $T_c$ and the surface atomic distance of substrates, therefore, it is unlikely that the strain effect plays a role in our experiments. The origin of the modification on *c*-axis parameter needs to be further identified, nevertheless, high-quality FeSe thin films with tunable $T_c$ may pave the way for understanding the nature of FeSe from bulk crystal to ultrathin film, and shed light on the applications of superconducting microelectronic devices, such as the hybrid Josephson junctions, single-photon detection superconducting nanowires, and so on.
This work was supported by National Key Basic Research Program of China (Grant Nos. 2015CB921000, and 2016YFA0300301), National Natural Science Foundation of China (Grant Nos. 11674374, 11474338, 11234006, and 61501220), the Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDY-SSW-SLH001 and QYZDY-SSW-SLH008), Strategic Priority Research Program of CAS (Grant Nos. XDB07020100 and XDB07030200), Beijing Municipal Science and Technology Project (Grant No. Z161100002116011), Jiangsu Provincial Natural Science Fund (No. BK20150561), and Opening Project of Wuhan National High Magnetic Field Center (No. 2015KF19).
Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. **130**, 3296 (2008). S. Haindl, M. Kidszun, S. Oswald, C. Hess, B. B$\ddot{u}$chner, S. K$\ddot{o}$lling, L. Wilde, T. Thersleff, V. V. Yurchenko, M. Jourdan, H. Hiramatsu, and H. Hosono, Rep. Prog. Phys. **77**, 046502 (2014). F. C. Hsu, J. Y. Luo, K. W. Yeh, T. K. Chen, T. W. Huang, P. M. Wu, Y. C. Lee, Y. L. Huang, Y. Y. Chu, D. C. Yan, and M. K. Wu, P. Natl. Acad. Sci. USA **105** (38), 14262-14264 (2008). J. Paglione and R. L. Greene, Nat. Phys. **6**, 645-658 (2010). T. K. Chen, C. C. Chang, H. H. Chang, A. H. Fang, C. H. Wang, W. H. Chao, C. M. Tseng, Y. C. Lee, Y. R. Wu, M. H. Wen, H. Y. Tang, F. R. Chen, M. J. Wang, M. K. Wu, and D. V. Dyck, P. Natl. Acad. Sci. USA **111** (1), 63-68 (2014). S. Kasaharaa, T. Watashigea, T. Hanagurib, Y. Kohsakab, T. Yamashitaa, Y. Shimoyamaa, Y. Mizukamia, R. Endoa, H. Ikedaa, K. Aoyamaa, T. Terashimae, S. Ujie, T. Wolff, H. v. L$\ddot{o}$hneysenf, T. Shibauchia, and Y. Matsudaa, P. Natl. Acad. Sci. USA **111** (46), 16309-16313 (2014). J. P. Sun, K. Matsuura, G. Z. Ye, Y. Mizukami, M. Shimozawa, K. Matsubayashi, M. Yamashita, T. Watashige, S. Kasahara, Y. Matsuda, J. Q. Yan, B. C. Sales, Y. Uwatoko, J. G. Cheng, and T. Shibauchi, Nat. Commun. **7**, 12146 (2016). S. Medvedev, T. M. McQueen, I. A. Troyan, T. Palasyuk, M. I. Eremets, R. J. Cava, S. Naghavi, F. Casper, V. Ksenofontov, G. Wortmann, and C. Felser, Nat. Mater. **8**, 630-633 (2009). Q. Y. Wang, Z. Li, W. H. Zhang, Z. C. Zhang, J. S. Zhang, W. Li, H. Ding, Y. B. Ou, P. Deng, K. Chang, J. Wen, C. L. Song, K. He, J. F. Jia, S. H. Ji, Y. Y. Wang, L. L. Wang, X. Chen, X. C. Ma, and Q. K. Xue, Chin. Phys. Lett. **29** (3), 037402 (2012). D. Liu, W. Zhang, D. Mou, J. He, Y. B. Ou, Q. Y. Wang, Z. Li, L. Wang, L. Zhao, S. He, Y. Peng, X. Liu, C. Chen, L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, J. Hu, X. Chen, X. Ma, Q. Xue, and X. J. Zhou, Nat. Commun. **3**, 931 (2012). R. Peng, H. C. Xu, S. Y. Tan, H. Y. Cao, M. Xia, X. P. Shen, Z. C. Huang, C. H. Wen, Q. Song, T. Zhang, B. P. Xie, X. G. Gong, and D. L. Feng, Nat. Commun. **5**, 5044 (2014). J. F. Ge, Z. L. Liu, C. Liu, C. L. Gao, D. Qian, Q. K. Xue, Y. Liu, and J. F. Jia, Nat. Mater. **14**, 285-289 (2015). J. J. Lee, F. T. Schmitt, R. G. Moore, S. Johnston, Y. T. Cui, W. Li, M. Yi, Z. K. Liu, M. Hashimoto, Y. Zhang, D. H. Lu, T. P. Devereaux, D. H. Lee, and Z. X. Shen, Nature **515**, 245-248 (2014). M. J. Wang, J. Y. Luo, T. W. Huang, H. H. Chang, T. K. Chen, F. C. Hsu, C. T. Wu, P. M. Wu, A. M. Chang, and M. K. Wu, Phys. Rev. Lett. **103**, 117002 (2009). S. Agatsuma, T. Yamagishi, S. Takeda, and M. Naito, Phys. C **470** (20), 1468-1472 (2010). A. Maeda, F. Nabeshima, H. Takahashi, T. Okada, Y. Imai, I. Tsukada, M. Hanawa, S. Komiya, and A. Ichinose, Appl. Surf. Sci. **312**, 43-49 (2014). Y. Han, W. Y. Li, L. X. Cao, S. Zhang, B. Xu and B. R. Zhao, J. Phys. Condens. Matter. **21** (23), 235702 (2009). Y. F. Nie, E. Brahimi, J. I. Budnick, W. A. Hines, M. Jain and B. O. Wells, Appl. Phys. Lett. **94** (24), 242505 (2009). T. K. Chen, J. Y. Luo, C. T. Ke, H. H. Chang, T. W. Huang, K. W. Yeh, C. C. Chang, P. C. Hsu, C. T. Wu, M. J. Wang, and M. K. Wu, Thin Solid Films **519** (5), 1540-1545 (2010). S. G. Jung, N. H. Lee, E. M. Choi, W. N. Kang, S. I. Lee, T. J. Hwang, and D. H. Kim, Phys. C **470** (22), 1977-1980 (2010). F. Nabeshima, Y. Imai, M. Hanawa, I. Tsukada and A. Maeda, Appl. Phys. Lett. **103** (17), 172602 (2013). S. Y. Tan, Y. Zhang, M. Xia, Z. R. Ye, F. Chen, X. Xie, R. Peng, D. F. Xu, Q. Fan, H. C. Xu, J. Jiang, T. Zhang, X. C. Lai, T. Xiang, J. P. Hu, B. P. Xie, and D. L. Feng, Nat. Mater. **12**, 634-640 (2013). C. L. Song, Y. L. Wang, Y. P. Jiang, Z. Li, L. Wang, K. He, X. Chen, X. C. Ma, and Q. K. Xue, Phys. Rev. B **84**, 020503 (2011). T. M. McQueen, Q. Huang, V. Ksenofontov, C. Felser, Q. Xu, H. Zandbergen, Y. S. Hor, J. Allred, A. J. Williams, D. Qu, J. Checkelsky, N. P. Ong, and R. J. Cava, Phys. Rev. B **79**,014522 (2009). Y. Fang, D. H. Xie, W. Zhang, F. Chen, W. Feng, B. P. Xie, D. L. Feng, X. C. Lai, and S. Y. Tan, Phys. Rev. B **93**, 184503 (2016). R. Schneider, A. G. Zaitsev, D. Fuchs, and H. v. L$\ddot{o}$hneysenf, Phys. Rev. Lett. **108**, 257003 (2012). X. L. Dong, H. X. Zhou, H. X. Yang, J. Yuan, K. Jin, F. Zhou, D. N. Yuan, L. L. Wei, J. Q. Li, X. Q. Wang, G. M. Zhang, and Z. X. Zhao, J. Am. Chem. Soc. **137**, 66 (2015).
|
---
abstract: 'The reversible jump algorithm is a useful Markov chain Monte Carlo method introduced by [@green1995reversible] that allows switches between subspaces of differing dimensionality, and therefore, model selection. Although this method is now increasingly used in key areas (e.g. biology and finance), it remains a challenge to implement it. In this paper, we focus on a simple sampling context in order to obtain theoretical results that lead to an optimal tuning procedure for the considered reversible jump algorithm, and consequently, to easy implementation. The key result is the weak convergence of the sequence of stochastic processes engendered by the algorithm. It represents the main contribution of this paper as it is, to our knowledge, the first weak convergence result for the reversible jump algorithm. The sampler updating the parameters according to a random walk, this result allows to retrieve the well-known $0.234$ rule for finding the optimal scaling. It also leads to an answer to the question: “with what probability should a parameter update be proposed comparatively to a model switch at each iteration?”'
address:
- 'Department of Statistics, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom'
- 'Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal, QC, H3C 3J7, Canada'
- 'Département de mathématiques, Université du Québec à Montréal, C.P. 8888, Succursale Centre-ville, Montréal, QC, H3C 3P8, Canada'
author:
- Philippe Gagnon
- Mylène Bédard
- Alain Desgagné
bibliography:
- '../reference.bib'
title: Weak Convergence and Optimal Tuning of the Reversible Jump Algorithm
---
Bayesian inference ,Markov chain Monte Carlo methods ,Metropolis-Hastings algorithms ,Model selection ,Optimal scaling ,Random walk Metropolis algorithms.
65C05 ,62F15
Introduction {#sec_intro_weak}
============
Markov chain Monte Carlo (MCMC) methods provide algorithms to generate Markov chains having an invariant measure that corresponds to the distribution with respect to which we are interested in computing integrals. The implementation of such samplers usually requires the specification of some functions. For instance, at each step of the Metropolis-Hastings (MH) algorithm ([@metropolis1953equation] and [@hastings1970monte]), the most commonly used method, a candidate for the next state of the Markov chain is generated from a proposal distribution (which has to be specified), and accepted according to a probability function (which is provided by the method). There are no guidelines that are valid in all situations for effectively achieving the specification step, at least no practical ones. The problem is that poor designs of the required functions may lead to inefficient algorithms, resulting in Markov chains that slowly explore their state space, and therefore, inadequate samples (see [@peskun1973optimum] and [@tierney1998note]). Researchers have therefore focused on finding optimal tuning procedures in special cases. [@roberts1997weak] studied the MH algorithm in the situation where the proposal distribution is a normal centered around the current state of the chain; algorithms of this type are called random walk Metropolis (RWM) algorithms. In their case, the specification step consists in selecting the variance of the normal distribution. This task is however not trivial as small variances lead to tiny movements of the Markov chain, while large variances induce high rejection rates of candidates. Assuming that the algorithm is used to sample from a distribution of $n$ independent and identically distributed (i.i.d.) random variables, the authors proved the existence of an asymptotically optimal variance for the random walk as $n\rightarrow\infty$ through a weak convergence argument. They also provided the following simple strategy to identify the (asymptotically) optimal scaling: tune the variance so that the acceptance rate of candidates is $0.234$. This leads to a straightforward implementation of the algorithm. A lot of research has been carried out to generalise this result to more elaborate target distributions (e.g. [@roberts2001optimal], [@neal2006optimal], [@bedard2007weak], [@bedard2008optimal], [@beskos2009optimal], [@bedard2012scaling], [@mattingly2012diffusion] and [@beskos2013optimal]). What has been observed is that the $0.234$ rule is robust, in the sense that it often leads to scalings that are (almost) optimal even when the i.i.d. assumption is violated.
A flaw of MH algorithms is that they do not allow subspace switches, i.e. switching from the parameter space of a model to that of another model. This gap was corrected by [@green1995reversible] with the introduction of the reversible jump (RJ) algorithm. This method has a tremendous potential because of its capability to deliver information on both the “good” models and their parameters from a single output. For instance, [@richardson1997bayesian] used it to estimate the number of components of mixtures, and their parameters, simultaneously. The price to pay is that many functions have to be specified in order to implement this algorithm. In this paper, we study the RJ algorithm in the same mindset as [@roberts1997weak]; we aim at providing guidelines to users and open new research directions towards an automatic RJ algorithm.
Existing research on the RJ algorithm has mainly focused on ways to facilitate subspace switches (e.g. [@brooks2003efficient], [@hastie2005towards], [@al2004improving] and [@karagiannis2013annealed]), and therefore, the exploration of the entire state space. The main drawback of the proposed approaches is the difficulty to implement them. This justifies the need for practical guidelines that will promote accessibility of the RJ algorithm. As a first step towards automated implementation of this method, we focus on a simple sampling context in order to obtain weak convergence results. The context is defined in Section \[sec\_context\_weak\]. We consider the natural generalisation of the target distribution in [@roberts1997weak], i.e. the case where the parameters are conditionally i.i.d. given any Model $K$. We assume additional structure on the target to diminish the number of functions required for the implementation. In particular, it makes the strategies proposed by the authors mentioned above unnecessary. We assume that Model $K+1$ is comprised of one more parameter than Model $K$, and that when switching from Model $K$ to Model $K+1$, the parameters that are in the former do not change distributions. Random walks are used to explore the model and parameter spaces. This context can correspond to variable selection in linear regression in the situation where the scale parameters of the error terms are known and the variables are orthogonal. Although the assumptions are strong, we believe that the tuning rules found under them are robust. We also believe that this is a good starting point for proving weak convergence results for more general target distributions and RJ algorithms.
The weak convergence of the sequence of stochastic processes engendered by the algorithm towards diffusion processes is established in Section \[sec\_problem\_weak\]. We explain in Section \[sec\_optimisation\] that this result allows to establish the validity of the $0.234$ rule of [@roberts1997weak], when the acceptance rate is computed considering only the iterations in which it is proposed to update the parameters. This result also allows to find the probability with which a model switch (and therefore a parameter update) should be proposed at each iteration. Essentially, the poorer is the design of the proposal distribution generating candidates for the additional parameter when switching from Model $K$ to Model $K+1$, the higher should be the probability to propose model switches. This is intuitive given that a poor design leads to poor candidates, and therefore, high rejection rates. The result indicates that the optimal probability for proposing parameter updates decreases as $1/\sqrt{A}$, where $1/A$ is equal (up to a known constant) to the probability to propose and accept model switches. Therefore, given that (in our case) the optimal probability for proposing parameter updates when $A=2$ is $0.415$, this provides a rule for constructing the proposal distribution for the different movement types. The proposed rules for tuning the RJ algorithm are described in detail in Section \[sec\_optimal\_impl\]. The advantage of considering a simple setting is that we obtain explicit solutions. In Section \[sec\_optimal\_impl\], we also present situations in which we conjecture that our results hold. They include, for instance, posterior distributions arising from robust principal component regressions. We show in Section \[sec\_simulation\_weak\] how the design of the RJ algorithm has an impact on the produced samples. We observe that, for moderate dimensions, selecting any probability between $0.2$ and $0.6$ for proposing parameter updates is almost optimal, which seems to be a valid general guideline.
Sampling Context {#sec_context_weak}
================
Let $$\pi_n(k,\mathbf{x}^k)=p_n(k)\prod_{i=1}^{n+k} f(x_i^k)$$ be the joint posterior distribution of $(K^n,\mathbf{X}^{K^n})$, where $K^n\in\{1,\ldots,\lfloor \sqrt{n}\log n\rfloor\}$ ($\lfloor \cdot \rfloor$ is the floor function), $\mathbf{X}^{K^n}:=(X_1^{K^n},\ldots, X_{n+K^n}^{K^n})\in\operatorname{\mathbb{R}}^{n+K^n}$, $n\in\{7,8,\ldots\}$, $f$ is a strictly positive one-dimensional probability density function (PDF), and $p_n$ is a probability mass function (PMF) such that $p_n(k)>0$ for all $k\in \{1,\ldots,\lfloor \sqrt{n}\log n\rfloor\}$. We consider $n$ to be an integer greater than or equal to 7 just to avoid technical complications in the proofs. The random variable $K^n$ represents the model indicator ($K^n=1$ implies that Model 1 is considered, for instance), and $\mathbf{X}^{K^n}$ is the parameter vector of Model $K^n$. Therefore, $X_1^1,\ldots,X_{n+1}^1$ are the $n+1$ parameters of Model 1, $X_1^2,\ldots,X_{n+2}^2$ are the $n+2$ parameters of Model 2, etc. To simplify the notation, we will denote $K:=K^n$ and then $\mathbf{X}^K:=\mathbf{X}^{K^n}$. As mentioned in the introduction, $\pi_n$ can correspond to the joint posterior of the models and their parameters in variable selection in the situation where the scale parameters of the error terms are known and the variables are orthogonal. In fact, it represents the situation where, in addition, the models comprised of $n$ variables or less have negligible probabilities (which is modelled by $p_n(k)=0$ for $k\leq n$).
The objective is to obtain adequate samples from the joint posterior distribution of $(K,\mathbf{X}^{K})$ through MCMC methods in order to approximate probabilities, expectations, or any other quantity we might be interested in. The following RJ algorithm is applied to sample from $\pi_n$:
1. Initialisation: set $(K(0),\mathbf{X}^{K(0)}(0))$.
2. 3. Generate $U\sim\mathcal{U}(0,1)$.
4. When $U\leq g(1)$ ($g$ is a PDF such that $g(j)>0$ for all $j\in\{1,2,3\}$), attempt an update of the parameters. Generate $\mathbf{Y}^{K(m)}(m+1)\sim \mathcal{N}(\mathbf{X}^{K(m)}(m),(\ell^2/(n+{K(m)}))\mathcal{I}_{n+K(m)})$, where $\mathbf{Y}^{K(m)}(m+1):=(Y_1^{K(m)}(m+1), \ldots,Y_{n+K(m)}^{K(m)}$ $(m+1))$, $\mathcal{I}_{n+K(m)}$ is the identity matrix of size $n+K(m)$ and $\ell$ is a positive constant. Generate $U_a\sim\mathcal{U}(0,1)$. When $$\label{eqn_prob_update}
U_a\leq 1 \wedge \frac{\prod_{i=1}^{n+K(m)} f(Y_i^{K(m)}(m+1))}{\prod_{i=1}^{n+K(m)} f(X_i^{K(m)}(m))},$$ set $(K(m+1),\mathbf{X}^{K(m+1)}$ $(m+1))=(K(m),\mathbf{Y}^{K(m)}(m+1))$.
5. When $g(1)<U\leq g(1)+g(2)$, attempt adding a parameter to switch from Model $K(m)$ to Model $K(m)+1$. Generate $U(m+1)\sim q$ ($q$ is a strictly positive PDF) and generate $U_a\sim\mathcal{U}(0,1)$. When $$\label{eqn_prob_add}
U_a \leq 1 \wedge \frac{f(U(m+1))p_n(K(m)+1)g(3)}{q(U(m+1))p_n(K(m))g(2)},$$ set $(K(m+1),\mathbf{X}^{K(m+1)}(m+1))=(K(m)+1,(\mathbf{X}(m)^{K(m)},U(m+1)))$.
6. When $U>g(1)+g(2)$, attempt withdrawing the last parameter to switch from Model $K(m)$ to Model $K(m)-1$. Generate $U_a\sim\mathcal{U}(0,1)$. When $$\label{eqn_prob_remove}
U_A \leq 1 \wedge \frac{q(X_{n+K(m)}^{K(m)}(m))p_n(K(m)-1)g(2)}{f(X_{n+K(m)}^{K(m)}(m))p_n(K(m))g(3)},$$ set $(K(m+1),\mathbf{X}^{K(m+1)}(m+1))=(K(m)-1,\mathbf{X}^{K(m)-}(m))$, where $\mathbf{X}^{K(m)-}(m)$ is the vector $\mathbf{X}^{K(m)}(m)$ without the last component (more precisely $\mathbf{X}^{K(m)-}(m):=(X_1^{K(m)},\ldots, X_{n+K(m)-1}^{K(m)})$).
7. In case of rejection, set $(K(m+1),\mathbf{X}^{K(m+1)}(m+1))=(K(m),\mathbf{X}^{K(m)}(m))$.
8. Go to Step 2.
The resulting stochastic process $\{(K(m),\mathbf{X}^K(m)): m\in\operatorname{\mathbb{N}}\}$ is a $\pi_n$-irreducible and aperiodic Markov chain. In addition, it is easily shown that this Markov chain satisfies the reversibility condition with respect to $\pi_n$ (see [@gagnon2016thesis]), and therefore, that it is ergodic, which guarantees that the Law of Large Numbers holds.
Regularity conditions are now imposed on the different functions. They allow to obtain the weak convergence result stated in Section \[sec\_result\_weak\].
First, we assume that the usual smoothness conditions on the function $f$ are satisfied: $f\in\mathcal{C}^2(\operatorname{\mathbb{R}})$ (the space of real-valued functions on $\operatorname{\mathbb{R}}$ with continuous second derivative), $(\log f(x))'$ is Lipschitz continuous and ${\mathbb{E}}[((\log f(X))')^4]<\infty$, where the expectation is computed with respect to $f$. This last condition can be replaced by ${\mathbb{E}}[(f''(X)/f(X))^2]<\infty$, which is slightly stronger. We additionally assume that there exists a constant $A^*\geq 1$ such that $$0<\frac{f}{q}\leq A^* \text{\quad and therefore \quad} \frac{1}{A^*}\leq \frac{q}{f}<\infty.$$ This condition corresponds to that required for the rejection sampling method. It ensures that the tails of $q$ are at least as heavy as those of $f$. A small value for the constant $A^*$ means that $q$ is similar to $f$, and therefore, that it is a good choice of proposal distribution. Note that, when we can directly sample from $f$, we can set $q=f$.
The distribution of $K$, which is $p_n$, also fulfills some conditions. The problem here is that $K$ is a discrete random variable, but the goal is to establish the weak convergence of its associated stochastic process towards a diffusion. A natural way of taking a step in this direction is to assume that the distribution of a suitable transformation of $K$ converges towards one having a PDF. In other words, it is to assume that, in the limit, this transformation of $K$ is a continuous random variable. We also have to consider that, contrarily to the acceptance probability for updating the parameters, it is not possible to use the Law of Large Numbers to obtain the converge of the acceptance probabilities for switching models, which would facilitate establishing the weak convergence. The following construction allows, when combined with additional structure on the PMF $g$, to reach our goal.
We assume that $p_n$ has its mode in the middle of the set $\{1,\ldots,\lfloor \sqrt{n}\log n\rfloor\}$ and is symmetric with respect to this mode. Two distinct cases thus have to be considered: when $\lfloor \sqrt{n}\log n\rfloor$ is even or odd. When $\lfloor \sqrt{n}\log n\rfloor$ is odd, the mode is $(\lfloor \sqrt{n}\log n\rfloor+1)/2$ and we assume that $$\begin{aligned}
&p_n(k+1)=a_{k,n}p_n(k), k\in\{(\lfloor \sqrt{n}\log n\rfloor+1)/2,\ldots,\lfloor \sqrt{n}\log n\rfloor-1\}, \cr
&p_n(k-1)=a_{k-1,n}p_n(k), k\in\{2,\ldots,(\lfloor \sqrt{n}\log n\rfloor+1)/2\},\end{aligned}$$ where $a_{k,n}:=(1-b_{k,n}/\sqrt{n})$ with $$b_{k,n}:={\left|\frac{k-\lfloor \sqrt{n}\log n\rfloor/2}{\sqrt{n}}\right|}.$$ Note that $a_{k,n}$ decreases with the distance between $k$ and the mode. This distribution is symmetric with respect to $(\lfloor \sqrt{n}\log n\rfloor+1)/2$ and is such that $$\begin{aligned}
p_n\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}+k\right)&=p_n\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}-k\right) =p_n\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\right)\prod_{i=1}^k\left(1-\frac{i-1/2}{n}\right),
\end{aligned}$$ where $k\in\left\{1,\ldots,(\lfloor \sqrt{n}\log n\rfloor-1)/2\right\}$.
When $\lfloor \sqrt{n}\log n\rfloor$ is even, the distribution $p_n$ is bimodal with modes at $\lfloor \sqrt{n}\log n\rfloor/2$ and $\lfloor \sqrt{n}\log n\rfloor/2+1$. Using the same definitions as above for $a_{k,n}$ and $b_{k,n}$, we assume that $$\begin{aligned}
&p_n(k+1)=a_{k,n}p_n(k), k\in\{\lfloor \sqrt{n}\log n\rfloor/2+1,\ldots,\lfloor \sqrt{n}\log n\rfloor-1\}, \cr
&p_n(k-1)=a_{k-1,n}p_n(k), k\in\{2,\ldots,\lfloor \sqrt{n}\log n\rfloor/2\},\end{aligned}$$ which implies that $$\begin{aligned}
\label{eqn_prob_k_even}
p_n\left(\frac{\lfloor \sqrt{n} \log n\rfloor}{2}+1+k\right)&=p_n\left(\frac{\lfloor \sqrt{n} \log n\rfloor}{2}-k\right)=p_n\left(\frac{\lfloor \sqrt{n} \log n\rfloor}{2}\right)\prod_{i=1}^k \left(1-\frac{i}{n}\right),
\end{aligned}$$ where $k\in\{1,\ldots,\lfloor \sqrt{n}\log n\rfloor/2-1\}$.
The function $p_n(k)$ decreases at an exponential rate which is bounded below by $1/2$ when the distance between $k$ and the mode increases. The ratios $p_n(k+1)/p_n(k)$ and $p_n(k-1)/p_n(k)$ are indeed essentially bounded below by $1/2$. The case $\lfloor \sqrt{n}\log n\rfloor=5$ ($n=7$), for instance, is such that the “best” model has $n+(\lfloor \sqrt{n}\log n\rfloor+1)/2=10$ parameters, because the mode of $p_n$ is $K=3$ and the models have $n+K$ parameters, and the model with an additional parameter is less appropriate (so is the model with one less parameter), in the sense that $p_n(4)/p_n(3)=p_n(2)/p_n(3)=0.93$. The more parameters we add (or withdraw), the less appropriate the models are. Although $p_n$ is a pure construction, similar structures may arise in situations in which the models can be ranked by number of parameters and the posterior reflects the existence of a balance between overfitting (which involves a lot of parameters) and stability, in the spirit of Occam’s razor. This is the case for instance in principal component regression. In this situation, the model indicator $K$ represents the number of components included in the model (i.e. Model $K=k$ is the model with the first $k$ components). It is easy to imagine situations where it would be optimal to include some, but not all, components. This would result in posterior distributions for the various models with structures as that described above (see, e.g., the real data analysis in [@gagnon2017robustPCR]). Note that, when $p_n$ has such a structure, it makes it easier for the RJ algorithm to explore the entire state space. Indeed the ratios $p_n(k+1)/p_n(k)$ and $p_n(k-1)/p_n(k)$ are never very small, which facilitates the transitions (see (\[eqn\_prob\_add\]) and (\[eqn\_prob\_remove\])).
Finally, we add structure on the PMF $g$. It will ease handling of the acceptance probabilities for switching models, because, as mentioned above, it is not possible to take advantage of the Law of Large Numbers in this case. This represents the last required step towards the weak convergence result. We consider that the function $g$ is as follows: $$\begin{aligned}
\label{def_g}
g(j):=\begin{cases}
\tau \text{ if } j=1, \cr
(1-\tau)A/(A+1) \text{ if } j=2, \cr
(1-\tau)/(A+1) \text{ if } j=3,
\end{cases}
\end{aligned}$$ where $0<\tau<1$ is a constant and $A:=2A^*$. The acceptance probability associated with the inclusion of an extra parameter (see (\[eqn\_prob\_add\])) becomes the minimum between 1 and $f(U)/q(U)\times 1/A\times p_n(K+1)/p_n(K)$. By assumption, $2f/q\leq A$ and $p_n(K+1)/p_n(K)\leq 2$; therefore, this acceptance probability is simply $f(U)/q(U)\times 1/A\times p_n(K+1)/p_n(K)$. Furthermore, the acceptance probability associated with the withdrawal of the last parameter (see (\[eqn\_prob\_remove\])) becomes the minimum between 1 and $q(X_{n+K}^{K})/f(X_{n+K}^{K})\times A\times p_n(K-1)/p_n(K)\geq 1$, which means that this type of movement is automatically accepted (whenever it is possible to withdraw a parameter, i.e. when $K>1$). Therefore, the probability to propose and accept switches from Model $K$ to Model $K-1$ is $(1-\tau)/(A+1)$. Proposition \[prop\_prob\_acc\_ajout\] indicates that this is asymptotically equal to the “average” probability to propose and accept switches from Model $K$ to Model $K+1$. The probability that $K(m)$ increases by one is therefore the same as that of decreasing by one, globally (and asymptotically).
\[prop\_prob\_acc\_ajout\] Consider the assumptions and the RJ algorithm described in this section. If we assume that $(K(0),\mathbf{X}^K(0))\sim\pi_n$, then for all $m\in\operatorname{\mathbb{N}}$, $${\mathbb{E}}\left[1 \wedge \frac{f(U(m+1))p_n(K(m)+1)g(3)}{q(U(m+1))p_n(K(m))g(2)}\right]\rightarrow \frac{1}{A} \text{ as } n\rightarrow\infty.$$
See Section \[sec\_proofs\_props\].
We believe that the tuning rule for the proposal distribution $g$ mentioned in the introduction (that will be described in detail in Section \[sec\_optimal\_impl\]) is valid when the structure of $g$ is different than that above. What allows us to establish our weak convergence results is that the acceptance probabilities for switching models are (roughly) constant.
Towards Optimal Implementation of the RJ algorithm {#sec_result_weak}
==================================================
In order to implement the RJ algorithm described in Section \[sec\_context\_weak\], we have to specify the PDF $q$ and values for the constants $A$, $\tau$ and $\ell$. In Section \[sec\_problem\_weak\], we present weak convergence results that are used in Section \[sec\_optimisation\] to find asymptotically optimal (as $n\rightarrow \infty$) values for $\tau$ and $\ell$. This is what allows deriving the rules for tuning the RJ algorithm.
Weak Convergence Results {#sec_problem_weak}
------------------------
In order to study the asymptotic behaviour of the algorithm, we consider the following rescaled stochastic process: $$\begin{aligned}
\label{def_Z}
&\mathbf{Z}^n(t):= \left(\frac{K(\lfloor nt \rfloor )-\lfloor \sqrt{n}\log n \rfloor/2}{\sqrt{n}},\mathbf{X}^{K(\lfloor nt \rfloor )}(\lfloor nt \rfloor )\right),
\end{aligned}$$ where $t\geq0$. The continuous-time stochastic process $\{\mathbf{Z}^n(t): t\geq0\}$ is a sped up and modified version of $\{(K(m),\mathbf{X}^K(m)): m\in\operatorname{\mathbb{N}}\}$. In any given iteration, the average jump size of the parameters $$\mathbf{X}^{K(\lfloor nt \rfloor )}(\lfloor nt \rfloor ):=\left(X_1^{K(\lfloor nt \rfloor)}(\lfloor nt \rfloor),\ldots,X_{n+K(\lfloor nt \rfloor)}^{K(\lfloor nt \rfloor)}(\lfloor nt \rfloor)\right)$$ decreases with $n$ because the variance of the random walk is proportional to $1/(n+{K(\lfloor nt \rfloor)}))$. The jump size of $\{K(\lfloor nt \rfloor )/\sqrt{n}: t\geq 0\}$ also decreases with $n$. In fact, each time it moves, its jump size is $1/\sqrt{n}$. The decreasing size of the jumps, combined with the acceleration of $\{\mathbf{Z}^n(t): t\geq0\}$, result in a continuous and non-trivial limiting process. We subtract $\lfloor \sqrt{n}\log n \rfloor/(2\sqrt{n})$ from $\{K(\lfloor nt \rfloor )/\sqrt{n}: t\geq 0\}$ in order to obtain a limiting process with components that take values on the real line. In particular, the asymptotic stationary distribution of this component of $\{\mathbf{Z}^n(t): t\geq0\}$, that we denote $\{Z_1^n(t):t\geq 0\}$, is a standard normal, as stated in Proposition \[prop\_z\_1\_to\_normal\].
\[prop\_z\_1\_to\_normal\] Consider the assumptions described in Section \[sec\_context\_weak\], the stochastic process $\{\mathbf{Z}^n(t): t\geq0\}$ defined in (\[def\_Z\]), and assume that $\mathbf{Z}^n(0)\sim\pi_n$. Then, as $n\rightarrow\infty$, $Z_1^n(t)$ converges in distribution towards a standard normal random variable, for all $t\geq 0$.
See Section \[sec\_proofs\_props\].
The main result is now stated.
\[main\_result\] Consider the assumptions and the RJ algorithm described in Section \[sec\_context\_weak\]. Consider the stochastic process $\{\mathbf{Z}^n(t): t\geq0\}$ defined in (\[def\_Z\]) and assume that $\mathbf{Z}^n(0)\sim\pi_n$. Then, as $n\rightarrow\infty$, the first two components of $\{\mathbf{Z}^n(t):t\geq0\}$ converge weakly towards a bidimensional Langevin diffusion, i.e. $$\{\mathbf{Z}_{1,2}^n(t): t\geq 0\}:=\{(Z_1^n(t),Z_2^n(t)): t\geq 0\}\Rightarrow \{\mathbf{Z}(t): t\geq 0\} \text{ as } n\rightarrow\infty,$$ where the process $\{\mathbf{Z}(t): t\geq 0\}$ is comprised of two independent components such that $Z_1(0)\sim \mathcal{N}(0,1)$, $Z_2(0)\sim f$, and $$dZ_1(t)=-(1-\tau)/(A+1)\times Z_1(t)dt+\sqrt{2(1-\tau)/(A+1)}dB_1(t),$$ $$dZ_2(t)=\tau \ell^2 \Phi(-\ell\sqrt{\Upsilon}/2)(\log f(Z_2(t)))' dt+\sqrt{2\tau \ell^2\Phi(-\ell\sqrt{\Upsilon}/2)}dB_2(t),$$ with $\{B_1(t): t\geq0\}$ and $\{B_2(t): t\geq0\}$ being two independent Wiener processes, and $\Upsilon:={\mathbb{E}}\left[(\log f(Z_2(0))')^2\right]$.
See Section \[sec\_proof\_thm\_1\].
The notation “$\Rightarrow$” represents weak convergence (or convergence in distribution) of processes in the Skorokhod topology (for more details about this type of convergence, see Section 3 of [@ethier1986markov]).
Optimisation {#sec_optimisation}
------------
The sample paths of $\{\mathbf{Z}(t): t\geq 0\}$ depend on $\tau, A,\ell$, $\Upsilon$ and $f$. In this section, we find values for $\ell$ and $\tau$ that are such that, for given $A$ and $f$ (and therefore $\Upsilon$), the state space exploration of this stochastic process is optimal. Optimising the asymptotic state space exploration of $\{(K(\lfloor nt \rfloor ),X_1^{K(\lfloor nt \rfloor)}(\lfloor nt \rfloor)): t\geq 0\}$ is sufficient to optimise that of $\{(K(\lfloor nt \rfloor ),\mathbf{X}^{K(\lfloor nt \rfloor)}(\lfloor nt \rfloor)): t\geq 0\}$. Indeed, in addition to optimising the exploration of the model space, we optimise the exploration of the first parameter space, and all parameters of all models share the same behaviour. During the theoretical optimisation, the constant $A$ is considered to be fixed because its value cannot be arbitrarily chosen. Indeed, it is tied to the ratio $f/q\leq A^*=A/2$. Note that the PDF $q$ only has an impact on the sample paths of $\{\mathbf{Z}(t): t\geq 0\}$ through the constant $A$.
We first optimise the algorithm with respect to $\ell$. The stochastic process $\{Z_2(t): t\geq 0\}$ can be written as $Z_2(t)=V( 2\tau \ell^2 \Phi(-\ell\sqrt{\Upsilon}/2)\times t)$, where $\{V(t): t\geq 0\}$ is the following Langevin diffusion: $$dV(t)=(\log f(V(t)))'/2 \times dt+dB_2(t).$$ The term $ 2\tau \ell^2 \Phi(-\ell\sqrt{\Upsilon}/2)$ that multiplies the time index of $\{V(t): t\geq 0\}$ to obtain $\{Z_2(t): t\geq 0\}$ is sometimes called the “speed measure” of $\{Z_2(t): t\geq 0\}$. Hereafter, when we discuss about to the “speed” of a stochastic process, we refer to this term. Viewed as a function of $\tau$ and $\ell$, the process $\{Z_2(t): t\geq 0\}$ that optimally explores its state space is thus the one with the largest speed. We can optimise the algorithm with respect to $\ell$ by maximising the speed of $\{Z_2(t): t\geq 0\}$ with respect to this variable, because the value of $\ell$ does not have an impact on the sample paths of $\{Z_1(t): t\geq 0\}$ (see the definition of $\{Z_1(t): t\geq 0\}$ in Theorem \[main\_result\]). The function $2\tau \ell^2 \Phi(-\ell\sqrt{\Upsilon}/2)$ is maximised with respect to $\ell$ by $\ell=2.38/\sqrt{\Upsilon}$, as stated in Corollary 1.2 of [@roberts1997weak]. We therefore obtain the same optimal value as these authors. This is not surprising because the considered RJ algorithm updates the parameters in the same way as the RWM algorithm studied by these authors. Furthermore, the conditional distribution of the parameters given any model is essentially the same as their target distribution. In fact, the latter can be seen as a special case of the target that we consider, and their RMW algorithm can be seen as a special case of the considered RJ algorithm. It is thus interesting to observe that their result holds in our context.
The optimisation with respect to $\ell$ tells us that the most efficient way to update the parameters is to set $\ell=2.38/\sqrt{\Upsilon}$. Therefore, the optimal variance for the random walk is $(2.38^2/(\Upsilon( n+K(m))))\mathcal{I}_{n+K(m)}$. As mentioned in [@roberts1997weak], $\Upsilon$ can be seen as a measure of “roughness” of $f$. Consider for instance the case $f=\mathcal{N}(\mu,\sigma^2)$, which implies that $\Upsilon=1/\sigma^2$. In this case, we observe that small values for $\sigma$ correspond to “rough” $f$ functions, and vice versa. The “rougher” $f$ is, the smaller should be $\ell$. The result under normality also suggests that larger values for $\ell$ should be used when the tails are thicker.
It could seem necessary to know $\Upsilon:={\mathbb{E}}[(\log f(Z_2(0))')^2]$ in order to apply this optimal scaling result. Fortunately, the practical $0.234$ rule provided by [@roberts1997weak] can be used, as stated in Corollary \[prop\_conv\_prob\_acc\]. This corollary is an adapted version of Corollary 1.2 of [@roberts1997weak]. Its proof is similar to the one given by these authors, and is thus omitted (it can nevertheless be found in [@gagnon2016thesis]).
\[prop\_conv\_prob\_acc\] Consider the assumptions and the RJ algorithm described in Section \[sec\_context\_weak\]. Assuming that $(K(0),\mathbf{X}^K$ $(0))\sim\pi_n$, then, for all $m\in\operatorname{\mathbb{N}}$, $${\mathbb{E}}\left[1 \wedge \prod_{i=1}^{n+K(m)} \frac{f(Y_i^{K(m)}(m+1))}{f(X_i^{K(m)}(m))}\right]\rightarrow 2\Phi(-\ell\sqrt{\Upsilon}/2) \text{ as } n\rightarrow\infty.$$ In addition, setting $\ell=2.38/\sqrt{\Upsilon}$ is equivalent to having $2\Phi(-\ell\sqrt{\Upsilon}/2)=2\Phi(-2.38/2)= 0.234$.
Therefore, in order to reach optimal efficiency with respect to $\ell$, RJ users can monitor the acceptance rate of candidates $\mathbf{Y}^{K(m)}(m+1)$, where $\mathbf{Y}^{K(m)}(m+1)\sim \mathcal{N}(\mathbf{X}^{K(m)}(m),(\ell^2/(n+{K(m)}))\mathcal{I}_{n+K(m)})$, and tune the value of $\ell$ such that this rate is approximatively $0.234$. Note that this rate must be computed by considering only iterations in which it is proposed to update the parameters.
We now optimise the algorithm with respect to $\tau$. We need a measure that takes into account the fact that an increase in the value of $\tau$ results in an increase in the speed of $\{Z_2(t): t\geq 0\}$, but also in a decrease in that of $\{Z_1(t): t\geq 0\}$, and vice versa. Essentially, when the value of $\tau$ is increased, more updates of the parameters (and therefore less model switches) are proposed. It would seem natural to consider the total speed of these two processes to optimise the algorithm with respect to $\tau$. The total speed is given by (using the optimal value for $\ell$) $$2\left[\tau(2.38^2/\Upsilon)\Phi(-2.38/2)+(1-\tau)/(A+1) \right].$$ However, it is not a suitable measure because if, for instance $(2.38^2/\Upsilon)\Phi(-2.38/2)=(5.66/\Upsilon)$ $\Phi(-1.19)>1/(A+1)$, it would be proposed to choose the value of $\tau$ as close as possible to 1. In such a situation, there would be very few model switches, which would result in a slow exploration of the entire state space. We need a measure that penalises such a behaviour. This is achieved using integrated linear combinations of the autocorrelation functions (ACFs) of $\{Z_1(t): t\geq 0\}$ and $\{Z_2(t): t\geq 0\}$. Indeed, if for instance we set $\tau$ close to 1, $\{Z_2(t): t\geq 0\}$ would be a “fast” process with an ACF that decreases rapidly towards 0, while $\{Z_1(t): t\geq 0\}$ would be a “slow” process with an almost constant ACF around the value 1, which is undesirable. The sum of these two ACFs would decrease rapidly towards 1, thereafter remaining almost constant around this value. There should therefore exist a value of $\tau$ between 0 and 1 that induces two relatively “fast” processes, with a sum of ACFs that decreases relatively rapidly towards 0. We thus consider the integral of the sum of the ACFs of $\{Z_1(t): t\geq 0\}$ and $\{Z_2(t): t\geq 0\}$ to optimise the algorithm with respect to $\tau$: $$\begin{aligned}
\label{eqn_opt_mea}
\int_0^\infty \left\{ \text{corr}[Z_1(t),Z_1(t+s)]+\text{corr}[Z_2(t),Z_2(t+s)]\right\}ds,\quad t\geq 0.
\end{aligned}$$ We drew inspiration from the effective sample size (see, e.g., Section 12.3.5 of [@robert2004monte]). The measure can be viewed as the sum of the (infinitesimally) integrated autocorrelation times of $\{Z_1(t): t\geq 0\}$ and $\{Z_2(t): t\geq 0\}$. It therefore represents a measure of the total “inefficiency” of these processes and the optimal value of $\tau$ is the one that minimises it.
We need to compute the ACFs of $\{Z_1(t): t\geq 0\}$ and $\{Z_2(t): t\geq 0\}$ in order to find the optimal value for $\tau$. The process $\{Z_1(t): t\geq 0\}$ satisfies the conditions of Theorem 2.1 in [@bibby2005diffusion], implying that $$\text{corr}[Z_1(t),Z_1(t+s)]=\exp\left\{-(1-\tau)s/(A+1)\right\},\quad s,t\geq 0.$$ The behaviour of $\{Z_2(t): t\geq 0\}$ depends on $f$ and consequently its ACF cannot be computed in all generality. A particular situation is now studied in order to obtain general information about the optimal values of $\tau$. We consider the case $f=\mathcal{N}(\mu,\sigma^2)$, $\mu\in\operatorname{\mathbb{R}},\sigma>0$, which allows to obtain an explicit solution to our optimisation problem. Using the optimal value for $\ell$, we have that $$\begin{aligned}
dZ_2(t)&=-\tau 5.66 \sigma^2 \Phi\left(-1.19\right)\times (Z_2(t)-\mu)/\sigma^2 dt+\sqrt{2\tau 5.66 \sigma^2\Phi\left(-1.19\right)}dB_2(t).
\end{aligned}$$ This process also satisfies the conditions of Theorem 2.1 in [@bibby2005diffusion], implying that $$\text{corr}[Z_2(t),Z_2(t+s)]=\exp\left\{-\tau 5.66 \Phi\left(-1.19\right) s\right\},\quad s,t\geq 0.$$ Thus, when $f=\mathcal{N}(\mu,\sigma^2)$ and $\ell$ is set to its optimal value, the optimal value for $\tau$ is $$\begin{aligned}
\label{eqn_tau_opt}
& \frac{\sqrt{5.66 \Phi\left(-1.19\right)(A+1)}-1}{5.66 \Phi\left(-1.19\right)(A+1)-1}.
\end{aligned}$$ The constant $A$ clearly has an impact on the optimal value for $\tau$, as intuitively explained in the introduction. The optimal value essentially decreases as $1/\sqrt{A}$ (see Figure \[fig\_int\_tauopt\_A\]). When $A=2$, the optimal value for $\tau$ is $0.415$. This situation corresponds to $f=q$, and therefore to the best choice of distribution $q$. When $A=5$, the optimal value for $\tau$ is $0.334$, and when $A=25$, it is $0.194$. The constant $A$ therefore has an indirect impact on the sample paths of $\{Z_2(t): t\geq 0\}$ through the optimal choice for $\tau$. In other words, the larger is $A$, the higher should be the probability to propose model switches, which implies less parameter updates. Notice from Figure \[fig\_int\_tauopt\_A\] that the integrals of the sum of the ACFs do not vary much when $\tau$ is between $0.2$ and $0.6$ for the cases $A=2$ and $A=5$. As will be observed in the numerical experiment in Section \[sec\_simulation\_weak\], this is also the case when $A=25$, but $n$ is not too large. This suggests that, when either $A$ or $n$ is not too large, setting $\tau$ to any value in this interval leads to algorithms having similar efficiencies. As $n$ increases, users should however narrow down to the optimal values for $\tau$ in the case where $A$ is moderate or large.
![(a) Integrals of the sum of the ACFs as a function of $\tau$ for $A=2,5,25$ (the diamonds represent the optimal value for $\tau$), (b) Optimal value for $\tau$ as a function of $A$; for both graphs, $f=\mathcal{N}(\mu,\sigma^2)$ and $\ell$ is set to its optimal value[]{data-label="fig_int_tauopt_A"}](graph_int_tauopt_A_v3.pdf){width="15cm"}
Different distances between $f$ and $q$ lead to different upper bounds on $f/q$ (which are given by $A/2$), and also different acceptance probabilities for switching models. In practice, the former are much more difficult to evaluate than the latter. In addition, we believe that in general the behaviours of the acceptance probabilities are not summarised by the upper bound on $f/q$. This is why we propose to tune the distribution $g$ according to the acceptance rates.
Practical Considerations
========================
Optimal Implementation and Generalisation {#sec_optimal_impl}
-----------------------------------------
Now that the framework is properly defined and the results presented, we are able to state the rules for tuning the RJ algorithm presented in the introduction more precisely. We first recommend to perform some trial runs. They shall be used to identify the value of $\ell$ that is such that the proposed parameter updates are accepted with a rate of $0.234$. At the same time, record the number of times that model switches are proposed and accepted, i.e. the number of times that $K$ moves, and compute the rate of these events. If this rate is not too low or the dimensions of the models are not too large, setting $\tau$ to any value in the interval $(0.2, 0.6)$ should be suitable. We recommend to compute its optimal value as a reference. As $n$ increases, users should increasingly consider the optimal value. In contexts as ours, considering that the rate is, theoretically (and asymptotically), $2(1-\tau)/(A+1)$ (see Section \[sec\_context\_weak\]), the optimal value can be determined through (\[eqn\_tau\_opt\]). We believe this result leads to appropriate values in more general settings. We therefore propose to consider the following rule in these: divide the rate by $1-\tau$ (denote the result $r$), and compute the “optimal” value for $\tau$ using $(\sqrt{1/r}-1)/(1/r-1)$. This rule has been derived from (\[eqn\_tau\_opt\]), in which $5.66\Phi(-1.19)=0.66$ has been replaced by $1/2$. The acceptance rate of each different type of model switches should be recorded as well. As mentioned in Section \[sec\_context\_weak\], it is preferable to avoid very small rates. The information gathered about $f$ can be used to improve the design of the proposal distribution $q$, which will lead to higher rates. The process is in this sense iterative.
As mentioned in the introduction, the optimal scaling result of [@roberts1997weak] is known to be relatively robust, in the sense that it holds under weaker assumptions (see, e.g., [@roberts2001optimal] and [@bedard2007weak]). Believing that our theoretical results are also robust, we conjecture that they are valid when $$\begin{aligned}
\label{eqn_post_gen}
\pi_n(k,\mathbf{x}^k)=p_n(k)\prod_{i=1}^{n+k} f_i(x_i^k),\end{aligned}$$ where $f_i(x_i^k):=(1/\sigma_i) f((x_i^k-\mu_i)/\sigma_i)$, $\mu_i\in\operatorname{\mathbb{R}}$ and $\sigma_i>0$ are constants, and $f$ and $p_n$ satisfy the assumptions described in Section \[sec\_context\_weak\]. In this case, the sampler described in Section \[sec\_context\_weak\] is applied, the only difference being that a proposal distribution $q_{K+1}$ is used to add a parameter to switch from Model $K$ to Model $K+1$, in order to accommodate for the different functions $f_i$. The corresponding assumption on $f/q$ is $f_i/q_i\leq A_i^*\leq A_n^*$ for all $i\in\{n+1,\ldots,n+\lfloor \sqrt{n}\log n\rfloor\}$ with $A_n^*:=\max_{i} A_i^*$, and $A_n^*\leq A^*$ for all $n$. A similar procedure as that described previously for optimising the algorithm would be applied. We thus expect the results in this paper to be applicable to models in which the parameters are independent but not identically distributed. In particular, they turn out to be useful in designing the RJ algorithm when the posterior distribution arises from a robust principal component regression, as shown in [@gagnon2017robustPCR]. This is explained by the fact that the posterior has a structure similar to that in (\[eqn\_post\_gen\]). The authors are consequently able to design an efficient sampler that allows identification of the relevant components and estimation of the parameters for prediction purpose. The generalised version of the algorithm (described above) is applied. It is observed that the optimal value for $\ell$ corresponds to an acceptance rate close to $0.234$ and the optimal value for $\tau$ is close to $0.6$, which makes sense considering that the dimensions are small in their situation. Note that the generalisation of our results to contexts like this one is not trivial.
A Numerical Experiment {#sec_simulation_weak}
----------------------
Samples produced by the RJ algorithm provide information about the joint posterior distribution of the models and their parameters. As a result, users can select models (usually those with the highest frequencies in the sample) and estimate their parameters (using sample means and intervals for instance). Ideally, the selected models, along with the parameter estimates, would be the same as if the “true” posterior distribution had been used. In this section, we show what is the impact of the design of the sampler on the quality of the approximations of the posterior model probabilities and parameter estimates. We consider the framework defined in this paper, under which the optimal design is known. This will also allow to understand how the theoretical results presented in Section \[sec\_result\_weak\] translate in practice.
Implementing the RJ algorithm described in Section \[sec\_context\_weak\] comes down to specifying the PDF $q$ and the values of the constants $A,\tau$ and $\ell$. In Section \[sec\_result\_weak\], it has been shown that the constants $\tau$ and $A$ have an impact on the estimation of the whole joint posterior distribution of $(K, \mathbf{X}^K)$. The constant $\ell$ has an impact on the $\mathbf{X}^K$ part only and this has been thoroughly studied by [@roberts2001optimal]. In Section \[sec\_result\_weak\], it has also been explained that, asymptotically, the PDF $q$ only has an impact through the constant $A$. We can therefore reproduce any distance between $f$ and $q$ by only changing $A$. In this section, we therefore focus on showing the impact of the constants $\tau$ and $A$ on the approximations. In other words, we focus on showing how different probabilities of proposing parameter updates (and therefore model switches) and different distances between $f$ and $q$ have an impact on the approximations. We consider $f=\mathcal{N}(\mu,\sigma^2)$, $\mu\in\operatorname{\mathbb{R}},\sigma>0$, and we set $\ell=2.38\sigma$ (the optimal value) and $q=f$. We evaluate the performance of the algorithm for every $\tau\in( 0, 1)$, in the cases where $A=2,5,25$. For a given $A$, the optimal value for $\tau$ can thus be determined using (\[eqn\_tau\_opt\]).
For fixed $\tau$ and $A$, we evaluate the performance of the RJ algorithm using mean absolute deviations (MADs) around quantities that are usually of interest for users: the posterior mode of $K$ (denoted by $k^*$), and the posterior mean and standard deviation of $X_i^K$, $i\in\{1,\ldots,n+K\}$ (we consider the first parameter $X_1^{K}$ for the experiment). For a given sample produced by the RJ algorithm, $k^*$ is estimated by $\widehat{k^*}$, the mode of the sample associated with the random variable $K$, and $\mu$ and $\sigma$ are estimated by $\hat{\mu}$ and $\hat{\sigma}$, which are respectively the mean and standard deviation of the sample associated with the random variable $X_i^K$. Adequate samples lead to accurate estimates, thus resulting in small absolute deviations. For fixed $\tau$ and $A$, we approximate the MADs by independently running $N$ times the RJ algorithm and by computing $\sum_{i=1}^N |\widehat{k^*_{i}}-k^*|/N$, $\sum_{i=1}^N{\left|\hat{\mu}_{i}-\mu\right|}/N$ and $\sum_{i=1}^N{\left|\hat{\sigma}_{i}-\sigma\right|}/N$, where $\widehat{k^*_{i}}$, $ \hat{\mu}_{i}$ and $\hat{\sigma}_{i}$ are respectively the mode, mean and standard deviation based on the sample produced by the $i$th run. We also compute a global measure that we expect to have the same behaviour as that used in Section \[sec\_result\_weak\] (and given in (\[eqn\_opt\_mea\])) to optimise the algorithm with respect to $\tau$. This global measure is a linear combination of a standardised version of the MADs: $$\frac{\frac{1}{N}\sum_{i=1}^N |\widehat{k^*_{i}}-k^*|}{\sqrt{\frac{1}{N-1}\sum_{i=1}^N (\widehat{k^*_{i}}-k^*)^2}}+\frac{1}{2}\left(\frac{\frac{1}{N}\sum_{i=1}^N{\left|\hat{\mu}_{i}-\mu\right|}}{\sqrt{\frac{1}{N-1}\sum_{i=1}^N(\hat{\mu}_{i}-\mu)^2}}+\frac{\frac{1}{N}\sum_{i=1}^N{\left|\hat{\sigma}_{i}-\sigma\right|}}{\sqrt{\frac{1}{N-1}\sum_{i=1}^N(\hat{\sigma}_{i}-\sigma)^2}}\right).$$In this experiment, $N=\,$1,000, $\mu=0$, $\sigma=1$, and each sample is of size 100,000. The results are presented in Figure \[fig\_performance\].
As expected, the quality of the sample associated with $K$ decreases as the value of $\tau$ increases due to fewer model switches. An increase in $\tau$ has the opposite effect regarding the quality of the sample associated with $\mathbf{X}^K$. The vertical lines represent the optimal values for $\tau$, which are $0.415$, $0.334$ and $0.194$, when $A=2, 5, 25$, respectively. These values are optimal in the sense that they allow to reach the appropriate balance between adequate samples for $K$ (but inadequate ones for $\mathbf{X}^K$) and adequate samples for $\mathbf{X}^K$ (but inadequate ones for $K$).
Figure \[fig\_performance\] also helps illustrate that the smaller is $A$, the better it is, an aspect that has been theoretically justified in Section \[sec\_result\_weak\]. Indeed, the value of $A$ has a direct impact on the quality of the samples for $K$ (it decreases as the value of $A$ increases), and it has an indirect impact on the quality of those for $\mathbf{X}^K$ through the optimal value for $\tau$.
$\begin{array}{ccc}
\multicolumn{3}{c}{\hskip-0.5cm \includegraphics[width=14cm]{graph_mesure_k_v7.pdf}} \cr
\multicolumn{3}{c}{\hskip-0.5cm \includegraphics[width=14cm]{graph_mesure_mu_v5.pdf}} \cr
\multicolumn{3}{c}{\hskip-0.5cm \includegraphics[width=14cm]{graph_mesure_s_v5.pdf}} \cr
\multicolumn{3}{c}{\hskip-0.5cm \includegraphics[width=14cm]{graph_mesure_synthese_v3.pdf}} \cr
\hspace{1cm} \textbf{(a) \textit{A = 2}} & \hspace{2.5cm} \textbf{(b) \textit{A = 5}} & \hspace{1cm}\textbf{(c) \textit{A = 25}} \cr
\end{array}$
We finally note that, as $n\rightarrow\infty$, the curves defined by the global measure as a function of $\tau$ look more and more like those in Figure \[fig\_int\_tauopt\_A\] (a). For moderate values of $n$, the curves defined by the global measure are however almost flat between 0.2 and 0.6.
Conclusion {#sec_conclusion_weak}
==========
In this paper, we have provided guidelines for implementing an efficient RJ algorithm. They rely on theoretical results that hold when the algorithm is applied to sample from a target distribution $\pi_n$ that satisfies the assumptions provided in Section \[sec\_context\_weak\]. Essentially, the target distribution $\pi_n$ must be a product of the PMF $p_n$ (the distribution of the model indicator $K$) and a $(n+K)$-product of PDFs $f$. Being aware that this sampling context is simple, our goal was to open new research directions in optimal tuning of RJ algorithms and provide rules that we believe are robust. They seem valid for tuning the sampler when the posterior arise from robust principal component regression, as explained in Section \[sec\_optimal\_impl\]. We conjectured that they are and that, more generally, our results hold when $\pi_n$ is comprised of a product of different functions $f_i$.
Proof of Theorem \[main\_result\] {#sec_proof_thm_1}
=================================
This section is dedicated to the demonstration of the main result of this paper, the weak convergence $\{\mathbf{Z}_{1,2}^n(t):t\geq 0\}\Rightarrow \{\mathbf{Z}(t):t\geq 0\}$ in the Skorokhod topology as $n\rightarrow\infty$ (the stochastic processes $\{\mathbf{Z}_{1,2}^n(t):t\geq0\}$ and $\{\mathbf{Z}(t):t\geq0\}$ have been defined in Theorem \[main\_result\]). Thus consider the sampling context described in Section \[sec\_context\_weak\].
In order to prove the result, we demonstrate the convergence of the finite-dimensional distributions of $\{\mathbf{Z}_{1,2}^n(t): t\geq0\}$ to those of $\{\mathbf{Z}(t): t\geq0\}$. To achieve this, we verify condition (c) of Theorem 8.2 from Chapter 4 of [@ethier1986markov]. The weak convergence then follows from Corollary 8.6 of Chapter 4 of [@ethier1986markov]. The remaining conditions of Theorem 8.2 and the conditions specified in Corollary 8.6 are either straightforward or easily derived from the proof given in this section. They are nevertheless explicitly verified in [@gagnon2016thesis].
The proof of the convergence of the finite-dimensional distributions relies on the convergence of (what we call) the “pseudo-generator”, a quantity that we now introduce. The proof follows in Section \[sec\_proof\_finite\_dim\].
Pseudo-Generator {#sec_generator}
----------------
In this section, we introduce a quantity that we call the “pseudo-generator" of $\{\mathbf{Z}_{1,2}^n(t): t\geq 0\}$ due to its similarity with the infinitesimal generator of stochastic processes. It is defined as follows: $$\varphi_{n}(t):=n{\mathbb{E}}[h(\mathbf{Z}_{1,2}^n(t+1/n))-h(\mathbf{Z}_{1,2}^n(t))\mid {\mathcal{F}}^{\mathbf{Z}^n}(t)],$$ where $h\in\mathcal{C}_c^\infty(\operatorname{\mathbb{R}}^2)$, the space of infinitely differentiable functions on $\operatorname{\mathbb{R}}^2$ with compact support. Theorem 2.5 from Chapter 8 of [@ethier1986markov] allows us to restrict our attention to this set of functions when studying the limiting behaviour of the pseudo-generator (see [@gagnon2016thesis] for more details).
Let $$R^K(\lfloor nt \rfloor):=\frac{K(\lfloor nt \rfloor )-\lfloor \sqrt{n}\log n \rfloor/2}{\sqrt{n}}=Z_1^n(t).$$ The pseudo-generator $\varphi_{n}(t)$ can be decomposed into three parts, each associated with a specific type of movement, as follows: $$\varphi_{n}(t)=\varphi_{1,n}(t)+\varphi_{2,n}(t)+\varphi_{3,n}(t),$$ where $\varphi_{1,n}(t)$ is associated with the update of the parameters, i.e. $$\begin{aligned}
\varphi_{1,n}(t)&:=n\tau{\mathbb{E}}\left[\left(h(R^K,Y_1^K)-h(R^K,X_1^K)\right)\left(1 \wedge \frac{\prod_{i=1}^{n+K} f(Y_i^{K})}{\prod_{i=1}^{n+K} f(X_i^{K})}\right)\mid R^K,\mathbf{X}^K\right],
\end{aligned}$$ $\varphi_{2,n}(t)$ is associated with the inclusion of an extra parameter, i.e. $$\begin{aligned}
\label{def_phi2}
\varphi_{2,n}(t)&:=\frac{n(1-\tau)A}{A+1}{\mathbb{E}}\left[\left(h(R^{K+1},X_1^K)-h(R^K,X_1^K)\right)\left(1 \wedge \frac{f(U)p_n(K+1) }{q(U)p_n(K)A}\right)\mid R^K,\mathbf{X}^K\right],
\end{aligned}$$ and $\varphi_{3,n}(t)$ is associated with the withdrawal of the last parameter, i.e. $$\begin{aligned}
\label{def_phi3}
\varphi_{3,n}(t)&:=\frac{n(1-\tau)}{A+1}{\mathbb{E}}\left[\left(h(R^{K-1},X_1^K)-h(R^K,X_1^K)\right)\left(1 \wedge \frac{q(X_{n+K}^{K})p_n(K-1)A }{f(X_{n+K}^{K})p_n(K)}\right)\mid R^K,\mathbf{X}^K\right].
\end{aligned}$$ Note that the Markov process $\{(R^K(m),\mathbf{X}^{K(m)}(m)): m\in\operatorname{\mathbb{N}}\}$ is time-homogeneous, and consequently, the time index has been omitted to simplify the notation. Also note that, when there is an update of the parameters, only the parameters $\mathbf{X}^K$ move (the model indicator remains the same). When an extra parameter is included or the last parameter withdrawn, only the model indicator changes, as a switch from Model $K$ to Model $K+1$ or $K-1$ is made.
Proof of the Convergence of the Finite-Dimensional Distributions {#sec_proof_finite_dim}
----------------------------------------------------------------
Condition (c) of Theorem 8.2 essentially reduces to the following convergence: $$\begin{aligned}
{\mathbb{E}}\left[{\left|\varphi_n(t)-Gh(\mathbf{Z}_{1,2}^n(t))\right|}\right]\rightarrow 0 \text{ as } n\rightarrow \infty,
\end{aligned}$$ where $G$ is the generator of a diffusion with $G=G_1+G_2$ and $$\begin{aligned}
G_1h(\mathbf{Z}_{1,2}^n(t))&=\tau \ell^2 \Phi\left(-\frac{\ell\sqrt{\Upsilon}}{2}\right)(\log f(Z_2^n(t)))' h_y(\mathbf{Z}_{1,2}^n(t))+\tau \ell^2\Phi\left(-\frac{\ell\sqrt{\Upsilon}}{2}\right)h_{yy}(\mathbf{Z}_{1,2}^n(t)), \cr
G_2h(\mathbf{Z}_{1,2}^n(t))&=\frac{1-\tau}{A+1}\times -Z_1^n(t)h_{x}(\mathbf{Z}_{1,2}^n(t))+\frac{1-\tau}{A+1}h_{xx}(\mathbf{Z}_{1,2}^n(t)).
\end{aligned}$$ The function $h$ above is the same function $h$ involved in the definition of the random variable $\varphi_{n}(t)$ (given in Section \[sec\_generator\]). In other words, the convergence has to be proved for an arbitrary function $h\in\mathcal{C}_c^\infty(\operatorname{\mathbb{R}}^2)$. The functions $h_x$ and $h_{xx}$ respectively represent the first and second derivatives of $h$ with respect to its first argument. Analogously, the functions $h_y$ and $h_{yy}$ respectively represent the first and second derivatives of $h$ with respect to its second argument. Note that it exists a positive constant $M$ such that $h$ and all its derivatives are bounded in absolute value by this constant.
Using the triangle inequality, we have $$\begin{aligned}
{\mathbb{E}}\left[{\left|\varphi_n(t)-Gh(\mathbf{Z}_{1,2}^n(t))\right|}\right]&\leq {\mathbb{E}}\left[{\left|\varphi_{1,n}(t)-G_1h(\mathbf{Z}_{1,2}^n(t))\right|}\right]+{\mathbb{E}}\left[{\left|\varphi_{2,n}(t)+\varphi_{3,n}(t)-G_2h(\mathbf{Z}_{1,2}^n(t))\right|}\right].
\end{aligned}$$ In this paper, we show that the second term on the right-hand side (RHS) converges towards 0 as $n\rightarrow\infty$. The proof that the first term converges towards 0 is similar to that of Theorem 1.1 of [@roberts1997weak], and is thus omitted (it can nevertheless be found in [@gagnon2016thesis]).
The key here is the use of Taylor expansions in order to obtain derivatives of $h$ as in generators of diffusions.
We first analyse $\varphi_{2,n}(t)$ as defined in (\[def\_phi2\]). As explained in Section \[sec\_context\_weak\], $0\leq p_n(K+1)/p_n(K)\leq 2$, and therefore, $$\frac{f(U)p_n(K+1) }{q(U)p_n(K)A}\leq \frac{2f(U)}{q(U)A}\leq 1.$$ Consequently, since $h(R^{K+1},X_1^K)=h(R^{K}+1/\sqrt{n},X_1^K)$, $$\begin{aligned}
\varphi_{2,n}(t)&
=\frac{n(1-\tau)}{A+1}\left(h(R^K+1/\sqrt{n},X_1^K)-h(R^K,X_1^K)\right)\frac{p_n(K+1)}{p_n(K)}{\mathbb{E}}\left[\frac{f(U) }{q(U)}\mid R^K,\mathbf{X}^K\right] \cr
&=\frac{n(1-\tau)}{A+1}\left(h(R^K+1/\sqrt{n},X_1^K)-h(R^K,X_1^K)\right)\frac{p_n(K+1)}{p_n(K)}.
\end{aligned}$$ In the last equality, we use the fact that $U$ is independent of $(K,\mathbf{X}^K)$, and therefore, $$\begin{aligned}
{\mathbb{E}}\left[ \frac{f(U) }{q(U)}\mid R^K,\mathbf{X}^K\right]={\mathbb{E}}\left[ \frac{f(U) }{q(U)} \right]=\int_{-\infty}^\infty \frac{f(u)}{q(u)}q(u)du=1.
\end{aligned}$$ Note that $\varphi_{2,n}(t)=0$ when $K=\lfloor \sqrt{n}\log n\rfloor$ since $p_n( \lfloor \sqrt{n}\log n\rfloor+1)=0$.
We now study $\varphi_{3,n}(t)$ as defined in (\[def\_phi3\]). As explained in Section \[sec\_context\_weak\], when $2\leq K\leq \lfloor\sqrt{n}\log n\rfloor$ we have that $p_n(K-1)/p_n(K)\geq 1/2$. Therefore, when $2\leq K\leq \lfloor\sqrt{n}\log n\rfloor$, $$\frac{q(X_{n+K}^{K})p_n(K-1)A }{f(X_{n+K}^{K})p_n(K)}\geq \frac{q(X_{n+K}^{K})A }{2f(X_{n+K}^{K})}\geq 1.$$ This means that the acceptance probability of withdrawing the last parameter is 1, when it is possible to withdraw a parameter. Consequently, since $h(R^{K-1},X_1^K)=h(R^{K}-1/\sqrt{n},X_1^K)$, $$\begin{aligned}
&\varphi_{3,n}(t)=\frac{n(1-\tau)}{A+1}\left(h(R^{K}-1/\sqrt{n},X_1^K)-h(R^{K},X_1^K)\right){\mathds{1}}(2\leq K\leq \lfloor\sqrt{n}\log n\rfloor).
\end{aligned}$$ Note that $\varphi_{3,n}(t)=0$ when $K=1$ since $p_n(0)=0$.
By using Taylor expansions of $h$ around $R^K$, we obtain that $$\begin{aligned}
h(R^K+1/\sqrt{n},X_1^K)-h(R^K,X_1^K)&=\frac{1}{\sqrt{n}}h_x(R^K,X_1^K)+\frac{1}{2n}h_{xx}(R^K,X_1^K)+\frac{1}{6n^{3/2}}h_{xxx}(W,X_1^K), \cr
h(R^K-1/\sqrt{n},X_1^K)-h(R^K,X_1^K)&=-\frac{1}{\sqrt{n}}h_x(R^K,X_1^K)+\frac{1}{2n}h_{xx}(R^K,X_1^K)-\frac{1}{6n^{3/2}}h_{xxx}(T,X_1^K),
\end{aligned}$$ where $W$ belongs to $(R^K,R^K+1/\sqrt{n})$, $T$ belongs to $(R^K-1/\sqrt{n},R^K)$, and $h_{xxx}$ represents the third derivative of $h$ with respect to its first argument.
Therefore, $$\begin{aligned}
\label{phi2_phi_3_gen}
&\varphi_{2,n}(t)+\varphi_{3,n}(t)-G_2h(\mathbf{Z}_{1,2}^n(t))={\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1) \cr
&\qquad\times \frac{1-\tau}{A+1}h_x(R^K,X_1^K) \left[\sqrt{n}\left( \frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right] \cr
&+{\mathds{1}}(K=1)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left(\sqrt{n}\times\frac{p_n(K+1) }{p_n(K)}+R^K\right) \cr
&-{\mathds{1}}(K=\lfloor\sqrt{n}\log n\rfloor)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left(\sqrt{n}-R^K\right) \cr
&+{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\frac{1-\tau}{2(A+1)}h_{xx}(R^K,X_1^K)\left( \frac{p_n(K+1) }{p_n(K)}-1\right) \cr
&+{\mathds{1}}(K=1)\frac{1-\tau}{2(A+1)}h_{xx}(R^K,X_1^K)\left(\frac{p_n(K+1) }{p_n(K)}-2\right) \cr
&-{\mathds{1}}(K=\lfloor\sqrt{n}\log n\rfloor)\frac{1-\tau}{2(A+1)}h_{xx}(R^K,X_1^K) \cr
&+\frac{1-\tau}{6\sqrt{n}(A+1)}h_{xxx}(W,X_1^K) \frac{p_n(K+1) }{p_n(K)}{\mathds{1}}(1\leq K\leq \lfloor\sqrt{n}\log n\rfloor-1) \cr
&-\frac{1-\tau}{6\sqrt{n}(A+1)}h_{xxx}(T,X_1^K){\mathds{1}}(2\leq K\leq \lfloor\sqrt{n}\log n\rfloor).
\end{aligned}$$ We now show that the expectation of the absolute value of each term on the RHS in (\[phi2\_phi\_3\_gen\]) converges towards 0 as $n\rightarrow\infty$. Consequently, using the triangle inequality we will obtain $${\mathbb{E}}\left[{\left|\varphi_{2,n}(t)+\varphi_{3,n}(t)-G_2h(\mathbf{Z}_{1,2}^n(t))\right|}\right]\rightarrow 0 \text{ as } n\rightarrow\infty.$$ We start with the last terms in (\[phi2\_phi\_3\_gen\]) and make our way up. It is clear that the expectation of the absolute value of each of the last two terms converges towards 0 as $n\rightarrow\infty$ since $|h_{xxx}|\leq M$ and $0\leq p_n(K+1)/p_n(K)\leq 2$.
We now analyse the fourth one (starting from the bottom). As $n\rightarrow\infty$, $$\begin{aligned}
&{\mathbb{E}}\left[\left|{\mathds{1}}(K=1)\frac{1-\tau}{2(A+1)}h_{xx}(R^K,X_1^K)\left(\frac{p_n(K+1) }{p_n(K)}-2\right)\right|\right] \cr
&\qquad\leq \frac{M(1-\tau)}{A+1} \times {\mathbb{E}}\left[\left|{\mathds{1}}(K=1)\right|\right] =\frac{M(1-\tau)}{A+1} \times {\mathbb{P}}(K=1)\rightarrow 0,
\end{aligned}$$ using ${\left|h_{xx}\right|}\leq M$ and $0\leq {\left|p_n(K+1)/p_n(K)-2\right|}\leq 2$ in the first inequality. Proposition \[prop\_bound\_dom\] in Section \[lemmas\] is then used to conclude that ${\mathbb{P}}(K=1)\rightarrow 0$ as $n\rightarrow\infty$. The proof for the third term (starting from the bottom) is similar.
Applying Lemmas \[lemma\_5\] to \[lemma\_8\] from Section \[lemmas\], each of the remaining terms is seen to converge towards 0 in $L^1$ as $n\rightarrow\infty$, and thus $${\mathbb{E}}\left[{\left|\varphi_{2,n}(t)+\varphi_{3,n}(t)-G_2h(\mathbf{Z}_{1,2}^n(t))\right|}\right]\rightarrow 0 \text{ as } n\rightarrow\infty.$$
Results Used in the Proof of Theorem \[main\_result\] {#lemmas}
=====================================================
\[lemma\_5\] As $n\rightarrow\infty$, we have $$\begin{aligned}
&{\mathbb{E}}\left[\left|{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\frac{1-\tau}{2(A+1)}h_{xx}(R^K,X_1^K)\left(\frac{p_n(K+1) }{p_n(K)}-1\right)\right|\right]\rightarrow 0.
\end{aligned}$$
First, $$\begin{aligned}
&{\mathbb{E}}\left[\left|{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\frac{1-\tau}{2(A+1)}h_{xx}(R^K,X_1^K)\left(\frac{p_n(K+1) }{p_n(K)}-1\right)\right|\right] \cr
&\qquad\leq \frac{M(1-\tau)}{2(A+1)}{\mathbb{E}}\left[{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1) \left|\frac{p_n(K+1) }{p_n(K)}-1\right|\right],
\end{aligned}$$ because ${\left|h_{xx}\right|}\leq M$.
Considering the case where $\lfloor \sqrt{n}\log n\rfloor$ is odd, we have $$\begin{aligned}
&{\mathbb{E}}\left[{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1){\left|\frac{p_n(K+1) }{p_n(K)}-1\right|}\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right){\left|\frac{p_n(K+1) }{p_n(K)}-1\right|}\right] \cr
&\qquad+{\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right){\left|\frac{p_n(K+1) }{p_n(K)}-1\right|}\right].
\end{aligned}$$ We analyse each term separately. The first one corresponds to the case where $K$ is at the mode or at its right. Therefore, $p_n(K+1)/p_n(K)=a_{K,n}$ and $$\begin{aligned}
&{\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right){\left|\frac{p_n(K+1) }{p_n(K)}-1\right|}\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right){\left|a_{K,n}-1\right|}\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right)\frac{{\left|K-\lfloor \sqrt{n}\log n\rfloor/2\right|}}{n}\right] \cr
&\quad\leq \frac{\lfloor \sqrt{n}\log n\rfloor/2-1}{n} \leq \frac{\log n}{2\sqrt{n}} \rightarrow 0,
\end{aligned}$$ because $a_{K,n}=1-b_{K,n}/\sqrt{n}$ and $b_{K,n}={\left|K-\lfloor \sqrt{n}\log n\rfloor/2\right|}/\sqrt{n}$. We now study the case where $K$ is at the left of the mode. Therefore, $p_n(K+1)/p_n(K)=a_{K,n}^{-1}$ and $$\begin{aligned}
&{\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right){\left|\frac{p_n(K+1) }{p_n(K)}-1\right|}\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right){\left|a_{K,n}^{-1}-1\right|}\right]\cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right){\left|\frac{b_{K,n}}{\sqrt{n}-b_{K,n}}\right|}\right]\leq \frac{(\log n)/2}{\sqrt{n}-(\log n)/2}\rightarrow0,
\end{aligned}$$ using similar mathematical arguments as above and the fact that $1/(2\sqrt{n})\leq b_{K,n}\leq (\log n)/2$ when $2\leq K\leq (\lfloor \sqrt{n}\log n\rfloor-1)/2$. The proof for the case where $\lfloor \sqrt{n}\log n\rfloor$ is even is similar.
\[lemma\_6\] As $n\rightarrow\infty$, we have $${\mathbb{E}}\left[\left|{\mathds{1}}(K=1)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left(\sqrt{n}\times \frac{p_n(K+1) }{p_n(K)}+R^K\right)\right|\right]\rightarrow 0,$$ and $${\mathbb{E}}\left[\left|{\mathds{1}}(K=\lfloor\sqrt{n}\log n\rfloor)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left(\sqrt{n}-R^K\right)\right|\right]\rightarrow 0.$$
Using Proposition \[prop\_bound\_dom\], ${\left|h_x\right|}\leq M$, $0\leq p_n(K+1)/p_n(K)\leq 2$ and $R^K:=(K-\lfloor \sqrt{n}\log n\rfloor/2)/\sqrt{n}$, we have $$\begin{aligned}
&{\mathbb{E}}\left[\left|{\mathds{1}}(K=1)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left(\sqrt{n}\times \frac{p_n(K+1) }{p_n(K)}+R^K\right)\right|\right] \cr
&\qquad\leq\frac{(1-\tau) M}{A+1}\left(2\sqrt{n}+\frac{ \log n }{2}\right){\mathbb{P}}(K=1)\rightarrow 0,
\end{aligned}$$ since $$\begin{aligned}
{\mathds{1}}(K=1) {\left| \sqrt{n}\times\frac{p_n(K+1) }{p_n(K)}+R^K\right|}&\leq {\mathds{1}}(K=1)\left(2\sqrt{n}+{\left|\frac{1-\lfloor \sqrt{n}\log n\rfloor /2 }{\sqrt{n}}\right|}\right) \cr
&\leq{\mathds{1}}(K=1)\left( 2\sqrt{n}+\frac{\log n }{2}\right). \hspace{18mm}
\end{aligned}$$ The proof that $${\mathbb{E}}\left[\left|{\mathds{1}}(K=\lfloor\sqrt{n}\log n\rfloor)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left(\sqrt{n}-R^K\right)\right|\right]\rightarrow 0$$ is similar.
\[lemma\_8\] As $n\rightarrow\infty$, we have $$\begin{aligned}
&{\mathbb{E}}\left[\left|{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left[\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right]\right|\right]\rightarrow 0.
\end{aligned}$$
First, $$\begin{aligned}
&{\mathbb{E}}\left[\left|{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\frac{1-\tau}{A+1}h_x(R^K,X_1^K)\left[\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right]\right|\right] \cr
&\quad\leq\frac{(1-\tau)M}{A+1}{\mathbb{E}}\left[{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\left|\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right|\right],
\end{aligned}$$ because ${\left|h_{x}\right|}\leq M$. Considering the case where $\lfloor \sqrt{n}\log n\rfloor$ is odd, we have $$\begin{aligned}
&{\mathbb{E}}\left[{\mathds{1}}(2\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1)\left|\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right|\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right)\left|\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right|\right] \cr
&\qquad+{\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right)\left|\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right|\right].
\end{aligned}$$ We analyse each term separately. The first one corresponds to the case where $K$ is at the mode or at its right. Therefore, $p_n(K+1)/p_n(K)=a_{K,n}$ and $$\begin{aligned}
&{\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right)\left|\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right|\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(\frac{\lfloor \sqrt{n}\log n\rfloor+1}{2}\leq K\leq \lfloor \sqrt{n}\log n\rfloor-1\right)\left|\sqrt{n}\left(a_{K,n} -1\right)+R^K\right|\right] \cr
&\quad= {\mathbb{E}}\left[{\mathds{1}}\left(\frac{1}{2\sqrt{n}}\leq R^K\leq \frac{\lfloor\sqrt{n}\log n\rfloor-2}{2\sqrt{n}}\right){\left|-b_{K,n}+R^K\right|}\right]=0,
\end{aligned}$$ because $a_{K,n}=1-b_{K,n}/\sqrt{n}$ and $b_{K,n}={\left|K-\lfloor \sqrt{n}\log n\rfloor/2\right|}/\sqrt{n}=R^K$ when $R^K\geq0$ ($R^K:=(K-\lfloor \sqrt{n}\log n\rfloor/2)/\sqrt{n}$). We now study the case where $K$ is at the left of the mode. Therefore, $p_n(K+1)/p_n(K)=a_{K,n}^{-1}$ and $$\begin{aligned}
&{\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right)\left|\sqrt{n}\left(\frac{p_n(K+1) }{p_n(K)}-1\right)+R^K\right|\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right)\left|\sqrt{n}\left(a_{K,n}^{-1}-1\right)+R^K\right|\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(2\leq K\leq \frac{\lfloor \sqrt{n}\log n\rfloor-1}{2}\right)\left|\frac{\sqrt{n}}{\sqrt{n}-b_{K,n}}\times b_{K,n}+R^K\right|\right] \cr
&\quad={\mathbb{E}}\left[{\mathds{1}}\left(\frac{4-\lfloor \sqrt{n}\log n\rfloor}{2\sqrt{n}}\leq R^K \leq \frac{-1}{2\sqrt{n}}\right)\times-R^K\left|\frac{\sqrt{n}}{\sqrt{n}-b_{K,n}}-1\right|\right] \cr
&\quad\leq {\mathbb{E}}\left[{\mathds{1}}\left(\frac{4-\lfloor \sqrt{n}\log n\rfloor}{2\sqrt{n}}\leq R^K \leq \frac{-1}{2\sqrt{n}}\right)\frac{\log n}{2}\times\frac{b_{K,n}}{\sqrt{n}-b_{K,n}}\right] \cr
&\quad\leq \frac{\log n}{2}\times\frac{(\log n)/2}{\sqrt{n}-(\log n)/2}\rightarrow 0,
\end{aligned}$$ using similar mathematical arguments as above, the fact that $b_{K,n}=-R^K$ when $R^K<0$, and that $1/(2\sqrt{n})\leq-R^K\leq (\log n)/2$ when $(4-\lfloor \sqrt{n}\log n\rfloor)/$ $(2\sqrt{n})\leq R^K \leq -1/(2\sqrt{n})$. The proof for the case where $\lfloor \sqrt{n}\log n\rfloor$ is even is similar.
\[prop\_bound\_dom\] The random variable $K$ with PMF $p$ defined in Section \[sec\_context\_weak\] is such that, for all $\rho\in\operatorname{\mathbb{R}}$, $$\lim_{n\rightarrow\infty}n^\rho{\mathbb{P}}(K=1)=\lim_{n\rightarrow\infty}n^\rho{\mathbb{P}}(K=\lfloor \sqrt{n}\log n\rfloor)=0.$$
Consider the case where $\lfloor \sqrt{n}\log n\rfloor$ is even. Using equation (\[eqn\_prob\_k\_even\]), we have $$p_n(1)=p_n(\lfloor \sqrt{n}\log n\rfloor)=p_n\left(\frac{\lfloor \sqrt{n}\log n\rfloor}{2}\right)\prod_{i=1}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1}\left(1-\frac{i}{n}\right).$$ In the proof of Proposition \[prop\_z\_1\_to\_normal\] in Section \[sec\_proofs\_props\], we show that $p_n(\lfloor \sqrt{n}\log n\rfloor/2)\rightarrow 1/\sqrt{2\pi}$ as $n\rightarrow\infty$. Also, using the fact that $1-x\leq\exp\{-x\}$ for all $x\in\operatorname{\mathbb{R}}$, we have for all $\rho\in\operatorname{\mathbb{R}}$ $$\begin{aligned}
n^\rho\prod_{i=1}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \left(1-\frac{i}{n}\right)&\leq n^\rho \exp\left\{-\sum_{i=1}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \frac{i}{n}\right\} \cr
&=n^\rho\exp\left\{-\frac{1}{2}\left(\frac{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1}{\sqrt{n}}\right)^2 -\frac{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1}{2n}\right\}
\end{aligned}$$ $$\begin{aligned}
&\hspace{20mm}\leq n^\rho \exp\left\{-\frac{1}{2}\left(\frac{\lfloor \sqrt{n}\log n\rfloor}{2\sqrt{n}}-1\right)^2\right\}\rightarrow 0,
\end{aligned}$$ as $n\rightarrow\infty$. Similarly, we can show the result for the case where $\lfloor \sqrt{n}\log n\rfloor$ is odd.
Proofs of Propositions \[prop\_z\_1\_to\_normal\] and \[prop\_prob\_acc\_ajout\] {#sec_proofs_props}
================================================================================
The random variable $Z_1^n(t)$ is defined as $(K(\lfloor nt \rfloor)-\lfloor \sqrt{n}\log n\rfloor/2)/\sqrt{n}$, and $K(\lfloor nt \rfloor)\sim p_n$ for all $t$ (see Section \[sec\_context\_weak\] for the assumptions on $p_n$). Therefore, to simplify the notation, the time index is omitted for the rest of the proof. Consider the constant $z<0$ and the case where $\lfloor \sqrt{n}\log n\rfloor$ is even. We have $$\begin{aligned}
{\mathbb{P}}((K-\lfloor \sqrt{n}\log n\rfloor/2)/\sqrt{n}\leq z)&={\mathbb{P}}(K\leq z\sqrt{n}+\lfloor \sqrt{n}\log n\rfloor/2) \cr
&=\sum_{k=\lceil -z\sqrt{n}\rceil}^{\lfloor \sqrt{n}\log n\rfloor/2-1} p_n(\lfloor \sqrt{n}\log n\rfloor/2-k) \cr
&=p_n\left(\frac{\lfloor \sqrt{n} \log n\rfloor}{2}\right)\sum_{k=\lceil -z\sqrt{n}\rceil}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \prod_{i=1}^k \left(1-\frac{i}{n}\right),
\end{aligned}$$ using $\lfloor z\sqrt{n}+\lfloor \sqrt{n}\log n\rfloor/2\rfloor=\lfloor z\sqrt{n}\rfloor+\lfloor \sqrt{n}\log n\rfloor/2= \lfloor \sqrt{n}\log n\rfloor/2-\lceil- z\sqrt{n}\rceil$ in the second equality ($\lceil \cdot \rceil$ is the ceiling function), and equation (\[eqn\_prob\_k\_even\]) in the last equality. The sum above is well-defined if $n\geq \exp(-2z+5)$ and we select $n$ large enough to ensure this. Using again equation (\[eqn\_prob\_k\_even\]) and the fact that $\sum_{k=1}^{\lfloor\sqrt{n}\log n\rfloor} p_n(k)=1$, we have $$p_n\left(\frac{\lfloor \sqrt{n} \log n\rfloor}{2}\right)=\left(2\left(1+\sum_{k=1}^{\lfloor \sqrt{n}\log n\rfloor/2-1} \prod_{i=1}^k \left(1-\frac{i}{n}\right)\right)\right)^{-1}.$$ Therefore, $${\mathbb{P}}\left(\frac{K-\lfloor \sqrt{n}\log n\rfloor/2}{\sqrt{n}}\leq z\right)=\frac{(1/\sqrt{n})\sum_{k=\lceil -z\sqrt{n}\rceil}^{\lfloor \sqrt{n}\log n\rfloor/2-1} \prod_{i=1}^k (1-i/n)}{\frac{2}{\sqrt{n}}+\frac{2}{\sqrt{n}}\sum_{k=1}^{\lfloor \sqrt{n}\log n\rfloor/2-1} \prod_{i=1}^k (1-i/n)}.$$ Using the fact that $1-x\leq\exp\{-x\}$ for all $x\in\operatorname{\mathbb{R}}$, we have $$\prod_{i=1}^k \left(1-\frac{i}{n}\right)\leq \exp\left\{-\sum_{i=1}^k \frac{i}{n}\right\}=\exp\left\{-\frac{1}{2}\left(\frac{k}{\sqrt{n}}\right)^2 -\frac{k}{2n}\right\}.$$ In addition, for all $\delta>0$, there exists $\epsilon>0$ such that $\exp\{-(1+\delta)x\}\leq 1-x$ for $0\leq x<\epsilon$. Therefore, since $0\leq i/n\leq \lfloor\sqrt{n}\log n\rfloor/(2n)-1/n\leq \log n/(2\sqrt{n})\rightarrow 0$ as $n\rightarrow\infty$ when $1\leq i\leq k\leq \lfloor \sqrt{n}\log n\rfloor/2-1$, for all $\delta>0$, there exists a constant $N>0$ such that for all $n\geq N$, $$\prod_{i=1}^k \left(1-\frac{i}{n}\right)\geq\exp\left\{-\frac{(1+\delta)}{2}\left(\frac{k}{\sqrt{n}}\right)^2 -\frac{k(1+\delta)}{2n}\right\}.$$ The objective is to use a “Riemann sum” argument, where the length of the subintervals of the partition is $1/\sqrt{n}$, to study the asymptotic behaviour of the numerator and denominator of ${\mathbb{P}}((K-\lfloor \sqrt{n}\log n\rfloor/2)/\sqrt{n}\leq z)$. More precisely, we now prove that the numerator of ${\mathbb{P}}((K-\lfloor \sqrt{n}\log n\rfloor/2)/\sqrt{n}\leq z)$ converges towards $\int_{-z}^\infty \exp(-x^2/2) dx$ and that the denominator converges towards $\int_{-\infty}^\infty \exp(-x^2/2) dx=\sqrt{2\pi}$. To achieve this, we use Lebesgue’s dominated convergence theorem. First, we rewrite the numerator as $$\frac{1}{\sqrt{n}}\sum_{k=\lceil -z\sqrt{n}\rceil}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \prod_{i=1}^k \left(1-\frac{i}{n}\right)= \int_{-z}^\infty \sum_{k=\lceil -z\sqrt{n}\rceil}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \prod_{i=1}^k \left(1-\frac{i}{n}\right){\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x) dx.$$ Now, we analyse the integrand. For all $x\in(-z,\infty)$ and for large enough $n$, there exists a unique $k'\in\{\lceil -z\sqrt{n}\rceil,\ldots, \lfloor \sqrt{n}\log n\rfloor/2-1\}$ with ${\mathds{1}}_{[k'/\sqrt{n},(k'+1)/\sqrt{n})}(x)=1$. Also, $0\leq x-k'/\sqrt{n}<1/\sqrt{n}$, which implies that $k'/\sqrt{n}\rightarrow x$ as $n\rightarrow\infty$. Consequently, using the upper bound and the lower bound on $\prod_{i=1}^k (1-i/n)$, $$\sum_{k=\lceil -z\sqrt{n}\rceil}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \prod_{i=1}^k \left(1-\frac{i}{n}\right){\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x)=\prod_{i=1}^{k'} \left(1-\frac{i}{n}\right)\rightarrow \exp\{-x^2/2\},$$ as $n\rightarrow\infty$, because $k'/n\leq \lfloor\sqrt{n}\log n\rfloor/(2n)-1/n\leq \log n/(2\sqrt{n})\rightarrow 0$. Now, we prove that the integrand is bounded by an integrable function that does not depend on $n$. For all $x\in(-z,\infty)$, $$\begin{aligned}
\prod_{i=1}^k \left(1-\frac{i}{n}\right){\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x)&\leq \exp\left\{-\frac{1}{2}\left(\frac{k}{\sqrt{n}}\right)^2 -\frac{k}{2n}\right\}{\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x) \cr
&\leq \exp\left\{-\frac{1}{2}(x-1)^2\right\}{\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x),
\end{aligned}$$ using the upper bound on $\prod_{i=1}^k (1-i/n)$ in the first inequality, and then $x\leq (k+1)/\sqrt{n}\leq (k/\sqrt{n})+1$. As a result, $$\begin{aligned}
\sum_{k=\lceil -z\sqrt{n}\rceil}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1} \prod_{i=1}^k \left(1-\frac{i}{n}\right){\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x)&\leq \exp\left\{-\frac{1}{2}(x-1)^2\right\} \sum_{k=\lceil -z\sqrt{n}\rceil}^{\frac{\lfloor \sqrt{n}\log n\rfloor}{2}-1}{\mathds{1}}_{\left[\frac{k}{\sqrt{n}},\frac{k+1}{\sqrt{n}}\right)}(x)\cr
&= \exp\left\{-\frac{1}{2}(x-1)^2\right\} {\mathds{1}}_{\left[\frac{\lceil -z\sqrt{n}\rceil}{\sqrt{n}},\frac{\lfloor \sqrt{n}\log n\rfloor/2}{\sqrt{n}}\right)}(x)\leq \exp\left\{-\frac{1}{2}(x-1)^2\right\},
\end{aligned}$$ which is integrable. Similarly, we can prove that the denominator of ${\mathbb{P}}((K-\lfloor \sqrt{n}\log n\rfloor/2)/$ $\sqrt{n}\leq z)$ converges towards $\int_{-\infty}^\infty \exp(-x^2/2) dx=\sqrt{2\pi}$, and we can show the result for $z\geq 0$ and for the case where $\lfloor \sqrt{n}\log n\rfloor$ is odd.
The random variables $K(m)$ and $U(m+1)$ are independent and such that $K(m)\sim p_n$ and $U(m+1)\sim q$ for all $m\in\operatorname{\mathbb{N}}$ (see Section \[sec\_context\_weak\] for the assumptions on $p_n$ and $q$). Therefore, to simplify the notation, the time index is omitted for the rest of the proof. As explained in Section \[sec\_proof\_finite\_dim\], $${\mathbb{E}}\left[1 \wedge \frac{f(U)p_n(K+1)}{q(U)p_n(K)A}\right]={\mathbb{E}}\left[ \frac{f(U)p_n(K+1)}{q(U)p_n(K)A}\right],$$ and ${\mathbb{E}}[f(U)/q(U)]=1$. Finally, using Proposition \[prop\_bound\_dom\], we have $$\begin{aligned}
&\frac{1}{A}{\mathbb{E}}\left[ \frac{p_n(K+1)}{p_n(K)}\right]=\frac{1}{A}\sum_{k=1}^{\lfloor \sqrt{n}\log n\rfloor-1}p_n(k+1)=\frac{1}{A}(1-p_n(1))\rightarrow \frac{1}{A} \text{ as } n\rightarrow\infty.
\end{aligned}$$
Acknowledgements
================
The authors acknowledge support from the NSERC (Natural Sciences and Engineering Research Council of Canada), the FRQNT (Le Fonds de recherche du Québec - Nature et technologies) and the SOA (Society of Actuaries).
|
---
abstract: 'In many physical applications solitons propagate on supports whose topological properties may induce new and interesting effects. In this paper, we investigate the propagation of solitons on chains with a topological inhomogeneity generated by the insertion of a finite discrete network on the chain. For networks connected by a link to a single site of the chain, we derive a general criterion yielding the momenta for perfect reflection and transmission of traveling solitons and we discuss solitonic motion on chains with topological inhomogeneities.'
address:
- '$^1$ I.N.F.M. and Dipartimento di Fisica, Università di Parma, parco Area delle Scienze 7A Parma, I-43100, Italy'
- '$^2$ Dipartimento di Fisica and Sezione I.N.F.N., Università di Perugia, Via A. Pascoli Perugia, I-06123, Italy'
author:
- 'R. Burioni$^1$, D. Cassi$^1$, P. Sodano$^2$, A. Trombettoni$^1$, and A. Vezzani$^1$'
title: Propagation of Discrete Solitons in Inhomogeneous Networks
---
[**In the last decades, a huge amount of work has been devoted to the study of the propagation of discrete solitons in regular, translational invariant lattices. However, in several systems, like networks of nonlinear waveguide arrays, Bose-Einstein condensates in optical lattices, arrays of superconducting Josephson junctions and silicon-based photonic crystals, one can engineer the shape (i.e., the topology) of the network. Correspondingly, an interesting task is the study of the propagation of solitons in inhomogeneous networks. The general idea of this work is that network topology strongly affects the soliton propagation. We provide a general argument giving the momenta of perfect transmission and reflection for a soliton scattering through a finite general network attached to a site of a chain: the momenta of perfect transmission and reflection are related in a simple way to the energy levels of the attached network. This criterion directly links the transmission coefficients with the network adjacency matrix, which encodes all the relevant informations on its topology. Such relation puts into evidence the topological effects on the soliton propagation. The situations where finite linear chains, Cayley trees and other simple structures are attached to a site of an unbranched chain are investigated in detail.**]{}
Introduction
============
The analysis of nonlinear models on regular lattices [@flach98; @scott99; @hennig99], as well as the investigation of linear models on inhomogeneous and fractal networks [@nakayama94] has attracted a great deal of attention in the last decades: while nonlinearity dramatically modifies the dynamics, allowing for soliton propagation, energy localization, and the existence of discrete breathers [@scott99], topology mainly affects the energy spectrum giving rise to interesting phenomena such as anomalous diffusion, localized states, and fracton dynamics [@nakayama94]. It is now both timely and highly desirable to begin a thorough investigation of nonlinear models on general inhomogeneous networks, since one expects not only interesting new phenomena arising from the interplay between nonlinearity and topology, but also an high potential impact for applications to biology [@peyrard89; @special] and to signal propagation in optical waveguides [@kivshar03]. Recently, the effects of uniformity break on soliton propagation [@christodoulides01; @kevrekidis03] and localized modes [@mcgurn02] has been investigated by considering $Y$-junctions [@christodoulides01; @kevrekidis03] (consisting of a long chain inserted on a site of a chain yielding a star-like geometry) or geometries like junctions of two infinite waveguides or the waveguide coupler [@mcgurn02]. Here, we consider general finite networks inserted on a chain.
The discrete nonlinear Schrödinger equation (DNLSE) is a paradigmatic example of a nonlinear equation on a lattice which has been successfully applied to several contexts [@kevrekidis01; @ablowitz04]: in particular, it has been used to describe the physics of arrays of coupled optical waveguides [@eisenberg98; @morandotti99] and arrays of Bose-Einstein condensates [@trombettoni01]. It is well known that, on a homogeneous chain, the DNLSE is not integrable [@ablowitz04]; nevertheless, soliton-like wavepackets can propagate for a long time and the stability conditions of soliton-like solutions can be derived within variational approaches [@malomed96]. Furthermore, the dynamics of traveling pulses has been investigated in detail in literature [@duncan93; @flach99; @gomez04] (more references are in [@kevrekidis01]). The simplest example of an inhomogeneous chain is provided by an external potential localized on a site of the chain [@forinash94; @konotop96; @krolikowski96; @aceves96; @kevrekidis01]: an experimental set up with a single defect has been recently realized with coupled optical waveguides [@mandelik03]. Another relevant example of an inhomogeneous chain is obtained adding an additional Fano degree of freedom coupled to the a site of the chain, which gives the so-called Fano-Anderson model (see [@miroshnichenko03; @flach03; @miroshnichenko05] and references therein).
As a first step in the investigation of the properties of nonlinear models on general inhomogeneous networks, we shall analyze the propagation of DNLSE solitons on a class of inhomogeneous networks built by suitably adding a finite topological inhomogeneity to an unbranched chain. The general framework where this analysis can be carried out is provided by graph theory [@harary69]; in particular, we shall consider networks where a finite discrete graph $G^0$ is attached by a link to a site of the homogeneous, unbranched chain (see Fig.1) while all the sites potentials $\epsilon_i$ are set to a constant. Such systems may be experimentally realized by placing the nonlinear waveguides in a suitable inhomogeneous arrangement, like the one depicted in Fig.1. We mention that in arrays of Bose-Einstein condensates one can build up geometries, which differ from the unbranched chain, by properly superimposing the laser beams creating the optical lattices [@oberthaler03]. In superconducting Josephson arrays [@fazio01], present-day technologies allow for to prepare the insulating support for the junctions in order to create structures of the form “chain + a topological defect”. In the context of coupled nonlinear waveguides [@hennig99], one should couple the waveguides according the geometry of the graph $G^0$, and couple this network of waveguides to a single waveguide of the unbranched chain; a similar engineering should be requested to realize photonic crystal circuits [@birner01] obtained merging the circuit $G^0$ to the unbranched chain.
As we shall discuss in Section IV, the shape of the attached graph $G^0$ affects the transmission and reflection coefficients as a function of the soliton momentum: as an example of this general phenomenon, we consider unbranched chains to which simple graphs, like finite chains and Cayley trees, are added. Our analysis points to the fact that the topology of the network (i.e., how its sites are connected) controls the transmission properties, and that one can modify soliton propagation by varying the topology of the inserted network. In particular, we shall show that the momenta of perfect transmission are determined by the energy levels of the inserted graph, i.e., by the eigenfrequencies of the $G^0$’s oscillation modes in the linear case (see Section IV).
On a chain, stable solitonic wavepackets can propagate for long times [@kevrekidis01]. When a graph $G^0$ is inserted, one can study how the presence of this topological inhomogeneity modifies the soliton propagation. We numerically evaluate the transmission coefficients and we compare the numerical results with analytical findings obtained for a relevant soliton class, to which we refer as [*large-fast solitons*]{}. For large fast solitons the transmission coefficients can be evaluated within a linear approximation [@miroshnichenko03]. Indeed the characteristic time for the soliton-topological defect collision are very small with respect to the soliton dispersion time; therefore, the soliton scattering can be approximated by the scattering of a plane-wave. However, as we numerically checked in the Figs.2-6, the nonlinearity still plays a role, giving long-lived solitons, especially near the momenta of perfect reflection or perfect transmission: it keeps the soliton shape during its propagation. Since in many experimental settings one can easily check if the reflected wavepacket is vanishing, this work could provide a basis for a topological engineering of solitonic propagation on inhomogeneous networks.
The plan of the paper is the following. In Section II we review the properties of the DNLSE on a chain and we introduce the variational approach to investigate the soliton dynamics. In Section III, by using graph theory [@harary69], we define the DNLSE on inhomogeneous networks built by adding a topological perturbation to an unbranched chain; furthermore, we explain the numerical techniques used in the paper for the study of the soliton scattering, and we discuss the range of validity of the linear approximation used in the analytical computations. In Section IV we present our analysis yielding the conditions on the spectrum of the finite graph $G^0$ in order to obtain total reflection and transmission of solitons. In Section V we study the relevant case when $G^0$ is a finite, linear chain and we show that the Fano-Anderson model [@miroshnichenko03; @flach03] can be realized within our approach by considering a single link attached to the unbranched chain. In Section VI we show that, for self-similar graphs $G^0$, the values of momenta for which perfect reflection occurs becomes perfect transmission momenta when the next generation of the graph is considered and we study in detail the case of Cayley trees as an example of self-similar structures. In Section VII we study the transmission coefficients for three different inserted finite graphs: loops, stars and complete graphs. Finally, Section VIII is devoted to our concluding remarks.
DNLSE on a chain
================
Besides its theoretical interest, the DNLSE describes the properties of interesting systems, such as arrays of coupled optical waveguides and arrays of Bose-Einstein condensates. On a chain the DNLSE reads $$\label{DNLS-retta}
i \frac{\partial \psi_n}{\partial \tau} = - \frac{1}{2} (\psi_{n+1}
+ \psi_{n-1}) + \Lambda \mid \psi_n \mid ^2 \psi_n + \epsilon_{n} \psi_n$$ where $n$ is an integer index denoting the site position and the normalization condition is $\sum_n \mid \psi_n \mid ^2=1$. In Eq.(\[DNLS-retta\]) one has a kinetic coupling term only between nearest-neighbour sites, but the effect of next- nearest-neighbour coupling and long-term coupling has been also often considered (see the reviews [@hennig99; @kevrekidis01] for more references): in the present paper we consider only a constant nearest-neighbour interaction, but from the next Section we allow for that the number of nearest-neighbours of a site is not constant across the network (like for the simple chain), but it can vary according the topology of the graph.
In condensate arrays, $\psi_n(\tau)$ is the wavefunction of the condensate in the $n$th well. Time $\tau$ is in units of $\hbar/2K$, where $K$ is the tunneling rate between neighbouring condensates; $\epsilon_n=E_n/2K$ where $E_n$ is an external on-site field superimposed to the optical lattice and the nonlinear coefficient is $\Lambda=U/2K$, where $U$ is due to the interatomic interaction and it is proportional to the scattering length ($U$ is positive for $^{87}Rb$ atoms and is negative for $^{7}Li$ atoms).
In arrays of one-dimensional coupled optical waveguides [@eisenberg98] $\psi_n(\tau)$ is the electric field in the $n$th-waveguide at the position $\tau$ and the DNLSE describes the spatial evolution of the field. The parameter $\Lambda$ is proportional to the Kerr nonlinearity and the on-site potentials $\epsilon_n$ are the effective refraction indices of the individual waveguides. As the light propagates along the array, the coupling induces an exchange of power among the single waveguides. In the low power limit (i.e. when the nonlinearity is negligible), the optical field spreads over the whole array. Upon increasing the power, the output field narrows until it is localized in a few waveguides, and discrete solitons can finally be observed [@eisenberg98; @morandotti99]. Experiments with defects (i.e., with particular waveguides different from the others) have been already reported [@mandelik03].
On a chain DNLSE soliton-like wavepackets can propagate for a long time even if the equation is not integrable [@hennig99]. Let us consider, at $\tau=0$, a gaussian wavepacket centered in $\xi(\tau=0) \equiv \xi_0$, with initial momentum $k$ and width $\gamma(\tau=0) \equiv \gamma_0$: its time dynamics are studied resorting to the Dirac time-dependent variational approach [@dirac30] which well reproduces the exact results in the continuum theory [@cooper93]. In its discrete version the wavefunction can be written as a generalized gaussian $$\label{gaussol}
\psi_{n}(\tau)=\sqrt{{\cal K}} \, \cdot \,
e^{ -\frac{(n-\xi)^2}{\gamma^2} + ik(n-\xi) +
i \frac{\delta}{2}(n-\xi)^2}$$ where $\xi(\tau)$ and $\gamma(\tau)$ are, respectively, the center and the width of the density $\rho_n = \mid \psi_n \mid^2$, and $k(\tau)$ and $\delta(\tau)$ are the momenta conjugate to $\xi(\tau)$ and $\gamma(\tau)$ respectively; ${\cal K}$ is just a normalization factor. The wave packet dynamical evolution is obtained from the Lagrangian ${\cal L}= \sum_n i \dot{\psi}_n \psi_n^\ast - \cal{H}$, with the equations of motion for the variational parameters $\xi,\gamma,k,\delta$. In the absence of external potential ($\epsilon_n=0$), one obtains the Lagrangian [@trombettoni01a] $${\cal L}={\cal K} \sum\limits_{n=-\infty}^{\infty}
e^{-(2n^2+2n-4n\xi+2\xi^2-2\xi+1)/\gamma^2}
\cos{[\delta(n+1/2-\xi)+k]} -
\frac{\Lambda {\cal K}^2}{2} \sum\limits_{n=-\infty}^{\infty}
e^{ - 4 (n-\xi)^2 / \gamma^2}$$ $$\label{LAG-N-finito}
+{\cal K} \sum\limits_{n=-\infty}^{\infty} \left\{
-\frac{\dot{\delta}}{2} (n-\xi)^2 + \delta \dot{\xi}(n-\xi) -
\dot{k}(n-\xi)+k \dot{\xi} \right\} \,
e^{-2(n-\xi)^2/\gamma^2}.$$ With $\gamma$ not too small ($\gamma \gg 1$), we can replace the sums over $n$ with integrals: to evaluate the error committed, we recall that [@knuth89] $$\frac{\sum\limits_{n=-\infty}^{\infty} e^{-\frac{(n-\xi)^2}
{\gamma^2}}}{\int\limits_{-\infty}^{\infty} dn \, \, \,
e^{-\frac{(n-\xi)^2}{\gamma^2}} }
= 1+ O(e^{- \pi^2 \gamma^2}).
\label{knuth}$$ In this limit the normalization factor becomes ${\cal K}=\sqrt{2/\pi\gamma^2}$. We finally get [@trombettoni01] $$\label{LAG}
{\cal L}=k \dot{\xi} - \frac{\gamma^2 \dot{\delta}}{8} -
\frac{\Lambda}{2 \sqrt{\pi \gamma^2} } + \cos{k} \, \cdot \,
e^{-\eta},$$ where $\eta = 1 / 2 \gamma^2 + \gamma^2 \delta^2 / 8$. The equations of motion are $$\begin{aligned}
\dot{k} & = & 0 \label{var1} \\
\dot{\xi} & = & \sin{k} \, \cdot \, e^{-\eta} \label{var2} \\
\dot{\delta} & = & \cos{k} \Big(4 / \gamma^4-\delta^2 \Big)
e^{-\eta} + 2 \Lambda / \sqrt{\pi} \gamma^3 \label{var3} \\
\dot{\gamma} & = & \gamma \delta \cos{k} \, \cdot \,
e^{-\eta}: \label{var4}\end{aligned}$$ $k(\tau)=k$ is conserved. Notice that, due to the discreteness, the group velocity cannot be arbitrarily large ($\dot{\xi} \approx \sin{k} \le 1$). As Eq.(\[knuth\]) clearly shows, the variational equations of motions (\[var4\]) are meaningful only for large solitons: the Peierls-Nabarro potential does not appear in (\[var4\]), and the equations feature momentum conservation. We mention that in uniform DNLSE chains a threshold condition for the soliton propagation appears: only if the soliton is sufficiently broad solitons may freely move [@papa03]. In the following we shall consider only large-fast solitons, so that the variational equations of motions (\[var4\]) are appropriate; however, to study the propagation of localized discrete breathers in inhomogeneous networks one should study the Lagrangian (\[LAG-N-finito\]).
When $\dot{\gamma}=0$ and $\dot{\delta}=0$, the shape of the wavefunction does not vary and one has a variational soliton-like solution where the center of mass move with a constant velocity $\dot{\xi}= constant$. If $\Lambda>0$ the conditions $\dot{\gamma}=0$ and $\dot{\delta}=0$ can be satisfied only if $\cos{k}<0$ (i.e., only when the effective mass is negative): for this reason in the following we take only momenta $\pi/2 \le k \le \pi$ (positive velocities) or $-\pi \le k \le - \pi/2$ (negative velocities). In particular, for $\delta(\tau=0) \equiv \delta_0=0$ and large enough solitons ($\gamma_0 \gg 1$), the condition on $\Lambda$ allowing for a soliton solution is [@trombettoni01] $$\label{lambda_sol}
\Lambda_{sol} \approx 2 \sqrt{\pi} \frac{\mid \cos{k} \mid}
{\gamma_0}.$$
The stability of variational solutions has been numerically checked showing that the shape of the solitons is preserved for long times. In the following, we use the term “solitons” to name the solutions of the variational equations (\[var1\])-(\[var4\]). One can have a similar criterion using other variational approaches [@malomed96]. One also expects that the integrable version of the DNLSE, the so-called Ablowitz-Ladik equation [@ablowitz76; @ablowitz04], provides results very similar to those obtained in this paper.
DNLSE on graphs
===============
The DNLSE (\[DNLS-retta\]) can be generalized to a general discrete network by means of graph theory. A graph $G$ is given by a set of sites $i$ connected pairwise by set of unoriented links $(i,j)$ defining a neighbouring relation between the sites. The topology of a graph is described by its adjacency matrix $A_{i,j}$ which is defined to be $1$ if $i$ and $j$ are nearest-neighbours, and $0$ otherwise. The DNLSE on a graph reads as $$\label{DNLS-gen}
i \frac{\partial \psi_i}{\partial \tau} = - \frac{1}{2} \sum_{j} A_{i,j}
\psi_{j}+ \Lambda \mid \psi_i \mid ^2 \psi_i + \epsilon_{i} \psi_i$$
Equation (\[DNLS-gen\]) describes the wavefunction dynamics in a wide range of discrete physical systems, and it can be applied to regular lattices as well as to inhomogeneous networks such as fractals, complex biological structures and glasses. The properties of Eq.(\[DNLS-gen\]) on small graphs has been investigated in [@eilbeck85]. The first term in the right-hand side of Eq.(\[DNLS-gen\]) represents the hopping between nearest-neighbours (with tunneling rate proportional to $A_{i,j}$), the second is the nonlinear term, and the third one describes superimposed external potentials. We remark that in Eq.(\[DNLS-gen\]) the numbers of nearest neighbours is site-dependent. Furthermore, since $A_{i,j}=1$ if $i$ and $j$ are nearest-neighbours, one is assuming that the tunneling rate between neighbouring sites entering Eq.(\[DNLS-gen\]) is constant across the array and it is (in the chosen units) equal to $1$. Below, we shall consider also the case of a tunneling rate which is not constant across the array.
We shall focus on the situation where the graph $G$ is obtained by attaching a finite graph $G^0$ to a single site of the unbranched chain (see Fig.1) and setting $\epsilon_i=0$ for all the sites. We denote the sites of the unbranched chain and of the graph $G^0$ with latin indices $m,n,\dots$ and greek indices $\alpha, \beta, \cdots$ respectively. A single link connects the site $n=0$ of the chain with the site $\alpha$ of the graph $G^0$.
The scattering of a soliton through the topological perturbation can be numerically studied as follows. At $\tau=0$ (hereafter, we refer to $\tau$ as a time even if for the optical waveguides it represents a spatial variable) one prepares a gaussian soliton (\[gaussol\]) centered well to the left of $0$ (i.e., $\xi_0<0$) moving towards $n=0$ ($\sin(k)>0$) with a width related to the nonlinear coefficient according to Eq.(\[lambda\_sol\]). From Eq.(\[DNLS-gen\]) the time evolution of the wavefunction may be numerically evaluated: when $\tau_s \approx \xi_0 /\sin(k)$ the soliton scatters through the finite graph $G^0$ ($\sin(k)$ being the group velocity of the soliton). At a time $\tau$ well after the soliton scattering (i.e. $\tau \gg \tau_s$), the reflection and transmission coefficients ${\cal R}$ and ${\cal T}$ are given by $${\cal R}=\sum_{n<0} \mid \psi_n(\tau)\mid^2 \label{R}$$ $${\cal T}=\sum_{n>0} \mid \psi_n(\tau)\mid^2.
\label{T}$$
Note that, while in the linear case ($\Lambda=0$) one has ${\cal
R}+{\cal T}=1$, in general, nonlinearity violates unitarity by allowing for phenomena such as soliton trapping; nevertheless, there are regimes where soliton trapping is negligible and ${\cal R}+{\cal T} \approx 1$. We numerically checked that in the time dynamics reported in the paper this condition is well satisfied. Situations corresponding to resonant scattering (i.e. ${\cal R}=0$ or ${\cal T}=0$) have a particular relevance: in fact, these situations can be easily experimentally detected, and the soliton-like solution is stable also well after the scattering, as it is numerically verified in different examples of resonant reflection and transmission.
For an important class of soliton solutions (to which we refer as large-fast solitons) the scattering through a topological inhomogeneity can be analytically studied using a linear approximation. The interaction between the soliton and the topological inhomogeneity is characterized by two time-scales: the time of the soliton-defect interaction $\tau_{int}=\gamma/\sin{k}$ and the soliton dispersion time (i.e. the time scale in which the wavepacket will spread in absence of interaction) $\tau_{disp}=
\gamma/ (4 \sin{(1/2\gamma) \cos{k})}$ [@miroshnichenko03]. For [*large*]{} ($\gamma \gg 1$, as in many relevant experimental settings) and [*fast*]{} solitons, i.e. $$\label{linearaprox}
v=\sin{k} \gg (2/ \gamma) \cos{k},$$ one has that the soliton may be considered as a set of non interacting plane waves while experiencing scattering on the graph; thus the soliton transmission may be studied by considering, in the linear regime (i.e., $\Lambda=0$), the transport coefficients of a plane wave across the topological defect. The use of the linear approximation for the analysis of the interaction of a fast soliton with a local defect in the continuous nonlinear Schrödinger equation is reported in [@cao95]. Later, we shall compare the analytical findings with a numerical solution of Eq.(\[DNLS-gen\]), namely with the reflection and transmission coefficients ${\cal R}$ and ${\cal T}$ given by Eqs.(\[R\])-(\[T\]).
A general argument for resonant transmission
============================================
In this Section we show that, if a large-fast soliton scatters through a topological perturbation of an unbranched chain, the soliton momenta for perfect reflection and transmission are completely determined by the spectral properties of the attached graph $G^0$: in particular one has ${\cal R}=1$ if $2 \cdot \cos{k}$ coincides with an energy level of $G^0$, while ${\cal T}=1$ if $2 \cdot \cos{k}$ is an energy level of the reduced graph $G^r$, i.e., of the graph obtained from $G^0$ by cutting the site $\alpha$ from $G^0$ (see Fig.1). In algebraic graph theory (see e.g. [@harary69; @biggs74]) the energy level of a graph is simply defined as an eigenvalue of its adjacency matrix. We will call $A^0$ and $A^r$ the adjacency matrices of $G^0$ and $G^r$, respectively.
For large-fast solitons, the pertinent eigenvalue equation to investigate is $$\label{DLS}
-\frac{1}{2} \sum_{j} A_{i,j} \psi_j=\mu \psi_i$$ (here $i$ is a generic site of the network). The solution corresponding to a plane wave coming from the left of the chain is $\psi_n=a e^{ikn}+be^{-ikn}$ for $n<0$ and $\psi_n=c e^{ikn}$ for $n>0$, so that $\mu=-\cos{k}$. The reflection coefficient is given by ${\cal R}=\mid b / a \mid ^2$ and the transmission coefficient by ${\cal T}=\mid c / a \mid ^2$. The continuity at $0$ requires $a+b=c$. The equation in $0$ is $$-\frac{1}{2} (a e^{-ik}+b e^{ik}+c e^{ik}+\psi_{\alpha})=-\cos{k}
\cdot (a+b)
\label{eq_0}$$ while in $\alpha$ one has $$-\frac{1}{2} (a+b+\sum_{\eta \in G^0} A^0_{\alpha,\eta} \psi_\eta)=
-\cos{k} \cdot \psi_{\alpha}.
\label{eq_alpha}$$ At the sites $\eta$ of $G^r$ one obtains $$-\frac{1}{2} \sum_{\eta' \in G^0} A^0_{\eta,\eta'} \psi_{\eta'}=-\cos{k}
\cdot \psi_\eta.
\label{eq_altri}$$
For perfect reflection, i.e. for the momenta $k$’s such that ${\cal R}(k)=0$, one has $c=0$, $a=-b$ and from Eq.(\[eq\_0\]) $\psi_\alpha=-2a \sin{k}$. Therefore Eqs.(\[eq\_alpha\]) and (\[eq\_altri\]) reduce to the eigenvalue equation for the adjacency matrix $A^0$ and, apart from the trivial case $\cos(k)=0$, they are satisfied only if $2\cos{k}$ coincides with an eigenvalue of ${ A^0}$.
At variance, in order to find the momenta $k$’s such that ${\cal T}(k)=0$ (perfect transmission), one has $b=0$, $a=c$ and from Eq.(\[eq\_0\]) $\psi_\alpha=0$. Therefore, Eqs.(\[eq\_altri\]) reduces to the eigenvalue equation for $A^r$ and it is satisfied only if $2\cos{k}$ coincides with an eigenvalue of $A^r$.
This general argument can be easily extended to the situation where $p$ identical graphs $G^0$ are attached to $n=0$: indeed, now one has only to replace in Eq. (\[eq\_0\]) $\psi_{\alpha}$ with $p\psi_{\alpha}$ and the conditions for ${\cal T}(k)=0$ and ${\cal R}(k)=0$ do not change.
The stated result holds for the case where the tunneling rates between the neighbour sites of $G^0$ are constant (and equal to 1 in the chosen units): however one can also consider in $G^0$ non-uniform tunneling rates $t^0_{\eta,\eta'}>0$, where $\eta$ and $\eta'$ are nearest-neighbour sites belonging to $G^0$ ($t_{\eta,\eta'}^0=0$ if $A^0_{\eta,\eta'}=0$). The DNLSE at a site $\eta$ of $G^0$ becomes $$i \frac{\partial \psi_{\eta}}{\partial \tau}= -
\frac{1}{2} \sum_{\eta'} t_{\eta,\eta'}^0 \psi_{\eta'}+\Lambda
\mid \psi_\eta \mid^2 \psi_\eta$$ while the DNLSE at the sites $n$ of the of the chain remain unchanged. Now, the criterion states that ${\cal R}=1$ if $2\cos{k}$ coincides with an eigenvalue of the matrix $t_{\eta,\eta'}^0$, while ${\cal T}=1$ if $2\cos{k}$ is an eigenvalue of the matrix $t_{\eta,\eta'}^r$ defined as $t_{\eta,\eta'}^r=t_{\eta,\eta'}^0$ if $\eta$ and $\eta'$ belong to the reduced graph $G^r$.
Finite Linear Chains
====================
As a first simple application of the argument given in Section IV, we consider a single site $\alpha$ attached via a single link to the site $0$ of the unbranched chain. For Bose-Einstein condensates in optical lattices [@oberthaler03] the setup “infinite chain + single link” or “infinite chain + finite chain” may be realized by using two pairs of counterpropagating laser beams to create a star-shaped geometry in the $x$-$y$ plan [@brunelli04] and manipulating the frequencies of the superimposed harmonic magnetic potential so that in the $y$ direction only few sites can be occupied. In an optic context, chains of coupled waveguides are routinely built and studied [@kivshar03]: one can obtain the configuration “infinite chain + single link” by coupling a further waveguide to a waveguide of the chain.
The DNLSE in the sites $0$ and $\alpha$ reads $$i \frac{\partial \psi_0}{\partial \tau}=-\frac{1}{2} (\psi_1+\psi_{-1}+
\psi_\alpha)+\Lambda \mid \psi_0 \mid^2 \psi_0
\label{singolo_link_0}$$ and $$i \frac{\partial \psi_\alpha}{\partial \tau}=-\frac{1}{2} \psi_0+
\Lambda \mid \psi_\alpha \mid^2 \psi_\alpha.
\label{singolo_link_alpha}$$ It is transparent from Eqs.(\[singolo\_link\_0\])-(\[singolo\_link\_alpha\]) that the wavefunction $\psi_\alpha(\tau)$ may be interpreted as an additional local Fano degree of freedom, yielding the so-called Fano-Anderson model [@miroshnichenko03; @flach03; @miroshnichenko05]. In our approach, such degree of freedom is interpreted as a single link attached to the unbranched chain. As it is well known, the Fano-Anderson model describes interesting scattering properties: adding a generic finite graph (instead of a single link) gives rise to a yet richer variety of behaviors. We mention that in [@miroshnichenko05] the Fano degree of freedom is coupled to several sites of the unbranched chain: this would correspond in our description to a site linked to several sites of the chain, and, in general, to graphs attached to several sites of the chain. For simplicity, in the following we limit ourself to graphs inserted in a single site of the unbranched chain.
For large-fast solitons, when a single link is added to the unbranched chain, the reflection coefficient ${\cal R}$ from Eqs.(\[singolo\_link\_0\])-(\[singolo\_link\_alpha\]) is found to be [@miroshnichenko03] $${\cal R}=\frac{1}{1+4 \sin^2{(2k)}}:
\label{R_singolo_link}$$ in the regime where $\tau_{disp} \gg \tau_{int}$, we verified that the numerical results of the soliton scattering against the link are in agreement with Eq.(\[R\_singolo\_link\]) (see Fig.2). One sees that, when $k$ is approaching $\pi$ (solitons becoming slower), the agreement becomes worse. From the general results, there are no fully transmitted momenta and ${\cal R}=1$ only for $k=\pi/2$ and $k=\pi$, as one can also see by a direct inspection of (\[R\_singolo\_link\]).
One can also attach a finite chain of length $L$ at the site $0$. In the linear approximation for large-fast solitons (i.e., Eq.(\[DLS\])), the solution corresponding to a plane wave coming from the left of the unbranched chain is $\psi_n=a e^{ikn}+be^{-ikn}$ for $n<0$ and $\psi_n=c e^{ikn}$ for $n>0$, while in the attached chain $\psi_{\alpha}=f e^{ik \alpha}+ge^{-ik \alpha}$ ($\alpha=1,\cdots,L$ denotes the sites of the attached chain, and $\alpha=1$ is the site linked to $n=0$). Eq.(\[DLS\]) for $n=0$, $\alpha=1$, and $\alpha=L-1$ yields respectively $$a+b=c=f+g
\label{links_1}$$ $$-\frac{1}{2} (a e^{-i k} + b e^{-i k} + c e^{i k} + f e^{i k} + g^{-i k})=
\mu (a+b)
\label{links_2}$$ and $$-\frac{1}{2} (f e^{i k (L-1)} + g e^{-i k (L-1)})=
\mu (f e^{i k L} + g e^{-i k L})
\label{links_3}$$ where $\mu=-\cos{k}$.
We have five unknowns ($a$, $b$, $c$, $f$, and $g$) and four equations (\[links\_1\])-(\[links\_2\]) (the remaining condition being provided by the normalization). One may easily determine $b/a$, $c/a$, $f/a$, and $g/a$, getting $$\frac{b}{a}=\frac{e^{2 i k}(e^{2ikL}-1)}{1-2e^{2ik}+e^{2ik(L+2)}},$$ which leads to $${\cal R}=\mid b/a \mid^2 =
\frac{\sin^2{(kL)}}{[\cos{(kL)}-\cos{(k(L+2))}]^2+\sin^2{(kL)}}.
\label{R_L_links}$$ and ${\cal T}=\mid c/a \mid^2 = 1 - {\cal R}$. For $L=1$, Eq.(\[R\_L\_links\]) reduces to Eq.(\[R\_singolo\_link\]). In agreement with the general argument of Section IV, the number of minima and maxima increases with $L$. Eq.(\[R\_L\_links\]) for $L=2$ is compared in Fig.2 with the numerical results.
We notice that in the limit $L \to \infty$ the considered problem corresponds to the propagation of a soliton in a the so-called [*star graph*]{}, which has been recently investigated in the context of two-dimensional networks of nonlinear waveguide arrays [@christodoulides01] and $Y$-junctions for matter waves [@kevrekidis03].
Cayley trees
============
Eq.(\[R\_L\_links\]) yields that the values of $k$ allowing for perfect reflection (${\cal R}=1$) for the length $L$ coincide with the momenta of full transmission (${\cal T}=1$) when $G^0$ is a chain of length $L+1$. This property is readily understood: in fact, if $G^0$ is a chain of length $L$ the $k$’s for which ${\cal R}=1$ correspond to the energy levels of $G^0$. If $G^0$ is a chain of length $L+1$, the values of $k$’s for which ${\cal T}=1$ correspond to energy levels of the reduced graph $G^r$, which, in this case, is again given by a chain of length $L$. This is clearly a general property of any self-similar graph. As an example, we study in this section the situation in which $G^0$ is a Cayley tree of branching rate $p$ and generation $L$ (see Fig.3).
Let us consider the linear approximation for large-fast solitons Eq.(\[DLS\]). The plane wave coming from the left of the unbranched chain is $\psi_n=a e^{ikn}+be^{-ikn}$ for $n<0$ and $\psi_n=c e^{ikn}$ for $n>0$. This fixes $\mu=-\cos{k}$. Furthermore, from the continuity in $0$ it follows $a+b=c$.
For a Cayley tree, the eigenfunction must have, by symmetry, the same value at all the sites belonging to the same generation. If we denote by $\psi_\beta$ the eigenfunction at the sites at distance $\beta=1,\cdots,L$ from $n=0$, the eigenvalue equation (\[DLS\]) at the site $\alpha=2,\cdots,L-1$ reads $$-\frac{1}{2}(\psi_{\alpha-1}+p\psi_{\alpha+1})=\mu \psi_{\alpha}.
\label{eig_eq_Cayley}$$ The plane wave solutions of Eq.(\[eig\_eq\_Cayley\]) can be written as $$\psi_{\alpha}=\frac{1}{p^{\alpha/2}}(f e^{ik' \alpha}+ge^{-ik' \alpha}),
\label{cayley}$$ where $\alpha=1,\cdots,L$: in this way one gets from Eq.(\[eig\_eq\_Cayley\]) $\mu=-\sqrt{p} \cos{k'}$, so that $k'=\arccos{(p^{-1/2} \cos{k})}$. Eq.(\[DLS\]) for $n=0$, $\alpha=1$, and $\alpha=L-1$ gives respectively $$-\frac{1}{2} (a e^{-i k} + b e^{i k} + c e^{i k})-\frac{1}{2 \sqrt{p}}
(f e^{i k'} + g e^{-i k'})=\mu(a+b)
\label{cayley_1}$$ $$-\frac{1}{2} (a + b)-\frac{1}{2}
(f e^{2 i k'} + g e^{-2 i k'})=\frac{\mu}{\sqrt{p}}
(f e^{i k'} + g e^{-i k'})
\label{cayley_2}$$ and $$-\frac{1}{2} (f e^{i k' (L-1)} + g e^{-i k' (L-1)})=
\frac{\mu}{\sqrt{p}} (f e^{i k' L} + g e^{-i k' L}).
\label{cayley_3}$$ Using Eqs.(\[cayley\_1\])-(\[cayley\_3\]) and the condition $a+b=c$, one can determine $b/a$, $c/a$, $f/a$, and $g/a$: the resulting expressions is rather involved and here we will not explicitly write them. In Fig.3 we plot the numerical and analytical results for the reflection coefficient ${\cal R}$ when two Cayley trees, respectively with $L=5$ and $L=6$, are attached to the unbranched chain. One sees that when one pass to the next generation, the momenta for which full reflection occurs becomes momenta of full transmission.
Further Examples of inserted graphs
===================================
In general, it is possible to consider the scattering of a large-fast soliton through a large variety of inhomogeneous networks. Here, we analyze three further examples of network topologies: stars, loops and complete graphs. In each case, the reflection coefficients for large-fast solitons are derived with a procedure analogous to the one adopted in the previous section for Cayley trees. The analytical findings are compared with numerical results.
[*Loops:*]{} Let us consider a loop graph, i.e., a finite chain of $L$ sites $\alpha_1, \cdots, \alpha_L$ such that the sites $\alpha_1$ and $\alpha_L$ are linked to the site $n=0$ of the unbranched chain. For large-fast solitons, the reflection coefficient ${\cal R}$ in the linear approximated regime is $${\cal R}=2 \, \, \, \frac{\left[ 1+\cos{(k(L-1))}\right]^2 +
\sin^2{(k(L-1))}}{6-2 \cos{(2k)}+5 \cos{k} \cos{(kL)}-\cos{(k(L+3))}
+\sin{k} \sin{(kL)}}.
\label{R_L_loop}$$ We note that the number of momenta of perfect reflection and perfect transmission increases with $L$. Furthermore, at large $L$, the transmission properties of the loops become similar to those of a finite (not closed) chain \[see Eq.(\[R\_L\_links\])\]. In Fig.4 the numerical and analytical results are compared and a figure of the loop graph with $L=3$ is provided.
[*Stars:*]{} The $p$-star is the graph composed by a central site linked to $p$ sites, which are in turn connected only to the central site. Let us consider the case where $G^0$ is a $p$-star graph and $\alpha$ is the center, linked to $n=0$. In the linear approximation we have $${\cal R}=\frac{1}{1+\left( \frac{\sin{(3k)}}{\cos{k}}-(p-1) \tan{k}\right)^2}
\label{R_L_star}.$$ In Fig.5 the numerical and analytical results are compared. Eq.(\[R\_L\_star\]) shows that perfect transmission (${\cal R}=0$) is obtained only for for the momentum $k=\pi/2$. This can be directly proved applying the criterion of Section IV: indeed $G^0$ is a $p$-star, while $G^r$ consists of $p$ disconnected sites. Therefore, all the eigenvalues of the adjacency matrix $A^r$ equal zero, and the only momentum of perfect transmission is $k=\pi / 2$.
[*Complete graphs:*]{} The complete graph $K_M$ of $M$ sites is the graph where every pair of sites in linked [@harary69]: e.g, $K_3$ is a triangle. Inserting $K_{L+1}$ at the site $n=0$ of the unbranched chain (so that $n=0$ is one of the sites of $K_{L+1}$), one gets: $${\cal R}=\frac{1}{1+4 \frac{(L-1-2 \cos{k})^2}{L^2} \sin^2{k}}.
\label{R_L_simpl}$$ For $L >> 1$, ${\cal R} \approx
1/ (1+4 \sin^2{k})$, therefore the complete graph behaves as a single effective defect \[compare with Eq.(\[R\_singolo\_link\])\]. The comparison between the numerical and analytical results is presented in Fig.6.
Conclusions
===========
As a first step in addressing the issue of the interplay between nonlinearity and topology, we studied the discrete nonlinear Schrödinger equation on a network built by attaching to a site of an unbranched chain a topological perturbation $G^0$. The relevant situation corresponding to the Fano-Anderson model is obtained when one considers a single link attached to the linear chain. We showed that, by properly selecting the attached graph, one is able to control the perfect reflection and transmission of traveling solitons. We derived a general criterion yielding - once the energy levels of the graph $G^0$ is known - the momenta at which the soliton is fully reflected or fully transmitted. For self-similar graphs $G^0$, we found that the values of momenta for which perfect reflection occurs become perfect transmission momenta when the next generation of the graph is considered. For finite linear chains, loops, stars and complete graphs, we studied the transmission coefficients and we compared numerical results form the discrete nonlinear Schrödinger equation with analytical estimates. Our results evidence the remarkable influence of topology on nonlinear dynamics and are amenable to interesting applications in optics since one may think of engineering inhomogeneous chains acting as a filter for the motion of soliton [@burioni05].
[*Acknowledgments:*]{} We thank M. J. Ablowitz, P. G. Kevrekidis and B. A. Malomed for discussions.
[99]{}
S. Flach and C.R. Willis, Phys. Rep. [**295**]{}, 181 (1998).
A. C. Scott, [*Nonlinear Science: Emergence and Dynamics of Coherent Structures*]{}, Oxford University Press (1999).
D. Hennig and G. P. Tsironis, Phys. Rep. [**307**]{}, 333 (1999).
T. Nakayama, K. Yakubo, and R. L. Orbach, Rev. Mod. Phys. [**66**]{}, 381 (1994).
M. Peyrard and A. R. Bishop, Phys. Rev. Lett. [**62**]{}, 2755 (1989).
edited by P. L. Christiansen, J. C. Eilbeck, and R. D. Parmentier \[Physica D [**68**]{}, 1-186 (1993)\].
Y. S. Kivshar and G. P. Agrawal, [*Optical Solitons: from Fibers to Photonic Crystals*]{}, San Diego, Academic Press (2003).
D. N. Christodoulides and E. D. Eugenieva, Phys. Rev. Lett. [**87**]{}, 233901 (2001); Opt. Lett. [**26**]{}, 1876 (2001).
P. G. Kevrekidis, D. J. Frantzeskakis, G. Theocharis, and I. G. Kevrekidis, Phys. Lett. A [**317**]{}, 513 (2003).
A. R. McGurn, Phys. Rev. B [**61**]{}, 13235 (2000); [*ibid.*]{} [**65**]{}, 075406 (2002).
P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, Int. J. Mod. B [**15**]{}, 2833 (2001).
M. J. Ablowitz, B. Prinari, and A. D. Trubatch, [*Discrete and Continuous Nonlinear Schrödinger Systems*]{}, University Press (2004).
H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. [**81**]{}, 3383 (1998).
R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. [**83**]{}, 2726 (1999).
A. Trombettoni and A. Smerzi, Phys. Rev. Lett. [**86**]{}, 2353 (2001).
B. A. Malomed and M. I. Weinstein. Phys. Lett. A [**220**]{}, 91 (1996).
D. B. Duncan, J. C. Eilbeck, H. Feddersen, and J. A. D. Wattis, Physica D [**68**]{}, 1 (1993).
S. Flach and K. Kladko, Physica D [**127**]{} 61 (1999).
J. Gomez-Gardenes, L. M. Floria, M. Peyrard, and A. R. Bishop, Chaos [**14**]{}, 1130 (2004).
K. Forinash, M. Peyrard and B. A. Malomed, Phys. Rev. E [**49**]{}, 3400 (1994).
V. V. Konotop, D. Cai, M. Salerno, A. R. Bishop and N. Grønbech-Jensen, Phys. Rev. E [**53**]{}, 6476 (1996).
W. Krolikowski, and Yu. S. Kivshar, J. Opt. Soc. Am. B [**13**]{}, 876 (1996).
A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E [**53**]{}, 1172 (1996).
R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, Opt. Lett. [**28**]{}, 834 (2003).
A. E. Miroshnichenko, S. Flach, and B. A. Malomed, Chaos [**13**]{}, 874 (2003).
S. Flach, A. E. Miroshnichenko, V. Fleurov, and M. V. Fistul, Phys. Rev. Lett. [**90**]{}, 084101 (2003).
A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Y. S. Kivshar, Phys. Rev. E [**71**]{}, 036626 (2005).
F. Harary, [*Graph Theory*]{}, Addison-Wesley, Reading (1969).
M. K. Oberthaler and T. Pfau, J. Phys.: Condens. Matter [**15**]{}, R233 (2003).
R. Fazio and H. van der Zant, Phys. Rep. [**355**]{}, 235 (2001).
A. Birner, R. B. Wehrspohn, U. M. Gösele, and K. Busch, Adv. Mater. [**13**]{}, 377 (2001).
P. A. M. Dirac, Proc. Camb. Phil. Soc. [**26**]{}, 376 (1930).
F. Cooper, H. Shepard, C. Lucheroni, and P. Sodano, Physica D [**68**]{}, 344 (1993).
A. Trombettoni and A. Smerzi, J. Phys. B [**34**]{}, 4711 (2001).
R. E. Graham, D. E. Knuth, and O. Patashnik, [*Concrete Mathematics: A Foundation for Computer Science*]{}, Addison-Wesley, 1989.
I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, Phys. Rev. E [**68**]{}, 046604 (2003).
M. J. Ablowitz and J. F. Ladik, J. Math. Phys. [**17**]{}, 1011 (1976).
J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, Physica D [**16**]{}, 318 (1985).
X. D. Cao and B. A. Malomed, Phys. Lett. A [**206**]{}, 177 (1995).
N. L. Biggs, [*Algebraic Graph Theory*]{}, Cambridge University Press (1974).
I. Brunelli, G. Giusiano, F. P. Mancini, P. Sodano, and A. Trombettoni, J. Phys. B [**37**]{}, S275 (2004).
R. Burioni, D. Cassi, P. Sodano, A. Trombettoni, and A. Vezzani, cond-mat/0502280.
|
---
author:
- Matthias Droth
- Guido Burkard
- 'Vitor M. Pereira'
title: 'Reply to “Comment on ‘Piezoelectricity in planar boron nitride via a geometric phase’"'
---
=1
In relation to our original paper \[M. Droth *et al.*, Phys. Rev. B **94**, 075404 (2016)\], the Comment by [Li *et al.*]{} \[Phys. Rev. B **98**, 167403 (2018)\] claims to have identified a “*mistake in constructing the adiabatic process of the piezoelectricity*”. More specifically, they write that in our original work *“the erroneous usage of the polarization difference formula in Eq. (4) leads to an invalid analytical expression of piezoelectric constant in Eq. (12) of Reference ”*. We explain below why these and other minor claims in the Comment are unwarranted, and why we maintain that our result is correct and physically sound.
Choice of adiabatic parameter
=============================
The procedure used to obtain the piezoelectric constant in our original paper [@Droth2016] hinges on a calculation of the bulk polarization as an explicit function of the strain tensor \[cf. the original Eq. (9)\]. As established by the seminal work of Resta, Vanderbilt and King-Smith [@Resta; @KingVanderbilt], the only well defined physical quantity that is amenable to a controlled calculation is a variation in bulk polarization. This led to the development of the modern theory of polarization according to which such polarization differences are calculated by integrating the varying Berry curvature along an adiabatic path connecting the two states. One of the most typical situations involves computing the absolute bulk polarization of a given system: to exploit this method, one sets up an adiabatic path that connects that target with a reference system having zero bulk polarization [@Resta]. Such an adiabatic path is frequently defined in terms of a parameter of the Hamiltonian that can be continuously tuned between the initial and target state [@Resta] and, in particular, *the adiabatic process does not have to necessarily reflect any real physical process*. For example, in first-principles implementations it is rather common to obtain the bulk polarization by an adiabatic process that continuously changes the atomic pseudopotentials, or which evolves the lattice from an inversion-symmetric configuration to the target state through a fictitious deformation path (for a specific example where this evolution of a “virtual crystal” is done for a BN-based system see Ref. \[\]).
In calculations made with effective Hamiltonians, the earlier paper in the context of nanotubes by Mele and Král [@Mele2002] relies on exactly the same approach that we used of evolving the gap parameter in an adiabatic path that begins with a strained carbon nanotube and ends with a strained BN nanotube. As our adiabatic parameter is the gap $\Delta$, our adiabatic process begins with strained graphene and ends with strained BN; strain is kept finite and constant throughout. The adiabatic process chosen by [Li *et al.*]{} [@Li2018] begins with relaxed BN and ends with strained BN. Since the bulk polarization of the initial state is zero in the two cases (in uniaxially strained graphene because of inversion symmetry, and in relaxed BN because of its $D_{3h}$ point group), they are, in principle, both legitimate adiabatic paths to determine the bulk polarization. It is therefore erroneous, and not in line with the modern formulation of the quantum theory of polarization, to claim that “*the correct adiabatic process of piezoelectricity should reflect the deformation-induced polarization difference from the initial state of undeformed h-BN to the final state of deformed h-BN*”, as stated in the Comment.
The analytical expression for $e_{222}$
=======================================
Reinstating the parameter $\kappa$ associated with the electron-phonon coupling [@Suzuura2002], as per our original parametrization, [Li *et al.*]{}’s Eq. (18) should read $$e_{222}=\frac{e|\beta|\kappa}{2\pi a}\left[\text{sign}\,(\Delta)-
\frac{\Delta}{\sqrt{\sqrt{3} \pi t^2+\Delta^2}}\right].
\label{e222-Lietal}$$ In the original paper, we obtain instead [@GapConvention] \[cf. our original Eq. (12)\] $$e_{222} = \frac{e|\beta|\kappa}{\pi^2 a}\tan^{-1}
\left[ \frac{\Delta}{\sqrt{2w^2+\Delta^2}} \right].
\label{e222-orig}$$
Before further discussion, it is instructive to verify whether these results satisfy simple limits and symmetries. In the model Hamiltonian that is used to describe the BN monolayer [@Droth2016; @Li2018], changing $\Delta\to-\Delta$ amounts to an inversion transformation, since it is equivalent to swapping B with N atoms everywhere. Being a rank-3 tensor, $e_{222}$ should change sign under an inversion transformation, which is indeed satisfied by both results and .
However, it is clear that when $\Delta\to 0$, our result decreases and ultimately becomes zero in the limit of graphene ($\Delta=0$). This is just as expected, because the six-fold rotational symmetry of the latter precludes a piezoelectric response ($e_{ijk} = 0$ in graphene by symmetry). In contrast, the expression obtained by [Li *et al.*]{} remains finite as $\Delta\to 0$. In fact, according to their result, $e_{222}$ *increases* monotonically when $\Delta$ decreases. In addition to failing to recover the graphene case in the limit $\Delta\to0$, this is contrary to physical intuition because, even if the gap remains finite, when it decreases, the charge transfer between B and N decreases as well. Even though in the modern theory the macroscopic dipole moment is not defined in terms of local properties, one expects its magnitude to qualitatively follow the trend of the bond polarity. Hence, the dipole moment induced by a finite deformation in this system is expected to decrease when the gap parameter $\Delta$ is brought to zero. Seeing this from another perspective, if one makes an infinitesimal perturbation to graphene that breaks its intrinsic sublattice symmetry, one is breaking the inversion symmetry of the system only by an infinitesimal amount as well, and expects the induced polarization to be small. But the result by [Li *et al.*]{} predicts otherwise, namely a gap-independent value of $\bm{P}$ (and, by extension, of $e_{222}$ as well) in the limit where the gap is much smaller than the bandwidth ($\Delta \ll t$), and a finite discontinuity in $e_{222}$ when $\Delta$ varies infinitesimally between $0^-$ and $0^+$.
Therefore, our original work is suitable to characterize, in an entirely analytical way, the behavior of both the electronic contribution to the bulk polarization and the piezoelectric constant in BN, as intended and stated in our original paper. Moreover, relying on an analytical model Hamiltonian, it is only useful if it captures the qualitative trends of these quantities, in particular when transitioning from BN to the graphene limit, which is not the case with the result in the Comment.
Two different analytical results
================================
There are two aspects related to the different results obtained for $e_{222}$ in Eqs. and . The first one is a superficial difference related to the approximation used for the effective Brillouin zone (BZ) in the momentum integrations. Whereas our original paper uses a square, total-state-conserving effective BZ, the Comment uses a circular one. If $e_{222}$ is computed in a circular BZ domain (using polar coordinates for the integrations) we obtain $$e_{222} = \frac{e|\beta|\kappa}{4\pi a} \frac{\Delta}{\sqrt{q_c^2+\Delta^2}}
.$$ As expected, this expression agrees with Eq. to leading order when $\Delta \gg \{q_c,w\}$, up to a factor $\sim 1$ arising from the circular-vs-square geometry difference \[note that $4w^2 = \pi q_c^2 =
4\pi^2/(3\sqrt{3} a^2)$, see the original paper\]. Moreover, it corresponds precisely [@GapConvention] to the second term in the result above, which is the proposed analytical form in the Comment.
The second aspect has to do with the essential difference that makes the result proposed in the Comment a constant as $\Delta\to 0$, thus failing to capture the correct graphene (gapless) limit. The method followed in the Comment relies on two steps that are mathematically unsafe when combined: using strain as the adiabatic parameter and, in practice, differentiating the integral that yields the polarization with respect to strain beforehand. As a result, in the Comment, $e_{222}$ is expressed as the integral stated in their Eq. (9), where the derivatives with respect to the strain components appear in the integrand. The problem with this approach is that the original integrand that yields the polarization $\bm{P}$ is not a continuous function over the domain of the multiple integral. Hence, “passing” the derivative from the outside to the inside of the multiple integral (Leibniz’s rule) is not warranted. In Appendix \[app:integrations\], we show explicitly that, doing so, we obtain the result stated in the Comment (still using the gap as adiabatic parameter), but it arises only because of a blind application of Leibniz’s rule.
In addition, to corroborate the mathematical robustness of our original result, we provide in the Supplemental Material [@Supplement] a *Mathematica* notebook that verifies the steps involved in the calculations reported in our original paper.
Symmetry constraints on $e_{ijk}$
=================================
The discussion presented in the Comment in connection to its Eqs. (19-20) implicitly suggests that Eq. (10) in our paper, which states $$e_{211} = e_{112} = e_{121} = - e_{222},
\label{symm-constraint}$$ is incorrect and should instead read as the Comment’s Eq. (20). However, the relations *must* be satisfied by *any* rank-3 physical tensor in a system containing a three-fold rotational axis, and with the mirror plane parallel to the direction $\mathbf{u}_2$ (refer to any textbook covering symmetry aspects of crystals, such as our original Ref. \[47\]). This is the case of monolayer BN in the orientation shown in our Fig. 1, which has point group symmetry $D_{3h}$. Our definition in Eq. (9) of the paper ensures $e_{ijk}$ is a Cartesian tensor. Hence, it is clear that Eq. (20) in the Comment violates the symmetry-imposed constraint among the nonzero components of $e_{ijk}$. The error stems from the flawed considerations in the text preceding Eq. (19) of the Comment, namely because $\varepsilon_{ij}$ defined there is not a physical tensor (i.e., its components do not transform as those of a Cartesian tensor) due to the factor of 2 included in the definition of $\varepsilon_{12}$. Eq. above is a relation involving components of a *tensor* which should be adapted if one is using a Voigt representation, and not the other way around as stated in the Comment.
Better agreement with the DFT result
====================================
The authors of the Comment emphasize that they obtain a very good numerical agreement between their numerical result for $e_{222}$ and the one arising from first-principles in a clamped-ion calculation of the electronic contributions associated with the $\pi$ and $\sigma$ bands [@Duerloo2012]. This is misleading because such an agreement is simply a numerical coincidence, especially in view of the approximations involved in the effective (Dirac) Hamiltonian and the uncertainty associated with the estimates of the logarithmic derivative of the hopping parameter under strain. Moreover, those estimates were not based on the specific bandstructure predicted by the quoted density functional theory (DFT) calculation.
The authors of the Comment also refer to Ref. \[\] as an alternative approach to check their calculation. However, piezoelectricity in BN is not discussed in that reference and it also does not offer an alternative approach. Instead, the calculation shown there is very similar to the one in their Comment, adapted to transition metal dichalcogenides.
Conclusion
==========
In summary, we trust that our methodology and calculations reported originally in Ref. \[\] are sound and physically well justified.
MD acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within Project No. . MD and GB thank the European Science Foundation and the DFG for support within the EuroGRAPHENE project CONGRAN and the DFG for funding within SFB 767 and FOR 912.
Differentiating beforehand {#app:integrations}
==========================
There is an apparent advantage in performing the derivatives $$\frac{\partial P_x}{\partial A_y}\biggr|_{\bm{A}=0}
\quad\text{and}\quad
\frac{\partial P_y}{\partial A_x}\biggr|_{\bm{A}=0}\,,$$ required to compute $e_{222}$, before performing the adiabatic and BZ integrals since, in this way, the integrand becomes a simpler algebraic function and the shifted Dirac point disappears. This is, in effect, what is implied by the procedure followed in the Comment by [Li *et al.*]{} when using strain as the adiabatic parameter.
However, there is a technical subtlety in doing so which we illustrate following our approach [@Droth2016] that uses the gap as the adiabatic parameter. The $i$-th Cartesian component of the polarization is obtained as $$P_i = \int_0^\Delta d\lambda \left[ 2e \sum_\tau \int_{\text{BZ/2}}
\frac{d\textbf{q}}{(2\pi)^2}\,\Omega_{q_i,\lambda}^{(\tau)} \right],
\label{P-def-orig}$$ which is our original Eq. (4). To obtain the piezoelectric constant we need to compute, for example, \[cf. Eq. (11) in the paper\] $$e_{222} \propto \frac{\partial}{\partial A_x}
\left[
\int_0^\Delta d\lambda \int_{\text{BZ}/2} \!\! d\bm{q} \,
\Omega^{(\tau)}_{q_y,\lambda}
\right]_{\bm{A}=0}.
\label{e222-dA-outside}$$ Whether or not one can safely employ Leibniz’s rule and pass the derivative to the inside of the integral depends on $\Omega^{(\tau)}_{q_y,\lambda}$ being a continuous function of $(q_x,q_y,\lambda)$ in the domain of the multiple integral. However, $$\Omega^{(\tau)}_{q_y,\lambda} = \frac{\tau}{2} \,
\frac{q_x-A_x}{[(q_x-A_x)^2+(q_y-A_y)^2+\lambda^2]^{3/2}}\label{BC_app}$$ has a clear discontinuity at the point $(q_x,q_y,\lambda)=(A_x,A_y,0)$ and, therefore, the conditions of Leibniz’s rule are not fulfilled \[the singularity is integrable though—to zero—, as we demonstrate in Appendix \[app:singularity\]\]. As a result of this discontinuity, even though the integral of $\partial
\Omega^{(\tau)}_{q_y,\lambda}/\partial A_x$ converges, *its value depends on the sequence of the partial integrations*. We now show this explicitly.
We are interested in the integral $$I \equiv \int_0^\Delta d\lambda \int_{\text{BZ}/2} d\bm{q} \,
\frac{\partial \Omega^{(\tau)}_{q_y,\lambda}}{\partial A_x}
\Biggr|_{\bm{A}=0},
\label{I-def}$$ where $$\frac{\partial \Omega^{(\tau)}_{q_y,\lambda}}{\partial A_x}
\biggr|_{\bm{A}=0} =
\frac{2q_x^2-q_y^2-\lambda^2}{(q_x^2+q_y^2+\lambda^2)^{5/2}}
.$$ Since we are focusing on the technical aspect here, for simplicity, we drop all the non-essential prefactors and, for definiteness, consider $\Delta>0$ in the remainder of this appendix. Computing by integrating first over the BZ and $\lambda$ last, we obtain
$$I_{\lambda\text{ last, }\square} = -2\,
\text{tan}^{-1} \!\! \left( \frac{\Delta}{ \sqrt{2w^2+\Delta^2} } \right).
\label{e222-cart}$$
With the prefactors, this is our result (12) in the original paper, where the BZ integral has been performed over a square-shaped, state-conserving BZ, as discussed in the paper. For completeness, if one opts to use a circularly shaped domain with cutoff momentum $q_c$ (the choice made in the Comment), the result for $I$ becomes $$I_{\lambda\text{ last, }\bigcirc} =
- \frac{\pi\Delta}{ 2\sqrt{q_c^2+\Delta^2} } .
\label{e222-polar}$$
On the other hand, if one performs the *adiabatic integral first* followed by that over the BZ, one obtains
\[lambda-first\] $$I_{\lambda\text{ first, }\square} =
\pi - 2\,\text{tan}^{-1} \left( \frac{\Delta}{ \sqrt{2w^2+\Delta^2} } \right)
,
\label{e222-cart-2}$$ for the square BZ, and $$I_{\lambda\text{ last, }\bigcirc} =
\frac{\pi}{2} - \frac{\pi\Delta}{ 2\sqrt{q_c^2+\Delta^2} }
\label{e222-polar-lambda}$$
for the circular BZ. Note that this last result is the one reported in Eq. (18) of the Comment once the prefactors are restored [@GapConvention]. Therefore, computing by performing the adiabatic integral *first* followed by that over the BZ, introduces a constant, independent of $\Delta$.
Mathematically, this dependence of the result on the order of the multiple integrations arises because Eq. does not converge absolutely and, consequently, it is sensitive to whether one performs the $q_x$ integration before or after the one over $\lambda$ (changing to polar coordinates does not change this, of course) [@Pathology]. This non-uniform convergence is, of course, a consequence of having erroneously used Leibniz’s rule when the integrand of is not a continuous function (incidentally, performing the $\lambda$ integral last, which is physically motivated, yields the correct result).
In order to avoid these pitfalls, the method of calculation used in our original paper keeps $A_x$ and $A_y$ finite until the triple integral that yields $\bm{P}$ is obtained. We only linearize the result in $\bm{A}$ afterwards. In this way, the result is mathematically well defined, irrespective of the sequence used for the triple integration [@Supplement].
Singularity of the Berry curvature {#app:singularity}
==================================
In spherical coordinates, $$\begin{aligned}
q_x{-}A_x&=&r\sin(\theta)\cos(\phi)\,,\nonumber\\
q_y{-}A_y&=&r\sin(\theta)\sin(\phi)\,,\\
\lambda&=&r\cos(\theta)\,,\nonumber\end{aligned}$$ the Berry curvature given in Eq. (\[BC\_app\]) reads $$\Omega_{q_y,\lambda}^{(\tau)}\propto
\frac{r\sin(\theta)\cos(\phi)}{r^3}=
\frac{\sin(\theta)\cos(\phi)}{r^2}\,.$$ This expression has a discontinuity at $r{=}0$, i.e., at $(q_x,\,q_y,\,\lambda){=} (A_x,\,A_y,\,0)$, which lies in the integration domain of Eq. (\[P-def-orig\]). The support of this point is $0$ but the Berry curvature diverges so one must, in principle, verify whether there is a finite contribution to the integral. This contribution can be checked as follows: $$\begin{aligned}
&&\lim_{{\delta}\to0}\int_0^{\delta}{\rm d}r\int_0^{\pi/2}
{\rm d}\theta\,r\int_0^{2\pi}{\rm d}\phi\,r\,\sin(\theta)\,
\Omega_{q_y,\lambda}^{(\tau)}\nonumber\\
&&\propto
\lim_{\delta\to0}\int_0^\delta\frac{r^2}{r^2}{\rm d}r
\cdot\int_0^{\pi/2}\sin^2(\theta)\,{\rm d}\theta\cdot\int_0^{2\pi}
\cos(\phi)\,{\rm d}\phi\nonumber\\
&&=\lim_{\delta\to0}(\delta-0)\cdot\frac{\pi}{4}\cdot0\nonumber\\
&&=0\,.\end{aligned}$$ This means that the singularity at the shifted Dirac points integrates to 0 and does not add any particular contribution to the result of Eq. (\[P-def-orig\]).
M. Droth, G. Burkard, and V. M. Pereira, Phys. Rev. B **94**, 075404 (2016).
R. Resta, Ferroelectrics **136**, 51 (1992), R. Resta, Rev. Mod. Phys. **66**, 899 (1994).
D. Vanderbilt, Phys. Rev. B **41**, 7892 (1990), R. D. King-Smith and D. Vanderbilt,Phys. Rev. B **47**, 1651 (1993).
S. M. Nakhmanson *et al.*, Phys. Rev. B **67**, 235406 (2003).
E. J. Mele and P. Král, Phys. Rev. Lett. **88**, 056803 (2002).
J. Li, Y. Wang, Z. Wang, J. Tan, B. Wang, and Y. Liu, Phys. Rev. B **98**, 167403 (2018).
H. Suzuura and T. Ando, Phys. Rev. B **65**, 235412 (2002).
When comparing, note that in our notation the gap is $\hbar v_F \Delta$, while it is simply represented by $\Delta$ in the notation of the Comment (we maintain the original notations of each).
See Supplemental Material for a verification of our calculations using *Mathematica*.
K.-A. N. Duerloo, M. T. Ong, and E. J. Reed, J. Phys. Chem. **3**, 2871 (2012).
Y. Wang, Z. Wang, J. Li, J. Tan, B. Wang, and Y. Liu, Phys. Rev. B **98**, 125402 (2018).
The integral in question has precisely the same pathology as the double integral $\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2}\,dx\,dy$, which is a common counterexample to the interchangeability in the order of integrations in the absence of absolute convergence. In this case, the result is $+\pi/4$ ($-\pi/4$) if $y$ ($x$) is integrated first.
$\text{}$
$\text{}$
$\text{}$
$\text{}$
$\text{}$
$\text{}$
$\text{}$
$\text{}$
|
---
abstract: 'In this work we incorporate the newest ISO results on the mid-infrared spectral-energy-distributions (MIR SEDs) of galaxies into models for the number counts and redshift distributions of MIR surveys. A three-component model, with empirically determined MIR SED templates of (1) a cirrus/PDR component (2) a starburst component and (3) an AGN component, is developed for infrared (3–120) SEDs of galaxies. The model includes a complete IRAS 25selected sample of 1406 local galaxies ($z \leq 0.1$; Shupe et al. 1998a). Results based on these 1406 spectra show that the MIR emission features cause significant effects on the redshift dependence of the K-corrections for fluxes in the WIRE 25band and ISOCAM 15band. This in turn will affect deep counts and redshift distributions in these two bands, as shown by the predictions of two evolution models (a luminosity evolution model with $L\propto (1+z)^3$ and a density evolution model with $\rho\propto (1+z)^4$). The dips-and-bumps on curves of MIR number counts, caused by the emission features, should be useful indicators of evolution mode. The strong emission features at $\sim 6$–8will help the detections of relatively high redshift ($z\sim 2$) galaxies in MIR surveys. On the other hand, determinations of the evolutionary rate based on the slope of source counts, and studies on the large scale structures using the redshift distribution of MIR sources, will have to treat the effects of the MIR emission features carefully. We have also estimated a 15local luminosity function from the predicted 15fluxes of the 1406 galaxies using the bivariate (15vs. 25luminosities) method. This luminosity function will improve our understanding of the ISOCAM 15surveys.'
author:
- 'Cong Xu, Perry B. Hacking, Fan Fang, David L. Shupe, Carol J. Lonsdale, Nanyao Y. Lu, George Helou,'
- 'Gordon J. Stacey'
- 'Matthew L. N. Ashby'
title: 'Emission Features and Source Counts of Galaxies in Mid-Infrared'
---
Introduction
============
Current research on cosmology is focused on the debate whether the star formation history of the universe has a sharp peak in the redshift range of $z \sim 1$ – 2 (Madau1996), as derived from the Hubble Space Telescope deep surveys (Williams1996; Griffiths1994) and from ground based redshift surveys (Lilly1995; Koo and Kron 1992), or the universe had been forming stars in a relatively constant rate since an early galaxy formation time ($z \sim 5$) until $z\sim 1$ (Franceschini1997; Rowan-Robinson1997), as suggested by the cosmic far-infrared (FIR) background studies (Puget1996; Guiderdoni1997) and studies on the metal abundance in Intra-Cluster Medium (Mushotzky and Loewenstein 1997).
The problem with the HST surveys and ground based redshift surveys is the dust extinction, which may hide much of the star formation in early universe from these UV/optical surveys. Dust extinction has little effect on mid-infrared (MIR, 4— 40) surveys. Compared to the FIR ($>$40) deep surveys, the MIR surveys also have the advantage of using larger detector arrays and having better resolutions; the latter is of particular importance because in these bands the depth of a survey is ultimately constrained by the confusion limit. Therefore, MIR surveys will play important roles in galaxy evolution studies. Indeed, currently several deep MIR surveys are either being carried out using ISO (Franceschini1996; Oliver1997), or soon to be launched such as WIRE (Hacking1996; Schember1996) and SIRTF (Cruikshank and Werner 1997).
On the other hand, it was known before ISO that in the wavelength range between 3–20there are several broad band features (see Puget and Léger 1989 for a review), which may contribute substantially to the IRAS 12flux of galaxies (Gillett1975; Phillips1984; Giard1989; Helou1991). The best candidates for the emitters of these features are some large molecules ($\sim$1 nm) of Polycyclic Aromatic Hydrocarbon (Léger and Puget 1984; Allamandola1985), and the features are often called PAH features. New ISO observations (Lu1996, 1998; Vigroux1996; Boulade1996) show further that these features are present in the MIR spectra of most galaxies, and their equivalent widths can be as large as $\sim$10, comparable to or even larger than typical band passes for MIR filters (e.g. the band pass of the ISOCAM 15filter is $\sim$6). This indicates that when any of these features moves in or out of the band pass of a MIR survey, because of the redshifts of galaxies, very significant K-corrections will occur. Interpretations of MIR survey counts have to take this fact into account. A pre-ISO attempt on this issue, based on the laboratory PAH spectra (instead of the spectra of galaxies), can be found in Maffei (1994).
It is the primary aim of this paper to develop a new, emiprical model for the MIR SED’s of galaxies, and incorprate it into the model for number counts of MIR surveys. The SED model is built upon available MIR spectra of galaxies taken from published ISO observations, and is applied to the 1406 galaxies (with $z<0.1$) of the 25sample of Shupe(1998a), selected from the IRAS Faint Point Source Catalog (Moshir1992). For each galaxy in the sample a model spectrum is obtained, which has the MIR emission features and which can reproduce the four observed IRAS fluxes. These 1406 spectra are used to determine the K-corrections and their dispersions in our model for predictions on number counts and redshift distributions of MIR surveys. The effects of the emission features on these predictions are investigated. Detailed comparisons of the model results with some ISOCAM MIR surveys (Oliver1997; Clements1998) and predictions for WIRE survey will be found in a separate paper (Hacking et al. 1998). The computer code of the model presented here is available upon request to CX.
The paper is arranged as follows: the SED model is developed in Section 2. A local luminosity function at 15, calculated from the predicted 15fluxes of the 1406 galaxies, is presented in section 3. In Section 4 we present the model for source counts and investigate the effects of the emission features on the MIR number counts. Section 5 is devoted to the discussion and Section 6 to the conclusion.
An empirical model for IR SEDs of galaxies
==========================================
Properties of MIR SEDs of galaxies
----------------------------------
The following properties of MIR SEDs of galaxies can be found from the literature:
1. The MIR SEDs of normal galaxies (without AGNs) in the $\sim$10region are not smooth. Indeed there has been evidence for this since the observations of Gillett(1975) of the 8–13spectra of M 82 and NGC 253, the two nearby prototype starburst galaxies. Later ground-based observations (Roche1991) and the IRAS low-resolution spectral observations (Cohen and Klein 1989) show further that the 11.3feature and the 8feature are very common in the 8 – 13spectra of ‘HII’ galaxies (i.e. starburst galaxies). Most recently, Lu(1996, 1998) obtained MIR spectra (3–11.7) of a sample of late-type galaxies using ISO. These galaxies are part of a larger set of star-forming galaxies, selected to cover a broad range of infrared and optical properties, in the ISO key project on normal galaxies (Helou1996). The spectra were taken with ISOPHT operated in its PHT-S spectroscopic mode (Lemke1996) with a triangular chopping on both sides of the galaxy on the sky. In the wavelength range covered by the observations, 3–11.7, several prominent broad band features (e.g. at 6.2, 7.7, 8.6and 11.3) are clearly visible in almost every spectrum in the sample. ISOCAM CVF spectrum of NGC 5195 (the small companion of M51) shows further that beyond the PHT-S long wavelength boundary there is still another broad feature at $\sim$12.7(close to the \[NII\] line at 12.8; Boulade1996). All of these features are presumably emission bands of PAH molecules corresponding to different C-C and C-H modes (Léger and Puget 1984; Allamandola1985).
2. Viewing through the gallery of ISO MIR spectra of galaxies (Lu1996, 1998; Boulade1996; Vigroux1996; Metcalfe1996; Acosta-Pulido1996), of Galactic star formation regions and reflection nebulae (Verstraete1996; Cesarsky1996a, 1996b; Boulanger1996), and of Galactic cirrus (Mattila1996), it appears that in general, the PAH features and the underlying continuum in the MIR band correlate tightly, indicating that they may have the same origin. Except for the HII regions and violent starburst regions in interacting galaxies, the shape of the MIR spectra of galaxies, of different Galactic nebulae and of Galactic cirrus are remarkably similar. However, in the core of Galactic HII regions (Cesarsky1996b; Verstraete1996) and in a violent starburst region in Arp 244 (the famous Antennae which is a local example of galaxy merger), another component of the continuum which increases sharply with wavelength appears (Cesarsky1996b; Vigroux1996; Verstraete1996). In the framework of the dust model of Désert et al. (1990), this latter component is likely to be due to the 3-dimensional very small grains heated by the combined heating of multiple photons (Boulanger1996). The characteristic temperature of this component is usually too low for it to be seen at wavelengths $<$20when the heating radiation intensity is below $\sim 10^4$ times the local ISRF (Cesarsky1996b). Noticeably, this component affects mostly the MIR emission at $\lambda >$12and not so much for the emission at shorter wavelengths as shown by the ISO PHT-S spectra (3 – 11.7) of galaxies in the sample of Lu(1998) which includes both quiescent and starburst galaxies.
3. From their ground based observations Roche(1991, see also Moorwood 1986) found that many active galactic nuclei (AGNs) show distinctively different MIR spectra from those of starburst galaxies, in particular the broad band emission features are absent. This is confirmed by new ISO observations (e.g. Lutz1997a; Genzel1998). Roche(1991) argued that it is the AGN itself, instead of other reasons such as the abundance of certain elements, that affects the dust emission process to suppress the MIR features either by alternating the grain composition within the nuclear region, perhaps by the selective destruction of grain species, or through very different excitation conditions.
Our model
---------
Our model is based on the following assumptions:
: The MIR SED (3–16) of galaxies has three components, each has a fixed SED given by the corresponding template (Figures 1–3): (1) a cirrus/PDR (Photo-Dissociation Region) component which is dominant in the MIR SEDs of most of normal galaxies, (2) a starburst component due to dust heated by the very intense radiation in starburst regions which is likely to be the dominant component for violent starburst galaxies such as Arp 220, and (3) an AGN component due to the dust emission associated with AGNs.
The adopted MIR spectrum (3–16) of the cirrus/PDR component is presented in Figure 1 (the solid curve). Lu(1998) gives two average spectra, one from a set of 10 “FIR-cold” galaxies with an IRAS 60–to–100flux density ratio, $f_{60\mu}/f_{100\mu}$, between 0.28 and 0.40; and one from a set of 9 “FIR-warm” galaxies with $0.60 <
f_{60\mu}/f_{100\mu} < 0.88$. The relative flux accuracy of these spectra should be good to about 20%. The shape of our cirrus/PDR template spectrum at wavelengths shorter than $\sim$12is identical to that of the average spectrum for the FIR-cold galaxies in Lu(1998). At longer wavelengths a tail is taken from the spectrum of NGC 5195 (Boulade1996). It can be seen that the two spectra agree reasonably well at the wavelengths covered by both.
In the wavelength range of 8.7–16the spectrum of the starburst component (Figure 2) is taken from the ISO-CAM CVF spectrum of the overlapping region (region A) in Arp 244 (Vigroux1996), the famous Antennae. This region is where the two disks crash, and where much of the star formation in this merging system is taking place. The emission features are still visible in the spectrum, but most of the energy is in the continuum which has a relatively steep slope. Shorter than 8.7, the shape of the spectrum of region A in Arp 244 is very close to that of the average spectrum for the FIR-warm galaxies in Lu(1998). Therefore shorter than 8.7the starburst component is taken from the latter (Fig.2) which extends to 3. Interestingly, longer than 8.7 the MIR continuum in the average spectrum of the FIR-warm galaxies of Luis not as strong as in the starburst region in Arp 244, suggesting that in these galaxies a significant fraction of the MIR emission is emitted outside the starburst regions (i.e. associated with the cirrus/PDR component).
It should be noted that in the short wavelength channel of ISOPHT-S ($<$5) the mean spectra of Lu(1998) are rather noisy, and show no significant evidence for any detection of features (e.g. the 3.3feature) at $\lambda <$5. Thus both cirrus/PDR template and the starburst template presented above are approximated by flat spectra in the wavelength range 3$ < \lambda <$5, determined by the average flux densities (over the same wavelength range) of the mean spectrum of the FIR-cold galaxies and of the mean spectrum of the FIR-warm galaxies of Lu(1998), respectively.
The template of the MIR SED of the AGN component (Figure 3) is based on the MIR spectrum of the Seyfert 2 nucleus of NGC 1068 (Lutz1997a). Because we are only interested in the broad band SED, the narrow emission lines in this wavelength range are not included.
: For a given galaxy with measured IRAS fluxes of $f_{25\mu}$ and $f_{60\mu}$, the 25flux of the AGN component can be determined using the following ‘color-method’: $$\rm f_{25\mu}^{AGN} = \cases{ f_{25\mu}
\;\; & $\rm (f_{60\mu}/f_{25\mu} \leq 2.5)$; \cr
f_{25\mu}\times {5 - f_{60\mu}/f_{25\mu}\over 5 - 2.5}
\;\; & $\rm (2.5 < f_{60\mu}/f_{25\mu} < 5)$; \cr
0 \;\; & $\rm (f_{60\mu}/f_{25\mu} \geq 5)$. \cr} \label{eq:agn1}$$
This assumption is based on the fact that galaxies with low $f_{60\mu}/f_{25\mu}$ ratios have a large chance of exhibiting Seyfert activity (de Grijp1985). In an optical spectroscopic survey for a FIR color selected sample of 358 extragalactic sources with $1 < f_{60\mu}/f_{25\mu} <3.72$ (i.e. $-1.5 < \alpha_{25,60} < 0$), Keel(1994) found that more than 60 percent of the sources are Active galaxies (including Seyferts, QSOs and BL Lacs). In Figure 4 is plotted the $f_{25\mu}/f_{12\mu}$ vs. $f_{25\mu}/f_{60\mu}$ color-color diagram of the IRAS 25selected sample of Shupe(1998a). Sources with known AGNs (identified using NED) are plotted with different symbols. Indeed we find that for $\log (f_{25\mu}/f_{60\mu})\geq -0.4$ (i.e. $f_{60\mu}/f_{25\mu}\leq 2.5$) the vast majority of the sources are Active galaxies. In particular, most of the QSOs are on the right side of this boundary. On the other hand, most of the normal galaxies (i.e. without AGN) have $\log (f_{25\mu}/f_{60\mu})\leq -0.3$ (i.e. $f_{60\mu}/f_{25\mu}\geq 5$). Interestingly, many galaxies with AGNs, in particular most of the LINERs, also have $\log (f_{25\mu}/f_{60\mu})\leq -0.3$. It is likely that for these sources the IR fluxes are mainly from the host galaxies and the AGN component is not significant.
Once $f_{25\mu}^{AGN}$ is determined, the 12flux of the AGN component is derived as follows:
$$\rm f_{12\mu}^{AGN} = \cases{0.6\; f_{25\mu}^{AGN}
\;\; & $\rm (f_{25\mu}^{AGN} < f_{25\mu}$ and
$\rm 0.6\; f_{25\mu}^{AGN} < f_{12\mu})$ \cr
f_{12\mu}
\;\; & (otherwise) \cr } \label{eq:agn2}
%(0.6\; f_{25\mu}^{AGN} \geq f_{12\mu})$. \cr } \label{eq:agn2}$$
The standard $f_{12\mu}/f_{25\mu}$ ratio of 0.6 for the AGN component is taken from that of the Seyfert 2 nucleus of NGC 1068 (Lutz 1997a).
Based on the IRAS data, Giuricin(1995) found that Seyfert 2 galaxies tend to have steeper continuum spectra between 12 and 25than those of Seyfert 1 galaxies. However, given the large beams of IRAS detectors, it is not clear whether this reflects the intrinsic difference between the two types of AGN or whether it is due to the difference in the star formation activity around Seyfert 1s and Seyfert 2s (Maiolino1995). In this work, we choose to use a single template (Figure 3) to model the MIR SED of both types of AGN.
: Define $$\rm f_{25\mu}^{G} = f_{25\mu} - f_{25\mu}^{AGN}$$ and $$\rm f_{12\mu}^{G} = f_{12\mu} - f_{12\mu}^{AGN}\;\; .$$ The 12fluxes of the starburst component and of the cirrus/PDR component can be determined by the color $f_{25\mu}^{G}/ f_{12\mu}^{G}$ through the following formulae: $$\rm f_{12\mu}^{SB} = \cases{f_{12\mu}^{G}
\;\; & ($f_{25\mu}^{G}/ f_{12\mu}^{G}\geq 6$); \cr
& \cr
{f_{25\mu}^{G}/ f_{12\mu}^{G} - 2\over
6 - 2} \;\;
& ($\rm 2 < f_{25\mu}^{G}/ f_{12\mu}^{G} < 6$); \hfil \cr
& \cr
0 \;\; & ($f_{25\mu}^{G}/ f_{12\mu}^{G} \leq 2 $) \cr} \label{eq:sb}$$ and $$\rm f_{12\mu}^{PDR} = f_{12\mu}^{G}-f_{12\mu}^{SB}. \label{eq:pdr}$$
In this assumption, we assign a significantly higher standard $f_{25\mu}/ f_{12\mu}$ color ratio to the starburst component ($f_{25\mu}^{SB}/ f_{12\mu}^{SB}=6$) than that of the cirrus/PDR component ($f_{25\mu}^{PDR}/ f_{12\mu}^{PDR}=2$), to be consistent with the fast-rising MIR SED of the starburst template (Figure 2). This is also consistent with the anti-correlation between $f_{12\mu}/ f_{25\mu}$ and $f_{60\mu}/ f_{100\mu}$ of galaxies (Helou 1986) given that the $f_{60\mu}/ f_{100\mu}$ ratio is a good star formation strength indicator (Bothun1989; Xu and De Zotti 1989). For example Arp 299 (NGC 3690/IC 694), a famous local example of galaxy mergers, has $f_{25\mu}/ f_{12\mu}=5.8$ and $f_{60\mu}/ f_{100\mu}=0.95$.
Given Equations \[eq:agn1\], \[eq:agn2\], \[eq:sb\], and \[eq:pdr\], we can calculate the 12fluxes of the three components for a given galaxy with measured $f_{12\mu}$, $f_{25\mu}$ and $f_{60\mu}$. These 12fluxes provide the absolute calibrations for the MIR SED templates of these components, the sum of which is the model MIR SED (3–16) for the galaxy in question.
There are no broad band features at wavelengths longer than 16in the IR emission of galaxies.
This assumption is based on the available IR spectra in the literature. In particular the ISOSWS spectrum of M 82 (Lutz1997b), which covers the wavelength range 3–40, does not show any broad band features beyond the 12.7feature. Consequently, for a given galaxy, the IR radiation spectrum at wavelengths longer than 16is determined in our SED model by fitting the 3 IRAS fluxes at 25, 60 and 100(taking into account the real responsivity functions of IRAS detectors), and the flux density at 16predicted by the sum of the three MIR components, using a simple cubic spline algorithm (obviously the continuum so determined joins at 16the model MIR SED). The advantage of this empirical model is that it preserves the observed flux measurements of each source and allows the inclusion of a large sample of galaxies in a wide range of observed SEDs.
Figure 5 is a flow chart showing how actually our model proceeds.
Tests of the SED model
----------------------
In Figure 6 the predicted MIR spectrum of M 82 and the observed ISOSWS spectrum of M 82 (Lutz1997b) are compared. Allowing for the small apertures of ISOSWS channels (ranging from $14''\times 20''$ to $17''\times 40''$), we have multiplied the IRAS fluxes of M 82 at 12, 25, 60, and 100by 0.38, 0.5, 0.5 and 0.38, respectively. Admittedly these factors are somewhat arbitrary because at these wavelengths no high resolution maps of M 82 are available for us to do aperture photometry. So they are chosen to give the best fit. Nevertheless they look reasonable. In particular, the two different factors (0.5 for $f_{25\mu}$and $f_{60\mu}$, 0.38 for $f_{12\mu}$ and $f_{100\mu}$) can be understood by means of the known anti-correlation in the $f_{12\mu}/f_{25\mu}$ vs. $f_{60\mu}/f_{100\mu}$ diagram (Helou 1986), which suggests that $f_{25\mu}$and $f_{60\mu}$ are more concentrated in the starburst nucleus of M 82. The good agreement between the predicted and observed MIR [*continuum*]{} is indeed due to the fine tuning of the aperture corrections, while the good agreements between the shapes of the MIR [*features*]{} (which are independent of the fitting of the continuum) on the predicted and observed spectra are truly remarkable.
We made a further test to determine whether the model reliably predicts galaxies’ MIR SEDs. Ashby(1999) observed with ISOPHT-P a sample of 29 FIR bright galaxies (f$_{60\mu}> 5.8$ Jy) in several MIR wavelength bands, including the PHT-P 16band ($52''$ aperture). Our SED model was applied to a subsample of 18 of the Ashbygalaxies. The subsample was selected based on four criteria: (1) the signal-to-noise ratio of the PHT-P 16flux $>3$; (2) detection of the galaxy by IRAS in both the 12 and 25bands; (3) optical extent $<2'$, or when in a galaxy pair the combined size of the pair is $<2'$, (4) the distance to the nearest neighbor (of similar optical brightness) $>4'$. The last two criteria ensure that the PHT-P observations include most of the emission from the region enclosed in the IRAS apertures ($\sim 4'$). Convolving the predicted spectra with the PHT-P 16 responsivity function, the 16fluxes are predicted and compared with the observed $f_{16\mu}$ (Figure 7). Given the uncertainty of the observed $f_{16\mu}$ (in most case $\sim 30\%$, Ashby et al. 1999), the agreement between the predicted and observed $f_{16\mu}$’s is quite good. The model spectra of the same galaxies are plotted in Figure 8.
The Sample
----------
The IRAS 25selected sample (complete down to $f_{25\mu}=0.25$ Jy) of Shupe(1998a) contains 1456 galaxies, all have measured redshifts and measured IRAS 25fluxes. In order to have a local sample free from significant evolutionary effects, our sample includes only the 1406 galaxies in Shupe et al’s sample which have redshifts $\leq 0.1$. Most of them are also detected by IRAS at 12, 60 and 100. At 12, 330 (out of 1406) galaxies are not detected by IRAS and have therefore only upper limits. For $f_{60\mu}$ and $f_{100\mu}$, 18 and 94 galaxies have upper limits, respectively. We assign a flux density $f_\nu =0.8\times$upper limit for these cases.
The SED model thus produces for each galaxy in the sample a SED from 3–120. As examples, the SEDs of 50 galaxies with $9 \leq
\log (\nu L_\nu (25\mu{\rm m})/L_\sun) \leq 10$ are plotted in Figure 9.
Local luminosity function at 15
===============================
Several MIR surveys (Oliver1997; Franceschini1996) are being carried out by ISOCAM in the 15band, the most sensitive one for starburst galaxies among ISOCAM filters. However, there is no 15local luminosity function (LLF) available in the literature, because there have been no complete samples of galaxies in 15band. Oliver(1997) used the 12LLF of Rush et al. (1993) in the interpretation of their 15deep survey. This may be problematic because different galaxies have different 15-to-12flux ratios.
We construct a 15LLF in this section exploiting the model SEDs of the 1406 galaxies in our sample. First, for each galaxy in the sample, an in-band flux at 15 is derived by convolving its SED with the responsivity functions of ISOCAM 15filter (13–18, ISOCAM User’s Manual). The monochromatic flux at 15is derived from this by assuming $\nu f_{\nu}=$constant within the bandpass. When there is only an upper limit at 12, instead of multiplying a by factor of 0.8 to get an approximation of the real flux as in the previous section, here we input the 12upper limit into the SED model and obtain an upper limit for the 15flux. The 15LLF can then be calculated from the predicted 15luminosities, and the corresponding upper limits, of these galaxies.
Since our sample is 25-flux limited, not 15-flux limited, the bivariate method is used in the calculation of the 15luminosity function. The basic formula is $$\Psi(L_{15\mu} > l_{15\mu}) =
\int \Theta(L_{15\mu} > l_{15\mu} |L_{25\mu})\; \rho(L_{25\mu})\;\;
d\, L_{25\mu}$$ here $\Psi$ is the 15 luminosity function, $\rho$ the 25 luminosity function (Shupe1998a), and $\Theta$ the conditional probability function for a galaxy with given 25luminosity $L_{25\mu}$ to have the 15$\mu$ luminosity $L_{15\mu}$ brighter than $l_{15\mu}$. The integration is over the entire range of the 25luminosity. The derivative of $\Psi$ gives the differential luminosity function: $$\phi(L_{15\mu}) =
\int {\partial\Theta\over \partial L_{15\mu}} \rho(L_{25\mu})\;\;
d\, L_{25\mu}\;\; . \label{eq:lf}$$
In practice, the luminosity functions are calculated in the domain of the logarithms of the luminosities. Hence the $L_{15\mu}$’s and the $L_{25\mu}$’s in above formulae should be replaced by the corresponding logarithms. The sample is binned for both $\log (L_{25\mu})$ and $\log (L_{15\mu})$, and Eq(\[eq:lf\]) can be approximated by $$\phi[\log (L_{15\mu})=L_i] =\sum_j P_{i,j} \rho[\log(L_{25\mu})=L_j]\;\;
\Delta_j/\delta_i$$ where $L_j$ and $\Delta_j$ are the center and the width of the j-th bin of $\log (L_{25\mu})$, $L_i$ and $\delta_i$ are the center and the width of the i-th bin of $\log (L_{15\mu})$, and $P_{i,j}$ is the probability of galaxies in the j-th bin of $\log (L_{25\mu})$ (i.e. $L_j-0.5\Delta_j \leq log(L_{25\mu}) < L_j+0.5\Delta_j$) to have $\log (L_{15\mu})$ in the bin $L_i$ ($L_i-0.5\delta_i \leq \log (L_{15\mu}) < L_i+0.5\delta_i$).
In order to take into account the information contained in the upper limits, the Kaplan-Meier estimator (Kaplan and Meier 1958) is applied in the calculation of $P_{i,j}$ and its variance. In what follows we give a brief description of the algorithm.
For the subsample of galaxies in the j-th bin of $\log (L_{25\mu})$, define $F_{i-1}$ as the probability function $$F_{i-1}=P[\log(L_{15\mu})<L_i+0.5\delta_i]$$ and $$F_{i}=P[\log(L_{15\mu})<L_i-0.5\delta_i]\;\; .$$ So, $$P_{i,j}=F_{i-1}-F_{i}$$ and $$Var(P_{i,j})=Var(F_{i-1})+Var(F_i)-2\times Covar(F_{i-1},F_i)\;\; .$$ The algorithm for calculating the KM estimator of $F_{i-1}, F_i, Var(F_{i-1})$ and $Var(F_i)$ can be found in Feigelson and Nelson (1985), while the KM estimator for the covariance between $F_{i-1}, F_i$ (for $F_i > 0$) is (Akritas and LaValley 1997) $$Covar(F_{i-1},F_i) = {F_{i-1}\over F_i} Var(F_i)\;\; .$$
Finally, the variance of $\phi$ is $$\begin{aligned}
Var[\phi(\log(L_{15\mu})=L_i)] = \sum_j
\{ & Var(P_{i,j})\rho[\log(L_{25\mu})=L_j] + \nonumber \\
& P_{i,j}Var[\rho(\log(L_{25\mu})=L_j)] \}
(\Delta_j/\delta_i)^2 \; .\end{aligned}$$ The resultant 15differential luminosity function is given in Table 1 and plotted in Figure 10. A least-squares fit to the [*integral*]{} 15luminosity function by a smooth function with the form (Yahil1991) $$\Psi(L) = C \left(L\over L_\star \right)^{-\alpha}
\left(1+L\over L_\star\right)^{-\beta}$$ is carried out. The parameters of this fitting function are given in Table 2. The [*differential*]{} 15luminosity function specified by these parameters is given in Figure 10 by the solid line.
MIR number counts model and effects of emission features
========================================================
Model for number counts
-----------------------
Our model for galaxy counts as a function of the limiting MIR flux and redshift follows the prescription of Condon (1984; see also Hacking1987; Hacking & Soifer 1991). Here we give a brief outline of the formulation. Details can be found in the above references.
We consider only one set of cosmological parameters, namely $H_0=75$, $\Omega = 1$ ($q=0.5$), and the cosmological constant $\Lambda=0$.
Let $\rho (L)$ be the local differential luminosity function (number of galaxies in unit comoving volume and unit $\log(L)$ interval). When the evolutionary effects are taken into account, the LF at redshift $z$ is $$\rho' (L,z) = G(z)\, \rho \left( {L\over F(z)} \right);$$ where $G(z)$ is the density evolution function and $F(z)$ the luminosity evolution function.
The monochromatic luminosity in the rest frame is: $$L_\nu = f_\nu 4\pi D_{L}^2 \times 10^{0.4K}, \label{eq:l}$$ where $f_\nu$ (hereafter f) is the observed flux density, $z$ the redshift, $D_L$ the luminosity distance: $$D_L={2c\over H_0}\; (1+z)[1-(1+z)^{-0.5}]\;\; ,$$ and $K$ the standard K-correction (Lang 1980): $$K=2.5\log(1+z)+2.5\log{\int_{\lambda_1}^{\lambda_2}
S(\lambda)R(\lambda)\; d\lambda\over \int_{\lambda_1}^{\lambda_2}
S(\lambda/(1+z))R(\lambda)\; d\lambda} \; . \label{eq:k}$$ The derivative of the comoving volume vs. redshift $z$ is $$\begin{aligned}
{dV\over dz} & = & 4\pi D_A^2 \;\; {d\; D_A\over d\; z} \nonumber \\
& = & 4\pi D_A^2 \; {c\over H_0(1+z)^{1.5}} \; . \label{eq:dv}\end{aligned}$$ where $D_A$ is the angular distance: $$D_A= {2c \, [1-(1+z)^{-0.5}] \over H_0} \;\; .$$
By definition the K-correction is a function of the redshift. However, for a given bandpass ($R(\lambda)$), the K correction also depends on the detailed spectrum ($S(\lambda)$), which varies from galaxy to galaxy. Therefore for the model it is more appropriate to define the K-correction as a random variable with its [*probability distribution*]{} depending on the redshift and on some other observables. Our prescription for the K-correction explicitly includes a luminosity dependence because IRAS data have shown that the $f_{12\mu}/f_{25\mu}$ color (and therefore the MIR SED) is a strong function of the MIR luminosity (Fang1998). In this work we choose to approximate the K distribution by a Gaussian: $$U(K,L,z) = {1\over \sqrt{2\pi}\sigma_K} \;
\exp{-(K-K_0)^2\over 2\sigma^2_K} \label{eq:u}$$ where the mean $K_0$ and the dispersion $\sigma_K$ are dependent on the [*luminosity*]{} $L$ and the redshift $z$. We have also tried a prescription where the K-correction is a function of the FIR-to-blue luminosity ratio instead of the MIR luminosity $L$. No significant changes are found in the results for the number counts.
Given the above definitions, we have: $$\eta (L,z,K)\; d\log (L)\; dz\; dK = \rho' (L,z)
\; U(K,L,z)\; {dV\over dz}\; d\log(L)\; dz \; dK$$ where $\eta$ is the counts per unit $\log(L)$ interval per unit z interval per unit $K$ interval. Substituting the variable $L$ by $f$ according to Eq(\[eq:l\]) for given $z$ and K, and integrating over K, one obtains the prediction for the differential counts per unit $\log(f)$ interval and per unit $z$ interval: $$\xi (f,z) = \int_{-\infty}^{\infty}
\rho' [(L(f,z,K),z]
\; U[K,L(f,z,K),z] {dV\over dz} \; dK \;\; . \label{eq:xi}$$
K-corrections
-------------
The narrower the bandpass[^1], the more significant the effect of the emission features on the K-correction. Here we compare the predicted K-corrections for three bandpasses: ISOCAM 12band, ISOCAM 15band (ISOCAM User’s Manual), and WIRE 25band (Shupe et al. 1998b). The responsivity of these filters are plotted in Figure 11.
For each of the three bands, K-corrections at different $z$ are calculated for each of the 1406 galaxies in our sample using its model spectrum and Eq(\[eq:k\]). Then the sample is binned according to the monochromatic luminosity (rest frame) at the effective wavelength of the band, and the means and the dispersions of the K-corrections at different $z$ are calculated for the subsample of galaxies in each bin, which are used as estimators of the $K_0(L,z)$ and $\sigma_K (L,z)$ in Eq(\[eq:u\]).
In Figures 12a, 12b and 12c the mean $K_0$ as function of redshift $z$ for galaxies with different luminosities are plotted for the three filters, respectively, and the dispersions $\sigma_K$ are plotted as error bars. And in Figures 12d, 12e and 12f, the dependences of $K_0$ and $\sigma_K$ on luminosity are plotted for $z=0.5$, $z=1$ and $z=2$ for the same filters. The K-$z$ curves for WIRE 25band flux demonstrate two dips near $z=1$ and $z=2$, which are due to the MIR emission features at $\sim 11$–12and at $\sim 6$–8(see Figures 1 and 2), respectively. The peak around $z=1.5$ is caused by the trough at $\sim 9$–10in MIR SEDs of galaxies, which may partially be due to the silicate absorption. For the ISOCAM 15band flux, the bump caused by this SED trough moves to $z\sim 0.5$ and the emission features at $7$–8cause a dip around $z=1$. The K-$z$ curves for the ISOCAM 12band (the most broad band among the three) are relatively smooth.
Number counts
-------------
Through K-corrections, the MIR features affect the number counts and the redshift distributions of deep surveys. Figures 13a, 13b and 13c show Euclidean normalized differential counts predicted by different models of the cosmological evolution of galaxies for ISOCAM 12, ISOCAM 15and WIRE 25bands. The $L\propto (1+z)^3$ model is chosen because it fits the IRAS 60counts (Pearson and Rowan-Robinson 1996) well, and the results from the $\rho\propto (1+z)^4$ model are presented to highlight the difference between a density evolution model and a luminosity evolution model. The latest local luminosity functions (LLFs) at 12(Fang1998) and 25(Shupe1998a) which improve upon the earlier LLFs in these bands (Soifer and Neugebauer 1991; Rush1993), and the 15LLF presented in Section 3 are used in the calculations of model predictions.
In order to demonstrate the effects of the MIR features on the number counts, in Figures 13d, 13e and 13f we show the Euclidean normalized differential counts predicted by evolution models based on model spectra of galaxies [**without**]{} the MIR features, which are calculated using the smooth fits (power-laws) of the templates of the three components in Section 2.
In all of the three bands considered, the effects of the MIR features on the number counts predicted by the density evolution model ($\rho\propto (1+z)^4$) are significant. In Figures 13d, 13e, and 13f (no MIR features) the curves of predicted number counts by the density evolution model are rather flat, while in Figures 13a, 13b, and 13c the corresponding curves show dips and bumps caused by the MIR features. For example in Figure 13b (15) the curve of number counts predicted by the density evolution model shows two shallow bumps which are due to the emission features in the 11–13wavelength range and those in the 6–8.5 wavelength range, respectively. The prominent bump at $f_{25\mu}
\sim 0.1$ mJy in Figure 13c (25) is due to the features in both the 6–8.5wavelength range and the 11–13wavelength range (see the discussion of Figure 14). It appears that in all three bands the MIR features cause only slight distortions on the number counts predicted by the no evolution models.
There is a broad peak on the number counts predicted by the luminosity evolution model ($L\propto (1+z)^3$) in all six panels including those for models without the emission features. Hence this peak is not due to the emission features. Instead it is a combined effect caused by other two factors, one related to the evolutionary model and the other to the adopted cosmological model ($q_0=0.5$): The strong evolution ($L\propto (1+z)^3$) ensures that before the depth of a survey getting close to the ‘edge’ of the visible universe (i.e. $z\lsim 1$) the number counts increase much faster than the Euclidean predictions. Then, when the flux level becomes so faint that the characteristic redshifts of galaxies are significantly larger than unity (for a given flux, the stronger the luminosity evolution, the larger the characteristic redshift), the slope of $dV/dz$ (Eq(\[eq:dv\])) starts to decrease rapidly with increasing $z$ (decreasing flux), and the number counts drop significantly below the Euclidean predictions.
The effects of the MIR features on the number counts predicted by the luminosity evolution model look moderate. The peaks in Figure 13a (12) and in Figure 13b (15) are slightly narrower than those in Figure 13d and Figure 13e, and the peak in Figure 13c (25) is slightly broader than that in Figure 13f.
In Figures 14a–f, we compare the redshift distributions of sources at the flux level of $f_\nu=0.5$ mJy, predicted by different evolutionary models. Note that these distributions are for galaxies [**at**]{} this flux level, rather than for galaxies in samples brighter than this flux. The flux $f_\nu=0.5$ mJy is about the location of the peaks on the predicted number counts of the luminosity evolution model (Figure 13), and it is close to the flux limits of all ISOCAM deep surveys (Rowan-Robinson1997; Franceschini1996) and that of the planned WIRE survey (Hacking1996).
In all the three bands considered, the no evolution model (both with and without MIR features) predicts the mode of redshift distribution of galaxies with $f_\nu=0.5$ mJy at $z\sim 0.2$–0.4. This highlights the reason why the effects of the MIR features are so weak on the number counts predicted by this model: because the high redshift galaxies never contribute significantly to the number counts, the K-correction (through which the MIR features affect number counts) is never significant.
On the other hand, the MIR features add prominent dips and bumps on the redshift distributions predicted by the density evolution model. For example the sharp peak at $z\sim 1$ on the dotted line (density evolution model) in Fig.14c (25$\mu m$) is due to the features in in the 11 — 13$\mu m$ wavelength range. This peak (after Euclidian normalization) reaches its maximum at the flux level when the 11 — 13$\mu m$ features of the $L_\star$ galaxies are redshifted into the 25filter ($L_\star$ does not change with redshift in the density evolution model, hence for a given flux the $L_\star$ galaxies have the same redshift). This happens at $f_{25\mu m}\sim 0.1 mJy$, and therefore a strong bump appears at this flux level on the 25number counts predicted by the density evolution model (Fig.13c). At fainter flux level ($f_{25\mu m}\sim 30 \mu Jy$) the bump is taken over by sources of $z\sim 2$ for which the strong emission features in the 6–8.5wavelength range are in the bandpass of the WIRE 25$\mu m$ camera.
In Figures 14d, 14e and 14f (no MIR features), redshift distributions predicted by the luminosity evolution model are much broader than those predicted by the density evolution model and by the no evolution model. The reason for this difference is the following: When the K-correction is insignificant, the shape of the redshift distribution is essentially determined by the product of the luminosity function and the $dV/dz$ (c.f. Eq(\[eq:xi\])). Thus the redshift distributions predicted both by the density evolution model and by the no evolution model resemble the ‘visibility function’ of a local sample (e.g. Shupe1998a), with a sharp peak centered at the redshift of the $L_\star$ galaxies (for the given flux). On the other hand, in the luminosity evolution model, the $L_\star$ increases rapidly with redshift, causing a much broader peak in the redshift distribution (e.g. Figure 14f). It is because of this broadness of the redshift distributions that the effects of the MIR features on the number counts predicted by the luminosity evolution model (Figure 13) are relatively weak, in particular compared to those predicted by the density evolution model (Figure 13): the effects are largely smeared out by including galaxies in a wide range of redshifts for a given flux.
Discussion
==========
MIR emission features, if they exist in the IR SEDs of distant galaxies as in those of the local galaxies, can cause significant effects on the K-corrections (Figure 12), especially when the bandpass is comparable to the widths of the features (e.g. ISOCAM 15and WIRE 25bands). This affects the predictions on MIR number counts and the redshift distributions of different evolution models. The dips-and-bumps caused by the MIR features on the curves of the Euclidean normalized number counts will provide very useful diagnostic indicators of the type of evolution that is occurring, which may help to distinguish the density evolution model from luminosity evolution model when the redshifts are not available. On the other hand, the fluctuations in the slope of the number counts caused by the emission features should be taken into account in any attempt of determining the evolutionary rate from the [*slope*]{}.
Dramatic features are caused by the MIR features on the redshift distribution, suggesting that certain selection effects in redshift space are expected in MIR surveys. This may indeed be beneficial rather than annoying. For example, because of the strong emission features at $\sim 6$–8, the WIRE 25survey may detect a large number of galaxies of $z\lsim 3$. However, in studies on large scale structures based on deep MIR samples, these features must be separated from those caused by large scale inhomogeneities.
Summary and Conclusions
=======================
Our main results are summarized as follows:
1. An empirical, three-component model for the MIR SEDs of galaxies is developed. The templates of (1) a cirrus/PDR component, (2) a starburst component and (3) an AGN component are constructed from published ISO observations of MIR spectra of galaxies. Tests show that the model can reproduce the MIR spectrum of M 82 and the ISOPHT-P 16fluxes of a sample of galaxies quite well.
2. The model is applied to a sample of 1406 local galaxies ($z \leq 0.1$; Shupe et al. 1988a). Treating the K-correction as a random variable, its probability distribution (approximated by Gaussian) as a function of the redshift $z$ and of the monochromatic luminosity $L_\nu$ are calculated for three different bandpasses (WIRE 25, ISOCAM 15and ISOCAM 12) using these 1406 spectra.
3. Effects of MIR emission features on the MIR source counts and on the redshift distributions are studied for the same MIR bands assuming three different evolution models (no-evolution, $L\propto (1+z)^3$ and $\rho\propto (1+z)^4$). It is found that predictions by luminosity evolution model ($L\propto (1+z)^3$) and density evolution model ($\rho\propto (1+z)^4$) for source counts and redshift distributions in the WIRE 25band and the ISOCAM 15band are affected by the MIR emission features significantly, showing dips and bumps which can be identified as being caused by specific MIR emission features.
4. We point out that the dips-and-bumps on curves of MIR number counts will be useful indicators of evolution mode. The strong emission features at $\sim 6$–8will help the detections (especially in the WIRE 25survey) of relatively high redshift ($z\sim 2$) galaxies. On the other hand determinations of the evolutionary rate based on the slope of source counts, and studies on the large scale structures using the redshift distribution of MIR sources will have to treat the effects of the MIR emission features carefully.
5. A 15local luminosity function is calculated from the predicted 15fluxes of the 1406 galaxies using the bivariate (15vs. 25luminosities) method. This luminosity function will improve our understanding of the ISOCAM 15surveys.
We are grateful to James Peterson for providing the responsivity curve of WIRE 25camera in electronic form. CX thanks Michael Akritas and Eric Feigelson for stimulating discussions on the Kaplan Meier estimator, and Francois-Xavier Désert for helpful comments on an earlier version of this paper. NED is supported by NASA at IPAC. This work is supported in part by a grant from NASA’s Astrophysics Data Program.
Acosta-Pulido, J.A., Klaas, U., Laureijs, R.J. 1996, , 315, L121.
Akritas, M.G. and LaValley, M.P. (1997), In [*Handbook of Statistics, Vol. 15*]{}, eds. G. S. Maddala and C. R. Rao, 551.
Allamandola, L.J., Tielens, A.G.G.M., Barker, J.R. 1985, , 290, L25.
Ashby, M.L.N.1999, in preparation.
Bothun, G.D., Lonsdale, C.J., Rice, W. 1989, , 341, 129.
Boulade, O., Sauvage, M., Altieri, B. 1996, , 315, L85.
Boulanger, F., Reach, W.T., Abergel, A. 1996, , 315, L325.
Cesarsky, D., Lequeux, J., Abergel, A. 1996a, , 315, L305.
Cesarsky, D., Lequeux, J., Abergel, A. 1996b, , 315, L309.
Clements, D.1998, preprint.
Cohen, M. and Volk, K. 1989, AJ, 98, 1563.
Condon, J.J. 1984, , 287, 461.
Désert, F.-X., Boulanger, F., and Puget, J.L. 1990, , 237, 215.
de Grijp, M.H.K., Miley, K.K., Lub, J., and de Jong, T. 1985, Nature, 314, 240.
Fang, F., Shupe, D. L., Xu, C., and Hacking, P. B. 1998, , in press (astro-ph/9803162).
Feigelson, E. D. and Nelson, P. I. 1985, , 293, 192.
Franceschini, A.,(1997), preprint (astro-ph/9707080)
Franceschini, A., Cesarsky, C., and Rowan-Robinson, M. 1996, Memorie della Societa Astronomica Italiana.
Genzel, R., D. Lutz, E. Sturn, 1998, ApJ, in press (in http://www.mpe-garching.mpg.de/ISO/preprint).
Giard, M., Pajot, F., Caux, E., Lamarre, J.M., and Serra, G. 1989, A&A, 215, 92.
Gillett, F.C., Kleinmann, D.E., Wright, E.L., and Capps, R.W. 1975, , 198, L65.
Giuricin, G., Madirossian, F., and Mezzutti, M. 1995, , 446, 550.
Griffiths, R.E.1994, , 437, 67.
Guiderdoni, B., Bouchet, F., Puget, J.-L., Lagache, G., and Hivon, E. 1997, Nature, 390, 257.
Hacking, P. B.1996, The Wide-Field Infrared Explorer (WIRE) Mission, to appear in International Symposium on Diffuse Infrared Radiation and the IRTS, Institute of Space and Astronomical Sciences, Sagamihara, Japan.
Hacking, P. B.1998, in preparation.
Hacking, P. B., Condon, J. J., and Houck, J. R. 1987, , 316, L15.
Hacking, P. B., and Soifer, B. T. 1991, , 367, L49.
Helou, G. 1986, , 311, L33.
Helou, G., Ryter, C., and Soifer, B.T. 1991, , 376, 505
Helou, G., Malhotra, S., Beichman, C.A. 1996, , 315, L157.
Kaplan, E.L. and Meier, P. 1958, [*J. Am. Stat. Assoc.*]{}, 53, 457.
Keel, W.C., de Grijp, M.H.K., Miley, K.K., and Zheng, W. 1994, A&A, 283, 791.
Koo, D.G., and Kron. R.G. 1992, , 30, 613
Lang, K.R. 1980, Astrophysical Formulae, Springer-Verlag Berlin Heidelberg.
Léger, A. and Puget, J. L. 1984, , 137, L5.
Lemke, D., Klaas, U., Abolins, J.1996, , 315, L64.
Lu, N.Y., Helou, G., Beichman, C.A. 1996, , 28, 1356.
Lu, N.Y,1998, in preparation.
Lutz, D., Genzel, R., Kuntze, D., 1997b, in [*Extragalactic Astronomy in the Infrared*]{}, Proceedings of the XVIIth Moriond Astrophysical Meeting, eds. G.A. Mamon, T.X. Thuan and J.T. Thanh Van; p131.
Lutz, D., Genzel, R., Sturm, E., 1997a, A&SS, in press (in http://www.mpe-garching.mpg.de/ISO/preprint.html).
Madau, P., Ferguson, H.C., Dickinson, M.E., Giavalisco, M., Steidel, C., and Fruchter, A. 1996, . 283, 1388.
Maiolino, R., Ruiz, M., Rieke, G.H., and Keller, L.D. 1995, , 446, 561.
Maffei, B. 1994, thesis, “Etude des galaxies dans le domaine submillimetrique et realisation d’un spectrometre refroidi”, Institut d’Astrophysique Spatiale, Univ. Paris XI.
Moorwood, A.F.M. (1986), , 137, L5.
Mattila, K., Lemke, D., Haikala, L.K. 1996, , 315, L353.
Metcalfe, L., Steel, S.J., Barr, P. et al. 1996, , 315, L105.
Moshir M.,1992, IRAS Faint Source Catalog Explanatory Supplement, Version 2.
Mushotzky, R.F, and Loewenstein, M. 1997, ApJL, 481L.
Oliver, S., Goldschmidt, P., Franceschini, A., et al. 1997, , 289, 471.
Pearson, G., and Rowan-Robinson, M. 1996, , 283, 174.
Phillips, M.M., Aitken, D.K., and Roche, P.F. 1984, , 207, 35.
Puget, J.L., and Léger, A.,1989, , 27, 161.
Planesas, P., Mirabel, I.F., Sanders, D.B. 1991, , 370, 172.
Roche, P.F., Aitken, D.K., Smith, C.H., and Martin J.W. 1991, , 248, 606.
Rowan-Robinson, M., Mann, R.G., Oliver, S.J., et al. 1997, MNRAS, 289, 490.
Rush, B., Malkan, M.A., Spinoglio, L. 1993, ApJS, 89, 1.
Schember, H., Kemp, J., Ames, H., Hacking, P., Herter, T., Fafaul, B. Everett, D., Sparr, L. 1996, “The Wide-Field Infrared Explorer (WIRE)”, SPIE Proceedings, 2744, Infrared Technology and Applications XXII, p. 751-760.
Shupe, D. L., Fang, F., Hacking, P. B., and Huchra, J. P. 1998a, , in press (astro-ph/9803149).
Shupe, D.L., Larsen, M., Sargent, S., Peterson, J., Tansock, J., Luchik, T., Hacking, P., Herter, T. 1998b, in Space Telescopes V, Vol 3356, Proc. SPIE, ed. P.Y. Bely and J.B. Breckinridge (Bellingham, WA:SPIE), in press.
Soifer, B. T., and Neugebauer, G., 1991, , 101, 354.
Vigroux, L., Mirabel, F., Altiéri, B. et al. 1996, , 315, L93.
Verstraete, L., Puget, J.L, Falgarone, E. et al. 1996, , 315, L337.
Cruikshank, D.P. and Werner, M.W. 1997, in “Planets Beyond the Solar System”, ed. D.R. Soderblom, Ast. Soc. Pacific Conference Series, 119, 223.
Williams, S.E.1996, , 112, 1335.
Xu, C. and De Zotti, G. 1989, A&A, 225, 12.
Yahil, A., Strauss, M., Davis, M., & Huchra, J. P. 1991, , 372, 380.
[ccc]{} 7.7 & -2.23 & -2.82 8.1 & -2.88 & -3.34 8.5 & -2.76 & -3.43 8.9 & -2.88 & -3.67 9.3 & -3.13 & -4.06 9.7 & -3.52 & -4.46 10.1 & -4.18 & -5.05 10.5 & -5.06 & -5.80 10.9 & -5.93 & -6.58 11.3 & -7.00 & -7.46 11.7 & -7.91 & -7.91
2truecm
[cccc]{} $ 0.47\pm 0.13 $ & $ 2.20\pm 0.13$ & $ 9.62 \pm 0.15 $ & $-2.93773\pm 0.23 $
[^1]: The width of a bandpass is better defined by $log(\lambda_2) - log(\lambda_1)$, which is not affected by the redshift.
|
---
abstract: 'The results of extensive histogram cluster heat-bath Monte Carlo simulations on the critical behavior of the quasi-one dimensional Ising antiferromagnet on a stacked triangular lattice are presented. A small applied field is shown to induce a crossover from XY universality to mean-field tricritical behavior. Experimental estimates of critical exponents suggest that these two types of phase transitions are observed in $S=1$ CsNiCl$_3$ and $S=1/2$ CsCoBr$_3$, respectively. The present results demonstrate that this difference can be explained by an unusual staggered magnetic field arising from quantum exchange mixing previously proposed to account for spin excitations in $S=1/2$ quasi-one-dimensional Ising antiferromagnets.'
author:
- 'E. Meloche and M. L. Plumer'
title: 'Field induced tricritical behavior in the $S=1/2$ quasi one-dimensional frustrated Ising antiferromagnet'
---
Meaningful comparisons between experimental and theoretical results for critical exponents can be notoriously difficult due to inadequacy of standard techniques to sample thermodynamic quantities close enough to the phase transition region. It is often the case that small perturbations such as weak anisotropy govern the universality of critical phenomena but only very close to the critical temperature ($T_N$), rendering access through simulations or measurements problematic. Such issues are exacerbated by critical behavior thought to be influenced by magnetic frustration [@diep]. Additional complications can arise in model systems where exchange interactions are strongly anisotropic or in cases where quantum effects govern spin dynamics. The Ising antiferromagnet on a stacked triangular lattice (ISTAF) with strong c-axis exchange and weak inter-chain exchange interactions is an example of such a model system. Many quasi-one-dimensional (1D) ABX$_3$ compounds exhibit strong short-range intra-chain magnetic order (SRO) at temperatures well above the onset of true three-dimensional (3D) long range order (LRO) [@collins]. The present work demonstrates that quantum effects which govern 1D spin dynamics can impact effective critical properties of the 3D system.
Exchange mixing due to an unquenched orbital moment and crystal field effects in quasi-1D $S=1/2$ ISTAF’s has been proposed to give rise to an unusual contribution to the effective Hamiltonian in the form of a staggered magnetic field with a periodicity of two along the c-axis. This term has been employed to explain low temperature neutron scattering spectra in CsCoCl$_3$ and CsCoBr$_3$ [@Nagler1] and Raman data in TlCoCl$_3$ [@kuroe]. Invoking such an effect relies on the well-developed SRO intrinsic to quasi-1D magnetic systems at low T. Its applicability has been questioned, however, due to the fact that the inclusion of such a term in the model Hamiltonian essentially assumes 1D LRO and is thus a mean-field approximation inappropriate for soliton dynamics [@goff]. The impact of this field term on the measurements of [*effective*]{} critical behavior observed close to $T_N$ is not subject to this restriction and deserves consideration even though it may be strictly irrelevant in the renormalization group sense [*at*]{} $T_N$. In this work, the effect of this quantum staggered field in the $S=1/2$ quasi-1D ISTAF is mimicked in Monte Carlo (MC) simulations by considering the influence of a small uniform magnetic field applied to a model with ferromagnetic intra-chain interactions with AF inter-chain coupling. No such effective quantum staggered field is predicted to occur in the case of non-$S=1/2$ STAF’s such as ($S=1$) CsNiCl$_3$ which has been shown to exhibit integer-spin Haldane-gap phenomena [@buyers].
Critical behavior of the ISTAF have been investigated using numerous theoretical approaches, MC simulations and experimentally. Despite the large number of studies there is still no clear consensus in the literature regarding the critical exponents characterizing the phase transition between the partially disordered phase (PD) where one of the three sublattices remains disordered, and the paramagnetic phase, at $T_N$. Most modeling results [@collins; @Plumer1] support the notion from symmetry arguments that the transition at $T_N$ belongs to the $XY$ universality class [@blankschtein]. There have been suggestions, however, from several MC simulations that critical exponents are close to mean-field tricritcal [@Heinonen1; @Koseki1], but limited statistics and data analysis were used in these investigations. From the experimental standpoint, a number of past neutron scattering studies of CsCoBr$_3$ and CsCoCl$_3$ [@collins] have obtained sets of critical exponents that largely support the idea of $XY$ critical behavior. Intriguingly, recent high resolution neutron scattering experiments on CsCoBr$_3$ by Mao [*et al.*]{} [@Mao1] revealed results which suggest with tricritical mean-field behavior and it was speculated that this could be attributed to the anisotropic nature of the exchange interaction or a consequence of the quantum nature of the $S=1/2$ spins. Experimental data on weakly axial $S=1$ CsNiCl$_3$ with a quenched orbital moment [@buyers] are consistent with $XY$ universality [@collins; @guy2]. Both types of experimental systems, strongly Ising and weakly Ising, exhibit magnetic transitions at $T_N$ with the same symmetry and thus should belong to the same universality class.
Motivated by the new experimental data of Mao [*et al.*]{}, extensive MC simulations using the efficient Cluster Heat Bath algorithm (CHB)[@Koseki2; @Matsubara1], combined with the histogram method[@Ferrenberg1], were used here to investigate the effects of anisotropic exchange as well as a small applied field on the critical properties of the classical ISTAF. This work serves to the extend previous MC simulations of the ISTAF mentioned above, but especially that of Koseki [*et al.*]{} [@Koseki1], where the CHB was used with strongly anisotropic exchange but with no applied field, as well as that of Netz and Berker [@Netz1] and Plumer and Mailhot [@Plumer1] who considered *isotropic* exchange interactions with a nonzero applied field. The present results also compliment MC simulations on the frustrated XY-STAF where anisotropy in the exchange interactions was shown to induce a first-order transition [@Plumer2] in agreement with recent experimental data [@guy].
MC simulations in the present work used the order-parameter culumant crossing method[@Peczak1] to determine the critical temperature of the system. Finite-size scaling analysis at $T_N$ was then used to extract estimates of the critical exponents $\mu, \gamma$ and $\nu$. Both of these techniques have been thoroughly tested on frustrated Ising and Heisenberg spin models [@Peczak1].
For the purposes of determining the classical critical behavior it is adequate to use a model system with ferromagnetic exchange interactions along the $c$-axis, $J_0$, with or without a small applied uniform field $H$ directed along the $c$-axis. In addition to near-neighbor AF interactions between chains, $J_1$, the effects of a small next-nearest-neighbor exchange coupling, $J_2$, was also considered as in previous MC simulations [@Plumer3]. These effects can be incorporated in the Ising Hamiltonian: $$\begin{aligned}
\mathcal{H}= -\sum_{<i,j>} J_{ij} \sigma_{i} \sigma_{j}-H\sum_{i} \sigma_{i}\end{aligned}$$ where the spin at any site is $\sigma_{i}=\pm 1$. In zero applied field 4 different cases were investigated with parameters listed in table \[cases4\]. For all cases $J_0$ and $J_2$ are ferromagnetic and $J_1$ is antiferromagnetic. Finite-size scaling analysis was also performed at 5 nonzero values of $H$ for case II.
Case $J_0$ $J_1$ $J_2$
------ ------- ------- ------- -- -- --
I 1.0 -1.0 0
II 10.0 -1.0 0
III 10.0 -1.0 0.1
IV 10.0 -0.3 0.001
: \[cases4\] Exchange parameters used for the 4 zero-field cases.
![\[figcumulant\] Results of applying the order-paramater cumulant crossing method to estimate the critical temperature $T_N$ for case II. The straight lines correspond to linear fits to the data with $\textrm{ln}^{-1}(L'/L)\leq 2.2$. ](OPcum106512.EPS "fig:"){width="40.00000%"}\
The isotropic case I ($|J_0| = |J_1|$) was previously investigated using the MC histogram method with periodic boundary conditions for an $N=L\times L \times L$ system [@Bunker1; @Plumer1; @Plumer3]. Different sets of critical exponents characteristic of different universality classes were obtained in these previous studies, serving to illustrate the sensitivity of analyzing MC simulation data for the frustrated Ising model. Cases II, III, and IV correspond to systems with anisotropic quasi-1D exchange interactions (with $|J_0| \gg |J_1|$) for different values of $J_2$. The anisotropic case IV was previously investigated[@Koseki2] using the CHB algorithm and estimates of the critical temperature and the critical exponents $\beta$ and $\nu$ were obtained using a data collapse method.
The CHB algorithm employs open boundary conditions along the $c$-axis and periodic boundary conditions in the other directions. For systems with quasi-1D exchange interactions the CHB algorithm is more efficient than the Metropolis algorithm and allows for better statistics when simulating larger lattice sizes. We considered anisotropic lattices with $N=L\times L \times 10 L$ and $L=9,12,...,33,36$. These system sizes are smaller than those employed in Ref.[@Koseki1]. However, a significantly larger number of MC steps were used in any particular run. Averages were also performed over 10 independent simulations using different random initial spin configurations. Estimates of the errors were obtained by taking the standard deviation from the different simulations. Runs of $1\times 10^5-8\times 10^5$ MC steps were used for equilibration, and $5\times 10^5-1.2\times 10^6$ MC steps were used to calculate averages of several thermodynamic quantities including the order parameter $O$, susceptibility $\chi_1$, specific heat $C$, energy cumulant $U_E$, order-parameter cumulant $U_M$ and first logarithmic derivative of the order parameter $V_1$, as defined in Ref. [@Peczak1]. The primary order-parameter $O$ is defined in terms of the $\mathbf{Q}_{th}$ Fourier component of the spin density as $O=\left|\sum_i \sigma_i \exp(i \mathbf{Q}\cdot\mathbf{R}_i)\right|/N$ with $\mathbf{Q}=(2\pi/3,2\pi/3,0)$ in units of the lattice constants associated with the simple hexagonal structure. In addition, the temperature dependence of the secondary order-parameter $O'$ with the wavevector $\mathbf{Q}=(0,0,0)$, corresponding to the uniform magnetization, was also examined. Relevance of this component of the spin density (in zero applied field) on the nature of the critical behavior in these systems has been previously speculated [@Heinonen1; @Koseki1].
Accurate estimation of the critical temperature is an essential first step in utilizing the histogram method for determining critical exponents. Temperature sweeps for the $L=24$ system were initially performed using fewer MC steps to obtain a rough estimate of the transition temperature $T_N$ by locating the maxima of the susceptibility $\chi_1$ and specific heat $C$. For example, for case IV, temperature scans from $T=6.8-8.0$ in increments of $\Delta T=0.05$ yielded an estimate of $T_N=7.40$. To determine the critical temperature more accurately, several histograms were generated at temperatures above and below $T_N$. In this case, histograms were generated at $T=$7.34, 7.36, 7.38, 7.40, 7.42 for each lattice size $L$.
![\[fig6\] Finite-size scaling of the order parameter $O$ for case II where error bars on the data represent the standard deviation of the ten runs.](Op22.EPS){width="40.00000%"}
The order-parameter cumulant crossing method was used for the sets of parameters of Table I and an illustrative example of the results for case II is presented in Fig. \[figcumulant\]. For each case we plot the temperature at which $U_M$ for lattice size $L'$ intersects with the cumulants for $L=9,12$ and $15$. Linear fits are made using the results in the asymptotic region (i.e., $\textrm{ln}^{-1}(L'/L)\leq 2.2$) and an estimate of the critical temperature $T_N$ is obtained from the average value of the crossing points. The error $\pm \Delta T_N$ represents the standard deviation of these values.
The fourth-order energy cumulant evaluated at $T_N$ as a function of system size $L$ was also calculated in each of the four cases. The results extrapolate to $U_E^*=0.666663(3)$ for $L\rightarrow \infty$ in all cases, as expected for a continuous phase transition.
$T_N$ $\beta$ $\gamma$ $\nu$
----------------------------- -- ------------ -- ---------- -- ---------- -- ----------
I 2.926(3) 0.344(7) 1.31(3) 0.671(9)
Bunker et al.[@Bunker1] 2.920(5) 0.311(4) 1.43(3) 0.685(3)
Plumer et al.[@Plumer1] 2.9298(10) 0.341(4) 1.31(3) 0.662(9)
Heinonen et al.[@Heinonen1] 2.88 0.19(1) 1.15(5)
II 10.651(4) 0.355(6) 1.33(3) 0.670(7)
III 11.998(4) 0.358(5) 1.28(3) 0.677(3)
IV 7.406(6) 0.362(7) 1.35(2) 0.673(4)
Koseki et al.[@Koseki2] 7.34(4) 0.21(1) 1.31(3) 0.70(3)
XY 0.345 1.316 0.671
Tricritical 1/4 1 1/2
: \[critexp\] Comparison of the critical temperature and critical exponents determined in this work with other MC studies.
Finite-size scaling analysis at the critical temperature $T_N$ of the various thermodynamic quantities $O \sim L^{-\beta/\nu}$, $\chi \sim L^{\gamma/\nu}$, and $V_1 \sim L^{1/\nu}$, were used to obtain estimates of critical exponent ratios. Representative results are shown in Fig. \[fig6\] for the order parameter in case II where the error on the exponent ratio is obtained from the robustness of the fit. Our results show that the secondary order parameter $O'$ is not relevant at $T_N$ for $H=0$.
In Table \[critexp\] the zero field estimates of the critical temperature and critical exponents determined in this work are compared with those obtained by other MC studies. For each case studied, our set of critical exponents indicate 3D XY universality. This is in agreement with symmetry arguments as well as the MC results of Ref. [@Plumer1] but contrast with the tricritical behaviour seen in Refs. [@Heinonen1] and [@Koseki2], where less intensive simulations were used.
In order to estimate errors due to the uncertainty in the critical temperature $T_N$, finite-size scaling analysis was performed at temperatures slightly above and below $T_N$. In Fig. \[fig2\], results are shown for the critical exponents $\beta$, $\gamma$ and $\nu$ versus the choice of critical temperature for cases I, II, and IV. We find that the cases with quasi-1D exchange interactions (cases II, and IV) are more sensitive to the choice of critical temperature than the case with isotropic exchange coupling (case I). These results indicate that the critical temperature region is extremely narrow for the quasi-1D models, highlighting the need for care in the analysis and interpretation of both computational and experimental data for such systems.
$H$ $T_N$ $\beta$ $\gamma$ $\nu$
------ -- ----------- -- --------- ---------- ----------
0 10.651(4) 0.35(2) 1.33(5) 0.670(7)
0.10 10.651(5) 0.31(3) 1.3(1) 0.63(1)
0.25 10.685(5) 0.28(3) 1.22(6) 0.56(1)
0.50 10.756(5) 0.27(4) 1.02(9) 0.50(2)
1.00 10.891(5) 0.27(7) 1.0(1) 0.48(4)
: \[critexp2\] Critical temperature and critical exponents for several points along the paramagnet phase boundary. The errors on the exponents are estimated from fits performed at $T=T_N \pm \Delta T_N$.
The effects of a nonzero applied field were examined here for the quasi-1D case II only. This extends the work of Refs. [@Netz1; @Plumer1] where isotropic exchange was assumed and where it is argued the effect of an applied field is to change the symmetry to that of the 3-state Potts model and hence the transition should be first order within mean field theory. However, a more complicated phase diagram emerged as a result of these earlier MC simulations. The same number of MC steps as in the zero field cases were used in the present study for equilibration and to calculate thermal averages. At field strengths $H=$ 0.01, 0.25, 0.50 and 1.0 simulation results for the energy cumulant extrapolate to $U_E^*=0.666663(3)$, suggesting that the phase transitions remain continuous at these lower values. Results for the critical exponents at five field strengths from $H=0$ to $H=1$ are shown in Table \[critexp2\]. The error for the critical temperature is estimated from the scatter of the crossing points of the order-parameter cumulant data. The errors on the critical exponents are obtained from fits performed at $T=T_N\pm\Delta T_N$. For example, for $H=0.25$, the fits performed at $T=10.680$ yielded $\beta=0.26(1)$, $\gamma=1.28(4)$, $\nu=0.56(1)$, whereas those at $T=10.690$ yielded $\beta=0.306(6)$, $\gamma=1.17(2)$, $\nu=0.57(1)$. These results illustrate the sensitivity of the critical exponents with respect to the choice of critical temperature. The largest source of error on the critical exponents comes from the uncertainty of the critical temperature. The errors from the robustness of the fits are typically smaller. For small values of the applied field ($H=0.1$) the estimated values of the critical exponents $\beta$ and $\nu$ are slightly lower than the zero field cases. At intermediate field values $H=0.5$ and $H=1.0$ the magnetic phase transition is characterized by a set of exponents that are consistent with tricritical mean-field values. Monte Carlo simulations results for $H=2.0$ reveal a weak first order phase transition where $U_E^*=0.666642(3)$.
![\[fig2\] Values of the critical exponents $\beta$ for cases I, II, and IV, obtained for different choices of the critical temperature. Results for case III are qualitatively similar and have been omitted for clarity.](b12.EPS){width="40.00000%"}
The results of these extensive MC simulations using the CHB algorithm combined with the histogram analysis technique serve to resolve long-standing questions regarding both experimental and previous modeling results on the critical properties of the quasi-1D ISTAF where both XY and tricritical behavior has been reported. This work illustrates the difficulty in extracting reliable estimates of exponents due to the enhanced sensitivity of the critical region when exchange interactions are frustrated and anisotropic. A major focus of the present work has been the careful estimation of errors due to uncertainty in the assumed critical temperature, a feature relevant for the analysis of both modeled and measured data. Our results make clear that the classical model system yields XY criticality in the case of zero applied field for a variety of assumed exchange interactions, in agreement with most previous MC simulations results and with most experimental results on Ising-like ABX$_3$ compounds such as $S=1$ CsNiCl$_3$ [@collins]. Quantum effects due to SRO inherent in the quasi-1D $S=1/2$ compounds are mimicked by the addition of a small applied field and are shown to induce a cross-over to tricritical, then first-order behavior. Our results explain recent high-resolution neutron scattering experiments indicating tricritical exponents in CsCoBr$_3$ [@Mao1] where the strength of the extra field term had previously been estimated to be $H/J_0 =0.05$ [@Nagler1] consistent with our simulations results ($H=0.5$). This conclusion illustrates that quantum spin effects can control the experimentally accessible effective critical behaviour of quasi-1D ISTAF’s. This result is relevant to other systems that exhibit strong SRO before the onset of LRO.
We thank B. Southern and S. Nagler for insightful discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Atlantic Computational Excellence Network (ACEnet).
, edited by H.T. Diep (World Scientific, Singapore, 2004); B. Delamotte, D. Mouhanna, and M. Tissier, Phys. Rev. B **69**, 134413 (2004). M.F. Collins and O.A. Petrenk, Can. J. Phys. **75**, 605 (1997). S. E. Nagler [*et al*]{}., Phys. Rev. B **27**, 1784 (1983). H. Kuroe [*et al*]{}., J. Phys. Soc. Japan, **76**, 043713 (2007). J.P. Goff, D.A. Tennant, and S. E. Nagler, Phys. Rev. B **52**, 15992 (1995); H. Shiba [*et al.*]{}, J. Phys. Soc. Japan **72**, 2326 (2003). R. M. Morra [*et al.*]{}, Phys. Rev. B **38**, 543 (1986). M. L. Plumer and A. Mailhot, Physica A **222**, 437 (1995). D. Blankschtein [*et al.*]{}, Phys. Rev. B **29**, 5250 (1984); A. N. Berker [*et al.*]{}, J. Appl. Phys. **55**, 2416 (1984). O. Heinonen and R. G. Petschek, Phys. Rev. B **40**, 9052 (1989). O. Koseki and F. Matsubara, J. Phys. Soc. Japan **69**, 1202 (2000). Ming Mao [*et al.*]{}, Phys. Rev. B **66**, 184432 (2002). G. Quirion, T. Taylor, and M. Poirier, Phys. Rev. B **72**, 084403 (2005). O. Koseki and F. Matsubara, J. Phys. Soc. Japan **66**, 322 (1997). F. Matsubara [*et al.*]{}, Phys. Rev. Lett. **78**, 3237 (1997). A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. **61**, 2635 (1988). R. R. Netz and A. N. Berker, Phys. Rev. Lett **66**, 377 (1991). M. L. Plumer and A. Mailhot, J. Phys.: Condens. Matter **9**, L165 (1997). G. Quirion [*et al.*]{}, Phys. Rev. Lett. **97**, 077202 (2006). P. Peczak, A. M. Ferrenberg, and D. P. Landau, Phys. Rev. B **43**, 6087 (1991). A. Bunker, B. D. Gaulin, and C. Kallin, Phys. Rev. B **48**, 15861 (1993). M.L. Plumer [*et al.*]{}, Phys. Rev. B. **47**, 14321 (1993); M.L. Plumer and A. Mailhot, Phys. Rev. B **52**, 1411 (1995).
|
---
abstract: 'These notes aim to provide a concise pedagogical introduction to some important applications of the renormalization group in statistical physics. After briefly reviewing the scaling approach and Ginzburg–Landau theory for critical phenomena near continuous phase transitions in thermal equilibrium, Wilson’s momentum shell renormalization group method is presented, and the critical exponents for the scalar $\Phi^4$ model are determined to first order in a dimensional $\epsilon$ expansion about the upper critical dimension $d_c = 4$. Subsequently, the physically equivalent but technically more versatile field-theoretic formulation of the perturbational renormalization group for static critical phenomena is described. It is explained how the emergence of scale invariance connects ultraviolet divergences to infrared singularities, and the renormalization group equation is employed to compute the critical exponents for the $O(n)$-symmetric Landau–Ginzburg–Wilson theory to lowest non-trivial order in the $\epsilon$ expansion. The second part of this overview is devoted to field theory representations of non-linear stochastic dynamical systems, and the application of renormalization group tools to critical dynamics. Dynamic critical phenomena in systems near equilibrium are efficiently captured through Langevin stochastic equations of motion, and their mapping onto the Janssen–De Dominicis response functional, as exemplified by the field-theoretic treatment of purely relaxational models with non-conserved (model A) and conserved order parameter (model B). As examples for other universality classes, the Langevin description and scaling exponents for isotropic ferromagnets at the critical point (model J) and for driven diffusive non-equilibrium systems are discussed. Finally, an outlook is presented to scale-invariant phenomena and non-equilibrium phase transitions in interacting particle systems. It is shown how the stochastic master equation associated with chemical reactions or population dynamics models can be mapped onto imaginary-time, non-Hermitian “quantum” mechanics. In the continuum limit, this Doi–Peliti Hamiltonian is in turn represented through a coherent-state path integral action, which allows an efficient and powerful renormalization group analysis of, e.g., diffusion-limited annihilation processes, and of phase transitions from active to inactive, absorbing states.'
address: 'Department of Physics, Virginia Tech, Blacksburg, VA 24061-0435, USA'
author:
- 'Uwe C. Täuber'
title: 'Renormalization Group: Applications in Statistical Physics'
---
renormalization group ,critical phenomena ,critical dynamics ,driven diffusive systems,\
diffusion-limited chemical reactions , non-equilibrium phase transitions
82-01 ,82B27 ,82B28 ,82C26 ,82C27 ,82C28 ,82C31
Introduction {#intro}
============
Since Ken Wilson’s seminal work in the early 1970s [@Wilson74], based also on the groundbreaking foundations laid by Leo Kadanoff, Ben Widom, Michael Fisher [@Fisher74], and others in the preceding decade, the renormalization group (RG) has had a profound impact on modern statistical physics. Not only do renormalization group methods provide a powerful tool to analytically describe and quantitatively capture both static and dynamic critical phenomena near continuous phase transitions that are governed by strong interactions, fluctuations, and correlations, they also allow us to address physical properties associated with the emerging generic scale invariance in certain entire thermodynamic phases, many non-equilibrium steady states, and in relaxation phenomena towards either equilibrium or non-equilibrium stationary states. In fact, the renormalization group presents us with a conceptual framework and mathematical language that has become ubiquitous in the theoretical description of many complex interacting many-particle systems encountered in nature. One may even argue that the fundamental RG notions of universality and relevance or irrelevance of interactions and perturbations, and the accompanying systematic coarse-graining procedures are of crucial importance for any attempt at capturing natural phenomena in terms of only a few meso- or macroscopic degrees of freedom, and thus also form the essential philosophical basis for any computational modeling, including Monte Carlo simulations.
In these lecture notes, I aim to give a pedagogical introduction and concise overview of first the classic applications of renormalization group methods to equilibrium critical phenomena, and subsequently to the study of critical dynamics, both near and far away from thermal equilibrium. The second half of this article will specifically explain how the stochastic dynamics of interacting many-particle systems, mathematically described either through (coupled) non-linear Langevin or more “microscopic” master equations, can be mapped onto dynamical field theory representations, and then analyzed by means of RG-improved perturbative expansions. In addition, it will be demonstrated how exploiting the general structure of the RG flow equations, fixed point conditions, and prevalent symmetries yields certain exact statements. Other authors contributing to this volume will discuss additional applications of renormalization group tools to a broad variety of physical systems and problems, and also cover more recently developed efficient non-perturbative approaches.
Critical Phenomena
==================
We begin with a quick review of Landau’s generic mean-field treatment of continuous phase transitions in thermal equilibrium, define the critical exponents that characterize thermodynamic singularities, and then venture to an even more general description of critical phenomena by means of scaling theory. Next we generalize to spatially inhomogeneous configurations, investigate critical infrared singularities in the two-point correlation function, and analyze the Gaussian fluctuations for the ensuing Landau–Ginzburg–Wilson Hamiltonian (scalar Euclidean $\Phi^4$ field theory). This allows us to identify $d_c = 4$ as the upper critical dimension below which fluctuations crucially impact the critical power laws. Finally, we introduce Wilson’s momentum shell renormalization group approach, reconsider the Gaussian model, discuss the general emerging structure, and at last perturbatively compute the fluctuation corrections to the critical exponents to first order in the dimensional expansion parameter $\epsilon = d_c - d$. Far more detailed expositions of the contents of this chapter can be found in the excellent textbooks [@Ma76]–[@Mazenko03] and in chap. 1 of Ref. [@Tauberxx].
Continuous phase transitions
----------------------------
Different thermodynamic phases are characterized by certain macroscopic, usually extensive state variables called order parameters; examples are the magnetization in ferromagnetic systems, polarization in ferroelectrics, and the macroscopically occupied ground-state wave function for superfluids and superconductors. We shall henceforth set our order parameter to vanish in the high-temperature disordered phase, and to assume a finite value in the low-temperature ordered phase. Landau’s basic construction of a general mean-field description for phase transitions relies on an expansion of the free energy (density) in terms of the order parameter, naturally constrained by the symmetries of the physical system under consideration. For example, consider a scalar order parameter $\phi$ with discrete inversion or $Z_2$ symmetry that in the ordered phase may take either of two degenerate values $\phi_\pm = \pm |\phi_0|$. We shall see that the following generic expansion (with real coefficients) indeed describes a [*continuous*]{} or second-order phase transition: $$f(\phi) = \frac{r}{2} \, \phi^2 + \frac{u}{4 !} \, \phi^4 + \ldots
- h \, \phi \ ,
\label{landex}$$ if the temperature-dependent parameter $r$ changes sign at $T_c$. For simplicity, and again in the spirit of a regular Taylor expansion, we let $r = a (T - T_c^0)$, where $T_c^0$ denotes the [*mean-field critical temperature*]{}. Stability requires that $u > 0$ (otherwise more expansion terms need to be added); near the critical point we can simply take $u$ to be a constant. Note that the external field $h$, thermodynamically conjugate to the order parameter, [*explicitly*]{} breaks the assumed $Z_2$ symmetry $\phi \to -\phi$.
Minimizing the free energy with respect to $\phi$ then yields the thermodynamic ground state. Thus, from $f'(\phi) = 0$ we immediately infer the [*equation of state*]{} $$h(T,\phi) = r(T) \, \phi + \frac{u}{6} \, \phi^3 \ ,
\label{stateq}$$ and the minimization or stability condition reads $0 < f''(\phi) = r + \frac{u}{2} \, \phi^2$. At $T = T_c^0$, (\[stateq\]) reduces to the [*critical isotherm*]{} $h(T_c^0,\phi) = \frac{u}{6} \, \phi^3$. For $r > 0$, the [*spontaneous order parameter*]{} at zero external field $h = 0$ vanishes; for $r < 0$, one obtains $\phi_\pm = \pm \phi_0$, where $$\phi_0 = (6 |r| / u)^{1/2} \ .
\label{spordp}$$ Note the emergence of characteristic power laws in the thermodynamic functions that describe the properties near the critical point located at $T = T_c^0$, $h = 0$.
The continuous, but non-analytic onset of spontaneous ordering is the hallmark of a second-order phase transition, and induces additional thermodynamic singularities at the critical point: The isothermal order parameter [*susceptibility*]{} becomes $V \chi_T^{-1} = (\partial h / \partial \phi)_T = r + \frac{u}{2} \, \phi_0^2$, whence $$\frac{\chi_T}{V} = \left\{ \begin{array}{cc} 1 / r & \ r > 0 \\ 1 / 2 |r| &
\ r < 0 \end{array} \right. \ ,
\label{opsusc}$$ diverging as $|T - T_c^0|^{-1}$ on both sides of the phase transition, with [*amplitude ratio*]{} $\chi_T(T \downarrow T_c^0) / \chi_T(T \uparrow T_c^0) = 2$. Inserting (\[spordp\]) into the Landau [*free energy*]{} (\[landex\]) one finds for $T < T_c^0$ and $h = 0$ $$f(\phi_\pm) = \frac{r}{4} \, \phi_0^2 = - \frac{3 r^2}{2 u} \, ,
\label{frenor}$$ and consequently for the [*specific heat*]{} $$C_{h=0} = - V T \left( \frac{\partial^2 f}{\partial T^2} \right)_{h=0}
= V T \, \frac{3 a^2}{u} \ ,
\label{spheat}$$ whereas per construction $f(0) = 0$ and $C_{h=0} = 0$ in the disordered phase. Thus, Landau’s mean-field theory predicts a critical point [*discontinuity*]{} $\Delta C_{h=0} = V T_c^0 \, \frac{3 a^2}{u}$ for the specific heat. Experimentally, one indeed observes singularities in thermodynamic observables and power laws at continuous phase transitions, but often with critical exponents that differ from the above mean-field predictions. Indeed, the divergence of the order parameter susceptibility (\[opsusc\]) indicates violent fluctuations, inconsistent with any mean-field description that entirely neglects such fluctuations and correlations.
Scaling theory
--------------
The emergence of scale-free power laws suggests the following general scaling hypothesis for the free energy, namely that its singular contributions near a critical point ($T = T_c$, $h = 0$) can be written as a generalized [*homogeneous*]{} function $$f_{\rm sing}(\tau,h) = |\tau|^{2 - \alpha} \, {\hat f}_\pm
\left( h / |\tau|^\Delta \right) \ ,
\label{fschyp}$$ where $\tau = \frac{T - T_c}{T_c}$ measures the deviation from the (true) critical temperature $T_c$. Thus, the free energy near criticality is not an independent function of the two intensive control parameters $T$ or $\tau$ and $h$, but satisfies a remarkable two-parameter scaling law, with analytic [*scaling functions*]{} ${\hat f}_\pm(x)$ respectively for $T > T_c$ and $T < T_c$ that only depend on the ratio $x = h / |\tau|^\Delta$, and satisfy ${\hat f}_\pm(0) = {\rm const.}$ In Landau theory, the corresponding [*critical exponents*]{} are $\alpha = 0$, compare (\[frenor\]), and $\Delta = 3/2$, as can be inferred by combining Eqs. (\[stateq\]) and (\[spordp\]). The associated [*specific heat*]{} singularity follows again via $$C_{h=0} = - \frac{V T}{T_c^2}
\left( \frac{\partial^2 f_{\rm sing}}{\partial \tau^2} \right)_{h=0}
= C_\pm \, |\tau|^{-\alpha} \ ,
\label{spsing}$$ indicating a divergence if $\alpha > 0$, and a cusp singularity for $\alpha < 0$. Similarly, one obtains the [*equation of state*]{} $$\phi(\tau,h) = - \left( \frac{\partial f_{\rm sing}}{\partial h}
\right)_\tau = - |\tau|^{2 - \alpha - \Delta} \
{\hat f}_\pm'\left( h / |\tau|^\Delta \right) \ ,
\label{sceqst}$$ and therefrom the [*coexistence line*]{} at $h = 0$, $\tau < 0$ $$\phi(\tau,0) = - |\tau|^{2 - \alpha - \Delta} \ {\hat f}_-'(0) \propto
|\tau|^\beta \ ,
\label{screl1}$$ where we have identified $\beta = 2 - \alpha - \Delta$.
Additional [*scaling relations*]{}, namely identities that relate different critical exponents, can be easily derived; for example, on the [*critical isotherm*]{} at $\tau = 0$, the $\tau$-dependence in ${\hat f}_\pm'$ on the r.h.s. of (\[sceqst\]) must cancel the singular prefactor, i.e., ${\hat f}_\pm'(x \to \infty) \sim x^{(2 - \alpha - \Delta) / \Delta}$, and $$\phi(0,h) \propto h^{(2 - \alpha - \Delta) / \Delta} = h^{1 / \delta} \ , \
{\rm with} \ \delta = \Delta / \beta \ .
\label{screl2}$$ Finally, the isothermal [*susceptibility*]{} becomes $$\frac{\chi_\tau}{V} = \left( \frac{\partial \phi}{\partial h}
\right)_{\tau, \, h=0} = \chi_\pm \, |\tau|^{- \gamma} \, , \
\gamma = \alpha + 2 (\Delta - 1) \ , \
\label{screl3}$$ and upon eliminating $\Delta = \beta \delta$, one arrives at the following set of scaling relations $$\alpha + \beta (1 + \delta) = 2 = \alpha + 2 \beta + \gamma \, , \
\gamma = \beta (\delta - 1) \ . \
\label{screls}$$ Clearly, as consequence of the two-parameter scaling hypothesis (\[fschyp\]), there can only be [*two independent*]{} thermodynamic critical exponents. In the framework of Landau’s mean-field approximation, the set of critical exponents reads $\alpha = 0$, $\beta = \frac12$, $\gamma = 1$, $\delta = 3$, and $\Delta = \frac32$; note that these integer or rational numbers really just follow from straightforward dimensional analysis. In both computer and real experiments, one typically measures different critical exponent values, yet these still turn out to be [*universal*]{} in the sense that at least for short-range interaction forces they depend only on basic symmetry properties of the order parameter and the spatial dimensionality $d$, but [*not*]{} on microscopic details such as lattice structure, nature and strength of interaction potentials, etc. Indeed, the Ising ferromagnetic and the liquid-gas critical points, both characterized by a scalar real order parameter, are governed by identical power laws, as is the critical behavior for planar magnets with a two-component vector order parameter and the normal- to superfluid transition in helium 4, with a complex scalar or equivalently, a real two-component order parameter. The striking emergence of thermodynamic self-similarity in the vicinity of $T_c$ has been spectacularly demonstrated in the latter system, with the $\Lambda$-like shape of the specific heat curve appearing identical on milli- and micro-Kelvin temperature scales.
Landau–Ginzburg–Wilson Hamiltonian
----------------------------------
In order to properly include the effects of fluctuations, we need to generalize the Landau expansion (\[landex\]) to spatially varying order parameter configurations $S(x)$, which leads us to the [*coarse-grained*]{} effective [*Landau–Ginzburg–Wilson (LGW) Hamiltonian*]{} $$\begin{aligned}
&&{\cal H}[S] = \int \! d^dx \, \biggl[ \frac{r}{2} \, S(x)^2
+ \frac{1}{2} \, [\nabla S(x)]^2 \nonumber \\
&&\qquad\qquad\qquad + \frac{u}{4 !} \, S(x)^4 - h(x) \, S(x) \biggr] \ ,
\label{lgwham}\end{aligned}$$ where $r = a (T - T_c^0)$ and $u > 0$ as before, and $h(x)$ now represents a local external field. Under the natural assumption that spatial inhomogeneities are energetically unfavorable, the gradient term $\sim [\nabla S(x)]^2$ comes with a positive coefficient that has been absorbed into the scalar order parameter field. Within the canonical framework of statistical mechanics, the [*probability density*]{} for a configuration $S(x)$ is given by the [*Boltzmann factor*]{} ${\cal P}_s[S] = \exp (- {\cal H}[S] / k_{\rm B} T) / {\cal Z}[h]$. Here, the [*partition function*]{} ${\cal Z}[h]$ and expectation values of observables $A[S]$ are represented through functional integrals: $$\begin{aligned}
&&{\cal Z}[h] = \int \! {\cal D}[S] \ e^{- {\cal H}[S] / k_{\rm B} T} \ ,
\label{canpar} \\
&&\langle A[S] \rangle = \int\! {\cal D}[S] \ A[S(x)] \, {\cal P}_s[S] \ .
\label{canobs} \end{aligned}$$ At $h = 0$, for example, the $k$th order parameter moments follow via functional derivatives $$\Big\langle \prod_{j=1}^k S(x_j) \Big\rangle = (k_{\rm B} T)^k \prod_{j=1}^k
\frac{\delta}{\delta h(x_j)} \, {\cal Z}[h] \Big\vert_{h = 0} , \
\label{opmoms}$$ and similarly the associated cumulants can be obtained from functional derivatives of $\ln {\cal Z}[h]$; the partition function thus also serves as a [*generating function*]{}. For explicit calculations, one requires the integral measure in (\[canobs\]), e.g., through discretizing $x \to x_i$ on, say, a $d$-dimensional cubic hyperlattice, whence simply ${\cal D}[S] = \prod_i dS(x_i)$. Alternatively, one may employ the Fourier transform $S(x) = \int \frac{d^dq}{(2 \pi)^d} \, S(q) \, e^{i q \cdot x}$; noting that $S(-q) = S(q)^*$ since $S(x)$ is real, and consequently the real and imaginary parts of $S(q)$ are not independent, one only needs to integrate over wave vector half-space, $${\cal D}[S] = \prod_{q , q_1 > 0}
\frac{d \, {\rm Re} \, S(q) \ d \, {\rm Im} \, S(q)}{V} \ .
\label{intmft}$$
In the [*Ginzburg–Landau approximation*]{}, one considers only the most likely configuration $S(x)$, which is readily found by the method of steepest descent for the path integrals in (\[canobs\]), leading to the classical field or Ginzburg–Landau equation $$0 = \frac{\delta {\cal H}[S]}{\delta S(x)} = \left[ r - \nabla^2 +
\frac{u}{6} \, S(x)^2 \right] S(x) - h(x) . \
\label{glaneq}$$ In the spatially homogeneous case, (\[glaneq\]) reduces to the mean-field equation of state (\[stateq\]). Let us next expand in the fluctuations $\delta S(x) = S(x) - \phi$ about the mean order parameter $\phi = \langle S \rangle$ and linearize, which yields $\delta h(x) \approx \left( r - \nabla^2 + \frac{u}{2}\, \phi^2 \right)
\delta S(x)$. Through Fourier transform one then immediately obtains the order parameter response function in the mean-field approximation, also known as [*Ornstein–Zernicke susceptibility*]{} $$\chi_0(q) = \frac{\partial S(q)}{\partial h(q)} \bigg\vert_{h = 0} =
\frac{1}{\xi^{-2}+ q^2} \, , \
\label{ornzer}$$ where we have introduced the characteristic [*correlation length*]{} $\xi = (r + \frac{u}{2} \, \phi_0^2)^{-1/2}$, i.e., $$\xi = \left\{ \begin{array}{cc} 1 / r^{1/2} & \ r > 0 \\ 1 / |2 r|^{1/2} &
\ r < 0 \end{array} \right. \ .
\label{corlen}$$ On the other hand, consider the connected zero-field two-point [*correlation function*]{} (cumulant) $$\begin{aligned}
&&C(x-x') = \langle S(x) \, S(x') \rangle - \langle S \rangle^2 \nonumber \\
&&\qquad\qquad = (k_{\rm B} T)^2 \, \frac{\delta^2 \, \ln {\cal Z}[h]}
{\delta h(x) \, \delta h(x')} \bigg\vert_{h=0} ;
\label{corfun}\end{aligned}$$ in a spatially translation-invariant system, we may define its Fourier transform as $C(x) = \int \frac{d^dq}{(2 \pi)^d} \, C(q) \, e^{i q \cdot x}$, and through comparison with the definition of the susceptibility in (\[ornzer\]) arrive at the [*fluctuation-response theorem*]{} $C(q) = k_{\rm B} T \, \chi(q)$, valid in thermal equilibrium.
Generalizing the Ginzburg–Landau mean-field result (\[ornzer\]), we may formulate the [*scaling hypothesis*]{} for the two-point correlation function in terms of the following scaling ansatz, which defines both the [*Fisher exponent*]{} $\eta$ and the critical exponent $\nu$ that describes the divergence of the correlation length $\xi$ at $T_c$: $$C(\tau,q) = |q|^{- 2 + \eta} \, {\hat C}_\pm(q \xi) \ , \quad
\xi = \xi_\pm \, |\tau|^{- \nu} \ .
\label{corsch}$$ The thermodynamic susceptibility then becomes $$\chi(\tau,q=0) \propto \xi^{2 - \eta} \propto |\tau|^{- \gamma} \, , \
{\rm with} \ \gamma = \nu (2 - \eta) \ , \
\label{screl4}$$ providing us with yet another scaling relation that connects the thermodynamic critical exponent $\gamma$ with $\eta$ and $\nu$. Consequently, we see that the thermodynamic critical point singularities are induced by the diverging spatial correlations. Fourier back-transform gives $$C(\tau,x) = |x|^{- (d - 2 + \eta)} \, {\widetilde C}_\pm(x / \xi) \propto
\xi^{- (d - 2 + \eta)}
\label{spcorr}$$ at large distances $|x| \to \infty$. In this limit, one expects $\langle S(x) \, S(0) \rangle \to \phi^2 \propto (-\tau)^{2 \beta}$, and comparison with (\[spcorr\]) therefore implies the [*hyperscaling*]{} relations $$\beta = \frac{\nu}{2} \, (d - 2 + \eta) \ \ {\rm and} \ \
2 - \alpha = d \nu \ .
\label{hypscr}$$ The Ornstein–Zernicke function (\[ornzer\]) satisfies the scaling law (\[corsch\]) with the mean-field values $\nu = \frac12$ and $\eta = 0$. Notice that the set of mean-field critical exponents obeys (\[hypscr\]) only in $d = 4$ dimensions.
Gaussian approximation
----------------------
We now proceed to analyze the LGW Hamiltonian (\[lgwham\]) in the Gaussian approximation, where non-linear fluctuation contributions are neglected. In the high-temperature phase, we have $\phi = 0$, and thus simply omit the terms $\sim u \, S(x)^4$, leaving the Gaussian Hamiltonian $${\cal H}_0[S] = \int_q \left[ \frac{r + q^2}{2} \, |S(q)|^2 - h(q) S(-q)
\right] \ ,
\label{gauham}$$ with the abbreviation $\int_q = \int \! \frac{d^dq}{(2 \pi)^d}$. The associated Gaussian partition function is readily computed by completing the square in (\[gauham\]), or the linear field transformation ${\widetilde S}(q) = S(q) - h(q) / (r + q^2)$, $$\begin{aligned}
&&{\cal Z}_0[h] = \int \! {\cal D}[S] \ e^{- {\cal H}_0[S] / k_{\rm B}T}
\nonumber \\
&&\quad = \exp \left( \frac{1}{2 k_{\rm B}T} \int_q \frac{|h(q)|^2}{r + q^2}
\right) \, {\cal Z}_0[h = 0] \ , \
\label{gaupar}\end{aligned}$$ which yields the Gaussian two-point correlator $$\begin{aligned}
&&\langle S(q) \, S(q') \rangle_0 = \frac{(k_{\rm B} T)^2}{{\cal Z}_0[h]} \,
\frac{(2 \pi)^{2d} \, \delta^2 {\cal Z}_0[h]}{\delta h(-q)\, \delta h(-q')}
\bigg\vert_{h=0} \nonumber \\
&&= C_0(q) \, (2 \pi)^d \delta(q + q') \, , \
C_0(q) = \frac{k_{\rm B} T}{r + q^2} \, . \
\label{gaucor}\end{aligned}$$
Gaussian integrations give the free energy $F_0[h] = - k_{\rm B} T \ln {\cal Z}_0[h]$ of the model (\[gauham\]), $$F_0[h] = - \frac{1}{2} \int_q \left( \frac{|h(q)|^2}{r + q^2}
+ k_{\rm B} T V \, \ln \frac{2 \pi \, k_{\rm B} T}{r + q^2} \right) \, .
\label{gaufre}$$ Let us explore the leading singularity near $T_c^0$ in the [*specific heat*]{} $C_{h=0} = -T (\partial^2 F_0 / \partial T^2)_{h=0}$ that originates from derivatives with respect to the control parameter $r$, $$\frac{C_{h=0}}{V} \approx \frac{k_{\rm B}}{2} \, (a T_c^0)^2
\int_q \frac{1}{(r + q^2)^2} \ .
\label{gausph}$$
- In high dimensions $d > 4$, the r.h.s. integral is UV-divergent, but can be regularized by a Brillouin zone boundary cutoff $\Lambda \sim 2 \pi / a_0$ stemming from the original underlying lattice. Consequently the fluctuation contribution (\[gausph\]) is finite as $r \to 0$ and $\alpha = 0$ as in mean-field theory.
- In low dimensions $d < 4$, we set $k = q / \sqrt{r} = q \xi$ to render the fluctuation integral, which is UV-finite, dimensionless. With the $d$-dimensional unit sphere surface area $K_d = 2 \pi^{d/2} / \Gamma(d/2)$, one finds: $$\frac{C_{h = 0}}{V} \approx
\frac{k_{\rm B} (a T_c^0)^2 \, \xi^{4-d}}{2^d \pi^{d/2} \, \Gamma(d/2)}
\int_0^\infty \! \frac{k^{d-1}}{(1 + k^2)^2} \ dk \, . \
\label{sphd<4}$$ As the critical temperature is approached, the correlation length prefactor diverges $\propto |T - T_c^0|^{- \frac{4-d}{2}}$; already the lowest-order fluctuation contribution contains a strong infrared (IR) singularity that dominates over the mean-field power law.
- At $d = d_c = 4$, the integral diverges logarithmically as either $\Lambda$ or $\xi \to \infty$: $$\int_0^{\Lambda \xi} \frac{k^3}{(1 + k^2)^2}\ dk \sim \ln (\Lambda \xi) \ ;
\label{sphd=4}$$ note that at this [*upper critical dimension*]{}, ultraviolet and infrared divergences are intimately coupled.
Above the critical dimension, we thus expect the mean-field scaling exponents to correctly describe the critical power laws. In dimensions $d \leq d_c$, however, fluctuation contributions become prevalent and modify the mean-field scaling laws. Given the strongly fluctuating and correlated nature of critical systems, standard theoretical approaches such as perturbation, cluster, or high- and low-temperature expansions typically fail to yield reliable approximations. Fortunately, the renormalization group provides a powerful method to tackle interacting many-particle systems dominated by fluctuations and correlations, especially in a scale-invariant regime.
Wilson’s momentum shell renormalization group
---------------------------------------------
The [*renormalization group program in statistical physics*]{} can be summarized as follows: The goal is to establish a mathematical framework that can properly capture the infrared (IR) singularities appearing in thermodynamic properties as well as correlation functions near a continuous phase transitions and in related situations that are [*not*]{} perturbatively accessible. To this end, one exploits a fundamental new symmetry that emerges at a critical point, namely [*scale invariance*]{}, induced by the divergence of the dominant characteristic correlation length scale $\xi$. Approximation schemes need to carefully avoid the region where the physical IR singularities become manifest; instead, one analyzes the theory in the ultraviolet (UV) regime, by means of either of various equivalent methods: In Wilson’s momentum shell RG approach, one integrates out short-wavelength modes; in the field-theoretic version of the RG, one explicitly renormalizes the UV divergences. Either method quantifies the weight of fluctuation contributions to certain coarse-grained or “renormalized” physical parameters and couplings. One then maps the resulting system back to the original theory given in terms of some “effective” Hamiltonian, which in Wilson’s scheme entails a rescaling of both control parameters and field degrees of freedom. Thus one obtains recursion relations for [*effective*]{}, now scale-dependent [*running couplings*]{}. Subject to a recursive sequence of such renormalization group transformations, these effective couplings will
- either grow, and ultimately tend to infinity: to access a scale-invariant regime, one therefore has to set these [*relevant*]{} parameters to zero at the outset, which defines the [*critical surface*]{} of the problem;
- or diminish, and eventually approach zero: these [*irrelevant*]{} couplings consequently do not affect the asymptotic critical scaling properties;
- certain [*marginal*]{} parameters may also approach an [*infrared-stable fixed point*]{}, provided their initial value is located in the fixed point’s basin of attraction: clearly, scale-invariant behavior thus emerges near an IR-stable fixed point, and the independence from a wide range of initial conditions along with the automatic disappearance of the irrelevant couplings constitute the origin of [*universality*]{}.
The central idea now is to take advantage of the emerging scale invariance at a critical fixed point as a means to infer the proper infrared scaling behavior from an at least approximative analysis of the ultraviolet regime, where, e.g., perturbation theory is feasible. Thus one may establish a solid theoretical foundation for scaling laws such as (\[fschyp\]) and (\[corsch\]), thereby derive scaling relations, and also construct a systematic approximation scheme to compute critical exponents and even scaling functions. We shall soon see that an appropriate small parameter for a perturbational expansion is given through a [*dimensional expansion*]{} in terms of the deviation from the upper critical dimension $\epsilon = d_c - d$.
Wilson’s momentum shell renormalization group approach consists of two RG transformation steps:
1. Carry out the partition integral over all Fourier components $S(q)$ with wave vectors residing in the spherical momentum shell $\Lambda / b \leq |q| \leq \Lambda$, where $b > 1$: this effectively [*eliminates*]{} the short-wavelength modes.
2. Perform a [*scale transformation*]{} with the same scale parameter $b > 1$: $x \to x' = x / b$, $q \to q' = b \, q$. Accordingly, one also needs to rescale the fields: $$\begin{aligned}
&&S(x) \to S'(x') = b^\zeta S(x) \ , \nonumber \\
&&S(q) \to S'(q') = b^{\zeta - d} S(q) \ ,
\label{fldrsc}
\end{aligned}$$ with a proper choice of $\zeta$ ensuring that the rescaled residual Hamiltonian assumes the original form.
Subsequent iterations of this procedure yield [*scale-dependent effective couplings*]{}, and the task will be to analyze their dependence on the scale parameter $b$. Notice the [*semi-group*]{} character of the above RG transformations: there obviously exists no unique inverse, since the elimination step (1) discards detailed information about fluctuations in the UV regime.
The mechanism and efficacy of the momentum shell RG is best illuminated by first considering the exactly tractable Gaussian model. Introducing the short-hand notations $\int_q^< = \int_{|q| < \Lambda/b} \frac{d^dq}{(2 \pi)^d}$ and $\int_q^> = \int_{\Lambda / b \leq |q| \leq \Lambda} \frac{d^dq}{(2 \pi)^d}$, one may readily decompose the Hamiltonian (\[gauham\]) into distinct additive Fourier mode contributions $${\cal H}_0[S] = \left( \int_q^<\! + \int_q^>\! \right)
\left[ \frac{r + q^2}{2} \, |S(q)|^2 - h(q) \, S(-q) \right] .$$ Integrating out the momentum shell fluctuations then just gives a constant contribution to the free energy. We now wish to achieve that ${\cal H}_0[S^<] \to {\cal H}_0[S']$ under the scale transformations in step (2). For the term $\sim q^2 \, |S(q)|^2$, this is accomplished through the choice $\zeta = \frac{d - 2}{2}$ in (\[fldrsc\]); the other contributions then immediately result in the following recursion relations for the control parameters $r$ and $h$: $r \to r' = b^2 r$, and $h(q) \to h'(q') = b^{- \zeta} h(q)$, whence $h(x) \to h'(x') = b^{d - \zeta} h(x)$. Both the temperature variable $r$ and the external field thus constitute [*relevant*]{} parameters, and the [*critical surface*]{} in parameter space is given by $r = 0 = h$. As any other length scale, the correlation length scales according to $\xi \to \xi' = \xi / b$; eliminating the scale parameter $b$ one arrives at the relation $\xi \propto r^{-1/2}$, or $\nu = \frac12$. Likewise, for the rescaled correlation function one finds $C'(x') = b^{2 \zeta} \, C(x)$, whence $\eta = 0$: for the Gaussian theory, we recover the mean-field scaling exponents.
We can gain additional non-trivial information by considering further couplings; e.g., imagine adding contributions of the form $c_s \int \! d^dx \, (\nabla^s S)^2$ to the Hamiltonian (\[gauham\]) that represent higher-order terms in a gradient expansion for spatial order parameter fluctuations, subject to preserving the $Z_2$ and spatial inversion symmetries. One readily confirms that $c_s \to c_s' = b^{d - 2 s - 2 \zeta} c_s = b^{-2 (s-1)} c_s$, which implies that all these additional couplings $c_s$ are [*irrelevant*]{} for $s > 1$ and scale to zero under repeated scale transformations. The inversion symmetry $S(x) \to -S(x)$ permits general local non-linearities of the from $u_p \int \! d^dx\, S(x)^{2p}$. Under Gaussian model RG transformation, these scale as $u_p \to u_p' = b^{d - 2 p \zeta} u_p = b^{2 p - (p - 1) d} u_p$; these couplings are consequently [*marginal*]{} at $d_c(p) = 2 p / (p - 1)$, [*relevant*]{} for $d < d_c(p)$, and [*irrelevant*]{} for $d > d_c(p)$. The upper critical dimension decreases monotonously for $p \geq 2$, with the asymptote $d_c(\infty) = 2$. For the quartic coupling in the LGW Hamiltonian (\[lgwham\]), this confirms $d_c(2) = 4$, while $d_c(3) = 3$ for a sixth-order term $v \int \! d^dx\, S(x)^6$: $v \to v' = b^{6 - 2 d} v$, which becomes [*irrelevant*]{} near the upper critical dimension of the quartic term: $v' = b^{-2} v$ at $d_c(2) = 4$. In general, the coupling ratio $\frac{u_{p+1}'}{u_p'} = b^{2 - d} \frac{u_{p+1}}{u_p}$ renormalizes to zero in dimensions $d > 2$. At two dimensions, the fields $S(x)$ become dimensionless, $\zeta = 0$, and consequently all these non-linearities scale identically. The LGW Hamiltonian thus does not represent the correct asymptotic field theory, and one must resort to other effective descriptions (e.g., the non-linear sigma model).
The above considerations already allow a discussion of the general structure of the momentum shell RG procedure. According to (\[spcorr\]), the general field rescaling (\[fldrsc\]) should contain Fisher’s exponent $\eta$, $\zeta = \frac{d - 2 + \eta}{2}$, whence $h'(x) = b^{(d + 2 - \eta)/2} h(x)$, and (\[corsch\]) implies $\tau' = b^{1 / \nu} \tau$ with the correlation length critical exponent $\nu$. In the simplest scenario, there are thus only [*two*]{} relevant parameters $\tau$ and $h$. Let us further assume the presence of (a few) [*marginal*]{} perturbations $u_i \to u_i' = u_i^* + b^{- x_i} u_i$, while other couplings are [*irrelevant*]{}: $v_i \to v_i' = b^{- y_i} v_i$, with both $x_i > 0$ and $y_i > 0$. After a single RG transformation step, the free energy density becomes $$\begin{aligned}
&&f_{\rm sing}(\tau,h,\{ u_i \},\{ v_i \}) =
\label{fenrg1} \\
&&\quad b^{-d} f_{\rm sing}\Bigl( b^{1/\nu} \tau, b^{d - \zeta} h, \Big\{
u_i^* + \frac{u_i}{b^{x_i}} \Big\}, \Big\{ \frac{v_i}{b^{y_i}} \Big\} \Bigr)
\ . \nonumber\end{aligned}$$ After sufficiently many ($\ell \gg 1$) RG transformations, the marginal couplings have reached their fixed point values $u_i^*$, whereas the irrelevant perturbations have scaled to zero, $$\begin{aligned}
&&f_{\rm sing}(\tau,h,\{ u_i \},\{ v_i \}) =
\label{fenrga} \\
&&\quad b^{-\ell d} f_{\rm sing}\Bigl( b^{\ell/\nu} \tau,
b^{\ell (d + 2 - \eta) / 2} h, \{ u_i^* \}, \{ 0 \} \Bigr) \ . \nonumber\end{aligned}$$ Upon choosing $b^\ell |\tau|^\nu = 1$ for the scale parameter $b^\ell$, one arrives at the [*scaling form*]{} $$f_{\rm sing}(\tau,h) = |\tau|^{d \nu} \, {\hat f}_\pm \left(
h / |\tau|^{\nu (d + 2 - \eta) / 2} \right) \ ,
\label{fenrgs}$$ with ${\hat f}_\pm(x) = f_{\rm sing}( \pm 1, x, \{ u_i^* \}, \{ 0 \})$. With the exponent identities (\[screls\]) and (\[hypscr\]), this is equivalent to the scaling hypothesis (\[fschyp\]). In a similar manner, one readily derives the correlation function scaling law (\[spcorr\]), employing the [*matching condition*]{} $b^\ell = \xi / \xi_\pm$ for $$C(\tau,x,\{ u_i \},\{ v_i \}) = b^{- 2 \ell \zeta} \, C\Bigl( b^{\ell / \nu}
\tau, \frac{x}{b^\ell}, \{ u_i^* \}, \{ 0 \} \Bigr) \, . \
\label{corrga}$$
Dimensional expansion and critical exponents
--------------------------------------------
We are now ready to treat the non-linear fluctuation corrections by means of a systematic [*perturbation expansion*]{}. The quartic contribution to the Hamiltonian reads in Fourier space $${\cal H}_{\rm int} = \frac{u}{4 !} \! \int_{|q_i| < \Lambda} \!\!\!\!\!\!\!\!
S(q_1) S(q_2) S(q_3) S(-q_1-q_2-q_3) \ . \
\label{nonlin}$$ Both the full partition function for the Hamiltonian (\[lgwham\]) and any associated $N$-point correlation functions can then be rewritten in terms of expectation values in the [*Gaussian ensemble*]{} (we shall henceforth set $k_{\rm B} T = 1$) $$\begin{aligned}
&&{\cal Z}[h] = {\cal Z}_0[h] \, \Big\langle e^{- {\cal H}_{\rm int}[S]}
\Big\rangle_0 \, , \nonumber \\
&&\Big\langle \prod_i S(q_i) \Big\rangle =
\frac{\big\langle \prod_i S(q_i) \, e^{- {\cal H}_{\rm int}[S]}
\big\rangle_0}{\big\langle e^{- {\cal H}_{\rm int}[S]} \big\rangle_0} \ .
\label{expgau}\end{aligned}$$ Note that all expectation values of an odd number of fields $S(q_i)$ obviously vanish in the symmetric high-temperature phase, when the external field $h = 0$. Defining the [*contraction*]{} of two fields as the Gaussian two-point function or [*propagator*]{} in Fourier space, $$\underbrac{S(q) S}(q') = \langle S(q) S(q') \rangle_0 =
C_0(q) \, (2 \pi)^d \delta(q + q') \ , \nonumber$$ we may write down [*Wick’s theorem*]{} for Gaussian correlators containing an even number of fields, here a straightforward property of Gaussian integrations: $$\begin{aligned}
&&\langle S(q_1) S(q_2) \ldots S(q_{N-1}) S(q_N) \rangle_0 =
\label{wickth} \\
&&\!\!\!\!\!\!\!\!\!\sum_{{\rm permutations} \atop i_1(1) \ldots i_N(N)} \!\!
\underbrac{S(q_{i_1(1)}) S}(q_{i_2(2)}) \, \ldots \,
\underbrac{S(q_{i_{N-1}(N-1)}) S}(q_{i_N(N)}) \ . \nonumber\end{aligned}$$ Consequently, any arbitrary expectation value (\[expgau\]) can now be perturbatively evaluated via a series expansion with respect to the non-linear coupling $u$, and by means of (\[wickth\]) expressed through sums and integrals of products of Gaussian propagators $C_0(q_i)$.
As an example, we consider the first-order fluctuation correction to the zero-field two-point function $\langle S(q) S(q') \rangle = C(q) \, (2 \pi)^d \delta(q + q')$: $\big\langle S(q) S(q') \, \bigl[ 1 - \frac{u}{4 !}$ $\int_{|q_i| < \Lambda}
S(q_1) S(q_2) S(q_3) S(-q_1-q_2-q_3) \bigr] \big\rangle_0$. According to Wick’s theorem (\[wickth\]), there are two types of contributions: (i) Contractions of external legs $\underbrac{S(q) S}(q')$ yield terms that precisely cancel with the denominator in (\[expgau\]), leaving just the Gaussian propagator $\langle S(q) S(q') \rangle_0$. (ii) The twelve remaining contributions are all of the form $\int_{|q_i| < \Lambda} \underbrac{S(q) \, S}(q_1) \,
\underbrac{S(q_2) \, S}(q_3) \, \underbrac{S(-q_1-q_2-q_3) \, S}(q') =
(2 \pi)^d \delta(q + q') \, C_0(q)^2 \int_{|p| < \Lambda} C_0(p)$. Collecting all terms, one obtains $$C(q) = C_0(q) \, \biggl[ 1 - \frac{u}{2} \, C_0(q) \! \int_{|p| < \Lambda}
\!\!\! C_0(p) + O(u^2) \biggr] \, ; \
\label{1lpcor}$$ interpreting the bracket as the lowest-order contribution in Dyson’s equation (see Chap.3.1 below), the integral turns out to be the associated self-energy to $O(u)$, and (\[1lpcor\]) can be recast in the form $$C(q)^{-1} = r + q^2 + \frac{u}{2} \int_{|p| < \Lambda} \frac{1}{r + p^2}
+ O(u^2) \ . \
\label{1lpsen}$$ Notice that to order $u$, fluctuations here merely renormalize the “mass” $r$, but there is no modification of the momentum dependence in the two-point correlation function $C(q)$, implying that $\eta$ will remain zero in this approximation. In a similar manner, one readily finds the first-order fluctuation correction to the four-point function at vanishing external wave vectors, i.e., the non-linear coupling $u$ to be $- \frac32 u^2 \int_{|p| < \Lambda} C_0(p)^2$.
It is now a straightforward task to translate these perturbation theory results into first-order recursion relations for the couplings $r$ and $u$ by means of Wilson’s RG procedure. To this end, we split the field variables in outer ($S_>$: $S(q)$ with $\Lambda / b \leq |q| \leq \Lambda$) and inner ($S_<$: $S(q)$ with $|q| < \Lambda / b$) momentum shell contributions; we then realize that there are four types of contributions:
- terms involving merely inner shell fields that are not integrated, e.g. $\sim u \int^> S_<^4 e^{-{\cal H}_0[S]}$ just need to be re-exponentiated;
- integrals such as $u \int^> S_<^3 \, S_> e^{-{\cal H}_0[S]}$ vanish;
- contributions $\sim u \int^> S_>^4 e^{-{\cal H}_0[S]}$ that contain only outer shell fields become constants that directly contribute to the free energy;
- for terms of the form $\sim u \int^> S_<^2 \, S_>^2 e^{-{\cal H}_0}$, one has to perform Gaussian integrations over the outer shell fields $S_>$, yielding corrections to the propagator for the inner shell modes.
Employing (\[1lpsen\]), using $\eta = 0$, and introducing $S_d = K_d / (2 \pi)^d = 1 / 2^{d-1} \pi^{d/2} \Gamma(d/2)$, one thus finds to $O(u)$: $$\begin{aligned}
&&r' = b^2 \, \Bigl[ r + \frac{u}{2} \, A(r) \Bigr] \nonumber \\
&&\quad = b^2 \, \biggl[ r + \frac{u}{2} \, S_d \int_{\Lambda/b}^\Lambda
\frac{p^{d-1}}{r + p^2} \, dp \biggr] \ ,
\label{rectem} \\
&&u' = b^{4-d} u \, \Bigl[ 1 - \frac{3 u}{2} \, B(r) \Bigr] \nonumber \\
&&\quad = b^{4-d} u \, \biggl[ 1 - \frac{3 u}{2} S_d
\int_{\Lambda/b}^\Lambda \frac{p^{d-1} \, dp}{(r + p^2)^2} \biggr] \ .
\label{reccpl}\end{aligned}$$ For $T \gg T_c$, or $r \to \infty$, the fluctuation corrections become suppressed, and one recovers the recursion relations $r' = b^2 r$ and $u' = b^{4-d} u$ of the Gaussian theory. Near the critical point, i.e. for $r \ll 1$, one may expand $$\begin{aligned}
&&A(r) = S_d \Lambda^{d-2} \frac{1 - b^{2-d}}{d-2} \nonumber \\
&&\qquad\quad - r \, S_d \Lambda^{d-4} \, \frac{1 - b^{4-d}}{d-4} + O(r^2)
\ , \\
&&B(r) = S_d \Lambda^{d-4} \, \frac{1 - b^{4-d}}{d-4} + O(r) \ .
\label{reccrt}\end{aligned}$$
It is useful to consider instead differential RG flow equations that result from infinitesimal RG transformations. Setting $b = e^{\delta \ell}$ with $\delta \ell \to 0$, (\[rectem\])-(\[reccrt\]) turn into $$\begin{aligned}
&&\frac{d {\tilde r}(\ell)}{d \ell} = 2 {\tilde r}(\ell)
+ \frac{{\tilde u}(\ell)}{2} \, S_d \Lambda^{d-2} \nonumber \\
&&\qquad\quad\ - \frac{{\tilde r}(\ell) {\tilde u}(\ell)}{2} \, S_d
\Lambda^{d-4} + O({\tilde u} {\tilde r}^2,{\tilde u}^2) \ ,
\label{difrcr} \\
&&\frac{d {\tilde u}(\ell)}{d \ell} = (4-d) \, {\tilde u}(\ell)
- \frac{3}{2} \, {\tilde u}(\ell)^2 S_d \Lambda^{d-4} \nonumber \\
&&\qquad\quad\ + O({\tilde u} {\tilde r},{\tilde u}^2) \ .
\label{difrcu}\end{aligned}$$ We specifically seek renormalization group [*fixed points*]{} $(r^*,u^*)$ that describe scale-invariant behavior, to be determined by the conditions $d {\tilde r}(\ell) / d \ell = 0 = d {\tilde u}(\ell) / d \ell$. There is obviously always the [*Gaussian*]{} fixed point $u_0^* = 0$; linearizing (\[difrcu\]) in terms of the deviation $\delta {\tilde u}_0(\ell) = {\tilde u}(\ell) - u_0^*$, one finds $d \delta {\tilde u}_0(\ell) / d \ell \approx (d-4) \delta {\tilde u}_0(\ell)$; $u_0^*$ is hence stable for $d > d_c = 4$, but unstable for $d < 4$. Below the upper critical dimension, there exists also a positive [*Ising*]{} fixed point $u_{\rm I}^* S_d = \frac{2}{3} \, (4-d) \Lambda^{4-d}$, which is then also stable since $d \delta {\tilde u}_I(\ell) / d \ell \approx (4 - d)
\delta {\tilde u}_{\rm I}(\ell)$ for $\delta {\tilde u}_{\rm I}(\ell) = {\tilde u}(\ell) - u_{\rm I}^*$. Correspondingly, the critical behavior is governed by the Gaussian fixed point and associated scaling exponents in dimensions $d > 4$, but by the non-trivial Ising fixed point in low dimensions $d < d_c = 4$. Notice also that the numerical value of the Ising fixed point becomes small near the upper critical dimension, and indeed $u_{\rm I}^*$ emerges continuously from $u_0^* = 0$ as $\epsilon = 4 - d$ is increased from zero. At the non-trivial RG fixed point, $\epsilon$ may serve to provide a small effective expansion parameter for the perturbation expansion.
To lowest order in the non-linear coupling, (\[difrcr\]) yields at the Ising fixed point $r_{\rm I}^* = - \frac{1}{4} \, u_{\rm I}^* S_d \Lambda^{d-2}
= - \frac{1}{6} \, \epsilon \Lambda^2$, which describes a non-universal, fluctuation-induced downward shift of the critical temperature. Introducing the deviation $\tau = r - r_{\rm I}^* = a (T - T_c)$ from the true critical temperature $T_c$, one may rewrite the flow equation (\[difrcr\]) to obtain the recursion relation for this modified relevant [*running coupling*]{}: $$\frac{d {\tilde \tau}(\ell)}{d \ell} = {\tilde \tau}(\ell)
\left[ 2 - \frac{{\tilde u}(\ell)}{2} S_d \Lambda^{d-4} \right] \ .
\label{difrct}$$ Its solution in the vicinity of the Ising fixed point reads ${\tilde \tau}(\ell) = {\tilde \tau}(0) \exp\bigl[ \bigl( 2 -
\frac{\epsilon}{3} \bigr) \ell \bigr]$. Combining this result with ${\widetilde \xi}(\ell) = \xi(0) \, e^{-\ell}$, one identifies the correlation length exponent $\nu^{-1} = 2 - \frac{\epsilon}{3}$, or, in a consistent expansion to first order in $\epsilon = 4 - d$: $$\nu = \frac{1}{2} + \frac{\epsilon}{12} + O(\epsilon^2) \ , \quad
\eta = 0 + O(\epsilon^2) \ .
\label{icrtex}$$ As anticipated, the critical exponents depend only on the spatial dimension, not on the strength of the non-linear coupling or other non-universal parameters. Note that at $d_c = 4$, (\[difrcu\]) is solved by ${\tilde u}(\ell) = {\tilde u}(0) / [1 + 3 {\tilde u}(0) \ell / 16 \pi^2]$, a very slow approach to the Gaussian fixed point that induces [*logarithmic corrections*]{} to the mean-field critical exponents, see (\[xilgcr\]) below. We finally remark that the RG procedure generates novel coupling terms $\sim S^6$, $\nabla^2 S^4$, etc. To order $\epsilon^3$, their feedback into the recursion relations can however be safely neglected.
In summary, the renormalization group procedure as outlined above in Wilson’s momentum shell formulation allows us to [*derive*]{} hitherto phenomenological scaling laws, and thereby gain deeper insights into scale-invariant features. We have also seen that the number of relevant couplings (two at standard critical points, namely $\tau$ and $h$) equals the number of [*independent*]{} critical exponents. Below the [*upper critical dimension*]{}, fluctuation corrections modify the critical scaling drastically as compared to the mean-field predictions. Finally, perturbative calculations that are safely carried out in the UV regime may be employed to systematically compute scaling exponents through a power series in the dimensional parameter $\epsilon = d_c - d$.
Field Theory Approach to Critical Phenomena
===========================================
While Wilson’s momentum shell scheme renders the basic philosophy of the renormalization group transparent, it becomes computationally quite cumbersome once nested momentum integrals appear beyond the first order in perturbation theory. Unnecessary technical complications in evaluating fluctuation loops can be avoided by extending the UV cutoff to $\Lambda \to \infty$, at the price of divergences in dimensions $d \geq d_c$. However, we already know that the Gaussian theory governs the infrared properties in that dimensional regime; thus in statistical physics theses UV singularities do not really pose a troublesome issue. We may however employ powerful various tools from quantum field theory, proceed to formally renormalize the UV divergences, and thereby gain crucial information about the desired IR scaling limit, provided an IR-stable RG fixed point can be identified that allows us to connect the UV and IR regimes. This chapter provides a succinct overview of how to construct the perturbation expansion in terms of [*Feynman diagrams*]{} for one-particle irreducible vertex functions, proceeds to analyze the resulting UV singularities, and finally utilizes the renormalization group equation to identify fixed points and determine the accompanying critical exponents. For more extensive treatments of the field-theoretic RG approach to critical phenomena, see Refs. [@Ramond81]–[@Zinn93] and other excellent texts.
Perturbation expansion and Feynman diagrams
-------------------------------------------
We now generalize our analysis to a LGW Hamiltonian with [*continuous*]{} $O(n)$ order parameter symmetry $$\begin{aligned}
&&{\cal H}[{\vec S}] = \int \! d^dx \sum_{\alpha = 1}^n \biggl[ \frac{r}{2}
\, S^\alpha(x)^2 + \frac{1}{2} \, [\nabla S^\alpha(x)]^2 \nonumber \\
&&\qquad\qquad\qquad + \frac{u}{4!} \sum_{\beta = 1}^n S^\alpha(x)^2 \,
S^{\beta}(x)^2 \biggr] \ ,
\label{lgwhmn}\end{aligned}$$ which encapsulates the critical behavior for the Heisenberg model for a three-component vector order parameter ($n = 3$), the planar XY model (and equivalently, superfluids with complex scalar order parameter) for $n = 2$, reduces to Ising $Z_2$ symmetry for $n = 1$, and in fact describes the scaling properties of self-avoiding polymers in the limit $n \to 0$. As in (\[expgau\]), one constructs the perturbation expansion for arbitrary $N$-point functions $\langle \prod_i S^{\alpha_i} \rangle$ in terms of averages within the Gaussian ensemble with $u = 0$ (keeping $k_{\rm B} T = 1$). Diagrammatically, the Gaussian two-point functions or [*propagators*]{} $C_0(q) \, \delta^{\alpha \beta} = \delta^{\alpha \beta} / (r + q^2)$, diagonal in the field component indices, are represented through lines, to be connected through the non-linear [*vertices*]{} $- \frac{u}{6}$:
{width="0.6\columnwidth"}
In the presence of external fields $h^\alpha$, the partition function serves as [*generating functional*]{} for correlation functions (cumulants): $$\begin{aligned}
&&{\cal Z}[h] = \Big\langle \exp \! \int \!\! d^dx \sum_\alpha h^\alpha
S^\alpha \Big\rangle \ , \nonumber \\
&&\Big\langle \prod_i S^{\alpha_i} \Big\rangle_{(c)} = \prod_i
\frac{\delta (\ln) {\cal Z}[h]}{\delta h^{\alpha_i}} \Big\vert_{h = 0} \ .
\label{genfcc}\end{aligned}$$ The [*cumulants*]{} are graphically represented through [*connected Feynman diagrams*]{}; e.g., for the propagator:
{width="\columnwidth"}
Note that the second graph is a mere repetition of the first; indeed, upon defining the [*self-energy*]{} $\Sigma$ as the sum of all [*one-particle irreducible*]{} Feynman graphs that cannot be split into lower-order contributions simply by cutting a propagator line, one infers the following general structure for the full propagator $C(q)$:
{width="\columnwidth"}
The second line is a graphical depiction of [*Dyson’s equation*]{} that reads in Fourier space $C(q) = C_0(q) + C_0(q) \, \Sigma(q) \, C(q)$, solved by $C(q)^{-1} = C_0(q)^{-1} - \Sigma(q)$.
In order to similarly eliminate redundancies for arbitrary $N$-point functions, one proceeds with a Legendre transformation to construct the generating functional for [*vertex functions*]{}: $$\begin{aligned}
&&\Gamma[\Phi] = - \ln {\cal Z}[h] + \int \! d^dx \sum_\alpha h^\alpha \,
\Phi^\alpha \ , \nonumber \\
&&\Gamma^{(N)}_{\{ \alpha_i \}} = \prod_i^N
\frac{\delta \Gamma[\Phi]}{\delta \Phi^{\alpha_i}} \Big\vert_{h=0} \ ,
\label{genfcv}\end{aligned}$$ where $\Phi^\alpha = \delta \ln {\cal Z}[h] / \delta h^\alpha$. Through appropriate functional derivatives, these vertex functions can be related to the corresponding cumulants, for example for the two- and four-point functions: $$\begin{aligned}
&&\Gamma^{(2)}(q) = C(q)^{-1} \ , \nonumber \\
&&\Big\langle \prod_{i=1}^4 S(q_i) \Big\rangle_c = - \prod_{i=1}^4 C(q_i) \,
\Gamma^{(4)}(\{ q_i \}) \ .
\label{cumver}\end{aligned}$$ By means of these relations one easily confirms that the perturbation series for the vertex functions precisely consist of the [*one-particle irreducible*]{} Feynman graphs for the associated cumulants. Moreover, the perturbative expansion with respect to the non-linear coupling $u$ diagrammatically translates to a [*loop expansion*]{}. Explicitly, one obtains for the [*two-point vertex function*]{} to two-loop order (resulting from the first, third, and fourth graph above): $$\begin{aligned}
&&\Gamma^{(2)}(q) = r + q^2 + \frac{n+2}{6} \, u \int_k \frac{1}{r+k^2}
\label{2lpgm2} \\
&&\quad - \left( \frac{n+2}{6} \, u \right)^2 \int_k \frac{1}{r+k^2}
\int_{k'} \frac{1}{(r+{k'}^2)^2} \nonumber \\
&&\!\!\!\!\!\!\! - \frac{n+2}{18} \, u^2 \! \int_k \! \frac{1}{r+k^2}
\int_{k'} \! \frac{1}{r+{k'}^2} \, \frac{1}{r+(q-k-k')^2} \, ; \nonumber\end{aligned}$$ for the [*four-point vertex function*]{} to one-loop order, we just have the single Feynman diagram
{width="0.2\columnwidth"}
which yields at vanishing external wave vectors $$\Gamma^{(4)}(\{ q_i = 0 \}) = u - \, \frac{n+8}{6} \, u^2 \int_k
\frac{1}{(r+k^2)^2} \ .
\label{1lpgm4}$$
UV and IR divergences, renormalization
--------------------------------------
Let us now investigate the fluctuation correction (\[1lpgm4\]) to the four-point vertex function. In dimensions $d < 4$, we can safely set the UV cutoff $\Lambda \to \infty$, and obtain after rendering the integral dimensionless: $$\begin{aligned}
&&u \int \! \frac{d^dk}{(2 \pi)^d} \, \frac{1}{(r + k^2)^2} = \nonumber \\
&&\quad \frac{u \, r^{-2 + d/2}}{2^{d-1} \pi^{d/2} \Gamma(d/2)}
\int_0^\infty \frac{x^{d-1}}{(1 + x^2)^2} \, dx \ .
\label{intd<4}\end{aligned}$$ Observe that the true [*effective*]{} coupling in the perturbation expansion is not $u$, but $u \, r^{(d-4)/2} \to \infty$ as $r \to 0$, indicating the emerging [*infrared*]{} divergences that render a direct perturbative approach meaningless in the critical regime. We conclude once again that for $d < d_c = 4$, fluctuation corrections are IR-singular, and consequently expect the critical power laws to be modified as compared to the mean-field or Gaussian approximations. In contrast, the integral remains regular in the infrared for $d > 4$, but becomes ultraviolet-divergent. Indeed, keeping the cutoff finite, we have in dimensions larger or equal to the [*upper critical dimension*]{} $d_c = 4$: $$\int_0^\Lambda \! \frac{k^{d-1}}{(r + k^2)^2} \, dk \sim \left\{
\begin{array}{cc} \ln (\Lambda^2 / r) & \ d = 4 \\ \Lambda^{d-4} & \ d > 4
\end{array} \right. \ ,
\label{intd>4}$$ which both diverge as $\Lambda \to \infty$. Notice that at $d_c = 4$, the logarithmic IR and UV divergences are coupled, signaling the scale invariance of the LGW Hamiltonian in four dimensions.
One may take a convenient shortcut to determine the critical dimension through simple [*power counting*]{} in terms of an arbitrary momentum scale $\mu$. Lengths then scale as $[x] = \mu^{-1}$, wave vectors as $[q] = \mu$, and fields consequently have the (naive) scaling dimension $[S^\alpha(x)] = \mu^\zeta = \mu^{-1+d/2}$. As noted before, the fields become dimensionless in two dimensions. For the LGW Hamiltonian (\[lgwhmn\]) with continuous rotational symmetry in order parameter space, that is also just the [*lower critical dimension*]{}: For $d \leq d_{lc} = 2$, the system cannot maintain spatially homogeneous long-range order if $n \geq 3$; and at $d_{lc} = 2$ and for $n = 2$ only quasi-long-range order may exist with algebraically decaying correlations and temperature-dependent decay exponent (Berezinskii–Kosterlitz–Thouless scenario). For the couplings in (\[lgwhmn\]) one infers $[r] = \mu^2$, which means that the temperature control parameter constitutes a [*relevant*]{} coupling, and $[u] = \mu^{4-d}$, which is relevant for $d < 4$, [*marginal*]{} at $d_c = 4$, and (dangerously) irrelevant for $d > 4$. Furthermore, dimensional analysis confirms that fluctuation loops become UV-divergent only for the vertex functions $\Gamma^{(2)}(q)$, but only up to order $q^2$ in a long-wavelength expansion, and $\Gamma^{(4)}(\{ q_i = 0 \})$. The following table summarizes the mathematical distinctions and their physical implications in the different dimensional regimes.
------------ ----------- ----------------- --------------------
dim. perturb. $O(n)$ $\Phi^4$ critical
range series field theory behavior
$d \leq 2$ IR-sing. ill-defined no long-range
UV-conv. $u$ relevant order ($n \geq 2$)
$d < 4$ IR-sing. super-ren. non-classical
UV-conv. $u$ relevant exponents
$d_c = 4$ log. IR-/ renorm. logarithmic
UV-div. $u$ marginal corrections
$d > 4$ IR-reg. non-renorm. mean-field
UV-div. $u$ irrelevant exponents
------------ ----------- ----------------- --------------------
It is useful to perform the loop integrations in [*dimensional regularization*]{}; i.e., to assign the following values to wave vector integrals, even for non-integer $d$ and $\sigma$: $$\begin{aligned}
&&\int \! \frac{d^dk}{(2 \pi)^d} \,
\frac{k^{2 \sigma}}{\left( \tau + k^2 \right)^s} =
\label{dimreg} \\
&&\quad \frac{\Gamma(\sigma + d/2) \,
\Gamma(s - \sigma - d/2)}{2^d \, \pi^{d/2} \, \Gamma(d/2) \, \Gamma(s)} \
\tau^{\sigma - s + d/2} \ . \nonumber\end{aligned}$$ For $d < d_c$, where no UV divergences appear, this result follows directly by introducing spherical coordinates in momentum space. Beyond the upper critical dimension, essentially divergent surface integrals are discarded in (\[dimreg\]). UV singularities become manifest as [*dimensional poles*]{} in Euler’s $\Gamma$ functions.
We may now proceed with the renormalization program. The goal is to absorb the UV divergences into [*renormalized*]{} couplings that through this procedure become scale-dependent. Beginning with the order parameter susceptibility, we naturally demand that $\chi^{-1} = C(q=0)^{-1} = \Gamma^{(2)}(q=0) = \tau = r - r_c$, i.e., $\chi$ diverges at the true critical temperature $T_c$. From (\[2lpgm2\]) we thus obtain to first order in $u$, $$\begin{aligned}
&&r_c = - \frac{n+2}{6}\, u \!\int_k \frac{1}{r_c + k^2} + O(u^2) \nonumber\\
&&\quad = - \frac{n+2}{6} \, u \, S_d\, \frac{\Lambda^{d-2}}{d-2} + O(u^2)\ ,
\label{1lptcs}\end{aligned}$$ which is to be interpreted as a non-universal fluctuation-induced downward shift of the critical temperature, the analog of $r^*$ in Wilson’s scheme. As $\Lambda \to \infty$, $r_c$ becomes quadratically UV-divergent near four dimensions; this divergence becomes absorbed into the new proper temperature variable $\tau$ by means of an [*additive renormalization*]{}. Inserting (\[1lptcs\]) into (\[2lpgm2\]) then yields to order $u$ $$\chi(q)^{-1} = q^2 + \tau \, \biggl[ 1 - \frac{n+2}{6} \, u \! \int_k \!
\frac{1}{k^2 (\tau + k^2)} \biggr] \ .
\label{1lpsus}$$
At the upper critical dimension, (\[1lpsus\]) and (\[1lpgm4\]) are logarithmically divergent as $\Lambda \to \infty$, showing up as $1/\epsilon$ poles in the dimensionally regularized integral values. These UV poles are subsequently absorbed into [*renormalized*]{} fields $S_R^\alpha$ and parameters through the following [*multiplicative renormalization*]{} prescription: $$\begin{aligned}
&&S_R^\alpha = Z_S^{1/2} \, S^\alpha \ \Rightarrow \
\Gamma_R^{(N)} = Z_S^{-N/2} \, \Gamma^{(N)} \ ;
\label{fldren} \\
&&\tau_R = Z_\tau \, \tau \, \mu^{-2} \ , \quad \
u_R = Z_u \, u \, A_d \, \mu^{d-4} \ ,
\label{cplren}\end{aligned}$$ which defines [*dimensionless renormalized couplings*]{} $\tau_R$ and $u_R$, and where $A_d = \Gamma(3-d/2) / 2^{d-1} \, \pi^{d/2}$ is a regular (near $d_c = 4$) geometric factor. To this end, one must carefully avoid the IR-singular regime, i.e., evaluate the fluctuation integrals at a safe [*normalization point*]{}, e.g., $\tau_R = 1$ (or $q = \mu$). In the [*minimal subtraction*]{} procedure, the renormalization constants $Z_S$ in (\[fldren\]) and $Z_\tau$, $Z_u$ in (\[cplren\]) are chosen to contain [*only*]{} the $1/\epsilon$ poles and their residua. This leads to the following $Z$ factors $$\begin{aligned}
&&O(u_R): \ Z_\tau = 1 - \frac{n + 2}{6} \, \frac{u_R}{\epsilon} \ ,
\label{1lpztt} \\
&&\qquad\quad\,\ Z_u = 1 - \frac{n + 8}{6} \, \frac{u_R}{\epsilon} \ ,
\label{1lpztu} \\
&&O(u_R^2): \ Z_S = 1 + \frac{n + 2}{144} \, \frac{u_R^2}{\epsilon} \ ,
\label{2lpzts}\end{aligned}$$ all calculated to first non-trivial order in $u_R$ by means of dimensional regularization (\[dimreg\]) and within the minimal subtraction prescription. $Z_\tau$ and $Z_u$ follow directly from the one-loop results (\[1lpsus\]) and (\[1lpgm4\]), whereas $Z_S = 1$ to $O(u_R)$ due to the absence of any wave vector dependence in the “Hartree” loop, whence field renormalization only ensues to two-loop order from the rightmost “sunset” Feynman diagram in the propagator self-energy, and the $1/\epsilon$ pole in the final expression in (\[2lpgm2\]).
RG equation and critical exponents
----------------------------------
Through the selection of a normalization point well outside the critical regime, the renormalized fields and parameters in (\[fldren\]), (\[cplren\]) explicitly carry the momentum scale $\mu$. On the other hand, the [*unrenormalized*]{} quantities, including the $N$-point vertex functions, naturally do [*not*]{} depend on this arbitrary scale $\mu$: $$0 = \frac{d}{d \mu} \, \Gamma^{(N)}(\tau,u) = \frac{d}{d \mu} \left[
Z_S^{N/2} \, \Gamma_R^{(N)}(\mu,\tau_R,u_R) \right] .
\label{indgmu}$$ Carrying out the derivative with respect to $\mu$ by taking into account the scale dependence of $Z_s$, $\tau_R$, and $u_R$, (\[indgmu\]) can be rewritten as a partial differential equation, $$\begin{aligned}
&&\biggl[ \mu \, \frac{\partial}{\partial \mu} + \frac{N}{2} \, \gamma_S
+ \gamma_\tau \, \tau_R \, \frac{\partial}{\partial \tau_R} + \beta_u \,
\frac{\partial}{\partial u_R} \biggr] \nonumber \\
&&\qquad\qquad \Gamma_R^{(N)}(\mu,\tau_R,u_R) = 0 \ .
\label{rgeqvf}\end{aligned}$$ This Gell-Mann–Low [*renormalization group equation*]{} carries crucial information on the fundamental scale dependence of the [*renormalized*]{} physical system, here rendered explicit for the vertex functions. In (\[rgeqvf\]) we have introduced [*Wilson’s flow functions*]{} defined as $$\begin{aligned}
&&\gamma_S = \mu \, \frac{\partial}{\partial \mu}\Big\vert_0 \, \ln Z_S
\nonumber \\
&&\quad\ = - \frac{n + 2}{72} \, u_R^2 + O(u_R^3) \ ,
\label{gamfld} \\
&&\gamma_\tau = \mu \, \frac{\partial}{\partial \mu}\Big\vert_0 \, \ln
\frac{\tau_R}{\tau} \nonumber \\
&&\quad\ = - 2 + \frac{n + 2}{6} \, u_R + O(u_R^2) \ ,
\label{gamtau}\end{aligned}$$ where the second lines follow from the lowest-order results (\[2lpzts\]) and (\[1lpztt\]), and the [RG beta function]{} for the non-linear coupling $u$, $$\begin{aligned}
&&\beta_u = \mu \, \frac{\partial}{\partial \mu}\Big\vert_0 u_R = u_R
\Bigl[ d - 4 + \mu \, \frac{\partial}{\partial \mu} \Big\vert_0 \, \ln Z_u
\Bigr] \nonumber \\
&&\quad\ = u_R \Bigl[ -\epsilon + \frac{n + 8}{6} \, u_R + O(u_R^2) \Bigr]\ ,
\label{betafu}\end{aligned}$$ where (\[1lpztu\]) has been inserted.
The first-order linear partial differential equation (\[rgeqvf\]) can next be formally solved via the standard [*method of characteristics*]{}; to this end, one lets $\mu \to \mu(\ell) = \mu \, \ell$, with a dimensionless scale parameter $\ell$; note that in contrast to the convention in Chap. 2.5, the IR regime is now reached in the limit $\ell \to 0$. Inserting this parametrization into the RG equation (\[rgeqvf\]), one obtains an equivalent set of coupled first-order ordinary differential equations, namely the [*RG flow equations*]{} for the [*running couplings*]{} $$\ell \, \frac{d{\tilde \tau}(\ell)}{d\ell} =
{\tilde \tau}(\ell) \, \gamma_\tau(\ell) \ , \quad
\ell \, \frac{d{\tilde u}(\ell)}{d\ell} = \beta_u(\ell) \ ,
\label{runcpl}$$ with initial values ${\tilde \tau}(1) = \tau_R$, ${\tilde u}(1) = u_R$, and similar differential equations for the $N$-point vertex functions, which involve their overall naive scaling dimensions and the anomalous contributions stemming from the field renormalization, as encoded in (\[gamfld\]).
For example, for the susceptibility $\chi(q) = \Gamma^{(2)}(q)^{-1}$, one has $\chi_R(\mu,\tau_R,u_R,q)^{-1} = \mu^2\, {\hat \chi}_R(\tau_R,u_R,q/\mu)^{-1}$, and its RG flow correspondingly integrates to $$\chi_R(\ell)^{-1} = \chi_R(1)^{-1} \, \ell^2 \, \exp\biggl[ \int_1^\ell
\gamma_S(\ell') \, \frac{d\ell'}{\ell'} \biggr] \ .
\label{susscl}$$ Near an [*infrared-stable*]{} RG fixed point $u^*$, i.e., a zero of the RG beta function $\beta_u(u^*) = 0$ with $\beta_u'(u^*) > 0$, the flow equation for the running temperature variable is readily solved: ${\tilde \tau}(\ell) \approx \tau_R \, \ell^{\gamma_\tau^*}$, where $\gamma_\tau^* = \gamma_\tau(u^*)$. Inserting this and the fixed point value $\gamma_S^* = \gamma_S(u^*)$ into (\[susscl\]) yields the following general scaling form $$\chi_R(\tau_R,q)^{-1} \approx \mu^2 \, \ell^{2 + \gamma_S^*} \,
{\hat \chi}_R(\tau_R \, \ell^{\gamma_\tau^*}, u^*, q / \mu \, \ell)^{-1} \ .
\label{gensus}$$ With the [*matching*]{} condition $\ell = |q| / \mu$, one thus recovers (\[corsch\]), with the identifications $\eta = - \gamma_S^*$ and $\nu = - 1 / \gamma_\tau^*$ for the critical exponents.
Our perturbative RG analysis of the $O(n)$-symmetric LGW Hamiltonian (\[lgwhmn\]) yielded the one-loop beta function (\[betafu\]), whose zeros are (i) the Gaussian fixed point $u_0^* = 0$, stable for $\epsilon < 0$ or $d > d_c = 4$, obviously resulting in the mean-field exponents $\eta = 0$ and $\nu = \frac12$; and (ii) the non-trivial [*Heisenberg fixed point*]{} $$u_H^* = \frac{6 \, \epsilon}{n + 8} + O(\epsilon^2) \ ,
\label{heisfp}$$ which exists and becomes IR-stable for $\epsilon > 0$, i.e., in dimensions $d < 4$. This allows us to compute the critical exponents in a systematic $\epsilon = 4 - d$ expansion, $$\begin{aligned}
&&\eta = \frac{n + 2}{2 \, (n + 8)^2} \, \epsilon^2 + O(\epsilon^3) \ ,
\label{2lpeta} \\
&&\nu^{-1} = 2 - \frac{n + 2}{n + 8} \, \epsilon + O(\epsilon^2) \ .
\label{1lpnur}\end{aligned}$$ Aside from the dimensionality, these values only depend on the number of order parameter components $n$. Note that (\[1lpnur\]) reduces to the Ising exponent values (\[icrtex\]) for $n = 1$; in the limit $n \to \infty$ one also finds the correct exponents for the exactly solvable [*spherical model*]{}, namely $\eta = 0$ and $\nu = 1 / (2-\epsilon) = 1 / (d-2)$, which diverges at the lower critical dimension $d_{lc} = 2$. At the upper critical dimension $d_c = 4$, the solution of the flow equation for the running non-linear coupling reads $${\tilde u}(\ell) = \frac{u_R}{1 - \frac{n + 8}{6} \, u_R \, \ln \ell} \ ,
\label{ufld=4}$$ whence approximately $${\tilde \tau}(\ell) \sim
\frac{\tau_R}{\ell^2 (\ln |\ell|)^{(n + 2) / (n + 8)}} \ ,
\label{tfld=4}$$ which in turn implies that the correlation length divergence picks up [*logarithmic*]{} corrections to the mean-field behavior, $$\xi \propto \tau_R^{-1/2} \, (\ln \tau_R)^{(n + 2) / 2 (n+8)} \ .
\label{xilgcr}$$
The field-theoretic formulation of the renormalization group provides us with an elegant and powerful tool to extract the proper infrared scaling properties in low dimensions $d \leq d_c$ from a continuum theory via a careful analysis of its ultraviolet singularities that appear in dimensions $d \geq d_c$. The renormalization group equation carries information on the scale dependence of physical parameters and correlation functions, and allows to make connections between the UV and IR limits provided a stable RG fixed point can be identified. One may then systematically derive scaling laws, and acquire a thorough understanding of the origin of universality and its realm of validity for a specific physical system. Moreover, we have seen that a perturbative analysis allows a controlled computation of critical exponents (and also of scaling functions) in the framework of a dimensional expansion near the upper critical dimension. This $\epsilon$ expansion certainly provides useful information on overall trends, and can in some instances even be rendered to a precision calculation if sufficiently high orders in the perturbation series can be evaluated and subsequently be refined through Borel resummations. In addition, modern non-perturbative “exact” numerical RG methods have succeeded in yielding very accurate results. It should also be stressed that it is of course the very concept of universality that also allows us to infer meaningful information from numerical simulations of simplified lattice or continuum models.
Critical Dynamics
=================
We now venture to investigate [*dynamic*]{} critical phenomena near continuous phase transitions, first in the vicinity of critical point in thermal equilibrium, and later at genuine non-equilibrium phase transitions in externally driven, non-isolated systems. The natural time scale separation between the slow kinetics of the order parameter (along with any other conserved fields) and fast, non-critical degrees of freedom suggests a phenomenological description in terms of non-linear stochastic Langevin-type differential equations, and allows a generalization of universal scaling laws to time-dependent phenomena. Distinct dynamical universality classes ensue dependent on the order parameter itself representing a conserved quantity or not, and potentially its dynamical coupling to other conserved hydrodynamic modes [@Hohenberg77]. We begin by writing down scaling laws for dynamical response and correlation functions, and subsequently introduce effective mesoscopic Langevin equations, with stochastic noise correlations that near thermal equilibrium are constrained by fluctuation-dissipation relations. It is then demonstrated how such non-linear stochastic partial differential equations can be mapped onto a field theory representation via the Janssen–De Dominicis response functional, which in turn may be analyzed by means of the field-theoretic renormalization group tools developed in the previous chapter. We will specifically construct a dynamic perturbation theory expansion and determine the universal scaling behavior along with the dynamic critical exponents for the relaxational models A and B with non-conserved and conserved order parameter, respectively; for more in-depth treatments, the reader is referred to Refs. [@Janssen79]–[@Tauber07] and [@Tauberxx chaps. 4,5]. In addition to purely relaxational kinetics, we shall also address the critical dynamics of isotropic ferromagnets [@Frey94], as well as generic scale invariance and non-equilibrium phase transitions in driven diffusive systems [@Schmittmann95].
Dynamical scaling hypothesis
----------------------------
Let us first recall the behavior of the static order parameter correlation function and susceptibility near a critical point ($h = 0$ and $\tau \to 0$), captured by the scaling laws (\[corsch\]) and (\[spcorr\]), and induced by the divergence of the characteristic [*correlation length*]{}, $\xi(\tau) \sim |\tau|^{- \nu}$. As spatially correlated regions grow tremendously upon approaching the phase transition, one expects the typical relaxation time associated with the order parameter kinetics to increase as well, $t_c(\tau) \sim \xi(\tau)^z \sim |\tau|^{- z \nu}$. This phenomenon of [*critical slowing-down*]{} is governed by the [*dynamic critical exponent*]{} $z = \nu_t / \nu$, which can also be visualized as the ratio between the exponents that describe the divergence of correlations in the temporal and spatial “directions”, respectively. We may thus write down a [*dynamic scaling*]{} ansatz for the corresponding wavevector-dependent [*characteristic frequency*]{} scale, $$\omega_c(\tau,q) = |q|^z \, {\hat \omega}_\pm(q \, \xi) \ ,
\label{dynscf}$$ with ${\hat \omega}_\pm(y \to \infty) \to {\rm const.}$, whence the [*critical dispersion relation*]{} becomes $\omega_c(0,q) \sim |q|^z$.
We are particularly interested in describing the time dependence for the order parameter response and correlation functions: $$\begin{aligned}
&&\chi(x-x',t-t') = \frac{\partial \langle S(x,t) \rangle}
{\partial h(x',t')} \bigg\vert_{h=0} \ ,
\label{dynres} \\
&&C(x,t) = \langle S(x,t) \, S(0,0) \rangle - \langle S \rangle^2 \ ,
\label{dyncor}\end{aligned}$$ where we are considering a stationary dynamical regime where spatial and temporal time translation invariance holds. In [*thermal equilibrium*]{} (only !), the spatio-temporal Fourier transforms of these functions are intimately connected through the [*fluctuation-dissipation theorem*]{} (FDT) $$C(q,\omega) = 2 k_{\rm B} T \, {\rm Im} \ \frac{\chi(q,\omega)}{\omega} \ .
\label{fldist}$$ Generalizing the static scaling laws (\[corsch\]) and (\[spcorr\]), we may then formulate the [*dynamical scaling hypothesis*]{} for the asymptotic critical properties of the time-dependent susceptibility and correlation function: $$\begin{aligned}
&&\chi(\tau,q,\omega) = |q|^{- 2 + \eta} \,
{\hat \chi}_\pm(q \, \xi , \omega \, \xi^z) \ ,
\label{dynscr} \\
&&C(\tau,x,t) = |x|^{-(d - 2 + \eta)} \, {\widetilde C}_\pm(x/\xi, t/t_c) \ .
\label{dynscc}\end{aligned}$$ As a consequence of the stringent constraints imposed by the FDT (\[fldist\]), the same [*three*]{} independent critical exponents $\nu$, $\eta$, and $z$ characterize the universal scaling regimes in both (\[dynscr\]) and (\[dynscc\]). Away from thermal equilibrium, where the FDT restrictions do not apply, the dynamic response and correlation functions are however in general characterized by distinct scaling exponents. We remark that appropriate variants of the dynamical scaling hypothesis may also be postulated for transport coefficients.
Langevin dynamics and Gaussian theory
-------------------------------------
The critical slowing-down of the order parameter kinetics produces an effective time-scale separation between the critical degrees of freedom, additional conserved hydrodynamic modes that might be present, and all other comparatively “fast” variables. This observation naturally calls for a [*mesoscopic Langevin description*]{} of critical dynamics, where the fast degrees of freedom are treated as mere white [*noise*]{} that randomly affects the few slow modes in the system. In such a [*coarse-grained*]{} picture, one writes down coupled stochastic equations of motion for the order parameter and perhaps any other conserved fields that reflect their intrinsic microscopic reversible dynamics as well as irreversible relaxation kinetics, the latter connected in thermal equilibrium with the noise strengths through [*Einstein relations*]{} or FDTs. Generally the various possible [*mode couplings*]{} of the order parameter to additional conserved, and consequently diffusively slow modes leads to a splitting of the static into several [*dynamic universality classes*]{} [@Hohenberg77; @Folk06; @Tauberxx].
Here we shall assume that the order parameter field is decoupled from any other slow modes, and first focus on its purely relaxational kinetics [@Tauberxx Chaps. 4,5]. If the order parameter itself is not a conserved quantity, any deviation from thermal equilibrium will just tend to relax back to the minimizing configuration of the free energy, e.g. given by the $O(n$)-symmetric LGW Hamiltonian (\[lgwhmn\]): $$\frac{\partial S^\alpha(x,t)}{\partial t} =
- D \, \frac{\delta {\cal H}[{\vec S}]}{\delta S^\alpha(x,t)}
+ \zeta^\alpha(x,t) \, ,
\label{relmda}$$ with Gaussian white noise that is fully characterized by its first two moments, $$\begin{aligned}
&&\langle \zeta^\alpha(x,t) \rangle = 0 \ , \nonumber \\
&&\langle \zeta^\alpha(x,t) \, \zeta^\beta(x',t') \rangle = \\
\label{stnmda}
&&\qquad 2 D k_{\rm B} T \delta(x-x') \delta(t-t') \delta^{\alpha \beta} \ .
\nonumber\end{aligned}$$ As can be inferred from the associated Fokker–Planck equation, the [*Einstein relation*]{} that connects the noise correlator strength with the relaxation constant $D$ and temperature guarantees that the probability distribution for the field $S^\alpha$ asymptotically approaches the canonical stationary distribution ${\cal P}[{\vec S},t] \to {\cal P}_s[{\vec S}] \propto
\exp (-{\cal H}[{\vec S}] / k_{\rm B} T)$ as $t \to \infty$.
If the order parameter is [*conserved*]{} under the dynamics, satisfying a continuity equation, its spatial fluctuations can only relax [*diffusively*]{}; as a consequence, one needs to replace the constant relaxation rate $D$ by the diffusion operator $- D \, \nabla^2$, both in the Langevin equation (\[relmda\]) and the noise correlation (\[stnmda\]). In the following, we shall treat both these situations simultaneously, letting $D \to D (i \nabla)^a$, where either $a = 0$, corresponding to the purely relaxational [*model A*]{} of critical dynamics as appropriate for a non-conserved order parameter; or $a = 2$, which describes [*model B*]{} with a conserved order parameter field.
Let us again begin with the [*Gaussian*]{} or [*mean-field approximation*]{}, where we set the non-linear coupling $u = 0$. A Fourier transform in space and time according to $S^\alpha(x,t) = \int \frac{d^dq}{(2 \pi)^d} \int \frac{d\omega}{2 \pi} \,
S(q,\omega) \, e^{i q \cdot x - i \omega t}$ of the thus linearized Langevin equation (\[relmda\]) yields $$\begin{aligned}
&&\left[ -i \omega + D q^a (r + q^2) \right] S^\alpha(q,\omega) = \nonumber\\
&&\qquad\qquad D q^a \, h^\alpha(q,\omega) + \zeta^\alpha(q,\omega) \ ,
\label{rellin}\end{aligned}$$ with $\langle \zeta^\alpha(q,\omega) \rangle = 0$ and $$\begin{aligned}
&&\langle \zeta^\alpha(q,\omega) \zeta^\beta(q',\omega') \rangle =
2 k_{\rm B} T \, D q^a \nonumber \\
&&\qquad\quad (2 \pi)^{d+1} \delta(q + q') \delta(\omega + \omega')
\delta^{\alpha\beta} \ ,
\label{stnfsp}\end{aligned}$$ and where the external field term $- \sum_\alpha h^\alpha S^\alpha$ has been added to the LGW Hamiltonian (\[lgwhmn\]). A derivative of (\[rellin\]) with respect to the external field then immediately gives the [*dynamic response function*]{} $$\begin{aligned}
&&\chi_0^{\alpha\beta}(q,\omega)= \frac{\partial \langle S^\alpha(q,\omega)
\rangle}{\partial h^\beta(q,\omega)} \Big\vert_{h = 0} \nonumber \\
&&\qquad\qquad = D q^a G_0(q,\omega) \delta^{\alpha\beta} \ ,
\label{relres} \\
&&G_0(q,\omega) = \frac{1}{-i \omega + D q^a (r + q^2)} \ . \nonumber \end{aligned}$$ Its temporal Fourier backtransform of course satisfies [*causality*]{}: $G_0(q,t)$ vanishes for $t < 0$, and reads $$G_0(q,t) = e^{-D q^a (r + q^2) t} \, \Theta(t) \ ,
\label{relret}$$ from which we infer the characteristic relaxation rate $t_c^{-1} = D q^a (r + q^2)$: For model A ($a = 0$), the order parameter relaxes diffusively at the critical point (i.e., $z = 2$), while for model B ($a = 2$) critical slowing-down induces a crossover from $D r q^2$ to $D q^4$ as $r \to 0$ ($z = 4$). For $h^\alpha = 0$, the dynamic correlation function is readily obtained from (\[rellin\]) and (\[stnfsp\]): $$\begin{aligned}
&&\langle S^\alpha(q,\omega) \, S^\beta(q',\omega') \rangle_0 = C_0(q,\omega)
\nonumber \\
&&\qquad (2 \pi)^{d+1} \delta(q + q') \delta(\omega + \omega')
\delta^{\alpha\beta} \ , \nonumber \\
&&C_0(q,\omega) = \frac{2 k_{\rm B} T \, D q^a}{\omega^2 +
[D q^a (r+q^2)]^2}
\label{relcor} \\
&&\qquad\quad\;\ = 2 k_{\rm B}T D q^a \, |G_0(q,\omega)|^2 \ , \nonumber \\
&&C_0(q,t) = \frac{k_{\rm B} T}{r + q^2} \, e^{- D q^a \, (r + q^2) |t|} \ .
\label{relcot}\end{aligned}$$ Comparing these results with (\[dynscr\]) and (\[dynscc\]), one again identifies the static [*Gaussian critical exponents*]{} $\nu = \frac12$ and $\eta = 0$, and the [*mean-field dynamic exponent*]{} $z = 2 + a$ for the purely relaxational models A and B.
Field theory representations of Langevin dynamics
-------------------------------------------------
This subsection describes how stochastic Langevin equations of motion can be mapped onto continuous field theory representations. To this end, we consider the following general coupled Langevin equations for some mesoscopic stochastic variables $S^\alpha$: $$\begin{aligned}
&&\frac{\partial S^\alpha(x,t)}{\partial t} = F^\alpha[S](x,t)
+ \zeta^\alpha(x,t) \ ,
\label{langen} \\
&&\langle \zeta^\alpha(x,t) \zeta^\beta(x',t') \rangle = \nonumber \\
&&\quad 2 L^\alpha[S] \, \delta(x-x') \delta(t-t') \delta^{\alpha \beta} \ .
\label{noigen}\end{aligned}$$ Naturally we assume $\langle \zeta^\alpha(x,t) \rangle = 0$ here, since a non-vanishing mean of the [*stochastic forces*]{} or [*noise*]{} could just be included in the [*systematic*]{} forces $F^\alpha[S]$. Note that the [*noise correlator*]{} $L^\alpha$ may be an operator, as is the case for conserved variables, and could also be a functional of the slow fields $S^\alpha$. The crucial input is that we assume the noise history to represent a [*Gaussian*]{} stochastic process, whose probability distribution if completely determined by the second moment (\[noigen\]): $${\cal W}[\zeta] \propto \exp \, \biggl[ - \frac{1}{4} \! \int \!\! d^dx \!\!
\int_0^{t_f} \!\! dt \sum_\alpha \zeta^\alpha (L^\alpha)^{-1} \zeta^\alpha
\biggr] . \
\label{noidis}$$ Switching dynamical variables from the noise $\zeta^\alpha$ to the slow stochastic fields $S^\alpha$ yields ${\cal W}[\zeta] \, {\cal D}[\zeta] =
{\cal P}[S] {\cal D}[S] \propto e^{- {\cal G}[S]} {\cal D}[S]$, with the [*Onsager-Machlup functional*]{} providing the associated exponential weight that may be viewed as a field theory action: $$\begin{aligned}
&&{\cal G}[S] = \frac{1}{4} \int \! d^dx \int_0^{t_f} \! dt \sum_\alpha
\left( \partial_t S^\alpha - F^\alpha[S] \right) \nonumber \\
&&\qquad\qquad\quad \left[ (L^\alpha)^{-1} \left( \partial_t S^\alpha
- F^\alpha[S] \right) \right] .
\label{onsmch}\end{aligned}$$ The observant reader will have noticed that the functional determinant stemming from the variable transformation has been ignored here; however, upon utilizing a [*forward*]{} time discretization, i.e., the Itô interpretation for non-linear stochastic processes, this functional determinant turns out to be constant, and can simply be absorbed into the functional integral measure. Notice also that the overall normalization $\int {\cal D}[\zeta] W[\zeta] = 1$ implies the corresponding “partition function” to be unity, and hence to carry no information, in stark contrast with thermal equilibrium statistical mechanics. While the Onsager–Machlup functional (\[onsmch\]) provides a desirable field theory representation of stochastic Langevin dynamics, it is also plagued by two technical problems: First, it contains $(L^\alpha)^{-1}$, which for conserved fields entails an inverse differential operator or Laplacian Green’s function; second, it includes the square of the systematic force terms $F^\alpha[S]$ and consequently highly non-linear contributions. It is thus beneficial to partially linearize the above action by means of a Hubbard–Stratonovich transformation, at the expense of introducing an [*additional dynamical field variable*]{}.
In order to completely avoid any possible singularities incorporated in the inverse operator $(L^\alpha)^{-1}$, we follow another more direct route here. The goal is to compute averages of observables $A$ that should be functionals of the slow modes $S^\alpha$ over noise “histories”: $\langle A[S]
\rangle_\zeta \propto \int {\cal D}[\zeta] \, A[S(\zeta)] \, W[\zeta]$. Inserting a rather involved representation of unity in terms of a product of Dirac delta distributions on each space-time point, and subsequently writing these as integrals over [auxiliary fields]{} ${\widetilde S}^\alpha$ along the imaginary axis, $1 = \int \! {\cal D}[S] \prod_\alpha \prod_{(x,t)} \delta \,
\bigl( \partial_t S^\alpha(x,t) - F^\alpha[S](x,t) - \zeta^\alpha(x,t) \bigr) =
\int \! {\cal D}[i {\widetilde S}] \! \int \! {\cal D}[S] \exp \, \bigl[
- \int \! d^dx \! \int \! dt \sum_\alpha {\widetilde S}^\alpha \bigl(
\partial_t \, S^\alpha - F^\alpha - \zeta^\alpha \bigr) \bigr]$, we arrive at $$\begin{aligned}
&&\langle A[S] \rangle_\zeta \propto \int \! {\cal D}[i{\widetilde S}]
\int \! {\cal D}[S] \, A[S] \int \! {\cal D}[\zeta] \nonumber \\
&&\quad \exp \, \biggl[ - \! \int \! d^dx \! \int \! dt \! \sum_\alpha
{\widetilde S}^\alpha \bigl( \partial_t S^\alpha - F^\alpha[S] \bigr) \biggr]
\nonumber \\
&&\ \exp \, \biggl( - \!\int \!\! d^dx \! \int \!\! dt \sum_\alpha \biggl[
\frac{\zeta^\alpha (L^\alpha)^{-1} \zeta^\alpha}{4} - {\widetilde S}^\alpha
\zeta^\alpha \biggr] \biggr) \ . \nonumber\end{aligned}$$ Performing the Gaussian integral over the noise $\zeta^\alpha$ finally yields $$\begin{aligned}
&&\langle A[S] \rangle_\zeta = \int {\cal D}[S] \, A[S] \, {\cal P}[S] \, ,
\nonumber \\
&&{\cal P}[S] \propto \int {\cal D}[i {\widetilde S}] \,
e^{- {\cal A}[{\widetilde S},S]} \ .
\label{noisav}\end{aligned}$$ with the [*Janssen–De Dominicis response functional*]{} $$\begin{aligned}
&&{\cal A}[{\widetilde S},S] = \int \! d^dx \int_0^{t_f} \! dt
\label{janded} \\
&&\qquad \sum_\alpha \Bigl[ {\widetilde S}^\alpha \bigl( \partial_t \,
S^\alpha - F^\alpha[S] \bigr) - {\widetilde S}^\alpha L^\alpha[S] \,
{\widetilde S}^\alpha \Bigr] \ . \nonumber\end{aligned}$$ The stochastic dynamics is now encoded in [*two*]{} distinct sets of mesoscopic fields, namely the original slow variables $S^\alpha$ and the associated auxiliary fields ${\widetilde S}^\alpha$. Once again, the Gaussian noise normalization implies $\int \! {\cal D}[i{\widetilde S}] \int \! {\cal D}[S] \,
e^{- {\cal A}[{\widetilde S},S]} = 1$; furthermore, integrating out the auxiliary fields ${\widetilde S}^\alpha$ recovers the Onsager–Machlup functional (\[onsmch\]).
Specifically for the purely relaxational models A and B, the response functional reads (we now set $k_{\rm B} T = 1$): $$\begin{aligned}
&&{\cal A}[{\widetilde S},S] = \nonumber \\
&& \int \! d^dx \! \int \! dt \sum_\alpha \, \Bigl( {\widetilde S}^\alpha
\left[ \partial_t + D (i \nabla)^a (r - \nabla^2) \right] S^\alpha
\nonumber \\
&&\qquad\qquad - D {\widetilde S}^\alpha (i \nabla)^a {\widetilde S}^\alpha
- D {\widetilde S}^\alpha (i \nabla)^a h^\alpha \nonumber \\
&&\qquad\quad + D \, \frac{u}{6} \sum_\beta {\widetilde S}^\alpha
(i \nabla)^a S^\alpha S^\beta S^\beta \Bigr) \, ,
\label{resfab}\end{aligned}$$ The first two lines here represent the Gaussian action ${\cal A}_0$, and the term $\sim u$ the non-linear contributions. By means of (\[noisav\]) and (\[resfab\]), the dynamical order parameter susceptibility becomes $$\begin{aligned}
&&\chi^{\alpha\beta}(x-x',t-t') = \frac{\delta \langle S^\alpha(x,t) \rangle}
{\delta h^\beta(x',t')} \Big\vert_{h = 0} \nonumber \\
&&\qquad = D \big\langle S^\alpha(x,t) (i \nabla)^a \,
{\widetilde S}^\beta(x',t') \big\rangle \, ;\end{aligned}$$ i.e., the response function can be expressed as an expectation value that involves both order parameter $S^\alpha$ and auxiliary fields ${\widetilde S}^\beta$, whence the latter are also referred to as [*“response” fields*]{}. Invoking causality and time inversion symmetry, it is a straightforward exercise to derive the [*fluctuation-dissipation theorem*]{}, equivalent to (\[fldist\]): $$\begin{aligned}
&&\chi^{\alpha \beta}(x-x',t-t') = \nonumber \\
&&\quad \Theta(t-t') \,
\frac{\partial}{\partial t'} \, \big\langle S^\alpha(x,t) S^\beta(x',t')
\big\rangle \, .
\label{resfdt}\end{aligned}$$ In analogy with static, equilibrium statistical field theory (\[genfcc\]), one defines the [*generating functional*]{} for correlation functions and cumulants, $$\begin{aligned}
&&{\cal Z}[{\tilde j},j] = \Big\langle e^{\int \! d^dx \! \int \! dt
\sum_\alpha \left( {\tilde j}^\alpha \, {\widetilde S}^\alpha + j^\alpha \,
S^\alpha \right)} \Big\rangle \, ,
\label{resgen} \\
&&\Big\langle \prod_{ij} S^{\alpha_i} {\widetilde S}^{\alpha_j}
\Big\rangle_{(c)} \! = \prod_{ij} \frac{\delta}{\delta j^{\alpha_i}}
\frac{\delta (\ln) {\cal Z}[{\tilde j},j]}{\delta {\tilde j}^{\alpha_j}}
\Big\vert_{{\tilde j} = j = 0} \, . \nonumber\end{aligned}$$
Dynamic perturbation theory
---------------------------
We may now proceed and treat the non-linear terms $\sim u$ by means of a [*perturbation expansion*]{}, $$\begin{aligned}
&&\Big\langle \prod_{ij} S^{\alpha_i} \, {\widetilde S}^{\alpha_j}
\Big\rangle = \frac{\langle \prod_{ij} S^{\alpha_i} \,
{\widetilde S}^{\alpha_j} \, e^{-{\cal A}_{\rm int}[{\widetilde S},S]}
\rangle_0} {\langle e^{-{\cal A}_{\rm int}[{\widetilde S},S]} \rangle_0}
\nonumber \\
&&= \Big\langle \prod_{ij} S^{\alpha_i} \, {\widetilde S}^{\alpha_j}
\sum_{l=0}^\infty \frac{1}{l!} \left( -{\cal A}_{\rm int}[{\widetilde S},S]
\right)^l \Big\rangle_0 \, .
\label{dynper}\end{aligned}$$ Since the denominator is one owing to noise normalization, there are [*no “vacuum” contributions*]{} in this dynamic field theory. Note furthermore that [*causality*]{} implies $\langle {\widetilde S}^\alpha(q,\omega) \, {\widetilde S}^\beta(q',\omega')
\rangle_0 = 0$. From the Gaussian action ${\cal A}_0$ (with $u = 0$), one immediately recovers the expressions (\[relres\]) and (\[relcor\]) for $G_0(q,\omega)$ and $C_0(q,\omega)$, respectively. Since the dynamic correlation function can be expressed in terms of the noise strength and the response function, the graphical representation in terms of Feynman diagrams can be based entirely on the causal [*response propagators*]{} $G_0(q,\omega)$, represented as [*directed*]{} lines (we use the convention that time propagates from right to left) that connect ${\widetilde S}^\beta$ to $S^\alpha$ fields, and join at either two-point noise or four-point non-linear relaxation [*vertices*]{}, all subject to wave vector and frequency conservation as a consequence of spatial and temporal time translation invariance:
{width="\columnwidth"}
Following standard field theory procedures, one next identifies the [*cumulants*]{} as represented by [*connected*]{} Feynman diagrams, and in complete analogy with the static theory establishes [*Dyson’s equation*]{} for the full response propagator, $G(q,\omega)^{-1} = G_0(q,\omega)^{-1} - \Sigma(q,\omega)$. By means of the fields ${\widetilde \Phi}^\alpha = \delta \ln {\cal Z} / \delta {\tilde j}^\alpha$ and $\Phi^\alpha = \delta \ln {\cal Z} / \delta j^\alpha$ one constructs the [*generating functional*]{} for dynamical [*vertex functions*]{} via the Legendre transform $$\begin{aligned}
&&\Gamma[{\widetilde \Phi},\Phi] = - \ln {\cal Z}[{\tilde j},j] \nonumber \\
&&\quad + \int \! d^dx \! \int \! dt \sum_\alpha \left( {\tilde j}^\alpha \,
{\widetilde \Phi}^\alpha + j^\alpha \, \Phi^\alpha \right) \ ,
\label{dynvfn} \\
&&\Gamma^{({\widetilde N},N)}_{\{ \alpha_i \};\{ \alpha_j \}} =
\prod_i^{\widetilde N} \! \frac{\delta}{\delta {\widetilde \Phi}^{\alpha_i}}
\prod_j^N \! \frac{\delta}{\delta \Phi^{\alpha_j}} \,
\Gamma[{\widetilde \Phi},\Phi] \Big\vert_{{\tilde j}=0=j} \ . \nonumber\end{aligned}$$ Functional derivatives then establish the following connections between two-point cumulants and vertex functions, $$\begin{aligned}
&&\Gamma^{(1,1)}(q,\omega) = G(-q,-\omega)^{-1} \ ,
\label{dyn2pt} \\
&&\Gamma^{(2,0)}(q,\omega) = - \, \frac{C(q,\omega)}{|G(q,\omega)|^2}
\nonumber \\
&&\qquad = - \frac{2 D q^a}{\omega} \, {\rm Im} \ \Gamma^{(1,1)}(q,\omega)\ ,
\label{corver} \end{aligned}$$ where the last relation follows from the equilibrium FDT (\[fldist\]). Moreover, $\Gamma^{(0,2)}(q,\omega) = 0$ as a consequence of causality. One thus easily sees that the vertex functions are graphically represented by the [*one-particle (1PI) irreducible*]{} Feynman diagrams.
We can now formulate the [*Feynman rules*]{} for the dynamical perturbation expansion for the $l$-th order contribution to the vertex function $\Gamma^{({\widetilde N},N)}$, illustrated here for the one-loop graphs for $\Gamma^{(1,1)}$ and $\Gamma^{(1,3)}$:
{width="0.9\columnwidth"}
1. Draw all topologically different, connected 1PI graphs with ${\widetilde N}$ out- / $N$ incoming lines connecting $l$ relaxation vertices. Do [*not*]{} allow closed response loops (since $\Theta(0) = 0$ in the Itô calculus).
2. Attach wave vectors $q_i$, frequencies $\omega_i$ / times $t_i$, and internal indices $\alpha_i$ to all directed lines, obeying [*“momentum- energy” conservation*]{} at each vertex.
3. Each [*directed line*]{} corresponds to a [*response propagator*]{} $G_0(-q,-\omega)$ or $G_0(q,t_i-t_j)$, the two-point vertex to the [*noise strength*]{} $2 D q^a$, and the four-point [*relaxation vertex*]{} to $- D q^a u / 6$. Closed loops imply integrals over the internal wave vectors and frequencies or times, subject to causality constraints, as well as sums over the internal vector indices. The residue theorem may be applied to evaluate frequency integrals.
4. Multiply with $-1$ and the [*combinatorial factor*]{} counting all possible ways of connecting the propagators, $l$ relaxation vertices, and $k$ two-point vertices leading to topologically identical graphs, including a factor $1 / l! \, k!$ originating in the expansion of $\exp(-{\cal A}_{\rm int}[{\widetilde S},S])$.
The perturbation series then graphically becomes a [*loop expansion*]{}. For example, the propagator self-energy diagrams up to two-loop order are
{width="\columnwidth"}
With the abbreviation $\Delta(q) = D q^a (r + q^2)$, the corresponding explicit analytical expressions read: $$\begin{aligned}
&&\Gamma^{(1,1)}(q,\omega) = i \omega + D q^a \, \biggl[ r + q^2 \nonumber \\
&&\quad + \frac{n+2}{6} \, u \int_k \frac{1}{r+k^2} - \Bigl( \frac{n+2}{6}
\, u \Bigr)^2 \int_k \frac{1}{r+k^2} \nonumber \\
&&\quad \int_{k'} \frac{1}{(r+{k'}^2)^2} - \frac{n+2}{18} \, u^2 \int_k
\frac{1}{r+k^2} \nonumber \\
&&\qquad \int_{k'} \frac{1}{r+{k'}^2} \, \frac{1}{r+(q-k-k')^2}
\label{gamm11} \\
&&\quad \left( 1 - \frac{i \omega}{i \omega + \Delta(k) + \Delta(k') +
\Delta(q-k-k')} \right) \biggr] \ . \nonumber\end{aligned}$$
The renormalized noise vertex is represented by the vertex function $\Gamma^{(2,0)}(q,\omega)$; the first non-vanishing fluctuation correction appears at two-loop order:
{width="0.28\columnwidth"}
which translates to $$\begin{aligned}
&&\Gamma^{(2,0)}(q,\omega) = - 2 D q^a \, \biggl[ 1 + D q^a \, \frac{n+2}{18}
\, u^2 \nonumber \\
&&\quad \int_k \frac{1}{r+k^2} \int_{k'} \frac{1}{r+{k'}^2}
\frac{1}{r+(q-k-k')^2} \nonumber \\
&&{\rm Re} \, \frac{1}{i \omega + \Delta(k) + \Delta(k') + \Delta(q-k-k')}
\biggr] \ .
\label{gamm20}\end{aligned}$$ Finally, we shall require the renormalized relaxation vertex to one-loop order, $$\begin{aligned}
&&\Gamma^{(1,3)}(-3{\underline k}/2;\{ {\underline k}/2 \}) =
D \Bigl( \frac{3q}{2} \Bigr)^a u \nonumber \\
&&\quad \biggl[ 1 - \frac{n+8}{6} \, u \int_k \frac{1}{r+k^2} \,
\frac{1}{r+(q-k)^2} \nonumber \\
&&\qquad\quad \left( 1 - \frac{i \omega}{i \omega + \Delta(k) + \Delta(q-k)}
\right) \biggr] \ ,
\label{gamm13}\end{aligned}$$ evaluated at equal incoming external wave vectors and frequencies ${\underline k} = (q,\omega)$.
Critical dynamics of the relaxational models
--------------------------------------------
We may now proceed with the perturbative renormalization of the purely relaxational models A and B, generalizing the methods outlined in Chap. 3 to the dynamical action (\[resfab\]). The quadratic UV divergence (near the upper critical dimension $d_c = 4$) in (\[gamm11\]) is taken care of by an appropriate [*additive renormalization*]{}; since we are concerned with near-equilibrium kinetics here, and $\chi(q,\omega = 0) = \chi(q)$, the result is precisely the $T_c$ shift (\[1lptcs\]) evaluated in the static theory. In addition to (\[fldren\]) and (\[cplren\]), we need two new [*multiplicative renormalization*]{} factors associated with the response fields and the relaxation rate, $$\begin{aligned}
&&{\widetilde S}_R^\alpha = Z_{\widetilde S}^{1/2} {\widetilde S}^\alpha\ , \
D_R = Z_D \, D \ ;
\label{dynren} \\
&&\Rightarrow \ \Gamma_R^{({\widetilde N},N)} =
Z_{\widetilde S}^{-{\widetilde N}/2} Z_S^{-N/2}\, \Gamma^{({\widetilde N},N)}
\ . \nonumber\end{aligned}$$ As a consequence of the FDT (\[corver\]), which must hold for both the unrenormalized and renormalized vertex functions, these $Z$ factors are [*not*]{} independent in thermal equilibrium, but connected via $Z_D = \left( Z_S / Z_{\widetilde S} \right)^{1/2}$.
For model A with non-conserved order parameter ($a = 0$), extracting the UV poles in the minimal subtraction scheme from $\Gamma^{(2,0)}_R(0,0)$ or $\frac{\partial}{\partial i \omega} \Gamma^{(1,1)}_R(0,\omega)$ yields $$Z_D = 1 - \frac{n + 2}{144} \, \Bigl( 6 \ln \frac43 - 1 \Bigr) \,
\frac{u_R^2}{\epsilon} \ .
\label{2lpztd}$$ For model B with conserved order parameter ($a = 2$), on the other hand, the momentum dependence $\propto q^2$ of the relaxation vertex implies that to [*all orders*]{} in the perturbation expansion $$\begin{aligned}
&&\Gamma^{(1,1)}(q=0,\omega) = i \omega \ , \nonumber \\
&&\partial_{q^2} \, \Gamma^{(2,0)}(q,\omega) \big\vert_{q=0} = - 2 D \ ,
\label{modelb}\end{aligned}$$ whence we arrive at the exact relations $Z_{\widetilde S} \, Z_S = 1$ and $Z_D = Z_S$. The conservation law thus allows us to reduce the dynamical multiplicative to static renormalizations.
With (\[dynren\]) taken into account, the [*renormalization group equation*]{} for the renormalized vertex functions $\Gamma_R^{({\widetilde N},N)}(\mu,D,\tau_R,u_R)$ becomes in analogy with (\[rgeqvf\]): $$\begin{aligned}
&&\biggl[ \mu \frac{\partial}{\partial \mu} +
\frac{{\widetilde N} \gamma_{\widetilde S} + N \gamma_S}{2} +
\gamma_D \, D_R \, \frac{\partial}{\partial D_R} \nonumber \\
&&\qquad\ + \gamma_\tau \, \tau_R \, \frac{\partial}{\partial \tau_R} +
\beta_u \, \frac{\partial}{\partial u_R} \biggr] \,
\Gamma_R^{({\widetilde N},N)} = 0 \, ,
\label{dynrge}\end{aligned}$$ with the static [*RG beta function*]{} (\[betafu\]) and [*Wilson’s flow functions*]{} (\[gamfld\]), (\[gamtau\]), supplemented with $$\begin{aligned}
&&\gamma_{\widetilde S} = \mu \, \frac{\partial}{\partial \mu} \Big\vert_0 \,
\ln Z_{\widetilde S} \ ,
\label{gamres} \\
&&\gamma_D = \mu \, \frac{\partial}{\partial \mu}\Big\vert_0 \, \ln
\frac{D_R}{D} = \frac{\gamma_S - \gamma_{\widetilde S}}{2}
\label{gamrel}\end{aligned}$$ owing to the FDT. Employing characteristics $\mu \to \mu \, \ell $ to solve the Gell-Mann–Low RG equation (\[dynrge\]), one has, in addition to (\[runcpl\]), $$\ell \, \frac{d{\widetilde D}(\ell)}{d\ell} = {\widetilde D}(\ell) \,
\gamma_D(\ell) \ ,
\label{runrel}$$ with ${\widetilde D}(1) = D_R$.
For the [*dynamic susceptibility*]{} near an infrared-stable RG fixed point, the static scaling law (\[gensus\]) generalizes to $$\begin{aligned}
&&\chi_R(\tau_R,q,\omega)^{-1} \approx \mu^2 \ell^{2 + \gamma_S^*} \nonumber
\\
&&\ {\hat \chi}_R \Bigl( \tau_R \, \ell^{\gamma_\tau^*}, u^*,
\frac{q}{\mu \, \ell}, \frac{\omega}{D_R \mu^{2+a} \,
\ell^{2 + a + \gamma_D^*}} \Bigr)^{-1} ,
\label{gndsus}\end{aligned}$$ which allows us to identify the static [*critical exponents*]{} as before, $\eta = - \gamma_S^*$, and $\nu = - 1 / \gamma_\tau^*$; and in addition $z = 2 + a + \gamma_D^*$ for the [*dynamic critical exponent*]{}. From the explicit two-loop result (\[2lpztd\]) one thus obtains for the $O(n)$-symmetric model A by inserting the Heisenberg fixed point (\[heisfp\]) to order $\epsilon^2$ in the $4-\epsilon$ expansion $${\rm model \ A:} \ z = 2 + c \, \eta \ , \
c = 6 \ln \frac43 - 1 + O(\epsilon) \ . \
\label{mdazex}$$ Yet if the order parameter is conserved, one has $\gamma_D^* = \gamma_S^*$, which implies the [*exact scaling relation*]{} $${\rm model \ B:} \ z = 4 - \eta \ .
\label{mdbzex}$$
Critical dynamics of isotropic ferromagnets
-------------------------------------------
So far we have only considered purely relaxational, dissipative kinetics. Often, however, the Langevin description of critical dynamics needs to take into account [*reversible*]{} systematic forces contributing to $F[S]$ in (\[langen\]). The Langevin dynamics of [*isotropic ferromagnets*]{} provides a prominent example [@Frey94], [@Tauberxx Chap. 6]. The order parameter here is a three-component vector field, namely the magnetization density $S^\alpha(x,t)$, which represent the coarse-grained mesoscopic counterpart of the microscopic local Heisenberg spins, also the generators of the rotation group $O(3)$. From the spin operator commutation relation $\bigl[ S^\alpha , S^\beta \bigr] =
i \hbar \sum_{\gamma=1}^3 \epsilon^{\alpha \beta \gamma} S^\gamma$ and Heisenberg’s equation of motion, or their corresponding classical counterparts with commutators replaced by Poisson brackets, one readily obtains a spin precession term in the dynamics, in addition to the diffusive relaxation of the conserved magnetization density, and conserved stochastic noise: $$\begin{aligned}
&&\frac{\partial {\vec S}(x,t)}{\partial t} = - g \, {\vec S}(x,t) \times
\frac{\delta {\cal H}[{\vec S}]}{\delta {\vec S}(x,t)} \nonumber \\
&&\qquad\qquad + D \nabla^2 \,
\frac{\delta {\cal H}[{\vec S}]}{\delta {\vec S}(x,t)} + {\vec \zeta}(x,t)\ ,
\label{mdjlan} \\
&&\big\langle \zeta^\alpha(x,t) \, \zeta^\beta(x',t') \big\rangle = - 2 D
\, k_{\rm B} T \nonumber \\
&&\qquad\quad \nabla^2 \delta(x-x') \delta(t-t') \delta^{\alpha \beta} \ .
\label{mdjnoi}\end{aligned}$$ The coupled Langevin equations (\[mdjlan\]) with the three-component LGW Hamiltonian (\[lgwhmn\]) and noise correlator (\[mdjnoi\]) define the [*model J*]{} dynamic universality class.
The associated Janssen–De Dominicis response functional comprises the model B terms, (\[resfab\]) with $a = 2$, and the additional reversible [*mode-coupling vertex*]{}, $$\begin{aligned}
&&{\cal A}_J[{\widetilde S},S] = - g \! \int \! d^dx \! \int \! dt
\sum_{\alpha,\beta,\gamma} \epsilon^{\alpha \beta \gamma} \,
{\widetilde S}^\alpha\nonumber \\
&&\qquad\qquad\qquad\qquad\ S^\beta \left( \nabla^2 S^\gamma + h^\gamma
\right) \ .
\label{resmdj}\end{aligned}$$ It is diagrammatically represented by a wave vector-dependent three-point vertex:
{width="0.55\columnwidth"}
Straightforward power counting gives $[g] = \mu^{3-d/2}$ for the mode coupling strength, which therefore becomes marginal at the [*dynamical critical dimension*]{} $d_c' = 6$, and irrelevant for $d > d_c'$. In principle, theories with competing upper critical dimension pose interesting non-trivial technical problems. Recall, however, that we are here considering near-equilibrium dynamics; thus the static critical properties completely decouple from the system’s dynamics. Indeed, the dynamical critical exponent can be determined exactly from the underlying rotational symmetry as follows: We first note that according to (\[mdjlan\]) an external field $h^\gamma$ induces a rotation of the magnetization density vector: $\bigl\langle S^\alpha(x,t) \bigr\rangle_h = g \int_0^t \! dt' \sum_\beta
\epsilon^{\alpha \beta \gamma} \bigl\langle S^\beta(x,t') \bigr\rangle_h \,
h^\gamma(t)$. Thus, we obtain an exact identity linking the [*nonlinear susceptibility*]{} $R^{\alpha;\beta\gamma} = \delta^2 \langle S^\alpha \rangle / \delta h^\beta \,
\delta h^\gamma \vert_{h=0}$ with the linear order parameter response function: $$\begin{aligned}
&&\int \! d^dx' \, R^{\alpha;\beta\gamma}(x,t;x-x',t-t') = \nonumber \\
&&\qquad\ g \, \epsilon^{\alpha \beta \gamma} \, \chi^{\beta\beta}(x,t) \,
\Theta(t) \, \Theta(t-t') \ .
\label{nlinrs}\end{aligned}$$
This identity provides crucial information for the renormalization of the UV singularities. In addition to (\[dynren\]), we define the renormalized dimensionless mode coupling strength $$g_R^2 = Z_g \, g^2 \, B_d \, \mu^{d-6} \ ,
\label{mcpren}$$ where $B_d = \Gamma(4 - d/2) / 2^d \, d \, \pi^{d/2}$. Just as for model B, one has to all orders in the perturbation expansion $\Gamma^{(1,1)}(q=0,\omega) = i \omega$, whence again $Z_{\widetilde S} \, Z_S = 1$. Since (\[nlinrs\]) must hold for the renormalized susceptibilities as well, one then infers that $Z_g = Z_S$. Inspection of diagrams shows that the [*effective coupling*]{} in the dynamic perturbation expansion is $f = g^2 / D^2$. The associated RG beta function becomes $$\beta_f = \mu \, \frac{\partial}{\partial \mu} \Big\vert_0 f_R =
f_R \left( d - 6 + \gamma_S - 2 \, \gamma_D \right) \ .
\label{betafj}$$ Consequently, at [*any non-trivial*]{}, stable RG fixed point $0 < f^* < \infty$ the terms in the bracket must cancel each other: $2 \, \gamma_D^* = d - 6 + \gamma_S^* $ to [*all orders*]{} in the perturbation series with respect to $f_R$. Rotation invariance and the conservation law thus fix the dynamic critical exponent in dimensions $d < d_c' = 6$ to be $${\rm model \ J:} \ z = 4 + \gamma_D^* = \frac{d + 2 - \eta}{2} \ .
\label{mdjzex}$$ Indeed, an explicit one-loop calculation yields $\gamma_D = - f_R + O(u_R^2,f_R^2)$, which leads to the non-trivial model J RG fixed point $f_J^* = \frac{\varepsilon}{2} + O(\varepsilon^2)$, where $\varepsilon = 6 - d$. Note that $\eta = 0$ for $d > d_c = 4$, and $z = 4$ for $d > d_c' = 6$. Since the mode-coupling vertex does not contribute genuinely new IR singularities, dynamic scaling functions for isotropic ferromagnets and related models can be computed to exquisit precision by means of the [*mode-coupling approximation*]{}, which essentially amounts to a self-consistent one-loop theory for the propagators, ignoring vertex corrections [@Frey94].
Driven diffusive systems
------------------------
This chapter on the application of field-theoretic RG methods to non-linear stochastic Langevin dynamics concludes with two paradigmatic non-equilibrium model systems that display [*generic scale invariance*]{} and a continuous phase transition, respectively. Both are driven lattice gases [@Schmittmann95] consisting of particles that propagate via nearest-neighbor hopping which is biased along a specified ‘drive’ direction, and is subject to an exclusion constraint, i.e., only at most a single particle is allowed on each lattice site. If the system is set up with periodic boundary conditions, the biased diffusion generates a non-vanishing stationary mean particle current. At long times, the kinetics thus reaches a [*non-equilibrium steady state*]{} which turns out to be governed by [*algebraic*]{} rather than exponential temporal correlations. If in addition nearest-neighbor attractive interactions are included, the system displays a genuine non-equilibrium continuous phase transition in dimensions $d \geq 2$, from a disordered phase to an ordered state characterized by phase separation into low- and high-density regions, with the phase boundary oriented parallel to the drive and particle current. As the hopping bias vanishes, the phase transition is naturally described by the $d$-dimensional ferromagnetic equilibrium Ising model, since one may map the occupation numbers $n_i = 0,1$ to binary spin variables $\sigma_i = 2 n_i - 1 = \mp 1$.
We first consider the driven lattice gas with pure exclusion interactions, in one dimension also called [*“asymmetric exclusion process”*]{}. In order to construct a coarse-grained description for the non-equilibrium steady state of this system of particles with conserved density $\rho(x,t)$ and hard-core repulsion, driven along the ‘$\parallel$’ spatial direction on a $d$-dimensional lattice, one first writes down the [*continuity equation*]{} $$\frac{\partial}{\partial t} \, S(x,t) + {\vec \nabla} \cdot {\vec J}(x,t) = 0
\ ,
\label{ddscon}$$ where the scalar field $S(x,t) = 2 \rho(x,t) - 1$ represents a local magnetization in the spin language, whose mean remains fixed at $\langle S(x,t) \rangle = 0$, or $\langle \rho(x,t) \rangle = \frac12$. Next the current density must be specified; in the $d_\perp$-dimensional transverse sector ($d_\perp = d - 1$), one may simply assert a noisy diffusion current, whereas along the drive, the bias and exclusion are crucial: $J_\parallel = - c \, D \, \nabla_\parallel S + 2 D g \rho (1 - \rho) + \zeta$, where $c$ measures the ratio of diffusivities parallel and transverse to the net current. In the comoving reference frame where $\langle J_\parallel(x,t) \rangle = 0$, therefore $$\begin{aligned}
&&{\vec J}_\perp(x,t) = - D \, {\vec \nabla}_\perp S(x,t) + {\vec \eta}(x,t)
\ ,
\label{ddscur} \\
&&J_\parallel(x,t) = - c D \, \nabla_\parallel S(x,t) - \frac12 \, D g \,
S(x,t)^2 \nonumber \\
&&\qquad\qquad\qquad + \zeta(x,t) \ , \nonumber\end{aligned}$$ with $\langle \eta_i \rangle = 0 = \langle \zeta \rangle$, and the noise correlations $$\begin{aligned}
&&\langle \eta_i(x,t) \, \eta_j(x',t') \rangle = \nonumber \\
&&\qquad\qquad 2 D \, \delta(x-x') \delta(t-t') \delta_{ij} \ ,
\label{ddsnoi} \\
&&\langle \zeta(x,t) \, \zeta(x',t') \rangle =
2 D {\tilde c} \, \delta(x-x') \delta(t-t') \, . \nonumber\end{aligned}$$ It is important to realize that Einstein’s relations which connect the noise strengths and the relaxation rates need [*not*]{} be satisfied in the non-equilibrium steady state. Through straightforward rescaling of the field $S(x,t)$, one may however formally enforce this connection in the transverse sector, say; the deviation from the Einstein relation in the parallel direction is then encoded in (\[ddscur\]) and (\[ddsnoi\]) through the ratio $0 < w = {\tilde c} / c \not= 1$.
The Janssen–De Dominicis response functional for this driven diffusive system becomes $$\begin{aligned}
&&{\cal A}[{\widetilde S},S] = \!\! \int \!\! d^dx \!\! \int \!\! dt \,
{\widetilde S} \, \Bigl[ \partial_t S - D \left( \nabla_\perp^2
+ c \nabla_\parallel^2 \right) S \nonumber \\
&&\qquad + D \left( \nabla_\perp^2
+ {\tilde c} \, \nabla_\parallel^2 \right) {\widetilde S}
- \frac{D \, g}{2} \, \nabla_\parallel S^2 \Bigr] \, ;
\label{ddsact} \end{aligned}$$ the action (\[ddsact\]) represents a “massless” field theory, which hence displays [*generic scale invariance*]{} with no specific tuning of any control parameters required. One clearly has to allow for [*anisotropic scaling*]{} behavior owing to the very different dynamics parallel to the hopping bias; for example, (\[dynscr\]) for the dynamic response function needs to be generalized to $$\chi(q_\perp,q_\parallel,\omega) = |q_\perp|^{-2+\eta} \, {\hat \chi}\left(
\frac{q_\parallel}{|q_\perp|^{1 + \Delta}}, \frac{\omega}{|q_\perp|^z}
\right) ,
\label{ddssus}$$ where $\Delta$ denotes the [*anisotropy exponent*]{} ($\Delta = 0$ in the mean-field approximation).
Following the renormalization procedures in the previous subsections, one first realizes that the drive generates a three-point vertex $\propto g \, i q_\parallel$; consequently no transverse fluctuations are affected by this non-linearity, and $Z_{\widetilde S} = Z_S = Z_D = 1$ [*to all orders*]{} in the perturbation expansion in $g$. The absence of any transverse propagator renormalization immediately implies that $\eta = 0$ and $z = 2$ in (\[ddssus\]), which leaves merely the value of $\Delta$ to be determined. In a similar manner as for model J, one may in fact compute this exponent [*exactly*]{}; to this end, one observes that a generalized [*Galilean transformation*]{} $$\begin{aligned}
&&S(x_\perp,x_\parallel,t) \to
\label{galtra} \\
&&\qquad S'(x_\perp',x_\parallel',t') = S(x_\perp,x_\parallel - D g v \, t,t)
- v \nonumber\end{aligned}$$ leaves the Langevin equation (\[ddscon\]), (\[ddscur\]) or equivalent action (\[ddsact\]) invariant. Thus the (arbitrary) speed $v$ must scale as the order parameter field $S$, and since $Z_D = 1 = Z_S$, neither can the coupling $g$ be altered by fluctuations, or (\[galtra\]) would be violated for the renormalized theory. Hence $Z_g = 1$ as well, and the only remaining non-trivial renormalizations are those for the dimensionless parameters $c_R = Z_c \, c$ and ${\tilde c}_R = Z_{\tilde c} \, {\tilde c}$.
An explicit one-loop calculation establishes the existence of an IR-stable RG fixed point for the coupling $$v = g^2 / c^{3/2} \ , \quad v_R = Z_c^{3/2} \, v \, C_d \, \mu^{d-2}\ ,
\label{ddscpl}$$ with $C_d = \Gamma(2-d/2) / 2^{d-1} \pi^{d/2}$, and identifying $d_c = 2$ as the upper critical dimension for this problem. Evaluating the one-loop fluctuation corrections to the longitudinal propagator, one finds $$\begin{aligned}
&&\gamma_c = - \frac{v_R}{16} \left( 3 + w_R \right) \ ,
\label{ddsgam} \\
&&\gamma_{\tilde c} = - \frac{v_R}{32} \,
\left( 3 w_R^{-1} + 2 + 3 w_R \right) \nonumber\end{aligned}$$ for the anomalous scaling dimensions of $c$ and ${\tilde c}$, or $$\begin{aligned}
&&\beta_w = w_R \left( \gamma_{\tilde c} - \gamma_c \right) \nonumber \\
&&\quad\ = - \frac{v_R}{32} (w_R - 1) (w_R - 3) \ ,
\label{ddsbew} \\
&&\beta_v = v_R \left( d - 2 - \frac32 \, \gamma_c \right)
\label{ddsbev} \end{aligned}$$ for the associated RG beta functions of the ratio $w = {\tilde c} / c$ and the non-linear coupling $v$. At [*any non-trivial*]{} RG fixed point $0 < v^* < \infty$, (\[ddsbew\]) implies that either $w_N^* = 3$ or $w_E^* = 1$, but the latter is obviously [*stable*]{}; in the asymptotic scale-invariant regime, the Einstein relation is evidently restored, and the system effectively equilibrated. Moreover, in dimensions $d < d_c = 2$, (\[ddsbev\]) leads to the [*exact*]{} scaling exponents $$\Delta = - \frac{\gamma_c^*}{2} = \frac{2 - d}{3} \ , \quad
z_\parallel = \frac{z}{1 + \Delta} = \frac{6}{5 - d} \ .
\label{ddsexp}$$ At $d = 1$, specifically, one has $z_\parallel = \frac32$, which captures the dynamic scaling for the asymmetric exclusion process. In one dimension, the driven lattice gas with exclusion in fact maps onto the [*noisy Burgers equation*]{} for equilibrium fluid hydrodynamics, and also to the [*Kardar–Parisi–Zhang equation*]{} for curvature-driven surface or interface growth.
We finally briefly summarize the RG analysis for the driven lattice gas with conserved total density and attractive Ising interactions between the particles (and “holes”) at its critical point. Since the system orders in stripes along the drive direction, only the [*transverse*]{} fluctuations become critical. Therefore one must amend the response functional (\[ddsact\]) with a higher-order gradient term and non-linearity akin to the scalar model B, see (\[resfab\]) with $a = 2$; yet the noise terms too need only be retained in the transverse sector. For this driven model B, the effective critical action thus becomes $$\begin{aligned}
&&{\cal A}[{\widetilde S},S] \!= \!\! \int \!\! d^dx \!\! \int \!\! dt \,
{\widetilde S} \, \Bigl[ \partial_t S - D \nabla_\perp^2 \left( r -
\nabla_\perp^2 \right) S \nonumber \\
&&\ - D c \nabla_\parallel^2 S + D \left( \nabla_\perp^2 {\widetilde S}
- \frac{g}{2} \, \nabla_\parallel S^2 - \frac{u}{6} \, \nabla_\perp^2 S^3
\right) \Bigr] \ , \nonumber \\ &&
\label{crddsa}\end{aligned}$$ and (\[ddssus\]) is further generalized by adding a relevant temperature variable (now $\Delta = 1$ in mean-field theory): $$\begin{aligned}
&&\chi(\tau_\perp,q_\perp,q_\parallel,\omega) = |q_\perp|^{-2+\eta} \nonumber
\\
&&\qquad {\hat \chi}\left( \frac{\tau}{|q_\perp|^{1/\nu}},
\frac{q_\parallel}{|q_\perp|^{1 + \Delta}}, \frac{\omega}{|q_\perp|^z}
\right) \ .
\label{crddss}\end{aligned}$$
Straightforward power counting for the non-linear couplings yields $[g^2] = \mu^{5-d}$ and $[u] = \mu^{3-d}$; therefore the upper critical dimension is raised to $d_c = 5$ (compared to both the non-critical driven lattice gas and the equilibrium model B), and fluctuation corrections are dominated by the drive, while the static coupling $u$ is (dangerously) [*irrelevant*]{} near $d_c$. The three-point vertex $\propto g \, i q_\parallel$ again does not allow any renormalizations in the transverse sector, whence $Z_{\widetilde S} = Z_S = Z_D = 1$; consequently $\eta = 0$, $\nu = \frac12$, and $z = 4$ in (\[crddss\]) to all orders in the perturbation series, leaving only the anisotropy exponent to be determined. As before $Z_g = 1$ follows from Galilean invariance, imposing a simple structure for the RG beta function for the effective coupling (\[ddscpl\]): $$\beta_v = v_R \left( d - 5 - \frac32 \, \gamma_c \right) \ .
\label{crddsv}$$ In dimensions $d < d_c = 5$, at any non-trivial and finite RG fixed point, the scaling exponents are thus forced to assume the values $$\Delta = 1 - \frac{\gamma_c^*}{2} = \frac{8-d}{3} \ ,
\ z_\parallel = \frac{4}{1 + \Delta} = \frac{12}{11 - d} \ . \
\label{crddse}$$ These last examples clearly demonstrate how the powerful field-theoretic RG approach can help to exploit the basic symmetries for a given problem, allowing to determine certain non-trivial scaling exponents [*exactly*]{}.
Scale Invariance in Interacting Particle Systems
================================================
This last chapter details how the stochastic kinetics of classical interacting (reacting) particle systems, defined through a [*microscopic*]{} master equation, can also be mapped onto a dynamical field theory in the continuum limit [@Cardy97]–[@Tauber05]. For at most binary reactions, one can thus [*derive*]{} a corresponding mescoscopic Langevin representation, typically with multiplicative noise terms. Furthermore, RG tools may be applied to extract the infrared properties in scale-invariant systems, as will be exemplified for diffusion-limited annihilation processes [@Cardy97; @Tauber05; @Tauber07], and for non-equilibrium phase transitions from active to inactive, absorbing states, where all stochastic fluctuations cease [@Tauber05]–[@Janssen05]. Stochastic models in population dynamics and ecology are naturally formulated in a chemical reaction language, and hence amenable to these field-theoretic tools [@Mobilia07; @Tauber11].
Chemical reactions and population dynamics
------------------------------------------
Let us thus consider (classical) particles of various species $A,B,\ldots$ on a $d$-dimensional lattice that propagate by hops to nearest-neighbor sites, and either spontaneously decay or produce offspring, and/or upon encounter with other particles, undergo certain “chemical” reactions with prescribed rates. Our goal is to systematically construct a continuum description of such stochastic particle systems that however faithfully encodes the associated [*intrinsic reaction noise*]{}, and consequently allows us to properly address the effects of statistical fluctuations and spatio-temporal correlations [@Cardy97; @Tauber05; @Tauber07].
To set the stage, we introduce three characteristic examples that we shall explore in more detail below. First, we address the general single-species [*irreversible annihilation*]{} reaction $k \, A \to m \, A$ with integers $m < k$, and rate $\lambda_k$. Assuming the system to be well-mixed, one may neglect spatial variations and focus on the mean particle density $a(t) = \langle a(x,t) \rangle$. Ignoring in addition any non-trivial correlations, one can write down the [*rate equation*]{} for this stochastic process, which in essence thus constitutes the simplest mean-field approximation: $$\frac{\partial a(t)}{\partial t} = - (k - m) \, \lambda_k \, a(t)^k \ .
\label{annreq}$$ For $k = 1$ (and $m = 0$), this just describes spontaneous exponential decay, $a(t) = a(0) \, e^{- \lambda_1 \, t}$; for $k \geq 2$, (\[annreq\]) is easily integrated with the result $$a(t) = \bigl[ a(0)^{1-k} + (k-m) (k-1) \, \lambda_k \, t \bigr]^{-1/(k-1)} .
\label{mfasol}$$ For the $k$th order annihilation reaction, the particle density decays algebraically $a(t) \sim (\lambda_k \, t)^{-1/(k-1)}$ at long times $t \gg \lambda_k^{-1}$, with an amplitude that does not even depend on the initial density $a(0)$ anymore. The replacement of an exponential decay by a power law signals scale invariance and indicates the potential importance of fluctuations and correlations. Indeed, the annihilation kinetics generates particle [*anti-correlations*]{}, whence the long-time kinetics is dominated by the ensuing [*depletion zones*]{} in low dimensions $d \leq d_c(k)$ that need to be traversed by any potentially reacting particles. As a consequence, one obtains [*slower*]{} decay power laws than predicted by the mean-field rate equation (\[mfasol\]).
Next, we allow [*competing reactions*]{}, namely decay $A \to \emptyset$ (the empty state) with rate $\kappa$, and the reversible process $A \rightleftharpoons A + A$ with forward / backward rates $\sigma$ and $\lambda$, respectively. Again, we begin with an analysis of the rate equation for this set of reactions, $$\frac{\partial a(t)}{\partial t} = (\sigma - \kappa) \, a(t)
- \lambda \, a(t)^2 \ ,
\label{dprreq}$$ which obviously predicts a [*continuous non-equilibrium phase transition*]{} at $\sigma_c = \kappa$: For $\sigma > \kappa$, the mean particle density approaches a finite value, $a(t \to \infty) \to a_\infty = (\sigma - \kappa) / \lambda$. One refers to this state as an [*active*]{} phase; ongoing reactions cause the particle number to fluctuate about its average. On the other hand, for $\sigma < \kappa$, the density can only decrease, whence ultimately $a(t) \to 0$; in this [*inactive*]{} phase, all reaction processes terminate since they all require the presence of a particle. Such a state is therefore called [*absorbing*]{}: once reached, the stochastic dynamics cannot escape from it anymore. Right at the critical point $\sigma = \kappa$, one recovers the long-time algebraic decay of the pair annihilation process, $a(t) \sim (\lambda \, t)^{-1}$; this suggests the interpretation of (\[mfasol\]) as a [*critical*]{} power law induced by the precise cancellation of the contributions from first-order reactions that enter linearly proportional to the particle concentration. The obvious issues to be addressed by a more refined theoretical treatment are: How can internal reaction noise and correlations be systematically incorporated? What is the upper critical dimension $d_c$ below which fluctuations crucially alter the mean-field power laws? Can certain [*universality classes*]{} be identified, and the associated critical exponents be computed, at least perturbatively in a dimensional expansion near $d_c$?
Finally, let us address a prominent textbook example from population dynamics, namely the classic [*Lotka–Volterra predator-prey competition*]{}. Invoking the stochastic chemical reaction framework, this model is defined via spontaneous death $A \to \emptyset$ (rate $\kappa$) and birth $B \to B + B$ (rate $\sigma$) processes for the “predators” $A$ and “prey” $B$; absent any interactions between these two species, the predators must go extinct, while the prey population explodes exponentially. Interesting species competition and potentially coexistence is created by the binary [*predation*]{} reaction $A + B \to A + A$ (with rate $\lambda$). The associated coupled rate equations for the presumed homogeneous population densities read $$\begin{aligned}
&&\frac{\partial a(t)}{\partial t} = \lambda \, a(t) b(t) - \kappa \, a(t)
\ , \nonumber \\
&&\frac{\partial b(t)}{\partial t} = \sigma \, b(t) - \lambda \, a(t) b(t)\ .
\label{lvmreq} \end{aligned}$$ In this mean-field approximation, one easily confirms the existence of a conserved first integral for the ordinary differential equations (\[lvmreq\]): The quantity $K(t) = \lambda [a(t) + b(t)] - \ln [a(t)^\sigma b(t)^\kappa] = K(0)$ remains unchanged under the temporal evolution. Consequently, the mean-field trajectories are closed orbits in the phase space spanned by the population densities, and the dynamics is characterized by regular [*population oscillations*]{}, determined by the [*initial*]{} state. This is clearly not a biologically realistic feature, and indeed represents an artifact of the implicit mean-field factorization for the non-linear predation process. Upon including the internal reaction noise and spatial degrees of freedom with diffusively spreading particles, as, e.g., in individual-based Monte Carlo simulations, one in fact observes striking [*“pursuit and evasion” waves*]{} in the species coexistence phase that generate complex dynamical patterns, locally discernible as [*erratic*]{} population oscillations which ultimately become overdamped in finite systems. Moreover, if the local “carrying capacity” is finite, i.e., only a certain maximum number of particles may occupy each lattice site, there emerges a predator [*extinction threshold*]{} which indicates a continuous phase transition to an absorbing state, namely the lattice filled with prey. Stochastic fluctuations as well as reaction-induced noise and correlations are thus crucial ingredients to properly describe the large-scale features of spatially extended Lotka–Volterra systems even and especially far away from the extinction threshold (for recent overviews, see Refs. [@Mobilia07; @Tauber11]).
Coherent-state path integral for master equations
-------------------------------------------------
In the following, reacting particle systems on a $d$-dimensional lattice shall be [*defined*]{} through the associated chemical [*master equation*]{} governing a Markovian stochastic process with prescribed, time-independent transition rates. Any possible configuration at time $t$ of the stochastic dynamics is then uniquely characterized by a list of the integer occupation numbers $n_i = 0,1,2,\ldots$ for each particle species at sites $i$. The master equation governs the temporal evolution of the corresponding probability distribution $P(\{ n_i \};t)$ through a [*balance*]{} of gain and loss terms induced by the reaction processes; for example, for the binary reactions $A + A \to \emptyset$ and $A + A \to A$ with rates $\lambda$ and $\lambda'$: $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\frac{\partial}{\partial t} \, P(n_i;t) =
\lambda \, (n_i + 2) \, (n_i + 1) \, P(\ldots,n_i+2,\ldots;t) \nonumber \\
&&\quad + \lambda' \, (n_i + 1) \, n_i \, P(\ldots,n_i+1,\ldots;t)
\label{masteq} \\
&&\quad - (\lambda + \lambda') \, n_i \, (n_i-1) \, P(\ldots,n_i,\ldots;t)
\ , \nonumber\end{aligned}$$ with, say, an uncorrelated [*initial Poisson distribution*]{} $P(\{ n_i \},0) =
\prod_i \left( {\bar n}_0^{n_i} \, e^{-\bar n_0} / n_i ! \right)$.
Since the dynamics merely consists of increasing or decreasing the particle occupation numbers on each site, it calls for a representation through [*second-quantized bosonic ladder operators*]{}, at least if arbitrary many particles are permitted per site, with standard commutation relations $[a_i, a_j] = 0$, $[a_i,a_j^\dagger] = \delta_{ij}$, and an empty vacuum state $| 0 \rangle$ that is annihilated by all operators $a_i$, $a_i | 0 \rangle = 0$. The Fock space of states $| \{ n_i \} \rangle$ with $n_i$ particles on sites $i$ is then constructed through multiple creation operators acting on the vacuum, $| \{ n_i \} \rangle = \prod_i \bigl( a_i^\dagger \bigr)^{n_i} \, | 0 \rangle$; note that a different normalization from standard many-particle quantum mechanics has been implemented here. Thus, $a_i \, |n_i \rangle = n_i \, |n_i-1 \rangle$ and $a_i^\dagger \, |n_i \rangle = |n_i + 1 \rangle$, whence the states $| \{ n_i \} \rangle$ are eigenstates of ${\hat n}_i = a_i^\dagger a_i$ with eigenvalues $n_i$. Next one defines the formal [*state vector*]{} $| \Phi(t) \rangle = \sum_{\{ n_i \}} P(\{ n_i \};t) \, | \{ n_i \} \rangle$, whose temporal evolution is determined by the master equation (\[masteq\]), and may be written in terms of a time-independent quasi-Hamiltonian or Liouvillian $H$ that can be decomposed into a sum of local operators: $$\frac{\partial}{\partial t} \, | \Phi(t) \rangle = - H \, | \Phi(t) \rangle
\ , \ H = \sum_i H_i(a_i^\dagger,a_i) \ .
\label{masham}$$ Note that (\[masham\]) constitutes a [*non-Hermitian imaginary-time Schrödinger equation*]{}, with the formal solution $| \Phi(t) \rangle = \exp (- H t) \, | \Phi(0) \rangle$.
For example, the quasi-Hamiltonian in this [*Doi–Peliti bosonic operator formulation*]{} reads for diffusion-limited annihilation and coagulation reactions $$\begin{aligned}
&&H = D \sum_{<ij>} \bigl( a_i^\dagger - a_j^\dagger \bigr) \,
\bigl( a_i - a_j \bigr)
\label{annham} \\
&&\quad - \sum_i \Bigl[ \lambda \, \bigl( 1 - {a_i^\dagger}^2 \bigr) \, a_i^2
+ \lambda' \bigl( 1 - a_i^\dagger \bigr) \, a_i^\dagger a_i^2 \Bigr] \, ,
\nonumber\end{aligned}$$ where the first line represents nearest-neighbor hopping, and the second encodes the processes in (\[masteq\]). For each stochastic reaction, $H$ contains two contributions: the first one directly reflects the physical [*process*]{} under considerations, i.e., annihilation and production of particles, whereas the second term carries information on the [*order*]{} of the reaction (which powers of the particle concentrations enter the rate equations). In order to access the desired [*statistical averages*]{} with the time-dependent probability distribution $P(\{ n_i \};t)$, one needs the [*projection*]{} state $\langle {\cal P} | = \langle 0 | \prod_i e^{a_i}$, with $\langle {\cal P} | 0 \rangle = 1$; the mean value for any observable $F$, necessarily a function of all occupation numbers $n_i$, at time $t$ then follows as $$\begin{aligned}
&&\langle F(t) \rangle = \sum_{\{ n_i \}} F(\{ n_i \}) \, P(\{ n_i \};t)
\nonumber \\
&&\qquad\;\ = \langle {\cal P} | \, F(\{ a_i^\dagger \, a_i \}) \, | \Phi(t)
\rangle \ .
\label{mastav}\end{aligned}$$ [*Probability conservation*]{} implies that $1 = \langle {\cal P} | \Phi(t) \rangle = \langle {\cal P} | e^{- H \, t} |
\Phi(0) \rangle$, and therefore $\langle {\cal P} | H = 0$. By means of $\bigl[ e^a , a^\dagger \bigr] = e^a$, one may commute the product $e^{\sum_i a_i}$ through the quasi-Hamiltonian $H$, which effectively results in the operator shifts $a_i^\dagger \to 1 + a_i^\dagger$. $H$ must consequently vanish if all creation operators are replaced with $1$, $H_i(a_i^\dagger \to 1,a_i) = 0$. In averages, one may thus also replace $a_i^\dagger a_i \to a_i$; e.g., for the particle density one obtains simply $a(t) = \langle a_i \rangle$, while the two-point occupation number operator product becomes $a_i^\dagger a_i \, a_j^\dagger a_j \to a_i \delta_{ij} + a_i a_j$.
Starting with the Hamiltonian (\[masham\]), and based on the expectation values (\[mastav\]), one may invoke standard procedures from quantum many-particle theory to construct a [*path integral representation*]{} based on [*coherent states*]{}, defined as eigenstates of the annihilation operators $a_i$ with arbitrary complex eigenvalues $\phi_i$: $a_i \, |\phi_i \rangle = \phi_i \, |\phi_i \rangle$. It is straightforward to confirm $$\begin{aligned}
&&|\phi_i \rangle = \exp \, \Bigl( - \frac12 \, |\phi_i|^2 + \phi_i \,
a_i^\dagger \Bigr) \, | 0 \rangle \ ,
\label{cohstt} \\
&&1 = \int \prod_i \frac{d^2 \phi_i}{\pi} \, |\{ \phi_i \} \rangle \,
\langle \{ \phi_i \}| \, .
\label{cohovc} \end{aligned}$$ The [*closure relation*]{} (\[cohovc\]) demonstrates that the coherent states for each site $i$ form an [*overcomplete*]{} basis of Fock space. Upon splitting the time evolution into infinitesimal steps, and inserting (\[cohovc\]) into (\[mastav\]) with the formal solution of (\[masham\]), one eventually arrives at $$\begin{aligned}
&&\langle F(t) \rangle \propto \int \! \prod_i {\cal D}[\phi_i] \,
{\cal D}[\phi_i^*] \, F(\{ \phi_i \}) \, e^{- {\cal A}[\phi_i^*,\phi_i]} ,
\nonumber \\
&&{\cal A}[\phi_i^*,\phi_i] = \sum_i \Bigl[ - \phi_i(t_f)
\label{masdpa} \\
&&\qquad + \int_0^{t_f} \! dt \bigl[ \phi_i^* \, \partial_t \phi_i
+ H(\phi_i^*,\phi_i) \bigr] \! - {\bar n}_0 \phi^*_i(0) \Bigr] . \nonumber\end{aligned}$$
In the end we take the [*continuum limit*]{} $\phi_i(t) \to a_0^d$ $\psi(x,t)$ (with lattice constant $a_0$), and $\phi_i^*(t) \to {\hat \psi}(x,t)$; for diffusively propagating particles, the ensuing “bulk” action becomes $$\begin{aligned}
&&{\cal A}[{\hat \psi},\psi] = \int \! d^dx \int_0^{t_f} \! dt \, \Bigl[
{\hat \psi} \left( \partial_t - D \, \nabla^2 \right) \psi \nonumber \\
&&\qquad\qquad\qquad\qquad\qquad + {\cal H}_r \bigl( {\hat \psi}, \psi \bigr)
\Bigr] \, .
\label{masact}\end{aligned}$$ Here, ${\cal H}_r$ denotes the contributions stemming from the stochastic reaction kinetics; e.g., for [*pair annihilation and coagulation*]{}, $${\cal H}_r \bigl( {\hat \psi},\psi \bigr) = - \lambda \bigl( 1 -
{\hat \psi}^2 \bigr) \psi^2 - \lambda' \bigl( 1 - {\hat \psi} \bigr)
{\hat \psi} \, \psi^2 \ .
\label{annact}$$ Appropriate factors of $a_0$ were absorbed into the continuum diffusion constant $D$ and reaction rates $\lambda$,$\lambda'$. It is worthwhile emphasizing that the actions (\[masdpa\]) based on a master equation should be viewed as [*microscopic*]{} stochastic field theories, which may well require additional coarse-graining steps. Yet the internal stochastic dynamics of the master equation is faithfully and consistently accounted for, since aside from the continuum limit no approximations have been invoked; specifically, no assumptions on the form or strength of any noise terms have been made. As exemplified next for the action (\[annact\]), the Doi–Peliti coherent-state path integral representation of stochastic master equations may serve as a convenient starting point for systematic analytical approaches such as field-theoretic RG studies.
Diffusion-limited annihilation processes
----------------------------------------
Pair annihilation $A + A \to \emptyset$ or coagulation $A + A \to A$ represent the perhaps simplest but non-trivial diffusion-limited reactions. In order to reach beyond the mean-field rate equation approximation (\[mfasol\]), we explore the corresponding Doi–Peliti field theory (\[masact\]) with the specific reaction Hamiltonian (\[annact\]). First we note that the associated [*classical field equations*]{} $\delta {\cal A} / \delta \psi = 0 = \delta {\cal A} / \delta {\hat \psi}$ are solved by ${\hat \psi} = 1$ (which just reflects probability conservation) and $$\frac{\partial \psi(x,t)}{\partial t} = D \, \nabla^2 \, \psi(x,t) -
(2 \lambda + \lambda') \, \psi(x,t)^2 \ ,
\label{andifr}$$ i.e., the rate equation for the local density field $\psi(x,t)$ augmented by diffusive spreading. It is convenient to shift the conjugate field about the mean-field solution, ${\hat \psi}(x,t) = 1 + {\widetilde \psi}(x,t)$, which turns the reactive action into $${\cal H}_r \bigl( {\widetilde \psi},\psi \bigr) = (2 \lambda + \lambda') \,
{\widetilde \psi} \, \psi^2 + (\lambda + \lambda') \, {\widetilde \psi}^2 \,
\psi^2 \ .
\label{anshac}$$ Since the annihilation and coagulation processes $\propto \lambda, \lambda'$ generate the very same vertices, we conclude that aside from non-universal amplitudes, both diffusion-limited reactions should follow [*identical*]{} scaling behavior. One may also formally interpret the ensuing field theory as a Janssen–De Dominicis response functional (\[janded\]) originating from a “Langevin equation” (\[andifr\]) with added white noise $\zeta(x,t)$, whose second moment (\[noigen\]) is given by the functional $L[\psi] = - (\lambda + \lambda') \, \psi^2 < 0$. This negative variance, which reflects the emerging [*anti-*]{}correlations for surviving particles that are induced by the annihilation reactions, of course implies that a Langevin representation is not truly feasible for this stochastic process. One must also keep in mind that the fields $\psi$ and ${\hat \psi}$ are complex-valued; indeed, the reaction noise can be recast as [*“imaginary” multiplicative noise*]{} $\propto i \psi(x,t) \, \zeta(x,t)$ in the associated stochastic differential equation.
As coagulation thus falls into the same universality class as annihilation, let us more generally study the Doi–Peliti action for [*$k$-particle annihilation*]{} $k \, A \to \emptyset$, $$\begin{aligned}
&&{\cal A}[{\hat \psi},\psi] = \int \! d^dx \! \int \! dt \, \Bigl[
{\hat \psi} \, \bigl( \partial_t - D \nabla^2 \bigr) \psi \nonumber \\
&&\qquad\qquad\qquad\qquad\ - \lambda_k \, \bigl( 1 - {\hat \psi}^k \bigr) \,
\psi^k \Bigr] \, .
\label{kannac}\end{aligned}$$ The corresponding mean-field rate equation (\[annreq\]) predicts algebraic decay $a(t) \sim (\lambda_k \, t)^{-1/(k-1)}$ at long times. Since the field ${\hat \psi}$ appears to higher than quadratic power for $k \geq 3$, [*no*]{} (obvious) equivalent Langevin description is possible for triplet and higher-order annihilation reactions. With $[{\hat \psi}(x,t)] = 1$ and $[\psi(x,t)] = \mu^d$, as $\langle \psi(x,t) \rangle = a(t)$ is just the particle density, power counting gives $[\lambda_k] = \mu^{2 - (k-1) d}$; the upper critical dimension thus is $d_c(k) = 2/(k-1)$ for $k$th order annihilation, and one expects the mean-field power laws (\[mfasol\]) to be accurate for $k > 3$ in all physical dimensions $d \geq 1$; for triplet reactions, one should encounter merely logarithmic corrections in one dimension, where genuine non-trivial exponents ensue only for pair annihilation.
Even for $k = 2$, one cannot construct any loop graphs from the vertices in (\[kannac\]) that would modify the massless diffusion propagator, implying that $\eta = 0$ and $z = 2$. This leaves the task to determine the reaction [*vertex renormalization*]{}, which can also be achieved to [*all orders*]{} by summing the diagrammatic geometric series (essentially a [*Bethe–Salpeter equation*]{}; here for $k = 3$):
{width="\columnwidth"}
With the factor $B_{kd} = k! \, \Gamma(2 - d/d_c) \, d_c / k^{d/2} \, (4 \pi)^{d/d_c}$, one finds the renormalized reaction rate $$\begin{aligned}
&&g_R = Z_g \, \frac{\lambda}{D} \, B_{kd} \, \mu^{- 2 (1 - d/d_c)} \ ,
\nonumber \\
&&Z_g^{-1} = 1 +
\frac{\lambda \, B_{kd} \, \mu^{- 2 (1 - d/d_c)}}{D \, (d_c - d)} \ ,
\label{renanr}\end{aligned}$$ and the exact RG beta function and stable fixed point $$\beta_g = - \frac{2 g_R}{d_c} \left( d - d_c + g_R \right) \ , \quad
g^* = d_c - d \ .
\label{annrgf}$$
Next we write down the Gell-Mann–Low RG equation for the particle density $a(t)$, applying the matching condition $(\mu \ell)^2 = 1 / D t$: $$\begin{aligned}
&&\left[ d + 2 D t \, \frac{\partial}{\partial (D \, t)} - d \, n_0 \,
\frac{\partial}{\partial n_0} + \beta_g \, \frac{\partial}{\partial g_R}
\right] \nonumber \\
&&\qquad\quad a(\mu,D,n_0,g_R,t) = 0 \ ,
\label{andrge}\end{aligned}$$ with the solution $$\begin{aligned}
&&a(\mu,D,n_0,g_R,t) = \nonumber \\
&&\qquad (D \mu^2 \, t)^{- d/2} \, {\hat a}\bigl( n_0 \,
(D \mu^2 \, t)^{d/2} , {\widetilde g}(t) \bigr) \ .
\label{andrgs}\end{aligned}$$ The particle density at time $t$ naturally depends on its initial value $n_0$, clearly a [*relevant*]{} parameter in the RG sense. One therefore needs to establish through explicit calculation that the (tree level) scaling function ${\hat a}$ remains finite to [*all orders*]{} as $n_0 \to \infty$. In the end, (\[andrgs\]) yields for pair annihilation, $$\begin{aligned}
k = 2: &&d < 2: \ a(t) \sim (D \, t)^{-d/2} \ , \nonumber \\
&&d = 2: \ a(t) \sim (D \, t)^{-1} \ln (D \, t) \ ,
\label{k=2ann} \\
&&d > 2: \ a(t) \sim (\lambda \, t)^{-1} \ ; \nonumber\end{aligned}$$ while for the triplet reaction $$\begin{aligned}
k = 3: &&d = 1 \, : \
a(t) \sim \left[ (D \, t)^{-1} \ln (D \, t) \right]^{1/2} \ , \nonumber \\
&&d > 1: \ a(t) \sim (\lambda \, t)^{-1/2} \ .
\label{k=3ann}\end{aligned}$$ At low dimensions $d \leq d_c(k) = 2/(k-1)$, the density decay is slowed down as compared to the mean-field power laws by the emergence of depletion zones around the surviving particles. Further annihilations require that the reactants traverse the diffusion length $L(t) \sim (D \, t)^{1/2}$, which sets the typical separation scale; the corresponding density must then scale as $L(t)^{-d}$. Beyond the upper critical dimension $d_c(k)$, the system becomes effectively well-mixed, diffusion plays no limiting role, and the time scale is set by the reaction rate.
Phase transitions from active to absorbing states
-------------------------------------------------
Turning to our second example in the introductory remarks, we now investigate diffusing particles subject to the [*competing reactions*]{} $A \to \emptyset$ and $A \rightleftharpoons A + A$; adding the diffusion term to (\[dprreq\]), we arrive at the rate equation for the local particle density, $$\frac{\partial a(x,t)}{\partial t} = - D \left( r - \nabla^2 \right) a(x,t)
- \lambda \, a(x,t)^2 \ ,
\label{fiskol}$$ where $r = (\kappa - \sigma) / D$; in mathematical biology and ecology, the partial differential equation (\[fiskol\]) is known as the [*Fisher–Kolmogorov equation*]{}, and for example has been used to study population invasion fronts into empty regions. We shall instead focus on the critical region where the control parameter $r \to 0$, and a [*continuous non-equilibrium phase transition*]{} from an active to an inactive and absorbing state occurs.
The Doi–Peliti field theory action (\[masact\]) capturing the above reactions reads $$\begin{aligned}
&&{\cal A}[{\hat \psi},\psi] = \int \! d^dx \! \int \! dt \, \Bigl[
{\hat \psi} \left( \partial_t - D \, \nabla^2 \right) \psi \quad
\label{fikoac} \\
&&\ - \kappa \bigl( 1 - {\hat \psi} \bigr) \psi + \sigma \bigl( 1 -
{\hat \psi} \bigr) {\hat \psi} \, \psi - \lambda \big( 1 - {\hat \psi} \bigr)
{\hat \psi} \, \psi^2 \Bigr] \, . \nonumber\end{aligned}$$ Upon shifting and rescaling the fields according to ${\hat \psi}(x,t) = 1 + \sqrt{\lambda / \sigma} \, {\widetilde S}(x,t)$ and $\psi(x,t) = \sqrt{\sigma / \lambda} \, S(x,t)$, one arrives at $$\begin{aligned}
&&{\cal A}[{\widetilde S},S] = \int \! d^dx \! \int \! dt \, \Bigl(
{\widetilde S} \left[ \partial_t + D \left( r - \nabla^2 \right) \right] S
\nonumber \\
&&\qquad\qquad - u \, \bigl( {\widetilde S} - S \bigr) \, {\widetilde S} \, S
+ \lambda \, {\widetilde S}^2 \, S^2 \Bigr) \, ,
\label{dpcrac} \end{aligned}$$ where the three-point vertices have been symmetrized and now are proportional to the coupling $u = \sqrt{\sigma \, \lambda}$ with scaling dimension $[u] = \mu^{2-d/2}$. The associated [*upper critical dimension*]{} is therefore $d_c = 4$, and the annihilation four-point vertex $[\lambda] = \mu^{2-d}$ consequently is [*irrelevant*]{} in the RG sense near $d_c$. In the effective critical action, one may set $\lambda \to 0$, whereupon (\[dpcrac\]) reduces to the familiar [*Reggeon field theory*]{} action, which is invariant under [*rapidity inversion*]{} $S(x,t) \leftrightarrow - {\widetilde S}(x,-t)$. Viewing (\[dpcrac\]) as a Janssen–De Dominicis functional (\[janded\]), it is equivalent to the Langevin equation that amends the Fisher–Kolmogorov equation (\[fiskol\]) with a noise term, $$\!\!\! \frac{\partial S(x,t)}{\partial t} = D \bigl( \nabla^2 - r \bigr)
S(x,t) - u S(x,t)^2 \nonumber + \zeta(x,t) \, , \,\
\label{dplang}$$ with $\langle \zeta(x,t) \rangle = 0$ and the multiplicative “square-root” noise correlator $$\langle \zeta(x,t) \zeta(x',t') \rangle = 2 u \, S(x,t) \, \delta(x-x')
\delta(t-t') \, . \
\label{dpnois}$$
Drawing a space-time plot (time running from right to left) for the branching $A \to A + A$, death $A \to \emptyset$, and coagulation $A + A \to A$ processes, starting from a single occupied site, as depicted below, one realizes that they generate a [*directed percolation*]{} (DP) cluster, with “time” playing the role of a specified “growth” direction. The field theory (\[dpcrac\]) (with $\lambda = 0$) thus also describes the universal scaling properties of critical DP.
Indeed, one expects that active to absorbing state phase transitions should [*generically*]{} be captured by this DP universality class, namely in the absence of coupling to other slow conserved modes, disorder, or special additional symmetries. The origin for this remarkable DP conjecture becomes evident in a complementary coarse-grained phenomenological approach that will be framed in the language of epidemic spreading [@Janssen05]. Consider the following [*simple epidemic process*]{}:
1. A [*“susceptible” medium*]{} is locally [*“infected”*]{}, depending on the density of “sick” neighbors. Infected regions may later recover.
2. The “disease” [*extinction*]{} state is [*absorbing*]{}.
3. The disease [*spreads diffusively*]{} via infection, see 1.
4. Other fast microscopic degrees of freedom are incorporated as random [*noise*]{}. Yet according to condition 2, noise alone cannot regenerate the disease.
These decisive features can be encoded in a mesoscopic Langevin stochastic differential equation for the local density $n(x,t)$ of “active” (infected) individuals, $$\frac{\partial n(x,t)}{\partial t} = D \left( \nabla^2 - R[n(x,t)] \right)
n(x,t) + \zeta(x,t) \, , \,\
\label{seplan}$$ with the reactive functional $R[n]$, $\langle \zeta(x,t) \rangle = 0$, and noise correlator $L[n] = n \, N[n]$. In the spirit of Landau theory, near extinction one may expand these functionals in a Taylor series for small densities, $$r \approx 0: \, R[n] = r + u \, n \, + \ldots , \ N[n] = v + \ldots , \quad
\label{seplex}$$ where higher-order terms are [*irrelevant*]{} in the RG sense. After rescaling, the corresponding Janssen–De Dominicis response functional (\[janded\]) becomes identical to the Reggeon field theory action.
We now proceed to analyze the dynamic perturbation theory and renormalization for the DP action (\[dpcrac\]) to one-loop order. The only singular vertex functions are the propagator self-energy $\Gamma^{(1,1)}(q,\omega)$ and the three-point functions $\Gamma^{(1,2)} = - \Gamma^{(2,1)}$, owing to rapidity inversion symmetry, with the lowest-order Feynman graphs:
{width="0.7\columnwidth"}
-0.4truecm
Explicit evaluation of the self-energy yields $$\begin{aligned}
&&\Gamma^{(1,1)}(q,\omega) = i \omega + D (r + q^2) \nonumber \\
&&\qquad + \frac{u^2}{D} \int_k \frac{1}{i \omega / 2 D + r + q^2/4 + k^2}\ .
\label{dpgm11}\end{aligned}$$ As in critical statics, one needs to first ensure the [*criticality condition*]{}, namely that $\Gamma^{(1,1)}(0,0) = 0$ at the true percolation threshold $r = r_c$. To one-loop order, (\[dpgm11\]) results in the shift (additive renormalization) $$r_c = - \frac{u^2}{D^2} \int_k \frac{1}{r_c + k^2} + O(u^4) \ ,
\label{dpthsh}$$ and inserting $\tau = r - r_c$ in (\[dpgm11\]) subsequently leads to $$\begin{aligned}
&&\Gamma^{(1,1)}(q,\omega) = i \omega + D \left( \tau + q^2 \right)
\label{dpsg11} \\
&&\quad - \, \frac{u^2}{D} \int_k \frac{i \omega / 2 D + \tau + q^2/4}
{k^2 \left( i \omega / 2 D + \tau + q^2/4 + k^2 \right)} \ . \nonumber\end{aligned}$$ The three-point vertex function at vanishing external wave vectors and frequencies finally becomes $$\Gamma^{(1,2)}(\{ 0 \},(\{ 0 \}) = - 2 u \left[ 1 - \frac{2 u^2}{D^2} \!
\int_k \frac{1}{\left( \tau + k^2 \right)^2} \right] . \nonumber
\label{dpgm12}$$
For the [*multiplicative renormalizations*]{}, we follow (\[fldren\]) and (\[cplren\]), but with the slight modification $u_R = Z_u \, u \, A_d^{1/2} \mu^{(d-4)/2}$, as well as (\[dynren\]) with $Z_{\widetilde S} = Z_S$ owing to rapidity inversion invariance. Thus one obtains Wilson’s RG flow functions to one-loop order, $$\begin{aligned}
&&\gamma_S = \frac{v_R}{2} + O(v_R^2) \ , \quad
\gamma_D = - \frac{v_R}{4} + O(v_R^2) \ , \nonumber \\
&&\gamma_\tau = - 2 + \frac{3 v_R}{4} + O(v_R^2) \ ,
\label{dpwilf}\end{aligned}$$ with the effective coupling and associated beta function $$\begin{aligned}
&&v_R = \frac{Z_u^2}{Z_D^2} \, \frac{u^2}{D^2}\, A_d\, \mu^{d-4} \ ,
\label{dpecpl} \\
&&\beta_v = v_R \Bigl[ - \epsilon + 3 v_R + O(v_R^2) \Bigr] \ .
\label{dpbeta}\end{aligned}$$ Below the upper critical dimension $d_c = 4$, the [*IR-stable RG fixed point*]{} $$v_{\rm DP}^* = \frac{\epsilon}{3} + O(\epsilon^2)
\label{dprgfp}$$ appears, and solving the RG equation for the two-point [*correlation function*]{} in its vicinity yields $$\begin{aligned}
&&C_R(\tau_R,q,\omega)^{-1} \approx q^2 \, \ell^{\gamma_S^*} \nonumber \\
&&\qquad {\hat C}_R\Bigl( \tau_R \, \ell^{\gamma_\tau^*},v^*,
\frac{q}{\mu \ell}, \frac{\omega}{D_R \, \mu^2 \ell^{2 + \gamma_D^*}}
\Bigr)^{-1} .
\label{dpcorf}\end{aligned}$$ This allows us to identify the three independent [*critical exponents*]{} for directed percolation to order $\epsilon = 4-d$: $$\begin{aligned}
&&\eta = - \gamma_S^* = - \frac{\epsilon}{6} + O(\epsilon^2) \ , \nonumber \\
&&\nu^{-1} = - \gamma_\tau^* = 2 - \frac{\epsilon}{4} + O(\epsilon^2) \ ,
\label{dpcrex} \\
&&z = 2 + \gamma_D^* = 2 - \frac{\epsilon}{12} + O(\epsilon^2) \ . \nonumber\end{aligned}$$
The DP universality class also applies to many active to absorbing state phase transitions with more than just one particle species. As an example, consider the predator extinction threshold in the spatially extended stochastic two-species Lotka–Volterra model with finite carrying capacity discussed in the chapter introduction. The associated Doi–Peliti field theory action reads $$\begin{aligned}
&&S[{\hat a},a;{\hat b},b] = \int \!\! d^dx \! \int \!\! dt \, \Bigl[
{\hat a} \, \bigl( \partial_t - D_A \nabla^2 \bigr) \, a \nonumber \\
&&\qquad + \kappa \, \bigl( {\hat a} - 1 \bigr) \, a
+ {\hat b} \, \bigl( \partial_t - D_B \nabla^2 \bigr) \, b
\label{lvdpft} \\
&&\quad + \sigma \, \bigl( 1 - {\hat b} \bigr) \, {\hat b} \, b \,
e^{-\rho^{-1} \, {\hat b} b} + \lambda \, \bigl( {\hat b} - {\hat a} \bigr)
\, {\hat a} \, a \, b \Bigr] \, , \nonumber\end{aligned}$$ where diffusive spreading has been assumed, and the exponential term in the prey production term takes into account the local restriction to a maximum particle density $\rho$. As usual, one applies the field shifts ${\hat a} = 1 + {\tilde a}$, ${\hat b} = 1 + {\tilde b}$; realizing that $\rho^{-1}$ constitutes an irrelevant perturbation (since the density scales as $[\rho] = \mu^d$), we furthermore expand to lowest order in $\rho^{-1}$, which yields $$\begin{aligned}
&&S[{\tilde a},a;{\tilde b},b] = \int \!\! d^dx \! \int \!\! dt \,
\Bigl[ {\tilde a} \, \bigl( \partial_t - D_A \nabla^2 + \kappa \bigr) \, a
\nonumber \\
&&\qquad + {\tilde b} \, \bigl( \partial_t - D_B \nabla^2 - \sigma \bigr) \,
b - \sigma \, {\tilde b}^2 \, b
\label{dpshft} \\
&&\ + \sigma \, \rho^{-1} \, \bigl( 1 + {\tilde b} \bigr)^2 \, {\tilde b}
\, b^2 - \lambda \, \bigl( 1 + {\tilde a} \bigr) \,
\bigl( {\tilde a} - {\tilde b} \bigr) \, a \, b \Bigr] . \nonumber\end{aligned}$$ Near the predator extinction threshold, the prey almost fill the entire system. We therefore define the properly [*fluctuating fields*]{} $c = b_s - b$ with $b_s \approx \rho$ and $\langle c \rangle = 0$, and ${\tilde c} = - {\tilde b}$. Rescaling to $\phi = \sqrt{\sigma} \, c$ and ${\tilde \phi} = \sqrt{\sigma} \, {\tilde c}$, and noting that asymptotically $\sigma \to \infty$ under the RG flow since $[\sigma] = \mu^2$, the ensuing action simplifies drastically. At last, we add the [*growth-limiting reaction*]{} $A + A \to A$ (with rate $\tau$); the fields $\phi$ and ${\tilde \phi}$ can then be integrated out, leaving Reggeon field theory (\[dpcrac\]) as the resulting effective action, with the non-linear coupling $u = \sqrt{\tau \, \lambda \, b_s}$.
Concluding Remarks
==================
These lecture notes can of course only provide a very sketchy and vastly incomplete introduction to the use of field theory tools and applications of the renormalization group in statistical physics. I have merely focused on continuous phase transitions in equilibrium, dynamic critical phenomena in simple relaxational models, and a few examples for universal scaling behavior in non-equilibrium dynamical systems. Among the many topics not covered or even mentioned here are critical phenomena in finite and disordered systems; universality classes of critical dynamics with reversible couplings to other conserved modes; universal short-time and non-equilibrium relaxation scaling properties in the aging regime; depinning transitions and driven interfaces in disordered media; spin glasses and structural glasses; and of course phase transitions and generic scale invariance in quantum systems. Nor have I addressed powerful representations through supersymmetric or conformally invariant quantum field theories, Monte Carlo algorithms, or numerical non-perturbative RG approaches, since the latter will be covered elsewhere in this volume; for their applications to non-equilibrium systems, see, e.g., Ref. [@Canet11].
Nevertheless, I hope to have conveyed the message that methods from field theory are ubiquitous in statistical physics, and the renormalization group has served as a remarkably powerful mathematical tool to address at least the universal scaling aspects of cooperative behavior governed by strong correlations and fluctuations. Thus, the RG has become a cornerstone of our understanding of complex interacting many-particle systems, and its language and basic philosophy now pervade the entire field, with applications that increasingly reach out beyond fundamental physics to material science, chemistry, biology, ecology, and even sociology.
Finally, I would like to express my gratitude to the organizers (and their funding agencies) for the opportunity to attend and lecture at the 49th Schladming Theoretical Physics Winter School. I thoroughly enjoyed the stimulating and informal atmosphere in the Styrian Alps, and profited from many discussions with colleagues and students. I can only hope that all attendants learned as much about new and exciting developments in theoretical physics from my fellow lecturers as I did.
[00]{}
K.G. Wilson and J. Kogut, [*The renormalization group and the $\epsilon$ expansion*]{}, Phys. Rep. [**12 C**]{}, 75–200 (1974).
M.E. Fisher, [*The renormalization group in the theory of critical behavior*]{}, Rev. Mod. Phys. [**46**]{}, 597–616 (1974).
S.-k. Ma, [*Modern theory of critical phenomena*]{}, Benjamin–Cummings (Reading, 1976).
A.Z. Patashinskii and V.L. Pokrovskii, [*Fluctuation theory of phase transitions*]{}, Pergamon Press (New York, 1979).
N. Goldenfeld, [*Lectures on phase transitions and the renormalization group*]{}, Addison–Wesley (Reading, 1992).
J.J. Binney, N.J. Dowrick, A.J. Fisher, and M.E.J. Newman, [*The theory of critical phenomena*]{}, Oxford University Press (Oxford, 1993).
J. Cardy, [*Scaling and renormalization in statistical physics*]{}, Cambridge University Press (Cambridge, 1996).
G.F. Mazenko, [*Fluctuations, order, and defects*]{}, Wiley–Interscience (Hoboken, 2003).
U.C. Täuber, [*Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior*]{}, under contract with Cambridge University Press (Cambridge); see [http://www.phys.vt.edu/\~tauber/utaeuber.html]{}.
P. Ramond, [*Field theory — A modern primer*]{}, Benjamin–Cummings (Reading, 1981).
D.J. Amit, [*Field theory, the renormalization group, and critical phenomena*]{}, World Scientific (Singapore, 1984).
G. Parisi, [*Statistical field theory*]{}, Addison–Wesley (Redwood City, 1988).
C. Itzykson and J.M. Drouffe, [*Statistical field theory*]{}, Vol. I, Cambridge University Press (Cambridge, 1989).
M. Le Bellac, [*Quantum and statistical field theory*]{}, Oxford University Press (Oxford, 1991).
J. Zinn-Justin, [*Quantum field theory and critical phenomena*]{}, Clarendon Press (Oxford, 1993).
P.C. Hohenberg and B.I. Halperin, [*Theory of dynamic critical phenomena*]{}, Rev. Mod. Phys. [**49**]{}, 435–479 (1977).
H.K. Janssen, [*Field-theoretic methods applied to critical dynamics*]{}, in: [*Dynamical critical phenomena and related topics*]{}, ed. C.P. Enz, Lecture Notes in Physics, Vol. [**104**]{}, Springer (Heidelberg), 26–47 (1979).
R. Folk and G. Moser, [*Critical dynamics: a field theoretical approach*]{}, J. Phys. A: Math. Gen. [**39**]{}, R207–R313 (2006).
U.C. Täuber, [*Field theory approaches to nonequilibrium dynamics*]{}, in: [*Ageing and the Glass Transition*]{}, eds. M. Henkel, M. Pleimling, and R. Sanctuary, Lecture Notes in Physics, Vol. [**716**]{}, Springer (Berlin), 295–348 (2007).
E. Frey and F. Schwabl, [*Critical dynamics of magnets*]{}, Adv. Phys. [**43**]{}, 577–683 (1994).
B. Schmittmann and R.K.P. Zia, [*Statistical mechanics of driven diffusive systems*]{}, in: [*Phase Transitions and Critical Phenomena*]{}, ed. C. Domb and J.L. Lebowitz, Vol. [**17**]{}, Academic Press (London, 1995).
J.L. Cardy, [*Renormalisation group approach to reaction-diffusion problems*]{}, in: [*Proceedings of Mathematical Beauty of Physics*]{}, ed. J.-B. Zuber, Adv. Ser. in Math. Phys. [**24**]{}, 113 (1997).
D.C. Mattis and M.L. Glasser, [*The uses of quantum field theory in diffusion-limited reactions*]{}, Rev. Mod. Phys. [**70**]{}, 979–1002 (1998).
U.C. Täuber, M.J. Howard, and B.P. Vollmayr-Lee, [*Applications of field-theoretic renormalization group methods to reaction-diffusion problems*]{}, J. Phys. A: Math. Gen. [**38**]{}, R79–R131 (2005).
H. Hinrichsen, [*Nonequilibrium critical phenomena and phase transitions into absorbing states*]{} Adv. Phys. [**49**]{}, 815–958 (2001).
G. Ódor, [*Phase transition universality classes of classical, nonequilibrium systems*]{}, Rev. Mod. Phys. [**76**]{}, 663–724 (2004).
H.K. Janssen and U.C. Täuber, [*The field theory approach to percolation processes*]{}, Ann. Phys. (NY) [**315**]{}, 147–192 (2005).
M. Mobilia, I.T. Georgiev, and U.C. Täuber, [*Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka–Volterra models*]{}, J. Stat. Phys. [**128**]{}, 447–483 (2007).
U.C. Täuber, [*Stochastic population oscillations in spatial predator-prey models*]{}, J. Phys.: Conf. Ser. [**319**]{}, 012019 - 1–14 (2011).
L. Canet, H. Chaté, and B. Delamotte, [*General framework of the non-perturbative renormalization group for non-equilibrium steady states*]{}, J. Phys. A: Math. Gen. [**44**]{}, 495001 - 1–26 (2011).
|
---
abstract: |
We consider a $SBS$ Josephson junction the superconducting electrodes $S$ of which are in contact with normal metal reservoirs ($B$ means a barrier). For temperatures near $T_{c}$ we calculate an effective critical current $%
I_{c}^{\ast }$ and the resistance of the system at the currents $I<$ $%
I_{c}^{\ast }$ and $I>>I_{c}^{\ast }$. It is found that the charge imbalance, which arises due to injection of quasiparticles from the $N$ reservoirs into the $S$ wire, affects essentially the characteristics of the structure. The effective critical current $I_{c}^{\ast }$ is always larger than the critical current $I_{c}$ in the absence of the normal reservoirs and increases with decreasing the ratio of the length of the $S$ wire $2L$ to the charge imbalance relaxation length $l_{Q}$. It is shown that a series of peaks arises on the $I-V$ characteristics due to excitation of the Carlson-Goldman collective modes. We find the position of Shapiro steps which deviates from that given by the Josephson relation.
address: |
$^{(1)}$Theoretische Physik III,\
Ruhr-Universität Bochum, D-44780 Bochum, Germany\
$^{(2)}$Institute for Radioengineering and Electronics of Russian Academy of\
Sciences,11-7 Mokhovaya str., Moscow 125009, Russia\
author:
- 'A.F. Volkov$^{1,2}$.'
title: 'Charge imbalance and Josephson effects in superconductor-normal metal mesoscopic structures. '
---
Introduction
============
In the theory of the Josephson effect in weak links [@Josephson] it is assumed that the superconducting electrodes are in equilibrium. This means, in particular, that a gauge-invariant potential $\mu $
$$\mu =\frac{1}{2}\partial \chi /\partial t+eV, \label{Mu}$$
is equal to zero [@Ft1](we set $\hbar =1$). The Josephson relation follows immediately from this condition
$$2e(V_{2}-V_{1})=\partial \varphi /\partial t, \label{JosRelation}$$where $(V_{2}-V_{1})$ is the voltage drop across the Josephson junction and $%
\varphi =\chi _{1}-\chi _{2}$ is the phase difference. Most works on studies of the Josephson effects were carried out under equilibrium conditions so that the potential $\mu $ is zero in the superconducting electrodes $S_{1,2}$, and Eq. (\[JosRelation\]) is satisfied [@Kulik; @Barone]. The potential $\mu $ is related to a so called charge imbalance arising due to a different population of the electronlike and holelike branches of the quasiparticle spectrum of a superconductor [@Ft2]. The conditions under which this charge imbalance may arise were studied experimentally [Cl,Yu,KlMoi,Dolan,VanHar,Mamin,Chien,Strunk]{} and theoretically [Rieger,Tin,SSJLTP,AV,AVUsp,Ovch]{}.
In the last decade a great progress in preparation of superconductor/normal metal ($S/N$) nanostructures has been achieved. Varies properties of these structures were studied experimentally. One can mention the study of transport [@Petrashov; @Pothier; @Charlat; @Chien; @Klapwijk; @Shapira] and thermoelectric [@ChandraTherm; @SosninTherm] properties, the measurements of the density-of-states [@EsteveDOS] etc. Recently a transition of a thin and short superconducting wire into a resistive state caused by a current $I$ was investigated [@Bezryadin]. In particular, an increase of the critical current $I_{c}$ has been observed when an external magnetic field $H$ is applied [@Bezryadin]. A possible reason for the observed increase of $I_{c}$ may be a polarization of magnetic impurities by the magnetic field $H$ [@Feigelman; @Bezryadin]. Another mechanism has been suggested in Ref.[@Vodolazov]. It has been proposed that an increase of the critical currents characterizing phase-slip centers in a thin $S$ wire may be due to a shortening of the charge imbalance relaxation length $l_{Q}$ under the influence of $H$. However, it remains unclear how this mechanism can work in a pure superconducting state when there is no charge imbalance. As is known (for review see [@AVUsp; @Rammer; @Kopnin]), the charge imbalance arises only under nonequilibrium conditions. For example, this mechanism may be realized in a superconductor-normal metal ($SN$) nanostructure with a bias current, where the charge imbalance is built up due to injection of quasiparticles into the $S$ wire from the $N$ reservoirs [Rieger,Tin,SSJLTP,AV,AVUsp,Ovch]{}.
In the present, paper we investigate the Josephson effect in a $NSBSN$ structure under nonequilibrium conditions, in the presence of a bias current $%
I$ flowing through $SN$ interfaces. In this case the potential $\mu $ arises due to injection of quasiparticles into the superconductors $S$ from the normal reservoirs $N$ ($B$ is a barrier of an arbitrary type). We show that if the length of the $S$ wire $2L$ is less than or comparable with the charge imbalance relaxation length $l_{Q},$ the critical current increases and may significantly exceed the critical current $I_{c}$ in the absence of the normal reservoirs. The resistance of the system also will be calculated for the currents $I\leq I_{c}^{\ast }$ and $I\gg I_{c}^{\ast }$, where $%
I_{c}^{\ast }$ is an effective critical current. We find the position of Shapiro steps and show that peaks associated with the excitation of collective modes of the Carlson-Goldman type arise on the $I-V$ characteristics.
Note that in recent publications [@Keizer; @Peeters] a $NSN$ system without the Josephson junction was considered and the $I-V$ curve of the system was calculated numerically.
Model and Basic Equations
=========================
We consider the structure shown in Fig.1. We assum that the barrier is located in the middle of the $S$ wire, but the results remains qualitatively unchanged for a system with the barrier located at the $S/N$ interface. The system consists of a superconducting wire ($S$ wire) connecting two normal reservoirs $N$. In the middle of the $S$ wire there is a barrier which provides the Josephson coupling. The transparency of the $%
SN$ interface is assumed to be high. The analysis can be easily generalized to the case of arbitrary $NS$ interface transparencies. We also assume that the temperature $T$ is close to the critical one $T_{c}$ and therefore the inequality
$$\Delta \ll T \label{Tdelta}$$
is satisfied ($\Delta $ is the order parameter in the $S$ wire). In this case the effects of the branch imbalance are most significant. The lengths $%
L $ and $l_{Q}$ are assumed large in comparison with the Ginsburg-Landau correlation length $\xi _{GL}$, i.e.,
$$\xi _{GL}\approx 1.2\sqrt{D/T_{c}}(T_{c}/\Delta )\ll \{L,l_{Q}\},
\label{KsiGL}$$
where $D$ is the diffusion coefficient in the $S$ wire. The charge imbalance relaxation length $l_{Q}$ is determined by inelastic scattering processes and may be rather long ([@Tin; @SSJLTP; @AV; @Ovch; @AVUsp]). This assumption allows us to consider the order parameter constant in the major part of the $%
S$ wire. The same limit was considered in Refs. [@AV; @Ovch] where the resistance of the superconductor in a $SN$ system was calculated. The relation between $L$ and $l_{Q}$ may be arbitrary.
{width="90.00000%"}
Our aim is to obtain a relationship between the current $I$ and the voltage $%
V_{L}\equiv V(x)$ at $x=L$ (the voltage difference between the $N$ reservoirs is $V_{NN}=-2V_{L}$). We restrict ourselves with voltages $V_{NN}$ small compared to the energy gap $\Delta :$ $eV_{NN}\ll \Delta $. In this limit the distribution function (longitudinal in terms of Ref. [@SSJLTP]), which determines the order parameter $\Delta ,$ is close to the equilibrium one. Another distribution function denoted by $f_{1}$ in Ref. [@LO] (transverse in terms of Ref. [@SSJLTP]) was found in Ref. [SSJLTP,AV,Ovch]{}. To be more exact, the so called anomalous Green’s function, $\hat{g}^{(a)}=$ $\hat{g}^{R}\hat{\tau}_{3}f_{1}-f_{1}\hat{\tau}%
_{3}\hat{g}^{A}$, was found in Refs. [@AV; @Ovch; @AVUsp], where $\hat{g}%
^{R(A)}$ are the retarded (advanced) Green’s functions. Using the functions $%
\hat{g}^{(a)}$ and $\hat{g}^{R(A)}$, one can obtain the expression for the current $I$ in the $S$ wire, which in the main approximation in the parameters $\Delta /T$ and $V/T$ is equal to $$I=\mathcal{S}\sigma (E+\frac{\pi \Delta ^{2}}{2Te}Q), \label{Icurrent}$$where $\mathcal{S}$ is the cross section area of the $S$ wire, $\sigma $ is the conductivity of the $S$ wire in the normal state. We assume that the cross section area is small compared to $\{\xi _{GL}^{2},\lambda _{L}^{2}\}$ so that all the vectors $\{\mathbf{I,E,Q}\}$ have only the $x-$component $%
\{I,E,Q\}$ and depend on $x$. The electric field $E=-\partial V(x)/\partial
x $ (we drop the vector potential $A$ because we consider only the longitudinal electric field choosing a corresponding gauge). The condensate momentum $Q$ is defined as
$$Q=(1/2)\partial \chi /\partial x-2\pi A/\Phi _{0} \label{Q}$$
where $\Phi _{0}=hc/2e$ is the magnetic flux quantum; the vector potential $%
A $ can be dropped because we do not consider the action of magnetic field. The momentum $Q$ obeys the equation
$$\partial Q/\partial t=eE+\partial \mu /\partial x, \label{Qeq}$$
The spatial and temporal variation of the gauge-invariant potential $\mu (x,t)$ is described by the equation [@AVUsp; @SchoenRev]
$$(\partial /\partial t+\gamma )\mu =v_{CG}^{2}\partial Q/\partial x,
\label{MuEq}$$
where $\gamma $ is a quantity which determines the charge imbalance relaxation rate. It is related to inelastic scattering processes ($\gamma =1/\tau _{\epsilon }$, where $\tau _{\epsilon
}$ is the inelastic relaxation time), the condensate momentum ($\gamma \sim DQ^{2}\Delta /T$) in the presence of condensate flow or the gap anisotropy [Tin,SSJLTP,AV,Ovch]{}. The velocity $v_{CG}=\sqrt{2D\Delta }$ is the velocity of propagation of the Carlson-Goldman collective mode [CG,SSCollModes,AVCollModes,AVUsp,SchoenRev]{}. In the stationary case Eq.([MuEq]{}) can be written in the form
$$l_{Q}^{-2}\mu =(e/\sigma )\partial j_{S}/\partial x=\partial ^{2}\mu
/\partial x^{2}, \label{EqMuSt}$$
where $j_{S}=\sigma (\pi \Delta ^{2}/2Te)Q$ is the density of the supercurrent; $l_{Q}^{{}}=\sqrt{4TD/(\pi \gamma \Delta )}$ is the penetration depth of the electric field (or the charge relaxation length). Eq.(\[EqMuSt\]) describes the conversion of the quasiparticle and superconducting currents. One can see that the potential $\mu $ arises if the divergence of the supercurrent (or quasiparticle current) differs from zero.
The current in the $S$ wire can also be written as the current through the Josephson junction. In the main approximation it is equal to (see Appendix)
$$I=-\frac{2V_{0}}{R_{B}}+I_{c}\sin \varphi , \label{CurrentJJ}$$
where $2V_{0}\equiv 2V(0+)=-2V(0-)$ is the voltage drop across the Josephson junction ($V_{0}$ is negative), $R_{B}$ is the barrier resistance, $I_{c}$ is the Josephson critical current, and $\varphi =\chi (0_{+})-\chi (0_{-})$ is the phase difference. For simplicity we drop here the displacement current $%
C\partial V_{0}/\partial t$ assuming that the parameter $%
(CR_{B})e(I_{c}R_{B})$ is small (the main results concerning the critical current and the resistances of the system do not depend on the presence of the displacement current). At $V(L)\ll \Delta $ the critical current equals $%
I_{c}=\pi \Delta ^{2}/4TeR_{B}.$ The first term in Eq.(\[CurrentJJ\]) is the quasiparticle current and the second term is the supercurrent.
Eqs.(\[Icurrent\]-\[CurrentJJ\]) describe the system. These equations should be complemented by boundary conditions. Since we consider the voltages smaller than $\Delta $, the distribution function $f_{l},$ which determines the supercurrent, is close to the equilibrium one: $f_{l}=$ $\tanh (\epsilon /2T)$. This implies the conservation of the quasiparticle (correspondingly superconducting) currents at the Josephson junction, i.e.,
$$\mathcal{S}\sigma E(0_{+})=-\frac{2V_{0}}{R_{B}}=I-I_{c}\sin \varphi .
\label{BC1}$$
Another boundary condition relates the electric field at the edge of the $S$ wire $E(L)$ to the electric field in the $N$ region $E_{N}$. As is shown in Refs.[@AV; @Ovch], the electric field $E(x)$ experiences a jump at the $SN$ interface
$$\lbrack E]_{SN}\equiv E_{N}-E(L)=rE(L) \label{BCforE}$$
where $r\approx 0.7((\Delta /T)l_{\epsilon }/\xi
_{GL})^{1/4}\approx 1.17[((T_{c}-T)/T_{c})^{2}\tau _{\epsilon
}/\tau ]^{1/4}$; $\tau _{\epsilon },\tau $ are the inelastic and elastic scattering times. This jump is small if the temperature $T$ is close to $T_{c}$ and therefore the electric field (accordingly the quasiparticle current) is continuous at the $SN$ interface. For a finite value of $r$ and equal conductivities in the $S$ and $N$ regions we get from Eq.(\[BCforE\])
$$E(L)=E_{N}(1-\frac{r}{1+r}) \label{BC2}$$
Here the second term on the right is small near $T_{c}$. Solving Eqs.([Icurrent]{},\[Qeq\],\[MuEq\],\[CurrentJJ\]) with the boundary conditions (\[BC1\]-\[BC2\]), we can calculate, in particular, the effective critical current $I_{c}^{\ast }$ and the resistance of the system $%
R$.
Resistance and Josephson critical current.
==========================================
Eqs.(\[Icurrent\],\[Qeq\],\[MuEq\],\[CurrentJJ\]) are nonlinear equations in partial differentials in which all functions depend on time $t$ and coordinate $x$. Therefore the solution to these equations can not be obtained in a general case. We consider limiting cases of small and large currents $I$.
*a) Stationary case;* $I<I_{c}^{\ast }.$
If the current does not exceed a critical value $I_{c}^{\ast }$ (the effective critical current $I_{c}^{\ast }$ will be found below), the phase difference $\varphi $ and other quantities do not depend on time. In particular, $\partial Q/\partial t=0$ and the electric field is
$$eE(x)=-\partial \mu /\partial x \label{StatE}$$
The momentum of the condensate is found from Eq.(\[Icurrent\])
$$\frac{\pi \Delta ^{2}}{2T}Q=eIR_{n}+\partial \mu /\partial x
\label{StatQ}$$
where $R_{n}=(\mathcal{S}\sigma )^{-1}$ is the resistance of the $S$ wire in the normal state per unit length. The potential $\mu $ is described by Eq.(\[EqMuSt\]). At small currents when the condensate flow does not contribute to the relaxation of the charge imbalance, the length $l_{Q}^{{}}$ equals: $l_{Q}=\sqrt{4(D\tau _{\epsilon })T/\pi
\Delta }$ [@SSJLTP; @AV; @Ovch]. The solution of Eq.(\[EqMuSt\]) is
$$\mu (x)=A\cosh (x/l_{Q})+B\sinh (x/l_{Q}) \label{Mux}$$
The electric field and the potential $V(x)$ equal $$eE(x)=-[A\sinh (x/l_{Q})+B\cosh (x/l_{Q})]/l_{Q},\text{ \ }eV(x)=A\cosh
(x/l_{Q})+B\sinh (x/l_{Q})+V_{\ast } \label{E+V}$$where $V_{\ast }$ is an integration constant.
First we neglect the second term on the right in Eq.(\[BC2\]). From Eqs.(\[BC1\],\[BC2\]) we find the coefficient $B$ and $A$
$$B=-eR_{Q}(I-I_{c}\sin \varphi ),\text{ \ }A=-(eR_{Q}I+BC)/S \label{BA}$$
where $R_{Q}=l_{Q}/(\mathcal{S}\sigma )$, $C\equiv \cosh \theta$ and $%
S\equiv \sinh \theta$ with $\theta = L/l_{Q}$.
From Eq.(\[E+V\]) we obtain
$$eV_{L}=eV_{0}+A(C-1)+BS$$
Writing $V_{0}$ in terms of $B$ (see Eqs.(\[BC1\],\[BA\]) ), we can represent $V_{L}$ in the form
$$eV_{L}=B(\frac{b}{2}+\frac{C-1}{S})-IR_{Q}\frac{C-1}{S} \label{V_L}$$
In the stationary case one has $eV_{0}=\mu (0)=A.$ Combining Eqs.(\[BC1\],\[BA\]), we obtain from this formula
$$I(\frac{b}{2}+\frac{C-1}{S})=I_{c}(\frac{b}{2}+\frac{C}{S})\sin \varphi
\label{EffI_c1}$$
where $b\equiv R_{B}/R_{Q}$. Eq.(\[EffI\_c1\]) determines the effective critical current $I_{c}^{\ast }$
$$I_{c}^{\ast }/I_{c}=\frac{bS+2C}{bS+2(C-1)}=\frac{b\tanh (\theta
/2)+1+\tanh ^{2}(\theta /2)}{\tanh (\theta /2)(b+2\tanh (\theta
/2))} \label{EffI_c}$$
It is seen that the effective critical current $I_{c}^{\ast }$ is always larger than the critical current $I_{c}$ in the absence of the charge imbalance. In the limit of large $b=R_{B}/R_{Q}$ or $\theta =L/l_{Q}$ the effective critical current coincides with $I_{c}$. Therefore in a Josephson junction with a weak coupling between superconductors ($b\gg 1$), the effects of charge imbalance are not important. For a short $S$ wire, we obtain
$$I_{c}^{\ast }/I_{c}=\frac{b\theta +2}{\theta (b+2\theta )} \label{EffI_c3}$$
One can see that $I_{c}^{\ast }$ diverges at $\theta \rightarrow
0$. This fact is in agreement with the results of Ref.[@Zaitsev], where a very short $NSS^{\prime }$ system was considered, and it was concluded that the stationary state exists at any currents $I.$ However our consideration is valid only for not too small $\theta $ ($\theta =L/l_{Q}\succsim \xi
_{GL}/l_{Q}$). The dependence of $I_{c}^{\ast }/I_{c}$ on $\theta
$ is shown in Fig.2 for different $b.$
{width="90.00000%"}
What is the reason for the increase of the effective critical current from the physical point of view? The critical current is defined as a maximum current $I_{\max }$ at which the stationary state is possible: $\partial \varphi /\partial t=0.$ The current $I_{\max }$ in an ordinary equilibrium Josephson junction coincides with the critical current $I_{c}$ because the first term in Eq.(\[CurrentJJ\]) in the stationary state is zero: -$%
2V_{0}=(\hbar /2e)\partial \varphi /\partial t=0.$ As we noted in the Introduction, this equation follows from the fact that the potential $\mu (x)$ is zero. The same situation takes place near the $S/B/S$ interfaces in a long ($L>>l_{Q}$) $S$ wire. The charge imbalance, which determines $\mu (x)$ arises only in the vicinity of $S/N$ interface and decays on the scale of order $l_{Q}$. Therefore the effective critical current is the maximum current at which the stationary state with nonzero potentials $V$and $\mu $ is possible. Here we find this current for the case of a short system: $L<<l_{Q}$. Then, the electric field $E(x)$ and potential $V(x)$ can be written as
$$eE(x)\cong -[A(x/l_{Q})+B]/l_{Q},eV(x)\cong A+B(x/l_{Q})\text{ }
\label{EshortL}$$
The first term in the square brackets of Eq.(\[EshortL\]) describes the conversion of the quasiparticle current into the condensate current. In the normal state it is zero because $l_{Q}\rightarrow \infty $. At $x=L$ the charge is transferred by quasiparticles (we ignore the jump in the electric field setting $r=0$): $S\sigma E(L)=I$. At $x=0$ the quasiparticle current is $S\sigma E(0_{+})=I-I_{c}\sin \varphi .$ Therefore the coefficient $A$ related to the conversion of the quasiparticle current into the condensate one is equal to $$-A(L/l_{Q})\cong eI_{c}R_{Q}\sin \varphi .\text{ } \label{AshortL}$$On the other hand, at currents $I$ large in comparison with $I_{c}$ one has
$$I\cong -2V_{0}/R_{B}\cong -2A/eR_{B}. \label{IshortL}$$
Thus, from Eqs. (\[AshortL\]-\[IshortL\]), we find the maximum current for the stationary state
$$I_{c}^{\ast }\cong I_{c}(2l_{Q}/bL)$$
This formula is valid if the ratio $(l_{Q}/bL)$ is large. If the current exceeds this value, the stationary state is not possible. One can say that the length of the S wire is too short to provide the conversion of the quasiparticle current into the condensate current $I_{c}\sin \varphi $.
If we take into account the second term in the boundary condition (\[BC2\]), we obtain for $I_{c}^{\ast }$
$$I_{c}^{\ast }/I_{c}=\frac{2+b\tanh \theta }{\tanh \theta \lbrack b+2\tanh
(\theta /2)+2r/(S(1+r))]} \label{I_Cr}$$
The resistance of the system $R_{0}=V_{NN}/I=|2V_{L}|/I$ at $I<I_{c}^{\ast }$ can be easily found with the aid of Eqs. (\[V\_L\],\[EffI\_c1\])
$$R_{0}=2R_{Q}\frac{2\tanh \theta +b}{2+b\tanh \theta } \label{R_0}$$
In limiting cases we find
$$\text{ }R_{0}=2R_{Q}\left\{
\begin{array}{ll}
(2\theta +b)/(2+b\theta ), & \theta \ll 1 \\
1, & \theta \gg 1%
\end{array}
\right.$$
The resistance $R_{0}$ is equal to $R_{B}$ at $\theta \rightarrow 0$ and to $2R_{Q}$ at $\theta \rightarrow \infty $. Thus, the resistance of a short system is a combination of the barrier resistance $R_{B}$ and $%
R_{Q} $ (the resistance of the $S$ wire near the $SN$ interface). The resistance of a long system equals the resistance of the $S$ wire near the $%
SN$ interface on the scale $l_{Q}.$ The dependence of the resistance $R_{0} $ on $\theta $ for different $b$ is shown in Fig.3.
{width="90.00000%"}
With account for a finite jump of the electric field at the $SN$ interface (the second term on the right in Eq. (\[BC2\])), we obtain for $R_{0}$
$$R_{0}=\frac{2}{1+r}R_{Q}\frac{2\tanh \theta +b}{2+b\tanh \theta }$$
Consider now the case of large currents $I$ when the phase difference $%
\varphi (t)$ is increasing in time ($\varphi (t)\sim t$) and its oscillating part is small.
*a) Quasistationary case;* $I\gg I_{c}^{\ast }.$
In this case the condensate momentum is almost time-independent so that $%
\partial Q/\partial t\cong 0.$ The potentials $\mu ,V$ and the electric field $E$ are described by Eqs.(\[Mux\],\[E+V\]), but the formula for the coefficient $B$ is changed: $B=-eR_{Q}I.$ From Eq.(\[V\_L\]) we find the voltage difference between the $N$ reservoirs
$$V_{NN}\equiv -2V_{L}=I[R_{B}+4R_{Q}\frac{C-1}{S}] \label{V_NN2}$$
and the resistance $R_{\infty }$ at large currents
$$R_{\infty }\equiv \lbrack R_{B}+4R_{Q}\tanh (\theta /2)]
\label{R_inf}$$
In the case of a long $S$ wire ($\theta \gg 1$) we obtain: $%
R_{\infty }\equiv R_{B}+4R_{Q}$; that is, the resistance of system is the sum of the barrier resistance and the resistance of the regions of the $S$ wire where the electric field penetrates (close to the $SN$ interface and to the barrier). In the case of a short $S$ wire ($\theta \ll 1$) the resistance is: $R_{\infty }\equiv
R_{B}+2\theta R_{Q},$ that is, the contribution of the superconducting regions to the resistance decreases.
With account for the second term in Eq.(\[BC2\]) the resistance $R_{\infty
}$ acquires the form
$$R_{\infty }\equiv \lbrack R_{B}+4R_{Q}(1-\frac{r}{2(1+r)})\tanh (\theta /2)]$$
That is, the contribution of the $S$ region near the $SN$ interface to the resistance decreases.
The I-V characteristics and Shapiro steps
=========================================
In this Section we analyze the form of the current-voltage (I-V) characteristics and calculate the voltage $V_{Sh}$ which determines the position of the first Shapiro step. At a finite value of the ratio $%
R_{Q}/R_{B}$ the voltage $V_{Sh}$ differs from that given by Eq.([JosRelation]{}). We restrict ourselves with the limit of large currents $I$ employing an expansion in the parameter $I_{c}/I$. In the considered non-stationary case all the quantities depend on time, and in the lowest approximation in the parameter $I_{c}/I$ this dependence is determined by terms of the type $\exp (i\Omega
_{J}t)$, where $\Omega _{J}$ is the frequency of the Josephson oscillations. Eqs.(\[MuEq\],\[EqMuSt\],\[Mux\],[E+V]{}) can be easily generalized for the non-stationary case. The potential $%
\mu $ is described again by the equation
$$\partial ^{2}\mu /\partial x^{2}=k_{\Omega }^{2}\mu \label{EqMuNonst}$$
where $k_{\Omega }^{2}=(i\Omega +\gamma )(i\Omega +\Omega _{\Delta
})/v_{CG}^{2}$, $ \Omega _{\Delta }=\pi \Delta ^{2}/2T$. The solution for this equation is
$$\mu (x)=A_{\Omega }\cosh (k_{\Omega }x)+B_{\Omega }\sinh (k_{\Omega }x)
\label{MuxNonst}$$
The electric field and potential have the form
$$eE(x)=-\frac{\Omega _{\Delta }}{i\Omega +\Omega _{\Delta }}\partial \mu
/\partial x,\text{ \ }eV(x)=\frac{\Omega _{\Delta }}{i\Omega +\Omega
_{\Delta }}\mu +eV_{\ast }, \label{E+Vnonst}$$
In deriving Eq.(\[E+Vnonst\]) we assume that the current $I$ does not depend on time (no external ac signal). From the boundary conditions ([BC1]{},\[BC2\]) we find
$$B_{\Omega }=-eR_{\Omega }\frac{i\Omega +\Omega _{\Delta }}{\Omega _{\Delta }}%
(I-I_{c}\sin \varphi ),\text{ \ }A_{\Omega }=-(eR_{\Omega
}I(1+r)^{-1}+B_{\Omega }C_{\Omega })/S_{\Omega } \label{B+Anonst}$$
where $R_{\Omega }=(k_{\Omega }\sigma )^{-1}$, $C_{\Omega }=\cosh (k_{\Omega
}L),$ $S_{\Omega }=\sinh (k_{\Omega }L)$. The electric potential at the $N$ reservoir is
$$eV_{L}=\frac{\Omega _{\Delta }}{i\Omega +\Omega _{\Delta }}[B_{\Omega }(\frac{%
R_{B}}{2R_{\Omega }}+S_{\Omega })+A_{\Omega }(C_{\Omega }-1)]
\label{V_Lnonst}$$
Here we took into account the finite value of $r$. The equation for the phase difference $\varphi $ is obtained from the definition of $\mu _{0}=(1/4)\partial \varphi /\partial t+eV_{0}.$ It has the form
$$\frac{1}{2e}(\frac{\partial \varphi }{\partial t})_{\Omega
}+[R_{B}+2R_{\Omega }\frac{i\Omega +\Omega _{\Delta }}{\Omega _{\Delta }}%
\frac{C_{\Omega }}{S_{\Omega }}]I_{c}(\sin \varphi )_{\Omega }=[R_{B}+2R_{Q}(%
\frac{C_{Q}-1}{S_{Q}}+\frac{r}{(1+r)S_{Q}}]I \label{JosEq}$$
We rearrange the terms in Eq.(\[JosEq\]) and rewrite it in a dimensionless form $$(\frac{\partial \varphi }{\partial \tau })_{\omega }=\omega _{J}+[j(1+\frac{2%
}{b}(\frac{C_{Q}-1}{S_{Q}}+\frac{r}{(1+r)S_{Q}})-\omega _{J}]-[1+\frac{2}{%
b_{\omega }}\frac{C_{\omega }}{S_{\omega }}\frac{i\omega +\omega _{\Delta }}{%
\omega _{\Delta }}](\sin \varphi )_{\omega } \label{JosEq1}$$where $j=I/I_{c}$, $\omega _{J}=\Omega _{J}/(2eI_{c}R_{B}),$ $\tau
=t(2eI_{c}R_{B})$ are the dimensionless current, Josephson frequency and time, respectively; $b_{\omega }=b_{L}\theta _{\omega },$ $b_{L}=b\theta ,$ $%
\theta _{\omega }=\theta _{0}\sqrt{(i\omega +\omega _{\Delta
})(i\omega +\omega _{\gamma })}$, $\theta
_{0}=(L/v_{CG})(2I_{c}R_{B})\cong 1.11(\Delta
/T)^{3/2}\sqrt{T/E_{Th}},$ $E_{Th}=D/L^{2},$ $\omega _{\Delta
}=\Omega _{\Delta }/(2eI_{c}R_{B})=1$, $\omega _{\gamma }=\gamma
/(2eI_{c}R_{B})$.
The solution is sought in the form $\varphi (\tau )=\varphi
_{0}(\tau )+\varphi _{1}(\tau )+...,$ where $\varphi _{n}(\tau
)\sim j^{-n}$, for $n\geq 1$. In zero order approximation we obtain from Eq.(\[JosEq1\])
$$\varphi _{0}=\omega _{J0}\tau ,\text{ }\omega _{J0}=j[1+\frac{2}{b}(\frac{%
C_{Q}-1}{S_{Q}}+\frac{r}{(1+r)S_{Q}})] \label{ZeroAppr}$$
The correction $\varphi _{1}(\tau )$ is equal to
$$\varphi _{1}(\tau )=\omega _{J0}^{-1}[\mathop{Re}\mathcal{L}(\omega )\cos
(\omega \tau )-\mathop{\ Im} \mathcal{L}(\omega )\sin (\omega \tau )]
\label{FirstCorr}$$
where $\mathcal{L}(\omega )=[1+(2/b_{\omega })(C_{\omega }/S_{\omega
})(i\omega +\omega _{\Delta })/\omega _{\Delta }].$
Knowing $\varphi _{1}(\tau )$, one can find a correction to the I-V curve due to the Josephson oscillations. From Eq.(\[V\_Lnonst\]) we find for the dimensionless voltage drop $<v_{NN}>\equiv -<2V_{L}>/(2I_{c}R_{B})$
$$<v_{NN}>=j[1+\frac{4}{b}(\frac{C_{Q}-1}{S_{Q}}+\frac{r}{2(1+r)S_{Q}})]-[1+%
\frac{2}{b}\tanh (\theta /2)]<\sin \varphi > \label{v_NN}$$
Here $<\sin \varphi >=<\varphi _{1}(\tau )\cos (\omega \tau
)>=(2\omega _{J0})^{-1}\mathop{Re}\mathcal{L}(\omega )$. The first term is the Ohm’s law at large currents. The second term is a contribution from the Josephson oscillating current. Therefore the correction to the Ohm’s law is equal to
$$<\delta v_{NN}>=-\frac{1}{2\omega _{J0}}[1+\frac{2}{b}\tanh (\theta /2)]%
\mathop{Re}\mathcal{L}(\omega ) \label{CorrV_L}$$
{width="90.00000%"}
In Fig.4 we plot the dependence of this correction $<\delta
v_{NN}>$ on the normalized frequency $\omega _{J0}$, which is proportional to the normalized current $j$ (see Eq.(\[ZeroAppr\]) ) for two values of damping $\gamma $. In plotting Fig.4 the following values of parameters are taken: $%
D=50cm^{2}/s$, $T_{c}=1.3K,$ $\Delta /T=0.3$, $b=10,$ $L=0.5mkm$, $\tau _{\epsilon }=2\cdot 10^{-10}s.$ For these values we have: $\omega _{\gamma }=(2eI_{c}R_{B}\tau _{\epsilon })^{-1}=(2/\pi
)(\tau _{\epsilon }T)^{-1}(T/\Delta )^{2}\cong 0.27;$ $\theta
_{0}=(\pi/2\sqrt{2})(\Delta ^{2}/T)L/\sqrt{D\Delta }\cong
1.11(\Delta /T)^{3/2}(T/E_{Th})^{1/2}\cong
0.46,$ $l_{Q}\cong 2mkm$, $\theta =L/l_{Q}\cong 0.25,$ where $%
E_{Th}=D/L^{2}.$
It is seen that this correction has a series of peaks. These peaks are related to excitation of a collective mode of the Carlson-Goldman type in the system [@CG; @SSCollModes; @AVCollModes; @AVUsp; @SchoenRev]. The excitation occurs if the frequency of the Josephson oscillations $\Omega
_{J}=\omega _{J0}(2eI_{c}R_{B})$ coincides with the frequency of the Carlson-Goldman modes $\Omega _{CG}=v_{CG}/L$. In this case the quantity $%
\mathop{Re}\mathcal{L}(\omega )\sim \tanh ^{-1}(k_{\Omega }L)$ has a peak. The possibility to observe such modes in another Josephson system was studied in Ref.[@AVZ].
Shapiro steps on the I-V characteristics arise if in addition to dc current an ac current flows in the system. In the presence of the ac current, $I_{\sim }(t)=I_{\sim }\sin (\Omega _{ex}t)$, a new term appears on the right in Eq.(\[JosEq\]), which is proportional to $I_{\sim }(t)$. The position of the first Shapiro step is determined by the equation [@Kulik; @Barone]
$$\Omega _{J}\equiv (2eI_{c}R_{B})\omega _{J}=\Omega _{ex} \label{Shap}$$
where $\Omega _{J}$ is the frequency of the Josephson oscillations. We obtain the frequency of the Josephson oscillations, $\omega _{J}$, from Eq.(\[JosEq1\]). At large currents $j$ in the main approximation, $\omega _{J}$ is given by Eq.(\[ZeroAppr\]). On the other hand, the normalized voltage $2<v_{NN}>$ corresponding to the same current $j$ can be easily obtained from Eq.(\[V\_NN2\])
$$2<v_{NN}>=j[1+\frac{2}{b}(\tanh (\theta /2)+\frac{1}{(1+r)S})]
\label{ZeroApprV}$$
Therefore the deviation of $\ 2<v_{NN}>$ from the value given by the Josephson relation is given by the formula
$$\delta v_{Sh} \equiv \frac{2<v_{NN}>-\omega _{ex}}{\omega _{ex}}\mathbf{=}\frac{2}{%
(1+r)}\frac{\tanh (\theta /2)-r/S}{b+2[\tanh (\theta
/2)+r/((1+r)S)]} \label{ShapDev}$$
with $\omega _{ex}=\omega _{J0}$. One can see that this deviation can be both negative and positive depending on the parameters $\theta $ and $r$. In the limit of large $b$ the deviation $\delta v_{Sh}=0$, i.e. the Josephson relation is fulfilled. At an arbitrary $b$, the parameter $\delta v_{Sh}$ depends on the quantities $\theta $ and $r$. Thus, by measuring the deviation from the Josephson relation and the resistances $R_{0}$ and $R_{\infty }$, one can determine the ratio $b=R_{B}/R_{Q}$, the parameter $r$ characterizing the jump of the electric field at the $SN$ interface, and the charge relaxation length $l_{Q}=L/\theta
.$
Conclusions
===========
On the basis of a simple model we have analyzed the Josephson effects in a $%
NSBSN$ nanostructure at temperatures close to $T_{c}$. It turns out that the charge imbalance taking place in this system essentially changes the characteristics of the Josephson junction. If the barrier resistance $R_{B}$ is not too large in comparison with the resistance of the $S$ wire $R_{Q}$ on the scale of the charge imbalance relaxation length $l_{Q},$ the Josephson critical current increases (see Eq.(\[EffI\_c3\])) and the positions of Shapiro steps deviate from their position in equilibrium Josephson junctions (see Eq.(\[ShapDev\])). We have also calculated the resistance of the system for small ($R_{0}$ for $%
I<I_{c}^{\ast }$) and large ($R_{\infty }$ for $I>>I_{c}^{\ast }$) currents. The ratio of these resistances equals
$$\frac{R_{\infty }}{R_{0}}=\frac{(b+4\tanh (\theta /2))(2+b\tanh \theta )}{%
2(b+2\tanh \theta )} \label{RatioCon}$$
Thus, by measuring the ratio $R_{\infty }/R_{0}$ and the deviation $\delta v_{Sh}$, one can determine the coefficient $b$ and the the charge imbalance relaxation length $l_{Q}=L/\theta $.
We have also shown that there is a series of peaks on the $I-V$ characteristics of the system related to excitation of the Carlson-Goldman collective mode. Note that even if the barrier resistance $R_{B}$ is ten times larger than the resistance $R_{Q}$, the peaks in Fig.4 caused by collective mode excitations are clearly visible.
Note that we have adopted several assumptions. One of them is that the voltage $V_{L\text{ }}$ should be smaller than the energy gap $\Delta $. If the voltage is higher than $\Delta $, new effects may arise in the system. In particular, the critical current $I_{c}$ may change sign [@Klapwijk3; @VolkovPRL; @Zaikin; @Yip]. It would be of interest to measure the $I-V$ characteristics of the considered system in a wide range of the applied voltages.
Acknowledgements
================
I am grateful to A. Anishchanka for a technical assistance and to M. Kharitonov for bringing Ref.[@Vodolazov] to my attention. I would like to thank SFB 491 for financial support.
Appendix
========
Here we derive the boundary condition (\[BC1\]) and the expression for the current (\[CurrentJJ\]). The latter formula in our case is not obvious because the potential $\mu $ differs from zero and the Josephson relation (\[JosRelation\]) is not valid. Consider the boundary condition at the barrier $B$ for $4\ast 4$ matrix quasiclassical Green’s function $\check{g}$ [@Zaitsev; @K+L]
$$\sigma (\check{g}\partial \check{g}/\partial x)=(2\mathcal{S}R_{B})^{-1}[%
\check{g}_{0+,}\check{g}_{0-}] \label{BCA}$$
where $\check{g}_{0\pm }=\check{g}(\pm 0)$ are the Green’s functions on the right and left from the barrier $B$. The (11) and (22) elements of the $%
\check{g}$ matrix are the retarded and advanced Green’s functions $\hat{g}%
^{R(A)}$ and the (12) element is the Keldysh function $\hat{g}^{K}$. If the voltage in the system $V$ is small in comparison with $\Delta $, the distribution function $f_{l}$ which determines the energy gap $\Delta $ is close to the equilibrium one ($f_{l}\approx \tanh (\epsilon /2T)$).
The current in the system is given by
$$I=\frac{1}{8}\mathcal{S}\sigma \int d\epsilon Tr\{\hat{\tau}_{3}[\hat{g}%
^{R}(\epsilon ,t;r)\partial \hat{g}^{K}(\epsilon ,t;r)/\partial x+\hat{g}%
^{K}(\epsilon ,t;r)\partial \hat{g}^{A}(\epsilon ,t;r)/\partial x]\}
\label{CurrentA}$$
If we calculate the current with the help of Eqs.(\[BCA\],\[CurrentA\]), we obtain Eq.(\[Icurrent\]). Let us calculate the current through the Josephson junction using the right hand side of Eq.(\[BCA\]). This current consists of three terms: the quasiparticle current $I_{qp}$, the Josephson current $I_{J}$ and the interference current $I_{int}$ [@Kulik; @Barone]. The Josephson current is
$$I_{J}=\frac{\pi iT}{4R_{B}}\sum_{\omega }Tr\{\hat{\tau}_{3}[\hat{f}_{+}\hat{f%
}_{-}-\hat{f}_{-}\hat{f}_{+}]\} \label{JosCurrentA}$$
where $\hat{f}_{\pm }\equiv \hat{f}(\pm 0)=(\hat{\tau}_{2}\cos (\varphi
/2)\pm \hat{\tau}_{1}\sin (\varphi /2))\Delta /\sqrt{(\omega _{n}^{2}+\Delta
^{2})}$ are the Gor’kov’s quasiclassical Green’s functions on the right (left) from the barrier. Calculating the sum, we obtain
$$I_{J}=I_{c}\sin \varphi \label{I_cA}$$
with $I_{c}=\pi \Delta ^{2}/4TeR_{B}$. The interference current is given by
$$I_{int}=\frac{\cos \varphi }{16R_{B}}\int d\epsilon (\beta
V_{0})(f_{+}^{R}+f_{+}^{A}))(f_{-}^{R}+f_{-}^{A})\cosh ^{-2}(\epsilon \beta )
\label{IntCurrentA}$$
In the main approximation $I_{int}\approx V_{0}\Delta /(4TR_{B})%
\mathcal{S}\cos \varphi \ln (\Delta /\Gamma ),$ where $\Gamma $ is a damping in the spectrum of the superconductor. We see that the interference current is small in comparison with the Josephson current if the voltage $V_{0}$ is smaller than the value $\Delta
/\ln (\Delta /\Gamma ).$ The quasiparticle current $I_{qp}$ consists of two parts: $I_{qp}=I_{qp}^{(0)}+\delta I_{qp}$, where $I_{qp}^{(0)}$ is determined by the formula
$$I_{qp}^{(0)}=-\frac{1}{8eR_{B}}\int d\epsilon (\beta \partial \chi
_{+}/\partial t)(g_{+}^{R}-g_{+}^{A}))(g_{-}^{R}-g_{-}^{A})\cosh
^{-2}(\epsilon \beta )=-R_{B}^{-1}(\partial \chi _{+}/\partial t)
\label{QpCurrentA}$$
The term $\beta \partial \chi _{+}/\partial t\cosh ^{-2}(\epsilon
\beta )$ is obtained from the equilibrium distribution function after the transformation of the Green’s functions [@AVUsp]: $\check{g}\rightarrow U%
\check{g}U^{+},$ where $U=\exp (i\hat{\tau}_{3}\chi /2).$ The correction $%
\delta I_{qp}$ is determined by the response of the system to a perturbation of the potential $\mu $ [@AVUsp]. It is equal to $$\delta I_{qp}^{{}}=-\frac{1}{2eR_{B}}\int d\epsilon \frac{\tanh
((\epsilon
-\Omega /2)\beta )-\tanh ((\epsilon +\Omega /2)\beta )}{\Omega }\mu _{0}=%
\frac{2\mu _{0}}{eR_{B}} \label{DeltaIqpA}$$
where $\mu _{0}=(1/2)\partial \chi _{0}/\partial t+eV_{0}$. Thus, for the quasiparticle current we obtain the first term in Eq.(\[CurrentJJ\]).
The boundary condition (\[BCA\]) may be written for the retarded (advanced) Green’s functions $\hat{g}^{R(A)}.$ Since the distribution function $f_{l}$ approximately coincide with the equilibrium function, one can easily obtain from this boundary condition the equation of continuity of the condensate current at the Josephson junction. Therefore, the quasiparticle current also is continuous at the $SBS$ junction.
[99]{} B. D. Josephson, Rev. Mod. Phys. **46**, 251 (1974).
Throughout the paper we set $\hbar =k_{B}=1$.
I.O. Kulik and I.K. Yanson, Josephson effect in superconducting tunnel structures, Keter Press, Jerusalem (1972).
A.Barone and G.Paterno, Physics and applications of the Josephson effect, Wiley, NY (1982).
Note that the charge imbalance is the charge of quasiparticles. This charge is compensated by the charge of the condensate so that the net charge is zero unless we are not interested in small corrections of the order of $(l_{TF}/l_{Q})$, where $l_{TF}$ and $l_{Q}$ are the Thomas-Fermi screening length and the length of the electric field penetration into a superconductor.
J. Clarke, Phys. Rev. Lett. **28**, 1363 (1972); J. Clarke and J. L. Paterson, J. Low Temp. Phys. **15,** 491 (1974);T. Y. Hsiang and J. Clarke, Phys. Rev. B **21**, 945 (1980).
M. L. Yu and J. E. Mercereau, Phys. Rev. Lett. **28**, 1117 (1972).
T. M. Klapwijk and J. E. Mooij, Physica **B 81**, 132 (1976).
G. J. Dolan and L. D. Jackel, Phys. Rev. Lett. **39**, 1628 (1979).
D. J. Van Harlingen, J. Low Temp. Phys. **44**, 163 (1981).
H. J. Mamin, J. Clarke, and D. J. Van Harlingen, Phys. Rev. B **29**, 3881 (1984).
C.-J. Chien and V. Chandrasekhar, Phys. Rev. B **60**, 3665 (1999).
C. Strunk et al., Phys. Rev. B **57**, 10854 (1998).
T. J. Rieger, D. J. Scalapino, and J. E. Mercereau, Phys. Rev. Lett. **27**, 1787 (1971).
M. Tinkham and John Clarke, Phys. Rev. Lett. **28**, 1366 (1972); M. Tinkham, Phys. Rev.** B 6**, 1747 (1972).
A. Schmid and G. Schoen, J. Low Temp. Phys. **20**, 267 (1975).
S. N. Artemenko and A. F. Volkov, JETP Lett. **21**, 313 (1975); Sov. Physics JETP 43, 548 (1976); S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev, J. Low Temp. Phys. **30,** 487 (1978).
Yu. N. Ovchinnikov, J. Low Temp. Phys. **31,** 785 (1978).
S.N. Artemenko and A.F. Volkov, Sov. Phys. Usp. **22**, 295 (1980).
V. T. Petrashov et al., Phys. Rev. Lett. **70**, 347 (1993); ibid **74**, 5268 (1995).
H. Pothier et al., Phys. Rev. Lett. **73**, 2488 (1994).
P. Charlat et al., Phys. Rev. Lett. **77**, 4950 (1996).
A. Dimoulas et al., Phys. Rev. Lett. **74**, 602 (1995).
S. Shapira et al., Phys. Rev. Lett. **84**, 159 (2000).
A. Anthore, H. Pothier, and D. Esteve, Phys. Rev. Lett. **90**, 127001 (2003).
A. Parsons, I. A. Sosnin, and V. T. Petrashov, Phys. Rev. **B 67**, 140502 (2003); *ibid* G. Srivastava, I. Sosnin, and V. T. Petrashov, **72**, 012514 (2005).
Z. Jiang and V. Chandrasekhar, Phys. Rev. Lett. **94**, 147002 (2005); Phys. Rev. **B 72**, 020502 (2005).
A. Rogachev, T.-C. Wei, D. Pekker, A. T. Bollinger, P. M. Goldbart, and A. Bezryadin, Phys. Rev. Lett. **97**, 137001 (2006).
M.Yu. Kharitonov and M.V. Feigel’man, JETP Lett., **82**, 421 (2005).
D. Y. Vodolazov, Phys. Rev. B **75**, 184517 (2007).
A.I. Larkin and Yu.N. Ovchinnikov, in *Nonequilibrium Superconductivity*, edited by D.N. Langenberg and A.I. Larkin (Elsevier, Amsterdam, 1984).
J. Rammer and H. Smith, Rev. Mod. Phys. **58**, 323 (1986).
N.B. Kopnin, *Theory of Nonequilibrium Superconductivity* (Clarendon Press, Oxford, 2001).
R. S. Keizer, M. G. Flokstra, J. Aarts, and T. M. Klapwijk, Phys. Rev. Lett. **96**, 147002 (2006).
D. Y. Vodolazov and F. M. Peeters, Phys. Rev. B **75**, 104515 (2007).
P.L. Carlson and A.M. Goldman, Phys. Rev. Lett. **34**, 11 (1975).
G. Schoen, in *Nonequilibrium Superconductivity*, edited by D.N. Langenberg and A.I. Larkin (Elsevier, Amsterdam, 1984).
A. Schmid and G. Schoen, Phys. Rev. Lett. **34**, 941 (1975).
S.N. Artemenko and A.F. Volkov, Sov. Phys. JETP **42**, 1130 (1975).
A.V. Zaitsev, Sov. Phys. JETP 59, 1015 (1984).
M.Yu. Kupriyanov and V.F. Lukichev, Sov. Phys. JETP 67, 1163 (1988).
S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev, JETP Lett. **27**, 113 (1978).
J. J. A. Baselmans, A. Morpurgo, B. J. van Wees, and T. M. Klapwijk, Nature 397, 43 (1999).
A. F. Volkov, Phys. Rev. Lett. **74**, 4730 (1995).
F. K. Wilhelm, G. Schoen, and A. D. Zaikin, Phys. Rev. Lett. **81**, 1682 (1998).
S. K. Yip, Phys. Rev. B **58**, 5803 (1998).
|
---
abstract: 'Carbon dioxide ice clouds are thought to play an important role for cold terrestrial planets with thick dominated atmospheres. Various previous studies showed that a scattering greenhouse effect by carbon dioxide ice clouds could result in a massive warming of the planetary surface. However, all of these studies only employed simplified two-stream radiative transfer schemes to describe the anisotropic scattering. Using accurate radiative transfer models with a general discrete ordinate method, this study revisits this important effect and shows that the positive climatic impact of carbon dioxide clouds was strongly overestimated in the past. The revised scattering greenhouse effect can have important implications for the early Mars, but also for planets like the early Earth or the position of the outer boundary of the habitable zone.'
author:
- 'D. Kitzmann'
bibliography:
- 'references.bib'
title: Revisiting the scattering greenhouse effect of ice clouds
---
Introduction
============
Clouds are common in the atmospheres of Solar System planets and are likely ubiquitous in those of extrasolar planets as well. They affect every aspect of a planetary atmosphere, from radiative transfer, to atmospheric chemistry and dynamics, and they influence – if not control – aspects such as the surface temperature and, thus, the potential habitability of a terrestrial planet [@Marley2013cctp.book..367M]. Understanding the impact of clouds is thus instrumental for the study of planetary climates.
Clouds composed of carbon dioxide () ice particles are important for cold, -rich atmospheres. This includes, for example, the present and early Mars [@Forget1997], the early Earth [@Caldeira1992Natur.359..226C], or terrestrial exoplanets at the outer boundary of the habitable zone [@Forget1997; @Kasting1993]. Provided that condensation nuclei are available, such clouds would form easily in a supersaturated atmosphere.
Like for water clouds (e.g. ), the presence of clouds will firstly result in an increase of the planetary albedo by scattering incident stellar radiation back to space which leads to a cooling effect (albedo effect). In contrast to liquid or solid water, ice is mostly transparent with respect to absorption in the infrared [@Hansen1997JGR; @Hansen2005JGRE]. Thus, as argued by @Forget1997 or @Kasting1993, a classical greenhouse effect by absorption and re-emission of thermal radiation is unlikely to occur for a cloud composed of dry ice. On the other hand, ice particles can efficiently scatter thermal radiation back to the planetary surface, thereby exhibiting a scattering greenhouse effect. In their seminal study, Forget & Pierrehumbert [@Forget1997] showed that this scattering greenhouse effect of ice clouds alone would have been efficient enough to allow for liquid water on the surface of the early Mars. This strong heating effect was also found in later studies by e.g. @Pierrehumbert1998JAtS, @Mischna2000Icar, or @Colaprete2003. However, by using three-dimensional general circulation models, it has been suggested, that a limiting factor for the impact of clouds on early Mars might be the cloud coverage, which could be well below 100% [@Forget2013Icar..222...81F].
Especially, the outer boundary of the classical habitable zone (HZ) might also be influenced by the formation of ice clouds and their corresponding climatic impact. For the Sun, @Forget1997 reported an outer HZ boundary of 2.4 au for a planet fully covered with clouds, which is far larger than the cloud-free value of 1.67 au @Kasting1993.
The competing radiative cooling and heating effects are individually large, but partly balance each other. That means that small errors in the description of one of these radiative interactions can lead to large errors in a cloud’s net effect. So far, all previous studies on the climatic impact of clouds used a simplified treatment for the numerical solution of the radiative transfer equation to determine the atmospheric and surface temperatures. In particular, two-stream methods [@Toon1989JGR....9416287T] have exclusively been used. With only two angular directions available to determine the radiation field, these methods provide only limited means to describe e.g. the scattering phase function of the ice particles well enough to yield accurate results for describing radiative effects based on anisotropic scattering.
In a numerical radiative transfer study by , it was shown that these simplified two-stream methods underestimate the albedo effect at short wavelengths and overestimate the back-scattering of thermal radiation which forms the basis of the scattering greenhouse effect. However, this previous study did not use an atmospheric model and, thus, was unable to estimate the impact of a more accurate description of the scattering greenhouse effect on the surface temperature.
In this study, I will therefore re-asses the climatic impact of clouds using a sophisticated radiative transfer scheme. Since most of the studies related to the climatic impact of ice clouds have been done for early Mars, I will use the same modeling scenario to compare my atmospheric model calculations with the previously published results.
Model description
=================
To investigate the impact of ice clouds, I use a one-dimensional radiative-convective climate model. The model incorporates a state-of-the-art radiative transfer treatment based on a discrete ordinate method [@Hamre2013AIPC.1531..923H] which is able to treat the anisotropic scattering by cloud particles accurately .
The atmospheric model is stationary, i.e. it doesn’t contain an explicit time dependence, and assumes hydrostatic equilibrium. About one hundred grid points are used to resolve the vertical extension of the atmosphere. The model currently considers , and . Of those, only and are, however, used in this study.
The temperature profile is calculated from the requirement of radiative equilibrium by a time-stepping approach, as well as performing a convective adjustment, if necessary. The convective lapse rate is assumed to be adiabatic, taking into account the condensation of and . A surface albedo of 0.215 based on measurements of present Mars is used [@Kieffer1977JGR....82.4249K]. A more detailed model description will be presented in @Kitzmann2016inprep.
Radiative transfer
------------------
In contrast to many other atmospheric models for terrestrial exoplanets, the radiative transfer of this study is not separated into two different wavelength regimes. Instead, one single, consistent radiative transfer scheme within the wavelength range from to is used. At each distinct wavelength point, the plane-parallel radiative transfer equation $$\mu \frac{\mathrm{d} I_\lambda}{\mathrm{d}\tau_\lambda} = I_\lambda - S_{\lambda,\mathrm{*}}(\tau_\lambda)
\label{eq:rte}$$ is solved, with the general source function $$\begin{split}
S_{\lambda}(\tau_\lambda) = & S_{\lambda,\mathrm{*}}(\tau_\lambda) + (1 - \omega_\lambda) B_\lambda\\
& + \frac{\omega_\lambda}{2} \int_{-1}^{+1} p_\lambda(\mu,\mu')I_\lambda(\mu') \mathrm{d}\mu' \ ,
\end{split}$$ where $S_{\lambda,\mathrm{*}}$ is the contribution of the central star, $B_\lambda$ the Planck function, $\omega_\lambda$ the single scattering albedo, and $p_\lambda(\mu,\mu')$ the scattering phase function.
The scattering phase function is represented as an infinite series of Legendre polynomials [@Chandrasekhar1960ratr] $$p_\lambda(\mu,\mu') = \sum_{n=0}^{\infty} (2 n + 1) P_n(\mu) P_n(\mu') \chi_{\lambda,n}$$ with the Legendre polynomials $P_n(\mu)$ and the phase function moments $\chi_{\lambda,n}$. In practice the series is truncated at a certain $n=N_\mathrm{max}$. For discrete ordinate methods the number of moments $N_\mathrm{max}$ is equal to the number of ordinates (streams) considered in the radiative transfer equation [@Chandrasekhar1960ratr].
The equation of radiative transfer is solved by the discrete ordinate solver C-DISORT [@Hamre2013AIPC.1531..923H]. In contrast to two-stream methods, it yields the mathematically exact solution of the transfer equation for a given set of transport coefficients and phase functions, provided that enough computational streams are included. For this study, eight streams are used for all model calculations. Doubling the number of streams has no impact on the resulting radiation fluxes and surface temperatures.
The wavelength-dependent absorption by atmospheric molecules and clouds is treated by the opacity sampling method, which is one of the standard methods employed in cool stellar atmospheres [@Sneden1976ApJ...204..281S]. In contrast to the $k$-distribution method with the correlated-$k$ assumption, opacity sampling still operates in the usual wavelength/wavenumber space [@Mihalas1978stat.book.....M]. This approach has the advantage, that absorption coefficients of all atmospheric constituents are fully additive. Essentially, opacity sampling can be regarded as a degraded line-by-line formalism. It is based on the fact that the wavelength integral of the radiation flux converges already well before all spectral lines are fully resolved. At each single wavelength, however, the solution with the opacity sampling method is identical to a corresponding line-by-line radiative transfer.
For this study, the distribution of the wavelength points at which the equation of radiative transfer is solved, is treated separately in three different wavelength regions. In the infrared, the points are sampled along the Planck black body curves for different temperatures. This method is adopted from and guarantees an accurate treatment of the thermal radiation with a small number of wavelength points. In the visible and near infrared region, 10000 wavelength points are distributed equidistantly in wavenumbers. This region is most important for the temperature profile in the upper atmosphere where the temperatures are essentially determined by absorption of stellar radiation within well-separated lines. Beyond the visible wavelength region, about 100 points are used to cover the smooth Rayleigh scattering slope and resolving the stellar Lyman alpha emission line. In total about 12500 discrete wavelength points are used here. Tests by increasing the wavelength resolution show virtually no change in the wavelength-integrated flux. In fact, the high number of integration points in the near infrared region could also be decreased if one is not interested in the temperatures near the top-of-the atmosphere.
Absorption cross-sections for and are calculated as described in @Wordsworth2013ApJ...778..154W, using the HITRAN 2012 database [@Rothman2013JQSRT.130....4R]. For these calculations, the open source *Kspectrum* code (version 1.2.0) is used. In case of , the sub-Lorentzian line profiles of @Perrin1989JQSRT..42..311P are employed, while the contribution of collision induced absorption is taken from @Baranov2003JMoSp.218..260B. The continuum absorption of is derived from the MT-CKD description [@Mlawer2012RSPTA.370.2520M]. Rayleigh scattering is considered for and .
Cloud description
-----------------
The atmospheric model also takes the radiative effect of clouds directly into account. Following @Forget1997, the size distribution of the ice particles is described by a modified gamma distribution $$f(a) = \frac{\left( a_\mathrm{eff} \nu \right)^{2-1/\nu}}{\Gamma\left(\frac{1-2\nu}{\nu}\right)} a^{\left(\frac{1}{\nu}-3\right)} \mathrm
e^{-\frac{a}{a_\mathrm{eff}\nu}} \,
\label{eq:gamma_distribution}$$ described by the effective radius $a_\mathrm{eff}$ and an effective variance of 0.1 [@Forget1997].
Assuming spherical particles, the optical properties are calculated via Mie theory, using the refractive index of ice from @Hansen1997JGR [@Hansen2005JGRE]. The resulting optical properties are shown in Fig. \[fig:optical\_properties\] for some selected values of the effective radius and an optical depth of one. The optical depth $\tau$ of the clouds refers to the particular wavelength of $\lambda = 0.1 \ \mathrm{\mu m}$ throughout this study.
![**Optical properties of CO$_2$ ice particles.** Results are shown for four different size distributions: $a_{\mathrm{eff}}=10 \ \mathrm{\mu m}$ (*blue line*), $a_{\mathrm{eff}}=50 \ \mathrm{\mu m}$ (*green line*), $a_{\mathrm{eff}}=100.0 \
\mathrm{\mu m}$ (*red line*), $a_{\mathrm{eff}}=1000 \ \mathrm{\mu m}$ (*black line*). *Upper diagram*: optical depth (for $\tau = 1$), *middle diagram*: single scattering albedo, *lower diagram*: asymmetry parameter.[]{data-label="fig:optical_properties"}](optical_properties4.pdf){width="45.00000%"}
The Henyey-Greenstein function [@Henyey1941ApJ....93...70H] is used to approximate the scattering phase function. Although it lacks the complicated structure and detailed features of the full Mie phase function, the Henyey-Greenstein function preserves its average quantities, such as the asymmetry parameter $g$. The Henyey-Greenstein phase function is usually a very good replacement of the Mie phase function, especially at higher optical depths or if one is only interested in angular averaged quantities, such as the radiation flux [@vandeHulst1968JCoPh...3..291V; @Hansen1969JAtS...26..478H].
Effect of clouds in the early Martian atmosphere
================================================
To compare the climatic impact of ice clouds with previous model studies, I use the same model set-up as in @Forget1997, who studied the influence of clouds in the atmosphere of the early Mars.
Thus, following @Forget1997 or @Mischna2000Icar, a surface pressure of 2 bar is used on a planet with the radius and mass of Mars. To simulate the conditions for early Mars with a less luminous Sun, the incident stellar flux is set to 75% of the present-day Martian Solar irradiance. A high-resolution spectrum of the present-day Sun from @Gueymard2004 is used for the spectral energy distribution of the incident stellar flux.
![**Impact of ice clouds on the surface temperature.** Surface temperatures are shown as a function of optical depth (at a wavelength of $\lambda = 0.1 \ \mathrm{\mu m}$) and effective particle size of the ice clouds. The black contour lines are given in steps of 5 K. The white contour line indicates the clear-sky case.[]{data-label="fig:surface_temp_2d"}](2d_single.pdf){width="45.00000%"}
Because a supersaturated -rich atmosphere would provide a huge amount of condensible material, dry ice particles would grow rapidly, thereby potentially reaching large particle sizes under these conditions. The only detailed microphysical study for particle formation on early Mars obtained mean particle sizes of the order of [@Colaprete2003]. Thus, to cover this very wide parameter range, the effective particle sizes are varied between and in the following. A total cloud coverage is assumed to obtain an upper limit for the clouds’ climatic impact. The cloud layer is placed into the supersaturated region of the cloud-free atmosphere around a pressure of 0.1 bar. This corresponds roughly to the position of the cloud layer in @Forget1997 and @Mischna2000Icar.
Since the publication of @Forget1997 or @Mischna2000Icar, several updates to the molecular line lists and description of the molecule’s continuum absorption were introduced. While the greenhouse effect of became less effective [@Wordsworth2010Icar..210..992W], the absorptivity of increased, especially due to changes in the continuum absorption [@Mlawer2012RSPTA.370.2520M] and the line parameters [@Rothman2013JQSRT.130....4R]. Therefore, to limit the impact of the different molecular line lists on the comparison, only is considered as an atmospheric gas in the following. This corresponds to the “dry case” in @Forget1997.
In the clear-sky case, the surface temperature is about 220 K and thus roughly 8 K smaller than reported by @Forget1997 and @Mischna2000Icar. This is caused by a revised collision induced absorption (CIA) of the gas, yielding a smaller greenhouse effect [@Wordsworth2010Icar..210..992W].
The resulting surface temperatures for the cloudy cases are shown in Fig. \[fig:surface\_temp\_2d\] as a function of the cloud particle sizes and optical depth. In total, several hundreds of individual model calculations have been performed to obtain the results in Fig. \[fig:surface\_temp\_2d\].
In agreement with previous studies [@Forget1997; @Pierrehumbert1998JAtS; @Mischna2000Icar; @Colaprete2003], particles with effective radii larger than about can result in a net greenhouse effect. An efficient greenhouse effect, however, is only possible within a certain particle size range and for medium values of the optical depth. Particles larger than roughly are more or less neutral or - at high optical depths - even exhibit a net cooling effect. These particles are too large to feature an efficient back-scattering of the upwelling thermal radiation. Their large asymmetry parameters at thermal infrared wavelengths results in thermal radiation being predominantly scattered in the upward direction and, thus, away from the surface. In order to obtain the highest possible greenhouse effect, the particle sizes must be comparable to the wavelength of the atmospheric thermal radiation below the cloud layer (cf. also ). This is only the case for $a_{\mathrm{eff}}
\approx \unit[25]{\micro m}$ which therefore provides the largest net greenhouse effect.
In no case studied here, the freezing point of water is reached - quite in contrast to previous studies where surface temperatures in excess of had been found for the same scenario [@Forget1997]. The highest surface temperatures obtained for a pure 2 bar atmosphere are about . Even considering the difference due to the revised CIA, the differences in the resulting surfaces temperatures between the different radiative transfer approaches are astonishingly large. However, it should be noted, that the lower collision induced absorption of the molecules might also influence the resulting scattering greenhouse effect of the ice particles to a certain extend.
A direct comparison of the climatic impact with the previous studies of @Forget1997 and @Mischna2000Icar is shown in Fig. \[fig:contour\_sizes\]. Note, that @Forget1997 only used two effective radii (10 and ), while @Mischna2000Icar is limited to a single value of and studies only atmospheres saturated with water vapor.
![**Comparison of the impact of clouds on the surface temperature with previous studies.** The diagram shows the changes of the surface temperature in the presence of ice clouds ($T_\mathrm{cloudy}$) with respect to the clear-sky case ($T_\mathrm{clear-sky}$), compared to the results of @Forget1997 and @Mischna2000Icar. Thicker lines refer to surface temperatures above the freezing point of water. The results are shown for two effective radii: $a_{\mathrm{eff}} =$ (blue) and $a_{\mathrm{eff}} =$ (red). Solid lines denote the results from this study, dashed-dotted lines the dry atmosphere results from @Forget1997 for comparison. The single three blue markers compare results for an atmosphere fully saturated with water (circle: this study, square: @Mischna2000Icar, triangle: @Forget1997).[]{data-label="fig:contour_sizes"}](contour_lines_sizes_dry.pdf){width="45.00000%"}
![**Atmospheric temperature-pressure profiles for the dry case influenced by clouds.** The resulting temperature profiles are shown for the clear-sky case (black line) and for several values of the effective radius of the particle size distribution (see legend). The optical depth is 8 in each case. The effect of clouds on the energy transport is marked by different line-styles. Solid lines indicate a temperature profile in radiative equilibrium, dashed lines dry convective regions, and dashed-dotted a moist adiabatic lapse rate. The vertical dotted line denotes the freezing point of water. The dashed-dotted line marks the saturation vapor pressure curve of . The position of the cloud layer is denoted by the grey-shaded area. The inset plot shows the planetary Bond albedos affected by the presence of the ice clouds. Colours refer to the same effective particle sizes mentioned above.[]{data-label="fig:temperture_profiles"}](temperature_profiles_albedo2.pdf){width="35.00000%"}
The results in Fig. \[fig:contour\_sizes\] clearly suggest that the climatic impact of clouds was strongly overestimated in the past. The cloud-induced temperature changes determined here are several tens of Kelvin smaller in most cases. For an optical depth of 10 and an effective radius of also the results for a water-saturated atmosphere are compared. Again, the temperature increases due to the scattering greenhouse effect are between 20 and 30 K smaller than those found in @Forget1997 or @Mischna2000Icar for this particular case. Additionally, @Colaprete2003 also claimed a strong greenhouse effect for particle sizes of several hundred , which, as already shown in Fig. \[fig:surface\_temp\_2d\], is clearly not the case.
The presence of clouds also has a profound impact on the temperature profile, as shown in Fig. \[fig:temperture\_profiles\]. Due to their scattering greenhouse effect, they strongly increase the atmospheric temperature *locally*, just below the cloud base. This inhibits convection and creates a temperature profile where the usual fully convective troposphere is changed into a lower convective region near the surface and a second one above the cloud layer. Both convective regions are separated by a temperature profile determined by radiative equilibrium (see also @Mischna2000Icar). The local heating due to the cloud particles is in fact strong enough to cause their evaporation. Thus, such a cloud layer would be unable to persist in stationary equilibrium [@Colaprete2003]. Placing the cloud layer even higher up in the atmosphere would also result in the strong local heating.
Overall, the Bond albedos in the cloudy cases are somewhat higher than the ones of the reported by the previous studies. This is caused by the underestimation of the cloud’s albedo effect by the two-stream radiative transfer schemes .
Summary
=======
In this study, the potential impact of scattering greenhouse effect of ice clouds on the surface temperature is revisited by using an atmospheric model with an accurate radiative transfer method. By comparison with previous model studies on the early Mars, the results suggest that the potential heating effect was strongly overestimated in the past.
Based on the results presented here, it is, therefore, strongly recommended that atmospheric models which include the climatic effect of ice clouds should employ more suitable radiative transfer schemes. Additionally, previous model calculations involving the effect of ice clouds, such as for the position of the outer boundary of the habitable zone or the atmosphere of early Mars for example, clearly require to be revisited. This study also emphasizes the importance of using detailed radiative schemes when studying phenomena based on anisotropic scattering.
While the results for the scattering greenhouse effect found in this study are by far smaller than those reported by previous studies, this doesn’t mean that ice clouds are overall ineffective in warming planetary atmospheres. On the contrary, they can still provide an important contribution to the greenhouse effect in a certain parameter range. Thus, the original idea brought up by Forget & Pierrehumbert in their pilot study about a net scattering greenhouse effect still prevails, albeit much smaller than originally anticipated.
D.K. would like to thank Y. Alibert, K. Heng, J. Lyons, and J. Unterhinninghofen for their suggestions and comments and especially B. Patzer for the fruitful discussions. D.K also gratefully acknowledges the support of the Center for Space and Habitability of the University of Bern.
|
---
abstract: 'The dark energy cosmology with the equation of state $w=constant$ is considered in this paper. The $\Omega_{DE}-\Omega_{M}$ plane has been used to study the present state and expansion history of the universe. Through the mathematical analysis, we give the theoretical constraint of cosmological parameters. Together with some observations such as the transition redshift from deceleration to acceleration, more precise constraint on cosmological parameters can be acquired.'
author:
- 'Yuan Qiang$^{1}$, Tong-Jie Zhang$^{1,2}$'
bibliography:
- 'ztjcos-lensbib.bib'
title: 'The $\Omega_{DE}-\Omega_{M}$ Plane in Dark Energy Cosmology'
---
Introduction
============
One of the greatest challenges in modern cosmology is understanding the nature of the observed evolution status of the universe in late-time phase. The type Ia supernovae (SNe Ia) searches [@1998AJ....116.1009R; @1999ApJ...517..565P], the cosmic microwave background (CMB) results from balloon and ground experiments and recent WMAP [@2003ApJS..148..175S] and recent WMAP [@2003ApJS..148..175S] observation all suggest that the universe is spatially flat and undergoing a phase of accelerating at the present time due to the current domination of some sort of negative-pressure dark energy (DE). The dark energy is usually characterized by a parameter of an equation-of-state (hereafter, EOS) $w\equiv p/\rho $, the ratio of the spatially-homogeneous dark-energy pressure $p$ to its energy density $\rho $. The cosmological constant of order (10$^{-3}$eV)$^{4}$, the EOS of which $w=-1$, is the simplest candidate for dark energy. However, it is 120 orders of magnitude smaller than the naive expectations from quantum field theory. Another widely explored possibility is quintessence [@1988PhRvD..37.3406R; @1997PhRvD..55.1851C; @1998PhRvL..80.1582C], which is described in terms of a cosmic scalar field $\phi $. Thus in such models, the EOS takes $-1<w<-1/3$, and the dark-energy density decreases with scale factor $a(t)$ as $\rho \propto a^{-3(1+w)}$.
The $\Omega_{DE}-\Omega_{M}$ plane is one of the most fundamental diagrams in modern observational cosmology, which has been used to study the present state and expansion history of the universe. To study the cosmological dynamics of the universe for different scenarios the phase plane analysis is used. In this paper, we focus on the case of EOS $w=constant$ for simplicity. Three choices of $w$ are taken to be examples of our analysis, however, this analysis method can be appropriate for arbitrary $w$. The case of $w=-1$, which represents the dark energy is a constant independent of cosmic time (the cosmological constant), is the simplest choice for dark energy. Furthermore, it was strongly supported by SNe Ia [@2004ApJ...607..665R] and CMB [@2003ApJS..148..175S] observations. Another candidate for dark energy is the cosmic topological defect [@2003RvMP...75..559P] such as cosmic string [@1994csot.book.....V] and domain wall [@1999astro.ph..8047B], and so on. The EOS of topological defect is $w=-n/3$, where $n$ is the dimension of defect. $n=1$ (accordingly $w=-1/3$) corresponds to cosmic string and $n=2$ ($w=-2/3$) corresponds to domain wall respectively. In addition, these choices of $w$ are easy to be calculated analytically.
This paper is organized as follows: in Sec. \[eos\] we discuss the restrictions which can be set on this cosmological model from the astronomical observations, and in Sec. \[plane\] the $\Omega_{DE} -\Omega_M$ plane of the three cases ($w=-1,\,-1/3$ and $-2/3$) is discussed in detail. We study the transit redshift from deceleration to acceleration in the $\Omega_{DE} -\Omega_M$ plane in Sec. \[redshift\]. The discussion of the results and their further possible generalization is presented in Sec. \[cd\].
The Equation of State for Dark Energy {#eos}
=====================================
For most purpose, we consider a general dark energy EOS $w(z)$ which varies with the cosmic time $t$ or redshift $z$ $$\label{H^2}
w(z)=\frac{p_{DE}(z)}{\rho_{DE}(z)},$$ where $p_{DE}(z)$ and $\rho_{DE}(z)$ are the time-dependent pressure and energy density respectively and redshift $z$ is defined by scale factor $a$, $1+z=a_0/a$. Using the conservation of energy, we can express the energy density of dark energy by $$f_{DE}(z)=\frac{\rho_{DE}(z)}{\rho_{DE0}}=\exp\,[\int_{0}^{z}3(1+w_{DE}(z))
\mathrm{d}\ln (1+z)]. \label{rhox}$$ Thus Friedmann equation neglecting cosmic radiation can be expressed as $$E^{2}(z)=\frac{H^{2}(z)}{H_{0}^{2}} =\Omega_{M}(1+z)^{3}+
\Omega_{DE}f_{DE}(z)+\Omega_{k}(1+z)^{2}, \label{E2z}$$ where $\Omega_{M}$, $\Omega_{DE}$ and $\Omega_{k}$ are respectively density parameters at the present epoch $t_0$. More generally, we can rewrite Eq.(\[E2z\]) in terms of cosmological density parameters at redshift $z$ $$\Omega^z_M+\Omega^z_{DE}+\Omega^z_{k}=1, \label{Omegazp1}$$ where $$\left \{
\begin{array}{ll}
\Omega^z_M=\frac{\rho_{M}}{\rho_{c}}=\frac{\rho_{M0}(1+z)^{3}} {
\rho_{c0}E^{2}(z)}=\frac{\Omega_{M}(1+z)^{3}}{E^{2}(z)} & \\
\Omega^z_{DE}=\frac{\rho_{DE}}{\rho_{c}}=\frac{\rho_{DE0}f_{DE}} {
\rho_{c0}E^{2}(z)}=\frac{\Omega_{DE}f_{DE}(z)}{E^{2}(z)} & \\
\Omega^z_k=-\frac{kc^{2}}{a^{2}(z)H^{2}(z)}=
\frac{\Omega_{k}(1+z)^2}{E(z)^2} &
\end{array}
\right. , \label{Omegaz3}$$ and the critical density at redshift $z$ is $$\rho_{c}=\frac{3H^{2}}{8\pi G}=\frac{3H^{2}_{0}}{8\pi G}E^{2}(z)=\rho_{c0}E^{2}(z).$$
If $w_{DE}(z)=w$ (constant) is independent of time, the expression for $\rho
_{DE}(z)$ above becomes $$f_{DE}(z)=\frac{\rho _{DE}(z)}{\rho _{DE0}}=(1+z)^{3(1+w)}\propto
a^{-3(1+w)},$$ and the expression of $E(z)$ reduces to $$E(z)=\sqrt{\Omega _{M}(1+z)^{3}+\Omega _{DE}(1+z)^{3(1+w)}+\Omega
_{k}(1+z)^{2}}.$$ Throughout this paper, we just consider constant EOS $w$ rather the variation of its $w(t)$. At the present epoch, Eq.(\[Omegazp1\]) satisfies $\Omega _{M}+\Omega _{DE}+\Omega
_{k}=1$. For spatially flat universe, $ \Omega _{k}=0$, i.e. $\Omega _{M}+\Omega _{DE}=1$, then $$\Omega _{T}^{z}=\Omega _{M}^{z}+\Omega _{DE}^{z}=\frac{\Omega
_{M}(1+z)^{3}}{ E^{2}(z)}+\frac{\Omega
_{DE}(1+z)^{3(1+w)}}{E^{2}(z)}=1,$$ which means $\Omega _{T}^{z}$ will equal unit at any resdshift $z$ if only the universe is spatially flat today. Fig.\[fig1eps\] draw the cosmological density parameters $\Omega ^{z}$ as a function of redshift $z$ in various spatially flat cosmologies. With the expansion of the universe from the Big Bang (redshift $z=\infty $) to the future ($z=-1$) theoretically, matter density $\Omega _{M}^{z}$ drops to zero from unit, while dark energy density $\Omega _{DE}^{z}$ reaches unit from zero. The universe start from Einstein-de Sitter model and end in a de-Sitter phase. This point can also be demonstrated theoretically by taking the limit of Eq.(\[Omegaz3\]) $$\begin{aligned}
& &\lim_{z \to \infty} \Omega _{M}^{z}=\lim_{z\to \infty}
\frac{\Omega _{M}(1+z)^{3}}{E^{2}(z)}=1,\ \lim_{z\to -1}
\Omega _{M}^{z}=0; \nonumber\\
& &\lim_{z\to \infty}\Omega _{DE}^{z}=\lim_{z\to \infty}
\frac{\Omega _{DE}\,\exp [3\int_{0}^{z}(1+w(z))\mathrm{d}\,\ln
(1+z)]} {E^{2}(z)}=0,\ \lim_{z\to -1}\Omega _{DE}^{z}=1.
\label{limtOmega}\end{aligned}$$
![The evolution of cosmological density parameters $
\Omega^z_{M,\,DE,\,k,\,T}$ as a function of redshift $z$, where we have assumed a spatially flat concordant cosmological model: $\Omega_M=0.3$ and $\Omega_{DE}=0.7$ with $w=-1 $ and $\Omega_k=0$. The solid and dashed lines denote the evolution of $\Omega^z_M$ and $\Omega^z_{DE}$, while the two horizonal straight lines correspond to that of $\Omega^z_T$ and $\Omega^z_k$ , respectively. From left to right for $z>0$ and from right to left for $z<0$, the solid lines representing $\Omega^z_M$ and the dashed lines representing $\Omega^z_{DE}$ correspond to the cases of $w =-3/2,-1,-2/3,-1/2$ and $-1/3$.[]{data-label="fig1eps"}](f1.eps){width="80.00000%"}
The $\Omega_{DE}-\Omega_{M}$ Plane and the Expansion History of the Universe {#plane}
============================================================================
Using Eq.(\[Omegaz3\]), we can relate the density parameters $\Omega^z_{M,\,DE,\,k}$ at $z$ and their current ones $\Omega_{M,\,DE,\,k}$ by $$\frac{(\Omega^z_M+\Omega^z_{DE}-1)^3}{(\Omega^z_M)^{(1+3w)/w}(\Omega^z_{DE})^{-1/w}}= \frac{(\Omega_M+\Omega_{DE}-1)^3}{\Omega^{(1+3w)/w}_M\Omega^{-1/w}_{DE}}=C. \label{omegaplanew}$$ Given the value of the constant $C$, one can represent the specific cosmological models by the second term and the expansion history of the universe by the first term in Eq.(\[omegaplanew\]) in the $\Omega_{DE}-\Omega_{M}$ plane respectively. This leads to the identity of the the $\Omega_{DE}-\Omega_{M}$ plane and the expansion history of the universe in the $\Omega_{DE}-\Omega_{M}$ plane.
The universe can be roughly divided into three types: open, flat and closed. In the absence of the exotic dark energy, the fate of the universe are easily to be understood. At the presence of dark energy, the situation becomes more complicated. Especially, the recollapse of the universe can exist in some types of cosmological models. From the definition of expansion rate of the universe, Eq.(\[E2z\]), one can guarantee the existence of recollapse by the criterion $\dot{a}(t)=0$ at some moment $t$ $$\frac{H^2(z)}{H_{0}^2}=\frac{(\dot{a}/a)^2}{H_{0}^2}=E^2(z)=
\Omega_{M}(1+z)^{3}+\Omega_{DE}f_{DE}(z)+\Omega_{k}(1+z)^{2}=0. \label{E2z0}$$ Setting $x=a/a_0=1/(1+z)$, we can see that when $z$ varies from $+\infty$ to $-1$, $x$ varies from 0 to $+\infty$. Defining a function $$F(x)=\Omega_{DE}x^{-3w}+\Omega_{k}x+\Omega_{M}, \label{Fx}$$ we can easily find that $E^2(z)\geq0\Leftrightarrow F(x)\geq0$, and if only there exists a positive root to $F(x)$, the universe can recollapse at some time $a(t)$.
The Case of $w=-1$
------------------
For the case of cosmological constant, $w=-1$, we have $$\frac{(\Omega^z_M+\Omega^z_{DE}-1)^3}{(\Omega^z_M)^2\Omega^z_{DE}}= \frac{(\Omega_M+\Omega_{DE}-1)^3}{\Omega^2_M\Omega_{DE}}=C, \label{omegaplane1}$$ for $\Omega_{DE}-\Omega_{M}$ plane and the expansion trajectory of the universe in the $\Omega_{DE}-\Omega_{M}$ plane, and $$F(x)=\Omega_{DE}x^3+\Omega_{k}x+\Omega_{M}. \label{Fx1}$$ Given the value of $C$, one can follow the expansion trajectory of the universe in terms of the density parameters $\Omega^z$ at any redshift $z$, which, essentially, is identical to the $
\Omega_{DE}-\Omega_{M}$ plane that represents the location of specific cosmological models in terms of current density parameters $\Omega$. Parameter $C$ cannot be arbitrary for the constrain condition $F(x)\geq 0$ (otherwise, $E^2(z)=H^2/H^2_0<0$, it is impossible). So we get $\Omega_{DE}\geq0$. Below we just consider the situation of $\Omega_{DE}>0$, the case of $\Omega_{DE}=0$ is just the limit of $C\rightarrow-\infty$(see below). Taking the derivation of $F(x)$, we have $$F^{\prime}(x)=3\Omega_{DE}x^2+\Omega_{k}. \label{Fx1p}$$
For the cases $\Omega_k>0\,(k=-1)$ and $\Omega_k=0\,(k=0)$, which correspond to $C<0$ and $C=0$ respectively, $F^{\prime}(x)>0$, so $F(x)> F(0)=\Omega_M>0$ will be satisfied for all values of $x$. There is something different for the case $\Omega_k<0\,(k=1)$ which corresponds to $C>0$. There is a positive root $x=x_0=\sqrt{-\Omega_k/3\Omega_{DE}}$ for equation $F^{\prime}(x)=0$. And we can further get that $x_0$ is the minimum point of $F(x)$ for $F^{\prime\prime}(x_0)>0$. If only $F(x_0)\geq0$, $F(x)\geq0$. So this condition leads to $C\leq27/4$. To sum up, the value of $C$ can’t be arbitrary. There is a basic constraint from the standard cosmology. For the case of $w=-1$, $C$ has to be in the range (-$\infty,27/4$\]. Only when $C=27/4$, $F(x_0)=0$, that is to say $F(x)$ has a positive root $x=x_0$, the universe can recollapse at some time; when $C<27/4$, the universe will expand forever; and when $C>27/4$, the universe is impossible to exist as a physical one, or to say it is not a Big Bang universe.
These results are displayed in Fig.\[fig2eps\]. The permitted region is the interior of the line labelled $C=27/4$.
![The $\Omega_{DE}-\Omega_{M}$ plane for the case of $w=-1$ (cosmological constant). The permitted region is the interior of the line labelled $C=27/4$.[]{data-label="fig2eps"}](f2.eps){width="80.00000%"}
The Case of $w=-1/3$
--------------------
In this case, Eq.(\[omegaplanew\]) describing the $\Omega_{DE}-
\Omega_{M}$ plane and the expansion trajectory of the universe reduces to $$\frac{\Omega^z_M+\Omega^z_{DE}-1}{\Omega^z_{DE}}= \frac{\Omega_M+
\Omega_{DE}-1}{\Omega_{DE}}=\sqrt[3]{C}, \label{omegaplane13}$$ and the function $F(x)$ becomes $$F(x)=(\Omega_{DE}+\Omega_{k})x+\Omega_{M}, \label{Fx2}$$ which can be further written as $$F(x)=(1-\Omega_M)x+\Omega_{M}. \label{Fx22}$$
From the condition $F(x)\geq0$, we get $0<\Omega_M\leq 1$ (it can be easily understood that $\Omega_M>0$ originates from the fact that the matter density can’t be negative). According to Eq.(\[omegaplane13\]), we have $0<\Omega_M=(\sqrt[3]{C}-1)\Omega_{DE}+1\leq1$, so $$\left\{
\begin{array}{ll}
-\infty<C\leq1,\ \ &\ \ \rm{for}\ \Omega_{DE}>0\\
C=\pm\infty,\ \ &\ \ \rm{for}\ \Omega_{DE}=0\\
1\leq C<+\infty,\ \ &\ \ \rm{for}\ \Omega_{DE}<0\\
\end{array}
\right.. \label{OmegaDE}$$
These results are displayed in Fig.\[fig3eps\]. And there isn’t a positive root for function $F(x)$, so this kind of universe will expand forever.
![The same as Fig.\[fig2eps\] but for the case of $w=-1/3$. The permitted region of $\Omega_{DE}$ and $\Omega_M$ is between the two vertical line.[]{data-label="fig3eps"}](f3.eps){width="80.00000%"}
The Case of $w=-2/3$
--------------------
In this case, Eq.(\[omegaplanew\])reduces to $$\frac{\Omega^z_M+\Omega^z_{DE}-1}{\sqrt{\Omega^z_M}\sqrt{\Omega^z_{DE}}}=
\frac{\Omega_M+\Omega_{DE}-1}{\sqrt{\Omega_M}\sqrt{\Omega_{DE}}}=\sqrt[3]{C},
\label{omegaplane23}$$ and the function $F(x)$ becomes $$F(x)=\Omega_{DE}x^2+\Omega_{k}x+\Omega_{M}. \label{Fx3}$$ $F(x)\geq0$ needs $\Omega_{DE}\geq0$ (also we just consider $\Omega_{DE}>0$). The same as done in the case $w=-1$, we study the derivation of $F(x)$ $$F^{\prime}(x)=2\Omega_{DE}x+\Omega_{k}. \label{Fx3p}$$
For the cases $\Omega_k>0\,(k=-1)$ and $\Omega_k=0\,(k=0)$,which correspond to $C<0$ and $C=0$ respectively, $F^{\prime}(x)>0$, so $F(x)> F(0)=\Omega_M>0$ will be satisfied for all values of $x$. For the case of $\Omega_k<0\,(k=1)$ which corresponds to $C>0$, there is a positive root $x=x_0={-\Omega_k/2\Omega_{DE}}$ of equation $F^{\prime}(x)=0$ and $x_0$ is also the minimum point of $F(x)$ for $F^{\prime\prime}(x_0)>0$. From $F(x_0)\geq0$ we get $C\leq8$. So $C$ has to be in the range (-$\infty,8$\]. Only when $C=8$, $F(x)$ has a positive root $x=x_0$, the universe can recollapse at some time; when $C<8$, the universe will expand forever; and when $C>8$, the universe is impossible to exist as a physical one.
It can be shown from Fig.\[fig4eps\] that the expansion history of the universe are very familiar with that of cosmological constant but with the boundary of $C=8$.
![The same as Fig.\[fig2eps\] but for the case of $w=-2/3$. The permitted region is the interior of the line labelled $C=8$.[]{data-label="fig4eps"}](f4.eps){width="80.00000%"}
The Transition Redshift from Deceleration to Acceleration in the $
\Omega_{DE}-\Omega_{M}$ Plane {#redshift}
==================================================================
The decelerating parameter $q(z)$ is defined by $$q(z)\equiv(-\frac{\ddot{a}}{a})/H^{2}(z)=\frac{1}{2E^2(z)}\frac{\mathrm{d}
E^2(z)}{\mathrm{d} z}(1+z)-1, \label{qz}$$ the present value $q_0=q\,(z=0)$ of which is called decelerating factor. At the transit redshift $z_T$, the universe reaches $\ddot{a}(z_T)=0$ or $q(z_T)=0$ and evolves from deceleration to acceleration expansion, thus we can get the relation between density parameters and transition redshift $z_T$ in $\Omega_{DE}-\Omega_{M}$ plane $$\Omega_{DE}=-\frac{1}{(1+3w)(1+z_T)^{3w}}\Omega_M, \label{qzt0}$$ which leads to $$\Omega_{DE}=-\frac{\Omega_M}{(1+3w)}, \label{qzt00}$$ at $z_T=0$ or $q_0=0$. Clearly, as the transition redshift increases with the decrease of $\Omega_m$. The $\Omega_{DE}-\Omega_{M}$ plane with the best fit estimate of transition redshift $1+z_T=1.46\pm0.13$, and together with the theoretical constraints discussed above for $w=-1,-1/3$ and $-2/3$ are shown in Fig.\[fig5eps\], \[fig6eps\] and \[fig7eps\] respectively.
![The $\Omega_{DE}-\Omega_{M}$ plane with the best fit estimate of transition redshift $z_T$ [@2004ApJ...607..665R] for the case of $w=-1$. The curve labelled $C=27/4$ is the boundary of this model.[]{data-label="fig5eps"}](f5.eps){width="80.00000%"}
![The same as Fig.\[fig5eps\] but for the case of $w=-1/3$. Only $\Omega_M=0$ can lead to $q(z_T)=0$.[]{data-label="fig6eps"}](f6.eps){width="80.00000%"}
![The same as Fig.\[fig5eps\] but for the case of $w=-2/3$. The curve labelled $C=8$ is the boundary of this model.[]{data-label="fig7eps"}](f7.eps){width="80.00000%"}
Conclusion and Discussion {#cd}
=========================
In this paper we examined the dynamical evolution of the universe filled with a dark energy of EOS $w=$ constant. Based on the Big Bang model, we give some theoretical constraints of cosmological parameters of three special cases in the $\Omega_{DE}-\Omega_{M}$ plane. The physical constrain condition of $E^2(z)\geq0$ makes the cosmological parameters not be arbitrary. It is shown in the $\Omega_{DE}-\Omega_{M}$ plane by a limited region rather than all of the plane. Together with the observational results of $z_T$, we can constrain the cosmological parameters more strictly. It is shown in the Figs.\[fig5eps\]—\[fig7eps\] that the observation of $z_T$ is compatible with the model of cases of $w=-1$ and $w=-2/3$ to great extent, but is not consistent with the one of $w=-1/3$. This has also been revealed in work on the supernova measurements [@1998ApJ...509...74G; @1999ApJ...517..565P]. Apparently, the dark energy model with $w<-1/3$ make a positive contribution to the acceleration of the universe, while the model with $w=-1/3$ has effect on neither the acceleration nor deceleration. In fact, the supernova observation supports an accelerating universe. On the other hand, because there still exists difficulty in the accurate treatment of the behavior of cosmological defect models partly, these models have not been very thoroughly studied [@1997ApJ...491L..67S]. Thus, due to this motivation we in this paper just make a try to test the cosmic defect models by the properties of $\Omega_{DE}-\Omega_{M}$ plane, which maybe provides an alternative method to test cosmological models.
In addition, when $-1<w<-1/3$, it is similar to the case of $w=-2/3$ but is difficult to be studied analytically. From $F(x)\geq0$ we have a general result $\Omega_{DE}\geq0$. While $-1/3<w<0$, there should be $\Omega_k\geq0$ generally. There would be some difficulties to get the range of $C$. For an arbitrary value of $w$, one can give the detailed results numerically from Eq.(\[omegaplanew\]) and Eq.(\[Fx\]).
We are very grateful to the anonymous referee for constructive suggestion. This work was supported by the National Science Foundation of China (Grants No.10473002), 985 Project, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and the Scientific Research Foundation for Undergraduate of Beijing Normal University (Grant No.127002). T.J.Zhang would also like to thank Xiang-Ping Wu, Da-Ming Chen, Bo Qin, Ue-Li Pen and Peng-Jie Zhang for their hospitality during my visits to the cosmology groups of the National Astronomical Observatories of P.R.China and the Canadian Institute for Theoretical Astrophysics(CITA), University of Toronto.
|
---
address: |
Dipartimento di Fisica, Università di Milano and INFN-MILANO, Via Celoria 16,\
IT-20133 Milan, Italy.
author:
- 'V. Verzi'
title: Four fermion processes at LEP II
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
[Talk presented in\
XXXVI Recontres de Moriond Electroweak Interactions and Unified Theories, 10-17 March 2001.]{}
Introduction
============
In its second phase, LEP has been operated in years from 1996 to 2000 at centre-of-mass energies ($\sqrt{s}$) between 161 GeV and 208 GeV. This has allowed to each of the four experiments to collect nearly 700 $pb^{-1}$ of data above the $W$-pair production threshold, as summarised in table \[tab:lum\].
$\sqrt{s}$ (GeV) 161 172 183 189 192 196 200 202 205 207
------------------------ ---------- ---------- ---------- ----------- ---------- ---------- ---------- ---------- ---------- ----------- --
$\int L dt\ (pb^{-1})$ $\sim$10 $\sim$10 $\sim$55 $\sim$160 $\sim$25 $\sim$75 $\sim$85 $\sim$40 $\sim$85 $\sim$140
year 1996 1996 1997 1998 1999 1999 1999 1999 2000 2000
: Approximated value of the integrated luminosity collected by the DELPHI experiment at LEP II.
\[tab:lum\]
The high centre-of-mass energy achieved at LEP II has allowed to study several four fermion processes and hence to test the Standard Model (SM) of electroweak interactions in sectors only poorly known before. The most important process is the $WW$ production because it allows to measure the $W$ mass [@m01:mw] and the triple gauge boson couplings $WWV$ ($V=Z/\gamma$) (TGC) which appear at the tree level. Other four fermion final state are attained with the production of a pair of neutral gauge bosons $ZZ$ and $Z\gamma^\star$. These processes may receive New Physics contributions from triple neutral gauge boson couplings $ZZZ$, $ZZ\gamma$ and $Z\gamma\gamma$ which are expected to be unobservably small in the SM. The remaining four fermion processes involve the production of a single gauge boson: $We\nu_e$ and $Zee$. Particularly important is $We\nu_e$ because it allows to further constrain the $WW\gamma$ vertex [@yellow:lep2-1].
This paper presents the results on four fermion processes with particular attention to the preliminary measurements obtained with the data collected in the year 2000. The second section deals with $WW$ and $We\nu$ processes while the third deals with $ZZ$ and $Z\gamma^\star$ production. No new preliminary results on $Zee$ process have been presented and the old ones are described elsewhere [@zee-osaka].
$WW$ and $We\nu$ production
===========================
With the full luminosity available at LEP, about 11000 $WW$ events are expected to be produced in each of the four experiments. According to the $W$ decay modes, one has three different topologies in $WW$ events: fully hadronic ($WW \rightarrow qqqq$), semileptonic ($WW \rightarrow qql\nu_l$) and fully leptonic ($WW \rightarrow l \nu_l l \nu_l$) events respectively with branching fraction of 46%, 43% and 11%. These events can be selected with high efficiencies and high purities.
The four experiments have measured the $WW$ cross section at all energies above 161 GeV [@csww:00]. The combined results are shown in figure \[fig:csww-br-cswen\], compared with the theoretical predictions of RacoonWW [@racoonww] and YFSWW [@yfsww] programs that treat the ${\cal O}(\alpha)$ radiative corrections in “Double Pole Approximation” (DPA). The theoretical error for DPA is $\le 0.5\%$ and it is small compared to the error of 2% wich characterize the calculation of ${\cal O}(\alpha)$ corrections in “Improved Born Approximation” (IBA) [@yellow:lep2-1].
The $W$ decay branching fractions $Br(W \rightarrow ff)$ have been determined from the cross section for the individual $WW \rightarrow 4f$ decay channels, and the results are shown in figure \[fig:csww-br-cswen\]. The measurement of $B(W \rightarrow qq)$ allows an indirect determination [@ewg:cs] of the Cabibbo-Kobayashy-Maskawa matrix element $|V_{cs}|$: $|V_{cs}|_{LEP} = 0.996 \pm 0.013$.
The most general Lorentz invariant Lagrangian for $WWV$ ($V=Z/\gamma$) vertex interactions, under some reasonable requirements [@yellow:lep2-1], can be described by three independent couplings $\{ \Delta g_1^Z, \Delta k_\gamma, \lambda_\gamma \}$ [^1] (TGC) and another two constrained couplings, $\Delta k_Z = \Delta g_1^Z + \Delta k_\gamma \tan^2\theta_W$ and $\lambda_Z = \lambda_\gamma$.
The TGCs are measured using the information from the total cross section and from the shape of the distributions for physical observables. Due to the high precision achieved by LEP experiments, these distributions should be predicted including the ${\cal O}(\alpha)$ corrections in DPA. However, the generator programs used by the LEP experiments, wich include the description of the detectors, treat the ${\cal O}(\alpha)$ corrections using the IBA. Only the ALEPH collaboration [@tgc-aleph] has estimated the systematic effect due to the missing DPA, showing that it contributes in a sizeable way to the total systematic error for the couplings $\Delta g_1^Z$ and $\lambda_\gamma$. ALEPH is the only collaboration that has updated the results [@tgc-aleph] on TGCs including the data collected in 2000 obtaining the following 95% confidence intervals: $[-0.048,+0.080]$ for $\Delta g_1^Z$, $[-0.164,+0.134]$ for $\Delta k_\gamma$ and $[-0.059,+0.065]$ for $\lambda_\gamma$. No new LEP combined values have been produced after the 2000 summer conferences. It’s wortwhile to note that the missing DPA should not change drammatically the LEP combined results [@lwg:tgc] because for $\Delta g_1^Z$ and $\lambda_\gamma$ the statistical error dominates.
A model independent way to test the SM in the $WW$ production consists in comparing the measured average $W$ polarization $f_\lambda$ ($\lambda$ indicates the $W$ polarization) with the theoretical expectations. The L3 collaboration has done this measurement in the semi-leptonic channel [@wpol-l3]. Using the data collected in 2000, they have measured $f_0 = (21.6 \pm 5.3)\%$, $f_{-1} = (64.7 \pm 6.6)\%$ and $f_{+1} = (13.7 \pm 3.4)\%$ that are in agreement with the SM values: $f_0^{SM} = 22.0 \%$, $f_{-1}^{SM} = 62.3 \%$ and $f_{+1}^{SM} = 15.7 \%$.
The LEP combined measurements results for $We\nu$ cross section are shown in figure \[fig:csww-br-cswen\]. Since the 2000 winter conferences, the only new preliminary results have been presented by the ALEPH collaboration [@wen-aleph] which has analysed the data collected in the year 2000.
$ZZ$ and $Z\gamma^\star$ production
===================================
According to the $Z$ decay modes there are five visible decay channels. The dominant ones are the four jets channel ($ZZ \rightarrow qqqq$), the two jets plus missing energy channel ($ZZ \rightarrow qq\nu\nu$) and the two jets plus two leptons ($ZZ \rightarrow qqll$) channel with an expected branching ratio of 49%, 28% and 14% respectively.
While $qqll$ events can be selected with high purity, the $qqqq$ and $qq\nu\nu$ channels are affected by a large amount of background. In these two channels, multivariate analyses are usually used to discriminate the signal from the background. In the four jets channel the $b$ tag techinque is used againts $WW$ events. The contamination is than mainly due to QCD events. This is clear in figure \[fig:zz\] wich shows the output of the multivariate analysis developed by the DELPHI collaboration compared with the SM expectations for each process.
The four experiments have measured the $ZZ$ cross section at all energies above 183 GeV. The combined LEP results are shown in figure \[fig:zz\] and they are compared with the theoretical predictions calculated with the YFSZZ [@yfszz] and ZZTO [@yellow:4-ferm] programs.
${\scriptscriptstyle e^+e^- \rightarrow q\bar{q}\mu^+\mu^-}$
${\scriptscriptstyle \gamma^\star\rightarrow \mu^+\mu^-}$ ${\scriptscriptstyle Z\rightarrow \mu^+\mu^-}$
The $ZZ$ production is sensitive to possible anomalous vertex interactions $ZZV$ ($V=Z/\gamma$). Following the parametrization suggested in [@hagiwara] there are four independent couplings: $f_i^V$ ($i=1,2$ and $V=Z/\gamma$) that for $i=4$ are CP-odd and for $i=5$ are CP-even. The LEP combined results [@lwg:tgc] at the 95% of confidence level are $[-0.21,+0.23]$ for $f_4^\gamma$, $[-0.40,+0.33]$ for $f_4^Z$, $[-0.37,+0.49]$ for $f_5^\gamma$ and $[-0.27,+0.29]$ for $f_5^Z$. They are in agreement with the SM predictions $f_i^V=0$. These measurements benefit of the higher energy run of the year 2000, only through the analyses performed by the ALEPH and DELPHI collaborations [@ntgc-f].
In the $Z\gamma^\star$ production the virtual photon decays into a pair of fermions wich are separated by a small angle. The DELPHI collaboration [@zgstar] has presented the $Z\gamma^\star$ cross section for three different decay channels averaged over energies above 183 GeV: $\sigma_{Z\gamma^\star\rightarrow qq\mu\mu} = (0.123 \pm 0.025)$ pb, $\sigma_{Z\gamma^\star\rightarrow qq\nu\nu} = (0.129 \pm 0.038)$ pb and $\sigma_{Z\gamma^\star\rightarrow qqqq} = (0.074 \pm 0.045)$ pb. They are in agreement with the SM expectations: 0.098 pb for $qq\mu\mu$, $(0.092 \div 0.084)$ pb for $qq\nu\nu$ and 0.082 pb for qqqq.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank my DELPHI colleagues working in these subjects for all the constructive discussions.
References {#references .unnumbered}
==========
[99]{} N. Watson, these proceedings. , CERN 96-01 (1996), vol.I. Carmen Palomares, proceeding of ICHEP 2000. ALEPH 2001-013 CONF 2001-010; DELPHI 2001-024 CONF 465; L3 Note 2638; OPAL Physics Note PN469. A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, [*Nucl. Phys.*]{} B [**587**]{} (2000) 67. S. Jadach [*et al.*]{}, [*Phys. Lett.*]{} B [**417**]{} (1998) 326, hep-ph/0103163 The LEP collaborations and the LEP $WW$ Working Group, LEPEWWG/XSEC/2001-01\
http://lepewwg.web.cern.ch/LEPEWWG/lepww/4f/Winter01/xsec$_-$moriond01.ps.gz . Particle Data Group, D.E. Groom [*et al.*]{}, [*Eur. Phys. J.*]{} C [**15**]{} (2000) 1. ALEPH 2001-027 CONF 2001-021. The LEP collaborations and the LEP TGC Working Group, LEPEWWG/TGC/2001-01\
http://lepewwg.web.cern.ch/LEPEWWG/lepww/tgc/note.ps.gz . L3 Note 2636. ALEPH 2001-017 CONF 2001-014. S. Jadach, W. Placzek, B.F.L. Ward, [*Phys. Rev.*]{} D [**56**]{} (1997) 6939. , CERN 2000-009 (2000). ALEPH 2001-006 CONF 2001-003; DELPHI 2001-015 CONF 456; L3 Note 2641; OPAL Physics Note PN469. K. Hagiwara, K. Hikasa, R.D. Peccei and D. Zeppenfeld, [*Nucl. Phys.*]{} B [**282**]{} (1987) 253. ALEPH 2001-014 CONF 2001-011; DELPHI 2001-014 CONF 455. DELPHI 2001-008 CONF 449.
[^1]: $\Delta c = c - 1$ indicates the deviation from the SM expectation for $c$ that is 1 while in SM $\lambda_\gamma = 0$.
|
---
abstract: |
We investigate the phenomenology of Effective Supersymmetry (ESUSY) models wherein electroweak gauginos and third generation scalars have masses up to about 1 TeV while first and second generation scalars lie in the multi-TeV range. Such models ameliorate the SUSY flavor and CP problems via a decoupling solution, while at the same time maintaining naturalness. In our analysis, we assume independent GUT scale mass parameters for third and first/second generation scalars and for the Higgs scalars, in addition to $m_{1/2}$, $\tan\beta$ and $A_0$, and require radiative electroweak symmetry breaking as usual. We analyse the parameter space which is consistent with current constraints, by means of a Markov Chain Monte Carlo scan. The lightest MSSM particle (LMP) is mostly, but not always the lightest neutralino, and moreover, the thermal relic density of the neutralino LMP is frequently very large. These models may phenomenologically be perfectly viable if the LMP before nucleosynthesis decays into the axino plus SM particles. Dark matter is then an axion/axino mixture. At the LHC, the most important production mechanisms are gluino production (for $m_{1/2} \alt 700$ GeV) and third generation squark production, while SUSY events rich in $b$-jets are the hallmark of the ESUSY scenario. We present a set of ESUSY benchmark points with characteristic features and discuss their LHC phenomenology.\
PACS numbers: 14.80.Ly, 12.60.Jv, 11.30.Pb
---
LPSC10105\
UH-511-1152-10
[ **Effective Supersymmetry at the LHC\
**]{}
[Howard Baer$^{1}$, Sabine Kraml$^{2}$, Andre Lessa$^{1}$,\
Sezen Sekmen$^3$ and Xerxes Tata$^{4,5}$]{}\
[ *$^1$Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA\
$^2$Laboratoire de Physique Subatomique et de Cosmologie, UJF Grenoble 1, CNRS/IN2P3, INPG, 53 Avenue des Martyrs, F-38026 Grenoble, France\
$^3$Dept. of Physics, Florida State University, Tallahassee, FL 32306, USA\
$^4$Dept. of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822,USA\
$^5$Dept. of Physics, University of Wisconsin, Madison, WI 53706, USA* ]{}
Introduction {#sec:intro}
============
Particle physics models that include weak scale softly broken supersymmetry (SUSY) are especially compelling in that they stabilize the weak scale, even in the presence of new physics at much higher energy scales, such as $M_{\rm Planck}$ or $M_{\rm GUT}$. The simplest of such models, the Minimal Supersymmetric Standard Model (MSSM) [@dg; @sakai] enjoys some compelling indirect experimental support in that the measured values of the gauge couplings at energy scale $Q=M_Z$, when run to high energies via renormalization group evolution (RGE), nearly unify at $Q=M_{\rm GUT}\simeq 2\times
10^{16}$ GeV, as is expected in the simplest grand unified theories (GUTs). On the astrophysical side, SUSY theories with a conserved $R$-parity offer several candidates for the dark matter particle, among these: the lightest neutralino, the gravitino and the axion/axino supermultiplet.
Along with these successes, [*generic*]{} SUSY models also lead to new puzzles not present in the Standard Model (SM). Additional (usually discrete) symmetries are necessary to prevent a too rapid rate for proton decay. Moreover, unless weak scale soft SUSY breaking (SSB) parameters of the MSSM are flavor-blind [@dg] or “aligned” [@seiberg], and nearly real, the model leads to unacceptably large rates for flavor-changing transitions and $CP$ violating effects. We stress that various (sometimes, admittedly [*ad hoc*]{}) mechanisms have been suggested to ameliorate these undesired effects which, we speculate, arise because of our lack of understanding of how SM superpartners feel the effects of SUSY breaking.
Another possibility to suppress unwanted flavor-changing and $CP$-violating effects is to arrange for matter sparticles to be heavy so that their effects are sufficiently suppressed [@dine]. Large matter scalar masses also serve to suppress proton decay processes in a SUSY GUT. This [*decoupling solution*]{} to the SUSY flavor problem usually requires SSB terms of order 10–100 TeV [@arkmur], well beyond expectations from naturalness, which favors weak scale soft terms. It was noted as far back as 1986 [@manuel], and later again in Ref. [@dgpt], that the stability of the weak scale to radiative corrections requires only the electroweak (EW) gauginos and third generation sparticles — these couple with large strength to the Higgs sector — to have masses up to ${\cal O}(1)$ TeV, while gluinos and superpartners of the first two generations (whose direct couplings to the Higgs sector are very small, so that these enter naturalness considerations only at the two-loop level) could well have multi-TeV masses. Since the most stringent constraints on flavour- and $CP$-violation come from the first two generations of quarks and leptons, such a mass spectrum potentially alleviates the SUSY flavor and $CP$ problems without the need for undue fine-tuning of parameters. This was subsequently developed into the framework referred to as “Effective Supersymmetry” (ESUSY), by Cohen, Kaplan and Nelson [@ckn], who suggested two different realizations of the split matter generations idea by introducing a SUSY-breaking sector to which the first two generations couple more strongly than the third generation. As a result, the matter scalars of the first two generations acquire larger SUSY-breaking masses than third generation scalars.
The ESUSY scenario seems in recent years to have become less favored due to two measurements. The first — the measured (2–3)$\sigma$ deviation in $(g-2)_\mu$ from its Standard Model (SM) expectation [@gm2] — seems to require smuons/muon sneutrinos in the sub-TeV range if the deviation is to be attributed to SUSY. The second — the increasingly precise measurement of the dark matter (DM) density of the Universe — is difficult to reconcile in ESUSY if the dark matter is assumed to be dominantly composed of thermal relic neutralinos left over in standard Big Bang cosmology. With scalars in the multi-TeV range, along with a bino-like neutralino, the relic density is calculated to be typically several orders of magnitude higher than the experimentally observed value [@Jarosik:2010iu], \_[DM]{} h\^2 = 0.11230.0035, although there are some special parameter regions where this need not be the case.
At the present time, it is not completely clear whether the $(g-2)_\mu$ anomaly is real. The current discrepancy arises if one adopts the (more direct to use) hadronic vacuum polarization amplitude from low energy $e^+e^-\to hadrons$ data. If instead the vacuum polarization is taken from $\tau$ lepton decay data, then the discrepancy is smaller (but recently growing!).
In the case of neutralino DM, it is possible to have a small superpotential $\mu$ term co-existing with large scale masses, as occurs [*e.g.*]{} in the hyperbolic branch/focus point region of the mSUGRA[^1] model. Then the lightest neutralino, instead of being typically bino-like with a small annihilation cross section and concomitantly large relic density, becomes mixed higgsino dark matter with thermal relic neutralinos from the Big Bang making up the observed cold dark matter relic density. Other possibilities for obtaining the right relic density are resonant annihilation through the pseudoscalar Higgs, or co-annihilation with third-generation sfermions.
Regions of parameter space where the neutralino relic density is too large cannot be unequivocally excluded in extensions of the model. For instance, if one invokes the Peccei-Quinn-Weinberg-Wilczek (PQWW) solution to the strong CP problem [@PQWW], then one expects the presence of an axion/axino supermultiplet in SUSY theories. If the axino $\ta$ is the lightest SUSY particle (LSP), then $\tz_1\to\ta\gamma$ and other decay modes are allowed, which can greatly reduce the DM abundance far below the level expected from neutralinos. The scenario of mixed axion/axino DM has been examined in the SUGRA context recently in Ref. [@bbs] in a general 19 parameter MSSM, and appears to be at least as viable as the case with neutralino DM.
In light of these considerations, we feel it would be fruitful to re-visit some of the phenomenological implications of ESUSY models at the beginning of the LHC era. In our analysis, we subsume the qualitative features of the ESUSY model: [*i.e.*]{} third generation, Higgs sector and EW gaugino masses at or below the TeV scale, with multi-TeV SSB parameters for the first two generations. For simplicity we also assume gaugino mass unification.[^2] While we adopt the qualitative picture of Effective SUSY, we do [*not*]{} assume the validity of the more speculative mechanisms that the authors of Ref. [@ckn] suggest for the hierarchy between the SSB parameters of the first two generations and the corresponding parameters for the third generation and the gaugino sector. For our phenomenological analysis, we simply assume that such a hierarchy occurs for SSB parameters renormalized at a high scale that we take to be $M_{\rm GUT}$. We also take the less ambitious view that the SUSY flavour and $CP$ problems are ameliorated, but not completely solved, by this hierarchy, and allow the first two generations of scalars to have masses in the 5–20 TeV range, so that some degree of universality/alignment (for the first two generations) is still necessary to satisfy the most stringent flavour constraints. For simplicity, we will assume GUT scale scalar mass universality in the subspace of the first two generations, but allow independent SSB parameters for the third generation and the Higgs sector. This assumption of diagonal GUT scale scalar SSB mass squared matrices undoubtedly has an effect on flavor physics [@bfac], but should have very limited impact on the implications for collider physics and cosmology of the ESUSY scenario — which are after all our main interest in this paper.
The ESUSY model parameter space we will examine is thus given by the set of parameters (renormalized at the GUT scale) m\_0(1,2), m\_0(3), m\_[H\_u]{}, m\_[H\_d]{}, A\_0, m\_[1/2]{}, , [sign]{}() \[eq:pspace\] along with the top quark mass $m_t=173.1\pm 1.3$ GeV. Here $m_0(1,2)$ and $m_0(3)$ are the masses of the first/second and of the third generation sfermions, respectively; $m_{H_u}\equiv m_{H_u}^2/\sqrt{|m_{H_u}^2|}$ and $m_{H_d}\equiv m_{H_d}^2/\sqrt{|m_{H_d}^2|}$, are the SSB mass parameters of the up- and down-type Higgs scalars; $A_0$ is a universal trilinear coupling (relevant mostly only for the third generation); $m_{1/2}$ a universal gaugino mass parameter; and $\tan\beta\equiv v_u/v_d$. Radiative electroweak symmetry breaking can be used to determine $\mu^2$ using the measured value of $M_Z$.
We wish to maintain the successful gauge coupling unification at $Q=M_{\rm GUT}$, so that we will assume here that the MSSM is the correct effective field theory between $M_{\rm GUT}$ and $\widetilde{M}=m_0(1,2)$, the scale at which first and second generation scalars decouple from the theory. Below $\widetilde{M}$, we have ESUSY — the SM with two Higgs boson doublets together with third generation scalars, gauginos and higgsinos — as the effective theory.
Models with third generation scalar and gaugino-higgsino sector at $\alt$TeV but multi-TeV first and second generation mass parameters have been investigated in the past. Shortly after Cohen [*et al.*]{} [@ckn] laid down their framework, it was noted that two loop RGE effects arising due to heavy first/second generation scalars act to suppress third generation scalar mass parameters (even to tachyonic values), and a variety of flavor and $CP$-violating constraints were examined [@arkmur; @hisano]. In Refs. [@feng; @imh] it was shown that the inverted scalar mass hierarchy which is the hallmark of the ESUSY scenario could emerge dynamically in models with Yukawa coupling unification, although the $\widetilde{M}:M_{\rm weak}$ ratio was found to be limited [@imh]. This class of models — requiring $t-b-\tau$ Yukawa coupling unification and $SO(10)$ like boundary conditions with $\tan\beta\sim 50$ — has been investigated in detail in Refs. [@yuk; @bdr]. In Ref. [@gsimh], the magnitude of the $\widetilde{M}:M_{\rm weak}$ hierarchy was investigated in models without Yukawa coupling unification where the matter scalar mass parameters are already taken to be split at the GUT scale. This study assumed equal GUT scale values of third generation and Higgs SSB masses, and as a result, was limited in scope compared to the results to be presented here. In addition, several authors have investigated other related aspects of supersymmetric models with heavy scalars [@biswarup]. The remainder of this paper is organized as follows. In Sec. \[sec:pspace\], we explore the parameter space in Eq. (\[eq:pspace\]) and map out ranges of parameters that potentially lead to ESUSY at the weak scale. In Sec. \[sec:mcmc\], we perform a Markov Chain Monte Carlo (MCMC) analysis to search for SUSY scenarios which fulfill the ESUSY conditions, subject to various experimental constraints. We first describe the setup of our MCMC, and then show results for posterior probability distributions both for input parameters that result in ESUSY, as well as for expectations for various sparticle masses and selected experimental observables. Moreover, we pick out some characteristic benchmark points for further study. In Sec. \[sec:lhc\], we first present the main particle production rates and decay patterns relevant for LHC phenomenology of the ESUSY scenarios, and then illustrate the diversity of LHC phenomena through more detailed discussion of the phenomenology of the benchmark points, including the possibility that the lightest MSSM sparticle (LMP, as distinct from LSP) may be charged or coloured. We conclude in Sec. \[sec:conclude\] with a summary of our results.
Viable spectra and parameter space of the ESUSY model {#sec:pspace}
=====================================================
Constrained supersymmetric models such as the mSUGRA model have, for a fixed gluino mass, an upper limit on how massive scalars can be. For a model with universal scalar mass soft terms at the GUT scale, given by $m_0$, with $m_{1/2}$ fixed, the weak scale value of $\mu^2$ diminishes as $m_0$ increases. Ultimately, when $m_0$ is large enough, $|\mu |$ becomes comparable to the weak scale value of the bino mass $M_1$, and one gains neutralinos of mixed higgsino-bino composition which make a good candidate for thermal WIMPs. This is the so-called hyperbolic branch/focus point (HB/FP) region of the SUGRA parameter space [@hb_fp]. As $m_0$ increases even further, $|\mu |$ decreases even more and the $\tz_1$ becomes nearly pure higgsino, and the higgsino-like lightest chargino may drop below limits from LEP2 searches. For yet higher values of $m_0$, $\mu^2$ becomes negative, signaling an inappropriate electroweak symmetry breaking (EWSB) pattern. Thus, in the mSUGRA model, for a given value of $m_{1/2}$, $m_0$ can become no larger than a few TeV. If we proceed to models with non-universal generations, taking $m_0(1,2)$ as independent of $m_0(3)$, with $m_0(3)=m_{H_{u}}=m_{H_{d}}\equiv m_H$ as in Ref. [@gsimh], we expect, using 1-loop RGEs, by the same reasoning, an upper bound on $m_0(3)$. The point is that the first and second generations essentially decouple from the RG evolution of the Higgs SSB parameters (which enter the electroweak potential minimization conditions, see Ref. [@wss]). At two loop order, however, very large scalar masses do affect the running of the other scalar masses and in fact lead to their suppression at the weak scale [@hisano; @gsimh]. As $m_0(3)$ decreases (or $m_0(1,2)$ increases), so $m_0(3) \ll m_0(1,2)$, one of the third generation scalars becomes the LSP, turning tachyonic for even smaller (larger) $m_0(3)$ ($m_0(1,2)$) values. Therefore, with $m_0^2(3)=m_{H_{u,d}}^2$ at the GUT scale, there are both an upper (from $\mu^2 >0$) and a lower (from $m_{\tst,\tb,\tilde{\tau}}^2 > 0$) bound on $m_0(3)$. In Fig. \[fig:mH\], we show regions of allowed parameter space in the $m_H$ versus $m_0(3)$ plane for $m_0(1,2)=5$, 10 and 20 TeV, with [*a*]{}) $m_{1/2}=300$ GeV and [*b*]{}) $m_{1/2}=800$ GeV, where the spectrum calculations are performed using [ Isajet7.80]{} [@isa78]. The green dashed line is where $m_H\equiv m_0(3)$. The region to the right of the blue lines is forbidden due to lack of EWSB, while the region below the red lines is forbidden because a third generation scalar becomes tachyonic. The region to the left and above the contours yields viable spectra.
For low $m_{1/2}$, we see from Fig. \[fig:mH\][*a*]{}) that if we restrict our attention to $m_H=m_0(3)$ (green dashed line) only the interval 1 TeV $\lesssim m_0(3) = m_{H} \lesssim 4$ TeV is allowed for $m_0(1,2) = 5$ TeV, while there are no allowed values for $m_0(1,2) =
10$ or 20 TeV, since for these cases the $m_0(3) = m_{H}$ line lies entirely in the excluded region. However, through 1-loop RGE running effects, an increase in $m_{1/2}$ increases $m_{\tq}$. Hence larger values of $m_0(1,2)$ become viable if $m_{1/2}$ is large enough. This is seen in Fig. \[fig:mH\][*b*]{}), where the green dashed line (again representing $m_0(3) = m_{H}$) shows that, for $m_{1/2} = 800$ GeV, the range of allowed $m_0(3) = m_{H}$ values is larger for $m_0(1,2) = 5$ TeV, where the tachyonic lower bound is gone, so $0 < m_0(3) = m_{H}
\lesssim 5$ TeV. Besides, if a neutralino LSP is required, we have $0.25
< m_0(3) = m_{H} \lesssim 5$ TeV. For the higher $m_0(1,2)$ cases we see that the increase in $m_{1/2}$ provides an allowed region for $m_0(1,2) = 10$ TeV, but still is insufficient for the $m_0(1,2) = 20$ TeV case. It is for this reason that the solutions found in Ref. [@gsimh] with very large $m_0(1,2)$ values, as needed to suppress FCNCs, also required rather high $m_{1/2}$ values, and consequently very heavy gluinos: quite beyond the reach of LHC.
Since there is no compelling theoretical argument to link the Higgs and third generation mass scales, it seems reasonable to adopt independent values of $m_0(3)$ and $m_{H_{u}}=m_{H_d}= m_H$ (NUHM1) or $m_{{H_u}}\ne m_{H_d}$ (NUHM2). In this case, for a large value of $m_0(3)$, we can take $m_H^2\ll m_0^2(3)$, and thus give $m_H$ a head start on running towards the necessary negative squared values which are needed for successful EWSB (recall that at the weak scale, $\mu^2\simeq -m_{H_u}^2$ for even modest values of $\tan\beta$). This is the well-known feature of NUHM1 models wherein increasing $m_H$ results in a decrease of the weak scale value of $|\mu
|$ [@nuhm]. Note that large negative values of $m_H$ are excluded because there $m_A^2<0$, signaling that EWSB is not correctly obtained.
If we adopt a low value of $|m_H|\alt 1$ TeV, and then move from high to low $m_0(3)$ values, the third generation scalars decrease in mass. The region for ESUSY with third generation scalars at $\alt$TeV is at $m_0(3)$ values just above the tachyonic third generation region. Thus, the strategy for gaining viable ESUSY spectra is to 1.) adopt a large value of $m_0(1,2)\sim 5-20$ TeV, then 2.) adopt a low value of $m_H\sim
0-2$ TeV, and finally, 3.) starting at a several TeV value of $m_0(3)$, decrease its value until third generation scalars dip below the TeV region. If we want low $|\mu |$ values as well, then increase $m_H$ as close to the EWSB boundary (on the right of the figure) as desired.
It is instructive to portray ESUSY in various parameter planes. Figure \[fig:plane\] shows the allowed region in the $m_0(3)\
vs.\ m_{1/2}$ plane with $m_0(1,2)=10$ TeV, $m_{H_u}= 0.5$ TeV, $m_{H_d}= 0$ TeV (along with $A_0=1$ TeV, $\tan\beta =10$ and $\mu >0$). The left-side gray shaded region gives tachyonic stop or sbottom masses, while in the lower-right gray region the chargino mass is below its LEP2 limit. The white region gives viable supersymmetric mass spectra. We plot contours of $m_{\tg}$ (gray dashed), $m_{\tst_1}$ (blue solid), $m_h$ (dashed cyan) and $\mu$ (magenta dot-dashed). The region to the lower left of the cyan dashed contour has $m_h<114$ GeV. The slim region adjacent to but right of the tachyonic stop region gives viable ESUSY spectra with top squark masses $m_{\tst_1}<1$ TeV. There is an almost-impossible-to-see green region of thickness $\alt 6$ GeV in $m_{1/2}$ hugging the boundary of the tachyonic stop region where the thermal neutralino abundance is in accord with measurement: $\Omega_{\tz_1}h^2<0.11$. This is, in fact, the top squark co-annihilation region [@stop] for ESUSY. Along the tachyonic boundary is another narrow region where the $\tst_1$ can become the lightest MSSM particle (LMP), though not necessarily the LSP if a non-MSSM $R$-odd sparticle (such as the axino) is lighter. We also show the regions where $\tb_1$, $\tilde{\tau}_1$ or $\tilde{\nu}_{\tau L}$ is the LMP.[^3]
In Fig. \[fig:planeB\][*a*]{}), we fix instead the values of $m_0(1,2)$, $m_0(3)$ and $m_{1/2}$ and plot contours of $m_{\tst_1}$, $\mu$ and $m_A$ in the $m_{H_u}\ vs.\ m_{H_d}$ plane. As discussed above, for intermediate positive ranges of $m_{H_d}$ in the plot, where $m_{H_u} \sim m_{H_d}$, as $m_{H_u}$ increases, the value of $\mu$ decreases until $\mu^2 < 0$ and EWSB is no longer realized. This can be seen by the $\mu$ contours in this part of the figure where we also see a significant green region where the thermal neutralino relic density is consistent with its observed value. In the lower portion (low $m_{H_d}$) of the grey area on the right and at the bottom in Fig. \[fig:planeB\][*a*]{}), we do not obtain radiative EWSB because $m_h^2$ or $m_A^2$ become negative. The green region in this vicinity is where the neutralino relic density is compatible because of higgs resonance annihilation [@Afunnel]. Finally, for large enough positive values of $m_{H_u}$ and $m_{H_d}$, $\tilde{t}_1$ becomes the LMP (shaded blue region), with $m_{\tilde{t}}^2 < 0$ above that (upper portion of the grey area on the right) and co-annihilation with stops leads to a compatible neutralino relic density. Finally, we present in Fig. \[fig:planeB\][*b*]{}) the allowed region in the $m_{H_u}\ vs.\
m_0(3)$ plane with $m_0(1,2)=10$ TeV, $m_{1/2} = 0.5$ TeV and $m_{H_d}=
1$ TeV. We see that the ESUSY scenario is realized at low $m_0(3)$ and $m_{H_u} \lesssim m_0(3)$.
Markov Chain Monte Carlo analysis of ESUSY {#sec:mcmc}
==========================================
In this section, we perform an MCMC analysis to map out the regions of ESUSY parameter space that are consistent with current experimental constraints from collider and flavour physics. The characteristics of ESUSY are enforced by means of theoretical priors. We first briefly describe our set up and list the likelihood functions and model priors that we use, and then show our results for the posterior probability distributions both for input parameters as well as various sparticle masses and selected experimental observables. In Sec. \[ssec:BM\], we present some benchmark points for more detailed study.
Setup of the MCMC scans {#ssec:scans}
-----------------------
The setup and procedure of our MCMC closely follow [@Belanger:2009ti], so we do not repeat these here but instead just give the key data of the analysis. For details on the MCMC method and Bayesian analysis, see [*e.g.*]{} [@de; @Austri:2006pe; @Allanach:2007qk; @Trotta:2008bp]. In addition to [Isajet7.80]{} [@isa78], which we use for the spectrum calculation, we use [SuperIso2.7]{} [@Mahmoudi:2008tp] for computing B-physics observables and [micrOMEGAs2.4]{} [@Belanger:2006is] for the calculation of the neutralino LMP abundance.
For each point $P$ in the scan of ESUSY parameter space, we first compute individual likelihoods $L(O_i)$ for each experimental observable $O_i$. Then the overall likelihood is given as the product of these individual likelihoods as $L_P=\prod L(O_i)$. The observables we use are listed in Table \[tab:observables\] along with the parameters of the corresponding likelihood functions used. The forms of these likelihood functions are given by, $$L_1(x,x_0,\sigma_x) = \frac{1}{1+\exp[-(x-x_0)/\sigma_x]}, \quad
L_2(x,x_0,\sigma_x) = \exp\left[-\,\frac{(x-x_0)^2}{2\,\sigma_x^2}\,\right] \,,$$ for observables for which there is only an upper or lower bound, and for observables for which a measurement is available, respectively. For the LEP limit on the Higgs mass, we use [@Buchmueller:2009fn] $$\chi^2(m_h) = \frac{(m_h-m_h^{\rm limit})^2}{(1.1~{\rm GeV})^2+(1.5~{\rm GeV})^2}$$ with $m_h^{\rm limit}=115$ GeV. The likelihood $L(m_h)$ is then given by $L(m_h)=e^{-\chi^2(m_h)/2}$ for $m_h<115$ GeV and $L(m_h)=1$ for $m_h\ge 115$ GeV.
Observable Limit Likelihood function Ref.
------------------------------- ---------------------------------- ---------------------------------------------------- ------------------------------------
${\rm BF}(b\to s\gamma)$ $(3.52 \pm 0.34) \times 10^{-4}$ $L_2(x,3.52 \times 10^{-4}, 0.34 \times 10^{-4})$ [@Barberio:2008fa; @Misiak:2006zs]
${\rm BF}(B_s\to \mu^+\mu^-)$ $\le 5.8 \times 10^{-8}$ $L_1(x,5.8 \times10^{-8}, -5.8 \times 10^{-10})$ [@Aaltonen:2007kv]
$R(B_u\to \tau\nu_\tau)$ $1.28\pm0.38$ $L_2(x,1.28,0.38)$ [@Barberio:2008fa]
$m_t$ $173.1\pm 1.3$ $L_2(x,173.1,1.3)$ [@tev:2009ec]
$m_h$ $\ge 114.5$ $1$ or $\exp(-\chi^2(m_h)/2)$ [@Buchmueller:2009fn]
SUSY mass limits LEP limits $1$ or $10^{-9}$ [@lepsusy]
: \[tab:observables\] Observables used in the likelihood calculation.
In order to favour (sub)TeV-scale electroweak gauginos, higgsinos, and third generation sfermions, we multiply the likelihood obtained from the experimental constraints with the following model prior: $$L_{M_{\rm eff}}(m_X)= \frac{1}{1+\exp((m_{X}-M_{\rm eff})/{170 \ {\rm GeV}} ) }
\label{eq:m3prior}$$ for each $X=\tilde\chi^+_1,\,\tilde\chi^+_2,\,\tilde t_1,\,\tilde t_2,\, \tilde b_1$. Choosing, for instance, $M_{\rm eff}=1.5$ TeV, we get $L_{M_{\rm eff}}=0.95$, $0.5$, $0.05$ for $m_X=1$, $1.5$ and $2$ TeV respectively; for $M_{\rm eff}=1$ TeV, we get $L_{M_{\rm eff}}=0.95$, $0.5$, $0.05$ for $m_X=0.5$, $1$, $1.5$ TeV respectively. We have run chains for different $M_{\rm eff}$ and checked that the effect is indeed only to vary the upper bound on the effective SUSY masses; the qualitative features of the parameter space remain unchanged.
Finally, regarding the model parameters, we allow $m_{1/2}=[0,\, 2]$ TeV, $m_{0}(3) = [0,\, 10]$ TeV, $m_{H_{u,d}}^{} =[-10,\, 10]$ TeV (with $m_{H_{u,d}}$ unrelated), $A_{0} = [-40,\, 40]$ TeV, and $\tan\beta = [2,\, 60]$, taking flat prior probability distributions for all these parameters. In addition we let the top mass vary within $m_t=173.1 \pm 1.3$ GeV [@Group:2009qk] with a Gaussian distribution. For $m_0(1,2)$, we show results for two different approaches:
1. we let $m_0(1,2)$ vary from 5 to 20 TeV with a flat prior probability distribution ($L_{\widetilde M}\equiv 1$), or
2. we let $m_0(1,2)$ vary without limits but apply a model prior of $$L_{\widetilde M}(m_0(1,2))= \frac{1}{1+\exp((10-m_0(1,2))/{1.7 \
{\rm TeV}})} \, ,
\label{eq:m0prior}$$ in order to favour $m_0(1,2)$ values beyond 10 TeV.
The total likelihood of a point is then taken to be $L_{\rm tot}=L_P\times L_{M_{\rm eff}}\times L_{\widetilde M}$.
Before turning to the results, a comment is in order regarding the nature of the LMP. If we do not impose any dark matter requirement on the LMP, the large majority of points have a neutralino LMP with a far too high relic density of up to $\Omega_\chi h^2\sim 10^3$. As noted in the Introduction, this can be in agreement with cosmological observations if the “true” LSP is actually an axino, [*i.e.*]{} in the case of mixed axion/axino dark matter [@bbs; @Baer:2008eq]. Moreover, about 12% (4%) of the points accepted by the chains with $M_{\rm
eff}=1$ ($1.5$) TeV have a stop, sbottom or stau LMP. Again these points may be viable if the true LSP is the axino. However, as we will discuss below, the phenomenology in these charged LMP cases is quite different from the neutralino LMP case: in particular, if the stop, sbottom or stau LMP does not decay promptly but gives a charged track. In what follows, we will confine our MCMC analysis to the case of a neutralino LMP, which indeed occurs most frequently. We will, however, comment on the phenomenology of the colored or charged LMP case at the end of the next section.
MCMC results with neutralino LMP {#subsec:res}
--------------------------------
Figure \[fig:mcmc1\] shows the posterior probability distributions of the input parameters of the ESUSY model from MCMC scans requiring a neutralino LMP. Here we have used $M_{\rm eff}=1$ TeV in Eq. (\[eq:m3prior\]). Unseen dimensions are marginalized over. The figure compares the two priors for $m_0(1,2)$: The thin black lines are for case 1. where $m_0(1,2)=5-20$ TeV with a flat distribution, while the thick red lines are for case 2. which has no limits on $m_0(1,2)$ but the choice of the prior in Eq. (\[eq:m0prior\]) favours higher values of $m_0(1,2)$. We see that, as anticipated, we find solutions where third generation sfermion and gaugino SSB parameters are typically 1–2 TeV, while the first/second generation SSB parameters are favoured to lie beyond 10 TeV. Moderate values of $\tan\beta$ are favoured.
As can be seen, the precise condition on $m_0(1,2)$ hardly influences the probability distributions of the other parameters. A possible exception is $m_{1/2}$, which features a double-peak distribution. This comes from the fact that at small $m_{1/2}$, the parameter space volume is constrained in the directions of the scalar mass parameters, while there is more space in the $\tan\beta$ direction. For large $m_{1/2}$, on the other hand, $\tan\beta$ is very much constrained by BR$(b\to
s\gamma)$, while there is more volume in the large $m_0(1,2)$ directions. The effect is more pronounced for case 1. which does not disfavour smaller values for $m_0(1,2)$. Moreover, we see that larger $m_0(1,2)$ prefers somewhat larger $m_{1/2}$.[^4] It is also interesting to note that very high $m_0(1,2)$ around 15–20 TeV suffers from a low probability. This is because as $m_0(1,2)$ increases, we are forced to increasing larger values of $m_0(3)$ (see Fig. \[fig:mH\]). For values of $m_0(3)$ in the multi-TeV range, third generation SSB parameters at $\alt$TeV can occur only when the large two loop RGE contributions rather precisely cancel the naturally multi-TeV contribution that we start with – too little cancellations leave large SSB parameters, while too much cancellation leads to tachyonic masses. As a result, the region of parameter space with a light third generation rapidly shrinks when $m_0(1,2)$ exceeds $\sim$ 15–20 TeV.
The posterior probability distributions of several relevant masses are shown in Fig. \[fig:mcmc2\]. We infer from the EW gaugino spectrum (and also the $\mu$) distribution that most of the time the lightest neutralino is bino-like.
Figure \[fig:mcmc3\] displays the probability distributions of $B$-decay observables, $\Delta a_\mu=(g-2)_\mu^{\rm SUSY}$, the higgsino fraction $f_H = v_1^{(1)2}+v_2^{(1)2}$ (in the notation of Ref. [@wss]) of the $\tilde Z_1$, and the relic abundance $\Omega h^2$ of the $\tilde
Z_1$. It is clear that if the muon anomalous moment is confirmed, it cannot arise within the ESUSY framework. We note that there is non-negligible fraction of the parameter space with a higgsino-like LMP. We also see that the neutralino relic density peaks around $\Omega
h^2\simeq 1-10$ and goes up to $\Omega h^2\sim 10^4$. Nevertheless, solutions with low values of $\Omega h^2\alt 0.1$, where the relic density of thermal neutralinos does not yield a universe that is too short-lived, also have non-negligible probability. They occur because of Higgs funnel annihilation, coannihilation with light stops and/or sbottoms (or, less likely, staus), or because of a large LMP higgsino component.
Effective SUSY benchmark points {#ssec:BM}
-------------------------------
To illustrate the types of SUSY spectra and the diverse phenomenology that can result in ESUSY models, we list five benchmark points in Table \[tab:bm\]. The first three points we list have a neutralino LMP, while points ES4 and ES5 have a sbottom and a stau LMP, respectively. Point ES1 adopts an $m_0(1,2)=10$ TeV, which is the most favored value in the probability distributions resulting from the MCMC. On the other hand, ES1 has a value of $m_{\tg}=524$ GeV, not far from the ESUSY minimum of $m_{\tg}\agt 460$ Gev. The chargino $\tw_1$ and the neutralino $\tz_{2}$ have masses of just $139$ GeV, and the $\mu$ parameter is 857 GeV. All third generation squarks except $\tb_2$, as well as all EW gauginos are $\alt$1 TeV scale, while the two staus are over 2 TeV.
In point ES2, $m_0(1,2)$ is increased to 12 TeV. ES2 has much higher value of $m_{\tg}\sim 2.4$ TeV with concomitantly heavier $\tw_1$ and $\tz_{1,2}$, although all EW gauginos along with $\tst_1$, $\tst_2$ and $\tb_1$ are below 1 TeV. The ${\tb_2}$ and staus all have masses $\sim 1.3-1.4$ TeV range. For this point, $\tst_1$ decays exclusively via the three-body mode $\tst_1\to bW\tz_1$, while other third generation quarks decay via two-body decays.
Point ES3, with a value of $m_0(1,2)\simeq 10$ TeV, again has all third generation sfermions, as well as all charginos, neutralinos and Higgs bosons essentially at or well below 1 TeV. The gluino mass is rather high: $\sim 2.1$ TeV. One characteristic feature of this point is that, because of the near degeneracy of $\tst_1$ and $\tz_1$, not only the two-body decays but even the three body decay $\tst_1 \to bW\tz_1$ is kinematically forbidden. In this case, $\tst_1$ will decay via $\tst_1 \to c\tz_1$ or $\tst_1 \to b\tz_1 f\bar{f}'$, though of course the small available phase space will favour the former decay [@andrew].
Point ES4 has a high $m_0(1,2)=20$ TeV, a heavy gluino with a mass of 2.5 TeV, and correspondingly heavy EW gauginos. However, $\tst_1$ and $\tb_1$ are very light with masses of 327 GeV and 291 GeV, while $m(\tst_2)=793$ GeV. In this case, $\tb_1$ is the LMP.
Finally, point ES5 has been chosen to illustrate that staus can also be very light. Here, $\ttau_1$ is the LMP, with a mass of just 289 GeV, while $m(\ttau_2)=380$ GeV. The lighter stop and the EW gauginos are in the sub-TeV range, while other third generation sfermions are 1.1-1.2 TeV.
[lccccc]{} parameter & ES1 & ES2 & ES3 & ES4 & ES5\
$m_0(1,2)\ [{\rm TeV}]$ & 10 & 12 & 10 & 20 & 6\
$m_0(3)\ [{\rm TeV}]$ & 2.4 & 1.5 & 1.1 & 3.3 & 0.3\
$m_{H_{d}}\ [{\rm TeV}]$& 1 & $-0.35\phantom{0}$ & $-0.6\phantom{0}$ & 1 & 0\
$m_{H_{u}}\ [{\rm TeV}]$& 2 & 1.2 & 0.5 & 1 & 0\
$m_{1/2}\ [{\rm TeV}]$ & 0.16 & 1 & 0.85 & 1 & 0.8\
$A_0$ \[[TeV]{}\] & 0 & 0.9 & 1 & 0 & 0\
$\mu$ & 857 & 902 & 925 & 2329 & 855\
$m_{\tg}$ & 524 & 2446 & 2103 & 2526 & 1941\
$m_{\tu_L}\ [{\rm TeV}]$ & 10.0 & 12.1 & 10.1 & 20.1 & 6.2\
$m_{\tu_R}\ [{\rm TeV}]$ & 10.0 & 12.2 & 10.1 & 20.1 & 6.2\
$m_{\te_L}\ [{\rm TeV}]$ & 10.0 & 12.0 & 10.0 & 20.0 & 6.0\
$m_{\te_R}\ [{\rm TeV}]$ & 10.0 & 12.0 & 10.0 & 20.0 & 6.0\
$m_{\tst_1}$& 646 & 608 & 398 & 327 & 867\
$m_{\tst_2}$& 1049 & 948 & 770 & 793 & 1147\
$m_{\tb_1}$ & 1039 & 830 & 586 & [**292**]{} & 1083\
$m_{\tb_2}$ & 1711 & 1313 & 958 & 1885 & 1176.3\
$m_{\ttau_1}$ & 2269 & 1341 & 944 & 2956 & [**289**]{}\
$m_{\ttau_2}$ & 2299 & 1388 & 1008 & 3152 & 381\
$m_{\tw_1}$ & 139 & 815 & 708 & 881 & 658\
$m_{\tz_2}$ & 139 & 815 & 708 & 878 & 657\
$m_{\tz_1}$ & [**69**]{} & [**441**]{} & [**372**]{} & 452 & 347\
$m_A$ & 1022 & 450 & 398 & 2042 & 875\
$m_h$ & 110.7 & 118.3 & 117.3 & 119.9 & 117.8\
$\Omega_{\tz_1}h^2$ & 320 & 0.789 & $0.036$ & – & –\
\
$\sigma\ [{\rm fb}]$ & $23.2\times 10^3$ & $157.8$ & $1618.0$ & $12.3\times 10^3$ & $51.4$\
$\tg\tg$ & 62.8% & 0.02% & 0.01% & – & 1.5%\
$EW-ino\ pairs$ & 36.6% & 5.1% & 0.8% & 0.06% & 35.3%\
$slep.\ pairs$ & – & 0.02% & 0.01% & – & 24.4%\
$\tst_1\bar{\tst}_1$ & 0.4% & 78.3% & 87.6% & 28.2% & 26.5%\
$\tst_2\bar{\tst}_2$ & 0.03% & 4.5% & 1.8% & 0.3% & 3.8%\
$\tb_1\bar{\tb}_1$ & 0.02% & 11.5% & 9.1% & 71.4% & 4.9%\
$\tb_2\bar{\tb}_2$ & – & 0.4% & 0.4% & – & 2.9%\
Two of the ESUSY points with a neutralino as LMP (ES1 and ES2) yield a thermal abundance of neutralinos far in excess of WMAP measured value for the cold DM relic density, while point ES3 is in the top-squark co-annihilation region and gives a thermal neutralino abundance of $\Omega_{\tz_1}h^2 = 0.036$. As mentioned previously, the points ES1 and ES2 can still be cosmologically viable if we invoke the PQWW solution to the strong CP problem, which necessitates the introduction of an axion/axino supermultiplet into the theory. In this case, the axino $\ta$, and not the neutralino, can be the LSP. If the $\ta$ is the LSP, then the neutralinos will decay via $\tz_1\to \ta\gamma$ with lifetime smaller than $\alt$ 0.1–1s, [*i.e.*]{} before the onset of Big Bang nucleosynthesis (BBN), for a PQWW scale $f_a\alt
10^{12}$ GeV [@ckr]. Since every neutralino decays to an axino, the would be thermal relic density of neutralinos is reduced by a factor $m_{\ta}/m_{\tz_1}$, and for small enough $m_{\ta}$ would be compatible with the relic density measurement. Thermal production of axinos in the early universe can still proceed, but the relic abundance of axinos then depends on the re-heat temperature $T_R$ after inflation. The dark matter will then consist of an axino/axion admixture [@bbs], with relic axions being produced via the vacuum mis-alignment mechanism.
In the cases where $\tst_1$, $\tb_1$ or $\ttau_1$ are LMPs, such scenarios are again allowed if, as before, we assume that the axino is the LSP. In these cases the sfermions dominant decay is $\tf_1\to f \ta$, via loop-mediated processes; the competing three-body decays, $\ttau_1 \to \tau\ta\gamma$ [@branden] or $\tq\to q\ta
g$ [@small] are argued to have a small branching fraction. The loop calculation is complicated by the fact that the non-renormalizable axino-bino-photon (and the axino-bino-Z and axino-gluino-gluon) coupling enters the calculation. Nevertheless, these authors find that for $m_{\ttau}=100$ GeV, the stau LMP lifetime ranges from about 0.01s to 10h, for the PQWW scale $f_a$ in the range $5\times 10^9$–$5\times
10^{12}$ GeV, and scales inversely as the stau mass. The corresponding squark lifetime is shorter: about $2\times 10^{-6}$s for 500 GeV squarks and 1 TeV gluinos with $f_a=10^{11}$ GeV. If the LMP lifetime indeed exceeds a few seconds, this could disrupt the successful predictions of Big Bang nucleosynthesis [@jedamzik], though it appears that if $f_a< 10^{12}$ GeV, a stau LMP is relatively safe [@freitas]. For a general discussion of charged LMPs, see Ref. [@berger].
Phenomenology of Effective SUSY at the LHC {#sec:lhc}
==========================================
How will effective SUSY manifest itself at the LHC? To answer this, we first show in Fig. \[fig:xsecs14\] total gluino and third generation squark pair production cross sections at a 14 TeV $pp$ collider in the $m_0(3)\ vs.\
m_{1/2}$ plane displayed in Fig. \[fig:plane\], but now with $m_0(1,2)=20$ TeV. In frame [*a*]{}), the $pp\to\tg\tg X$ reaction (where $X$ denotes assorted hadronic debris) reaches to over $10^4$ fb for gluino masses as low as $m_{\tg}\alt 500$ GeV. The gluino pair cross section drops continuously as $m_{1/2}$, or alternatively $m_{\tg}$, increases.
The ESUSY region with third generation squarks having masses $\alt$TeV lies adjacent to the boundary of the gray tachyonic region. In frame [*b*]{}), we see the $pp\to\tst_i\bar{\tst}_i$ plus $\tb_i\bar{\tb}_i$ (for $i=1-2$) summed cross sections. These are typically in the 1-100 fb range all along the tachyonic boundary, with portions reaching cross sections as high as $10^3$ fb. Thus, the low $m_{1/2}$ region will consist of mainly gluino pair production, where the main influence of light third generation squarks will be upon the gluino branching fractions. As we move to higher $m_{1/2}$ values, the gluino pair cross section will diminish, and the total SUSY production cross section will be dominated by pair production of third generation squarks.
Unless the gluino is very heavy we would expect that, especially after selection cuts, LHC phenomenology would largely be determined by gluino pair production. SUSY event topologies would thus be sensitive to gluino decay patterns. In Fig. \[fig:gluinodec\], we show the gluino branching fractions for the MCMC scan points with $m_{\tg}\le 1$ TeV. Of course, when gluino two-body decays are kinematically accessible, these would dominate. We see from the figure that these are accessible for relatively few points when $m_{\tg}< 500$ GeV. We will return to this later when we discuss how one might distinguish the ESUSY model with light gluinos from the model with $t$-$b$-$\tau$ Yukawa unification where $m_{\tg}$ is bounded from above [@yuk]. For the bulk of the scanned points, the gluino decays via three body modes into top and bottom quarks plus EW gauginos; corresponding decays to the first two generations are suppressed because these squarks are very heavy. Bottom-jet tagging will clearly provide an effective way for enriching the ESUSY event sample at the LHC [@btag; @so10lhc1]. As in many SUSY models, direct decays to the neutralino LMP never have a very large branching fraction. The fact that gluino decays have large branching fractions to $\tw_1,\,\tz_2$ and top quarks implies that the ESUSY gluino event sample will include multi-lepton events from gluino decay cascades. Finally, we see from the last frame that while the branching fraction for the radiative gluino decays [@radglu] $\tg\to g\tz_i$ is usually small, it can reach 40–50%. We have checked that the neutralino in question is mostly $\tz_{3,4}$. This occurs when we have large splittings among third generation squarks with stops being lighter than sbottoms, when a light neutralino has a significant up-higgsino component and the decay $\tg\to
t\bar{t}\tz_1$ is kinematically forbidden. The branching fraction for decays to light quarks is very small for $m_0(1,2)>5$ TeV, although larger values are possible with the second $L_{\widetilde M}$ prior (case 2. in Sec. \[ssec:scans\]), which disfavours lower values of $m_0(1,2)$ but does not exclude them.
![*Results for the branching fractions for gluino decays from MCMC scan points with $m_{\tg}\le 1$ TeV versus the gluino mass. The upper two frames show the branching fractions for three-body decays of gluinos to third generation quarks plus EW gauginos. The bottom-left frame shows the branching fraction for two-body decays to third generation quarks and squarks, while the bottom-right frame illustrates that the radiative decays $\tg\to g\tz_i$ can sometimes be substantial. All results are for the $L_{\widetilde M}= 1$ case with 5 TeV $< m_0(1,2)<$ 20 TeV.*[]{data-label="fig:gluinodec"}](BFgluino2a.ps "fig:"){height="8cm" width="5.6cm"} ![*Results for the branching fractions for gluino decays from MCMC scan points with $m_{\tg}\le 1$ TeV versus the gluino mass. The upper two frames show the branching fractions for three-body decays of gluinos to third generation quarks plus EW gauginos. The bottom-left frame shows the branching fraction for two-body decays to third generation quarks and squarks, while the bottom-right frame illustrates that the radiative decays $\tg\to g\tz_i$ can sometimes be substantial. All results are for the $L_{\widetilde M}= 1$ case with 5 TeV $< m_0(1,2)<$ 20 TeV.*[]{data-label="fig:gluinodec"}](BFgluino2b.ps "fig:"){height="8cm" width="5.6cm"} ![*Results for the branching fractions for gluino decays from MCMC scan points with $m_{\tg}\le 1$ TeV versus the gluino mass. The upper two frames show the branching fractions for three-body decays of gluinos to third generation quarks plus EW gauginos. The bottom-left frame shows the branching fraction for two-body decays to third generation quarks and squarks, while the bottom-right frame illustrates that the radiative decays $\tg\to g\tz_i$ can sometimes be substantial. All results are for the $L_{\widetilde M}= 1$ case with 5 TeV $< m_0(1,2)<$ 20 TeV.*[]{data-label="fig:gluinodec"}](BFgluino2c.ps "fig:"){height="8cm" width="5.6cm"} ![*Results for the branching fractions for gluino decays from MCMC scan points with $m_{\tg}\le 1$ TeV versus the gluino mass. The upper two frames show the branching fractions for three-body decays of gluinos to third generation quarks plus EW gauginos. The bottom-left frame shows the branching fraction for two-body decays to third generation quarks and squarks, while the bottom-right frame illustrates that the radiative decays $\tg\to g\tz_i$ can sometimes be substantial. All results are for the $L_{\widetilde M}= 1$ case with 5 TeV $< m_0(1,2)<$ 20 TeV.*[]{data-label="fig:gluinodec"}](BFgluino2d.ps "fig:"){height="8cm" width="5.6cm"}
Light gluinos with $\tz_1$ as LMP (ES1)
---------------------------------------
The benchmark point ES1 features a rather light gluino with mass $m_{\tg}=524$ GeV, correspondingly light charginos and neutralinos, along with three sub-TeV third generation squarks. From Table \[tab:bm\], we see that total sparticle production cross section, summed over all reactions, is 23.2 pb. Of this, 62.8% comes from $pp\to\tg\tg X$ production, while 36.6% comes from EW gaugino production (sum of all chargino and neutralino production cross sections). Only a tiny fraction of the production cross section comes from third generation squarks, so the rate for $\tst_i\bar{\tst}_i$ and $\tb_i\bar{\tb}_i$ production (for $i=1,2$) is very small.
The gluino decays via three body modes as $\tg\to b\bar{b}\tz_2$ (37%), $\tg\to t\bar{t}\tz_1$ (19%) and $\tg\to t\bar{b}\tw_1+ c.c$ (41%). Thus, each gluino pair producton event will contain at least four $b$-jets, plus other jets, isolated leptons (from top quarks and EW gauginos) and $\eslt$. The $\tw_1$ decays via three-body modes $\tw_1\to\tz_1
f\bar{f}'$ in accord with $W^*$ propagator dominance ([*e.g.*]{} $\tw_1\to\tz_1 e\bar{\nu}_e$ at 11%). The $\tz_2$ decays to $b\bar{b}\tz_1$ 20% of the time, while leptonic decays such as $\tz_2\to\tz_1 e^+e^-$ occur at the 3% level. In this case, the $\tb_1$ is quite light and dominantly $\tb_L$, which enhances the $\tz_2$ decay to $b$ quarks at the expense of first and second generation fermions. Clean trilepton signals from $\tw_1\tz_2$ production may also be observable [@trilep_lhc].
The ES1 scenario will have much the same LHC phenomenology as the Yukawa-unified SUSY with a 500 GeV gluino [@so10lhc1], which is also dominated by gluino pair production followed by three-body gluino decays into mainly $b$-quarks. A natural question to ask is whether it is possible to distinguish these scenarios at the LHC. One difference is that Yukawa-unified SUSY requires a very large $A_0$ parameter with $A_0\sim -2m_{16}$, where $m_{16}$ is the matter GUT scale scalar mass parameter, which lies typically in the range 8–20 TeV. This large value of $A_0$ feeds into gaugino mass evolution at two-loops in the RGEs, and suppresses $m_{\tg}$ with respect to $m_{\tz_1}$ and $m_{\tz_2,\tw_1}$.
To illustrate this, we show in Fig. \[fig:diff\] a scatter plot of $m_{\tg}$ versus $m_{\tz_2}-m_{\tz_1}$ obtained by scanning the parameter space of (a) the ESUSY model (red pluses) and (b) the Yukawa-unified (YU) model with “just-so” GUT scale Higgs soft mass splitting (HS, dark blue stars) and the YU model with full $D$-term splitting, right-hand neutrinos and 3rd generation scalar splitting (the DR3 model [@dr3], light blue squares). Here, we require in YU models that $R$, the ratio of the largest to the smallest of the GUT scale Yukawa couplings, is smaller than $1.05$. While there is significant overlap of the two classes of Yukawa-unified models, the ESUSY model lies on a distinct band. We expect that the neutralino mass difference edge will be rather well determined by the $m(\ell^+\ell^-)$ distribution. Then, even a crude measure of the gluino mass at the 10–20% level via $M_{eff}$ [@meff] or total cross section [@gabe] would suffice for this distinction. Also, in Ref. [@so10lhc1], it is shown that the mass difference $m_{\tg}-m_{\tz_2}$ might be extracted via the kinematic edge in the $m(bb)$ distribution.[^5] If $m_{\tg}$ can be determined even more precisely using $m_{T2}$ [@mt2], its generalizations [@mtgen], or other methods that have recently been suggested, the distinction between the models would be even sharper.
Light top and bottom squarks, heavy gluinos and $\tz_1$ as LMP (ES2 & ES3)
--------------------------------------------------------------------------
For point ES2, since $m_{1/2}$ is large, the value of $m_{\tg}$ is also large, and so $\tg\tg$ will not be produced at an appreciable rate at LHC. Instead, the total SUSY cross section of $\sigma_{SUSY}=157.8$ fb is dominated by 78.3% $\tst_1\bar{\tst}_1$ production, and 11.5% $\tb_1\bar{\tb}_1$ production. A few percent of $\tst_2\bar{\tst}_2$ also contribute. The $\tst_1\to bW\tz_1$ decay occurs at nearly 100% branching fraction, so the $\tst_1\bar{\tst}_1$ production results in a $b\bar{b}W^+W^- +\eslt$ final state. This should be separable from $t\bar{t}$ background if large $\eslt$ or $M_{eff}$ is required. Another possible background is $Zt\bar{t}$ production. Because gluinos are heavy, SUSY contamination to the signal [@kadala] is not an issue. The $\tb_1\to
W\tst_1$ decays at nearly 100% branching fraction, so this component of the production cross section will result in a $b\bar{b}W^+W^-W^+W^- +\eslt$ final state.
The case of ES3 has four sub-TeV third generation squarks plus tau-sleptons near a TeV or below. The total production cross section of $\sigma_{SUSY}=1618$ fb comes from 87.6% $\tst_1\bar{\tst}_1$ production, 9.1% $\tb_1\bar{\tb}_1$ production plus a few percent of heavier top and bottom squark pairs. The visible decay products from $\tst_1$ decay will be soft since there is only a 26 GeV mass gap between the $\tst_1$ and the $\tz_1$; the visible decay products of $\tst_1$ may not reliably detectable unless they are boosted. The $\tb_1\to W^-\tst_1$ at nearly 100% branching fraction, so from $\tb_1\bar{\tb}_1$ production we expect a $W^+W^- +\eslt$ final state, accompanied by soft charm jets or other soft hadronic debris. The possible background comes from $W^+W^-Z$ production.
Before turning to the next benchmark case we remark that, although we have not examined an example of such a benchmark point, it has been shown [@nojpop] that if the gluinos and stops both have sub-TeV masses and that the decays $\tg\to t\tst_1 \to tb\tw_1$ and/or $\tg \to
b\tb_1 \to bt\tw_1$ are kinematically accessible and occur with large branching fractions, a partial reconstruction of the event using techniques to tag the secondary top quark may be possible. Moreover, in [@Kraml:2005kb] it has been shown that if $\tg\to t\tst_1 \to tc\tz_1$ dominates, the signature of 2 b-jets plus 2 same-sign leptons plus additional jets and missing energy is an excellent discovery channel for gluino masses up to about 900 GeV.
Bottom or top squarks as LMP: the case of heavy quasi-stable colored particles (ES4)
------------------------------------------------------------------------------------
Benchmark point ES4 illustrates an ESUSY model with $\tst_1$ and $\tb_1$ lighter than $\tz_1$, and in fact the bottom squark $\tb_1$ is the LMP. This is viable if the axino is the true LSP so that $\tb_1\to b\ta$, and the dark matter consists of an axion/axino admixture. For case ES4, with a PQ breaking scale of order $f_a \sim 10^{11}$ GeV, we find a bottom squark lifetime of $\sim 10^{-6}$ sec, [*i.e.*]{} the $\tb_1$ is stable as far as collider searches go, but it decays well before the start of Big Bang nucleosynthesis. Previous searches for quasi-stable squark hadrons $Q$ at ALEPH find that $m_Q>95\ (92)$ GeV for up-type (down-type) squark hadrons [@aleph]. Searches by CDF evidently require $m_Q\agt 241.5$ GeV [@cdf; @raklev]. At the LHC, for case ES4, sparticles will be produced with total cross section $\sigma_{SUSY}\simeq 1.2\times 10^4$ fb, of which 71.4% is $\tb_1\bar{\tb}_1$ production, and 28.2% is $\tst_1\bar{\tst}_1$ production. The $\tst_1\to \tb_1 W$ branching fraction is essentially 100%, and so augments the $\tb_1$ production rates. The $\tb_1$ is a quasi-stable colored particle, and will hadronize to form an $R$-meson or baryon $B^{\pm ,0}=\tb_1 \bar{q}$ or $\tb_1 qq'$, where $q$ is usually $u$ or $d$. The properties of squark hadrons have been reviewed in [@fairbairn; @raklev]. The heavy $B$ hadron will be produced and propagate through the detector with minimal energy loss due to hadronic interactions. It can be of charge $\pm 1$ or $0$, and in fact as it propagates, it has charge exchange reactions with nuclei in the detector material, so that the path of the $B$-hadron can thus be intermittently charged or neutral [@dt; @bcg].
Stable squark hadrons can be detected as muon-like events, albeit with the possibility of intermittently appearing and disappearing tracks. Detection can be made using $dE/dx$ measurments in the tracking system, or time-of-flight (ToF) measurements in the muon system. Measuring both the $B$ hadron momentum and velocity should allow a $B$ mass measurement to about 1-2 GeV [@raklev]. The reach of LHC for quasi-stable squark hadrons has been estimated in Ref. [@raklev]. Using those results, we estimate the LHC reach with $\sqrt{s}=7$ TeV and 1 fb$^{-1}$ of integrated luminosity to be up to $m_{\tb_1} \sim 315$ GeV, which would already test point ES4! The LHC reach with $\sqrt{s}=14$ TeV and 100 fb$^{-1}$ is estimated as $m_{\tb_1}\sim 800$ GeV, from just $\tb_1\bar{\tb}_1$ production alone. Additional contributions to $\tb_1$ production from cascade decays would increase the reach, and also make clear that $\tb_1$-pair production is not the only new physics process occuring at the LHC.
We note also that many cases with a $\tst_1$ LMP can also be generated in ESUSY. In these cases, with a quasi-stable top-squark hadron, the lifetime and reach discussion is qualitatively similar to that given above.
Staus as LMP: the case of heavy quasi-stable charged particles (ES5)
--------------------------------------------------------------------
For ESUSY benchmark point ES5, we have $\ttau_1$ as LMP. As with point ES4, we may again invoke a lighter axino to escape constraints on charged stable exotics and also mixed axion/axino CDM. In this case, the stau decays through loop diagrams via $\ttau_1\to \tau\ta$ with a lifetime typically of order 1 sec for $f_a\sim 10^{11}$ GeV, and does not significantly disrupt BBN [@freitas]. Searches for quasi-stable charged particles at OPAL require $m_{\ttau_1}\agt 98.5$ GeV [@opal].
The quasi-stable stau will propagate slowly though LHC detectors much like a heavy muon, and leave a highly-ionizing track [@dt]. The $dE/dx$, ToF and track bending measurements should allow for momentum and velocity, and hence mass determinations.
For case ES5, the total SUSY production cross section is $\sigma_{SUSY}\simeq 51.4$ fb. Of this, just about a quarter comes from stau pair production. However, since all produced sparticles now cascade down into the stau LMP, the total $\ttau_1$ production is augmented, in this case, by a factor four. For instance, $\tst_1\bar{\tst}_1$ is produced at 26.5% rate, and $\tst_1\to \tz_1 t$ at 53.3%, $\tst_1\to t\tz_2$ at 7.4% and $\tst_1\to b\tw_1$ at 39.3%. Then, $\tz_1\to
\tau^\pm\ttau_1^\mp$ at 100% branching fraction. Also, $\tw_1\to
\ttau_1\nu_\tau$ at 100%, and $\tz_2\to \tau^\pm\ttau_2^\mp$ at 42% and $\tnu_\tau\bar{\nu}_\tau +c.c$ at 49%, while $\tnu_\tau\to
\tz_1\nu_\tau$, followed by further $\tz_1$ decay. Thus, we expect the cascade decay events to each contain at least two $\ttau_1$ tracks, frequently in company of an assortment of $\tau$-jets, together with $b$ quarks and leptons from $W$-bosons. If the energy of the LMP (as measured from the velocity and momentum of the track) can be included, $\eslt$ in such SUSY events arises only from neutrinos, and so is not especially large.
The LHC reach for quasi-stable staus has been estimated in [@raklev] in the case where only stau pair production contributes. The reach for LHC at $\sqrt{s}=14$ TeV with 100 fb$^{-1}$ of integrated luminosity is found to be $m_{\tau_1}\sim 250$ GeV. This is conservative for the present case, since $\ttau_1$ production will be augmented by a factor of four from production via cascade decays of heavier sparticles. Moreover, as in the ES4 case, detection of more complicated events other than just stau pair production will bring home the richness of the new physics being detected.
Summary and Conclusions {#sec:conclude}
=======================
Effective SUSY has been suggested as a model for ameliorating the flavour and $CP$ problems that are endemic to generic SUSY models, while maintaining naturalness in the EWSB sector. The general idea is that SUSY states that have large couplings to the Higgs sector — third generation sfermions and EW gauginos — have masses at or below the TeV scale, while first and second generation sparticles that directly affect the most stringently constrained flavour-changing processes are heavy. The typical ESUSY spectrum consists of multi-TeV first and second generation sfermions along with EW gauginos, Higgs bosons and third generation sfermions at or below the TeV scale; gluinos may be as light as 450 GeV or as heavy as several TeV.
In this paper, we have examined the LHC phenomenology of ESUSY models. Toward this end, we set up the ESUSY model with parameters defined at the GUT scale in Sec. \[sec:intro\]. For simplicity of incorporting flavour constraints without greatly impacting LHC physics, we have assumed GUT scale degeneracy of the SSB mass parameters of the first two generations, but allowed the corresponding parameters for the third generation and Higgs scalars to be independent.
We have delineated the viable parameter space in Sec. \[sec:pspace\]. In Sec. \[ssec:scans\], we have described the MCMC setup that was used for finding the portion of the entire parameter space consistent with various low energy constraints as well as lower limits on sparticle and Higgs boson masses along with theory priors that were used to obtain ESUSY spectra. Our results for the favoured ranges of the input parameters, assuming that the LMP is a neutralino (this is true for the bulk of the points), are shown by the posterior probability distributions in Fig. \[fig:mcmc1\]. The corresponding distributions for sparticle masses, illustrated in Fig. \[fig:mcmc2\], indeed show the expected qualitative features of the ESUSY spectrum: $\alt$TeV third generation and EW gaugino masses, 10-20 TeV first/second generation squarks (and sleptons, that we did not show), and gluinos ranging from 0.5-4 TeV. The posterior probability distributions for various low energy observables and the thermal neutralino relic density are shown in Fig. \[fig:mcmc3\]. While the former generally lie close to their SM values (because of the nature of the ESUSY spectra), the neutralino relic density $\Omega h^2$ ranges from below 0.01 to $10^3$ with a peak value around $1-10$.
We note that models with large values of $\Omega_{\tz_1} h^2$ are phenomenologically every bit as viable as models with a thermal neutralino LSP provided the LMP neutralino is not the true LSP, but decays into the LSP before the advent of Big Bang nucleosynthesis. The axino of the axion-axino multiplet which is present in SUSY models with the PQWW solution to the strong $CP$ problem is an excellent example of an LSP that is not the LMP, and indeed models where the measured CDM is comprised of axions and axinos have been studied in the literature. Moreover, if the axino is the LSP, models where the LMP is charged or coloured are also allowed provided again that the LMP decays before nucleosynthesis.
Our main results on the LHC phenomenology of ESUSY models were presented in Sec. \[sec:lhc\]. The most important sparticle production mechansims at the LHC are gluino pair production for values of $m_{1/2}\alt
700$ GeV, and third generation squark pair production. A very high $b$-jet multiplicity is the hallmark of multi-jet+multilepton+$\eslt$ events within the ESUSY framework. Gluinos mostly decay via (real or virtual) third generation squarks (see Fig. \[fig:gluinodec\]) to third generation quarks and EW gauginos. While chargino decay branching fractions tend to follow those of the $W$-boson, the branching fraction for neutralinos to $b$ quarks is often enhanced in these scenarios. Third generation squarks frequently also have $b$-quarks as their decay products. Finally, ESUSY events are also often rich in $t$ quarks whose decays contribute to the multi-lepton component of the signal.
We have discussed the phenomenology of five ESUSY benchmark points introduced in Sec. \[ssec:BM\]. Of these, just the point ES3 has a thermal neutralino relic density in accord with the measured cold DM relic density, although here the $\tz_1$ would just make up about $1/3$ of the DM. For the other points, the LMP (which need not even be a neutralino) must decay into the true LSP (which might be the axino) plus SM particles. If the LMP is a neutralino, this decay typically occurs well after the neutralino has passed through the LHC detectors, and the LHC phenomenology is essentially the same as for a stable neutralino. The LMP may also be a coloured squark or an electrically charged stau, as illustrated by the last two benchmark points that we discuss. Assuming again that the axino is the LSP, the LMP is quasi-stable and typically traverses the detector before it decays. In these cases, the presence of a (possibly intermittent, in the case of squark LMP) track of a slowly moving particle (whose velocity is obtained through timing information), rather than $\eslt$, will be the hallmark of SUSY events.
To conclude, effective supersymmetry with third generation squarks and EW gauginos at or below the TeV scale, but 10–20 TeV first/second generation sfermions ameliorates the flavour and $CP$ problems that plague many SUSY models. We have examined the LHC phenomenology of relic-density-consistent ESUSY scenarios and shown that it may differ qualitatively from that in the most frequently examined mSUGRA scenario. The ESUSY picture may also be tested in low energy measurements. In particular, if the branching fraction for $B\to
\tau\nu$ decay, and especially the muon magnetic moment show significant deviation from their SM values, the ESUSY picture would be strongly disfavoured.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A. Pukhov for his help in implementing the [Isajet]{} – [micrOMEGAs]{} interface for SUSY scenarios with non-universal scalars. XT thanks the UW IceCube collaboration for making his visit to the University of Wisconsin where this work was done, possible. We also thank the Galileo Galilei Institute (GGI), Florence, Italy for hospitality at the time of the Workshop on Dark Matter: Its origin, nature and prospects for detection during which many of the ideas presented in this paper came together. This work was funded in part by the US Department of Energy, and by the French ANR project ToolsDMColl, BLAN07-2-194882.
[99]{} S. Dimopoulos and H. Georgi, [[*Nucl. Phys. *]{}[**B 193**]{} (1981) 150]{}. N. Sakai, [*Z. Phys. C*]{}, [**11**]{} (1981) 153. Y. Nir and N. Seiberg, [[*Phys. Lett. *]{}[**B 309**]{} (1993) 337]{}. M. Dine, A. Kagan and S. Samuel, [[*Phys. Lett. *]{}[**B 243**]{} (1990) 250]{}. N. Arkani-Hamed and H. Murayama, [[*Phys. Rev. *]{}[**D 56**]{} (1997) R6733]{}. M. Drees [[*Phys. Rev. *]{}[**D 33**]{} (1986) 1468]{}. S. Dimopoulos and G. Giudice, [[*Phys. Lett. *]{}[**B 357**]{} (1995) 573]{}; A. Pomarol and D. Tomassini, [[*Nucl. Phys. *]{}[**B 466**]{} (1996) 3]{}. A. Cohen, D. B. Kaplan and A. Nelson, [[*Phys. Lett. *]{}[**B 388**]{} (1996) 588]{}. For an update on $(g-2)_\mu$, see T. Teubner, K. Hagiwara, R. Liao, A.D. Martn and D. Nomura, arXiv:1001.5401. N. Jarosik [*et al.*]{}, arXiv:1001.4744 \[astro-ph.CO\]; E. Komatsu [*et al.*]{}, arXiv:1001.4538 \[astro-ph.CO\]. R. Peccei and H. Quinn, [[*Phys. Rev. Lett. *]{}[**38**]{} (1977) 1440]{} and [[*Phys. Rev. *]{}[**D 16**]{} (1977) 1791]{}; S. Weinberg, [[*Phys. Rev. Lett. *]{}[**40**]{} (1978) 223]{}; F. Wilczek, [[*Phys. Rev. Lett. *]{}[**40**]{} (1978) 279]{}. H. Baer, A. Box and H. Summy, [[*J. High Energy Phys. *]{}[**0908**]{} (2009) 080]{}; H. Baer and A. Box, arXiv:0910.0333 [*Eur. Phys. J.* ]{}, in press; H. Baer, A. Box and H. Summy, arXiv:1005.2215 (2010). A. Cohen, D. B. Kaplan, F. Lepeintre and A. Nelson, [[*Phys. Rev. Lett. *]{}[**78**]{} (1997) 2300]{}. K. Agashe and M. Grasser, [[*Phys. Rev. *]{}[**D 59**]{} (1999) 015007]{}; J. Hisano, K. Kurosawa and Y. Nomura, [[*Nucl. Phys. *]{}[**B 584**]{} (2000) 3]{}. J. Bagger, J. Feng, N. Polonsky and R. J. Zhang, [[*Phys. Lett. *]{}[**B 473**]{} (2000) 264]{}. H. Baer, P. Mercadant and X. Tata, [[*Phys. Lett. *]{}[**B 475**]{} (2000) 289]{}; H. Baer, C. Balazs, M. Brhlik, P. Mercadante, X. Tata and Y. Wang, [[*Phys. Rev. *]{}[**D 64**]{} (2001) 015002]{}. H. Baer and J. Ferrandis, [[*Phys. Rev. Lett. *]{}[**87**]{} (2001) 211803]{}; D. Auto, H. Baer, C. Balazs, A. Belyaev, J. Ferrandis and X. Tata, [[*J. High Energy Phys. *]{}[**0306**]{} (2003) 023]{}; H. Baer, S. Kraml, S. Sekmen and H. Summy, [[*J. High Energy Phys. *]{}[**0803**]{} (2008) 056]{}; H. Baer, S. Kraml and S. Sekmen, [[*J. High Energy Phys. *]{}[**0909**]{} (2009) 005]{}. T. Blazek, R. Dermisek and S. Raby, [[*Phys. Rev. Lett. *]{}[**88**]{} (2002) 111804]{}; T. Blazek, R. Dermisek and S. Raby, [[*Phys. Rev. *]{}[**D 65**]{} (2002) 115004]{}; R. Dermisek, S. Raby, L. Roszkowski and R. Ruiz de Austri, [[*J. High Energy Phys. *]{}[**0304**]{} (2003) 037]{}; R. Dermisek, S. Raby, L. Roszkowski and R. Ruiz de Austri, [[*J. High Energy Phys. *]{}[**0509**]{} (2005) 029]{}; W. Altmannshofer, D. Guadagnoli, S. Raby and D. Straub, [[*Phys. Lett. *]{}[**B 668**]{} (2008) 385]{}. H. Baer, C. Balazs, P. Mercadante, X. Tata and Y. Wang, [[*Phys. Rev. *]{}[**D 63**]{} (2001) 015011]{}. N. Desai and B. Mukhopadhyaya, [[*Phys. Rev. *]{}[**D 80**]{} (2009) 055019]{}; S. G. Kim, N. Maekawa, K. Nagao, M. Nojiri and K. Sakurai, [[*J. High Energy Phys. *]{}[**0910**]{} (2009) 005]{}; N. Bernal, A. Djouadi and P. Slavich, [[*J. High Energy Phys. *]{}[**0707**]{} (2007) 016]{}; N. Bernal, [[*JCAP*]{}[**0908**]{} (2009) 022]{}; M. J. White and F. Feroz, arXiv:1002.1922. K. L. Chan, U. Chattopadhyay and P. Nath, [[*Phys. Rev. *]{}[**D 58**]{} (1998) 096004]{}; J. Feng, K. Matchev and T. Moroi, [[*Phys. Rev. Lett. *]{}[**84**]{} (2000) 2322]{} and [[*Phys. Rev. *]{}[**D 61**]{} (2000) 075005]{}; see also H. Baer, C. H. Chen, F. Paige and X. Tata, [[*Phys. Rev. *]{}[**D 52**]{} (1995) 2746]{} and [[*Phys. Rev. *]{}[**D 53**]{} (1996) 6241]{}; H. Baer, C. H. Chen, M. Drees, F. Paige and X. Tata, [[*Phys. Rev. *]{}[**D 59**]{} (1999) 055014]{}; for a model-independent approach, see H. Baer, T. Krupovnickas, S. Profumo and P. Ullio, [[*J. High Energy Phys. *]{}[**0510**]{} (2005) 020]{}. H. Baer and X. Tata, [*Weak Scale Supersymmetry: From Superfields to Scattering Events*]{}, (Cambridge University Press, 2006). H. Baer, F. Paige, S. Protopopescu and X. Tata, [hep-ph/0312045]{}; see also H. Baer, J. Ferrandis, S. Kraml and W. Porod, [[*Phys. Rev. *]{}[**D 73**]{} (2006) 015010]{}. The Isajet manual is available at [http://www.nhn.ou.edu/ isajet/]{}. See H. Baer, A. Mustafayev, S. Profumo, A. Belyaev and X. Tata, [[*Phys. Rev. *]{}[**D 71**]{} (2005) 095008]{} and [[*J. High Energy Phys. *]{}[**0507**]{} (2005) 065]{}, and references therein. C. Böhm, A. Djouadi and M. Drees, [[*Phys. Rev. *]{}[**D 30**]{} (2000) 035012]{}; J. R. Ellis, K. A. Olive and Y. Santoso, [[*Astropart. Phys. *]{}[**18**]{} (2003) 395]{}; J. Edsjö, [*et al.*]{}, JCAP [**0304**]{} (2003) 001. M. Drees and M. Nojiri, [[*Phys. Rev. *]{}[**D 47**]{} (1993) 376]{}; H. Baer and M. Brhlik, [[*Phys. Rev. *]{}[**D 57**]{} (1998) 567]{}; H. Baer, M. Brhlik, M. Diaz, J. Ferrandis, P. Mercadante, P. Quintana and X. Tata, [[*Phys. Rev. *]{}[**D 63**]{} (2001) 015007]{}; J. Ellis, T. Falk, G. Ganis, K. Olive and M. Srednicki, [[*Phys. Lett. *]{}[**B 510**]{} (2001) 236]{}; L. Roszkowski, R. Ruiz de Austri and T. Nihei, [[*J. High Energy Phys. *]{}[**0108**]{} (2001) 024]{}; A. Djouadi, M. Drees and J. L. Kneur, [[*J. High Energy Phys. *]{}[**0108**]{} (2001) 055]{}; A. Lahanas and V. Spanos, [[*Eur. Phys. J.* ]{}[**C 23**]{} (2002) 185]{}. G. Belanger, F. Boudjema, A. Pukhov and R. K. Singh, JHEP [**0911**]{}, 026 (2009) \[arXiv:0906.5048 \[hep-ph\]\]. R. R. de Austri, R. Trotta and L. Roszkowski, JHEP [**0605**]{}, 002 (2006) \[arXiv:hep-ph/0602028\]. B. C. Allanach, K. Cranmer, C. G. Lester and A. M. Weber, JHEP [**0708**]{}, 023 (2007) \[arXiv:0705.0487 \[hep-ph\]\]. R. Trotta, F. Feroz, M. P. Hobson, L. Roszkowski and R. Ruiz de Austri, JHEP [**0812**]{}, 024 (2008) \[arXiv:0809.3792 \[hep-ph\]\]. F. Mahmoudi, Comput. Phys. Commun. [**180**]{}, 1579 (2009) \[arXiv:0808.3144 \[hep-ph\]\]. G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. [**176**]{}, 367 (2007) \[arXiv:hep-ph/0607059\]. O. Buchmueller [*et al.*]{}, Eur. Phys. J. C [**64**]{}, 391 (2009) \[arXiv:0907.5568 \[hep-ph\]\]. E. Barberio [*et al.*]{} \[Heavy Flavor Averaging Group\], arXiv:0808.1297 \[hep-ex\]. M. Misiak [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 022002 (2007) \[arXiv:hep-ph/0609232\]. T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**100**]{}, 101802 (2008) \[arXiv:0712.1708 \[hep-ex\]\]. Tevatron Electroweak Working Group (CDF and D0 collaborations), arXiv:0903.2503 \[hep-ex\]. LEP2 SUSY Working Group (ALEPH, DELPHI, L3 and OPAL collaborations), [http://lepsusy.web.cern.ch/lepsusy/]{} T. E. W. Group \[CDF Collaboration and D0 Collaboration\], arXiv:0908.2171 \[hep-ex\]. H. Baer and H. Summy, Phys. Lett. B [**666**]{}, 5 (2008) \[arXiv:0803.0510 \[hep-ph\]\]. A. Box and X. Tata, [[*Phys. Rev. *]{}[**D 79**]{} (2009) 035004]{}. L. Covi, J. E. Kim and L. Roszkowski, [[*Phys. Rev. Lett. *]{}[**82**]{} (1999) 4180]{}; L. Covi, H. B. Kim, J. E. Kim and L. Roszkowski, [[*J. High Energy Phys. *]{}[**0105**]{} (2001) 033]{}. A. Brandenburg [*et al.*]{} [[*Phys. Lett. *]{}[**B 617**]{} (2005) 99]{}.
L. Covi, L. Roszkowski and M. Small, [[*J. High Energy Phys. *]{}[**0207**]{} (2002) 023]{}. K. Jedamzik, [[*Phys. Rev. *]{}[**D 74**]{} (2006) 103509]{}. A. Freitas, F. Steffen, N. Tajuddin and D. Wyler, [[*Phys. Lett. *]{}[**B 682**]{} (2009) 193]{}. C. Berger, L. Covi, S. Kraml and F. Palorini, [[*JCAP*]{}[**0810**]{} (2008) 005]{}. U. Chattopadhyay, A. Datta, A. Datta, A. Datta, and D. P. Roy, [[*Phys. Lett. *]{}[**B 493**]{} (2000) 127]{}; P. Mercadante, J. K. Mizukoshi and X. Tata, [[*Phys. Rev. *]{}[**D 72**]{} (2005) 035009]{}; S. P. Das, A. Datta, M. Guchait, M. Maity and S. Mukherjee, [[*Eur. Phys. J.* ]{}[**C 54**]{} (2008) 645]{}; R. Kadala J. K. Mizukoshi, P. Mercadante and X. Tata, [[*Eur. Phys. J.* ]{}[**C 56**]{} (2008) 511]{}.
H. Baer, S. Kraml, S. Sekmen and H. Summy, [[*J. High Energy Phys. *]{}[**0810**]{} (2008) 079]{}; H. Baer, S. Kraml, A. Lessa and S. Sekmen, [[*J. High Energy Phys. *]{}[**1002**]{} (2010) 055]{}. H. Baer, X. Tata and J. Woodside, [[*Phys. Rev. *]{}[**D 42**]{} (1990) 1568]{}. H. Baer, C. H. Chen, F. Paige and X. Tata, [[*Phys. Rev. *]{}[**D 50**]{} (1994) 4508]{}. H. Baer, S. Kraml, S. Sekmen. [[*J. High Energy Phys. *]{}[**0909**]{} (2009) 005]{} I. Hinchliffe [*et al.*]{}[[*Phys. Rev. *]{}[**D 55**]{} (1997) 5520]{} and [[*Phys. Rev. *]{}[**D 60**]{} (1999) 095002]{}; H. Bachacou, I. Hinchliffe and F. Paige, [[*Phys. Rev. *]{}[**D 62**]{} (2000) 015009]{}. H. Baer, V. Barger, G. Shaughnessy, H. Summy and L. T. Wang, [[*Phys. Rev. *]{}[**D 75**]{} (2007) 095010]{}. C. Lester and D. Summers, [[*Phys. Lett. *]{}[**B 463**]{} (1999) 99]{}; A. Barr, C. Lester and D. Summers, J. Phys. [**G29**]{} (2003) 2343; W. S. Cho, K. Choi, Y. Kim and C. Park, [[*Phys. Rev. Lett. *]{}[**100**]{} (2008) 171801]{} and [[*J. High Energy Phys. *]{}[**0802**]{} (2008) 035]{}; H-C. Cheng and Z. Han, [[*J. High Energy Phys. *]{}[**0812**]{} (2008) 063]{}. C. Lester and a Barr, [[*J. High Energy Phys. *]{}[**0712**]{} (2007) 102]{}; B. Gripaios, [[*J. High Energy Phys. *]{}[**0802**]{} (2008) 053]{}; A. Barr, B. Gripaios and C. Lester, [[*J. High Energy Phys. *]{}[**0911**]{} (2009) 096]{}; K. Matchev, F. Moortgat, L. Pape and M. Park, [[*J. High Energy Phys. *]{}[**0908**]{} (2009) 104]{}. R. Kadala [*et al.*]{} Ref.[@btag]. J. Hisano, K. Kawagoe and M. Nojiri, [[*Phys. Rev. *]{}[**D 68**]{} (2003) 035007]{}; J. Hisano, K. Kawagoe, R. Kitano and M. Nojiri, [[*Phys. Rev. *]{}[**D 66**]{} (2002) 115004]{}. S. Kraml and A. R. Raklev, [[*Phys. Rev. *]{}[**D 73**]{} (2006) 075002]{}. A. Heister [*et al*]{} (ALEPH Collaboration), [[*Eur. Phys. J.* ]{}[**C 31**]{} (2003) 327]{}. T. Aaltonen [*et al.*]{} (CDF Collaboration), arXiv:0902.1266. A. Raklev, arXiv:0908.0315. M. Fairbairn [*et al.*]{}, [[*Phys. Rept. *]{}[**438**]{} (2007) 1]{}. M. Drees and X. Tata, [[*Phys. Lett. *]{}[**B 252**]{} (1990) 695]{}. H. Baer, K. Cheung and J. Gunion, [[*Phys. Rev. *]{}[**D 59**]{} (1999) 075002]{}. G. Abbiendi [*et al.*]{} (OPAL Collaboration), [[*Phys. Lett. *]{}[**B 572**]{} (2003) 8]{}.
[^1]: Minimal Supersymmetric model with Universal soft terms at the Gut scale and RAdiative electroweak symmetry breaking.
[^2]: While the gluino mass parameter can, in principle, be hierarchically larger, RGE effects due to a very heavy gluino would raise third generation squark mass parameters to high values. Gauge coupling unification also prefers that the gluino not be too heavy.
[^3]: With a common value of $m_0(3)$, the $\tnu_{\tau}$ LMP may be surprising since at the one-loop level Yukawa interactions drive $m_{\ttau_R}^2$ to smaller values than $m_{\ttau_L}^2$. In this case, however, two-loop effects due to heavy scalars are large, and for smaller values of $m_{1/2}$ can lead to $m_{\ttau_L}< m_{\ttau_R}$ and a sneutrino LMP because of the $D$-term. Since one-loop effects increase with $m_{1/2}$, ultimately the usual situation with stau LMP is obtained. Incidently, the compensation between the one and two-loop contributions implies that for certain parameters we can have a near equality of the left- and right-stau mass parameters, and so large stau mixing even though the tau Yukawa coupling is not particularly large.
[^4]: The shape of the posterior probability distribution of $m_{1/2}$, and $\tan\beta$ are likely the most sensitive to the $b\to s\gamma$ constraint, and hence to our assumption that there are no off-diagonal pieces in the GUT scale squark mass matrices.
[^5]: We have also checked that a scatter plot of $m_{\tg}-m_{\tz_2}$ vs, $m_{\tz_2}-m_{\tz_1}$ also serves to separate the ESUSY model from the Yukawa-unified SUSY scenarios, but do not show it here for brevity.
|
---
abstract: 'We extend Langton’s valuative criterion for families of coherent algebraic sheaves to a complex analytic set-up. As a consequence we derive a set of sufficient conditions for the compactness of a moduli space of semistable sheaves over a compact complex manifold. This applies also to some cases appearing in complex projective geometry not covered by previous results.'
address: |
Matei Toma\
Institut de Mathématiques Élie Cartan, Université de Lorraine\
B.P. 70239\
54506 Vandoeuvre-lès-Nancy Cedex\
France
author:
- Matei Toma
title: Properness criteria for families of coherent analytic sheaves
---
Introduction
============
There is a variety of situations when moduli spaces of semistable coherent sheaves over projective schemes are known to exist, see [@HL] for an extensive treatment of this topic. Such moduli spaces typically turn out to be projective. This is often proved by using a result of Langton who checked that Grothendieck’s valuative criterion for properness applies to families of semistable sheaves [@Langton], [@HL Theorem 2.B.1]. Sometimes even if the base variety $X$ is projective, in order to define stability one may be led to consider arbitrary real ample classes as polarizations, cf. [@GT17], [@GRT1], [@GRTsurvey], [@CampanaPaun]. However when the polarization is irrational Langton’s result doesn’t directly apply and its proof cannot be adapted in a straightforward way to such a case. When the base variety is compact complex analytic, one may still speak of semistability with respect to Gauduchon metrics. But even over compact Kähler manifolds an analogous result to Langton’s is not available and complex analytic moduli spaces for semistable sheaves have only been constructed in special cases.
It is the purpose of this paper to provide replacements of properness valuative criteria as in Langton’s result in a complex analytic set-up and to show how they may be applied to prove compactness of moduli spaces of semistable sheaves. For coherent sheaves over a compact complex manifold a fairly general semistability notion will be described in Section \[section:degrees\]. (We define semistability only for pure sheaves but we later give analogues of our statements in the non-pure case.) Then in Section \[section:one-dimensional-case\] we deal with families of semistable sheaves over one dimensional bases. In this context we prove Theorem \[thm:dimension one\] which already provides a solution to the case of irrational polarizations mentioned above. This theorem basically says that for a flat family of coherent sheaves with general semistable members over a curve one may replace the special members in such a way that the resulting family has only semistable fibres. When the base manifold is not projective however, such a result on one dimensional families is no longer sufficient. This is because the two current definitions of meromorphic mapppings do not coincide. Stoll’s definition which uses extensions along curves is weaker than Remmert’s which requires a factorization through a proper modification, [@Hirschowitz]. In Section \[section:arbitrary-dimension\] we prove Theorem \[thm:dimension>1\] which provides a replacement for Langton’s valuative criterion also for families over higher dimensional bases, albeit under a more restrictive semistabilty condition. Using this result we give in Section \[section:application\] sufficient conditions for a moduli space of semistable sheaves over a compact complex manifold to be compact.
For Theorem \[thm:dimension one\] we follow Langton’s original line of proof with two new inputs, one of combinatorial nature which compensates the lack of rationality of the polarizations involved, and one application of Artin approximation which avoids non-properness issues of the relative Douady space. Theorems \[thm:dimension>1\] and \[thm:application\] are essentially new and they are especially pertinent to the complex analytic context. Their proofs depend on a notion of boundedness for sets of isomorphy classes of coherent analytic sheaves, which was introduced in [@Toma16].
[*Acknowledgements:*]{} I wish to thank Sergei Ivashkovich for pointing out to me Hirschowitz’ paper [@Hirschowitz] which discusses the two definitions of meromorphic mappings.
Degree functions and stability {#section:degrees}
==============================
Various definitions of semistability for coherent sheaves on projective manifolds are in use and many recent papers aim at a formalization of their properties, see e.g. [@JoyceIII], [@Andre].
Here we content ourselves with the presentation of the stability notion which will be appear in the results of this paper. It will use a generalization of the Hilbert polynomial of a coherent sheaf which will make sense in a complex geometric (not necessarily projective) framework. This notion finds applications even when the base space $X$ is a smooth projective variety; cf. [@GT17], [@GRT1], [@GHT]. In order to introduce it one may choose to work either on a category $\operatorname{Coh}_d(X)$ of coherent sheaves of dimension at most $d$ on an analytic space $X$ or on a quotient category $\operatorname{Coh}_{d,d'}(X):=\operatorname{Coh}_d(X)/\operatorname{Coh}_{d'-1}(X)$ as in [@HL Section 1.6]. In this paper we chose the first but the statements can be easily rephrased to stay valid for the second approach.
In the sequel $X$ will be a compact analitic space of dimension $n$ and $d$, $d'$ will denote integers satisfying $n\ge d\ge d'\ge 0$. In particular $X$ may be the associated analytic space of a proper algebraic space over ${\mathbb{C}}$.
We denote by $K_0(X)=K_0(\operatorname{Coh}(X))$ the Grothendieck group of coherent sheaves on $X$ and by $[F]$ the class in $K_0(X)$ of a coherent sheaf $F$. If $F$ has dimension at most $p$, we write $\operatorname{cycle}_p(F)$ for the $p$-cycle associated to $F$.
Degrees. \[def:degrees\] Consider a group morphism $\deg_p:K_0(X)\to{\mathbb{R}}$ inducing a positive map on non-zero $p$-cycles when putting $\deg_p(Z):=\deg_p([\cO_Z])$ for irreducible $p$-cycles $Z$, and the following properties:
1. $\deg_p([F])=\deg_p(\operatorname{cycle}_p(F))$ for any $F\in\operatorname{Coh}_p(X)$,
2. if a set of positive $p$-cycles is such that $\deg_p$ is bounded on it, then $\deg_p$ takes only finitely many values on this set,
3. $\deg_p$ is continuous on flat families of sheaves.
4. $\deg_p$ is locally constant on flat families of sheaves.
We will call such a function a [*degree function in dimension $p$*]{} if it has properties (1), (2) and (3). If only the first property is satisfied, we will call it a [*weak degree function in dimension $p$*]{} and if $\deg_p$ has all four properties, we will call it a [*strong degree function in dimension $p$*]{} . We will write for simplicity $\deg_p(F)=\deg_p([F])$ for any $F\in \operatorname{Coh}(X)$.
A collection of degree functions $(\deg_d,...,\deg_{d'})$ in dimensions $d$ to $d'$ on $X$ will be called a [*$(d,d')$-degree system*]{} and similarly for weak or strong degree functions.
Note that for any weak degree function $\deg_p$ in dimension $p$ on $X$ one has $\deg_p(F)>0$ if $F$ is $p$-dimensional, and $\deg_p(F)=0$ if $F\in\operatorname{Coh}_{p-1}(X)$.
Strong degree functions appear naturally if $X$ is endowed with differential forms $\omega_{p}$ of degree $2p$ which are $\d$-closed and such that their $(p,p)$-component $\omega_p^{p,p}$ are strictly positive in Lelong’s sense; cf. [@BM III.2.4, IV.10.6]. In this case for $F\in \operatorname{Coh}(X)$ one defines $\deg_p(\cF):=\int_{\tau_p(\cF)}\omega_p$, where the integral is computed on a semianalytic representative of $\tau_p(\cF)$, cf. [@BlHe69], [@DoPo77], [@AG Thm. 7.22], [@Gor81 8.4],[@Her66]. These functions satisfy condition (2) of Definition \[def:degrees\] since they are constant on connected components of the corresponding cycle spaces and since no analytic space can have an infinite number of irreducible components accumulating at a point. When $(X,\omega)$ is a Kähler space, cf. [@Var89 II.1.2], we obtain such $2p$-forms as $p$-powers of the Kähler form $\omega$. When $(X, \cO_X(1))$ is a projective variety endowed with an ample line bundle, by taking $\omega$ a strictly positive curvature form of $\cO_X(1)$, we recover the coefficient of the Hilbert polynomial of $F$ in degree $p$ as $$\frac{\deg_p(F)}{p!}.$$ However even on projective manifolds one is naturally led to consider degree functions which are not associated to an ample polarization, cf. [@GT17], [@GRT1], [@GRTsurvey], [@CampanaPaun].
In the above situation the condition $\d\omega^{p,p}_p=0$ implies that the corresponding degree function is locally constant on flat families of sheaves. An example having found applications in the literature, where only continuity holds is that of degree functions on non-Kählerian compact manifolds. Such a manifold always carries a Gauduchon form $\omega_{n-1}$, i.e. such that $\omega_{n-1}$ is positive of type $(n-1,n-1)$ and $\partial\bar\partial\omega_{n-1}=0$. One then defines degree functions in dimensions $n$ and $n-1$ by setting $\deg_n(F):=\operatorname{rk}(F)$ and $\deg_{n-1}(F):=\int_X [\omega_{n-1}]_A\cdot c_{1}(F)_{BC}$, where the classes are computed in Aeppli cohomology and in Bott-Chern cohomology respectively, cf. [@LuTe], [@Tel10]. More generally, any strictly positive $\partial\bar\partial$-closed $(p,p)$-form on $X$ gives rise to a a degree function in dimension $p$. One shows that Condition (2) of Definition \[def:degrees\] is satisfied by the same argument as in the Kähler case. Indeed such a function is pluriharmonic on the cycle space by [@Barlet78 Proposition 1] and attains its minimum on any closed subset of the cycle space by Bishop’s theorem. It is therefore constant on any irreducible component of this space. Note that any compact complex manifold $X$ admits a degree function in dimension zero defined by $\deg_0(F):=\int_X\ch_n(F)$. In this way Gauduchon surfaces $(X,\omega_1)$ get a $(2,0)$-degree system.
For the following definition the order relation we will consider on ${\mathbb{R}}^{d-d'+1}$ will be the lexicographic order.
Semistability.
Suppose that $X$ is equipped with a weak $(d,d')$-degree system $(\deg_{d},...,\deg_{d'})$. For any $d$-dimensional coherent sheaf $F$ we define its slope vector with respect to this system as $$\mu(F):=(\frac{\deg_{d}(F)}{\deg_d(F)},\frac{\deg_{d-1}(F)}{\deg_d(F)},..., \frac{\deg_{d'}(F)}{\deg_d(F)})\in {\mathbb{R}}^{d-d'+1}.$$ A $d$-dimensional sheaf $F$ will be called [*slope-semistable*]{} or just [*semistable*]{} if it is pure and if for any non-trivial subsheaf $E\subset F$ we have $\mu(E)\le\mu(F)$.
Note that in the case $d'=d$, semistability on $\operatorname{Coh}_d(X)$ just means $d$-dimensional purity.
With literally the same proof as in [@HL Section 1.3] one checks the existence of a Harder-Narasimhan filtration for the above semistability notion:
With respect to a weak $(d,d')$-degree system on $X$ any pure $d$-dimensional sheaf $F$ admits a unique increasing filtration $$0=HN_0(F)\subset HN_1(F)\subset...\subset HN_l(F)=F,$$ with semistable factors $HN_i(F)/HN_{i-1}(F)$ for $1\le i\le l$ and such that $$\mu( HN_1(F)/HN_{0}(F))>...>\mu( HN_l(F)/HN_{l-1}(F)).$$
In particular under the above hypotheses $HN_1(F)$ has the properties of a [*maximal destabilizing subsheaf*]{} of $F$, i.e. for all subsheaves $E\subset F$ one has $\mu(E)\le \mu(HN_1(F))$, and in case of equality $E\subset HN_1(F)$. Moreover, $HN_1(F)$ is a [*saturated subsheaf of $F$*]{}, i.e the quotient $F/HN_1(F)$ is either zero or pure $d$-dimensional.
Before we go on to the relative case let us remark that purity is a Zariski open property in flat families of coherent sheaves. Indeed if $S$ is any analytic space and if $E$ is a flat family of $d$-dimensional coherent sheaves on the fibres of $X$ parameterized by $S$, then one can adapt Maruyama’s approach in [@Maruyama-Construction Proposition 1.13] to prove that the set of points $s\in S$ such that $E_s$ is not pure is a closed analytic subset of $S$. It suffices to work in loc. cit. with local resolutions and apply the purity criterion from [@Maruyama-Construction Lemma 1.12].
By a [*relative Harder-Narasimhan filtration*]{} for a flat family $E$ of $d$-dimensional coherent sheaves on $X$ parametrized by an irreducible analytic space $S$ we mean a proper bimeromorphic morphism of irreducible analytic spaces $T\to S$ together with a filtration $$0=HN_0(E)\subset HN_1(E)\subset...\subset HN_l(E)=E_T$$ such that the factors $HN_i(E)/HN_{i-1}(E)$ are flat over $T$ for $1\le i\le l$ and which induces the absolute Harder-Narasimhan filtrations fibrewise over some dense Zariski open subset of $S$, cf. [@TomLimII] for a more general situation and [@HL Section 2.3] for the projective algebraic case.
In order to obtain a relative Harder-Narasimhan filtration for a family of sheaves we need stronger assumptions on the degree functions. Using the techniques of [@Toma16] the following result is obtained in [@TomLimII].
\[thm:relativeHNfiltration\] Suppose that $X$ is endowed with a strong $(d,d')$-degree system induced by a system of strictly positive $\partial\bar\partial$-closed differential forms. Then with respect to the corresponding semistability notion every flat family $E$ of $d$-dimensional coherent sheaves on $X$ with pure general members parametrized by an irreducible analytic space $S$ has a relative Harder-Narasimhan filtration $(T\to S, HN_\bullet(E))$. Moreover this filtration has the following universal property: if $f:T'\to S$ is some morphism of irreducible analytic spaces and if $F_\bullet$ is a filtration of $E_{T'}$ with flat factors, which coincides fibrewise with the absolute Harder-Narasimhan filtration over some point $s\in S$, then $f$ factorizes over $T$ and $F_\bullet=HN_\bullet(E)_{T'}$.
A consequence of this theorem is the fact that semistability is a Zariski open property in flat families of sheaves.
Another way to look at the relative Harder-Narasimhan filtration is to consider its direct image over $X\times S$ and the filtration which this induces on $E$. In particular, if $E$ is a family as in the theorem’s statement and whose general fibres are not semistable and if $(f:T\to S, HN_\bullet(E))$ is the relative Harder-Narasimhan filtration of $E$, then the fibers over general points $s\in S$ of the image $F_1$ of the composition of sheaf homomorphisms $$(\id_X\times f)_*(HN_1(E)) \to (\id_X\times f)_*(E_T)\to E$$ coincide with $HN_1(E_s)$. We shall call the sheaf $F_1$ the [*relative maximal destabilizing sheaf*]{} of $E$.
One dimensional families {#section:one-dimensional-case}
========================
In this section we deal with the case of one-dimensional families in its analytic formulation. The attentive reader will be able to translate the argument in terms of families over the spectrum of a discrete valuation ring, when the base space is algebraic, with some care however when applying Artin approximation. We will denote by ${\mathbb{D}}$ the open unit disc in ${\mathbb{C}}$ and write ${\mathbb{D}}^*:={\mathbb{D}}\setminus\{0\}$.
\[thm:dimension one\] Let $X$ be a compact complex manifold endowed with a $(d,d')$-degree system and let $F$ be a ${\mathbb{D}}$-flat family of $d$-dimensional sheaves on $X$. Suppose that for $s\in{\mathbb{D}}\setminus\{0\}$ the fibres $F_s$ are semistable. Then there exists a coherent subsheaf $F'\subset F$ coinciding with $F$ over ${\mathbb{D}}\setminus\{0\}$ and such that the fibre $F'_0$ over zero is also semistable.
Before starting the proof, note that $\cO_{{\mathbb{D}},s}$ is a principal ideal domain for any $s\in{\mathbb{D}}$, so for an $\cO_{{\mathbb{D}},s}$-module being flat boils down to being torsion-free. Thus, since $F$ has no ${\mathbb{D}}$-torsion, any coherent subsheaf of $F$ continues to be flat over ${\mathbb{D}}$.
The proof follows the line of [@HL Section 2B] with an essential expansion due to the lack of discreteness of the degree functions in our set-up. A smaller change appears at the end where we avoid the issue on the use of the properness of the relative Douady space and replace it by a short argument using Artin approximation.
For any integer $\delta\in[d',d]$ we will consider semistability with respect to the $(d,\delta)$-degree system $(\deg_d,...,\deg_\delta)$ obtained by restricting the given $(d,d')$-degree system on $X$. We will call a semistable sheaf with respect to such a restricted system shortly [*$(d,\delta)$-semistable*]{}. Note that a sheaf is $(d,d)$-semistable if and only if it is pure of dimension $d$. Thus for $d'=d$ Theorem \[thm:dimension one\] just says that if $F$ is flat and with pure fibers over ${\mathbb{D}}^*$, then a subsheaf $F'\subset F$ exists with $F'_{{\mathbb{D}}^*}=F_{{\mathbb{D}}^*}$ and $F'_0$ pure as well. This is the case $\delta=d$ of the following Claim and a proof is suggested in [@HL Exercise 2.B.2]. Since this special case works under weaker hypotheses and has an easier proof we provide a separate statement for it as Proposition \[prop:dimension one\].
\[Claim\] Let $X$, $F$ be as in the theorem’s statement and $d\ge\delta \ge d'$. Then, if $F_0$ is moreover $(d,\delta+1)$-semistable, there exists a subsheaf $F'\subset F$ with $F'_{{\mathbb{D}}^*}=F_{{\mathbb{D}}^*}$ and $(d,\delta)$-semistable fibre $F'_0$ over zero.
The theorem will follow from this Claim by descending induction on $\delta$.
We now proceed to the proof of the Claim. The case $\delta=d$ has already been discussed, so we will work under the hypothesis $d>\delta\ge d'$. Assuming by contradiction that the Claim is false we first construct an infinite descending filtration $F=F^0\supset F^1 \supset F^2 \supset F^3 \supset...$ such that $F^n_{{\mathbb{D}}^*}= F_{{\mathbb{D}}^*}$ and $F^n_0$ not $(d,\delta)$-semistable for every $n\in{\mathbb{N}}$, as follows: supposing that $F^n$ has already been constructed, let $B^n\subset F_0^n$ be the maximal $(d,\delta)$-destabilizing subsheaf of $F_0^n$, let $G^n:=F_0^n/B^n$ and let $F^{n+1}$ be the kernel of the composition $F^n\to F_0^n\to G^n$. These sheaves are related through two exact sequences $$\label{seq:1}
0\to B^n\to F_0^n\to G^n\to 0$$ $$\label{seq:2}
0\to G^n\to F_0^{n+1}\to B^n\to 0.$$ To explain the second we will tensorize over $\cO_{\mathbb{D}}$ the exact sequence $0\to F^{n+1}\to F^{n}\to G^n\to 0$ by $0\to \gm\to \cO_{\mathbb{D}}\to\cO_{\mathbb{D}}/\gm\to0$, where $\gm$ is the ideal sheaf of the origin in ${\mathbb{D}}$, to get the following commutative diagram with exact rows and columns. $$\label{diagr:snake}
\begin{gathered}
\xymatrix{
& & &0\ar[d] & \\
& 0 \ar[d] & 0 \ar[d]\ar[r] & Tor_1^{\cO_{\mathbb{D}}}(G^n,\cO_{\mathbb{D}}/\gm) \ar[d]& \\
0 \ar[r]& F^{n+1} \otimes_{\mathbb{D}}\gm \ar[r]\ar[d] & F^{n+1} \ar[d]\ar[r] & F_0^{n+1}\ar[r] \ar[d] & 0 \\
0 \ar[r]& F^{n}\otimes_{\mathbb{D}}\gm \ar[r]\ar[d] & F^{n} \ar[d]\ar[r] & F_0^{n}\ar[r] \ar[d] & 0 \\
& G^n\otimes_{\mathbb{D}}\gm \ar[r]^0\ar[d] & G^n \ar[d]\ar[r]^{\cong} & G^n\otimes_{\mathbb{D}}\cO_{\mathbb{D}}/\gm\ar[r] \ar[d] & 0 \\
& 0 & 0 & 0 &
}
\end{gathered}$$ Then we use the Snake Lemma and the fact that $\gm\cong\cO_{\mathbb{D}}$.
Combining the exact sequences \[seq:1\], \[seq:2\] we get a self explanatory commutative diagram with exact rows and columns: $$\label{diagr:3X3}
\begin{gathered}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d]& \\
0 \ar[r]& A^{n+1} \ar[r]\ar[d] & B^{n+1} \ar[d]\ar[r] & C^{n+1}\ar[r] \ar[d] & 0 \\
0 \ar[r]& G^{n} \ar[r]\ar[d] & F^{n+1}_0 \ar[d]\ar[r] & B^{n}\ar[r] \ar[d] & 0 \\
0 \ar[r] & K^{n+1} \ar[r]\ar[d] & G^{n+1} \ar[d]\ar[r] & L^{n+1}\ar[r] \ar[d] & 0 \\
& 0 & 0 & 0 &
}
\end{gathered}$$ By continuity $\deg_p(F_0^n)=\deg_p(F_0)$ for all $n\in{\mathbb{N}}$ and $p$ with $d\ge p\ge d'$. If $C^{n+1}\neq 0$, then $C^{n+1}$ is pure $d$-dimensional as a submodule of the $(d,\delta)$-semistable module $B^{n}$ and $$\label{ineq:1}
\mu_{d,\delta}(B^{n+1})\le \mu_{d,\delta}(C^{n+1})\le \mu_{d,\delta}(B^{n}).$$ If $C^{n+1}=0$, we have $$\label{ineq:2}
\mu_{d,\delta}(B^{n+1})=\mu_{d,\delta}(A^{n+1})\le \mu_{d,\delta}(G^{n})<\mu_{d,\delta}(B^{n})$$ by the choice of $B^n\subset F^n_0$. We also have $\mu_{d,\delta}(B^{n})>\mu_{d,\delta}(F_0)$, hence the sequence $(\mu_{d,\delta}(B^{n}))_n$ is descending and bounded from below. In fact only the $\frac{\deg_\delta}{\deg_d}$ may vary on this sequence. At this point if we knew the degree functions to be discrete, we would conclude that the sequence $(\mu_{d,\delta}(B^{n}))_n$ is stationary. In our situation we need to construct further objects in order to get the stationary behaviour of this sequence.
The idea is to introduce at each formation step of the subsheaves $F^n$ a decomposition of $F_0^n$ into smaller and smaller building blocks as $n$ increases. We will get at each step a collection $\cC^n$ of $2^{n+1}$ building blocks. We show that this crumbling process must eventually stop, leading to the desired stationary behaviour. We indicate below the first three steps of this process.
Step 0: The decomposition is given by the exact sequence \[seq:1\] for $n=0$. We set $\cC^0:=(B^0,G^0)$.
Step 1: The case $n=1$ for the sequences \[seq:1\] and \[seq:2\] lead to a diagram of type \[diagr:3X3\]. We set $\cC^1:=(A^1, C^1, K^1, L^1 )$. We may view $\cC^1$ as the result of cutting $\cC^0$ into pieces by using \[seq:2\]. The reconstruction of $B^0,B^1,G^0,G^1,F_0^0,F_0^1$ is possible starting from $\cC^1$.
Step 2: We use again \[seq:2\] this time for $n=2$ to cut each component of $\cC^1$ into two further pieces. These give the eight vertices denoted by $\ast$ in the following diagram. We allow to count isomorphic components of $\cC^n$ several times if they appear at different places in the decomposition process. Note that $B^2$ cuts off subobjects of components of $\cC^1$; these appear represented in the (back) plane of $B^2$. $$\xymatrix{
& & \ast\ar[ddd]\ar[ddl] & & & & & & \ast\ar[ddd]\ar[ddl]\\
& & & & & & & & \\
& K^1\ar[ddd]\ar[ddl] & & & & & & A^1\ar[ddd]\ar[ddl]&\\
& & A^2\ar[ddd]\ar[ddl]\ar[rrr] & & &B^2\ar[rrr]\ar[ddl] & & & C^2\ar[ddd]\ar[ddl]\\
\ast\ar[ddd]& & & & & & \ast\ar[ddd]& & \\
& G^1\ar[ddd]\ar[ddl]\ar[rrr] & & &F_0^2\ar[rrr]\ar[ddl] & & & B^1\ar[ddd]\ar[ddl]&\\
& & \ast\ar[ddl]& & & & & & \ast\ar[ddl]\\
K^2\ar[ddd]\ar[rrr] & & &G^2\ar[rrr] & & & L^2\ar[ddd]& &\\
& L^1\ar[ddl] & & & & & & C^1\ar[ddl]&\\
& & & & & & & & \\
\ast& & & & & & \ast& & \\
}$$
For $n>2$ the elements of $\cC^n$ will appear as vertices of the $n+1$-dimensional hypercube in a diagram constructed in a recursive manner as above.
Note that modulo $\operatorname{Coh}_\delta$ all components of $\cC^n$ vanish or are $(d,\delta+1)$-semistable. Note also that for any component $E$ of $\cC^n$, we have $0\le \operatorname{cycle}_d(E)\le\operatorname{cycle}_d(F_0)$. In fact the sum of these cycles over all components of $\cC^n$ equals $\operatorname{cycle}_d(F_0)$. Thus there exists some threshold $n_0\in {\mathbb{N}}$, such that the set of $d$-cycles of components of $\cC^n$ is constant for $n\ge n_0$. For $n>n_0$ the decomposition into building blocks from $\cC^n$ of components $E$ of $\cC^{n_0}$ shows that $B^n$ cuts off subobjects $E'$ in such components $E$ and these subobjects are to be used in the reconstruction of $B^n$ itself; in fact they will be the quotients of a suitable filtration of $B^n$. If $E$ is $d$-dimensional then such a subobject either vanishes or is pure $d$-dimensional since $B^n$ is pure $d$-dimensional itself. In this second case $\operatorname{cycle}_d(E')=\operatorname{cycle}_d(E)$ by our assumption on $\cC^{n_0}$, and in particular $E/E'\in\operatorname{Coh}_\delta(X)$ and $\deg_\delta(E/E')=\deg_\delta(\operatorname{cycle}_\delta(E/E')\ge 0$. If $E$ is not $d$-dimensional then $E$ has at most dimension $\delta$ and we have $\deg_\delta(E/E')=\deg_\delta(\operatorname{cycle}_\delta(E/E')\ge 0$ in this case too. Consider now a subsequence $(B^{n_k})_{k>1}$ of $(B^n)_{n\ge n_0}$ such that all its terms cut off non-zero subobjects on the same subcollection of $d$-dimensional components of $\cC^{n_0}$. It follows that the sequence $(\operatorname{cycle}_d(B^{n_k}))_{k>1}$ is constant. On the other hand using the above notations and taking sums over all components $E$ of $\cC^{n_0}$ we find $\deg_\delta(B^{n_k})=\sum_E \deg_\delta(E')=\sum_E\deg_\delta(E)-\sum_E\deg_\delta(E/E')=\deg_\delta(F_0)-\sum_E\deg_\delta(\operatorname{cycle}_\delta(E/E'))=\deg_\delta(F_0)-\deg_\delta(\sum_E\operatorname{cycle}_\delta(E/E'))$. By our assumption on the degree functions it follows that the cycles $\sum_E\operatorname{cycle}_\delta(E/E')$ are bounded and may attain only a finite number of values when $k$ varies. This implies that the sequence $(\mu_{d,\delta}(B^{n_k}))_k$is stationary, hence also $(\mu_{d,\delta}(B^{n}))_n$ is stationary.
We continue now the proof of the Claim following again [@HL]. By the above we may assume that the sequence $(\mu_{d,\delta}(B^{n}))_n$ is even constant. Then the inequalities \[ineq:2\] show that $C^{n+1}\neq0$ for all $n$ and using \[ineq:1\] we further find $\mu_{d,\delta}(B^{n+1})= \mu_{d,\delta}(C^{n+1})= \mu_{d,\delta}(B^{n})$. Thus either $A^{n+1}$ vanishes or it is $(d,\delta)$-semistable with $\mu_{d,\delta}(A^{n+1})= \mu_{d,\delta}(B^{n+1})= \mu_{d,\delta}(C^{n+1})$. In the latter case we would have $\mu_{d,\delta}(B^{n+1})=\mu_{d,\delta}(A^{n+1})\le \mu_{d,\delta}(G^{n})<\mu_{d,\delta}(B^{n})$ which is impossible. Hence and from diagram \[diagr:3X3\] we get $A^{n+1}=0$, $B^{n+1}\cong C^{n+1}\subset
B^{n}$, $G^n\cong K^{n+1}\subset G^{n+1}$ for all $n\in{\mathbb{N}}$. Moreover since the sequence $(\operatorname{cycle}_d(B^n))_n$ is stationary we may as well suppose that it is constant. It follows that $L^{n+1}\in\operatorname{Coh}_{\delta-1}$. In particular the ascending sequence of pure $d$-dimensional sheaves $G^n$ is constant modulo $\operatorname{Coh}_{d-2}(X)$, thus their reflexive hulls $(G^n)^{DD}$ are all the same and in particular the ascending sequence $(G^n)_n$ of subsheaves of $(G^0)^{DD}$ is stationary. We assume again for simplicity that this sequence too is constant. So the central vertical and horizontal exact sequences of diagram \[diagr:3X3\] are split. We will write from now on $G:=G^n$, $B:=B^n$ and $Q^n:=F/F^n$. Via the splittings $F_0^n\cong B\oplus G$ the morphisms $F_0^{n+1}\to F_0^n$ from diagram \[diagr:snake\] become compositions $B\oplus G\to B\to B\oplus G$ of the natural projections and injections. Hence the cokernel of the composition $F_0^n\to F^{n-1}_0\to...\to F_0^0$ is isomorphic to $G$. All in all we obtain that $Q_0^n\cong G$. From the diagram $$\xymatrix{
0 \ar[r] & F^{n+1} \ar[r]\ar[d] & F\ar[r]\ar[d] & Q^{n+1}\ar[r] \ar[d] & 0 \\
0 \ar[r]& F^{n} \ar[r] & F\ar[r] & Q^{n}\ar[r] & 0
}$$ we see that $\Ker(Q^{n+1}\to Q^n)\cong\Coker(F^{n+1}\to F^n)\cong G$. We will next show by induction on $n$ that $Q^n$ is flat over $\cO_{\mathbb{C}}/\gm^n$. The assertion is clear for $n=1$ so we assume it true for $n$ and start proving it for $n+1$. For this we tensorize the exact sequence $$0\to G\to Q^{n+1}\to Q^n\to 0$$ by $$0\to \gm/\gm^{n+1}\to\cO_{\mathbb{D}}/\gm^{n+1}\to \cO_{\mathbb{D}}/\gm\to0$$ over $\cO_{\mathbb{D}}$ to get $$\begin{gathered}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d]& \\
0 \ar[r]& G\otimes\gm/\gm^{n+1} \ar[r]\ar[d] & G\otimes\cO_{\mathbb{D}}/\gm^{n+1} \ar[d]\ar[r] & G\otimes\cO_{\mathbb{D}}/\gm\ar[r] \ar[d] & 0 \\
& Q^{n+1}\otimes\gm/\gm^{n+1} \ar[r]\ar[d] & Q^{n+1}\otimes\cO_{\mathbb{D}}/\gm^{n+1} \ar[d]\ar[r] & Q^{n+1}\otimes\cO_{\mathbb{D}}/\gm\ar[r] \ar[d] & 0 \\
0\ar[r] & Q^n\otimes\gm/\gm^{n+1} \ar[r]\ar[d] & Q^n \otimes\cO_{\mathbb{D}}/\gm^{n+1} \ar[d]\ar[r] & Q^n\otimes\cO_{\mathbb{D}}/\gm\ar[r] \ar[d] & 0 \\
& 0 & 0 & 0 &
}
\end{gathered}$$ where the exactness of the first two vertical sequences follows from the induction hypothesis and from the fact that $\gm/\gm^{n+1} $ is a $\cO_{\mathbb{C}}/\gm^n$-module. Thus the morphism $Q^{n+1}\otimes\gm/\gm^{n+1} \to Q^{n+1}\otimes\cO_{\mathbb{D}}/\gm^{n+1}$ is injective and the local flatness criterion [@HL Lemma 2.1.3] shows that $Q^{n+1}$ is a flat $\cO_{\mathbb{D}}/\gm^{n+1}$-module. This gives a projective system of maps over ${\mathbb{D}}$ from the spaces $\Spec(\cO_{\mathbb{D}}/\gm^{n})$ to the relative Douady space $D_{F/X\times{\mathbb{D}}/{\mathbb{D}}}$ over ${\mathbb{D}}$ of quotients of $F$ which may be seen as a formal section of the natural projection of germs $(D_{F/X\times{\mathbb{D}}/{\mathbb{D}}}, F\twoheadrightarrow Q^1)\to ({\mathbb{D}},0)$. Such a section admits an analytic approximation to order one by Artin approximation [@Artin68 Theorem 1.4.ii] showing that the projection $(D_{F/X\times{\mathbb{D}}/{\mathbb{D}}}, F\twoheadrightarrow Q^1)\to ({\mathbb{D}},0)$ is surjective. This contradicts the stability of the fibres of $F$ over ${\mathbb{D}}^*$.
\[prop:dimension one\] Let $X$ be a compact complex manifold and let $F$ be a ${\mathbb{D}}$-flat family of $d$-dimensional coherent sheaves on $X$ whose fibres over ${\mathbb{D}}\setminus\{0\}$ are pure of dimension $d$. Then there exists a coherent subsheaf $F'\subset F$ coinciding with $F$ over ${\mathbb{D}}\setminus\{0\}$ and such that its fibre $F'_0$ over $0$ is also pure.
We follow the strategy of proof of Theorem \[thm:dimension one\] but we take this time $B_n:=T_{d-1}(F_0^n)$, the maximal subsheaf of $F_0^n$ of dimension at most $d-1$, cf. [@HL Definition 1.1.4]. Then $G^n:=F_0^n/B^n$ is pure and diagram \[diagr:3X3\] takes the form $$\label{diagr:3X3prop}
\begin{gathered}
\xymatrix{
& & 0 \ar[d] & 0 \ar[d]& \\
& & B^{n+1} \ar[d]\ar[r]^\cong & B^{n+1} \ar[d] & \\
0 \ar[r]& G^{n} \ar[r]\ar[d]^\cong & F^{n+1}_0 \ar[d]\ar[r] & B^{n}\ar[r] \ar[d] & 0 \\
0 \ar[r] & G^{n} \ar[r] & G^{n+1} \ar[d]\ar[r] & L^{n+1}\ar[r] \ar[d] & 0 \\
& & 0 & 0 &
}
\end{gathered}$$
We have inequalities for associated $(d-1)$-cycles: $$0\le\operatorname{cycle}_{d-1}(B^{n+1})\le\operatorname{cycle}_{d-1}(B^{n}),$$ hence the sequence $(\operatorname{cycle}_{d-1}(B^{n}))_n$ must be stationary and we may assume that $\dim(L^n)\le d-2$ for all $n$. We immediately get then that the ascending sequence of subsheaves of $(G_0)^{DD}$ is stationary and as before we assume that this sequence is constant and write $G:=G^n$, $B:=B^n$. The rest of the proof follows ad litteram the proof of Theorem \[thm:dimension one\] but for its last sentence where we get a contradiction to purity instead of semistability.
Families of arbitrary dimension {#section:arbitrary-dimension}
===============================
We now turn our attention to the case of higher dimensional parameter spaces. For simplicity we will only consider smooth parameter spaces. As in the previous section we give a separate purity statement. This is the content of Proposition \[prop:dimension>1\]. For the main result of the section a stronger assumption on the degree functions will be made which will guarantee the existence of a relative maximal destabilizing subsheaf and in particular that semistability is a Zariski open property in flat families of coherent sheaves, see Section \[section:degrees\].
\[thm:dimension>1\] Let $X$ be a compact complex manifold endowed with a strong $(d,d')$-degree system induced by a system of strictly positive $\partial\bar\partial$-closed differential forms, where $0\leq d'\leq d$. Let further $F$ be a family over $S$ of $d$-dimensional sheaves on $X$, where $S$ is a connected smooth parameter space. Suppose that general fibres of $F$ are $(d,d')$-semistable and let $Z\subset S$ be the union of the non-flatness locus of $F$ with the closed analytic subset of $S$ parametrizing non-$(d,d')$-semistable sheaves. Let further $K\subset S$ be a compact subset. Then there exist a proper modification $S'\to S$ and a coherent sheaf $F'$ on $X\times S'$ such that $F'$ is flat over $S'$, all fibers of $F'$ over $K\times_SS'$ are $(d,d')$-semistable and $F'$ coincides with $F_{S'}$ over $(S\setminus Z)\times_SS'$.
It is clear that we only need to deal with the finitely many irreducible components of $Z$ which meet the compact set $K$. In the sequel we will assume for simplicity of notation that all irreducible components of $Z$ meet $K$. The idea of the proof is on one hand to try to reduce the dimension of the bad set $Z$ for a suitable subsheaf of $F$ constructed by a similar procedure to that which is used in the proof of Theorem \[thm:dimension one\]. On the other hand it will be convenient to work in the case when $Z$ is a simple normal crossings divisor and $F$ is flat over $S$. We may reduce ourselves to this situation by repeatedly blowing up $S$ at smooth centers by Hironaka’s flattening theorem [@Hir75]. Since under this requirement the dimension of $Z$ is maximal, we introduce a “badness index” $b$ in order to control the induction process in the following way: For any irreducible component $Z_i$ of $Z$ we say that a proper holomorphic map $\pi_i:Z_i\to B_i$ is [*good (for $F$)*]{} if the restrictions of $F$ to the fibers of $\pi_i$ are constant families. Then we set $b_i$ to be lowest possible dimension of a base $B_i$ of such a good map, and $b$ to be the maximum among the $b_i$. The strategy will be the following: we start with the flat family $F$ whose non-semistable locus is a divisor with simple normal crossings $Z=\sum_iZ_i$ and with badness index $b$. From this data we produce a subfamily $F'\subset F$, with $F'_{S\setminus Z}= F_{S\setminus Z}$ and with strictly smaller bad locus. If this bad locus is empty, then $F'$ is the family we were looking for and we stop. If not, after a suitable proper modification $S'\to S$ the pull-back of this non-semistable locus becomes a divisor with simple normal crossings $D'$ on $S'$ with badness index $b'$ for the family $F'_{S'}/\Tors_{S'}(F'_{S'})$ and such that $b'<b$. We work now with the family $F'_{S'}/\Tors_{S'}(F'_{S'})$ which may be supposed in addition to be flat over $S'$, by Hironaka’s flattening theorem [@Hir75] again. It is clear that this process eventually stops.
The proof of the existence of the desired subfamily $F'\subset F$ on $S$ will follow the same path as in the one-dimensional case by descending induction on $\delta$. The corresponding claim will be
\[Claim>1\] Let $X$, $F$ be as in the theorem’s statement and moreover such that $F$ is flat over $S$ and $Z$ is a divisor with simple normal crossings. Suppose that $d\ge\delta \ge d'$. the general fibers of $F_Z$ are $(d,\delta+1)$-semistable for some $\delta$ with $d\ge\delta \ge d'$. Then there exists a subsheaf $F'\subset F$ with $F'_{S\setminus Z}=F_{S\setminus Z}$ with $(d,\delta)$-semistable fibres over general points of $Z$ and with non-flatness locus which is nowhere dense in $Z$.
We take first a relative maximal (semi)destabilizing subsheaf $B^0$ of $F_Z$ and put $G^0:=F_Z/B^0$ and $F^1:=\Ker(F\to G^0)$. Note that $B^0_s$ is the maximal $(d,\delta)$-(semi)destabilizing subsheaf only for general points $s\in Z$. Flatness of $B^0$ and $G^0$ likewise only holds generically over $Z$. But we can work at such general points and see that the whole proof of Claim \[Claim\] goes through in this new relative setting. (We will consider of course relative support cycles for the components of the collections $\cC^n$ this time. We will also use the corresponding relative statements at each moment of the proof, such as [@Artin68 Theorem 1.3] for instance.) In this way we obtain a subsheaf $F'\subset F$ with $F'_{S\setminus Z}=F_{S\setminus Z}$ and with $(d,\delta)$-semistable fibres over general points of $Z$. At each step of the proof the non-flatness locus of the sheaves $F^n$ is nowhere dense in $Z$. Indeed, assuming this to be true for $F^n$, we check it for $F^{n+1}$ by using an analogue of diagram \[diagr:snake\] around a point of $Z$ where both $F^n$ and $G^n$ are flat: $$\label{diagr:snake>1}
\begin{gathered}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & Tor_1^{\cO_{S}}(G^n,\cO_{Z}) \ar[d]& \\
& F^{n+1}(-Z) \ar[r]\ar[d] & F^{n+1} \ar[d]\ar[r] & F^{n+1}_{Z}\ar[r] \ar[d] & 0 \\
0 \ar[r]& F^n_{Z}(-Z) \ar[r]\ar[d] & F^n_{Z} \ar[d]\ar[r] & F^n_{Z}\ar[r] \ar[d] & 0 \\
Tor_1^{\cO_{Z}}(G^n,\cO_{Z}) \ar[r]^{\cong} & G^n(-Z) \ar[r]^0\ar[d] & G^n \ar[d]\ar[r]^{\cong} & G^n\ar[r] \ar[d] & 0 \\
& 0 & 0 & 0 &
}
\end{gathered}$$ from which we immediately obtain exact sequences $$0\to G^n(-Z)\to F^{n+1}_{Z}\to F^n_{Z}\to G^n\to0$$ and $$0 \to F^{n+1}(-Z) \to F^{n+1}\to F^{n+1}_{Z}\to 0.$$ From the first sequence we infer that $F^{n+1}_{Z}$ is flat over $Z$ around the chosen point and from the second combined with [@BaSt Corollaire 5.1.4] that $F^{n+1}$ is flat over $S$ around the chosen point again. This proves Claim \[Claim>1\].
If the bad locus $Z'$ of $F'$ on $S$ is not empty, we perform a proper modification $S'\to S$ on $S$ so that $F'':=F'_{S'}/\Tors_{S'}(F'_{S'})$ is flat over $S'$ with divisorial bad locus $Z''$. Let $b$, $b'$ and $b''$ be the badness indices of $F$, $F'$ and $F''$, respectively. Let $Z_i\subset Z$ be an irreducible component of $Z$ and $\pi_i:Z_i\to B_i$ be a good map for $F$. By construction of $F'$ it follows that the restriction of the family $F'$ is constant on the fibres of $Z'_i\to B_i$, hence the intersection $Z_i\cap Z'$ fibres over a proper Zariski subset $B'_i$ of $B_i$. This shows that $b'<b$. When passing to the pair $(S',F'')$ it is clear that good maps for $F''$ are obtained from good maps for $F'$ by composing with $S'\to S$. In particular one has $b''\le b'<b$. We have thus completed the induction argument and the proof of the theorem.
As in the case of one dimensional bases the previous arguments adapt to yield the following
\[prop:dimension>1\] Let $X$ be a compact complex manifold and let $F$ be a family over $S$ of $d$-dimensional sheaves on $X$, where $S$ is a connected smooth parameter space. Suppose that general fibres of $F$ are pure and let $Z\subset S$ be the union of the non-flatness locus of $F$ with the closed analytic subset of $S$ parametrizing non-pure sheaves. Let further $K\subset S$ be a compact subset. Then there exist a proper modification $S'\to S$ and a coherent sheaf $F'$ on $X\times S'$ such that $F'$ is flat over $S'$, all fibres of $F'$ over $K\times_SS'$ are pure and $F'$ coincides with $F_{S'}$ over $(S\setminus Z)\times_SS'$.
Application to moduli spaces of semistable sheaves {#section:application}
==================================================
In this section we give an application of Theorem \[thm:dimension>1\] to compactness of moduli spaces of semistable sheaves. For such a result to hold one needs boundedness for the class of sheaves one is considering. This is typically attained by fixing the topological type or the Hilbert polynomial of these sheaves. This is the way to think of property $\cP$ in the statement below.
\[thm:application\] Let $X$ be a compact complex manifold endowed with a weak $(d,d')$-degree system and let $\cP$ be an open and closed property on $d$-dimensional coherent sheaves on $X$. Suppose that the corresponding semistability property satisfies the following properties:
1. openness in flat families of coherent sheaves,
2. boundedness when restricted to the class of sheaves with the property $\cP$,
3. the conclusion of Theorem \[thm:dimension>1\] when restricted to the class of sheaves with the property $\cP$,
4. the existence of a coarse moduli space $M^{ss}_\cP$ for semistable sheaves with the property $\cP$.
Then $M^{ss}_\cP$ is compact.
Let $K\subset S$, $F$ be a compact subset of a smooth complex space space and a family of coherent sheaves on $X$ over $S$ giving the boundedness of the set of isomorphism classes of semistable sheaves having property $\cP$ on $X$, cf. [@Toma16], [@TomLimII]. By restricting $S$ to a finite number of its connected components, we may suppose that all the sheaves in the corresponding family over $S$ have the property $\cP$. Let $S^{ss}\subset S$ be the open subset which parametrizes semistable sheaves, let $D:=S\setminus S^{ss}$ and let $S'\to S$ be the proper modification given by Theorem \[thm:dimension>1\]. Then the family $F'$ given by the conclusion of Theorem \[thm:dimension>1\], the universal property of $M^{sst}_\cP$ and the choice of $K$ and $F$ show the existence of a surjective morphism $K\times_SS'\to M^{ss}_\cP$. This proves our statement.
\#1[0=]{} \#1[0=]{} \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[GRT16b]{}
Vincenzo Ancona and Bernard Gaveau, *Differential forms on singular varieties*, Pure and Applied Mathematics (Boca Raton), vol. 273, Chapman & Hall/CRC, Boca Raton, FL, 2006, De Rham and Hodge theory simplified.
Yves André, *Slope filtrations*, Confluentes Math. **1** (2009), no. 1, 1–85.
M. Artin, *On the solutions of analytic equations*, Invent. Math. **5** (1968), 277–291.
Daniel Barlet, *Le faisceau [$\omega ^{\cdot }_{X}$]{} sur un espace analytique [$X$]{} de dimension pure*, Fonctions de plusieurs variables complexes, [III]{} ([S]{}ém. [F]{}rançois [N]{}orguet, 1975–1977), Lecture Notes in Math., vol. 670, Springer, Berlin, 1978, pp. 187–204.
Thomas Bloom and Miguel Herrera, *De [R]{}ham cohomology of an analytic space*, Invent. Math. **7** (1969), 275–296.
Daniel Barlet and Jón Magnússon, *Cycles analytiques complexes. [I]{}. [T]{}héorèmes de préparation des cycles*, Cours Spécialisés \[Specialized Courses\], vol. 22, Société Mathématique de France, Paris, 2014.
C. B[ă]{}nic[ă]{} and O. St[ă]{}n[ă]{}[ş]{}il[ă]{}, *Méthodes algébriques dans la théorie globale des espaces complexes*, Gauthier-Villars, Paris, 1977.
Frédéric Campana and Mihai Păun, *Foliations with positive slopes and birational stability of orbifold cotangent bundles*, `arXiv:1508.02456`, 2015.
Pierre Dolbeault and Jean Poly, *Differential forms with subanalytic singularities; integral cohomology; residues*, Several complex variables ([P]{}roc. [S]{}ympos. [P]{}ure [M]{}ath., [V]{}ol. [XXX]{}, [W]{}illiams [C]{}oll., [W]{}illiamstown, [M]{}ass., 1975), [P]{}art 1, Amer. Math. Soc., Providence, R.I., 1977, pp. 255–261.
Daniel Greb, Peter Heinzner, and Matei Toma, *Good moduli for semistable sheaves with respect to [K]{}ähler polarizations*, in preparation.
R. Mark Goresky, *Whitney stratified chains and cochains*, Trans. Amer. Math. Soc. **267** (1981), no. 1, 175–196.
Daniel Greb, Julius Ross, and Matei Toma, *Moduli of vector bundles on higher-dimensional base manifolds—construction and variation*, Internat. J. Math. **27** (2016), no. 6, 1650054, 27.
, *Variation of [G]{}ieseker moduli spaces via quiver [GIT]{}*, Geom. Topol. **20** (2016), no. 3, 1539–1610.
Daniel Greb and Matei Toma, *Compact moduli spaces for slope-semistable sheaves*, Algebr. Geom. **4** (2017), no. 1, 40–78.
Miguel E. Herrera, *Integration on a semianalytic set*, Bull. Soc. Math. France **94** (1966), 141–180.
Heisuke Hironaka, *Flattening theorem in complex-analytic geometry*, Amer. J. Math. **97** (1975), 503–547.
André Hirschowitz, *Les deux types de méromorphie différent*, J. Reine Angew. Math. **313** (1980), 157–160.
Daniel Huybrechts and Manfred Lehn, *The geometry of moduli spaces of sheaves*, second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.
Dominic Joyce, *Configurations in abelian categories. [III]{}. [S]{}tability conditions and identities*, Adv. Math. **215** (2007), no. 1, 153–219.
Stacy G. Langton, *Valuative criteria for families of vector bundles on algebraic varieties*, Ann. of Math. (2) **101** (1975), 88–110.
Martin L[ü]{}bke and Andrei Teleman, *The [K]{}obayashi-[H]{}itchin correspondence*, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
Masaki Maruyama, *Construction of moduli spaces of stable sheaves via [S]{}impson’s idea*, Moduli of vector bundles ([S]{}anda, 1994; [K]{}yoto, 1994), Lecture Notes in Pure and Appl. Math., vol. 179, Dekker, New York, 1996, pp. 147–187.
Andrei Teleman, *Instantons and curves on class [VII]{} surfaces*, Ann. of Math. (2) **172** (2010), no. 3, 1749–1804.
Matei Toma, *Bounded sets of sheaves on [K]{}ähler manifolds ii*, in preparation.
, *Bounded sets of sheaves on [K]{}ähler manifolds*, J. Reine Angew. Math. **710** (2016), 77–93.
Jean Varouchas, *Kähler spaces and proper open morphisms*, Math. Ann. **283** (1989), no. 1, 13–52.
------------------------------------------------------------------------
|
---
abstract: 'In this paper we study the well-posedness for the inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in Sobolev spaces $H^s$, $s\geq0$. The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical NLS model where $\alpha=0$. The conventional approach does not work particularly for the critical cases $\beta=\frac{4-2\alpha}{d-2s}$. It is still an open problem. The main contribution of this paper is to develop the well-posedness theory in this critical case (as well as non-critical cases). To this end, we approach to the matter in a new way based on a weighted $L^p$ setting which seems to be more suitable to perform a finer analysis for this model. This is because it makes it possible to handle the singularity $|x|^{-\alpha}$ in the nonlinearity more effectively. This observation is a core of our approach that covers the critical case successfully.'
address: 'Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea'
author:
- 'Jungkwon Kim, Yoonjung Lee and Ihyeok Seo'
title: 'On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case'
---
[^1]
Introduction
============
In this paper we are concerned with the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation (INLS) $$\label{INLS}
\left\{
\begin{aligned}
&i \partial_{t} u + \Delta u =\lambda |x|^{-\alpha} |u|^{\beta} u,\quad (x,t) \in \mathbb{R}^d \times \mathbb{R}, \\
&u(x, 0)=u_0(x),
\end{aligned}
\right.$$ where $\alpha,\beta>0$ and $ \lambda = \pm 1$. The equation is called *focusing* INLS when $\lambda=-1$ and *defocusing* INLS when $\lambda=1$. This type of equation arises naturally in nonlinear optics and plasma physics for the propagation of laser beams in an inhomogeneous medium ([@B; @TM]). The case $\alpha = 0$ is the classical nonlinear Schrödinger equation (NLS) whose well-posedness theory in Sobolev spaces $H^s$ has been extensively studied over the past several decades and is well understood ([@CW; @CW2; @GV3; @GV4; @K; @Ts]). However, much less is known about the INLS equation which has drawn attention in recent years.
Before reviewing known results for the Cauchy problem , we recall the critical Sobolev index from which one can divide the matter into three cases. Note first that if $u(x,t)$ is a solution of so is $u_{\lambda}(x,t)= \lambda^{\frac{2-\alpha}{\beta}} u(\lambda x, \lambda^2 t)$, with the initial data $u_{\lambda,0}(x)=u_\lambda(x,0)$ for all $\lambda>0$. We then easily see $$\|u_{\lambda,0}\|_{\dot{H}^s} = \lambda^{s+\frac{2-\alpha}{\beta}-\frac{d}{2}}\|u_0 \|_{\dot{H}^s}$$ from which the critical Sobolev index is given by $s_{c}= d/2-(2-\alpha)/\beta$ which determines the scale-invariant Sobolev space $\dot{H}^{s_c}$. In this regard, the case $s_{c}=0$ (alternatively $\beta =\frac{4-2\alpha}{d}$) is referred to as the mass-critical (or $L^2$-critical). If $s_{c}=1$ (alternatively $\beta =\frac{4-2\alpha}{d-2}$) the problem is called the energy-critical (or $H^1$-critical), and finally it is known as the mass-supercritical and energy-subcritical if $0 < s_{c} < 1$. Therefore, we restrict our attention to the cases where $0\leq s_c\leq1$. We also assume $d\geq3$ to make the review shorter.
The well-posedness for the INLS equation was first studied by Genoud-Stuart [@GS] in the sense of distribution. Using the abstract theory developed by Cazenave [@C], they showed that the focusing INLS equation with $0<\alpha<2$ is locally well-posed in $H^1(\mathbb{R}^d)$ if $0<\beta<\frac{4-2\alpha}{d-2}$. In this case, Genoud [@Ge] and Farah [@Fa] also showed how small should be the initial data to have global well-posedness, respectively, in the spirit of Weinstein [@We] and Holmer-Roudenko [@HR] for the classical case $\alpha=0$.
In all of the previous results mentioned above, the solution has been constructed in the energy space $H^1$ smaller than $L^2$. By comparison, Guzmán [@GU] used the standard contraction mapping argument based on the known classical Strichartz estimates to establish the well-posedness for the INLS equation in various Sobolev spaces $H^s$, $s\leq1$. His result in $H^1$ was also improved by Dinh [@D] extending the validity of $\alpha$ in two and three dimensions. More importantly, Guzmán showed that with $0<\alpha<2$ is locally well-posed in $L^2$ larger than the energy space $H^1$ if $0<\beta<\frac{4-2\alpha}{d}$. He also treated the local well-posedness in $H^s$ for $\max\{0,s_c\}<s\leq1$ if $0<\beta<\frac{4-2\alpha}{d-2s}$. But here, the critical case $\beta=\frac{4-2\alpha}{d-2s}$ with $0\leq s\leq 1$ and $d\geq3$ is left unsolved. Unlike the NLS equation where $\alpha=0$ ([@CW; @CW2]), the conventional approach does not work for this case. It is still an open problem.
The main contribution of this paper is to develop the well-posedness theory in this critical case as well as the non-critical case. To this end, we approach to the matter in a new way based on a weighted $L^p$ setting which does seem to be more suitable to perform a finer analysis for the INLS model. This is because it makes it possible to deal with the singularity $|x|^{-\alpha}$ in the nonlinearity more effectively. This observation is a core of our approach that covers the critical case successfully.
Before stating our results, we introduce the following weighted space-time norms $$\|f\|_{L_{t}^q L_{x}^r(|x|^{-r\gamma})}= \left( \int_{\mathbb{R}} \left(\int_{\mathbb{R}^d} |x|^{-r\gamma} |f(x)|^r dx \right)^{\frac{q}{r}} dt\right)^{\frac{1}{q}} \nonumber$$ where $1\leq r< \infty$ and $\gamma\geq0$. Our first result is then the following local well-posedness for the INLS equation in $L^2$ up to the $L^2$-critical case, $\beta=(4-2\alpha)/d$.
\[Thm1\] Let $d \geq 3$, $0 < \alpha < 2 $ and $0 < \beta \leq (4-2\alpha)/d$. Then for $u_0\in L^2(\mathbb{R}^d)$ there exist $T>0$ and a unique solution $u \in C([0,T] ; L^{2}) \cap L^{q}([0,T] ; L^{r}(|x|^{-r \gamma}))$ of the problem for $$\label{condition23}
\max \lbrace 0,\, \frac{\alpha-1}{\beta+1} \rbrace <\gamma < \min \{ 1,\, \frac{\alpha}{\beta+1} \}$$ and $(q, r)$ satisfying $$\label{condition21}
\frac{2}{q}=d( \frac{1}{2}-\frac{1}{r}) +\gamma, \quad \frac{\gamma}{2} < \frac{1}{q} \leq \frac{1}{2},$$ $$\label{condition22}
\max \{\frac{1}{2(\beta+1)},\, \frac{d-2}{2d}+\frac{\gamma}{d}\}<\frac{1}{r}
<\min \{\frac{d-2(\alpha-1)}{2d(\beta+1)}+\frac{\gamma}{d},\,\frac{1}{2}\}.$$ Furthermore, the continuous dependence on initial data holds.
As we shall see in the proof of the theorem, we can give a precise estimate for the life span of the solution according to the size of the initial data, $T\sim\|u_0\|_{L^2}^{-4\beta/(4-2\alpha-d\beta)}$, in the subcritical case. Thanks to the mass conservation for the INLS equation, $$\label{mass}
\textit{Mass} \equiv M[u(t)] = \int_{\mathbb{R}^d} |u(x, t)|^2 dx = M[ u_0 ],$$ we can then apply the local result repeatedly, preserving the length of the time interval to get a global solution. However, the situation for the critical case is quite different. In this case, the local solution exists in a time interval depending on the data $u_0$ itself and not on its norm. Thus, the conservation does not guarantee the existence of a global solution any more. For this reason, $\|u_0\|_{L^2}$ is assumed to be small for the critical case in the following global result.
\[cor\] Under the same conditions as in Theorem \[Thm1\], the local solution extends globally in time with $u \in C([0,\infty) ; L^2) \cap L^{q}([0,\infty) ; L^{r}(|x|^{-r \gamma}))$ for $u_0\in L^2(\mathbb{R}^d)$. Particularly in the critical case $\beta=(4-2\alpha)/d$, $\|u_0\|_{L^2}$ is assumed to be small and the solution scatters in $L^2$, i.e., there exists $\varphi\in L^2$ such that $$\lim_{t\rightarrow\infty}\|u(t)-e^{it\Delta}\varphi\|_{L_x^2}=0.$$
When $\alpha = 0$, the INLS model becomes the classical NLS equation which has been the subject of intensive work for a long time. In this case, the $L^2$ theory was obtained by Tsutsumi [@Ts] when $0<\beta<4/d$. The $L^2$-critical case $\beta=4/d$ was later treated by Cazenave-Weissler [@CW]. Particularly when $\alpha=0$ in our approach, it is deduced that $\gamma=0$ (see Remark \[opl\]), and hence resulting results naturally cover this classical results for the NLS equation.
We also obtain the following local well-posedness in $H^s$ for $s>0$ up to the critical case $\beta=(4-2\alpha)/(d-2s)$ which is equivalent to $s_c=s$.
\[Thm2\] Let $d \geq 3$ and $0<s<1/3$. Assume that $$\max\{\frac{26-3d}{12},\, \frac{12s+4ds-8s^2}{d+4s}\}<\alpha<2$$ and $$\max\{0,\, \frac{10s-2\alpha}{d-6s}\}< \beta \leq \frac{4-2\alpha}{d-2s}.$$ Then for $u_0\in H^s(\mathbb{R}^d)$ there exist $T>0$ and a unique solution $u \in C([0,T] ; H^{s}) \cap L^{q}([0,T] ; L^{r}(|x|^{-r \gamma}))$ of the problem if $$\label{condition33}
\max \{3s,\, \frac{\alpha+s-1}{\beta+1}\} < \gamma < \min \{1+s,\, \frac{\alpha-s}{\beta+1},\ \frac{d\beta+2\alpha-4s}{2(\beta+1)} \}$$ and $(q, r)$ satisfies $$\label{condition31}
\frac{2}{q}=d( \frac{1}{2}-\frac{1}{r}) +\gamma-s, \quad \frac{\gamma-s}{2}< \frac{1}{q} \leq \frac{1}{2},$$ $$\label{condition32}
\max \lbrace \frac{1}{2(\beta+1)},\, \frac{d-2s-2}{2d}+\frac{\gamma}{d} \rbrace < \frac{1}{r} < \min \lbrace \frac{d-2s-2(\alpha-1)}{2d(\beta+1)} + \frac{\gamma}{d},\, \frac{1}{2}\rbrace,$$ $$\frac{1}{2(\beta+1)} + \frac{2s-\alpha}{d(\beta+1)} + \frac{\gamma}{d} < \frac{1}{r}.$$
When $\beta\leq(4-2\alpha)/d$, we already have the well-posedness result in $L^2$ which implies automatically a result in $H^s$. Hence an essential part in the latter case is when $\beta>(4-2\alpha)/d$. The range of $\beta$ in the theorem contains this range when $\alpha\geq5s$, and when $\alpha<5s$ if we further assume $\alpha\geq\frac{5ds-2d+12s}{6s}$.
One of the most basic tools for the well-posedness of nonlinear dispersive equations is the contraction mapping principle. The key ingredient in this argument is the availability of Strichartz estimates. Indeed, Guzmán [@GU] makes use of the fractional product and chain rules to derive a contraction from the nonlinearity in based on the classical Strichartz estimates. Unfortunately, this approach does not work for the critical case. In our approach we make use of the weighted Strichartz estimates (see and ) which deal with the singularity $|x|^{-\alpha}$ in the nonlinearity more effectively. More interestingly, we take advantage of the smoothing estimates (see or ) which makes it possible to apply more directly the contraction mapping principle without using the fractional product and chain rules. As consequences, we can handle the $H^s$-critical cases, and even $u \in L_t^{q}L_x^{r}(|x|^{-r \gamma}))$ in which $L_x^r$ is less restrictive than the usual $H_x^{s,r}$. Indeed, there are no less restrictive alternatives than $L_x^r$ within the framework of Strichartz estimates. These aspects are a core of our approach which establishes the well-posedness theory up to the critical case successfully.
\[Prop1\] Let $d \geq 3 $ and $0\leq s<1/2$. Assume that $(q,r)$ and $(\tilde{q}, \tilde{r})$ satisfy $$\label{adm}
\frac{2}{q}=d\left(\frac{1}{2}-\frac{1}{r}\right)+\gamma -s,\quad \frac{\gamma-s}{2}< \frac{1}{q} \leq \frac{1}{2},
\quad \frac{\gamma-s}{2} \leq \frac{1}{r} <\frac{1}{2},$$ $$\label{adm2}
\frac{2}{\tilde{q}}=d\left(\frac{1}{2}-\frac{1}{\tilde{r}}\right)+\tilde{\gamma}+s,
\quad \frac{\tilde{\gamma}+s}{2}< \frac{1}{\tilde{q}} \leq \frac{1}{2},
\quad \frac{\tilde{\gamma}+s}{2}\leq \frac{1}{\tilde{r}} < \frac{1}{2},$$ for $3s < \gamma < 1+s$ and $s < \tilde{\gamma}<1-s$. Then we have $$\label{T}
\big\| e^{it\Delta} f \big\|_{L_t^{q} L_{x}^{r} (|x|^{-r\gamma})} \lesssim \| f\|_{\dot{H}^s},$$ $$\label{T*}
\bigg\| \,|\nabla|^s\int_{-\infty}^\infty e^{-i\tau\Delta} F(\cdot, \tau)d\tau \bigg\|_{L^2 } \lesssim \| F \|_{L_t^{\tilde{q}'}L_x^{\tilde{r}'} (|x|^{\tilde{r}'\tilde{\gamma}})},$$ and if we additionally assume $q>\tilde{q}'$ $$\label{TT*}
\bigg\| \int_{0} ^{t} e^{i(t-\tau)\Delta} F(\cdot,\tau)d\tau \bigg\|_{L_t^{q}L_x^{r} (|x|^{-r\gamma})} \lesssim
\| F \|_{L_t^{\tilde{q}'}L_x^{\tilde{r}'} (|x|^{\tilde{r}'\tilde{\gamma}})}.$$
\[opl\] Particularly when $s=0$, the above three estimates also hold for the case where $\gamma=0$ and $\tilde{\gamma}=0$, including $(q,r)=(\infty,2)$ and $(\tilde{q},\tilde{r})=(\infty,2)$. The first two estimates and also hold for $(q,r)=(2,2)$ with $\gamma=1+s$ and $(\tilde{q},\tilde{r})=(2,2)$ with $\tilde{\gamma}=1-s$, respectively. Using these cases in our approach, one is led to have the theorems up to the boundary points in some ranges of $q,r,\gamma$. But this is an insignificant part, and hence we omit the details.
The first conditions in and are just the scaling conditions for which the estimates hold.
It should be also noted that is equivalent to $$\label{smoothing}
\big\|\,|\nabla|^s e^{it\Delta} f \big\|_{L_t^{\tilde{q}} L_{x}^{\tilde{r}} (|x|^{-\tilde{r}\tilde{\gamma}})} \lesssim \| f \|_{L^2}$$ by duality. This shows the smoothing effect of the Schrödinger propagator. To obtain the well-posedness in $H^s$, $s>0$, we make use of this type of smoothing estimates because it allows us to have the inhomogeneous estimate without derivative from which it is possible to apply directly the contraction mapping principle to the nonlinearity in without using the fractional product and chain rules. Indeed, by the standard $TT^*$ argument, and with the same $s$ implies without derivative. In this regard, we need to have the common range of $s$, $0\leq s<1/2$, for which both and hold, although holds more widely for $0\leq s<(d-2)/2$ (see Section \[sec2\] for details).
*Outline of paper.* In Section \[sec2\], we prove Proposition \[Prop1\] by making use of the complex interpolation between the classical Strichartz estimates and the Kato-Yajima smoothing estimates appealing to the complex interpolation space identities. In Section \[sec3\], we obtain some weighted estimates for the nonlinear term $|x|^{-\alpha}|u|^{\beta}u$ of the INLS equation. These nonlinear estimates will play a crucial role in later sections \[sec4\] and \[sec5\] to obtain the well-posedness results by applying the contraction mapping argument to the nonlinearity along with the estimates in Proposition \[Prop1\].
Throughout this paper, the letter $C$ stands for a positive constant which may be different at each occurrence. We also denote $A\lesssim B$ to mean $A\leq CB$ with unspecified constants $C>0$.
Proof of Proposition \[Prop1\] {#sec2}
==============================
In this section we prove the estimates in Proposition \[Prop1\]. Let $d\geq3$. We then first recall the classical Strichartz estimate $$\label{clastr}
\| e^{it\Delta} f \|_{L_t^{a} L_x^{b}} \lesssim \| f \|_{L^{2}}$$ which holds if and only if $\frac{2}{a}=d \left( \frac{1}{2}-\frac{1}{b} \right)$ and $2\leq a, b\leq \infty$. This estimate was first established by Strichartz [@St] in the diagonal case $a=b$ and then extended to mixed space-time norms as above (see [@GV4; @KT]). It is also necessary for us to make use of the Kato-Yajima smoothing estimate $$\label{weism}
\| \,|\nabla|^\rho e^{it\Delta} f \|_{L_{t,x}^{2}(|x|^{-2(1-\rho)})} \lesssim \| f \|_{L^{2}}$$ which holds if and only if $-\frac{d-2}{2} < \rho < \frac{1}{2}$. This estimate was first discovered by Kato and Yajima [@KY] for $0\leq \rho<\frac12$ and Ben-Artzi and Klainerman [@BK] gave an alternate proof of this result. Since then, the full range was obtained by Sugimoto [@Su], although it was later shown by Vilela [@V] that the range is indeed optimal (see also [@W]).
Now we make use of the complex interpolation between these two estimates and by appealing to the following complex interpolation space identities (see [@BL]).
\[idd\] Let $0<\theta<1$, $1\leq p_0,p_1<\infty$ and $s_0,s_1\in\mathbb{R}$. Then the following identities hold:
- Given two complex Banach spaces $A_0,A_1$, $$(L^{p_0}(A_0),L^{p_1}(A_1))_{[\theta]}=L^p((A_0,A_1)_{[\theta]}),$$ and $$(L^{p_0}(w_0),L^{p_1}(w_1))_{[\theta]}=L^p(w),$$ with $1/p=(1-\theta)/p_0+\theta/p_1$ and $w=w_0^{p(1-\theta)/p_0}w_1^{p\theta/p_1}$.
- $$(\dot{H}^{s_0},\dot{H}^{s_1})_{[\theta]}=\dot{H}^s$$ with $s=(1-\theta)s_0+\theta s_1$, $s_0\neq s_1$.
Here, $(\cdot\,,\cdot)_{[\theta]}$ denotes the complex interpolation functor.
Using the complex interpolation between and , we first see $$\| e^{it\Delta} f \|_{(L_t^{a}L_x^{b},L_t^{2}L_x^{2}(|x|^{-2(1-\rho)}))_{[\theta]}}
\lesssim \| f \|_{(\dot{H}^0,\dot{H}^{-\rho})_{[\theta]}},$$ and then we make use of Lemma \[idd\] to get $$\label{inter}
\| e^{it\Delta} f \|_{L_t^{q} L_x^{r}(|x|^{-r\gamma})} \lesssim \left\| f \right\|_{\dot{H}^{-\sigma}}$$ where $$\label{cdo}
\frac1q=\frac{1-\theta}{a}+\frac{\theta}{2},\quad \frac1r=\frac{1-\theta}{b}+\frac{\theta}{2},
\quad\gamma=(1-\rho)\theta,\quad \sigma=\rho\theta$$ under the conditions $$\label{under}
2\leq a,b\leq \infty,\ (a,b)\neq(\infty,2),\ \frac{2}{a}=d \left( \frac{1}{2}-\frac{1}{b} \right),\ 0<\theta<1,\ -\frac{d-2}{2} < \rho < \frac{1}{2}.$$ By eliminating the redundant exponents $a,b,\rho,\theta$ here, all the conditions on $q,r,\gamma,\sigma$ for which holds are summarized as follows: $$\label{inttt0}
0< \gamma+\sigma < 1,\quad -\frac{\gamma(d-2)}{d} < \sigma < \gamma,$$ $$\label{intecon0}
\frac{2}{q}=d \left( \frac{1}{2}-\frac{1}{r} \right)+\gamma+\sigma,
\quad \frac{\gamma+\sigma}{2} \leq \frac{1}{q}, \frac{1}{r} \leq \frac{1}{2},\quad(\frac1q,\frac1r)\neq(\frac{\gamma+\sigma}{2},\frac12).$$ Indeed, from the last two conditions in it follows that $\theta=\gamma+\sigma$. The third condition in implies $2/q=d(1/2-1/r)+\theta$. Hence the first conditions in and are concluded. Combining the last conditions in and , we see $-(d-2)\theta/2<\sigma<\theta/2$, and then substituting $\theta=\gamma+\sigma$ implies the second condition in . Finally, applying the first two conditions in with $\theta=\gamma+\sigma$ to the first two ones in , the last two conditions in are concluded.
Now let $s\geq0$. By substituting $-s$ for $\sigma$ in , we obtain the first estimate in the proposition because implies directly the condition and implies $\frac{d}{d-2}s<\gamma<1+s$ which is wider than $3s<\gamma<1+s$. On the other hand, we substitute $s$, $\tilde{q}$, $\tilde{r}$, $\tilde{\gamma}$ for $\sigma$, $q$, $r$, $\gamma$, respectively in to obtain which is equivalent to the second estimate in the proposition. In this case, implies directly the condition and it is not difficult to see that implies the condition $s<\tilde{\gamma}<1-s$. Hence the common range of $s$ for which both and hold is given by $0\leq s<1/2$. This is the reason why we choose $3s<\gamma<1+s$ instead of $\frac{d}{d-2}s<\gamma<1+s$ as the range of $\gamma$ for which holds.
It remains to prove . By the $TT^*$ argument, and with the same value of $s$ imply $$\bigg\| \int_{-\infty}^\infty e^{i(t-\tau)\Delta} F(\cdot,\tau)d\tau \bigg\|_{L_t^{q}L_x^{r} (|x|^{-r\gamma})} \lesssim
\| F \|_{L_t^{\tilde{q}'}L_x^{\tilde{r}'} (|x|^{\tilde{r}'\tilde{\gamma}})}.$$ By applying the following Christ-Kiselev lemma [@CK], we now get for $q>\tilde{q}'$ as desired.
Let $X$ and $Y$ be two Banach spaces and let $T$ be a bounded linear operator from $L^\alpha(\mathbb{R};X)$ to $L^\beta(\mathbb{R};Y)$ such that $$Tf(t)=\int_{\mathbb{R}} K(t,s)f(s)ds.$$ Then the operator $$\widetilde{T}f(t)=\int_{-\infty}^t K(t,s)f(s)ds$$ has the same boundedness when $\beta>\alpha$, and $\|\widetilde{T}\|\lesssim\|T\|$.
In particular, if $s=0$, the estimate for the case $\gamma=0$ is just the same as , and for the case $\tilde{\gamma}=0$ follows from its dual estimate. By the $TT^*$ argument together with these cases, follows for the case where $s=0$ and $\gamma,\tilde{\gamma}=0$. It follows also directly from that and hold for $(q,r)=(2,2)$ with $\gamma=1+s$ and $(\tilde{q},\tilde{r})=(2,2)$ with $\tilde{\gamma}=1-s$, respectively. This shows Remark \[opl\].
Nonlinear estimates {#sec3}
===================
In this section we obtain some weighted estimates for the nonlinearity of the INLS equation using the same spaces as those involved in the weighted Strichartz estimates in Proposition \[Prop1\]. These nonlinear estimates will play a crucial role in later sections to obtain the well-posedness results applying the contraction mapping principle.
The $L^2$ case
--------------
We first establish the nonlinear estimates which are necessary for us to obtain the well-posedness in $L^2$ in the next section.
\[le1\] Let $d\ge 3$, $0< \alpha < 2 $ and $0 < \beta \leq (4-2\alpha)/d$. Assume that the exponents $q,r,\gamma$ satisfy all the conditions given as in Theorem \[Thm1\]. Then there exist certain $\tilde{q},\tilde{r},\tilde{\gamma}$ satisfying all the conditions given as in Proposition \[Prop1\] with $s=0$ for which $$\begin{aligned}
\label{Festi}
\left\| |x|^{-\alpha} |u|^{\beta} v \right\|_{L_t^{\tilde{q}'}(I ; L_{x}^{\tilde{r}'}(|x|^{\tilde{r}' \tilde{\gamma}}))}
\leq T^{\theta_0} \left\| u \right\|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r\gamma}))} ^{\beta} \left\| v \right\|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r\gamma}))}\end{aligned}$$ holds for any finite interval $I=[0,T]$ and $$\label{th1}
\theta_0 = -\frac{d\beta}{4} + 1 - \frac{\alpha}{2}.$$
It should be noted that $\theta_0\geq0$ if and only if $\beta\leq(4-2\alpha)/d$.
Let us first consider the exponent pairs $(q, r,\gamma)$ and $(\tilde{q}, \tilde{r},\widetilde{\gamma})$ satisfying $q>\tilde{q}'$, $$\label{r}
\frac{2}{q} = d(\frac{1}{2} - \frac{1}{r}) +\gamma,
\quad \frac{\gamma}{2} < \frac{1}{q} \leq \frac{1}{2},\quad\frac{\gamma}{2} \leq \frac{1}{r} <\frac{1}{2},
\quad 0< \gamma < 1,$$ $$\label{r2}
\frac{2}{\tilde{q}} = d(\frac{1}{2} - \frac{1}{\tilde{r}}) + \tilde{\gamma},
\quad \frac{\tilde{\gamma}}{2} < \frac{1}{\tilde{q}} \leq \frac{1}{2},
\quad\frac{\tilde{\gamma}}{2} \leq \frac{1}{\tilde{r}} < \frac{1}{2},
\quad 0< \tilde{\gamma} < 1,$$ which are just the same given as in Proposition \[Prop1\] with $s=0$ for which the estimates therein hold. We then let $$\label{setting}
\frac{1}{\tilde{q}'}= \theta_0 + \frac{\beta+1}{q},\quad \frac{1}{\tilde{r}'}=\frac{\beta+1}{r} \quad \textnormal{and} \quad \tilde{\gamma}-\alpha=-\gamma(\beta+1).$$ By combining the first conditions in and together with , it is not difficult to see that $\theta_0$ is determined by $$\theta_0 = \frac{1}{\tilde{q}'} - \frac{\beta+1}{q} = -\frac{d\beta}{4} + 1 - \frac{\alpha}{2}$$ as in .
Next we use Hölder’s inequality repeatedly along with to get $$\begin{aligned}
\left\| |x|^{-\alpha} |u|^{\beta} v \right\|_{L_t^{\tilde{q}'}(I ; L_{x}^{\tilde{r}'}(|x|^{\tilde{r}' \tilde{\gamma}}))}
&=\left\||x|^{\tilde{\gamma}-\alpha}|u|^{\beta} v \right\|_{L_t^{\tilde{q}'}(I;L_x^{\tilde{r}'})} \\
& = \left\| |x|^{-\gamma(\beta+1)} |u|^{\beta} v \right\|_{L_t^{\tilde{q}'}(I ; L_{x}^{\frac{r}{\beta+1}})} \\
& \leq T^{\theta_0} \left\| |x|^{-\gamma(\beta+1)} |u|^{\beta} v \right\|_{L_t^{\frac{q}{\beta+1}}(I ; L_{x}^{\frac{r}{\beta+1}})} \\
& \leq T^{\theta_0} \left\| u \right\|_{L_t^{q}(I; L_{x}^{r} (|x|^{-r\gamma}))}^{\beta} \left\| v \right\|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r\gamma}))}\end{aligned}$$ as desired in .
Now it remains to show the requirements on $q$, $r$ and $\gamma$ for which holds. We first use the last two conditions in to transform the exponents $\tilde{r}$ and $\tilde{\gamma}$ in to $r$ and $\gamma$, as follows: $$\label{r11}
\frac{2}{\tilde{q}} = d(\frac{\beta+1}{r}-\frac{1}{2}) + \alpha-\gamma(\beta+1),$$ $$\label{r111}
\frac{\alpha-\gamma(\beta+1)}{2} < \frac{1}{\tilde{q}}\leq \frac{1}{2},\quad \frac{1}{2(\beta+1)} < \frac{1}{r} \leq \frac{2-\alpha}{2(\beta+1)}+\frac{\gamma}{2},$$ and $$\label{rrt}
0< \alpha-\gamma(\beta+1) < 1.$$ Next we substitute into the first one in to get $$\label{r11'}
\frac{1}{2(\beta+1)} < \frac{1}{r} \leq \frac{d-2(\alpha-1)}{2d(\beta+1)} +\frac{\gamma}{d}.$$ Similarly we get $$\label{rlower}
\frac{d-2}{2d} + \frac{\gamma}{d} \leq \frac{1}{r} < \frac{1}{2}$$ by substituting the first one into the second one in . Notice here that the range of $1/r$ in may be replaced by because $\frac{\gamma}{2}\leq\frac{d-2}{2d}+\frac{\gamma}{d}$ when $\gamma<1$, and similarly using , it is easy to see that the second one in may be replaced by . Finally, using the last two conditions in and the first conditions in and , one can easily see that $q>\tilde{q}'$ is transformed to $\frac1r<\frac{2-\alpha}{d\beta}+\frac{\gamma}{d}$, but this may be replaced by . From the first one in , we also require $1/\tilde{q}'-1/q=\theta_0+\beta/q>0$, but this is satisfied since $\frac{1}{q}>\frac{\gamma}{2}>0$.
Hence all the conditions on $q$, $r$ and $\gamma$ which we require are summarized as follows: $$\frac{2}{q} = d(\frac{1}{2} - \frac{1}{r}) + \gamma,\quad \frac{\gamma}{2} < \frac{1}{q} \leq \frac{1}{2},$$ $$\max{\{\frac{1}{2(\beta+1)},\frac{d-2}{2d}+\frac{\gamma}{d}\}}<\frac{1}{r}
<\min{\{\frac{d-2(\alpha-1)}{2d(\beta+1)}+\frac{\gamma}{d},\frac{1}{2}\}},$$ and $$\max \{0,\ \frac{\alpha-1}{\beta+1} \} < \gamma < \min \{ 1, \frac{\alpha}{\beta+1} \},$$ which are the same as in Theorem \[Thm1\].
The $H^s$ case
--------------
Now we obtain some weighted estimates for the nonlinear term which will be used for the well-posedness in $H^s$ in Section \[sec5\]. In comparison with the previous lemma, it is more delicate to obtain Lemma \[le2\] below in which we need to get an additional estimate having the term $|\nabla|^{-s}$ that is quite difficult to handle in the weighted spaces. For this, we will make use of a weighted version of the Sobolev embedding.
\[le2\] Let $d \geq 3$ and $0<s<1/3$. Assume that $$\label{102}
\max\{\frac{26-3d}{12},\frac{12s+4ds-8s^2}{d+4s}\}<\alpha<2$$ and $$\label{1012}
\max\{0,\frac{10s-2\alpha}{d-6s}\}< \beta \leq \frac{4-2\alpha}{d-2s}.$$ If the exponents $q,r,\gamma$ satisfy all the conditions given as in Theorem \[Thm2\], then there exist certain $\tilde{q}_i,\tilde{r}_i,\tilde{\gamma}_i$, $i=1,2$, satisfying all the conditions given as in Proposition \[Prop1\] with $s >0$ for which $$\label{F1}
\left \||x|^{-\alpha} |u|^{\beta} v \right\|_{L_t^{\tilde{q}_1'}(I;L_x^{\tilde{r}_1'}(|x|^{\tilde{r}_1'\tilde{\gamma}_1}))}\lesssim T^{\theta_1} \|u\|_{L_t^q(I;L_x^r(|x|^{-r \gamma}))}^{\beta}\|v\|_{L_t^q(I;L_x^r(|x|^{-r \gamma}))}$$ and $$\begin{aligned}
\label{F2}
\nonumber\big\| |\nabla|^{-s} (|x|^{-\alpha} |u|^{\beta} v)&\big\|_{L_t^{\tilde{q}_2'}(I;L_x^{\tilde{r}_2'}(|x|^{\tilde{r}_2'\tilde{\gamma}_2}))}\\
&\lesssim T^{\theta_2} \|u\|_{L_t^q(I;L_x^r(|x|^{-r \gamma}))}^{\beta} \|v\|_{L_t^q(I;L_x^r(|x|^{-r \gamma}))}\end{aligned}$$ hold for any finite interval $I=[0, T]$ and $$\label{th2}
\theta_1= -\frac{d\beta}{4}+1-\frac{\alpha}{2}+\frac{s\beta}{2},\quad \quad \theta_2=-\frac{d\beta}{4} +1-\frac{\alpha}{2} +\frac{s(\beta+1)}{2}.$$
We note that $\beta \leq (4-2\alpha)/(d-2s)$ if and only if $\theta_1 \ge 0 $ (or $\theta_2 \ge s/2$).
First we consider the exponent pairs $(q,r,\gamma)$ and $(\tilde{q}_i, \tilde{r}_i, \tilde{\gamma}_i)$ for $i=1,2$ satisfying $q>\tilde{q}_i'$, $$\label{Hs2}
\frac{2}{q}=d ( \frac{1}{2}- \frac{1}{r} )+\gamma-s,\quad
\frac{\gamma-s}{2}< \frac{1}{q} \leq \frac{1}{2},\quad
\frac{\gamma-s}{2}\leq \frac{1}{r} < \frac{1}{2},
\quad 3s < \gamma < 1+s,$$ $$\label{Hs3}
\frac{2}{\tilde{q}_i} = d ( \frac{1}{2}- \frac{1}{\tilde{r}_i} ) + \tilde{\gamma}_i+s,
\ \frac{\tilde{\gamma}_i+s}{2}< \frac{1}{\tilde{q}_i} \leq \frac{1}{2},
\ \frac{\tilde{\gamma}_i+s}{2}\leq \frac{1}{\tilde{r}_i} < \frac{1}{2},
\ s < \tilde{\gamma}_i < 1-s,$$ which are just the same given as in Proposition \[Prop1\] with $0<s<1/2$ for which the estimates therein hold.
### Proof of
The first estimate is obtained in a similar way as in the previous subsection. Let us first set $$\label{set2}
\frac{1}{{\tilde{q}_1}'}=\theta_1+\frac{\beta+1}{q},\quad \frac{1}{{\tilde{r}_1}'}=\frac{\beta+1}{r} \quad \textnormal{and} \quad \tilde{\gamma}_1 -\alpha = -\gamma(\beta+1).$$ By combining the first conditions in and for $i=1$ together with , it is then easy to see that $\theta_1$ is determined by $$\theta_1=\frac{1}{{\tilde{q}_1}'}-\frac{\beta+1}{q}=-\frac{d\beta}{4}+1-\frac{\alpha}{2}+\frac{s\beta}{2}$$ as in .
Next we apply Hölder’s inequality repeatedly along with to get $$\begin{aligned}
\left \||x|^{-\alpha} |u|^{\beta} v \right\|_{L_t^{\tilde{q}_1'}(I;L_x^{\tilde{r}_1'}(|x|^{\tilde{r}_1'\tilde{\gamma}_1}))}
&= \left\||x|^{\tilde{\gamma}_1-\alpha}|u|^{\beta} v \right\|_{L_t^{\tilde{q}_1'}(I;L_x^{\tilde{r}_1'})} \cr
&= \left\||x|^{ -\gamma(\beta+1)} |u|^{\beta} v \right\|_{L_t^{\tilde{q}_1'}(I;L_x^{\frac{r}{\beta+1}})} \cr
& \leq T^{\theta_1} \left\||x|^{ -\gamma(\beta+1)}|u|^{\beta} v \right\|_{L_t^{\frac{q}{\beta+1}}(I;L_x^{\frac{r}{\beta+1}})} \cr
& \leq T^{\theta_1} \left\|u \right\|_{L_t^{q}(I;L_x^{r}(|x|^{-r\gamma}))}^{\beta} \left\| v \right\|_{L_t^{q}(I;L_x^{r}(|x|^{-r\gamma}))}
\end{aligned}$$ as desired in .
Now we only need to show the requirements on $q$, $r$ and $\gamma$ for which holds. Using the last two conditions in , we first change the exponents $\tilde{r}_1$ and $\tilde{\gamma}_1$ in for $i=1$ into $r$ and $\gamma$, as follows: $$\begin{gathered}
\label{q1} \frac{2}{\tilde{q}_1} = -\frac{d}{2}+ \frac{d(\beta+1)}{r}+\alpha+s-\gamma(\beta+1), \\
\label{q2} \frac{\alpha-\gamma(\beta+1)+s}{2}< \frac{1}{\tilde{q}}_1 \leq \frac{1}{2},
\quad \frac{1}{2(\beta+1)}< \frac{1}{r} \leq \frac{2-\alpha-s}{2(\beta+1)}+\frac{\gamma}{2},\end{gathered}$$ $$\begin{gathered}
\label{g1}
\frac{\alpha+s-1}{\beta+1}< \gamma <\frac{\alpha-s}{\beta+1}.\end{gathered}$$ Next we insert into the first one in to get $$\label{rrr}
\frac{1}{2(\beta+1)}< \frac{1}{r} \leq \frac{1 - \alpha - s}{d(\beta+1)}+\frac{1}{2(\beta+1)}+\frac{\gamma}{d}$$ from which the second one in can be removed using . Finally, all the requirements deduced from $q>\tilde{q}_1'$ are already satisfied by the other requirements similarly as before.
In summary, the requirements on $q$, $r$ and $\gamma$ for which holds are , and which are less restrictive than those in Theorem \[Thm2\]. But we will show that the common requirements for which both and hold are given exactly by those in Theorem \[Thm2\].
### Proof of
Now we have to obtain under the requirements , and on $q$, $r$ and $\gamma$ for which holds. Let us first set $$\label{H1}
\frac{1}{\tilde{q}_2'}=\theta_2+\frac{\beta+1}{q}$$ and then use Hölder’s inequality in $t$ to the left-hand side of to get $$\begin{aligned}
\left\|\, |\nabla|^{-s} (|x|^{-\alpha}|u|^{\beta}v) \right\|_{L_{t}^{\tilde{q}_2'}(I;L_x^{\tilde{r}_2'}(|x|^{\tilde{r}_2'\tilde{\gamma}_2}))}
&=\left\|\,|x|^{\tilde{\gamma}_2} |\nabla|^{-s} (|x|^{-\alpha}|u|^{\beta}v) \right\|_{L_{t}^{\tilde{q}_2'}(I;L_x^{\tilde{r}_2'})} \cr
&\leq T^{\theta_2} \left\|\,|x|^{\tilde{\gamma}_2} |\nabla|^{-s} (|x|^{-\alpha}|u|^{\beta}v)\right\|_{L_{t}^{\frac{q}{\beta+1}}(I;L_x^{{\tilde{r}_2}'})}.\end{aligned}$$ To handle the term $|\nabla|^{-s}$ here, we make use of the following lemma which is a weighted version of the Sobolev embedding.
\[le3\] Let $d\ge 1$ and $0<s<d$. If $$1<p\leq q<\infty,\quad -d/q<b\leq a<d/p' \quad\text{and}\quad a-b-s=d/q-d/p,$$ then $$\label{soso}
\||x|^{b}f\|_{L^q} \leq C_{a, b, p, q} \||x|^{a} |\nabla|^s f\|_{L^p}.$$
Indeed, applying with $a=\alpha-\gamma(\beta+1)$, $b=\tilde{\gamma}_2$, $p=\frac{r}{\beta+1}$, $q=\tilde{r}_2'$, and $f=|x|^{-\alpha}|u|^{\beta}v$, and then using Hölder’s inequality, we get for $0<s<d$ $$\begin{aligned}
\left\|\,|x|^{\tilde{\gamma}_2} |\nabla|^{-s} (|x|^{-\alpha}|u|^{\beta}v) \right\|_{L_{t}^{\frac{q}{\beta+1}}(I;L_x^{\tilde{r}_2'})}
&\lesssim \left\| |x|^{-\gamma(\beta+1)}|u|^{\beta}v \right\|_{L_{t}^{\frac{q}{\beta+1}}(I;L_x^{\frac{r}{\beta+1}})} \cr
&\lesssim \left\| |x|^{-\gamma} u\right\|_{L_{t}^{q}(I;L_x^{r})} ^{\beta} \left\| |x|^{-\gamma} v\right\|_{L_{t}^{q}(I;L_x^{r})}\end{aligned}$$ if $$\label{H3}
1< \frac{r}{\beta+1} \leq \tilde{r}_2' < \infty,$$ $$\label{H5}
-\frac{d}{\tilde{r}_2'}<\tilde{\gamma}_2\leq\alpha-\gamma(\beta+1)<d-\frac{d(\beta+1)}{r},$$ and $$\label{H4}
\tilde{\gamma}_2= \alpha-\gamma(\beta+1) -s -\frac{d}{\tilde{r}_2'}+\frac{d(\beta+1)}{r}.$$ Since $\tilde{\gamma}_2>0$, the first inequality in is redundant, and since the upper bound of $1/r$ in is less than the one which follows from the third inequality in , it is also redundant. Hence is reduced to $$\label{H55}
\tilde{\gamma}_2\leq\alpha-\gamma(\beta+1).$$
Now we replace $\tilde{\gamma}_2$ in the first condition of for $i=2$ with to get $$\label{q2tilde}
\frac{2}{\tilde{q}_2}= -\frac{d}{2}+\alpha-\gamma(\beta+1)+\frac{d(\beta+1)}{r}.$$ Then by inserting and the first one of into , we see that $$\theta_2 =\frac{1}{\tilde{q}_2'}-\frac{\beta+1}{q}=-\frac{d\beta}{4}+1-\frac{\alpha}{2}+\frac{s(\beta+1)}{2}.$$ as in .
By using and , the exponents $\tilde{q}_2$ and $\tilde{\gamma}_2$ in all the inequalities in for $i=2$ can be removed as follows: $$\label{H11}
\frac{1}{r}\leq \frac{1-\alpha}{d(\beta+1)}+\frac{1}{2(\beta+1)}+\frac{\gamma}{d},\quad \frac{1}{\tilde{r}_2} < \frac12,$$ $$\label{H12}
\alpha-\gamma(\beta+1) -\frac{d}{\tilde{r}_2'}+\frac{d(\beta+1)}{r} \leq \frac{2}{\tilde{r}_2}<1,$$ $$\label{H22}
2s < \alpha-\gamma(\beta+1) -\frac{d}{\tilde{r}_2'}+\frac{d(\beta+1)}{r} < 1.$$ Here, and come from the second and third ones in , respectively, while is derived from the last one in . Similarly, and are transformed to $$\label{H33}
0\leq \frac{\beta+1}{r}-\frac{1}{\tilde{r}_2'} <\frac{1}{\tilde{r}_2}\quad\text{and}\quad \frac{d(\beta+1)}{r}-\frac{d}{\tilde{r}_2'} \leq s,$$ respectively. It is immediate that the first inequality in is redundant by . Since $\tilde{r}_2 > 2$, and are reduced to $$\label{rt0}
2s < \alpha-\gamma(\beta+1) -\frac{d}{\tilde{r}_2'}+\frac{d(\beta+1)}{r} \leq \frac{2}{\tilde{r}_2}$$ which is possible for $s<1/\tilde{r}_2$. If $s<1/\tilde{r}_2$, is then reduced to $$\label{rt1}
0 \leq \frac{\beta+1}{r}-\frac{1}{{\tilde{r}_2'}} \leq \frac{s}{d}.$$ In summary, we are reduced to and , which are equivalent to $$\label{rt2}
\frac{2s-\alpha+\gamma(\beta+1)}{d}-\frac{\beta+1}{r}+1 < \frac{1}{\tilde{r}_2} \leq \frac{d-\alpha+\gamma(\beta+1)}{d-2}-\frac{d(\beta+1)}{(d-2)r}$$ and $$\label{rt3}
1-\frac{\beta+1}{r}\leq \frac{1}{\tilde{r}_2} \leq \frac{s}{d} + 1 -\frac{\beta+1}{r}$$ respectively, under the condition $$\label{rt45}
s<\frac1{\tilde{r}_2}<\frac12.$$
Now, it suffices to check that there exists $\tilde{r}_2$ satisfying , and under the conditions , and . To do so, we first make each lower bound of $1/\tilde{r}_2$ in , and less than all the upper bounds in turn. Then we are reduced to $$\label{r22}
\frac{1}{r}< \frac{1-s}{\beta+1} + \frac{2s-\alpha}{d(\beta+1)} + \frac{\gamma}{d}$$ and $$\label{rrr2}
\frac{1}{2(\beta+1)}+\frac{2s-\alpha}{d(\beta+1)}+\frac{\gamma}{d} <\frac{1}{r}.$$ Indeed, starting the process from the lower bound in , we arrive at , $s-\alpha+\gamma(\beta+1)<0$, and , but the second one is redundant by . Similarly from the lower bound in , we arrive at $\frac1r\leq\frac{2-\alpha}{2(\beta+1)}+\frac\gamma2$, $0\leq \frac sd$, and $\frac{1}{2(\beta+1)}<\frac1r$. But here the first and third ones are replaced by under , and the second one is redundant. Finally from the lower bound in , we arrive at , $\frac{1}{r}< \frac{1-s}{\beta+1}+\frac{s}{d(\beta+1)}$, and $s<1/2$, but the second one is replaced by using and the last one is redundant. Nextly, we note that may be also eliminated by using the fact that $d-2-2sd+6s =(d-2)(1-2s)+2s>0$, and we combine the first two conditions in to conclude $$\label{rt4}
\frac{1}{2}-\frac{1-\gamma+s}{d} \leq \frac{1}{r} < \frac{1}{2},$$ which replaces the third condition in since $\gamma-s<1$. Finally, all the requirements deduced from $q>\tilde{q}_2'$ are already satisfied by the other requirements similarly as before.
Hence, all the requirements on $r$ are , and for which both and hold. In summary, $$\label{r3}
\max \lbrace \frac{1}{2(\beta+1)},\ \frac{d-2s-2}{2d}+\frac{\gamma}{d} \rbrace < \frac{1}{r}
< \min \lbrace \frac{d-2s-2(\alpha-1)}{2d(\beta+1)} + \frac{\gamma}{d}, \frac{1}{2} \rbrace$$ and $$\label{r4}
\frac{1}{2(\beta+1)} + \frac{2s-\alpha}{d(\beta+1)} + \frac{\gamma}{d} < \frac{1}{r},$$ which are the same as in Theorem \[Thm2\].
Next we check that there exist $r$ satisfying and . For this, we make each lower bound of $1/r$ in them less than all the upper bounds in in turn. Then the restrictions on $\gamma$ and $s$ are deduced as $$\label{g2}
\quad \gamma<\frac{d\beta+2\alpha-4s}{2(\beta+1)}$$ and $0<s<1/3$. In fact, staring the process from the first lower bound in , we arrive at $\alpha + s -1 -\gamma(\beta+1) < 0$ and $\beta > 0$, but the former is redundant by as well as $\beta>0$. Similarly from the second lower bound in , we arrive at $\beta < \frac{4-2\alpha}{d-2-2s}$ and $\gamma < 1+s$ which are obviously redundant. Finally from the lower bound in , we arrive at $s<1/3$ and .
Hence all the requirements on $\gamma$ for which both and hold are $3s< \gamma < 1+s$, and . In summary, $$\label{g4}
\max \{3s,\, \frac{\alpha+s-1}{\beta+1}\} < \gamma < \min \{1+s,\, \frac{\alpha-s}{\beta+1},\ \frac{d\beta+2\alpha-4s}{2(\beta+1)} \}$$ as in Theorem \[Thm2\].
To guarantee $\gamma$ satisfying all the requirements in under $0<s<1/3$, we make each lower bound of $\gamma$ in less than all the upper bounds in turn. Then we are reduced to $$\label{beta1}
\beta < \frac{\alpha-4s}{3s} \quad \textnormal{and} \quad \frac{10s-2\alpha}{d-6s} < \beta.$$ Indeed, starting from the first lower bound, we see $s<1/2$ and . Nextly, from the second lower bound, we have $\alpha-2 < \beta + s\beta$, $s<1/2$ and $6s-2 < d\beta$ which are clearly redundant.
Now the first condition in is satisfied if $\frac{4-2\alpha}{d-2s} < \frac{\alpha-4s}{3s}$, which is equivalent to $$\frac{12s+4ds-8s^2}{d+4s} < \alpha$$ as in Theorem \[Thm2\]. Then we need to check that there exist $\alpha \in (\frac{12s+4ds-8s^2}{d+4s}, 2)$. This is possible if $$\label{funcs}
-4s^2+2s+2ds<d$$ holds for $0<s<1/3$. To show this, we consider a quadratic function $f_1(s) = -4s^2 +2(1+d)s$ which is concave downward. Since $f_1(s)$ is an increasing function for $s \in (0,\frac{d+1}{4})$ including $s \in (0, \frac{1}{3})$, the inequality follows clearly from the fact that $f_1(\frac{1}{3})<d$. Next we check that there exist $\beta$ satisfying the second condition of under $0<s<1/3$. For this, we need to show $ \frac{10s-2\alpha}{d-6s} < \frac{4-2\alpha}{d-2s} $ which is equivalent to $$\label{mcv}
-10s^2+(5d+12-4\alpha)s < 2d.$$ Similarly, we consider a quadratic function $f_2(s)=-10s^2+(5d+12-4\alpha)s$. It is also concave downward and increases for $s < \frac{5d+12-4\alpha}{20}$. Since $\frac{1}{3} < \frac{5d+12-4\alpha}{20}$, we require $f_2(\frac{1}{3}) < 2d$ to show , which is equivalent to $$\frac{26-3d}{12} < \alpha$$ as in Theorem \[Thm2\]. Hence, $\alpha$ and $\beta$ are required as in and . Since $\frac{d}{2} - \frac{2-\alpha}{\beta}\leq s$ (i.e., $\beta \leq \frac{4-2\alpha}{d-2s}$), we finally require $$\frac{d}{2} - \frac{2-\alpha}{\beta} <\frac13$$ which is equivalent to $\beta < \frac{12-6\alpha}{3d-2}$. But here, $\frac{4-2\alpha}{d-2s} < \frac{12-6\alpha}{3d-2}$ for $0<s<1/3$. No more requirement occurs. This completes the proof.
The well-posedness in $L^2$ {#sec4}
===========================
In this section we prove Theorem \[Thm1\] and Corollary \[cor\] by applying the contraction mapping principle along with the weighted Strichartz estimates in Proposition \[Prop1\] with $s=0$. The nonlinear estimates in Lemma \[le1\] play a key role in this step.
The subcritical case
--------------------
By Duhamel’s principle, we first write the solution of the Cauchy problem as $$\label{Duhamel}
\Phi_{u_0}(u) = e^{it \Delta} u_0 - i\lambda \int_{0}^{t} e^{i(t-\tau)\Delta}F(u) d\tau$$ where $F(u)=|\cdot|^{-\alpha} |u(\cdot,\tau)|^{\beta}u(\cdot, \tau)$. For appropriate values of $T,M>0$, we shall show that $\Phi$ defines a contraction map on $$\begin{aligned}
\label{contset}
X(T,M)=\Big\lbrace u\in &C_t(I;L_x^2)\cap L_t^q(I;L_x^r(|x|^{-r\gamma})) :\\
&\sup_{t\in I}\|u\|_{L_x^2}+\|u\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))} \leq M \Big\rbrace.\end{aligned}$$ equipped with the distance$$d(u,v)=\sup_{t\in I}\|u-v\|_{L_x^2}+\|u-v\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}.$$ Here, $I=[0,T]$ and $(q,r,\gamma)$ is given as in Theorem \[Thm1\].
For this, we first show that $\Phi$ is well defined on $X$. Namely, for $u\in X$ $$\label{fgj}
\sup_{t\in I}\|\Phi(u)\|_{L_x^2}+\|\Phi(u)\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))} \leq M.$$ Using and combined with , we obtain $$\| \Phi(u)\|_{L_t^{q}(I ; L_{x}^{r}(|x|^{-r \gamma}))}
\leq C\| u_0 \|_{L^2} + C\left\||x|^{-\alpha}|u|^{\beta}u \right\|_{L_{t}^{\tilde{q}'} (I; L_{x}^{\tilde{r}'}(|x|^{\tilde{r}' \tilde{\gamma}}))}.$$ By applying Lemma \[le1\], it follows then that $$\begin{aligned}
\nonumber\left\| \Phi(u) \right\|_{L_{t}^{q}(I; L_{x}^{r}(|x|^{-r\gamma}))}
&\leq C\| u_0 \|_{L^2} + CT^{\theta_0} \| u \|_{L_t^{q}(I; L_{x}^{r} (|x|^{-r\gamma}))} ^{\beta+1} \\
\label{1j}&\leq C\| u_0 \|_{L^2} + CT^{\theta_0} M^{\beta+1}.\end{aligned}$$ On the other hand, applying Plancherel’s theorem, and in turn, we see $$\begin{aligned}
\label{conv}
\nonumber\sup_{t\in I}\|\Phi(u)\|_{L_x^2} &\leq C\|u_0\|_{L^2} +C\bigg\|\int_{-\infty}^{\infty} e^{-i\tau\Delta}\chi_{[0,t]}(\tau)F(u) d\tau\bigg\|_{L_x^2}\\
\nonumber&\leq C\|u_0\|_{L^2}+C\|F(u)\|_{L_t^{\tilde{q}'}(I; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \widetilde{\gamma}}))}\\
&\leq C\|u_0\|_{L^2}+CT^{\theta_0} M^{\beta+1}.\end{aligned}$$ Hence, if we fix $M=4C\| u_0 \|_{L^2}$ and take $T>0$ such that $$\label{dgh0}
CT^{\theta_0}M^{\beta}\leq\frac 18,$$ we get . In the subcritical case where $\beta<(4-2\alpha)/d$, $T$ carries a positive power $\theta_0$ here. Thus one can give a precise estimate for the life span of the solution according to the size of the initial data, $T\sim\|u_0\|_{L^2}^{-\beta/\theta_0}$.
Next we show that $\Phi$ is a contraction. Namely, for $u,v\in X$ $$\label{alk}
d(\Phi(u),\Phi(v)) \leq \frac{1}{2} d(u,v ).$$ Using the same arguments used in , we see $$\label{cont24}
\sup_{t\in I}\|\Phi(u)-\Phi(v)\|_{L_x^2}
\leq C\|F(u)-F(v)\|_{L_t^{\tilde{q}'}(I; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \widetilde{\gamma}}))}.$$ Then we will show $$\label{cont3}
\| F(u)-F(v) \|_{L_t^{\tilde{q}'}(I; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \tilde{\gamma}}))} \leq \frac{1}{4C} \| u-v \|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}.$$ Indeed, using the following simple inequality $$\label{sin}
\left||u|^{\beta}u-|v|^{\beta}v \right| \leq C \big(|u|^{\beta}|u-v|+|v|^{\beta}|u-v|\big),$$ we are reduced to showing $$\label{cont}
\left\| |x|^{-\alpha} |u|^{\beta} |u-v| \right\|_{L_t^{\tilde{q}'}(I; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \tilde{\gamma}}))} \leq \frac{1}{8C} \left\| u-v \right\|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}$$ by symmetry. For this we apply Lemma \[le1\] with $v$ replaced by $|u-v|$ so that $$\begin{aligned}
\left\| |x|^{-\alpha} |u|^{\beta}|u-v| \right\|_{L_t^{\tilde{q}'}(I; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \tilde{\gamma}}))}
& \leq T^{\theta_0} \| u \|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}^{\beta} \| u-v \|_{L_t^{q}(I; L_{x}^{r} (|x|^{-r \gamma}))} \cr
& \leq T^{\theta_0} M^{\beta} \| u-v \|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))},\end{aligned}$$ which implies because we have chosen $T,M>0$ in so that $T^{\theta_0}M^{\beta}\leq1/(8C)$. On the other hand, $$\|\Phi(u)-\Phi(v)\|_{L_{t}^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}
\leq C\|F(u)-F(v)\|_{L_t^{\tilde{q}'}(I; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \tilde{\gamma}}))}.$$ Now we obtain combining this, and .
Therefore, we have proved that there exists a unique local solution $$u \in C(I ; L^{2}) \cap L^{q}(I ; L^{r}(|x|^{-r \gamma}))$$ with $T\sim\|u_0\|_{L^2}^{-\beta/\theta_0}$. The continuous dependence of the solution $u$ with respect to the initial data $u_0$ follows clearly in the same way: $$\begin{aligned}
d(u,v)&\lesssim d\big(e^{it \Delta}u_0, e^{it\Delta} v_0\big) +d\bigg(\int_{0}^{t} e^{i(t-\tau)\Delta}F(u)d\tau, \int_{0}^{t} e^{i(t-\tau)\Delta}F(v) d\tau\bigg)\\
&\lesssim\|u_0-v_0\|_{L^2}.\end{aligned}$$ Here, $u,v$ are the corresponding solutions for initial data $u_0,v_0$, respectively. Thanks to the mass conservation , the above process can be also iterated on translated time intervals, preserving the length of the time interval comparable to $\|u_0\|_{L^2}^{-\beta/\theta_0}$ to extend the above local solution globally in time. The proof is now complete.
The critical case
-----------------
The critical case requires slightly different arguments and it yields different conclusions. This is due to the fact that the power $\theta_0$ in the above argument becomes zero in this case. This time we cannot gain a small power of $T$ and the smallness must have a different source. For this reason, the global result will follow from the smallness of the initial data. We give the main lines of the proof.
We start from showing that $\Phi$ defines a contraction on $$\begin{aligned}
\label{contset}
\widetilde{X}(T,M,N)=\Big\lbrace u\in &C_t(I;L_x^2)\cap L_t^q(I;L_x^r(|x|^{-r\gamma})) :\\
&\sup_{t\in I}\|u\|_{L_x^2}\leq N,\quad \|u\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))} \leq M \Big\rbrace\end{aligned}$$ equipped with the distance $$d(u,v)=\sup_{t\in I}\|u-v\|_{L_x^2}+\|u-v\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}.$$ First, we see as in and that $$\sup_{t\in I}\|\Phi(u)\|_{L_x^2}\leq C\|u_0\|_{L^2}+C M^{\beta+1}$$ and $$\|\Phi(u)\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}\leq \|e^{it\Delta}u_0\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}+C M^{\beta+1},$$ respectively. Observe here that $$\|e^{it\Delta}u_0\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}\leq \varepsilon$$ for some $\varepsilon>0$ small enough which will be chosen later, provided that either $\|u_0\|_{L^2}$ is small (see with $s=0$) or it is satisfied for some $T>0$ small enough by the dominated convergence theorem. Hence, one can take $T=\infty$ in the first case and $T$ to be this small time in the second. We therefore get $\Phi(u)\in \widetilde{X}$ for $u\in\widetilde{X}$ if $$\label{az}
C\|u_0\|_{L^2}+C M^{\beta+1}\leq N\quad\text{and}\quad \varepsilon+C M^{\beta+1}\leq M.$$ On the other hand, using the same argument employed to show , we see $$\begin{aligned}
d(\Phi(u),\Phi(v))
&=\sup_{t\in I}\|\Phi(u)-\Phi(v)\|_{L_x^2}+\|\Phi(u)-\Phi(v)\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}\\
&\leq CM^\beta\| u-v \|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}\\
&\leq CM^\beta d(u,v).\end{aligned}$$ Now by setting $N=2C\|u_0\|_{L^2}$ and $M=2\varepsilon$ and then choosing $\varepsilon>0$ small enough so that holds and $CM^\beta \leq1/2$, it follows that $\widetilde{X}$ is stable by $\Phi$ and $\Phi$ is a contraction on $\widetilde{X}$.
Therefore, there exists a unique local solution $u \in C(I ; L^{2}) \cap L^{q}(I ; L^{r}(|x|^{-r \gamma}))$ in the time interval $[0,T]$ with a small $T$. Recall from the above argument that when $\|u_0\|_{L^2}$ is small enough, we can take $T=\infty$ to obtain a global solution. The continuous dependence on the initial data $u_0$ follows clearly in the same way as before. It remains to prove the scattering property. Following the argument above, one can easily see that $$\begin{aligned}
\big\|e^{-it_2\Delta}u(t_2)-e^{-it_1\Delta}u(t_1)\big\|_{L_x^2}&=\bigg\|\int_{t_1}^{t_2}e^{-i\tau\Delta}F(u)d\tau\bigg\|_{L_x^2}\\
&\lesssim\|F(u)\|_{L_t^{\tilde{q}'}([t_1,t_2]; L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \tilde{\gamma}}))}\\
&\lesssim\| u \|_{L_t^{q}([t_1,t_2]; L_{x}^{r} (|x|^{-r\gamma}))} ^{\beta+1} \quad\rightarrow\quad0\end{aligned}$$ as $t_1,t_2\rightarrow\infty$. This implies that $$\varphi:=\lim_{t\rightarrow\infty}e^{-it\Delta}u(t)$$ exists in $L^2$. Furthermore, one has $$u(t)-e^{it\Delta}\varphi= i\lambda \int_{t}^{\infty} e^{i(t-\tau)\Delta}F(u) d\tau,$$ and hence $$\begin{aligned}
\big\|u(t)-e^{it\Delta}\varphi\big\|_{L_x^2}=\bigg\|\int_{t}^{\infty} e^{i(t-\tau)\Delta}F(u) d\tau\bigg\|_{L_x^2}
&\lesssim\|F(u)\|_{L_t^{\tilde{q}'}([t,\infty); L_{x}^{\tilde{r}'} (|x|^{\tilde{r}' \tilde{\gamma}}))}\\
&\lesssim\| u \|_{L_t^{q}([t,\infty); L_{x}^{r} (|x|^{-r\gamma}))} ^{\beta+1} \quad\rightarrow\quad0\end{aligned}$$ as $t\rightarrow\infty$. This completes the proof.
The well-posedness in $H^s$ {#sec5}
===========================
This final section is devoted to the proof of Theorem \[Thm2\]. The proof based on Proposition \[Prop1\] and Lemma \[le2\] is similar to the one given in the previous section for the $L^2$ case; therefore, we shall give only a sketch of it.
The subcritical case
--------------------
For appropriate values of $T,M>0$, we show that $\Phi$ defines a contraction map on $$\begin{aligned}
X(T,M)=\Big\lbrace u\in &C_t(I;H_x^s)\cap L_t^q(I;L_x^r(|x|^{-r\gamma})) :\\
&\sup_{t\in I}\|u\|_{H_x^s}+\|u\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))} \leq M \Big\rbrace\end{aligned}$$ equipped with the distance $$d(u,v)=\sup_{t\in I}\|u-v\|_{H_x^s}+\|u-v\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}.$$ Here, $I=[0,T]$ and the exponent pair $(q,r,\gamma)$ is given as in Theorem \[Thm2\]. Firstly, $\Phi(u) \in X$ for $u \in X$, i.e., $$\label{fgjs}
\sup_{t\in I}\|\Phi(u)\|_{H_x^s}+\|\Phi(u)\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))} \leq M.$$ Indeed, using with $(\tilde{q}, \tilde{r}, \tilde{\gamma}) = (\tilde{q}_1, \tilde{r}_1, \tilde{\gamma}_1)$ followed by in Lemma \[le2\] and , we see that $$\begin{aligned}
\nonumber \| \Phi(u)\|_{L_t^{q}(I ; L_{x}^{r}(|x|^{-r \gamma}))}
&\leq C\|u_0\|_{H^s} + C\left\||x|^{-\alpha}|u|^{\beta}u \right\|_{L_{t}^{\tilde{q}_1'} (I; L_{x}^{\tilde{r}_1'}(|x|^{\tilde{r}_1' \tilde{\gamma}_1}))}\\
\nonumber&\leq C\| u_0 \|_{{H}^s} + CT^{\theta_1} \left\| u \right\|_{L_t^{q}(I; L_{x}^{r} (|x|^{-r\gamma}))} ^{\beta+1} \\
\label{1js}&\leq C\| u_0 \|_{{H}^s} + CT^{\theta_1} M^{\beta+1}\end{aligned}$$ because $\| f \|_{\dot{H}^s}\lesssim\| f \|_{H^s}$. On the other hand, from Plancherel’s Theorem combined with the smoothing estimate , we see $$\begin{aligned}
\sup_{t\in I}\|u\|_{H_x^s} &\leq C\|u_0\|_{H^s} + C\sup_{t\in I} \left\| \int_{0}^t e^{-i\tau\Delta} F(u)\ d\tau \right\|_{H^s}\\
&\leq C\|u_0\|_{H^s} +C\big(\| F(u)\|_{L_t^{\tilde{q}_1'}(I; L_x^{\tilde{r}_1'}(|x|^{\tilde{r}_1'\tilde{\gamma}_1}))}+\big\| |\nabla|^{-s} F(u) \big\|_{L_{t}^{{\tilde{q}_2}'}(I;L_x^{{\tilde{r}_2}'}(|x|^{{\tilde{r}_2}' \tilde{\gamma}_2}))}\big).\end{aligned}$$ Here we also used, for the second inequality, that $\| f \|_{H^s}\lesssim\| f \|_{\dot{H}^s}+\| f \|_{L^2}$. By using the nonlinear estimates and , it follows then that $$\begin{aligned}
\label{cr}
\nonumber\sup_{t\in I}\|u\|_{H_x^s} &\leq C\| u_0\|_{H^s} +C (T^{\theta_1}+T^{\theta_2}) \|u\|_{L_t^q(I; L_x^r(|x|^{-r\gamma}))}^{\beta+1}\\
&\leq C\| u_0\|_{H^s} +C (T^{\theta_1}+T^{\theta_2}) M^{\beta+1}.\end{aligned}$$ Thus, if we set $M=4C\| u_0 \|_{H^s}$ and take $T>0$ such that $$\label{dgh1}
C(T^{\theta_1}+T^{\theta_2})M^{\beta}\leq\frac 18,$$ we obtain . Nextly, $\Phi$ is a contraction on $X$, i.e., for $u,v\in X$ $$\label{alks}
d(\Phi(u),\Phi(v)) \leq \frac{1}{2} d(u,v).$$ Indeed, we first see as above that $$\begin{aligned}
\sup_{t\in I}\|\Phi(u)-\Phi(v)\|_{H_x^s}
\leq\ & C \left\| F(u)-F(v) \right\|_{L_t^{\tilde{q}_1'}(I; L_x^{\tilde{r}_1'}(|x|^{\tilde{r}_1'\tilde{\gamma}_1}))} \cr
& + C\left\| |\nabla|^{-s} (F(u)-F(v)) \right\|_{L_{t}^{{\tilde{q}_2}'}(I;L_x^{{\tilde{r}_2}'}(|x|^{{\tilde{r}_2}' \tilde{\gamma}_2}))}.\end{aligned}$$ Using the simple inequality as before combined with Lemma \[le2\], the right-hand side here is bounded by $$2C(T^{\theta_1}+T^{\theta_2}) M^{\beta}\left\| u-v \right\|_{L_t^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}.$$ Meanwhile, by we have $$\|\Phi(u)-\Phi(v)\|_{L_{t}^{q}(I; L_{x}^{r}(|x|^{-r \gamma}))}
\leq C\|F(u)-F(v)\|_{L_t^{\tilde{q}_1'}(I; L_{x}^{\tilde{r}_1'} (|x|^{\tilde{r}_1' \tilde{\gamma}_1}))}.$$ Consequently, we obtain because of .
We have proved that there exists a unique solution $u \in C(I ; H^s) \cap L^{q}(I ; L^{r}(|x|^{-r \gamma}))$ with $T=T(\|u_0\|_{H^s}, \alpha, \beta)$. The continuous dependence on the data follows clearly as before.
The critical case
-----------------
The situation in this case is similar to the $L^2$ case because the power $\theta_1$ in the above argument becomes zero in this case. But the other power $\theta_2=s/2>0$ yields a slight different conclusion as shown in the proof.
We start from showing that $\Phi$ defines a contraction on $$\begin{aligned}
\label{contsets}
\widetilde{X}(T,M,N)=\Big\lbrace u\in &C_t(I;H^s)\cap L_t^q(I;L_x^r(|x|^{-r\gamma})) :\\
&\sup_{t\in I}\|u\|_{H_x^s}\leq N,\quad \|u\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))} \leq M \Big\rbrace\end{aligned}$$ equipped with the distance $$d(u,v)=\sup_{t\in I}\|u-v\|_{H_x^s}+\|u-v\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}.$$ As in and , we see that $$\sup_{t\in I}\|\Phi(u)\|_{H_x^s}\leq C\|u_0\|_{H^s}+C(1+T^{\theta_2}) M^{\beta+1}$$ and $$\|\Phi(u)\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}\leq \|e^{it\Delta}u_0\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}+C M^{\beta+1},$$ respectively. By the dominated convergence theorem we observe that $$\label{oax}
\|e^{it\Delta}u_0\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}\leq \varepsilon$$ for given $\varepsilon>0$ which will be chosen later, if $T>0$ is small enough. Hence, $\Phi(u)\in \widetilde{X}$ for $u\in\widetilde{X}$ if $$\label{azs}
C\|u_0\|_{H^s}+C (1+T^{\theta_2})M^{\beta+1}\leq N\quad\text{and}\quad \varepsilon+C M^{\beta+1}\leq M.$$ Meanwhile, using the same argument employed to show , we see $$\begin{aligned}
d(\Phi(u),\Phi(v))
&=\sup_{t\in I}\|\Phi(u)-\Phi(v)\|_{H_x^s}+\|\Phi(u)-\Phi(v)\|_{L_t^q(I;L_x^r(|x|^{-r\gamma}))}\\
&\leq 2C(1+T^{\theta_2}) M^\beta d(u,v).\end{aligned}$$ Now we take $N=2C\|u_0\|_{H^s}$ and $M=2\varepsilon$, and then choose $\varepsilon>0$ small enough so that holds and $2C(1+T^{\theta_2})M^\beta \leq1/2$. It follows now that $\widetilde{X}$ is stable by $\Phi$ and $\Phi$ is a contraction on $\widetilde{X}$.
Therefore, there exists a unique solution $u \in C(I ; H^s) \cap L^{q}(I ; L^{r}(|x|^{-r \gamma}))$ in the time interval $[0,T]$ with $T=T(u_0,\alpha,\beta)$. Here we cannot take $T=\infty$ even if $\|u_0\|_{H^s}$ is small enough to guarantee , because $\theta_2=s/2>0$. The continuous dependence on the data follows clearly in the same way as before.
[9]{}
M. Ben-Artzi and S. Klainerman, *Decay and regularity for the Schrödinger equation*, J. Anal. Math. 58 (1992), 25-37.
L. Bergé, *Soliton stability versus collapse*, Phys. Rev. E, 62 (2000), 3071-3074.
J. Bergh and J. Löfström, *Interpolation Spaces, An Introduction*, Springer-Berlag, Berlin-New York, 1976.
T. Cazenave, *Semilinear Schrödinger equations*, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
M. Christ and A. Kiselev, *Maximal functions associated to filtrations*, J. Funct. Anal. 179 (2001), 409-425.
T. Cazenave and F. B. Weissler, *Some remarks on the nonlinear Schrödinger equation in the critical case*, Nonlinear Semigroups, Partial differential equations and attractors (Washington, DC, 1987), Lecture Notes in Math. 1394, Springer, Berlin, 1989, 18-29.
T. Cazenave and F. B. Weissler, *The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$*, Nonlinear Anal., 14 (1990), 807-836.
V. D. Dinh, *Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation*, arXiv:1710.01392v1.
L. G. Farah, *Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation*, J. Evol. Equ. 16 (2016), 193-208.
F. Genoud, *An inhomogeneous, $L^2$-critical, nonlinear Schrödinger equation*, Z. Anal. Anwend. 31 (2012), 283-290.
F. Genoud and C. A. Stuart, *Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves*, Discrete Contin. Dyn. Syst. 21 (2008), 137-186.
C. M. Guzmán, *On well posedness for the inhomongeneous nonlinear Schrödinger equation*, Nonlinear Anal. Real World Appl. 37 (2017), 249-286.
J. Ginibre and G. Velo, *On a class of nonlinear Schrödinger equations I. The Cauchy problem, general case*, J. Funct. Anal. 32 (1979), 1-32.
J. Ginibre and G. Velo, *The global Cauchy problem for the nonlinear Schrödinger equation revisited*, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
J. Holmer, S. Roudenko, *A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation*, Comm. Math. Phys. 282 (2008), 435-467.
T. Kato, *On nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), 113-129.
T. Kato and K. Yajima, *Some examples of smooth operators and the associated smoothing effect*, Rev. Math. Phys. 1 (1989), 481-496.
M. Keel and T. Tao, *Endpoint Strichartz estimates*, Amer. J. Math 120 (1998), 955-980.
E. M. Stein and G. Weiss, *Fractional Integrals on $n$-Dimensional Euclidean Space*, J. Math. Mech. 7 (1958), 503-514.
R. S. Strichartz, *Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations*, Duke Math. J. 44 (1977), 705-714.
M. Sugimoto, *Global smoothing properties of generalized Schrödinger equations*, J. Anal. Math. 76 (1998), 191-204.
I. Towers and B. A. Malomed, *Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity*, J. Opt. Soc. Amer. B Opt. Phys. 19 (2002), 537-543.
Y. Tsutsumi, *$L^{2}$-solutions for nonlinear Schrödinger equations and nonlinear groups*, Funkcial. Ekvac. 30 (1987), 115-125.
M. C. Vilela, *Regularity of solutions to the free Schrödinger equation with radial initial data*, Illinois J. Math. 45 (2001), 361-370.
K. Watanabe, *Smooth perturbations of the selfadjoint operator $|\Delta|^{\alpha/2}$*, Tokyo J. Math. 14 (1991), 239-250.
M. Weinstein, *Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys. 87 (1983), 567-576.
[^1]: This research was supported by NRF-2019R1F1A1061316.
|
---
abstract: 'In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a local-to-global optimality criterion, the steepest descend algorithm by successive $k$-submodular function minimizations, the persistency property, and the proximity theorem. Our theory is motivated by minimum cost free multiflow problem. To this problem, Goldberg and Karzanov gave two combinatorial weakly polynomial time algorithms based on capacity and cost scalings, without explicit running time. As an application of our theory, we present a new simple polynomial proximity scaling algorithm to solve minimum cost free multiflow problem in $O( n \log (n AC) {\rm MF}(kn, km))$ time, where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $A$ is the maximum of edge-costs, $C$ is the total sum of edge-capacities, and ${\rm MF}(n'',m'')$ denotes the time complexity to find a maximum flow in a network of $n''$ nodes and $m''$ edges. Our algorithm is designed to solve, in the same time complexity, a more general class of multiflow problems, minimum cost node-demand multiflow problem, and is the first combinatorial polynomial time algorithm to this class of problems. We also give an application to network design problem.'
author:
- |
Hiroshi HIRAI\
Department of Mathematical Informatics,\
Graduate School of Information Science and Technology,\
University of Tokyo, Tokyo, 113-8656, Japan.\
`hirai@mist.i.u-tokyo.ac.jp`
title: 'L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem'
---
Introduction
============
An [*L$^\natural$-convex function*]{} (Favati-Tardella [@FT90], Murota [@MurotaMPA], Fujishige-Murota [@FM00]) is a function $g$ on integer lattice $\ZZ^n$ satisfying so-called [*discrete midpoint convexity inequality*]{}: $$\label{eqn:midpoint_org}
g(x) + g(y) \geq g(\lfloor (x + y)/2\rfloor) + g(\lceil (x + y)/2\rceil)
\quad (x,y \in \ZZ^n),$$ where $\lfloor \cdot \rfloor$ (resp. $\lceil \cdot \rceil$) is an operation on $\RR^n$ that rounds down (resp. up) the decimal fraction of each component. L$^\natural$-convex functions may be viewed as a $\ZZ^n$-generalization of [*submodular functions*]{}, and constitute a fundamental class of discrete convex functions in [*discrete convex analysis*]{} (Murota [@MurotaBook]). A representative example of L$^\natural$-convex function is a function $g$ represented as the following form: $$\label{eqn:typical}
g(x) = \sum_{i} g_i (x_i) + \sum_{i,j} h_{ij}(x_i - x_j) \quad (x = (x_1,x_2,\ldots,x_n) \in \ZZ^n),$$ where $g_i$ and $h_{ij}$ are one-dimensional convex functions. The minimization of such a function has both theoretical and practical interests; it is the dual of a minimum cost network flow problem, and has important applications in computer vision [@KS09]. Thus theory of L$^\natural$-convex functions provides a unified treatment for optimizing these important classes of functions. Let us mention some of particular features of L$^\natural$-convex functions $g$. (1) An optimality criterion of a local-to-global type [@MurotaBook [Theorem 7.14]{}]: For each point $x \in \ZZ^n$, a local minimization problem around $x$ is defined, and if $x$ is local optimal, then $x$ is global optimal. Moreover this local minimization problem is a [*submodular function minimization*]{}, and is solvable in polynomial time [@GLS; @IFF; @Schrijver]. In particular, if $x$ is not local optimal, then we can find another point $y$ with $g(y) < g(x)$; we naturally obtain $y$ with smallest $g(y)$. The resulting descent algorithm, called the [*steepest descent algorithm*]{}, correctly outputs a global minimizer of $g$ [@MurotaBook Section 10.3.1]. The number of the descent steps is the $l_{\infty}$-distance between the initial point and global minimizers [@KS09; @MurotaShioura14]. (2) Proximity theorem (Iwata-Shigeno [@IS02], see [@MurotaBook Theorem 7.18]): For a minimizer $x$ over the set $(2\ZZ)^n$ of all even integral vectors, there is a global minimizer $y$ in the $l_{\infty}$-ball around $x$ with radius $n$. This intriguing property is the basis of the proximity scaling algorithm for L$^\natural$-convex functions [@MurotaBook Section 10.3.2].
Recently the L-convexity is considered for functions on general graph structures other than $\ZZ^n$. Observe that $\ZZ$ is naturally identified with the vertex set of a directed path (of infinite length), and a function on $\ZZ^n$ is regarded as a function on the $n$-product of these paths. Observe that operations $\lceil, \rceil$ and $\lfloor, \rfloor$ are definable in a graph-theoretical way. Hence L$^\natural$-convex functions are well-defined functions on the Cartesian product of directed paths. Based on this observation, Kolmogorov [@Kolmogorov11MFCS] considered an analogue of L$^{\natural}$-convex functions defined on the product of rooted trees, called [*tree-submodular functions*]{}. Hirai [@HH13SODA; @HH_HJ13] considered an analogue of L$^{\natural}$-convex functions on a more general structure, a [*modular complex*]{}, which is a structure obtained by gluing of modular lattices. His motivation comes from the tractability classification of minimum 0-extension problems and a combinatorial duality theory of multicommodity flows.
In this paper, we continue this line of research. We introduce the notion of [*L-extendability*]{} for functions on (the vertex set of) the $n$-fold Cartesian product $T^n$ of trees $T$. This notion is inspired by the idea of [*submodular relaxation*]{} [@GK13; @IWY14; @Kolmogorov12DAM] and related half-integral relaxations of NP-hard problems, such as vertex cover and multiway cut. We first introduce a variation of a tree-submodular function, called an [*alternating L-convex function*]{}. Alternating L-convex functions are also defined by a variation of the discrete midpoint convexity inequality (\[eqn:midpoint\]), and coincide with Fujishige’s [*UJ-convex functions*]{} [@Fujishige14] if $T$ is a path and is identified with $\ZZ$. As an analogue of half-integral integer lattice $(\ZZ/2)^n$, we consider the product $(T^*)^{n}$ of the [*edge-subdivision*]{} $T^*$ of $T$. Then an [*L-extendable function*]{} is defined as a function $g$ on $T^n$ such that there is an alternating L-convex function $\bar g$ on $(T^*)^n$ such that its restriction to $T^n$ is equal to $g$, where $\bar g$ is called an [*L-convex relaxation*]{} of $g$. The first half of our main contribution is to establish basic properties of alternating L-convex functions and L-extendable functions. We show that alternating L-convex functions admit an optimality criterion of a local-to-global type. Here the local problem is the problem of minimizing a [*$k$-submodular function*]{}, a generalization of submodular and bisubmodular functions introduced by Huber and Kolmogorov [@HK12]. This optimality criterion is an immediate consequence of the definition, and may not be precisely new; it is expected from [@HH13SODA; @Kolmogorov11MFCS]. We further prove the $l_{\infty}$-geodesic property for the steepest descent algorithm for alternating L-convex functions: The number of the iterations is equal to the $l_{\infty}$-distance from the initial point to minimizers. We establish the proximity theorem for L-extendable functions: Regard $T$ as a bipartite graph with two color classes $B,W$. For a minimizer $x$ over $B^n$, there is a global minimizer of $g$ within the $l_{\infty}$-ball around $x$ with radius $n$. We prove the persistency property for L-convex relaxations: a minimizer of an L-extendable function is obtained by [*rounding*]{} any minimizer of its L-convex relaxation. This property is known for the cases of bisubmodular and $k$-submodular relaxations [@GK13; @IWY14; @Kolmogorov12DAM]. We introduce a useful special class of L-extendable functions, called [*2-separable convex functions*]{}, as an analogue of a class of functions with form (\[eqn:midpoint\]). We give explicit L-convex relaxations for which the steepest descent algorithm is implementable by a maximum flow algorithm. In fact, the local problem is a minimization of a special $k$-submodular function, which is a sum of (binary) [*basic $k$-submodular functions*]{}, introduced by Iwata, Wahlström, and Yoshida [@IWY14]. They showed that this class of $k$-submodular functions can be minimized by maximum flow computation. Therefore the L-convex relaxation of a $2$-separable convex function is efficiently minimized. For some cases, an optimal solution of this relaxation can easily be rounded to a $2$-approximate solution of the original 2-separable convex function. This approximation algorithm can be viewed as a generalization of the classical $2$-approximation algorithm for multiway cut [@DJPSY94].
These results are motivated by the design of a new simple polynomial scaling algorithm for [*minimum cost free multiflow problem*]{}, which is the second half of our main contribution. For an undirected (integer-)capacitated network with terminal set $S$, a [*multiflow*]{} is a pair of a set of paths connecting terminals in $S$ and its flow-value function satisfying capacity constraints. A [*maximum free multiflow*]{} is a multiflow of a maximum total flow-value. Suppose that each edge has a nonnegative cost. The minimum cost free multiflow problem asks to find a maximum free multiflow with the minimum total cost. Karzanov [@Kar79] proved that there always exists a half-integral minimum cost maximum free multiflow, and presented a pseudo-polynomial time algorithm to find it. Later he [@Kar94] gave a strongly polynomial time algorithm by using a generic polynomial time LP solver (the ellipsoid method or the interior point method). Currently no [*purely combinatorial*]{} strongly polynomial time algorithm is known. Goldberg and Karzanov [@GK97] presented two purely combinatorial weakly polynomial time algorithms: one of them is based on capacity scaling and the other one is based on cost scaling. However the description and analysis of their algorithms (in each scaling phase) are not easy; they did not give an explicit polynomial running time. As an application of the theory of L-extendable functions, we present a new simple purely combinatorial polynomial time scaling algorithm. We hope that our algorithm will be a step toward the design of a purely combinatorial strongly polynomial time algorithm for minimum cost multiflow problem. We formulate a dual of our problem to the minimization of a 2-separable L-convex function on the product $T^n$ of a subdivided-star $T$, where a [*subdivided star*]{} is a tree obtained from a star by successive edge-subdivisions. Then we can apply a domain scaling technique. The scaled problem in each phase is again a minimization of a 2-separable L-convex function, and is solved by the steepest descent algorithm implemented by max-flow computations. The number of iterations is estimated by the proximity theorem the $l_{\infty}$-geodesic property, and the persistency property. In the last phase, we obtain an optimal dual solution, and from this we can construct a desired minimum cost free multiflow. Our algorithm may be viewed as a multiflow version of a proximity scaling (or domain scaling) algorithm for the convex dual of minimum cost flow problem [@AHO04; @KS09], and is the first combinatorial algorithm to this problem having an explicit polynomial running time. The total time is $O( n \log (n AC) {\rm MF}(k n, km))$, where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $A$ is the maximum of an edge-cost, $C$ is the total sum of edge-capacities, and ${\rm MF}(n',m')$ is the time complexity of finding a maximum flow in a network of $n'$ nodes and $m'$ edges. Our algorithm is designed to solve, in the same time complexity, a more general class of multiflow problems, [*minimum cost node-demand multiflow problems*]{}, and is the first combinatorial polynomial time algorithm for this class of the problems. This multiflow problem arises as an LP-relaxation of a class of network design problems, called [*terminal backup problems*]{} [@AK11; @BKM13; @XAC08INFOCOM]. Recently Fukunaga [@Fukunaga14] gave a 4/3-approximation algorithm for capacitated terminal backup problem, based on rounding an LP solution (obtained by a generic LP-solver). Our algorithm gives a practical and combinatorial implementation of his algorithm.
We present results for L-extendable functions in Section \[sec:L-extendable\] and results for minimum cost multiflow problem in Section \[sec:multiflow\].
#### Notation.
Let $\RR$, $\RR_+$, $\ZZ$, and $\ZZ_+$ denote the sets of reals, nonnegative reals, integers, and nonnegative integers, respectively. Let $\overline{\RR} := \RR \cup \{\infty\}$, where $\infty$ is an infinity element and is treated as: $x < \infty$ $(x \in \RR)$ and $\infty + x = \infty$ $(x \in \overline{\RR})$.
For a function $f: E \to \barRR$ on a set $E$, let ${\mathop{\rm dom} }f$ denote the set of elements $x \in E$ with $f(x) \neq \infty$. For a subset $X \subseteq E$, let $f(X)$ denote $\sum_{x \in X} f(x)$. For an undirected (directed) graph $G = (V,E)$, an edge between nodes $i$ and $j$ (from $i$ to $j$) is denoted by $ij$. For a node subset $X$, let $\delta X$ denote the set of edges $ij$ with $i \in X$ and $j \not \in X$. For nodes $s,t$, an [*$(s,t)$-cut*]{} is a node subset $X$ with $s \in X \not \ni t$. For a node subset $A$, an $(s,A)$-cut is a node subset $X$ with $s \in X \subseteq V \setminus A$.
L-extendable functions {#sec:L-extendable}
======================
In this section, we introduce alternating L-convex functions and L-extendable functions on the product of trees. In Section \[subsec:k-submodular\], we give preliminary arguments on $k$-submodular functions. In Section \[subsec:alternating\], we introduce alternating L-convex functions, and establish the L-optimality criterion (Theorem \[thm:L-optimality\]) and the $l_\infty$-geodesic property (Theorem \[thm:bound\]) of the steepest descent algorithm. In Section \[subsec:L-extendable\], we introduce L-extendable functions, and establish the proximity theorem (Theorem \[thm:proximity\]) and the persistency property (Theorem \[thm:persistency\]). In Section \[subsec:2-separable\], we introduce 2-separable convex functions. We show that the steepest descent algorithm for their L-convex relaxations is implementable by minimum cut computations (Theorems \[thm:2separable\] and \[thm:approx\]). Less obvious theorems are proved in Section \[subsec:proof\].
Preliminaries on $\pmb k$-submodular function {#subsec:k-submodular}
---------------------------------------------
For a nonnegative integer $k$, let $S_k$ be a $(k+1)$-element set with specified element $0$. Define a partial order $\preceq$ on $S_k$ by $0 \prec u$ for $u \in S_{k} \setminus \{0\}$ with no other relations. Then $S_k$ is a meet-semilattice of minimum element $0$; in particular meet $\wedge$ exists. For $u,v \in S_{k}$, define $u \sqcup v$ by $u \sqcup v := u \vee v$ if $u,v$ are comparable, and $u \sqcup v := 0$ otherwise. For an $n$-tuple $\pmb{k} = (k_1, k_2,\ldots, k_n)$ of nonnegative integers, let $S_{\pmb k}$ denote the direct product $S_{k_1} \times S_{k_2} \times \cdots \times S_{k_n}$ of posets $S_{k_i}$ $(i=1,2,\ldots,n)$. For $x = (x_1,x_2,\ldots,x_n), y = (y_1,y_2,\ldots,y_n) \in S_{\pmb k}$, let $x \wedge y := (x_1 \wedge y_1, x_2 \wedge y_2, \ldots, x_n \wedge y_n)$ and $x \sqcup y := (x_1 \sqcup y_1, x_2 \sqcup y_2, \ldots, x_n \sqcup y_n)$.
$f: S_{\pmb k} \to \barRR$ is [*$\pmb k$-submodular*]{} if it satisfies $$f(x) + f(y) \geq f(x \wedge y) + f(x \sqcup y) \quad (x,y \in S_{\pmb k}).$$
The notion of $\pmb k$-submodularity is introduced by Huber-Kolmogorov [@HK12]. If $\pmb{k} = (k,k, \ldots,k)$, then $\pmb{k}$-submodular functions are particularly called $k$-submodular. Then $1$-submodular functions are submodular functions, and $2$-submodular functions are bisubmodular functions. Although both classes of functions can be minimized in polynomial time (under the oracle model) [@FI05; @GLS; @IFF; @MF10; @Qi88; @Schrijver], it is not known whether general $\pmb k$-submodular functions can be minimized in polynomial time. Recently, Thapper and Živný [@TZ12FOCS] discovered a powerful criterion for solvability of [*valued CSP*]{}, that is, a minimization of a function represented as a sum of functions with constant arity ($=$ the number of variables). As a consequence of their criterion, if $\pmb k$-submodular function $f$ is represented and given as $$f(x) = \sum_{i} f_{i} (x_{i_1}, x_{i_2},\ldots,x_{i_m}) \quad (x = (x_1,x_2,\ldots,x_n) \in S_{\pmb k}),$$ where each $f_{i}$ is $\pmb k$-submodular and the number $m$ of variables is constant, then $f$ can be minimized by solving a certain polynomial size linear program (BLP); see also [@KTZ13]. We will deal with a further special class of $\pmb k$-submodular functions, considered by [@IWY14], in which this class of $\pmb k$-submodular functions can be efficiently minimized by a maximum flow algorithm. Let $k,k'$ be nonnegative integers. For $a \in S_k$, let $\epsilon_a$ and $\theta_a$ be one-dimensional $k$-submodular functions on $S_k$ defined by $$\epsilon_a (u) :=
\left\{
\begin{array}{ll}
1 & {\rm if}\ u = a \neq 0, \\
0 & {\rm otherwise},
\end{array} \right. \\
\quad
\theta_a(u) := \left\{
\begin{array}{ll}
-1 & {\rm if}\ u = a \neq 0, \\
0 & {\rm if}\ u= 0,\\
1 & {\rm otherwise}.
\end{array} \right.
\quad (u \in S_{k}).$$ It is not difficult to see that any one-dimensional $k$-submodular function $f$ is a nonnegative sum of $\epsilon_{a}$ and $\theta_a$ plus a constant: $$\label{eqn:unary}
f = f(0) + (f(0) - f(a)) \theta_a + \sum_{b \in S_k \setminus \{ 0, a\}} (f(b) - 2 f(0) + f(a))\epsilon_b,$$ where $a$ is a minimizer of $f$ over $S_k$. For $(a,a') \in S_{k} \times S_{k'}$, let $\mu_{a,a'}$ be a function on $S_k \times S_{k'}$ defined by $$\mu_{a,a'}(u,v) := \left\{
\begin{array}{ll}
0 & {\rm if} \ \mbox{$u = a \neq 0$ or $v = a' \neq 0$ or $u = v = 0$}, \\
1 & {\rm if}\ \mbox{$v = 0 \neq u \neq a$ or
$u= 0 \neq v \neq a'$},\\
2 & {\rm otherwise},
\end{array}\right. \quad ((u,v) \in S_k \times S_{k'}).$$
For a poset isomorphism $\sigma: S_k \to S_{k'}$ (a bijection from $S_{k}$ to $S_{k'}$ with $\sigma(0) = 0$), let $\delta_{\sigma}$ be a function on $S_k \times S_{k'}$ defined by $$\delta_{\sigma}(u,v) := \left\{
\begin{array}{ll}
0 &{\rm if} \ v = \sigma (u), \\
1 & {\rm if}\ |\{u,v\} \cap \{0\}| = 1, \\
2 & {\rm otherwise},
\end{array}\right. \quad ((u,v) \in S_k \times S_{k'}).$$ If $\sigma$ is the identity map ${\rm id}$, then $\delta_{\rm id}$ is denoted by $\delta$.
Observe that both $\mu_{a,a'}$ and $\delta_{\sigma}$ are $\pmb k$-submodular. A (binary) [*basic $\pmb k$-submodular function*]{} $f$ on $S_{\pmb k}$ for $\pmb{k}= (k_1,k_2,\ldots, k_n)$ is a function represented as
Type I:
: $f(x) = f' (x_i)$ for some $i$ and one-dimensional $k$-submodular function $f' :S_{k_i} \to \overline\RR$,
Type II:
: $f(x) = \delta_{\sigma}(x_i,x_j)$ for distinct $i,j$ and an isomorphism $\sigma: S_{k_i} \to S_{k_j}$, or
Type III:
: $f(x) = \mu_{a,a'}(x_i, x_j)$ for distinct $i,j$ and $(a,a') \in S_{k_i} \times S_{k_j}$.
Iwata, Wahlström, and Yoshida [@IWY14] showed that a sum of basic $k$-submodular function can be efficiently minimized by any maximum flow algorithm.
\[thm:IWY14\] A nonnegative sum of $m$ basic $\pmb{k}$-submodular functions for $\pmb{k} = (k_1,k_2,\ldots,k_n)$ can be minimized in $O ({\rm MF}(k n, km))$ time, where $k := \max k_i$.
Their algorithm is sketched as follows. Suppose that $S_{k_i} =\{0,1,2,\ldots,k_i\}$ for $1 \leq i \leq n$. We consider a directed network ${\cal N}$ with vertex set $U$, edge set $A$, and edge capacity $c: A \to \RR_+$, where $U$ consists of source $s$, sink $t$, and $v_i^{1}, v_i^{2},\ldots,v_{i}^{k_i}$ $(1 \leq i \leq n)$. Let $U_i := \{v_i^{1},v_i^{2},\ldots,v_{i}^{k_i}\}$. A [*legal cut*]{} is an $(s,t)$-cut $X$ such that $|X \cap U_i| \leq 1$ for $1 \leq i \leq n$. There is a natural bijection $\phi$ from $S_{\pmb k}$ to the set of legal cuts, where $\phi$ is given by $$\phi(x) := \{s\} \cup \{ v_i^{x_i} \mid 1 \leq i \leq n, x_i \neq 0\} \quad (x = (x_1,x_2,\ldots,x_n) \in S_{\pmb k}).$$ For an $(s,t)$-cut $X$, let $\check{X}$ denote the legal cut obtained from $X$ by deleting $U_i \cap X$ with $|U_i \cap X| \geq 2$ for $i=1,2,\ldots,n$. We say that network ${\cal N}$ [*represents*]{} function $f: S_{\pmb k} \to \overline\RR$ if
- for some constant $K$, it holds $f(x) = c(\delta \phi(x)) + K$ for every $x \in S_{\pmb k}$, and
- $c(\delta \check{X}) \leq c(\delta X)$ for every $(s,t)$-cut $X$.
Suppose that ${\cal N}$ represents $f$. By (1) and (2), the minimum of $f + K$ is equal to the minimum $(s,t)$-cut capacity, and some legal cut is a minimum $(s,t)$-cut. By $\check{X} \subseteq X$ and (2), any inclusion-minimal minimum $(s,t)$-cut is necessarily a legal cut. Recall the fundamental fact in network flow theory that there is a unique inclusion-minimal mincut $X^*$, which is equal to the set of vertices reachable from $s$ in the residual network for any maximum $(s,t)$-flow. Then $x^* := \phi^{-1}(X)$ is a minimizer of $f$. Namely $f$ is minimized by a single max-flow computation on ${\cal N}$.
As shown in [@IWY14], any nonnegative sum of basic $\pmb k$-submodular functions is represented by some network ${\cal N}$. Notice that if $f$ is represented by ${\cal N} = (U,A,c)$, then for constants $\alpha > 0$ and $\beta$, $\alpha f + \beta$ is represented by ${\cal N}' = (U, A, \alpha c)$, and that if $f$ and $f'$ are represented by networks ${\cal N}$ and ${\cal N'}$ (on the same vertex set $U$), respectively, then $f + f'$ is represented by the union of ${\cal N}$ and ${\cal N'}$. Also notice that a unary $\pmb k$-submodular function $f$ is decomposed to $\theta_a$ and $\epsilon_b$ as (\[eqn:unary\]). If $f(0) = \infty$, then there is at most one $u \in S_k$ with $u \in {\mathop{\rm dom} }f$, and the coordinate $x_i$ of term $f(x_i)$ may be fixed to $u$. So we can assume that $f(0) \in {\mathop{\rm dom} }f$. Therefore it suffices to construct a network for the following cases of $f$: (i$_0$) $f(x) = \infty$ if $x_i = u \neq 0$ and $0$ otherwise, (i) $f(x) = \epsilon_a(x_i)$ $(a \neq 0)$, (ii) $f(x) = \theta_a (x_i)$, (iii) $f(x) = \delta_\sigma (x_i, x_j)$, and (iv) $f(x) = \mu_{a,a'}(x_i, x_j)$. Case (i$_0$): Consider the network consisting of a single edge $v_i^u t$ with infinite capacity.
Case (i): Consider the network consisting of a single edge $v_i^{a}t$ with unit capacity.
Case (ii): If $a = 0$, then $f$ is the sum of $\epsilon_{a'}$, and reduces to case (i). Suppose that $a \neq 0$. Consider the network consisting of edges $s v_i^{a}$ and $v_i^{j}t$ for $j \in \{1,2,\ldots,k_i\} \setminus \{a\}$.
Case (iii): Consider the network consisting of edges joining $v_i^{u}$ and $v_{j}^{\sigma(u)}$ in both directions for $u = 1,2,\ldots,k_i (= k_j)$.
Case (iv): If $a = a' = 0$, then $f$ is the sum of two unary $k$-submodular functions, and reduces to case (i). So suppose $a \neq 0$. Consider the network consisting of edges $v_j^u v_i^{a}$ for $u \in \{1,2,\ldots,k_j\} \setminus \{a'\}$ and $v_i^u v_j^{a'}$ for $u \in \{1,2,\ldots,k_i\} \setminus \{a\}$ if $a' \neq 0$.
In all four cases, it is easy to check that the network represents $f$, i.e., it satisfies (1) and (2). In particular, each basic $\pmb k$-submodular function is represented by $O(k)$ edges, where $k = \max_i k_i$. Thus a nonnegative sum of $m$ basic $\pmb k$-submodular functions is represented by a network with $O(kn)$ vertices and $O(km)$ edges. Hence it can be minimized by $O({\rm MF}(kn,km))$ time.
Alternating L-convex functions {#subsec:alternating}
------------------------------
Here we introduce the notion of an alternating L-convex function on the product of trees. Alternating L-convex functions may be viewed as a natural variant of [*strongly tree-submodular functions*]{} due to Kolmogorov [@Kolmogorov11MFCS]; see Remark \[rem:midpoint\] for a detailed relation. In addition, when $\ZZ^n$ is identified with (the vertex set) of the $n$-fold product of a path with infinite length, alternating L-convex functions are equal to [*UJ-convex functions*]{} considered by Fujishige [@Fujishige14]. Also alternating L-convex functions constitutes a useful special class of [*L-convex functions on modular complexes*]{} [@HH13SODA; @HH_HJ13], in which a modular complex are taken to be the product of zigzag oriented trees. Let $T$ be a tree. We will use the following convention:
> [*The vertex set of a tree $T$ is also denoted by $T$.*]{}
The exact meaning will always be clear in this context. Regard $T$ as a bipartite graph. Let $B$ and $W$ be the color classes of $T$, where a vertex in $B$ is called [*black*]{} and a vertex in $W$ is called [*white*]{}. Define a partial order $\preceq$ on $T$ by: $u \prec v$ if $u$ and $v$ are adjacent with $(u,v) \in W \times B$. Then the resulting poset has no chain of length $2$; $B$ and $W$ are the sets of maximal and minimal elements, respectively. Let $d$ denote the path metric of $T$, i.e., $d(u,v)$ is the number of edges in the unique path between $u$ and $v$. For vertices $u, v \in T$, there uniquely exists a pair $(a, b)$ of vertices such that $d(u,v) = d(u,a) + d(a, b) + d(b,v)$, $d(u,a) = d(b, v)$, and $d(a,b) \leq 1$. In particular, $a = b$ or $a$ and $b$ are adjacent. Define $u \bullet v$ and $u \circ v$ so that $\{u \bullet v, u \circ v\} = \{a,b\}$ and $u \circ v \preceq u \bullet v$. Namely if $a= b$ then $u \bullet v = u \circ v = a = b$. Otherwise $u \bullet v$ and $u \circ v$ are the black and white vertices in $\{a,b\}$, respectively.
Let $n$ be a positive integer. We consider the $n$-fold Cartesian product $T^n$ of $T$. We will use the $l_{\infty}$-metric on $T^n$, which is also denoted by $d$: $$d (x,y) := \max_{1 \leq i \leq n} d(x_i,y_i) \quad (x,y \in T^n).$$
A function $g: T^n \to \barRR$ is called [*alternating L-convex*]{}, or simply, [*L-convex*]{} if it satisfies $$\label{eqn:midpoint}
g(x) + g(y) \geq g(x \bullet y) + g( x \circ y) \quad (x,y \in T^n).$$
The defining inequality (\[eqn:midpoint\]) can be viewed as a variation of the discrete midpoint convexity (\[eqn:midpoint\_org\]); see also Remark \[rem:midpoint\]. As the direct product of posets $T$, we regard $T^n$ as a poset, where the partial order is also denoted by $\preceq$. For $x = (x_1,x_2,\ldots,x_n) \in T^n$, the principal ideal ${\cal I}(x) := \{ y \in T^n \mid y \preceq x\}$ and filter ${\cal F}(x) := \{ y \in T^n \mid y \succeq x\}$ of $x$ are given as follows: $$\begin{aligned}
{\cal I}(x) & = & {\cal I}(x_1) \times {\cal I}(x_2) \times \cdots \times {\cal I}(x_n), \\
{\cal F}(x) & = & {\cal F}(x_1) \times {\cal F}(x_2) \times \cdots \times {\cal F}(x_n), \\
{\cal I}(x_i) & = &
\left\{
\begin{array}{ll}
\{x_i\} \cup \{ \mbox{all neighbors of $x_i$} \} & {\rm if}\ x_i \in B, \\
\{x_i\} & {\rm if}\ x_i \in W,
\end{array} \right. \\
{\cal F}(x_i) & = &
\left\{
\begin{array}{ll}
\{x_i\} \cup \{ \mbox{all neighbors of $x_i$} \} & {\rm if}\ x_i \in W, \\
\{x_i\} & {\rm if}\ x_i \in B.
\end{array} \right. \end{aligned}$$ Regard ${\cal I}(x_i)$ as poset $S_{k_i}$ with $k_i := |{\cal I}(x_i)| - 1$ and the minimum element $x_i$, and also regard ${\cal F}(x_i)$ as poset $S_{k'_i}$ with $k'_i := |{\cal F}(x_i)|-1$ and the minimum element $x_i$. Then ${\cal I}(x) \simeq S_{\pmb k}$ for $\pmb{k} = (k_1,k_2,\ldots,k_n)$ and ${\cal F}(x) \simeq S_{\pmb k}$ for $\pmb{k} = (k'_1,k'_2,\ldots,k'_n)$. By this correspondence, it is easy to see that $\bullet = \sqcup$ and $\circ = \wedge$ on ${\cal F}(x)$, and that $\circ = \sqcup$ and $\bullet = \wedge$ on ${\cal I}(x)$. Therefore we have:
\[lem:locally\_k\_submo\] For each $x \in T^n$, an L-convex function $g$ is $\pmb k$-submodular on ${\cal I}(x)$ and on ${\cal F}(x)$.
In particular, in the case where $T$ is a star with $k$ leaves and center $x_0 \in W$, under the correspondence $T^n = {\cal F}(x_0)^n \simeq {S_k}^n$, L-convex functions and $k$-submodular functions are the same. The following is an analogue of the L-optimality criterion of L$^{\natural}$-convex functions [@MurotaBook Theorem 7.5] (and strongly-tree submodular function [@Kolmogorov11MFCS]):
\[thm:L-optimality\] Let $g$ be an L-convex function on $T^n$. If $x \in {\mathop{\rm dom} }g$ is not a minimizer of $g$, then there exists $x' \in {\cal I}(x) \cup {\cal F}(x)$ with $g(x') < g(x)$.
There is $y$ with $g(y) < g(x)$. Take such $y$ with $d(x, y)$ minimum. By inequality (\[eqn:midpoint\]), we have $2 g(x) > g(x) + g(y) \geq g(x \bullet y) + g(x \circ y)$. Thus $g(x \bullet y) < g(x)$ or $g(x \circ y) < g(x)$ holds. By the minimality, it must hold $d(x, y) \leq 1$ (since $d(y, x \bullet y) \leq \lceil d(x,y)/2 \rceil$ and $d(y, x \circ y) \leq \lceil d(x,y)/2 \rceil$). Then $x \bullet y \in {\cal F}(x)$, $x \circ y \in {\cal I}(x)$, and the claim holds.
This directly implies the following descent algorithm, which is an analogue of the steepest descent algorithm of L$^\natural$-convex functions [@MurotaBook Section 10.3.1].
Steepest descent algorithm:
:
Input:
: An L-convex function $g$ and a vertex $x \in {\mathop{\rm dom} }g$.
Step 1:
: Let $y$ be a minimizer of $g$ over ${\cal I}(x)\cup {\cal F}(x)$.
Step 2:
: If $g(x) = g(y)$, then stop. Otherwise $x := y$, and go to step 1.
By Theorem \[thm:L-optimality\], if the algorithm terminates in step 2, then $x$ is a minimizer of $g$. Also the step 1 is conducted by minimizing $g$ over ${\cal I}(x)$ and $g$ over ${\cal F}(x)$. By Lemma \[lem:locally\_k\_submo\], $g$ is $\pmb k$-submodular on ${\cal I}(x)$ and on ${\cal F}(x)$, and step 1 can be conducted by $\pmb k$-submodular function minimization. Therefore, if $g$ is represented as a sum of $\pmb k$-submodular functions of bounded arity, then each iteration is conducted in polynomial time. To obtain a complexity bound, we need to estimate the total number of iterations. In the case of L$^{\natural}$-convex functions, the number of iterations is bounded by the constant of the $l_{\infty}$-diameter of the effective domain [@KS09]. A recent analysis [@MurotaShioura14] showed that the number of iterations is [*exactly*]{} equal to the minimum of a certain directed analogue of $l_{\infty}$-distance from the initial point $x$ to the set of minimizers. We will establish an analogous result for our L-convex function. Let ${\rm opt} (g)$ denote the set of all minimizers of $g$. Obviously the total number of the iterations is at least the minimum $l_{\infty}$-distance $d({\rm opt}(g),x)
:= \min \{ d(y,x) \mid y \in {\rm opt}(g) \}$ from $x$ to ${\rm opt} (g)$. This lower bound is almost tight, as follows.
\[thm:bound\] For an L-convex function $g$ on $T^n$ and a vertex $x \in {\mathop{\rm dom} }g$, the total number $m$ of the iterations of the steepest descent algorithm applied to $(g, x)$ is at most $d({\rm opt}(g), x) + 2$. If $g(x) = \min_{y \in {\cal F}(x)} g(y)$ or $g(x) = \min_{y \in {\cal I}(x)} g(y)$, then $m = d({\rm opt}(g), x)$.
In the case where $x \in B^n$ or $W^n$, the condition $g(x) = \min_{y \in {\cal F}(x)} g(y)$ or $g(x) = \min_{y \in {\cal I}(x)} g(y)$ is automatically satisfied. In fact, [@HH_HJ13] announced (a slightly weaker version of) this result for general L-convex functions on modular complexes. However, the proof needs a deep geometric investigation on modular complex, and will be given in a future paper [@HHprepar]. We give a self-contained proof of Theorem \[thm:bound\] in Section \[subsec:bound\].
L-extendable functions {#subsec:L-extendable}
----------------------
Next we introduce the notion of the L-extendability. This notion was inspired by the idea of [*($k$-)submodular relaxation*]{} used in [@GK13; @IWY14; @Kolmogorov12DAM].
A function $h: B^n \to \barRR$ is called [*L-extendable*]{} if there exists an L-convex function $g$ on $T^n$ such that the restriction of $g$ to $B^n$ is equal to $h$.
We also define the L-extendability of a function on $T^n$ via the edge-subdivision. The [*edge-subdivision*]{} $T^*$ of $T$ is obtained by adding a new vertex $w_{uv}$ for each edge $e = uv$, and replacing $e$ in $T$ by two edges $uw_{uv}, w_{uv} v$. The new vertex $w_{uv}$ is called the [*midpoint*]{} of $uv$. The original $T$ is a subset of $T^*$ and is one of the color classes of $T^*$. In $T^*$, vertices in $T$ are supposed to be black and midpoints are supposed to be white.
A function $h: T^n \to \barRR$ is called [*midpoint L-extendable*]{}, or simply, [*L-extendable*]{} if there exists an L-convex function $g$ on $(T^*)^n$ such that the restriction of $g$ to $T^n$ is equal to $h$.
We call $g$ an [*L-convex relaxation*]{} of $h$. If $\min_{x \in T^n} g(x) = \min_{x \in (T^*)^n} h(x)$, then $g$ is called an [*exact*]{} L-convex relaxation. In fact, any L-convex function admits an exact L-convex relaxation, and is (midpoint) L-extendable [@HHprepar]; see Remark \[rem:midpoint\] for related arguments. We will see that vertex-cover problem and multiway cut problem admits $k$-submodular relaxation (L-convex relaxation in our sense). This means that it is NP-hard to minimize L-extendable functions in general. However L-extendable functions have several useful properties. The main results in this section are following three properties of L-extendable functions. These properties will play crucial roles in the proximity scaling algorithm for minimum cost multiflow problem in Section \[sec:multiflow\]. Proofs of the three theorems are given in Section \[subsec:proof\].
The first property is an optimality criterion analogous to Theorem \[thm:L-optimality\]:
\[thm:optimality\_h\] Let $h: B^n \to \barRR$ be an L-extendable function. For $x \in {\mathop{\rm dom} }h$, if $x$ is not a minimizer of $h$, then there exists $y \in B^n$ such that $d(x,y) \leq 2$ and $h(y) < h(x)$.
The second property is so-called the [*persistency*]{}. This notion was introduced by Kolmogorov [@Kolmogorov12DAM] for bisubmodular relaxation, and was extended to $k$-submodular relaxation [@GK13; @IWY14]. The persistency property says that from a minimizer $x$ of a relaxation $g$, we obtain a minimizer $y$ of $h$ by [*rounding*]{} each white component of $x$ to an adjacent (black) vertex.
\[thm:persistency\] Let $h:B^n \to \barRR$ be an L-extendable function and $g: T^n \to \barRR$ its L-convex relaxation. For a minimizer $x$ of $g$, then there is a minimizer $y$ of $h$ with $y \in {\cal F}(x) \cap B^n$.
The third one is a proximity theorem. The proximity theorem of L$^{\natural}$-convex function $g:\ZZ^n \to \barRR$ says that for any minimizer $x$ of $g$ over $(2\ZZ)^n$, there is a minimizer $y$ of $g$ with $\|x - y\|_{\infty} \leq n$ (Iwata-Shigeno [@IS02]; see [@MurotaBook Theorem 7.6] and [@FujiBook Theorem 20.10]). We establish an analogous result for L-extendable functions.
\[thm:proximity\] Let $h: T^n \to \barRR$ be a midpoint L-extendable function, and let $x$ be a minimizer of $h$ over $B^n$. Then there exists a minimizer $y$ of $h$ with $d(x, y) \leq 2 n$. In addition, if $h$ admits an exact L-convex relaxation, then there exists a minimizer $y$ of $h$ with $d(x, y) \leq n$
As noted in [@Kolmogorov12DAM], vertex cover problem is a representative example admitting a bisubmodular relaxation (L-convex relaxation in our sense). Let $G = (V,E)$ be an undirected graph with (nonnegative) cost $a$ on $V = \{1,2,\ldots,n\}$. A [*vertex cover*]{} is a set $X$ of vertices meeting every edge. The vertex cover problem asks to find a vertex cover $X$ of minimum cost $a(X)$. The well-known IP formulation of this problem is: Minimize $\sum_{i \in V} a(i) x_i$ over $x \in \{0,1\}^{n}$ satisfying $x_i + x_j \geq 1$ for $ij \in E$. Define $I_{\geq 1}: \RR \to \overline\RR$ by $I_{\geq 1}(z) = 0$ if $z \geq 1$ and $\infty$ otherwise, and define $\omega: \{0,1\}^n \to \overline{\RR}$ by $\omega(x) := \sum_{i \in V} a(i) x_i + \sum_{ij \in E} I_{\geq 1}(x_i + x_j)$. Then the vertex cover problem is the minimization of $\omega$. This function $\omega$ is midpoint L-extendable (if $\{0,1\}$ is identified with the vertex set of the graph of single edge). Indeed, the natural extension $\bar \omega: \{0,1/2,1\}^n \to \overline{\RR}$ is a bisubmodular relaxation (if $\{0,1/2,1\}$ is identified with $S_2$ with $0 \succ 1/2 \prec 1$). The [*submodular vertex cover problem*]{} [@IN09] is to minimize submodular function $f: \{0,1\}^n \to \overline{\RR}$ over $x \in \{0,1\}^n$ satisfying $x_i + x_j \geq 1$ for $ij \in E$. Namely this is the minimization of $\omega$ defined by $\omega(x) = f(x) + \sum_{ij \in E} I_{\geq 1} (x_i + x_j)$. Again $\omega$ is midpoint L-extendable. Indeed a function $x \mapsto (f(\lceil x \rceil) + f(\lfloor x \rfloor))/2 + \sum_{ij \in E} I_{\geq 1} (x_i + x_j)$ is a bisubmodular relaxation of $\omega$.
\[rem:midpoint\] We can consider several variants of the discrete midpoint convexity inequality and associated discrete convex functions. Suppose that each edge of $T$ has an orientation. Let ${\rm mid}: T \times T \to T^*$ be defined by: ${\rm mid}(p,q)$ is the unique vertex $u \in T^*$ with $d(p,u) = d(u,q)$ and $d(p,u) + d(u,q) = d(p,q)$. For $u \in T^* \setminus T$, let $\overline u$ and $\underline u$ denote the vertices of $T$ such that $\overline u \underline u$ is an edge with midpoint $u$, and is oriented from $\overline u$ to $\underline u$. For $u \in T$, let $\overline u = \underline u := u$. Extend these operations to operations on $T^n$ in componentwise, as above. Consider function $g$ satisfying $$\label{eqn:midpoint'}
g(x) + g(y) \geq g( \overline{{\rm mid} (x, y)}) + g( \underline{{\rm mid} (x,y)}) \quad (x,y \in T^n).$$ In the case where $T$ is a path on $\ZZ$ obtained by joining $i$ and $i+1$ and by orienting $i \to i+1$ $(i \in \ZZ)$, the above inequality (\[eqn:midpoint’\]) is equal to (\[eqn:midpoint\_org\]), and $g$ is L$^\natural$-convex. In the case where there is a unique sink in $T$, i.e., $T$ is a rooted tree, the operations $\overline{\rm mid}$ and $\underline{\rm mid}$ are equal, respectively, to $\sqcup$ and $\sqcap$ in the sense of [@Kolmogorov11MFCS], and $g$ is strongly tree submodular. Also notice that alternating L-convex functions correspond to the zigzag orientation.
So different orientations of $T$ define different classes of discrete convex functions. Theory of L-extendable functions captures all these discrete convex functions by the following fact:
If $g: T^n \to \overline\RR$ satisfies (\[eqn:midpoint’\]), then $g$ is midpoint L-extendable, where its exact L-convex relaxation $\bar g: (T^*)^n \to \overline\RR$ is given by $$\bar g(u) := ( g(\overline u ) + g(\underline u ))/2 \quad (u \in (T^*)^n).$$ \[eqn:fact\]
We will give the proof of this fact in [@HHprepar] (since it is bit tedious). In particular the minimization of $g$ over $T^n$ can be solved by the minimization of $\bar g$ over $(T^*)^n$. Instead of $g$, we can apply our results to $\bar g$ (and obtain results for original $g$). This is another reason why we consider alternating L-convex functions and L-extendable functions.
$2$-separable convex functions {#subsec:2-separable}
------------------------------
In this section, we introduce a special class of L-extendable functions, called [*2-separable convex functions*]{}. This class is an analogue of a class of functions $f$ on $\ZZ^n$ represented as the following form: $$\label{eqn:2separable_ZZ}
\sum_{i} f_i (x_i) + \sum_{i,j} g_{ij} ( x_i - x_j ) + \sum_{i,j} h_{ij} ( x_i + x_j ) \quad (x \in \ZZ^n),$$ where $f_i, g_{ij}$, and $h_{ij}$ are $1$-dimensional convex functions on $\ZZ$. Hochbaum [@Hochbaum02] considers minimization of functions with this form, and provides a unified framework to NP-hard optimization problems admitting half-integral relaxation and 2-approximation algorithm. Recall (\[eqn:typical\]) that a function without terms $h_{ij}(x_i+ x_j)$ is a representative example of L$^\natural$-convex functions. It is known that the half-integral relaxation of (\[eqn:2separable\_ZZ\]) can be efficiently minimized by a maximum flow algorithm [@Hochbaum02]; also see [@AHO04; @KS09].
In this section, we show analogous results: a 2-separable convex function admits an L-convex relaxation each of whose local $\pmb k$-submodular function is a sum of basic $\pmb k$-submodular functions. Hence this L-convex relaxation can be efficiently minimized by successive applications of max-flow min-cut computations. Moreover, for some special cases, a solution of the L-convex relaxation can be rounded to a 2-approximation solution of the original 2-separable convex function.
We start with the (one-dimensional) convexity on a tree. A function $h$ on $\ZZ$ is said to be [*nondecreasing*]{} if $
\Delta h(t) := h(t) - h(t-1) \geq 0$ for $t \in \ZZ$, an is said to be [*convex*]{} if $
\Delta^2h(t) := h(t+1) - 2h(t) + h(t-1) \geq 0$ for $t \in \ZZ$, and is said to be [*even*]{} if $(h(t - 1) + h(t + 1))/2 = h(t)$ for every odd integer $t$. We can naturally define convex functions on a tree $T$. It should be noted that this notion of convexity was considered in the classical literature of facility location analysis [@DFL76; @Kolen; @TFL83]. For $u,v \in T$ and an integer $t$ with $0 \leq t \leq d(u,v)$, let $[u,v]_t$ denote the unique vertex $s$ satisfying $d(u,v) = d(u,s) + d(s,v)$ and $t = d(u,s)$. A function $h$ on $T$ is said to be [*convex*]{} if for any vertices $u,v$ in $T$, a function on $\ZZ$, defined by $t \mapsto h([u,v]_t)$ for $0 \leq t \leq d(u,v)$ (and $t \mapsto +\infty$ otherwise), is convex.
\[lem:tree\_convexity\] For a function on $T$, the convexity, L-convexity, and L-extendability are equivalent. For convex functions $f,g$ on $T$ and $\alpha, \beta \in \RR_+$, $\alpha f+ \beta g$ is convex, and $\max (f, g)$, defined by $u \mapsto \max \{f(u), g(u)\}$, is convex.
Let $h$ be a function on $\ZZ$. For vertices $z,w \in T$, we consider three functions $h_T, h_{T;z}, h_{T;z,w}$ defined by $$\begin{aligned}
h_T (u,v) & := & h(d(u,v)) \quad (u,v \in T), \\
h_{T;z} (u) & := & h(d(u,z)) \quad (u \in T), \\
h_{T;z,w} (u,v) & := & h(d(u,z) + d(v,w)) \quad (u,v \in T).\end{aligned}$$ We will see below that $h_{T;z}$ is L-convex, and the other two functions are (midpoint) L-extendable; notice that they are not L-convex in general.
\[thm:2separable\] Let $h$ be a non-decreasing convex function on $\ZZ$, and let $z,w$ be vertices of $T$.
- $h_{T;z}$ is convex on $T$.
- Suppose that $h$ is even. Then $h_{T}$ is L-convex. Moreover, for $(u,v) \in T \times T$, function $h_T$ on ${\cal F}(u) \times {\cal F}(v)$ is a sum of basic $\pmb k$-submodular functions. Namely, for $(s,t) \in {\cal F}(u) \times {\cal F}(v)$, it holds $$h_T (s,t) - h (D) =
\left\{ \begin{array}{ll}
\Delta h(1) \delta (s,t) & {\rm if}\ u = v \in W, \\
\Delta h(D) \theta_{a}(s) & {\rm if}\ u \in W, v \in B, \\
\Delta h(D) \theta_{b}(t) & {\rm if}\ u \in B, v \in W, \\
\Delta h(
D) (\theta_{a}(s) + \theta_{b}(t)) +
\Delta^2 h(D) \mu_{a,b}(s,t) & {\rm if}\ u,v \in W: u \neq v,\\
0 & {\rm if}\ u,v \in B.
\end{array}\right.$$ where $D := d(u,v)$, and $a$ and $b$ are vertices in ${\cal F}(u)$ and in ${\cal F}(v)$ nearest to $v$ and $u$, respectively.
- Suppose that $h$ is even, and $z,w$ belong to the same color class. Then $h_{T;z,w}$ is L-convex. Moreover, for $(u,v) \in T \times T$, function $h_{T;z,w}$ on ${\cal F}(u) \times {\cal F}(v)$ is a sum of basic $\pmb k$-submodular functions. Namely, for $(s,t) \in {\cal F}(u) \times {\cal F}(v)$, it holds $$h_{T;z,w} (s,t) - h(D)
=
\left\{ \begin{array}{ll}
\Delta h (D) \theta_a(s) & {\rm if}\ u \in W, v \in B, \\
\Delta h (D) \theta_b(t) & {\rm if}\ u \in B, v \in W, \\
\Delta h(D) (\theta_a(s) + \theta_b(t)) +
\Delta^2 h(D) \mu_{a,b}(s,t)
& {\rm if}\ u,v \in W, \\
0 & {\rm if}\ u,v \in B,
\end{array}\right.$$ where $D:= d(u,z) + d(v,w)$, and $a$ and $b$ are vertices in ${\cal F}(u)$ and ${\cal F}(v)$ nearest to $z$ and $w$, respectively.
The local expressions of $h_{T}$ and $h_{T;z,w}$ on ${\cal I}(u) \times {\cal I}(v)$ are obtained by replacing roles of $B$ and $W$.
A function $\omega$ on $T^n$ is said to be [*$2$-separable L-convex*]{} if $\omega$ is a sum of functions given in Theorem \[thm:2separable\]: $$\label{eqn:omega}
\omega (x) := \sum_{i} f_i (x_i) + \sum_{i,j} g_{ij} (d(x_i, x_{j}))
+ \sum_{i,j} h_{ij} (d(x_i, z_i) + d(x_j, w_j)) \quad (x \in T^n),$$ where each $f_i$ is a convex function on $T$, each $g_{ij}$ and $h_{ij}$ are nondecreasing even convex function on $\ZZ$, and $z_i$ and $w_j$ are vertices in the same color class. A function $\omega$ on $B^n$ is said to be [*$2$-separable convex*]{} if $\omega$ is the form of (\[eqn:omega\]) where each $g_{ij}$ and $h_{ij}$ are (not necessarily even) nondecreasing convex functions on $\ZZ$. A $2$-separable convex function on $B^n$ is L-extendable, and its L-convex relaxation $\bar \omega$ on $T^n$ is explicitly given by $$\bar \omega (x) := \sum_{i} f_i (x_i) + \sum_{i,j} \bar g_{ij} (d(x_i, x_{j}))
+ \sum_{i,j} \bar h_{ij} (d(x_i, z_i) + d(x_j, w_j)) \quad (x \in T^n),$$ where $\bar g_{ij}$ and $\bar h_{ij}$ are even functions obtained from $g_{ij}$ and $h_{ij}$ by replacing $g_{ij}(z)$ and $h_{ij}(z)$ by $(g_{ij}(z - 1) + g_{ij}(z+1))/2$ and $(h_{ij}(z - 1) + h_{ij}(z+1))/2$, respectively, for each odd integer $z$. A function $\omega$ on $T^n$ is also said be $2$-separable convex if $\omega$ is the form of (\[eqn:omega\]) where each $g_{ij}$ and $h_{ij}$ are (not necessarily even) nondecreasing convex functions on $\ZZ$. Then $\omega$ is midpoint L-extendable.
By Theorem \[thm:2separable\], the L-convex relaxation $\bar \omega$ is locally a sum of basic $k$-submodular functions. Hence $\bar \omega$ is efficiently minimized by successive applications of maximum flow (minimum cut) computations. We will see in Section \[sec:multiflow\] that 2-separable convex functions arise from minimum cost multiflow problems.
A [*convex multifacility location function*]{} is a special 2-separable convex function represented as $$\omega (x) = \sum_{i,j} f_{ij} (d(x_i, z_j)) + \sum_{i,j} g_{ij} (d(x_i, x_{j})) \quad (x \in B^n),$$ where $f_{ij}$ and $g_{ij}$ are nonnegative-valued nondecreasing convex functions, and $z_j$ are black vertices. In this case, we take an L-convex relaxation $$\label{eqn:bar_omega}
\bar \omega (x) = \sum_{i,j} \bar f_{ij} (d(x_i, z_j)) + \sum_{i,j} \bar g_{ij} (d(x_i, x_{j})) \quad (x \in T^n).$$ For the case where all terms are linear functions $b_{ij} d(x_i, z_j)$, $c_{ij} d(x_i, x_j)$ with nonnegative coefficients $b_{ij}$, $c_{ij}$, the problem of minimizing (\[eqn:bar\_omega\]) is known as a [*multifacility location problem*]{} on a tree; see [@Kolen; @TFL83] and also its recent application to computer vision [@FPTZ10; @GK13], where a faster algorithm in [@GK13] is applicable to the case where only $g_{ij}$ are linear (since our notion of convexity is the same as [*$T$-convexity*]{} in [@GK13] for the case of uniform edge-length).
There is a natural rounding scheme from $T^n$ to $B^n$. In some cases, we can construct a good approximate solution for $\omega$ from a minimizer of $\bar \omega$. For $y \in B$ and $x \in T^n$, let $x_{\rightarrow y}$ denote the vertex $z \in B^n$ such that for each $i=1,2,\ldots,n$, $z_i = x_i$ if $x_i \in B$, and $z_i$ is the unique neighbor of $x_i$ close to $y$ (i.e., $d(x_i,y) = 1 + d(z_i,y)$) if $x_{i} \in W$.
\[thm:approx\] Let $\omega: B^n \to \RR$ be a $2$-separable convex function consisting of $m$ terms, and $\bar \omega: T^n \to \RR$ be its L-convex relaxation.
- For a given $x \in B^n$, there is an $O(d({\rm opt}(\bar \omega), x) {\rm MF}(k n, k m))$ time algorithm to find a global minimizer $x^*$ of $\bar \omega$ over $T^n$, where $k$ is the maximum degree of $T$.
- Suppose that $\omega$ is a convex multifacility location function. For any $y \in B$, the rounded solution $(x^*)_{\rightarrow y}$ is a $2$-approximate solution of $\omega$.
\[rem:multiway\] As discussed in [@IWY14], multiway cut problem is a representative example of NP-hard problems admitting a $k$-submodular relaxation (an L-convex relaxation in our sense). Let $G = (V,E)$ be an undirected graph with a set $S \subseteq V$ of terminals and an edge-capacity $c:E \to \RR_+$. Let $V \setminus S = \{1,2,\ldots,n\}$. A [*multiway cut*]{} ${\cal X}$ is a partition of $V$ such that each part contains exactly one terminal. The capacity $c({\cal X})$ of ${\cal X}$ is the sum of $c(ij)$ over all edges $ij$ whose ends $i$ and $j$ belong to distinct parts in ${\cal X}$. The multiway cut problem in $G$ is the problem of finding a multiway cut with minimum capacity. The multiway cut problem is formulated as minimization of a multifacility location function on a star. Let $T$ be a star with leaf set $S$ and center vertex $0$. Suppose that $B = S$ and $W = \{0\}$. Consider the following 2-separable convex function minimization: $$\begin{aligned}
\label{eqn:multiway}
\mbox{Min.} && \frac{1}{2} \sum_{1 \leq i \leq n} \sum_{s \in S: si \in E}
c(si) d(s, x_i) + \frac{1}{2} \sum_{ij \in E: 1 \leq i,j \leq n} c(ij) d (x_i, x_j), \\
\mbox{s.t.} && (x_1,x_2,\ldots,x_n) \in B^n. \nonumber \end{aligned}$$ This is equivalent to the multiway cut problem on $G$. To see this, for a multiway cut ${\cal X}$, define $x = (x_1,x_2,\ldots,x_n)$ by: $x_i := s$ if $i$ and $s$ belong to the same part of ${\cal X}$. Then the objective of the resulting $x$ is equal to the cut capacity of ${\cal X}$. Conversely, for a solution $x$ of (\[eqn:multiway\]), define $X_s$ by the set of all vertices $i$ with $x_i = s$ for $s \in S$. Then ${\cal X} := \{X_s\}_{s \in S}$ is a multiway cut whose capacity is equal to the objective value of $x$.
An L-convex relaxation problem relaxes the constraint $x \in B^n$ into $x \in T^n = (B \cup \{ 0\})^n$. If $T^n$ is identified with ${S_k}^n$, then this is a $k$-submodular function minimization, where $d$ is equal to $\delta$. An optimum $x^*$ of the relaxed problem can be efficiently obtained by $(s, S \setminus \{s\})$-mincut computations for $s \in S$; see Section \[subsub:LC\]. Take some $s \in B$. Consider $y := (x^*)_{\rightarrow s}$. This 2-approximation is essentially equal to the classical 2-approximation algorithm of multiway cut problem [@DJPSY94]; see [@Vazirani Algorithm 4.3].
Proofs {#subsec:proof}
------
In this section, we give proofs of results. We will often use the following variation of $\pmb k$-submodularity inequality.
\[lem:k-submo’\] If $f$ is a $\pmb k$-submodular function on $S_{\pmb k}$, then $$\label{eqn:k-submo'}
f(x) + f(y) \geq f(x \wedge y) + \frac{1}{2} f(x \sqcup (x \sqcup y)) + \frac{1}{2} f((x \sqcup y) \sqcup y)
\quad (x,y \in S_{\pmb k}).$$
Observe that $u \wedge (u \sqcup v)$ is $0$ if $0 \neq u \neq v \neq 0$ and is $u$ otherwise $(u,v \in S_k)$. From this we have $$\begin{aligned}
((x \sqcup y) \wedge x) \sqcup (( x \sqcup y) \wedge y) & = & x \sqcup y, \\
((x \sqcup y) \wedge x) \wedge (( x \sqcup y) \wedge y) & = & x \wedge y.\end{aligned}$$ Similarly $u \sqcup (u \sqcup v)$ is $v$ if $u = 0$ and is $u$ otherwise. From this we have $$(x \sqcup (x \sqcup y)) \sqcup ((x \sqcup y) \sqcup y) =
(x \sqcup (x \sqcup y)) \wedge ((x \sqcup y) \sqcup y) = x \sqcup y.$$ Thus we have $$\begin{aligned}
&& f((x \sqcup y) \wedge x) + f(( x \sqcup y) \wedge y) \geq f(x \wedge y) + f(x \sqcup y), \\
&& f(x) + f(x \sqcup y) \geq f(x \wedge (x \sqcup y)) + f(x \sqcup (x \sqcup y) ), \\
&& f(x \sqcup y) + f(y) \geq f((x \sqcup y) \wedge y ) + f((x \sqcup y) \sqcup y), \\
&& f(x \sqcup (x \sqcup y)) +
f((x \sqcup y) \sqcup y) \geq 2 f(x \sqcup y).\end{aligned}$$ Adding the first three inequalities and one half of the forth inequality, we obtain (\[eqn:k-submo’\]).
The binary operation $(x,y) \mapsto x \sqcup (x \sqcup y)$ plays important roles in the subsequent arguments. We note the following properties:
$x \sqcup (x \sqcup y) \succeq x$.
If $y$ has no zero component, then so does $x \sqcup (x \sqcup y)$.
$(x \sqcup (x \sqcup y)) \sqcup y = x \sqcup y$ and $(x \sqcup (x \sqcup y)) \sqcup (x \sqcup y) = x \sqcup (x \sqcup y)$. \[eqn:sqcup\]
These properties immediately follow from the behavior of each component: $$u \sqcup (u \sqcup v) = \left\{
\begin{array}{ll}
v & {\rm if}\ u = 0, \\
u & {\rm otherwise},
\end{array}\right. \quad (u,v \in S_{k}).$$
### Proof of Theorem \[thm:bound\] {#subsec:bound}
Let $g$ be an L-convex function on $T^n$, and let $x$ be a vertex in ${\mathop{\rm dom} }g$. We first show the latter part. Suppose (w.l.o.g.) that $$\label{eqn:g(x)=min_g(y)}
g(x) = \min_{y \in {\cal F}(x)} g(y).$$ Let $x = x^0,x^1,x^2,\ldots,x^m$ be a sequence of vertices in generated by the steepest descent algorithm applied to $(g,x)$. Then it holds $$\label{eqn:zigzag}
x = x^0 \succ x^1 \prec x^2 \succ x^3 \prec x^4 \succ \cdots.$$ Indeed, $x^0 \succ x^1$ follows from (\[eqn:g(x)=min\_g(y)\]). Also $x^{i} \prec x^{i+1} \prec x^{i+2}$ (or $x^{i} \succ x^{i+1} \succ x^{i+2}$) never occurs. Otherwise $x^{i+2} \in {\cal F}(x^i)$ and $g(x^{i+2}) < g(x^{i+1}) < g(x^i)$; this is impossible from the definition of the steepest descent algorithm.
\[lem:geodesic\] For $z \in {\cal I}(x^k) \cup {\cal F}(x^k)$ with $g(z) < g(x^k)$, we have $$d(x,z) = k+1 \quad (k=1,2,\ldots).$$
By (\[eqn:zigzag\]), it holds that $x^{k-1}, x^{k+1} \in {\cal F}(x^{k})$ if $k$ is odd and $x^{k-1}, x^{k+1} \in {\cal I}(x^{k})$ if $k$ is even. We use the induction on $k$. Then we can assume that $d(x^{k-1}, x) = k-1$ and $d(x^{k}, x) = k$. Suppose for the moment that $k$ is odd. We are going to show that $d(z, x) = k+1$. By Lemma \[lem:locally\_k\_submo\], $g$ is $\pmb k$-submodular on ${\cal F}(x^k)$. Notice that both $x^{k-1}$ and $z$ belong to ${\cal F}(x^k)$. By Lemma \[lem:k-submo’\], we have $$g(z) + g(x^{k-1}) \geq g(z \wedge x^{k-1}) + \frac{1}{2} g(z \sqcup (z \sqcup x^{k-1}))
+ \frac{1}{2} g((z \sqcup x^{k-1}) \sqcup x^{k-1}).$$
Since $x^k$ is a minimizer of $g$ over ${\cal I}(x^{k-1})$, we have $$g(z \wedge x^{k-1}) \geq g(x^{k}) > g(z).$$ Hence we necessarily have $$2 g(x^{k-1}) > g(z \sqcup (z \sqcup x^{k-1})) + g((z \sqcup x^{k-1}) \sqcup x^{k-1}).$$ This implies $g(x^{k-1}) > g(z \sqcup (z \sqcup x^{k-1}))$ or $g(x^{k-1}) > g((z \sqcup x^{k-1}) \sqcup x^{k-1})$. The second case is impossible. This follows from: $x \preceq (z \sqcup x) \sqcup x$ (see (\[eqn:sqcup\]) (1)) and (\[eqn:g(x)=min\_g(y)\]) for $k=1$, and $x^{k-2} \preceq x^{k-1} \preceq (z \sqcup x^{k-1}) \sqcup x^{k-1}$ and $g(x^{k-1}) = \min_{y \in {\cal F}(x^{k-2})} g(y)$ for $k > 1$. Thus we have $$\label{eqn:=>>}
g((z \sqcup x^{k-1}) \sqcup x^{k-1}) \geq g(x^{k-1}) > g(z \sqcup (z \sqcup x^{k-1})).$$ Let $z' := x^{k-1} \wedge (z \sqcup (z \sqcup x^{k-1}))$. Then $x^{k} \preceq z' \preceq x^{k-1}$. By Lemma \[lem:k-submo’\], we have $$g(z \sqcup (z \sqcup x^{k-1})) + g(x^{k-1}) \geq g(z') +
\frac{1}{2} g(z \sqcup (z \sqcup x^{k-1})) + \frac{1}{2} g( (z \sqcup x^{k-1}) \sqcup x^{k-1}),$$ where we use (\[eqn:sqcup\]) (3) to obtain the second and third terms in the right hand side. By (\[eqn:=>>\]) we have $
g(x^{k-1}) > g(z').
$ Notice $z' \in {\cal I}(x^{k-1})$. By induction, we have $$\label{eqn:|p-q'|=i}
d(x,z') = k.$$ Hence we can take an index $j$ with $d(x_j,z'_j) = k$. By $x^k_j \preceq z'_j \preceq x^{k-1}_j$ (in ${\cal F}(x^{k}_j)$), we have $x^k_j = z'_j \prec x^{k-1}_j$ and $d(x_j,x^{k-1}_j) = k-1$. Since $x^{k}_j = ((x^{k-1}_j \sqcup z_j) \sqcup z_j) \wedge x^{k-1}_j$, we have $x^{k-1}_j \neq z_j \neq x^{k}_j$; otherwise $x^{k-1}_j = z_j$ or $x^{k}_j = z_j$ implies a contradiction $x^{k-1}_j = x^{k}_j$. Thus $x^{k-1}_j$ and $z_j$ are distinct neighbors of $x^k_j$ in $T$ with $d(x_j, x^{k-1}_j) = k-1$ and $d(x_j, x^{k}_j) = k$. Since $T$ is a tree, we have $d(x_j, z_j) = k + 1$, and $d(x, z) = k + 1$. The argument for even $k (\geq 2)$ is same; reverse the partial order $\preceq$.
Let ${\rm opt}(g)$ be the set of minimizers of $g$, and let $m^* := \min_{z \in {\rm opt}(g)} d(x,z)$. Since $x^m$ is optimal, we have $m^* \leq m$. Our goal is to show $m = m^*$. Let $\tilde g$ be a function defined by $\tilde g(y) := g(y)$ if $d(x,y) \leq m^*$ and $\tilde g(y) := \infty$ otherwise. Then $\tilde g$ is also L-convex since $d(x, y) \leq m^*$ and $d(x, y') \leq m^*$ imply $d(x, y \bullet y') \leq m^*$ and $d(x, y \circ y') \leq m^*$. The subsequence $x = x^1,x^2,\ldots,x^{m^*}$ is also obtained by applying the steepest descent algorithm to $\tilde g$ from $x$. By Lemma \[lem:geodesic\], no vertex $z$ with $d(x,z) > m^*$ is produced. Hence $x^{m^*}$ is necessarily a minimizer of $\tilde g$, and is also a minimizer of $g$. Thus $m = m^*$. This completes the latter part of the proof of Theorem \[thm:bound\]. The former part is now immediately obtained. Suppose that $x$ does not satisfy (\[eqn:g(x)=min\_g(y)\]). But $x^1$ always satisfies $g(x^1) = \min_{y \in {\cal F}(x^1)} g(y)$ or $g(x^1) = \min_{y \in {\cal I}(x^1)} g(y)$. The sequence $(x^1,x^2,\ldots,x^m)$ is also obtained by the steepest descent algorithm. By the latter claim and the triangle inequality, we have $m - 1 = d(x^1, {\mathop{\rm opt} }(g)) \leq d(x, {\mathop{\rm opt} }(g)) + 1$.
\[prop:sequence\] Let $g$ be an L-convex function on $T^n$. For $y \in {\rm opt}(g)$ and $x \in {\mathop{\rm dom} }g$ with $g(x) = \min_{y \in {\cal F}(x)} g(y)$, there is a sequence $x = x^0,x^1,x^2,\ldots, x^m = y$ such that
- $m = d(x,y)$,
- $g(x^{i}) > g(x^{i+1})$ for $i < d({\rm opt}(g),x)$ and $g(x^{i}) = g(x^{i+1})$ for $i \geq d({\rm opt}(g),x)$, and
- $g(x^{i+1}) = \min \{ g(z) \mid z \in {\cal I}(x^{i})\}$ for even $i$ and $g(x^{i+1}) = \min \{ g(z) \mid z \in {\cal F}(x^{i})\}$ for odd $i$.
Take a sufficiently small $\epsilon > 0$. Consider the function $g'$ defined by $$g'(x) = g(x) + \epsilon \sum_{i=1}^n d(x_i, y_i) \quad (x \in T^n).$$ By Theorem \[thm:2separable\] (that will be proved independently), the second term is L-convex, and hence $g'$ is also L-convex, and has the unique minimizer $y$. Apply the steepest descent algorithm to $(g',x)$. By Theorem \[thm:bound\], we obtain a sequence $x = x^0,x^1,\ldots,x^m = y$ with (1), (2), and (3) for $g'$. Since $\epsilon$ is sufficiently small, any steepest direction $x^{i+1}$ for $g'$ at $x^i$ is also a steepest direction for $g$. Thus the sequence also satisfies (1),(2), and (3) for $g$.
### Proof of Theorem \[thm:persistency\]
Let $x$ be a minimizer of an L-convex relaxation $g$ of an L-extendable function $h$. Take a minimizer $y \in B^n$ of $h$ such that $d(x,y)$ is minimum. We can take a sequence $y = y^0,y^1,y^2,\ldots,y^m = x$ satisfying the conditions in Proposition \[prop:sequence\], where $m = d(x,y)$. If $m = 1$, then $x \in {\cal I}(y)$ ($y \in {\cal F}(x)$) and we are done. Suppose (indirectly) $m \geq 2$. Let $z := y^1$ and $w := y^2$. Notice that $y,w \in {\cal F}(z)$. Applying Lemma \[lem:k-submo’\] to $\pmb k$-submodular function $g$ on ${\cal F}(z)$, we have $$g(y) + g(w) \geq g(y \wedge w) + \frac{1}{2} g(y \sqcup (y \sqcup w)) + \frac{1}{2} g(w \sqcup (w \sqcup y)).$$ Since $y$ is a maximal element in ${\cal F}(z)$ and $y \preceq y \sqcup (y \sqcup w)$, we have $y = y \sqcup (y \sqcup w)$, and $g(y \sqcup (y \sqcup w)) = g(y)$. Also it holds $g(y \wedge w) \geq g(z) \geq g(w)$ (by Proposition \[prop:sequence\] (3)). This implies $$g(y) \geq g(w \sqcup (w \sqcup y)).$$ Here $w':= w \sqcup (w \sqcup y)$ is also maximal (in ${\cal F}(z)$) and has no zero components (see (\[eqn:sqcup\]) (2)). Thus $w'$ belongs to $B^n$, and is also a minimizer of $h$. Since $d(w',w) =1$ (by $w \preceq w'$) and $d(w,x) = m - 2$, we have $d(x,y) > d(x,w')$. A contradiction to the minimality.
### Proof of Theorem \[thm:optimality\_h\]
Let $g$ be an L-convex relaxation of $h$. Suppose that $x$ is not a minimizer of $h$. Then $x$ is not a minimizer of $g$. By the L-optimality criterion (Theorem \[thm:L-optimality\]), there is $z \in {\cal I}(x)$ such that $g(z) < g(x)$. If $z$ is a minimizer of $g$, then by Theorem \[thm:persistency\] there is a minimizer $y \in B^n \cap {\cal F}(z)$ of $h$, as required. Suppose that $z$ is not a minimizer of $g$. There is $w \in {\cal F}(z)$ such that $g(w) < g(z)$. By Lemma \[lem:k-submo’\] with $x \sqcup (x \sqcup w) = x$, we have $$g(x) + g(w) \geq g(x \wedge w) + \frac{1}{2} g(x) + \frac{1}{2} g(w \sqcup (w \sqcup x)).$$ Notice $g(w) < g(x \wedge w)$. Hence $g(x) > g(w \sqcup (w \sqcup x))$, and $w \sqcup (w \sqcup x)$ is a required vertex in $B^n$ (by (\[eqn:sqcup\]) (2)).
### Proof of Theorem \[thm:proximity\]
We start with preliminary arguments. For a quarter integer $u \in \ZZ/4$, define half-integers $[u]_1, [u]_{1/2} \in \ZZ/2$ by $$[u]_1 := \left\{
\begin{array}{ll}
u & {\rm if} \ u \in \ZZ/2, \\
\mbox{the integer nearest to $u$} & {\rm otherwise},
\end{array}\right.$$ $$[u]_{1/2} := \left\{
\begin{array}{ll}
u & {\rm if}\ u \in \ZZ/2, \\
\mbox{the non-integral half-integer nearest to $u$} & {\rm otherwise}.
\end{array}\right.$$
\[lem:1/4\] For $u,v \in \ZZ/4$, we have $$\begin{aligned}
&& \lfloor u - v \rfloor \leq [u]_1 - [v]_1 \leq \lceil u - v \rceil, \\
&& \lfloor u - v \rfloor \leq [u]_{1/2} - [v]_{1/2} \leq \lceil u - v \rceil.\end{aligned}$$
It suffices to consider the cases $([u]_1,[v]_1) = (u \pm 1/4, v)$, $(u, v \pm 1/4)$, $(u + 1/4, v - 1/4)$, or $(u - 1/4, v + 1/4)$. Suppose that the first two cases occur. Then $u - v$ is not a half-integer, and hence $\lfloor u - v \rfloor \leq u - v \pm 1/4 \leq \lceil u - v \rceil$. Consider the last two cases. Then $u \in \ZZ - 1/4$ and $v \in \ZZ + 1/4$ or $u \in \ZZ + 1/4$ and $v \in \ZZ - 1/4$. Hence $u - v \in \ZZ + 1/2$. This means that $u - v$ is not an integer but a half-integer. Thus $\lfloor u - v \rfloor \leq u - v \pm 1/2 \leq \lceil u - v \rceil$. The second inequality follows from the same argument.
Let $x,y \in T^n$. Let $P_i$ denote the unique path connecting $x_i$ and $y_i$ in $T$. We regard vertices of $P_i$ as integers $0,1,\ldots, d_i := d(x_i,y_i)$ by the following way. Associate vertex $u$ in $P_i$ with integer $d(x_i,u) \in \{0,1,\ldots, d_{i}\}$. Then $x_i = 0$ and $y_i = d_i$. Similarly, let $P_i^*$ denote the unique path connecting $x_i$ and $y_i$ in $T^{*}$. Associate the midpoint of each edge $uv$ in $P_i$ with half-integer $(u + v)/2 (= (d(x_i,u) + d(x_i,v))/2)$. Then the vertices of the product $P := P_1 \times P_2 \times \cdots \times P_n$ are integer vectors $z$ with $0 \leq z_i \leq d_i$, and the vertices of $P^* := P^*_1 \times P^*_2 \times \cdots \times P^*_n$ are half-integer vectors $z$ with $0 \leq z_i \leq d_i$. Under this correspondence, it holds $$z \bullet z' = [ (z+ z')/2 ]_1, \quad z \circ z' = [ (z+ z')/2]_{1/2} \quad (z,z' \in P^* \subseteq (\ZZ/2)^n ),$$ where $[\cdot]_{1}$ and $[\cdot]_{1/2}$ are extended on $(\ZZ/4)^n$ in componentwise.
For $i \in \{1,2,\ldots,n\}$, let $e_i$ denote the $i$-th unit vector, and let $\pi_{i} := e_1 + e_2 + \cdots + e_i$. We can assume that $d_1 \geq d_2 \geq \cdots \geq d_n$. Then $y$ is represented as $$\label{eqn:expression}
y = x + \sum_{i=1}^{n} (d_{i}- d_{i+1}) \pi_{i},$$ where we let $d_{n+1} = 0$. Let $((x,y))$ and $((x,y))^*$ be the sets of integral points and half-integral points, respectively, in the polytope $$Q(x,y) := \{ z \in \RR^n \mid 0 \leq z_{i} - z_{i+1} \leq d_{i} - d_{i+1} \ (1 \leq i \leq n) \},$$ where we let $z_{n+1} := 0$. Observe that the polytope $Q(x,y)$ is the set of points $z$ represented as $$\label{eqn:represented}
z = x + \sum_{i=1}^{n} \alpha_i \pi_{i}$$ for $\alpha_i \in [0, d_{i} - d_{i+1}]$ $(i=1,2,\ldots,n)$. Note that this representation is unique.
\[lem:z+z’\] For $z,z' \in ((x,y))^*$, both $[(z + z')/2]_1$ and $[(z + z')/2]_{1/2}$ belong to $((x,y))^*$.
We show that half-integer vectors $[(z +z')/2]_1$ and $[(z + z')/2]_{1/2}$ belong to $Q(x,y)$. By convexity, $w := (z+ z')/2 \in (\ZZ/4)^n$ belong to $Q(x,y)$. Hence $$0 \leq w_i - w_{i+1} \leq d_{i} - d_{i+1}.$$ Notice that $d_{i} - d_{i+1}$ is integral. By Lemma \[lem:1/4\], we have $$0 \leq [w_i]_1 - [w_{i+1}]_1 \leq d_{i} - d_{i+1}, \quad
0 \leq [w_i]_{1/2} - [w_{i+1}]_{1/2} \leq d_{i} - d_{i+1}.$$ This means that both $[w]_1$ and $[w]_{1/2}$ belong to $Q(x,y)$.
Let $h: T^n \to \overline{\RR}$ be an L-extendable function, and let $g: (T^*)^n \to \overline{\RR}$ be its L-convex relaxation. The discrete midpoint convexity inequality on $P^*$ is given by $$\label{eqn:midpoint_P*}
g(z) + g(z') \geq g( [(z+z')/2]_{1}) + g([(z+z')/2 ]_{1/2}) \quad (z,z' \in P^*).$$ In particular, for $i,j \in \{1,2,\ldots,n\}$, we have $$\begin{aligned}
g(\pi_{j}/2) + g(\pi_{i}/2 + \pi_{j}) & \geq & g(\pi_{j}) + g( \pi_{i}/2 + \pi_{j}/2), \label{eqn:pi1} \\
g(0) + g(\pi_{i}/2 + \pi_{j}/2) & \geq & g(\pi_{i}/2) + g(\pi_{j}/2), \label{eqn:pi2} \\
g(\pi_i/2 + \pi_j/2) + g(\pi_i + \pi_j) & \geq & g(\pi_i + \pi_j/2) + g(\pi_i/2 + \pi_j), \label{eqn:pi3}\\
g(\pi_i/2) + g(\pi_i + \pi_j/2) & \geq & g(\pi_i/2 + \pi_j/2) + g(\pi_i). \label{eqn:pi4}\end{aligned}$$ For example, $(\pi_i+3\pi_j)/4 = \pi_{i} + 3(e_{i+1} + e_{i+2} + \cdots + e_{j})/4$ for $i < j$ and $(\pi_i+3\pi_j)/4 = \pi_{j} + (e_{j+1} + e_{j+2} + \cdots + e_{i})/4$ for $i > j$. Thus $[(\pi_i + 3 \pi_j)/4]_1 = \pi_j$ and $[(\pi_i + 3 \pi_j)/4]_{1/2} = \pi_i/2 + \pi_j/2$. From (\[eqn:midpoint\_P\*\]), we see the first equality. The remaining are obtained in the same way.
\[lem:dom\] For $x,y \in {\mathop{\rm dom} }h$, it holds $((x,y)) \subseteq {\mathop{\rm dom} }h$.
We use the induction on $d(x,y) = k$. We can assume that ${\mathop{\rm dom} }h$ belongs to $((x,y))$. Indeed, modify $g$ and $h$ so that they take $\infty$ on points not belonging to $Q(x,y)$. By Lemma \[lem:z+z’\], $g$ is still L-convex, and hence is an L-convex relaxation of $h$. In particular $h$ is L-extendable. We may assume that one of $x,y$, say $x$, is not a minimizer of $h$ (by adding a 2-separable L-convex function $z \mapsto \sum_{i=1}^n d(z_i, y_i)$ to $h, g$ if necessarily).
Use expression (\[eqn:expression\]) to represent $x, y$. By induction it suffices to show that each $\pi_i$ with $d_i > d_{i+1}$ belongs to ${\mathop{\rm dom} }h$, since $d(\pi_i, y) = k-1$ and $y = \pi_i + (d_{i} - d_{i+1} - 1) \pi_i + \sum_{j \neq i} (d_j - d_{j+1}) \pi_j$. By Theorem \[thm:optimality\_h\] with expression (\[eqn:represented\]), there is $j$ such that $d_{j} > d_{j+1}$ and $h(x) = h(0) > h(\pi_{j}) < \infty$. Consider an index $i \neq j$ with $d_i > d_{i+1}$. We show that $\pi_i \in {\mathop{\rm dom} }h$. By induction for $(\pi_j, y)$, we have $\pi_i + \pi_j \in {\mathop{\rm dom} }h \subseteq {\mathop{\rm dom} }g$. By applying the midpoint convexity (\[eqn:midpoint\_P\*\]) for $g$ at $(0, \pi_j)$ and at $(\pi_j, \pi_i + \pi_j)$, we have $\pi_j/2, \pi_j + \pi_i/2 \in {\mathop{\rm dom} }g$. By (\[eqn:pi1\]), we have $\pi_i/2 + \pi_j /2 \in {\mathop{\rm dom} }g$. Similarly, by (\[eqn:pi2\]), we have $\pi_i/2 \in {\mathop{\rm dom} }g$. By (\[eqn:pi3\]) we have $\pi_i + \pi_j /2 \in {\mathop{\rm dom} }g$. Finally, by (\[eqn:pi4\]), we have $\pi_i \in {\mathop{\rm dom} }h$, as required.
The essence of the proximity theorem is the following, where this lemma may be viewed as an analogue of [@FujiBook (20.37)].
\[lem:pi\_i\] For $x \in {\mathop{\rm dom} }h$ and a minimizer $y$ of $h$ with $d(x, {\mathop{\rm opt} }(h)) = d(x,y)$, we have $$h(x) > g(x + \pi_i /2) > h(x + \pi_{i}) \quad (i: d_{i} - d_{i+1} \geq 2).$$
As above, we can assume that ${\mathop{\rm dom} }h$ belongs to $((x,y))$, and ${\mathop{\rm dom} }h = ((x,y))$ (by Lemma \[lem:dom\]). Then $y$ is a unique minimizer of $h$ over $((x,y))$. We use the induction on $d(x,y)$. Consider first the case where $y = x + (d_i - d_{i+1}) \pi_i$ for some $i$ with $d_i - d_{i+1} \geq 2$. By Theorem \[thm:optimality\_h\] with induction, we have $h(0) > h(\pi_i) > h(2\pi_i)$. By $h(0) + h(\pi_i) = g(0) + g(\pi_i) \geq 2 g(\pi_i/2)$ we have $h(0) > g(\pi_i/2).$ By $h(\pi_i) + h(2 \pi_i) = g(\pi_i) + g(2\pi_i) \geq 2 g(3\pi_i/2)$, we have $g(\pi_i) > g(3\pi_i/2)$. By $g(\pi_i/2) + g(3\pi_i/2) \geq 2 g(\pi_i)$, we have $g(\pi_i/2) > g(\pi_i) = h(\pi_i)$. Thus $h(0) > g(\pi_i/2) > h(\pi_i)$ holds.
Consider the general case. Take $i$ with $d_i - d_{i+1} \geq 2$. We may assume that there is $j \neq i$ with $d_j - d_{j+1} \geq 1$. By induction for $(\pi_j, y)$, we have $$\label{eqn:j>i+j}
g(\pi_j) = h(\pi_j) > g(\pi_j + \pi_i/2) > g(\pi_i + \pi_j) = h(\pi_i + \pi_j).$$ By (\[eqn:j>i+j\]) and (\[eqn:pi1\]), we have $g(\pi_j/2) > g(\pi_i/2 + \pi_j/2)$, and, by (\[eqn:pi2\]), $g(0) > g(\pi_{i}/2)$. Similarly, by (\[eqn:j>i+j\]) and (\[eqn:pi3\]), we have $g(\pi_i/2 + \pi_j/2) > g(\pi_i + \pi_j/2)$, and, by (\[eqn:pi4\]), $g(\pi_i/2) > g(\pi_i)$. Thus we have $h(0) = g(0) > g(\pi_i/2) > g(\pi_i) = h(\pi_i)$, as required.
We are ready to prove Theorem \[thm:proximity\]. Let $h$ be a midpoint L-extendable function on $T^n$, and let $x$ be a minimizer of $h$ over $B^n$. Let $y$ be a minimizer of $h$ over $T^n$ with $d(x, {\rm opt}(h)) = d(x,y)$. We can assume that $h(y) < h(x)$. Consider $((x,y))$ as above. Since $x$ is a minimizer of $h$ over $B^n$, we have $h(x) \leq h(x + \alpha \pi_i)$ for an even integer $\alpha$. On the other hand, by Lemma \[lem:pi\_i\], we have $h(x) > h(x + \alpha \pi_{i})$ for $\alpha \in 1,2,\ldots, d_{i} - d_{i+1} - 1$ if $d_i - d_{i+1} \geq 2$. This means that $d_{i} - d_{i+1} \geq 3$ is impossible. Thus $d_i - d_{i+1} \leq 2$ for each $i$. Hence $d(x,y) \leq 2 n$.
Consider the case where $h$ admits as an exact L-convex relaxation. In the proof of Lemma \[lem:pi\_i\], we can assume that $y$ is also a unique minimizer of $g$ (by perturbing $h,g$ if necessarily). Consequently the statement of Lemma \[lem:pi\_i\] holds for index $i$ with $d_i - d_{i+1} \geq 1$. Therefore, in the above argument, $d_{i} - d_{i+1} \geq 2$ is impossible. Thus we obtain the latter statement of Theorem \[thm:proximity\].
### Proof of Lemma \[lem:tree\_convexity\]
It suffices to consider the case where $T$ is a path. Hence $T$ is naturally identified with $\ZZ$, and $B = 2\ZZ$. Suppose that $f$ is convex on $\ZZ$. Then it is easy to see that $f(u) + f(v) \geq f( \lfloor (u + v)/2 \rfloor)
+ f( \lceil (u + v)/2 \rceil) = f( u \circ v) + f(u \bullet v)$. Thus $f$ is (alternating) L-convex. The converse is also easy. If $f:\ZZ \to \RR$ is convex, then $\bar f: \ZZ/2 \to \RR$ defined by $u \mapsto (f(\lfloor u \rfloor) + f( \lceil u\rceil))/2$ is also convex (and L-convex) on $T^*$, and $f$ is L-extendable. The converse is also easy: the restriction of convex function on $\ZZ/2$ to $\ZZ$ is also convex on $\ZZ$. The latter part is straightforward to be verified.
### Proof of Theorem \[thm:2separable\]
We begin with preliminary arguments on the convexity of a tree. Let $\bar T$ be the set of all formal combinations of vertices of form $\lambda u + \mu v$, where $uv$ is an edge, and $\lambda$ and $\mu$ are nonnegative reals with $\lambda + \mu = 1$. Informally speaking, $\bar T$ is a “tree" obtained by filling the “unit segment" to each edge. We can naturally regard $T$ and $T^*$ as subsets of $\bar T$ (by $T^{*} \ni w_{uv} \mapsto (1/2) u + (1/2) v$). Also the metric $d$ on $T$ is extended to $\bar T$ as follows. For two points $p= \lambda u + \mu v$, $p' = \lambda' u' + \mu' v' \in \bar T$, if $(u,v) = (u',v')$, then $d(p,p') := |\lambda - \lambda'| = |\mu - \mu'|$. Otherwise we can assume that $d(v,v') = d(v,u) + d(u,u') + d(u',v')$. Define $d(p,p') := \mu + d(u,u') + \mu'$. For points $p,q \in \bar T$ and $t \in [0,1]$, there is a unique point $r$, denoted by $p \circ_{t} q$, such that $d(p,q) = d(p, r) + d(r,q)$, $d(p,r) := t d(p,q)$, and $d(r,q) := (1- t) d(p,q)$. Consider the Cartesian product $\bar T^n$. For $x,y \in \bar T^n$, define $x \circ_t y := (x_1 \circ_t y_1, x_2 \circ_t y_2,\ldots, x_n \circ_t y_n)$. A function $f$ on $\bar T^n$ is said to be [*convex*]{} if it satisfies $$(1 - t) f(x) + t f(y) \geq f(x \circ_t y) \quad (t \in [0,1], x,y \in \bar T^n).$$ In the case where $T$ is a path of infinite length, $\bar T$ is isometric to $\RR$, and this convexity coincides with the ordinary Euclidean convexity. An old theorem in location theory, due to Dearing, Francis, and Lowe [@DFL76], says that the distance function $d$ is convex on $\bar T^2$.
\[lem:DFL\] $d$ is convex on $\bar T^2$.
We note local expressions of functions $(s,t) \mapsto h(d(s,t))$ and $(s,t) \mapsto h(d(s,a) + d(t,b))$.
\[lem:123\] Let $h$ be an even function on $\ZZ$ and let $u, v \in W$.
- For $s,t \in {\cal F}(u)$, we have $$h(0) + \Delta h(1) \delta (s,t) = h(d(s,t)).$$
- For $s,a \in {\cal F}(u)$, we have $$h(1) + \Delta h(1) \theta_{a}(s)
=
\left\{ \begin{array}{ll}
h(d(s,a)) & {\rm if}\ a \neq u,\\
h(d(s,a) + 1) & {\rm if}\ a = u.
\end{array}\right.$$
- For $(s,t),(a,b) \in {\cal F}(u) \times {\cal F}(v)$, we have $$\begin{aligned}
&& h(2) + \Delta h(2) (\theta_a(s) + \theta_b(t)) + \Delta^2 h(2) \mu_{a,b}(s,t) \\[0.5em]
&& \ = \left\{ \begin{array}{ll}
h(d(s,a) + d(t,b) + 2) & {\rm if }\ (a,b) = (u,v), \\
h(d(s,a) + d(t,b) +1) & {\rm if }\ a = u, b \neq v\ {\rm or}\ a \neq u, b = v,\\
h(d(s,a)+ d(t,b)) & {\rm if}\ a \neq u, b \neq v.
\end{array}\right. \\\end{aligned}$$
(1). Observe $d(s,t) = \delta (s,t)$. Thus $d(s,t) = 0$ implies $h(0) + \Delta h(1) \cdot 0 = h(0)$. Also $d(s,t) = 1$ implies $h(0) + \Delta h(1) \cdot 1 = h(1)$, and $d(s,t) = 2$ implies that $h(0) + \Delta h(1) \cdot 2
= 2 h(1) - h(0) = h(2)$ (by the evenness of $h$).
(2). Consider the case where the right hand side is equal to $h(2)$. Then $a \neq s \neq u$ must hold. Therefore the left hand side is $h(1) + \Delta h(1) \cdot 1 = h(2)$. Suppose that the right hand side is equal to $h(1)$. Then $s = u$ holds. Therefore the left hand side is $ h(1) + \Delta h(1) \cdot 0 = h(1)$. Suppose that the right hand side is equal to $h(0)$. Then $s = a \neq u$ holds. Therefore the left hand side is $h(1) + \Delta h(1) \cdot (-1) = h(0)$.
(3). Consider the case where the right hand side is equal to $h(4)$. Then $u \neq s \neq a$ and $v \neq t \neq b$ must hold. The left hand side is equal to $h(2) + \Delta h(2) \cdot 2 + \Delta^2 h(2) \cdot 2= - h(2) + 2h(3) = h(4)$. Consider the case where the right hand side is equal to $h(3)$. Then $u \neq s \neq a$ and $v = t$ or $u = s$ and $v \neq t \neq b$ must hold. The left hand side is equal to $h(2) + \Delta h(2) \cdot 1 + \Delta^2 h(2) \cdot 1= h(3)$.
Consider the case where the right hand side is equal to $h(2)$. Then $s = u$ and $t = v$ must hold. The left hand side is equal to to $h(2) + \Delta h(2) \cdot 0 + \Delta^2 h(2) \cdot 0= h(2)$.
Consider the case where the right hand side is equal to $h(1)$. Then $s = a = u$ and $t = b \neq v$, $s = a \neq u$ and $t = b = v$, $s = a \neq u$ and $b \neq t = v$, or $a \neq s = u$ and $t = b \neq v$. The left hand side is equal to $h(2) + \Delta h(2) \cdot (-1) + \Delta^2 h(2) \cdot 0= h(1)$.
Consider the case where the right hand side is equal to $h(0)$. Then $s= a \neq u$ and $t = b \neq v$. The left hand side is equal to $h(2) + \Delta h(2) \cdot (-2) + \Delta^2 h(2) \cdot 0= - h(2) + 2 h(1) = h(0)$.
#### Proof of (1).
Take any vertex $x$ of $T$ and its two distinct neighbors $y,y'$. It suffices to show $h(d(y,z)) + h(d(y',z)) \geq 2 h(d(x,z))$. Observe that $\{ d(y,z), d(y',z)\} = \{ d(x,z) + 1, d(x,z) -1\}$ or $d(y,z) = d(y',z) = d(x,z) +1$ holds. For the first case, $h(d(y,z)) + h(d(y',z)) = h(d(x,z) +1) + h(d(x,z)-1)
\geq 2h(d(x,z))$ by the convexity of $h$. For the second case, $h(d(y,z)) + h(d(y',z)) \geq h(d(x,z)) + h(d(x,z)) = 2h(d(x,z))$ by the monotonicity of $h$.
#### Proof of (2).
Extend $h: \ZZ \to \RR$ to $\ZZ/2 \to \RR$ by $h(z) := h(z)$ if $z \in \ZZ$ and $h(z) := (h(z - 1/2) + h(z+ 1/2))/2$ otherwise. Take $(u,v), (u',v') \in T^2$. We are going to show the discrete midpoint convexity for $h_{T}$: $$h(d(u,v)) + h (d(u',v')) \geq h(d(u \circ u', v \circ v'))
+ h (d(u \bullet u', v \bullet v')).$$
Since $h$ is convex, we have $$h(d(u,v)) + h (d(u',v')) \geq 2 h \left(\frac{d(u,v) + d(u',v')}{2} \right).$$ By Lemma \[lem:DFL\], we have $d(u,v) + d(u',v') \geq 2 d(u \circ_{1/2} u', v \circ_{1/2} v')$. Since $h$ is nondecreasing, we have $$2 h \left(\frac{d(u,v) + d(u',v')}{2} \right) \geq 2 h (d(u \circ_{1/2} u', v \circ_{1/2} v')).$$
$
2 h (d(u \circ_{1/2} u', v \circ_{1/2} v')) = h(d(u \circ u', v \circ v'))
+ h(d(u \bullet u', v \bullet v')). $
Let $\bar u := u \circ_{1/2} u'$ and $\bar v := v \circ_{1/2} v'$. The claim is obvious when $\bar u = \bar v$ or both $\bar u$ and $\bar v$ belong to $T$. So we consider the other cases.
Case 1: $\bar u\in T$ and $\bar v \not \in T$. Then $d(\bar u, \bar v)$ is a half-integer, and hence we have $$2 h(d(\bar u, \bar v)) = h(d(\bar u, \bar v) - 1/2)
+ h(d(\bar u, \bar v) + 1/2).$$ Since $\bar u \in T$, we have $\bar u = u \circ u' = u \bullet u'$, and $$\{ d(\bar u, v \circ v'), d(\bar u, v \bullet v')\} = \{ d(\bar u, \bar v) - 1/2, d(\bar u, \bar v) +1/2\}.$$ Hence we have the claim.
Case 2: $\bar u \not \in T$ and $\bar v \not \in T$. We can take edges $ss'$ and $tt'$ of $T$ such that $\bar u$ and $\bar v$ are the midpoints of $ss'$ and $tt'$, respectively. In particular, $d(\bar u,s) = d(t, \bar v) =d(\bar u, s') = d(t', \bar v) = 1/2$. We can assume that $$d(\bar u, \bar v) = d(\bar u, s) + d(s,t) + d(t, \bar v).$$ Then $d(\bar u, \bar v) = d(s',t) = d(s,t')$, and $d(s',t') = d(s,t) + 2$. Suppose that $d(s,t)$ is odd. Then $s$ and $t$ belong to different color classes; so $(s', t) = (u \circ u', v \circ v')$ and $(s,t') = (u \bullet u', v \bullet v')$ or $(s, t') = (u \circ u', v \circ v')$ and $(s',t) = (u \bullet u', v \bullet v')$. Thus $d(\bar u, \bar v) = d(u \circ u', v \circ v') = d(u \bullet u', v \bullet v')$, and the claim is true. Suppose that $d(s,t)$ is even. Then $(s, t) = (u \circ u', v \circ v')$ and $(s', t') = (u \bullet u', v \bullet v')$ or $(s', t') = (u \circ u', v \circ v')$ and $(s, t) = (u \bullet u', v \bullet v')$. Thus we have $$\{ d(u \circ u', v \circ v'), d(u \bullet u', v \bullet v') \} = \{d(s,t), d(s,t) + 2\}.$$ By $d(\bar u, \bar v) = d(s,t) + 1$ that is odd, we have $$2 h (d(\bar u, \bar v)) = h(d(s,t)) + h(d(s,t) + 2).$$ Thus we have the claim.
Therefore $h_T$ is L-convex. We verify the latter part of (2). For the case for $u = v \in W$, we obtain the formula from Lemma \[lem:123\] (1). For other cases, we have $$h(d(s,t)) = h(d(s,a) + d(a,b) + d(b, t)).$$ If $u \in W,v \in B$, then $t = b =v$, and $d(a,b) = D - 1$ is even. By Lemma \[lem:123\] (2), we obtain $h_T(s,t) = h(d(s,a) + D -1) = h(D) + \Delta h(D) \theta_a(s)$. The argument for the case $u \in B,v \in W$ is the same. Suppose that $u,v \in W$ with $u \neq v$. Then $d(s,a) + d(t,b) + D - 2$, and $d(a,b) = D - 2$ is even. By Lemma \[lem:123\] (3) applied to $h_T(s,t) = h(d(s,a) + d(t,b) + D -2)$, we obtain the required formula.
#### Proof of (3).
We show the discrete midpoint convexity for $h_{T;z,w}$. Take vertices $u,v,u',v' \in T$. Then we have $$\begin{aligned}
&& h(d(u,z) + d(v,w)) + h(d(u',z) + d(v',w)) \\
&& \geq 2 h \left( \frac{d(u,z) + d(v,w) + d(u',z) + d(v',w)}{2} \right) \\
&& \geq 2 h ( d(u \circ_{1/2} u',z) + d(v \circ_{1/2} v',w)),\end{aligned}$$ where we use convexity of $h$ in the first inequality, and use the monotonicity and Lemma \[lem:DFL\] in the second.
$2 h ( d(u \circ_{1/2} u',z) + d(v \circ_{1/2} v',w))
= h_{T;z,w} (u \bullet u', v \bullet v')
+ h_{T;z,w} (u \circ u', v \circ v')$.
Let $\bar u := u \circ_{\mbox{\tiny$1/2$}} u'$ and $\bar v := v \circ_{\mbox{\tiny$1/2$}} v'$.
Case 1: $\bar u \in T$ and $\bar v \in T$. In this case, $\bar u = u \bullet u' = u \circ u'$ and $\bar v = v \bullet v' = v \circ v'$ hold, and hence the claim holds.
Case 2: $\bar u \not \in T$ and $\bar v \in T$. In this case, there is an edge $st$ of $T$ such that $\bar u$ is the midpoint of $st$. We can assume that $d(z,s) = d(z,t) + 1$ and $d(z, \bar u) = d(z,t) + 1/2$. Since $d(\bar u,z) + d(\bar v, w)$ is not an integer, we have $$\begin{aligned}
2 h( d(\bar u,z) + d(\bar v, w)) & = & h(d( \bar u ,z) + d(\bar v, w) +1/2) +
h(d(\bar u,z) + d(\bar v,w) -1/2) \\
&= & h(d(s,z) + d(\bar v,w)) + h(d(t, z) + d(\bar v,w)). \end{aligned}$$ Then the claim follows from $\bar v = v \bullet v' = v \circ v'$, and $\{ u \bullet u', u \circ u'\} = \{s, t\}$.
Case 3: $\bar u \not \in T$ and $\bar v \not \in T$. Let $ab$ denote the edge such that $\bar u$ is the midpoint of $ab$, and let $st$ denote the edge such that $\bar v$ is the midpoint of $st$. We can assume that $(a,b) = (u \bullet u', u \circ u')$ and $(s,t) = (v \bullet v', v \circ v')$. Case 3.1: $d(a,z) - d(b,z) = d(s,w) - d(t, w) \in \{-1,1\}$. Since $z,w$ have the same color and $a,s$ have the same color, $d(\bar u,z) + d(\bar v, w) = d(a,z) + d(s,w) \pm 1$ must be odd. Hence we have $$\begin{aligned}
2 h( d(\bar u,z) + d(\bar v,w))
& = & h(d(\bar u,z)+ d(\bar v, w) - 1) + h(d(\bar u,z)+ d(\bar v,w) + 1) \\
& = & h(d(a,z)+ d(s, w)) + h(d(b,z)+ d(t,w))\end{aligned}$$ as required.
Case 3.2: $d(a,z) - d(b,z) \neq d(s,w) - d(t,w)$. In this case, $d(\bar u,z) + d(\bar v,w) = d(a,z) + d(s,w) = d(b, z)+ d(t, w)$, and hence we have the claim.
We show the latter part of (3). It holds that $d(s,z) + d(t,w) = d(s,a) + d(t,b) + d(a,z) + d(b,w)$. Suppose that $u \in W, v \in B$. Then ${\cal F}(v) = \{v\}$, $b = v = t$, and $D$ is odd. Suppose that $z \neq u$. Then $a \neq u$, and $d(s,z) + d(t,w) = d(s,a) + D - 1$. Thus by Lemma \[lem:123\] (2), we have $h_{T;z,w} (s,t) = h(d(s,a) + D - 1) =
h(D) + \Delta h(D) \theta_a(s)$. Suppose that $z = u$. Then $a = u$, and $d(s,z) + d(t,w) = d(s,a) + 1 + (D - 1)$. By Lemma \[lem:123\] (2), we have $h_{T;z,w} (s,t) = h(d(s,a) + 1+ D - 1) =
h(D) + \Delta h(D) \theta_a(s)$. The argument for $u \in B, v \in W$ is similar.
Suppose that $u,v \in W$. If $u \neq z$ and $v \neq w$, then $a \neq u$ and $b \neq v$. Also $d(s,z) + d(t,w) = d(s,a) + d(t,b) + D - 2$, and $D$ is even. Apply Lemma \[lem:123\] (3) to $h_{T;z,w} (s,t) = h(d(s,a) + d(t,b) + D - 2)$, we obtain the formula.
If $u = z$ and $v \neq w$ or $u \neq z$ and $v = w$, then $u = a$ and $v \neq b$ or $u \neq a$ and $v = b$, and we have $d(s,z) + d(t,w) = d(s,a) + d(t,b) + D - 1$ with $D$ even. By Lemma \[lem:123\] (2) to $h_{T;z,w} (s,t) = h(d(s,a) + d(t,b) + 1 + (D - 2))$, we obtain the formula. If $u = z$ and $v = w$, then $u = a$, $v = b$, and $d(s,z) + d(t,w) = d(s,a) + d(t,b)$ with $D = 0$. By Lemma \[lem:123\] (2) to $h_{T;z,w} (s,t) = h(d(s,a) + d(t,b))$, we obtain the formula.
### Proof of Theorem \[thm:approx\]
(1). By Theorem \[thm:2separable\], all functions $h_T, h_{T,z}, h_{T,z,w}$ are locally basic $k$-submodular. Hence, by Theorem \[thm:IWY14\], the minimization of $\bar \omega$ over ${\cal I}(x)$ and ${\cal F}(x)$ can be done in $O({\rm MF}(kn, km))$ time. By Theorem \[thm:bound\], the steepest descent algorithm for $\omega$ with initial point $x \in B^n$ iterates $d({\rm opt} (\bar \omega), x)$ steps to obtain an optimal solution $x^*$ of $\bar \omega$. Hence the total time is $O( d({\rm opt} (\bar \omega), x){\rm MF}(k n, k m))$.
(2). Let $u:= (x^*)_{\rightarrow y}$. It suffices to show $$f_{ij}(d(u_i, z_j)) \leq 2 \bar f_{ij}(x^*_i, z_j), \quad g_{ij}(d(u_i, u_j)) \leq 2 \bar g_{ij}(x^*_i, x^*_j)$$ for each $i,j$.
We may suppose that $x^*_i \neq u_i$. Namely $x^*_i$ is a white vertex, and $d(x^*_i, z_j)$ is odd. Thus $\bar f_{ij} (d(x^*_i, z_j)) = \{f_{ij} (d(x^*_i, z_j) - 1) + f_{ij} (d(x^*_i, z_j) +1)\}/2$. If $d(u_i, z_j) = d(x^*_i, z_j) - 1$, then by monotonicity and nonnegativity we have $\bar f_{ij}(d(u_i, z_j)) \leq \bar f_{ij} (d(x^*_i, z_j)) \leq 2 \bar f_{ij} (d(x^*_i, z_j))$. If $d(u_i, z_j) = d(x^*_i, z_j) + 1$, then by nonnegativity we have $$f_{ij}(d(u_i, z_j)) = 2 \bar f_{ij} (d(x^*_i, z_j)) - f_{ij} (d(x^*_i, z_j) - 1) \leq 2 \bar f_{ij} (d(x^*_i, z_j)).$$
Next consider $\bar g_{ij}(d(x^{*}_i, x^{*}_j))$. Suppose that $x^*_j$ is black. Then $x^*_j = u_j$ and $d(x^{*}_i, x^{*}_j)$ is odd. Thus $\bar g_{ij}(d(x^{*}_i, x^{*}_j)) = \{ g_{ij}(d(x^{*}_i, x^{*}_j) - 1) + g_{ij}(d(x^{*}_i, x^{*}_j) + 1)\}/2$. By the same argument above, we have $g_{ij}(d(u_i,u_j)) \leq 2 \bar g_{ij}(d(x^{*}_i, x^{*}_j))$. Suppose that $x^*_j$ is also white; $d(x^{*}_i, x^{*}_j)$ is even. If the unique path between $x^*_i$ and $y$ contains $x^*_j$, then $d(x^*_i,x^*_j) = d(u_i, u_j)$, and $g_{ij}(d(u_i,u_j))
= \bar g_{ij}(d(x^{*}_i, x^{*}_j)) \leq 2 \bar g_{ij}(d(x^{*}_i, x^{*}_j))$. The same holds for the case where the unique path between $x^*_j$ and $y$ contains $x^*_i$. Suppose not. There is a vertex $m$ in the path between $x^*_i$ and $x^*_j$ such that $m$ belongs to the path between $y$ and $x^*_i$ and the path between $y$ and $x^*_j$. Therefore the rounded $u_i$ and $u_j$ are closer to $m$. Hence $d(u_i,u_j) \leq d(x^{*}_i, x^{*}_j)$, and by monotonicity we have $g_{ij}(d(u_i,u_j)) \leq \bar g_{ij}(d(x^{*}_i, x^{*}_j)) \leq 2 \bar g_{ij}(d(x^{*}_i, x^{*}_j))$.
Minimum cost multiflow {#sec:multiflow}
======================
In this section, as an application of the results in the previous section, we provide a new simple combinatorial (weakly) polynomial time algorithm to solve minimum cost maximum free multiflow problem. The design of such an algorithm was the original motivation of developing the theory of L-extendable functions.
Let ${\cal N}$ be an undirected network on node set $V$, edge set $E$, terminal set $S \subseteq V$, and edge-capacity $c: E \to \ZZ_+$. An [*$S$-path*]{} in ${\cal N}$ is a path connecting distinct terminals in $S$. A [*multiflow*]{} $f$ is a pair $({\cal P}, \lambda)$ of a (multi-)set ${\cal P}$ of $S$-paths and a nonnegative-valued function $\lambda: {\cal P} \to \RR_+$ satisfying the capacity constraint: $$f(e) := \sum \{ \lambda (P) \mid \mbox{$P \in {\cal P}$: $P$ contains $e$ } \} \leq c(e) \quad (e \in E).$$ If $2 \lambda$ is integer-valued, then $f$ is called [*half-integral*]{}. For distinct terminals $s,t \in S$, let $f(s,t)$ denote the total value of the $(s,t)$-flow in $f$, i.e., $$f(s,t) := \sum \{ \lambda (P) \mid \mbox{$P \in {\cal P}$: $P$ is an $(s,t)$-path }\}.$$ Let $f(s)$ denote the total value on flows connecting $s$, i.e., $f(s) := \sum_{t \in S \setminus \{s\}} f(s,t)$. Then the total flow-value $v_f$ is defined by $$v_f := \sum_{P \in {\cal P}} \lambda(P) = \sum_{s,t \in S: s \neq t} f(s,t) = \frac{1}{2} \sum_{s \in S} f(s).$$ The [*maximum free multiflow problem*]{} (MF) is:
(MF)
: Find a multiflow $f$ having the maximum total flow value $v_f$.
A [*maximum free multiflow*]{} is a multiflow having the maximum total flow-value.
Observe that $f(s)$ is the total flow value of the $(s, S \setminus \{s\})$-flow in $f$, and hence is at most the minimum value $\kappa_s$ of an $(s, S \setminus \{s\})$-cut. Thus the total flow-value of any multiflow is at most $\sum_{s \in S} \kappa_s/2$. A classical theorem by Lovász [@Lov76] and Cherkassky [@Cher77] says that this bound is always attained by a half-integral multiflow.
\[thm:LC\] The maximum flow-value of a free multiflow is equal to $$\frac{1}{2} \sum_{s \in S} \kappa_s,$$ and there exists a half-integral maximum free multiflow.
Karzanov [@Kar79] considered a minimum cost version of (MF). Now the network ${\cal N}$ has a nonnegative edge-cost $a: E \to \ZZ_+$. The total [*cost*]{} $a_f$ of multiflow $f$ is defined by $a_f := \sum_{e \in E} a(e) f(e).$ The [*minimum cost maximum free multiflow problem*]{} is:
(MCMF)
: Find a maximum free multiflow having the minimum total cost.
There are two approaches to solve this problem. The first one is based on the following auxiliary maximum multiflow problem with a positive parameter $\mu > 0$:
(M)
: Find a multiflow $f$ maximizing $\mu v_f - a_f$.
If $\mu$ is sufficiently large, then an optimal multiflow in (M) is a minimum cost maximum free multiflow. Karzanov [@Kar79] showed the half-integrality of (M).
\[thm:halfintegral\] For any $\mu \geq 0$, there exists a half-integral optimal multiflow in $(M)$.
In particular there always exists a half-integral minimum cost free multiflow. The previous algorithms [@GK97; @Kar79; @Kar94] are based on this formulation.
The second approach, which we will mainly deal with, is based on a node-demand multiflow formulation. We are further given a nonnegative [*demand*]{} $r: S \to \ZZ_+$ on terminal set $S$. A multiflow $f$ is said to be [*feasible*]{} (to $r$) if $f(s) \geq r(s)$ for $s \in S$. The [*minimum cost feasible free multiflow problem*]{} (N) is:
(N)
: Find a feasible multiflow having the minimum total cost.
This problem (N) can solve (MCMF). Indeed, for each $s \in S$, add new non-terminal node $\bar s$ and new edge $s\bar s$ with capacity $\kappa_s$ and zero cost, and replace each edge $is$ by $i \bar s$. Let $r(s) := \kappa_s$ for $s \in S$. Any feasible multiflow $f$ for the new network must satisfy $f(s) = r(s) = \kappa_s$. After contracting edges $s\bar s$, the resulting $f$ is necessarily a maximum free multiflow in the original network. Also all maximum free multiflows are obtained in this way. Indeed, by Theorem \[thm:LC\], a maximum free multiflow is simultaneously a maximum single commodity $(s, S \setminus \{s\})$-flow for $s \in S$. Hence $f(s) = \kappa_s$ must hold, and $f$ is extended to a feasible multiflow in the new network. The problem (N) itself seems natural and fundamental, but has not been well-studied so far. Also we do not know whether (N) reduces to (M), and whether (M) reduces to (N). Recently Fukunaga [@Fukunaga14] addressed the problem (N) in connection with a class of network design problems, called (generalized) [*terminal backup problems*]{} [@AK11; @BKM13; @XAC08INFOCOM]. As was noted by him, the problem (N) is also formulated as the following cut-covering linear program. $$\begin{aligned}
\mbox{\bf (L)} \quad {\rm Min.} && \sum_{e \in E} a(e) x(e) \nonumber \\
{\rm s. t. } && x(\delta X) \geq r(s) \quad (s \in S, X \in {\cal C}_s), \nonumber \\
&& 0 \leq x(e) \leq c(e) \quad (e \in E),\end{aligned}$$ where ${\cal C}_s$ denotes the set of node subsets $X \subseteq V$ such that $X$ contains $s$ and does not contain other terminals. The problems (N) and (L) are equivalent in the following sense.
\[[see [@Fukunaga14]]{}\] \[lem:NvsL\]
- For an optimal solution $f$ of (N), the flow-support $x: E \to \RR_+$ defined by $x(e) := f(e)$ $(e \in E)$ is an optimal solution of (L).
- For an optimal solution $x: E \to \RR_+$ of (L), a feasible multiflow $f$ in ${\cal N}$ with the capacity $x$ exists, and is optimal to (N).
In particular the optimal values of the two problems are the same.
A feasible multiflow $f$ contains an $(s, S \setminus \{s\})$-flow with the total flow-value at least $r(s)$. The capacity of any $(s, S \setminus \{s\})$-cut under capacity $x$ is at least $r(s)$ for $s \in S$. This means that $x$ is feasible to (L). Hence the optimal value of (L) is at least that of (N). Conversely, for a feasible solution $x$ of (L), consider a maximum free multiflow $f$ under capacity $x$. By Theorem \[thm:LC\], $f(s)$ is equal to the minimum capacity of an $(s, S \setminus \{s\})$-cut under capacity $x$, which is at least $r(s)$. Thus $f$ is feasible, and $a_f \leq \sum_{e \in E} a(e) x(e)$. This means that the optimal value of (N) is at least that of (L).
Notice that a feasible multiflow exists (or (L) is feasible) if and only if $$\label{eqn:feasible}
c(\delta X) \geq r(s) \quad (s \in S, X \in {\cal C}_s).$$ We will assume this condition in the sequel. Fukunaga [@Fukunaga14] proved the half-integrality for (N) and (L).
\[thm:fuku\] There exist half-integral optimal solutions in (N) and in (L). They can be obtained in strongly polynomial time.
The polynomial time solvability depends on a generic LP-solver for solving (L); observe that the separation of the feasible region of (L) is done by minimum cut computations, and thus (L) is solved by the ellipsoid method. Also (L) has an extended formulation of polynomial size [^1], and thus is solved by the interior point method.
As an application of results in the previous section, we present a purely combinatorial polynomial time scaling algorithm to obtain half-integral optimal solutions in (N), in (L), and in (MCMF). The main result in this section is as follows:
\[thm:multiflow\_main\] There exists an $O(n \log (n AC)\, {\rm MF}(kn, km))$ time algorithm to solve (N), (L), and (MCMF), where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $A$ is the maximum of edge-costs, $C$ is the total sum of edge-capacities.
To the best of our knowledge, our algorithm is the first combinatorial polynomial time algorithm for (N) and (L), and the first combinatorial algorithm for (MCMF) with an explicit polynomial running time.
In Section \[subsec:multiflow\_duality\], we formulate a dual of (N) as a convex location problem on a (topological) tree ${\cal T}$ that is the union of the coordinate axises in $\RR^S_+$, and establish the half-integrality (Proposition \[prop:min-max\]). Then we give an optimality criterion (Lemma \[lem:optimality\]) for (N). In Section \[subsec:double\_covering\], we explain an algorithm to construct an optimal multiflow in (N) from a given optimal dual solution. This algorithm is a slight modification of the algorithm of [@Kar94] devised for (M). In Section \[subsec:proximity\_scaling\], we present an algorithm to solving a dual of (N), providing the proof of Theorem \[thm:multiflow\_main\]. By the half-integrality, the dual of (N) is the minimization of a 2-separable L-convex function on a tree obtained by joining half-integral points in ${\cal T}$. We will design a proximity scaling algorithm by considering a $2$-separable L-convex function minimization over the tree of $2^{\sigma}$-integral points in each scaling phase $\sigma$. The time complexity will be estimated by the results of the previous section. We also sketch how to adapt our algorithm to solve (M). In Section \[subsec:additional\], we give additional results and remarks. In particular, we explain that our combinatorial algorithm gives a practical implementation of Fukunaga’s $4/3$-approximation algorithm [@Fukunaga14] for capacitated terminal backup problem. We also give a further simple and instructive but pseudo-polynomial time algorithm to solve (N).
Duality {#subsec:multiflow_duality}
-------
We first formulate a dual of (N) as a continuous location problem on a tree (in topological sense). Let $\RR^S$ denote the set of functions on $S$. For each terminal $s \in S$, let $e_s$ denote the function defined by $e_s(s) := 1$ and $e_{s}(t) := 0$ for $t \neq s$. Namely $e_s$ is the $s$th unit vector of $\RR^S$. Let ${\cal T}_s := \RR_+ e_s$, and let ${\cal T} := \bigcup_{s \in S} {\cal T}_s \subseteq \RR^S$. The metric $D$ on ${\cal T}$ is defined by $$D(p,q) := \left\{
\begin{array}{ll}
|p(s) - q(s)| & {\rm if}\ \mbox{$p,q \in {\cal T}_s$ for $s \in S$}, \\
|p(s)| + |q(t)| & {\rm if}\ \mbox{$p \in {\cal T}_s, q \in {\cal T}_t$ for distinct $s,t \in S$},
\end{array} \right. \quad (p,q \in {\cal T}).$$ The space ${\cal T}$ is isometric to a [*star*]{} obtained by gluing half-lines $\RR_+$ along the origin. Notice that $D$ is not equal to an induced metric on $\RR^S$.
Let $V = \{1,2,\ldots,n\}$. Consider the following continuous location problem on ${\cal T}$: $$\begin{aligned}
\mbox{{\bf (D)}: Max.} && \sum_{s \in S} r(s) D(0, p_s) - \sum_{ij \in E} c(ij) (D(p_i, p_j) - a(ij))^+ \\
\mbox{s.t.} && p = (p_1,p_2,\ldots, p_n) \in
{\cal T} \times {\cal T} \times \cdots \times {\cal T}, \\
&& p_s \in {\cal T}_s \quad (s \in S), \end{aligned}$$ where $(z)^+$ denotes $\max (0, z)$. A feasible solution $p$ of (D) is called a [*potential*]{}, and called [*half-integral*]{} if each $p_i$ is a half-integral vector in $\RR^S$. A potential $p$ is called [*proper*]{} if $D(0, p_i) \leq D(0,p_s)$ for each $s \in S$ and $i \in V$ with $p_i \in {\cal T}_s$.
\[prop:min-max\] The minimum value of (N) is equal to the maximum value of (D). Moreover there exists a proper half-integral optimal potential in (D).
We will give an algorithmic proof in Section \[subsub:simple\], and here give a sketch of the proof.
For a (half-integral) potential $p$ and a terminal $s \in S$, let $p'$ be a (half-integral) potential defined by $p'_i := p_s$ if $p_i \in {\cal T}_s$ and $D(0, p_i) > D(0, p_s)$, and $p'_i := p_i$ otherwise. Then the objective value of (D) does not decrease. Therefore there always exists a proper optimal potential in (D).
Let ${\cal C} := \bigcup_{s \in S} {\cal C}_s$. The LP-dual of (L) is equivalent to: $$\begin{aligned}
\label{eqn:LP-dual'}
\mbox{Max.} && \sum_{s \in S} r(s)\sum_{X \in {\cal C}_s} \pi (X)
- \sum_{e \in E} c(e) \left(\sum_{X \in {\cal C}: e \in \delta X} \pi (X) - a(e) \right)^+\\
\mbox{s.t.} && \pi: {\cal C}\to \RR_+. \nonumber\end{aligned}$$ By the standard uncrossing argument, one can show that there always exists an optimal solution $\pi$ such that for $X,Y \in {\rm supp}\, \pi := \{ X \in {\cal C} \mid \pi(X) > 0 \}$, it holds $X \subseteq Y$ or $X \supseteq Y$ if $X,Y \in {\cal C}_{s}$ for $s \in S$, and $X \cap Y = \emptyset$ if $X \in {\cal C}_s$ and $Y \in {\cal C}_{s'}$ for distinct $s, s' \in S$. Such a solution is called [*laminar*]{}.
Thus it suffices to show that for a proper potential $p$ there is $\pi: {\cal C} \to \RR_+$ satisfying $$\begin{aligned}
\label{eqn:suffice}
\sum_{X \in {\cal C}: e \in \delta X} \pi(X) & = & D(p_i, p_j) \quad (ij \in E), \\
\sum_{X \in {\cal C}_s} \pi(X) & = & D(0, p_s) \quad (s \in S), \nonumber\end{aligned}$$ and that for a laminar solution $\pi: {\cal C} \to \RR_+$, there is a proper potential $p$ satisfying (\[eqn:suffice\]).
Let $p = (p_1,p_2,\ldots,p_n)$ be a proper potential. For $s \in S$ with $p_s \neq 0$, suppose that $\{p_1,p_2,\ldots,p_n\} \cap ({\cal T}_s \setminus \{0\}) = \{q_1,q_2,\ldots, q_{k_s} = p_s\}$ with $0 < D(0, q_1) < D(0, q_{2}) < \cdots < D(0, q_{k_s})$. For $j=1,2,\ldots,k_s$, define $X^s_j$ and $\pi^s_j$ by $$X^s_j := \{ i \in V \mid p_i \in \{q_j, q_{j+1},\ldots,q_{k_s}\}\}, \
\pi^s_j := D(q_{j-1}, q_{j}),$$ where we let $q_0 := 0$. Then $X^s_j \in {\cal C}_s$. Define $\pi: {\cal C} \to \RR_+$ by $\pi(X^s_j) := \pi^s_j$ and $\pi(X) := 0$ for other $X$. Then (\[eqn:suffice\]) holds.
Conversely, let $\pi$ be a laminar solution of (\[eqn:LP-dual’\]). Then we can assume that ${\rm supp}\, \pi \cap {\cal C}_s = \{ X^s_1,X^s_2,\ldots X^s_{k_s}\}$ with $X^s_1 \supset X^s_2 \supset \cdots \supset X^s_{k_s} \ni s$. For each node $i$, if $i$ does not belong to any member of ${\rm supp}\, \pi$, then define $p_i := 0$. Otherwise there uniquely exist $s \in S$ and $j \in \{1,2,\ldots,k_s\}$ such that $i \in X_j^s$ and $i \not \in X_{j+1}^s$, where $X_{k_s+1}^s := \emptyset$. Define $p_i := (\sum_{l= 1}^j \pi(X_l^s)) e_s \in {\cal T}_s$. Then we obtain a proper potential $p = (p_1,p_2,\ldots,p_n)$ of (D) satisfying (\[eqn:suffice\]).
By Theorem \[thm:fuku\], for every cost vector $a$ (not necessarily nonnegative) there exists a half-integral optimal solution in (L). By the total dual (half-)integrality, there exists a half-integral laminar optimal solution in (\[eqn:LP-dual’\]), and there exists a half-integral optimal potential in (D).
We next provide an optimality criterion for (N) and (D). For a potential $p$, an $(s,t)$-path $P = (s = i_0,i_1,\ldots,i_l = t)$ is said to be [*$p$-geodesic*]{} if $$\sum_{k = 0}^{l-1} D(p_{i_k}, p_{i_{k+1}}) = D(p_s, p_t).$$ Observe from the triangle inequality that $(\geq)$ always holds.
\[lem:optimality\] A feasible flow $f = ({\cal P}, \lambda)$ and a potential $p$ are both optimal if and only if they satisfy the following conditions:
- For each edge $ij$, if $D(p_i, p_j) > a(ij)$, then $f(ij) = c(ij)$.
- For each edge $ij$, if $D(p_i, p_j) < a(ij)$, then $f(ij) = 0$.
- For each path $P$ in ${\cal P}$, if $\lambda(P) > 0$, then $P$ is $p$-geodesic.
- For each terminal $s$, if $D(0, p_s) > 0$, then $f(s) = r(s)$.
For a path $P = (i_0,i_1,\ldots,i_l)$, let $D(p(P)) := \sum_{k = 0}^{l-1} D(p_{i_k}, p_{i_{k+1}})$. The statement follows from the previous proposition, and $$\begin{aligned}
&& \sum_{ij \in E} a(ij)f(ij) - \sum_{s \in S} r(s) D(0, p_s) + \sum_{ij \in E} c(ij) (D(p_i, p_j) - a(ij))^+ \\
& & = \sum_{ij \in E} a(ij)f(ij) + \sum_{ij \in E} c(ij) (D(p_i, p_j) - a(ij))^+ - \sum_{ij \in E} f(ij) D(p_i, p_j) \\
&& \quad + \sum_{ij \in E} f(ij) D(p_i, p_j) - \sum_{st} f(s,t) D(p_{s}, p_{t}) + \sum_{st} f(s,t) D(p_{s}, p_{t}) - \sum_{s \in S} r(s) D(0, p_s) \\
&& = \sum_{ij \in E} (D(p_i, p_j) - a(ij))^+ \{ c(ij) - f(ij) \} +
\sum_{ij \in E} (a(ij) - D(p_i, p_j))^+ f(ij) \\
&& \quad + \sum_{st}\sum_{P \in {\cal P}: \scriptsize\mbox{$P$ connects $s,t$}} \lambda(P) \{ D(p(P)) - D(p_{s}, p_{t}) \} + \sum_{s \in S} (f(s) - r(s)) D(0, p_s),\end{aligned}$$ where $st$ is taken over all unordered pairs of distinct terminals, and we use $$\begin{aligned}
&& \sum_{ij \in E} f(ij) D(p_i, p_j) = \sum_{ij \in E} \sum_{P \in {\cal P}:e \in P} \lambda(P) D(p_i, p_j) = \sum_{P \in {\cal P}} \lambda(P) D(p(P)), \\
&&
\sum_{st} f(s,t)D(p_{s}, p_{t}) =
\sum_{st} f(s,t) \left\{ D(p_{s},0) + D(0, p_{t}) \right\} = \sum_{s \in S} f(s) D(0, p_s).\end{aligned}$$
Double covering network {#subsec:double_covering}
-----------------------
Here we describe an algorithm to construct an optimal multiflow in (N) from an optimal potential $p$ in (D) under the condition that each edge-cost is positive:
(CP)
: $a(e) > 0$ for each edge $e \in E$.
We will see in Remark \[rem:howto\] that we can assume (CP) by a perturbation technique. As Karzanov [@Kar94] did for (M), a half-integral optimal multiflow $f$ in (N) is also obtained by an integral circulation of a certain directed network ([*double covering network*]{}) ${\cal D}_p$ associated with an optimal potential $p$.
![Double covering network[]{data-label="fig:double_covering"}](double_covering.pdf)
Let $p$ be a (proper) potential. Let $U_0$ denote the set of non-terminal nodes $i$ with $p_i = 0$. For each terminal $s \in S$, let $U_s$ denote the set of nodes consisting of terminal $s$ and non-terminal nodes $i$ with $p_i \in {\cal T}_s \setminus \{0\}$. Then $V$ is the disjoint union of $U_0$ and $U_s$ for $s \in S$. Let $E_{=}$ denote the set of edges $ij$ with $D(p_i,p_j) = a(ij)$, and let $E_{>}$ denote the set of edges $ij$ with $D(p_i,p_j) > a(ij)$. The [*double covering network*]{} ${\cal D}_p$ relative to $p$ is a directed network constructed as follows. For each terminal $s$, consider two nodes $s^+, s^-$. For each non-terminal node $i$ not in $U_0$, consider two nodes $i^+, i^-$. For each (non-terminal) node $i$ in $U_0$, consider $2|S|$ nodes $i^{s+}, i^{s-}$ $(s \in S)$. The node set of ${\cal D}_p$ consists of these nodes. Next we define the edge set $A$ of ${\cal D}_p$. For each edge $ij \in E_{=} \cup E_{>}$, define the edge set $A_{ij}$ by: $$A_{ij} := \left\{ \begin{array}{ll}
\{ j^+i^+, i^-j^- \} & {\rm if}\ i,j \in U_s, D(0,p_i) < D(0,p_j), \\
\{j^+i^{s+}, i^{s-}j^-\} & {\rm if}\ i \in U_0, j \in U_s, \\
\{ i^+j^-, j^+i^- \} & {\rm if}\ i \in U_s, j \in U_t, s \neq t.
\end{array}
\right.$$ Notice that for $ij \in E_{=} \cup E_{>}$, potentials $p_i$ and $p_j$ are different points in ${\cal T}$ since $a(ij)$ is positive. The upper capacity of the two edges in $A_{ij}$ is defined as $c(ij)$. The lower capacity is defined as $0$ if $ij \in E_{=}$ and $c(ij)$ if $ij \in E_{>}$. For each (non-terminal) node $i$ in $U_0$, the edge set $B_i$ is defined as $\{ i^{s+} i^{t-} \mid s,t \in S, s \neq t\}$. The lower capacity and the upper capacity of these edges are defined as $0$ and $\infty$, respectively. For terminal $s \in S$, add edge $s^- s^+$. The lower capacity is defined as $r(s)$, and the upper capacity is defined as $\infty$ if $p_s = 0$ and $r(s)$ otherwise. The edge set of ${\cal D}_{p}$ is the (disjoint) union of all edge sets $A_{ij}$ $(ij \in E_{=} \cup E_{>})$, $B_i$ $(i \in U_0)$, $\{ s^-s^+\}$ $(s \in S)$ (as a multiset). As in Figure \[fig:double\_covering\], readers may imagine that ${\cal D}_p$ is embedded into ${\cal T}$ by the map $i^{\pm} \mapsto p_i$,
Consider an integral feasible circulation $\phi: A \to \ZZ_+$ of this network (if it exists). Decompose $\phi$ into the sum of characteristic vectors of directed cycles $C_1,C_2,\ldots, C_{m'}$ with positive integral coefficients $q_1,q_2,\ldots,q_{m'}$, where $m'$ is at most the number of edges of ${\cal D}_p$. By construction of ${\cal D}_p$, any directed cycle must meet $s^-s^+$ for some terminal $s$, and next meets $t^-t^+$ for other terminal $t$ after meeting $s^-s^+$. Delete all terminal edges $s^-s^+$ from each $C_l$, and obtain directed paths $P_{l}^{1},P_{l}^{2},\ldots, P_l^{n_l}$ $(n_l \leq |S|)$. Then each $P_l^j$ is a path from $s^+$ to $t^-$ for distinct $s,t \in S$. Let $\bar P_l^{r}$ be the $S$-path in the original network ${\cal N}$ obtained from $P_l^{r}$ by replacing $i^{\pm}$ or $i^{s \pm}$ by $i$ (and removing $i^{s+} i^{t -}$). Let ${\cal P}$ be the union of $S$-paths $\bar P_l^{r}$ over $l =1,2,\ldots,m', r = 1,2,\ldots, n_l$. Let $\lambda(\bar P_l^{r}) := q^r_l/2$. Then $f_{\phi} := ({\cal P},\lambda)$ is a half-integral multiflow; we see in the proof of the next lemma that $f_{\phi}$ indeed satisfies the capacity constraint.
\[prop:f\_phi\] A potential $p$ is optimal if and only if there exists a feasible circulation in ${\cal D}_p$. Moreover, for any (integral) feasible circulation $\phi$, the (half-integral) multiflow $f_{\phi}$ is optimal to (P).
(Only if part). Let $f = ({\cal P},\lambda)$ be an optimal multiflow. Then $f$ satisfies the conditions of Lemma \[lem:optimality\]. Consider an path $P = (s = i_0,i_1,\ldots,i_k = t)$ in ${\cal P}$ with $\lambda(P) > 0$. By condition (2) with (CP), each edge in $P$ belongs to $E_{=} \cup E_{>}$. By condition (4), there are an index $l$ such that $i_0,i_1,\ldots,i_l \in U_s$ and $i_{l+1}, i_{l+2}, \ldots,i_{k} \in U_t$ with $D(0,p_{i_0}) > D(0,p_{i_1}) > \cdots > D(0,p_{i_l}) > 0 <
D(0,p_{i_{l+1}}) < D(0,p_{i_{l+2}}) < \cdots < D(0,p_{i_k})$, or $i_0,i_1,\ldots,i_{l-1} \in U_s$, $i_l \in U_0$, and $i_{l+1}, i_{l+2}, \ldots,i_{k} \in U_t$ with $D(0,p_{i_0}) > D(0,p_{i_1}) > \cdots > D(0,p_{i_{l-1}}) > 0 = D(0,p_{i_l}) <
D(0,p_{i_{l+1}}) < D(0,p_{i_{l+2}}) < \cdots < D(0,p_{i_k})$. For the former case, the union of $A_{ij}$ over edges $ij$ in $P$ forms an $(s^+, t^-)$-path and an $(t^+, s^-)$-path. For the latter case, the union of $\{ i_l^{s+}i_l^{t-},
i_l^{t+}i_l^{s-}\}$ and $A_{ij}$ over edges $ij$ in $P$ forms an $(s^+, t^-)$-path and an $(t^+, s^-)$-path. Hence a feasible circulation $\phi_{f}$ is constructed as follows. For each terminal $s$, define $\phi_f(s^-s^+) := f(s)$. For each edge $ij \in E_{=} \cup E_{>}$, define $\phi_f(\vec e) := f(ij)$ for $\vec e \in A_{ij}$. For each non-terminal node $i \in U_0$ and distinct $s,t \in S$, define $\phi_f(i^{s+}i^{t-})$ as the total flow-value of $(s,t)$-flows in $f$ using node $i$. Then the resulting $\phi_f$ is a feasible circulation in ${\cal D}_p$.
(If part). We verify that $p$ and $f_{\phi}$ satisfy the conditions of Lemma \[lem:optimality\]. Since there is no edge in ${\cal D}_p$ coming from $ij \in E$ with $a(ij) - D(p_i,p_j) > 0$, the multiflow $f_\phi$ does not use edge $ij$ with $a(ij) - D(p_i,p_j) > 0$, and hence satisfies the condition (2). Observe that $f_{\phi}(e) = (\phi(e^+) + \phi(e^{-}))/2 (\leq c(e))$ for an edge $e = ij \in E_{=} \cup E_{>}$ with $A_{ij} = \{e^+,e^-\}$. From this, if $e \in E_{>}$, then $f_{\phi}(e) = (\phi(e^+) + \phi(e^{-}))/2 = c(e)$, proving the condition (1). For terminal $s$, $\phi(s^-s^+)$ is the sum of $q_j$ over indices $j$ such that the cycle $C_j$ contains $s^-s^+$, which is equal to the sum of $\lambda(\bar P_l^r)$ over $S$-paths $\bar P_l^r$ connecting terminal $s$, i.e., $f_{\phi}(s)$. Thus $f_{\phi}(s) = \phi(s^-s^+) \geq r(s)$; in particular $f_{\phi}$ is feasible to $r$. Moreover $f_{\phi}(s) = r(s)$ if $s \in U_s$, proving the condition (4).
Finally consider condition (3) for $P = (s = i_0, i_1,\ldots, i_{l} = t) \in {\cal P}$. Observe from the construction of ${\cal D}_p$ that $p_{i_k} \neq p_{i_{k+1}}$, and $D(p_{i_{k-1}},p_{i_{k+1}}) = D(p_{i_{k-1}},p_{i_{k}}) + D(p_{i_{k}},p_{i_{k+1}})$. Since the metric space ${\cal T}$ is a tree, we obtain $D(p (P)) = D(p_s,p_t)$; see the next lemma.
In the last part of the proof, we use the following distance property of a tree, which we can easily prove (by an inductive argument).
Let $G$ be a tree (with a positive edge-length), and let $x = x_0,x_1,\ldots,x_l = y$ be a sequence of vertices in $G$. Suppose that
- $x_i \neq x_{i+1}$ for $i=0,1,2,\ldots,l-1$, and
- $d(x_{i-1}, x_{i+1}) = d(x_{i-1}, x_{i}) + d(x_{i}, x_{i+1})$ for $i=1,2,\ldots,l-1$.
Then $\sum_{i=0}^{l-1} d(x_{i},x_{i+1}) = d(x,y)$.
A simple example $(x,z,z,x)$ shows that the condition (1) is necessary.
One may wonder why the edge-cost positivity (CP) is needed. Consider the case where some of edges have zero cost. There may exist edges $ij \in E_{=}$ with $D(p_i, p_j) = 0$. Therefore we need to add edges to ${\cal D}_p$ corresponding to those edges. Even if we manage to construct a set ${\cal P}$ of paths from a feasible circulation in a modified network, consecutive nodes in some path $P$ may have the same potential, and does not guarantee that $P$ is $p$-geodesic ($P$ may connect the same terminal).
\[rem:howto\] The modification is the same as that given in [@GK97; @Kar94] used for (M). Let $Z$ denote the set of edges $e$ with $a(e) = 0$. Define a positive edge-cost $a'$ by $$\label{eqn:a'}
a'(e) := \left\{ \begin{array}{ll}
1 & {\rm if}\ e \in Z, \\
(2C(Z) + 1) a(e) & {\rm otherwise},
\end{array}\right. \quad (e \in E).$$ Then any half-integral optimal solution $x$ in (L) with edge-cost $a'$ is also optimal to (L) with edge-cost $a$. Indeed, by the half-integrality theorem (Theorem \[thm:fuku\]), it suffices to show that for every half-integral solution $y$ in (L) with cost $a$ it holds $$a x - a y \leq 0,$$ where we simply denote $\sum a(e) x(e)$ by $a x$. Indeed, we have $$(2C(Z) + 1) a x - (2C(Z) +1) a y = a' x - a' y - x(Z) + y(Z) \leq C(Z).$$ This implies that $ax - ay \leq C(Z)/(2C(Z) + 1) < 1/2$. Since $ax$ and $ay$ are half-integers, we have $ax - ay \leq 0$, as required.
Proximity scaling algorithm {#subsec:proximity_scaling}
---------------------------
In this section we present an algorithm to prove Theorem \[thm:multiflow\_main\]. By the arguments in the previous section, it suffices to solve (D). Let $\omega: {\cal T}^n \to \overline{\RR}$ be defined by $$\omega(p) := \sum_{ij \in E} c(ij) (D(p_i,p_j) -a(ij))^+ +
\sum_{s \in S} I_s (p_s) - r(s) D(0,p_s) \quad (p \in {\cal T}^n),$$ where $I_s$ denotes the indicator function of ${\cal T}_s$: $$I_s (q) := \left\{\begin{array}{ll}
0 & {\rm if}\ q \in {\cal T}_s, \\
\infty & {\rm otherwise},
\end{array} \right. (q \in {\cal T}).$$ Then (D) is equivalent to the minimization of $\omega$. The range in which an optimum exists is given as follows, where $A := \max \{ a(e) \mid e \in E\}$.
\[lem:region\] There exists a proper half-integral optimal potential $p$ such that $D(0,p_i) \leq n A$ for $i=1,2,\ldots,n$.
Take a proper half-integral optimal potential $p$. Suppose that $D(0, p_s) > n A$ for $s \in S$, and that $\{p_i \mid i \in U_s\} = \{q_1,q_2,\ldots, q_l = p_s\}$ with $D(0,q_j) < D(0, q_{j+1})$. Let $q_0 := 0$. Then $l \leq n$ and $\sum_{j=1}^{l} D(q_{j-1}, q_{j}) = D(0, p_s) > n A$. Thus there is an index $k (\geq 1)$ with $D(q_{k-1}, q_{k}) > A$. Let $X := \{ i \in U_s \mid p_i \in \{q_k, q_{k+1},\ldots,q_l\}\}$. For each $ij \in \delta X$, it holds $D(p_{i},p_{j}) \geq D(q_{k-1}, q_{k}) > A \geq a(ij)$. Hence $\delta X \subseteq E_{>}$. Let $\alpha := D(q_{k-1}, q_{k}) - A > 0$, which is a half-integer. Define proper half-integral potential $p'$ by $$p'_i := \left\{ \begin{array}{ll}
p_i - \alpha e_s & {\rm if}\ i \in X (\ni s), \\
p_i & {\rm otherwise},
\end{array} \right. \quad (i \in V).$$ Then $D(p'_i, p'_j) = D(p_i, p_j) - \alpha$ if $ij \in \delta X$ with $i \in X$, and $D(p'_i, p'_j) = D(p_i,p_j)$ otherwise. Also $D(0, p'_s) = D(0, p_s) - \alpha$ and $D(0, p'_t) = D(0, p_t)$ for other terminal $t \neq s$. By feasibility (\[eqn:feasible\]), we obtain $$\begin{aligned}
\omega(p') - \omega(p) & = &\sum_{ij \in \delta X} c(ij) \{ D(p'_i, p'_j) - D(p_i,p_j) \} - r(s) \{ D(0, p'_s) - D(0,p_s) \} \\
& = & - \alpha \{ c(\delta X) - r(s) \} \leq 0. \end{aligned}$$ Thus $p'$ is also optimal. Let $p := p'$. Repeat this procedure to obtain an optimal potential $p$ as required.
Let $L := \lceil \log n A \rceil$, and let ${\cal T}'$ be the subset of points $q$ of ${\cal T}$ with $D(0, q) \leq 2^{L}$. By the above lemma, (D) is equivalent to the minimization of $\omega$ over $({\cal T}')^n$. For $\sigma = -1, 0,1,2,\ldots, L$, let $T_\sigma$ denote the tree on ${\cal T}' \cap (2^{\sigma} \ZZ^S)$ such that vertices $u,v$ are adjacent if $D(u,v) = 2^{\sigma}$. In particular, $T_\sigma$ is a (graph-theoretical) tree discretizing ${\cal T}'$. The graph metric of $T_{\sigma}$ is denoted by $d_{\sigma}$. Then it holds $$2^{\sigma} d_{\sigma}(u,v) = D(u,v).$$ The two color classes of $T_\sigma$ are denoted by $B_\sigma$ and $W_\sigma$, and suppose $0 \in B_\sigma$. Then $T_{\sigma-1}$ is naturally identified with the subdivision of $T_\sigma$. Hence $$\begin{aligned}
T_{\sigma-1} = (T_\sigma)^*, \quad B_{\sigma-1} = T_\sigma.\end{aligned}$$ For $s \in S$, define $f_{s, \sigma}: T_{\sigma} \to \overline{\RR}$ by $$f_{s, \sigma}(p) := I_s (p) - r(s) 2^{\sigma} d_\sigma(0,p) \quad (p \in T_{\sigma}).$$ For each edge $ij \in E$, define $g_{ij, \sigma}: \ZZ \to \RR$ by $$g_{ij, \sigma}(z) := c(ij) (2^{\sigma} z - a(ij))^+ \quad (z \in \ZZ).$$ Let $\omega_{\sigma}: {T_{\sigma}}^{n} \to \overline{\RR}$ be the restriction of $\omega$ to ${T_{\sigma}}^n$, which is given by $$\omega_{\sigma}(p) = \sum_{s \in S} f_{s, \sigma}(p_s) + \sum_{ij \in E} g_{ij, \sigma}(d_\sigma(p_i,p_j)) \quad (p \in {T_{\sigma}}^{n}).$$ For each edge $ij$, consider even function $\bar g_{ij, \sigma}: \ZZ \to \RR$ defined as in Section \[subsec:2-separable\]. Namely let $\bar g_{ij, \sigma}(z) := (g_{ij, \sigma}(z-1) + g_{ij, \sigma}(z+1))/2$ if $z$ is odd and $\bar g_{ij, \sigma}(z) := g_{ij, \sigma}(z)$ if $z$ is even. Define $\bar \omega_{\sigma}: {T_{\sigma}}^{n} \to \overline{\RR}$ by $$\bar \omega_{\sigma}(p) = \sum_{s \in S} f_{s, \sigma}(p_s) + \sum_{ij \in E} \bar g_{ij, \sigma}(d_\sigma(p_i,p_j)) \quad (p \in {T_{\sigma}}^{n}).$$
\[lem:omega\]
- $\omega_{\sigma}$ and $\bar \omega_{\sigma}$ are (2-separable) L-extendable and L-convex on ${T_\sigma}^n$, respectively.
- $\bar \omega_{\sigma}$ is an L-convex relaxation of $\omega_{\sigma+1}$.
- Any minimizer of $\bar \omega_{-1}$ is optimal to (D).
(1). Observe that $f_{s, \sigma}$ is convex on $T_{\sigma}$. Obviously $g_{ij, \sigma}$ is convex on $\ZZ$. Apply Lemma \[lem:tree\_convexity\] and Theorem \[thm:2separable\] to obtain the claim.
(2). Let $p \in {T_{\sigma+1}}^n = {B_{\sigma}}^n$. Then $d_{{\sigma}}(p_i,p_j)$ is an even integer, and $\bar g_{ij, \sigma}(d_{\sigma}(p_i,p_j)) = g_{ij, \sigma}(d_{\sigma}(p_i,p_j))$. Hence $\omega_{\sigma}(p) = \omega(p) = \omega_{\sigma+1}(p)$.
(3). We show $\bar \omega_{-1} = \omega_{-1}$. From the view of the proof of (2), it suffices to show that $\bar g_{ij,-1}(z)= g_{ij,-1}(z)$ for any odd integer $z$. Since $a(ij)$ is an integer, either $(z-1)/2, (z+1)/2 \leq a(ij)$ or $(z-1)/2, (z+1)/2 \geq a(ij)$ holds. From this, we see $\bar g_{ij,-1}(z)= g_{ij,-1}(z)$. Notice that $T_{-1}$ is the set of half-integral potentials. The claim follows from the half-integrality (Proposition \[prop:min-max\]).
Thus our goal is to minimize the L-convex function $\omega_{-1}$. We are now ready to describe our scaling algorithm to solve (D):
Proximity scaling algorithm:
:
Step 0:
: Replace $a$ by $a'$ defined by (\[eqn:a’\]) if $a$ is not positive. Let $\sigma := L = \lceil \log n A \rceil$ and $p^{\sigma+1}:= (0,0,\ldots,0) \in {B_{\sigma}}^n$.
Step 1:
: Find a minimizer $p^{\sigma} \in {T_\sigma}^{n}$ of $\bar \omega_{\sigma}$ by the steepest descent algorithm with initial point $p^{\sigma + 1}$.
Step 2:
: If $\sigma = -1$, then $p = p^{-1}$ is an optimal solution of (D), and go to step 3. Otherwise, let $\sigma \leftarrow \sigma - 1$ and go to step 1.
Step 3:
: Construct ${\cal D}_p$, and find an integral feasible circulation $\phi$. Then $f_{\phi}$ is a half-integral optimal multiflow in (N) as required.
The time complexity of step 1 is estimated as follows.
$d_{\sigma} ({\rm opt}(\bar \omega_{\sigma}), p^{\sigma + 1}) \leq 6n + 4$.
We show the existence of a minimizer $q^*$ of $\bar \omega_{\sigma}$ with $d_{\sigma} (q^*, p^{\sigma + 1}) \leq 6n + 4$. First consider the case where $\sigma = L$ or $L-1$. In this case, the diameter of $T_{\sigma}$ is $2$ or $4$. Hence the inequality obviously holds. Consider the case $\sigma \leq L - 2$. Then $p^{\sigma+1}$ is a minimizer of an L-convex relaxation $\bar \omega_{\sigma+1}$ of $\omega_{\sigma+2}$ (Lemma \[lem:omega\] (2)). By the persistency (Theorem \[thm:persistency\]), there exists a minimizer $q$ of $\omega_{\sigma+2}$ (over ${T_{\sigma+2}}^{n}$) with $d_{\sigma+1} (p^{\sigma+1}, q) \leq 1$. Since $q$ is also a minimizer of L-extendable function $\omega_{\sigma+1}$ (on ${T_{\sigma+1}}^{n} =(B_{\sigma+1} \cup W_{\sigma+1})^{n}$) over ${B_{\sigma+1}}^{n}$, by proximity theorem (Theorem \[thm:proximity\]), there is a minimizer $q'$ of $\omega_{\sigma+1}$ (over ${T_{\sigma+1}}^{n})$ with $d_{\sigma+1}(q',q) \leq 2 n$. Since $\bar \omega_{\sigma}$ is an L-convex relaxation of $\omega_{\sigma+1}$, the restriction of $\bar \omega_{\sigma}$ to ${B_{\sigma}}^n = {T_{\sigma+1}}^n$ is equal to $\omega_{\sigma+1}$. Hence $q'$ is a minimizer of $\bar \omega_{\sigma}$ over ${B_{\sigma+1}}^n$. Since $\bar \omega_{\sigma}$ is also midpoint L-extendable on ${T_\sigma}^{n} = (B_{\sigma} \cup W_\sigma)^{n}$, by the proximity theorem, there is a minimizer $q^*$ of $\bar \omega_{\sigma}$ over ${T_{\sigma}}^{n}$ with $d_{\sigma}(q^*, q') \leq 2 n$. Notice $2 d_{\sigma +1} = d_{\sigma}$. Thus we have $$d_{\sigma}(p^{\sigma+1},q^*) \leq d_{\sigma}(p^{\sigma+1}, q) + d_{\sigma}(q, q') + d_{\sigma}(q', q^*)
\leq 2 + 4 n + 2 n$$ as required.
Therefore, by Theorem \[thm:bound\], the number of iterations of the steepest descent algorithm is at most $6n + 4$. Since $\bar \omega_{\sigma}$ is a 2-separable L-convex function consisting of $O(m)$ terms, and the maximum degree of $T_{\sigma}$ is the number $k$ of terminals, by Theorem \[thm:approx\] we can find an optimal solution in $O( n {\rm MF}(kn, km))$ time. The total step is $O( n L \, {\rm MF}(kn,km))$, where $L$ can be taken to be $\lceil \log 2 (C(Z) + 1) n A \rceil = O(\log n A C)$. This proves Theorem \[thm:multiflow\_main\].
#### Algorithm for (M).
Let us sketch a proximity scaling algorithm to solve (M). Corresponding to (D), consider the following location problem on ${\cal T}$. $$\begin{aligned}
\mbox{(D$'$): Max.} && \sum_{ij \in E} c(ij) (D(p_i, p_j) - a(ij))^+ \\
\mbox{s.t.} && p = (p_1,p_2,\ldots, p_n) \in
{\cal T}^n, \\
&& p_s = \mu e_s/2 \quad (s \in S).\end{aligned}$$ Again a feasible solution of (D$'$) is called a potential, and called half-integral if each $p_i$ is half-integral. The following duality is implicit in [@HHMPA; @Kar94].
The maximum value of (M) is equal to the minimum value of (D$'$). Moreover, if $\mu$ is an integer, then there exists a half-integral optimal potential in (D$'$).
The edge-capacitated formulation is transformed to the node-capacitated formulation, discussed in [@HHMPA], as follows. Replace each edge $e = ij$ by the series of two edges $iu$ and $uj$. Define the node-capacity and the node-cost of the new node $u$ by $c(e)$ and $a(e)$. No edge-capacity and edge-cost are given. The node-capacities of the original nodes are $\infty$. Then (M) becomes a node-capacitated problem, and the results in [@HHMPA Section 4] are applicable. In particular, the dual of (M) is given by the problem (4.6) of [@HHMPA] in setting $\bar \mGamma := {\cal T}$ and $\bar R_s := \{ \mu e_s /2\}$. Subtree $F(i)$ for the original node $i$ is a single point $p_i$ (by $b(x) = \infty$), and hence $F(u)$ for new node $u$ replacing original edge $ij$ is a path between $p_i$ and $p_j$ with length (diameter) $D(p_i, p_j)$. Thus (4.6) of [@HHMPA] becomes (D$'$). The half-integrality follows from [@HHMPA Remark 4.7].
By the same argument in the proof of Lemma \[lem:optimality\], one can prove that a multiflow $f$ and a potential $p$ are both optimal if and only if they satisfy conditions (1), (2), and (3) in Lemma \[lem:optimality\]. The corresponding double covering network ${\cal D}'_p$ is obtained by replacing the lower bound and the upper bound capacities of $s^-s^+$ of ${\cal D}_p$ by $0$ and $\infty$, respectively. Then we obtain an analogue of Proposition \[prop:f\_phi\] that $p$ is optimal if and only if there exists a feasible circulation $\phi$ in ${\cal D}'_p$, and $f_{\phi}$ is an optimal multiflow in (M). We may consider that the variables of (D$'$) are $p_i$ for non-terminal nodes $i \in V \setminus S$ (since a potential $p_s$ of each terminal $s$ is fixed to $\mu e_s /2$ in (D$'$)). For non-terminal node $i$, define $f_i: {\cal T} \to \RR_+$ by $f_i (q) := \sum_{s \in S: si \in E} c(si) D(\mu e_s/2, q)$. We may assume that the set of non-terminal nodes are $\{1,2,\ldots,n-k\}$. Then (D$'$) is the minimization of $\omega (p) := \sum_{1 \leq i \leq n-k} f_i( p_i)
+ \sum_{ij \in E: 1 \leq i,j \leq n-k} c(ij) (D(p_i, p_j)- a(ij))^+$. Again it is easy to see that there is an optimal potential $p$ with $D(0,p_i) \leq \mu/2$. For $\sigma = -1, 0,1,2,\ldots$, define $T_{\sigma}$, $g_{ij,\sigma}$, $\bar g_{ij,\sigma}$, $\omega_{\sigma}$, and $\bar \omega_{\sigma}$ as above. Then Lemma \[lem:omega\] holds in this setting. Let $L := \lceil \log \mu \rceil$. By the proximity scaling algorithm, we can minimize $\omega'$ in $O((n-k) \lceil \log \mu \rceil {\rm MF}(kn, km))$ time. It is shown in [@GK97; @Kar94] that if $\mu \geq 2A_1C + 1$ for $A_1 := \sum_{e \in E} a(e)$ and $C = \sum_{e \in E} c(e)$, then every half-integral optimal multiflow in (M) is a minimum cost multiflow. Also it is shown in [@GK97; @Kar94] that cost $a$ is replaced by $a'$ (defined by (\[eqn:a’\])) to satisfy the cost positivity. Any half-integral optimal multiflow in (M) with cost $a'$ is optimal for original cost $a$. Thus, letting $\mu := 2 A'_1C + 1 = O(A_1 C)$, we obtain a minimum cost half-integral multiflow in $O( (n-k) \log (A_1 C ) {\rm MF} (kn, km))$ time.
Additional results and remarks {#subsec:additional}
------------------------------
### Lovász-Cherkassky formula, $k$-submodular function, and multiway cut {#subsub:LC}
We note a relation between Lovász-Cherkassky formula (Theorem \[thm:LC\]), $k$-submodular function minimization, and multiway cut. Suppose that $S$ consists of $k$ terminals, and the set of non-terminal nodes is $\{1,2,\ldots,n\}$. Recall notions in Section \[subsec:k-submodular\]. Adding $0$ to $S$, we obtain poset $S_k$, and consider the following $k$-submodular function minimization: $$\begin{aligned}
\label{eqn:LS}
\mbox{Min.} && \frac{1}{2} \sum_{1 \leq i \leq n} \sum_{s \in S: si \in E}
c(si) \delta (s, p_i) + \frac{1}{2} \sum_{ ij \in E: 1 \leq i,j \leq n} c(ij) \delta (p_i, p_j)
\\
\mbox{s.t.} && p = (p_1,p_2, \ldots,p_n) \in {S_k}^n.\nonumber\end{aligned}$$ Recall Example \[rem:multiway\] with $d = \delta$ that this problem is nothing but a $k$-submodular relaxation (or an L-convex relaxation) of multiway cut.
Furthermore this problem is also a dual of maximum free multiflow problem (MF), and hence the optimal value of this problem is equal to $\sum_{s \in S} \kappa_s/2$. To see this, for $p \in {S_k}^n$ and $s \in S$, let $X^p_s := \{ s\} \cup \{i \mid p_i = s\}$. Then $X^p_s$ is an $(s, S \setminus \{s\})$-cut. Observe $\sum_{s \in S} c(\delta X^{p}_s)/2$ is equal to the objective value of (\[eqn:LS\]) at $p$. Conversely, take a minimum $(s, S \setminus \{s\})$-cut $X_s$ for each $s \in S$. We can assume that $X_s$ $(s \in S)$ are disjoint. If $X_s \cap X_t \neq \emptyset$, then, by submodularity, we can replace $X_s$ by $X_s \setminus X_t$ and $X_t$ by $X_t \setminus X_s$ without increasing the cut capacity. Define $p$ by $p_i = s$ if $i \in X_s$ for some $s \in S$ and $p_i = 0$ otherwise. Then the objective value at $p$ is equal to $\sum_{s \in S} c(\delta X_s)/2 = \sum_{s \in S} \kappa_s/2$. In particular, this $k$-submodular function minimization can be solved by $(s, S\setminus\{s\})$-mincut computation for each $s \in S$.
### Application to terminal backup problem
The linear program (L) arises as an LP-relaxation of a class of network design problems, called [*terminal backup problems*]{} [@AK11; @XAC08INFOCOM]. Given a graph $G = (V,E)$ with terminal set $S$ and edge-cost $a: E \to \ZZ_+$ the terminal backup problem asks to find a minimum cost subgraph $F$ with the property that each terminal $s$ is reachable to other terminal in $F$. Anshelevich and Karagiozova [@AK11] proved that this problem is solvable in polynomial time. Bern[á]{}th, Kobayashi, and Matsuoka [@BKM13] considered the following weighted generalization. Given a graph $G = (V,E)$ with terminal set $S$, edge-cost $a: E \to \ZZ_+$, and a requirement $r: S \to \ZZ_+$, find a minimum cost integral edge-capacity $x: E \to \ZZ_+$ such that for each terminal $s$ there is an integral $(s, S \setminus \{s\})$-flow in $(V,E,x)$ with flow-value at least $r(s)$. They proved that this generalization is solvable in (strongly) polynomial time, and asked whether a natural capacitated version of this problem is tractable or not. The capacitated version is to impose the condition $x(e) \leq c(e)$ $(e \in E)$ for $c: E \to \ZZ_+$, and is formulated as the following integer program: $$\begin{aligned}
\mbox{(CTB)}\quad {\rm Min.} && \sum_{e \in E} a(e) x(e) \nonumber \\
{\rm s. t. } && x(\delta X) \geq r(s) \quad (s \in S, X \in {\cal C}_s), \nonumber \\
&& x(e) \in \{0,1,2,\ldots, c(e)\} \quad (e \in E).\end{aligned}$$ The problem (L) is nothing but a natural LP-relaxation of (CTB).
Fukunaga [@Fukunaga14] studied (CTB), and proved the half-integrality (Theorem \[thm:fuku\]) of the LP-relaxation (L). As was noted by him, a $2$-approximation solution is immediately obtained from a half-integral optimal solution $x$ in (L) by rounding each non-integral component $x(e)$ to $x(e) +1/2$. He devised a clever rounding algorithm to obtain a $4/3$-approximation solution. Our algorithm gives a practical and combinatorial implementation of his $4/3$-approximation algorithm, as follows. A half-integral optimal solution $x$ of (L) and an optimal potential $p$ of (D) are obtained by our combinatorial algorithm. Fukunaga’s algorithm rounds a special half-integral optimal solution $\tilde x$ obtained from $x$ by the following fixing procedure. Let $E_1$ be the set of edges $e$ with $x(e) \in \ZZ$. Let $\tilde x (e):= x(e)$ for $e \in E_1$. For an edge $e\in E \setminus E_1$ (with non-integral $x(e)$), check whether there is an optimal solution $y$ in (L) such that $y(e) \in \{\lfloor x(e) \rfloor, \lceil x(e) \rceil\}$, $\lfloor x(e') \rfloor \leq y(e') \leq \lceil x(e') \rceil$ for $e' \in E \setminus (E_1 \cup \{e\})$, and $y(e') = \tilde x(e')$ for $e' \in E_1$. If such $y$ exists, then let $\tilde x(e) := y(e)$. Otherwise let $\tilde x(e) := x(e)$. Add $e$ to $E_1$, and repeat until $E_1 = E$ to obtain $\tilde x$. This procedure can be implemented on the double covering network ${\cal D}_p$. By Lemma \[lem:NvsL\] with (CP), $y$ is optimal to (L) if and only if $y$ is the flow-support of some optimal multiflow $f$ in (N). From view of (the proof of) Proposition \[prop:f\_phi\], $y$ is optimal if and only if there is a circulation $\phi$ of ${\cal D}_p$ with $y(ij) = \phi(\vec e)$ for $ij \in E, \vec e \in A_{ij}$. Therefore the above procedure reduces to checking the existence of a circulation in ${\cal D}_p$ with changing the lower and upper capacities of edges in $A_{ij}$ to $\lfloor x(e) \rfloor$ or $\lceil x(e) \rceil$ appropriately. Thus $\tilde x$ is obtained by at most $m$ max-flow computations on ${\cal D}_p$.
### Simple descent algorithm by double covering network {#subsub:simple}
We here present a simple and instructive but pseudo-polynomial time algorithm solving (N) and (D) of the following description:
> For a potential $p$, find a feasible circulation $\phi$ in ${\cal D}_p$. If $\phi$ exists, then $f_{\phi}$ is an optimal multiflow, and stop. Otherwise, from an infeasibility certificate of ${\cal D}_p$, we obtain another potential $p'$ with $\omega(p') < \omega(p)$. Let $p \rightarrow p'$ and repeat.
The presented algorithm can always keep $p$ half-integral, providing an algorithmic proof of Proposition \[prop:min-max\]. Assume the cost positivity (CP). For a (proper) half-integral potential $p$, construct the double covering network ${\cal D}_p$, as above. We reduce the circulation problem on ${\cal D}_p$ to the maximum flow problem on a modified network $\tilde {\cal D}_{p}$. Consider supper source $a^{+}$ and sink $a^{-}$. For each $ij \in E_{>0}$ modify edge set $A_{ij}$ as follows. For each $uv \in A_{ij}$ replace $uv$ by two edges $ua^{-}, a^{+} v$ with (upper-)capacity $c(ij)$ (and lower-capacity $0$). For each terminal $s \in S$, add new two edges $s^- a^+$ and $s^+ a^{-}$ with capacity $r(s)$. For edge $s^{-}s^{+}$, change the lower-capacity to $0$ and the upper-capacity to $\infty$ if $p_s = 0$ and to $0$ otherwise. The resulting network is denoted by $\tilde {\cal D}_p$. Consider the maximum $(a^+,a^-)$-flow problem on the new network $\tilde {\cal D}_{p}$. This is a standard reduction of a circulation problem to a max-flow problem. In particular, ${\cal D}_p$ has a feasible circulation if and only if a maximum $(a^+, a^-)$-flow $\tilde {\cal D}_p$ saturates all edges leaving $a^{+}$ (entering $a^-$), i.e., $\{a^+\}$ is a minimum $(a^+,a^-)$-cut.
Let $\tilde V_1$ (resp. $\tilde V_{2}$) be the set of nodes $i^+$, $i^-$, $i^{s+}$, or $i^{s-}$ such that $i$ has an integral potential $p_i$ (resp. non-integral potential $p_i$). By the integrality of $a(ij)$, there is no edge $ij$ in $E_{=}$ such that $i$ has an integral potential and $j$ has a non-integral potential. Hence there is no edge connecting between $\tilde V_{1}$ and $\tilde V_{2}$.
An $(a^+, a^{-})$-cut $X$ in $\tilde {\cal D}_p$ is called [*legal*]{} if
- $X \cap \tilde V_1$ or $X \cap \tilde V_2$ is empty,
- for each $i \in U_0$, $X \cap B_{i}$ is empty or $\{ i^{s+}\} \cup \{i^{t-} \mid t \in S \setminus \{s\}\}$ for some $s \in S$, and
- for other node $i$, $X \cap \{i^+,i^-\}$ is empty, $\{i^+\}$, or $\{i^-\}$.
For a legal cut $X$ of $\tilde D_{p}$, the potential $p^X$ is defined by: $$p^X_i := \left\{ \begin{array}{ll}
e_s/2 & {\rm if}\ \mbox{$i \in U_0$ and $i^{s+} \in X$ for $s \in S$},\\
p_i + e_s/2 & {\rm if}\ \mbox{$i^{+} \in X$ and $i \in U_s$ for $s \in S$}, \\
p_i - e_s/2 & {\rm if}\ \mbox{$i^{-} \in X$ and $i \in U_s$ for $s \in S$},\\
p_i & {\rm otherwise},
\end{array} \right. \quad (i \in V).$$ Then the following lemma holds; the proofs are given in the end of this section.
\[lem:legal\]
- For a legal cut $X$ in $\tilde {\cal D}_p$, we have $$\omega(p^X) - \omega(p) = \frac{1}{2} \{ c(\delta X) - c(\delta \{a^{+}\}) \}.$$
- Let $X$ be a (unique) inclusion-minimal minimum $(a^+,a^-)$-cut in $\tilde D_p$, and let $X_1 := X \setminus \tilde V_2$ and $X_2:= X \setminus \tilde V_1$. Then both $X_1$ and $X_2$ are legal cuts with $$c(\delta X) = c(\delta X_1) + c(\delta X_2) - c( \delta \{a^+\}).$$ In particular, if $c(\delta X) < c(\delta \{a^+\})$, then $c(\delta X_1) < c(\delta \{a^+\})$ or $c(\delta X_2) < c(\delta \{a^+\})$.
Therefore we can check the optimality of $p$ by solving the maximum-flow problem on $\tilde {\cal D}_p$. If $p$ is not optimal, then we obtain another half-integral potential $p^X$ having a smaller objective value. This naturally provides the following algorithm:
Descent algorithm by double covering network
:
Step 0:
: Replace $a$ by $a'$ defined by (\[eqn:a’\]) if $a$ is not positive. Let $p := (0,0,\ldots,0)$.
Step 1:
: Construct $\tilde D_{p}$, and find a minimal minimum $(a^+,a^-)$-cut $X$ and a maximum $(a^+,a^-)$-flow $\tilde \phi$.
Step 2:
: If $X = \{a^+\}$, then $p$ is optimal, and construct a feasible circulation $\phi$ on ${\cal D}_p$ from $\tilde \phi$ and an optimal multiflow $f_{\phi}$ from $\phi$; stop. Otherwise go to step 3.
Step 3:
: Let $X_1 := X \setminus \tilde V_2$ and $X_2 := X \setminus \tilde V_1$. Choose $j \in \{1,2\}$ with $c(\delta X_j) < c(\delta \{a^+\})$. Let $p := p^{X_j}$ and go to step 1.
Observe that this algorithm coincides with the steepest descent algorithm applied to L-convex function $\omega_{-1}$ on ${T_{-1}}^n$, where $p^{X_j}$ is a steepest direction of ${\cal I}(p)$ for even iterations and of ${\cal F}(p)$ for odd iterations. Thus, by Theorem \[thm:bound\] and Lemma \[lem:region\], the number of the iteration is $O(n AC )$. The numbers of nodes and edges of $\tilde {\cal D}_p$ are $O(kn)$ and $O(m+ k^2n)$, respectively. Thus we have:
The above algorithm runs in $O(n AC\, {\rm MF}( kn, m + k^2 n ))$ time.
In the above algorithm, each step minimizes a sum of basic ${\pmb k}$-submodular functions of type I and III (thanks to (CP)). The above network $\tilde {\cal D}_p$ may be viewed as yet another representation of ${\pmb k}$-submodular functions. In fact, an arbitrary sum of basic ${\pmb k}$-submodular functions of type I and III admits this kind of a network representation (Yuta Ishii, Master Thesis, University of Tokyo, 2014). However this representation seems not to capture basic ${\pmb k}$-submodular functions of type II. In each scaling phase of the proximity scaling algorithm, the objective functions of local problems may contain a ${\pmb k}$-submodular term of type II. This is why we need an algorithm in Theorem \[thm:IWY14\].
(1). Let $X$ be a legal cut (with finite cut capacity). Observe that $$c(\delta X) = \sum_{ij \in E_{=} \cup E_{>}} c(ij) | \delta X \cap A_{ij}|
+ \sum_{s \in S} r(s) |\delta X \cap \{ a^+ s^+, s^-a^- \}|.$$ In particular we have $$c(\delta \{a^+\}) = \sum_{ij \in E_{>}} 2 c(ij) + \sum_{s \in S} r(s).$$ Let $p' := p^X$. For an edge $ij \in E$, if $p_i$ and $p_j$ are integral and non-integral potentials, respectively, then $ij \not \in E_{=}$ and $D(p_i,p_j) -1/2 \leq D(p'_i, p'_j) \leq D(p_i,p_j) +1/2$. Therefore $a(ij) > D(p_i,p_j)$ implies $a(ij) \geq D(p'_i,p'_j)$ and $a(ij) < D(p_i,p_j)$ implies $a(ij) \leq D(p'_i,p'_j)$. Consequently we have $$\begin{aligned}
\omega (p')- \omega (p) &= &\sum_{ij \in E_{>}} c(ij) (D(p'_i, p'_j)- D(p_i, p_j) ) + \sum_{ij \in E_{=}} c(ij) (D(p'_i, p'_j) - a(ij))^+ \\
&& - \sum_{s \in S} r(s) ( D(0, p'_s) - D(0, p_s) ). \\
& = & \sum_{ij \in E_{>}} c(ij) ( D(p'_i, p'_j)- D(p_i, p_j) + 1) + \sum_{ij \in E_{=}} c(ij) (D(p'_i, p'_j) - a(ij))^+ \\
&& + \sum_{s \in S} r(s) ( D(0, p_s) - D(0, p'_s) + 1/2) - c( \delta \{a^+\})/2.\end{aligned}$$ It suffices to show that $$\begin{aligned}
|\delta X \cap A_{ij}|/2 & = & D(p'_i, p'_j)- D(p_i, p_j) + 1 \quad (ij \in E_{>}), \label{eqn:1} \\
|\delta X \cap A_{ij}|/2 & = & (D(p'_i, p'_j) - a(ij))^+ \quad (ij \in E_=), \label{eqn:2}\\
|\delta X \cap \{ a^+s^+, s^- a^- \}|/2 & = & D(0, p_s) - D(0, p'_s) + 1/2 \quad (s \in S). \label{eqn:3}\end{aligned}$$ Pick $ij \in E_{=} \cup E_{>}$.
Case 1: $i \in U_s$, $j \in U_s \cup U_0$ and $D(0,p_i) > D(0,p_j)$. If $j \in U_0$, then $j^{s\pm}$ is denoted by $j^{\pm}$. Suppose $ij \in E_{=}$ (to show (\[eqn:2\])). If $\{i^+,i^-,j^+,j^-\} \cap X$ is empty or contains $i^-$ or $j^+$, then $\delta X \cap A_{ij}$ is empty, $D(p'_i,p'_j) - D(p_i,p_j) \leq 0$ and hence $(D(p'_i,p'_j) - a(ij))^+ = 0$. If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^+\}$ or $\{j^-\}$, then $|\delta X \cap A_{ij}| = 1$, $D(p'_i,p'_j) - D(p_i,p_j) = 1/2$, and $(D(p'_i,p'_j) - a(ij))^+ = 1/2$. If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^+,j^-\}$, then $|\delta X \cap A_{ij}| = 2$, $D(p'_i,p'_j) - D(p_i,p_j) = 1$, and $(D(p'_i,p'_j) - a(ij))^+ = 1$. Suppose $ij \in E_{>}$ (to show (\[eqn:1\])). If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^+,j^+\}$, $\{i^-,j^-\}$ or empty, then $|\delta X \cap A_{ij}| = 2$, $D(p'_i,p'_j) = D(p_i,p_j)$, and $D(p'_i,p'_j) - D(p_i,p_j) +1 = 1$. If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^+\}$ or $\{j^-\}$, then $|\delta X \cap A_{ij}| = 3$, $D(p'_i,p'_j) = D(p_i,p_j) + 1/2$, and $D(p'_i,p'_j) - D(p_i,p_j) +1 = 3/2$. If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^-\}$ or $\{j^+\}$, then $|\delta X \cap A_{ij}| = 1$, $D(p'_i,p'_j) = D(p_i,p_j) - 1/2$, and $D(p'_i,p'_j) - D(p_i,p_j) +1 = 1/2$. If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^+, j^-\}$, then $|\delta X \cap A_{ij}| = 4$, $D(p'_i,p'_j) = D(p_i,p_j) + 1$, and $D(p'_i,p'_j) - D(p_i,p_j) +1 = 2$. If $\{i^+,i^-,j^+,j^-\} \cap X = \{i^-, j^+\}$, then $|\delta X \cap A_{ij}| = 0$, $D(p'_i,p'_j) = D(p_i,p_j) - 1$, and $D(p'_i,p'_j) - D(p_i,p_j) +1 = 0$.
Case 2: $i \in U_s$ and $j \in U_{s'}$. In this case, (\[eqn:1\]) and (\[eqn:2\]) are obtained by replacing roles of $j^+$ and $j^-$ in the above case 1.
Next consider terminal $s \in S$ (to show (\[eqn:3\])). If $X \cap \{s^+,s^-\} = \{s^+\}$, then $D(0,p'_s) = D(0,p_s) + 1/2$ and $\delta X \cap \{a^+s^+, s^-a^-\}$ is empty. If $X \cap \{s^+,s^-\} = \{s^-\}$, then $D(0,p'_s) = D(0,p_s) - 1/2$ and $|\delta X \cap \{a^+s^+, s^-a^-\}| = 2$. If $X \cap \{s^+,s^-\}$ is empty, then $D(0,p'_s) = D(0,p_s)$ and $|\delta X \cap \{a^+s^+, s^-a^-\}| = 1$. For all cases, (\[eqn:3\]) holds.
(2). The equality $c(\delta X) = c(\delta X_1) + c(\delta X_2) - c( \delta \{a^+\})$ follows from the fact that there is no edge between $\tilde V_1$ and $\tilde V_2$. So it suffices to show that minimal minimum $(a^+,a^-)$-cut $X$ satisfies the conditions (2) and (3) of legal cuts.
Suppose (for contradiction) that $X$ contains $\{i^+,i^-\}$ or $\{i^{s+}, i^{s-}\}$. Remove all such pairs of nodes from $X$ to obtain another $(a^+,a^-)$-cut $X'$. Observe that $|A_{ij} \cap \delta X| \geq |A_{ij} \cap \delta X'|$ for each $ij \in E_{=} \cup E_{>}$, and $|\{a^+s^+, s^-a^-\} \cap \delta X| \geq |\{a^+s^+, s^-a^-\} \cap \delta X'|$ for each terminal $s$. Moreover $\delta X'$ does not contain edge $i^{s +}i^{s'-}$ (of infinite capacity). Otherwise $i^{s+}, i^{s'-}, i^{s'+} \in X \not \ni i^{s-}$, and $\delta X$ has edge $i^{s'+}i^{s-}$ of infinite capacity; a contradiction. Thus $X'$ is also a minimum cut, contradicting the minimality of $X$. Thus $\delta X$ cannot contain both $i^{+}$ ($i^{s+}$) and $i^{-}$ ($i^{s-}$). Suppose for contradiction that $X$ contains $i^{s-}$ and does not contain $i^{s'+}$ for each $s' \in S \setminus \{s\}$. In this case, remove $i^{s-}$ from $X$. Then the cut capacity does not increase, contradicting the minimality of $X$. Hence $X$ satisfies (2) and (3), as required.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Takuro Fukunaga for communicating [@Fukunaga14], Vladimir Kolmogorov for helpful comments on the earlier version of this paper, and the referee for helpful comments. The work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004.
[1]{}
R. K. Ahuja, D. S. Hochbaum, and J. B. Orlin, A cut-based algorithm for the nonlinear dual of the minimum cost network flow problem, [*Algorithmica*]{} [**39**]{} (2004), 189–208.
E. Anshelevich and A. Karagiozova, Terminal backup, 3D matching, and covering cubic graphs, [*SIAM Journal on Computing*]{} [**40**]{} (2011), 678–708.
A. Bern[á]{}th, Y. Kobayashi, and T. Matsuoka, The generalized terminal backup problem, EGRES, TR-2013-07, 2013 (the extended abstract in SODA’13).
B. V. Cherkasski, A solution of a problem of multicommodity flows in a network. [*Ekonomika i Matematicheskie Metody*]{} [**13**]{} (1977), 143–151 (in Russian).
P. M. Dearing, R. L. Francis, and T. J. Lowe, Convex location problems on tree networks, [*Operations Research*]{} [**24**]{} (1976), 628–642.
E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis, The complexity of multiterminal cuts, [*SIAM Journal on Computing*]{} [**23**]{} (1994), 864–894.
P. Favati and F. Tardella, Convexity in nonlinear integer programming, [*Ricerca Operativa*]{} [**53**]{} (1990), 3–44.
P. Felzenszwalb, G. Pap, [É]{}. Tardos, and R. Zabih, Globally optimal pixel labeling algorithms for tree metrics, In: [*Proceedings of The Twenty-Third [IEEE]{} Conference on Computer Vision and Pattern Recognition (CVPR’10)*]{} (2010), pp. 3153–3160.
T. Fukunaga, Approximating the generalized terminal backup problem via half-integral multiflow relaxation, In: [*32nd International Symposium on Theoretical Aspects of Computer Science (STACS’15)*]{} (2015), pp. 316–328. [arXiv:1409.5561]{}.
S. Fujishige, [*Submodular Functions and Optimization, 2nd Edition*]{}, Elsevier, Amsterdam, 2005.
S. Fujishige, Bisubmodular polyhedra, simplicial divisions, and discrete convexity, [*Discrete Optimization*]{} [**12**]{} (2014) 115–120.
S. Fujishige and S. Iwata, Bisubmodular function minimization, [*SIAM Journal on Discrete Mathematics*]{} [**19**]{} (2005), 1065–1073.
S. Fujishige and K. Murota, Notes on L-/M-convex functions and the separation theorems, [*Mathematical Programming, Series A*]{} [**88**]{} (2000), 129–146.
A. V. Goldberg and A. V. Karzanov, Scaling methods for finding a maximum free multiflow of minimum cost, [*Mathematics of Operations Research*]{} [**22**]{} (1997), 90–109.
I. Gridchyn and V. Kolmogorov, Potts model, parametric maxflow and $k$-submodular functions, In: [*Proceedings of IEEE International Conference on Computer Vision (ICCV’13)*]{}, 2013, 2320–2327. [arXiv:1310.1771]{}. M. Grötschel, L. Lovász, and A. Schrijver, [*Geometric Algorithms and Combinatorial Optimization*]{}, Springer-Verlag, Berlin, 1988.
H. Hirai, Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees, [*Mathematical Programming, Series A*]{} [**137**]{} (2013), 503–530.
H. Hirai, Discrete convexity and polynomial solvability in minimum 0-extension problems, [*Mathematical Programming, Series A*]{}, to appear (the extended abstract appeared SODA’13).
H. Hirai, Discrete convexity for multiflows and 0-extensions, in: [*Proceeding of 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications*]{}, 2013, pp. 209–223. H. Hirai, L-convexity on graph structures, in preparation.
D. S. Hochbaum, Solving integer programs over monotone inequalities in three variables: a framework for half integrality and good approximations, [*European Journal of Operational Research*]{} [**140**]{} (2002), 291–321.
A. Huber and V. Kolmogorov, Towards minimizing $k$-submodular functions, in: [*Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO’12)*]{}, LNCS 7422, Springer, Berlin, 2012, pp. 451–462. S. Iwata, L. Fleischer, and S. Fujishige, A combinatorial strongly polynomial algorithm for minimizing submodular functions, [*Journal of the ACM*]{} [**48**]{} (2001), 761–777.
S. Iwata and K. Nagano, Submodular function minimization under covering constraints, In: [*Proceeding of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09)*]{} (2009), pp. 671–680.
S. Iwata and M. Shigeno, Conjugate scaling algorithm for Fenchel-type duality in discrete convex optimization, [*SIAM Journal on Optimization*]{} [**13**]{} (2002), 204–211.
Y. Iwata, M. Wahlström, and Y. Yoshida, Half-integrality, LP-branching and FPT Algorithms, [arXiv:1310.2841]{}, the conference version appeared in SODA’14.
A. W. J. Kolen, [*Tree Network and Planar Rectilinear Location Theory*]{}, CWI Tract 25, Center for Mathematics and Computer Science, Amsterdam, 1986.
A. V. Karzanov, A minimum cost maximum multiflow problem, in: [*Combinatorial Methods for Flow Problems*]{}, Institute for System Studies, Moscow, 1979, pp. 138–156 (Russian).
A. V. Karzanov, Minimum cost multiflows in undirected networks, [*Mathematical Programming, Series A*]{} [**66**]{} (1994), 313–324.
V. Kolmogorov, Generalized roof duality and bisubmodular functions, [*Discrete Applied Mathematics*]{} [**160**]{} (2012), 416–426.
V. Kolmogorov, Submodularity on a tree: Unifying L$^\natural$-convex and bisubmodular functions, in: [*Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS’11)*]{}, LNCS 6907, Springer, Berlin, 2011, pp. 400–411
V. Kolmogorov and A. Shioura, New algorithms for convex cost tension problem with application to computer vision, [*Discrete Optimization*]{} [**6**]{} (2009), 378–393.
V. Kolmogorov, J. Thapper, and S. Živný, The power of linear programming for general-valued CSPs, [*SIAM Journal on Computing*]{} [**44**]{} (2015), 1–36.
L. Lovász, On some connectivity properties of Eulerian graphs, [*Acta Mathematica Academiae Scientiarum Hungaricae*]{} [**28**]{} (1976), 129–138.
S. T. McCormick and S. Fujishige, Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization, [*Mathematical Programming, Series A*]{} [**122**]{} (2010), 87–120.
K. Murota, Discrete convex analysis, [*Mathematical Programming*]{} [**83**]{} (1998), 313–371.
K. Murota, [*Discrete Convex Analysis*]{}, SIAM, Philadelphia, 2003.
K. Murota and A. Shioura, Exact bounds for steepest descent algorithms of L-convex function minimization, [*Operations Research Letters*]{} [**42**]{} (2014), 361–366.
L. Qi, Directed submodularity, ditroids and directed submodular flows, [*Mathematical Programming*]{} [**42**]{} (1988), 579–599.
A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time, [*Journal of Combinatorial Theory, Series B*]{} [**80**]{} (2000), 346–355.
B. C. Tansel, R. L. Francis, and T. J. Lowe, Location on networks I, II, [*Management Science*]{} [**29**]{} (1983), 498–511.
J. Thapper and S. Živný, The power of linear programming for valued CSPs, in: [*Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12)*]{}, 2012, pp. 669–678. D. Xu, E. Anshelevich, and M. Chiang, On survivable access network design: complexity and algorithms, in: [*Proceedings of the 27th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies (INFOCOM’08)*]{}, 2008, pp. 186–190. V. V. Vazirani, [*Approximation Algorithms*]{}, Springer-Verlag, Berlin, 2001.
[^1]: Instead of exponentially many conditions $x(\delta X) \geq r(s)$, consider a single commodity $(s, S\setminus \{s\})$-flow $\varphi_s$ under capacity $x$ with total flow value at least $r(s)$.
|
---
abstract: '[*Hubble Space Telescope*]{} Advanced Camera for Surveys (ACS) images are used to identify and study star cluster candidates in the nearby spiral galaxy M101. About 3000 round, slightly-resolved cluster candidates are identified in 10 ACS pointings covering an area of 106 arcmin$^2$. The cluster candidates’ color and size distributions are consistent with those of star clusters in other nearby spirals. The majority of the M101 candidates are blue and more likely to be associated with the galaxy’s spiral arms, implying that they are young. The galaxy-luminosity-normalized number of ‘young massive clusters’ in M101 is similar to that found in other spirals, as is the cluster density at a fiducial absolute magnitude. We confirm a previous finding that M101 has a large number of faint, red star clusters: if these are old globular clusters then this galaxy has a very large globular cluster population. More plausible is that the faint red clusters are reddened young clusters; their colors and luminosities are also consistent with this explanation.'
author:
- Pauline Barmby
- 'K.D. Kuntz'
- 'John P. Huchra'
- 'Jean P. Brodie'
title: 'Hubble Space Telescope Observations of Star Clusters in M101[^1] '
---
Introduction
============
Understanding the stellar populations of galaxies is key to untangling the mysteries of their formation and evolution. Because of our location in the Milky Way, getting an overall picture of the stellar populations is difficult. The nearby spiral galaxy M101 (NGC 5457) is the closest face-on spiral [Hubble type SAB(rs)cd; @rc3] and provides an excellent opportunity for resolution of stellar population details. Previous studies of stellar populations in M101 include work on the Cepheid variable stars [@kel96; @ste98], X–ray binaries [@kdk05; @mukai03], and novae [@scp00], among many others.
Star clusters are excellent tracers of stellar populations: they are much brighter than single stars and usually have small internal spreads in age and metallicity. Globular clusters in particular are believed to be indicators of galaxy history [e.g., @str05]. Globular cluster systems (GCSs) in spirals have been studied much less than their counterparts in ellipticals: spirals have fewer clusters per unit of galaxy light, and their irregular background light makes detection of clusters more challenging. The Milky Way GCS is the ‘gold standard’ for comparison to all others, but it is difficult to say whether it is a truly typical GCS since the sample of spiral galaxy GCSs available for comparison is small. Younger star cluster populations are better-studied in other galaxies [e.g., @lr99] than in the Milky Way; due to extinction, the census of young cluster populations in the Galaxy is far from complete [@bica03].
Studying clusters in edge-on spirals [e.g., @pg03] ameliorates the background light issue, but the nearest such galaxies are distant enough that their star clusters are unresolved; confirmation that candidates are true clusters is difficult. Recently, [*Hubble Space Telescope*]{} (HST) imaging has been used to search for star clusters in a number of low-inclination nearby spiral galaxies, including M81 [@cft1 Nantais et al., 2006 in preparation], M51 [@bik03; @lcw05], and these two galaxies plus M83, NGC 6946 and M101 [@cwl04]. The high spatial resolution of HST images aids in distinguishing true clusters from contaminants such as background galaxies and multiple star blends and improves photometric precision. The HST studies mentioned above have generally covered only small regions of these nearby galaxies using one or two HST pointings. Here we complement these works by presenting initial results from a study of star clusters found in ten HST/ACS fields in M101, covering a much larger fraction of the galaxy. Throughout this work we assume a distance to M101 of 6.7 Mpc [$m-M=29.13$; @hst_kp]. At this distance, one arcsecond subtends 32.5 pc.
Observations and cluster selection
==================================
Images of 10 fields in M101 (see Figure \[m101-fields\]) were taken with the Wide Field Camera of the Advanced Camera for Surveys [@acs] on HST, during the period 2002 November 13–16. Each field was observed for 900 s in the F435W filter and 720 s each in F555W and F814W; the exposures were ‘cosmic-ray-split’ but not dithered. The pipeline-processed, drizzled versions of the data retrieved in 2003 February were used in this analysis. The $S/N=10$ detection limit for point sources in these data is given by the ACS exposure time calculator as $B=25.7$, $V=25.4$, $I=25.0$. The total area covered by our survey is about 106 arcmin$^2$; this accounts for the overlap between the ACS pointings and the missing area in the ACS inter-chip gaps.
Even a casual inspection of the images reveals a wealth of detail (e.g., see Figure \[m101-detail\]), and an initial attempt at source extraction yielded tens of thousands of detected objects per field. The expected population of M101 star clusters includes a few hundred globular clusters and a few thousand younger clusters, so finding these requires care and attention. Although automatic selection can reduce the number of cluster candidates to a manageable one, we found that human judgment was required in the final selection.
Star clusters in M101 were expected to be slightly resolved in HST images: a typical Milky Way globular cluster half-light radius of 3.2 pc corresponds to about 2 ACS/WFC pixels. Unfortunately there are many other sources which are slightly resolved in the images, including background galaxies, H II regions, blended stars, etc. To understand what star clusters in M101 might look like, we modeled their appearance using HST images of M31 globular clusters. The M31 cluster images were taken with STIS as part of program GO-8640, designed to measure structural parameters for the clusters. We chose six clusters with a range of sizes and shapes, and modeled their appearance in ACS images of M101 using version 6.0 of TinyTIM [@ttim]. M31 is about 10 times closer than M101, so the STIS images were input to TinyTIM as 10-times oversampled images, then convolved with the HST/ACS point spread function using TinyTIM’s “scene-generation” facility. Appropriate PSFs for the three ACS filters were used, but no attempt was made to model the color-dependence of cluster appearance; the wide bandpass of the STIS 50CCD filter made this unnecessary. We computed PSFs and modeled artificial M101 clusters on a grid of 144 positions on the ACS WFC detectors.
The artificial clusters were used first to define the cluster-selection procedure. We chose $V$ magnitudes randomly from a Gaussian with mean 22.0 and dispersion 1.2, $B-V$ and $V-I$ colors from Gaussians with means of 0.72 and 0.96 and dispersion 1.2, and reddening from a uniform distribution with $0<E(B-V)<0.2$. The clusters were then added into the ACS images of field 1 (the innermost pointing) on their PSF grid positions. After experimenting with object detection and measurement schemes, we decided to use SExtractor [@ba96] for object detection and measurement,[^2] followed by FWHM measurements with the [iraf]{} task [apphot.radprof]{}. Final cluster candidate selection criteria were: objects which appear in all three of $B$, $V$, and $I$, with MAG\_BEST$<23.5$, CLASS\_STAR$<0.05$, and ELLIP$<0.5$ as measured on the $I$ images, and $2.2 < {\rm FWHM} < 6$ pixels as measured by [radprof]{}. Adding the magnitude criterion ensured that objects would be bright enough for shape measurements and star/galaxy classifications to be meaningful. Adding the FWHM criterion removed many extended sources which appeared to be galaxies, reducing the number of candidates to a manageable level while retaining most of the inserted artificial clusters. The upper limit on FWHM of 6 pixels corresponds to an effective radius $R_e=14.4$ pc (see Section \[sec:sizedist\]) at the distance of M101.
The actual cluster-finding procedure consisted of applying the above criteria to each of the ten fields. Before finding clusters, we inserted 20 artificial clusters in each field. The same scheme described above was used, except that the cluster locations were chosen randomly and the cluster image used was the one with the closest PSF location. After doing object detection and applying the cluster criteria, the remaining candidates were visually examined to remove objects which appeared to be background galaxies or blended stars. The visual inspection imparts some subjectivity to the selection procedure; however it proved to be necessary since we were unable to define an automatic selection procedure which was both complete (picking out most of the artificial clusters) and reliable (not picking too many obvious contaminants). The fraction of candidates removed by the visual selection ranged from almost 50% in the central field to about 30% in the outer fields.
The completeness of the selection procedure is not well-defined because of the subjective visual inspection. However, it can be estimated by considering the fraction of artificial clusters which survived the selection procedure. This ranged from 70% in the central field to 90% in several of the outer fields, with an average of 83%. A few clusters were missed because they were inserted at magnitudes beyond the cutoff; most of the rest were deleted by the visual inspection procedure, usually because they were not quite compact enough to look like distinct objects. The final result of the cluster selection procedure was a list of 2920 candidates. Images of a few candidates chosen at random from the final list are shown in Figure \[samp-clust\]. The cluster candidates are reasonably round and appear resolved compared to nearby stars; further analysis of their light distribution is in Section \[sec:sizedist\].
Photometry of the cluster candidates was extracted from the SExtractor output. Total magnitudes were measured with the SExtractor MAG\_AUTO estimator. Colors were measured using aperture magnitudes in 2.5 pixel radius apertures. The aperture magnitudes were corrected to large apertures using aperture corrections derived from the clusters themselves; since the clusters are slightly resolved, corrections based on the instrument point spread function alone are inappropriate. Corrections used were $-0.905$, $-0.941$ , and $-1.055$ magnitudes in the F435W, F555W, and F814W filters, respectively. Transformations to the standard Johnson-Cousins system were made using the equations and parameters in @acs_cal; no corrections for charge transfer efficiency were made, but these are expected to be small since the observations were made early in the life of ACS.
We used the H I column density map given by @kdk03 and a dust-to-gas ratio of $E(B-V) = 1.72\times10^{-22}$N(HI+H$_2$) [@boh78] to de-redden the colors of the M101 cluster candidates. The column density map reflects the total absorbing column in the M101 disk, so this correction will over-estimate the reddening for objects in front of the disk plane (i.e., some globular clusters.) Without further information, there is no way to tell which clusters are in front of the disk and which are behind, and the reddening corrections are mostly small (the median value in the area covered by the ACS images is $E(B-V) = 0.05$, with a maximum value of 0.26). Galactic extinction in the direction of M101 is very low [$E(B-V)<0.01$; @cwl04] so we do not correct for it.
We can compare the results of our selection with the study of @bks96 [hereafter BKS]. These authors performed a visual search for clusters in one HST/WFPC2 field which is contained within our central ACS pointing.[^3] Their 43 cluster candidates have a median $V$ magnitude of 21.5. Two of their candidates fall in the gap between the ACS detectors, and two are not identifiable on the WFPC2 image. In the same area searched by BKS, our cluster selection procedure picked 232 candidates with a median magnitude of $V=22.8$. Twenty-two of the remaining 39 BKS candidates (56%) are in our candidate list. Visual examination of the 17 BKS clusters not in our list shows them to have a range of morphologies, but they are generally less round and less isolated than the clusters on our list (see also Figure \[m101-detail\]). As indicated by the median magnitudes, our candidates are generally fainter than those of BKS. Comparison of photometry for objects in common shows a small offset: median differences between their photometry and ours are $\Delta V = 0.16\pm 0.04$, $\Delta(B-V) = -0.14\pm 0.03$, and $\Delta(V-I) = +0.08\pm 0.03$. BKS state that their magnitudes are uncertain by $\geq 0.1$ mag, so we do not believe the offset indicates a serious problem in our photometry.
Analysis
========
Star cluster system studies have generally focused on two separate populations: the old, globular clusters or the young, massive clusters. Elliptical galaxies, of course, have only the first type of cluster, while the clusters studied in galaxy mergers exemplify the second. The color distribution of the M101 cluster candidates is expected to reflect their age distribution: while colors of globular clusters are often used as indicators of metallicity [e.g. @kw98], many of the M101 candidates are far too blue to be old, metal-poor clusters. In the following analysis, we will use color as a simple observational distinction between older and younger clusters, to ease comparison with other star cluster studies. $B-V=0.45$ is the red limit used by @lr99 for their studies of ‘Young Massive Clusters’, and is also close to the blue limit of the Milky Way globular cluster $(B-V)_0$ distribution, so we adopt $(B-V)_0=0.45$ as the dividing line between old and young clusters. It should be remembered, however, that the age distribution of star clusters in M101 is not necessarily bimodal, and this division is somewhat arbitrary. More precise age estimates will be enabled with the use of UV and/or H$\alpha$ imaging; such data already exist for a portion of the observed field and will be utilized in a future paper.
Of the 2920 star cluster candidates in our sample, 1877 (65%) are blue, with $(B-V)_0<0.45$. The effect of our $I=23.5$ magnitude limit is such that our cluster sample contains faint clusters only if they are red. The bluest clusters in our sample have $V-I \approx -0.5$, so the sample is not color-biased above $V=23$ $(M_V=-6.1)$. To this limit, there are 1715 cluster candidates of which 1260 (73%) are blue. Figure \[color-spat\] shows the spatial distribution of the M101 candidates, both the complete sample and the bright ($V<23$) red and blue subsamples. The spiral pattern of the galaxy is more apparent in the blue clusters. This is consistent with them being younger and associated with star formation in the spiral arms, and indeed the bluest clusters ($(B-V)<0.2$) trace the arms even more clearly.
Color and luminosity distributions
----------------------------------
Observed color distributions for the candidates are shown in Figure \[2color\]. The left panel of this figure compares the colors for M101 clusters to those in other nearby spirals: M81 [@cft1], M33 [@m98], and M51 [@bik03]; these colors have not been corrected for reddening. The color distribution of M101 candidates is most similar to that of the M33 candidates, which is unsurprising given that the two galaxies are the same Hubble type and have a similar specific star formation rate. M101 has fewer blue clusters than M51 and fewer red clusters than M81. This is broadly consistent with a picture in which the proportion of blue to red clusters reflects the recent star formation rate (M51 might be expected to have enhanced star formation due to encounters with its companion), and the number of red clusters is proportional to the galaxy bulge mass (which is larger for M81, an earlier-type galaxy than M101). @cft2 interpret the separation of M81 cluster colors into two groups (at $B-V\approx0.5$ and $V-I\approx1.0$) as a separation in age. No such clear separation is apparent in the M101 cluster candidates.
In the right panel of Figure \[2color\] we compare colors of red M101 cluster candidates to those of globular clusters in the Milky Way [@h96], M31 [@b00], and globular cluster candidates in M101 itself from @cwl04. The median colors of M101 cluster candidates are clearly consistent with those of the globulars in the other galaxies, indicating that we have indeed detected a population of old globular clusters in M101. The larger scatter of the M101 colors presumably reflects the larger photometric or reddening-correction errors.
Figure \[cmd\] compares the joint distribution of colors and absolute magnitudes for M101, M81, and the M101 clusters found by @cwl04. @cwl04 were specifically attempting to select old, globular clusters in M101, while @cft1 were searching only for ‘compact’ star clusters in M81 with no selection on age. The effect of our $I$-band magnitude cut is particularly clear in the right-hand panel of this figure: our cluster sample is missing faint, blue clusters. It is clear that the excess of red clusters in M81 compared to M101 is mostly at brighter cluster luminosities: brighter than $M_V=-7.4$ (the expected peak of the globular cluster luminosity function), the M81 clusters are mostly red, while the M101 clusters are mostly blue. We also confirm the detection by @cwl04 of a number of faint, red clusters in M101. Some of these could be faint background galaxies, especially ellipticals or the bulges of spirals whose disks are too faint to observe. Such contamination is unlikely to account for all of the faint red clusters, whose nature is discussed further in Section \[sec:spatdist\].
The cluster candidate luminosity distribution is shown in Figure \[lum-dist\], for the full sample and the red and blue subsamples. The strong fall-off at $V>23$ ($M_V>-6.1$) is due to our imposed magnitude limit (see above). Brighter than this limit, we find that the blue clusters are about 0.25 mag brighter in the median than the red clusters. From population synthesis models, such a difference is consistent with the blue clusters being younger, by a few Gyr if both populations are $\gtrsim 8$ Gyr old, or less if the clusters are younger. Inferred ages are of course strongly dependent on additional factors such as metallicity, reddening, and initial mass function. As discussed below, there are many more red clusters than the expected number of globular clusters for a galaxy of M101’s luminosity: the smooth curve in the left panel of the figure shows a ‘standard’ globular cluster luminosity function (a Gaussian with mean $M_V=-7.4$ and standard deviation $\sigma=1.3$) scaled to the number of clusters with $M_V<-7.4$ (also see Section \[sec:spatdist\]). The bright end of the LF for the full sample is consistent with the power-law distribution of luminosity $dN(L)/dL \propto L^{-2}$ ($N(L) \propto L^{-1}$), similar to the distributions seen for young clusters in mergers [@whi99; @mwsf97] and [*HST*]{}-identified (not necessarily young) clusters in other spirals [@lar02b], as well as the bright end of the GCLF for the Milky Way and M31 [@hp94; @mcl94].
Cluster populations and spatial distribution\[sec:spatdist\]
------------------------------------------------------------
Do the blue and red subsamples of M101 correspond to ‘young’ and ‘globular’ cluster groups? One way to find out is to compare the number of clusters per unit galaxy luminosity in M101 to values for other galaxies. The total magnitudes of M101 as given by @rc3 are $B_T=8.31$, $V_T=7.86$, corresponding to luminosities of $M_B=-20.82$, $M_V=-21.27$. The ‘young massive cluster’ (YMC) criterion used by @lr99 in their analysis of such clusters in nearby spirals includes both color ($B-V<0.45$) and faint magnitude limits, $M_V<-8.5$ (the latter to avoid contamination by individual massive stars). 140 of our candidates meet these criteria, giving a value of $T_N=N_{\rm YMC} \times 10^{0.4(M_B+15)}=0.66$. This is a fairly typical value for spirals in the @lr99 survey: although M101 has more YMCs than all but one of the @lr99 galaxies, it is also more luminous than any of them. Only the brightest M101 blue clusters are YMCs. To compare the overall cluster population to other spirals we can use ${\Sigma}_{\rm cl}^{-8}$, the number density of clusters per kpc$^2$ per 1 magnitude bin at $M_V=-8$ [defined by @lar02b]. M101 has ${\Sigma}_{\rm cl}^{-8}=1.0$, which is quite typical of spirals.
Globular cluster populations in different galaxies are usually compared by means of the specific frequency, defined as $S_N= N_{\rm GC} 10^{0.4(M_V+15)}$. If we assume that all 1043 red objects are globular clusters, the corresponding globular cluster specific frequency of M101 is $S_N=3.0$. This value is quite high compared to other spiral galaxies, which typically have values in the range 0.5–1.0 [@pmb03; @cwl04], so assuming M101 to be a typical spiral would tend to support the argument that the faint red clusters are not globulars. As another estimate of $S_N$ in M101, we can [*assume*]{} that the M101 globular clusters have the usual Gaussian luminosity function with a peak at $M_V=-7.4$ [following @cwl04], that our cluster selection is complete to this magnitude limit, and that all red clusters brighter than this limit are true globulars. There are 160 such clusters, so if the luminosity function assumption is correct, $S_N=(320/1043)\times3.0 = 0.9$, a much more typical value for spiral galaxies, although still higher than the $0.5\pm0.2$ found for late-type spirals by @cwl04. These authors found $S_N= 0.4\pm0.1$ for M101 specifically; however, this is an extrapolation from a small number of detected globulars (29).
Are M101’s ‘excess’ faint red clusters low-mass old globulars which (unlike in earlier-type spirals) have not been destroyed by dynamical effects? Or are they instead reddened young clusters? At least some are likely to be true old clusters, since @cwl04 found a similar population after using $U$-band information to eliminate younger clusters from their sample. However, since we use only $BVI$ colors, we suspect that the majority of our faint red clusters are likely to be reddened younger clusters. The HI map upon which the column density map is based has a resolution of $\sim15\arcsec$ (the beam was $13\arcsec\times16\arcsec$) while the resolution of the 12m CO map was $\sim55\arcsec$. Since young clusters are often associated with current star-formation and localized concentrations of neutral and molecular gas, the low spatial resolution of the column density map could cause the reddening of younger clusters to be underestimated. If we have underestimated the reddening of some clusters, their absolute magnitudes would also be underestimated: correcting for this effect would then imply that the faint red clusters were intrinsically brighter than the faint blue clusters. This is a selection effect rather than a physical difference, however: more heavily-reddened clusters must be intrinsically brighter to be observable [for a discussion of this effect in M31, see @bpb02]. Given the similarity of the luminosity functions of faint red and blue clusters (see previous section), we believe it more likely that the faint red clusters are mostly reddened younger clusters.
Next we consider the spatial distribution of the M101 clusters. Using the results of the artificial cluster experiments described above, we made a ‘completeness map’, which accounts for both the spatially-varying completeness and the area on the sky actually observed as a function of distance from the center of M101. As Figure \[m101-fields\] shows, our observations cover most of the central 5 (9.75 kpc) radius of the galaxy, but only a small fraction of the region between 5 and 10 arcmin. In a set of annular bins, we counted the observed number of clusters and divided by the completeness-corrected, observed area of the annuli. The cluster candidate surface density distribution is shown in Figure \[surf-dens\]. For comparison we also show the surface density distributions of young massive clusters in four spirals from @lr99 [their Figure 8], and of Milky Way globular clusters, using positions from @h96 projected on to the plane of the Galaxy.
The overall normalization of the cluster surface density distribution in M101 is higher than in the Milky Way. This is not unexpected since the latter includes only globular clusters. However, the shape of the distribution is also different in M101: while the Milky Way globulars show a power-law distribution $n(R)\propto R^{-\alpha}, \alpha\sim-1.5$, the M101 and young massive cluster distributions look much more like King profiles, with a constant density core and rapid fall-off beyond. This is consistent with the M101 and other young massive clusters mostly belonging to the galaxy disks rather than the more extended halo usually identified with globular clusters.
Size distribution\[sec:sizedist\]
---------------------------------
We measured cluster candidate sizes using the [ishape]{} program [@lar99]. This software fits a number of PSF-convolved model light distributions to the images of individual objects. We fit a King model with concentration index $c=30$, for the most direct comparison with the results of @lar99, although the derived effective radius is not particularly sensitive to the model chosen. We fit only circular models, as experiments showed that using elliptical light models did not significantly change the measured sizes but did increase the computing time by a factor of $\sim 3$. The distribution of cluster candidate effective radii $R_e$ (the radius which contains half of the integrated light) is shown in Figure \[size-dist\]. The median $R_e$ was 2.2 ACS pixels, or 3.5 pc, which confirms that our selection procedure did indeed select mostly slightly extended objects. Fewer than 10% of our candidates were flagged as ‘possibly stellar’ by [ishape]{}.
There is a difference in the size distributions of red and blue clusters. The red clusters, with a median $R_e=4.1$ pc, are slightly larger than the blue clusters, whose median size is $3.2$ pc. Dynamical studies of star cluster evolution [e.g. @go97] show that large clusters are more easily destroyed, so we expected that the red (older) clusters would be slightly smaller on average than the blue (younger). However, since the clusters are only marginally resolved, the difference between the two groups is not of major significance and could reflect contamination of the red cluster sample by younger clusters or galaxies. The range of blue cluster candidate sizes is similar to the range found by @lar99 for ‘young massive clusters’ and is also comparable to the median half-light radius for Milky Way globular clusters $R_e=3.2$ pc [@h96].
Summary
=======
We have presented the results from an initial search for star clusters in Hubble Space Telescope/Advanced Camera for Surveys images of the nearby giant spiral M101. Defining a reliable sample of cluster candidates in these complex, crowded images proved to be a challenge. Our final cluster candidate list contains nearly 3000 objects of which a majority are blue and associated with the galaxy’s spiral arms. These cluster candidates have color distributions similar to candidates and confirmed clusters in other spirals including the Milky Way, M31, M81, M33, and M51. M101 has fewer bright, red clusters than the earlier-type spiral M81, but many more faint red clusters [as originally noted by @cwl04]. If all of these faint red clusters were globulars, M101 would have a much higher globular cluster specific frequency and a much different globular cluster luminosity function from other galaxies. We suggest that many of the red M101 clusters are likely to be reddened young clusters, which is supported by the consistency of the overall cluster luminosity distribution and fiducial cluster density with those found in other spirals by @lar02b. The spatial distribution of the M101 clusters shows a relatively flat core and a steep drop-off at larger galactocentric distances, suggesting that the clusters are associated with the galaxy disk rather than its halo. The size distributions of the M101 cluster candidates show them to be slightly resolved, as expected, and are consistent with cluster sizes measured in other galaxies.
Support for program GO-9490 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.
Barmby, P. 2003, in Extragalactic Globular Cluster Systems, ed. M. Kissler-Patig (Berlin: Springer-Verlag), 143
Barmby, P., Huchra, J. P., Brodie, J. P., Forbes, D. A., Schroder, L. L., & Grillmair, C. J. 2000, , 119, 727
, P., [Perrett]{}, K. M., & [Bridges]{}, T. J. 2002, , 329, 461
, E. & [Arnouts]{}, S. 1996, , 117, 393
, E., [Dutra]{}, C. M., [Soares]{}, J., & [Barbuy]{}, B. 2003, , 404, 223
, A., [Lamers]{}, H. J. G. L. M., [Bastian]{}, N., [Panagia]{}, N., & [Romaniello]{}, M. 2003, , 397, 473
Bohlin, R., Savage, B., & Drake, J. 1978, , 224, 132
Bresolin, F., Kennicutt, R. C., & Stetson, P. B. 1996, , 112, 1009
, R., [Ford]{}, H. C., & [Tsvetanov]{}, Z. 2001a, , 122, 1330
, R., [Tsvetanov]{}, Z., & [Ford]{}, H. C. 2001b, , 122, 1342
, R., [Whitmore]{}, B., & [Lee]{}, M. G. 2004, , 611, 220
, G., [de Vaucouleurs]{}, A., [Corwin]{}, H. G., [Buta]{}, R. J., [Paturel]{}, G., & [Fouque]{}, P. 1991, [Third Reference Catalogue of Bright Galaxies (RC3)]{} (New York: Springer-Verlag)
Ford, H. [et al.]{} 2002, SPIE, 4854, 81
, W. L. [et al.]{} 2001, , 553, 47
, O. Y. & [Ostriker]{}, J. P. 1997, , 474, 223
, P., [Strader]{}, J., [Brenneman]{}, L., [Kissler-Patig]{}, M., [Minniti]{}, D., & [Huizinga]{}, J. 2003, , 343, 665
Harris, W. E. 1996, , 112, 1487
, W. E. & [Pudritz]{}, R. E. 1994, , 429, 177
, D. D. [et al.]{} 1996, , 463, 26
, J. 1995, in Astronomical Data Analysis Software and Systems IV, ed. R. Shaw, H. Payne, & J. Hayes, ASP Conf. Ser., 349
, A. & [Whitmore]{}, B. C. 1998, , 116, 2841
, K. D., [Gruendl]{}, R. A., [Chu]{}, Y.-H., [Chen]{}, C.-H. R., [Still]{}, M., [Mukai]{}, K., & [Mushotzky]{}, R. F. 2005, , 620, L31
, K. D., [Snowden]{}, S. L., [Pence]{}, W. D., & [Mukai]{}, K. 2003, , 588, 264
, S. S. 1999, , 139, 393
, S. S. 2002, , 124, 1393
, S. S. & [Richtler]{}, T. 1999, , 345, 59
, M. G., [Chandar]{}, R., & [Whitmore]{}, B. C. 2005, , 130, 2128
, D. E. 1994, , 106, 47
, B. W., [Whitmore]{}, B. C., [Schweizer]{}, F., & [Fall]{}, S. M. 1997, , 114, 2381
Mochejska, B. J., Kaluzny, J., Krockenberger, M., Sasselov, D. D., & Stanek, K. Z. 1998, Acta Astronomica, 48, 455
, K., [Pence]{}, W. D., [Snowden]{}, S. L., & [Kuntz]{}, K. D. 2003, , 582, 184
, A. W., [Ciardullo]{}, R., & [Pritchet]{}, C. J. 2000, , 530, 193
, M. [et al.]{} 2005, , 117, 1049
Stetson, P. B. [et al.]{} 1998, , 508, 491
, J., [Brodie]{}, J. P., [Cenarro]{}, A. J., [Beasley]{}, M. A., & [Forbes]{}, D. A. 2005, , 130, 1315
, B. C., [Zhang]{}, Q., [Leitherer]{}, C., [Fall]{}, S. M., [Schweizer]{}, F., & [Miller]{}, B. W. 1999, , 118, 1551
[^1]: Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs \#8640 and \#9490.
[^2]: SExtractor parameters were set to the following values: SEEING\_FWHM 01, DETECT\_MINAREA=4, DETECT\_THRESH=1.5, and the images were convolved with a $5\times5$ pixel Gaussian filter before object detection.
[^3]: We discovered that the coordinates given for BKS’ clusters 30–43 do not fall on any obvious objects in the ACS image; it appears that these objects fall on the WF4 chip, but their RA and Dec were computed using the world coordinate system for the PC1 image. With the correct transformations all except two of the coordinate pairs fall on obvious cluster candidates in the WF4 image.
|
---
abstract: 'Quantum error correction is required to compensate for the fragility of the state of a quantum computer. We report the first experimental implementations of quantum error correction and confirm the expected state stabilization. In NMR computing, however, a net improvement in the signal-to-noise would require very high polarization. The experiment implemented the 3-bit code for phase errors in liquid state state NMR.'
address: |
$^1$Dept. of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139\
$^2$Los Alamos National Laboratory,Los Alamos, NM 87545\
$^3$BCMP, Harvard Medical School, 240 Longwood Ave., Boston, MA 02115
author:
- 'D. G. Cory$^1$, W. Mass$^1$, M. Price$^1$ E. Knill$^2$, R. Laflamme$^2$ W. H. Zurek$^2$ T. F. Havel$^3$ and S. S. Somaroo$^3$'
title: Experimental Quantum Error Correction
---
Quantum computers exploit the superposition principle to solve some problems much more efficiently than any known algorithm for their classical counterparts. These problems include factoring large numbers[@shor:qc1994a] (thereby breaking public key cryptography), combinatorial searching[@grover:qc1995a] and simulations of quantum systems[@feynman:qc1986a; @lloyd:qc1996a; @zalka:qc1996b]. Exploiting the power of quantum computation was thought to be physically impossible due to the extreme fragility of quantum information[@landauer:qc1995a; @unruh:qc1995a]. This judgment was demonstrated to be overly pessimistic when quantum error-correction techniques[@shor:qc1995b; @steane:qc1995a; @shor:qc1996a] were found to protect quantum information against corruption due to imperfect control and decoherence. It is now known that for physically reasonable models of decoherence a quantum computation can be as long as desired with arbitrarily accurate answers, provided the error rate is below a threshold value[@knill:qc1998a; @aharonov:qc1996a; @preskill:qc1998a; @kitaev:qc1996a]. Thus decoherence and imprecision are no longer considered insurmountable obstacles to realizing a quantum computer.
The chief remaining obstacle to quantum computing is the difficulty of finding suitable physical systems whose quantum states can be accurately controlled. Devices based on ion traps [@cirac:qc1995a] have so far been limited to two bits[@monroe:qc1995a]. Recently, liquid state NMR techniques have been shown to be capable of quantum computations with at least three bits [@cory:qc1997b; @laflamme:qc1997a]. This makes it possible, for the first time, to experimentally implement the simplest quantum error-correcting codes, and so test these ideas in physical systems with a variety of decoherence, dephasing and relaxation phenomena.
In room temperature liquid state NMR, one can coherently manipulate the internal states of the coupled spin $1\over 2$ nuclei in each of an ensemble of molecules subject to a large external magnetic field. Although the ensemble nature of the system and the small energy of each spin imply that the set of accessible states is highly mixed, it has been shown that experimental methods exist that can be used to isolate the pure state behavior of the system, thus permitting limited application of NMR to quantum computation[@cory:qc1997a; @chuang:qc1997a].
Here we describe the implementation of a quantum error-correcting code which compensates for small phase errors due to fluctuations in the local magnetic field. The behavior of this code was measured for two systems: The ${}^{13}$C labeled carbons in alanine subject to the correlated phase errors induced by diffusion in a pulsed magnetic field gradient, and the proton and two labeled carbons in trichloroethylene (TCE) subject to its natural relaxation processes. In alanine, we observed correction of first-order errors for a given input state, while in TCE we demonstrated state preservation of an arbitrary state by error correction.
In NMR loss of phase coherence may be both due to bona fide irreversible decoherence[@zurek:qc1991] and due to dephasing which can be reversed by spin echo. Both lead to symptoms which, in an ensemble, are identical from the standpoint of quantum computation. Both can be corrected using the same scheme we shall demonstrate below.
It is important to comment on the applicability of this technique to ensemble quantum computing. Although our experiments validate the usefulness of error correction for quantum computing with pure states, there is a substantial loss of signal associated with the use of ancilla spins in weakly polarized systems. We argue that in this setting, the loss of signal involved in exploiting ancillas removes any advantage for computation gained by error correction, at least unless the system is sufficiently polarized to enable the generation of nearly pure states. Nevertheless, our experiments demonstrate that error-correcting codes can be implemented, and that they behave as predicted. The simple three-bit quantum error-correcting code used here is designed to compensate to first order for small random phase fluctuations. These fluctuations constitute a random evolution of the state $$\begin{aligned}
\label{eq:phase}
{| b_1b_2b_3 \rangle} & ~\longrightarrow~ &
e^{-i(\theta_1{\mbox{\boldmath$\scriptstyle\sigma$}}_{\rm z}^{1}+\theta_2{\mbox{\boldmath$\scriptstyle\sigma$}}_{\rm
z}^{2}+\theta_3{\mbox{\boldmath$\scriptstyle\sigma$}}_{\rm z}^{3})}
{| b_1b_2b_3 \rangle}
\\ && =~ \nonumber
e^{i( (-1)^{b_1}\theta_1+(-1)^{b_2}\theta_2+(-1)^{b_3}\theta_3)}
{| b_1b_2b_3 \rangle} ~,\end{aligned}$$ where $b_i$ is $0$ or $1$, $\theta_i$ is a random phase variable, and ${\mbox{\boldmath$\sigma$}}_z^{i}$ is the Pauli matrix acting on the $i$’th spin. The $\theta_i$ depend on the error rates in the model, which is described in detail below.
The error-correcting code itself is a phase variant of the classical three bit majority code with a decoding technique that preserves the quantum information in the encoded state. Let ${| + \rangle} = ({| 0 \rangle}+{| 1 \rangle})/\sqrt{2}$ and ${| - \rangle} = ({| 0 \rangle}-{| 1 \rangle})/\sqrt{2}$. The state $(\alpha{| 000 \rangle}+\beta{| 100 \rangle})$ is encoded as $\alpha{| +\,+\,+ \rangle}+\beta{| -\,-\,- \rangle}$ by a unitary transformation. The first-order expansion of the operator in Eq. (\[eq:phase\]) in the small random phases is $${\bf 1} - i\theta_1{\mbox{\boldmath$\sigma$}}_z^{1} - i\theta_2{\mbox{\boldmath$\sigma$}}_z^{2}
- i\theta_3{\mbox{\boldmath$\sigma$}}_z^{3} ~,$$ which evolves the encoded state to $$\begin{aligned}
\alpha{| ++\,+ \rangle}+\beta{| --\,- \rangle}&\rightarrow& ~~
\alpha{| ++\,+ \rangle}+\beta{| --\,- \rangle}\\ \nonumber && - i \theta_1
(\alpha{| -+\,+ \rangle}+\beta{| +-\,- \rangle})\\ \nonumber && - i \theta_2
(\alpha{| +-\,+ \rangle}+\beta{| -+\,- \rangle})\\ \nonumber && - i \theta_3
(\alpha{| ++\,- \rangle}+\beta{| --\,+ \rangle}) ~. \nonumber\end{aligned}$$
The different errors map the encoded state into orthogonal subspaces. Thus one can determine which error occurred without destroying the encoded quantum information. This is accomplished by a measurement which reveals the subspace the state has moved into. After decoding, the original state of the first spin can then be restored by a unitary transformation, while the other two spins contain information (the syndrome) about the error which occurred. A network which accomplishes the encoding, decoding and error-correction steps is shown in Figure \[figure:dec\_net\].
In NMR experiments, non-unitary processes are classified as spin-lattice and spin-spin relaxation [@OldTestiment; @NewTestiment]. The former involves an exchange of energy with the non-spin degrees of freedom (the lattice), and returns the spins to thermal equilibrium. The latter leads to a loss of phase coherence. For spin $\frac{1}{2}$ nuclei, both processes are due to fluctuating local magnetic fields. The three spin code corrects for errors due to locally fluctuating fields along the z axis.
We focus on a weakly coupled three-spin system where the strongest contribution to coherence loss is from external fields which contribute the Hamiltonian $${\cal H}_{\rm R}~\equiv~\gamma^1 {\bf I}^1 \cdot {\bf B}^1(t) +
\gamma^2 {\bf I}^2 \cdot {\bf B}^2(t) +
\gamma^2 {\bf I}^3 \cdot {\bf B}^3(t)~,$$ where ${\bf I} = ({\bf I}_{\rm x},{\bf I}_{\rm y},{\bf I}_{\rm z})$ and ${\bf I}_u = {1\over 2}{\mbox{\boldmath$\sigma$}}_u$ ($u={\rm x,y,z}$). In the case of slowly varying external fields the induced random phase fluctuations are identical to those described in Eq. (\[eq:phase\]). As a result, the off-diagonal elements of the density matrix decay exponentially at a rate which depends on the fields ${\bf B}^k$ at each spin, their gyromagnetic ratios $\gamma^k$, the coherence order and the zero frequency components of the spectral densities of the fields. The “coherence order” is the difference between the total angular momenta along the z-axis of the two states ${| b \rangle}$, ${| b' \rangle}$ (in units of $\hbar/2$)[@gradientRef]. To obtain a clean demonstration of error correction, a simple error model was implemented precisely in the case of alanine. This implementation used the random molecular motion induced by diffusion in a constant field gradient to mimic the effect of a slowly varying random field. This is achieved by turning on an external field gradient $\nabla_{\rm z} {\bf B} = \partial B_{\rm z} / \partial z$ across the sample for a time $\delta$. This modifies the magnetization in the sample with a phase varying linearly along the z direction according to $\partial\phi/\partial z = n \delta \gamma \partial B_{\rm z}/\partial z$, where $n$ is the coherence order of the density matrix element and $\gamma$ is the gyromagnetic ratio. A reverse gradient is used to refocus the magnetization after allowing molecular diffusion to take place for amount of time $t$. As a result of random spin displacement $\Delta z$, the phases of the spins are not returned to their original values but are randomly modified by $(n\delta\gamma\partial B_{\rm z}/\partial z)\Delta z$. For a gaussian displacement profile with a width of $\sqrt{2Dt}$, the effective decoherence time of this process is proportional to the diffusion constant $D$ as well as to the square of the coherence order $n$ [@gradientRef]: $$\frac{1}{\tau}
~=~ \left( \frac{\partial\phi}{\partial z}\right)^2 D
~=~\gamma^2 (\nabla_{\rm z} {\bf B})^2 n^2 \delta^2 D ~.$$ This artificially induced “decoherence” in the alanine experiments is an example of completely correlated phase scrambling. This occurs naturally if all the spins have equal gyromagnetic ratios in the slow motion regime. We used TCE to demonstrate error-correction in the presence of the natural decoherence and dephasing of a molecule where $T_2\ll T_1$.
Most NMR experiments are described using the product operator formalism [@SoEiLeBoEr:83]. This formalism describes the state as a sum of products of the operators ${\bf I}_x^k$, ${\bf I}_y^k$, ${\bf I}_z^k$. The identity component of such a sum is the same for any state and is usually suppressed to yield the “deviation” (traceless) density matrix. The effect of error-correction can be understood from the point of view of this formalism. As an example, consider encoding the state ${\bf I}_{\rm z}^1$ using two ancillas initially in their ground states. Up to a constant factor, the initial state is described by $$\begin{aligned}
\mbox{\boldmath$\rho$}_{\sf A} &=&
{\bf I}_{\rm z}^1
({\mbox{$\frac{1}{2}$}} {\bf 1} + {\bf I}_{\rm z}^2)
({\mbox{$\frac{1}{2}$}} {\bf 1} + {\bf I}_{\rm z}^3)\nonumber\\
&=& {\mbox{$\frac{1}{4}$}} {\bf I}_{\rm z}^1 +
{\mbox{$\frac{1}{2}$}} {\bf I}_{\rm z}^1 {\bf I}_{\rm z}^2 +
{\mbox{$\frac{1}{2}$}} {\bf I}_{\rm z}^1 {\bf I}_{\rm z}^3 +
{\bf I}_{\rm z}^1 {\bf I}_{\rm}^2 {\bf I}_{\rm z}^3 ~.
\label{eq:input_states}\label{eq:rhoA}\end{aligned}$$ The density matrix ${\mbox{\boldmath$\rho$}}_{\sf A}$ consists of an incoherent sum of four terms.
After encoding the state is $${\mbox{\boldmath$\rho$}}_{\mathsf B} ~\equiv~ {\mbox{$\frac{1}{4}$}} \left(
{\bf I}_{\mathrm x}^1 + {\bf I}_{\mathrm x}^2 +
{\bf I}_{\mathrm x}^3 + 4\, {\bf I}_{\mathrm x}^1
{\bf I}_{\mathrm x}^2 {\bf I}_{\mathrm x}^3 \right) ~.$$ In the case of completely correlated phase errors, this decays as $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_{\sf C} &~\equiv~& {\mbox{$\frac{1}{4}$}} \left(
\left( {\bf I}_{\rm x}^1 + {\bf I}_{\rm x}^2 +
{\bf I}_{\rm x}^3 \right) e^{-t / \tau} \right.\\&&\left.\mbox{}\hspace{3.5pt} +
\left( 3 {\bf I}_{\rm x}^1 {\bf I}_{\rm x}^2 {\bf I}_{\rm x}^3
+ {\bf I}_{\rm x}^1 {\bf I}_{\rm y}^2 {\bf I}_{\rm y}^3
+ {\bf I}_{\rm y}^1 {\bf I}_{\rm x}^2 {\bf I}_{\rm y}^3
+ {\bf I}_{\rm y}^1 {\bf I}_{\rm y}^2 {\bf I}_{\rm x}^3 \right)
e^{-t / \tau} \right. \nonumber\\ && \left. \mbox{}\hspace{3.5pt} +
\left( {\bf I}_{\rm x}^1 {\bf I}_{\rm x}^2 {\bf I}_{\rm x}^3
- {\bf I}_{\rm x}^1 {\bf I}_{\rm y}^2 {\bf I}_{\rm y}^3
- {\bf I}_{\rm y}^1 {\bf I}_{\rm x}^2 {\bf I}_{\rm y}^3
- {\bf I}_{\rm y}^1 {\bf I}_{\rm y}^2 {\bf I}_{\rm x}^3 \right)
e^{-9 t / \tau} \right) ~. \nonumber\end{aligned}$$ Decoding and error correction mixes these states together so as to cancel the initial decay of the first spin. The reduced density matrix for the first spin becomes $$\label{eq:rhoE}
{\mbox{\boldmath$\rho$}}_{\sf E}^1 ~\equiv~
{\mbox{$\frac{1}{8}$}} {\bf I}_{\mathrm z}^1
( 9 e^{-t/\tau} - e^{-9t/\tau} ) ~\approx~
{\bf I}_{\mathrm z}^1
( 1 - {9\over 2} t^2 / \tau^2 + \cdots ) ~.$$ The effect of error correction can be seen from the absence of terms depending linearly on $t$. Similar results can be obtained for other initial states and also for uncorrelated phase errors.
In the alanine experiments, each of the four product operators in the sum of Eq. \[eq:input\_states\] was realized in a separate experiment, and the final state after encoding and decoding inferred by adding the results. The loss of polarization over time in each product operator was measured explicitly in each experiment. The results were added computationally to simulate the effect of the Toffoli gate and are shown in Figure \[figure:ala\_res\]. The components of ${\bf
I}_{\mathrm z}^1 {\bf I}_{\mathrm z}^2 {\bf I}_{\mathrm z}^3$ which evolved as a single and a triple coherence when encoded were separated by gradient labeling [@gradientRef]. The initial slopes of the decay curves for each operator were estimated by linear square fit to the logarithm of the measured intensities. The values obtained are $-0.0034$ ( ${\bf
I}_{\mathrm z}^1$), $-0.0030$ ($2 {\bf I}_{\mathrm z}^1 {\bf
I}_{\mathrm z}^2$), $-0.0047$ ($2 {\bf I}_{\mathrm z}^1 {\bf
I}_{\mathrm z}^3$), $-0.0038$ ($4 {\bf I}_{\mathrm z}^1 {\bf
I}_{\mathrm z}^2 {\bf I}_{\mathrm z}^3$, single coherence) and $-0.0395$ ($4 {\bf I}_{\mathrm z}^1 {\bf I}_{\mathrm z}^2 {\bf
I}_{\mathrm z}^3$, triple coherence) When the slopes are added as required for error-correction with a pseudopure input, we get a measured net slope of $0.00081$, which should be compared to the slopes for the single coherences whose average is $-0.0037$. The net curve has quadratic behavior for small delays to within experimental error.
The goal of our experiments with TCE was to establish the behavior of encoding/decoding and error correction on all possible initial states subject to the natural decoherence and dephasing. The spins were prepared in the states $$\rho({1\over 2}{\bf 1}+{\bf I}_z^{2})({1\over 2}{\bf 1}+{\bf
I}_z^{3}),
\label{eq:tceinput}$$ with $\rho$ one of the four inputs ${1\over 2}{\bf 1}$, ${\bf I}_z^1$, ${\bf I}_x^1$, ${\bf I}_y^1$. Any possible input is just a linear combination of these four states. We used gradient methods to directly generate the four states of Eq. \[eq:tceinput\] with a series of experiments. They were then subjected to pulse sequences for encoding and decoding (experiment I) with a variable delay between the two operations. Decoherence and dephasing take place during the delay. The reduced density matrix on the first spin (the output) was measured. In the second experiment (II) decoding was followed by error correction before the output was determined. Ideally the output would be identical to the input. The measured outputs were compared to the ideal ones by computing the “entanglement fidelity” [@entanglementfid]. This is a useful measure of how well the quantum information in the input is preserved. Entanglement fidelity is the sum of the correct polarization left in the output state for each input. More precisely, given input ${\bf I}_a^1$, let $f_a$ be the relative polarization of ${\bf I}_a^1$ in the output compared to the input. Then $f={1\over 4}(1+f_x+f_y+f_z)$, this formula is correct for processes which do not affect the completely mixed state ${1\over 2}{\bf 1}$. The results for nine different delays are shown in Figure \[figure:tce\_fids\]. The curves show that error correction decreases the initial slope by a factor of $\sim 10$ (by square fit to the logarithm).
Our demonstration of error-correction does not imply that error-correction can be used to overcome the problems of high temperature ensemble quantum computing. In this model of quantum computing, the initial state can be described as a small, linear deviation from the infinite temperature equilibrium. Thus, the deviation is proportional to a Hamiltonian of $n$ weakly interacting particles. In this limit no method of error-correction based on externally applied, time dependent fields can improve the polarization of any particle by more than a factor proportional to $\sqrt{n}$ [@sorensen:qc1989]. If one wishes to use error-correction an even bigger problem is encountered: The initial state of the ancillas used for each encoding/decoding cycle must be pure. In the high temperature regime, the best we can do to ensure that is to generate a pseudopure deviation in the ancillas. Unfortunately, this deviation has to be created [*simultaneously*]{} on all ancillas, leading to an exponential reduction in polarization proportional to the total number of ancillas required [@warren:qc1997a]. This always exceeds the loss of polarization. Another problem is the inability to reuse ancilla bits. This has two consequences. The first is that decoherence rapidly removes information in the state, leading to computations which are logarithmically bounded in time [@aharonov:qc1996b]. Second, the total number of ancillas required is proportional to the time-space product of the computation, rather than to a power of it’s logarithm.
Our work shows that liquid state NMR can be used to test fundamental ideas in quantum computing and communication involving non-trivial numbers of bits subject to a variety of errors. Our experiments demonstrate for the first time the state preserving effect of the three bit phase error-correcting code. The first-order behavior was established to high accuracy for a specific state in alanine, while the overall effect was observed and the improvement in state recovery verified in TCE. These experiments confirm not only the validity of theories of quantum error correction in a simple case, but also demonstrate the ability, in liquid state NMR, to control the state of three spin-half particles. This is an important advance for quantum computing, as this is the first system where this degree of control has been successfully implemented.
**Acknowledgments**
We thank Jeff Gore for his help with the simulations. This work was supported by, or in part by, the U. S. Army Research Office under contract/grant number DAAG 55-97-1-0342 from the DARPA Ultrascale Computing Program. E. K., R. L. and W. H. Z thank the National Security Agency for support.
[10]{}
P. W. Shor. In [*Proceedings of the 35’th Annual Symposium on Foundations of Computer Science*]{}, pages 124–134, Los Alamitos, California, 1994. IEEE Press.
L. K. Grover. quant-ph/9605043.
R. P. Feynman. , 16:507–531, 1986.
S. Lloyd. , 273:1073–1078, 1996.
C. Zalka. , in press.
R. Landauer. , 353:367–376, 1995.
W. G. Unruh. , 51:992–997, 1995.
P. W. Shor. , 52:2493–, 1995.
A. Steane. , 452:2551–, 1996.
P. W. Shor. In [*Proceedings of the Symposium on the Foundations of Computer Science*]{}, pages 56–65, Los Alamitos, California, 1996. IEEE press. quant-ph/9605011.
E. Knill, R. Laflamme, and W. Zurek. , vol 279, p.342, 1998.
D. Aharonov and M. Ben-Or. quant-ph/9611025, to appear in STOC’97, 1996.
J. Preskill. , in press.
A. Yu. Kitaev. preprint, 1996.
J. Cirac and P. Zoller. , 74:4091–, 1995.
C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland. , 75:4714–, 1995.
D. G. Cory, M. P. Price, and T. F. Havel. , 1998. In press (quant-ph/9709001).
R. Laflamme, E. Knill, W.H. Zurek, P. Catasti and S. Marathan. quant-ph/9709025, 1997.
D. G. Cory, A. F. Fahmy, and T. F. Havel. , 94:1634–1639, 1997.
N. A. Gershenfeld and I. L. Chuang. , 275:350–356, 1997.
W. H. Zurek .
A. Abragam. . Clarendon Press, Oxford, U.K., 1961.
R. R. Ernst, G. Bodenhausen, and A. Wokaun. . Clarendon Press, Oxford, U.K., 1987.
C. P. Slichter. Principles of Magnetic Resonance (3rd ed.). Springer Verlag, 1990.
O. W. [Sörensen]{}, G. W. Eich, M. H. Levitt, G. Bodenhausen, and R. R. Ernst. , 16:163–, 1983.
B. Schumacher. 54, 2614-2628, 1996.
O. W. [Sörensen]{}. , 21:503–569, 1989.
W. S. Warren. , 277:1688–1698, 1997.
D. Aharonov and M. Ben-Or. quant-ph/9611028, 1996.
|
---
abstract: 'Non-Markovian corrections to the Markovian quantum master equation of an open quantum system are investigated up to the second order of the interaction between the system of interest and a thermal reservoir. The concept of “natural correlation” is discussed. When the system is naturally correlated with a thermal reservoir, the time evolution of the reduced density matrix looks Markovian even in a short-time regime. If the total system was initially in an “unnatural” state, the natural correlation is established during the time evolution, and after that the time evolution becomes Markovian in a long-time regime. It is also shown that for a certain set of reduced density matrices, the naturally correlated state does not exist. If the initial reduced density matrix has no naturally correlated state, the time evolution is inevitably non-Markovian in a short-time regime.'
author:
- Takashi Mori
title: Natural correlation between system and thermal reservoir
---
The time evolution of the reduced density matrix $\rho_{\rm S}(t)$ of the system of interest in contact with a large thermal reservoir is described by the quantum master equation [@Breuer_text; @Weiss_text; @Kubo_text]. When the coupling constant $\lambda$ between the system and the reservoir is very small, the time scale of the relaxation is well separated from the time scale of the microscopic motion of the system and that of the reservoir, and hence the Markov approximation is justified. Indeed, the Markovian quantum master equation is rigorously derived in the van Hove limit, $\lambda\rightarrow 0$ with $\lambda^2 t$ fixed [@Davies_text]. However, in some real situations, such as atoms in photonic band gap media [@John1995; @Quang1997; @Vats1998; @Chen2012] and an optical cavity coupled to a structured reservoir [@Longhi2006], the deviation from the van Hove limit is important and the non-Markov effect is not negligible.
In this paper, the non-Markovian corrections are investigated up to the order of $\lambda^2$. The key concept discussed in this work is the [*natural correlation*]{} between the system of interest and the thermal reservoir. The following properties are to be clarified. When the total system starts from a [*naturally correlated state*]{}, the time evolution of the system of interest obeys the Markovian quantum master equation even if there is deviation from the van Hove limit. On the other hand, if the initial state is not a naturally correlated state, the time evolution must be non-Markovian in short times $t\lesssim\tau_{\rm R}$, where $\tau_{\rm R}$ is a characteristic time of the motion of the thermal reservoir. Even in this case, after a sufficiently long time $t\gg\tau_{\rm R}$, the time evolution approximately becomes Markovian since the natural correlation is established during the time evolution.
The further observation of this work is that there is a set ${\cal U}$ of reduced density matrices such that for any $\rho_{\rm S}\in{\cal U}$, the naturally correlated state does [*not*]{} exist. Therefore, if some $\rho_{\rm S}\in{\cal U}$ is chosen as an initial state of the system of interest, the time evolution is inevitably non-Markovian in the short time regime $t\lesssim\tau_{\rm R}$. After some time $t$, $\rho_{\rm S}(t)$ will get out of ${\cal U}$ and the natural correlation will develop there.
Now the setup is explained. The Hamiltonian of the total system is given by $H_{\rm T}=H_{\rm S}+H_{\rm R}+\lambda H_{\rm I}$, where $H_{\rm S}$ is the Hamiltonian of the system of interest, $H_{\rm R}$ is the Hamiltonian of the thermal reservoir, and $H_{\rm I}$ is the interaction Hamiltonian. The coupling constant $\lambda$ is relatively small but not vanishingly small, and hence we consider the non-Markovian corrections up to $O(\lambda^2)$. The density matrix of a whole system is denoted by $\rho_{\rm T}(t)$ and the reduced density matrix is then given by $\rho_{\rm S}(t)={\rm Tr}_{\rm R}\rho_{\rm T}(t)$. The initial state of the reservoir is assumed to be close to an equilibrium state $\rho_{\rm R}^{\rm eq}=e^{-\beta H_{\rm R}}/{\rm Tr}_{\rm R}e^{-\beta H_{\rm R}}$ where $\beta>0$ is the inverse temperature of the reservoir. At an initial time $t_0$, it is assumed that $\lim_{\lambda\rightarrow 0}\rho_{\rm T}(t_0)=\rho_{\rm S}(t_0)\rho_{\rm R}^{\rm eq}$. Without loss of generality, it is assumed that ${\rm Tr}_{\rm R}H_{\rm I}\rho_{\rm R}^{\rm eq}=0$. By applying the standard Nakajima-Zwanzig projection operator method [@Nakajima1958; @Zwanzig1960; @Kubo_text], the time evolution of $\rho_{\rm S}(t)$ for $t\geq t_0$ is given by $$\begin{aligned}
\dot{\rho}_{\rm S}(t)=&-iL_{\rm S}\rho_{\rm S}(t)
-\lambda^2\int_{t_0}^tdt'{\rm Tr}_{\rm R}e^{-(t-t')QiL_{\rm T}}QL_{\rm I}\rho_{\rm R}^{\rm eq}\rho_{\rm S}(t')
\nonumber \\
&-i\lambda{\rm Tr}_{\rm R}L_{\rm I}e^{-(t-t_0)QiL_{\rm T}}Q\rho_{\rm T}(t_0),
\label{eq:gQME}\end{aligned}$$ which is formally exact. Here the projection operators $P$ and $Q=1-P$ are defined as $P:=\rho_{\rm R}^{\rm eq}{\rm Tr}_{\rm R}$. The Liouvillian $L_{X}$ ($X=$T, S, R, I) is defined as $L_X(\cdot):=[H_X,(\cdot)]$. Up to $O(\lambda^2)$, the second term of the right-hand side of Eq. (\[eq:gQME\]) is simplified as $-\Lambda_0\rho_{\rm S}(t)+\Lambda_{t-t_0}\rho_{\rm S}(t)$ with \_t:=\^2\_t\^dt’[Tr]{}\_[R]{}L\_[I]{}L\_[I]{}(-t’), \[eq:Lambda\] where $L_{\rm I}(t)(\cdot)=[H_{\rm I}(t),(\cdot)]$ in the interaction picture, $H_{\rm I}(t)=e^{it(H_{\rm S}+H_{\rm R})}H_{\rm I}e^{-it(H_{\rm S}+H_{\rm R})}$. If the terms up to $O(\lambda^2)$ of the last term of Eq. (\[eq:gQME\]) are denoted by ${\cal I}_{t-t_0}\rho_{\rm T}(t_0)$, we have \_[S]{}(t)=-(iL\_[S]{}+\_0)\_[S]{}(t)+\_[t-t\_0]{}\_[S]{}(t)+[I]{}\_[t-t\_0]{}\_[T]{}(t\_0). \[eq:QME\] In the Markov approximation, we take the limit of $t_0\rightarrow-\infty$. As a result, the following Markovian quantum master equation (the Redfield equation [@Redfield1957]) is obtained: \_[S]{}\^[(M)]{}(t)=-(iL\_[S]{}+\_0)\_[S]{}\^[(M)]{}(t). \[eq:Markov\] In this paper we call $\Lambda_{t-t_0}\rho_{\rm S}(t)$ and ${\cal I}_{t-t_0}\rho_{\rm T}(t_0)$ in Eq. (\[eq:QME\]) non-Markovian contributions. In particular, the former is referred to as the “non-Markovian relaxation term” and the latter is referred to as the “initial correlation term”. These terms are negligible in the van Hove limit, but here we consider finite corrections up to $O(\lambda^2)$. It is noted that the initial correlation is often omitted by considering a product initial state, but here we do not make such an assumption.
Up to $O(\lambda^2)$, we can evaluate the non-Markovian contributions by the perturbation theory. The result is $$\begin{aligned}
\rho_{\rm S}(t)=e^{-(t-t_0)(iL_{\rm S}+\Lambda_0)}
\{\rho_{\rm S}(t_0)+\delta\rho_1[t-t_0;\rho_{\rm S}(t_0)]
\nonumber \\
+\delta\rho_2[t-t_0;\rho_{\rm T}(t_0)]\},
\label{eq:solution}\end{aligned}$$ where $$\begin{aligned}
\delta\rho_1[t;\rho_{\rm S}]&:=\int_0^tdt'e^{iL_{\rm S}t'}\Lambda_{t'}e^{-iL_{\rm S}t'}\rho_{\rm S}
\nonumber \\
=&\lambda^2\int_0^tdt'\int_0^{\infty}dt''
{\rm Tr}_{\rm R}L_{\rm I}(t')L_{\rm I}(-t'')\rho_{\rm R}^{\rm eq}\rho_{\rm S},
\label{eq:delta1}
\\
\delta\rho_2[t;\rho_{\rm T}]&:=\int_0^tdt'e^{iL_{\rm S}t'}{\cal I}_{t'}\rho_{\rm T}.
\label{eq:delta2}\end{aligned}$$ The non-Markovian corrections are expressed by $\delta\rho_1[t-t_0;\rho_{\rm S}(t_0)]$ and $\delta\rho_2[t-t_0;\rho_{\rm T}(t_0)]$. Here let us write the interaction Hamiltonian as $H_{\rm I}=\sum_iX_iY_i$, where $X_i$ and $Y_i$ are operators acting on the Hilbert space of the system of interest and the reservoir, respectively. We define the correlation functions of the reservoir as $C_{ij}(t):={\rm Tr}_{\rm R}\rho_{\rm R}^{\rm eq}Y_i(t)Y_j(0)$ and the characteristic decay time of $\{C_{ij}(t)\}$ is denoted by $\tau_{\rm R}$. Each correction, $\delta\rho_1[t-t_0;\rho_{\rm S}(t_0)]$ and $\delta\rho_2[t-t_0;\rho_{\rm T}(t_0)]$, then converges to some constant matrix for $t-t_0\gg\tau_{\rm R}$. Therefore, when $t-t_0\gg\tau_{\rm R}$, the non-Markovian contribution is taken into account as a slippage of the initial condition of the Markovian quantum master equation, \_[S]{}\^[(M)]{}(t\_0)=\_[S]{}(t\_0)+\_1\[;\_[S]{}(t\_0)\]+\_2\[;\_[T]{}(t\_0)\]. Suárez, Silbey, and Oppenheim [@Suarez1992] originally pointed out that an initial condition of the Markovian quantum master equation slips due to the non-Markovian effect. Gaspard and Nagaoka [@Gaspard-Nagaoka1999] analyzed this slippage of the initial condition and derived the same expression of $\delta\rho_1[\infty;\rho_{\rm S}(t_0)]$, but they did not consider the contribution from the initial correlation, $\delta\rho_2$. As for calculations of equilibrium fluctuations, van Kampen took into account the effect of the initial correlation [@van_Kampen_text; @van_Kampen2004]. In the present work, we shall show that a new insight can be obtained if [*both*]{} the non-Markovian relaxation term and the initial correlation term are treated in a unified way for [*general*]{} initial conditions (not restricted to equilibrium states or product states).
Let us consider a time $t_1$ with $t_1-t_0\gg\tau_{\rm R}$. In this case, $\delta\rho_i[t-t_0;\cdot]\simeq\delta\rho_i[\infty;\cdot]$ ($i=1,2$) and $\rho_{\rm S}(t)$ approximately obeys the Markovian quantum master equation (\[eq:Markov\]) for $t\geq t_1$. Now let us choose $t_1$ as a new initial time. Then $\rho_{\rm S}(t)=e^{-(t-t_1)\{ iL_{\rm S}+\Lambda_0)}(\rho_{\rm S}(t_1)
+\delta\rho_1[t-t_1;\rho_{\rm S}(t_1)]+\delta\rho_2[t-t_1;\rho_{\rm T}(t_1)]\}$ for $t\geq t_1$. On the other hand, as was explained above, $\rho_{\rm S}(t)$ should obey the Markovian quantum master equation for $t>t_1$ if we regard $t_0$ as an initial time. Since the time evolution of $\rho_{\rm S}(t)$ should be independent of our choice of an initial time, the above observation yields that $\delta\rho_1[t-t_1;\rho_{\rm S}(t_1)]+\delta\rho_2[t-t_1;\rho_{\rm T}(t_1)]$ is very small for $t>t_1$.
In general, $\delta\rho_1[t-t_1;\rho_{\rm S}(t_1)]+\delta\rho_2[t-t_1;\rho_{\rm T}(t_1)]$ is not exactly equal to zero as long as $t_1-t_0$ is finite. However, if \_1\[t-t\_1;\_[S]{}(t\_1)\]+\_2\[t-t\_1;\_[T]{}(t\_1)\]=0 tt\_1 \[eq:condition\] is satisfied, the time evolution of the system of interest is Markovian for $t\geq t_1$ exactly up to $O(\lambda^2)$.
When $\rho_{\rm T}$ satisfies Eq. (\[eq:condition\]), we say that the system is [*naturally correlated*]{} with a thermal reservoir. More precisely, for a given $\rho_{\rm S}$, we define the [*naturally correlated state*]{} $\rho_{\rm T}^{\rm (N)}$ as a matrix satisfying (i) $\rho_{\rm T}^{\rm (N)}\geq 0$, (ii) ${\rm Tr}_{\rm R}\rho_{\rm T}^{\rm (N)}=\rho_{\rm S}$, (iii) $\lim_{\lambda\rightarrow 0}Q\rho_{\rm T}^{\rm (N)}=0$, and (iv) $\delta\rho_1[t;\rho_{\rm S}]+\delta\rho_2[t;\rho_{\rm T}^{\rm (N)}]=0$ for any $t\geq 0$. The state $\rho_{\rm T}^{\rm (N)}$ satisfying the above conditions is a special one, but this special correlation is automatically generated during the time evolution even if we do not prepare a naturally correlated initial state. In this sense, $\rho_{\rm T}^{\rm (N)}$ is “natural”.
For example, if the initial state is given by a product state $\rho_{\rm T}=\rho_{\rm S}\rho_{\rm R}^{\rm eq}$, Eq. (\[eq:condition\]) does not hold unless $\delta\rho_1[t;\rho_{\rm S}]=0$ for all $t>0$, which is generally not the case. In this sense, for a given $\rho_{\rm S}$, the product form $\rho_{\rm T}=\rho_{\rm S}\rho_{\rm R}^{\rm eq}$ is “unnatural”. A trivial example of naturally correlated states is an equilibrium state of the total system, $\rho_{\rm T}^{\rm eq}=e^{-\beta H_{\rm T}}/{\rm Tr}_{\rm SR}e^{-\beta H_{\rm T}}$. We can show $\delta\rho_1+\delta\rho_2=0$ for $\rho_{\rm T}^{\rm eq}$; see also Ref. [@Mori2008]. It is stressed that it is highly nontrivial whether nonequilibrium naturally correlated states exist. The condition (iv) is required for all $t>0$ and that is a very strong condition.
In this paper, as for the naturally correlated states, the following two results are obtained. The first result is that for any fixed $\rho_{\rm S}$, the conditions (ii), (iii), and (iv) are satisfied if and only if \_[T]{}\^[(N)]{}=\_[S]{}\_[R]{}\^[eq]{}+i\_0\^dtL\_[I]{}(-t)\_[S]{}\_[R]{}\^[eq]{}+D. \[eq:natural\] This result is obtained by Eqs. (\[eq:delta1\]) and (\[eq:delta2\]). Here, $D$ is an operator satisfying ${\rm Tr}_{\rm R}D=0$ and ${\rm Tr}_{\rm R}L_{\rm I}(t)D=0$ for all $t$ [^1]. If Eq. (\[eq:natural\]) satisfies condition (i), it gives an explicit expression of the naturally correlated state for a given $\rho_{\rm S}$. Because $\rho_{\rm T}^{\rm (N)}\geq 0$ for $\lambda\rightarrow 0$, it is expected that $\rho_{\rm T}^{(N)}\geq 0$ at least for small values of $\lambda$. The second result, however, insists that for any fixed $\lambda\neq 0$, there is a set ${\cal U}$ such that $\rho_{\rm T}^{\rm (N)}$ given by Eq. (\[eq:natural\]) does not satisfy condition (i) for any $\rho_{\rm S}\in{\cal U}$; there is no naturally correlated state for $\rho_{\rm S}\in{\cal U}$. Therefore, if the initial state of the system of interest is chosen from ${\cal U}$, the time evolution is inevitably non-Markovian in a short-time regime $t\lesssim\tau_{\rm R}$. After a sufficiently long time $t\gg\tau_{\rm R}$, $\rho_{\rm S}(t)$ will get out of ${\cal U}$ and the density matrix of the total system will approach a naturally correlated state.
According to the Pechukas theorem [@Pechukas1994], a linear assignment of $\rho_{\rm S}\rightarrow\rho_{\rm T}\geq 0$ for all $\rho_{\rm S}\geq 0$ is impossible except for the product assignment $\rho_{\rm S}\rightarrow\rho_{\rm S}\rho_{\rm R}$. If only $D=0$ were allowed, Eq. (\[eq:natural\]) would give a linear assignment, and hence this assignment would not work for all $\rho_{\rm S}$; ${\cal U}$ is not empty. In general, however, $D\neq 0$ is allowed. In that case, $D$ may depend on $\rho_{\rm S}$ nonlinearly, and the Pechukas theorem does not ensure that ${\cal U}$ is not empty. Our second result shows that ${\cal U}$ is not empty even in that case.
So far we have assumed that a characteristic correlation time $\tau_{\rm R}$ exists. However, it is remarked that even if $\{C_{ij}(t)\}$ decays algebraically and thus there is no characteristic time $\tau_{\rm R}$, the above results are valid as long as $\delta\rho_1[t;\rho_{\rm S}]$ has the limit of $t\rightarrow\infty$. If this limit does not exist, the non-Markovian effect cannot be treated perturbatively. This point will be explained later.
Although we have defined the set ${\cal U}$, it is hard to judge whether a given $\rho_{\rm S}$ is in ${\cal U}$. Therefore, we explicitly construct a subset ${\cal U}'\subset{\cal U}$, which is much easier to calculate numerically. The argument relies on the variational consideration. If $\rho_{\rm T}^{\rm (N)}$ is positive semidefinite, $\<\Psi|\rho_{\rm T}^{\rm (N)}|\Psi\>\geq 0$ for any state vector $|\Psi\>$. Here we restrict state vectors into the following set of trial state vectors $\{|\Psi_{\alpha}\>\}_{\alpha}$: |\_=|\_0\^[(S)]{}|\_\^[(R)]{} -i\_0\^tdt’H\_[I]{}(t’)|\_0\^[(S)]{}|\_\^[(R)]{}, where $\xi\in\mathbb{R}$, $t\geq 0$, $|\phi_0^{\rm (S)}\>$ is the eigenvector of $\rho_{\rm S}$ with the smallest eigenvalue $p_0$, and $|\psi_{\alpha}^{\rm (R)}\>$ is an energy eigenstate of the reservoir, $H_{\rm R}|\psi_{\alpha}^{\rm (R)}\>=E_{\alpha}|\psi_{\alpha}^{\rm (R)}\>$. For $\xi=1$, the above form of $|\Psi_{\alpha}\>$ corresponds to the time-evolved quantum state obtained by the first order perturbation theory in the interaction picture starting from a product initial state. Because the natural correlation is established during the time evolution of the total system, this choice of trial state vectors seems to be natural. Here $\xi$ and $t$ play the role of variational parameters.
Here we consider the quantity $\sum_{\alpha}\<\Psi_{\alpha}|\rho_{\rm T}^{\rm (N)}|\Psi_{\alpha}\>$, which is independent of $D$. Substituting Eq. (\[eq:natural\]), we obtain \_\_|\_[T]{}\^[(N)]{}|\_=p\_0+\^2\[\^2A(t)-B(t)\], where A(t):=\_0\^tdt’\_0\^tdt”\_0\^[(S)]{}|[Tr]{}\_[R]{}H\_[I]{}(t’)\_[S]{}\_[R]{}\^[eq]{}H\_[I]{}(t”)|\_0\^[(S)]{}, which is non-negative $A(t)\geq 0$ for $t\geq 0$, and \^2B(t):=\_0\^[(S)]{}|\_1\[t;\_[S]{}\]|\_0\^[(S)]{}. Since $\xi\in\mathbb{R}$ and $t\geq 0$ are arbitrary, we choose them so that $\sum_{\alpha}\<\Psi_{\alpha}|\rho_{\rm T}^{\rm (N)}|\Psi_{\alpha}\>$ becomes minimum. Putting $\xi=B(t)/2A(t)$ and minimizing with respect to $t>0$, we have \_\_|\_[T]{}\^[(N)]{}|\_=p\_0-\^2\_[t>0]{}, \[eq:U’\] which is strictly lower than $p_0$. If the right-hand side of Eq. (\[eq:U’\]) is negative, it contradicts the non-negativity of $\rho_{\rm T}^{\rm (N)}$. Hence, we define ${\cal U}'$ as ’:={\_[S]{}:p\_0-\^2\_[t>0]{}<0}. \[eq:def\_U’\] Clearly ${\cal U}'\subset{\cal U}$ and it is not an empty set. When the system of interest is in a pure state, $p_0=0$ and thus the right-hand side of Eq. (\[eq:U’\]) is negative. That is, all the pure states do not have their naturally correlated states. It is because the density matrix of the total system must be a product state $\rho_{\rm S}\rho_{\rm R}$ in this case and there is no room for building up any correlation between the system of interest and the thermal reservoir.
![(Color online) The regions of ${\cal U}'$ (dotted region) and ${\cal N}$ (shaded region). Up to $O(\lambda^2)$, ${\cal N}\subset{\cal U}'$. All the physical states are in the unit circle (solid line).[]{data-label="fig:region"}](fig1.eps){width="6cm"}
As a demonstration, let us consider the spin-boson model, in which a two level system is in contact with an infinitely large boson bath. The Hamiltonian is given by H\_[S]{}=S\^z, H\_[R]{}=\_r\_rb\_r\^b\_r, H\_[I]{}=X\_1Y\_1 \[eq:spin-boson\] with $X_1=S^x$ and $Y_1=\sum_r\nu_r(b_r^{\dagger}+b_r)$. Here $b_r$ and $b_r^{\dagger}$ are the annihilation and the creation operators of the boson, $[b_r,b_{r'}^{\dagger}]=\delta_{r,r'}$ and $[b_r,b_{r'}]=[b_r^{\dagger},b_{r'}^{\dagger}]=0$. The vector $\bm{S}$ denotes the spin-1/2 operator. The spectral density of the boson bath is assumed to be the ohmic one with the Lorentz-Drude cutoff [@Breuer_text], $J(\omega):=\sum_r\nu_r^2\delta(\omega_r-\omega)=\omega\Omega^2/(\omega^2+\Omega^2)$, for which we can numerically evaluate $A(t)$, $B(t)$, and also the right-hand side of Eq. (\[eq:U’\]). The reduced density matrix is parametrized by $x$, $y$, and $z$ with $\sqrt{x^2+y^2+z^2}\leq 1$ so that $\rho_{\rm S}=(1/2)\hat{I}+xS^x+yS^y+zS^z$, where $\hat{I}$ is the identity matrix. In this Bloch-sphere representation, $p_0=(1-\sqrt{x^2+y^2+z^2})/2$.
The region of ${\cal U}'$ is shown for $z=0$ on the $xy$-plane in Fig. \[fig:region\]. In this figure, the parameters are set to $\varepsilon=\Omega=\beta=1$ and $\lambda=0.5$. This parameter choice is just for demonstration of the result. The domain of ${\cal U}'$ is not so large; the length of this domain from the surface of the Bloch sphere is proportional to $\lambda^2$. Although we cannot obtain the region of ${\cal U}$ explicitly, we expect that this ${\cal U}'$ gives a good approximation of ${\cal U}$. Figure \[fig:region\] implies that naturally correlated states exist for a large part of physically allowed reduced density matrices $\{\rho_{\rm S}\}$.
It is a well known problem that the Markovian quantum master equation (\[eq:Markov\]) sometimes violates the non-negativity of the reduced density matrix. It depends on the initial state $\rho_{\rm S}$ whether or not the non-negativity is violated during the Markovian time evolution. In Fig. \[fig:region\], we also show the domain ${\cal N}$ of initial states $\{\rho_{\rm S}\}$ for which the non-negativity of $\{\rho_{\rm S}^{\rm (M)}(t)\}_{t\geq 0}$ with $\rho_{\rm S}^{\rm (M)}(0)=\rho_{\rm S}$ is violated. We numerically find ${\cal N}\subset{\cal U}'$ up to $O(\lambda^2)$; all the initial conditions for which the non-negativity is violated during the Markovian time evolution do not have their naturally correlated states [^2]. The violation of non-negativity is due to the improper choice of an initial state of the Markovian equation. This statement was already indicated by the previous work [@Suarez1992] and the present study gives a theoretical ground for this statement. Moreover, the fact that ${\cal U}'$ is larger than ${\cal N}$ means that we should take special care to use the Markovian quantum master equation even if the Markovian time evolution does not violate the non-negativity and therefore it looks physical. The general criterion for making use of the Markovian master equation is whether $\rho_{\rm S}(0)$ is in ${\cal U}$ or not. For $\rho_{\rm S}\in{\cal U}$, the time evolution cannot be Markovian in a short-time regime. Therefore, an initial state of the Markovian quantum master equation must be chosen from the outside of ${\cal U}$.
As was mentioned just before presenting the main results, our perturbative treatment of the non-Markovian effect is invalid for the case where $\delta\rho_1[t;\rho_{\rm S}]$ does not have the limit of $t\rightarrow\infty$. This limitation excludes the case where an ohmic or sub-ohmic Bson bath [@Weiss_text], i.e. $J(\omega)\sim\omega^s$ with $s\leq 1$ at low frequencies, is at zero temperature and, moreover, $\{X_i\}$ have non-zero diagonal elements in the basis diagonalizing $H_{\rm S}$. In the spin-boson model calculated above, $X_1=S^x$ and it does not have diagonal elements in the basis diagonalizing $S^z$, and hence we can safely treat the non-Markovian effect perturbatively. However, if we consider another coupling with $X_2=S^z$, the perturbative treatment of the non-Markovian effect becomes invalid for an ohmic or sub-ohmic reservoir at zero temperature. In that case, very long time correlations of the boson bath, which originate from the singularity around $\omega=0$, cause the breakdown of the perturbative treatment of the non-Markovian effect.
Even if the perturbation analysis works well, of course, there are very small Markovian and non-Markovian effects coming from higher-order terms than $\lambda^2$. Higher-order terms might become important for very long times. For instance, the terms of $O(\lambda^4)$ may change the stationary state in $O(\lambda^2)$ [@Mori2008; @Fleming2011]. However, they do not play important roles for the transient relaxation dynamics, which is the major interest of this work.
In conclusion, it has been argued that there are special states called the “naturally correlated states” by the evaluation of the non-Markovian contributions up to $O(\lambda^2)$, and moreover it has been shown that there is no naturally correlated state for some reduced density matrices $\rho_{\rm S}\in{\cal U}$. If the system is in a naturally correlated state, the time evolution is Markovian up to $O(\lambda^2)$ regardless of the deviation from the van Hove limit. On the other hand, if the system is not in a naturally correlated state, the later time evolution is non-Markovian until the natural correlation develops for $\rho_{\rm S}(t)$ being outside of ${\cal U}$. The subset ${\cal U}'$ of ${\cal U}$ is given by Eq. (\[eq:def\_U’\]), which allows explicit numerical computation. The domain of ${\cal U}'$ is demonstrated for the spin-boson model. As a result, it is numerically shown that the domain of ${\cal U}'$ is wider than that of ${\cal N}$, in which the Markovian time evolution violates the non-negativity of the density matrix. It confirms the previous qualitative argument given in Ref. [@Suarez1992] on the violation of non-negativity due to the neglected non-Markovian effect in the short-time regime. It is made clear that the underlying reason why the non-Markovian effect is important for some initial conditions is the impossibility of developing the natural correlation between the system and the thermal reservoir at short times.
The author thanks Seiji Miyashita for helpful discussions during the early stages of this work and Tomotaka Kuwahara for valuable comments. This work was supported by the Sumitomo foundation.
[19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ** (, ).
, **, vol. (, ).
, , , ** (, ).
, ** (, ).
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, **, vol. (, ), ed.
, ****, ().
, ****, ().
, ****, ().
, ****, ().
[^1]: For a sufficiently complex reservoir, it is expected that only $D=0$ is allowed. In this case, the reservoir is called “ergodic”. The boson bath given by Eq. (\[eq:spin-boson\]) is not ergodic in this sense although non-linear couplings between boson modes might make the reservoir ergodic.
[^2]: Strictly speaking, ${\cal N}$ is not completely a subset of ${\cal U}'$ for a finite value of $\lambda$. However, a typical length of the region ${\cal N}\backslash{\cal U}'$ shrinks faster than $\lambda^2$. The statement “${\cal N}\subset{\cal U}'$ up to $O(\lambda^2)$” should be interpreted in this sense.
|
---
abstract: 'A modification of the standard product used in local field theory by means of an associative deformed product is proposed. We present a class of deformed products, one for every spin $S=0,1/2,1$, that induces a nonlocal theory, displaying different form for different fields. This type of deformed product is naturally supersymmetric and it has an intriguing duality.'
author:
- 'Elisabetta Di Grezia$^{1,2}$[^1], Giampiero Esposito$^3$[^2], Gennaro Miele$^{4,3}$[^3]'
---
Introduction
============
The main problem of any local quantum field theory is the presence of ultra-violet divergences. In fact the S-matrix is expressed in terms of the products of causal functions of the field operators. Since the causal functions have fairly strong singularities on the light cone, the products of such functions are not mathematically defined. This problem arises from the ill-defined nature of the product of two local field operators at the same space-time point. This generates one of the main problems of quantum field theory - the so-called problem of ultraviolet divergences. There are different regularization procedures to deal with such divergences, making the S-matrix elements mathematically meaningful. These are, for example, the subtraction procedure interpreted in various manners in a local field theory, the summation of asymptotic series for the Green functions and super-propagators, a nonlocal generalization of the theory. In particular, the nonlocal quantum field theory which replaces local quantum field theory is very old, dating from 1950’s, starting with Pais and Uhlenbleck (1950), Efimov and coworkers [@efim] (1970-onwards), Moffat, Woodard and coworkers (1990) [@kleppe], [@moffat]. The basic idea to try to avoid “infinities” was to assume a nonlocal interaction and thus to provide a natural cut-off. One has to build a nonlocal quantum field theory which is a self-consistent scheme satisfying all principles of conventional quantum field theory (unitarity, causality, relativistic invariance, etc.) and providing the basis for correct description of nonlocality effects. There are some problems with the gauge invariance because of nonlocality in gauge field interactions. Nevertheless, there are different nonlocal approaches which give a good possibility of building a completely ultraviolet finite theory of fundamental interactions.
\(i) One way is to introduce nonlocality in the interaction term [@nonloc0] writing down the Lagrangian of scalar fields in the form $$L=\phi(x)^\dag(\partial^2+m^2)\phi(x) + \lambda\Phi(x)^\dag\Phi(x),
\label{(1)}$$ where the nonlocal field $\Phi(x)$ is obtained from the local one $\phi(x)$ by “smearing” over the nonlocality domain with the characteristic scale $l_0$. Without specifying the nature of this nonlocality, and introducing the phenomenological form factor $K$, the nonlocal field $\Phi(x)$ is defined as $$\Phi(x) \equiv \int dy K(x-y)\phi(y)=K(l_0^2\partial^2)\phi(x),$$ where the nonlocal operator $K(l_0^2\partial^2)$ can be written in the form $$K(l_0^2\partial^2)=\sum_{n=0}^{\infty}\frac{c_n}{(2n)!}
(l_0^2\partial^2)^{n},$$ $K$ being an entire function without any zeros. Then the generalized function $K(x-y)=
K(l_0^2\partial^2)\delta(x-y)$ belongs to one of the spaces of nonlocal generalized functions which was introduced and explored in the works of Efimov [@efim]. One rewrites the Lagrangian (\[(1)\]) in terms of the nonlocal fields $\Phi(x)$ as $$L=\Phi(x)^\dag(\partial^2+m^2)\Phi(x) + \lambda\Phi(x)^\dag\Phi(x),
\label{(2)}$$ in this way the $\phi(x)$-propagator (smeared propagator) is obtained by taking the Fourier transform of $$\frac{\exp{\left[\frac{p^2-m^2}{l_0^2}\right]}}{p^2-m^2+i\epsilon}.$$ This suggests interpreting the nonlocal quantum field theory as an effective theory valid up to an energy scale $l_0$; and for energy scales beyond $l_0$, one has to replace the nonlocal quantum field theory by a more fundamental theory of constituents having its own larger mass scale and coupling constant. In a sense, this formulation can be viewed either as a regularization, or as a physical theory with a finite mass parameter $l_0$. Such a theory preserves causality at tree-level in the S-matrix (it is the same as the local one), but it suffers from quantum causality violations, which are a serious limitation.
\(ii) Another way to have a finite quantum field theory was proposed in [@Nonloc] on the basis of infinite-component fields, which results in the introduction of a special form of nonlocality.
\(iii) Another way to introduce nonlocality in the theory is to consider a special class of field theories with higher derivatives. In the canonical formulation, one usually considers Lagrangians with only first derivatives. However, higher derivative theories, including nonlocal theories, also have many physical applications. For example, when one integrates out high energy degrees of freedom in a local field theory, the low-energy effective action is generically nonlocal [@[1.1]]. Higher derivative theories were also considered in order to find a finite quantum field theory [@[2.2]], before the advent of renormalization. Moreover, theories with infinitely many derivatives are unavoidable in string theory [@[3.3]; @[4.4]]. There are other examples, such as higher derivative gravity [@[5.5]], meson-nucleon interactions [@[6.6]], and spacetime noncommutative field theory [@[7.7]; @[8.8]], and so on. In most cases, higher derivative terms appear as higher-order corrections in the effective Lagrangian, hence a perturbative approximation scheme would already be very useful.
\(iv) Another example of nonlocality is in quantum mechanics where it has long been an intriguing topic in the past decades, and so far there has been no experiment contradicting nonlocality. It refers to the correlation between two particles separated in space, e.g. the entanglement derived from the Bell theory [@bell] and well confirmed in many experiments [@bell1]. All these experiments used massless photons as carriers of the states, and the nonlocality is of the Bell type.\
Finally we would like to mention our approach, in which nonlocality is introduced through the deformation of the product, and in a perturbative approximation it is reduced to a field theory with higher derivatives.
We shall now introduce the plan of the paper. In Sec. 2 we review the deformed products, i.e. deformed field theories and their nonlocal properties. In Sec. 3 we propose a class of deformed products which are associative, with different expressions for spin $S=0,1/2,1$, respectively. In Sec. 4 we exhibit their properties. In Sec. 5 we study the deformed interaction term. In Secs. 6 and 7 we show that the scalar field theory with our deformed product is the same as found by Moffat [@moffat1]. Concluding remarks are presented in Sec. 8.
Review of deformed field theory
===============================
Deformation quantization was born as an attempt to interpret the quantization of a classical system as an associative deformation (i.e. via star-products) of the algebra of classical observables. This idea was behind the mind of many mathematical physicists and physicists [@weyl; @wigner] as illustrated by the historical developments which led to deformation quantization. By deformed it is meant that the standard point-wise multiplication of functions has been replaced by a new product which may or may not be commutative. Recently, algebras of functions with a deformed product have been studied intensively [@cit]. These are deformed (star-) products which remain associative but not commutative. There is a class of K-deformed products which generate deformed associative products, see for example [@pvm]. Before talking about deformed field theory we first summarize the concept of deformed coordinate spaces as quantum spaces. Deformed coordinate spaces are defined in terms of coordinates $x_\mu$ and their commutation relations. The $\theta$, $k$, $q$-deformations are the best known examples [@dcs]. They are: the canonical relations $[x_\mu,x_\nu]=\theta_{\mu\nu}$, which for constant $\theta_{\mu\nu}$ leads to the so-called $\theta_{\mu\nu}$-deformed coordinate space; the Lie-type relations where the coordinates form a Lie algebra $[x_\mu,x_\nu]=C_{\mu\nu}^\lambda x_\lambda$ and $C_{\mu\nu}^\lambda$ are the structure constants, this framework leads to the $k$-deformed coordinate space; and then the quantum group relations, $x^\mu x^\nu= R_{\rho\sigma}^{\mu\nu}x^\rho
x^\sigma\theta_{\mu\nu}$ where the R-matrix defines a quantum group, this leads to the q-deformed spaces.
Deformations of mathematical structures have been used at different moments in physics. When Galilean transformations between inertial systems were seen not to describe adequately the physical world, a deformation of the group law arose as the solution to this paradox. The Lorentz group is a deformation of the Galilei group in terms of the parameter $c$ . In this deformation scheme, the old structure is seen as a limit or contraction when the parameter takes a preferred value. Hence a deformation, an inverse of contraction (in the sense of Segal–Wigner–Inonu contraction), is one of the methods of generalization of a physical theory [@[1]]. The undeformed theory can be recovered from the deformed one when taking a limit of deformation parameter to some value, e.g., nonrelativistic, classical physics, the undeformed theory, is recovered from relativistic physics when taking the velocity of light $c\rightarrow \infty$, and, from the point of view of quantum physics, when taking the Planck constant $h \rightarrow 0$. The mathematical structure of quantum mechanics has also an ingredient of deformation with respect to classical mechanics. This naive concept has been applied to field theories on noncommutative spaces considered as deformations of flat Euclidean or Minkowski spaces. Since noncommutative geometry generalizes standard geometry in using a noncommutative algebra of functions, it is naturally related to the simpler context of deformation theory. The star product is a product in the space of formal power series in $\hbar$ whose coefficients are functions on the phase space. Thus, a product of fields on NC spaces can be expressed as a deformed product or star-product [@[2]; @[3]] of fields on commutative spaces [@[6]; @[8]]. The star product can be seen as a higher-order $f$-dependent differential operator acting on the function $g$. The noncommutativity is governed by a parameter such that the commutative case appears in the limit where this parameter approaches zero. The simplest and most well known example of $\star$-product is the Moyal–Weyl product, and it first appeared in quantum mechanics [@m]. It was first introduced by H. Weyl for his quantization procedure and later by Moyal [@[1]] to relate functions on phase space to quantum mechanical operators in Hilbert space. For this reason the $\star$-product is called in the literature as Moyal–Weyl product. In the Moyal–Weyl $\star$-product representation the noncommutative coordinates ${\hat x}_{\mu}$ (and their functions) are mapped to commutative coordinates $x_\mu$ with commutative pointwise product replaced by deformed (nonlocal) $\star$-product defined as $$\begin{aligned}
&&f\star g(x) \equiv \exp{(i\theta^{\mu\nu}\partial_\mu^y\partial_\nu^z
f(y)g(z))}|_{y=z=x}\nonumber \\
&=&\sum_{n=1}^{\infty}{\left(\frac{i}{2}\right)}^n
\frac{1}{n!}\theta^{\rho_1\sigma_1}\cdot\theta^{\rho_n\sigma_n}
(\partial_{\rho_1}\cdot\partial_{\rho_n}f(x))
(\partial_{\sigma_1}\cdot\partial_{\sigma_n}g(x)).\end{aligned}$$ This implies the presence of (infinitely many) derivatives in the action, hence the theory becomes nonlocal, and the noncommutative quantum field theories are a special case of a nonlocal quantum field theory. Other $\star$-products will occur for different orderings. The way of looking at noncommutative geometry in terms of deformed products can give different insights. In fact a deformed gauge theory leads to a theory with a larger symmetry structure, i.e. the enveloping algebra structure, and it exhibits its nonlocal nature. Nevertheless, the commutative field theories can be recovered from their noncommutative counterparts when the noncommutativity tensor approaches zero: $\theta_{\mu\nu}\rightarrow 0$. A property of noncommutative field theories is the presence of nonlocal interaction terms, which explicitly breaks Lorentz invariance. In fact under the integration, the star-product of fields does not affect the quadratic parts of the Lagrangian, whereas it gives rise to a nonlocal interaction part.
Hence, Feynman rules in momentum space are modified with respect to the commutative ones, in fact the vertices are modified by a phase factor. The deformed vertices differ from the nondeformed ones by a factor of type $\cos( 1/2 p_\mu\theta^{\mu\nu}p_\nu)$. When $\theta_{\mu\nu}\rightarrow 0$, the deformed vertex reduces to the nondeformed one.
One can give a star-product quantization scheme following [@marm; @marm1], and see that there is a class of star products ($K$-star products) which are obtained via a specific deformation procedure [@marm1]. Much more recently, noncommutative geometry has entered physics in different contexts.
One context is string theory. In their pioneering paper, Connes, Douglas and Schwarz [@4] introduced noncommutative spaces (tori) as possible compactification manifolds of space-time. Non commutative geometry arises as a possible scenario for short-distance behaviour of physical theories. In the framework of open string theory [@12], Seiberg and Witten in [@[7.7]] identified limits in which the whole string dynamics, in presence of a $B$-field, is described by a deformed gauge theory in terms of a Moyal–Weyl star product on space-time. The field theory associated to string theory, in the low-energy limit, is nonlocal, because the fields in the action are multiplied by a (deformed) star-product. The deformed theories enjoy renormalization properties as well as UV/IR connection reminiscent of string theory. Other approaches connecting deformation theory to theories of gravity have also appeared in the literature. Among others, there is the deformation quantization of M-theory [@19], quantum anti-de Sitter spacetime [@20], q-gravity [@21] and gauge theories of quantum groups [@22]. Another area covered by noncommutativity is supersymmetric theory. The deformation aspects of supersymmetric field theories were investigated in [@sc; @FL; @KL; @KP]. Analogous to noncommutative field theories on bosonic spacetime, noncommutative superfield theories can be formulated in ordinary superspace by multiplying functions given on it via a $\star$-product which is generated by some bi-differential operator or Poisson structure $P$. It defines a deformed superspace and leads to deformed products for general superfields. There will be symmetries of the undeformed (local) field theory which are explicitly broken in the deformed (nonlocal) case. In this case only free actions preserve all supersymmetries while interactions get deformed and are not invariant under all standard supersymmetry transformations, because the integral of the star product of two superfields is not deformed, while in the case of three or more superfields the integral is deformed.
In [@FL] the authors present a variety of deformations, both for N = 1 and extended (N = 2) supesymmetry in D = 4, which vary according to the differential operators chosen to construct the Poisson bracket that afterwards becomes quantized with a star product of Moyal–Weyl type. For example in [@FL], the first deformation has the advantage of being manifestly supersymmetric, while the second, although it explicitly breaks half of the supersymemtry, allows the definition of chiral and antichiral superfields, which form subalgebras of the star product. Another way to construct a deformed field theory is with derivatives which are an essential input for the construction of deformed field equations such as the deformed Klein–Gordon or Dirac equations [@gord].
At the end we would like to emphasize some properties of deformed field theory as a nonlocal theory. The quantum deformation modifies the behavior of relativistic theories at distances comparable to and smaller than the length $l$ corresponding to the deforming parameter. It appears that, by virtue of the deformation of local product of fields in the interaction, the vertex will be replaced by a deformed nonlocal product, with the nonlocality extending to distances of order $l$.
Such a quantized space-time geometry can provide additional convergence factors or even a finite quantum field theory. Indeed, if one introduces a masslike deformation parameter, it occurs also as a regularizing parameter.
There are also attempts to remove the ultraviolet divergences by introducing nonlocality into the interaction Lagrangian. Hence the advantage of the nonlocal character of the deformed product is the following: first, one has succeeded in introducing into the interaction Lagrangian all the ambiguity in the choice of the shape and the value of the “elementary” length; second, the amplitudes of the physical processes have no additional singularities in the finite region of change of the invariant momentum variables as compared to the local theory.
Nonlocal quantum field theory faces, however, many difficulties. One of the main difficulties in constructing the non-local quantum field theory appears to be the formulation of macro-causality of the $S$-matrix. Then it seems that a reasonable macro-causality condition imposed on the $S$-matrix would be a generalization of the micro-causality condition [@efim]. However, as one can see in Efimov’s paper [@efim], there is indeed a causality violation but, from the physical point of view, the problem may be formulated in such a way that the amount of causality violation would satisfy the usual requirements imposed on nonlocal theories. Indeed, using the Lagrangian of the quantized field system, Efimov expands the $S$-matrix in the small coupling constant. Therefore, in the case of the small coupling constant interaction, the violation of causality at large distances is rather small. Another property is the unitarity. The postulate of unitarity of the $S$-matrix in quantum field theory is one of the principal requirements for the theory to be regarded as self-consistent and physically acceptable. Efimov for example, in Ref. [@efim1], proves the unitarity of the S-matrix in the $n$-th order of perturbation theory in a nonlocal quantum field theory.
A class of deformed products
============================
In this section we propose a class of associative deformed products, with different expressions for spin $S=0,1/2,1$, respectively. In [@marm1] a deformed operator product was introduced ($K$-product) in the form $\hat{f} ·_K \hat{g} =
\hat{f} \hat{K} \hat{g}$ where $\hat{K}$ is a generic operator. It satisfies the associativity condition $$(\hat{f} \bigtriangleup_{K} \hat{g})
\bigtriangleup_{K} \hat{h}=\hat{f}\bigtriangleup_{K}
(\hat{g}\bigtriangleup_{K} \hat{h}).$$ As emphasized in [@pvm] the K-deformed products are a way to generate new associative products. To construct our new deformed product we take into consideration this one and that the star-product in Quantum Mechanics, because of its nonlocal nature, can be described through an integral kernel [@star]. This integral kernel plays the role of the structure function for the product and the star-product reduces to the more familiar asymptotic expansion with a particular choice of this kernel. Inspired by all these properties, and bearing in mind that the star product is a particular associative deformed product, we import this formalism used in Quantum Mechanics to Field Theory to define a new deformed product $(A\diamondsuit_\theta B)(x)$ according to $$(A\diamondsuit_\theta B)(x)\equiv \int_{{{\mathbb{R}}}^4}\int_{{{\mathbb{R}}}^4}\,
A(y)L(x,y,z)\,B(z) \,dy\,dz, \label{Moyal-prodint}$$ where the integral kernel $L$ has different shape for each spin. The associativity condition for operator symbols implies that the kernel $L(x,y,z)$ satisfies the nonlinear equation $$\int L(x_1,x_2,y)L(y,x_3,x_4)dy= \int L(x_1,y, x_4)L(x_2,x_3,y)dy.
\label{ass}$$ Our kernel is of the type $\delta(x,z)[\exp{\theta
f(\partial})]\delta(x,y)$, hence it fulfils the associativity condition (\[ass\]) thanks to the properties of a Dirac $\delta$-functional.
For every spin we have a different choice of integral kernel. In the case of $S=0$, i.e. a scalar field, we start with a theory with a static term $$\int d^4 x \frac{1}{2}m_2^{2}.
\phi\diamondsuit\phi .$$ With a particular choice of kernel, we can write $$\delta L_{{\rm scal}}=\delta(x,z)\left[-1 + \theta\Box\right]\delta(x,y),$$ and we obtain a dynamical theory. It can be seen as a leading order term of expansion, in the parameter $\theta$, of the following general definition: $$L_{{\rm scal}}:= \delta(x,z)[\exp{\theta\Box}]\delta(x,y).
\label{freesca}$$ In this form it actually displays a “static nature”, as we will show later.
In the case of $S=1$, i.e. a vector field, we start with a theory with a static term $$\int
d^4 x \frac{1}{2}m_1^2
A_\mu \diamondsuit A^{\mu}.$$ The particular choice of kernel is now such that $$\begin{aligned}
&&\delta L_{{\rm
vect}}(x,y,z)=\delta(x,z)[D_\theta]_\mu^\nu\delta(x,y)
=\delta(x,z)\left[g_\mu^\nu +
\theta\Delta_\mu^\nu\right]\delta(x,y)\nonumber
\\
&& =\delta(x,z)\left[g_\mu^\nu + \theta \left(\Box_y
\delta_\mu^\nu- \left(1-\frac{1}{\alpha}\right)
\nabla^\nu\nabla_\mu\right)\right]\delta(x,y),\end{aligned}$$ which leads to a dynamical theory. It can be seen as a leading-order term in the expansion, in the parameter $\theta$, of the following general definition: $$L_{{\rm vect}} := \delta(x,z)[\exp{\theta\Delta^\mu_\nu}]\delta(x,y).$$
In the case of spin $S=1/2$, i.e. a spinor field, we start with a theory with a static term $$\int
d^4 x \frac{1}{2}m_3 \psi\diamondsuit\overline{\psi}.$$ Our particular choice of kernel is such that $$\delta
L_{{\rm matter}}(x,y,z)=\delta(x,z)\left[
-1+i\sqrt{\theta}\gamma_\mu\partial^\mu\right]\delta(x,y).$$ It can be seen as a leading-order term in the expansion, in the parameter $\theta$, of the following general definition: $$L_{{\rm matter}}:=
\delta(x,z)[\exp{-i\sqrt{\theta}\gamma_\mu\partial^\mu}]\delta(x,y).
\label{fermionic}$$
Properties of the deformed product
==================================
In this section we analyze some peculiar properties of our deformed product, outlining the possible implications.
Supersymmetric nature of the deformed product and its apparent dynamical nature
-------------------------------------------------------------------------------
To achieve correspondence to lowest-order theory we must impose the condition: $m_i\sqrt{\theta}=1$, which then implies $m_i=m=\frac{1}{\sqrt{\theta}}$. This requires, of course, supersymmetry, i.e. that the fields $\phi, \psi, A_\mu$ belong to a massive vector $N=1$ superfield. Hence, to lowest order, the “static” massless theory in the corresponding deformed product for scalar, vector, fermionic fields is “equivalent” to a dynamical massive supermultiplet, i.e., if we restrict ourselves to the leading term, linear in the deformation parameter, we get naturally a supersymmetric formulation, as in a dual deformation.
In principle one can think of having a class of deformed products and, corresponding to different choices of kernel, to build a deformed supersymmetryc Wess–Zumino model, as in a supersymmetric U(1) theory, introducing the dynamics to lowest order in the kernel and not in the supersymmetric formulation, as in [@sup]. In our case, with suitable choice of kernel in the scalar, spinor and vector Lagrangian, we can reproduce in a natural way a supersymmetric action which suffers from nonlocality at subsequent orders in $\theta$. The Lagrangian for a globally supersymmetric matter multiplet is [@mukhi] $$\begin{aligned}
L^{N=1}_{{\rm matter}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} +
\frac{1}{2}\bar{\psi}\gamma_\mu\partial^\mu\psi -
\frac{1}{2}m\bar{\psi}\psi + \frac{1}{2}m^2A_\mu A^\mu -
\frac{1}{2}m^2\phi^{2},\end{aligned}$$ where $A^{\mu}$ is a vector field, $\psi$ is a Majorana spinor field, and $\phi$ is a pseudoscalar field.
All fields must have the same mass. The above Lagrangian in superspace formalism is $$L^{N=1}_{{\rm matter}}= [\Phi\dag \Phi]_D + m[\Phi \Phi]_F
+m[V_{WZ}^2]_D\nonumber + \frac{1}{32}[W^\alpha W_\alpha]_{F}.$$ With our prescription, it becomes $$L^{N=1}_{{\rm matter}}= \frac{1}{2}m^2
A_\mu \diamondsuit_{S=1} A^\mu
+ \frac{1}{2}m \psi\diamondsuit_{S=1/2}\overline{\psi }+\frac{1}{2}m^2
\phi\diamondsuit_{S=0}\phi.$$ What seems to happen is that the dynamics, which is put at the level of superfields in a supersymmetric Lagrangian, is found in a nonlocal theory to first order non vanishing in the $\theta$ parameter.
Hence one can think of using the deformed product, to lowest order in the $\theta$ parameter, to obtain the dynamics, i.e., one can encode dynamics in a product and the other way around. Changing deformed product means having a different dynamics. We can have infinitely many derivatives in order to reproduce, at different orders in $\theta$, the dynamics of the system. This is a first step towards obtaining a more elaborate model and the dynamics in the deformed product, where the $\theta$ parameter is essential, and its smallness is important to make sure that higher-order (derivative) terms are of no importance. Thus, [*the dynamics can be seen as a “perturbative” effect which disappears in the global “nonperturbative” static expression*]{}.
Nonlocal nature of the deformed product
---------------------------------------
Hence, starting with the free static action and using the (bi)-differential operator to lowest order in the parameter, one obtains a known massive free-field theory, that is a lower order approximation of a more complicated nonlocal theory. In this way we have a nonlocal field theory from a local theory in which we deformed the product on a nonlocality domain with a characteristic scale $\sqrt{\theta}$. While in [@kovalenko], [@moffat], [@prokhorov], [@joglekar] one replaces a local field by a “smearing” field, in our model we put the nonlocality in the product, as in the star product. Thus, the deformed quantum field theory is a particular case of a nonlocal quantum field theory.
Duality of the Kalb–Ramond field in this deformed product
---------------------------------------------------------
In this section we analyze an intriguing property of a type of deformed product. We show how the duality property of a Kalb–Ramond is changed by using a particular deformed product at lower order in $\theta$. A well-known result is that a massless field $H_{\nu\rho\sigma}=\partial_{[\nu}B_{\rho\sigma]}$ in an undeformed product is equivalent to a massless scalar field, i.e., the degrees of freedom of the antisymmetric tensor field $B_{\rho\sigma}$ are only one: $$\partial^{\mu}\phi=\frac{1}{6}\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma},$$ while the massive field is equivalent to a massive vector field, i.e., the degrees of freedom of $B_{\rho\sigma}$ are three: $$H^{\mu}=\frac{1}{6}\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma}.$$ In view of these properties, we want to generalize the duality of Kalb–Ramond in the deformed case. We show that a massless deformed Kalb–Ramond theory is dual to a U(1)-breaking theory, i.e., it is equivalent to a massive vector field theory, with the above choice of duality and choosing a particular kernel for the deformed product. The presence of deformation leaves nontrivial transverse and longitudinal modes, unlike the classical undeformed massless case. For this purpose, we are interested in a model ruled by the deformed action $$S_{H}=\frac{1}{3!}\int d^{4}x H \star_\theta H(x) := \int d^{4}x
\left[\int_{{{\mathbb{R}}}^4}\int_{{{\mathbb{R}}}^4}\,
H_{\alpha\beta\gamma}(y)
L^{\alpha\beta\gamma}_{\rho\sigma\tau}(x,y,z)\,H^{\rho\sigma\tau}(z)
\,dy\,dz\right],
\label{Moyal-prodint}$$ while the action $S_{B}=\frac{1}{3!}\int
d^{4}xH_{\mu\nu\rho}(B)H^{\mu\nu\rho}(B)$ is deformed through the integral kernel $L$ chosen to recover the U(1) gauge theory, and is given by $$L^{\alpha\beta\gamma}_{\rho\sigma\tau}(x,y,z):=
\frac{1}{6}\epsilon^{\mu\alpha\beta\gamma}
\epsilon_{\nu\rho\sigma\tau}[D_\theta]_\mu^\nu,
\label{Moyalkernel}$$ and $[D_\theta]_\mu^\nu$ reads as $$\begin{aligned}
[D_\theta]_\mu^\nu
& := & \delta(x,z)\left[g_\mu^\nu +
\frac{\theta^2}{m^2}\Delta_\mu^\nu\right]\delta(x,y) \nonumber \\
& = & \delta(x,z)\left[g_\mu^\nu
+ {\theta^{2}\over m^{2}} \left(\Box_y \delta_\mu^\nu -
\left(1-\frac{1}{\alpha}\right)
\nabla^\nu\nabla_\mu\right)\right]\delta(x,y).
\label{Moyalkernel21}\end{aligned}$$ The presence of $\alpha$ in (\[Moyalkernel21\]) reflects what we know about QED in its formulation with functional integrals in the Lorenz gauge to obtain an invertible operator on the potential $A_\mu$. Actually, its inclusion for a massive vector field model is not compelling, it is a redundant term, i.e., we are summing a vanishing term ($\alpha=\infty$). With this choice of $L$ and with the following duality transformation: $$H_{\mu\nu\rho}=\frac{1}{\sqrt{\theta}}\epsilon_{\mu\nu\rho\sigma}A^{\sigma},
\label{dual}$$ to first order in $\theta$, the fundamental (at high energy) deformed action (\[Moyal-prodint\]) is dual to the action of a massive spin-$1$ field with mass $m_\theta
=\frac{1}{\sqrt{\theta}}\sim M_{pl}$ (effective theory at low energy), i.e. $$S_{\rm dual}=S_H=\int d^{4}x\left[
\frac{1}{2}m_{\theta}^2 A_\mu A^\mu
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{2\alpha}(\partial_\mu
A^\mu)^2\right],
\label{azioneqed}$$ where $m=\frac{1}{\sqrt{\theta}}$ and $\partial_\mu A^\mu$ can be shown to vanish.
In general we can give a formal expression of $L$ to every order in $\theta$: $$[D_{\rm global}]_\mu^\nu:= \delta(x,z)[\exp{{\theta^{2}\over
m^{2}}\Delta_\mu^\nu}]\delta(x,y). \label{Moyalkernelglobal1}$$ Its expansion in powers of $\theta$ is in terms of derivatives of increasing order.
To zeroth order in $\theta$ (or $\theta=0$), we recover the duality of the massless Kalb–Ramond to a massless scalar field $\phi$, putting $A^{\sigma}=\partial^\sigma \phi$. To first order in ${\theta^{2}\over m^{2}}=\tilde{\theta}$, a deformed massless Kalb–Ramond (gauge invariant) is dual to a massive spin-$1$ field.
The operator $\Delta_\mu^\nu$ in Eq. (\[Moyalkernelglobal1\]) is a hyperbolic operator. To have a meaningful expression, we have to make a Wick rotation, so that it becomes an elliptic operator. In this way the action of the massive vector field can be seen as the lowest order (effective theory) of a (broader) nonlocal theory, which is reduced at zeroth order to a free scalar theory. A massive spin-$1$ theory may be regarded as the low-energy limit of a fundamental deformed theory, where the low-energy limit is set by the massive term $\frac{1}{\sqrt{\theta}}$.
On using different representations of deformed product it is possible to find a “dual” deformed product, i.e., we can use duality in the reverse order. Then we start with $\int d^4 x \frac{1}{2}m_1^2
A_\mu \diamondsuit A^\mu$, and after the duality transformation we have $\int d^4 x \frac{1}{2}H
\diamondsuit H$, hence we are able to write a dual deformed product to first perturbative order. In this way the duality and the deformation are connected, and when $\theta\rightarrow 0$, the deformed product and the duality disappear.
Deformed interaction term
=========================
In this section we analyze the deformed interaction term between matter and a vector field, and in its dual version.
Deformed interaction term between matter and vector field
---------------------------------------------------------
We have a U(1) theory of an unknown particle at high energy which can decay (because we do not observe it) and in principle it can produce an observable physics. Thus, we can imagine a coupling with matter and we study the decay. We construct a model in which the deformed (high energy) action of matter and its interaction term with the vector field corresponds to a low-energy action of massive fermions plus the vector-matter interaction plus correction derivative terms. The general Lagrangian density for a vector field $L(V_{\mu },\psi)$ describing all their interactions is given by $$\begin{aligned}
L &=&-\frac{1}{4}G_{\mu \nu }G^{\mu \nu }
+\frac{1}{2}M^{2}V_{\mu }V^{\mu }
+\beta\partial _{\nu }V_{\mu} V^{\mu }V^{\nu } \nonumber \\
&&+\gamma V_{\mu }V_{\nu }V^{\mu }V^{\nu }+i \overline{\psi}
\gamma_\mu \partial^\mu \psi -\overline{\psi }m\psi +V_{\mu}
\overline{\psi }\gamma ^{\mu }\psi.
\label{LN1}\end{aligned}$$ To obtain the correspondence with a vector theory we can consider different combinations, i.e., $(\bar{\psi}\diamondsuit\gamma^{\mu}\psi) A_{\mu}$, or $(\bar{\psi}\gamma^{\mu}\diamondsuit\psi) A_{\mu}$. We can take account of two possibilities in the form $$\begin{aligned}
S_{\psi\bar{\psi}A}&=&\frac{1}{2}\int d^{4}x
\,\left[\bar{\psi}\diamondsuit\left(m -\gamma^{\mu} A_{\mu}
\right)\psi + \bar{\psi}\left(m -\gamma^{\mu}
A_{\mu}\right)\diamondsuit\psi\right] \nonumber \\
& :=& \int d^{4}x \,\left\{\int_{{{\mathbb{R}}}^4}\int_{{{\mathbb{R}}}^4}\,
\left[\bar{\psi}(y) L_{{\rm matter}}(x,y,z)\left(m
-\gamma^{\mu}A_{\mu}\right)\,\psi(z)
\right.\right. \nonumber \\
&&+\bar{\psi}(y) \left.\left.\left(m
-\gamma^{\mu}A_{\mu}\right)L_{{\rm matter}}(x,y,z)\,\psi(z)
\,dy\,dz\right]\right\}, \label{Moyal-prodmatter1}\end{aligned}$$ hence with the integral kernel $L_{{\rm matter}}$ chosen to recover the vector-fermion action $$L_{{\rm matter}}(x,y,z):= \delta(x,z)(-1
+\sqrt{\theta}\gamma_\mu\partial^\mu)\delta(x,y),
\label{Moyalkernel1}$$ to first order in $\theta$, the fundamental deformed action (\[Moyal-prodmatter1\]) is an action of a massive spin-$1$ in interaction with matter plus a correction term, i.e. $$\begin{aligned}
&& S_{\rm}^{{\rm total}}
=S_A + S_{\psi\bar{\psi}A}\nonumber \\ &=&\int d^{4}x\left[
\frac{1}{2}m_{\theta}^2 A_\mu A^\mu
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ i\,m\,\sqrt{\theta}
\overline{\psi }\gamma_\mu
\partial^\mu \psi -\overline{\psi }m\psi\right.
\nonumber \\ &&-\left. \frac{1}{\sqrt{\theta}\,M}A_{\mu }
\overline{\psi }\gamma ^{\mu }\psi - \frac{i}{M} \overline{\psi }
\partial^\rho(A_{\rho }\psi)\right].
\label{azionematter1}\end{aligned}$$ To have a correct correspondence we have the condition $m\sqrt{\theta}=1$, i.e., at high energy the only possible mass is ”driven” by the $\theta$ term, and the charge of coupling is $Q=\frac{1}{M\sqrt{\theta}}$.
In general we can give a formal expression of $L$ to every order in $\theta$ as in (\[fermionic\]): $$[L_{{\rm matter-global}}]:=
\delta(x,z)[\exp{-i\sqrt{\theta}\gamma_\mu\partial^\mu}]\delta(x,y).
\label{Moyalkernelglobal}$$ We thus find that the vector field combines with matter to become, at low energy, a massive vector field, and it interacts through the standard coupling described by the previous Eq. (\[azionematter1\]). The presence of deformation “seems” to introduce a [*dynamics*]{} in a [*static*]{} system to zeroth order.
Deformed interaction term between matter and Kalb–Ramond field
--------------------------------------------------------------
We construct a model in which we show that the deformed (high energy) action of matter and its interaction term with Kalb–Ramond corresponds, after a duality transformation, to a low-energy action of massive fermions plus the vector-matter interaction plus correction derivative terms. Such a tensor field, which appears, for example, in the massless sector of a heterotic string theory, is assumed to coexist with gravity in the bulk, in a five-dimensional Randall–Sundrum scenario [@rs]. It has a well-known geometric interpretation as the spacetime torsion. We consider the most general gauge-invariant action of a second-rank antisymmetric Kalb–Ramond tensor gauge theory, including the coupling with matter modes [@rs]: $${\cal L}_{\psi\bar{\psi}H}= -\frac{1}{M_{Pl}}
\bar{\psi}[i\gamma^{\mu}\sigma^{\nu\lambda}
H_{\mu\nu\lambda}]\psi.$$ The general Lagrangian density for a vector field $L(V_{\mu},\psi)$ describing all their interactions is given by $$\begin{aligned}
L &=&-\frac{1}{4}G_{\mu \nu }G^{\mu \nu }
+\frac{1}{2}M^{2}V_{\mu }V^{\mu }
+\beta\partial _{\nu }V_{\mu} V^{\mu }V^{\nu } \nonumber \\
&&+\gamma V_{\mu }V_{\nu }V^{\mu }V^{\nu }+i \overline{\psi
}\gamma_\mu \partial^\mu \psi -\overline{\psi }m\psi +V_{\mu }
\overline{\psi }\gamma ^{\mu }\psi.
\label{LN}\end{aligned}$$
To obtain the correspondence with a vector theory we can have different combinations, i.e., $\frac{1}{M_{Pl}}(\bar{\psi}\diamondsuit\gamma^{\mu}
\sigma^{\nu\lambda}\gamma_5\psi)
H_{\mu\nu\lambda}$, or $\frac{1}{M_{Pl}}(\bar{\psi}\gamma^{\mu}\sigma^{\nu\lambda}
\gamma_5\diamondsuit\psi)
H_{\mu\nu\lambda}$. We can take account of two possibilities by writing $$\begin{aligned}
S_{\psi\bar{\psi}H}^{{\rm dual}}&=&\frac{1}{2}\int d^{4}x
\,\left[\bar{\psi}\diamondsuit\left(m
-\frac{1}{M_{Pl}}\gamma^{\mu}\sigma^{\nu\lambda}\gamma_5
H_{\mu\nu\lambda} \right)\psi + \bar{\psi}\left(m
-\frac{1}{M_{Pl}}\gamma^{\mu}\sigma^{\nu\lambda}
H_{\mu\nu\lambda}\right)\diamondsuit\psi\right] \nonumber \\
& :=&
\int d^{4}x \,\left\{\int_{{{\mathbb{R}}}^4}\int_{{{\mathbb{R}}}^4}\,
\left[\bar{\psi}(y) L_{{\rm matter}}(x,y,z)\left(m
-\frac{1}{M_{Pl}}\gamma^{\mu}\sigma^{\nu\lambda}
\gamma_5H_{\mu\nu\lambda}\right)\,\psi(z)
\right.\right.\nonumber \\
&&+\bar{\psi}(y) \left.\left.\left(m
-\frac{1}{M_{Pl}}\gamma^{\mu}\sigma^{\nu\lambda}
H_{\mu\nu\lambda}\right)L_{{\rm matter}}(x,y,z)\,\psi(z)
\,dy\,dz\right]\right\},
\label{Moyal-prodmatter}\end{aligned}$$ hence with the integral kernel $L_{{\rm matter}}$ chosen to recover the vector-fermion action $$L_{{\rm matter}}(x,y,z):= \delta(x,z)(-1
+\sqrt{\theta}\gamma_\mu\partial^\mu)\delta(x,y),
\label{Moyalkernel}$$ and with the duality transformation (\[dual\]) to first order in $\theta$, the fundamental noncommutative action (\[Moyal-prodmatter\]) is dual to the action of a massive spin-$1$ pseudo-vector field in interaction with matter plus a correction term, i.e. $$\begin{aligned}
&& S_{\rm dual}^{{\rm total}}=S_H
+ S_{\psi\bar{\psi}H}^{{\rm dual}}\nonumber \\ &=&\int d^{4}x\left[
\frac{1}{2}m_{\theta}^2 A_\mu A^\mu
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ i\,m\,\sqrt{\theta}
\overline{\psi }\gamma_\mu
\partial^\mu \psi -\overline{\psi }m\psi\right.
\nonumber \\ &&-\left. \frac{1}{\sqrt{\theta}\,M_{Pl}}A_{\mu }
\overline{\psi }\gamma ^{\mu }\psi - \frac{i}{M_{Pl}}
\overline{\psi }
\partial^\rho(A_{\rho }\psi)\right],
\label{azionematter}\end{aligned}$$ where we have used the identity $\gamma_\lambda\Sigma_{\mu\nu}=i[g_{\lambda\mu}\gamma_\nu-
g_{\lambda\nu}\gamma_\mu+ i
\epsilon_{\lambda\mu\nu\rho}\gamma_5\gamma_\rho]$. To have a correct correspondence we have the condition $m\sqrt{\theta}=1$, i.e., at high energy the only possible mass is “driven” by the $\theta$ term, and the charge of coupling is $Q=\frac{1}{M_{Pl}\sqrt{\theta}}$.
In general we can give a formal expression of $L$ to every order in $\theta$: $$[L_{{\rm matter-global}}]:=
\delta(x,z)[\exp{-i\sqrt{\theta}\gamma_\mu\partial^\mu}]\delta(x,y).
\label{Moyalkernelglobal}$$ We thus find that the Kalb–Ramond field combines with matter to become at low energy a massive vector filed, and it interacts through the standard coupling described by the previous Eq. (\[azionematter\]).
Application of the deformed product to a free scalar field
==========================================================
In this section we evaluate the Green function and dispersion relation for the free scalar action with Lagrangian density (\[freesca\]), showing its static nature. Then we analyze the dynamical scalar field theory with our deformed product, showing that, in a sense, it is equivalent to the one found by Moffat [@moffat1].
Green function and dispersion relation of the nonlocal model. A fictitious dynamical theory
-------------------------------------------------------------------------------------------
Given any differential operator $D$ on $R^4$ one can define a map $\sigma(D)$ called the [*symbol*]{} of $D$: $\sigma:D\rightarrow \sigma(D)\equiv e^{-\alpha k_\mu x^\mu }D
e^{\alpha k_\mu x^\mu }$. The associated equation of motion is $D\psi =0$, to which there corresponds the dispersion relation $\sigma(D;k,\omega)=0$, e.g. $\omega=\omega(k)$. For $D=\Box$ one obtains $$\sigma(\Box+m^2)=\alpha^2(\vec{k}\cdot\vec{k}-\omega^2) +m^{2}.$$ By setting $\sigma(D;k,\omega)=0$ one obtains the dispersion relation $$E^2=\vec{k}\cdot\vec{k} +m^{2},$$ which leads to the following wave equation: $$(\Box+m^2)\phi=0.$$ The Green function is defined by $$G(x,x')=\int d^4k
\frac{e^{ik_\mu(x^\mu-x'^\mu)}}{\sigma(\Box+m^2)},$$ and it satisfies the equation $$(\Box+m^2)G(x,x')=-\delta(x-x')=-\int d^4{k}
e^{ik_\mu(x^\mu-x'^\mu)}.$$ In our nonlocal model for a free scalar field, the symbol is $\sigma(m^2 \exp{\theta\Box};k,\omega)=m^2\exp{(-k_\mu
k^\mu\theta)}=m^2\sum_{n=0}^\infty (-k_\mu k^\mu\theta)^{n}$. If we instead consider a finite approximation, at small $\theta$ we have correction terms. Taking into account that the symbol maps $\exp{\theta\Box}\rightarrow\exp{-k_\mu k^\mu\theta}$, the Green function is $$\begin{aligned}
&&G(x,x')=m^2\int d^4k
\frac{1}{(2\pi)^4}e^{ik_\mu(x^\mu-x'^\mu)}{\exp{(-k_\mu
k^\mu\theta)}} \nonumber \\
&=& m^2\int d^4k\frac{1}{(2\pi)^4}
e^{\delta^{\mu\nu}(\sqrt{\theta}k_\mu-\frac{i}{2\sqrt{\theta}}(x_\mu-x'_\mu))
(\sqrt{\theta}k_\nu-\frac{i}{2\sqrt{\theta}}(x_\nu-x'_\nu))}
\exp{[-\frac{1}{4\theta}\delta^{\mu\nu}(x_\mu-x'_\mu)(x_\nu-x'_\nu)]}
\nonumber \\
&=& m^2\prod_{m=1}^{4}\int dk_{m}
\frac{1}{(2\pi)^4}e^{\delta^{jl}(\sqrt{\theta}k_{j}
-\frac{i}{2\sqrt{\theta}}(x_{j}-x'_{j}))
(\sqrt{\theta}k_{l}-\frac{i}{2\sqrt{\theta}}(x_{l}-x'_{l}))}
\exp{[-\frac{1}{4\theta}\delta^{\mu\nu}(x_\mu-x'_\mu)(x_\nu-x'_\nu)]}
\nonumber \\
&=& m^2
{\left(\frac{1}{64\pi\theta}\right)}^{2}
\exp{[-\frac{1}{4\theta}
\delta^{\mu\nu}(x_\mu-x'_\mu)(x_\nu-x'_\nu)]},\end{aligned}$$ where we have used the Gaussian integral, and $G(k)= (m^2\exp{-k_\mu
k^\mu\theta})^{-1}=\frac{1}{-m^2 +k^2 + \sum_{n=2}^\infty (-k_\mu
k^\mu\theta)^n}$, ($\,\,\theta=m^{-2}$), while in [@moffat1] the modified Feynman propagator in momentum space is $i\Delta_F(k)=\frac{i\exp{1/2k_\mu \tau^{\mu\nu}k_\nu\theta}}{ k^2
-m^2+i\epsilon}$. We note that the exact expression of the Green function does not show any pole, corresponding to the static nature of the global theory, while dynamics can be recovered from a perturbative expansion.
Application of the deformed product to a free scalar field theory. Analogy with the Moffat model
================================================================================================
In this section we show that the free scalar field theory with our deformed product, in a sense, is the same as that found by Moffat [@moffat1]. As we saw in the above subsection the product does not induce a dynamics, because the dispersion relation obtained by setting to zero the symbol does not have a solution. This means that we have to start with an ordinary dynamical action rewritten through this deformed product.
Our deformed product can be seen as an application of the product proposed by Moffat [@moffat1] in which we find supersymmetry in the different shape of the deformed product, but not in the superspace formalism: $$\label{diamondproduct}
({\hat\phi}_1\diamondsuit{\hat\phi}_2)(\rho)=\biggl[\exp
\biggl(-\frac{1}{2}\tau^{\mu\nu}
\frac{\partial}{\partial\rho^\mu}\frac{\partial}{\partial\eta^\nu}\biggr)
\phi_1(\rho)\phi_2(\eta)\biggr]_{\rho=\eta} $$ $$
=\phi_1(\rho)\phi_2(\rho)-\frac{1}{2}
\tau^{\mu\nu}\frac{\partial}{\partial\rho^\mu}
\phi_1(\rho)\frac{\partial}{\partial\rho^\nu}\phi_2(\rho)+O(\tau^2).$$ Here, by comparison with the Moffat product in [@moffat1], $\tau^{\mu\nu}=\delta^{\mu\nu}\theta$ and $\rho=\eta$. Hence our product is a particular application of it and it is associative and commutative. In our case the modified Feynman propagator ${\bar\Delta}_F$ is defined by the vacuum expectation value of the time-ordered $\star$-product $$i{\bar\Delta}_F(x-y)\equiv\langle 0\vert
T(\phi(x)\diamondsuit\phi(y))\vert 0 \rangle
=\frac{i}{(2\pi)^4}\int\frac{d^4k\exp[-ik(x-y)]
\exp[\frac{1}{2}(k^2\theta)]}{k^2-m^2+i\epsilon}.$$ In momentum space this gives $$i{\bar\Delta}_F(k)=\frac{i\exp[\frac{1}{2}(k^2\theta)]}
{k^2-m^2+i\epsilon},$$ which reduces to the standard commutative field theory form for the Feynman propagator $$i\Delta_F(k)=\frac{i}{k^2-m^2+i\epsilon}$$ in the limit $\vert\theta^{\mu\nu}\vert\rightarrow 0$. The free-field $\phi^2$ theory is nonlocal, unlike the corresponding one in ordinary local field theory, resulting in a modified Feynman propagator ${\bar\Delta}_F(k)$ and modified dispersion relation. It turns out to be a particular case of that treated by Moffat in [@moffat1].
Concluding remarks
==================
In our paper, a modification of the standard product used in local field theory by means of an associative deformed product has been proposed. We have built a class of deformed products, one for every spin $S=0,1/2,1$, that induces a nonlocal theory, displaying different form for different fields. This type of deformed product is naturally supersymmetric and it has an intriguing duality.
It now remains to be seen whether a suitable variant of our construction can lead to a product different from the one used by Moffat [@moffat1].
G. Esposito is grateful to the Dipartimento di Scienze Fisiche of Federico II University, Naples, for hospitality and support. Conversations with Fedele Lizzi have been very helpful.
G. V. Efimov, [*Commun. Math. Phys.*]{} [**5**]{} (1967) 42; Nonlocal Interactions of Quantized Fields (Moscow, 1977). G. Kleppe and R. P. Woodard, [*Nucl. Phys. B*]{} [**388**]{} (1992) 81. J. W. Moffat, [*Phys. Rev. D*]{} [**41**]{} (1990) 1177; B. J. Hand and J. W. Moffat, [*Phys. Rev. D*]{} [**43**]{} (1991) 1896; D. Evens, J. W. Moffat, G. Kleppe and R. P. Woodard, [*Phys. Rev. D*]{} [**43**]{} (1991) 499. J. W. Moffat, [*Phys. Rev. D*]{} [**37**]{} (1990) 1177. N. V. Krasnikov, [*Phys. Lett. B*]{} [**195**]{} (1987) 377. A. O. Barvinsky and G. A. Vilkovisky, [*Nucl. Phys. B*]{} [**282**]{} (1987) 163; [*Nucl. Phys. B*]{} [**333**]{} (1990) 471. A. Pais and G. E. Uhlenbeck, [*Phys. Rev.*]{} [**79**]{} (1950) 145. D. A. Eliezer and R. P. Woodard, [*Nucl. Phys. B*]{} [**325**]{} (1989) 389. H. Hata, [*Phys. Lett. B*]{} [**217**]{} (1989) 438; [*Nucl. Phys. B*]{} [**329**]{} (1990) 698. K. S. Stelle, [*Phys. Rev. D*]{} [**16**]{} (1977) 953; J. Julve and M. Tonin, [*Nuovo Cim. B*]{} [**46**]{} (1978) 137. P. Kristensen and C. Moller, [*K. Dan. Vidensk. Selsk. Mat-Fys. Medd.*]{} [**27**]{} (1952) 7. N. Seiberg and E. Witten, [*J. High Energy Phys.*]{} 99[**09**]{}: 032 (1999). N. Seiberg, L. Susskind and N. Toumbas, [*J. High Energy Phys.*]{} 00[**06**]{}: 044 (2000). J. S. Bell, [*Physics (N.Y.)*]{} [**1**]{} (1964) 195. A. Aspect, P. Grangier and G. Roger, [*Phys. Rev. Lett*]{} [**47**]{} (1981) 460; ibid. [**49**]{} (1982) 91; P.G. Kwiat, K. Mattle, H. Weinfurter and A. Zeilinger, ibid. [**75**]{} (1995) 4337; W. Tittel, J. Brendel, H. Zbinden and N. Gisin, ibid. [**81**]{} (1998) 3563; G. Weihs, T. Jennewein, C. Simon, H. Weinfurter and A. Zeilinger, ibid. [**81**]{} (1998) 5039; A. Aspect, Nature (London) [**398**]{} (1999) 189. J. W. Moffat, [*Phys. Lett. B*]{} [**506**]{} (2001) 193. H. Weyl, [*The Theory of Groups and Quantum Mechanics*]{} (Dover, New York, 1931). E. Wigner, [*Phys. Rev.*]{} [**40**]{} (1932) 749. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, [*Ann. Phys.*]{} [**111**]{} (1978) 61; D. Sternheimer, [*AIP Conf. Proc.*]{} [**453**]{} (1998) 107; M. Kontsevich, [*Lett. Math. Phys.*]{} [**66**]{} (2003) 157. O. V. Man’ko, V. I. Man’ko, G. Marmo and P. Vitale, [*Phys. Lett. A*]{} [**360**]{} (2007) 522. C. S. Chu and P. M. Ho, [*Nucl. Phys. B*]{} [**550**]{} (1999) 151; V. Schomerus, [*J. High Energy Phys.*]{} 99[**06**]{}: 030 (1999); J. Lukierski, A. Nowicki, H. Ruegg and V. N. Tolstoy, [*Phys. Lett. B*]{} [**264**]{} (1991) 331; J. Lukierski, A. Nowicki and H. Ruegg, [*Phys. Lett. B*]{} [**293**]{} (1992) 344; B. Jurco, S. Schraml, P. Schupp and J. Wess, [*Eur. Phys. J. C*]{} [**17**]{} (2000) 521; A. Klimyk and K. Schmudgen, [*Quantum Groups and Their Representations*]{} (Springer, Berlin, 1997). J. E. Moyal, [*Proc. Cambridge Phil. Soc.*]{} [**45**]{} (1949) 99. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, [*Ann. Phys. (N.Y.)*]{} [**110**]{} (1978) 111. C. Zachos, Geometrical evaluation of star products, hep-th/9912238. T. Filk, [*Phys. Lett. B*]{} [**376**]{} (1996) 53. S. Minwalla, M. Van Raamsdonk and N. Seiberg, [*J. High Energy Phys.*]{} 00[**02**]{}: 020 (2000). H. Weyl, [*Z. Phys.*]{} [**46**]{} (1927) 1. O. V. Man’ko, V.I. Man’ko and G. Marmo, [*Phys. Scr.*]{} [**62**]{} (2000) 446. O. V. Man’ko, V. I. Man’ko and G. Marmo, [*J. Phys. A*]{} [**35**]{} (2002) 699. A. Connes, M. R. Douglas and A. Schwarz, [*J. High Energy Phys.*]{} 98[**02**]{}: 003 (1998). M. R. Douglas and C. Hull, [*J. High Energy Phys.*]{} 98[**02**]{}: 008 (1998). D. Minic, hep-th/9909022. A. Jevicki and S. Ramgoolam, [*J. High Energy Phys.*]{} 99[**04**]{}: 032 (1999); Z. Chang, [*Eur. Phys. J. C*]{} [**17**]{} (2000) 527; H. Steinacker, [*Adv. Theor. Math. Phys.*]{} [**4**]{} (2000) 155. R. J. Finkelstein, [*Lett. Math. Phys.*]{} [**38**]{} (1996) 53. L. Castellani, [*Phys. Lett. B*]{} [**292**]{} (1992) 93. L. Brink and J. H. Schwarz, [*Phys. Lett. B*]{} [**100**]{} (1981) 310. S. Ferrara and M. A. Lledó, [*J. High Energy Phys.*]{} 00[**05**]{}: 008 (2000). P. Kosianski, J. Lukierski and P. Maalanka, Quantum deformations of space-time SUSY and noncommutative superfield theory, hep-th/0011053. D. Klemm, S. Penati and L. Tamassia, [*Class. Quant. Grav.*]{} [**20**]{} (2003) 2905. A. Nowicki, E. Sorace and M. Tarlini, [*Phys. Lett. B*]{} [**302**]{} (1993) 419. V. A. Alebastrov and G. V. Efimov, [*Commun. Math. Phys.*]{} [**31**]{} (1973) 1. J. M. Gracia Bondia, F. Lizzi, G. Marmo and P. Vitale, [*J. High Energy Phys.*]{} 02[**04**]{}: 026 (2002). A. F. Ferrari et al., [*Phys. Rev. D*]{} [**74**]{} (2006) 125016; A. F. Ferrari, H. O. Girotti, M. Gomes, A. Yu. Petrov, A. A. Ribeiro and A. J. da Silva, [*Phys. Lett. B*]{} [**577**]{} (2003) 83. S. Mukhi, [*Phys. Rev. D*]{} [**20**]{} (1979) 1839. S. G. Kovalenko, in [*Dubna 1992, Weak and Electromagnetic Interactions in Nuclei*]{}, 505–514. L. V. Prokhorov, [*JETP*]{} [**45**]{} (1963) 791. S. D. Joglekar, hep-th/0601006; Hai-Jun Wang, quant-ph/0512162. B. Mukhopadhyaya, S. Sen, S. Sen and S. SenGupta, [*Phys. Rev. D*]{} [**70**]{} (2004) 066009.
[^1]: Electronic address: digrezia@na.infn.it
[^2]: Electronic address: giampiero.esposito@na.infn.it
[^3]: Electronic address: gennaro.miele@na.infn.it
|
---
abstract: 'In this paper we deal with a free boundary problem modeling the growth of nonnecrotic tumors. The tumor is treated as an incompressible fluid, the tissue elasticity is neglected and no chemical inhibitor species are present. We re-express the mathematical model as an operator equation and by using a bifurcation argument we prove that there exist stationary solutions of the problem which are not radially symmetric.'
address:
- 'Institut f[ü]{}r Angewandte Mathematik, Leibniz Universit[ä]{}t Hannover, Welfengarten 1, 30167 Hannover, Germany. '
- 'Institut f[ü]{}r Angewandte Mathematik, Leibniz Universit[ä]{}t Hannover, Welfengarten 1, 30167 Hannover, Germany. '
author:
- Joachim Escher
- 'Anca-Voichita Matioc'
title: Bifurcation analysis for a free boundary problem modeling tumor growth
---
Introduction and the main result
================================
Cristini et all. obtained in [@VCris] a new mathematical formulation of an existing model (see [@BC; @FR; @Gr]) which describes the evolution of nonnecrotic tumors in both vascular and avascular regimes. As widely used in the modelling, the tumor is treated as an incompressible fluid and tissue elasticity is neglected. Cell-to-cell adhesive forces are modeled by surface tension at the tumor-tissue interface. The growth of the tumor is governed by a balance between cell-mitosis and apoptosis (programed cell-death). The rate of mitosis depends on the concentration of nutrient and no inhibitor chemical species are present. This new model is obtained by considering different intrinsic time and length scales for the tumor evolution which are integrated by means of algebraic manipulations into the model. The model presented in [@BC; @FR; @Gr], has been studied extensively by different authors [@BF05; @Cui; @CE; @CE1; @FR]. It is known that the moving boundary problems associated to it are well-posed locally in time [@CE1; @FR] and, as a further common characteristic, there exists, for parameters in a certain range, a unique radially symmetric equilibrium [@Cui; @CE; @CE1; @FR01]. This results have been verified to hold true also for the model deduced in [@VCris], cf. [@VCris; @EM; @EM1]. The authors of [@CE; @CE1; @CEZ; @FR01; @ZC] show by using the theorem on bifurcation from simple eigenvalues due to Crandall and Rabinowitz that in the situations they consider there exists besides the unique radially symmetric equilibrium other nontrivial equilibria. Though the problems they consider are different, these nontrivial steady-state solutions are asymptotically identical near the circular equilibrium. Numerical experiments suggest also for the model [@VCris] that there may exist stationary solutions which are no longer radially symmetric.
In this paper we focus on the general, i.e. non-symmetric situation, when the tumor domain is arbitrary and look for nontrivial steady-states of the model [@VCris]. Additionally to the well-posedness of the associated moving boundary problem, stability properties of the unique radially symmetric solution are established in [@EM1]. Particularly, it is shown that if $G$, the rate of mitosis relative to the relaxation mechanism is large, then the circular equilibrium is unstable, which also suggests existence of nontrivial stationary solutions. When studying the set of equilibria we deal with a free boundary problem which is reduced to an operator equation between certain subspaces of the small Hölder spaces over the unit circle $h^{m+\beta}({\mathbb{S}})$. We apply then the theorem on bifurcations from simple eigenvalues to this equation and obtain infinitely many bifurcation branches consisting only of stationary solutions of our model. Near the circular equilibrium these solutions match perfectly the ones found in [@CE; @CE1; @CEZ; @FR01].
The outline of the paper is as follows: we present in the subsequent section the mathematical model and the main result, Theorem \[BifTh\]. Section 3 is dedicated to the proof of the Theorem \[BifTh\].
The mathematical model and the main result
==========================================
The two-dimensional system associated to the model [@VCris] is described in detail in [@EM]. The steady-state solution of the moving boundary problem presented there are precisely the solutions of the free boundary problem $$\label{eq:problem}
\left \{
\begin{array}{rllllll}
\Delta \psi &=& f(\psi ) &\text{in} & \Omega, \\[1ex]
\Delta p &=& 0 & \text{in}& \Omega, \\[1ex]
\psi &=& 1 & \text{on}& \partial \Omega, \\[1ex]
p&=& \kappa_{{\partial}{\Omega}}- AG \displaystyle\frac{ |x|^2}{4} &\text{on}& \partial\Omega,\\[1ex]
G\displaystyle\frac{{\partial}\psi}{{\partial}n} -\displaystyle\frac{{\partial}p}{{\partial}n} -AG \displaystyle\frac{n\cdot x}{2} &=&0
& \text{on}& \partial \Omega. \\[1ex]
\end{array}
\right.$$ The fully nonlinear system consists of two decoupled Dirichlet problems, one for the rate $\psi$ at which nutrient is added to the tumor domain ${\Omega}$, and one for the pressure $p$ inside the tumor. These two variables are coupled by the fifth equation of . Hereby $\kappa_{{\partial}{\Omega}}$ stands for the curvature of ${\partial}{\Omega}$ and $A$ describes the balance between the rate of mitosis (cell proliferation) and apoptosis (naturally cell death). The function $f\in C^\infty([0,\infty))$ has the following properties $$\label{eq:conditions}
f(0)=0 \qquad\text{and}\qquad f'(\psi)>0 \quad\text{for}\quad \psi\geq0.$$ We already know from [@EM Theorem 1.1] that
\[T:2\] Given $(A,G) \in(0, f(1))$, there exists a unique radially symmetric solution $D(0,R_A)$ to problem . The radius $R_A$ of the stationary tumor depends only on the parameter A and decreases with respect to this variable.
Hence $D(0,R_A)$ is a solution of for all $G\in{\mathbb{R}}.$ Thus, we may use $G$ as a bifurcation parameter to obtain also other solutions of . This is in accordance with the numerical simulation [@VCris].
In order to determine steady-states of we introduce a parametrisation for the unknown tumor domain ${\Omega}.$ Therefore we define the small Hölder spaces $h^{r}({\mathbb{S}})$, $r\geq 0$, as closure of the smooth functions $C^\infty({\mathbb{S}})$ in the Hölder space $C^{r}({\mathbb{S}})$, whereby, ${\mathbb{S}}$ stands for the unit circle and we identify functions on ${\mathbb{S}}$ with $2\pi$-periodic functions on ${\mathbb{R}}.$ Furthermore, we fix $\alpha\in(0,1)$ and use functions belonging to the open neighbourhood $${\mathcal{V}}:=\{ \rho\in h^{4+\alpha}({\mathbb{S}})\,:\, \|\rho\|_{C({\mathbb{S}})}<1/4 \},$$ of the zero function in $ h^{4+\alpha}({\mathbb{S}})$ to parametrise domains close to the discus $D(0,R_A)$. Given $\rho\in{\mathcal{V}},$ we define domain $${\Omega}_\rho:=\left\{x\in{\mathbb{R}}^2\, :\, |x|<R_A\left(1+\rho\left(x/|x|\right)\right)\right\}\cup\{0\},$$ with boundary ${\partial}{\Omega}_\rho=\Gamma_\rho:=\left\{R\left(1+\rho(x)\right)x\,:\,x\in{\mathbb{S}}\right\}.$ Given $x\in\Gamma_\rho,$ the real number $\rho(x/|x|)$ is the ratio of the signed distance from $x$ to the circle $R_A\cdot {\mathbb{S}}$ and $R_A$. If ${\Omega}={\Omega}_{\rho}$ for some ${\rho}\in{\mathcal{V}}$, problem re-writes With this notation is equivalent to the following system of equations $$\label{3}
\left \{
\begin{array}{rlcllll}
\Delta \psi &=& f(\psi ) &\text{in} &\Omega _{\rho}, \\[1ex]
\Delta p &=& 0 & \text{in}& \Omega _{\rho}, \\[1ex]
\psi &=& 1 & \text{on}& \Gamma _{\rho}, \\[1ex]
p&=& \kappa_{\Gamma_{\rho}}- AG \displaystyle\frac{ |x|^2}{4} &\text{on}&\Gamma _{\rho}, \\[1ex]
\left<G\nabla \psi -\nabla p- AG\displaystyle \frac{ x}{2}, \nabla N_{\rho}\right> &=&0 & \text{on}& \Gamma _{\rho},
\end{array}
\right.$$ where $N_{\rho}:A(3R_A/4,5R_A/4)\to{\mathbb{R}}$ is the function defined by $N_\rho(x):=|x|-R_A-R_A\rho(x/|x|)$ for all $x$ in the annulus $$A(3R_A/4,5R_A/4):=\{x\in{\mathbb{R}}^2\,:\, 3R_A/4<|x|<5R_A/4\}.$$
We re-expressed now problem as an abstract operator equation on unit circle ${\mathbb{S}}$. To this scope we introduce for each ${\rho}\in{\mathcal{V}}$ the Hanzawa diffeomorphism $\Theta_\rho:{\mathbb{R}}^2\to{\mathbb{R}}^2$ by $$\Theta_\rho(x)=Rx+\frac{Rx}{|x|}{\varphi}(|x|-1)\rho\left(\displaystyle\frac{x}{|x|}\right),$$ where the cut-off function ${\varphi}\in C^\infty({\mathbb{R}},[0,1])$ satisfies $${\varphi}(r)=\left\{
\begin{array}{llll}
&1,& |r|\leq 1/4,\\[2ex]
&0,& |r|\geq 3/4,
\end{array}
\right.$$ and additionally $\max|{\varphi}'(r)|<4.$ In can be easily seen that $\Theta_\rho $ is a diffeomorphism mapping ${\Omega}:=D(0,1)$ onto $ {\Omega}_{\rho}$, i.e. $\Theta_\rho\in \mbox{\it{Diff}}\,^{4+\alpha}({\Omega},{\Omega}_\rho)\cap \mbox{\it{Diff}}\,^{4+\alpha}({\mathbb{R}}^2,{\mathbb{R}}^2).$ As we did in [@EM1] we define for each ${\rho}\in{\mathcal{V}}$ the function ${\mathcal{T}}({\rho}):=\psi\circ\Theta_{\rho}$, whereby $\psi$ is the solution of the semilinear Dirichlet problem $$\label{7}
\left \{
\begin{array}{rlcllll}
\Delta \psi &=& f(\psi ) &\text{in} &\Omega _{\rho}, \\[1ex]
\psi &=& 1 & \text{on}& \Gamma _{\rho},
\end{array}
\right.$$ respectively for ${\rho}\in{\mathcal{V}}$ and $G\in{\mathbb{R}}$ we set ${\mathcal{S}}(G,{\rho}):=p\circ\Theta_{\rho}$, where $p$ solves $$\label{8}
\left \{
\begin{array}{rlcllll}
\Delta p &=& 0 & \text{in}& \Omega _{\rho}, \\[1ex]
p&=& \kappa_{\Gamma_{\rho}}- AG \displaystyle\frac{ |x|^2}{4} &\text{on}&\Gamma _{\rho}.
\end{array}
\right.$$ With this notation, our problem reduces to the operator equation $$\label{9}
\text{$\Phi(G,{\rho})=0 $ in $h^{1+\alpha}({\mathbb{S}})$}$$ where $\Phi:{\mathbb{R}}\times {\mathcal{V}}\to h^{1+\alpha}({\mathbb{S}})$ is the nonlinear and nonlocal operator defined by $$\label{D:P}
\Phi(G,{\rho}):=\left<G\nabla \left({\mathcal{T}}({\rho})\circ\Theta^{-1}_{\rho}\right) -\nabla \left({\mathcal{S}}(G,{\rho})\circ\Theta^{-1}_{\rho}\right)- AG\displaystyle \frac{ x}{2}, \nabla N_{\rho}\right>\circ\Theta_{\rho}.$$ The function $\Phi$ is smooth $\Phi\in C^\infty({\mathbb{R}}\times{\mathcal{V}}, h^{1+\alpha}({\mathbb{S}}))$, cf. [@EM1], and the steady-state $D(0,R_A)$ corresponds to the function ${\rho}=0$ which is solution of for all $G\in{\mathbb{R}}.$
Therefore, we shall refer to $\Sigma:=\{(G,0)\,:\, G\in{\mathbb{R}}\} $ as the set of trivial solutions of . The main result of this paper, Theorem \[BifTh\] states that there exist infinitely many local bifurcation branches emerging from $\Sigma$ and which consist only of solutions of the original problem . Our analysis is based on the fact that we could determine an explicit formula for the partial derivative ${\partial}_{\rho}\Phi(G,0)$ cf. [@EM1]. Given ${\rho}\in h^{4+\alpha}({\mathbb{S}}),$ we let ${\rho}=\sum_{k\in{\mathbb{Z}}}{\widehat}{\rho}(k)x^k$ denote its associated Fourier series. Then we have: $$\label{eq:PHI}
{\partial}\Phi(G,0)\left[\sum_{k\in{\mathbb{Z}}}{\widehat}{\rho}(k)x^k\right]
=\underset{k\in {\mathbb{Z}}}\sum \mu_k(G) \widehat {\rho}(k) x^k,$$ where the symbol $(\mu_k(G))_{k\in{\mathbb{Z}}}$ is given by the relation $$\label{eq:symbol}
\mu_k(G):= -\frac{1}{R_A^3}|k|^3+ \frac{1}{R_A^3}|k| - G \left(\frac{A}{2}\frac{u_{|k|}'(1)}{u_{|k|}(1)}+A-f(1)\right),$$ and $u_{|k|}\in C^\infty([0,1])$ is the solution of the initial value problem $$\label{uniculu}
\left\{
\begin{array}{rlll}
u''+\displaystyle\frac{2n+1}{r}u' &=& R_A^2f'(v_0)u, \quad & 0<r<1,\\[2ex]
u(0) &=& 1, \\[2ex]
u'(0)&=&0,
\end{array}
\right.$$ when $n=|k|$ and $v_0:={\mathcal{T}}(0).$ By the weak maximum principle $v_0$ is radially symmetric, so that we identified in $v_0$ with its restriction to the segment $[0,1].$ Particularly, and imply that $$\label{eq:spectr}
\sigma({\partial}\Phi(G,0))=\{\mu_k(G)\,:\, k\in{\mathbb{Z}}\}.$$
Before stating our main result we study first the properties of the symbol $(\mu_k(G))_{k\in{\mathbb{Z}}}.$
\[L:pre\] There exists $M>0$ such that $$\label{eq:est}
\text{$u_k'(1)\leq \frac{M}{2k+2}$ and $u_k(1)\leq1+\frac{M}{(2k+1)(2k+3)}$}$$ for all $k\in{\mathbb{N}}.$
Let $k\in {\mathbb{N}}$ be fixed. Since $$\begin{aligned}
\label{50}
u_k(r)=1+ \int_0^r \frac{R_A^2}{s^{2k+1}}\int_0^s \tau^{2k+1}f'(v_0(\tau))u_k(\tau)\, d\tau \, ds, \quad 0\leq r\leq 1,\end{aligned}$$ we deduce that $u_k$ is strictly increasing for all $k\in{\mathbb{N}}.$ Let now $v:=u_{k+1}-u_k$. From we obtain $$\left \{
\begin{array}{rlll}
v''+\displaystyle\frac{2k+1}{r}v'&=& R^2 f'(v_0)v-\displaystyle\frac{2}{r}u_{k+1}', &0< r< 1,\\[1ex]
v'(0)&=& 0,\\[1ex]
v(0)&=& 0.
\end{array}
\right.$$ Furthermore, we have $$\begin{aligned}
\underset{t\to 0}\lim \frac{v'(t)}{t} &= \underset{t\to 0}\lim \frac{u_{k+1}'(t)-u_k'(t)}{t}= R_A^2f'(v_0(0))\left( \frac{1}{2k+4}-\frac{1}{2k+2}\right)\\[1ex]
&= -R_A^2 f'(v_0(0)) \frac{2}{(2k+4)(2k+2)} < 0,\end{aligned}$$ which implies by that $v'(t) <0$ for $t\in (0, \delta)$ and some $\delta<1.$ Thus, $v$ is decreasing on $(0, \delta).$ Let now $t\in [0,1]$ and set $m_t:= \min_{[0,t]} v \leq 0$. A maximum principle argument shows that the nonpositive minimum must be achieved at $t$, $m_t=v(t)$ which implies $u_{k+1}(t)\leq u_k(t)$ for all $t\in [0,1]$. Particularly, $u_{k+1}(1) \leq u_k(1)$.
We prove now the estimate for $u_k'(1).$ Setting $M:= R_A^2 u_0(1) \max_{[0,1]} f'(v_0)$ we obtain, due to , that $$\begin{aligned}
u_k'(1)&= \int_0^1 R_A^2 \int_0^s \tau^{2k+1}f'(v_0(\tau))u_k(\tau) \, d\tau \\[1ex]
&\leq M \int_0^1 \, \int_0^s \tau^{2k+1} \, d\tau= \frac{M}{(2k+1)(2k+3)}.\end{aligned}$$ The estimate for $u_k$ follows similarly.
By Lemma \[L:pre\] we know that $u_k'(1)/u_k(1)\to_{k\to\infty}0.$ Therefore, we may define for $k\in{\mathbb{N}}$ with $$\frac{A}{2} \frac{u_k'(1)}{u_k(1)}+ A-f(1) \neq 0,$$ the constant $$\begin{aligned}
\label{Gk}
G_k:=\displaystyle\frac{\displaystyle{-\frac{1}{R^3}k^3+\frac{1}{R^3}k}}{\displaystyle{\frac{A}{2} \frac{u_k'(1)}{u_k(1)}+ A-f(1)}},\end{aligned}$$ which is the only natural number such that $\mu_k(G_k)=0.$ We do this since we cannot estimate whether $\mu_k(G)$, with $k$ small are zero or not. For example, we know from [@EM1] that $\mu_1(G)=0$ for all $G\in{\mathbb{R}},$ which makes the things difficult when trying to apply bifurcation theorems to . In virtue of Lemma \[L:pre\] we have:
\[propGk\] There exists $k_1\in{\mathbb{N}}$ with the property that $0<G_k <G_{k+1}$ for all $k\geq k_1.$
From we obtain by partial integration that $$\begin{aligned}
k\cdot (u'_{k}(1)-u'_{k+1}(1)) =& \,R_A^2 \int_0^1 k \tau^{2k+1} f'(v_0(\tau)) [u_k(\tau)-\tau^2 u_{k+1}( \tau)] \, d\tau \\[1ex]
= &\, R_A^2 k\left.\frac{\tau^{2k+2}}{2k+2}f'(v_0(\tau)) [u_k(\tau)-\tau^2 u_{k+1}( \tau)] \right|_0^1\\[1ex]
&- \int_0^1 k\frac{\tau^{2k+2}}{2k+2}\left\{ f''(v_0(\tau)) \left [u_k(\tau)-\tau^2 u_{k+1}( \tau)\right] \phantom{\int}\right.\\[1ex]
&+ \left. f'(v_0(\tau))\phantom{\int} \hspace{-0.2cm}[u_k'(\tau)-2\tau u_{k+1}(\tau) -\tau^2 u_{k+1}'(\tau)] \right\}\, d\tau \\[1ex]
\leq & \,R_A^2f'(1) \frac{k}{2k+2} (u_k(1)- u_{k+1}( 1))+ L\frac{k}{(k+1)^2},\end{aligned}$$ with a constant $L$ independent of $k$. Letting now $k \to \infty$ we get $$\begin{aligned}
\label{anan+1}
k\cdot (u'_{k}(1)-u'_{k+1}(1))\underset {k \to \infty }\longrightarrow 0.\end{aligned}$$ Having this estimates at hand we can prove now the assertion of the lemma. For simplicity we set $C:= 2(f(1)-A)/A >0$, $a_k:=u_k'(1),$ and $b_k:=1+u_k(1).$ In virtue of $$G_k = \frac{2}{AR^3} \frac{\displaystyle{k^3-k}}{\displaystyle{C- \frac{u_k'(1)}{u_k(1)}}}.$$ we compute $$\begin{aligned}
G_{k+1}> G_k \Leftrightarrow \, & \frac{(k+1)^3-(k+1)}{C- \displaystyle\frac{a_{k+1}}{1+b_{k+1}}} > \frac{k^3-k}{C- \displaystyle\frac{a_k}{1+b_k}}\\[1ex]
\Leftrightarrow \,& C(k+1)^3-C(k+1)-Ck^3+Ck \\[1ex]
&> \frac{a_{k+1}}{1+b_{k+1}}(k-k^3)+ \frac{a_k}{1+b_k} [(k+1)^3-(k+1)]\\[1ex]
\Leftrightarrow \, & C(3k^2+3k) > \frac{a_{k+1}(k-k^3)+a_k(k^3+3k^2+2k)}{(1+b_k)(1+b_{k+1})}\\[1ex]
&\phantom{aaaaaaaaaa} +\frac{a_{k+1}b_k (k-k^3)+a_k b_{k+1}(k^3+3k^2+2k)}{(1+b_k)(1+b_{k+1})}\\[1ex]
\Leftrightarrow \, & C(3k^2+3k) > \frac{k^3(a_k-a_{k+1})+a_{k+1}k+a_k(3k^2+2k)}{(1+b_k)(1+b_{k+1})}\\[1ex]
&\phantom{aaaaaaaaaa} +\frac{a_{k+1}b_k (k-k^3)+a_k b_{k+1}(k^3+3k^2+2k)}{(1+b_k)(1+b_{k+1})}.\end{aligned}$$ Taking into consideration the relations and , we find a positive integer $k_1$ such that strict inequality holds in the last relation above for all $k\geq k_1.$
Finally, we set $$G_\bullet:= \frac{\displaystyle\frac{1}{R_A^3 }k_1^3
-\frac{1}{R_A^3 }k_1}{\underset{0\leq k\leq k_1}\min \left \{\left |f(1)-A-\displaystyle\frac{A}{2}
\displaystyle\frac{u_k'(1)}{u_k(1)}\right|:f(1)-A-\displaystyle\frac{A}{2} \frac{u_k'(1)}{u_k(1)}\not =0\right\}}.$$
The main result of this paper is the following theorem:
\[BifTh\] Assume $A\in(0,f(1))$ and that $$\label{eq:feri}
\frac{A}{2} \frac{u_0'(1)}{u_0(1)}+A-f(1) \not = 0$$ holds true. Let $l\geq 2$ be fixed and $k\in {\mathbb{N}},$ $k\geq 1$ such that $G_{kl}>G_\bullet.$ The pair $(G_{kl}, 0)$ is a bifurcation point from the trivial solution $\Sigma$. More precisely, in a suitable neighbourhood of $(G_{kl}, 0),$ there exists a smooth branch of solutions $\left(G^{kl}(\varepsilon), \rho^{kl}(\varepsilon )\right)$ of problem . For ${\varepsilon}\to0$, we have the following asymptotic expressions: $$\begin{aligned}
&&G^{kl}(\varepsilon )= G_{kl}+ O({\varepsilon}), \\[1ex]
&&\rho^{kl}(\varepsilon )={\varepsilon}\cos(kls)+O({\varepsilon}^2).\end{aligned}$$ Moreover, any other $G> G_\bullet,$ $G\not\in \{G_{kl}:k\geq1\},$ is not a bifurcation point.
Condition is not to restrictive. Particularly, if $R_A=1$ and $f={\mathop{\rm id}\nolimits}_{[0,\infty)}$ we have shown in [@EM1] that is satisfied.
Proof of the main result
========================
The main tool we use when proving Theorem \[BifTh\] is the classical bifurcation result on bifurcations from simple eigenvalues due to Crandall and Rabinowitz:
\[ThmCR\]Let $X, $ $Y$ be real Banach spaces and $G(\lambda,u)$ be a $C^q$ $(q\geq3)$ mapping from a neighbourhood of a point $(\lambda_0,u_0)\in {\mathbb{R}}\times X$ into $Y$. Let the following assumptions hold:
- $G(\lambda_0,u_0)=0, \, \partial_\lambda G(\lambda_0,u_0)=0,$
- ${\mathop{\rm Ker}\nolimits}{\partial}_uG(\lambda_0,u_0)$ is one dimensional, spanned by $v_0,$
- ${\mathop{\rm Im}\nolimits}{\partial}_uG(\lambda_0,u_0)$ has codimension 1,
- $ {\partial}_\lambda{\partial}_\lambda G(\lambda_0,u_0)\in {\mathop{\rm Im}\nolimits}{\partial}_uG(\lambda_0,u_0)$, ${\partial}_\lambda{\partial}_u G(\lambda_0,u_0)v_0\notin {\mathop{\rm Im}\nolimits}{\partial}_uG(\lambda_0,u_0).$
Then $(\lambda_0,u_0)$ is a bifurcation point of the equation $$\label{CR}
G(\lambda,u)=0$$ in the following sense: In a neighbourhood of $(\lambda_0,u_0)$ the set of solutions of equation consists of two $C^{q-2}$ curves $\Sigma_1$ and $\Sigma_2$, which intersect only at the point $(\lambda_0,u_0).$ Furthermore, $\Sigma_1$, $\Sigma_2$ can be parameterised as follows:
- $(\lambda, u(\lambda)),$ $|\lambda-\lambda_0| \, \mbox{is} \, small, \, u(\lambda_0)=u_0, \, u'(\lambda_0)=0,$
- $(\lambda({\varepsilon}),u({\varepsilon})),$ $|{\varepsilon}| \, \mbox{is} \, small, \, (\lambda(0),u(0))=(\lambda_0,u_0), \, u'(0)=v_0.$
We want to apply Theorem \[ThmCR\] to the particular problem . As we already mentioned $\Phi(G,0)=0$ for all $G\in{\mathbb{R}}.$ However, since we can not estimate the eigenvalues $\mu_k(G), $ $1\leq k\leq k_1$ we have to eliminate them from the spectrum of ${\partial}\Phi(G,0),$ and also to reduce the dimensions of the eigenspace corresponding to an eigenvalue $\mu_k(G)$, $k\geq k_1,$ which we may chose to be equal to $0$ if $G$ is large enough, to one. This is due to the fact that the dimension of the eigenspace corresponding to an arbitrary eigenvalue $\mu_k(G)$ is larger then $2,$ since $x^k $ and $x^{-k}$ are eigenvectors of this eigenvalue.
This may be done by restricting the operator $\Phi$ to spaces consisting only of $2\pi/l-$periodic and even functions. Given $k\in{\mathbb{N}}$ and $l\in{\mathbb{N}}, l\geq2$, we define $$h^{k+\alpha}_{e,l}({\mathbb{S}}):=\{ \text{${\rho}\in h^{k+\alpha}({\mathbb{S}})\,:\, {\rho}(x)={\rho}({\overline}x)$ and $ {\rho}(x)={\rho}(e^{2\pi i/l}x)$ for all $x\in{\mathbb{S}}$}\},$$ where for $x\in {\mathbb{C}},$ ${\overline}x$ denotes its complex conjugate. Set further ${\mathcal{V}}_{e,l}:={\mathcal{V}}\cap h^{4+\alpha}_{e,l}({\mathbb{S}}).$ By identifying functions on ${\mathbb{S}}$ with $2\pi-$periodic functions on ${\mathbb{R}}$ we can expand ${\rho}\in h^{k+\alpha}_{e,l}({\mathbb{S}})$ in the following way $${\rho}(s)=\sum_{k=0}^\infty a_k \cos(kls),$$ where $a_k=2{\widehat}{\rho}(kl)$ for $k\geq0.$ With this notation we have:
\[L:rest2\] Given $l\geq 2$, the operator $\Phi$ maps smoothly ${\mathbb{R}}\times {\mathcal{V}}_{e,l}$ into $h^{1+\alpha}_{e,l}({\mathbb{S}}),$ i.e. $$\phi\in C^\infty({\mathbb{R}}\times {\mathcal{V}}_{e,l}, h^{1+\alpha}_{e,l}({\mathbb{S}})).$$
The proof is similar to that of [@AM Lemma 5.5.2] and therefore we omit it.
Finally, we come to the proof of our main result.
Fix now $l\geq 2$. We infer from relation that the partial derivative of the smooth mapping $\Phi:{\mathbb{R}}\times {\mathcal{V}}_{e,l}\to h^{1+\alpha}_{e,l}({\mathbb{S}})$ with respect to ${\rho}$ at $(G,0)$ is the Fourier multiplier $$\label{drhophi}
{\partial}_{\rho}\Phi(G, 0)\left[ \sum_{k=0}^\infty a_k \cos(kls)\right] = \sum_{k=0}^\infty \mu_{kl} (G)a_k \cos(kls)$$ for all ${\rho}= \sum_{k=0}^\infty a_k \cos(klx)\in h^{4+\alpha}_{e,l}({\mathbb{S}}),$ where $\mu_{kl}(G)$, $k\in{\mathbb{N}},$ is defined by . Our assumption implies that $\mu_0(G) \not = 0.$
The proof is based on the following observation: if $G>G_\bullet$ and $\mu_{kl}(G)=0$ then it must hold that $G=G_{kl} $ and $kl> k_1.$ Indeed, we notice that if $\mu_{kl}(G)=0,$ for some $kl\leq k_1$, then $$\begin{aligned}
G_\bullet < G=
\frac{\displaystyle \frac{1}{R_A^3 }(kl)^3 -\displaystyle\frac{1}{R_A^3 }kl}
{ \left |f(1)-A-\displaystyle\frac{A}{2}\displaystyle \frac{u_{kl}'(1)}{u_{kl}(1)}\right|}
\leq G_\bullet,\end{aligned}$$ since, by and $l\geq2$, $kl\geq2.$ If $kl> k_1$, then $\mu_{kl}(G)=0 $ iff $$\begin{aligned}
G =\frac{-\displaystyle\frac{1}{R_A^3 }(kl)^3 +\displaystyle\frac{1}{R_A^3 }kl}
{\displaystyle\frac{A}{2} \displaystyle \frac{u_{kl}'(1)}{u_{kl}(1)}+A-f(1)}=G_{kl}.\end{aligned}$$
Let $1\leq k\in {\mathbb{N}}$ be given such that $G_{kl}>G_\bullet$. From Lemma \[propGk\] and the previous observation we get $\mu_{ml}(G_{kl})\neq 0$ for $m\neq k.$ Our assumption ensures that the Fr' echet derivative ${\partial}_{\rho}\Phi(G_{ml}, 0)$ of the restriction $\Phi:{\mathbb{R}}\times {\mathcal{V}}_{e,l} \subset h^{4+\alpha}_{e,l}({\mathbb{S}})\to h^{1+\alpha}_{e,l}({\mathbb{S}}),$ which is given by relation , has a one dimensional kernel spanned by $\cos(kls).$ We also observe that its image is closed and has codimension equal to one.
We are left now to prove that the transversality condition $(iv)$ of Theorem \[ThmCR\] holds. Since $G_{kl}>G_\bullet$, our observation implies $kl\geq k_1+1$ and so $$-\left( \frac{A}{2}\frac{u_{kl}'(1)}{u_{kl}(1)}+ A-f(1)\right)>0.$$ Further on, we get from relation and that $${\partial}_G{\partial}_{\rho}\Phi(0)\left[\cos(kls)\right]=- \left( \frac{A}{2}\frac{u_{kl}'(1)}{u_{kl}(1)}+ A-f(1)\right) \cos(kls),$$ and since $\cos(kls)\not \in {\mathop{\rm Im}\nolimits}{\partial}_{\rho}\Phi(G_{kl}, 0)$ we deduce that the assumption of Theorem \[ThmCR\] are all verified. By applying Theorem \[ThmCR\] we obtain the bifurcation result stated in Theorem \[BifTh\] and the asymptotic expressions for the bifurcation branches $\left(G^{kl}(\varepsilon), \rho^{kl}(\varepsilon )\right).$
Moreover, if $G> G_\bullet$ and $G\neq G_{kl}$ for all $k$ with $kl\geq k_1+1$, then it must hold that $\mu_{kl}(G)\neq 0$ for all $k\in {\mathbb{N}},$ and we may apply [@AM Theorem 4.5.1] to obtain that ${\partial}_{\rho}\Phi(G, 0)$ is an isomorphisms. The Implicit function theorem then states that $(G, 0)$ is not a bifurcation point and the proof is completed.
The possible steady-states of problem are depicted in Figure $1$.
\[F:bifu\] $$\includegraphics[width=0.9\linewidth]{bifu2D.eps}$$
[9999]{}
: [*Symmetric-breaking bifurcation for free boundary problems*]{}, [[Indiana Univ. Math. J.]{}]{} [**[54]{}**]{}, 927–947 (2005).
: “Analytic Theory of Global Bifurcation: An Introduction”, Princeton, New Jersey, 2003.
: [*Growth of nonnecrotic tumors in the presence and absence of inhibitors*]{} [ Math. Biosci.]{}, [**[130]{}**]{}, 151–181 (1995).
: [*Bifurcation from simple eigenvalues*]{}, [ Journal of Functional Analysis ]{}, [**[8]{}**]{}, 321–340 (1971).
: [*Nonlinear simulation of tumor growth*]{}, [ Journal of Mathematical Biology]{}, [**46**]{}, 191–224 (2003).
: [*Analysis of a free boundary problem modeling tumor growth*]{}, [ Acta Mathematica Sinica, English Series]{}, [**21**]{} (5), 1071–1082 (2005).
: [*Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors*]{}, [ SIAM J. Math. Anal.]{}, [**39**]{} (1), 210–235 (2007).
: [*Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth*]{}, [ Comm. Part. Diff. Eq.]{}, [**33**]{} (4), 636–655 (2008).
: [*Bifurcation for a free boundary problem with surface tension modelling the growth of multi-layer tumors*]{}, [ J. Math. Anal. Appl.]{}, [**337**]{} (1), 443–457 (2008).
: [*Advection diffusion models for solid tumors [*in vivo*]{} and related free-boundary problems*]{}, [ Math. Mod. Meth. Appl. Sci.]{}, [**10**]{}, 379–408 (2000).
: [*Radially symmetric growth of nonnecrotic tumors*]{}, to appear in Nonlinear Differential Equations and Applications.
: [*Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors*]{}, submitted.
: [*Analysis of a mathematical model for the growth of tumors*]{}, [ J. Math. Biol.]{}, [**38**]{}, 262–284 (1999).
: [*Symmetry-breaking bifurcation of analytic solutions to free boundary problems*]{}, [[Trans. Amer. Math. Soc.]{}]{}, [**[353]{}**]{}, 1587–1634 (2001).
: [*On the growth and stability of cell cultures and solid tumors*]{}, [ J. Theor. Biol.]{}, [**56**]{}, 229–242 (1976).
: “Modelling and analysis of nonnecrotic tumors”, S" udwestdeutcher Verlag f" ur Hochschulschriften, Saarbrücken, 2009.
: [*Bifurcations for a multidimensional free boundary problem modeling the growth of tumor cord*]{}, [ Nonlinear Analysis: real Word Applications]{}, [**10**]{}, 2990–3001 (2009).
|
---
abstract: 'Let $R$ be the free $\CC$-algebra on $x$ and $y$ modulo the relations $x^5=yxy$ and $y^2=xyx$ endowed with the $\ZZ$-grading $\deg x=1$ and $\deg y=2$. Let $\BB_3$ denote the blow up of $\CC\PP^2$ at three non-colliear points. The main result in this paper is that the category of quasi-coherent $\cO_{\BB_3}$-modules is equivalent to the quotient of the category of $\ZZ$-graded $R$-modules modulo the full subcategory of modules $M$ such that for each $m \in M$, $(x,y)^nm=0$ for $n \gg 0$. This reduces almost all representation-theoretic questions about $R$ to algebraic geometric questions about the del Pezzo surface $\BB_3$. For example, the generic simple $R$-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, and Van den Bergh, imply that $R$ is a noetherian domain of global dimension three'
address: ' Department of Mathematics, Box 354350, Univ. Washington, Seattle, WA 98195'
author:
- 'S. Paul Smith'
title: 'A non-commutative homogeneous coordinate ring for the third del Pezzo surface'
---
[^1]
Introduction
============
We will work over the field of complex numbers.
The surface obtained by blowing up $\PP^2$ at three non-colinear points is, up to isomorphism, independent of the points. It is called the [third del Pezzo surface]{} and we will denote it by $\BB_3$.
Let $R$ be the free $\CC$-algebra on $x$ and $y$ modulo the relations $$\label{relations}
x^5=yxy \qquad \hbox{and } \qquad y^2=xyx.$$ Give $R$ a $\ZZ$-grading by declaring that $$\deg x=1 \qquad \hbox{and } \qquad \deg y=2.$$
In this paper we prove there is a surprisingly close relationship between the non-commutative algebra $R$ and the third del Pezzo surface. This relationship can be exploited to obtain a deep understanding of the representation theory of $R$.
\[thm.main1\] Let $R$ be the non-commutative algebra $\CC[x,y]$ with relations (\[relations\]). Let $\Gr R$ be the category of $\ZZ$-graded left $R$-modules. Then there is an equivalence of categories $$\Qcoh \BB_3 \equiv {{\Gr R}\over{\sT}}$$ where the left-hand side is the category of quasi-coherent $\cO_{\BB_3}$-modules and the right-hand side is the quotient category modulo the full subcategory $\sT$ consisting of those modules $M$ such that for each $m \in M$, $(x,y)^nm=0$ for $n \gg 0$.
Theorem \[thm.main1\] is a consequence of the following result.
\[thm.main2\] Let $R$ be the non-commutative algebra $\CC[x,y]$ with relations (\[relations\]). There is an automorphism $\s$ of $\BB_3$ having order six, and a line bundle $\cL$ on $\BB_3$ such that $R$ is isomorphic to the twisted homogeneous coordinate ring $$B(\BB_3,\cL,\s) : = \bigoplus_{n \ge 0} H^0(\BB_3,\cL_n)$$ where $$\cL_n:= \cL \otimes (\s^*)\cL \otimes \cdots \otimes (\s^*)^{n-1}\cL.$$
Results of Artin, Tate, and Van den Bergh now imply that $R$ is a 3-dimensional Artin-Schelter regular algebra and therefore has the following properties.
Let $R$ be the algebra $\CC[x,y]$ with relations (\[relations\]). Then
1. $R$ is a left and right noetherian domain;
2. $R$ has global homological dimension 3;
3. $R$ is Auslander-Gorenstein and Cohen-Macaulay in the non-commutative sense;
4. the Hilbert series of $R$ is the same as that of the weighted polynomial ring on three variables of weights 1, 2, and 3;
5. $R$ is a finitely generated module over its center [@ST94 Cor. 2.3];
6. $R^{(6)}:=\oplus_{n=0}^\infty R_{6n}$ is isomorphic to $\bigoplus_{n=0}^\infty H^0(\BB_3\cO(-nK_{\BB_3})$;
7. $\Spec R^{(6)}$ is the anti-canonical cone over $\BB_3$, i.e., the cone obtained by collapsing the zero section of the total space of the anti-canonical bundle over $\BB_3$.
This close connection between $R$ and $\BB_3$ means that almost all aspects of the representation theory of $R$ can be expressed in terms of the geometry of $\BB_3$. We plan to address this question in another paper.
The justification for calling $R$ a non-commutative homogeneous coordinate ring for $\BB_3$ is the similarity between the equivalence of categories in Theorem \[thm.main1\] and following theorem of Serre:
> if $X \subset \PP^n$ is the scheme-theoretic zero locus of a graded ideal $I$ in the polynomial ring $S=\CC[x_0,\ldots,x_n]$ with its standard grading, and $A=S/I$, then there is an equivalence of categories $$\label{eq.Serre}
> \Qcoh X \equiv {{\Gr A}\over{\sT}}$$ where the right-hand side is the quotient category of $\Gr A$, the category of graded $A$-modules, by the full subcategory $\sT$ of modules supported at the origin.
The author is grateful to the following people: Amer Iqbal for directing the author to some papers in high energy physics where the algebra $R$ appears in a hidden form; Paul Hacking and Sándor Kovács for passing on some standard results about del Pezzo surfaces and algebraic geometry; and Darin Stephenson for telling the author that the algebra $R$ is an iterated Ore extension, and for providing some background and guidance regarding his two papers cited in the bibliography.
The non-commutative algebra $R=\CC[x,y]$ with $x^5=yxy$ and $y^2=xyx$ {#sect.nc.ring}
=====================================================================
The following result is a straightforward calculation. The main point of it is to show that $R$ has the same Hilbert series as the weighted polynomial ring on three variables of weights 1, 2, and 3.
\[Stephenson\] \[prop.steph\] The ring $R:=\CC[x,y]$ with defining relations $$x^5=yxy \qquad \hbox{and} \qquad y^2=xyx$$ is an iterated Ore extension of the polynomial ring $\CC[w]$. Explicitly, if $\zeta$ is a fixed primitive $6^{\th}$ root of unity, then:
1. $R= \CC[w][z;\s][x;\tau,\d]$ where $\s \in \Aut \CC[z]$, $\tau \in \Aut \CC[w,z]$, and $\d$ is the $\tau$-derivation defined by $$\begin{aligned}
\s(w)& =\zeta w,
\\
\tau(w) &= - \zeta^2 w, \quad \tau(z) = \zeta z,
\\
\d(w) & = z, \quad \d(z) =- w^2;
\end{aligned}$$
2. A set of defining relations of $R=\CC[z,w,x]$ is given by $$\begin{aligned}
zw = & \zeta wz,
\\
xw = & -\zeta^2wx+z ,
\\
xz = & \zeta zx- w^2;\end{aligned}$$
3. $R$ has basis $\{ w^iz^jx^k \; | \; i,j,k \ge 0\}$;
4. $R$ is a noetherian domain;
5. The Hilbert series of $R$ is $(1-t)^{-1}(1-t^2)^{-1}(1-t^3)^{-1}$.
Define the elements $$\begin{aligned}
w: & = \, y-x^2
\\
z: & = \, xw + \zeta^2 wx
\\
&= \, xy +\zeta^2 yx - \zeta x^3\end{aligned}$$ of $R$. Since $y$ belongs to the ring generated by $x$ and $w$, $\CC[x,y]=\CC[x,w]=\CC[x,w,z]$. It is easy to check that $$\label{steph.relns}
zw=\zeta wz, \quad xw= z- \zeta^2 wx, \quad xz=\zeta zx - w^2.$$
Let $R'$ be the free algebra $\CC\langle w,x,z \rangle$ modulo the relations in (\[steph.relns\]). We want to show $R'$ is isomorphic to $R$. We already know there is a homomorphism $R' \to R$ and we will now exhibit a homomorphism $R \to R'$ by showing there are elements $x$ and $Y$ in $R'$ that satisfy the defining relations for $R$. Define the element $Y:=w+x^2$ in $R'$. A straightforward computation in $R'$ gives $$xwx-x^2w = w^2+wx^2$$ so $$Y^2= w^2+x^2w + wx^2 + x^4 = xwx+x^4=xYx.$$ In the next calculation we make frequent use of the fact that $1-\xi+\xi^2=0$. Deep breath... $$\begin{aligned}
YxY& = (w+x^2)xw +wx^3 + x^5
\\
& = (w+x^2) (z-\zeta^2wx)+ \big[ wx^3 + x^5 \big]
\\
& = x^2 z -\zeta^2 x^2wx + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big]
\\
& = x(\zeta zx -w^2) -\zeta^2 x(z-\zeta^2wx)x + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big]
\\
& =(\zeta-\zeta^2)xzx - xw^2 - \zeta xwx^2 + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big]
\\
& =(\zeta-\zeta^2)(\zeta zx -w^2)x - (z-\zeta^2wx) w -\zeta(z- \zeta^2 wx)x^2
\\
& \qquad \qquad + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big]
\\
& =(\zeta^2-\zeta^3)zx^2 -(\zeta-\zeta^2)w^2x - zw +\zeta^2 wxw -\zeta zx^2 - wx^3
\\
& \qquad \qquad + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big]
\\
& =(\zeta^2-\zeta^3-\zeta)zx^2 +\zeta^2 wxw + \big[ (1-\zeta) wz -\zeta w^2x + x^5 \big]
\\
& = \zeta^2 wxw + \big[ (1-\zeta) wz -\zeta w^2x + x^5 \big]
\\
& = \zeta^2 w(z-\zeta^2wx) + \big[ -\zeta^2 wz -\zeta w^2x + x^5 \big]
\\
& = x^5.
\end{aligned}$$ Since $YxY=x^5$, $R$ is isomorphic to $R'$. Hence $R$ is an iterated Ore extension as claimed. The other parts of the proposition follow easily.
It is an immediate consequence of the relations that $x^6=y^3$. Hence $x^6$ is in the center of $R$.
The del Pezzo surface $\BB_3$
=============================
Let $\BB_3$ be the surface obtained by blowing up the complex projective plane $\PP^2$ at three non-collinear points. We will write $E_1$, $E_2$, and $E_3$ for the exceptional curves associated to the blowup.
The $-1$-curves on $\BB_3$ lie in the following configuration $$\label{six.lines}
\qquad
\UseComputerModernTips
\xymatrix{
&& & \save []+<0cm,0.1cm>*\txt<4pc>{$\scriptstyle{E_2}$} \restore \ar@{-}[dddlll]
& \ar@{-}[dddrrr] \save []+<0cm,0.1cm>*\txt<4pc>{$\scriptstyle{E_3}$} \restore
\\
& \save []+<-0.3cm,0cm>*\txt<4pc>{$\scriptstyle{L_1}$} \restore
\ar@{-}[rrrrr] &&&&& \save []+<0.5cm,0cm>*\txt<4pc>{$\scriptstyle{X=0}$} \restore &
\\
\save []+<-0.2cm,0.1cm>*\txt<4pc>{$\scriptstyle{Z=0}$} \restore &&&&&&&
\save []+<0.3cm,0.1cm>*\txt<4pc>{$\scriptstyle{Y=0}$} \restore
&&&&&
\\
\save []+<-0.2cm,-0.1cm>*\txt<4pc>{$\scriptstyle{s=0}$} \restore &&&&&&& \save []+<0.3cm,-0.1cm>*\txt<4pc>{$\scriptstyle{u=0}$} \restore &
\\
&
\save []+<-0.3cm,0cm>*\txt<4pc>{$\scriptstyle{E_1}$} \restore
\ar@{-}[rrrrr] &&&&& \save []+<0.5cm,0cm>*\txt<4pc>{$\scriptstyle{t=0}$} \restore
\\
&&& \ar@{-}[uuulll] \save []+<0cm,-0.1cm>*\txt<4pc>{$\scriptstyle{L_3}$} \restore
& \save []+<0cm,-0.1cm>*\txt<4pc>{$\scriptstyle{L_2}$} \restore \ar@{-}[uuurrr] }$$ where $L_1$, $L_2$, and $L_3$ are the strict transforms of the lines in $\PP^2$ spanned by the points that are blown up. (The labeling of the equations for the $-1$-curves will be justified shortly.)
The union of the $-1$-curves is an anti-canonical divisor and is, of course, ample.
The Picard group of $\BB_3$ {#sect.Pic}
---------------------------
The Picard group of $\BB_3$ is free abelian of rank four. We will identify it with $\ZZ^4$ by using the ordered basis $$H,\; -E_2, \; -E_1, \; -E_3$$ where the $E_i$s are the exceptional curves over the points blown up and $H$ is the strict transform of a line in $\PP^2$ in general position, i.e., missing the points being blown up. Thus $$H=(1,0,0,0), \quad E_1=(0,0,-1,0), \quad E_2=(0,-1,0,0), \quad E_3=(0,0,0,-1).$$ The canonical divisor $K$ is $-3H+E_1+E_2+E_3$ so the anti-canonical divisor is $$-K=(3,1,1,1).$$
Cox’s homogeneous coordinate ring
---------------------------------
By definition, Cox’s homogeneous coordinate ring [@Co] for a complete smooth toric variety is $$S:= \bigoplus_{[\cL] \in \Pic X} H^0(X,\cL).$$ For the remainder of this paper $S$ will denote Cox’s homogeneous coordinate ring for $\BB_3$.
Let $X,Y,Z,s,t,u$ be coordinate functions on $\CC^6$. One can present $\BB_3$ as a toric variety by defining it as the orbit space $$\BB_3:={{\CC^6 -W}\over{(\CC^\times)^4}}$$ where the irrelevant locus, $W$, is the union of nine codimension two subspaces, namely $$\label{irrel.locus}
\begin{array}{ccc}
\begin{array}{c}
X=t=0\\
Y=s=0\\
Z=u=0
\end{array}
\qquad
\begin{array}{c}
X=Y=0\\
Y=Z=0\\
Z=X=0
\end{array}
\qquad
\begin{array}{c}
s=t=0\\
u=t=0\\
s=u=0
\end{array}
\end{array}$$ and $(\CC^\times)^4$ acts with weights $$\begin{array}{rrrrrr}
X & \phantom{x} & 1 & 1 & 0 & 1 \\
Y & \phantom{x} & 1 & 0 & 1 & 1 \\
Z & \phantom{x} & 1 & 1 & 1& 0 \\
s & \phantom{x} & 0 & -1 & 0 & 0\\
t & \phantom{x} & 0 & 0 & -1 & 0 \\
u & \phantom{x} & 0 & 0 & 0 & -1
\end{array}.$$ Therefore $S$ is the $\ZZ^4$-graded polynomial ring $$S=\CC[X,Y,Z,s,t,u]$$ with the degrees of the generators given by their weights under the $(\CC^\times)^4$ action. It follows from Cox’s results [@Co Sect. 3] that $$\Qcoh \BB_3 \equiv {{\Gr(S,\ZZ^4)}\over{\sT}}$$ where $\Gr(S,\ZZ^4)$ is the category of $\ZZ^4$-graded $S$-modules and $\sT$ is the full subcategory consisting of all direct limits of modules supported on $W$.
The labelling of the $-1$-curves the diagram (\[six.lines\]) is explained by the existence of the morphisms $$\UseComputerModernTips
\xymatrix{
& \BB_3 \ar[dl] \ar[dr]
\\
\PP^2 && \PP^2
}
\qquad
\UseComputerModernTips
\xymatrix{
& \save []+<-0.2cm,0.2cm>*\txt<4pc>{$\text{ $(X,Y,Z,s,t,u) $}$} \restore \ar[dl] \ar[dr]
\\
\save []+<-0.6cm,-0.2cm>*\txt<4pc>{$\text{$(Xsu,Ytu,Zst)$}$} \restore&&
\save []+<+0.3cm,-0.2cm>*\txt<4pc>{$\text{$(YZt,XZs,XYu)$}$} \restore
}$$ that collapse the $-1$-curves.
An order six automorphism $\s$ of $\BB_3$
-----------------------------------------
The cyclic permutation of the six $-1$-curves on $\BB_3$ extends to a global automorphism of $\BB_3$ of order six. We now make this explicit.
The category of graded rings consists of pairs $(A,\G)$ consisting of an abelian group $\G$ and a $\G$-graded ring $A$. A morphism $(f,\theta):(A,\G) \to (B,\Upsilon)$ consists of a ring homomorphism $f:A \to B$ and a group homomorphism $\theta:\G \to \Upsilon$ such that $f(A_i) \subset B_{\theta(i)}$ for all $i \in \G$.
Let $\tau:S \to S$ be the automorphism induced by the cyclic permutation $$\label{defn.tau}
\UseComputerModernTips
\xymatrix{
X \ar[r] & u \ar[r] & Y \ar[r]_\tau & t \ar[r] & Z \ar[r] & s \ar@{}@/_1pc/[lllll] | {<}
}$$ and let $\theta:\ZZ^4 \to \ZZ^4$ be left multiplication by the matrix $$\theta= \begin{pmatrix}
2 & -1 & -1 & -1 \\
1 & -1 & -1 & 0 \\
1 & 0 & -1 & -1 \\
1 & -1 & 0 & -1
\end{pmatrix}.$$ Then $(\tau,\theta):(S,\ZZ^4) \to (S,\ZZ^4)$ is an automorphism in the category of graded rings. Because the irrelevant locus (\[irrel.locus\]) is stable under the action of $\tau$, $\tau$ induces an automorphism $\s$ of $\BB_3$. It follows from the definition of $\tau$ that $\s$ cyclically permutes the six $-1$-curves
Since $(\tau,\theta)^6=\id_{(S,\ZZ^4)}$ the order of $\s$ divides six. But the action of $\s$ on the set of $-1$-curves has order six, so $\s$ has order six as an automorphism of $\BB_3$.
Fix a primitive cube root of unity $\omega$. The left action of $\theta$ on $\ZZ^4=\Pic \BB^3$ has eigenvectors $$v_1=\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ \end{pmatrix}, \quad
v_2=\begin{pmatrix} 3 \\ 1 \\ 1 \\ 1 \\ \end{pmatrix}, \quad
v_3=\begin{pmatrix} 0 \\ 1 \\ \omega \\ \omega^2 \\ \end{pmatrix}, \quad
v_4=\begin{pmatrix} 0 \\ 1 \\ \omega^2 \\ \omega \\ \end{pmatrix},$$ with corresponding eigenvalues $-1$, $+1$, $\omega^2$, $\omega$.
A twisted hcr for $\BB_3$
=========================
A sequence of line bundles on $\BB_3$ {#sect.Dn}
-------------------------------------
We will blur the distinction between a divisor $D$ and the class of the line bundle $\cO(D)$ in $\Pic\BB_3$.
We define a sequence of divisors: $D_0$ is zero; $D_1$ is the line $L_1$; for $n \ge 1$ $$D_n:=(1+\theta + \cdots + \theta^{n-1})(D_1).$$ We will write $\cL_n:=\cO(D_n)$. Therefore $$\cL_n = \cL_1 \otimes \s^* \cL_1 \otimes \ldots \otimes (\s^*)^{n-1} \cL_1.$$ For example, $$\begin{array}{cc}
\begin{array}{l}
\cO(D_1)=\cL_1 = \cO(1,1,0,1) \\
\cO(D_2)=\cL_2 = \cO(1,1,0,0) \\
\cO(D_3)=\cL_3 = \cO(2,1,1,1) \\
\cO(D_4)=\cL_4 = \cO(2,1,0,1) \\
\cO(D_5)=\cL_5 = \cO(3,2,1,1) \\
\cO(D_6)=\cL_6 = \cO(3,1,1,1) \\
\phantom{xyz} \\
\phantom{xyz}
\end{array}
\begin{array}{l}
=\cO(L_1 )\\
=\cO(L_1 - E_3 )\\
=\cO(L_1 + L_2 - E_3) \\
=\cO(L_1 + L_2 -E_1 - E_3) \\
=\cO(L_1 + L_2 + L_3 -E_1 - E_3) \\
=\cO(L_1 + L_2 + L_3 -E_1 -E_2 - E_3) \\
=\cO( 3H -E_1 -E_2 - E_3)\\
=\cO(-K).
\end{array}
\end{array}$$
\[lem.Dn\] Suppose that $m \ge 0$ and $0 \le r \le 5$. Then $$D_{6m+r} = D_{r}-mK.$$
Since $\theta^6=1$, $$\begin{aligned}
\sum_{i=0}^{6m+r-1}\theta^i & =(1+\theta+\cdots+\theta^5)\sum_{j=0}^{m-1}\theta^{6j} + \theta^{6m}(1+\theta + \cdots+\theta^{r-1})
\\
&=(1+\theta + \cdots+\theta^{r-1}) + m(1+\theta+\cdots+\theta^5)\end{aligned}$$ where the sum $1+\theta+\cdots +\theta^{r-1}$ is empty and therefore equal to zero when $r=0$. Therefore $D_{6m+r}= D_{r} + mD_6 = D_{r}-mK$, as claimed.
Vanishing results
-----------------
For a divisor $D$ on a smooth surface $X$, we write $$h^i(D):=\dim H^i(X,\cO_{X}(D)).$$ We need to know that $h^1(D)=h^2(D)=0$ for various divisors $D$ on $\BB_3$.
If $D-K$ is ample, then the Kodaira Vanishing Theorem implies that $h^0(K-D)=h^1(K-D)=0$ and Serre duality then gives $h^2(D)=h^1(D)=0$.
The notational conventions in section \[sect.Pic\] identify $\Pic \BB_3$ with $\ZZ^4$ via $$aH-cE_1-bE_2-dE_3 \equiv (a,b,c,d)$$ where $H$ is the strict transform of a line in $\PP^2$. The intersection form on $\BB_3$ is given by $$H^2=1, \qquad E_i.E_j=-\d_{ij}, \qquad H.E_i=0,$$ so the induced intersection form on $\ZZ^4$ is $$(a,b,c,d) \cdot(a',b',c',d')=aa'-bb'-cc'-dd'.$$
\[lem.van\] Let $D=(a,b,c,d) \in \Pic\BB_3\equiv \ZZ^4$. Suppose that $$\label{eq.ample2}
(a+3)^2> (b+1)^2+(c+1)^2+(d+1)^2$$ and $$\label{eq.ample3}
b,\, c,\, d >-1, \qquad \hbox{and} \qquad a+1>b+c,\, b+d, \, c+d.$$ Then $D-K$ is ample, whence $h^1(D)=h^2(D)=0$.
The effective cone is generated by $L_1$, $L_2$, $L_3$, $E_1$, $E_2$, and $E_3$ so, by the Nakai-Moishezon criterion, $D-K$ is ample if and only if $(D-K)^2>0$ and $D.L_i>0$ and $D.E_i>0$ for all $i$. Now $D-K=(a+3,b+1,c+1,d+1)$, so $(D-K)^2>0$ if and only if (\[eq.ample2\]) holds and $(D-K).D'>0$ for all effective $D'$ if and only if (\[eq.ample3\]) holds.
Hence the hypothesis that (\[eq.ample2\]) and (\[eq.ample3\]) hold implies that $D-K$ is ample. The Kodaira Vanishing Theorem now implies that $h^0(K-D)=h^1(K-D)=0$. Serre duality now implies that $h^2(D)=h^1(D)=0$.
\[lem.h1D\] For all $n \ge 0$, $h^1(D_n)=h^2(D_n)=0$.
The value of $D_n$ for $0 \le n \le 6$ is given explicitly in section \[sect.Dn\]. We also note that $D_7 =
D_1+D_6=(4,2,1,2)$. It is routine to check that conditions (\[eq.ample2\]) and (\[eq.ample3\]) hold for $D=D_n$ when $n=0,2,3,4,5,6,7$. Hence $h^1(D_n)=h^2(D_n)=0$ when $n=0,2,3,4,5,6,7$.
We now consider $D_1$ which is the $-1$-curve $X=0$. (Since $(D_1-K).D_1 = 0$, $D_1-K$ is not ample so we can’t use Kodaira Vanishing as we did for the other small values of $n$.) It follows from the exact sequence $0 \to \cO_{\BB_3} \to \cO_{\BB_3}(D_1) \to \cO_{D_1}(D_1) \to 0$ that $H^p(\BB_3,\cO_{\BB_3}(D_1)) \cong H^p(\BB_3,\cO_{D_1}(D_1))$ for $p=1,2$. However, $\cO_{D_1}(D_1)$ is the normal sheaf on $D_1$ and as $D_1$ can be contracted to a smooth point on the second del Pezzo surface, $\cO_{D_1}(D_1) \cong \cO_{D_1}(-1)$. But $D_1 \cong \PP^1$ so $H^p(\BB_3, \cO_{D_1}(D_1))
\cong H^p(\PP^1,\cO(-1))$ which is zero for $p=1,2$. It follows that $h^1(D_1)=h^2(D_1)=0$.
Thus $h^1(D_n)=h^2(D_n)=0$ when $0 \le n \le 7$. We have also shown that $D_n-K$ is ample when $2 \le n \le 7$. We now argue by induction. Suppose $n \ge 8$ and $D_{n-6}-K$ is ample. Now $D_n-K=D_{n-6}-K -K$. Since a sum of ample divisors is ample, $D_n-K$ is ample. It follows that $h^1(D_n)=h^2(D_n)=0$.
The twisted homogeneous coordinate ring $B(\BB_3,\cL,\s)$
---------------------------------------------------------
We assume the reader is somewhat familiar with the notion of twisted homogeneous coordinate rings. Standard references for that material are [@AV], [@ATV1], [@ATV2], and [@AZ].
The notion of a $\s$-ample line bundle [@AV] plays a key role in the study of twisted homogeneous coordinate rings. Because $\cL_6$ is the anti-canonical bundle and therefore ample, $\cL_1$ is $\s$-ample. This allows us to use the results of Artin and Van den Bergh in [@AV] to conclude that the twisted homogeneous coordinate ring $$B(\BB_3,\cL,\s) = \bigoplus_{n=0}^\infty B_n = \bigoplus_{n=0}^\infty H^0(\BB_3,\cL_n)$$ is such that $$\Qcoh \BB_3 \equiv {{\Gr B}\over{\sT}}$$ where $\sT$ is the full subcategory of $\Gr B$ consisting of those modules $M$ such that for each $m \in M$, $B_nm=0$ for $n \gg 0$. It then follows that $B$ has a host of good properties—see [@AV] for details.
We will now compute the dimensions $h^0(D_n)$ of the homogeneous $B_n$ of $B$. We will show that $B$ has the same Hilbert series as the non-commutative ring $R$, i.e., the same Hilbert series as the weighted polynomial ring with weights 1, 2, and 3. As usual we write $\chi(D) =h^0(D)-h^1(D)+h^2(D)$. The Riemann-Roch formula is $$\chi(\cO(D))=\chi(\cO)+\hbox{${{1}\over{2}}$} D\cdot (D-K)$$ where $K$ denotes the canonical divisor. We have $\chi(\cO_{\BB_3})=1$ and $K_{\BB_3}^2= 6$.
\[lem.h0D\] Suppose $0 \le r \le 5$. Then $$h^0(D_{6m+r})=\begin{cases} (m+1)(3m+r) & \text{if $r \ne 0$,}
\\
3m^2+3m+1 & \text{if $r=0$}
\end{cases}$$ and $$\sum_{n=0}^\infty h^0(D_n) t^n ={{1}\over{(1-t)(1-t^2)(1-t^3)}} .$$
Computations for $1 \le r \le 5$ give $D_{r}^2=r-2$ and $D_{r}\cdot K= -r $. Hence $$\begin{aligned}
\label{T.series}
\chi(D_{6m+r}) &= 1 + \hbox{${{1}\over{2}}$} (D_{r}-mK)\cdot (D_{r}-(m+1) K)
\\
&= 1 + \hbox{${{1}\over{2}}$} \big(D_r^2-(2m+1)D_r.K +6m(m+1)^2\big)
\\
& = (3m+r)(m+1)\end{aligned}$$ for $m \ge 0$ and $1 \le r \le 5$. When $r=0$, $D_r=0$ so $$\chi(D_{6m}) = 3m^2+3m+1.$$ By Lemma \[lem.h1D\], $\chi(D_n)=h^0(D_n)$ for all $n \ge 0$ so it follows from the formula for $\chi(D_n)$ that $$\label{eq.chi.diff}
h^0(D_{n+6})-h^0(D_n)=n+6$$ for all $n \ge 0$.
To complete the proof of the lemma, it suffices to show that $h^0(D_n)$ is the coefficient of $t^n$ in the Taylor series expansion $$f(t):={{1}\over{(1-t)(1-t^2)(1-t^3)}} = \sum_{n=0}^\infty a_n t^n.$$ Because $$(1-t^6)f(t) = (1-t+t^2)(1-t)^{-2} = 1 + \sum_{n=1}^\infty nt^n,$$ it follows that $$\begin{aligned}
(1-t^6)f(t) & = a_0+a_1t+\cdots+a_5t^5 + \sum_{n=0}^\infty (a_{n+6}-a_n)t^{n+6}
\\
& = 1+t+2t+\cdots+5t^5 + \sum_{n=6}^\infty nt^n
\\
& = 1+t+2t+\cdots+5t^5 + \sum_{n=0}^\infty (n+6)t^{n+6}.\end{aligned}$$ In particular, if $0 \le r \le 5$, $a_r=h^0(D_r)$. We now complete the proof by induction. Suppose we have proved that $a_i=h^0(D_i)$ for $i \le n+5$. By comparing the expressions in the Taylor series we see that $$a_{n+6}= a_n + (n +6)= h^0(D_n)+n+6= h^0(D_{n+6})$$ where the last equality is given by (\[eq.chi.diff\]).
### Remark.
It wasn’t necessary to compute $\chi(D_n)$ in the previous proof. The proof only used the fact that $\chi(D_{n+6})-\chi(D_n)=n+6$ which can be proved directly as follows: $$\begin{aligned}
\chi(D_{n+6})-\chi(D_n) &= \hbox{${{1}\over{2}}$} D_{n+6}\cdot(D_{n+6}-K) - \hbox{${{1}\over{2}}$} D_n\cdot(D_{n}-K)
\\
& = \hbox{${{1}\over{2}}$} (D_{n+6}-D_n)\cdot(D_{n+6}+D_n -K)
\\
&=-K\cdot(D_r-(m+1)K)
\\
& = 6(m+1) - K\cdot D_r
\\
&=n+6.\end{aligned}$$
The isomorphism $R \to B(\BB_3,\cL,\s)$
---------------------------------------
The ring $B$ has the following basis elements in the following degrees: $$\begin{array}{ccccccc}
\begin{array}{l}
\cO(1,1,0,1) \\
\cO(1,1,0,0) \\
\cO(2,1,1,1) \\
\cO(2,1,0,1) \\
\cO(3,2,1,1) \\
\cO(3,1,1,1)
\end{array}
\begin{array}{l}
X \\
Xu \\
XYu\\
XYtu\\
XYZtu\\
XYZstu
\end{array}
\begin{array}{l}
\\
Zt \\
YZt\\
YZt^2\\
YZ^2t^2\\
YZ^2st^2
\end{array}
\begin{array}{l}
\\
\\
XZs\\
XZst\\
XZ^2st\\
XZ^2s^2t
\end{array}
\begin{array}{l}
\\
\\
\\
X^2su\\
X^2Zsu\\
X^2Zs^2u
\end{array}
\begin{array}{l}
\\
\\
\\
\\
X^2Yu^2\\
X^2Ysu^2
\end{array}
\begin{array}{l}
\\
\\
\\
\\
\\
XY^2tu^2 \, \; Y^2Zt^2u
\end{array}
\end{array}$$ Although $B$ is a graded subspace of Cox’s homogeneous coordinate ring $S$,
> [ *the multiplication in $B$ is not that in $S$.* ]{}
The multiplication in $B$ is Zhang’s twisted multiplication [@Ztwist] with respect to the automorphism $\tau$ defined in (\[defn.tau\]): the product in $B$ of $a \in B_m$ and $b \in B_n$ is $$\label{eq.Z.tw.mult}
a *_B b := a\tau^m(b).$$ To make it clear whether a product is being calculated in $B$ or $S$ we will write $x$ for $X$ considered as an element of $B$ and $y$ for $Zt$ considered as an element of $B$. Therefore, for example, $$\begin{aligned}
x^5 & =X\tau(X)\tau^2(X)\tau^3(X)\tau^4(X)\tau^5(X)
\\
&
= XuYtZ
\\
&= (Zt) Y(uX)
\\
&
= Zt\tau^2(X)\tau^3(Zt)
\\
& = yxy\end{aligned}$$ and $$y^2=Zt\tau^2(Zt)=Zt(sX) = X(sZ)t = X\tau(zt)\tau^3(X)=xyx.$$ The following proposition is an immediate consequence of these two calculations.
\[prop.R->B\] Let $R$ be the free algebra $\CC\langle x,y\rangle$ modulo the relations $x^5=yxy$ and $y^2=xyx$. Then there is a $\CC$-algebra homomorphism $$R=\CC[x,y] \to B(\BB_3,\cL,\s), \qquad x \mapsto X, \; y \mapsto Zt.$$
\[lem.low.degs\] The homomorphism in Proposition \[prop.R->B\] is an isomorphism in degrees $\le 6$.[^2]
By Proposition \[prop.steph\], $R$ has Hilbert series $(1-t)^{-1}(1-t^2)^{-1}(1-t^3)^{-1}$, so the dimension of $R_n$ in degrees $1,2,3,4,5,6$ is $1,2,3,4,5,7$.
The $n^{\th}$ row in the following table gives a basis for $B_n$, $1 \le n \le 6$. One proceeds down each column by multiplying on the right by $x$. There wasn’t enough room on a single line for $B_6$ so we put the last two entries for $B_6$ on a new line. $$\begin{array}{cccccc}
\begin{array}{l}
x=X \\
x^2=Xu \\
x^3=XYu\\
x^4=XYtu\\
x^5=XYZtu\\
x^6=XYZstu\\
\phantom{x}
\end{array}
\!\!\!\!
\begin{array}{l}
\\
y=Zt \\
yx=YZt\\
yx^2=YZt^2\\
yx^3=YZ^2t^2\\
yx^4=YZ^2st^2\\
\phantom{x}
\end{array}
\!\!
\begin{array}{l}
\\
\\
xy=XZs\\
y^2=XZst\\
y^2x=XZ^2st\\
y^2x^2=XZ^2s^2t\\
\phantom{x}
\end{array}
\!\!
\begin{array}{l}
\\
\\
\\
x^2y=X^2su\\
xy^2=X^2Zsu\\
xy^2x=X^2Zs^2u\\
yx^2y=Y^2Zt^2u
\end{array}
\!\!
\begin{array}{l}
\\
\\
\\
\\
x^3y=X^2Yu^2\\
x^2y^2=X^2Ysu^2\\
x^4y=XY^2tu^2
\end{array}
\end{array}$$ The products involving $x$ and $y$ were made by using the formula (\[eq.Z.tw.mult\]) in the same way that it was used to show that $x^5=yxy$.
\[lem.gend.gl.sects\] $\cL_2$ is generated by its global sections.
A line bundle $\cL$ on a variety is generated by its global sections if and only if for each point $p$ on the variety there is a section of $\cL$ that does not vanish at $p$. In this case, $H^0(\BB_3,\cL_2)$ is spanned by $Xu$ and $Zt$. One can see from the diagram (\[six.lines\]) that the zero locus of $Xu$ does not meet the zero locus of $Zt$, so the common zero locus of $Xu$ and $Zt$ is empty.
As a $\CC$-algebra, $B$ is generated by $B_1$ and $B_2$.
It follows from the explicit calculations in Lemma \[lem.low.degs\] that the subalgebra of $B$ generated by $B_1$ and $B_2$ contains $B_m$ for all $m \le 6$. It therefore suffices to prove that the twisted multiplication map $B_2 \otimes B_n \to B_{n+2}$ is surjective for all $n \ge 5$.
By definition, $B_2=H^0(\cL_2)$ and this has dimension two so, by Lemma \[lem.gend.gl.sects\], there is an exact sequence $0 \to \cN \to B_2 \otimes \cO_{\BB_3} \to \cL_2 \to 0$ for some line bundle $\cN$. In fact, $\cN \cong \cL_2^{-1}$.
By definition, $\cL_{n+2}=\cL_2 \otimes \cM$ where $\cM \cong \cO(D_{n+2}-D_2)$, and the twisted multiplication map $B_2 \otimes B_n \to B_{n+2}$ is the ordinary multiplication map $$B_2 \otimes H^0(\cM) = H^0(\cL_2) \otimes H^0(\cM) \to H^0(\cL_2 \otimes \cM).$$
Applying $-\otimes \cM$ to the exact sequence $0 \to \cL_2^{-1} \to B_2 \otimes \cO_{\BB_3} \to \cL_2 \to 0$ and taking cohomology gives an exact sequence $$0 \to H^0(\cL_2^{-1} \otimes \cM) \to B_2 \otimes H^0(\cM) \to H^0(\cL_2 \otimes \cM) \to H^1(\cL_2^{-1} \otimes \cM).$$ Hence, if $h^1(\cL_2^{-1} \otimes \cM)=0$, then $B_2B_n=B_{n+2}$. But $$\cL_2^{-1} \otimes \cM \cong \cO(-D_2 +D_{n+2}-D_2)$$ so we need to show that $D_{6m+r}-2D_2-K$ is ample whenever $6m+r \ge 7$ and $0 \le r \le 5$. By Lemma \[lem.Dn\], $D_{6m+r}=D_r-mK$. We therefore need to check that conditions (\[eq.ample2\]) and (\[eq.ample3\]) in Lemma \[lem.van\] hold for the divisors $D$ in the following table: $$\begin{array}{lll}
&& D:=D_r-2D_2 - (m+1)K\\
r=0 \quad & m \ge 2 \quad & (3m+1,m-1,m+1,m+1) \\
r=1 \quad & m \ge 1 \quad& (3m+2,m,m+1,m+2) \\
r=2 \quad & m \ge 1 \quad & (3m+2,m,m+1,m+1) \\
r=3 \quad & m \ge 1 \quad& (3m+3,m,m+2,m+2) \\
r=4 \quad & m \ge 1 \quad & (3m+3,m,m+1,m+2) \\
r=5 \quad & m \ge 1 \quad & (3m+4,m+1,m+2,m+2).
\end{array}$$ This is a routine though tedious task.
\[thm.isom\] Let $R$ be the free algebra $\CC\langle x,y\rangle$ modulo the relations $x^5=yxy$ and $y^2=xyx$. Then the $\CC$-algebra homomorphism $$\Phi: R=\CC[x,y] \to B(\BB_3,\cL,\s), \qquad x \mapsto X, \; y \mapsto Zt,$$ is an isomorphism of graded algebras.
By Lemma \[lem.low.degs\], $B_1$ and $B_2$ are in the image of $\Phi$. It follows that $\Phi$ is surjective because $B$ is generated by $B_1$ and $B_2$. But $\Phi(R_n) \subset B_n$, and $R$ and $B$ have the same Hilbert series, so $\Phi$ is also surjective.
Consider $R^{(3)}\supset \CC[x^3,xy,yx]$. Since $\dim R_6=7=(\dim R_3)^2 -2$ there is a 2-dimensional space of quadratic relations among the elements $x^3$, $xy$, and $yx$. Hence $R^{(3)}$ is not a 3-dimensional Artin-Schelter regular algebra. The relations in the degree two component of $R^{(3)}$ are generated by $$(x^3)^2=(xy)^2=(yx)^2.$$
[12]{}
M. Artin, J. Tate and M. Van den Bergh, Some algebras related to automorphisms of elliptic curves, in [*The Grothendieck Festschrift, Vol.1,*]{} 33-85, Birkhauser, Boston 1990.
M. Artin, J. Tate and M. Van den Bergh, Modules over regular algebras of dimension 3, [*Invent. Math.,*]{} [**106**]{} (1991) 335-388.
M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, [*J. Alg.,*]{} [**133**]{} (1990) 249-271.
M. Artin and J.J. Zhang, Noncommutative Projective Schemes, [*Adv. Math.*]{}, [**109**]{} (1994) 228-287.
D.A. Cox, The homogeneous coordinate ring of a toric variety, [*J. Alg. Geom.,*]{} [**4**]{} (1995) 17-50.
Serre, J.-P., Faisceaux algébriques cohérents, [*Ann. Math.,*]{} [**61**]{} (1955) 197-278.
S.P. Smith and J.J. Zhang, Homogeneous coordinate rings in noncommutative algebraic geometry, In preparation.
S.P. Smith and J. Tate, The centers of the 3- and 4-dimensional Sklyanin algebras, [*K-Theory*]{} [**8**]{} (1994), 19-63.
D. R. Stephenson, Artin-Schelter Regular Algebras of Global Dimension Three, [*J.Alg.,*]{} [**183**]{} (1996)55-73.
D. R. Stephenson, Algebras Associated to Elliptic Curves, [*Trans. Amer. Math. Soc.,*]{} [**349**]{} (1997) 2317-2340.
J.J. Zhang, Twisted graded algebras and equivalences of graded categories, [*Proc. Lond. Math. Soc.,*]{} [**72**]{} (1996) 281-311.
[^1]: The author was supported by the National Science Foundation, Award No. 0602347
[^2]: We will eventually prove that the homomorphism is an isomorphism in all degrees but the low degree cases need to be handled separately.
|
---
abstract: 'Modularity is a key organizing principle in real-world large-scale complex networks. Many real-world networks exhibit modular structures such as transportation infrastructures, communication networks and social media. Having the knowledge of the shortest paths length distribution (DSPL) between random pairs of nodes in such networks is important for understanding many processes, including diffusion or flow. Here, we provide analytical methods which are in good agreement with simulations on large scale networks with an extreme modular structure. By extreme modular, we mean that two modules or communities may be connected by maximum one link. As a result of the modular structure of the network, we obtain a distribution showing many peaks that represent the number of modules a typical shortest path is passing through. We present theory and results for the case where inter-links are weighted, as well as cases in which the inter-links are spread randomly across nodes in the community or limited to a specific set of nodes.'
author:
- Eitan Asher
- Hillel Sanhedrai
- 'Nagendra K. Panduranga'
- Reuven Cohen
- Shlomo Havlin
bibliography:
- 'communities.bib'
title: Distance Distribution in Extreme Modular Networks
---
[^1]
[^2]
Introduction
============
The study of complex networks gains extensive interest in the last years as networks successfully model and lead to better understanding of many real world systems and processes in which interacting objects are involved. In these models, objects are represented as nodes, and the interactions by links [@albert2002statistical; @caldarelli2007scale; @cohen2010complex; @watts1998collective; @newman2018networks; @barabasi2016network; @estrada2012structure].
Many real world networks exhibit a modular or community structure [@Eriksen_internet_2003; @Guimer2005; @garas2008structural; @bullmore2012economy]. That is, a network is comprised of smaller networks (called communities, or modules) that are highly connected within themselves (by intra-links), and have a lower number of links between them (inter-links), which is a key to their structure and function. For demonstration, see Fig. \[fig: m comm illustration\]. Knowing the distances distribution within networks with such topology is important for many reasons such as designing fast-communication, navigation, disease spreading and for optimizing processes on large graphs.
![[**Illustration of the model.**]{} We study networks that comprise $M$ modules, each of which has the same degree distribution, same topology and same size, $n$. Thus, we study a modular network with a total of $M \times n=N$ nodes. The upper picture shows the whole network, where nodes of each module have different color. The lower picture, is a projection of the upper network, showing the outer network, where each node represents a module. We start by constructing each module according to a given topology, mean degree ($k_{in}$) and links distribution, and then construct the outer network which has its own topology, mean degree ($k_{out}$) and links distribution, where we consider each module as a node. Two modules can be connected by a maximum of one link. In section \[sec: results\] we provide an analysis of different network topologies. []{data-label="fig: m comm illustration"}](illustration_model){width="0.99\linewidth"}
For each random pair of nodes $i$ and $j$ in the network, many paths can exist, or non at all. The distance between a pair of nodes is naturally defined as the shortest path length among all the paths existing between them. Distribution of shortest paths are expected to depend on the network structure and size. However, apart from a few studies [@NewmanWattsStog2001; @DOROGOVTSEV2003; @Hofstad2005RandomStruct; @vandenEsker2005Extremes; @Hofstad_distPL_2008JMP; @van2001paths], the shortest paths length distribution (DSPL) despite its importance, attracted little attention. Recent studies developed analytical methods to compute the DSPL in [Erdős-Rényi]{}and configuration-model networks [@katzav2015analytical; @nitzan2016distance]. Another paper studied the DSPL in modular random networks [@dorogovtsev2008organization], testing the conditions in which the number of inter-links between two or more modules control the network topology. This means, answering the question “how many links between two modules are needed in order to unite them into one?”. Adding more inter-links results in a change of the SPL distribution, which approaches a $\delta$ function as we add more inter-links. Still, the case where the connections between the modules is itself a complex network, meaning that the inter-links are determined according to a given outer network, an analytical approach for finding the DSPL has not been developed yet.
As a motivation for the present study, we analyzed the distance distribution (DSPL) in the Internet routers-IP autonomous systems (AS) network. Each AS functions as a community, and contains routers IPs which are the nodes inside the community. The data were obtained from the center of applied Internet analysis (Caida) [@Caida]. Several studies have been performed on distances in the Internet [@albert1999internet; @canbaz2017analysis; @zhou2004accurately], here we want to point out a specific phenomena which occurs when more and more inter-links are removed. In this case, a wavy distribution emerges. In Fig. \[fig: AS\_dist\] we show the distance distribution (DSPL) of our data for different values of maximal inter-links degrees. By limiting the number of inter-links of an AS, we mimic a situation in which the Internet network undergoes an attack or power shortage. We can observe in Fig. \[fig: AS\_dist\] multiple peaks for the DSPL after such an attack, representing the modules passed by the shortest path. This phenomena motivates us here to develop a simplified model of extreme community structure which exhibits a wavy DSPL, and we study this analytically in order to better understand this phenomena.
In this paper we develop an analytical approach for obtaining the DSPL of a modular network. Our theory calculates the DSPL of the network given the shortest path distribution within one module (in net), and the DSPL of the network that connects the modules by treating each module as a node (out net). Our method holds for any inner and outer network topologies, and not only for random networks. The model we suggest assumes an extreme community condition where each module $I$ is connected to module $J$ with a maximum of one link that connects two randomly chosen nodes in both modules. Another condition we assume here is that the outer network has no small loops as explained in detail below. In order to better simulate real world phenomena, such as routing and transportation between cities or countries, our model assumes a weight $w$ for inter-links, where intra-links weight is set to 1. We further include analysis of various cases in which inter-links are limited to a specific set of nodes rather than being chosen randomly from the inner network. Analytic analysis of specific network topologies is also included.
The paper is organized as follows: In Sec. \[basicModel\] we present the basic model and theory we use to find the DSPL. In sections \[sec: one interconnected node\] and \[sec: DifferentCasesOfConnectionsTours\] we extend the theory for different inter-links configurations. In Sec. \[sec: results\] we present our results of DSPL, from both our theory and simulations of selected network topologies. More comprehensive mathematical analysis of some specific cases of network of networks is presented in more detail in the Appendix.
![[**Analysis of autonomous systems (AS) distances distribution for different values of maximal number of inter-links outgoing from one module (max degree).**]{} AS data was obtained from Caida and was collected using MIDAR-iff, where router topology based on aliases discovered by MIDAR, iffinder, and kapar. The data contained approximately 100 million nodes which are assigned into 47,000 communities (AS). a node that connects two different ASs is an inter-linked node. Here we consider a case in which the Internet network undergoes a deliberate attack or experience node failures due to lack of electricity supply on inter-links between AS. We expect that inter-linked nodes fall with higher probability, due to their high betweenness centrality (in case of a deliberate attack) or high power demand in the case of power outage). We examine the distance distribution for those cases, allowing different maximal inter-link degree of an AS. Here, distance is defined as the shortest path length between two nodes in the graph. Note the wavy pattern of the distribution.[]{data-label="fig: AS_dist"}](AS_dist.png){width="0.99\linewidth"}
Model and Theory {#modelTheory}
================
Basic model {#basicModel}
-----------
Let a network consist of $m$ communities, or modules. Each module is assumed to be of the same size and constructed in the same fashion (or just with the same distances distribution), e.g., [Erdős-Rényi]{}, scale-free (SF), random regular (RR), lattice or any other structure. An outer network, which also can take any structure, regards every module as a node. Therefore we obtain a “large” network which comprises modules, and another network on top of it which connects those modules as illustrated in Fig. \[fig: m comm illustration\].\
Our model assumes the following:\
$\mathit{1}$. There is at most one inter-link between two modules.\
$\mathit{2}$. The inter-links connect between pairs of *random* nodes of two modules.\
$\mathit{3}$. inter-links have a weight $w$ (integer), while the weight of intra-links is 1.\
$\mathit{4}$. The outer network has no small loops.\
$\mathit{5}$. As a consequence of $\mathit{4}$, an outer shortest path between modules in the outer network is single and the second outer shortest path is much longer than it. Therefore, the shortest path in the whole network, in most cases, will pass through the shortest path of the outer network.\
It is important to notice that while assumptions 4 and 5 hold for short distances, they partially fail for the long distances in the network. Hence, we expect slight deviations at the end of the distribution, as seen in general in the figures. Random sparse networks, for instance, exhibit locally tree-like behavior [@newman2018networks382]. The range of this behavior is up to the average distance of the network approximately [@cohen2010complex12], therefore for these networks our theory is accurate up to the average distance of the network, and then it has slight deviations as we show below. For 1D lattice, for example, assumptions 4 and 5 are valid up to the longest distances, whereas for a 2D lattice, the assumptions fail.\
Now, the shortest path length (SPL) distribution in each module (inner paths) is $P_l^{in}$, and has the generating function $G_{in}(x) = \sum_{l=0}^{\infty}P_l^{in}x^l $. Likewise, the SPL distribution of the outer network is $P_l^{out}$ and has the generating function $G_{out}(x)= \sum_{l=0}^{\infty}P_l^{out}x^l$.\
According to the above assumptions, one can find that the SPL between two random nodes in the network satisfies $$d = \sum_{i=1}^{l^{out}+1}l_i^{in}+wl^{out},
\label{eq: d first}$$ which yields $$d + w = \sum_{i=1}^{l^{out}+1}(l_i^{in}+w) ,$$ where $d$ is the total distance between two random nodes, $l^{out}$ is the external shortest path length between the communities those nodes reside in, and $l_i^{in}$ is the internal distance between nodes in the same community which function as the connecting nodes between the communities in $l^{out}$. See illustration in Fig. \[fig: Theory illustration\].\
![[**Illustration of the problem and the theory.**]{} Consider two random nodes $i$ and $j$, which reside in different modules $m_{i} $ and $m_{j}$. In order to reach via the shortest path from node $i$ (source) to node $j$ (target), one has to walk as follows. First, to find the outer shortest path that connects the modules ($l^{out}$). Next, to look for the shortest path within $m_i$ to the node that connects the source node to a node that resides in the next module of $l^{out}$, which is denoted by $l_1^{in}$ in the figure. We iterate this process again in the next modules on the path, until we finally land in our target node. Our total path length will be $d = \sum{l_i^{in} + wl^{out} }$. []{data-label="fig: Theory illustration"}](illustration_theory.png){width="0.99\linewidth"}
This is a sum of independent random variables where the number of elements of the sum itself, is also a random variable. Then we can use known theorems [@johnson2005univariate] to conclude the following results.\
First, from Wald’s identity, $${\langle d + w \rangle} = {\langle l^{out}+1 \rangle}{\langle l^{in}+w \rangle},$$ which gives $${\langle d \rangle} = ({\langle l^{out} \rangle}+1)({\langle l^{in} \rangle}+w) - w.
\label{eq: average distance}$$ This result suggests that in small world networks the extreme modularity condition makes the average distance much longer. Furthermore, for the generating functions one can write $$\begin{gathered}
\big[x^wG_d(x)\big] = \big[xG_{out}(x)\big]\circ \big[x^wG_{in}(x)\big] \\
x^wG_d(x) = x^wG_{in}(x)G_{out}(x^wG_{in}(x)),\end{gathered}$$ where $G_d(x)$ is the generating function of $P_d$, the probability distribution of $d$, and $\circ$ is a composition of functions.\
Thus, we get $$G_d(x) = G_{in}(x)G_{out}(x^wG_{in}(x)).
\label{eq: composition}$$ Since we have the generating function of the shortest path distribution we are consequently able to find $P_d$ by derivation or integration (Cauchy formula) numerically by $$\begin{gathered}
P_d = \frac{G_d^{(d)}(0)}{d!} = \frac{1}{2\pi i}\oint \frac{G_d(z)}{z^{d+1}} \mathrm{d}z
\label{eq: integral}\end{gathered}$$ where the integral is performed on a close path around $z=0$ in the complex plain. This integral is far more simple to compute numerically than computing high derivatives. A simple contour can be a canonical circle with $r=1$.\
See Appendix \[sec: appendix specific\] where we analyze analytically few specific cases of network of networks topologies. Including, 1-2D lattices, Poisson distance distribution, two modules and star graph. We find for these cases explicitly all or part of the following expressions. $G_{in/out}(x)$, $G_d(x)$ and $P_d$. For two modules with Poisson DSPL we find analytically also a condition for the appearance of two peaks rather than one peak, see Eqs. and and Figs. \[fig: ratio\] and \[fig: two poisson\].
One node has all the inter-links in each module {#sec: one interconnected node}
-----------------------------------------------
When analyzing the internet data (AS) that was mentioned above, we noticed the fact that many inter-connected nodes have multiple inter-links. In order to cover other realistic cases such as this, we consider also the scenario in which all the inter-links of a module go out and in from the same single inter-connected node, rather than from random nodes as the above model, see Fig. \[fig: illustration p=0\]. This situation changes the distance significantly, $$d = l_1^{in} + l_2^{in} + wl^{out},$$ and if $l^{out}=0$ then $d = l^{in}$ (because the source and the target reside in the same module). Hence $$\begin{aligned}
G_d(x) & = G_{out}(0)G_{in}(x) +
\\
&\left[ G_{out}(x^w)-G_{out}(0) \right] \left[ G_{in}(x) \right]^2.
\end{aligned}
\label{eq: same inter}$$
![ [**Illustration of the model in section \[sec: one interconnected node\].**]{} Here, each module has only one interconnected node to which all the inter-links of this module are connected.[]{data-label="fig: illustration p=0"}](illustration_theory_con2){width="0.4\linewidth"}
Different cases of inter-links connections {#sec: DifferentCasesOfConnectionsTours}
------------------------------------------
In this section, we consider the case in which, when entering a module via an interconnected node $i$, we leave this module via different interconnected node $j$ with probability $p$, or, when departing the module via the same node with probability $1-p$. See Fig. \[fig: illustration lcon\].
![**Illustration of the model in section \[sec: DifferentCasesOfConnectionsTours\].** (a) In this scenario we enter the community through node $i$ and leave the community with probability $p$ through a different node $j$, with addition of a tour inside the community. (b) Here, with probability $1-p$, the arrival and the departure to and from the community is from the same node. []{data-label="fig: illustration lcon"}](illustration_theory_con){width="0.7\linewidth"}
{width="99.00000%"}
In appendix \[sec: DifferentCasesOfConnectionsTours appendix\] we find that for this case $$\begin{aligned}
G_d(x) & = G_{out}(0)G_{in}(x) +
\\
& \left[ G_{in}(x) \right]^2 \frac{ G_{out}\left( x^w \big( 1-p + pG_{in}(x) \big) \right)-G_{out}(0) }{ 1-p + pG_{in}(x) }.
\end{aligned}
\label{eq: p-model}$$ One can see that the last equation converges nicely to those of chapters \[basicModel\] (Eq. ) and \[sec: one interconnected node\] (Eq. ) at the limits $p=1$ and $p=0$ respectively.
Results {#sec: results}
=======
Fig. \[fig: m comm\] compares between theory and simulations for different network layouts and parameters where the inter-links have the same length as the inner links, i.e. $w=1$. In general the figure shows a good agreement between theory and simulations.\
It is important to notice the distance distribution exhibits a wavy behavior on top of a hill envelope. The intuitive explanation for this is that each hill represents paths between nodes in two modules that have the same outer distance. The first hill comes from paths between nodes inside the same module, while the second hill comes from paths between neighboring modules, which are about twice longer due to their consistency of two inner paths - the first, in the source module, from the source node to the interconnected node inside the source module, and the second, from the interconnected node in the target module, to the target node. The second hill is higher because there are more paths between neighboring modules than paths within a single module. In other words, in the outer network (in between modules), there are more shortest paths with $l^{out}=1$ than with $l^{out}=0$. The same holds for the third hill ($l^{out}=2$) and so on. That is to say, what rules the hills’ heights is the outer SPl distribution, therefore we get a bell shaped envelope which comes from the outer network distribution, and upon it hills which come from inner networks distribution.\
Note that, for the long distances there is a slight deviation between the theory and the simulations results. This can be explained by the fact that in theory we neglect loops in the outer network, while in practice, for finite networks, there are long loops (the short ones are negligible). The long loops causes that there are modules far from each other have few outer similar paths between them. This multiplicity of similar outer paths shortens the distance from a source node to a target node because the shortest path is chosen among them. In this case, we will need to find the minimum of similar independent random variables, which is different (lower) than the expectation value relative to the random variables.
Fig. \[fig: m comm with w\] shows the results for the distance distribution for the extreme modular network, both theory and simulations, where inter-links are weighted with $w=10$. It can be seen that the separation between the hills becomes more significant because paths between modules with different outer distances have dramatically different lengths as a result of the length of the inter-links. Within a single 2D lattice there is a broad distance distribution because the system is not a small world network. As a result when the inter-links are not much longer than the inner ones, the wavy behavior vanishes because the widths of hills are large so they become blended together. However, when $w$ is sufficiently large, the waves are very distinct.
Fig. \[fig: r results\] shows the results of Eq., for various values of $p$. This model suits a more realistic case, in which there is a probability $p$ of accessing and leaving a community through a different or the same ($1-p$) interconnected node. Note the reduction in the number of waves when $p$ approaches 0, which is the case in which no intra-module paths were taken.
In order to examine the emergence of the wavy distribution, we regulate the parameter $n$, modules size. We show in Fig. \[fig: waves emergence\]a that where $n$ is very small the network acts as a single network, of course. However, when we increase $n$ more and more, at some point the wavy pattern appears and becomes more and more clear. In the Appendix Sec. \[sec: two modules app\], we find analytically a criterion for the emergence of a wavy distribution in a network of two modules. In Fig. \[fig: waves emergence\]b we show by changing the outer average degree, $k_{out}$, how the sparsity of the outer network affects the waviness of the distribution.
{width="0.49\linewidth"} {width="0.49\linewidth"}
discussion
==========
In this paper we develop a framework to find analytically the distance distribution within networks with extreme community structure given the distributions of the inner and outer networks. We study here a model where we assume there is at most a single inter-link between modules. We showed that the SPL distribution has a wavy pattern in good agreement with simulations. Future work can investigate the validation of this model for real networks, where multiple links between modules exist.
Acknowledgments
===============
We thank the Italian Ministry of Foreign Affairs and International Cooperation jointly with the Israeli Ministry of Science, Technology, and Space (MOST); the Israel Science Foundation, ONR, the Japan Science Foundation with MOST, BSF-NSF, ARO, the BIU Center for Research in Applied Cryptography and Cyber Security, and DTRA (Grant no. HDTRA-1-10-1-0014) for financial support.
Specific networks {#sec: appendix specific}
=================
1D lattice
----------
Consider a 1D lattice with periodic boundaries with size $L$. For simplicity we take odd $L$. Then, the distances of each node from all other nodes have the following frequency $$\begin{gathered}
N_l =
\begin{cases}
1, \hspace{0.7cm} l=0
\\
2, \hspace{0.7cm} 1 \leq l \leq (L-1)/2
\\
0, \hspace{0.72cm} l>L/2
\end{cases},
\label{eq: Nl 1D lattice}\end{gathered}$$ where $N_l$ is the number of nodes in distance $l$ from the source node. Then, the distance distribution, $P_l$, is obtained by $$P_l = N_l/L.$$ The generating functions of $N_l$ and $P_l$ satisfy $$\begin{aligned}
&
\begin{aligned}
G_{N}(x) &= -1+2\left(1+x+...+x^{(L-1)/2}\right)
\\
&= -1+2 \frac{1-x^{(L+1)/2}}{1-x} ,
\end{aligned} \\[8pt]
&G_P(x)=G_N(x)/L.
\end{aligned}$$ Thus, $$G_{1D}(x) = G_P(x) = \frac{1}{L} \left(-1+2 \frac{1-x^{(L+1)/2}}{1-x}\right).
\label{eq: G(x) 1d lattice}$$
Comment: In Eq. we counted twice the distances between different nodes $i$ and $j$ ($(i,j)$ and $(j,i)$), and only once the distance (0) between a node $i$ to itself. The reason is that we define $P_l$ as the probability of the distance between two random nodes to be $l$. Indeed the probability to choose different nodes $i$ and $j$ is twice as large as the probability to choose the same node $i$ twice. Note, it matters only for the value of $P_0$.
2D lattice
----------
Consider a 2D square lattice with periodic boundaries and size $L \times L$. We assume for simplicity that $L$ is odd. Then, the distances of each node from all other nodes have the following frequency $$N_l =
\begin{cases}
1, & l=0
\\
4l, & 1 \leq l < L/2
\\
4(L-l), & L/2 < l \leq L
\\
0, & l>L
\end{cases},$$ and the distance distribution is $$P_l = N_l/L^2.$$ We note that $N_l$ is obtained by a convolution of the series $a_l$ and $b_l$, where $$a_l=
\begin{cases}
1, & 0 \leq l < L/2-1
\\
0, &l > L/2-1
\end{cases} , \quad
b_l=
\begin{cases}
1, & 0 \leq l < L/2
\\
0, & l > L/2
\end{cases} ,$$ such that $$\begin{cases}
N_0 = 1
\\
N_{l+1} = 4(a_l * b_l)
\end{cases}.$$ As a result, the generating functions of these sequences ($a_l,~b_l,~N_l,~P_l$) satisfy $$\begin{cases}
G_{N}(x)=1+4xG_{a}(x)G_b(x)
\\
G_P(x)=G_N(x)/L^2
\end{cases}.$$ But we note that $$\begin{aligned}
\begin{aligned}
& G_{a}(x)= 1+x+...+x^{(L-3)/2} = \frac{1-x^{(L-1)/2}}{1-x}.
\\
& G_{b}(x)= 1+x+...+x^{(L-1)/2} = \frac{1-x^{(L+1)/2}}{1-x}.
\end{aligned}\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
& G_{2D}(x) = G_P(x)
\\
& = \frac{1}{L^2} \left(1+4x \frac{(1-x^{(L-1)/2})(1-x^{(L+1)/2})}{(1-x)^2}\right).
\end{aligned}
\label{eq: G(x) 2d lattice}$$
1D lattice of 2D lattices
-------------------------
Consider a circle of square lattices that are interconnected with one inter-link between two random nodes from neighboring lattices. The inter-links have weight $w$ while the intra-links have weight 1. Then, the distance distribution is obtained according to Eq. , and by $$G_{1D\times 2D}(x) = G_{2D}(x)G_{1D}(x^w G_{2D}(x)) .$$ This result is shown in Fig. \[fig: m comm with w\]c.
Poisson distance distribution
-----------------------------
To obtain insight into the wavy distribution, we assume here that the distances in a network have a Poissonian distribution. This will enable us to obtain analytically the distance distribution in our model of extreme modular networks. Indeed random networks have in certain parameters range a distance distribution which can be approximated by a Poissonian distribution, as shown in Fig. \[fig: ER vs poisson\]. This changes with the degree and the network size very much. For higher degrees it does not work so well, while for small degrees it works well.
![ [**Test of Poisson approximation for ER DSPL.**]{} The simulations are on ER network with $k=2$ (top) and $k=4$ (bottom). The lines with markers are from simulations and the lines without markers are Poisson distribution with the same average. One can see that for $k=2$ in this region of sizes the approximation is good, while if $n$ increases further it becomes less accurate. For $k=4$ it is much less accurate. Generally, we see a difference that Poisson distribution has a standard deviation $\sqrt{\lambda}$ where the average is $\lambda$, while the DSPL does not change its standard deviation while changing its average for large $n$. []{data-label="fig: ER vs poisson"}](ERk2_vs_poisson "fig:"){width="0.95\linewidth"} ![ [**Test of Poisson approximation for ER DSPL.**]{} The simulations are on ER network with $k=2$ (top) and $k=4$ (bottom). The lines with markers are from simulations and the lines without markers are Poisson distribution with the same average. One can see that for $k=2$ in this region of sizes the approximation is good, while if $n$ increases further it becomes less accurate. For $k=4$ it is much less accurate. Generally, we see a difference that Poisson distribution has a standard deviation $\sqrt{\lambda}$ where the average is $\lambda$, while the DSPL does not change its standard deviation while changing its average for large $n$. []{data-label="fig: ER vs poisson"}](ERk4_vs_poisson "fig:"){width="0.95\linewidth"}
Thus, under proper conditions, if $$P_l = \frac{\lambda^l}{l!}e^{-\lambda},$$ where $\lambda= {\langle l \rangle}$, then the generating function is as known $$G(x) = e^{\lambda(x-1)}.
\label{eq: poisson gf}$$ Now, if both inner and outer networks have approximately Poisson distribution, then for $P_d$ it is satisfied according to Eq. and that $$G_d(x) = e^{\lambda_{in}(x-1)}e^{\lambda_{out}\left(x^we^{\lambda_{in}(x-1)}-1\right)},
\label{eq: Poisson on Poisson}$$ where $\lambda_{in} = {\langle l^{in} \rangle}$ and $\lambda_{out} = {\langle l^{out} \rangle}$.\
Still, it is difficult to find an explicit formula for the Taylor coefficients, which are $P_d$, in order to find some criteria for the emergence of wavy distribution. However, numerical calculation shows that if ${\langle l^{in} \rangle}$ is large enough relative to ${\langle l^{out} \rangle}$, then the waves appear. See Fig. \[fig: poisson\].
![ [**Both inner and outer networks have Poisson distance distribution.**]{} Analytical results from Eq. and where ${\langle l^{out} \rangle}=5$. []{data-label="fig: poisson"}](fig_poisson_of_poisson){width="0.95\linewidth"}
Two modules {#sec: two modules app}
-----------
To better understand the transition from a single peak to wavy distribution of distances, we study here a simple case which can be fully analyzed analytically. To this end, we study a network of two connected nodes that satisfies $P_0=1/2,~P_1=1/2$ and $P_l=0$ for any other $l$. Hence, the generating function is $$G_{\text{two}}(x) = \frac{1}{2}(1+x).
\label{eq: two}$$ Let a network of two modules which have a Poisson DSPL, then according to Eqs. , and we obtain $$G_{\text{two}\times \text{Poisson}}(x)
= \frac{1}{2} e^{\lambda(x-1)} \left[1+x^we^{\lambda(x-1)}\right],$$ where $\lambda={\langle l^{in} \rangle}$.\
Then we can find the coefficients of the Taylor series $$G_{\text{two}\times \text{Poisson}}(x)
= \frac{1}{2}\sum_{d=0}^{\infty}\frac{\lambda^d}{d!}e^{-\lambda}x^d + \frac{1}{2}\sum_{d=0}^{\infty}\frac{(2\lambda)^d}{d!}e^{-2\lambda}x^{d+w} ,$$ what yields $$P_d =
\begin{cases}
\frac{1}{2}\frac{\lambda^d}{d!}e^{-\lambda}, & d<w
\\
\frac{1}{2}\frac{\lambda^d}{d!}e^{-\lambda}
+ \frac{1}{2}\frac{(2\lambda)^{d-w}}{(d-w)!}e^{-2\lambda}, & d\geq w
\end{cases}.
\label{eq: Pd poisson of poisson}$$ If $w=1$ then for $d\geq 1$ $$P_d =
\frac{1}{2}\frac{\lambda^d}{d!}e^{-\lambda} \left( 1+\frac{d2^d}{2\lambda} e^{-\lambda} \right).$$ Next we find the ratio $$\phi(d,\lambda) \coloneqq \frac{P_{d+1}}{P_d} =
\frac{\lambda}{d+1} \frac{2\lambda+(d+1)2^{d+1}e^{-\lambda}}{2\lambda+d2^de^{-\lambda}}
\label{eq: ratio}$$ because this ratio indicates whether the series $P_d$ increases ($\phi>1$) or decreases ($\phi<1$).
![[**Analysis of the behavior of $P_d$ to find out when it increases and decreases for two modules having Poissonian DSPL with average $\lambda$.**]{} The plot presents the ratio $P_{d+1}/P_d$ according to Eq. . If the ratio is greater than 1 then $P_d$ increases, and if the ratio is lower than 1 then $P_d$ decreases. In this analysis the inter-links weight $w=1$. []{data-label="fig: ratio"}](ratio){width="0.95\linewidth"}
From Fig. \[fig: ratio\] one can see that for small $\lambda$ (blue) $P_d$ increases up to some value and then decreases. In contrast, for large $\lambda$ (orange) $P_d$ increases again after decreasing, which indicates a wavy pattern. However, in the transition (red) two conditions are satisfied. $$\begin{array}{ll}
{ \textup{\uppercase\expandafter{\romannumeral1}}} \quad
& \phi(d_c,\lambda_c) = 1
\\
{ \textup{\uppercase\expandafter{\romannumeral2}}}
&\frac{\partial \phi}{\partial d} (d_c,\lambda_c) = 0
\end{array}.
\label{eq: 2eqs criticality}$$ Numerical solution of these equations yields $\lambda_c \approx 6.27$. Namely, for two modules which have Poisson DSPL, if ${\langle l^{in} \rangle}>6.27$, then two peaks will appear. Assuming each module is ER with $k_{in}=2$ (See fig. \[fig: ER vs poisson\]), we find numerically that the required size should be approximately $n=160$ in order to satisfy ${\langle l \rangle}>6.27$. See Fig. \[fig: two poisson\] where the simulations results are consistent with this prediction.\
For different values of $w$ a similar analysis can be done. Higher values of $w$ yield lower values of $\lambda_c$. As example, we find numerically that $\lambda_c\approx3.31$ where $w=2$.\
In contrast, where $w=0$, then $\lambda_c\approx8.38$.
![[**Emergence of multiple peaks in two modules with Poisson DSPL.**]{} The upper panel is from theory (Eq. ), and we see that the emergence of two peaks is for $5<{\langle l^{in} \rangle}<7$ which is in agreement with the value found $\lambda_c=6.27$. In the lower panel we show simulations of ER with $k=2$, and one can see that the transition is slightly above $n=150$, consistent with the finding that approximately $n=160$ gives ${\langle l \rangle}=6.27$. It works well for $k=2$ because for this range the Poisson approximation holds well as shown in Fig. \[fig: ER vs poisson\]. []{data-label="fig: two poisson"}](fig_two_poisson_theory "fig:"){width="0.95\linewidth"} ![[**Emergence of multiple peaks in two modules with Poisson DSPL.**]{} The upper panel is from theory (Eq. ), and we see that the emergence of two peaks is for $5<{\langle l^{in} \rangle}<7$ which is in agreement with the value found $\lambda_c=6.27$. In the lower panel we show simulations of ER with $k=2$, and one can see that the transition is slightly above $n=150$, consistent with the finding that approximately $n=160$ gives ${\langle l \rangle}=6.27$. It works well for $k=2$ because for this range the Poisson approximation holds well as shown in Fig. \[fig: ER vs poisson\]. []{data-label="fig: two poisson"}](fig_two_poisson_simulation "fig:"){width="0.95\linewidth"}
Star graph
----------
Star graph with $n$ nodes has the following distance distribution $$N_l =
\begin{cases}
n, & l=0
\\
2(n-1), & l=1
\\
2\binom{n-1}{2}, & l=2
\end{cases},$$ and $P_l=N_l/ n^2$. Then $$G_{\text{star}}(x) = P_0+P_1x+P_2x^2.$$ $P_1$ is about twice $P_0$, and $P_2$ is about $n$ times $P_0$. Thus, for outer network star graph, if the inner network is such that there are separated peaks, there will be three peaks where the third one is much higher depending on $n$.
Different cases of inter-links connections {#sec: DifferentCasesOfConnectionsTours appendix}
==========================================
In this section, we analyze in detail the case of section \[sec: DifferentCasesOfConnectionsTours\] for which, when entering a module via an interconnected node $i$, we leave this module via different interconnected node $j$ with probability $p$, or, when departing from the module via the same node with probability $1-p$. See Fig. \[fig: illustration lcon\].
We denote $l^{con}$ as the length of the path within the module which was taken during the course, excluding the first and the last modules. Thus, with probability $1-p$, $l^{con}=0$ (when entering and exiting were via the same node), and with probability $p$, $l^{con}=l^{in}$ (when entering and exiting the module has been done via different nodes). In the latter case, the distance between the two interconnected nodes is the typical random distance within the module. Therefore, $$\begin{aligned}
\begin{aligned}
d &= l_1^{in} + l_2^{in} + wl^{out} + \sum_{i=1}^{l_{out}-1}l_i^{con}
\\
& = l_1^{in} + l_2^{in} + w + \sum_{i=1}^{l_{out}-1} \left( l_i^{con}+w \right),
\end{aligned}\end{aligned}$$ and if $l^{out}=0$ then $d = l^{in}$. Hence, $$\begin{gathered}
G_d(x) = G_{out}(0)G_{in}(x) + \left[ G_{in}(x) \right]^2 x^w \times
\\
\Big(\left[ G_{out}(x)-G_{out}(0) \right] / x \Big) \circ
\Big( x^w \big( 1-p + pG_{in}(x) \big) \Big).\end{gathered}$$ As a result $$\begin{aligned}
G_d(x) & = G_{out}(0)G_{in}(x) +
\\
& \left[ G_{in}(x) \right]^2 \frac{ G_{out}\left( x^w \big( 1-p + pG_{in}(x) \big) \right)-G_{out}(0) }{ 1-p + pG_{in}(x) }.
\end{aligned}
\label{eq: p-model appendix}$$ One can note that the last equation converges nicely to those of chapters \[basicModel\] (Eq. ) and \[sec: one interconnected node\] (Eq. ) at the limits $p=1$ and $p=0$ respectively.
[^1]: These two authors contributed equally
[^2]: These two authors contributed equally
|
---
abstract: 'An existing single resonance model with S11, P11 and P13 Breit-Wiegner resonances in the s-channel has been re-applied to the old data. It has been shown that the standard set of resonant parameters fails to reproduce the shape of the differential cross section. The resonance parameter determination has been repeated retaining the most recent knowledge about the nucleon resonances. The extracted set of parameters has confirmed the need for the strong contribution of a P11(1710) resonance. The need for any significant contribution of the P13 resonance has been eliminated. Assuming that the Baker. et al data set[@Bak78] is a most reliable one, the P11 resonance can not but be quite narrow. It emerges as a good candidate for the non-strange counter partner of the established pentaquark anti-decuplet.'
author:
- |
S. Ceci, A. Švarc and B. Zauner\
Rudjer Bošković Institute,\
Bijenička c. 54,\
10 000 Zagreb, Croatia\
E-mail: Alfred.Svarc@irb.hr
title: Nucleon resonances and processes involving strange particles
---
In spite of the fact that the experimental data for the process , which show a distinct peeking around the energy range of 1700 MeV, are available for quite some time[@Bak78; @Dat78] the existence of a P$_{11}$(1710) resonance in that energy range is not generally accepted, but even questioned[@Arn85; @Arn04]. However, all coupled channel models[@PWA] do accept the P$_{11}$(1710) state as a legal and needed state, and the general agreement is that it is strongly inelastic.
The confirmation and the additional proof for the existence of the P$_{11}$(1710) resonance turned out to be critically needed quite recently in light of reported observations of exotic pentaquark $\Theta$(1539) and $\Xi$(1862) states[@Pen04], as the mentioned state turns out to belong quite naturally into the pentaquark anti-decuplet configuration predicted by recent chiral soliton[@Dia97] and $q^4 \bar{q}$ strong color-spin correlated[@Jaf03] models.
Keeping in mind the accumulated knowledge about nucleon resonances[@PWA] we have applied the existing, single resonance model of $\pi$N $\rightarrow$ K$ \Lambda$ process[@Tsu00] to the existing data set[@Dat78], and allowed for the explicit presence of a narrow P$_{11}$ resonance. The single resonance model calculation has been repeated using the standard set of parameters for the S11(1650), P11(1710) and P13(1720) resonances of ref.[@PDG], and they are given in Table 1 denoted as “PDG”. As it is shown in Figs.1 and 2. (thin solid line) that choice of parameters reproduces only the absolute value of the total cross section quite well, and manages to reproduce the shape of the angular distribution [**only**]{} at w=1683 MeV. It fails miserably in reproducing the shape of the differential cross section at other energies. To eliminate the problem we have fitted the $K \Lambda$ branching ratios in the Breit-Wigner parameterization of the afore discussed resonances to the available experimental data set, enforcing a good description of the absolute value of the total cross section [**simultaneously**]{} keeping the shape of the [*differential*]{} cross sections of the process linear in $cos(\theta)$ (indicated by experimental data of ref.[@Bak78]).\
The following resonance parameter extraction method has been applied:\
we have started with the belief that when a set of resonance parameters (masses, widths and branching fractions) is once established in any analysis using channels other then $K \Lambda$ channel, the only parameter which we are allowed to vary is a [*branching fraction*]{} to $K \Lambda$ channel, [*while everything else (masses, widths, branching fractions to other channels) can not be changed*]{}. Hence, as starting values we have used the resonant parameters for the $S_{11}$, $P_{11}$ and $P_{13}$ resonances obtained in the coupled channel analysis of $\pi$-nucleon scattering based on the $\pi$-elastic and channels[@Bat98]. The only parameter which was allowed to vary was the branching fraction to the $K \Lambda$ channel.
Resonance parameters for the single resonance model.
-------------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------------
\[-1.5ex\] [$M$]{} $\Gamma[MeV]$ $x_{\pi N}$\[%\] $x_{K \Lambda}$\[%\]
$S_{11}$ $P_{11}$ $P_{13}$ $S_{11}$ $P_{11}$ $P_{13}$ $S_{11}$ $P_{11}$ $P_{13}$ $S_{11}$ $P_{11}$ $P_{13}$
\[1ex\]
\[-1.5ex\] $PDG$ 1650 1710 1720 150 100 150 70 15 15 7 15 6.5
\[1ex\] $Sol \ 1$ 1652 1713 1720 202 180 244 79 22 18 [*2.4*]{} [*23*]{} [*0.16*]{}
\[1ex\] $Sol \ 2$ 1652 1713 1720 202 180 244 79 22 18 [*2.4*]{} [*35*]{} [*0.16*]{}
\[1ex\] $Sol \ 3$ 1652 [**1700**]{} 1720 202 [**60**]{} 244 79 22 18 [**4**]{} [**30**]{} [*0.16*]{}
\[1ex\]
-------------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------------
\[table2\]
The three solutions for the choice of resonant parameters are obtained, and are together with the “standard” (PDG) solution given in Table 1. The agreement with the experimental data is given in Figs. 1. and 2. In extracting new resonance parameters we have kept in mind that the overall data set[@Bak78; @Dat78] is mutually inconsistent. In addition, as the latest measured set of data[@Bak78] shows a surprisingly narrow width when compared to the overall trend, we have treated it separately with special care.
We fit the “lower” and “upper” band of the total cross section imposing the correct angular dependence at the same time, and obtain $Sol \ 1$ (dotted line) and $Sol \ 2$ (dashed line). To obtain the $Sol \ 3$ (thick solid line) we fit [*only*]{} Baker et al data[@Bak78].\
“Standard” (PDG) solution introduces “too much curvature” in the differential cross section throughout the whole energy range indicating too big P-waves contribution relative to the S-wave. If the shape of the angular dependence is to be reproduced, contrary to the standard belief[@PWA; @PDG], the contribution of $P_{13}$ partial wave is negligible for all obtained solutions. The branching ratio of $S_{11}$ resonance to $K \Lambda$ channel is somewhat smaller then previously believed. The branching ratio of $P_{11}$ resonance to $K \Lambda$ channel is significantly bigger. If the latest Baker et al data[@Bak78] are to be taken very seriously the best agreement with the experiment is obtained for [*very narrow*]{} $P_{11}$ resonance, not observed in other processes and strongly inelastic - $Sol \ %%@
3$; hence a candidate for a non-strange pentaquark counter-partner.
The re-measuring of the differential cross section for the $\pi$N $\rightarrow$ K$ \Lambda$ process in the energy range 1600 MeV $<$ w $<$ 1800 MeV is badly needed. The decisive conclusion about the existence of the $P_{11}$ non-strange pentaquark counter-partner will be possible only when the improved set of data is fully incorporated in one of the existing coupled channel partial wave analyses[@PWA].
[0]{}
R.D. Baker et. al., [*Nucl. Phys.*]{} [**B141**]{}, 29 (1978). Landolt-Börnstein, New Series, ed. H. Schopper, [**8**]{} (1973). R.A. Arndt, J.M. Ford and L.D. Roper, [*Phys. Rev.*]{} [**D32**]{}, 1085 (1985). R. A. Arndt et. al., [*Phys. Rev.*]{} [**C69**]{}, 035213 (2004). R.E. Cutkosky, C.P. Forsyth, R.E. Hendrick and R.L. Kelly, [*Phys. Rev.*]{} [**D20**]{}, 2839 (1979); M. Batinić, I. Šlaus, A. Švarc and B.M.K. Nefkens, [*Phys. Rev*]{} [**C51**]{}, 2310 (1995); T.P. Vrana, S.A. Dytman and T.S.-H- Lee, [*Phys. Rep.*]{} [**328**]{}, 181 (2000). T. Nakano et al., [*Phys. Rev. Lett.*]{} [**91**]{}, 012002 (2003); : NA49 Collaboration, [*Phys.Rev.Lett.*]{} [**92**]{}, 042003 (2004). D. Diakonov, V. Petrov and M. Polyakov, [*Z. Physik*]{} [**A359**]{}, 305 (1997). R. Jaffe and F. Wilczek, [*Phys. Rev. Lett.*]{} [**91**]{}, 232003 (2003). K. Tsushima, A. Sibirtsev and A.W. Thomas, [*Phys.Rev.*]{} [**C62**]{}, 064904 (2000). K. Hagiwara et.al., [*Phys. Rev.*]{} [**D66**]{}, 010001 (2002.) M. Batinić, I. Dadić, I. Šlaus, A. Švarc, B.M.K. Nefkens and T.S.-H. Lee, [*Physica Scripta*]{} [**58**]{}, 15 (1998)
|
---
abstract: 'We report a neutron diffraction study of the magnetic phase transitions in the charge-density-wave (CDW) TbTe$_3$ compound. We discover that in the paramagnetic phase there are strong 2D-like magnetic correlations, consistent with the pronounced anisotropy of the chemical structure. A long-range incommensurate magnetic order emerges in TbTe$_3$ at $T_{mag1}$ = 5.78 K as a result of continuous phase transitions. We observe that near the temperature $T_{mag1}$ the magnetic Bragg peaks appear around the position (0,0,0.24) (or its rational multiples), that is fairly close to the propagation vector $(0,0,0.29)$ associated with the CDW phase transition in TbTe$_3$. This suggests that correlations leading to the long-range magnetic order in TbTe$_3$ are linked to the modulations that occur in the CDW state.'
address: |
$^1$ Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland\
$^2$ Laboratory for Neutron Scattering, PSI, CH-5232 Villigen, Switzerland\
$^3$ Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, Stanford, California 94305, USA\
$^4$ Stanford Institute of Energy and Materials Science, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park 94025, California 94305, USA
author:
- 'F. Pfuner$^1$, S.N. Gvasaliya$^{1,2}$, O. Zaharko$^2$, L. Keller$^2$, J. Mesot$^{1,2}$, V. Pomjakushin$^2$, J.-H. Chu $^{3,4}$, I.R. Fisher$^{3,4}$, L. Degiorgi$^1$'
title: 'Incommensurate Magnetic Order in TbTe$_3$'
---
Introduction
============
The RTe$_3$ family of compounds (R=rare earth) have recently attracted renewed interest as a model system for layered (two-dimensional) charge-density-wave (CDW) materials. RTe$_3$ crystallizes in an orthorhombic structure, composed of double layers of planar Te- sheets separated by corrugated R-Te layers. The average chemical structure can be described using the $Cmcm$ space group [@norling1966] (in this setting the lattice parameters $a$ and $c$ are close to each other, whereas $b$ is approximately 6 times longer), although in these materials an incommensurate lattice modulation with a wave vector $q_c \sim 2/7c^*$ is observed below the phase transition into the CDW state [@dimasi1995; @malliakas2005]. Such a CDW state with modulations along the $c$-axis is present in all RTe$_3$ materials. However, the members with the heaviest rare-earth elements exhibit a second CDW phase transition at lower temperatures that is characterized by a modulation wavevector $q_a \sim 1/3a^*$ [@ru2008; @fang2007]. The temperatures of these two CDW transitions show opposite trends as a function of the R-atom mass. Whereas $T_{CDW1}$ decreases towards heavier members of the family ([*i.e.*]{}, going from La to Tm), the $T_{CDW2}$ associated with the $a$- axis incommensurability is higher for heavier compounds [@ru2008; @ru2008_2]. For the specific compound studied in this report, TbTe$_3$, transport and diffraction experiments yield $T_{CDW1}$ = 336 K. Although transport measurements do not clearly reveal the presence of a second CDW transition, STM measurements performed at 6 K do show an ordering pattern that is associated with the wavevector $\sim2/3a^*$ [@fang2007]. More recent high resolution x-ray diffraction measurements confirm the presence of a transverse CDW at low temperatures, with wave-vector $\sim 0.683 a^*$, though $T_{CDW2}$ is at present unknown [@banerjee].
The RTe$_3$ family appears to be a classic CDW system, for which the CDW wavevector corresponds to an enhancement of the generalized susceptibility $\chi(q)$, which is largely driven by FS nesting [@brouet2008; @laverock2005; @mazin2008]. The enhancement is, however, far from being a divergence, suggesting an important role for strong electron-phonon coupling [@mazin2008]. The Fermi surface is composed of states associated with the double Te layers, and is therefore similar for all members of the series. The principal role of the rare earth ion R appears to be chemical pressure, which only modestly affects the resulting CDW wavevectors [@dimasi1995; @ru2008].
In addition to hosting an incommensurate CDW, for magnetic rare earth ions, RTe3 also hosts long range antiferromagnetic order at low temperatures. The magnetic phase transitions of RTe$_3$ have been studied using magnetisation [@ru2008_2; @ru2006; @iyeiri2003; @deguchi2009], calorimetry [@ru2008_2; @ru2006; @deguchi2009] and electrical resistivity [@ru2008_2; @ru2006; @iyeiri2003; @deguchi2009]. Using these macroscopic techniques the temperatures of the magnetic transitions have been determined and a phase diagram summarizing the temperatures of CDW and magnetic phase transition of RTe$_3$ has been established in Ref. [@ru2008_2]. However, the nature of the long-range magnetic order in this family of compounds has not yet been studied by diffraction techniques, and the effect of the incommensurate lattice modulation on the magnetic structure is unknown. To bridge this gap we have performed neutron scattering experiments in the representative compound TbTe$_3$. The choice of this material was dictated by the following reasons. In order to establish possible relationship of the propagation vectors associated to the CDW state and of the long-range magnetic order one needs to study the material that has reasonably close values between $T_{CDW2}$ and $T_{N}$. Based on this condition, the DyTe$_3$ and TbTe$_3$ are both attractive candidates according to their phase diagram [@ru2008_2]. DyTe$_3$ undergoes two magnetic phase transitions at T$\sim$3.6 K and T$\sim$3.45 K [@ru2008_2]. However, due to high absorption of Dy the neutron measurements on DyTe$_3$ are too challenging [@comment1]. TbTe$_3$ exhibits three closely located magnetic phase transitions at the following temperatures: $T_{mag1}$ = 5.78 K, $T_{mag2}$ = 5.56 K and $T_{mag3}$ = 5.38 K [@ru2008_2]. Here, we present a study of the magnetic phase transitions in TbTe$_3$ using powder and single crystal neutron diffraction.
Experimental techniques
=======================
Single crystals of TbTe$_3$ were grown by slow cooling a binary melt, as described elsewhere [@ru2006]. As the material is rather air-sensitive, the TbTe$_3$ powder was obtained by crushing the single crystals in He-atmosphere. The powder was immediately sealed in a cylindric vanadium can of 6 mm in diameter used for the measurements.
Powder neutron diffraction experiments were performed at the DMC cold-neutron diffractometer [@DMC](neutron spallation source SINQ [@SINQ], PSI) in the temperature range 1.5 – 60 K. Most of the data were collected with neutron wavelengths of $\lambda=2.46$ Å, giving access to the wave vectors range 0.1–3.4 Å$^{-1}$. Typical exposure times were $\sim12$ hours.
The experiments on the single crystals of TbTe$_3$ with size of roughly $\sim 7 \times 8 \times 2.5$ mm$^3$ were performed at the TriCS single crystal diffractometer [@TriCS]. A pyrolytic graphite (PG) monochromator was used with neutron wavelength $\lambda=2.32$ Å. A 120 mm long PG filter was installed in the beam in order to ensure full suppression of higher-order neutrons. The temperature range of interest for these measurements is $T\sim2 - 8$ K, because the three magnetic phase transitions are close to each other in the range $5-6$ K. This interval was scanned with steps of $\sim 0.05$ K. To ensure the required accuracy in temperature an “orange”-type He flow cryostat was used. One of the samples was aligned in the \[H,K,0\] plane and a second sample was aligned in the \[0,K,L\]. The normal beam geometry was employed, tilting the detector in the range of -10...10 deg. This allowed to access reflections with L=$\pm$1 and H=$\pm$1, respectively.
Powder Diffraction Results
==========================
![Temperature dependence of the difference-patterns with respect to the data at the reference temperature of 60 K. The inset emphasizes the changes of the diffraction pattern in the range of wavevectors 0.2 - 2Å[ ]{}. []{data-label="differences"}](figure1.eps)
![Indexed magnetic reflections for TbTe$_3$ at 2 K in a powder diffraction pattern (the contribution from the chemical structure is eliminated by subtracting the powder spectrum taken at T=60 K). The indexed peaks were identified by single-crystal neutron diffraction experiments. Labels in the brackets are the ($H,K,L$) indices of the peaks given in terms of the average $Cmcm$ structure. The peaks that are associated with the magnetic propagation vector q$^{mag}_1=(0,0,0.21)$ are labeled in red, those associated with the magnetic propagation vector q$^{mag}_2=(0,0,0.5)$ are shown in black, and finally, with q$^{mag}_3=(0,1,0)$ - in green. Note that the Bragg peaks that are labeled for simplicity as ($0,0,0.21$), in fact have a non-zero component of modulation along the ($<0,K,0>$) direction and more precisely should be described as ($0,0.01,0.21$). See text and Fig. \[fits\_000p21\] for details. []{data-label="Fig_DMC_diff"}](figure2.eps)
The data at 60 K was taken as a reference for the scattering in the paramagnetic phase of TbTe$_3$. The average chemical structure of TbTe$_3$ was refined with Fullprof [@fullprofref] within the $Cmcm$ space group following a structural model of the RTe$_3$-type materials developed in Ref. [@malliakas_2006]. This analysis indicates the presence of several preferred orientations in the sample, impeding high-quality refinements of both chemical and magnetic structures. Nevertheless, the lattice parameters obtained from the data $a=4.3050\pm0.00015$Å[ ]{}, $b= 25.33757\pm0.0012$Å[ ]{}, and $c= 4.28024\pm0.00012$Å[ ]{} are in agreement with those published earlier [@malliakas_2006]. In what follows we concentrate on the differences of the data taken at temperatures lower than 60 K. In this experimental setup the low-temperature changes in the chemical structure of TbTe$_3$ are not resolved.
Figure \[differences\] shows difference-patterns at selected temperatures, obtained by eliminating the contribution from the chemical structure with the subtraction of the diffraction pattern at 60 K. As expected from the macroscopic properties of TbTe$_3$, magnetic Bragg peaks appear in the diffraction patterns collected below $T_{mag1}$= 5.78 K. Furthermore, relatively sharp asymmetric peaks appear in the diffraction patterns even at temperatures slightly higher then $T_{mag1}$ (e.g., the patterns taken at 6.2 K and 7 K). As shown in Fig. \[differences\], these unusual peaks have an extended shoulder towards higher wavevectors. Such scattering profiles from powders are typical for systems with pronounced quasi-two-dimensional (quasi-2D) properties, as observed [*e.g.*]{} in graphite [@warren_1941] or in 2D magnetic materials [@roessli_1993]. Therefore, one may expect that in a relatively broad temperature range above $T_{mag1}$ TbTe$_3$ exhibits quasi-2D behavior. The positions of the magnetic Bragg peaks are temperature-dependent and may be clearly distinct in different magnetic phases. This suggests that the magnetic phases occurring in TbTe$_3$ are modulated and the propagation vector is locked-in at incommensurate position only below $T_{mag3}$. In addition, the asymmetric peaks above $T_{mag1}$ are located at a different wavevector as compared to magnetic Bragg peaks observed just below $T_{mag1}$. This is best seen for the scattering profiles shown in Fig. \[differences\] in the vicinity of ${\bf Q} \sim 0.7^{-1}$Å[ ]{}.
Figure \[Fig\_DMC\_diff\] shows the magnetic diffraction pattern measured from TbTe$_3$ at 2 K. Some of the magnetic peaks could be indexed in terms of the average chemical cell and are labeled in Fig. \[Fig\_DMC\_diff\] (due to the complexity of the pattern the use of single-crystal neutron diffraction data was essential for this step). Clearly, most of the magnetic Bragg peaks are at positions incommensurate with the average chemical structure of the material. It may be speculated that the broad magnetic peaks observed in the vicinity of $Q \simeq 0.9 $Å[ ]{} and $Q \simeq 1.2$Å[ ]{} are composed by many magnetic Bragg peaks which are located close to each other in wave-vector space, due to the rather large value of the lattice parameter $b$. We note here that among the magnetic Bragg peaks we could not identify any with non-zero value of $H$.
Single-Crystal Diffraction Results
==================================
![The temperature evolution of the intensity distribution taken in the vicinity of the (0,0,0.21) position of TbTe$_3$. Vertical solid lines on both plots denote $T_{mag3}$ = 5.38 K. Note that below $T_{mag3}$ the reflection is incommensurate in both $<0,K,0>$ and $<0,0,L>$ directions. []{data-label="maps_000p21"}](figure3a.eps "fig:") ![The temperature evolution of the intensity distribution taken in the vicinity of the (0,0,0.21) position of TbTe$_3$. Vertical solid lines on both plots denote $T_{mag3}$ = 5.38 K. Note that below $T_{mag3}$ the reflection is incommensurate in both $<0,K,0>$ and $<0,0,L>$ directions. []{data-label="maps_000p21"}](figure3b.eps "fig:")
Taking into account the complexity of the powder diffraction patterns observed in magnetically ordered phases of TbTe$_3$, there are considerable difficulties not only in developing a proper model of the arrangement of the Tb magnetic moments, but even in determining the propagation vectors in the magnetically ordered phases. In a trial to overcome these difficulties a series of single-crystal experiments were performed at the base temperature $T$=2 K, below $T_{mag3}$. Those reflections, which were successfully indexed at $T$=2 K, are labeled in Fig. \[Fig\_DMC\_diff\]. There was no attempt to index the peaks observed in powder diffraction patterns at higher temperatures. Instead, the temperature evolution of those peaks that were identified at $T$= 2 K was traced up to $T\sim7$ K.
![The temperature evolution of the propagation vectors (left column) and of the intensity (right column) measured around the position (0,0,21). Green symbols denote the position and the intensity of the (0,$0+\delta$,0.21) satellite, whereas black squares stand for the (0,$0-\delta$,0.21). Vertical dashed lines denote the temperatures of the magnetic phase transitions. []{data-label="fits_000p21"}](figure4.eps)
Figure \[maps\_000p21\] shows false-color maps of neutron intensities taken in a close vicinity of the ($0,0,0.21$) incommensurate magnetic reflection in the temperature range $1.5 - 6$ K. At the base temperature this reflection has incommensurate components both along the $<0,K,0>$ and $<0,0,L>$ directions. However, the incommensurate component along the $<0,K,0>$ direction appears to be small and diminishes above $T_{mag3}$ in a step-like way. Above $T_{mag3}$ the incommensurate components along the $<0,0,L>$ direction is also temperature-dependent. For a more quantitative analysis the scans for each temperature are fitted assuming a Gaussian shape for a Bragg reflection. The resulting temperature dependence of the position and intensity of the peak ($0,0,0.21$) are shown in Fig. \[fits\_000p21\]. The modulation vector along the $<0,0,L>$ direction stays constant within the precision of the measurements below $T_{mag3}$. Above $T_{mag3}$ the propagation vector increases smoothly and reaches the value of $(0, 0, 0.242\pm0.001)$ at $T_{mag1}$. The modulation vector along the $<0,K,0>$ direction is also constant $(0, 0.01\pm0.001, 0)$ below $T_{mag3}$, while above this temperature the component of the modulation vanishes. The intensity measured around the $(0,0,0.21)$ position also appears to be nearly temperature-independent below $T_{mag3}$. Above this temperature however, the overall intensity observed in the vicinity of the $(0,0,0.21)$ position drops by a factor of 2, as can be deduced from Fig. \[fits\_000p21\]b-d. Further the intensity gradually decreases and above $T_{mag1}$ we could not detect any scattering above the background level around that position. In the whole studied temperature range the width of the peaks is limited by resolution. Gradual decrease of the Bragg intensity suggest a continuous character of the phase transition that the crystal undergoes at $T_{mag1}$, whereas an abrupt change in magnetic intensity at $T_{mag3}$ points to a first order phase transition.
The magnetic Bragg peaks, characterized by the propagation vector (0,0,0.5), exhibit a qualitatively different behavior. Figure \[fits\_040p5\] shows the temperature dependence of the $(0,4,0.5)$ Bragg peak in the temperature range $4 - 6$ K as a representative example. Whereas below $T_{mag3}$ the propagation vector is (0,0,0.5), above this temperature the position of the magnetic Bragg peaks becomes incommensurate in a step-like fashion as (0,0,0.5$\pm\delta$) with $\delta$ displaying a temperature dependence. The intensity of the $(0,4,0.5)$ magnetic peak decreases abruptly at $T_{mag3}$ and smoothly diminishes towards $T_{mag1}$. Weak magnetic peaks around the $(0,4,0.5)$ position are still observed in paramagnetic phase up to $\sim6$ K.
![The temperature evolution of the intensity distribution taken in the vicinity of the (0,4,0.5) position of TbTe$_3$. Red triangles show the position and the intensity of (0,0,$0.5+\delta$) satellite, while blue circles denote the position and the intensity of (0,0,$0.5-\delta$) satellite. Vertical dashed lines denote the temperatures of the magnetic phase transitions. []{data-label="fits_040p5"}](figure5a.eps "fig:") ![The temperature evolution of the intensity distribution taken in the vicinity of the (0,4,0.5) position of TbTe$_3$. Red triangles show the position and the intensity of (0,0,$0.5+\delta$) satellite, while blue circles denote the position and the intensity of (0,0,$0.5-\delta$) satellite. Vertical dashed lines denote the temperatures of the magnetic phase transitions. []{data-label="fits_040p5"}](figure5b.eps "fig:")
Discussion and Conclusions
==========================
We have performed a first study of magnetic phase transitions in TbTe$_3$ by powder and single-crystal neutron diffraction. We find that in the paramagnetic phase, slightly above $T_{mag1}$ = 5.78 K, there are pronounced 2D-like magnetic correlations. At $T_{mag1}$ long-range magnetic order emerges as a result of a continuous phase transition. In all three magnetically ordered phases, incommensurate modulations are present. This observation is at variance with the typical behavior of unmodulated rare-earth intermetallic compounds (see [*e.g.*]{} [@gignoux1993; @gignoux1994]). Incommensurate magnetic structures often appear in these materials just below the phase transition into the magnetically ordered state. However, in most cases magnetoelastic coupling and crystal-field effects typically result in a lock-in transition to a commensurate magnetic structure. The behavior found in TbTe$_3$ only partially resembles these general expectations. As shown in Fig. \[fits\_040p5\] the magnetic propagation vector of the $(0,0,0.5\pm\delta)$-type observed just below $T_{mag1}$ turns out to be locked-in below $T_{mag3}$ into the simple antiferromagnetic position $(0,0,0.5)$. However, the magnetic Bragg peaks observed below $T_{mag1}$ near the position (0,0,0.24), stabilize at low temperature at the incommensurate position (0,$0+\delta$,0.21). This effect is possibly due to the incommensurate lattice modulation present in TbTe$_3$ in the CDW state.
The propagation vectors of the magnetic structures, which we succeeded to identify in TbTe$_3$, have components either along the $<0,K,0>$ or $<0,0,L>$ directions, or a linear combination of both. We could not assign any of the magnetic peaks, observed in powder diffraction pattern, as a Bragg reflection with non-zero component $<H,0,0>$ of the propagation vector. Therefore, our results do not point towards a link of the propagation vector associated with the second CDW phase transition and of the magnetic order in TbTe$_3$. Nonetheless, we discover that near the temperature $T_{mag1}$ the magnetic Bragg peaks appear around the positions (0,0,0.24) (or its rational multiples). This value is fairly close to the propagation vector $(0,0,0.29)$ [@ru2008] associated with the high-temperature CDW phase transition, raising the possibility that correlations leading to the long-range magnetic order in TbTe$_3$ might be linked to the modulated chemical structure in the CDW state. These results stimulate further work to refine the magnetic structure in this and related magnetic RTe$_3$ compounds, with the ultimate goal of establishing how the CDW and magnetic order interact and coexist.
Acknowledgments
===============
This work is based on the experiments performed at the Paul Scherrer Institut, Switzerland and has been supported by the Swiss National Foundation for the Scientific Research within the NCCR MaNEP pool. Work at Stanford University was supported by the Department of Energy, Office of Basic Energy Sciences under contract DE-AC02-76SF00515.
References {#references .unnumbered}
==========
[99]{} Norling B K and Steinfink H 1966 Inorg. Chem. [**5**]{}, 1488 DiMasi E, Aronson M C, Mansfield J F, Foran B and Lee S 1995 Phys. Rev. B [**52**]{}, 14516 Malliakas C, Billinge S J L, Kim H J and Kanatzidis M G 2005 J. Am. Chem. Soc. [**127**]{}, 6510 Ru N, Condron C L, Margulis G Y, Shin K Y, Laverock J, Dugdale S B, Toney M F and Fisher I R 2008 Phys. Rev. B [**77**]{}, 035114 Fang A, Ru N, Fisher I R and Kapitulnik A 2007 Phys. Rev. Lett. [**99**]{}, 046401 Ru N, Chu J H and Fisher I R 2008 Phys. Rev. B [**78**]{}, 012410 A. Banerjee and T. Rosenbaum, private communication Brouet V, Yang W L, Zhou X J, Hussain Z, Moore R G, He R, Lu D H, Shen Z X, Laverock J, Dugdale S B, Ru N, and Fisher I R 2008 Phys. Rev. B [**77**]{}, 235104 Laverock J, Dugdale S B, Major Zs, Alam M A, Ru N, Fisher I R, Santi G and Bruno E 2005 Phys. Rev. B [**71**]{}, 085114 Johannes M D and Mazin I I 2008 Phys. Rev. B [**77**]{}, 165135 Ru N and Fisher I R 2006 Phys. Rev. B [**73**]{}, 033101 Yuji Iyeiri, Teppei Okumura, Chishiro Michioka and Kazuya Suzuki 2003 Phys. Rev. B [**67**]{}, 144417 Deguchi K, Okada T, Chen G F, Ban D, Aso N and Sato N K 2009 Journal of Physics Conference Series [**150**]{}, 042023 In fact we gave a trial and performed a neutron powder diffraction experiment on DyTe$_3$ packed into a double wall vanadium container. However, with the exception of two most intense peaks, the signal to noise ratio was not favorable for reliable conclusions. Therefore, we do not report the neutron scattering results obtained for DyTe$_3$ and concentrate here on TbTe$_3$. Fischer P, Keller L, Schefer J and Kohlbrecher J 2000 Neutron News [**11**]{}, 19 Blau B et al., 2009 Neutron News [**20**]{}, 5 Schefer J, Könnecke M, Murasik A, Czopnik A, Strässle Th, Keller P and Schlump N 2000 Physica B [**276-278**]{}, 168 Rodríguez-Carvajal J 1993 Physica B [**192**]{}, 55 Malliakas C and Kanatzidis M 2006 J. Am. Chem. Soc. [**128**]{}, 12612 Warren B E 1941 Phys. Rev. [**59**]{}, 693 Roessli B, Fischer P, Zolliker M, Allenspach P, Mesot J, Staub U, Furrer A, Kaldis E, Bucher B, Karpinski J, Jilek E and Mutka H 1993 Z. Phys. B [**91**]{}, 149 Gignoux D and Schmitt D 1993 Phys. Rev. B [**48**]{}, 12682 Gignoux D and Schmitt D 1994 JMMM [**129**]{}, 53
|
---
abstract: 'Recent models for unsupervised representation learning of text have employed a number of techniques to improve contextual word representations but have put little focus on discourse-level representations. We propose [[Conpono]{}]{}[^1], an inter-sentence objective for pretraining language models that models discourse coherence and the distance between sentences. Given an anchor sentence, our model is trained to predict the text $k$ sentences away using a sampled-softmax objective where the candidates consist of neighboring sentences and sentences randomly sampled from the corpus. On the discourse representation benchmark DiscoEval, our model improves over the previous state-of-the-art by up to 13% and on average 4% absolute across 7 tasks. Our model is the same size as BERT-Base, but outperforms the much larger BERT-Large model and other more recent approaches that incorporate discourse. We also show that [[Conpono]{}]{} yields gains of 2%-6% absolute even for tasks that do not explicitly evaluate discourse: textual entailment (RTE), common sense reasoning (COPA) and reading comprehension (ReCoRD).'
author:
- '**Dan Iter**'
- '**Kelvin Guu**'
- '**Larry Lansing**'
- '**Dan Jurafsky**'
bibliography:
- 'anthology.bib'
- 'acl2020.bib'
date: April 2020
title: Pretraining with Contrastive Sentence Objectives Improves Discourse Performance of Language Models
---
Discussion
==========
In this paper we present a novel approach to encoding discourse and fine-grained sentence ordering in text with an inter-sentence objective. We achieve a new state-of-the-art on the DiscoEval benchmark and outperform BERT-Large with a model that has the same number of parameters as BERT-Base. We also observe that, on DiscoEval, our model benefits the most on ordering tasks rather than discourse relation classification tasks. In future work, we hope to better understand how a discourse model can also learn fine-grained relationship types between sentences from unlabeled data. Our ablation analysis shows that the key architectural aspects of our model are cross attention, an auxiliary MLM objective and a window size that is two or greater. Future work should explore the extent to which our model could further benefit from initializing with stronger models and what computational challenges may arise.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank the Stanford NLP group for their feedback. We gratefully acknowledge support of the DARPA Communicating with Computers (CwC) program under ARO prime contract no. W911NF15-1-0462
Appendix
========
We include some fine-grained DiscoEval results that were reported as averages, as well as implementation and reproduction details for our experiments.
SP, BSO and DC breakdown
------------------------
------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
Model Wiki arxiv ROC Wiki arxiv ROC Wiki Ubuntu
BERT-Large 50.7 47.3 63.4 70.4 66.8 70.8 65.1 54.2
RoBERTa-Base 38.35 33.73 44.00 60.19 55.16 60.66 62.80 53.89
BERT-Base BSO 49.23 50.92 60.80 74.67 68.56 72.22 88.80 56.41
[[Conpono]{}]{} - MLM 50.95 51.90 61.92 77.98 71.45 76.68 86.70 50.00
[[Conpono]{}]{} Small 44.90 41.23 50.10 65.03 58.89 61.19 78.10 57.32
[[Conpono]{}]{} *isolated* 49.33 44.60 56.53 59.16 57.48 56.94 71.60 54.71
[[Conpono]{}]{} *uni-encoder* 54.30 58.58 66.75 78.25 71.65 73.99 86.00 57.90
k=4 54.07 58.30 67.15 79.04 72.21 76.89 88.38 **58.85**
k=3 54.65 **59.55** 67.22 **79.34** **73.61** **77.08** **89.48** 56.00
k=2 **54.83** 58.77 **68.40** 79.24 74.16 76.84 89.22 56.41
k=1 44.05 40.98 57.65 68.47 62.40 67.24 89.03 56.20
------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
Table \[table:finegrained\] shows the scores for each model per each dataset domain for the SP, BSO and DC tasks in DiscoEval.
[[Conpono]{}]{} pretraining details
-----------------------------------
[[Conpono]{}]{} is pretrained on 1.6 million examples randomly sampled from Wikipedia and BooksCorpus. We use the same number of training examples for all the ablations and training BERT-Base BSO. On example consists of a single anchor and 32 candidate targets, 4 losses (1 for each of the 4 randomly chosen true targets (ie. $k$)). We use a 25% warm up rate and a learning rate of 5e-5. The model is initialized with BERT-Base weights. We add a square interaction weight matrix that is the same size as model output dimensions (ie. 756) that is referred to as $W_k$ in Section \[sec:model\]. There is one such matrix for each $k$. The maximum sequence length of the input is 512, though do to some preprocessing constraints, the maximum input seen by the model is 493.
Our [[Conpono]{}]{} small model has a hidden size of 128, an intermediate size 512, and has 2 hidden layers. We train it on 38.4 million examples, including examples from CCNews. Samples are drawn from each source proportional to the size of the source, meaning that about 70% of training examples come from CCNews. Otherwise, we use all the same parameters as [[Conpono]{}]{}.
Parameter counts
----------------
Table \[table:param\_counts\] shows the number of parameters in each model used.
Model Parameters
---------------------------------- ------------
BERT-Base 110M
RoBERTa-Base 110M
[[Conpono]{}]{} \[All Variants\] 110M
BERT-Large 335M
: []{data-label="table:param_counts"}
RTE, COPA and ReCoRD details
----------------------------
**RTE** is trained for 3240 steps, with checkpoints every 750 steps and a learning rate of 8e-6. The warm-up proportion is 10% and the a maximum sequence length of 512
**COPA** is trained for 300 steps, with checkpoints every 50 steps and a learning rate of 1e-5. The warm-up proportion is 10% and the maximum sequence length of 512.
**ReCoRD** is trained for 8 epochs over the training data with a learning rate of 2e-5, warm-up proportion of 10% and a maximum sequence length of 512.
[^1]: Code is available at https://github.com/google-research/language/tree/master/language/conpono and https://github.com/daniter-cu/DiscoEval
|
---
author:
- |
Alexandru Nica [^1]\
Department of Mathematics\
University of Michigan\
Ann Arbor, MI 48109-1003, USA\
(e-mail: andu@math.lsa.umich.edu)
- |
Roland Speicher [^2]\
Institut für Angewandte Mathematik\
Universität Heidelberg\
Im Neuenheimer Feld 294\
D-69120 Heidelberg, Germany\
(e-mail: roland.speicher@urz.uni-heidelberg.de)
date: November 1995
title: |
**$R$-diagonal pairs - a common approach\
to Haar unitaries and circular elements**
---
[**Introduction**]{}
The Haar unitary and the circular element provide two of the most frequently used $*$-distributions in the free probability theory of Voiculescu, and in its remarkable applications to the study of free products of von Neumann algebras (see e.g. \[4,5,11,12,19,20\]). The starting point of the present paper is that if the $R$-transform (i.e. the free analogue of the logarithm of the Fourier transform) of these $*$-distributions is considered, then expressions of a similar nature are obtained. The formulas are: $$\begin{array}{lc}
(I) &
{ [ R( \mu_{u, { u^{*} }} ) ] ( z_{1} , z_{2} ) \ = \
\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2k-2)!}{(k-1)!k!}
(z_{1} z_{2} )^{k} +
\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2k-2)!}{(k-1)!k!}
(z_{2} z_{1} )^{k} }
\end{array}$$ and $$\begin{array}{lccc}
(II) & & &
{ [ R( \mu_{c, c^{*} } ) ] ( z_{1} , z_{2} ) \ = \
z_{1} z_{2} + z_{2} z_{1} , }
\end{array}$$ where $u$ is a Haar unitary and $c$ is a circular element (in some non-commutative probability spaces). Hence, a class of pairs of elements which contains both $( u, { u^{*} })$ and $(c, { c^{*} })$ is the following: for $x,y$ elements (random variables) in a non-commutative probability space, the pair $(x,y)$ is said to be [*$R$-diagonal*]{} if the $R$-transform $R( \mu_{x,y} )$ has the form: $$\begin{array}{lccc}
(III) & & &
{ [R( \mu_{x,y} )] ( { z_{1} , z_{2} }) \ = \
\sum_{k=1}^{\infty} \alpha_{k} (z_{1} z_{2})^{k} +
\sum_{k=1}^{\infty} \alpha_{k} (z_{2} z_{1})^{k} }
\end{array}$$ for some sequence $( \alpha_{k} )_{k=1}^{\infty}$ of complex coefficients.
The main result of the paper (stated in Theorem 1.5, Corollary 1.8 below) is that the class of $R$-diagonal pairs has a remarkable property of “absorption” under the operation of “nested” multiplication of free pairs. That is, for $a_{1}, a_{2}, p_{1},
p_{2}$ in a non-commutative probability space: if the sets $\{ { a_{1}, a_{2} }\}$ and $\{ { p_{1} , p_{2} }\}$ are free, and if the pair $( { a_{1}, a_{2} })$ is $R$-diagonal, then so is $( a_{1}p_{1} ,
p_{2}a_{2} );$ and moreover, there exists a simple formula relating the $\alpha$’s of Eqn.($III$) written for the two pairs $( { a_{1}, a_{2} })$ and $( a_{1}p_{1} , p_{2}a_{2} ).$
As an immediate consequence, $R$-diagonal pairs exist in abundance, even if we only want to consider pairs of the form $(x, x^{*}),$ and in the W$^{*}$-probabilistic context. We hope that, while not as fundamental as $(u, { u^{*} })$ and $(c, { c^{*} })$ from Eqns.($I$), ($II$), other pairs in this class will also find their role in the theory, and in its applications.
A special interest is presented by the $R$-diagonal pairs of the form $(up, (up)^{*} ),$ where $u$ is a Haar unitary and $p$ is $*$-free from $u$ (in a $C^{*}$-probability space, say). A couple of applications of the main result to this class is presented in Sections 1.9, 1.10 below. A situation when several such free pairs $(up_{1} , ( up_{1} )^{*} ), \ldots , (up_{k} , ( up_{k} )^{*} )$ are considered at the same time is addressed in Theorem 1.13.
>From the technical point of view, our approach to the $R$-diagonal pairs is based on the combinatorial description of the $R$-transform, via the lattice $NC(n)$ of non-crossing partitions of ${ \{ 1, \ldots ,n \} },$ $n \geq 1.$ More than once the proofs depend in an essential way on considerations involving a certain operation ${ \framebox[7pt]{$\star$} }$ on formal power series, introduced in our previous paper [@NS2]; this operation represents in some sense the combinatorial facet of the $R$-transform approach to the multiplication of free $n$-tuples of non-commutative random variables.
A detailed description of the results of the paper is made in the next-coming Section 1. The rest of the paper is organized as follows. In Section 2 we review some facts about non-crossing partitions, and in Section 3 we review the $R$-transform and the operation ${ \framebox[7pt]{$\star$} }$. Section 4 is devoted to pointing out a certain canonical bijection between the set of intervals of $NC(n)$ and the set of 2-divisible partitions in $NC(2n),$ which plays an important role in our considerations; we also note in Section 4 how, as a consequence of this combinatorial fact, one of the applications of ${ \framebox[7pt]{$\star$} }$ presented in [@NS2] can be improved. The proofs of the results on $R$-diagonal pairs announced in Section 1 are divided between the remaining Sections 5-8 of the paper.
$\ $
$\ $
$\ $
[**1.1 Basic definitions**]{} In this section we briefly review some basic free probabilistic terminology used throughout the paper (for a more detailed treatment, we refer to the monograph [@VDN]).
[**The framework**]{} We will call [*non-commutative probability space*]{} a pair ${ ( {\cal A} , \varphi ) }$, where ${ {\cal A} }$ is a unital algebra (over ${ \mbox{\bf C} }$), and $\varphi : { {\cal A} }\rightarrow { \mbox{\bf C} }$ is a linear functional normalized by $\varphi (1) =1.$ If we require in addition that ${ {\cal A} }$ is a $C^{*}$-algebra, and $\varphi$ is positive, then ${ ( {\cal A} , \varphi ) }$ is called a [*$C^{*}$-probability space.*]{}
[**Freeness**]{} A family of unital subalgebras ${ {\cal A} }_{1} , \ldots , { {\cal A} }_{n} \subseteq { {\cal A} }$ is said to be free in ${ ( {\cal A} , \varphi ) }$ if for every $k \geq 1,$ $1 \leq i_{1}, \ldots , i_{k} \leq n,$ $a_{1} \in { {\cal A} }_{i_{1}}, \ldots , a_{k} \in { {\cal A} }_{i_{k}}$, we have the implication: $$\left\{ \begin{array}{c}
i_{1} \neq i_{2}, i_{2} \neq i_{3}, \ldots , i_{k-1} \neq i_{k} \\
\varphi ( a_{1} ) = \varphi ( a_{2} ) = \cdots = \varphi ( a_{k} ) =0
\end{array} \right\}
\ \Rightarrow \ \varphi ( a_{1} a_{2} \cdots a_{k} ) =0.$$ The notion of freeness in ${ ( {\cal A} , \varphi ) }$ extends to arbitrary subsets of ${ {\cal A} }$, by putting ${ {\cal X} }_{1} , \ldots , { {\cal X} }_{n} \subseteq { {\cal A} }$ to be free if and only if the unital subalgebras generated by them are so. The freeness of a family of elements ${ x_{1}, \ldots ,x_{n} }\in { {\cal A} }$ is defined as the one of the family of subsets $\{ x_{1} \} , \ldots ,
\{ x_{n} \} \subseteq { {\cal A} }.$
If ${ ( {\cal A} , \varphi ) }$ is a ${ \mbox{$C^{*}$-probability space} }$, then the fact that ${ x_{1}, \ldots ,x_{n} }\in { {\cal A} }$ are $*$-free means by definition that the subsets $\{ x_{1} , x_{1}^{*} \} , \ldots , \{ x_{n} , x_{n}^{*} \}$ are free.
[**Joint distributions**]{} The [*joint distribution*]{} of the family of elements ${ a_{1}, \ldots ,a_{n} }\in { {\cal A} }$, in the non-commutative probability space ${ ( {\cal A} , \varphi ) }$, is by definition the linear functional $\mu_{{ a_{1}, \ldots ,a_{n} }} : { { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle }\rightarrow { \mbox{\bf C} }$ given by: $$\left\{ \begin{array}{l}
\mu_{{ a_{1}, \ldots ,a_{n} }} (1) \ = 1, \\
\mu_{{ a_{1}, \ldots ,a_{n} }} ( { X_{i_{1}} \cdots X_{i_{k}} }) \ = \ \varphi ( { a_{i_{1}} \cdots a_{i_{k}} }) \ \ \mbox{ for }
k \geq 1 \mbox{ and } 1 \leq { i_{1} , \ldots ,i_{k} }\leq n,
\end{array} \right.$$ where ${ { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle }$ is the algebra of polynomials in $n$ non-commuting indeterminates ${ X_{1}, \ldots ,X_{n} }$.
If we only have one element $a = a_{1} \in { {\cal A} }$, then the functional in (1.2) is just $\mu_{a} : { \mbox{\bf C} }[X] \rightarrow { \mbox{\bf C} },$ $\mu_{a} (f) = \varphi (f(a))$ for $f \in { \mbox{\bf C} }[X],$ and is called the [*distribution*]{} of $a$ in ${ ( {\cal A} , \varphi ) }.$
If ${ ( {\cal A} , \varphi ) }$ is a ${ \mbox{non-commutative probability space} }$, then when speaking about the [*$*$-distribution*]{} of an element $x \in { {\cal A} }$, one usually refers to the joint distribution $\mu_{x, x^{*}};$ another (equivalent) approach goes by looking at the joint distribution $\mu_{Re(x), \ Im(x)}$ of the real and imaginary parts $Re(x) = (x+x^{*})/2,$ $Im(x) = (x-x^{*})/2i.$
The definitions of a Haar unitary and of a circular element are made by prescribing (for the circular in an indirect way) what is their $*$-distribution. We will consider the framework of a ${ \mbox{$C^{*}$-probability space} }$ ${ ( {\cal A} , \varphi ) }$. Recall that:
- an element $u \in { {\cal A} }$ is called a [*Haar unitary*]{} in ${ ( {\cal A} , \varphi ) }$ if it is unitary, and if $\varphi (a^{n}) =0$ for every $n \in { \mbox{\bf Z} }\setminus \{ 0 \} ;$
- an element $a \in { {\cal A} }$ is called [*semicircular*]{} in ${ ( {\cal A} , \varphi ) }$ if it is selfadjoint and its distribution $\mu_{a}$ is $\frac{1}{2 \pi} \sqrt{4-t^{2}} dt$ on \[-2,2\] (in other words, if $\varphi (a^{n} ) = \frac{1}{2 \pi} \int_{-2}^{2} t^{n}
\sqrt{4-t^{2}} dt,$ $ n \geq 0);$
- an element $c \in { {\cal A} }$ is called [*circular*]{} in ${ ( {\cal A} , \varphi ) }$ if it is of the form $(a+ib)/ \sqrt{2}$ with $a,b$ semicircular and free in ${ ( {\cal A} , \varphi ) }$.
We take this occasion to review one more remarkable distribution:
- an element $a \in { {\cal A} }$ is called [*quarter-circular*]{} in ${ ( {\cal A} , \varphi ) }$ if it is positive and its distribution $\mu_{a}$ is $\frac{1}{\pi} \sqrt{4-t^{2}} dt$ on \[0,2\] (i.e., if $\varphi (a^{n} ) = \frac{1}{\pi} \int_{0}^{2} t^{n}
\sqrt{4-t^{2}} dt,$ $ n \geq 0).$
[**The $R$-transform**]{} Given a functional of the kind appearing in Eqn.(1.2) (i.e. $\mu : { { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle }\rightarrow { \mbox{\bf C} }$, such that $\mu (1) =1),$ its $R$-transform $R( \mu )$ is a certain formal power series in $n$ “non-commuting complex variables” ${ z_{1}, \ldots ,z_{n} }$: $$[ R( \mu ) ] ( { z_{1}, \ldots ,z_{n} }) \ = \ { \sum_{k=1}^{\infty} \
\sum_{i_{1}, \ldots ,i_{k} =1}^{n} \alpha_{(i_{1}, \ldots ,i_{k}) }
z_{i_{1}} \cdots z_{i_{k}} }.$$ The coefficients $( \alpha_{( { i_{1} , \ldots ,i_{k} })} )_{k \geq 1, 1 \leq { i_{1} , \ldots ,i_{k} }\leq n}$ of $R( \mu )$ are also called the [*free*]{} (or [*non-crossing) cumulants*]{} of $\mu .$ The precise definition of $R( \mu )$ (i.e. of how the free cumulants are constructed from $\mu$) will be reviewed in Section 3 below.
If ${ a_{1}, \ldots ,a_{n} }$ are elements in the non-commutative probability space ${ ( {\cal A} , \varphi ) }$, then the $R$-transform $R( \mu_{ { a_{1}, \ldots ,a_{n} }} )$ contains the information about the joint distribution $\mu_{{ a_{1}, \ldots ,a_{n} }}$ rearranged in such a way that the freeness or non-freeness of ${ a_{1}, \ldots ,a_{n} }$ becomes transparent. More precisely, as proved in \[14,8\] ${ a_{1}, \ldots ,a_{n} }$ are free in ${ ( {\cal A} , \varphi ) }$ if and only if the coefficient of $z_{i_{1}} z_{i_{2}} \cdots z_{i_{k}}$ in $[ R( \mu_{{ a_{1}, \ldots ,a_{n} }} )] ( { z_{1}, \ldots ,z_{n} })$ vanishes whenever we don’t have $i_{1} = i_{2} = \cdots = i_{k} ;$ i.e., if and only if $R( \mu_{{ a_{1}, \ldots ,a_{n} }} )$ is of the form $$[ R( \mu_{{ a_{1}, \ldots ,a_{n} }} ) ] ( { z_{1}, \ldots ,z_{n} }) \ = \
f_{1} ( z_{1} ) + \cdots + f_{n} ( z_{n} ),$$ for some formal power series of one variable ${ f_{1}, \ldots ,f_{n} }.$
We will refer to the coefficient-vanishing condition presented in the preceding paragraph by saying that “the series $R( \mu_{{ a_{1}, \ldots ,a_{n} }} )$ has no mixed coefficients”. If this happens, then ${ f_{1}, \ldots ,f_{n} }$ of (1.4) can only be the 1-dimensional $R$-transforms $R( \mu_{a_{1}} ) , \ldots , R( \mu_{a_{n}} )$, respectively.
$\ $
[**1.2 Haar unitaries and circular elements**]{} If $F_{k}$ denotes the free group on generators ${ g_{1} , \ldots ,g_{k} }$, then the left-translation operators ${ u_{1} , \ldots ,u_{k} }$ with ${ g_{1} , \ldots ,g_{k} }$ on $l^{2} (F_{k})$ form a family of $*$-free Haar unitaries in $(L(F_{k} ) , \tau )$, where $L(F_{k})$ is the von Neumann II$_{1}$ factor of $F_{k}$ and $\tau$ is the unique normalized trace on $L(F_{k});$ ${ u_{1} , \ldots ,u_{k} }$ is in some sense “the obvious system of generators” for $L(F_{k}).$
In recent work of Voiculescu, Radulescu, Dykema (see e.g. \[4,5,11,12,20\]) it was shown that very powerful results on $L(F_{k})$ can be obtained by using a different family of generators, consisting of free semicircular elements. The Haar unitaries are also appearing in this picture, but in a more subtle way, either via asymptotic models (for instance in [@V4], Section 3), or via the theorem of Voiculescu [@V5] on the polar decomposition of the circular element. This theorem states that if ${ {\cal A} }$ is a von Neumann algebra, with $\varphi : { {\cal A} }\rightarrow { \mbox{\bf C} }$ a faithful normal trace, and if $c$ is circular in ${ ( {\cal A} , \varphi ) }$, then by taking the polar decomposition $c=up$ of $c$ one gets that: $u$ is a Haar unitary, $p$ is quarter-circular, and $u,p$ are $*$-free. The original proof given by Voiculescu in [@V5] for this fact depends on the asymptotic matrix model for free semicircular families developed in [@V4]. A direct, combinatorial proof was recently found by Banica [@Ba].
The starting point of this work was the observation that the $R$-transforms of the $*$-distributions of the Haar unitary and of the circular element have similar forms. In fact, the goal of the present paper is in some sense to understand the relation between these two elements, from the point of view of the $R$-transform. Although this will not be our main concern, a new proof for the polar decomposition of the circular element will also follow (see the discussion in 1.9, 1.10 below.)
If $c$ is a circular element in the ${ \mbox{$C^{*}$-probability space} }$ ${ ( {\cal A} , \varphi ) }$, then from the fact that its real and imaginary parts are multiples of free semicirculars, one gets immediately that: $$[ R( \mu_{Re (c), \ Im (c)} ) ] ( z_{1} , z_{2} ) \ = \
\frac{1}{2} ( z_{1}^{2} + z_{2}^{2} );$$ then by using a result from [@N] concerning linear changes of coordinates, this implies: $$[ R( \mu_{c, c^{*} } ) ] ( z_{1} , z_{2} ) \ = \
z_{1} z_{2} + z_{2} z_{1}.$$ On the other hand, it was shown in \[15, Section 3.4\] that for $u$ a Haar unitary in some $C^{*}$-probability space, one has: $$[ R( \mu_{u, { u^{*} }} ) ] ( z_{1} , z_{2} ) \ = \
\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2k-2)!}{(k-1)!k!}
(z_{1} z_{2} )^{k} +
\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2k-2)!}{(k-1)!k!}
(z_{2} z_{1} )^{k} .$$
$\ $
Thus a class of pairs of elements which contains both $(c, { c^{*} })$ and $(u, { u^{*} })$ is given by the following
$\ $
[**1.3 Definition:**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, and let ${ a_{1}, a_{2} }$ be in ${ {\cal A} }.$ We will say that $( { a_{1}, a_{2} })$ is an [*$R$-diagonal pair*]{} if
\(i) the coefficients of $\underbrace{z_{1} z_{2} \cdots z_{1} z_{2}}_{2n}$ and $\underbrace{z_{2} z_{1} \cdots z_{2} z_{1}}_{2n}$ in $[R( \mu_{ { a_{1}, a_{2} }} )] ( { z_{1} , z_{2} })$ are equal, for every $n \geq 1;$
\(ii) every coefficient of $R( \mu_{{ a_{1}, a_{2} }} )$ not of the form mentioned in (i) is equal to 0.
$\ $
In other words, the pair $( { a_{1}, a_{2} })$ is $R$-diagonal if and only if $R( \mu_{ { a_{1}, a_{2} }} )$ has the form $$[R( \mu_{{ a_{1}, a_{2} }} )] ( { z_{1} , z_{2} }) \ = \
\sum_{k=1}^{\infty} \alpha_{k} (z_{1} z_{2})^{k} +
\sum_{k=1}^{\infty} \alpha_{k} (z_{2} z_{1})^{k}$$ for some sequence $( \alpha_{k} )_{k=1}^{\infty}$. If this happens, then the series of one variable $f(z) = \sum_{k=1}^{\infty}
\alpha_{k} z^{k}$ will be called the [*determining series*]{} of the pair $( { a_{1}, a_{2} }).$
$\ $
[**1.4 Remark**]{} From (1.6) it follows that the determining series of $(c, { c^{*} })$ is just $f(z)=z.$ The determining series for $(u, { u^{*} })$, coming out from (1.7), will be denoted by $Moeb:$ $$Moeb (z) \ = \ \sum_{k=1}^{\infty}
\frac{(-1)^{k+1} (2k-2)!}{(k-1)!k!} z^{k} ,$$ and will be called the Moebius series (of one variable - compare also to Eqn.(3.8) below).
$\ $
The main result of the paper can then be stated as follows.
$\ $
[**1.5 Theorem**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, such that $\varphi$ is a trace (i.e. $\varphi (xy) = \varphi (yx),$ $x,y \in { {\cal A} }$), and let ${ a_{1}, a_{2} }, { p_{1} , p_{2} }\in { {\cal A} }$ be such that $( { a_{1}, a_{2} })$ is an $R$-diagonal pair, and such that $\{ { p_{1} , p_{2} }\}$ is free from $\{ { a_{1}, a_{2} }\} .$ Then $( a_{1} p_{1} , p_{2} a_{2} )$ is also an $R$-diagonal pair.
$\ $
Moreover, there exists a simple formula which connects the determining series of $( { a_{1}, a_{2} })$ and $( { a_{1} p_{1} , p_{2} a_{2} });$ this formula is presented in Corollary 1.8 below.
$\ $
[**1.6 The operation ${ \framebox[7pt]{$\star$} }$**]{} The formula announced in the previous phrase involves a certain binary operation ${ \framebox[7pt]{$\star$} }$ on the set of ${ \mbox{formal power series} }$ $\{ f \ |$ $f(z) = \sum_{k=1}^{\infty} \alpha_{k} z^{k} ; \ $ $\alpha_{1} , \alpha_{2} , \alpha_{3} , \ldots \in { \mbox{\bf C} }\} .$ One possible way of defining ${ \framebox[7pt]{$\star$} }$ is via the equation $$R( \mu_{ab} ) \ = \ R( \mu_{a} ) \ { \framebox[7pt]{$\star$} }\ R( \mu_{b} ),$$ holding whenever $a$ is free from $b$ in some ${ \mbox{non-commutative probability space} }$ ${ ( {\cal A} , \varphi ) }.$ (This definition makes sense because $R( \mu_{ab} )$ is completely determined by $R( \mu_{a} )$ and $R( \mu_{b} ),$ and because any two series $f,g$ of the considered type can be realized as $R( \mu_{a} )$ and $R( \mu_{b} )$ with $a,b$ free in some ${ ( {\cal A} , \varphi ) }.)$ The operation ${ \framebox[7pt]{$\star$} }$ also has an alternative combinatorial definition, which will be reviewed in Section 3.3 below. The best point of view seems to be to consider both approaches to ${ \framebox[7pt]{$\star$} }$, and switch from one to the other as needed. This operation appeared (under a different name) in \[14,9\], in connection to the work of Voiculescu [@V3] on products of free elements. The name ${ \framebox[7pt]{$\star$} }$ was first used in [@NS2], where the multivariable versions of the operation were introduced and applied. We mention that an important feature distinguishing the 1-dimensional instance of ${ \framebox[7pt]{$\star$} }$ from the others is that in this (and only this) case ${ \framebox[7pt]{$\star$} }$ is commutative.
$\ $
The formula announced immediately after Theorem 1.5 comes out in the following way.
$\ $
[**1.7 Proposition**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, such that $\varphi$ is a trace, and let $( { a_{1}, a_{2} })$ be an $R$-diagonal pair in ${ ( {\cal A} , \varphi ) }$. If $f$ is the determining series of $( { a_{1}, a_{2} }),$ then: $f \ = \ R( \mu_{ a_{1} a_{2} } ) \ { \framebox[7pt]{$\star$} }\ Moeb,$ where $Moeb$ is the Moebius series, as in Eqn.(1.9) (and of course, $\mu_{ a_{1} a_{2} } : { \mbox{\bf C} }[X] \rightarrow { \mbox{\bf C} }$ denotes the distribution of the product $a_{1} \cdot a_{2} ).$
$\ $
[**1.8 Corollary**]{} In the context of Theorem 1.5, if $f$ and $g$ are the determining series of the $R$-diagonal pairs $( { a_{1}, a_{2} })$ and $( { a_{1} p_{1} , p_{2} a_{2} }),$ respectively, then we have the relation $$g \ = \ f \ { \framebox[7pt]{$\star$} }\ R( \mu_{ p_{1} p_{2} } ).$$
$\ $
[**Proof**]{} We can write that $$g \ = \ R( \mu_{ a_{1} p_{1} p_{2} a_{2} } ) \ { \framebox[7pt]{$\star$} }\ Moeb
\mbox{ (by Proposition 1.7) }$$ $$= \ R( \mu_{ a_{2} a_{1} p_{1} p_{2} } ) \ { \framebox[7pt]{$\star$} }\ Moeb
\mbox{ (because $\varphi$ is a trace) } $$ $$= \ R( \mu_{ a_{2} a_{1} } ) \ { \framebox[7pt]{$\star$} }\ R( \mu_{ p_{1} p_{2} } )
\ { \framebox[7pt]{$\star$} }\ Moeb \mbox{ (by (1.10)) }$$ $$= \ \left( R( \mu_{ a_{2} a_{1} } ) \ { \framebox[7pt]{$\star$} }\ Moeb \right)
\ { \framebox[7pt]{$\star$} }\ R( \mu_{ p_{1} p_{2} } )
\mbox{ (because ${ \framebox[7pt]{$\star$} }$ is commutative) } $$ $$= \ f \ { \framebox[7pt]{$\star$} }\ R( \mu_{ p_{1} p_{2} } )
\mbox{ (again by Proposition 1.7). {\bf QED} }$$
$\ $
[**1.9 Application**]{} Note that the Moebius series appears in two different ways in the above discussion (in Remark 1.4 and Proposition 1.7, respectively). This has a consequence concerning $R$-diagonal pairs of the form $(x, { x^{*} }),$ in the $C^{*}$-context. Let us denote by ${\cal R}_{c}$ the set of formal power series of one variable which occur as $R( \mu ),$ with $\mu$ a probability measure on ${ \mbox{\bf R} }$ having compact support contained in $[0, \infty )$ (in connection to ${\cal R}_{c},$ see also [@V2], Section 3). We have the following
[**Fact:**]{} A formal power series $f$ of one variable can appear as determining series for an $R$-diagonal pair $(x, { x^{*} })$ in some $C^{*}$-probability space if and only if it is of the form $f = g { \framebox[7pt]{$\star$} }Moeb$, with $g \in {\cal R}_{c}.$ If this happens, then $f$ can be in fact written as the determining series of an $R$-diagonal pair $(up , (up)^{*} ),$ with $u$ Haar unitary, $p$ positive, and such that $u$ is $*$-free from $p.$
[**Proof**]{} Implication “$\Rightarrow$” follows from Proposition 1.7 ($f= R( \mu_{xx^{*}} ) { \framebox[7pt]{$\star$} }Moeb$, and $R( \mu_{xx^{*}} ) \in {\cal R}_{c} ).$ Conversely, assume that $f=g { \framebox[7pt]{$\star$} }Moeb,$ with $g \in {\cal R}_{c}.$ We can always find a $C^{*}$-probability space ${ ( {\cal A} , \varphi ) }$ and $u,p \in { {\cal A} }$, $*$-free, such that $u$ is Haar unitary, $p$ is positive, and $R( \mu_{p^{2}} )=g.$ (For instance we can take ${ {\cal A} }= L^{\infty} ( \mu ) \star L^{\infty} ( { \mbox{\bf T} }),$ endowed with the free product of $\mu$ with the Lebesgue measure on ${ \mbox{\bf T} }$, where $\mu$ is such that $R( \mu ) =g.)$ The pair $(up , (up)^{*} )$ is $R$-diagonal by Theorem 1.5, and has determining series $Moeb { \framebox[7pt]{$\star$} }R(
\mu_{p^{2}} ) = Moeb { \framebox[7pt]{$\star$} }g =f,$ by Corollary 1.8. [**QED**]{}
We thus see that, from the point of view of the $*$-distribution, any $R$-diagonal pair $(x, { x^{*} })$ in a $C^{*}$-probability space can be replaced with one of the form $( up , (up)^{*} ).$ This can be pushed to a “polar decomposition result”, if we consider the von Neumann algebra setting, with a normal faithful trace, and if we also assume that Ker $x$ = $\{ 0 \} .$ Indeed, in such a situation we get that the von Neumann subalgebras generated by $x$ and $up$ (in their $W^{*}$-probability spaces) are canonically isomorphic, by an isomorphism which sends $x$ into $up$ (this is the same type of argument as, for instance, in [@V5], Remark 1.10). We obtain in this way a product decomposition $x =u'p',$ with $u'$ Haar unitary, $p'$ positive, and $u'$ $*$-free from $p',$ and the uniqueness of the polar decomposition shows that $u'p'$ is necessarily the polar decomposition of $x.$ (The needed fact that Ker $p'$ = $\{ 0 \}$ is obtained by verifying that the distribution of $p'^{2}$ has no atom at 0.)
$\ $
[**1.10 Application**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{$C^{*}$-probability space} }$, with $\varphi$ a trace, and let $u,p \in { {\cal A} }$ be such that $u$ is a Haar unitary, $*$-free from $p.$ We look for necessary and sufficient conditions for the real and imaginary parts of $up$ to be free. Without loss of generality, we can assume that $p$ is normalized in such a way that $\varphi (pp^{*} ) =1.$
By using 1.5, 1.8 and also Eqn.(1.7) of 1.2 we infer that $(up, p^{*} u^{*} )$ is an $R$-diagonal pair, with determining series $g$ given by: $$g \ = \ Moeb \ { \framebox[7pt]{$\star$} }\ R( \mu_{ pp^{*} } ) .$$ We write explicitly $g(z) = \sum_{k=1}^{\infty} \beta_{k} z^{k} ;$ the assumption that $\varphi (pp^{*} ) =1$ plugged into (1.12) implies that $\beta_{1} =1.$ Remembering how the determining series was defined in 1.3, we have that $$[ R( \mu_{up , p^{*} u^{*} } ) ] ( { z_{1} , z_{2} }) \ = \
\sum_{k=1}^{\infty} \beta_{k} ( z_{1} z_{2} )^{k} \ + \
\sum_{k=1}^{\infty} \beta_{k} ( z_{2} z_{1} )^{k} ;$$ then by doing a linear change of coordinates (as in [@N], Section 5) we get $$[ R( \mu_{Re(up) , Im(up) } ) ] ( { z_{1} , z_{2} }) \ = \
\sum_{k=1}^{\infty} \frac{\beta_{k}}{4^{k}}
( ( z_{1} + i z_{2} )( z_{1} - i z_{2}) )^{k} \ + \
\sum_{k=1}^{\infty} \frac{\beta_{k}}{4^{k}}
( ( z_{1} - i z_{2} )( z_{1} + i z_{2}) )^{k} .$$
By the result stated in (1.4), $Re(up)$ and $Im(up)$ are free if and only if the series in (1.13) has no mixed coefficients; but a direct analysis of the right-hand side of (1.13) shows that this can happen if and only if $\beta_{2} = \beta_{3} = \cdots =0.$ Hence $Re(up)$ and $Im(up)$ are free if and only if the series $g$ of (1.12) is just $g(z) =z.$ Finally, the equation $[ Moeb \ { \framebox[7pt]{$\star$} }\ R( \mu_{pp^{*}} ) ] (z) =z$ is easily solved “in the unknown” $R( \mu_{ pp^{*} } ),$ and is found to be equivalent to $[R( \mu_{ pp^{*} } )] (z) = z/(1-z).$ We thus obtain the following
[**Fact:**]{} With $u$ and $p$ as in the first paragraph of 1.10, the necessary and sufficient condition for the real and imaginary part of $up$ to be free is $$[R( \mu_{ pp^{*} } )] (z) = z/(1-z).$$
Note that if $\beta_{1} =1$ and $\beta_{2} = \beta_{3} = \cdots
= 0,$ then the right-hand side of (1.13) is just $(z_{1}^{2} + z_{2}^{2} )/2 ,$ and by comparing (1.13) against (1.5) we find that $up$ is circular. Hence $up$ is circular whenever (1.14) holds.
We also note that (1.14) does hold if $p=p^{*} =$ quarter-circular (the square of the quarter-circular is the same thing as the square of the semicircular, and the $R$-transform of the latter square is well-known - see e.g. [@NS2], Lemma 4.2 or Lemma 1.1 in the Appendix). Here again the uniqueness of the polar decomposition yields from this point a proof for the polar decomposition of the circular element.
$\ $
[**1.11 Case of several pairs**]{} Another aspect which can be studied in the context of 1.10 is: what happens if instead of looking just at $up,$ we look at a family $up_{1} , up_{2}, \ldots , up_{k},$ where $u, p_{1}, p_{2} , \ldots , p_{k}$ are $*$-free? It is shown by Banica in [@Ba] that $up_{1} , up_{2}, \ldots , up_{k}$ are also $*$-free if the following happens: every $p_{j}$ $(1 \leq j \leq k)$ is in some sense $*$-modeled by an operator of the form $S^{*} + S^{m_{j}},$ $m_{j} \geq 1,$ where $S$ is the unilateral shift on $l^{2} ( { \mbox{\bf N} }).$ We have found a condition expressed in terms of “moments of pairs”, which still implies the $*$-freeness of $up_{1} , up_{2}, \ldots , up_{k}.$ This condition, which is satisfied by the pair $( S^{*} + S^{m} , S + {S^{m}}^{*} )$ for every $m \geq 1 ,$ is described as follows.
$\ $
[**1.12 Definition**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, and let ${ a_{1}, a_{2} }$ be in ${ {\cal A} }.$ We will say that $( { a_{1}, a_{2} })$ is a [*diagonally balanced pair*]{} if $$\varphi ( \underbrace{a_{1}a_{2} \cdots a_{1}a_{2}a_{1} }_{2n+1} ) =
\varphi ( \underbrace{a_{2}a_{1} \cdots a_{2}a_{1}a_{2} }_{2n+1} ) =
0, \ \mbox{ for every } n \geq 0.$$
$\ $
Then we have:
$\ $
[**1.13 Theorem**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, with $\varphi$ a trace, and let $u, p_{1,1}, p_{1,2} , \ldots , p_{k,1},
p_{k,2}$ be in ${ {\cal A} }$ such that:
\(i) $u$ is invertible and $\varphi (u^{n}) =0$ for every $n \in { \mbox{\bf Z} }\setminus \{ 0 \};$
\(ii) the pairs $( p_{1,1} , p_{1,2} ), \ldots ,
( p_{k,1} , p_{k,2} )$ are diagonally balanced;
\(iii) the sets $\{ u, { u^{-1} }\} ,
\{ p_{1,1}, p_{1,2} \} , \ldots , \{ p_{k,1} , p_{k,2} \}$ are free. Then the sets $\{ up_{1,1}, p_{1,2} { u^{-1} }\} , \ldots ,
\{ up_{k,1}, p_{k,2} { u^{-1} }\}$ are also free.
$\ $
The techniques used for proving the results announced in 1.5, 1.7, 1.13 above are based on the combinatorial approach to the $R$-transform, via the lattice $NC(n)$ of non-crossing partitions of ${ \{ 1, \ldots ,n \} }$, $n \geq 1.$ In some instances, we are able to use an elegant idea of Biane [@Bi] which reduces assertions on non-crossing partitions to calculations in the group algebra of the symmetric group (but there are also situations when this mechanism is apparently not applying, and we have to use “geometric” arguments). It seems that from the combinatorial point of view, a certain canonical bijection between the set of intervals of $NC(n)$ and the set of 2-divisible partitions in $NC(2n)$ has a significant role in the considerations. We have incidentally noticed that, as a consequence of this combinatorial fact, we can improve one of the applications of the operation ${ \framebox[7pt]{$\star$} }$ that were presented in [@NS2]. Namely, we have:
$\ $
[**1.14 Theorem**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, with $\varphi$ a trace, and let ${ a_{1}, \ldots ,a_{n} }, b' , b'' \in { {\cal A} }$ be such that the pair $(b',b'')$ is diagonally balanced, and such that $\{ b',b'' \}$ is free from $\{ { a_{1}, \ldots ,a_{n} }\} .$ Then $\{ b'a_{1}b'' , \ldots , b'a_{n}b'' \}$ is free from $\{ { a_{1}, \ldots ,a_{n} }\} .$
$\ $
The particular case of 1.14 when $b'$ = $b''$ is a semicircular element is stated in Application 1.10 of [@NS2]. The Theorem 1.14 sounds quite “elementary”, and it is not impossible that it also has a simple direct proof, using only the definition of freeness (we weren’t able to find one, though).
$\ $
$\ $
$\ $
$\ $
[**2.1 Definition of NC(n)**]{} If $\pi = \{ B_{1} , \ldots , B_{r} \}$ is a partition of ${ \{ 1, \ldots ,n \} }$ (i.e. $B_{1} , \ldots , B_{r}$ are pairwisely disjoint, non-void sets, such that $B_{1} \cup \cdots \cup B_{r}$ = ${ \{ 1, \ldots ,n \} }$), then the equivalence relation on ${ \{ 1, \ldots ,n \} }$ with equivalence classes $B_{1} , \ldots , B_{r}$ will be denoted by ${ \stackrel{\pi}{\sim} }$; the sets $B_{1} , \ldots , B_{r}$ will be also referred to as the [*blocks*]{} of $\pi$. The number of elements in the block $B_k$, $1 \leq k \leq r,$ will be denoted by $|B_{k}|$.
A partition $\pi$ of ${ \{ 1, \ldots ,n \} }$ is called [*non-crossing*]{} if for every $1 \leq i < j < i' < j' \leq n$ such that $i { \stackrel{\pi}{\sim} }i'$ and $j { \stackrel{\pi}{\sim} }j'$, it necessarily follows that $i { \stackrel{\pi}{\sim} }j { \stackrel{\pi}{\sim} }i' { \stackrel{\pi}{\sim} }j'$. The set of all non-crossing partitions of ${ \{ 1, \ldots ,n \} }$ will be denoted by $NC(n).$ On $NC(n)$ we will consider the [*refinement order,*]{} defined by $\pi \leq \rho \ { \stackrel{def}{ \Leftrightarrow } }$ each block of $\rho$ is a union of blocks of $\pi$ (in other words, $\pi \leq \rho$ means that the implication $i { \stackrel{\pi}{\sim} }j \Rightarrow i { \stackrel{\rho}{\sim} }j$ holds, for $1 \leq i,j \leq n).$ The partially ordered set $NC(n)$ was introduced by G. Kreweras in [@K], and its combinatorics has been studied by several authors (see e.g. [@SU], and the list of references there).
$\ $
[**2.2 The circular picture**]{} of a partition $\pi = \{ B_{1},
\ldots , B_{r} \}$ of ${ \{ 1, \ldots ,n \} }$ is obtained by drawing $n$ equidistant and clockwisely ordered points ${ P_{1} , \ldots ,P_{n} }$ on a circle, and then by drawing for each block $B_{k}$ of $\pi$ the inscribed convex polygon with vertices $\{ P_{i} \ | \ i \in B_{k} \}$ (this polygon may of course be reduced to a point or a line segment). It is immediately verified that $\pi$ is non-crossing if and only if the $r$ convex polygons obtained in this way are disjoint.
We take the occasion to mention that when $B$ is a block of the partition $\pi \in NC(n),$ we will use for $i<j$ in $B$ the expression “$i$ and $j$ are consecutive in $B$” to mean that either $B \cap \{ i+1, \ldots , j-1 \} = \emptyset$ or $i$ = min $B,$ $j$ = max $B.$ On the circular picture, the fact that $i,j$ are consecutive in $B$ means that $P_{i} P_{j}$ is an edge (rather than a diagonal) of the inscribed polygon with vertices $\{ P_{h} \ | \ h \in B \} .$
$\ $
[**2.3 The Kreweras complementation map**]{} is a remarkable order anti-isomorphism $K:NC(n) \rightarrow NC(n)$, introduced in $\cite{K}$, Section 3, and described as follows.
Let $\pi$ be in $NC(n),$ and consider the circular picture of $\pi ,$ involving the points $P_{1} , \ldots , P_{n},$ as in 2.2. Denote the midpoints of the arcs of circle $P_1 P_2 , \ldots ,
P_{n-1} P_n , \ P_n P_1$ by $Q_{1}, \ldots , Q_{n-1},$ $ Q_{n}$, respectively. Then the “complementary” partition $K( \pi )
\in NC(n)$ is given, in terms of the corresponding equivalence relation ${ \stackrel{ K( \pi ) }{\sim} }$ on ${ \{ 1, \ldots ,n \} }$, by putting for $1 \leq i,j \leq n:$ $$i { \stackrel{ K( \pi ) }{\sim} }j \ { \stackrel{def}{ \Leftrightarrow } }\ \left\{ \begin{array}{l}
{ \mbox{ there are no $1 \leq h,k \leq n$ such that
$h { \stackrel{\pi}{\sim} }k$ and such that} } \\
{ \mbox{ the line segments $Q_{i} Q_{j}$ and $P_{h} P_{k}$ have
non-void intersection.} }
\end{array} \right.$$ It is easily verified that ${ \stackrel{ K( \pi ) }{\sim} }$ of (2.1) is indeed an equivalence relation on ${ \{ 1, \ldots ,n \} }$, and that the partition $K( \pi )$ corresponding to it is non-crossing (for the latter thing, we just have to look at the circular picture of $K( \pi ),$ with respect to the points ${ Q_{1} , \ldots ,Q_{n} }).$
In order to verify that $K: NC(n) \rightarrow NC(n)$ defined in this way really is an order anti-isomorphism, one can first remark that $K^{2} ( \pi )$ is (for every $\pi \in NC(n)$) a rotation of $\pi$ with $360^{o}/n$; this shows in particular that $K$ is a bijection. The implication $\pi \leq \rho \Rightarrow K( \pi ) \geq K( \rho )$ is a direct consequence of (2.1), and the converse must also hold, since $K^{2}$ is an order-preserving isomorphism of $NC(n).$
As a concrete example, the circular-picture verification for $K( \{ \{ 1,4,5 \} , \{ 2,3 \} ,$ $\{ 6,8 \} ,$ $\{ 7 \} \} )$ = $\{ \{ 1,3 \} , \{ 2 \} , \{ 4 \} , \{ 5,8 \} , \{ 6,7 \} \}
\in NC(8)$ is shown in Figure 1.
$\ $
$$\mbox{\bf Figure 1.}$$
By examining the common circular picture for $\pi$ and $K( \pi )$, it becomes quite obvious that $K( \pi )$ could be alternatively defined as the biggest $\rho$ in $(NC(n) , \leq )$ with the property that the partition of ${ \{ 1,2, \ldots ,2n \} }$ obtained by interlacing $\pi$ and $\rho$ is still non-crossing. This fact is formally recorded in the next proposition.
$\ $
[**2.4 Proposition**]{} Let $\pi$ and $\rho$ be in $NC(n)$. Denote by $\pi '$ and $\rho '$ the partitions of $\{ 2,4, \ldots ,$ $2n \}$ and $\{ 1,3, \ldots , 2n-1 \}$, respectively, which get identified to $\pi$ and $\rho$ via the order-preserving bijections $ { \{ 1, \ldots ,n \} }\rightarrow \{ 2,4, \ldots ,2n \}$ and $ { \{ 1, \ldots ,n \} }\rightarrow \{ 1,3, \ldots ,2n-1 \}$. Denote by $\sigma$ the partition of $\{ 1,2, \ldots , 2n \}$ formed by putting $\pi '$ and $\rho '$ together. Then $\sigma$ is ${ \mbox{non-crossing} }$ if and only if $\pi \leq K( \rho ).$
$\ $
[**2.5 The relative Kreweras complement**]{} Given $\rho \in NC(n),$ one can define a relativized version of the Kreweras complementation map, $K_{\rho},$ which is an order anti-isomorphism of $\{ \pi \in NC(n) \ | \ \pi \leq \rho \} .$ If we write explicitly $\rho = \{ B_{1}, \ldots , B_{r} \} ,$ then an arbitrary element of $\{ \pi \in NC(n) \ | \ \pi \leq \rho \}$ can be written as $\{ A_{1,1}, \ldots , A_{1,s_{1}}, \ldots , A_{r,1}, \ldots ,
A_{r,s_{r}} \} ,$ where $A_{k,1} \cup \cdots \cup A_{k,s_{k}} =
B_{k},$ $1 \leq k \leq r.$ The relative Kreweras complement $K_{\rho} ( \pi )$ is obtained by looking for every $1 \leq k \leq r$ at the non-crossing partition $\{ A_{k,1} , \ldots , A_{k,s_{k}} \}$ of $B_{k},$ and by taking its Kreweras complement, call it $\theta_{k} ,$ in the sense of Section 2.3 above (of course, in order to do this, we need to identify canonically $B_k$ with $\{ 1, \ldots , |B_{k}| \} )$. Then $K_{\rho} ( \pi )$ is obtained by putting together the partitions $\theta_{1}$ of $B_{1},$ $\ldots$ , $\theta_{r}$ of $B_{r}$ (see also [@NS2], Sections 2.4, 2.5).
Note that if $\rho = \{ { \{ 1, \ldots ,n \} }\}$ is the maximal element of $(NC(n), \leq )$, then $K_{\rho}$ coincides with $K:NC(n) \rightarrow NC(n)$ discussed in 2.3.
$\ $
[**2.6 Relation with permutations**]{} A useful way of “encoding” non-crossing partitions by permutations was introduced by Ph. Biane in [@Bi]. Let ${\cal S}_n$ denote the group of all permutations of ${ \{ 1, \ldots ,n \} }$. For $B = \{ i_1 < i_2 < \cdots < i_m \} \subseteq { \{ 1, \ldots ,n \} }$ we denote by $\gamma_{B} \in {\cal S}_n$ the cycle given by $$\left\{ \begin{array}{l}
{ \gamma_{B} ( i_1 ) = i_2 , \ldots , \gamma_{B} ( i_{m-1} ) = i_m ,
\gamma_{B} ( i_{m} ) = i_1 ,} \\
{ \gamma_{B} (j) = j \mbox{ for } j \in { \{ 1, \ldots ,n \} }\setminus B }
\end{array} \right.$$ (if $|B| =1,$ we take $\gamma_{B}$ to be the unit of ${\cal S}_n$). Then for every $\pi \in NC(n),$ the permutation associated to it is $$Perm ( \pi ) \ = \ \prod_{B \ block \ of \ \pi} \gamma_{B}$$ (the cycles $( \gamma_{B} \ ; \ B$ block of $\pi$) commute, so the order of the factors in the product (2.3) does not matter).
It was shown in [@Bi] how the (obviously injective) map $Perm : NC(n) \rightarrow {\cal S}_{n}$ can be used for an elegant analysis of the skew-automorphisms (i.e. automorphisms or anti-automorphisms) of $(NC(n) , \leq ).$ We will only need here the “$Perm$” characterization of the Kreweras complementation map, which goes as follows: if $\gamma \in {\cal S}_{n}$ denotes the cycle ${ ( 1 \rightarrow 2 \rightarrow \cdots \rightarrow
n \rightarrow 1 ) }$, then $$Perm (K( \pi )) \ = \ Perm ( \pi )^{-1} \ \gamma ,
\mbox{ for every } \pi \in NC(n).$$ This characterization can be extended without difficulty to the situation of the relative Kreweras complement, we have: $$Perm ( K_{\rho} ( \pi )) \ = \ Perm ( \pi )^{-1} \ Perm ( \rho ),$$ for every $\pi , \rho \in NC(n)$ such that $\pi \leq \rho$ (see [@NS2], Section 2.5).
$\ $
$\ $
$\ $
${ \framebox[7pt]{$\star$} }$ [**and of the**]{} $R$[**-transform**]{}
$\ $
We will follow the presentation of [@NS2], Section 3. The approach to the $R$-transform taken here is based on elements of Moebius inversion theory for non-crossing partitions, on the lines of [@S1]. We mention that the connection between this approach and the original one of Voiculescu in [@V2] is made via an alternative description of the $n$-dimensional $R$-transform, which goes by “modeling on the full Fock space over ${ \mbox{\bf C} }^{n}$” (see [@N]).
$\ $
[**3.1 Notation**]{} Let $n$ be a positive integer. We denote by $\Theta_{n}$ the set of formal power series without constant coefficient in $n$ non-commuting variables $z_1 , \ldots , z_n .$ An element of $\Theta_{n}$ is thus a series of the form $$f( { z_{1}, \ldots ,z_{n} }) \ = \ { \sum_{k=1}^{\infty} \
\sum_{i_{1}, \ldots ,i_{k} =1}^{n} \alpha_{(i_{1}, \ldots ,i_{k}) }
z_{i_{1}} \cdots z_{i_{k}} },$$ where $( \alpha_{( { i_{1} , \ldots ,i_{k} })} \ ; \ k \geq 1, \ 1 \leq { i_{1} , \ldots ,i_{k} }\leq n)$ is a family of complex coefficients.
$\ $
[**3.2 Notations for coefficients**]{} The following conventions for denoting coefficients of formal power series will be used throughout the whole paper.
$1^{o}$ For $f \in \Theta_{n}$ and $k \geq 1$, $1 \leq { i_{1} , \ldots ,i_{k} }\leq n,$ we will denote $$[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }] (f) \ { \stackrel{def}{=} }\ \mbox{the coefficient of ${ z_{i_{1}} \cdots z_{i_{k}} }$ in $f$} .$$
$2^{o}$ Restrictions of $k$-tuples: let $k \geq 1$ and $1 \leq { i_{1} , \ldots ,i_{k} }\leq n$ be integers, and let $B = \{ h_{1} <
h_{2} < \cdots < h_{r} \}$ be a non-void subset of ${ \{ 1, \ldots ,k \} }.$ Then by “$( { i_{1} , \ldots ,i_{k} }) | B$” we will understand the $r$-tuple $(i_{h_{1}} , i_{h_{2}} , \ldots , i_{h_{r}} ).$ An expression like $$[ \mbox{coef } ( { i_{1} , \ldots ,i_{k} })|B ] (f)$$ for $f \in \Theta_{n},$ $k \geq 1, \ 1 \leq { i_{1} , \ldots ,i_{k} }\leq n$ and $\emptyset \neq B \subseteq { \{ 1, \ldots ,k \} }$, will hence mean that the convention of notation (3.2) is applied to the $|B|$-tuple $( { i_{1} , \ldots ,i_{k} })|B.$
$3^{o}$ Given $f \in \Theta_{n},$ $k \geq 1, \ 1 \leq { i_{1} , \ldots ,i_{k} }\leq n$ integers, and $\pi$ a non-crossing partition of ${ \{ 1, \ldots ,k \} },$ we will denote $$[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }; \pi ] (f) \ { \stackrel{def}{=} }\ \prod_{B \ block \ of \ \pi}
[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }|B ] (f) .$$ Thus if $n,k, \ 1 \leq { i_{1} , \ldots ,i_{k} }\leq n$ and $\pi$ are fixed, then $[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }; \pi ] : \Theta_{n} \rightarrow { \mbox{\bf C} }$ is a functional, generally non-linear (it is linear if and only if $\pi$ is the partition into only one block, $\{ { \{ 1, \ldots ,k \} }\} ,$ in which case $[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }; \pi ] = [ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }]$ of (3.2)).
$4^{o}$ The 1-dimensional case: If $n=1,$ then the convention of notation in (3.2) can (and will) be abridged from $[ \mbox{coef } ( \underbrace{1, \ldots ,1}_{k} ) ] (f)$ to just $[ \mbox{coef } (k)] (f).$ A similar abbreviation will be used for the convention in (3.4); note that the restriction of $k$-tuples gets in this case the form “$(k)|B = ( |B| )$”, for $\emptyset \neq B \subseteq { \{ 1, \ldots ,k \} },$ hence (3.4) becomes: $$[ \mbox{coef } (k); \pi ] (f) \ = \
\prod_{B \ block \ of \ \pi} [ \mbox{coef } (|B|) ] (f),$$ for $f \in \Theta_{1}, \ k \geq 1$ and $\pi \in NC(k).$
$\ $
[**3.3 The operation [ ]{}**]{} Let $n$ be a positive integer. We denote by ${ \framebox[7pt]{$\star$} }\ (= { \framebox[7pt]{$\star$} }_{n} )$ the binary operation on the set $\Theta_{n}$ of 3.1, determined by the formula $$[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }] (f { \framebox[7pt]{$\star$} }g) \ = \ \sum_{\pi \in NC(k)}
[ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }; \pi ] (f) \cdot [ { \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) }; K( \pi ) ] (g) ,$$ holding for every $f,g \in \Theta_{n}, \ k \geq 1, \ 1 \leq
{ i_{1} , \ldots ,i_{k} }\leq n,$ and where $K: NC(k) \rightarrow NC(k)$ is the Kreweras complementation map reviewed in Section 2.3.
The operation ${ \framebox[7pt]{$\star$} }$ on $\Theta_{n}$ is associative (see [@NS2], Proposition 3.5) and is commutative when (and only when) $n=1$ (see e.g. [@NS1], Proposition 1.4.2). The unit for ${ \framebox[7pt]{$\star$} }$ is the series which takes the sum of the variables, $Sum ( { z_{1}, \ldots ,z_{n} }) { \stackrel{def}{=} }z_{1} + \cdots + z_{n}.$
An important role in the considerations related to ${ \framebox[7pt]{$\star$} }$ is played by the series $$Zeta ( { z_{1}, \ldots ,z_{n} }) \ = \ \sum_{k=1}^{\infty} \
\sum_{{ i_{1} , \ldots ,i_{k} }=1}^{n} { z_{i_{1}} \cdots z_{i_{k}} }$$ and $$Moeb ( { z_{1}, \ldots ,z_{n} }) \ = \ \sum_{k=1}^{\infty} \ \sum_{{ i_{1} , \ldots ,i_{k} }=1}^{n}
(-1)^{k+1} \frac{ (2k-2)! }{ (k-1)! k! } \ { z_{i_{1}} \cdots z_{i_{k}} },$$ which are called the ($n$-variable) Zeta and Moebius series, respectively. (The names are coming from the combinatorial interpretation of these series. The relation with the Moebius inversion theory in a poset, as developed in [@DRS], is particularly clear in the case when $n=1$ - see [@NS1].) $Zeta$ and $Moeb$ are inverse to each other with respect to ${ \framebox[7pt]{$\star$} },$ i.e. $Zeta { \framebox[7pt]{$\star$} }Moeb$ = $Sum$ = $Moeb { \framebox[7pt]{$\star$} }Zeta.$ Moreover, they are central with respect to ${ \framebox[7pt]{$\star$} }$ (i.e. $Zeta { \framebox[7pt]{$\star$} }f = f { \framebox[7pt]{$\star$} }Zeta$ for every $f \in \Theta_{n},$ and similarly for $Moeb).$
$\ $
Although chronologically the $R$-transform preceded the ${ \framebox[7pt]{$\star$} }$-operation, it is convenient to define it here in the following way.
$\ $
[**3.4 Definition**]{} Let $n$ be a positive integer. To every functional $\mu : { { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle }\rightarrow { \mbox{\bf C} },$ normalized by $\mu (1)=1$ (where ${ { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle }$ is the algebra of polynomials in $n$ non-commuting indeterminates, as in (1.2)), we attach two formal power series $M( \mu ), R( \mu ) \in \Theta_{n},$ by the equations: $$[M( \mu )] ( { z_{1}, \ldots ,z_{n} }) \ = \ \sum_{k=1}^{\infty} \
\sum_{{ i_{1} , \ldots ,i_{k} }=1}^{n} \mu ( { X_{i_{1}} \cdots X_{i_{k}} }) { z_{i_{1}} \cdots z_{i_{k}} }$$ and $$R( \mu ) \ = \ M( \mu ) \ { \framebox[7pt]{$\star$} }\ Moeb.$$ The series $R( \mu )$ is called the $R$-transform of $\mu .$
$\ $
[**3.5 The moment-cumulant formula**]{} Since $Moeb$ is the inverse of $Zeta$ under ${ \framebox[7pt]{$\star$} },$ the Equation (3.10) is clearly equivalent to $$M( \mu ) \ = \ R( \mu ) \ { \framebox[7pt]{$\star$} }\ Zeta;$$ (3.11) is in some sense “the formula for the $R^{-1}$-transform”, since the transition from $M( \mu )$ back to $\mu$ is trivial. The formula (3.11) is more frequently used than (3.10), for the reason that the coefficients of $Zeta$ are easier to handle than those of $Moeb.$ The equation obtained by plugging (3.6) into (3.11) and taking into account that $[ \mbox{coef } ( { i_{1} , \ldots ,i_{k} }); \pi ] (Zeta)$ is always equal to 1 was first observed in [@S1], and looks like this: $$[ \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) ] (M( \mu )) \ = \
\sum_{\pi \in NC(k)}
[ \mbox{coef } ( { i_{1} , \ldots ,i_{k} }); \pi ] (R( \mu )),$$ for $k \geq 1$ and $1 \leq { i_{1} , \ldots ,i_{k} }\leq n.$ We will call (3.12) the [*moment-cumulant formula,*]{} because it connects the coefficient $[ \mbox{coef } ( { i_{1} , \ldots ,i_{k} }) ] (M( \mu )) $ - which is just $\mu ( { X_{i_{1}} \cdots X_{i_{k}} }),$ a “moment” of $\mu ,$ with the coefficients of $R( \mu )$ - which were called in [@S1] the “free (or non-crossing) cumulants” of $\mu .$
$\ $
The multi-variable $R$-transform has the fundamental property that it stores the information about a joint distribution (in the sense of Eqn.(1.2)) in such a way that the freeness or non-freeness of the non-commutative random variables involved becomes very transparent. More precisely, we have the following
$\ $
[**3.6 Theorem \[14,8\]:**]{} The families of elements $\{ a_{1,1}, \ldots ,a_{1,m_{1}} \} ,$ $\ldots$ , $\{ a_{n,1}, \ldots ,a_{n,m_{n}} \}$ are free in the non-commutative probability space ${ ( {\cal A} , \varphi ) }$ if and only if the coefficient of $z_{i_{1},j_{1}} z_{i_{2},j_{2}} \cdots z_{i_{k},j_{k}}$ in $[ R( \mu_{a_{1,1}, \ldots ,a_{1,m_{1}} , \ldots , a_{n,1},
\ldots ,a_{n,m_{n}}} )]$ $( z_{1,1}, \ldots ,z_{1,m_{1}} , \ldots , z_{n,1},
\ldots ,z_{n,m_{n}} )$ vanishes whenever we don’t have $i_{1} = i_{2} = \cdots = i_{k};$ i.e. if and only if $R( \mu_{a_{1,1}, \ldots ,a_{1,m_{1}} , \ldots , a_{n,1},
\ldots ,a_{n,m_{n}}} )$ is of the form $$f_{1} ( z_{1,1}, \ldots ,z_{1,m_{1}} ) + \cdots +
f_{n} ( z_{n,1}, \ldots ,z_{n,m_{n}} )$$ for some formal power series ${ f_{1}, \ldots ,f_{n} }.$
$\ $
We will refer to the coefficient-vanishing condition in Theorem 3.6 by saying that “the $R$-series has no mixed coefficients”. If this happens, then ${ f_{1}, \ldots ,f_{n} }$ of (3.13) can only be the $R$-transforms $R( \mu_{a_{1,1}, \ldots ,a_{1,m_{1}}} )$, $\ldots$ , $R( \mu_{a_{n,1}, \ldots ,a_{n,m_{n}}} ),$ respectively.
$\ $
The Equation (3.6) of 3.3 will be called in the sequel “the combinatorial definition of ${ \framebox[7pt]{$\star$} }$”. In an approach where the $R$-transform is defined first, another possible definition of ${ \framebox[7pt]{$\star$} }$ would be provided by the following fact (which comes here as a theorem).
$\ $
[**3.7 Theorem ([@NS2])**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, and let $a_{1} , \ldots , $ $a_{n}, b_{1} , \ldots b_{n} \in { {\cal A} }$ be such that $\{ { a_{1}, \ldots ,a_{n} }\}$ is free from $\{ { b_{1}, \ldots ,b_{n} }\} .$ Then $$R( \mu_{a_{1}b_{1} , \ldots , a_{n}b_{n}} ) \ = \
R( \mu_{{ a_{1}, \ldots ,a_{n} }} ) \ { \framebox[7pt]{$\star$} }\ R( \mu_{{ b_{1}, \ldots ,b_{n} }} ).$$
$\ $
We mention that it is often handier to use the Equation (3.14) after performing a ${ \framebox[7pt]{$\star$} }$-operation with $Zeta$ on the right on both sides, and invoking (3.11); this leads to the equivalent equations: $$M( \mu_{a_{1}b_{1} , \ldots , a_{n}b_{n}} ) \ = \
R( \mu_{{ a_{1}, \ldots ,a_{n} }} ) \ { \framebox[7pt]{$\star$} }\ M( \mu_{{ b_{1}, \ldots ,b_{n} }} )$$ or $$M( \mu_{a_{1}b_{1} , \ldots , a_{n}b_{n}} ) \ = \
M( \mu_{{ a_{1}, \ldots ,a_{n} }} ) \ { \framebox[7pt]{$\star$} }\ R( \mu_{{ b_{1}, \ldots ,b_{n} }} )$$ (obtained by attaching the $Zeta$ on the right-hand side to $R( \mu_{{ b_{1}, \ldots ,b_{n} }} )$ and to $R( \mu_{{ a_{1}, \ldots ,a_{n} }} ),$ respectively).
$\ $
Finally, since in this paper we are dealing only with tracial non-commutative probability spaces, it is useful to recall the following simple fact concerning cyclic permutations in the $R$-transform.
$\ $
[**3.8 Proposition**]{} Let $n$ be a positive integer and let $\mu : { { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle }\rightarrow { \mbox{\bf C} }$ be a linear functional, such that $\mu (1)=1.$ Assume that $\mu$ is a trace (i.e. $\mu (p'p'') =
\mu (p''p')$ for every two polynomials $p',p'' \in { { \mbox{\bf C} }\langle X_{1} , \ldots ,X_{n} \rangle };$ this is for instance the case whenever $\mu = \mu_{{ a_{1}, \ldots ,a_{n} }}$ for some elements ${ a_{1}, \ldots ,a_{n} }$ in some ${ ( {\cal A} , \varphi ) }$, with $\varphi$ a trace). Then the coefficients of $R( \mu )$ are invariant under cyclic permutations, i.e. $$[ \mbox{coef } ( { i_{1} , \ldots ,i_{k} }, { j_{1} , \ldots ,j_{l} }) ] (R( \mu )) \ = \
[ \mbox{coef } ( { j_{1} , \ldots ,j_{l} }, { i_{1} , \ldots ,i_{k} }) ] (R( \mu ))$$ for every $k,l \geq 1$ and $1 \leq { i_{1} , \ldots ,i_{k} }, { j_{1} , \ldots ,j_{l} }\leq n.$
$\ $
The proof of Proposition 3.8 comes out immediately by using the moment-cumulant formula, see e.g. [@S2], Section 2.4.
$\ $
$\ $
$\ $
$\ $
[**4.1 Definition**]{} Let $n$ be a positive integer, let $\sigma$ be in $NC(2n),$ and consider the permutation $Perm ( \sigma )
\in {\cal S}_{2n}$ associated to $\sigma$ as in Section 2.6. We will say that $\sigma$ is [*parity-alternating*]{} (respectively [*parity-preserving*]{}) if the difference $i-[Perm( \sigma )] (i)
\in { \mbox{\bf Z} }$ is odd (respectively even) for every $1 \leq i \leq 2n.$ We will use the notations $$\left\{ \begin{array}{l}
{ \{ \sigma \in NC(2n) \ | \ \sigma \mbox{ parity-alternating } \}
\ { \stackrel{def}{=} }\ NC_{p-alt} (2n), } \\
{ \{ \sigma \in NC(2n) \ | \ \sigma \mbox{ parity-preserving} \}
\ { \stackrel{def}{=} }\ NC_{p-prsv} (2n). }
\end{array} \right.$$
$\ $
[**4.2 Remark**]{} It is clear that a partition $\sigma \in NC(2n)$ is parity-preserving if and only if each of its blocks either is contained in $\{ 1,3, \ldots , 2n-1 \}$ or is contained in $\{ 2,4, \ldots , 2n \} .$ For parity-alternating partitions, an equivalent description is: $$\sigma \in NC_{p-alt} (2n) \ \Leftrightarrow \ \{ \mbox{ every block
of $\sigma$ has an even number of elements} \} .$$ Implication “$\Rightarrow$” in (4.2) is immediate, while “$\Leftarrow$” uses the fact that if $B$ is a block of $\sigma$ and if $i<j$ in $B$ are such that $\{ i+1, \ldots ,j-1 \} \cap B = \emptyset ,$ then the interval $\{ i+1, \ldots ,j-1 \}$ is a union of other blocks of $\sigma$ (hence the named interval has an even number of elements, and hence $i$ and $j$ have different parities).
$\ $
[**4.3 Remark**]{} The Kreweras complementation map $K:NC(2n)
\rightarrow NC(2n)$ puts in bijection parity-alternating and parity-preserving partitions; we have in fact both equalities: $$\left\{ \begin{array}{l}
{ K( NC_{p-alt} (2n) ) \ = \ NC_{p-prsv} (2n), } \\
{ K( NC_{p-prsv} (2n) ) \ = \ NC_{p-alt} (2n). }
\end{array} \right.$$ Indeed, denoting the cycle ${ ( 1 \rightarrow 2 \rightarrow \cdots \rightarrow
2n \rightarrow 1 ) }$ by $\gamma$, we know (Eqn.(2.4)) that $Perm(K( \sigma )) = Perm( \sigma )^{-1} \gamma ,$ $\sigma \in NC(2n);$ this equality and the fact that $\gamma$ itself is parity-alternating imply together the inclusion “$\subseteq$” of both Equations (4.3). But then, since $K$ is one-to-one, we get that both inequalities between $| NC_{p-alt} (2n) |$ and $| NC_{p-prsv} (2n) |$ are holding; hence $| NC_{p-alt} (2n) | = | NC_{p-prsv} (2n) |,$ and in (4.3) we must really have equalities.
$\ $
The combinatorial remark mentioned in the title of the section is the following.
$\ $
[**4.4 Proposition**]{} Let $n$ be a positive integer. In order to distinguish the notations, we will write ${ K^{(2n)} }$ for the Kreweras complementation map on $NC(2n)$ (while the Kreweras map on $NC(n)$ will be simply denoted by $K$).
$1^{o}$ We have a canonical bijection $$\{ ( \pi , \rho ) \ | \ \pi , \rho \in NC(n), \ \pi \leq \rho \ \}
\ni ( \pi , \rho ) \ \longrightarrow \ \sigma \in NC_{p-prsv} (2n),$$ where $\sigma$ of (4.4) is determined as follows: the restriction $\sigma | \{ 2,4, \ldots , 2n \}$ is obtained by transporting $\pi$ from ${ \{ 1, \ldots ,n \} }$ to $\{ 2,4, \ldots , 2n \}$, while the restriction $\sigma | \{ 1,3, \ldots , 2n-1 \}$ is obtained by transporting $K^{-1} ( \rho )$ from ${ \{ 1, \ldots ,n \} }$ to $\{ 1,3, \ldots , 2n-1 \}$. (In terms of equivalence relations associated to partitions, we have $2i { \stackrel{\sigma}{\sim} }2j \Leftrightarrow i { \stackrel{\pi}{\sim} }j$ and $2i-1 { \stackrel{\sigma}{\sim} }2j-1 \Leftrightarrow i { \stackrel{K^{-1} ( \rho )}{\sim} }j$, for every $1 \leq i,j \leq n.)$
$2^{o}$ If $\pi \leq \rho$ in $NC(n)$ and $\sigma \in NC_{p-prsv} (2n)$ are as above, then the relative Kreweras complement $K_{\rho} ( \pi ) \in NC(n)$ (described in Section 2.5) is given by the formula: $$K_{\rho} ( \pi ) \ = \ \left\{ \ \{ \frac{b}{2} \ | \ b \in
B \cap \{ 2,4, \ldots ,2n \} \} \ ; \ B
\mbox{ block of } { K^{(2n)} }( \sigma ) \right\} .$$
$\ $
[**Proof**]{} $1^{o}$ is an immediate consequence of Proposition 2.4.
For $2^{o}$, we will compare the permutations associated to the two partitions appearing in (4.5) (the right-hand side of (4.5), which could be written as $\frac{1}{2} [ { K^{(2n)} }( \sigma ) |
\{ 2,4, \ldots , 2n \} ]$, is also in $NC(n);$ this is because the naturally defined operation of restriction preserves the quality of a partition of being non-crossing - easy verification).
Let us denote the permutations $Perm( \pi )$ and $Perm( \rho )$ by $\alpha$ and $\beta$, respectively. Then we know (Eqn.(2.5)) that $Perm( K_{\rho} ( \pi )) = \alpha^{-1} \beta .$ On the other hand, let us denote the partition in the right-hand side of (4.5) by $\theta$. We have that ${ K^{(2n)} }( \sigma ) \in NC_{p-alt} (2n)$ (because $\sigma \in NC_{p-prsv} (2n)$ and by Remark 4.3), and this immediately implies the following formula, expressing $Perm ( \theta )$ in terms of $Perm( { K^{(2n)} }( \sigma )):$ $$2 [ Perm( \theta ) ] (i) \ = \ \left( Perm( { K^{(2n)} }( \sigma ))
\right) ^{2} (2i), \ \ 1 \leq i \leq n.$$ Hence what we have to do is calculate $Perm( { K^{(2n)} }( \sigma ))$ in terms of $\alpha$ and $\beta$, and verify that the right-hand side of (4.6) equals $2 \alpha^{-1} ( \beta (i)).$
Now, the definition of $\sigma$ in terms of $\pi$ and $\rho$ is converted at the level of the associated permutations by saying that $Perm ( \sigma ) \in {\cal S}_{2n}$ acts by: $$\left\{ \begin{array}{l}
{ [Perm( \sigma )] (2i) \ = \ 2[ Perm( \pi )] (i) \ = \
2 \alpha (i), } \\
{ [Perm( \sigma )] (2i-1) \ = \ 2[ Perm( K^{-1} ( \rho ) )] (i) -1, }
\end{array} \ \ 1 \leq i \leq n. \right.$$ It is more convenient to look at $Perm( \sigma )^{-1} ,$ which is thus given by $$\left\{ \begin{array}{l}
{ [Perm( \sigma )]^{-1} (2i) \ = \ 2 \alpha^{-1} (i), } \\
{ [Perm( \sigma )]^{-1} (2i-1) \ = \ 2[ Perm( K^{-1} ( \rho ) )]^{-1}
(i) -1, }
\end{array} \ \ 1 \leq i \leq n. \right.$$
Denoting the cycles ${ ( 1 \rightarrow 2 \rightarrow \cdots \rightarrow
n \rightarrow 1 ) }\in {\cal S}_{n}$ and ${ ( 1 \rightarrow 2 \rightarrow \cdots \rightarrow
2n \rightarrow 1 ) }\in {\cal S}_{2n}$ by $\gamma_{n}$ and $\gamma_{2n}$, respectively, we get (by appropriately substituting in (2.4)) that $Perm( K^{-1} ( \rho ))^{-1} = \beta \gamma_{n}^{-1}$ and that $Perm( { K^{(2n)} }( \sigma )) = Perm( \sigma )^{-1} \gamma_{2n} .$ Hence for every $1 \leq i \leq n:$ $$[Perm( { K^{(2n)} }( \sigma )) ] (2i-1) \ = \
[ Perm( \sigma )^{-1} \gamma_{2n} ] (2i-1) \ = \
Perm( \sigma )^{-1} (2i) \ \stackrel{(4.7)}{=} \ 2 \alpha^{-1} (i);$$ and for every $1 \leq i \leq n-1$, a similar calculation leads to $$[Perm( { K^{(2n)} }( \sigma )) ] (2i) \ = \ 2 \beta (i) -1.$$ Equation (4.9) must also hold for $i=n,$ because $Perm( { K^{(2n)} }( \sigma ))$ is a bijection. From (4.8) and (4.9) it is easily verified that the right-hand side of (4.6) is indeed $2 \alpha^{-1} ( \beta (i)),$ and this concludes the proof. [**QED**]{}
$\ $
[**4.5 Corollary**]{} There exists a bijection between $\{ ( \pi , \rho ) \ | \ \pi , \rho \in NC(n),$ $\pi \leq \rho \}$ and $NC_{p-alt} (2n)$, such that, for $( \pi , \rho )$ in the first set corresponding to $\tau$ in the second:
\(i) $\pi$ and $\tau$ have the same number, say $k$, of blocks, and moreover,
\(ii) we can write $\pi = \{ A_{1}, \ldots , A_{k} \}$ and $\tau = \{ B_{1}, \ldots , B_{k} \}$ in such a way that $|A_{j}| = \frac{1}{2} |B_{j}|$ for every $1 \leq j \leq k.$
$\ $
(Note: The less involved fact that $\{ ( \pi , \rho ) \ |$ $\pi, \rho \in NC(n), \ \pi \leq \rho \}$ and $NC_{p-alt} (2n)$ have the same number - namely $(3n)!/n!(2n+1)!$ - of elements, was well-known, see e.g. [@E] or Section 2 of [@S1].)
$\ $
[**Proof**]{} The desired bijection is the diagonal arrow of the diagram $$\begin{array}{cccccccc}
{ \{ ( \pi , \rho ) \ | \ \pi , \rho \in NC(n), \ \pi \leq \rho \} } &
{ -- } & { - \rightarrow } & { NC_{p-prsv} (2n) } &
{ - } & \stackrel{{ K^{(2n)} }}{--} & { \rightarrow } & { NC_{p-alt} (2n) } \\
{ | } & & & & & & & \\
{ | } & & & & & & & \\
{ | } & & & & & & & \\
{ | } & & & & & & & \\
{ \downarrow } & & & & & & & \\
{ \{ ( \pi , \rho ) \ | \ \pi , \rho \in NC(n), \ \pi \leq \rho \} } &
& & & & & &
\end{array}$$ where the leftmost horizontal arrow is the bijection described in Proposition 4.4.$1^{o}$, and the vertical arrow is the bijection $( \pi , \rho ) \rightarrow ( K_{\rho} ( \pi ), \rho ).$ Indeed, if $( \pi , \rho ) \rightarrow \tau$ by the diagonal arrow of (4.10), then from Proposition 4.4.$2^{o}$ it follows that $\pi$ = $\left\{ \ \{ \frac{b}{2} \ | \ b \in B \cap \{ 2,4, \ldots ,2n
\} \} \ ; \right.$ $\left. B \mbox{ block of } \tau \right\} ;$ but since $\tau$ is parity-alternating, it is clear that $| \{ B \cap \{ 2,4, \ldots ,2n \} | = |B|/2$ for every block $B$ of $\tau .$ [**QED**]{}
$\ $
We now pass to the proof of the Theorem 1.14. We use the same line as for the mentioned Application 1.10 of [@NS2], which relies on the following
$\ $
[**4.6 Freeness Criterion:**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, such that $\varphi$ is a trace, let ${ {\cal C} }\subseteq { {\cal A} }$ be a unital subalgebra, and let ${ {\cal X} }\subseteq { {\cal A} }$ be a non-void subset. Assume that for every $m \geq 1$ and $c_1 , \ldots , c_m \in {\cal C}$, $x_1 , \ldots , x_m \in { {\cal X} }$, it is true that $$\varphi ( c_1 x_1 c_2 x_2 \cdots c_m x_m ) \ =
\ [ \mbox{coef } (1, \ldots , m) ]
(R( \mu_{ c_1 , \ldots , c_m } ) { \framebox[7pt]{$\star$} }M( \mu_{ x_1 , \ldots , x_m }))$$ (where, according to the notations set in Section 3.2, the right-hand side of (4.11) denotes the coefficient of $z_1 \cdots z_m$ in the formal power series $R( \mu_{ c_1 , \ldots , c_m } ) { \framebox[7pt]{$\star$} }M( \mu_{ x_1 , \ldots , x_m })).$ Then ${ {\cal X} }$ is free from ${\cal C}$ in ${ ( {\cal A} , \varphi ) }$.
$\ $
For the proof of 4.6, see [@NS2], Proposition 4.7. In addition to this criterion, we will use the following simple lemma.
$\ $
[**4.7 Lemma**]{} Given $m \geq 1$ and $\pi \in NC(m),$ the following are equivalent:
$1^{o}$ there exists a block $B$ of $\pi$ such that $|B|$ is odd;
$2^{o}$ there exists a block $B = \{ i_{1} < i_{2} < \cdots
< i_{k} \}$ of $\pi$ such that either $k=1$, or $k \geq 3$ is odd and all the differences $i_{2} - i_{1}, i_{3} - i_{2},
\ldots , i_{k} - i_{k-1}$ are also odd.
$\ $
[**Proof**]{} We only need to show that $1^{o} \Rightarrow 2^{o}.$ We proceed by induction on $m.$ The case $m=1$ is clear, we prove the induction step $\{ 1, \ldots , m-1 \} \Rightarrow m.$ We fix $\pi \in NC(m)$ which satisfies $1^{o}$, and we also fix a block $B$ of $\pi$ such that $|B|$ is odd. Clearly, we can assume that $|B| \neq 1$ and that there exist two elements $i<j$ in $B$ such that $\{ i+1, \ldots , j-1 \} \cap B = \emptyset$ and such that $j-i$ is even (otherwise we are done). We also fix $i$ and $j.$ The interval $\{ i+1, \ldots, j-1 \}$ is a union of blocks of $B;$ let $\pi_{o} \in NC(j-i-1)$ be the partition obtained from $\pi | \{ i+1, \ldots , j-1 \}$ by translation with $i$ downwards. Then $\pi_{o}$ satisfies $1^{o}$ (because $j-i-1$ is odd), hence also $2^{o}$ (by the induction hypothesis). It only remains to pick a block of $\pi_{o}$ which satisfies $2^{o},$ and shift it back (with $i$ upwards) into a block of $\pi .$ [**QED**]{}
$\ $
[**Proof of Theorem 1.14**]{} Let ${ ( {\cal A} , \varphi ) }$ and ${ a_{1}, \ldots ,a_{n} },b' ,b'' \in { {\cal A} }$ be as in the statement of the theorem. Let ${\cal C}$ denote the unital subalgebra of ${ {\cal A} }$ generated by ${ a_{1}, \ldots ,a_{n} }$, and put ${ {\cal X} }= \{ b'a_{1}b'', \ldots , b'a_{n}b'' \}$. The sufficient freeness condition of 4.6 becomes here: $$\varphi( c_{1} (b'a_{i_{1}}b'') c_{2} (b'a_{i_{2}}b'') \cdots
c_{m} (b'a_{i_{m}}b''))$$ $$\ = \ [ \mbox{coef } (1, \ldots ,m) ]
\left( R( \mu_{ c_{1} , \ldots , c_{m} } ) { \framebox[7pt]{$\star$} }M( \mu_{ b'a_{i_{1}}b'' , \ldots , b'a_{i_{m}}b'' } ) \right) ,$$ to be verified for every $m \geq 1, \ c_{1} , \ldots , c_{m} \in {\cal C}$ and $1 \leq i_{1} , \ldots , i_{m} \leq n.$
We start with the left-hand side of (4.12), and rewrite it as $$\varphi( (c_{1} b')(a_{i_{1}}b'')(c_{2} b')(a_{i_{2}}b'') \cdots
(c_{m} b')(a_{i_{m}}b''))$$ $$= \ [ \mbox{coef } (1,2, \ldots ,2m) ]
( M( \mu_{ c_{1} b', a_{i_{1}}b'', \ldots , c_{m} b', a_{i_{m}}b'' } )$$ $$= \ [ \mbox{coef } (1,2, \ldots ,2m) ]
( R( \mu_{ c_{1}, a_{i_{1}}, \ldots , c_{m}, a_{i_{m}} } )
{ \framebox[7pt]{$\star$} }M( \mu_{ \underbrace{b',b'', \ldots , b',b''}_{2m} } ) ),$$ (because $\{ b' ,b'' \}$ is free from $\{ c_{1} , \ldots , c_{m},
a_{i_{1}}, \ldots , a_{i_{m}} \}$ and by the formula (3.15)) $$= \ \sum_{\sigma \in NC(2m)}
[ \mbox{coef } (1,2, \ldots ,2m); \sigma ]
( R( \mu_{ c_{1}, a_{i_{1}}, \ldots , c_{m}, a_{i_{m}} } ))
\cdot$$ $$[ \mbox{coef } (1,2, \ldots ,2m); { K^{(2m)} }( \sigma ) ]
( M( \mu_{b',b'', \ldots , b',b''} ) )$$ (by the combinatorial definition of ${ \framebox[7pt]{$\star$} }$ in 3.3).
Now, let us remark that the quantity $[ \mbox{coef } (1,2, \ldots ,2m); { K^{(2m)} }( \sigma ) ]
( M( \mu_{b',b'', \ldots , b',b''} ) )$ appearing in (4.13) vanishes whenever ${ K^{(2m)} }( \sigma )$ is not parity-alternating. Indeed, if ${ K^{(2m)} }( \sigma ) \not\in NC_{p-alt} (2m),$ then ${ K^{(2m)} }( \sigma )$ must have at least one block $B$ with $|B|$ odd (by Remark 4.2), hence also at least one block $B$ with the property stated in $4.7.2^{o}$ (by the Lemma 4.7). But for any block $B$ with the latter property we have $[ \mbox{coef } (1,2, \ldots ,2m)| B ]
( M( \mu_{b',b'', \ldots , b',b''} ) )$ =0, because $(b',b'')$ is diagonally balanced; this implies that $[ \mbox{coef } (1,2, \ldots ,2m); { K^{(2m)} }( \sigma ) ]$ $( M( \mu_{b',b'', \ldots , b',b''} ) )$ =0, by Eqn.(3.4).
By also invoking the Remark 4.3, we hence see that in (4.13) we can in fact sum only over $\sigma \in NC_{p-prsv} (2m)$ (the contribution of a $\sigma \not\in NC_{p-prsv} (2m)$ is always zero).
But then we can perform in (4.13) the “substitution” indicated by the bijection of Proposition 4.4.$1^{o}$, i.e. we can pass to a sum indexed by $\{ ( \pi , \rho ) \ | \ \pi , \rho \in NC(m),$ $\pi \leq \rho \} .$ Of course, in order to do this we need to rewrite the general summand of (4.13) not in terms of $\sigma \in NC_{p-prsv} (2m),$ but in terms of the pair $( \pi , \rho )$ corresponding to it. We leave it to the reader to verify that
\(a) the part $[ \mbox{coef } (1,2, \ldots ,2m); \sigma ]
( R( \mu_{ c_{1}, a_{i_{1}}, \ldots , c_{m}, a_{i_{m}} } ))$ of the general summand gets converted into the product $$[ \mbox{coef } (1, \ldots ,m); \pi ]
( R( \mu_{ a_{i_{1}}, \ldots , a_{i_{m}} } )) \cdot
[ \mbox{coef } (1, \ldots ,m); K^{-1} ( \rho ) ]
( R( \mu_{ c_{1}, \ldots , c_{m} } ));$$ and
\(b) due to the description of $K_{\rho} ( \pi )$ in Proposition 4.4.$2^{o}$, the part $[ \mbox{coef } (1,2, \ldots ,2m);$ ${ K^{(2m)} }( \sigma ) ]$ $( M( \mu_{ \underbrace{b',b'', \ldots , b',b''}_{2m} } ) )$ of the general summand gets converted into the coefficient $[ \mbox{coef } (1, \ldots ,m); K_{ \rho} ( \pi ) ]
( M( \mu_{ \underbrace{b''b', \ldots , b''b'}_{m} } ) ).$ (Indeed, if we write ${ K^{(2m)} }( \sigma )$ = $\{ B_{1} , \ldots , B_{k} \}$ and $K_{\rho} ( \pi )$ = $\{ A_{1} , \ldots , A_{k} \}$ in such a way that $|B_{j}| =
2|A_{j}|,$ $1 \leq j \leq k,$ then the both $[ \mbox{coef } \ldots ]$ quantities claimed to be equal are seen to be just $\prod_{j=1}^{k} \varphi ( (b''b')^{|A_{j}|} ).$ We are of course using here the fact that $\varphi ( (b'b'')^{l} )$ = $\varphi ( (b''b')^{l} )$ for every $l \geq 0,$ which holds because $\varphi$ is a trace.)
The conclusion of (a) and (b) above is that (4.13) can be continued with: $$\sum_{ \begin{array}{c}
{\scriptstyle \pi , \rho \in NC(n)} \\
{\scriptstyle such \ that} \\
{\scriptstyle \pi \leq \rho }
\end{array} }
[ \mbox{coef } (1, \ldots ,m); K^{-1} ( \rho ) ]
( R( \mu_{ c_{1}, \ldots , c_{m} } )) \cdot
[ \mbox{coef } (1, \ldots ,m); \pi ]
( R( \mu_{ a_{i_{1}}, \ldots , a_{i_{m}} } )) \cdot$$ $$[ \mbox{coef } (1, \ldots ,m); K_{ \rho} ( \pi ) ]
( M( \mu_{ b''b', \ldots , b''b' } ) ) .$$ It is not hard to check that the quantity in (4.14) is exactly $$[ \mbox{coef } (1, \ldots ,m) ]
( R( \mu_{ c_{1}, \ldots , c_{m} } ) { \framebox[7pt]{$\star$} }R( \mu_{ a_{i_{1}}, \ldots , a_{i_{m}} } ) { \framebox[7pt]{$\star$} }M( \mu_{ b''b' , \ldots , b''b' } ) )$$ (indeed, one has just to look at the combinatorial formula for the ${ \framebox[7pt]{$\star$} }$-product of three series $f,g,h,$ calculated in the order $f \ { \framebox[7pt]{$\star$} }\ ( g \ { \framebox[7pt]{$\star$} }\ h))$ - compare for instance to [@NS2], Eqn.(3.7)).
Finally, since $b''b'$ is free from $\{ a_{i_{1}}, \ldots , a_{i_{m}}
\}$ we have (by applying again (3.15)) that $R( \mu_{ a_{i_{1}}, \ldots , a_{i_{m}} } ) { \framebox[7pt]{$\star$} }M( \mu_{ b''b' , \ldots , b''b' } )$ = $M( \mu_{ a_{i_{1}} b''b' , \ldots , a_{i_{m}} b''b' } )$; the latter series can also be written as $M( \mu_{ b' a_{i_{1}} b'', \ldots , b' a_{i_{m}} b'' } )$, because we are working in a tracial non-commutative probability space. We thus arrive to the fact that (4.15) coincides with the right-hand side of (4.12), and this concludes the proof. [**QED**]{}
$\ $
$\ $
$\ $
$\ $
We first show that Definition 1.12 could have been equally well formulated in terms of the free cumulants of $a_{1}$ and $a_{2}$ (rather than their moments).
$\ $
[**5.1 Proposition**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, and let ${ a_{1}, a_{2} }$ be in ${ {\cal A} }.$ The pair $( { a_{1}, a_{2} })$ is diagonally balanced if and only if the coefficients of $\underbrace{z_{1} z_{2} \cdots z_{1} z_{2} z_{1} }_{2n+1}$ and $\underbrace{z_{2} z_{1} \cdots z_{2} z_{1} z_{2} }_{2n+1}$ in $[ R( \mu_{{ a_{1}, a_{2} }} ) ] ( { z_{1} , z_{2} })$ are equal to zero for every $n \geq 0.$
$\ $
[**Proof**]{} This is an immediate consequence of the Lemma 4.7 and the Equations (3.10), (3.11) connecting the $M$- and $R$-series of $\mu_{{ a_{1}, a_{2} }} .$ For instance, when proving the “$\Rightarrow$” part, we write $$[ \mbox{coef } (1,2, \ldots ,1,2,1) ] (R( \mu_{{ a_{1}, a_{2} }} )) \ = \
[ \mbox{coef } (1,2, \ldots ,1,2,1) ] (M( \mu_{{ a_{1}, a_{2} }} )
{ \framebox[7pt]{$\star$} }Moeb )$$ $$= \ \sum_{\pi \in NC(2n+1)}
[ \mbox{coef } (1,2, \ldots ,1,2,1) ; \pi ] (M( \mu_{{ a_{1}, a_{2} }} ))
\cdot [ \mbox{coef } (1,2, \ldots ,1,2,1) ; K( \pi ) ] ( Moeb )$$ (and a similar formula for $[ \mbox{coef } (2,1, \ldots ,2,1,2) ] (R( \mu_{{ a_{1}, a_{2} }} ))$ ). Then we note that in the sum (5.1) every term is actually vanishing; indeed, the Lemma 4.7 and the hypothesis that $( { a_{1}, a_{2} })$ is a diagonally balanced pair imply together that $[ \mbox{coef } (1,2, \ldots ,1,2,1) ; \pi ] (M( \mu_{{ a_{1}, a_{2} }} ))$ = 0 for every $\pi \in NC(2n+1).$ [**QED**]{}
$\ $
[**5.2 Remark**]{} It is useful to record here that if $( { a_{1}, a_{2} })$ is a diagonally balanced pair in ${ ( {\cal A} , \varphi ) }$, where $\varphi$ is a trace, then the coefficient of $z_{i_{1}} z_{i_{2}} \cdots z_{i_{2n+1}}$ in $[ R( \mu_{ { a_{1}, a_{2} }} ) ] ( { z_{1} , z_{2} })$ is also vanishing whenever $(i_{1}, i_{2}, \ldots , i_{2n+1})$ is a cyclic permutation of $(1,2, \ldots ,1,2,1)$ or of $(2,1, \ldots , 2,1,2).$ This follows from Propositions 5.1 and 3.8.
$\ $
Since (as it is clear from 1.3 and 5.1) every $R$-diagonal pair is diagonally balanced, the next result contains the Proposition 1.7 as a particular case.
$\ $
[**5.3 Proposition**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$ such that $\varphi$ is a trace, and let $( { a_{1}, a_{2} })$ be a diagonally balanced pair in ${ ( {\cal A} , \varphi ) }.$ For every $k \geq 1$ we denote by $\alpha_{k}$ the coefficient of $(z_{1} z_{2} )^{k}$ (equivalently, of $(z_{2} z_{1} )^{k}$ ) in the series $[ R( \mu_{{ a_{1}, a_{2} }} ) ] ( { z_{1} , z_{2} }),$ and we consider the series of one variable $f(z) = \sum_{k=1}^{\infty} \alpha_{k} z^{k} .$ Then $f = R( \mu_{a_{1} a_{2}} ) { \framebox[7pt]{$\star$} }Moeb,$ where $Moeb$ is the Moebius series, as in Eqn.(1.9) (and of course $\mu_{a_{1} a_{2} } : { \mbox{\bf C} }[X] \rightarrow { \mbox{\bf C} }$ denotes the distribution of the product $a_{1} \cdot a_{2} ).$
$\ $
[**Proof**]{} For every $n \geq 1$ we have: $$\varphi ( (a_{1}a_{2} )^{n} ) \ = \
[ \mbox{coef } ( \underbrace{1,2, \ldots , 1,2}_{2n} ) ]
(M( \mu_{{ a_{1}, a_{2} }} ))$$ $$\stackrel{(3.12)}{=} \ \sum_{\tau \in NC(2n)}
[ \mbox{coef } ( \underbrace{1,2, \ldots , 1,2}_{2n} ) ; \tau ]
(R( \mu_{{ a_{1}, a_{2} }} ) ).$$ If the partition $\tau \in NC(2n)$ appearing in (5.2) is not in $NC_{p-alt}(2n),$ then it has at least one block $B$ with $|B|$ odd (by Remark 4.2), hence it also has a block $B$ satisfying condition $2^{o}$ of Lemma 4.7. This and the description of the diagonally balanced pair $( { a_{1}, a_{2} })$ in terms of cumulants (Proposition 5.1) imply together that $[ \mbox{coef } ( \underbrace{1,2, \ldots , 1,2}_{2n} ) ; \tau ]
(R( \mu_{{ a_{1}, a_{2} }} ) ) =0.$ On the other hand, if $\tau =
\{ B_{1} , \ldots , B_{h} \}$ is in $NC_{p-alt} (2n),$ then by the very definition of $NC_{p-alt} (2n)$ we have that $$[ \mbox{coef } ( \underbrace{1,2, \ldots , 1,2}_{2n} ) ; \tau ]
(R( \mu_{{ a_{1}, a_{2} }} ) ) \ = \ \alpha_{|B_{1}|/2} \cdots
\alpha_{|B_{h}|/2}$$ (where the $\alpha$’s are as in the statement of the proposition). This argument shows that the sum in (5.2) is equal to: $$\sum_{ \begin{array}{c}
{\scriptstyle \tau \in NC_{p-alt} (2n) } \\
{\scriptstyle \tau = \{ B_{1} , \ldots , B_{h} \} }
\end{array} }
\ \alpha_{|B_{1}|/2} \cdots \alpha_{|B_{h}|/2} .$$ We next transform the sum in (5.3) by using the bijection between $NC_{p-alt} (2n)$ and $\{ ( \pi , \rho ) \ |$ $\pi , \rho \in NC(n),$ $\pi \leq \rho \}$ indicated in Corollary 4.5; we obtain: $$\sum_{ \begin{array}{c}
{\scriptstyle \pi , \rho \in NC(n) } \\
{\scriptstyle such \ that \ \pi \leq \rho ,} \\
{\scriptstyle \pi = \{ A_{1} , \ldots , A_{h} \} }
\end{array} }
\ \alpha_{|A_{1}|} \cdots \alpha_{|A_{h}|}$$ $$= \ \sum_{ \begin{array}{c}
{\scriptstyle \pi \in NC(n) } \\
{\scriptstyle \pi = \{ A_{1} , \ldots , A_{h} \} }
\end{array} }
\ \alpha_{|A_{1}|} \cdots \alpha_{|A_{h}|} \cdot
\mbox{ card } \{ \rho \in NC(n) \ | \ \rho \geq \pi \} .$$ Now, for a given $\pi = \{ A_{1}, \ldots , A_{h} \} \in NC(n),$ the product $\alpha_{|A_{1}|} \cdots \alpha_{|A_{h}|}$ is just $[ \mbox{coef } (n); \pi ] (f)$ (in the sense of the notations in 3.2.$4^{o}$), while on the other hand it is an easy exercise to identify $\mbox{ card } \{ \rho \in NC(n) \ | \ \rho \geq \pi \}$ as $[ \mbox{coef } (n) ; K( \pi )] (Zeta { \framebox[7pt]{$\star$} }Zeta).$ Hence (5.4) can be continued with: $$\sum_{\pi \in NC(n)}
[ \mbox{coef } (n); \pi ] (f) \cdot
[ \mbox{coef } (n) ; K( \pi )] (Zeta { \framebox[7pt]{$\star$} }Zeta) \ = \
[ \mbox{coef } (n) ] (f { \framebox[7pt]{$\star$} }Zeta { \framebox[7pt]{$\star$} }Zeta).$$
But the expression we had started with was $\varphi ( (a_{1} a_{2} )^{n} )$, i.e. $[ \mbox{coef } (n) ] (M ( \mu_{ a_{1} \cdot a_{2} } )).$ We have in other words proved the equality $$M( \mu_{a_{1} \cdot a_{2}} ) \ = \ f { \framebox[7pt]{$\star$} }Zeta { \framebox[7pt]{$\star$} }Zeta$$ (since the two series have identical coefficients). It only remains to take the ${ \framebox[7pt]{$\star$} }$ product with $Moeb { \framebox[7pt]{$\star$} }Moeb$ on both sides of (5.5), and take into account that $Moeb$ is the inverse of $Zeta$ under ${ \framebox[7pt]{$\star$} }$, and that $M( \mu_{a_{1} \cdot a_{2}} )
{ \framebox[7pt]{$\star$} }Moeb = R( \mu_{a_{1} \cdot a_{2}} ).$ [**QED**]{}
$\ $
$\ $
$\ $
$\ $
We begin by proving the particular case when the pair $\{ p_{1} , p_{2} \}$ in Theorem 1.5 is of the form $\{ p,1 \} ;$ that is:
$\ $
[**6.1 Proposition**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, such that $\varphi$ is a trace, and let $a_{1} , a_{2} , p \in { {\cal A} }$ be such that $( { a_{1}, a_{2} })$ is an $R$-diagonal pair, and $p$ is free from $\{ { a_{1}, a_{2} }\} .$ Then $( a_{1} p, a_{2} )$ is also an $R$-diagonal pair.
$\ $
[**Proof**]{} We fix $m \geq 1$ and a string ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} }) \in { \{ 1,2 \} }^{m}$ for some $m \geq 1,$ such that $[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }) ] (R( \mu_{a_{1}p,a_{2}}))
\neq 0.$ Our task is to show that $m$ is even, and that the string ${ \varepsilon }$ is either $( \underbrace{1,2,1,2, \ldots ,1,2}_{m} )$ or $( \underbrace{2,1,2,1, \ldots ,2,1}_{m} ).$ (This would verify Condition (ii) in Definition 1.3; note that Condition (i) of 1.3 is automatically verified, since it is assumed that $\varphi$ is a trace.)
The couples $( { a_{1}, a_{2} })$ and $(p,1)$ are free, hence we can use Theorem 3.7 to infer that $R( \mu_{a_{1}p,a_{2}} ) =
R( \mu_{{ a_{1}, a_{2} }} ) { \framebox[7pt]{$\star$} }R( \mu_{p,1} );$ the combinatorial definition of ${ \framebox[7pt]{$\star$} }$ applied to this situation yields then the formula: $$[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} })] (R( \mu_{a_{1}p,a_{2}} )) \ = \
\sum_{\tau \in NC(m)} [ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }) ; \tau ]
(R( \mu_{a_{1},a_{2}} )) \cdot$$ $$[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }) ; K( \tau )] (R( \mu_{p,1} )).$$ Since the left-hand side of (6.1) is assumed to be non-zero, we can choose (and fix) a partition $\tau \in NC(m)$ such that $$\left\{ \begin{array}{l}
{ [ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }) ; \tau ]
(R( \mu_{a_{1},a_{2}} )) \neq 0 } \\
{ [ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }) ; K( \tau )]
(R( \mu_{p,1} )) \neq 0. }
\end{array} \right.$$
The first condition (6.2) together with the hypothesis that $( { a_{1}, a_{2} })$ is an $R$-diagonal pair imply together that for every block $B= \{ i_{1} < i_{2} < \cdots < i_{k} \}$ of $\tau :$ $k$ is even, and $l_{i_{1}} \neq l_{i_{2}},$ $l_{i_{2}} \neq l_{i_{3}}, \ldots , l_{i_{k-1}} \neq l_{i_{k}}.$ This already implies that $m$ is even, and that among ${ l_{1} , \ldots ,l_{m} }$ there are $m/2$ occurrences of 1 and $m/2$ occurrences of 2.
In order to get the interpretation for the second condition (6.2), we note that, since $p$ is always free from 1, we have (by Theorem 3.6, in fact its particular case stated in Eqn.(1.4)): $$[R( \mu_{p,1} )] ( { z_{1} , z_{2} }) \ = \
[R( \mu_{p} )] ( z_{1} ) + [R( \mu_{1} )] ( z_{2} ) \ = \
[R( \mu_{p} )] ( z_{1} ) + z_{2} .$$ By taking this into account, we see that if the second condition (6.2) is satisfied, then $\{ i \}$ has to be a one-element block of $K( \tau )$ whenever we have $l_{i} =2.$ A quick look at how the Kreweras complement $K:NC(m) \rightarrow NC(m)$ is defined shows that $\{ i \}$ is a one-element block of $K( \tau )$ if and only if $i$ and $i+1$ are in the same block of $\tau$ (where if $i=m$ we read “1” instead of “$i+1$”). Hence whenever we have $l_{i} =2,$ we also have that $i$ and $i+1$ lie in the same block of $\tau$, and then from the discussion in the preceding paragraph we deduce that $l_{i+1} =1.$
We have thus proved that: $m$ is even; among ${ l_{1} , \ldots ,l_{m} }$ there are $m/2$ of 1 and $m/2$ of 2; and $l_{i}=2 \Rightarrow l_{i+1} =1,$ for $1 \leq i \leq m-1,$ and $l_{m} =2 \Rightarrow l_{1} =1.$ We leave it as an elementary exercise to the reader to verify that a string $( { l_{1} , \ldots ,l_{m} })$ with all these properties can only be $(1,2,1,2, \ldots ,1,2)$ or $(2,1,2,1, \ldots ,2,1).$ [**QED**]{}
$\ $
[**6.2 Remark**]{} With exactly the same argument, we could have shown that $( { a_{1}, a_{2} })$ $R$-diagonal $\Rightarrow$ $( a_{1}, pa_{2} )$ $R$-diagonal (i.e. the particular case $\{ { p_{1} , p_{2} }\} = \{ 1,p \}$ of Theorem 1.5). Since (obviously) $R$-diagonality is not affected by reversing the order of the elements of a pair, it follows that in Proposition 6.1 the element $p$ could have been in fact inserted anywhere (left or right of either $a_{1}$ or $a_{2}).$ We take the occasion to mention here that (as the reader can easily verify by experimenting on coefficients of small length) the Theorem 1.5 itself becomes false if in its statement the pair $( a_{1}p_{1} , p_{2}a_{2} )$ is replaced for instance with $( a_{1}p_{1} , a_{2}p_{2} ).$
$\ $
In view of what has been proved up to now, the Theorem 1.5 is equivalent to the following statement.
$\ $
[**6.3 Proposition**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, such that $\varphi$ is a trace, and let ${ a_{1}, a_{2} }, { p_{1} , p_{2} }\in { {\cal A} }$ be such that $( { a_{1}, a_{2} })$ is an $R$-diagonal pair, and such that $\{ { p_{1} , p_{2} }\}$ is free from $\{ { a_{1}, a_{2} }\} .$ Then $\mu_{a_{1}p_{1} , p_{2}a_{2}} =
\mu_{a_{1}p_{1}p_{2}, a_{2}} .$
$\ $
Indeed, Theorem 1.5 follows from Propositions 6.1 and 6.3, while conversely, Proposition 6.3 is implied by Theorem 1.5 and Corollary 1.8.
Proposition 6.3 simply says that if we put (in the notations used there) $x_{1} = a_{1}p_{1}, x_{2} = p_{2}a_{2},$ $y_{1} = a_{1}p_{1}p_{2}, y_{2} = a_{2},$ then we have $$\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} ) \ = \
\varphi ( y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}} )$$ for every $m \geq 1$ and ${ l_{1} , \ldots ,l_{m} }\in \{ 1,2 \} .$ The point of view we will take concerning (6.4) is the following: when we write back $x_{1} = a_{1}p_{1}, x_{2} = p_{2}a_{2},$ $y_{1} = a_{1}p_{1}p_{2}, y_{2} = a_{2},$ then both $x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}}$ and $y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}}$ are converted into monomials of length $2m$ in ${ a_{1}, a_{2} }, { p_{1} , p_{2} };$ and moreover, both these monomials of length $2m$ can be regarded as the result obtained by starting with $a_{l_{1}} a_{l_{2}} \cdots a_{l_{m}}$ and inserting $p_{1}$’s and $p_{2}$’s in between the $a$’s, according to a certain pattern:
- in order to obtain $x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}}$, we insert a $p_{1}$ immediately to the right of each occurrence of $a_{1} ,$ and a $p_{2}$ immediately to the left of each occurrence of $a_{2};$
- in order to obtain $y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}}$, we insert a $p_{1}p_{2}$ immediately to the right of each occurrence of $a_{1}.$
So in some sense what we have to do is compare the two insertions patterns described above. Due to its length, this will be done separately in the next section.
We only make now one simple remark: since $\varphi$ is a trace, both monomials appearing in (6.4) can be permuted cyclically; hence if we assume in (6.4) that $l_{1} =1,$ we are in fact missing only the case when $l_{1} = l_{2} = \cdots = l_{m} =2.$ But the latter case can be easily settled if we note that $\varphi ( b_{1}^{n} ) = \varphi ( b_{2}^{n} ) =0$ for every $R$-diagonal pair $( { b_{1}, b_{2} })$ and for every $n \geq 1$ (this in turn is an immediate consequence of the moment-cumulant formula (3.12), combined with the particular form of the series $R( \mu_{{ b_{1}, b_{2} }} )).$ Indeed, if $l_{1} = l_{2} = \cdots = l_{m} =2,$ then the right-hand side of (6.4) becomes $\varphi ( a_{2}^{m} )=0.$ In order to verify that the left-hand side of (6.4) is also vanishing, we can invoke the same argument, where we use in addition the Remark 6.2 ($p_{2}a_{2}$ enters the $R$-diagonal pair $( a_{1} , p_{2}a_{2} ),$ hence $ \varphi ( ( p_{2}a_{2} )^{m} ) =0).$
$\ $
$\ $
$\ $
[**7.1 Notations**]{} Throughout this section we fix a positive integer $m \geq 1,$ and a string ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} }) \in \{ 1,2
\} ^{m} .$ We make the assumption that $l_{1} =1.$ The number of occurrences of 1 in the string will be denoted by $n$ ($1 \leq n \leq m).$
Also, for the whole section we fix a circle of radius 1 in the plane, and the points ${ P_{1} , \ldots ,P_{m} }, { Q_{1} , \ldots ,Q_{m} }, { R_{1} , \ldots ,R_{n} }$ on the circle, positioned as follows. First we draw $P_{1}, \ldots ,$ $P_{m}$ around the circle, equidistant and in clockwise order. Then we draw ${ Q_{1} , \ldots ,Q_{m} },$ according to the following rule: if $l_{i} =1,$ we put $Q_{i}$ on the arc of circle going from $P_{i}$ to $P_{i+1}$, such that the length of the arc from $P_{i}$ to $Q_{i}$ is 1/3 of the length of the arc from $P_{i}$ to $P_{i+1}$ (i.e. $2 \pi /3m);$ and if $l_{i} =2,$ we put $Q_{i}$ on the arc of circle from $P_{i-1}$ to $P_{i}$, such that the length of the arc $Q_{i} P_{i}$ is $2 \pi /3m$ (all arcs described clockwisely). It is convenient to view the points ${ Q_{1} , \ldots ,Q_{m} }$ as colored, namely we will say that $Q_{i}$ is red whenever $l_{i} =1,$ and that it is blue whenever $l_{i} =2.$ (Note that $Q_{1}$ is red, since it is assumed that $l_{1} =1.)$ Finally, we inspect the $n$ points $Q_{i}$ that are red, clockwisely and starting from $Q_{1},$ and we second-name as ${ R_{1} , \ldots ,R_{n} }$, in this order (every $Q_{i}$ which is red is hence at the same time an $R_{j}$ for some $j).$
For example, the next figure shows how our circular picture looks if $m=8$ and ${ \varepsilon }=(1,2,2,1,1,2,1,2).$
$\ $
$$\mbox{\bf Figure 2.}$$
[**7.2 The “complementation” maps $C_{Q}$ and $C_{R}$**]{} Using the circular picture associated to the string ${ \varepsilon },$ we will define two maps ${ C_{Q} }: NC(m) \rightarrow NC(m),$ and ${ C_{R} }: NC(m) \rightarrow NC(n).$ It is convenient to formulate the definition in terms of equivalence relations associated to partitions (recall from 2.1 that if $\pi$ is a partition of ${ \{ 1, \ldots ,k \} },$ then the equivalence relation ${ \stackrel{\pi}{\sim} }$ corresponding to $\pi$ is simply “$i { \stackrel{\pi}{\sim} }j \Leftrightarrow i,j$ belong to the same block of $\pi$”, $1 \leq i,j \leq k).$ Given $\sigma \in NC(m),$ the equivalence relations corresponding to the partitions ${ C_{Q} }( \sigma )$ of ${ \{ 1, \ldots ,m \} }$ and ${ C_{R} }( \sigma )$ of ${ \{ 1, \ldots ,n \} }$ are defined as follows: $$\left\{ \begin{array}{l}
{ \mbox{ - For $1 \leq i,j \leq m$ we have $i { \stackrel{{ C_{Q} }( \sigma )}{\sim} }j$
if and only if there are no $1 \leq h,k \leq m$} } \\
{ \mbox{ such that $h { \stackrel{\sigma}{\sim} }k$ and such that the line
segments $Q_{i} Q_{j}$ and $P_{h} P_{k}$ have } } \\
{ \mbox{ non-void intersection;} } \\
{ } \\
{ \mbox{ - For $1 \leq i,j \leq n$ we have $i { \stackrel{{ C_{R} }( \sigma )}{\sim} }j$
if and only if there are no $1 \leq h,k \leq m$} } \\
{ \mbox{ such that $h { \stackrel{\sigma}{\sim} }k$ and such that the line
segments $R_{i} R_{j}$ and $P_{h} P_{k}$ have } } \\
{ \mbox{ non-void intersection;} }
\end{array} \right.$$
We leave it to the reader to verify that ${ \stackrel{{ C_{Q} }( \sigma )}{\sim} }$ and ${ \stackrel{{ C_{R} }( \sigma )}{\sim} }$ defined in (7.1) really are equivalence relations on ${ \{ 1, \ldots ,m \} }$ and ${ \{ 1, \ldots ,n \} }$, respectively, and moreover, that ${ C_{Q} }( \sigma)$ and ${ C_{R} }( \sigma )$ are indeed non-crossing partitions. All these verifications reduce to the same geometric argument, that can be stated as follows: let $X,Y$ be points on the circle, and let $Z$ be either on the circle or in the open disk enclosed by it, such that $\{ X,Y,Z \}
\cap \{ { P_{1} , \ldots ,P_{m} }\} = \emptyset ;$ if the segment $P_{h} P_{k}$ (for some $1 \leq h < k \leq m)$ intersects $XY$, then it must also intersect $XZ \cup YZ.$
$\ $
The relevance of the complementation maps ${ C_{Q} }$ and ${ C_{R} }$ for our discussion comes from the following
$\ $
[**7.3 Proposition:**]{} Let ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, and let ${ a_{1}, a_{2} }, p_{1},$ $p_{2} \in { {\cal A} }$ be such that $\{ { a_{1}, a_{2} }\}$ is free from $\{ { p_{1} , p_{2} }\}$ in ${ ( {\cal A} , \varphi ) }.$ Define $x_{1} = a_{1}p_{1}, x_{2} = p_{2}a_{2}, y_{1} = a_{1}p_{1}p_{2},
y_{2} =a_{2}.$ Then we have: $$\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} ) \ = $$ $$= \ \sum_{\sigma \in NC(m)}
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \sigma ] ( R( \mu_{{ a_{1}, a_{2} }} ))
\cdot [ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); { C_{Q} }( \sigma ) ]
( M( \mu_{{ p_{1} , p_{2} }} ))$$ and $$\varphi ( y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}} ) \ = $$ $$= \ \sum_{\sigma \in NC(m)}
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \sigma ] ( R( \mu_{{ a_{1}, a_{2} }} ))
\cdot [ \mbox{coef } ( n ); { C_{R} }( \sigma ) ]
( M( \mu_{p_{1} \cdot p_{2}} ))$$ (where ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ is the string fixed in the beginning of the section; the notations for the series $M( \mu_{\ldots} ), R( \mu_{\ldots} ),$ and for their coefficients are as in Section 3.2).
$\ $
[**Proof**]{} We will only show the proof of (7.2), the one for (7.3) is similar. This kind of argument has been used several times before, in connection to the Kreweras complementation map $K$ (instead of ${ C_{Q} }, { C_{R} }$) - see 3.4 in [@S2], 3.4 in [@NS1], 3.11 in [@NS2].
If in $x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}}$ we write back each $x_{1}$ as $a_{1}p_{1}$ and each $x_{2}$ as $p_{2}a_{2}$, we obtain a monomial of length $2m$ in ${ a_{1}, a_{2} }, { p_{1} , p_{2} }$. By looking just at the $a$’s in this monomial, we see that they are $a_{l_{1}}, a_{l_{2}}, \ldots , a_{l_{m}},$ exactly in this order, and placed on a certain set of positions $I \subseteq { \{ 1,2, \ldots ,2m \} },$ with $|I| =m.$ It is also clear that on the complementary set of positions $J = { \{ 1,2, \ldots ,2m \} }\setminus I$ of our monomial of length $2m$ we have $p_{l_{1}}, p_{l_{2}}, \ldots , p_{l_{m}},$ exactly in this order. So we can say that $x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}}$ is obtained by shuffling together $a_{l_{1}} a_{l_{2}} \cdots a_{l_{m}}$ and $p_{l_{1}} p_{l_{2}} \cdots p_{l_{m}}$, where the $a$’s have to sit on the positions indicated by $I$, and the $p$’s have to sit on the positions indicated by $J = { \{ 1,2, \ldots ,2m \} }\setminus I.$
Now, the value of $\varphi$ on the monomial of length $2m$ discussed in the preceding paragraph can be viewed as a coefficient of length $2m$ of the series $M( \mu_{{ a_{1}, a_{2} }, { p_{1} , p_{2} }} ).$ We write this coefficient as a summation over $NC(2m),$ by using the moment-cumulant formula (3.12). Because of the assumption that $\{ { a_{1}, a_{2} }\}$ is free from $\{ { p_{1} , p_{2} }\}$ (which implies that we have $[R( \mu_{{ a_{1}, a_{2} }, { p_{1} , p_{2} }} )] ( { z_{1} , z_{2} }, z_{3} , z_{4} )$ = $[R( \mu_{{ a_{1}, a_{2} }} )] ( { z_{1} , z_{2} })$ + $[R( \mu_{{ p_{1} , p_{2} }} )] ( z_{3} , z_{4} )),$ the summation over $NC(2m)$ we have arrived to has a lot of vanishing terms; a partition $\theta \in NC(2m)$ can in fact bring a non-zero contribution to the sum if and only if each block of $\theta$ either is contained in $I$ or is contained in $J$ (with $I,J$ the sets of positions of the preceding paragraph). This brings us to the formula: $$\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} ) \ =$$ $$= \ \sum_{ \begin{array}{c}
{\scriptstyle \sigma , \tau \in NC(m) } \\
{\scriptstyle I,J - compatible}
\end{array} }
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \sigma ] (R( \mu_{{ a_{1}, a_{2} }} ))
\cdot
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \tau ] (R( \mu_{{ p_{1} , p_{2} }} ));$$ in (7.4), the fact that $\sigma , \tau \in NC(m)$ are $I,J$-compatible has the meaning that if we transport $\sigma$ from ${ \{ 1, \ldots ,m \} }$ onto $I$ and we transport $\tau$ from ${ \{ 1, \ldots ,m \} }$ onto $J = { \{ 1,2, \ldots ,2m \} }\setminus I$, then the partition of ${ \{ 1,2, \ldots ,2m \} }$ which is obtained in this way is still non-crossing.
If we now look back at how the complementation map ${ C_{Q} }: NC(m) \rightarrow NC(m)$ was defined, it is immediate that the $I,J$-compatibility of $\sigma , \tau \in NC(m)$ is equivalent to the condition $\tau \leq { C_{Q} }( \sigma ).$ This means that (7.4) can be continued with: $$\sum_{\sigma \in NC(m)}
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \sigma ] (R( \mu_{{ a_{1}, a_{2} }} ))
\cdot \{ \sum_{ \begin{array}{c}
{\scriptstyle \tau \in NC(m), } \\
{\scriptstyle \tau \leq { C_{Q} }( \sigma )}
\end{array} }
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \tau ] (R( \mu_{{ p_{1} , p_{2} }} )) \ \} .$$ Finally, we note that $$\sum_{ \begin{array}{c}
{\scriptstyle \tau \in NC(m), } \\
{\scriptstyle \tau \leq { C_{Q} }( \sigma )}
\end{array} }
[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \tau ] (R( \mu_{{ p_{1} , p_{2} }} ))
\ = \ [ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); { C_{Q} }( \sigma ) ]
(M( \mu_{{ p_{1} , p_{2} }} )),$$ as it follows by a repeated utilization of the moment-cumulant formula (3.12). Substituting (7.6) in (7.5) brings us to the right-hand side of (7.2), and concludes the proof. [**QED**]{}
$\ $
[**7.4 Corollary**]{} Let ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ be the string of 1’s and 2’s fixed in 7.1, and recall that $1 \leq n \leq m$ is the number of occurrences of 1 in the string. Let on the other hand ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, with $\varphi$ a trace, and consider ${ a_{1}, a_{2} }, { p_{1} , p_{2} }\in { {\cal A} }$ such that $( { a_{1}, a_{2} })$ is an $R$-diagonal pair, and such that $\{ { p_{1} , p_{2} }\}$ is free from $\{ { a_{1}, a_{2} }\}$. Define $x_{1} = a_{1}p_{1}, x_{2} = p_{2}a_{2}, y_{1} = a_{1}p_{1}p_{2},
y_{2} =a_{2}.$ If it is not true that $m$ is even and $n=m/2,$ then $\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} )$ = $\varphi ( y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}} )$ = 0.
$\ $
[**Proof**]{} If it is not true that $m$ is even and $n=m/2,$ then due to the particular form of $R( \mu_{{ a_{1}, a_{2} }} )$ we have that $[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \sigma ] (R( \mu_{{ a_{1}, a_{2} }} ))$ = 0 for every $\sigma \in NC(m).$ But then all the terms of the summations on the right-hand side of (7.2), (7.3) are equal to zero. [**QED**]{}
$\ $
Corollary 7.4 shows that the equality (6.4), whose proof is the goal of the present section, takes place in a trivial way unless we impose the following
$\ $
[**7.5 Supplementary hypothesis:**]{} From now on, until the end of section, we will assume that the string ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ fixed in 7.1 contains an equal number of 1’s and of 2’s (i.e., $m$ is even and $n=m/2).$
$\ $
In this case, the quantities $\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} )$ and $\varphi ( y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}} )$ mentioned in 7.4 are in general non-zero, and in order to verify their equality we will need to prove some facts concerning “${ \varepsilon }$-alternating partitions”.
$\ $
[**7.6 Definition**]{} A partition $\sigma \in NC(m)$ will be called [*${ \varepsilon }$-alternating*]{} (where ${ \varepsilon }$ = $(l_{1} , \ldots ,$ $l_{m} )$ is the string of 7.1) if for every block $B= \{ i_{1} < i_{2} < \cdots < i_{k} \}$ of $\sigma$ we have that $l_{i_{1}} \neq l_{i_{2}} , \ldots ,
l_{i_{k-1}} \neq l_{i_{k}} , l_{i_{k}} \neq l_{i_{1}} .$ We denote the set of all ${ \varepsilon }$-alternating partitions in $NC(m)$ by ${ NC_{{ \varepsilon }- alt} (m) }$.
$\ $
Other two (obviously equivalent) ways of stating that $\sigma \in
NC(m)$ is ${ \varepsilon }$-alternating are:
- in the terminology of 2.2: we have $l_{i} \neq l_{j}$ whenever $1 \leq i < j \leq m$ belong to the same block of $\sigma ,$ and are consecutive in that block; or
- in the terminology of 3.2.$2^{o}$: for every block $B$ of $\sigma ,$ the $|B|$-tuple $( { l_{1} , \ldots ,l_{m} })|B$ is of the form $(1,2,1,2, \ldots ,1,2)$ or $(2,1,2,1, \ldots ,2,1).$
Note that from 7.6 (or any of the two equivalent reformulations stated above) it follows that every block of $\sigma \in { NC_{{ \varepsilon }- alt} (m) }$ has an even number of elements; i.e., ${ NC_{{ \varepsilon }- alt} (m) }$ is a subset of $NC_{p-alt} (m)$ discussed in Section 4.
$\ $
[**7.7 Proposition**]{} The complementation map ${ C_{Q} }: NC(m) \rightarrow NC(m)$ sends ${ NC_{{ \varepsilon }- alt} (m) }$ into itself.
$\ $
[**Proof**]{} Consider the picture with $2m$ points ${ P_{1} , \ldots ,P_{m} },
{ Q_{1} , \ldots ,Q_{m} }$ sitting on a circle of radius 1, which was used to define ${ C_{Q} }$ in 7.2. We denote by ${ \underline{D} }$ the closed disk enclosed by the circle. During this proof we will be particularly interested in a certain type of convex subsets of ${ \underline{D} },$ described as follows. Let $i_{1}, j_{1}, \ldots , i_{k}, j_{k}$ $(1 \leq k \leq m/2)$ be a family of $2k$ distinct indices in ${ \{ 1, \ldots ,m \} }$, with the property that for every $1 \leq h \leq k,$ all the points $P_{i_{1}}, P_{j_{1}}, \ldots , P_{i_{h-1}}, P_{j_{h-1}},$ $P_{i_{h+1}}, P_{j_{h+1}}, \ldots , P_{i_{k}}, P_{j_{k}}$ lie in the same open half-plane ${ \underline{S} }_{h}$ determined by the line through $P_{i_{h}}$ and $P_{j_{h}}.$ Then the set ${ \underline{X} }{ \stackrel{def}{=} }{ \underline{D} }\cap { \underline{S} }_{1} \cap \cdots \cap { \underline{S} }_{k}$ will be called the [*trimmed disk*]{} determined by $i_{1}, j_{1}, \ldots , i_{k}, j_{k}$. A picture of a trimmed disk (exemplified for $m=12)$ is shown in Figure 3:
$$\mbox{\bf Figure 3:}
\mbox{ An example of trimmed disk, when $m=12.$}$$
The trimmed disk ${ \underline{X} }$ determined by $i_{1}, j_{1}, \ldots , i_{k}, j_{k}$ is a convex set (neither open nor closed). The points $P_{i_{1}}, P_{j_{1}}, \ldots , P_{i_{k}}, P_{j_{k}}$ will be called the [*vertices*]{} of ${ \underline{X} }$ (and can be identified as those $P_{i} , \ 1 \leq i \leq m,$ which lie in the closure of ${ \underline{X} }$ but not in ${ \underline{X} }$ itself). The boundary of ${ \underline{X} }$ consists of $2k$ “edges”; $k$ of these edges are the line segments $P_{i_{h}} P_{j_{h}}, \ 1 \leq h \leq k,$ and the other $k$ edges are arcs of the circle. Note that when we travel around the boundary of ${ \underline{X} },$ the rectilinear and curvilinear edges alternate.
Let us now fix for the rest of the proof a partition $\sigma \in
{ NC_{{ \varepsilon }- alt} (m) },$ about which we want to show that ${ C_{Q} }( \sigma )$ is also in ${ NC_{{ \varepsilon }- alt} (m) }.$
For every block $B$ of $\sigma$ we denote by ${ \underline{H} }_{B}$ the closed convex hull of the points $\{ P_{i} \ | \ i \in B \} ;$ ${ \underline{H} }_{B}$ is thus a closed convex polygon with $|B|$ vertices, inscribed in the circle. The polygons ${ ( { \underline{H} }_{B} \ ; \ B \mbox{ block of } \sigma ) }$ are disjoint, due to the fact that $\sigma$ is non-crossing. We look at the connected components of the complement ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} }$. Each of these connected components is a trimmed disk; the proof of this fact is most easily done by taking the polygons ${ \underline{H} }_{B}$ out of ${ \underline{D} }$ one by one, and using an induction argument.
Given $1 \leq i,j \leq m,$ we have that $i$ and $j$ are in the same block of ${ C_{Q} }( \sigma )$ if and only if the points $Q_{i}$ and $Q_{j}$ lie in the same connected component of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} }$; this follows immediately from the definition of the map ${ C_{Q} }$ in (7.1), where at “$\Leftarrow$” we also use the fact that the connected components of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} }$ are convex. The blocks of ${ C_{Q} }( \sigma )$ are thus found by looking at the connected components ${ \underline{X} }$ of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} }$, with the property that ${ \underline{X} }\cap \{ { Q_{1} , \ldots ,Q_{m} }\} \neq \emptyset .$ Hence in order to show that ${ C_{Q} }( \sigma )$ is ${ \varepsilon }$-alternating, we have to consider such a connected component ${ \underline{X} },$ and prove that when we travel around the boundary of ${ \underline{X} }$ we meet an even number of points $Q_{i} ,$ which are of alternating colors. (Recall from 7.1 that the points ${ Q_{1} , \ldots ,Q_{m} }$ are colored - $Q_{i}$ is red when $l_{i}=1,$ and is blue when $l_{i}=2.)$
We fix for the rest of the proof a connected component ${ \underline{X} }$ of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} }$, such that ${ \underline{X} }\cap \{ { Q_{1} , \ldots ,Q_{m} }\} \neq \emptyset .$ As remarked earlier, ${ \underline{X} }$ is a trimmed disk. Note that all the curvilinear edges of ${ \underline{X} }$ have length $2 \pi /m$ (because otherwise ${ \underline{X} }$ would also contain some of the points $P_{i}, \ 1 \leq i \leq m;$ but all the points $P_{i}$ are in $\cup_{B} { \underline{H} }_{B} ).$ The points $Q_{i}$ that are to be found in $X$ are all lying on the curvilinear edges of the boundary of ${ \underline{X} }.$ On the other hand, if the arc from $P_{i}$ to $P_{i+1}$ is such a curvilinear edge, then it contains zero, one or two points from $\{ { Q_{1} , \ldots ,Q_{m} }\} ,$ according to what are $l_{i}$ and $l_{i+1}:$
- if $l_{i} = l_{i+1} =1,$ then it contains $Q_{i} ;$
- if $l_{i} = l_{i+1} =2,$ then it contains $Q_{i+1} ;$
- if $l_{i} =1, l_{i+1} =2,$ then it contains $Q_{i}$ and $Q_{i+1} ;$
- if $l_{i} =2, l_{i+1} =1,$ then it contains no point from $\{ { Q_{1} , \ldots ,Q_{m} }\} .$
The conclusion of the preceding paragraph is that if we want to find the points from $\{ { Q_{1} , \ldots ,Q_{m} }\}$ that lie in ${ \underline{X} },$ what we have to do is inspect the vertices of ${ \underline{X} },$ and see for each such vertex $P_{i}$ whether the corresponding $Q_{i}$ is on a curvilinear edge of ${ \underline{X} }$ or not.
But if we are to inspect the vertices of ${ \underline{X} },$ the way we want to do this is by pairing them along the rectilinear edges of ${ \underline{X} }.$ (Recall that the rectilinear and curvilinear edges of ${ \underline{X} }$ are alternating, so that the rectilinear edges contain all the vertices of ${ \underline{X} },$ without repetitions.) So let us go around the boundary of ${ \underline{X} },$ clockwisely, and take one by one the rectilinear edges which occur. Always for such an edge, call it $P_{i} P_{j}$ (where $P_{i}$ comes first in clockwise order), we have that $l_{i} \neq l_{j}$ - here is the place where we are using the hypothesis $\sigma \in { NC_{{ \varepsilon }- alt} (m) }.$ It is easily seen that:
- if $l_{i} =1, l_{j} =2,$ then $P_{i}$ and $P_{j}$ do not produce points $Q_{i},Q_{j}$ in ${ \underline{X} };$
- if $l_{i} =2, l_{j} =1,$ then both $Q_{i}$ and $Q_{j}$ are in ${ \underline{X} };$ moreover, when recording clockwisely what is $\{ { Q_{1} , \ldots ,Q_{m} }\} \cap { \underline{X} },$ these two points $Q_{i}, Q_{j}$ will be consecutive, of different colors, and the blue one will come first.
Hence we can organize the points in $\{ { Q_{1} , \ldots ,Q_{m} }\} \cap { \underline{X} }$ in pairs, such that when we travel clockwisely around the boundary of ${ \underline{X} }$ the points from each pair are consecutive, of different colors, and with the blue one coming first. But this clearly implies that $\{ { Q_{1} , \ldots ,Q_{m} }\} \cap { \underline{X} }$ has an even number of points, and of alternating colors - as it was to be shown. [**QED**]{}
$\ $
[**7.8 Corollary**]{} If $\sigma \in { NC_{{ \varepsilon }- alt} (m) }$, then the partitions ${ C_{Q} }( \sigma ) \in NC(m)$ and ${ C_{R} }( \sigma ) \in NC(n)$ defined in 7.2 have the same number of blocks, say $k;$ moreover, we can write them as ${ C_{Q} }( \sigma ) = \{ B_{1} , \ldots , B_{k} \}$ and ${ C_{R} }( \sigma ) = \{ A_{1} , \ldots , A_{k} \}$, in such a way that $|B_{j}| =2|A_{j}|$ for every $1 \leq j \leq k$ (recall that $m=2n,$ due to the supplementary hypothesis made in 7.5).
$\ $
[**Proof**]{} Consider the closed convex polygons ${ ( { \underline{H} }_{B} \ ; \ B \mbox{ block of } \sigma ) }$ defined as in the proof of Proposition 7.7. As pointed out in the named proof, the blocks of ${ C_{Q} }( \sigma )$ are in one-to-one correspondence with the connected components ${ \underline{X} }$ of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} },$ having the property that ${ \underline{X} }\cap \{ { Q_{1} , \ldots ,Q_{m} }\} \neq \emptyset .$ In exactly the same way it is seen that the blocks of ${ C_{R} }( \sigma )$ are in one-to-one correspondence with the connected components ${ \underline{X} }$ of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} },$ having the property that ${ \underline{X} }\cap \{ { R_{1} , \ldots ,R_{n} }\} \neq \emptyset .$ So the proof will be over if we can show that for an arbitrary connected component ${ \underline{X} }$ of ${ { \underline{D} }\setminus \cup_{B} { \underline{H} }_{B} }$ we have: $$\left\{ \begin{array}{l}
{ \mbox{ (a) }
{ \underline{X} }\cap \{ { Q_{1} , \ldots ,Q_{m} }\} \neq \emptyset \Leftrightarrow
{ \underline{X} }\cap \{ { R_{1} , \ldots ,R_{n} }\} \neq \emptyset ; } \\
{ } \\
{ \mbox{ (b) if the equivalent statements of (a) are holding, } } \\
{ \mbox{ then } | { \underline{X} }\cap \{ { Q_{1} , \ldots ,Q_{m} }\} |=2| { \underline{X} }\cap \{ { R_{1} , \ldots ,R_{n} }\} |. }
\end{array} \right.$$ But recall now that the set $\{ { R_{1} , \ldots ,R_{n} }\}$ is nothing else than the set of those points from $\{ { Q_{1} , \ldots ,Q_{m} }\}$ that are colored in red. Hence (7.7) can be restated as: $$\left\{ \begin{array}{l}
{ \mbox{ (a) there are some points $Q_{i}, \ 1 \leq i \leq m$
in ${ \underline{X} }$ if and only if} } \\
{ \mbox{ there are some red points
$Q_{i}, \ 1 \leq i \leq m$ in ${ \underline{X} }$;} } \\
{ } \\
{ \mbox{ (b) if the equivalent statements of (a) are holding,
then the total} } \\
{ \mbox{ number of points $Q_{i}$ in ${ \underline{X} }$ is twice the number
of red points $Q_{i}$ in ${ \underline{X} }$.} }
\end{array} \right.$$ But (7.8) is a clear consequence of the fact that ${ C_{Q} }( \sigma )
\in { NC_{{ \varepsilon }- alt} (m) },$ proved in 7.7. [**QED**]{}
$\ $
The next proposition concludes the proof of (6.4) (and hence of Proposition 6.3 and of Theorem 1.5).
$\ $
[**7.9 Proposition**]{} Let ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ be the string of 1’s and 2’s fixed in 7.1; recall that according to 7.5, $m$ is even, and the number of occurrences of both 1 and 2 in the string is $n=m/2.$ Let on the other hand ${ ( {\cal A} , \varphi ) }$ be a ${ \mbox{non-commutative probability space} }$, with $\varphi$ a trace, and consider ${ a_{1}, a_{2} }, { p_{1} , p_{2} }\in { {\cal A} }$ such that $( { a_{1}, a_{2} })$ is an $R$-diagonal pair, and such that $\{ { p_{1} , p_{2} }\}$ is free from $\{ { a_{1}, a_{2} }\}$. Define $x_{1} = a_{1}p_{1}, x_{2} = p_{2}a_{2},
y_{1} = a_{1}p_{1}p_{2}, y_{2} =a_{2}.$ Then $\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} )$ = $\varphi ( y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}} ).$
$\ $
[**Proof**]{} We consider the expressions of $\varphi ( x_{l_{1}} x_{l_{2}} \cdots x_{l_{m}} )$ and $\varphi ( y_{l_{1}} y_{l_{2}} \cdots y_{l_{m}} )$ via summations over $NC(m),$ as shown in Eqns.(7.2),(7.3); we will prove that for every $\sigma \in NC(m),$ the terms indexed by $\sigma$ in the two sums of (7.2) and (7.3) coincide. If $\sigma \not\in { NC_{{ \varepsilon }- alt} (m) }$ this is clear, because $[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); \sigma ] (R( \mu_{{ a_{1}, a_{2} }} ))$ = 0 (due to the particular form of $R( \mu_{{ a_{1}, a_{2} }} )$), hence the terms indexed by $\sigma$ in (7.2) and (7.3) are both equal to zero. For $\sigma \in { NC_{{ \varepsilon }- alt} (m) }$, it suffices to show that $$[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); { C_{Q} }( \sigma ) ]
(M( \mu_{{ p_{1} , p_{2} }} )) \ = \
[ \mbox{coef } ( m/2 ); { C_{R} }( \sigma ) ]
(M( \mu_{p_{1} \cdot p_{2}} )) .$$ According to Corollary 7.8, we can write ${ C_{Q} }( \sigma ) = \{ B_{1}, \ldots , B_{k} \} ,$ ${ C_{R} }( \sigma ) = \{ A_{1}, \ldots , A_{k} \} ,$ in such a way that $|B_{j}| = 2|A_{j}|$ for every $1 \leq j \leq k.$ On the other hand, due to the fact that ${ C_{Q} }( \sigma )$ is also ${ \varepsilon }$-alternating (by Proposition 7.7), we see that $$[ \mbox{coef } ( { l_{1} , \ldots ,l_{m} }); { C_{Q} }( \sigma ) ]
(M( \mu_{{ p_{1} , p_{2} }} )) \ = \
\varphi ( (p_{1}p_{2})^{|B_{1}|/2} ) \cdots
\varphi ( (p_{1}p_{2})^{|B_{k}|/2} )$$ (in (7.10) the fact that $\varphi$ is a trace is also used). But it is clear that $$[ \mbox{coef } ( m/2 ); { C_{R} }( \sigma ) ]
(M( \mu_{p_{1} \cdot p_{2}} )) \ = \
\varphi ( (p_{1}p_{2})^{|A_{1}|} ) \cdots
\varphi ( (p_{1}p_{2})^{|A_{k}|/2} );$$ the right-hand sides of (7.10) and (7.11) coincide, which establishes (7.9). [**QED**]{}
$\ $
$\ $
$\ $
$\ $
[**8.1 Notations**]{} In this section ${ ( {\cal A} , \varphi ) }$ is a fixed non-commutative probability space, such that $\varphi$ is a trace, and $u, p_{1,1} , p_{1,2} , \ldots , p_{k,1} , p_{k,2} \in { {\cal A} }$ are elements satisfying the conditions (i), (ii), (iii) of the Theorem 1.13. It will be handier to prove that the sets $\{ p_{1,1} { u^{-1} }, up_{1,2} \} , \ldots ,
\{ p_{k,1} { u^{-1} }, up_{k,2} \}$ are free (this is equivalent to the statement of Theorem 1.13, by swapping $p_{j,1}$ with $p_{j,2}$ for every $1 \leq j \leq k).$
We make the notations $p_{j,1} { u^{-1} }{ \stackrel{def}{=} }a_{j,1},$ $u p_{j,2} { \stackrel{def}{=} }a_{j,2},$ $1 \leq j \leq k.$ In order to start the proof that the sets $\{ a_{1,1} , a_{1,2} \} ,
\ldots , \{ a_{k,1} , a_{k,2} \}$ are free in ${ ( {\cal A} , \varphi ) }$, we will use an abstract nonsense construction: we consider (and fix for the whole section) another non-commutative probability space $( { {\cal B} }, \psi ),$ with $\psi$ a trace, and elements $b_{1,1} , b_{1,2} , \ldots , b_{k,1} , b_{k,2} \in { {\cal B} }$ such that:
\(j) $\mu_{b_{j,1} , b_{j,2}} = \mu_{a_{j,1} , a_{j,2}} ,$ for every $1 \leq j \leq k$ (where the joint distributions $\mu_{b_{j,1} , b_{j,2}}$ and $\mu_{a_{j,1} , a_{j,2}}$ are considered in $( { {\cal B} }, \psi )$ and ${ ( {\cal A} , \varphi ) },$ respectively).
(jj) the sets $\{ b_{1,1} , b_{1,2} \} , \ldots , \{ b_{k,1} , b_{k,2} \}$ are free in $( { {\cal B} }, \psi ).$ Such a construction can of course be done, for instance we can take $( { {\cal B} }, \psi )$ to be the free product $ \star_{j=1}^{k} ( { {\cal A} }_{j} , \varphi | { {\cal A} }_{j} ),$ where ${ {\cal A} }_{j}$ is the unital algebra generated by $\{ a_{j,1} , a_{j,2} \}$ in ${ {\cal A} },$ and then we can take $b_{j,1} , b_{j,2}$ to be just $a_{j,1} , a_{j,2},$ but viewed in ${ {\cal B} }.$ (Note that $\star_{j=1}^{k} ( \varphi | { {\cal A} }_{j} )$ is a trace, because each $\varphi | { {\cal A} }_{j}$ is so, and by Proposition 2.5.3 of [@VDN] - this justifies why we could assume that $\psi$ is a trace.)
In this setting, our goal is to show that $$\mu_{a_{1,1} , a_{1,2} , \ldots , a_{k,1} , a_{k,2}} \ = \
\mu_{b_{1,1} , b_{1,2} , \ldots , b_{k,1} , b_{k,2}} .$$ Indeed, as it is clear from its very definition, the freeness of a family of subsets can be read from the joint distribution of the union of those subsets; so in the presence of (8.1), the freeness of $\{ a_{1,1} , a_{1,2} \} , \ldots , \{ a_{k,1} , a_{k,2} \}$ is implied by the one of $\{ b_{1,1} , b_{1,2} \} , \ldots , \{ b_{k,1} , b_{k,2} \} .$
$\ $
[**8.2 Remark**]{} For every $1 \leq j \leq k,$ the pair $( a_{j,1} , a_{j,2} )$ - and hence $( { b_{j,1} , b_{j,2} })$ too - is $R$-diagonal with determining series $$f_{j} \ { \stackrel{def}{=} }\ R( \mu_{p_{j,1} \cdot p_{j,2}} ) \
{ \framebox[7pt]{$\star$} }\ Moeb.$$ This follows from Theorem 1.5 and Proposition 1.7 (and the fact that $a_{j,1} \cdot a_{j,2} = p_{j,1} \cdot p_{j,2} ).$
We have, in other words, that $$[ R( \mu_{{ a_{j,1} , a_{j,2} }} )] ( { z_{j,1} , z_{j,2} }) \ = \
[ R( \mu_{{ b_{j,1} , b_{j,2} }} )] ( { z_{j,1} , z_{j,2} }) \ = \
f_{j} ( z_{j,1} \cdot z_{j,2} ) +
f_{j} ( z_{j,2} \cdot z_{j,1} ),$$ for every $1 \leq j \leq k.$ The freeness of $\{ { b_{1,1}, b_{1,2} }\} , \ldots , \{ { b_{k,1} , b_{k,2} }\}$ enables us to obtain from (8.3) the formula for the $R$-transform $R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} ),$ this is the series of $2k$ variables $$f( { z_{1,1} , z_{1,2} }, \ldots , { z_{k,1} , z_{k,2} }) \ { \stackrel{def}{=} }\ \sum_{j=1}^{k}
f_{j} ( z_{j,1} \cdot z_{j,2} ) + f_{j} ( z_{j,2} \cdot z_{j,1} ).$$ (Of course, we don’t have the analogue of this fact for the $a$’s - if we would, the proof would be finished.)
$\ $
A way of re-interpreting (8.2) which will be used later on is the following:
$\ $
[**8.3 Lemma**]{} For every $1 \leq j \leq k$ and every $n \geq 1,$ the coefficients of $( z_{j,1} \cdot z_{j,2} )^{n},$ $( z_{j,2} \cdot z_{j,1} )^{n}$ in $[ R( \mu_{{ b_{j,1} , b_{j,2} }} )] ( { z_{j,1} , z_{j,2} })$ and $[ R( \mu_{{ p_{j,1} , p_{j,2} }} )] ( { z_{j,1} , z_{j,2} })$ are (all four of them) equal to the coefficient of order $n$ in the series $f_{j}$ of (8.2).
$\ $
[**Proof**]{} For $R( \mu_{{ b_{j,1} , b_{j,2} }} )$ this has been explicitly written in (8.3), while for $R( \mu_{{ p_{j,1} , p_{j,2} }} )$ we use Proposition 5.3, the hypothesis that the pair $( { p_{j,1} , p_{j,2} })$ is diagonally balanced, and the form of $f_{j}$ in (8.2). [**QED**]{}
$\ $
[**8.4 The approach to the proof**]{} of Theorem 1.13 will follow from now on the same pattern as the one used for the proof of Theorem 1.5. Indeed, Eqn.(8.1) simply says that for every $m \geq 1$ and ${ l_{1} , \ldots ,l_{m} }\in \{ 1,2 \} ,$ ${ h_{1} , \ldots ,h_{m} }\in { \{ 1, \ldots ,k \} },$ we have $$\varphi ( a_{h_{1},l_{1}} \cdots a_{h_{m},l_{m}} ) \ = \
\psi ( b_{h_{1},l_{1}} \cdots b_{h_{m},l_{m}} ) .$$ In the rest of the section we will work on proving (8.5), in a way which parallels the development of Section 7. We will first obtain formulas expressing the two sides of (8.5) as summations over $NC(m).$ From these formulas it will be clear that both sides of (8.5) are equal to zero, unless $( { l_{1} , \ldots ,l_{m} }) \in
\{ 1,2 \} ^{m}$ satisfies a certain balancing condition - in fact the same as the one stated in 7.5. Starting from that point, we will assume that the balancing condition holds, and in order to complete the proof we will need to throw in a “geometrical” argument concerning the circular picture of a non-crossing partition.
$\ $
Both sides of (8.5) are invariant under cyclic permutations of the monomials involved (because $\varphi$ and $\psi$ are traces). Thus if we assume in (8.5) that $l_{1} =1,$ we are in fact only missing the case when $l_{1} = l_{2} = \cdots = l_{m} =2.$ But in the latter case we have:
$\ $
[**8.5 Lemma**]{} $\varphi ( a_{h_{1},2} a_{h_{2},2} \cdots a_{h_{m},2} )$ = $\psi ( b_{h_{1},2} b_{h_{2},2} \cdots b_{h_{m},2} )$ = 0.
$\ $
[**Proof**]{} For the $a$’s: $\varphi ( a_{h_{1},2} a_{h_{2},2} \cdots a_{h_{m},2} )$ = $\varphi ( up_{h_{1},2} up_{h_{2},2} \cdots up_{h_{m},2} )$ can be written as a coefficient of the series $M( \mu_{up_{h_{1},2} , up_{h_{2},2} , \ldots , up_{h_{m},2}} ).$ But this series is identically zero; indeed, by using Eqn.(3.16) and the fact that $u$ is free from $\{ p_{1,2} , \ldots , p_{k,2} \}$ ), we get: $$M( \mu_{up_{h_{1},2} , up_{h_{2},2} , \ldots , up_{h_{m},2}} )
\ = \ M( \mu_{\scriptstyle
\underbrace{u, \ldots ,u}_{m} } ) { \framebox[7pt]{$\star$} }R( \mu_{p_{h_{1},2} , p_{h_{2},2} , \ldots , p_{h_{m},2}} ).$$ It is obvious, however, that $M( \mu_{u, \ldots , u} )=0,$ and that in general we have $0 { \framebox[7pt]{$\star$} }f =0.$
For the $b$’s: $b_{h,2}$ is a part of the $R$-diagonal pair $( { b_{h,1} , b_{h,2} }),$ and therefore, as remarked in the final paragraph of Section 6, we must have $\psi ( b_{h,2}^{n} )=0$ for every $1 \leq h \leq k$ and $n \geq 1.$ But then, if we also take into account that $b_{1,2} , b_{2,2} , \ldots , b_{k,2}$ are free, the equality $\psi ( b_{h_{1},2} b_{h_{2},2} \cdots b_{h_{m},2} )$ = 0 follows directly from the definition of freeness in (1.1). [**QED**]{}
$\ $
[**8.6 Notations**]{} From now on, and until the end of the section, we fix: $m \geq 1;$ ${ l_{1} , \ldots ,l_{m} }\in \{ 1,2 \} ^{m}$ such that $l_{1} =1;$ and ${ h_{1} , \ldots ,h_{m} }\in { \{ 1, \ldots ,k \} }.$ Our goal is to prove (8.5) for this fixed set of data.
We denote the $m$-tuple $( { l_{1} , \ldots ,l_{m} }) \in \{ 1,2 \} ^{m}$ by ${ \varepsilon },$ and we denote by $n$ $( 1 \leq n \leq m)$ the number of occurrences of 1 in ${ \varepsilon }.$
We will use the geometrical objects constructed in Sections 7.1, 7.2 above. That is, we consider again the circle of radius 1 and the points ${ P_{1} , \ldots ,P_{m} }, { Q_{1} , \ldots ,Q_{m} }$ sitting on it, and positioned in the way described in 7.1; and we consider again the complementation map ${ C_{Q} }: NC(m) \rightarrow NC(m)$ described in 7.2.
$\ $
[**8.7 Remark**]{} Consider the product $a_{h_{1},l_{1}} a_{h_{2},l_{2}} \cdots a_{h_{m},l_{m}}$ appearing in the right-hand side of (8.5). If in this product we write back each $a_{h_{i},1}$ as $p_{h_{i},1} { u^{-1} }$ and each $a_{h_{i},2}$ as $u p_{h_{i},2}$, we obtain an expression (monomial) of length $2m$ in ${ p_{1,1} , p_{1,2} }, \ldots ,
{ p_{k,1} , p_{k,2} }, u, { u^{-1} }.$ By looking just at the $p$’s in the latter monomial, we see that they are $p_{h_{1},l_{1}} , p_{h_{2},l_{2}} , \ldots , p_{h_{m},l_{m}} ,$ exactly in this order, and placed on a certain set of positions $I \subseteq { \{ 1,2, \ldots ,2m \} },$ with $|I|=m.$ On the complementary set of positions $J = { \{ 1,2, \ldots ,2m \} }\setminus I$ of our monomial of length $2m$ we have factors of $u$ and ${ u^{-1} };$ we denote them as $u^{\lambda_{1}}, u^{\lambda_{2}}, \ldots ,u^{\lambda_{m}},$ in the order in which they appear from left to right. It is clear that $\lambda_{i}, \ 1 \leq i \leq m,$ is entirely determined by $l_{i},$ via the formula: $l_{i} =1 \Rightarrow \lambda_{i} = -1,$ $l_{i} =2 \Rightarrow \lambda_{i} = +1.$
We can thus say that the product $a_{h_{1},l_{1}} a_{h_{2},l_{2}} \cdots a_{h_{m},l_{m}}$ is obtained by shuffling together $p_{h_{1},l_{1}} p_{h_{2},l_{2}} \cdots p_{h_{m},l_{m}}$ and $u^{\lambda_{1}} u^{\lambda_{2}} \cdots u^{\lambda_{m}},$ where the $p$’s have to sit on the positions indicated by $I$, and the $u$’s have to sit on the positions indicated by $J = { \{ 1,2, \ldots ,2m \} }\setminus I.$ Note that $I$ and $J$ are exactly the same as the ones appearing in the proof of Proposition 7.3.
$\ $
[**8.8 Proposition**]{} In the notations established in 8.1, 8.6, 8.7, we have: $$\varphi ( a_{h_{1},l_{1}} a_{h_{2},l_{2}} \cdots
a_{h_{m},l_{m}} ) \ =$$ $$= \ \sum_{\sigma \in NC(m)}
[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]
( R( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }} )) \cdot$$ $$[ \mbox{coef } ( { \lambda_{1} , \ldots , \lambda_{m} }); { C_{Q} }( \sigma ) ]
( M( \mu_{u, { u^{-1} }} ))$$ and $$\psi ( b_{h_{1},l_{1}} b_{h_{2},l_{2}} \cdots
b_{h_{m},l_{m}} ) \ =$$ $$= \ \sum_{\sigma \in NC(m)}
[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]
( R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} )) .$$ (In (8.6), (8.7) the series $R( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }} )$ and $R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} )$ are acting in the variables ${ z_{1,1} , z_{1,2} }, \ldots , { z_{k,1} , z_{k,2} }$ - same as in (8.4), for instance. The series $M( \mu_{u, { u^{-1} }} )$ is viewed as acting in the two variables $z_{+1}$ and $z_{-1}.)$
$\ $
[**Proof**]{} Equation (8.7) is just the moment-cumulant formula, see Section 3.5 above.
The argument proving (8.6) is very similar to the one used in the proof of Proposition 7.3, and for this reason we will only outline its main steps. We view $\varphi ( a_{h_{1},l_{1}} a_{h_{2},l_{2}} \cdots
a_{h_{m},l_{m}} )$ as a coefficient of length $2m$ of the series $M( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }, u, { u^{-1} }} ),$ and we then expand it as a summation over $NC(2m)$ by using the moment-cumulant formula (i.e., the appropriate version of Eqn.(3.12) in 3.5). Due to the fact that $\{ { p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }\}$ is free from $\{ u, { u^{-1} }\} ,$ what we arrive to is the formula (paralleling (7.4) in the proof of 7.3): $$\varphi ( a_{h_{1},l_{1}} a_{h_{2},l_{2}} \cdots
a_{h_{m},l_{m}} ) \ =$$ $$= \ \sum_{ \begin{array}{c}
{\scriptstyle \sigma , \tau \in NC(m)} \\
{\scriptstyle I,J - compatible}
\end{array} }
[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]
( R( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }} )) \cdot$$ $$[ \mbox{coef } ( { \lambda_{1} , \ldots , \lambda_{m} }); \tau ) ] ( R( \mu_{u, { u^{-1} }} )).$$ Then we process the right-hand side of (8.8) exactly in the way the right-hand side of (7.4) was processed in the proof of Proposition 7.3, and this leads to the right-hand side of (8.6). [ **QED**]{}
$\ $
[**8.9 Corollary**]{} Let ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ be the string of 1’s and 2’s of Notations 8.6. If the number $n$ of occurrences of 1 in ${ \varepsilon }$ does not equal the number $m-n$ of occurrences of 2 in ${ \varepsilon },$ then $\varphi ( a_{h_{1},l_{1}} a_{h_{2},l_{2}} \cdots
a_{h_{m},l_{m}} )$ = $\psi ( b_{h_{1},l_{1}} b_{h_{2},l_{2}} \cdots
b_{h_{m},l_{m}} )$ = 0.
$\ $
[**Proof**]{} For the $a$’s: Due to how ${ \lambda_{1} , \ldots , \lambda_{m} }$ are determined by ${ l_{1} , \ldots ,l_{m} }$ (see Remark 8.7), we have in this case that the number $n$ of $(-1)$’s in $( { \lambda_{1} , \ldots , \lambda_{m} })$ is different from the number $m-n$ of $1$’s in $( { \lambda_{1} , \ldots , \lambda_{m} })$. But then it is immediate that $[ \mbox{coef } ( { \lambda_{1} , \ldots , \lambda_{m} }); \tau] ( M( \mu_{u, { u^{-1} }} ))$ = 0 for every $\tau \in NC(m),$ and hence all the terms in the sum of (8.6) are vanishing.
For the $b$’s: in (8.4) we have an explicit formula for $R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} ),$ the inspection of which makes clear (in view of the hypothesis of the present corollary) that all the terms of the sum in (8.7) are vanishing. [**QED**]{}
$\ $
We therefore make, exactly as we did in 7.5, the following
$\ $
[**8.10 Supplementary hypothesis:**]{} From now on we will assume that the string ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ of 8.6 contains an equal number of 1’s and 2’s (i.e., $m$ is even and $n=m/2).$
$\ $
Moreover, we will consider the notion of an [*${ \varepsilon }$-alternating*]{} partition in $NC(m),$ which is exactly the one defined in 7.6. Recall that the set of all the ${ \varepsilon }$-alternating partitions in $NC(m)$ is denoted by ${ NC_{{ \varepsilon }- alt} (m) }.$
$\ $
[**8.11 Proposition**]{} A partition $\sigma \in NC(m)$ is ${ \varepsilon }$-alternating if and only if it has the following properties:
$1^{o}$ every block of $\sigma$ has at least two elements;
$2^{o}$ there exists no block $B$ of $\sigma$ such that: $|B|$ is odd, and $( { l_{1} , \ldots ,l_{m} })|B$ is a cyclic permutation of $( \underbrace{1,2, \ldots ,1,2,1}_{|B|} )$ or of $( \underbrace{2,1, \ldots ,2,1,2}_{|B|} )$ (where the notation $( { l_{1} , \ldots ,l_{m} })|B$ is in the sense of 3.2.$2^{o});$
$3^{o}$ every block of the complementary partition ${ C_{Q} }( \sigma )$ has an even number of elements.
$\ $
[**Proof**]{} “$\Rightarrow$” As remarked immediately after the Definition 7.6, every block of $\sigma$ has an even number of elements (this implies $1^{o}$ and $2^{o}).$ The same holds for ${ C_{Q} }( \sigma ),$ because ${ C_{Q} }( \sigma )$ is also in ${ NC_{{ \varepsilon }- alt} (m) },$ by Proposition 7.7.
“$\Leftarrow$” By contradiction, we assume that $\sigma$ is not ${ \varepsilon }$-alternating. This means that we can find $1 \leq i<j \leq m$ such that $i$ and $j$ are in the same block of $\sigma ,$ and consecutive in that block (in the sense of 2.2), and such that moreover $l_{i} =l_{j}.$ From all the pairs $(i,j)$ with these properties, we choose one, $( { i_{o} }, { j_{o} }),$ such that the length of the segment $P_{{ i_{o} }}P_{{ j_{o} }}$ is minimal. (We are using in this proof the circular picture involving the points ${ P_{1} , \ldots ,P_{m} }, { Q_{1} , \ldots ,Q_{m} }$ introduced in 7.1, 7.2.)
We denote by ${ \underline{S} }$ the open half-plane that is determined by the line through $P_{{ i_{o} }}$ and $P_{{ j_{o} }}$, and that does not contain the center of the circle. (If the line $P_{{ i_{o} }}P_{{ j_{o} }}$ goes through the center of the circle, any of the two open half-planes determined by it can be chosen as ${ \underline{S} }.)$ The minimality assumption on $( { i_{o} }, { j_{o} })$ implies that: $$\left\{ \begin{array}{l}
{ \mbox{ if $1 \leq i < j \leq m$ are in the same block of
$\sigma ,$ and consecutive in that block, } } \\
{ \mbox{ and if in addition we have that $P_{i},P_{j} \in
{ \underline{S} },$ then necessarily $l_{i} \neq l_{j}$} }
\end{array} \right.$$ (this is simply because the length of $P_{i}P_{j}$ is strictly smaller than the one of $P_{{ i_{o} }}P_{{ j_{o} }}).$ Note that this argument gives in fact that $l_{i} \neq l_{j}$ even if we only assume that $P_{i},$ $P_{j}$ lie in the closure of ${ \underline{S} }$, but $(i,j) \neq ( { i_{o} }, { j_{o} }).$
Let $B$ be the block of $\sigma$ containing ${ i_{o} }$ and ${ j_{o} }.$ We claim that $\{ P_{j} \ | \ j \in B \} \cap { \underline{S} }= \emptyset .$ Indeed, this is obvious if $B$ is reduced to $\{ { i_{o} }, { j_{o} }\}$, so let us assume that $|B| \geq 3.$ Since ${ i_{o} }, { j_{o} }$ are consecutive in $B,$ we have that $\{ P_{j} \ | \ j \in B \}$ is contained in one of the closed half-planes determined by $P_{{ i_{o} }}P_{{ j_{o} }} ;$ so if we would assume $\{ P_{j} \ | \ j \in B \} \cap { \underline{S} }\neq \emptyset ,$ then it would follow that $\{ P_{j} \ | \ j \in B , \ j \neq { i_{o} }, { j_{o} }\} \subseteq { \underline{S} }.$ But then the remark concluding the preceding paragraph would imply that $l_{i} \neq l_{j}$ whenever $i<j$ are consecutive in $B$, with the exception of the case when $(i,j) = ( { i_{o} }, { j_{o} }),$ and this would violate property $2^{o}$ of $\sigma .$
Now, the set $\{ j \ | \ 1 \leq j \leq m, \ P_{j} \in { \underline{S} }\}$ is either void or a union of blocks of $\sigma ,$ because of the non-crossing character of $\sigma$ (and because of what was proved about the block $B \ni { i_{o} }, { j_{o} }).$ Remark that for every block $B'$ of $\sigma$ which is contained in $\{ j \ | \ 1 \leq j \leq m, \ P_{j} \in { \underline{S} }\} ,$ we must have that $|B'|$ is even; indeed, from (8.9) it follows that $l_{i} \neq l_{j}$ for every $i,j \in B'$ which are consecutive in $B' ,$ and this couldn’t happen if $|B'|$ would be odd. Consequently, the set $\{ j \ | \ 1 \leq j \leq m, \ P_{j} \in { \underline{S} }\}$ has an even cardinality (possibly zero).
By recalling how the points ${ Q_{1} , \ldots ,Q_{m} }$ were constructed, we next note that $\{ j \ | \ 1 \leq j \leq m, \ P_{j} \in { \underline{S} }\}$ $\subset$ $\{ j \ | \ 1 \leq j \leq m, \ Q_{j} \in { \underline{S} }\} ,$ and moreover that the set-difference $\{ j \ | \ 1 \leq j \leq m,$ $Q_{j} \in { \underline{S} }, \ P_{j} \not\in { \underline{S} }\}$ has exactly one element, which is either ${ i_{o} }$ or ${ j_{o} }.$ The latter assertion amounts to the following two facts, both obvious: (a) if $1 \leq j \leq m$ is such that $P_{j}$ does not belong to the closure of ${ \underline{S} },$ then $Q_{j} \not\in { \underline{S} };$ and (b) $Q_{{ i_{o} }}$ and $Q_{{ j_{o} }}$ lie in opposite open half-planes determined by the line $P_{{ i_{o} }} P_{{ j_{o} }},$ hence exactly one of them is in ${ \underline{S} }$ (the assumption $l_{{ i_{o} }} = l_{{ j_{o} }}$ is of course crucial for (b)). The conclusion of this paragraph is that the cardinality $| \{ j \ | \ 1 \leq j \leq m, \ Q_{j} \in { \underline{S} }\} |$ = 1 + $| \{ j \ | \ 1 \leq j \leq m, \ P_{j} \in { \underline{S} }\} |$ is an odd number.
But because of how the definition of the complementation map ${ C_{Q} }$ was made in 7.2, the set $\{ j \ | \ 1 \leq j \leq m, \ Q_{j} \in { \underline{S} }\}$ is a union of blocks of ${ C_{Q} }( \sigma ).$ Hence, in view of the property $3^{o}$ of $\sigma ,$ $| \{ j \ | \ 1 \leq j \leq m, \ Q_{j} \in { \underline{S} }\} |$ is an even number - contradiction. [**QED**]{}
$\ $
[**8.12 The conclusion of the proof**]{} Recall that what we need to prove is the equality (8.5) of 8.4, for the $m \geq 1,$ ${ l_{1} , \ldots ,l_{m} }\in \{ 1,2 \} ,$ ${ h_{1} , \ldots ,h_{m} }\in { \{ 1, \ldots ,k \} }$ that were fixed in 8.6, and where ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ satisfies the additional hypothesis 8.10.
Let us express both quantities $\varphi ( a_{h_{1},l_{1}} \cdots a_{h_{m},l_{m}} )$ and $\psi ( b_{h_{1},l_{1}} \cdots b_{h_{m},l_{m}} )$ appearing in (8.5) as summations over $NC(m),$ in the way shown in Proposition 8.8 (Eqns.(8.6) and (8.7), respectively). We will prove that for every $\sigma \in NC(m),$ the terms indexed by $\sigma$ in the two sums of (8.6) and (8.7) coincide - this will of course imply that the sums are equal.
Up to now, the $m$-tuple $( { h_{1} , \ldots ,h_{m} }) \in { \{ 1, \ldots ,k \} }^{m} ,$ which is part of our data, didn’t play any role. Let us view this $m$-tuple as a function from ${ \{ 1, \ldots ,m \} }$ to ${ \{ 1, \ldots ,k \} },$ and denote by ${ L_{1} , \ldots ,L_{k} }$ its level sets; i.e., $L_{h} { \stackrel{def}{=} }\{ i \ |$ $1 \leq i \leq m, \ h_{i} =h \} ,$ for $1 \leq h \leq k.$ We will say about a partition $\sigma \in NC(m)$ that it is [*$h$-acceptable*]{} if every block of $\sigma$ is contained in one of the level sets ${ L_{1} , \ldots ,L_{k} };$ and we will denote by ${ NC_{h - acc} }(m)$ the set of all $h$-acceptable partitions in $NC(m).$
The general term of the sum in (8.7) is a product of coefficients of $R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} ),$ made in the way dictated by $\sigma \in NC(m)$ (according to the recipe of 3.2.$3^{o}).$ By using the freeness of the sets $\{ { b_{1,1}, b_{1,2} }\} , \ldots , \{ { b_{k,1} , b_{k,2} }\}$ and Theorem 3.6, it is easily seen that this product is zero whenever $\sigma$ is not $h$-acceptable; while for $\sigma \in { NC_{h - acc} }(m)$ we get the formula: $$[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]
( R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} )) \ =$$ $$= \ \prod_{ \begin{array}{c}
{\scriptstyle 1 \leq h \leq k} \\
{\scriptstyle such \ that} \\
{\scriptstyle L_{h} \neq \emptyset}
\end{array} } \ \{
\ \prod_{ \begin{array}{c}
{\scriptstyle B \ block \ of \ \sigma} \\
{\scriptstyle such \ that} \\
{\scriptstyle B \subseteq L_{h}}
\end{array} } \
[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }) | B]
( R( \mu_{{ b_{h,1} , b_{h,2} }} )) \ \} ,$$ (where the notation for the restricted $|B|$-tuple $( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }) | B$ is taken from 3.2.$2^{o}).$ We next recall from 8.2 that the pair $( { b_{h,1} , b_{h,2} })$ is $R$-diagonal, for every $1 \leq h \leq k;$ hence even if we assume that $\sigma \in { NC_{h - acc} }(m),$ the particular form of $R( \mu_{{ b_{h,1} , b_{h,2} }} )$ will force the product in (8.10) to still be equal to zero, unless we also assume that the $|B|$-tuple $( { l_{1} , \ldots ,l_{m} })|B$ is $(1,2,1,2, \ldots ,1,2)$ or $(2,1,2,1, \ldots ,
2,1)$ for every block $B$ of $\sigma .$ In other words, if we want the contribution of $\sigma \in NC(m)$ to the sum (8.7) to be non-zero, then we must also impose (besides the condition $\sigma \in { NC_{h - acc} }(m))$ that $\sigma$ is ${ \varepsilon }$-alternating.
To conclude the discussion concerning (8.7), let us consider $\sigma \in { NC_{h - acc} }(m) \cap { NC_{{ \varepsilon }- alt} (m) }.$ Then the product in (8.10) can be prelucrated by using Lemma 8.3, and this yields the formula $$[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]
( R( \mu_{{ b_{1,1}, b_{1,2} }, \ldots , { b_{k,1} , b_{k,2} }} )) \ =$$ $$= \ \prod_{ \begin{array}{c}
{\scriptstyle 1 \leq h \leq k} \\
{\scriptstyle such \ that} \\
{\scriptstyle L_{h} \neq \emptyset}
\end{array} } \ \{
\ \prod_{ \begin{array}{c}
{\scriptstyle B \ block \ of \ \sigma} \\
{\scriptstyle such \ that} \\
{\scriptstyle B \subseteq L_{h}}
\end{array} } \
[ \mbox{coef } ( |B|/2 ) ] ( f_{h} ) \ \}$$ (where ${ f_{1} , \ldots ,f_{k} }$ are the series of one variable discussed in Remark 8.2).
We now start looking at the term indexed by $\sigma$ in the sum (8.6). From the parallel discussion made above, we see that we have to consider three cases: (a) $\sigma \in NC(m) \setminus
{ NC_{h - acc} }(m);$ (b) $\sigma \in { NC_{h - acc} }(m) \setminus { NC_{{ \varepsilon }- alt} (m) };$ and (c) $\sigma \in { NC_{h - acc} }(m) \cap { NC_{{ \varepsilon }- alt} (m) }.$ More precisely, we have to show that the term indexed by $\sigma$ in (8.6) is zero in the cases (a) and (b), and is given by the right-hand side of (8.11) in the case (c).
Case (a) is easy. Indeed, by using the freeness of $\{ { p_{1,1} , p_{1,2} }\} , \ldots , \{ { p_{k,1} , p_{k,2} }\}$ and Theorem 3.6, it is easily seen that in this case $[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]$ $( R( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }} ))$ equals zero.
Case (c) is also easy. Indeed, in this case the $h$-acceptability of $\sigma$ implies that $[ \mbox{coef } ( (h_{1},l_{1}),$ $\ldots , (h_{m},l_{m}) ); \sigma ]$ $( R( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }} ))$ has an expression of the kind shown in the right-hand side of (8.10) (with the $b$’s replaced by $p$’s); and after that, due to the assumption that $\sigma \in { NC_{{ \varepsilon }- alt} (m) },$ this expression can be prelucrated by using Lemma 8.3, and yields exactly the right-hand side of (8.11). On the other hand, in case (c) we also have ${ C_{Q} }( \sigma ) \in
{ NC_{{ \varepsilon }- alt} (m) }$ (by Proposition 7.7); by using this fact, and by following how $( { \lambda_{1} , \ldots , \lambda_{m} }) $ is determined by ${ \varepsilon }= ( { l_{1} , \ldots ,l_{m} })$ (see Remark 8.7), one sees immediately that we have $[ \mbox{coef } ( \lambda_{1}, \ldots , \lambda_{m} ); { C_{Q} }( \sigma )]$ $(M ( \mu_{u, { u^{-1} }} ))$ =1.
Finally, let us consider the case (b). The $h$-acceptability of $\sigma$ implies that (similarly to (8.10)) we have the formula $$[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }); \sigma ]
( R( \mu_{{ p_{1,1} , p_{1,2} }, \ldots , { p_{k,1} , p_{k,2} }} )) \ =$$ $$= \ \prod_{ \begin{array}{c}
{\scriptstyle 1 \leq h \leq k} \\
{\scriptstyle such \ that} \\
{\scriptstyle L_{h} \neq \emptyset}
\end{array} } \ \{
\ \prod_{ \begin{array}{c}
{\scriptstyle B \ block \ of \ \sigma} \\
{\scriptstyle such \ that} \\
{\scriptstyle B \subseteq L_{h}}
\end{array} } \
[ \mbox{coef } ( { (h_{1},l_{1}), \ldots ,(h_{m},l_{m}) }) | B]
( R( \mu_{{ p_{h,1} , p_{h,2} }} )) \ \} .$$ We have to show that either the product in (8.12) is zero, or $[ \mbox{coef } ( { \lambda_{1} , \ldots , \lambda_{m} }); { C_{Q} }( \sigma ) ]$ $(M ( \mu_{u, { u^{-1} }} ))$ is zero. We will obtain this from the hypothesis that $\sigma \not\in { NC_{{ \varepsilon }- alt} (m) },$ in the equivalent formulation that comes out by negating Proposition 8.11. That is, we know that because $\sigma \not\in { NC_{{ \varepsilon }- alt} (m) },$ one of the following three things must happen:
$1^{o}$ either: $\sigma$ has a block $B$ with one element. In this case the product in (8.12) contains a factor which is a coefficient of length one of the series $R( \mu_{{ p_{h,1} , p_{h,2} }} ),$ $1 \leq h \leq k;$ but any such coefficient is zero, because the pairs $( { p_{h,1} , p_{h,2} })$ are diagonally balanced.
$2^{o}$ or: $\sigma$ has a block $B$ such that $|B|$ is odd and $( { l_{1} , \ldots ,l_{m} })|B$ is a cyclic permutation of $(1,2, \ldots ,1,2,1)$ or of $(2,1, \ldots ,2,1,2)$. Let $h \in { \{ 1, \ldots ,k \} }$ be such that $B$ is contained in the level set $L_{h}.$ Then $[ \mbox{coef } ( (h_{1},l_{1}), \ldots , (h_{m},l_{m}) )|B ]
( R( \mu_{{ p_{h,1} , p_{h,2} }} ))$ (which is the same thing as $[ \mbox{coef } ( (h,l_{1} ), \ldots ,(h, l_{m} ))|B ]
( R( \mu_{{ p_{h,1} , p_{h,2} }} ))$, by the definition of $L_{h})$, is zero - because $( { p_{h,1} , p_{h,2} })$ is diagonally balanced, and by Remark 5.2.
$3^{o}$ or: ${ C_{Q} }(\sigma )$ has a block containing an odd number of elements. But then it is clear that $[ \mbox{coef } ( { \lambda_{1} , \ldots , \lambda_{m} }); { C_{Q} }( \sigma ) ]
(M ( \mu_{u, { u^{-1} }} ))$ =0. This concludes the discussion of case (b), and the proof. [**QED**]{}
$\ $
$\ $
$\ $
[99]{}
T. Banica. On the polar decomposition of circular variables, preprint.
Ph. Biane. Some properties of crossings and partitions, preprint.
P. Doubilet, G.-C. Rota, R. Stanley. On the foundations of the combinatorial theory (VI): the idea of generating function, in Proc. of the 6th Berkeley symposium on mathematical statistics and probability, Lucien M. Le Cam et al. editors, University of California Press, 1972, 267-318.
K. Dykema. Interpolated free group factors, Pacific J. Math. 163(1994), 123-135.
K. Dykema. Free products of hyperfinite von Neumann algebras and free dimension, Duke J. Math. 69(1993), 97-119.
P. Edelman. Chain enumeration and non-crossing partitions, Discrete Math. 31(1980), 171-180.
G. Kreweras. Sur les partitions non-croisees d’un cycle, Discrete Math. 1(1972), 333-350.
A. Nica. $R$-transforms of free joint distributions, and non-crossing partitions, to appear in the Journal of Functional Analysis.
A. Nica, R. Speicher. A “Fourier transform” for multiplicative functions on non-crossing partitions, preprint.
A. Nica, R. Speicher. On the multiplication of free $n$-tuples of non-commutative random variables, preprint.
F. Radulescu. The fundamental group of the von Neumann algebra of a free group with infinitely many generators is ${ \mbox{\bf R} }_{+} \setminus \{ 0 \} ,$ J. Amer. Math. Soc. 5(1992), 517-532.
F. Radulescu. Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Inventiones Math. 115(1994), 347-389.
R. Simion, D. Ullman. On the structure of the lattice of non-crossing partitions, Discrete Math. 98(1991), 193-206.
R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Annalen 298(1994), 611-628.
R. Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Habilitationsschrift, Heidelberg 1994.
D. Voiculescu. Symmetries of some reduced free product C\*-algebras, in Operator algebras and their connection with topology and ergodic theory, H. Araki et al. editors (Springer Lecture Notes in Mathematics, volume 1132, 1985), 556-588.
D. Voiculescu. Addition of certain non-commuting random variables, J. Functional Analysis 66(1986), 323-346.
D. Voiculescu. Multiplication of certain non-commuting random variables, J. Operator Theory 18(1987), 223-235.
D. Voiculescu. Limit laws for random matrices and free products, Inventiones Math. 104(1991), 201-220.
D. Voiculescu. Circular and semicircular systems and free product factors, in Operator algebras, unitary representations, enveloping algebras, and invariant theory, A. Connes et al. editors, Birkhäuser, 1990, 45-60.
D. Voiculescu, K. Dykema, A. Nica. Free random variables, CRM Monograph Series, volume 1, AMS, 1992.
[^1]: Research done while this author was on leave at the Fields Institute, Waterloo, and the Queen’s University, Kingston, holding a Fellowship of NSERC, Canada.
[^2]: Supported by a Heisenberg Fellowship of the DFG.
|
---
abstract: |
Many infrastructure networks have a modular structure and are also interdependent with other infrastructures. While significant research has explored the resilience of interdependent networks, there has been no analysis of the effects of modularity. Here we develop a theoretical framework for attacks on interdependent modular networks and support our results through simulations. We focus, for simplicity, on the case where each network has the same number of communities and the dependency links are restricted to be between pairs of communities of different networks. This is particularly realistic for modeling infrastructure across cities. Each city has its own infrastructures and different infrastructures are dependent only within the city. However, each infrastructure is connected within and between cities. For example, a power grid will connect many cities as will a communication network, yet a power station and communication tower that are interdependent will likely be in the same city. It has previously been shown that single networks are very susceptible to the failure of the interconnected nodes (between communities) [@shai2014resilience] and that attacks on these nodes are even more crippling than attacks based on betweenness [@da2015complex]. In our example of cities these nodes have long range links which are more likely to fail. For both treelike and looplike interdependent modular networks we find distinct regimes depending on the number of modules, $m$. (i) In the case where there are fewer modules with strong intraconnections, the system first separates into modules in an abrupt first-order transition and then each module undergoes a second percolation transition. (ii) When there are more modules with many interconnections between them, the system undergoes a single transition. Overall, we find that modular structure can significantly influence the type of transitions observed in interdependent networks and should be considered in attempts to make interdependent networks more resilient.\
\
*Keywords:* modular networks, networks of networks, percolation, interdependent networks
author:
- 'Louis M. Shekhtman'
- Saray Shai
- Shlomo Havlin
bibliography:
- 'paper.bib'
title: Resilience of Networks Formed of Interdependent Modular Networks
---
Introduction
============
Many recent studies have explored interdependent and multilayer networks [@buldyrev-nature2010; @gao-naturephysics2012; @parshani-prl2010; @leichtdsouza2009; @cellai-pre2013; @brummitt-pnas2012; @PhysRevE.90.012803; @gao-prl2011; @hu-pre2011; @bashan-pre2011; @parshani-pnas2011; @bashan-jsp2011; @PhysRevE.85.066134; @vespignani-nature2010; @rinaldi-ieee2001; @peerenboom-proceedings2001; @radicchi-naturephysics2013; @son-epl2012; @zhao-jstatmech2013; @donges-epjb2011; @baxter-prl2012; @gomez-prl2013; @rosato-criticalinf2008; @shao2015local; @danziger-ndes2014; @boguna-PhysRevX; @yanqing-PhysRevX; @radicchi-PhysRevX; @danziger2015resistors; @stochastic2012; @amir-naturecomm2012]. Further studies have also explored more realistic structures such as spatially embedded networks [@bashan-naturephysics2013; @wei-prl2012; @danziger-jcomnets2014; @shekhtman-pre2014; @berezin2013spatially; @danziger2015two], and different types of realistic failures such as response under degree-based attacks [@dong-pre2013; @huang-pre2011] and localized attacks [@berezin2013spatially; @shao2015local]. Nevertheless, all previous studies on interdependent networks ignored the realistic effect of modularity on the resilience of interdependent networks.
Many real world networks have a modular structure including biological networks [@bullmore2012economy], infrastructure such as the power grid, internet [@eriksen2003modularity] and airport networks [@guimera2005worldwide], and financial networks [@garas2008structural]. Several studies have explored the robustness of individual modular systems (i.e. single networks) [@bagrow2011robustness; @babaei2011cascading] yet no study has considered the effect of interdependence in modular networks. It is well known that many of these systems are also interdependent and thus it is crucial to understand how the modular structure affects the resilience of interdependent networks.
There has also been considerable work on understanding various types of attacks on networks [@cohen-prl2001; @callaway-prl2000; @shao2015local; @berezin2013spatially; @gallos2005stability]. Recent work by Shai et al. [@shai2014resilience] developed an analytical method where the attack is carried out on interconnected nodes, i.e. nodes that connect communities. Further work by da Cunha et al. [@da2015complex] showed that in real networks, attacks on interconnected nodes are even more damaging than attacks based on betweenness. It is particularly important to consider the effectiveness of this type of attack in interdependent networks since the researchers in [@da2015complex] showed that the US power grid is among the most susceptible networks to this type of attack and it is well known that power grids are interdependent with other infrastructures.
In our model we assume modular networks composed of $m$ modules, and a fixed ratio, $\alpha$, between the probability for an intramodule link and intermodule link. We further fix the total average degree, $k_\text{tot}$, of the network. Using these three parameters we can determine the average intramodule degree, $k_\text{intra}$ and the average intermodule degree $k_\text{inter}$. We obtain $$\begin{aligned}
\frac{k_\text{intra}}{k_\text{inter}}&=\frac{\alpha}{m-1} \\
k_\text{tot}&=k_\text{intra}+k_\text{inter}.\end{aligned}$$
Note that $k_\text{inter}$ increases with $m$, since networks with more modules have more interlinks [@shai2014resilience]. We generate $n$ of these modular networks, and for simplicity we assume that each network has the same $m$, $\alpha$ and $k$. We then create dependency links between the nodes in the different networks. The fraction of nodes in network $i$ which depend on nodes in network $j$ is defined as $q_{ij}$. The dependency links are either bidirectional (in our analysis of treelike dependencies, Sec. III, Fig. \[fig:4mod-tree\]) or unidirectional (in our analysis of looplike dependencies, Sec. IV, Fig. \[fig:4mod-loop\]). In both cases we further restrict the dependency links such that a node in community $m_a$ in network $i$ will depend on a node in community $m_a$ in network $j$, i.e. the dependency links are within the same community. This restriction is particularly reasonable in our example of cities where each city has its own tightly connected, interdependent infrastructure with relatively few connectivity links between the cities.
We focus on the case of attack on the interconnected nodes, i.e. nodes with at least one connectivity link to a different module. Many studies have shown that these links are particularly susceptible to failure in biological networks and serve as efficient targets in attacking infrastructure [@da2015complex]. Further in our example of cities, these nodes have the longer distance links that are more likely to fail [@mcandrew2015robustness].
Our attack randomly removes a fraction $1-r$, of the interconnected nodes in the network until there are no remaining interconnected nodes and then continues to remove nodes randomly. The fraction of unremoved interconnected nodes, $r$, is related to the overall fraction of unremoved nodes, $p$, by $$r=\frac{p-\bar{p}_{inter}}{1-\bar{p}_{inter}}
\label{eq:r-p-general}$$ where $\bar{p}_{inter}$ is the probability that a node is *not* interconnected.
[0.33]{}
[0.33]{} {width="0.7\linewidth"}
[0.32]{} {width="1.0\linewidth"}
We study this model on both treelike networks of networks (NoNs) and looplike NoNs (Fig. \[fig:diagram-NoN\]) and find that there are two distinct regimes depending on the number of modules, $m$. In the first regime, for small $m$, the network first separates into isolated, interdependent, yet functional modules, i.e. there is a transition at $r=0$. However, for larger $m$ the network collapses due to the removal of a finite nonzero fraction of interconnected nodes. We provide an analytic solution that predicts the critical point, $m^*$, where the system has a transition from one behavior to the other, as well as solutions for the size of the giant component as a function of the fraction of removed nodes. We support our theory by simulations.
Failure and Attack on Interdependent Modular Networks
=====================================================
We now extend the method of Callaway et al. [@callaway-prl2000] to the case of interdependent networks or network of networks (NoN). We will then specifically apply this framework to the case of interdependent modular networks.
We begin by recalling the derivation in [@callaway-prl2000] for a single network $i$ with a given degree distribution $P_i(k)$ described by the generating function $$G(x)=\sum_{k=0}^{\infty}P_i(k)x^k.$$ For degree based attacks where nodes with degree $k$ are removed with probability $1-r_k$, the generating function is [@callaway-prl2000] $$F_0(x)=\sum_{k=0}^\infty r_kP_i(k)x^k.
\label{eq:F-single}$$ The generating function of the branching process is then $$F_1(x)=F_0'(x)/G'(x).$$ Following Callaway et al. [@callaway-prl2000] we obtain the probability that a randomly chosen edge leads to a cluster of given size, $H_1(x)$, as $$H_1(x)=1-F_1(x)+xF_1\left[H_1(x)\right]
\label{eq:cluster1}$$ and the probability that a randomly chosen node leads to a cluster of a given size, $H_0(x)$, $$H_0(x)=1-F_0(1)+xF_0\left[H_1(x)\right].
\label{eq:cluster2}$$ The fraction of nodes in the giant component is $$P_\infty(x)=1-H_0(1)=F_0(1)-F_0(u),
\label{eq:final-single}$$ with $u$ being the smallest non-negative real solution of the self consistency equation $$u=1-F_1(1)+F_1(u).
\label{eq:u}$$
In order to generalize this framework for the case of attack on a NoN we must include the fact that nodes have an additional uniform likelihood to fail, $1-p_\text{dep}$, where $p_\text{dep}$ comes from the effect of the dependency links and depends on the specific topology of the NoN.
In such a case we have that $$\tilde{F}_0(x)=p_\text{dep}\sum_{k=0}r_kP_i(k)x^k
\label{eq:NoN-gen}$$ where $\tilde{F}_0(x)$ is the anologue of Eq. (\[eq:F-single\]) for interdependent networks and $p_\text{dep}$ can be extracted from the sum since it does not depend on the degree of the node. The rest of the derivation continues the same for Eqs. (\[eq:F-single\])-(\[eq:final-single\]). We note that Eq. (\[eq:u\]) is the same as before only now with the extra factor of $p_\text{dep}$, giving $$1=p_\text{dep}\frac{F_1(u)-F_1(1)}{u-1},
\label{eq:NoN-final}$$ and Eq. (\[eq:final-single\]) also will have the same factor $p_\text{dep}$. This pair of equations, Eqs. (\[eq:final-single\]) and (\[eq:NoN-final\]), can be used to find the giant component of a NoN under any sort of degree based attack and can be used as an alternative method to that of Huang et al. [@huang-pre2011] and Dong et al. [@dong-pre2013].
We will now generalize this framework to the case where each network in the NoN is a modular network with the parameters $m$, $\alpha$, and $k$, defined above. For the case of a modular network where the modules are made up of Erdős-Rényi structures, Eq. (\[eq:r-p-general\]) becomes $$r=\frac{p-e^{-k_\text{inter}}}{1-e^{-k_\text{inter}}}.
\label{eq:r-p}$$ We recall the results from Shai et al. [@shai2014resilience] for the generating functions $$\begin{aligned}
G(x)&=e^{-(k_{intra}+k_{inter})(x-1)} \\
F_0(x)&=e^{k_{intra}(x-1)-k_{inter}}(1-r)+rG(x) \\
F_1(x)&=F'_0(x)/G'(x).\end{aligned}$$ We now combine the results from Shai et al. [@shai2014resilience] for a single network with Eqs. (\[eq:final-single\]) and (\[eq:NoN-final\]) for a NoN. We note that these equations not only describe the case of an interdependent modular NoN but also the case where we have both random failure with probability $1-p_\text{rand}$ and targeted attack. For our purposes we will assume that $p_\text{rand}=p_\text{dep}$, i.e. the random damage is solely the result of the dependencies. Our new generating functions are thus, $$\begin{aligned}
& \tilde{F}_0(x) = p_\text{dep} F_0(x) \nonumber \\
& \tilde{F}_{1}(x) = p_\text{dep} F_{1}(x).\end{aligned}$$ The average connected component size is given by Leicht and D’Souza [@leichtdsouza2009], $$\langle s \rangle =p_\text{dep} F_0(1)+p_\text{dep}rk_\text{intra}F_0(1)j_0+p_\text{dep}rk_\text{inter} j_1$$ where $$\begin{aligned}
j_0&=p_\text{dep}F_0(1)+p_\text{dep}rk_\text{intra}F_0(1)j_0+p_\text{dep}rk_\text{inter}j_1 \\
j_1&=p_\text{dep}r+p_\text{dep}rk_\text{intra}j_0+p_\text{dep}rk_\text{inter}j_1.\end{aligned}$$
Combining the above equations, we find that there is a giant component that spans the system when
$$\begin{aligned}
& \big( 1 - p_\text{rand}k_\text{intra} (e^{-k_\text{inter}} + r_c- r_c e^{-k_\text{inter}} ) \big) (1 - p_\text{rand} r_c k_\text{inter}) - p_\text{rand}^2 r_c^2 k_\text{intra} k_\text{inter} = 0 \nonumber \\
& \Rightarrow r_c^2 \big( p_\text{rand}^2 k_\text{intra} k_\text{inter} e^{-k_\text{inter}} \big) + r_c \big( p_\text{rand} k_\text{inter} + p_\text{rand} k_\text{intra} - p_\text{rand} k_\text{intra} e^{-k_\text{inter}} - p_\text{rand}^2 k_\text{intra} k_\text{inter} e^{-k_\text{inter}} \big) + \big( p_\text{rand} k_\text{intra} e^{-k_\text{inter}} -1 \big) = 0
\label{eq:r-c}\end{aligned}$$
which can be solved using the quadratic formula for $r_c$.
Treelike network of interdependent modular networks
====================================================
We now consider the case of a treelike NoN (see Fig. \[fig:4mod-tree\]) with full dependency and no-feedback, i.e. if network $i$ depends on network $j$ then each node in $i$ depends on a single node in $j$ and vice versa. As stated earlier, this can serve as a good model of infrastructure across cities. We remove a fraction $1-r$ of the interconnected nodes from one of the networks and aim to obtain the mutual giant connected component. The fraction of nodes which fail due to the effect of the dependencies is given by $$p_\text{dep}=(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})^{n-1}.
\label{eq:dep-trees}$$ This can be understood by noting that each node is a part of a $n$-tuple of interdependent nodes and we require that all these $n$ nodes be in their networks respective giant components. Therefore, besides the node itself, there are $n-1$ dependent nodes that must all be in their networks’ respective giant components after the initial attack.
If we combine Eqs. (\[eq:NoN-gen\]), (\[eq:NoN-final\]), and (\[eq:dep-trees\]) we obtain the mutual giant connected component in the NoN,
$$P_\infty=
\begin{cases}
\Big ( e^{-k_\text{inter}}(1-r)(1-e^{-k_\text{intra}P_\infty})+r(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})\Big ) (1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})^{n-1} & 0<r<1 \\
\tfrac{p}{m}(1-e^{-mk_\text{intra}P_\infty})^{n} & r<0.
\end{cases}
\label{eq:mod_intra-dep}$$
Since the dependency links are within modules, Eq. (\[eq:mod\_intra-dep\]) for $r<0$ is simply the equation for a treelike NoN with $k=k_\text{intra}$ and a giant component that is smaller by a factor of $m$. Nonetheless the value of $p_c$ will be the same as for a regular treelike NoN with $k=k_\text{intra}$. We show analytic solutions of these equations along with simulations in Fig. \[fig:modular-intra-dep-all\] for varying $m$ and varying $n$. By varying either of these parameters we may have either one or two abrupt percolation transitions. Note that for the case of varying $n$ the point of the first abrupt drop is the same for all curves. This is because the point where the modules separate depends only on $m$, $\alpha$ and $k_\text{tot}$, but not on $n$ (see Fig. \[fig:pic\]). It is worth noting though that for larger $n$, the second regime, where modules exist independently (see Fig. \[fig:after-separation\]), may not be present if the individual modules do not have enough intralinks to survive the damage from the dependencies.
We demonstrate that in the case where there are two abrupt transitions, the first transition (at higher $p$) represents the separation of the modules and we therefore have $m$ individual modules functioning separately. This is indicated by the spike in the size of the second largest component, $P_{\infty2}$, in Fig. \[fig:gcc2\] and is demonstrated visually in Fig. \[fig:pic\]. In general for a network composed of $m$ modules, there are $m$ equal components functioning independently. After the separation into modules, we observe a second abrupt transition due to the interdependence.
As $m$ increases, the network begins to act more like a regular network of $n$ interdependent Erdős-Rényi networks. It is clear from the framework developed until now that there exists a critical value of $m$, $m^*$ above which the system undergoes a single transition whereas below $m^*$ the system first separates into separate modules before collapsing entirely.
[0.49]{} {width="1.0\linewidth"}
[0.49]{} {width="1.0\linewidth"}
[0.49]{} {width="1.0\linewidth"}
[0.49]{} {width="1.0\linewidth"}
[0.45]{}
[0.45]{} {width="0.9\linewidth"}
We will now determine $p_c$ and $m^*$ for this system using the framework developed in Chap. II. For certain regimes there exist two first-order, discontinuous jumps in the size of the giant component (see Fig. \[fig:modular-intra-dep\]). We define $p_c$ as the point where the first (higher $p$) jump occurs, regardless of whether there is a second jump afterwards. If desired, the point of the second jump can be found by solving for the critical threshold for a typical Erdős-Rényi NoN with $k=k_\text{intra}$. We will now solve for $p_c$ by first finding $r_c$ using Eq. (\[eq:r-c\]) and then converting it to $p_c$ using Eq. (\[eq:r-p\]). We recall that for the case of trees we use $p_\text{dep}=(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})^{n-1}$ in Eq. (\[eq:r-c\]) as was done in Eq. (\[eq:mod\_intra-dep\]). This gives us the following system of equations
$$\begin{aligned}
p_\text{dep}&=(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})^{n-1} \\
P_\infty&= \Big ( e^{-k_\text{inter}}(1-r_c)(1-e^{-k_\text{intra}P_\infty})+r_c(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})\Big ) (1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})^{n-1} \\
r_c&=\frac{-b+\sqrt{b^2-4ac}}{2a}
\label{eq:pc-tree-mod}\end{aligned}$$
with the values of $a$, $b$ and $c$ being the coefficients from Eq. (\[eq:r-c\]).
We see in Fig. \[fig:pc1\] that the simulations fit the theory well except near the point where we switch to having a single abrupt transition. This small discrepancy is because there are large fluctuations in the fraction of individual modules that are part of the giant component, which are not considered in the mean-field approach. Once we move away from this critical point the theory again fits the simulations.
In order to find $m^*$ we compare the value of $p$ when $r=0$ to the $p_c$ for a treelike network of networks with $k=k_\text{intra}$. Essentially this means comparing the fraction of nodes needed to remove all interconnected nodes and the value of $p_c$ where the modules themselves break apart.
We can trivially find the value of $p$ when $r=0$ by noting that $r=\tfrac{p-e^{-k_\text{inter}}}{1-e^{-k_\text{inter}}}$. We obtain in terms of $m$ $$p(r=0)=e^{-k_\text{tot}\tfrac{m-1}{\alpha+m-1}}.
\label{eq:pc_r}$$ We note that the value of $p_c$ for a treelike network of networks (without communities) was determined by Gao et al. [@gao-prl2011]. In order to find $p_c$ we first find the value of $\bar{k}_{min}$, the degree at which the system undergoes a transition. This is done using the system of equations $$\begin{aligned}
f_c=-[nW(-\frac{1}{n}e^{-1/n}]^{-1} \\
\bar{k}_{min}=[nf_c(1-f_c)^{(n-1)}]^{-1}\end{aligned}$$ where $W(x)$ is the Lambert function.
Then $p_c=\bar{k}_{min}/<k>$ where $<k>$ is the average degree of each network, in our case $<k>=k_\text{intra}$. We obtain $$p_c=\tfrac{\bar{k}_{min}(\alpha+m-1)}{k_\text{tot}\alpha}.
\label{eq:pc1}$$
The value of $m^*$ is found by setting Eq. (\[eq:pc\_r\]) equal to Eq. (\[eq:pc1\]) to obtain $$\tfrac{\bar{k}_{min}(\alpha+m^*-1)}{k_\text{tot}\alpha}=e^{-k_\text{tot} \tfrac{m^*-1}{\alpha+m^*-1}}.
\label{eq:m*-trees}$$ This equation can be solved numerically for any value of $n$ and $\alpha$. Essentially, there are two competing effects in this equation. The left side represents the value of $p_c$ for an interdependent network of networks with $k=k_\text{intra}$ and the right side represents the point where we have removed all nodes with an interlink. As $m$ increases, $k_\text{intra}$ decreases, meaning that the modules become more vulnerable and $p_c$ from the left side of the equation increases. On the other hand, as $m$ increases, $k_\text{inter}$ increases so more nodes have an interlink and $p_c$ on the right side of the equation decreases (more resilient). We demonstrate these two effects and show the point where they intersect in Fig. \[fig:m\*\_det\]. Further we show the value of $m^*$ for various values of $\alpha$ and $n$ in Fig. \[fig:m\*\_vals\]. Also, as $n$ increases, $\bar{k}_{min}$ increases, meaning the affect of the interdependence is stronger. Essentially it causes the dashed curve in Fig. \[fig:m\*\_det\] to increase, thus decreasing $m*$, the point where the curves intersect. Thus for larger $n$ we have a lower $m^*$, as seen in Fig. \[fig:m\*\_vals\].
[0.45]{} {width="1.0\linewidth"}
[0.45]{} {width="1.0\linewidth"}
Loop of interdependent modular networks
========================================
We will now demonstrate our approach on another example where we consider the specific case of a directed loop of modular networks. In this case, for simplicity, we perform the initial attack on all of the networks rather than just one of them. Also, whereas the dependency links in Chap III. were bidirectional here we use unidirectional links as part of a loop (see Fig. \[fig:4mod-loop\]). The fraction of nodes which are interdependent is defined as $q$, i.e. $1-q$ is the fraction of autonomous nodes, and is assumed here, for simplicity, to be the same for all pairs of networks. Here again, we will only consider dependency links which are restricted to being between two networks but within the same module. It has been shown that for looplike NoNs the number of networks in the loop is not relevant to the calculation [@gao-naturephysics2012; @gao-general-net; @shekhtman-pre2014].
The equation governing this system is once again Eq. (\[eq:NoN-final\]) with $p_\text{dep}=(1-q+qP_\infty)$. Substituing this into Eq. (\[eq:NoN-final\]) gives
$$P_\infty=
\begin{cases}
\left(e^{-k_\text{inter}}(1-r)(1-e^{-k_\text{intra}P_\infty})+r(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})\right)(1-q+qP_\infty) & 0<r<1 \\
\frac{p}{m}(1-e^{-k_\text{intra}mP_\infty})(1-q+qmP_\infty) & r<0.
\end{cases}
\label{eq:loop-pinf}$$
We compare simulations and theory according to Eq. (\[eq:loop-pinf\]) in Fig. \[fig:RR-pinf\]. We note that here if the system undergoes two transitions, the first one (at a higher $p$) is abrupt and the second one is either continuous or abrupt. For the parameters we used, looplike networks of networks undergo a second order transition [@gao-prl2011; @gao-general-net], yet for different values of $k_\text{tot}$ and $q$ they can undergo an abrupt transition. For looplike NoNs, the number of networks, $n$, does not play a role but increasing $q$ the fraction of interdependent nodes, weakens the resilience of the system (see Fig. \[fig:RR-varyq\]). We further note that for looplike NoNs there is a maximum coupling, $q_\text{max}$ above which the entire network collapses even for $p\rightarrow 1$. Our value for $q_\text{max}$ will be the same as that of Gao et al. [@gao-general-net], namely $q_\text{max}=1-1/k_\text{tot}$ since for $p=1$ our networks respond to random percolation like ER networks. Indeed, in Fig. \[fig:RR-varyq\] we do not show higher values of $q$ since the system does not survive even for $p=1$ if $q>1-1/k_\text{tot}$. For example, with $k_\text{tot}=4$ (the value in our plots) we obtain a maximum coupling of $q=0.75$, above which the system collapses even for $p\rightarrow1$.
[0.49]{} {width="1.0\linewidth"}
[0.49]{} {width="1.0\linewidth"}
[0.45]{} {width="1.0\linewidth"}
[0.45]{} {width="1.0\linewidth"}
Following the analysis in Chap. II we can find $p_c$ using Eq. (\[eq:r-c\]). In this case $p_\text{dep}=(1-q+qP_\infty)$, which we will substitute into Eq. (\[eq:r-c\]). Our system corresponding to Eq. (\[eq:pc-tree-mod\]) becomes,
$$\begin{aligned}
p_\text{dep}&=(1-q+qP_\infty) \\
P_\infty&=\left(e^{-k_\text{inter}}(1-r)(1-e^{-k_\text{intra}P_\infty})+r(1-e^{-(k_\text{intra}+k_\text{inter})P_\infty})\right)\\
r_c&=\frac{-b+\sqrt{b^2-4ac}}{2a}
\label{eq:pc-loop}\end{aligned}$$
with the values of $a$, $b$ and $c$ being the coefficients from Eq. (\[eq:r-c\]). We plot the $p_c$ obtained based on the theory and compare with simulations in Fig. \[fig:RR-pc\].
To find $m^*$ we use the same method as was done in Chap. III for treelike networks of networks. Specifically we compare the $p_c$ at which the modules become separated and the $p_c$ for a network of networks with $k=k_\text{intra}$. We have from Gao et al. [@gao-general-net] that $p_c$ for a looplike NoN is $p_c=\frac{1}{k(1-q)}$. If we solve this and use the representation with $\alpha$ and $m$ rather than $k_\text{intra}$, we obtain $$\frac{\alpha+m^*-1}{k_\text{tot}\alpha(1-q)}=e^{-\frac{k_\text{tot}(m^*-1)}{\alpha+m-1}}.
\label{eq:mstar-RR}$$ The values of $m^*$, as a function of $\alpha$, based on Eq. (\[eq:mstar-RR\]) can be seen in Fig. \[fig:m\*\_vals-RR\]. Increasing $q$ decreases the robustness of the individual modules and thus decreases the value of $m^*$ since the system is more likely to collapse before the modules are separated.
Discussion
==========
In summary, we have developed a framework for studying attacks on interdependent modular networks. Our results show that modular NoNs can behave significantly differently from random NoNs or spatially embedded NoNs in that they may undergo two separate percolation phase transitions. One transition occurs (at a higher $p$) where the modules become separated, and a second transition occurs when the individual modules collapse. For the case of a fully interdependent treelike NoN with bidirectional dependency links, both of these transitions are first order, whereas for a looplike NoN with unidirectional dependency links the first transition (higher $p$) is abrupt while the second is either abrupt or continuous depending on the parameters. These results might be relevant for many interdependent systems such as financial networks, biological networks and are particularly relevant for models of city infrastructure where most of the interdependence presumably occurs within a single city (regarded here as a community) even though there are connections to other cities. Another reason our attack is realistic for city infrastructure is because the interconnected nodes contain the interconnected links that are longer and therefore more likely to fail [@mcandrew2015robustness].
We also note that the theoretical approach developed here can be used for many other types of targeted attack and for other network of network structures as well.
Acknowledgments {#acknowledgments .unnumbered}
===============
LS and SH acknowledge the LINC (no. 289447 funded by the EC’s Marie-Curie ITN program (FP7-PEOPLE-2011-ITN)) and MULTIPLEX (EU-FET project 317532) projects, the Deutsche Forschungsgemeinschaft (DFG), the Israel Science Foundation, ONR and DTRA for financial support. SS thanks the James S. McDonnell Foundation 21st Century Science Initiative - Complex Systems Scholar Award (grant 220020315) for financial support.
|
---
abstract: |
For $m \in {\mathbb N}$, let $S_m$ be the Suzuki curve defined over ${\mathbb{F}}_{2^{2m+1}}$. It is well-known that $S_m$ is supersingular, but the $p$-torsion group scheme of its Jacobian is not known. The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. In this paper, we compute a closed formula for the $a$-number of $S_m$ using the action of the Cartier operator on ${H^0(S_m,\Omega^1)}$.\
Keywords: Suzuki curve, maximal curve, Jacobian, p-torsion, a-number.\
MSC: 11G20, 14G50, 14H40.
author:
- Holley Friedlander
- Derek Garton
- Beth Malmskog
- Rachel Pries
- Colin Weir
bibliography:
- 'summergradSuzuki.bib'
title: 'The $a$-numbers of Jacobians of Suzuki curves'
---
Introduction {#Introduction}
============
Let $m\in {\mathbb N}$, $q=2^{2m+1}$, and $q_0=2^m$. The *Suzuki curve* $\mathcal{S}_m\subset\mathbb{P}^2$ is defined over ${\mathbb{F}}_q$ by the homogeneous equation: $$W^{q_0}(Z^q + ZW^{q-1}) = Y^{q_0}(Y^q + YW^{q-1}).$$ This curve is smooth and irreducible with genus $g=q_0(q-1)$ and it has exactly one point at infinity [@HansenStich Proposition 1.1]. The number of points on the Suzuki curve over ${\mathbb{F}}_q$ is $\#S_m\left({\mathbb{F}}_q\right)=q^2+1$; this number is optimal in that it reaches Serre’s improvement to the Hasse-Weil bound [@HansenStich Proposition 2.1].
In fact, $S_m$ is the unique $\mathbb{F}_q$-optimal curve of genus $g$ [@FTunique]. This shows that $S_m$ is the Deligne-Lusztig variety of dimension $1$ associated with the group $Sz(q)= {}^2B_2(q)$ [@H92 Proposition 4.3]. The curve $S_m$ has the Suzuki group $Sz(q)$ as its automorphism group; the order of $Sz(q)$ is $q^2(q-1)(q^2+1)$ which is very large compared with $g$. Because of the large number of rational points relative to their genus, the Suzuki curves provide good examples of Goppa codes [@GK08 Section 4.3], [@GKT], [@HansenStich].
The $L$-polynomial of $S_m$ is $(1+\sqrt{2q} t +q t^2)^g$ [@H92 Proposition 4.3]. It follows that $S_m$ is supersingular for each $m \in {\mathbb N}$. This fact implies that the Jacobian ${\rm Jac}(S_m)$ is isogenous to a product of supersingular elliptic curves and that ${\rm Jac}(S_m)$ has no $2$-torsion points over $\overline{{\mathbb{F}}}_2$. However, there are still open questions about ${\rm Jac}(S_m)$. In this paper, we address one of these by computing a closed formula for the $a$-number of ${\rm Jac}(S_m)$.
The $a$-number is an invariant of the $2$-torsion group scheme ${\rm Jac}(S_m)[2]$. Specifically, if $\alpha_{2}$ denotes the kernel of Frobenius on the additive group $\mathbb{G}_{a}$, then the $a$-number of $S_m$ is $a(m) = {\rm dim}_{\overline{{\mathbb{F}}}_2}\text{Hom}(\alpha_{2},\text{Jac}(S_m)[2])$. It equals the dimension of the intersection of ${\rm Ker}(F)$ and ${\rm Ker}(V)$ on the Dieudonné module of ${\rm Jac}(S_m)[2]$. Having a supersingular Newton polygon places constraints upon the $a$-number but does not determine it. The $a$-number also gives partial information about the decomposition of ${\rm Jac}(S_m)$ into indecomposable principally polarized abelian varieties, Lemma \[decompose\], and about the Ekedahl-Oort type of ${\rm Jac}(S_m)[2]$, see Section \[Corollaries\].
In Section \[Derek\], we prove that the $a$-number of $S_m$ is $a(m)=q_0(q_0+1)(2q_0+1)/6$, see Theorem \[maintheorem\]. The proof uses the action of the Cartier operator on ${H^0(S_m,\Omega^1)}$ as computed in Section \[Colin\].
Author Pries was partially supported by NSF grant DMS-11-01712. We would like to thank the NSF for sponsoring the research workshop for graduate students at Colorado State University in June 2011 where the work on this project was initiated. We would like to thank Amy Ksir and the other workshop participants for their insights.
The $a$-number {#preliminaries}
==============
Suppose $A$ is a principally polarized abelian variety of dimension $g$ defined over an algebraically closed field $k$ of characteristic $p > 0$. For example, $A$ could be the Jacobian of a $k$-curve of genus $g$. Consider the multiplication-by-$p$ morphism $[p]:A \to A$ which is a finite flat morphism of degree $p^{2g}$. It factors as $[p]=V \circ F$. Here, $F:A \to A^{(p)}$ is the relative Frobenius morphism coming from the $p$-power map on the structure sheaf; it is purely inseparable of degree $p^g$. The Verschiebung morphism $V:A^{(p)} \to A$ is the dual of $F$.
The kernel of $[p]$ is $A[p]$, the $p$-torsion of $A$, which is a quasi-polarized $BT_1$ group scheme. In other words, it is a quasi-polarized finite commutative group scheme annihilated by $p$, again having morphisms $F$ and $V$. The rank of $A[p]$ is $p^{2g}$. These group schemes were classified independently by Kraft (unpublished) [@Kraft] and by Oort [@O:strat]. A complete description of this topic can be found in [@M:group] or [@O:strat].
Two invariants of (the $p$-torsion of) an abelian variety are the $p$-rank and $a$-number. The *$p$-rank* of $A$ is $r(A)=\dim_{\mathbb{F}_p}\left(\operatorname{Hom}\left(\mu_p,A[p]\right)\right)$, where $\mu_p$ is the kernel of Frobenius on the multiplicative group $\mathbb{G}_m$. Then $p^{r(A)}$ is the cardinality of $A[p]\left(\overline{\mathbb{F}}_p \right)$. The *$a$-number* of $A$ is $a(A)=\dim_k\left(\operatorname{Hom}\left(\alpha_p, A[p]\right)\right)$, where $\alpha_p$ is the kernel of Frobenius on the additive group $\mathbb{G}_a$. It is well-known that $1 \leq a(A) +r(A) \leq g$. Another definition for the $a$-number is $$a(A)={\rm dim}_{{\mathbb F}_p}({\rm Ker}(F) \cap {\rm Ker}(V)).$$
If $X$ is a (smooth, projective, connected) $k$-curve, then the $a$-number of $A={\rm Jac}(X)$ equals the dimension of the kernel of the Cartier operator $\operatorname{\mathcal{C}}$ on $H^0(X,\Omega^1)$ [@LO 5.2.8]. The reason for this is that the action of $\operatorname{\mathcal{C}}$ on $H^0(X,\Omega^1)$ is the same as the action of $V$ on $V{\rm Jac}(X)[p]$. This is the property that we use to calculate the $a$-number $a(m)$ of the Jacobian of the Suzuki curve $S_m$.
Regular 1-forms for the Suzuki curves {#Colin}
=====================================
In this section, we compute the action of the Cartier operator on the vector space of regular 1-forms for the Suzuki curves.
Geometry of the Suzuki curves
-----------------------------
Let $m\in {\mathbb N}$, $q=2^{2m+1}$, and $q_0=2^m$. Consider the Suzuki curve $\mathcal{S}_m\subset\mathbb{P}^2$ defined over ${\mathbb{F}}_q$ by the homogeneous equation: $$W^{q_0}(Z^q + ZW^{q-1}) = Y^{q_0}(Y^q + YW^{q-1}).$$
The curve $S_m$ is smooth and irreducible and has one point $P_\infty$ at infinity (when $W=Y=0$ and $Z=1$). Consider the irreducible affine model of $S_m$ defined by the equation $$\label{Suzuki}
z^q + z = y^{q_0}(y^q + y)$$ where $y:=Y/W$ and $z := Z/W$.
The following result is well-known, see e.g., [@HansenStich Proposition 1.1]. We include an alternative proof that illustrates the geometry of some of the quotient curves of $S_m$ and an important point about the $a$-number.
\[Lgenus\] The curve $S_m$ has genus $g=q_0(q-1)$.
The set ${\mathbb{F}}_q^*=\left\{\mu_1, \ldots, \mu_{q-1}\right\}$ can be viewed as a set of representatives for the $q-1$ cosets of ${\mathbb F}^*_2$ in ${\mathbb F}^*_q$. The Suzuki curve has affine equation $z^q-z=f(y)$ where $f(y)=y^{q_0+q}+ y^{q_0+1} \in {\mathbb{F}}_2(y)$. For $1 \leq i \leq q-1$, let $Z_{i}$ be the Artin-Schreier curve with equation $z_i^2-z_i = \mu_i f(y)$. As seen in [@GS91 Proposition 1.2], the set $\{Z_i \to {\mathbb P}^1_y \ \mid \ 1 \leq i \leq q-1\}$ is exactly the set of degree $2$ covers $Z \to {\mathbb P}^1_y$ which are quotients of $S_m \to {\mathbb P}^1_y$. By [@GP05 Proposition 3], an application of [@KR89 Theorem C], there is an isogeny $$\operatorname{Jac}{(S_m)} \sim \oplus_{i=1}^{q-1}{\operatorname{Jac}{(Z_{i})}}.$$
By Artin-Schreier theory, $\mu_i f(y)$ can be modified by any polynomial of the form $T^2- T$ for $T \in {\overline {\mathbb{F}}}_2[y]$ without changing the ${\overline {\mathbb{F}}}_2$-isomorphism class of the Artin-Schreier cover $Z_i \to {\mathbb P}^1_y$. Thus $Z_i$ is isomorphic to an Artin-Schreier curve with equation $z_i^2- z_i=h_i(y)$ for some $h_i(y) \in \overline{{\mathbb{F}}}_2[y]$ with degree $2q_0+1={\rm max}\{(q_0+q)/q_0, q_0+1\}$. For $1 \leq i \leq q-1$, the curve $Z_i$ is a $\mathbb{Z}/2$-cover of the projective line branched only at $\infty$, where it is totally ramified. Moreover, the break in the filtration of higher ramification groups in the lower numbering is at index ${\rm deg}(h_i(y))=2q_0+1$. By [@Stich VI.4.1], the genus of $Z_i$ is $q_0$. Thus $g={\rm dim}(\operatorname{Jac}(S_m))=(q-1){\rm dim}(\operatorname{Jac}(Z_i))=q_0(q-1)$.
Consider the Artin-Schreier curve $Z_i: z_i^2- z_i = h_i(y)$ from the proof of Lemma \[Lgenus\]. By [@elkinpries Proposition 3.4], since ${\rm deg}(h_i)=2q_0+1 \equiv 1 \bmod 4$, the $a$-number of $Z_i$ is $q_0/2$. Thus the $a$-number of $\oplus_{i=1}^{q-1}{\rm Jac}(Z_i)$ is $q_0(q-1)/2$, exactly half of the genus of $S_m$. The fact that ${\rm Jac}(S_m)$ is isogenous to $\oplus_{i=1}^{q-1}{\rm Jac}(Z_i)$ gives little information about the $a$-number of $S_m$ since the $a$-number is not an isogeny invariant.
The Hasse-Weil bound states that a (smooth, projective, connected) curve $X$ of genus $g$ defined over $\mathbb{F}_q$ must satisfy $$q+1-2g\sqrt{q}\leq\#X(\mathbb{F}_q)\leq q+1+2g\sqrt{q}.$$ A curve that meets the upper bound is called an $\mathbb{F}_q$-maximal curve.
It is easy to check that the number of ${\mathbb{F}}_{q^2}$-points on the Suzuki curve is $\#S_m\left({\mathbb{F}}_{q^2}\right)=q^2+1$ and so $S_m$ is not maximal over ${\mathbb{F}}_{q^2}$. Analyzing powers of the eigenvalues of Frobenius shows the following.
The Suzuki curve $S_m$ is ${\mathbb{F}}_{q^4}$-maximal.
The $L$-polynomial of $S_m$ is $L(S_m, t)=(1+\sqrt{2q} t +q t^2)^g$ [@H92 Proposition 4.3]. This factors as $L(S_m,t)=(1 - \alpha t)^g (1 - \overline{\alpha} t)^g$ where $\alpha = −q_0 (1 + i)$. That implies that $\#S_m ({\mathbb{F}}_{q^4} ) = q^4 + 1 - (-q_0)^4 (\alpha^4 + \overline{\alpha}^4)g = q^4 + 1 + 2q^2g$ which shows that $S_m$ is ${\mathbb{F}}_{q^4}$-maximal.
A curve which is maximal over a finite field is supersingular, in that the slopes of the Newton polygon of its $L$-polynomial all equal $1/2$. Thus $S_m$ is supersingular. The supersingularity condition is equivalent to the condition that $\operatorname{Jac}(S_m)$ is isogenous to a product of supersingular elliptic curves. A supersingular curve in characteristic $2$ has $2$-rank $0$. This implies, a priori, that the $a$-number of $S_m$ is at least one.
Regular $1$-forms
-----------------
To compute a basis for the vector space $H^0(S_m, \Omega^1)$ of regular $1$-forms on $S_m$, consider the functions $h_1,h_2\in{\mathbb{F}}\left(S_m\right)$ given by: $$\begin{aligned}
h_1:&=z^{2q_0} + y^{2q_0+1}, \\
h_2:&=z^{2q_0}y+h_1^{2q_0}.\end{aligned}$$
For any $f\in{\mathbb{F}}(S_m)$, let $v_\infty(f)$ denote the valuation of $f$ at $P_\infty$.
\[poleorder\] The functions $y,z,h_1,h_2 \in {\mathbb{F}}(S_m)$ have no poles except at $P_\infty$ where
$$\begin{aligned}
v_y:&=-v_\infty(y) =q, & v_z:&=-v_\infty(z)=q+q_0, \\
v_{h_1}:&=-v_\infty(h_1) =q+2q_0, & v_{h_2}:&=-v_\infty(h_2)=q+2q_0+1. \\\end{aligned}$$
The function $\pi = h_1/h_2$ is a uniformizer at $P_\infty$.
See [@HansenStich Proposition 1.3].
The function $y$ is a separating variable so $dy$ is a basis of the $1$-dimensional vector space of differential $1$-forms. The next lemma shows that $dy$ is regular.
\[dy\] The differential $1$-form $dy$ satisfies $$v_\infty(dy) = 2g-2 \quad \mbox{and} \quad v_P(dy) = 0$$ for all points $P \in S_m(\overline{{\mathbb{F}}}_q)$.
Recall that $\pi$ is a uniformizer at $P_\infty$. To take the valuation of $dy$ at $P_\infty$, we first rewrite $dy=f(x,y) d\pi$ for some $f(y,z) \in {\mathbb{F}}_q(y,z)$. Note that $$\begin{aligned}
d\pi &= d\left(\frac{h_1}{h_2}\right)
= \frac{h_2\,dh_1 - h_1 \,dh_2}{h_2^2}
= \frac{h_2\,y^{2q_0} - h_1 \,z^{2q_0}}{h_2^2} dy.\end{aligned}$$ Since $v_\infty(h_2^2) = -2(q+2q_0+1)$ and $$\begin{aligned}
v_\infty(h_2\,y^{2q_0} - v_{h_1} \,z^{2q_0}) &= \min \{ -2q_0v_y - v_{h_2}, -2q_0v_z - v_{h_1}\}\\
&= -2q_0v_z - v_{h_1} \\
&= -4q_0^3 - 2q_0,\end{aligned}$$ we see that $$\begin{aligned}
v_\infty(dy) &= v_\infty \left(\frac{h_2^2}{h_2^{2q_0} - h_1 z^{2q_0}} d\pi \right) \\
&= -2q + 2q_0 -2 - \left( -4q_0^3 - 2q_0 \right) \\
&= 4q_o^3 -2q_0 -2 \\
&= 2g-2.\end{aligned}$$ We next show that $dy$ has no zero or pole at any affine point of $S_m$. Note that, for any $a \in \overline{{\mathbb{F}}}_q$, the polynomial $z^q + z + a$ splits into distinct factors in $\overline{{\mathbb{F}}}_q(z)$, so there are exactly $q$ points of $S_m(\overline{{\mathbb{F}}}_q)$ lying over any $y_0 \in \mathbb{A}^1_y(\overline{{\mathbb{F}}}_q)$. Since $$\left[\overline{{\mathbb{F}}}_q(y,z):\overline{{\mathbb{F}}}_q(y)\right]=q,$$ the $\overline{{\mathbb{F}}}_q$-Galois cover $S_m \to {\mathbb P}^1_y$ is unramified at all affine points of $S_m$. Consequently, for any point $P\in S_m(\overline{{\mathbb{F}}}_q)$ lying over $a\in\mathbb{A}^1_y(\overline{{\mathbb{F}}}_q)$, we see that $v_P(y-a) = 1$. Thus, $y-a$ is a uniformizer at $P$ and $v_P(dy)=0$, proving the proposition.
By Lemma \[dy\], finding a basis for $H^0(S_m, \Omega^1)$ is equivalent to finding a basis for $L((dy))$, since $(dy) = (2g-2) P_\infty$ is the canonical divisor. To do this, we make use of the relations: $$\label{reduction}
z^2 = yh_1 +h_2, \quad h_1^{q_0} = z + y^{q_0+1}, \quad h_2^{q_0} = h_1 + zy^{q_0},$$ which can be verified by direct substitution, and the following proposition.
[@HansenStich Proposition 1.5] \[semithm\] Let $SG$ be the semigroup $\left\langle q,q+q_0,q+2q_0,q+2q_0+1 \right\rangle$. Then $\#\left\{n\in SG\mid 0\leq n\leq 2g-2\right\}=g$.
We now have all the required information to find a basis of $H^0(S_m, \Omega^1)$.
\[basis\] The following set is a basis of $H^0(S_m, \Omega^1)$: $$\mathcal{B}:=\left\{y^a z^b h_1^c h_2^d \, dy \, \mid \, (a,b,c,d) \in \mathcal{E} \right\}$$ where $\mathcal{E}$ is the set of $(a,b,c,d) \subset \mathbb{Z}^4$ satisfying $$\begin{array}{c}
0 \leq b \leq 1,\quad 0 \leq c \leq q_0-1,\quad 0 \leq d \leq q_0-1,\\
av_y+bv_z+cv_{h_1}+dv_{h_2} \leq 2g-2.
\end{array}$$
To prove linear independence, it suffices to prove that all elements in our basis have distinct valuations at $P_\infty$. Suppose that $y^a z^b h_1^c h_2^d \, dy \in \mathcal{B}$ and $y^{a'} z^{b'} h_1^{c'} h_2^{d'} \, dy\in\mathcal{B}$ have the same valuation at $P_\infty$; we will show they are equal. Comparing their valuations at $P_\infty$, we must have that $$\label{poleeq}
(a-a')v_y + (b-b')v_z + (c-c')v_{h_1} + (d-d')v_{h_2} =0$$
Now consider equation (\[poleeq\]) modulo $q_0$. As $q_0$ divides $v_y,v_z$ and $v_{h_1}$, $$(d-d') \equiv 0 \mod q_0.$$ As $0 \leq d,d' < q_0$, it must be the case that $d = d'$. Substituting $d-d'=0$ into equation (\[poleeq\]) and reducing modulo $2q_0$ yields that $$(b-b')q_0 \equiv 0 \mod 2q_0.$$ However, as $0 \leq b,b' \leq 1$, it must also be the case that $b = b'$. Simplifying (\[poleeq\]) and reducing modulo $q = 2q_o^2$ yields that $$(c-c')(q-2q_0) = (c-c')2q_0 \equiv 0 \mod 2q_0^2 .$$ Since $0 \leq c,c' \leq q_0-1$, we find that $c = c'$; so $a= a'$ as well.
We claim that the above set also spans $L\left((dy)\right)$. Clearly the valuations at $P_{\infty}$ of $$\begin{aligned}
\left\{y^az^bh_1^ch_2^c\mid(a-a')v_y + (b-b')v_z + (c-c')v_{h_1} + (d-d')v_{h_2} \leq 2g-2 \right\}\end{aligned}$$ are equal to $\left\{n\in SG\mid 0\leq n\leq 2g-2\right\}$, which is a set of size $g$ by Proposition \[semithm\]. Rewriting elements of the above set in terms of our basis will not change their valuation at $P_\infty$. Thus we can use the relations in equation (\[reduction\]) to see that $\mathcal{B}$ also contains an element for each of the $g$ possible valuations at $P_{\infty}$. By the previous paragraphs, each valuation occurs exactly once. By Riemann-Roch, $\ell\left((dy)\right)= g$, so $\mathcal{B}$ is a basis.
Action of the Cartier operator {#Holley}
------------------------------
In characteristic $2$, the Cartier operator $\operatorname{\mathcal{C}}$ acts on differential $1$-forms according to the following properties: (see e.g., [@TVN Section 2.2.5]).
1. $\operatorname{\mathcal{C}}$ is $1/2$-linear; i.e., $\operatorname{\mathcal{C}}$ is additive and $\operatorname{\mathcal{C}}{(f^2\omega)}=f\operatorname{\mathcal{C}}{(\omega)}$.
2. $\operatorname{\mathcal{C}}{(y^j\,dy)}=
\begin{cases}
0, & \textrm{if }j\not\equiv 1 \mod 2\\
y^{e-1}\,dy&\textrm{if }j=2e-1.
\end{cases}$
3. $\operatorname{\mathcal{C}}{(\omega)}=0$ if and only if $\omega$ is exact; i.e., if and only if $\omega=df$ for some $f \in {\mathbb{F}}_q\left(S_m\right)$.
4. $\operatorname{\mathcal{C}}{(\omega)}=\omega$ if and only if $\omega=df/f$ for some $f \in{\mathbb{F}}_q\left(S_m\right)$.
Any 1-form $\omega \in H^0(S_m, \Omega^1)$ can be written in the form $\omega=(f^2+g^2y) dy$, as $\operatorname{char}({\mathbb{F}}_q)=2$. Then $$\label{Ecartier}
\operatorname{\mathcal{C}}{((f^2+g^2y) dy)}=g \, dy.$$ By these properties, it is clear that $$\label{eq:evenc}
\operatorname{\mathcal{C}}(y^{2e_1+r_1}z^{2e_2+r_2}h_1^{2e_3+r_3}h_2^{2e_4+r_4}\,dy)=y^{e_1}z^{e_2}h_1^{e_3}h_2^{e_4}\operatorname{\mathcal{C}}(y^{r_1}z^{r_2}h_1^{r_3}h_2^{r_4} \,dy).$$
Hence to compute the action of $\operatorname{\mathcal{C}}$ on $H^0(S_m, \Omega^1)$, we need only compute $\operatorname{\mathcal{C}}$ on the 16 monomials in $y,z,h_1,h_2$ of degree less than or equal to one in each variable. The table below shows this action, where each $\operatorname{\mathcal{C}}(f\, dy)$ is written in terms of the original basis using the curve equation (\[Suzuki\]).
$$\displaystyle \begin{array}{|l|l|}
\hline
f&\operatorname{\mathcal{C}}{(f dy)}\\
\hline
\hline
1&0\\
\hline
y&dy\\
\hline
z&y^{q_0/2}\, dy\\
\hline
h_1&y^{q_0}\, dy\\
\hline
h_2&\left((yh_1)^{q_0/2}+h_2\right)\, dy\\
\hline
yz&h_1^{q_0/2}\,dy\\
\hline
yh_1&\left((yh_1)^{q_0/2}+h_2\right)\, dy\\
\hline
zh_1&(yh_2)^{q_0/2}\, dy\\
\hline
zh_2&(h_1h_2)^{q_0/2}\, dy\\
\hline
h_1h_2&\left(h_1+zy^{q_0}\right)\, dy\\
\hline
yzh_1&\left(y^{q_0/2}z+(h_1h_2)^{q_0/2}\right)\, dy\\
\hline
yzh_2&\left(zh_1^{q_0/2}+y^{q_0/2+1}h_2^{q_0/2}\right)\, dy\\
\hline
zh_1h_2&\left(zy^{q_0/2}h_2^{q_0/2}+h_1^{q_0/2+1}\right)\, dy\\
\hline
yh_1h_2&\left((yh_1)^{q_0/2}z+h_2^{q_0/2}z\right)\, dy\\
\hline
yzh_1h_2&\left(y^{q_0/2}h_2+zh_1^{q_0/2}h_2^{q_02}\right)\, dy\\
\hline
\end{array}$$
We illustrate the computation for $zh_1h_2\, dy$. Direct computation yields $$\begin{aligned}
\operatorname{\mathcal{C}}{(zh_1h_2\, dy)}&=\operatorname{\mathcal{C}}{\left(zh_1\left(yz^{2q_0}+h_1^{2q_0}\right)\, dy\right)}\\
&=z^{q_0}\operatorname{\mathcal{C}}{\left(zyh_1\, dy\right)}+h_1^{q_0}\operatorname{\mathcal{C}}{\left(zh_1\, dy\right)}\\
&=\left(y^{q_0/2}z^{q_0+1}+h_1^{q_0/2}h_2^{q_0/2}z^{q_0}+h^{q_0/2}h_1^{q_0/2}h_2^{q_0/2}\right)\, dy.\end{aligned}$$ To write this expression in terms of the original basis, we identify the monomials with the highest pole order at infinity. Since $$v_\infty(h_1^{q_0/2}h_2^{q_0/2}z^{q_0})=v_\infty(h^{q_0/2}h_1^{q_0/2}h_2^{q_0/2})=-4q_0^3-3q_0^2-q_0/2<-\left(2g-2\right),$$ these two terms may be simplified. Using Section \[Colin\], $$\begin{aligned}
h_1^{q_0/2}h_2^{q_0/2}z^{q_0}+&h^{q_0/2}h_1^{q_0/2}h_2^{q_0/2}\\
&=h_1^{q_0/2}h_2^{q_0/2}\left(y^{q_0/2}h_1^{q_0/2}+h_2^{q_0/2}\right)+y^{q_0/2}h_1^{q_0}h_2^{q_0/2}
=h_1^{q_0/2}h_2^{q_0}.\end{aligned}$$ The final expression follows by rewriting $z^{q_0+1}$ and $h_2^{q_0}$ in terms of lower order basis elements using equations (\[reduction\]).
To compute $\operatorname{\mathcal{C}}{(\omega)}$ for a general element $\omega\in\mathcal{B}$, simply apply equation (\[eq:evenc\]) and use the table above; in nearly all cases the direct result will again be in terms of the basis $\mathcal{B}$. The only exception is when $\omega=zh_1^{q_0-1}h_2\,dy$. In this case we have: $$\begin{aligned}
\operatorname{\mathcal{C}}{\left(h_1^{q_0-2}\cdot zh_1h_2\, dy\right)}&=h_1^{q_0/2-1}\operatorname{\mathcal{C}}{\left(zh_1h_2\right)}\\
&=\left(zy^{q_0/2}h_1^{q_0/2-1}h_2^{q_0/2}+h_1^{q_0}\right)\, dy.\end{aligned}$$ Using equations (\[reduction\]), one can obtain an expression in terms of the original basis.
The $a$-number for Suzuki curves {#Derek}
================================
A closed form formula for the $a$-number
----------------------------------------
We now have the tools to compute $a(m)$. The calculation amounts to counting lattice points in polytopes in $\mathbb{R}^3$, which is a hard problem in general. In our case, however, the values $v_y$, $v_{h_1}$, and $v_{h_2}$ are so similar that the polytopes in question are nearly regular; this makes our counting problem much easier.
\[maintheorem\] Let $a(m)$ and $g(m)$ be the $a$-number and genus of $S_m$ respectively. Then $$a(m)=\frac{q_0(q_0+1)(2q_0+1)}{6}.$$ In particular, $$\frac{1}{6}<\frac{a(m)}{g(m)}<\frac{1}{6}+\frac{1}{2^{m+1}}.$$
Recall from Section \[preliminaries\] that $a(m)$ is the dimension of the kernel of $\operatorname{\mathcal{C}}$ on ${H^0(S_m,\Omega^1)}$. By equation (\[Ecartier\]), $a(m)$ is the dimension of the vector space of regular differentials of the form $f^2\,dy$. Since $f^2$ can have a pole only at $P_{\infty}$, and since the order of the pole can be at most $2g-2$, we see that $a(m)=\ell((g-1)P_{\infty})$. Moreover, squaring is a homomorphism, so $$\ell((g-1)P_{\infty})=\#\left\{\omega\in\mathcal{B}\mid v_{\infty}(\omega)\leq g-1\right\}.$$ By Section \[Colin\], this number is exactly $$\#\left\{(a,b,c,d) \in \mathcal{E} \mid v_ya+v_zb+v_{h_1}c+v_{h_2}d\leq g-1
\right\};$$ here we use the notation $\mathcal{E}$ as we did in Proposition \[basis\]. Recall that $b \in \{0,1\}$. When $b = 0$, we must count $\{(a,c,d)\in\mathbb{N}^3\mid a+c+d\leq q_0-1\}$. This follows from the fact that that $$q_0-1=\frac{g-1}{v_{h_2}}<\frac{g-1}{v_{h_1}}<\frac{g-1}{v_y}<q_0.$$ For $b = 1$, we must count $\{(a,c,d)\in\mathbb{N}^3\mid a+c+d\leq q_0-2\}$ since $$q_0-2<\frac{g-1-v_z}{v_{h_2}}<\frac{g-1-v_z}{v_{h_1}}<\frac{g-1-v_z}{v_y}<q_0-1.$$ Using these two facts, we obtain $$\begin{aligned}
a(m)
=&\#\left\{(a,c,d)\in\mathbb{N}^3\mid a+c+d\leq q_0-1\right\} \\
& \quad + \#\left\{(a,c,d)\in\mathbb{N}^3\mid a+c+d\leq q_0-2\right\}\\
=&\sum_{i=2}^{q_0+1}{\binom{i}{2}}+\sum_{i=2}^{q_0}{\binom{i}{2}}\\
=&1+\sum_{i=2}^{q_0}{\left(\binom{i+1}{2}+\binom{i}{2}\right)}\\
=&\sum_{i=1}^{q_0}{i^2},\end{aligned}$$ as desired.
To prove the second statement, simply note that $$\frac{1}{6}<\frac{q_0(q_0+1)(2q_0+1)}{6q_0(q-1)}=\frac{1}{6}\cdot\frac{q_0^2+\frac{3}{2}q_0+\frac{1}{2}}{q_0^2-\frac{1}{2}}<\frac{1}{6}\left(1+\frac{3}{q_0}\right).$$
Open questions {#Corollaries}
--------------
Here are two open questions about $\operatorname{Jac}(S_m)$.
\[Q1\] What is the decomposition of $\operatorname{Jac}(S_m)$ into indecomposable principally polarized abelian varieties?
Theorem \[maintheorem\] gives partial information about Question \[Q1\], namely an upper bound on the number of factors appearing in the decomposition, because of the following fact.
\[decompose\] Suppose $A$ is a principally polarized abelian variety with $p$-rank $0$ and $a$-number $a$. If $A$ decomposes as the direct sum of $t$ principally polarized abelian varieties, then $t \leq a$.
Write $A \simeq \oplus_{i=1}^t A_i$ where each $A_i$ is a principally polarized abelian variety. For $1 \leq i \leq t$, consider the $p$-torsion group scheme $A_i[p]$. The $a$-number of $A_i[p]$ is at least $1$ since its $p$-rank is $0$. Thus the $a$-number of $A$ is at least $t$.
To state the second question, we need some more notation.
The Ekedahl-Oort type of a principally polarized abelian variety $A$ over $k$ is defined by the interaction between the Frobenius $F$ and Verschiebung $V$ operators on the $p$-torsion group scheme $A[p]$. It determines the isomorphism class of $A[p]$ and its invariants such as the $a$-number. To define the Ekedahl-Oort type, recall that the isomorphism class of a symmetric $BT_1$ group scheme ${\mathbb G}$ over $k$ can be encapsulated into combinatorial data. This topic can be found in [@O:strat]. If ${\mathbb G}$ has rank $p^{2g}$, then there is a *final filtration* $N_1 \subset N_2 \subset \cdots \subset N_{2g}$ of ${\mathbb G}$ as a $k$-vector space which is stable under the action of $V$ and $F^{-1}$ such that $i=\dim{(N_i)}$. The [*Ekedahl-Oort type*]{} of ${\mathbb G}$, also called the [*final type*]{}, is $\nu=[\nu_1, \ldots, \nu_r]$ where ${\nu_i}=\dim{(V(N_i))}$. The Ekedahl-Oort type of ${\mathbb G}$ is canonical, even if the final filtration is not.
There is a restriction $\nu_i \leq \nu_{i+1} \leq \nu_i +1$ on the final type. Moreover, all sequences satisfying this restriction occur. This implies that there are $2^g$ isomorphism classes of symmetric $BT_1$ group schemes of rank $p^{2g}$. The $p$-rank is $\max{\{i \mid \nu_i=i\}}$ and the $a$-number equals $g-\nu_g$.
\[Q2\] What is the Ekedahl-Oort type of $\operatorname{Jac}(S_m)[2]$? Equivalently, what is the covariant Dieudonné module of $\operatorname{Jac}(S_m)[2]$?
Theorem \[maintheorem\] gives partial information about Question \[Q2\], by limiting the possible final types. For the group scheme $\operatorname{Jac}{\left(S_m\right)}[2]$, the Ekedahl-Oort type satisfies that $\nu_1 = 0$ and $\nu_g = q_0(10q_0+7)(q_0-1)/6$. In particular, $\operatorname{Jac}(S_m)$ is not superspecial since $a(m) \not = g(m)$. This implies that $\operatorname{Jac}(S_m)$ is not isomorphic to the product of supersingular elliptic curves; it is only isogenous to the product of supersingular elliptic curves.
In the next example, we give some more information about the Ekedahl-Oort type of $\operatorname{Jac}(S_1)[2]$ (the case $m=1$).
If $m=1$ then $q_0=2$, $q=8$, and $g=14$. By Section \[Holley\], the image of $\operatorname{\mathcal{C}}$ on ${H^0(S_m,\Omega^1)}$ is spanned by the nine $1$-forms $$\{dy, ydy, h_1dy, y^2dy, yh_1dy, yh_2, (z+y^3)dy, h_1h_2dy, y^2zdy \}.$$ The image of $\operatorname{\mathcal{C}}^2$ on ${H^0(S_m,\Omega^1)}$ is spanned by the four $1$-forms $$\{dy, y^2dy, (z+y^3)dy, (h_1+y^2z)dy\}.$$ Also $\operatorname{\mathcal{C}}^3$ trivializes ${H^0(S_m,\Omega^1)}$. Thus $\nu_1=\nu_2=\nu_3=\nu_4=0$, and $\nu_9=4$, and $\nu_{14}=9$. The combinatorial restrictions on the final type imply that $\nu_{10}=5$, $\nu_{11}=6$, $\nu_{12}=7$, and $\nu_{13}=8$. This leaves only five possibilities for the final type, and thus for the isomorphism class of $\operatorname{Jac}(S_1)[2]$.
Holley Friedlander\
University of Massachusetts, Amherst\
Amherst, MA 01003\
holleyf@math.umass.edu
Derek Garton\
University of Wisconsin–Madison\
Madison, WI 53706\
garton@math.wisc.edu
Beth Malmskog\
Wesleyan University\
Middletown, CT 06457\
emalmskog@wesleyan.edu
Rachel Pries\
Colorado State University\
Fort Collins, CO 80521\
pries@math.colostate.edu
Colin Weir\
University of Calgary\
Calgary, AB, Canada\
cjweir@ucalgary.ca
|
---
abstract: 'We numerically explore how the subhalo mass-loss evolution is affected by the tidal coherences measured along different eigenvector directions. The mean virial-to-accretion mass ratios of the subhalos are used to quantify the severity of their mass-loss evolutions within the hosts, and the tidal coherence is expressed as an array of three numbers each of which quantifies the alignment between the tidal fields smoothed on the scales of $2$ and $30\,h^{-1}$Mpc in each direction of three principal axes. Using a Rockstar halo catalog retrieved from a N-body simulation, we investigate if and how the mass-loss evolutions of the subhalos hosted by distinct halos at fixed mass scale of \[$1$-$3$\]$10^{14}\,h^{-1}\,M_{\odot}$ are correlated with three components of the tidal coherence. The tides coherent along different eigenvector directions are found to have different effects on the subhalo mass-loss evolution, which cannot be ascribed to the differences in the densities and ellipticities of the local environments. It is shown that the substructures surrounded by the tides highly coherent along the first eigenvector direction and highly [*incoherent*]{} along the third eigenvector direction experience the least severe mass-loss evolution, while the tides highly [*incoherent*]{} only along the first eigenvector direction is responsible for the most severe mass-loss evolution of the subhalos. Explaining that the coherent tides have an obstructing effect on the satellite infalls onto their hosts and that the strength of the obstruction effect depends on which directions the tides are coherent or [*incoherent*]{} along, we suggest that the multidimensional dependence of the substructure evolution on the tidal coherence should be deeply related to the complex nature of the large-scale assembly bias.'
author:
- Jounghun Lee
title: A Multidimensional Dependence of the Substructure Evolution on the Tidal Coherence
---
Introduction {#sec:intro}
============
The classical excursion set theory based on the standard $\Lambda$CDM (cosmological constant $\Lambda$ and cold dark matter) model provided an analytical framework within which the formation and evolution of DM halos, the building blocks of the large- scale structure in the universe, can be physically tracked down [@PS74; @bbks86; @bon-etal91; @BM96; @SMT01]. According to this theory, the hierarchical accretion and merging events, which are the dominant driver of the halo growth, owe their frequencies solely to the halo masses. N-body simulations that were performed to complement the theory with desired accuracy and precision, however, invalidated this simple picture, discovering a puzzling phenomenon, so called the “halo assembly bias”: The clustering strength of the DM halos affect their formation epochs and growth rates on the same mass scale [@GW07]. Although the discovery of this phenomenon baffled for long the community of the large-scale structure, it is now generally accepted that the cosmic web, anisotropic large-scale tidal environments surrounding DM halos [@bon-etal96], must be mainly responsible for the deviation of the simple prediction of the excursion set theory on the halo growths from the reality [e.g., @san-etal07; @hah-etal09; @wan-etal11; @zomg1; @toj-etal17; @yan-etal17; @mus-etal18; @MK19; @ram-etal19]. Thus, a key to understanding the halo assembly bias is to figure out what aspect of the anisotropic tidal fields affects the halo growths.
The cosmic web is further classified into four different types each of which has a distinct geometrical shape and dimension: zero dimensional knots, one dimensional filaments, two dimensional walls and three dimensional voids [@hah-etal07]. Among them, the most anisotropic web-type, the filament, turned out to embed the majority of DM halos [e.g., @gan-etal19] which were believed to grow via the preferential merging and accretion of satellites along the narrow one-dimensional channels [e.g., @wes-etal95; @PB02; @ver-etal11]. A recent numerical work of @zomg1 based on a high-resolution N-body simulation, however, revealed that the motions of satellites confined in the filamentary environments could have opposite effects on the growths of galactic halos, depending on the filament thickness [see also @GP16]. If multiple fine filaments cross one another at some nodes, the radial motions of the satellites along the filaments facilitate their infalls onto the galactic halos located at the nodes, enhancing the growths of the hosts. Whereas, in the bulky filaments thicker than the sizes of the constituent galactic halos, the satellites preferentially move in the tangential directions orthogonal to the filament axes, which lead to the deterrence of the satellite infalls and the retarded growths of their hosts. Quantifying the filament thickness in terms of the ellipticity of the surrounding large-scale structure and incorporating it into the conditions for the halo formation, @zomg1 proposed a new extension of the excursion set theory which could accommodate the opposite effects of the large-scale tidal environments on the growths of the galactic halos [see also @zomg3].
Motivated by the insightful work of @zomg1, several attempts were made to improve their model by incorporating more realistic conditions from the halo growths or by extending the model to the larger scales or to the other web types [@laz-etal17; @mus-etal18; @lee19]. For instance, @lee19 introduced a new concept of the “tidal coherence” for a quantitative explicit description of the filament thickness, suggesting that bulky thick (multiple fine) filaments should be outcomes of the highly coherent ([*incoherent*]{}) tides defined as the strong (weak) alignments between the first eigenvectors corresponding to the larges eigenvalues of the tidal fields smoothed on two widely separated scales. With the numerical analysis on the cluster scales, @lee19 indeed found that the radial (tangential) motions of the infall-zone satellites around host clusters are obstructed (facilitated) by the highly coherent tides, which implies that the halo growth sensitively depends on the degree of the tidal coherence.
Yet, the prime focus of @lee19 was the future evolution of the cluster halos rather than their past evolutions, dealing with the infall-zone satellites which have yet to fall into the halos. It is necessary to treat the real satellites for the investigation of the effect of the tidal coherence on the past growths of the DM halos. Besides, the original definition of the tidal coherence in terms only of the first eigenvector direction may neglect the possibilities that the coherence in the second and third eigenvector directions corresponding to the second largest and smallest eigenvalues are not evinced by the coherence in the first eigenvector direction and that the simultaneous coherence of the tides in multiple eigenvector directions may have different effects on the halo growths.
In this Paper, we attempt to incorporate the multi-dimensional aspect of the tidal coherence into the idea of @lee19 and to explore how it affects the halo growths by measuring a correlation between the mass-loss evolution of the halo satellites and the multi-dimensional tidal coherence. In Section \[sec:data\] the definition of the multi-dimensional tidal coherence as well as the description of the numerical data sets utilized for this analysis are presented. In Sections \[sec:1d\]-\[sec:3d\], the effects of the simultaneous coherence of the tides along one, two and three eigenvector directions on the subhalo mass-loss evolutions are presented. In Section \[sec:sum\] the final results are summarized and its implication on the halo assembly bias is discussed. Throughout this analysis, we will assume a concordance cosmology with initial conditions prescribed by the Planck result [@planck13].
Dependence of the Satellite Mass-Loss on the Tidal Coherence
============================================================
Tidal Coherence as a Multi-Component Array {#sec:data}
------------------------------------------
For this analysis, we utilize the catalog of the Rockstar halos [@rockstar] and density field at $z=0$ retrieved from the website of the Small MultiDark Planck simulation[^1][SMDPL, @smdpl], a DM-only N-body simulation performed on a periodic box of linear size $400\,h^{-1}$Mpc, containing $3840^{3}$ DM particles of individual mass $m_{p}=9.63\times 10^{7}\,h^{-1}\,M_{\odot}$ for the Planck cosmology [@planck13]. The catalog contains both of the distinct halos and the subhalos, which can be distinguished by their parent ID (pId): The former has pId$=-1$ while the pId of the latter is nothing but the ID of its parent halo, a least massive distinct halo which gravitationally hosts the latter. Selecting as the hosts the massive cluster-size distinct halos in the mass range of $(1-3)10^{14}\,h^{-1}\,M_{\odot}$, we identify their subhalos whose pId’s match their ID’s.
For each subhalo belonging to each host, we determine the ratio, $\xi_{m}\equiv M_{\rm vir}/M_{\rm acc}$, of its virial mass, $M_{\rm vir}$, to its accretion mass, $M_{\rm acc}$, defined as the subhalo mass at the moment of its accretion to its host. The majority of the subhalos are to lose their masses after their infalls via various processes like the tidal stripping/heating and dynamical frictions [@bos-etal05], for which cases we expect $\xi_{m}<1$. The lower value of $\xi_{m}$ below unity indicates that the given subhalo must have experienced the severe mass-loss processes for longer time after the infall. Yet, in some rare occasions, the subhalos can gain masses through merging inside the hosts for which case $\xi_{m}$ can exceed unity. From here on, two terms, [*subhalos*]{} and [*satellites*]{}, will be interchangeablly used to refer to the non-distinct Rockstar halos gravitationally bound to some larger distinct halos.
As done in @lee19, we compute the tidal field, $T_{ij}({\bf x})$, from the density field defined on the $512^{3}$ grid points, $\rho({\bf x})$, by taking the following steps: (i) Calculating the density contrast field as $\delta({\bf x})\equiv (\rho({\bf x})-\bar{\rho})/\bar{\rho}$ where $\bar{\rho}$ is the mean density averaged over the grid points. (ii) Performing the Fourier transformation of $\delta({\bf x})$ into $\tilde{\delta}({\bf k})$. (iii) Smoothing the density field in the Fourier space with a Gaussian filter on the scale of $R_{f}=30\,h^{-1}$Mpc as $\tilde{\delta}_{s}({\bf k})\equiv \tilde{\delta}({\bf k})\exp(-k^{2}R^{2}_{f}/2)$. (iv) Computing the Fourier amplitude of the tidal field as $\tilde{T}_{ij}\equiv k_{i}k_{j}\tilde{\delta}_{s}({\bf k})/k^{2}$. (v) Performing the inverse Fourier transformation of $\tilde{T}_{ij}({\bf k})$ into $T_{ij}({\bf x})$. At the grid point, ${\bf x}_{h}$, where each of the selected hosts is located, we diagonalize $T_{ij}({\bf x}_{h})$ to find a set of three eigenvalues $\{\lambda_{i}\}_{i=1}^{3}$ (with a decreasing order) and the corresponding eigenvectors $\{{\bf e}_i\}_{i=1}^{3}$. Then, we repeat the whole process but with a smaller filtering scale of $R^{\prime}_{f}=2\,h^{-1}$Mpc to obtain a new set of $\{\lambda^{\prime}_{i}\}_{i=1}^{3}$ and $\{{\bf e}^{\prime}_i\}_{i=1}^{3}$.
As mentioned in Section \[sec:intro\], the tidal coherence, $q$, was originally defined as $q\equiv \vert{\bf e}_{1}\cdot{\bf e}^{\prime}_{1}\vert$ [@lee19]. In the current work, we redefine $q$ as a multi-component array as $$\label{eqn:multi_q}
q_{i}=\vert{\bf e}_{i}\cdot{\bf e}^{\prime}_{i}\vert\, \qquad {\rm for}\ {\rm each}\ i\in \{1,2,3\}\, .$$ If $q_{i}$ is equal to or higher than $0.9$ (lower than $0.2$) at a given region, the tides is said to be highly coherent ([*incoherent*]{}) along the $i$th eigenvector direction at the region. A critical question to which we would like to find an answer in the following Subsections is whether or not the subhalos located in the regions where the tides are highly coherent or [*incoherent*]{} in different eigenvector directions exhibit different mass-loss evolutions.
One Dimensional Dependence {#sec:1d}
--------------------------
In this Subsection, we are going to study how the mean value of the subhalo virial-to-accretion mass ratios depends on each of the three components of the tidal coherence, $\{q_{i}\}_{i=1}^{3}$, calling it one-dimensional (1D) dependence of the subhalo mass-loss evolution on the tidal coherence. We first divide the sample of the selected host halos into two subsamples: One contains those hosts surrounded by the tides highly coherent along the first eigenvector direction, satisfying the condition of $q_{1}\ge 0.9$. The other consists of those surrounded by the tides not so strongly coherent along the first eigenvector direction with $q_{1}< 0.9$. Table \[tab:1d\] lists the mean masses ($\langle M_{h}\rangle$) and numbers ($N_{h}$) of the hosts contained in each subsample. As can be seen, although the latter subsample (i.e., $q_{1}< 0.9$) contains three times larger number of hosts, no significant difference in $\langle M_{h}\rangle$ between the two subsamples is noted, which assures that if the values of $\langle\xi_{m}\rangle$ from the two subsamples are significantly different from each other, then it should not be ascribed to the mass difference.
For each host contained in each subsample, we select only those subhalos which experienced the [*mass-loss*]{} process, i.e., $\xi_{m}< 1$, excluding those few subhalos which experienced the mass-gain process, $\xi_{m}\ge 1$. Then, we calculate the mean virial-to-accretion mass ratio, $\langle \xi_{m}\rangle$, averaged over the selected subhalos of the hosts contained in each subsample. The errors, $\sigma_{\xi_{m}}$, in the measurement of $\langle\xi_{m}\rangle$, is calculated as its standard deviation as $\sigma_{\xi_{m}}\equiv [\langle(\xi_{m}-\langle\xi_{m}\rangle)^{2}\rangle/(N_{\rm sub}-1)]^{1/2}$ where $N_{\rm sub}$ is the total number of the subhalos of the hosts contained in each subsample.
Figure \[fig:1d\] plots the values of $\langle\xi_{m}\rangle$ from the two subsamples with $q_{1}\ge 0.9$ and $q_{1}< 0.9$ as thick red and blue bars, respectively, with the associated errors $\sigma_{\xi_{m}}$ in its left panel, explicitly demonstrating that the former yields a significantly higher value of $\langle\xi_{m}\rangle$ than the latter. This trend implies that the satellites located in the regions surrounded by the tides highly coherent along the first eigenvector direction experience less severe mass-loss evolution after their infalls onto their hosts than the other counterparts with $q_{1}< 0.9$. Based on the insights from @lee19, we put forth the following explanation to understand this phenomenon: As the satellites surrounded by highly coherent tides along the first eigenvector direction develop velocities in the tangential direction, which deter their infalls onto the hosts, reducing the amount of time during which the subhalos are exposed to the effects of the tidal stripping/heating or dynamical fraction inside their hosts.
Repeating the above procedure but with the subsamples obtained by contraining the value of $q_{2}$ ($q_{3}$) instead of $q_{1}$ with the same threshold of $0.9$, we also investigate how $\langle\xi_{m}\rangle$ differs between the cases of $q_{2}\ge 0.9$ and $q_{2}<0.9$ ($q_{3}\ge 0.9$ and $q_{3}<0.9$). The middle (right) panel of Figure \[fig:1d\] plots the same as the left panel but for the case that the subsample is divided by imposing the threshold condition on the value of $q_{2}$ ($q_{3}$). As can be seen, the subhalos of the hosts located in the regions with $q_{2}\ge 0.9$ ($q_{3}\ge 0.9$) yield a larger value of $\langle\xi_{m}\rangle$ than those with $q_{2}< 0.9$ ($q_{3}< 0.9$), the same trend as that shown in the left panel of Figure \[fig:1d\]. Note, however, that the larger (smaller) difference in $\langle \xi_{m}\rangle$ between the two subsamples are found for the case that the threshold condition is imposed on the value of $q_{3}$ ($q_{2})$ rather than on the value of $q_{1}$.
To see whether or not this difference in $\langle\xi_{m}\rangle$ witnessed in Figure \[fig:1d\] is a secondary effect induced by any differences in the local density ($\delta$) or ellipticity ($e$) between the two subsamples, we determine the values $\delta$,and $e$ at the grid point of each host. The three tidal eigenvalues, $\{\lambda^{\prime}_{i}\}_{i=1}^{3}$ on the scale of $2\,h^{-1}$Mpc obtained in Subsection \[sec:data\] is used to calculate $\delta$ and $e$: $\delta=\sum_{i=1}^{3}\lambda^{\prime}_{i}$, and $e\equiv [(1+\delta)^{-1}\sum_{i<j}(\lambda^{\prime}_{i}-\lambda^{\prime}_{j})^{2}]^{1/2}$ . This definition of $e$, was devised by @ram-etal19 to eliminate any correlation between $e$ and $\delta$.
Taking the mean values, $\langle\delta\rangle$ and $\langle e\rangle$, averaged over all hosts contained in each of the subsamples, we plot them in the top and bottom panels of Figure \[fig:den\_1d\], respectively. As can be seen, when the value of $q_{2}$ or $q_{3}$ are constrained by using a threshold of $0.9$, no significant differences are found in $\langle\delta\rangle$ and $\langle e\rangle$ between the two subsamples. Whereas, the subsample with $q_{1}\ge 0.9$ is found to have substantially larger values of $\langle\delta\rangle$ and $\langle e\rangle$ than the other subsample with $q_{1}<0.9$. That is, the regions surrounded by the tides highly coherent along the first eigenvectors tend to be more overdense and more anisotropic due to the simultaneous compression of matter along the coherent first eigenvector direction. This result brings out a suspicion that the higher value of $\langle\xi_{m}\rangle$ found in the subsample with $q_{1}\ge 0.9$ may be caused by the higher values of $\langle\delta\rangle$ and $\langle e\rangle$.
Now that the tides highly coherent along the eigenvector direction are found to have an obstruction effect on the satellite infalls, the next quest is to investigate whether the tides highly [*incoherent*]{} along any eigenvector direction have the opposite effect or not. For this quest, we use two thresholds: an upper-bound threshold of $0.2$ and a lower-bound threshold of $0.9$ to construct two subsamples (i.e., $q_{i}\ge 0.9$ and $q_{i}< 0.2$ for each $i\in \{1,2,3\}$) and then conduct the same analysis. Figures \[fig:1d\_hl\]-\[fig:den\_1d\_hl\] plot the same as Figures \[fig:1d\]-\[fig:den\_1d\], respectively, but with the conditions of $q_{i}\ge 0.9$ and $q_{i}< 0.2$ instead of $q_{i}\ge 0.9$ and $q_{i}<0.9$. The left panel of Figure \[fig:1d\_hl\] reveals that the difference in $\langle\xi_{m}\rangle$ between the two subsamples obtained by putting two thresholds of $0.9$ and $0.2$ on the value of $q_{1}$ is larger than that by putting one threshold of $0.9$. This result indicates that the tides highly [*incoherent*]{} along the first eigenvector direction indeed have the opposite effect on the satellite infalls: it facilitates the satellite infalls onto the hosts, leading them to undergo the more severe mass-loss evolution after the infalls. Meanwhile, the left panel of Figure \[fig:den\_1d\_hl\] shows that the difference in $\langle\delta\rangle$ and $\langle e\rangle$ between the two subsamples obtained by putting two thresholds of $0.9$ and $0.2$ on the value of $q_{1}$ is smaller than that by using one threshold of $0.9$, which proves that the larger values of $\langle\delta\rangle$ and $\langle e\rangle$ are not mainly responsible for the more severe mass-loss evolution of the subhalos found from the subsample with $q_{1}\ge 0.9$.
It is interesting, however, to discover in the right panel of Figure \[fig:1d\_hl\] that the tides highly [*incoherent*]{} along the third eigenvector direction does not have the expected opposite effect, compared to that coherent along the same direction. The difference in $\langle\xi_{m}\rangle$ between the subsamples obtained by putting two thresholds of $0.9$ and $0.2$ on $q_{3}$ is smaller than that between the subsamples obtained by putting one threshold of $0.9$ on $q_{3}$. This result indicates that the tides highly [*incoherent*]{} along the third eigenvector direction have an obstructing effect on the satellite infalls rather than facilitating it unlike the tides highly [*incoherent*]{} along the first eigenvector direction. This phenomenon may be closely linked with the larger mean ellipticity, $\langle e\rangle$, found in the subsample with $q_{3}<0.2$ than in the subsample with $q_{3}<0.9$, shown in the right panel of Figure \[fig:den\_1d\_hl\]. The tides highly [*incoherent*]{} along the third eigenvector direction can increase the tidal anisotropy of a region, which in turn makes it harder for the satellites in the region to fall onto their hosts. As the satellite infalls are deterred, they must go through less severe mass-loss evolution after the infalls till the present epochs. Note also in the middle panels of Figures \[fig:1d\]-\[fig:den\_1d\_hl\] that the tidal coherence measured along the second eigenvector direction have the weakest effect on the subhalo mass-loss evolution, showing no significant differences in $\langle\xi_{m}\rangle$, $\langle\delta\rangle$, and $\langle e\rangle$ among three samples with $q_{2}\ge 0.9$, $q_{2}< 0.9$ and $q_{2}<0.2$.
Two Dimensional Dependence {#sec:2d}
--------------------------
Now that the surrounding tides coherent along different eigenvector directions are found to have different effects on the mass-loss evolution of the subhalos, we would like to explore the effects of the tides coherent simultaneously along two eigenvector directions. Since the tidal coherence measured along the second eigenvector direction is found to have the weakest effect on the subhalo mass-loss evolution in Subsection \[sec:1d\], we will focus on the tidal coherence measured simultaneously along the first and third eigenvector directions (i.e., $q_{1}$ and $q_{3}$) in this Subsection.
We first separate the selected host halos into four subsamples by simultaneously constraining the values of $q_{1}$ and $q_{3}$ with a single threshold of $0.9$ (see Table \[tab:2d\]). Then, we calculate ($\langle \xi_{m}\rangle,\ \sigma_{\xi_{m}}$), ($\langle\delta\rangle,\ \sigma_{\delta}$) and ($\langle e\rangle,\ \sigma_{e}$) by taking the same steps described in Subsection \[sec:1d\] for each of the four subsamples, the results of which are displayed in Figures \[fig:2d\]-\[fig:den\_2d\]. As can be seen in Figure \[fig:2d\], the two subsamples satisfying the conditions of $(q_{1}\ge 0.9,\ q_{3}< 0.9)$ and $(q_{1}<0.9,\ q_{3}\ge 0.9)$ yield significantly higher values of $\langle\xi_{m}\rangle$ than the other two subsamples. A crucial implication of this result is that the tides highly coherent along the first (third) eigenvector direction but not along the third (first) eigenvector directions have a stronger obstructing effect on the satellite infalls than the tides highly coherent along both of the first and third eigenvector directions.
It is interesting to see that while the two subsamples with of $(q_{1}\ge 0.9,\ q_{3}< 0.9)$ and $(q_{1}<0.9,\ q_{3}\ge 0.9)$ show no significant difference in the values of $\langle\xi_{m}\rangle$ and $\langle\delta\rangle$ from each other, a substantial difference in the value of $\langle e\rangle$ is found between them (see Figure \[fig:den\_2d\]): the regions surrounded by the tides highly coherent along the first eigenvector direction but not along the third ones are more anisotropic than those surrounded by the tides highly coherent along the third eigenvector direction but not along the first ones. Given that the tidal anisotropy can also have an effect of obstructing the satellite infalls, the larger value of $\langle\xi_{m}\rangle$ found from the subsample with $q_{1}\ge 0.9$ and $q_{3}< 0.9$ may be partly caused by its larger value of $\langle e\rangle$ than that from the subsample with $q_{1}< 0.9$ and $q_{3}\ge 0.9$. The lowest value of $\langle\xi_{m}\rangle$ is found from the subsample with $q_{1}< 0.9$ and $q_{2}< 0.9$, which indicates that the tides highly coherent along none of the first nor third eigenvector directions have the weakest obstructing and/or strongest facilitating effects of the satellite infalls.
We also investigate the effect of the highly [*incoherent*]{} tides on the subhalo mass-loss evolution and on the local density and ellipticity as well by constraining the value of $q_{1}$ and $q_{3}$ with double thresholds of $0.9$ and $0.2$, the results of which are shown in Figures \[fig:2d\_hl\]-\[fig:den\_2d\_hl\]. As can be seen in Figure \[fig:2d\_hl\], the subsample with $q_{1}\ge 0.9$ and $q_{3}<0.2$ yields the highest value of $\langle\xi_{m}\rangle$ among the four, while its lowest value is found in the subsample with $q_{1}<0.2$ and $q_{3}\ge 0.9$. This result indicates that the tides highly coherent along the first eigenvector direction and highly [*incoherent*]{} along the third eigenvector direction are most effective in obstructing the satellite infalls, while the tides highly coherent along the third eigenvector direction and [*incoherent*]{} along the first eigenvector direction are most effective in facilitating the infalls among the four. Given that the subsample with ($q_{1}\ge 0.9,\ q_{3}<0.2$) yields the highest value of $\langle e\rangle$ among the four, the largest value of $\langle\xi_{m}\rangle$ from the subsample with $q_{1}\ge 0.9$ and $q_{3}<0.2$ should be partially caused by a larger value of $\langle e\rangle$.
It is worth recalling that in Subsection \[sec:1d\] the tides highly coherent only along the third eigenvector direction have been already found to obstruct the satellite infalls rather than facilitate them (see the right panel of Figure \[fig:1d\]). Nevertheless, if the tides are simultaneously incoherent along the first eigenvector direction, then the facilitating effect of the tidal [*incoherence*]{} along the first eigenvector direction seem to overwhelm the obstructing effect of the tidal coherence along the third eigenvector direction, according to the result shown in Figure \[fig:1d\_hl\]. In other words, it is the tidal [*incoherence*]{} along the first eigenvector direction that plays the most decisive dominant role of facilitating the satellite infalls, driving the largest amount of mass-loss of the subhalos in the post-infall stages.
Meanwhile, the high coherence of the tides along the first eigenvector direction seems to be synergetic with its simultaneous [*incoherence*]{} along the third eigenvector direction (see Figure \[fig:2d\_hl\]). The subsample with ($q_{1}\ge 0.9,\ q_{2}<0.2$) yields the lowest value of $\langle\xi_{m}\rangle$ not only among the subsamples obtained by simultaneously constraining both of $q_{1}$ and $q_{3}$ but also among the subsamples obtained by constraining only one of three components of $\{q_{i}\}$ (see Figure \[fig:1d\_hl\]). Our interpretation is that the high tidal anisotropy associated with the tides highly [*incoherent*]{} along the third eigenvector direction tends to magnify the obstructing effect of the high tidal coherence along the first eigenvector direction.
Three Dimensional Dependence {#sec:3d}
----------------------------
Now that the simultaneous constraints of $q_{1}$ and $q_{3}$ uncovers the complex two-dimensional dependence of the subhalo mass-loss evolution on the tidal coherence, it should be legitimate to investigate how $\langle\xi_{m}\rangle$ depends on all of the three components of $\{q_{i}\}_{i=1}^{3}$, calling it three dimensional (3D) dependence of the subhalo mass-loss evolution on the tidal coherence. We first separate the host halos into eight subsamples by constraining simultaneously the values of $(q_{1},\ q_{2},\ q_{3})$ with a single threshold of $0.9$ (see Table \[tab:3d\]).
Through the same procedure described in Subsection \[sec:1d\], we determine the values of ($\langle \xi_{m}\rangle,\ \sigma_{\xi_{m}}$), ($\langle\delta\rangle,\ \sigma_{\delta}$) and ($\langle e\rangle,\ \sigma_{e}$), for each of the eight subsamples, which are plotted in Figures \[fig:3d\]-\[fig:den\_3d\]. As can be seen in Figure \[fig:3d\], we find the highest and lowest values of $\langle\xi_{m}\rangle$ from the subsamples with $(q_{1}\ge 0.9, q_{2}\ge 0.9, q_{3}<0.9)$ and $(q_{1}< 0.9, q_{2}\ge 0.9, q_{3}\ge 0.9)$, respectively, among the eight. In Figure \[fig:den\_3d\] where the eight subsamples exhibit little difference in $\langle\delta \rangle$ but substantial difference in $\langle e\rangle$, we find the highest and lowest values of $\langle e\rangle$ from the same two subsamples, which implies that the large difference in $\langle e\rangle$ among the two subsamples should be linked with the large difference in $\langle\xi_{m}\rangle$.
For the case that $q_{1}\ge 0.9$ and $q_{3}<0.9$, the simultaneous constraint of $q_{2}\ge 0.9$ gives the highest value of $\langle\xi_{m}\rangle$ (scarlet bar). Whereas, for the case that $q_{1}<0.9$ and $q_{3}\ge 0.9$, the same constraint of $q_{2}\ge 0.9$ yields the opposite signal, i.e., the lowest value of $\langle\xi_{m}\rangle$ (green bar). Note also that the subsample with $(q_{1}<0.9,\ q_{2}\ge 0.9,\ q_{3}<0.9)$ corresponding to the tides highly coherent only along the second eigenvector direction but not along the first and third ones yields relatively low value of $\langle\xi_{m}\rangle$. This result indicates that the effect of the high tidal coherence along the second eigenvector direction shifts from the obstruction to the facilitation of the satellite infalls, depending on which eigenvector direction between the first and third the tides are simultaneously coherent. If the tides are highly coherent along none of the first and third eigenvector direction, then the high tidal coherence along the second eigenvector direction does not have a strong effect on the satellite infalls.
It is interesting to see that the tides highly coherent along all of the three eigenvector directions (red bar) are less effective in obstructing the satellite infalls than the tides highly coherent along the first and second eigenvector directions but not highly coherent along the third eigenvector direction (scarlet bar). It is even not so effective in obstructing the satellite infalls as the tides highly coherent only along the third eigenvector direction but not along the first and second eigenvector direction (thick violet bar). Note also that the second highest value of $\langle\xi_{m}\rangle$ is found from the subsample with $q_{1}<0.9,\ q_{2}<0.9,\ q_{3}\ge 0.9$ (violet bar). Given that the mean ellipticity from this subsample is relatively low compared with the other seven cases (see Figure \[fig:den\_3d\]), this result implies that the net obstructing effect of the tides highly coherent only along the third but not along the first and second eigenvector directions may be stronger than that of the tides highly coherent only along the first and second eigenvector directions but not along the third eigenvector direction (scarlet bar).
As done in Subsections \[sec:1d\] and \[sec:2d\], we also investigate how the degree of the tidal [*incoherence*]{} measured along all of three eigenvector directions is linked with the subhalo mass-loss evolution, creating seven new subsamples by constraining simultaneously all of the three components, $(q_{1},\ q_{2},\ q_{3})$ with double thresholds of $0.9$ and $0.2$ (see Table \[tab:3d\]): It turns out that no hosts satisfy the conditions of $(q_{1}\ge0.9,\ q_{2}< 0.2,\ q_{3}< 0.2)$, $(q_{1}<0.2,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9)$ and $(q_{1}< 0.2,\ q_{2}\ge 0.9,\ q_{3}< 0.2)$, leaving three subsamples empty. The values of $(\langle\xi_{m}\rangle,\ \sigma_{\xi_{m}})$, $(\langle\delta\rangle,\ \sigma_{\delta})$ and $(\langle e\rangle,\ \sigma_{e})$ obtained from the rest four non-empty subsamples as well as from the subsample with $(q_{1}\ge 0.9,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9)$ are shown in Figures \[fig:3d\_hl\]-\[fig:den\_3d\_hl\].
The subsample with $(q_{1}\ge 0.9,\ q_{2}< 0.2,\ q_{3}< 0.2)$ yields the highest value of $\langle\xi_{m}\rangle$ (olive green bar), while the lowest value (violet bar) is found from the subsample with $(q_{1}< 0.2,\ q_{2}< 0.2,\ q_{3}\ge 0.9)$. Since the difference in $\langle e\rangle$ between the two subsamples is not so large enough to explain their difference in $\langle\xi_{m}\rangle$ (see Figure \[fig:den\_3d\_hl\]), the different mean ellipticities between the two subsamples should not be the main cause of the significant difference in the mean virial-to-accretion mass ratios between them. The tides highly coherent along the first eigenvector direction but highly incoherent along the second and third eigenvector directions are much more effective in obstructing the satellite infalls than the tides highly coherent along the third eigenvector direction but highly incoherent along the first and second eigenvector directions.
The comparison of the result shown in Figures \[fig:3d\] and \[fig:3d\_hl\] reveals that the subsample with $(q_{1}<0.9,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9)$ yield a lower value of $\langle\xi_{m}\rangle$ than the subsample with $(q_{1}<0.2,\ q_{2}< 0.2,\ q_{3}\ge 0.9)$. The tides highly coherent along the third eigenvector direction but highly [*incoherent*]{} along the first and second eigenvector direction are less effective in facilitating the satellite infalls than the tides highly coherent along the second and third eigenvector direction but not so highly coherent along the first eigenvector direction.
Another interesting fact revealed by the comparison between the two Figures is that the value $\langle\xi_{m}\rangle$ from the subsample with $(q_{1}<0.2,\ q_{2}\ge 0.9,\ q_{3}< 0.2)$ is as high as that from the subsample with $(q_{1}\ge 0.9,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9)$. This result indicates that the tides highly coherent only along the second eigenvector direction but highly [*incoherent*]{} along the first and third eigenvector directions are as effective in obstructing the satellite infalls as the tides highly [*coherent*]{} along all of the three eigenvector directions. It is a rather surprising unexpected result since we have already found in Subsection \[sec:1d\] that the tides highly coherent along the first eigenvector direction have an obstructing effect on the satellite infalls and that the tides highly [*incoherent*]{} along the same direction have the opposite effect, i.e., facilitating the satellite infalls. The slightly larger value of $\langle e\rangle$ from the subsample with $(q_{1}<0.2,\ q_{2}\ge 0.9,\ q_{3}< 0.2)$ than that from the subsample with $(q_{1}\ge 0.9,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9)$ should be related to this puzzling phenomenon (see Figures \[fig:den\_3d\]-\[fig:den\_3d\_hl\]). The tidal [*incoherence*]{} along the third eigenvector direction tends to increases the tidal anisotropy (i.e., mean ellipticity) which plays a role in increasing the value of $\langle\xi_{m}\rangle$, as shown in Subsection \[sec:1d\]. The obstructing effect of the high tidal anisotropy caused by the tidal [*incoherence*]{} along the third eigenvector direction compensates the facilitating effect of the high [*incoherence*]{} of the tides along the first eigenvector direction.
Summary and Discussion {#sec:sum}
======================
We have systematically studied the dependence of the subhalo mass-loss evolution on the multi-dimensional aspect of the tidal coherence by using the numerical datasets retrieved from the SMPDL [@smdpl]. For this study, we have quantified the subhalo mass-loss evolution in terms of the mean virial-to-accretion mass ratios averaged over the subahlos, and expressed the tidal coherence as an array of three numbers, $\{q_{i}\}_{i=1}^{3}$, where $q_{i}$ represents the alignments between the $i$th eigenvectors of the tidal fields smoothed on two widely separated scales of $2\,h^{-1}$Mpc and $30\,h^{-1}$Mpc. To eliminate the well known strong dependence of the subhalo mass-loss evolution on the masses of their hosts [@bos-etal05], we select only those subhalos belonging to the hosts whose masses lie in the narrow range of $1\le M_{h}/(10^{14}\,h^{-1}\,M_{\odot})\le 3$.
It has been found that the subhalos surrounded by the tides highly coherent along a eigenvector direction ($q_{i}\ge 0.9,\ ^{\forall} i\in\{1,2,3\}$) tend to have higher mean values of the virial-to-accretion mass ratios than their counterparts ($q_{i}< 0.9$), no matter what eigenvector direction is chosen. Our interpretation of this result is that the tides highly coherent along any eigenvector direction has an effect of obstructing the satellite infalls onto the hosts, which leads the satellites to experience the least severe mass-loss evolution in their post-infall stages. It has also been shown that the high tidal coherence along the third (second) eigenvector direction has the strongest (weakest) obstructing effect on the satellite infalls.
The tides highly [*incoherent*]{} along a different eigenvector direction, however, has turned out to have a different effect. The tides highly [*incoherent*]{} along the first eigenvector direction ($q_{1}< 0.2$) have an effect opposite to the tides coherent along the same direction ($q_{1}\ge 0.9$) on the subhalo mass-loss evolution: the former facilitates the satellite infalls while the latter obstructs them, leading the subhalos surrounded by the former to lose much larger amount of masses after the infalls than those surrounded by the latter. In fact, the subhalos surrounded by the tides highly [*incoherent*]{} along the first eigenvector direction have been found to yield the lowest mean virial-to-accretion mass ratios. Whereas, the tides highly [*incoherent*]{} along the third eigenvector direction ($q_{3}< 0.2$) have an effect of obstructing rather than facilitating the satellite infalls, similar to the tides highly coherent along the same direction ($q_{3}\ge 0.9$).
It is shown that the simultaneous coherence or [*incoherence*]{} of the tides along two or three eigenvector directions have more complex effects on the subhalo mass-loss evolution. The high tidal coherence along the first eigenvector direction has been found to be synergic with the high tidal [*incoherence*]{} along the minor eigenvector direction ($q_{1}\ge 0.9$ and $q_{3}< 0.2$) in obstructing the satellite-infalls, yielding the highest mean virial-to-accretion mass ratios of the subhalos. Whereas, the high tidal [*incoherence*]{} along the first eigenvector direction has turned out to be discordant with both of the high tidal coherence and [*incoherence*]{} along the third eigenvector direction in facilitating the satellite infalls. The high tidal coherence along the second eigenvector direction have turned out to be synergic with the high tidal coherence along the first eigenvector direction in obstructing the satellite infalls, provided that the tides are not so coherent along the third eigenvector direction. Meanwhile, provided that the tides are not so coherent along the first eigenvector direction, the high tidal coherence along the second eigenvector direction has been found synergic with the high tidal coherence along the third eigenvector direction in facilitating the satellite infalls.
Although the tides highly coherent along one of the three eigenvector direction have an obstructing effect on the satellite infalls, the simultaneous coherence of the tides along all of the three eigenvector directions have been found not to reinforce the obstructing effect. The tides highly coherent along the first eigenvector direction and [*incoherent*]{} along the second and third eigenvector directions have been found more effective in obstructing the satellite infalls than the tides simultaneously coherent along all of the three eigenvector directions. The same is true for the simultaneous [*incoherence*]{} of the tides along all of the three eigenvector directions, which have been found not to reinforce the effect of facilitating the satellite infalls. The tides highly coherent along the third eigenvector direction and simultaneously [ *incoherent*]{} along the first and second eigenvector direction have been found more effective in facilitating the satellite infalls than the tides simultaneously [*incoherent*]{} along all of the three eigenvector directions.
Determining the mean values of the local density contrasts, $\langle\delta\rangle$, and tidal anisotropies, $\langle e\rangle$, averaged over the regions with different tidal coherences, we have found negligible differences in $\langle\delta\rangle$ and substantial differences in $\langle e\rangle$ among the regions. Noting that the simultaneous coherence along all of the three eigenvector directions plays a significant role of reducing the tidal anisotropy, and recalling that the high tidal anisotropy has been known to obstruct the satellite infalls [e.g., @zomg1], we have explained that the higher tidal anisotropy should be contributed to the stronger obstructing effect of the tides highly coherent along the first eigenvector direction but highly [*incoherent*]{} along the second and third eigenvector directions than the tides highly coherent along all of the three eigenvector directions. Yet, we have also shown that the multi-dimensional tidal coherence have an independent net effect on the subhalo-mass loss evolution, which cannot be ascribed simply to the differences in the tidal anisotropy.
Given that the mean virial-to-accretion mass ratios of the subhalos reflect not only their mass-loss evolutions but also how fast their host clusters have grown as well as in what dynamical states they are [@bos-etal05], the bottom line of our work is as follows: The formation and evolution of the cluster halos at fixed mass scales located in the environments with similar densities and tidal anisotropies still show variations with the multi-dimensional effects of the tidal coherence. We suspect that this result may be responsible for the large scatters around the spherical critical density contrast of $\delta_{c}\equiv 1.68$ required for the formation of a cluster halo, which could not be entirely explained by the scale-dependence of the non-spherical counter-part, $\delta_{ec}$ [e.g., @MR10; @CA11]. Our result may be also closely related to the elusive nature of the large-scale assembly bias, whose existence have so far gained no observational confirmations [e.g., see @SM19]. It is not only the density and tidal strengths but also the multi-dimensional tidal coherence that we must take into account to detect the large-scale assembly bias. We plan to work on finding a direct link between the tidal coherence and the large-scale assembly bias as well as on extending the excursion set model by incorporating the tidal coherence, hoping to report the results elsewhere in the near future.
I acknowledge the support of the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education (NO. 2016R1D1A1A09918491). I was also partially supported by a research grant from the NRF of Korea to the Center for Galaxy Evolution Research (No.2017R1A5A1070354).
[000]{} Bardeen, J. M., Bond, J. R., Kaiser, N., et al. 1986, , 304, 15 Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, , 762, 109 Bond, J. R., Cole, S., Efstathiou, G., et al. 1991, , 379, 440 Bond, J. R., & Myers, S. T. 1996, , 103, 1 Bond, J. R., Kofman, L., & Pogosyan, D. 1996, , 380, 603 Borzyszkowski, M., Porciani, C., Romano-D[í]{}az, E., & Garaldi, E. 2017, , 469, 594 Corasaniti, P. S., & Achitouv, I. 2011, , 84, 023009 Desjacques, V. 2008, , 388, 638 Ganeshaiah Veena, P., Cautun, M., Tempel, E., et al. 2019, , 487, 1607 Gao, L., & White, S. D. M. 2007, , 377, L5 Garaldi, E., Romano-D[í]{}az, E., Borzyszkowski, M., & Porciani, C. 2017, arXiv:1707.01108 Gonz[á]{}lez, R. E., & Padilla, N. D. 2016, , 829, 58 Hahn, O., Porciani, C., Carollo, C. M., & Dekel, A. 2007, , 375, 489 Hahn, O., Porciani, C., Dekel, A., et al. 2009, , 398, 1742 Klypin, A., Yepes, G., Gottl[ö]{}ber, S., Prada, F., & He[ß]{}, S. 2016, , 457, 4340 Lazeyras, T., Musso, M., & Schmidt, F. 2017, JCAP, 2017, 059 Lee, J. 2019, , 872, L6 Maggiore, M., & Riotto, A. 2010, , 717, 515 Mansfield, P., & Kravtsov, A. V. 2019, arXiv e-prints, arXiv:1902.00030 Musso, M., Cadiou, C., Pichon, C., et al. 2018, , 476, 4877 Obuljen, A., Dalal, N., & Percival, W. J. 2019, arXiv e-prints, arXiv:1906.11823 Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, , 571, A16 Plionis, M., & Basilakos, S. 2002, , 329, L47 Press, W. H., & Schechter, P. 1974, , 187, 425 Ramakrishnan, S., Paranjape, A., Hahn, O., et al. 2019, arXiv e-prints, arXiv:1903.02007 Sandvik, H. B., M[ö]{}ller, O., Lee, J., et al. 2007, , 377, 234 Sheth, R. K., Mo, H. J., & Tormen, G. 2001, , 323, 1 Sunayama, T., & More, S. 2019, arXiv e-prints, arXiv:1905.07557 Tojeiro, R., Eardley, E., Peacock, J. A., et al. 2017, , 470, 3720 Wang, H., Mo, H. J., Jing, Y. P., et al. 2011, , 413, 1973 West, M. J., Jones, C., & Forman, W. 1995, , 451, L5 van den Bosch, F. C., Tormen, G., & Giocoli, C. 2005, , 359, 1029 Vera-Ciro, C. A., Sales, L. V., Helmi, A., et al. 2011, , 416, 1377 Yang, X., Zhang, Y., Lu, T., et al. 2017, , 848, 60
![Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts surrounded by the tides highly coherent along one of three eigenvector directions as red bar (major, intermediate and third eigenvector directions in the left, middle and right panels, respectively.) In each panel, the complement case of the tides not so highly coherent along the same direction is plotted as blue bar.[]{data-label="fig:1d"}](f1.eps)
![Mean values of the density contrast and ellipticity averaged over the regions surrounded by the tides highly coherent along one of three eigenvector directions as red bar in the top and bottom panels, respectively. In each panel, the blue bar correspond to the complement case. []{data-label="fig:den_1d"}](f2.eps)
![Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts surrounded by the tides highly coherent ([*incoherent*]{}) along one of three eigenvector directions as red (blue) bar.[]{data-label="fig:1d_hl"}](f3.eps)
![Mean values of the density contrast and ellipticity averaged over the regions surrounded by the tides highly coherent ([*incoherent*]{}) along one of three eigenvector directions as red (blue) bar in the top and left panels, respectively.[]{data-label="fig:den_1d_hl"}](f4.eps)
![Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts surrounded by the tides highly coherent along both of the first and third (thick red bar), along the first but not along the third (thick ocher bar), highly coherent along the third but not along the first (thick green bar), and highly coherent along none of the first and third eigenvector directions (thick blue bar).[]{data-label="fig:2d"}](f5.eps)
![Mean values of the density contrast and ellipticity averaged over the regions surrounded by the tides for the four different cases described in the caption of Figure \[fig:2d\] in the top and bottom panels, respectively.[]{data-label="fig:den_2d"}](f6.eps)
![Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts surrounded by the tides highly coherent along both of the first and third (red bar), highly coherent along the first but highly [*incoherent*]{} along the third (ocher bar), highly coherent along the third but highly [*incoherent*]{} along the first (green bar), and highly [*incoherent*]{} along the first and third eigenvector directions (blue bar).[]{data-label="fig:2d_hl"}](f7.eps)
![Mean values of the density contrast and ellipticity averaged over the regions surrounded by the tides for the four different cases described in the caption of Figure \[fig:2d\_hl\] in the top and bottom panels, respectively.[]{data-label="fig:den_2d_hl"}](f8.eps)
![Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts surrounded by the tides highly coherent along all of the three eigenvector directions are plotted as carmine bar. The seven complement cases corresponding to the tides coherent along not all of the three eigenvector directions are plotted as different color bars.[]{data-label="fig:3d"}](f9.eps)
![Mean values of the density contrast and ellipticity averaged over the regions surrounded by the tides for the eight different cases described in the caption of Figure \[fig:3d\] in the top and bottom panels, respectively.[]{data-label="fig:den_3d"}](f10.eps)
![Mean virial-to-accretion mass ratios of the subhalos belonging to the hosts surrounded by the tides highly [*incoherent*]{} along all of the three eigenvector directions are plotted as black bar. For comparison, the case of the tides coherent along all of the three directions are also plotted as carmine bar. The results from the three cases of the tides highly [*incoherent*]{} along two of the three eigenvector directions but highly coherent along the other directions are plotted as different color bars.[]{data-label="fig:3d_hl"}](f11.eps)
![Mean values of the density contrast and ellipticity averaged over the regions surrounded by the tides for the five different cases described in the caption of Figure \[fig:3d\_hl\] in the top and bottom panels, respectively.[]{data-label="fig:den_3d_hl"}](f12.eps)
[ccc]{} $q_{1}\ge 0.9$ & $1.63\pm 0.03$ & $306$\
$q_{1}<0.9$ & $1.58\pm 0.02$ & $1096$\
$q_{1}<0.2$ & $1.78\pm 0.01$ & $179$\
$q_{2}\ge 0.9$ &$1.60\pm 0.04$ & $174$\
$q_{2}<0.9$ & $1.59\pm 0.01$ & $1228$\
$q_{2}<0.2$ & $1.75\pm 0.01$ & $275$\
$q_{3}\ge 0.9$ & $1.60\pm 0.03$ & $254$\
$q_{3}<0.9$ & $1.59\pm 0.02$ & $1148$\
$q_{3}<0.2$ & $1.70\pm 0.01$ & $203$\
\[tab:1d\]
[ccc]{} $q_{1}\ge 0.9,\ q_{2}\ge 0.9$ & $1.62\pm 0.06$ & $92$\
$q_{1}\ge 0.9,\ q_{2}<0.9$ & $1.63\pm 0.04$ & $214$\
$q_{1}< 0.9,\ q_{2}\ge 0.9$ & $1.59\pm 0.04$ & $162$\
$q_{1}< 0.9,\ q_{2}<0.9$ & $1.58\pm 0.02$ & $934$\
$q_{1}\ge 0.9,\ q_{2}<0.2$ & $1.58\pm 0.08$ & $26$\
$q_{1}< 0.2,\ q_{2}\ge 0.9$ & $1.42\pm 0.19$ & $9$\
$q_{1}< 0.2,\ q_{2}<0.2$ & $1.63\pm 0.06$ & $78$\
\[tab:2d\]
[ccc]{} $q_{1}\ge 0.9,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9$ & $1.61\pm 0.06$ & $82$\
$q_{1}\ge 0.9,\ q_{2}\ge0.9,\ q_{3}< 0.9$ & $1.65\pm 0.17$ & $11$\
$q_{1}\ge 0.9,\ q_{2}< 0.9,\ q_{3}\ge 0.9$ & $1.73\pm 0.19$ & $10$\
$q_{1}\ge 0.9,\ q_{2}<0.9,\ q_{3}< 0.9$ & $1.64\pm 0.04$ & $203$\
$q_{1}< 0.9,\ q_{2}\ge 0.9,\ q_{3}\ge 0.9$ & $1.61\pm 0.16$ & $10$\
$q_{1}< 0.9,\ q_{2}\ge0.9,\ q_{3}< 0.9$ & $1.62\pm 0.06$ & $71$\
$q_{1}< 0.9,\ q_{2}< 0.9,\ q_{3}\ge 0.9$ & $1.59\pm 0.04$ & $152$\
$q_{1}< 0.9,\ q_{2}<0.9,\ q_{3}< 0.9$ & $1.57\pm 0.02$ & $863$\
$q_{1}\ge 0.9,\ q_{2}<0.2,\ q_{3}< 0.2$ & $1.54\pm 0.09$ & $22$\
$q_{1}< 0.2,\ q_{2}\ge 0.9,\ q_{3}< 0.2$ & $1.42\pm 0.19$ & $9$\
$q_{1}< 0.2,\ q_{2}< 0.2,\ q_{3}\ge 0.9$ & $1.58\pm 0.12$ & $19$\
$q_{1}< 0.2,\ q_{2}<0.2,\ q_{3}< 0.2$ & $1.61\pm 0.17$ & $5$\
\[tab:3d\]
[^1]: https://www.cosmosim.org
|
---
abstract: |
We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $\rho$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $\rho \ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.
[**Keywords:**]{} diffusion processes, Euler scheme, weak error analysis.
author:
- 'O. Bencheikh and B. Jourdain[^1]'
title: 'Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise'
---
[[**AMS Subject Classification (2010):**]{} *60H35, 60H10, 65C30, 65C05*]{}
Introduction
============
In numerous areas such as mathematical finance when, for example, modelling a stock price process whose trend dramatically changes when a factor goes down a threshold value, or in stochastic control theory when choosing a control process that minimizes the expected discounted cost, we end-up dealing with diffusions that do not show a smooth behavior which results in Stochastic Differential Equations with discontinuous coefficients. In the present paper, we are interested in the Euler-Maruyama discretization of the stochastic differential equation $$X_t = X_0 + W_t + \int_0^t b(s,X_s)\,ds,\quad t\in[0,T]\label{edsnl}$$ where $(W_t)_{t\ge 0}$ is a $d$-dimensional Brownian motion independent from the initial ${{\mathbb R}}^d$-valued random vector $X_0$, $T\in(0,+\infty)$ is a finite time horizon and the drift coefficient $b:[0,T]\times{{\mathbb R}}^d\to{{\mathbb R}}^d$ is merely measurable and bounded.\
While the convergence properties of the Euler-Maruyama scheme are well understood for SDEs with smooth coefficients, the case of irregular coefficients is still an active field of research. Concerning the strong error, the additive noise case is investigated in [@HalKloe] where Halidias and Kloeden only prove convergence and in [@DarGer; @NeuSzo] where rates are derived. Dareiotis and Gerencsér [@DarGer] obtain convergence with $L^2$-order $1/2-$ (meaning $1/2-\epsilon$ for arbitrarily small $\epsilon >0$) in the time-step for bounded and Dini-continuous time-homogeneous drift coefficients and check that this order is preserved in dimension $d=1$ when the Dini-continuity assumption is relaxed to mere measurability. In the scalar $d=1$ case, Neuenkirch and Szölgyenyi [@NeuSzo] assume that the drift coefficient is the sum of a $ \mathcal{C}^{2}_b$ part and a bounded integrable irregular part with a finite Sobolev-Slobodeckij semi-norm of index $\kappa \in (0,1)$. They prove $L^2$-convergence with order $\frac{3}{4}\wedge\frac{1 + \kappa}{2} -$ for the equidistant Euler-Maruyama scheme, the cutoff of this order at $\frac{3}{4}$ disappearing for a suitable non-equidistant time-grid. Note that an exact simulation algorithm has been proposed by Étoré and Martinez [@EtoMArt] for one-dimensional SDEs with additive noise and time-homogeneous and smooth except at one discontinuity point drift coefficient.\
More papers have been devoted to the strong error of the Euler scheme for SDEs with a non constant diffusion coefficient. The first result goes back to Gyöngy and Krylov [@GyoKryl] who established convergence in probability (without any rate) when the coefficients are continuous in space and pathwise uniqueness holds for the stochastic differential equation. Gyöngy [@Gyo] proves almost sure convergence with order $1/4-$ when the diffusion coefficient is locally Lipschitz in space and the drift coefficient locally one-sided Lipschitz in space uniformly in time and some Lyapunov condition holds. Yan [@Yan] investigates conditions under which the Euler scheme converges to the unique weak solution of the SDE. In dimension $d=1$, this author also shows the $L^1$-order $\beta_1\wedge\frac{\alpha}{2}\wedge\frac{\alpha}{1+\alpha}\beta_2$ when the drift coefficient is Lipschitz in the spatial variables and $\beta_1$-Hölder in time while the diffusion coefficient is $(\frac{1}{2}+\alpha)$-Hölder in space and $\frac{\beta_2}{2}$-Hölder in time. Still in dimension one, Gyongy and Rasonyi [@GR] obtain the $L^1$-order $\alpha\wedge\frac{\gamma}{2}$ when the diffusion coefficient is $(\frac{1}{2}+\alpha)$-Hölder continuous in space and the drift coefficient is the sum of a function Lipschitz continuous in space and a function non-increasing and $\gamma$-Hölder continuous in space. Like in [@GyoKryl; @Gyo], the discretization only concerns the spatial variable of the coefficients while the time variable still moves continuously in the scheme analysed. In [@NgTag1], Ngo and Taguchi prove $L^1$-order $1/2$ when (resp. $\alpha\in (0,1/2]$, when $d=1$,) the diffusion coefficient is uniformly elliptic, bounded, Lipschitz (resp. $(\frac{1}{2}+\alpha)$-Hölder) in space and the drift coefficient one-sided Lipschitz, bounded and with bounded variation in space with respect to some Gaussian measure. The coefficients are assumed to be $1/2$-Hölder with respect to the time variable. In a second paper [@NgTag2] specialized to dimension $d=1$ with time-homogeneous coefficients, they show $L^1$-order $\frac{\beta}{2}\wedge \alpha$ under the same assumption on the diffusion coefficient and when the drift is the sum of a bounded $\beta$-Hölder function and a bounded function with bounded variation with respect to some Gaussian measure. When the diffusion coefficient is uniformly elliptic, Lipschitz continuous in space, Dini-continuous in time and the drift coefficient is Dini-continuous in both variables, Bao, Huang and Yuan [@BHY] prove $L^2$-convergence with an order expressed in terms of the Dini modulus of continuity.\
Recent attention has been paid to the Euler-Maruyama discretization of SDEs with a piecewise Lipschitz drift coefficient and a globally Lipschitz diffusion coefficient which satisfies some non-degeneracy condition on the discontinuity hypersurface of the drift coefficient. Leobacher and Szölgyenyi [@LeoSzo3] prove convergence with $L^2$-order $1/4-$ of the Euler-Maruyama method. This result is proved by comparison with a scheme with $L^2$-order $1/2$ [@LeoSzo1; @LeoSzo2] obtained by the Euler discretization of a transformation of the original SDE which permits to remove the discontinuity of the drift. In dimension $d=1$, Müller-Gronbach and Yaroslavtseva [@MGYaro1] recover for each $p\in[1,\infty)$ the $L^p$-order of convergence $1/2$ valid when the drift coefficient is globally Lipschitz. In higher dimension, this order $1/2$ (up to some logarithmic factor) is proved by Neuenkirch, Szölgyenyi and Szpruch [@NeunSzoSpr] for the $L^2$-error of an Euler-Maruyama scheme with adaptive time-stepping.\
Concerning the weak error, Mikulevicius and Platen [@MikP] prove that, under uniform ellipticity, when the coefficients are $\alpha$-Hölder with respect to the spatial variables and $\frac{\alpha}{2}$-Hölder with respect to the time variable for $\alpha\in(0,1)$, then ${{\mathbb E}}[f(X_T)]$ is approximated with order $\frac{\alpha}{2}$ by replacing $X_T$ by the Euler-Maruyama scheme at time $T$ when the test function $f:{{\mathbb R}}^d\to {{\mathbb R}}^d$ is twice continuously differentiable with $\alpha$-Hölder second order derivatives. In the additive noise case, Kohatsu-Higa, Lejay and Yasuda [@KohY], prove that for $f$ thrice continuously differentiable with polynomially growing derivatives, the convergence holds with order $1/2-$ when $d\ge 2$ (resp. $1/3-$ when $d=1$) and the drift coefficient is time homogeneous, bounded and Lipschitz except on a set $G$ such that $\varepsilon^{-d}$ times the Lebesgue measure of $\{x\in{{\mathbb R}}^d:\inf_{y\in G}|x-y|\le\varepsilon\}$ is bounded. Their approach, which consists in regularizing the drift coefficient and considering both the stochastic differential equation and the Euler scheme for the regularized coefficient, is also applied in [@Koh] to the case of time-dependent, bounded, uniformly elliptic and continuous diffusion coefficients. In [@MenoKona], Konakov and Menozzi regularize both coefficients to obtain that the absolute difference between the densities with respect to the Lebesgue measure of the solution and its Euler-Maruyama discretization is bounded from above by a Gaussian density multiplied by a factor with order $\frac{\alpha}{2}-$, when these coefficients are uniformly elliptic, bounded and $\frac{\alpha}{2}$-Hölder continuous in time and $\alpha$-Hölder continuous in space. The order of the factor is $\frac{1}{2d}-$ when the coefficients are bounded, uniformly elliptic, continuously differentiable with bounded derivatives up to the order $2$ in time and the order $4$ in space at the possible exception, for the drift coefficient, of a finite union of time-independent smooth submanifolds where it can be discontinuous. In [@NFrik], Frikha deals with time-homogeneous one-dimensional stochastic differential equations possibly involving a local time term in addition to a bounded measurable drift coefficient and a bounded uniformly elliptic and $\alpha$-Hölder diffusion coefficient. He proves that the absolute difference between the densities is smaller than a Gaussian density multiplied by a factor with order $\frac{\alpha}{2}$ in the time-step. The latest work in this field is by Suo, Yuan and Zhang [@SuYuZ] who study in a multidimensional setting stochastic differential equations with additive noise and time-homogeneous drift coefficients with at most linear growth and satisfying an integrated against some Gaussian measure $\alpha$-Hölder type regularity condition. When this coefficient has sublinear growth (and under some restriction on the time-horizon when it has linear growth), they prove convergence in total variation with order $\frac{\alpha}{2}$.\
In the current paper, we consider the stochastic differential equation with additive noise and bounded and measurable drift function $b:[0,T]\times{{\mathbb R}}^d\to{{\mathbb R}}^d$. We are interested in estimating the spatial integral of the absolute difference between the densities with respect to the Lebesgue measure of the solution and its Euler-Maruyama discretization with time-step $h\in(0,T]$. This integral is equal to the total variation distance between the probability measures that admit these densities with respect to the Lebesgue measure. Note that the approximation of a Markovian semi-group in total variation distance has been investigated by Bally and Rey [@Bally] who apply their results to the Ninomiya discretization scheme.
To get rid with any assumption stronger than mere measurability concerning the regularity of the drift coefficient with respect to the time variable, we consider the Euler-Maruyama discretization with randomized time variable of . It evolves inductively on the regular time-grid $(kh)_{k \in \left\llbracket0,\left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket}$ by: $$X^{h}_{(k+1)h}=X^h_{kh}+\left(W_{(k+1)h}-W_{kh}\right)+b\left(\delta_{k}, X^h_{kh}\right)h, \label{euler}$$ where the random variables $(\delta_k)_{k \in \left\llbracket0,\left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket}$ are independent, respectively distributed according to the uniform law on $[kh,(k+1)h]$ and are independent from $\left(X_0,(W_t)_{t\ge 0}\right)$. Notice that this sequence is of course not needed to randomize the time variable when $b$ is time-homogeneous. To our knowledge, such randomization techniques have been proposed so far to improve the strong convergence properties of discretizations of ordinary differential equations [@Stengle1; @Stengle2; @HeinMil; @Daun] or stochastic differential equations [@CruW]. They also happen to be quite efficient in terms of weak error. Indeed, the above randomization turns out to enable convergence in total variation distance with order $1$ up to some logarithmic factor. For $s\in[0,T)$, we denote by $\ell_s = \lfloor s/h\rfloor$ the index of the corresponding time interval $s \in [kh,(k+1)h)$. s.t. $k \le \lfloor T/h\rfloor -1$. We consider then the following continuous time interpolation of the scheme: $$X^h_t = X_0 + W_t + \int_0^t b\left(\delta_{\ell_s},X^h_{\tau^h_s}\right)\,ds,\quad t\in[0,T]\label{conteuler}.$$ For $t\ge 0$, we denote by $\mu_t$ the law of $X_t$ and by $\mu^h_t$ the law of $X^h_t$. We have $\mu_0=\mu^h_0 = m$. For $t>0$, according to Proposition \[propspde\] below, $\mu_t$ and $\mu^h_t$ admit densities $p(t,.)$ and $p^h(t,.)$ with respect to the Lebesgue measure. Therefore, our approach amounts to study the rate of convergence of the $L^1$-norm of the difference between $p(t,.)$ and $p^h(t,.)$. We assume, in what is next, that $X_0$ is distributed according to a probability measure $m$ on ${{\mathbb R}}^d$ and the drift $b=(b_{i})_{1 \le i \le d}:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is a measurable function bounded by $B<+\infty$ when ${{\mathbb R}}^d$ is endowed with the $L^\infty$-norm.\
The paper is organized as follows. In Section \[MainRes\], we state our main results. We first obtain the convergence of the weak error in total variation in $\mathcal{O}(\sqrt h)$ when $b$ is measurable and bounded. When assuming more regularity on $b$ with respect to the space variables, namely that the divergence in the sense of distributions of $b$ with respect to these variables is in $L^{\rho}\left({{\mathbb R}}^d\right)$ for some $\rho \ge d$ uniformly with respect to the time variable, the weak rate of convergence $\left\|\mu_{kh} - \mu^h_{kh}\right\|_{\rm TV}$ is $\frac{1}{\sqrt{kh}}\left(1 + \ln\left(k\right)\right)h$ and it improves, at the terminal time, to $1$ up to a logarithmic factor. Furthermore, when assuming more regularity on the probability measure $m$ in addition to the spatial regularity on $b$, we improve the previous weak rate of convergence by eliminating the prefactor $\frac{1}{\sqrt{kh}}$. We obtain these results by comparing the mild equation satisfied by $p(t,.)$ and the perturbed mild equation satisfied by $p^h(t,.)$. We investigate through the Lamperti transform the application of those theorems to one-dimensional SDEs with a non-constant diffusion coefficient. Sections \[preuve\] and \[preuveB\] are dedicated to the proofs of the main results in Section \[MainRes\]. We finally provide numerical experiments in Section \[NumExp\] to illustrate our results.\
Beforehand, note that our results also apply to the more general case of SDEs with constant and non degenerate diffusion coefficient $\sigma \in {{\mathbb R}}^d \times {{\mathbb R}}^d$: $$\begin{aligned}
\displaystyle Y_t = Y_0 + \sigma W_t + \int_0^t \tilde b \left(s,Y_s\right)\,ds, \quad t\in[0,T]\end{aligned}$$ where the ${{\mathbb R}}^d$-valued random variable $Y_0$ is independent from $\left(W_t\right)_{t \ge 0}$ and $\tilde b:[0,T]\times {{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded. Indeed, our results remain true for this type of diffusions since the transformation $\left(X_t = \sigma^{-1}Y_t\right)_{t \in [0,T]}$ is solution to the dynamics initialized by $X_0 = \sigma^{-1}Y_0$ for the choice of $b(t,x) = \sigma^{-1} \tilde b(t,\sigma x)$. The associated Euler scheme evolving inductively on the time grid $\left(kh\right)_{k \in \left \llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right \rrbracket}$ is defined by: $$\begin{aligned}
\displaystyle X^h_{(k+1)h} = X^h_{kh} + \left(W_{(k+1)h}-W_{kh}\right) + \sigma^{-1} \tilde b\left(\delta_k, \sigma X^h_{kh} \right)h,\end{aligned}$$ and we can clearly see that $\left(\sigma X^h_{kh}\right)_{k \in \left \llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right \rrbracket}$ coincides exactly with the Euler scheme $\left(Y^h_{kh}\right)_{k \in \left \llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right \rrbracket}$ of the process $\left(Y_t\right)_{t \in [0,T]}$. Denoting by $\tilde \mu_t$ the law of $Y_t$ and by $\tilde \mu^h_t$ the law of $Y^h_t$ the continuous time interpolation of the Euler scheme, we have that $\left\|\tilde \mu_t - \tilde \mu^h_t \right\|_{\rm TV} = \left\| \mu_t - \mu^h_t \right\|_{\rm TV}$. Moreover, Lemma \[lema1\] in the appendix, which relates the spatial divergences in the sense of distributions of $y\mapsto \tilde b(t,y)$ and $x\mapsto \sigma^{-1}\tilde b(t,\sigma x)$, ensures that when the drift coefficient $\tilde b(t,y)$ satisfies the strengthened hypotheses in Theorem \[cvgenceB\] below, then so does $\sigma^{-1}\tilde b(t,\sigma x)$.\
Before going any further, we introduce some additional notation.
Notation: {#notation .unnumbered}
---------
- For $x \in {{\mathbb R}}^d$, we denote by $|x| = \left(\sum \limits_{i=1}^d x_{i}^2 \right)^{1/2}$ the euclidean norm of $x$.
- For $ 1 \le p < \infty$, we denote by $L^p\left({{\mathbb R}}^d\right)$ the space of measurable functions on ${{\mathbb R}}^d$ which are $L^p$-integrable for the Lebesgue measure i.e. $f\in L^p$ if $\displaystyle \left\Vert f\right\Vert_{L^p} = \left( \int_{{{\mathbb R}}^d} |f(x)|^p \; dx \right)^{\frac{1}{p}}<+\infty$.
- The space $L^{\infty}\left({{\mathbb R}}^d\right)$ refers to the space of almost everywhere bounded measurable functions on ${{\mathbb R}}^d$ endowed with the norm $\left\Vert f\right\Vert_{L^{\infty}} = \inf \left\{ C \ge 0: |f(x)|\le C \; dx \text{ a.e. on } {{\mathbb R}}^d \right\} $.
- For notational simplicity, when a function $g$ is defined on $[0,T] \times {{\mathbb R}}^d$ and $x \in {{\mathbb R}}^d$, we may use sometimes the notation $g_0(x) := g(0,x)$.
- We denote by $W^{1,1}\left({{\mathbb R}}^d\right)$ the Sobolev space over ${{\mathbb R}}^d$ defined as $W^{1,1}\left({{\mathbb R}}^d\right) \equiv \left\{u \in L^1\left({{\mathbb R}}^d\right): \nabla u \in L^1\left({{\mathbb R}}^d\right)^d \right\}$ where $\nabla u$ refers to the spatial derivative of $u$ in the sense of distributions. The space is endowed with the norm $\left\| u\right\|_{W^{1,1}} = \left\| u\right\|_{L^1} + \sum\limits_{i=1}^{d}\left\|\partial_{x_i}u \right\|_{L^1}$.
- For any open subset $A \subset {{\mathbb R}}^d$, we denote by $\mathcal{C}^k_c(A)$ the space of real functions continuously differentiable in $A$ up to the order $k \in {{\mathbb N}}$, with compact support on $A$.
- We denote by $BV({{\mathbb R}})$ the space of functions with bounded variation on ${{\mathbb R}}$. For a function $f \in L^1_{\rm loc}\left({{\mathbb R}}\right)$, if $f \in BV({{\mathbb R}})$ then the derivative of $f$ in the sense of distributions is a finite measure in ${{\mathbb R}}$.
- Let $\left(\mu_1, \dots, \mu_{d^*}\right)$ be signed bounded measures on ${{\mathbb R}}^d$ and $f:{{\mathbb R}}^d \to {{\mathbb R}}^{d^*}$ a $\mathcal{C}^0$-integrable function, with $d^* \in \{ 1,d\}$. We define the convolution product of $f$ and $\mu = \left(\mu_1, \dots, \mu_{d^*}\right)$ by: $$\displaystyle \bigg(f * \mu\bigg)(x) = \sum_{i=1}^{d^*}\int_{{{\mathbb R}}^d} f_{i}(x-y)d\mu_{i}(y) \; \text{ for } x \in {{\mathbb R}}^d.$$ When each $\mu_{i}$ admits a density $g_{i}$ with respect to the Lebesgue measure, we also denote by $\Big(f*g\Big)$ this convolution product.
Main results {#MainRes}
============
In this section, we give the main results concerning the convergence of $\mu^h_t$, the law of the Euler discretization with time-step $h$ towards its limit $\mu_t$. We will make an intensive use of the interpretation of the total variation norm of their difference as the $L^1$-norm of the difference between their respective densities $p^h(t,.)$ and $p(t,.)$.\
We recall that the total variation norm for a signed measure $\mu$ on ${{\mathbb R}}^d$ is defined as: $$\begin{aligned}
\label{totVar}
\displaystyle \left\|\mu \right\|_{\rm TV} = \sup_{\varphi \in \mathcal{L}} \int_{{{\mathbb R}}^d}\varphi(x) d\mu(x)\end{aligned}$$ where $\mathcal{L}$ denotes the set of all measurable functions $\varphi:{{\mathbb R}}^d \to [-1,1]$. Moreover, when $\mu$ admits a density $f_{\mu}$ with respect to a reference non-negative measure $\lambda$, we have the following equality: $$\begin{aligned}
\label{caract}
\displaystyle \left\|\mu\right\|_{\rm TV} = \int_{{{\mathbb R}}^d} \left|f_{\mu}(x)\right|\lambda(dx) .\end{aligned}$$ Let us now state our estimation of the weak convergence rate of the Euler scheme towards its limit.
\[cvgence\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. Then: $$\begin{aligned}
\exists \, C<+\infty,\;\forall h \in (0,T],\; \forall k \in \left\llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket, \quad \left\|\mu_{kh}-\mu^h_{kh} \right\|_{\rm TV}\le C \sqrt h.
\end{aligned}$$
- In dimension $d=1$, when specialized to the constant diffusion coefficient and absence of local time term case, Theorem 2.2 in [@NFrik] gives a finer estimation of the absolute difference between the densities by $C\sqrt{h}$ times some Gaussian density. Our result is recovered by spatial integration. It implies that for any bounded and measurable test function $f:{{\mathbb R}}^d\to{{\mathbb R}}$, ${{\mathbb E}}\left[f\left(X^h_{kh}\right)\right]-{{\mathbb E}}\left[f\left(X_{kh}\right)\right]={\mathcal O}(\sqrt{h})$.
- Up to some factor $h^{-\varepsilon}$ with $\varepsilon$ arbitrarily small, this behaviour was proved in dimension $d\ge 2$ for thrice continuously differentiable test functions (with polynomial growth together with their derivatives) $f$ by Kohatsu-Higa, Lejay and Yasuda [@KohY], when the drift coefficient is time-homogeneous and Lipschitz outside some sufficiently small set.
- Suo, Yuan and Zhang [@SuYuZ] proved convergence with order $\frac{\alpha}{2}$ in total variation when the drift coefficient is time-homogeneous and satisfies some integrated against a Gaussian measure $\alpha$-Hölder type of regularity condition. Since $\alpha$ appears to take values smaller than one, we obtain the better order of convergence $1/2$ without any regularity assumption. On the other hand, we need boundedness of the drift coefficient whereas Suo, Yan and Zhang get rid of this assumption and only assume sublinear growth and even linear growth but with some restriction on the time-horizon $T$.
Now, when assuming more regularity on $b$ with respect to the space variables, we obtain a better rate of convergence:
\[cvgenceB\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. If $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or for $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$; where $\nabla \cdot b(t,.)$ and $\partial_x b(t,.)$ are respectively the spatial divergence and the spatial derivative of $b$ in the sense of distributions. Then: $$\begin{aligned}
\displaystyle \exists \, \tilde{C}<+\infty, \;\forall h \in (0,T],\; \forall k \in \left\llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket, \quad \left\|\mu_{kh}-\mu^h_{kh} \right\|_{\rm TV} \le \frac{\tilde{C}}{\sqrt{kh}}\Big(1 + \ln\left(k\right)\Big)h.
\end{aligned}$$
As a consequence, when $n$ is a positive integer, we have that $ \left\|\mu_{T}-\mu^{T/n}_{T} \right\|_{\rm TV} \le \tilde C\sqrt{T}\left(\frac{1+\ln(n)}{n}\right)$. Therefore, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. This in particular applies to the bounded one-dimensional time-homogeneous drift coefficient with bounded variation defined by Suo, Yuan and Zhang in Example 2.3 [@SuYuZ] using some Cantor set, for which they obtain convergence with order $1/4$ uniformly in time.
- Of course, the regularity assumption on the drift coefficient $b$ is satisfied when it is Lipschitz in space (which is equivalent to the boundedness of its spatial gradient in the sense of distributions). When $d\ge 2$, the regularity assumption only involves the spatial divergence and not the full spatial gradient $\nabla b(t,.)$ in the sense of distributions. Note that if we suppose the stronger assumption $\sup_{t \in [0,T]} \|\nabla b(t,.)\|_{L^\rho}<\infty$ for $\rho\in(d,+\infty]$, then, according to the boundedness assumption and Corollary $IX.14$ [@Brezis], the drift is locally $\left(1 - \frac{d}{\rho} \right)$-Hölder continuous in space.
- Theorem 1.6 [@MenoKona] deals with a drift coefficient bounded, continuously differentiable with bounded derivatives up to the order $2$ in time and the order $4$ in space outside a finite union of time-independent smooth submanifolds where it can be discontinuous. Its constant diffusion statement says that when $m$ is a Dirac mass, then the absolute difference between the densities is bounded from above by a Gaussian density multiplied by a factor sum of a term with order $\frac{1}{d}-$ in our total variation rate $\frac{h}{\sqrt{kh}}$ and a term with order $1-$ in $h$ over the distance to the discontinuity set.
Furthermore, if we assume more regularity on $m$ in addition to the spatial regularity on $b$, we obtain the following result:
\[propregM\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. Moreover, assume that $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or that for $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$; where $\nabla \cdot b(t,.)$ and $\partial_x b(t,.)$ are respectively the spatial divergence and the spatial derivative of $b$ in the sense of distributions. If $m$ admits a density w.r.t. the Lebesgue measure that belongs to $W^{1,1}\left({{\mathbb R}}^d\right)$ then: $$\begin{aligned}
\displaystyle \exists \, \hat{C}<+\infty, \;\forall h \in (0,T],\; \forall k \in \left\llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket, \quad \left\|\mu_{kh}-\mu^h_{kh} \right\|_{\rm TV} \le \hat{C}\Big(1 + \ln\left(k\right)\Big)h.
\end{aligned}$$
- We can see in Theorem \[cvgenceB\] that when $b$ is more regular with respect to the space variables, the weak convergence rate in total variation is bounded by $\left(1 + \ln\left(k\right)\right)h$ times a prefactor $\frac{1}{\sqrt{kh}}$ that decreases over time and explodes in small time. According to Proposition \[propregM\], this prefactor is removed when assuming more regularity on $m$.
- For $\varphi:{{\mathbb R}}^d\to{{\mathbb R}}$, measurable and bounded and using Equation , we deduce from Theorem \[cvgence\] that $\left|{{\mathbb E}}\left[\varphi\left(X^h_{kh}\right)\right]-{{\mathbb E}}\left[\varphi\left(X_{kh}\right)\right]\right|\le C \|\varphi\|_\infty \sqrt h$, from Theorem \[cvgenceB\] that $\left|{{\mathbb E}}\left[\varphi\left(X^h_{kh}\right)\right]-{{\mathbb E}}\left[\varphi\left(X_{kh}\right)\right]\right|\le \frac{\tilde{C} \|\varphi\|_\infty}{\sqrt{kh}}\left(1 + \ln\left(k\right)\right)h$ and from Proposition \[propregM\] that $\left|{{\mathbb E}}\left[\varphi\left(X^h_{kh}\right)\right]-{{\mathbb E}}\left[\varphi\left(X_{kh}\right)\right]\right|\le \hat{C} \|\varphi\|_\infty \left(1 + \ln\left(k\right)\right)h$.
The proofs of the two theorems and the proposition, that we will detail in Sections \[preuve\] and \[preuveB\], rely on the propositions that we present in Section \[mild\]. Before going any further, we investigate through the Lamperti transform the application of those theorems to SDEs with non-constant diffusion coefficient in dimension $d=1$.
Application to one-dimensional SDEs with non-constant diffusion coefficient
---------------------------------------------------------------------------
Let us consider the one-dimensional stochastic differential equation: $$\begin{aligned}
\label{diffusY}
Y_t = Y_0 + \int_0^t \sigma\left(Y_s \right)\,dW_s + \int_0^t \beta\left(s,Y_s \right)\,ds, \quad t \in [0,T]\end{aligned}$$ where $(W_t)_{t \ge 0}$ is a one-dimensional Brownian motion independent from $Y_0$ which is some $(l,r)$-valued random variable with $-\infty \le l < r \le +\infty$. Let $z \in (l,r)$, we assume that $\sigma: (l,r) \to {{\mathbb R}}^{*}_{+}$ is a $\mathcal{C}^1$ function with $\displaystyle \lim_{y \to \substack{r \\ l} } \int_{z}^y \frac{dw}{\sigma(w)}= \pm \infty$, $\beta: [0,T]\times (l,r) \to {{\mathbb R}}$ is measurable and that $(t,x) \mapsto \left( \frac{\beta(t,x)}{\sigma(x)} - \frac{\sigma^{'}(x)}{2}\right)$ is bounded on $[0,T] \times (l,r)$. We introduce the Lamperti transform $\Big(X_t = \psi\left(Y_t\right)\Big)_{t\in [0,T]}$ where $\psi: (l,r) \to {{\mathbb R}}$ is defined by $\displaystyle \psi(y) = \int_z^y \frac{dw}{\sigma(w)}$ . By Itô’s formula, $\left(X_t\right)_{t\in [0,T]}$ is solution to the dynamics for the choice $d=1$, $\displaystyle b(t,.)= \left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1}$ and initialized by $\psi(Y_0)$: $$\begin{aligned}
\label{diffusX}
X_t = \psi(Y_0) + W_t + \int_0^t \left( \frac{\beta(s,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ\psi^{-1}\left(X_s \right) \,ds, \quad t \in [0,T].\end{aligned}$$ Existence and uniqueness for can be deduced from the existence and uniqueness for . Indeed, according to [@Veret], the SDE admits a pathwise unique strong solution $\left(X_t\right)_{t \in [0,T]}$. By Itô’s formula, $\left(\psi^{-1}\left(X_t \right)\right)_{t \in [0,T]}$ is a solution to . As for the uniqueness, the images of any two solutions of by $\psi$ coincide by uniqueness for . Since $\psi: (l,r) \to {{\mathbb R}}$ is one to one, these two solutions coincide. For $t \ge 0$, we denote by $\nu_t$ the probability distribution of $Y_t$. The law $\mu_t$ of $X_t$ is then the pushforward of $\nu_t$ by $\psi$ i.e. $\mu_t = \psi \# \nu_t$ and conversely $\nu_t = \psi^{-1} \# \mu_t$.\
We are going to approximate $\left(Y^h_{kh}\right)_{k \in \left\llbracket0,\left\lfloor \frac{T}{h} \right\rfloor\right\rrbracket}$ by $\left(\psi^{-1}\left(X^h_{kh}\right)\right)_{k \in \left\llbracket0,\left\lfloor \frac{T}{h} \right\rfloor\right\rrbracket}$ where $\left(X^h_{kh}\right)_{k \in \left\llbracket0,\left\lfloor \frac{T}{h} \right\rfloor\right\rrbracket}$ is the Euler scheme of with time-step $h \in (0,T]$ initialized by $X^h_0 = \psi(Y_0)$ and evolving inductively on the time grid $(kh)_{k \in \left\llbracket0,\left\lfloor \frac{T}{h} \right\rfloor\right\rrbracket}$ by: $$\begin{aligned}
\label{eulerX}
X^h_{(k+1)h}=X^h_{kh}+\left(W_{(k+1)h}-W_{kh}\right)+\left( \frac{\beta(\delta_k,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1}\left(X^h_{kh}\right)h, \end{aligned}$$ where the random variables $\delta_k$ are independent, distributed according to the uniform law on $[kh,(k+1)h]$ and are independent from $\left(Y_0,(W_t)_{t\ge 0}\right)$.\
We denote by $\nu^h_t$ the law of $\psi^{-1}\left(X^h_t \right)$. Since for $\varphi \in \mathcal{L}$ defined right after , $\varphi \circ \psi^{-1} \in \mathcal{L}$, we have: $$\begin{aligned}
\displaystyle \left\|\nu_{kh}-\nu^h_{kh} \right\|_{\rm TV} = \left\|\psi^{-1} \#\mu_{kh}-\psi^{-1} \#\mu^h_{kh} \right\|_{\rm TV} &= \sup_{\varphi \in \mathcal{L}} \int_{{{\mathbb R}}}\varphi\left(\psi^{-1}(x)\right)\left(\mu_{kh}(dx) - \mu_{kh}^h(dx)\right) \le \left\|\mu_{kh}-\mu^h_{kh} \right\|_{\rm TV}.\end{aligned}$$ On the other hand, for $\mathcal{\hat L}$ denoting the set of all measurable functions $\hat \varphi:(l,r) \to [-1,1]$, if $\varphi \in \mathcal{L}$ then $\hat \varphi = \varphi \circ \psi \in \mathcal{\hat L}$ and: $$\begin{aligned}
\displaystyle \left\|\mu_{kh}-\mu^h_{kh} \right\|_{\rm TV} = \sup_{\varphi \in \mathcal{L}}\int_{{{\mathbb R}}} \varphi(x)\Big(\mu_{kh}(dx) - \mu_{kh}^h(dx)\Big) &= \sup_{\varphi \in \mathcal{L}}\int_{{{\mathbb R}}} \varphi\circ \psi(x)\left(\nu_{kh}(dx) - \nu_{kh}^h(dx)\right)\\
&\le \sup_{\hat \varphi \in \mathcal{\hat L}}\int_{(l,r)} \hat \varphi(x)\left(\nu_{kh}(dx) - \nu_{kh}^h(dx)\right) = \left\|\nu_{kh}-\nu^h_{kh}\right\|_{\rm TV}.\end{aligned}$$
Therefore, $\left\|\nu_{kh}-\nu^h_{kh} \right\|_{\rm TV} = \left\|\mu_{kh}-\mu^h_{kh} \right\|_{\rm TV}$ and we obtain directly from Theorem \[cvgence\] the following result for the weak convergence rate of $\nu^h_t$ towards $\nu_t$:
\[cvgenceY\] Assume $\sigma:(l,r) \to {{\mathbb R}}^{*}_{+}$ is $\mathcal{C}^1$ with $\displaystyle \lim_{y \to \substack{r \\ l} } \int_{z}^y \frac{dw}{\sigma(w)}= \pm \infty$, $\beta:[0,T]\times(l,r) \to {{\mathbb R}}$ is measurable and $(t,x) \mapsto \left(\frac{\beta(t,x)}{\sigma(x)} - \frac{\sigma^{'}(x)}{2}\right)$ is bounded on $[0,T] \times (l,r)$. Then: $$\begin{aligned}
\exists \, C<+\infty,\;\forall h \in (0,T],\; \forall k \in \left\llbracket 0, \left\lfloor \frac{T}{h}\right\rfloor \right\rrbracket, \quad \left\|\nu_{kh}-\nu^h_{kh} \right\|_{\rm TV}\le C \sqrt h.
\end{aligned}$$
Let us now discuss the assumptions on $\beta$ and $\sigma$ in order to apply Theorem \[cvgenceB\]. According to Definition $3.4$ [@AmbFusPal], the variation $V\Big(\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1},{{\mathbb R}}\Big)$ of $\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1}$ in ${{\mathbb R}}$ is defined by: $$\begin{aligned}
\displaystyle V\left(\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1},{{\mathbb R}}\right) &:= \sup\left\{ \int_{{{\mathbb R}}}\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1}(x)\varphi^{'}(x)\,dx :\; \varphi \in \mathcal{C}^1_c({{\mathbb R}}), \; \|\varphi \|_{\infty}\le 1 \right\} \\
&:= \sup\left\{ \int_{(l,r)}\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)(y)\left(\varphi\circ\psi\right)^{'}(y)\,dy :\; \varphi \in \mathcal{C}^1_c({{\mathbb R}}), \; \|\varphi \|_{\infty}\le 1 \right\}. \end{aligned}$$ Since $\psi$ is a $\mathcal{C}^1$-diffeomorphism from $(l,r)$ to ${{\mathbb R}}$: $$\begin{aligned}
\displaystyle V\left(\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1},{{\mathbb R}}\right) &:= \sup\left\{ \int_{(l,r)}\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)(y)\tilde \varphi^{'}(y)\,dy :\; \tilde \varphi \in \mathcal{C}^1_c((l,r)), \; \|\tilde \varphi \|_{\infty}\le 1 \right\} \\
&= V\left(\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right), (l,r) \right).\end{aligned}$$ Moreover, using Proposition $3.6$ [@AmbFusPal], we have that: $$\displaystyle \left\|\partial_x b(t,.) \right\|_{\rm TV} = V\left(\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\circ \psi^{-1},{{\mathbb R}}\right) = V\left(\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right), (l,r) \right) = \left\|\partial_x \left(\frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\right\|_{\rm TV}$$ where the spatial derivatives are defined in the sense of distributions on ${{\mathbb R}}$ and $(l,r)$. Therefore, when assuming more regularity on $\left( \frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)$ with respect to the space variables, we obtain a better rate of convergence:
\[cvgenceBY\] Assume $\sigma:(l,r) \to {{\mathbb R}}^{*}_{+}$ is $\mathcal{C}^1$ with $\displaystyle \lim_{y \to \substack{r \\ l} } \int_{z}^y \frac{dw}{\sigma(w)}= \pm \infty$, $\beta:[0,T]\times(l,r) \to {{\mathbb R}}$ is measurable and $(t,x) \mapsto \left(\frac{\beta(t,x)}{\sigma(x)} - \frac{\sigma^{'}(x)}{2}\right)$ is bounded on $[0,T] \times (l,r)$. Moreover, assume that $\sup_{t \in [0,T]}\left\|\partial_x \left(\frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\right\|_{\rm TV}<+\infty$ where the spatial derivative is defined in the sense of distributions on $(l,r)$. Then: $$\begin{aligned}
\displaystyle \exists \, \tilde{C}<+\infty, \;\forall h \in (0,T],\; \forall k \in \left\llbracket 0, \left\lfloor \frac{T}{h} \right \rfloor \right\rrbracket, \quad \left\|\nu_{kh}-\nu^h_{kh} \right\|_{\rm TV} \le \frac{\tilde{C}}{\sqrt{kh}}\Big(1 + \ln\left(k\right)\Big)h.
\end{aligned}$$
We will now discuss the additional assumptions on the law of $Y_0$ and $\sigma$ in order to apply Proposition \[propregM\]. Let us assume that $Y_0$ admits a density $q_0$ w.r.t. the Lebesgue measure. By a change of variables, $X_0$ admits the density $(\sigma q_0)\circ \psi^{-1}$ w.r.t. the Lebesgue measure. Since $q_0 \in L^1\left((l,r)\right)$ and $\sigma$ is $\mathcal{C}^1$ so locally bounded on $(l,r)$, $(\sigma q_0)$ defines a distribution. Moreover, we assume that $\Big(\sigma q_0\Big)' \in L^1\left((l,r)\right)$ where the derivative of $(\sigma q_0)$ is defined in the sense of distributions. Now, let $I$ be a compact subinterval of $(l,r)$ and let $\varphi$ be a $\mathcal{C}^{\infty}_c$ function on ${{\mathbb R}}$ such that $(\varphi \circ \psi)$ is null outside $I$. We have, through a change of variables, that: $$\begin{aligned}
\displaystyle \int_{{{\mathbb R}}} \varphi'(x)(\sigma q_0)\circ \psi^{-1}(x)\,dx = \int_{(l,r)} \Big(\varphi \circ \psi \Big)'(y) (\sigma q_0)(y)\,dy.\end{aligned}$$ Since $\left(\varphi \circ \psi\right) \in W^{1,1}\left(I\right)$ and $(\sigma q_0) \in W^{1,1}\left(I\right)$, we can apply Corollary $VIII.9$ [@Brezis] and obtain that: $$\begin{aligned}
\displaystyle \int_{{{\mathbb R}}} \varphi'(x)(\sigma q_0)\circ \psi^{-1}(x)\,dx = - \int_{(l,r)} (\varphi \circ \psi)(y) \Big(\sigma q_0\Big)'(y)\,dy = -\int_{{{\mathbb R}}} \varphi(x) \Big(\sigma q_0\Big)'\circ \psi^{-1}(x) \sigma \circ \psi^{-1}(x)\,dx.\end{aligned}$$ Hence, we get, in the sense of distributions, that $\Big((\sigma q_0) \circ \psi^{-1}\Big)'=(\sigma q_0)'\circ \psi^{-1} \times \sigma \circ \psi^{-1}$. Through a change of variables, we obtain that $\displaystyle \left\|\Big((\sigma q_0)\circ \psi^{-1} \Big)'\right\|_{L^1({{\mathbb R}})}= \left\|\Big(\sigma q_0\Big)'\right\|_{L^1((l,r))} $ and since $(\sigma q_0)' \in L^1\left((l,r)\right)$, we conclude that the density of $X_0$ is in $W^{1,1}\left({{\mathbb R}}\right)$.
Assume $\sigma:(l,r) \to {{\mathbb R}}^{*}_{+}$ is $\mathcal{C}^1$ with $\displaystyle \lim_{y \to \substack{r \\ l} } \int_{z}^y \frac{dw}{\sigma(w)}= \pm \infty$, $\beta:[0,T]\times(l,r) \to {{\mathbb R}}$ is measurable and $(t,x) \mapsto \left(\frac{\beta(t,x)}{\sigma(x)} - \frac{\sigma^{'}(x)}{2}\right)$ is bounded on $[0,T] \times (l,r) $. Moreover, assume that $\sup_{t \in [0,T]}\left\|\partial_x \left(\frac{\beta(t,.)}{\sigma} - \frac{\sigma^{'}}{2}\right)\right\|_{\rm TV}<+\infty$ where the spatial derivative is defined in the sense of distributions on $(l,r)$. If $Y_0$ admits a density $q_0$ such that $\left\|(\sigma q_0)'\right\|_{L^1\left((l,r)\right)} < +\infty$, where the spatial derivative is defined in the sense of distributions, then: $$\begin{aligned}
\displaystyle \exists \, \hat{C}<+\infty, \;\forall h \in (0,T],\; \forall k \in \left\llbracket 0, \left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket, \quad \left\|\nu_{kh}-\nu^h_{kh} \right\|_{\rm TV} \le \hat{C}\Big(1 + \ln\left(k\right)\Big)h.
\end{aligned}$$
Existence of densities and mild equations {#mild}
-----------------------------------------
We are going to state, in the next result, the existence for $t>0$ of the densities $p(t,.)$ and $p^h(t,.)$ by showing that $p(t,.)$ solves a mild equation and $p^{h}(t,.)$ solves a perturbed version of this mild equation. Let $G_t(x) = \exp\left(-\frac{|x|^2}{2t}\right) \Big/ \sqrt{(2\pi t)^d } $ denote the heat kernel in ${{\mathbb R}}^{d}$, we have:
\[propspde\] For each $t\in(0,T]$, $\mu_t$ and $\mu^h_t$ admit densities $p(t,.)$ and $p^h(t,.)$ with respect to the Lebesgue measure on ${{\mathbb R}}^d$ s.t. we have $dx$ a.e.: $$\begin{aligned}
\displaystyle p(t,x) &= G_t*m(x) - \int_0^t \nabla G_{t-s}*\Big(b(s,.)p(s,.)\Big)(x)\,ds, \label{mildf}\\
\forall h \in (0,T], \quad p^h(t,x) &= G_t*m(x)-\int_0^t {{\mathbb E}}\left[\nabla G_{t-s}\left(x-X^h_s\right)\cdot b\left(\delta_{\ell_s}, X^h_{\tau^h_s}\right)\right]\,ds. \label{mildfn1}
\end{aligned}$$
Let $t>0$, $f$ be a $\mathcal{C}^2$ and compactly supported function on ${{\mathbb R}}^d$. We set $\varphi(s,x) = G_{t-s} * f(x)$ for $(s,x) \in [0,t) \times {{\mathbb R}}^d$ and $\varphi(t,x) = f(x)$. The function $\varphi(s,x)$ is continuously differentiable w.r.t. $s$ and twice continuously differentiable w.r.t. $x$ on $[0,t]\times{{\mathbb R}}^d$ with bounded derivatives and solves $$\label{edpvarphi}
\partial_s \varphi(s,x)+\frac{1}{2}\Delta \varphi(s,x)=0\mbox{ for }(s,x)\in[0,t]\times{{\mathbb R}}^d.$$ We compute ${{\mathbb E}}\left[\varphi(t,X_t)\right]$ where $(X_s)_{s\ge 0}$ solves . Using , applying Ito’s formula and taking expectations, we obtain that: $${{\mathbb E}}\left[\varphi(t,X_t)\right]={{\mathbb E}}\left[\varphi(0,X_0)+\int_0^t \nabla \varphi (s,X_s).b(s,X_s) \,ds\right].$$ By Fubini’s Theorem and since $G_t$ is even, we obtain: $$\begin{aligned}
\displaystyle {{\mathbb E}}\left[f(X_t)\right] &= \int_{{{\mathbb R}}^d} G_t * f(x) m(x)\,dx + \int_{(0,t] \times {{\mathbb R}}^d} \sum\limits_{i=1}^d \int_{{{\mathbb R}}^d} \partial_{x_i}G_{t-s}(x-y)f(y)\,dy \left(b_i(s,.) \mu_s\right)(x)\,dx\,ds \\
&= \int_{{{\mathbb R}}^d} f(x) \bigg(G_t * m(x) - \int_0^t \nabla G_{t-s} * \left(b(s,.)\mu_s\right)(x) \,ds\bigg) \,dx.\end{aligned}$$ Since $f$ is arbitrary, we conclude that $X_t$ admits a density that we denote by $p(t,.)$ and that satisfies the mild formulation .\
Let us establish that $X^h_t$ admits a density $p^{h}(t,.)$ that satisfies a perturbed version of the previous equation. Using similar arguments, we get: $${{\mathbb E}}\left[\displaystyle \varphi\left(t,X^h_t\right)\right] = {{\mathbb E}}\left[\varphi(0,X_0)+ \int_0^t \nabla \varphi \left(s,X^h_s\right).b\left(\delta_{\ell_s},X^h_{\tau^h_s}\right)\,ds\right].$$ Once again, by Fubini’s Theorem and since $G_t$ is even, we obtain: $$\begin{aligned}
\displaystyle {{\mathbb E}}\left[f\left(X^h_t\right)\right] &= \int_{{{\mathbb R}}^d} G_t * f(x) m(x)\,dx + {{\mathbb E}}\left[\int_{(0,t] \times {{\mathbb R}}^d} \left(\nabla G_{t-s}(X^h_s-x)f(x)\,dx\right) \cdot b\left(\delta_{\ell_s}, X^h_{\tau^h_s}\right)\,ds \right] \\
&= \int_{{{\mathbb R}}^d} f(x) \bigg(G_t * m(x) - \int_0^t {{\mathbb E}}\left[\nabla G_{t-s}(x -X^h_s)\cdot b\left(\delta_{\ell_s}, X^h_{\tau^h_s}\right) \right] \,ds \bigg) \,dx.\end{aligned}$$ The function $f$ being arbitrary, we can conclude.
Now, let us put in a form closer to but with an additional perturbation term that we control in the following proposition.
\[mildfn2\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. Then: $$\begin{aligned}
\displaystyle \forall h \in (0,T],\forall k \in \left\llbracket 1, \left\lfloor \frac{T}{h} \right\rfloor \right\rrbracket,\quad &p^h\left(kh,.\right) = G_{kh}*m - \int_{0}^{kh} \nabla G_{kh-\tau^h_s} * \bigg(b(s,.)\mu^h_{\tau^h_s} \bigg)\,ds + R^h(k,.),\\
\text{ where } \quad &\left\| R^h(k,.) \right\|_{L^1} \le 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left(\frac{1}{2} + \ln\left(k\right) \right)h.
\end{aligned}$$
Let $k \in \left\llbracket 1, \left\lfloor\frac{T}{h} \right\rfloor\right\rrbracket$. By written for $t=kh$ and since $X^h_s = X^h_{jh} + \left(W_s-W_{jh}\right) + b\left(\delta_{j}, X^h_{jh}\right)\left(s- jh\right)$ for $s \in \left[jh, (j+1)h\right)$, we have $dx$ a.e.: $$\begin{aligned}
\displaystyle &p^h(kh,x) \\
&= G_{kh}*m(x)- \sum_{j=0}^{k-1} \int_{jh}^{(j+1)h} {{\mathbb E}}\bigg.\bigg[{{\mathbb E}}\Big[\nabla G_{kh-s}\left(x-X^h_s\right)\cdot b\left(\delta_{j}, X^h_{jh}\right) \Big\vert X^h_{jh}, \delta_{j}\bigg] \bigg]\,ds \\
&= G_{kh}*m(x)- \sum_{j=0}^{k-1} \int_{jh}^{(j+1)h}{{\mathbb E}}\Bigg[ {{\mathbb E}}\left. \left[\nabla G_{kh-s}\Big(x- X^h_{jh} - \left(W_s - W_{jh} \right) - b\left(\delta_{j}, X^h_{jh}\right)\left(s- jh\right)\Big)\right\vert X^h_{jh}, \delta_{j} \right]\cdot b\left(\delta_{j}, X^h_{jh}\right)\Bigg] \,ds. \end{aligned}$$ Using the independence between the increments $\left(W_s - W_{jh}\right)_{s \ge jh}$ and $\left(X^h_{jh}, \delta_{j}\right)$ as well as the fact that the heat kernel is a convolution semi-group s.t. for $0 \le u < s < t, \; \nabla G_{t-s}*G_{s-u} = \nabla G_{t-u}$, we deduce: $$\begin{aligned}
\displaystyle p^h(kh,x)= G_{kh}*m(x)- \sum_{j=0}^{k-1} \int_{jh}^{(j+1)h} {{\mathbb E}}\Bigg[\nabla G_{kh-jh}\bigg(x - X^h_{jh}- b\left(\delta_{j}, X^h_{jh}\right)\left(s- jh\right)\bigg)\cdot b\left(\delta_{j}, X^h_{jh}\right)\Bigg] \,ds.\end{aligned}$$ Using Taylor’s formula with integral reminder at first order, we obtain: $$\begin{aligned}
&\displaystyle \nabla G_{kh-jh}\bigg(x- X^h_{jh} - b\left(\delta_{j},X^h_{jh}\right)\left(s- jh\right)\bigg) \\
&= \nabla G_{kh-jh}\bigg(x- X^h_{jh}\bigg) - (s - jh) \int_0^1 \sum_{i=1}^d \partial_{x_i} \nabla G_{kh-jh}\bigg(x-X^h_{jh}-\alpha\, b\left(\delta_{j},X^h_{jh}\right)(s-t_j) \bigg) b_i\left(\delta_{j},X^h_{jh}\right)\,d\alpha.\end{aligned}$$ We plug this equality in the previous equation. We can easily see by induction using Equation that $\delta_{j}$ is independent from $X^h_{jh}$ for $j \le \left\lfloor\frac{T}{h}\right\rfloor$. Therefore, we obtain: $$\begin{aligned}
\displaystyle &p^h(kh,x) - G_{kh}*m(x) + \sum_{j=0}^{k-1} \int_{jh}^{(j+1)h} \nabla G_{kh-jh} * \Big(b(s,.)\mu^h_{jh}\Big) \,ds \\
&= \sum_{j=0}^{k-1} \int_{jh}^{(j+1)h}(s-jh) {{\mathbb E}}\left[\int_0^1 \sum_{i=1}^d \sum_{l=1}^d \frac{\partial^2 }{\partial x_i x_l} G_{kh-jh}\Big(x-X^h_{jh}-\alpha\, b\left(\delta_{j},X^h_{jh}\right)(s-jh) \Big)b_i\left(\delta_{j},X^h_{jh}\right)b_l\left(\delta_{j},X^h_{jh}\right)\,d\alpha \right] \,ds.\end{aligned}$$ We denote by $R^h(k,x)$ the right-hand side of the previous equation. To upper-bound the $L^1$-norm of $R^h$, we use the estimates and from Lemma \[EstimHeatEq\], and the boundedness of $b$ to obtain: $$\begin{aligned}
\displaystyle \left\|R^h(k,.)\right\|_{L^1} &\le \int_0^{kh} (s-\tau^h_s) \sum_{i=1}^d \sum_{l=1}^d \left\|\frac{\partial^2 }{\partial x_i x_l} G_{kh-\tau^h_s} \right\|_{L^1}\left\|b_i \right\|_{L^{\infty}}\left\|b_l \right\|_{L^{\infty}} \,ds \\
&\le 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \int_0^{kh} \frac{s-\tau^h_s}{kh - \tau^h_s}\,ds \\
&\le 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left( \int_0^{(k-1)h} \frac{h}{kh - s}\,ds + \int_{(k-1)h}^{kh} \frac{s-(k-1)h}{h}\,ds \right) \\
&= 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left(-h\ln\left(h\right) + h \ln(kh) + \frac{h}{2} \right).\end{aligned}$$ One can easily conclude.
Proof of the convergence rate in total variation {#preuve}
================================================
To prove Theorem \[cvgence\], we need the following Lemma that gives an estimation of the regularity of $p(t,.)$ with respect to the time variable.
\[evoldens\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. $$\displaystyle \exists \, Q<+\infty, \forall \, 0 < r\le s\le T,\quad \left\|p(s,.) - p(r,.) \right\|_{L^1} \le Q\bigg(\ln(s/r)+\sqrt{s-r}\bigg).$$
Let $0 < r \le s \le T$. We have: $$\begin{aligned}
\displaystyle p(s,.) - p(r,.) = \left( G_s - G_r \right) * m - \int_0^s \nabla G_{s-u} *\Big(b(u,.)p(u,.)\Big)\,du + \int_0^r \nabla G_{r-u} *\Big(b(u,.)p(u,.)\Big)\,du.\end{aligned}$$ Using the estimates , and from Lemma \[EstimHeatEq\], we obtain: $$\begin{aligned}
\displaystyle &\left\|p(s,.) - p(r,.) \right\|_{L^1} \\
&\le \left\|\left( G_s - G_r \right) * m \right\|_{L^1} + \left\|\int_0^r \left(\nabla G_{s-u} - \nabla G_{r-u} \right)*\Big(b(u,.)p(u,.)\Big)\,du \right\|_{L^1} + \left\|\int_r^s \nabla G_{s-u} *\Big(b(u,.)p(u,.)\Big)\,du \right\|_{L^1} \\
&\le \left\|\int_r^s \left(\partial_u G_u* m\right)\,du \right\|_{L^1} + \left\|\int_0^r \int_{r-u}^{s-u} \partial_{\theta} \nabla G_{\theta} *\Big(b(u,.)p(u,.)\Big)\,d\theta\,du\right\|_{L^1}+ dB\int_r^s \sum\limits_{i=1}^d \left\| \partial_{x_i} G_{s-u}\right\|_{L^1} \,du \\
&\le \int_r^s \left\|\partial_u G_u \right\|_{L^1} \,du + \frac{B}{2} \int_0^r \int_{r-u}^{s-u} \sum\limits_{i=1}^d \sum\limits_{j=1}^d \left\| \frac{\partial^3G_{\theta}}{ \partial x_i\partial x_j^2} \right\|_{L^1}d\theta \,du + 2\sqrt{\frac{2}{\pi}}dB \sqrt{s-r} \\
&\le d\ln(s/r) + 2\sqrt{\frac{2}{\pi}} (2d+3)dB \Big(\sqrt{s-r} - \left(\sqrt{s} - \sqrt{r}\right)\Big)+ 2\sqrt{\frac{2}{\pi}}dB \sqrt{s-r}.\end{aligned}$$ The conclusion holds with $\displaystyle Q = d\max\left(1,4\sqrt{\frac{2}{\pi}}(d+2)B\right)$.
We are now ready to prove Theorem \[cvgence\]. Since $\mu_0 = \mu_0^h = m$, using Equality and Proposition \[propspde\], to prove the theorem amounts to prove that: $$\begin{aligned}
\exists \, C<+\infty, \forall h \in (0,T],\; \forall k \in \left\llbracket 1, \left \lfloor \frac{T}{h} \right \rfloor\right\rrbracket, \quad \left\|p^h\left(kh,.\right)-p\left(kh,.\right)\right\|_{L^1} \le C \sqrt h.\end{aligned}$$ Let $k \in \left\llbracket 1, \left \lfloor \frac{T}{h} \right \rfloor \right\rrbracket$, we have: $$\begin{aligned}
\displaystyle p^h\left(kh,.\right) - p\left(kh,.\right) = V^h(k,.) + R^h(k,.) - \int_0^{h} \bigg(\nabla G_{kh-s} * \Big(b(s,.)p(s,.)\Big) - \nabla G_{kh} * \Big(b(s,.)m\Big)\bigg)\,ds\end{aligned}$$ where $R^h(k,.)$ is defined in Proposition \[mildfn2\] and $V^h(k,.)$ is defined by: $$\displaystyle V^h(k,.) = \int_{h}^{kh}\bigg(\nabla G_{kh-s} * \Big(b(s,.)p(s,.) \Big)-\nabla G_{kh-\tau^h_s} * \Big(b(s,.)p^h\left(\tau^h_s,.\right)\Big)\bigg)\,ds.$$ Since $V^h(1,.) = 0$, we suppose $k \ge 2$ and express $V^h(k,.)$ as $V^h(k,.)= \sum\limits_{p=1}^{3}V^h_p(k,.)$ where: $$\begin{aligned}
V_1^h(k,.) &= \int_{h}^{kh}\bigg(\nabla G_{kh-s}- \nabla G_{kh-\tau^h_s} \bigg)* \Big(b(s,.)p(s,.) \Big)\,ds,\\
V_2^h(k,.) &= \int_{h}^{kh}\nabla G_{kh-\tau^h_s} * \bigg(b(s,.)\Big[p(s,.)-p\left(\tau^h_s,.\right) \Big]\ \bigg)\,ds,\\
V_3^h(k,.) &= \int_{h}^{kh}\nabla G_{kh-\tau^h_s} * \bigg(b(s,.)\Big[p(\tau^h_s,.)-p^h\left(\tau^h_s,.\right) \Big]\ \bigg)\,ds.\end{aligned}$$ On the one hand, using the estimate from Lemma \[EstimHeatEq\], we obtain immediately that: $$\begin{aligned}
\label{v3}
\displaystyle \left\|V_3^h(k,.)\right\|_{L^1} &\le \sqrt{\frac 2 \pi} dB\int_{h}^{kh}\frac{1}{\sqrt{kh-\tau^h_s}}\left\|p(\tau^h_s,.) - p^h(\tau^h_s,.) \right\|_{L^1}\,ds = \sqrt{\frac 2 \pi}dB\sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1}.\end{aligned}$$ On the other hand, using the estimate from Lemma \[EstimHeatEq\], we have for the first time-step that:
$$\begin{aligned}
\label{firstStep}
\left\| \int_0^{h} \bigg(\nabla G_{kh-s} * \Big(b(s,.)p(s,.)\Big) - \nabla G_{kh} * \Big(b(s,.)m\Big)\bigg)\,ds \right\|_{L^1} &\le B \int_0^h \sum_{i=1}^d \Big(\left\|\partial_{x_i} G_{kh-s} \right\|_{L^1} + \left\|\partial_{x_i} G_{kh} \right\|_{L^1} \Big)\,ds \nonumber \\
&= \sqrt{\frac 2 \pi}dB \left( 2\left(\sqrt{kh} - \sqrt{(k-1)h}\right)+\frac{h}{\sqrt{kh}}\right)\nonumber\\
&\le \sqrt{\frac 2 \pi}dB\frac{3h}{\sqrt{kh}}.\end{aligned}$$
Now, using Inequality and Proposition \[mildfn2\] for the first inequality, Inequality for the second inequality and finally the fact that $\ln(k) \le \ln\left(\frac T h\right)$ with $\sup_{x >0}\left\{ \left(\frac{1}{2}+\ln\left(\frac{T}{x}\right) \right)\sqrt{x} \right\}$ is attained for $x = T e^{-\frac 3 2}$ for the third inequality, we obtain:
$$\begin{aligned}
\displaystyle &\left\|p(kh,.)-p^h\left(kh,.\right)\right\|_{L^1} \nonumber\\
&\le \left\|V^h_1(k,.)\right\|_{L^1}+ \left\|V^h_2(k,.)\right\|_{L^1}+ \sqrt{\frac 2 \pi}dB\sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1} \nonumber \\
&\phantom{\|V}+ 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left(\frac{1}{2} + \ln\left(k\right) \right)h + \left\| \int_0^{h} \bigg(\nabla G_{kh-s} * \Big(b(s,.)p(s,.)\Big) - \nabla G_{kh} * \Big(b(s,.)m\Big)\bigg)\,ds \right\|_{L^1} \label{firstintegB} \\
&\le \left\|V^h_1(k,.)\right\|_{L^1}+ \left\|V^h_2(k,.)\right\|_{L^1}+ \sqrt{\frac 2 \pi}dB\sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1} \nonumber \\
&\phantom{\|V\|}+ 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left(\frac{1}{2} + \ln\left(k \right) \right)h + \sqrt{\frac 2 \pi}dB\frac{3h}{\sqrt{kh}} \label{firstinteg}\end{aligned}$$
$$\begin{aligned}
&\le \left\|V^h_1(k,.)\right\|_{L^1}+ \left\|V^h_2(k,.)\right\|_{L^1}+ \sqrt{\frac 2 \pi}dB \sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1} \nonumber \\
&\phantom{\|V\|}+ \sqrt{\frac 2 \pi}dB\left(2\sqrt{\frac{2T}{\pi}}B(\pi + d-1)e^{-\frac{3}{4}} + 3 \right) \sqrt h \label{ici1}.\end{aligned}$$
Let us now estimate $\left\|V^h_1(k,.)\right\|_{L^1}$ and $\left\|V^h_2(k,.)\right\|_{L^1}$ for $k \ge 2$.\
\
$\bullet$ For $p=1$, using the estimates and from Lemma \[EstimHeatEq\], we obtain: $$\begin{aligned}
\displaystyle \left\|V_1^h(k,.)\right\|_{L^1} &\le \frac{B}{2} \int_{h}^{(k-1)h} \int_{kh - s}^{kh - \tau^h_s}\sum_{i=1}^d \sum_{j=1}^d\left\|\frac{\partial^3 G_r}{\partial x_i \partial x^2_j}\right\|_{L^1}dr\,ds + B\int_{(k-1)h}^{kh}\bigg(\left\|\nabla G_{kh-s} \right\|_{L^1} + \left\|\nabla G_{h} \right\|_{L^1}\bigg)\,ds\\
&\le \frac{1}{2}\sqrt{\frac{2}{\pi}}(2d+3)dB\int_{h}^{(k-1)h} \int_{kh - s}^{kh - \tau^h_s}\frac{dr}{r^{3/2}}\,ds + dB\sqrt{\frac{2}{\pi}}\int_{(k-1)h}^{kh}\bigg(\frac{1}{\sqrt{kh-s}} + \frac{1}{\sqrt{ h}} \bigg)\,ds\\
&\le \sqrt{\frac{2}{\pi}}(2d+3)dB \int_{h}^{(k-1)h} \frac{h}{2\left(kh-s\right)^{3/2}}\,ds + 3dB\sqrt{\frac{2}{\pi}}\sqrt{h} \le 2\sqrt{\frac{2}{\pi}}(d+3)dB\sqrt{h}.\end{aligned}$$ $\bullet$ For $p=2$, using the estimate from Lemma \[EstimHeatEq\] and Lemma \[evoldens\], we obtain: $$\begin{aligned}
\displaystyle \left\|V_2^h(k,.)\right\|_{L^1} &\le \sqrt{\frac 2 \pi}dB\int_{h}^{kh}\frac{1}{\sqrt{kh-\tau^h_s}} \left\|p(s,.) - p(\tau^h_s,.) \right\|_{L^1}\,ds \\
&\le \sqrt{\frac 2 \pi}dBQ\Bigg( \int_{h}^{kh} \frac{\ln(s/ \tau^h_s) }{\sqrt{kh-\tau^h_s}}\,ds + \int_{h}^{kh} \frac{\sqrt{s-\tau^h_s}}{\sqrt{kh-\tau^h_s}}\,ds \Bigg).\end{aligned}$$ Using the fact that for $\displaystyle s \ge h,\; \ln\left(\frac{s}{\tau^h_s}\right) \le \frac{s-\tau^h_s}{\tau^h_s} \le \frac{2h}{s}$ and Lemma \[integ\], we have that: $$\begin{aligned}
\label{integLog}
\displaystyle \int_{h}^{kh} \frac{\ln(s/ \tau^h_s) }{\sqrt{kh-\tau^h_s}}\,ds &\le \int_{h}^{kh}\frac{2h}{s\sqrt{kh-s}}\,ds \le \frac{2h}{\sqrt{kh}} \ln\left(4k\right).\end{aligned}$$ Moreover, using the fact that $\displaystyle \frac{\sqrt{s-\tau^h_s}}{\sqrt{kh - \tau^h_s}} \le \frac{\sqrt h}{\sqrt{kh -s}}$, we deduce that: $$\begin{aligned}
\displaystyle \left\|V_2^h(k,.)\right\|_{L^1} &\le 2\sqrt{\frac 2 \pi}dBQ\Bigg(\sup_{x\ge 1}\frac{\ln(4x)}{\sqrt x} + \sqrt{(k-1)h}\Bigg) \sqrt h \le 2\sqrt{\frac 2 \pi}dBQ\Bigg(\frac{4}{e} +\sqrt T\Bigg) \sqrt h.\end{aligned}$$ Hence, using and the estimates $\left\|V_1^h(k,.)\right\|_{L^1}$ and $\left\|V_2^h(k,.)\right\|_{L^1}$, we obtain: $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \le L \sqrt{h} + \sqrt{\frac{2}{\pi}}dB \sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1}\end{aligned}$$ where $\displaystyle L = 2\sqrt{\frac 2 \pi}dB\Bigg(\left(d + \frac 9 2 \right)+ Q\left(\frac{4}{e} + \sqrt T \right)+ \sqrt{\frac{2T}{\pi}}B(\pi + d-1)e^{-\frac 3 4} \Bigg)$. We iterate this inequality to obtain: $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \le L\left(1 + 2\sqrt{\frac{2}{\pi}}dB \sqrt{(k-1)h} \right)\sqrt{h}+ \frac{2d^2B^2}{\pi} \sum_{j=1}^{k-1} \sum_{l=1}^{j-1}\frac{h}{\sqrt{k-j}\sqrt{j-l}}\left\|p(lh,.) - p^h(lh,.) \right\|_{L^1}.\end{aligned}$$ We re-write the double-sum the following way: $$\begin{aligned}
\displaystyle \sum_{j=1}^{k-1} \sum_{l=1}^{j-1}\frac{h}{\sqrt{k-j}\sqrt{j-l}}\left\|p(lh,.) - p^h(lh,.) \right\|_{L^1} &= \sum_{l=1}^{k-2} \sum_{j=l+1}^{k-1}\frac{h}{\sqrt{k-j}\sqrt{j-l}}\left\|p(lh,.) - p^h(lh,.) \right\|_{L^1} \\
&= h \sum_{l=1}^{k-2} \Bigg(\left\|p(lh,.) - p^h(lh,.) \right\|_{L^1} \sum_{i=1}^{k-l-1} \frac{1}{\sqrt{i}\sqrt{(k-l) - i} }\Bigg)\\
&\le \pi \int_{h}^{kh}\left\|p\left(\tau^h_s,.\right) - p^h\left(\tau^h_s,.\right) \right\|_{L^1} \,ds,\end{aligned}$$ where we used Lemma \[sumPi\] for the last inequality. Therefore, $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \le L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\sqrt{h} + 2d^2B^2 h \sum_{j=1}^{k-1}\left\|p\left(jh,.\right) - p^h\left(jh,.\right) \right\|_{L^1}.\end{aligned}$$ We apply Lemma \[Gronw\] and obtain: $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} &\le L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\sqrt{h} + 2d^2B^2 L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\sqrt{h} \sum_{j=1}^{k-1} h \exp\Big(2d^2B^2(kh -(j+1)h) \Big) \\
&\le L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\sqrt{h} + 2d^2B^2 L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\sqrt{h}\exp\Big(2d^2B^2T\Big)(kh-h)\\
&\le L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\left(1 + 2d^2B^2T\exp\Big(2d^2B^2T\Big) \right)\sqrt{h}.\end{aligned}$$ The conclusion holds with $\displaystyle C = L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right)\left(1 + 2d^2B^2T\exp\Big(2d^2B^2T\Big) \right)$.
Proof of the convergence rate in total variation when assuming more regularity on $b$ w.r.t. to the space variables {#preuveB}
===================================================================================================================
The following proposition, developed in Subsection \[hypH\], enables to establish an estimate of the total variation norm of the divergence of $b(t,.)p(t,.)$ for $t>0$ from the regularity assumed on $b$ w.r.t. to the space variables. When assuming extra regularity on $m$, the estimate is improved.
\[hyp\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. If $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or for $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$; where $\nabla \cdot b(t,.)$ and $\partial_x b(t,.)$ are respectively the spatial divergence and the spatial derivative of $b$ in the sense of distributions. Then: $$\begin{aligned}
\label{hyp1}
\displaystyle \exists\, M <+\infty, \forall t \in(0,T], \quad \left\|\nabla \cdot \Big(b(t,.)p(t,.) \Big) \right\|_{\rm TV} \le \frac{M}{\sqrt t},
\end{aligned}$$ and: $$\begin{aligned}
\label{nablaP}
\forall t \in (0,T], \quad p(t,.) = G_t*m - \int_0^t G_{t-s} * \nabla \cdot \Big(b(s,.)p(s,.)\Big)\,ds.
\end{aligned}$$ Moreover, if $m$ admits a density w.r.t. the Lebesgue measure that belongs to $W^{1,1}\left({{\mathbb R}}^d\right)$, we obtain: $$\begin{aligned}
\label{hyp2}
\displaystyle \exists \, \tilde{M} <+\infty, \forall t \in [0,T], \quad \left\|\nabla \cdot \Big(b(t,.)p(t,.) \Big) \right\|_{\rm TV} \le \tilde{M}.
\end{aligned}$$
In fact, when $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$, we will prove that for $t \in (0,T]$, $\nabla \cdot \Big(b(t,.)p(t,.) \Big) \in L^1\left({{\mathbb R}}^d\right)$ and $\left\|\nabla \cdot \Big(b(t,.)p(t,.) \Big) \right\|_{\rm TV} = \left\|\nabla \cdot \Big(b(t,.)p(t,.) \Big) \right\|_{L^1}$.
The results of Proposition \[hyp\] will be used in the proof of Theorem \[cvgenceB\] detailed in Subsection \[demoo\], and in the proof of Proposition \[propregM\] detailed in Subsection \[demooo\].\
\
We bring to attention that, in this subsection, all the derivatives and divergence are defined in the sense of distributions.
Proof of Proposition \[hyp\] {#hypH}
----------------------------
Let $\theta \in (0,T]$, we define the Banach spaces $\displaystyle \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right) = \left\{q \in \mathcal{C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right): \sup_{t \in (0,\theta]}\left\|q(t,.)\right\|_{L^1}<+\infty \right\}$, $\displaystyle \mathcal{\tilde{C}}\left((0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right) = \left\{q \in \mathcal{C}\left((0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right): {\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{q} = \sup_{t \in (0,\theta]}\left\|q(t,.)\right\|_{L^1} + \sup_{t \in (0,\theta]}\sqrt{t}\left\|\nabla q(t,.)\right\|_{L^1}<+\infty \right\}$ and $\mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$ endowed respectively with the norms $\displaystyle \sup_{t \in (0,\theta]}\left\|q(t,.)\right\|_{L^1}$, ${\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{q}$ and $\displaystyle \sup_{t \in [0,\theta]}\left\|q(t,.)\right\|_{W^{1,1}}$. One has $\mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right) \subset \mathcal{\tilde{C}}\left((0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right) \subset \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$.\
The next theorem states regularity properties of the density $\left(p(t,.)\right)_{t \in (0,T]}$ when assuming more regularity on $b$ w.r.t. the space variables.
\[sobolP\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. If $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or for $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$ then $p \in \mathcal{\tilde C}\left((0,T],W^{1,1}\left({{\mathbb R}}^d \right) \right)$. Moreover, if $m$ admits a density w.r.t. the Lebesgue measure in $W^{1,1}\left({{\mathbb R}}^d\right)$ then $p\in \mathcal{C}\left([0,T],W^{1,1}\left({{\mathbb R}}^d\right)\right)$.
The proof of Theorem \[sobolP\] relies on the uniqueness of the mild equation . This latter can be proved by a fixed-point method. To do so, for $\theta \in (0,T]$, we define on the space $\mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$ the map $\Phi$: $$\begin{aligned}
\displaystyle \Phi:q \mapsto \Bigg(\Phi_t(q) = G_t * m - \int_0^t \nabla G_{t-s} * \Big( b(s,.)q(s,.)\Big)\,ds \Bigg)_{t \in (0,\theta]}.\end{aligned}$$ By a slight abuse of notation, we do not make explicit the dependence of the map $\Phi$ on the time horizon $\theta$. Let us check that $\Phi$ is well-defined. For $t \in (0,\theta]$, we have, using the estimate from Lemma \[EstimHeatEq\], that: $$\begin{aligned}
\label{estimL1}
\displaystyle \left\|\Phi_t(q)\right\|_{L^1} \le \left\|G_t*m\right\|_{L^1} + dB\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}}\left\|q(s,.)\right\|_{L^1}\,ds \le 1 + 2dB\sqrt{\frac{2 \theta}{\pi}}\sup_{u \in (0,\theta]}\|q(u,.)\|_{L^1}.\end{aligned}$$ Hence, since $q \in \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$, we have that $\sup_{t \in (0,\theta]}\left\|\Phi_t(q)\right\|_{L^1} <+\infty$.\
The following result ensures that the map $\Phi$ admits a unique fixed-point in $\displaystyle \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$.
\[uniqFP\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. For all $\theta \in (0,T]$, $\left(p(t,.)\right)_{t \in (0,\theta]}$ is the unique fixed-point of the map $\Phi$ in $\mathcal{\bar C} \left((0,\theta], L^1\left({{\mathbb R}}^d \right)\right)$.
Let $q \in \mathcal{\bar C}\left((0,\theta], L^1\left({{\mathbb R}}^d \right)\right)$. Using Inequality , we have that $\sup_{t \in (0,\theta]}\left\|\Phi_t(q)\right\|_{L^1} <+\infty$.\
For $0 < r \le s \le \theta$, adapting the proof of Lemma \[evoldens\], we obtain that: $$\begin{aligned}
\label{timeCtyPhi}
\displaystyle \left\|\Phi_s(q) - \Phi_r(q) \right\|_{L^1} \le \max\left(1,\sup_{u \in [r,s]}\|q(u,.)\|_{L^1}\right)Q \Big(\ln(s/r) + \sqrt{s-r} \Big).\end{aligned}$$ Therefore, $t \mapsto \Phi_t(q)$ is continuous on $(0,\theta]$ with values in $L^1\left({{\mathbb R}}^d\right)$ and $\Phi(q) \in \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$.\
Now, let $q,\tilde q \in \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$. Using the same reasoning as for Inequality , we obtain: $$\begin{aligned}
\label{equA}
\displaystyle \left\|\Phi_t(q) - \Phi_t(\tilde q)\right\|_{L^1} &\le dB\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}}\left\|q(s,.) - \tilde q(s,.)\right\|_{L^1}\,ds. \end{aligned}$$ Let $n \in {{\mathbb N}}^{*}$, we define $\Phi^{n+1} = \Phi^n \circ \Phi = \Phi \circ \Phi^n$ and iterate Inequality $2n$-times to obtain: $$\begin{aligned}
\displaystyle \left\|\Phi^{2n}_t(q) - \Phi^{2n}_t(\tilde q)\right\|_{L^1} &\le \left(dB\sqrt{\frac 2 \pi}\right)^{2n} \pi^n \int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \left\|q(s,.) - \tilde q(s,.)\right\|_{L^1}\,ds \\
&\le (dB)^{2n}\frac{(2\theta)^n}{n!}\; \sup_{u \in (0,\theta]}\left\|q(u,.) - \tilde q(u,.)\right\|_{L^1}.\end{aligned}$$ Therefore, $\sup_{t \in (0,\theta]}\left\|\Phi^{2n}_t(q) - \Phi^{2n}_t(\tilde q)\right\|_{L^1} \le (dB)^{2n}\frac{(2T)^n}{n!}\; \sup_{t \in (0,\theta]}\left\|q(t,.) - \tilde q(t,.)\right\|_{L^1} $ and for $n$ big enough, $\Phi^{2n}$ is a contraction on $\mathcal{\bar{C}}\left((0,\theta], L^1\left({{\mathbb R}}^d \right)\right)$. By Picard’s Theorem, $\Phi^{2n}$ admits then a unique fixed-point $q$ in $\mathcal{\bar{C}} \left((0,\theta], L^1\left({{\mathbb R}}^d \right)\right)$. We have that $\Phi(q) = \Phi\left(\Phi^{2n}(q)\right) = \Phi^{2n}\left(\Phi(q)\right)$ making $\Phi(q)$ a fixed-point of $\Phi^{2n}$, but since this latter is unique, we conclude that $\Phi(q) = q$. For $t>0$, $p(t,.)$ is solution to the mild equation and using Lemma \[evoldens\], $p \in \mathcal{\bar C}\left((0,\theta],L^1\left({{\mathbb R}}^d\right)\right)$. Consequently, $\left(p(t,.)\right)_{t \in (0,\theta]}$ is the unique fixed-point of the map $\Phi$ in $\mathcal{\bar C} \left((0,\theta], L^1\left({{\mathbb R}}^d\right)\right)$.
Now, we seek to establish more regularity on the fixed-point of the map $\Phi$.
\[regFixpoint\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$. Moreover, assume that $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or that for $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$.
- If $m$ admits a density w.r.t the Lebesgue measure in $W^{1,1}\left({{\mathbb R}}^d\right)$ then for all $\theta \in (0,T]$, $\Phi$ admits a unique fixed-point in the space $\mathcal{C}\left([0,\theta], W^{1,1}\left({{\mathbb R}}^d \right)\right)$.
- Otherwise, there exists $\theta_0 \in (0,T]$ s.t. $\Phi$ admits a unique fixed-point in the space $\mathcal{\tilde C} \left((0,\theta_0], W^{1,1}\left({{\mathbb R}}^d \right)\right)$.
Let us deduce Theorem \[sobolP\] before giving the proof of Proposition \[regFixpoint\].
$\bullet$ For $\theta = \theta_0$ given by Proposition \[regFixpoint\]:
- Let $t \in (0, \theta_0]$. According to Proposition \[regFixpoint\], the map $\Phi$ admits a unique fixed-point in $\mathcal{\tilde C}\left((0,\theta_0], W^{1,1}\left({{\mathbb R}}^d \right)\right)$. With the inclusion $\mathcal{\tilde C}\left((0,\theta_0], W^{1,1}\left({{\mathbb R}}^d \right)\right) \subset \mathcal{\bar C} \left((0,\theta_0], L^1\left({{\mathbb R}}^d \right)\right)$, this fixed-point coincides with the unique fixed-point of $\Phi$ in $ \mathcal{\bar C} \left((0,\theta_0], L^1\left({{\mathbb R}}^d \right)\right)$ which is $(p(t,.))_{t \in (0,\theta_0]}$ according to Lemma \[uniqFP\]. Therefore, we have that $(p(t,.))_{t \in (0,\theta_0]} \in \mathcal{\tilde C}\left((0,\theta_0], W^{1,1}\left({{\mathbb R}}^d \right)\right)$ and $\sup_{t \in (0,\theta_0]}\sqrt{t}\|p(t,.) \|_{W^{1,1}}\le \max\left(1,\sqrt{\theta_0}\,\right){\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{p} <+\infty$.
- Now, let $t \in [\theta_0,T]$. Using the fact that the heat kernel is a convolution semi-group and that $$\displaystyle p(\theta_0,.) = G_{\theta_0}*m - \int_0^{\theta_0}\nabla G_{\theta_0 - r}*\Big(b(r,.)p(r,.)\Big)\,dr,$$ we obtain: $$\displaystyle p(t,.) = G_{t-\theta_0}*p(\theta_0,.) - \int_0^{t-\theta_0}\nabla G_{(t-\theta_0) - s}*\Big(b(\theta_0 + s,.)p(\theta_0 + s,.)\Big)\,ds$$ such that for $u=(t - \theta_0) \in [0,T-\theta_0]$: $$\displaystyle p(\theta_0+u,.) = G_{u}*p(\theta_0,.) - \int_0^{u}\nabla G_{u - s}*\Big(b(\theta_0 + s,.)p(\theta_0 + s,.)\Big)\,ds.$$ Hence, by Lemma \[sobolP\], $\left(p(\theta_0+u,.)\right)_{u \in [0,T-\theta_0]}$ is the unique fixed-point in $\mathcal{\bar C}\left((0,T-\theta_0], L^{1}\left({{\mathbb R}}^d \right)\right)$ of the functional defined like $\Phi$ but with $m$ replaced by $p(\theta_0,.)$ and $b$ shifted by $\theta_0$ in the time variable. Since $p(\theta_0,.) \in W^{1,1}\left({{\mathbb R}}^d\right)$ and according to Proposition \[regFixpoint\], this functional admits a unique fixed-point in $\mathcal{C} \left([0,T-\theta_0], W^{1,1}\left({{\mathbb R}}^d \right)\right)$ and this fixed-point coincides with $\left(p(\theta_0+u,.)\right)_{u \in [0,T-\theta_0]}$. Therefore, $\left(p(t,.)\right)_{t \in [\theta_0,T]} \in \mathcal{C}\left([\theta_0,T], W^{1,1}\left({{\mathbb R}}^d \right)\right)$ and $\sup_{t \in [\theta_0,T]}\sqrt{t}\| p(t,.)\|_{W^{1,1}} \le \sqrt{T}\sup_{t \in [\theta_0,T]}\| p(t,.)\|_{W^{1,1}} <+\infty$.
We can conclude.\
$\bullet$ Now, we assume that $m$ admits a density w.r.t. the Lebesgue measure in $W^{1,1}\left({{\mathbb R}}^d\right)$. For $\theta = T$ and according to Proposition \[regFixpoint\], the map $\Phi$ admits a unique fixed-point in $\mathcal{C}\left([0,T], W^{1,1}\left({{\mathbb R}}^d \right)\right)$. With the inclusion $\mathcal{C}\left([0,T], W^{1,1}\left({{\mathbb R}}^d \right)\right) \subset \mathcal{\bar C} \left((0,T], L^1\left({{\mathbb R}}^d \right)\right)$, this fixed-point coincides with the unique fixed-point of $\Phi$ in $ \mathcal{\bar C} \left((0,T], L^1\left({{\mathbb R}}^d \right)\right)$ which is $(p(t,.))_{t \in (0,T]}$ according to Lemma \[uniqFP\]. Therefore, $(p(t,.))_{t \in [0,T]} \in \mathcal{C}\left([0,T], W^{1,1}\left({{\mathbb R}}^d \right)\right)$.
To prove Proposition \[regFixpoint\], we need the following convolution and derivation in the sense of distributions result.
\[Sobol\] Let $q:{{\mathbb R}}^d \to {{\mathbb R}}$ be a function in $W^{1,1}\left({{\mathbb R}}^d\right)$ and $g:{{\mathbb R}}^d \to {{\mathbb R}}^d$ a bounded measurable function. We assume either that $\|\nabla \cdot g\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or that for $d=1$, $\left\|g'\right\|_{\rm TV}<+\infty$.\
$\bullet$ Under the first assumption, for any $\varphi:{{\mathbb R}}^d \to {{\mathbb R}}\;$ $\mathcal{C}^{\infty}$-bounded together with its first order derivatives, we have: $$\begin{aligned}
\displaystyle \int_{{{\mathbb R}}^d} \nabla \varphi(x) \cdot \Big(q(x)g(x)\Big)\,dx &= - \int_{{{\mathbb R}}^d}\varphi(x) \Big(\nabla q(x) \cdot g(x) + q(x) \nabla \cdot g(x) \Big)\,dx
\end{aligned}$$ so that in the sense of distributions, $\nabla \cdot \Big(qg \Big)= q\nabla \cdot g + \nabla q\cdot g$.\
Moreover, for $\displaystyle \check{C} = \sup_{\substack{f \in W^{1,1} \\ f \neq 0 }} \frac{\|f\|_{L^{\frac{\rho}{\rho-1}}({{\mathbb R}}^d)}}{\|f\|_{W^{1,1}({{\mathbb R}}^d)}}$ which is finite according to Corollary $IX.10$ [@Brezis], we have that: $$\begin{aligned}
\label{normMaj}
\displaystyle \left\|\nabla \cdot \Big(qg \Big) \right\|_{L^1} &\le \check{C}\left\|q \right\|_{W^{1,1}} \left\| \nabla \cdot g\right\|_{L^{\rho}} + \left\|\nabla q \right\|_{L^1} \left\| g\right\|_{L^{\infty}}.
\end{aligned}$$ $\bullet$ Under the second assumption, for any $\varphi:{{\mathbb R}}\to {{\mathbb R}}\;$ $\mathcal{C}^{\infty}$-bounded together with its first order derivative, we have: $$\begin{aligned}
\displaystyle \int_{{{\mathbb R}}} \varphi'(x)\Big(q(x)g(x)\Big)\,dx &= - \int_{{{\mathbb R}}}\varphi(x)q'(x)g(x)\,dx - \int_{{{\mathbb R}}}\varphi(x)q(x)g'(dx)
\end{aligned}$$ where the continuous representative of $q$ which exists according to Theorem $VIII.2$ [@Brezis], is chosen to define $q(x)g'(dx)$. Moreover, the derivative of $qg$ in the sense of distributions is a bounded measure on the real line and for $\displaystyle \check{C} = \sup_{\substack{f \in W^{1,1} \\ f \neq 0 }} \frac{\|f\|_{\infty}}{\|f\|_{W^{1,1}({{\mathbb R}})}}$ which is finite according to Theorem $VIII.7$ [@Brezis], we have that: $$\begin{aligned}
\label{normMaj1}
\displaystyle \left\|(qg)' \right\|_{\rm TV} &\le \check{C} \left\|q \right\|_{W^{1,1}} \left\| g'\right\|_{\rm TV} + \left\|q' \right\|_{L^1} \left\| g\right\|_{\infty}.
\end{aligned}$$
According to Theorem $IX.2$ [@Brezis], there exists a sequence of functions $\left(q_n\right)_{n}$ in $\mathcal{C}^{\infty}_c\left({{\mathbb R}}^d\right)$ such that when $n \to +\infty$, $q_n \to q$ and $\nabla q_n \to \nabla q$ respectively in $L^1\left({{\mathbb R}}^d\right)$ and $L^1\left({{\mathbb R}}^d\right)^d$. We have: $$\begin{aligned}
\label{convLim}
\displaystyle \int_{{{\mathbb R}}^d} \nabla\varphi(x)\cdot \Big(q_n(x)g(x)\Big)\,dx &= \int_{{{\mathbb R}}^d}\nabla\Big(\varphi(x)q_n(x)\Big)\cdot g(x)\,dx - \int_{{{\mathbb R}}^d}\varphi(x)\nabla q_n(x)\cdot g(x)\,dx.\end{aligned}$$ Since $\nabla \varphi$, $g$ and $\varphi$ are bounded functions on ${{\mathbb R}}^d$, we have $\displaystyle \int_{{{\mathbb R}}^d} \nabla\varphi(x)\cdot \Big(q_n(x)g(x)\Big)\,dx \underset{n \to +\infty}{\longrightarrow} \int_{{{\mathbb R}}^d} \nabla\varphi(x)\cdot \Big(q(x)g(x)\Big)\,dx$ and $\displaystyle \int_{{{\mathbb R}}^d} \varphi(x)\nabla q_n(y)\cdot g(x)\,dx \underset{n \to +\infty}{\longrightarrow} \int_{{{\mathbb R}}^d} \varphi(x)\nabla q(x)\cdot g(x)\,dx$.\
$\bullet$ Under the first assumption, since $\varphi q_n \in C^{\infty}_c\left({{\mathbb R}}^d\right)$, we have that $\displaystyle \int_{{{\mathbb R}}^d}\nabla\Big(\varphi(x)q_n(x)\Big)\cdot g(x)\,dx = - \int_{{{\mathbb R}}^d}\varphi(x)q_n(x)\nabla \cdot g(x)\,dx$. Using Hölder’s inequality, we obtain that: $$\begin{aligned}
\displaystyle \left|\int_{{{\mathbb R}}^d}\varphi(x)q_n(x)\nabla \cdot g(x)\,dx - \int_{{{\mathbb R}}^d}\varphi(x)q(x)\nabla \cdot g(x)\,dx\right| &\le \sup_{x \in {{\mathbb R}}^d}|\varphi(x)| \left\| q_n - q\right\|_{L^{\frac{\rho}{\rho-1}}} \left\| \nabla \cdot g\right\|_{L^{\rho}} \\
&\le \check{C} \sup_{x \in {{\mathbb R}}^d}|\varphi(x)| \left\| q_n - q\right\|_{W^{1,1}} \left\| \nabla \cdot g\right\|_{L^{\rho}}\\
&\underset{n \to +\infty}{\longrightarrow} 0.\end{aligned}$$ Hence, taking the limit $n \to +\infty$ in Equation , we get, in the sense of distributions, that $\nabla \cdot \left(qg\right) = q\nabla \cdot g + \nabla q \cdot g$. Now, using once again Hölder’s inequality and Corollary $IX.10$ [@Brezis], one has $$\left\| q \nabla \cdot g\right\|_{L^1} \le \left\|q\right\|_{L^{\frac{\rho}{\rho-1}}} \left\| \nabla \cdot g\right\|_{L^{\rho}} \le \check{C} \left\| q\right\|_{W^{1,1}} \left\| \nabla \cdot g\right\|_{L^{\rho}}$$ and one deduces that $$\left\|\nabla \cdot \Big(qg \Big) \right\|_{L^1} \le \check{C} \left\|q \right\|_{W^{1,1}} \left\| \nabla \cdot g\right\|_{L^{\rho}} + \left\|\nabla q \right\|_{L^1} \left\| g\right\|_{L^{\infty}}.$$ $\bullet$ Under the second assumption, since $\varphi q_n \in \mathcal{C}^{\infty}_c({{\mathbb R}})$, we have that $\displaystyle \int_{{{\mathbb R}}}\Big(\varphi q_n\Big)'(x)g(x)\,dx = - \int_{{{\mathbb R}}} \varphi(x)q_n(x)g'(dx)$. According to Theorem $VIII.2$ and Theorem $VIII.7$ [@Brezis], $q$ admits a bounded and continuous representative, and for this representative the integral $\displaystyle \int_{{{\mathbb R}}}\varphi(x)q(x) g'(dx)$ makes sense. Using Hölder’s inequality, we have: $$\begin{aligned}
\displaystyle \left|\int_{{{\mathbb R}}}\varphi(x)q_n(x)g'(dx) - \int_{{{\mathbb R}}}\varphi(x)q(x)g'(dx)\right| &\le \sup_{x \in {{\mathbb R}}}|\varphi(x)| \sup_{x \in {{\mathbb R}}}\left| q_n(x) - q(x) \right| \left\| g'\right\|_{\rm TV} \\
&\le \check{C} \sup_{x \in {{\mathbb R}}}|\varphi(x)| \left\| q_n - q\right\|_{W^{1,1}} \left\| g'\right\|_{\rm TV}\\
&\underset{n \to +\infty}{\longrightarrow} 0.\end{aligned}$$ Hence, taking the limit $n \to +\infty$ in Equation , we get, in the sense of distributions, that $\left(qg\right)' = q g' + q'g$. Now, using once again Hölder’s inequality and Theorem $VIII.7$ [@Brezis], one has $$\left\| qg'\right\|_{\rm TV} \le \sup_{x \in {{\mathbb R}}}\left|q(x) \right| \left\| g'\right\|_{\rm TV} \le \check{C} \left\|q\right\|_{W^{1,1}} \left\| g'\right\|_{\rm TV}$$ and one deduces that $$\left\|(qg)' \right\|_{\rm TV} \le \check{C} \left\|q \right\|_{W^{1,1}} \left\| g'\right\|_{\rm TV} + \left\|q' \right\|_{L^1} \left\| g\right\|_{\infty}.$$
We are now ready to prove Proposition \[regFixpoint\].
We are going to suppose that $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$. When $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$, the proof is analogous and the estimations remain valide when replacing $\left\|\nabla \cdot b(t,.) \right\|_{L^{\rho}}$ by $\left\|\partial_x b(t,.)\right\|_{\rm TV}$.\
For $\theta \in (0,T]$, let $ 0 \le s < t \le \theta $. If $q(s,.)$ is in $W^{1,1}\left({{\mathbb R}}^d\right)$, we apply Lemma \[Sobol\] with $\varphi = G_{t-s}$ which is $\mathcal{C}^{\infty}$-bounded together with its first order derivatives and $g = b(s,.)$ to obtain that: $$\begin{aligned}
\nabla G_{t-s} * \Big(b(s,.)q(s,.)\Big) = G_{t-s} * \nabla \cdot \Big(b(s,.)q(s,.)\Big).\end{aligned}$$ If $q \in \mathcal{\tilde C}\left( (0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$, we first have: $$\begin{aligned}
\label{nablaq}
\forall t \in (0,\theta], \quad \Phi_t(q) = G_t * m - \int_0^t G_{t-s} * \nabla \cdot \left(b(s,.)q(s,.)\right)\,ds.\end{aligned}$$
$\bullet$ We start by proving that there exists $\theta_0$ s.t. $\Phi$ admits a unique fixed-point in $\mathcal{\tilde C}\left((0,\theta_0],W^{1,1}\left({{\mathbb R}}^d\right)\right)$:\
Let $t \in (0,\theta]$ and $q \in \mathcal{\tilde C}\left((0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$. We have, using the estimate from Lemma \[EstimHeatEq\] and Inequality from Lemma \[Sobol\], that: $$\begin{aligned}
\displaystyle &\int_{0}^t \left\|\nabla G_{t-s} * \nabla \cdot \Big(b(s,.)q(s,.)\Big)\right\|_{L^1}\,ds \\
&\le d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \left(B + \check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right) \Big(\|q(s,.)\|_{L^1} + \|\nabla q(s,.)\|_{L^1} \Big)\,ds \\
&\le d\sqrt{2 \pi}\max(1,\sqrt{T})\left(B + \check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\right)\left(\sup_{u \in (0,\theta]}\|q(u,.)\|_{L^1} + \sup_{u \in (0,\theta]}\sqrt u\|\nabla q(u,.)\|_{L^1} \right).\end{aligned}$$ Therefore, $\displaystyle \int_{0}^t \left\|\nabla G_{t-s} * \nabla \cdot \Big(b(s,.)q(s,.)\Big)\right\|_{L^1}\,ds \le d\sqrt{2 \pi}\max(1,\sqrt{T})\left(B + \check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\right){\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{q}$ which is finite. We can then apply Fubini’s theorem and obtain that, in the sense of distributions, the gradient of $\Phi_t(q)$ defined in , is equal to: $$\begin{aligned}
\label{gradPhi}
\displaystyle \nabla \Phi_t(q) = \nabla G_t* m - \int_0^t \nabla G_{t-s}*\nabla \cdot \Big(b(s,.)q(s,.) \Big)\,ds.\end{aligned}$$ We can now estimate ${\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Phi(q)}$. Using the same arguments as before, we have that: $$\displaystyle {\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{G*m} = \sup_{t \in [0,\theta]}\|G_t*m \|_{L^1} + \sup_{t \in [0,\theta]}\sqrt{t}\sum_{i=1}^d\|\partial_{x_i} G_t*m\|_{L^1} \le 1 + d\sqrt{\frac 2 \pi} <+\infty,$$ and that: $$\begin{aligned}
\displaystyle &\|\Phi_t(q)\|_{L^1} + \sqrt{t}\|\nabla \Phi_t(q)\|_{L^1} \\
&\le \left(1 + d\sqrt{\frac 2 \pi}\right) + d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \left\{ B\|q(s,.)\|_{L^1} + \sqrt{t} \Big(\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\|q(s,.)\|_{W^{1,1}} + B\|\nabla q(s,.)\|_{L^1}\Big) \right\}\,ds \\
&\le \left(1 + d\sqrt{\frac 2 \pi}\right) + d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \bigg\{ \left(B + \sqrt{T}\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\|q(s,.)\|_{L^1} \\
&\phantom{\left(1 + d\sqrt{\frac 2 \pi}\right) + d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \bigg\{\Big(B + \sqrt{T}}+ \sqrt{\theta}\left(B + \check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\|\nabla q(s,.)\|_{L^1} \bigg\}\,ds \end{aligned}$$ $$\begin{aligned}
&\le \left(1 + d\sqrt{\frac 2 \pi}\right) + \sqrt{\theta}d\sqrt{\frac 2 \pi}\Bigg\{ 2\left(B + \sqrt{T}\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sup_{u\in (0,\theta]}\|q(u,.)\|_{L^1} \\
&\phantom{\left(1 + d\sqrt{\frac 2 \pi}\right) + \sqrt{\theta}\bigg\{ 2d\sqrt{\frac 2 \pi}\Big(B + \sqrt{T}\check{C} }+ \pi\left(B + \check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right) \sup_{u\in (0,\theta]}\sqrt{u}\|\nabla q(u,.)\|_{L^1} \Bigg\} \\
&= \left(1 + d\sqrt{\frac 2 \pi}\right) + \sqrt{\theta}d\sqrt{\frac 2 \pi}\Bigg\{ B\left(2\sup_{u\in (0,\theta]}\|q(u,.)\|_{L^1} + \pi \sup_{u\in (0,\theta]}\sqrt{u}\|\nabla q(u,.)\|_{L^1} \right) \\
&\phantom{\left(1 + d\sqrt{\frac 2 \pi}\right) + \sqrt{\theta}\bigg\{ 2d\sqrt{\frac 2 \pi}\Big(B + }+ \check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\left(2\sqrt{T}\sup_{u\in (0,\theta]}\|q(u,.)\|_{L^1} + \pi \sup_{u\in (0,\theta]}\sqrt{u}\|\nabla q(u,.)\|_{L^1} \right) \Bigg\}\\
&\le \left(1 + d\sqrt{\frac 2 \pi}\right) + d\sqrt{2 \pi \theta}\left(B + \max(1,\sqrt{T})\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right) \left(\sup_{u\in (0,\theta]}\|q(u,.)\|_{L^1} + \sup_{u\in (0,\theta]}\sqrt{u}\|\nabla q(u,.)\|_{L^1} \right).\end{aligned}$$ Hence, $\displaystyle {\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Phi(q)} \le \left(1 + d\sqrt{\frac 2 \pi}\right) + d\sqrt{2 \pi \theta}\left(B + \max(1,\sqrt{T})\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right){\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{q}$ and ${\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Phi(q)} <+\infty$. Now, let $0 <r\le s\le \theta$, adapting the proof of Lemma \[evoldens\] and using Inequality from Lemma \[Sobol\], we obtain that: $$\begin{aligned}
\displaystyle &\left\|\nabla \Phi_s(q) - \nabla \Phi_r(q) \right\|_{L^1} \\
&\le d(2d+3)\sqrt{\frac 2 \pi}\left(\frac{1}{\sqrt{r}} - \frac{1}{\sqrt s} \right) + 4d(d+2)\sqrt{\frac 2 \pi}\left(B + \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sup_{u \in [r,s]}\|q(u,.)\|_{W^{1,1}}\sqrt{s-r}.\end{aligned}$$ Since ${\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Phi(q)} = \sup_{u \in (0,\theta]}\|q(u,.)\|_{L^1} + \sup_{u \in (0,\theta]}\sqrt{u}\|\nabla q(u,.)\|_{L^1}$ and using Inequality , we can conclude that $t\mapsto \Phi_t(q)$ is continuous on $(0,\theta]$ with values in $W^{1,1}\left({{\mathbb R}}^d\right)$. Hence, $\Phi(q) \in \mathcal{\tilde C}\left((0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$.\
Now, let $q,\tilde q \in \mathcal{\tilde C} \left((0,\theta], W^{1,1}\left({{\mathbb R}}^d \right)\right)$, we obtain, with the same reasoning above, that $$\displaystyle {\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{\Phi(q) - \Phi(\tilde q)} \le d\sqrt{2 \pi \theta}\left(B + \max(1,\sqrt{T})\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right){\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{q-\tilde q}.$$ If: $$\displaystyle \theta < \frac{1}{2\pi d^2 \left(B + \max(1,\sqrt{T})\check{C} \sup_{u\in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)^2} =: 2\theta_0$$ then the map $\Phi$ is a contraction on the space $\mathcal{\tilde C} \left((0,\theta_0], W^{1,1}\left({{\mathbb R}}^d \right) \right)$. Using Picard’s Theorem, the map $\Phi$ admits then a unique fixed-point on the space $\mathcal{\tilde C} \left((0,\theta_0], W^{1,1}\left({{\mathbb R}}^d \right)\right)$.\
$\bullet$ Now, we assume that $m$ admits a density w.r.t. the Lebesgue measure in $W^{1,1}\left({{\mathbb R}}^d\right)$. Let us prove that the map $\Phi$ admits a unique-fixed point in $\mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$ for all $\theta \in (0,T]$:\
Let $t \in (0,\theta]$ and $q \in \mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$ for all $\theta \in (0,T]$. Using Equation and the fact that $m$ admits a density w.r.t. the Lebesgue measure in $W^{1,1}\left({{\mathbb R}}^d\right)$, we obtain: $$\begin{aligned}
\displaystyle \nabla \Phi_t(q) = G_t* \nabla m - \int_0^t \nabla G_{t-s}*\nabla \cdot \Big(b(s,.)q(s,.) \Big)\,ds.\end{aligned}$$ Using the estimate from Lemma \[EstimHeatEq\] and Inequality from Lemma \[Sobol\], we have: $$\begin{aligned}
\displaystyle &\left\|\Phi_t(q)\right\|_{W^{1,1}} \le \left\|G_t*m\right\|_{W^{1,1}} + d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \left(B\left\|q(s,.)\right\|_{L^1} + \left\|\nabla. \Big(b(s,.)q(s,.)\Big)\right\|_{L^1} \right)\,ds \\
&\le \left(1 + \sum_{i=1}^d\left\|\partial_{x_i} m\right\|_{L^1} \right) + d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \Big(B\left\|q(s,.)\right\|_{L^1} + \check{C}\|\nabla \cdot b(s,.) \|_{L^{\rho}}\left\|q(s,.)\right\|_{W^{1,1}} + B \left\|\nabla q(s,.) \right\|_{L^1} \Big)\,ds\end{aligned}$$
$$\begin{aligned}
&\le \left(1 + \sum_{i=1}^d\left\|\partial_{x_i} m\right\|_{L^1} \right) + d\sqrt{\frac 2 \pi}\int_0^t \frac{1}{\sqrt{t-s}} \Big(B\left\|q(s,.)\right\|_{L^1} + \check{C}\|\nabla \cdot b(s,.) \|_{L^{\rho}}\left\|q(s,.)\right\|_{W^{1,1}} + B \left\|\nabla q(s,.) \right\|_{L^1} \Big)\,ds\\
&\le \left(1 + \sum_{i=1}^d\left\|\partial_{x_i} m\right\|_{L^1} \right) + d\sqrt{\frac 2 \pi} \left(B + \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\int_0^t \frac{1}{\sqrt{t-s}}\left\|q(s,.)\right\|_{W^{1,1}}\,ds \\
&\le \left(1 + \sum_{i=1}^d\left\|\partial_{x_i} m\right\|_{L^1} \right) + 2d\sqrt{\frac{ 2 t}{\pi}} \left(B + \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sup_{u \in [0,\theta]}\left\|q(u,.)\right\|_{W^{1,1}}.\end{aligned}$$
Hence, $\displaystyle \sup_{t \in [0,\theta]}\left\|\Phi_t(q)\right\|_{W^{1,1}} \le \left(1 + \sum_{i=1}^d\left\|\partial_{x_i} m\right\|_{L^1} \right) + 2d\sqrt{\frac{ 2 \theta}{\pi}} \left(B + \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sup_{t \in [0,\theta]}\left\|q(t,.)\right\|_{W^{1,1}}$ is finite. Now, concerning the continuity of $t \mapsto \Phi_t(q)$ on $[0,\theta]$ with values in $W^{1,1}\left({{\mathbb R}}^d\right)$, we already proved above the continuity on $(0,\theta]$. As for the continuity at $t=0$, we denote by $\tau_y w$ the translation of $w \in L^1\left({{\mathbb R}}^d\right)$ by $y \in {{\mathbb R}}^d$ defined by $\tau_y w(x) = w(x -y)$ for $x \in {{\mathbb R}}^d$. We have: $$\begin{aligned}
\displaystyle \left\|G_t*w - w\right\|_{L^1} &= \int_{{{\mathbb R}}^d}\left|\int_{{{\mathbb R}}^d}G_t(y)w(x-y)\,dy - w(x)\right|\,dx = \int_{{{\mathbb R}}^d}\left|\int_{{{\mathbb R}}^d}G_1(y)w(x-\sqrt{t}y)\,dy - w(x)\right|\,dx \\
&\le \int_{{{\mathbb R}}^d}G_1(y)\left\|\tau_{\sqrt{t}y}w-w\right\|_{L^1}\,dy.
\end{aligned}$$ Using Lemma $IV.4$ [@Brezis], we have that $\left\|\tau_{\sqrt{t}y}w-w\right\|_{L^1} \to 0$ when $t \to 0$ and since $\left\|\tau_{\sqrt{t}y}w-w\right\|_{L^1} \le 2 \left\|w\right\|_{L^1}$, by dominated convergence, we obtain that $G_t*w \to w$ when $t \to 0$ in $L^1\left({{\mathbb R}}^d\right)$. We replace $w$ by $m$ and $\nabla m$ since $m$ admits a density in $W^{1,1}\left({{\mathbb R}}^d\right)$ and conclude that $\left\|G_t*m -m \right\|_{W^{1,1}} \to 0$ when $t \to 0$. Moreover, we have that: $$\begin{aligned}
\displaystyle \left\|\int_0^t G_{t-s} * \nabla \cdot \Big(b(s,.)q(s,.)\Big)\,ds \right\|_{W^{1,1}} &\le \left(B + \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right) \sup_{u \in [0,\theta]}\left\|q(u,.)\right\|_{W^{1,1}}\, \left(t + d\sqrt{\frac 2 \pi}\sqrt{t} \right) \end{aligned}$$ which converges to $0$ when $t \to 0$. Hence, $\displaystyle \left\|\Phi_t(q) - m \right\|_{W^{1,1}}$ converges to $0$ when $t \to 0$ and $\Phi \in \mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$.\
Now, let $q,\tilde q \in \mathcal{C} \left([0,\theta], W^{1,1}\left({{\mathbb R}}^d \right)\right)$. With the same reasoning above, we obtain that: $$\begin{aligned}
\label{machin}
\displaystyle \left\|\Phi_t(q) - \Phi_t(\tilde q)\right\|_{W^{1,1}} &\le d\sqrt{\frac 2 \pi} \left(B + \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\int_0^t \frac{1}{\sqrt{t-s}}\left\|q(s,.) - \tilde q(s,.)\right\|_{W^{1,1}}\,ds.\end{aligned}$$ As done in the proof of Lemma \[uniqFP\], for $n \in {{\mathbb N}}^{*}$, we iterate Inequality $2n$-times and deduce that for $n$ big enough, $\Phi^{2n}$ is a contraction on $\mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$. We conclude, through Picard’s Theorem, the existence of a unique fixed-point of the map $\Phi$ on $\mathcal{C}\left([0,\theta],W^{1,1}\left({{\mathbb R}}^d\right)\right)$.
We are now ready to prove Proposition \[hyp\].
We are going to suppose that $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$. When $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$, the proof is analogous and the estimations remain valide when replacing $\left\|\nabla \cdot b(t,.) \right\|_{L^{\rho}}$ by $\left\|\partial_x b(t,.)\right\|_{\rm TV}$.\
The proof is an immediate consequence of Theorem \[sobolP\] and Lemma \[Sobol\]. Indeed, assuming the regularity on $b$ w.r.t. the space variables, we have from Theorem \[sobolP\] that $\left(p(t,.) \right)_{t \in (0,T]} \in \mathcal{\tilde C}\left((0,T],W^{1,1}\left({{\mathbb R}}^d\right)\right)$. We can then apply Lemma \[Sobol\] to obtain, for $t \in (0,T]$, that: $$\begin{aligned}
\displaystyle \sqrt{t}\left\|\nabla \cdot \Big(b(t,.)p(t,.) \Big) \right\|_{\rm TV} &\le \left( B+ \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sqrt{t} \left\| \nabla p(t,.) \right\|_{L^1} + \check{C} \sqrt{T}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\left\|p(t,.) \right\|_{L^1} \\
&\le \max\left(B+ \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}, \check{C} \sqrt{T}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\right){\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{p} <+\infty.\end{aligned}$$ The conclusion holds with $M = \max\left(B+ \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}, \check{C} \sqrt{T}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}}\right){\@ifstar{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
s
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{ \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
}
\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{p}$. Since $p \in \mathcal{\tilde C}\left((0,T],W^{1,1}\left({{\mathbb R}}^d\right)\right)$, using Lemma \[Sobol\], we have as in Equality that: $$\displaystyle \forall t \in (0,T], \quad p(t,.) = G_t * m - \int_0^t G_{t-s}*\nabla \cdot \Big(b(s,.)p(s,.)\Big)\,ds.$$ Moreover, when $m$ admits a density w.r.t. the Lebesgue measure in $W^{1,1}\left({{\mathbb R}}^d\right)$, according to Theorem \[sobolP\], $\left(p(t,.) \right)_{t \in [0,T]} \in \mathcal{C}\left([0,T],W^{1,1}\left({{\mathbb R}}^d\right)\right)$. Therefore, using once again Lemma \[Sobol\], we obtain for $t \in [0,T]$ that: $$\begin{aligned}
\displaystyle \left\|\nabla \cdot \Big(b(t,.)p(t,.) \Big) \right\|_{\rm TV} &\le \left( B+ \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sup_{u \in [0,T]}\left\| p(u,.) \right\|_{W^{1,1}} <+\infty.\end{aligned}$$ The conclusion holds with $\tilde M = \left( B+ \check{C}\sup_{u \in [0,T]}\|\nabla \cdot b(u,.) \|_{L^{\rho}} \right)\sup_{u \in [0,T]}\left\| p(u,.) \right\|_{W^{1,1}}$.
Proof of Theorem \[cvgenceB\] {#demoo}
-----------------------------
We first use Inequality from Proposition \[hyp\] to obtain a stronger regularity of $p(t,.)$ with respect to the time variable.
\[evoldensB\] Assume Inequality . We have: $$\displaystyle \exists \, \tilde Q<+\infty, \forall \, 0 < r\le s\le T,\quad \left\|p(s,.) - p(r,.) \right\|_{L^1} \le \tilde Q\bigg( \ln(s/r) + \frac{s-r}{2\sqrt r} \ln\left(\frac{4s}{s-r} \right) +\left(\sqrt{s} - \sqrt{r}\right) \bigg).$$
We will adapt the proof of Lemma \[evoldens\]. Let $0 < r \le s \le T$. Using Equality from Proposition \[hyp\], we have that: $$\begin{aligned}
\label{diffP}
\displaystyle p(s,.) - p(r,.) = \left( G_s - G_r \right) * m - \int_0^s G_{s-u} * \nabla \cdot \Big(b(u,.)p(u,.)\Big)\,du + \int_0^r G_{r-u} * \nabla \cdot \Big(b(u,.)p(u,.)\Big)\,du.
\end{aligned}$$ Therefore, using the estimates and from Lemma \[EstimHeatEq\], the fact that $\ln(1+x) \le x, \; \forall x >0$ and Lemma \[intg\], we obtain:
$$\begin{aligned}
\displaystyle &\left\|p(s,.) - p(r,.) \right\|_{L^1} \\
&\le \left\|\left( G_s - G_r \right) * m \right\|_{L^1} + \left\|\int_0^r \left( G_{s-u} - G_{r-u} \right)*\nabla \cdot \Big(b(u,.)p(u,.)\Big)\,du \right\|_{L^1} + \left\|\int_r^s G_{s-u} * \nabla \cdot \Big(b(u,.)p(u,.)\Big)\,du \right\|_{L^1} \\
&\le \int_r^s \left\|\partial_u G_u \right\|_{L^1} \,du +\int_0^s \int_{r-u}^{s-u} \left\|\partial_{\theta} G_{\theta} \right\|_{L^1} \left\|\nabla \cdot \Big(b(u,.)p(u,.) \Big) \right\|_{\rm TV}\,d\theta\,du + M \int_r^s \frac{du}{\sqrt u}\\
&\le d\ln(s/r) + dM \int_0^r \int_{r-u}^{s-u} \frac{d\theta}{\theta \sqrt{u}}\,du + 2M \left(\sqrt{s} - \sqrt{r} \right) \\
&= d\ln(s/r) + 2dM \left(\frac{s-r}{\sqrt s + \sqrt r} \ln\left(\frac{\left(\sqrt s + \sqrt r \right)^2}{s-r} \right) + 2\sqrt r \ln\left(1 + \frac{\sqrt s - \sqrt r}{2 \sqrt r} \right) \right) + 2M \left(\sqrt{s} - \sqrt{r}\right)\\
&\le d\ln(s/r) + dM \; \frac{s-r}{\sqrt r} \ln\left(\frac{4s}{s-r} \right) + 2M(1+d)\left(\sqrt{s} - \sqrt{r}\right).
\end{aligned}$$
The conclusion holds with $\tilde Q = \max\left(d,2M(1+d)\right)$.
We are now ready to prove Theorem \[cvgenceB\]. Once again, using Equality and Proposition \[propspde\], to prove the theorem amounts to prove that: $$\exists \, \tilde{C}<+\infty, \forall h \in (0,T], \, \forall k \in \left\llbracket 1, \left\lfloor\frac{T}{h}\right\rfloor \right\rrbracket, \quad \left\| p(kh,.) - p^h(kh,.)\right\|_{L^1} \le \frac{\tilde C}{\sqrt{kh}}\left(1 + \ln\left(k\right) \right)h.$$ For $h \in (0,T], \, k \in \left\llbracket 1, \left\lfloor\frac{T}{h}\right\rfloor \right\rrbracket$, we recall Inequality : $$\begin{aligned}
\displaystyle \Big\|p(kh,.)-p^h\left(kh,.\right)\Big\|_{L^1} &\le \left\|V^h_1(k,.)\right\|_{L^1}+ \left\|V^h_2(k,.)\right\|_{L^1} + \sqrt{\frac 2 \pi}dB\sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1} \\
&+ 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left(\frac{1}{2} + \ln\left(k \right) \right)h + \sqrt{\frac{2}{\pi}} dB \frac{3h}{\sqrt{kh}}.\end{aligned}$$ Let us estimate $\left\|V^h_1(k,.)\right\|_{L^1}$ and $\left\|V^h_2(k,.)\right\|_{L^1}$ for $k \ge 2$ by taking advantage of the additional regularity of $b$ and using Equality from Proposition \[hypH\]:\
\
$\bullet$ We use Lemma \[Sobol\] and the additional regularity of $b$ to transfer the gradient from $G$ to $bp$ and rewrite $V_1^h(k,.)$ as:
$$\begin{aligned}
\label{v1}
\displaystyle &V_1^h(k,.) = \int_{h}^{kh}\bigg(G_{kh-s}- G_{kh-\tau^h_s} \bigg)* \nabla \cdot \Big(b(s,.)p(s,.) \Big)\,ds \nonumber \\
&= \int_h^{(k-1)h}\left(\int_{kh-s}^{kh-\tau^h_s}\partial_u G_u \,du\right)*\nabla \cdot \Big(b(s,.)p(s,.) \Big)\,ds + \int_{(k-1)h}^{kh}\bigg(G_{kh-s}- G_{kh-\tau^h_s} \bigg)* \nabla \cdot \Big(b(s,.)p(s,.) \Big)\,ds.\end{aligned}$$
Therefore, using Estimate from Lemma \[EstimHeatEq\], Inequality and Lemma \[integ\], we obtain: $$\begin{aligned}
\displaystyle \left\|V_1^h(k,.)\right\|_{L^1} &\le dM \int_{h}^{(k-1)h}\frac{1}{\sqrt s} \ln\left(1 + \frac{s-\tau^h_s}{kh -s}\right)\,ds + 2M \int_{(k-1)h}^{kh}\frac{ds}{\sqrt s}\\
&\le dM \int_{h}^{(k-1)h} \frac{h}{(kh-s)\sqrt s}\,ds + 4M \left(\sqrt{kh}-\sqrt{(k-1)h} \right) \\
&\le dM\frac{h}{\sqrt{kh}}\ln\left(4k\right) + 4M \frac{h}{\sqrt{kh} + \sqrt{(k-1)h}} \\
&\le \frac{2M\left( 2 + d\ln(2)\right)}{\sqrt{kh}}\left(1 + \ln\left(k\right)\right)h.\end{aligned}$$ $\bullet$ Using Lemma \[evoldensB\] and the estimate from Lemma \[EstimHeatEq\], we have: $$\begin{aligned}
\displaystyle \left\|V_2^h(k,.) \right\|_{L^1} &\le \sqrt{\frac 2 \pi}dB\tilde Q \Bigg\{ \int_{h}^{kh} \frac{\ln(s/\tau^h_s)}{\sqrt{kh - \tau^h_s}}\,ds + \int_{h}^{kh} \frac{s-\tau^h_s}{2\sqrt{\tau^h_s}\sqrt{kh - \tau^h_s}} \ln\left(\frac{4s}{s-\tau^h_s} \right)\,ds + \int_{h}^{kh}\frac{\sqrt{s} - \sqrt{\tau^h_s}}{\sqrt{kh - \tau^h_s}} \,ds \Bigg\}.\end{aligned}$$ We use Inequality , the fact that $\displaystyle \sup_{0<x\le h}\left( x\ln\left(\frac{4kh}{x}\right)\right) $ is attained for $x=h$ since $h \le kh$ and Lemma \[sumPi\] to obtain: $$\begin{aligned}
\displaystyle \left\|V_2^h(k,.) \right\|_{L^1} &\le \sqrt{\frac 2 \pi} dB\tilde Q\left\{ \frac{2h}{\sqrt{kh}}\ln(4k) + h \ln\left( 4k\right)\int_{h}^{kh} \frac{ds}{2\sqrt{\tau^h_s}\sqrt{kh - \tau^h_s}} + \int_{h}^{kh} \frac{s -\tau^h_s}{\left(\sqrt{s}+\sqrt{\tau^h_s} \right)\sqrt{kh - \tau^h_s}} \,ds \right\}\\
&\le \sqrt{\frac 2 \pi}dB\tilde Q \left\{\frac{2h}{\sqrt{kh}}\ln(4k) + \frac{h}{2} \ln\left(4k\right) \sum_{j=1}^{k-1} \frac{1}{\sqrt{j}\sqrt{k-j}} + \frac{h}{2} \sum_{j=1}^{k-1} \frac{1}{\sqrt{j}\sqrt{k-j}} \right\} \\
&\le \sqrt{\frac 2 \pi}dB\tilde Q\left\{\frac{2}{\sqrt{kh}}\ln\left(4k\right) + \frac{\pi}{2}\Big(1 +\ln\left(4k\right) \Big) \right\}h.\end{aligned}$$ Therefore, $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} &\le \frac{\tilde L}{\sqrt{kh}}\Big(1 + \ln\left(k\right)\Big)h+ \sqrt{\frac{2}{\pi}}dB \sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1}\end{aligned}$$ where $\displaystyle \tilde L = 2 M\Big(2+d\ln(2)\Big) + dB \left\{\sqrt{\frac 2 \pi}\left(3 + 4\ln(2)\tilde Q \right) + \left(2B\left(1+\frac{d-1}{\pi} \right)+ \sqrt{\frac{\pi}{2}}\Big(1+2\ln(2)\Big)\tilde Q \right)\sqrt{T} \right\}$. Iterating this inequality, using the fact that for $j \le k, \ln(j) \le \ln(k)$ and using Lemma \[sumPi\], we obtain: $$\begin{aligned}
\displaystyle &\left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \\
&\le \frac{\tilde L}{\sqrt{kh}}\Big(1 + \ln\left(k\right)\Big)h+ \sqrt{\frac{2}{\pi}}dB \sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left(\frac{\tilde L}{\sqrt{jh}}\Big(1 + \ln\left(j\right)\Big)h+ \sqrt{\frac{2}{\pi}}dB \sum_{l=1}^{j-1} \frac{h}{\sqrt{jh - lh}}\left\|p(lh,.) - p^h(lh,.) \right\|_{L^1} \right) \end{aligned}$$
$$\begin{aligned}
&\le \tilde{L}\left(\frac{1}{\sqrt{kh}} + \sqrt{\frac 2 \pi}dB \sum_{j=1}^{k-1}\frac{h}{\sqrt{kh-jh}\sqrt{jh}} \right)\Big(1 + \ln\left(k\right)\Big)h+ 2d^2B^2 h\sum_{j=1}^{k-1}\left\|p\left(jh,.\right) - p^h\left(jh,.\right) \right\|_{L^1}\\
&\le \tilde L\sqrt{h}\left(1 + dB\sqrt{2\pi T }\right)\frac{1 + \ln\left(k\right)}{\sqrt{k}}+ 2d^2B^2 h\sum_{j=1}^{k-1}\left\|p\left(jh,.\right) - p^h\left(jh,.\right) \right\|_{L^1}.\end{aligned}$$
We apply Lemma \[Gronw\] and obtain that: $$\begin{aligned}
\displaystyle &\left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \\
&\le \tilde L\sqrt{h}\left(1 + dB\sqrt{2\pi T }\right)\frac{1 + \ln\left(k\right)}{\sqrt{k}} + 2d^2B^2 \tilde L\sqrt{h}\left(1 + dB\sqrt{2\pi T }\right) h \sum_{j=1}^{k-1}\frac{1+\ln(j)}{\sqrt{j}}\exp\Big(2d^2B^2\left(kh -(j+1)h\right)\Big).\end{aligned}$$ Now, using once again the fact that for $j \le k, \ln(j) \le \ln(k)$, we have: $$\begin{aligned}
\displaystyle \sqrt{h}\sum_{j=1}^{k-1}\frac{1+\ln(j)}{\sqrt{j}}\exp\Big(2d^2B^2\left(kh -(j+1)h\right)\Big) &\le (1+\ln(k))\exp\Big(2d^2B^2 T \Big) \int_h^{kh}\frac{ds}{\sqrt{s}} \\
&= 2(1+\ln(k))\exp\Big(2d^2B^2 T \Big)\sqrt{T}.\end{aligned}$$ Therefore, we deduce that: $$\displaystyle \forall h \in (0,T], \forall k \in \left \llbracket 1, \left \lfloor \frac{T}{h}\right \rfloor \right \rrbracket , \quad \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \le \frac{\tilde C}{\sqrt{kh}}\Big(1 + \ln\left(k\right)\Big)h$$ where $\displaystyle \tilde C = \tilde L \left(1 + 4d^2B^2 T\left(1 + dB\sqrt{2\pi T }\right) \exp\Big(2d^2B^2T \Big)\right)$.
Proof of Proposition \[propregM\] {#demooo}
---------------------------------
We first use Inequality from Proposition \[hyp\] to obtain a stronger regularity of $p(t,.)$ with respect to the time variable.
\[evoldensM\] Assume $b:[0,T]\times{{\mathbb R}}^d \to {{\mathbb R}}^d$ is measurable and bounded by $B<+\infty$ such that $\,\sup_{t \in [0,T]} \|\nabla \cdot b(t,.)\|_{L^\rho}<+\infty$ for some $\rho \in [d,+\infty]$ or that for $d=1$, $\sup_{t \in [0,T]}\left\|\partial_x b(t,.)\right\|_{\rm TV}<+\infty$; where $\nabla \cdot b(t,.)$ and $\partial_x b(t,.)$ are respectively the spatial divergence and the spatial derivative of $b$ in the sense of distributions. Moreover, assume that $m$ admits a density w.r.t. the Lebesgue measure that belongs to $W^{1,1}\left({{\mathbb R}}^d\right)$. We have: $$\displaystyle \exists \, \hat Q<+\infty, \forall \, 0 < r\le s\le T,\quad \left\|p(s,.) - p(r,.) \right\|_{L^1} \le \hat Q \Bigg((\sqrt s - \sqrt r)+ (s-r)\ln\left(\frac{s}{s-r}\right) + r\ln\left(s/r\right) + (s-r)\Bigg).$$
We will adapt, once again, the proof of Lemma \[evoldens\]. Let $0 < r \le s \le T$, using Equality , the estimates , and from Lemma \[EstimHeatEq\], Inequality from Proposition \[hyp\] and Lemma \[intg\], we obtain: $$\begin{aligned}
\displaystyle &\left\|p(s,.) - p(r,.) \right\|_{L^1} \\
&\le \left\|\left( G_s - G_r \right) * m \right\|_{L^1} + \left\|\int_0^r \left( G_{s-u} - G_{r-u} \right)*\nabla \cdot \Big(b(u,.)p(u,.)\Big)\,du \right\|_{L^1} + \left\|\int_r^s G_{s-u} * \nabla \cdot \Big(b(u,.)p(u,.)\Big)\,du \right\|_{L^1} \\
&\le \left\|\int_r^s \left(\partial_u G_u* m\right)\,du \right\|_{L^1} + \tilde M \int_0^s \int_{r-u}^{s-u} \left\|\partial_{\theta} G_{\theta} \right\|_{L^1}\,d\theta\,du + \tilde M (s-r)\\
&\le \frac{1}{2}\sum_{i=1}^d\|\partial_{x_i} m\|_{L^1}\int_r^s \left\| \partial_{x_i} G_{u}\right\|_{L^1}\,du + d \tilde M\int_0^r \ln\left(\frac{s-u}{r-u}\right)\,du + \tilde M (s-r)\\
&\le 2\sqrt{\frac 2 \pi} \sum_{i=1}^d\|\partial_{x_i} m\|_{L^1} \left(\sqrt{s} - \sqrt{r} \right) + d \tilde M \left((s-r)\ln\left(\frac{s}{s-r}\right) + r\ln\left(s/r\right) \right) + \tilde M (s-r).
\end{aligned}$$ The conclusion holds with $\displaystyle \hat Q = \max\left(2\sqrt{\frac 2 \pi} \sum_{i=1}^d\|\partial_{x_i} m\|_{L^1}, d\tilde M \right)$.
We are now ready to prove Proposition \[propregM\]. Once more, using Equality and Proposition \[propspde\], to prove the theorem amounts to prove that: $$\exists \, \hat{C}<+\infty, \forall h \in (0,T], \, \forall k \in \left\llbracket 1, \left\lfloor\frac{T}{h}\right\rfloor \right\rrbracket, \quad \left\| p(kh,.) - p^h(kh,.)\right\|_{L^1} \le \hat{C}\Big(1 + \ln\left(k\right) \Big)h.$$ For $h \in (0,T], \, k \in \left\llbracket 1, \left\lfloor\frac{T}{h}\right\rfloor \right\rrbracket$, we recall Inequality : $$\begin{aligned}
\displaystyle \Big\|p(kh,.)-&p^h\left(kh,.\right)\Big\|_{L^1} \le \left\|V^h_1(k,.)\right\|_{L^1}+ \left\|V^h_2(k,.)\right\|_{L^1} + \sqrt{\frac 2 \pi}dB\sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1}\\
&+ 2dB^2 \left(1 + \frac{d-1}{\pi}\right) \left(\frac{1}{2} + \ln\left(k \right) \right)h + \left\| \int_0^{h} \bigg(\nabla G_{kh-s} * \Big(b(s,.)p(s,.)\Big) - \nabla G_{kh} * \Big(b(s,.)m\Big)\bigg)\,ds \right\|_{L^1}.\end{aligned}$$ Concerning the last term of the right-hand side of this previous inequality, we use Lemma \[Sobol\] and the additional regularity of $b$ to transfer the gradient from $G$ to $bp$; and using Inequality from Proposition \[hyp\], we obtain that: $$\begin{aligned}
\displaystyle \bigg\| \int_0^{h} \bigg(G_{kh-s} * \nabla \cdot \Big(b(s,.)p(s,.)\Big) &- G_{kh} * \nabla \cdot \Big(b(s,.)m\Big)\bigg)\,ds \bigg\|_{L^1} \\
&\le \int_0^{h} \left(\left\|\nabla \cdot \Big(b(s,.)p(s,.)\Big) \right\|_{\rm TV} + \left\|\nabla \cdot \Big(b(s,.)m\Big) \right\|_{\rm TV}\right)\,ds \le 2\tilde M h.\end{aligned}$$ Let us estimate $\left\|V^h_1(k,.)\right\|_{L^1}$ and $\left\|V^h_2(k,.)\right\|_{L^1}$ for $k \ge 2$ by taking advantage of the additional regularity of $b$ and $m$, and using Equality from Proposition \[hypH\]:\
\
$\bullet$ We recall Equality : $$\begin{aligned}
\displaystyle &V_1^h(k,.) = \int_h^{(k-1)h}\left(\int_{kh-s}^{kh-\tau^h_s}\partial_u G_u \,du\right)*\nabla \cdot \Big(b(s,.)p(s,.) \Big)\,ds + \int_{(k-1)h}^{kh}\bigg(G_{kh-s}- G_{kh-\tau^h_s} \bigg)* \nabla \cdot \Big(b(s,.)p(s,.) \Big)\,ds.\end{aligned}$$ Therefore, using the fact that $\ln(1+x) \le x, \, \forall x>0$ and Estimate from Lemma \[EstimHeatEq\], we obtain: $$\begin{aligned}
\displaystyle \left\|V_1^h(k,.)\right\|_{L^1} &\le d\tilde M \int_{h}^{(k-1)h}\ln\left(1 + \frac{s-\tau^h_s}{kh -s}\right)\,ds + 2 \tilde M h\\
&\le d\tilde M \int_{h}^{(k-1)h} \frac{h}{kh-s}\,ds + 2 \tilde M h = \tilde M\left( d\ln(k-1) + 2\right)h \\
&\le \tilde M \Big(2 + d\ln\left(k\right) \Big)h.\end{aligned}$$ $\bullet$ Using Lemma \[evoldensM\], the estimate from Lemma \[EstimHeatEq\] and the fact that $\ln(1+x) \le x, \, \forall x>0$, we have: $$\begin{aligned}
\displaystyle \left\|V_2^h(k,.)\right\|_{L^1} &\le \sqrt{\frac 2 \pi}dB\hat Q \bigg\{ \int_{h}^{kh} \frac{\sqrt s - \sqrt{\tau^h_s}}{\sqrt{kh - \tau^h_s}}\,ds + \int_{h}^{kh} \frac{s-\tau^h_s}{\sqrt{kh-\tau^h_s}}\ln\left(\frac{s}{s-\tau^h_s} \right)\,ds \\
&\phantom{\sqrt{\frac 2 \pi}dB\hat Q \bigg\{ \int_{h}^{kh} \frac{\sqrt s - \sqrt{\tau^h_s}}{\sqrt{kh - \tau^h_s}}\,ds}+ \int_{h}^{kh} \frac{\tau^h_s}{\sqrt{kh-\tau^h_s}}\ln\left(1 + \frac{s-\tau^h_s}{\tau^h_s} \right)\,ds + \int_{h}^{kh}\frac{s - \tau^h_s}{\sqrt{kh - \tau^h_s}} \,ds \bigg\}\\
&\le \sqrt{\frac 2 \pi}dB\hat Q \bigg\{\int_{h}^{kh} \frac{h}{\sqrt{kh - s}\sqrt{s}}\,ds+ \int_{h}^{kh}\frac{s-\tau^h_s}{\sqrt{kh-s}}\ln\left(\frac{kh}{s-\tau^h_s} \right)\,ds + 2\int_{h}^{kh}\frac{h}{\sqrt{kh - s}} \,ds\bigg\}.\end{aligned}$$ The function $x \mapsto x\ln\left(kh/x\right)$ is increasing on the interval $(0,kh/e]$, attains its maximum at $x = kh/e$ and non-increasing on the interval $[kh/e, +\infty)$. Therefore, we get that: $$\displaystyle \left(s-\tau^h_s\right)\ln\left(\frac{kh}{s-\tau^h_s} \right)\le \left( h \ln\left(k\right) \mathbf{1}_{\left\{ h \le \frac{kh}{e}\right\}} + \frac{kh}{e}\mathbf{1}_{\left\{ h > \frac{kh}{e}\right\}} \right) \le \Big(1+\ln\left(k\right)\Big)h.$$ We then deduce that: $$\begin{aligned}
\displaystyle \left\|V_2^h(k,.)\right\|_{L^1} &\le \sqrt{\frac 2 \pi}dB\hat Q \left(\pi + 2\sqrt{T}\Big(1+\ln\left(k\right)\Big) + 4\sqrt{T} \right)h.\end{aligned}$$ Therefore, $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} &\le \hat L \Big(1+\ln\left(k\right)\Big)h+ \sqrt{\frac{2}{\pi}}dB \sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1}\end{aligned}$$ where $\displaystyle \hat L = \max(2,d)\tilde M + dB \left(2B\left(1 + \frac{d-1}{\pi} \right) + \sqrt{\frac 2 \pi}\hat Q \left(\pi + 6\sqrt{T}\right) \right)$. Iterating this inequality, using the fact that for $j \le k, \ln(j) \le \ln(k)$ and using Lemma \[sumPi\], we obtain: $$\begin{aligned}
\displaystyle &\left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} \\
&\le \hat L \Big(1+\ln\left(k\right)\Big)h + \sqrt{\frac{2}{\pi}}dB \sum_{j=1}^{k-1} \frac{h}{\sqrt{kh - jh}}\left(\hat L \Big(1+\ln\left(j\right)\Big)h + \sqrt{\frac{2}{\pi}}dB \sum_{l=1}^{j-1} \frac{h}{\sqrt{jh - lh}} \left\|p(lh,.) - p^h(lh,.) \right\|_{L^1}\right) \\
&\le \hat L\left(1 + \sqrt{\frac 2 \pi}dB\int_h^{kh} \frac{ds}{\sqrt{kh - s}} \right) \Big(1+\ln\left(k\right)\Big)h + 2d^2B^2h \sum_{j=1}^{k-1}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1} \\
&\le \hat L\left(1 + 2\sqrt{\frac{2T}{\pi}}dB \right) \Big(1+\ln\left(k\right)\Big)h + 2d^2B^2h \sum_{j=1}^{k-1}\left\|p(jh,.) - p^h(jh,.) \right\|_{L^1}.\end{aligned}$$ Finally using Lemma \[Gronw\] and for $\hat C = \hat L \left(1+ 2\sqrt{\frac{2T}{\pi}}dB \right)\left(1 + 2d^2B^2T\exp\Big(2d^2B^2T \Big) \right)$, we conclude that: $$\begin{aligned}
\displaystyle \left\|p(kh,.) - p^h(kh,.) \right\|_{L^1} &\le \hat C \Big(1+\ln\left(k\right)\Big)h.\end{aligned}$$
Numerical Experiments {#NumExp}
=====================
In order to confirm our theoretical estimates for the convergence rate in total variation of $\mu^h$ to its limit $\mu$, we study SDEs with a piecewise constant drift coefficient and additive noise, as done by Göttlich, Lux and Neuenkirch in [@GotNeue]. We consider the special case of one-dimensional SDEs with one drift change at zero: $$\begin{aligned}
X_t = x + W_t + \int_0^t\bigg( \alpha \mathbf{1}_{(-\infty,0)}(X_s) + \beta \mathbf{1}_{[0,+\infty)}(X_s) \bigg)\,ds \label{generalSDE}\end{aligned}$$ where $X_0=x \in {{\mathbb R}}$ is the initial value and $\alpha, \beta \in {{\mathbb R}}$. The difference $(\beta - \alpha)$ represents the height of the jump at the discontinuity point zero. Here, the drift satisfies the reinforced hypothesis of Theorem \[cvgenceB\] where the derivative of the drift in the sense of distributions is equal to $(\beta-\alpha)\delta_0$.\
We analyze how the initial value $x$ affects the error and how the jump height influences the empirical rate of convergence. We also observe how the drift direction towards or away from the discontinuity point zero influences the error. When $\alpha > 0 > \beta$, we speak about inward pointing drift coefficient. Inversely, when $\alpha < 0 < \beta$, it is about outward pointing drift coefficient.
The specific case $\alpha = -\beta = \theta >0$
-----------------------------------------------
For $\theta >0$, we study SDEs with inward pointing drift coefficient of the form: $$\displaystyle X_t = x + W_t - \int_0^t \operatorname{sgn}(X_s)\,\theta \,ds.$$ This process is called a Brownian motion with two-valued, state-dependent drift. This example was also used by Kohatsu-Higa, Lejay and Yasuda in [@Koh] to estimate the weak convergence rate of the Euler-Maruyama scheme. According to [@KaratShre], the transition density function of the process $(X_t)_{t \ge 0}$ starting at $x \ge 0$ is the following: $$\begin{aligned}
p_t(x,z) = \left\{
\begin{aligned}
&\frac{1}{\sqrt{2\pi t}}\exp\left(-\frac{\left(x-z-\theta t\right)^2}{2t} \right) + \frac{\theta e^{-2\theta z}}{\sqrt{2\pi t}}\int_{x+z}^{+ \infty} \exp\left(-\frac{(y-\theta t)^2}{2t} \right)\,dy \quad \text{when } z > 0, \\
&\frac{e^{2\theta x}}{\sqrt{2\pi t}}\exp\left(-\frac{\left(x-z+\theta t\right)^2}{2t} \right) + \frac{\theta e^{2\theta z}}{\sqrt{2\pi t}}\int_{x-z}^{+ \infty} \exp\left(-\frac{(y-\theta t)^2}{2t} \right)\,dy \quad \text{when } z \le 0.\\
\end{aligned}
\right. \end{aligned}$$ For $x \le 0$, the transition density can be deduced from the symmetry of the Brownian motion that gives $p_t(x,z) = p_t(-x,-z)$.\
We seek to observe the dependence of the error in total variation at terminal time $T$: $\left\| \mu_{T} - \mu^h_{T}\right\|_{\rm TV}$ on the time step $h$ that we choose s.t. $\frac{T}{h}$ is an integer. To do so, we estimate $ \left\|p(T,.) - p^h(T,.) \right\|_{L^1}$ using a kernel density estimator for $p^h(T,.)$. We denote by $N$ the number of random variables $\left( X^{i,h}_T\right)_{1\le i \le N}$ that are i.i.d. with density $p^h(T,.)$. The kernel density estimator of this latter is defined by: $$p_{\epsilon, N}^h(T,x) = \frac{1}{\epsilon N} \sum\limits_{j=1}^N K\left( \frac{x-X^{j,h}_T}{\epsilon}\right)$$ where $K$ represents the kernel and $\epsilon >0$ is a smoothing parameter called the bandwidth. The kernel is a non-negative and integrable even function that ensures the required normalization of a density i.e. $\int_{-\infty}^{+\infty} K(x)\,dx = 1$. As for the smoothing parameter, its influence is critical since a very small $\epsilon$ makes the estimator show insignificant details and a very large $\epsilon$ causes oversmoothing and may mask some characteristics. So a compromise is needed. The optimal smoothing parameter can be chosen through a minimisation of the asymptotic mean integrated squared error. For an explicit known density, as it is the case here, we have that: $$\begin{aligned}
\label{epsilon}
\epsilon = c N^{-1/5} \quad \text{with} \quad c = \frac{R(K)^{1/5}}{m_2(K)^{2/5}R\left(\partial^2_{xx} p_t\right)^{1/5}}\end{aligned}$$ where for a given function $g$, $\displaystyle R(g) = \int_{{{\mathbb R}}}g(x)^2\,dx$ and $\displaystyle m_2(g) = \int_{{{\mathbb R}}}x^2 g(x)\,dx$. We will choose, in what follows, the Epanechnikov kernel defined by: $$K(x) = \frac{3}{4}\left(1 - x^2\right) \mathbf{1}_{\left\{|x| \le 1\right\}}$$ which is known to be theoretically optimal in a mean square error sense with $R(K) = 3/5$ and $m_2(K) = 1/5$. For $\left( X^{(i),h}_T\right)_{1\le i \le N}$ denoting the increasing reordering of $\left( X^{i,h}_T\right)_{1\le i \le N}$, we make the following trapezoidal approximation: $$\begin{aligned}
\displaystyle \left\|p^h(T,.) - p(T,.) \right\|_{L^1} &\simeq \sum_{i=1}^{N-1} \frac{1}{2} \left(X^{(i+1),h}_T - X^{(i),h}_T\right)\Bigg\{\left|p_{\epsilon, N}^h\left(T,X^{(i+1),h}_T\right) - p\left(T,X^{(i+1),h}_T \right) \right| \\
&\phantom{\sum_{i=1}^{N-1} \frac{1}{2} \left(X^{(i+1),h}_t - X^{(i),h}_t\right)\Bigg\{p_{\epsilon, N}^h\left(X^{(i),h}_t\right) - p\left(X\right)} + \left|p_{\epsilon, N}^h\left(T,X^{(i),h}_T\right) - p\left(T,X^{(i),h}_T \right) \right|\Bigg\}.\end{aligned}$$
We also define the precision of this estimation as half the width of the $95 \%$ confidence interval of the empirical error i.e. $\text{Precision} = 1.96 \times \sqrt{\text{Variance}/R}$ where $R$ denotes the number of Monte-Carlo runs and Variance denotes the empirical variance over these runs of the empirical error.
### Illustration of the theoretical order of convergence in total variation
To observe the convergence rate in total variation for the case $\theta = 1.0$ and $x=0.0$, we fix the time horizon $T=1$ and the number $N=500000$ of i.i.d. samples in the kernel density estimator large enough in order to observe the effect of the time-step $h$ on the error. The simulation is done with $R = 20$ Monte-Carlo runs. We obtain the following results for the estimation of the error and its associated precision:
[ |c|c|c|c|c|]{}\
Time-step $h$ & Estimation & Precision & Ratio of decrease & Theoretical Ratio\
$T/4$ & $0.2903$ & $ 5.48 \times 10^{-4}$ & $\times$ & $\times$\
$T/8$ & $0.1680$ & $ 6.96 \times 10^{-4}$ & $1.73$ & $1.63$\
$T/16$ & $0.0956$ & $ 4.88 \times 10^{-4}$ & $1.76$ & $1.69$\
$T/32$ & $0.0543$ & $ 5.26 \times 10^{-4}$ & $1.76$ & $1.73$\
$T/64$ & $0.0314$ & $ 6.53 \times 10^{-4}$ & $1.73$ & $1.76$\
$T/128$ & $0.0191$ & $ 3.55 \times 10^{-4}$ & $1.64$ & $1.79$\
$T/256$ & $0.0133$ & $ 2.71 \times 10^{-4}$ & $1.43$ & $1.80$\
$T/512$ & $0.0101$ & $ 3.10 \times 10^{-4}$ & $1.31$ & $1.82$\
0.4cm
- We observe that the ratio of successive estimations $\frac{\text{Estimation}(h)}{\text{Estimation}(h/2)}$ is roughly around $1.72$. But when $h$ becomes small, the ratio decreases towards $1$ (a constant error) because for so small discretizations steps, the effect of the kernel density estimation parameter $N$ cannot be neglected unless $N$ is extremely large.
- The last column refers to the theoretical ratios equal to $2\left(\frac{1 + \ln(T/h)}{1+\ln(2T/h)}\right)$ which is the expected behaviour of the error. On the range of values $\left\{\frac{T}{8},\frac{T}{16},\frac{T}{32},\frac{T}{64},\frac{T}{128} \right\}$, both the empirical and the theoretical ratios are equal to $1.72$ in average.
- Moreover, the order of convergence in total variation of the Euler scheme is here equal to $0.76$. This order is given by the slope of the regression line, which we obtain when plotting $\log\left\|p(T,.) - p^h(T,.)\right\|_{L^1}$ versus $\log(h)$.
### Dependence of the order of convergence on the initial value $x$ for fixed $\theta=1$
To underline the influence of the initial value of the SDE, we start by generating plots of the explicit transition density function for various initializations $x \in \{-1, 0, 1, 2.5, 5 \}$ and different time horizons $T \in \{1,3,6\}$. We choose a fixed $\theta=1$. We also plot the kernel transition density estimation for the different values of $x$ at $T=1$ for $N=100000$ and a time-step $h = 0.0001$.\
We first observe from Figures \[fig:fig1\] and \[fig:fig2\] that the kernel density estimator catches the discontinuity and reproduces well the expected distribution. We also see from Figures \[fig:fig2\], \[fig:fig3\] and \[fig:fig4\] that when the process starts from the discontinuity point $x=0.0$ or close to it $x \in \{-1,1\}$, it visits the discontinuity point several times. When we increase the time horizon $T$, the inward pointing drift allows the process to visit the discontinuity point zero when starting far from it.
[0.4]{} ![The transition density function for various initializations $x$[]{data-label="fig:densities"}](densities_kernel.png "fig:"){width="\linewidth"}
[0.4]{} ![The transition density function for various initializations $x$[]{data-label="fig:densities"}](densities.png "fig:"){width="\linewidth"}
[0.4]{} ![The transition density function for various initializations $x$[]{data-label="fig:densities"}](densities_3.png "fig:"){width="\linewidth"}
[0.4]{} ![The transition density function for various initializations $x$[]{data-label="fig:densities"}](densities_6.png "fig:"){width="\linewidth"}
We also generate an example of a solution sample path with time-step $h=0.006$ for various initializations $x$ to confirm that point.
{width="8cm"}
Now, we give the empirical convergence orders obtained for various initializations $x$. These orders are given by the slopes of the regression lines in a log-log scale. The parameters used here are $T=5$, $N=500000$ and step sizes $h \in \left\{\frac{T}{8},\frac{T}{16},\frac{T}{32},\frac{T}{64},\frac{T}{128},\frac{T}{256},\frac{T}{512}\right\}$.\
Initial value $x$ $-1.0$ $0.0$ $1.0$ $2.5$ $5.0$
----------------------------- -------- -------- -------- -------- --------
Empirical convergence order $0.77$ $0.77$ $0.77$ $0.77$ $0.70$
0.4cm
We can see from the above table that the empirical convergence orders are stable with respect to the initial value for this type of diffusion. A small difference is observed for the initial value $x=5.0$: then the process starts far from the discontinuity point zero and the time horizon is not long enough for the process to visit it with high probability. When the process does not reach the discontinuity, the Euler scheme is exact making the error smaller and the influence of the kernel estimation error stronger.
### Dependence of the order of convergence on the jump height
To underline the influence of the jump height equal to $2\theta$, we start by generating an example of a sample path for two values of $\theta \in \{1,10\}$ with fixed time-horizon $T=1$, time-step $h=0.003$ and initial value $x=0.0$.
{width="8cm"}
We observe that when $\theta$ is big, the process is more likely to visit the discontinuity multiple times than for a smaller value of $\theta$. We can explain this by introducing the process $Y_t = \frac{1}{\theta}X_t$ that starts from $Y_0 = \frac{x}{\theta}$ and has the following dynamics: $$Y_t = Y_0 + \frac{1}{\theta}W_t - \int_0^t \operatorname{sgn}\left(Y_t\right)\,dt.$$ The diffusion coefficient equal to $1/\theta$ becomes very small when $\theta$ becomes large so that the process has an almost deterministic behaviour and is sticked to the discontinuity point zero by the drift.\
Also, the explicit transition density tends to the Laplace density $\theta e^{-2\theta|x|}$ when $t \to +\infty$. We generate the plots of the kernel density estimate and the explicit transition density function for various $\theta = \{1,3,5,10,20\}$. We choose $T=1$, $x=0.0$, $N=100000$ and $h=0.0001$.
[0.4]{} ![The transition density function for various $\theta$[]{data-label="fig:theta_dens_b"}](densities_theta_estim.png "fig:"){width="\linewidth"}
[0.4]{} ![The transition density function for various $\theta$[]{data-label="fig:theta_dens_b"}](theta.png "fig:"){width="\linewidth"}
We can see that when $\theta$ is big, the density converges quickly towards the Laplace density. In Figure \[cettefigure\], we confirm this behaviour by giving the $L^1$-error between the explicit transition densities and the Laplace densities for each $\theta$.\
Now, we give the empirical convergence orders obtained for different values of $\theta$. The parameters used here are $T=1$, $x=0.0$ and $N=800000$. We choose ranges of step-sizes $h$ depending on $\theta$ since for small $\theta$, the discretization error are smaller and the kernel estimation error comparatively more influent. For large $\theta$, a large time-step implies a very large error because on each time-step, when starting close to the discontinuity point zero, the Euler scheme will move far away to the other side of this discontinuity, a behaviour forbidden for the limiting SDE by the large inward pointing drift.
Jump-height $\theta$ $1.0$ $3.0$ $5.0$ $10.0$ $20.0$
----------------------------- ---------------------------------------------------------------- --------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------
time-step range $h$ $\displaystyle \left\{\frac{1}{2^3},...,\frac{1}{2^8}\right\}$ $\displaystyle\left\{\frac{1}{2^4},...,\frac{1}{2^9}\right\}$ $\displaystyle\left\{\frac{1}{2^5},...,\frac{1}{2^{10}}\right\}$ $\displaystyle\left\{\frac{1}{2^7},...,\frac{1}{2^{12}}\right\}$ $\displaystyle\left\{\frac{1}{2^8},...,\frac{1}{2^{13}}\right\}$
Empirical convergence order $0.76$ $0.77$ $0.77$ $0.77$ $0.72$
0.4cm
We can see from the above table that the empirical convergence orders are relatively stable with respect to the jump-height for this type of diffusion. A small difference is observed for $\theta=20.0$ since the error is still large for the time-steps considered.
General case
------------
To our knowledge, no closed-form of a density of $X_t$ solving is available for general $\alpha, \beta \in {{\mathbb R}}$. The idea is still to estimate the $L^1$-norm of the difference of the densities at maturity $T$ but this time, instead of comparing $p^h(T,.)$ to $p(T,.)$, we will compare $p^{h}(T,.)$ to $p^{h/2}(T,.)$ and the expected behaviour is: $$\begin{aligned}
\displaystyle \left\|p^h(T,.) - p^{h/2}(T,.) \right\|_{L^1} \le \left\|p^h(T,.) - p(T,.) \right\|_{L^1} + \left\|p^{h/2}(T,.) - p(T,.) \right\|_{L^1} \le \left(\frac{3}{2} + \ln(2) \right) \tilde C \left(1 + \ln\left(\frac T h \right) \right)h.\end{aligned}$$ In order to estimate $p^h(T.)$, we use, once again, a kernel density estimator but this time, we choose the Gaussian kernel defined by: $$K(x) = \frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^2}{2} \right) \quad \text{ for } x \in {{\mathbb R}}.$$ We make this choice since no explicit density is available to estimate the bandwith and for Gaussian kernels we can obtain use the so-called Silverman’s rule of thumb [@Silv]: $$\epsilon = c N^{-1/5} \quad \text{ with } \quad c= 0.9 \times \min\left(\hat \sigma, \frac{IQR}{1.34}\right)$$ where the standard deviation $\hat \sigma$ and the interquantile range $IQR$ are easily computed from the sample of size $N$. When the density to estimate is a bimodal mixture, we apply Silverman’s rule of thumb on each mode.\
For $\left(X^{i,{h/2}}_T\right)_{1 \le i \le N}$ i.i.d. variables with density $p^{h/2}(T,.)$, the kernel density estimator of this latter is then defined by: $$p_{\epsilon, N}^{h/2}(T,x) = \frac{1}{\epsilon N} \sum\limits_{j=1}^N K\left( \frac{x-X^{j,h/2}_T}{\epsilon}\right)$$ and we make the following trapezoidal approximation: $$\begin{aligned}
\displaystyle \left\|p^h(T,.) - p^{h/2}(T,.) \right\|_{L^1} &\simeq \sum_{i=1}^{N-1} \frac{1}{2} \left(X^{(i+1),h}_T - X^{(i),h}_T\right)\Bigg\{\left|p_{\epsilon, N}^h\left(T,X^{(i+1),h}_T\right) - p_{\epsilon, N}^{h/2}\left(T,X^{(i+1),h}_T \right) \right| \\
&\phantom{\sum_{i=1}^{N-1} \frac{1}{2} \left(X^{(i+1),h}_t - X^{(i),h}_t\right)\Bigg\{p_{\epsilon, N}^h\left(X^{(i),h}_t\right) - } + \left|p_{\epsilon, N}^h\left(T,X^{(i),h}_T\right) - p_{\epsilon, N}^{h/2}\left(T,X^{(i),h}_T \right) \right|\Bigg\}.\end{aligned}$$
In what follows, we will study the case of outward pointing diffusions i.e. $\alpha < 0 < \beta$ and observe how the initial value and the jump-height influences the error. Beforehand, we observe the convergence rate in total variation when varying the time-step $h$.
### Illustration of the theoretical order of convergence in total variation
To observe the convergence rate in total variation for the case $\alpha = - 3.0$, $\beta = 4.0$ and $x=0.0$, we fix the time horizon $T=1$ and the number $N=250000$ of i.i.d. samples in the kernel density estimator large enough in order to observe the effect of the time-step $h$ on the error. The simulation is done with $R = 20$ Monte-Carlo runs. We obtain the following results for the estimation of the error and the associated precision:
[ |c|c|c|c|c|]{}\
Time-step $h$ & Estimation & Precision & Ratio of decrease & Theoretical Ratio\
$T/32$ & $0.1105$ & $3.58 \times 10^{-4}$ & $\times$ & $\times$\
$T/64$ & $0.0763$ & $3.17 \times 10^{-4}$ & $1.45$ & $1.76$\
$T/128$ & $0.0478$ & $2.31 \times 10^{-4}$ & $1.60$ & $1.79$\
$T/256$ & $0.0279$ & $1.91 \times 10^{-4}$ & $1.71$ & $1.80$\
$T/512$ & $0.0156$ & $1.92 \times 10^{-4}$ & $1.79$ & $1.82$\
$T/1024$ & $0.0081$ & $1.15 \times 10^{-4}$ & $1.93$ & $1.84$\
$T/2048$ & $0.0043$ & $1.21 \times 10^{-4}$ & $1.87$ & $1.85$\
0.4cm
- We observe that the ratio of successive estimations $\frac{\text{Estimation}(h)}{\text{Estimation}(h/2)}$ is roughly around $1.72$.
- The last column refers to the theoretical ratios equal to $2\left(\frac{1 + \ln(T/h)}{1+\ln(2T/h)}\right)$ which is the expected behaviour of the error. On the range of values $\left\{\frac{T}{256},\frac{T}{512},\frac{T}{1024},\frac{T}{2048}\right\}$, both the empirical and the theoretical ratios are equal to $1.82$ in average.
- Moreover, the order of convergence in total variation of the Euler scheme is here equal to $0.79$. This order is given, once again, by the slope of the regression line in a log-log scale.
### Dependence of the order of convergence on the initial value $x$ for fixed $\alpha = -3.0$ and $\beta = 4.0$
To underline the influence of the initial value of the SDE, we start by generating plots of the estimated transition density function for various initializations $x \in \{-1.4,-0.4,-0.2, -0.15, 0.0, 0.6 \}$. We choose $\alpha = -3.0$, $\beta=4.0$, $T=1$, $N=100000$ and $h=0.0001$.
[0.45]{} ![Transition density functions and examples of a solution sample paths for various initializations $x$[]{data-label="iciFig"}](densities_alpha_beta.png "fig:"){width="\linewidth"}
[0.45]{} ![Transition density functions and examples of a solution sample paths for various initializations $x$[]{data-label="iciFig"}](sample_path_alpha.png "fig:"){width="\linewidth"}
We see from Figure \[iciFig\] that when the process starts from the discontinuity point $x=0.0$ or close to it $x \in \{-0.2,-0.15\}$, we have a bimodal mixture. When starting far from the discontinuity point zero, we are less likely to visit it and the distribution is Gaussian-like. We also confirm this point by an example of a solution sample path with time-step $h=0.001$ for various initializations $x$.\
Now, we give the empirical convergence orders obtained for various initializations $x$. These orders are given, once again, by the slopes of the regression lines in log-log scales. The parameters used here are $T=1$, $N=500000$ and step sizes $h \in \left\{\frac{T}{16},\frac{T}{32},\frac{T}{64},\frac{T}{128},\frac{T}{256},\frac{T}{512},\frac{T}{1024},\frac{T}{2048}\right\}$.\
Initial value $x$ $-1.4$ $-0.4$ $-0.2$ $0.0$ $0.6$
----------------------------- -------- -------- -------- -------- --------
Empirical convergence order $0.28$ $0.68$ $0.82$ $0.71$ $0.51$
0.4cm
We can see from the above table that the empirical convergence orders seem to depend on the initial value and the spectrum of orders obtained for different initial values is very broad with values between $0.28$ and $0.82$. When starting further and further from the discontinuity point zero, we first obtain a better order of convergence but when $|x|$ becomes large it deteriorates since the kernel estimation error becomes more influent.
### Dependence of the order of convergence on the jump height
To underline the influence of the jump height equal to $(\beta-\alpha)$, we start by generating a sample path for two values of $(\alpha,\beta) \in \{(-4.0,3.0),(-0.6,1.0)\}$ with fixed time-horizon $T=1$ and initial value $x=0.0$.
{width="8cm"}
We observe that the solution drifts away from the discontinuity point zero and therefore, there are not many chances for a drift correction to take place.\
Now, we give the empirical convergence orders obtained for various $(\alpha,\beta)$. These orders are given, once again, by the slopes of the regression lines in log-log scales. The parameters used here are $T=1$, $N=500000$, $x=0.0$ and step-sizes $h \in \left\{\frac{T}{8},\frac{T}{16},\frac{T}{32},\frac{T}{64},\frac{T}{128},\frac{T}{256},\frac{T}{512}\right\}$.\
$(\alpha,\beta)$ $(-6.0,8.0)$ $(-3.0,4.0)$ $(-1.5,2.0)$ $(-0.75,1.0)$ $(-0.375,0.5)$
----------------------------- -------------- -------------- -------------- --------------- ----------------
Empirical convergence order $0.51$ $0.66$ $0.90$ $1.02$ $0.78$
0.4cm
We can see from the table above that the empirical convergence orders are not stable with respect to the jump-height for outward pointing drift diffusions. Enlarging the jump height makes the solution to drift away from the discontinuity point zero and increases the error.
Conclusion
----------
We were able, through our numerical experiments, to confirm our theoretical estimates for the convergence rate in total variation of $\mu^h_T$ to its limit $\mu_T$ since the order $1$ up to a logarithmic factor was recovered.
Moreover, the study conducted when varying the type of drift (inward pointing or outward pointing) has highlighted several features. Our results and interpretations coincide with those obtained by Göttlich, Lux and Neuenkirch in [@GotNeue] when they estimate the root mean-squared strong error. As them, we show that for inward pointing drift coefficients, the convergence order is independent of the initial value and the jump-height. This is not the case for outward pointing drift coefficients: the numerical orders are less stable in the initial value and the jump-height. A possible explanation is that the inward pointing drift coefficient engender many drift changes, while only few drift changes occur in the case of an outward pointing drift coefficient. The solution, in the latter case, can quickly drift away from the discontinuity and because of a small probability of a drift change, the empirical convergence rate might be subject to rare event effects and the linear regression estimates become questionable.
Appendix
========
\[lema1\] Let $\sigma\in{{\mathbb R}}^{d\times d}$ be a non-degenerate matrix and $\tilde g:{{\mathbb R}}^d\to{{\mathbb R}}^d$ be a measurable and locally integrable function, the spatial divergence in the sense of distributions of which is a Radon measure denoted by $\nabla\cdot\tilde g(dy)$. Then, the function $g:{{\mathbb R}}^d\to{{\mathbb R}}^d$ defined by $g(x)=\sigma^{-1}\tilde g(\sigma x)$ is locally integrable and its spatial divergence $\nabla \cdot g(dx)$ in the sense of distributions is the image of $\left|{\rm det}(\sigma^{-1})\right|\nabla \cdot \tilde g$ by $y\mapsto\sigma^{-1} y$. In particular, the total mass of $\nabla \cdot g(dx)$ is equal to $\left|{\rm det}(\sigma^{-1})\right|$ times the total mass of $\nabla \cdot \tilde g(dy)$ and when $\nabla \cdot \tilde g(dy)$ admits the density $f(y)$ with respect to the Lebesgue measure, then $\nabla\cdot g(dx)$ admits the density $f(\sigma x)$.
The local integrability of $g$ is easily obtained by the change of variables $y=\sigma x$. For any ${\cal C}^\infty$ function $\varphi:{{\mathbb R}}^d\to{{\mathbb R}}$ with compact support, we obtain using the same change of variables that $$\begin{aligned}
\int_{{{\mathbb R}}^d}g(x).\nabla_x\varphi(x)dx&=\int_{{{\mathbb R}}^d}\sigma^{-1}\tilde g(\sigma x).\nabla_x\varphi(x)dx=
|{\rm det}(\sigma^{-1})|\int_{{{\mathbb R}}^d}\tilde g(y).(\sigma^{-1})^*\nabla_x\varphi(\sigma^{-1}y)dy\\&=|{\rm det}(\sigma^{-1})|\int_{{{\mathbb R}}^d}\tilde g(y).\nabla_y[\varphi(\sigma^{-1}y)]dy=-|{\rm det}(\sigma^{-1})|\int_{{{\mathbb R}}^d}\varphi(\sigma^{-1}y)\nabla\cdot\tilde g(dy),
\end{aligned}$$ which implies the first statement. The one concerning the total masses immediately follows and the one concerning the densities is obtained by the inverse change of variables $x=\sigma^{-1}y$.
For $t>0$, let $G_t$ denote the heat kernel in ${{\mathbb R}}^{d}$: $\displaystyle G_t(x) = \frac{1}{\sqrt{(2\pi t)^d }}\exp\left(-\frac{| x |^2}{2t}\right)$. The following lemma provides a set of estimates that are very useful:
\[EstimHeatEq\] The function $G_t(x)$ solves the heat equation: $$\begin{aligned}
\label{heateq}
\partial_tG_t(x) - \frac{1}{2} \Delta G_t(x) = 0, \quad (t,x) \in [0,+\infty)\times {{\mathbb R}}^d.\end{aligned}$$ We have estimates of the $L^1$-norm of the first order time derivative and the spatial derivatives of $G$ up to the third order: $$\begin{aligned}
&\left\Vert \partial_t G_t \right\Vert_{L^1} \le \frac d t ,\label{timeDerivG} \\
&\left\Vert \partial_{x_i} G_t \right\Vert_{L^1} = \sqrt{\frac{2}{\pi t}}, \label{FirstDerivG} \\
&\left\Vert \frac{\partial^2 G_{t}}{\partial x_i^2} \right\Vert_{L^1} \le \frac{2}{t}, \label{SecondDerivG1}\\
&\left\Vert \frac{\partial^2 G_{t}}{\partial x_i x_j} \right\Vert_{L^1} = \frac{2}{\pi t}\quad \text{when } j \neq i, \label{SecondDerivG2} \\
& \left\Vert \frac{\partial^3 G_{t}}{\partial x_j x_i^2} \right\Vert_{L^1} \le \left\{
\begin{aligned}
&\frac{2}{t^{3/2}}\sqrt{\frac{2}{\pi}} \quad \text{when } j \neq i, \\
&\frac{5}{t^{3/2}}\sqrt{\frac{2}{\pi}} \quad \text{when } j = i.\\
\end{aligned}
\right. \label{ThirdDerivG}\end{aligned}$$
Let us compute the estimate . To do so, we use Fubini’s theorem and obtain: $$\begin{aligned}
\left\Vert \partial_{x_i} G_t \right\Vert_{L^1} &= \int_{{{\mathbb R}}^d} \frac{\left|x_i \right|}{t}\frac{1}{\left(2\pi t\right)^{d/2}}\exp\left(-\sum\limits_{j=1}^{d}\frac{x_j^2}{2t} \right)\,dx_1\dots dx_d \\
&= \left(\int_{{{\mathbb R}}} \frac{\left|y \right|}{t}\frac{1}{\sqrt{2\pi t}}\exp\left(-\frac{y^2}{2t}\right)\,dy\right)\times\left(\int_{{{\mathbb R}}} \frac{1}{\sqrt{2\pi t}}\exp\left(-\frac{y^2}{2t}\right)\,dy\right)^{d-1} \\
&= \sqrt{\frac{2}{\pi t}}.
\end{aligned}$$ We can express the second and third spatial derivatives of $G$ as: $$\begin{aligned}
&\frac{\partial^2}{\partial x_i x_j} G_t(x) = \left\{
\begin{aligned}
&\frac{x_i x_j}{t^2}G_t(x) \quad \text{when } j \neq i, \\
&\left(-1 + \frac{x_i^2}{t}\right)\frac{G_t(x)}{t} \quad \text{when } j = i.\\
\end{aligned}
\right. \; \text{and } \; \frac{\partial^3}{\partial x_j \partial x_i^2} G_t(x) = \left\{
\begin{aligned}
&\left(1 - \frac{x_i^2}{t}\right)\frac{x_j}{t^2}G_t(x) \quad \text{when } j \neq i, \\
&\left(3 - \frac{x_i^2}{t}\right)\frac{x_i}{t^2}G_t(x) \quad \text{when } j = i.\\
\end{aligned}
\right.
\end{aligned}$$ Using Fubini’s theorem as for the estimate , $\displaystyle \int_{{{\mathbb R}}} \frac{y^2}{t^2}\frac{e^{-\frac{y^2}{2t}}}{\sqrt{2 \pi t}}\,dy = \frac{1}{t}$ and $\displaystyle \int_{{{\mathbb R}}} \frac{|y|^3}{t^3}\frac{e^{-\frac{y^2}{2t}}}{\sqrt{2 \pi t}}\,dy = \frac{2}{t^{3/2}}\sqrt{\frac{2}{\pi}}$, we obtain the estimates , and . As for the estimate , we deduce it from the heat equation and the estimate .
\[sumPi\] We have: $$\displaystyle \forall n \ge 2, \quad \sum\limits_{k=1}^{n-1} \frac{1}{\sqrt{k}\; \sqrt{n-k}} \le \pi - \frac{2}{n}.$$
We define the function $f(x) = \frac{1}{\sqrt{x}\sqrt{1-x}}$ on $(0,1)$. We easily check that $\forall x \in (0,1), f(x) \ge f\left(1/2\right) = 2$ and that $\displaystyle \int_0^1f(x)\,dx = \pi$. Using the monotonicity of $f$ on $(0,1/2]$ and $[1/2,1)$, we obtain: $$\displaystyle \left\{ \begin{aligned}
&\int_{\frac{k-1}{n} }^{ \frac{k}{n}} f(x)\,dx \ge \frac{1}{n} f\left(\frac{k}{n}\right) \quad \text{when } 1 \le k \le \frac{n}{2}, \\
&\int_{\frac{k}{n} }^{ \frac{k+1}{n}} f(x)\,dx \ge \frac{1}{n} f\left(\frac{k}{n}\right) \quad \text{when } \frac{n}{2} \le k \le (n-1).\\
\end{aligned}
\right.$$ Therefore,\
$\bullet$ When $n$ is even: $$\begin{aligned}
\displaystyle \frac{1}{n}\sum_{k=1}^{\frac{n}{2}}f\left(\frac{k}{n}\right) + \frac{1}{n}\sum_{k=\frac{n}{2}+1}^{n-1}f\left(\frac{k}{n}\right) \le \sum_{k=1}^{\frac{n}{2}}\int_{\frac{k-1}{n} }^{ \frac{k}{n}} f(x)\,dx + \sum_{k=\frac{n}{2}+1}^{n-1}\int_{\frac{k}{n} }^{ \frac{k+1}{n}} f(x)\,dx &= \int_0^{\frac 1 2}f(x)\,dx + \int_{\frac 1 2 + \frac 1 n}^1f(x)\,dx \\
&= \pi - \int_{\frac 1 2}^{\frac 1 2 + \frac 1 n}f(x)\,dx \le \pi - \frac 2 n .
\end{aligned}$$ $\bullet$ When $n$ is odd: $$\begin{aligned}
\displaystyle \frac{1}{n}\sum_{k=1}^{\frac{n-1}{2}}f\left(\frac{k}{n}\right) + \frac{1}{n}\sum_{k=\frac{n+1}{2}}^{n-1}f\left(\frac{k}{n}\right) \le \sum_{k=1}^{\frac{n-1}{2}}\int_{\frac{k-1}{n} }^{ \frac{k}{n}} f(x)\,dx + \sum_{k=\frac{n+1}{2}}^{n-1}\int_{\frac{k}{n} }^{ \frac{k+1}{n}} f(x)\,dx &= \int_0^{\frac{1}{2} - \frac{1}{2n}}f(x)\,dx + \int_{\frac{1}{2} + \frac{1}{2n}}^1f(x)\,dx \\
&= \pi - \int_{\frac{1}{2} - \frac{1}{2n}}^{\frac{1}{2} + \frac{1}{2n}}f(x)\,dx \le \pi - \frac 2 n .
\end{aligned}$$ We can conclude.
\[integ\] For $0 < a \le x \le T$, $$\begin{aligned}
\displaystyle \int_a^x \frac{dy}{y\sqrt{x-y}} &= \frac{1}{\sqrt x} \ln\left(\frac{\left(\sqrt x + \sqrt{x-a} \right)^2}{a} \right)\le \frac{1}{\sqrt x} \ln\left(\frac{4x}{a} \right).
\end{aligned}$$
Using the change of variable $u=\sqrt{x-y}$ then a partial fraction decomposition, we obtain:\
$ \displaystyle \int_a^x \frac{dy}{y\sqrt{x-y}} = 2\int_0^{\sqrt{x-a}} \frac{du}{u^2 - x} = \left[\frac{1}{\sqrt x} \ln\left(\frac{\sqrt x + u}{\sqrt x - u}\right)\right]^{\sqrt{x-a}}_0$.
\[intg\] For $0 < r \le s \le T$, $$\begin{aligned}
\displaystyle \frac{1}{2}\int_0^r \frac{\ln(s-u) - \ln(r-u)}{\sqrt u}\,du = \frac{s-r}{\sqrt s + \sqrt r}\ln\left(\frac{\left(\sqrt s + \sqrt r \right)^2}{s-r} \right) + 2 \sqrt r \ln\left(1 + \frac{\sqrt s - \sqrt r}{2 \sqrt r} \right).
\end{aligned}$$
We start by applying the change the variable $\theta = \sqrt u$ and obtain $$\begin{aligned}
\displaystyle \frac{1}{2}\int_0^r \frac{\ln(s-u) - \ln(r-u)}{\sqrt u}\,du &= \int_0^{\sqrt r} \bigg(\ln\left(s-\theta^2 \right) - \ln\left(r -\theta^2\right)\bigg)\,d\theta \\
&= \int_0^{\sqrt r} \bigg(\ln\left(\sqrt s-\theta\right)+\ln\left(\sqrt s+\theta\right)-\ln\left(\sqrt r-\theta\right)-\ln\left(\sqrt r+\theta\right) \bigg)\,d\theta.
\end{aligned}$$ A simple integration of $\ln(x)$ permits us to conclude.
The next lemma is a discrete version of Gronwall’s lemma and was proved by Holte [@Holt].
\[Gronw\] If $\left(y_n \right)_{n \in {{\mathbb N}}}$, $\left(f_n \right)_{n \in {{\mathbb N}}}$ and $\left(g_n \right)_{n \in {{\mathbb N}}}$ are non-negative sequences and $$y_n \le f_n + \sum_{i= 0}^{n-1}g_i y_i \quad \text{ for } n \in {{\mathbb N}}$$ then $$y_n \le f_n + \sum_{i= 0}^{n-1}f_i g_i \exp \left(\sum_{j=i+1}^{n-1}g_j \right) \quad \text{ for } n \in {{\mathbb N}}.$$
[plain]{} L. Ambrosio, N. Fusco and D. Pallara, *Functions of Bounded Variation and Free Discontinuity Problems*. Oxford Mathematical Monographs, pp.119–120 ,2000.
V. Bally and C. Rey, *Approximation of Markov semigroups in total variation distance*. Electronic Journal of Probability, Vol.21(12), 2016.
J. Bao, X. Huang and C. Yuan, *Convergence Rate of Euler-Maruyama Scheme for SDEs with Hölder-Dini Continuous Drifts*. Journal of Theoretical Probability, Vol.32, pp.848–871, 2019.
H. Brezis, *Analyse Fonctionelle: Théorie et applications*. Collection Mathématiques appliquées pour la maîtrise - Masson, $2^{e}$ tirage, 1987.
K. Dareiotis and M. Gerencsér, *On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift*. ArXiv preprint arXiv:1812.04583, 2020.
T. Daun, *On the randomized solution of initial value problems*. Journal of Complexity, Vol.27, pp.300–311, 2011.
P. Etore and M. Martinez, *Exact simulation for solutions of one-dimensional Stochastic Differen- tial Equations with discontinuous drift*. ESAIM: Probability and Statistics, EDP Sciences, Vol.18, pp.686–702, 2014.
N. Frikha, *On the weak approximation of a skew diffusion by an Euler-type scheme*. Bernoulli, Vol.24(3), pp.1653–1691, 2018.
S. Göttlich, K. Lux and A. Neuenkirch, *The Euler scheme for stochastic differential equations with discontinuous drift coefficient: A numerical study of the convergence rate*. Advances in Difference Equation, Vol.429, 2019.
I. Gyöngy and N. Krylov, *Existence of strong solutions for Ito’s stochastic equations via approximations*. Probability Theory and Related Fields, Vol.105, pp.143–158, 1996.
I. Gyöngy, *A note on Euler’s approximations*. Potential Analysis, Vol.8, pp.205–216 , 1998.
I. Gyöngy and M. Rasonyi, *A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients*. Stochastic Processes and their Applications, Vol.121, pp.2189–2200, 2011.
N. Halidias and P.E. Kloeden, *A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient*. BIT Numerical Mathematics, Vol.48, pp.51–59 ,2008.
S. Heinrich and B. Milla, *The randomized complexity of initial value problems*. Journal of Complexity, Vol.24, pp.77–88, 2008.
J. M. Holte, *Discrete Gronwall Lemma And Applications*. Mathematical Association of America North Central Section Meeting at UND, 24 October 2009.
I. Karatzas and S. Shreve, *Brownian Motion and Stochastic Calculus*. Graduate Texts in Mathematics, Vol.113(2), p.441, 1998.
A. Kohatsu-Higa, A. Lejay and K. Yasuda, *On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient*. Mathematical Economics, Kyoto, Japan, pp.94–106, 2012.
A. Kohatsu-Higa, A. Lejay and K. Yasuda, *Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with non-regular drift*. Journal of Computational and Applied Mathematics, Vol.326, pp.138–158, 2017.
V. Konakov and S. Menozzi, *Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients*. Electronic Journal of Probability, Vol.22(46), pp.1–47, 2017.
R. Kruse and Y. Wu, *A randomized milstein method for stochastic differential equations with non-differentiable drift coefficients*. Discrete and Continuous, Vol.24(8), pp.3475–3502, 2019.
G. Leobacher and M. Szölgyenyi, *A numerical method for SDEs with discontinuous drift*. BIT Numerical Mathematics, Vol.56, pp.151–162, 2016.
G. Leobacher and M. Szölgyenyi, *A strong order $1/2$ method for multidimensional SDEs with discontinuous drift*. Annals of Applied Probability, Vol.27(4), pp.2382–2418, 2017.
G. Leobacher and M. Szölgyenyi, *Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient*. Numerische Mathematik, Vol.138(1), pp.219–239, 2017.
R. Mikulevicius and E. Platen, *Rate of convergence of the Euler approximation for diffusion processes*. Mathematische Nachrichten, Vol.151, pp.233–239, 1991.
R. Mikulevicius, *On the rate of convergence of simple and jump-adapted weak Euler schemez for Lévy driven SDEs*. Stochastic Processes and their Applications, Vol.122(7), pp.2730–2757, 2012.
R. Mikulevicius and C. Zhang, *Weak Euler approximation for Itô’s diffusion and jump processes*. Stochastic Analysis and Applications, Vol.33(3), pp.549–571, 2015.
T. Müller-Gronbach and L.Yaroslavtseva, *On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient*. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, Vol.56(2), pp.1162–1178, 2020.
T. Müller-Gronbach and L.Yaroslavtseva, *A strong order $3/4$ method for SDEs with discontinuous drift coefficient*. ArXiv preprint arXiv:1904.09178, 2019.
A. Neuenkirch, M. Szölgyenyi and L. Szpruch, *An Adaptive Euler-Maruyama Scheme for Stochastic Differential Equations with Discontinuous Drift and its Convergence Analysis*. SIAM Journal on Numerical Analysis, Vol.57(1), pp.378–403, 2019.
A. Neuenkirch and M. Szölgyenyi, *The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem*. ArXiv preprint arXiv:1904.07784, 2020.
H. Ngo and D. Taguchi, *Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients*. Mathematics of Computation, Vol.85, pp.1793–1819, 2016.
H. Ngo and D. Taguchi, *On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients*. IMA Journal of Numerical Analysis, Vol.37(4), pp.1864–1883, 2017.
B. W. Silverman, *Density estimation for statistics and data analysis*. London: Chapman & Hall/CRC. p.45, 1998.
G. Stengle, *Numerical Methods for Systems with Measurable Coefficients*. Appl. Math. Lett., Vol.3(4), pp.25–29, 1990.
G. Stengle, *Error analysis of a randomized numerical method*. Numerische Mathematik, Vol.70, pp.119–128, 1995.
Y. Suo, C. Yuan and S. Zhang, *Weak convergence of Euler scheme for SDEs with singular drift*. ArXiv preprint arXiv:2005.04631, 2020.
L. Yan, *The Euler scheme with irregular coefficients*. The Annals of Probability, Vol.30(3), pp.1172–1194, 2002.
A. J. Veretennikov, *On strong solutions and explicit formulas for solutions of stochastic integral equations*. Mathematics of the USSR-Sbornik, Vol.39(3), pp.387–403, 1981.
[^1]: Université Paris-Est, Cermics (ENPC), INRIA, F-77455 Marne-la-Vallée, France. E-mails : benjamin.jourdain@enpc.fr, oumaima.bencheikh@enpc.fr
|
---
title: Hadron attenuation at HERMES and Jefferson Lab
---
Introduction {#sec:intro}
============
In deep inelastic scattering experiments the reaction products hadronize long before they reach the detector. Thus by using elementary nucleon targets one cannot obtain information on the space-time picture of hadronization. A simple estimate of the hadron formation proper time via the hadronic radius $r_h$ yields hadron formation lengths of the order $\gamma\cdot r_h$ in the laboratory frame. At high energies the Lorentz factor $\gamma$ leads to formation lengths that may easily exceed typical nuclear dimensions. By using nuclear targets one, therefore, has the unique possibility to investigate the final-state interactions (FSI) of the prehadronic system and to study the dynamics of the hadronization process.
In the recent past the HERMES collaboration has carried out an extensive study of hadron production in deep inelastic lepton-nucleus scattering using 27.6 GeV and 12 GeV positron beams [@HERMESDIS; @HERMESDIS_new]. The observed attenuation of hadrons –compared to a deuterium target– has basically led to two different interpretations: The authors of Refs. [@ArleoWang] assume that hadronization occurs far outside the nucleus and that the attenuation is caused by a partonic energy loss prior to hadronization. On the other hand color neutral prehadrons might form rather early after the initial deep inelastic scattering event and undergo FSI on their way out of the nucleus [@Falteralt; @Falterneu; @Kopeliovich; @Accardi]. The latter effect should be even more pronounced in the kinematical regime of the ongoing Jefferson Lab experiment which uses a lower energy (5.4 GeV) electron beam [@Brooks].
In Ref. [@Falterneu] we have given a thorough theoretical investigation of hadron attenuation in lepton-nucleus scattering at HERMES and EMC energies in the framework of a semi-classical Boltzmann-Uehling-Uhlenbeck (BUU) transport model. In this work we apply our model (with the same parameter set) to lower energies, i.e. HERMES at 12 GeV beam energy and the kinematical regime of the Jefferson Lab experiment.
Model {#sec:model}
=====
In Refs. [@Falteralt; @Falterneu; @EffeFalter] we have developed a method to combine a quantum mechanical coherent treatment of nuclear shadowing –as observed in high energy photonuclear reactions– with an incoherent coupled-channel description of the (pre)hadronic FSI using a BUU transport model. This is achieved by splitting the electron-nucleus reaction into two parts. In the first step the high-energy virtual photon interacts with a bound nucleon inside the nucleus and produces a final state which is determined by using the Lund Monte Carlo generators [<span style="font-variant:small-caps;">Pythia</span>]{} and [<span style="font-variant:small-caps;">Fritiof</span>]{}. In addition we account for nuclear effects such as binding energies, Fermi motion, Pauli-blocking of final state nucleons and nuclear shadowing. In the second step the final state is propagated through the nucleus within the coupled-channel transport model.
The strength of the shadowing effect strongly depends on the coherence lengths of the photon’s hadronic fluctuations. The coherence length can be understood as the distance that the virtual photon travels as a hadronic fluctuation which has the quantum numbers of the photon, e.g. a vector meson. If the coherence length exceeds the mean-free path of the hadronic fluctuation inside the nucleus, the photon-nucleus interaction will get shadowed just like an ordinary hadron-induced nuclear reaction. In Fig. \[fig:profile\] we show the probability distribution for nucleons inside a $^{84}$Kr nucleus to participate in the primary photon-nucleon interaction. The kinematics of the virtual photon with momentum along the $z$-axis corresponds to that of a typical HERMES event. For such a photon the coherence length of the $\rho^0$-meson fluctuation is of the order of the nuclear radius and the nucleons on the front side of the $^{84}$Kr nucleus shadow the downstream nucleons.
![Profile function for shadowing. Shown is the probability distribution for the interaction of an incoming photon (from left) with given virtuality $Q^2$ and energy $\nu$ with nucleons in a $^{84}$Kr nucleus.[]{data-label="fig:profile"}](fig1.eps)
The photon-nucleon interaction leads to the excitation of one or more strings which fragment into color-neutral prehadrons due to the creation of quark-antiquark pairs from the vacuum. As discussed in Ref. [@Kopeliovich] the production time of these prehadrons is very short. For simplicity we set the production time to zero in our numerical realization. These prehadrons are then propagated using our coupled-channel transport theory.
After a formation time, which we assume to be a constant $\tau_f$ in the restframe of the hadron, the hadronic wave function has built up and the reaction products propagate and interact like usual hadrons. The prehadronic cross sections $\sigma^*$ during the formation time are determined by a simple constituent quark model $$\begin{aligned}
\label{eq:prehadrons} \sigma^*_\mathrm{prebaryon}&=
&\frac{n_\mathrm{org}}{3}\sigma_\mathrm{baryon} , \nonumber\\
\sigma^*_\mathrm{premeson}&=&\frac{n_\mathrm{org}}{2}\sigma_\mathrm{meson}
,\end{aligned}$$ where $n_\mathrm{org}$ denotes the number of (anti-)quarks in the prehadron stemming from the beam or target. As a consequence the prehadrons that solely contain (anti-)quarks produced from the vacuum in the string fragmentation do not interact during $\tau_f$. Using this recipe the total effective cross section of the final state rises like in the approach of Ref. [@Ciofi] each time when a new hadron has formed.
The FSI of the reaction products are described within our BUU transport model. The latter is based on a set of generalized transport equations for each particle species $i$, $$\label{eq:BUU}
\left(\frac{\partial}{\partial t}+\vec \nabla_{\vec p} H\vec \nabla_{\vec r}
-\vec \nabla_{\vec r} H\vec \nabla_{\vec p}\right) F_i(\vec r,\vec p,\mu;t)=
I_\mathrm{coll}(\{F_j\})$$ where $H$ is a relativistic Hamilton function which contains a mean-field potential in case of baryons. The transport equations (\[eq:BUU\]) describe the time-evolution of the spectral phase-space densities $F_i$ that are coupled via the collision integral $I_\mathrm{coll}$. The latter accounts for changes in the spectral phase-space density due to particle creation and annihilation in binary collisions. This means that the final hadron in an electron-nucleus reaction does not necessarily need to be produced in the primary virtual photon-nucleon interaction but can be created later on via side-feeding in the FSI. We note that this (probabilistic) coupled-channel treatment of the FSI goes far beyond the usual single-channel Glauber approach.
![Multiplicity ratios of $\pi^{\pm,0}$, $K^\pm$, $p$ and $\bar{p}$ for a $^{84}$Kr nucleus (when using a 27.6 GeV positron beam at HERMES) as a function of the hadron energy fraction $z_h=E_h/\nu$ and the photon energy $\nu$. The solid line represents the result of a simulation, where we use the constituent quark concept (\[eq:prehadrons\]) for the prehadronic cross sections and a formation time $\tau_f=0.5$ fm/$c$. The dotted line in the proton spectrum indicates the result of a simulation where all $\gamma^*N$ events are created by PYTHIA. The data are taken from Ref. [@HERMESDIS_new].[]{data-label="fig:Krid"}](fig2.eps)
Results {#sec:results}
=======
Due to our coupled channel-treatment of the FSI the (pre)hadrons might not only be absorbed in the nuclear medium but can produce new particles in an inelastic interaction, thereby shifting strength from the high to the low energy part of the hadron spectrum. In addition our event-by-event simulation allows us to account for all kinematic cuts and the acceptance of the detector. In Ref. [@Falterneu] we have demonstrated that our calculations are in excellent agreement with the experimental HERMES data [@HERMESDIS_new] taken on various nuclear targets at a beam energy $E_\mathrm{beam}=27.6$ GeV if one assumes a formation time $\tau_f=0.5$ fm/c for all hadron species. As an example we show in Fig. \[fig:Krid\] the multiplicity ratios $$\label{eq:multiplicity-ratio}
R_M^h(z_h,\nu)=\frac{\frac{N_h(z_h,\nu)}
{N_e(\nu)}\big|_A}{\frac{N_h(z_h,\nu)}{N_e(\nu)}\big|_D}
,$$ for identified hadrons on a $^{84}$Kr target. Here $N_h$ is the yield of semi-inclusive hadrons in a given $(z_h,\nu)$-bin and $N_e$ the yield of inclusive deep inelastic scattering leptons in the same $\nu$-bin. For the deuterium target, i.e. the nominator of Eq. (\[eq:multiplicity-ratio\]), we simply use the isospin averaged results of a proton and a neutron target. Thus in the case of deuterium we neglect the FSI of the produced hadrons and also the effect of shadowing and Fermi motion.
![Calculated multiplicity ratio of positively and negatively charged hadrons and pions for a $^{14}$N and $^{84}$Kr target when using a 12 GeV positron beam at HERMES. For the calculation we use the formation time $\tau_f=0.5$ fm/$c$ and the constituent-quark concept (\[eq:prehadrons\]) for the prehadronic cross sections. The data are taken from Ref. [@Nez04].[]{data-label="fig:HERMES12"}](fig3.eps)
In Fig. \[fig:HERMES12\] we show that our approach is also capable to describe the observed multiplicities of charged hadrons in the HERMES experiment at $E_\mathrm{beam}=$12 GeV. This success suggests that our model can also be applied for the electron beam energies that will be used at Jefferson Lab.
![Calculated multiplicity ratio of identified $\pi^\pm$, $\pi^0$, $K^\pm$, $K^0$ and $\bar{K}^0$ for $^{12}$C (dotted lines), $^{56}$Fe (dashed lines) and $^{208}$Pb nuclei (solid lines). The simulation has been done for a 5 GeV electron beam and the CLAS detector. The dash-dotted line represents a calculation for $^{56}$Fe without Fermi motion. In all calculations we use the formation time $\tau_f=0.5$ fm/$c$ and the constituent-quark concept (\[eq:prehadrons\]) for the prehadronic cross sections.[]{data-label="fig:JLab"}](fig4.eps)
In Fig. \[fig:JLab\] we present our predictions for the multiplicity ratios of identified hadrons at 5 GeV electron beam energy. Besides the considerably lower beam energy used at Jefferson Lab the major difference to the HERMES experiment is the much larger geometrical acceptance of the CLAS detector. The latter leads to an increased detection of low energy secondary particles that are produced in the FSI and that lead to a strong increase of the multiplicity ratio at low fractional hadron energies $z_h=E_h/\nu$. In addition, the relatively small average photon energy leads to a visible effect of Fermi motion on the multiplicity ratio of more massive particles. Since the virtual photon cannot produce antikaons without an additional strange meson, e.g. $\gamma^*N\to
K\bar{K}N$, the maximum fractional energy $z_h$ is limited to 0.9 for antikaons. Because of the energy distribution in the three body final state and the finite virtuality of the photon – set by the kinematic cut $Q^2>1$ GeV$^2$ – the maximum fractional energy of antikaons is further reduced. As a result, the $z_h$ spectra for $K^-$ and $\bar{K}^0$ in Fig. \[fig:JLab\] do not exceed $z_h\approx0.8$. For the same reason the production of antikaons with $z_h>0.2$ is reduced at the lower end of the photon spectrum. The Fermi motion in the nucleus enhances the yield of antikaons in these two extreme kinematic regions as can be seen by comparison with the dash-dotted line in Fig. \[fig:JLab\] which represents the result of a calculation for $^{54}$Fe where Fermi motion has been neglected. Certainly, the kaons can be produced in a two-body final state (e.g. $\gamma^*N\to K\Lambda$), however, the accompanying hyperon has a relatively large mass. Therefore, similar effects, although less pronounced, show also up for the kaons. Beside the effects of Fermi motion the multiplicity ratios of kaons and antikaons show the same features as for higher energies.
Conclusions
===========
In this work we have presented a model that allows for a clean-cut separation of the initial state interactions of a high energy (virtual) photon –giving rise to nuclear shadowing– and the nuclear FSI of the reaction products. This allows to apply a coupled-channel transport code to perform realistic event-by-event simulations of high energy lepton-nucleus scattering and to account for experimental cuts and detector efficiencies.
From our comparison with the experimental data we conclude that a large part of the observed hadron attenuation at HERMES may be attributed to prehadronic FSI of the reaction products in the nuclear environment. Furthermore, we also expect a strong effect of these prehadronic FSI at the considerably lower Jefferson Lab energies.
Our implemented space-time picture of hadronization is still simplistic and needs improvements. In Ref. [@neu] we have therefore started to extract the four-dimensional prehadron production points from the Lund fragmentation routine [<span style="font-variant:small-caps;">Jetset</span>]{} in [<span style="font-variant:small-caps;">Pythia</span>]{}. This information will be used in future transport simulations of deep inelastic lepton-nucleus scattering.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work has been supported by BMBF.
[99]{}
A. Airapetian et al. \[HERMES Collaboration\], [*Eur. Phys. J. C*]{} [**20**]{} (2001) 479.
A. Airapetian et al. \[HERMES Collaboration\], [*Phys. Lett. B*]{} [**577**]{} (2003) 37.
E. Wang and X.-N. Wang, [*Phys. Rev. Lett.*]{} [**89**]{} (2002) 162301; F. Arleo, [*Eur. Phys. J.*]{} [**C30**]{} (2003) 213.
T. Falter, W. Cassing, K. Gallmeister and U. Mosel, (2004) 61.
T. Falter, W. Cassing, K. Gallmeister and U. Mosel, (2004) 054609.
B. Z. Kopeliovich, J. Nemchik, E. Predazzi and A. Hayashigaki, , 211 (2004).
A. Accardi, V. Muccifora and H.-J. Pirner, [*Nucl. Phys.*]{} [**A720**]{} (2003) 131.
W. K. Brooks, [*Fizika*]{} [**B13**]{} (2004) 321.
M. Effenberger and U. Mosel, (2000) 014605; T. Falter and U. Mosel, (2002) 024608; T. Falter, K. Gallmeister and U. Mosel, (2003) 054606, Erratum-ibid. [**C68**]{} (2003) 019903.
C. Ciofi degli Atti, B. Z. Kopeliovich, [*Eur. Phys. J.*]{} [**A17**]{}, 133 (2003).
P. Di Nezza \[HERMES Collaboration\], J. Phys. G. [**30**]{}, S783 (2004).
K. Gallmeister and T. Falter, nucl-th/0502015.
|
---
abstract: 'The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the emergence of space-time patterns in the form of coherent structures, whose profiles asymptote to wavetrains. In particular, we give a complete existence and stability analysis of wavetrains and prove existence of various coherent structures, including soliton- and domain wall-type solutions. Decisive for the solution structure is the size of anisotropy compared with the difference of field intensity and current intensity normalized by the damping.'
bibliography:
- 'wavetrain.bib'
---
[Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations]{}
C. Melcher[^1] & J.D.M. Rademacher[^2]
Introduction {#s:intro}
============
This paper concerns the analysis of spatio-temporal pattern formation for the axially symmetric Landau-Lifshitz-Gilbert-Slonczeweski equation in one space dimension $$\label{e:llg}
\partial_t \m = \m\times \Big[ \alpha \partial_t \m - \partial_x^2 \m + (\mu m_3 - h)\3 + \beta \m \times \3 \Big]$$ as a model for the magnetization dynamics $\m=\m(x,t) \in \mathbb{S}^2$ (i.e. $\m$ is a direction field) driven by an external fields $\boldsymbol{h}=h\3$ and currents $\boldsymbol{j}=\beta\3$. The parameters $\alpha > 0$ and $\mu \in \R$ are the Gilbert damping factor and the anisotropy constant, respectively. A brief overview of the physical background and interpretation of terms is given below in Section \[s:model\]. While the combination of field and current excitations gives rise to a variety of pattern formation phenomena, see e.g. [@Bertotti:10; @Hoefer:05; @Hoefer:13; @Kravchuk:13], not much mathematically rigorous work is available so far, in particular for the dissipative case $\alpha>0$ that we consider. The case of axial symmetry is not only particularly convenient from a technical perspective. It offers at the same time valuable insight in the emergence of space-time patterns and displays strong similarities to better studied dynamical systems such as real and complex Ginzburg-Landau equations. In this framework we examine the existence and stability of wavetrain solutions of , i.e., solutions of the form $$\m(x,t) = e^{\rmi \varphi(kx-\omega t)} \m_0,$$ where the complex exponential acts on $\m_0 \in \mathbb{S}^2$ by rotation about the $\3$-axis (cf. Figure \[f:wavetrain\] for an illustration). We give a complete characterization of wavetrains and their $L^2$-stability. We also investigate the existence of their spatial connections, i.e. coherent structures of the form $$\m(x,t) = e^{i \varphi(x,t)} \m_0(x-st) \quad \text{where} \quad \varphi(x,t)= \phi(x- s t) + \Omega t$$ and such that $\m_0(\xi) = (\sin \theta(\xi) ,0, \cos \theta(\xi) )$. We completely characterize the existence of stationary ($s=0$) and small amplitude coherent structures with $|s|$ above an explicit threshold, and identify fast ($|s|\gg 1$ larger than an abstract threshold) coherent structures. The analysis of these kinds of solutions is inspired by and bears similarities with that of the real Ginzburg-Landau equation, which we also briefly discuss. The interested reader is referred to the review [@AransonKramer:02] and the references therein.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illustration of a wavetrain profile $\m(\varphi)$ (a) in the 2-sphere and (b) as a space-time plot of, e.g., $m_2$. In (a) the thick arrow represents $\m=(m,m_3)$, the thin line its trajectory as a function of $\varphi=kx-\omega t$.[]{data-label="f:wavetrain"}](3Dwavetrain.eps "fig:"){width="30.00000%"} ![Illustration of a wavetrain profile $\m(\varphi)$ (a) in the 2-sphere and (b) as a space-time plot of, e.g., $m_2$. In (a) the thick arrow represents $\m=(m,m_3)$, the thin line its trajectory as a function of $\varphi=kx-\omega t$.[]{data-label="f:wavetrain"}](wavetrain.eps "fig:"){width="35.00000%"}
(a) (b)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Part of a coherent structure profile for supercritical anisotropy in spherical coordinates. In the reduced equations it maps to a long period solution. Here $\mu=7$, $h-\Omega=-1, \Omega=\beta/\alpha$ and first integral $C=1$, cf. . (a) $m_2(x)$, (b) $(m_1,m_2)$-plane.[]{data-label="f:cohex"}](coh-m2-mu7-hbeal-1C1 "fig:"){width="40.00000%"} ![Part of a coherent structure profile for supercritical anisotropy in spherical coordinates. In the reduced equations it maps to a long period solution. Here $\mu=7$, $h-\Omega=-1, \Omega=\beta/\alpha$ and first integral $C=1$, cf. . (a) $m_2(x)$, (b) $(m_1,m_2)$-plane.[]{data-label="f:cohex"}](coh-m12-mu7-hbeal-1C1 "fig:"){width="40.00000%"}
(a) (b)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The small amplitude and fast coherent structures are forms of magnetic domain walls, with the spatially asymptotic states being the constant up- or down-magnetization states ($\m=\pm \3$), or a wavetrain. The connections with the wavetrain thus form a spatial interface between constant and precessional states.
More specifically, the parameter space for existence of wavetrains and coherent structures is largely organized by the difference of anisotropy $\mu$ and the force balance $|h-\beta/\alpha |$, of magnetic field strength minus the ratio of current intensity and damping factor, which also appears in the ODE for solutions that are homogeneous in space. Motivated by the nature of bifurcations we introduce the following notions:
- ‘supercritical anistropy’ if $\mu>|h-\beta/\alpha|$
- ‘subcritical anisotropy’ if $0< |\mu| < |h-\beta/\alpha|$,
- ‘subsubcritical anisotropy’ if $-\mu>|h-\beta/\alpha|$.
From a physical viewpoint $\mu$ and $\alpha$ are material specific, while $h,\beta$ are control parameters.
Notably, supercritical anisotropy is ‘easy-plane’, while subsubcritical anistropy is ‘easy-axis’ (see §\[s:model\]). In this framework the main results may be summarized somewhat informally as follows:
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
. Bottom row: Ranges of these profiles in the sphere, which rotate in time with frequency $\Omega=\alpha/\beta$, about the $\3$-axis (black arrows). (a) Half-period of a profile that crosses $\pm\3$, and passes close to a homogeneous wavetrain, which generates the plateau around $\xi=0$. (b) superposition of two periodic profiles crossing $\3$ and $-\3$, respectively.[]{data-label="f:profiles-c0-q0"}](3D-meridian "fig:"){width="20.00000%"} . Bottom row: Ranges of these profiles in the sphere, which rotate in time with frequency $\Omega=\alpha/\beta$, about the $\3$-axis (black arrows). (a) Half-period of a profile that crosses $\pm\3$, and passes close to a homogeneous wavetrain, which generates the plateau around $\xi=0$. (b) superposition of two periodic profiles crossing $\3$ and $-\3$, respectively.[]{data-label="f:profiles-c0-q0"}](3D-meridian2 "fig:"){width="20.00000%"}
(a) (b)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### The homogeneous up- and down-magnetization equilibria $\pm\3$.
Both of these are unstable for the supercritical anisotropy, and both are stable in the subsubcritical regime. If $h>\beta/\alpha$ then $\3$ is unstable for subcritical anisotropy, while $-\3$ is stable, and the situation is reversed for $h<\beta/\alpha$. The transition from sub- to supercritical is a supercritical Hopf-type bifurcation, while the transition from sub- to subsubcritical is a subcritical one. (Lemma \[l:conststab\] and Theorem \[t:wt-ex\].)
#### Wavetrains.
1. *Supercritical anisotropy:* There exist unique wavetrains for all wavenumbers with absolute value in $[0,\sqrt{\mu-|h-\beta/\alpha|})$ and above $\sqrt{\mu+|h-\beta/\alpha|}$. The only stable wavetrains have wavenumber $k$ near zero and become sideband unstable at a critical wavenumber below $\sqrt{\mu-|h-\beta/\alpha|}$.
2. *Subcritical anisotropy:* There exist unique wavetrains for all wavenumbers larger than $\sqrt{\mu+|h-\beta/\alpha|}$, and these are all unstable.
3. *Subsubcritical anisotropy:* There exist unique wavetrains for all wavenumbers, and these are all unstable.
(Theorems \[t:wt-ex\], \[t:wt\])
#### Stationary coherent structures.
1. *Supercritical anisotropy:* For fixed parameters there exist various stationary coherent structures ($s=0$) including a class of ‘homogeneous’ ones, whose range lies within a meridian on the sphere that rotates in time azimuthally with constant frequency $\beta/\alpha$. An interesting case of the latter is a symmetric pair of ‘phase slip’ soliton-type coherent structures, whose spatially asymptotic states are the same spatially homogeneous oscillation ($k=0$), but the intermediate profile crosses either $\3$ or $-\3$, so that the asymptotic states differ azimuthally by $180^\circ$. There also exists a non-homogeneous soliton-type solution with asymptotic state being a wavetrain.
2. *Sub- and subsubcritical anisotropy:* All stationary coherent structures have periodic profiles except a homogeneous phase slip soliton to $\pm\3$ for $\sgn(h-\beta/\alpha)=\pm1$.
(Theorems \[t:coh-c0-q0\], \[t:coh-c0\])
#### Fast and small amplitude coherent structure.
For each sufficiently large speed there exist a family of front-type coherent structures parametrized by the azimuthal frequency. Their profiles connect $\pm\3$ with each other or, if it exists, the unique wavetrain. Small amplitude coherent structure above an explicit lower speed bound are of front type and exist only for super- and subcritical anistropy. (Theorems \[t:coh-fast\], \[t:coh-small\].)
While the stability analysis of wavetrains shows that the fast and small coherent structures are unstable in an $L^2$-sense, they could be stable in weighted spaces, and knowing unstable solutions may also be useful for stabilizing control schemes. However, we do not investigate this further here.
#### Higher space dimensions.
In $N$ space dimensions, the second derivative with respect to $x$ in would be replaced by a Laplace operator $\sum_{j=1}^N \partial_{x_j}^2$. Wavetrain type solutions are then of the form $$\m(x,t) = \m_*(k \cdot x-\omega t),$$ where $k=(k_1,\ldots,k_N)$. Notably, for $k_j=0$, $2\leq j \leq N$ these are solutions from one space dimension extended trivially (constant) in the additional directions.
Conveniently, the rotation symmetry (gauge invariance in the Ginzburg-Landau context), means that the analyses of $\pm\3$ and these wavetrains is already covered by that of the one-dimensional case: the linearization is space-independent and therefore there is no symmetry breaking due to different $k_j$. Indeed, all relevant quantities are rotation symmetric, depending only on $k^2=\sum_{j=1}^N k_j^2$ or $\ell^2=\sum_{j=1}^N \ell_j^2$, where $\ell=(\ell_1,\ldots,\ell_N)$ is the Fourier wavenumber vector of the linearization. In particular, the only nontrivial stability boundary are sideband instabilities, which occur simultaneously for all directions.
Concerning coherent structures, turns into an elliptic PDE in general so that the analysis in this paper only allows for the trivial extension into higher dimensions.
This article is organized as follows. In Section \[s:model\], the terms in the model equation and its well-posedness are discussed. Section \[s:hopf\] concerns the stability of the trivial steady states $\pm\3$ and in §\[s:wt\] existence and stability of wavetrains are analyzed. Section \[s:coh\] is devoted to coherent structures.
**Acknowledgement.** JR has been supported in part by the NDNS+ cluster of the Dutch Science Fund (NWO).
Landau-Lifshitz equations {#s:model}
=========================
The classical equation of magnetization dynamics, the conservative Landau-Lifshitz equation [@Landau_Lifshitz:35], corresponds to a torque balance for the magnetic moment, represented by a unit vector field $\m=\m(t) \in \mathbb{S}^2$, $$\dot{\m} +\gamma \, \m \times \boldsymbol{h}_{\rm eff}=0.$$ It features a free precession of $\m$ about the effective field $\boldsymbol{h}_{\rm eff}= - \mathcal{E}'(\m)$, i.e., minus the variation of the underlying interaction energy $\mathcal{E}=\mathcal{E}(\m)$, which is conserved in time. The gyromagnetic ratio $\gamma>0$ is a parameter which appears as the typical precession frequency. By rescaling time, one can always assume $\gamma=1$. Adding a dissipative counter-field, which is proportional to the velocity $\dot{\m}$, yields the Landau-Lifshitz-Gilbert equation (LLG) [@Gilbert:04; @Hubert_Schaefer] $$\label{eq:Gilbert_form}
\dot{\m} = \m \times \left( \alpha \, \dot{\m} - \boldsymbol{h}_{\rm eff} \right).$$ The Gilbert damping factor $\alpha>0$ is a constant that can be interpreted dynamically as the inverse of the typical relaxation time. It is useful to take into account that there are several equivalent forms of LLG. Carrying out another cross multiplication of by $\m$ from the left and taking into account that $
- \m \times \m \times \boldsymbol{\xi} = \boldsymbol{\xi} - (\m \cdot \boldsymbol{\xi}) \m
$ is the orthonormal projection to the tangent plane of $\mathbb{S}^2$ at $\m$, one obtains $$\label{eq:mixed_form}
\alpha \, \dot{\m} + \m \times \dot{\m} = - \m \times \m \times \boldsymbol{h}_{\rm eff},$$ where the right hand side can also be written as $\boldsymbol{h}_{\rm eff} - (\m \cdot \boldsymbol{h}_{\rm eff} ) \, \m$. Multiplying this equation by $\alpha$ and adding it to the original one, one obtains the dissipative Landau-Lifshitz equation[^3], i.e. $$\label{eq:LL_form}
(1+ \alpha^2) \dot{\m} =- \m \times \left( \alpha \, \m \times \boldsymbol{h}_{\rm eff} + \boldsymbol{h}_{\rm eff} \right),$$ which has been introduced in the original work [@Landau_Lifshitz:35]. In case of LLG, the energy $\mathcal{E}(\m)$ is not conserved but a Liapunov functional, i.e., more precisely (recall $\boldsymbol{h}_{\rm eff}= - \mathcal{E}'(\m)$) $$\frac{d}{dt} \mathcal{E}(\m(t))= - \alpha \|\dot{\m}(t)\|^2 \quad \text{or equivalently} \quad
\frac{d}{dt} \mathcal{E}(\m(t))= - \frac{\alpha}{1+\alpha^2} \| \m \times \boldsymbol{h}_{\rm eff} \|^2.$$ Gilbert damping enables the magnetization to approach (spiral down to) a steady state, i.e. satisfying $\m \times \boldsymbol{h}_{\rm eff}=0$ (Browns equation), as $t \to \infty$.
#### Spin-torque interaction.
The system can be driven out of equilibrium conventionally by external magnetic fields $\boldsymbol{h}$ which appears as part of the effective field. In modern spintronic applications, magnetic systems are excited by spin polarized currents (with direction of polarization $\boldsymbol{\hat{e}}_p \in \mathbb{S}^2$) giving rise to a spin torque $$\label{eq:spin_torque}
\m \times \m \times {\bf j} \quad \text{where} \quad \boldsymbol{j}= \beta \frac{\boldsymbol{\hat{e}}_p}{1+c \; \m \cdot \boldsymbol{\hat{e}}_p},$$ which has been introduced in [@Berger:96; @Slonczewski:96]. Here, the parameters $\beta>0$ and $c \in (-1,1)$ depend on the intensity of the current and ratio of polarization [@Bertotti:08]. Typically we have $ \boldsymbol{\hat{e}}_p = \3$. Accordingly, the modified Landau-Lifshitz-Gilbert equation, also called Landau-Lifshitz-Gilbert-Slonczewski equation (LLGS), reads $$\label{eq:LLG_ODE}
\dot{\m} = \m \times \left( \alpha \, \dot{\m} - \boldsymbol{h}_{\rm eff} + \m \times \boldsymbol{j} \right).$$ One may extend the notion of effective field to include current interaction by letting $$\boldsymbol{H}_{\rm eff} =\boldsymbol{h}_{\rm eff} - \m \times \boldsymbol{j},$$ where the second term is usually called Slonczewski term. In this framework can also be written in the form and with $\boldsymbol{h}_{\rm eff}$ repalced by $\boldsymbol{H}_{\rm eff}$. Observe, however, that the Slonczewski term (and hence $\boldsymbol{H}_{\rm eff}$) is non-variational and that the energy is no longer a Liapunov functional. Introducing the potential $\Psi(\m) = \frac{\beta}{c} \ln(1+c \, \m \cdot \boldsymbol{\hat{e}}_p)$ of $\boldsymbol{j}$ (for $c\neq 0$) reveals the *skew variational* structure $$\m \times \left[ \alpha \, \dot{\m}+ \mathcal{E}'(\m) \right] =- \m \times \m \times \left[ \dot{\m} +\Psi'(\m) \right].$$
#### The continuum equation.
Continuum theories are concerned with magnetization fields $\m=\m(x,t)$ such that $|\m|=1$. In the micromagnetic model the underlying interaction energies are integral functionals in $\m$ containing in particular exchange (Dirichlet) interaction, dipolar stray-field interaction, crystal anisotropy and Zeeman interaction with external magnetic field, see e.g. [@Hubert_Schaefer]. The effective field is computed from the $L^2$ gradient of the energy, i.e. $\boldsymbol{h}_{\rm eff}= - \nabla_{\small L^2} \mathcal{E}(\m)$, and becomes a partial differential equation. In this paper we shall mainly focus on the spatially one-dimensional situation and consider energies of the form $$\label{eq:energy}
\mathcal{E}(\m) = \frac{1}{2} \int \left( |\partial_x \m|^2 + \mu m_3^2\right) \, dx - \int \boldsymbol{h} \cdot \m \; dx.$$ Here, $\boldsymbol{h} \in \R^3$ is a constant applied magnetic field. The parameter $\mu \in \R$ features *easy plane* anisotropy for $\mu>0$ and *easy axis* anisotropy for $\mu <0$, respectively, according to energetically preferred subspaces. This term comprises crystalline and shape anisotropy effects. Shape anisotropy typically arises from stray-field interactions which prefer magnetizations tangential to the sample boundaries. Hence $\mu >0$ corresponds to a thin-film perpendicular to the $\3$-axis whereas $\mu <0$ corresponds to a thin wire parallel to the $\3$-axis. The effective field corresponding to reads $$\boldsymbol{h}_{\rm eff}= \partial_x^2 \m - \mu m_3 \3 + \boldsymbol{h}.$$ With the choices $\boldsymbol{h}= h\, \3$ and $\hat{e}_p= \3$, the Landau-Lifshitz-Gilbert-Slonczewski equation exhibits the aforemented rotation symmetry about the $\boldsymbol{\hat e}_3$-axis. Our analysis is valid for small $|c|$ in and we focus on $c=0$ so that $\boldsymbol{j} =\beta \, \3$. This is equation , i.e., $$\partial_t \m = \m\times \Big[ \alpha \partial_t \m - \partial_x^2 \m + (\mu m_3 - h)\3 + \beta \m \times \3 \Big].$$ The presence of a spin torque $\m \times \m \times \boldsymbol{j}$ exerted by a constant current may induce switching between magnetization states or magnetization oscillation, [@Berkov_Miltat:08; @Bertotti:05; @Bertotti:08]. For the latter effect, the energy supply due to the electric current compensates the energy dissipation due to damping enabling a stable oscillation, called *precessional states*. In applications the typical frequency is in the range of GHz, so that a precessional state would basically act as a microwave generator. In the class of spatially homogeneous states, precessional states are periodic orbits with $m_3=const.$ and of constant angular velocity $ \beta/\alpha$. It is more subtle, however, to understand the occurrence and stability of spatially non-homogeneous precessional states.
#### Extensions and related work.
Non-symmetric variants of our equation have been used e.g. in the description of the field driven motion of a flat domain wall connecting antipodal steady states $m_3 = \pm 1$ as $x_1 \to \pm \infty$. A prototypical situation is the field driven motion of a flat Bloch wall in an uniaxial the bulk magnet governed by $$\partial_t \m = \m\times \left(\alpha \partial_t \m - \partial_x^2 \m + \mu_1 m_1 \boldsymbol{\hat e}_1 + ( \mu_3 m_3 \boldsymbol - h ) \boldsymbol{\hat e}_3 \right).$$ In this case $\mu_1>0> \mu_3$, where $\mu_1$ corresponds to stray-field and $\mu_3$ to crystalline anisotropy. Explicit traveling wave solutions were obtained in unpublished work by Walker, see e.g. [@Hubert_Schaefer], and reveal interesting effects such as the existence of a terminal velocity (called Walker velocity) and the notion of an effective wall mass. A mathematical account on Walker’s explicit solutions and investigations on their stability, possible extensions to finite layers and curved walls can be found e.g. in [@Carbou:10; @Melcher:04; @Podio_Tomassetti:04]. Observe that our axially symmetric model is obtained in the limit $\mu_1 \searrow 0$. On the other hand, the singular limit $\mu_3 \to + \infty$ leads to trajectories confined to the $\{m_3=0\}$ plane (equator map), and can be interpreted as a thin-film limit. In suitable parameter regimes it can be shown that the limit equation is a dissipative wave equation governing the motion of Néel walls [@Capella:07; @Melcher:03; @Melcher:10].
#### Well-posedness of LLGS.
It is well-known that Landau-Lifshitz-Gilbert equations and its variants have the structure of quasilinear parabolic systems. In the specific case of , one has the extended effective field $\boldsymbol{H}_{\rm eff} =\boldsymbol{h}_{\rm eff} - \m \times \boldsymbol{j}$, more precisely $$\label{eq:H_ext}
\boldsymbol{H}_{\rm eff} = \partial^2_x \m - f(\m) \quad \text{where} \quad f(\m)= (\mu m_3 - h) \3 + \beta \m \times \3.$$ Hence the corresponding Landau-Lifshitz form of reads $$\label{eq:LL_form_1}
(1+\alpha^2) \partial_t \m =- \m\times \Big[ \partial_x^2 \m - f(\m) \Big]- \alpha \m \times \m\times \Big[ \partial_x^2 \m - f(\m) \Big].$$ Taking into account[^4] $$\label{eq:div}
\m\times \partial_x^2 \m = \partial_x( \m\times \partial_x \m) \quad \text{and} \quad - \m \times \m \times \partial_x^2 \m = \partial_x^2 \m + |\partial_x \m|^2 \m$$ valid for $\m$ sufficiently smooth and $|\m|=1$, one sees that has the form $$\begin{aligned}
\label{e:quasi}
\partial_t \m = \partial_x \left( A(\m) \partial_x \m \right) + B(\m, \partial_x \m)\end{aligned}$$ with analytic functions $A:\R^3 \to \R^{3 \times 3}$ and $B:\R^3 \times \R^{3} \to \R^3$ such that $A(\m)$ is uniformly elliptic for $\alpha>0$, in fact $$\boldsymbol{\xi} \cdot A(\m) \boldsymbol{\xi} = \frac{\alpha}{1+\alpha^2} | \boldsymbol{\xi}|^2
\quad \text{for all} \quad \boldsymbol{\xi} \in \R^3.$$
Well-posedness results for $\alpha>0$ can now be deduced from techniques based on higher order energy estimates as in [@Melcher:11; @Melcher_Ptashnyk] or maximal regularity and interpolation as in [@users_guide]. In particular, we shall rely on results concerning perturbations of wavetrains, traveling waves, and steady states. Suppose $\m_\ast=\m_\ast(x,t)$ is a smooth solution of with bounded derivatives up to all high orders (only sufficiently many are needed) and $\m_0:\R \to \mathbb{S}^2$ is such that $\m_0- \m_\ast(\cdot,0) \in H^2(\R)$. Then there exist $T>0$ and a smooth solution $\m: \R \times (0,T) \to \mathbb{S}^2$ of such that $\m-\m_\ast \in C^0([0,T); H^2(\R)) \cap C^1([0,T); L^2(\R))$ with $$\lim_{t \searrow 0} \| \m(t)- \m_0\|_{H^2} =0 \quad \text{and} \quad \lim_{t \nearrow T} \| \m(t)- \m_\ast(\cdot,t)\|_{H^2} =\infty
\quad \text{if $T<\infty$}.$$ The solution is unique in its class and the flow map depends smoothly on initial conditions and parameters.
Given the smoothness of solutions, we may compute pointwise $\partial_t |\m|^2 = 2\m \cdot \partial_t \m$, so that for $|\m|=1$ the cross product form of the right hand side of gives $\partial_t |\m|^2=0$. Hence, the set of unit vector fields, $\{|m|=1\}$, is an invariant manifold of consisting of the solutions to that we are interested in.
In addition to well-posedness, also stability and spectral theory for , see, e.g., [@users_guide], carry over to . In particular, the computations of $L^2$-spectra in the following sections are justified and yield nonlinear stability for strictly stable spectrum and nonlinear instability for the unstable (essential) spectrum.
#### Landau-Lifshitz-Gilbert-Slonczewski versus complex Ginzburg-Landau equations.
Stereographic projection of yields $$(\alpha + {\rm i}) \zeta_t = \partial_x^2 \zeta -
\frac{2 \bar{\zeta} (\partial_x \zeta)^2}{1+|\zeta|^2} + \mu \frac{(1- |\zeta|^2) \zeta }{1+ |\zeta|^2} - (h + i \beta) \zeta
\quad \text{where} \quad
\zeta = \frac{m_1+{\rm i} m_2}{1+m_3},$$ valid for magnetizations avoiding the south pole.
There is also a global connection between LLG and CGL in the spirit of the classical Hasimoto transformation [@Hasimoto:72], which turns the (undamped) Landau-Lifshitz equation in one space dimension ($\boldsymbol{h}_{\rm eff}=\partial_x^2$) into the focussing cubic Schrödinger equation. The idea is to disregard the customary coordinates representation and to introduce instead a pull-back frame on the tangent bundle along $\m$. In the case of $\mu=\beta=h=0$, i.e. $\boldsymbol{h}_{\rm eff}= \partial_x \m$, this leads to $$\label{eq:gauged_LLS}
(\alpha + \rmi) \mathcal{D}_t u = \mathcal{D}_{x}^2 u$$ where $u=u(x,t)$ is the complex coordinate of $\partial_x \m$ in the moving frame representation, and $\mathcal{D}_x$ and $\mathcal{D}_t$ are covariant derivatives in space and time giving rise to cubic and quintic nonlinearities, see [@Melcher:11; @Melcher_Ptashnyk] for details.
Hopf instabilities of the steady states $\m=\pm \3$ {#s:hopf}
===================================================
As a starting point and to motivate the subsequent analysis of more complex patterns, let us consider the stability of the constant magnetizations $\pm\3$. Substituting $\m= \pm\3 +\delta \boldsymbol{n} + o(\delta)$, where $\boldsymbol{n}=(n,0) \in T_{\m}\mathbb{S}^2$, into gives, at order $\delta$, the linear equation $$\partial_t \boldsymbol{n} =(\pm\mu-h) \boldsymbol{n} \times \3
\pm \3\times (\alpha \partial_t \boldsymbol{n} - \partial_x^2 \boldsymbol{n} + \beta \boldsymbol{n}\times \3),$$ which may be written in complex form as $$\partial_t n \pm \beta n = i \left( \alpha \partial_t n - \partial_x^2 n - (h \mp \mu) n \right).$$ Its eigenvalue problem diagonalizes in Fourier space (for $x$) and yields the matrix eigenvalue problem $$\left |
\begin{array}{ccc}
\pm\beta -\lambda & \Lambda \\
-\Lambda & \pm\beta -\lambda
\end{array}
\right | = 0,$$ where $\Lambda = \pm\mu - h \mp \alpha \lambda \mp \ell^2$ with $\ell$ the Fourier wave number. The determinant reads $$(\pm\beta- \lambda)^2 = - \Lambda^2 \quad \Leftrightarrow \quad
\pm\beta -\lambda = \sigma \rmi \Lambda, \; \sigma\in\{\pm 1\}.$$ Considering real and imaginary parts this leads to $$\begin{aligned}
(1+\alpha^2)\Re(\lambda) &= \pm\beta - \alpha(\ell^2\pm h-\mu)= \alpha(\mu \mp(h-\beta/\alpha)-\ell^2) \\
\Im(\lambda) &= \sigma(\mp\Re(\lambda) +\beta/\alpha), \end{aligned}$$ so that the maximal real part has $\ell=0$. At criticality, where $\Re(\lambda)=0$ the imaginary parts are $\pm\beta/\alpha$, which corresponds to a Hopf-instability of the spectrum, and we expect the emergence of oscillating solutions whose frequency at onset is $\beta/\alpha$.
The formulas for real- and imaginary parts immediately give the results mentioned in §\[s:intro\] and
\[l:conststab\] The constant magnetizations $\m=\pm\3$ are (strictly) $L^2$-stable if and only if the anisotropy is subsubcritical ($-\mu>|h-\beta/\alpha|$), and both are unstable if and only if it is supercritical ($\mu>|h-\beta/\alpha|$). The instabilities are of Hopf-type with frequency $\beta/\alpha$.
The result extends to the more general situation of with $|c|<1$. In fact, linearization leads to the analogous equation $$\partial_t \boldsymbol{n} =(\pm\mu-h) \boldsymbol{n} \times \3
\pm \3\times (\alpha \partial_t \boldsymbol{n} - \partial_x^2 \boldsymbol{n} + \beta^\pm \boldsymbol{n}\times \3)$$ with $\beta^\pm=\beta/(1\pm c)$ at $m_3 = \pm 1$, respectively.
Wavetrains {#s:wt}
==========
To exploit the roation symmetry about the $\3$-axis, we change to polar coordinates in the planar components $m=(m_1,m_2)$ of the magnetization $\m = (m, m_3)$. With $m=r \exp(\rmi \varphi)$ equation changes to
$$\label{e:llg-mod}
\left(\begin{array}{cc}
\alpha & -1\\
1 & \alpha
\end{array}\right)
\left(\begin{array}{c}
r^2 \partial_t \varphi\\
\partial_t m_3
\end{array}\right)
=
\left(\begin{array}{c}
\partial_x(r^2\partial_x\varphi) \\
\partial_x^2 m_3 + |\partial_x \m|^2 m_3
\end{array}\right)
+
r^2
\left(\begin{array}{c}
\beta\\
h-\mu m_3
\end{array}\right),$$
where $$|\partial_x \m|^2 = (\partial_x r)^2 + r^2 (\partial_x \varphi)^2 + (\partial_x m_3)^2 \quad \text{and} \quad r^2 + m_3^2 =1.$$ This can be seen as follows. In view of , with $\boldsymbol{h}_{\rm eff}$ replaced by the extended effective field $\boldsymbol{H}_{\rm eff}= \boldsymbol{h}_{\rm eff} - \m \times \boldsymbol{j}$ as is and taking into account , reads $$\alpha \partial_t \m + \m\times \partial_t \m = \partial_x^2 \m +(h- \mu m_3) \boldsymbol{\hat e}_3 + \left( |\partial_x \m|^2 + \mu m_3^2 - h m_3 \right) \m
- \beta \m\times \3.$$ The third component of the above equation is the second component of , whereas the first component of is obtained upon inner multiplication by $\m^\perp = (m^\perp,0)=(ie^{i \varphi},0)$ and taking into account that $\m \times \3= - \m^\perp$.
The rotation symmetry has turned into the shift symmetry $\varphi \mapsto \varphi$ + const. In full spherical coordinates $\m=\binom{e^{i \varphi} \sin \theta}{\cos \theta}$, further simplifies to $$\label{e:llg-cyl}
\left(\begin{array}{cc}
\alpha & -1\\
1 & \alpha
\end{array}\right)
\left(\begin{array}{c}
\sin \theta \partial_t \varphi\\
-\partial_t \theta
\end{array}\right)
=
\left(\begin{array}{c}
2\cos \theta \partial_x \theta \, \partial_x \varphi + \sin \theta\partial_x^2\varphi \\ -\partial_x^2 \theta + \sin\theta\cos\theta (\partial_x\varphi)^2\end{array}\right)
+
\sin \theta
\left(\begin{array}{c}
\beta\\
h-\mu \cos \theta
\end{array}\right).$$
Existence of wavetrains
-----------------------
Wavetrains are solutions of the form $\m(x,t) = \m_*(kx-\omega t)$, where $k$ is referred to as the wavenumber and $\omega$ as the frequency. A natural type of wavetrains are relative equilibria with respect to the phase shift symmetry for which $\varphi = k x - \omega t$ and $m_3, r$ are constant. See Figure \[f:wavetrain\] for an illustration.
Substituting this ansatz into yields the algebraic equations $$\left(\begin{array}{cc}
\alpha & -1\\
1 & \alpha
\end{array}\right)
\left(\begin{array}{c}
-\sin(\theta) \omega\\
0
\end{array}\right)
=
\left(\begin{array}{c}
0\\
\sin(\theta)\cos(\theta)k^2
\end{array}\right)
+
\sin(\theta)
\left(\begin{array}{c}
\beta\\
h-\mu \cos(\theta)
\end{array}\right).$$ Thus either $\theta\equiv0 \mod \pi$ or $$\begin{aligned}
-\alpha\omega &= \beta\\
-\omega &= (k^2-\mu)m_3 + h.\end{aligned}$$ In the first case we have $r=0$, which corresponds to the constant upward or downward magnetizations, $(r,m_3)=(0,\pm1)$ with unspecified $k$ and $\omega$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Plots of equilibrium locations in the $(k,m_3)$-plane including the trivial equilibria $\pm\3$ plotted with thick line if stable (when not intersecting wavetrain parameters). Compare Figure \[f:wt-ex\]. See . (a) supercritical (easy plane) anisotropy ($\mu=1$, $h-\beta/\alpha=1/2$), (b) subcritical anisotropy ($\mu=1$, $h-\beta/\alpha=2$) no homogeneous oscillations ($k=0$) exist, and (c) subsubcritical (easy axis) anisotropy $\mu=-1$, $h-\beta/\alpha=0.9$.[]{data-label="f:wavetrains"}](wt-cyl.eps "fig:"){width="30.00000%"} ![Plots of equilibrium locations in the $(k,m_3)$-plane including the trivial equilibria $\pm\3$ plotted with thick line if stable (when not intersecting wavetrain parameters). Compare Figure \[f:wt-ex\]. See . (a) supercritical (easy plane) anisotropy ($\mu=1$, $h-\beta/\alpha=1/2$), (b) subcritical anisotropy ($\mu=1$, $h-\beta/\alpha=2$) no homogeneous oscillations ($k=0$) exist, and (c) subsubcritical (easy axis) anisotropy $\mu=-1$, $h-\beta/\alpha=0.9$.[]{data-label="f:wavetrains"}](wt-cyl-nonzero.eps "fig:"){width="30.00000%"} ![Plots of equilibrium locations in the $(k,m_3)$-plane including the trivial equilibria $\pm\3$ plotted with thick line if stable (when not intersecting wavetrain parameters). Compare Figure \[f:wt-ex\]. See . (a) supercritical (easy plane) anisotropy ($\mu=1$, $h-\beta/\alpha=1/2$), (b) subcritical anisotropy ($\mu=1$, $h-\beta/\alpha=2$) no homogeneous oscillations ($k=0$) exist, and (c) subsubcritical (easy axis) anisotropy $\mu=-1$, $h-\beta/\alpha=0.9$.[]{data-label="f:wavetrains"}](wt-cyl-instab.eps "fig:"){width="30.00000%"}
(a) (b) (c)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In the second case, first notice that absence of dissipation ($\alpha=0$) implies absence of magnetic field ($\beta=0$) and there is a two-parameter set of wavetrains given by the second equation. The case we are interested in is $\alpha>0$ and the existence conditions can be conveniently written as $$\begin{aligned}
\omega &= -\frac{\beta}{\alpha} \label{e:freq}\\
\cos(\theta) &= \frac{h-\beta/\alpha}{\mu-k^2}, \quad (\mu\neq k^2). \label{e:psi}\end{aligned}$$ Here $\mu=k^2$ happens only in the easy plane case $\mu\geq 0$, and $\alpha h=\beta$ (with $\theta$ unspecified). Notably, all wavetrains oscillate with frequency given by the ratio of applied current and dissipation.
An involution symmetry involving parameters is $$\label{e:parasym}
(h-\beta/\alpha, \theta) \to (\beta/\alpha-h, \theta + \pi),$$ so that the sign of $h-\beta/\alpha$ is irrelevant for the qualitative picture.
Solvability of requires that $|\mu-k^2| > |h-\beta/\alpha|$ (unless $r=0$), so that only for super- and subsubcritical anisotropy, $$\label{e:muex}
|\mu|>|h-\beta/\alpha|,$$ there are wavetrains with wavenumber in an interval around $k=0$. In other words, non-trivial spatially nearly homogeneous oscillations require sufficiently small (in absolute value) difference between applied magnetic field and oscillation frequency (ratio of applied current and dissipation). The transition into this regime goes via the ‘Hopf’ instability from §\[s:hopf\]. Combining with Lemma \[l:conststab\] and straightforward analysis of gives the following lemma. The three types of solution sets are plotted in Figure \[f:wavetrains\]. Clearly, the solution sets are symmetric with respect to the signs of $k$ and $\theta$, respectively.
\
\[t:wt-ex\] There are three types of wavetrain parameter sets solving :
1. For supercritical anisotropy there is one connected component of wavetrain parameters including $k=0$, and two connected components with unbounded $|k|$, each with constant sign of $k$.
2. For subcritical anisotropy there are two connected components with unbounded $|k|$, each with constant sign of $k$.
3. For subsubcritical anisotropy there are two connected components, each a graph over the $k$-axis.
The Hopf-type instabilities of $\pm\3$ noted in §\[s:hopf\] at the transition from sub- to supercritical anisotropy is a supercritical bifurcation in the sense that solutions emerge at the loss of stability of the basic solution, here $\pm\3$, while that from sub- to subsubcritical is subcritical in the sense that solutions emerge at the gain of stability.
A fruitful viewpoint on existence conditions for wavetrains is to consider the frequency as a function of the wavenumber, which is referred to as the nonlinear dispersion relation. This is degenerate here, because the wavetrain frequency $\omega$ is independent of the wavenumber $k$ in . As a consequence, the group velocity $\rmd \omega/\rmd k$, which describes the motion of perturbations by localized wave packets, equals the phase velocity $\omega/k$ for all wavetrains.
With the nonlinear dispersion relation in mind, it is customary to consider the existence region of wavetrains in wavenumber-parameter space. This is given by and is periodic in $\theta$, with $r=0$ (zero amplitude) at $\theta\equiv 0 \mod \pi$, which gives the boundary for nontrivial amplitude, $$\begin{aligned}
\label{e:exbdry}
\mu= k^2 \pm(h-\beta/\alpha),\end{aligned}$$ a pair quadratic parabolas in $(k,\mu)$-space. The solution set in this projection is sketched in Figure \[f:wt-ex\]. Remark that this set is non-empty for any parameter set $\alpha, \beta, h\in \R$ of . However, not all wavenumbers are possible due to the geometric constraint. Notably, the existence region consists of two disjoint sets, one contained in $\{\mu>0\}$ with convex boundary and one extending into $\{\mu<0\}$ with concave boundary. This again highlights the super- and subcritical nature of the wavetrain emergence noted in Theorem \[t:wt-ex\]. From this, we would expect an exchange of stability, so that only in the supercritical case stable wavetrains emerge.
Before proving this, let us briefly discuss the origins of this expectation. The fact that frequencies are constant means that in co-rotating (‘detuned’) coordinates $\theta = \varphi -\frac\beta\alpha t$ the wavetrains have frequency zero and are thus stationary. The situation is then reminiscent of the (cubic) real Ginzburg-Landau equation mentioned in the introduction, $$\begin{aligned}
\label{e:GL}
\partial_t A = \partial_x^2 A + \tilde\mu A \mp A|A|^2, \; A(x,t) \in \C,\end{aligned}$$ which describes the dynamics near pattern forming Turing instabilities and possesses the gauge-symmetry $A\to \rme^{\rmi\varphi}A$. The solution $A=0$ becomes unstable as $\tilde\mu$ becomes positive, which is the analogue of $\pm\3$ losing stability as $\mu$ crosses $\pm(h-\beta/\alpha)$.
Concerning wavetrains, the analogue of the symmetry group ansatz above is $A=r\rme^{kx-\omega t}$, $r\in\R$ the amplitude, which yields $$\omega = 0, \; \mu=k^2\pm r^2.$$ In particular, all these wavetrains are stationary and the existence region, bounded by $r=0$, is a parabola in $(k,\mu)$-space, convex in the ‘+’-case and concave otherwise. A stability analysis of these wavetrains (which we omit) shows that only in the convex (supercritical) case stable wavetrains emerge, and these lie in the socalled Eckhaus region $\mu\geq 3k^2$. Hence, the aforementioned expectation, which is fully justified in the next section. The overall picture for is thus a combination of the scenarios from a supercritical ($\mu>0$) and a subcritical ($\mu<0$) real Ginzburg-Landau equation, connected in parameter space via $\mu$.
We have only considered the special case of with $c=0$, for otherwise and give rise to a quadratic equation for $m_3$. For a parameter set away from special points, however, there exists for $0<|c|\ll1 $ a unique solution $m_3$ which is compatible with the geometric constraint. In this generic situation the considerations on spectral stability to be carried out in Section \[sec:spectral\] hold true by continuous differentiability.
Stability of wavetrains {#sec:spectral}
-----------------------
In this section spectral stability of wavetrains is determined. For convenience, the comoving frame $y=x-\cph t$ is considered with $\cph=\omega/k$ the wavespeed. In this variable the wavetrain is an equilibrium of . Again for convenience, time is rescaled to $t = (1+\alpha^2)\tau$. The explicit formulation of then reads
$$\label{e:llg-mod-ex}
\partial_\tau\left(\begin{array}{c}
\! \varphi \!\\
\! m_3 \!
\end{array}\right)
=
\left(\begin{array}{c}
\alpha(\partial_y(r^2\partial_y\varphi)/r^2 +\beta) +
(\partial_y^2 m_3 + |\partial_y \m|^2 m_3)/r^2 + h - \mu m_3 + \cph \partial_y\varphi \\
\alpha(\partial_y^2 m_3 + |\partial_y \m|^2 m_3+r^2(h-\mu m_3)) - \partial_y(r^2\partial_y\varphi) - r^2\beta + \cph\partial_y m_3
\end{array}\right),$$
where $r^2 = 1-m_3^2$.
Let $\calF=(\calF_1,\calF_2)^\rmt$ denote the right hand side of . Wavetrains have constant $r$ and $m_3$ so that quadratic terms in their derivatives can be discarded for the linearization $\calL$ of $\calF$ in a wavetrain, and from $|\partial_y \m|^2$ only $r^2(\partial_y \varphi)^2$ is relevant. The components of $\calL$ are $$\begin{aligned}
\partial_\varphi \calF_1 &= \alpha \partial_y^2 + \cph \partial_y + 2k m_3 \partial_y\\
\partial_{m_3} \calF_1 &= r^{-2}\partial_y^2 + k^2 - \mu - 2\alpha k m_3r^{-2} \partial_y\\
\partial_{\varphi} \calF_2 &= 2\alpha k m_3 r^2 \partial_y - r^2 \partial_y^2\\
\partial_{m_3} \calF_2 &= \alpha \partial_y^2 + \alpha r^2(k^2-\mu) + \cph \partial_y - 2\alpha m_3(m_3 k^2 + h - \mu m_3) + 2m_3(k\partial_y + \beta)\\
& = \alpha \partial_y^2 + \alpha r^2(k^2-\mu) + \cph \partial_y + 2m_3 k\partial_y,\end{aligned}$$ where the last equation is due to . Since all coefficients are constant, the eigenvalue problem $$\calL u = \lambda u$$ is solved by the characteristic equation arising from an exponential ansatz $u=\exp(\nu y)u_0$, which yields the matrix $$\begin{aligned}
A(\nu, \cph) :=
\left(\begin{array}{cc}
\alpha \nu^2 + (\cph+2 k m_3)\nu & -r^{-2} \nu(-\nu + 2\alpha k m_3) + k^2-\mu\\
r^2\nu(-\nu + 2\alpha k m_3) & \alpha \nu^2 + (\cph + 2k m_3)\nu + \alpha r^2(k^2-\mu)
\end{array}\right).\\\end{aligned}$$ The characteristic equation then reads $$\begin{aligned}
d_{\cph}(\lambda,\nu) & := |A(\nu,\cph) - \lambda| = |A(\nu,0) -(\lambda - \rmi \nu \cph)|
= d_0(\lambda-\cph \nu,\nu) \nonumber \\
d_0(\lambda,\nu) & = \lambda^2 - \tr A(\nu,0) \lambda + \det A(\nu,0) \label{e:disp}\end{aligned}$$ with trace and determinant of $A(\nu,0)$ $$\begin{aligned}
\tr A(\nu,0) &= 2\nu(\alpha \nu +2 k m_3) + \alpha r^2(k^2-\mu)\\
\det A(\nu,0) & = (1+\alpha ^2)\nu^2(\nu^2+ 4k^2 m_3^2 + r^2(k^2-\mu))\\
& = (1+\alpha ^2)\nu^2\left(\nu^2+ (3k^2 + \mu)\frac{(\beta/\alpha - h)^2}{(k^2-\mu)^2} + k^2 - \mu\right).\end{aligned}$$ In the last equation was used.
The characteristic equation is also referred to as the *complex dispersion relation*. The spectrum of $\calL$, for instance in $L^2(\R)$, consists of solutions for $\nu=\rmi \ell$ and is purely essential spectrum (in the sense that $\lambda-\calL$ is not a Fredholm operator with index zero). Indeed, setting $\nu=\rmi\ell$ corresponds to Fourier transforming in $y$ with Fouriermode $\ell$. Note that the solution $d(0,0)=0$ stems from spatial translation symmetry in $y$.
The real part of solutions $\lambda$ of does not depend on $\cph$, which means that spectral stability is independent of $\cph$ and is therefore completely determined by $d_0(\lambda, \rmi \ell)=0$.
First note the eigenvalues $A(0,0)$ are $0$ and $\alpha r^2(k^2-\mu)$ so that *in the easy axis case $\mu<0$ all wavetrains are unstable*. In the easy plane case $\mu>0$ the wavetrains for $k^2 \leq \mu$ have a chance to be spectrally stable. In the following we therefore restrict to $\mu>0$ and $k^2<\mu$ and check the possible marginal stability configurations case by case.
#### Sideband instability.
A sideband instability occurs when the curvature of the curve of essential spectrum attached to the origin changes sign so that the essential spectrum extends into positive real parts. Let $\tilde\lambda_0(\ell)$ denote the curve of spectrum of $A(\rmi\ell,0)$ attached to the origin, that is $\tilde\lambda_0(0)=0$, and let $'$ denote the differentiation with respect to $\ell$. Derivatives of $\tilde\lambda_0$ can be computed by implicit differentiation of $$\det(A(\rmi\ell,0) - \lambda)=\det A(\rmi\ell,0) +\lambda^2- \tr(A(\rmi\ell,0))\lambda = 0.$$ Since $\partial_\nu \det A(0,0) = 0$ we have $\tilde\lambda_0'(0) =0$ and therefore $$\tilde\lambda_0''(0) = -\frac{\partial_\nu^2 \det A(0,0)}{\tr\, A(0,0)}= \frac{1+\alpha^2}{\alpha r^2(\mu-k^2)}C,$$ where $D = (3k^2+\mu)\frac{(\beta/\alpha-h)^2}{(k^2-\mu)^2} + k^2-\mu$.
At $k=0$ (which means $\mu>\beta/\alpha-h)$ we have $D = (\beta/\alpha-h)^2/\mu-\mu = (m_3^2-1)\mu<0$ so that $\tilde\lambda_0''(0) <0$, which implies stable sideband. Since $k^2\neq \mu$, sideband instabilities are sign changes of $$\begin{aligned}
f(K) &:= (K-\mu)^2D = (3K+\mu)(\beta/\alpha-h)^2 + (K-\mu)^3\\
&= K^3 - 3\mu K^2 + 3(\mu^2+(\beta/\alpha-h)^2)K + \mu( -\mu^2 + (\beta/\alpha-h)^2),\end{aligned}$$ where $K=k^2$.
For $\mu>0$, $f$ is monotone increasing for positive $K$: $f'(0)>0$ and $f'\neq 0$ for real $K$ by direct calculation. Since $f(0) < 0$ and $f(\mu)>0$ (recall $\mu>\beta/\alpha-h$ for existence) there is precisely one real root to $f(k^2) = 0$ in $k^2<\mu$. Therefore, for all $\mu>0$ precisely one sideband instability occurs in $k^2<\mu$ for increasing $k^2$ from zero. In Figure \[f:wavetrainstab\] we plot the resulting stability region.
For later purposes note that the non-zero solutions $\nu$ to $$d_0(0,\nu)=\det A(\nu,0)=(1+\alpha^2)\nu^2(\nu^2+D)=0$$ change at the sideband instability $D=0$ from positive and negative real to a complex conjugate pair.
[cc]{}  and \[f:wt-ex\] with stable range of wavetrains in (a) bold line, and in (b) the dark shaded region.[]{data-label="f:wavetrainstab"}](wt-cyl-stab.eps "fig:"){width="40.00000%"} &
\
(a) & (b)
#### Hopf instability.
A Hopf instability occurs when the essential spectrum touches the imaginary axis at nonzero values, that is, there is $\gamma\neq0$ so that $d_0(\rmi\gamma,\rmi\ell)=0$. The imaginary part reads $(-2\alpha\ell^2 + \alpha r^2(k^2-\mu))\gamma=0$ and $\gamma\neq$ implies $2\ell^2 = r^2(k^2-\mu)$. Hence, there is no Hopf instability in the range $k^2<\mu$. Recall that wavetrains with $k^2>\mu$ are already unstable.
#### Turing instability.
A Turing instability occurs when the spectrum touches the origin for nonzero $\ell$, that is, there is $\ell\neq0$ so that $d_0(0,\rmi\ell)=0$, which means $\det A(\rmi\ell,0)=0$. Touching the origin means in addition $\partial_\ell \det A(\rmi\ell) = 0$. However, $\det A(\rmi\ell,0)=(1+\alpha^2)\ell^2(\ell^2 - D)$ for $D\neq 0$ away from sideband instabilities. Since $\ell^2 = D>0$ implies unstable sideband there are no relevant solutions.
#### Fold.
A fold means that a curve of spectrum crosses the origin upon parameter variation. However, the sideband calculation shows that there is always a unique curve of spectrum attached to the origin for $k^2\neq \mu$, which holds except at $h=\beta/\alpha$.
This exhausts the list of possible marginally stable spectral configuration and we conclude that, in analogy to the real Ginzburg-Landau equation, only sideband instabilities occur.
Taking into account that $k^2= \mu$ for a wavetrain can occurs only if $h=\alpha/\beta$ we thus have
\[t:wt\] All wavetrains whose wavenumber $k$ satisfies $k^2>\mu$ are unstable. For $\mu>0$ there is $k_*>0$ such that wavetrains with wavenumber $|k|<k_*$ are spectrally stable, while those with $|k|>k_*$ are unstable. In case $h\neq \alpha/\beta$ a sideband instability occurs at $k=\pm k_*$. There is no secondary instability for $k^2<\mu$. Nontrivial spectrally stable wavetrains exist only for supercritical anisotropy, $|h-\beta/\alpha|<\mu$.
Coherent structures {#s:coh}
===================
The coexistence of wavetrains and constant magnetizations raises the question how these interact. In this section we study solutions that spatially connect wavetrains in a coherent way. In order to locate such solutions induced by symmetry we make the ansatz $$\label{e:ansatz}
\begin{array}{rl}
\xi &= x - st\\
\varphi &= \phi(\xi) + \Omega t\\
\theta &= \theta(\xi),
\end{array}$$ with constant $s, \Omega$. Solutions of of this form are generalized travelling waves to with speed $s$ that have a superimposed oscillation about $\3$ with frequency $\Omega$. This ansatz is completely analogous to that used in the study of the real and complex Ginzburg-Landau equations. See, e.g., the review [@AransonKramer:02]. As for the existence and stability of wavetrains the results in this section near the instabilities of $\pm\3$ are analogous to those for the real Ginzburg-Landau equation. However, this is not the case globally.
Substituting ansatz into with $' = \rmd/\rmd \xi$ and $q=\phi'$ gives, after division by $\sin(\theta)$, the ODEs $$\label{e:llg-coh}
\left(\begin{array}{cc}
\alpha & -1\\
1 & \alpha
\end{array}\right)
\left(\begin{array}{c}
\sin(\theta) (\Omega-s q)\\
-s \theta'
\end{array}\right)
=
\left(\begin{array}{c}
2\cos(\theta)\theta' q + \sin(\theta)q' \\
-\theta'' + \sin(\theta)\cos(\theta)q^2
\end{array}\right)
+
\sin(\theta)
\left(\begin{array}{c}
\beta\\
h-\mu \cos(\theta)
\end{array}\right),$$ on the cylinder $(\theta,q)\in S^1\times \R$, which is the same as $\{(m_3,r,q)\in\R^3: m_3^2+r^2=1\}$.
#### Wavetrains.
Steady states with vanishing $\xi$-derivative correspond to the wavetrains discussed in §\[s:wt\] which have constant wavenumber $k=q=\rmd \phi/\rmd \xi$, frequency $\omega=sq - \Omega$ and amplitude $r$ in $m$-space. Notably, for $s\neq 0$, steady states of have the selected wavenumber $q$ given by $$\label{e:qeq}
q= \frac{\Omega-\beta/\alpha}{s}.$$ The $\theta$-value of steady states solves $\sin(\theta)=0$, i.e., $r=0$, or $\cos(\theta)(q^2-\mu) + h-\beta/\alpha=0$, where the latter is the same as the wavetrain existence condition with $k$ replaced by $q$. In other words, the ansatz for $s\neq 0$ removes all equilibria with wavenumber $k\neq\frac{\Omega-\beta/\alpha}{s}$.
#### Coherent structure ODEs.
Writing as an explicit ODE gives $$\label{e:ode}
\begin{array}{rl}
\theta' &= p\\
p' &= \sin(\theta) \left(h + (q^2-\mu) \cos(\theta) -(\Omega - sq) \right) + \alpha s p \\
q' &= \displaystyle \alpha (\Omega-sq) - \beta + \frac{s -2 \cos(\theta)q}{\sin(\theta)}p,
\end{array}$$ whose study is the subject of the following sections. For later use we also note the ‘desingularization’ by the (singular) coordinate change $p=\sin(\theta)\tp$ so that $\tp' = p'/\sin(\theta) - \tp^2 \cos(\theta)$, which gives $$\label{e:dode}
\begin{array}{rl}
\theta' &= \sin(\theta)\tp\\
\tp' &= h + (q^2-\mu) \cos(\theta) -(\Omega - sq) + \alpha s\tp - \cos(\theta)\tp^2 \\
q' &= \alpha (\Omega-sq) - \beta + (s -2 \cos(\theta)q)\tp.
\end{array}$$ Hence, is equivalent to except at zeros of $\sin(\theta)$. In particular, for $\tp=0$ the equilibria of are those of , but $\theta= n\pi$, $n\in \mathbb{Z}$ are invariant subspaces which may contain equilibria with $\tp\neq 0$.
Stationary coherent structures ($s=0$)
--------------------------------------
In this section we consider the case $s=0$ (which does not imply time-independence), where equations reduce to $$\label{e:ode-c0}
\begin{array}{rl}
\theta'' &= \sin(\theta) \left(h-\Omega + (q^2-\mu) \cos(\theta)\right) \\
q' &= \alpha \Omega- \beta -2\cot(\theta)\theta'q,
\end{array}$$ where $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.
In case $\Omega\neq\alpha/\beta$ there are no equilibria and it will be shown at the end of this section that there are no coherent structure-type solutions in that case. For $\Omega=\alpha/\beta$, system is invariant under the reflection $q\to -q$ so that $\{q=0\}$ is an invariant plane which separates the three dimensional phase space. In particular, *there cannot be connections between equilibria (=wavetrains) with opposite signs of $q$, that is, sign reversed spatial wavenumbers*.
### Homogeneous solutions ($q=0$)
![Phase plane streamplots of with <span style="font-variant:small-caps;">Mathematica</span> ($\theta$ horizontal, $\theta'$ vertical axes). (a)-(c) have $h-\Omega=1/2$. (a) supercritical anisotropy ($\mu=1$), (b) subcritical ($\mu=0$), (c) subsubcritical ($\mu=-1$), (d) degenerate subsubcritical case $h=\Omega$, $\mu=-1$.[]{data-label="f:phaseplane-c0-q0"}](coh-c0-q0-hOm0p5-mu1 "fig:"){width="24.00000%"} ![Phase plane streamplots of with <span style="font-variant:small-caps;">Mathematica</span> ($\theta$ horizontal, $\theta'$ vertical axes). (a)-(c) have $h-\Omega=1/2$. (a) supercritical anisotropy ($\mu=1$), (b) subcritical ($\mu=0$), (c) subsubcritical ($\mu=-1$), (d) degenerate subsubcritical case $h=\Omega$, $\mu=-1$.[]{data-label="f:phaseplane-c0-q0"}](coh-c0-q0-hOm0p5-mu0 "fig:"){width="24.00000%"} ![Phase plane streamplots of with <span style="font-variant:small-caps;">Mathematica</span> ($\theta$ horizontal, $\theta'$ vertical axes). (a)-(c) have $h-\Omega=1/2$. (a) supercritical anisotropy ($\mu=1$), (b) subcritical ($\mu=0$), (c) subsubcritical ($\mu=-1$), (d) degenerate subsubcritical case $h=\Omega$, $\mu=-1$.[]{data-label="f:phaseplane-c0-q0"}](coh-c0-q0-hOm0p5-mu-1 "fig:"){width="24.00000%"} ![Phase plane streamplots of with <span style="font-variant:small-caps;">Mathematica</span> ($\theta$ horizontal, $\theta'$ vertical axes). (a)-(c) have $h-\Omega=1/2$. (a) supercritical anisotropy ($\mu=1$), (b) subcritical ($\mu=0$), (c) subsubcritical ($\mu=-1$), (d) degenerate subsubcritical case $h=\Omega$, $\mu=-1$.[]{data-label="f:phaseplane-c0-q0"}](coh-c0-q0-hOm0-mu-1 "fig:"){width="24.00000%"}\
(a)(b)(c)(d)
Solutions in the invariant set $\{q=0\}$ have the form $m(\xi) = r(\xi)\exp(\rmi t \Omega )$ and turns into a second order ODE on the circle $\{m_3^2+r^2=1\}$. The ODE for $\theta$ from is given by the nonlinear pendulum equation $$\label{e:ode-c0-q0}
\theta'' = \sin(\theta) \left(h-\Omega-\mu \cos(\theta)\right),$$ which is invariant under $\theta\to-\theta$ and is Hamiltonian with potential energy $$P_0(\theta)=\cos(\theta)(h-\Omega- \frac\mu 2 \cos(\theta)).$$ The symmetry applies and we therefore assume in the following that $\Omega=\beta/\alpha <h$.
We plot the qualitatively different vector fields of in Figure \[f:phaseplane-c0-q0\] and some profiles in Figure \[f:profiles-c0-q0\]. Coherent structure solutions are completely characterized as in the figures, which we formulate next explicitly for the original PDE with $\m=(m,m_3)$, $m=r \rme^{\rmi\varphi}$, $r=\sin(\theta)$, $m_3 = \cos(\theta)$. Homoclinic profiles may be interpreted as (dissipative) solitons. The heteroclinic connections in item 2(a) can be viewed as (dissipative) solitons with ‘phase slip’.
\[t:coh-c0-q0\] Let $s=0$ and $\Omega=\alpha/\beta$ and consider solutions to of the form with $\varphi$ constant in $\xi$, i.e., $q=0$. These oscillate in time pointwise about the $\3$-axis with frequency $\Omega=\beta/\alpha$. Assume without loss of generality, due to , that $h>\Omega$.
1. Consider subcritical anisotropy $h-\Omega >|\mu|>0$. There exist no nontrivial wavetrains with $q=0$, and the coherent structure solutions with $q=0$ are a pair of homoclinic profiles to $\hat e_3$, and three one-parameter families of periodic profiles, one bounded and two semi-unbounded. The homoclinic profiles each cross once through $-\3$ in opposite $\theta$-directions. The limit points of the bounded curve of periodic profiles are $-\3$ and the union of homoclinic profiles. Each of the homoclinics is the limit point of one of the semi-unbounded families, each of which has unbounded $\theta$-derivatives. The profiles from the bounded family each cross $-\3$ once during a half-period, the profiles of the unbounded family cross both $\pm\3$ during one half-period.
2. Suppose super- or subsubcritical anisotropy $|\mu|>h-\Omega$. There exists a wavetrain with $k=0$, which is stable in the supercritical case ($\mu>0$) and unstable in the subsubcritical case ($\mu<0$). In this appears in the form of two equilibria being the symmetric pair of intersection points of the wavetrain orbit and a meridian on the sphere, phase shifted by $\pi$ in $\varphi$-direction. Details of the following can be read off Figure \[f:phaseplane-c0-q0\] analogous to item 1.
1. In the supercritical case ($\mu>h-\Omega$) the coherent structure solutions with $q=0$ are two pairs of heteroclinic connections between the wavetrain and its phase shift, and four curves of periodic profiles; two bounded and two semi-unbounded.
2. In the subsubcritical case ($\mu<h-\Omega$) the coherent structure solutions with $q=0$ are two pairs of homoclinic connections to $\pm\3$, respectively, and five curves of periodic profiles, three bounded and two semi-unbounded.
3. The degenerate case $h=\Omega$, $\mu<0$ is the only possibility for profiles of stationary coherent structures to connect between $\pm\3$, which then come in a pair as in the corresponding panel of Figure \[f:phaseplane-c0-q0\]. The remaining coherent structures with $q=0$ are analogous to the supercritical case with $\pm\3$ and the pair of wavetrain and its phase shift interchanged.
As for wavetrains discussed in §\[s:wt\], the condition $\cos(\theta) = (h-\Omega)/\mu$ yields the existence criterion $|h-\Omega|<|\mu|$ for an equilibrium to in $(0,\pi)$. The derivative of the right hand side of at $\theta=0$ is $h-\Omega-\mu$, which dictates the type of all equilibria and only saddles generate heteroclinic or homoclinic solutions.
It remains to study the connectivity of stable and unstable manifolds of saddles, which is given by the difference in potential energy $P_0(\theta)$. Since $P_0(0)-P_0(\pi) = 2(h-\Omega)$ the claims follow.
### Non-homogeneous solutions ($q\neq 0$)
+:---------------------------------:+:---------------------------------:+
|  | Figure \[f:wavetrainstab\](a) |
| with solutions to for | with solutions to for |
| $\theta\in(0,\pi)$ and $C=0.1$ | $\theta\in(0,\pi)$ and $C=0.1$ |
| (thick solid line), and $C=0.4$ | (thick solid line), and $C=0.4$ |
| (thick dashed line). (b) Upper | (thick dashed line). (b) Upper |
| panel: potential $P(\theta)$ for | panel: potential $P(\theta)$ for |
| parameters as in (a) and $C=0.1$. | parameters as in (a) and $C=0.1$. |
| Lower panel: same with | Lower panel: same with |
| $C=0.4$.[]{data-label="f:wt-cyl-q | $C=0.4$.[]{data-label="f:wt-cyl-q |
| "}](wt-cyl-q){width="\textwidth"} | "}](coh-c0-pot0p1-hOm-0p5-mu1 "fi |
| | g:"){width="100.00000%"}\ |
| |  |
| | with solutions to for |
| | $\theta\in(0,\pi)$ and $C=0.1$ |
| | (thick solid line), and $C=0.4$ |
| | (thick dashed line). (b) Upper |
| | panel: potential $P(\theta)$ for |
| | parameters as in (a) and $C=0.1$. |
| | Lower panel: same with |
| | $C=0.4$.[]{data-label="f:wt-cyl-q |
| | "}](coh-c0-pot0p4-hOm-0p5-mu1 "fi |
| | g:"){width="100.00000%"} |
+-----------------------------------+-----------------------------------+
| (a) | (b) |
+-----------------------------------+-----------------------------------+
In order to study for $q\neq 0$, we note the following first integral. Since $\Omega=\beta/\alpha$, the equation for $q$ can be written as $$(\log|q|)' = -2(\log|\sin(\theta)|)',$$ and therefore explicitly integrated. With integration constant $C = \sin(\theta(0))^2 |q(0)|$ this gives $$\label{e:coh-c0-q}
q = \frac{C}{\sin(\theta)^2}.$$ Substituting this into the equation for $\theta$ yields the nonlinear pendulum $$\label{e:ode-c0-red}
\theta'' = \sin(\theta) \left(h-\Omega-\mu \cos(\theta)\right) + C^2\frac{\cos(\theta)}{\sin(\theta)^3},$$ with singular potential energy $$P(\theta) = P_0(\theta) + \frac 1 2 C^2 \cot(\theta)^2.$$ The energy introduces barriers at multiples of $\pi$ so that solutions for $q\neq 0$ cannot pass $\pm\hat e_3$, and as $C$ increases the energy landscape becomes qualitatively independent of $\Omega, h, \mu$.
As an immediate consequence of the energy barriers and the fact that only stable wavetrains are saddle points we get
\[t:coh-c0\] Let $s=0$ and $\Omega=\alpha/\beta$. Consider solutions to of the form . Intersections in $(q,\theta)$-space of the curve $\mathcal{C}$ given by with the wavetrain existence curves $\mathcal{W}$ from (with $k$ replaced by $q$) are in one-to-one correspondence with equilibria of .
Consider supercritical anisotropy $0\leq h-\Omega < \mu$ and assume $C$ is such that $\mathcal{C}$ transversely intersects the component of $\mathcal{W}$ which intersects $\{q=0\}$. Then the intersection point with smaller $q$-value corresponds to a spectrally stable wavetrain, and in there is a pair of homoclinic solutions to this wavetrain. All other intersection points are unstable wavetrains.
All other non-equilibrium solutions of the form with $s=0$ are periodic in $\xi$, and this is also the case for all other parameter settings.
The homoclinic orbit is a soliton-type solution to with asymptotic state a wavetrain. See Figure \[f:cohex\] for a nearby solution with periodic (reduced profile.
Notably, the tangential intersection of $\mathcal{C}$ and $\mathcal{W}$ is at the sideband instability.
In Figure \[f:wt-cyl-q\] we plot an illustration in case of supercritical anisotropy $\mu>\Omega-h>0$. In the upper panel of (b) the local maxima each generate a pair of homoclinic solutions to the stable wavetrain it represents. The values of $q$ that it visits lie on the bold curve in (a), whose intersections with the curve of equilibria are the local maxima and minima. The lower panel in (b) has $C=0.4$ and the local maxima disappeared. All solutions are periodic and lie on the thick dashed curve in (a).
### Absence of wavetrains ($\Omega\neq \alpha/\beta$)
In this case the first integral $Q= \log(|q|\sin(\theta)^2)$ is monotone, $$Q' = \frac{\Omega-\alpha/\beta}{q} \neq 0,$$ and therefore $|q|$ is unbounded as $\xi\to\infty$ or $\xi\to-\infty$. In view of , we also infer that $\theta$ oscillates so that there are no relevant solutions of the form .
Moving coherent structures
--------------------------
Using the desingularized system , we prove existence of some non-stationary coherent structures.
### Fast front-type coherent structures ($|s|\gg 1$)
In this section we prove existence of fast front-type coherent structures with at least one asymptotic state of the profile being the constant up- or down-magnetization $\pm\3$.
As a first step, it is convenient to consider such solutions in the desingularized system .
\[t:coh-fast\] For any bounded set of $(\alpha, \beta, \mu, \Omega_0, \Omega_1)$ there exists $s_0>0$ and neighborhood $U$ of $M_0:=\{\theta\in[0,\pi], \tp=q-\Omega_1=0\}$ such that for all $|s|\geq s_0$ the following holds for with $\Omega=\Omega_0 + \Omega_1 s$. There exists a pair of heteroclinic trajectories $(\theta_{\pm h}, \tp_{\pm h}, q_{\pm h})(\xi)$ in $U$ with $\frac{\rmd}{\rmd\xi}\theta_h(\xi) = O(s^{-1})$, and $\lim_{\xi\to -\infty}\theta_{-h}(\tau \xi) = 0$, $\lim_{\xi\to \infty}\theta_{+h}(\tau \xi) = \pi$, where $\tau = \mathrm{sgn} (s(h-\beta/\alpha -\mu))$.
The local wavenumbers are always nontrivial, $q_{\pm h}\not\equiv 0$, and the heteroclinics form a smooth family in $s^{-1}$ with $$\lim_{|\xi|\to\infty} q_{\pm h}(\xi) =\frac 1 s \left(\frac{h+\alpha\beta\mp\mu}{1+\alpha^2}-\Omega\right) + O(s^{-2}).$$
On the spatial scale $\eta = \xi/s$ the heteroclinics solve, to leading order in $s^{-1}$, the ODE $$\label{e:superslow}
\frac{\rmd}{\rmd\eta} \theta = \frac{\alpha}{1+\alpha^2}\sin(\theta)\left(h-\frac{\beta}{\alpha}+(\Omega_1^2-\mu)\cos(\theta)\right).$$
Before the proof we note the consequences of this for coherent structures in .
\[c:fast\] The heteroclinic solutions of Theorem \[t:coh-fast\] are in one-to-one correspondence with heteroclinic solutions to that lie in $U$ and connect $\theta=0,\pi$ and/or the possible equilibrium at $\theta_1:=\arccos((h-\beta/\alpha)/\mu)$. For $\theta\in(0,\pi)$ all properties carry over to with the bijection given by $p=\sin(\theta)\tp$.
Some remarks are in order.
- The ODE is the spatial variant of the temporal heteroclinic connection in : setting all space derivatives to zero, the $\theta$-equation of reads $$-\partial_t \theta = \frac{\alpha}{1+\alpha^2}\sin(\theta)\left(h-\frac{\beta}{\alpha}-\mu\cos(\theta)\right),$$ which is with $\mu$ replaced by $\Omega_1^2-\mu$ and up to possible direction reversal. Since $\Omega_1=q$ on the slow manifold (i.e. at leading order), the reduced flow equilibria reproduce the wavetrain existence condition . This kind of relation between temporal dynamics and fast travelling waves formally holds for any evolution equation in one space dimension.
- For increasing speeds these solutions are decreasingly localized, hence far from a sharp transition. For $\Omega_1\neq 0$ the azimuthal frequency increases with $s$.
- The uniqueness statement is limited, since in the $(\theta,p,q)$-coordinates the neighborhood $U$ from the theorem is ‘pinched’ near $\theta=0,\pi$: a uniform neighborhood in $(\theta,\tp)$ has a sinus boundary in $(\theta,p)$.
Concerning stability, we infer from Lemma \[l:conststab\] and Theorem \[t:wt\] that none of these fronts can be stable in an $L^2$-sense: always one of the asymptotic states is unstable. In the supercritical case $\pm\3$ are unstable, in the subcritical case one of $\pm\3$ is unstable and in the subsubcritical case the wavetrain with $\theta=\theta_1\in(0,\pi)$ is unstable. However, it is not ruled out that these fronts are stable in a suitable weighted sense as invasion fronts into an unstable state, which we do not pursue further in this paper.
Let us set $s=\eps^{-1}$ so that the limit to consider is $\eps\to 0$. Since we will rescale space with $\eps$ and $\eps^{-1}$ this means sign changes of $s$ reflect the directionality of solutions.
The existence proof relies on a geometric singular perturbation argument and we shall use the terminology from this theory, see [@Fenichel; @Jones], and also sometimes suppress the $\eps$-dependence of $\theta,\tp,q$.
Upon multiplying the $\tp$- and $q$-equations of by $\eps$ we obtain the, for $\eps\neq 0$ equivalent, ‘slow’ system $$\label{e:s-dode}
\begin{aligned}
\theta' &= \sin(\theta)\tp\\
\eps\tp' &= \alpha \tp + q - \Omega_1 + \eps(h + (q^2-\mu) \cos(\theta) -\Omega_0 - \cos(\theta)\tp^2) \\
\eps q' &= \tp - \alpha (q- \Omega_1) + \eps(\alpha\Omega_0 - \beta -2 \cos(\theta)q \tp).
\end{aligned}$$
Setting $\eps=0$ gives the algebraic equations $$A\begin{pmatrix}\tp\\q\end{pmatrix}=-\Omega_1\begin{pmatrix}-1\\\alpha\end{pmatrix}, \;
A =
\begin{pmatrix}
\alpha & 1\\
1 & - \alpha
\end{pmatrix}.$$ Since $\det A = -(1+\alpha^2)<0$ the unique solution is $\tp=q-\Omega_1=0$ and thus the ‘slow manifold’ is $M_0$ as defined in the theorem, with ‘slow flow’ given by $$\theta' = \sin(\theta)\tp.$$ Since $\tp = 0$ at $\eps=0$, $M_0$ is a manifold (a curve) of equilibria at $\eps=0$, so that the slow flow is in fact ‘superslow’ and will be considered explicitly below. Since the slow manifold is one-dimensional (and persists for $\eps>0$ as shown below) it suffices to consider equilibria for $\eps>0$. These lie on the one hand at $\theta=\theta_0\in\{0,\pi\}$, if $$A\begin{pmatrix}\tp\\q\end{pmatrix} + \Omega_1\begin{pmatrix}-1\\\alpha\end{pmatrix} + \eps F(\tp,q)=0\;,\quad F(\tp,q) :=
\begin{pmatrix}
h + \sigma(q^2-\mu) -\Omega_0 - \sigma\tp^2 \\
\alpha\Omega_0 - \beta -2 \sigma q\tp
\end{pmatrix},$$ where $\sigma = \cos(\theta_0)\in \{-1,1\}$. Since $\det A = -(1+\alpha^2)<0$ the implicit function theorem provides a locally unique curve of equilibria $(\tp_\eps,q_\eps)$ for sufficiently small $\eps$, where $$\left.\frac{\rmd}{\rmd\eps}\right|_{\eps=0}
\begin{pmatrix}
\tp_\eps\\
q_\eps
\end{pmatrix}
= -A^{-1}F(0,0) = -A^{-1}
\begin{pmatrix}
h-\sigma\mu-\Omega_0\\
\alpha\Omega_0 - \beta
\end{pmatrix}.$$ This proves the claimed location of asymptotic states.
On the other hand, for $\theta\neq 0$ system is equivalent to . From the previous considerations of equilibria (=wavetrains) we infer that the unique equilibria in an $\eps$-neighborhood of $M_0$ are those at $\theta=\theta_0$, $(\tp,q) = (\tp_\eps,q_\eps)$ together with the single solution $\theta_1$ to , if it exists, with $k$ replaced by $q=\eps(\Omega_0-\beta/\alpha)$.
Towards the persistence of $M_0$ as a perturbed invariant manifold for $|\eps|>0$, let us switch to the ‘fast’ system by rescaling the time-like variable to $\zeta = \xi/\eps$. With $\dot \theta = \rmd \theta /\rmd \zeta$ etc., this gives $$\label{e:f-dode}
\begin{aligned}
\dot\theta &= \eps\sin(\theta)\tp\\
\dot\tp &= \alpha \tp + q-\Omega_1 + \eps(h + (q^2-\mu) \cos(\theta) -\Omega_0 - \cos(\theta)\tp^2) \\
\dot q &= \tp - \alpha (q-\Omega_1) + \eps(\alpha\Omega_0 - \beta -2 \cos(\theta)q \tp).
\end{aligned}$$ Note that $M_0$ is (also) a manifold of equilibria at $\eps=0$ in this system and the linearization of in $M_0$ for transverse directions to $M_0$ is given by $A$. Since the eigenvalues of $A$, $\pm\sqrt{(1+\alpha^2)}$, are off the imaginary axis, $M_0$ is normally hyperbolic and therefore persists as an $\eps$-close invariant one-dimensional manifold $M_\eps$, smooth in $\eps$ and unique in a neighborhood of $M_0$. See [@Fenichel]. The aforementioned at least two and at most three equilibria lie in $M_\eps$, and, $M_\eps$ being one-dimensional, these must be connected by heteroclinic orbits.
For the connectivity details it is convenient to derive an explicit expression of the leading order flow. We thus switch to the superslow time scale $\eta = \eps\xi$ and set $\op = \tp/\eps$, $\oq=(q-\Omega_1)/\eps$, which changes to (subdindex $\eta$ means $\rmd/\rmd\eta$) $$\label{e:ss-dode}
\begin{aligned}
\theta_\eta &= \sin(\theta)\op\\
\eps\op_\eta &= \alpha \op + \oq + h +(\Omega_1^2-\mu)\cos(\theta) -\Omega_0 + \eps\cos(\theta)(\eps (\oq^2- \op^2) +2\oq\Omega_1) \\
\eps \oq_\eta &= \op - \alpha \oq + \alpha\Omega_0 - \beta -2\eps \cos(\theta)(\eps \oq \op + \Omega_1\op).
\end{aligned}$$ At $\eps=0$, solving the algebraic equations for $\op$ provides the flow for $\theta$ given by $$A^{-1}
\begin{pmatrix}
h+(\Omega_1^2-\mu)\cos(\theta)-\Omega_0\\
\alpha\Omega_0-\beta
\end{pmatrix}
= \frac{1}{1+\alpha^2}
\begin{pmatrix}
\alpha h- \alpha(\Omega_1^2-\mu)\cos(\theta) - \beta\\
h - (\Omega_1^2-\mu)\cos(\theta) - (1+\alpha^2)\Omega_0+ \alpha\beta.
\end{pmatrix}$$ so that the leading order superslow flow on the invariant manifold is given by .
Recall that and are equivalent for $\theta\in(0,\pi)$. Next, we note that the limit of the vector field of along such a heteroclinic from is zero by construction. Hence, for each of the heteroclinic orbits in of Theorem \[t:coh-fast\], there exist a heteroclinic orbit in in the sense of the corollary statement.
### Small amplitude moving coherent structures
In this section we consider all possible small amplitude coherent structures for speeds $$\begin{aligned}
\label{e:lows}
s^2>\frac{4q^2}{1+\alpha^2},\end{aligned}$$ where $q\neq 0$ is an equilibrium $q$-value . Small amplitude means $q$ must lie near a bifurcation point, which are the intersection points of the solution curves from with $\theta=\theta_0=0,\pi$. This gives $$\label{e:eps0}
\cos(\theta_0) \left(q^2-\mu\right) = \frac{\beta}{\alpha} - h$$ and is only possible for super- and subcritical anisotropy.
We show that such coherent structures must be of front-type. Recall that for $s=0$ there are also coherent structures with periodic and homoclinic (soliton-type) profiles. In the previous section we found front-type solutions when $h-\beta/\alpha$ enters $(-\mu,\mu)$, that is, the anisotropy changes from subcritical to super- or subsubcritical, and when the speed $s$ is above a theoretical threshold. In this section, we give an explicit bound for this threshold in the small amplitude limit. We shall prove that for speeds satisfying , these points are pitchfork-type bifurcations in , which give rise to front-type coherent structures. As in the previous section, we locate such solutions in from an analysis of . Remark that in the PDE the pitchfork-type bifurcation corresponds to a Hopf-type bifurcation due to the superimposed oscillation about the $\3$-axis.
\[t:coh-small\] Let $S_\eps=(\alpha_\eps,\beta_\eps, h_\eps,\mu_\eps,\Omega_\eps,s_\eps)$ be a curve in the parameter space of with $\eps\in(-\eps_0,\eps_0)$ for some $\eps_0>0$ sufficiently small such that $\alpha_\eps>0$ and $q_\eps, S(\eps)$ solve , with $q_\eps$ strictly monotone increasing in $\eps$, and at $\eps=0$.
Then with parameters $S_\eps$ has equilibria at $(\theta,\tp,q)=(\theta_0,0,q_\eps)$ and at $\eps=0$ the following occur.
1. There is a pitchfork-type bifurcation in case $q_0\neq\mu_0$ and $\theta_0=0$ or $\pi$: If $q_0>\mu_0$ ($q_0<\mu_0$) two equilibria bifurcate as $\eps$ changes from negative to positive (positive to negative), which are connected to $(\theta_0,0,q_\eps)$ by a heteroclinic connection, respectively. These are unique in a neighborhood $U$ of $(\theta_0,0,q_0)$.
The heteroclinics converge to $(\theta_0,0,q_\eps)$ as $\mathrm{sgn}(s(q^2-\mu))\xi\to \infty$ so that a bifurcation in $\{q^2<\mu\}$ is supercritical and subcritical in $\{q^2>\mu\}$.
These bifurcations are not generic in the coordinates of , because its linearization in equilibria with $\theta=0$ or $\pi$ always has a kernel.
2. There is no local bifurcation in case $q_0-\mu_0=h_0-\beta_0/\alpha_0=0$ and $\theta_0=\pi/2$: for $\eps\neq 0$ there is at most one equilibrium and there are no small amplitude solutions to in its vicinity.
Analogous to Corollary \[c:fast\] we have
\[c:coh-small\] The heteroclinic solutions of Theorem \[t:coh-small\] are in one-to-one correspondence with heteroclinic solutions to , connecting to $\theta=0$ or $\theta=\pi$ in $U$. Bounded solutions for $\theta\not\in\{0,\pi\}$ are also in one-to-one correspondence.
The existence analysis of equilibria already implies the emergence of two symmetric equilibria under the stated conditions in item (1), and that there is at most one in item (2). Hence, it remains to show that the center manifold associated to the bifurcation is one-dimensional and to obtain the directionality of heteroclinics.
For the former it suffices to show that the linearization at the bifurcation point has only a simple eigenvalue on the imaginary axis, namely at zero. The linearization of in any point with $\tp=0$ (in particular for all relevant equilibria) gives the $3\times 3$ matrix $$\tilde A=
\left(\begin{array}{c|c}
0 & \sin(\theta)\; 0\\
\hline
\begin{array}{c}(\mu-q^2)\sin(\theta)\\0\end{array} & B
\end{array}\right), \;
B =
\left(\begin{array}{cc}
\alpha s & 2q \cos(\theta)-s\\
s-2q\cos(\theta) & \alpha s
\end{array}\right).$$ In particular, for $\theta=\theta_0=0,\pi$ and for $\mu_0=q_0^2$, the matrix $\tilde A$ has a kernel with eigenvector $(1,0,0)^t$. The remaining eigenvalues are those of $B$. Since $B$ has vanishing trace, its eigenvalues are either a complex conjugate pair (for $\det B>0$) or a pair of sign reversed real numbers (for $\det B<0$) or a double zero ($\det B=0$). Since $\det B=4q^2\cos(\theta_0)^2-(1+\alpha^2)s^2$ the constraints in the theorem statement imply a simple zero eigenvalue and no further eigenvalues on the imaginary axis.
The simple eigenvalue on the imaginary axis implies existence of a one-dimensional center manifold that includes all equilibria near $(\theta_0,0,q_0)$ for nearby parameters.
In case (1) the uniqueness of bifurcating equilibria on either side of the symmetry axes and invariance of the one-dimensional center manifold implies presence and local uniqueness of the claimed heteroclinic connections. In order to determine the directionality of these, we compute the leading order correction to the eigenvalues of $\tilde A$ for $\theta=\theta_0+\delta$. The straightforward calculation yields the eigenvalue of the bifurcation branch as $$\frac{\alpha}{(1+\alpha^2)s}(q^2-\mu)\delta^2 + \calO(\delta^4),$$ which gives the claimed directionality.
[^1]: Lehrstuhl I für Mathematik and JARA-FIT, RWTH Aachen University, 52056 Aachen, Germany, melcher@rwth-aachen.de
[^2]: Fachbereich 3 – Mathematik, Universität Bremen, Postfach 33 04 40, 28359 Bremen, Germany, rademach@math.uni-bremen.de
[^3]: original form $\dot{\m} =- \m \times \boldsymbol{h}_{\rm eff} - \lambda \, \m \times \m \times \boldsymbol{h}_{\rm eff} $ with Landau-Lifshitz damping $\lambda>0$
[^4]: $-\m \times \m \times \boldsymbol{\xi} = (\boldsymbol{1}-\m \otimes \m) \boldsymbol{\xi}$ is the orthogonal projection of $\boldsymbol{\xi}$ onto the tanget space $T_{\m}\mathbb{S}^2$ at $\m$.
|
---
abstract: 'We investigate how hierarchical models for the co-evolution of the massive black hole (MBH) and AGN population can reproduce the observed faint X-ray counts. We find that the main variable influencing the theoretical predictions is the Eddington ratio of accreting sources. We compare three different models proposed for the evolution of AGN Eddington ratio, $f_{\rm Edd}$: constant $f_{\rm Edd}=1$, $f_{\rm Edd}$ decreasing with redshift, and $f_{\rm Edd}$ depending on the AGN luminosity, as suggested by simulations of galactic mergers including MBHs and AGN feedback. We follow the full assembly of MBHs and host halos from early times to the present in a $\Lambda$CDM cosmology. AGN activity is triggered by halo major mergers and MBHs accrete mass until they satisfy the observed correlation with velocity dispersion. We find that all three models can reproduce fairly well the total faint X-ray counts. The redshift distribution is however poorly matched in the first two models. The Eddington ratios suggested by merger simulations predicts no turn-off of the faint end of the AGN optical luminosity function at redshifts $z\ga1$, down to very low luminosity.'
author:
- |
Marta Volonteri$^{1}$, Ruben Salvaterra$^{2}$ & Francesco Haardt$^{2}$\
$^{1}$Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK\
$^{2}$Dipartimento di Fisica e Matematica, Universitá dell’Insubria, Via Valleggio 11, 22100 Como, Italy
bibliography:
- 'xrc.bib'
title: 'Constraints on the Accretion History of Massive Black Holes from Faint X-ray Counts'
---
cosmology: theory – black holes – galaxies: evolution – quasars: general
Introduction
============
Several hierarchical models for the evolution of the MBH and AGN populations [see,e.g., @haehnelt1993; @haiman2000; @hatziminaoglou2001; @wyithe2003; @Cattaneo1999; @kauffmann2000; @cavaliere2000; @VMH; @Granatoetal; @lapi2006] have proved successful in reproducing the AGN optical luminosity function (OLF) in a large redshift range ($1\la z\la6$). Typically, these models assume that AGN activity is triggered by major mergers. Galactic interactions trigger gas inflows, and the cold gas may be eventually driven into the very inner regions, fueling an accretion episode and the growth of the nuclear MBH. Hydrodynamic simulations of major mergers have shown that a significant fraction of the gas in interacting galaxies falls to the center of the merged system [@mihos94; @mihos96]: the cold gas may be eventually driven into the very inner regions, fueling an accretion episode and the growth of the nuclear BH. This last year has been especially exciting, as the first high resolution simulations of galactic mergers including BHs and AGN feedback, showed that the merger scenario is generally correct [@Springel2005; @dimatteo2005]. In hierarchical models of galaxy formation major mergers are responsible for forming bulges and elliptical galaxies. Support for merger driven activity therefore comes from the observed correlation between bulge luminosity - or stellar velocity dispersion - and black hole mass, suggesting a single mechanism for assembling black holes and forming spheroids in galaxy halos [@magorrian1998; @Marconietal2004; @Gebhardt2000; @ferrarese2000].
Notwithstanding the success at high redshift, hierarchical models struggle to match the AGN OLF at low redshift ($z\la1$) by overpredicting the bright end, and underpredicting the faint end of the OLF, as the decrease of the halo merger rate with time is less dramatic than the observed fall of the AGN population. The overprediction of bright AGN can be imputed to inefficient cooling in large halos. Imposing an upper limit to the AGN host halo mass ($\sim10^{13.5} M_{\odot}$, [@wyithe2003; @Marulli2006]) in fact significantly improves the match at the bright end of the OLF.
The under-abundance of faint AGN can be instead attributed to the assumption, common to most models, of very efficient accretion, at rates close to the Eddington rate. [@merloni2003] and [@merloni2004] have shown that low-redshift AGN are probably accreting inefficiently, i.e. both at an accretion rate much smaller than the Eddington rate and with a low radiative efficiency. These considerations suggest that successful predictions for the evolution of AGN luminosity should include more sophisticated models for accretion. A first step in this direction can be taken by considering the results of the recent merger simulations, which track also accretion on a central MBH. Although these simulations lack the necessary resolution for resolving the accretion process in the vicinity of the MBH, empirical models [@hopkins2005a], based on coupling results from the above simulations with the observed LF in the hard X-ray band (HXLF), have been shown to reproduce simulaneously several optical and X-ray observations. The [@hopkins2005a] empirical models, however, are not embedded in a cosmological evolutionary framework, that is they derive the MBH population properties at a given time, but not how the population of black holes at an earlier time evolves into the MBHs present at a later time. From the observed HXLF [@hopkins2005a] derive the rate at which AGN of a given luminosity at the peak of activity must be created. This information is then used as a proxy for the galaxy merger rate which should provide the boundary conditions for determining the evolution of the MBH population.
We here couple the predictions from [@hopkins2005a] with the merger rate expected in the currently favoured cold dark matter scenario. The main novelties of the present investigation with respect to [@hopkins2005a], are therefore that (i) the rate of mergers, which trigger AGN activity, is directly derived in the currently favoured cold dark matter (CDM) cosmology. In principle, the empirical merger rate derived by [@hopkins2005a] is not granted to correspond to the CDM one. Also, (ii), we grow MBHs in a self-consistent way, that is we trace the whole accretion history of MBHs from early times to the present, requiring continuity in the population.
Our aim is to investigate here to which extent hierarchical models, coupled with the prescriptions based on the above simulations (§1), can reproduce the low-redshift evolution of AGN. We show that reproducing the HXLF is a necessary, but not sufficient, condition for matching the redshift distribution of faint X-ray counts (§2). We identify the redshift distribution of faint X-ray counts as the most sensitive observational result to discriminate between models (§3, §4). Finally, in §5 we summarize the results and discuss their implications.
Formation of massive black holes and growth by mass accretion
=============================================================
In our framework pregalactic ‘seed’ holes form at early times. In most of our calculations we follow [@VHM; @Volonterietal2005], assuming that seed MBHs form with intermediate masses ($m_{\rm seed}\lid 600\,{\,{\rm M_\odot}}$) in halos collapsing at $z=20$ from rare 3.5-$\sigma$ peaks of the primordial density field [@MadauRees2001] as end-product of the very first generation of stars. The assumed ‘bias’ assures that almost all halos above $10^{11}\,{\,{\rm M_\odot}}$ actually host a BH at all epochs. We also check the influence of the initial conditions by considering seed MBH formation as in [@kous]. In this model seed MBH form from the low angular momentum tail of material in halos with efficient gas cooling. In first approximation, seed MBH form in halos with mass above the threshold $M_H\simeq10^7{\,{\rm M_\odot}}(1+z/18)^{-3/2}$, with a mass $m_{\rm seed}\simeq5\times10^4{\,{\rm M_\odot}}(M_H/10^7{\,{\rm M_\odot}})(1+z/18)^{3/2}$ [@kous]. We have dropped here the dependency on the halo spin parameter and gas fraction, as we are not interested in the detailed seed formation process, but only in testing an alternative model for seed formation which predicts much larger seed masses.
Nuclear activity is triggered by halo mergers: in each major merger the hole in the more massive halo accretes gas until its mass scales with the fifth power of the circular velocity of the host halo with a normalization which reproduces the observed local correlation between MBH mass and velocity dispersion ($m_{\rm BH}-\sigma_*$ relation).
The rate at which mass is accreted scales with the Eddington rate for the MBH. In model I the accretion rate is set exactly to be the Eddington rate. Defining the Eddington ratio as $f_{\rm Edd}=\dot {M}/\dot{M}_{\rm Edd}$, model I has a constant $f_{\rm Edd}=1$. [@Shankar2004] suggest that if the Eddington ratio evolves with redshift the MBH mass function derived from a deconvolution of the AGN LF agrees better with the local MBH mass function [@aller2002; @Shankar2004; @Marconietal2004]. [@Shankar2004] suggest the following parameterization: $$\label{eq:shankar1}
f_{\rm Edd}(z)=\left\{
\begin{array}{ll}
f_{\rm Edd,0} & z\geqslant3 \\
f_{\rm Edd,0}[(1+z)/4]^{1.4} & z<3
\end{array}
\right.$$ with $f_{\rm Edd,0}=0.3$. Face value, Equation \[eq:shankar1\] underpredicts the LFs of AGN at high redshift, in our framework. As we start from small high redshift seeds, for our model II we modify Equation \[eq:shankar1\] as follows:
$$\label{eq:shankar2 }
f_{\rm Edd}(z)=\left\{
\begin{array}{ll}
1 & z\geqslant6 \\
0.078(1+z)^2-0.623(1+z)+1.545 & 3\lid z<6 \\
f_{\rm Edd,0}[(1+z)/4]^{1.4} & z<3,
\end{array}
\right.$$
where the quadratic form smoothly joins the $z<3$ and $z>6$ functional forms.
Finally, we test models where the Eddington ratio scales with the AGN luminosity. In model IIIa we parameterize the accretion rate following [@hopkins2005b]. The results of simulations are presented in [@hopkins2005b] in terms of AGN luminosity. As our main variable is the black hole mass, we introduce some simplifications to the model, but the general trend is preserved (see [@hopkins2006] for a thorough discussion). The time spent by a given AGN per logarithmic interval is approximated by [@hopkins2005b] as: $$\label{eq:dtdlogL }
\frac{{d}t}{{d}L}=|\alpha|t_Q\, L^{-1}\, \left(\frac{L}{10^9L_\odot}\right)^\alpha,$$ where $t_Q\simeq10^9$ yr, and $\alpha=-0.95+0.32\log(L_{\rm peak}/10^{12} L_\odot)$. Here $L_{\rm peak}$ is the luminosity of the AGN at the peak of its activity. [@hopkins2006] show that approximating $L_{\rm peak}$ with the Eddington luminosity of the MBH at its final mass (i.e., when it sets on the $m_{\rm BH}-\sigma_*$ relation) the difference in their results is very small. If we write the accretion rate in terms of the time-varying Eddington rate: $\dot {M}=f_{\rm Edd}(t)\dot{M}_{\rm Edd}=f_{\rm Edd}(t) m_{\rm BH}/t_{\rm Edd}$ ($t_{\rm Edd}=0.45\,{\rm Gyr} $), the AGN luminosity can be written as $L=\epsilon f_{\rm Edd}(t) \dot{M}_{\rm Edd} c^2$, where $\epsilon$ is the radiative efficiency[^1]. Differentiating with respect to the Eddington ratio, we can write a simple differential equation for $\dot f_{\rm Edd}(t)$: $$\label{eq:doteddratio }
\frac{{d}f_{\rm Edd}(t)}{{d}t}=\frac{ f_{\rm Edd}^{1-\alpha}(t) }{|\alpha| t_Q}\left(\frac{\epsilon \dot{M}_{\rm Edd} c^2}{10^9L_\odot}\right)^{-\alpha}.$$ Solving this equation gives us the instantaneous Eddington ratio for a given MBH at a given time, and we self-consistently grow the MBH mass: $$\label{eq:bhgrowth }
M(t+\Delta t)=M(t)\,\exp\left(\int_{\Delta t}\frac{dt}{t_{\rm Edd}} f_{\rm Edd}(t)\frac{1-\epsilon}{\epsilon}\right).$$ We define a lower limit to the Eddington ratio $f_{\rm Edd}=10^{-3}$.
As will be discussed in Section §4, we also consider a modification of model IIIa, where we include a much stronger dependence of $f_{\rm Edd}$ on the galaxy velocity dispersion, in practice we modify the exponent $\alpha$ in Equation \[eq:dtdlogL \] as follows: $$\label{eq:doteddalpha}
\alpha=-0.5\left(\frac{V_c}{320}\right)^2+1.5\left(\frac{V_c}{320}\right)^{1/3}\log_{10}\left(1.46\frac{V_c}{320}\right).$$ Although not physically motivated, as Equation \[eq:dtdlogL \] is not either, Equation \[eq:doteddalpha\] was inspired by the trend in accretion rates shown by [@dimatteo2005]. The Eddington ratios found with this modification (model IIIb) are typically lower than in model IIIa. Again we set $f_{\rm Edd}>10^{-3}$. Model IIIb is therefore representative of a simple attempt to decrease further the typical accretion rate of MBHs at low redshift.
The main assumptions regarding the dynamical evolution of the MBH population in our models can be found in [@VHM; @Volonterietal2005; @Vrees2006].
Luminosity functions
====================
We have calculated the luminosity functions by implementing the different accretion models within a comprehensive model for black holes evolution in a Cold Dark Matter universe. The history of dark matter halos and their associated black holes is traced by merger trees (Volonteri, Haardt & Madau 2003). The evolution of the massive black hole population traces the accretion and dynamical processes involving black holes. We have assumed that accretion onto nuclear black holes is triggered by halo mergers, and we have then computed the accretion rate and luminosity of the active systems as described in the previous section. At every step of the simulations we apply the appropriate Eddington rate to accreting MBHs. For model I $f_{\rm Edd}=1$ for all MBHs at all times. For model II, $f_{\rm Edd}$ is only redshift dependent (see Eq. \[eq:shankar2 \]), while for models IIIa and IIIb we determine $f_{\rm Edd}$ as a function of the black hole mass at the beginning of the timestep, and of the host velocity dispersion.
The luminosity functions are computed selecting the black holes which are active at the chosen output redshifts ($z=0.5, 1, 2, 3$), and weighting each of them according to the Press & Schechter function. We derive the AGN bolometric luminosity as $L=\epsilon f_{\rm Edd}(t) \dot{M}_{\rm Edd} c^2$. We apply the bolometric corrections and spectrum of [@Marconietal2004] to model the SED in the blue band. The spectrum of unabsorbed (here dubbed Type I) AGN is described by a power–law with photon index $\Gamma=-1.9$, exponentially cut–off at $E_c=500$ keV. The averaged SED of absorbed (Type II) sources (i.e., sources with absorbing column $\log{(N_{\rm H}/{\rm cm}^{-2})}>22$) is described by the same Type I spectrum for $E> 30$ keV, and by a power–law (continuously matched) with photon index $\Gamma=-0.2$ [@Sazonov2004] at lower energies. The Type II/Type I ratio is, in general, function of luminosity and redshift. Here, we adopt the model \#4 of [@lafranca2005], which explicity allows for the redshift and luminosity evolution of the $N_H$ distribution, providing the best fit model to the HXLF of the HELLAS2XMM sample [@fiore]. Error bars, at 1–$\sigma$, for the theoretical LFs have been computed assuming Poisson statistics.
We have compared our theoretical OLF at different redshifts to the OLF by [@croom2004] obtained by merging the 2dF QSO Redshift Survey (2QZ), with the 6dF QSO Redshift Survey (6QZ). Figures \[fig1\] and \[fig2\] show an absorption-corrected OLF against the best fit models by [@croom2004]. In order to guide the eye, we have extrapolated the OLF at the faint end below the observational limit (dashed lines) adopting the same [@croom2004] fit. We note, however that the theoretical OLF flattens towards lower luminosities, as has been observed [@richards2005; @hunt2004].
The agreement between the theoretical and observed OLFs is good for all models (I, II, III), which result almost undistinguishable in the observed luminosity range (solid line in Figures \[fig1\] and \[fig2\]). Model I overpredicts the bright-end of the OLF at low redshift ($z<1$) more substantially than models II and III; on the other hand models with subEddington accretion tend to underpredict the bright end at high redshift. In fact, we cannot form MBHs massive enough to power bright AGN by $z=3$ if models II or III are considered at all times[^2].
![Luminosity function of AGN in the B-band corrected for absorption. Clockwise: z=0.5, 1, 2, 3. Green squares show model I ($f_{\rm Edd}$=1), magenta pentagons model IIIa ($f_{\rm Edd}$ luminosity dependent). [*Solid lines :*]{} 2QZ/6QZ LF. The dashed lines show the extrapolation to faint magnitudes. []{data-label="fig1"}](LFB_abscorr1_col.ps){width="8cm"}
![Luminosity function of AGN in the B-band corrected for absorption. Clockwise: z=0.5, 1, 2, 3. Blue triangles show model II ($f_{\rm Edd}$ redshift dependent), red circles model IIIb ($f_{\rm Edd}$ luminosity dependent).[]{data-label="fig2"}](LFB_abscorr2_col.ps){width="8cm"}
The OLF is known to be a biased tracer of the accretion history of MBHs, missing the vast majority of Type II objectes. Moreover it spans a smaller range in luminosity compared to the HXLF [@ueda2003].
When we compare the theoretical and observed HXLFs of unabsorbed AGN (Figures \[fig3\] and \[fig4\]), large differences at the faint end become apparent. Model I largely underestimates the faint end at $z\la1$, while at $z\ga2$ it agrees very well in the luminosity range probed by current surveys. Model II underestimates the normalization of HXLF at all redshift, although the shape is satisfactorily matched. In model II accretion is simply not enough to grow black holes massive enough to account for the bright end of the HXLF. If the normalization is changed from, $f_{\rm Edd,0}=0.3$ to $f_{\rm Edd,0}=1$ (cfr. Lapi et al. 2006), the model fares much better at high-z (basically corresponding to model I at $z>2$), but incurs in the same issues of model I at lower-z.
Models IIIa and IIIb fare better in reproducing the low redshift HXLF, but predicts a large population of faint AGN at $z\ga2$ and slightly underestimate the bright end at high redshift ($z>2$). We note here that our approach differs from that by [@hopkins2005b]. The starting point of [@hopkins2005b] is the HXLF, from which they derive the quasar birth rate, and consequently the OLF and other diagnostics. Our approach instead follows the evolutionary path of MBHs and AGN, that is the population evolves self-consistently along the cosmic epochs, according to the accretion properties stated in §2. The HXLF becomes therefore a constraint, rather than an input of the model as in [@hopkins2005b].
We have investigated the impact of the initial conditions, by applying model IIIa to a scenario in which seeds are much more massive, as in [@kous]. The resulting luminosity functions are negligibly different with respect to the corresponding models assuming smaller seeds. This is because observable properties are mainly determined by the accretion history rather than by the initial conditions. Differences arise only at luminosity around $10^{42}$ erg s$^{-1}$, as a bump at $10^{42}$ erg s$^{-1}$, where the MBHs with a mass around that of the initial seeds are clustered, and a sharp decrease faint-ward of $10^{42}$ erg s$^{-1}$.
![Luminosity function of AGN in the hard X-ray band \[2-10 Kev\]. Symbols as in Figure 1. The dashed lines show the Ueda et al. (2003) HXLF in the observationally constrained luminosity range.[]{data-label="fig3"}](LFX1_col.ps){width="8cm"}
![Luminosity function of AGN in the hard X-ray band \[2-10 Kev\]. Symbols as in Figure 2. The dashed lines show the Ueda et al. (2003) HXLF in the observationally constrained luminosity range.[]{data-label="fig4"}](LFX2_col.ps){width="8cm"}
Faint X-ray counts
==================
The luminosity functions (OLF and HXLF) are the most sophisticated analysis of the evolution of AGN as a function of luminosity and redshift. On the other hand the available surveys do not probe yet the extreme faint end where theoretical models mostly differ, except at very low redshift. Figures \[fig3\] and \[fig4\] show that theoretical models predictions, at $z> 0.5$ branch off at luminosities not yet sampled by the HXLF. Number counts are the results of integrating over intrinsic luminosity and distance. Number counts are a weaker test than the luminosity function, as AGN with a wide range of intrinsic luminosities are included at each flux. Nevertheless, they are the most direct probe of the AGN population. They are independent of cosmology, allow to probe further the faint population, where the HXLF is still prohibitive because of spectroscopic flux limits. We therefore compute the expected X-ray counts for the same AGN population that we used to determine the luminosity functions and compare the model results to the most recent determinations of X-ray counts and their redshift distribution.
Basic Equations
---------------
The number of sources (per unit solid angle) seen in the flux range $S$, $S+dS$ by an observer located at $z_0$, is
$$\frac{dN}{d\Omega dS}(z_0,S)=\int^{\infty}_{z_0}
\left( \frac{dV_c}{dz d\Omega} \right) n_c(z, S) \, dz,$$
where $dV_c/dz d\Omega$ is the comoving volume element per unit redshift per unit solid angle, and $n_c(z,S)$ is the comoving density of sources at redshift $z$, with observed flux in the range $[S, S + dS]$. The integrated flux of a source observed at $z_0$ is given by $$\label{eq:flux}
S=\frac{1}{4\pi \, d_L^2(z_0,z)}\int_{\Delta \nu} \tilde L_\nu (M) d\nu ,$$ where $\nu=\nu_0 (1+z)/(1+z_0)$, $d_L(z_0,z)$ is the luminosity distance between redshift $z_0$ and $z$, $\tilde L_\nu(M)$ is the specific luminosity [*averaged over the source lifetime*]{} (assumed to be only a function of the BH mass, $M$), and $\Delta \nu$ is the [*rest-frame*]{} frequency bandwidth.
The background specific intensity $J_{\nu_0}(z_0)$ observed at redshift $z_0$ at frequency $\nu_0$, is $$\label{eq:J}
J_{\nu_0}(z_0)= \frac{(1+z_0)^3}{4\pi}\int^{\infty}_{z_0}
\epsilon_\nu(z) \frac{dl}{dz}dz,$$ where $dl/dz$ is the proper line element, and the comoving specific emissivity $\epsilon_\nu(z)$ is given by $$\begin{aligned}
\label{eq:eps}
\epsilon_\nu(t)&=&\int dM\,\, \int_0^t L_{\nu}(t-t',M)\frac{dn_c}{dt'dM}dt' \nonumber \\
&\simeq& \int dM\,\, \tau \tilde L_{\nu}(M) \frac{dn_c}{dt dM}. \end{aligned}$$ The second approximated equality holds once we consider the source light curve averaged over the typical source lifetime $\tau$, assuming the formation rate of sources per unit mass as constant over such timescale.
Number counts
-------------
![Predicted $\log(N)/\log(S)$ in the observed soft–X band \[0.5–2 keV\] for the different models. Dotted lines show the contribution of sources with $z<2$, whereas dashed lines the contribution of sources with $z>2$. Solid line is the sum of the two components. Points report data obtained with [*Chandra*]{} (dots; Moretti et al. 2003) and [*XMM*]{} (squares; Baldi et al. 2002), and the bow-tie indicates the result of the fluctuation analysis of the [*Chandra*]{} deep field [@Bauer2004]. []{data-label="fig5"}](number_sxr.ps){width="8cm"}
![Predicted $\log(N)/\log(S)$ in the observed hard–X band \[2–10 keV\] for the different models. Lines and points are the same of Fig. \[fig4\]. []{data-label="fig6"}](number_hxr.ps){width="8cm"}
We computed the number counts predicted by our different assumptions concerning the evolution of the Eddington–parameter along the cosmic history. The soft and hard X–ray logN/logS are shown in fig. \[fig5\] and fig. \[fig6\], respectively. The total counts (solid lines) are divided into the contribution of sources at $z<2$ (dotted lines), and $z>2$ (dashed lines). Model results are compared to a compilation of X–ray data from the [*Chandra*]{} (dots, [@Moretti2003]), and [*XMM*]{} (squares; [@Baldi2002]) deep field surveys. The bow–tie indicates results of the fluctuation analysis of the [*Chandra*]{} deep field [@Bauer2004].
Model I fails to reproduce the slope of the observed logN/logS, falling short in the number of bright objects, and slightly overpredicting faint AGN. Moreover, the counts are dominated by high redshift sources for fluxes below $\log S<-14.6$. In Model II, the redshift distribution of AGN is somewhat pushed towards lower redshift, because of the relatively longer accretion time involved. The model underpredicts the counts both in the soft and hard X–ray bands, as BHs do not have enough time to grow. Model IIIa matches well the observed logN/logS in the soft band, but underpredicts the counts in the hard band. Finally, Model IIIb gives a reasonable good description in both bands, though it slightly overpredicts counts at very faint fluxes. AGN number counts are dominated, in the entire observed flux range, by low redshift objects, the contribution of sources at $z>2$ becoming significant only at fluxes as faint as the limits of the more recent surveys.
Redshift distribution of X–ray selected AGN
-------------------------------------------
Aiming at constraining further the 4 different models employed, we compare the predicted redshift distributions to the results of the [*Serendipitous Extragalactic X-ray Source Identification*]{} (SEXSI) program, a survey designed to resolve a large fraction of the 2–10 keV cosmic X–ray background [@Eckart2006]. The survey covers 1 deg$^2$ for fluxes $>1\times 10^{-14}\;\Xflux$, and 2 deg$^2$ for fluxes $>3\times 10^{-14}\;\Xflux$. Given the large survey area, the SEXSI program minimizes the effects of cosmic variance. The catalog contains a total of 477 spectra, among which 438 have redshift and optical identification [@Eckart2006]. The Type I AGN redshift distribution of the SEXSI program and of our selected models are shown in fig. \[fig7\]. Note that we splitted the original SEXSI data in order to match our definition of Type I/Type II sources. Moreover, we have convolved the predicted number counts with the sky coverage of the survey for different flux limits [@Harrison2003].
Model I fails completely to reproduce the observed redshift distribution. In particular, the model largely underestimates the number of sources at $z<1.5$: the distribution peaks at redshift higher than observed. A better match to data is achieved by Model II. The general shape of the distribution is reproduced, although the model largely underestimates the total number of sources, as already pointed out. Model IIIa and IIIb are in reasonable agreement with the data. Model IIIa falls short to the data at $z<1$, while Model IIIb overpredicts the number of sources observed in the range $0.5<z<1.5$. In conclusion, the best agreement with the observed logN/logS, and with the redshift distribution of sources is found assuming a luminosity dependent Eddington rate [@hopkins2005a].
We also tested that, in a model in which BH seeds are more massive, as in [@kous], the resulting $\log N/\log S$ and redshift distribution do not differ significantly with respect to models with earlier, smaller seeds. In conclusions, different formation scenarios for seed BHs are difficult to discriminate on the basis of available X–ray deep field surveys.
![Distribution of Type I AGN as function of redshift for different models (points) compared to the result of Eckart et al. (2005; solid line) from the SEXSI program. In order to compare our results to the observed distribution, we have convolved the predicted number counts with the sky coverage of the survey at different flux limits [@Harrison2003], and we have split the data so to match our definition of Type I/Type II sources.[]{data-label="fig7"}](red_dist_le22.ps){width="8cm"}
Unresolved X–ray Background
---------------------------
According to [@Moretti2003], the intensity of the total X–ray background (XRB) is $7.53\pm 0.35 \times 10^{-12}$ and $2.02\pm 0.11 \times 10^{-11}\;\Xback$ in the 0.5–2 keV, and 2–10 keV energy bands, respectively. A large fraction, $\simeq 94$%, of the soft XRB (SXRB) has been attributed to sources with fluxes exceeding $2.4\times 10^{-17}\;\Xflux$, while $\sim 89$% of the hard XRB (HXRB) is resolved into sources whose flux is $\geq 2.1\times 10^{-16}\;\Xflux$ (Moretti et al. 2003). More recently, [@Hickox2005] estimated the unaccounted fraction of the XRB due to extragalactic unresolved sources as $1.77\pm 0.31\times 10^{-12}\;\Xback$ in the soft X–ray energy band (0.5–2 keV) and $3.4\pm 1.7\times 10^{-12}\;\Xback$ in the hard X–ray energy band (2–8 keV).
Using our different models, we compute the contribution to the unresolved XRB due to faint AGN lying below sensitivity limits of current X–ray surveys. The cumulative contribution from sources with flux above a given threshold is shown in Fig. \[fig8\], where it is compared to the recent estimate of Hickox & Markevitch (2005; shaded area). Different line styles refer to different models: Model I (long-dashed line), Model II (dotted line), Model IIIa (dot-dashed line), and Model IIIb (short-dashed line). The unresolved SXRB and HXRB are shown in the top and bottom panel, respectively. Although significant differences (within a factor of $\simeq 2$) are found between different models, all predict a contribution to the unresolved XRB consistent with available limits. Note that all models can account for the whole unresolved HXRB, while they give at most 50% of the unresolved SXRB. Our results imply the existence of a further population of faint X–ray sources in the soft band. Indeed, Salvaterra et al. (2006) have found that a significant contribution to the unresolved XRB may come from accreting BHs at very high redshift ($z>6$), a redshift range not considered here. By means of a dedicated model of the SMBH assembly at early times, consistent with the SDSS OLF at $z=6$ and with ultra–deep X–ray constraints, they found that the contribution to the SXRB of very high redshift, undetected AGN could be as high as $\sim 0.4\times 10^{-12}\;\Xback$, providing the residual unresolved flux. The contribution of such population to the HXRB is still consistent with the available limits, being only $\sim 0.9 \times 10^{-12}\;\Xback$.
![Predicted cumulative contribution to the unresolved XRB from different models as function of the X–ray flux. Different lines refer to different model: Model I (long-dashed line), Model II (dotted line), Model IIIa (dot-dashed line), and Model IIIb (short-dashed line). The shaded area show the measured unaccounted background as reported by Hickox & Markevitch (2005). Top panel: unaccounted XRB in the observed soft–X band \[0.5-2 keV\]. Bottom panel: unaccounted XRB in the observed hard–X band \[2–8 keV\].[]{data-label="fig8"}](Xback2.ps){width="8cm"}
Discussion
==========
We have attempted in this paper to place constraints on the global accretion properties of the MBH population at $z<3$. We consider the full cosmological evolution of MBH embedded in their host halos, rather than adopting an empirical approach which takes as a starting point the observed LF in a given band in order to explain the properties of AGN in other bands. We focus here on the strength of accretion, parameterizing the accretion rate as a function of the Eddington rate, $f_{\rm Edd}$. We show that simple models which assume $f_{\rm Edd}=1$ (model I), although highly idealized, still are able to explain satisfactorily the growth of MBH, as traced by the OLF, in a large redshift range, as claimed previously by various investigations. Even at low redshift, where hierarchical models start to fail in their predictions, the OLF is very well reproduced. The HXLF probes fainter sources than the OLF, and the simplistic model I is shown to provide a poor match with the HXLF at $z<1$. A decreasing average Eddington ratio [@Shankar2004 model II] provides a better agreement with the *shape* of the HXLF, but underestimates the *normalization*, as there is not enough time to grow high mass MBHs if the Eddington ratio is not large at $z>3$. A model (model III) with an Eddington ratio depending on luminosity [@hopkins2005a; @Shankar2004] seems to be the best match at low redshift. The difference among the models is magnified when the faint X-ray counts, and in particular their redshift distribution, are calculated. Model I has a strictly hierarchical growth of MBHs, and moreover, a univocal relationship between MBH mass and AGN luminosity. Model II allows massive black holes at low redshift to shine at lower luminosity, as the accretion rate, in units of the Eddington one, decreases with time. A rigid redshift dependence, however, creates two problems: first, subEddington accretion at $z>3$ implies a much slower growth for high redshift AGN, thus causing an underestimate of the LFs *normalization* at all the considered redshifts. Second, the evolution of the Eddington rate is not fast enough, at $z<3$, to account for the faint end of the LFs, in particular the HLF. The accretion rates, and Eddington ratios, predicted by simulations are here embedded into a cosmological framework (models IIIa and IIIb). The resulting AGN population provides a satisfactory match with the OLF, HXLF and faint X-ray counts at low redshift, while sources at high redshift are more problematic. The low accretion rates predicted by simulations imply very long growth timescales for black holes, and therefore underestimate the occurrence of bright quasars powered by billion solar masses black holes at high redshift ($z>2$).
It is not clear, however, if the accretion rate found in simulations, which relates to model III, i.e., strongly subEddington for low-luminosity sources (and therefore in all cases for small MBHs) indeed applies at very high-redshift. The simulations on which [@hopkins2005a] model is based assume mergers between “normal” galaxies. How “normal” are galaxies at $z>3$? It might indeed be possible that the conditions in pre-galactic structures at high-redshift (e.g., the disturbed morphological state of galaxies) cannot be studied with simulations of mergers of evolved discs and bulges, but would require different initial conditions for the merging galaxies. Future deep surveys can help us distinguish between various models in the high redshift Universe (see Figure \[fig3\]), and can probably locate the time, if any, for a transition between messy mergers with on average efficient accretion onto the MBHs, and standard galactic mergers, predicting long periods of inefficient accretion.
Finally, we computed the contribution to the XRB of faint AGN lying below the sensitivity limits of current X–ray surveys. We found that all model predict a contribution to the unresolved XRB consistent with the available limits, accounting for the whole unresolved XRB in the 2–8 keV band and for $\sim 50$% in the 0.5–2 keV. The residual background intensity in the soft band may be provided by AGN shining at $z\ga 6$ (Salvaterra et al. 2006).
Acknowledgments
===============
We would like to thank Dave Alexander for interesting discussions and helpful comments. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949. NSF-KITP-06-88 pre-print.
[^1]: We determine the radiative efficiency self-consistently tracking the evolution of black hole spins throughout our calculations [@Volonterietal2005]. We adopt an upper limit to the radiative efficiency of $\epsilon=0.16$, as this corresponds, adopting the standard conversion for accretion from a thin disc, to a maximum spin parameter of the BH $\hat a=0.9$. This value was chosen in agreement with [@Gammieetal2004] simulations, which suggest that the maximum spin MBHs can achieve by coupling with discs in magneto-hydrodynamical simulations is $\hat a\simeq 0.9$.
[^2]: We have modified model III, assuming $f_{\rm Edd}$=1 at $z>12$ in order to obtain significant growth of MBHs before z=6. See [@Vrees2006] for a discussion on the constraints set by the OLF of bright AGN at $z=6$.
|
---
abstract: 'We show that surgery on a connected clover (or clasper) with at least one loop preserves the concordance class of a knot. Surgery on a slightly more special class of clovers preserves invertible concordance. We also show that the converse is false. Similar results hold for clovers with at least two loops vs. $S$-equivalence.'
address: |
Department of Mathematics, University of Warwick\
Coventry, CV4 7AL, UK
author:
- |
Stavros Garoufalidis\
Jerome Levine
title: 'Concordance and 1-loop clovers'
---
[http://www.math.gatech.edu/\\char'176stavros,
http://www.math.brandeis.edu/Faculty/jlevine/](http://www.math.gatech.edu/\char'176stavros,
http://www.math.brandeis.edu/Faculty/jlevine/)
Introduction
============
History
-------
M. Goussarov and K. Habiro have independently studied links and 3-manifolds from the point of view of surgery on objects called [*Y-graphs, claspers or clovers*]{}, respectively by [@Gu; @H] and [@GGP]. Following the notation of [@GGP], given a pair $(M,K)$ consisting of a knot $K$ in an $M$, and a clover $G \subset M-K$, surgery on the framed link associated to $G$ produces a new pair $(M,K)_G$. Thus, by specifying a class of clovers $\mathfrak{c}$ we can define an equivalence relation (also denoted by $\mathfrak{c}$) on the set $\K\M$ of knots in s and sometimes on its subset $\K$ of knots in $S^3$.
It is often the case that for certain classes of clovers $\mathfrak{c}$, the equivalence relation is related to some natural topological equivalence relation. In this paper we will be particularly interested in [*concordance*]{} (in the smooth category) but will also discuss $S$-equivalence.
We begin by discussing some known facts. Using the terminology of [@GGP], let $\mathfrak{c}^{\D\D}$ denote the class of clovers $G \subset
S^3 -K$ of degree $1$ (that is, the class of s) whose leaves form a $0$-framed unlink which bounds disks disjoint from $G$ that intersect $K$ geometrically twice and algebraically zero times. Surgery on such clovers was called a [*double $\D$elta*]{} move by Naik-Stanford, who showed that
[[@NS]]{} $\mathfrak{c}^{\D\D}$ coincides with $S$-equivalence on $\K$.
Relaxing the above condition, let $\mathfrak{c}^{\mathrm{loop}}$ denote the class of clovers $G \subset M-K$ whose leaves have zero linking number with $K$. Surgery on such clovers was called a [*loop move*]{} by G.-Rozansky who showed that
[[@GR]]{} $\mathfrak{c}^{\mathrm{loop}}$ coincides with $S$-equivalence on $\K\M$.
Let us make the following definition. If $G$ is a clover in $M-K$ and $\L$ a set of leaves of $G$, we say $\L$ is [*simple*]{} if the elements of $\L$ bound disks in $M$ each of which intersects $K$ exactly once but whose interiors otherwise are disjoint from $K$, $G$ and each other. Consider now for every non-negative integer $n$, the class $\mathfrak{c}^n$ of clovers $G \subset S^3-K$ whose entire set of leaves is simple, and such that each connected component of $G$ is a graph with at least $n$ loops (i.e., whose first betti number is at least $n$). Kricker and Murakami-Ohtsuki showed that
[[@Kr; @MO]]{} $\mathfrak{c}^{2}$ implies $S$-equivalence on $\K$.
In fact, if we let $\mathfrak{c}^{\rm iv}$ denote the class of clovers $G$ such that each component of $G$ has at least one internal trivalent vertex, and $G$ has a simple set of leaves containing one leaf from each component, then it is not hard to check that $\mathfrak{c}^2\sub\mathfrak{c}^{\rm iv}$ and [@Kr; @MO] actually proved that $\mathfrak{c}^{\rm
iv}$ implies $S$-equivalence. Combining this with a recent result of Conant-Teichner [@CT] we actually have:
[[@CT]]{}$\mathfrak{c}^{\rm iv}$ coincides with $S$-equivalence on $\K$.
Statement of the results
------------------------
In the present paper we will prove the following results.
$\mathfrak{c}^{1}$ implies concordance on $\K$.
An different proof of Theorem \[thm.GL\] has been obtained by Conant-Teichner [@CT] relying on the notion of [*grope cobordism*]{}. This result was also announced by the first author in [@Le2], where an analogous statement was proved, and our proof will follow the lines of that argument. The result was also known to Habiro, according to private communication.
A slight refinement of the class $\mathfrak{c}^1$ relates to a classical refinement of concordance known as [*invertible concordance*]{}. Recall that a knot in $S^3$ is called [*double-slice*]{} if it can be exhibited as the intersection of a $3$-dimensional hyperplane in $\BR^4$ with an [*unknotted*]{} imbedding of $S^2$ in $\BR^4$; see e.g. [@Su]. Such knots are obviously slice, and it is shown in [@Su] that, for any knot $K$, the connected sum $K \sharp (-K)$ is double-slice, where $-K$ denotes the mirror image of $K$. On the other hand the Stevedore knot is slice but not double-slice (see [@Su]). More generally, following [@Su], we say that $K$ is [*invertibly concordant*]{} to $K'$ if there is a concordance $V$ from $K$ to $K'$ and a concordance $W$ from $K'$ to $K$ so that if we stack $W$ on top of $V$, the resulting concordance from $K$ to itself is diffeomorphic to the product concordance $(I\times S^3 ,I\times K)$. If we write $K\le K'$, then $\le$ is transitive and reflexive and perhaps even a partial ordering. It is easy to see that $0\le K$, where $0$ denotes the trivial knot, if and only if $K$ is double-slice.
Let $\mathfrak{c}^{1,\mathrm{nf}}$ denote the subclass of $\mathfrak{c}^1$ consisting of clovers with [*no forks*]{}— a fork is a trivalent vertex two of whose incident edges contain a univalent vertex. Then, we will prove:
If $G$ is a clover in the class $\mathfrak{c}^{1,\mathrm{nf}}$ and $K'$ is obtained from $K$ by surgery on $G$ then $K\le K'$.
It is natural to ask whether the converses to Theorems \[thm.Kr\], \[thm.GL\] and \[thm.double\] are true. If that were the case, one could extract from the rational functions invariants of [@GK] many concordance invariants of knots. It was a bit of a surprise for us to show that the converses are all false.
First of all, it will follow easily from a recent result of Livingston that:
There are $S$-equivalent knots which are not $\mathfrak{c}^2$-equivalent.
Then we will generalize some techniques of Kricker to prove:
There are double-slice knots which are not $\mathfrak{c}^1$-equivalent to the unknot.
The proofs of Proposition \[thm.5\] and Theorem \[thm.6\] allow one to easily construct specific knots with the desired properties. See [@Li Theorem 10.1] for knots that satisfy Proposition \[thm.5\]. For the $(5,2)$-torus knot $T_{5,2}$, we have that $T_{5,2} \sharp
(-T_{5,2})$ is a knot that satisfies Theorem \[thm.6\].
Plan of the proof
-----------------
Theorems \[thm.GL\] and \[thm.double\] follow from an analysis of the surgery link corresponding to a clover.
Proposition \[thm.5\] follows easily from the fact (proven recently by Livingston [@Li], using Casson-Gordon invariants) that $S$-equivalence does not imply concordance.
Theorem \[thm.6\] follows from the fact that under surgery on $\mathfrak{c}^1$-clovers, the Alexander polynomial changes under a more restrictive way than under a concordance.
Proofs
======
Proof of Theorem \[thm.GL\]
---------------------------
Suppose that $G$ is a connected clover of class $\mathfrak{c}^1$ and $L$ is its associated framed link, [@Gu; @H; @GGP]. We want to show that the knot $K'$ obtained from $K$ by surgery on $L$ is concordant to $K$. Note that the manifold $M$ obtained from $S^3$ by surgery on $L$ is diffeomorphic to $S^3$, see [@Gu; @H; @GGP].
We can express $L$ as a union of two sublinks $L'$ and $L''$ such that:
- $L'$ is a trivial $0$-framed link in $S^3 -K$,
- $L''$ is a trivial $0$-framed link in $S^3$.
Assuming this lemma we can complete the proof of Theorem \[thm.GL\] as follows.
Consider $I\times K\sub I\times S^3$ and $\frac{1}{2}\times
L\sub\frac{1}{2}\times (S^3 -K)$. Consider a union of disjoint disks $D'$ in $\frac{1}{2}\times (S^3 -K)$ bounded by $L'$ and push their interiors into $[0,\frac{1}{2})\times (S^3-K)$. Also consider a union of disjoint disks $D''$ in $\frac{1}{2}\times S^3$ bounded by $L''$ and push their interiors into $(\frac{1}{2},1]\times S^3$. Now let $X\sub I\times S^3$ be obtained from $[0,\frac{1}{2}]\times S^3$ by removing a tubular neighborhood of $D'$ and adjoining a tubular neighborhood of $D''$. The boundary of $X$ consists of $0\times S^3$ and a copy of $M$, which is diffeomorphic to $S^3$. Thus $X$ is diffeomorphic to $I\times
S^3$ ( indeed, add a $D^4$ to $X$ along $0 \times S^3$ and observe that any two imbeddings of a $4$-disk in a fixed $4$-disk are isotopic). Moreover $X$ contains $[0,\frac{1}{2}]\times K$, which is a concordance from $0\times K\sub 0\times S^3$ to $\frac{1}{2}\times K\sub M$, which is just $K'$.
This is a generalization of the argument used to prove Theorem 2 in [@Le2]. Recall (eg. from [@GGP Section 2.3]) that surgery on a clover $G$ with $n$ edges corresponds to surgery on a link $L$ of $2n$ components. Given an orientation of the edges of $G$, we can split $L$ into the disjoint union of $n$-component sublinks $L'$ and $L''$, where $L'$ (resp. $L''$) consists of the sublink of $L$ assigned to the tails of the edges of $G$ (resp. of the heads of the edges of $G$, together with the leaves of $G$). As long as we avoid assigning all three of the components at a trivalent vertex to $L'$ or $L''$, we will have the desired decomposition of $L$. The corresponding conditions imposed on the orientation of the edges of $G$ are:
1. No trivalent vertex is a source or a sink,
2. Every edge with a univalent vertex is oriented toward the univalent vertex.
These are the same conditions as (i) and (ii) in the proof of Theorem 2 in [@Le2] except that we now require no trivalent sinks also. But this will follow by the same argument as in [@Le2] except that we need to choose the orientations of the cut edges more carefully. In particular we need to avoid choosing the orientation of two cut edges which share a trivalent vertex so that they both point into that vertex. But it is not hard to see that this can be done.
The next two remarks are an addendum to Theorem \[thm.GL\].
Observe that the sublinks $L'$ and $L''$ of $L$ which are constructed from $G$ have the same number of components, and that the linking matrix of $L$ is hyperbolic. Lemma \[lem.split\] is analogous to the case of a knot which bounds a Seifert surface with a metabolic Seifert surface. In that case, the knot is algebraically slice, and if a metabolizer can be chosen to be bands of the Seifert surface that form a slice link, then the knot is slice.
Suppose that a knot $K'$ is obtained from the unknot $K$ by surgery on a connected clover of class $\mathfrak{c}^1$. It follows from Theorem \[thm.GL\] that $K'$ is slice. Using the calculus of clovers, one can show that $K'$ is actually ribbon, as observed also by Kricker and Habiro.
Proof of Theorem \[thm.double\]
-------------------------------
We need a refinement of Lemma \[lem.split\]. Consider a connected clover $G$ of class $\mathfrak{c}^{1,\mathrm{nf}}$ and let $L$ be its associated framed link.
There is a link $\bar L$ in $S^3 -K$, Kirby equivalent to $L$ in $S^3 -K$, so that $\bar L$ is a union of two sublinks $\bar L' ,\bar L''$, each of which is trivial in $S^3 -K$.
Assuming this lemma, we finish the proof following the lines of the argument following Lemma \[lem.split\]. The only difference is that we now use $\bar L$ instead of $L$ and that $X'=\overline{I\times S^3 -X}$, which is also diffeomorphic to $I\times S^3$, now also contains $[\frac{1}{2} ,1]\times K$. Thus $M$ splits the trivial concordance from $K$ to itself. This, by definition, means $K\le K'$.
For each univalent vertex of $G$, there is a corresponding part of $L$ which looks like the left part of Figure \[fig.nf\].
$${\begin{array}{c} \hspace{-1.3mm}
\raisebox{-4pt}{\epsfig{figure=leaf.eps,width=3in}}
\hspace{-1.9mm}\end{array}}$$
Now we can perform a Kirby move (see [@Kr],[@MO]) so that the four component link $\{L_1,\dots,L_4\}$ in Figure \[fig.nf\] is replaced by two component link $\{\overline{L}_3,\overline{L}_4\}$. If we do this at every univalent vertex of $G$ we obtain the link $\bar L$. Now consider the partition $L=L'\cup L''$ given by Lemma \[lem.split\]. The corresponding partition of $\overline{L}$ is given by $\bar L'=\{\bar K | K \in L'-\{L_1,L_2\}\}$ and $\bar L''=\{\bar K | K \in L''-\{L_1,L_2\}\}$. It is easy to see that both $\bar L'$ and $\bar L''$ are trivial in $S^3 -K$. This completes the proof.
Proof of Proposition \[thm.5\]
------------------------------
Assume that $S$-equivalence implies $\mathfrak{c}^2$ on $\K$. Since $\mathfrak{c}^2$ implies $\mathfrak{c}^1$, and $\mathfrak{c}^1$ implies concordance (by Theorem \[thm.GL\]), it follows that $S$-equivalence implies concordance. This is false. Livingston using Casson-Gordon invariants, shows that there are $S$-equivalent knots which are algebraically slice, but not slice, [@Li Theorem 0.4]. Since Livingston uses Casson-Gordon invariants, his examples have nontrivial Alexander module.
Proof of Theorem \[thm.6\]
--------------------------
We show that the Alexander polynomial $\D$ of a knot changes in a more restrictive way under $\mathfrak{c}^1$-equivalence than under concordance. Recall that if $K$ and $K'$ are concordant knots, then their Alexander polynomials satisfy $\D_{K'}(t)\th '(t)\th '(t\i )=\D_{K}(t)
\theta(t)\theta(t^{-1})$ for some $\theta(t),\th '(t) \in \BZ[t,t^{-1}]$ satisfying $\theta(1)=\th
'(1)=\pm 1$. Moreover, there are double-slice knots with Alexander polynomial $\theta(t)\theta(t^{-1})$ for any such $\theta$. On the other hand,
Let $K$ and $K'$ be $\mathfrak{c}^1$-equivalent knots. Then, $$\Delta_{K'}(t)\th '(t)\th '(t\i )=\Delta_{K}(t)\theta (t)\theta (t^{-1} )$$ where $\theta (t)$ and $\th '(t)$ are products of polynomials of the form $1\pm
t^k (t-1)^n$ for some integers $k, n$ with $n>0$.
We prove this using a generalization of an argument of Kricker [@Kr]. Consider a connected clover $G$ of the class $\mathfrak{c}^1$. Suppose that $K'$ is obtained from $K$ by surgery on $G$. If $G$ has at least one internal trivalent vertex, then $K$ and $K'$ are $S$-equivalent (see the discussion following Theorem \[thm.Kr\]); in particular $\Delta_K
(t)=\Delta_{K'}(t)$. Otherwise, $G$ must be a [*wheel*]{} with a certain number $n$ of legs and with a total of $2n$ edges. Thus, the associated link $L'$ in $S^3 -K$ has $4n$ components (see Figure below). Using the Kirby move in Figure \[fig.nf\] at every leaf of $G$ we see that $L'$ is Kirby-equivalent in $S^3 -K$ to a link $L$ with $2n$ components, whose components can be numbered in pairs $l_1 ,r_1 ,\dots ,l_n ,r_n$ so that:
1. $l_i$ (resp. $r_i$) bounds a disk $d_i$ (resp. $e_i$) in $S^3 -K$,
2. $d_i\cap e_i$, for $1\le i\le n$, each consists of two oppositely oriented clasps,
3. $e_i\cap d_{i+1}$, for $1\le i<n$ and $e_n\cap d_1$ each consists of a single clasp, and
4. there are no other intersections among the disks.
An example for $n=2$ is shown below: $${\begin{array}{c} \hspace{-1.3mm}
\raisebox{-4pt}{\epsfig{figure=kricker.eps,width=4in}}
\hspace{-1.9mm}\end{array}}$$ We can now lift $d_i$ and $e_i$ to disks, $\ti d_i$ and $\ti e_i$, in the infinite cyclic cover $\ti X$ of $X=S^3
-K$. The lifts of $l_i ,r_i$ form a link $\ti L$ in $\ti X$ which has a linking matrix $B$ with entries in $\BZ [t,t\i ]$. To compute $B$ note that we can choose the lifts $\ti d_i$ and $\ti e_i$ so that:
1. $\ti d_i\cap\ti e_i$ consists of a single clasp, for every $i$,
2. $\ti d_i\cap t(\ti e_i )$ consists of a single clasp, oriented opposite to that in (1), for every $i$,
3. $\ti e_i\cap\ti d_{i+1}$, for $1\le i<n$, consists of a single clasp, and
4. $\ti e_n\cap t^k (\ti d_1 )$, for some integer $k$, consists of a single clasp.
In (4), $k$ (up to sign) is just the linking number of $K$ with the imbedded wheel of $G$.
Now it follows from this intersection data and the fact that $L$ is $0$-framed that we can orient $L$ so that the linking matrix $B$ is given by $$B=\mat 0 D {D^\star} 0
\hspace{.5cm} \text{ where } \hspace{.5cm}
D=\left( \begin{array}{ccccc}t-1&1&0&\ldots&0\\
0&t-1&1&0&\vdots\\
\vdots&\ddots&\ddots&\ddots&0\\
0&\ldots&0&t-1&1\\
\pm t^k&0&\ldots&0&t-1
\end{array}\right) .$$ For any matrix $A$ over $\BZ [t,t\i ]$, $A^\star$ denotes the conjugate (under the involution $t\leftrightarrow t\i$) transpose of $A$. The desired result $\D_{K'}(t)=\D_K
(t)\th (t)\th (t\i )$ is now a consequence of the following lemma, which is proved by a standard argument going back to Kervaire-Milnor, generalized to covering spaces (see for example [@Le1 p.140]).
Suppose $K\sub S^3$ is a knot, $L$ a framed link in $X=S^3 -K$, and $K'\sub
S^3_L$ the knot produced from $K$ by surgery on $L$. Assume that the components of $L$ are null-homologous in $X$ and the components of $\ti L\sub\ti X$, the lift of $L$ into $\ti X$, are null-homologous. In this case we have well-defined linking numbers of the components of $\ti L$ which are organized into a matrix $B$ with entries in $\zt$ in the usual way. Let $A(K)=H_1(\ti X)$ and $A(K')=
H_1(\ti Y)$ denote the [*Alexander modules*]{} of $K, K'$, where $Y=S^3_L-K'$.
There is an exact sequence of $\zt$-modules $$0\to M\to A(K')\to A(K)\to 0$$ where $M$ is a module with presentation matrix $B$. In particular, $\D_{K'}=\D_K \det(B)$.
Observe that $\ti Y=\ti X_{\ti L}$. Consider the following diagram of exact sequences of $\zt$-modules. [$$\divide\dgARROWLENGTH by2
\begin{diagram}
\node[3]{H_2 (\ti Y, \ti X \mskip-1.9mu-\mskip-1.9mu \ti L )}\arrow{s,r}{\bd_*}\\
\node{H_2 (\ti X)}\arrow{e}\node{H_2(\ti X,\ti X \mskip-1.9mu-\mskip-1.9mu\ti
L)}\arrow{e}\node{H_1 (\ti X\mskip-1.9mu-\mskip-1.9mu\ti L)}\arrow{s}\arrow{e,t}{i_*}\node{H_1 (\ti
X)}\arrow{e}\node{H_1 (\ti X,\ti X\mskip-1.9mu-\mskip-1.9mu\ti L)}\\
\node[3]{H_1 (\ti Y)}\arrow{s}\\
\node[3]{H_1 (\ti Y,\ti X\mskip-1.9mu-\mskip-1.9mu\ti L)}
\end{diagram}$$]{}Notice that $H_1 (\ti X,\ti X-\ti L)=H_1 (\ti Y,\ti X-\ti L)=0$. Moreover, $H_2 (\ti X,\ti X-\ti L)$ is freely generated by the meridian disks of $L$, lifted to $\ti X$, and $H_2 (\ti Y,\ti X-\ti L)$ is freely generated by the disks attached by the surgeries. Thus, since the components of $\ti L$ are null-homologous in $\ti X$, $i_*\circ\bd_* =0$. Also note that $H_2
(\ti X)=0$ and so we have a mapping $$H_2 (\ti Y,\ti X-\ti L)\to H_2 (\ti X,\ti X-\ti L)$$ induced by $\bd_*$, which can be interpreted as expressing the longitudes of $\ti L$ as linear combinations of the meridians of $\ti L$ in $H_1 (\ti X-\ti
L)$. Therefore this map is given by the linking numbers of $\ti L$ and has $B$ as a representative matrix. This completes the proof of Lemma \[lem.surg\] and, as a consequence, Lemma \[lem.Kr\].
To complete the proof of Theorem \[thm.6\] we need the following lemma.
Let $f(t)$ be a polynomial of the form $1\pm t^k (t-1)^n$, for any integers $k,n$ with $n\not= 0$. Then any root of $f(t)$ which lies on the unit circle must be of the form $e^{\pm\pi i/3}$.
If $z$ is a root of $f(t)$ then $|z|^k |z-1|^n =1$. Thus we have $|z|=|z-1|=1$, from which the conclusion follows.
Now choose some $\th (t)$ with a root on the unit circle different from $e^{\pm\pi i/3}$ but with $\th (1)=1$—for example any cyclotomic polynomial of composite order not equal to $6$. Let $K$ be a double-slice knot with Alexander polynomial $\th (t)\th (t\i )$ (see [@Su Theorem 3.3]). Then it follows from Lemmas \[lem.Kr\] and \[lem.root\] that $K$ is not $\mathfrak{c}^1$ equivalent to the trivial knot.
We end with a remark concerning the inverse of surgery on a wheel.
Recall that if a knot $K'$ is obtained from a knot $K$ by surgery on a $G$, then there exists a $G'$ such that $K$ is obtained from $K'$ by surgery on $G'$, see [@GGP Theorem 3.2]. Recall also that surgery on a wheel is described in terms of surgery on a union of s, as explained in [@GGP Section 2.3]; in particular the inverse of surgery on a wheel can be described in terms of surgery on a union of s. One might guess that the inverse of surgery on a wheel can be described in terms of surgery on a wheel. This is false, since the proof of Lemma \[lem.Kr\] implies that if $K'$ is obtained from $K$ by surgery on a wheel $G$, then $\D_K$ always divides (and it can happen that it is not equal to) $\D_{K'}$.
[**Acknowledgements**]{}The authors were partially supported by NSF grants DMS-98-00703 and DMS-99-71802 respectively, and by an Israel-US BSF grant.
[\[EMSS\]]{}
J. Conant and P. Teichner, [*Grope cobordism of classical knots*]{}, preprint 2000, [math.GT/0012118]{}.
S. Garoufalidis, M. Goussarov and M. Polyak, [*Calculus of clovers and finite type invariants of 3-manifolds*]{}, Geometry and Topology, [**5**]{} (2001) 75–108.
[to3em]{}and L. Rozansky, [*The loop expansion of the Kontsevich integral, abelian invariants of knots and $S$-equivalence*]{}, preprint 2000, [math.GT/0003187]{}.
[to3em]{}and A. Kricker, [*A rational noncommutative invariant of boundary links*]{}, preprint 2001, [math.GT/0105028]{}.
M. Goussarov, [*Finite type invariants and $n$-equivalence of 3-manifolds*]{}, C. R. Acad. Sci. Paris Ser. I. Math. [**329**]{} (1999) 517–522.
K. Habiro, [*Claspers and finite type invariants of links*]{}, Geometry and Topology [**4**]{} (2000) 1-83.
A. Kricker, [*Covering spaces over claspered knots*]{}, preprint 1999, [arxiv:math.GT/9901029]{}.
J. Levine, [*A characterization of knot polynomials*]{}, Topology [**4**]{} (1965) 135–141.
[to3em]{}, [*Homology cylinders; an expansion of the mapping class group*]{}, preprint 2000, [math.GT/0010247]{}.
C. Livingston, [*Examples in concordance*]{}, preprint 2001, [math.GT/0101035]{}.
H. Murakami and T. Ohtsuki, [*Finite type invariants of knots via their Seifert matrices*]{}, preprint 1999, [math.GT/9903069]{}.
S. Naik and T. Stanford, [*Double $\Delta$-moves and $S$-equivalence*]{}, preprint 1999, [math.GT/9911005]{}.
D. W. Sumners, [*Invertible knot cobordisms*]{}, Commentarii Math. Helveticii, [**46**]{} (1971) 240–256.
|
---
abstract: 'We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.'
author:
- 'Bea Adam-Day [^1]'
- 'John Howe [^2]'
- 'Rosario Mennuni [^3]'
title: |
On Double-Membership Graphs\
of Models of Anti-Foundation
---
This paper is concerned with the model-theoretic study of a class of graphs arising as reducts of a certain non-well-founded set theory.
Ultimately, models of a set theory are digraphs, where a directed edge between two points denotes membership. To such a model, one can associate various graphs, such as the *membership graph*, obtained by symmetrising the binary relation $\in$, or the *double-membership graph*, which has an edge between $x$ and $y$ when $x\in y$ and $y\in x$ hold simultaneously. We also consider the structure equipped with the two previous graph relations, which we call the *single-double-membership graph*. In [@adamdaycameron] the first author and Peter Cameron investigated this kind of object in the non-well-founded case. We continue this line of study, and answer some questions regarding such graphs which were left open in the aforementioned work.
It is well-known that every membership graph of a countable model of $\mathsf{ZFC}$ is isomorphic to the Random Graph (see e.g. [@cameron]). The usual proof of this fact goes through for set theories much weaker than $\mathsf{ZFC}$, but uses the Axiom of Foundation in a crucial way, hence the interest in (double-)membership graphs of non-well-founded set theories.
Perhaps the most famous of these is $\mathsf{ZFA}$[^4], obtained from $\mathsf{ZFC}$ by replacing the Axiom of Foundation with the *Anti-Foundation Axiom*. This axiom was explored in, amongst others, [@fortihonsell; @aczel; @barwisemoss]. It provides a rich class of non-well-founded sets, the structure of which reflects that of the well-founded sets. In every model of Anti-Foundation there will be, for example, unique sets $a$ and $b$ such that $a=\set{b, \emptyset}$ and $b=\set{a,\set{\emptyset}}$, and a unique $c=\set{c, \emptyset, \set{\emptyset}}$. These are pictured in Figure \[figure:afaexample\].
\(a) at (0,0)[$a$]{}; (b) at (1,0)[$b$]{}; (em) at (0,-0.75)[$\emptyset$]{}; (one) at (1,-0.75)[$\set\emptyset$]{};
\(c) at (3,0)[$c$]{}; (em1) at (3,-0.75)[$\emptyset$]{}; (one1) at (4,-0.75)[$\set\emptyset$]{};
\(a) edge\[bend left, postaction=decorate\] (b) (b) edge\[bend left, postaction=decorate\] (a) (em) edge\[postaction=decorate\] (a) (one) edge\[postaction=decorate\] (b) (em) edge\[postaction=decorate\] (one) ; (em1) edge\[postaction=decorate\] (c) (one1) edge\[postaction=decorate\] (c) (em1) edge\[postaction=decorate\] (one1) ;
\(c) ++ (-0.01, 0.08) arc \[start angle=30, end angle=310, radius=0.13\];
In [@adamdaycameron] it was proven that all membership graphs of countable models of $\mathsf{ZFA}$ are isomorphic to the ‘Random Loopy Graph’: the Fraïssé limit of finite graphs with self-edges. This structure is easily seen to be $\aleph_0$-categorical, ultrahomogeneous, and supersimple of SU-rank $1$. On the other hand, double-membership graphs of models of $\mathsf{ZFA}$ are, in a number of senses, much more complicated. For instance, [@adamdaycameron Theorem 3] shows that they are not $\aleph_0$-categorical, and we show further results in this direction.
The structure of the paper is as follows. After setting up the context in Section \[sec:setup\], we answer [@adamdaycameron Question 3] in Section \[sec:connectedcomponents\] by characterising the connected components of double-membership graphs of models of $\mathsf{ZFA}$. In the same section, we show that if we do not assume Anti-Foundation, but merely drop Foundation, then double-membership graphs can be almost arbitrary. Section \[sec:cmany\] answers [@adamdaycameron Questions 1 and 2] by proving the following theorem.
There are, up to isomorphism, continuum-many countable (single-)double-membership graphs of models of $\mathsf{ZFA}$, and continuum-many countable models of each of their theories.
In Section \[sec:theory\] we study the common theory of double-membership graphs, which we show to be incomplete. Then, by using methods more commonly encountered in finite model theory, we characterise the completions of said theory in terms of consistent collections of consistency statements.
The double-membership graphs of two models $M$ and $N$ of $\mathsf{ZFA}$ are elementarily equivalent precisely when $M$ and $N$ satisfy the same consistency statements.
We also show that all of these completions are wild in the sense of neostability theory, since each of their models interprets (with parameters) arbitrarily large finite fragments of $\mathsf{ZFC}$. Our final result, below — obtained with similar techniques — answers [@adamdaycameron Question 5] negatively. The analogous statement for double-membership graphs holds as well.
For every single-double-membership graph of a model of $\mathsf{ZFA}$, there is a countable elementarily equivalent structure which is not the single-double-membership graph of any model of $\mathsf{ZFA}$.
Set-Up {#sec:setup}
======
Since Anti-Foundation allows for sets that are members of themselves, in what follows we will need to deal with graphs where there might be an edge between a point and itself. These are called *loopy graphs* in [@adamdaycameron] but, for the sake of concision, we depart from common usage by adopting the following convention.
By *graph* we mean a first-order structure with a single relation which is binary and symmetric (it is not required to be irreflexive).
There are a number of equivalent formulations of Anti-Foundation. The form which we shall be using is known in the literature (e.g. [@barwisemoss p. 71]) as the *Solution Lemma*. Other formulations are in terms of *homomorphism onto transitive structures* (axiom $X_1$ from [@fortihonsell]), or *decorations*. For the equivalence between them, see e.g. [@aczel p. 16].
\[defin:flatsystem\] Let $X$ be a set of ‘indeterminates’, and $A$ a set of sets. A *flat system of equations* is a set of equations of the form $x = S_x$, where $S_x$ is a subset of $X \cup A$ for each $x \in X$. A *solution* $f$ to the flat system is a function taking elements of $X$ to sets, such that after replacing each $x\in X$ with $f(x)$ inside the system, all of its equations become true.
The *Anti-Foundation Axiom* ($\mathsf{AFA}$) is the statement that every flat system of equations has a unique solution.
Consider the flat system with $X=\set{x,y}$, $A=\set{\emptyset,\set{\emptyset}}$ and the following equations. $$\begin{split} x&=\{y,\emptyset\}\\
y&=\{x,\set\emptyset\}
\end{split}$$ The image of its unique solution $x\mapsto a, y\mapsto b$ is pictured in Figure \[figure:afaexample\].
Note that solutions of systems need not be injective, and in fact uniqueness sometimes prevents injectivity. For instance, if $x\mapsto a$ is the solution of the flat system consisting of the single equation $x=\set{x}$, then $x\mapsto a, y\mapsto a$ solves the system with equations $x=\set y$ and $y=\set x$, whose unique solution is therefore not injective.
There exists a weak form of $\mathsf{AFA}$ that only postulates the existence of solutions to flat systems, but not necessarily their uniqueness, known as axiom $X$ in [@fortihonsell] or $\mathsf{AFA}_1$ in [@aczel]. In what follows and in [@adamdaycameron] uniqueness is never used, hence all the results go through for models of $\mathsf{ZFC}$ with Foundation replaced by $\mathsf{AFA}_1$. For brevity, we still state everything for $\mathsf{ZFA}$.
\[fact:equicon\] $\mathsf{ZFC}$ without the Axiom of Foundation proves the equiconsistency of $\mathsf{ZFC}$ and $\mathsf{ZFA}$.
In one direction, from a model of $\mathsf{ZFA}$ one obtains one of $\mathsf{ZFC}$ by restricting to the well-founded sets. In the other direction, see [@fortihonsell Theorem 4.2] for a class theory version, or [@aczel Chapter 3] for the $\mathsf{ZFC}$ statement.
Since we are interested in studying (reducts of) models of $\mathsf{ZFA}$, we need to assume they exist in the first place, since otherwise the answers to the questions we are studying are trivial. Therefore, in this paper we work in a set theory which is slightly stronger than usual.
The ambient metatheory is $\mathsf{ZFC}+\operatorname{Con}(\mathsf{ZFC})$.
\[defin:symmetrisation\] Let $L=\set{\in}$, where $\in$ is a binary relation symbol, and $M$ an $L$-structure. Let $S$ and $D$ be the definable relations $$\begin{gathered}
S(x,y)\coloneqq x\in y\lor y\in x\\
D (x,y)\coloneqq x\in y\land y\in x\end{gathered}$$ The *single-double-membership graph*, or *SD-graph*, $M_0$ of $M$ is the reduct of $M$ to $L_0\coloneqq\set{S,D}$. The *double-membership graph*, or *D-graph*, $M_1$ of $M$ is the reduct of $M$ to $L_1\coloneqq\set{D}$.
So, given an $L$-structure $M$, i.e. a digraph (possibly with loops) where the edge relation is $\in$, we have that $M_0{\vDash}S(x,y)$ if and only if in $M$ there is at least one $\in$-edge between $x$ and $y$. Similarly $M_0{\vDash}D(x,y)$ means that in $M$ we have both $\in$-edges between $x$ and $y$. The idea is that, if $M$ is a model of some set theory, then $M_0$ is a symmetrisation of $M$ which keeps track of double-membership as well as single-membership, and $M_1$ only keeps track of double-membership.
In [@adamdaycameron], $M_0$ is called the *membership graph (keeping double-edges)* of $M$ and $M_1$ is called the *double-edge graph* of $M$. Note that, strictly speaking, SD-graphs are not graphs, according to our terminology.
For the majority of the paper we are concerned with D-graphs, since most of the results we obtain for them imply the analogous versions for SD-graphs. This situation will reverse in Theorem \[thm:noinfdiam\].
Let $M{\vDash}\mathsf{ZFA}$. We say that $A\subseteq M$ is an *$M$-set* iff there is $a\in M$ such that $A=\set{b\in M\mid M{\vDash}b\in a}$.
So an $M$-set $A$ is a definable subset of $M$ which is the extension of a set in the sense of $M$, namely the $a\in M$ in the definition. We will occasionally abuse notation and refer to an $M$-set $A$ when we actually mean the corresponding $a\in M$.
Connected Components {#sec:connectedcomponents}
====================
Let $M{\vDash}\mathsf{ZFA}$. It was proven in [@adamdaycameron Theorem 4] that, for every finite connected graph $G$, the D-graph $M_1$ has infinitely many connected components isomorphic to $G$. It was asked in [@adamdaycameron Question 3] if more can be said about the infinite connected components of $M_1$. In this section we characterise them in terms of the graphs inside $M$.
Let $G$ be a graph in the sense of $M{\vDash}\mathsf{ZFA}$, i.e. a graph whose domain and edge relation are $M$-sets, the latter as, say, a set of Kuratowski pairs. If $G$ is such a graph and $M{\vDash}\text{`$G$ is connected'}$, then $G$ need not necessarily be connected. This is due to the fact that $M$ may have non-standard natural numbers, hence relations may have non-standard transitive closures. We therefore introduce the following notion.
\[defin:region\] Let $a\in M{\vDash}\mathsf{ZFA}$. Let $b\in M$ be such that $$M{\vDash}\text{`$b$ is the transitive closure of $\set{a}$ under $D$'}$$ The *region of $a$ in $M$* is $\set{c\in M\mid M{\vDash}c\in b}$. If $A\subseteq M$, we say that $A$ is a *region of $M$* iff it is the region of some $a\in M$.
\[rem:compdef\] For each $a\in M$, the region of $a$ in $M$ is an $M$-set.
For $a\in M$, if $A$ is the region of $a$ and $B$ is the transitive closure of $\set{a}$ under $D$ computed in the metatheory, i.e. the connected component of $a$ in $M_1$, then $B\subseteq A$. In particular, regions of $M$ are unions of connected components of $M_1$. If $M$ contains non-standard natural numbers and the diameter of $B$ is infinite then the inclusion $B\subseteq A$ may be strict, and $B$ may not even be an $M$-set. From now on, the words ‘connected component’ will only be used in the sense of the metatheory.
Most of the appeals to $\mathsf{AFA}$ in the rest of the paper will be applications of the following proposition. In fact, after proving it, we will only deal directly with flat systems twice more.
\[pr:embedgraphs\] Let $M_1$ be the D-graph of $M{\vDash}\mathsf{ZFA}$, and let $G$ be a graph in $M$. Then there is $H\subseteq M_1$ such that
1. $(H, D^{M_1}{\upharpoonright}H)$ is isomorphic to $G$,
2. $H$ is a union of regions of $M$, and
3. $H$ is an $M$-set.
Work in $M$ until further notice. Let $G$ be a graph in $M$, say in the language $\set{R}$. Let $\kappa$ be its cardinality, and assume up to a suitable isomorphism that $\operatorname{dom}G=\kappa$. In particular, note that every element of $\operatorname{dom}G$ is a well-founded set. Consider the flat system $$\set{x_i=\set{i, x_j\mid j\in\kappa, G{\vDash}R(i,j)}\mid i\in \kappa}$$ Let $s{\colon}x_i\mapsto a_i$ be a solution to the system. If $i\ne j$, then $i\in a_i\setminus a_j$, and therefore $s$ is injective. Observe that
1. \[point:iso\] since $R$ is symmetric, we have $a_i\in a_j\in a_i\iff G{\vDash}R(i,j)$, and
2. \[point:ucc\] for all $b\in M$ and all $i\in \kappa$, we have $b\in a_i\in b$ if and only if there is $j<\kappa$ such that $b=a_j$ and $G{\vDash}R(i,j)$.
Now work in the ambient metatheory. Consider the $M$-set $$H\coloneqq\set{a_i\mid M{\vDash}i\in \kappa}=\set{b\in M\mid M{\vDash}b\in \operatorname{Im}(s)}\subseteq M_1$$ By \[point:iso\] above, $(H, D^{M_1}{\upharpoonright}H)$ is isomorphic to $G$ and, by \[point:ucc\] above, $H$ is a union of regions of $M$.
We can now generalise [@adamdaycameron Theorem 4], answering [@adamdaycameron Question 3]. The words ‘up to isomorphism’ are to be interpreted in the sense of the metatheory, i.e. the isomorphism need not be in $M$.
\[thm:aq3\] Let $M{\vDash}\mathsf{ZFA}$. Up to isomorphism, the connected components of $M_1$ are exactly the connected components (in the sense of the metatheory) of graphs in the sense of $M$. In particular, there are infinitely many copies of each of them.
Let $C$ be a connected component of a graph $G$ in $M$. By Proposition \[pr:embedgraphs\] there is an isomorphic copy $H$ of $G$ which is a union of regions of $M$, hence, in particular, of connected components of $M_1$. Clearly, one of the connected components of $H$ is isomorphic to $C$.
In the other direction, let $a\in M_1$ and consider its connected component. Inside $M$, let $G$ be the region of $a$. Using Remark \[rem:compdef\] it is easy to see that $(G, D{\upharpoonright}G)$ is a graph in $M$, and one of its connected components is isomorphic to the connected component of $a$ in $M_1$.
For the last part of the conclusion take, inside $M$, disjoint unions of copies of a given graph.
If one does not assume some form of $\mathsf{AFA}$ and for instance merely drops Foundation, then double-membership graphs can be essentially arbitrary, as the following proposition shows.
Let $M{\vDash}\mathsf{ZFC}$ and let $G$ be a graph in $M$. There is a model $N$ of $\mathsf{ZFC}$ without Foundation such that $N_1$ is isomorphic to the union of $G$ with infinitely many isolated vertices, i.e. points without any edges or self-loops.
Note that the isolated vertices are necessary, as $N$ will always contain well-founded sets.
Let $G$ be a graph in $M$, say in the language $\set{R}$. Assume without loss of generality that $G$ has no isolated vertices, and that $\operatorname{dom}G$ equals its cardinality $\kappa$. For each $i\in \kappa$ choose $a_i\subseteq \kappa$ which has foundational rank $\kappa$ in $M$, e.g. let $a_i\coloneqq\kappa\setminus \set i$. Let $b_j\coloneqq\set{a_i\mid G{\vDash}R(i,j)}$ and note that, since no vertex of $G$ is isolated, $b_j$ is non-empty, thus has rank $\kappa+1$. Define $\pi{\colon}M\to M$ to be the permutation swapping each $a_i$ with the corresponding $b_i$ and fixing the rest of $M$. Let $N$ be the structure with the same domain as $M$, but with membership relation defined as $$N{\vDash}x\in y\iff M{\vDash}x\in \pi(y)$$ By [@rieger Section 3][^5], $N$ is a model of $\mathsf{ZFC}$ without Foundation. To check that $N_1$ is as required, first observe that $$N{\vDash}a_i\in a_j\iff M{\vDash}a_i\in \pi(a_j)=b_j\iff G{\vDash}R(i,j)$$ so $\set{a_i\mid M{\vDash}i\in \kappa}$, equipped with the restriction of $D^{N_1}$, is isomorphic to $G$. To show that there are no other $D$-edges in $N_1$, assume that $N_1{\vDash}D(x,y)$, and consider the following three cases (which are exhaustive since $D$ is symmetric).
1. $x$ and $y$ are both fixed points of $\pi$. This contradicts Foundation in $M$.
2. $y=a_i$ for some $i$, so $N{\vDash}x\in a_i$, hence $M{\vDash}x\in \pi(a_i)=b_i$. Then $x=a_j$ for some $j$ by construction.
3. $y=b_i$ for some $i$. From $N{\vDash}x\in b_i$ we get $M{\vDash}x\in a_i\subseteq \kappa$, thus $x$ has rank strictly less than $\kappa$. Therefore, $x$ is not equal to any $a_j$ or $b_j$, hence $\pi(x)=x$. Again by rank considerations, it follows that $M{\vDash}b_i\notin x=\pi(x)$, so $N{\vDash}b_i\notin x$, a contradiction.
Continuum-Many Countable Models {#sec:cmany}
===============================
We now turn our attention to answering [@adamdaycameron Questions 1 and 2]. Namely, we compute, via a type-counting argument, the number of non-isomorphic D-graphs of countable models of $\mathsf{ZFA}$ and the number of countable models of their complete theories. The analogous results for SD-graphs also hold.
\[defin:flowers\] Let $n\in \omega\setminus\set{0}$. Define the $L_1$-formula $$\begin{gathered}
{\varphi}_n(x)\coloneqq \neg D(x,x)\land \exists {{z}_{0},\ldots,{z}_{n-1}} \Bigl(\bigl(\bigwedge_{0\le i<j<n}z_i\ne z_j\bigr)\\
\land \bigl(\bigwedge_{0\le i<n}D(z_i, x)\bigr)\land \bigl(\forall z\; D(z,x){\rightarrow}\bigvee_{0\le i< n} z=z_i\bigr)
\Bigr)
\end{gathered}$$ For $A$ a subset of $\omega\setminus\set{0}$, define the set of $L_1$-formulas $$\begin{gathered}
\beta_{A}(y)\coloneqq \set{\neg D(y,y)}\cup \set{\exists x_n\; {\varphi}_n(x_n)\land D(y,x_n)\mid n\in A}\\\cup \set{\neg(\exists x_n\; {\varphi}_n(x_n)\land D(y,x_n))\mid n\in \omega\setminus(\set{0}\cup A)}
\end{gathered}$$ We say that $a\in M_1$ is an *$n$-flower* iff $M_1{\vDash}{\varphi}_n(a)$. We say that $b\in M_1$ is an *$A$-bouquet* iff for all $\psi(y)\in \beta_{A}(y)$ we have $M_1{\vDash}\psi(b)$.
So $a$ is an $n$-flower if and only if, in the D-graph, it is a point of degree $n$ without a self-loop, while $b$ is an $A$-bouquet iff it has no self-loop, it has $D$-edges to at least one $n$-flower for every $n\in A$, and it has no $D$-edges to any $n$-flower if $n\notin A$.
\(y) at (0,0)[$a$]{}; (0) at (0.59,0.81)[$\set{a,0}$]{}; (1) at (-0.59,0.81)[$\set{a,1}$]{}; (2) at (-0.95,-0.309)[$\set{a,2}$]{}; (3) at (0,-1)[$\set{a,3}$]{}; (4) at (0.95,-0.309)[$\set{a,4}$]{};
\(y) edge\[postaction=decorate\] (0) (0) edge\[postaction=decorate\] (y) (y) edge\[postaction=decorate\] (1) (1) edge\[postaction=decorate\] (y) (y) edge\[postaction=decorate\] (2) (2) edge\[postaction=decorate\] (y) (y) edge\[postaction=decorate\] (3) (3) edge\[postaction=decorate\] (y) (y) edge\[postaction=decorate\] (4) (4) edge\[postaction=decorate\] (y) ;
\[lemma:bouquetsexist\] Let $A_0$ be a finite subset of $\omega\setminus\set{0}$ and let $M{\vDash}\mathsf{ZFA}$. Then $M_1$ contains an $A_0$-bouquet.
It suffices to find a certain finite graph as a connected component of $M_1$, so this follows from Proposition \[pr:embedgraphs\] (or directly from [@adamdaycameron Theorem 4]).
If $M$ is a structure, denote by $\operatorname{Th}(M)$ its theory.
\[pr:contrtypes\] Let $M{\vDash}\mathsf{ZFA}$. Then in $\operatorname{Th}(M_1)$ the $2^{\aleph_0}$ sets of formulas $\beta_A$, for $A\subseteq \omega\setminus \set0$, are each consistent, and pairwise contradictory. In particular, the same is true in $\operatorname{Th}(M)$.
If $A, B$ are distinct subsets of $\omega\setminus\set 0$ and, without loss of generality, there is an $n\in A\setminus B$, then $\beta_A$ contradicts $\beta_B$ because $\beta_A(y){\vdash}\exists x_n\; ({\varphi}_n(x_n)\land D(y,x_n))$ and $\beta_B(y){\vdash}\neg \exists x_n\; ({\varphi}_n(x_n)\land D(y,x_n))$.
To show that each $\beta_A$ is consistent it is enough, by compactness, to show that if $A_0$ is a finite subset of $A$ and $A_1$ is a finite subset of $\omega\setminus(\set{0}\cup A)$ then there is some $b\in M$ with a $D$-edge to an $n$-flower for every $n\in A_0$ and no $D$-edges to $n$-flowers whenever $n\in A_1$. Any $A_0$-bouquet will satisfy these requirements and, by Lemma \[lemma:bouquetsexist\], an $A_0$-bouquet exists inside $M_1$.
For the last part, note that all the theories at hand are complete (in different languages), and whether or not an intersection of definable sets is empty does not change after adding more definable sets.
To conclude, we need the following standard fact from model theory.
\[fact:manytypes\] Every partial type over $\emptyset$ of a countable theory can be realised in a countable model.
\[co:manymodels\] Let $M$ be a model of $\mathsf{ZFA}$. There are $2^{\aleph_0}$ countable models of $\mathsf{ZFA}$ such that their D-graphs (resp. SD-graphs) are elementarily equivalent to $M_1$ (resp. $M_0$) and pairwise non-isomorphic.
Consider the pairwise contradictory partial types $\beta_A$. By Fact \[fact:manytypes\], $\operatorname{Th}(M)$ has $2^{\aleph_0}$ distinct countable models, as each of them can only realise countably many of the $\beta_A$. The reducts to $L_1$ (resp. $L_0$) of models realising different subsets of $\set{\beta_A\mid A\subseteq \omega\setminus\set 0}$ are still non-isomorphic, since the $\beta_A$ are partial types in the language $L_1$.
The previous Corollary answers affirmatively [@adamdaycameron Questions 1 and 2].
For the results in this section to hold, it is not necessary that $M$ satisfies the whole of $\mathsf{ZFA}$. It is enough to be able to prove Lemma \[lemma:bouquetsexist\] for $M$, and it is easy to see than one can provide a direct proof whenever in $M$ it is possible to define infinitely many different well-founded sets, e.g. von Neumann natural numbers, and to ensure existence of solutions to flat systems of equations. This can be done as long as $M$ satisfies Extensionality, Empty Set, Pairing, and $\mathsf{AFA}_1$[^6]. If we replace, in Definition \[defin:flatsystem\], ‘$x=S_x$’ with ‘$x$ and $S_x$ have the same elements’, then we can even drop Extensionality.
Common Theory {#sec:theory}
=============
The main aim of this section is to study the common theory of the class of D-graphs of $\mathsf{ZFA}$. We show in Corollary \[co:concon\] that it is incomplete, and in Corollary \[co:completions\] characterise its completions in terms of collections of consistency statements. Furthermore, we show that each of these completions is untame in the sense of neostability theory (Corollary \[co:untame\]) and has a countable model which is not a D-graph, and that the same holds for SD-graphs (Corollary \[co:q5\]), therefore solving negatively [@adamdaycameron Question 5].
Let $K_1$ be the class of D-graphs of models of $\mathsf{ZFA}$. Let $\operatorname{Th}(K_1)$ be its common $L_1$-theory.
Let ${\varphi}$ be an $L_1$-sentence. We define an $L_1$-sentence $\mu({\varphi})$ as follows. Let $x$ be a variable not appearing in ${\varphi}$. Let $\chi(x)$ be obtained from ${\varphi}$ by relativising $\exists y$ and $\forall y$ to $D(x,y)$. Let $\mu({\varphi})$ be the formula $\exists x\; (\neg D(x,x)\land \chi(x))$.
In other words, $\mu({\varphi})$ can be thought of as saying that there is a point whose set of neighbours is a model of ${\varphi}$.
Suppose ${\varphi}$ is a ‘standard’ sentence, i.e. one which is a formula in the sense of the metatheory, say in the finite language $L'$. Let $M{\vDash}\mathsf{ZFA}$, and let $N$ be an $L'$-structure in $M$. Then, whether $N{\vDash}{\varphi}$ or not is absolute between $M$ and the metatheory. Every formula we mention is of this kind, and this fact will be used tacitly from now on.
Let $\Phi$ be the set of $L_1$-sentences which imply $\forall x,y\;(D(x,y){\rightarrow}D(y,x))$.
\[lemma:muworks\] For every $L_1$-sentence ${\varphi}\in \Phi$ and every $M{\vDash}\mathsf{ZFA}$ we have $$M{\vDash}\operatorname{Con}({\varphi})\iff M_1{\vDash}\mu({\varphi})$$ Moreover, if this is the case, then there is $H\subseteq M_1$ such that
1. $(H, D^{M_1}{\upharpoonright}H)$ satisfies ${\varphi}$,
2. $H$ is a union of regions of $M$, and
3. $H$ is an $M$-set.
Note that the class of graphs in $M$ is closed under the operations of removing a point or adding one and connecting it to everything. Now apply Proposition \[pr:embedgraphs\].
Define $L_{\mathsf{NBG}}\coloneqq\set{E}$, where $E$ is a binary relational symbol. We think of $L_1$ as ‘the language of graphs’ and of $L_{\mathsf{NBG}}$ as ‘the language of digraphs’, specifically, digraphs that are models of a certain class theory (see below), hence the notation. It is well-known that every digraph is interpretable in a graph, and that such an interpretation may be chosen to be uniform, in the sense below. See e.g. [@hodges Theorem 5.5.1].
\[fact:interpretation\] Every $L_{\mathsf{NBG}}$-structure $N$ is interpretable in a graph $N'$. Moreover, for every $L_{\mathsf{NBG}}$-sentence $\theta$ there is an $L_1$-sentence $\theta'$ such that
1. $\theta$ is consistent if and only if $\theta'$ is, and
2. for every $L_{\mathsf{NBG}}$-structure $N$ we have $N{\vDash}\theta\iff N'{\vDash}\theta'$.
\[co:muphiprime\] For every $L_{\mathsf{NBG}}$-sentence $\theta$, let $\theta'$ be as in Fact \[fact:interpretation\]. For all $M{\vDash}\mathsf{ZFA}$ $$M{\vDash}\operatorname{Con}(\theta)\iff M_1{\vDash}\mu(\theta')$$
Apply Lemma \[lemma:muworks\] to ${\varphi}\coloneqq \theta'$.
\[co:untame\] Let $M{\vDash}\mathsf{ZFA}$. Then every model of $\operatorname{Th}(M_1)$ interprets with parameters arbitrarily large finite fragments of $\mathsf{ZFC}$. In particular $\operatorname{Th}(M_1)$ has $\mathsf{SOP}$, $\mathsf{TP_2}$, and $\mathsf{IP}_k$ for all $k$.
If $\theta$ is the conjunction of a finite fragment of $\mathsf{ZFC}$, it is well-known that $\mathsf{ZFA}{\vdash}\operatorname{Con}(\theta)$. Since a model of $\theta$ is a digraph, we can apply Corollary \[co:muphiprime\]. If $a$ witnesses the outermost existential quantifier in $\mu(\theta')$, then $\theta$ is interpretable with parameter $a$.
We now want to use Corollary \[co:muphiprime\] to show that the common theory $\operatorname{Th}(K_1)$ of the class of D-graphs of models of $\mathsf{ZFA}$ is incomplete. Naively, this could be done by choosing $\theta$ to be a finite axiomatisation of some theory equiconsistent with $\mathsf{ZFA}$, and then invoking the Second Incompleteness Theorem. For instance, one could choose von Neumann-Bernays-Gödel class theory $\mathsf{NBG}$, axiomatised in the language $L_\mathsf{NBG}$[^7], as this is known to be equiconsistent with $\mathsf{ZFC}$ (see [@felgner]), hence with $\mathsf{ZFA}$. The problem with this argument is that, in order for it to work, we need a further set-theoretical assumption in our metatheory, namely $\operatorname{Con}(\mathsf{ZFC}+\operatorname{Con}(\mathsf{ZFC}))$. This can be avoided by using another sentence whose consistency is independent of $\mathsf{ZFA}$, provably in $\mathsf{ZFC}+\operatorname{Con}(\mathsf{ZFC})$ alone. We would like to thank Michael Rathjen for pointing out to us the existence of such a sentence.
Let $\mathsf{NBG}^-$ denote $\mathsf{NBG}$ without the axiom of Infinity. We will use special cases of a classical theorem of Rosser and of a related result. For proofs of these, together with their more general statements, we refer the reader to [@smorynskitw Chapter 7, Application 2.1 and Corollary 2.6].
There is a $\Pi^0_1$ arithmetical statement $\psi$ which is independent of $\mathsf{ZFA}$.
\[fact:eqcon\] Let $\psi$ be a $\Pi^0_1$ arithmetical statement. There is another arithmetical statement $\widetilde \psi$ such that $\mathsf{ZFA}{\vdash}\psi{\leftrightarrow}\operatorname{Con}(\mathsf{NBG}^-+\widetilde\psi)$.
\[co:concon\] $\operatorname{Th}(K_1)$ is not complete.
Let $\psi$ be given by Rosser’s Theorem, and let $\widetilde\psi$ be given by Fact \[fact:eqcon\] applied to $\psi$. Apply Corollary \[co:muphiprime\] to $\theta\coloneqq\mathsf{NBG}^-+\widetilde\psi$.
It is therefore natural to study the completions of $\operatorname{Th}(K_1)$, and it follows easily from $K_1$ being pseudoelementary that all of these are the theory of some actual D-graph $M_1$. We provide a proof for completeness.
\[pr:completions\] Let $T$ be an $L$-theory, and let $K$ be the class of its models. Let $L_1\subseteq L$, and for $M\in K$ denote $M_1\coloneqq M{\upharpoonright}L_1$. Let $K_1\coloneqq\set{M_1\mid M\in K}$ and $N{\vDash}\operatorname{Th}(K_1)$. Then there is $M\in K$ such that $M_1 \equiv N$.
We are asking whether there is any $M{\vDash}T\cup \operatorname{Th}(N)$, so it is enough to show that the latter theory is consistent. If not, there is an $L_1$-formula ${\varphi}\in \operatorname{Th}(N)$ such that $T{\vdash}\neg {\varphi}$. In particular, since $\neg{\varphi}\in L_1$, we have that $\operatorname{Th}(K_1){\vdash}\neg {\varphi}$, and this contradicts that $N{\vDash}\operatorname{Th}(K_1)$.
In order to characterise the completions of $\operatorname{Th}(K_1)$, we will use techniques from finite model theory, namely Ehrenfeucht-Fraïssé games and $k$-equivalence. For background on these concepts, see [@ebbinghausflum].
\[lemma:duog\] Let $G=G_0\sqcup G_1$ be a graph with no edges between $G_0$ and $G_1$, and let $H=H_0\sqcup H_1$ be a graph with no edges between $H_0$ and $H_1$. If $(G_0, {{a}_{1},\ldots,{a}_{m-1}})\equiv_k (H_0,{{b}_{1},\ldots,{b}_{m-1}})$ and $(G_1, a_m)\equiv_k (H_1, b_m)$, then $(G,{{a}_{1},\ldots,{a}_{m}})\equiv_k (H, {{b}_{1},\ldots,{b}_{m}})$.
This is standard, see e.g. [@ebbinghausflum Proposition 2.3.10].
\[thm:completions\] Let $M$ and $N$ be models of $\mathsf{ZFA}$. The following are equivalent.
1. \[point:m1n1\] $M_1\equiv N_1$.
2. \[point:Phi\] $M_1$ and $N_1$ satisfy the same sentences of the form $\mu({\varphi})$, as ${\varphi}$ ranges in $\Phi$.
3. \[point:consistency\]$M$ and $N$ satisfy the same consistency statements.
For statements about graphs, the equivalence of \[point:Phi\] and \[point:consistency\] follows from Lemma \[lemma:muworks\]. For statements in other languages, it is enough to interpret them in graphs using [@hodges Theorem 5.5.1].
For the equivalence of \[point:m1n1\] and \[point:Phi\], we show that for every $n\in \omega$ the Ehrenfeucht-Fraïssé game between $M_1$ and $N_1$ of length $n$ is won by the Duplicator, by describing a winning strategy. The idea behind the strategy is the following. Recall that, for every finite relational language and every $k$, there is only a finite number of $\equiv_{k}$-classes, each characterised by a single sentence (see e.g. [@ebbinghausflum Corollary 2.2.9]). After the Spoiler plays a point $a$, the Duplicator replicates the $\equiv_k$-class of the region of $a$ using Lemma \[lemma:muworks\].
Fix the length $n$ of the game and denote by ${{a}_{1},\ldots,{a}_{m}}\in M_1$ and ${{b}_{1},\ldots,{b}_{m}}\in N_1$ the points chosen at the end of turn $m$. The Duplicator defines, by simultaneous induction on $m$, sets $G^{m}_0\subseteq M_1$ and $H^{m}_0\subseteq N_1$, and makes sure that they satisfy the following conditions.
1. \[point:trivial\] ${{a}_{1},\ldots,{a}_{m}}\in G^m_0$ and ${{b}_{1},\ldots,{b}_{m}} \in H^m_0$.
2. \[point:selfcontained\] $G^m_0$ and $H^m_0$ are unions of regions of $M$ and $N$ respectively.
3. \[point:aresets\] $G_0^m$ and $H_0^m$ are respectively an $M$-set and an $N$-set.
4. \[point:stategycondition\] When $G^m_0$ and $H^m_0$ are equipped with the $L_1$-structures induced by $M$ and $N$ respectively, we have $(G^m_0,{{a}_{1},\ldots,{a}_{m}})\equiv_{n-m} (H^m_0, {{b}_{1},\ldots,{b}_{m}})$.
Before the game starts (‘after turn $0$’) we set $G^0_0=H^0_0=\emptyset$ and all conditions trivially hold. Assume inductively that they hold after turn $m-1$. We deal with the case where the Spoiler plays $a_m\in M_1$; the case where the Spoiler plays $b_m \in N_1$ is symmetrical.
Let $G^{m}_1$ be the region of $a_m$ in $M$. If $G_1^m\subseteq G_0^{m-1}$ then, since by inductive hypothesis condition \[point:stategycondition\] held after turn $m-1$, the Duplicator can find $b_m\in H_0^{m-1}$ such that $(G_0^{m-1},{{a}_{0},\ldots,{a}_{m}})\equiv_{n-m}(H_0^{m-1},{{b}_{0},\ldots,{b}_{m}})$. It is then clear that all conditions hold after setting $G_0^m=G_0^{m-1}$ and $H_0^m=H_0^{m-1}$.
Otherwise, by \[point:selfcontained\], we have $G^m_1\cap G^{m-1}_0=\emptyset$. Let ${\varphi}$ characterise the $\equiv_{n-m+1}$-class of $G^m_1$. Note that, if $n-m+1\ge 2$, then ${\varphi}\in \Phi$ automatically. Otherwise, replace ${\varphi}$ with ${\varphi}\land \forall x\forall y\; (D(x,y){\rightarrow}D(y,x))$. By Remark \[rem:compdef\], $G^m_1$ is an $M$-set, hence $M{\vDash}\operatorname{Con}({\varphi})$. By Lemma \[lemma:muworks\] and assumption, there is a union $H^m_1$ of regions of $N$ which is an $N$-set and such that $G^m_1\equiv_{n-m+1} H^m_1$. By inductive hypothesis, $H_0^{m-1}$ is also an $N$-set by \[point:aresets\]. Therefore, up to writing a suitable flat system in $N$, we may replace $H_1^m$ with an isomorphic copy which is still a union of regions and an $N$-set, but with $H_1^m\cap H_0^{m-1}=\emptyset$.
Let $b_m\in H_1^m$ be the choice given by a winning strategy for the Duplicator in the game of length $n-m+1$ between $G_1^m$ and $H_1^m$ after the Spoiler plays $a_m\in G_1^m$ as its first move. Set $G^m_0=G^{m-1}_0\cup G^m_1$ and $H^m_0=H^{m-1}_0\cup H^m_1$. Note that $G^{m-1}_0, G^m_1, H^{m-1}_0, H^m_1$ are all unions of regions and $M$-sets or $N$-sets, hence \[point:selfcontained\] and \[point:aresets\] hold (and \[point:trivial\] is clear). Moreover both unions are disjoint, so the hypotheses of Lemma \[lemma:duog\] are satisfied and $(G^m_0,{{a}_{1},\ldots,{a}_{m}}) \equiv_{n-m} (H^m_0, {{b}_{1},\ldots,{b}_{m}})$, i.e. \[point:stategycondition\] holds.
To show that this strategy is winning, note that the outcome of the game only depends on the induced structures on ${{a}_{1},\ldots,{a}_{n}}$ and ${{b}_{1},\ldots,{b}_{n}}$ at the end of the final turn. These do not depend on what is outside $G_0^n$ and $H_0^n$ since they are unions of regions, hence unions of connected components. As \[point:stategycondition\] holds at the end of turn $n$, the structures induced on ${{a}_{1},\ldots,{a}_{n}}$ and ${{b}_{1},\ldots,{b}_{n}}$ are isomorphic.
\[co:completions\] Let $N{\vDash}\operatorname{Th}(K_1)$. Then $\operatorname{Th}(N)$ is axiomatised by $$\operatorname{Th}(K_1)\cup \set{\mu({\varphi})\mid {\varphi}\in \Phi, N{\vDash}\mu({\varphi})}\cup \set{\neg \mu({\varphi})\mid {\varphi}\in \Phi, N{\vDash}\neg\mu({\varphi})}$$
Let $N'$ satisfy the axiomatisation above. Since $N$ and $N'$ are models of $\operatorname{Th}(K_1)$ we may, by Proposition \[pr:completions\], replace them with D-graphs $M_1\equiv N$ and $M_1'\equiv N'$ of models of $\mathsf{ZFA}$. By Theorem \[thm:completions\] $M_1\equiv M_1'$.
By the previous corollary, combined with Lemma \[lemma:muworks\], theories of double-membership graphs correspond bijectively to consistent (with $\mathsf{ZFA}$, equivalently with $\mathsf{ZFC}$) collections of consistency statements.
The reader familiar with finite model theory may have noticed similarities between the proof of Theorem \[thm:completions\] and certain proofs of the theorems of Hanf and Gaifman (see [@ebbinghausflum Theorems 2.4.1 and 2.5.1]). In fact one could deduce a statement similar to Theorem \[thm:completions\] directly from Gaifman’s Theorem. This would characterise the completions of $\operatorname{Th}(K_1)$ in terms of *local formulas*, of which the $\mu({\varphi})$ form a subclass, yielding a less specific result than Corollary \[co:completions\]. Moreover, we believe that the correspondence with collections of consistency statements provides a conceptually clearer picture.
Similar ideas can be used to study [@adamdaycameron Question 5], which asks whether a countable structure elementarily equivalent to the SD-graph $M_0$ of some $M{\vDash}\mathsf{ZFA}$ must itself be the SD-graph of some model of $\mathsf{ZFA}$. We provide a negative solution in Corollary \[co:q5\]. Again, Gaifman’s Theorem could be used directly to deduce its second part.
\[thm:noinfdiam\] Let $M{\vDash}\mathsf{ZFA}$. There is a countable $N\equiv M_0$ such that $N{\upharpoonright}L_1$ has no connected component of infinite diameter.
Before the proof, we show how this solves [@adamdaycameron Question 5].
\[co:q5\] For every $M{\vDash}\mathsf{ZFA}$ there are a countable $N\equiv M_0$ which is not the SD-graph of any model of $\mathsf{ZFA}$ and a countable $N'\equiv M_1$ which is not the D-graph of any model of $\mathsf{ZFA}$.
Let $N$ be given by Theorem \[thm:noinfdiam\] and $N'\coloneqq N{\upharpoonright}L_1$. Now observe that, as follows easily from Proposition \[pr:embedgraphs\], any reduct to $L_1$ of a model of $\mathsf{ZFA}$ has a connected component of infinite diameter.
Note that this proves slightly more: a negative solution to the question would only have required to find a single pair $(M_0, N)$ satisfying the conclusion of the corollary.
Up to passing to a countable elementary substructure, we may assume that $M$ itself is countable. Let $N$ be obtained from $M_0$ by removing all points whose connected component in $M_1$ has infinite diameter. We show that $M_0\equiv N$ by exhibiting, for every $n$, a sequence $(I_j)_{j\le n}$ of non-empty sets of partial isomorphisms between $M_0$ and $N$ with the back-and-forth property (see [@ebbinghausflum Definition 2.3.1 and Corollary 2.3.4]). The idea is to adapt the proof of [@otto Lemma 2.2.7] (essentially Hanf’s Theorem) by considering the Gaifman balls with respect to $L_1$, while requiring the partial isomorphisms to preserve the richer language $L_0$.
On an $L_0$-structure $A$, consider the distance $d{\colon}A\to \omega\cup \set \infty$ given by the graph distance in the reduct $A{\upharpoonright}L_1$ (where $d(a,b)=\infty$ iff $a,b$ lie in distinct connected components). If ${{a}_{1},\ldots,{a}_{k}}\in A$ and $r\in \omega$, denote by $\operatorname{dom}(B(r,{{a}_{1},\ldots,{a}_{k}}))$ the union of the balls of radius $r$ (with respect to $d$) centred on ${{a}_{1},\ldots,{a}_{k}}$. Equip $\operatorname{dom}(B(r,{{a}_{1},\ldots,{a}_{k}}))$ with the $L_0$-structure induced by $A$, then expand to an $L_0\cup\set{{{c}_{1},\ldots,{c}_{k}}}$-structure $B(r, {{a}_{1},\ldots,{a}_{k}})$ by interpreting each constant symbol $c_i$ with the corresponding $a_i$. We stress that, even though $B(r, {{a}_{1},\ldots,{a}_{k}})$ carries an $L_0\cup\set{{{c}_{1},\ldots,{c}_{k}}}$-structure, and we consider isomorphisms with respect to this structure, the balls giving its domain are defined with respect to the distance induced by $L_1$ alone.
Set $r_j\coloneqq (3^{j}-1)/2$ and fix $n$. Define $I_n\coloneqq\set{\emptyset}$, where $\emptyset$ is thought of as the empty partial map $M_0\to N$. For $j<n$, let $I_j$ be the following set of partial maps $M_0\to N$: $$I_j\coloneqq\set{{{a}_{1},\ldots,{a}_{k}}\mapsto {{b}_{1},\ldots,{b}_{k}}\mid k\le n-j, B(r_j, {{a}_{1},\ldots,{a}_{k}})\cong B(r_j, {{b}_{1},\ldots,{b}_{k}})}$$ We have to show that for every map ${{a}_{1},\ldots,{a}_{k}}\mapsto {{b}_{1},\ldots,{b}_{k}}$ in $I_{j+1}$ and every $a\in M_0$ \[resp. every $b\in N$\] there is $b\in N$ \[resp. $a\in M_0$\] such that ${{a}_{1},\ldots,{a}_{k}}, a\mapsto {{b}_{1},\ldots,{b}_{k}}, b$ is in $I_j$.
Denote by $\iota$ an isomorphism $B(r_{j+1}, {{a}_{1},\ldots,{a}_{k}})\to B(r_{j+1}, {{b}_{1},\ldots,{b}_{k}})$ and let $a\in M_0$. If $a$ is chosen in $B(2\cdot r_j+1, {{a}_{1},\ldots,{a}_{k}})$, then by the triangle inequality and the fact that $2\cdot r_j+1+r_j=r_{j+1}$ we have $B(r_j, a)\subseteq B(r_{j+1}, {{a}_{1},\ldots,{a}_{k}})$, and we can just set $b\coloneqq \iota(a)$.
Otherwise, again by the triangle inequality, $B(r_j, a)$ and $B(r_{j}, {{a}_{1},\ldots,{a}_{k}})$ are disjoint and there is no $D$-edge between them. Note, moreover, that they are $M$-sets. This allows us to write a suitable flat system, which will yield the desired $b$.
Working inside $M$, for every $d\in B(r_j, a)$ choose a well-founded set $h_d$ such that for all $d,d_0,d_1\in B(r_j,a)$ we have
1. \[point:noedge\] $h_{d_0}\notin h_{d_1}$,
2. \[point:distinct\]if $d_0\ne d_1$ then $h_{d_0}\ne h_{d_1}$,
3. \[point:nocup\]$h_d\notin B(r_{j}, {{b}_{1},\ldots,{b}_{k}})$,
4. \[point:cup\]$h_d\notin \bigcup B(r_{j}, {{b}_{1},\ldots,{b}_{k}})$, and
5. \[point:cupcup\]$h_d\notin \bigcup \bigcup B(r_{j}, {{b}_{1},\ldots,{b}_{k}})$.
Let $\set{x_d\mid d\in B(r_j,a)}$ be a set of indeterminates. Define $$\begin{aligned}
P_d&\coloneqq\set{x_{e}\mid e\in B(r_j, a), M{\vDash}e\in d} \\
Q_d&\coloneqq\set{\iota(f)\mid f\in B(r_j, {{a}_{1},\ldots,{a}_{k}}), M{\vDash}S(d,f)}\end{aligned}$$ and consider the flat system $$\set{x_d=\set{h_d}\cup P_d \cup Q_d\mid d\in B(r_j, a)}\label{eq:lastflatsystem}\tag{$*$}$$ Intuitively, the terms $P_d$ ensure that the image of a solution is an isomorphic copy of $B(r_j, a)$, while the terms $Q_d$ create the appropriate $S$-edges between the image and $B(r_j,{{b}_{1},\ldots,{b}_{k}})$ (note that we do not need any $D$-edges because there are none between $B(r_j,a)$ and $B(r_j, {{a}_{1},\ldots,{a}_{k}})$). The $\set{h_d}$ are needed for bookkeeping reasons, in order to avoid pathologies. We now spell out the details; keep in mind that each $P_d$ consists of indeterminates, and each $Q_d$ is a subset of $B(r_j, {{b}_{1},\ldots,{b}_{k}})$.
Let $s$ be a solution of , guaranteed to exist by $\mathsf{AFA}$. By \[point:noedge\] and the fact that each member of $\operatorname{Im}(s)$ contains some $h_d$, we have $\set{h_d\mid d\in B(r_j,a)}\cap \operatorname{Im}(s)=\emptyset$. Using this together with \[point:distinct\] and \[point:nocup\] we have $h_d\in s(x_e)\iff d=e$, hence $s$ is injective.
Let $s'\coloneqq d\mapsto s(x_d)$ and $b\coloneqq s'(a)$. By \[point:cup\] we have that $\operatorname{Im}(s)$ does not intersect $B(r_j, {{b}_{1},\ldots,{b}_{k}})$, and we already showed that it does not meet $\set{h_d\mid d\in B(r_j,a)}$. By looking at and at the definition of the terms $P_d$, we have that $\operatorname{Im}(s)=B(r_j, b)$ and that $s'$ is an isomorphism $B(r_j,a)\to B(r_j, b)$.
Note that the only $D$-edges involving points of $\operatorname{Im}(s)$ can come from the terms $P_d$: the $h_d$ are well-founded, and there are no $g\in \operatorname{Im}(s)$ and $\ell\in B(r_{j}, {{b}_{1},\ldots,{b}_{k}})$ such that $g\in \ell$, since $g$ contains some $h_d$ but this cannot be the case for any element of $\ell$ because of \[point:cupcup\]. Hence $\operatorname{Im}(s)$ is a connected component of $M_1$ and it has diameter not exceeding $2\cdot r_j$, so is included in $N$.
Set $\iota'\coloneqq s'\cup(\iota {\upharpoonright}B(r_j,{{a}_{1},\ldots,{a}_{k}}))$. This map is injective because it is the union of two injective maps whose images $B(r_j, b)$ and $B(r_j, {{b}_{1},\ldots,{b}_{k}})$ are, as shown above, disjoint. Moreover, there are no $D$-edges between $B(r_j, b)$ and $B(r_j, {{b}_{1},\ldots,{b}_{k}})$, since the former is a connected component of $M_1$. By inspecting the terms $Q_d$, we conclude that $\iota'$ is an isomorphism $B(r_j, {{a}_{1},\ldots,{a}_{k}},a)\to B(r_j, {{b}_{1},\ldots,{b}_{k}},b)$, and this settles the ‘forth’ case.
The proof of the ‘back’ case, where we are given $b\in N$ and need to find $a\in M_0$, is analogous (and shorter, as we do not need to ensure that the new points are in $N$): we can consider statements such as $e\in d$ when $e,d\in N$ since the domain of the $L_0$-structure $N$ is a subset of $M$.
[Problems]{} We leave the reader with some open problems.
1. Axiomatise the theory of D-graphs of models of $\mathsf{ZFA}$.
2. Axiomatise the theory of SD-graphs of models of $\mathsf{ZFA}$.
3. Characterise the completions of the theory of SD-graphs of models of $\mathsf{ZFA}$.
[Acknowledgements]{} We are grateful to Michael Rathjen for pointing out to us Fact \[fact:eqcon\], and to Dugald Macpherson and Vincenzo Mantova for their guidance and feedback. The first author is supported by a Leeds Doctoral Scholarship. The second and third authors are supported by Leeds Anniversary Research Scholarships.
[ADC17]{}
P. <span style="font-variant:small-caps;">Aczel</span>. *Non-Wellfounded Sets*, volume 14 of *[CSLI]{} Lecture Notes*. Publications (1988).
B. <span style="font-variant:small-caps;">Adam-Day</span> and P. <span style="font-variant:small-caps;">Cameron</span>. *Undirecting membership in models of [ZFA]{}*. Preprint available at <https://arxiv.org/abs/1708.07943> (submitted, 2017).
J. <span style="font-variant:small-caps;">Barwise</span> and L. S. <span style="font-variant:small-caps;">Moss</span>. *Vicious Circles*, volume 60 of *[CSLI]{} Lecture Notes*. Publications (1996).
P. <span style="font-variant:small-caps;">Cameron</span>. *The [Random]{} [Graph]{}*. In *The Mathematics of [Paul]{} [Erdős]{}*, edited by R. L. <span style="font-variant:small-caps;">Graham</span>, J. <span style="font-variant:small-caps;">[Nešetřil]{}</span> and S. <span style="font-variant:small-caps;">Butler</span>, volume 2, pages 353–378. Springer (2013).
H.-D. <span style="font-variant:small-caps;">Ebbinghaus</span> and J. <span style="font-variant:small-caps;">Flum</span>. *Finite model theory*. Springer Monographs in Mathematics. Springer-Verlag (1995).
U. <span style="font-variant:small-caps;">Felgner</span>. *Comparison of the axioms of local and universal choice*. *Fundamenta Mathematicae*, 71:43–62 (1971).
M. <span style="font-variant:small-caps;">Forti</span> and F. <span style="font-variant:small-caps;">Honsell</span>. *Set theory with free construction principles*. *Annali della [Scuola]{} [Normale]{} [Superiore]{} di [Pisa]{} — [Classe]{} di [Scienze]{}*, 10:493–522 (1983).
W. <span style="font-variant:small-caps;">Hodges</span>. *Model Theory*, volume 42 of *Encyclopedia of Mathematics and its Applications*. Cambridge University Press (1993).
K. <span style="font-variant:small-caps;">Kunen</span>. *Set theory*, volume 102 of *Studies in logic and the foundations of mathematics*. North Holland (1980).
M. <span style="font-variant:small-caps;">Otto</span>. *Finite model theory*. <https://www.karlin.mff.cuni.cz/~krajicek/otto.pdf> (2006).
L. <span style="font-variant:small-caps;">Rieger</span>. *A contribution to [Gödel’s]{} axiomatic set theory, [I]{}*. *Czechoslovak Mathematical Journal*, 07:323–357 (1957).
C. <span style="font-variant:small-caps;">Smorynski</span>. *Self-reference and modal logic*. Universitext. Springer (1985).
[^1]: email: [B.Adam-Day@leeds.ac.uk](B.Adam-Day@leeds.ac.uk) <span style="font-variant:small-caps;">orcid</span>: <https://orcid.org/0000-0002-7891-916X>
[^2]: email: [J.A.Howe@leeds.ac.uk](J.A.Howe@leeds.ac.uk) <span style="font-variant:small-caps;">orcid</span>: <https://orcid.org/0000-0001-7580-6349>
[^3]: email: [R.Mennuni@leeds.ac.uk](R.Mennuni@leeds.ac.uk) <span style="font-variant:small-caps;">orcid</span>: <https://orcid.org/0000-0003-2282-680X>
[^4]: Note that, in the literature, the name $\mathsf{ZFA}$ is also used for a certain variant of $\mathsf{ZFC}$ which admits urelements.
[^5]: Strictly speaking, [@rieger] works in class theory. The exact statement we use is that of [@kunen Chapter IV, Exercise 18].
[^6]: Stated using a sensible coding of flat systems, which can be carried out using Pairing.
[^7]: The reader may have encountered an axiomatisation using two sorts; this can be avoided by declaring sets to be those classes that are elements of some other class.
|
---
abstract: 'We construct and apply Strominger-Yau-Zaslow mirror transformations to understand the geometry of the mirror symmetry between toric Fano manifolds and Landau-Ginzburg models.'
address:
- 'The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong'
- '*Current address: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA*'
- 'The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong'
author:
- Kwokwai Chan
- Naichung Conan Leung
title: Mirror symmetry for toric Fano manifolds via SYZ transformations
---
Introduction
============
Mirror symmetry has been extended to the non-Calabi-Yau setting, notably to Fano manifolds, by the works of Givental [@Givental94], [@Givental96], [@Givental97a], Kontsevich [@Kontsevich98] and Hori-Vafa [@HV00]. If $\bar{X}$ is a Fano manifold, then its mirror is conjectured to be a pair $(Y,W)$, where $Y$ is a non-compact Kähler manifold and $W:Y\rightarrow\mathbb{C}$ is a holomorphic Morse function. In the physics literature, the pair $(Y,W)$ is called a *Landau-Ginzburg model*, and $W$ is called the *superpotential* of the model. One of the very first mathematical predictions of this mirror symmetry is that there should be an isomorphism between the small quantum cohomology ring $QH^*(\bar{X})$ of $\bar{X}$ and the Jacobian ring $Jac(W)$ of the function $W$. This has been verified (at least) for toric Fano manifolds by the works of Batyrev [@Batyrev93], Givental [@Givental97a] and many others. A version of the *Homological Mirror Symmetry Conjecture* has also been formulated by Kontsevich [@Kontsevich98], which again has been checked in many cases [@Seidel00], [@Ueda04], [@AKO04], [@AKO05], [@Abouzaid05], [@Abouzaid06]. However, no direct geometric explanation for the mirror symmetry phenomenon for Fano manifolds had been given, until the works of Cho-Oh [@CO03], which showed that, when $\bar{X}$ is a toric Fano manifold, the superpotential $W$ can be computed in terms of the counting of Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers.
On the other hand, the celebrated *Strominger-Yau-Zaslow (SYZ) Conjecture* [@SYZ96] suggested that mirror symmetry for Calabi-Yau manifolds should be understood as a *T-duality*, i.e. dualizing special Lagrangian torus fibrations, modified with suitable quantum corrections. This will explain the geometry underlying mirror symmetry [@Morrison96]. Recently, Gross and Siebert [@GS07] made a breakthrough in the study of this conjecture, after earlier works of Fukaya [@Fukaya02] and Kontsevich-Soibelman [@KS04]. It is expected that their program will finally provide a very explicit and geometric way to see how mirror symmetry works for *both* Calabi-Yau and non-Calabi-Yau manifolds (more precisely, for varieties with effective anticanonical class). On the other hand, in [@Auroux07], Auroux started his program which is aimed at understanding mirror symmetry in the non-Calabi-Yau setting by applying the SYZ approach. More precisely, he studied the mirror symmetry between a general compact Kähler manifold equipped with an anticanonical divisor and a Landau-Ginzburg model, and investigated how the superpotential can be computed in terms holomorphic discs counting on the compact Kähler manifold. In particular, this includes the mirror symmetry for toric Fano manifolds as a special case.\
In this paper, we shall again follow the SYZ philosophy and study the mirror symmetry phenomenon for toric Fano manifolds by using T-duality. The main point of this work, which is also the crucial difference between this and previous works, is that, explicit transformations, which we call *SYZ mirror transformations*, are constructed and used to understand the results (e.g. $QH^*(\bar{X})\cong Jac(W)$) implied by mirror symmetry. From this perspective, this paper may be regarded as a sequel to the second author’s work [@Leung00], where semi-flat SYZ mirror transformations (i.e. fiberwise real Fourier-Mukai transforms) were used to study mirror symmetry for semi-flat Calabi-Yau manifolds. While in that case, quantum corrections do not arise because the Lagrangian torus fibrations are smooth (i.e. they are fiber bundles), we will have to deal with quantum corrections in the toric Fano case.
However, we shall emphasize that the quantum corrections which arise in the toric Fano case are only due to contributions from the anticanonical toric divisor (the toric boundary); correspondingly, the Lagrangian torus fibrations do *not* have proper singular fibers (i.e. singular fibers which are contained in the complement of the anticanonical divisor), so that their bases are affine manifolds with boundaries but *without singularities*. This is simpler than the general non-Calabi-Yau case treated by Gross-Siebert [@GS07] and Auroux [@Auroux07], where further quantum corrections could arise, due to the fact that general Lagrangian torus fibrations do admit proper singular fibers, so that their bases are affine manifolds with *both* boundaries and singularities. Hence, the toric Fano case is in-between the semi-flat case, which corresponds to nonsingular affine manifolds without boundary, and the general case. In particular, in the toric Fano case, we do not need to worry about wall-crossing phenomena, and this is one of the reasons why we can construct the SYZ mirror transformations explicitly as fiberwise Fourier-type transforms, much like what was done in the semi-flat case [@Leung00]. (Another major reason is that holomorphic discs in toric manifolds with boundary in Lagrangian torus fibers are completely classified by Cho-Oh [@CO03].) It is interesting to generalize the results here to non-toric settings, but, certainly, much work needs to be done before we can see how SYZ mirror transformations are constructed and used in the general case. For more detailed discussions of mirror symmetry and the wall-crossing phenomena in non-toric situations , we refer the reader to the works of Gross-Siebert [@GS07] and Auroux [@Auroux07].\
What follows is an outline of our main results. We will focus on one half of the mirror symmetry between a complex $n$-dimensional toric Fano manifold $\bar{X}$ and the mirror Landau-Ginzburg model $(Y,W)$, namely, the correspondence between the symplectic geometry (A-model) of $\bar{X}$ and the complex geometry (B-model) of $(Y,W)$.
To describe our results, let us fix some notations first. Let $N\cong\mathbb{Z}^n$ be a lattice and $M=N^\vee=\textrm{Hom}(N,\mathbb{Z})$ the dual lattice. Also let $N_\mathbb{R}=N\otimes_\mathbb{Z}\mathbb{R}$, $M_\mathbb{R}=M\otimes_\mathbb{Z}\mathbb{R}$, and denote by $\langle\cdot,\cdot\rangle:M_\mathbb{R}\times
N_\mathbb{R}\rightarrow\mathbb{R}$ the dual pairing. Let $\bar{X}$ be a toric Fano manifold, i.e. a smooth projective toric variety such that the anticanonical line bundle $K_{\bar{X}}$ is ample. Let $v_1,\ldots,v_d\in N$ be the primitive generators of the 1-dimensional cones of the fan $\Sigma$ defining $\bar{X}$. Then a polytope $\bar{P}\subset M_\mathbb{R}$ defined by the inequalities $$\langle x,v_i\rangle\geq\lambda_i,\quad i=1,\ldots,d,$$ and with normal fan $\Sigma$, associates a Kähler structure $\omega_{\bar{X}}$ to $\bar{X}$. Physicists [@HV00] predicted that the mirror of ($\bar{X}$, $\omega_{\bar{X}}$) is given by the pair $(Y,W)$, where $Y$, which we call *Hori-Vafa’s mirror manifold*, is biholomorphic to the non-compact Kähler manifold $(\mathbb{C}^*)^n$, and $W:Y\rightarrow\mathbb{C}$ is the Laurent polynomial $$e^{\lambda_1}z^{v_1}+\ldots+e^{\lambda_d}z^{v_d},$$ where $z_1,\ldots,z_n$ are the standard complex coordinates of $Y\cong(\mathbb{C}^*)^n$ and $z^v$ denotes the monomial $z_1^{v^1}\ldots z_n^{v^n}$ if $v=(v^1,\ldots,v^n)\in
N\cong\mathbb{Z}^n$.
The symplectic manifold $(\bar{X},\omega_{\bar{X}})$ admits a Hamiltonian action by the torus $T_N=N_\mathbb{R}/N$, and the corresponding moment map $\mu_{\bar{X}}:\bar{X}\rightarrow\bar{P}$ is naturally a Lagrangian torus fibration. While this fibration is singular (with collapsed fibers) along $\partial\bar{P}$, the restriction to the open dense $T_N$-orbit $X\subset\bar{X}$ is a Lagrangian torus bundle $$\mu=\mu_{\bar{X}}|_X:X\rightarrow P,$$ where $P=\bar{P}\setminus\partial\bar{P}$ is the interior of the polytope $\bar{P}$.[^1] Applying T-duality and the *semi-flat SYZ mirror transformation* (see Definition \[def3.1\]) to this torus bundle, we can, as suggested by the SYZ philosophy, obtain the mirror manifold $Y$ (see Proposition \[prop3.1\] and Proposition \[prop3.2\]).[^2] However, we are not going to get the superpotential $W:Y\rightarrow\mathbb{C}$ because we have ignored the anticanonical toric divisor $D_\infty=\bigcup_{i=1}^d
D_i=\bar{X}\setminus X$, and hence quantum corrections. Here, for $i=1,\ldots,d$, $D_i$ denotes the toric prime divisor which corresponds to $v_i\in N$. To recapture the quantum corrections, we consider the (trivial) $\mathbb{Z}^n$-cover $$\pi:LX=X\times N\rightarrow X$$ and various functions on it.[^3] Let $\mathcal{K}(\bar{X})\subset H^2(\bar{X},\mathbb{R})$ be the Kähler cone of $\bar{X}$. For each $q=(q_1,\ldots,q_l)\in\mathcal{K}(\bar{X})$ (here $l=d-n=\textrm{Picard number of $\bar{X}$}$), we define a $T_N$-invariant function $\Phi_q:LX\rightarrow\mathbb{R}$ in terms of the counting of Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers of $\mu:X\rightarrow P$ (see Definition \[def2.1\] and Remark \[rmk2.1\]). If we further assume that $\bar{X}$ is a product of projective spaces, then this family of functions $\{\Phi_q\}_{q\in\mathcal{K}(\bar{X})}\subset
C^\infty(LX)$ can be used to compute the small quantum cohomology ring $QH^*(\bar{X})$ of $\bar{X}$ as follows (see Section \[sec2\] for details).
\[prop1.1\]$\mbox{}$
1. The logarithmic derivatives of $\Phi_q$, with respect to $q_a$, for $a=1,\ldots,l$, are given by $$q_a\frac{\partial\Phi_q}{\partial q_a}=\Phi_q\star\Psi_{n+a}.$$ Here, for each $i=1,\ldots,d$, the function $\Psi_i:LX\rightarrow\mathbb{R}$ is defined, in terms of the counting of Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers which intersect the toric prime divisor $D_i$ at an interior point (see the statement of Proposition \[prop2.1\] and the subsequent discussion), and $\star$ denotes a convolution product of functions on $LX$ with respect to the lattice $N$.
2. We have a natural isomorphism of $\mathbb{C}$-algebras $$\label{isom1.1}
QH^*(\bar{X})\cong\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L},$$ where $\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]$ is the polynomial algebra generated by $\Psi_1^{\pm1},\ldots,\Psi_n^{\pm}$ with respect to the convolution product $\star$, and $\mathcal{L}$ is the ideal generated by linear relations that are defined by the linear equivalence among the toric divisors $D_1,\ldots,D_d$, provided that $\bar{X}$ is a product of projective spaces.
The proof of the above isomorphism (\[isom1.1\]) given in Subsection \[subsec2.1\] will be combinatorial in nature and is done by a simple computation of certain Gromov-Witten invariants. While this result may follow easily from known results in the literature, we choose to include an elementary proof to make this paper more self-contained. Our proof relies on the assumption that $\bar{X}$ is a product of projective spaces. However, the more important reason for us to impose such a strong assumption is that, when $\bar{X}$ is a product of projective spaces, there is a better way to understand the geometry underlying the isomorphism (\[isom1.1\]) by using *tropical geometry*. A brief explanation is now in order. More details can be found in Subsection \[subsec2.2\].
Suppose that $\bar{X}$ is a product of projective spaces. We first define a tropical analog of the small quantum cohomology ring of $\bar{X}$, call it $QH^*_{trop}(\bar{X})$. The results of Mikhalkin [@Mikhalkin03] and Nishinou-Siebert [@NS04] provided a one-to-one correspondence between those holomorphic curves in $\bar{X}$ which have contribution to the quantum product in $QH^*(\bar{X})$ and those tropical curves in $N_\mathbb{R}$ which have contribution to the tropical quantum product in $QH^*_{trop}(\bar{X})$. From this follows the natural isomorphism $$QH^*(\bar{X})\cong QH^*_{trop}(\bar{X}).$$ Next comes a simple but crucial observation: *each tropical curve which contributes to the tropical quantum product in $QH^*_{trop}(\bar{X})$ can be obtained by gluing tropical discs*.[^4] Now, making use of the fundamental results of Cho and Oh [@CO03] on the classification of holomorphic discs in toric Fano manifolds, we get a one-to-one correspondence between the relevant tropical discs and the Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers of $\mu:X\rightarrow P$. The latter were used to define the functions $\Psi_i$’s. So we naturally have another canonical isomorphism $$QH^*_{trop}(\bar{X})\cong\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L}.$$
Hence, by factoring through the tropical quantum cohomology ring $QH^*_{trop}(\bar{X})$ and using the correspondence between symplectic geometry (holomorphic curves and discs) of $\bar{X}$ and tropical geometry (tropical curves and discs) of $N_\mathbb{R}$, we obtain a more conceptual and geometric understanding of the isomorphism (\[isom1.1\]). This is in line with the general philosophy advocated in the Gross-Siebert program [@GS07].
Notice that all these can only be done for products of projective spaces because, as is well known, tropical geometry cannot be used to count curves which have irreducible components mapping to the toric boundary divisor, and if $\bar{X}$ is not a product of projective spaces, those curves *do* contribute to $QH^*(\bar{X})$ (see Example 3 in Section \[sec4\]). This is the main reason why we confine ourselves to the case of products of projective spaces, although the isomorphism (\[isom1.1\]) holds for all toric Fano manifolds (see Remark \[rmk2.3\]).\
Now we come to the upshot of this paper, namely, we can explicitly construct and apply SYZ mirror transformations to understand the mirror symmetry between $\bar{X}$ and $(Y,W)$. We shall define the SYZ mirror transformation $\mathcal{F}$ for the toric Fano manifold $\bar{X}$ as *a combination of the semi-flat SYZ mirror transformation and taking fiberwise Fourier series* (see Definition \[def3.2\] for the precise definition). Our first result says that the SYZ mirror transformation of $\Phi_q$ is precisely the exponential of the superpotential $W$, i.e. $\mathcal{F}(\Phi_q)=\exp(W)$. Then, by proving that the SYZ mirror transformation $\mathcal{F}(\Psi_i)$ of the function $\Psi_i$ is nothing but the monomial $e^{\lambda_i}z^{v_i}$, for $i=1,\ldots,d$, we show that $\mathcal{F}$ exhibits a natural and canonical isomorphism between the small quantum cohomology ring $QH^*(\bar{X})$ and the Jacobian ring $Jac(W)$, which takes the quantum product $\ast$ (which can now, by Proposition \[prop1.1\], be realized as the convolution product $\star$) to the ordinary product of Laurent polynomials, just as what classical Fourier series do. This is our main result (see Section \[sec3\]):
\[main\_thm\]$\mbox{}$
1. The SYZ mirror transformation of the function $\Phi_q\in C^\infty(LX)$, defined in terms of the counting of Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers, is the exponential of the superpotential $W$ on the mirror manifold $Y$, i.e. $$\mathcal{F}(\Phi_q)=e^W.$$ Furthermore, we can incorporate the symplectic structure $\omega_X=\omega_{\bar{X}}|_X$ on $X$ to give the holomorphic volume form on the Landau-Ginzburg model $(Y,W)$ through the SYZ mirror transformation $\mathcal{F}$, in the sense that, $$\mathcal{F}(\Phi_q e^{\sqrt{-1}\omega_X})=e^W\Omega_Y.$$
2. The SYZ mirror transformation gives a canonical isomorphism of $\mathbb{C}$-algebras $$\label{isom1.2}
\mathcal{F}:QH^*(\bar{X})\overset{\cong}{\longrightarrow} Jac(W),$$ provided that $\bar{X}$ is a product of projective spaces.
Here we view $\Phi_q e^{\sqrt{-1}\omega_X}$ as the *symplectic structure corrected by Maslov index two holomorphic discs*,[^5] and $e^W\Omega_Y$ as the holomorphic volume form of the Landau-Ginzburg model $(Y,W)$.
As mentioned at the beginning, the existence of an isomorphism $QH^*(\bar{X})\cong Jac(W)$ is not a new result, and was established before by the works of Batyrev [@Batyrev93] and Givental [@Givental97a]. However, we shall emphasize that the key point here is that there is an isomorphism which is realized by an explicit Fourier-type transformation, namely, the SYZ mirror transformation $\mathcal{F}$. This hopefully provides a more conceptual understanding of what is going on.
In [@FOOO08a] (Section 5), Fukaya-Oh-Ohta-Ono studied the isomorphism $QH^*(\bar{X})\cong Jac(W)$ from the point of view of Lagrangian Floer theory. They worked over the Novikov ring, instead of $\mathbb{C}$, and gave a proof (Theorem 1.9) of this isomorphism (over the Novikov ring) for all toric Fano manifolds basing on Batyrev’s formulas for presentations of the small quantum cohomology rings of toric manifolds and Givental’s mirror theorem [@Givental97a]. Their proof was also combinatorial in nature, but they claimed that a geometric proof would appear in a sequel of [@FOOO08a]. A brief history and a more detailed discussion of the proof of the isomorphism were also contained in Remark 1.10 of [@FOOO08a]. See also the sequel [@FOOO08b].\
The rest of this paper is organized as follows. In the next section, we define the family of functions $\{\Phi_q\}_{q\in\mathcal{K}(\bar{X})}$ in terms of the counting of Maslov index two holomorphic discs and give a combinatorial proof Proposition \[prop1.1\], which is followed by a discussion of the role played by tropical geometry. The heart of this paper is Section \[sec3\], where we construct explicitly the SYZ mirror transformation $\mathcal{F}$ for a toric Fano manifold $\bar{X}$ and show that it indeed transforms the symplectic structure of $\bar{X}$ to the complex structure of $(Y,W)$, and vice versa. This is the first part of Theorem \[main\_thm\]. We then move on to prove the second part, which shows how the SYZ mirror transformation $\mathcal{F}$ can realize the isomorphism $QH^*(\bar{X})\cong
Jac(W)$. Section \[sec4\] contains some examples. We conclude with some discussions in the final section.
Maslov index two holomorphic discs and $QH^*(\bar{X})$ {#sec2}
======================================================
In the first part of this section, we define the functions $\Phi_q$, $q\in\mathcal{K}(\bar{X})$, and $\Psi_1,\ldots,\Psi_d$ on $LX$ in terms of the counting of Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers of the moment map $\mu:X\rightarrow P$, and show how they can be used to compute the small quantum cohomology ring $QH^*(\bar{X})$ in the case when $\bar{X}$ is a product of projective spaces. In particular, we demonstrate how the quantum product can be realized as a convolution product (part 2. of Proposition \[prop1.1\]). In the second part, we explain the geometry of these results by using tropical geometry.
Computing $QH^*(\bar{X})$ in terms of functions on $LX$ {#subsec2.1}
-------------------------------------------------------
Recall that the primitive generators of the 1-dimensional cones of the fan $\Sigma$ defining the toric Fano manifold $\bar{X}$ are denoted by $v_1,\ldots,v_d\in N$. Without loss of generality, we can assume that $v_1=e_1,\ldots,v_n=e_n$ is the standard basis of $N\cong\mathbb{Z}^n$. The map $$\partial:\mathbb{Z}^d\rightarrow N,\ (k_1,\ldots,k_d)\mapsto\sum_{i=1}^d k_iv_i$$ is surjective since $\bar{X}$ is compact. Let $K$ be the kernel of $\partial$, so that the sequence $$\label{seq2.1}
0\longrightarrow K\overset{\iota}{\longrightarrow}\mathbb{Z}^d
\overset{\partial}{\longrightarrow}N\longrightarrow0$$ is exact (see, for example, Appendix 1 in the book of Guillemin [@Guillemin94]).
Now consider the Kähler cone $\mathcal{K}(\bar{X})\subset
H^2(\bar{X},\mathbb{R})$ of $\bar{X}$, and let $q_1,\ldots,q_l\in\mathbb{R}_{>0}$ ($l=d-n$) be the coordinates of $\mathcal{K}(\bar{X})$. For each $q=(q_1,\ldots,q_l)\in\mathcal{K}(\bar{X})$, we choose $\bar{P}$ to be the polytope defined by $$\bar{P}=\{x\in M_\mathbb{R}:\langle x,v_i\rangle\geq\lambda_i,\quad i=1,\ldots,d\}$$ with $\lambda_i=0$, for $i=1,\ldots,n$, and $\lambda_{n+a}=\log
q_a$, for $a=1,\ldots,l$. This associates a Kähler structure $\omega_{\bar{X}}$ to $\bar{X}$.
\[rmk2.0\] Let $\bar{P}$ be the polytope defined by the inequalities $$\langle x,v_i\rangle\geq\lambda_i,\quad i=1,\ldots,d.$$ Also let $$Q_1=(Q_{11},\ldots,Q_{d1}),\ldots,Q_l=(Q_{1l},\ldots,Q_{dl})\in\mathbb{Z}^d$$ be a $\mathbb{Z}$-basis of $K$. Then the coordinates $q=(q_1,\ldots,q_l)\in\mathcal{K}(\bar{X})$ of the Kähler cone are given by $q_a=e^{-r_a}$, where $$r_a=-\sum_{i=1}^d Q_{ia}\lambda_i,$$ for $a=1,\ldots,l$. Hence, different choices of the $\mathbb{Z}$-basis of $K$ and the constants $\lambda_1,\ldots,\lambda_d$ can give rise to the same Kähler structure parametrized by $q\in\mathcal{K}(\bar{X})$. We choose the $\mathbb{Z}$-basis $\{Q_1,\ldots,Q_l\}$ of $K$ such that $(Q_{n+a,b})_{1\leq a,b\leq l}=\textrm{Id}_{l\times l}$, and the constants $\lambda_1,\ldots,\lambda_d$ such that $\lambda_1=\ldots=\lambda_n=0$.
Recall that $\mu:X\rightarrow P$ is the restriction of the moment map $\mu_{\bar{X}}:\bar{X}\rightarrow\bar{P}$ to the open dense $T_N$-orbit $X\subset\bar{X}$, where $P$ is the interior of the polytope $\bar{P}$. For a point $x\in P$, we let $L_x=\mu^{-1}(x)\subset X$ be the Lagrangian torus fiber over $x$. Then the groups $H_2(\bar{X},\mathbb{Z})$, $\pi_2(\bar{X},L_x)$ and $\pi_1(L_x)$ can be identified canonically with $K$, $\mathbb{Z}^d$ and $N$ respectively, so that the exact sequence (\[seq2.1\]) above coincides with the following exact sequence of homotopy groups associated to the pair $(\bar{X},L_x)$: $$0\longrightarrow H_2(\bar{X},\mathbb{Z})\overset{\iota}{\longrightarrow}
\pi_2(\bar{X},L_x)\overset{\partial}{\longrightarrow}\pi_1(L_x)\longrightarrow0.$$ To proceed, we shall recall some of the fundamental results of Cho-Oh [@CO03] on the classification of holomorphic discs in $(\bar{X},L_x)$:
\[cho-oh\] $\pi_2(\bar{X},L_x)$ is generated by $d$ Maslov index two classes $\beta_1,\ldots,\beta_d\in\pi_2(\bar{X},L_x)$, which are represented by holomorphic discs with boundary in $L_x$. Moreover, given a point $p\in L_x$, then, for each $i=1,\ldots,d$, there is a unique (up to automorphism of the domain) Maslov index two holomorphic disc $\varphi_i:(D^2,\partial D^2)\rightarrow(\bar{X},L_x)$ in the class $\beta_i$ whose boundary passes through $p$,[^6] and the symplectic area of $\varphi_i$ is given by $$\label{area}
\textrm{Area}(\varphi_i)=\int_{\beta_i}\omega_{\bar{X}}=\int_{D^2}\varphi_i^*\omega_{\bar{X}}=2\pi(\langle
x,v_i\rangle-\lambda_i).$$
Furthermore, for each $i=1,\ldots,d$, the disc $\varphi_i$ intersects the toric prime divisor $D_i$ at a unique interior point. (We can in fact choose the parametrization of $\varphi_i$ so that $\varphi_i(0)\in D_i$.) Indeed, a result of Cho-Oh (Theorem 5.1 in [@CO03]) says that, the Maslov index of a holomorphic disc $\varphi:(D^2,\partial D^2)\rightarrow(\bar{X},L_x)$ representing a class $\beta\in\pi_2(\bar{X},L_x)$ is given by twice the algebraic intersection number $\beta\cdot D_\infty$, where $D_\infty=\bigcup_{i=1}^dD_i$ is the toric boundary divisor (see also Auroux [@Auroux07], Lemma 3.1).
Let $LX$ be the product $X\times N$. We view $LX$ as a (trivial) $\mathbb{Z}^n$-cover over $X$: $$\pi:LX=X\times N\rightarrow X,$$ and we equip $LX$ with the symplectic structure $\pi^*(\omega_X)$, so that it becomes a symplectic manifold. We are now in a position to define $\Phi_q$.
\[def2.1\] Let $q=(q_1,\ldots,q_l)\in\mathcal{K}(\bar{X})$. The function $\Phi_q:LX\rightarrow\mathbb{R}$ is defined as follows. For $(p,v)\in LX=X\times N$, let $x=\mu(p)\in P$ and $L_x=\mu^{-1}(x)$ be the Lagrangian torus fiber containing $p$. Denote by $$\pi_2^+(\bar{X},L_x)=\Big\{\sum_{i=1}^d
k_i\beta_i\in\pi_2(\bar{X},L_x):k_i\in\mathbb{Z}_{\geq0},\
i=1,\ldots,d\Big\}$$ the positive cone generated by the Maslov index two classes $\beta_1,\ldots,\beta_d$ which are represented by holomorphic discs with boundary in $L_x$. For $\beta=\sum_{i=1}^d
k_i\beta_i\in\pi_2^+(\bar{X},L_x)$, we denote by $w(\beta)$ the number $k_1!\ldots k_d!$. Then set $$\Phi_q(p,v)=\sum_{\beta\in\pi_2^+(\bar{X},L_x),\ \partial\beta=v}
\frac{1}{w(\beta)}e^{-\frac{1}{2\pi}\int_\beta\omega_{\bar{X}}}.$$
\[rmk2.1\]$\mbox{}$
1. We say that $\Phi_q$ is defined by the counting of Maslov index two holomorphic discs because of the following: Let $(p,v)\in
LX, x=\mu(p)\in P, L_x\subset X$ and $\beta_1,\ldots,\beta_d\in\pi_2(\bar{X},L_x)$ be as before. For $i=1,\ldots,d$, let $n_i(p)$ be the (algebraic) number of Maslov index two holomorphic discs $\varphi:(D^2,\partial
D^2)\rightarrow(\bar{X},L_x)$ in the class $\beta_i$ whose boundary passes through $p$. This number is well-defined since $\bar{X}$ is toric Fano (see Section 3.1 and Section 4 in Auroux [@Auroux07]). Then we can re-define $$\Phi_q(p,v)=\sum_{\beta\in\pi_2^+(\bar{X},L_x),\ \partial\beta=v}
\frac{n_\beta(p)}{w(\beta)}e^{-\frac{1}{2\pi}\int_\beta\omega_{\bar{X}}},$$ where $n_\beta(p)=n_1(p)^{k_1}\ldots n_d(p)^{k_d}$ if $\beta=\sum_{i=1}^d k_i\beta_i$. Defining $\Phi_q$ in this way makes it explicit that $\Phi_q$ carries enumerative meaning. By Theorem \[cho-oh\], we have $n_i(p)=1$, for all $i=1,\ldots,d$ and for any $p\in X$. So this definition reduces to the one above.
2. By definition, $\Phi_q$ is invariant under the $T_N$-action on $X\subset\bar{X}$. Since $X=T^*P/N=P\times\sqrt{-1}T_N$ (and the moment map $\mu:X\rightarrow
P$ is nothing but the projection to the first factor), we may view $\Phi_q$ as a function on $P\times N$.
3. The function $\Phi_q$ is well-defined, i.e. the infinite sum in its definition converges. To see this, notice that, by the symplectic area formula (\[area\]) of Cho-Oh, we have $$\Phi_q(p,v)=\Bigg(\sum_{\substack{k_1,\ldots,k_d\in\mathbb{Z}_{\geq0},\\ \sum_{i=1}^d k_iv_i=v}}
\frac{q_1^{k_{n+1}}\ldots q_l^{k_d}}{k_1!\ldots
k_d!}\Bigg)e^{-\langle x,v\rangle},$$ and the sum inside the big parentheses is less than $e^{n+q_1+\ldots+q_l}$.
For $T_N$-invariant functions $f,g:LX\rightarrow\mathbb{R}$, we define their *convolution product* $f\star
g:LX\rightarrow\mathbb{R}$ by $$(f\star g)(p,v)=\sum_{v_1,v_2\in N,\ v_1+v_2=v}f(p,v_1)g(p,v_2),$$ for $(p,v)\in LX$. As in the theory of Fourier analysis, for the convolution $f\star g$ to be well-defined, we need some conditions for both $f$ and $g$. We leave this to Subsection \[subsec3.2\] (see Definition \[def3.3\] and the subsequent discussion). Nevertheless, if one of the functions is nonzero only for finitely many $v\in N$, then the sum in the definition of $\star$ is a finite sum, so it is well-defined. This is the case in the following proposition.
\[prop2.1\]\[=part 1. of Proposition \[prop1.1\]\] The logarithmic derivatives of $\Phi_q$, with respect to $q_a$ for $a=1,\ldots,l$, are given by $$q_a\frac{\partial\Phi_q}{\partial q_a}=\Phi_q\star\Psi_{n+a}$$ where $\Psi_i:LX\rightarrow\mathbb{R}$ is defined, for $i=1,\ldots,d$, by $$\Psi_i(p,v)=\left\{ \begin{array}{ll}
e^{-\frac{1}{2\pi}\int_{\beta_i}\omega_{\bar{X}}} & \textrm{if $v=v_i$}\\
0 & \textrm{if $v\neq v_i$,}
\end{array}\right.$$ for $(p,v)\in LX=X\times N$, and with $x=\mu(p)\in P$, $L_x=\mu^{-1}(x)$ and $\beta_1,\ldots,\beta_d\in\pi_2(\bar{X},L_x)$ as before.
We will compute $q_l\frac{\partial\Phi_q}{\partial q_l}$. The others are similar. By using Cho-Oh’s formula (\[area\]) and our choice of the polytope $\bar{P}$, we have $$e^{\langle x,v\rangle}\Phi_q(p,v)=
\sum_{\substack{k_1,\ldots,k_d\in\mathbb{Z}_{\geq0},\\ \sum_{i=1}^d
k_iv_i=v}}\frac{q_1^{k_{n+1}}\ldots q_l^{k_d}}{k_1!\ldots k_d!}.$$ Note that the right-hand-side is independent of $p\in X$. Differentiating both sides with respect to $q_l$ gives $$\begin{aligned}
e^{\langle x,v\rangle}\frac{\partial\Phi_q(p,v)}{\partial q_l} & = &
\sum_{\substack{k_1,\ldots,k_{d-1}\in\mathbb{Z}_{\geq0},\
k_d\in\mathbb{Z}_{\geq1},\\ \sum_{i=1}^d
k_iv_i=v}}\frac{q_1^{k_{n+1}}\ldots
q_{l-1}^{k_{d-1}}q_l^{k_d-1}}{k_1!\ldots k_{d-1}!(k_d-1)!}\\
& = & \sum_{\substack{k_1,\ldots,k_d\in\mathbb{Z}_{\geq0},\\
\sum_{i=1}^d k_iv_i=v-v_d}}\frac{q_1^{k_{n+1}}\ldots
q_l^{k_d}}{k_1!\ldots
k_d!}\\
& = & e^{\langle x,v-v_d\rangle}\Phi_q(p,v-v_d).\end{aligned}$$ Hence, we obtain $$\begin{aligned}
q_l\frac{\partial\Phi_q(p,v)}{\partial q_l}=q_l e^{-\langle
x,v_d\rangle}\Phi_q(p,v-v_d).\end{aligned}$$ Now, by the definition of the convolution product $\star$, we have $$\begin{aligned}
\Phi_q\star\Psi_d(p,v)=\sum_{v_1,v_2\in N,\
v_1+v_2=v}\Phi_q(p,v_1)\Psi_d(p,v_2)=\Phi_q(p,v-v_d)\Psi_d(p,v_d),\end{aligned}$$ and $\Psi_d(p,v_d)=e^{-\frac{1}{2\pi}\int_{\beta_d}\omega_{\bar{X}}}=e^{\lambda_d-\langle
x,v_d\rangle}=q_le^{-\langle x,v_d\rangle}$. The result follows.
In the previous proposition, we introduce the $T_N$-invariant functions $\Psi_1,\ldots,\Psi_d\in C^\infty(LX)$. Similar to what has been said in Remark \[rmk2.1\](1), these functions carry enumerative meanings, and we should have defined $\Psi_i(p,v)$, $i=1,\ldots,d$ in terms of the counting of Maslov index two holomorphic discs in $(\bar{X},L_{\mu(p)})$ with boundary $v$ which pass through $p$, i.e. $$\Psi_i(p,v)=\left\{ \begin{array}{ll}
n_i(p)e^{-\frac{1}{2\pi}\int_{\beta_i}\omega_{\bar{X}}} & \textrm{if $v=v_i$}\\
0 & \textrm{if $v\neq v_i$,}
\end{array}\right.$$ for $(p,v)\in LX=X\times N$, where $x=\mu(p)\in P,
L_x=\mu^{-1}(x)\subset X$ and $\beta_1,\ldots,\beta_d\in\pi_2(\bar{X},L_x)$ are as before. Again, since the number $n_i(p)$ is always equal to one, for any $p\in X$ and for all $i=1,\ldots,d$, this definition of $\Psi_i$ is the same as the previous one. But we should keep in mind that the function $\Psi_i\in C^\infty(LX)$ encodes the following enumerative information: for each $p\in X$, there is a unique Maslov index two holomorphic disc $\varphi_i$ in the class $\beta_i$ with boundary in the Lagrangian torus fiber $L_{\mu(p)}$ whose boundary passes through $p$ and whose interior intersects the toric prime divisor $D_i$ at one point. In view of this, we put the $d$ functions $\{\Psi_i\}_{i=1}^d$, the $d$ families of Maslov index two holomorphic discs $\{\varphi_i\}_{i=1}^d$ and the $d$ toric prime divisors $\{D_i\}_{i=1}^d$ in one-to-one correspondences: $$\label{1-1}
\{\Psi_i\}_{i=1}^d\ \overset{1-1}{\longleftrightarrow}\
\{\varphi_i\}_{i=1}^d\ \overset{1-1}{\longleftrightarrow}\
\{D_i\}_{i=1}^d.$$ Through these correspondences, we introduce linear relations in the $d$-dimensional $\mathbb{C}$-vector space spanned by the functions $\Psi_1,\ldots,\Psi_d$ using the linear equivalences among the divisors $D_1,\ldots,D_d$.
Two linear combinations $\sum_{i=1}^d a_i\Psi_i$ and $\sum_{i=1}^d
b_i\Psi_i$, where $a_i,b_i\in\mathbb{C}$, are said to be linearly equivalent, denoted by $\sum_{i=1}^d a_i\Psi_i\sim\sum_{i=1}^d
b_i\Psi_i$, if the corresponding divisors $\sum_{i=1}^d a_iD_i$ and $\sum_{i=1}^d b_iD_i$ are linearly equivalent.
We further define $\Psi_i^{-1}:LX\rightarrow\mathbb{R}$, $i=1,\ldots,d$, by $$\Psi_i^{-1}(p,v)=\left\{ \begin{array}{ll}
e^{\frac{1}{2\pi}\int_{\beta_i}\omega_X} & \textrm{if $v=-v_i$}\\
0 & \textrm{if $v\neq-v_i$,}
\end{array}\right.$$ for $(p,v)\in LX$, so that $\Psi_i^{-1}\star\Psi_i=\mathbb{1}$, where $\mathbb{1}:LX\rightarrow\mathbb{R}$ is the function defined by $$\mathbb{1}(p,v)=\left\{ \begin{array}{ll}
1 & \textrm{if $v=0$}\\
0 & \textrm{if $v\neq 0$.}
\end{array}\right.$$ The function $\mathbb{1}$ serves as a multiplicative identity for the convolution product, i.e. $\mathbb{1}\star f=f\star\mathbb{1}=f$ for any $f\in C^\infty(LX)$. Now the second part of Proposition \[prop1.1\] says that
\[prop2.2\] We have a natural isomorphism of $\mathbb{C}$-algebras $$\label{isom2.3}
QH^*(\bar{X})\cong\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L},$$ where $\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]$ is the polynomial algebra generated by $\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}$ with respect to the convolution product $\star$ and $\mathcal{L}$ is the ideal generated by linear equivalences, provided that $\bar{X}$ is a product of projective spaces.
In the rest of this subsection, we will give an elementary proof of this proposition by simple combinatorial arguments and computation of certain Gromov-Witten invariants.
First of all, each toric prime divisor $D_i$ ($i=1,\ldots,d$) determines a cohomology class in $H^2(\bar{X},\mathbb{C})$, which will be, by abuse of notations, also denoted by $D_i$. It is known by the general theory of toric varieties that the cohomology ring $H^*(\bar{X},\mathbb{C})$ of the compact toric manifold $\bar{X}$ is generated by the classes $D_1,\ldots,D_d$ in $H^2(\bar{X},\mathbb{C})$ (see, for example, Fulton [@Fulton93] or Audin [@Audin04]). More precisely, there is a presentation of the form: $$H^*(\bar{X},\mathbb{C})=\mathbb{C}[D_1,\ldots,D_d]/(\mathcal{L}+\mathcal{SR}),$$ where $\mathcal{L}$ is the ideal generated by linear equivalences and $\mathcal{SR}$ is the *Stanley-Reisner ideal* generated by *primitive relations* (see Batyrev [@Batyrev91]). Now, by a result of Siebert and Tian (Proposition 2.2 in [@ST94]), $QH^*(\bar{X})$ is also generated by $D_1,\ldots,D_d$ and a presentation of $QH^*(\bar{X})$ is given by replacing each relation in $\mathcal{SR}$ by its quantum counterpart. Denote by $\mathcal{SR}_Q$ the quantum Stanley-Reisner ideal. Then we can rephrase what we said as: $$QH^*(\bar{X})=\mathbb{C}[D_1,\ldots,D_d]/(\mathcal{L}+\mathcal{SR}_Q).$$ The computation of $QH^*(\bar{X})$ (as a presentation) therefore reduces to computing the generators of the ideal $\mathcal{SR}_Q$.
Let $\bar{X}=\mathbb{C}P^{n_1}\times\ldots\times\mathbb{C}P^{n_l}$ be a product of projective spaces. The complex dimension of $\bar{X}$ is $n=n_1+\ldots+n_l$. For $a=1,\ldots,l$, let $v_{1,a}=e_1,\ldots,v_{n_a,a}=e_{n_a},v_{n_a+1,a}=-\sum_{j=1}^{n_a}e_j\in
N_a$ be the primitive generators of the 1-dimensional cones in the fan of $\mathbb{C}P^{n_a}$, where $\{e_1,\ldots,e_{n_a}\}$ is the standard basis of $N_a\cong\mathbb{Z}^{n_a}$. For $j=1,\ldots,n_a+1$, $a=1,\ldots,l$, we use the same symbol $v_{j,a}$ to denote the vector $$(0,\ldots,\underbrace{v_{j,a}}_{\textrm{$a$-th}},\ldots,0)
\in N=N_1\oplus\ldots\oplus N_l,$$ where $v_{j,a}$ sits in the $a$th place. These $d=\sum_{a=1}^l(n_a+1)=n+l$ vectors in $N$ are the primitive generators of the 1-dimensional cones of the fan $\Sigma$ defining $\bar{X}$. In the following, we shall also denote the toric prime divisor, the relative homotopy class, the family of Maslov index two holomorphic discs with boundary in Lagrangian torus fibers and the function on $LX$ corresponding to $v_{j,a}$ by $D_{j,a}$, $\beta_{j,a}$, $\varphi_{j,a}$ and $\Psi_{j,a}$ respectively.
There are exactly $l$ primitive collections given by $$\mathfrak{P}_a=\{v_{j,a}:j=1,\ldots,n_a+1\},\ a=1,\ldots,l,$$ and hence the Stanley-Reisner ideal of $\bar{X}=\mathbb{C}P^{n_1}\times\ldots\times\mathbb{C}P^{n_l}$ is given by $$\mathcal{SR}=\langle D_{1,a}\cup\ldots\cup D_{n_a+1,a}:a=1\ldots,l\rangle.$$
Let $\mathfrak{P}$ be any primitive collection. By definition, $\mathfrak{P}$ is a collection of primitive generators of 1-dimensional cones of the fan $\Sigma$ defining $\bar{X}$ such that for any $v\in\mathfrak{P}$, $\mathfrak{P}\setminus\{v\}$ generates a $(|\mathfrak{P}|-1)$-dimensional cone in $\Sigma$, while $\mathfrak{P}$ itself does not generate a $|\mathfrak{P}|$-dimensional cone in $\Sigma$. Suppose that $\mathfrak{P}\not\subset\mathfrak{P}_a$ for any $a$. For each $a$, choose $v\in\mathfrak{P}\setminus(\mathfrak{P}\cap\mathfrak{P}_a)$. By definition, $\mathfrak{P}\setminus\{v\}$ generates a cone in $\Sigma$. But all the cones in $\Sigma$ are direct sums of cones in the fans of the factors. So, in particular, $\mathfrak{P}\cap\mathfrak{P}_a$, whenever it’s nonempty, will generate a cone in the fan of $\mathbb{C}P^{n_a}$. Since $\mathfrak{P}=\bigsqcup_{a=1}^l\mathfrak{P}\cap\mathfrak{P}_a$, this implies that the set $\mathfrak{P}$ itself generates a cone, which is impossible. We therefore conclude that $\mathfrak{P}$ must be contained in, and hence equal to one of the $\mathfrak{P}_a$’s.
Hence, to compute the quantum Stanley-Reisner ideal $\mathcal{SR}_Q$, we must compute the expression $D_{1,a}\ast\ldots\ast D_{n_a+1,a}$, for $a=1,\ldots,l$, where $\ast$ denotes the small quantum product of $QH^*(\bar{X})$. Before doing this, we shall recall the definitions and properties of the relevant Gromov-Witten invariants and the small quantum product for $\bar{X}=\mathbb{C}P^{n_1}\times\ldots\times\mathbb{C}P^{n_l}$ as follows.\
For $\delta\in H_2(\bar{X},\mathbb{Z})$, let $\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)$ be the moduli space of genus 0 stable maps with $m$ marked points and class $\delta$. Since $\bar{X}$ is convex (i.e. for all maps $\varphi:\mathbb{C}P^1\rightarrow\bar{X}$, $H^1(\mathbb{C}P^1,\varphi^*T\bar{X})=0$), the moduli space $\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)$, if nonempty, is a variety of pure complex dimension $\textrm{dim}_\mathbb{C}(\bar{X})+c_1(\bar{X})\cdot\delta+m-3$ (see, for example, the book [@Aluffi97], p.3). For $k=1,\ldots,m$, let $ev_k:\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)\rightarrow\bar{X}$ be the evaluation map at the $k$th marked point, and let $\pi:\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)\rightarrow\overline{\mathcal{M}}_{0,m}$ be the forgetful map, where $\overline{\mathcal{M}}_{0,m}$ denotes the Deligne-Mumford moduli space of genus 0 stable curves with $m$ marked points. Then, given cohomology classes $A\in
H^*(\overline{\mathcal{M}}_{0,m},\mathbb{Q})$ and $\gamma_1,\ldots,\gamma_m\in H^*(\bar{X},\mathbb{Q})$, the Gromov-Witten invariant is defined by $$GW_{0,m}^{\bar{X},\delta}(A;\gamma_1,\ldots,\gamma_m)=
\int_{[\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)]}\pi^*(A)\wedge
ev_1^*(\gamma_1)\wedge\ldots\wedge ev_m^*(\gamma_m),$$ where $[\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)]$ denotes the fundamental class of $\overline{\mathcal{M}}_{0,m}(\bar{X},\delta)$. Let $\ast$ be the small quantum product of $QH^*(\bar{X})$. Then it is not hard to show that, for any classes $\gamma_1,\ldots,\gamma_r\in H^*(\bar{X},\mathbb{Q})$, the expression $\gamma_1\ast\ldots\ast\gamma_r$ can be computed by the formula $$\gamma_1\ast\ldots\ast\gamma_r=\sum_{\delta\in H_2(\bar{X},\mathbb{Z})}\sum_{i}
GW_{0,r+1}^{\bar{X},\delta}(\textrm{PD(pt)};\gamma_1,\ldots,\gamma_r,t_i)t^iq^\delta,$$ where $\{t_i\}$ is a basis of $H^*(\bar{X},\mathbb{Q})$, $\{t^i\}$ denotes the dual basis of $\{t_i\}$ with respect to the Poincaré pairing, and $\textrm{PD(pt)}\in
H^{2m-6}(\overline{\mathcal{M}}_{0,m},\mathbb{Q})$ denotes the Poincaré dual of a point in $\overline{\mathcal{M}}_{0,m}$ (see, e.g. formula (1.4) in Spielberg [@Spielberg01]). Moreover, since $\bar{X}$ is homogeneous of the form $G/P$, where $G$ is a Lie group and $P$ is a parabolic subgroup, the Gromov-Witten invariants are enumerative, in the sense that $GW_{0,r+1}^{\bar{X},\delta}(\textrm{PD(pt)};\gamma_1,\ldots,\gamma_r,t_i)$ is equal to the number of holomorphic maps $\varphi:(\mathbb{C}P^1;x_1,\ldots,x_r,x_{r+1})\rightarrow\bar{X}$ with $\varphi_*([\mathbb{C}P^1])=\delta$ such that $(\mathbb{C}P^1;x_1,\ldots,x_r,x_{r+1})$ is a given point in $\overline{\mathcal{M}}_{0,r+1}$, $\varphi(x_k)\in\Gamma_k$, for $k=1,\ldots,r$, and $\varphi(x_{r+1})\in T_i$, where $\Gamma_1,\ldots,\Gamma_r,T_i$ are representatives of cycles Poincaré duals to the classes $\gamma_1,\ldots,\gamma_r,t_i$ respectively (see [@Aluffi97], p.12).\
We shall now use the above facts to compute $D_{1,a}\ast\ldots\ast
D_{n_a+1,a}$, which is given by the formula $$D_{1,a}\ast\ldots\ast D_{n_a+1,a}=\sum_{\delta\in H_2(\bar{X},\mathbb{Z})}\sum_{i}
GW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)t^iq^\delta.$$ First of all, since $H_2(\bar{X},\mathbb{Z})$ is the kernel of the boundary map $\partial:\pi_2(\bar{X},L_x)=\mathbb{Z}^d\rightarrow\pi_1(L_x)=N$, a homology class $\delta\in H_2(\bar{X},\mathbb{Z})$ can be represented by a $d$-tuple of integers $$\delta=(c_{1,1},\ldots,c_{n_1+1,1},\ldots,c_{1,b},\ldots,c_{n_b+1,b},\ldots,c_{1,l},
\ldots,c_{n_l+1,l})\in\mathbb{Z}^d$$ satisfying $\sum_{b=1}^l\sum_{j=1}^{n_b+1}c_{j,b}v_{j,b}=0\in N$. Then we have $c_1(\bar{X})\cdot\delta=\sum_{b=1}^l\sum_{j=1}^{n_b+1}c_{j,b}$. For the Gromov-Witten invariant $GW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)$ to be nonzero, $\delta$ must be represented by irreducible holomorphic curves which pass through all the divisors $D_{1,a},\ldots,D_{n_a+1,a}$. This implies that $c_{j,a}\geq1$, for $j=1,\ldots,n_a+1$, and moreover, $\delta$ lies in the cone of effective classes $H_2^{\textrm{eff}}(\bar{X},\mathbb{Z})\subset
H_2(\bar{X},\mathbb{Z})$. By Theorem 2.15 of Batyrev [@Batyrev91], $H_2^{\textrm{eff}}(\bar{X},\mathbb{Z})$ is given by the kernel of the restriction of the boundary map $\partial|_{\mathbb{Z}_{\geq0}^d}:\mathbb{Z}_{\geq0}^d\rightarrow
N$. So we must also have $c_{j,b}\geq0$ for all $j$ and $b$, and we conclude that $$c_1(\bar{X})\cdot\delta=\sum_{b=1}^l\sum_{j=1}^{n_b+1}c_{j,b}\geq n_a+1.$$ By dimension counting, $GW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)\neq0$ only when $$2(\textrm{dim}_\mathbb{C}(\bar{X})+c_1(\bar{X})\cdot\delta+(n_a+2)-3)=
2((n_a+2)-3)+2(n_a+1)+\textrm{deg}(t_i).$$ The above inequality then implies that $\textrm{deg}(t_i)\geq2\textrm{dim}(\bar{X})$. We therefore must have $t_i=\textrm{PD(pt)}\in
H^{2\textrm{dim}(\bar{X})}(\bar{X},\mathbb{Q})$ and $\delta\in
H_2(\bar{X},\mathbb{Z})$ is represented by the $d$-tuple of integers $\delta_a:=(c_{1,1},\ldots,c_{n_l+1,l})\in\mathbb{Z}^d$, where $$c_{j,b}=\left\{\begin{array}{ll}
1 & \textrm{if $b=a$ and $j=1,\ldots,n_a+1$}\\
0 & \textrm{otherwise,}
\end{array}\right.$$ i.e., $\delta=\delta_a$ is the pullback of the class of a line in the factor $\mathbb{C}P^{n_a}$. Hence, $$D_{1,a}\ast\ldots\ast D_{n_a+1,a}=
GW_{0,n_a+2}^{\bar{X},\delta_a}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},\textrm{PD(pt)})q^{\delta_a}.$$ By Theorem 9.3 in Batyrev [@Batyrev93] (see also Siebert [@Siebert95], section 4), the Gromov-Witten invariant on the right-hand-side is equal to 1. Geometrically, this means that, for any given point $p\in X\subset\bar{X}$, there is a unique holomorphic map $\varphi_a:(\mathbb{C}P^1;x_1,\ldots,x_{n_a+2})\rightarrow\bar{X}$ with class $\delta_a$, $\varphi_a(x_j)\in D_{j,a}$, for $j=1,\ldots,n_a+1$, $\varphi_a(x_{n_a+2})=p$ and such that $(\mathbb{C}P^1;x_1,\ldots,x_{n_a+2})$ is a given configuration in $\overline{\mathcal{M}}_{0,n_a+2}$. Also note that, for $a=1,\ldots,l$, $q^{\delta_a}=\exp(-\frac{1}{2\pi}\int_{\delta_a}\omega_{\bar{X}})=e^{-r_a}=q_a$, where $(q_1,\ldots,q_l)$ are the coordinates of the Kähler cone $\mathcal{K}(\bar{X})$. Thus, we have the following lemma.
\[lem2.2\] For $a=1,\ldots,l$, we have $$D_{1,a}\ast\ldots\ast D_{n_a+1,a}=q_a.$$ Hence, the quantum Stanley-Reisner ideal of $\bar{X}=\mathbb{C}P^{n_1}\times\ldots\times\mathbb{C}P^{n_l}$ is given by $$\mathcal{SR}_Q=\langle D_{1,a}\ast\ldots\ast D_{n_a+1,a}-q_a:a=1\ldots,l\rangle,$$ and the quantum cohomology ring of $\bar{X}$ has a presentation given by $$\begin{aligned}
QH^*(\bar{X})=&\frac{\mathbb{C}[D_{1,1},\ldots,D_{n_1+1,1},\ldots,D_{1,l},\ldots,D_{n_l+1,l}]}
{\big\langle D_{j,a}-D_{n_a+1,a}:j=1,\ldots,n_a,\
a=1,\ldots,l\big\rangle
+\big\langle\prod_{j=1}^{n_a+1}D_{j,a}-q_a:a=1,\ldots,l\big\rangle}.\end{aligned}$$
Proposition \[prop2.2\] now follows from a simple combinatorial argument:
For a general toric Fano manifold $\bar{X}$, recall, from Remark \[rmk2.0\], that we have chosen a $\mathbb{Z}$-basis $$Q_1=(Q_{11},\ldots,Q_{d1}),\ldots,Q_l=(Q_{1l},\ldots,Q_{dl})\in\mathbb{Z}^d$$ of $K=H_2(\bar{X},\mathbb{Z})$ such that $(Q_{n+a,b})_{1\leq a,b\leq
l}=\textrm{Id}_{l\times l}$. So, by Cho-Oh’s symplectic area formula (\[area\]), for $a=1,\ldots,l$, we have $$\begin{aligned}
\Big(\sum_{i=1}^nQ_{ia}\int_{\beta_i}\omega_{\bar{X}}\Big)+\int_{\beta_{n+a}}\omega_{\bar{X}}
& = & 2\pi\Big(\sum_{i=1}^nQ_{ia}(\langle
x,v_i\rangle-\lambda_i)\Big)+2\pi(\langle x,v_{n+a}\rangle-\lambda_{n+a})\\
& = & 2\pi\langle
x,\sum_{i=1}^nQ_{ia}v_i+v_{n+a}\rangle-2\pi(\sum_{i=1}^aQ_{ia}\lambda_a+\lambda_{n+a})\\
& = & 2\pi r_a.\end{aligned}$$ Then, by the definition of the convolution product of functions on $LX$, we have $$\begin{aligned}
\Psi_1^{Q_{1a}}\star\ldots\star\Psi_n^{Q_{na}}\star\Psi_{n+a}(x,v)
& = & \left\{
\begin{array}{ll}
e^{-\frac{1}{2\pi}(\sum_{i=1}^nQ_{ia}\int_{\beta_i}\omega_{\bar{X}})
-\frac{1}{2\pi}\int_{\beta_{n+a}}\omega_{\bar{X}}} & \textrm{if $v=0$}\\
0 & \textrm{if $v\neq0$}
\end{array} \right.\\
& = & \left\{ \begin{array}{ll}
e^{-r_a} & \textrm{if $v=0$}\\
0 & \textrm{if $v\neq0$}
\end{array} \right.\\
& = & q_a\mathbb{1},\end{aligned}$$ or $\Psi_{n+a}=q_a(\Psi_1^{-1})^{Q_{1a}}\star\ldots\star(\Psi_n^{-1})^{Q_{na}}$, for $a=1,\ldots,l$.
Suppose that the following condition: $Q_{ia}\geq0$ for $i=1,\ldots,n$, $a=1,\ldots,l$, and for each $i=1,\ldots,n$, there exists $1\leq a\leq l$ such that $Q_{ia}>0$, is satisfied, which is the case when $\bar{X}$ is a product of projective spaces. Then the inclusion $$\mathbb{C}[\Psi_1,\ldots,\Psi_n,\Psi_{n+1},\ldots,\Psi_d]\hookrightarrow
\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]$$ is an isomorphism. Consider the surjective map $$\rho:\mathbb{C}[D_1,\ldots,D_d]\rightarrow\mathbb{C}[\Psi_1,\ldots,\Psi_d]$$ defined by mapping $D_i$ to $\Psi_i$ for $i=1,\ldots,d$. This map is not injective because there are nontrivial relations in $\mathbb{C}[\Psi_1,\ldots,\Psi_d]$ generated by the relations $$\Psi_1^{Q_{1a}}\star\ldots\star\Psi_n^{Q_{na}}\star\Psi_{n+a}-q_a\mathbb{1}=0,\ a=1,\ldots,l.$$ By Lemma \[lem2.2\], the kernel of $\rho$ is exactly given by the ideal $\mathcal{SR}_Q$ when $\bar{X}$ is a product of projective spaces. Thus, we have an isomorphism $$\mathbb{C}[D_1,\ldots,D_d]/\mathcal{SR}_Q\overset{\cong}{\longrightarrow}\mathbb{C}[\Psi_1,\ldots,\Psi_d].$$ Since $(\mathbb{C}[D_1,\ldots,D_d]/\mathcal{SR}_Q)/\mathcal{L}=\mathbb{C}[D_1,\ldots,D_d]/(\mathcal{L}+\mathcal{SR}_Q)
=QH^*(\bar{X})$, we obtain the desired isomorphism $$QH^*(\bar{X})\cong\mathbb{C}[\Psi_1,\ldots,\Psi_d]/\mathcal{L}\cong
\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L},$$ provided that $\bar{X}$ is a product of projective spaces.
\[rmk2.3\]$\mbox{}$
1. In [@Batyrev93], Theorem 5.3, Batyrev gave a “formula” for the quantum Stanley-Reisner ideal $\mathcal{SR}_Q$ for any compact toric Kähler manifolds, using his own definition of the small quantum product (which is different from the usual one because Batyrev counted only holomorphic maps from $\mathbb{C}P^1$). By Givental’s mirror theorem [@Givental97a], Batyrev’s formula is true, using the usual definition of the small quantum product, for all toric Fano manifolds. Our proof of Lemma \[lem2.2\] is nothing but a simple verification of Batyrev’s formula in the case of products of projective spaces, without using Givental’s mirror theorem.
2. In any event, Batyrev’s formula in [@Batyrev93] for a presentation of the small quantum cohomology ring $QH^*(\bar{X})$ of a toric Fano manifold $\bar{X}$ is correct. In the same paper, Batyrev also proved that $QH^*(\bar{X})$ is canonically isomorphic to the Jacobian ring $Jac(W)$, where $W$ is the superpotential mirror to $\bar{X}$ (Theorem 8.4 in [@Batyrev93]). Now, by Theorem \[thm3.3\] in Subsection \[subsec3.3\], the inverse SYZ transformation $\mathcal{F}^{-1}$ gives a canonical isomorphism $\mathcal{F}^{-1}:Jac(W)\overset{\cong}{\rightarrow}
\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L}$. Then, the composition map $QH^*(\bar{X})\rightarrow\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L}$, which maps $D_i$ to $\Psi_i$, for $i=1,\ldots,d$, is an isomorphism. This proves Proposition \[prop2.2\] all toric Fano manifolds. We choose not to use this proof because all the geometry is then hid by the use of Givental’s mirror theorem.
The role of tropical geometry {#subsec2.2}
-----------------------------
While our proof of the isomorphism (\[isom2.3\]) in Proposition \[prop2.2\] is combinatorial in nature, the best way to understand the geometry behind it is through the correspondence between holomorphic curves and discs in $\bar{X}$ and their tropical counterparts in $N_\mathbb{R}$. Indeed, this is the main reason why we confine ourselves to the case of products of projective spaces. Our first task is to define a tropical analog $QH^*_{trop}(\bar{X})$ of the small quantum cohomology ring of $\bar{X}$, when $\bar{X}$ is a product of projective spaces. For this, we shall recall some notions in tropical geometry. We will not state the precise definitions, for which we refer the reader to Mikhalkin [@Mikhalkin03], [@Mikhalkin06], [@Mikhalkin07] and Nishinou-Siebert [@NS04].\
A *genus 0 tropical curve with $m$ marked points* is a connected tree $\Gamma$ with exactly $m$ unbounded edges (also called leaves) and each bounded edge is assigned a positive length. Let $\overline{\mathcal{M}}_{0,m}^{trop}$ be the moduli space of genus 0 tropical curves with $m$ marked points (modulo isomorphisms). The combinatorial types of $\Gamma$ partition $\overline{\mathcal{M}}_{0,m}^{trop}$ into disjoint subsets, each of which has the structure of a polyhedral cone $\mathbb{R}_{>0}^e$ (where $e$ is the number of bounded edges in $\Gamma$). There is a distinguished point in $\overline{\mathcal{M}}_{0,m}^{trop}$ corresponding to the (unique) tree $\Gamma_m$ with exactly one ($m$-valent) vertex $V$, $m$ unbounded edges $E_1,\ldots,E_m$ and *no* bounded edges. See Figure 2.1 below. We will fix this point in $\overline{\mathcal{M}}_{0,m}^{trop}$; this is analog to fixing a point in $\overline{\mathcal{M}}_{0,m}$.
(100,30) (30,11.5, 12,27.5) (15,25.5)[$E_2$]{} (30,11.5, 12,-1.5) (16.5,-1)[$E_3$]{} (30,11.5, 40,30.5) (39,26.5)[$E_1$]{} (30,11.5, 48,0.5) (41,-0.2)[$E_4$]{} (29.2,10.7)[$\bullet$]{} (32,11)[$V$]{} (55,11.5)[Figure 2.1: $\Gamma_4\in\overline{\mathcal{M}}_{0,4}^{trop}.$]{}
Let $\Sigma$ be the fan defining $\bar{X}=\mathbb{C}P^{n_1}\times\ldots\times\mathbb{C}P^{n_l}$, and denote by $\Sigma[1]=\{v_{1,1},\ldots,v_{n_1+1,1},\ldots,v_{1,a},\ldots,v_{n_a+1,a},\ldots,
v_{1,l},\ldots,v_{n_l+1,l}\}\subset N$ the set of primitive generators of 1-dimensional cones in $\Sigma$. Let $h:\Gamma_m\rightarrow N_\mathbb{R}$ be a continuous embedding such that, for each $k=1,\ldots,m$, $h(E_k)=h(V)+\mathbb{R}_{\geq0}v(E_k)$ for some $v(E_k)\in\Sigma[1]$, and the following *balancing condition* is satisfied: $$\sum_{k=1}^m v(E_k)=0.$$ Then the tuple $(\Gamma_m;E_1,\ldots,E_m;h)$ is a *parameterized $m$-marked, genus 0 tropical curve in $\bar{X}$*. The *degree* of $(\Gamma_m;E_1,\ldots,E_m;h)$ is the $d$-tuple of integers $\delta(h)=(c_{1,1},\ldots,c_{n_1+1,1},\ldots,c_{1,a},\ldots,c_{n_a+1,a},\ldots,
c_{1,l},\ldots,c_{n_l+1,l})\in\mathbb{Z}^d$, where $$c_{j,a}=\left\{\begin{array}{ll}
1 & \textrm{if $v_{j,a}\in\{v(E_1),\ldots,v(E_m)\}$}\\
0 & \textrm{otherwise.}
\end{array}\right.$$ By the balancing condition, we have $\sum_{a=1}^l\sum_{j=1}^{n_a+1}c_{j,a}v_{j,a}=0$, i.e. $\delta(h)$ lies in the kernel of $\partial:\mathbb{Z}^d\rightarrow N$, and so $\delta(h)\in H_2(\bar{X},\mathbb{Z})$.
We want to consider the tropical counterpart, denoted by $$TGW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i),$$ of the Gromov-Witten invariant $GW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)$. [^7] Since a general definition is not available, we introduce a tentative definition as follows.
We define $TGW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)$ to be the number of parameterized $(n_a+1)$-marked, genus 0 tropical curves of the form $(\Gamma_{n_a+1};E_1,\ldots,E_{n_a+1};h)$ with $\delta(h)=\delta$ such that $h(E_j)=h(V)+\mathbb{R}_{\geq0}v_{j,a}$, for $j=1,\ldots,n_a+1$, and $h(V)\in\textrm{Log}(T_i)$, where $T_i$ is a cycle Poincaré dual to $t_i$, whenever this number is finite. We set $TGW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)$ to be 0 if this number is infinite. Here, $\textrm{Log}:X\rightarrow
N_\mathbb{R}$ is the map, after identifying $X$ with $(\mathbb{C}^*)^n$, defined by $\textrm{Log}(w_1,\ldots,w_n)=(\log|w_1|,\ldots,\log|w_n|)$, for $(w_1,\ldots,w_n)\in X$.
We then define the *tropical small quantum cohomology ring* $QH^*_{trop}(\bar{X})$ of $\bar{X}=\mathbb{C}P^{n_1}\times\ldots\times\mathbb{C}P^{n_l}$ as a presentation: $$QH^*_{trop}(\bar{X})=\mathbb{C}[D_{1,1},\ldots,D_{n_1+1,1},\ldots,D_{1,l}\ldots,D_{n_l+1,l}]/
(\mathcal{L}+\mathcal{SR}_Q^{trop}),$$ where $\mathcal{SR}_Q^{trop}$ is the tropical version of the quantum Stanley-Reisner ideal, defined to be the ideal generated by the relations $$D_{1,a}\ast_T\ldots\ast_T D_{n_a+1,a}=\sum_{\delta\in H_2(\bar{X},\mathbb{Z})}\sum_{i}
TGW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)t^iq^\delta,$$ for $a=1,\ldots,l$. Here $\ast_T$ denotes the product in $QH^*_{trop}(\bar{X})$, which we call the tropical small quantum product. It is not hard to see that, as in the holomorphic case, we have $$TGW_{0,n_a+2}^{\bar{X},\delta}(\textrm{PD(pt)};D_{1,a},\ldots,D_{n_a+1,a},t_i)
=\left\{\begin{array}{ll}
1 & \textrm{if $t_i=\textrm{PD(pt)}$ and $\delta=\delta_a$}\\
0 & \textrm{otherwise.}
\end{array}\right.$$ Indeed, as a special case of the *correspondence theorem* of Mikhalkin [@Mikhalkin03] and Nishinou-Siebert [@NS04], we have: For a given point $p\in X$, let $\xi:=\textrm{Log}(p)\in
N_\mathbb{R}$. Then the unique holomorphic curve $\varphi_a:(\mathbb{C}P^1;x_1,\ldots,x_{n_a+2})\rightarrow\bar{X}$ with class $\delta_a$, $\varphi_a(x_j)\in D_{j,a}$, for $j=1,\ldots,n_a+1$, $\varphi_a(x_{n_a+2})=p$ and such that $(\mathbb{C}P^1;x_1,\ldots,x_{n_a+2})$ is a given configuration in $\overline{\mathcal{M}}_{0,n_a+2}$, is corresponding to the unique parameterized $(n_a+1)$-marked tropical curve $(\Gamma_{n_a+1};E_1,\ldots,E_{n_a+1};h_a)$ of genus 0 and degree $\delta_a$ such that $h_a(V)=\xi$ and $h_a(E_j)=\xi+\mathbb{R}_{\geq0}v_{j,a}$, for $j=1,\ldots,n_a+1$. It follows that $$\mathcal{SR}_Q^{trop}=\langle D_{1,a}\ast_T\ldots\ast_T D_{n_a+1,a}-q_a:a=1\ldots,l\rangle,$$ and there is a canonical isomorphism $$\label{step1}
QH^*(\bar{X})\cong QH^*_{trop}(\bar{X}).$$
All these arguments and definitions rely, in an essential way, on the fact that $\bar{X}$ is a product of projective spaces, so that Gromov-Witten invariants are enumerative and all the (irreducible) holomorphic curves, which contribute to $QH^*(\bar{X})$, are not mapped into the toric boundary divisor $D_\infty$. Remember that tropical geometry cannot be used to count nodal curves or curves with irreducible components mapping into $D_\infty$.
Next, we take a look at tropical discs. Consider the point $\Gamma_1\in\overline{\mathcal{M}}_{0,1}^{trop}$. This is nothing but a half line, consisting of an unbounded edge $E$ emanating from a univalent vertex $V$. See Figure 2.2 below.
(100,20) (20,3, 63,20) (19,2)[$\bullet$]{} (16,1)[$V$]{} (41,7)[$E$]{} (64,10)[Figure 2.2: $\Gamma_1\in\overline{\mathcal{M}}_{0,1}^{trop}.$]{}
A *parameterized Maslov index two tropical disc in $\bar{X}$* is a tuple $(\Gamma_1,E,h)$, where $h:\Gamma_1\rightarrow
N_\mathbb{R}$ is an embedding such that $h(E)=h(V)+\mathbb{R}_{\geq0}v$ for some $v\in\Sigma[1]$.[^8] For any given point $\xi\in N_\mathbb{R}$, it is obvious that, there is a unique parameterized Maslov index two tropical disc $(\Gamma_1,E,h_{j,a})$ such that $h_{j,a}(V)=\xi$ and $h_{j,a}(E)=\xi+\mathbb{R}_{\geq0}v_{j,a}$, for any $v_{j,a}\in\Sigma[1]$. Comparing this to the result (Theorem \[cho-oh\]) of Cho-Oh on the classification of Maslov index two holomorphic discs in $\bar{X}$ with boundary in the Lagrangian torus fiber $L_\xi:=\textrm{Log}^{-1}(\xi)\subset X$, we get a one-to-one correspondence between the families of Maslov index two holomorphic discs in $(\bar{X},L_\xi)$ and the parameterized Maslov index two tropical discs $(\Gamma_1,E,h)$ in $\bar{X}$ such that $h(V)=\xi$. We have the holomorphic disc $\varphi_{j,a}:(D^2,\partial D^2)\rightarrow(\bar{X},L_\xi)$ corresponding to the tropical disc $(\Gamma_1,E,h_{j,a})$.[^9] Then, by (\[1-1\]), we also get a one-to-one correspondence between the parameterized Maslov index two tropical discs $(\Gamma_1,E,h_{j,a})$ in $\bar{X}$ and the functions $\Psi_{j,a}:L_X\rightarrow\mathbb{R}$: $$\{\varphi_{j,a}\}\ \overset{1-1}{\longleftrightarrow}\
\{(\Gamma_1,E,h_{j,a})\}\ \overset{1-1}{\longleftrightarrow}\
\{\Psi_{j,a}\}.$$
Now, while the canonical isomorphism $$\label{step2}
QH^*_{trop}(\bar{X})\cong\mathbb{C}[\Psi_{1,1}^{\pm1},\ldots,\Psi_{n_1,1}^{\pm1},\ldots,
\Psi_{1,l}^{\pm1},\ldots,\Psi_{n_l,l}^{\pm1}]/\mathcal{L}$$ follows from the same simple combinatorial argument in the proof of Proposition \[prop2.2\], the geometry underlying it is exhibited by a simple but crucial observation, which we formulate as the following proposition.
\[prop2.3\] Let $\xi\in N_\mathbb{R}$, then the unique parameterized $(n_a+1)$-marked, genus 0 tropical curve $(\Gamma_{n_a+1};E_1,\ldots,E_{n_a+1};h_a)$ such that $h_a(V)=\xi$ and $h_a(E_j)=\xi+\mathbb{R}_{\geq0}v_{j,a}$, for $j=1,\ldots,n_a+1$, is obtained by gluing the $n_a+1$ parameterized Maslov index two tropical discs $(\Gamma_1,E,h_{1,a}),\ldots,(\Gamma_1,E,h_{n_a+1,a})$ with $h_{j,a}(V)=\xi$, for $j=1,\ldots,n_a+1$, in the following sense: The map $h:(\Gamma_{n_a+1};E_1,\ldots,E_{n_a+1})\rightarrow
N_\mathbb{R}$ defined by $h|_{E_j}=h_{j,a}|_E$, for $j=1,\ldots,n_a+1$, gives a parameterized $(n_a+1)$-marked, genus 0 tropical curve, which coincides with $(\Gamma_{n_a+1};E_1,\ldots,E_{n_a+1};h_a)$.
Since $\sum_{j=1}^{n_a+1}v_{j,a}=0$, the balancing condition at $V\in\Gamma_{n_a+1}$ is automatically satisfied. So $h$ defines a parameterized $(n_a+1)$-marked, genus 0 tropical curve, which satisfies the same conditions as $(\Gamma_{n_a+1};E_1,\ldots,E_{n_a+1};h_a)$.
For example, in the case of $\bar{X}=\mathbb{C}P^2$, this can be seen in Figure 2.3 below.
(100,35) (20,17, 38,17) (20,17, 20,35) (20,17, 5,2) (10,1)[$(\Gamma_3,h)$]{} (19.1,16.1)[$\bullet$]{} (16,16)[$V$]{} (42,16)[glued from]{} (74,17, 93,17) (73,16.1)[$\bullet$]{} (83,13.5)[$(\Gamma_1,h_1)$]{} (72,19, 72,37) (71.1,18)[$\bullet$]{} (72.5,29)[$(\Gamma_1,h_2)$]{} (71,16, 56,1) (70,15)[$\bullet$]{} (60,2)[$(\Gamma_1,h_3)$]{} (35,-2)[Figure 2.3]{}
The functions $\Psi_{j,a}$’s could have been defined by counting parameterized Maslov index two tropical discs, instead of counting Maslov index two holomorphic discs. So the above proposition indeed gives a geometric reason to explain why the relation $$D_{1,a}\ast_T\ldots\ast_T D_{n_a+1,a}=q_a$$ in $QH^*_{trop}(\bar{X})$ should coincide with the relation $$\Psi_{1,a}\star\ldots\star\Psi_{n_a+1,a}=q_a\mathbb{1}$$ in $\mathbb{C}[\Psi_{1,1}^{\pm1},\ldots,\Psi_{n_1,1}^{\pm1},\ldots,
\Psi_{1,l}^{\pm1},\ldots,\Psi_{n_l,l}^{\pm1}]/\mathcal{L}$. The convolution product $\star$ may then be thought of as a way to encode the gluing of tropical discs.
We summarize what we have said as follows: In the case of products of projective spaces, we factor the isomorphism $QH^*(\bar{X})\cong\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L}$ in Proposition \[prop2.2\] into two isomorphisms (\[step1\]) and (\[step2\]). The first one comes from the correspondence between holomorphic curves in $\bar{X}$ which contribute to $QH^*(\bar{X})$ and tropical curves in $N_\mathbb{R}$ which contribute to $QH^*_{trop}(\bar{X})$. The second isomorphism is due to, on the one hand, the fact that each tropical curve which contributes to $QH^*_{trop}(\bar{X})$ can be obtained by gluing Maslov index two tropical discs, and, on the other hand, the correspondence between these tropical discs in $N_\mathbb{R}$ and Maslov index two holomorphic discs in $\bar{X}$ with boundary on Lagrangian torus fibers. See Figure 2.4 below.
(100,31) (5,30)[$QH^*(\bar{X})$]{} (11,16)[(0,1)[12.5]{}]{} (11,16)[(0,-1)[11]{}]{} (4,1)[$QH^*_{trop}(\bar{X})$]{} (28.5,26)[(2,-1)[13]{}]{} (28.5,26)[(-2,1)[9]{}]{} (31,25.5)[Prop \[prop2.2\]]{} (28.5,6)[(2,1)[13]{}]{} (28.5,6)[(-2,-1)[7]{}]{} (31,4)[Prop \[prop2.3\]]{} (41,14.5)[$\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L}$]{} (80,15.5)[(1,0)[9]{}]{} (80,15.5)[(-1,0)[8]{}]{} (80,16.5)[$\mathcal{F}$]{} (90,14.5)[$Jac(W)$]{} (50,-2)[Figure 2.4]{}
Here $\mathcal{F}$ denotes the SYZ mirror transformation for $\bar{X}$, which is the subject of Section \[sec3\].
SYZ mirror transformations {#sec3}
==========================
In this section, we first derive Hori-Vafa’s mirror manifold using semi-flat SYZ mirror transformations. Then we introduce the main character in this paper: the SYZ mirror transformations for toric Fano manifolds, and prove our main result.
Derivation of Hori-Vafa’s mirror manifold by T-duality {#subsec3.1}
------------------------------------------------------
Recall that we have an exact sequence (\[seq2.1\]): $$\label{seq3.1}
0\longrightarrow K\overset{\iota}{\longrightarrow}\mathbb{Z}^d
\overset{\partial}{\longrightarrow}N\longrightarrow0,$$ and we denote by $$Q_1=(Q_{11},\ldots,Q_{d1}),\ldots,Q_l=(Q_{1l},\ldots,Q_{dl})\in\mathbb{Z}^d$$ a $\mathbb{Z}$-basis of $K$. The mirror manifold of $\bar{X}$, derived by Hori and Vafa in [@HV00] using physical arguments, is the complex submanifold $$Y_{HV}=\Big\{(Z_1,\ldots,Z_d)\in\mathbb{C}^d:\prod_{i=1}^d Z_i^{Q_{ia}}=q_a,\ a=1,\ldots,l\Big\},$$ in $\mathbb{C}^d$, where $q_a=e^{-r_a}=\exp(-\sum_{i=1}^d
Q_{ia}\lambda_i)$, for $a=1,\ldots,l$. As a complex manifold, $Y_{HV}$ is biholomorphic to the algebraic torus $(\mathbb{C}^*)^n$. By our choice of the $\mathbb{Z}$-basis $Q_1,\ldots,Q_l$ of $K$ in Remark \[rmk2.0\], $Y_{HV}$ can also be written as $$Y_{HV}=\Big\{(Z_1,\ldots,Z_d)\in\mathbb{C}^d:Z_1^{Q_{1a}}\ldots Z_n^{Q_{na}}Z_{n+a}=q_a,\ a=1,\ldots,l\Big\}.$$ Note that, in fact, $Y_{HV}\subset(\mathbb{C}^*)^d$. In terms of these coordinates, Hori and Vafa predicted that the superpotential $W:Y_{HV}\rightarrow\mathbb{C}$ is given by $$\begin{aligned}
W & = & Z_1+\ldots+Z_d\\
& = & Z_1+\ldots+Z_n+\frac{q_1}{Z_1^{Q_{11}}\ldots Z_n^{Q_{n1}}}+
\ldots+\frac{q_l}{Z_1^{Q_{1l}}\ldots Z_n^{Q_{nl}}}.\end{aligned}$$ The goal of this subsection is to show that the SYZ mirror manifold $Y_{SYZ}$, which is obtained by applying T-duality to the open dense orbit $X\subset\bar{X}$, is contained in Hori-Vafa’s manifold $Y_{HV}$ as a bounded open subset. The result itself is not new, and can be found, for example, in Auroux [@Auroux07], Proposition 4.2. For the sake of completeness, we give a self-contained proof, which will show how T-duality, i.e. fiberwise dualizing torus bundles, transforms the *symplectic quotient space* $X$ into the *complex subspace* $Y_{SYZ}$.\
We shall first briefly recall the constructions of $\bar{X}$ and $X$ as symplectic quotients. For more details, we refer the reader to Appendix 1 in Guillemin [@Guillemin94].
From the above exact sequence (\[seq3.1\]), we get an exact sequence of real tori $$\label{seq3.2}
1\longrightarrow T_K\overset{\iota}{\longrightarrow}T^d
\overset{\partial}{\longrightarrow}T_N\longrightarrow1,$$ where $T^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ and we denote by $K_\mathbb{R}$ and $T_K$ the real vector space $K\otimes_\mathbb{Z}\mathbb{R}$ and the torus $K_\mathbb{R}/K$ respectively. Considering their Lie algebras and dualizing give another exact sequence $$\label{seq3.2'}
0\longrightarrow
M_\mathbb{R}\overset{\check{\partial}}{\longrightarrow}(\mathbb{R}^d)^\vee
\overset{\check{\iota}}{\longrightarrow}K_\mathbb{R}^\vee\longrightarrow0.$$ Denote by $W_1,\ldots,W_d\in\mathbb{C}$ the complex coordinates on $\mathbb{C}^d$. The standard diagonal action of $T^d$ on $\mathbb{C}^d$ is Hamiltonian with respect to the standard symplectic form $\frac{\sqrt{-1}}{2}\sum_{i=1}^d dW_i\wedge
d\bar{W}_i$ and the moment map $h:\mathbb{C}^d\rightarrow(\mathbb{R}^d)^\vee$ is given by $$h(W_1,\ldots,W_d)=\frac{1}{2}(|W_1|^2,\ldots,|W_d|^2).$$ Restricting to $T_K$, we get a Hamiltonian action of $T_K$ on $\mathbb{C}^d$ with moment map $h_K=\check{\iota}\circ
h:\mathbb{C}^d\rightarrow\check{K}_\mathbb{R}$. In terms of the $\mathbb{Z}$-basis $\{Q_1,\ldots,Q_l\}$ of $K$, the map $\check{\iota}:(\mathbb{R}^d)^\vee\rightarrow K^\vee_\mathbb{R}$ is given by $$\label{iota*}
\check{\iota}(X_1,\ldots,X_d)=\Bigg(\sum_{i=1}^d
Q_{i1}X_i,\ldots,\sum_{i=1}^d Q_{il}X_i\Bigg),$$ for $(X_1,\ldots,X_d)\in(\mathbb{R}^d)^\vee$, in the coordinates associated to the dual basis $\check{Q}_1,\ldots,\check{Q}_l$ of $K^\vee=\textrm{Hom}(K,\mathbb{Z})$. The moment map $h_K:\mathbb{C}^d\rightarrow K^\vee_\mathbb{R}$ can thus be written as $$h_K(W_1,\ldots,W_d)=\frac{1}{2}\Bigg(\sum_{i=1}^d
Q_{i1}|W_i|^2,\ldots,\sum_{i=1}^d Q_{il}|W_i|^2\Bigg)\in
K^\vee_\mathbb{R}.$$ In these coordinates, $r=(r_1,\ldots,r_l)=-\check{\iota}(\lambda_1,\ldots,\lambda_d)$ is an element in $K^\vee_\mathbb{R}=H^2(\bar{X},\mathbb{R})$, and $\bar{X}$ and $X$ are given by the symplectic quotients $$\bar{X}=h_K^{-1}(r)/T_K\textrm{ and }X=(h_K^{-1}(r)\cap(\mathbb{C}^*)^d)/T_K$$ respectively.
In the above process, the image of $h_K^{-1}(r)$ under the map $h:\mathbb{C}^d\rightarrow(\mathbb{R}^d)^\vee$ lies inside the affine linear subspace $M_\mathbb{R}(r)=\{(X_1,\ldots,X_d)\in(\mathbb{R}^d)^\vee:\check{\iota}(X_1,\ldots,X_d)=r\}$, i.e. a translate of $M_\mathbb{R}$. In fact, $h(h_K^{-1}(r))=\check{\iota}^{-1}(r)\cap\{(X_1,\ldots,X_d)\in(\mathbb{R}^d)^\vee:X_i\geq0,\textrm{
for $i=1,\ldots,d$}\}$ is the polytope $\bar{P}\subset
M_\mathbb{R}(r)$, and $h(h_K^{-1}(r)\cap(\mathbb{C}^*)^d)=
\check{\iota}^{-1}(r)\cap\{(X_1,\ldots,X_d)\in(\mathbb{R}^d)^\vee:X_i>0,\textrm{
for $i=1,\ldots,d$}\}$ is the interior $P$ of $\bar{P}$. Now, restricting $h$ to $h_K^{-1}(r)\cap(\mathbb{C}^*)^d$ gives a $T^d$-bundle $h:h_K^{-1}(r)\cap(\mathbb{C}^*)^d\rightarrow P$ (which is trivial), and $X$ is obtained by taking the quotient of this $T^d$-bundle fiberwise by $T_K$. Hence, $X$ is naturally a $T_N$-bundle over $P$, which can be written as $$X=T^*P/N=P\times\sqrt{-1}T_N$$ (cf. Abreu [@Abreu00]).[^10] The reduced symplectic form $\omega_X=\omega_{\bar{X}}|_X$ is the standard symplectic form $$\omega_X=\sum_{j=1}^n dx_j\wedge du_j$$ where $x_1,\ldots,x_n\in\mathbb{R}$ and $u_1,\ldots,u_n\in\mathbb{R}/2\pi\mathbb{Z}$ are respectively the coordinates on $P\subset M_\mathbb{R}(r)$ and $T_N$. In other words, the $x_j$’s and $u_j$’s are *symplectic coordinates* (i.e. action-angle coordinates). And the moment map is given by the projection to $P$ $$\mu:X\rightarrow P.$$
We define the *SYZ mirror manifold* by T-duality as follows.
The SYZ mirror manifold $Y_{SYZ}$ is defined as the total space of the $T_M$-bundle, where $T_M=M_\mathbb{R}/M=(T_N)^\vee$, which is obtained by fiberwise dualizing the $T_N$-bundle $\mu:X\rightarrow
P$.
In other words, we have $$Y_{SYZ}=TP/M=P\times\sqrt{-1}T_M\subset M_\mathbb{R}(r)\times\sqrt{-1}T_M.$$ $Y_{SYZ}$ has a natural complex structure, which is induced from the one on $M_\mathbb{R}(r)\times\sqrt{-1}T_M\cong(\mathbb{C}^*)^n$. We let $z_j=\exp(-x_j-\sqrt{-1}y_j)$, $j=1,\ldots,n$, be the *complex coordinates* on $M_\mathbb{R}(r)\times\sqrt{-1}T_M\cong(\mathbb{C}^*)^n$ and restricted to $Y_{SYZ}$, where $y_1,\ldots,y_n\in\mathbb{R}/2\pi\mathbb{Z}$ are the coordinates on $T_M=(T_N)^\vee$ dual to $u_1,\ldots,u_n$. We also let $\Omega_{Y_{SYZ}}$ be the following nowhere vanishing holomorphic $n$-form on $Y_{SYZ}$: $$\Omega_{Y_{SYZ}}=\bigwedge_{j=1}^n(-dx_j-\sqrt{-1}dy_j)
=\frac{dz_1}{z_1}\wedge\ldots\wedge\frac{dz_n}{dz_n},$$ and denote by $$\nu:Y_{SYZ}\rightarrow P$$ the torus fibration dual to $\mu:X\rightarrow P$.
\[prop3.1\] The SYZ mirror manifold $Y_{SYZ}$ is contained in Hori-Vafa’s mirror manifold $Y_{HV}$ as an open complex submanifold. More precisely, $Y_{SYZ}$ is the bounded domain $\{(Z_1,\ldots,Z_d)\in
Y_{SYZ}:|Z_i|<1,\ i=1,\ldots,d\}$ inside $Y_{HV}$.
Dualizing the sequence (\[seq3.2\]), we get $$1\longrightarrow T_M\overset{\check{\partial}}{\longrightarrow}
(T^d)^\vee\overset{\check{\iota}}{\longrightarrow}(T_K)^\vee\longrightarrow1,$$ while we also have the sequence (\[seq3.2’\]) $$0\longrightarrow
M_\mathbb{R}\overset{\check{\partial}}{\longrightarrow}(\mathbb{R}^d)^\vee
\overset{\check{\iota}}{\longrightarrow}K^\vee_\mathbb{R}\longrightarrow0.$$ Let $T_i=X_i+\sqrt{-1}Y_i\in\mathbb{C}/2\pi\sqrt{-1}\mathbb{Z}$, $i=1,\ldots,d$, be the complex coordinates on $(\mathbb{R}^d)^\vee\times\sqrt{-1}(T^d)^\vee\cong(\mathbb{C}^*)^d$. If we let $Z_i=e^{-T_i}\in\mathbb{C}^*$, $i=1,\ldots,d$, then, by the definition of $Y_{HV}$ and by (\[iota\*\]), we can identify $Y_{HV}$ with the following complex submanifold in $(\mathbb{C}^*)^d$: $$\check{\iota}^{-1}(r\in
K^\vee_\mathbb{R})\cap\check{\iota}^{-1}(1\in
(T_K)^\vee)\subset(\mathbb{R}^d)^\vee\times\sqrt{-1}(T^d)^\vee=(\mathbb{C}^*)^d.$$ Hence, $$Y_{HV}=M_\mathbb{R}(r)\times\sqrt{-1}T_M\cong(\mathbb{C}^*)^n$$ as complex submanifolds in $(\mathbb{C}^*)^d$. Since $Y_{SYZ}=TP/M=P\times\sqrt{-1}T_M$, $Y_{SYZ}$ is a complex submanifold in $Y_{HV}$. In fact, as $P=\check{\iota}^{-1}(r)\cap\{(X_1,\ldots,X_d)\in(\mathbb{R}^d)^*:X_i>0,i=1,\ldots,d\}$, we have $$Y_{SYZ}=\{(Z_1,\ldots,Z_d)\in Y_{HV}:|Z_i|<1,\textrm{ for $i=1,\ldots,d$}\}.$$ So $Y_{SYZ}$ is a bounded domain in $Y_{HV}$.
We remark that, in terms of the complex coordinates $z_j=\exp(-x_j-\sqrt{-1}y_j)$, $j=1,\ldots,n$, on $Y_{HV}=M_\mathbb{R}(r)\times\sqrt{-1}T_M\cong(\mathbb{C}^*)^n$, the embedding $\check{\partial}:Y_{HV}\hookrightarrow(\mathbb{C}^*)^d$ is given by $$\check{\partial}(z_1,\ldots,z_n)=(e^{\lambda_1}z^{v_1},\ldots,e^{\lambda_d}z^{v_d}),$$ where $z^v$ denotes the monomial $z_1^{v^1}\ldots z_n^{v^n}$ if $v=(v^1,\ldots,v^n)\in N=\mathbb{Z}^n$. So the coordinates $Z_i$’s and $z_i$’s are related by $$Z_i=e^{\lambda_i}z^{v_i},$$ for $i=1,\ldots,d$, and the SYZ mirror manifold $Y_{SYZ}$ is given by the bounded domain $$Y_{SYZ}=\{(z_1,\ldots,z_n)\in Y_{HV}=(\mathbb{C}^*)^n:|e^{\lambda_i}z^{v_i}|<1,\ i=1,\ldots,d\}.$$ Now, the superpotential $W:Y_{SYZ}\rightarrow\mathbb{C}$ (or $W:Y_{HV}\rightarrow\mathbb{C}$) is of the form $$W=e^{\lambda_1}z^{v_1}+\ldots+e^{\lambda_d}z^{v_d}.$$
From the above proposition, the SYZ mirror manifold $Y_{SYZ}$ is strictly *smaller* than Hori-Vafa’s mirror manifold $Y_{HV}$. This issue was discussed in Hori-Vafa [@HV00], Section 3 and Auroux [@Auroux07], Section 4.2, and may be resolved by a process called *renormalization*. We refer the interested reader to those references for the details. In this paper, we shall always be (except in this subsection) looking at the SYZ mirror manifold, and the letter $Y$ will also be used exclusively to denote the SYZ mirror manifold.
SYZ mirror transformations as fiberwise Fourier transforms {#subsec3.2}
----------------------------------------------------------
In this subsection, we first give a brief review of semi-flat SYZ mirror transformations (for details, see Hitchin [@Hitchin97], Leung-Yau-Zaslow [@LYZ00] and Leung [@Leung00]). Then we introduce the SYZ mirror transformations for toric Fano manifolds, and prove part 1. of Theorem \[main\_thm\].\
To begin with, recall that the dual torus $T_M=(T_N)^\vee$ of $T_N$ can be interpreted as the moduli space of flat $U(1)$-connections on the trivial line bundle $T_N\times\mathbb{C}\rightarrow T_N$. In more explicit terms, a point $y=(y_1,\ldots,y_n)\in
M_\mathbb{R}\cong\mathbb{R}^n$ corresponds to the flat $U(1)$-connection $\nabla_y=d+\frac{\sqrt{-1}}{2}\sum_{j=1}^n y_j
du_j$ on the trivial line bundle $T_N\times\mathbb{C}\rightarrow
T_N$. The *holonomy* of this connection is given, in our convention, by the map $$\textrm{hol}_{\nabla_y}:N\rightarrow U(1),\ v\mapsto
e^{-\sqrt{-1}\langle y,v\rangle}.$$ $\nabla_y$ is gauge equivalent to the trivial connection $d$ if and only if $(y_1,\ldots,y_n)\in(2\pi\mathbb{Z})^n=M$. So, in the following, we will regard $(y_1,\ldots,y_n)\in
T_M\cong\mathbb{R}^n/(2\pi\mathbb{Z})^n$. Moreover, this construction gives all flat $U(1)$-connections on $T_N\times\mathbb{C}\rightarrow T_N$ up to unitary gauge transformations. The universal $U(1)$-bundle $\mathcal{P}$, i.e. the Poincaré line bundle, is the trivial line bundle over the product $T_N\times T_M$ equipped with the $U(1)$-connection $d+\frac{\sqrt{-1}}{2}\sum_{j=1}^n (y_jdu_j-u_jdy_j)$, where $u_1,\ldots,u_n\in\mathbb{R}/2\pi\mathbb{Z}$ are the coordinates on $T_N\cong\mathbb{R}^d/(2\pi\mathbb{Z})^d$. The curvature of this connection is given by the two-form $$F=\sqrt{-1}\sum_{j=1}^n dy_j\wedge du_j\in\Omega^2(T_N\times T_M).$$
From this perspective, the SYZ mirror manifold $Y$ is the moduli space of pairs $(L_x,\nabla_y)$, where $L_x$ ($x\in P$) is a Lagrangian torus fiber of $\mu:X\rightarrow P$ and $\nabla_y$ is a flat $U(1)$-connection on the trivial line bundle $L_x\times\mathbb{C}\rightarrow L_x$. The construction of the mirror manifold in this way is originally advocated in the SYZ Conjecture [@SYZ96] (cf. Hitchin [@Hitchin97] and Sections 2 and 4 in Auroux [@Auroux07]).
Now recall that we have the dual torus bundles $\mu:X\rightarrow
P\textrm{ and }\nu:Y\rightarrow P$. Consider their fiber product $X\times_P Y=P\times\sqrt{-1}(T_N\times T_M)$. $$\begin{CD}
X\times_P Y @>\pi_Y>> Y \\
@VV\pi_X V @VV \nu V \\
X @>\mu>> P
\end{CD}\\$$ By abuse of notations, we still use $F$ to denote the fiberwise universal curvature two-form $\sqrt{-1}\sum_{j=1}^n dy_j\wedge
du_j\in\Omega^2(X\times_P Y)$.
\[def3.1\] The semi-flat SYZ mirror transformation $\mathcal{F}^{\textrm{sf}}:\Omega^*(X)\rightarrow\Omega^*(Y)$ is defined by $$\begin{aligned}
\mathcal{F}^{\textrm{sf}}(\alpha) & = &
(-2\pi\sqrt{-1})^{-n}\pi_{Y,*}(\pi_X^*(\alpha)\wedge e^{\sqrt{-1}F})\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{T_N}\pi_X^*(\alpha)\wedge
e^{\sqrt{-1}F},\end{aligned}$$ where $\pi_X:X\times_P Y\rightarrow X$ and $\pi_Y:X\times_P
Y\rightarrow Y$ are the two natural projections.
The key point is that, the semi-flat SYZ mirror transformation $\mathcal{F}^{\textrm{sf}}$ transforms the (exponential of $\sqrt{-1}$ times the) symplectic structure $\omega_X=\sum_{j=1}^ndx_j\wedge du_j$ on $X$ to the holomorphic $n$-form $\Omega_Y=\frac{dz_1}{z_1}\wedge\ldots\wedge\frac{dz_n}{z_n}$ on $Y$, where $z_j=\exp(-x_j-\sqrt{-1}y_j)$, $j=1,\ldots,n$. This is probably well-known and implicitly contained in the literature, but we include a proof here because we cannot find a suitable reference.\
\[prop3.2\] We have $$\mathcal{F}^{\textrm{sf}}(e^{\sqrt{-1}\omega_X})=\Omega_Y.$$ Moreover, if we define the inverse SYZ transformation $(\mathcal{F}^{\textrm{sf}})^{-1}:\Omega^*(Y)\rightarrow\Omega^*(X)$ by $$\begin{aligned}
(\mathcal{F}^{sf})^{-1}(\alpha) & = &
(-2\pi\sqrt{-1})^{-n}\pi_{X,*}(\pi_Y^*(\alpha)\wedge e^{-\sqrt{-1}F})\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{T_M}\pi_Y^*(\alpha)\wedge
e^{-\sqrt{-1}F},\end{aligned}$$ then we also have $$(\mathcal{F}^{\textrm{sf}})^{-1}(\Omega_Y)=e^{\sqrt{-1}\omega_X}.$$
The proof is by straightforward computations. $$\begin{aligned}
\mathcal{F}^{\textrm{sf}}(e^{\sqrt{-1}\omega_X}) & = &
(-2\pi\sqrt{-1})^{-n}\int_{T_N}\pi_{X}^*(e^{\sqrt{-1}\omega_X})\wedge e^{\sqrt{-1}F}\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{T_N}e^{\sqrt{-1}\sum_{j=1}^n (dx_j+\sqrt{-1}dy_j)\wedge du_j}\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{T_N} \bigwedge_{j=1}^n\big(1+\sqrt{-1}(dx_j+\sqrt{-1}dy_j)\wedge du_j\big)\\
& = & (2\pi)^{-n}\int_{T_N}
\Bigg(\bigwedge_{j=1}^n(-dx_j-\sqrt{-1}dy_j)\Bigg)\wedge
du_1\wedge\ldots\wedge du_n\\
& = & \Omega_Y,\end{aligned}$$ where we have $\int_{T_N}du_1\wedge\ldots\wedge du_n=(2\pi)^n$ for the last equality. On the other hand, $$\begin{aligned}
(\mathcal{F}^{\textrm{sf}})^{-1}(\Omega_Y) & = &
(-2\pi\sqrt{-1})^{-n}\int_{T_M}\pi_Y^*(\Omega_Y)\wedge e^{-\sqrt{-1}F}\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{T_M}
\Bigg(\bigwedge_{j=1}^n(-dx_j-\sqrt{-1}dy_j)\Bigg)\wedge
e^{\sum_{j=1}^n dy_j\wedge du_j}\\
& = & (2\pi\sqrt{-1})^{-n}\int_{T_M}\bigwedge_{j=1}^n\big((dx_j+\sqrt{-1}dy_j)\wedge e^{dy_j\wedge du_j}\big)\\
%& = & i^{-n}\int_{T_M} \bigwedge_{j=1}^n\Big((dx_j+idy_j)\wedge(1+idu_j\wedge dy_j)\Big)\\
& = & (2\pi\sqrt{-1})^{-n}\int_{T_M}\bigwedge_{j=1}^n\big(dx_j+\sqrt{-1}dy_j-dx_j\wedge du_j\wedge dy_j\big)\\
& = & (2\pi)^{-n}\int_{T_M}\bigwedge_{j=1}^n \Big(1+\sqrt{-1}dx_j\wedge du_j\Big)\wedge dy_j\\
& = &
(2\pi)^{-n}\int_{T_M}\bigwedge_{j=1}^n\big(e^{\sqrt{-1}dx_j\wedge
du_j}\wedge dy_j\big)\end{aligned}$$ $$\begin{aligned}
& = & (2\pi)^{-n}\int_{T_M}e^{\sqrt{-1}\sum_{j=1}^n dx_j\wedge du_j}\wedge dy_1\wedge\ldots\wedge dy_n\\
& = & e^{\sqrt{-1}\omega_X},\end{aligned}$$ where we again have $\int_{T_M}dy_1\wedge\ldots\wedge dy_n=(2\pi)^n$ in the last step.
One can also apply the semi-flat SYZ mirror transformations to other geometric structures and objects. For details, see Leung [@Leung00].\
The semi-flat SYZ mirror transformation $\mathcal{F}^{\textrm{sf}}$ can transform the symplectic structure $\omega_X$ on $X$ to the holomorphic $n$-form $\Omega_Y$ on $Y$. However, as we mentioned in the introduction, we are not going to obtain the superpotential $W:Y\rightarrow\mathbb{C}$ in this way because we have ignored the toric boundary divisor $\bar{X}\setminus X=D_\infty=\bigcup_{i=1}^d
D_i$. Indeed, it is the toric boundary divisor $D_\infty$ which gives rise to the quantum corrections in the A-model of $\bar{X}$. More precisely, these quantum corrections are due to the existence of holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers which have intersections with the divisor $D_\infty$. To restore this information, our way out is to look at the (trivial) $\mathbb{Z}^n$-cover $$\pi:LX=X\times N\rightarrow X.$$ Recall that we equip $LX$ with the symplectic structure $\pi^*(\omega_X)$; we will confuse the notations and use $\omega_X$ to denote either the symplectic structure on $X$ or that on $LX$. We will further abuse the notations by using $\mu$ to denote the fibration $$\mu:LX\rightarrow P,$$ which is the composition of the map $\pi:LX\rightarrow X$ with $\mu:X\rightarrow P$.\
We are now ready to define the SYZ mirror transformation $\mathcal{F}$ for the toric Fano manifold $\bar{X}$. It will be constructed as *a combination of the semi-flat SYZ transformation $\mathcal{F}^{\textrm{sf}}$ and taking fiberwise Fourier series*.
Analog to the semi-flat case, consider the fiber product $$LX\times_P Y=P\times N\times\sqrt{-1}(T_N\times T_M)$$ of the maps $\mu:LX\rightarrow P$ and $\nu:Y\rightarrow P$. $$\begin{CD}
LX\times_P Y @>\pi_Y>> Y \\
@VV\pi_{LX} V @VV \nu V \\
LX @>\mu>> P
\end{CD}\\$$ Note that we have a covering map $LX\times_P Y\rightarrow X\times_P
Y$. Pulling back $F\in\Omega^2(X\times_P Y)$ to $LX\times_P Y$ by this covering map, we get the fiberwise universal curvature two-form $$F=\sqrt{-1}\sum_{j=1}^n dy_j\wedge du_j\in\Omega^2(LX\times_P Y).$$ We further define the *holonomy function* $\textrm{hol}:LX\times_P Y\rightarrow U(1)$ as follows. For $(p,v)\in LX$ and $z=(z_1,\ldots,z_n)\in Y$ such that $\mu(p)=\nu(z)=:x\in P$, we let $x=(x_1,\ldots,x_n)$, and write $z_j=\exp(-x_j-\sqrt{-1}y_j)$, so that $y=(y_1,\ldots,y_n)\in
(L_x)^\vee:=\nu^{-1}(x)\subset Y$. Then we set $$\textrm{hol}(p,v,z):=\textrm{hol}_{\nabla_y}(v)=e^{-\sqrt{-1}\langle y,v\rangle},$$ where $\nabla_y$ is the flat $U(1)$-connection on the trivial line bundle $L_x\times\mathbb{C}\rightarrow L_x$ over $L_x:=\mu^{-1}(x)$ corresponding to the point $y\in(L_x)^\vee$.
\[def3.2\] The SYZ mirror transformation $\mathcal{F}:\Omega^*(LX)\rightarrow\Omega^*(Y)$ for the toric Fano manifold $\bar{X}$ is defined by $$\begin{aligned}
\mathcal{F}(\alpha) & = &
(-2\pi\sqrt{-1})^{-n}\pi_{Y,*}(\pi_{LX}^*(\alpha)\wedge e^{\sqrt{-1}F}\textrm{hol})\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{N\times
T_N}\pi_{LX}^*(\alpha)\wedge e^{\sqrt{-1}F}\textrm{hol},\end{aligned}$$ where $\pi_{LX}:LX\times_P Y\rightarrow LX$ and $\pi_Y:LX\times_P
Y\rightarrow Y$ are the two natural projections.
Before stating the basic properties of $\mathcal{F}$, we introduce the class of functions on $LX$ relevant to our applications.
\[def3.3\] A $T_N$-invariant function $f:LX\rightarrow\mathbb{C}$ is said to be admissible if for any $(p,v)\in LX=X\times N$, $$f(p,v)=f_v e^{-\langle x,v\rangle},$$ where $x=\mu(p)\in P$ and $f_v\in\mathbb{C}$ is a constant, and the fiberwise Fourier series $$\widehat{f}:=\sum_{v\in N}f_v e^{-\langle
x,v\rangle}\textrm{hol}_{\nabla_y}(v)=\sum_{v\in N}f_vz^v,$$ where $z^v=\exp(\langle-x-\sqrt{-1}y,v\rangle)$, is convergent and analytic, as a function on $Y$. We denote by $\mathcal{A}(LX)\subset
C^\infty(LX)$ set of all admissible functions on $LX$.
Examples of admissible functions on $LX$ include those $T_N$-invariant functions which are not identically zero on $X\times\{v\}\subset LX$ for only finitely many $v\in N$. In particular, the functions $\Psi_1,\ldots,\Psi_d$ are all in $\mathcal{A}(LX)$. We will see (in the proof of Theorem \[thm3.2\]) shortly that $\Phi_q$ is also admissible.
Now, for functions $f,g\in\mathcal{A}(LX)$, we define their *convolution product* $f\star g:LX\rightarrow\mathbb{C}$, as before, by $$(f\star g)(p,v)=\sum_{v_1,v_2\in N,\ v_1+v_2=v}f(p,v_1)g(p,v_2).$$ That the right-hand-side is convergent can be seen as follows. By definition, $f,g\in\mathcal{A}(LX)$ implies that for any $p\in X$ and any $v_1,v_2\in N$, $$f(p,v_1)=f_{v_1}e^{-\langle x,v_1\rangle},\ g(p,v_2)=g_{v_2}e^{-\langle x,v_2\rangle},$$ where $x=\mu(p)$ and $f_{v_1},g_{v_2}\in\mathbb{C}$ are constants; also, the series $\widehat{f}=\sum_{v_1\in N}f_{v_1}z^{v_1}$ and $\widehat{g}=\sum_{v_2\in N}g_{v_2}z^{v_2}$ are convergent and analytic. Then their product, given by $$\begin{aligned}
\widehat{f}\cdot\widehat{g}=\Bigg(\sum_{v_1\in
N}f_{v_1}z^{v_1}\Bigg)\Bigg(\sum_{v_2\in
N}g_{v_2}z^{v_2}\Bigg)=\sum_{v\in N}\Bigg(\sum_{\substack{v_1,v_2\in
N,\\ v_1+v_2=v}}f_{v_1}g_{v_2}\Bigg)z^v,\end{aligned}$$ is also analytic. This shows that the convolution product $f\star g$ is well defined and gives another admissible function on $LX$. Hence, the $\mathbb{C}$-vector space $\mathcal{A}(LX)$, together with the convolution product $\star$, forms a $\mathbb{C}$-algebra.\
Let $\mathcal{O}(Y)$ be the $\mathbb{C}$-algebra of holomorphic functions on $Y$. Recall that $Y=TP/M=P\times\sqrt{-1}T_M$. For $\phi\in\mathcal{O}(Y)$, the restriction of $\phi$ to a fiber $(L_x)^\vee=\nu^{-1}(x)\cong T_M$ gives a $C^\infty$ function $\phi_x:T_M\rightarrow\mathbb{C}$ on the torus $T_M$. For $v\in N$, the $v$-th Fourier coefficient of $\phi_x$ is given by $$\widehat{\phi}_x(v)=\int_{T_M}\phi_x(y)e^{\sqrt{-1}\langle y,v\rangle}dy_1\wedge\ldots\wedge dy_n.$$ Then, we define a function $\widehat{\phi}:LX\rightarrow\mathbb{C}$ on $LX$ by $$\widehat{\phi}(p,v)=\widehat{\phi}_x(v),$$ where $x=\mu(p)\in P$. $\widehat{\phi}$ is clearly admissible. We call the process, $\phi\in\mathcal{O}(Y)\mapsto\widehat{\phi}\in\mathcal{A}(LX)$, taking *fiberwise Fourier coefficients*. The following lemma follows from the standard theory of Fourier analysis on tori (see, for example, Edwards [@Edwards79]).
\[lem3.1\] Taking fiberwise Fourier series, i.e. the map $$\mathcal{A}(LX)\rightarrow\mathcal{O}(Y),\quad f\mapsto\widehat{f}$$ is an isomorphism of $\mathbb{C}$-algebras, where we equip $\mathcal{A}(LX)$ with the convolution product and $\mathcal{O}(Y)$ with the ordinary product of functions. The inverse is given by taking fiberwise Fourier coefficients. In particular, $\widehat{\widehat{f}}=f$ for any $f\in\mathcal{A}(LX)$.
The basic properties of the SYZ mirror transformation $\mathcal{F}$ are summarized in the following theorem.
\[thm3.1\] Let $\mathcal{A}(LX)e^{\sqrt{-1}\omega_X}:=\{fe^{\sqrt{-1}\omega_X}:f\in
\mathcal{A}(LX)\}\subset\Omega^*(LX)$ and $\mathcal{O}(Y)\Omega_Y:=\{\phi\Omega_Y:\phi\in\mathcal{O}(Y)\}\subset\Omega^*(Y)$.
1. For any admissible function $f\in\mathcal{A}(LX)$, $$\mathcal{F}(fe^{\sqrt{-1}\omega_X})=\widehat{f}\Omega_Y\in\mathcal{O}(Y)\Omega_Y.$$
2. If we define the inverse SYZ mirror transformation $\mathcal{F}^{-1}: \Omega^*(Y)\rightarrow\Omega^*(LX)$ by $$\begin{aligned}
\mathcal{F}^{-1}(\alpha) & = &
(-2\pi\sqrt{-1})^{-n}\pi_{LX,*}(\pi_Y^*(\alpha)\wedge e^{-\sqrt{-1}F}\textrm{hol}^{-1})\\
& = & (-2\pi\sqrt{-1})^{-n}\int_{T_M}\pi_Y^*(\alpha)\wedge
e^{-\sqrt{-1}F}\textrm{hol}^{-1},\end{aligned}$$ where $\textrm{hol}^{-1}:LX\times_P Y\rightarrow\mathbb{C}$ is the function defined by $\textrm{hol}^{-1}(p,v,z)=1/\textrm{hol}(p,v,z)=e^{\sqrt{-1}\langle
y,v\rangle}$, for any $(p,v,z)\in L_X\times_P Y$, then $$\mathcal{F}^{-1}(\phi\Omega_Y)=\widehat{\phi}e^{\sqrt{-1}\omega_X}\in\mathcal{A}(LX)e^{\sqrt{-1}\omega_X},$$ for any $\phi\in\mathcal{O}(Y)$.
3. The restriction map $\mathcal{F}:\mathcal{A}(LX)e^{\sqrt{-1}\omega_X}
\rightarrow\mathcal{O}(Y)\Omega_Y$ is a bijection with inverse $\mathcal{F}^{-1}:\mathcal{O}(Y)\Omega_Y\rightarrow\mathcal{A}(LX)e^{\sqrt{-1}\omega_X}$, i.e. we have $$\mathcal{F}^{-1}\circ\mathcal{F}=\textrm{Id}_{\mathcal{A}(LX)e^{\sqrt{-1}\omega_X}},\
\mathcal{F}\circ\mathcal{F}^{-1}=\textrm{Id}_{\mathcal{O}(Y)\Omega_Y}.$$ This shows that the SYZ mirror transformation $\mathcal{F}$ has the inversion property.
Let $f\in\mathcal{A}(LX)$. Then, for any $v\in N$, $f(p,v)=f_ve^{-\langle x,v\rangle}$ for some constant $f_v\in\mathbb{C}$. By observing that both functions $\pi_{LX}^*(f)$ and $\textrm{hol}$ are $T_N$-invariant functions on $LX\times_P Y$, we have $$\begin{aligned}
\mathcal{F}(fe^{\sqrt{-1}\omega_X}) & = &
(-2\pi\sqrt{-1})^{-n}\int_{N\times
T_N}\pi_{LX}^*(fe^{\sqrt{-1}\omega_X})\wedge
e^{\sqrt{-1}F}\textrm{hol}\\
& = & (-2\pi\sqrt{-1})^{-n}\sum_{v\in
N}\pi_{LX}^*(f)\cdot\textrm{hol}
\int_{T_N}\pi_{LX}^*(e^{\sqrt{-1}\omega_X})\wedge e^{\sqrt{-1}F}\\
& = & (-2\pi\sqrt{-1})^{-n}\Bigg(\sum_{v\in
N}\pi_{LX}^*(f)\cdot\textrm{hol}\Bigg)
\Bigg(\int_{T_N}\pi_X^*(e^{\sqrt{-1}\omega_X})\wedge
e^{\sqrt{-1}F}\Bigg).\end{aligned}$$ The last equality is due to the fact that the forms $\pi_{LX}^*(e^{\sqrt{-1}\omega_X})=\pi_X^*(e^{\sqrt{-1}\omega_X})$ and $e^F$ are independent of $v\in N$. By Proposition \[prop3.2\], the second factor is given by $$\int_{T_N}\pi_X^*(e^{\sqrt{-1}\omega_X})\wedge e^{\sqrt{-1}F}
=(-2\pi\sqrt{-1})^n\mathcal{F}^{\textrm{sf}}(e^{\sqrt{-1}\omega_X})=(-2\pi\sqrt{-1})^n\Omega_Y,$$ while the first factor is the function on $Y$ given, for $x=(x_1,\ldots,x_n)\in P$ and $y=(y_1,\ldots,y_n)\in T_M$, by $$\begin{aligned}
\Bigg(\sum_{v\in N}\pi_{LX}^*(f)\cdot\textrm{hol}\Bigg)(x,y) & = &
\sum_{v\in N} f_ve^{-\langle x,v\rangle}e^{-\sqrt{-1}\langle y,v\rangle}\\
& = & \sum_{v\in N} f_vz^v\\
& = & \widehat{f}(z),\end{aligned}$$ where $z=(z_1,\ldots,z_n)=(\exp(-x_1-\sqrt{-1}y_1),\ldots,\exp(-x_n-\sqrt{-1}y_n))\in
Y$. Hence $\mathcal{F}(fe^{\sqrt{-1}\omega_X})=\widehat{f}\Omega_Y\in\mathcal{O}(Y)\Omega_Y$. This proves (i).
For (ii), expand $\phi\in\mathcal{O}(Y)$ into a fiberwise Fourier series $$\phi(z)=\sum_{w\in N}\widehat{\phi}_x(w)e^{-\sqrt{-1}\langle
y,w\rangle},$$ where $x,y,z$ are as before. Then $$\begin{aligned}
\mathcal{F}^{-1}(\phi\Omega_Y) & = &
(-2\pi\sqrt{-1})^{-n}\int_{T_M}\pi_Y^*(\phi\Omega_Y)\wedge e^{-\sqrt{-1}F}\textrm{hol}^{-1}\\
& = & (-2\pi\sqrt{-1})^{-n}\sum_{w\in
N}\Bigg(\widehat{\phi}_x(w)\int_{T_M}e^{\sqrt{-1}\langle
y,v-w\rangle}\pi_Y^*(\Omega_Y)\wedge e^{-\sqrt{-1}F}\Bigg).\end{aligned}$$ Here comes the key observation: If $v-w\neq0\in N$, then, using (the proof of) the second part of Proposition \[prop3.2\], we have $$\begin{aligned}
& & \int_{T_M} e^{\sqrt{-1}\langle
y,v-w\rangle}\pi_Y^*(\Omega_Y)\wedge e^{-\sqrt{-1}F}\\
& = & \int_{T_M} e^{\sqrt{-1}\langle
y,v-w\rangle}\Bigg(\bigwedge_{j=1}^n(-dx_j-\sqrt{-1}dy_j)\Bigg)\wedge
e^{\sum_{j=1}^n dy_j\wedge du_j}\\
& = & (-\sqrt{-1})^n
e^{\sqrt{-1}\omega_X}\int_{T_M}e^{\sqrt{-1}\langle
y,v-w\rangle} dy_1\wedge\ldots\wedge dy_n\\
& = & 0.\end{aligned}$$ Hence, $$\begin{aligned}
\mathcal{F}^{-1}(\phi\Omega_Y) & = &
(-2\pi\sqrt{-1})^{-n}\widehat{\phi}_x(v)\int_{T_M}\pi_Y^*(\Omega_Y)\wedge
e^{-\sqrt{-1}F}\\
& = &
\widehat{\phi}\Big((\mathcal{F}^{\textrm{sf}})^{-1}(\Omega_Y)\Big)
=\widehat{\phi}e^{\omega_X}\in\mathcal{A}(LX)e^{\omega_X},\end{aligned}$$ again by Proposition \[prop3.2\].
\(iii) follows from (i), (ii) and Lemma \[lem3.1\].
We will, again by abuse of notations, also use $\mathcal{F}:\mathcal{A}(LX)\rightarrow\mathcal{O}(Y)$ to denote the process of taking fiberwise Fourier series: $\mathcal{F}(f):=\widehat{f}$ for $f\in\mathcal{A}(LX)$. Similarly, we use $\mathcal{F}^{-1}:\mathcal{O}(Y)\rightarrow\mathcal{A}(LX)$ to denote the process of taking fiberwise Fourier coefficients: $\mathcal{F}^{-1}(\phi):=\widehat{\phi}$ for $\phi\in\mathcal{O}(Y)$. To which meanings of the symbols $\mathcal{F}$ and $\mathcal{F}^{-1}$ are we referring will be clear from the context.
We can now prove the first part of Theorem \[main\_thm\], as a corollary of Theorem \[thm3.1\].
\[thm3.2\] The SYZ mirror transformation of the function $\Phi_q\in
C^\infty(LX)$, defined in terms of the counting of Maslov index two holomorphic discs in $\bar{X}$ with boundary in Lagrangian torus fibers, is the exponential of the superpotential $W$ on the mirror manifold $Y$, i.e. $$\mathcal{F}(\Phi_q)=e^W.$$ Conversely, we have $$\mathcal{F}^{-1}(e^W)=\Phi_q.$$ Furthermore, we can incorporate the symplectic structure $\omega_X=\omega_{\bar{X}}|_X$ on $X$ to give the holomorphic volume form on the Landau-Ginzburg model $(Y,W)$ through the SYZ mirror transformation $\mathcal{F}$, and vice versa, in the following sense: $$\mathcal{F}(\Phi_q e^{\sqrt{-1}\omega_X})=e^W\Omega_Y,\
\mathcal{F}^{-1}(e^W\Omega_Y)=\Phi_q e^{\sqrt{-1}\omega_X}.$$
By Theorem \[thm3.1\], we only need to show that $\Phi_q\in
C^\infty(LX)$ is admissible and $\mathcal{F}(\Phi_q)=\widehat{\Phi}_q=e^W\in\mathcal{O}(Y)$. Recall that, for $(p,v)\in LX=X\times N$ and $x=\mu(p)\in P$, $$\Phi_q(p,v)=\sum_{\beta\in\pi_2^+(\bar{X},L_x),\
\partial\beta=v}\frac{1}{w(\beta)}e^{-\frac{1}{2\pi}\int_\beta\omega_{\bar{X}}}.$$ For $\beta\in\pi_2^+(\bar{X},L_x)$ with $\partial\beta=v$, by the symplectic area formula (\[area\]) of Cho-Oh, we have $\int_\beta\omega_{\bar{X}}=2\pi\langle x,v\rangle+\textrm{const}$. So $\Phi_q(p,v)$ is of the form $\textrm{const}\cdot e^{-\langle
x,v\rangle}$. Now, $$\begin{aligned}
\sum_{v\in N}\Phi_q(p,v)\textrm{hol}_{\nabla_y}(v) & = & \sum_{v\in
N}\Bigg(\sum_{\beta\in\pi_2^+(\bar{X},L_x),\
\partial\beta=v}\frac{1}{w(\beta)}e^{-\frac{1}{2\pi}\int_\beta\omega_{\bar{X}}}\Bigg)e^{-\sqrt{-1}\langle y,v\rangle}\\
& = & \sum_{k_1,\ldots,k_d\in\mathbb{Z}_{\geq0}}\frac{1}{k_1!\ldots
k_d!}e^{-\sum_{i=1}^d k_i(\langle
x,v_i\rangle-\lambda_i)}e^{-\sum_{i=1}^d k_i\sqrt{-1}\langle
y,v_i\rangle}\\
& = & \prod_{i=1}^d\Bigg(\sum_{k_i=0}^\infty\frac{1}{k_i!}
\big(e^{\lambda_i-\langle
x+\sqrt{-1}y,v_i\rangle}\big)^{k_i}\Bigg)\\
& = & \prod_{i=1}^d \exp(e^{\lambda_i}z^{v_i})=e^W.\end{aligned}$$ This shows that $\Phi_q$ is admissible and $\widehat{\Phi}_q=e^W$.
The form $\Phi_q e^{\sqrt{-1}\omega_X}\in\Omega^*(LX)$ can be viewed as *the symplectic structure modified by quantum corrections from Maslov index two holomorphic discs in $\bar{X}$ with boundaries on Lagrangian torus fibers*. That we call $e^W\Omega_Y$ the holomorphic volume form of the Landau-Ginzburg model $(Y,W)$ can be justified in several ways. For instance, in the theory of singularities, one studies the complex oscillating integrals $$I=\int_{\Gamma} e^{\frac{1}{\hbar}W}\Omega_Y,$$ where $\Gamma$ is some real $n$-dimensional cycle in $Y$ constructed by the Morse theory of the function $\textrm{Re($W$)}$. These integrals are reminiscent of the periods of holomorphic volume forms on Calabi-Yau manifolds, and they satisfy certain Picard-Fuchs equations (see, for example, Givental [@Givental97b]). Hence, one may think of $e^W\Omega_Y$ as playing the same role as the holomorphic volume form on a Calabi-Yau manifold.
Quantum cohomology vs. Jacobian ring {#subsec3.3}
------------------------------------
The purpose of this subsection is to give a proof of the second part of Theorem \[main\_thm\]. Before that, let us recall the definition of the Jacobian ring $Jac(W)$. Recall that the SYZ mirror manifold $Y$ is given by the bounded domain $$Y=\{(z_1,\ldots,z_n)\in(\mathbb{C}^*)^n:|e^{\lambda_i}z^{v_i}|<1,\ i=1,\ldots,d\},$$ in $(\mathbb{C}^*)^n$, and the superpotential $W:Y\rightarrow\mathbb{C}$ is the Laurent polynomial $$W=e^{\lambda_1}z^{v_1}+\ldots+e^{\lambda_d}z^{v_d},$$ where, as before, $z^v$ denotes the monomial $z_1^{v^1}\ldots
z_n^{v^n}$ if $v=(v^1,\ldots,v^n)\in N=\mathbb{Z}^n$. Let $\mathbb{C}[Y]=\mathbb{C}[z_1^{\pm1},\ldots,z_n^{\pm1}]$ be the $\mathbb{C}$-algebra of Laurent polynomials restricted to $Y$. Then the Jacobian ring $Jac(W)$ of $W$ is defined as the quotient of $\mathbb{C}[Y]$ by the ideal generated by the logarithmic derivatives of $W$: $$\begin{aligned}
Jac(W) & = & \mathbb{C}[Y]\Big/\Big\langle z_j\frac{\partial
W}{\partial z_j}:j=1,\ldots,n\Big\rangle\\
& = &\mathbb{C}[z_1^{\pm1},\ldots,z_n^{\pm1}]\Big/\Big\langle
z_j\frac{\partial W}{\partial z_j}:j=1,\ldots,n\Big\rangle.\end{aligned}$$ The second part of Theorem \[main\_thm\] is now an almost immediate corollary of Proposition \[prop2.2\] and Theorem \[thm3.3\].
\[thm3.3\] The SYZ mirror transformation $\mathcal{F}$ gives an isomorphism $$\mathcal{F}:\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]/\mathcal{L}\rightarrow
Jac(W)$$ of $\mathbb{C}$-algebras. Hence, $\mathcal{F}$ induces a natural isomorphism of $\mathbb{C}$-algebras between the small quantum cohomology ring of $\bar{X}$ and the Jacobian ring of $W$: $$\begin{aligned}
\mathcal{F}:QH^*(\bar{X})\overset{\cong}{\longrightarrow}Jac(W),\end{aligned}$$ provided that $\bar{X}$ is a product of projective spaces.
The functions $\Psi_1,\Psi_1^{-1},\ldots,\Psi_n,\Psi_n^{-1}$ are all admissible, so $\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]$ is a subalgebra of $\mathcal{A}(LX)$. It is easy to see that, for $i=1,\ldots,d$, the SYZ mirror transformation $\mathcal{F}(\Psi_i)=\widehat{\Psi}_i$ of $\Psi_i$ is nothing but the monomial $e^{\lambda_i}z^{v_i}$. By our choice of the polytope $\bar{P}\subset M_\mathbb{R}$, $v_1=e_1,\ldots,v_n=e_n$ is the standard basis of $N=\mathbb{Z}^n$ and $\lambda_1=\ldots=\lambda_n=0$. Hence, $$\mathcal{F}(\Psi_i)=z_i,$$ for $i=1,\ldots,n$, and the induced map $$\mathcal{F}:\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]\rightarrow\mathbb{C}[z_1^{\pm1},\ldots,z_n^{\pm1}]$$ is an isomorphism of $\mathbb{C}$-algebras. Now, notice that $$z_j\frac{\partial W}{\partial z_j}
=\sum_{i=1}^d z_j\frac{\partial}{\partial
z_j}(e^{\lambda_i}z_1^{v_i^1}\ldots z_n^{v_i^n})=\sum_{i=1}^d v_i^j
e^{\lambda_i}z_1^{v_i^1}\ldots z_n^{v_i^n}=\sum_{i=1}^d v_i^j
e^{\lambda_i}z^{v_i},$$ for $j=1,\ldots,n$. The inverse SYZ transformation of $z_j\frac{\partial W}{\partial z_j}$ is thus given by $$\mathcal{F}^{-1}(z_j\frac{\partial W}{\partial z_j})
=\widehat{\sum_{i=1}^d v_i^je^{\lambda_i}z^{v_i}}=\sum_{i=1}^d
v_i^j\Psi_i.$$ Thus, $$\mathcal{F}^{-1}\Big(\Big\langle
z_j\frac{\partial W}{\partial
z_j}:j=1,\ldots,n\Big\rangle\Big)=\mathcal{L},$$ is the ideal in $\mathbb{C}[\Psi_1^{\pm1},\ldots,\Psi_n^{\pm1}]$ generated by linear equivalences. The result follows.
Examples {#sec4}
========
In this section, we give some examples to illustrate our results.\
**Example 1. $\bar{X}=\mathbb{C}P^2$.** In this case, $N=\mathbb{Z}^2$. The primitive generators of the 1-dimensional cones of the fan $\Sigma$ defining $\mathbb{C}P^2$ are given by $v_1=(1,0),v_2=(0,1),v_3=(-1,-1)\in N$, and the polytope $\bar{P}\subset M_\mathbb{R}\cong\mathbb{R}^2$ we chose is defined by the inequalities $$x_1\geq0,\ x_2\geq0,\ x_1+x_2\leq t,$$ where $t>0$. See Figure 4.1 below.
(100,35) (10,2)[(0,1)[35]{}]{} (10,2)[(1,0)[35]{}]{} (10,32, 40,2) (40,2.8, 40,1.2) (40,-1)[$t$]{} (10.8,32, 9.2,32) (7.8,31)[$t$]{} (8,0)[0]{} (13,11)[$\bar{P}\subset M_\mathbb{R}$]{} (5.3,14.5)[$D_1$]{} (23,-1.3)[$D_2$]{} (24.9,17.9)[$D_3$]{} (75,17, 93,17) (75,17, 75,35) (75,17, 60,2) (74.15,16.15)[$\bullet$]{} (71,16)[$\xi$]{} (87,18)[$E_1$]{} (75.5,30)[$E_2$]{} (67,6.3)[$E_3$]{} (75,4)[$(\Gamma_3,h)$ in $N_\mathbb{R}$]{} (44,-3)[Figure 4.1]{}
The mirror manifold $Y$ is given by $$\begin{aligned}
Y & = & \{(Z_1,Z_2,Z_3)\in\mathbb{C}^3:Z_1Z_2Z_3=q, |Z_i|<1,\ i=1,2,3\}\\
& = & \{(z_1,z_2)\in(\mathbb{C}^*)^2:|z_1|<1, |z_2|<1,
|\frac{q}{z_1z_2}|<1\},\end{aligned}$$ where $q=e^{-t}$ is the Kähler parameter, and, the superpotential $W:Y\rightarrow\mathbb{C}$ can be written, in two ways, as $$W=Z_1+Z_2+Z_3=z_1+z_2+\frac{q}{z_1z_2}.$$ In terms of the coordinates $Z_1,Z_2,Z_3$, the Jacobian ring $Jac(W)$ is given by $$\begin{aligned}
Jac(W) & = & \mathbb{C}[Z_1,Z_2,Z_3]\big/\big\langle
Z_1-Z_3,Z_2-Z_3,Z_1Z_2Z_3-q\big\rangle\\
& \cong & \mathbb{C}[Z]\big/\big\langle Z^3-q\big\rangle.\end{aligned}$$ There are three toric prime divisors $D_1,D_2,D_3$, which are corresponding to the three admissible functions $\Psi_1,\Psi_2,\Psi_3:LX\rightarrow\mathbb{R}$ defined by $$\begin{aligned}
\Psi_1(p,v) & = & \left\{ \begin{array}{ll}
e^{-x_1} & \textrm{if $v=(1,0)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\\
\Psi_2(p,v) & = & \left\{ \begin{array}{ll}
e^{-x_2} & \textrm{if $v=(0,1)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\\
\Psi_3(p,v) & = & \left\{ \begin{array}{ll}
e^{-(t-x_1-x_2)} & \textrm{if $v=(-1,-1)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\end{aligned}$$ for $(p,v)\in LX$ and where $x=\mu(p)\in P$, respectively. The small quantum cohomology ring of $\mathbb{C}P^2$ has the following presentation: $$\begin{aligned}
QH^*(\mathbb{C}P^2) & = & \mathbb{C}[D_1,D_2,D_3]\big/\big\langle
D_1-D_3,D_2-D_3,D_1\ast D_2\ast D_3-q\big\rangle\\
& \cong & \mathbb{C}[H]\big/\big\langle H^3-q\big\rangle,\end{aligned}$$ where $H\in H^2(\mathbb{C}P^2,\mathbb{C})$ is the hyperplane class. Quantum corrections appear only in one relation, namely, $$D_1\ast D_2\ast D_3=q.$$ Fix a point $p\in X$. Then the quantum correction is due to the unique holomorphic curve $\varphi:(\mathbb{C}P^1;x_1,x_2,x_3,x_4)\rightarrow\mathbb{C}P^2$ of degree 1 (i.e. a line) with 4 marked points such that $\varphi(x_4)=p$ and $\varphi(x_i)\in D_i$ for $i=1,2,3$. The parameterized 3-marked, genus 0 tropical curve corresponding to this line is $(\Gamma_3;E_1,E_2,E_3;h)$, which is glued from three half lines emanating from the point $\xi=\textrm{Log}(p)\in N_\mathbb{R}$ in the directions $v_1$, $v_2$ and $v_3$. See Figure 4.1 above. These half lines are the parameterized Maslov index two tropical discs $(\Gamma_1,h_i)$, where $h_i(V)=\xi$ an $h_i(E)=\xi+\mathbb{R}_{\geq0}v_i$, for $i=1,2,3$ (see Figure 2.3). They are corresponding to the Maslov index two holomorphic discs $\varphi_1,\varphi_2,\varphi_3:(D^2,\partial
D^2)\rightarrow(\mathbb{C}P^2,L_{\mu(p)})$ which pass through $p$ and intersect the corresponding toric divisors $D_1,D_2,D_3$ respectively.\
**Example 2. $\bar{X}=\mathbb{C}P^1\times\mathbb{C}P^1$.** The primitive generators of the 1-dimensional cones of the fan $\Sigma$ defining $\mathbb{C}P^1\times\mathbb{C}P^1$ are given by $v_{1,1}=(1,0),v_{2,1}=(-1,0),v_{1,2}=(0,1),v_{2,2}=(0,-1)\in
N=\mathbb{Z}^2$. We choose the polytope $\bar{P}\subset
M_\mathbb{R}=\mathbb{R}^2$ to be defined by the inequalities $$0\leq x_1\leq t_1,\ 0\leq x_2\leq t_2$$ where $t_1,t_2>0$. See Figure 4.2 below.
(100,36) (2,6)[(0,1)[31]{}]{} (2,6)[(1,0)[42]{}]{} (2,32, 39,32) (39,6, 39,32) (39,6.8, 39,5.2) (38,2.4)[$t_1$]{} (3.8,32, 6.2,32) (-1,31)[$t_2$]{} (0,4)[0]{} (13.8,17.3)[$\bar{P}\subset M_\mathbb{R}$]{} (-4.5,18)[$D_{1,1}$]{} (39.5,18)[$D_{2,1}$]{} (17.5,3.1)[$D_{1,2}$]{} (17.5,33.2)[$D_{2,2}$]{} (57,23, 103,23) (85,6, 85,35) (84.1,22)[$\bullet$]{} (82.8,19.7)[$\xi$]{} (80,2.3)[$(\Gamma_2,h_2)$]{} (58.5,20)[$(\Gamma_2,h_1)$]{} (92,30)[$N_\mathbb{R}$]{} (42,-2)[Figure 4.2]{}
The mirror Landau-Ginzburg model $(Y,W)$ consists of $$\begin{aligned}
Y\!\!\!&=&\!\!\!\{(Z_{1,1},Z_{2,1},Z_{1,2},Z_{2,2})\in
\mathbb{C}^4:Z_{1,1}Z_{2,1}=q_1,Z_{1,2}Z_{2,2}=q_2,
|Z_{i,j}|<1,\textrm{ all $i,j$}\}\\
\!\!\!&=&\!\!\!\{(z_1,z_2)\in(\mathbb{C}^*)^2:|z_1|<1,|z_2|<1,
|\frac{q_1}{z_1}|<1,|\frac{q_2}{z_2}|<1\},\end{aligned}$$ where $q_1=e^{-t_1}$ and $q_2=e^{-t_2}$ are the Kähler parameters, and $$W=Z_{1,1}+Z_{2,1}+Z_{1,2}+Z_{2,2}=z_1+\frac{q_1}{z_1}+z_2+\frac{q_2}{z_2}.$$ The Jacobian ring $Jac(W)$ is given by $$\begin{aligned}
Jac(W) & = &
\frac{\mathbb{C}[Z_{1,1},Z_{2,1},Z_{1,2},Z_{2,2}]}{\big\langle
Z_{1,1}-Z_{2,1},Z_{1,2}-Z_{2,2},Z_{1,1}Z_{2,1}-q_1,Z_{1,2}Z_{2,2}-q_2\big\rangle}\\
& \cong & \mathbb{C}[Z_1,Z_2]\big/\big\langle
Z_1^2-q_1,Z_2^2-q_2\big\rangle.\end{aligned}$$ The four toric prime divisors $D_{1,1},D_{2,1},D_{1,2},D_{2,2}$ correspond respectively to the four admissible functions $\Psi_{1,1},\Psi_{2,1},\Psi_{1,2},\Psi_{2,2}:LX\rightarrow\mathbb{C}$ defined by $$\begin{aligned}
\Psi_{1,1}(p,v) & = & \left\{ \begin{array}{ll}
e^{-x_1} & \textrm{if $v=(1,0)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\\
\Psi_{2,1}(p,v) & = & \left\{ \begin{array}{ll}
e^{-(t_1-x_1)} & \textrm{if $v=(0,-1)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\\
\Psi_{1,2}(p,v) & = & \left\{ \begin{array}{ll}
e^{-x_2} & \textrm{if $v=(0,1)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\\
\Psi_{2,2}(p,v) & = & \left\{ \begin{array}{ll}
e^{-(t_2-x_2)} & \textrm{if $v=(0,-1)$}\\
0 & \textrm{otherwise,}
\end{array} \right.\end{aligned}$$ for $(p,v)\in LX$ and where $x=\mu(p)\in P$. The small quantum cohomology ring of $\mathbb{C}P^1\times\mathbb{C}P^1$ is given by $$\begin{aligned}
QH^*(\mathbb{C}P^1\times\mathbb{C}P^1)\!\!\!&=&\!\!\!
\frac{\mathbb{C}[D_{1,1},D_{2,1},D_{1,2},D_{2,2}]}{\big\langle
D_{1,1}-D_{2,1},D_{1,2}-D_{2,2},D_{1,1}\ast D_{2,1}-q_1,D_{1,2}\ast
D_{2,2}-q_2\big\rangle}\\
\!\!\!&\cong&\!\!\!\mathbb{C}[H_1,H_2]\big/\big\langle
H_1^2-q_1,H_2^2-q_2\big\rangle\end{aligned}$$ where $H_1,H_2\in H^2(\mathbb{C}P^1\times\mathbb{C}P^1)$ are the pullbacks of the hyperplane classes in the first and second factors respectively. Quantum corrections appear in two relations: $$D_{1,1}\ast D_{2,1}=q_1\textrm{ and }D_{1,2}\ast D_{2,2}=q_2.$$ Let us focus on the first one, as the other one is similar. For any $p\in X$, there are two Maslov index two holomorphic discs $\varphi_{1,1},\varphi_{2,1}:(D^2,\partial
D^2)\rightarrow(\mathbb{C}P^1\times\mathbb{C}P^1,L_{\mu(p)})$ intersecting the corresponding toric divisors. An interesting feature of this example is that, since the sum of the boundaries of the two holomorphic discs is zero as a *chain*, instead of as a class, in $L_{\mu(p)}$, they glue together *directly* to give the unique holomorphic curve $\varphi_1:(\mathbb{C}P^1:x_1,x_2,x_3)\rightarrow\mathbb{C}P^1\times\mathbb{C}P^1$ of degree 1 with $\varphi_1(x_1)\in D_{1,1}$, $\varphi_1(x_2)\in
D_{2,1}$ and $\varphi_1(x_3)=p$. So the relation $D_{1,1}\ast
D_{2,1}=q_1$ is directly corresponding to $\Psi_{1,1}\star\Psi_{2,1}=q_1\mathbb{1}$, without going through the corresponding relation in $QH^*_{trop}(\bar{X})$. In other words, we do not need to go to the tropical world to see the geometry of the isomorphism $QH^*(\mathbb{C}P^1\times\mathbb{C}P^1)\cong
\mathbb{C}[\Psi_{1,1}^{\pm1},\Psi_{1,2}^{\pm1}]/\mathcal{L}$ (although in Figure 4.2 above, we have still drawn the tropical lines $h_1$ and $h_2$ passing through $\xi=\textrm{Log}(p)\in
N_\mathbb{R}$).\
**Example 3. $\bar{X}$ is the toric blowup of $\mathbb{C}P^2$ at one point.** Let $\bar{P}\subset\mathbb{R}^2$ be the polytope defined by the inequalities $$x_1\geq0,\ 0\leq x_2\leq t_2,\ x_1+x_2\leq t_1+t_2,$$ where $t_1,t_2>0$.
(100,32) (18,4)[(0,1)[30]{}]{} (18,4)[(1,0)[60]{}]{} (18,28, 44,28) (18,28.1, 44,28.1) (18,27.9, 44,27.9) (17.1,27.1)[$\bullet$]{} (43.1,27.1)[$\bullet$]{} (44,28, 68,4) (18.8,28, 17.2,28) (14.1,27)[$t_2$]{} (68,3.2, 68,4.8) (63.5,0.6)[$t_1+t_2$]{} (16,1)[0]{} (28,14)[$\bar{P}\subset M_\mathbb{R}$]{} (13.5,15.5)[$D_1$]{} (38,1)[$D_2$]{} (56,17)[$D_3$]{} (28,29)[$D_4$]{} (48,30.5)[(-1,0)[15]{}]{} (49,29.5)[exceptional curve]{} (80,15)[Figure 4.3]{}
The toric Fano manifold $\bar{X}$ corresponding to this trapezoid (see Figure 4.3 above) is the blowup of $\mathbb{C}P^2$ at a $T_N$-fixed point. The primitive generators of the 1-dimensional cones of the fan $\Sigma$ defining $\bar{X}$ are given by $v_1=(1,0),v_2=(0,1),v_3=(-1,-1),v_4=(0,-1)\in N=\mathbb{Z}^2$. As in the previous examples, we have the mirror manifold $$\begin{aligned}
Y & = &
\{(Z_1,Z_2,Z_3,Z_4)\in\mathbb{C}^4:Z_1Z_3=q_1Z_4,Z_2Z_4=q_2,|Z_i|<1\textrm{for
all $i$}\}\\
& = &
\{(z_1,z_2)\in(\mathbb{C}^*)^2:|z_1|<1,|z_2|<1,|\frac{q_1q_2}{z_1z_2}|<1,|\frac{q_2}{z_2}|<1\},\end{aligned}$$ and the superpotential $$W=Z_1+Z_2+Z_3+Z_4=z_1+z_2+\frac{q_1q_2}{z_1z_2}+\frac{q_2}{z_2},$$ where $q_1=e^{-t_1},q_2=e^{-t_2}$. The Jacobian ring of $W$ is $$Jac(W)=\frac{\mathbb{C}[Z_1,Z_2,Z_3,Z_4]}{\big\langle
Z_1-Z_3-Z_4,Z_2-Z_4,Z_1Z_3-q_1Z_4,Z_2Z_4-q_2\big\rangle}$$ and the small quantum cohomology ring of $\bar{X}$ is given by $$QH^*(\bar{X})=\frac{\mathbb{C}[D_1,D_2,D_3,D_4]}{\big\langle
D_1-D_3-D_4,D_2-D_4,D_1\ast D_3-q_1D_4,D_2\ast
D_4-q_2\big\rangle}.$$ Obviously, we have an isomorphism $QH^*(\bar{X})\cong Jac(W)$ and, the isomorphism $$QH^*(\bar{X})\cong\mathbb{C}[\Psi_1^{\pm1},\Psi_2^{\pm1}]/\mathcal{L}$$ in Proposition \[prop2.2\] still holds, as we have said in Remark \[rmk2.3\]. However, the geometric picture that we have derived in Subsection \[subsec2.2\] using tropical geometry breaks down. This is because there is a *rigid* holomorphic curve contained in the toric boundary $D_\infty$ which *does* contribute to $QH^*(\bar{X})$. Namely, the quantum relation $$D_1\ast D_3=q_1D_4$$ is due to the holomorphic curve $\varphi:\mathbb{C}P^1\rightarrow\bar{X}$ such that $\varphi(\mathbb{C}P^1)\subset D_4$. This curve is exceptional since $D_4^2=-1$, and thus cannot be deformed to a curve outside the toric boundary. See Figure 4.3 above. Hence, it is *not* corresponding to any tropical curve in $N_\mathbb{R}$. This means that tropical geometry cannot “see” the curve $\varphi$, and it is not clear how one could define the tropical analog of the small quantum cohomology ring in this case.
Discussions
===========
In this final section, we speculate the possible generalizations of the results of this paper. The discussion will be rather informal.\
The proofs of the results in this paper rely heavily on the classification of holomorphic discs in a toric Fano manifold $\bar{X}$ with boundary in Lagrangian torus fibers, and on the explicit nature of toric varieties. Nevertheless, it is still possible to generalize these results, in particular, the construction of SYZ mirror transformations, to non-toric situations. For example, one may consider a *complex flag manifold* $\bar{X}$, where the Gelfand-Cetlin integrable system provide a natural Lagrangian torus fibration structure on $\bar{X}$ (see, for example, Guillemin-Sternberg [@GS83]). The base of this fibration is again an affine manifold with boundary but without singularities. In fact, there is a *toric degeneration* of the complex flag manifold $\bar{X}$ to a toric variety, and the base is nothing but the polytope associated to that toric variety. Furthermore, the classification of holomorphic discs in a complex flag manifold $\bar{X}$ with boundary in Lagrangian torus fibers was recently done by Nishinou-Nohara-Ueda [@NNU08], and, at least for the full flag manifolds, there is an isomorphism between the small quantum cohomology ring and the Jacobian ring of the mirror superpotential (cf. Corollary 12.4 in [@NNU08]). Hence, one can try to construct the SYZ mirror transformations for a complex flag manifold $\bar{X}$ and prove results like Proposition \[prop1.1\] and Theorem \[main\_thm\] as in the toric Fano case.\
Certainly, the more important (and more ambitious) task is to generalize the constructions of SYZ mirror transformations to the most general situations, where the bases of Lagrangian torus fibrations are affine manifolds with both boundary and singularities. To do this, the first step is to make the construction of the SYZ mirror transformations become a local one. One possible way is the following: Suppose that we have an $n$-dimensional compact Kähler manifold $\bar{X}$, together with an anticanonical divisor $D$. Assume that there is a Lagrangian torus fibration $\mu:\bar{X}\rightarrow\bar{B}$, where $\bar{B}$ is a real $n$-dimensional (possibly) singular affine manifold with boundary $\partial\bar{B}$. We should also have $\mu^{-1}(\partial\bar{B})=D$. Now let $U\subset
B:=\bar{B}\setminus\partial\bar{B}$ be a small open ball contained in an affine chart of the nonsingular part of $B$, i.e. $\mu^{-1}(b)$ is a nonsingular Lagrangian torus in $\bar{X}$ for any $b\in U$, so that we can identify each fiber $\mu^{-1}(b)$ with $T^n$ and identify $\mu^{-1}(U)$ with $T^*U/\mathbb{Z}^n\cong
U\times T^n$. Let $N\cong\mathbb{Z}^n$ be the fundamental group of any fiber $\mu^{-1}(b)$, and consider the $\mathbb{Z}^n$-cover $L\mu^{-1}(U)=\mu^{-1}(U)\times N$. Locally, the mirror manifold should be given by the dual torus fibration $\nu:U\times(T^n)^\vee\rightarrow U$. Denote by $\nu^{-1}(U)$ the local mirror $U\times(T^n)^\vee$. Then we can define the *local SYZ mirror transformation*, as before, through the fiber product $L\mu^{-1}(U)\times_U\nu^{-1}(U)$. $$\begin{CD}
L\mu^{-1}(U)\times_U\nu^{-1}(U) @> >> \nu^{-1}(U) \\
@VV V @VV \nu V \\
L\mu^{-1}(U) @>\mu>> U
\end{CD}\\$$ Now, fix a reference fiber $L_0=\mu^{-1}(b_0)$. Given $v\in N$, define a function $\Upsilon_v:L\mu^{-1}U\rightarrow\mathbb{R}$ as follows. For any point $p\in\mu^{-1}(U)$, let $L_b=\mu^{-1}(b)$ be the fiber containing $p$, where $b=\mu(p)\in U$. Regard $v$ as an element in $\pi_1(L_b)$. Consider the 2-chain $\gamma$ in $\mu^{-1}(U)$ with boundary in $v\cup L_0$, and define $$\Upsilon_v(p,v)=\exp(-\frac{1}{2\pi}\int_\gamma\omega_{\mu^{-1}(U)}),$$ where $\omega_{\mu^{-1}(U)}=\omega_{\bar{X}}|_{\mu^{-1}(U)}$ is the restriction of the Kähler form to $\mu^{-1}(U)$. Also set $\Upsilon_v(p,w)=0$ for any $w\in N\setminus\{v\}$ (cf. the discussion after Lemma 2.7 in Auroux [@Auroux07]). This is analog to the definitions of the functions $\Psi_1,\ldots,\Psi_d$ in the toric Fano case, and it is easy to see that the local SYZ mirror transformations of these kind of functions give local holomorphic functions on the local mirror $\nu^{-1}(U)=U\times(T^n)^\vee$. We expect that these constructions will be sufficient for the purpose of understanding quantum corrections due to the boundary divisor $D$. However, to take care of the quantum corrections which arise from the proper singular Lagrangian fibers (i.e. singular fibers contained in $X=\mu^{-1}(B)$), one must modify and generalize the constructions of the local SYZ mirror transformations to the case where $U\subset B$ contains singular points. For this, new ideas are needed in order to incorporate the wall-crossing phenomena.\
**Acknowledgements.** We thank the referees for very useful comments and suggestions. The work of the second author was partially supported by RGC grants from the Hong Kong Government.
[99]{}
M. Abouzaid, *Homogeneous coordinate rings and mirror symmetry for toric varieties*, Geom. Topol. 10 (2006), 1097–1157 (math.SG/0511644).
, *Morse homology, tropical geometry, and homological mirror symmetry for toric varieties*, Selecta Math. (N.S.) 15 (2009), no. 2, 189–270 (math.SG/0610004).
M. Abreu, *Kähler geometry of toric manifolds in symplectic coordinates*, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), 1–24, Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, 2003 (math.DG/0004122).
*Quantum cohomology at the Mittag-Leffler Institute*, Edited by P. Aluffi, Scuola Normale Superiore, Pisa, 1997.
M. Audin, *Torus actions on symplectic manifolds*, Second revised edition. Progress in Mathematics, 93. Birkhäuser Verlag, Basel, 2004.
D. Auroux, *Mirror symmetry and T-duality in the complement of an anticanonical divisor*, J. Gokova Geom. Topol. 1 (2007), 51–91 (arXiv:0706.3207).
D. Auroux, L. Katzarkov and D. Orlov, *Mirror symmetry for weighted projective planes and their noncommutative deformations*, Ann. of Math. (2) 167 (2008), no. 3, 867–943 (math.AG/0404281).
, *Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves*, Invent. Math. 166 (2006), no. 3, 537–582 (math.AG/0506166).
V. Batyrev, *On the classification of smooth projective toric varieties*, Tohoku Math. J. (2) 43 (1991), no. 4, 569–585.
, *Quantum cohomology rings of toric manifolds*, Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). Asterisque No. 218 (1993), 9–34 (alg-geom/9310004).
K.-W. Chan and N.-C. Leung, *On SYZ mirror transformations*, to appear in Advanced Studies in Pure Mathematics, “New developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry” (arXiv:0808.1551).
C.-H. Cho and Y.-G. Oh, *Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds*, Asian J. Math. 10 (2006), no. 4, 773–814 (math.SG/0308225).
R. E. Edwards, *Fourier series. A modern introduction. Vol. 1*, Graduate Texts in Mathematics, 64. Springer-Verlag, New York-Berlin, 1979.
K. Fukaya, *Multivalued Morse theory, asymptotic analysis and mirror symmetry*, Graphs and patterns in mathematics and theoretical physics, 205–278, Proc. Sympos. Pure Math., 73, Amer. Math. Soc., Providence, RI, 2005.
K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds I, to appear in Duke Math. J. (arXiv:0802.1703).
, Lagrangian Floer theory on compact toric manifolds II: Bulk deformations, preprint, 2008 (arXiv:0810.5654).
W. Fulton, *Introduction to toric varieties*, Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993.
A. Givental, *Homological geometry and mirror symmetry*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 472–480, Birkhäuser, Basel, 1995.
, *Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture*, Topics in singularity theory, 103–115, Amer. Math. Soc. Transl. Ser. 2, 180, Amer. Math. Soc., Providence, RI, 1997 (alg-geom/9612001).
, *A mirror theorem for toric complete intersections*, Topological field theory, primitive forms and related topics (Kyoto, 1996), 141–175, Progr. Math., 160, Birkhäuser Boston, Boston, MA, 1998 (alg-geom/9701016).
, *A tutorial on quantum cohomology*, Symplectic geometry and topology (Park City, UT, 1997), 231–264, IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999.
M. Gross, Mirror symmetry for $\mathbb{P}^2$ and tropical geometry, preprint, 2009 (arXiv:0903.1378).
M. Gross and B. Siebert, *From real affine geometry to complex geometry*, preprint, 2007 (math.AG/0703822).
V. Guillemin, *Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces*, Progress in Mathematics, 122. Birkhäuser Boston, Inc., Boston, MA, 1994.
V. Guillemin and S. Sternberg, *The Geĺfand-Cetlin system and quantization of the complex flag manifolds*, J. Funct. Anal. 52 (1983), no. 1, 106–128.
N. Hitchin, The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) [**25**]{} (1997), no. 3-4, 503–515 (dg-ga/9711002).
K. Hori and C. Vafa, *Mirror symmetry*, preprint, 2000 (hep-th/0002222).
M. Kontsevich, *Lectures at ENS*, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona.
M. Kontsevich and Y. Soibelman, *Affine structures and non-Archimedean analytic spaces*, The unity of mathematics, 321–385, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006 (math.AG/0406564).
N.-C. Leung, *Mirror symmetry without corrections*, Comm. Anal. Geom. 13 (2005), no. 2, 287–331 (math.DG/0009235).
N.-C. Leung, S.-T. Yau and E. Zaslow, *From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform*, Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319–1341 (math.DG/0005118).
G. Mikhalkin, *Enumerative tropical algebraic geometry in $\mathbb{R}^2$*, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377 (math.AG/0312530).
, *Tropical geometry and its applications*, International Congress of Mathematicians. Vol. II, 827–852, Eur. Math. Soc., Zürich, 2006 (math.AG/0601041).
, *Moduli spaces of rational tropical curves*, Proceedings of Gokova Geometry-Topology Conference 2006, 39–51, Gokova Geometry/Topology Conference (GGT), Gokova, 2007 (arXiv:0704.0839).
D. Morrison, *The geometry underlying mirror symmetry*, New trends in algebraic geometry (Warwick, 1996), 283–310, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999 (alg-geom/9608006).
T. Nishinou, *Disc counting on toric varieties via tropical curves*, preprint, 2006 (math.AG/0610660).
T. Nishinou, Y. Nohara and K. Ueda, *Toric degenerations of Gelfand-Cetlin systems and potential functions*, preprint, 2008 (arXiv:0810.3470).
T. Nishinou and B. Siebert, *Toric degenerations of toric varieties and tropical curves*, Duke Math. J. 135 (2006), no. 1, 1–51 (math.AG/0409060).
P. Seidel, *More about vanishing cycles and mutation*, Symplectic geometry and mirror symmetry (Seoul, 2000), 429–465, World Sci. Publ., River Edge, NJ, 2001 (math.SG/0010032).
B. Siebert, *An update on (small) quantum cohomology*, Mirror symmetry, III (Montreal, PQ, 1995), 279–312, AMS/IP Stud. Adv. Math., 10, Amer. Math. Soc., Providence, RI, 1999.
B. Siebert and G. Tian, *On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator*, Asian J. Math. 1 (1997), no. 4, 679–695 (alg-geom/9403010).
H. Spielberg, *Multiple quantum products in toric varieties*, Int. J. Math. Math. Sci. 31 (2002), no. 11, 675–686 (math.SG/0107030).
A. Strominger, S.-T. Yau and E. Zaslow, *Mirror symmetry is T-duality*, Nuclear Phys. B 479 (1996), no. 1-2, 243–259 (hep-th/9606040).
K. Ueda, *Homological mirror symmetry for toric del Pezzo surfaces*, Comm. Math. Phys. 264 (2006), no. 1, 71–85 (math.AG/0411654).
[^1]: $\mu:X\rightarrow P$ is a special Lagrangian fibration if we equip $X$ with the standard holomorphic volume form on $(\mathbb{C}^*)^n$, so that $X$ becomes an almost Calabi-Yau manifold. See Definition 2.1 and Lemma 4.1 in Auroux [@Auroux07].
[^2]: In fact, we prove that T-duality gives a bounded domain in the Hori-Vafa mirror manifold. This result also appeared in Auroux’s paper ([@Auroux07], Proposition 4.2).
[^3]: In the expository paper [@CL08], we interpreted LX as a finite dimensional subspace of the free loop space $\mathcal{L}\bar{X}$ of $\bar{X}$.
[^4]: More recently, Gross [@Gross09] generalized this idea further to give a tropical interpretation of the *big* quantum cohomology of $\mathbb{P}^2$.
[^5]: In [@CL08], we rewrote the function $\Phi_q$ as $\exp(\Psi_1+\ldots+\Psi_d)$, so that $\Phi_q
e^{\sqrt{-1}\omega_X}=\exp(\sqrt{-1}\omega_X+\Psi_1+\ldots+\Psi_d)$ and the formula in part 1. of Theorem \[main\_thm\] becomes $\mathcal{F}(e^{\sqrt{-1}\omega_X+\Psi_1+\ldots+\Psi_d})=e^W\Omega_Y$. May be it is more appropriate to call $\sqrt{-1}\omega_X+\Psi_1+\ldots+\Psi_d\in\Omega^2(LX)\oplus\Omega^0(LX)$ the symplectic structure corrected by Maslox index two holomorphic discs.
[^6]: Another way to state this result: Let $\mathcal{M}_1(L_x,\beta_i)$ be the moduli space of holomorphic discs in $(\bar{X},L_x)$ in the class $\beta_i$ and with one boundary marked point. In the toric Fano case, $\mathcal{M}_1(L_x,\beta_i)$ is a smooth compact manifold of real dimension $n$. Let $ev:\mathcal{M}_1(L_x,\beta_i)\rightarrow L_x$ be the evaluation map at the boundary marked point. Then we have $ev_*[\mathcal{M}_1(L_x,\beta_i)]=[L_x]$ as $n$-cycles in $L$. See Cho-Oh [@CO03] and Auroux [@Auroux07] for details.
[^7]: “TGW” stands for “tropical Gromov-Witten”.
[^8]: For precise definitions of general tropical discs (with higher Maslov indices), we refer the reader to Nishinou [@Nishinou06]; see also the recent work of Gross [@Gross09].
[^9]: This correspondence also holds for other toric manifolds, not just for products of projective spaces.
[^10]: We have, by abuse of notations, used $N$ to denote the family of lattices $P\times\sqrt{-1}N$ over $P$. Similarly, we denote by $M$ the family of lattices $P\times\sqrt{-1} M$ below.
|
---
abstract: 'According to the widely accepted statistical interpretation of black hole entropy the mean separation between energy levels of black hole should be exponentially small. But this sharply disagrees with the value obtained from the quantization of black hole area. It is shown that the new statistical interpretation of black hole entropy proposed in my paper arXiv:0911.5635 gives the correct value.'
---
[**The entropy and mean separation between energy levels of black hole**]{}\
.5cm K. Ropotenko\
*State Administration of Communications,*
*Ministry of Transport and Communications of Ukraine,*
*22, Khreschatyk, 01001, Kyiv, Ukraine*
`ro@stc.gov.ua`
According to Bekenstein [@bek], quantization of the black hole area means that the area spectrum of black hole is of the form $$\label{bek} A_n=\Delta A \cdot n,\quad n=0,1,2,...,$$ where $\Delta A $ is the quantum of black hole area. Despite this, there is still no general agreement on the precise value of $\Delta
A $; in the literature there are several alternative proposals for $\Delta A $ (see, for example, [@med] and references therein). From (\[bek\]) the mass spectrum follows; for example, in the case where $\Delta A =8\pi l_P^{2}$ (here the precise value of $\Delta A
$ is not crucial for the form of the spectrum) the mass (energy) of a black hole is $$\label{mass1} M_n=m_P \sqrt{\frac{n}{2}}$$ which for large $n$ ($n\gg 1$) gives the energy spacing $$\label{mass2} \Delta M_n=\frac{M_n}{2n}=\frac{m_P^{2}}{4M_n}.$$ This value however does not agree with an estimation obtained from the definition of entropy. As is well known, the entropy of an ordinary system $S$, by definition, is the logarithm of the number of states $W$ with energy between $E$ and $E+\delta{E}$ $$\label{ent1} S = \ln W.$$ The width $\delta{E}$ is some energy interval characteristic of the limitation in our ability to specify absolutely precisely the energy of a macroscopic system. Dividing $\delta{E}$ by the number of states $\exp(S)$ we obtain the mean separation between energy levels of the system [@lan]: $$\label{main1}\langle \Delta{M} \rangle \sim \delta{E} \exp(-S).$$ The interval $\delta{E}$ is equal in order of magnitude to the mean canonical-ensemble fluctuation of energy of a system. However, there exist problems with the description of a black hole in a canonical ensemble. For example, because a black hole has negative specific heat $C_{v}$, $C_{v} = -8\pi G M^{2}$, energy fluctuations calculated in the canonical ensemble have formally negative variance: $\langle(\delta{E})^{2}\rangle=C_{v}T_H^{2} \sim -
m_P^{2}$, where $T_H$ is the Hawking temperature. The situation is quite different if a black hole is part of a thermodynamical system which has a finite size. For example, if a black hole is placed in a reservoir of radiation and the total energy of the system is fixed, a stable equilibrium configuration can exist. It appears that the equilibrium is stable if the radiation and black hole temperatures coincide, $T_{rad}=T_H\equiv T$, and $E_{rad}< M/4$, where $E_{rad}$ is the energy of radiation. The latter condition can be reformulated as the restriction on the volume of reservoir $V$, $4aVT^{5}< 1$, where $a$ is the radiation constant. According to this condition Pavon and Rubï found [@pav] that the mean square fluctuations of the black hole energy (mass) is given by $$\label{delta1} \langle (\delta E)^{2}\rangle =(1/8\pi) T^{2} Z,$$ $Z$ being the quantity $4aVT^{3}/(1-4aVT^{5})$, $G=c=\hbar =1$ and the Boltzmann constant $k_B=(8\pi)^{-1}$. It is clear that (restoring $G$, $c$, $\hbar$, and $k_B$) $$\label{delta2} \langle (\delta E)^{2}\rangle \sim
\frac{m_P^{4}}{M^{2}}$$ and $$\label{delta3} \delta E \sim \frac{m_P^{2}}{M}.$$ So we obtain the mean separation between energy levels for a black hole [@sred] $$\label{main2}\langle \Delta{M} \rangle \sim \frac{m_P^{2}}{M}
\exp(-S).$$ This value is however exponentially smaller than (\[mass2\]). Thus a problem arises. It has not yet received attention in the literature.
In this note I propose a solution of the problem. The point is as follows. In the quantum mechanical description, the accuracy with which the mass of a black hole can be defined by a distant observer is limited by the time-energy uncertainty relation as well as by the decrease of the mass of the black hole due to transition from a higher energy level to a lower one. The lifetime of a state $M_n$ is proportional to the inverse of the imaginary part of the effective action [@muk]; less formally, it is the time needed to emit a single Hawking quantum and this is proportional to the gravitational radius $R_g$. So $\delta{E}_q \sim 1/R_g$, where I have added the subscript “q” to refer to the quantum uncertainty. On the other hand, $\delta{E}_q\sim T_H$ for the transition from state $n$ to state $n-1$. It is obvious that the energy interval $\delta{E}_q$ contains only a single state. As is well known, in this case statistics is not applicable; by definition the statistical treatment is possible only if $\delta{E}$ contains many quantum states. It is clear that $\delta{E}_q$ is of the same order of magnitude as (\[delta3\]). This means that the formula (\[main2\]) is not applicable.
Nevertheless, since the mean separation between energy levels is nothing but the energy per unit state, we can define it as follows. In [@ro] it was noticed that in contrast to alternative values, the quantum of area $\Delta A = 8\pi l_P^{2}$ does not follow from the accepted statistical interpretation of black hole entropy of the form (\[ent1\]); on the contrary, a new statistical interpretation follows from it. Namely, in [@ro] it was shown that if the number of microstates accessible to a black hole is to be an integer and the entropy of the black hole is to satisfy the generalized second law of thermodynamics, the black hole entropy should be of the form $$\label{ent2} S =2\pi W,$$ that is, in contrast to (\[ent1\]), without logarithm, and the number of microstates is $W \equiv n= A/8\pi l_P^{2}$. So we can define $$\label{main4} \langle \Delta{M} \rangle \sim \frac{\Delta M}{\Delta
n}\sim \frac{dM}{dS}=T_H \sim \frac{m_P^{2}}{M};$$ this agrees with (\[mass2\]), as it should. Thus the first law of thermodynamics as well as entropy defines the energy spacing of a black hole.
[99]{}
J.D. Bekenstein, Phys. Rev. D [**7**]{}, 2333 (1973); Lett. Nuovo Cimento [**11**]{}, 467 (1974); “Quantum Black Holes as Atoms”, arXiv:gr-qc/9710076 (1997).
A.J.M. Medved, “A brief commentary on black hole entropy”, arXiv:gr-qc/0410086 (2004); “On the ”Universal“ Quantum Area Spectrum”, arXiv:0906.2641 (2009).
L. Landau and E. Lifshitz, *Statistical Physics* (Pergamon Press, Oxford, 1980).
D. Pavon and J.M. Rubï, Phys. Lett. A **99**, 214 (1983).
M. Srednicki, “On the Observability of Quantum Information Radiated from a Black Hole”, arXiv:hep-th/0207090 (2002); T. Damour, “The entropy of black holes: a primer”, arXiv:hep-th/0401160 (2004); V. Balasubramanian, D. Marolf, and M. Rozali, Gen. Rel. Grav. **38**, 1529 (2006).
V.F. Mukhanov, JEPT Lett. **44**, 63 (1986).
K. Ropotenko, “The quantum of area $\Delta A = 8\pi l_P^{2}$ and a statistical interpretation of black hole entropy”, arXiv:0911.5635 (2009).
|
[**Comment on “Duality of $x$ and $\psi$ in Quantum Mechanics” by A.E. Faraggi and M. Matone**]{}\
[**J.L. Lucio$^{(1)}$ and M. Ruiz–Altaba$^{(2)}$**]{}
[*$^{(1)}$ Instituto de Física, Universidad de Guanajuato, León, Gto., México*]{}
[*$^{(2)}$ Instituto de Física, UNAM, A.P. 20-364, 01000 México, D.F.*]{}
Recently, Faraggi and Matone introduced \[1\] an equation “dual" to the time–independent Schrödinger equation namely which they derived \[1\] in a rather roundabout way with the help of a “prepotential” $\cal F(\psi)$. They went on to speculate on a possible dual quantization of space based on this formula, employing also this formalism to study duality transformations in the pilot–wave version of quantum mechanics \[2\].
We would like to point out that, regardless of the conceptual interest and difficulties of interpreting space in terms of a wave–function instead of the opposite, equation (2) can be derived very easily from the trivial observation that = \^[-1]{} , and thus
Plugging (4) into (1) gives (2) directly. A more interesting truly dual version of (1) should involve a dual potential $\tilde V (\psi)$; we have no idea how to derive such an equation, nor what its interpretation would be.
[**Acknowledgements**]{}
Work partially supported by CONACyT under contract 3979PE-9608 and Catedra Patrimonial de Excelencia Nivel II.
\[1\] [A.E. Faraggi and M. Matone, [*Phys. Rev. Lett.*]{} [**78**]{} (1997) 163; [hep-th/9606063]{}.]{}
\[2\] [A.E. Faraggi and M. Matone, [hep-th/9705108]{}.]{}.
|
---
abstract: 'Market equilibrium is a solution concept with many applications such as digital ad markets, fair division, and resource sharing. For many classes of utility functions, equilibria are captured by convex programs. We develop simple first-order methods that are suitable for solving these programs for large-scale markets. We focus on three practically-relevant utility classes: linear, quasilinear, and Leontief utilities. Using structural properties of a market equilibrium under each utility class, we show that the corresponding convex programs can be reformulated as optimization of a structured smooth convex function over a polyhedral set, for which projected gradient achieves linear convergence. To do so, we utilize recent linear convergence results under weakened strong-convexity conditions, and further refine the relevant constants, both in general and for our specific setups. We then show that proximal gradient (a generalization of projected gradient) with a practical version of linesearch achieves linear convergence under the [Proximal-[[PŁ]{}]{}]{} condition. For quasilinear utilities, we show that Mirror Descent applied to a specific convex program achieves sublinear last-iterate convergence and recovers the Proportional Response dynamics, an elegant and efficient algorithm for computing market equilibrium under linear utilities. Numerical experiments show that proportional response is highly efficient for computing an approximate solution, while projected gradient with linesearch can be much faster when higher accuracy is required.'
author:
- |
Yuan Gao\
Department of IEOR, Columbia University\
New York, NY, 10027\
`gao.yuan@columbia.edu`\
Christian Kroer\
Department of IEOR, Columbia University\
New York, NY, 10027\
`christian.kroer@columbia.edu`
bibliography:
- 'references.bib'
title: 'First-order methods for large-scale market equilibrium computation\'
---
|
---
abstract: 'With the developments of dual-lens camera modules, depth information representing the third dimension of the captured scenes becomes available for smartphones. It is estimated by stereo matching algorithms, taking as input the two views captured by dual-lens cameras at slightly different viewpoints. Depth-of-field rendering (also be referred to as synthetic defocus or bokeh) is one of the trending depth-based applications. However, to achieve fast depth estimation on smartphones, the stereo pairs need to be rectified in the first place. In this paper, we propose a cost-effective solution to perform stereo rectification for dual-lens cameras called direct self-rectification, short for DSR[^1] . It removes the need of individual offline calibration for every pair of dual-lens cameras. In addition, the proposed solution is robust to the slight movements, [*e.g.*]{}, due to collisions, of the dual-lens cameras after fabrication. Different with existing self-rectification approaches, our approach computes the homography in a novel way with zero geometric distortions introduced to the master image. It is achieved by directly minimizing the vertical displacements of corresponding points between the original master image and the transformed slave image. Our method is evaluated on both realistic and synthetic stereo image pairs, and produces superior results compared to the calibrated rectification or other self-rectification approaches.'
author:
- 'Ruichao Xiao[^2]'
- Wenxiu Sun
- Jiahao Pang
- Qiong Yan
- Jimmy Ren
- |
SenseTime Research\
[{xiaoruichao, sunwenxiu, pangjiahao, yanqiong, rensijie}@sensetime.com]{}
bibliography:
- 'ref.bib'
title: 'DSR: Direct Self-rectification for Uncalibrated Dual-lens Cameras'
---
Introduction {#sec:intro}
============
Related Works {#sec:related}
=============
Methodology {#sec:method}
===========
Experiments {#sec:results}
===========
Discussion and Conclusion {#sec:conclude}
=========================
[^1]: Code is avaliable at [github.com/garroud/self-rectification](https://github.com/garroud/self-rectification)
[^2]: R. Xiao and W. Sun contributed equally to this work.
|
---
abstract: 'We perform some experimental simulations in spherical symmetry and axisymmetry to understand the post-shock-revival evolution of core-collapse supernovae. Assuming that the stalled shock wave is relaunched by neutrino heating and employing the so-called light bulb approximation, we induce shock revival by raising the neutrino luminosity by hand up to the critical value, which is also determined by dynamical simulations. A 15${\rm M}_{\odot}$ progenitor model is employed. We incorporate nuclear network calculations with a consistent equation of state in the simulations to account for the energy release by nuclear reactions and their feedback to hydrodynamics. Varying the shock-relaunch time rather arbitrarily, we investigate the ensuing long-term evolutions systematically, paying particular attention to the explosion energy and nucleosynthetic yields as a function of this relaunch time, or equivalently the accretion rate at shock revival. We study in detail how the diagnostic explosion energy approaches the asymptotic value and which physical processes contribute to the explosion energy in what proportions as well as their dependence on the relaunch time and the dimension of dynamics. We find that the contribution of nuclear reactions to the explosion energy is comparable to or greater than that of neutrino heating. In particular, recombinations are dominant over burnings in the contributions of nuclear reactions. Interestingly 1D models studied in this paper cannot produce the appropriate explosion energy and nickel mass simultaneously, overproducing nickels, whereas this problem is resolved in 2D models if the shock is relaunched at 300-400ms after bounce.'
author:
- 'Yu Yamamoto, Shin-ichiro Fujimoto, Hiroki Nagakura and Shoichi Yamada'
title: 'Post-shock-revival evolutions in the neutrino-heating mechanism of core-collapse supernovae'
---
Introduction
============
Setup {#sec:setup}
=====
Outline
-------
Basic equations, input physics & numerical methods\[sec:method\]
----------------------------------------------------------------
Step 1: 1D simulation of the infall of envelope\[sec:step1\]
------------------------------------------------------------
Step 2: search of critical luminosities\[sec:step2\]
----------------------------------------------------
Step 3: computations of post-relaunch evolutions\[sec:step3\]
-------------------------------------------------------------
Results {#sec:results}
=======
Spherically symmetric 1D models {#sec:1d}
-------------------------------
Axisymmetric 2D models {#sec:2d}
----------------------
Summary and discussion {#sec:discussion}
======================
|
---
abstract: 'We present a computational random walk method of measuring the isothermal pressure of the lattice gas with and without the excluded volume interaction. The method is based on the discretization of the exact thermodynamic relation for the pressure. The simulation results are in excellent agreement with the theoretical predictions.'
address: ' Department of Physics, Lewis Laboratory, Lehigh University, Bethlehem, PA 18015\'
author:
- |
Daniel C. Hong$^{}
$[^1] and Kevin Mc Gouldrick
title: '**Measurement of Isothermal Pressure of Lattice Gas By Random Walk**'
---
1.0 true cm [**I. Introduction**]{} 0.2 true cm The pressure is defined as the force per unit area and is one of the measurable macroscopic thermodynamic quantities. It is a macroscopic manifestation of the microscopic phenomena of atomic collisions, by which atoms or molecules transfer the momentum and thus exert the force to the wall. At the macroscopic level, the pressure can be directly measured by monitoring or recording the force on the wall of a container. At the microscopic level, a conventional way is to design a method that keeps track of the momentum transfered by atoms to the wall via collisions, and this may be done by Molecular Dynamics(MD) simulations.$^1$ However, developing and learning MD simulations is time consuming, and is often well beyond the scope of undergraduate or introductory graduate level statistical mechanics. To the best of the authors’ knowledge, no conventional introductory statistical mechanics text books have yet dealt with the computational methods of [*measuring*]{} the pressure. Considering the surge of computer-assisted course developments in recent years, it may be desirable to design a computational method of measuring the pressure. The purpose of this paper is to introduce such a Monte Carlo method, which is conceptually simple and extremely easy to implement with a minimal knowledge of comutation, because it only deals with the random walk of point particles on a lattice. We first explain the mathematical basis of the method in section II and III, and then present simulation results for the lattice gas with and without excluded volume interaction in section IV. 0.2 true cm [**II. Background**]{} 0.2 true cm Our statring point is the mathematical expression of the pressure P of a container of volume V at a given temperature T, which can be found in any standard text book of statistical mechanics$^2$, $$P = -\partial F/\partial V = kT{\frac{\partial lnZ}{\partial V}}\eqno (1)$$ where $F=-kTlnZ$ is the free energy of the system with Z the partition function and k the Boltzmann constant. The contribution to the partition function comes from two sources, one from the kinetic energy and the other from the potential energy. The latter is known as the configurational integral, which invovles the integration of interaction energy among particles. The former is a function of temperature, and is independent of the volume. Therefore, at equilibrium, the contribution coming from the kinetics is decoupled, and is separated out from the configurational statistics, and the equilibrium pressure can be determined by considering only the configurational properties of the system. The nature of interactions among particles is determiend by the form of the potential energy. In this paper, we consider the interaction potential U to be either nonexistent, in which case the pressure is given by the ideal gas law, or the particles interact with each other via hard sphere potential. Even with hard sphere potential, the exact expression of the pressure for the continuum system is not yet known, even though a few approximate closed forms for the pressure have been reported in the literature. $^{3,4,5}$ However, for the lattice gas, the situation becomes much simpler, because in this case, the free energy can be computed exactly. In the lattice version, the hard sphere interaction may be equivalent to introducing the excluded volume interaction by prohibiting the multiple occupancy of particles on a given lattice site. Without such restriction, the system will follow the ideal gas statistics. Note that in both cases, the contribution to the free energy comes only from entropic part because the interaction energy is zero. Hence, $ F = - TS = -kT lnZ$, with S the entropy of the system. We now compute the entropy of the lattice gas, the result of which will then be used to test the validity of the random walk method.
Consider now a lattice model, where N particles are confined to move only on d dimensional discrete lattice points of a container with the volume $V=H\bullet L^{d-1}$ with H the height and $L^{d-1}$ the area of the side wall, where the pressure is being monitored. Note that we set the lattice spacing to the unit value. In the absence of excluded volume interactions, the entropy of the system is given by:
$$S = kln\Omega = kln(\omega^N) \eqno (2)$$ where $kln\omega$ is the entropy of a single particle, which is simply the volume V. Hence, we obtain the ideal gas law. $$P= -T({\frac{\partial S}{\partial V}}) = k{\frac{N}{V}} T = k\phi T
\eqno (3)$$ with $\phi=N/V$ the density of the particle. 0.2 true cm In the presence of the excluded volume interaction, the entropy of the system is no longer the product of one particle entropy. Rather, it is the total number of ways of putting N particles in a volume V. In the discrete version, it becomes,
$$S = kln\Omega(V,N) \approx -V[\phi ln\phi + (1-\phi)ln(1-\phi)] \eqno (4)$$ where $\Omega(V,N) = V!/N!(V-N)!$ and the Sterling’s formula has been invoked. The equilibrium pressure is then given by:
$$P = -T{\frac{\partial S}{\partial V}} = -kTln(1-\phi) \eqno (5)$$ One may use the grand partition function approach to obtain the similar result$^2$. In this case, the grand partition function, Q, becomes: $ Q=\sum_{N=0}^{\infty}z^NZ_N = (1+z)^V$, where the fugacity $z=exp(\beta\mu)$ and the canonical partition function $Z_N=V!/N!(V-N)!$. Now, from the relations (see Chapter 7 of Huang in ref.2) $PV/kT=lnQ$, and $\bar N = z\partial{logQ}{/\partial
z} = zV/(1+z)$, and $\phi = \bar N/V$, we can easily obtain eq.(2). Note that the pressure of the lattice gas exhibits the logarithmic singularity at the closed packed density. For the continuum system, it has been proved that the pressure exhibits the algebraic singularity.$^6$ 0.2 true cm [ **III. The Mathematical Basis for a Random Walk Monte Carlo Technique**]{} 0.2 true cm We now describe a mathematical basis for a Monte Carlo technique that computes the pressure of the lattice gas numerically. If the method is correct, then it should reproduce Eqs. (3) and (5). This is based on a method derived first for the continuum system by Percus$^{7}$, and then extended to a lattice system to measure the pressure in polymeric fluids simulations.$^8$ For the sake of completeness, we will briefly describe the essence of the formalism here, but the reader is referred to ref.7 and especially ref. 8 where the essential idea has been advanced. For a d dimensional lattice gas in a container of a height H along the z direction and the wall area $L^{d-1}$, the derivative becomes the difference equation, and thus the discrete expression of the pressure is given by: $$P = -{\frac{\partial F}{\partial V}} = k(T/L^{d-1})ln(Z(H)/Z(H-1))
\eqno (6)$$ where $Z(H)$ is the partition function with the height H and $Z(H-1)$ is that with H-1. Note that $Z(H-1)$ is equivalent to $Z(H)$ with an infinite repulsive potential at the wall, in which case the particles cannot appoarch the wall at z=H and thus with U the interaction potential between the wall and particles, the original partition function, $Z(H, U=\infty)$, will reduce to $Z(H-1,U=0).$ In order to utilize this observation in designing the Monte Carlo method, we introduce a parameter $\lambda = exp(-U/kT)$ with k the Boltzmann constant. Note that $\lambda=0$ corresponds to $U=\infty$, and $\lambda=1$ to $U=0$. Let the total number of particles in the system be $N$, and $N_w$ be the particles at the wall. The partition function, $\bar Z$, for this modified system becomes:
$$\bar
Z(H,\lambda) = \sum_{E}exp(-E/kT)=
\sum_{N_w =0} ^{L^{(d-1)}} \lambda^{N_w} \Omega(N_w,N-N_w)
\eqno (7)$$ where the summation is taken over all possible energy configurations and $\Omega$ is the total number of configurations with $N_w$ particles at the wall and $N-N_w$ particles in the bulk. The probablity, $P(N_w)$, of finding $N_w$ particles at the wall in the presence of the potential U is, $P(N_w) = \lambda^{N_w}\Omega(N_w,N-N_w)/\bar Z$. Further, it is obvious that $Z(H)=\bar Z(H,\lambda=1)$ and $Z(H-1)=\bar Z(H,\lambda=0)$. Therefore, if the mean number of particles at the wall is denoted by $<N_w>$, then it is given by: $$<N_w> = \lambda {\frac{\partial ln\bar Z}{\partial \lambda}} \eqno (8)$$ Next, we use a simple mathematical identity to express the ratio $ln(Z(H)/Z(H-1))$ in terms of $<N_w>$ as: $$ln(Z(H)/Z(H-1)) = ln(\bar Z(H,\lambda=1)/\bar Z(H,\lambda=0)) =
\int_o^1 d\lambda {\frac{\partial ln \bar Z(H,\lambda)}
{\partial \lambda}} \eqno (9)$$ Hence, we obtain the desired expression for the pressure$^{8}$:
$$P/kT = \int_0^1 {\frac{d\lambda}{\lambda}}\rho_w \eqno (10)$$ where $\rho_w\equiv <N_w>/L^{d-1}$ is the density at the wall.
0.3 true cm We now demonstrate here that Eq.(10) is indeed identical to Eqs.(3) and (5). Consider first the case where the excluded volume interactions are present. Since the partition function $\bar Z$ (Eq.(7)) contains only the configurational term, it is straightforward to write down the expression for $\bar Z$ for particles in a volume $V=HA \equiv H\bullet L^{(d-1)}$ with $A \equiv L^{d-1}$ the area of the container. We first consider the case with excluded volume. The partition function becomes, $$\bar Z(H,\lambda) = \sum_{N_w =0} ^{L^{(d-1)}} \lambda^{N_w} \Omega(A,N_w)
\Omega(B,N-N_w) \eqno (11)$$ where $B \equiv (H-1)L^{(d-1)}$. The first term,$\lambda^{N_w}$ in (A-1) is due to the wall potential, and the second and the third terms are the total number of ways of putting $N_w$ particles in the wall of area A and $N-N_w$ particles in the bulk of a volume $B$. Both $\Omega 's$ in the summation are given by the binomial coefficient, i.e.namely $\Omega(Q,R) = Q!/R!(Q-R)!$. Let the density of the particle, $N/B = \phi < 1$. Then, in the thermodynamic limit, $N_w<<N$. Using the Sterling’s formula, we find:
$$ln\Omega(B, N-N_w) \approx -B[ \phi ln \phi + (1-\phi) ln (1-\phi)]
+ N_w ln[\phi/(1-\phi)] + O(N_w^2/B)$$ Hence, $$\Omega(B,N-N_w) \approx exp(Bs_o) [\phi/(1-\phi)]^{N_w} \eqno (12)$$
where $s_o$ is the bulk entropy per site; i.e. $s_o = -[\phi ln \phi + (1-\phi)ln(1-\phi)]$. By putting $(12)$ into $(11)$, we now obtain the closed expression for the partition function $\bar Z$:
$$\bar Z = exp(Bs_o) [1 + \lambda \phi/(1-\phi)]^A$$
from which we obtain the exact formula for the density at the wall, $\rho_w$: $$\rho_w = (\lambda/A){\frac{\partial ln\bar Z}{\partial \lambda}} =
\lambda \alpha(\phi)/(1+\lambda\alpha(\phi))\eqno (13)$$ with $\alpha(\phi) = \phi/(1-\phi)$. Note that $\rho_w(\lambda=1) = \phi$ as expected. Hence, we find
$$P/kT = \int _0 ^1 d \lambda \rho_w/\lambda = -ln(1-\phi) \eqno (14)$$
In the case when the excluded volume interaction is not present, the particion function becomes much simpler:
$$\bar Z = \sum_{N_w=0}^{N}\lambda^{N_w}\Omega(N,N_w)
A^{N_w} B^{N-N_w} = B^N(1+\lambda A/B)^N \eqno (15)$$ Hence, in the thermodynamic limit of $A/B<<1$, the density at the wall, $\rho_w$, becomes:
$$\rho_w = (\lambda/A) {\frac{\partial ln\bar Z}{\partial \lambda}} =
{\frac{\lambda \phi}{1+\lambda A/B}} \rightarrow \lambda \phi \eqno (16)$$
from which follows the ideal gas law:
$$P/kT = \int_o^1 {\frac{d\lambda}{\lambda}}\rho_w = \phi \eqno (17)$$
[**IV. Random Walk Simulations**]{} 0.2 true cm We now carry out Monte Calro random walk simulations to measure the density at the wall and compare them with the formulas (13) and (16). 0.2 true cm We take a two dimensional lattice of size $H=20$ and $L=10$ with the volume $V=HL$. We now turn on the repulsive wall potential $0 \le U \le \infty$ at z=H, and introduce a parameter $0 \le \lambda = exp(-U/T) \le 1$. The wall potential is short range and thus acts only when the particle is right next to the wall, i.e., at $z=H-1$. At $z=H-1$, the particle moves to the wall at $z=H$ with the probability $p=\lambda$. This is the standard metropolitan algorithm. Otherwise, the particles are free to move in the bulk (with the exception of the excluded volume interaction). We also impose periodic boundary conditions along the vertical axis. With these simple rules, the Monte Carlo simulations proceed as follows. Initially N particles with the bulk density $\phi=N/V$ are randomly placed on a lattice and they undergo random walk. With the excluded volume interaction, multiple occupancy on a lattice point is prohibited. Without the excluded volume interaction, multiple occupancy is allowed. When a particle is right next to the right wall, then it moves there with the probability $\lambda$. When an attempt has been made to move all the N particles once, then one Monte Carlo(MC) step is elapsed. At each MC step, the number of particles at $z=H-1$ is registered. Then, after M MC time steps, the density of the particles near the wall, $\rho_w$ at $z=H-1$ is computed as the average over MC time t and configurations, namely: $\rho_w= [\sum_{t} N_w(t)/M]/L$ with $N_w(t)$ the muber of particles at $z=H-1$ at a given MC time t. $\rho_w$ is then measured as a function of $\lambda$ for different bulk density $\phi 's$. We now present simulation data. 0.2 true cm We measured the particle density at the wall, $\rho_w(\lambda,\phi)$, with and without excluded interactions, for the bulk density $\phi=0.1,0.3,0.5,0.7, 0.9$ and for $\lambda = 0.1,0.3,0.5,0.7, 0.9,1.0$. By symmetry, $\rho_{wall}(1,\phi) =\phi$. The simulation data are presented in Fig.1 for the excluded volume interaction and in Fig.2 without the excluded volume interactions. The dotted lines in both Figures are the predicted formulas, $(13)$ and $(16)$, and the squares are the simulation data. Even for a small size of lattice used in the simulations, the agreement between simulation data and the predicted forumla appear to be quite remarkable. The typical error bars are less than a few percent and the error bars are expected to vanish in the thermodynamic limit as the size increases. The good agreement between the simulations and the prediction, even for a small size of lattice, is precisely because of the fact that the ratio $A/B$, in equation (16) is essentially the ratio of one dimensional lattice sites over those of two dimensions, which is in the presence case of order $1/20=0.05<<1$. Even for a modest size of two dimensional lattice, this ratio is small and its contribution is almost insignificant in higher dimensions. Further, the logarithmic singularity in the pressure at the closed packed density is also well captured in the lattice simulations with the typical error bars are less than a few percent. We now conclude with two comments.
First, while this method appears to produce excellent results, the method becomes a little problematic when there is an external bias, say the gravity toward the wall. In this case, unless the bias is small, it will eventually pull all the particles to the wall, giving the wall density essentially close to one, and largely independent of $\lambda$. In this case, it becomes quite difficult to exploit the relation (10) for diffent $\lambda$’s. We have not come up with a method to overcome this difficulty. Such a method, if successful, will be quite useful because it will enable one to directly measure the force profile of granular materials in a container, which is the subject of current studies \[9-12\].
Second, the method, however, is expected to work quite well in the absence of bias for other interacting systems. An example would be the charged or uncharged lattice gas with either long or short range interactions such as Coulomb interactions, or other Ising type interactions near the critical point. Such studies will require elaborate finite size scaling analysis and extensive large scale simulations, which will be reported in future communications. 1.0 true cm [**VI. Acknowledgements**]{} 0.2 true cm This work was supported by NSF as a part of Research Experiences for Undergraduate Students program at Lehigh University. DCH wishes to express his thanks to Ronald Dickman for helpful discussions as well as for his independently obtaining eq.(14).
[**References**]{} 0.3 true cm 1. For example, M. P. Allen and D. J. Tildesley, [*Computer Simulations of Liquids*]{}, (Ocford Scientific Publications, 1987) 0.2 true cm 2. For example, F. Reif, [*Fundamentals of Statistical and Thermal Physics,*]{} (McGraw-Hill, New York, 1965) p.163. A little advanced one is: K. Huang, [*Statistical Mechanics*]{}, (John Wiley and Sons, 1987). 0.2 true cm 3. J.K. Percus and G.J. Yevick, “Analysis of Classical Statistical Mechanics by Means of Collective Coordinates”, Phys. Rev. [**110**]{}, 527 (1958) 0.2 true cm 4. N.F. Carnahan and K.E. Sterling, “Equation of State for Non-attracting Rigid Spheres”, J. Chem. Phys. [**51**]{}, 6232 (1970) 0.2 true cm 5. J. P. Hansen and I.R. McDonalds, [*Theory of simple liquids*]{}, (Academic Press, 1986) p. 95. 0.2 true cm 6. M.E. Fisher, “ Bounds for the Derivative of the Free Energy and the Pressure of a Hard-Core Systems near Close Packing,” J. Chem. Phys. [**42**]{}, 3852 (1965) 0.2 true cm 7. J. K. Percus, “Models for Density Variation at a Fluid Surface”, J. Stat. Phys. [**15**]{}, 423 (1976) 0.2 true cm 8. R. Dickmann, “On the Equation of State of Athermal Lattice Chains: Test of Mean Field and Scaling Theories in Two Dimensions”, J. Chem. Phys. 91, 454 (1989); [*ibid*]{}, [*Mumerical Methods for Polymer Systems,*]{} (Springer-Verlag, New York 1997) p.1. 0.2 true cm 9. C.-h Liu et al, Science [**269**]{}, 513 (1995) 0.2 true cm 10. S. N. Coppersmith et al, Phys. Rev. E [**53**]{}, 4673 (1996) 0.2 true cm 11. S. Luding, Phys. Rev. E [**55**]{}, 4720 (1997) 0.2 true cm 12. D. Mueth, H. Jaeger, and S. R. Nagel, “Force Distribution In a Granular Medium” (preprint)
0.2 true cm Figure captions: 1.0 true cm Fig. 1. Comparison between the Monte Carlo data and the exact formula (13) for the lattice gas with the excluded volume interaction for given bulk density $\phi$’s for different $\lambda 's$. The vertical axis is the particle density at the wall, $\rho_w$, and the horizontal axis is $\lambda$. From the bottom to top, $\phi =
0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$. Dotted lines are the exact formula(13). 1.0 true cm Fig. 2. Same as in Fig.1 except that multiple occupancy is allowed at the lattice sites. Dotted lines are the exact formula(16).
[^1]: E-mail address: dh09@lehigh.edu
|
---
abstract: 'We study the one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise, which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in [1/2,1)$ in the spatial variable. We show that the normalized spatial average of the solution over $[-R,R]$ converges in total variation distance to a normal distribution, as $R$ tends to infinity. We also provide a functional central limit theorem.'
address:
- 'Conacyt Research Fellow - Universidad Nacional Autónoma de México. Instituto de Matemáticas, Oaxaca, México'
- 'University of Kansas, Department of Mathematics, USA'
- 'University of Kansas, Department of Mathematics, USA'
author:
- 'Francisco Delgado-Vences'
- David Nualart
- Guangqu Zheng
title: A Central Limit Theorem for the stochastic wave equation with fractional noise
---
[^1]
[**Mathematics Subject Classifications (2010)**]{}: 60H15, 60H07, 60G15, 60F05.
[**Keywords:**]{} Stochastic wave equation, central limit theorem, Malliavin calculus, Stein’s method.
Introduction
============
We consider the one-dimensional stochastic wave equation $$\label{eq:heat-equation}
\frac{\partial^2 u}{\partial t^2} = \frac {\partial ^2 u} {\partial x^2} + \sigma(u) \frac { \partial ^2 W} {\partial t \partial x},$$ on ${\mathbb R}_+\times {\mathbb R}$, where $W(t,x)$ is a Gaussian random field that is a Brownian motion in time and behaves as a fractional Brownian motion with Hurst parameter $H \in [1/2,1)$ in the spatial variable. For $H=1/2$, the random field $W$ is just a two-parameter Wiener process on ${\mathbb R}_+\times {\mathbb R}$. We assume $u(0,x)=1$, $\frac {\partial } {\partial t} u(0,x)=0$ and $\sigma$ is a Lipschitz function with Lipschitz constant $L\in(0,\infty)$.
It is well-known (see, for instance, [@Dalang]) that equation has a unique *mild solution*, which is adapted to the filtration generated by $W$, such that $\sup\big\{ {\mathbb E}\big[ \vert u(t,x)\vert^2\big]\,:\, x\in {\mathbb R}, t\in[0,T] \big\} < \infty$ and $$\label{eq: mild}
u(t,x) = 1+ \frac 12 \int_0^t \int_{{\mathbb R}} \mathbf{1}_{\{ |x-y| \le t-s\}} \sigma(u(s,y))W(ds,dy)\,,$$ where the above stochastic integral is defined in the sense of Itô-Walsh.
In this paper, we are interested in the asymptotic behavior as $R$ tends to infinity of the spatial averages $$\label{1}
\int_ {-R}^R u(t,x)dx,$$ where $t>0$ is fixed and $u(t,x)$ is the solution to . We remark that, for each fixed $t>0$, the process $\{ u(t,x), x\in {\mathbb R}\}$ is *strictly stationary*[^2], meaning that the finite-dimension distributions of the process $\{u(t,x+y),x\in {\mathbb R}\}$ do not depend on $y$. Furthermore, $u(t,x)$ is measurable with respect to the $\sigma$-field generated by the random variables $\{W(s,z): |x-z| \le t-s\}$. As a consequence,
1. for $H=1/2$, the random variables $u(t,x)$ and $u(t,y)$ are independent if $|x-y| > 2t$;
2. for $H\in (1/2,1)$, $u(t,x)$ and $u(t,y)$ have a correlation that decays like $|x-y-2t| ^{2H-2}$ when $|x-y|\to +\infty$, which is a consequence of Gebelein’s inequality (see, for instance, [@V]).
Therefore, we expect the Gaussian fluctuation of the spatial averages .
Our first goal is to apply the methodology of Malliavin-Stein to provide a quantitative central limit theorem for , which will be described in total variation distance.
Define the normalized averages by $$\label{eq:quantity-of-interest}
F_R(t):= \frac{1}{\sigma_R}\left(\int_ {-R}^R u(t,x)dx- 2R\right),$$ where $u(t,x)$ is the solution to and $\sigma_R^2 ={\displaystyle {\rm Var}\left(\int_{-R}^R u(t,x)dx\right) }$.
To avoid triviality, throughout this paper, we assume that $\sigma(1)\not =0$, which guarantees that $\sigma_R>0$ for all $R>0$ and also that $\sigma_R$ is of order $R^{H}$; see Lemma \[lema1\] and Propositions \[pro:covariance1\], \[pro:covariance2\] below.
Our first result is the following quantitative central limit theorem.
\[thm:TV-distance\] Let $d_{\rm TV}$ denote the total variation distance $($see $)$ and let $Z\sim \mathcal{N}(0,1)$. For any fixed $t> 0$, there exists a constant $C= C_{t,H,\sigma}$, depending on $t, H$ and $\sigma$, such that $$d_{\rm TV}\left( F_R(t),Z\right) \leq C R^{H-1} \,.$$
Our second objective is to provide the functional version of Theorem \[thm:TV-distance\].
\[thm:functional-CLT\] For any $s>0$, we set $\eta(s)= {\mathbb E}\big[ \sigma(u(s,y)) \big]$ and $\xi(s)= {\mathbb E}\big[ \sigma^2(u(s,y)) \big]$, which do not depend on $y$ due to the stationarity. Then, for any $T>0$, as $R\to+\infty$,
- if $H= 1/2$, then $$\left\{ \frac{1}{\sqrt{R}}\left(\int_{-R}^R u(t,x)dx - 2R\right) \right\}_{t\in [0,T]} \Rightarrow \left\{ \sqrt{2} \int_0^t (t-s) \sqrt{\xi(s)} dB_s\right\}_{t\in [0,T]};$$
- if $H\in (1/2,1)$, then $$\left\{ R^{-H} \left(\int_{-R}^R u(t,x)dx - 2R\right) \right\}_{t\in [0,T]} \Rightarrow \left\{ 2^H \int_0^t (t-s) \eta(s) dB_s\right\}_{t\in [0,T]} \,.$$
Here $B$ is a standard Brownian motion and the above weak convergence takes place in the space of continuous functions $C([0,T])$.
Theorem \[thm:TV-distance\] is proved using a combination of Stein’s method for normal approximation and Malliavin calculus, following the ideas introduced by Nourdin and Peccati in [@NP]. The main idea is as follows. The total variation distance $d_{\rm TV}\left( F_R(t),Z\right) $ is bounded by $2 \sqrt{ {\rm Var} \langle DF_R(t), v_R \rangle_{\mathfrak H}}$, where $D$ is the derivative in the sense of Malliavin calculus, ${\mathfrak H}$ is the Hilbert space associated to the noise $W$ and $v_R$ is an ${\mathfrak H}$-valued random variable such that $F_R(t) =\delta(v_R)$, $\delta$ being the adjoint of the derivative operator, called the divergence or the Skorohod integral. A key new ingredient in the application of this approach is to use the representation of $F_R(t)$ as a stochastic integral of $v_R$, taking into account that the Itô-Walsh integral is a particular case of the Skorohod integral.
A similar problem for the stochastic heat equation on ${\mathbb R}$ has been recently considered in [@HNV18], but only in the case of a space-time white noise. In this case, it was proved in [@HNV18] that the limiting process in the functional central limit theorem is a martingale, which is not true for our wave equation. Moreover, in the colored case $H\in (1/2,1)$ considered here, we have found the surprising result that the square moment $ {\mathbb E}[ \sigma^2(u(s,y)) ]$ in the white noise case is replaced by the square of the first moment $({\mathbb E}[ \sigma(u(s,y)) ])^2$. Furthermore, the rate of convergence depends on the Hurst parameter $H$.
When $\sigma(u)=u$, the solution has an explicit Wiener chaos expansion. A natural question in this case is [*whether the central limit is chaotic*]{}, meaning that the projection on each Wiener chaos contributes to the limit. Such a phenomenon has been observed in other cases (see, for instance, [@HN]). We will show that for $H>1/2$ only the first chaos contributes to the limit, where as for $H=1/2$, we will see in Remark \[rem:nonchaotic\] that the first chaos is not the only contributor in the limit and to check whether or not this central limit is chaotic, one shall go through the usual arguments for chaotic central limit theorem (see [@Eulalia Section 8.4]).
The rest of the paper is organized as follows. In Section \[sec:prel\] we recall some preliminaries on Malliavin calculus and Stein’s method. Sections \[sec:thm1\] and \[sec:thm2\] are devoted to the proofs of our main theorems. We put the proof of a technical lemma (Lemma \[lemma: iteration\]) in the appendix. This lemma, which has an independent interest, states that the $p$-norm of the Malliavin derivative $D_{s,y} u(t,x)$ can be estimated, up to constant that depends on $p$ and $t$, by the fundamental solution of the wave equation $\frac 12\mathbf{1}_{\{ |x-y| \le t-s\}}$.
Along the paper we will denote by $C$ a generic constant that might depend on the fixed time $t$, the Hurst parameter $H$ and the non-linear coefficient $\sigma$, and it can vary from line to line.
Preliminaries {#sec:prel}
=============
We denote by $W= \{ W(t,x), t\ge 0, x\in {\mathbb R}\}$ a centered Gaussian family of random variables defined in some probability space $(\Omega, \mathcal{F}, P)$, with covariance function given by $${\mathbb E}\left[W(t,x) W(s,y)\right] = \frac{s\wedge t }{ 2} \big( |x|^{2H} + |y|^{2H} - |x-y| ^{2H}\big),$$ where $ H \in [1/2, 1)$.
Let ${\mathfrak H}_0$ be the Hilbert space defined as the completion of the set of step functions on ${\mathbb R}$ equipped with the inner product $$\begin{aligned}
\label{corfctH}
\langle \varphi, \phi \rangle_{{\mathfrak H}_0}
= \begin{cases}
H(2H-1) {\displaystyle \int_{{\mathbb R}^2} \varphi(x) \phi(y) | x-y| ^{2H-2} \,dxdy }\qquad\text{if $H\in (1/2 ,1)$,} \\
{\displaystyle \int_{{\mathbb R}} \varphi(x) \phi(x) \, dx}, \quad\qquad\qquad \qquad\qquad\qquad\qquad \text{if $H=1/2$.}
\end{cases}\end{aligned}$$ Set ${\mathfrak H}= L^2( {\mathbb R}_+ ; {\mathfrak H}_0)$ and notice that $${\mathbb E}\left[W(t,x) W(s,y)\right] = \langle \mathbf{1}_{[0,t]\times [0,x]} , \mathbf{1}_{[0,s]\times [0,y]} \rangle_{{\mathfrak H}},$$ where, by convention, $[0,x] = [-|x|, 0]$ if $x$ is negative. Therefore, the mapping $(t,x) \rightarrow W(t,x)$ can be extended to a linear isometry between ${\mathfrak H}$ and the Gaussian subspace of $L^2(\Omega)$ generated by $W$. We denote this isometry by $\varphi \longmapsto W(\varphi)$.
When $H= 1/2$, the space ${\mathfrak H}$ is simply $L^2( {\mathbb R}_+ \times {\mathbb R})$ and $W(\varphi)$ is the Wiener-Itô integral of $\varphi$: $$W(\varphi)= \int_{{\mathbb R}_+ \times {\mathbb R}} \varphi(t,x) W(dt,dx).$$ For $H \in ( 1/2, 1)$, the space $L^{1/H} ({\mathbb R})$ is known to be continuously embedded into ${\mathfrak H}_0$; see [@MMV01; @PipirasTaqqu].
For any $t\ge 0$, we denote by ${\mathcal F}_t$ the $\sigma$-field generated by the random variables $\{ W(s,x) :0\le s\le
t, x\in {\mathbb R}\}$. Then, for any adapted ${\mathfrak H}_0$-valued stochastic process $\{X(t),\; t\ge 0 \}$ such that $$\label{inte}
\int_0^\infty {\mathbb E}[ \| X(t) \|_{{\mathfrak H}_0} ^2] d t <\infty,$$ the following stochastic integral $$\label{DW}
\int_0^\infty \int_{{\mathbb R}} X(s,y) W(d s, d y)$$ is well-defined and satisfies the isometry property $${\mathbb E}\left[ \left( \int_0^\infty \int_{{\mathbb R}} X(s,y) W(d s, d y) \right)^2 \right] = {\mathbb E}\left ( \int_0^\infty \| X(t) \|_{{\mathfrak H}_0} ^2 d t \right)\, .$$
We will make use of the following lemma and the notation $\alpha_H = H(2H-1)$.
\[lem1\] For any $H\in (1/2, 1)$, $s,t \ge 0$ and $x,\xi \in {\mathbb R}$, we have $$\begin{aligned}
\notag
&2\alpha_H \int_{{\mathbb R}^2} \mathbf{1}_{\{|x-y| \le t\}} \mathbf{1}_{\{| \xi-z| \le s \}} | y-z| ^{2H-2} dydz\\ \notag
&\qquad = \big\vert x-\xi -t- s \big\vert ^{2H} + \big\vert x-\xi+t+s \big\vert ^{2H} \\ \label{ecu2}
&\quad \quad\quad\quad\quad - \big\vert x-\xi+ t - s \big\vert ^{2H} - \big\vert x-\xi - t+s \big\vert ^{2H}.
\end{aligned}$$
Let $B^H$ be a two-sided fractional Brownian motion with Hurst parameter $H$. That is, $B^H=\{ B^H_t, t \in {\mathbb R}\}$ is a centered Gaussian process with covariance $${\mathbb E}[B^H_t B^H_s] = \frac 12 \left( |t|^{2H} + |s|^{ 2H} - |t-s| ^{2H} \right)\,, \,\, s,t\in{\mathbb R}\,.$$ Notice that both sides of are equal to $
2 {\mathbb E}\big[ (B^H_{x+t} - B^H_{x-t})( B^H_{\xi+s} - B^H_{\xi-s}) \big]
$, in view of and the above covariance structure. So the desired equality follows immediately.
The proof of our main theorems relies on a combination of Malliavin calculus and Stein’s method. We will introduce these tools in the next two subsections.
Malliavin calculus
------------------
Now we recall some basic facts on Malliavin calculus associated with $W$. For a detailed account of the Malliavin calculus with respect to a Gaussian process, we refer to Nualart [@Nualart].
Denote by $C_p^{\infty}({\mathbb R}^n)$ the space of smooth functions with all their partial derivatives having at most polynomial growth at infinity. Let $\mathcal{S}$ be the space of simple functionals of the form $$F = f(W(h_1), \dots, W(h_n))$$ for $f\in C_p^{\infty}({\mathbb R}^n)$ and $h_i \in {\mathfrak H}$, $1\leq i \leq n$. Then, $DF$ is the ${\mathfrak H}$-valued random variable defined by $$\begin{aligned}
DF=\sum_{i=1}^n \frac {\partial f} {\partial x_i} (W(h_1), \dots, W(h_n)) h_i\,.\end{aligned}$$ The derivative operator $D$ is closable from $L^p(\Omega)$ into $L^p(\Omega; {\mathfrak H})$ for any $p \geq1$ and we let $\mathbb{D}^{1,p}$ be the completion of $\mathcal{S}$ with respect to the norm $$\|F\|_{1,p} = \left({\mathbb E}\big[ |F|^p \big] + {\mathbb E}\big[ \|D F\|^p_{\mathfrak H}\big] \right)^{1/p} \,.$$ We denote by $\delta$ the adjoint of $D$ given by the duality formula $${\mathbb E}(\delta(u) F) = {\mathbb E}( \langle u, DF \rangle_{\mathfrak H})$$ for any $F \in \mathbb{D}^{1,2}$ and $u\in{\rm Dom} \, \delta \subset L^2(\Omega; {\mathfrak H})$, the domain of $\delta$. The operator $\delta$ is also called the Skorohod integral, because in the case of the Brownian motion, it coincides with an extension of the Itô integral introduced by Skorohod (see [@GT; @NuPa]). More generally, in the context of our Gaussian noise $W$, any adapted random field $X$ that satisfies (\[inte\]) belongs to the domain of $\delta$ and $\delta (X)$ coincides with the Dalang-Walsh-type stochastic integral (\[DW\]): $$\delta (X) =
\int_0^\infty \int_{{\mathbb R}} X(s,y) W(d s, d y).$$ As a consequence, the mild formulation equation can also be written as $$\label{eq: mild Skorohod}
u(t,x) = 1 + \frac{1}{2} \delta\Big( \mathbf{1}_{ \{\vert x- \ast \vert \le t -\cdot\} } \sigma\big( u(\cdot, \ast) \big)\Big).$$
It is known that for any $(t,x)\in{\mathbb R}_+\times {\mathbb R}$, the solution $u(t,x)$ to equation (\[eq:heat-equation\]) belongs to $\mathbb{D}^{1,p}$ for any $p\ge 2$ and the derivative satisfies the following linear stochastic integral differential equation for $t\ge s$, $$\begin{aligned}
\notag
D_{s,y}u(t,x) &= \frac 12 \mathbf{1}_{\{ |x-y| \le t-s\}} \sigma(u(s,y)) \\ \label{ecu1}
& \qquad + \frac 12 \int_s^t \int_{{\mathbb R}} \mathbf{1}_{\{ |x-z| \le t-r \}} \Sigma(r,z) D_{s,y} u(r,z) W(dr,dz),\end{aligned}$$ where $\Sigma(r,z)$ is an adapted process, bounded by the Lipschitz constant of $\sigma$ (we refer to the appendix for more details on the properties of the derivative). If $\sigma $ is continuously differentiable, then $\Sigma(r,z)= \sigma'(u(r,z))$. This result is proved in [@Nualart Proposition 2.4.4] in the case of the stochastic heat equation with Dirichlet boundary conditions on $[0,1]$ driven by a space-time white noise. Its proof can be easily extended to the wave equation on ${\mathbb R}$ driven by the colored noise $W$. We also refer to [@CHN18; @NQ] for additional references, where this result is used for $\sigma\in C^1({\mathbb R})$.
In the end of this subsection, we record a technical result that is essential for our arguments, and we postpone its proof to the Appendix.
\[lemma: iteration\] For any $p\in[ 2,+\infty)$, $0 \leq t \le T $ and $x\in {\mathbb R}$, we have for almost every $(s,y)\in[0,T]\times {\mathbb R}$, $$\label{ecu3}
\| D_{s,y} u(t,x) \|_p \le C \mathbf{1}_{\{ |x-y| \le t-s\}}$$ for some constant $C = C_{T,p,H,\sigma}$ that depends on $T,p,H$ and the function $\sigma$.
Stein’s method
--------------
Stein’s method is a probabilistic technique that allows one to measure the distance between a probability distribution and a target distribution, notably the normal distribution. Recall that the total variation distance between two real random variables $F$ and $G$ is defined by $$d_{\rm TV}(F,G) := \sup_{B \in \mathcal{B}({\mathbb R})} \big\vert P(F \in B) - P(G \in B) \big\vert\,, \label{TVDIST}$$ where $\mathcal{B}({\mathbb R})$ is the collection of all Borel sets in ${\mathbb R}$.
The following theorem provides the well-known Stein’s bound in the total variation distance; see [@NP Chapter 3].
\[thm:Stein\] For $Z \sim \mathcal{N}(0,1)$ and for any integrable random variable $F$, $$d_{\rm TV}(F, Z) \leq \sup_{f \in \mathscr{F}_{\rm TV} } \big\vert {\mathbb E}[ f'(F)] - {\mathbb E}[F f(F)]\big\vert \,,$$ where $\mathscr{F}_{\rm TV}$ is the class of continuously differentiable functions $f:{\mathbb R}\to{\mathbb R}$ such that $\|f\|_{\infty} \leq \sqrt{\pi/2} $ and $ \|f'\|_{\infty} \leq 2 $.
For a proof of this theorem, see [@NP Theorem 3.3.1]. Theorem \[thm:Stein\] can be combined with Malliavin calculus to get a very useful estimate (see [@HNV18; @Eulalia; @Zhou]).
\[lem: dist\] Let $F=\delta (v)$ for some ${\mathfrak H}$-valued random variable $v\in{\rm Dom }\delta$. Assume $F\in \mathbb{D}^{1,2}$ and ${\mathbb E}[F^2] = 1$ and let $Z \sim \mathcal{N}(0,1)$. Then we have $$d_{\rm TV}(F, Z) \leq 2 \sqrt{{\rm Var}\big[ \langle DF, v\rangle_{{\mathfrak H}} \big] }\,.$$
In the course of proving Theorem \[thm:functional-CLT\], we also need the following lemma, which is a generalization of [@NP Theorem 6.1.2]; see [@HNV18 Proposition 2.3].
\[lemma: NP 6.1.2\] Let $F=( F^{(1)}, \dots, F^{(m)})$ be a random vector such that $F^{(i)} = \delta (v^{(i)})$ for $v^{(i)} \in {\rm Dom}\, \delta$ and $ F^{(i)} \in \mathbb{D}^{1,2}$, $i = 1,\dots, m$. Let $Z$ be an $m$-dimensional centered Gaussian vector with covariance $(C_{i,j})_{ 1\leq i,j\leq m} $. For any $C^2$ function $h: {\mathbb R}^m \rightarrow {\mathbb R}$ with bounded second partial derivatives, we have $$\big| {\mathbb E}[ h(F)] -{\mathbb E}[ h(Z)] \big| \le \frac{m}{2} \|h ''\|_\infty \sqrt{ \sum_{i,j=1}^m {\mathbb E}\Big[ \big(C_{i,j} - \langle DF^{(i)}, v^{(j)} \rangle_{{\mathfrak H}}\big)^2 \Big] } \,,$$ where $\|h ''\|_\infty : = \sup\big\{ \big\vert \frac{\partial^2}{\partial x_i\partial x_j} h(x) \big\vert\,:\, x\in{\mathbb R}^m\, , \, i,j=1, \ldots, m \big\}$.
Proof of Theorem \[thm:TV-distance\] {#sec:thm1}
====================================
We begin with the asymptotic variance of $F_R(t)$, as $R$ tends to infinity. We need some preliminary results and notation. We fix $t > 0$ and define $$\varphi_R(s,y) =\frac 12 \int_{-R} ^R \mathbf{1}_{\{|x-y| \le t-s\}} dx.$$ Notice that $2\varphi_R(s,y)$ is the length of $[-R,R] \cap [ y-t+s, y+ t-s] $, so $$\varphi_R(s,y) = \frac{1}{2} \Big( \big[ R\wedge (y+ t-s)\big] - \big[ (-R) \vee (y-t+s)\big] \Big)_+.$$ As a consequence, we deduce that
$ \varphi_R(s,y) =0$, if $|y| \geq R+t-s$; and $ \varphi_R(s,y) \le R \wedge (t-s)$.
Set ${\displaystyle
G_R =G_R(t): = \int_{-R}^Ru(t,x)dx - 2R.
}$ With this notation, we can write $$G_R= \int_0^t \int_{{\mathbb R}} \varphi_R(s,y) \sigma (u(s,y)) W(ds,dy).$$
The next lemma provides a useful formula.
\[rem1\] Let $0 < a \leq b$ and define $\varphi_{a,R}(y ) = \frac{1}{2} \int_{-R}^R \mathbf{1}_{\{ \vert x - y \vert \leq a \}} \, dx$, then we have, for any $R\ge 2b$, $$\int_{{\mathbb R}} \frac{1}{R} \varphi_{a,R}(y) \varphi_{b,R}(y) dy= 2ab - R^{-1} \Big( \frac{1}{2}ab^2 + \frac{1}{6}a^3 \Big) .$$ Therefore, $\lim_{R\to +\infty} \int_{{\mathbb R}} \frac{1}{R} \varphi_{a,R}(y) \varphi_{b,R}(y) dy =2ab$.
We can write $$\begin{aligned}
&\quad \int_{{\mathbb R}} \frac{1}{R} \varphi_{a,R}(y) \varphi_{b,R}(y) \, dy = \frac{1}{4R} \int_{\mathbb R}\int_{[-R, R]^2} \mathbf{1}_{\{|\tilde{x}-y| \le a\}} \mathbf{1}_{\{|x-y| \le b\}} \, d\tilde{x} dx dy \\
&= \frac{1}{4R} \int_{[-R, R]^2}d\tilde{x} dx \Big( \mathbf{1}_{\{|\tilde{x}-x| \le b-a\}} + \mathbf{1}_{\{ b-a < |\tilde{x}-x| \leq b+a\}} \Big) \int_{{\mathbb R}} \mathbf{1}_{\{|\tilde{x}-y| \le a ,|x-y| \le b\}} \, dy \\
&= \frac{1}{4R} \int_{[-R, R]^2} \Big\{ \mathbf{1}_{\{|\tilde{x}-x| \le b-a\}} (2a) + \mathbf{1}_{\{b-a <|\tilde{x}-x| \leq b+a\}} \big( a+b - \vert x- \tilde{x} \vert \big) \Big\} dx d\tilde{x} \,,
\end{aligned}$$ which is equal to $
2ab - R^{-1} \Big( \frac{1}{2}ab^2 + \frac{1}{6}a^3 \Big)$ for any $R\geq 2b$, as one can verify.
The next result provides the asymptotic variance of $G_R(t)$ for $H=1/2$.
\[pro:covariance1\] Suppose $H= 1/2$. Denote $\xi(s)= {\mathbb E}\big[\sigma^2(u(s,x)) \big] $, which does not depend on $x$ as a consequence of stationarity. Then $$\lim_{R\to\infty} \frac{1}{R} {\mathbb E}\big[ G_R ^2\big] = 2 \int_0^t (t-s)^2\xi(s) ds.$$ and ${\mathbb E}\big[ G_R^2\big] \geq {\displaystyle \left( \frac{5}{3} \int_0^t (t-s)^2\xi(s) ds\right)R}$ for any $R\geq 2t$.
Thanks to the Itô isometry, we have $${\mathbb E}[ G_R^2] = \int_0^t \int_{{\mathbb R}} \varphi ^2 _R(s,y) {\mathbb E}\big[ \sigma^2 (u(s,y))\big] dyds =
\int_0^t \xi(s) \int_{{\mathbb R}} \varphi ^2 _R(s,y) \, dy\,ds.$$ If $R \geq 2t$, we can see from Lemma \[rem1\] that $$\begin{aligned}
\label{use1}
\frac{1}{R} \int_{{\mathbb R}} \varphi ^2 _R(s,y) dy = 2(t-s)^2 \Big( 1- \frac{t-s}{3R} \Big) \in \Big[ \,\, \frac{5}{3}(t-s)^2 , 2(t-s)^2 \Big] \,.\end{aligned}$$ This leads easily to the results.
Surprisingly, in the case $H>1/2$, we obtain a different formula for the asymptotic variance of $G_R$.
\[pro:covariance2\] Suppose $H\in (1/2,1)$. Denote $\eta(s)= {\mathbb E}[\sigma(u(s,x))] $, which does not depend on $x$ as a consequence of stationarity. Then $$\lim_{R\to\infty} R^{-2H} {\mathbb E}[ G_R ^2] = 2^{2H} \int_0^t (t-s)^2 \eta^2(s)\, ds.$$
Thanks to the Itô isometry, we have $${\mathbb E}[ G_R^2] = \alpha_H \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,y) \varphi_R(s,z) {\mathbb E}\big[ \sigma (u(s,y))\sigma (u(s,z)) \big] | y-z| ^{2H-2} dydzds,$$ where $\alpha_H = H(2H-1)$. Keeping in mind that $ \big\{\sigma\big(u(t,x)\big), x\in{\mathbb R}\big\}$ is stationary, we write $
{\mathbb E}\big[ \sigma (u(s,y))\sigma (u(s,z)) \big] =: \Psi (s, y-z).
$ Then, $${\mathbb E}[ G_R^2] = \alpha_H \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \Psi(s,\xi) | \xi| ^{2H-2} d\xi dzds.$$ We claim that $$\label{ecu11}
\lim_{|\xi|\to +\infty} \sup_{0\le s\le t } | \Psi(s,\xi) -\eta^2(s)|=0.$$ In order to show (\[ecu11\]), we apply a two-parameter version of the Clark-Ocone formula (see *e.g.* [@CKNP19 Proposition 6.3]). We can write $$\sigma (u(s,y)) = {\mathbb E}[ \sigma (u(s,y))]
+ \int_0^s \int_{{\mathbb R}} {\mathbb E}\Big[ D_{r,\gamma}\big( \sigma (u(s,y)) \big) | \mathcal{F}_r\Big] \, W(dr, d\gamma)$$ and $$\sigma (u(s,z)) = {\mathbb E}[ \sigma (u(s,z))]
+ \int_0^s \int_{{\mathbb R}} {\mathbb E}\Big[ D_{r,\beta}\big( \sigma (u(s,z))\big) | \mathcal{F}_r\Big] \, W(dr, d\beta).$$ As a consequence, $$\label{ecu4}
{\mathbb E}\Big[ \sigma \big(u(s,y)\big)\sigma \big(u(s,z)\big) \Big] = \eta^2(s)+ T(s, y,z),$$ where $$\begin{aligned}
T(s, y,z) \notag
&=\int_0^s \int_{{\mathbb R}^2} {\mathbb E}\Big\{ {\mathbb E}\Big[ D_{r,\gamma}\big( \sigma (u(s,y)) \big) | \mathcal{F}_r\Big] {\mathbb E}\Big[ D_{r,\beta}\big( \sigma (u(s,z))\big) | \mathcal{F}_r\Big] \Big\} \\ \label{fb2}
&\qquad\qquad\qquad\qquad\times | \gamma-\beta|^{2H-2} d\gamma d\beta dr.\end{aligned}$$ By the chain-rule for the derivative operator (see [@Nualart Proposition 1.2.4]), $$D_{r,\gamma}\big( \sigma (u(s,y)) \big) = \Sigma(s,y) D_{r,\gamma} u(s,y)$$ and $$D_{r,\beta}\big( \sigma (u(s,z)) \big)= \Sigma(s,z) D_{r,\beta} u(s,z)$$ with $\Sigma(s,y)$ an adapted random field uniformly bounded by the Lipschitz constant of $\sigma$, denoted by $L$. This implies, using (\[ecu3\]), $$\begin{aligned}
\notag
&\quad \Big\vert {\mathbb E}\Big\{ {\mathbb E}\Big[ D_{r,\gamma}\big( \sigma (u(s,y)) \big) | \mathcal{F}_r\Big] {\mathbb E}\Big[ D_{r,\beta}\big( \sigma (u(s,z))\big) | \mathcal{F}_r\Big] \Big\} \Big\vert \\ \label{fb1}
& \leq L^2 \|D_{r,\gamma} u(s,y)\|_2 \|D_{r,\beta} u(s,z)\|_2 \leq C \mathbf{1}_{\{ |\gamma-y| \le s-r\}}\mathbf{1}_{\{ |\beta-z| \le s-r\}},
\end{aligned}$$ for some constant $C$. Therefore, substituting (\[fb1\]) into (\[fb2\]), we can write $$\begin{aligned}
\vert T(s,y,z) \vert \leq C \int_0^s \int_{{\mathbb R}^2} \mathbf{1}_{\{ |\gamma-y| \le s-r\}}\mathbf{1}_{\{ |\beta-z| \le s-r\}}
| \gamma-\beta|^{2H-2} d\gamma d\beta dr. \label{exp1}
\end{aligned}$$ If $ |y-z| > 2s$, we have $$\mathbf{1}_{\{ |\gamma-y| \le s-r\}}\mathbf{1}_{\{ |\beta-z| \le s-r\}} | \gamma-\beta|^{2H-2} \leq \mathbf{1}_{\{ |\gamma-y| \le s-r\}}\mathbf{1}_{\{ |\beta-z| \le s-r\}} \big( \vert y-z\vert -2s \big)^{2H-2}$$ and therefore deduce from that (for $ |y-z| > 2s$) $$\begin{aligned}
\vert T(s,y,z) \vert & \leq C \int_0^s \int_{{\mathbb R}^2} \mathbf{1}_{\{ |\gamma-y| \le s-r\}}\mathbf{1}_{\{ |\beta-z| \le s-r\}} \big( \vert y-z\vert -2s \big)^{2H-2} d\gamma d\beta dr \\
&\leq 4Ct^3\big( \vert y-z\vert -2t \big)^{2H-2} \xrightarrow{\vert y-z\vert\to+\infty} 0 \,.
\end{aligned}$$ Thus, claim is established in view of formula (\[ecu4\]).
Let us continue our proof of Proposition \[pro:covariance1\]. We first show that the quantity $$\begin{aligned}
\label{quan}
\frac{1}{R^{2H}} \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \big[ \Psi(s,\xi) - \eta^2(s)\big] \vert \xi \vert ^{2H-2} d\xi dzds
\end{aligned}$$ converges to zero, as $R\to+\infty$.
By , we can find $K = K_\varepsilon > 0$ for any given $\varepsilon > 0$ such that $$\sup\Big\{ | \Psi(s, \xi) -\eta^2(s)| : s\in[0,t], | \xi| > K \Big\} < \varepsilon.$$ Now we divide the above integration domain into two parts $\vert \xi \vert\leq K$ and $\vert \xi \vert > K$.
[*Case (i):*]{} On the region $| \xi | \le K$, by Cauchy-Schwarz inequality and , we get for $R\ge 2t$ $$\begin{aligned}
\int_{\mathbb R}\varphi_R(s,\xi+z )\varphi_R(x,z) \, dz& \leq \left( \int_{\mathbb R}\varphi_R^2(s,\xi+z )\, dz \right)^{1/2} \left( \int_{\mathbb R}\varphi^2_R(s,z )\, dz \right)^{1/2} \\
&= \int_{\mathbb R}\varphi^2_R(s,z )\, dz = 2R(t-s)^2 \Big( 1- \frac{t-s}{3R} \Big) \,.\end{aligned}$$ Since $\Psi(s,y-z) - \eta^2(s) = T(s,y,z )$ is uniformly bounded for $(s,y,z)\in[0,t]\times{\mathbb R}^2$, $$\begin{aligned}
&\quad R^{-2H} \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \big| \Psi(s,\xi) - \eta^2(s) \big| \mathbf{1}_{\{ \vert\xi\vert \leq K\}} \vert \xi \vert ^{2H-2} \, d\xi dzds \\
&\leq C R^{-2H} \int_0^t \int_{-K}^{K} \left( \int_{\mathbb R}\varphi_R(s,\xi+z )\varphi_R(x,z) \, dz \right) \vert \xi \vert ^{2H-2} \, d\xi ds \\
&\leq C R^{1-2H} \int_0^t (t-s)^2 \int_{-K}^{K} \vert \xi \vert ^{2H-2} \, d\xi ds \xrightarrow{R\to+\infty} 0 \,.\end{aligned}$$
[*Case (ii):*]{} On the region $| \xi | > K$, we know $| \Psi(s, \xi) -\eta^2(s)| < \varepsilon$ for $s\leq t$. Thus, $$\begin{aligned}
& \quad R^{-2H} \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \big| \Psi(s,\xi) - \eta^2(s) \big| \mathbf{1}_{\{ \vert\xi\vert > K\}} \vert \xi \vert ^{2H-2} \, d\xi dzds\\
&\leq \frac{\varepsilon}{R^{2H}} \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \vert \xi \vert ^{2H-2} \, d\xi dzds \\
& = \frac{\varepsilon}{4R^{2H}} \int_{-R}^R \int_{-R}^R dx dx' \,\, \int_0^t \int_{{\mathbb R}^2} \mathbf{1}_{\{ \vert x-z\vert \leq t-s \}} \mathbf{1}_{\{ \vert x'-z'\vert \leq t-s \}} \vert z-z'\vert^{2H-2} ds dzdz' \\
& \leq \frac{\varepsilon t}{4R^{2H}} \int_{-R}^R \int_{-R}^R dx dx' \,\, \int_{{\mathbb R}^2} \mathbf{1}_{\{ \vert x-z\vert \leq t \}} \mathbf{1}_{\{ \vert x'-z'\vert \leq t \}} \vert z-z'\vert^{2H-2} dzdz' \\
&=: \varepsilon t^3 \mathcal{A}(R) \,.\end{aligned}$$ We can rewrite $ \mathcal{A}(R)$, after a change of variables and supposing $R>t$, in the following form $$\begin{aligned}
\mathcal{A}(R) &= \int_{[-1,1]^2} \int_{{\mathbb R}^2} \frac{R}{2t} \mathbf{1}_{\{ \vert x-z\vert \leq tR^{-1} \}} \frac{R}{2t} \mathbf{1}_{\{ \vert x'-z'\vert \leq tR^{-1} \}} \vert z-z'\vert^{2H-2} \, dzdz' dxdx' \\
&= \int_{[-1,1]^2} \big(\zeta_R\ast g_2\big)(x, x')dx\, dx' \xrightarrow{R\to+\infty} \| g_1\| _{L^1({\mathbb R}^2)} \,, \end{aligned}$$ where $g_m(z,z') := \vert z-z'\vert^{2H-2} \mathbf{1}_{\{z\neq z'\}} \mathbf{1}_{\{z, z'\in[-m,m]\}}$, for $m=1,2$, are integrable functions on ${\mathbb R}^2$ and $$\Big\{ \zeta_R(x,x') = \frac{R}{2t} \mathbf{1}_{\{ \vert x\vert \leq tR^{-1} \}} \frac{R}{2t} \mathbf{1}_{\{ \vert x'\vert \leq tR^{-1} \}} \,,\,\, R > t \Big\}$$ defines an approximation of the identity. This leads to the asymptotic negligibility of the quantity , as $\varepsilon > 0$ is arbitrary.
Therefore, it suffices to show that $$\begin{aligned}
\notag
& \quad \lim_{R\rightarrow \infty} \frac{\alpha_H}{R^{2H}} \int_0^t \eta^2(s) \int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \vert \xi \vert ^{2H-2} d\xi dzds \\
& = 2^{2H} \int_0^t(t-s)^2 \eta^2(s)ds. \label{quan2}
\end{aligned}$$ The previous computations imply that $$\frac{1}{R^{2H}}\int_{{\mathbb R}^2} \varphi_R(s,\xi+z )\varphi_R(s,z) \vert \xi \vert ^{2H-2} d\xi dz \leq 4t^2 \mathcal{A}(R)$$ is uniformly bounded over $s\in[0,t]$. Moreover, we can get $$\begin{aligned}
&\quad \lim_{R\to+\infty} \frac{\alpha_H}{R^{2H}}\int_{{\mathbb R}^2} \varphi_R(s,\xi )\varphi_R(s,z) \vert \xi -z \vert ^{2H-2} d\xi dz \notag \\
&= (t-s)^2 \alpha_H \| g_1\|_{L^1({\mathbb R}^2)} = 2^{2H}(t-s)^2 \,, \label{for:nonchaotic}\end{aligned}$$ where the last equality follows from Lemma \[lem1\]. Hence (\[quan2\]) follows by the dominated convergence theorem and this concludes our proof.
It follows from the above two propositions that for fixed $t>0$, the variance of $G_R(t)$, denoted by $\sigma_R^2$, is $O(R^{2H})$. The next lemma states that $R^{2H}$ is the exact order under our standing assumption $\sigma(1) \not =0$, which is also a necessary condition to have this order. Moreover, $\sigma(1) \not =0$ is equivalent to $\sigma_R >0$ for all $R>0$.
\[lema1\] The following four conditions are equivalent:
- $\sigma(1) =0$.
- $ \sigma_R =0$ for all $R>0$.
- $ \sigma_R =0$ for some $R>0$.
- $\lim_{R\to\infty}\sigma_R^2 R^{-2H} =0$.
If $\sigma(1)=0$, then writing the solution as the limit of the Picard iterations starting with the constant solution $1$, we obtain that $u(t,x)=1$ for all $(t,x)$. As a consequence, $\sigma_R=0$ for all $R>0$ and (i) implies (ii). Clearly (ii) implies (iii) and (iv). Now suppose that (iv) holds. Then Propositions \[pro:covariance1\] and \[pro:covariance2\] imply that for almost every $s\in[0,t]$,
- ${\mathbb E}\big[ \sigma^2( u(s,y) ) \big] =0$ in the case $H=1/2$,
- ${\mathbb E}\big[ \sigma( u(s,y) ) \big] =0$ in the case $H\in (1/2,1)$.
By the $L^2(\Omega)$-continuity of the process $(s,y)\in{\mathbb R}_+\times{\mathbb R}\longmapsto u(s,y)$ (see *e.g.* [@Dalang99 Theorem 13]), letting $s$ tend to $0$, we deduce that $\sigma(1) = 0$ in both cases $H=1/2$ and $H\in(1/2, 1)$.
Finally, suppose that (iii) holds and assume that $H\in (1/2,1) $ (the proof in the case $H= 1/2$ is similar). By $L^2$-continuity, we can see that the function $\Psi(s,y):= {\mathbb E}\big[ \sigma (u(s,0)) \sigma (u(s,y)) \big] $ is continuous on ${\mathbb R}_+\times{\mathbb R}$. Note that, for almost all $s \in [0,t]$, $$\label{a11}
\int_{{\mathbb R}^2} \varphi_R(s,y)\varphi_R(s,z) \sigma (u(s,y)) \sigma (u(s,z)) |y-z| ^{2H-2} dydz =0$$ almost surely. In the above integral, the variables $y$ and $z$ have support contained in the interval $[-R-t, R+t]$. If $\sigma(1) \neq 0$, there exists a sufficiently small $\delta >0$ such that for $(s, y, z)\in[0,\delta] \times [ -R-t, R+t]^2$ $${\mathbb E}\big[ \sigma (u(s,y)) \sigma (u(s,z)) \big] = \Psi(s, y-z) \geq \vert \sigma(1)\vert^2/2 \,,$$ which is a contradiction to (\[a11\]). Therefore, $\sigma(1) =0$ and (iii) implies (i).
\[rem:nonchaotic\] It follows from Proposition \[pro:covariance2\] that, if $H\in (1/2,1)$, the random variable $G_R$ is [*not chaotic*]{} in the linear case. More precisely, when $\sigma(x)= x$, the above proposition gives us $${ \rm Var}\big( G_R) \sim R^{2H} 2^{2H} \int_0^t (t-s)^2 ds = \frac{ 4^Ht^3}{3} R^{2H} \quad \text{as $R\to+\infty$.}$$ Due to linearity, one can obtain the Wiener-chaos expansion of $G_R$ easily: $$G_R = \int_0^t\int_{\mathbb R}\varphi_R(s,y) \, W(ds, dy) + \text{\it higher-order chaoses.}$$ Then, the variance of the first chaos is equal to $$\int_0^t \alpha_H \int_{{\mathbb R}^2} \varphi_R(s,y) \varphi_R(s,z) \vert y -z\vert^{2H-2} \, dydzds \sim \frac{ 4^Ht^3}{3} R^{2H} \quad \text{as $R\to+\infty$,}$$ which is a consequence of and dominated convergence. This shows that only the first chaos contribute to the limit, that is, there is a non-chaotic behavior of the spatial average of the *linear* stochastic wave equation, when $H\in (1/2,1)$.
For $H=1/2$ and $\sigma (x)=x$, we obtain from Proposition \[pro:covariance1\] $${ \rm Var}\big( G_R) \sim 2R \int_0^t (t-s)^2 {\mathbb E}[u^2(s,x)] ds \quad \text{as $R\to+\infty$,}$$ whereas the variance of the projection on the first chaos is, using Lemma \[rem1\], $$\int_0^t\int_{\mathbb R}\varphi_R^2(s,y) dyds = \frac 23 Rt^3- \frac {t^4}{6}.$$ Notice that $ {\mathbb E}[u^2(s,x)] \ge ( {\mathbb E}[u(s,x) ])^2 =1$ and the inequality is strict for all $s\in (0,t]$ (otherwise $u(s,x)$ would be a constant). This implies that the first chaos is not the only contributor to the limiting variance.
Before we give the proof of Theorem \[thm:TV-distance\], by using the same argument as in the proof of Propositions \[pro:covariance1\] and \[pro:covariance2\], we obtain an asymptotic formula for ${\mathbb E}\big[ G_R(t_i) G_R(t_j) \big]$ with $t_i, t_j\in{\mathbb R}_+$, which is a useful ingredient for our proof of functional central limit theorem.
\[rem2\] Suppose $t_i , t_j\in{\mathbb R}_+$. If $H=1/2$, we have $$\begin{aligned}
{\mathbb E}\big[ G_R(t_i) G_R(t_j) \big] = \int_0^{t_i\wedge t_j} \int_{{\mathbb R}} \varphi^{(i)}_R(s,y) \varphi^{(j)}_R(s,y) \xi(s) dyds\,,\end{aligned}$$ where $ \varphi^{(i)}_R(s,y) = \frac{1}{2} \int_{-R}^R \mathbf{1}_{\{|x-y| \le t_i -s\}} \, dx$ and we obtain $$\begin{aligned}
\lim_{R\to+\infty} \frac 1R {\mathbb E}\big[ G_R(t_i) G_R(t_j) \big] =2 \int_0^{t_i\wedge t_j} (t_i-s)(t_j-s) \xi(s) \, ds \,.\end{aligned}$$ In the case $H\in (1/2,1)$, we have ${\mathbb E}\big[ G_R(t_i) G_R(t_j) \big] $ equal to $$\begin{aligned}
\alpha_H \int_0^{t_i\wedge t_j} \int_{{\mathbb R}^2} \varphi^{(i)}_R(s,y) \varphi^{(j)}_R(s,z) \Psi(s,y-z) | y-z| ^{2H-2} dydzds\,,\end{aligned}$$ and we obtain $$\begin{aligned}
&\quad \lim_{R\to+\infty} R^{-2H} {\mathbb E}\big[ G_R(t_i) G_R(t_j) \big] \\
&= \lim_{R\to+\infty} \alpha_H \int_0^{t_i\wedge t_j} ds~ \eta^2(s) \int_{{\mathbb R}^2} \varphi^{(i)}_R(s,y) \varphi^{(j)}_R(s,z) | y-z| ^{2H-2} dydz \\
&= 2^{2H} \int_0^{t_i\wedge t_j} (t_i-s)(t_j-s) \eta^2(s) \, ds \,.\end{aligned}$$
Now let us prove Theorem \[thm:TV-distance\].
By Proposition \[lem: dist\], if $F = \delta(v) \in \mathbb{D}^{1,2}$ with ${\mathbb E}(F^2)=1$, we have $$d_{\rm TV}(F,Z)\leq 2\sqrt{{\rm Var}\big[\langle DF,v\rangle_{{\mathfrak H}}\big]}.$$ Recall that in our case we have, as a consequence of Fubini’s theorem, that $$\begin{split}
F_R:=F_R(t) &= \frac{1}{\sigma_R} \int_{-R}^R \big[ u(t,x) - 1 \big]dx \\
&= \frac{1}{\sigma_R} \int_0^t \int_{{\mathbb R}} \varphi_R(s,y) \sigma (u(s,y)) W(ds,dy).
\end{split}$$ Similarly as in , we can write, for any fixed $t > 0$, $
F_R =\delta(v_R)$ with $v_R(s,y) = \sigma_R^{-1}\mathbf{1} _{[0,t]}(s) \varphi_R(s,y) \sigma (u(s,y)).
$ Moreover, $$D_{s,y}F_R =\mathbf{1} _{[0,t]}(s) \frac{1}{\sigma_R}\int_{-R}^RD_{s,y}u(t,x)dx.$$ Then, it follows from and Fubini’s theorem that $$\begin{aligned}
&\quad \int_{-R}^RD_{s,y} u(t,x) \, dx \\
&= \varphi_R(s,y) \sigma(u(s,y)) + \int_s^t \int_{{\mathbb R}} \varphi_R(r,z) \Sigma(r,z) D_{s,y} u(r,z) W(dr,dz).\end{aligned}$$ In what follows, we separate our proof into two cases: $H= 1/2$ and $H> 1/2$.
[*Case $H= 1/2$.*]{} We write $$\langle DF_R,v_R\rangle_{{\mathfrak H}} := B_1 +B_2,$$ where $$B_1= \frac { 1} {\sigma_R^2} \int_0^t \int_{{\mathbb R}} \varphi_R^2(s,y) \sigma ^2(u(s,y)) \,ds\, dy$$ and $$\begin{aligned}
B_2 &= \frac { 1 } {\sigma^2_R} \int_0^t \int_{{\mathbb R}} \varphi_R(s,y) \sigma(u(s,y)) \\
&\qquad \qquad \times \left( \int_s^t\int_{\mathbb R}\varphi_R(r,z) \Sigma(r,z) D_{s,y} u(r,z) W(dr,dz) \right) \, dyds.\end{aligned}$$ Notice that for any process $\Phi= \{\Phi(s), s\in [0,t]\}$ such that $ \sqrt{{\rm Var} (\Phi_s)} $ is integrable on $[0,t]$, it holds that $$\begin{aligned}
\sqrt{ {\rm Var} \left( \int_0^t \Phi_s ds \right)} \le \int_0^t \sqrt{{\rm Var} (\Phi_s)} ds \,. \label{FACT00}
\end{aligned}$$ So we can write $$\sqrt{ {\rm Var} \big[ \langle DF_R,v_R\rangle_{{\mathfrak H}} \big] } \le \sqrt{2}\Big( \sqrt{ {\rm Var}(B_1) } + \sqrt{ {\rm Var}(B_2) } \Big) \leq \sqrt{2}(A_1+A_2) \,,$$ with $$\begin{aligned}
A_1 &=\frac{1}{\sigma^2_R} \int_0^t \left( \int_{{\mathbb R}^2} \varphi^2_R(s,y)\varphi^2_R(s,y')
{\rm Cov} \Big[ \sigma^2 \big(u(s,y)\big) , \sigma ^2\big(u(s, y')\big) \Big] dydy'
\right)^{1/2} ds
\intertext{and}
A_2 &= \frac{1} {\sigma_R^2} \int_0^t \Bigg( \int_{{\mathbb R}^3} \int_s^t \varphi^2_R(r,z) \varphi_R(s,y) \varphi_R(s,y') \\
&\times
{\mathbb E}\Big[ \Sigma^2(r,z) D_{s,y} u(r,z) D_{s,y'} u(r,z)
\sigma\big(u(s,y)\big)\sigma\big(u(s,y')\big) \Big] \, dydy' dz dr
\Bigg)^{ 1/2} ds \,.
\end{aligned}$$ Then the rest of the proof for this case ($H=1/2$) consists in estimating $A_2$ and $A_1$. The proof will be done in two steps.
[*Step 1:*]{} Let us proceed with the estimation of $A_2$. As before, denote by $L$ the Lipschitz constant of $\sigma$ and for $p\ge 2$, as a consequence of stationarity, we write $$\label{Kp}
K_p(t) =\sup_{0\le s\le t} \sup_{y\in {\mathbb R}} \| \sigma(u(s,y)) \|_p = \sup_{0\le s\le t} \| \sigma(u(s,0)) \|_p.$$ Then, $$\begin{aligned}
& \quad \Big\vert {\mathbb E}\Big( \Sigma^2(r,z) D_{s,y} u(r,z) D_{s,y'} u(r,z) \sigma\big(u(s,y)\big)\sigma\big(u(s,y')\big) \Big) \Big\vert \\
& \le K_4^2(t) L^2 \| D_{s,y} u(r,z) \|_4 \| D_{s, y'} u(r,z) \|_4 \leq CK_4^2(t) L^2 \mathbf{1}_{\{|y-z| \le r-s\}} \mathbf{1}_{\{|y'-z| \le r-s\}} \,,
\end{aligned}$$ where the last inequality follows from Lemma \[lemma: iteration\]. This implies, together with Proposition \[pro:covariance1\], that, for any $R\ge 2t$, $$\begin{aligned}
A_2 \leq& \frac{C} {R} \int_0^t \Bigg( \int_{{\mathbb R}^3} \int_s^t \varphi_R^2(r,z) \varphi_R(s,y) \varphi_R(s,y') \\
&\qquad \qquad\times
\mathbf{1}_{\{|y-z| \le r-s\}}\mathbf{1}_{\{|y' -z| \le r-s\}} dydy' dz dr
\Bigg)^{ 1/2} ds .
\end{aligned}$$ Using first $\varphi_R(s,y) \varphi_R(s,y') \le (t-s)^2$ and then integrating in $y$ and $y'$, we obtain $$A_2 \leq \frac{C} {R} \int_0^t \left( \int_s^t \int_{{\mathbb R}} \varphi_R^2(r,z)
dz dr
\right)^{ 1/2} ds \leq \frac{C} {R} \int_0^t \left( \int_s^t 2R(t-r)^2 dr
\right)^{ 1/2} ds \,,$$ where the last inequality follows from . Therefore, we have $A_2
\le C/ \sqrt{R}$ for any $R\geq 2t$.
[*Step 2:*]{} Consider now the term $A_1$. We begin with a bound for the covariance $${\rm Cov} \Big[ \sigma^2 \big(u(s,y)\big) , \sigma ^2\big(u(s, y')\big) \Big] \,.$$ Using a version of Clark-Ocone formula for two-parameter processes, we write $$\sigma^2 (u(s,y)) = {\mathbb E}\big[\sigma^2 (u(s,y)) \big] + \int_0^s \int_{{\mathbb R}} {\mathbb E}\big[ D_{r,z} \big(\sigma^2(u(s,y))\big) | \mathcal{F}_r \big] \, W(dr,dz).$$ Then, ${\rm Cov} \Big[ \sigma^2 \big(u(s,y)\big) , \sigma ^2\big(u(s, y')\big) \Big]$ is equal to $$\begin{aligned}
\int_0^s \int_{{\mathbb R}} {\mathbb E}\Big\{ {\mathbb E}\big[ D_{r,z} \big(\sigma^2(u(s,y)) \big) | \mathcal{F}_r \big]
{\mathbb E}\big[ D_{r,z} \big( \sigma^2(u(s,y'))\big) | \mathcal{F}_r \big] \Big\}
dzdr .
\end{aligned}$$ By the chain rule, we have $
D_{r,z} \big(\sigma^2(u(s,y)) \big) = 2 \sigma(u(s,y)) \Sigma(s,y) D_{r,z} u(s,y)
$, thus $
\left\| {\mathbb E}[ D_{r,z} \big(\sigma^2(u(s,y)) \big) | \mathcal{F}_r ] \right\| _2 \le 2K_4(t) L \left\| D_{r,z} u(s,y)\right\|_4 .
$ Then, using Lemma \[lemma: iteration\], we can write $$\begin{aligned}
& \left| {\rm Cov} \Big[ \sigma^2 \big(u(s,y)\big) , \sigma ^2\big(u(s, y')\big) \Big] \right| \le C \int_0^s \int_{{\mathbb R}} \left\| D_{r,z} u(s,y)\right\|_4 \left\| D_{r,z} u(s,y')\right\|_4 dz dr \\
& \quad \le C \int_0^s \int_{{\mathbb R}} \mathbf{1}_{\{ |y-z| \le s-r\}} \mathbf{1}_{\{ |y'-z| \le s-r\}} dz dr \le C \mathbf{1}_{\{|y-y'| \le 2s\}} \,.
\end{aligned}$$ This leads to the following estimate for $A_1$, for any $R\geq 2t$: $$A_1 \le \frac{C}{R} \int_0^t \left( \int_{{\mathbb R}^2} \varphi^2_R(s,y)\varphi^2_R(s,y')
\mathbf{1}_{\{|y-y'| \le 2s\}} dydy'
\right)^{ 1/2} ds.$$ Since $ \varphi^2_R(s,y)\varphi^2_R(s,y') \le (t-s)^4 \mathbf{1}_{\{|y| \vee |y'| \le R+t-s\}}$, we get $A_1 \leq C /\sqrt{R}$ for $R\geq 2t$. This concludes our proof for the case $H=1/2$.
The proof for the other case is more involved but we can proceed in similar steps.
[*Case $H > 1/2$*]{}. In this case, we write $
\langle DF_R,v_R\rangle_{{\mathfrak H}} := B_1 +B_2,
$ where $$\begin{aligned}
B_1 &= \frac { \alpha_H } {\sigma_R^2} \int_0^t \int_{{\mathbb R}^2} \varphi_R(s,y)\varphi_R(s, y') \sigma \big(u(s,y)\big) \sigma\big(u(s,y')\big) |y-y'| ^{2H-2} \, dydy'ds
\intertext{and}
B_2 &= \frac { \alpha_H } {\sigma_R^2} \int_0^t \int_{{\mathbb R}^2}
\left( \int_s^t \varphi_R(r,z) \Sigma(r,z) D_{s,y} u(r,z) \, W(dr,dz) \right) \\
&\qquad \qquad\qquad\qquad\qquad \times \varphi_R(s,y') \sigma\big(u(s,y')\big) | y-y'| ^{2H-2}~ dydy' ds.\end{aligned}$$ This decomposition implies $ \sqrt{ {\rm Var} \big[\langle DF_R,v_R\rangle_{{\mathfrak H}} \big] } \le \sqrt{2}(A_1+A_2),
$ with $$\begin{aligned}
A_1&=\frac{\alpha_H}{\sigma^2_R} \int_0^t \Bigg( \int_{{\mathbb R}^4} \varphi_R(s,y)\varphi_R(s,y')
\varphi_R(s, \tilde{y})\varphi_R(s, \tilde{y}') |y-y'| ^{2H-2} |\tilde{y}-\tilde{y}'| ^{2H-2} \\
&\quad \times {\rm Cov} \Big[ \sigma \big(u(s,y)\big) \sigma\big(u(s,y')\big) , \sigma \big(u(s,\tilde{y})\big) \sigma\big(u(s,\tilde{y}')\big) \Big] ~ dy dy' d\tilde{y}d\tilde{y}'
\Bigg)^{1/2} ds
\intertext{and}
A_2 &= \frac{\alpha_H^{3/2}} {\sigma_R^2} \int_0^t \Bigg( \int_{{\mathbb R}^6} \int_s^t \varphi_R(r,z)\varphi_R(r,\tilde{z} ) \varphi_R(s,y') \varphi_R(s,\tilde{y}') \\
&\qquad\quad\times
{\mathbb E}\Big\{ \Sigma(r,z) D_{s,y} u(r,z) \Sigma(r,\tilde{z}) D_{s,\tilde{y}} u(r,\tilde{z}) \sigma\big(u(s,\tilde{y}')\big)\sigma\big(u(s,y')\big) \Big\} \\
& \qquad\qquad \times | y-y'| ^{2H-2} | \tilde{y}-\tilde{y}'| ^{2H-2} |z-\tilde{z}| ^{2H-2} dydy' d\tilde{y} d\tilde{y}' dz d\tilde{z} dr
\Bigg)^{1/2} ds \,.
\end{aligned}$$ The proof will be done in two steps:
[*Step 1:*]{} Let us first estimate the term $A_2$. Recall that $L$ denotes the Lipschitz constant of $\sigma$ and recall the notation $K_p(t)$ ($p\geq 2$) introduced in (\[Kp\]). We can write $$\begin{aligned}
&\quad \Big\vert {\mathbb E}\Big\{ \Sigma(r,z) D_{s,y} u(r,z) \Sigma(r,\tilde{z}) D_{s,\tilde{y}} u(r,\tilde{z}) \sigma\big(u(s,\tilde{y}')\big)\sigma\big(u(s,y')\big) \Big\} \Big\vert \\
& \le K_4^2(t) L^2 \| D_{s,y} u(r,z) \|_4 \| D_{s, \tilde{y}} u(r,\tilde{z}) \|_4 \leq C ~ \mathbf{1}_{\{|y-z| \le r-s\}} \mathbf{1}_{\{|\tilde{y}-\tilde{z}| \le r-s\}}
\end{aligned}$$ where the last inequality follows from Lemma \[lemma: iteration\].
Now we derive from Proposition \[pro:covariance2\] the following estimate: For fixed $t>0$, there exists a constant $R_t$ that depends on $t$ such that for any $R\geq R_t$, $$\begin{aligned}
A_2 &\leq \frac{C} {R^{2H}} \int_0^t \Bigg( \int_{{\mathbb R}^6} \int_s^t \varphi_R(r,z)\varphi_R(r,\tilde{z} ) \varphi_R(s,y') \varphi_R(s,\tilde{y}')
\mathbf{1}_{\{|y-z| \le r-s, |\tilde{y}-\tilde{z}| \le r-s\}} \\
&\qquad\qquad \quad \times | y-y'| ^{2H-2} | \tilde{y}-\tilde{y}'| ^{2H-2} |z-\tilde{z}| ^{2H-2} dydy' d\tilde{y} d\tilde{y}' dz d\tilde{z} dr
\Bigg)^{1/2} ds,
\end{aligned}$$ where $C$ is a constant that depends on $t,p,H$ and $\sigma$.
The integral in the spatial variable term can be rewritten as $$\begin{aligned}
\mathbf{I}&:= \frac{1}{16} \int_{[-R,R]^4} \int_{{\mathbb R}^6} \mathbf{1}_{\{ \vert x-z\vert\leq t-r, \vert \tilde{x} -\tilde{z}\vert \leq t-r, |x'-y'| \le t-s, |\tilde{x}'-\tilde{y}'| \le t-s, |y-z| \le r-s, |\tilde{y}-\tilde{z}| \le r-s\}} \\
&\qquad\qquad \times | y-y'| ^{2H-2} | \tilde{y}-\tilde{y}'| ^{2H-2} |z-\tilde{z}| ^{2H-2} ~ dx d\tilde{x} dx' d\tilde{x}'dydy' d\tilde{y} d\tilde{y}' dz d\tilde{z} \\
&= \frac{R^{6H+4}}{16} \int_{[-1,1]^4} \int_{{\mathbb R}^6} \mathbf{1}_{\{ \vert x-z\vert\leq \frac{t-r}{R}, \vert \tilde{x} -\tilde{z}\vert \leq \frac{t-r}{R}, |x'-y'| \le \frac{t-s}{R}, |\tilde{x}'-\tilde{y}'| \le \frac{t-s}{R}, |y-z| \le \frac{r-s}{R}, |\tilde{y}-\tilde{z}| \le \frac{r-s}{R}\}} \\
&\qquad\qquad \times | y-y'| ^{2H-2} | \tilde{y}-\tilde{y}'| ^{2H-2} |z-\tilde{z}| ^{2H-2} ~ dx d\tilde{x} dx' d\tilde{x}'dydy' d\tilde{y} d\tilde{y}' dz d\tilde{z},
\end{aligned}$$ where the second equality follows from a simple change of variables. Assuming $R\ge t$ and integrating in the variables $x, x', \tilde{x}, \tilde{x}'\in[-1,1]$, we have $$\begin{aligned}
\mathbf{I} & \le R^{6H} \int_{{\mathbb R}^6} \mathbf{1}_{\{ |y-z| \le \frac{t}{R}, |\tilde{y}-\tilde{z}| \le \frac{t}{R}\}} \mathbf{1}_{[-2,2]} (z) \mathbf{1}_{[-2,2]} (\tilde{z})
\mathbf{1}_{[-2,2]} (y') \mathbf{1}_{[-2,2]} (\tilde{y}')
\\
&\qquad\qquad \times | y-y'| ^{2H-2} | \tilde{y}-\tilde{y}'| ^{2H-2} |z-\tilde{z}| ^{2H-2} ~ dydy' d\tilde{y} d\tilde{y}' dz d\tilde{z}.
\end{aligned}$$ If $K= \sup_{y\in [-3,3]} \int_{-2}^2 | y-y'| ^{2H-2} dy'$, then for $R\geq R_t + t$, $$\mathbf{I} \le K^2 R^{6H} \int_{{\mathbb R}^4} \mathbf{1}_{\{ |y-z| \le \frac{t}{R}, |\tilde{y}-\tilde{z}| \le \frac{t}{R}\}} \mathbf{1}_{[-2,2]} (z) \mathbf{1}_{[-2,2]} (\tilde{z})
|z-\tilde{z}| ^{2H-2} ~ dy d\tilde{y} dz d\tilde{z}.$$ Finally, integrating in $y$ and $\tilde{y}$, yields for $R\geq R_t + t$, $$\mathbf{I} \le 36 K^2 R^{6H-2} \int_{[-2,2]^2}
|z-\tilde{z}| ^{2H-2} ~ dz d\tilde{z}.$$ As a consequence, $$A_2 \le C R^{H-1}$$ for $R$ big enough.
[*Step 2:*]{} It remains to estimate the term $A_1$. We will show $A_1 \le C R^{H-1}$ for $R$ big enough. We begin with a bound for the covariance $${\rm Cov} \Big[ \sigma \big(u(s,y)\big) \sigma\big(u(s,y')\big) , \sigma \big(u(s,\tilde{y})\big) \sigma\big(u(s,\tilde{y}')\big) \Big] \,.$$ According to a version of Clark-Ocone formula for two-parameter processes, we write $$\begin{aligned}
\sigma \big(u(s,y)\big) \sigma\big(u(s,y')\big) &= {\mathbb E}\big[\sigma \big(u(s,y)\big) \sigma\big(u(s,y')\big)\big] \\
&\quad+ \int_0^s \int_{{\mathbb R}} {\mathbb E}\Big[ D_{r,z} \Big(\sigma\big(u(s,y)\big) \sigma\big(u(s,y')\big) \Big) | \mathcal{F}_r \Big] W(dr,dz).
\end{aligned}$$ Then, $$\begin{aligned}
&\quad {\rm Cov} \Big[ \sigma \big(u(s,y)\big) \sigma\big(u(s,y')\big) , \sigma \big(u(s,\tilde{y})\big) \sigma\big(u(s,\tilde{y}')\big) \Big] \\
& = \alpha_H \int_0^s \int_{{\mathbb R}^2} {\mathbb E}\Big\{ {\mathbb E}\Big[ D_{r,z} \Big(\sigma\big(u(s,y)\big) \sigma\big(u(s,y')\big) \Big)| \mathcal{F}_r \Big] \\
&\qquad\qquad\qquad \times {\mathbb E}\Big[ D_{r,z'} \Big( \sigma\big(u(s,\tilde{y})\big) \sigma\big(u(s,\tilde{y}')\big)\Big) | \mathcal{F}_r \Big] \Big\} |z-z'|^{2H-2} ~dz dz'dr .
\end{aligned}$$ Applying the chain rule for Lipschitz functions (see [@Nualart Proposition 1.2.4]), we have $$\begin{aligned}
D_{r,z} \Big(\sigma\big(u(s,y)\big) \sigma\big(u(s,y')\big)\Big) &= \sigma\big(u(s,y)\big) \Sigma(s,y') D_{r,z} u(s,y')\\
& \qquad + \sigma\big(u(s,y')\big) \Sigma(s,y) D_{r,z} u(s,y)
\end{aligned}$$ and therefore, $\Big\| {\mathbb E}\big[ D_{r,z} \big(\sigma (u(s,y) ) \sigma (u(s,y') ) \big)| \mathcal{F}_r \big] \Big\| _2$ is bounded by $$2K_4(t) L \big\{ \| D_{r,z} u(s,y) \|_4+ \| D_{r,z} u(s,y') \|_4\big\}\, .$$ Applying Lemma \[lemma: iteration\], we get $ | {\rm Cov} \big[ \sigma (u(s,y) ) \sigma (u(s,y') ) , \sigma (u(s,\tilde{y}) ) \sigma (u(s,\tilde{y}') ) \big] |
$ bounded by $$\begin{aligned}
& 4L^2 K_4^2(t) \int_0^s \int_{{\mathbb R}^2 } ( \left\| D_{r,z} u(s,y)\right\|_4 +\left\| D_{r,z} u(s,y')\right\|_4 ) \\
&\qquad\quad \qquad\qquad \times \Big( \| D_{r,z'} u(s,\tilde{y}) \|_4 + \| D_{r,z'} u(s,\tilde{y}') \|_4 \Big) |z-z'| ^{2H-2} dz dz'dr \\
& \le C \int_0^s \int_{{\mathbb R}^2 } \Big( \mathbf{1}_{\{ |y-z| \le s-r\}} + \mathbf{1}_{\{ |y'-z| \le s-r\}} \Big) \\
& \qquad\qquad\qquad\qquad \times \Big( \mathbf{1}_{\{ |\tilde{y}-z'| \le s-r\}} + \mathbf{1}_{\{ |\tilde{y}'-z'| \le s-r\}} \Big) |z-z'| ^{2H-2} dz dz'dr .
\end{aligned}$$ So the spatial integral in the expression of $A_1$ can be bounded by $$\begin{aligned}
&\quad \mathbf{J}:=C\int_0^s\int_{{\mathbb R}^6} \varphi_R(s,y)\varphi_R(s,y')
\varphi_R(s, \tilde{y})\varphi_R(s, \tilde{y}') \\
& \quad\quad\quad\times |y-y'| ^{2H-2} |\tilde{y}-\tilde{y}'| ^{2H-2} |z-z'| ^{2H-2} \Big( \mathbf{1}_{\{ |y-z| \le s-r\}} + \mathbf{1}_{\{ |y'-z| \le s-r\}} \Big) \\
&\qquad \qquad \times\Big( \mathbf{1}_{\{ |\tilde{y}-z'| \le s-r\}} + \mathbf{1}_{\{ |\tilde{y}'-z'| \le s-r\}} \Big) dy dy' d\tilde{y}d\tilde{y}' dz dz'dr \\
&= C\int_0^s \int_{[-R, R]^4} \int_{{\mathbb R}^6} \mathbf{1}_{\{ \vert x - y \vert \vee \vert x' - y' \vert \vee \vert \tilde{x} - \tilde{y} \vert \vee \vert \tilde{x}' - \tilde{y}' \vert \leq t-s \} } \\
& \quad\quad\quad\times |y-y'| ^{2H-2} |\tilde{y}-\tilde{y}'| ^{2H-2} |z-z'| ^{2H-2} \Big( \mathbf{1}_{\{ |y-z| \le s-r\}} + \mathbf{1}_{\{ |y'-z| \le s-r\}} \Big) \\
&\qquad \qquad \times\Big( \mathbf{1}_{\{ |\tilde{y}-z'| \le s-r\}} + \mathbf{1}_{\{ |\tilde{y}'-z'| \le s-r\}} \Big) dx dx' d\tilde{x} d\tilde{x}' dy dy' d\tilde{y}d\tilde{y}' dz dz'dr \\
&\leq 4Ct \int_{[-R, R]^4} \int_{{\mathbb R}^6} \mathbf{1}_{\{ \vert x - y \vert \vee \vert x' - y' \vert \vee \vert \tilde{x} - \tilde{y} \vert \vee \vert \tilde{x}' - \tilde{y}' \vert \leq t \} } \mathbf{1}_{\{ |y-z| \le t\}} \mathbf{1}_{\{ |\tilde{y}-z'| \le t\}} \\
& \quad\quad\qquad\times |y-y'| ^{2H-2} |\tilde{y}-\tilde{y}'| ^{2H-2} |z-z'| ^{2H-2} dx dx' d\tilde{x} d\tilde{x}' dy dy' d\tilde{y}d\tilde{y}' dz dz' \,,\end{aligned}$$ due to symmetry. Then, it follows from the exactly the same argument as in the estimation of $\mathbf{I}$ in the previous step that $
\mathbf{J} $ is bounded by $C R^{6H-2}$ for $R$ big enough. This gives us the desired estimate for $A_1$ and finishes the proof.
Proof of Theorem \[thm:functional-CLT\] {#sec:thm2}
=======================================
We begin with the following result that ensures tightness.
\[pro:tightness\] Let $u(t,x)$ be the solution to equation . Then for any $0\leq s < t\leq T$ and any $p\ge 2$, there exists a constant $C_{p,T}$, depending on $T$ and $p$, such that for any $R\geq T$, $$\begin{aligned}
\label{tendue}
{\mathbb E}\left(~ \left|\int_{-R}^R u(t,x)dx - \int_{-R}^Ru(s,x)dx\right|^p ~ \right) \leq C_{p,T}R^{pH}(t-s)^{p}~ .\end{aligned}$$
Let us assume that $s < t$. We can write $$\begin{aligned}
2\int_{-R}^R \big[ u(t,x) - u(s,x)\big]~ dx =
\int_0^T \int_{\mathbb R}(\varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) )\sigma(u(r,y))W(dr,dy),\end{aligned}$$ where $
\varphi_{t,R}(r,y) = \mathbf{1}_{\{r\le t\}} \int_{-R} ^R \mathbf{1} _{\{ |x-y| \le t-r\}} dx .
$ The rest of our proof consists of two parts.
[*Step 1:*]{} Suppose that $H= 1/2$. Using Burkholder-Davis-Gundy inequality and Minkowski’s inequality, we get, for some absolute constant $c_p\in(0,+\infty)$, $$\begin{aligned}
&\quad {\mathbb E}\left(\left|\int_{-R}^R u(t,x)dx - \int_{-R}^Ru(s,x)dx\right|^p \right)\\
& \leq c_p {\mathbb E}\left[ \left( \int_0^T \int_{{\mathbb R}} \Big( \varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \Big) ^2 \sigma^2\big(u(r,y)\big) \, dy dr\right)^{p/2} ~\right]\\
& \leq c_p \left(\int_0^T\int_{{\mathbb R}} \Big( \varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \Big)^2 \big\|\sigma(u(r,y)) \big\|_p^2 ~dy dr\right)^{p/2}\\
& \leq c_p K^p_p(T) \left(\int_0^T\int_{{\mathbb R}} \Big( \varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \Big)^2 dy
dr\right)^{p/2}\,,
\end{aligned}$$ where $K_p(T)$ has been defined in (\[Kp\]). Now we notice that $$\begin{aligned}
&\quad \big\vert \varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big\vert \notag \\
&\leq \mathbf{1}_{\{r\le s\}} \int_{-R} ^R \left|\mathbf{1}_{\{|x-y| \le t-r\}}
- \mathbf{1}_{\{|x-y| \le s-r\}} \right| dx +
\mathbf{1}_{\{s< r \le t \}}\int_{-R} ^R \mathbf{1}_{\{|x-y| \le t-r\}}dx \notag \\
&\le 2\Big( t-s + (t-r)\mathbf{1}_{\{s< r \le t \}} \Big) \mathbf{1}_{ \{ |y| \le R+t\}} \leq 4(t-s) \mathbf{1}_{ \{ |y| \le R+t\}} \,. \label{est11}
\end{aligned}$$ This implies for $R\geq T$, $$\int_0^T\int_{{\mathbb R}} \big( \varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big)^2 dyds \le 64T R(t-s)^2 ,~\text{and thus establishes \eqref{tendue}.}$$
[*Step 2:*]{} Suppose that $H\in ( 1/2,1)$. In the same way, we write $$\begin{aligned}
\notag
&\quad {\mathbb E}\left(\left|\int_{-R}^R u(t,x)dx - \int_{-R}^Ru(s,x)dx\right|^p \right)\\
& \leq c_p \label{a2}
{\mathbb E}\Bigg[ \Big( \int_0^T \left\| \big(\varphi_{t,R}(r,\cdot) - \varphi_{s,R}(r,\cdot) \big)\sigma\big(u(r,\cdot)\big) \right\|_{{\mathfrak H}_0}^2 dr\Bigg)^{p/2}\Bigg].
\end{aligned}$$ As mentioned in Section \[sec:prel\], for $H\in(1/2, 1)$, the space $L^{1/H}({\mathbb R})$ is continuously embedded into $\mathfrak{H}_0$. Consequently, there is a constant $C_H>0$, depending on $H$, such that $$\begin{aligned}
\notag
& \quad \left\| \big(\varphi_{t,R}(r,\cdot) - \varphi_{s,R}(r,\cdot) \big)\sigma\big(u(r,\cdot)\big) \right\|_{{\mathfrak H}_0}^2 \\ \label{a1}
& \le C_H \left( \int_{{\mathbb R}} \big|\varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big| ^{1/H} |\sigma\big(u(r,y)\big) |^{1/H} dy \right)^{2H}.
\end{aligned}$$ Substituting (\[a1\]) into (\[a2\]) and applying Hölder’s and Minkowski’s inequalities, we can write $$\begin{aligned}
&\quad {\mathbb E}\left(\left|\int_{-R}^R u(t,x)dx - \int_{-R}^Ru(s,x)dx\right|^p \right)\\
& \le c_p C_H^{p/2} T^{p/2-1}
\int_0^T {\mathbb E}\left[ \left( \int_{{\mathbb R}} \big|\varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big| ^{1/H} |\sigma\big(u(r,y)\big) |^{1/H} dy \right)^{pH} \right]dr\\
& \le c_p C_H^{p/2} T^{p/2-1}
\int_0^T \left( \int_{{\mathbb R}} \big|\varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big| ^{1/H} \|\sigma\big(u(r,y)\big) \|_{p} ^{1/H} dy \right)^{pH} dr\\
&\le c_p C_H^{p/2} T^{p/2-1} K^p_p(T)
\int_0^T \left( \int_{{\mathbb R}} \big|\varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big| ^{1/H} dy \right)^{pH} dr.
\end{aligned}$$ Finally, from , which holds true for any $R\geq T$, we can write $$\left( \int_{{\mathbb R}} \big|\varphi_{t,R}(r,y) - \varphi_{s,R}(r,y) \big| ^{1/H} dy \right)^{pH}
\le 4^{p(1+H)} (t-s)^p R^{pH}.$$ It is then straightforward to get .
We need to prove tightness and the convergence of the finite-dimensional distributions. Notice that tightness follows from Proposition \[pro:tightness\] and the well-known criterion of Kolmogorov.
Let us now show the convergence of the finite-dimensional distributions. We fix $0\le t_1< \cdots <t_m \le T$ and consider $$F_R(t_i) := \frac{1}{R^H} \left(\int_{-R}^R u(t_i,x)dx - 2R\right) = \delta\big( v_R^{(i)}\big) ~\text{for $i=1, \dots, m$,}$$ where $$v_R^{(i)} (s,y) = \mathbf{1} _{[0,t_i]}(s) \frac{\sigma\big(u(s,y)\big)}{ R^H }\varphi^{(i)}_R(s,y) ~ \text{with}~ \varphi^{(i)}_R(s,y) = \frac{1}{2} \int_{-R}^R \mathbf{1}_{\{|x-y| \le t_i -s\}} \, dx \,.$$ Set $\mathbf{F}_R=\big( F_R(t_1), \dots, F_R(t_m) \big)$ and let $Z$ be a centered Gaussian vector on ${\mathbb R}^m$ with covariance $(C_{i,j})_{1\leq i,j\leq m}$ given by $$C_{i,j} := \begin{cases} {\displaystyle 2 \int _0^{t_i \wedge t_j} (t_i-r) (t_j-r) \xi(r) ~dr}, \,\,\,\quad\quad\text{if $H= 1/2$;}\\
{\displaystyle 2^{2H} \int _0^{t_i \wedge t_j} (t_i-r) (t_j-r) \eta^2(r) ~dr}, \quad\text{if $H\in (1/2,1)$. }
\end{cases}$$ We recall here that $\xi(r)= {\mathbb E}\big[ \sigma^2\big(u(r,y)\big)\big]$ and $\eta(r)= {\mathbb E}\big[\sigma\big(u(r,y)\big)\big]$. Then, we need to show $\mathbf{F}_R$ converges in distribution to $Z$ and in view of Lemma \[lemma: NP 6.1.2\], it suffices to show that for each $i,j$, $\langle DF_R(t_i), v_R^{(j)} \rangle_{{\mathfrak H}}$ converges to $C_{i,j} $ in $L^2(\Omega)$, as $R\to+\infty$. The case $i=j$ has been tackled before and the other case can be dealt with by using arguments similar to those in the proof of Theorem \[thm:TV-distance\]. For the convenience of readers, we only sketch these arguments as follows.
We consider two cases: $H=1/2$ and $H\in(1/2, 1)$. In each case, we need to show (i) ${\mathbb E}\big[ F_R(t_i) F_R(t_j) \big] \to C_{i,j}$ and (ii) ${\rm Var}\big( \langle DF_R(t_i), v_R^{(j)} \rangle_{{\mathfrak H}} \big)\to 0$, as $R\to+\infty$. Point (i) has been established in [*Remark*]{} \[rem2\]. To see point (ii) for the case $H=1/2$, we begin with the decomposition $\langle DF_R(t_i), v_R^{(j)} \rangle_{{\mathfrak H}} = B_1(i,j) +B_2(i,j) $ with $$\begin{aligned}
B_1(i,j): = \frac{1}{R} \int_0^{t_i\wedge t_j} \int_{{\mathbb R}} \varphi_R^{(i)}(s,y) \varphi_R^{(j)}(s,y) \sigma ^2(u(s,y)) \,ds\, dy \end{aligned}$$ and $$\begin{aligned}
B_2(i,j):&= \frac { 1 }{R} \int_0^{t_i\wedge t_j} \int_{{\mathbb R}} \varphi^{(j)}_R(s,y) \sigma(u(s,y)) \\
&\qquad \times\left( \int_s^{t_i}\int_{\mathbb R}\varphi^{(i)}_R(r,z) \Sigma(r,z) D_{s,y} u(r,z) W(dr,dz) \right) \, dyds.\end{aligned}$$ Then using and going through the same lines as for the estimation of $A_1, A_2$, we can get $$\begin{aligned}
&\quad \sqrt{{\rm Var}\big( B_2(i,j) \big) } \leq \frac{1}{R} \int_0^{t_i\wedge t_j} ds\Bigg( \int_{{\mathbb R}^3} \int_s^{t_i} \varphi^{(i)}_R(r,z)^2 \varphi^{(j)}_R(s,y) \varphi^{(j)}_R(s,y') \\
&\qquad \times
{\mathbb E}\Big[ \Sigma^2(r,z) D_{s,y} u(r,z) D_{s,y'} u(r,z)
\sigma\big(u(s,y)\big)\sigma\big(u(s,y')\big) \Big] \, dydy' dz dr
\Bigg)^{ 1/2} \\
&\leq \frac{C} {R} \int_0^{t_i\wedge t_j} \Bigg( \int_{{\mathbb R}^3} \int_s^{t_i} \varphi_R^{(i)}(r,z)^2 \varphi^{(j)}_R(s,y) \varphi^{(j)}_R(s,y') \\
&\qquad \qquad\qquad\qquad\times
\mathbf{1}_{\{|y-z| \vee |y' -z| \le r-s\}} dydy' dz dr
\Bigg)^{ 1/2} ds \leq \frac{C} {\sqrt{R}}.
\end{aligned}$$ That is, we have ${\rm Var}\big( B_2(i,j) \big) \to 0$, as $R\to+\infty$. We can also get $$\begin{aligned}
&\quad \sqrt{{\rm Var}\big( B_1(i,j) \big) } \leq \frac{1}{R} \int_0^{t_i\wedge t_j} \Bigg( \int_{{\mathbb R}^2} \varphi^{(i)}_R(s,y) \varphi^{(j)}_R(s,y) \varphi^{(i)}_R(s,y')\varphi^{(j)}_R(s,y')\\
&\qquad\qquad\qquad\qquad\qquad \qquad \quad \times {\rm Cov} \Big[ \sigma^2 \big(u(s,y)\big) , \sigma ^2\big(u(s, y')\big) \Big] dydy' \Bigg)^{1/2} ds\\
&\leq \frac{C}{R} \int_0^{t_i\wedge t_j} \Bigg( \int_{{\mathbb R}^2} \varphi^{(i)}_R(s,y) \varphi^{(j)}_R(s,y) \varphi^{(i)}_R(s,y')\varphi^{(j)}_R(s,y') \mathbf{1}_{\{ \vert y - y' \vert \leq 2s \}} dydy' \Bigg)^{1/2} ds \\
&\leq \frac{C}{R} \int_0^{t_i\wedge t_j} \Bigg( \int_{{\mathbb R}^2} (t_i + t_j)^4 \mathbf{1}_{\{ \vert y\vert \vee \vert y'\vert \leq R + t_i + t_j \}} \mathbf{1}_{\{ \vert y - y' \vert \leq 2s \}} dydy' \Bigg)^{1/2} ds \leq \frac{C}{\sqrt{R}} \,.
\end{aligned}$$ That is, we have ${\rm Var}\big( B_1(i,j) \big) \to 0$, as $R\to+\infty$.
To see point (ii) for the case $H\in(1/2, 1)$, one can begin with the same decomposition and then use to arrive at similar estimations as those for $\mathbf{I}$ and $\mathbf{J}$. Therefore the same arguments ensure $
{\rm Var}\big(\langle DF_R(t_i), v_R^{(j)} \rangle_{{\mathfrak H}}\big) \leq CR^{2H-2}
$. Now the proof of Theorem \[thm:functional-CLT\] is completed.
Appendix: Proof of Lemma \[lemma: iteration\]
==============================================
This appendix provides the proof of our technical Lemma and it consists of two parts. The first part proceeds assuming $$\begin{aligned}
\label{claimbdd}
\mathfrak{L}:= \sup_{(r,z)\in [0,t]\times {\mathbb R}} \| D_{s,y} u(r, z) \|_p < +\infty ~\text{for almost every $(s,y)\in{\mathbb R}_+\times{\mathbb R}$}
\end{aligned}$$ and the second part is devoted to establishing the above bound. Note that *a priori*, we do not know whether $D_{s,y}u(r,z)$ is a function of $(s,y)$ or not in the case where $H\in(1/2, 1)$, so the assumption also guarantees that $D_{s,y}u(r,z)$ is indeed a random function in $(s,y)$; see Section \[Sec52\] for more explanation.
Proof of Lemma \[lemma: iteration\] assuming
---------------------------------------------
The proof will be done in two steps.
[*Step 1: Case $H= 1/2$*]{}. From (\[ecu1\]), using Burkholder’s and Minkowski’s inequality, we can write $$\begin{aligned}
&\quad \| D_{s,y}u(t,x) \|_p \\
& \leq \frac{ K_{p}(t)}{2} \mathbf{1}_{\{ |x-y| \le t-s\}} + \frac{Lc_p}{2} \left( \int_s^t \int_{{\mathbb R}} \mathbf{1}_{\{ |x-z| \le t-r \}} \| D_{s,y} u(r,z) \|_p^2 drdz \right)^{1/2}\end{aligned}$$ with $c_p$ a constant that only depends on $p$. It follows from the elementary inequality $(a+b)^2\leq 2a^2+2b^2$ that $$\begin{aligned}
\| D_{s,y}u(t,x) \|_p^2 \le \frac{K^2_{p}(t)}{2} \mathbf{1}_{\{ |x-y| \le t-s\}} + \frac{ L^2c^2_p}{2} \int_s^t \int_{{\mathbb R}} \mathbf{1}_{\{ |x-z| \le t-r \}} \| D_{s,y} u(r,z) \|^2_p drdz.\end{aligned}$$ Iterating this inequality yields, for any positive integer $M$, $$\begin{aligned}
&\quad \| D_{s,y}u(t,x) \|_p^2 \label{tt1} \leq \frac { K^2_{p}(t) } 2 \mathbf{1}_{\{ |x-y| \le t-s\}} \notag \\
& +\frac { K^2_{p}(t) } 2 \sum_{N=1}^M \frac { c_p^{2N} L^{2N}} {2^N} \int_{\Delta_N(s,t)} \int_{{\mathbb R}^{N}}
\left(\prod_{n=1} ^{N} \mathbf{1}_{\{ |z_{n-1}-z_n| \le r_{n-1}-r_n \}} \right) \mathbf{1}_{\{ |z_{N}-y| \le r_{N}-s \}} d{\bf r}d{\bf z} \notag \\
& + \frac { c_p^{2M+2} L^{2M+2}} {2^{M+1}} \int_{\Delta_{M+1}(s,t)} \int_{{\mathbb R}^{M+1}}
\left(\prod_{n=1} ^{M} \mathbf{1}_{\{ |z_{n-1}-z_n| \le r_{n-1}-r_n \}} \right) \mathbf{1}_{\{ |z_{M}- z_{M+1}| \le r_{M}- r_{M+1} \}} \notag \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \times \| D_{s,y} u(r_{M+1}, z_{M+1}) \|^p_2\, d{\bf r}d{\bf z} \,, \notag\end{aligned}$$ where $\Delta_N(s,t) := \big\{ ( r_1, \dots, r_N)\in{\mathbb R}^N\vert s<r_N < r_{N-1} < \cdots <r_1<t\big\}$, $d{\bf r}= dr_1 \cdots dr_{N}$, $d{\bf z} = dz_1 \cdots dz_{N}$ and with the convention $r_0=t$ and $z_{0}=x$.
Notice that if $$\left(\prod_{n=1} ^{N} \mathbf{1}_{\{ |z_{n-1}-z_n| \le r_{n-1}-r_n \}} \right) \mathbf{1}_{\{ |z_{N}-y| \le r_{N}-s \}} \not =0,$$ then on $\Delta_N(s,t)$, $
|x-y| = |z_{0} -y | \le \sum_{n=1} ^{N} |z_{n-1} -z_n| + |z_N-y|\le t-s
$ and similarly on $\Delta_N(s,t)$, $| z_n-y| \le t-s$ for[^3] $n=1, \dots, N$.
Now we deduce from that $$\begin{aligned}
\| D_{s,y}u(t,x) \|_p^2 &\le& \mathbf{1}_{\{ |x-y| \le t-s\}} \frac { K^2_{p}(t) }2 \left(
1 + \sum_{N=1} ^\infty c_p^{2N} L^{2N}
\frac {(t-s)^{2N} }{N!} \right)\\
&\le & \mathbf{1}_{\{ |x-y| \le t-s\}} \frac { K^2_{p}(t) }2 \exp \big( c_p^2 L^2 t^2\big),\end{aligned}$$ which provides the desired estimate.
[*Step 2: Case $H\in( 1/2, 1)$*]{}. Proceeding as before, and using the inequality $$\| D_{s,y} u(r,z)D_{s,y} u(r,\tilde{z}) \|_{p/2}
\le \frac 12 \left( \| D_{s,y} u(r,z) \|_p^2 + \| D_{s,y} u(r,\tilde{z}) \|_p^2 \right),$$ we obtain $$\begin{aligned}
&\| D_{s,y}u(t,x) \|_p^2 \le \frac{ K^2_{p}(t) } 2 \mathbf{1}_{\{ |x-y| \le t-s\}} \\
& \quad + \frac { c^2_p L^2} 2 \alpha_H \int_s^t \int_{{\mathbb R}^2} \mathbf{1}_{\{ |x-z| \le t-r \}}
\mathbf{1}_{\{ |x-\tilde{z}| \le t-r \}} \| D_{s,y} u(r,z) \|^2_p |z- \tilde{z} | ^{2H-2} drdz d\tilde{z} .\end{aligned}$$ By iteration, this leads to the following estimate. For any positive integer $M$, $$\begin{aligned}
&\| D_{s,y}u(t,x) \|_p^2 \le \frac { K^2_{p}(t) } 2 \mathbf{1}_{\{ |x-y| \le t-s\}} + \frac { K^2_{p}(t) } 2 \sum_{N=1} ^M \frac { c_p^{2N} L^{2N}} {2^N} \int_{\Delta_N(s,t)} \alpha_H^N \int_{{\mathbb R}^{2N}} \\
& \quad
\left(\prod_{n=1} ^{N}
\mathbf{1}_{\{ |z_{n-1}-\tilde{z}_n| \vee|z_{n-1}-z_n| \le r_{n-1}-r_n \}} |z_n- \tilde{z}_n | ^{2H-2} \right) \mathbf{1}_{\{ |z_{N}-y| \le r_{N}-s \}} d{\bf r}d{\bf z}d{\bf \tilde{z}} \\
&+ \frac{(Lc_p)^{2M+2}}{2^{M+1}} \int_{\Delta_{M+1}(s,t)} d\mathbf{r} ~ \alpha_H^{M+1} \int_{{\mathbb R}^{2M+2}} d\mathbf{z}d\tilde{\mathbf{z}} \\
& \times \left( \prod_{n=1} ^{M+1}
\mathbf{1}_{\{ |z_{n-1}-\tilde{z}_n| \vee|z_{n-1}-z_n| \le r_{n-1}-r_n \}} |z_n- \tilde{z}_n | ^{2H-2} \right)\| D_{s,y} u(r_{M+1}, z_{M+1})\|_p^2\end{aligned}$$ with the same convention as before. Note that Lemma \[lem1\] implies that on $\Delta_N(s,t)$, $$\begin{aligned}
& \alpha_H \int_{{\mathbb R}^2} \mathbf{1}_{\{ \vert z_{n-1} - z_{n} \vert \vee \vert z_{n-1} - \tilde{z}_{n} \vert \leq r_{n-1} - r_{n} \}} \vert z_{n} - \tilde{z}_n \vert^{2H-2} dz_{n}d\tilde{z}_n \notag \\
&\leq \alpha_H \int_{{\mathbb R}^2} \mathbf{1}_{\{ \vert z_{n-1} - z \vert \vee \vert z_{n-1} - z' \vert \leq t \}} \vert z - z' \vert^{2H-2} dzdz' \leq 4^H t^{2H} \,; \label{ele3}
\end{aligned}$$ and note that we again have the following implication: $$\prod_{n=1} ^{N} \mathbf{1}_{\{ |z_{n-1}-z_n| \le r_{n-1}-r_n \}} \mathbf{1}_{\{ |z_{N}-y| \le r_{N}-s \}} \neq0 \Longrightarrow \vert x - y\vert \leq t -s \,,$$ which, together with , implies $$\begin{aligned}
\| D_{s,y}u(t,x) \|_p^2& \le \frac { K^2_{p}(t) } 2 \mathbf{1}_{\{ |x-y| \le t-s\}} + \frac { K^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s\}} \sum_{N=1} ^M \frac { (Lc_p 2^H t^{H})^{2N} } {2^N} \frac{t^N}{N!} \\
& \qquad + \mathfrak{L}^2 ~ \frac { (Lc_p 2^H t^{H})^{2M+2} } {2^{M+1}} \frac{t^{M+1}}{(M+1)!} \qquad \text{($\mathfrak{L}$ is defined in \eqref{claimbdd})} \,.\end{aligned}$$ Letting $M\to+\infty$ leads to $$\begin{aligned}
\| D_{s,y}u(t,x) \|_p^2 & \leq \frac{ K^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s\}} + \frac { K^2_{p}(t) }{ 2}\mathbf{1}_{\{ |x-y| \le t-s\}} \sum_{N=1} ^\infty \frac { (Lc_p 2^H t^{H})^{2N} } {2^N} \frac{t^N}{N!} \\
&\leq \frac { K^2_{p}(t) }{2}\exp\big( 2L^2c_p^2 t^{2H+1} \big) \mathbf{1}_{\{ |x-y| \le t-s\}} \,.
\end{aligned}$$ This concludes our proof of Lemma \[lemma: iteration\] assuming .
Proof of {#Sec52}
---------
The proof will be done in two steps.
[*Step 1: Case $H=1/2$*]{}. It is well known in the literature that for any $p\ge 2$, $u(t,x)\in\mathbb{D}^{1,p}$ and $$\begin{aligned}
\label{shallimp}
\sup_{(t,x)\in[0,T]\times {\mathbb R}} {\mathbb E}\big[ \| Du(t,x)\| _{\mathfrak{H}}^p \big] < +\infty \,;\end{aligned}$$ indeed, in the Picard iteration scheme (see *e.g.* ), one can first prove the iteration $u_n$ converges to the solution $u$ in $L^p(\Omega)$ uniformly in $[0,T]\times {\mathbb R}$, then we derive the uniform bounded for ${\mathbb E}\big[ \| Du_n(t,x) \| _\mathfrak{H}^p\big]$, so that by standard Malliavin calculus argument, we can get the convergence of $Du_n(t,x)$ to $Du(t,x)$ with respect to the weak topology on $L^p(\Omega ; \mathfrak{H})$ and hence the desired uniform bound . We omit the details for this case ($H=1/2$) and refer to the arguments for the other case ($H>1/2$).
Consider an approximation of the identity $\big( M_\varepsilon,\varepsilon > 0\big) $ in $L^1({\mathbb R}_+\times{\mathbb R})$ satisfying $M_\varepsilon(s,y) = \varepsilon^{-2} M(s/\varepsilon,y/\varepsilon)$ for some nonnegative $M\in C_c({\mathbb R}_+\times{\mathbb R})$. Taking into account that $(\omega, s,y) \to D_{s,y}u(t,x)$ belongs to $L^2( {\mathbb R}_+\times{\mathbb R}; L^2(\Omega))$, we deduce that the convolution $Du(t,x) * M_\varepsilon$ converges to $ Du(t,x)$ in $L^2( {\mathbb R}_+\times{\mathbb R}; L^2(\Omega))$, as $\varepsilon$ tends to zero. Therefore, there exist a sequence $\{\varepsilon_n\}$ such that $\varepsilon_n \downarrow 0$ and $(Du(t,x) * M_{\varepsilon_n})(s,y)$ converges almost surely to $D_{s,y}u(t,x)$ for almost all $(s,y) \in {\mathbb R}_+\times {\mathbb R}$, as $n\to+\infty$. By Fatou’s Lemma, this implies that for almost all $(s,y) \in {\mathbb R}_+\times{\mathbb R}$, $$\| D_{s,y} u(t,x) \| _p \le \sup_{n\in{\mathbb N}} \big\| \big(Du(t,x) \ast M_{\varepsilon_{n}} \big)(s,y) \big\| _p \label{further0}.$$
Now we fix $(s,y)$ that satisfies and put for $\varepsilon > 0$ $$\begin{aligned}
Q_\varepsilon(t) :& = \sup_{z\in{\mathbb R}} \big\| \big(Du(t,z) \ast M_{\varepsilon} \big)(s,y) \big\| _p^2 \label{Q(t)}\\
&= \sup_{z\in{\mathbb R}} \left\| \int_{{\mathbb R}_+\times {\mathbb R}} D_{s',y'} u(t, z) M_{\varepsilon}(s'-s, y'-y) \, ds'dy' \right\|_p^2\,,~ t\in [0,T]. \notag
\end{aligned}$$ In the following,
1. we will prove for each $\varepsilon > 0$, $Q_\varepsilon$ is uniformly bounded on $[0,T]$;
2. we will obtain an integral inequality for $Q_\varepsilon$;
3. we will conclude with the classic Gronwall’s lemma.
Recall from and we can write $$\begin{aligned}
&\big(Du(t,z) \ast M_{\varepsilon} \big)(s,y) = \frac{1}{2} \int_{{\mathbb R}_+\times {\mathbb R}}{\bf 1}_{\{ | z - y' | \leq t -s' \}} \sigma\big(u(s',y') \big)M_{\varepsilon}(s'-s, y'-y) ds'dy' \\
&\qquad\qquad + \frac{1}{2} \int_0^t\int_{\mathbb R}{\bf 1}_{\{ | z - \xi | \leq t -a \}}\Sigma(a,\xi) \big( Du(a,\xi)\ast M_{\varepsilon}\big)(s,y) W(da,d\xi) .
\end{aligned}$$ Then, using Burkholder’s inequality and Minkowski’s inequality in the same way as before, we can arrive at $$\begin{aligned}
Q_\varepsilon(t) & \leq \frac{K_p^2(t)}{2} \left(\sup_{z\in{\mathbb R}} \int_{{\mathbb R}_+\times {\mathbb R}} \mathbf{1}_{\{ \vert z - y'\vert \leq t-s' \}} M_{\varepsilon}(s'-s, y'-y) \, ds'dy' \right)^2 \\
&\qquad + L^2c_p^2 t \int_0^t Q_\varepsilon(a)\, da \\
&\leq \frac{K_p^2(t)}{2} \| M \| _{L^1({\mathbb R}_+\times{\mathbb R})} ^2 + L^2c_p^2 t \int_0^t Q_\varepsilon(a)\, da, ~ t\in[0,T].
\end{aligned}$$ We know from and Cauchy-Schwarz inequality that $$Q_\varepsilon(t) \leq \| M_\varepsilon\|_\mathfrak{H}^2 \sup_{z\in{\mathbb R}} \Big( {\mathbb E}\big[ \| Du(t,z)\|_\mathfrak{H}^p \big] \Big)^{2/p} \,,$$ which is uniformly bounded on $[0,T]$. Then it follows from Gronwall’s lemma that $$Q_\varepsilon(t) \leq \frac{K_p^2(t)}{2}e^{L^2c_p^2 t} \| M \| _{L^1({\mathbb R}_+\times{\mathbb R})} ^2, ~ \forall t\in[0,T].$$ The above bound is independent of $\varepsilon$, thus we can further deduce that $$\sup_{(r,z)\in[0,t]\times{\mathbb R}}\| D_{s,y} u(r, z) \|_p^2 \leq \frac{K_p^2(t)}{2}e^{L^2c_p^2 t} \| M \| _{L^1({\mathbb R}_+\times{\mathbb R})} ^2 < +\infty \,.$$ That is, claim is established for the case $H=1/2$.\
[*Step 2: Case $H\in(1/2,1)$*]{}. In this case we have first to show that $Du(t,x)$ is an element of $L^2( \Omega \times {\mathbb R}_+ \times {\mathbb R})$ and for this we will use the Picard iterations. Let $u_0(t,x) =1$ and for $n\ge 0$, set $$\begin{aligned}
u_{n+1}(t,x) = 1 + \frac{1}{2} \int_0^t \int_{\mathbb R}\mathbf{1}_{\{ \vert x - y \vert \leq t-s\}} \sigma\big( u_n(s,y) \big) \, W(ds, dy) \,. \label{Picardit}
\end{aligned}$$ It is routine to show that for any given $T\in{\mathbb R}_+$, $$\begin{aligned}
\label{UNILP}
\lim_{n\to+\infty} \sup_{(t,x)\in[0,T]\times{\mathbb R}} \| u(t,x) - u_{n}(t,x) \|_p =0 \,.
\end{aligned}$$ We know that for each $n\geq 0$, $u_n(t,x)\in \mathbb{D}^{1,p}$ with $$\begin{aligned}
&\quad D_{s,y}u_{n+1}(t,x)= \frac{1}{2} \mathbf{1}_{\{ \vert x - y \vert \leq t - s \} } \sigma\big( u_{n}(s,y) \big) \\
&\qquad\qquad\qquad\qquad \qquad + \frac{1}{2} \int_s^t \int_{\mathbb R}\mathbf{1}_{\{ \vert x - z \vert \leq t - r \} } \Sigma_n(r,z) D_{s,y} u_n(r,z) \, W(dr, dz) \,,\end{aligned}$$ with $\Sigma_n(r,z) $ being an adapted process bounded by $L$. Thus, using Burkholder’s inequality, Minkowski’s inequality and the easy inequality $$\begin{aligned}
\| XY\|_{p/2} \leq \frac{1}{2} \| X \| _p^2 +\frac{1}{2} \| Y \| _p^2 \label{ele2}
\end{aligned}$$ for any $X, Y\in L^p(\Omega)$, we get $ \| D_{s,y}u_{n+1}(t,x) \|_p^2 $ bounded by $$\begin{aligned}
\frac{ \widetilde{K}^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s\}} + \frac{L^2c_p^2}{2} \int_s^t \alpha_H \int_{{\mathbb R}^2} & \mathbf{1}_{\{ |x-z| \vee \vert x - z' \vert \le t-r \}} \vert z- z'\vert^{2H-2} \\
&\qquad\qquad\qquad \times \| D_{s,y} u_n(r,z) \|_p^2 ~ drdzdz' \,,\end{aligned}$$ where $\widetilde{K}_p(t):=\sup\big\{ \| \sigma ( u_n(s,x) ) \| _p \,: n\geq 0, (s,x)\in[0, t]\times {\mathbb R}\big\} $. Iterating this procedure gives us $$\begin{aligned}
& \| D_{s,y}u_{n+1}(t,x) \|_p^2 \leq \frac { \widetilde{K}^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s\}} +\frac {\widetilde{K}^2_{p}(t) }{2} \sum_{\ell=1}^n \frac { c_p^{2\ell} L^{2\ell}} {2^\ell} \int_{\Delta_\ell(s,t)} d{\bf r} \alpha_H^\ell \int_{{\mathbb R}^{2\ell}} \\
& \quad\times
\left(\prod_{k=0} ^{\ell-1} \mathbf{1}_{\{ |z_{k}-z_{k+1}| \vee |z_k-z_{k+1}'| \le r_{k-1}-r_k \}} \vert z_{k+1} - z_{k+1}' \vert^{2H-2} \right) \mathbf{1}_{\{ |z_{\ell}-y| \le r_{\ell}-s \}} d{\bf z}' d{\bf z} \,.\end{aligned}$$ Again, it is easy to see the following implication holds: $$\mathbf{1}_{\{ |z_{\ell}-y| \le r_{\ell}-s \}} \prod_{k=0} ^{\ell-1} \mathbf{1}_{\{ |z_{k}-z_{k+1}| \vee |z_k-z_{k+1}'| \le r_{k-1}-r_k \}} \neq 0 \Longrightarrow \vert x - y\vert \leq t-s \,,$$ therefore $$\begin{aligned}
& \| D_{s,y}u_{n+1}(t,x) \|_p^2 \leq \frac { \widetilde{K}^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s\}} +\frac {\widetilde{K}^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s \}} \sum_{\ell=1}^n \frac { c_p^{2\ell} L^{2\ell}} {2^\ell} \int_{\Delta_\ell(s,t)} d{\bf r} \\
& \qquad\qquad \times \alpha_H^\ell \int_{{\mathbb R}^{2\ell}}
\left(\prod_{k=0} ^{\ell-1} \mathbf{1}_{\{ |z_{k}-z_{k+1}| \vee |z_k-z_{k+1}'| \le r_{k-1}-r_k \}} \vert z_{k+1} - z_{k+1}' \vert^{2H-2} \right) d{\bf z}' d{\bf z} \\
&\leq \frac { \widetilde{K}^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s\}} +\frac {\widetilde{K}^2_{p}(t) }{2} \mathbf{1}_{\{ |x-y| \le t-s \}} \sum_{\ell=1}^n \frac { c_p^{2\ell} L^{2\ell}} {2^\ell} (4^H t^{2H+1})^\ell \frac{1}{\ell!},\end{aligned}$$ where the last inequality is a consequence of . We conclude that $$\begin{aligned}
\| D_{s,y}u_{n+1}(t,x) \|_p^2 \leq \frac { \widetilde{K}^2_{p}(t) }{2} e^{ 2 t^{2H+1} c_p^2 L^2 } \mathbf{1}_{\{ |x-y| \le t-s\}} =: C \mathbf{1}_{\{ |x-y| \le t-s\}} . \label{DnBDD}
\end{aligned}$$ It follows immediately from Minskowski’s inequality and that $${\mathbb E}\big[ \| Du_n(t,x) \| _{L^2({\mathbb R}_+\times{\mathbb R})}^p \big] \leq \left( \int_{{\mathbb R}_+\times{\mathbb R}} \| D_{s,y} u_n(t,x) \| _p^2 \, dsdy \right)^{p/2} \leq \big( C t^2 \big)^{p/2}$$ [*uniformly*]{} in $n\geq 1$ and [*uniformly*]{} in $x\in{\mathbb R}$. In particular, $\big\{ Du_n(t,x) , n\geq 1 \big\}$ is uniformly bounded in $L^p(\Omega ; L^2({\mathbb R}_+\times{\mathbb R}))$. Note that the convergence in and standard Malliavin calculus arguments can lead us to the fact that up to some subsequence, $Du_n(t,x)$ converges to $Du(t,x)$ in the weak topology of $L^p(\Omega ; L^2({\mathbb R}_+\times{\mathbb R}))$, so we can conclude that $D_{s,y}u(t,x)$ is indeed a function in $(s,y)$ and for any fixed $T \in{\mathbb R}_+$, $$\sup_{(t,x)\in[0,T]\times{\mathbb R}} {\mathbb E}\big[ \| Du(t,x) \| _{L^2({\mathbb R}_+\times{\mathbb R})}^p \big] < +\infty \,.$$ Now we use the same approximation of the identity $( M_\varepsilon )$ and obtain for almost every $(s,y)\in{\mathbb R}_+\times{\mathbb R}$, $$\| D_{s,y} u(t,x) \| _p^2 \leq \sup_{\varepsilon > 0} \left\| \int_{{\mathbb R}_+\times {\mathbb R}} D_{s',y'} u(t, x) M_{\varepsilon}(s'-s, y'-y) \, ds'dy' \right\|_p^2 \,.$$ Let $\varepsilon > 0$ be fixed and let $Q_\varepsilon(t)$ be defined as in , we have in this case, applying Lemma \[lem1\], $$\begin{aligned}
Q_\varepsilon(t)
&\leq \frac{1}{2} K^2_p(t) \| M \|^2_{L^1({\mathbb R}_+\times{\mathbb R})} \\
& \quad + \frac{L^2c_p^2}{2} \int_0^t\alpha_H\int_{{\mathbb R}^2} \mathbf{1}_{\{ \vert x -z \vert\vee \vert x-z'\vert\leq t-r\}} \vert z-z'\vert^{2H-2} Q_\varepsilon(r)\, dr dzdz' \\
&\leq \frac{1}{2} K^2_p(t) \| M \|^2_{L^1({\mathbb R}_+\times{\mathbb R})} + \frac{L^2c_p^2 4^H t^{2H}}{2} \int_0^tQ_\varepsilon(r)\, dr.\end{aligned}$$ Similarly as in previous case, we have $$Q_\varepsilon(t) \leq \| M_\varepsilon \| _{L^2({\mathbb R}_+\times{\mathbb R})}^2 \Big( {\mathbb E}\big[ \| Du(t,x) \| _{L^2({\mathbb R}_+\times{\mathbb R})}^p \big] \Big)^{2/p}$$ so that the same application of Gronwall’s lemma gives $$\sup_{(r,z)\in[0,t]\times{\mathbb R}}\| D_{s,y} u(r, z) \|_p^2 \leq \frac{K_p^2(t)}{2}e^{2L^2c_p^2 t^{2H}} \| M \| _{L^1({\mathbb R}_+\times{\mathbb R})} ^2 < +\infty \,.$$ That is, claim is also established for the case $H\in(1/2,1)$.$\square$
[**Acknowledgement:**]{} We would like to thank two anonymous referees for their helpful remarks and in particular to one of them for detecting a mistake in the proof of Theorem \[thm:TV-distance\] and for providing a generous amount of comments that improve our paper.
[999]{}
L. Chen, Y. Hu and D. Nualart: Regularity and strict positivity of densities for the nonlinear stochastic heat equation. To appear in: *Mem. Amer. Math. Soc.* 2018.
L. Chen, D. Khoshnevisan, D. Nualart and F. Pu. Spatial ergodicity for SPDEs via Poincaré-type inequalities. (2019) arXiv:1907.11553
R. C. Dalang: Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.’s. *Electron. J. Probab.* Volume 4 (1999), paper no. 6, 29 pp.
R. C. Dalang: The Stochastic wave equation. In: Khoshnevisan D., Rassoul-Agha F. (eds) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol 1962. Springer, Berlin, Heidelberg (2009)
B. Gaveau and P. Trauber: L’intégrale stochastique comme opérateur de divergence dans l’espace founctionnel. [*J. Funct. Anal.*]{} 46 (1982), 230-238.
Y. Hu and D. Nualart: Renormalized self-intersection local time for fractional Brownian motion. [*Ann. Probab.*]{} 33 (2005), 948-983.
J. Huang, D. Nualart and L. Viitasaari: A central limit theorem for the stochastic heat equation. *arXiv preprint*, 2018
J. Mémin, Y. Mishura and E. Valkeila: Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. [*Stat. Probab. Lett.*]{} Volume 51, Issue 2, 2001, Pages 197-206
I. Nourdin and G. Peccati: Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge Tracts in Mathematics, 192. [*Cambridge University Press*]{}, Cambridge, 2012. xiv+239 pp.
D. Nualart: The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). [*Springer-Verlag, Berlin,*]{} 2006. xiv+382 pp.
D. Nualart and E. Nualart: *Introduction to Malliavin Calculus*. IMS Textbooks, Cambridge University Press, 2018.
D. Nualart and E. Pardoux: Stochastic calculus with anticipating integrands. [*Probab. Theory Re. Fields*]{} 78 (1988), 535-581.
D. Nualart and L. Quer-Sardanyons: Existence and smoothness of the density for spatially homogeneous SPDEs. [*Potential Analysis*]{} 27 (2007), 281-299.
D. Nualart and H. Zhou: total variation estimates in the Breuer-Major theorem. *arXiv preprint*, 2018
V. Pipiras and M.S. Taqqu: Integration questions related to fractional Brownian motion. *Probab Theory Relat Fields* (2000) Volume 118, Issue 2, pp 251–291.
M. Veraar: Correlation inequalities and applications to vector-valued Gaussian random variables and fractional Brownian motion. *Potential Anal.* 30 (2009), 341-370. J. B. Walsh: . In: École d’été de probabilités de Saint-Flour, XIV—1984, 265–439. Lecture Notes in Math. 1180, Springer, Berlin, 1986.
[^1]: D. Nualart is supported by NSF Grant DMS 1811181.
[^2]: To see the strict stationarity, we fix $y\in{\mathbb R}$ and put $v(t,x) = u(t,x+y)$: It is clear that $v$ solves the stochastic heat equation driven by the shifted noise $\{ W(t,x+y), t\in{\mathbb R}_+, x\in{\mathbb R}\} $, which has stationary increments in the spatial variable.
[^3]: This in particular implies that the contribution of the integration with respect to $dz_n$ is at most $2(t-s)$.
|
---
abstract: 'A visual-relational knowledge graph (KG) is a multi-relational graph whose entities are associated with images. We explore novel machine learning approaches for answering visual-relational queries in web-extracted knowledge graphs. To this end, we have created <span style="font-variant:small-caps;">ImageGraph</span>, a KG with 1,330 relation types, 14,870 entities, and 829,931 images crawled from the web. With visual-relational KGs such as <span style="font-variant:small-caps;">ImageGraph</span> one can introduce novel probabilistic query types in which images are treated as first-class citizens. Both the prediction of relations between unseen images as well as multi-relational image retrieval can be expressed with specific families of visual-relational queries. We introduce novel combinations of convolutional networks and knowledge graph embedding methods to answer such queries. We also explore a zero-shot learning scenario where an image of an entirely new entity is linked with multiple relations to entities of an existing KG. The resulting multi-relational grounding of unseen entity images into a knowledge graph serves as a semantic entity representation. We conduct experiments to demonstrate that the proposed methods can answer these visual-relational queries efficiently and accurately.'
author:
- |
Daniel Oñoro-Rubio daniel.onoro@neclab.eu\
Mathias Niepert mathias.niepert@neclab.eu\
Alberto García-Durán alberto.duran@neclab.eu\
Roberto González-Sánchez roberto.gonzalez@neclab.eu\
NEC Labs Europe Roberto J. López-Sastre robertoj.lopez@uah.es\
University of Alcalá
bibliography:
- 'image-graph\_bib.bib'
title: |
Answering Visual-Relational Queries in\
Web-Extracted Knowledge Graphs
---
Introduction
============
Numerous applications can be modeled with a knowledge graph representing entities with nodes, object attributes with node attributes, and relationships between entities by directed typed edges. For instance, a product recommendation system can be represented as a knowledge graph where nodes represent customers and products and where typed edges represent customer reviews and purchasing events. In the medical domain, there are several knowledge graphs that model diseases, symptoms, drugs, genes, and their interactions (cf. [@genonto; @WishartKGCSTGH:2008]). Increasingly, entities in these knowledge graphs are associated with visual data. For instance, in the online retail domain, there are product and advertising images and in the medical domain, there are patient-associated imaging data sets (MRIs, CTs, and so on). In addition, visual data is a large part of social networks and, in general, the world wide web.
Knowledge graphs facilitate the integration, organization, and retrieval of structured data and support various forms of search applications. In recent years KGs have been playing an increasingly crucial role in fields such as question answering [@Das_2017_CVPR], language modeling [@ahn2016], and text generation [@serban2016generating]. Even though there is a large body of work on constructing and maintaining KGs, the setting of visual-relational KGs, where entities are associated with visual data, has not received much attention. A visual-relational KG represents entities, relations between these entities, and a large number of images associated with the entities (see Figure \[fig:KG-sample\] for an example). While <span style="font-variant:small-caps;">ImageNet</span> [@imagenet_cvpr09] and the <span style="font-variant:small-caps;">VisualGenome</span> [@Krishna2016] datasets are based on KGs such as WordNet they are predominantly used as either an object classification data set as in the case of <span style="font-variant:small-caps;">ImageNet</span> or to facilitate scene understanding in a single image. With this work, we address the problem of reasoning about visual concepts across a large set of images organized in a knowledge graph. We want to explore to what extent web-extracted visual data can be used to enrich existing KGs so as to facilitate complex visual search applications going beyond basic image retrieval.
The core idea of our work is to treat images as first-class citizens both in KGs and visual-relational queries. The main objective of our work is to understand to what extent visual data associated with entities of a KG can be used in conjunction with deep learning methods to answer these visual-relational queries. Allowing images to be arguments of queries facilitates numerous novel query types. In Figure \[fig:KG-queries\] we list some of the query types we address in this paper. In order to answer these queries, we built on KG embedding methods as well as deep representation learning approaches for visual data. This allows us to answer these visual queries both accurately and efficiently.
There are numerous application domains that could benefit from query answering in visual KGs. For instance, in online retail, visual representations of novel products could be leveraged for zero-shot product recommendations. Crucially, instead of only being able to retrieve similar products, a visual-relational KG would support the prediction of product attributes and more specifically what attributes customers might be interested in. For instance, in the fashion industry visual attributes are crucial for product recommendations [@Liu_2016_CVPR; @veit2015]. Being able to ground novel visual concepts into an existing KG with attributes and various relation types is a reasonable approach to zero-shot learning.
We make the following contributions. First, we introduce <span style="font-variant:small-caps;">ImageGraph</span>, a visual-relational web-extracted KG with 1,330 relations where 829,931 images are associated with 14,870 different entities. Second, we introduce a new set of visual-relational query types. Third, we propose a novel set of neural architectures and objectives that we use for answering these novel query types. These query types generalize image retrieval and link prediction queries. This is the first time that deep CNNs and KG embedding learning objectives are combined into a joint model. Fourth, we show that the proposed class of deep neural networks are also successful for zero-shot learning, that is, creating relations between entirely unseen entities and the KG using only visual data at query time.
{width="100.00000%"}
Related Work
============
We discuss the relation of our contributions to previous work with an emphasis on relational learning, image retrieval, object detection, scene understanding, existing data sets, and zero-shot learning.
### Relational Learning {#relational-learning .unnumbered}
There has been a flurry of approaches tailored to specific problems such as link prediction in multi-relational graphs. Examples are knowledge base factorization and embedding approaches [@bordes2013translating; @nickel2011three; @guu2015traversing] and random-walk based ML models [@lao2011random; @gardner2015efficient]. More recently, the focus has been on integrating additional attribute types such as text [@Yahya2016_WSDM; @li2017_ICML], temporal graph dynamics [@Rakshit2017_ICML], and multiple modalities [@pezeshkpour_emnlp18]. Another line of research is concerned with extensions of the link prediction problem to multi-hop reasoning [@Zhang2018_AAAI]. We cannot list all prior link prediction methods here and instead refer the reader to two survey papers [@nickel2016review; @al2011survey]. Contrary to existing approaches, we address the problem of answering visual-relational queries in knowledge graphs where the entities are associated with web-extracted images. We also address the zero-shot learning scenario, a problem that has not been addressed in the context of link prediction in multi-relational graphs.
### Image ranking {#image-ranking .unnumbered}
Image retrieval is a popular problem and has been addressed by several authors [@tang2013_PAMI; @yang2016_IP; @Jiang2017_WSDM; @Niu2018_WSDM; @Guy2018_WSDM]. In [@yang2016_IP] a re-ranking of the output of a given search engine by learning a click-based multi-feature similarity is proposed. The authors performed spectral clustering and obtained the final ranked results by computing click-based clusters. In [@Guy2018_WSDM] the authors fine-tune a DNN to rank photos a user might like to share in social media as well as a mechanism to detect duplicates. In [@Niu2018_WSDM] a joint user-image embedding is learned to generate a ranking based on user preferences. Contrary to these previous approaches we introduce a set of novel visual query types in a web-extracted KG with images and provide methods to answer these queries efficiently.
### Relational and Visual Data {#relational-and-visual-data .unnumbered}
Previous work on combining relational and visual data has focused on object detection [@felzenszwalb2010object; @Girshick:2014; @RussakovskyICCV13; @marino2017; @li2017learning] and scene recognition [@Doersch:2013; @pandey2011scene; @Sadeghi:2012; @Xiao:2010; @Teney_2017_CVPR] which are required for more complex visual-relational reasoning. Recent years have witnessed a surge in reasoning about human-object, object-object, and object-attribute relationships [@Gupta2009ObservingHI; @Farhadi2009; @Malisiewicz2009BeyondCT; @yao2010modeling; @felzenszwalb2010object; @chen2013neil; @izadinia2014incorporating; @zhu2014reasoning]. The VisualGenome project [@Krishna2016] is a knowledge base that integrates language and vision modalities. The project provides a knowledge graph, based on <span style="font-variant:small-caps;">WordNet</span>, which provides annotations of categories, attributes, and relation types for each image. Recent work has used the dataset to focus on scene understanding in single images. For instance, Lu [@Lu2016] proposed a model to detect relation types between objects depicted in an image by inferring sentences such as “man riding bicycle." Veit [@veit2015] propose a siamese CNN to learn a metric representation on pairs of textile products so as to learn which products have similar styles. There is a large body of work on metric learning where the objective is to generate image embeddings such that a pairwise distance-based loss is minimized [@schroff:2015; @bell2015learning; @oh2016deep; @sohn2016improved; @wang:2017]. Recent work has extended this idea to directly optimize a clustering quality metric [@song2017deep]. In Vincent [@vincent_akbc17] they proposed a mutual embedding space for images and knowledge graphs so the relationships between an image and known entities in a knowledge graph are jointly encoded. Zhou [@Zhou_2016_CVPR] propose a method based on a bipartite graph that links depictions of meals to its ingredients. Johnson [@Johnson2015] propose to use the VisualGenome data to recover images from text queries. In the work of Thoma [@thomaRB17], they merge in a joint representation the embeddings from images, text, and KG and use the representation to perform link prediction on DBpedia [@dbpedia-swj]. <span style="font-variant:small-caps;">ImageGraph</span> is different from these data sets in that the relation types hold between different images and image annotated entities. This defines a novel class of problems where one seeks to answer queries such as “*How are these two images related?*" With this work, we address problems ranging from predicting the relation types for image pairs to multi-relational image retrieval.
### Zero-shot Learning {#zero-shot-learning .unnumbered}
We focus on exploring ways in which KGs can be used to find relationships between visual data of unseen entities, that is, entities not part of the KG during training, and visual data of known KG entities. This is a form of zero-shot learning (ZSL) where the objective is to generalize to novel visual concepts. Generally, ZSL methods ([@Romera-Paredes2015; @Zhang2015]) rely on an underlying embedding space, such as one based on visual attributes, to recognize unseen categories. With this paper, we do not assume the availability of such a common embedding space but we assume the existence of an external visual-relational KG. Similar to our approach, when this explicit knowledge is not encoded in the underlying embedding space, other works rely on finding the similarities through linguistic patterns ([@Ba2015; @Lu2016]), leveraging distributional word representations so as to capture a notion of similarity. These approaches, however, address scene understanding in a single image, these models are able to detect the visual relationships in one given image. Our approach, on the other hand, finds relationships between different images and entities.
<span style="font-variant:small-caps;">ImageGraph</span>: A Web-Extracted Visual Knowledge Graph
================================================================================================
<span style="font-variant:small-caps;">ImageGraph</span> is a visual-relational KG whose relational structure is based on <span style="font-variant:small-caps;">Freebase</span> [@Bollacker:2008] and, more specifically, on <span style="font-variant:small-caps;">FB15k</span>, a subset of <span style="font-variant:small-caps;">FreeBase</span> and a popular benchmark data set [@nickel2016review]. Since <span style="font-variant:small-caps;">FB15k</span> does not include visual data, we perform the following steps to enrich the KG entities with image data. We implemented a web crawler that is able to parse query results for the image search engines Google Images, Bing Images, and Yahoo Image Search. To minimize the amount of noise due to polysemous entity labels (for example, there are more than 100 <span style="font-variant:small-caps;">Freebase</span> entities with the text label “Springfield") we extracted, for each entity in <span style="font-variant:small-caps;">FB15k</span>, all Wikipedia URIs from the $1.9$ billion triple <span style="font-variant:small-caps;">Freebase</span> RDF dump. For instance, for Springfield, Massachusetts, we obtained such URIs as and `Springfield_(MA)`. These URIs were processed and used as search queries for disambiguation purposes. We used the crawler to download more than 2.4M images (more than 462Gb of data). We removed corrupted, low quality, and duplicate images and we used the 25 top images returned by each of the image search engines whenever there were more than 25 results. The images were scaled to have a maximum height or width of 500 pixels while maintaining their aspect ratio. This resulted in 829,931 images associated with 14,870 different entities ($55.8$ images per entity). After filtering out triples where either the head or tail entity could not be associated with an image, the visual KG consists of 564,010 triples expressing 1,330 different relation types between 14,870 entities. We provide three sets of triples for training, validation, and testing plus three more image splits also for training, validation and test. Table \[tab:StatsKB\] lists the statistics of the resulting visual KG. Any KG derived from <span style="font-variant:small-caps;">FB15k</span> such as <span style="font-variant:small-caps;">FB15k-237</span>[@toutanova2015observed] can also be associated with the crawled images. Since providing the images themselves would violate copyright law, we provide the code for the distributed crawler and the list of image URLs crawled for the experiments in this paper.
The distribution of relation types is depicted in Figure \[fig:rel-freq:a\]. It plots for each relation type the number of triples it occurs in. Some relation types such as $\mathtt{award\_nominee}$ or $\mathtt{profession}$ occur quite frequently while others such as $\mathtt{ingredient}$ have only few instances. $4\%$ of the relation types are symmetric, $88\%$ are asymmetric, and $8\%$ are others (see Table \[tab:relation\_types\_samples\]). Table \[fig:rel\_types\_cake\] lists specific instances of some relation types. There are 585 distinct entity types such as $\mathtt{Person, Athlete}$, and $\mathtt{City}$. Figure \[fig:ent-types:a\] shows the most frequent entity types. Figure \[fig:ent-freq:a\] visualizes the distribution of entities in the triples of <span style="font-variant:small-caps;">ImageGraph</span> and some example entities. Table \[tab:StatsKB\] lists some statistics of the <span style="font-variant:small-caps;">ImageGraph</span> KG and other KGs from related work. First, we would like to emphasize the differences between <span style="font-variant:small-caps;">ImageGraph</span> and the Visual Genome project (VG) [@Krishna2016]. With <span style="font-variant:small-caps;">ImageGraph</span> we address the problem of learning a representation for a KG with canonical relation types and not for relation types expressed through text. On a high level, we focus on answering visual-relational queries in a web-extracted KG. This is related to information retrieval except that in our proposed work, images are first-class citizens and we introduce novel and more complex query types. In contrast, VGD is focused on modeling relations between objects in images and the relation types are expressed in natural language. Additional differences between <span style="font-variant:small-caps;">ImageGraph</span> and <span style="font-variant:small-caps;">ImageNet</span> are the following. <span style="font-variant:small-caps;">ImageNet</span> is based on <span style="font-variant:small-caps;">WordNet</span> a lexical database where synonymous words from the same lexical category are grouped into synsets. There are $18$ relations expressing connections between synsets. In <span style="font-variant:small-caps;">Freebase</span>, on the other hand, there are two orders of magnitudes more relations. In <span style="font-variant:small-caps;">FB15k</span>, the subset we focus on, there are 1,345 relations expressing location of places, positions of basketball players, and gender of entities. Moreover, entities in <span style="font-variant:small-caps;">ImageNet</span> exclusively represent entity types such as $\mathtt{Cats}$ and $\mathtt{Cars}$ whereas entities in <span style="font-variant:small-caps;">FB15k</span> are either entity types or instances of entity types such as $\mathtt{Albert\ Einstein}$ and $\mathtt{Paris}$. This renders the computer vision problems associated with <span style="font-variant:small-caps;">ImageGraph</span> more challenging than those for existing datasets. Moreover, with <span style="font-variant:small-caps;">ImageGraph</span> the focus is on learning relational ML models that incorporate visual data both during learning and at query time.
Representation Learning for Visual-Relational Graphs {#sec:representation_learning}
====================================================
A knowledge graph (KG) $\mathcal{K}$ is given by a set of triples $\mathbf{T}$, that is, statements of the form $(\mathtt{h}, \mathtt{r}, \mathtt{t})$, where $\mathtt{h}, \mathtt{t} \in \mathcal{E}$ are the head and tail entities, respectively, and $\mathtt{r} \in \mathcal{R}$ is a relation type. Figure \[fig:KG-sample\] depicts a small fragment of a KG with relations between entities and images associated with the entities. Prior work has not included image data and has, therefore, focused on the following two types of queries. First, the query type $(\mathtt{h}, \mathtt{r?}, \mathtt{t})$ asks for the relations between a given pair of head and tail entities. Second, the query types $(\mathtt{h}, \mathtt{r}, \mathtt{t?})$ and $(\mathtt{h?}, \mathtt{r}, \mathtt{t})$, asks for entities correctly completing the triple. The latter query type is often referred to as knowledge base completion. Here, we focus on queries that involve visual data as query objects, that is, objects that are either contained in the queries, the answers to the queries, or both.
Visual-Relational Query Answering
---------------------------------
When entities are associated with image data, several completely novel query types are possible. Figure \[fig:KG-queries\] lists the query types we focus on in this paper. We refer to images used during training as *seen* and all other images as *unseen*.
1. Given a pair of *unseen* images for which we do *not* know their KG entities, determine the *unknown* relations between the underlying entities.
2. Given an *unseen* image, for which we do *not* know the underlying KG entity, and a relation type, determine the *seen* images that complete the query.
3. Given an *unseen* image of an *entirely new entity* that is not part of the KG, and an *unseen* image for which we do *not* know the underlying KG entity, determine the *unknown* relations between the two underlying entities.
4. Given an *unseen* image of an *entirely new entity* that is not part of the KG, and a known KG entity, determine the *unknown* relations between the two entities.
For each of these query types, the sought-after relations between the underlying entities have never been observed during training. Query types (3) and (4) are a form of zero-shot learning since neither the new entity’s relationships with other entities nor its images have been observed during training. These considerations illustrate the novel nature of the visual query types. The machine learning models have to be able to learn the relational semantics of the KG and not simply a classifier that assigns images to entities. These query types are also motivated by the fact that for typical KGs the number of entities is orders of magnitude greater than the number of relations.
Deep Representation Learning for Visual-Relational Query Answering
------------------------------------------------------------------
We first discuss KG completion methods and translate the concepts to query answering in visual-relational KGs. Let $\mathbf{raw}_{\mathtt{i}}$ be the raw feature representation for entity $\mathtt{i} \in \mathcal{E}$ and let $f$ and $g$ be differentiable functions. Most KG completion methods learn an embedding of the entities in a vector space via some *scoring function* that is trained to assign high scores to correct triples and low scores to incorrect triples. Scoring functions have often the form $f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}})$ where $\mathtt{r}$ is a relation type, $\mathbf{e}_{\mathtt{h}}$ and $\mathbf{e}_{\mathtt{t}}$ are $d$-dimensional vectors (the *embeddings* of the head and tail entities, respectively), and where $\mathbf{e}_{\mathtt{i}} = g(\mathbf{raw}_{\mathtt{i}})$ is an *embedding function* that maps the raw input representation of entities to the embedding space. In the case of KGs without visual data, the raw representation of an entity is simply its one-hot encoding.
Existing KG completion methods use the embedding function $g(\mathbf{raw}_{\mathtt{i}}) = \mathbf{raw}_{\mathtt{i}}^{\intercal} \mathbf{W}$ where $\mathbf{W}$ is a $|\mathcal{E}| \times d$ matrix, and differ only in their scoring function, that is, in the way the embeddings of the head and tail entities are combined with the parameter vector $\bm{\phi}_{\mathtt{r}}$:
- Difference (<span style="font-variant:small-caps;">TransE</span>[@bordes2013translating]): $f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}}) = -||\mathbf{e}_{\mathtt{h}} + \bm{\phi}_{\mathtt{r}} - \mathbf{e}_{\mathtt{t}}||_2$ where $\bm{\phi}_{\mathtt{r}}$ is a $d$-dimensional vector;
- Multiplication (<span style="font-variant:small-caps;">DistMult</span>[@yang2014learning]): $f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}}) = (\mathbf{e}_{\mathtt{h}} * \mathbf{e}_{\mathtt{t}}) \cdot \bm{\phi}_{\mathbf{r}}$ where $*$ is the element-wise product and $\bm{\phi}_{\mathbf{r}}$ a $d$-dimensional vector;
- Circular correlation (<span style="font-variant:small-caps;">HolE</span>[@Nickel:2016]): $f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}}) = (\mathbf{e}_{\mathtt{h}} \star \mathbf{e}_{\mathtt{t}}) \cdot \bm{\phi}_{\mathbf{r}}$ where $[\textbf{a} \star \textbf{b}]_k = \sum_{i = 0}^{d-1} \textbf{a}_i \textbf{b}_{(i+k) \mbox{\ \texttt{mod}\ } d}$ and $\bm{\phi}_{\mathbf{r}}$ a $d$-dimensional vector; and
- Concatenation: $f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}}) = (\mathbf{e}_{\mathtt{h}} \odot \mathbf{e}_{\mathtt{t}}) \cdot \bm{\phi}_{\mathbf{r}}$ where $\odot$ is the concatenation operator and $\bm{\phi}_{\mathbf{r}}$ a $2d$-dimensional vector.
For each of these instances, the matrix $\mathbf{W}$ (storing the entity embeddings) and the vectors $\bm{\phi}_{\mathtt{r}}$ are learned during training. In general, the parameters are trained such that $f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}})$ is high for true triples and low for triples assumed not to hold in the KG. The training objective is often based on the logistic loss, which has been shown to be superior for most of the composition functions [@trouillon2016complex],
\[loss-sigmoid\] \_\_[(,,) \_]{} (1 + (-f\_(\_, \_)) + \_[(,,) \_]{} (1 + (f\_(\_, \_))) + ||||\^[2]{}\_[2]{},
where $\mathbf{T}_{\mathtt{pos}}$ and $\mathbf{T}_{\mathtt{neg}}$ are the set of positive and negative training triples, respectively, $\Theta$ are the parameters trained during learning and $\lambda$ is a regularization hyperparameter. For the above objective, a process for creating corrupted triples $\mathbf{T}_{\mathtt{neg}}$ is required. This often involves sampling a random entity for either the head or tail entity. To answer queries of the types $(\mathtt{h}, \mathtt{r}, \mathtt{t?})$ and $(\mathtt{h?}, \mathtt{r}, \mathtt{t})$ after training, we form all possible completions of the queries and compute a ranking based on the scores assigned by the trained model to these completions.
For the queries of type $(\mathtt{h}, \mathtt{r?}, \mathtt{t})$ one typically uses the softmax activation in conjunction with the categorical cross-entropy loss, which does not require negative triples $$\label{loss-softmax}
\hspace{-2mm} \min_{\Theta} \hspace{-3mm} \sum_{ (\mathtt{h},\mathtt{r},\mathtt{t}) \in \mathbf{T}_{\mathtt{pos}}} \hspace{-4mm} -\log\left( \frac{\exp(f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}}))}{\sum_{\mathtt{r} \in \mathcal{R}} \exp(f_{\mathtt{r}}(\mathbf{e}_{\mathtt{h}}, \mathbf{e}_{\mathtt{t}})) } \right) + \lambda ||\Theta||^{2}_{2},$$ where $\Theta$ are the parameters trained during learning.
For visual-relational KGs, the input consists of raw image data instead of the one-hot encodings of entities. The approach we propose builds on the ideas and methods developed for KG completion. Instead of having a simple embedding function $\mathtt{g}$ that multiplies the input with a weight matrix, however, we use deep convolutional neural networks to extract meaningful visual features from the input images. For the composition function $\mathtt{f}$ we evaluate the four operations that were used in the KG completion literature: difference, multiplication, concatenation, and circular correlation. Figure \[fig:KG-architecture\] depicts the basic architecture we trained for query answering. The weights of the parts of the neural network responsible for embedding the raw image input, denoted by $\mathtt{g}$, are tied. We also experimented with additional hidden layers indicated by the dashed dense layer. The composition operation $\mathbf{op}$ is either difference, multiplication, concatenation, or circular correlation. To the best of our knowledge, this is the first time that KG embedding learning and deep CNNs have been combined for visual-relationsl query answering.
Experiments
===========
We conduct a series of experiments to evaluate the proposed approach. First, we describe the experimental set-up that applies to all experiments. Second, we report and interpret results for the different types of visual-relational queries.
General Set-up
--------------
We used <span style="font-variant:small-caps;">Caffe</span>, a deep learning framework [@jia2014caffe] for designing, training, and evaluating the proposed models. The embedding function $\mathtt{g}$ is based on the VGG16 model introduced in [@cimonyan14c_vgg16]. We pre-trained the VGG16 on the ILSVRC2012 data set derived from <span style="font-variant:small-caps;">ImageNet</span> [@imagenet_cvpr09] and removed the softmax layer of the original VGG16. We added a 256-dimensional layer after the last dense layer of the VGG16. The output of this layer serves as the embedding of the input images. The reason for reducing the embedding dimensionality from 4096 to 256 is motivated by the objective to obtain an efficient and compact latent representation that is feasible for KGs with billion of entities. For the composition function $\mathtt{f}$, we performed either of the four operations difference, multiplication, concatenation, and circular correlation. We also experimented with an additional hidden layer with ReLu activation. Figure \[fig:KG-architecture\] depicts the generic network architecture. The output layer of the architecture has a softmax or sigmoid activation with cross-entropy loss. We initialized the weights of the newly added layers with the Xavier method [@xavier2010].
We used a batch size of $45$ which was the maximal possible fitting into GPU memory. To create the training batches, we sample a random triple uniformly at random from the training triples. For the given triple, we randomly sample one image for the head and one for the tail from the set of training images. We applied SGD with a learning rate of $10^{-5}$ for the parameters of the VGG16 and a learning rate of $10^{-3}$ for the remaining parameters. It is crucial to use two different learning rates since the large gradients in the newly added layers would lead to unreasonable changes in the pretrained part of the network. We set the weight decay to $5 \times 10^{-4}$. We reduced the learning rate by a factor of $0.1$ every 40,000 iterations. Each of the models was trained for 100,000 iterations.
Since the answers to all query types are either rankings of images or rankings of relations, we utilize metrics measuring the quality of rankings. In particular, we report results for hits@1 (hits@10, hits@100) measuring the percentage of times the correct relation was ranked highest (ranked in the top 10, top 100). We also compute the median of the ranks of the correct entities or relations and the Mean Reciprocal Rank (MRR) for entity and relation rankings, respectively, defined as follows:\
$$\hspace{-3mm} \mbox{MRR} = \frac{1}{2|\mathbf{T}|} \sum_{(\mathtt{h},\mathtt{r},\mathtt{t})\in \mathbf{T}}
\left(\frac{1}{\mathtt{rank}_{\mathtt{img}(\mathtt{h})}} + \frac{1}{\mathtt{rank}_{\mathtt{img}(\mathtt{t})}}\right)$$ $$\mbox{MRR} = \frac{1}{|\mathbf{T}|} \sum_{(\mathtt{h},\mathtt{r},\mathtt{t})\in \mathbf{T}} \frac{1}{\mathtt{rank}_{\mathtt{r}}},$$ where $\mathbf{T}$ is the set of all test triples, $\mathtt{rank}_{\mathtt{r}}$ is the rank of the correct relation, and $\mathtt{rank}_{\mathtt{img}(\mathtt{h})}$ is the rank of the highest ranked image of entity $\mathtt{h}$. For each query, we remove all triples that are also correct answers to the query from the ranking. All experiments were run on commodity hardware with 128GB RAM, a single 2.8 GHz CPU, and a NVIDIA 1080 Ti.
Visual Relation Prediction
--------------------------
Model Median Hits@1 Hits@10 MRR
-------------------------------------- -------- ---------- ---------- -----------
$\mathtt{VGG16}$+$\mathtt{DistMult}$ 94 6.0 11.4 0.087
Prob. Baseline 35 3.7 26.5 0.104
DIFF 11 21.1 50.0 0.307
MULT 8 15.5 54.3 0.282
CAT **6** **26.7** **61.0** **0.378**
DIFF+1HL 8 22.6 55.7 0.333
MULT+1HL 9 14.8 53.4 0.273
CAT+1HL **6** 25.3 60.0 0.365
: \[tab:relationship\_pred\] Results for the relation prediction problem.
Given a pair of unseen images we want to determine the relations between their underlying unknown entities. This can be expressed with $(\mathtt{img}_{\mathtt{h}}, \mathtt{r?}, \mathtt{img}_{\mathtt{t}})$. Figure \[fig:KG-queries\] illustrates this query type which we refer to as visual relation prediction. We train the deep architectures using the training and validation triples and images, respectively. For each triple $(\mathtt{h}, \mathtt{r}, \mathtt{t})$ in the training data set, we sample one training image uniformly at random for both the head and the tail entity. We use the architecture depicted in Figure \[fig:KG-architecture\] with the softmax activation and the categorical cross-entropy loss. For each test triple, we sample one image uniformly at random from the test images of the head and tail entity, respectively. We then use the pair of images to query the trained deep neural networks. To get a more robust statistical estimate of the evaluation measures, we repeat the above process three times per test triple. Again, none of the test triples and images are seen during training nor are any of the training images used during testing. Computing the answer to one query takes the model 20 ms.
We compare the proposed architectures to two different baselines: one based on entity classification followed by a KB embedding method for relation prediction ($\mathtt{VGG16}$+$\mathtt{DistMult}$), and a probabilistic baseline (Prob. Baseline). The entity classification baseline consists of fine-tuning a pretrained VGG16 to classify images into the $14,870$ entities of <span style="font-variant:small-caps;">ImageGraph</span>. To obtain the relation type ranking at test time, we predict the entities for the head and the tail using the VGG16 and then use the KB embedding method <span style="font-variant:small-caps;">DistMult</span>[@yang2014learning] to return a ranking of relation types for the given (head, tail) pair. $\mathtt{DistMult}$ is a KB embedding method that achieves state of the art results for KB completion on FB15k [@kadlec2017knowledge]. Therefore, for this experiment we just substitute the original output layer of the VGG16 pretrained on <span style="font-variant:small-caps;">ImageNet</span> with a new output layer suitable for our problem. To train, we join the train an validation splits, we set the learning rate to $10^{-5}$ for all the layers and we train following the same strategy that we use in all of our experiments. Once the system is trained, we test the model by classifying the entities of the images in the test set. To train $\mathtt{DistMult}$, we sample $500$ negatives triples for each positive triple and used an embedding size of $100$. Figure \[fig:methods\] illustrates the $\mathtt{VGG16}$+$\mathtt{DistMult}$ baseline and contrasts it with our proposed approach. The second baseline (probabilistic baseline) computes the probability of each relation type using the set of training and validation triples. The baseline ranks relation types based on these prior probabilities.
{width="95.00000%"}
Table \[tab:relationship\_pred\] lists the results for the two baselines and the different proposed architectures. The probabilistic baseline outperforms the $\mathtt{VGG16}$+$\mathtt{DistMult}$ baseline in 3 of the metrics. This is due to the highly skewed distribution of relation types in the training, validation, and test triples. A small number of relation types makes up a large fraction of triples. Figure \[fig:rel-freq:a\] and \[fig:ent-freq:a\] depicts the plots of the counts of relation types and entities. Moreover, despite <span style="font-variant:small-caps;">DistMult</span> achieving a hits@1 value of 0.46 for the relation prediction problem between entity pairs the baseline $\mathtt{VGG16}$+$\mathtt{DistMult}$ performs poorly. This is due to the poor entity classification performance of the VGG (accurracy: 0.082, F1: 0.068). In the remainder of the experiments, therefore, we only compare to the probabilistic baseline. In the lower part of Table \[tab:relationship\_pred\], we lists the results of the experiments. DIFF, MULT, and CAT stand for the different possible composition operations. We omitted the composition operation circular correlation since we were not able to make the corresponding model converge, despite trying several different optimizers and hyperparameter settings. The post-fix 1HL stands for architectures where we added an additional hidden layer with ReLu activation before the softmax. The concatenation operation clearly outperforms the multiplication and difference operations. This is contrary to findings in the KG completion literature where MULT and DIFF outperformed the concatenation operation. The models with the additional hidden layer did not perform better than their shallower counterparts with the exception of the DIFF model. We hypothesize that this is due to difference being the only linear composition operation, benefiting from an additional non-linearity. Each of the proposed models outperforms the baselines.
---------- --------- --------- ---------- ---------- ----------- -----------
Model Head Tail Head Tail Head Tail
Baseline 6504 2789 11.9 18.4 **0.065** **0.115**
DIFF 1301 877 19.6 26.3 0.051 0.094
MULT 1676 1136 16.8 22.9 0.040 0.080
CAT 1022 727 21.4 27.5 0.050 0.087
DIFF+1HL 1644 1141 15.9 21.9 0.045 0.085
MULT+1HL 2004 1397 14.6 20.5 0.034 0.069
CAT+1HL 1323 919 17.8 23.6 0.042 0.080
CAT-SIG **814** **540** **23.2** **30.1** 0.049 0.082
---------- --------- --------- ---------- ---------- ----------- -----------
: \[tab:image\_ret\]Results for multi-relational image retrieval.
Multi-Relational Image Retrieval
--------------------------------
Given an unseen image, for which we do not know the underlying KG entity, and a relation type, we want to retrieve existing images that complete the query. If the image for the head entity is given, we return a ranking of images for the tail entity; if the tail entity image is given we return a ranking of images for the head entity. This problem corresponds to query type (2) in Figure \[fig:KG-queries\]. Note that this is equivalent to performing multi-relational metric learning which, to the best of our knowledge, has not been done before. We performed experiments with each of the three composition functions $\mathtt{f}$ and for two different activation/loss functions. First, we used the models trained with the softmax activation and the categorical cross-entropy loss to rank images. Second, we took the models trained with the softmax activation and substituted the softmax activation with a sigmoid activation and the corresponding binary cross-entropy loss. For each training triple $(\mathtt{h}, \mathtt{r}, \mathtt{t})$ we then created two negative triples by sampling once the head and once the tail entity from the set of entities. The negative triples are then used in conjunction with the binary cross-entropy loss of equation \[loss-sigmoid\] to refine the pretrained weights. Directly training a model with the binary cross-entropy loss was not possible since the model did not converge properly. Pretraining with softmax and categorical cross-entropy loss was crucial to make the binary loss work.
During testing, we used the test triples and ranked the images based on the probabilities returned by the respective models. For instance, given the query $(\mathtt{img}_{\mathtt{Sens\overline{o}}\mbox{-}\mathtt{ji}}, \mathtt{locatedIn}, \mathtt{img_{\mathtt{t}}?})$, we substituted $\mathtt{img}_{\mathtt{t}}?$ with all training and validation images, one at a time, and ranked the images according to the probabilities returned by the models. We use the rank of the highest ranked image belonging to the true entity (here: $\mathtt{Japan}$) to compute the values for the evaluation measures. We repeat the same experiment three times (each time randomly sampling the images) and report average values. Again, we compare the results for the different architectures with a probabilistic baseline. For the baseline, however, we compute a distribution of head and tail entities for each of the relation types. For example, for the relation type $\mathtt{locatedIn}$ we compute two distributions, one for head and one for tail entities. We used the same measures as in the previous experiment to evaluate the returned image rankings.
Table \[tab:image\_ret\] lists the results of the experiments. As for relation prediction, the best performing models are based on the concatenation operation, followed by the difference and multiplication operations. The architectures with an additional hidden layer do not improve the performance. We also provide the results for the concatenation-based model with softmax activation where we refined the weights using a sigmoid activation and negative sampling as described before. This model is the best performing model. All neural network models are significantly better than the baseline with respect to the median and hits@100. However, the baseline has slightly superior results for the MRR. This is due to the skewed distribution of entities and relations in the KG (see Figure \[fig:ent-freq:a\] and Figure \[fig:rel-freq:a\]). This shows once more that the baseline is highly competitive for the given KG. Figure \[fig:examples\] visualizes the answers the CAT-SIG model provided for a set of four example queries. For the two queries on the left, the model performed well and ranked the correct entity in the top 3 (green frame). The examples on the right illustrate queries for which the model returned an inaccurate ranking. To perform query answering in a highly efficient manner, we precomputed and stored all image embeddings once, and only compute the scoring function (involving the composition operation and a dot product with $\bm{\phi}_{\mathtt{r}}$) at query time. Answering one multi-relational image retrieval query (which would otherwise require 613,138 individual queries, one per possible image) took only 90 ms.
Zero-Shot Visual Relation Prediction
------------------------------------
The last set of experiments addresses the problem of zero-shot learning. For both query types, we are given an new image of an entirely new entity that is not part of the KG. The first query type asks for relations between the given image and an unseen image for which we do not know the underlying KG entity. The second query type asks for the relations between the given image and an existing KG entity. We believe that creating multi-relational links to existing KG entities is a reasonable approach to zero-shot learning since the relations to existing visual concepts and their attributes provide a characterization of the new entity/category.
For the zero-shot experiments, we generated a new set of training, validation, and test triples. We randomly sampled 500 entities that occur as head (tail) in the set of test triples. We then removed all training and validation triples whose head or tail is one of these 1000 entities. Finally, we only kept those test triples with one of the 1000 entities either as head or tail but not both. For query type (4) where we know the target entity, we sample 10 of its images and use the models 10 times to compute a probability. We use the average probabilities to rank the relations. For query type (3) we only use one image sampled randomly. As with previous experiments, we repeated procedure three times and averaged the results. For the baseline, we compute the probabilities of relation in the training and validation set (for query type (3)) and the probabilities of relations conditioned on the target entity (for query type (4)). Again, these are very competitive baselines due to the skewed distribution of relations and entities. Table \[tab:zero-shot\_pred\] lists the results of the experiments. The model based on the concatenation operation (CAT) outperforms the baseline and performs surprisingly well. The deep models are able to generalize to unseen images since their performance is comparable to the performance in the relation prediction task (query type (1)) where the entity was part of the KG during training (see Table \[tab:relationship\_pred\]). Figure \[fig:zsl-qualitative\] depicts example queries for the zero-shot query type (3). For the first query example, the CAT model ranked the correct relation type first (indicated by the green bounding box). The second example is more challenging and the correct relation type was not part of the top 10 ranked relation types. Figure \[fig:zsl-kg-example\] shows one concrete example of the zero-shot learning problem. In green, visual data from an unknown entity is linked with visual data from KG entities (blue) by ranking the most probable relation types. This problem cannot be addressed with standard relation prediction methods since entities need to be part of the KG during training for these models to work.
{width="0.6\columnwidth"} \[fig:zsl-kg-example\]
Conclusion
==========
KGs are at the core of numerous AI applications. Research has focused either on link prediction working only on the relational structure or on scene understanding in a single image. We present a novel visual-relational KG where the entities are enriched with visual data. We proposed several novel query types and introduce neural architectures suitable for probabilistic query answering. We propose a novel approach to zero-shot learning as the problem of visually mapping an image of an entirely new entity to a KG.
|
---
abstract: |
One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is $\beta/2$ times the inverse of particle distance. When $\beta=2$, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson’s model with $\beta=2$ and $N$ particles, $\X(t)=(X_1(t), \dots, X_N(t)),
t \in [0,\infty), 2 \leq N < \infty$, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function $\Ai(z)$ is an entire function with zeros all located on the negative part of the real axis $\R$. We consider Dyson’s model with $\beta=2$ starting from the first $N$ zeros of $\Ai(z)$, $0 > a_1 > \cdots > a_N$, $N \geq 2$. In order to properly control the effect of such initial confinement of particles in the negative region of $\R$, we put the drift term to each Brownian motion, which increases in time as a parabolic function : $Y_j(t)=X_j(t)+t^2/4+\{d_1+\sum_{\ell=1}^{N}(1/a_{\ell})\}t,
1 \leq j \leq N$, where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the $N \to \infty$ limit of $\Y(t)=(Y_1(t), \dots, Y_N(t)), t \in [0, \infty)$, we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of $\Ai(z)$ on the negative $\R$ is occupied by one particle, to the stationary state $\mu_{\Ai}$. The stationary state $\mu_{\Ai}$ is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on $\R$ and in which the Tracy-Widom distribution describes the rightmost particle position.\
[**KEY WORDS:**]{} Zeros of Airy function; Relaxation process; Dyson’s model; Determinantal point process; Entire function; Weierstrass canonical product
author:
- 'Makoto Katori [^1] and Hideki Tanemura [^2]'
date: 29 September 2009
title: Zeros of Airy Function and Relaxation Process
---
Introduction
============
\[chap: Introduction\]
Dyson’s model: One-Dimensional Brownian Particle System Interacting through Pair Force $1/x$
--------------------------------------------------------------------------------------------
To understand the time-evolution of distributions of interacting particle systems on a large space-time scale ([*thermodynamic and hydrodynamic limits*]{}) is one of the main topics of statistical physics. If the interactions among particles are short ranged, the standard theory is useful. If they are long ranged, however, general theory has not yet been established and thus detailed study of model systems is required [@Spo91].
In the present paper, we consider Brownian particles in one dimension with long-ranged repulsive forces acting between any pair of particles, where the strength of force is exactly equal to $1/x$ when the particle distance is $x$. If the number of particles is finite $N < \infty$, the system is described by $\Xi(t)=\sum_{j=1}^{N} \delta_{X_j(t)}$, $2 \leq N < \infty$, where $\X(t)=(X_1(t), \dots, X_N(t))$ satisfies the following system of stochastic differential equations (SDEs); $$dX_j(t)=dB_j(t)
+
\sum_{\substack{1 \leq k \leq N
\\ k \not= j}}
\frac{1}{X_j(t)-X_k(t)} dt, \quad
1 \leq j \leq N, \quad
t \in [0, \infty)
\label{eqn:Dyson}$$ with independent one-dimensional standard Brownian motions $B_j(t), 1 \leq j \leq N$. The SDEs obtained by replacing the $1/x$ force in (\[eqn:Dyson\]) by $\beta/(2x)$ with a parameter $\beta >0$ were introduced by Dyson [@Dys62] to understand the statistics of eigenvalues of hermitian [*random matrices*]{} as particle distributions of interacting Brownian motions in $\R$. Corresponding to the special values $\beta=1,2$ and 4, hermitian random matrices are in the three statistical ensembles with different symmetries, called the Gaussian orthogonal ensemble (GOE), the Gaussian unitary ensemble (GUE), and the Gaussian symplectic ensemble (GSE), respectively [@Dys62b]. In particular for $\beta=2$, that is the case of (\[eqn:Dyson\]), if the eigenvalue distribution of $N \times N$ hermitian random matrices in the GUE with variance $\sigma^2$ is denoted by $\mu^{\rm GUE}_{N, \sigma^2}$, we can show $$\lim_{N \to \infty} \mu^{\rm GUE}_{N, 2N/\pi^2}
(\cdot)
= \mu_{\sin}(\cdot),
\label{eqn:lim_sin}$$ where $\mu_{\sin}$ denotes the [*determinantal (Fermion) point process*]{} [@Sos00; @ST03] with the so-called [*sine kernel*]{} $$K_{\sin}(x)=
\frac{1}{2 \pi} \int_{|k| \leq \pi} dk \,
e^{\sqrt{-1} k x}
= \frac{\sin (\pi x) }{\pi x},
\quad x \in \R.
\label{eqn:Ksin}$$ That is, $\mu_{\sin}$ is a spatially homogeneous particle distribution, in which the particle density is given by $\rho_{\sin}=\lim_{x \to 0} K_{\sin}(x)=1$ and any $N_1$-point correlation function $\rho_{\sin}(\x_{N_1}), \x_{N_1}=(x_1, \dots, x_{N_1}) \in \R^{N_1},
N_1 \geq 2$, is given by a determinant of an $N_1 \times N_1$ real symmetric matrix; $$\rho_{\sin}(\x_{N_1})= \det_{1 \leq j, k \leq N_1}
\Big[ K_{\sin}(x_j-x_k) \Big].$$ Based on this fact known in the random matrix theory [@Meh04], Spohn [@Spo87] studied the equilibrium dynamics obtained in the infinite-particle limit $N \to \infty$ of Dyson’s model (\[eqn:Dyson\]). Since the $1/x$ force is not summable, in the infinite-particle limit $N \to \infty$ the sum in (\[eqn:Dyson\]) should be regarded as an improper sum, in the sense that for $X_j(t) \in [-L,L]$ the summation is restricted to $k$’s such that $X_k(t) \in [-L, L]$ and then the limit $L \to \infty$ is taken. It is expected that the dynamics with an infinite number of particles can exist only for initial configurations having the same asymptotic density to the right and left [@Spo87; @KT08].
The problem, which we address in the present paper, is how we can control Dyson’s model with an infinite number of particles starting from [*asymmetric*]{} initial configurations. The motivation is again coming from the random matrix theory as follows. Consider the [*Airy function*]{} [@AS65; @VS04] $$\Ai(z) = \frac{1}{2 \pi} \int_{\R} dk \,
e^{\sqrt{-1}(z k+k^3/3)}.
\label{eqn:Airy1}$$ It is a solution of Airy’s equation $f''(z)-z f(z)=0$ with the asymptotics on the real axis $\R$: $$\begin{aligned}
&& \Ai(x) \simeq \frac{1}{2 \sqrt{\pi} x^{1/4}}
\exp\left( - \frac{2}{3} x^{3/2} \right),
\nonumber\\
&& \Ai(-x) \simeq \frac{1}{\sqrt{\pi} x^{1/4}}
\cos \left( \frac{2}{3}x^{3/2}-\frac{\pi}{4} \right)
\quad \mbox{in} \quad x \to + \infty.
\label{eqn:Aiasym}\end{aligned}$$ In the GUE random matrix theory, the following scaling limit has been extensively studied: $$\lim_{N \to \infty} \mu^{\rm GUE}_{N, N^{1/3}}
(2N^{2/3}+\cdot)=\mu_{\Ai}(\cdot),
\label{eqn:lim_Ai}$$ where $\mu_{\Ai}$ is the determinantal point process such that the correlation kernel is given by [@For93; @TW94], $$\begin{aligned}
K_{\Ai}(y|x)
&=& \int_{0}^{\infty} du \,
\Ai(u+x) \Ai(u+y) \nonumber\\
\label{eqn:AiryKer}
&=& \left\{ \begin{array}{ll}
\displaystyle{
\frac{\Ai(x) \Ai'(y)-\Ai'(x) \Ai(y)}{x-y}},
& x \not= y \in \R \cr
(\Ai'(x))^2-x (\Ai(x))^2,
& x=y \in \R. \end{array} \right.\end{aligned}$$ It is another infinite-particle limit different from (\[eqn:lim\_sin\]) and is called the [*soft-edge scaling limit*]{}, since $x^2/2t \simeq (2 N^{2/3})^2/(2N^{1/3})=2N$ marks the right edge of semicircle-shaped profile of the GUE eigenvalue distribution (see, for example, [@KT07]). The particle distribution $\mu_{\Ai}$ with the [*Airy kernel*]{} (\[eqn:AiryKer\]) is highly asymmetric: As a matter of fact, the particle density $\rho_{\Ai}(x)=K_{\Ai}(x|x)$ decays rapidly to zero as $x \to \infty$, but it diverges $$\rho_{\Ai}(x)
\simeq \frac{1}{\pi} (-x)^{1/2} \to \infty
\quad \mbox{as} \quad x \to - \infty.
\label{eqn:rhoAi}$$ Let $R$ be the position of the rightmost particle on $\R$ in $\mu_{\Ai}$. Then its distribution is given by the celebrated [*Tracy-Widom distribution*]{} [@TW94] $$\mu_{\Ai}(R < x)=\exp \left[
-\int_{x}^{\infty}(y-x) (q(y))^2 dy \right],$$ where $q(x)$ is the unique solution of the Painlevé II equation $$q''=xq + 2 q^3$$ satisfying the boundary condition $q(x) \simeq \Ai(x)$ in $x \to \infty$. Prähofer and Spohn [@PS02] and Johansson [@Joh03] studied the equilibrium fluctuation of this rightmost particle and called it the [*Airy process*]{}. Tracy and Widom derived a system of partial differential equations, which govern the Airy process [@TW03]. See also [@AM99; @AM05]. How can we realize $\mu_{\Ai}$ as the equilibrium state of Brownian infinite-particle system interacting through pair force $1/x$ ? The initial configurations should be asymmetric, but what kinds of conditions should be satisfied by them ? How should we modify the SDEs of original Dyson’s model (\[eqn:Dyson\]), when we provide finite-particle approximations for such asymmetric infinite particle systems ?
In the present paper, as an explicit answer to the above questions, we will present a relaxation process with an infinite number of particles converging to the stationary state $\mu_{\Ai}$ in $t \to \infty$. Its initial configuration is given by $$\xi_{\cA}(\cdot)=\sum_{a \in {\cA} } \delta_a (\cdot)
=\sum_{j=1}^{\infty} \delta_{a_{j}}(\cdot),
\label{eqn:xiA}$$ in which every zero of the Airy function (\[eqn:Airy1\]) is occupied by one particle. This special choice of the initial configuration is due to the fact that the zeros of the Airy function are located only on the negative part of the real axis $\R$, $$\cA \equiv \Ai^{-1}(0)
=\Big\{ a_{j}, j \in \N \, : \, \Ai(a_{j})=0, \,
0 > a_1 > a_2 > \cdots \Big\},
\label{eqn:AiryZero}$$ with the values [@AS65] $
a_1 = -2.33 \dots,
a_2 = -4.08 \dots,
a_3 = -5.52 \dots,
a_4 = -6.78 \dots,
$ and that they admit the asymptotics [@AS65; @VS04] $$a_{j} \simeq - \left( \frac{3 \pi}{2} \right)^{2/3} j^{2/3} \quad
\mbox{in} \quad j \to \infty.
\label{eqn:ainf}$$ Then the average density of zeros of the Airy function around $x$, denoted by $\rho_{\Ai^{-1}(0)}(x)$, behaves as $$\rho_{\Ai^{-1}(0)}(x) \simeq \frac{1}{\pi} (-x)^{1/2}
\to \infty
\quad \mbox{as} \quad x \to -\infty,$$ which coincides with (\[eqn:rhoAi\]). The approximation of our process with a finite number of particles $N < \infty$ is given by $\Xi_{\cA}(t)=\sum_{j=1}^{N} \delta_{Y_j(t)}$ with $$Y_j(t) = X_j(t)+\frac{t^2}{4} + D_{\cA_N} t, \quad
1 \leq j \leq N, \quad
t \in [0, \infty),
\label{eqn:finite_Airy}$$ associated with the solution $\X(t)=(X_1(t), \dots, X_N(t))$ of Dyson’s model (\[eqn:Dyson\]), where $$D_{\cA_N}=d_1+\sum_{\ell=1}^N \frac{1}{a_\ell}.
\label{eqn:DA}$$ Here $d_1=\Ai'(0)/\Ai(0)$ and $
{\cA}_N \equiv \Big\{0 > a_1 > \dots > a_N \Big\}
\subset {\cA}
$ is the sequence of the first $N$ zeros of the Airy function. In other words, $\Y(t)=(Y_1(t), Y_2(t), \dots, Y_N(t))$ satisfies the following SDEs ; $$\begin{aligned}
dY_j(t) &=& dB_j(t)+
\left( \frac{t}{2}+D_{\cA_N} \right) dt
+\sum_{\substack{1 \leq k \leq N
\\ k \not= j}}
\frac{dt}{Y_j(t)-Y_k(t)}
\nonumber\\
&=& dB_j(t)+\sum_{\substack{1 \leq k \leq N
\\ k \not= j}}
\left( \frac{1}{Y_j(t)-Y_k(t)}+\frac{1}{a_k}\right)dt
+ \left(\frac{t}{2} +d_1+\frac{1}{a_j}\right)dt,
\nonumber\\
&& \hskip 5cm 1 \leq j \leq N, \quad
t \in [0, \infty),
\label{eqn:YN}\end{aligned}$$ where $B_j(t)$’s are independent one-dimensional standard Brownian motions.
For $\Y(0)=\x \in \R^N$, set $\xi^{N}(\cdot)=\sum_{j=1}^{N} \delta_{x_j}(\cdot)$ and consider the process $\Xi_{\cA}(t)$ starting from the configuration $\xi^N$. We consider a set of initial configurations $\xi^N$ such that they are in general different from the $N$-particle approximation of (\[eqn:xiA\]), $$\xi_{\cA}^N(\cdot)
= \sum_{a \in {\cA}_N} \delta_a(\cdot)
=\sum_{j=1}^{N} \delta_{a_j}(\cdot),
\label{eqn:xiAN}$$ but the particle density $\rho(x)$ of $\lim_{N \to \infty} \xi^N$ will show the same asymptotic in $x \to -\infty$ as (\[eqn:rhoAi\]). Because of the strong repulsive forces acting between particle pairs in (\[eqn:Dyson\]), such confinement of particles in the negative region of $\R$ at the initial time causes strong [*positive drifts*]{} of Brownian particles. The coefficient (\[eqn:DA\]) of the drift term $D_{\cA_N} t$ added in (\[eqn:finite\_Airy\]), however, [*negatively diverges*]{} $$D_{{\cA}_N} \simeq - \left(\frac{12}{\pi^2} \right)^{1/3}
N^{1/3} \to - \infty
\quad \mbox{as} \quad N \to \infty.
\label{eqn:Dinf}$$ We will determine a class of asymmetric initial configurations denoted by $\mX^{\cA}_0$, which includes $\xi_{\cA}$ as a typical one, such that the effect on dynamics of asymmetry in configuration will be compensated by the additional drift term $D_{\cA_N} t$ in the infinite-particle limit $N \to \infty$ and the dynamics with an infinite number of particles exists. Note that we should take $N \to \infty$ limit for finite $t < \infty$ in our process (\[eqn:finite\_Airy\]) to discuss non-equilibrium dynamics with an infinite number of particles. In the class $\mX^{\cA}_0$, when the initial configuration is specially set to be $\xi_{\cA}$, (\[eqn:xiA\]), we can prove that the dynamics shows a relaxation in the long-term limit $t \to \infty$ to the equilibrium dynamics in $\mu_{\Ai}$. In the proof we use the special property of the systems (\[eqn:Dyson\]) and (\[eqn:YN\]) such that the processes have space-time determinantal correlations. This feature comes from the fact that if and only if the strength of pair force is exactly equal to $1/x$ when the particle distance is $x$, [*i.e.*]{}, iff $\beta=2$, Dyson’s model is realized as the Brownian motions [*conditioned never to collide with each other*]{} [@Gra99; @KT07].
In order to explain the importance of the notion of [*entire functions*]{} for the present problem, we rewrite the results reported in our previous paper [@KT08] for Dyson’s model with symmetric initial configurations below. Then the changes which we have to do for the systems with asymmetric initial configurations are shown. There the origin of the quadratic term $t^2/4$ in (\[eqn:finite\_Airy\]) will be clarified.
Processes with Space-time Determinantal Correlations and Entire Functions
-------------------------------------------------------------------------
In an earlier paper [@KT08], we studied a class of a non-equilibrium dynamics of Dyson’s model with $\beta=2$ and an infinite number of particles. As an example in the class, we reported a relaxation process, denoted here by $(\Xi(t), \P_{\sin})$, which starts from a configuration $$\xi_{\Z}(\cdot)=\sum_{a \in \Z} \delta_a(\cdot),
\label{eqn:xiZ}$$ in which every point of $\Z$ is occupied by one particle, and converges to the stationary state $\mu_{\sin}$. This process $(\Xi(t), \P_{\sin})$ is [*determinantal*]{}, in the sense that there is a function $\mbK_{\sin}(s,x ; t, y)$ called the [*correlation kernel*]{} such that it is continuous with respect to $(x, y) \in \R^2$ for any fixed $(s,t) \in [0, \infty)^2$, and that, for any integer $M \geq 1$, any sequence $(N_m)_{m=1}^{M}$ of positive integers, and any time sequence $0 < t_1 < \cdots < t_M < \infty$, the $(N_1, \dots, N_M)$-[*multitime correlation function*]{} $\rho_{\sin}(t_1, \x^{(1)}_{N_1}; \dots; t_M, \x^{(M)}_{N_M}),
\x^{(m)}_{N_m}= (x^{(m)}_1, \dots, x^{(m)}_{N_m} ) \in \R^{N_m},
1 \leq m \leq M$, is expressed by a determinant of a $\sum_{m=1}^{M} N_m \times \sum_{m=1}^{M}N_m$ asymmetric real matrix; $$\rho_{\sin} \Big( t_1, \x^{(1)}_{N_1}; \dots ;
t_M, \x^{(M)}_{N_M} \Big)=
\det_{
\substack{1 \leq j \leq N_{m},
1 \leq k \leq N_{n} \\ 1 \leq m, n \leq M}
}
\Big[
\mbK_{\sin} (t_m, x_{j}^{(m)}; t_n, x_{k}^{(n)} )
\Big].
\label{eqn:rho0}$$ The finite dimensional distributions of the process $(\Xi(t), \P_{\sin})$ are determined by $\mbK_{\sin}$ through (\[eqn:rho0\]). It is expected that the correlation kernel $\mbK_{\sin}$ is described by using the sine function as is the correlation kernel $K_{\sin}$ of the stationary distribution $\mu_{\sin}$ given by (\[eqn:Ksin\]). It is indeed true. Set $$f(z)= \sin (\pi z), \quad z \in \C,
\label{eqn:sine1}$$ and $$p_{\sin}(t, x)=
\frac{e^{-x^2/2t}}{\sqrt{2 \pi |t|}}, \quad
t \in \R \setminus \{0\}, \quad x \in \C.
\label{eqn:psin}$$ When $t >0$, $p_{\sin}(t,y-x)$ is the heat kernel: the solution of the heat equation $\partial u(t,x)/\partial t
= (1/2) \partial^2 u(t,x) /\partial x^2$ with $\lim_{t \to 0} u(t,x)dx = \delta_y(dx)$, and is expressed using (\[eqn:sine1\]) as $$p_{\sin}(t, y-x) =
\frac{1}{2} \int_{\R} du \,
e^{-\pi^2 u^2 t/2}
\Big\{ f(ux) f(uy)
+ f(ux+1/2) f(uy+1/2) \Big\}.$$ For $0 < s < t$, by setting (\[eqn:psin\]), the Chapman-Kolmogorov equation $$\int_{\R}dy \, p_{\sin}(t-s, z-y) p_{\sin}(s, y-x)
=p_{\sin}(t, z-x)
\label{eqn:CK1}$$ can be extended to $$\int_{\R}dy \, p_{\sin}(-t, z-y)
p_{\sin}(t-s, y-x)
= p_{\sin}(-s, z-x).
\label{eqn:CK2}$$ Then $\mbK_{\sin}(s,x;t,y)$ is given by $$\begin{aligned}
\mbK_{f}(s,x; t,y)
&=& \sum_{a \in f^{-1}(0)} \int_{\sqrt{-1} \, \R}
\frac{dz}{\sqrt{-1}} \,
p_f(0,a; s,x)
\frac{1}{z-a} \frac{f(z)}{f'(a)}
p_f(t, y; 0, z)
\nonumber\\
&&
- {\bf 1}(s>t) p_f(t, y; s, x),
\qquad s,t \geq 0, \quad x, y \in \R
\label{eqn:mbKf}\end{aligned}$$ with setting (\[eqn:sine1\]) and $p_f(s,x;t,y)=p_{\sin}(t-s, y-x)$ with (\[eqn:psin\]), where $f^{-1}(0)$ denotes the [*zero set*]{} of the function $f$; $f^{-1}(0)=\{z: f(z)=0\}$, $f'(a)=df(z)/dz|_{z=a}$, and ${\bf 1}(\omega)$ is the indicator of a condition $\omega$; ${\bf 1}(\omega)=1$ if $\omega$ is satisfied, and ${\bf 1}(\omega)=0$ otherwise. In this paper $\int_{\sqrt{-1} \, \R} dz \, \cdot$ means the integral on the imaginary axis in $\C$ from $-\sqrt{-1} \infty$ to $\sqrt{-1} \infty$. The well-definedness of the correlation kernel $\mbK_{\sin}$ and thus of the process $(\Xi(t), \P_{\sin})$ is guaranteed [@KT08] by the fact that the sine function (\[eqn:sine1\]) is an [*entire function*]{} ([*i.e.*]{}, analytic in the whole complex plane $\C$), and the [*order of growth*]{} $\rho_f$, which is generally defined for an entire function $f$ by $$\rho_f= \limsup_{r \to \infty}
\frac{\log \log M_f(r)}{\log r}
\quad \mbox{for} \quad
M_f(r)=\max_{|z|=r} |f(z)|,$$ is one. (The [*type*]{} defined by $
\sigma_{f}=\limsup_{r \to \infty}
\log M_f(r) / r^{\rho_f}
$ is equal to $\pi$ for (\[eqn:sine1\]). That is, the sine function (\[eqn:sine1\]) is an [*entire function of exponential type*]{} $\pi$ [@Lev96] ; $
M_f(r) \sim e^{\pi r}
$ as $r \to \infty$.) We can show that $$\mbK_{\sin}(t, x; t, y)\mbK_{\sin}(t, y; t, x) dx dy
\to
\xi_{\Z}(dx) {\bf 1}(x=y) \qquad
\mbox{as} \quad t \to 0,$$ since $f^{-1}(0)=\sin^{-1}(0)/\pi=\Z$. It implies that the initial configuration (\[eqn:xiZ\]) of the relaxation process $(\Xi(t), \P_{\sin})$ shall be regarded as the point-mass distribution on the zero set of the sine function (\[eqn:sine1\]). Moreover, we showed in [@KT08], by noting $$\frac{1}{z-a} \frac{f(z)}{f'(a)}
=K_{\sin}(z-a),$$ if $a \in f^{-1}(0)=\Z$ and $z \not=a$ for (\[eqn:sine1\]), that $$\mbK_{\sin}(s+\theta, x; t + \theta, y)
\to
\bK_{\sin}(t-s, y-x)
\quad \mbox{as} \quad \theta \to \infty
\label{eqn:convKsine}$$ with the so-called [*extended sine kernel*]{}, $$\bK_{\sin}(t, x)
= \left\{ \begin{array}{ll}
\displaystyle{
\int_{0}^{1} du \, e^{\pi^2 u^2 t/2}
\cos ( \pi u x ) }
& \mbox{if $t>0 $} \cr
& \cr
K_{\sin}(x)
& \mbox{if $t=0$} \cr
& \cr
\displaystyle{
- \int_{1}^{\infty} du \,
e^{\pi^2 u^2 t/2} \cos (\pi u x)}
& \mbox{if $t<0$},
\end{array} \right.
\label{eqn:sine-kernel}$$ $x \in \R$. The equilibrium dynamics in $\mu_{\sin}$, first studied by Spohn [@Spo87], has been shown to be determinantal with the correlation kernel (\[eqn:sine-kernel\]) by Nagao and Forrester [@NF98]. This process is realized in the long-term limit of the relaxation process $(\Xi(t), \P_{\sin})$. See also the Dirichlet form approach by Osada to the reversible process with respect to $\mu_{\sin}$ [@Osa96; @Osa08].
Now we set $$f(z)=\Ai(z), \quad z \in \C.
\label{eqn:f=Ai}$$ The Airy function $\Ai(z)$, (\[eqn:Airy1\]), is another entire function, whose order of growth is $\rho_f=3/2$ with type $\sigma_f=2/3$; $
\max_{|z|=r} |f(z)| \sim \exp [ (2/3) r^{3/2} ]
$ as $r \to \infty$. For $t>0$, we consider $$p_{\Ai}(t, y|x)=\int_{\R} du \,
e^{ut/2} f(u+x) f(u+y), \quad
x, y \in \R,
\label{eqn:pAi0}$$ which is the solution of the differential equation; $$\frac{\partial}{\partial t} u(t,x)
= \frac{1}{2} \left(
\frac{\partial^2}{\partial x^2} - x \right) u(t,x)
\quad \mbox{with} \quad
\lim_{t \to 0} u(t, x)dx= \delta_y (dx).$$ The integral $\displaystyle{\int_{\R} dz \, p_{\Ai}(t, z|x)}$ is given by $$g(t,x) =
\exp \left( - \frac{tx}{2}+\frac{t^3}{24} \right).
\label{eqn:g1}$$ We find, for $s < t < 0$, $g(s, x) p_{\Ai}(t-s,y|x)/g(t, y)$ is equal to the transition probability density of $$B(t)+\frac{t^2}{4}
\label{eqn:tsquare}$$ from $x$ at time $s$ to $y$ at time $t$, $x, y \in \R$, where $B(t), t \in [0, \infty)$ is the one-dimensional standard Brownian motion. (See also [@FS05] and references therein.) Then for $s, t \in \R, s \not= t, x, y \in \C$, we set $$\begin{aligned}
&& q(s, t, y-x) =
p_{\sin}\left(t-s, \left(y-\frac{t^2}{4} \right)-
\left( x-\frac{s^2}{4} \right) \right)
\nonumber\\
\label{eqn:q}
&& = \frac{1}{\sqrt{2 \pi |t-s|}}
\exp \left[
-\frac{(y-x)^2}{2(t-s)}+\frac{(t+s)(y-x)}{4}
-\frac{(t-s)(t+s)^2}{32} \right],\end{aligned}$$ and as an extension of (\[eqn:pAi0\]) we define $$\begin{aligned}
&& p_{\Ai}(t-s, y|x)
= \frac{g(t,y)}{g(s,x)} q(s, t, y-x)
\nonumber\\
\label{eqn:pAi}
&& = \frac{1}{\sqrt{2\pi|t-s|}}
\exp \left[ -\frac{(y-x)^2}{2(t-s)}
-\frac{(t-s)(y+x)}{4} + \frac{(t-s)^3}{96} \right].\end{aligned}$$ Corresponding to (\[eqn:CK1\]) and (\[eqn:CK2\]), we have the two sets of equalities $$\begin{aligned}
\label{eqn:CKq1}
&& \int_{\R}dy \, q(s,t, z-y) q(0,s,y-x)=q(0,t,z-x) \\
\label{eqn:CKq2}
&& \int_{\R} dy \, q(t,0,z-y) q(s,t, y-x)=q(s,0,z-x)\end{aligned}$$ and $$\begin{aligned}
\label{eqn:CKAi1}
&& \int_{\R} dy \, p_{\Ai}(t-s,z|y) p_{\Ai}(s,y|x)
=p_{\Ai}(t, z|x) \\
\label{eqn:CKAi2}
&& \int_{\R} dy \,
p_{\Ai}(-t,z|y) p_{\Ai}(t-s,y|x)
= p_{\Ai}(-s,z|x)\end{aligned}$$ for $0 < s < t$.
Let $\mbK_{\Ai}$ be the function given by (\[eqn:mbKf\]) with setting (\[eqn:f=Ai\]) and $p_f(s,x;t,y)=p_{\Ai}(t-s,y|x)$ with (\[eqn:pAi\]). We will prove that $\mbK_{\Ai}$ is well-defined as a correlation kernel and it determines finite dimensional distributions of an infinite particle system through a similar formula to (\[eqn:rho0\]). We denote this system by $(\Xi_{\cA}(t), \P_{\Ai})$. The fact that $p_{\Ai}$ used in $\mbK_{\Ai}$ is a transform (\[eqn:pAi\]) of the transition probability density $q$ of (\[eqn:tsquare\]) is the origin of the quadratic term $t^2/4$ in (\[eqn:finite\_Airy\]). We can show that $$\mbK_{\Ai}(t, x; t, y) \mbK_{\Ai}(t, y; t, x) dx dy
\to
\xi_{\cA}(dx) {\bf 1}(x=y) \quad \mbox{as} \quad
t \to 0.$$ By using the integral formula for (\[eqn:f=Ai\]) $$\frac{1}{z-a} \frac{f(z)}{f'(a)}
=\frac{1}{(\Ai'(a))^2}
\int_{0}^{\infty} du \, \Ai(u+z) \Ai(u+a)$$ for $a \in {\cA}, z \not= a$, and the fact that $
\{ \Ai(x+a)/\Ai'(a),
a \in {\cA} \}
$ forms a complete orthonormal basis for the space $L^2(0, \infty)$ of square integrable functions on the interval $(0, \infty)$ [@Tit62], we will prove that $$\mbK_{\Ai}(s+\theta, x; t + \theta, y)
\to
{\bK}_{\Ai}(t-s, y|x)
\quad \mbox{as} \quad \theta \to \infty,
\label{eqn:convKAi}$$ where ${\bK}_{\Ai}$ is the so-called [*extended Airy kernel*]{}, $${\bK}_{\Ai}(t, y|x)
= \left\{
\begin{array}{cc}
\displaystyle{
\int_{0}^{\infty} d u \,
e^{-ut/2} \Ai(u+x) \Ai(u+y)}
& \mbox{if $t \geq 0$} \cr
& \cr
\displaystyle{
- \int_{-\infty}^{0} d u \,
e^{-ut/2} \Ai(u+x) \Ai(u+y) }
& \mbox{if $t < 0$},
\end{array}
\right.
\label{eqn:AiryK}$$ $x, y \in \R$. We denote by $(\Xi_{\cA}(t), {\bP}_{\Ai})$ the infinite particle system, which is determinantal with the correlation kernel ${\bK}_{\Ai}$ [@FNH99; @NKT03; @KNT04; @KT07]. The Airy kernel (\[eqn:AiryKer\]) of $\mu_{\Ai}$ is given by $K_{\Ai}(y|x)={\bf K}_{\Ai}(0, y|x)$ and thus $(\Xi_{\cA}(t), {\bP}_{\Ai})$ is a reversible process with respect to $\mu_{\Ai}$. The process $(\Xi_{\cA}(t), \P_{\Ai})$, which is determinantal with the correlation kernel $\mbK_{\Ai}$, is a non-equilibrium infinite particle system exhibiting the relaxation phenomenon from the initial configuration $\xi_{\cA}$ to the stationary state $\mu_{\Ai}$.
Then consider the finite-particle system (\[eqn:finite\_Airy\]) again. Let $\P_{\cA}^{\xi^N}$ be the distribution of the process $\Xi_{\cA}(t)=\sum_{j=1}^{N} \delta_{Y_j(t)}$ starting from a configuration $\xi^N$. We denote by $\mM$ the space of nonnegative integer-valued Radon measures on $\R$, which is a Polish space with the [*vague topology*]{}: we say $\xi_n$ converges to $\xi$ vaguely, if $\displaystyle{\lim_{n \to \infty} \int_{\R} \varphi(x) \xi_n(dx)
=\int_{\R} \varphi(x) \xi(dx)}$ for any $\varphi \in {\rm C}_0(\R)$, where ${\rm C}_0(\R)$ is the set of all continuous real-valued functions with compact supports. Any element $\xi$ of $\mM$ can be represented as $\xi(\cdot) = \sum_{j\in \Lambda}\delta_{x_j}(\cdot)$ with an index set $\Lambda$ and a sequence of points in $\R$, $\x =(x_j)_{j \in \Lambda}$ satisfying $\xi(I)=\sharp\{x_j: x_j \in I \} < \infty$ for any compact subset $I \subset \R$. For $A \subset \R$, we write the [*restriction*]{} of $\xi$ on $A$ as $(\xi\cap A) (\cdot)= \displaystyle{\sum_{j \in \Lambda : x_j \in A}}
\delta_{x_j}(\cdot)$. We put $
\mM_0= \Big\{ \xi\in\mM :
\xi(\{x\})\le 1 \mbox { for any } x\in\R\Big\}.
$ We will prove that the finite particle process $(\Xi_{\cA}(t), \P_{\cA}^{\xi^N})$ is determinantal for any initial configuration $\xi^N \in \mM_0$ and give the correlation kernel $\mbK_{\cA}^{\xi^N}$ (Proposition \[Proposition:Finite\]). For $\xi \in \mM$ with an infinite number of particles $\xi(\R)=\infty$, when $\mbK_{\cA}^{\xi \cap [-L, L]}$ converges to a continuous function as $L \to \infty$, the limit is written as $\mbK_{\cA}^{\xi}$. If $\P^{\xi \cap [-L, L]}_{\cA}$ converges to a probability measure $\P^{\xi}_{\cA}$ on $\mM^{[0, \infty)}$, which is determinantal with the correlation kernel $\mbK_{\cA}^{\xi}$, weakly in the sense of finite dimensional distributions as $L \to \infty$ in the vague topology, we say that the process $(\Xi_{\cA}(t), \P_{\cA}^{\xi})$ is [*well defined with the correlation kernel*]{} $\mbK_{\cA}^{\xi}$. (The regularity of the sample paths of $\Xi_{\cA}(t)$ will be discussed in the forthcoming paper [@KT09].) We will give sufficient conditions for initial configurations $\xi \in \mM_0$ so that the process $(\Xi_{\cA}(t), \P_{\cA}^{\xi})$ is well defined (Theorem \[Theorem:Infinite\]). We denote by $\mX^{\cA}$ the set of configurations $\xi$ satisfying the conditions and put $\mX^{\cA}_0=\mX^{\cA} \cap \mM_0$. It is clear that the configuration $\xi_{\cA} \in \mX^{\cA}_0$. Then, if we consider the finite particle systems $\Xi_{\cA}(t)=\sum_{j=1}^{N} \delta_{Y_j(t)}, N \geq 2$, with (\[eqn:finite\_Airy\]) starting from the $N$-particle approximation of $\xi_{\cA}$, (\[eqn:xiAN\]), we can prove $(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}^N})
\to (\Xi_{\cA}(t), \P_{\Ai})$ as $N \to \infty$ in the sense of finite dimensional distributions (Theorem \[Theorem:from\_roots\] (i)). That is, $(\Xi_{\cA}(t), \P_{\Ai})=(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}})$ with (\[eqn:xiA\]). Moreover, we will show (\[eqn:convKAi\]) and prove the relaxation phenomenon $(\Xi_{\cA}(t+\theta), \P_{\Ai})
\to (\Xi_{\cA}(t), \bP_{\Ai})$ (Theorem \[Theorem:from\_roots\] (ii)).
The paper is organized as follows. In Sect. 2 preliminaries and main results are given. Some remarks on extensions of the present results are also given there. In Sect. 3 the properties of the Airy function used in this paper are summarized. Section 4 is devoted to proofs of results.
Preliminaries and Main Results
==============================
For $\xi(\cdot)=\sum_{j\in \Lambda}
\delta_{x_j}(\cdot) \in\mM$, we introduce the following operations;
(shift)
: for $u \in \R$, $\tau_u \xi(\cdot) =\displaystyle{\sum_{j \in \Lambda}}
\delta_{x_j+u}(\cdot)$,
(square)
: $\displaystyle{
\xi^{\langle 2 \rangle}(\cdot)
=\sum_{j \in \Lambda} \delta_{x_j^2} (\cdot)}$.
We use the convention such that $$\prod_{x\in\xi}f(x) =\exp
\left\{\int_\R \xi(dx) \log f(x) \right\}
=\prod_{x \in \supp \xi}f(x)^{\xi(\{x\})}$$ for $\xi\in \mM$ and a function $f$ on $\R$, where $\supp \xi = \{x \in \R : \xi(\{x\}) > 0\}$. For a multivariate symmetric function $g$ we write $g((x)_{x \in \xi})$ for $g((x_j)_{j \in \Lambda})$.
Determinantal processes
-----------------------
As an $\mM$-valued process $(\Xi(t), \P^{\xi})$, we consider the system such that, for any integer $M \geq 1$, $f_m \in {\rm C}_{0}(\R), \theta_m \in \R,
1 \leq m \leq M$, $0 < t_1 < \cdots < t_M < \infty$, the expectation of $\displaystyle{\exp \Big\{ \sum_{m=1}^{M} \theta_m
\int_{\R} f_m(x) \Xi(t_m, dx) \Big\}}$ can be expanded by $$\chi_{m}(x)=e^{\theta_{m} f_{m}(x)}-1, \quad
1 \leq m \leq M,$$ as $$\begin{aligned}
{\cG}^{\xi}[\chi] &\equiv&
\E^{\xi} \left[
\exp \left\{ \sum_{m=1}^{M} \theta_m
\int_{\R} f_m(x) \Xi(t_m, dx) \right\} \right]
\nonumber\\
&=&
\sum_{N_{1} \geq 0} \cdots
\sum_{N_{M} \geq 0}
\prod_{m=1}^{M}\frac{1}{N_m !}
\int_{\R^{N_{1}}} \prod_{j=1}^{N_1} d x_{j}^{(1)}
\cdots
\int_{\R^{N_{M}}}
\prod_{j=1}^{N_{M}} d x_{j}^{(M)} \nonumber\\
&& \qquad \times \prod_{m=1}^{M} \prod_{j=1}^{N_{m}}
\chi_{m} \Big(x_{j}^{(m)} \Big)
\rho\Big( t_{1}, \x^{(1)}; \dots ; t_{M}, \x^{(M)} \Big).
\label{eqn:Gxi}\end{aligned}$$ Here $\rho$’s are locally integrable functions, which are symmetric in the sense that $\rho(\dots; t_m, \sigma(\x^{(m)}); \dots)
=\rho(\dots; t_m, \x^{(m)}; \dots)$ with $\sigma(\x^{(m)})
\equiv (x^{(m)}_{\sigma(1)}, \dots, x^{(m)}_{\sigma(N_m)})$ for any permutation $\sigma \in {\cS}_{N_m},
1 \leq \forall m \leq M$. In such a system $\rho( t_{1}, \x^{(1)};
\dots ; t_{M}, \x^{(M)})$ is called the $(N_1, \dots, N_{M})$-[*multitime correlation function*]{} and ${\cG}^{\xi}[\chi]$ the [*generating function of multitime correlation functions*]{}. There are no multiple points with probability one for $t > 0$. Then we assume that there is a function $\mbK(s,x;t,y)$, which is continuous with respect to $(x,y) \in \R^2$ for any fixed $(s,t) \in [0, \infty)^2$, such that $$\rho \Big(t_1,\x^{(1)}; \dots;t_M,\x^{(M)} \Big)
=\det_{
\substack{1 \leq j \leq N_{m}, 1 \leq k \leq N_{n}
\\ 1 \leq m, n \leq M}
}
\Bigg[
\mbK(t_m, x_{j}^{(m)}; t_n, x_{k}^{(n)} )
\Bigg]$$ for any integer $M \geq 1$, any sequence $(N_m)_{m=1}^{M}$ of positive integers, and any time sequence $0 < t_1 < \cdots < t_M < \infty$. Let $\T=\{t_1, \dots, t_M\}$. We note that $\Xi^{\T}=\sum_{t \in \T} \delta_t \otimes \Xi(t)$ is a determinantal (Fermion) point process on $\T \times \R$ with an operator ${\cK}$ given by $${\cK}f(s,x)=\sum_{t \in \T} \int_{\R} dy \,
\mbK(s,x;t, y) f(t,y),
\quad f(t, \cdot) \in {\rm C}_0(\R),\, t \in \T.$$ When ${\cK}$ is symmetric, Soshnikov [@Sos00] and Shirai and Takahashi [@ST03] gave sufficient conditions for $\mbK$ to be a correlation kernel of a determinantal point process. Though such conditions are not known for asymmetric cases, a variety of processes, which are determinantal with asymmetric correlation kernels, have been studied. See, for example, [@TW03; @KT07]. If there exists a function $\mbK$, which has the above properties and determines the finite dimensional distributions of the process $(\Xi(t), \P^{\xi})$, we say the process $(\Xi(t), \P^{\xi})$ is [*determinantal with the correlation kernel*]{} $\mbK$ [@KT08].
For $N \in \N$, the determinant of an $N \times N$ matrix ${\rm M}=(m_{jk})_{1 \leq j, k \leq N}$ is defined by $\displaystyle{\sum_{\sigma \in {\cS}_N}
{\rm sgn}(\sigma) \prod_{j=1}^{N} m_{j \sigma(j)}}$, where ${\rm sgn}(\sigma)$ denotes the sign of permutation $\sigma$. Any permutation $\sigma$ consists of exclusive cycles. If we write each cyclic permutation as $${\sf c}=\left( \begin{array}{cccc}
a & b & \cdots & \omega \cr
b & c & \cdots & a \end{array} \right)$$ and the number of cyclic permutations in a given $\sigma$ as $\ell(\sigma)$, then the determinant of ${\rm M}$ is expressed as $$\det {\rm M}=\sum_{\sigma \in {\cS}_N}
{\rm sgn}(\sigma)
\prod_{{\sf c}_j: 1 \leq j \leq \ell(\sigma)}
\Big( m_{ab} m_{bc} \dots m_{\omega a} \Big).$$ It implies that, with given $a_1, a_2, \dots, a_N$, even if each element $m_{jk}$ of the matrix ${\rm M}$ is replaced by $m_{jk} \times (a_j/a_k)$, the value of determinant is not changed. The above observation will lead to the following lemma.
\[thm:gauge\] Let $(\Xi(t), \P)$ and $(\widetilde{\Xi}(t), \widetilde{\P})$ be the processes, which are determinantal with correlation kernels $\mbK$ and $\widetilde{\mbK}$, respectively. If there is a function $G(s,x)$, which is continuous with respect to $x \in \R$ for any fixed $s \in [0, \infty)$, such that $$\mbK(s,x;t,y)=\frac{G(s,x)}{G(t,y)}
\widetilde{\mbK}(s,x;t,y),
\quad s, t \in [0, \infty), \quad x, y \in \R,
\label{eqn:gauge1}$$ then $$(\Xi(t), \P)=(\widetilde{\Xi}(t), \widetilde{\P})
\label{eqn:gauge2}$$ in the sense of finite dimensional distributions.
0.3cm
In literatures, (\[eqn:gauge1\]) is called the [*gauge transformation*]{} and (\[eqn:gauge2\]) is said to be the [*gauge invariance*]{} of the determinantal processes.
The Weierstrass canonical product and entire functions
------------------------------------------------------
For $\xi^N \in \mM_0, \xi^N(\R)=N < \infty$, with $p \in \N_0 \equiv \N \cup \{0\}$ we consider the product $$\Pi_{p}(\xi^N, z)=
\prod_{x \in \xi^{N} \cap \{0\}^{\rm c}}
G \left( \frac{z}{x}, p \right),
\quad z \in \C,$$ where $$G(u,p) = \left\{
\begin{array}{ll}
\displaystyle{1-u}
& \mbox{if} \quad p=0 \\
& \\
\displaystyle{(1-u)\exp
\left [ u+\frac{u^2}{2}+\cdots +\frac{u^p}{p} \right]}
& \mbox{if} \quad p\in\N.
\end{array} \right.
\label{Weierstrass}$$ The functions $G(u,p)$ are called the [*Weierstrass primary factors*]{}. With $\alpha > 0$ we put $$M_\alpha(\xi^N)
=\left( \int_{\{0\}^{\rm c}}
\frac{1}{|x|^\alpha}\xi^N(dx)
\right)^{1/\alpha}.$$ For $\xi \in \mM_0$ with $\xi(\R)=\infty$, we write $M_\alpha(\xi,L)$ for $M_\alpha(\xi\cap [-L,L]), L > 0$, and put $\displaystyle{
M_\alpha(\xi)= \lim_{L\to\infty}M_\alpha(\xi,L),
}$ if the limit finitely exists. If $M_{p+1}(\xi) < \infty$ for some $p \in \N_0$, the limit $$\Pi_{p}(\xi, z)= \lim_{L \to \infty}
\Pi_{p}(\xi \cap [-L, L], z)
= \prod_{x \in \xi \cap \{0\}^{\rm c}}
G \left( \frac{z}{x}, p \right),
\quad z \in \C
\label{eqn:Pip}$$ finitely exists. This infinite product is called the [*Weierstrass canonical product of genus*]{} $p$ [@Lev96]. The [*Hadamard theorem*]{} [@Lev96] claims that any entire function $f$ of finite order $\rho_f < \infty$ can be represented by $$f(z)=z^m e^{P_{q}(z)}
\Pi_{p}(\xi_f, z),
\label{eqn:Hadamard}$$ where $p$ is a nonnegative integer less than or equal to $\rho_f$, $P_q(z)$ is a polynomial in $z$ of degree $q\le \rho_f$, $m$ is the multiplicity of the root at the origin, and $
\xi_f=\sum_{x \in f^{-1}(0) \cap \{0\}^{\rm c}}
\delta_x.
$ We give two examples ; $$\begin{aligned}
\label{eqn:WH1a}
\sin( \pi z) &=& \pi z \Pi_0(\xi_{\Z}, z), \\
\label{eqn:WH1}
\Ai(z) &=& e^{d_0+d_1 z} \Pi_1(\xi_{\cA}, z)\end{aligned}$$ with (\[eqn:xiZ\]), (\[eqn:xiA\]) and $$\begin{aligned}
d_0 &=&\log \Ai (0) = - \log \Big( 3^{2/3} \Gamma(2/3) \Big),
\nonumber\\
d_1 &=& \frac{\Ai'(0)}{\Ai(0)}
=-\frac{3^{1/3} \Gamma(2/3)}{\Gamma(1/3)}
=- \frac{3^{5/6} (\Gamma(2/3))^2}{2 \pi}.
\label{eqn:d0d1}\end{aligned}$$
For $\xi^N\in\mM_0$ with $\xi^N(\R)=N$ we put $$\Phi_p(\xi^N, a, z) \equiv \Pi_p(\tau_{-a}\xi^N, z-a)
=\prod_{x\in\xi^N \cap \{a\}^{\rm c}}
G\left(\frac{z-a}{x-a},p \right),
\quad a, z \in \C.
\label{entire2}$$ With (\[eqn:xiAN\]) we set $$\begin{aligned}
\Phi_{\cA}(\xi^N,z)
&\equiv& e^{d_1 z} \exp \Bigg[
\int_{\R} \frac{z}{x} \xi_{\cA}^N(dx) \Bigg]
\Pi_0(\xi^N, z)
\nonumber\\
\label{eqn:entire_A0}
&=& e^{d_1 z}
\exp \Bigg[
\int_{\{0\}^{\rm c}} \frac{z}{x} (\xi_{\cA}^N-\xi^N)(dx)
\Bigg]
\Pi_1(\xi^N, z), \quad z \in \C, \\
\Phi_{\cA}(\xi^N,a, z) &\equiv& \Phi_{\cA}(\tau_{-a}\xi^N,z-a)
\nonumber\\
\label{eqn:entire_A}
&=& e^{d_1(z-a)}
\exp \Bigg[
\int_{\R} \frac{z-a}{x} \xi_{\cA}^N(dx) \Bigg]
\Phi_0(\xi^N, a, z),
\quad a, z \in \C. \nonumber\\\end{aligned}$$
\[thm:2\_1\] Let $\xi^N\in\mM_0$ with $\xi^N(\R)=N < \infty$ and $\xi^N(\{0\})=0$. Then for $a \in \supp \xi^{N}, z \not= a$, $$\Phi_{\cA}(\xi^N,a,z)
= \frac{1}{z-a}\frac{\Phi_{\cA}(\xi^N,z)}{\Phi_{\cA}'(\xi^N,a)},
\label{eqn:relation2_A}$$ where $\Phi_{\cA}'(\cdot,a)=\partial \Phi_{\cA}(\cdot,z)/
\partial z \Big|_{z=a}$.
[*Proof.*]{} Since $1- (z-a)/(x-a) = (x-z)/(x-a)
= (1-z/x)/(1-a/x)$, $$\Phi_{\cA}(\xi^N,a,z)=
\frac{\displaystyle{e^{d_1 z} \exp \left[
\int_{\R} \frac{z}{x} \xi^{N}_{\cA}(dx) \right]
\Pi_0(\xi^{N}-\delta_a,z)}}
{\displaystyle{e^{d_1 a} \exp \left[
\int_{\R} \frac{a}{x} \xi^{N}_{\cA}(dx) \right]
\Pi_0(\xi^{N}-\delta_a,a)}},
\label{eqn:eqQ}$$ where the numerator is equal to $\Phi_{\cA}(\xi^N,z)/(1-z/a)$. From (\[eqn:entire\_A0\]), we have $$\begin{aligned}
&& \frac{\partial}{\partial z}
\Phi_{\cA}(\xi^N, z) = d_1 \Phi_{\cA}(\xi^N, z) \nonumber\\
&& \qquad + e^{d_1 z}
\int_{\R} \frac{1}{x} \xi_{\cA}^{N}(dx)
\exp \left[ \int_{\R} \frac{z}{y} \xi_{\cA}^{N}(dy) \right]
\Pi_{0}(\xi^N, z)
\nonumber\\
&& \qquad + e^{d_1 z}
\exp \left[ \int_{\R} \frac{z}{y} \xi_{\cA}^{N}(dy) \right]
\int_{\R} \left(-\frac{1}{x}\right)
\Pi_{0}(\xi^N-\delta_x, z) \xi^{N}(dx).
\nonumber\end{aligned}$$ Since $a \in \supp \xi^N$ is assumed, $$\Phi'_{\cA}(\xi^N, a)=
e^{d_1 a} \exp \left[ \int_{\R} \frac{a}{x} \xi^{N}(dx) \right]
\left(-\frac{1}{a}\right)
\Pi_{0}(\xi^N-\delta_a, a).$$ It implies that the denominator of (\[eqn:eqQ\]) is equal to $-a \Phi'_{\cA}(\xi^N, a)$. Then (\[eqn:relation2\_A\]) is obtained. 0.5cm For $\xi^N\in\mM_0$ with $\xi^N(\R)=N$ we put $$M_{\cA}(\xi^N)
=\int_{\{0\}^{\rm c}} \frac{1}{x} (\xi_{\cA}^N-\xi^N)(dx).
\label{eqn:MAN}$$ For $\xi\in\mM_0$ with $\xi(\R)=\infty$ we write $M_{\cA}(\xi, L)$ for $M_{\cA}(\xi \cap [-L,L]), L > 0$, and put $\displaystyle{
M_{\cA}(\xi)=\lim_{L\to\infty}M_{\cA}(\xi,L),
}$ if the limit finitely exists. For $\xi \in \mM_0$, $p\in\N_0$, $a\in\R$, and $z\in\C$ we define $\displaystyle{
\Phi_p (\xi,a,z)=\lim_{L\to\infty}
\Phi_p(\xi \cap [a-L, a+L], a, z)
}$ and $\displaystyle{
\Phi_{\cA} (\xi,a,z)=\lim_{L\to\infty}
\Phi_{\cA}(\xi \cap [a-L, a+L], a, z)
}$, if the limits finitely exist. We note that $\Phi_p (\xi,a,z)$ finitely exists and is not identically $0$, if $M_{p+1}(\tau_{-a}\xi)<\infty$, and $\Phi_{\cA} (\xi,a,z)$ does and $\Phi_{\cA}(\xi,a,z) \not \equiv0$, if $|M_{\cA}(\tau_{-a}\xi)|<\infty$ and $M_2(\tau_{-a} \xi) < \infty$. For $\xi \in \mM_0, a \in \supp \xi$, the following equalities will hold, if all the entries of them finitely exist ; $$\begin{aligned}
&& \Phi_0(\xi, a, z) = \Pi_0(\xi, z)
\Phi_0(\xi \cap \{0\}^{\rm c}, a, 0)
\left(\frac{z}{a}\right)^{\xi(\{0 \})}
\frac{a}{a-z},
\nonumber\\
&& \Phi_0(\xi \cap \{0\}^{\rm c},a,0)
=\Pi_0(\xi \cap \{-a\}^{\rm c},-a)
\Phi_0(\xi^{\langle 2 \rangle} \cap \{0\}^{\rm c}, a^2,0)
2^{1-\xi(\{-a\})},
\nonumber\end{aligned}$$ and then $$\begin{aligned}
\Phi_1(\xi,a,z) &=& e^{S(\xi,a,z)} \Pi_1(\xi,z)
\Pi_1(\xi\cap \{-a\}^{\rm c},-a)
\nonumber\\
\label{eqn:finite_zeta}
&& \, \times
\Phi_0(\xi^{\langle 2 \rangle} \cap \{0\}^{\rm c}, a^2,0)
\left(\frac{z}{a}\right)^{\xi(\{0 \})}\frac{a}{a-z},\end{aligned}$$ where $$S(\xi,a,z)= \int_{\{a\}^{\rm c}}
\frac{z-a}{x-a} \xi(dx)
-\int_{\{0\}^{\rm c}}
\frac{z}{x} \xi(dx)
+ \int_{\{0,-a\}^{\rm c}}
\frac{a}{x} \xi(dx).
\label{eqn:S1}$$
\[thm:2\_2\] For $a \in \cA, z \not= a$ $$\frac{1}{z-a}
\frac{\Ai(z)}{\Ai'(a)}
= \Phi_{\cA}(\xi_{\cA}, a, z).
\label{eqn:relation1b}$$
[*Proof.*]{} By (\[eqn:WH1\]) and the definition (\[eqn:entire\_A0\]), $$\Ai (z)
=e^{d_0} \Phi_{\cA}(\xi_{\cA},z), \quad z \in \C.
\label{eqn:relation1}$$ As approximations of the Airy function we introduce functions $$\Ai_N(z) = e^{d_0+d_1 z}
\prod_{\ell=1}^{N} \left( 1- \frac{z}{a_{\ell}} \right)
e^{z/a_{\ell}}, \quad N \in \N,
\label{eqn:WH1app}$$ where $0 >a_1 > \cdots > a_N$ are the first $N$ zeros of $\Ai(z)$. Since $\xi_{\cA}^N=\sum_{j=1}^{N} \delta_{a_j}$ satisfies the condition of Lemma \[thm:2\_1\], (\[eqn:relation2\_A\]) with (\[eqn:relation1\]) and $\Ai_N'(a_j)=e^{d_0}
\Phi_{\cA}'(\xi_{\cA}^{N}, a_j)$ gives $$\frac{1}{z-a_{j}}
\frac{\Ai_N(z)}{\Ai_N'(a_j)}
= \Phi_{\cA}(\xi_{\cA}^N, a_j, z),
\quad 1 \leq j \leq N.$$ Taking $N\to\infty$, we have (\[eqn:relation1b\]).
Statement of results
--------------------
For the solution $\X(t)=(X_1(t),X_2(t),\dots,X_N(t))$ of Dyson’s model (\[eqn:Dyson\]) with $\beta=2$ with the initial state $\X(0)=\x$, we denote the distribution of the process $\Xi(t)=\sum_{j=1}^N \delta_{X_j(t)}$ by $\P^{\xi^N}$ with $\xi^N=\sum_{j=1}^{N} \delta_{x_j}$. In [@KT08] we proved that Dyson’s model (\[eqn:Dyson\]) with $\beta=2$ starting from any fixed configuration $\xi^N \in \mM$ is determinantal with the correlation kernel $\mbK^{\xi^N}$ given by $$\begin{aligned}
&& \mbK^{\xi^N}(s, x; t, y)
= \frac{1}{2 \pi \sqrt{-1}}
\oint_{\Gamma(\xi^N)} dz
\int_{\sqrt{-1} \, \R} \frac{dw}{\sqrt{-1}}
\,
\nonumber\\
&& \hskip 3cm \times p_{\sin}(s, x-z)
\frac{1}{w-z}
\prod_{x' \in \xi^{N}}
\left( 1- \frac{w-z}{x'-z} \right)
p_{\sin}(-t, w-y)
\nonumber\\
&&\qquad\qquad\qquad - {\bf 1}(s > t)p_{\sin}(s-t, y-x),
\label{eqn:KN1a}\end{aligned}$$ where $\Gamma(\xi^N)$ is a closed contour on the complex plane $\C$ encircling the points in $\supp \xi^N$ on $\R$ once in the positive direction, and $p_{\sin}$ is given by (\[eqn:psin\]). If $\xi^N\in \mM_0$, by performing the Cauchy integrals (\[eqn:KN1a\]) is written as $$\begin{aligned}
\mbK^{\xi^N}(s, x; t, y)&=& \int_{\R} \xi^N(d x') \,
\int_{\sqrt{-1} \, \R}
\frac{dy'}{\sqrt{-1}} \,
p_{\sin}(s, x-x')
\Phi_0(\xi^{N}, x', y') p_{\sin}(-t, y'-y)
\nonumber\\
&& - {\bf 1}(s > t)p_{\sin}(s-t, x-y).
\label{eqn:KN1}\end{aligned}$$ Then the following is obtained for the process $(\Xi_{\cA}(t), \P_{\cA}^{\xi^N})$ with $\Xi_{\cA}(t)=\sum_{j=1}^{N} \delta_{Y_j(t)}$, where $\Y(t)=(Y_1(t), \dots, Y_N(t))$ is given by (\[eqn:finite\_Airy\]).
\[Proposition:Finite\] The process $(\Xi_{\cA}(t), \P_{\cA}^{\xi^N})$, starting from any fixed configuration $\xi^N\in \mM_0$ with $\xi^N(\R) = N < \infty$, is determinantal with the correlation kernel $\mbK_{\cA}^{\xi^N}$ given by $$\begin{aligned}
\mbK^{\xi^{N}}_{\cA}(s,x ;t,y)
&=&\int_{\R}\xi^N(dx')
\int_{\sqrt{-1} \, \R}
\frac{dy'}{\sqrt{-1}} \,
q(0,s, x-x')
\Phi_{\cA}(\xi^N, x', y')
q(t,0, y'-y)
\nonumber\\
&& - {\bf 1}(s>t)q(t, s, x-y),
\label{eqn:K3}\end{aligned}$$ where $q$ is given by (\[eqn:q\]).
0.3cm We introduce the following conditions:
([**C.1**]{}) there exists $C_0 > 0$ such that $|M_{\cA}(\xi)| < C_0$,
([**C.2**]{}) (i) there exist $\alpha\in (3/2,2)$ and $C_1>0$ such that $
M_\alpha(\xi) \le C_1,
$\
(ii) there exist $\beta >0$ and $C_2 >0$ such that $$M_1(\tau_{-a^2} \xi^{\langle 2 \rangle}) \leq C_2
(|a| \vee 1)^{-\beta}
\quad \mbox{for all} \quad a \in \supp \xi.$$ We denote by $\mX^{\cA}$ the set of configurations $\xi$ satisfying the conditions ([**C.1**]{}) and ([**C.2**]{}), and put $\mX^{\cA}_0 = \mX^{\cA} \cap \mM_0$.
\[Theorem:Infinite\] If $\xi\in \mX^{\cA}_0$, the process $(\Xi_{\cA}(t), \P_{\cA}^{\xi})$ is well defined with the correlation kernel $$\begin{aligned}
&& \mbK^{\xi}_{\cA}(s,x;t,y)
=\int_{\R}\xi(dx')
\int_{\sqrt{-1} \, \R}
\frac{dy'}{\sqrt{-1}} \,
q(0,s, x-x')
\Phi_{\cA}(\xi, x', y')
q(t, 0, y'-y)
\nonumber\\
&& \qquad\qquad\qquad - {\bf 1}(s>t)q(t, s, x-y).
\label{eqn:Kinfinite}\end{aligned}$$
In the proof of this theorem, a useful estimate of $\Phi_{\cA}$ in (\[eqn:Kinfinite\]) is obtained (Lemma \[thm:4\_3\] [(ii)]{}). By virtue of it, we can see $$\mbK_{\cA}^{\xi}(t, x; t, y)
\mbK_{\cA}^{\xi}(t, y; t, x)
dxdy \to \xi(dx){\bf 1}(x=y)
\quad \mbox{as} \quad t \to 0
\label{eqn:strongC}$$ in the vague topology. Then Theorem \[Theorem:Infinite\] gives an infinite particle system starting form the configuration $\xi$.
The main result of the present paper is the following.
\[Theorem:from\_roots\] [(i)]{} Let $\xi_{\cA}^N(\cdot)$ be the configuration (\[eqn:xiAN\]). Then $$(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}^N})
\to
(\Xi_{\cA}(t), \P_{\Ai})
\quad \mbox{as} \quad N \to \infty$$ in the sense of finite dimensional distributions. Here the process $(\Xi_{\cA}(t), \P_{\Ai})$ is determinantal with the correlation kernel (\[eqn:mbKf\]) with setting (\[eqn:f=Ai\]) and $p_f(s, x; t, y)=p_{\Ai}(t-s, y|x)$ with (\[eqn:pAi\]), that is $$\begin{aligned}
\mbK_{\Ai}(s,x;t,y)
&=&
\sum_{a \in \Ai^{-1}(0)}
\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}} \,
p_{\Ai}(s,x|a)
\frac{1}{z-a} \frac{\Ai(z)}{\Ai'(a)}
p_{\Ai}(-t,z|y)
\nonumber\\
&& \quad - {\bf 1}(s>t)p_{\Ai}(s-t,x|y).
\label{eqn:K3a}\end{aligned}$$
[(ii)]{} Let $(\Xi_{\cA}(t), \bP_{\Ai})$ be the process, which is determinantal with the extended Airy kernel (\[eqn:AiryK\]). Then $$(\Xi_{\cA}(t+\theta), \P_{\Ai}) \to (\Xi_{\cA}(t), \bP_{\Ai})
\quad \mbox{as} \quad \theta\to\infty
\label{relax}$$ weakly in the sense of finite dimensional distributions.
Remarks on Extensions of the Results
------------------------------------
\(1) By definition (\[eqn:MAN\]), $M_{\cA}(\xi_{\cA})=0$. The asymptotic property of the zeros (\[eqn:ainf\]) implies $$\zeta^{\cA}(\alpha) \equiv
\Big( M_{\alpha}(\xi_{\cA}) \Big)^{\alpha}
=\sum_{a \in \cA} \frac{1}{|a|^{\alpha}} < \infty,
\quad \mbox{if} \quad
\alpha > \widehat{\rho}_f=\frac{3}{2}.
\label{eqn:Azeta}$$ In general, order of growth $\rho_f$ of a canonical produce (\[eqn:Pip\]) is equal to the [*convergence exponent*]{} $\widehat{\rho}_f$ of the sequence of its zeros [@Lev96]. For $\Ai(z)$, $\rho_f=3/2$. The function $\zeta^{\cA}(\alpha)$ may be called the [*Airy zeta function*]{} [@VS04], which is meromorphic in the whole of $\C$ [@FL01]. Moreover, we know $$\zeta^{\cA}(2)=M_1(\xi_{\cA}^{\langle 2 \rangle})
=\sum_{a \in {\cA}} \frac{1}{a^2}
=d_1^2 < \infty
\label{eqn:M1A}$$ with (\[eqn:d0d1\]). Then $\xi_{\cA}$ satisfies the conditions ([**C.1**]{}) and ([**C.2**]{}) : $\xi_{\cA} \in \mX^{\cA}$. Since $\xi_{\cA} \in \mM_0$, Theorem \[Theorem:Infinite\] guarantees the well-definedness of the infinite particle system $(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}})$. (Its equivalence with $(\Xi(t), \P_{\Ai})$ is stated in Theorem \[Theorem:from\_roots\] (i).) Note that the negative divergence (\[eqn:Dinf\]) of the drift term $D_{\cA_N} t$ of (\[eqn:finite\_Airy\]) in $N \to \infty$ for $t < \infty$ corresponds to that $\zeta^{\cA}(1)=-\sum_{a \in \cA}(1/a)=\infty$. This fact and (\[eqn:M1A\]) mean that the Airy function has [*genus*]{} 1 [@Lev96; @VS04].
Examples of infinite particle configurations in $\mX^{\cA}_0$ other than $\xi_{\cA}$ are given as follows. For $\kappa>0$, we put $$g^\kappa(x) ={\rm sgn}(x) |x|^{\kappa}, \ x\in\R,
\mbox{ and }
\eta^{\kappa}(\cdot)=\sum_{\ell\in\Z}
\delta_{g^\kappa(\ell)}(\cdot).$$ For any $\kappa > 1/2$ we can confirm by simple calculation that any configuration $\xi \in \mM_0$ with $\supp \xi \subset \supp \eta^{\kappa}
=\{g^{\kappa}(\ell): \ell \in \Z\}$ satisfies ([**C.2**]{})(i) with any $\alpha \in (1/\kappa, 2)$ and some $C_1=C_1(\alpha)>0$ depending on $\alpha$ and ([**C.2**]{})(ii) with any $\beta \in (0, 2 \kappa-1)$ and some $C_2=C_2(\beta) >0$ depending on $\beta$. Assume that $\xi \in \mM_0$ is chosen so that $\supp \xi \subset \supp \eta^{\kappa}$ for some $\kappa > 1/2$ and $|M_{\cA}(\xi)| < \infty$. Then $\xi \in \mX^{\cA}_0$. The fact (\[eqn:ainf\]) implies that this assumption can be satisfied only if $\kappa \in (1/2, 2/3]$.
0.3cm (2) If there exists, however, $\beta' < (\beta-1) \wedge (\beta/2)$ for $\xi \in \mM_0$ such that $\sharp \{ x \in \xi:
\xi([x-|x|^{\beta'}, x+|x|^{\beta'}]) \geq 2\} = \infty$, then $\xi$ does not satisfy the condition ([**C.2**]{}) (ii). In order to include such initial configurations as well as those with multiple points in our study of the process $(\Xi_{\cA}(t), \P_{\cA}^{\xi})$ with $\xi(\R)=\infty$, we introduce another condition for configurations:
([**C.3**]{}) there exists $\kappa \in (1/2,2/3]$ and $m\in\N$ such that $$m(\xi,\kappa)\equiv
\max_{k\in\Z} \xi\bigg( [ g^\kappa(k), g^\kappa(k+1)] \bigg) \le m.$$
We denote by $\mY^{\cA}_{\kappa, m}$ the set of configurations $\xi$ satisfying ([**C.1**]{}) and ([**C.3**]{}) with $\kappa\in (1/2,2/3]$ and $m\in\N$, and put $$\mY^{\cA} = \bigcup_{\kappa\in (1/2,2/3]}\bigcup_{m\in\N}
\mY^{\cA}_{\kappa, m}.$$ Noting that the set $\{\xi \in \mM: m(\xi,\kappa)\le m\}$ is relatively compact for each $\kappa\in (1/2,2/3]$ and $m\in \N$, we see that $\mY^{\cA}$ is locally compact.
In the present paper, we report our study of the relaxation process $(\Xi(t), \P_{\Ai})$ from a special initial configuration $\xi_{\cA}$ to the stationary state $\mu_{\Ai}$. We expect that $\mu_{\Ai}$ is an attractor in the configuration space $\mY^{\cA}$ and $\xi_{\cA}$ is a point included in the basin. Motivated by such consideration, we are interested in the continuity of the process with respect to initial configuration. We have found, however, that if $\xi(\R)=\infty$, the weak convergence of processes in the sense of finite dimensional distributions can not be concluded from the convergence of initial configurations in the vague topology. Following the idea given by our previous paper [@KT08], we introduce a stronger topology for $\mY^{\cA}$.
Suppose that $\xi, \xi_n \in \mY^{\cA}, n \in \N$. We say that $\xi_n$ converges $\Phi_{\cA}$-[*moderately*]{} to $\xi$, if $$\lim_{n\to\infty} \Phi_{\cA}(\xi_n,\sqrt{-1},\cdot)
= \Phi_{\cA}(\xi,\sqrt{-1},\cdot)
\mbox{ uniformly on any compact set of $\C$.}
\label{eqn:THC2}$$ It is easy to see that (\[eqn:THC2\]) is satisfied, if the following two conditions hold: $$\begin{aligned}
&&\lim_{L\to\infty}\sup_{n>0 }\lim_{M\to\infty}
\Bigg|
M_{\cA}(\xi_n, M)- M_{\cA}(\xi_n, L)
\Bigg|=0,
\label{eqn:THC3}
\\
&&\lim_{L\to\infty} \sup_{n>0 }
\Bigg| M_2(\xi_n)- M_2(\xi_n, L)
\Bigg|=0.
\label{eqn:THC4}\end{aligned}$$ By the similar argument given in [@KT08], the following statements are proved.\
[(i)]{} If $\xi\in\mY^{\cA}$, the process $(\Xi_{\cA}, \P_{\cA}^{\xi})$ is well defined.
[(ii)]{} Suppose that $\xi, \xi_n \in \mY^{\cA}_{\kappa, n}, n \in\N$, for some $\kappa\in (1/2,2/3]$ and $m\in\N$. If $\xi_n$ converges $\Phi_{\cA}$-moderately to $\xi$, then $
(\Xi_{\cA}, \P_{\cA}^{\xi_n})
\to
(\Xi_{\cA}, \P_{\cA}^{\xi})$ weakly in the sense of finite dimensional distributions as $n \to \infty$ in the vague topology.
Moreover, we can show $\mu_{\Ai}(\mY^{\cA})=1$. By this fact and the above mentioned continuity with respect to initial configurations, we can prove that the stationary process $(\Xi_{\cA}(t), \bP_{\Ai})$, which is determinantal with the extended Airy kernel (\[eqn:AiryK\]), is [*Markovian*]{} [@KT09].
0.3cm (3) As mentioned in Introduction, the purpose of the present paper is to give a method for [*asymmetric*]{} initial configurations to construct infinite particle systems of Brownian motions interacting through pair force $1/x$. In order to clarify the results, we have concentrated on the case in this paper such that the initial configuration is $\xi_{\cA}$ (Theorem \[Theorem:from\_roots\]) or its modification $\xi \in \mX^{\cA}_0$; see the condition [**(C.1)**]{} with (\[eqn:MAN\]) (Theorem \[Theorem:Infinite\]). In the former case the constructed infinite particle system $(\Xi_{\cA}(t), \P_{\Ai})$ has the stationary measure $\mu_{\Ai}$, which is obtained in the soft-edge scaling limit of the eigenvalue distribution in GUE well-studied in the random matrix theory. Thus we have specified the entire function used in our analysis in the form $\Phi_{\cA}(\xi,z)$ given by (\[eqn:entire\_A0\]), which is suitable for the Airy function (see (\[eqn:relation1\])). The point of our method is to put the relationship between the entire function appearing in the correlation kernel $\mbK^{\xi}$, the “typical" initial configuration $\xi_{f}$, and the drift term in the SDEs providing finite-particle approximations. By the same argument as reported here, the following will be proved. Let $f$ be the entire function such that $f(0) \not=0$, it is expressed by the Weierstrass canonical product of genus one, $\Pi_1(\xi_f,z)$, and the zeros can be labelled as $0 < |x_1| < |x_2| < \cdots$. Then with $\xi_f=\sum_{j=1}^{\infty} \delta_{x_j}$ we put $$D(\xi_f,N)=\sum_{j=1}^N \frac{1}{x_j}$$ and introduce the $N$-particle system $$Y^f_j(t)=X_j(t)+D(\xi_f, N) t, \quad
1 \leq j \leq N, \quad t \in [0, \infty),$$ where $\X(t)=(X_1(t), \dots, X_N(t))$ is the solution of Dyson’s model (\[eqn:Dyson\]) starting from the first $N$ zeros of $f$, $X_j(0)=x_j, 1 \leq j \leq N$. Then $\Xi_f(t)=\sum_{j=1}^{N} \delta_{Y^f_j(t)}$ converges to the dynamics in $N \to \infty$, in the sense of finite dimensional distribution, which is determinantal with the correlation kernel $$\begin{aligned}
\mbK_f^{\xi_f}(s,x ;t,y)
&=&\int_{\R}\xi_f(dx')
\int_{\sqrt{-1} \, \R}
\frac{dy'}{\sqrt{-1}}
p_{\sin}(s, x-x')
\Phi_1(\xi_f,x',y')
p_{\sin}(-t, y'-y)
\nonumber\\
&& - {\bf 1}(s>t)p_{\sin}(s-t, x-y).\end{aligned}$$ Moreover, even if the initial configuration $\xi$ is different from $\xi_f$, but it satisfies the condition $$\left| \int_{-L}^{L} \frac{1}{x}
(\xi_f-\xi)(dx) \right| < C_0$$ for any $L > 0$ with a positive finite $C_0$ independent of $L$, then the process starting from $\xi$, is well-defined. In general, the obtained dynamics with an infinite number of particles is not stationary, while Theorem \[Theorem:from\_roots\] gave the example which converges to a stationary dynamics $(\Xi_{\cA}(t), \bP_{\Ai})$ in the long-term limit.
Properties of the Airy Functions
================================
\[chap: Airy\]
Integrals
---------
By the fact $\Ai''(x)=x \Ai(x)$, the following primitive is obtained for $c \not=0$ [@VS04], $$\begin{aligned}
\label{eqn:prim1}
&&\int du \, ( \Ai(c(u+x)) )^2
=(u+x) ( \Ai(c(u+x)) )^2
-\frac{1}{c} ( \Ai'(c(u+x)) )^2, \\
&&\int du \, \Ai(c(u+x)) \Ai(c(u+y))
\nonumber\\
\label{eqn:prim2}
&& \qquad
= \frac{\Ai'(c(u+x)) \Ai(c(u+y))
- \Ai(c(u+x)) \Ai'(c(u+y))}
{c^2(x-y)}.\end{aligned}$$ By setting $c=1$ and integral interval be $[0, \infty)$ in (\[eqn:prim2\]), we obtain the integral $$\int_{0}^{\infty} du \, \Ai(u+x) \Ai(u+y)
=\frac{\Ai(x) \Ai'(y)-\Ai'(x) \Ai(y)}
{x-y},$$ since $\lim_{x \to \infty} \Ai(x)=\lim_{x \to \infty} \Ai'(x)=0$ by (\[eqn:Aiasym\]). If we set $y = a \in {\cA}$ and $x=z \not=a$, then $$\int_{0}^{\infty} du \,
\Ai(u+z) \Ai(u+a)
=\frac{\Ai(z) \Ai'(a)}{z-a},$$ since $\Ai(a)=0$. Then we have the expression $$\frac{1}{z-a}
\frac{\Ai(z)}{\Ai'(a)}
=\frac{1}{(\Ai'(a))^2}
\int_{0}^{\infty} du \,
\Ai(u+z) \Ai(u+a)
\label{eqn:exp1}$$ for $a \in {\cA}, z \not=a$.
Airy transform
--------------
The following integral formulas are proved [@Joh03].
\[thm:eAiAi\] For $c > 0, x, y \in \R$ $$\begin{aligned}
\label{eqn:eAiAi}
&& \int_{\R} du \, e^{c u} \Ai(u+x) \Ai(u+y)
=\frac{1}{\sqrt{4 \pi c}}
e^{-(x-y)^2/(4c)-c(x+y)/2+c^3/12}, \\
\label{eqn:eAiAi2}
&& \int_{\R} d y \int_{\R} du \,
e^{c u} \Ai(u+x) \Ai(u+y)
= e^{-cx+c^3/3}.\end{aligned}$$
[*Proof.*]{} Consider the integral $$\begin{aligned}
I &=& \int_{\R} du \, e^{c x}
\Ai(u+x) \Ai(u+y) \nonumber\\
&=& \int_{0}^{\infty} du \, e^{c u}
\Ai(u+x) \Ai(u+y)
+ \int_{-\infty}^{0} du \, e^{c u}
\Ai(u+x) \Ai(u+y).
\nonumber\end{aligned}$$ By the definition of the Airy function (\[eqn:Airy1\]), for any $\eta >0$, $$\begin{aligned}
I &=& \int_{0}^{\infty} du \, e^{c u}
\frac{1}{(2 \pi)^2} \int_{\Im z = \eta} dz \,
\int_{\Im w > c - \eta} d w \,
e^{\sqrt{-1} \{ z^3/3+(u+x)z +w^3/3+(y+u)w \}}
\nonumber\\
&+& \int_{-\infty}^{0} du \, e^{c u}
\frac{1}{(2 \pi)^2} \int_{\Im z = \eta} dz \,
\int_{\Im w < c - \eta} d w
e^{\sqrt{-1}\{ z^3/3+ (u+x)z+w^3/3+(u+y)w \}}
\nonumber\\
&=& \frac{1}{(2\pi)^2} \int_{\Im z = \eta} dz
\int_{\Im w > c - \eta} dw \,
e^{\sqrt{-1}(z^3/3+ x z + w^3/3+ y w)}
\int_{0}^{\infty} du \,
e^{ \{c+\sqrt{-1}(z+w)\} u } \nonumber\\
&+& \frac{1}{(2\pi)^2} \int_{\Im z = \eta} dz
\int_{\Im w < c - \eta} dw \,
e^{\sqrt{-1}(z^3/3+x z+ w^3/3+yw)}
\int_{-\infty}^{0} du \,
e^{ \{ c+\sqrt{-1}(z+w)\} u } \nonumber\\
&=& - \frac{1}{(2\pi)^2} \int_{\Im z = \eta} dz
\int_{\Im w > c - \eta} dw \,
e^{\sqrt{-1}(z^3/3+xz + w^3/3+ yw)}
\frac{1}{c+\sqrt{-1}(z+w)} \nonumber\\
&+& \frac{1}{(2\pi)^2} \int_{\Im z = \eta} dz
\int_{\Im w < c - \eta} dw \,
e^{\sqrt{-1}(z^3/3+xz + w^3/3+yw)}
\frac{1}{c+\sqrt{-1}(z+w)}
\nonumber\\
&=& \frac{1}{(2 \pi)^2} \int_{\Im z =\eta} dz \,
e^{\sqrt{-1}(z^3/3+xz)}
\left\{ \int_{\Im w < c-\eta} d w
- \int_{\Im w > c-\eta} dw \right\}
\frac{e^{\sqrt{-1}(w^3/3+yw)}}{c+\sqrt{-1}(z+w)}.
\nonumber\end{aligned}$$ It is equal to the integral $$\frac{1}{2\pi} \int_{\Im z = \eta} dz \,
e^{\sqrt{-1}(z^3/3+xz)}
\frac{1}{2 \pi \sqrt{-1}}
\oint_{C} dw \, \frac{e^{\sqrt{-1}(w^3/3+\sqrt{-1}yw)}}
{w-(\sqrt{-1} c-z)},$$ where $C$ is a closed contour on $\C$ encircling a pole at $w=\sqrt{-1} c-z$ once in the positive direction. By performing the Cauchy integral, we have $$\begin{aligned}
I &=& \frac{1}{2 \pi} \int_{\Im z = \eta} dz \,
e^{\sqrt{-1}(z^3/3+xz)}
e^{\sqrt{-1} (\sqrt{-1} c-z)^3/3
-y (c+\sqrt{-1}z)}
\nonumber\\
&=& \frac{1}{2\pi} e^{c^3/3-c y}
\int_{\Im z = \eta} dz \,
e^{ - c z^2 + \sqrt{-1} (x-y+c^2) z }.
\nonumber\end{aligned}$$ By performing a Gaussian integral, we have (\[eqn:eAiAi\]). Since $$-\frac{1}{4 c}(x-y)^2 - \frac{c}{2}(x+y)
+\frac{c^3}{12}
= - \frac{1}{4 c} \Big\{ y -(x-c^2) \Big\}^2
- c x + \frac{c^3}{3},$$ the Gaussian integration of (\[eqn:eAiAi\]) with respect to $y$ gives (\[eqn:eAiAi2\]). 0.5cm By setting $c=t/2 >0$ in (\[eqn:eAiAi2\]) and $c=(t-s)/2 >0$ in (\[eqn:eAiAi\]), respectively, we obtain the equalities $$\begin{aligned}
&& \int_{\R} dy \int_{\R} du \, e^{ut}
\Ai(u+x) \Ai(u+y)= g(t,x),
\nonumber\\
&& \int_{\R} du \,
e^{u(t-s)/2} \Ai(u+x) \Ai(u+y)
= \frac{g(t,y)}{g(s,x)} q(s,t, y-x)
\nonumber\end{aligned}$$ with (\[eqn:g1\]) and (\[eqn:q\]). Thus we have defined $g(t,x)$ for any $ t \in \R$ by (\[eqn:g1\]) and $p_{\Ai}(s,x;t,y)$ for any $s, t \in \R, s \not= t$ by (\[eqn:pAi\]). As special cases, we have $$\begin{aligned}
\label{eqn:equalM1}
p_{\Ai}(t,y|x)
&=& g(t,y) q(0,t,y-x), \\
\label{eqn:equalM2}
p_{\Ai}(-t,y|x) &=& \frac{1}{g(t,x)} q(t,0,y-x),
\quad t > 0, x,y, \in \R.\end{aligned}$$
If we take the $c \to 0$ limit in (\[eqn:eAiAi2\]), we obtain $$\int_{\R} d \xi \,
\int_{\R} d x \,
\Ai(\xi-x) \Ai(\xi'-x) = 1.$$ The expression (\[eqn:eAiAi\]) and the above result implies the orthonormality of the Airy function in the sense; $$\left(\int_{\R} du \,
\Ai(u+x) \Ai(u+y) \right) dy =\delta_x(dy).
\label{eqn:ortho}$$ The [*Airy transform*]{} $f(x) \mapsto \varphi(\xi)$ is then defined by $$\varphi(\xi)=\int_{\R} dx \,
f(x) \Ai(\xi+x),
\label{eqn:Atran}$$ and the inverse transform is given by $\displaystyle{
f(x)=\int_{\R} d \xi \, \varphi(\xi)
\Ai(\xi+x).
}$ Now a parameter $c \in \C$ is introduced and the family of functions are defined as $\{
w_{c}(x)= \Ai(x/c)/|c|
\}$. The Airy transform (\[eqn:Atran\]) is then generalized as $$\varphi_{c}(\xi) =
\int_{\R} d \xi \, f(x) w_{c}(\xi+x)
= \frac{1}{|c|} \int_{\R} dx f(x)
\Ai \left( \frac{\xi+x}{c} \right).$$
\[thm:AiryTrans\] The Airy transform with $c$ of the normalized Gaussian function $
f(x)=e^{-x^2}/\sqrt{\pi}
$ is given by $\varphi_{c}(\xi)=|c|^{-1}
e^{\{\xi+1/(24 c^3)\}/(4c^3)}
\Ai(\xi/c+1/(16 c^4))$. That is, $$\int_{\R} dx \,
\frac{1}{\sqrt{\pi}} e^{-x^2}
\Ai \left( \frac{\xi+x}{c} \right)
= \exp \left\{ \frac{1}{4 c^3}
\left( \xi + \frac{1}{24 c^3} \right) \right\}
\Ai \left( \frac{\xi}{c}
+ \frac{1}{16 c^4} \right).
\label{eqn:AiTrans}$$
[*Proof.*]{} By the definition of the Airy function (\[eqn:Airy1\]), $$\begin{aligned}
I &=& \int_{\R} dx \,
\frac{1}{\sqrt{\pi}} e^{-x^2}
\Ai \left( \frac{\xi+x}{c} \right)
\nonumber\\
&=& \int_{\R} dx \,
\frac{1}{\sqrt{\pi}} e^{-x^2}
\frac{1}{2 \pi} \int_{\R} dk \,
e^{\sqrt{-1}\{k^3/3+(\xi+x)k/c\}}
\nonumber\\
&=& \frac{1}{2\pi} \int_{\R} dk \,
e^{\sqrt{-1} k^3/3+ \sqrt{-1} \xi k/c}
\frac{1}{\sqrt{\pi}} \int_{\R} dx \,
\exp \left( - x^2 - \sqrt{-1} \frac{k}{c} x \right).
\nonumber\end{aligned}$$ By performing the Gaussian integral we have $$I=\frac{1}{2 \pi } \int_{\R} dk \,
\exp \left( \sqrt{-1} \frac{k^3}{3}
+ \sqrt{-1} \frac{\xi k}{c} - \frac{k^2}{4 c^2} \right).$$ By completing a cube, we find the equality $$\begin{aligned}
&& \sqrt{-1} \frac{k^3}{3}
+ \sqrt{-1} \frac{\xi k}{c} - \frac{k^2}{4 c^2}
=\sqrt{-1} \frac{1}{3} \left(
k+\sqrt{-1} \frac{1}{4 c^2} \right)^3
\nonumber\\
&& \qquad
+ \sqrt{-1} \left( \frac{\xi}{c}+\frac{1}{16 c^4} \right)
\left(k + \sqrt{-1} \frac{1}{4 c^2} \right)
+ \frac{1}{4 c^3}
\left( \xi + \frac{1}{24 c^3} \right).
\nonumber\end{aligned}$$ By using the definition of the Airy function (\[eqn:Airy1\]), (\[eqn:AiTrans\]) is obtained. 0.3cm For $t>0, y, u \in \R$ we will obtain the equality $$\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}} \,
\Ai(u+z) q(t, 0, z-y)
=\int_{\R} dx \, \frac{1}{\sqrt{\pi}}
e^{-x^2} \Ai \left( \sqrt{-2t} \, x +
u+y-\frac{t^2}{4} \right)$$ by changing the integral variable as $z \mapsto x=(z-y+t^2/4)/\sqrt{-2t}$. If we set $\xi=(u+y-t^2/4)/\sqrt{-2t}$ and $c=1/\sqrt{-2t}$, the RHS is identified with the LHS of (\[eqn:AiTrans\]). Since $$\frac{\xi}{c}+\frac{1}{16c^4}=u+y, \quad
\frac{1}{4 c^3} \left( \xi+\frac{1}{24 c^3} \right)
=\left( -\frac{ty}{2}+\frac{t^3}{24} \right) - \frac{ut}{2},$$ (\[eqn:AiTrans\]) of Lemma \[thm:AiryTrans\] with (\[eqn:g1\]) gives $$\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}} \,
\Ai(u+z) q(t, 0, z-y)
= g(t, y) e^{-ut/2} \Ai(u+y),
\quad t >0, y, u \in \R.$$ Combination with (\[eqn:equalM2\]) gives $$\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}} \,
\Ai(u+z) p_{\Ai}(-t, z|y)
= e^{-ut/2} \Ai(u+y),
\quad t >0, y, u \in \R
\label{eqn:equalM3}$$
Fourier-Airy series
-------------------
Let us consider the integral $$I_{\ell \ell'} = \int_{0}^{\infty} dx \,
\Ai(x+a_{\ell}) \Ai(x+a_{\ell'}),
\quad a_{\ell}, a_{\ell'} \in {\cA}.$$ In the case $\ell \not= \ell'$, the formula (\[eqn:prim2\]) gives $$I_{\ell \ell'}
=\frac{\Ai'(a_{\ell}) \Ai(a_{\ell'})
-\Ai(a_{\ell}) \Ai'(a_{\ell'})}{a_{\ell}-a_{\ell'}}=0,$$ whereas if $\ell =\ell'$, the formula (\[eqn:prim1\]) gives $
I_{\ell \ell}= ( \Ai'(a_{\ell}) )^2.
$ Therefore the functions $$\left\{ \frac{\Ai(x+a_{\ell})}{\Ai'(a_{\ell})},
\ell \in \N \right\}
\label{eqn:Aiset}$$ form an orthogonal basis for $f \in L^2(0, \infty)$ (see Sect. 4.12 in [@Tit62]). The completeness of (\[eqn:Aiset\]) is also established : $$\sum_{\ell \in \N}
\frac{\Ai(x+a_{\ell}) \Ai(y+a_{\ell})}
{( \Ai'(a_{\ell}) )^2} dy
=\delta_{x} (dy),
\quad x,y \in (0,\infty).
\label{eqn:complete}$$ Then for any $f \in L^2(0, \infty)$, we can write the expression $$f(x)=\sum_{\ell \in \N} c_{\ell}
\frac{\Ai(x+a_{\ell})}{\Ai'(a_{\ell})},
\quad x\in [0,\infty),$$ and call it the [*Fourier-Airy series expansion*]{}. The coefficients $c_{\ell}$ of this expansion are determined by $
c_{\ell}=\{
\int_{0}^{\infty} dx \, f(x) \Ai(x+a_{\ell})\}
/\Ai'(a_{\ell})
$.
Proof of Results
================
Proof of Proposition \[Proposition:Finite\]
-------------------------------------------
With (\[eqn:DA\]) we put $$\widehat{g}(s,x)=\exp\left\{
-D_{\cA_N} \left(
\frac{D_{\cA_N} s}{2}+\frac{s^2}{4}-x \right)\right\},
\, s,x\in\R.$$ By the definition (\[eqn:q\]) of $q$, for $s >0, x, x' \in \R$, we have $$\begin{aligned}
&&\frac{p_{\sin}(s,(x-D_{\cA_N}s-s^2/4)-x')}
{q(0, s, x-x')}
\nonumber\\
&& \qquad =\exp\left[
-\frac{1}{2s}
\left\{
\left(x-D_{\cA_N}s -\frac{s^2}{4}-x' \right)^2
-\left(x-\frac{s^2}{4}-x' \right)^2 \right\} \right]
\nonumber\\
&& \qquad =\exp\left[
-D_{\cA_N} s \left(
\frac{D_{\cA_N}s}{2} + \frac{s^2}{4}-x+x' \right) \right]
\nonumber\\
&& \qquad =\widehat{g}(s,x) e^{-D_{\cA_N} x'},
\nonumber\end{aligned}$$ and for $t > 0, y, y' \in \R$, we have $$\frac{p_{\sin}(-t, y'-(y-D_{\cA_N}t-t^2/4))}
{q(t,0, y'-y)}
=\frac{1}{\widehat{g}(t,y)} e^{D_{\cA_N} y'}.
\nonumber$$ Similarly, we have $$\begin{aligned}
&&\frac{p_{\sin}(s-t,(x-D_{\cA_N} s-s^2/4)-
(y-D_{\cA_N} t-t^2/4))}{q(t, s, y-x)}
\nonumber\\
&&=\exp\left[
-\frac{1}{2(s-t)}
\left\{
\left(x-y-D_{\cA_N}(s-t)-\frac{s^2-t^2}{4} \right)^2
\right. \right. \nonumber\\
&& \qquad \qquad \left. \left.
-\left( \left(x-\frac{s^2}{4} \right)
-\left(y-\frac{t^2}{4} \right) \right)^2
\right\}
\right]
\nonumber\\
&&=\exp\left[- D_{\cA_N}
\left\{ \frac{D_{cA_N}}{2}(s-t)
+\frac{s^2-t^2}{4}-(x-y) \right\} \right]
=\frac{\widehat{g}(s,x)}{\widehat{g}(t,y)}.
\nonumber\end{aligned}$$ Then we have $$\begin{aligned}
&& \mbK^{\xi^N}\left(s, x-D_{\cA_N}s-\frac{s^2}{4};
t, y-D_{\cA_N}t- \frac{t^2}{4} \right)
\nonumber\\
&&
= \frac{\widehat{g}(s,x)}{\widehat{g}(t,y)}
\left[
\int_{\R} \xi^{N}(dx')
\int_{\sqrt{-1} \, \R}
\frac{dy'}{\sqrt{-1}} \, \right.
\nonumber\\
&& \qquad \qquad \times
q(0, s, x-x')
e^{-D_{\cA_N}x'} \Phi_0(\xi^N, x', y') e^{D_{\cA_N}y'}
q(t, 0, y'-y) \nonumber\\
&& \hskip 6cm
-{\bf 1}(s > t) q(t, s, x-y) \Bigg].
\nonumber\end{aligned}$$ The identity (\[eqn:entire\_A\]) implies $$e^{-D_{\cA_N}x'} \Phi_0(\xi^N,x',y')
e^{D_{\cA_N}y'}
=\Phi_{\cA}(\xi^N,x',y').$$ By the gauge invariance of determinantal processes (Lemma \[thm:gauge\]), the proof is completed.
Proof of Theorem \[Theorem:Infinite\]
-------------------------------------
First we prepare some lemmas for proving Theorem \[Theorem:Infinite\].
\[thm:4\_1\] Let $\alpha\in (1,2)$ and $\delta> \alpha-1$. Suppose that $M_\alpha(\xi)<\infty$ and put $L_0=L_0(\alpha, \delta, \xi)=
(2M_\alpha(\xi))^{\alpha/(\delta-\alpha+1)}$. Then $$M_1 (\xi,L) \le L^\delta, \quad L\ge L_0.$$
Since this lemma was proved as Lemma 4.3 in [@KT08], here we omit the proof.
\[thm:4\_2\] If $\xi$ satisfies [([**C.2**]{})]{} [(i)]{} and [(ii)]{}, for any $\theta\in (\alpha \vee (2-\beta), 2)$ there exists $C=C(C_1,C_2,\theta)>0$ such that $$\left|
\int_{\{0,a\}^{\rm c}} \
\left( \frac{1}{x} -\frac{1}{x-a}\right)
\xi(dx)\right|
\le C|a \vee 1|^{\theta-1},
\quad a\in \supp(\xi-\delta_0).
\label{Difference}$$
[*Proof.*]{} We divide $\{0,a\}^{\rm c}$ into three sets $A_1= \{x\in\{0,a\}^{\rm c} : |x|\le |a|/2 \}$, $A_2= \{x\in\{0,a\}^{\rm c} : |a|/2 <|x|\le 2|a| \}$, and $A_3= \{x\in\{0,a\}^{\rm c} : 2|a|<|x|\}$, and put $$I_j= \int_{A_j}\left| \frac{1}{x} -\frac{1}{x-a}
\right|\xi(dx)
\quad j=1,2,3.$$ When $x\in A_1$, $|x^2-a^2|\ge 3 a^2/4$ and $|x+a|\le 3 |a|/2$, and then $$I_1 = \int_{A_1}\frac{|a||x+a|}{|x||x^2-a^2|}\xi(dx)
\le 2M_1\left(\xi,\frac{|a|}{2}\right).$$ By Lemma \[thm:4\_1\] for any $\delta>\alpha-1$, we can take $C>0$ such that $$I_1\le C|a \vee 1|^\delta.
\label{I_1}$$ When $x\in A_2$, $|x+a| \leq 3|a|$ and $|a|/|x|\le 2$, and then $$I_2 = \int_{A_2}\frac{|a||x+a|}{|x||x^2-a^2|}\xi(dx)
\le 6|a|M_1(\tau_{-a^2}\xi^{\langle 2 \rangle}).$$ From the condition ([**C.2**]{}) [(ii)]{} $$I_2\le 6 C_2 |a\vee 1|^{1-\beta}.
\label{I_2}$$ When $x\in A_3$, $|x-a|> |x|/2$, and then $$I_3 = \int_{A_3}\frac{|a|}{|x||x-a|}\xi(dx)
\le 2^{\alpha-1} |a|^{\alpha-1} M_{\alpha}(\xi)^{\alpha}.$$ From the condition ([**C.2**]{}) [(i)]{} $$I_3\le 2^{\alpha-1} C_1 |a\vee 1|^{\alpha-1}.
\label{I_3}$$ Combining the estimates (\[I\_1\]), (\[I\_2\]) and (\[I\_3\]), we have (\[Difference\]).
\[thm:4\_3\] [(i)]{}If $\xi$ satisfies the conditions [([**C.2**]{})]{} [(i)]{} and [(ii)]{}, for any $\theta\in (\alpha \vee (2-\beta), 2)$ there exists $C=C(C_1,C_2,\theta)>0$ such that $$|\Phi_1(\xi,a,\sqrt{-1} y)|\le
\exp\left[ C\{(|y|^{\theta} \vee 1) + (|a|^{\theta} \vee 1) \}\right],$$ for $y\in\R$ and $a\in\supp \xi$.
[(ii)]{}If $\xi$ satisfies the conditions [([**C.1**]{}), ([**C.2**]{})]{} [(i)]{} and [(ii)]{}, for any $\theta\in (\alpha \vee (2-\beta), 2)$ there exists $C=C(C_0, C_1,C_2,\theta)>0$ such that $$|\Phi_{\cA}(\xi,a, \sqrt{-1} y)|\le
\exp\left[ C\{(|y|^{\theta} \vee 1)
+ (|a|^{\theta} \vee 1) \}\right],$$ for $y\in\R$ and $a\in\supp \xi$.
[*Proof.*]{} We prove [(i)]{} of this lemma. From the condition ([**C.1**]{}) and the relation (\[eqn:entire\_A\]), [(ii)]{} is easily derived from [(i)]{}. We first consider the case that $a=0\in\supp \xi$. Remind that $$\Phi_1(\xi, 0, z)
= \Pi_{1}(\xi, z)
= \exp\left[
\int_{\{0\}^{\rm c} } \left\{
\log \left(1-\frac{z}{x}\right) + \frac{z}{x} \right\}
\xi(dx)\right].$$ When $2|z|<|x|$, by using the expansion $$\log \left(1-\frac{z}{x}\right)
= - \sum_{k \in \N} \frac{1}{k}
\left(\frac{z}{x}\right)^k,$$ we have $$\left|\log \left(1-\frac{z}{x}\right) + \frac{z}{x}\right|
\le \left|\frac{z}{x}\right|^2.$$ Then $$|\Pi_1(\xi\cap [-2|z|,2|z|]^{\rm c},z)|
\le \exp\left\{|z|^2 \int_{|x|>2|z|} \frac{1}{x^2}\xi(dx) \right\}
\leq \exp \Big\{|z|^{\alpha} M_{\alpha}(\xi)^\alpha \Big\}.
\label{estimate:Phi_1_1}$$ On the other hand $ |1-z/x| \leq e^{|z|/|x|}$. Then $$|\Pi_1(\xi\cap [-2|z|,2|z|],z)|
\le \exp\left\{2|z| \int_{|x|\le 2|z|} \frac{1}{|x|}\xi(dx) \right\}
=\exp \Big\{2|z| M_1(\xi,2|z|) \Big\}.
\label{estimate:Phi_1_2}$$ From (\[estimate:Phi\_1\_1\]) and (\[estimate:Phi\_1\_2\]), with the condition ([**C.2**]{}) [(i)]{} and Lemma \[thm:4\_1\], we see that for any $\theta\in (\alpha \vee (2-\beta), 2)$ there exists $C=C(C_1,\theta)>0$ such that $$|\Phi_1(\xi, 0, z)|\le
\exp\left[ C\{(|z|^{\theta} \vee 1) \}\right],
\label{estimata:a=0}$$ for $z\in\C$ and $a\in\supp \xi$.
Next we consider the case that $a\in\supp \xi$ and $a\not=0$. By the conditions ([**C.1**]{}) and ([**C.2**]{}) the equality (\[eqn:finite\_zeta\]) is valid. By (\[estimata:a=0\]) $$|\Pi_1(\xi,z)\Pi_1(\xi\cap \{-a\}^{\rm c},-a)|
\le \exp \Big[ C\{(|z|^{\theta} \vee 1)
+ (|a|^{\theta} \vee 1) \} \Big].$$ By the condition ([**C.2**]{}) [(ii)]{} $$|\Phi_0(\xi^{\langle 2 \rangle} \cap \{0\}^{\rm c}, a^2,0)|
\le \exp \Big\{|a|^2 M_1(\tau_{-a^2}\xi^{\langle 2 \rangle}) \Big\}
\le \exp \Big\{C_2 (|a|\vee 1)^{2-\beta} \Big\},$$ and $|(\sqrt{-1} y/a)^{\xi(\{0\})} a/(a-\sqrt{-1} y)| \leq 1$. Now we evaluate $S(\xi,a,z)$. $$\begin{aligned}
&&S(\xi,a,z)=
\int_{\{0,a\}^{\rm c}}
\left( \frac{z-a}{x-a}-\frac{z-a}{x}\right) \xi(dx)
\nonumber\\
&& \hskip 4cm
+\frac{(z-a)}{-a}\xi(\{0\})
-\frac{z}{a}+\frac{a}{a}-\frac{a}{-a} \xi(\{-a\})
\nonumber\\
&&=(z-a) \int_{\{0,a\}^{\rm c}}
\left(\frac{1}{x-a}-\frac{1}{x}\right) \xi(dx)
-\frac{(1+\xi(\{0\}))z}{a}+1+\xi(\{0\})+\xi(\{-a\}).
\nonumber\end{aligned}$$ From Lemma \[thm:4\_2\] and the fact $1/a^2 \le C_2$ and then $|2z/a|\le 2\sqrt{C_2}|z|$, we have $$|S(\xi,a,z)|\le C|z-a||a\vee 1|^{\theta-1} +2\sqrt{C_2}|z|+3
\le C' \Big\{(|y|^{\theta} \vee 1)
+ (|a|^{\theta} \vee 1) \Big\}$$ for some $C'>0$. This completes the proof. [*Proof of Theorem \[Theorem:Infinite\].*]{} Note that $\xi\cap [-L,L]$, $L>0$ and $\xi$ satisfy ([**C.1**]{}) and ([**C.2**]{}) with the same constants $C_0, C_1, C_2$ and indices $\alpha, \beta$. By virtue of Lemma \[thm:4\_3\] [(ii)]{} we see that there exists $C >0$ such that $$|\Phi_{\cA} (\xi\cap [-L,L],a, \sqrt{-1} y)|
\le \exp \Big[ C \Big\{(|y|\vee 1)^{\theta}
+ (|a|\vee 1)^{\theta} \Big\} \Big],$$ $\forall L>0, \; \forall a\in \supp \xi, \forall y \in \R$. Since for any $y\in\R$ $$\Phi_{\cA} (\xi\cap [-L,L],a,\sqrt{-1} y)
\to \Phi_{\cA} (\xi,a,\sqrt{-1} y), \quad L\to\infty,$$ we can apply [*Lebesgue’s convergence theorem*]{} to (\[eqn:K3\]) and obtain $$\lim_{L\to\infty} \mbK_{\cA}^{\xi\cap [-L,L]}
\left(s, x; t, y \right)
=\mbK_{\cA}^{\xi}\left(s, x; t, y\right).$$ Since for any $(s,t) \in (0, \infty)^{2}$ and any finite interval $I \subset \R$ $$\sup_{x, y \in I} \Big|
\mbK_{\cA}^{\xi \cap [-L, L]}(s, x; t, y) \Big| < \infty,$$ we can obtain the convergence of generating functions for multitime correlation functions (\[eqn:Gxi\]); ${\cG}^{\xi \cap [-L, L]}[\chi] \to
{\cG}^{\xi}[\chi]$ as $L \to \infty$. It implies $\P_{\cA}^{\xi \cap [-L, L]} \to
\P_{\cA}^{\xi}$ as $L \to \infty$ in the sense of finite dimensional distributions. Then the proof is completed.
Proof of Theorem \[Theorem:from\_roots\]
-----------------------------------------
\(i) It is clear that $\xi_{\cA}$ is an element of $\mX_{\cA}^0$. (See the item (1) of Sect. 2.4.) Then by Theorem \[Theorem:Infinite\] $(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}^{N}})
\to (\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}})$ as $N \to \infty$ in the sense of finite dimensional distributions, where $(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}})$ is the determinantal with the correlation kernel $$\begin{aligned}
\mbK^{\xi_{\cA}}_{\cA}(s,x;t,y)
&=&
\int_{\R}\xi_{\cA}(da)
\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}}
\, q(0,s, x-a)
\Phi_{\cA}(\xi_{\cA}, a, z) q(t,0, z-y)
\nonumber\\
&& \quad - {\bf 1}(s>t)q(t,s, x-y).
\nonumber\end{aligned}$$ Using the equalities (\[eqn:equalM1\]) and (\[eqn:equalM2\]) and the definition (\[eqn:pAi\]), we have $$\begin{aligned}
\mbK_{\cA}^{\xi_{\cA}}(s,x;t,y)
&=& \int_{\R} \xi_{\cA}(da)
\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}} \,
\frac{p_{\Ai}(s, x|a)}{g(s,x)}
\frac{1}{z-a} \frac{\Ai(z)}{\Ai'(a)}
g(t,y) p_{\Ai}(-t,z|y)
\nonumber\\
&& \quad
-{\bf 1}(s>t) \frac{g(t,y)}{g(s,x)}
p_{\Ai}(s-t,x|y)
\nonumber\\
&=& \frac{g(t,y)}{g(s,x)}
\mbK_{\Ai}(s,x;t,y)
\nonumber\end{aligned}$$ with (\[eqn:K3a\]), where we have used (\[eqn:relation1b\]) of Lemma \[thm:2\_2\]. By the gauge invariance, Lemma \[thm:gauge\], $(\Xi_{\cA}(t), \P_{\cA}^{\xi_{\cA}})
=(\Xi_{\cA}(t), \P_{\Ai})$ in the sense of finite dimensional distributions.\
(ii) If we use the expression (\[eqn:exp1\]), (\[eqn:K3a\]) becomes $$\begin{aligned}
\mbK_{\Ai}(s,x;t,y) &=&
\sum_{a \in \Ai^{-1}(0)}
\int_{\sqrt{-1} \, \R} \frac{dz}{\sqrt{-1}}
\, p_{\Ai}(s, x|a)
\nonumber\\
&& \qquad \times
\frac{1}{(\Ai'(a))^2}
\int_{0}^{\infty} du \, \Ai(u+z) \Ai(u+a)
p_{\Ai}(-t, z|y)
\nonumber\\
&& - {\bf 1}(s>t) p_{\Ai}(s-t,x|y).
\nonumber\end{aligned}$$ By (\[eqn:equalM3\]), the first term of the RHS equals $$\sum_{a \in \Ai^{-1}(0)} p_{\Ai}(s,x|a)
\frac{1}{(\Ai'(a))^2} \int_{0}^{\infty} du \,
e^{-ut/2} \Ai(u+y) \Ai(u+a).$$ Since $s>0$, we can use the expression (\[eqn:pAi0\]) for $p_{\Ai}(s, x|a)$ and the above is written as $$\int_{0}^{\infty} du \int_{\R} dw \,
e^{-ut/2+ws/2}
\Ai(u+y) \Ai(w+x)
\sum_{\ell \in \N}
\frac{\Ai(u+a_{\ell}) \Ai(w+a_{\ell})}
{( \Ai'(a_{\ell}) )^2}.$$ From the completeness (\[eqn:complete\]), the above gives $$\mbK_{\Ai}(s,x; t,y)
={\bK}_{\Ai}(t-s, y|x)+R(s,x;t,y)$$ with the extended Airy kernel $\bK_{\Ai}$ given by (\[eqn:AiryK\]) and $$\begin{aligned}
R(s,x;t,y) &=& \int_{0}^{\infty} du \int_{-\infty}^0 dw \,
e^{-ut/2+ws/2}
\Ai(u+y) \Ai(w+x)
\nonumber\\
&& \quad \times
\sum_{\ell \in \N}
\frac{\Ai(u+a_{\ell}) \Ai(w+a_{\ell})}
{( \Ai'(a_{\ell}) )^2}.
\nonumber\end{aligned}$$ Since for any fixed $s,t >0$ $\lim_{\theta\to\infty}|R(s+\theta,x;t+\theta,y)|\to 0$ uniformly on any compact subset of $\R^2$, (\[eqn:convKAi\]) holds in the same sense. Hence we obtain (\[relax\]). This completes the proof.
[*Acknowledgments.*]{} M.K. is supported in part by the Grant-in-Aid for Scientific Research (C) (No.21540397) of Japan Society for the Promotion of Science. H.T. is supported in part by the Grant-in-Aid for Scientific Research (KIBAN-C, No.19540114) of Japan Society for the Promotion of Science.
[99]{} Abramowitz, M., Stegun, I. : Handbook of Mathematical Functions. Dover, New York (1965)
Adler, M., van Moerbeke, P. : The spectrum of coupled random matrices. Ann. Math. [**149**]{}, 921-976 (1999)
Adler, M., van Moerbeke, P. : PDF’s for the joint distributions of the Dyson, Airy and Sine processes. Ann. Probab. [**33**]{}, 1326-1361 (2005)
Dyson, F. J. : A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. [**3**]{}, 1191-1198 (1962)
Dyson, F. J. : The three fold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. [**3**]{}, 1199-1215 (1962)
Ferrari, P. L., Spohn, H. : Constrained Brownian motion : Fluctuations away from circular and parabolic barriers. Ann. Probab. [**33**]{}, 1302-1325 (2005)
Flajolet, P., Louchard, G. : Analytic variations on the Airy distribution. Algorithmica [**31**]{}, 361-377 (2001)
Forrester, P. J. : The spectrum edge of random matrix ensembles. Nucl. Phys. B [**402**]{}, 709-728 (1993)
Forrester, P. J., Nagao, T., Honner, G. : Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B [**553**]{}(PM), 601-643 (1999)
Grabiner, D. J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincaré, Probab. Stat. [**35**]{}, 177-204 (1999)
Johansson, K. : Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. [**242**]{}, 277-329 (2003)
Katori, M., Tanemura, H.: Noncolliding Brownian motion and determinantal processes. J. Stat. Phys. [**129**]{}, 1233-1277 (2007)
Katori, M., Tanemura, H.: Non-equilibrium dynamics of Dyson’s model with an infinite number of particles. Commun. Math. Phys. doi:10.1007/s00220-009-0912-3. arXiv:0812.4108\[math.PR\]
Katori, M., Tanemura, H.: in preparation
Katori, M., Nagao, T., Tanemura, H. : Infinite systems of non-colliding Brownian particles. In: Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. in Pure Math., vol. 39, pp.283-306. Mathematical Society of Japan, Tokyo (2004). arXiv:math.PR/0301143
Levin, B. Ya.: Lectures on Entire Functions. Translations of Mathematical Monographs, vol.150. Amer. Math. Soc., Providence (1996)
Mehta, M. L. : Random Matrices, 3rd edn. Elsevier, Amsterdam (2004)
Nagao, T., Forrester, P. J. : Multilevel dynamical correlation functions for Dyson’s Brownian motion model of random matrices. Phys. Lett. [**A247**]{}, 42-46 (1998)
Nagao, T., Katori, M., H. Tanemura, H. : Dynamical correlations among vicious random walkers. Phys. Lett. [**A 307**]{}, 29-35 (2003)
Osada, H. : Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions. Commun. Math. Phys. [**176**]{}, 117-131 (1996)
Osada, H. : Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. arXiv:0902.3561 \[math.PR\]
Prähofer, M., Spohn, H. : Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. [**108**]{}, 1071-1106 (2002)
Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point process. J. Funct. Anal. [**205**]{}, 414-463 (2003)
Soshnikov, A. : Determinantal random point fields. Russian Math. Surveys [**55**]{}, 923-975 (2000)
Spohn, H. : Interacting Brownian particles: A study of Dyson’s model. In: Papanicolaou, G. (ed.) Hydrodynamic Behavior and Interacting Particle Systems, IMA Volumes in Mathematics and its Applications, vol. 9, pp.151-179. Springer, Berlin (1987)
Spohn, H. : Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)
Titchmarsh, E. C. : Eigenfunction Expansions Associated with Second-order Differential Equations. Part I, 2nd edn. Clarendon Press, Oxford (1962)
Tracy, C. A., Widom, H. : Level-spacing distributions and the Airy kernel. Commun. Math. Phys. [**159**]{}, 151-174 (1994)
Tracy, C. A., Widom, H. : A system of differential equations for the Airy process. Elect. Commun. Probab. [**8**]{}, 93-98 (2003)
Vallée, O., Soares, M. : Airy Functions and Applications to Physics. Imperial College Press, London (2004)
[^1]: Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: katori@phys.chuo-u.ac.jp
[^2]: Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan; e-mail: tanemura@math.s.chiba-u.ac.jp
|
---
abstract: 'Graph-based models have been widely used to fraud detection tasks. Owing to the development of Graph Neural Networks (GNNs), recent works have proposed many GNN-based fraud detectors, which are based on either homogeneous or heterogeneous graphs. These works design some GNNs, aggregating neighborhood information to learn the node embeddings. The aggregation relies on the assumption that neighbors share similar context, features, and relations. However, the inconsistency problem incurred by fraudsters is hardly investigated, i.e., the context inconsistency, feature inconsistency, and relation inconsistency. In this paper, we introduce these inconsistencies and design a new GNN framework, $\mathsf{GraphConsis}$, to tackle the inconsistency problem: (1) for the context inconsistency, we propose to combine the context embeddings with node features; (2) for the feature inconsistency, we design a consistency score to filter the inconsistent neighbors and generate corresponding sampling probability; (3) for the relation inconsistency, we learn the relation attention weights associated with the sampled nodes. Empirical analyses demonstrate that the inconsistency problem is critical in fraud detection tasks. Extensive experiments show the effectiveness of $\mathsf{GraphConsis}$. We also released a GNN-based fraud detection toolbox with implementations of SOTA models. The code is available at .'
author:
- 'Zhiwei Liu, Yingtong Dou, Philip S. Yu'
- Yutong Deng
- Hao Peng
bibliography:
- 'sample-base.bib'
title: Alleviating the Inconsistency Problem of Applying Graph Neural Network to Fraud Detection
---
=4
<ccs2012> <concept> <concept\_id>10002978.10003022.10003026</concept\_id> <concept\_desc>Security and privacy Web application security</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
INTRODUCTION
============
There are various kinds of fraudulent activities on the Internet [@jiang2016suspicious], e.g., fraudsters disguise as regular users to post fake reviews [@kaghazgaran2019wide] and commit download fraud [@dou2019uncovering]. By modeling entities as nodes and the corresponding interactions between entities as edges [@peng2019fine], we can design graph-based algorithms to detect suspicious patterns and therefore spot the fraudsters. Along with the development of Graph Neural Networks (GNNs) [@kipf2016semi; @velivckovic2017graph; @hamilton2017inductive], previous endeavors propose many GNN-based fraud detection frameworks [@wang2019fdgars; @li2019spam; @zhong2020financial; @liu2018heterogeneous; @wang2019semi; @zhang2019key].
Among those frameworks, [@wang2019fdgars; @li2019spam] detect opinion fraud in the online review system, [@zhong2020financial; @liu2018heterogeneous; @wang2019semi] aim at financial fraud, and [@zhang2019key] targets cyber-criminal in online forums. They proposed new GNNs upon either homogeneous [@wang2019fdgars; @li2019spam] or heterogeneous [@liu2018heterogeneous; @zhong2020financial; @li2019spam; @wang2019semi; @zhang2019key] graphs. Regarding base GNN models, FdGars [@wang2019fdgars] and GAS[@li2019spam] adopt GCN [@kipf2016semi], SemiGNN and Player2Vec [@zhang2019key] adopt GAT [@velivckovic2017graph], and other works [@liu2018heterogeneous; @li2019spam; @zhong2020financial] devise new aggregators to aggregate the neighborhood information. Those GNN-based fraud detectors learn node representation iteratively and predict the node suspiciousness in an end-to-end and semi-supervised fashion.
{width="81.00000%"}
However, all existing methods ignore the inconsistency problem when designing a GNN model regarding the fraud detection task. The inconsistency problem is associated with the aggregation process of GNN models. The mechanism of aggregation is based on the assumption that the neighbors share similar features and labels [@hou2020measure]. When the assumption diminishes, we can no longer aggregate neighborhood information to learn node embeddings. For example, as Figure \[fig:consis\] (*Left*) shows, the inconsistency in fraud detection problem comes from three perspectives:
\(1) **Context inconsistency**. Smart fraudsters can connect themselves to regular entities as camouflage [@kaghazgaran2018combating; @sun2018adversarial]. Meanwhile, the amount of fraudsters is much less than that of regular entities. Directly aggregating neighbors by the GNNs can only help fraudulent entities aggregate the information from regular entities and thus prevent themselves from being spotted by fraud detectors. For example, in Figure \[fig:consis\] (*Left*), the fraudster $v_1$ connects to $3$ benign entities under relation II.
\(2) **Feature inconsistency**. Take the opinion fraud (e.g., spam reviews) detection problem as an example [@li2019spam]. Assuming there are two reviews posted from the same user but regarding products in distinct categories, these two reviews have an edge since they share the same user. However, their review content (features) are far from each other, as they are associated with different products. Direct aggregation makes the GNN hard to distinguish the unique semantic characteristics of reviews and finally affects its ability to detect spam reviews. For example, in Figure \[fig:consis\] (*Left*), we can observe that the feature of node $v_1$ is inconsistent with that of nodes $v_4$, $v_6$, and $v_7$.
\(3) **Relation inconsistency**. Since entities are connected with multiple types of relations, equally treating all relations results in a relation inconsistency problem. For example, two reviews may either be connected by the same user or the same product, which are respectively *common-user* relation and *common-product* relation. Assuming that one review is suspicious, the other one should have a higher suspiciousness if they are connected by *common-user* relation, since fraudulent users tend to post more than one fraudulent reviews. For example, in Figure \[fig:consis\] (*Left*), we find that under relation II, the fraudster $v_1$ is connected to two other fraudsters. However, under relation I, the fraudster is connected to only one fraudster but three benign entities.
To tackle all above inconsistencies, we design a novel GNN framework, named as **GraphConsis**. The framework of $\mathsf{GraphConsis}$ is shown in Figure \[fig:consis\] (*Right*). $\mathsf{GraphConsis}$ is built upon a heterogeneous graph with multiple relations. $\mathsf{GraphConsis}$ differs existing GNNs from the aggregation process. Instead of directly aggregating neighboring embeddings, we design three techniques to resolve three inconsistency problems simultaneously. Firstly, to handle the context inconsistency of neighbors, $\mathsf{GraphConsis}$ assigns each node a trainable context embedding, which is illustrated as the gray block aside nodes in Figure \[fig:consis\] (*Right*). Secondly, to aggregate consistent neighbor embeddings, we design a new metric to measure the *embedding consistency* between nodes. By incorporating the embedding consistency score into the aggregation process, we ignore the neighbors with a low consistency score (e.g. the node $v_4$ is dropped in Figure \[fig:consis\] (*Right*)) and generate the sampling probability. Last but not least, we learn relation attention weights associated with neighbors in order to alleviate the relation inconsistency problem. The contributions of this paper are:
- To the best of our knowledge, we are the first work addressing the inconsistency problem in GNN models.
- We empirically analyze three inconsistency problems regarding applying GNN models to fraud detection tasks.
- We propose $\mathsf{GraphConsis}$ to tackle three inconsistency problems, which combines context embedding, neighborhood information measure, and relational attention.
PRELIMINARIES
=============
We detect fraud entities in the graph by using node representations. Hence, we first introduce node representation learning. A heterogeneous graph $G=\{V, \mathbf{X}, \{E_{r}\}|_{r=1}^{R}\big\}$, where $V$ denotes the nodes, $\mathbf{X}$ is the feature matrix of nodes, and $E_{r}$ denotes the edges w.r.t. the relation $r$. We have $R$ different types of relations. To represent the nodes as vectors, we need to learn a function $f:V\rightarrow \mathbb{R}^{d}$ that maps nodes to a $d$ dimensional space, where $d \ll |V|$. The function $f$ should preserve both the structural information of the graph and the original feature information of the nodes. With the learned node embeddings, we can train a classifier $C:\mathbb{R}^{d}\rightarrow \{0,1\}$ to detect whether a given node is a fraudster, where $1$ denotes fraudster, and $0$ denotes benign entity. In this paper, we adopt the GNN framework to learn the node representation through neighbor aggregation. GNN framework can train the mapping function $f$ and the classifier $C$ simultaneously. We only need to input the graph and the labels of nodes to a GNN model. The general framework of a GNN model is: $$\label{eq:gnn_framework}
\mathbf{h}_{v}^{(l)}= \mathbf{h}_{v}^{(l-1)} \oplus \operatorname{AGG}^{(l)}\left(\left\{\mathbf{h}_{v^{\prime}}^{(l-1)}: v^{\prime} \in \mathcal{N}_{v}\right\}\right),$$ where $\mathbf{h}_{v}^{(l)}$ is the hidden embedding of $v$ at $l$-th layer, $\mathcal{N}_{v}$ denotes the neighbors of node $v$, and the $\operatorname{AGG}$ represents the aggregation function that maps the neighborhood information into a vector. Here, we use $\oplus$ to denote the combination of neighbor information and the center node information, it can be direct addition or concatenation then passed to a neural network. For the $\operatorname{AGG}$ function, we first assign a sampling probability to the neighboring nodes. Then we sample $Q$ nodes and average[^1] them as a vector. The calculation of probability is introduced later in Eq. (\[eq:sampling prob\]). Note that the framework of GNN is a $L$-layer structure, where $1\leq l \leq L$. At $l$-th layer, it aggregates the information from $l-1$-th layer.
PROPOSED MODEL
==============
Context Embedding
-----------------
The aggregator combines the information of neighboring nodes according to Eq. (\[eq:gnn\_framework\]). When $k=1$, the hidden embedding $\mathbf{h}^{(0)}_{v}$ is equivalent to the node feature. To tackle the context inconsistency problem, we introduce a trainable context embedding $\mathbf{c}_{v}$ for node $v$., instead of only using its feature vector $\mathbf{x}_{v}$. The first layer of the aggregator then becomes: $$\label{eq:gnn_framework}
\mathbf{h}_{v}^{(1)}= \{\mathbf{x}_{v}\|\mathbf{c}_{v}\} \oplus \operatorname{AGG}^{(1)}\left(\left\{\mathbf{x}_{v^{\prime}} \|\mathbf{c}_{v^{\prime}}: v^{\prime} \in \mathcal{N}_{v}\right\}\right),$$ where $\|$ denotes the concatenation operation. The context embedding is trained to represent the local structure of the node, which can help to distinguish the fraud. If we use addition operation for $\oplus$, then $\mathbf{h}_{v}\in \mathbb{R}^{2d}$.
Neighbor Sampling {#sec:neighbor sampling}
-----------------
Since there exists a feature inconsistency problem, we should sample related neighbors rather than assign equal probabilities to them. Thus, we compute the **consistency score** between embeddings: $$s^{(l)}(u, v) = \exp\left(-\|\mathbf{h}_{u}^{(l)} - \mathbf{h}_{v}^{(l)}\|_{2}^{2}\right),$$ where $s^{(l)}(\cdot,\cdot)$ denotes the consistency score for two nodes at $l$-th layer, and $\|\cdot\|_2$ is the $l_2$-norm[^2] of vector. We first apply a threshold $\epsilon$ to filter neighbors far away from consistent. Then, we assign each node $u$ to the filtered neighbors $\Tilde{\mathcal{N}}_{v}$ of node $v$ with a sampling probability by normalizing its consistency score: $$~\label{eq:sampling prob}
p^{(l)}(u; v) = s^{(l)}(u, v)/{\sum_{u\in\Tilde{\mathcal{N}}_{v}}s^{(l)}(u, v)}.$$ Note that the probability is calculated at each layer for the $\operatorname{AGG}^{(l)}$.
Relation Attention
------------------
We have $R$ different relations in the graph. The relation information should also be included in the aggregation process to tackle the relation inconsistency problem. Hence, for each relation $r$, we train a relation vector $\mathbf{t}_r$, where $r=\{1,2,\dots,R\}$, to represent the relation information that should be incorporated. Since the relation information should be aggregated along with the neighbors to center node $v$, we adopt the self-attention mechanism [@velivckovic2017graph] to assign weights for $Q$ sampled neighbor nodes: $$\alpha_{q}^{(l)} = \exp\left(\sigma\left(\{\mathbf{h}_{q}^{(l)}\|\mathbf{t}_{r_q}\}\mathbf{a}^{\top}\right)\right)/{\sum_{q=1}^{Q}\exp\left(\sigma\left(\{\mathbf{h}_{q}^{(l)}\|\mathbf{t}_{r_q}\}\mathbf{a}^{\top}\right)\right)},
$$ where $r_q$ denotes the relation of $q$-th sample with node $v$, $\sigma$ is the activation function, and $a\in\mathbb{R}^{4d}$ represents the attention weights that is shared for all attention layer. The final $\operatorname{AGG}^{(l)}$ is: $$\operatorname{AGG}^{(l)}\left(\left\{\mathbf{h}_{q}^{(l-1)}\right\}\Big|_{q=1}^{Q}\right) = \sum_{q=1}^{Q}\alpha_{q}^{(l)}\mathbf{h}_{q}^{(l)},$$ where $\mathbf{h}_{q}^{(l)}$ is the embedding of $q$-th node sampled based on Eq. (\[eq:sampling prob\]).
EXPERIMENTS
===========
Experimental Setup
------------------
### Dataset and Graph Construction
We utilize the YelpChi spam review dataset [@Rayana2015], along with three other benchmark datasets [@kipf2016semi; @hamilton2017inductive] to study the graph inconsistency problem in the fraud detection task. The YelpChi spam review dataset includes hotel and restaurant reviews filtered (spam) and recommended (legitimate) by Yelp. In this paper, we conduct a spam review classification task on the YelpChi dataset which is a binary classification problem. We remove products with more than 800 reviews to restrict the size of the computation graph. The pre-processed dataset has 29431 users, 182 products, and 45954 reviews (%14.5 spams).
Based on previous studies [@Rayana2015] which show the spam reviews have connections in user, product, rating, and time, we take reviews as nodes in the graph and design three relations denoted by *R-U-R*, *R-S-R*, and *R-T-R*. *R-U-R* connects reviews posted by the same user; *R-S-R* connects reviews under the same product with the same rating; *R-T-R* connects two reviews under the same product posted in the same month. We take the 100-dimension Word2Vec embedding of each review as its feature like previous work [@li2019spam].
### Baselines
To show the ability of $\mathsf{GraphConsis}$ in alleviating inconsistency problems, we compare its performance with a non-GNN classifier, vanilla GNNs, and GNN-based fraud detectors.
- **Logistic Regression**. A non-GNN classifier that makes predictions only based on the reviews features.
- **FdGars (GCN)** [@wang2019fdgars]. A spam review detection algorithm using GCN [@kipf2016semi].
- **GraphSAGE** [@hamilton2017inductive]. A popular GNN framework which samples neighboring nodes before aggregation.
- **Player2Vec** [@zhang2019key]. A state-of-the-art fraud detection model which uses GCN to encode information in each relation, and uses GAT to aggregate neighbors from different relations.
### Experimental Settings
We use Adam optimizer to train our model based on the cross-entropy loss. For the hyper-parameters, we choose $2$-layer structure, and the number of samples is set as $10$ and $5$ for the first layer and second layer, respectively. The embedding dimension of the hidden layer is $200$ and $100$ for the first layer and second layer, respectively. We use F1-score to measure the overall classification performance and AUC to measure the performance of identifying spam reviews.
The Inconsistency Problem
-------------------------
We first take the Yelpchi dataset to demonstrate the inconsistency problem in applying GNN to fraud detection tasks. Table \[tab:inconsistency\] shows the statistics of graphs built on YelpChi comparing to node classification benchmark datasets used by [@kipf2016semi; @hamilton2017inductive]. *Yelp-ALL* is composed of three single-relation graphs. Comparing to three widely-used benchmark node classification datasets, we find that a multi-relation graph constructed on YelpChi has a much higher density (the average node degree is greater than 100). It demonstrates that the real-world fraud graphs usually incorporate complex relations and neighbors, and thus render inconsistency problems. Before we compare the graph characteristics and analyze three inconsistency problems, similar to [@hou2020measure], we design two characteristic scores. One is the context characteristic score: $$\label{eq:context_chara}
\gamma_{r}^{(c)} = \sum_{(u,v)\in{E}_{r}}\left(1-\mathbb{I}\left(u\sim v\right)\right)/|{E}_{r}|,$$ where $\mathbb{I}(\cdot)\in\{0,1\}$ is an indicator function to indicate whether node $u$ and node $v$ have the same label. We sum all the indication w.r.t. all the edges and normalized by the total number of edges $|{E}_{r}|$. The context characteristic measures the label similarity between neighboring nodes under a specific relation $r$. The other one is the feature characteristic score: $$\label{eq:feature_chara}
\gamma_{r}^{(f)}=\sum_{(u,v) \in{E}_{r}}\exp\left(-\left\|\mathbf{x}_{u}-\mathbf{x}_{v}\right\|^{2}_{2}\right)/{|{E}_{r}|\cdot d},
$$ where we employ the RBF kernel function[^3] as the similarity measurement between two connected nodes. The overall feature characteristic score is normalized by the product of the total number of edges $|E_{r}|$ and the feature dimension $d$. Normalizing the similarity by feature dimension is to fairly compare the feature characteristics of different graphs, which may have different feature dimensions.
**Context Inconsistency.** We compute the context characteristic $\gamma_{r}^{(c)}$ based on Eq. (\[eq:context\_chara\]), which measures the context consistency. For the graph *R-T-R*, *R-S-R* and *Yelp-ALL*, there are less than $10\%$ of neighboring nodes have similar labels. It shows that fraudsters may hide themselves among regular entities under some relations.
**Feature Inconsistency.** We calculate the feature characteristic $\gamma_{r}^{(f)}$ using Eq. (\[eq:feature\_chara\]). The graph constructed by *R-U-R* relation (reviews posted by the same user) has higher feature characteristic than the other two relations. Thus, we need to sample the neighboring nodes not only based on their relations but also the feature similarities.
**Relation Inconsistency.** For graphs constructed by three different relations, the neighboring nodes also have different feature/label inconsistency score. Thus, we need to treat different relations with different attention weights during the aggregation.
\[tab:inconsistency\]
Performance Evaluation
----------------------
Table \[tab:exp\_result\] shows the experiment results of the spam review detection task. We could see that $\mathsf{GraphConsis}$ outperforms other models under $80\%$ and $60\%$ of training data on both metrics, which suggests that we can alleviate the inconsistency problem. Compared with other GNN-based models, LR performs stably and better on AUC. It indicates that the node feature is useful, but the aggregator in GNN undermines the classifier in identifying fraudsters. This observation also proves that the inconsistency problem is critical and should be considered when applying GNNs to fraud detection tasks. Compared to Player2Vec which also learns relation attention, $\mathsf{GraphConsis}$ performs better. It suggests that solely using relation attention cannot alleviate the feature inconsistency. The neighbors should be filtered and then sampled based on our designed methods. FdGars directly aggregates neighbors’ information and GraphSAGE samples neighbors with equal probability. Both of them perform worse than $\mathsf{GraphConsis}$, which shows that our neighbor sampling techniques are useful.
\[tab:exp\_result\]
Conclusion and Future Works
===========================
In this paper, we investigate three inconsistency problems in applying GNNs in fraud detection problem. To address those problems, we design three modules respectively and propose $\mathsf{GraphConsis}$. Experiment results show the effectiveness of $\mathsf{GraphConsis}$. Future work includes devising an adaptive sampling threshold for each relation to maximize the receptive field of GNNs. Investigating the inconsistency problems under other fraud datasets is another avenue of future research.
This work is supported by the National Key R&D Program of China under grant 2018YFC0830804, and in part by NSF under grants III-1526499, III-1763325, III-1909323, and CNS-1930941. For any correspondence, please refer to Hao Peng.
[^1]: Other pooling techniques can also be applied.
[^2]: Other metrics, such as $l_1$-norm, are also applicable.
[^3]: Other kernel functions can also be applied.
|
---
abstract: 'In this work we discuss the occurrence of ferromagnetism in transition-like metals. The metal is represented by two hybridized($V$) and shifted $(\epsilon_s$) bands one of which includes Hubbard correlation whereas the other is uncorrelated. The starting point is to transform the original Hamiltonian into an effective one. Only one site retains the full correlation (U) while in the others the correlations are represented by an effective field, the self-energy(single-site approximation). This field is self-consistently determined by imposing the translational invariance of the problem. Thereby one gets an exchange split quasi-particle density of states and then an electron-spin polarization for some values of the parameters $(U,V, \alpha, \epsilon_s)$, $\alpha$ being the ratio of the effective masses of the two bands and of the occupation number $n$.'
address:
- 'Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150,Rio de Janeiro, 22290-180, RJ, Brazil'
- 'Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro, RJ, Brazil'
author:
- 'C.M. Chaves'
- 'A.Troper'
title: Cross effect of Coulomb correlation and hybridization in the occurrence of ferromagnetism in two shifted band transition metals
---
,
Ferromagnetic metal; Correlation; Single-site approximation;
71.10.-w, 71.10.Fd, 71.20.Be
The model
=========
In recent years the study of magnetism in itinerant ferromagnets such as Fe, Co, Ni has been the subject of a great deal of efforts by several approaches. Examples are the dynamical mean field theory (DMFT) [@gabi] and the modified alloy analogy (MAA)[@sch]. In an previous work [@sces07] we have developed a two band model, consisting of a Hubbard like narrow band( band $a$) with intrasite Coulomb interaction U, hybridized with another band, which is broad and uncorrelated (band $b$), through the hybridization coupling $V_{ab}$. The two bands had the same center (symmetric regime). Now we treat a more general situation , with a shift between the centers of the two bands.
We review briefly the method [@sces07]: The initial Hamiltonian we adopt is then $$\begin{aligned}
\label{H}
\mathcal{H}&=&\sum_{i,j,\sigma }t_{ij}^{a}a_{i\sigma }^{+}a_{j\sigma
}+\sum_{i,j,\sigma }t_{ij}^{b}b_{i\sigma }^{+}b_{j\sigma
}\\
&+&\sum_{i}Un_{i\uparrow }^{(a)}n_{i\downarrow }^{(a)}
+\sum_{i,j,\sigma }(V_{ab}b_{i\sigma }^{+}a_{j\sigma }+V_{ba}^{+}a_{i\sigma}^{+}b_{j\sigma })~,\nonumber\\end{aligned}$$ where $n_{i\sigma }^{a}=a_{i\sigma }^{+}a_{i\sigma }$ ; $\sigma$ denotes spin. $t_{ij}$ denotes the tunneling amplitudes between neighboring sites $i$ and $j$ , in each band. As in Roth’s approach[@roth1], we reduce the presence of the correlation to only one site (the origin, say ), while in the others acts an effective spin and energy dependent but site independent field, the self-energy $\Sigma^{\sigma}$. This field is self-consistently determined by imposing the vanishing of the scattering $T$ matrix associated to the origin. We thus arrive at the effective Hamiltonian $$\begin{aligned}
\label{Heff}
\mathcal{H}_{eff}&=&\sum_{i,j,\sigma }t_{ij}^{a}a_{i\sigma }^{+}a_{j\sigma
}+\sum_{i,j,\sigma }t_{ij}^{b}b_{i\sigma }^{+}b_{j\sigma
}\\
&+&\sum_{i,\sigma}n_{i\sigma}^{a}\Sigma^{\sigma}+Un^{a}_{0\uparrow }n^{a}_{0\downarrow}
+\sum_{i,j,\sigma }(V_{ab}b_{i\sigma }^{+}a_{j\sigma }+ \nonumber\
h.c.)\\
&-&\sum_{\sigma}n_{0\sigma}^{a}\Sigma^{\sigma},\nonumber\\end{aligned}$$ $\mathcal{H}_{eff}$ still includes the difficulty of dealing with the Coulomb intra-atomic term at the origin. We use the Green function method [@zuba] ; the equations of motion for the corresponding Green functions $G^{cd}_{ij\sigma}(w)= <<c_{i\sigma},d^+_{j\sigma}>>_w$ , where $c,d=a,b$, are $$\begin{aligned}
\label{gaaba}
&&wG^{aa}_{ij,\sigma}(w)=\delta_{ij}+\sum_l t^{a}_{il}G^{aa}_{lj,\sigma}(w)+ \Sigma^\sigma G^{aa}_{ij,\sigma}(w)\nonumber\\
&&+ \sum_l V_{ab}(R_i-R_l)G^{ba}_{lj,\sigma}(w)
+\delta_{i0}[U G^{aa,a}_{0j,\sigma}(w)+\nonumber \\
&&-\Sigma^\sigma G^{aa}_{0j,\sigma}(w)] ; \\
&&wG^{ba}_{ij,\sigma}(w)= \sum_l t^{b}_{il}G^{ba}_{lj,\sigma}(w)+ \sum_l V_{ba}(R_i-R_l)G^{aa}_{lj,\sigma}(w)\nonumber\end{aligned}$$ where $G^{aa,a}_{0j,\sigma}(w)=<<n^â_{0-\sigma}a_{0\sigma};a^+_{j\sigma}>>_w \equiv \Gamma^{aa}_{0j\sigma}(w)$ is a higher order Green function, whose equation of motion , after the neglecting of the broadening correction[@hubIII; @herr] reduces to $$\begin{aligned}
\label{gama}
\nonumber\
&& w\Gamma^{aa}_{ij,\sigma}(w) = <n^a_{0-\sigma}>\delta_{ij}+\sum_l t^{a}_{il}\Gamma^{aa}_{lj,\sigma}(w)+ \Sigma^\sigma \Gamma^{aa}_{ij,\sigma}(w) \nonumber\\
&&+ \sum_l V_{ab}(R_i-R_l)<<n^a_{0-\sigma} b_{l\sigma};a^+_{j\sigma} >>\\
&&+\delta_{i0}(U-\Sigma^\sigma )G^{aa,a}_{0j,\sigma}(w). \nonumber\end{aligned}$$ The ressonance broadening occurs, in the terminology of the alloy analogy (AA), when the opposite spin direction are not kept frozen. Some remarks are in order about Eq.(\[gama\]): The scattering correction is already included; in the Hubbard terminology[@hubIII; @herr] of an AA of up and down spins, this correction would correspond to disorder scattering and produces a damping of the quasi-particles. Secondly, the hybridization generates a new function $<<n^a_{0-\sigma}b_{l\sigma};a^+_{j\sigma}>>_w$ and its equation of motion, again after neglecting the broadening correction, reduces to $$\begin{aligned}
\label{ncd}
w<<n^a_{0-\sigma} b_{i\sigma};a^+_{j\sigma}>> &=&\sum_l(t^b_{il}<<n^a_{0-\sigma} b_{l\sigma};a^+_{j\sigma}>>\nonumber\\
&+& V_{ab}(R_i-R_l)\Gamma^{aa}_{lj\sigma}(w))\end{aligned}$$ At this point we have an effective impurity problem; the direction of the impurity spin is not fixed.
We solve explicitly the problem defined by Eq.(\[gaaba\]), Eq.(\[gama\])and Eq.(\[ncd\]), obtaining, after imposing $T=0$, the following Green function for the $a$ band: $$G^{a}_{kk',\sigma}(w)={\frac{\delta_{kk'}}{w-\tilde{\epsilon}^a_k -\Sigma^\sigma(w)}},
\label{gd}$$ In this equation $$\tilde {\epsilon}^{a}_{k}=\epsilon^{a}_k +{\frac{|V_{ab}|^2(k)}{w-{\epsilon}^b_k }},
\label{etild}$$ is the recursion relation of the $a$ band modified by the hybridization $V$ and $\epsilon^{a}_k$ and $\epsilon^{b}_k$ denote the bare bands, with $$\epsilon^{a}_{k}={\frac{t_{a}(cos(k_xa)+\cos(k_ya)+\cos(k_za))}{A}},
\label{ed}$$ In this paper we use $t_a =1$ and $A=3$, in arbitrary energy units. All energy magnitudes are taken in units of $t_a$, making them dimensionless. The bare $a$ band width is then $W=2$. For simplicity we adopt homothetic bands $$\label{homo}
\epsilon^{b}_{k}=\epsilon_s +\alpha \epsilon^{a}_{k}.$$ $\epsilon_s$ is the center of the $b$ band; as the $a$ band is centered at the origin, this parameter represents a shift in the bands. $\alpha$ is a phenomenological parameter describing the ratio of the effective masses of the $a$ and the $b$ electrons. From now on we take $k_ia \rightarrow k_{i}, i=x,y,z$ and $V_{ab}=V_{ba}\equiv V=$ real and constant independent of $k_i$.
The vanishing of the T-matrix gives further a self-consistent equation for the self-energy: $$\label{elivre}
\Sigma^\sigma= U<n^a_{0-\sigma}>+(U-\Sigma^\sigma)F^\sigma(w,\Sigma^\sigma)\Sigma^\sigma,$$ with $$F^\sigma(w,\Sigma^\sigma)= N^{-1}\sum_k G^{a}_{kk,\sigma}
\label{F}$$
Numerical Results
=================
We perform the self-consistency in both $\Sigma^\sigma$ and in $<\nolinebreak n_{0,\sigma}^a>$, for each total occupation $n=<n^a> +<n^b>$. The total number of electrons per site, is fixed at $n=1.6$ (but see below), a little less than half-filling. We want now to exhibit the combined effect of $U$, $V$, $\alpha$, $n$, and $\epsilon_s$ at $T=0K$.
In fig (\[mV\]) we plot magnetization versus V. It is clear that small values of $V$ help stabilize the ferromagnetic order but larger ones tend to inhibit it [@sch]. This is because hybridization, apart from changing the occupations of the $a$ and $b$ bands, together with the $\epsilon_s$ increases (small $V$) and decreases (large $V$) the $a$-density of states at the Fermi level.
In fig (\[mepsilons\]) the magnetization is plot versus $\epsilon_s$. We see that the shift then tends to favor ferromagnetism.
In fig (\[do\]) we plot the charge transference $a->b$ or vice-versa as function of $\epsilon_s$ and it is seen that from $\epsilon_s$ $\approx 0.8$ on, this transference increases the number of $a$ electrons thus tending to favor ferromagnetism.
![Magnetization versus hybridization $V$ for $U=3$, $\alpha=1.5$, $n=1.6$ and $\epsilon_s = 1.0$. Small values of hybridization tend to favor ferromagnetism.[]{data-label="mV"}](mVU3eps1.eps){width="45.00000%"}
![Magnetization versus band shift for $U=3$, $V=0.4$, $\alpha=1.5$ and $n=1.6$. Larger values of the band shift tend to favor ferromagnetism.[]{data-label="mepsilons"}](mepsilons.eps){width="45.00000%"}
![Charge transference versus band shift for $U=3$, $V=0.4$ $\alpha=1.5$ and $n=1.6$ []{data-label="do"}](transfer.eps){width="45.00000%"}
In fig (\[malfa\]) one exhibits the dependence of the magnetization on the ratio of the effective masses beween the correlated and the uncorrelated bands. We argue that the increasing of $\alpha$ is proportional to a decreasing of the effective mass of the correlated $a$ band with respect to the free electron $b$ band and hence the magnetization should also decrease.
![Magnetization versus $\alpha$, the band width relation for $U=3$, $V=0.4$, $n=1.6$ and $\epsilon_s = 1.0$. Small values of hybridization tend to favor ferromagnetism.[]{data-label="malfa"}](MalfaU3V04.eps){width="45.00000%"}
In fig (\[mn\]) one displays the magnetization as function of the total occupation $n$. We notice that small values of $n$ favors paramagnetism while after some occupation, here $n\approx 1.6$, the magnetization drops down.
![Magnetization versus total occupation $n$ for $U=3$, $V=0.4$, $\alpha=1.5$ and $\epsilon_s = 1.0$.[]{data-label="mn"}](Mversusn.eps){width="45.00000%"}
![Density of states of the correlated $a$ band for $U=3$, $V=0.4$ ,$\alpha=1.5$, $n=1.6$ and $\epsilon_s=1.0$. The Fermi level is at $E_F=0.443$ and a magnetization of $0.174$ develops. The combined effect of hybridization and the band shift produces a band broadening.[]{data-label="dosd"}](DosdU3V04eps1.eps){width="45.00000%"}
In fig (\[dosd\]) we present the density of states (DOS) of the $a$-band for $U=3$, $V=0.4$ and $\epsilon_s = 1.0$ . The Fermi level is at $E_F=0.443$ and a magnetization of $0.174$ arises. The combined effect of hybridization and the band shift produces a band broadening[@sch]. The DOS here obtained exhibits a bimodal structure caracterizing a Hubbard strongly correlated regime.
In fig (\[dosb\]) we show the density of state (DOS) of the uncorrelated $b$ band, for the same set of parameters, namely $U=3$, $V=0.4$, $\alpha=1.5$ and $\epsilon_s = 1.0$ . We verify that the renormalized band remains almost unchanged when compared with the bare one. In fact, hybridization affects this band , enlarging it , but no noticeable $b$ magnetic moment arises. Moreover, it does not present a bimodal structure.
![Density of states of the $b$ band for $U=3$, $V=0.4$, $\alpha=1.5$ and $\epsilon_s=1.0$. The Fermi level is at $E_F=0.443$.[]{data-label="dosb"}](DOSb.eps){width="45.00000%"}
For the sake of completeness we display in fig (\[DosaU01\]) a situation envolving the weak correlation regime, $U/W<<1$. Now the $a$ band is renormalized as a typical Hartree-Fock (HF) band without exhibiting the Hubbard bimodal structure. Moreover, from fig (\[sigU01\]), where we plot the real part of the self-energy, we see a trend of the usual HF regime, namely an almost constant value of the self-energy. For comparison we show in fig (\[sigU03\]) the self-energy for a strong correlated limit.
![Density of states of the $a$ band in the weak coupling regime, $U=0.1$, for $V=0.4$, $\alpha=1.5$, $n=1.6$ and $\epsilon_s = 1.0$. Notice the absence of bimodal structure.[]{data-label="DosaU01"}](DosaU01V04a15e1.eps){width="45.00000%"}
![Real part of the self-energy $\Sigma^\sigma$ for $U=0.1$, $V=0.4$, $\alpha=1.5$, $n=1.6$ and $\epsilon_s = 1.0$. In this regime $\Sigma^\sigma$ shows a very weak dependence on the energy.[]{data-label="sigU01"}](sigdU01V04a15e1.eps){width="45.00000%"}
![Real part of the self-energy $\Sigma^\sigma$ for $U=3$, $V=0.4$, $\alpha=1.5$, $n=1.6$ and $\epsilon_s = 1.0$.[]{data-label="sigU03"}](sigdU3V04a15e1.eps){width="45.00000%"}
Final comments
==============
The traditional view of the origin of ferromagnetism in metals has been under intense scrutiny recently [@gabi; @sch; @bat; @voll]. Conventional mean-field calculations favor ferromagnetism but corrections tend to reduce the range of validity of that ground state [@voll]. In this paper, using the single site approximation, we obtain ferromagnetic solution for a set of parameters (e.g. $U/W=1.5$ ,$V/W=0.2$, $\epsilon_s=1.0$ and $\alpha=1.5$).
As a continuation of this systematic study, the generation of the phase diagram [@ch] for the model is in progress.
Acknowledgement
===============
CMC and AT aknowledge the support from the brazilian agencies $PCI/MCT$ and $CNP_q$.
[99]{}
A Georges, G Kotliar, W. Krauth and M Rozenberg, Rev. Mod. Phys. 68 (1996) 13 .
S Schwieger and W Nolting, Phys. Rev. B64 (2001) 144415 .
C. M. Chaves, A. Gomes and A. Troper, Physica B403 (2008)1459.
L.M.Roth, in AIP Conference Proceedings 18 Magnetism and Magnetic Materials, (1973) 668. Zubarev D N 1960 [*Sov. Phys.-Usp*]{} [**3**]{} 320. Hubbard J 1964 [*Proc. Royal Soc.*]{} [**A281**]{} 401-419 . Herrmann T and Nolting W 1996-II [*Phys. Rev.*]{} B[**53**]{} 10579-10588. C. D. Batista *et al*, Phys. Rev. Letters 88 (2002) 187203-1. D. Vollhardt *et al*, Adv. Solid State Phys. 38 (1999) 383. C. M. Chaves and A. Troper, to be published.
|
---
abstract: 'Recent X-ray observations of galaxy clusters show that the distribution of intra-cluster medium (ICM) metallicity is remarkably uniform in space and time. In this paper, we analyse a large sample of simulated objects, from poor groups to rich clusters, to study the dependence of the metallicity and related quantities on the mass of the systems. The simulations are performed with an improved version of the Smoothed-Particle-Hydrodynamics `GADGET-3` code and consider various astrophysical processes including radiative cooling, metal enrichment and feedback from stars and active galactic nuclei (AGN). The scaling between the metallicity and the temperature obtained in the simulations agrees well in trend and evolution with the observational results obtained from two data samples characterised by a wide range of masses and a large redshift coverage. We find that the iron abundance in the cluster core ($r<0.1R_{500}$) does not correlate with the temperature nor presents a significant evolution. The scale invariance is confirmed when the metallicity is related directly to the total mass. The slope of the best-fitting relations is shallow ($\beta\sim-0.1$) in the innermost regions ($r<0.5R_{500}$) and consistent with zero outside. We investigate the impact of the AGN feedback and find that it plays a key role in producing a constant value of the outskirts metallicity from groups to clusters. This finding additionally supports the picture of early enrichment.'
author:
- |
N. Truong$^1$[^1], E. Rasia$^{2,3}$, V. Biffi$^{2,4}$, F. Mernier$^{1,8,9}$, N. Werner$^{1,11,12}$, M. Gaspari$^{5}$[^2], S. Borgani$^{2,4,6}$, S. Planelles$^{7}$, D. Fabjan$^{10,2}$, and G. Murante$^{2}$\
\
$^1$ MTA-Eötvös University Lendület Hot Universe Research Group, Pázmány Péter sétány 1/A, Budapest, 1117, Hungary\
$^2$ INAF, Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34131, Trieste, Italy\
$^3$ Department of Physics, University of Michigan, 450 Church St., Ann Arbor, MI 48109\
$^4$ Dipartimento di Fisica dell’ Università di Trieste, Sezione di Astronomia, via Tiepolo 11, I-34131 Trieste, Italy\
$^{5}$ Department of Astrophysical Sciences, Princeton University, 4 Ivy Ln, Princeton, NJ 08544-1001, USA\
$^{6}$ INFN, Instituto Nazionale di Fisica Nucleare, Trieste, Italy\
$^{7}$ Departamento de Astronomía y Astrofísica, Universidad de Valencia, c/Dr. Moliner, 50, 46100 Burjassot, Valencia, Spain\
$^{8}$ Institute of Physics,Eötvös University, Pázmány P. s. 1/A, Budapest, 1117, Hungary\
$^{9}$ SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands\
$^{10}$ Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia\
$^{11}$ Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic\
$^{12}$ School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan\
bibliography:
- 'ref.bib'
title: 'Mass-Metallicity Relation from Cosmological Hydrodynamical Simulations and X-ray Observations of Galaxy Groups and Clusters'
---
[galaxies: clusters: general — galaxies: clusters: intracluster medium — X-ray: galaxy: clusters — methods: numerical]{}
Introduction {#sec:intro}
============
Being the largest gravitationally-bound objects in the Universe, galaxy clusters can in first approximation be considered as closed boxes which contain a fair representation of the cosmic baryon content (@white.etal.1993, @frenk.etal.1999). Therefore they are ideal laboratories to study the cosmic cycle of the baryonic matter in its various phases: hot intra-cluster medium (ICM), cold gas, and stellar component. The first component, the ICM, emits predominantly in the X-ray band due to its high temperature ($T\sim10^{7}-10^{8}$ K), making X-ray observations a key tool to study thermodynamics of the ICM (see @bohringer.werner.2010 for a review). Furthermore, emission-line features of the X-ray observed spectra also reveal a wealth of information about the chemical composition of the intra-cluster gas, thereby offering a unique window to study the ICM metal enrichment (@werner.etal.2008 [@mernier18_REVIEW]).
The ICM metal enrichment involves numerous astrophysical processes (see, e.g., @borgani.etal.2008 [@biffi18_REVIEW]). Metals, elements that are heavier than H and He, are created via stellar nucleosynthesis and released by means of stellar mass loss of low- and intermediate-mass stars or of supernova (SN) explosions. Different types of supernovae produce different elements. The core-collapse or Type II supernovae (SNII) produce mainly light elements, e.g. O, Ne, Mg, or Si, while Type-Ia supernovae (SNIa) are source of heavy metals: Fe and Ni. The intermediate-mass elements are produced by both types of supernovae, and lighter elements (C, N) originate from low- and intermediate-mass stars during their Asymptotic Giant Branch (AGB) phase. The produced metals subsequently enrich the surrounding astrophysical environment thanks to multi-scale mixing processes, such as galactic winds, active galactic nuclei (AGN) feedback, ram-pressure stripping, and mergers (see @schindler.diaferio.2008 for a review). In particular, X-ray observations show evidences that AGN outflows can eject metals up to several 100 kpc even in massive galaxy clusters and groups (e.g., @kirkpatrick.etal.2011 [@ettori.etal.2013]). This phenomenon is reproduced in high-resolution hydrodynamical simulations as well [e.g., @gaspari.etal.2013].
Despite the astrophysically complex nature of the ICM metal enrichment, recent X-ray observations have shown that the distribution of metals in the ICM is remarkably homogeneous in space (@werner.etal.2013 [@urban.etal.2017]) and in time (@ettori.etal.2015 [@McDonald.etal.2016]). [@werner.etal.2013] and [@urban.etal.2017] show that the iron profile in the ICM of Perseus and another ten clusters observed by Suzaku is flat at radii $r>0.25R_{200}$[^3] with an iron abundance level of $\sim0.3\ Z_{Fe,\odot}$. In another work, [@McDonald.etal.2016], X-ray observations of $153$ clusters observed by [*Chandra*]{}, [*XMM-Newton*]{}, and [*Suzaku*]{} telescopes, reveal no evidence for strong redshift evolution of the global ICM metallicity ($r<R_{500}$) (see also @ettori.etal.2015). Even more interestingly, a recent observation of the Perseus cluster with [*Hitomi*]{} (@hitomi.2017) shows that the chemical composition of its ICM is consistent with the solar chemical composition in terms of abundance ratios.
Numerical modelling also supports the uniform picture of ICM metal enrichment. In recent studies based on simulations, [@biffi.etal.2017; @biffi.etal.2018] and [@vogelsberger.etal.2018] also find a flat metallicity profile in cluster outskirts that is constant over time. The uniformity of ICM metallicity in space and time supports the picture of early metal enrichment. In this scenario, early-time pristine gas in galaxies was enriched at high redshift ($z\sim5-6$) and then spread widely mainly by feedback processes. The gas was subsequently accreted into massive halos and heated by gravitational compression. Since the gas was already metal-rich at the time of accretion, the ICM metallicity profile appears to be flat at large cluster-centric distances.
In addition to a flat and constant metal distribution, the early enrichment scenario can be tested by investigating the mass dependence of the metallicity. For this reason, there is recently interest from both theoretical (@dolag.etal.2017 [@yates.etal.2017; @barnes.etal.2017; @vogelsberger.etal.2018]) and observational (@renzini.andreon.2004 [@mantz.etal.2017; @yates.etal.2017]) sides in investigating how the ICM metallicity varies with cluster scale. Up to date, the most complete observed sample, for the study of averaged ICM metallicity in clusters, has been carried out by [@mantz.etal.2017]. This work shows that iron abundance in the ICM anti-correlates with ICM temperature, however, the study is limited to a massive sample only ($T>5$ keV). An earlier work from [@yates.etal.2017], based on a compilation of various observational datasets from the literature and on a sample obtained from semi-analytical models, reports a similar anti-correlation between observed iron abundance and temperature in high-temperature systems, while in low-temperature systems ($T<1.7$ keV) the compiled sample exhibits a drop in the iron abundance (see also, @sun.etal.2012 [@mernier.etal.2016]). This drop is not present in recent theoretical studies (e.g., @yates.etal.2017).
In this paper we carry out a thorough study of how ICM metallicity varies (1) from group to cluster scales, (2) in various radial ranges and (3) in time, by employing cosmological hydrodynamical simulations.
The simulated sample that we analysed is an extended dataset with respect to the one presented in [@biffi.etal.2017; @biffi.etal.2018]. Those simulated clusters have also been shown to reproduce various realistic thermodynamical properties of the ICM (@rasia.etal.2015 [@Villaescusa-Navarro.etal.2016; @biffi.etal.2016; @planelles.etal.2017; @truong.etal.2018]).
In the first part of this study, we present a comparison between simulations and observations on the dependence of the ICM metallicity on the gas temperature, in which specific attention is payed to potential biases that might affect the observational results. In particular, the observational datasets that we use include the CHEERS sample observed by the [*XMM-Newton*]{} telescope (@de_plaa.etal.2017), in which the X-ray spectra are analysed with the last up-dated model of ICM emission for metallicity estimation. The detailed description of the CHEERS data as well as its X-ray analysis is provided in a companion paper (@mernier.etal.2018). In addition, we also compare our simulation results against observational data by [@mantz.etal.2017].
In the second part, we theoretically investigate how the metallicity varies as a function of cluster mass as well as how this relation evolves over redshift. In order to enhance the role played by AGN feedback on the early enrichment, we study how the iron mass, hydrogen mass and stellar fraction relate with the total mass in simulations with and without AGN feedback.
The paper is organized in the following way. We describe the main features of the numerical simulation and the method of analysis in Section 2. In Section 3, we briefly introduce the observed datasets: the CHEERS (@de_plaa.etal.2017), and [@mantz.etal.2017] samples. This is followed by a detailed comparison between simulations and observations. In Section 4, we present a theoretical investigation of the iron and oxygen abundance-mass relations as well as their evolution over time. The effects of AGN feedback on the mass-metallicity relation, and related quantities, are also discussed in this Section. We devote Section 5 to a detailed discussion of the systematics of simulation results and to a comparison with other theoretical studies. Finally, we summarise the main results and conclude in Section 6.
Throughout this study, we adopt the solar metallicity values provided by [@asplund.etal.2009] for both the theoretical and the observational analysis and we consider $h\equiv H_0/(100\ \rm{km}\ \rm{s}^{-1}\ \rm{Mpc}^{-1})=0.72$, where $H_0$ is the Hubble constant.
\[sec:cmrr\]
Analysis of Simulations {#Simulation}
=======================
The Simulated Clusters
----------------------
The simulations employed in this study have been described in recent works (@rasia.etal.2015 [@Villaescusa-Navarro.etal.2016; @biffi.etal.2016; @planelles.etal.2017; @biffi.etal.2017; @biffi.etal.2018; @truong.etal.2018]), where we show that they reasonably reproduce chemo- and thermo-dynamical cluster properties in comparison to observational data.
In this Section we summarise only the main features and refer the reader to the aforementioned references for more details, and specifically to [@biffi.etal.2017] for the description of how chemical enrichment is included in the simulations.
The simulated galaxy clusters are obtained from $29$ zoomed-in Lagrangian regions re-simulated with an upgraded version of the `GADGET-3` code (@springel.2005). Starting from an initial Dark-Matter (DM) only simulation with a volume of $(1 h^{-1}\rm{Gpc})^3$, 24 regions around most massive clusters with mass $M_{200}>8\times10^{14}h^{-1}M_\odot$, and 5 regions surrounding isolated groups with $M_{200}\sim[1-4]\times10^{14}h^{-1}M_\odot$ are selected and re-simulated at higher resolution and with added baryonic components. Each high-resolution region has a radius that extends at least 5 times the virial radius of the central object covering a box size of about $25-30 h^{-1}$ Mpc (see @bonafede.etal.2011 for more details on the initial conditions). The mass resolutions for the DM particles and the initial gas particles are $m_{DM}=8.47\times10^{8}h^{-1}M_\odot$, $m_{gas}=1.53\times10^{8}h^{-1}M_\odot$, respectively. The simulation is performed with cosmological parameters consistent with results from the WMAP-7 (@komatsu.etal.2011): $\Omega_m=0.24$, $\Omega_b=0.04$, $n_s=0.96$ for the primodial spectral index, $\sigma_8=0.8$ for the amplitude of the density fluctuations power spectrum, and $H_0=72\ \rm{km}\ \rm{s^{-1}}\ \rm{Mpc^{-1}}$ for the Hubble parameter. The Plummer-equivalent softening length is fixed equal to $\epsilon=3.75 h^{-1}\rm{kpc}$ for DM and gas particles, and $\epsilon=2h^{-1}\rm{kpc}$ for black hole and star particles. The DM softening length is fixed in comoving units for $z>2$ and in physical units at lower redshifts. For other types of particles, it is always given in comoving units.
Regarding the hydrodynamical scheme, we employ the improved Smoothed-Particle-Hydrodynamics (SPH) formulation described in [@beck.etal.2016]. Comparing to the standard `GADGET` code, the new SPH scheme incorporates a number of advanced features including: the choice of a higher-order Wendland $C^4$ kernel function, the implementation of a time-dependent artificial viscosity scheme, and artificial conduction. These advanced features improve the scheme’s ability in treating contact discontinuities and gas-dynamical instabilities, thereby overcoming several limitations of standard SPH schemes.
For the study of metallicity of the intra-cluster gas, we adopt two different prescriptions for the ICM physics which have been used to produce two simulated samples from the same initial conditions. The run called CSF (cooling-star-formation) includes the following physical processes:
- Heating/cooling from Cosmic Microwave Background (CMB) and from a UV/X-ray time-dependent uniform ionising background included as in [@haardt.madau.2001].
- Metallicity-dependent radiative gas cooling as in [@wiersma.etal.2009], and star formation [@springel.hernquist.2003].
- Metal enrichment, as described in [@tornatore.etal.2007], accounting for three different channels of enrichment, namely from SNII, SNIa, and AGB stars. We follow the production and evolution of 15 different chemical species: H, He, C, Ca, O, N, Ne, Mg, S, Si, Fe, Na, Al, Ar, Ni. We assume the stellar initial mass function (IMF) by [@chabrier.2003], and the mass-dependent lifetimes of [@padovani.matteucci.1993]. The stellar yields used in the model are the set provided in [@thielemann.etal.2003] for SNIa stars and the one from [@karakas.2010] for AGB stars. For SNII, we use the metal-dependent yields taken from [@woosley.weaver.1995] and [@romano.etal.2010]. The heavy elements produced by stellar particles in the simulations, according to this model, are then distributed to the surrounding gas particles by smoothing them onto the SPH kernel, as it is done for the other thermodynamical quantities. Once gas particles are enriched, the diffusion of metals is essentially due to the motion of the gas particles themselves. This allows metals to be circulated on large cluster scales over time.
- Thermal feedback from supernovae as originally prescribed by [@springel.hernquist.2003] with the value of the mass-loading parameter equal to 2. Kinetic feedback from the SN is also included, in the form of galactic winds with a velocity of $350\ \rm{km}\ \rm{s}^{-1}$.
The other run is called AGN. It has all the features of the CSF run but additionally includes the treatment of gas accretion onto super-massive black holes (SMBH) powering AGN feedback, for which we employ the model by [@steinborn.etal.2015] (an upgrade from the original model proposed by @springel.dimatteo.hernquist.2005) that considers both hot and cold accretion (e.g., @gaspari.etal.2018). In this updated model, we consider both radiative and mechanical feedback that are produced via gas accretion onto SMBH, and both are released into the surrounding environment in form of thermal energy. The radiated energy then couples with the surrounding gas with feedback efficiency $\epsilon_f=0.05$. In this model, we treat cold and hot gas accretion separately, in particular for the cold gas accretion, we boost the Bondi rate by a factor of $100$ thereby mimicking the effect of the cold accretion mode (see more detailed discussion in @gaspari.temi.brighenti.2017). For this study, we select a mass-limited sample with $M_{500}>10^{13}M_\odot$ with at least $100$ gas particles in the core regions ($r<0.1R_{500}$)[^4]. In Fig. \[fig1\] we show the cumulative distribution of cluster total mass for the AGN and CSF simulations, in terms of $M_{500}$, at the four redshifts, $z=0, 0.5, 1$, and $1.5$.
Computing simulated ICM quantities
----------------------------------
In the following we briefly describe how quantities of interest are computed within the simulation snapshots. In computing X-ray quantities, we select only gas particles with a low fraction of cold gas ($<10\%$) to avoid the inclusion of star-forming particles.
- *Projected quantities.* For the purpose of comparing simulation results with observational data, we employ projected quantities. Namely, we compute the temperature and the metallicity within a cylindrical volume with the length of $2\times R_{vir}$, where $R_{vir}$ is the virial radius[^5], and the area confined by two circular apertures. In order to be consistent with X-ray observations, which are typically centred on the surface brightness peak, we centre the cylindrical regions on the centre of the gas mass calculated from the gas particles within $R_{2500}$ (which is typically $\sim 0.3 R_{500}$). We verify that the impact of using the gas centre of mass instead of the X-ray peak is negligible in the case of the metallicity-temperature scaling (see Appendix A2).
For the temperature, we adopt the spectroscopic-like formula proposed by [@mazzotta.etal.2004]: $$T_{\rm sl}=\frac{\Sigma_i \rho_i m_i T_i^{0.25}}{\Sigma_i \rho_i m_i T_i^{-0.75}},$$ where $m_i$, $\rho_i$, and $T_i$ are the mass, density, and temperature of the $i^{th}$ gas element, respectively. For the computation of spectroscopic-like temperature, we apply a temperature cut $T_{i}>0.3$ keV to select particles that should emit in the X-ray band.
For the metallicity, we adopt the [*emission-weighted*]{} estimate. In case of iron (and similarly for oxygen) this quantity is defined as: $$Z_{\rm Fe,ew}=\frac{\Sigma_i n_{e,i}n_{H,i}\Lambda(T_i, Z_i)\times Z_{i,{\rm Fe}}}{\Sigma_i n_{e,i}n_{H,i}\Lambda(T_i, Z_i)},$$ where $Z_i$ and $Z_{i,{\rm Fe}}$ are respectively the global and the iron metallicities of the gas particle, and $n_{e,i}$, $n_{H,i}$ are the electron and hydrogen number densities, respectively. $\Lambda$ is the cooling function computed based on particle temperature and metallicity by assuming the APEC model (@smith.etal.2001) in XSPEC[^6] v.12.9.0 (@arnaud.1996) for radiative emission and by integrating over the $[0.01-100]$ keV energy band.
- *Theoretical estimates.* For the theoretical investigation of the simulated clusters, we use three-dimensional measurements, computed within spherical shells centred on the minimum of the system potential well.
For the iron and oxygen abundances we employ the [*mass-weighted*]{} formula, defined as: $$Z_{\rm X,mw}=\frac{\Sigma_i m_i\times Z_{i,{\rm X}}}{\Sigma_i m_i},$$ where $m_i$ is the gas particle mass, and $\rm X$ represents one of the two elements.
Similarly, to explore the intrinsic dependence of ICM metallicity on the cluster scale and related quantities, we also measure total, iron and hydrogen masses, as well as stellar fractions, evaluated within spherical three-dimensional regions.
Fitting Method
--------------
We model the relations between metallicity, total mass or temperature, and redshift with power-law functional forms. The specific function adopted will be specified, case by case, in the following sections. In order to characterise the slope and normalisation of the relations, we always perform a log-log linear regression fit. For this task, we employ the IDL routines `linmix_err.pro` and `mlinmix_err.pro` which adopt a Bayesian approach to linear regression, in which best-fit parameters are determined via a Monte Carlo Markov Chains method, as described in [@kelly.2007]. This method enables us to treat the intrinsic scatter as a free parameter, like the slope and normalisation, and to account for possible correlation between measurement errors and intrinsic scatter (see @kelly.2007 for detailed discussion). However, it is worth noticing that for simulation data, which do not have any associated statistical uncertainty, there are simpler methods that can be used for the fitting. For instance, we find that, by using the non-Bayesian fitting routine `robust_linefit.pro` in IDL to fit the mass-metallicity relation ($Z_{Fe}-M_{500}$) in the central region ($r<0.1R_{500}$), the results are consistent with the Bayesian approach within $1\sigma$.
Comparison to observations
==========================
In the first part of this study we investigate the iron abundance in simulated and observed clusters as a function of the ICM temperature. In fact, even though the mass is the optimal description of the system scale, this quantity is not directly observable and temperature measurements, which are easier to derive, are typically used in X-ray studies. For the purpose of a more faithful comparison to observational data, we therefore employ projected spectroscopic-like temperature estimates of the ICM in simulated clusters as well. We note that the relation between temperature and mass for the clusters in our AGN simulations is in reasonable agreement with observational findings, both at low and intermediate redshifts [@truong.etal.2018].
Observational Data Sets {#Observation}
-----------------------
In the following we briefly describe the main features of the two observational datasets used for the comparison with simulation results, namely the CHEERS sample and the sample presented in [@mantz.etal.2017]. Specifically, we will take advantage of the large temperature range spanned by the former, a local cluster sample, in order to investigate in detail the dependence of central metallicity on core temperature in local clusters. For the purpose of exploring the redshift evolution of the metallicity-temperature relation, in different radial ranges, we will use instead the latter, which contains a larger number of massive clusters.
- The CHEERS sample includes 43 nearby ($z<0.1$) cool core clusters, groups, and massive ellipticals observed by [*XMM-Newton*]{}. The iron abundance is constrained by the [*XMM-Newton*]{} EPIC (MOS 1, MOS 2, and pn) instruments, as they can access both the Fe-L and the Fe-K line complexes of the X-ray spectral window. We use the Fe measurements from [@mernier.etal.2018], which are derived within $0.1R_{500}$ and obtained from fitting X-ray spectra using the up-to-date version of the spectral code used to model the ICM emission, namely SPEXACT[^7]. A major update has been recently released (i.e., from SPEXACT v2 to SPEXACT v3, @de_plaa.etal.2017) and, compared to the previous observational results, SPEXACT v3 revises the Fe abundance in groups significantly higher and makes them on average consistent with that in clusters. In order to minimise the impact of the Fe-bias, all the spectra were fitted with three single-temperature components that mimic a Gaussian temperature distribution. The complete details and discussion on the data analysis methods and the effects of the latest spectral model improvements are presented in [@mernier.etal.2018].
- The sample by [@mantz.etal.2017] is the largest cluster sample available up-to-date. It consists of 245 massive systems ($T>5\,$keV) selected from X-ray and Sunyaev-Zel’dovich effect surveys, with X-ray observations obtained from [*Chandra*]{}. The sample encompasses a broad redshift range: $0<z<1.2$. Metallicity analysis is performed for three radial ranges: $[0-0.1]R_{500}$, $[0.1-0.5]R_{500}$, and $[0.5-1]R_{500}$. The spectrum extracted in each annular bin is fitted using XSPEC assuming a single-temperature component APEC model (ATOMDB version 2.0.2).
\[Comparison to observations\]
The $Z_{\rm Fe}-T$ relation for groups and clusters
---------------------------------------------------
In Fig. \[fig2\] we show the $Z_{\rm Fe}-T$ relation measured within $0.1R_{500}$ for our simulated samples in comparison to the CHEERS dataset. Since the observational sample includes only nearby objects, we consider only $z=0$ simulated clusters. In addition, to be consistent with the data, we select simulated systems that have temperature greater than $0.7$ keV at $z=0$. For the sake of clarity, we show the individual data points for the AGN (black asterisks) and CHEERS (red circles) samples, while for the CSF sample we present only the median $Z_{\rm Fe}-T$ relation and the associated $68.3\%$ confidence region (blue dashed line and shaded area).
To characterise the correlation between $Z_{\rm Fe}$ and $T$, we compute the Spearman’s rank correlation coefficients ($r_s$) for simulated and observed samples. Additionally, we also quantify the dependence of iron abundance on temperature by fitting the simulated and the observed data with the following formula: $$Z_{\rm Fe}=Z_{0T}\bigg(\frac{T}{1.7\ \rm{keV}}\bigg)^{\beta_T}, \label{eq4}$$ which is characterised by two parameters: the normalisation $Z_{0T}$ and the slope $\beta_T$. We choose the pivot point for the temperature equal to $1.7$ keV, which is close to the mean values of the simulated and observed sample temperatures. The $Z_{\rm Fe}-T$ in the CSF run cannot be simply described by a single power law (see the discussion in the next paragraph), therefore we fit Eq. (\[eq4\]) separately to high-temperature ($T>1.7$ keV) and low-temperature ($T<1.7$ keV) subsamples. The correlation coefficients $r_s$ and the best-fitting parameters of Eq. (\[eq4\]) are reported in Table \[tb1\].
Sample $\log_{10}(Z_{0T}\ [Z_{\rm Fe,\odot}])$ $\beta_T$ $\sigma\log_{10}Z_{\rm Fe}|T$ $r_s$
--------------------- ----------------------------------------- ---------------- ------------------------------- ---------------------------
AGN full $-0.178\pm0.011$ $-0.10\pm0.04$ $0.16\pm0.01$ $-0.21\ (2\times10^{-3})$
AGN NCC $-0.190\pm0.013$ $-0.14\pm0.04$ $0.16\pm0.01$ $-0.27\ (4\times10^{-4})$
AGN CC $-0.157\pm0.023$ $0.00\pm0.07$ $0.15\pm0.02$ $-0.08\ (6\times10^{-1})$
CSF ($T<1.7$ keV) $0.035\pm0.039$ $0.33\pm0.15$ $0.19\pm0.01$ $0.21\ (5\times10^{-3})$
CSF ($T\ge1.7$ keV) $0.082\pm0.032$ $-0.06\pm0.09$ $0.14\pm0.01$ $-0.04\ (8\times10^{-1})$
CHEERS data $-0.127\pm0.019$ $0.00\pm0.06$ $0.11\pm0.01$ $+0.03\ (8\times10^{-1})$
The AGN and CHEERS samples show almost no correlation between the iron abundance and the temperature in the cluster core. The Spearman’s correlation coefficients obtained from simulations reveal either no correlation ($|r_s| <0.1$ with 60-80 per cent probability of null hypothesis) or a very low-level of correlation ($|r_s| \sim 0.2-0.3$) with null-hypothesis probability of order of $10^{-3}$-$10^{-4}$ . The CHEERS data shows a flat distribution of iron abundance across the temperature range with the slope $\beta_T$ consistent with zero within the uncertainty. Also, in this case the two quantities are uncorrelated ($r_s\sim 0$) at higher significance (80 per cent consistent with no correlation). The relation for our AGN data also presents a very shallow slope $\beta_{T}\sim-0.1$, which implies that, on average, a cluster that is 10 times hotter than a group has only about $20\%$ lower metallicity. This indicates that the ICM iron abundance is statistically constant from groups to clusters in the entire sample. On the other hand, the CSF run exhibits a flat distribution of iron abundance only in high-temperature systems ($T>1.7$ keV), while in the low-temperature regime the CSF $Z_{\rm Fe}$ falls as temperature decreases with the slope $\beta_T\sim0.3$.
Compared to the CHEERS sample, the iron abundance of the AGN simulated objects with $T\geq 2$ keV is slightly lower than the observed value at a given temperature. At the pivot point of temperature $1.7$ keV, the observed $Z_{\rm Fe}$ is higher than the simulated one by $13$ per cent, yet the two values are consistent within the intrinsic scatters. While, the CSF $Z_{\rm Fe}$ is about $70\%$ higher than the CHEERS value at the pivot temperature. The offset in normalisation between simulations and observations can be ascribed to several factors including the sample selection as discussed in the following section and the details of the model for chemical enrichment used in the simulations, e.g., the stellar initial mass function and the sets of yields (see discussion in Section 5.1). Therefore, the slight offset between simulated and observed data should be not considered in itself a reason of serious concern. It is, however, relevant to remark that the offset between the AGN and the CSF samples reflects an intrinsic difference in the metal production predicted by these two models since they are both obtained with the same description of stellar and chemical evolution.
We note that the simulated data exhibit slightly higher scatter (e.g., in AGN simulation, $\sigma\log_{10}Z_{\rm Fe}|T=0.16$) than the observed CHEERS data ($\sigma\log_{10}Z_{\rm Fe}|T=0.11$). It is evident from Fig. \[fig2\] that the larger scatter in simulated data is mostly due to the statistics of low-temperature groups ($T<1.7$ keV). It is important to notice that in this low-temperature range, there is about $70$ per cent of the AGN objects, while the number of low-temperature objects in the CHEERS sample is $\sim42$ per cent. Observationally, it is more challenging to both detect and observe groups rather than clusters, particularly due to the depth required to obtain a good measurement of temperature and metal abundances. The observational X-ray selected sample thus tends to favor the inclusion of massive cool core clusters that are easier to detect and observe, whereas simulations have a larger fraction of groups. This is mostly due to the steepness of the mass function that enhances the statistics of smaller objects in quite large volume-limited zoom-in regions[^8]. These low-mass simulated systems have small radius (the mean $R_{500}\sim450$ kpc) so that the level of metallicity in the central region ($<0.1R_{500}$) is very sensitive to AGN feedback, the efficiency of radiative cooling, and the dynamical state of the system. To conclude, the larger scatter found in simulations compared to observations is mostly due to the CHEERS small statistics of low-temperature systems ($T<1.7$ keV). Further contributions may come from the observational difficulties of detecting groups with a low-metallicity core since they are likely less peaked in the X-rays.
### Cool-core and non-cool-core $Z_{\rm Fe}-T$ relations
Cool-core clusters are shown to host a larger amount of metals in the core region compared to non cool-core clusters (e.g, @DeGrandi.etal.2004). Therefore we might expect a difference in the $Z_{\rm Fe}-T$ relations derived from each separate population that needs to be evaluated to estimate any potential bias in observed samples. This is particularly relevant for the CHEERS sample as its member clusters are selected by RGS (the Reflection Grating Spectrometer onboard [*XMM-Newton*]{}) which is selectively sensitive to centrally-peaked clusters (@de_plaa.etal.2017) and peakiness can be considered as a “proxy” for the cool-coreness (e.g., @mantz.etal.2017). To address this issue, we divide the simulated AGN dataset into cool-core and non cool-core subsamples, and study the iron abundance-temperature relation for each of them separately. The same task cannot be done for the CSF dataset since in our CSF simulation, the diversity of CC and NCC systems is not present due to the lack of an efficient form of feedback, e.g., feedback from AGN (see @rasia.etal.2015 and @biffi.etal.2017 for detailed discussions). For the selection of simulated CC systems, we compute the pseudo-entropy defined as: $$\sigma=\frac{(T_{\rm IN}/T_{\rm OUT})}{(EM_{\rm IN}/EM_{\rm OUT})^{1/3}}, \label{eq5}$$ where $T$ is the spectroscopic-like temperature and $EM$ is the emission measure, computed in the IN ($r<0.05\times R_{180}$) and OUT ($0.05\times R_{180}<r<0.2\times R_{180}$) regions. Following [@rasia.etal.2015] we define the CC systems as those that have a pseudo-entropy value lower than $0.55$, whereas systems with $\sigma > 0.55$ are classified as NCC. This definition of CC clusters was originally introduced to compare the statistics of CCs in the simulated sample of [@rasia.etal.2015] with the observational ratio found by [@rossetti.etal.2011]. We keep the same criterion here because it appropriately mimics that of the CHEERS sample for the following two reasons:
- The pseudo-entropy depends on both the emission-measure ratio and the temperature ratio. However, the former is the dominant quantity as CCs have a steeply declining emission profiles with radius. This propriety is common with the CHEERS clusters that, indeed, have been selected for having a clear X-ray peak in their core.
- Our AGN simulations exhibit a strong correlation between the pseudo-entropy and the central entropy, namely all simulated CCs have low pseudo-entropy (by definition) and at the same time they all have a low value of the central entropy. Similarly, also the CHEERS objects are characterised by a low central entropy (below $30\ \rm{keV}\rm{cm}^2$ accordingly to table A.2 in @pinto.etal.2015).
For these reasons, a cool-core object is similarly identified in our simulations and in the observational sample that we compare with. Therefore, we do not investigate other criteria used in literature (@cavagnolo.etal.2009 [@hudson.etal.2010; @leccardi.etal.2010; @McDonald.etal.2013; @pascut.ponman.2015; @barnes.etal.2018]).
We show in Fig. \[fig3\] the $Z_{\rm Fe}-T$ relation for both the CC and NCC subsamples obtained from the AGN simulation along with the CHEERS data. In general the CC systems have slightly higher central iron abundance, e.g., $\sim8\%$ at 1.7 keV, than the NCC ones, which is consistent with recent numerical and observational studies showing that the ICM iron profile in CC clusters is steeper and peaked in the central regions (@leccardi.etal.2010 [@rasia.etal.2015; @ettori.etal.2015; @biffi.etal.2017]). Whereas the difference is more prominent for clusters with $T>2$ keV, for the low-temperature systems, the separation between the metallicity level in CCs and NCCs is not well-established and the scatter is large. As a consequence, the slope of the $Z_{\rm Fe}-T$ of CCs is flatter than that of NCCs. Hence, for the study of $Z_{\rm Fe}-T$ relation, the effect of solely including CC systems mainly affects the slope of the high-temperature end, whereas the effect is not very significant in the low-T regime.
[{width="\textwidth"}]{}
The CC subsample is in better agreement with the CHEERS dataset than the NCC subsample, not only in terms of the slope, $\beta_T\sim0$, but also in terms of the normalisation. The simulated and observed normalisations are consistent at $1\sigma$ and the relative offset is less than 0.05 $Z_{\rm Fe,\odot}$. The cores of both simulated CC and observed clusters present a constant iron abundance equal to $Z_{\rm Fe}\simeq0.75 Z_{\rm Fe,\odot}$ over the considered range of temperature with a dispersion of $\sim40$ per cent and $\sim30$ per cent, respectively. However, we remind that the absolute value of the metallicity-temperature normalisation depends on metal enrichment model and its assumptions (discussed in more detail in Section 5.1), therefore the matching (or mismatching) value of the normalisations for simulations and observations should be not over-interpreted. Instead, it is worth stressing how the simulated CC subsample is in better agreement with the CHEERS objects.
[{width="\textwidth"}]{}
$Z_{\rm Fe}-T$ evolution in massive clusters
--------------------------------------------
In this subsection we compare the AGN and CSF simulated $Z_{\rm Fe}-T$ relations with the results obtained by [@mantz.etal.2017] for massive clusters. To be consistent with them, we select simulated clusters with $T_{[0.1-0.5]R_{500}}>5$ keV, where $T_{[0.1-0.5]R_{500}}$ is the temperature measured within the region $[0.1-0.5]R_{500}$. We consider 6 different snapshots corresponding to redshifts: $z=0, 0.25, 0.5, 0.6, 0.8$, and $1$, where we, respectively, identify $26(30), 19(23), 10(15), 11(15), 6(8),$ and $2(3)$ objects in the AGN (CSF) simulation. For this comparison, we use the projected emission-weighted iron abundance and the spectroscopic-like temperature computed within three radial ranges: $[0-0.1]R_{500}$, $[0.1-0.5]R_{500}$, and $[0.5-1]R_{500}$. Following the approach by Mantz et al., we quantify the correlation between $Z_{\rm Fe}$ and $T$ in addition to study the evolution of the relation by simultaneously fitting all simulated data with the formula: $$Z_{\rm Fe}=Z_{0T}\times\bigg(\frac{1+z}{1+z_{\rm{piv}}}\bigg)^{\gamma_z}\times\bigg(\frac{T}{T_{\rm{piv}}}\bigg)^{\beta_T},\label{eq7}$$ where the free parameters are: the normalisation $Z_{0T}$, the slope $\beta_T$, and the redshift evolution of the iron abundance, $\gamma_{z}$. We fix the pivot values of redshift and temperature, $z_{\rm{piv}}$ and $T_{\rm{piv}}$, to the same values used in the observational analysis reported in Table \[tb2\], where we also list the best-fitting parameters of the $Z_{\rm Fe}-T$ relation from our analysis as well as from the work of [@mantz.etal.2017] (see their Table 1).
$Aperture$ $z_{\rm{piv}}$ $T_{\rm{piv}}[\rm{keV}]$ $\log_{10}(Z_{0T}\ [Z_{\rm Fe,\odot}])$ $\beta_{\rm{T}}$ $\gamma_{\rm{z}}$ $\sigma_{\log_{10}Z_{\rm Fe}|T}$ $r_s$
---------------------------- ---------------- -------------------------- ----------------------------------------- ------------------ ------------------- ---------------------------------- --------------------------- -- --
[**AGN**]{}
$[0.0-0.1]R_{500}:$ $0.23$ $6.4$ $-0.230\pm0.011$ $-0.38\pm0.11$ $-0.32\pm0.13$ $0.09\pm0.01$ $-0.40\ (4\times10^{-4})$
$[0.1-0.5]R_{500}:$ $0.19$ $8.0$ $-0.440\pm0.009$ $-0.05\pm0.08$ $-0.18\pm0.06$ $0.045\pm0.004$ $+0.01\ (9\times10^{-1})$
$[0.5-1]R_{500}:$ $0.17$ $6.7$ $-0.720\pm0.016$ $-0.47\pm0.09$ $-0.34\pm0.10$ $0.07\pm0.01$ $-0.38\ (9\times10^{-4})$
[**CSF**]{}
$[0.0-0.1]R_{500}:$ $0.23$ $6.4$ $-0.003\pm0.021$ $-0.38\pm0.10$ $-0.46\pm0.17$ $0.16\pm0.01$ $-0.30\ (4\times10^{-3})$
$[0.1-0.5]R_{500}:$ $0.19$ $8.0$ $-0.350\pm0.014$ $0.03\pm0.13$ $-0.37\pm0.12$ $0.10\pm0.01$ $+0.17\ (1\times10^{-1})$
$[0.5-1]R_{500}:$ $0.17$ $6.7$ $-0.843\pm0.041$ $-0.72\pm0.20$ $-0.93\pm0.24$ $0.19\pm0.01$ $-0.04\ (7\times10^{-1})$
[**[@mantz.etal.2017]**]{}
$[0.0-0.1]R_{500}:$ $0.23$ $6.4$ $-0.217\pm0.009$ $-0.35\pm0.06$ $-0.14\pm0.17$ $0.08\pm0.01$ N/A
$[0.1-0.5]R_{500}:$ $0.19$ $8.0$ $-0.384\pm0.007$ $0.10\pm0.07$ $-0.71\pm0.15$ $0.04\pm0.01$ N/A
$[0.5-1]R_{500}:$ $0.17$ $6.7$ $-0.622\pm0.040$ $0.22\pm0.34$ $-0.30\pm0.91$ $0.00^{+0.07}_{-0.00}$ N/A
As shown in Fig. \[fig4\], the AGN simulated radial trend of the averaged iron abundance is consistent with the data showing a decrease of similar amplitude from the cluster core to the outskirts. On average, both simulated and observed iron abundance are higher in the innermost region, where $Z_{\rm Fe}\sim0.6 Z_{\rm Fe,\odot}$, and they gradually decrease to the level of $0.2 Z_{\rm Fe,\odot}$ in the most external radial range. This result is in line with previous works on ICM metallicity profiles from both simulations and observations (e.g., @werner.etal.2013 [@rasia.etal.2015; @urban.etal.2017; @biffi.etal.2017; @biffi.etal.2018]). On the other hand, the CSF simulated clusters are largely inconsistent with the data: the iron abundance is about $70$ per cent higher than the observed value in the central region ($r<0.1R_{500}$), but rapidly decreasing with temperature in the most external regions ($r>0.5R_{500}$).
The AGN simulation also agrees well with the analysis by Mantz et al. on the $Z_{\rm Fe}-T$ intrinsic scatter. Among the three considered radial ranges, both simulations and observations present the largest scatter in the core (see Table \[tb2\]), that is strongly affected by several astrophysical processes such as feedback from the AGN, or intense stellar activity. The CSF clusters exhibit higher scatter among the three considered ranges in comparison to the AGN and Mantz et al. results.
In terms of the slopes, we notice a good agreement between both simulations and observations in the two regions within half of $R_{500}$, while the best-fitting lines seem to have opposite trends in the cluster outskirts ($\beta_{\rm AGN}=-0.47$, $\beta_{\rm CSF}=-0.72$, and $\beta_{\rm obs}=0.22$). However, the observational constraints are weak and considering the 1$\sigma$ error associated with $\beta_{\rm obs}$, equal to 0.34, the AGN and observational slopes are consistent within $2\sigma$. In addition, we notice that most of the AGN simulated points are well within the shaded area which shows the observational 1$\sigma$ dispersion around the best-fitting line.
It is interesting to note that the pronounced steepness of the AGN data is caused by the specific selection of the hottest clusters as illustrated in Fig. \[fig5\], where the best-fitting relation of the most massive sample (red line) is compared with the overall trend (black line). In particular, the steep relation that characterises the core of the massive simulated clusters is biased by the segregation between the CC and NCC systems: the CCs have both higher metallicity and lower temperature compared to the NCCs. This separation disappears when the groups are added to the sample (see also Fig. \[fig3\]). According to [@barnes.etal.2018], in their simulations this result might depend on the operational definition of cool-core systems. Addressing this issue with our simulated clusters is nevertheless beyond the scope of this work.
Finally, the evolution of the simulated $Z_{\rm Fe}-T$ relations is positive in all radial ranges: at fixed temperature, the iron abundance increases with time (see $\gamma_z$ values in Table \[tb2\]). The amplitude of this variation in the AGN simulation is however limited, as the iron abundance on average grows by less than 30% from $z=1$ to $z=0$. This result is consistent with the results from Mantz et al. (except in the intermediate range) and from other observational analyses (@ettori.etal.2015 [@McDonald.etal.2016]). Differently, the CSF iron abundance in general evolves more rapidly in comparison to the observational data, except in the intermediate range $[0.1-0.5]R_{500}$, due to highly efficient star formation in the CSF simulation.
As for the radial trend of the evolution, the simulated data show that the normalisation of the $Z_{\rm Fe}-T$ relation evolves almost equally in the three radial ranges, except for the enhancement of the CSF $Z_{\rm Fe}$ in the range $[0.5-1]R_{500}$, while the observed data by [@mantz.etal.2017] exhibit a stronger evolution of the iron abundance, with $\gamma_{z}=-0.71\pm0.15$, in the intermediate radial range. The authors suggest that the late-time increase of iron abundance of the gas in the intermediate radial range could be due to the mixing with enriched gas from the cluster centres caused by mergers or AGN outflows. Observational results on the spatial pattern of metallicity evolution is, however, still matter of debate. At variance with the conclusions by [@mantz.etal.2017], [@ettori.etal.2015] show that the ICM metallicity slightly evolves in the central region of the CC clusters only and, even in these objects, it remains constant at larger radii. On the other hand, [@McDonald.etal.2016] point out that the ICM metallicity is consistent with no evolution outside of the core.
[{width="\textwidth"}]{}
[{width="\textwidth"}]{}
Mass-Metallicity Relation and Evolution {#Mass_Metallicity Relation and Evolution}
=======================================
After verifying that our numerical model generally reproduces observational findings, we provide here a detailed prediction on the scale invariance of the metallicity distribution and its evolution. In observational data the ICM temperature is used as mass-proxy, however, since both temperature and metallicity are derived from the same spectra there is a certain degeneracy between the two quantities. Furthermore, the measurements of the temperature in the central regions can be biased low because of multi-temperature gas. In the previous section we mimic the spectral measurements of metallicity and temperature by using projected quantities and we showed agreement between simulated and observed data. In this section, we take advantage of the precise knowledge of the mass from simulations and we study how the distribution and evolution of metallicity depends on the total mass ($M_{500}$) and how this trend is influenced by AGN feedback. At first, we will analyse the mass dependence of both iron and oxygen in AGN and CSF simulations, while in the second part we will focus exclusively on the iron because the two metal elements present very similar behaviours. All relations are extended to poor groups, $M_{500} > 1\times 10^{13} M_{\odot}$, satisfying the condition reported at the end of Section 2.1.
The $Z_{\rm Fe}-M_{500}$ and $Z_{\rm O}-M_{500}$ relations
----------------------------------------------------------
The relationships between $Z_{\rm Fe}$, $Z_{\rm O}$ and $M_{500}$ are computed within the same apertures as before with the addition of one more external region: $[1-2]R_{500}$. The relations are evaluated separately at four fixed redshifts: $z=0,\ 0.5,\ 1,$ and $1.5$. At a second stage, we quantify the evolution of the relation combining the samples.
[{width="\textwidth"}]{}
Fe O
--------------------- ----------------------- ---------------- ---------------------- ----------------
Radial Range $\log_{10}A_{\rm Fe}$ $B_{\rm Fe}$ $\log_{10}A_{\rm O}$ $B_{\rm O}$
$[0.0-0.1]R_{500}$: $-0.090\pm0.016$ $-0.22\pm0.03$ $-0.175\pm0.003$ $-0.05\pm0.01$
$[0.1-0.5]R_{500}$: $-0.404\pm0.012$ $-0.18\pm0.02$ $-0.528\pm0.008$ $-0.08\pm0.02$
$[0.5-1.0]R_{500}$: $-0.623\pm0.008$ $-0.29\pm0.02$ $-0.762\pm0.006$ $-0.29\pm0.01$
$[1.0-2.0]R_{500}$: $-0.765\pm0.007$ $-0.06\pm0.02$ $-0.911\pm0.007$ $-0.05\pm0.01$
In Fig. \[fig6\] we show the results on the iron (upper panel) and oxygen (lower panel) mass relations for both AGN and CSF simulations. As the mass-metallicity relations in the CSF run cannot be simply described by a single power law, we opt to represent the CSF results by showing the median relations of $Z_{\rm Fe}-M_{500}$ and $Z_{\rm O}-M_{500}$ in 10 logarithmic mass bins. To characterise the mass-metallicity relations of the AGN clusters, we fit the simulated data extracted in a given radial range and at a particular time, to a formula similar to Eq. (\[eq4\]): $$Z_{X}=Z_{0,X}\times\bigg(\frac{M_{500}}{10^{14}M_\odot}\bigg)^{\beta_{X}}, \label{eq2}$$ where $X$ stands for either iron or oxygen. The best-fitting parameters of the $Z_{\rm Fe}-M_{500}$ and $Z_{\rm O}-M_{500}$ relations as well as the Spearman’s rank correlation coefficients are reported in Table \[tb3\]. We include the results for all four apertures and for all times from $z=0$ to $z=1.5$.
Confirming the results shown in Section 3, the CSF simulation shows a significantly steeper metallicity profile compared to the AGN. In the central region, the CSF clusters on average exhibit $Z_{\rm Fe}$ and $Z_{\rm O}$ which is $\sim1.5$ times higher than the AGN objects while in the outskirts it is lower by a factor $\sim2.3$. We notice that the CSF metallicity appears to drop in low-mass systems not only in the cluster core ($r<0.1R_{500}$), as seen in Section 3.1, but also at larger radii. We provide a more quantitative discussion on the comparison between AGN and CSF mass-metallicity relations later in Section 4.2.
Unlike the CSF case, the mass-metallicity relation for AGN simulations is well described by a single power law at all the considered radius and redshift ranges. In general, the behaviour of the two mass-abundance relations, $Z_{\rm Fe}-M_{500}$ and $Z_{\rm O}-M_{500}$, is very similar and almost independent of the radial range and redshift at which they are computed. The Spearman’s rank correlation coefficients (Table \[tb3\]) for both relations indicate an anti-correlation between the metal abundances and the mass in the innermost region ($r<0.5R_{500}$), especially strong at the lowest redshifts ($z \leq 0.5$) where the correlation coefficients are around $0.5-0.6$ with extremely low null-hypothesis probability — below $<10^{-11}$. At higher redshifts, the anti-correlation between the inner quantities reduces to mild ($|r_s| \sim 0.3-0.4$) but still with high significance (null-hypothesis below $10^{-3}$). Considering, instead, the metallicity abundances in the cluster outskirts at low redshifts, i.e. $z \leq 0.5$, we verify that they do not correlate with the total mass ($|r_s|$ between 0.05 and $0.2$ with a high probability — above 40 per cent — that the correlation value is obtained by chance). However, the slopes of all relations are very shallow ($\beta <-0.15$) everywhere and at anytime (see also Fig. \[fig7\]). The net difference between a $10^{14} M_{\odot}$ group and a $10^{15} M_{\odot}$ cluster is limited to an abundance decrease of 20-30 per cent. This mild trend is present at any time since the slope $\beta$ does not substantially vary with redshift as shown in the top panels of Fig. \[fig7\].
In the bottom panels of Fig. \[fig7\], we report the evolution of the scatter. We notice that the $Z_{\rm O}-M_{500}$ relation always presents a higher intrinsic scatter than the $Z_{\rm Fe}-M_{500}$. The difference is stronger in the core ($r<0.1R_{500}$) where it reaches approximately $0.02-0.03$ dex. With the exception of the difference in amplitude, the trends with radii and time of both scatters are almost identical: the largest values are detected in the core and at most recent times ($\sigma_{\log_{10}Z|M}=0.14-0.16$ dex). Outside that region, both scatters promptly decrease and they remain constant out to the virial region ($\sim 2 R_{500}$). The redshift-dependence of the intrinsic scatters is present only in the core where the most important astrophysical processes take place. Specifically, for the iron and the oxygen abundances we respectively find a scatter increase of $37$ per cent and $18$ per cent from $z=1.5$ to $z=0$.
Since the slopes of the AGN mass-metallicity relations are constant with time at all radii with the exception of a minimal variation in the most external radial bin, we can study the evolution of the relations by simply measuring the shift in the normalisation at a fixed mass that we chose to be equal to $M_{500}=10^{14}M_\odot$. We quantify the evolution in each radial range by fitting the best-fitting normalisations with the following relation: $$Z_{0}(z)=A\times(1+z)^B\label{eq3}.$$ The results are reported in Table \[tb4\] and shown in Fig. \[fig8\]. The overall metallicity exhibits an extremely weak evolution within $R_{500}$ and no evolution at all in the region between $R_{500}$ and $2R_{500}$. The largest evolution regards the increase by about 23 per cent of the iron and oxygen abundance with respect to the present values in the $[0.5-1] R_{500}$ region. At redshift $1.5$ the normalisations of the iron and oxygen are already very close to the $z=0$ levels. To be precise, in the four radial ranges from the core to the virial radius, the iron normalisation at $z=1.5$ amounted to 82, 85, 77, 95 per cent of its value at $z=0$. The level of the oxygen is even less evolving since its normalisation is almost constant (in the four regions the $z=1.5$ value is 96, 93, 77, 96 per cent the $z=0$ value). Considering the shallow slope of the relation this result implies that the metal level was already built up in high$-z$ clusters.
### Evolution of the stellar fraction
In the previous section, we found a mild variation in the oxygen abundance in the AGN simulation only in the $[0.5-1] R_{500}$ radial range. We check, here, whether this could be related to an increase of stellar content in that region. We, thus, consider how the stellar fraction, $f_{*}\equiv M_{\rm star}/M_{\rm tot}$, depends on the total mass, $M_{500}$, and how it evolves with time from $z=1.5$ to $z=0$. As found for the metal abundances and as expected, the stellar fraction exhibits negative gradient with radius: the stellar content is mostly concentrated in the central region ($r<0.1R_{500}$) and rapidly decreases towards the outskirts (see also, e.g., @planelles.etal.2013 [@battaglia.etal.2013]). In the core $r<R_{500}$ there is a trend with the cluster mass but no significant evolution from $z=1.5$ to $z=0$. Precisely, the stellar fraction is 10-15 per cent in the core of the smallest systems and about $4-5$ per cent in the largest clusters. The change of the relation normalisation, measured at $10^{14} M_{\odot}$, is less than 8 per cent. The result found in our simulations is in line with the recent observational study by [@chiu.etal.2018] (see also, e.g., @lin.etal.2012), who also found an anti-correlation between stellar fraction and cluster mass within $R_{500}$ (see also, @planelles.etal.2013) and no evidence of evolution in the redshift range between $z=0.2$ and $z=1.25$.
However, if we restrict to the region $[0.5-1] R_{500}$ we do see that $10^{14} M_{\odot}$ clusters have a $30$ per cent reduction in their stellar fraction (see Fig. \[fig9\]). The amplitude of this variation is still small but, noticeably, goes in the opposite direction with respect to the evolution of the oxygen abundance that we discuss in the previous section: while, at fixed mass, the abundance grows from z=1.5 to z=0, the stellar content decreases. The increase in metallicity therefore cannot be associated to fresh stellar formation, instead, it is related to the accretion of already enriched gas, in addition to the accretion of pristine gas. In the next section, we investigate the role played by the AGN in raising the metal level in the outskirts of small groups.
[![The stellar fraction-mass relation ($f_*-M_{500}$) in the AGN simulation shown at the radial range $[0.5-1]R_{500}$ for different redshifts. The solid lines represent the best-fit relations at those redshifts, while the shaded areas specify the $68.3\%$ confidence regions.[]{data-label="fig9"}](fig9 "fig:"){width="50.00000%"}]{}
[{width="\textwidth"}]{}
The effects of AGN feedback on the mass-metallicity relation
------------------------------------------------------------
Accounting that the iron and oxygen behaviours are very similar, in this Section we focus only on the former and we discuss the effect of the AGN on the mass-metallicity relation by comparing two sets of simulations obtained with and without the AGN.
In Fig. \[fig10\], we show how the iron abundance, $Z_{\rm Fe}$, the hydrogen mass, $M_{\rm H}$, the iron mass, $M_{\rm Fe}$, and the stellar fraction vary with the total mass, $M_{500}$, in both simulations at $z=0$. The four columns correspond to the different radial apertures from the core to the virial regions (from left to right). In the bottom panels, we enlighten the differences between the CSF and AGN results. The most striking difference on the abundance-mass relations derived in the two runs is that without AGN the relation cannot be represented by a single power-law since there is a break at around $ 2\times 10^{14} M_{\odot}$.
At the cluster scale ($M_{500}>2\times 10^{14}M_\odot$), the abundance dependence on mass shows the same trend in the two runs at all radii, but there is a clear offset between the normalisations of the two relations. In the core, $r<0.1R_{500}$, the CSF $Z_{\rm Fe}$ is about 60 per cent higher than in the AGN run while in the two most external regions, $[0.5-1]R_{500}$ and $[1-2]R_{500}$, it is about 25 and 50 per cent lower, respectively. To understand the origin of these ratios, we look at the two separate contributions from the hydrogen mass and the iron mass in the second and third row, respectively. The gas fraction of the CSF clusters is always lower than that of the AGN clusters because the efficient radiative cooling, not regulated by the AGN activity, cools down a greater amount of hot gas that is subsequently converted into stars and thus is removed from the ICM. The phenomenon is in place at all radii but it is particularly strong in the core where the CSF runs host unrealistically massive bright central galaxies. The result of the excessive overall star formation is to drastically enhance the stellar fraction, as can be seen in the fourth row. In the core and at fixed mass, the CSF runs can have three times more stars than in the AGN runs outside the core. In the CSF runs the star formation is actually still active at low-redshifts, however, the process does not lead to larger iron mass of the ICM with respect to the AGN clusters (third row), because the freshly formed metals are immediately locked back into newly formed stars. In this way the efficient stellar production of the CSF runs prevents the circulation of the metals from the star forming regions to the ICM. For this reason the iron mass is lower than in the AGN outside the core. On the other hand, AGN feedback peaks at high redshift, when the potential well of the (proto-)cluster is still relatively shallow and star formation is also quite intense. As a consequence, this feedback channel plays a key role on spreading the metals created at high redshift. The process has the twofold effect of removing metals from star forming regions and of enriching the pristine gas that surrounded the small potential well of the early galaxies and that subsequently accrete into the low-$z$ clusters [@biffi.etal.2017; @biffi.etal.2018].
Both radiative cooling and AGN heating have a stronger impact in the lowest mass regime. The iron abundance in the CSF groups is largely reduced everywhere outside the core, while it agrees with the value in the AGN run in the innermost region. There, the hydrogen and iron masses depart from the AGN runs but with similar amplitude and sign. Indeed, generally the early activity of the AGN produces less concentrated groups with a reduced gas contribution. In addition to reducing the gas available for producing new stars, the AGNs also heat the medium. Both phenomena quench star formation, thereby reducing the accretion of already enriched gas, so that iron abundance can only grow through the explosions of long-lived SNIa. Outside the core, the reduction in the gas mass in the CSF groups is comparable to that in the CSF clusters. However, the iron mass decreases even further in groups without AGN because the radiative cooling, which is more efficient in smaller systems, selectively removes highly enriched gas.
Discussion
==========
Systematics
-----------
Cosmological simulations are useful to study the trend of metallicity with radius, mass, or temperature as well as its scatter and redshift evolution. The level of metals produced, instead, depends on the assumptions and simplifications underlying the stellar evolution and chemical models, and the measuring procedure. We dedicate this section to the discussion of systematics that might affect our conclusions.
[*Model of chemical enrichment.*]{} The chemical distribution and evolution derived from numerical simulations depend on the assumed models of star formation and stellar evolution. In [@tornatore.etal.2007], the authors present the chemical model employed in our simulations and thoroughly investigate the effect of changing the stellar initial mass function, the yields for SN, the SN explosion rate (or equivalently the lifetimes), and the SN feedback efficiency. We briefly summarise here the main results of that work. Changing the IMF has the strongest effect on the pattern of chemical enrichment. Using an IMF that is top-heavier than Salpeter IMF produces a higher value of iron abundance (by a factor of 2), to a level that exceeds the observed amount, and also increases the $[{\rm O/Fe}]$ relative abundance ($\sim60\%$). On the other hand, the SN yields and explosion rate have marginal effect on the overall pattern of chemical enrichment. Finally, increasing the SN feedback strength results in suppressing the star formation rate thereby decreasing the level of metal enrichment.
[*Emission-weighted versus spectral metallicity.*]{} To compare with the observational data, we consider the emission-weighted metallicity. In [@rasia.etal.2008] it was shown that the metallicity derived from [*XMM-Newton*]{} spectra of simulated mock observations was generally in good agreement with the emission-weighted metallicity. In particular, the emission-weighted estimation was proven to well reproduce the iron abundance for objects with temperature $T<2$ keV and $T>3$ keV where the Fe abundance is solidly measured via either Fe-L or Fe-K lines. A possible overestimate of order of 15-40 per cent was found in systems with intermediate temperature (with the highest discrepancy due to low signal-to-noise ratio). However, the small detected bias was found for the spectroscopic metallicity obtained under the assumption of a single temperature emitting plasma. Instead, the observational measures that we compare with (@mernier.etal.2018) use a multi-temperature approach. With this approach, Mernier et al., appropriately capture the level of the continuum close to the line, therefore, suppress any residual bias in the equivalent width of the lines related to an incorrect determination of the continuum. Moreover, this approach allows one to reproduce the spectra of systems that behave as single-temperature objects, since the temperature parameters associated with their different model components are independent and free to converge if necessary (see Section 2.2 in @mernier.etal.2018).
[*Projection effects.*]{} Observational measurements of metallicity are rarely de-projected. Therefore, when we compare simulations with the observational data, we need to project the simulated quantities. The main effect of projecting the abundances along a certain direction is that the normalisation of the metallicity-temperature relation becomes slightly lower than in the case of 3D abundance. This effect is stronger in the cluster core ($r<0.1R_{500}$), while it becomes less important in the outer regions (see details in Appendix A1). Due to the negative radial gradient of metallicity (@biffi.etal.2017), projection reduces the metal abundance with respect to the value computed within a sphere. This effect becomes more significant in low-mass objects as the majority of metals is concentrated in the very central regions. As a result, the 2D metallicity-temperature relation appears to be slightly shallower than the 3D relation, yet the two slopes are consistent within $1\sigma$. In conclusion, projection effects do not alter the trends discussed here.
Comparison to other numerical studies
-------------------------------------
Other authors recently also investigated the ICM metallicity-temperature relation using semi-analytical models (@yates.etal.2017) as well as cosmological simulations (@dolag.etal.2017 [@barnes.etal.2017; @vogelsberger.etal.2018]).
[@yates.etal.2017] compiled 10 different observational datasets taken from literature and homogenised the datasets to study the $Z_{\rm Fe}-T$ relation within $R_{500}$ and compare it with numerical results obtained from the semi-analytical L-GALAXIES galaxy evolution model. The semi-analytical model predicts a weak anti-correlation between the ICM iron abundance and temperature for groups and clusters as observed in our study (with the slope of the $Z_{500}-T_{500}$ relation of $\sim-0.1$), while this behaviour is present only in observed clusters of the homogenised dataset with $T>1.7$ keV. In the group regime, instead, the observed $Z_{\rm Fe}$ appears to drop as the temperature decreases. The authors suggest that the discrepancy between simulated and observed results can be solved by requiring an efficient mechanism to remove metal-rich gas, e.g., via AGN feedback, out of the central cluster regions. Our results do not support this interpretation. Indeed, our AGN simulations shows that the metallicity level in groups and clusters is similar in regions outside the cluster core ($r>0.1R_{500}$). Further on, the agreement between our AGN simulations and the CHEERS sample on the trend of the $Z_{\rm Fe}-T$ relation in the innermost regions suggests that the discrepancy between simulated and observed data reported in [@yates.etal.2017] might be due to a bias on the spectroscopic metallicity derived by fitting the observational spectra with an old atomic data code. [@mernier.etal.2018] discuss in detail how the old version of atomic data might significantly bias low the inferred metallicity in low-temperature systems.
There are also studies of the $Z_{\rm Fe}-T$ relation using cosmological simulations: Magneticum (@dolag.etal.2017), Cluster-EAGLE (@barnes.etal.2017), and IllustrisTNG (@vogelsberger.etal.2018). Those simulations include a wide range of astrophysical processes: radiative cooling, star formation and chemical enrichment, stellar and AGN feedback. [@barnes.etal.2017] and [@vogelsberger.etal.2018] predict relatively flat trend of the $Z_{\rm Fe,500}-T_{500}$ relation with $1\sigma$ dispersion less than $0.1$ solar unit. Similarly, @dolag.etal.2017 show very mild temperature trend of the Fe abundance in the central regions ($r<R_{2500}$). Those results are in line with our study on the trend of the $Z_{\rm Fe}-T$ relation.
The fact that our AGN simulations reproduce a trend for the iron abundance-temperature relation which is consistent with the latest observational results by [@mernier.etal.2018] supports current models of cosmological simulations, in particular it further emphasises the role of AGN feedback in shaping not only ICM X-ray properties but also its metal enrichment.
Summary and Conclusions
=======================
X-ray observations have shown that the intra-cluster gas metallicity appears to be distributed uniformly at large radii and exhibits no significant trend over cosmic time. We confirm these findings in our simulations (@biffi.etal.2017 [@biffi.etal.2018]) and we investigate, here, how the ICM metallicity depends on cluster scale (expressed both as temperature and mass) and how these relations evolve. The simulations are performed with an updated version of the `GADGET-3` code which includes radiative cooling, star formation, metal enrichment, stellar and AGN feedback. Results from observational data are taken from [@mantz.etal.2017] and from the CHEERS sample (@de_plaa.etal.2017). The latter is analysed with the last updated atomic model for the metallicity estimation (@mernier.etal.2018). In the first part of our analysis, we compare our simulation results to the observational datasets and investigate how the ICM iron abundance varies as a function of the gas temperature. We particularly focus on potential biases, regarding the diversity of the core properties (CC and NCC) and selection effects, that might affect the observational analyses. In the second part, we carry out a study of the mass-metallicity-redshift relation for iron and oxygen abundances. In addition, we also investigate how the mass-metallicity relation behaves when we remove the AGN feedback, in order to investigate its effect on the ICM metal enrichment. In our discussion we highlight some of the model systematics that might affect the comparison between simulated and observed data and stress how the level of metallicity is sensitive to the numerical choices linked to stellar and chemical sub-grid models. For this reason, the analysis presented here is devoted to predicting and comparing general trends between global quantities involving metal abundance, rather than emphasising the absolute enrichment level. Throughout the study, we express both simulated and observed metallicity in terms of solar metallicity as obtained from [@asplund.etal.2009]. The main results of our study can be summarised as follows:
1. We compared the simulated $Z_{\rm Fe}-T$ relation of the AGN simulations to the CHEERS sample which spans from groups to hot clusters. Both datasets show no evidence for a significant correlation between the ICM iron abundance and the ICM temperature measured in the core ($r<0.1R_{500}$) of systems with $T_{[0-0.1]R_{500}}>0.7$ keV. In particular, we did not find any significant break or feature that was present in earlier X-ray analysis. When we split our simulated sample in CC and NCC clusters, the former subsample better agrees with the CHEERS data. In particular both simulated and observed cluster cores consistently show a mean value of $Z_{\rm Fe}$ of about $0.75\ Z_{\rm Fe,\odot}$ with a dispersion of $40\%$ and $30\%$, respectively. We remind that even if the agreement between simulated and observed values of the iron abundance in the core might depend on the chemical model assumptions, the different metallicity levels between the simulated CC and NCC is a solid result and confirms the often claimed trend found in observational samples. Fitting the data with a power-law, we find a very shallow slope ($\sim-0.1$) implying an extremely small variation in the abundances of groups and clusters (20-30 per cent).
2. When compared to the observed sample of massive clusters, with $T_{[0.1-0.5]R_{500}}>5$ keV and $0<z<1.2$, from [@mantz.etal.2017], we find that the AGN simulations consistently reproduce the radial dependence of the $Z_{\rm Fe}-T$ relation for different radial ranges. For this sample of hot clusters, both simulated and observed data show a stronger correlation between iron abundance and temperature. However, we find that the correlation is significantly reduced when including smaller temperature systems. Our simulations reveal no significant trend with redshift (with a variation lower than $20\%$ since $z=1$) in the $Z_{\rm Fe}-T$ normalisation among all the considered radial ranges. This is at variance with respect to the results by [@mantz.etal.2017], that present a stronger increase ($\sim40\%$) of iron abundance at intermediate radii, $0.1R_{500}<r<0.5R_{500}$. We notice, however, that among various observational works there is no agreement on this trend (see @ettori.etal.2015 [@McDonald.etal.2016]).
3. Both the iron and oxygen abundances of the AGN clusters exhibit an anti-correlation with cluster mass for regions within $R_{500}$, while no correlation is found in the cluster outskirts ($R_{500}<r<2R_{500}$). However, fitting the relation with a power-law, we found that even in the core the slope is shallow ($|\beta| < 0.15$) implying that the metallicity is only few tens of per cent different from poor groups to rich clusters. We do not detect any significant evolution for the relation since $z=1.5$. Considering that the metallicity varies little across the mass scale and does not change in time, we can conclude that the majority of the iron and oxygen were already reaching the current levels since high-$z$ throughout the clusters.
4. The effect of the AGN feedback is studied by comparing the $Z_{\rm Fe}-M_{500}$ relation obtained in runs performed with and without the AGN feedback. Without the AGN, the $z=0$ systems present an higher stellar fraction at all radii but particularly in the core. The increase of the stellar fraction at each distance is almost independent on the mass of the system. The much higher stellar production, counter intuitively, is not associated to an increase of the iron mass of the ICM due to the fact that freshly produced metals are locked back into newly formed stars. Instead, the iron mass is comparable to the AGN case only in the core, but it is always reduced in the outskirts. This gap is scale-dependent being more extreme in the group regime, whereby causing the non-AGN iron abundance outside of the core to drop for systems with mass below $M_{500}\approx2\times10^{14}\ M_\odot$.
Our study shows that simulations and observations agree in supporting a weak variation of the ICM metallicity from groups to clusters of galaxies and that the metal content does not substantially varies with time. In addition, we confirm that when AGN feedback is included the level of metallicity in the outskirts is flat and at a relatively high level (about 20-30 per cent of the solar value). All these findings further support the early enrichment scenario.
There is still ample space for future works from both simulation and observation sides to consolidate the results. Future observations will be needed to improve the current statistics in the group regime. With the current X-ray telescopes, it is challenging to increase the number of observed small systems (e.g., those with temperature below 1 keV) as they are extremely faint in the X-ray band. In this regard, the next generation of X-ray telescopes, such as [*ATHENA*]{}[^9], would be of extremely utility in observing low-mass systems (@nandra.etal.2013 [@pointecouteau.etal.2013]). The [*ATHENA*]{} telescope, with large effective area and high-resolution spectroscopy, is expected to provide robust estimation of the gas-phase metallicity in the group regime.
From the numerical point of view, a direction of improvement would be represented by a more sophisticated metal diffusion model that now is implemented by spreading the metals to gas particles within the kernel. In addition, we have not included dust formation and destruction which should be considered for an accurate description of the metal content. Also, the current model of AGN feedback is exclusively thermal, while the mechanical AGN feedback (e.g., jets outflows) is relevant for anisotropically ejecting metals at larger radii from the core.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We would like to thank the anonymous referee for constructive comments and suggestions that improve the paper. This work was supported by the Lendület LP2016-11 grant awarded by the Hungarian Academy of Sciences. We acknowledge financial support from the agreement ASI-INAF n.2017-14-H.0, the INFN INDARK grant, and �Consorzio per la Fisica� of Trieste. M.G. is supported by NASA through Einstein Postdoctoral Fellowship Award Number PF5-160137 issued by the Chandra X-ray Observatory Center, which is operated by the SAO for and on behalf of NASA under contract NAS8-03060. Support for this work was also provided by Chandra grant GO7-18121X. SP is “Juan de la Cierva” fellow (ref. IJCI-2015-26656) of the [*Spanish Ministerio de Econom[í]{}a y Competitividad*]{} (MINECO) and acknowledges additional support from the MINECO through the grant AYA2016-77237-C3-3-P and the Generalitat Valenciana (grant GVACOMP2015-227). DF acknowledges financial support from the Slovenian Research Agency (research core funding No. P1-0188).
Projection and Centring Effects
===============================
In this section, we quantify how the metallicity-temperature relation can be affected by projection effects or by the adopted centre definition.
Projection Effects
------------------
We show in Fig. \[figa1\] the comparison between the 3D and the 2D iron abundance as a function of the gas temperature in the AGN simulation at $z=0$. For this comparison, we employ the spectroscopic-like estimate of the temperature, while for the iron abundance we use both mass-weighted and emission-weighted estimates, as described in Section 2.2. We fit simulated data, for those systems that have a minimum number of 100 gas particles in the central region ($r<0.1R_{500}$) and with the mass $M_{500}>10^{13}\ M_\odot$, to Eq. (\[eq4\]) to obtain the best-fit parameters for both $Z_{\rm Fe}-T$ relations. The best-fit parameters are reported in Table \[tba1\]. We find that the emission-weighted $Z_{\rm Fe}-T$ normalisation is always higher than the mass-weighted one. The difference is higher in the 2D comparison reaching a level of $20-35$ per cent, while it is reduced for the 3D case to values of $15-20$ percent in the intermediate and outer regions, and to a minimum of 5 per cent in the innermost region of the core. The slopes are consistent at $1\sigma$. We limit the following discussion on the projection effects to the emission-weighted $Z_{\rm Fe}-T$ relations only.
Compared to the 3D $Z_{\rm Fe}-T$ relation, the 2D relation has slightly lower normalisation with a more marked difference in the central regions ($r<0.1R_{500}$) where the 3D normalisation is about $1.2$ times larger than the 2D value at the fixed temperature $T=1.7\ \rm{keV}$. While in the outer region ($0.5R_{500}<r<R_{500}$), the corresponding ratio of the 2D to the 3D values is $\sim1.07$. In particular, we notice that the offset between 2D and 3D iron abundance is more visible in low-mass systems than in more massive ones. As a consequence, the slope of the 2D $Z_{\rm Fe}-T$ relation is slightly shallower than the one of the 3D relation, yet the two slopes are consistent at $1\sigma$. We verify that the effect comes primarily from the discrepancy between 2D and 3D iron abundance with the 2D $Z_{\rm Fe}$ being lower than the 3D value due to the contribution of metal-poor gas particles along the projected direction. On the other hand, there is no significant difference between 2D and 3D spectroscopic-like temperatures.
[{width="\textwidth"}]{}
Radial Range $\log_{10}(Z_{0,{\rm Fe}}\ [Z_\odot])$ $\beta_{\rm Fe}$ $\sigma_{\log_{10}Z_{\rm Fe}|M}$
--------------------------------------------- ---------------------------------------- ------------------ ----------------------------------
[**2D emission-weighted $Z_{\rm Fe}-T$**]{}
$[0.0-0.1]R_{500}$: $-0.132\pm0.015$ $-0.19\pm0.04$ $0.132\pm0.010$
$[0.1-0.5]R_{500}$: $-0.360\pm0.009$ $-0.12\pm0.03$ $0.076\pm0.006$
$[0.5-1]R_{500}$: $-0.620\pm0.008$ $-0.06\pm0.02$ $0.076\pm0.006$
[**2D mass-weighted $Z_{\rm Fe}-T$**]{}
$[0.0-0.1]R_{500}$: $-0.265\pm0.012$ $-0.15\pm0.03$ $0.104\pm0.008$
$[0.1-0.5]R_{500}$: $-0.492\pm0.007$ $-0.11\pm0.02$ $0.057\pm0.004$
$[0.5-1]R_{500}$: $-0.701\pm0.006$ $-0.04\pm0.02$ $0.056\pm0.004$
[**3D emission-weighted $Z_{\rm Fe}-T$**]{}
$[0.0-0.1]R_{500}$: $-0.069\pm0.019$ $-0.19\pm0.05$ $0.161\pm0.012$
$[0.1-0.5]R_{500}$: $-0.306\pm0.009$ $-0.16\pm0.03$ $0.081\pm0.006$
$[0.5-1]R_{500}$: $-0.589\pm0.007$ $-0.09\pm0.02$ $0.071\pm0.005$
[**3D mass-weighted $Z_{\rm Fe}-T$**]{}
$[0.0-0.1]R_{500}$: $-0.081\pm0.016$ $-0.22\pm0.04$ $0.140\pm0.011$
$[0.1-0.5]R_{500}$: $-0.381\pm0.008$ $-0.17\pm0.02$ $0.067\pm0.005$
$[0.5-1]R_{500}$: $-0.644\pm0.006$ $-0.08\pm0.02$ $0.061\pm0.005$
Miscentring Effects
-------------------
We show in Fig. \[figa2\] the comparison among the metallicity-temperature relations obtained by employing three different centre definitions: the minimum of the cluster potential well, the centre of the gas mass within $R_{2500}$, and the X-ray emission maximum within the same radius. For this comparison, we used the 2D emission-weighted $Z_{\rm Fe}-T$ relations in the AGN simulation at $z=0$. Fig. \[fig2\] and the best-fit parameters reported in Table \[tb1\] (the full AGN sample) refer to the centre defined as the centre of the gas mass. We also characterise the two other $Z_{\rm Fe}-T$ relations by fitting them to Eq. (\[eq4\]) and report the best-fit parameters in Table [\[tba2\]]{}. As shown in Fig. \[figa2\], the three metallicity-temperature relations are in total agreement among each other. From the tables, we can notice that normalisation, slope and scatter are consistent within the 1$\sigma$ uncertainties. We conclude that the results on the metallicity-temperature relation shown in the paper are robust against the centre definition used.
[{width="\textwidth"}]{}
Centre $\log_{10}(Z_{\rm 0,Fe}\ [Z_\odot])$ $\beta_{\rm Fe}$ $\sigma_{\log_{10}Z_{\rm Fe}|M}$
----------------------------- -------------------------------------- ------------------ ----------------------------------
[**True centre of mass**]{} $-0.175\pm0.011$ $-0.10\pm0.03$ $0.155\pm0.008$
[**X-ray emission peak**]{} $-0.185\pm0.012$ $-0.14\pm0.04$ $0.167\pm0.008$
[^1]: truongnhut@caesar.elte.hu
[^2]: Einstein and Spitzer Fellow
[^3]: The mass enclosed within a sphere that has averaged density equal to $200$ times the critical density of the Universe ($\rho_{\rm{crit}}$). In general, the mass $M_\Delta$ is related to the radius $R_\Delta$ by the equation: $M_\Delta=\frac{4\pi}{3}\Delta\rho_{\rm{crit}}R_\Delta^3$. We also use $R_{500}$ and $R_{2500}$ in this work.
[^4]: The fraction of AGN (CSF) clusters with at least $100$ gas particles in the core with respect to the total sample are: $34 (39)$, $39(53)$, $34(64)$, and $36(67)$ per cent, at $z=0$, $0.5$, $1$, and $1.5$, respectively.
[^5]: For the cosmology used in our simulations, the virial radius, according to [@bryan.norman.1998], corresponds to $\Delta\approx93$ at $z=0$.
[^6]: https://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/
[^7]: SPEX Atomic Code and Tables, as part of the fitting package SPEX (https://www.sron.nl/astrophysics-spex).
[^8]: We note that the simulated sample is not strictly volume-limited as the objects are obtained from the high-resolution zoomed-in regions. However, as each region’s radius extends the size of at least 5 times $R_{vir}$ of the central object, this volume is large enough to contain several other groups.
[^9]: http://www.the-athena-x-ray-observatory.eu/
|
---
abstract: 'We analyze, within a minimal model that allows analytical calculations, the electronic structure and Landau levels of graphene multi-layers with different stacking orders. We find, among other results, that electrostatic effects can induce a strongly divergent density of states in bi- and tri-layers, reminiscent of one-dimensional systems. The density of states at the surface of semi-infinite stacks, on the other hand, may vanish at low energies, or show a band of surface states, depending on the stacking order.'
author:
- 'F. Guinea'
- 'A. H. Castro Neto'
- 'N. M. R. Peres'
bibliography:
- 'graphite0\_15.bib'
title: Electronic states and Landau levels in graphene stacks
---
Introduction
============
Recent experiments show that single layer graphene, and stacks of graphene layers, have unusual electronic properties [@Netal04; @Zetal05], that may be useful in the design of new electronic devices [@Betal04; @Betal05; @Netal05b]. Among them, we can mention the Dirac-like dispersion relation of a single graphene layer [@Netal05; @Zetal05b], the chiral parabolic bands in bilayers that lead to a new type of Quantum Hall effect [@Ketal06], or the possibility of confining charge to the surface in systems with a few graphene layers [@Metal05c]. While the study of graphene multilayers is still in its infancy [@JCGN06], the current experimental techniques allow for the extraction/production of multi-layers with the accuracy of a single atomic layer. The ability of creating stacks of graphene layers can provide an extra dimension to be explored in terms of electronic properties with functionalities that cannot be obtained with other materials.
The nature of the stacking order in graphene multilayers has its origins in the single graphene plane, a two-dimensional (2D) honeycomb lattice with two inequivalent sub-lattices, A and B (two atoms per 2D unit cell). The staggered stacking occurs in highly oriented pyrolytic graphite (HOPG) where the graphene layers are arranged so that there two graphene layers per three-dimensional (3D) unit cell in a $ABAB \cdots$ sequence (see Fig. \[lattice\]). In this case, one of the atoms in one of the planes (say B) is exactly in the center of the hexagon in the other plane. Hence, only one of the sublattices (A) is on the top of the same sublattice in the other layer. Nevertheless, this is not the only possible stacking observed in these systems [@BCP88]. Rhombohedral graphite with stacking sequence $ABCABC
\cdots$ has three layers per 3D unit cell (in the third layer the sublattice B becomes on top of sublattice A of the second layer). Furthermore, stacking defects have been repeatedly observed in natural graphitic samples [@B50; @G67], and also in epitaxially grown graphene films [@Retal05]. The richness of possible stacking configurations is due to the weak van der Waals forces that keep the layers together. Furthermore, the electronic structure of the conduction band in graphene and bulk graphite is well described by a tight binding model which includes hopping between the $\pi$ orbitals in Carbon atoms, neglecting the remaining atomic orbitals which give rise to the $\sigma$ bands in graphite [@W47; @M57; @BN58; @SW58; @M64; @DSM77; @BCP88]. It is known that the low energy electronic structure depends on the stacking order in bulk samples [@H58; @M69; @TL88; @CMGV91; @CML92; @CGM94].
In the following, we analyze the electronic structure and Landau levels of graphitic structure with different stacking orders. We use a minimal tight binding model that only includes interlayer hopping between nearest neighbor Carbon atoms in contiguous layers, and the ${\bf k} \cdot {\bf p}$ expansion for the dependence on the momentum parallel to the layers. Within the ${\bf k} \cdot {\bf p}$ approximation, the two inequivalent corners of the Brillouin zone can be studied separately, and we will analyze only one of them. We do not consider the effects of disorder, already discussed in other publications [@PGN06; @JCGN06], but focus instead on the analytical solution of the model that provide information on the unique properties of these systems.
The model used here does not take into account hopping which may be relevant in order to describe the fine details of the band structure, such as the trigonal distortions, that are also difficult to estimate using more demanding local density functional (DFT) methods. Moreover, recent angle resolved photoemission experiments in crystalline graphite [@lanzara_new] show that those effects are very small. On the other hand, the methods used here, based on a mapping onto a set of simple nearest-neighbor one-dimensional (1D) tight binding Hamiltonians is quite feasible, allowing us to study a variety of situations, with and without an applied magnetic field. It is worth noting that when the energy scales associated to the electron-electron interaction and/or to in plane disorder are larger than the hopping neglected here, the description presented below will be a reasonable ansatz for the calculation of the effects of the electron-electron interaction.
The model used is described in the Section \[model\], as well as a simple scheme that allows the mapping of the problem of an arbitrary number of coupled graphene layers onto a 1D tight binding model with nearest neighbor hopping only. Section \[stacks\] describes the main properties of few layer systems, also in the presence of a magnetic field. In Section \[semi\] we discuss the bulk and surface electronic structure of semi-infinite stacks of graphene layers, with different stacking order. Section \[conclusions\] contains the main conclusions of our work.
The model. {#model}
==========
We consider the staggered stacking, where the layer sequence can be written as $1212 \cdots$, and rhombohedral stacking, $123123 \cdots$ and combinations of the two. We do not consider hexagonal stacking, where all atoms in a given plane are on top of atoms in the neighboring layers, $111 \cdots$. In all stacking considered, the hopping between a pair neighboring layers takes place through half of the atoms in each plane, as schematically shown in Fig. \[lattice\].
![Top view of the arrangement of the Carbon atoms in two neighboring graphene planes in the staggered stacking.[]{data-label="lattice"}](lattice.eps){width="5cm"}
We describe the in-plane electronic properties of graphene using a low energy, long wavelength, expansion around the $K$ and $K'$ corners of the 2D Brillouin Zone. This description uses as only input parameter the Fermi velocity, ${v_{\rm F}}= 3 t a/2$, where $t$ ($\approx 3$ eV) is the hopping between $\pi$ orbitals located at nearest neighbor atoms, and $a = 1.42$Å is the distance between Carbon atoms. We assume that hopping between planes takes place only between atoms which are on top of each other in neighboring layers. We denote the hopping integral as ${t_\perp}$ ($\approx 0.1 t$). The whole model is defined by the parameters ${v_{\rm F}}$ and ${t_\perp}$. For instance, for the graphene bilayer (the 3D unit cell of the staggered stacking) the Hamiltonian has the form: $H = \sum_{{\bf k}} \Psi^{\dag} ({\bf k}) H_0({\bf k}) \Psi({\bf k})$, where ${{{\bf k}}}=(k_x,k_y)$ is the 2D momentum measured relative to the $K$ ($K'$) point (we use units such that $c=1=\hbar$), $$\label{eq:Hkin0bilayer}
H_0({{\bf k}}) = {v_{\rm F}}\begin{pmatrix}
0 & k e^{i \phi({{\bf k}})} & {t_\perp}/ {v_{\rm F}}& 0 \\
k e^{-i \phi({{\bf k}})} & 0 & 0 & 0 \\
{t_\perp}/ {v_{\rm F}}& 0 & 0 & k e^{-i \phi({{\bf k}})} \\
0 & 0 & k e^{i \phi({{\bf k}})} & 0
\end{pmatrix},$$ $\Psi^{\dag}({\bf k}) = \bigl( c^{\dag}_{\text{A},1,{\bf k}} \;
c^{\dag}_{\text{B},1,{\bf k}} \;
c^{\dag}_{\text{A},2,{\bf k}} \; c^{\dag}_{\text{B},2,{\bf k}} \bigr)$ is the electron spinor creation operator, and $\phi({\bf k}) =
\tan^{-1}(k_{y}/k_{x})$ is the 2D angle in momentum space.
In the case of the staggered stacking with $2N$ planes, the full Hamiltonian consists of $2N \times 2N$ matrix with $N/2$ blocks with size $4 \times 4$ given by (\[eq:Hkin0bilayer\]). The spinor operator is given by creation and annihilation operators for electrons in different planes: $c^{\dag}_{\alpha,n,{\bf k}}$ where $\alpha=A,B$ labels the sublattices in each plane, and $n=1, \cdots, N$ labels the planes. We define the Green’s functions: $$\begin{aligned}
G^n_{\alpha,\beta}({{\bf k}},t) = -i \langle {\cal T} c_{\alpha,n,{{\bf k}}}(t) c^{\dag}_{\beta,n,{{\bf k}}}(0) \rangle \, ,
\label{greenfunc}\end{aligned}$$ where ${\cal T}$ is the time ordering operator, and its Fourier transform, $G^n_{\alpha,\beta}({{\bf k}},\omega)$, that can be used to calculate the properties of these systems.
It is easy to see from (\[eq:Hkin0bilayer\]) that ${G_A^n ( {| {\bf k} |}, \omega )}$ ($\alpha=\beta=\text{A})$ and ${G_B^n ( {| {\bf k} |}, \omega )}$ ($\alpha=\beta=\text{B}$), satisfy the equations: $$\begin{aligned}
\omega G_A^{2n} ( {| {\bf k} |}, \omega ) &= &t_\perp [ G_A^{2n-1} ( {| {\bf k} |}, \omega )
+ G_A^{2n+1} ( {| {\bf k} |}, \omega )]
\nonumber \\
&+& {v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}G_B^{2n} ( {| {\bf k} |}, \omega ) \, ,
\nonumber \\
\omega G_B ^{2n} ( {| {\bf k} |}, \omega ) &= &{v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}G_A^{2n} ( {| {\bf k} |}, \omega ) \, ,
\nonumber \\
\omega G_A^{2n+1} ( {| {\bf k} |}, \omega ) &= &t_\perp [ G_A^{2n} ( {| {\bf k} |}, \omega ) +
G_A^{2n+2} ( {| {\bf k} |}, \omega )]
\nonumber \\
&+& {v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}G_B^{2n+1} ( {| {\bf k} |}, \omega ) \, ,
\nonumber \\
\omega G_B^{2n+1} ( {| {\bf k} |}, \omega ) &= &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}G_A^{2n+1} ( {| {\bf k} |}, \omega ) \, .
\label{g_HOPG}\end{aligned}$$ The phase factors ${e^{i {\phi_{\bf k}}}}, {e^{-i {\phi_{\bf k}}}}$ can be gauged away by a gauge transformation, leading to a set of equations where the odd and even numbered layers are indistinguishable. We can integrate out the $B$ sites in eq.(\[g\_HOPG\]), and write an effective equation for ${G_A ( {| {\bf k} |}, \omega )}$: $$\left[ \omega - \frac{( {v_{\rm F}}{| {\bf k} |})^2}{\omega} \right] {G_A^n ( {| {\bf k} |}, \omega )}= t_\perp [ {G_A^{n+1} ( {| {\bf k} |}, \omega )}+ {G_A^{n-1} ( {| {\bf k} |}, \omega )}] \, ,$$ which is formally equivalent to the tight binding equations which describe a 1D chain with hopping energy $t_{\perp}$ and effective on site energy $( {v_{\rm F}}{| {\bf k} |})^2/\omega$. The Green’s function for the $B$ atoms can be obtained from ${G_A ( {| {\bf k} |}, \omega )}$ using the expression: $$\left[G^n_{\alpha,\beta}({| {\bf k} |},\omega)\right]
\equiv \left( \begin{array}{cc} {{G_A^n ( {| {\bf k} |}, \omega )}}^{-1} + \frac{( {v_{\rm F}}{| {\bf k} |})^2}{\omega} &
{v_{\rm F}}{| {\bf k} |}\\ {v_{\rm F}}{| {\bf k} |}& \omega \end{array} \right)^{-1} \, ,$$ so that, $${G_B^n ( {| {\bf k} |}, \omega )}= \frac{1}{\omega} + \frac{( {v_{\rm F}}{| {\bf k} |})^2}{\omega^2} {G_A^n ( {| {\bf k} |}, \omega )}\, .$$ A sketch of the resulting 1D tight binding model is given in Fig. \[TB\_1D\].
![Schematic view of the mapping of the problem of a stack of graphene layers onto a set of one dimensional problems. The hopping between sites are ${t_\perp}$ and $k e^{i \pm \phi}$, where $\phi = \arctan(k_y/k_x)$. Left: staggered stacking. Right: Rhombohedral stacking. Note that the phases $\phi$ can be gauged away in both cases.[]{data-label="TB_1D"}](TB_1D.eps){width="8cm"}
In the case of the rhombohedral stacking ($123123 \cdots$), a suitable gauge transformation can be used in order to make all layers equivalent, although the two sublattices $A$ and $B$ within each layer remain different, leading to the following set of equations: $$\begin{aligned}
\omega G^n_A ( \omega ) &= &t_\perp G^{n-1}_B ( \omega ) + {v_{\rm F}}{| {\bf k} |}G^n_B (
\omega ) \, ,
\nonumber \\
\omega G^n_B ( \omega ) &= &t_\perp G^{n+1}_A ( \omega ) + {v_{\rm F}}{| {\bf k} |}G^n_A (
\omega ) \, ,
\label{g_turbo}\end{aligned}$$ that determine the properties of the rhombohedral stacking.
In a finite magnetic field $B$ the above equations can be modified with the use of a Peierls substitution ${{\bf k}}\to {{\bf k}}+ e {\bf A}$, where ${\bf A}$ is the vector potential ($\nabla \times {\bf A} = {\bf B}$). In this case, the system breaks into Landau levels that, for a single graphene layer, have energy [@PGN06]: $$\begin{aligned}
E(s,j) = s \omega_c \sqrt{j + 1} \, ,\end{aligned}$$ where $j = 0,1, \cdots$ labels the Landau levels, $s=\pm 1$ labels the electron and hole bands, $$\begin{aligned}
\omega_c = \sqrt{2} v_F/{\ell_{\rm B}}= v_F \sqrt{2 e B} \, ,
\label{wc}\end{aligned}$$ is the cyclotron frequency, and $l_B$ is the cyclotron length.
In the presence of magnetic field we can replace the momentum label ${{\bf k}}$ by the Landau label $j$ (${| {\bf k} |}\to \sqrt{j}/{\ell_{\rm B}}$) and the extension of the equations (\[g\_HOPG\]) become: $$\begin{aligned}
\omega G_A^{2n} ( j,\omega ) = t_\perp [ G_A^{2n-1} (j,\omega )
+ G_A^{2n+1} (j,\omega )]
\nonumber \\
+ \frac{{v_{\rm F}}\sqrt{m}}{{\ell_{\rm B}}} G_B^{2n} (j-1,\omega) \, ,
\nonumber \\
\omega G_B ^{2n} (j-1,\omega ) = \frac{{v_{\rm F}}\sqrt{m}}{{\ell_{\rm B}}} G_A^{2n} (j,\omega) \, ,
\nonumber \\
\omega G_A^{2n+1} (j,\omega ) = t_\perp [ G_A^{2n} ( m , \omega ) +
G_A^{2n+2} ( m , \omega )]
\nonumber \\
+ \frac{{v_{\rm F}}\sqrt{j+1}}{{\ell_{\rm B}}} G_B^{2n+1} (j+1,\omega) \, ,
\nonumber \\
\omega G_B^{2n+1} (j+1,\omega ) = \frac{{v_{\rm F}}\sqrt{j+1}}{{\ell_{\rm B}}}
G_A^{2n+1} (j,\omega) \, .
\label{g_HOPG_B}\end{aligned}$$ Note that, in this case, the two inequivalent layers in the unit cell cannot be made equivalent by a gauge transformation as in (\[g\_HOPG\]).
In rhombohedral case, the presence of a magnetic field, (\[g\_turbo\]) become: $$\begin{aligned}
\omega G^n_A ( \omega ) &= &t_\perp G^{n-1}_B ( \omega ) + \frac{{v_{\rm F}}}{{\ell_{\rm B}}} \sqrt{j} G^n_B (\omega ) \, ,
\nonumber \\
\omega G^n_B ( \omega ) &= &t_\perp G^{n+1}_A ( \omega ) + \frac{{v_{\rm F}}}{{\ell_{\rm B}}}
\sqrt{j+1} G^n_A ( \omega ) \, .
\label{g_turbo_B}\end{aligned}$$
Stacks of a few graphene layers. {#stacks}
================================
Generic energy bands.
---------------------
Using the 1D tight binding mapping discussed in the preceding section, we can write an implicit equation for the eigenenergies of a system with $N$ layers with the staggered stacking as: $$\epsilon_{{{\bf k}}} = \frac{( {v_{\rm F}}{| {\bf k} |})^2}{\epsilon_{{{\bf k}}}} + 2 t_\perp \cos \left(
\frac{\pi n}{N+1} \right) \, ,$$ where $n=1, \cdots N$, so that the dispersion becomes: $$\epsilon_{{{\bf k}}} \!=\! t_\perp \!\cos \left(\!\frac{\pi n}{N+1}\!\right)\!\pm\!
\sqrt{( {v_{\rm F}}{| {\bf k} |})^2 \!+\! t_\perp^2 \!\cos^2 \left(\!\frac{\pi n}{N+1}\!\right)} \, .
\label{dispersion_ML}$$
Bilayer.
--------
A number of properties of the present model applied to a graphene bilayer can be found in ref. \[\]. We extend those results here to the case where there is an electrostatic potential which makes the two layers inequivalent. The Hamiltonian reads: $${\cal H}_{2L} ( {{\bf k}}) \!\!\equiv\!\! \left(\!\!\! \begin{array}{cccc} \Delta &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}&{t_\perp}&0 \\ {v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&\Delta &0 &0 \\ {t_\perp}&0 &- \Delta &{v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}\\ 0
&0 &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}&- \Delta \end{array} \!\!\!\right) \, ,
\label{hamil_2L}$$ where the difference in the electrostatic potentials in the two layers is $2
\Delta$. For ${v_{\rm F}}{| {\bf k} |}\ll \Delta \ll {t_\perp}$, the bands at low energy are given by $\pm {\epsilon_{\bf k}}$ where: $${\epsilon_{\bf k}}\approx \Delta - \frac{\Delta {v_{\rm F}}^2 {| {\bf k} |}^2}{{t_\perp}^2} + \frac{{v_{\rm F}}^4 {| {\bf k} |}^4}{2 \Delta {t_\perp}^2} \, .$$ Hence, the bands show an unusual “mexican hat” dispersion, with extrema at $k_0
\sim \Delta / {v_{\rm F}}$ with energy $\epsilon_{vH} = \Delta - \Delta^3 / {t_\perp}^2$. Near these special points, the electronic density of states diverges as: $$D_\pm ( \epsilon ) \propto \frac{{t_\perp}\sqrt{\Delta}}{{v_{\rm F}}^2 \sqrt{\epsilon \mp
\epsilon_{vH}}} \, ,
\label{dos_2L}$$ which is the divergence seen in 1D systems. In order to understand the reason for this 1D behavior consider the situation where the chemical potential is above (or below) $\epsilon_{vH}$. In this case the Fermi surface looks like a ring (a Fermi ring), as shown in Fig. \[ring\]. At large radius $k_0$ the ring approaches a 1D dispersion with nested Fermi surfaces, characteristic of 1D systems.
![The Fermi ring: the Fermi sea of a biased bilayer.[]{data-label="ring"}](ring_2.eps){width="5cm"}
In the presence of a magnetic field, the Hamiltonian becomes: $${\cal H}_{2L} (j) \!\!\equiv\!\! \left(\!\! \begin{array}{cccc} \Delta &\frac{{v_{\rm F}}\sqrt{j}}{{\ell_{\rm B}}} &{t_\perp}&0 \\ \frac{{v_{\rm F}}\sqrt{j}}{{\ell_{\rm B}}} &\Delta &0 &0 \\ {t_\perp}&0 &- \Delta
&\frac{{v_{\rm F}}\sqrt{j+1}}{{\ell_{\rm B}}} \\ 0
&0 &\frac{{v_{\rm F}}\sqrt{j+1}}{{\ell_{\rm B}}} &- \Delta \end{array} \!\!\right) \, ,
\label{hamil_2L_B}$$ where $j$ is the index of the Landau level. For values of $j$ such that $(
{v_{\rm F}}\sqrt{j} ) / {\ell_{\rm B}}\ll \Delta \ll {t_\perp}$, we obtain: $$\epsilon ( j ) \!\!=\!\! - \frac{{v_{\rm F}}^2 \Delta}{{\ell_{\rm B}}^2 {t_\perp}^2} \left(\!j\!+\!\frac{1}{2}\!
\right)
\!\pm\! \sqrt{\Delta^2 \left(\!1\!-\!\frac{{v_{\rm F}}^2}{2 {\ell_{\rm B}}^2 {t_\perp}^2}\!\right)^2
\!+\! \frac{{v_{\rm F}}^4 j ( j+1 )}{{\ell_{\rm B}}^4 {t_\perp}^2}} \, .$$
Trilayer.
---------
### HOPG stacking (ABA).
The model used here leads to six energy bands in a trilayer. In the absence of electrostatic potentials that break the equivalence of the three layers, we can use eq.(\[dispersion\_ML\]), to obtain: $$\begin{aligned}
{\epsilon_{\bf k}}&= & \pm {v_{\rm F}}{| {\bf k} |}\, ,
\nonumber
\\
{\epsilon_{\bf k}}&= &\frac{{t_\perp}\sqrt{2}}{2} \pm \sqrt{\frac{{t_\perp}^2}{2} + {v_{\rm F}}^2 {| {\bf k} |}^2} \, .\end{aligned}$$ There is a band with Dirac-like linear dispersion, and two more bands which disperse quadratically near $\epsilon = 0$, as in a bilayer. The Landau levels, at low energies are: $$\begin{aligned}
\epsilon ( j ) &= &\pm \frac{{v_{\rm F}}\sqrt{j}}{{\ell_{\rm B}}} \, , \nonumber \\
\epsilon ( j ) &= &\pm \frac{{v_{\rm F}}^2 \sqrt{j (j+1 )}}{{\ell_{\rm B}}^2 {t_\perp}}\end{aligned}$$ Hence, the spectrum can be viewed as a superposition of the corresponding spectrum for the Dirac equation, and that obtained for a bilayer.
The calculation of the energy levels in the presence of electrostatic fields which break the symmetry between the three layers is more complex, and cannot be performed fully analytically. We consider the case when the upper layer is at potential $\Delta$, the lower layer is at potential $- \Delta$, and the middle layer is at zero potential. The Hamiltonian is:
$${\cal H}_{3L} ( {{\bf k}})\!\!=\!\! \left(\!\!\begin{array}{cccccc} \Delta &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}&0 &0 &0 &0 \\ {v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&\Delta &{t_\perp}&0 &0 &0 \\ 0 &{t_\perp}&0 &{v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&{t_\perp}&0 \\ 0 &0 &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}&0 &0 &0 \\ 0 &0 &{t_\perp}&0 &- \Delta
&{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}\\ 0 &0 &0 &0 &{v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&- \Delta \end{array}\!\!\right)
\, ,$$
In the limit ${v_{\rm F}}{| {\bf k} |}\ll \Delta \ll {t_\perp}$, one can use perturbation theory to obtain: $${\epsilon_{\bf k}}\approx \frac{\Delta {v_{\rm F}}{| {\bf k} |}}{\sqrt{2} {t_\perp}} \left( 1 - \frac{{v_{\rm F}}^2
{| {\bf k} |}^2}{\Delta^2} \right) \, .$$ This equation, although approximate, describes correctly the existence of states at zero energy when ${v_{\rm F}}{| {\bf k} |}= \Delta$. The dispersion shows extrema at momenta $k_0 \propto \Delta / {v_{\rm F}}$ with energy $\epsilon_{vH} \propto \Delta^2 /
{t_\perp}$, leading to a divergent density of states: $$D_\pm ( \epsilon ) \propto \frac{\Delta \sqrt{{t_\perp}}}{{v_{\rm F}}^2 \sqrt{\epsilon \mp
\epsilon_{vH}}}
\label{dos_3L}$$ This expression, as in the case of eq. (\[dos\_2L\]), is typical of 1D systems. The low energy bands of a trilayer with $ABA$ stacking are shown in the upper panel of Fig.\[\[3L\]\].
![Low energy bands of a trilayer with an electrostatic field which hreaks the equivalence of the three layers (see text). Top: HOPG stacking, $ABA$. Bottom: rhombohedral stacking, $ABC$. The parameters used are ${v_{\rm F}}= 1, {t_\perp}= 0.1 , \Delta = 0.04$.[]{data-label="3L"}](fig_math.ps "fig:"){width="6cm"} ![Low energy bands of a trilayer with an electrostatic field which hreaks the equivalence of the three layers (see text). Top: HOPG stacking, $ABA$. Bottom: rhombohedral stacking, $ABC$. The parameters used are ${v_{\rm F}}= 1, {t_\perp}= 0.1 , \Delta = 0.04$.[]{data-label="3L"}](fig_math_2.ps "fig:"){width="6cm"}
### Rhombohedral stacking (ABC).
The low energy bands of a trilayer with the $ABC$ stacking differ significantly from those for the $ABA$ stacking discussed previously. The $6
\times 6$ hamiltonian is:
$${\cal H}_{3L} ( {{\bf k}}) \equiv \left( \begin{array}{cccccc} \Delta &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}&0 &0 &0 &0 \\ {v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&\Delta &{t_\perp}&0 &0 &0 \\ 0 &{t_\perp}&0 &{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}&0 &0 \\ 0 &0 &{v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&0 &{t_\perp}&0 \\ 0 &0 &0 &{t_\perp}&- \Delta
&{v_{\rm F}}{| {\bf k} |}{e^{i {\phi_{\bf k}}}}\\ 0 &0 &0 &0 &{v_{\rm F}}{| {\bf k} |}{e^{-i {\phi_{\bf k}}}}&- \Delta \end{array} \right)
\label{hamil_3L_eh}$$
Assuming that $\Delta \ll {t_\perp}$, only two out of the six $\pi$ bands lie at energies $| \epsilon | \ll {t_\perp}$. These bands are derived from the orbitals at the $B$ sublattice in the two outermost layers, given by the first and sixth entry in eq.(\[hamil\_3L\_eh\]). One can define an effective $2 \times 2$ hamiltonian: $${\cal H}_{3L \, \, eff} \equiv \left( \begin{array}{cc} - \Delta
+ \frac{\Delta {v_{\rm F}}^2
{| {\bf k} |}^2}{ {t_\perp}^2} &\frac{{v_{\rm F}}^3 {| {\bf k} |}^3}{{t_\perp}^2}
\\ \frac{{v_{\rm F}}^3 {| {\bf k} |}^3}{{t_\perp}^2} & \Delta - \frac{\Delta {v_{\rm F}}^2
{| {\bf k} |}^2}{{t_\perp}^2} \end{array} \right)$$ The dispersion is cubic at low momenta, ${\epsilon_{\bf k}}\approx ( {v_{\rm F}}{| {\bf k} |})^3 /
{t_\perp}^2$. The density of states is $D ( \epsilon ) \propto {t_\perp}^{4/3} / (
{v_{\rm F}}^{2} \epsilon^{1/3} )$. The low energ bands of a trilayer with $ABC$ stacking is shown in the lower panel of Fig.\[\[3L\]\].
The effective $2 \times 2$ hamiltonian in the presence of an applied magnetic field is, for $\Delta = 0$: $${\cal H}_{3L \, \, rhombo} \equiv \left( \begin{array}{cc} 0 &\frac{{v_{\rm F}}^3
\sqrt{n ( n+1 ) ( n+2 )}}{{\ell_{\rm B}}^3 {t_\perp}^3} \\ \frac{{v_{\rm F}}^3 \sqrt{n ( n+1 ) (
n+2 )}}{{\ell_{\rm B}}^3 {t_\perp}^3} &0 \end{array} \right)$$ There are, in addition, states localized in the outermost layers, with Landau index $n=0$, and in two layers, with Landau index $n=1$. It is interesting to note that Landau levels associated to different $K$ points are localized in different layers.
Semi-infinite stacks of graphene layers. {#semi}
========================================
In this section we consider the case of a graphene multilayer with a surface termination. In the case of the staggered stacking, we can use eq.(\[g\_HOPG\]), and obtain for the Green’s function in the bulk as: $${G_A ( {| {\bf k} |}, \omega )}= \frac{\omega}{\sqrt{[ \omega^2 - ( {v_{\rm F}}{| {\bf k} |})^2 ] - 4 t_\perp^2
\omega^2}} \, .$$
We can obtain the local density of states by integrating ${G_A ( {| {\bf k} |}, \omega )}$ and ${G_B ( {| {\bf k} |}, \omega )}$ over the parallel momentum ${{\bf k}}$. This integral can be performed analytically, and we obtain, for the bulk: $$\begin{aligned}
G_A ( \omega ) \!&=&\!\frac{\omega}{{v_{\rm F}}^2} \ln \left[ \frac{ ( {v_{\rm F}}\Lambda )^2}{
\omega ( \omega \!+\! \sqrt{\omega^2 \!-\! 4 t_\perp^2} )} \right] \, ,
\nonumber \\
G_B ( \omega ) \!\!\!&= &\!\!\! \frac{\sqrt{\omega^2 \!-\! 4 t_\perp^2}}{{v_{\rm F}}^2} \!+\!
\frac{\omega}{{v_{\rm F}}^2} \!
\ln \!\!\left[\! \frac{ ( {v_{\rm F}}\Lambda )^2}{
\omega ( \omega + \sqrt{\omega^2 - 4 t_\perp^2} )} \!\right] \! \!.
\label{green}\end{aligned}$$ The local density of states is given by (see also ref. \[\]): $$\begin{aligned}
{\rm Im} G_A ( \omega ) &= & \frac{\omega}{{v_{\rm F}}^2} \left\{
\left[\frac{\pi}{2} + \arctan \left( \frac{\omega}{\sqrt{4 t_\perp^2 -
\omega^2}} \right)\right] \right.
\nonumber
\\
&\times& \left. \Theta(2 {t_\perp}-\omega) + \pi \Theta(\omega-2 {t_\perp})\right\} \, ,
\nonumber
\\
{\rm Im} G_B ( \omega )\!\!\!&= &\!\!\!\! \left\{\!\!\frac{\sqrt{4 t_\perp^2 -
\omega^2}}{{v_{\rm F}}^2} \!\!+\!\! \frac{\omega}{{v_{\rm F}}^2} \left[ \frac{\pi}{2} \!+\! \arctan
\left(\!\frac{\omega}{\sqrt{4 t_\perp^2 - \omega^2}} \right)\!\right]\!\right\}
\nonumber
\\
&\times& \Theta(2 {t_\perp}-\omega) + \frac{\pi}{{v_{\rm F}}^2} \Theta(\omega-2 {t_\perp}) \, .\end{aligned}$$ The density of states in the bulk of the staggered stacking is shown in Fig. \[DOS\].
![Density of states for the staggered stacking. Continuous line: ${\rm Im} G_B ( \omega)$. Dashed line: ${\rm Im} G_A ( \omega)$. []{data-label="DOS"}](g_bulk.eps){width="8cm"}
We can use the equivalence to a 1D model for each value of ${{\bf k}}$, (\[g\_HOPG\]) and (\[g\_turbo\]), in order to obtain the Green’s function at the surface layer of a semi-infinite system: $$\begin{aligned}
G^{\rm surface}_A ( \omega ) &= &\frac{\sqrt{[ \omega^2 - ( {v_{\rm F}}{| {\bf k} |})^2 ] - 4 t^2 \omega^2}}{2 t^2
\omega} - \frac{\omega}{2 t_\perp^2} \, , \nonumber \\
G^{\rm surface}_B ( \omega ) &= &\frac{1}{\omega} - \frac{{v_{\rm F}}^2 {| {\bf k} |}^2}{\omega^2} G^{\rm surface}_A ( \omega ) \, ,\end{aligned}$$ for staggered stacking. The local density of states, obtained after integrating these expressions over ${{{\bf k}}}$ is shown in Fig. \[DOS\_surface\].
![Density of states at the surface layer. Continuous line: ${\rm Im} G_B ( \omega)$. Dashed line: ${\rm Im} G_A ( \omega)$. []{data-label="DOS_surface"}](g_surface.eps){width="8cm"}
The 1D tight binding model which describes the electronic bands in rhombohedral graphite (stacking order $123123 \cdots$) is given in (\[g\_turbo\]). Using this equation, the Green’s function, integrated over the perpendicular momentum, $k_\perp$, is given by: $$G^n ( {{{\bf k}}} , \omega ) \!\!=\!\! \frac{\omega}{\sqrt{\left[ \omega^2 \!-\! ( {v_{\rm F}}{| {\bf k} |}\!+\! {t_\perp})^2
\right] \left[ \omega^2 \!-\! ( {v_{\rm F}}{| {\bf k} |}\!+\! {t_\perp})^2 \right]}} .$$ The local density of states can be obtained by integrating this expression over ${{\bf k}}$: $${\rm Im} G^n ( \omega ) = \frac{| \omega |}{{v_{\rm F}}^2} \, .
\label{g_local_turbo}$$ The material, within this approximation, is a semi-metal, with vanishing density of states at the Fermi level at half filling, $\omega = 0$. Note that the density of states at low energies is independent of the value of ${t_\perp}$.
In the presence of a magnetic field, we use (\[g\_turbo\_B\]). We can integrate out the Green’s functions for the sites in a given sublattice and obtain: $$\begin{aligned}
\left( \omega - \frac{{t_\perp}^2}{\omega} - \frac{{v_{\rm F}}^2 m}{{\ell_{\rm B}}^2 \omega} \right)
G^n_A ( \omega ) = \frac{{t_\perp}{v_{\rm F}}\sqrt{m}}{{\ell_{\rm B}}\omega} G^{n-1}_A ( \omega )
\nonumber
\\
+ \frac{{t_\perp}{v_{\rm F}}\sqrt{n+1}}{{\ell_{\rm B}}\omega} G^{n+1}_A ( \omega ) \, .\end{aligned}$$ These equations are formally equivalent to those obtained for the wavefunctions of a displaced 1D harmonic oscillator: $${\cal H} \equiv \epsilon_0 + \omega_0 b^\dag b + g ( b^\dag + b ) \, ,
\label{h_eff}$$ with the correspondence: $\epsilon_0 \leftrightarrow {t_\perp}^2 / \omega$, $\omega_0
\leftrightarrow {v_{\rm F}}^2 / ( {\ell_{\rm B}}^2 \omega )$, $g \leftrightarrow ( {t_\perp}{v_{\rm F}}) / (
{\ell_{\rm B}}\omega )$. The eigenenergies of (\[h\_eff\]) are $\epsilon_m =
\epsilon_0 - g^2 / \omega_0 + m \omega_0$, and we can write the eigenenergies associated to eq.(\[g\_turbo\_B\]) as: $$\epsilon_m = \frac{{t_\perp}^2}{\epsilon_m} -
\frac{({t_\perp})^2}{\epsilon_m} + m \frac{{v_{\rm F}}^2}{{\ell_{\rm B}}^2 \epsilon_m} \, ,$$ and, finally, we obtain: $$\epsilon_m = \frac{{v_{\rm F}}\sqrt{m}}{{\ell_{\rm B}}} \, .$$ The spectrum, in an applied magnetic field, is discrete, and equal to that in graphene, as well as the local density of states in the absence of the field, eq. (\[g\_local\_turbo\]). The discreteness of the spectrum survives when a stack of rhombohedral graphene is embedded into the staggered stacking, as shown in Fig. \[Landau\_levels\].
Using the model described in Section \[model\], we obtain for the projected density of states: $$\begin{aligned}
G^{\rm surface}_A ( \omega ) &=& \frac{\omega^2 - {t_\perp}^2 - ( {v_{\rm F}}{| {\bf k} |})^2}{2 {t_\perp}{v_{\rm F}}{| {\bf k} |}\omega}
\nonumber
\\
&-&\!\! \frac{\sqrt{\left[ \omega^2 \!-\! ( {v_{\rm F}}{| {\bf k} |}\!-\!
{t_\perp})^2 \right] \left[ \omega^2 \!-\! ( {v_{\rm F}}{| {\bf k} |}\!+\! {t_\perp})^2 \right]}}{2 {t_\perp}{v_{\rm F}}{| {\bf k} |}\omega} \, ,
\nonumber
\\
G^{\rm surface}_B ( \omega ) &= & \frac{1}{\omega - ( {v_{\rm F}}{| {\bf k} |})^2 G^{\rm surface}_A ( \omega )} \, .\end{aligned}$$ The real part of these Green’s functions has a pole at $\omega = 0$ for ${| {\bf k} |}\le {t_\perp}/ {v_{\rm F}}$, indicating the presence of a surface state. The projected density of states, as function of ${| {\bf k} |}$, is sketched in Fig. \[dos\_surface\].
![Sketch of the projected density of states at the surface, as function of parallel momentum. Left: staggered stacking. Right: rhombohedral stacking (the line at $\epsilon=0$ stands for a dispersionless band of surface states).[]{data-label="dos_surface"}](dos_surface.eps){width="8cm"}
We can also analyze modifications of the surface, induced by electrostatic potentials or local changes of the stacking order. A shift of the topmost layer by a potential $\epsilon$ does not change qualitatively the projected band structure shown in Fig. \[dos\_surface\], unless $\epsilon \ge
{t_\perp}$. On the other hand, a stacking of the type $1232323 \cdots$ leads to a surface band, which can be shifted by an external potential, $\epsilon$, applied to the topmost layer, labeled $1$. As the structure looks like a perfect staggered stacking beyond this layers, and the couplings are local, the Green’s function in the topmost layer is given by: $$\left[G_{\alpha,\beta}(\omega)\right] \equiv \left( \begin{array}{cc}
\omega - \epsilon &{v_{\rm F}}{| {\bf k} |}\\ {v_{\rm F}}{| {\bf k} |}&\left( G_A^{{\rm stag.}} \right)^{-1} + \frac{( {v_{\rm F}}{| {\bf k} |})^2}{\omega} \end{array} \right)^{-1}$$ where the labels $A , B$ correspond to the two inequivalent sites in the topmost layer. A sketch of the resulting projected electronic structure is shown in Fig. \[surface\_states\]. This result is consistent with the observation of at least two frequencies in the Shubnikov-de Haas experiments in graphene multilayers with induced carriers at the surface reported in ref. \[\].
![Sketch of the projected density of states of a stack of graphene layers with ordering $CABAB \cdots$, in presence of a potential shift in the topmost layer. The lines outside the continuum of states stand for a two bands of of surface states.[]{data-label="surface_states"}](surface_states.eps){width="6cm"}
Conclusions
===========
We have analyzed a minimal model for the electronic structure of stacks of graphene planes, which allows us to obtain a number of interesting results analytically. In the following, we outline some of the most relevant ones:
- Bilayers and trilayers, in the presence of electrostatic fields that break the equivalence of the layers, develop van Hove singularities where the density of states behaves in a quasi 1D fashion, $D ( \epsilon )
\propto ( \epsilon - \epsilon_{vH} )^{-1/2}$.
- The Landau levels of a trilayer, in the absence of electrostatic effects, are given by a set of levels which depend on the field in the same way as for a single layer, $\epsilon ( j ) = \pm ( {v_{\rm F}}\sqrt{j} ) / {\ell_{\rm B}}$, and another set which is equivalent to the levels in a bilayer, $\epsilon ( j ) = \pm (
( {v_{\rm F}}^2 \sqrt{j(j+1)} ) / {t_\perp}{\ell_{\rm B}}^2 )$. A comparison of the two sets will allow for an independent measurement of the value of ${t_\perp}$.
- The surface density of states in staggered stacking vanishes at zero energy at the sites with a nearest neighbor in the next layer. This result may explain the difference in the images of the atoms at the two sublattices observed in STM measurements of graphite surfaces[@TL88].
- The bulk local density of states of rhombohedral stacking ($ABCABC \cdots$) vanishes at zero energy, and is independent of the value of the interlayer hopping, ${t_\perp}$. The spectrum of bulk Landau levels is discrete, unlike most three dimensional systems, and shows the same field dependence as that of a single graphene layer, with independence of the value of ${t_\perp}$. Hence, discrete, quasi-2D Landau levels may exist in nominally 3D samples, helping to explain the experiments in ref. \[\].
- The surface of rhombohedral, and of staggered stacking with a rhombohedral termination ($CABAB \cdots$) has surface states, with a well defined dispersion as function of the parallel momentum. This result may help to explain the observation of Shubnikov-de Haas oscillations in graphitic systems with a highly doped surface [@Metal05c].
Acknowledgments.
================
We have benefited from interesting discussions on these topics with C. Berger, P. Esquinazi, W. A. de Heer, A. K. Geim, A. F. Morpurgo, J. Nilsson, K. Novoselov, J. G. Rodrigo, and S. Vieira. A.H.C.N. was supported through NSF grant DMR-0343790. N. M. R. P. thanks ESF Science Programme INSTANS 2005-2010, and FCT under the grant POCTI/FIS/58133/2004. F. G. acknowledges funding from MEC (Spain) through grant FIS2005-05478-C02-01 and the European Union Contract 12881 (NEST).
|
---
abstract: 'The multi-loop amplitudes for the closed, oriented superstring are represented by finite dimensional integrals of explicit functions calculated through the super-Schottky group parameters and interaction vertex coordinates on the supermanifold. The integration region is proposed to be consistent with the group of the local symmetries of the amplitude and with the unitarity equations. It is shown that, besides the $SL(2)$ group, super-Schottky group and modular one, the total group of the local symmetries includes an isomorphism between sets of the forming group transformations, the period matrix to be the same. The singular integration configurations are studied. The calculation of the integrals over the above configurations is developed preserving all the local symmetries of the amplitude, the amplitudes being free from divergences. The nullification of the 0-, 1-, 2- and 3-point amplitudes of massless states is verified. Vanishing the amplitudes for a longitudinal gauge boson is argued.'
author:
- |
G.S. Danilov [^1]\
Petersburg Nuclear Physics Institute,\
Gatchina, 188300, St.-Petersburg, Russia
title: |
Preprint PNPI-2449, 2001
Manifest calculation and the finiteness of the superstring Feynman diagrams
---
=16.5cm
Introduction
============
Superstrings [@gsw] are currently considered for the unified interaction particle theory. Nevertheless, despite great efforts [@ver; @as; @momor; @vec; @pst; @ntw; @mand] during years, the multi-loop superstring interaction amplitudes were not calculated explicitly so as to be suitable for applications.
It has been attempted [@ver] to construct the integration measures (partition functions) from modular forms on the Riemann surface that requires a complicated module description. Further still, the world-sheet supersymmetry is lost as far as in this case the amplitudes depend on the choice of a basis of the gravitino zero modes [@as; @momor]. The 0-, 1-, 2- and 3-point massless state amplitudes of higher genera are not seen to be nullified contrary to the requirement [@martpl] following from the space-time supersymmetry. In the supercovariant scheme [@bshw] the calculation is complicated because of Grassmann moduli, especially, in the Ramond case. As the result, for an extended period only the Neveu-Schwarz spin structures where known explicitly [@vec; @pst; @dan90].
All the contributions to the amplitude, the Ramond sector included, have been explicitly calculated in [@danphr; @dannph; @dan96]. The superstring amplitudes have been given by integrals of a sum over super-spin structures [@danphr; @dannph] defined by super-Schottky groups on the complex $(1|1)$ supermanifold [@bshw]. Every term of the sum is the integration measure times the vacuum value of the vertex product calculated through the superfield vacuum correlators. The super-spin structures [@danphr; @dannph] are superconformal extensions of ordinary spin ones [@swit]. Being an $SL(2)$ transformation, the super-Schottky group element “$s$” is, generically, determined [@vec; @pst; @volkov] by its multiplier $k_s$ along with limiting points $U_s=(u_s|\mu_s)$ and $V_s=(v_s|\nu_s)$ where $\mu_s$ and $\nu_s$ each are the Grassmann partner of $u_s$ or, respectively, of $v_s$. The integrations are performed over the above parameters and over coordinates of the interaction vertices on the complex $(1|1)$ supermanifold, any $(3|2)$ variables being fixed due to the $SL(2)$ symmetry.
The integration region over the Schottky parameters was not, however, fully known. Moreover, the calculation of the integrals remained to be solved because of degenerated configurations where the leading approximation integrand is proportional to either a vacuum function, or to the 1-point one. Unlike the genus-1 case, the higher genus functions under discussion do not vanish locally in the space of super-Schottky group parameters. In this case the amplitude seemingly diverges like the boson string amplitude [@bk]. In the superstring, due to Grassmann integrations, the result is, however, finite or divergent being dependent on the choice of the integration variables (see a discussion of this point in the beginning of Section 7 of the present paper). In this case one must be guided by the requirement to preserve the local symmetries of the amplitude. In particular, if divergences appear, the $SL(2)$ symmetry is broken due to the cut-off parameter. As was observed [@dan99], the divergences have an evident tendency for the cancellation, if the calculation preserves the explicit $SL(2)$ symmetry. Nevertheless, in [@dan99] the total cancellation of the divergences was not verified. Moreover, an explicit regularization procedure [@dan99] for the integrals is not necessary as far as the result is expected to be finite.
At present paper the integration region over the Schottky parameters is given to be restricted by the group of the local symmetries of the amplitude and by the unitarity. The above group of the local symmetries is considered. The singular configurations are studied. The calculation of the integrals over the singular configurations is proposed. In this case we, step-by-step, integrate over variables of every handle in such a way that for every step, the integral is convergent. So the multi-loop superstring amplitudes are obtained to be finite. We argue that the calculation preserves all the local symmetries of the amplitude. The finiteness of the superstring amplitudes has been expected [@gsw; @mand; @berk] from years, but till now it was clearly seen only for the one-loop case. Unlike one-loop amplitudes, the finiteness of the multi-loop ones is provided not by local sum rules but by the vanishing of certain convergent integrals of a sum over super-spin structures. We verify the required nullification of the vacuum amplitude, vacuum-dilaton transition constant and of the 2- and 3- point amplitudes for the massless boson states. We show the vanishing of the amplitudes for the emission of a longitudinal boson due to the gauge invariance.
The present paper provides rich opportunities for applications of the superstring perturbation series. In particular, the obtained expressions can be used for the calculation of the infinity genus amplitudes and for the summation of the perturbation series under various asymptotics conditions. Being calculated in the infra-red energy limit, they give explicit and rather compact amplitudes in any perturbative order for the 10-dimensional $IIA$ super-gravity corresponding to the superstring considered (to be given in another place). So, among other things, the superstring perturbation series is the wonderfully effective instrument for the explicit calculation in the quantum field theory [@kapl]. In this paper the closed, oriented superstring theory is considered. An extension of the results to the open and/or non-oriented superstring will be given elsewhere. The perturbation series can also be constructed in heterotic and compactified superstring theories. Moreover, the series can be built for the Dirichlet boundary conditions. The perturbation series for relevant compactified (super)string can be used for calculations in four-dimensional quantum field theories [@kapl].
As it is known, the group of the local symmetries of the amplitude includes the $SL(2)$ group, changes of any given interaction vertex coordinate by super-Schottky group transformations and the modular group. We show that, in addition, the total genus-$n$ group ${\cal R}_n$ contains also a group $\{\tilde
G\}$ of isomorphic replacements $\Gamma_s\to G_s\Gamma_sG_s^{-1}$ of the given $\{\Gamma_s\}$ set of forming super-Schottky group transformations by the $\{G_s\Gamma_sG_s^{-1}\}$ set. Here $\{G_s\}$ is a relevant set of the super-Schottky group transformations (not every $\{G_s\}$ set originates the isomorphism, and, therefore, is relevant!) Evidently, the considered isomorphism does not touch the period matrix.
For the Riemann surface theory, there are known[@siegal] the constraints making the period matrix to be inside the fundamental domain of the modular group. In the present paper we extend the constraints [@siegal] to the superstring case where the period matrix depends on the super- spin structure [@pst; @danphr]. The period matrix being given through the super-Schottky group parameters, the above constraints bound a region for the Schottky variables. The discussed constraints are, however, invariant under the above $\{\tilde G\}$ group transformations preserving the period matrix. Moreover, we demonstrate module configurations destroying the unitarity equations (see Section 4 of the present paper). In this case one of the limiting points of the forming group transformation $\Gamma_s$ penetrates deep enough into the Schottky circle for the forming transformation $\Gamma_r$ with $r\neq s$ while another point lies outside both Schottky circles assigned to $\Gamma_r$. The considered configurations need to be excluded from the integration region. Evidently, they can not be reduced by the $\{\tilde G\}$ transformations to those configurations, which are consistent with the unitarity equations. Hence the fundamental domains for the direct product of $SL(2)$ group times $\{\tilde G\}$ do not all are equivalent to each other, but they are divided into the classes of the equivalent domains.
The unitarity equations are already saturated due to the region where all the Schottky circles of the forming group transformations are separated from each other, but it is not a fundamental region for the local symmetry group. We argue that the integration region can be restricted by configurations having no group limiting points inside the common interior of any pair of Schottky circles of the forming group transformations. Really the discussed region can be varied within a certain range, as it is usually for a group fundamental region.
The modular symmetry constraints [@siegal], along with the $\{\tilde G\}$ symmetry, bounds the absolute value of the multiplier of any Schottky group transformation by a magnitude smaller than unity. In this case, for not very high genera, we show that the Schottky circles of the same forming transformation are separated from each other. For the sake of simplicity we assume the same to be in the general case though now we have not a full proof for this.
When the $k$ multiplier for the boson loop is nullified, the divergence appear for the single spin structure. The divergence is, however, canceled for the sum over relevant spin structures, just as in the Neveu-Schwarz sector [@vec]. Divergences might be also due to singularities in the $\{u_s,v_s\}$ limiting points of the forming group transformations. As has been noted above, generally, the discussed singularities for the genus-$n>1$ amplitudes are not canceled locally even for the total sum over the spin structures. Nevertheless, we give the integration procedure calculating the amplitude to be finite and consistent with its local symmetries.
When all the limiting group points lay within a finite domain, the singularities in $\{u_s,v_s\}$ do not appear in the integration region until certain of $(v_s-u_s)$ differences go to zero. Generally, the integrand is, however, singular when all the limiting points of $n_1\leq n$ transformations go to the same point $z_0$. For the amplitudes having more than three legs, the basic configurations dangerous for the divergences are the configurations with no more than one interaction vertex coordinate going to $z_0$.
When limiting points $u_1$ and $v_1$ of the sole transformation go to each other, the integrand is singular in $\tilde u_1=v_1-u_1$ (or in $w_1=\tilde v_1-\nu_1\mu_1$) at $\tilde v_1\to0$ (or at $w_1\to0$). By a direct calculation using manifest expressions, one can, however, verify that the singularity is canceled. The total cancellation of the singularity is achieved after the summation over the spin structures of the considered handle and, if the interaction vertex coordinate goes to $z_0$, once certain integrations to be performed, $u_1$ or $v_1$ being fixed (see the Section 6). Therefore, the contribution to the amplitude of the $n_1=1$ configuration is finite provided that the integral over $\tilde v_1$ (or $w_1$) is calculated after integrations over other variables of the considered configuration to be taken.
For the $n_1=2$ we consider the integral over variables of the discussed configuration keeping $\tilde v_2$ (or $w_2$) to be fixed. So the integral is a function of $\tilde v_2$ (or $w_2$). The integration over $\tilde v_1$ (or $w_1$) is taken after the integration over the remaining variables of the first handle to be performed. In this case the integral of the sum over the spin structures is convergent due to the cancellation of the singularity at $\tilde v_1\to0$ or $w_1\to0$ for the $n_1=1$ integral with fixed $\tilde v_1$ of the sum over the spin structure of the handle. Indeed, at $\tilde
v_1=0$ (or $w_1=0$) the integrand of the $n_1=2$ integral is a sum of terms bi-linear in the $n_1=1$ integrals (see the proof in the beginning of Section 7). Moreover, we show (Section 7) that the $n_1=2$ integral is convergent being taken of the sum over the spin structures either of two handles. In the calculation of the amplitude we have deal only with these convergent integrals and with the integral of the total sum over spin structures of both handles, which is convergent, too. As we show, the last integral being a function of $\tilde v_2$ (or $w_2$), is finite at $\tilde v_2=0$ (or $w_2=0$). Thus the contribution to the amplitude from the discussed configuration is finite, as far as this contribution to the amplitude is just obtained by the additional integration over $\tilde v_2=0$ (or $w_2=0$) of the discussed integral with $\tilde v_2$ (or $w_2$) to be fixed.
The required finiteness of the integral with fixed $\tilde
v_2\to0$ (or $w_2\to0$) at $\tilde v_2\to0$ ($w_2\to0$) is manifested through the change of the integration variables by relevant transformations from $SL(2)$ and $\{\tilde G\}$ groups, as well as by a super-Schottky group transformation of the interaction vertex coordinate (if it goes to $z_0$). The discussed finiteness of the integral with fixed $\tilde v_2\to0$ (or $w_2\to0$) at $\tilde
v_2\to0$ ($w_2\to0$) is again shown to be due to the cancellation of the singularity for the $n_1=1$ integrals. Apart from $SL(2)$ transformations, the remaining transformations depend, however, on the spin structure because of fermion-boson mixing, which is present for non-zero $\{\mu_s,\nu_s\}$ Grassmann parameters. At the same time, we can not perform the transformations of the integration variables for the integral of the single spin structure since this integral is divergent. Nevertheless, the transformations dependent on the spin structure of the first handle can be made for the convergent integral of the sum over the spin structures of the second handle. And the transformations dependent on the spin structure of the second handle can be made for the convergent integral of the sum over the spin structures of the first one. Being a group product of the above transformations, an arbitrary $\{\tilde G\}$ or super-Schottky group transformation can be performed only for the integral of the total sum over the spin structures of the configuration. Hence the cancellation of the singularity at $\tilde v_2\to0$ ($w_2\to0$) is verified only for the integral of the total sum over the spin structures. Correspondingly, the finiteness of the contribution to the amplitude from the discussed $n_1=2$ configuration is achieved only for the integrals of the total sum over the spin structures of the configuration. In this case one calculates the integral over the variable of the 1-st handle keeping $\tilde v_1$ (or $w_1$) to be fixed. Then one calculate the integral over $\tilde v_1$ (or $w_1$). After this one calculates the integral over the variable of the 2-nd handle keeping $\tilde v_2$ (or $w_2$) to be fixed, and after this the integral over $\tilde v_2$ (or $w_2$) is calculated. The calculation preserves all the local symmetries of the amplitude including the modular symmetry. Being, generally, dependent on the spin structure [@dannph], modular transformations can be performed only for the integral of the total sum over the spin structures, like the above discussed $\{\tilde G\}$ and super-Schottky group transformations.
In the general case $n_1>2$ one integrates, step-by-step, over the limiting points of every forming transformation $s$ (except the points fixed due to the $SL(2)$ symmetry) once the summation over its spin structures to be performed. The integral is first calculated with fixed $\tilde v_s= v_s-u_s$ or, on equal terms, with fixed $w_s=\tilde v_s-\nu_s\mu_s$. For every step, the integral being a function of $\tilde v_s$ (or $w_s$), is shown to be non-singular at $\tilde v_s=0$ (or $w_s=0$). Hence it can be further integrated over $\tilde v_s$ (or $w_s$), the result being finite. When some $(u_s,v_s)$ pairs go to the infinity the integral is finite again since the infinite point can be reduced to the finite one by the relevant $L(2)$ transformation. As far as the calculation preserves all the local symmetries, the amplitude is independent of the fundamental region of the ${\cal R}_n$ group, which is employed as the integration region. In particular, the amplitude is independent of those the $(3|2)$ variables, which are fixed.
The 0-, 1-, 2- and 3-point massless state amplitude is given by the integrals with the $(3|2)$ fixed parameters $(u_1|\mu_1)$ and $(v_1|\nu_1)$ assigned to any given handle along with one of two local limiting points ($u_2$ or $v_2$) of any one of the remaining handles. We show this integral is convergent, and it is zero at $u_1\to v_1$. Thus, being invariant under the $SL(2)$ and $\{\tilde G\}$ group and under the super-Schottky group (for 1-, 2- and 3-point amplitudes), the integral vanishes identically for any $v_1$ and $u_1$, as it is required.
There are different super-extensions of ordinary spin structures, but not all they are suitable for the superstring, especially, because the space of half-forms does not necessarily have a basis when there are odd moduli [@hodkin]. The super-Schottky groups for all superspin structures have been constructed in [@danphr; @dannph; @dan93]. In the Neveu-Schwarz case they have been given before [@vec; @pst; @martnp]. Due to the fermion-boson agitation, the genus-$n>1$ super-spin structures are different from ordinary spin structures [@swit] where boson fields are single-valued on Riemann surfaces while fermion fields being twisted about $(A,B)$-cycles, may only receive the sign. Really the super-Schottky group description of the supermanifold is non-split in the sense of [@crrab]. The transition to a split description is singular [@dan97], the superstring being non-invariant under the transition discussed. As the result, the world-sheet supersymmetry is lost in [@ver] where a split module description was implied. Contrary to [@ver], our calculation preserves the world-sheet supersymmetry, as well as other local symmetries of the amplitude.
In the calculation we use partial functions and superfield vacuum correlators obtained [@danphr] by equations, which were derived from the requirement that multi-loop superstring amplitudes are independent of a choice of both the [*vierbein*]{} and the gravitino field. As it is usually, the superfield vacuum correlators is calculated through the holomorphic Green functions. For the Ramond type handle, the round about the Schottky circle being given by a non-split transformation [@danphr; @dannph; @dan93], the holomorphic Green functions in the Ramond sector can not be represented by the Poincar[é]{} series [@ales]. Correspondingly, the integration measure is not a product over known expressions [@vec; @dan90] in terms of super-Schottky group multipliers. Nevertheless, the discussed genus-$n$ functions can be given [@danphr] by a series over integrals of relevant genus-1 function products. For the Neveu-Schwarz sector the above series can be reduced to the Poincar[é]{} series. Now we continue the study of the holomorphic Green functions and integration measures. In particular, we present them (Appendices B and C of the paper ) in the form, which is more convenient for application than the expressions in [@danphr]. We also derive them through the functions of lower genera $n_i\geq1$. We employ these expressions for the calculation of the integrals over singular configurations.
The paper is organized as it follows. Sections 2 contains a brief review of the super-Schottky group parameterization [@danphr; @dannph] using in the paper. The expression for the multi-loop superstring amplitude is given. In Sections 3 and 4 the constraints on the integration region are discussed. The integration region over the Schottky parameters is proposed. In Section 5 the integration measures and the superfield vacuum correlators are derived through the lower genus functions. The integration measures and the superfield vacuum correlators are calculated for the degenerated configurations dangerous for divergences. In Section 6 the strategy calculating the integrals is discussed. Vanishing the amplitude of the emission of a longitudinally polarized boson is argued. Configurations dangerous for divergences are collected. The cancellation of divergences for easy configurations is demonstrated. In Section 7 the finiteness of the amplitudes is shown. The vanishing of the 0-, 1-, 2- and 3-point functions is verified. The preservation of the local symmetries of the amplitudes is argued.
Expression for the multi-loop superstring amplitude
===================================================
As it was noted, we employ super-Schottky groups variables. The super-Schottky group determines the super-spin structure on the complex $(1|1)$ supermanifold mapped by the $t=(z|\vartheta)$ coordinate. The genus-$n$ super-spin structure presents a superconformal extension of the relevant genus-$n$ spin one given by the set of transformations $\Gamma^{(0)}_{a,s}(l_{1s})$ and $\Gamma^{(0)}_{b,s}(l_{2s})$ (where $s=1,\dots n$), which correspond to the round of $A_s$-cycle and, respectively, of the $B_s$-cycle on the Riemann surface. They depend on the theta function characteristics $l_{1s}$ and $l_{2s}$ assigned to the given handle $s$. A discrimination is made only between those (super-)spin structures, for which field vacuum correlators are distinct. So $l_{1s}$ and $l_{2s}$ can be restricted by 0 and 1/2. In this case $l_{1s}=0$ is assigned to the Neveu-Schwarz handle while $l_{1s}=1/2$ is reserved for the Ramond one. In doing so $$\Gamma^{(0)}_{b,s}(l_{2s})=\left\{z\to \frac{a_sz+b_s}
{c_sz+d_s}\,,\quad\vartheta\to -\frac{(-1)^{2l_{2s}}}{c_sz+d_s}
\vartheta\right\},\,\, \Gamma^{(0)}_{a,s}(l_{1s})=\left\{z\to
z\,,\quad\vartheta\to (-1)^{2l_{1s}}\vartheta \right\}
\label{zgama}$$ where $a_sd_s-b_sc_s=1$. Furthermore, $$a_s=\frac{u_s-k_sv_s}{\sqrt
{k_s}(u_s-v_s)}\,,\quad d_s=\frac{k_su_s-v_s}
{\sqrt{k_s}(u_s-v_s)}\quad{\rm and}\quad
c_s=\frac{1-k_s}{\sqrt{k_s}(u_s-v_s)}
\label{uvk}$$ where $k_s$ is a complex multiplier and $|k_s|\leq1$. Further, $u_s$ is the attractive limiting (unmoved) point of (\[zgama\]) while $v_s$ is the repulsive limiting one. The set of transformations (\[zgama\]) along with their group products form the Schottky group. The $\Gamma^{(0)}_{b,s}(l_{2s})$ transformation in (\[zgama\]) turns the boundary of the Schottky circle $C_{v_s}$ into the boundary of $C_{u_s}$ where $$C_{v_s}=\{z:|c_sz+d_s|=1\}\quad{\rm
and}\quad C_{u_s}=\{z:|-c_sz+a_s|=1\}
\label{circ}$$ for $s=1,\dots n$. Using (\[uvk\]), one can see that $v_s$ lies inside $C_{v_s}$ and outside $C_{u_s}$. Correspondingly, $u_s$ is inside $C_{u_s}$ and outside $C_{v_s}$. Since $\Gamma^{(0)}_{a,s}(l_{1s})$ in (\[zgama\]) corresponds to the round of the Schottky circle, in the Ramond case a square root cut appears on $z$-plane between $u_s$ and $v_s$. The fundamental region of a group is that one, which does not contain points congruent[^2] under the group, and such that the neighborhood of any point on the boundary contains points congruent to the points of the region [@ford]. In particular, a fundamental region of the Schottky group on the complex $z$ plane is the exterior of all the Schottky circles associated with the given group. The group invariant integral of the conformal $(1,1)$ tensor being calculated over the fundamental region of the group, does not depend on the choice of the fundamental region above [@ford].
In the superstring theory the forming transformations (\[zgama\]) are replaced by $SL(2)$ transformations $\Gamma_{a,s}(l_{1s})$ and $\Gamma_{b,s}(l_{2s})$ with [@danphr; @dannph; @dan93] $$\Gamma_{a,s}(l_{1s})=\tilde\Gamma^{-1}_s
\Gamma^{(0)}_{a,s}(l_{1s})\tilde\Gamma_s\,,
\qquad \Gamma_{b,s}(l_{2s})=
\widetilde\Gamma^{-1}_s\Gamma^{(0)}_{b,s}
(l_{2s})\widetilde\Gamma_s
\label{gamab}$$ where $\Gamma^{(0)}_{b,s}(l_{2s})$ and $\Gamma^{(0)}_{a,s}(l_{1s})$ are given by (\[zgama\]) while $\widetilde\Gamma_s$ depends, among other things, on two Grassmann parameters $(\mu_s,\nu_s)$ as it follows $$\tilde\Gamma_s=\left\{
z=z^{(s)} +\vartheta^{(s)}\varepsilon_s(z^{(s)})\,,\quad
\vartheta=\vartheta^{(s)}\left(1+
\frac{\varepsilon_s\varepsilon'_s}2\right)+
\varepsilon_s(z^{(s)})
\right\}\,,
\label{tgam}$$ $$\varepsilon_s(z)=\frac{\mu_s(z-v_s)-\nu_s(z-u_s)}
{u_s-v_s}\,,\qquad
\varepsilon'_s=\partial_z\varepsilon_s(z)
\label{teps}$$ Thus $(u_s|\mu_s)$ and $(v_s|\nu_s)$ are limiting points of transformations (\[gamab\]). The set of the transformations (\[gamab\]) for $s=1,\dots,n$ together with their group products forms the genus-$n$ super-Schottky group. If $l_{1s}=1/2$, then both transformations (\[gamab\]) are non-split, as it was already discussed in the Introduction. The $\Gamma_{a,s}(l_{1s})$ transformation relates superconformal $p$-tensor $T_p(t)$ with its value $T_p^{(s)}(t)$ obtained from $T_p(t)$ by $2\pi$-twist about $C_{v_s}$-circle (\[circ\]). So, $T_p(t)$ is changed under the $\Gamma_{a,s}(l_{1s})=
\{t\rightarrow t_s^a\}$ and $\Gamma_{b,s}=\{t\rightarrow
t_s^b\}$ transformations as follows $$T_p(t_s^a)=T_p^{(s)}(t)Q_{\Gamma_{a,s}(l_{1s})}^p(t),\qquad
T_p(t_s^b)=T_p(t)Q_{\Gamma_{b,s}(l_{2s})}^p(t).
\label{stens}$$ Here $Q_G(t)$ is the factor, which the spinor left derivative $D(t)$ receives under the $SL(2)$ transformation $G(t)=
\{t\rightarrow t_G=(z_G(t)|\vartheta_G(t))\}$: $$Q_G^{-1}(t)=D(t)\vartheta_G(t)\,;\qquad
D(t_G)= Q_G(t)D(t)\,,\qquad
D(t)=\vartheta\partial_z+\partial_\vartheta
\label{supder}$$ It follows from (\[supder\]) that for the group product $G=G_1G_2$, $$Q_{G_1G_2}(t)=Q_{G_1}(G_2(t))Q_{G_2}(t)
\label{suprel}$$ Furthermore, $\Gamma_{b,s}(l_{2s})$ turns the boundary of the $\hat C_{u_s}$ “circle” to the boundary of $\hat C_{v_s}$ where $$\hat C_{v_s}=\{t:|c_sz^{(s)}+d_s|^2=1\}\quad{\rm and}\quad
\hat C_{u_s}=\{t:|-c_sz^{(s)}+a_s|^2=1\}
\label{hcirc}$$ and $z^{(s)}$ is defined by (\[tgam\]). Moreover, the same is true for the “circles” $$\hat C_{v_s}'=\{t:|Q_{\Gamma_{b,s}}(t)|^2=1\}\quad{\rm and}\quad
\hat C_{u_s}'=\{t:|Q_{\Gamma_{b,s}^{-1}}(t)|^2=1\}
\label{hcirc1}$$ where the super-derivative factors (\[supder\]) correspond to $\Gamma_{b,s}(l_{2s})$ and, respectively, its inverse transformation. “Circles” (\[hcirc\]) and (\[hcirc1\]) differ from (\[circ\]) only in terms proportional to the Grassmann quantities. Being constructed for every group transformation, both the “circles” can be used to define the boundary of the fundamental region. For applications it is useful solely to keep in mind that the fundamental region can be given by the step function factor through relevant functions $\ell_G(t)$ and $\ell_G(G(t))$ as it follows $$B_{L,L'}^{(n)}(t,\bar t;\{q,\bar q\})=
\prod_G\theta(|\ell_G(t)|^2-1) \theta(1-|\ell_G(G(t))|^2)
\label{zbound}$$ where $\theta(x)$ is step function defined to be $\theta(x)=1$ at $x>0$ and $\theta(x)=0$ at $x<0$. Further, $L=\{l_{1s},l_{2s}\}$ is the super-spin structure for the right movers while $L'$ is the same for the left ones. The product is taken over all group products $G$ of the $\Gamma_{b,s}(l_{1s})$ transformations except $G=I$. Evidently, one can exclude from $\{G\}$ a transformation inverse to the given one of the set. It is implied that all the group limiting points lay exterior to the region, and the region does not contain points related with each other by the group transformation. The region (\[zbound\]) is relevant for the integration region for the group invariant integral of the $(1/2,1/2)$ super-tensor. Generically, the argument of step function (\[zbound\]) depends on Grassmann parameters. The expansion in a series over the above Grassmann ones originates $\delta$-functions and their derivatives, which give rise to the boundary terms in the integral. The integral is independent of the boundary sharp that can be directly verified for infinitesimal variations of the boundary. As it is usually [@ford], one can replace any part of the fundamental region by a congruent part and still have a fundamental region. The integral discussed is independent of the fundamental region, which it is taken over.
The superstring amplitude (\[ampl\]) is calculated by the integration over a fundamental region of the total group ${\cal R}$ of local symmetries, for details see Sections 3 and 4. In this case the $\{N_0\}$ set of $(3|2)$ variables among the group limiting points and interaction vertex coordinates are fixed due to the $SL(2)$ symmetry. Simultaneously, the integrand is multiplied by a factor $H(\{N_0\})$. For the sake of simplicity, we assume to be fixed two any variables $z_1^{(0)}$ and $z_2^{(0)}$ along with their Grassmann partners $\vartheta_1^{(0)}$ and $\vartheta_2^{(0)}$, and one more variable $z_3^{(0)}$, as well. In this case [@danphr] $$H(\{N_0\})=(z_1^{(0)}-z_3^{(0)})(z_2^{(0)}-z_3^{(0)})
\left[1-\frac{\vartheta_1^{(0)}\vartheta_3}
{2(z_1^{(0)}-z_3^{(0)})}-\frac{\vartheta_2^{(0)}\vartheta_3}
{2(z_2^{(0)}-z_3^{(0)})}\right]\,.
\label{factorg}$$ So, generically, (\[factorg\]) depends on the integration variable $\vartheta_3$, which is the Grassmann partner of $z_3^{(0)}$. The $n$-loop, $m$-point amplitude $A_m^{(n)}(\{p_j,\zeta^{(j)}\})$ for the interaction states carrying 10-dimensional momenta $\{p_j\}$ and the polarization tensors $\zeta^{(j)}$, is given by[^3] $$\begin{aligned}
A_m^{(n)}(\{p_j\},\zeta^{(j)}\})=\frac{g^{2n+m-2}}{2^nn!}\int
|H(\{N_0\})|^2\sum_{L,L'}
Z_{L,L'}^{(n)}(\{q,\overline q\})
<\prod_{r=1}^mV(t_r,\overline t_r;p_r;\zeta^{(r)})>
\nonumber\\
\times
\hat
B_{L,L'}^{(n)}(\{q,\overline q\})\tilde
B_{L,L'}^{(n)}(\{q,\overline
q\})
\prod_{j=1}^mB_{L,L'}^{(n)}(t_j,\bar
t_j;\{q,\bar q\})
(dqd\overline qdtd\overline t)'
\label{ampl}\end{aligned}$$ where $\{q\}=\{k_s,u_s,v_s,\mu_s,\nu_s\}$ is the set of the super-Schottky group parameters, $g$ is the coupling constant, $H(\{N_0\})$ is defined by (\[factorg\]) and $L$ ($L'$) labels the super-spin structures of right (left) movers. Further, $Z_{L,L'}^{(n)}(\{q,\overline q\})$ is the partition function, and $<...>$ denotes the vacuum expectation of the product of the interaction vertices $V(t_r,\bar t_r;p_r;\zeta^{(r)})$. The integration is performed over those variables, which do not belong to the $\{N_0\}$ set. The step function factor $\hat
B_{L,L'}^{(n)}(\{q,\overline q\})$ keeps the period matrix to be interior to the fundamental region of the modular group. As far as the period matrix is given through the super-Schottky group parameters $\{q\}$, this factor restricts the integration region over $\{q\}$. The discussed factor depends on the spin structures as far as the period matrix depends on the spin structure by terms proportional to Grassmann parameters. Further, $\tilde
B_{L,L'}^{(n)}(\{q,\overline q\})$ more bounds the integration region due to the $\{\tilde G\}$ symmetry and the unitarity as it has been mentioned in the Introduction. Both factors are discussed in Sections 3 and 4. The step multiplier $B_{L,L'}^{(n)} (t_j;\{q,\overline q\})$ is given by (\[zbound\]) at $t=t_j$. Generically, $\ell_G(t_j)$ can depend on $j$. The discussed multiplier is assigned every $t_j$ including $t_j$ of the $\{N_0\}$ set, as well. Indeed, due to the invariance under the super-Schottky group changes of any one vertex coordinate, $t_j$ can be fixed insides region (\[zbound\]). Really the discussed step factor must be assigned to fixed $t_j$ of the $\{N_0\}$ set for the unitarity equations to be true, see Section 4.
The $1/2^n$ factor in (\[ampl\]) is due to the symmetry of the integrand under the interchange between $(u_s|\mu_s)$ and $(v_s|\nu_s)$, and $1/n!$ is due to the symmetry under the interchange of the handles. Both symmetries are particular cases of the modular symmetry [@siegal] (see the next Section). The above factors are consistent with the unitarity equations (as an example, see Appendix A of the paper). For any boson variable $x$ we define $dxd\overline
x=d(Re\,x)d(Im\,x)/(4\pi)$. For any Grassmann variable $\eta$ we define $\int d\eta\eta=1$. The super-spin structure being odd[^4], the integrand has some features due to the spinor zero modes, which present in this case. We mainly discuss the even super-spin structures as far as the odd super-spin one can be obtained by a factorization of relevant even super-spin structure [@dan96].
Our classification over the super-spin structures implies that $$|\arg
k_s|\leq\pi\,,\quad|\arg(u_s-u_r)|\leq\pi\,,
\,\,|\arg(u_s-v_r)|\leq\pi\,,\,\,|\arg(v_s-v_r)|\leq\pi\,,
\quad(s\neq r).
\label{argum}$$ To be accurate, (\[argum\]) are given for the “bodies” of the corresponding quantities, as far as they may have “soul” parts proportional to Grassmann parameters. Using the Green functions [@danphr; @dannph], one can see that $|\arg k_s|\leq\pi$ discriminates between the Green functions for $(l_{1s}=0,l_{2s}=0)$ and for $(l_{1s}=0,l_{2s}=1/2)$. The rest constraints discriminate between $(l_{1s}=l_{1r}=1/2,
l_{2s}=l_{2r}=0)$ and $(l_{1s}=l_{1r}=1/2, l_{2s}=l_{2r}=1/2)$. The adding of $\pm2\pi$ to any one of the quantities in (\[argum\]) presents a modular transformation [@dannph], see Section 3 below. We use (\[argum\]) instead of known constrains [@siegal] for the real part of the period matrix.
We consider the massless boson interaction amplitudes. Thus, for the normalization used, the known expression [@fried] of the interaction vertex through the string superfields $X^N(t,t')$ for $N=0,\dots 9$ is as follows $$V(t,\overline
t;p;\zeta)=4\zeta_{MN} [D(t)X^M(t,\overline
t)][\overline{D(t)}X^N(t,\overline t)]
\exp[ip_RX^R(t,\overline
t)]
\label{vert}$$ where $p=\{p^M\}$ is 10-momentum of the interacting boson while $\zeta_{MN}$ is its polarization tensor, $p^M\zeta_{MN}=p^N\zeta_{MN}=0$, and $p^2=0$. The spinor derivative $D(t)$ is defined in (\[supder\]). The summation over twice repeated indexes is implied. We use the “mostly plus” metric. The dilaton $\zeta_{MN}$ tensor is equal to the transverse Kronecker symbol $\delta_{MN}^\perp$. The vacuum expectation in (\[ampl\]) is calculated in term of the genus-$n$ scalar superfield vacuum correlator given through the holomorphic Green function $R_L^{(n)}(t,t';\{q\})$ and super-holomorphic functions $J_r^{(n)}(t;\{q\};L)$ having periods (here $r=1,\dots n$). In this case $$\begin{aligned}
J_r^{(n)}(t_s^b;\{q\};L) = J_r^{(n)}(t;\{q\};L)+2\pi i
\omega_{sr}(\{q\},L)\,,
\nonumber\\
J_r^{(n)}(t_s^a;\{q\};L)=J_r^{(n)(s)}(t;\{q\};L)+
2\pi i\delta_{rs}
\label{trjs}\end{aligned}$$ where $t_s^a$ and $t_s^b$ are the same as in (\[stens\]) and $\omega_{sr}(\{q\},L)$ is the corresponding element of the period matrix. The Green function is changed under the transformations (\[stens\]) as $$\begin{aligned}
R_L^{(n)}(t_r^b,t';\{q\})=R_L^{(n)}(t,t';\{q\})+
J_r^{(n)}(t';\{q\};L)\,,
\nonumber\\
R_L^{(n)}(t_r^a,t';\{q\})=R_L^{(n)(r)}(t,t';\{q\})\,.
\label{rtrans}\end{aligned}$$ In this case the Green function is normalized as it follows $$R_L^{(n)}(t,t';\{q\})=\ln(z-z'-\vartheta\vartheta')+
\tilde R_L^{(n)}(t,t';\{q\})
\label{lim}$$ where $\tilde R_L^{(n)}(t,t';\{q\})$ has no a singularity at $z=z'$. The scalar superfield vacuum correlator $\hat
X_{L,L'}(t,\overline t;t',\overline t';\{q\})$ is given by[^5] $$4\hat X_{L,L'}(t,\overline t;t',\overline t';\{q\})=
R_L^{(n)}(t,t';\{q\})+\overline{R_{L'}^{(n)}(t,t';\{q\})}
+I_{LL'}^{(n)}(t,\overline
t;t',\overline t';\{q\})\,,
\label{corr}$$ $$\begin{aligned}
I_{LL'}^{(n)}(t,\overline
t;t',\overline t';\{q,\bar q\})= [J_s^{(n)}(t;\{q\};L) +
\overline{J_s^{(n)}(t;\{q\};L')}]
[\Omega_{L,L'}^{(n)}(\{q,\overline q \})]_{sr}^{-1}
\nonumber\\ \times
[J_r^{(n)}(t';\{q\};L)+\overline{J_r^{(n)}(t';\{q\};L')}]
\label{illin}\end{aligned}$$ where $\Omega_{L,L'}^{(n)}(\{q,\overline q\})$ being calculated in terms of the $\omega^{(n)}(\{q\},L)$ period matrix, is $$\Omega_{L,L'}^{(n)}(\{q,\overline q\})=
2\pi
i[\overline{\omega^{(n)}(\{q\},L')}-
\omega^{(n)}(\{q\},L)].
\label{grom}$$ As it is usually, the Green function at the same point $z=z'$ is defined to be the $\tilde R_L^{(n)}(t,t';\{q\})$ at $z=z'$. The dilaton emission amplitudes include the vacuum pairing $\tilde I_{L,L'}^{(n)}(t,\bar t;\{q\})$ of the superfields in front of the exponential in (\[vert\]). In line with aforesaid $$\tilde
I_{L,L'}^{(n)}(t,\bar t;\{q,\bar q\})=
2D(t)D(\bar t')I_{L,L'}^{(n)}
(t,\overline t;t',\overline t';\{q\})|_{t=t'}
\label{ngcor}$$ where the definitions are given in (\[corr\]) and in (\[illin\]). The dilaton-vacuum transition constant is determined by the integral of (\[ngcor\]) over the supermanifold. Integrating by parts the right side of (\[ngcor\]) with the following using of the relations (\[trjs\]), one obtains the discussed constant to be $n$ times the vacuum amplitude [@gsw].
Due to a separation in right and left movers, the integration measure in (\[ampl\]) is represented as $$Z_{L,L'}^{(n)}(\{q,\overline
q\})=(8\pi)^{5n}[\det\Omega_{L,L'}^{(n)}(\{q,\overline q
\})]^{-5} Z_L^{(n)}(\{q\}) \overline {Z_{L'}^{(n)}(\{q\})}
\label{hol}$$ where $Z_L^{(n)}(\{q\})$ is a holomorphic function of the $q$ moduli and the $\Omega_{L,L'}^{(n)}(\{q,\overline q \})$ matrix is given by (\[grom\]). The holomorphic partition function in (\[hol\]) is given by $$Z_L^{(n)}(\{q\})=\hat Z_L^{(n)}(\{q\})
\prod_{s=1}^n(u_s-v_s-\mu_s\nu_s)^{-1}
\label{zinv}$$ where $\hat Z_L^{(n)}(\{q\})$ is invariant under the $SL(2)$ transformations as far as $$du_sdv_sd\mu_s d\nu_s/(u_s-v_s-\mu_s\nu_s)
\label{sliv}$$ is $SL(2)$ invariant. Explicit $\hat Z_L^{(n)}(\{q\})$ and other functions of interest are given in Section 5. In two following Sections we consider the integration region.
Modular symmetry constraints
============================
The modular transformation of the supermanifold, generically, presents a globally defined, holomorphic superconformal change $t\to\hat t$ of the coordinate along with holomorphic changes $q\to\hat q$ of the super-Schottky group parameters and by a change $L\to\hat L$ of the super-spin structure. Like the modular transformation of the Riemann surface[@siegal], it determines the going to a new basis of non-contractable cycles. So the period matrix $\omega(\{q\},L)$ is changed, as it is usually, by $$\omega(\{q\},L)=[A\omega(\{\hat
q\},\hat L)+B] [C\omega(\{\hat q\},\hat L)+D]^{-1}
\label{modtr}$$ where integer $A$, $B$, $C$ and $D$ matrices obey [@siegal] the relations $$C^TA=A^TC\,,\qquad
D^TB=B^TD\,,\qquad D^TA-B^TC=I\,.
\label{mrel}$$ The right-top “$T$” symbol labels the transposing. In this case $t(\hat t;\{\hat q\};\hat L)$ and $q(\{\hat q\};\hat L)$ both depend on the superspin structure by terms proportional to Grassmann parameters [@dannph]. Then the period matrix also depends on the super-spin structure, as has been noted in the Introduction. For zero Grassmann parameters the period matrix is reduced to $\omega^{(0)}(\{q\})$ whose matrix elements $\omega_{sp}^{(0)}(\{q\})$ are given through the Schottky parameters $\{q\}$ by [@vec; @danphr] $$2\pi i\omega_{sp}^{(0)}(\{q\})= \delta_{sp}\ln k_p+
{\sum_\Gamma}^{''}\ln\frac{[u_s-g_\Gamma(u_p)]
[v_s-g_\Gamma(v_p)]}{[u_s-g_\Gamma(v_p)][v_s-g_\Gamma(u_p)]}
\label{omega0}$$ where $\delta_{sp}$ is the Kronecker symbol. The summation in (\[omega0\]) is performed over all those the transformations $g_\Gamma$ of the Schottky group, whose leftmosts are not group powers of $g_s$, or the rightmosts are not group powers of $g_r$. Besides, $g_\Gamma\neq I$, if $s=p$. So the addition of $\pm2\pi$ to $\arg k_s$ adds $\pm1$ to $\omega_{ss}^{(0)}(\{q\})$. For $r\neq s$, due to the term with $\Gamma=I$ in (\[omega0\]), the addition of $\pm2\pi$ to the argument of the difference between limiting points in (\[argum\]) adds $\pm1$ to $\omega_{sr}^{(0)}(\{q\})$. Hence the discussed changes of arguments (\[argum\]) just correspond to transformations of the period matrix by (\[modtr\]) with $C=0$ and $A=D=I$. Evidently, it is true for non-zero Grassmann parameters too, and we do not enlarge on this matter. We note only that for non-zero Grassmann parameters, the addition of $\pm2\pi$ to the argument of the difference between limiting points in (\[argum\]) is accompanied [@dannph] by a certain change of $t$ and of $\{q\}$. Relations (\[argum\]) replace constraints $|Re\,\omega_{sp}^{(0)}(\{q\})|\leq1/2$ for the period matrix [@siegal]. In the both cases the sum in (\[ampl\]) includes all the distinct spin structures without a double counting, but constraints (\[argum\]) are much more convenient for applications than constraints $|Re\,\omega_{sp}^{(0)}(\{q\})|\leq1/2$.
Further constraints appear due to transformations (\[modtr\]) with $$B=C=0\,,\quad A=F\,,\quad D^{-1}=
F^T\,,\quad \det F=\pm1\,,
\label{parttr}$$ $F$ being an integer matrix. For the diagonal $F$ matrix, $F_{ss}=-1$ corresponds to the replacement of the corresponding group transformation by its inverse that interchanges between $(u_s|\mu_s)$ and $(v_s|\nu_s)$. Indeed, from (\[rtrans\]), under the above replacement, $J_s^{(n)}(t;\{q\};L)$ receives the sign. Hence from (\[trjs\]), the period matrix elements $\omega_{rs}(\{q\},L)$ with $r\neq s$ also receive the sign that just correspomds to modular transformation discussed (for zero Grassmann parameters this follows directly from (\[omega0\])). The $F$ matrix having a sole non-zero non-diagonal matrix element $F_{s_1s_2}=F_{s_1s_2}=1$ and $F_{s_1s_1}=F_{s_2s_2} =0$, corresponds to the interchange $s_1\rightleftharpoons s_2$ between the handles. Thus the Schottky parameters is bounded by constraints [@siegal], which discriminate between repulsive and attractive limiting points, as well as between the handles. In their stead we prefer to introduce the $1/(2^nn!)$ factor in (\[ampl\]). Remaining $F$ matrices correspond to transitions to new basic cycles, which are certain sums over the former basic ones. For non-zero Grassmann parameters, unlike the case of zero Grassmann ones, both $t$ and $\{q\}$ are changed. Indeed, in this case $\Gamma_{as}(l_{1s}=1/2)$ does not commute with transformations (\[gamab\]) assigned to another handle. Hence the group transformations for rounds about resulted $(A,B)$-cycles being a sum of the former ones, do not commute with each other, if $t$ and $\{q\}$ do not changed. The required changes of $t$ and $\{q\}$ could be calculated by the method developed in [@dannph], but it is not a subject of the present paper.
For zero Grassmann parameters, among period matrices related by (\[parttr\]), one takes [@siegal] the matrix having the smallest imaginary part $y_{jj}(\{q,\bar q\})$. In this case $$[Fy(\{q,\bar q\}) F^T]_{jj}\geq y_{jj}(\{q,\bar q\})\,.
\label{imagin}$$ As it is usually, $y_{jj}(\{q,\bar q\})$ is non-negative. Starting with $j=1$, one, step-by-step, constructs a constraints for $j=2,\dots,n$. Calculating the integral, one sums with the $1/n!$ factor over permutations of the handles. Further constraints are due to transformations with $C\neq0$. In this case, among the $\omega^{(0)}(\{q\})$ period matrices related by (\[modtr\]), one takes the matrix, which gives the greatest magnitude of $\det y(\{q,\bar q\})$. Due to (\[modtr\]) and (\[mrel\]), the corresponding constraints are given [@siegal] by $$|\det
[C\omega^{(0)}(\{q\})+D]|^2\geq1
\label{bound}$$ where $C$ and $D$ are the matrices in (\[modtr\]). Important particular constraints are obtained from (\[imagin\]) when, for given $r$ and $j$, one takes $F_{jr}=F_{rj}=\pm1$ and either $F_{jj}=0$, or $F_{rr}=0$. All other non-diagonal elements of $F$ are assumed to be zeros. Then $$min[y_{jj}(\{q,\bar
q\}),\,y_{rr}(\{q,\bar
q\})]\pm2y_{rs}(\{q,\bar
q\})\geq 0\,.
\label{imnd}$$ Also, using special $C$ and $D$ in (\[bound\]), one derives important constraints for the principal minors $[\det(F\omega^{(0)}(\{q\})F^T+\tilde B)]_{s_1\dots s_k}$ of the $\det(F\omega^{(0)}(\{q\})F^T+\tilde B)$ as it follows $$[\det(F\omega^{(0)}(\{q\})F^T+\tilde
B)]_{s_1\dots s_k}[\det(
\overline{F\omega^{(0)}(\{q\})F^T+
\tilde B)}]_{s_1\dots s_k}\geq1\
\label{minors}$$ where $F$ is an integer matrix, $\det F=\pm1$. There is no summation over the $s_1\dots s_k$ indices, and the integer matrix $\tilde B$ is chosen from the condition that $Re\,|F\omega^{(0)}(\{q\})F^T+\tilde B|\leq1/2$.
For non-zero Grassmann parameters, $\omega^{(0)}(\{q\})$ is replaced by $\omega(\{q\},L)$ while $\overline{\omega^{(0)}(\{q\})}$ is replaced by $\overline{\omega(\{q\},L')}$. All the other extensions contain $\overline{\omega(\{q\},L)}$ and/or $\omega(\{q\},L')$. Hence they are not consistent with the modular symmetry. Indeed, (\[modtr\]) relates $\overline{\omega(\{q\},L)}$ with $\overline{\omega(\{q_L\},\hat L)}$, which is different from $\overline{\omega(\{q_{L'}\}, \hat L)}$ due to terms proportional to the Grassmann parameters [@dannph]. So the imaginary part of the period matrix is replaced by $$y(\{q,\bar
q\},L,L')=\frac{1}{2i}[\omega(\{q\},L)-
\overline{\omega(\{q\},L')}]\,.
\label{yll}$$ In this case the corresponding step factor in (\[ampl\]) is given by $$\begin{aligned}
\hat B_{L,L'}^{(n)}(\{q,\overline q\})=\left[\prod_{C,D}
\theta(\det\{[C\omega(\{q\},L+D]
[C(\overline{\omega(\{q\},L')}+\tilde B)+D]\}-1)\right]
\nonumber\\
\times
\left[\sum\prod_{ F}\theta([ Fy(\{q,\bar q\},L,L')
F^T]_{jj}
-y_{jj}(\{q,\bar q\},L,L'))\right]\prod_{j=1}^n
\theta(y_{jj}(\{q,\bar q\},L,L'))
\label{sbound}\end{aligned}$$ where the sum is performed over permutations of the handles while the products are calculated over all the matrices $C$, $D$ and $F$ in (\[modtr\]) and (\[parttr\]). The step function $\theta(x)$ is the same as in (\[zbound\]).
Integration region
==================
The period matrix is preserved under isomorphic changes $$\Gamma_{a,s}(l_{1s})\to
G_s\Gamma_{a,s}(l_{1s})G_s^{-1}\,,\quad \Gamma_{b,s}(l_{2s})\to
G_s\Gamma_{b,s}(l_{2s})G_s^{-1}
\label{morph}$$ of the set of the forming group transformations (\[gamab\]) where $G_s$ is a relevant transformation [^6] from the super-Schottky group. Generically, $G_s$ depends on $s$. The discussed isomorphism only replaces the limiting points $U_s=(u_s|\mu_s)$ and $V_s=(v_s|\nu_s)$ of the transformation (\[gamab\]) by $G_sU_s$ and by $G_sV_s$. As far as this is globally defined holomorphic transformation, the amplitude integral (\[ampl\]) is invariant under the $\{\tilde G\}$ group of the isomorphic changes discussed. In particular, the holomorphic Green function and the scalar functions are not changed since (\[morph\]) does not touch (\[trjs\]) and (\[rtrans\]). Moreover, from the equations [@danphr] for the partition functions, one can derive that the invariant partition function $\hat
Z_L^{(n)}(\{q\})$ in (\[zinv\]) is also unchanged. Only the multiplier behind $\hat Z_L^{(n)}(\{q\})$ receives a factor, which is just canceled by the Jacobian of the transformation, as it also follows from the below constructing of the considered group.
In doing so, for given $s_1$ and $s_2\neq s_1$, one changes the transformations (\[gamab\]) for $s=s_1$ by (\[morph\]) with $G_{s_1}$ to be any one from transformations (\[gamab\]) for $s=s_2$. The resulted set $S(s_1|s_2)$ is evidently isomorphic to the former set $S_0$ of transformations (\[gamab\]) for $s=1,\dots,n$. Moreover, $U_{s_1}$ and $V_{s_1}$ are changed by the $SL(2)$ transformation independent of $U_{s_1}$ and $V_{s_1}$. Thus factor (\[sliv\]) is unchanged. In this way one obtains the set of different $S(s_1|s_2)$ sets. Applying this procedure to every set, one builds further sets. As an example, one obtains the $S(s_1,s_2|s_2,s_1)$ set where the transformations for $s=s_2$ from the $S(s_1|s_2)$ set are changed by (\[morph\]) with $G_{s_2}$ to be the $s=s_1$ transformation from the same $S(s_1|s_2)$ set. In this case the $s_2$ transformation of the $S(s_1,s_2|s_2,s_1)$ set depends, among other things, on the starting transformation (\[gamab\]) for the same $s=s_2$. By construction, all the sets are isomorphic to each other, the amplitude being invariant under the transformations discussed. So every set can be used as the set of forming group transformations. In this way one constructs the desired infinite (super)-group $\{\tilde G\}$ generating forming group transformation sets by action of $\{\tilde G\}$ on the given set (\[gamab\]).
Since (\[zbound\]) and (\[sbound\]) are preserved under the $\tilde G$ transformations, a further constraint of the integration region is necessary to exclude domains related by $\tilde G$ each to other. In particular, due to the $\tilde G$ symmetry, either one of two limiting points of every forming transformation (\[gamab\]) can be restricted to be the exterior of Schottky circles assigned to all the other forming transformations. For certain configurations, both limiting point of every given forming transformation appear exterior to the Schottky circles above. These configurations can not be obtained by $\tilde G$ transformations of configurations where this is not so. Moreover, the integral over the last configurations destroys the unitarity equations, as demonstrated for the case when all the Schottky multipliers go to zero.
In this case, from (\[uvk\]), the boundaries of Schottky circles (\[circ\]) are given by the conditions $$|z-v_s|^2=|k_s||u_s-v_s|^2\,,\qquad
|z-u_s|^2=|k_s||u_s-v_s|^2\,.
\label{circlim}$$ Period matrix elements (\[omega0\]) are reduced to $$2\pi
i\omega_{jr}^{(0)}\to
\ln\frac{(u_j-u_r)(v_j-v_r)}{(u_j-v_r)(v_j-u_r)}\,,
\qquad \omega_{jj}\to\ln k_j
\label{kzero}$$ Moreover, from (\[imnd\]), one derives that $$2\left|\ln\bigg|\frac{(u_j-u_r)(v_j-v_r)}{(u_j-v_r)(v_j-u_r)}
\bigg|\right|\leq min\bigg(-\ln|k_j|,\,-\ln|k_r|\bigg)
\label{ndgbound}$$ Below we assume $j=1$ and $r=2$. Constraint (\[ndgbound\]) allows, for instance, the discussed configuration $$|u_1-u_2|\sim |k_2|^{1-\delta}\,,\quad
|v_1-u_2|\sim|k_2|^{1/2-\delta}\,,\quad|k_2|>|k_1|\,,
|v_1-v_2|\sim|u_1-v_2|\sim1
\label{antun}$$ where $k_2\to0$ and $\delta<<1$. In this case $v_2$ is exterior to $C_{u_1}$ and to $C_{v_1}$ while $u_2$ lies inside $C_{u_1}$. Other constraints (\[sbound\]) also do not prevent the above configuration. In parallel with (\[antun\]), the boundary of (\[ndgbound\]) is reached for only two limited points, say $u_1$ and $u_2$, to go to each other. In this case $|u_1-u_2|
\sim\sqrt{|k_2|}$, all the other distances being of order of the unity. Both regions originate discontinuities. To demonstrate this, we discuss the discontinuities in the energy invariant $s=-(p_1+p_2)^2=-(p_1+p_2)^2$ for the two-loop scattering amplitude. We ignore specifics due to the Grassmann moduli that is not of very importance for the matter discussed. We fix the complex coordinates $z_1$, $z_2$ and $z_3$ of the vertices to be exterior to the Schottky circles. The integration is performed over the Schottky parameters and over $z_4=z$, certain group limiting points being assumed to go to $z_3$. In addition, $z$ is assumed nearby one of Schottky circles. If the circle is not nearby $z_3$, then $z$ is changed by a relevant Schottky group transformation moving $z$ to the interior of a circle to be nearby $z_3$. First, we consider the configuration $$|u_1-z_3|\sim|u_2-z_3|\sim
min(|k_1|^{1/2-\delta}\,,|k_2|^{1/2-\delta})\,,\quad
|v_1-z_3|\sim|v_2-z_3|\sim|v_1-v_2|\sim1
\label{crclm}$$ where $\delta$ being positive, goes to zero. In this case the Schottky circles have no a common interior. We define variables $y_1\sim y\sim1$ to be given by $(u_1-z_3)= y_1(u_2-u_1)$ and $(z-u_1)=y(u_2-u_1)$. The integration over the “small” variables being taken, the discontinuity is represented by an integral over four complex $y_1$, $y$, $(v_1-z_3)$ and $(v_2-z_3)$. The integral just corresponds to the unitarity equation. Indeed, this unitarity equation is by-linear in $2\to3$ tree amplitudes, every amplitude being the integral over two complex variables (an example is given in Appendix A). More discontinuities would appear, if $z_3$ is allowed to penetrate into the Schottky circles. For instance, due to configuration where $z_3$ being interior to $C_{v_1}$, $g_1(z_3)$, is, simultaneously, to be exterior to all the Schottky circles. And $C_{u_1}$ along with $C_{u_2}$ both go to $g_1(z_3)$. In this case the unitarity equations would be broken. Hence constraints (\[zbound\]) for the fixed vertex coordinates hold the unitarity equations.
For the configuration (\[antun\]), there are only $(u_1-z_3)$ and $(z-z_3)$ to be of the same order magnitudes. So only two variables of order the unity can be constructed. They are $(v_1-z_3)$ and $(\tilde z-z_3)$ where $z-z_3=(\tilde
z-z_3)(u_1-z_3)$. Once the integration over the small variables being performed, the discontinuity is given by the integral over $v_1\sim1$ and $\tilde z\sim1$, which is not a product of two $2\to3$ tree amplitudes. Moreover (see Appendix A), for the tachyon-tachyon scattering amplitude from the boson string theory the configuration (\[antun\]) originates the false threshold at $s=6m_{th}^2$ where $m_{th}^2=-8$ is the square of the tachyon mass. A like discontinuity appears, if $u_2$ and $v_2$ in (\[antun\]) both are the $\{\tilde G\}$ image of the limiting points of a forming transformation (instead of to be the limiting points of the forming one). Along with (\[antun\]), the discussed configuration must be removed from the integration region.
By above, the unitarity equations are saturated due to the region when the Schottky circles of the forming transformations do not overlap each other. But this region being no a fundamental region for the symmetry group, can not be the total integration region. The relevant region for the integration one seems the ${\cal G}_n$ region where all the limiting group points of $\{\tilde G\}$ lay inside the Schottky circles of the forming transformation (\[gamab\]) and outside the common interior of any pair of the circles above. Indeed, we demonstrate that for the integrals of the group covariant expression over the boundary of the region, the boundary can be replaced by pieces, which are congruent to each other under the $\{\tilde G\}$ group. Hence the group invariant integral over the region discussed is the same under those infinitesimal variations of the circles, which are related by the corresponding group transformation (\[gamab\]). In addition, the interior of the region does not contain the points related by the $\{\tilde G\}$ transformations. So the region possesses properties of the fundamental region desired. We demonstrate a range, which the boundary of the region can be varied within.
For this purpose we consider the boundary “1” with $u_2=u_2^{(1)}$ laying on $C_{u_1}$, and the boundary “2” including the $u_2=u_2^{(2)}$ point on $C_{v_1}$. The $u_2^{(2)}$ point is obtained by the $\Gamma_{b,1}^{-1}(l_{21})$ transformation (\[gamab\]) of $u_2^{(1)}$. On the boundary “1” the remaining variables cover a domain ${\cal C}[u]$ and, respectively, over ${\cal C}[v]$. We add ${\cal C}[u]$ by the ${\cal C}^{(1)}[v]$ domain, which is the image of ${\cal C}[v]$ under that $\tilde G=\tilde G_2$ transformation. The transformation only changes both limiting points $U_2=(u_2|\mu_2)$ and $V_2=(v_2|\nu_2)$ by $\Gamma_{b,1}(l_{21})$. Respectively, we add ${\cal C}[v]$ by the ${\cal C}^{(-1)}[u]$ domain to be the image of ${\cal C}[u]$ under the inverse transformation $\tilde G_2^{-1}$. Due to the $\tilde G$ symmetry, the integral over ${\cal
C}[v]+{\cal C}[u]$ differs only by the factor $1/2$ from the integral over ${\cal C}'[v]+ {\cal C}'[u]$ where ${\cal
C}'[v]={\cal C}[v]+{\cal C}^{(-1)}[u]$ and ${\cal C}'[u]={\cal
C}[u]+{\cal C}^{(1)}[v]$. So we can replace ${\cal C}[u]$ by ${\cal C}'[u]$ and ${\cal C}[v]$ by ${\cal C}'[v]$ with dividing the integral by 2. By construction, the boundaries ${\cal
C}'[u]$ and ${\cal C}'[v]$ are just transformed to each other under the above transformation $\tilde G_2$, as it required for the fundamental region boundaries. The kindred consideration can be performed for those boundaries, which include the group limiting points obtained by a $\tilde G$ change of the forming set. So the boundary of the region can be represented by pieces, which are congruent to each other under the $\{\tilde
G\}$ group. Thus the boundary integral is nullified, the integral over ${\cal G}_n$ the region discussed is the same under those infinitesimal variations of the circles, which are related by group transformations.
To clarify a range, which the boundary of the region can be varied within, we consider the genus-2 configuration (\[crclm\]) with $|k_2|<|k_1|$. Then the ${\cal
C}[u]$ region restricts $v_2$ to lay between $C_{u_1}$ and some closed curve ${\ell}_u$ rounding $C_{u_1}$. The above curve is determined by (\[ndgbound\]). Respectively, ${\cal C}[v]$ restricts $v_2$ to lay between $C_{v_1}$ and a closed curve ${\ell}_v$ around $C_{u_1}$, the curve being determined by (\[ndgbound\]). When $u_2^{(1)}$ penetrates into $C_{u_1}$, the $u_2^{(2)}$ point goes away from $C_{v_1}$. The penetration of $u_2^{(1)}$ into $C_{u_1}$ is allowed until $u_2^{(2)}$ meets ${\ell}_v$. At this moment, an integration region over $v_2$ can not be deformed continuously that gives a natural restriction for the penetration $u_2^{(1)}$ into $C_{u_1}$. One can see that configuration (\[antun\]) is not reached. The considered example holds that the ${\cal G}_n$ region is relevant for the integration one.
The limiting points of any super-Schottky group transformation $G$ are obtained by the action $G^n$ at $n\to\pm\infty$ on an arbitrary point including the limiting points of the transformations (\[gamab\]) to be among them. So, in the ${\cal G}_n$ region all the group limiting points lay outside the overlapping of Schottky circles assigned to the forming transformations. Using (\[suprel\]), one shows that, if the leftmost of the transformation is a positive (negative) power of $\Gamma_{b,s}(l_{2s})$, the attractive limiting point lies inside the $\hat C_{u_s}$ circle[^7] (the $\hat C_{v_s}$ one). The attractive (repulsive) limiting point of $\Gamma_{b,s}(l_{2s})$ lies exterior to the $\hat
C_{u_G}$ circle[^8] of any group transformation $G$ having the leftmost no to be a positive (negative) power of $\Gamma_{b,s}(l_{2s})$. So, we take $\tilde
B^{(n)}(\{q,\overline q\})$ in (\[ampl\]) as $$\begin{aligned}
\tilde
B^{(n)}(\{q,\overline q\})= \prod_{s=1}^n
\Biggl(\prod_{G\in\{G_s\}}
\theta(|\tilde\ell_s
(U_G)|^2-1)
\theta(1-|\tilde\ell_s
(\Gamma_s[U_G])|^2)\Biggl)
\nonumber\\
\times
\Biggl(\prod_{G\in\{G_s'\}}
\theta(1-|\tilde\ell_s
(U_G)|^2)\Biggl)
\prod_{G\in\{G_s''\}}
\theta(|\tilde\ell_s
(\Gamma_s[U_G])|^2-1)
\label{bunit}\end{aligned}$$ where $\tilde\ell_s(t)$ is circle (\[hcirc\]) or (\[hcirc1\]), or $\tilde\ell_s(t)$ is obtained by a continuous deformation of the circle within the region allowed by (\[sbound\]). In this case $\Gamma_s[U_G]$ denotes the $\Gamma_{b,s}(l_{2s})$ transformation (\[gamab\]) of $U_G$. Furthermore, $\{G_s\}$ is formed by those group products of $\Gamma_{b,r}(l_{2r})$, whose leftmost is not a power of $\Gamma_{b,s}(l_{2s})$. Leftmost of any transformation from the $\{G_s'\}$ set is a positive power of $\Gamma_{b,s}(l_{2s})$. Every $\{G_s''\}$ transformation has leftmost to be a negative power of $\Gamma_{b,s}(l_{2s})$. The sets include the given transformation along with its inverse one. Hence in (\[bunit\]) only the attractive limiting points $U_G$ present. They are defined by the condition that $G^n[t]\to
U_G$ at $n\to\infty$. In place of (\[bunit\]), one can use any region obtained through a replacement of a part of (\[bunit\]) by the part congruent to it under the $\{\tilde
G\}$ group transformation.
In the considered region the radius of the Schottky circle of any group transformation $G$ is finite, if its limiting points lay at a finite distance from limiting points of the forming group transformation. Then (\[minors\]) forbids the multiplier $|k_G|\to1$. Indeed, choosing relevant $F$ in (\[minors\]), one obtains that $\omega_{GG}^{(0)}(\{q\})\geq1$ where $\omega_{GG}^{(0)}$ is given by (\[omega0\]) for $j=r=G$. Furthermore, $(u_G-v_G)\to0$ at $k_G\to1$, as far as the radius of the circles assigned to $G$ is finite. Thus the sum in (\[omega0\]) for the matrix element discussed is nullified forcing $k_G$ to be small. The limiting points going to infinity can be moved to the same finite point by a relevant $L(2)$ transformation that does not change the sum (\[omega0\]). So the sum vanishes, the multiplier being small again. For any integer $\tilde n$, the multiplier $k_G\to\exp[2\pi/\tilde n]$ is too excluded since the multiplier of $G^{\tilde n}$ goes to the unity that is forbidden by above. As far as $\tilde n$ can be arbitrary large, $|k_G|\to1$ is excluded at all. Along with the $k\to0$, the $k\to 1$ region contributes to the unitarity equations. So it is naturally that this region is a copy of the region where $k\to0$.
When all the multipliers are small, one see from (\[omega0\]) that $$2\pi i\omega_{jj}^{(0)}(\{q\})\approx \ln k_j
+2\sum\limits_{s\neq j}\frac{k_s(u_s-v_s)^2(u_j-v_j)^2}
{(u_j-u_s)(v_j-u_s)(u_j-v_s)(v_j-v_s)}+\dots
\label{omkzer}$$ Only the leading term being considered, constraints (\[minors\]) require $|k_j|\leq1/230$. In this case Schottky circles (\[circ\]) of the same transformation to be separated from each other. The leading correction is most when the boundary of (\[circlim\]) is achieved. For $n$ limiting points to be closed to each other, the correction is roughly $\sim n|k|d_{uv}/\hat d \sim n\sqrt{|k|}$. Here $\hat d$ is a characteristic distance between closed limiting points of distinct forming transformations while $d_{uv}$ is a characteristic size for the $|u_s-v_s|$ distances. For the genera being not much high, the correction is small. In this case the Schottky circles $C_{u_s}$ and $C_{v_s}$ have no the common interior. When the genus is increased, $\hat d$ grows that might reduce the correction. In the general case we have not the estimation for the correction term, but it seems natural to expect that constraints (\[sbound\]) and (\[bunit\]) always forbid for $C_{u_s}$ and $C_{v_s}$ to be overlapped.
Constraints (\[sbound\]) and (\[bunit\]) along with (\[zbound\]) fully determine the integration region in (\[ampl\]). One can also use any region obtained by a replacement of an arbitrary part of the given region by a part congruent under the group ${\cal R}$ of the local symmetries of the amplitude. Subtle details due to divergences in the particular spin structure are discussed in Section 7. The $\{N_0\}$ set being changed, the integral is reduced to the initial form due to due to the $SL(2)$ symmetry, the symmetry under the $\{\tilde G\}$ group and the symmetry under the super-Schottky group transformations of any given interaction vertex coordinate. So the amplitude is independent of the choice of the $\{N_0\}$ set. The local symmetries are employed in the following calculation of the integrals over the degenerated configurations, but details of the integration region will not be important for this purpose.
Green functions and the integration measures
============================================
In [@danphr; @dannph] the Green function $R_L^{(n)}(t,t';\{q\})$, the period matrix and the scalar functions have been obtained in terms of genus-1 functions. Now we represent them through the genus-$n_i$ functions where $\sum_i n_i=n$ and $n_i\geq1$. In this case $n$ handles are divided into groups of $n_i$ ones, every group being given by the set $\{q\}_i$ of the super-Schottky group parameters and by its super-spin structure $L_i$ assumed to be even. So $\{q\}=\{\{q\}_i\}$ and $L=\{L_i\}$. Using the obtained formulas, we derive convenient expressions of the above quantities for degenerated configurations mentioned in the Introduction. In the following Sections these expressions will be applied to the calculation of integrals over the degenerated configurations of interest.
Along with $R_L^{(n)}(t,t';\{q\})$ of Section 2, we consider $K_L^{(n)}(t,t';\{q\})$ defined to be $$K_L^{(n)}(t,t';\{q\})=D(t')R_L^{(n)}(t,t';\{q\})
\label{kr}$$ where the spinor derivative is defined by (\[supder\]). Furthermore, we build (see also [@danphr]) a matrix operator $\hat K=\{\hat K_{sr}\}$ where $\hat K_{sr}$ is an integral operator vanishing at $s=r$. For $s\neq r$, the kernel of $\hat K_{sr}$ is $\tilde
K_{L_s}^{(n_s)}(t,t';\{q\}_s)dt'$. Here $\tilde
K_{L_s}^{(n_s)}(t,t';\{q\}_s)$ is related by (\[kr\]) with $\tilde R_{L_s}^{(n_s)}(t,t';\{q\}_s)$, which is the non-singular part (\[lim\]) of the Green function. So $$K_{L_s}^{(n_s)}(t,t';\{q\}_s)=
\frac{\vartheta-\vartheta'}
{z-z'}+\tilde K_{L_s}^{(n_s)}(t,t';\{q\}_s)\,.
\label{polk}$$ Like [@danphr], we define kernels together with the differential $dt'=dz'd\vartheta'/2\pi i$. The discussed operator performs the integration with $\tilde
K_{L_s}^{(n_s)}(t,t';\{q\}_s)$ over $t'$ along $C_r$-contour, which surrounds the limiting points associated with the considered group $r$ of the handles and the cuts between limiting points for the Ramond handles. The desired relation for the Green function is $$\begin{aligned}
R_L^{(n)}(t,t';\{q\})=\ln(z-z'-\vartheta\vartheta')+
\sum_{s}\tilde
R_{L_s}^{(n_s)}(t,t';\{q\}_s)
\nonumber\\
+\sum_{r,s}\int_{C_s}[(1-\hat K)^{-1}\hat
K]_{rs}(t,t_1)dt_1\tilde R_{L_r}^{(n_r)}(t_1,t';\{q\}_r)
\label{rprt}\end{aligned}$$ where $[(1-\hat K)^{-1}\hat K]_{rs}(t,t_1)dt_1$ is the kernel of the operator $$(1-\hat K)^{-1}\hat K=
\hat K+\hat K\hat K+\dots
\label{opr}$$ In the Neveu-Schwarz sector where Green function (\[kr\]) has the poles solely [@vec; @pst; @dan90], the integrals in (\[rprt\]) are calculated without difficulties. In this case, using for Green functions (\[polk\]) Poincar[é]{} series [@vec; @pst; @dan90], one reproduces the series for $R_L^{(n)}(t,t';\{q\})$. In fact, to prove (\[rprt\]), one needs only check that (\[rprt\]) satisfies to (\[rtrans\]). In particular, to verify (\[rtrans\]) under the transformations assigned to the $r$-th group of the handles, we represent (\[rprt\]) as $$\begin{aligned}
R_L^{(n)}(t,t';\{q\})=
R_{L_r}^{(n_r)}(t,t';\{q\}_r)+
\sum_{s\neq r}\int_{C_s}
K_{L_r}^{(n_r)}(t,t_1;\{q\}_r)
\tilde R_{L_s}^{(n_s)}(t_1,t';\{q\}_s)dt_1
\nonumber\\
+\sum_{p\neq r}\sum_{s}
\int_{C_p}K_{L_r}^{(n_r)}(t,t_1;\{q\}_r)dt_1\int_{C_s}
[(1-\hat K)^{-1}\hat
K]_{ps}(t_1,t_2)dt_2 \tilde R_{L_s}^{(n_s)}(t,t';\{q\}_s)
\label{rpart}\end{aligned}$$ where $R_{L_r}^{(n_r)}(t,t';\{q\}_r)$ and $K_{L_r}^{(n_r)}(t,t_1;\{q\}_r)$ are total Green functions including the singular term in (\[lim\]) and (\[polk\]). Indeed, once, using (\[opr\]), one calculates the contribution to (\[rpart\]) of the pole term in (\[polk\]), eq.(\[rprt\]) appears. Relations (\[rtrans\]) for the transformations of the $r$-th group discussed are evidently satisfied for (\[rpart\]), the scalar function $J_{j_r}^{(n)}(t;\{q\};L)$ associated with the $j_r$ handle of the $r$-th supermanifold being $$\begin{aligned}
J_{j_r}^{(n)}(t;\{q\};L)=J_{j_r}^{(n_r)}(t;\{q\}_r;L_r)+
\sum_{s\neq r}\int_{C_s}D(t_1)
J_{j_r}^{(n_r)}(t_1;\{q\}_r;L_r)dt_1
\tilde R_{L_s}^{(n_s)}(t_1,t;\{q\}_s)
\nonumber\\
+\sum_{p\neq r}\sum_{s}
\int_{C_p}D(t_1)
J_{j_r}^{(n_r)}(t_1;\{q\}_r;L_r)dt_1\int_{C_s}
[(1-\hat K)^{-1}\hat
K]_{ps}(t_1,t_2)dt_2 \tilde R_{L_s}^{(n_s)}(t_2,t;\{q\}_s)\,.
\label{tjr}\end{aligned}$$ Hence (\[rpart\]) and (\[rprt\]) both are the correct expressions for $R_L^{(n)}(t,t'\{q\})$. The period matrix is calculated from (\[tjr\]). For this purpose one considers the difference $J_{j_r}^{(n)}(t;\{q\};L)-J_{j_r}^{(n)}(t_0;\{q\};L)$ where $t_0$ is a fixed parameter. The above difference is given by $$\begin{aligned}
J_{j_r}^{(n)}(t;\{q\};L)-J_{j_r}^{(n)}(t_0;\{q\};L)=
J_{j_r}^{(n_r)}(t;t_0\{q\}_r;L_r)
+\int_{C_s'}D(t_1)J_{j_r}^{(n_r)}
(t_1;\{q\}_r;L_r)
dt_1
\nonumber\\
\times
R_{L_s}^{(n_s)}(t_1,t;t_0;\{q\}_s)
\nonumber\\
+\sum_{p\neq r}
\int_{C_p}D(t_1)
J_{j_r}^{(n_r)}(t_1;\{q\}_r;L_r)dt_1\int_{C_s'}
[(1-\hat K)^{-1}\hat
K]_{ps}(t_1,t_2)dt_2R_{L_s}^{(n_s)}(t_2,t;t_0;\{q\}_s)
\label{jar}\end{aligned}$$ where both $z$ and $z_0$ lay inside the $C_s'$ contour while $$\begin{aligned}
R_{L_s}^{(n_s)}(t_1,t;t_0;\{q\}_s)=
R_{L_s}^{(n_s)}(t_1,t;\{q\}_s)-
R_{L_s}^{(n_s)}(t_1,t_0;\{q\}_s)\,,
\nonumber\\
J_{j_r}^{(n_r)}(t;t_0\{q\}_r;L_r)=
J_{j_r}^{(n_r)}(t;\{q\}_r;L_r)
-J_{j_r}^{(n_r)}(t_0;\{q\}_r;L_r)
\label{jar1}\end{aligned}$$ where $R_{L_s}^{(n_s)}(t_1,t;\{q\}_s)$ is the total Green function (\[lim\]) including the singular term. To prove (\[jar\]), one, using (\[opr\]), calculates the contribution from the $\ln[(z_2-z-\vartheta_2\vartheta)/
(z_2-z_0-\vartheta_2\vartheta_0)]$ term due to the singularity of the Green function. The corresponding integral is transformed to the one along the cut between $z_2=z-\vartheta_2\vartheta$ and $z_2=z_0-\vartheta_2\vartheta_0$. Then it is found to be $$\int
D(t_2)f(t_2)[\theta(z_2-z-\vartheta_2\vartheta)-
(z_2-z_0-\vartheta_2\vartheta_0)]dz_2d\vartheta_2=
f(t)-f(t_0)
\label{calom}$$ where $f(t)$ denotes either the Green function, or $J_{j_r}^{(n_r)}(t_1;\{q\}_r;L_r)$. As the results, one obtains (\[tjr\]). From (\[jar\]), the $\omega_{j_rj_s}^{(n)}(\{q\};L)$ element of the period matrix is found to be $$\begin{aligned}
2\pi
i\omega_{j_rj_s}^{(n)}(\{q\};L)=\delta_{j_rj_s}\ln k_{j_r}
+(1-\delta_{j_rj_s})\int_{C_s}D(t)
J_{j_r}^{(n_r)}(t;\{q\}_r;L_r)dt
J_{j_s}^{(n_s)}(t;\{q\}_s;L_s)
\nonumber\\
+\sum_{p}\int_{C_p}D(t)
J_{j_r}^{(n_r)}(t;\{q\}_r;L_r)dt\int_{C_s}[(1-\hat K)^{-1}\hat
K]_{pr}(t,t')dt'
J_{j_s}^{(n_s)}(t';\{q\}_s;L_s)\,.
\label{omjr}\end{aligned}$$ For the odd super-spin structure, due to the spinor zero mode, there is no the Green function satisfying (\[rtrans\]) and, at the same time, having no non-physical poles. In this case further terms need to be added in (\[rprt\]) providing true properties of $R_L^{(n)}(t,t';\{q\})$. In particular, these terms appear in the genus-$n$ Green function given in terms of the genus-1 functions when $L$ includes genus-1 odd spin structures, see Appendix B of the present paper.
The integration measure is given by (\[hol\]) and (\[zinv\]) with $\hat
Z_L^{(n)}(\{q\})$ to be [@danphr] $$\hat
Z_L^{(n)}(\{q\})= \tilde Z^{(n)}(\{q\},L)\prod_{s=1}^n
\frac{ Z^{(1)}(k_s;l_{1s},l_{2s})}{k_s^{(3-2l_{1s})/2}}
\label{zhol}$$ where the $(l_{1s},l_{2s})$ theta characteristics are either 0, or 1/2, while [@danphr] $$Z^{(1)}(k;l_{1},l_{2})=(-1)^{2l_{1s}+2l_{2s}-1}16^{2l_{1s}}
\prod_{p=1}^\infty
\frac{[1+(-1)^{2l_2}k^pk^{(2l_1-1)/2}]^8}{[1-k^p]^8}\,.
\label{z1h}$$ The expression of $\tilde Z_L^{(n)}(\{q\})$ through the genus-$n_i$ functions is derived using Appendix C along with the Green functions given in the Appendix B. When all the $L_i$ super-spin structures are even, the desired expression is given by (see Appendix C for more details) $$\ln\hat Z_L^{(n)}(\{q\}))=\sum_{i}\ln
\hat Z_{L_i}^{(n_i)}(\{q\}_i))-5trace\ln(I-\hat K) +
trace\ln(I-\hat G)
\label{ipf}$$ where $\hat K$ is the same as in (\[rprt\]). The $\hat G$ operator is constructed similar to $\hat K$, the non-singular part $\tilde G_{L_s}^{(n_s)}(t,t';\{q\}_s)$ of the $G_{L}^{(n)}(t,t';\{q\})$ ghost superfield Green function [@danphr] being employed instead of $\tilde
K_{L_s}^{(n_s)}(t,t';\{q\}_s)$. In this case $$G_{L}^{(n)}(t,t';\{q\})= \frac{\vartheta-\vartheta'}{z-z'}
+\tilde G_{L_s}^{(n_s)}(t,t';\{q\}_s)
\label{greg}$$ where the last term on the right side has no singularity at $z=z'$. The Green function is a superconformal 3/2-tensor under transformations (\[stens\]) of $t'$, but it is not a superconformal (-1) tensor under the transformations of $t$. Indeed, in the last case the Green function receives additional terms being the sum of a polynomials in $t$ multiplied by a relevant superconformal 3/2-tensor in $t'$ (see eq.(63) in [@danphr] and Appendix C of the present paper).
The obtained expressions can be applied to the degenerated configurations where all the limiting points of $n_i$ forming group transformations go to the same point. As the basic case, we consider $n_1<n$ forming group ones, the limiting points going to $z_0$. We say that they form the degenerated $n_1$ configuration, its super-spin structure being $L_1$. The remaining $n_2=n-n_1$ handles form the $n_2$ configuration, its super-spin structure being $L_2$. We assume no more than one interaction vertex to be near $z_0$. As was noted in the Introduction, the considered configuration is the main one, which might originate divergences in the group limiting points of the amplitude integral (\[ampl\]).
By Section 4, the Schottky multipliers are not closely to the unity in their absolute values. Moreover, the $\tilde G$ symmetry bounds $z_0$ to lay exterior to Schottky circles assigned to the $n_2$ configuration. When $z_0$ lies at a finite distances from the group limiting points of the $n_2$ configuration, we imply that $\rho_1<<\rho$. In this case $\rho_1$ is the maximal size of the $n_1$ configuration while $\rho$ is the minimal distance between $z_0$ and any point of essence assigned to the $n_2$ one. So $\rho\leq\rho_2$ where $\rho_2$ is the maximal size of the $n_2$ configuration. If $z_0\to\infty$, then $\rho_1\leq\rho$ and $\rho>>\rho_2$. This case is, however, reduced to the finite $z_0$ case by a relevant $L(2)$ transformation. Hence, for brevity, we discuss finite $z_0$. The super-spin structures $L_1$ and $L_2$ are taken to be even. We obtain for this configuration the leading approximated integration measure, vacuum correlator, scalar functions and period matrix, as well as the leading corrections for the above quantities. To clarify the method, we present a more detailed calculation of the leading corrections for the holomorphic partition function (\[ipf\]). In this case the sum on the right side of (\[ipf\]) gives the factorized partition function while two rest terms are corrections. In particular, the correction due to the second term is given by $$J_{cor}^n=-trace\ln(I-\hat K)=
\int_{C_I}\Biggl(\int_{ C_{II}}\tilde
K_{L_1}^{(n_1)}(t_1,t_2)dt_2 \tilde
K_{L_2}^{(n_2)}(t_2,t_1)\Biggr) dt_1+\dots
\label{kcor}$$ where $dt=dzd\vartheta/2\pi$. In this case $\tilde
K_{L_1}^{(n_1)}(t,t')$ and $\tilde K_{L_2}^{(n_2)}(t,t')$ are defined by (\[kr\]) on the genus-$n_1$ supermanifold and, respectively, on the genus-$n_2$ one. The $C_I$ contour bounds the domain occupied by the degenerated handles while the $C_{II}$ contour bounds the domain of the rest handles. To calculate the first term on the right side of (\[kcor\]), the $C_I$ contour is gone on a distance $\sim\rho$ from the degenerated handles. Thus $\tilde K_{L_1}^{(n_1)}(t,t')$ can be approximated by its asymptotics, $z$ and $z'$ both being far from $z_0$. Due to (\[kr\]), the above asymptotics is related with the asymptotics of $\tilde R_L^{(n_1)}(t,t')$ in (\[lim\]), which are given by $$\tilde R_L^{(n)}(t,t';\{q\})\approx
\frac{\tilde a_L^{(n)}(\{q\})+
\tilde\alpha_L^{(n)}(\{q\})(\vartheta+\vartheta')}{(z-z_0)
(z'-z_0)}
-\frac{\tilde b_L^{(n)}(\{q\})\vartheta\vartheta'}
{(z-z_0)(z'-z_0)}
\Biggl[\frac{1}
{z-z_0}-\frac{1}
{z'-z_0}\Biggl]\,.
\label{asr}$$ We assign a smallness $\sim\sqrt{\rho_1}$ to each of the integrated Grassmann parameters of the degenerated $n_1$ configuration. Along with the estimation $\sim1/\sqrt{\rho_1}$ for its differential, it correctly determines the magnitude of the integral over the variables discussed. Hence $$\tilde a_L^{(n)}(\{q\})\sim\rho_1^2\,,\qquad
\tilde b_L^{(n)}(\{q\})\sim\rho_1^2\,,\qquad
\tilde\alpha_L^{(n)}(\{q\})\sim\rho_1\sqrt{\rho_1}
\label{coef}$$ The asymptotics of $K_{L_1}^{(n_1)}(t,t')$ is given by (\[kr\]) and (\[asr\]) at $n=n_1$ and $L=L_1$. The integrals over $z_1$ and $z_2$ are calculated by the Cauchy theorem upon going the contours to surround the poles on the right side of (\[asr\]). Thus $$J_{cor}^n \approx [\tilde a_{L_1}^{(n_1)}(\{q\}_1)
\partial_z\partial_{z'}-
2\tilde b_{L_1}^{(n_1)}(\{q\}_1)
\partial_ z\partial_{\vartheta}
\partial_{\vartheta'}
-2\tilde\alpha_{L_1}^{(n_1)}(\{q\}_1)
\partial_z\partial_{\vartheta'}]
\tilde
R_{L_2}^{(n_2)}(t,t';\{q\}_2)
\label{rescor}$$ where $z=z'=z_0$ and $\vartheta=\vartheta'=0$. The indices “1” and “2” are assigned to the corresponding configuration. Due to (\[coef\]), the terms in (\[rescor\]) are $\sim
\rho_1^2\rho_2^2/\rho^4$ and $\sim
\rho_1\sqrt{\rho_1}\rho_2\sqrt{\rho_2}/\rho^3$. The last term in (\[ipf\]) is calculated in the same fashion using Appendix C. Being no more than $\sim
\rho_1^3\sqrt{\rho_1}\rho_2^3\sqrt{\rho_2}/\rho^7$, it can be neglected. Thus $$Z_L^{(n)}(\{q\}))\approx Z_{L_1}^{(n_1)}(\{q\}_1))
Z_{L_2}^{(n_2)}(\{q\}_2))[1+5J_{cor}^n]
\label{ipfas}$$ where $J_{cor}^n$ is given by (\[rescor\]). Other corrections are calculated in the kindred manner. In particular, from (\[omjr\]), the corrections for the $\omega_{j_2j_2'}^{(n)}(\{q\};L)$ period matrix elements are also proportional to the coefficients in (\[asr\]). Here $j_2$ labels the cycles assigned to the $n_2$ configuration. Really we shall see in Section 6 that corrections proportional to the coefficients in (\[asr\]) are too small to originate divergences. More large corrections are determined by the asymptotics at $z\to\infty$ of the scalar function $$J_r^{(n)}(t;\{q\};L)\approx
\frac{\hat S_r^{(n)}(\{q\};L)+
\hat\Sigma_r^{(n)}(\{q\};L)\vartheta}{z-z_0}
\equiv \frac{\hat J_r^{(n)}(\{q\};L;\vartheta)}{z-z_0}\,.
\label{asj}$$ In this case $\hat S_r^{(n_1)}(\{q\}_1;L_1)
\sim\rho_1$ and $\hat\Sigma_r^{(n_1)}(\{q\}_1;L_1)\sim\sqrt\rho_1$. Below $j_1$ is reserved for the cycles assigned to the degenerated $n_1$ configurations. Corrections for $\omega_{j_1j_1'}^{(n)}(\{q\};L)$ are quadratic in the coefficients of (\[asj\]) since the sole $\sim\rho_1$ term $\sim\hat\Sigma_{j_1}^{(n_1)}(\{q\}_1;L_1)
\hat\Sigma_{j_1'}^{(n_1)}(\{q\}_1;L_1)$ vanishes. Indeed, the above term is multiplied by $\partial_\vartheta
\partial_{\vartheta'}\tilde
R_{L_2}^{(n_2)}(t,t';\{q\}_2)$ at $z=z'$, which is nullified due to the bose symmetry of the Green function. So $$\begin{aligned}
\omega_{j_1j_1'}^{(n)}(\{q\};L)\approx
\omega_{j_1j_1'}^{(n_1)}(\{q_1\};L_1)\,,\qquad
\omega_{j_2j_2'}^{(n)}(\{q\};L)\approx
\omega_{j_2j_2'}^{(n_2)}(\{q_2\};L_2)\,,
\nonumber\\
2\pi
i\omega_{j_1j_2}^{(n)}(\{q\};L)\approx-\int
\hat J_{j_1}^{(n_1)}(\{q\}_1;L_1;\vartheta_1)
d\vartheta_1
D(t_1)J_{j_2}^{(n_2)}(t_1;\{q\}_2;L_2)
\label{omas}\end{aligned}$$ where the integral is calculated at $z_1=z_0$, for further definitions see (\[asj\]). The Green functions and the scalar functions for $|z-z_0|>>\rho_1$ and $|z'-z_0|>>\rho_1$ are found to be $$\begin{aligned}
J_{j_2}^{(n)}(t;\{q\};L)\approx
J_{j_2}^{(n_2)}(t;\{q\}_2;L_2)\,,\quad
R_L^{(n)}(t,t';\{q\})\approx R_{L_2}^{(n_2)}(t,t';\{q\}_2)\,,
\nonumber\\
J_{j_1}^{(n)}(t;\{q\};L)\approx-\int
\hat J_{j_1}^{(n_1)}(\{q\}_1;L_1;\vartheta_1)
d\vartheta_1 D(t_1)
R_{L_2}^{(n_2)}(t_1,t;\{q\}_2)
\label{scal}\end{aligned}$$ where $z_1=z_0$. The indices “1” and “2” mark to the corresponding configurations. When $|z-z_0|\sim\rho_1$ and $|z'-z_0|>>\rho_1$ the correction for the Green function is calculated through the asymptotics of $\tilde
R_L^{(n)}(t,t';\{q\})$ at $z'\to\infty$. The above asymptotics is given by $$\tilde R_L^{(n)}(t,t';\{q\})\approx
\frac{\tilde S^{(n)}(t;\{q\};L)+
\tilde\Sigma^{(n)}(t;\{q\},L)\vartheta'}{(z'-z_0)}
\equiv
\frac{\tilde E_L^{(n)}(t;\{q\};\vartheta')}{(z'-z_0)}\,.
\label{asymr}$$ In more details, the coefficients are represented as $$\begin{aligned}
\tilde S^{(n)}(t;\{q\},L)=
\vartheta\tilde\Sigma_1^{(n)}(z;\{q\};L)+
\tilde S_1^{(n)}(t;\{q\},L)\,,
\nonumber\\
\tilde\Sigma^{(n)}(t;\{q\};L)=\vartheta\tilde S_2^{(n)}
(z;\{q\};L)+
\tilde\Sigma_2^{(n)}(z;\{q\};L);
\label{cfnt}\end{aligned}$$ where the “tilde” terms determine the asymptotics of the “tilde” Green function in (\[lim\]). By dimensional reasons, $\tilde S_2^{(n_1)}(z;\{q\}_1;L_1)\sim1$, $\tilde\Sigma_1^{(n_1)}(z;\{q\}_1;L_1)\sim
\tilde\Sigma_2^{(n_1)}(z;\{q\}_1;L_1)
\sim\sqrt{\rho_1}$ and $\tilde
S_1^{(n_1)}(t;\{q\}_1;L_1)\sim\rho_1$. Further, at $z\to\infty$ the functions discussed are related with the coefficients in (\[asr\]) by $(z-z_0)\tilde
S_1^{(n)}(t;\{q\},L)\to\tilde a_L^{(n)}(\{q\})$, $(z-z_0)^2\tilde S_2^{(n)}(t;\{q\},L)\to\tilde
b_L^{(n)}(\{q\})$, $(z-z_0)\tilde\Sigma_1^{(n)}(t;\{q\},L)\to\tilde
\alpha_L^{(n)}(\{q\})$ and $\tilde\Sigma_2^{(n)}(t;\{q\},L)\to
\tilde\Sigma_1^{(n)}(t;\{q\},L)$. In the case discussed, $$R_L^{(n)}(t,t';\{q\})=
R_{L_2}^{(n_2)}(t_0,t';\{q\}_2)
-\int E_{L_1}^{(n_1)}(t;\{q\}_1;\tilde\vartheta)
d\tilde\vartheta
D(\tilde t)R_{L_2}^{(n_2)}(\tilde t,t';\{q\}_2)
+\dots
\label{sgras}$$ where $\tilde z=z_0$, $t_0=(z_0|0)$ and, in addition, $$\begin{aligned}
E_{L_1}^{(n_1)}(t;\{q\}_1;\tilde\vartheta)=
S^{(n_1)}(t;\{q\}_1;L_1)+\Sigma^{(n_1)}(t;\{q\}_1;L_1)
\tilde\vartheta\,,\quad
S^{(n_1)}(t;\{q\}_1;L_1)=
\nonumber\\
-(z-z_0)+\tilde
S^{(n_1)}(t;\{q\}_1;L_1)\,,\quad
\Sigma^{(n_1)}(t;\{q\}_1;L_1)=\vartheta
+\tilde\Sigma^{(n_1)}(t;\{q\}_1;L_1)
\label{etild}\end{aligned}$$ with definitions given in (\[asymr\]) and in (\[cfnt\]). The sum $[\ln(z'-z_0)+
E_{L_1}^{(n_1)}(t;\{q\}_1;\tilde\vartheta)]$ gives the asymptotics at $\tilde z\to\infty$ of the Green function $R_{L_1}^{(n_1)}(t,\tilde t;\{q\}_1)$ related with $\tilde R_{L_1}^{(n_1)}(t,\tilde t;\{q\}_1)$ by (\[lim\]). The scalar functions at $|z-z_0|\sim\rho_1$ are given by $$\begin{aligned}
J_{j_1}^{(n)}(t;\{q\};L)=
J_{j_1}^{(n_1)}(t;\{q\}_1;L_1)
- \int
\hat J_{j_1}^{(n_1)}(\{q\}_1;L_1;\tilde\vartheta)
d\tilde\vartheta D(\tilde t)
\tilde R_{L_2}^{(n_2)}(\tilde t,t;\{q\}_2)+\dots\,,
\nonumber\\
J_{j_2}^{(n)}(t;\{q\};L)=
J_{j_2}^{(n_2)}(t_0;\{q\}_2;L_2)-
\int
E_{L_1}^{(n_1)}(t;\{q\}_1;\tilde\vartheta)
d\tilde\vartheta D(\tilde t)
J_{j_2}^{(n_2)}(\tilde t;\{q\}_2;L_2)\dots
\label{nscal}\end{aligned}$$ with the definitions given in (\[sgras\]). The correction $\sim \hat\Sigma_{j_1}^{(n_1)}(\{q\}_1;L_1)
\tilde\Sigma_{j_1}^{(n_1)}(\{q\}_1;L_1)\sim\rho_1$ in the first line is absent because, being proportional to $\partial_\vartheta\partial_{\vartheta'}\tilde
R_{L_2}^{(n_2)}(t,t';\{q\}_2)$ at $z=z'$, it is nullified. The vacuum correlator (\[corr\]) is given through ${\cal X}_{L_1,L_1'}^{(n_1)}(t,\overline
t;\vartheta_1;\{q,\bar q\}_1)$ defined to be $$\begin{aligned}
{\cal X}_{L_1,L_1'}^{(n_1)}(t,\overline
t;\vartheta_1;\{q,\bar q\}_1)=
E_{L_1}^{(n_1)}(t;\{q\}_1;\vartheta_1)+
[J_s^{(n_1)}(t;\{q\}_1;L_1)+
\overline{J_s^{(n_1)}(t;\{q\}_1;L_1')}]
\nonumber\\
\times
[\Omega_{L_1,L_1'}^{(n_1)}(\{q,\bar q \}_1)]_{sr}^{-1}
\hat
J_r^{(n_1)}(\{q\}_1;L_1;\vartheta_1)
\equiv X_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)+\Xi_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)\vartheta_1\,;
\nonumber\\
X_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)=-(z-z_0)+
\tilde X_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)\,,
\nonumber\\
\Xi_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)=\vartheta+
\tilde\Xi_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)\,.
\label{corhx}\end{aligned}$$ In this case the “tilted” quantities are calculated through the “tilted” ones in (\[etild\]). From (\[corr\]), (\[sgras\]) and (\[nscal\]) the correlator is given by $$\begin{aligned}
\hat X_{L,L'}(t,\overline t;t',\overline
t';\{q\})\approx \hat X_{L_2,L_2'}(t_0,\overline t_0;t',\overline
t';\{q\}_2) -\int[{\cal
X}_{L_1,L_1'}^{(n_1)}(t,\overline
t;\vartheta_1;\{q,\bar q\}_1)d\vartheta_1
\nonumber\\
\times
D(t_1)
\hat X_{L_2,L_2'}(t_1,\overline{t_1};t',\overline
t';\{q\}_2)
\nonumber\\
-\int\overline{
[{\cal
X}_{L_1',L_1}^{(n_1)}(t,\overline
t;\vartheta_1;\{q,\bar q\}_1)}d\bar\vartheta_1
\overline{D( t_1)} \hat
X_{L_2,L_2'}(t_1,\overline{t_1};t',\overline t';\{q\}_2)
+\dots
\label{corl}\end{aligned}$$ where $t_0=(z_0|0)$, $t_1=(z_0|\vartheta_1)$ and the “dots” encode the terms at $t=t_0$ due to the leading corrections for the $(j_2,j_2')$ matrix elements of (\[grom\]) and for the scalar functions carrying the $j_1$ index. The above terms are easy found using (\[grom\]), (\[omas\]) and (\[nscal\]). From (\[rescor\]) and (\[omas\]), the integration measure is mainly factorized. For $n_1=1$, the genus-1 factor $$Z_{L_1,L_1'}^{(1)}(\{q,\overline
q\}_1)=Z_{tore}(k,\bar k;L_1,L_1')|u-v-\mu\nu|^{-2}
\label{torus}$$ differs from the integration measure $Z_{tore}(k,\bar
k;L_1,L_1')$ on the torus [@gsw] by the $|u-v-\mu\nu|^{-2}$ multiplier, which is due to the integration over Killing genus-1 modes. Furthermore, only even $L_1$ super-spin structures might originate the divergences. Otherwise the integrand is not singular, see Appendix C. Moreover, the contribution to (\[ampl\]) of odd super-spin structures is obtained by the factorization of the relevant even super-spin ones, see the end of Appendix C. So, it is sufficient to check the cancellation of divergences for the even super-spin structures. In doing so we use eqs. (\[rescor\]) and (\[ipfas\]) for the holomorphic partition function, expression (\[omas\]) for the period matrix, eqs. (\[scal\]) and (\[nscal\]) for the scalar functions, eq. (\[sgras\]) for the Green function and eq.(\[corl\]) for the vacuum correlator.
Integrals of the superstring theory
===================================
The amplitude (\[ampl\]) includes Grassmann integrations along with the ordinary ones. In this case the result is finite or divergent depending on the used integration variables, as this is seen for an easy integral $$I_{(ex)}=\int\frac{dxdyd\alpha
d\beta d\bar\alpha d\bar\beta}
{|z-\alpha\beta|^p}\theta(1-|z|^2)
\label{examp}$$ where $z=x+iy$ while $\alpha$ and $\beta$ are complex Grassmann variables. The complex parameter $p$ characterizes the strength of the singularity. Integrals of this kind really appear in (\[ampl\]). In particular, from (\[zinv\]), the integration measure contains singularity (\[examp\]) with $p=2$. The kindred expression is originated by the singularity (\[lim\]) at $z=z'$ of the Green function. In this case $p=s_{jl}/4+2$ where $s_{jl}=-(p_j+p_l)^2$ is the square center mass energy in the given reaction channel and the add 2 is due to the vacuum contractions of the fields in front of the exponential in (\[vert\]). For the sake of simplicity, we bound the integration region in (\[examp\]) by $|z|^2\leq1$. Once the Grassmann integrations being performed, one obtains the integral $$I_{(ex)}=p^2\int\frac{dxdy} {4|z|^{p+2}}\theta(1-|z|^2)\,,
\label{exam}$$ which is divergent at $z=0$, if $Re\,p>0$. On the other side, in (\[examp\]) one can turn to the variable $\tilde
z=z-\alpha\beta$. Then the Grassmann variables will present only in the step function $\theta(|\tilde z+\alpha\beta|^2)$. After the integration over the Grassmann variables, the integral is turned to the boundary integral at $|z|^2=1$. Thus for any $p$ the result of the integration is finite being $$I_{(ex)}=-\int\frac{d\tilde
xd\tilde yd\alpha d\beta d\bar\alpha d\bar\beta} {|\tilde
z|^p}\alpha\beta\bar\alpha\bar\beta \left[\delta(|\tilde
z|^2-1)+|\tilde z|^2\frac{d\delta(|\tilde z|^2-1)}{d|\tilde
z|^2}\right]=-\frac{\pi p}{2}\,.
\label{ex}$$ So (\[examp\]) depends on the integration variables, at least for $Re\,p>0$. For $Re\,p<0$ when (\[exam\]) is convergent, (\[exam\]) and (\[ex\]) both give the same result. One could, however, change the integration variable $z$ by $z+\sum_{i=1}^N\delta_i\delta_i^{(1)}$ where $\delta_i$ and $\delta_i^{(1)}$ are arbitrary Grassmann numbers. When $p$ is not an negative even number, the resulted integrand has the singularity $\sim|z|^{-(p+2+2N)}$. So, for $2N>-p$, the integral is divergent. For a negative even $p$ the integral can be reduced to the singular integral by a change $z=\tilde
z+\alpha\beta$ with the following $|\tilde z|=|\hat z|^{p_1}$ and $\arg \tilde z= \arg\hat z$ where $p_1>0$ is no integer or half-integer, the strength of the resulted singularity being $(p_1p-1)$. This integral can be transformed to the divergent one. The result is, however, the same, if the integration variable change remains the integral to be convergent.
Calculating the amplitude, we are guided by preserving its local symmetries. Divergences break the conformal symmetry due to a cutoff parameter, the amplitude depends on $\{N_0\}$ set (\[factorg\]) that falls the theory. In the proposed calculation all the divergences are cancelled and the local symmetries are preserved.
Due to the singularity (\[lim\]) at $z=z'$ of the Green function, the integrand in (\[ampl\]) is singular when $m_1>1$ interaction vertices go the same point on the complex $z$ plane. When the vertices are accompanied by degenerated Schottky circles, singularities present also for the integration measure. If $1<m_1<(m-1)$ for the $m>3$ point amplitude, then the strength of the singularity depends on the energy 10-invariant of the given reaction channel. The integral is calculated [@gsw] for those energies below the reaction threshold, where it is convergent. The result is analytically continued to energies above the threshold. In this case the amplitude receives singularities required by the unitarity equations (as an example, see Appendix A). Due to the energy-momentum conservation, there is no a domain for the energy 10-invariants where the integrals over all the singular regions are convergent simultaneously. Hence the amplitude is obtained by the summing of the pieces obtained by the analytical continuation from the distinct regions of the 10-invariants. For instance, the calculation of the scattering amplitude includes an analytical continuation from the relevant energy region of the integrals over nodal regions. Each of the integrals gives rise to the cut in $s$, $t$ and $u$-Mandelstam invariant. Every region contains a pair of the corresponding vertices going to each other (see the discussion of the unitarity equations in Section 4 and in Appendix A). The analytical continuation procedure discussed is evidently consistent with the local symmetries of the amplitude.
Moreover, the amplitudes for the emission of a longitudinal gauge boson are nullified as it is required. Indeed, from (\[vert\]), in this case the integrated function is the super-derivative in $t_j$ of a local function of the $t_j$ coordinate assigned to the longitudinal boson discussed. When $z_j$ and certain other vertex coordinates both go to the same point, the integral is calculated for those energy invariants, which it is convergent for. Hence the integral is reduced to the integral over boundary of a singular region discussed. The boundary integral is analytically continued to considered energies. Having no singularity discussed, the boundary integral is independent from which energy region it was be continued. Hence the boundary integrals being collected together, are canceled.
By aforesaid, the divergences of the $m$-point amplitude may appear only when a number $m_1$ of the vertices at the same point is $m_1=0$, $m_1=1$ and $m_1\geq(m-1)$. The $m_1\geq(m-1)$ case is, however, out of the integration region, if $\{N_0\}$ set in (\[ampl\]) is formed by $(3|2)$ ones from the vertex coordinates. So first we consider configurations of degenerated Schottky circles, no more than one vertex being nearby. From (\[hol\]), (\[zinv\]) and (\[zhol\]), the integration measure is singular in Schottky multipliers $k_s$ and in group limiting points, as well.
Due to (\[sbound\]) and (\[bunit\]), the singularity in $k_s$ appears only at $k_s\to0$. In this case, from (\[zhol\]) and (\[z1h\]), the holomorphic integration measure is $\sim
k_s^{-(3-2l_{1s})/2}$. Half-integer powers of $k_s$ at $l_{1s}=0$ are cancelled after the summation over $2l_{2s}=0$ and $2l_{2s}=1$ since the sum is unchanged when $\sqrt
k_s\to-\sqrt k_s$, see Section 3 (for the vacuum amplitude this directly follows from (\[z1h\])). From (\[grom\]) and (\[omjr\]), the non-holomorphic factor in (\[hol\]) is $\sim1/(\ln|k_s|)^5$ at $k_s\to0$. So the integrand (\[ampl\]) at $k_s\to0$ is $\sim 1/[|k_s|^2(\ln|k_s|)^5]$, the integral (\[ampl\]) over small $|k_s|$ being finite as it has been observed in [@vec] for the Neveu-Schwarz sector.
Singularities in the group limiting points are due to the configurations discussed in Section 5 where $n_1$ degenerated handles (carrying the even super-spin structure $L_1$) go to the same point $z_0$. As far as $z_0\to\infty$ is reduced to a finite $z_0$ by a relevant $L(2)$ transformation, we mainly consider finite $z_0$. By aforesaid, we assume no more than one vertex nearby $z_0$. When no to be the vertex, we say this is the vacuum configuration. If the $z_j$ coordinate of the corresponding interaction vertex $j$ goes to $z_0$, we say it is the $j$-th configuration. With our convention for the $\{N_0\}$ set (see Section 2) all the Grassmann module parameters of the $n_1$ configuration are the integration variables while, for the $j$-th configuration, the $\vartheta_j$ Grassmann partner of $z_j$ may be fixed. So we discuss the integral over the group limiting points of the $n_1$ configuration with given $z_0$ and $\vartheta_j$ (for the $j$-th one). We bound from top the size of the $n_1$ configuration by the $\Lambda<<\rho$ cut-off where (see Section 5) $\rho$ is the characteristic distance from $z_0$ to points assigned to the $n_2$ configuration. Due to (\[bunit\]), and (\[zbound\]), the integrand is non-singular until no one of the differences $\tilde v_s=(v_s-u_s)$ is equal to zero. Hence we take the above differences as the integration variables. On equal terms one can also take the super-difference $w_s=\tilde
v_s-\nu_s\mu_s$ instead of $\tilde v_s$. To be detailed, the remaining variables are chosen to be $u_s$, $\mu_s$ and $\nu_s$. When $\{\tilde v_s\}$ (or $\{w_s\}$) are fixed, the integrals over the above variables are non-singular. By dimensional reasons, the result of the integration may, however, be singular at $\tilde v_s\to0$ (or at $w_s\to0$) that originates the divergences for the integral over $\{\tilde
v_s\}$ (or over $\{w_s\}$). All $\tilde v_s$ (or $w_s$) being of the same order $\sim\rho_1\to0$, the singularity is due to the integration over the region in a size $\sim\rho_1$. The easiest way to estimate the integrals is, as has been noted already, to assign the $\sim\sqrt{\rho_1}$ smallness to each one of the integrated Grassmann variable of the degenerated genus-$n_1$ supermanifold. Simultaneously, the differential of the variable is $\sim1/\sqrt{\rho_1}$. So leading terms in (\[cont\]) and (\[jint\]) might originate the divergence $\sim(\rho/\rho_1)^2$ where $\rho>>\rho_1$ characterizes distances from $z_0$ to points associated with the $n_2$ non-degenerated configuration. The corrections terms might originate the divergences $\sim(\rho/\rho_1)$ and $\sim\ln(\rho/\rho_1)$. We propose, however, a calculation, which avoids the divergences. Before we need to discuss in more details the singular configurations of interest, which are the vacuum configuration and the $j$-th one.
For the vacuum configuration the integrand (\[ampl\]) can be represented as $${\cal F}_0^{n_1,n_2}(\{q,\bar q\};\{p_r,\zeta^{(r)}\})=
\sum_{P,\overline P'}O_0^{(n_1)}(P,\overline
P';\{q,\bar q\}_1)Y_{P,\overline
P'}^{(n_2)}
\label{cont}$$ where $O^{(n_1)}(P,\overline P';\{q\}_1)$ is calculated for the degenerated configuration while $Y_{P,\overline P'}^{(n_2)}$ depends only on $z_0$, parameters of the $n_2$ configuration and on characteristics of the interaction states. As above, $\{q\}_1$ is the set of the module variables assigned to the degenerated configuration. The sum over corrections $(P)$ for holomorphic functions and over corrections $(P')$ for the anti-holomorphic ones, includes $P=1$ (and $P'=1$) corresponding to the leading term. Eq.(\[cont\]) follows directly from the calculation of the holomorphic functions in the previous Section. Among other things, the sum in (\[cont\]) includes terms due to the corrections for the boundary (\[sbound\]). For the vacuum configuration (\[cont\]) the leading term $O_0^{(n_1)}(1,\bar 1 ;\{q,\bar
q\}_1)$ is $$O_0^{(n_1)}(1,\bar 1
;\{q,\bar q\}_1)= \sum_{L_1,L_1'}\tilde
Z_{L_1,L_1'}^{(n_1)}(\{q,\overline q\}_1)
\label{lidvac}$$ where zero point function $\tilde Z_{L,L'}^{(n)}(\{q,\overline
q\})$ including step functions (\[sbound\]) and (\[bunit\]) is given by $$\tilde
Z_{L,L'}^{(n)}(\{q,\overline q\})=\frac{g^{2n}}{2^nn!}
Z_{L,L'}^{(n)}(\{q,\overline q\})
\hat B_{L,L'}^{(n)}(\{q,\overline q\})\tilde
B_{L,L'}^{(n)}(\{q,\overline q\})\,,
\label{tilpf}$$ integration measure $Z_{L,L'}^{(n)}(\{q,\overline q\})$ being given by (\[hol\]) and (\[zinv\]). For the $j$-th configuration containing the dilaton emission vertex, the leading term $O^{(n_1)}(DJ,\overline{DJ};1,\bar1;\{q,\bar q\}_1)$ is due to pairing (\[ngcor\]) of fields (\[vert\]) in front of the exponential calculated for the $n_1$ configuration. In this case $$O_j^{(n_1)}(DJ,\overline{DJ};1,\bar1;\{q,\bar q\}_1)=
\sum_{L_1,L_1'}\tilde
Z_{L_1,L_1'}^{(n_1)}(\{q,\overline q\}_1)
\tilde I_{L_1,L_1'}^{(n_1)}(t_j,\bar
t_j;\{q,\bar q\}_1)
B_{L_1,L_1'}^{(n_1)}(t_j,\bar t_j;\{q,\bar q\}_1)
\label{lidj}$$ where $t_j=(z_j|\vartheta_j)$ is the vertex coordinate going to $z_0$, for other definitions see also (\[ngcor\]) and (\[tilpf\]). We define the integrand together with the step factor (\[zbound\]). The integrand of (\[ampl\]) for the $j$-th configuration (including the corrections) is given by $$\begin{aligned}
{\cal F}_{(j)}^{n_1,n_2}(t_j,\bar t_j;\{q,\bar
q\};\{p_r,\zeta^{(r)}\})=
\sum_{P,\bar P'}O_j^{(n_1)}(DJ,\overline{DJ};P,\bar
P';\{q,\bar q\}_1)\tilde
Y_{P,\bar P'}^{(n_2)}(j)
\nonumber\\
+\sum_{P,\bar P'}
\tilde O_j^{(n_1)}(
P,\bar P';\{q,\bar q\}_1)
\tilde Y_{P,\bar P'}^{(n_2)}(j)
\label{jint}\end{aligned}$$ where $t_0=(z_0|0)$. As above, $P$ lists holomorphic corrections, $\bar P'$ lists anti-holomorphic ones, and $\tilde
Y_{P,\bar P'}^{(n_2)}(j)$ depends only on the parameters of the $n_2$ configuration, the interaction particle characteristics and on $z_0$. Every term for the first sum on the right side is proportional to $\tilde
I_{L_1,L_1'}^{(n_1)}(t_j,\bar t_j;\{q,\bar q\}_1)$. So the first sum presents only when the configuration includes the dilaton emission vertex. The remaining terms are included in the second sum. In the second sum the term $P=1$ (and $P'=1$) is absent. When no one of $\{q\}_1$ belongs to the $\{N_0\}$ set, one can take $z_0=z_j$. Otherwise we identify $z_0$ with a group limiting point. In more details, the $(P,\bar P')$ term in (\[cont\]) is represented as $$O_0^{(n_1)}(P,\overline P';\{q,\bar
q\}_1)= \sum_{L_1,L_1'} P(\{q\}_1;L_1)\hat A_{P,P'}
(\{q,\bar q\}_1;L_1,L')
\overline{P'(\{q\}_1;L_1')}
\label{vaccor}$$ where $\hat A_{P,P'}(\{q,\bar q\}_1;L_1,L')$ offers the $SL(2)$ and $\{\tilde G\}$ symmetries while $P(\{q\}_1;L_1)$ is the function of $\{q\}_1$ in (\[asr\]) or in (\[asj\]), or products constructed using the functions above. As it has been noted, $P=1$ or $P'=1$ is assigned to the leading term for the corresponding movers. In a like fashion, the term for the first sum on the right side of (\[jint\]) is given by $$\begin{aligned}
O_j^{(n_1)}(DJ,\overline{DJ};P,\bar
P';\{q,\bar q\}_1)=
\sum_{L_1,L_1'}\sum_{r,s}
B_{L_1,L_1'}^{(n_1)}(t_j,\bar t_j;\{q,\bar q\}_1)
D(t_j)J_r^{(n_1)}(t_j;\{q\}_1;L_1)
\nonumber\\
\times
\tilde A_{P,P'}^{(r,s)}(\{q,\bar q\}_1;L_1,L')
P(t_j;\{q\}_1;L_1)
\overline{D(t_j)J_s^{(n_1)}(t_j;\{q\}_1;L_1')
P'(t_j;\{q\}_1;L_1')}
\label{corj1}\end{aligned}$$ where $\tilde A_{P,P'}^{(r,s)}(\{q,\bar q\}_1;L_1,L')$ has the $SL(2)$ and $\{\tilde G\}$ symmetries. For the second sum one obtains $$\begin{aligned}
\tilde O_j^{(n_1)}( P,\bar P';\{q,\bar
q\}_1)= \sum_{L_1,L_1'} P(t_j;\{q\}_1;L_1) \hat A_{P,P'}^{(j)}
(\{q,\bar q\}_1;L_1,L')
\nonumber\\
\times
\overline{
P'(t_j;\{q\}_1;L_1')}
B_{L_1,L_1'}^{(n_1)}(t_j,\bar t_j;\{q,\bar q\}_1)
\label{corj2}\end{aligned}$$ where $\hat A_{P,P'}^{(j)}(\{q,\bar q\}_1;L_1,L')$ possesses the $SL(2)$ and $\{\tilde G\}$ symmetries. The $P(t_j;\{q\}_1;L_1)$ function in (\[corj1\]) and in (\[corj2\]) is the one of $P(\{q\}_1;L_1)$ in (\[vaccor\]), or one of functions (\[etild\]), or it is a certain product of the functions above. For $P$ and $P'$ in (\[cont\]) and (\[jint\]) we use the same symbol as for the corresponding function on the right side of (\[vaccor\]), (\[corj1\]) or of (\[corj2\]). As an example, $O_0^{(n_1)}(\hat S_r,\bar P';\{q,\bar q\}_1)$ is given by (\[vaccor\]) for $P(\{q\}_1;L_1)=\hat S_r^{(n_1)}(\{q\}_1;L_1)$, which is defined in (\[asj\]). The term with $P=\Xi$ in (\[jint\]) is given by (\[corj1\]) for $P(t_j;\{q\}_1;L_1)=
\tilde\Xi_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)$ where the right side is defined in (\[corhx\]). The term with $P=DX$ is given by (\[corj2\]) for $P(t_j;\{q\}_1;L_1)=D(t_j)
X_{L_1,L_1'}^{(n_1)}(t,\overline t;\{q,\bar
q\}_1)$ where $X$ is defined in (\[corhx\]). And so on. For the following, below we collect terms due to holomorphic and anti-holomorphic corrections each are no less than $\sim\rho_1$ in respect to the leading term. Since $P'$ repeats $P$, only $P$ are listed.
The sum in (\[cont\]) and the first sum in (\[jint\]) include $P=1$, $P=\hat S_r$, $P=\hat\Sigma_r$ and $P=\hat\Sigma_r\Sigma_s$ where $\Sigma_r$ and $S_r$ are defined by (\[asj\]). These terms are due to corrections for the period matrix (\[omas\]) and corrections for the correlator (\[corr\]) when $z$ and $z'$ both are not nearby $z_0$. In (\[jint\]) the discussed terms are due to, in addition, by the “dots” terms in (\[corl\]). Evidently, $O_0^{(n_1)}(\hat\Sigma_r,\overline P';\{q,\bar q\}_1)$ in (\[cont\]) being odd function of $\{\mu,\nu\}$, is nullified after the integration over the Grassmann variables. The first sum in (\[jint\]) contains also terms with $P=X$ and $P=\Xi$ due to corrections in (\[corl\]). Also, it includes the by-linear term with $P=\hat\Sigma_r\Xi$. Generally, in (\[jint\]) we do not assume the integration over $\vartheta_j$, but for the estimation, it is convenient to assign the $\sim\sqrt{\rho_1}$ smallness also to $\vartheta_j$. Finally a proportional to $\vartheta_j$ term must be multiplied by $\sim 1/\sqrt{\rho_1}$. From Section 5, the correction for the pairing (\[ngcor\]) at $t=t_j$ is $\sim D(t_j) {\cal
X}_{L_1,L_1'}^{(n_1)}(t,\overline t;\vartheta_1;\{q,\bar q\}_1)$ times $\overline{D(t_j){\cal X}_{L_1,L_1'}^{(n_1)}(t,\overline
t;\vartheta_2;\{q,\bar q\}_1)}$, see eq.(\[corl\]). Hence $P$ for the second sum in (\[jint\]) runs $P=DX$, $P=D\Xi$, $P=\hat\Sigma_rD\Xi$ and $P=\Xi D\Xi$. In this case $DX$ and $D\Xi$ each denote the super-derivative in respect to $t_j$ of the corresponding function. All the rest holomorphic (anti-holomorphic) corrections are less than $\sim\rho_1$.
For $n_1=1$ the singularity of (\[cont\]) at $\tilde
v_1=v_1-u_1=0$ is canceled locally due to the summation over the spin structures. Indeed, the terms with $P=1$ vanish since, from (\[torus\]), each a term is proportional to the torus partition function, the sum over the torus partition functions being nullified [@gsw]. Terms with $P=\hat S_1$ and $P=\hat\Sigma_1$ disappear for the same reason since the genus-1 scalar function (\[jr1\]) is independent of the spin structure. As far as the singularity is canceled separately for the right movers and for the left ones, spin dependent corrections proportional to the coefficients in (\[asr\]) are not singular. For the integrals over $(\mu_1,\nu_1)$ with $w_1=v_1-u_1-\nu_1\mu_1$ to be given, the leading singularity is canceled already for every spin structure since the leading terms do not depend on $(\mu_1,\nu_1)$, but the cancellation of the non-leading singularity occurs only for the sum over the spin structures.
For the $j$-th configuration one can define boundary (\[zbound\]) of the fundamental region on the complex $z_j$ plane using the “circles” (\[hcirc\]). In this case $\ell_s(t)$ is the same for all spin structures. Then the spin structure independent terms are nullified locally due to the summation over the spin structures. If one uses “circles” (\[hcirc1\]) dependent on the spin structure, the spin structure independent terms are nullified after the integration over module variables (but $\tilde
v_1$). Corrections due to the coefficients in (\[asr\]) are too small to originate singular terms. Further spin structure dependent terms are due to those corrections for the vacuum correlator (\[corl\]), which include the $\tilde E_{L_1}^{(1)}(t;\{q\}_1;\vartheta')$ function (\[asymr\]), see eq.(\[corhx\]). For $n_1=1$ the above function is given by (\[assgr\]) of Appendix B. From (\[assgr\]), the spin structure dependent part of is $\sim(\vartheta-\varepsilon(z))$. At the same time, from (\[jr1\]), it follows that $(\vartheta-\varepsilon(z))D(t)J(t)=0$. So the first sum on the right side of (\[jint\]) is non-singular. From (\[assgr\]), the singularity of the second sum disappears for every spin structure after the integration over the Grassmann variables (with an exception of any one from them) and after the following integration over either $u_1$, or over $z_j$. The integral over $u_1$ can be replaced by the integral over $(z_j-u_1)$ since the integrand depends on $u_1$ solely through $(u_1-z_j)$.
Instead of $\tilde v_1$, one can use the $w_1=(v_1-u_1-\nu_1\mu_1)$ variable. To obtain (\[assgr\]) in the $(u,w,z)$ variables, the partial derivative $(\partial_w)_{(u,z)}$ when $(u,z)$ are fixed, is calculated through $(\partial_z)_{(u,w)}$ with $(u,w)$ are fixed. For this purpose one can use the invariance of (\[assgr\]) under the special $SL(2)$ transformations $$z=\tilde
z+\tilde\vartheta\vartheta_0\,,\quad \vartheta=
\tilde\vartheta-\vartheta_0
\label{trco}$$ where $\vartheta_0$ is a parameter common for all the variables. The above invariance of (\[assgr\]) follows from the invariance under (\[trco\]) of Green function (\[zgrin\]). In this case functions (\[cfnt\]), which determine the corrections, are found to be $$\begin{aligned}
\tilde\Sigma^{(1)}(t;\{q\}_1;L_1)= (\vartheta-\varepsilon(z))
\hat W_1-
w^{-1}\biggl[\hat W_b(\mu-\nu)
-\mu\nu\vartheta\partial_z[(z-u)\hat W_1
+\hat W_b]\biggl]\,,
\nonumber\\
\tilde S^{(1)}(t;\{q\}_1;L_1)=\tilde S_{inv}^{(1)}
(t;\{q\}_1;L_1)
-\tilde\Sigma^{(1)}(t;L_1)\vartheta\,,
\nonumber\\
\tilde S_{inv}^{(1)}(t;L_1)=\hat W_b
-w^{-1}\biggl[[(z-u)\hat W_1-\hat W_2
\hat W_b](\mu-\nu)-
\varepsilon(z)\partial_z \hat W_1\biggl](\vartheta-\mu)
\label{exmpl}\end{aligned}$$ with $\{q\}_1=(u,w,k)$, $\hat W_1\equiv W_1(z,u,w;L_1)$, $\hat W_2\equiv W_2(z,u,w;L_1)$ and $\hat W_b\equiv W_b(z,u,w)$. So $\hat W_b$ does not depend on $L_1$. Both $ \tilde\Sigma^{(1)}(t;\{q\}_1;L_1)$ and $\tilde
S_{inv}^{(1)}(t;\{q\}_1;L_1)$ are invariant under (\[trco\]). For the second sum in (\[jint\]) the singularity at $w_1=0$ disappears for every spin structure after the integration over $u_1$, $\mu_1$ and $\nu_1$ (if we fix $z_j$ and $\vartheta_j$). Really (\[exmpl\]) is required only to verify the vanishing of the term proportional to $D(t_j)\tilde\Sigma^{(1)}(t;\{q\}_1;L_1)$. The rest terms have fermi statistics and, in addition, they are invariant under (\[trco\]) along with step factor (\[zbound\]). After the integration over $\mu_1$ and $\nu_1$ the integrand appears to be the derivative in $u_1$ of the local function (see Appendix D). Thus the singularity disappears after the integration over $u_1$. (Generically, the cancellation of the non-leading singularity in $w$ forces the cancelation of the singularity in $\tilde v$ and vice versa, see the next Section). The leading singularity of the first sum in (\[jint\]) disappears for the same reasons while the non-leading singularities disappear only after the summation over the spin structures. One can define boundary (\[zbound\]) of the fundamental region using either the “circles” (\[hcirc\]), or (\[hcirc1\]). The cancellation of the singular terms occurs for both cases.
In addition, the singularity of (\[jint\]) for $n_1=1$ disappears after the integration over $t_j$, the summation over the spin structures being performed. To see this one transforms the integration variables by (\[tgam\]) reducing $\mu_1$ and $\nu_1$ to zeros. The above change of the variables is correct since the integral is non-singular. The integral vanishes locally in the super-Schottky group parameters due to the known nullification of the 1- and 2-point genus-1 function [@gsw].
Finiteness of the multi-loop superstring amplitudes
===================================================
To clarify the calculation of the integrals for $n_1>1$, first we consider the $n_1=2$ configuration, the group limiting points being $U_s=(u_s|\mu_s)$ and $V_s=(v_s|\nu_s)$. Here $s=1$ and $s=2$ mark the first handle and the 2-nd one. In the vacuum configuration (\[cont\]) we take $z_0=u_2$, while in (\[jint\]) we take $z_0=z_j$ assuming $t_j$ to be fixed. As before, the size of the configuration is restricted from top by a cut-off $\Lambda<<\rho$ where $\rho$ is the characteristic distance from $z_0$ to points associated with the $n_2$ configuration. As has been discussed in the Introduction, we consider the integrals of functions (\[vaccor\]), (\[corj1\]) or (\[corj2\]), every function being the sum over the spin structures of the configuration discussed. The integrals are taken over the group limiting points of the configuration keeping $z_0$ and $\tilde v_2$ or $w_2$ to be fixed. Calculating the integral over the variables of the 1-st handle, we first integrate over $u_1$, $\mu_1$ and $\nu_1$. Then the singularity at $\tilde v_1\to0$ disappears due to vanishing the integrals of the $n_1=1$ functions. Indeed, at $\tilde v_1\to0$, the integrand is given by (\[cont\]) or (\[jint\]) with $n_1=n_2=1$ where $Y_{P,\overline P'}^{(n_2=1)}$ is equal to the corresponding expression among (\[vaccor\]), (\[corj1\]) and (\[corj2\]) calculated for the handle “2”. Therefore, the considered $n_1=2$ integrals are convergent. And they are functions of $\tilde v_2$ or $w_2$. As it was announced in the Introduction, it will be shown that the integrals have no the singularity at $\tilde
v_2=0$ or $w_2=0$. Thus the contribution to the amplitude from the considered configuration is finite since it is just given by the additional integration over $\tilde v_2=0$ (or $w_2=0$) of the integrals considered. Also we show the vanishing of the 0,- 1-, 2- and 3-point genus-2 amplitudes.
For the $j$-th configuration we perform $u_2$-boost of the integration variables. Then the integral over $u_2$ with fixed $t_j$ is turned to the integral over $z_j$ with fixed $u_2=0$ and fixed $\vartheta_j$, as well. So we shall consider the integral with fixed $u_2=0$ and $\vartheta_j$, the integration over $z_j$ being performed. Often, as it explained below, we shall transform these integrals into the integrals with $\vartheta_j$ to be the integration variable.
By the reasons of the previous paragraph, the integral with fixed $\tilde v_2$ (or $w_2$) of the sum over the spin structures of the first handle is convergent even without the summation over the spin structures of the second handle. Moreover, due to the nullification of the integrals of the $Y_{P,\overline
P'}^{(n_2=1)}$ functions associated with the 2-nd handle, the integral of the sum over the spin structures of the second handle is convergent for the particular spin structure of the first handle. Hence the integral of either of the partial spin structure sums discussed is convergent, as it has been announced in the Introduction. By the above reasons, the integral with fixed both $U_2=(u_2|\mu_2)$ and $V_2=(v_2|\nu_2)$ of the sum over the spin structures of the 1-st handle is convergent, as well. If, additionally, the integration over $t_j=(z_j|\vartheta_j)$ is performed, the integral with fixed $U_2$ and $V_2$ of the sum over the spin structures of the 2-nd one is convergent, too. Indeed, in this case the divergence at $\tilde v_1=0$ (or $w_1=0$) is canceled due to the nullification of the 1- and 2-point genus-1 function (see the end of Section 6). The integrals of the discussed partial spin structure sums both to be convergent, the integral of the total spin structure sum admits changes of the integration variables by the spin structure dependent transformations from the group of its local symmetries can be performed (see a discussion of this point in the Introduction).
First we consider the integrals of terms in (\[vaccor\]) and (\[corj1\]) with $P=1$. For convenience, these terms in (\[jint\]) can be additionally integrated over $\vartheta_j$. Indeed, solely the $\sim\vartheta_j$ piece of the integrand contributes to the integral while the remaining part disappears due to the integration over the Grassmann module variables. By aforesaid, the integral of the sum over the spin structures of either of two handles is convergent. Furthermore, the $P=1$ terms in both (\[cont\]) and (\[jint\]) are invariant under transformations (\[trco\]) of the holomorphic variables (the anti-holomorphic ones may be unchanged). Thus for $w_2=v_2-u_2-\nu_2\mu_2$ to be fixed and once the integration over the holomorphic Grassmann variables to be performed, the integrand is transformed to a sum of derivatives in respect to the boson integration variables, see (\[prp\]) in the Appendix D. Hence the corresponding integral is nullified apart from the non-singular at $w_2\to0$ terms originated by the $\Lambda$ cut-off. One could also change the integration variables but $\mu_2$ by transformation (\[trco\]) with $\vartheta_0=\mu_2$ (it remains the same both $w_2$ and $u_2$). Then $\mu_2$ is removed from the integrand (with an exception of the $\Lambda$ cut-off boundary). In this case the singularity at $w_2=0$ vanishes after the integration over $\mu_2$. Both calculations give the same result since the integral is not singular. At the same time, the singular integral of the single spin structure of every handle is divergent or finite depending on the variables used. In particular, the last integral appears to be the finite once the above change the integration variables to be performed. For the integral of the sum over the spin structures of any handle the discussed ambiguity is absent.
When $\tilde v_2$ is fixed instead of $w_2$ the integral of function (\[jint\]) with $P=1$ considered is, for convenience, again additionally integrated over $\vartheta_j$. To see for no to be the singularity at $\tilde v_2=0$, we represent the discussed integral by the integral over $u_1$, $\mu_2$ and $\nu_2$ of the integral ${\cal
A}_{1,P'}^{(2)}(u_1,u_2,\tilde v_2,\mu_2,\nu_2)$ calculated with fixed $u_1$, $u_2$, $\tilde v_2$, $\mu_2$ and $\nu_2$. By reasons given in the third paragraph of this Section, this integral is convergent for the sum over the spin structures of either of two handles. The discussed singularity at $\tilde v_2=0$ might appear solely due to the region where $u_1$, $u_2$ and $v_2$ both are closely to each other since the $n_1=1$ integral appearing when $u_2\to v_2$ and $u_1\neq u_2$, has not the singularity discussed. Hence we assume both $u_1$ and $v_2$ to be fixed closely to $u_2$. In this case we remove the $\Lambda$ cut-off because, owing to (\[bunit\]), the integration $v_1$ variable is bounded inside a small region nearby $u_2$. Then, due to the symmetry of the integrand under $SL(2)$, $\{\tilde G\}_2$ and super-Schottky group transformations, ${\cal
A}_{1,P'}^{(2)}(u_1,u_2,\tilde v_2,\mu_2,\nu_2)$ is related with its magnitude ${\cal A}_{1,P'}^{(2)}(u_1,u_2,\tilde v_2,0,0)$ at $\mu_2=\nu_2=0$ by (the proof is given in the next paragraph) $${\cal
A}_{1,P'}^{(2)}(u_1,u_2,\tilde v_2,\mu_2,\nu_2)=
\biggl(1-\mu_2\nu_2/\tilde v_2
\biggl){\cal
A}_{1,P'}^{(2)}(u_1,u_2,\tilde v_2,0,0)\,.
\label{linvc}$$ For the same integral $\tilde{{\cal
A}}_{1,P'}^{(2)}(u_1,u_2,w_2,\mu_2,\nu_2)$ to be considered as function of $w_2$ (instead of $\tilde v_2$) one obtains that $${\tilde{\cal
A}_{1,P'}}^{(2)}(u_1,u_2,w_2,\mu_2,\nu_2)=
\biggl(1-\mu_2\nu_2/w_2
-\mu_2\nu_2
\partial_{w_2}
\biggl){\cal
A}_{1,P'}^{(2)}(u_1,u_2,w_2,0,0)\,,
\label{lnvc}$$ Eq.(\[lnvc\]) is obtained by substituting $\tilde v_2=w_2-\mu_2\nu_2$ into (\[linvc\]). Integrating (\[lnvc\]) over $u_1$, $\mu_2$ and $\nu_2$, one sees that the absence of the singularity at $w_2=0$ for the integral on the left side (proving in the previous paragraph) forces the vanishing at $w_2=0$ of the integral of ${\cal
A}_{1,P'}^{(2)}(u_1,u_2,w_2,0,0)$ over $u_1$. Besides, from (\[linvc\]) and (\[lnvc\]), the singularity at $\tilde v_2=0$ disappears for the integral of ${\cal
A}_{1,P'}^{(2)}(u_1,u_2,\tilde v_2,\mu_2,\nu_2)$ over $u_1$, $\mu_2$ and $\nu_2$, as it is required. Relation (\[linvc\]) will be proved for the integral of the total spin structure sum. Hence the discussed cancellation of the singularity in $\tilde v_2$ is shown only for the integrals of the sum over all the spin structures of the configuration considered.
To prove (\[linvc\]), we represent the integrand ${\cal O}
(1,\bar P';\{q,\bar q\}_1)$ of the discussed integral as $[{\cal O}(1,\bar P';\{q,\bar
q\})H(U_2,V_2,U_1)]$ times $H^{-1}(U_2,V_2,U_1)$ where $H(U_2,V_2,U_1)$ is the factor (\[factorg\]) for $t_1^{(0)}=U_2$, $t_2^{(0)}=V_2$ and $t_3^{(0)}=U_1$. As before, $U_2=(u_2|\mu_2)$, $V_2=(v_2|\nu_2)$ and $U_1=(u_1|\mu_1)$. Then we change the holomorphic integration variables by the $SL(2)$ transformation (\[trnsf\]) (see Appendix E), which reduces $\mu_2$ and $\nu_2$ to zeros, but it does not change both $u_1$, $u_1$ and $v_2$. Since the expression inside the square brackets is $SL(2)$ co-variant, the transformation remains it the same, but for $\mu_2=\nu_2=0$. Moreover, we obtain that $H^{-1}(U_2,V_2,U_1)$ in front of the square brackets receives only the $[1+\mu_2\nu_2/(u_2-v_2)]$ multiplier. The $\sim\mu_1$ terms in $H^{-1}(U_2,V_2,U_1)$ does not contribute to the integral since $[H(U_2,V_2,U_1) {\cal O}_0^{(2)}(1;\bar
P';\{q,\bar q\})]$ at $\mu_2=\nu_2=0$ is even function of $(\mu_1,\nu_1)$ (after the integration over $\vartheta_j$ the $j$-th configuration). The integration region boundary (\[zbound\]) and (\[bunit\]) is non-invariant under the transformation that originates additional boundary integrals. They, however, cancel each other since, on equal terms, the integration region can be bounded by (\[zbound\]) and (\[bunit\]) taken as for the former variables, so for the resulted ones. Indeed, each of the integration regions is the fundamental region of the local symmetry group of the integral. Hence the integral is the same in both cases. As the result, (\[linvc\]) appears. Really the boundary integrals are canceled to be reduced to each other by the change of the integration variables by means spin dependent transformations of the $\{\tilde G\}_2$ group and of the super-Schottky group (for the $j$-configuration). As it has been discussed, these transformations can be surely performed only for the integrals of the whole spin structure sum. Hence relation (\[linvc\]) is established only for the integral of the total sum over the spin structures of the discussed configuration.
The ${\cal
A}_{1,1}^{(2)}(u_1,u_2,\tilde v_2,0,0)$ integral assigned to function (\[vaccor\]) (for $P=P'=1$) times $|(u_1-u_2)(u_1-v_2)|^2$, is just the genus-2 vacuum amplitude, while the corresponding integral of (\[corj1\]) is the 1-point, genus-2 one. Indeed, every discussed amplitude is given by an integral (\[ampl\]) over $\mu_1$, $\nu_1$ and $v_1$ and their complex conjugated with $u_1$, $u_2$, $v_2$, $\mu_2$ and $\nu_2$ to be fixed. The integral does not depend on the fixed variables due to $SL(2)$, $\tilde G$ and super-Schottky group symmetries. Taking $\mu_2=\nu_2=0$, one just obtains ${\cal
A}_{1,1}^{(2)}(u_1,u_2,\tilde v_2,0,0)$ required. Calculating its limit at $v_2\to u_2$ under the integral sign, one finds that ${\cal A}_{1,1}^{(2)}(u_1,u_2,\tilde v_2=0,0,0)$ is nullified owing to the above discussed properties of the $n_1=1$ functions. Indeed, at $v_2\to u_2$ it is expressed through the $n_1=1$ integrals (see the first paragraph of this Section). The limit under the integral sign is correct since (see Section 4) at $v_2\to u_2$ both $u_2$ and $v_2$ lay exterior to the Schottky circles assigned to the handle “1”. Thus the integral vanishes identically since it is independent of $\tilde v_2$. So the 0- and 1-point amplitudes both are nullified. As it has been discussed above, the independence of the integral on the fixed variables implies a possibility to perform spin dependent transformations $\{\tilde G\}_2$ and, for the 1-point amplitude, the super-Schottky group transformations of the interaction vertex coordinate. These transformations can be surely performed only for the integrals of the whole spin structure sum. Hence the nullification is shown only for the integral of the whole sum over the spin structures. To directly verify the vanishing of the discussed amplitudes $v_2=u_2$ for arbitrary $(\mu_2,\nu_2)$, it requires the consideration of corrections $\sim |\tilde v_2|^2$ for each one of movers, which we did not performed.
The integrals of terms with $P=1$ being non-singular at $w_2=0$ (or $\tilde v_2=0$), corrections might be singular when both $P\neq1$ and $P'\neq1$. In this case only corrections listed in the previous Sections need to be examined. For $\tilde v_2$ to be fixed, the integral differs from the corresponding integral with fixed $w_2$ by the additional term $\mu_2\nu_2\partial_{\tilde v_2}{\cal
O}_0^{(2)}(P, \overline P' ;\{q,\bar q\})$. By the dimensional reasons, the singular part of the integral of ${\cal
O}_0^{(2)}(P, \overline P' ;\{q,\bar q\})$ at $\mu_2=\nu_2=0$ (when $P\neq1$ and $P'\neq1$) is $(\overline
u_2-\overline v_2)^{-1}$ times a non-singular factor. This singularity is week to give the divergence in the amplitude. So it is sufficient to verify the absence of the singularity for only one of $w_2$ and $\tilde v_2$ to be fixed.
The terms of (\[jint\]) with $P=\hat\Sigma_r\hat\Sigma$, $P=\hat\Sigma_r\tilde\Xi$, $P=\hat\Sigma D\Xi$ and $P=\tilde\Xi D\Xi$ (for definitions, see the text just below eq.(\[corj2\])) can be additionally integrated over $\vartheta_j$ since, like the terms with $P=1$, the remaining part disappears due to the integration over the Grassmann module variables. The integrals are invariant under (\[trco\]). Indeed, due to the invariance under (\[trco\]) of $\tilde
R_L^{(n)}(t,t';\{q\})$, both $\tilde\Xi_{L_1,L_1'}^{(n_1)}(t_j,\overline
t_j;\{q,\bar q\}_1)$ and $D(t_j)\Xi_{L_1,L_1'}^{(n_1)}(t_j,\overline
t_j;\{q,\bar q\}_1)$ (see (\[corhx\])) are invariant under (\[trco\]) that, in turn, provides the discussed invariance of the considered terms. Thus the singularity at $w_2=0$ is nullified for the integral of the sum over the spin structures of any one of the two handles, just as for the $P=1$ terms. So, by the previous paragraph, the singularity at $\tilde v_2$ disappears, too. Only the integrals (see the previous Section) linear in functions (\[asj\]) and (\[sgras\]) need to be more examined.
In doing so, for convenience we again transform the integrals of functions (\[jint\]) with fixed $\vartheta_j$ to integrals where $\vartheta_j$ is the integration variable. For $P=X$, $P=DX$ and $P=\hat S_r$ we can additionally integrate over $\vartheta_j$ as far as the $\sim\mu_1\nu_1\mu_2\nu_2$ term of the integrand includes $\vartheta_j$, as well. With $\vartheta_j$ no to be fixed, the terms with $P=\hat\Sigma_r$, $P=D\Xi$ and $P=\Xi$ disappear after the integration over Grassmann variables as far as there is no a piece proportional to all the Grassmann variables including $\vartheta_j$. When $\vartheta_j$ is fixed (this is just assumed), we reduce $\mu_2$ to zero by transformation (\[trco\]) with $\vartheta_0=\mu_2$. In this case we omit the change of the $\Lambda$ cut-off boundary since it does not originate the singularity at $w_2=0$. When $\tilde v_2$ is fixed instead of $w_2$, additional terms appear due to this transformation, but they can be omitted. Indeed, by aforesaid, they are non-singular in $\tilde v_2$. Hence the integrals of terms with $P=\hat\Sigma_r$, $P=D\Xi$ or $P=\Xi$ are turned into the integrals over $\vartheta_j$ with $\mu_2=0$ to be fixed. The integral of the $\vartheta_j$ term in $\Xi$ (defined by (\[corhx\])) is nullified being proportional to a vanishing integral of a term with $P=1$.
For definiteness, we discuss the integrals with $\tilde
v_2$ to be fixed. As above, $u_2$ is fixed, too (see the second paragraph of this Section). Really in this case the third fixed point being $\infty$, presents. Indeed, the integrand contains $P$ to be the asymptotics of the corresponding function. We shall prove that every considered integral ${\cal
A}_{P,P'}(u_2,\tilde v_2,\infty)$ is related to an integral $\tilde{{\cal A}}(u_2,\tilde v_2,u_1^{(0)},0,0; \tilde P,\tilde
P')$ with the fixed $u_1=u_1^{(0)}$, $u_2$, $\tilde v_2$ and $\mu_2=\nu_2=0$ as it follows $${\cal
A}_{P,P'}(u_2,\tilde v_2,\infty)=\frac{1}{|\tilde v_2|^2}
\tilde{{\cal A}}(u_2,\tilde v_2,u_1^{(0)},0,0; \tilde P,\tilde
P')|(u_1^{(0)}-u_2)(u_1^{(0)}-v_2)|^2 +\dots
\label{rel}$$ where the “dots” denote terms, which are non-singular in $\tilde v_2$ at $\tilde v_2=0$. The integrand for $\tilde{{\cal A}}(u_2,\tilde v_2,u_{1^(0)},0,0;
\tilde P,\tilde P')$ contains $D(t_0')P(t_0')$ to be the spinor derivative (\[supder\]) of the $P(t_0')$ function whose asymptotics at $z_0'\to\infty$ is proportional to the $P$ function in (\[vaccor\]), (\[corj1\]) or (\[corj2\]). In the above integral the integration over $t_0'$ is implied. It will be shown that $\tilde{{\cal A}}(u_2,\tilde v_2,u_{1^(0)},0,0;
\tilde P,\tilde P')$ is zero at $\tilde v_2=0$ due to vanishing of the $n_1=1$ integrals discussed in Section 6. Hence the singularity at $\tilde v_2$ disappears for the considered integral ${\cal
A}_{P,P'}(u_2,\tilde v_2,\infty)$. Thus (see the Introduction and the beginning of this Section) the contribution to the amplitude from the considered configuration is finite.
Eq.(\[rel\]) is derived with using a change of the integration variables by the relevant $SL(2)$ transformation. In doing so the integration region (\[zbound\]) and (\[bunit\]) is changed. To reduce it to the former region, the spin structure dependent transformations of the $\{\tilde G\}_2$ group and of the super-Schottky group transformations of $t_j$ (for the $j$-th configuration) are necessary. As it has been discussed above, these spin structure dependent transformations can be surely performed only for the integrals of the total sum over the spin structures. Hence eq.(\[rel\]) is established only for the integrals of the total sum over the spin structures of the $n_1=2$ configuration discussed.
To derive eq.(\[rel\]) we consider the integral ${\cal
A}(u_2,\tilde v_2;\tilde P(t_0'),
\tilde P'(t_0'))$, which is obtained by the replacement $P\to D(t_0')P(t_0')$ and $P'\to D(t_0')P'(t_0')$ in ${\cal A}_{P,P'}(u_2,\tilde v_2,\infty)$. Here $D(t_0')P(t_0')$ is the spinor derivative of a relevant function $P(t_0')$ proportional to $P$ at $z_0'\to\infty$. In particular, the integrals of terms with $P=\hat
S_r$ and $P=\hat\Sigma_r$ are calculated from the integral of term with $P(t_0')=D(t_0')J_r(t_0')$ obtained by the replacing of $P$ in (\[vaccor\]) and (\[corj1\]) by $D(t_0')J_r^{(2)}(t_0';\{q\};L_1)$. The integrals of $P=X$ and $P=\Xi$ are calculated from the integral of terms with $P(t_0')=D(t_0') {\cal X}(t_0')$. In this case the corresponding $P$ in (\[corj1\]) and (\[corj2\]) are replaced by $D(t_0')\hat X_{L_1,L_1'}^{(2)}(t_j,\overline
t_j;t_0',\bar t_0';\{q,\bar q\}_1)$ where the vacuum correlator $\hat X_{L_1,L_1'}^{(2)}(t_j,\overline t_j;t_0',\bar
t_0';\{q,\bar q\}_1)$ is given by (\[corr\]). Correspondingly, the integrals of terms with $P=DX$ and $P=D\Xi$ are calculated from the integral containing $P(t_0')=DD(t_0'){\cal X}(t_0')$. In doing so the integrals of terms with $P=\hat S_r$, $P=X$ and $P=DX$ are calculated as the $z_0'\to\infty$ limit of the corresponding integral multiplied by $(z_0'-u_2)(z_0'-v_2)$, the integration over $\vartheta_0'$ being performed. And the integral of terms with $P=\hat\Sigma_r$, $P=\tilde\Xi$ or $P=D\Xi$ is the $z_0'\to\infty$ limit of the corresponding integral with $\mu_2=\vartheta_0'=0$ multiplied by $(z_0'-v_2)$. We want to relate the above integrals to the integrals with $u_1$ to be fixed instead of $z_0'$. AS the first step, we calculate the considered integrals in terms of the integrals with fixed $\mu_2=\nu_2=0$ and with the same $u_2$, $v_1$ and $z_0'$ (in the last integrals the integration over $\vartheta_0'$ is performed). Then we use the relevant $L(2)$ transformation $g(z)$ to fix $u_1$ instead of $z_0'$. To do the first step above, we represent ${\cal
A}(u_2,\tilde v_2;\tilde P(t_0'),
\tilde P'(t_0'))$ by the integral over either $(\mu_2,\nu_2)$, or $\nu_2$ (with $\mu_2=0$) of the ${\cal
A}(u_2,\tilde v_2,\mu_2,\nu_2;\tilde P(t_0'),
\tilde P'(t_0'))$ integral with $(\mu_2,\nu_2)$ to be fixed. By the third paragraph of this Section, all the integrals of the sum over the spin structures either of the handles are convergent. Once we calculate the $(\mu_2,\nu_2)$ dependence of the integrals, the desired ${\cal
A}(u_2,\tilde v_2;\tilde P(t_0'), \tilde P'(t_0'))$ integrals appear to be given though the ${\cal A}(u_2,\tilde
v_2,\mu_2,\nu_2;\tilde P(t_0'), \tilde P'(t_0'))$ integral with $\mu_2=\nu_2=0$, as it is required. To calculate the $(\mu_2,\nu_2)$ dependence of the ${\cal A}(u_2,\tilde
v_2,\mu_2,\nu_2;\tilde P(t_0'), \tilde P'(t_0'))$ integral, we change the integration variables by transformation (\[trnsf\]) (see Appendix E), which does not change $u_2$, $v_1$ and $z_0'$. In doing so we proceed like the deriving of (\[lnvc\]).
The considered transformation (\[trnsf\]) to be performed for the term including $P(t_0')=D(t_0')\hat
X_{L_1,L_1'}^{(2)}(t_j,\overline t_j;t_0',\bar t_0';\{q,\bar
q\}_1)$, the integral receives the add since the vacuum correlator (\[corr\]) is, generally, not invariant under $SL(2)$ transformations. Really, the above correlator receives two additional terms, every term being dependent on only one of the points. For arbitrary Grassmann parameters these terms are rather tremendous. To avoid the direct calculation of the additional terms, we replaces the integral by the integral of the difference $$\begin{aligned}
\Delta_{L_1,L_1'} (t_j,\overline
t_j;t_0',\bar t_0';\{q,\bar q\}_1)= D(t_0'){\cal
X}_{L_1,L_1'}^{(2)}(t_j,\overline t_j;t_0';\{q,\bar q\}_1)
\nonumber\\
-\biggl[D(t_0'){\cal X}_{L_1,L_1'}^{(2)}(t_j,\overline t_j;
t_0';\{q,\bar q\}_1)\biggl]_{t_j=t_0',\bar t_j=\bar t_0'}
\label{difx}\end{aligned}$$ where the correlator at the same point is defined as usually, see Section 2. Indeed, being at $\tilde z\to\infty$ smaller than $\sim\rho_1$, the last term of the difference does not originate the singularity at $\tilde v_2=0$ in the integral. The add to (\[difx\]) under the $SL(2)$ change is due to only the singular term due to is absent for the correlator at the same points. So the add to (\[difx\]) is $D(t_0')\ln Q(t_0')$ where $Q(t_0')$ is given by (\[supder\]) for the transformation (\[trnsf\]) considered. Being independent of the module variables, this addition term originates the $P=1$ integral, which vanishes. Finally, the correlator at the same point $t_0'$ can be omitted since it is not contribute to the singular term. Hence all the specifics due to the non-invariance of the vacuum correlator can be neglected.
As before, the $\Lambda$ cut-off boundary does not contributes to the singularity due to the vanishing of the integrals of the $n_1=1$ functions. Thus the singular part of the integral associated with $P=\hat S_r$, $P=X$ or $P=DX$ is, like the $P=1$ case, related by (\[lnvc\]) with the corresponding integral at $\mu_2=\nu_2=0$. To calculate ${\cal
A}(u_2,\tilde v_2\tilde P(t_0'),
\tilde P'(t_0'))$ desired, one integrates the above relation over $(\mu_2,\nu_2)$. Thus ${\cal
A}(u_2,\tilde v_2\tilde P(t_0'),
\tilde P'(t_0'))$ appears to be ${\cal A}(u_2,\tilde v_2,0,0;\tilde P(t_0'),
\tilde P'(t_0'))$ times $1/|v_2|^2$. For the integrals with $\vartheta_0'=\mu_2=0$ we reduce $\nu_2$ to zero and $\vartheta_0'$ to $\vartheta'$. In this case the integration over $\nu_2$ is replaced by the integration over $\vartheta'$ and, simultaneously, the integral receives factor $(z_0'-u_2)/(u_2-v_2)$. In any case the singular at $v_2=0$ part of the desired ${\cal A}_{P,P'}(u_2,\tilde
v_2,\infty)$ integral in (\[rel\]) is the limit at $z_0'\to\infty$ of the integral over $\vartheta'$ of ${\cal A}(u_2,\tilde v_2,0,0;\tilde P(t_0'),
\tilde P'(t_0'))$ times $|(z_0'-u_2)(z_0'-v_2)|^2/|v_2|^2$. The $(z_0'-u_2)(z_0'-v_2)$ factor is just the factor (\[factorg\]) for fixed $z_0'$, $u_2$, $v_2$ and $\mu_2=\nu_2=0$. Furthermore, we can remove the cut-off and, simultaneously, to restrict the integration region by the $B_0^{(2)}(t_0',\bar t_0';\{q,\bar
q\}_1)$ step factor (\[zbound\]). Indeed, like the $P=1$ case, the integral may be non-vanishing at $\tilde v_2=0$ solely due to the region where $u_1$, $v_1$ and $v_2$ both go to $u_2$. The resulted integral with the above constraint of the integration region is convergent. Truly, when either $\tilde v_1\to0$, or $u_1$ and $v_1$ both go to the infinity, the singular part of the integrand disappears since it is proportional to the vanishing integral of $n_1=1$ function assigned to the handle “2”. When $z_j\to z_0'$, the integral is again convergent due to the vanishing of the $P=1$ integrals. We fix $u_1=u_1^{(0)}$ (nearby $u_2$) by a relevant $L(2)$ transformation $g(z)$ of the integration variables. Then $z_0'$ turns into $z'$, and the integration over $u_1$ is replaced by the integration over $z'$. Once the $L(2)$ transformation being performed, the $(z_0'-u_2)(z_0'-v_2)$ factor in the integral is replaced by $(u_1^{(0)}-u_2)(u_1^{(0)}-v_2)$, and eq.(\[rel\]) appears.
Except of the integral of the term with either $P'(t_0')$, or $P(t_0')$ to be $D(t_0'){\cal X}(t_0')$, the integrand for the right side integral in (\[rel\]) is obtained by the $P\to
P(t_0')$ replacement in the integrand for the left side one. Taking the $\tilde v_2\to0$ limit, one can see that the right side integral vanishes at $\tilde v_2=0$ due to the nullification of the integrals of the $n_1=1$ function associated with $(u_2,v_2)$ limiting points. The integral is invariant under $SL(2)$ and $\tilde G$ transformations as well the super-Schottky transformations of the interaction vertex coordinate. Hence it vanishes identically in $\tilde v_2$.
The integral of the term with $P(t_0')=D(t_0'){\cal X}(t_0')$ is non-invariant under the considered transformation $g(z)=(az+b)/(cz+d)$. Truly, it receives the additional term $-D(t')\hat
X_{L_1,L_1'}^{(2)}(t(\infty),\overline t(\infty);t',\bar
t';\{q,\bar q\}_1)$, which appears due to the corresponding addition to the vacuum correlator, see Appendix E. In this case $t'=(z'|\vartheta')$ and $t(\infty)=(z(\infty)|\vartheta(\infty))$ with $z(\infty)=-d/c$ and $\vartheta(\infty)=0$. The discussed $g(z)$ transformation has $u_2$ and $v_2$ to be the limiting points, and the multiplier $k$ to be $k=(z_0'-u_2)(z'-v_2)/(z_0'-v_2)(z'-u_2)$. So $z(\infty)=
z'+(z'-u_2)(z'-v_2)/(z_0'-z')$. The integration region is changed under the $g(z)$ transformation. To calculate the right side integral in (\[rel\]) at $\tilde v_2=0$ by taking the $\tilde
v_2=0$ limit under the integral sign, the integration region needs to be is reduced to the (\[zbound\]) and (\[bunit\]). In doing so $t(\infty)$ is changed. Nevertheless, since the addition term discussed is invariant under the super-Schottky transformations of $t(\infty)$, one can replace it by its super-Schottky group image laying interior to region (\[zbound\]). In this case the desired $\tilde v_2=0$ limit under the integral sign can be performed. Moreover, the discussed additional term is not singular at $\tilde v_2\to0$. Thus the integral is nullified at $\tilde v_2\to0$ due the nullification of the $n_1=1$ function, just as the integrals discussed above. Since considered integral is not $SL(2)$-invariant, it does not vanish identically, unlike the integrals considered before. In any case, from (\[rel\]) the desired integrals of (\[cont\]) and of (\[jint\]) have no the singularity at $\tilde v_2\to0$. So the contribution to the amplitude of the discussed $n_1=2$ configuration is finite.
The nullification of the 2- and 3-point amplitudes is verified as for the 0- and 1-point ones discussed above. Every amplitude is given by the corresponding integral (\[ampl\]) of the 2- or 3-point function. The integral is taken over $\mu_1$, $\nu_1$ and $v_1$ and their complex conjugated, both $\mu_e$, $\nu_2$, $u_1$, $u_2$ and $v_2$ being fixed. The integral of the sum over the spin structures of either of two handles is convergent for the given spin structure of the remaining handle. In this case the integral of the spin structure sum of the 2-nd handle is convergent due properties 1-, 2- and 3- point genus-1 functions. The 3-point genus-1 function are examined just as the $n_1=1$ functions in the previous Section. Since the desired amplitude are independent of the fixed parameters (the total sum over the spin structures is implied), they can be calculated at $\mu_2=\nu_2=0$. It allows to avoid a large number of the corrections $\sim |\tilde v_2|^2$ in each one of movers. The amplitude is nullified at $v_2\to u_2$ due to vanishing the integrals of $n_1=1$ functions associated with the handle “2”. Being independent of the fixed variables, the considered amplitude is zero for any $v_2$.
For general $n_1$, the cancellation of the divergences can be derived by the mathematical induction. Assuming the $n_1'$ integrals to be convergent for all $n_1'<n_1$, one verifies the cancellation of the singularity at $\tilde v_{n_1}=0$ or $w_{n_1}=0$ for the integrals over variables of the remaining $(n_1-1)$ handles. In doing so the consideration like given for the $n_1=2$ case, is performed. In particular, it is verified that the integrals of the spin structures of the $(n_1-1)$ handles are convergent for every spin structure of the $n_1$-th handle. And the integrals of the sum over the spin structures of the $n_1$-th handle are convergent for every spin structure of the remaining $(n_1-1)$ handles. As for $n_1=2$, the integrals with $P(t_0')$ instead of $P$ are considered. Like the $n_1=2$ case, the total cancellation of the singularity is verified for the whole sum over the spin structures. Step-by-step, the nullification of the 0-, 1, 2- and 3-point amplitudes is verified.
Due to $L(2)$ symmetry, the cancellation of the divergences for the finite $z_0$ forces the same for $z_0\to\infty$. This case could also be considered like the finite $z_0$ case. When the $\{N_0\}$ set in (\[ampl\]) for the $m>3$ point amplitude is formed by the limiting group points, further singular configurations appear to be with either $(m-1)$, or $m$ vertices go to the same point. In this case the leading approximated integrand is proportional either to the 1- function or, respectively, to the 0-point one. By aforesaid, the above 0- and1-point integrals are nullified along with the integrals due to the leading corrections. Hence the considered configurations originate no divergences in the amplitude.
So, the divergences do not appear when one integrate, step-by-step, the sum over the spin structures of the given handle. over its limiting points. For the considered handle, the integration over the difference $\tilde
v$ or over the super-difference $w$ between the limiting points is performed in the last turn. In this case the $SL(2)$ symmetry is preserved along with the $\{\tilde G\}$ symmetry and with the symmetry under super-Schottky group changes of every particular interaction vertex coordinate. These spin structure dependent transformations can be performed due to the convergence of the integrals over $n_1>1$ configurations of the above discussed partial sums over the spin structures.
The total group of the local symmetries of the amplitude contains, in addition, the modular group. Naively, the modular symmetry is provided due in this case to terms of the sum over the spin structures in amplitude (\[ampl\]) are correctly transformed into each other [@dannph]. So the invariance under the $\sqrt k_s\to-\sqrt k_s$ change ($k_s$ is the super-Schottky group multiplier) is evident since the integration variables are not touched. For non-zero Grassmann moduli, all the other modular transformations (including the addition $\pm2\pi$ to each of the remaining arguments in (\[argum\])) are, however, accompanied by the spin structure dependent change [@dannph] of the integration variables. The integral of the single spin structure over the $n_1$ configurations being divergent, both a possibility to perform the modular transformations and the modular invariance need an argumentation. Like $\{\tilde G\}$ and super-Schottky group transformations. the modular transformations can be performed due to the convergence of the integrals of the above discussed partial sums over the spin structures. And the modular symmetry is preserved. The argumentation is outlined below.
The transition functions for modular transformations depend on the super-spin structure by terms proportional to Grassmann super-Schottky group parameters [@dannph]. The discussed terms are calculated [@dannph] from the set of integral equations, the kernels being given through ghost Green functions for zero module parameters. The integration is performed along contours where every contour rounds the Schottky circles of the given handle together with the cut between the circles. For the $n_1$ configuration the leading approximated transition functions coincide with the transition functions of the relevant modular transformation of the genus-$n_1$. And the remaining $n_2=n-n_1$ configuration is changed by the relevant transition functions of the modular genus-$n_2$ transformation. The corrections for the transition functions are due to corrections for the ghost Green functions, which are calculated either using the expressions [@dannph], or using representation of the Green functions in terms of the genus-1 ones at zero Grassmann module parameters, see Appendix C of the present paper. In the last case the method of Section 4 can be employed. For the $n_2$ configuration, the spin structure dependent corrections are no larger than $\sim\rho_1^2$ for the Grassmann transition function and no larger than $\sim\rho_1^3$ for the boson transition one, $\rho_1\to0$ (for definitions, see the previous Section). So, they are negligible.
For the $n_1=1$ case ($\tilde v\to0$) the leading approximated transition functions are independent of the spin structure. The spin structure dependent corrections for the Grassmann transition function are not larger than $\sim|\tilde v|^3$, and they are not more than $\sim|\tilde
v|^4$ for the boson transition one. Before the modular transformation to be performed, we can previously cut with below the integration region $\tilde v\to0$ by some cut-off $\tilde\rho$ and then to perform the required change of the variables. At $\tilde\rho\to0$ the additional terms due to spin structure dependent corrections are nullified since the integral of these terms is found to be convergent. So the resulted integral coincides with the former one. For $n_1=2$ we, as before, consider the integral of the given spin structure of the first handle over its variables, $U_2$ and $V_2$ being fixed. We transform the variables of the 1-st handle for the sum over the spin structures of the second one. Then we transform the variables of the 2-nd handle for the integral of the sum over the spin structures of the first one. Hence for the integral of the total sum over the spin structures of the configuration the modular transformation can be performed, the modular symmetry being preserved. For $n_1>2$ the mathematical induction is employed So all the local symmetries of the amplitude are preserved.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The research described in this publication was made possible in part by Award No. RP1-2108 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF), and in part by Grants No. 00-02-16691 and No. 00-15-96610 from the Russian Fundamental Research Foundation.
Integration region and the unitarity
====================================
As an example, we consider the genus-2 forward scattering amplitude. The vacuum expectation of the vertex product is proportional to $\exp S$ where, being calculated for zero Grassmann parameters, $S$ is given through the field vacuum correlators (\[corr\]) of the boson string [@divfl; @dan89]. The Schottky multipliers to go to zeros, the leading approximated holomorphic Green function $R(z,z')$ is $\ln(z-z')$ while the scalar function $J_r^{(2)}(z;\{q\})$ is $\ln[(z-u_r)/(z-v_r)]$. The period matrix is given by (\[kzero\]). Then for the tachyon-tachyon forward scattering amplitude in the boson string theory $$\begin{aligned}
S=-\frac{s}{4}\left[\ln\left|\frac{(z_1-z_2)(z_3-z)}
{(z_1-z_3)(z_2-z_3)}\right|-\ln\left|
\frac{(z_1-u_j)(z_3-v_j)}
{(z_1-v_j)(z_3-u_j)}\right|
\frac{\hat\omega_{jl}}{\det\hat\omega}
\ln\left|\frac{(z_2-u_l)(z-v_l)}
{(z_2-v_l)(z-u_l)}\right|\right]
\nonumber\\
-2\ln\left|\frac{(z_1-u_j)(z-u_j)(z_2-v_j)
(z_3-v_j)}{(z_1-v_j)
(z-v_j)(z_2-u_j)(z_3-u_j)}\right|
\frac{\hat\omega_{jl}}{\det\hat\omega}
\ln\left|\frac{(z_1-u_l)(z-u_l)(z_2-v_l)
(z_3-v_l)}{(z_1-v_l)
(z-v_l)(z_2-u_l)(z_3-u_l)}\right|
\nonumber\\
-4\ln|(z_1-z_2)(z_3-z)|
\label{sapp}\end{aligned}$$ where $s=-(p_1+p_2)^2=-(p_3+p_4)^2$ and $\hat\omega_{jl}$ is given by $$\hat\omega_{11}=\ln|k_2|\,,\quad\hat\omega_{22}=
\ln|k_1|\,,\quad
\hat\omega_{12}=\hat\omega_{21}=-\ln\left|\frac{(u_1-u_2)
(v_1-v_2)}{(u_1-v_2)(v_1-u_2)}\right|\,.
\label{hatom}$$ For the massless boson scattering amplitude only the proportional to $s$ term presents on the right side of (\[sapp\]). First we discuss the case where both $v_1$ and $v_2$ does not go to $z_3$. The configurations $|k_1|\geq|k_2|$ and $|k_1|\leq|k_2|$ give the same contribution to the amplitude. So we consider $|k_1|\geq|k_2|$. We define new variables by $$\begin{aligned}
\ln|k_1|=x,\quad\ln|k_2|=x\alpha,\quad
|u_2-u_1|=|k_1|^{\beta}|v_1-z_3|,
\nonumber\\
u_1-z_3=
y_1(u_2-u_1),\quad z-u_1=y(u_2-u_1)
\label{new}\end{aligned}$$ where $\alpha\geq1$. From (\[ndgbound\]), it follows that $0\leq\beta\leq1/2$. Generically, $y\sim y_1\sim1$. In the boson string theory [@divfl; @dan89] at $k_1\to0$ the holomorphic partition function is $\sim|k_1k_2|^{-4}$ while in the superstring theory (see Section 5 of the present paper) it is $\sim|k_1k_2|^{-2}$. Terms of the expansion of the integrand (\[ampl\]) over the powers of the small variables corresponds to different thresholds while the expansion over powers of $1/x$ corresponds to the expansion over powers of the center mass space momentum of the intermediate state. Near the given threshold the leading approximated amplitude $A_4^{(2)}(s)$ discussed is found to be $$A_4^{(2)}(s)=\frac{(4\pi)^{D}}{8} \int\limits_1^\infty
d\alpha\int\limits_0^{1/2}d\beta
\frac{\hat
A_1(\alpha,\beta)\hat A_2(\alpha,\beta)}
{[\alpha-\beta^2]^{D/2}}
\int\limits_{-\infty}^{-\kappa}\frac{e^{x\tilde S}}{x^{D-2}}dx
\label{sngamp}$$ where the variables are defined by (\[new\]). In this case $D=10$ for the superstring and $D=26$ for the boson string. The cutoff $\kappa>>1$ bounds the region of small $|k_1|$. Furthermore, $\hat A_1(\alpha,\beta)$ is an integral over real and imaginary parts of both $y$ and $y_1$ while $\hat A_2(\alpha,\beta)$ is an integral over real and imagery parts of $v_1$ and of $v_2$. By using (\[sapp\]) one find that $$\tilde
S=\biggl(\beta-\frac{\alpha\beta^2+\beta^2
-2\beta^3}{\alpha-\beta^2}\biggl)
\biggl[-\frac{s}{4} +\tilde
p(\alpha,\beta)\biggl(\beta-\frac{\alpha\beta^2+\beta^2
-2\beta^3}{\alpha-\beta^2}\biggl)^{-1}\biggl]
\label{tildes}$$ where $\tilde p(\alpha,\beta)$ is a linear function of its arguments with the coefficients depending on the threshold discussed. In the superstring theory all the coefficients are non-negative. At $\tilde S\geq0$ the integral (\[sngamp\]) is divergent. Thus the cut begin with that $s$, which is the minimal value of $s$ where $\tilde S$ is nullified in the integration region. To calculate the discontinuity, we go to $\tilde x=-x\tilde S$ integrated from $\kappa\tilde S$ till $\infty$. When $\tilde S$ rounds the $\tilde S=0$ point, the initial point $\kappa\tilde S$ of the integration contour gets about the pole at $\tilde x=0$ to be either above the $\tilde
x=0$ point, or below it, depending on the sign of the $Im\,\tilde S$. Thus the discontinuity $[A_4^{(2)}(s)]_{disc}$ of $\tilde A_4^{(2)}(s)$ is given by the integral over $\tilde
x$ along the closed contour surrounding the $\tilde x=0$ point to be $$[A_4^{(2)}(s)]_{disc}=\frac{(4\pi)^{D}}{8} \int\limits_1^\infty
d\alpha\int\limits_0^{1/2}d\beta
\frac{\hat
A_1(\alpha,\beta)\hat A_2(\alpha,\beta)}
{[\alpha-\beta^2]^{D/2}}\theta(-\tilde S)\frac{2\pi
\tilde S^{(D-3)}}{(D-3)!}
\label{disamp}$$ Every threshold $s=s_i$ determine the minimum of the second term in the square brackets on the right side of (\[tildes\]) at corresponding $\alpha=alpha_i$ and $\beta=\beta_i$, which, among other, can be on the boundary of the region. At $(s-s_i)\to0$ only small $(\alpha-\alpha_i)$ and $\beta-\beta_i$ contribute to (\[disamp\]). Being smooth functions of $\alpha$ and $\beta$, in the leading approximation $\hat A_1(\alpha,\beta)$ and $\hat
A_2(\alpha,\beta)$ both are replaced by $\hat
A_1(\alpha_i,\beta_i)$ and $\hat A_2(\alpha_i,\beta_i)$. It is naturally to expect that every one from $\hat
A_1(\alpha_i,\beta_i)$ and $\hat A_2(\alpha_i,\beta_i)$ present the threshold values of the corresponding $2\to3$ amplitude as this is for three tachyon cut of the tachyon-tachyon amplitude in the boson string theory. In this case $\tilde
p(\alpha,\beta)=m_{th}^2(1+\alpha-\beta)/4$ where $m_{th}^2=-8$ is the square of the tachyon mass. As far as $m_{th}^2<0$, the calculation includes rather subtle matters, which are not discussed here. Instead we calculate the discontinuity for a $m_{th}^2>0$ continuing the obtained result to $m_{th}^2=-8$. There is only one threshold to be at $s=9m_{th}^2$. In this case $\beta\approx1/2$ and $\alpha\approx1$. So (\[tildes\]) is approximated by $$\tilde S\approx
\frac{1}{3}\biggl(-\frac{s-9m_{th}^2}{6}
+8m_{th}^2[(\alpha-1)^2-2(\alpha-1)(\frac{1}{2}-\beta)+
2(\frac{1}{2}-\beta)^2]\biggl)
\label{tilds}$$ Other factors in (\[disamp\]) are taken at $\alpha=1$ and $\beta=1/2$. One can check that $\hat A_1(1,1/2)=\hat
A_2(1,1/2) =A_5^{(0)}$ is really the $2\rightarrow3$ tree tachyon interaction amplitude [@gsw] at $s=9m_{th}^2$. In this case $\hat A_1(1,1/2)$ given by the integral where three vertex coordinates are fixed to be 0, 1 and $\infty$. In $\hat
A_2(1,1/2)$ the fixed coordinates are $z_1$, $z_2$ and $z_3$. The integral is none other than $[A_5^{(0)}]^2$ times the phase volume.
For the configuration $v_2\to z_3$, $u_1\to z_3$ and $u_2\to z_3$ we discuss, as an example, the case $k_1\geq k_2$. We define new integration variables as it follows $$\begin{aligned}
\ln|k_1|=x,\quad\ln|k_2|=x\alpha,\quad
|u_2-u_1|=|k_1|^{\beta+\delta}|v_1-z_3|,\quad
|v_2-u_1|=|k_1|^\delta|v_1-z_3|,
\nonumber\\
|u_1-z_3|=
|k_1|^\eta|v_1-z_3|,\quad z-u_1=|k_1|^\eta y(v_1-z_3)
\label{newvar}\end{aligned}$$ where $1\leq\alpha$ and $0\leq\delta\leq\eta$. It follows from (\[ndgbound\]) that $0\leq\beta\leq1/2$. In addition, $0\leq\eta\leq1/2$, as far as $z_3$ lies out of the $C_{u_1}$ circle. Then, by using eq.(\[sapp\]), for the tachyon-tachyon forward scattering amplitude the expression of $\tilde S$ in (\[disamp\]) is found to be $$\begin{aligned}
\tilde
S=\biggl(\eta-\frac{\alpha\eta^2-(\eta-\delta)^2
-2\beta\eta(\eta-\delta)}{\alpha-\beta^2}\biggl)
\biggl[-\frac{s}{4}+\frac{m_{th}^2}{4}(1+\alpha-\beta)
\nonumber\\
\times
\biggl(\eta-\frac{\alpha\eta^2-(\eta-\delta)^2
-2\beta\eta(\eta-\delta)}{\alpha-\beta^2}\biggl)^{-1}\biggl]
\label{tlds}\end{aligned}$$ In this case the false threshold appears at $s=6m_{th}^2$, which corresponds to the minimum of the last term on the right side of (\[tlds\]) to be at $\alpha=1$, $\beta=\eta=1/2$ and $\delta=\eta(1-\beta)=1/4$. The discussed configuration is removed from the integral as it is proposed in Section 4.
Explicit Scalar superfield Green functions
==========================================
Explicitly $R_L^{(n)}(t,t';\{q\})$ can be given through the genus-1 Green functions $R_{l_s}^{(1)}(t,t';s)$ calculated for the Schottky parameters $\hat m_s=(k_s,u_s,v_s)$ along with the Grassmann ones $\mu_s$ and $\nu_s$, the spin structure being $l_s=(l_{1s},l_{2s})$. Due to (\[gamab\]), the above genus-1 function is written through $z_s$ and $\vartheta_s$ in (\[tgam\]) using the boson Green function $R_b^{(1)}(z,z';\hat
m_s)$ and the fermion Green one $R_f^{(1)}(z,z';\hat
m_s;l_{1s},l_{2s})$ as it follows [@danphr] $$\begin{aligned}
R_{l_s}^{(1)}(t,t';s)=R_b^{(1)}(z_s,z_s';\hat m_s)
-\vartheta_s\vartheta_s'
R_f^{(1)}(z_s,z_s';\hat m_s;l_{1s},l_{2s})
\nonumber\\
+\tilde\varepsilon_s'\vartheta_s'\Upsilon_s(\infty,z_s')
+\tilde\varepsilon_s'\vartheta_s\Upsilon_s(z_s,\infty)\,,
\nonumber\\
\Upsilon_s(z,z')=(z-z')R_f^{(1)}(z,z';\hat m_s;l_{1s},l_{2s})
\label{zgrin}\end{aligned}$$ The proportional to $\Upsilon_s$ terms are added to provide decreasing $K_{l_s}^{(1)}(t,t';s)$ at $z\to\infty$ or $z'\to\infty$. The above $K_{l_s}^{(1)}(t,t';s)$ is related with $R_{l_s}^{(1)}(t,t';s)$ by (\[kr\]). The Poincar[é]{} series for boson and fermion Green functions are given in the end of this Appendix. For the odd spin structure $l_m=(1/2,1/2)$ we use [@danphr] the Green function with the property that $$R_{l_m}^{(1)}(t_m^b,t';m)=R_{l_m}^{(1)}(t,t';m)+
J_m^{(1)}(t')-
\varphi_m(t)\varphi_m(t')
\label{trans1}$$ where $J_m^{(1)}(t)$ the genus-1 scalar function and $\varphi_m(t)$ is the spinor zero mode given by $$\varphi_m(t)=\frac{\vartheta_m(u_m-v_m)^{1/2}}
{[(z_m-u_m)(z_m-v_m)]^{1/2}}+
\varepsilon_m'(u_m-v_m)^{1/2}
\label{phi}$$ Here $(z_m|\theta_m)$ variables are determined by (\[tgam\]). The last term in (\[phi\]) provides vanishing the spinor zero mode at $z\to\infty$. The genus-1 scalar function $J_s^{(1)}(t)$ is $$J_s^{(1)}(t)=\ln\frac{z_s-u_s}{z_s-v_s}
\label{jr1}$$ where $z_r$ is given by (\[tgam\]).
For the genus-$n$ superspin structure without the odd genus-1 spin structures, the desired Green function is directly given by (\[rpart\]) $\tilde R_{L_r}^{(n_r)}(t,t';\{q\}_r)$ are replaced by $\tilde R_{l_s}^{(1)}(t,t';s)$. The kernel of the operator $\hat K=\hat K_{sr}$ for $s\neq r$ is $\tilde
K_{l_s}^{(1)}(t,t';s)dt'$, which is the non-singular part of $K_{l_s}^{(1)}(t,t';s)$ related with $\tilde R_{l_s}^{(1)}(t,t';s)$ by (\[kr\]). The non-singular part of the Green function is defined by (\[lim\]). The integration over $t'$ is performed along $C_r$-contour surrounding the limiting points $u_r$ and $v_r$ and, for the Ramond handle, the cut between them.
Below we denote $R_L^{[n]}(t,t';\{q\})$ the Green function given it terms of the $R_{l_s}^{(1)}(t,t';s)$ genus-1 functions by (\[rpart\]). If odd spin structure handles present, the change of $R_{l_s}^{(1)}(t,t';s)$ under $t\to t_s^{(b)}$ is different from (\[rtrans\]) due to the last term in (\[trans1\]). We show that in this case the Green function satisfying (\[rtrans\]), is given by $$R_L^{(n)}(t,t';\{q\})= R_L^{[n]}(t,t';\{q\}) -\frac{1}{2}
\sum_{m,m'}\Phi_m(t;L;\{q\})
\hat V_{mm'}^{-1}\Phi_{m'}(t';L;\{q\})
\label{rprta}$$ where the last term appears, if genus-1 odd spin structures present. The $\hat V_{mm'}$ matrix elements are defined only for those $(m,m')$ that label the odd genus-1 spin structures. Both $\hat V_{mm'}$ and $\Phi_m(t;L;\{q\})$ are calculated in terms of the genus-1 zero spinor modes $\varphi_m(t)$ defined by (\[phi\]). In so doing $$\Phi_m(t;L;\{q\})=
\varphi_m(t)+\sum_p\int_{C_m}[(1-\hat K)^{-1}\hat
K]_{pm}(t,t')dt'\varphi_m(t')
\label{bphi}$$ where $\delta_{mm'}$ is the Kronecker symbol while the $\hat V_{mm'}$ matrix is found to be[^9] $$\begin{aligned}
\hat V_{mm'}=-\frac{1}{2}\sum_{p\neq m}\int_{C_p}D(t)\varphi_m(t)dt
\int_{C_{m'}}[(1-\hat K)^{-1}\hat
K]_{pm'}(t,t')dt'\varphi_{m'}(t') \nonumber\\
-\frac{1}{2}(1-\delta_{mm'})\int_{C_{m'}}D(t)\varphi_m(t)dt
\varphi_{m'}(t)\,,
\label{vmatr}\end{aligned}$$ To check that (\[rprta\]) satisfies eqs.(\[rtrans\]), the first term on the right side of (\[rprta\]) is presented by (\[rpart\]), and $\Phi_m(t;L;\{q\})$ in (\[rpart\]) is presented in the like way as $$\begin{aligned}
\Phi_m(t;L;\{q\})=
\sum_{p\neq r}\int_{C_p}K_{l_r}^{(1)}(t,t_1;r)dt_1\int_{C_m}
[(1-\hat K)^{-1}\hat
K]_{pm}(t_1,t_2)dt_2\varphi_m(t_2)
\nonumber\\
+\delta_{mr}\varphi_m(t)+(1-\delta_{mr})
\int_{C_m}K_{l_r}^{(1)}(t,t_1;r)dt_1\varphi_m(t_1)
\label{bigphi}\end{aligned}$$ Indeed, calculating the contribution to (\[bigphi\]) of the pole term in $K_{l_r}^{(1)}(t,t_1;r)$, one obtains (\[bphi\]) using (\[opr\]). One can see from (\[bigphi\]) the relations (\[rtrans\]) to be true, the scalar function $J_r^{(n)}(t;\{q\};L)$ in (\[rtrans\]) being $$J_r^{(n)}(t;\{q\};L)=\tilde J_r^{(n)}(t;\{q\};L)
-\frac{1}{2}\sum_{m,m'}\Phi_m^{(r)}(L;\{q\})
\hat V_{mm'}^{-1}\Phi_{m'}(t;L;\{q\})
\label{jr}$$ where $\tilde J_r^{(n)}(t;\{q\};L)$ is given by (\[tjr\]) through the genus-1 functions, and $$\begin{aligned}
\Phi_{m'}^{(r)}(L;\{q\})=
\Phi_{m'}(t_r^b;L;\{q\})-\Phi_{m'}(t;L;\{q\})=
(1-\delta_{mr})
\int_{C_m}D(t_1)J_r^{(1)}(t_1)dt_1\varphi_m(t_1)
\nonumber\\
+\sum_{p\neq r}\int_{C_p}D(t_1)J_r^{(1)}(t_1)dt_1\int_{C_m}
[(1-\hat K)^{-1}\hat
K]_{pm}(t_1,t_2)dt_2\varphi_m(t_2)\,.
\label{phir}\end{aligned}$$ The genus-1 function $J_r^{(1)}(t)$ is given by (\[jr1\]). To calculate the period matrix, one chooses a fixed parameter $t_0$, as it is discussed in Section 3. Then one can check that $$\begin{aligned}
J_r^{(n)}(t;\{q\};L)-J_r^{(n)}(t_0;\{q\};L)=
[\tilde J_r^{(n)}(t;\{q\};L)-\tilde J_r^{(n)}(t_0;\{q\};L)]
\nonumber\\
-\frac{1}{2}\sum_{m,m'}\Phi_m^r(L\{q\})
\hat V_{mm'}^{-1}\{\Phi_{m'}(t;L;\{q\})-
\Phi_{m'}(t_0;L;\{q\})\}
\label{jara}\end{aligned}$$ where the term in square brackets is calculated by (\[jar\]) for $j_r=r$ through the genus-1 functions. To prove (\[jara\]), one calculates the contribution to (\[jara\]) of the singular term in the Green function by the method given in Section 3. In addition, eq.(\[bphi\]) for $\Phi_{m'}(t;L;\{q\})$ is used. From (\[jara\]), the period matrix elements are found to be $$\begin{aligned}
2\pi i\omega_{rs}^{(n)}(\{q\};L)=
2\pi i\tilde\omega_{rs}^{(n)}(\{q\};L)
-\frac{1}{2}\sum_{m,m'}\Phi_m^{(r)}(L;\{q\})
\hat V_{mm'}^{-1}\Phi_{m'}^{(s)}(L;\{q\})
\nonumber\\
+(1-\delta_{rs})\int_{C_s}D(t_1)J_r^{(1)}(t)dtJ_s^{(1)}(t)\,.
\label{omjra}\end{aligned}$$ where $\tilde\omega_{rs}^{(n)}(\{q\};L)$ is presented by (\[omjr\]) at $j_r=r$ and $j_s=s$ in terms of the genus-1 functions. The boson Green function $R_b^{(1)}(z,z';\hat m)$ in (\[zgrin\]) is given by (the symbol “$s$” is omitted) $$\partial_{z'}R_b^{(1)}(z,z';\hat m)=
-\sum_n\frac{1}{(z-g_n(z'))(c_nz'+d_n)^2}
\label{bsc}$$ where the sum is performed over the group products of the Schottky transformation $g(z)$. In so doing $g_0(z)=z$, and that negative values of $n$ are associated with the inverse transformations. The fermion Green functions are given by $$\begin{aligned}
R_f^{(1)}(z,z';\hat m;l_1=1/2,l_2)=
\sum_{n=0}^\infty(-1)^{(2l_2+1)n}
\left[\frac{1}{1-k^n\frac{(z-v)(z'-u)}{(z-u)(z'-v)}}-
\frac{1}{1-k^n\frac{(z-u)(z'-v)}{(z-v)(z'-u)}}\right]
\nonumber\\ \times
\frac{(u-v)}
{2\sqrt{(z-u)(z-v)(z'-u)(z'-v)}}\,,
\nonumber\\
R_f^{(1)}(z,z';\hat m;l_1=0,l_2)=
\sum_n\frac{(-1)^{(2l_2+1)n}}{(z-g_n(z'))(c_nz'+d_n)}.
\label{fsc}\end{aligned}$$ where like (\[bsc\]), the summation is performed over the group products of $g(z)$. At $z'\to\infty$ $$R_f^{(1)}(z,z';\hat m;l_1,l_2)\to \frac{1}{z-z'}+
\frac{\hat W_1(z;\hat m;l_1,l_2)}{z'-u}+
\frac{\hat W_2(z;\hat m;l_1,l_2)}{(z'-u)^2}\,.
\label{asfgf}$$ with corresponding $\hat W_1(z;\hat m;l_1,l_2)$ and $\hat W_2(z;\hat m;l_1,l_2)$. At $z\to\infty$ $$\hat W_2(z';\hat m;l_1,l_2)\to-(z-u)
\hat W_1(z';\hat m;l_1,l_2)\,,
\qquad
\hat W_1(z;\hat m;l_1,l_2)\to\frac{\hat
a(k;l_1,l_{2})(u-v)^2}{(z'-u)^2}
\label{assig}$$ where $\hat a(k;l_1,l_2)$ depends on the multiplier and on the spin structure. Up to the unessential constant term, the non-singular part (\[lim\]) of the genus-1 Green function (\[zgrin\]) is given at $z'\to\infty$ by (see eq.(\[tgam\]) for definitions) $$\begin{aligned}
\tilde R_{l_s}^{(1)}(t,t';s)(z'-u)=(\vartheta-\varepsilon(z))
(1+\varepsilon\varepsilon')[
\hat W_1(z;\hat m;l_1,l_2)(\vartheta'-\mu)
+\hat W_2(z;\hat m;l_1,l_2)\varepsilon']
\nonumber\\
+[\hat W_b(z;\hat m)-\vartheta\varepsilon(z)\partial_z
\hat W_b(z;\hat m)]
(1-\varepsilon'\vartheta')
\label{assgr}\end{aligned}$$ where $\hat W_b(z;\hat m)$ determines the asymptotics of the boson Green function (\[bsc\]).
Integration measures
====================
In this Appendix we reduce to the convenient for application form eq.(127) from [@danphr] expressing $\tilde Z^{(n)}(\{q\},L)$ through genus-1 functions. First, we transform the scalar superfield contribution on the right side of eq.(112) of [@danphr] as it follows (in notations of [@danphr]) $$\begin{aligned}
trace\ln(I-\hat K^{(1)}+\hat\varphi\hat f)
=trace\ln(I-\hat
K^{(1)})
-trace\ln[I+\hat f_1(I-\hat K^{(1)})^{-1}\varphi]
\nonumber\\
=trace\ln(I-\hat K^{(1)}) -\ln\det \hat V \,.
\label{dfln}\end{aligned}$$ We have used that for any operators $A_1$, $A_2$ and for $A$ with $\det A\neq0$, $$\begin{aligned}
trace\ln[A+A_1A_2]=trace\ln A+trace\ln[1+A^{-1}A_1A_2]
\nonumber\\
=trace\ln A+(-1)^Ptrace\ln[1+A_2A^{-1}A_1]
\label{trace}\end{aligned}$$ where $P=1$ when both $A_1$ and $A_2$ are the Fermi operators, otherwise $P=0$. In addition, eq.(50) and eq.(52) of [@danphr] are used. The $K^{(1)}$ operator in (\[dfln\]) is the same as Appendix B. In addition the ghost contribution we express now in terms of function $\hat
G^{(1)}$ given below instead of $G_\sigma^{(1)}(z,z')$ in eq.(127) from [@danphr]. The desired factor in (\[zhol\]) is given by $$\ln\tilde
Z^{(n)}(\{q\},L)=-5trace\ln(I-\hat K)+5\ln\det \hat V
+trace\ln(I-\hat G)-\ln\det \hat U
\label{tzh}$$ where the $\hat K$ operator is the same as in (\[rprt\]) and the $\hat V$ matrix is defined by (\[vmatr\]). The matrix operator $\hat G$ is defined in terms of a genus-1 ghost correlator $G_{l_s}^{(1)}(t,t';s)$ defined below in the same manner as $\hat K$ is given in terms of $K_{l_s}^{(1)}(t,t';s)$. So $\hat G=\{\hat G_{sr}\}$ where $\hat G_{sr}$ is an integral operator vanishing at $s=r$. For $s\neq r$, the kernel of $\hat G_{sr}$ is $\tilde
G_{l_s}^{(1)}(t,t';s)dt'$ defined by (\[greg\]) for the genus-1 case. Like $\hat V$, the elements $\hat
U_{mn}$ of $\hat U$ are defined only for $(m,m')$ assigned to the odd genus-1 spin structures. They are given it terms of 3/2 zero modes $\chi_m^{(1)}(t)$ and in terms of -1/2 genus-1 zero modes $\phi_m^{(1)}(t)$ as it follows $$\begin{aligned}
\hat
U_{mm'}=-\frac{1}{2}\sum_{p\neq m}\int_{C_p}\chi_m^{(1)}(t)dt
\int_{C_{m'}}[(1-\hat
G)^{-1}\hat G]_{pm'}(t,t')dt'\phi_{m'}^{(1)}(t')
\nonumber\\
-\frac{1}{2}(1-\delta_{mm'})\int_{C_{m'}}\chi_m^{(1)}(t)dt
\phi_{m'}^{(1)}(t)\,.
\label{umatr}\end{aligned}$$ The above genus-1 zero modes are given by $$\chi_m^{(1)}(t)=-\frac{(u_m-v_m)^2}
{[(z_m-u_m)(z_m-v_m)Q_m^2(t)]^{3/2}}\,,\quad
\phi_m^{(1)}(t)=
\frac{\vartheta_mQ_m^2(t)\sqrt{(z_m-u_m)(z_m-v_m)]}}
{(u_m-v_m)}
\label{uf}$$ where $Q_m$ is defined in (\[supder\]) and the $(z_m|\theta_m)$ variables are defined by (\[tgam\]) at $s=m$. The genus-1 function $G_{l_s}^{(1)}(t,t';s)$ is given through the boson Green function $G_b^{(1)}(z,z';\hat m_s)$ and the fermion Green one $G_f^{(1)}(z,z';\hat
m_s;l_{1s},l_{2s})$ as [@danphr] $$\begin{aligned}
G_{l_s}^{(1)}(t,t';s)=Q_s^2(t)
[G_b^{(1)}(z_s,z_s';\hat m_s)
\theta_s'+
\theta G_f^{(1)}(z_s,z_s';\hat m_s;l_{1s},l_{2s})
\nonumber\\
-\tilde\varepsilon_s'\Upsilon_s^{(gh)}(\infty,z_s')]Q_s^{-3}(t')\,,
\nonumber\\
\Upsilon_s^{(gh)}(z,z')=
(z-z')G_f^{(1)}(z,z';\hat m_s;l_{1s},l_{2s})
\label{zghgri}\end{aligned}$$ where $z_s$ and $\vartheta_s$ are defined by (\[tgam\]) while $\hat m_s=(k_s,u_s,v_s)$. The proportional to $\Upsilon_s^{(gh)}$ terms provide decreasing $G_{l_s}^{(1)}(t,t';s)$ at $z\to\infty$ or at $z'\to\infty$. The boson part of the ghost Green function in (\[zghgri\]) is (see eq.(68) in [@danphr]) $$G_b^{(1)}(z,z';\hat m)=
-\sum_n\frac{1}{(z-g_n(z'))(c_nz'+d_n)^4}
\label{bgsc}$$ where the summation is performed over the group products of $g(z)$. For even spin structures the fermion part in (\[zghgri\]) expressed in terms of (\[fsc\]) as $$\begin{aligned}
G_f^{(1)}(z,z';\hat m;l_1,l_2)=
\frac{(z-u)(z-v)}{(z'-u)(z'-v)}R_f^{(1)}(z,z';\hat m;l_1,l_2)
\nonumber\\
-\frac{(z-v)\Sigma_1(z';\hat m;0,l_2)+
\Sigma_2(z';\hat m;0,l_2)}{(z'-u)(z'-v)}
\label{fgsc}\end{aligned}$$ where the last term is calculated in terms of (\[asfgf\]). In this case $$\Sigma_1(z;\hat m;l_1,l_2)=1+
\hat\Sigma_1(z;\hat m;l_1,l_2)\,,\qquad
\Sigma_2(z;\hat m;l_1,l_2)=z-u+
\hat\Sigma_2(z;\hat m;l_{1},l_{2})\,.
\label{asrel}$$ One can check that the function (\[fgsc\]) goes to zero at $z\to\infty$. Furthermore, at $z'\to\infty$ $$G_f^{(1)}(z,z';\hat m;l_1,l_2)\to \frac{1}{z-z'}+
\frac{\Sigma_{gh}(z;\hat m;l_1,l_2)}{(z'-u)^3}
\label{azpeg}$$ where the numerator in the last term is a function os $z$. If both $z\to\infty$ and $z'\to\infty$, then $$G_f^{(1)}(z,z';\hat m;l_1,l_2)\to \frac{1}{z-z'}-
\frac{a_{gh}(k;l_1,l_2)(u-v)^4}{(z-u)(z'-u)^3}
\left[\frac{3}{z'-u}-
\frac{1}{z-u}\right]
\label{azzeg}$$ where $a_{gh}(k;l_1,l_2)$ depends on the multiplier and on the spin structure. Moreover, $$\begin{aligned}
(cz+d)G_f^{(1)}(g(z),z';\hat m;l_1,l_2)-
G_f^{(1)}(z,z';\hat m;l_1,l_2)+\frac{1-\sqrt k}{\sqrt k}p_\mu(z)
\chi_\mu(z';\hat m;l_1,l_2)
\nonumber\\
-(1-\sqrt k)
\chi_\nu(z';\hat m;l_1,l_2)
\label{evfcon}\end{aligned}$$ where $p_\mu(z)$ and $p_\nu(z)$ are given by $$p_\mu(z)=\frac{2(z-v)}{u-v}\,,\quad
p_\nu(z)=-\frac{2(z-u)}{u-v}
\label{pol}$$ while the depending on $z'$ functions are defined to be $$\begin{aligned}
\chi_\mu(z;\hat m;l_1,l_2))=-
\frac{(u-v)\Sigma_1(z;\hat m;l_1,l_2)+
\Sigma_2(z;\hat m;l_1,l_2)}{2(z-u)(z-v)}\,,
\nonumber\\
\chi_\nu(z;\hat m;l_1,l_2))=-
\frac{\Sigma_2(z;\hat m;l_1,l_2)}{2(z-u)(z-v)}\,.
\label{chistev}\end{aligned}$$ Eq.(\[evfcon\]) is non other than eq.(63) of [@danphr] in the genus-1 case. For the odd spin structure, due to the $(-1/2)$ mode, there is no the Green function obeying (\[evfcon\]). In this case we define the ghost Green function by $$\begin{aligned}
G_f^{(1)}(z,z';\hat m;1/2,1/2)=G_{(\sigma=1)}^{(1)}(z,z')
-2\left(\sqrt{\frac{z-u}{z-v}}-1\right)
\chi_\nu(z';\hat m;1/2,1/2)\,,
\nonumber\\
\chi_\nu(z;\hat m;1/2,1/2)=-\frac{1}{2\sqrt{(z-u)(z-v)}}\sum_n
\frac{1}{(c_nz'+d_n)^2}
\label{ghostodd}\end{aligned}$$ where $G_{(\sigma=1)}^{(1)}(z,z')$ is defined by eq.(69) in [@danphr]. At $z'\to\infty$ $$G_f^{(1)}(z,z';\hat m;1/2,1/2)\to\frac{1}{z-z'}+
\frac{\Sigma_{gh}(z;\hat m;1/2,1/2)}{(z'-u)^2}\,.
\label{azpog}$$ Moreover, at $z\to\infty$ $$\Sigma_{gh}(z;\hat m;1/2,1/2)\to\frac{(u-v)^2}{8(z-u)}
\label{azzpog}$$ Under the Schottky transformation $g(z)$ the above function (\[ghostodd\]) is changed as $$\begin{aligned}
(cz+d)G_{(f)}^{(1)}(g(z),z';\hat m;1/2,1/2)-
G_f^{(1)}(z,z';\hat m;1/2,1/2)=
\nonumber\\
\left[\frac{1-\sqrt k}{\sqrt k}p_\mu(z)
-(1-\sqrt k)p_\nu(z)\right]
\chi_\nu(z';\hat m;1/2,1/2)
\nonumber\\
-\frac{\sqrt{
(z-u)(z-v)}(u-v)}{[(z'-v)(z'-u)]^{\frac{3}{2}}}\,.
\label{trghodd}\end{aligned}$$ Eq.(\[trghodd\]) follows from (\[ghostodd\]) along with eq.(82) in [@danphr]. We show that in this case the left side of (\[trghodd\]) is obtained in a form of series, which are given in terms of (\[trghodd\]) as it follows $$\begin{aligned}
-\frac{(u-v)^2}{[(z-v)(z-u)]^{\frac{3}{2}}}=
c\sqrt{\frac{z-v}{z-u}}\sum_{n=-\infty}^{\infty}
\frac{1}{k^{(n+1)/2}Q_n(z)Q_{n+1}^2(z)}+
\frac{1}{k^{n/2}Q_n^2(z)Q_{n+1}(z)}\,,
\nonumber\\
\chi_\nu(z;\hat m;1/2,1/2)=
\frac{c}{2}\sqrt{\frac{z-v}{z-u}}\sum_{n=-\infty}^{\infty}
\frac{1}{k^{(n-1)/2}Q_n(z)Q_{n+1}^2(z)}+
\frac{1}{k^{n/2}Q_n^2(z)Q_{n+1}(z)}\,.
\label{munuod}\end{aligned}$$ Here $Q_n(z)=c_nz+d_n$ for the group product $g^n$. Indeed, the right side in the first line of (\[munuod\]) is $\sim
1/z^{3/2}$ at $z\to\infty$, and so it is proportional to 3/2-zero mode on the left side. To verify the coefficient, the both parts are multiplied by $\sqrt{(z-u)(z-v)}$. Then the equation is integrated along the corresponding Schottky circle. In the second line, the leading at $z\to\infty$ term on the left side is equal to corresponding term on the right side. So the right side of the discussed relation may differ from its left side only by the term proportional to $\chi_\mu(z;\hat
m;1/2,1/2)$. To verify the equation, it is again multiplied by $\sqrt{(z-u)(z-v)}$ and then it is integrated along the Schottky circle. For even spin structures the ghost function (\[fgsc\]) is related to $G_{(\sigma=1)}^{(1)}(z,z')$ in [@danphr] by $$\begin{aligned}
G_{(\sigma=1)}^{(1)}(z,z')=
G_f^{(1)}(z,z';\hat m;l_1,l_2)
-[p_\mu(z)\chi_\mu(z';\hat m;l_1,l_2)+
p_\nu(z)\chi_\nu(z';\hat m;l_1,l_2)]
\nonumber\\
+\frac{1}{2}\sqrt{\frac{z-u}{z-v}}
[p_\mu(z)(3\chi_\mu(z';\hat m;l_1,l_2)
-\chi_\nu(z';\hat m;l_1,l_2))
\nonumber\\
+p_\nu(z)(\chi_\mu(z';\hat m;l_1,l_2)+
\chi_\nu(z';l_1,l_2))]
\label{ghostev}\end{aligned}$$ with $p_\mu(z)$ and $p_\nu(z)$ being defined by (\[pol\]).
To derive the ghost terms in (\[tzh\]) one represents $\hat
S_\sigma^{(1)}$ in eq.(127) of [@danphr] through $\hat
G^{(1)}$. For this purpose one uses eq.(81) of [@danphr] along with eqs. (\[ghostodd\]) and (\[ghostev\]) of the present paper. The result expression being arranged by (\[trace\]), the desired terms in (\[tzh\]) are obtained.
To verify eq.(\[ipf\]), we represent $G_L^{(n)}(t,t';\{q\})$ in (\[ipf\]) as it follows $$\begin{aligned}
G_L^{(n)}(t,t';\{q\})=
\sum_{s=1}^n\tilde
G_{l_s}^{(1)}(t,t';s)
+\sum_{r,s}\int_{C_s}[(1-\hat G)^{-1}\hat
G]_{rs}(t,t_1)dt_1\tilde G_{l_s}^{(1)}(t_1,t';s)
\nonumber\\
+\frac{\vartheta-\vartheta'}{z-z'}
-\sum_{m,m'}\phi_m^{(n)}(t;L;\{q\})
\hat U_{mm'}^{-1}\chi_{m'}^{(n)}(t';L;\{q\})
\label{gprt}\end{aligned}$$ where the sum in the last term is performed over genus-1 odd spin handles. The $\hat U_{mm'}$ matrix elements are defined by (\[umatr\]) and the functions in the last term on the right side are calculated in terms of the genus-1 zero modes (\[uf\]) by $$\begin{aligned}
\chi_m^{(n)}(t;L;\{q\})=\chi_m^{(1)}(t)+
\sum_{p,p'}\int_{C_p}\chi_m^{(1)}(t')dt'\int_{C_p'}[(1-\hat
G)^{-1}]_{pp'}(t',t_1)dt_1\tilde G(t_1,t)\,,
\nonumber\\
\phi_m^{(n)}(t;L;\{q\})=\tilde\phi_m^{(1)}(t)+
\sum_p\int_{C_m}[(1-\hat G)^{-1}\hat
G]_{pm}(t,t')dt'\phi_m^{(1)}(t')
\label{bphig}\end{aligned}$$ where $\tilde\phi_m^{(1)}(t)$ is as it follows $$\tilde\phi_m^{(1)}(t)=\phi_m^{(1)}(t)
-\frac{1}{2}\vartheta_mQ_m^2(t)[2z_m-u-v]-\frac{\varepsilon'}{8}
\label{phireg}\,.$$ Due to (\[tgam\]) and (\[uf\]), $\tilde\phi_m^{(1)}(t)$ vanishes at $z\to\infty$. To prove (\[gprt\]) one uses the same trick as in (\[rprta\]). Then one can verify that $\phi_m^{(n)}(t;L;\{q\})$ is the superconformal 3/2 tensor under the transformations of the super-Schottky group, and that (\[gprt\]) obeys the conditions (63) of the paper [@danphr]. Thus (\[gprt\]) is the correct expression for $G_L^{(n)}(t,t';\{q\})$. In (\[umatr\]) and (\[bphig\]) the $\phi_m^{(1)}(t)$ zero mode can be replaced by $\tilde\phi_m^{(1)}(t)$. Indeed the difference of the above quantities has not singularities inside the $C_r$ contour, the integral of it along $C_r$ being equal to zero. So one substitutes in eq.(\[ipf\]) the Green functions given by (\[rprta\]) and by (\[gprt\]) through the genus-1 functions. Further one uses (\[trace\]) considering the $(m,m')$ sum in (\[rprta\]) and in (\[gprt\]) as the separable operator $A_1A_2$. In so doing (\[tzh\]) appears. So (\[ipf\]) is proved.
Applying (\[trace\]) to the non-holomorphic factor in (\[hol\]), one can obtains that $$5trace\ln \hat V-5\ln\det
\Omega_{L,L'}^{(n)}(\{q,\overline q
\})= -5\ln\det\tilde
\Omega_{L,L'}^{(n)}(\{q,\overline q
\})
+5trace\ln\tilde V
\label{trvm}$$ where $V$ is the same as in (\[tzh\]) and the summation over $(r,s)$ is implied. The $\tilde
\Omega_{L,L'}^{[n]}(\{q,\overline q \})$ matrix is calculated for the period one $\tilde\omega_{rs}^{(n)}(\{q\};L)$ in (\[omjra\]). The $\tilde V_{mm'}$ of $\tilde V$ is defined by $$\tilde V_{mm'}=
\hat V_{mm'}+\Phi_m^{(r)}(L;\{q\})
[\tilde\Omega_{L,L'}^{(n)}(\{q,\overline q
\})]_{rs}^{-1}
\Phi_{m'}^{(s)}(L;\{q\})
\label{tildv}$$ where $\Phi_m^{(r)}(L;\{q\})$ is defined by (\[phir\]). From (\[vmatr\]) one can see that $\hat V^T=-\hat V$ and therefore, $\tilde V^T=-\tilde V$. So $\hat V_{mm}=\tilde V_{mm}=0$. Then for the degenerated configuration of Section 5 one obtains using (\[vmatr\]) (\[phir\]), that $det\tilde
V\sim\rho_1/\rho$ when both $L_1$ and $L_2$ are odd super-spin structures. From (\[umatr\]) $\det\hat U$ in (\[tzh\]) is not nullified when $1<n_1<n-1$, all the rest factors being finite. In this case the integration measure is $\sim(\rho_1/\rho)^5$. When both $L_1$ and $L_2$ are odd and either $n_1=1$, or $n_2=n-n_1=1$, then $\det\hat U$ is nullified due to a presence of the ghost (-1/2) zero genus-1 mode. Then $\det\hat U\sim(\rho_1/\rho)^3$ when $n_1=1$, and $\det\hat U\sim(\rho_1/\rho)^2$ for $n_2=1$ and $n>2$. In this case the integration measure is $\sim (\rho_1/\rho)^2$ and, respectively, $\sim
(\rho_1/\rho)^3$. Hence the integration measure is nullified when $L_1$ and $L_2$ are odd. Due to the second term on the right side of (\[rprta\]), the Green function is $\sim1/\sqrt{\rho_1}$ when one of its argument lies near $z_0$ (see Section 5) another argument being at a finite distance from $z_0$. So the integrand decreases also for the configuration where one from the vertex coordinates goes to $z_0$. So $L_1$ being odd, the degenerated configurations discussed are not able to originate divergences, as it has already been noted in the end of Section 5. In the $1<n_1<n-1$ case the $\sim(\rho_1/\rho)^5$ smallness in the integration measure is compensated when a number of the vertices is $\geq10$ and 5 vertices lay near $z_0$. This case is relevant for obtaining the contribution to (\[ampl\]) of odd super-spin structures [@dan96].
Property of the function invariant under the super-boosts
==========================================================
We consider the function $\psi(\{(x_r|\xi_r)\},\{(w_s|\iota_s)\})$ invariant under change (\[trco\]) of its $p$ arguments $(x_r|\xi_r)$ where $r=1,\dots,p$. Here $\iota_s=\nu_s-\mu_s$ while $w_s=v_s-u_s-\nu_s\mu_s$. Due to the invariance under the boosts, it depends on the differences $\{(x_r-x_s)\}$. We presents the above function as it follows $$\begin{aligned}
\psi(\{(x_r|\xi_r)\},\{(w_s|\iota_s)\})
=\xi_i\left(\prod_{j=2}^p
(\xi_j-\xi_1)\right)
\psi_0(\{x_r\},\{(w_s|\iota_s)\})
\nonumber\\
+\sum_{i=2}^p
\left(\prod_{j=2,j\neq i}^p
(\xi_j-\xi_1)\right)
\psi_i(\{\tilde x_r\},\{(w_s|\iota_s)\})
+\dots
\label{func}\end{aligned}$$ where $\tilde x_r=x_r-x_1$ while the dots encode lower powers of $\{\xi_r\}$. Really $\{(x_r|\xi_r)\}$ are identified with $(u_r|\mu_r)$ and with the vertex coordinates of interest. Applying (\[trco\]) to (\[func\]), one obtains the desired relation $$\psi_0(\{(x_r)\},\{(w_s|\iota_s)\})
=\sum_{j=2}^p(-1)^j\partial_{\tilde
x_j}\psi_j(\{(\tilde x_r)\}),\{(w_s|\iota_s)\}\,.
\label{prp}$$
SL(2) transformations
=====================
The general super-conformal transformation is given by $$z={\it f}(\hat z)+{\it f}'(\hat z)\hat\vartheta\xi(\hat z)\,,
\quad
\vartheta=\sqrt{{\it f}'(\hat z)}[(1+\frac{1}{2}\xi\xi')
\hat\vartheta+\xi(\hat z)]
\label{trn}$$ where ${\it f}'(z)=\partial_z {\it f}(z)$ while ${\it f}(z)$ is the transition function and $\xi(z)$ is the Grassmann partner. For the transformation, which preserving $z_1$, $z_2$ and $z_3$, reduces $\vartheta_1$ and $\vartheta_2$ to zeros, $${\it f}(\hat z)=\hat z-\frac{(\hat z-z_1)(\hat z-z_2)}
{(z_3-z_1)(z_3-z_2)}\hat\vartheta_3\xi_0(z_3)\,,
\quad
\xi(\hat z)=\frac{\vartheta_1(\hat z-z_2)}{(z_1-z_2)\sqrt{{\it
f}'(z_1)}} -\frac{\vartheta_2(\hat z-z_1)}{(z_1-z_2)\sqrt{{\it
f}'(z_2)}}
\label{trnsf}$$ where $\xi_0(z)=[\vartheta_1(z-z_2)-\vartheta_2(z-z_1)]/(z_1-z_2)$. Evidently, ${\it f}'(z_1){\it f}'(z_2)=1$. If $\vartheta_1=\vartheta_3=0$, then ${\it f}(\hat z)=\hat z$ and $\xi(\hat z)=-\vartheta_2(\hat z-z_1)/(z_1-z_2)$.
Under the $L(2)$ transformation given by (\[trn\]) with ${\it f}(\hat z)=g(\hat z)$ and $\xi(\hat z)\equiv0$, the spinor derivative of the vacuum correlator (\[corr\]) is changed as $$\begin{aligned}
D(t')\hat X_{L,L'}(t,\overline t;t',\overline t';\{q\})=
[g'(\hat z)]^{-1/2}D(\hat t')\biggl[
\hat X_{L,L'}(\hat t,\overline{\hat t};t',\overline{
\hat t'};\{q\})
\nonumber\\
-\hat X_{L,L'}(\hat t(\infty),\overline{\hat
t(\infty)};t',\overline{ \hat t'};\{q\})\biggl]
\label{trcorr}\end{aligned}$$ where $\hat t(\infty)=(\hat z=g^{(-1)}(\infty)|
\hat\vartheta=0)$ while $g^{(-1)}$ is the transformation inverse to $g$. The addition is determined by the requirement that the right side of (\[trcorr\]) vanishes at $z\to\infty$, just as its left part.
[99]{} M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, vols.I and II ( Cambridge Univesity Press, England, 1987). E. Verlinde and H. Verlinde, Phys. Lett. B192 (1987) 95. J. Atick and A. Sen, Nucl.Phys. B. 296 (1988) 157;\
J. Atick, J. Rabin and A. Sen, Nucl. Phys. B 299 (1988) 279. G. Moore and A. Morozov, Nucl. Phys. B 306 (1988) 387; P. Di Vecchia, K. Hornfeck, M. Frau, A. Ledra and S. Sciuto, Phys. Lett. B211 (1988) 301. J.L. Petersen, J.R. Sidenius and A.K. Tollst[é]{}n, Phys Lett. B 213 (1988) 30; Nucl. Phys. B 317 (1989) 109. B.E.W. Nilsson, A.K. Tollst[é]{}n and A. W[ä]{}tterstam, Phys Lett. B 222 (1989) 399. S. Mandelstam, Phys.Lett. B 277 ( 1992 ) 82. E. Martinec, Phys. Lett. B171 (1986) 189. G.S. Danilov, Sov. J. Nucl. Phys. 52 (1990) 727 \[ Jadernaja Fizika 52 (1990) 1143 \]; Phys. Lett. B 257 (1991) 285. G.S. Danilov, Phys. Rev. D51 (1995) 4359 \[Erratum-ibid. D52 (1995) 6201\]. G.S. Danilov, Nucl. Phys. B463 (1996) 443. G.S. Danilov, Phys. Atom. Nucl. 59 (1996) 1774 \[Yadernaya Fizika 59 (1996) 1837\]. M.A. Baranov and A.S. Schwarz, Pis’ma ZhETF 42 (1985) 340 \[JETP Lett. 49 (1986) 419\]; D. Friedan, Proc. Santa Barbara Workshop on Unified String theories, eds. D. Gross and M. Green ( World Scientific, Singapore, 1986). N. Seiberg and E. Witten, Nucl. Phys. B276 (1986) 272. D.V. Volkov, A.A. Zheltukhin and A.I. Pashnev, Sov. J. Nucl. Phys. 27 (1978) 131 \[Jadernaya Fizika 27 (1978) 243\]. A.A. Belavin and V.G. Knizhnik, Phys.Lett. B 168 (1986) 201; ZhETF 91 (1986) 364. G.S. Danilov, Surveys in High Energy Physics 14 (1999) 205 (Talk given on the $XXX$ PNPI Winter School, 1998). N. Berkovits, Nucl. Phys. B408 (1993) 43. V.S. Kaplunovsky, Nucl. Phys. B307 (1988) 145; Z. Bern and D.A. Kosower, Phys. Rev. D38 (1988) 1888; M. Frau, I. Pesando, S. Sciuto, A. Lerda and R. Russo, Phys. Lett. B400 (1997) 52. Z. Bern, L. Dixon and D.A. Kosower, Phys. Rev. Lett 70 (1993) 2677; P. Di Vecchia, A. Lerda, L. Magnea and R. Margotta, Phys Lett. B351 (1995) 445; P. Di Vecchia, A. Lerda, L. Magnea R. Margotta and R. Russo, Nucl. Phys. B469 (1996) 235. C.L. Siegal, Topics in Complex Function Theory, Vol. 3 (New York, Willey, 1973). L. Hodkin, J. Physics and Gravity, 6 (1989) 333. G.S. Danilov, JETP Lett. 58 (1993) 796 \[Pis’ma JhETF 58 (1993) 790.\] E. Martinec, Nucl. Phys. B 281 (1986) 157. L. Crane and J.M. Rabin , Commun. Math. Phys. 113 (1988) 601;\
J.D. Cohn, Nucl. Phys. B306 (1988) 239. G.S. Danilov, Phys. Atom. Nucl. 60 (1997) 1358 \[Yadernaya Fizika 60 (1997) 1495\]. V. Alessandrini, Nuovo Cim. 2A (1971) 321; L. Ford, Automorphic Functions (New York, Chelsea 1951). D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271 (1986) 93. P. Di Vecchia, M. Frau, A. Ledra and S. Sciuto, Phys. Lett. B 199 (1987) 49. G.S. Danilov, Sov. J. Nucl. Phys. 49 (1989) 1106 \[ Jadernaja Fizika 49 (1989) 1787 \].
[^1]: E-mail address: danilov@thd.pnpi.spb.ru
[^2]: The congruent points, or curves, or domains are those related by a group transformation other that the identical transformation [@ford].
[^3]: Through the paper the overline denotes the complex conjugation.
[^4]: The (super)-spin structure is even (odd), if $4l_1l_2=4\sum_{s=1}^ nl_{1s}l_{2s}$ is even (odd).
[^5]: The string tension is taken to be $1/\pi$
[^6]: Not every $\{\tilde G_s\}$ set is relevant to determine the isomorphism. As an example, the set formed by $G_1G_2G_1G_2^{-1}G_1^{-1}$ and $G_2$ is not isomorphic to the $(G_1,G_2)$ set since $G_1$ can not be represented by a group product constructed by using $G_1G_2G_1G_2^{-1}G_1^{-1}$ and $G_2$.
[^7]: As an example, $u_{g_1g_2}$ being the attractive point of the $g_1g_2$ transformation, $|Q_{g_1^{-1}}(u_{g_1g_2})|=
|Q_{g_1^{-1}}(g_1g_2(u_{g_1g_2}))|=|Q_{g_1}^{-1}(g_2u_{g_1g_2})|
\leq1$ due to the constraints above, $Q_G(z)=c_Gz+d_G$. Thus $u_{g_1g_2}$ lies inside $C_{v_1}$.
[^8]: As an example, $|Q{g_1g_2}(u_{g_3})|=
|Q_{g_1}(g_2(u_{g_3}))Q_{g_2}(u_{g_3})|\geq1$.
[^9]: This matrix is slightly different from the corresponding matrix in [@danphr]. Eq.(\[rprta\]) can be also obtained from eq.(59) of [@danphr] using the second of eqs.(52) in [@danphr].
|
---
abstract: 'Results will be presented from a study of the QCD transition region using $2+1$ flavors of fermions and a dislocation suppressing gauge action on a lattice with temporal extent of 8 and spatial extent 16 ($1.9-2.7$ fm). A series of temperatures from 140 through 200 MeV, separated by 10 MeV have been studied. All the simulations lie on a line of constant physics with 200 MeV pions, realized using domain wall fermion, a chirally symmetric fermion formulation. The chiral condensates, susceptibility, anomalous symmetry breaking and a detailed study of the Dirac spectrum will be described and compared with earlier staggered results.'
author:
- Zhongjie Lin
bibliography:
- 'panic\_proc.bib'
title: |
The QCD Phase Transition Region\
with Domain Wall Quarks
---
[ address=[Physics Department, Columbia University\
(HotQCD Collaboration)]{} ]{}
The chiral phase transition of strongly-interacting matter has been an intriguing topic for decades. Many significant efforts, both theoretical and experimental, have been dedicated to understanding this phenomena. Lattice Quantum Chromodynamics, since the day of its birth, has offered a powerful technique for an [*ab initio*]{}, non-perturbative study of the transition region. Here we present a preliminary study of the transition region using a chiral formulation of lattice fermions in order to more accurately realize the chiral symmetry responsible for the transition.
Gauge and Fermion actions
=========================
While the gauge field can be implemented on the lattice in a straightforward manner, the usual lattice fermion schemes suffer from problems such as breaking of chiral symmetry or creating doublers [*etc.*]{}, which interfere with the exploration of the QCD chiral phase transition using lattice methods. The domain wall fermion (DWF) formulation [@Kaplan:1992bt; @Furman:1994ky], a variant of Wilson fermions, avoids these problems at the cost of introducing an extra, $s$ dimension. The left and right-handed chiral states are bound to the four dimensional boundaries (walls) at the two ends of the 5-D volume. Unphysical, massive degrees of freedom from the fifth dimension are to a large extent canceled by the introduction of a Pauli-Villars term in the simulation. The low modes produce a discrete, four-dimensional version of QCD.
The DWF formulation has chiral breaking symmetry under good control. The residual chiral symmetry breaking is characterized by a residual, additive quark mass: $ m_\mathrm{res}(L_s) = c_1\exp{(-\lambda_cL_s)}/L_s +c_2/L_s$. If the extension in fifth dimension, $L_s$ is taken to infinity, one will obtain the overlap fermion formulation which exactly preserves chiral symmetry. The infinite $L_s$ limit commutes with the continuum extrapolation, allowing us to choose an appropriate $L_s$ to circumvent the heavy computational burden of overlap fermions while keeping the chiral symmetry breaking minimal. Another benefit of the DWF formalism is that $U(1)_A$ symmetry is only broken by axial anomaly in contrast to staggered fermions, where the anomalous symmetry is also broken by $a^2$ artifacts.
In the simulation of QCD thermodynamics on a lattice with constant physical quark masses, one naturally needs a small bare quark mass. Since the residual mass is an additive correction to the bare quark mass in DWF formalism, [*viz.*]{} $m_q=m_\mathrm{res}+m_\mathrm{input}$, we must also minimize $m_\mathrm{res}$. From expression for $m_{\rm res}$, the residual mass can be decomposed into two parts. The first part is produced by standard five dimensional states and is exponentially suppressed; the other, comes from gauge field dislocations associated with changing topology and is only suppressed by $1/L_s$. Therefore, naively increasing $L_s$ will not only be computationally expensive but also inefficient. Fortunately, the topology change associated with the second term also leads to zero modes of an unphysical 4-dimensional Dirac operator, $D_W(-M_5)$ so we could multiply a factor ([*e.g.*]{} determinant of that Dirac operator) to suppress those configurations. Since such a factor will freeze topological tunneling, we add a twisted mass term to the determinant and introduce the ratio: $$\label{eqn:dsdr}
\mathcal{W}(M_0, \epsilon_b, \epsilon_f) =
\frac{\det\left[D^\dagger_W(-M_0)D_W(-M_0) + \epsilon^2_f\right]}
{\det\left[D^\dagger_W(-M_0)D_W(-M_0) + \epsilon^2_b\right]}.$$ This is the dislocation suppression determinant ratio (DSDR) [@Vranas:1999rz; @Fukaya:2006vs; @Renfrew:2009wu]. This alters predominately the ultraviolet part of the theory, so the physical quantities in which we are interested are minimally affected.
$T\,(\mathrm{MeV})$ $\beta$ $L_s$ $m_{\mathrm{res}}a$ $m_la$ $m_sa$ ${\rule{0pt}{2.6ex}}{\rule[-1.2ex]{0pt}{0pt}}\left<{\bar{\psi}\psi}\right>_l/T^3$ $\Delta{\bar{\psi}\psi}/T^3$ $\chi_{l,disc.}/T^2$
--------------------- --------- ------- --------------------- ---------- -------- ----------------------------------------------------------------------------------- ------------------------------ ----------------------
140 1.633 48 0.00612 -0.00136 0.0519 6.26(12) 7.74(12) 36(3)
150 1.671 48 0.00296 0.00173 0.0500 6.32(29) 6.10(29) 41(2)
150 1.671 32 0.00648 -0.00189 0.0464 8.39(10) 7.06(10) 44(3)
160 1.707 32 0.00377 0.000551 0.0449 5.25(17) 4.83(17) 43(4)
170 1.740 32 0.00209 0.00175 0.0427 4.03(18) 2.78(18) 35(5)
180 1.771 32 0.00132 0.00232 0.0403 3.16(15) 1.56(15) 25(4)
190 1.801 32 0.00076 0.00258 0.0379 2.44(9) 0.71(9) 11(4)
200 1.829 32 0.00046 0.00265 0.0357 2.19(8) 0.47(8) 10(3)
: Summary of the ensembles.[]{data-label="tab:conf"}
Table \[tab:conf\] summarizes the ensembles in our simulation. We use $2+1$ flavors of DWF with Iwasaki and DSDR gauge action. The spatial extent of the lattice is 16 (about 1.9 - 2.7 fm). The temporal extent is fixed to be 8 and the $\beta$ values are chosen so that we have a range of temperatures from 140 to 200 MeV. The bare quark masses are also adjusted to make our ensembles lie on a line of constant physics with a pion mass of about 200 MeV.
Chiral Symmetry
===============
The first quantity of interest is the chiral condensate shown as a function of Monte Carlo time in fig. \[fig:pbp\]. As compared to those below 160MeV, the evolutions of the chiral condensate for temperatures above 160MeV display a distinct character, fluctuating above a base line. The graph of the disconnected susceptibilities gives a more direct indication of a transition temperature around 160 MeV, which agrees quite well with the staggered results. However, since our spatial volume is quite limited (an aspect ratio 2), a quantitative comparison is likely premature.
\[fig:pbp\]
The chiral condensate is closely related to the small eigenvalues of the Dirac operator, which thus provides an alternative perspective into the symmetries associated with the QCD phase transition. For instance, the well-known Banks-Casher relation [@Banks:1979yr], $\lim_{m_l\to0}\lim_{V\to\infty}
\langle\bar{\psi}_l\psi_l\rangle =-\pi\lim_{\lambda\to0}\lim_{m_l\to0}
\lim_{V\to\infty}\rho(\lambda)$, relates the chiral condensate and the $\lambda=0$ value of the density $\rho(\lambda)$ of Dirac eigenvalues.
The Dirac spectrum also provides unique insights into the anomalous $U(1)_A$ symmetry. Specifically, the $U(1)_A$-breaking difference of pseudoscalar and scalar susceptibilities is given by: $$\label{eqn:delta}
\Delta_{\pi-\delta} \equiv \frac{\chi_{\pi} - \chi_{\delta}}{T^2}
= \int\mathrm{d}\lambda\ \rho(\lambda)
\frac{4m_l^2}{\left(m_l^2+\lambda^2\right)^2}.$$ While $\Delta_{\pi-\delta}$ is expected to receive a small, dilute instanton gas contribution from $\rho(\lambda) \propto m^2\delta(\lambda)$ at large temperatures, there may also be larger, lower-temperature non-semi-classical contribution from $\rho(\lambda) \propto \lambda$. We used Kalkreuter-Simma’s method [@Kalkreuter:1995mm] to calculate the lowest 100 eigenvalues of the hermitian version of DWF Dirac operator. The eigen-spectrum is then renormalized using an method similar to Giusti and Luscher [@Giusti:2008vb].
-0.2in
\[fig:eig2\]
As can be seen in fig. \[fig:eig2\], the renormalization aligns the light quark mass (dashed vertical line) with the near-zero mode peak. It also compensates for the discrepancy in the Banks-Casher relation, which however still does not agree at about $50\%$ level below the transition. We attribute the discrepancy to the finite volume or finite mass effects. Above 170MeV, the chiral condensate and eigendensity both start to vanish as expected for temperatures above the transition. But it remains unclear whether the slope of the eigendensity vanishes at 180MeV. At even higher temperature, a possible gap starts to emerge, indicating an effective restoration of $U(1)_A$ symmetry.
We conclude that DWF provides a well-suited tool to study the QCD phase transition. From the chiral susceptibility, we find indication of a transition temperature around 160MeV. Well above the transition, we observe possible signals of $U(1)_A$ restoration. Larger volume results and chiral extrapolation will be needed for a more definite conclusion.
I appreciate help and advice from N. Christ, F. Karsch and R. Mawhinney and thank M. Cheng, P. Hegde and all members of HotQCD as well as Columbia colleagues H. Yin and Q. Liu for help and discussions. This work was supported in part by U.S. DOE grant DE-FG02-92ER40699. The simulations were carried out on the BG/P machine at LLNL, the DOE- and RIKEN-funded QCDOC machines and NYBlue machine at Brookhaven National Lab.
|
---
abstract: 'We present three-dimensional kinematics of Sagittarius (Sgr) trailing tidal debris in six fields located 70-130$\arcdeg$ along the stream from the Sgr dwarf galaxy core. The data are from our proper-motion (PM) survey of Kapteyn’s Selected Areas, in which we have measured accurate PMs to faint magnitudes in $\sim40\arcmin\times40\arcmin$ fields evenly spaced across the sky. The radial velocity (RV) signature of Sgr has been identified among our follow-up spectroscopic data in four of the six fields and combined with mean PMs of spectroscopically-confirmed members to derive space motions of Sgr debris based on $\sim$15-64 confirmed stream members per field. These kinematics are compared to predictions of the @lm10a model of Sgr disruption; we find reasonable agreement with model predictions in RVs and PMs along Galactic latitude. However, an upward adjustment of the Local Standard of Rest velocity ($\Theta_{\rm LSR}$) from its standard 220 km s$^{-1}$ to at least $232\pm14$ km s$^{-1}$ (and possibly as high as $264\pm23$ km s$^{-1}$) is necessary to bring 3-D model debris kinematics and our measurements into agreement. Satisfactory model fits that simultaneously reproduce known position, distance, and radial velocity trends of the Sgr tidal streams, while significantly increasing $\Theta_{\rm LSR}$, could only be achieved by increasing the Galactic bulge and disk mass while leaving the dark matter halo fixed to the best-fit values from @lm10a. We derive low-resolution spectroscopic abundances along this stretch of the Sgr stream and find a constant \[Fe/H\] $\sim$ -1.15 (with $\sim0.5$ dex scatter in each field – typical for dwarf galaxy populations) among the four fields with reliable measurements. A constant metallicity suggests that debris along the $\sim60\arcdeg$ span of this study was all stripped from Sgr on the same orbital passage.'
author:
- 'Jeffrey L. Carlin, Steven R. Majewski, Dana I. Casetti-Dinescu, David R. Law, Terrence M. Girard, and Richard J. Patterson'
title: Kinematics and Chemistry of Stars Along the Sagittarius Trailing Tidal Tail and Constraints on the Milky Way Mass Distribution
---
Introduction
============
With the profusion of data provided in recent years by deep, large-area photometric surveys such as the Two Micron All Sky Survey (2MASS) and Sloan Digital Sky Survey (SDSS), a wealth of stellar substructure has been uncovered in the Milky Way (MW) halo. The finding and subsequent mapping of numerous stellar tidal streams and overdensities (e.g., Sagittarius — @ili+01 [@msw+03; @bze+06]; Monoceros — @nyr+02 [@iil+03; @yng+03]; other SDSS streams — @g09 [@bei+07; @g06; @g06a; @gd06]) has borne out the idea [@sz78; @m93; @mmh96] that remnants of accreted dwarf galaxies make up much of the stellar halo of the Milky Way. The direct confirmation of the accretion of late-infalling subhalos via discovery of ubiquitous long-lived, coherent tidal debris streams has provided strong constraints on models of small-scale hierarchical structure formation under the prevailing $\Lambda$-Cold Dark Matter ($\Lambda$CDM) cosmology (e.g., @ans+03a [@bj05; @fjb+06]). Furthermore, because the tidal streams retain the kinematical signatures of the orbits of their progenitors (i.e., angular momentum and energy), stellar debris in the streams can be used as sensitive probes of the underlying Galactic gravitational potential (e.g., @jzs+99 [@ili+01; @h04; @mga+04; @jlm05; @ljm05; @mlp+06]).
{width="4.25in"}
The best-known and [*only*]{} widely agreed-upon case of a presently visible dwarf galaxy undergoing tidal disruption in the Milky Way halo is the Sagittarius (Sgr) dwarf spheroidal (dSph)[^1]. The core of this galaxy was first discovered by @igi94 in a kinematical study of the outer Galactic bulge, with the first large-scale mapping of the Sgr leading and trailing tidal arms done by @msw+03 using 2MASS M-giant stars. Various studies have reported the discovery of stars (e.g., @msw+03 [@mga+04; @bze+06; @ynj+09; @cbi+10]; a comprehensive summary of the earlier detections appears in @msw+03) or star clusters (e.g., Pal 12: @dmg+00; Whiting 1: @czm07; many clusters: @bfi03; a summary of Sgr clusters appears in @lm10b) plausibly associated with debris from Sgr, either trailing or leading it along its orbit. Line-of-sight velocities (i.e., [*radial velocities*]{}, or RVs) of Sgr members have been determined at a few positions along the stream (e.g., @dhm+01 [@mkl+04; @mbb+07]), and, along with the spatial distribution of these stars, provide constraints on models of the Sgr-Milky Way interaction (e.g., @jsh95 [@hw01; @ili+01; @h04; @mga+04]). A comprehensive effort at modeling the Sgr disruption constrained by all observations available after about a decade of study was done by @ljm05, who were able to reproduce most extant data, but were unable to completely reconcile the apparent need for a prolate MW halo potential to produce the leading arm [*radial velocities*]{} on the one hand, and an oblate halo to match the [*positions*]{} of leading debris on the other. This contradiction has apparently been recently resolved by @lmj09, who propose that the Milky Way might have a triaxial halo; a comprehensive $N$-body model based on the best-fitting triaxial halo [@lm10a hereafter LM10] reasonably matches nearly all existing constraints (spatial [*and*]{} kinematical) of Sgr tidal debris.[^2] The Sagittarius dwarf and its tidal debris are thus proving to be an excellent laboratory for studying both the dynamics of tidally disrupting dwarf galaxies and star stream formation, as well as the shape and strength of the Galactic gravitational potential that is the cause of this disruption. It is this model of to which we shall compare our data throughout this work.
Our Proper Motion Survey
------------------------
To date, no systematic survey has addressed the tangential velocities (derived from proper motions) of the identified major Galactic tidal streams. Only a few studies (e.g., @dmg+02 [@ccg+08; @cgm+09a; @ccg+10; @krh10]) have published [*any*]{} proper motion results for major Galactic substructures, and typically not at a level of precision that is useful for constraining dynamical models of tidal stream production and evolution. In an effort to detect and characterize halo substructures, we have been working on a project to obtain full phase-space information (positions and full 3-D space motions) for individual stars in Kapteyn’s Selected Areas (SAs; see @cmg+06 for an overview of this project). The sky positions of the Selected Areas were chosen by Jacobus @k06 to provide evenly spaced coverage for a systematic exploration of Milky Way structure. We have attempted to carry on at least part of this legacy by taking advantage of Mt. Wilson 60-inch telescope photographic plate material taken by Kapteyn and collaborators [@skv+30] for their survey to make up the first-epoch data of our survey (in particular, near-equatorial fields at $\delta = 0\arcdeg,
+15\arcdeg$, and $-15\arcdeg$). The distribution on the sky of those SAs that make up our survey is shown in an Aitoff projection in Figure \[fig:radec\]. Some of the equatorial SA fields lie along the orbit of the Sagittarius dwarf galaxy (which is approximated by the blue, solid curve in Figure \[fig:radec\]) – it is a subset of these fields (in particular, six fields along the trailing tidal tail; see the right panel of Figure \[fig:radec\]) that are the focus of the present work.
Constraints on the Local Standard of Rest Velocity
--------------------------------------------------
The positions of tidal debris that have been found over a large stretch of the Sagittarius orbit place fairly strong constraints on the three-dimensional motions of the Sgr dwarf. It is, however, important to confirm and refine the models by measuring space velocities of stream stars (especially proper motions, which are difficult to measure for stars in distant Galactic substructures). In the case of the Sgr trailing tail, however, proper motions measured for debris stars are also rather sensitive to the Sun’s motion through the Galaxy. This arises because much of the Sagittarius trailing tidal tail is positioned at a roughly constant distance below the Galactic plane, with the Sgr orbital plane nearly coincidental with the Galactic $X_{\rm GC}-Z_{\rm GC}$ plane.[^3] @mlp+06 [hereafter “MLPP”] noted that because of this orientation, longitudinal proper motions of Sgr trailing debris located sufficiently far away from the South Galactic Pole contain virtually no contribution from Sagittarius motions, and almost entirely reflect the solar motion.
Efforts to measure fundamental dynamical properties of the Milky Way, such as its rotation curve, $\Theta(R)$, are complicated by our Sun’s own (poorly known) motion within the Galaxy. Measurements of $\Theta_{\rm LSR}$, the Galactic rotation speed at the solar circle (the [*Local Standard of Rest*]{}, “LSR”), vary by 25%, despite many efforts at its determination. The value adopted by the IAU in 1985 of $\Theta_{\rm LSR}$ = 220 km s$^{-1}$ (see @kl86) has long represented a reasonable approximation to existing measurements (note, however, that prior to the 1985 IAU adoption of $\Theta_{\rm
LSR}$ = 220 km s$^{-1}$, the 1964 IAU general assembly adopted 250 km s$^{-1}$; see a listing of pre-1985 measurements of $\Theta_{\rm LSR}$ in @kl86). Constraints taking into account the ellipticity of the disk have suggested the LSR velocity could be as low as $\sim$180 km s$^{-1}$ [@kt94]. A similarly low value of $184 \pm 8$ km s$^{-1}$ was found by @om98, who modified previous methods of determining the Oort constants by including radial variations of gas density in their mass modeling of the Galactic rotation curve. Proper motions of Galactic Cepheids from $Hipparcos$ [@fw97] yield a result of $\Theta_{\rm LSR} =(217.5 \pm 7.0) (R_0/8)$ km s$^{-1}$ (where $R_0$ is the distance from the Sun to the Galactic center; the IAU adopted value is $R_0 = 8.5$ kpc), in line with the IAU standard. Using re-reduced $Hipparcos$ data [@v07] with improved systematic errors, @yzk08 found $\Theta_{\rm LSR} =(243 \pm 9)
(R_0/8)$ km s$^{-1}$ based on thin-disk O-B5 stars. Estimates based on absolute PMs of Galactic bulge stars in the field of view of globular cluster M4 using the [*Hubble Space Telescope (HST)*]{} yield $(202.4\pm20.8) (R_0/8)$ km s$^{-1}$ [@krh+04] and $(220.8 \pm
13.6) (R_0/8)$ km s$^{-1}$ [@bpk+03] (with both studies using data from the same $HST$ observations). Long-term $VLBA$ monitoring of Sgr A\*, the radio source at the Galactic center, led to a proper motion of Sgr A\* from which @rb04 revised the LSR velocity upward to $(235.6\pm1.2) (R_0/8)$ km s$^{-1}$. @gsw+08 combined stellar kinematics near the Galactic center with the proper motion of Sgr A\* to derive $(229
\pm 18) (R_0/8.4)$ km s$^{-1}$. More recently, $\Theta_{\rm LSR}$ has been suggested to be even higher, $(254\pm16) (R_0/8.4)$ km s$^{-1}$, based on trigonometric parallaxes of Galactic star-forming regions [@rmz+09]. Reanalysis of these same data, including the Sgr A\* proper motion, by @bhr09 found a similar $(244\pm13)$ km s$^{-1}$. @krh10 provided constraints on the MW halo potential by analysing the GD-1 [@gd06] stellar stream, combining SDSS photometry, USNO-B+SDSS proper motions (see @mml+04 [@mml+08]), and spectroscopy to obtain 6-D phase-space data over a large stretch of the stream, which they used to estimate $\Theta_{\rm LSR} =
(224\pm13) (R_0/8.4)$ km s$^{-1}$ (though this result is made somewhat more uncertain due to a systematic dependence on the flattening of the disk+halo potential). Finally, a combined estimate including many of the above results as priors finds a value of $(236\pm11) (R_0/8.2)$ km s$^{-1}$ [@bhr09]. Most of the estimates discussed here rely on the Oort constants, and thus are dependent on our incomplete knowledge of $R_0$. Despite numerous attempts at determining the circular velocity at the solar circle, this constant remains poorly constrained. It is clear that independent methods would be valuable to obtain alternative estimates of $\Theta_{\rm LSR}$.
Here we use a new, independent method for ascertaining $\Theta_{\rm
LSR}$ that has the advantage over most of the previously mentioned methods in that the results have virtually complete decoupling from an assumed value of $R_0$. As discussed in , the trailing arm of the Sagittarius tidal stellar stream is ideally placed to serve as an absolute velocity reference for the LSR. With an orbital pole of $(l_p,b_p) = (274,-14)\arcdeg$, Sgr is almost on a polar orbit, and the line of nodes of the intersection of the Galactic midplane and the Sgr debris plane is almost coincident with the Galactic $X_{\rm GC}$ axis (the axis containing the Sun and Galactic center). This is illustrated in Figure \[fig:sgr\_xyz\], which shows the projection of Sgr debris from the model onto the Galactic $X_{\rm GC}-Z_{\rm GC}$, $Y_{\rm GC}-Z_{\rm GC}$, and $X_{\rm GC}-Y_{\rm GC}$ planes. In the upper left panel (the $X_{\rm
GC}-Z_{\rm GC}$ plane), the Sgr orbital plane is nearly face-on, while in the other two panels, few Sgr debris points are seen more than $\sim5$ kpc on either side of the $X_{\rm GC}-Z_{\rm GC}$ plane (i.e., $|Y_{\rm GC}| \lesssim 5$ kpc for nearly all Sgr debris). The motions of Sgr stars [*within*]{} its (virtually non-precessing; @jlm05) debris plane, as observed from the LSR, are therefore almost entirely in the Galactic $U$ and $W$ velocity components (i.e., in the $X_{GC}$-$Z_{GC}$ plane), whereas $V$ motions of Sgr tidal tail stars almost entirely reflect [*solar motion*]{} — i.e., $\Theta_{\rm LSR}$ (plus the Sun’s peculiar motion in $V$, established to be in the range $\sim$ +5 to +12 km s$^{-1}$; e.g., @db98). The Sgr trailing tail is positioned fairly equidistantly from the Galactic disk for a substantial fraction of its stretch across the Southern MW hemisphere [@msw+03]. This band of stars arcing almost directly “beneath” us within the $X_{GC}$-$Z_{GC}$ plane (see the upper panel of Figure \[fig:sgr\_xyz\]) provides a remarkable, stationary zero-point reference against which to make direct measurement of the solar motion [*almost completely independent of the Sun’s distance from the GC.*]{}
Because of the fortuitous orientation of the Sgr debris, the majority of $\Theta_{\rm LSR}$ motion (i.e., $V$) is seen in the proper motions of these stars, with the reflex solar motion almost entirely contained in the $\mu_l \cos($b) component for Sgr trailing arm stars (at least for those stream stars away from the South Galactic Pole (SGP) coordinate “discontinuity", where the $\mu_l \cos($b) of Sgr stream stars switches sign). Fig. 4 of shows the essence of the proposed experiment via measurement of $\mu_l \cos($b) for Sgr trailing arm stars, which shows a trend with debris longitude, $\Lambda_{\sun}$[^4], that reflects the solar motion. In the region from $100\arcdeg \lesssim \Lambda_{\sun}
\lesssim 200\arcdeg$, $\mu_l \cos($b) is nearly constant, because the motion of Sgr debris contributes little to the $V$-component of velocity. Thus accurate measurement of $\mu_l \cos($b) for Sgr trailing tail stars along this stretch of the stream will provide a means of estimating $\Theta_{\rm LSR}$ with almost no dependence on $R_0$.
Five of the Kapteyn fields for which we have precise ($\sim$1 mas yr$^{-1}$) proper motions lie squarely on the Sgr trailing arm in this $\Lambda_{\sun}$ range, and one other (SA 92) is on the periphery of the stream. In Section \[nbodyresults.sec\], we will use the mean Sgr debris proper motions derived in four of these six fields to derive constraints on $\Theta_{\rm LSR}$.
Metallicities and Detailed Abundances of the Sagittarius System
---------------------------------------------------------------
@cmc+07 presented one of the first studies of high-resolution spectroscopic metallicities derived for Sgr debris. Their work showed that M-giants along the Sagittarius leading stream exhibit a significant metallicity gradient (which had previously been suggested to be present over smaller separations from the Sgr core based on photometric techniques; e.g., @a01 [@mga+04; @bnc+06]), decreasing from a mean \[Fe/H\] = -0.4 in the core to $\sim -0.7$ between $\sim 60-120\arcdeg$ from the core (i.e., between $300 > \Lambda_{\sun} > 240\arcdeg$) , and to $\sim
-1.1$ at $\gtrsim300\arcdeg$ from the main Sgr body. Such a population gradient along the stream likely arose due a strong metallicity gradient being present in the dSph before its tidal disruption; thus the outer, more metal-poor populations were preferentially lost as tidal stripping progressed at earlier times relative to the more intermediate-age (and higher metallicity) populations remaining in the core (a mechanism for this process has been demonstrated in the context of an $N-$body model by ). In addition, apparently some younger populations were formed even after Sgr began disrupting. The existence of a population gradient has also been seen by @bnc+06, who found that the relative numbers of blue horizontal branch (BHB) stars to red clump (RC, or red horizontal branch) stars are much higher in a leading stream field than in the Sgr core. Since BHB stars arise in older, more metal-poor populations than the RC stars, this must indicate that the stripped population was made up of predominantly older, less-enriched stars than remain in the core today. @kyd10 extended the search for chemical evolutionary signatures to the trailing tail of Sgr, observing a handful of stars selected from the 2MASS M-giant catalogs of @msw+03 at high resolution in each of two fields at distances of $66\arcdeg$ and $132\arcdeg$ from the core. Keller et al. combined the mean metallicities in these two fields with the \[Fe/H\] = -0.4 result for the Sgr core from @mbb+05, and derived a metallicity fit as a function of $\Lambda_{\odot}$ of $\Delta$\[Fe/H\] = (-2.4$\pm$0.3) $\times 10^{-3}$ dex degree$^{-1}$. This trend (seen in their Figure 4) also passes through the mean metallicity of \[Fe/H\] $\approx$ -0.6 derived by @mbb+07 in a narrow region centered at $\Lambda_{\odot} = 100\arcdeg$.
For consistency, all of the data included in the @kyd10 study (including those from @mbb+05 [@mbb+07] and @ccm+10) were derived from M giants, which are, however, biased toward metal-rich, and therefore relatively younger, stars. In the current study, we explore the metallicity in fields between $75 <
\Lambda_{\sun} < 130\arcdeg$ from the Sgr core along the trailing tail using predominantly main sequence stars. Such stars near the main sequence turnoff are much less prone to metallicity biases than M giants, because MSTO stars are present in all stellar populations. An additional advantage of focusing on MSTO stars is that the number density of turnoff stars is much higher than both young, M giant tracers and older horizontal-branch stars; this provides us a much larger sample with which to characterize the Sgr trailing tail metallicity. Older trailing debris populations have recently been studied by @sig+10, who used SDSS Stripe 82 data to develop a new technique for estimating metallicity from photometric data where both RR Lyrae variables and main-sequence stars from the same structure can be identified. Their work found a constant \[Fe/H\] = -1.20$\pm$0.1 for Sgr debris along much of the same region of the trailing tail we are studying. In Section 5 we explore the Sgr trailing tail metallicity based on our samples of predominantly MSTO stars.
An important diagnostic of the star-formation timescale in a system is the $\alpha$-element abundance; the $\alpha$ elements (e.g., Mg, Ca, Ti) are produced mainly in Type II supernovae (SNe), which are the evolutionary endpoints of massive stars that dominate the chemical evolution at early times. Once Type Ia SNe begin to occur, the \[$\alpha$/Fe\] ratio will decrease, because $\alpha$-elements are less effectively produced by these supernova progenitors, while the overall metallicity, \[Fe/H\], will continue to increase. This produces a “knee” in the \[$\alpha$/Fe\] vs. \[Fe/H\] diagram, which acts essentially as a chronometer for a given system, since the \[Fe/H\] of the knee indicates the transition from SNII-dominated evolution to SNIa contributions. This phenomenon has been seen in a number of dSph systems, which typically show lower \[$\alpha$/Fe\] at a given \[Fe/H\] than Galactic populations because of a slower enrichment (e.g., @scs01 [@svt+03; @vis+04; @tvs+03; @gsw+05]; see also a recent review by @tht09). However, at the lowest metallicities, the \[$\alpha$/Fe\] of dSphs more closely resemble those of the MW halo. Studies by @sbb+07 and @mbb+05 found the same underabundance of $\alpha$-elements relative to the Milky Way for the core of the Sagittarius dSph. This finding has been extended into the Sgr trailing stream by @mbb+07, and into the leading arm by @ccm+10, with both studies using M-giants from the catalog of @msw+03. However, M giants are biased to higher metallicity and more recently star-forming population(s) of Sgr, so a natural next step in understanding the evolution of the original, pre-disruption Sgr dSph is to derive detailed abundances (especially for s-process and $\alpha$-elements) for a significant sample of the more metal-poor, older stars populating the core or, more accessibly, in Sgr’s more nearby streams. In Section 5 we present relative Mg abundances derived from our spectra. We show that the majority of confirmed old, metal-poor Sgr stream members appear to have distinct Mg abundances from those of the Milky Way stellar populations along the lines of sight probed.
Goals of This Paper
-------------------
Here, we present data in six of the Kapteyn’s Selected Areas from our deep proper-motion survey [@cmg+06]. In these six fields intersecting the trailing tidal tail of the Sgr system, we have augmented our proper-motion catalogs with follow-up spectroscopy. Sgr debris has been identified from among the stars with measured radial velocities, and these Sgr candidates are used to derive the mean three-dimensional kinematics and chemistry of the Sgr trailing stream.
In Section \[photom\_pm.sec\], we briefly introduce the proper motion survey (a more detailed discussion of the survey appears in @cmg+06), and discuss in depth the spectroscopic observations with the WIYN+Hydra and MMT+Hectospec multifiber instruments that yielded a total of $>1500$ radial velocities among proper motion-selected stars within the six fields of view. Section \[rv\_meas.sec\] will detail the final selection of candidate Sgr debris in each of our fields based on RV, proper motion, and color-magnitude selection. Section \[sgr\_kinematics.sec\] presents maximum-likelihood estimates of the most precise proper motions ($\sim1-2$ mas yr$^{-1}$ per star, or $\sim 0.2-0.7$ mas yr$^{-1}$ mean for each field) yet measured for Sagittarius debris. These measured kinematics are compared to the models of @lm10a, and found to agree rather well with the predictions for Sgr debris motions. However, we follow in Section \[mw\_constraints.sec\] with an analysis of the residual [*disagreement*]{} between our measurements and the models, or more accurately, we use the discrepancy to reassess the magnitude of the solar reflex motion, which is the dominant contributor to the proper motions in the direction of Galactic longitude. We show that our proper motion data (specifically, $\mu_l \cos $b) are inconsistent with the standard IAU value of 220 km s$^{-1}$ for the Local Standard of Rest motion at the $\sim1-2\sigma$ level and favor a significantly higher value, consistent with several of the most recent $\Theta_{\rm
LSR}$ studies using radio techniques. In Section \[sgr\_abund.sec\] we apply a software pipeline designed to derive stellar abundances from low-resolution spectra to the numerous spectra we have obtained for this project. While the metallicities show a hint of a gradient among the metal-poor stars in our study consistent with previous work, we cannot rule out a constant \[Fe/H\] over the range of stream longitude covered. We also examine the relative magnesium and iron index strengths for information on $\alpha$-abundance patterns of Sgr debris. We find Mg abundances of Sgr members are typically lower at a given \[Fe/H\] than field stars, consistent with the behavior seen in most MW dSphs. Finally, Section \[summary.sec\] concludes with a brief summary of our work, and future avenues these data can be used to explore.
The Data
========
Field Locations
---------------
The data discussed here are part of our ongoing deep proper-motion survey in a subset of Kapteyn’s Selected Areas (@m92; @cmg+06) at declinations of $\pm15\arcdeg$ and $0\arcdeg$. The survey as designed by @k06 consists of $\sim1\arcdeg$ fields evenly spaced at $\sim15\arcdeg$ intervals along strips of constant declination (see Fig. 1 in @cmg+06). A handful of our near-equatorial survey fields (see the left panel of Figure \[fig:radec\]) fall on or near the location of Sgr trailing tidal debris as mapped by @msw+03 using M-giants from 2MASS. The location of our fields relative to the models (constrained by the Majewski et al. data, among others) of @lm10a can be seen in the right panel of Figure \[fig:radec\], which suggests that we can expect a significant contribution from Sgr debris to the stellar populations along these lines of sight. In @cmg+06 and @ccg+08 we showed that faint ($V$ or $g \gtrsim 20$), blue ($B-V < 0.8$ or $g-r < 0.6$) overdensities (at colors and magnitudes consistent with the expected Sgr main sequence turnoff) in the color-magnitude diagrams of those SA fields intersecting the Sgr orbital path also show clumping in the distributions of their proper motions; these excesses and their clumping in proper motion space suggest that a distant, common-motion population, likely to be Sgr tidal debris, is present in these fields. In this work, we focus on six fields: SAs 116, 117, 92, 93, 94, and 71 (listed in order of increasing $\Lambda_{\sun}$). Coordinates for these fields are given in Table \[tab:sa\_tab\], which includes equatorial and Galactic positions as well as the longitude and latitude in the Sagittarius coordinate system. In this study, we are discussing fields that are $\sim74\arcdeg - 128\arcdeg$ from the core of the Sgr dSph, along its trailing stream.
The positions of the SAs in this study in the Galactic Cartesian $X_{\rm GC}-Z_{\rm GC}$ plane are shown in the upper left panel of Figure \[fig:sgr\_xyz\], overlaid atop simulated Sgr debris from the model. Of course, to place points on this figure for the SAs requires an estimate of heliocentric distance. Where needed throughout this work, we use mean distances to Sgr debris in each SA field estimated from the model debris along corresponding lines of sight. We have chosen to do this rather than measure Sgr debris distances because (a) our data in most fields don’t reach much fainter than the main sequence turnoff of Sgr, and (b) the expected line-of-sight depth of the Sgr stream in this portion of the trailing tail is $\sim10$ kpc, which “smears” the main sequence out by as much as $\sim0.5$ magnitudes. Both of these factors make isochrone fitting to derive distances rather unconstrained, so for all analysis requiring a distance estimate we adopt the mean model debris distances from at each position, with the line-of-sight depth of the stream in each field defining the uncertainty in the distance. These values are given in Table \[tab:sgr\_kinematics\] below.
[cccccccc]{}\
\
& & & & & & &\
& (J2000.0) & (J2000.0) & (degrees) & (degrees) & (degrees) & (degrees) &\
\
71 & 03:17:11.5 & 15:24:57.6 & 167.1 & -34.7 & 128.2 & -5.6 & 0.19\
94 & 02:55:58.1 & 00:30:03.6 & 175.3 & -49.3 & 116.3 & 4.8 & 0.09\
93 & 01:54:52.1 & 00:46:40.8 & 154.2 & -58.2 & 103.2 & -3.2 & 0.03\
92 & 00:55:03.8 & 00:47:13.2 & 124.9 & -62.1 & 90.1 & -10.6 & 0.03\
117 & 01:17:04.1 & -14:11:13.2 & 149.0 & -75.7 & 87.6 & 5.1 & 0.02\
116 & 00:18:08.4 & -14:19:19.2 & 90.1 & -75.0 & 74.9 & -1.4 & 0.02\
Photometry and Proper Motions {#photom_pm.sec}
-----------------------------
For those fields (SAs 92, 93, 94, and most of SA 116) that lie within the Sloan Digital Sky Survey (SDSS) footprint, we have used photometry (shown in Figure \[fig:gr\_cmd\_vpd\]) from SDSS Data Release 7 (DR7; @aaa+09) for our analyses. The remaining photometric data for this survey (for SAs 71 and 117) are photographic and derived from the late epoch (du Pont 2.5-m telescope) plates from which the proper motions were measured. Calibration of the photographic magnitudes in the blue (IIIa-J+GG385) and visual (IIIa-F+GG495) passbands onto the standard Johnson-Cousins system was achieved using CCD photometry taken in 1997-1998 with the Swope 1-m at Las Campanas Observatory. However, that UBV CCD photometry only covers a small portion ($\sim20-30\%$) of each field, and is shallower than the magnitude limit of the photographic plates by 1-1.5 magnitudes for red stars, and $\sim0.3$ magnitudes for blue stars, yielding poor photometric calibration (for details, see @cmg+06). Thus, for SAs 71 and 117, those aspects of our analysis that rely on the photometry (e.g., spectroscopic target selection, photometric parallax distance estimates, and calibration of color-dependent systematics in the proper motions) are subject to the uncertainties in calibration of the photographic photometry propagated by up to 0.05 magnitudes in the $B$ and $V$ photometry.
Details of the proper motion reductions appear in @cmg+06, so here we provide only an overview. For all of the near-equatorial ($-15\arcdeg \le \delta \le 15\arcdeg$) Selected Areas in our study, proper motions are derived from plates taken with the Mt. Wilson 60-inch between 1909-1912, combined with deliberately matched plates (in approximate area and plate scale) taken by S. Majewski with the 2.5-m Las Campanas du Pont telescope between 1996-1998. All plates were digitized with the Yale PDS microdensitometer. Most background QSOs and galaxies are near the limiting magnitude of the proper motion catalogs derived from solely the Mt. Wilson and du Pont plates; therefore if we used only these data, the correction to an absolute proper motion frame would be determined by only a handful of poorly-measured faint objects. To extend the proper motion limiting magnitude beyond the limit imposed by the Mt. Wilson plates, we augmented the Kapteyn survey data with measurements of plates from the first Palomar Observatory Sky Survey (POSS-I), which were taken in the early 1950s. These plates, while of much coarser plate scale ($67\farcs2$ mm$^{-1}$; compare to $10\farcs92$ mm$^{-1}$ for the du Pont, and $27\farcs12$ mm$^{-1}$ for the 60”), are deeper than the 60” plates, and provide a $\sim40$-year baseline with both the Mt. Wilson plates [*and*]{} the du Pont plates, allowing us to extend the limiting magnitude of the survey (at least for the POSS-I/du Pont proper motion baseline) to $V\gtrsim20.5$.
In addition, for two of the SA fields in this study (SAs 71 and 94) we have included plates from the Kitt Peak National Observatory Mayall 4-meter telescope, taken at prime focus in the mid-1970s by A. Sandage and in the mid-1990s by S. Majewski. This additional high-quality plate material extends the limiting magnitude in SAs 71 and 94 to $\sim50\%$ completeness at SDSS $r$ magnitudes of $\sim22$ (compared to $\sim50\%$ completeness at $r\sim19.5$ in fields without 4-meter plates; the difference in depth between SA 94 and the other fields can be seen in Figure \[fig:gr\_cmd\_vpd\]). The precision of the proper motion measurements for the “deeper" fields (i.e., those with 4-meter plates) is improved by roughly a factor of 2 at the same magnitude over those with no 4-m plate material; this arises because of both more plate material with fine (186 mm$^{-1}$) plate scale at intermediate epochs and the increased depth provided by the 4-m plates.
[cccccc]{}\
\
& & & & &\
& & & (seconds) & &\
\
[**71**]{} & Dec 2002 & WIYN+Hydra & 4 x 1800 & 37 & 18-19\
- & Nov 2003 & WIYN+Hydra & 10 x 1800, 4 x 1800 & 74 & 18-19\
- & Dec 2005 & WIYN+Hydra & 6 x 1800 & 27 & 19.5\
- & Oct 2006 & WIYN+Hydra & 6 x 1800 & 47 & 19.0\
- & Dec 2006 & WIYN+Hydra & 4 x 1800 & 58 & 19.3\
- & Dec 2007 & WIYN+Hydra & 6 x 2400 & 53 & 19.7\
- & Dec 2008 & MMT+Hectospec & 6 x 1800, 6 x 1800 & 176, 193 & 22.0\
& [**503**]{} &\
\
[**92**]{} & Oct 2006 & WIYN+Hydra & 6 x 1800 & 39 & 19.4\
- & - & SDSS & - & 211 & 21.5\
& [**254**]{} &\
\
[**93**]{} & Oct 2006 & WIYN+Hydra & 6 x 1800 & 55 & 19.8\
- & Oct 2007 & WIYN+Hydra & 6 x 1800 & 51 & 19.9\
- & Dec 2008 & MMT+Hectospec & 4 x 1500 & 195 & 21.5\
- & - & SDSS & - & 101 & 21.8\
& [**292**]{} &\
\
[**94**]{} & Oct 2007 & WIYN+Hydra & 8 x 1800 & 15 & 19.0\
- & Dec 2007 & WIYN+Hydra & 6 x 2400 & 53 & 19.9\
- & Dec 2008 & MMT+Hectospec & 8 x 1800, 6 x 1800 & 189, 165 & 22.5\
- & - & SDSS & - & 88 & 22.5\
& [**432**]{} &\
\
[**116**]{} & Oct 2007 & WIYN+Hydra & 6 x 1800 & 25 & 16.7\
- & Dec 2007 & WIYN+Hydra & 4 x 2700 & 59 & 17.7\
- & Nov 2008 & WIYN+Hydra & 7 x 1800 & 60 & 19.9\
& [**122**]{} &\
\
[**117**]{} & Oct 2007 & WIYN+Hydra & 8 x 1800 & 47 & 19.6\
- & Dec 2007 & WIYN+Hydra & 4 x 2400 & 49 & 19.7\
- & Dec 2008 & MMT+Hectospec & (4 x 1800)+(4 x 1500) & 182 & 20.5\
& [**206**]{} &\
\
Radial Velocities {#rv_meas.sec}
-----------------
The survey fields in which we focus this Sgr study fall on or near the portion of the trailing stream in which @mkl+04 and @mbb+07 have identified a clear Sgr radial velocity signature. These fields can be seen relative to the orbital path of the Sgr dSph in the left panel of Figure \[fig:radec\], and with respect to the expected location of Sgr trailing tidal debris according to the best-fitting models of @lm10a in the right panel of Figure \[fig:radec\]. We have already shown evidence [@cmg+06; @ccg+08] that the overdensities of faint, blue stars that are tightly clumped in proper motions in a few of these fields are likely made up of Sgr debris. It was these apparent overdensities that guided our target selection for spectroscopic follow-up. We began with spectroscopy from the Hydra multifiber spectrograph on the WIYN 3.5-m telescope; in most of the shallower fields of this survey, moderate-resolution spectra can be obtained with this instrument in a reasonable amount of observing time. For the deep fields (and some of the shallower fields as well), we used another multi-object spectrograph, the Hectospec instrument on the MMT 6.5-m, which allowed us to observe $\gtrsim200$ Sgr stream candidates simultaneously per setup down to faint ($g$ or $V\gtrsim21.5$) magnitudes. We describe the observations and data reduction for each instrument separately below.
### Sample Selection
Targets for spectroscopic follow-up were selected to lie within the locus of the suspected Sgr main sequence turnoff (MSTO) at faint ($g$ or $V\gtrsim19.5$) magnitudes and blue ($g-r$ or $B-V \lesssim 0.8$) colors. @cmg+06 showed that the proper motions of these Sgr MSTO candidates clump more tightly than those of the predominantly nearby M-dwarfs at red colors. The tight clumping in the proper motion vector point diagram (VPD) of stars in the MSTO feature was used to define a selection box in proper motion space which should contain any Sgr debris that is present in each field, and eliminate a good fraction of unrelated stars of similar color and magnitude. Care was taken not to be too stringent with either the proper motion or photometric criteria, to preserve as many potential Sgr stars in the wings of the distributions as possible. Because the quality and depth of the photometry and proper motions varies between fields, different candidate selection criteria were adopted for each field.
For WIYN+Hydra observations, only stars brighter than 20th magnitude (either $V$ or $g$, depending on whether a given field had SDSS photometry) were included in the multifiber setups, because fainter stars than this require rather long exposures with a 3.5-meter telescope to achieve adequate signal-to-noise for radial velocity measurement. After all available fibers were filled with MSTO candidates the remaining fibers were assigned to targets at relatively bright magnitudes ($\lesssim 18$) that also reside within the VPD selection criteria (i.e., potential Sgr RGB stars). Stars fainter than 20th magnitude were targeted with the 6.5-meter MMT telescope, which easily reaches Sgr MSTO candidates in these fields.
### WIYN+Hydra Observations {#wiyn_obs.sec}
Spectroscopic data were obtained during a total of eight observing runs with the WIYN 3.5-m telescope[^5] between December 2002 and November 2008. We used the Hydra multi-fiber spectrograph in two different setups. The first one (Dec. 2002, Nov. 2003 observing runs) used the 800@30.9 grating with the red fiber cables and an order centered in the neighborhood of the Mg triplet (5170 Å) and covering about 980 Å of the spectrum. This setup delivered a dispersion of 0.478 Å$~$pix$^{-1}$ and a resolving power $R \sim 5400$ (resolution $\sim1$ Å). The second spectrograph configuration (Dec. 2005, Oct./Dec. 2006, Oct./Dec. 2007, and Nov. 2008) used the 600@10.1 grating with the red fiber cable to yield a wavelength coverage $\lambda$ = 4400–7200 Å at a dispersion of 1.397 Å$~$pix$^{-1}$, for a spectral resolution of 3.35 Å ($R
\sim 1500$ at $\lambda$ = 5200 Å). This spectral region was selected to include the H$\beta$, Mg triplet, Na D, and H$\alpha$ spectral features. Typically 60-70 targets were placed on Hydra fibers, with the remaining 15-20 fibers placed on blank sky regions to allow for accurate sky subtraction. Each of the 2005-6 datasets was obtained in less than optimal conditions, including substantial scattered moonlight in Dec. 2005 and cloudy conditions in both 2006 runs. The majority of the 2007 and 2008 data were obtained under favorable conditions. We further note that the Nov. 2008 observing run occurred after the WIYN Bench Spectrograph Upgrade, which included the implementation of a new collimator into the Bench configuration, as well as a new CCD that delivers greatly increased throughput.
Table \[tab:sgr\_obs\_tab\] summarizes the observations. Each Hydra configuration was exposed multiple times (usually in sets of 30 min. exposures) to enable cosmic ray removal. Standard pre-processing of the initial two-dimensional spectra used the CCDRED package in IRAF.[^6] Frames were summed, then 1-D spectra were extracted using the DOHYDRA multifiber data reduction utilities (also in IRAF). Dispersion solutions were fitted using 30–35 emission lines from CuAr arc lamp exposures taken at each Hydra configuration. On each observing run we targeted a few bright radial velocity standards covering spectral types from F through early K (both dwarfs and giants), each through multiple fibers, to yield multiple individual cross-correlation template spectra. These RV standard spectra were cross-correlated against each other using the IRAF tool FXCOR to determine the accuracy of the velocities and remove any outliers (i.e., those stellar spectra that yield unreasonable cross-correlation results due to template mismatch or some defect, such as a poorly-removed cosmic ray, bad CCD column, or other unknown culprit). Measured velocities of the RV standards typically agreed with published IAU standard values to within 1-2 km s$^{-1}$. Radial velocities for program stars were derived by cross-correlating all object spectra against all of the standards taken on the same observing run. To maximize the $S/N$ in faint, metal-poor stars, only $\sim200$ Å-wide regions centered on the H$\beta$, Mg triplet, and H$\alpha$ absorption lines were used for cross-correlation.
Radial velocity uncertainties were derived using the @vmo+95 method, as described in @mcf+06 and @fmp+06. The Tonry-Davis ratio (TDR; @td79) scales with $S/N$, such that individual RV errors can be calculated directly from the TDR, provided you have multiple observations at varying $S/N$ of some particular standard star to map the dependence. We have used this technique for all datasets except those from the Dec. 2006 observing run, when only a total of four RV standard spectra were taken. For the SA 71 configuration observed on this run, the RV uncertainty for each stars is derived as the standard deviation of the RV results from that star’s spectrum using cross-correlation against each of the four standards. Typical RV uncertainties for individual measurements for all fields were $\sigma_{V} \approx 5-10$ km s$^{-1}$, with most spectra having $S/N\sim$15–20 per Angstrom. The uncertainty depends on $S/N$ (essentially magnitude) for all stars from a single Hydra pointing; however, the varying exposure times between Hydra setups and changing observing conditions mean that $\sigma_{V}$ is not strictly a function of magnitude in our final catalogs.
### MMT+Hectospec Observations
To obtain spectra of fainter Sgr MSTO candidates, we were granted three nights of queue-scheduled NOAO observing time on the MMT 6.5-m.[^7] A total of six observing configurations were observed with the 300-fiber Hectospec multifiber spectrograph [@ffr+05] mounted at the f/5 focus of the MMT. Targets were selected from among the apparent stellar overdensities of blue MSTO canidates at magnitudes too faint ($g \gtrsim 20$) to be reasonably observed with WIYN+Hydra, but using the same proper motion criteria used to choose Hydra targets. In each of these configurations, a few targets previously observed with Hydra were included for a radial velocity consistency check, and any fibers unable to be filled with faint stars were assigned to brighter ($g <
20$) Sgr RGB candidates.
We used the 270 gpm grating, centered at $\sim$6400 Å, to give a working wavelength range of 3700-9150 Å at a dispersion of $\sim$1.21 Å$~$pix$^{-1}$ (4.85 Å resolution). This low resolution allows us to obtain adequate signal-to-noise ($S/N \gtrsim 10$ per Å) spectra of stars as faint as $g=22.5$ to measure radial velocities in 3-4 hours of total exposure time. We observed two multifiber setups each in SAs 71 and 94, the two fields in this study that have 4-meter plates (and thus deeper proper-motion catalogs). The rest of the time was devoted to one configuration each in SAs 93 and 117. Each configuration had 200-230 targets assigned (essentially as many as the fiber assignment software would allow), with the remainder designated as sky fibers. The sky fibers were chosen in areas with the nearest $g
< 22$ star at least 15$\arcsec$ away, distributed throughout the field so that a number of them would fall within each of the two CCD chips of the Hectospec system. The number of exposures in each field, each exposure time, the number of stars with measured radial velocities, and the limiting magnitude of each spectroscopic field are given in Table \[tab:sgr\_obs\_tab\].
The Hectospec data were reduced using an external version of the SAO “SPECROAD” reduction pipeline [@mwc+07] written by Juan Cabanela and called ESPECROAD.[^8] The pipeline automates many reduction steps, including bias-correction, flat-fielding, cosmic-ray rejection, fiber-to-fiber throughput adjustments, and sky subtraction. Wavelength calibration was performed manually using sets of three combined HeNeAr calibration lamp exposures from each night of observing.
We derived RVs using the IRAF task FXCOR to cross-correlate object spectra against fourteen template RV standard spectra of nine different stars ranging in spectral type from F through K. We first correlated the standard spectra against each other, and found that our measurements agree with published RVs[^9] to within 4.9 km s$^{-1}$ for all of these stars, with zero mean offset, and $\sigma_{\Delta V} =
3.0$ km s$^{-1}$. To minimize the effects of noise for spectra with lower $S/N$, the cross-correlation was restricted to the regions around the H$\alpha$, Mg triplet, and H$\beta$ lines.
Each of the object spectra was cross-correlated against all 14 standards, and the mean RV from each of these 14 measurements was adopted as the final result. For the queue-scheduled Hectospec observations, we relied on the queue to provide radial velocity standards. We were thus unable to obtain repeated exposures of the same RV standard stars to allow us to use the “Vogt method” (as described in Section \[wiyn\_obs.sec\]) to determine RV errors. Instead, uncertainties were estimated as the standard deviation of the 14 independent RV measurements thus derived, and vary (essentially as a function of $S/N$) from $\sigma_{V\rm helio} \sim 3$ km s$^{-1}$ at $g = $18.0 to $\sim$15 km s$^{-1}$ at $g =$ 21.5-22.
From repeat measures of a handful of stars, including multiple Hydra or Hectospec observations as well as many observed with both systems, we found mean systematic offsets of $<5$ km s$^{-1}$ between observing runs (including both Hydra and Hectospec data). These offsets were applied to all RVs from a given run to place all measurements on the same system as the Dec. 2007 WIYN+Hydra velocities.
We also note that because we selected most Hectospec targets to be faint, blue objects with miniscule proper motions, a large number of obvious QSOs and AGN spectra appeared in our data. These were added to the samples of QSOs and galaxies that provided the fixed absolute proper motion frame in each field, improving the zero points in these fields.
### SDSS spectra
We supplemented our database of radial velocities by matching our proper motion catalogs to the SDSS spectroscopic database. The number of additional RVs contributed by SDSS in each field is noted in Table \[tab:sgr\_obs\_tab\]. The majority of SDSS stars are red, nearby M-dwarfs, so very few additional Sagittarius candidates were contributed by the addition of these spectroscopic data. However, the handful of Sgr stars that are present, as well as any other stars in common with our observations, were used for a consistency check. From the stars in common between SDSS and our observations in SAs 94 and 93, we find mean offsets of $\leq$ 5 km s$^{-1}$. The accuracies of SDSS radial velocities are $\sim4$ km s$^{-1}$ at $g < 18$, decreasing to $\sim15$ km s$^{-1}$ at $g \sim 20$ [@yrn+09], so we choose not to offset the RVs .
Sagittarius Tidal Debris Kinematics {#sgr_kinematics.sec}
===================================
Radial Velocities {#sgr_rvs.sec}
-----------------
Figure \[fig:rvhist\] shows all measured velocities in each of the six survey fields (the total number of stellar radial velocities in each field is given in Table \[tab:sgr\_obs\_tab\]) in the Galactocentric ($V_{\rm GSR}$[^10]) frame. These consist of all RVs from WIYN+Hydra, MMT+Hectospec, and where available, SDSS. Where multiple measurements exist, the final catalog reflects the error-weighted mean radial velocity. In each of these fields, a broad peak is seen at $V_{\rm GSR} \sim 0-100$ km s$^{-1}$ which is made up primarily of Milky Way stellar populations in these high-latitude fields. An additional velocity peak is evident in SAs 71, 94, 93, and 117, well separated (in all fields except SA 117) from the Galactic distribution in each of these fields; it is this additional peak we shall show to consist of mainly Sagittarius trailing tidal debris. There is no readily apparent peak at lower $V_{\rm GSR}$ values in SAs 92 and 116 – this arises for different reasons in the two cases. SA 116 is the field in which we have the fewest measured radial velocities, and even the stars for which we do have data are not optimally selected to find Sgr debris. Because of limitations on exposure times due to weather, the observed stars were all at relatively bright magnitudes ($g <
18$), where only very few Sgr red giants should be found. More spectra of stars at faint ($g \gtrsim 19$) magnitudes and blue colors will be necessary to identify Sgr debris among the SA 116 data. In SA 92, on the other hand, nearly twice as many spectra are available than in SA 116, and mostly at relatively faint magnitudes. The paucity of obvious Sgr debris in this field is because of the location of SA 92 on the periphery of the stream, where Sgr stellar densities are rather low. There are a handful of stars at low ($V_{\rm GSR} < 0$ km s$^{-1}$) velocities, but hardly enough candidates to assert that a clear Sgr presence is indicated in SA 92.
To assess whether these apparent velocity overdensities are expected among Galactic populations in each line of sight, we compared the radial velocity distributions to those from the Besançon Galaxy model [@rrd+03].[^11] In each SA field, the model query was run five times to smooth out the finite sampling statistics in each individual model run. The five catalogs were concatenated, then for each Kapteyn field the measured velocities were compared to the expected radial velocities of smooth Galactic populations by scaling the summed Besançon model to match the total number of stars in the broad peak in each RV histogram. This was done separately for “bright” and “faint” samples in each field, since stars in different magnitude ranges preferentially sample different Galactic populations with different velocity distributions (e.g., “faint” blue Galactic stars in the region of the CMD where Sgr MSTO stars reside will be predominantly halo stars, and thus have a much higher velocity dispersion than stars of similar color, but much brighter magnitude, where thin/thick disk MSTO stars predominate).
The resulting scaled model distributions are shown as grey filled histograms in Figure \[fig:sa7194\_vhelhist\] for SAs 71 and 94, with the measured heliocentric radial velocities given as solid-lined histograms. The broad peak is reproduced well by the model populations, suggesting that (a) there are no large global velocity offsets present in our data, and (b) the prominent peaks are indeed due to foreground/background Milky Way stars. The additional peaks at $V_{\rm helio} \approx -170$ km s$^{-1}$ (SA 71) and $V_{\rm helio}
\approx -150$ km s$^{-1}$ (SA 94) are clearly not due to any expected Galactic populations along these lines of sight.
Similar histograms are shown for SAs 93 and 117 in Figure \[fig:sa93117\_vhelhist\], which again clearly show that the broad, prominent peak in each field is made up of Galactic populations, and the peaks at $V_{\rm helio}
\approx -160$ km s$^{-1}$ (SA 93) and $V_{\rm helio} \approx -100$ km s$^{-1}$ (SA 117) are inconsistent with expected Milky Way velocities. Note that the peak in SA 117 overlaps the wings of the Galactic distribution, making it slightly more difficult to isolate bona fide Sgr members in this field on the basis of radial velocities alone.
Finally, we performed the same examination in SAs 92 and 116, with the results shown in Figure \[fig:sa92116\_vhelhist\]. Nearly all of the velocities shown in SA 92 are from the SDSS database, and are predominantly very red M-dwarfs. For this reason, the long tail of the RV distribution at negative velocities, which is due to thick disk and halo MSTO stars, is not well reproduced by our data set. There are a small number of stars at Sgr-like velocities in Figure \[fig:sa92116\_vhelhist\], but these fall within the expected locus of MW stars, so we cannot definitively say that Sgr members are present among our SA 92 sample. In SA 116, no excess peak of measured RVs relative to the model predictions is apparent. This is not surprising given (a) the caveats in the first paragraph of this section regarding the data in SA 116, and (b) the fact that the @lm10a model predicts Sgr debris in this field to have RVs of $-100 \lesssim V_{\rm helio} \lesssim -50$ km s$^{-1}$, overlapping the wings of the Galactic distribution in this field. A handful of Sgr members may thus be present among our velocities, but they are difficult to distinguish from the Milky Way halo stars by their RVs.
With RVs in hand, a next culling for Sgr stream candidates was obtained by simply taking all stars within a generously defined range around the evident associated radial velocity peak. Such a selection will include a few Milky Way interlopers, so we examined the samples thus selected to remove non-Sgr stars. We first removed all stars with proper motions $|\mu| > 10$ mas yr$^{-1}$ in either dimension; such stars, if actually at the distance of the Sgr trailing tail in this region of sky ($\sim$25-40 kpc), would have unreasonably large ($>
1000$ km s$^{-1}$) tangential velocities ($V_{\rm tan} = 4.74d\mu$ km s$^{-1}$, where d is the distance in kpc and $\mu$ the proper motion in mas yr$^{-1}$). Faint, blue stars with proper motions of this magnitude must therefore be nearby (foreground) MW white dwarfs or metal-poor subdwarfs. After removing these stars, we then examine the positions of all selected candidates in the color-magnitude diagram. We reject faint stars that are well redward of the readily apparent Sgr main sequence, and at brighter magnitudes, we remove only stars at positions obviously inconsistent with being Sgr red giants or horizontal branch stars. In SA 93, a clear offset was visible between mean proper motions of bright ($g < 19.0$) Sgr candidates and fainter ones, so we chose to keep only candidates at $g > 19.0$, on the assumption that the density of stream stars should be much greater at fainter magnitudes near the lower RGB and MSTO than along the upper RGB. This yields fewer total Sgr candidates, but the ones that remain have much higher probability of being Sgr members than do the brighter candidates.
Selecting Final Sgr Candidates
------------------------------
We now discuss how we pared down the samples of Sgr candidates from the initial broad RV and proper motion selections in each field to the final, more securely-identified samples used for analysis. For brevity, we will show detailed examples for only two of the six fields in the study. The first of these is SA 94, which is the “best-case” field in our study, because it has deep proper motions due to the availability of 4-meter plates in its data set. For comparison, we follow the SA 94 discussion with details of SA 93, which has a much shallower proper motion catalog than SA 94, but also has high-quality SDSS photometry. These two fields were chosen simply to give the reader an idea of the type and quality of the data included in this study, and the process we followed to select Sgr candidates.
### Selecting Sgr Candidates in SA 94
Figure \[fig:sa94\_cmd\] shows the SDSS $g - r$ vs. $g$ color-magnitude diagram (CMD) for SA 94. The left panel shows all stars for which we have measured proper motions, with SDSS-classified galaxies removed from the sample. On the right-hand side, open squares depict all stars observed spectroscopically to illustrate the candidate selection. The majority of spectroscopic targets in this deep proper-motion field were selected from the Sgr MSTO feature of faint, blue stars, and observed with the large-aperture MMT 6.5-m telescope. The remaining targets are either (a) bright stars observed with WIYN+Hydra, or (b) targets chosen to fill unused fibers after all possible MSTO candidates had been assigned. The final culled sample of Sgr candidates (based initially on RV selection, with further interactive proper-motion and CMD selection performed as discussed below) is shown by the large, filled black squares. As expected, these concentrate at the Sgr MSTO locus, with a handful of brighter stars having properties consistent with Sgr membership as RGB or red clump stars. We have overlaid an isochrone from @ggo+04 for an old (10 Gyr), metal-poor (\[Fe/H\] = -1.3) population at the expected distance (d = 29.5 kpc; @lm10a) of Sgr debris in SA 94; the age and metallicity of this isochrone is chosen to match the Sgr metal-poor population identified by @sdm+07, which should be the dominant contributor to debris in this portion of the trailing tail. The final set of Sgr candidates in this field concentrate near this ridgeline; most of the scatter about the isochrone is likely due to the $\pm$5 kpc line-of-sight depth of the Sgr stream as well as the intrinsic population dispersion in Sgr. The proper motion vector point diagram (VPD) is displayed as Figure \[fig:sa94\_vpd\], with panels and symbols the same as in Figure \[fig:sa94\_cmd\]. Sgr candidates clump more tightly in the VPD than the overall population of spectroscopic targets, which is consistent with the notion that we are indeed measuring a distant, common-motion grouping of stars.
To further cull the sample, we turn to the reduced proper motion diagram (RPMD). Reduced proper motion, defined as $H_g \equiv g + 5$ log $\mu + 5$, where $g$ is the apparent magnitude and $\mu$ is the total proper motion in arcsec yr$^{-1}$, compresses stars with common tangential velocity into coherent features in the RPMD. Thus, a common-motion population should form a sequence in the RPMD, even if the population has a significant line-of-sight depth (see @m99, especially Figure 4). In Figure \[fig:sa94\_rpmd\] we show such a diagram for SA 94, with the same isochrone as in Figure \[fig:sa94\_cmd\], shifted to the measured tangential velocity of SA 94 Sgr debris (to be discussed in Section \[pm\_meas.sec\]). After an initial calculation of the mean motion, candidates that were obviously inconsistent with a broadly-defined region ($\gtrsim$0.3 magnitudes in $H_g$ and/or $g-r$) about the ridgeline were manually removed from the sample.
### Selecting Sgr Candidates in SA 93
SA 94 is one of only two fields (with SA 71) of the six in this study that have KPNO 4-m plates, and thus have 1-1.5 magnitude deeper proper motions. Furthermore, of those two, SA 94 has higher-quality photometry (from SDSS) than the photographic magnitudes used for SA 71. Thus SA 94 is our best field in terms of overall data quality. To illustrate how a more typical field compares to the highest-quality SA 94 data, we show in Figures \[fig:sa93\_cmd\], \[fig:sa93\_vpd\], and \[fig:sa93\_rpmd\] the same type of plots as in Figures \[fig:sa94\_cmd\], \[fig:sa94\_vpd\], and \[fig:sa94\_rpmd\], but for SA 93. This field is located in a high stellar density region of the stream, as is SA 94, but has shallower proper motion data (see Fig. \[fig:gr\_cmd\_vpd\]), providing far fewer Sgr MSTO candidates for follow-up spectroscopy. Because the available proper-motion data do not sample the MSTO as robustly as in SA 94, this field was given lower priority for spectroscopy, with only one relatively short MMT+Hectospec configuration observed (see Table \[tab:sgr\_obs\_tab\]). However, in this one Hectospec setup, nearly all available $g \lesssim 20.2$ Sgr candidates satisfying the (somewhat relaxed relative to SA 94) target selection criteria were observed. The overlaid @ggo+04 isochrone for an old (10 Gyr), metal-poor (\[Fe/H\] = -1.3) population at the expected distance ($d$ = 28 kpc) of Sgr debris in SA 93 (blue curve in Figure \[fig:sa93\_cmd\]) is seemingly consistent with all but the brightest of the identified candidates. We have also overplotted (dashed gray curve) an older ($\sim$15.8 Gyr), more metal-poor (\[Fe/H\] = -1.7) isochrone with the same distance as the blue curve; this curve passes through the positions of the bright, blue stars, suggesting that they may also be Sgr members of an old horizontal-branch/asymptotic giant branch population. However, the MSTO of such an old population is beyond the magnitude limit of our proper-motion survey.[^12] The RPMD (Figure \[fig:sa93\_rpmd\]) for SA 93 shows the same ridgeline as in the CMD, shifted by the final measured tangential velocity for the Sgr candidates. Because the uncertainty in the mean proper motion is much larger in this field than in SA 94 (on the order of $\sim$100 km s$^{-1}$ in tangential velocity for SA 93, compared to $\sim$50 km s$^{-1}$ in SA 94), it is difficult to conclude much from the RPMD. The benefits of the additional 4-meter plates in SAs 94 and 71 are clearly illustrated by the relatively fewer identified Sgr stream members in the shallower SA 93 field compared to the deeper data sets. Zoomed-in versions of the VPDs for SAs 94 and 93 are given in Figure \[fig:sa9493\_vpd\_members\], with error bars shown on all points to illustrate the quality of the proper-motion data (note that the error bars on individual stars in each of the panels of Fig. \[fig:sa9493\_vpd\_members\] are of comparable size, in spite of the higher-quality proper motion data in SA 94 than in SA 93, because the majority of Sgr candidates in SA 94 are faint \[$g > 21$\] MSTO candidates, while SA 93 candidates are mostly $>1$ magnitude brighter than this). As previously mentioned, a proper motion offset was seen between faint ($g > 19.0$) stars in SA 93 and brighter candidates; we included only the faint ($g > 19.0$) stars in our proper motion measurement, because these are more likely to be true Sgr members. The maximum likelihood estimate of the absolute proper motion of Sgr debris in each field is represented by the large red asterisks, which have $1\sigma$ uncertainty smaller than the size of the point.
### Summary of Sgr Candidate Selection
In summary, the basic Sgr candidate selection in each Selected Area began with a broad RV selection centered on the apparent Sgr velocity peak (e.g., $-250 < V_{\rm hel} < -100$ km s$^{-1}$ in SAs 94 and 93). This was followed by removing high proper motion stars ($|\mu| >
10$ mas yr$^{-1}$ in either dimension), which would have tangential velocities much greater than the Milky Way escape velocity if those stars were at the 25-40 kpc distances of Sgr debris along the SA lines of sight. We then used our knowledge of the distance and metallicity expected for Sgr debris in these fields to remove stars at faint magnitudes that are more than $\sim0.3$ magnitudes from the expected color-magnitude and RPMD loci in each field (note that in some cases we did not apply this at bright magnitudes, since there is considerable uncertainty about the exact CMD locus for Sgr debris above the turnoff). We calculated mean proper motions from the selected stars, then iteratively removed outliers ($>3\sigma$) in proper motion. Finally, we manually inspected the remaining candidates to remove any stars that were $\sim1-2\sigma$ outliers from the identified Sgr locus in [*all observables*]{} (i.e., RV, $g-r$ color, magnitude, [*and*]{} proper motion).
Once Sgr candidates were selected using color, magnitude, RV, proper motion, and RPMD criteria, the kinematical properties of Sgr debris in each SA field were estimated using a maximum likelihood method (e.g., @pm93 [@hgi+94; @kwe+02]). The final measured radial velocities and velocity dispersions, along with uncertainties in these values, are given in Table \[tab:sgr\_kinematics\] in both the heliocentric and Galactocentric (GSR) frames. The estimates of the mean radial velocity (in the GSR frame) of Sgr debris can be seen in the top panels of Figure \[fig:lambda\_3dkinematics\], which overlays our measurements atop the best-fitting debris model of @lm10a. The colors in the figure were chosen to match those used by @lm10a, who color-coded points along the stream according to the orbital passage on which they became unbound. Gold points correspond to debris stripped during the two most recent perigalactic passages, and magenta during the previous two passages. This portion of the trailing tail has a well-constrained radial velocity trend, which was measured by @mkl+04 and @mbb+07, and was one of the constraints on the @lm10a models. Sgr model trailing tail debris within $\pm3.0\arcdeg$ of the position of each Kapteyn line of sight is shown in the right-hand panels of Figure \[fig:lambda\_3dkinematics\] as small open squares. This illustrates that not only do our measured radial velocities agree quite well with the model, but that the dispersion in each field appears to match the distribution of model points. However, we caution that the velocity dispersions we find (see Table \[tab:sgr\_kinematics\]) are higher than those derived by @mkl+04 and @mbb+07 for the trailing tail. It is unclear whether the dispersion is truly intrinsically higher for Sgr trailing tail MSTO populations we sampled than for the M giants previously studied, or whether our derived dispersions are inflated by the large measurement uncertainties for our RVs, velocity zero-point offsets between the many data sets we have combined, or Milky Way foreground/background contamination in our Sgr candidate samples. While we have endeavored to account for all of these factors, a robust conclusion likely requires high-resolution spectra of trailing-tail MSTO stars.
[ccccccccccccc]{} 71 & 33 & -172.9$\pm$4.9 & -141.4$\pm$4.9 & 25.6$\pm$3.5 & 0.39$\pm$0.44 & -0.55$\pm$0.42 & 0.66$\pm$0.43 & -0.17$\pm$0.43 & 138.1$\pm$ 47.1 & 80.7$\pm$ 77.1 & 80.1$\pm$ 63.3 & 38.0$\pm$4.5\
94 & 64 & -142.4$\pm$2.3 & -141.3$\pm$2.3 & 17.4$\pm$1.8 & 0.35$\pm$0.27 & -2.32$\pm$0.29 & 1.88$\pm$0.28 & -1.40$\pm$0.28 & 227.9$\pm$ 34.1 & -49.5$\pm$ 54.0 & -12.6$\pm$ 30.8 & 29.5$\pm$4.0\
93 & 15 & -157.2$\pm$4.7 & -114.1$\pm$4.7 & 16.1$\pm$3.3 & 0.42$\pm$0.73 & -2.66$\pm$0.56 & 1.60$\pm$0.70 & -2.16$\pm$0.60 & 210.6$\pm$ 81.3 & -101.6$\pm$ 88.4 & -10.7$\pm$ 45.4 & 28.0$\pm$3.0\
117 & 43 & -90.0$\pm$3.9 & -69.3$\pm$3.9 & 24.0$\pm$2.8 & 0.02$\pm$0.64 & -3.11$\pm$0.72 & 1.28$\pm$0.65 & -2.84$\pm$0.71 & 229.3$\pm$ 83.7 & -76.8$\pm$ 94.6 & 11.5$\pm$ 24.9 & 25.0$\pm$4.0\
\
92 & 6 & -158.9$\pm$6.3 & -78.4$\pm$6.3 & 10.6$\pm$8.7 & 1.34$\pm$1.07 & -3.84$\pm$0.66 & 1.46$\pm$1.06 & -3.80$\pm$0.66 & 145.6$\pm$123.3 & -296.9$\pm$107.9 & -84.6$\pm$ 45.6 & 27.5$\pm$2.5\
116 & 10 & -75.4$\pm$6.7 & -22.3$\pm$6.7 & 20.3$\pm$4.9 & 1.12$\pm$1.04 & -3.30$\pm$1.15 & -0.67$\pm$1.07 & -3.42$\pm$1.12 & 87.3$\pm$124.3 & -170.8$\pm$129.6 & -22.7$\pm$ 35.3 & 24.5$\pm$2.0\
Proper Motions {#pm_meas.sec}
--------------
With the sample of Sgr debris candidates identified in each field, we used a maximum-likelihood method to estimate the Sgr absolute proper motions in both spatial directions ($\mu_{\alpha} \cos (\delta$), $\mu_{\delta}$). These results are seen in Table \[tab:sgr\_kinematics\] with their uncertainties, which include the uncertainty in the proper motion zero point in each field added in quadrature to the maximum likelihood error estimate. The uncertainties for each field depend on many factors, including the depth and quality of the plates, the number of available reference objects (i.e., background QSOs and galaxies) used to convert from relative to absolute proper motions, the number and depth of spectroscopic targets obtained (and thus the number of Sgr candidates identified), and the position of each field relative to the highest-density regions of the stream (i.e., the number of Sgr candidates expected in each field).
Mean proper motions along Galactic coordinates (i.e., $\mu_l \cos b,
\mu_b$) in each field are compared to the model of @lm10a as a function of $\Lambda_{\odot}$ in the middle and lower panels of Figure \[fig:lambda\_3dkinematics\], with points once again color-coded by the orbital passage in which they became unbound, and model points within $\pm3.0\arcdeg$ of each SA line of sight highlighted for guidance. The results for SAs 94, 93, and 117 agree nicely with the model predictions within the uncertainties in $\mu_b$, but show a $\sim1-2\sigma$ offset from the main trend in $\mu_l \cos
(b$) (left middle panel). The mean proper motions along both directions for SA 71 (the leftmost data points in Figure \[fig:lambda\_3dkinematics\]) are slightly shifted (by $\sim
1-1.5\sigma$) from the mean of the model prediction for this field. A number of factors contribute to the difficulty in selecting a “pure” sample of Sgr debris in SA 71, and there is thus an additional uncertainty (besides the formal errors) in the mean Sgr debris proper motions in this field. First, this field is at somewhat low latitude ($b=-34.7\arcdeg$), and thus suffers greater contamination from Galactic populations, as evidenced by the extended tail of low-velocity stars in the lower panel of Figure \[fig:sa7194\_vhelhist\]. This lower latitude also means that SA 71 suffers significantly more reddening than higher-latitude fields – nearly 0.6 magnitudes of extinction using the @sfd98 maps. Secondly, the distance of Sgr debris increases with $\Lambda_{\odot}$ along the portion of the trailing tail in this study, so that stream members in SA 71 are nearly 40 kpc away, making them fainter than in the other fields. Finally, the poor-quality photographic photometry we are limited to in this field renders inscrutable the typical CMD features such as the blue edge of the Galactic disk MSTO and the Sgr upper main sequence.
Assuming distances to Sgr debris in each field as given in Table \[tab:sgr\_kinematics\] and values of $V_{\rm circ}$ = 220 km s$^{-1}$ and $R_0$ = 8.0 kpc, we converted the measured Sgr debris motions to Galactocentric $UVW_{\rm GC}$ velocities (i.e., Cartesian velocities such that the Sun is moving at 9.0, 232.0, 7.0 km s$^{-1}$ assuming $V_{\rm circ}$ = 220 km s$^{-1}$ and ($U_0$, $V_0$, $W_0$) = (9.0, 12.0, 7.0) km s$^{-1}$ [@mb81] for the solar motion relative to the Local Standard of Rest; to facilitate direct comparison, values used for these constants are the same as those in the model), which are shown in Table \[tab:sgr\_kinematics\] (note that we placed SAs 92 and 116 – the two fields with no securely identified Sgr members – in a separate section in Table \[tab:sgr\_kinematics\]. These data are given for completeness, but are not used for subsequent analysis.). The $U_{\rm GC}$ component should dominate the total space velocity of Sgr debris in each of these trailing arm fields. This can be surmised from Figure \[fig:sgr\_xyz\], which shows that the motion of the Sgr trailing tail is oriented almost parallel to the Galactic $X$-axis in the $X_{\rm GC}-Z_{\rm GC}$ plane. This, in addition to the fact that the Sagittarius orbital plane is only slightly misaligned with the Galactic $X_{\rm GC}-Z_{\rm GC}$ plane [@msw+03; @mlp+06], suggests that most of the motion in this part of the trailing tail is inward toward the Galactic center and roughly parallel to the Galactic disk (i.e., the $X_{\rm GC}-Y_{\rm GC}$ plane) at a distance of $\sim$20-25 kpc below the disk. This is borne out by the measured $UVW_{\rm GC}$ velocities in our SA fields – the $U_{\rm GC}$ component in the four fields with quality measurements is by far the largest component of the 3-D motion. This can be seen even more clearly by considering the proper motions along Galactic coordinates, but in a Galactic rest frame (designated as $\mu'_{l}
\cos($b) and $\mu'_{b}$). These proper motions, given in Table \[tab:sgr\_pms\_gsr\], show $\mu'_{l}
\cos($b) proper motions of nearly zero in each field – as expected for streaming motions confined to the $X_{\rm GC}-Z_{\rm GC}$-plane. As we will show in Section 4, the offset of these derived longitudinal proper motions (reflected in the $V_{\rm GC}$ Galactic velocity component) from zero can be used to reevaluate the velocity of the Local Standard of Rest (under the assumption that the longitudinal motions [*should be*]{} zero).
[cccccc]{}\
\
& & & & &\
& & & & &\
\
71 & -141.4 & -0.47 & 0.39 & -0.61 & 0.00\
94 & -141.3 & -0.77 & -1.09 & 0.22 & -1.31\
93 & -114.1 & -0.71 & -1.36 & 0.00 & -1.54\
117 & -69.3 & -1.17 & -1.57 & -0.44 & -1.91\
Constraints on Milky Way Stucture {#mw_constraints.sec}
=================================
As discussed in Section 1.1, the opportune orientation of the Sgr trailing tidal tail means that the observed motion of tidal stream stars in the Galactic $Y$ direction (i.e., towards $[l,b] =
[90^{\circ}, 0^{\circ}]$) is dominated by the solar reflex motion, which consists of the solar peculiar motion and the Galactic rotational motion at the solar circle (i.e., the Local Standard of Rest $\Theta_{\rm LSR}$). As shown by @mlp+06, the intrinsic motion of Sgr debris along the $Y$ direction ($V_{\rm GC}$, contained primarily in the $\mu_l \cos($b) component of proper motion) varies only slowly across the region of the trailing tail between $70\arcdeg
\le \Lambda_{\odot} \le 130\arcdeg$, making the fields of view in which we have deep proper motion data ideal for constraining $\Theta_{\rm LSR}$. It can be seen in Table \[tab:sgr\_kinematics\] that $V_{\rm GC}$ for Sgr debris in each of the four fields (SAs 71, 94, 93, and 117) with reliable data is non-zero at the $\sim1\sigma$ level. Setting aside SA 71, in which it is difficult to securely identify Sgr debris, the remaining three fields exhibit $V_{\rm GC}$ systematically offset to negative values. If indeed the expected $V_{\rm GC}$ for Sgr debris in these fields is zero, this suggests that the value of $\Theta_{\rm LSR}$ that was subtracted from the $V$-component of these velocities was [*lower*]{} than it should be – i.e., $\Theta_{\rm LSR}$ should be greater than the canonical 220 km s$^{-1}$.
In this section, we use variations on the numerical model of the Sgr tidal stream to isolate the contribution to $\mu_l
\cos($b) from the solar reflex motion and identify the value of $\Theta_{\rm LSR}$ favored by our proper motion data.
$N$-body models {#nbody.sec}
---------------
Though the measured Galactic Cartesian $V-$velocity (i.e., motion along the Galactic $Y$-component) of Sgr trailing tidal debris is dominated by Solar reflex motion, there is some contribution of intrinsic Sgr motion to the $V-$component of debris velocities. In particular, we must consider the following effects when trying to back out $\Theta_{\rm LSR}$ from measured $V_{\rm GC}$ for Sgr debris: (1) the slight inclination of the Sgr debris plane to the Galactic $XZ_{\rm GC}$ plane means a small fraction of Sgr space motion is projected onto the measured motions (i.e., the $V$ velocity is in fact a function of both $\Theta_{\rm LSR}$ [*and*]{} intrinsic Sgr motion); (2) the Galactic Standard of Rest (GSR) frame radial velocities used to constrain the Sgr model were derived assuming a value of $\Theta_{\rm LSR}$; (3) changing $\Theta_{\rm LSR}$ correspondingly changes the Milky Way mass scale, which thus affects the space velocity of the Sgr dSph in the models. Therefore, taking into account these dependencies on the assumed value of $\Theta_{\rm LSR}$, we repeat the analysis, changing $\Theta_{\rm LSR}$ to construct self-consistent models for the Sgr tidal stream in each of four choices for the Local Standard of Rest speed, namely $\Theta_{\rm
LSR} = 190, 250, 280, 310$ km s$^{-1}$ (in addition to the original value of $\Theta_{\rm LSR} = 220$ km s$^{-1}$ used in earlier sections of this paper).
Our methodology is described in detail by . In brief, we constrain the model Sgr dwarf to lie at the observed location $(l,b)=(5.6^{\circ}, -14.2^{\circ})$, distance $D_{\rm Sgr} =
28$ kpc (@siegel2011, @sdm+07), and radial velocity $v_{\rm Sgr} = 142.1$ km s$^{-1}$ in the heliocentric frame. The orbital plane is constrained to be that defined by the trailing arm tidal debris, which has experienced minimal angular precession [@jlm05], and the speed of Sgr tangential to the line of sight ($v_{\rm tan}$; see Section 3.3 of ) is constrained by a $\chi^2$ minimization fit to the radial velocities of trailing arm tidal debris. We fix the mass and radial scalelength of the Sgr progenitor so that the fractional mass loss history of the dwarf is similar in all models to that of .
The adopted Milky Way Galactic mass model consists of three components: a Hernquist spheroid (representing the Galactic bulge), a @mn75 disk, and a logarithmic dark matter halo. The Local Standard of Rest in this model is given by: $$\Theta_{\rm LSR} =
\sqrt{R_{\odot} (a_{\rm bulge} + a_{\rm disk} + a_{\rm
halo})}$$ where $a_{\rm bulge}$, $a_{\rm disk}$, and $a_{\rm halo}$ respectively represent the gravitational acceleration exerted on a unit-mass at the location of the Sun due to the Galactic bulge, disk, and halo components. In the model (for which $\Theta_{\rm LSR} = 220$ km s$^{-1}$), the bulge/disk/halo respectively contribute 32%/49%/19% of the total centripetal acceleration at the position of the Sun, corresponding to bulge/disk masses $M_{\rm bulge} = 3.4 \times 10^{10} M_{\odot}$ and $M_{\rm
disk} = 1.0 \times 10^{11} M_{\odot}$, and a total mass within 50 kpc of $4.5 \times 10^{11} M_{\odot}$.
[cccc]{}\
\
& & &\
& & &\
\
190 & $6.8\times10^{10}$ & $2.3\times10^{10}$ & 0.68\
220 & $1.0\times10^{11}$ & $3.4\times10^{10}$ & 1.00\
250 & $1.4\times10^{11}$ & $4.6\times10^{10}$ & 1.35\
280 & $1.8\times10^{11}$ & $6.0\times10^{10}$ & 1.76\
310 & $2.2\times10^{11}$ & $7.5\times10^{10}$ & 2.21\
264 & $1.5\times10^{11}$ & $5.2\times10^{10}$ & 1.53\
232 & $1.1\times10^{11}$ & $3.9\times10^{10}$ & 1.14\
Since the baryonic Galactic disk and bulge components are the dominant factors in determining $\Theta_{\rm LSR}$ (together comprising $>
80$% of the total centripetal force), we therefore scale the total bulge $+$ disk mass as necessary to normalize the rotation curve at the solar circle ($R_{\odot} = 8$ kpc) to the chosen value of $\Theta_{\rm LSR}$. The masses of the disk and bulge components in each of the models are given in Table \[tab:bulgediskmasses\]. We leave the [*ratio*]{} of bulge/disk mass fixed in order to preserve the shape of the rotation curve interior to the solar circle. In addition, we fix the Galactic dark matter halo parameters (mass, axis ratios, and scalelength) to the best-fit values derived by since these authors found that these values were relatively insensitive to factors of $\sim 2$ variation in the mass scale of the baryonic Galactic components.
Results {#nbodyresults.sec}
-------
Constraints on $\Theta_{\rm LSR}$ were derived in two ways. In the first of these methods, we assumed (as argued previously in this paper, as well as in ) that the dominant contribution to the measured $\mu_l \cos($b) component of Sgr motion is due to the solar rotation, and that the largest component of $\Theta_{\rm LSR}$ is along $\mu_l \cos($b). We have shown that these are reasonable first-order assumptions, and thus use only the longitudinal proper motions as constraints on fitting $\Theta_{\rm
LSR}$ in our first attempt. After doing so, however, we performed a similar analysis, but using all three dimensions of Sgr debris motions as constraints to determine $\Theta_{\rm LSR}$. In the following, we present both results, which come out somewhat different from each other (though consistent within 1$\sigma$). Fits using only $\mu_l
\cos($b) tend to prefer relatively high values of $\Theta_{\rm LSR}$, while those constrained by full 3-D kinematics tend toward lower values more in line with the IAU standard of 220 km s$^{-1}$.
### $\Theta_{\rm LSR}$ Constraints Using Only $\mu_l \cos($b) Motions of Sgr Debris
We quantify the agreement of our proper motions with those of simulated Sgr tidal debris from each of our grid of models using a $\chi^2$ statistic. The $\chi^2$ fitting was performed using mean Sgr debris proper motions in only SAs 71, 94, 93, and 117 – as discussed in Section \[sgr\_rvs.sec\], the results in SAs 92 and 116 are unreliable for a variety of reasons. For our model comparison, we first select all Sgr model points within $\pm3.0\arcdeg$ in both $\Lambda_{\odot}$ and $(\alpha,\delta)$ of each SA field. The large area (relative to the $40\arcmin \times
40\arcmin$ coverage of each SA field) used to select model debris corresponding to each SA position ensures that enough $N-$body particles are selected for robust measurement of model debris motions at each position. This also makes the fitting less sensitive to small-scale differences in positions and densities of debris stars between the models and the actual stream that arise due to the vagaries of the modeling and our incomplete knowledge of the Sgr trailing tail properties. Figure \[fig:mul\_varyvcirc\] shows the model debris $\mu_l \cos($b) as a function of $\Lambda_{\odot}$ for each of the five Sgr simulations, with points corresponding to each SA field shown as small open gray squares. It is clear that $\mu_l
\cos($b) changes very little over the $6\arcdeg$ ranges in $(\alpha,\delta)$ used. Furthermore, the small number of selected model points, even in such a large selection region, shows that these broad selection criteria are necessary to have sufficient model points for comparison. The maximum likelihood proper motion results in SAs 71, 94, 93, and 117 are shown in Figure \[fig:mul\_varyvcirc\] as open black diamonds, with error bars reflecting $1\sigma$ uncertainties. It can be seen in the figure that the models with $\Theta_{\rm LSR} > 220$ km s$^{-1}$ tend to reproduce the longitudinal proper motions for most of the fields better than the standard 220 km s$^{-1}$ value of this fundamental constant. Note that SA 71, at $\Lambda_{\odot} = 128\arcdeg$, is the exception to this trend – as discussed in Section 3.3, identifying bona fide Sgr debris in this field is more difficult than the others and so these data are more suspect.
{height="6.0in"}
Each of the Sgr model simulations provides predicted kinematics of Sgr debris for a Galactic potential constrained by a given value of $\Theta_{\rm LSR}$. As discussed previously, the Galactic $V$-component of the Sgr debris space velocity along the trailing tail contains little contribution due to the Sgr motion; nearly all the $V$ velocity (as measured by the $\mu_l \cos($b) component of the proper motion) is reflected Solar motion. Thus, to first order, we can simply compare our mean Sgr proper motions along Galactic longitude in the trailing tidal tail to our models of Sgr debris for different values of $\Theta_{\rm LSR}$ and determine the value of $\Theta_{\rm LSR}$ that best reproduces the measured PMs. To do so, we defined a $\chi^2$ residual:
$$\chi^2_\mu = \sum_i \frac{\mu_{l,\rm SA} [\rm i] - \mu_{l,\rm mod} [\rm i]}{\sigma_{\mu_{l,\rm SA}[\rm i]}}$$
where $\mu_{l,\rm SA} [\rm i]$ represents the mean $\mu_l \cos($b) proper motion in each of the four SA fields, and $\mu_{l,\rm mod} [\rm
i]$ the mean proper motion of the corresponding model debris for each field. The residuals are weighted by the uncertainty, $\sigma_{\mu_l,\rm SA [\rm i]}$, in each SA proper motion. This $\chi^2$ statistic was initially calculated for the proper motions presented in Table \[tab:sgr\_kinematics\] relative to each of the five Sgr debris models (corresponding to $\Theta_{\rm LSR} = 190, 220,
250, 280, 310$ km s$^{-1}$). Rather than running new (and laborious) $N$-body simulations for many intermediate values of $\Theta_{\rm
LSR}$ and calculating $\chi^2$ for each of them, we choose to find the minimum $\chi^2$ by fitting a parabola to the $\chi^2$ results for each of the five modeled values of $\Theta_{\rm LSR}$. The results of the $\chi^2$ calculation and the parabolic fit are seen in Figure \[fig:chisq\_vs\_theta\] for the proper motions of the “best” SA samples given in Table \[tab:sgr\_kinematics\]. The minimum of the parabola yields $\Theta_{\rm LSR,min} = 270.4$ km s$^{-1}$.
To estimate the uncertainty in $\Theta_{\rm LSR}$, we choose a bootstrap (resampling with replacement) method (see @a10 and references therein). This technique uses the entire sample of individual Sgr candidate star proper motions, and thus yields an estimate of the errors in $\Theta_{\rm LSR}$ including the effects of proper motion measurement errors and “contamination” of the proper-motion samples by Milky Way stars. Our selected samples of Sgr candidates in SAs 71, 94, 93, and 117 should contain mostly Sgr debris, plus some amount of contamination by MW stars that will vary depending on the depth of the proper-motion and radial-velocity catalogs, the local Sgr stream density, and the Galactic latitude of each field. From the Sgr samples in each field, we performed 100,000 bootstrap resamplings, wherein $N$ random selections were made from the $N$ original stars in each field (i.e., the catalogs of candidates were resampled with replacement). Iteratively $3\sigma$-clipped mean proper motions of these resampled Sgr candidates were measured, and the mean proper motions used in an identical $\chi^2$ fitting routine to that described above. Assuming that the contaminants in each sample are somewhat uniformly distributed in their kinematical quantities, this method should yield a statistically robust result for $\Theta_{\rm LSR}$ and its uncertainty (due to both the intrinsic measurement errors and the MW contamination). The best-fitting values of $\Theta_{\rm LSR}$ for these 100,000 samples are given as a histogram in the left panel of Figure \[fig:theta\_hist\], along with a Gaussian fit (red curve) to the results. From this Gaussian, we derive a final value of $\Theta_{\rm LSR} = 264 \pm 23$ km s$^{-1}$.
### $\Theta_{\rm LSR}$ Constraints Using Three-Dimensional Motions of Sgr Debris
The constraints we derived on $\Theta_{\rm LSR}$ using only the longitudinal proper motions assume that the contributions of $\Theta_{\rm LSR}$ to $\mu_b$ and $V_{\rm GSR}$ are negligible. If the Sgr orbital plane was exactly coincident with the Galactic $XZ_{\rm
GC}$ plane, then indeed the rotation velocity at the solar circle would only be reflected in the $\mu_l \cos($b) motions. In reality, the Sgr orbital plane is [*not*]{} perfectly aligned with the Milky Way $XZ$-plane, so there is some projection of $V_{\rm circ}$ onto $\mu_b$ and $V_{\rm GSR}$. In fact, for Sgr debris in the four fields of view comprising this study (SAs 71, 94, 93, and 117), only (75%, 87%, 61%, 58%, respectively) of the total value of $\Theta_{\rm
LSR}$ is projected onto $\mu_l \cos($b). We ran the $\chi^2$ fitting again, but this time including all three dimensions of the motion as constraints. The bootstrap analysis gave a result of $\Theta_{\rm LSR}
= 232 \pm 14$ km s$^{-1}$ – a histogram of the bootstrap results is seen in the right panel of Figure \[fig:theta\_hist\]. This mean value is lower by $\sim1.4\sigma$ than the result using only $\mu_l
\cos($b). Formally, this is a better fit than the one-dimensional result (with uncertainty of 14 km s$^{-1}$ compared to an uncertainty of 23 km s$^{-1}$ from the fits using only longitudinal proper motions), but the two are consistent within their 1$\sigma$ uncertainties.
Finally, we performed the same exercise using all three dimensions of Sgr debris motions, but excluding the less reliable SA 71 field. The uncertain identification of Sgr debris in SA 71 is likely the reason this field (at $\Lambda_{\odot} = 128\arcdeg$) is an outlier from the predicted kinematical trends in Figures \[fig:lambda\_3dkinematics\] and \[fig:mul\_varyvcirc\]. The bootstrap fit using only SAs 94, 93, and 117 yields $\Theta_{\rm LSR} = 244 \pm 17$ km s$^{-1}$. This slightly higher value for $\Theta_{\rm LSR}$ suggests that (as is evident in Figures \[fig:lambda\_3dkinematics\] and \[fig:mul\_varyvcirc\]) the Sgr candidates in SA 71 skew our results toward lower $\Theta_{\rm LSR}$.
Ultimately, we have derived three estimates of $\Theta_{\rm LSR}$ – one based on a simple one-dimensional analysis (using all four fields) that gave $264 \pm 23$ km s$^{-1}$, another based on three-dimensional data yielding $232 \pm 14$ km s$^{-1}$, and a final 3-D result with SA 71 excluded, which gave $244 \pm 17$ km s$^{-1}$. It is likely that the true result is somewhere between the two extremes (264 km s$^{-1}$ and 232 km s$^{-1}$) from our methods.
### Sgr Disruption Models For Best-Fitting $\Theta_{\rm LSR}$
We now repeat the $N$-body analysis described in Section \[nbody.sec\] two times, first taking $\Theta_{\rm LSR} =
264$ km s$^{-1}$ as a constraint on the models, then again using $\Theta_{\rm LSR} = 232$ km s$^{-1}$. The resulting $N$-body model for the 264 km s$^{-1}$ case matches the angular position, distance, and radial velocity trends of the observed Sgr tidal streams (using all of the observational constraints included in the original model) almost equally as well as did the LM10 model (formally, $\chi = 3.1$ for the $\Theta_{\rm LSR} = 264$ km s$^{-1}$ model, compared to $\chi = 3.4$ for the model; see discussion in Section 4.3 of ). In addition, as demonstrated in the left panel of Figure \[sgrpm.fig\] the proper motion of the remnant core of the Sgr dwarf in this revised model ($\mu_l \cos($b)$ = -2.54$ mas yr$^{-1}$, $\mu_b = 1.92$ mas yr$^{-1}$) is a substantially better match to observations (e.g., @dgv+05 [@ppo10]) than was the LM10 model.
However, the $N$-body model in a Milky Way halo with $\Theta_{\rm LSR}
= 232$ km s$^{-1}$ fits equally well as does the 264 km s$^{-1}$ case, $\chi = 3.1$. For this model, the Sgr core proper motion (seen in the right panel of Figure \[sgrpm.fig\]) is intermediate between those of the model and the 264 km s$^{-1}$ result, as might be expected. In this case, the proper motions are discrepant with both the @dgv+05 and @ppo10 results at the $\sim1.5\sigma$ level. In the following subsection we discuss the ramifications of these $\Theta_{\rm LSR}$ results for the Milky Way halo.
Robustness of the Galactic Mass Models
--------------------------------------
The results for $\Theta_{\rm LSR}$ from our analysis in the previous subsection suggest that the true value of $\Theta_{\rm LSR}$ as constrained by Sgr trailing tail debris likely lies between 232-264 km s$^{-1}$, but that more (or more sensitive) proper motion measurements of Sgr trailing debris are needed to resolve this issue. In this subsection, we will discuss the implications of the $\Theta_{\rm LSR}$ constraints resulting from our two methods; we remind the reader that the upper end of the range (i.e., the 264 km s$^{-1}$ result) is less robustly determined than the results that produced lower values of the circular velocity. However, we include this value in our discussion to present the reader with the range of possible ramifications of what, in either case, represents an upward revision of $\Theta_{\rm LSR}$ from the accepted value.
The value of $\Theta_{\rm LSR} = 232 \pm 14$ km s$^{-1}$ we found using all three dimensions of Sgr debris kinematics is consistent with the canonical 220 km s$^{-1}$ value at the roughly 1$\sigma$ level. However, this value is also consistent (within the 1$\sigma$ uncertainties) with recent determinations of $\Theta_{\rm LSR}$ that have found the rotation speed to be higher than the standard 220 km s$^{-1}$ value \[e.g., @rmz+09 – $\Theta_{\rm LSR} = (254\pm16)
(R_0/8.4)$ km s$^{-1}$; @bhr09 – $\Theta_{\rm LSR} = 244\pm13$ km s$^{-1}$\]. For a change in $\Theta_{\rm LSR}$ of only about 10 km s$^{-1}$, it is difficult to make any conclusions about whether the additional mass required to increase the rotation speed must reside in the Galactic halo or the disk/bulge. We note that placing the additional mass in the disk and bulge (with the halo fixed) yields $M_{\rm bulge} = 3.9 \times 10^{10} M_{\odot}$ and $M_{\rm disk} = 1.1
\times 10^{11} M_{\odot}$ for the 232 km s$^{-1}$ model – an increase of $\sim10\%$ over the disk and bulge mass from the model of . The relatively high value of $\Theta_{\rm LSR} =
264 \pm 23$ km s$^{-1}$ found by our analysis using only the $\mu_l
\cos($b) motions would require that the mass of the Galactic bulge and disk components be increased by $\sim50\%$ from the values assumed by to $M_{\rm bulge} = 5.2 \times 10^{10} M_{\odot}$ and $M_{\rm disk} = 1.53 \times 10^{11} M_{\odot}$. This disk mass is near the peak of the probability distribution ($M_{\rm disk} = 1.35
\times 10^{11} M_{\odot}$) found by @krh10 based on fitting the GD-1 stream in a three-component gravitational potential similar to our own. We caution however that the orbit of Sgr is largely insensitive to the [*distribution*]{} of the excess mass between the two baryonic components, and solutions that yield similar $\chi^2$ can be found by ascribing all or part of the needed adjustment in $\Theta_{\rm LSR}$ to changes in the mass of either the disk or bulge components alone. We do note, however, that the relative fraction of the total disk+bulge mass in each component is constrained by the need to reproduce the shape of the observed Milky Way rotation curve interior to $R_{\odot}$ (see Figure \[RotCurve.fig\]).
![Model Milky Way rotation curves as a function of radius from the Galactic center. Solid blue/green/red lines respectively represent rotation curves with $\Theta_{\rm LSR} = 264$ km s$^{-1}$ achieved via scaling the Galactic bulge$+$disk, Galactic disk alone, and Galactic halo alone. Included for comparison is the rotation curve of the original LM10 model (solid black line) normalized to $\Theta_{\rm LSR} = 220$ km s$^{-1}$. The vertical dotted line represents the location of the Sun at $R_{\odot} = 8$ kpc.[]{data-label="RotCurve.fig"}](f20.eps){width="2.25in"}
The total mass of the Milky Way interior to 50 kpc in the 264 km s$^{-1}$ model is $5.2 \times 10^{11} M_{\odot}$, similar to the value of $4.5 \times 10^{11} M_{\odot}$ in the model. Since we have accounted for the increased $\Theta_{\rm LSR} = 264$ km s$^{-1}$ by increasing the disk+bulge mass (which is a relatively small component of the total virial mass), the mass of the Milky Way interior to 200 kpc is $M_{\rm vir} = 1.6
\times 10^{12} M_{\odot}$, similar to the value of $1.5 \times 10^{12}
M_{\odot}$ derived by assuming that $\Theta_{\rm LSR} = 220$ km s$^{-1}$.
We note that it was not possible to obtain a satisfactory model (within our parameterization of the Milky Way components; exploration of different dark halo models is beyond the scope of this work) for the Sgr stream by leaving both the bulge and disk masses fixed at their values and accounting for changes in $\Theta_{\rm LSR}$ by scaling the dark matter halo. The dark matter halo profile is characterized by the parameters $v_{\rm halo}$ and $r_{\rm halo}$ (see Eqn. 3 of )[^13], which describe the total mass normalization and radial scalelength of the halo respectively. Since dark matter in the model contributes only 19% of the total centripetal acceleration in the solar neighborhood, $v_{\rm halo}$ must be scaled up drastically (by a factor of $\sim 3$ in total halo mass) to increase $\Theta_{\rm LSR}$ from 220 km s$^{-1}$ to 264 km s$^{-1}$, and necessitates a large increase in the space velocity of Sgr along its orbit (to $\sim 400$ km s$^{-1}$) to produce a leading arm debris stream at an observed peak distance of $\sim 50$ kpc (see Figure 6 of ). However, such a rapidly moving satellite model yields radial velocities along the tidal debris streams that are systematically discrepant from observations by $\sim
75$ km s$^{-1}$. Similarly, it is neither possible to obtain a satisfactory fit for larger $\Theta_{\rm LSR}$ values ($\Theta_{\rm
LSR} \gtrsim 280$ km s$^{-1}$, for which the halo mass scaling problem is even more extreme), nor for much lower values ($\Theta_{\rm LSR}
\sim 190$ km s$^{-1}$, because the baryonic bulge+disk mass component alone require $\Theta_{\rm LSR} > 190$ km s$^{-1}$).
Another possibility we considered was to again fix the baryonic mass (i.e., the bulge+disk component), but to a smaller value than the model, and allow the halo mass to vary. In particular, we attempted to fit a model with $\Theta_{\rm LSR} = 232$ km s$^{-1}$, but with the bulge+disk mass decreased by 10% from the values. Even this small change in the baryonic mass required scaling up the dark matter halo mass by $\sim50$% to keep $\Theta_{\rm LSR} = 232$ km s$^{-1}$. The best-fit $N$-body model in this case fit the Sgr trailing-tail velocities, but was a poor fit to the leading arm SDSS distances because of the increased speed of the Sgr core necessitated by the much larger halo. This illustration highlights the large changes in the halo mass effected by even small changes in the baryonic mass or the LSR velocity when fitting to observational data on the Sgr system. In fact, it is a testament to how well-constrained the Sgr system is by the current observational data that we are unable to fit the data if we change the dark halo model substantially.
One possible way to construct an $N$-body model of the Sgr dwarf that fits the observational data relatively well while dramatically changing $\Theta_{\rm LSR}$ is to adopt a Galactic halo whose scalelength is a factor of $\sim 10$ shorter than commonly adopted (from $r_{\rm halo} = 12$ kpc in the LM10 model to $r_{\rm halo} = 1$ kpc). However, the Galactic rotation curve implied by such a short halo scalelength declines steeply outside the solar circle (Figure \[RotCurve.fig\]), in conflict with observations (e.g., @sho09). We therefore conclude that it is not possible to satisfactorily model the Sgr dwarf in a Milky Way model with the bulge and disk masses fixed at the values of $M_{\rm
bulge} = 3.4 \times 10^{10} M_{\odot}$, $M_{\rm disk} = 1.0 \times
10^{11} M_{\odot}$, and the Milky Way halo scaled to produce $\Theta_{\rm LSR}$ much higher than 220 km s$^{-1}$. Thus our (and other recent) suggestions that $\Theta_{\rm LSR}$ is due an upward revision implies that the disk and/or bulge components – but [*not*]{} the halo – of the Milky Way are more massive than previously thought.
Abundances {#sgr_abund.sec}
==========
While the spectra in the Selected Areas were obtained primarily with kinematics in mind, they have sufficient resolution and, for a large fraction of stars, sufficient $S/N$, to obtain information not only on metallicity but also abundance patterns. This allows us an independent estimate of abundance distributions that, while of lower precision than the echelle work of @mbb+05 [@mbb+07] and @cmc+07 [@ccm+10], is derived for many Sgr stars, and is less biased than those M-giant studies.
Sgr Metallicity
---------------
Metallicities were measured for all stars using a software pipeline entitled “EZ\_SPAM” (Easy Stellar Parameters and Metallicities), details of which will appear in a forthcoming paper (Carlin et al. 2011, [*in prep.*]{}). EZ\_SPAM relies on the well-understood and calibrated Lick spectral indices (see, e.g., @wfg+94 [@f87]) to measure stellar properties from low-resolution spectra. In particular, estimates of \[Fe/H\] are derived for target stars using eight Fe indices combined with the H$\beta$ index. Calibration of these multi-dimensional data comes from fits of known \[Fe/H\] values as a function of the Lick Fe and H$\beta$ indices for stars in the atlas of @s07 [based on the spectra of @j98]. The EZ\_SPAM code yields \[Fe/H\] measurements with $1\sigma$ precision of $\sim0.3$ dex at $S/N \approx 20$, decreasing to $\sim0.1$ dex at higher signal-to-noise ($S/N \gtrsim 50$).
[ccccc]{}\
\
& & & &\
& & & & (degrees)\
\
71 & -1.14$\pm$0.19 & 0.97$\pm$0.14 & 24 & 128.2\
94 & -1.13$\pm$0.08 & 0.61$\pm$0.06 & 57 & 116.3\
93 & -1.25$\pm$0.11 & 0.47$\pm$0.08 & 23 & 103.2\
92 & -0.97$\pm$0.16 & ... & 2 & 90.1\
117 & -1.08$\pm$0.08 & 0.52$\pm$0.06 & 43 & 87.6\
116 & -1.21$\pm$0.21 & 0.64$\pm$0.15 & 10 & 74.9\
The metallicity distribution for all well-measured stars (i.e., those with spectra having $S/N > 20$) in each of the four fields (SAs 71, 94, 93, and 117) in which Sgr debris are reliably identified is given in Figure \[fig:feh\_hist\_sas\]. For each field, the solid (black) histogram shows \[Fe/H\] of non-Sgr stars, and the hashed (red) histogram gives the distribution of \[Fe/H\] for stars selected to be Sgr members. The bottom panel represents the distribution of metallicities for all Sgr members from the four trailing-tail fields, normalized by the total number of stars (147) in the sample to produce a fractional distribution. In each field, Sgr members are typically more metal-poor than the field stars, with the possible exception of those in SA 117.
For each SA field in the survey, a maximum likelihood estimate for \[Fe/H\] was derived from all well-measured stars in the final Sgr candidate sample. The resulting values for Sgr debris metallicities in each field are given in Table \[tab:sa\_feh\_tab\] along with $\sigma_{\rm [Fe/H]}$, the dispersion in \[Fe/H\] about the mean. As was the case for the kinematics in SAs 92 and 116, we regard the \[Fe/H\] results in these fields (and, to a lesser degree, those in SA 71) with some skepticism, because the identification of Sgr debris in these fields is rather unreliable. The mean metallicities for Sgr stars are displayed in Figure \[fig:feh\_lambda\] as a function of $\Lambda_{\odot}$; solid squares depict SAs 71, 94, 93, and 117 (i.e., the “well-measured“ fields), with open symbols included for SAs 92 and 116. Error bars represent the uncertainties in the mean value from the maximum likelihood estimator; however, the scatter of \[Fe/H\] for Sgr candidates in each field is rather large. Typical fields have $\sigma_{\rm [Fe/H]} = 0.5-0.6$ dex about the quoted mean values, similar to the broad metallicity distribution function for Sgr stars seen by, e.g., @sm02, @zbb+04, @sdm+07, and @mbb+05. The scatter is even larger in SA 71 (at $\Lambda_{\odot} = 128\arcdeg$); this may arise for a number of reasons. As can be discerned from Figure \[fig:radec\], SA 71 may be sampling Sgr debris stripped on multiple pericentric passages (i.e., both gold and magenta debris may be present in this field). Furthermore, this is the lowest-latitude field among those in this study, and thus may be also suffering more contamination from Galactic thick disk stars. Finally, we note that SA 71 has been shown by @ccg+08 to contain a significant number of stars from the ”Monoceros stream" overdensity, which could contribute to the inflation of the metallicity dispersion in this field, though it is unlikely that many Monoceros stars would lie within our Sgr radial velocity criteria for this field. Also shown in Figure \[fig:feh\_lambda\] is a solid line at constant \[Fe/H\] = -1.15, which is the mean value from the four well-measured fields; the tight correspondence of the mean values of each field to this line is consistent with the notion that debris along this narrow stretch of the trailing tail has constant metallicity. However, there is a hint of a shallow gradient, which we confirm by fitting a linear trend to the four good data points. This fit, overlaid as a dashed line in Figure \[fig:feh\_lambda\], is \[Fe/H\] = -0.991$\pm$0.003 - (0.0014$\pm$0.0036) $\Lambda_{\odot}$. While suggestive of a slight gradient, the slope given is also consistent with zero within the errors of the fit. This is not surprising considering that nearly all debris in the portion of the stream contained within this study is expected to have been stripped on the same pericentric passage of the Sgr core, as evidenced by the fact that all of our fields overlap gold-colored debris in Figure \[fig:radec\] (i.e., debris that became unbound during the last two perigalactic passages; see @lm10a for more detail about the color scheme used).
Our measured metallicity of \[Fe/H\] $\sim$ -1.2 for Sgr trailing debris is $\sim0.6$ dex more metal-poor than the result obtained by @kyd10 for trailing-tail M-giants. This is not surprising, as M giants are biased toward relatively younger, more enriched stellar populations. In spite of this difference between the older, metal-poor Sgr stars in our SA fields and the more metal-rich M-giants, we measure a shallow gradient in \[Fe/H\] as a function of $\Lambda_{\odot}$, with a slope consistent with the @kyd10 measurement, and just a simple offset in the zero-point metallicity. A more apt comparison for the mean metallicity of Sgr debris in our SA fields is the work of @sig+10, who used SDSS Stripe 82 data to develop a new technique for estimating metallicity from photometric data, which relies on combined information from both RR Lyrae variables and main-sequence stars from the same structure. Because SAs 94, 93, and 92 are within Stripe 82 (and, in fact, we have used those SDSS data in our analysis), the Sesar et al. study probes identical stellar populations from the Sgr trailing tail as our work. This is borne out by the fact that our measured \[Fe/H\] = -1.15 is in excellent agreement with the value of \[Fe/H\] = -1.20$\pm$0.1 derived by @sig+10 for Sgr debris along the same portion of the trailing stream.
Metallicity Distribution Function
---------------------------------
Previous attempts to measure the metallicity distribution function (MDF) of the Sgr stream have suffered from both small number statistics and stellar tracers that have an inherent metallicity bias. For example, @cmc+07 measured the MDF for M giant stars at several points along the Sgr leading stream as well as its core; using these data they attempted to reconstruct the MDF that the Sgr stream progenitor would have had several Gyr ago. However, because M giants only form in metal rich stellar populations, this analysis, while able to show that the mean metallicity of stars varies along the stream, was inadequate to assess the MDF across the full metallicity range of the system. In addition, the analysis was based on a relatively small sample (about 7 dozen stars), nearly half of which are in the core of the Sgr dSph. @mbb+07 derived abundances of Sgr M giants along the [*trailing*]{} tidal tail, from which they derived $\langle$\[Fe/H\]$\rangle = -0.61 \pm 0.13$ between $80\arcdeg <
\Lambda_{\sun} < 100\arcdeg$ (from 6 M giants), and $\langle$\[Fe/H\]$\rangle = -0.83 \pm 0.11$ (mean of 4 stars) for debris even further along the trailing tail. The existence of a metallicity gradient among Sgr stream M giants along both the trailing and leading tails was confirmed by @kyd10, who combined their additional measurements of 5 stars at $\Lambda_{\sun} = 66\arcdeg$ ($\langle$\[Fe/H\]$\rangle \sim -0.5$) and 6 stars at $\Lambda_{\sun} =
132\arcdeg$ ($\langle$\[Fe/H\]$\rangle \sim -0.7$) with the Chou et al. and Monaco et al. results to confirm the gradient in \[Fe/H\] among Sgr stream M giants. However, all of these M-giant studies suffer an inherent bias toward metal-rich stellar populations, and are likely not showing the true MDF of the Sgr system.
Blue horizontal branch stars (BHBs) are another easily-identified and rather unambiguous tracer of halo substructure that has been used to probe the Sgr stream. However, BHB stars arise only in old, metal-poor populations, and are thus not ideal tracers of the global MDF of a system consisting of multiple stellar populations. @ynj+09 performed an extensive study of the Sgr tails using SDSS and SEGUE spectroscopy of BHB stars in both the northern and southern Galactic caps. BHB stars in the portions of both the leading ($200\arcdeg <
\Lambda_{\sun} < 300\arcdeg$) and trailing ($70\arcdeg <
\Lambda_{\sun} < 110\arcdeg$) tails in this study have MDFs peaking at $\langle$\[Fe/H\]$\rangle \sim -1.7$, with significant numbers of stars as low as \[Fe/H\] $\sim$ -2.5. Although this result turns up metal-poor populations not seen in M giants, it is difficult to make conclusions about the overall MDF of the Sgr stream or progenitor based on biased metallicity tracers such as BHB stars and M giants.
Fortunately, because the present analysis makes use of MSTO stars, it is far less susceptible to metallicity biases and can provide new insights into the MDF (particularly at the intermediate to metal-poor end) of the stream (and therefore the progenitor) MDF. Of course, our spectra have $\sim10\times$ worse resolution than the various echelle resolution studies, but our sample of Sgr stream stars is significantly larger, including 147 with good enough S/N ($>20$) for \[Fe/H\] measurements to the approximately $\lesssim$ 0.3 dex level. As shown in Figure \[fig:feh\_hist\_sas\] and \[fig:feh\_lambda\], the mean \[Fe/H\] for that portion of the stream probed by our SA data is about -1.1, but with a significant tail to both solar metallicity as well as very metal poor ($< -2.0$) stars. Indeed, our sample includes some rather metal-poor stars associated with the Sgr system, with stars as metal-poor as the -2.5 dex BHBs seen by @ynj+09.
Figure \[fig:sgr\_mdf\] shows the MDF derived from our data in the $88\arcdeg < \Lambda_{\sun} < 128\arcdeg$ portion of the Sgr trailing tail. The MDF we derive is significantly broader and extending to much more metal-poor stars than indicated by the biased, M giant studies (shown in the upper panel of Figure \[fig:sgr\_mdf\] as a black dashed histogram for the Sgr core and green dot-dashed lines for the leading arm), encompassing both the M-giant and BHB results. A comparison of our MDF to the @cmc+07 reconstruction of the Sgr M-giant MDF from several Gyr ago based on their core and leading-arm samples is given in the lower panel of Figure \[fig:sgr\_mdf\]. Clearly our Sgr trailing-tail sample is lacking the metal-rich component seen in the present-day core, but shows a similar distribution to the metal-poor tail of the reconstructed MDF. Additional metal-poor stars are present in our sample that are not seen in the M giant samples; these are likely drawn from similar populations to those in the @ynj+09 study.
Obviously, as has been shown by Chou et al. and others, the total MDF of the entire Sgr system will include a higher contribution of metal-rich stars when the core is included, but we also expect more metal-poor stars from those parts of the tails with larger separation from the core than we explore. Thus, while we cannot yet accurately reconstruct the total MDF of the Sgr system, at least we now have a better feel of the [*breadth*]{} of the Sgr MDF from the data shown in Figures \[fig:feh\_hist\_sas\] and \[fig:sgr\_mdf\]. Comparison of the latter MDF with those of other MW dSphs (summarized, e.g., in @kls+11) shows Sgr to be more typical of other MW satellites. In particular, the Sgr MDF resembles even more that of the LMC (as has been previously suggested by, e.g., @mbf+03, @ctg+05, and @mbb+05), which has been argued to be a chemical analog to the Sgr progenitor by @ccm+10 and a morphological analog by @lkm+10.
“Alpha” Abundances
------------------
As shown in Section 5.1, we have identified metal-poor populations in (at least) four of the fields from our study, which explore a different segment of the stellar populations in the Sgr stream than previous M-giant studies. We have observed stars in these fields only at low resolution, and thus cannot do detailed element-by-element chemical analysis such as that enabled by high-resolution spectroscopy. However, we can use the low-resolution Lick indices to explore relative $\alpha$-abundances for the stars in our study. Specifically, we explore the relative Mg abundances using the Lick Mg b index centered at 5160-5190 Å. Calibrating the Mg b index to an actual \[Mg/Fe\] abundance is difficult, because the strength of Mg lines is highly sensitive to surface gravity, with some additional sensitivity to effective temperature and \[Fe/H\]. Disentangling these effects is difficult with low-resolution spectra, but we can still explore a subset of the stars in our samples in a way that is relatively free of the effects of surface gravity and temperature of individual stars. To do so, we select only blue ($0.2 < g - r < 0.7$, or $B - V < 0.9$) stars, which should be mostly main-sequence dwarfs (thus, with similar surface gravity), since no giants are found at such blue colors. Furthermore, the temperature sensitivity of the Mg line strength, which is already much smaller than the log $g$ sensitivity, is mitigated by concentrating on a limited color range. In Figure \[fig:mgfe\_feh\] we show a “pseudo-\[Mg/Fe\]“ ratio, given as the logarithm of the ratio of the Lick Mg b index to the mean of all eight Lick Fe indices (after transforming them to a common scale), for all of the blue stars in SAs 71, 94, 93, and 117 for which we have high enough signal-to-noise ($>30$) to precisely measure indices and \[Fe/H\]. Black points in this diagram are all stars with non-Sgr radial velocities, while our samples of all stars with Sgr-like RVs in each field are shown as colored points. It is readily apparent that the black (MW) points mostly occupy a different region of the diagram than the colored (Sgr) dots, which suggests an intrinsic chemical difference between the populations (though some overlap is expected, especially at low metallicities, where many MW halo stars likely resemble dSphs in their abundance patterns). Indeed, the behavior seen in Figure \[fig:mgfe\_feh\] is exactly that seen for many MW dSphs – for more metal-rich dSph stars, the Mg (or $\alpha$) abundance is lower (on average) at a given \[Fe/H\] than in the Galactic populations, with the two populations converging at lower metallicities (i.e., at the ”knee" in the dSph’s distribution). Among the more metal-rich (and younger) M-giant populations of the Sgr stream, there is some indication that the knee in \[$\alpha$/Fe\] occurs at -1.2 $\lesssim$ \[Fe/H\] $\lesssim$ -1.0 [@ccm+10; @mbb+07], but this is difficult to assess because of the lack of M-giants at lower metallicity. Thus the apparent convergence of Sgr trailing tail \[Mg/Fe\] with the plateau seen in Galactic stars at \[Fe/H\] $\lesssim$ -1.5 may be an extension of the same behavior seen in the M-giant studies. Alternatively, since we’ve already shown that the mean \[Fe/H\] along the trailing tail differs between the M-giant sample of @mbb+07, who find \[Fe/H\] $\sim$ -0.6, and our result of \[Fe/H\] $\sim$ -1.2 (which is also consistent with the findings of @sig+10), our study may be sampling a distinctly older, more metal-poor population of Sgr debris than the M-giant tracers. Further characterization of the $\alpha$-element behavior along the Sgr trailing tail would benefit from either a calibration of our Mg b/$<$Fe$>$ index onto \[Mg/Fe\] abundance or the identification of bona fide stream giant stars that are bright enough for echelle-resolution spectroscopic follow-up.
Summary {#summary.sec}
=======
We have presented the first large-scale study of the 3-D kinematics of the Sagittarius trailing tidal stream, with data spanning $\sim60\arcdeg$ along the trailing tail. The data include deep, precise proper motions derived from photographic plates with a $\sim$90-year baseline, and radial velocities from more than 1500 low-resolution stellar spectra, of which $>150$ have been identified as Sgr debris stars. Mean absolute proper motions of these Sgr stars in four of the six 40$\arcmin \times 40\arcmin$ fields from our survey have been derived with $\sim0.25-0.7$ mas yr$^{-1}$ per field precision in each dimension (depending on the quality and depth of plate material and the number of spectra obtained in each field). Mean three-dimensional kinematics in each of these four fields have been shown to agree with the predicted $V_{\rm GSR}$ and $\mu_b$ from the Sagittarius disruption models of . However, there is a systematic disagreement in the $\mu_l \cos($b) proper motions (with the exception of the somewhat problematical SA 71 field), which we use to assess refinements to the mass scale of the Milky Way (particularly its disk and bulge components).
While proper motions along the portion of the trailing tail in this study provide constraints on Sgr tidal disruption models, the fortuitous orientation of the Sgr plane also allows us to use the measured proper motions to derive the circular velocity at the Solar circle (or “Local Standard of Rest”), $\Theta_{\rm LSR}$. Our first-order approximation using only the $\mu_l \cos($b) proper motions as constraints yields $\Theta_{\rm LSR} = 264 \pm 23$ km s$^{-1}$. From our measured 3-D kinematics, we find this fundamental Milky Way parameter to be $\Theta_{\rm LSR} = 232 \pm 14$ km s$^{-1}$, or $\sim1\sigma$ higher than the IAU standard value of 220 km s$^{-1}$. When we remove SA 71, a field in which it is more difficult to unambiguously identify Sgr debris, from the sample we find $\Theta_{\rm LSR} = 244 \pm 17$ km s$^{-1}$. We suggest that the true value of $\Theta_{\rm LSR}$ lies somewhere between 232-264 km s$^{-1}$, while noting that all three of these estimates are consistent with each other within their 1$\sigma$ uncertainties.
Our general result that the circular velocity at the Solar radius is higher than the IAU standard of 220 km s$^{-1}$ agrees with the recent derivation of $\Theta_{\rm LSR} = 254 \pm 16$ km s$^{-1}$ by @rmz+09 using trigonometric parallaxes of star forming regions in the outer disk. The same maser data from the Reid et al. study were reanalyzed by @bhr09, and yield a result of 246 $\pm$ 30 km s$^{-1}$ (244 $\pm$ 13 km s$^{-1}$ if priors on the proper motion of Sgr A\* are included, and 236 $\pm$ 11 km s$^{-1}$ if the additional contribution of orbital fitting to the GD-1 stellar stream is included). Again, our independent result is consistent with these studies, and inconsistent with the IAU accepted value of 220 km s$^{-1}$ for this fundamental constant at the 1-2$\sigma$ level. Identification of additional Sgr candidates could increase the accuracy of our determination of $\Theta_{\rm LSR}$, as could the addition of another epoch of accurate data to the proper motion measurements.
We note that while @rmz+09 argued that their measurement of 254 km s$^{-1}$ would imply an upward revision of the total Milky Way mass by a factor of $\sim2$ (to a mass similar to that of M31), we have shown that this is not required to produce $\Theta_{\rm LSR}$ even higher than that of Reid et al. Scaling the Milky Way dark matter halo up in mass to a level that yields $\Theta_{\rm LSR} = 264$ km s$^{-1}$ while simultaneously reproducing known leading arm debris requires the Sgr core to have a high ($\sim400$ km s$^{-1}$) space velocity, resulting in RVs in the tidal streams that are discrepant by $\sim75$ km s$^{-1}$ from measured values. Instead, we show that because $>
80\%$ of the centripetal force at the location of the Sun is contributed by mass in the Galactic disk and bulge, an increase of $\sim50\%$ in the mass of the disk+bulge accounts for the additional acceleration needed to produce 264 km s$^{-1}$ rotation at the solar circle, while contributing only a small ($\sim7\%$) increase to the total virial mass of the Milky Way. With the additional constraint on the disk+bulge mass provided by our measurement of $\Theta_{\rm LSR}$, we have found a satisfactory model of Sgr disruption that matches all of the constraints used in fitting the model, while additionally predicting a proper motion for the Sgr dwarf that is in much better agreement with observations than the model.
Stellar metallicities have been derived from the low-resolution spectra of Sgr candidates, and the mean metallicity of Sgr tidal debris derived in each field. We find that a constant \[Fe/H\] = -1.15 is consistent with the observations of all four fields for which Sgr members were reliably identified. However, a linear fit to these four data points suggests that a gradient of ($1.4 \times 10^{-3}$) dex degree$^{-1}$ is also reasonable (though this value is consistent with zero slope within the uncertainty), in line with previous findings (e.g., @cmc+07 [@kyd10]) of a metallicity gradient among M-giants along both the leading and trailing tidal tails. The scatter of \[Fe/H\] in each of the survey fields is $\gtrsim$0.5 dex, which is typical of the stellar populations seen in the core of the Sgr dSph (e.g., @sm02 [@zbb+04; @sdm+07; @mbb+05]). We show the metallicity distribution function for the trailing tail that is free from the biases inherent in previous studies of the Sgr MDF. We find that the MDF of trailing debris is similar to MDFs of typical classical Milky Way dwarf spheroidals. A “pseudo-\[Mg/Fe\]” was measured based on the ratio of Lick Mg b and $<$Fe$>$ indices; the behavior of log (Mg b/$<$Fe$>$) with \[Fe/H\] for Sgr main-sequence candidates is markedly different from Galactic stars of similar photometric colors (identified by radial velocity) from among the same datasets. Furthermore, the trend is similar to that typically seen for dSphs, in that \[Mg/Fe\] is deficient at a given \[Fe/H\] for Sgr stars relative to the Milky Way field populations, and converges to a “knee” in Figure \[fig:mgfe\_feh\] at lower (\[Fe/H\] $\sim$ -1.5) metallicity. This is a lower \[Fe/H\] than previously reported for Sgr M giants, and may reflect a bias intrinsic to those earlier M-giant studies. High-resolution spectroscopic follow-up will be necessary to confirm this trend among the old, metal-poor populations of recently-stripped Sgr debris.
We appreciate the useful comments provided by the anonymous referee. We thank Mei-Yin Chou for kindly sharing the Sgr MDF data used in Figure 23, and Heidi Newberg for many useful discussions. JLC acknowledges support from National Science Foundation grant AST-0937523, and observing travel support from the NOAO thesis student program for proposal ID 2008B-0448. JLC and SRM acknowledge partial funding of this work from NSF grant AST-0807945 and NASA/JPL contract 1228235. DIC and TG acknowledge NSF grant AST-0406884. DRL acknowledges support provided by NASA through Hubble Fellowship grant \# HF-51244.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.
SRM is grateful to Allan Sandage for alerting him to the existence of the Mt. Wilson Selected Areas plates, for suggesting their use as a first epoch to expand our proper motion work in Kapteyn Selected Areas, and for his contribution of KPNO plates to our survey.
[*Facilities:*]{} , , , , , ,
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
[113]{} natexlab\#1[\#1]{}
, M. G., [Navarro]{}, J. F., [Steinmetz]{}, M., & [Eke]{}, V. R. 2003, , 591, 499
, K. N., [Adelman-McCarthy]{}, J. K., [Ag[ü]{}eros]{}, M. A., [et al.]{} 2009, , 182, 543
, C. 2001, , 377, 389
, R. 2010, ArXiv:1009.2755
, L. R., [Piotto]{}, G., [King]{}, I. R., & [Anderson]{}, J. 2003, , 126, 247
, M., [Ferraro]{}, F. R., & [Ibata]{}, R. 2003, , 125, 188
, M., [Newberg]{}, H. J., [Correnti]{}, M., [Ferraro]{}, F. R., & [Monaco]{}, L. 2006, , 457, L21
, V., [Zucker]{}, D. B., [Evans]{}, N. W., [et al.]{} 2006, , 642, L137
, V., [Evans]{}, N. W., [Irwin]{}, M. J., [et al.]{} 2007, , 658, 337
, J., [Hogg]{}, D. W., & [Rix]{}, H. 2009, , 704, 1704
, J. S., & [Johnston]{}, K. V. 2005, , 635, 931
, J. L., [Casetti-Dinescu]{}, D. I., [Grillmair]{}, C. J., [Majewski]{}, S. R., & [Girard]{}, T. M. 2010, , 725, 2290
, G., [Zinn]{}, R., & [Moni Bidin]{}, C. 2007, , 466, 181
, D. I., [Carlin]{}, J. L., [Girard]{}, T. M., [et al.]{} 2008, , 135, 2013
, D. I., [Girard]{}, T. M., [Majewski]{}, S. R., [et al.]{} 2009, , 701, L29
, D. I., [Majewski]{}, S. R., [Girard]{}, T. M., [et al.]{} 2006, , 132, 2082
, M., [Cunha]{}, K., [Majewski]{}, S. R., [et al.]{} 2010, , 708, 1290
, M., [Majewski]{}, S. R., [Cunha]{}, K., [et al.]{} 2007, , 670, 346
, A. A., [Tolstoy]{}, E., [Gallagher]{}, III, J. S., & [Smecker-Hane]{}, T. A. 2005, , 129, 1465
, M., [Bellazzini]{}, M., [Ibata]{}, R. A., [Ferraro]{}, F. R., & [Varghese]{}, A. 2010, , 721, 329
, W., & [Binney]{}, J. J. 1998, , 298, 387
, D. I., [Girard]{}, T. M., [van Altena]{}, W. F., & [L[ó]{}pez]{}, C. E. 2005, , 618, L25
, D. I., [Majewski]{}, S. R., [Girard]{}, T. M., & [Cudworth]{}, K. M. 2000, , 120, 1892
, D. I., [Majewski]{}, S. R., [Girard]{}, T. M., [et al.]{} 2002, , 575, L67
, R. C., [Helmi]{}, A. [Morrison]{}, H. L., [et al.]{} 2001, , 555, L37
, D., [Fata]{}, R., [Roll]{}, J., [et al.]{} 2005, , 117, 1411
, M., & [Whitelock]{}, P. 1997, , 291, 683
, M., [Belokurov]{}, V., [Evans]{}, N. W., [et al.]{} 2006, , 651, 167
, A. S., [Johnston]{}, K. V., [Bullock]{}, J. S., & [Robertson]{}, B. E. 2006, , 638, 585
, E. D. 1987, , 93, 1388
, P. M., [Mu[ñ]{}oz]{}, R. R., [Phelps]{}, R. L., [Majewski]{}, S. R., & [Kunkel]{}, W. E. 2006, , 131, 922
, D., [Smith]{}, V. V., [Wallerstein]{}, G., [Gonzalez]{}, G., & [Charbonnel]{}, C. 2005, , 129, 1428
, A. M., [Salim]{}, S., [Weinberg]{}, N. N., [et al.]{} 2008, , 689, 1044
, L., [Grebel]{}, E. K., [Odenkirchen]{}, M., & [Chiosi]{}, C. 2004, , 422, 205
, C. J. 2006, , 645, L37
—. 2006, , 651, L29
—. 2009, , 693, 1118
, C. J., & [Dionatos]{}, O. 2006, , 643, L17
, J. C., [Gilmore]{}, G., [Irwin]{}, M. J., & [Carter]{}, D. 1994, , 269, 957
, A. 2004, , 610, L97
, A., & [White]{}, S. D. M. 2001, , 323, 529
, R., [Lewis]{}, G. F., [Irwin]{}, M., [Totten]{}, E., & [Quinn]{}, T. 2001, , 551, 294
, R. A., [Gilmore]{}, G., & [Irwin]{}, M. J. 1994, , 370, 194
, R. A., [Irwin]{}, M. J., [Lewis]{}, G. F., [Ferguson]{}, A. M. N., & [Tanvir]{}, N. 2003, , 340, L21
, K. V., [Law]{}, D. R., & [Majewski]{}, S. R. 2005, , 619, 800
, K. V., [Spergel]{}, D. N., & [Hernquist]{}, L. 1995, , 451, 598
, K. V., [Zhao]{}, H., [Spergel]{}, D. N., & [Hernquist]{}, L. 1999, , 512, L109
, L. A. 1998, PhD thesis, The University of North Carolina at Chapel Hill
, J. S., [Richer]{}, H. B., [Hansen]{}, B. M., [et al.]{} 2004, , 601, 277
, J. C. 1906, [Plan of selected areas]{} (Groningen, Hoitsema brothers, 1906.)
, S. C., [Yong]{}, D., & [Da Costa]{}, G. S. 2010, , 720, 940
, F. J., & [Lynden-Bell]{}, D. 1986, , 221, 1023
, E. N., [Lanfranchi]{}, G. A., [Simon]{}, J. D., [Cohen]{}, J. G., & [Guhathakurta]{}, P. 2011, , 727, 78
, J., [Wilkinson]{}, M. I., [Evans]{}, N. W., [Gilmore]{}, G., & [Frayn]{}, C. 2002, , 330, 792
, S. E., [Rix]{}, H., & [Hogg]{}, D. W. 2010, , 712, 260
, K., & [Tremaine]{}, S. 1994, , 421, 178
, D. R., [Johnston]{}, K. V., & [Majewski]{}, S. R. 2005, , 619, 807
, D. R., & [Majewski]{}, S. R. 2010, , 714, 229 (LM10)
—. 2010, , 718, 1128
, D. R., [Majewski]{}, S. R., & [Johnston]{}, K. V. 2009, , 703, L67
, E. L., [Kazantzidis]{}, S., [Majewski]{}, S. R., [et al.]{} 2010, , 725, 1516
, S. R. 1992, , 78, 87
—. 1993, , 31, 575
, S. R. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 165, The Third Stromlo Symposium: The Galactic Halo, ed. [B. K. Gibson, R. S. Axelrod, & M. E. Putman]{}, 76–+
, S. R., [Law]{}, D. R., [Polak]{}, A. A., & [Patterson]{}, R. J. 2006, , 637, L25 (MLPP)
, S. R., [Munn]{}, J. A., & [Hawley]{}, S. L. 1996, , 459, L73+
, S. R., [Skrutskie]{}, M. F., [Weinberg]{}, M. D., & [Ostheimer]{}, J. C. 2003, , 599, 1082
, S. R., [Kunkel]{}, W. E., [Law]{}, D. R., [et al.]{} 2004, , 128, 245
, D., [G[ó]{}mez-Flechoso]{}, M. [Á]{}., [Aparicio]{}, A., & [Carrera]{}, R. 2004, , 601, 242
, D., & [Binney]{}, J. 1981, [Galactic astronomy: Structure and kinematics /2nd edition/]{} (W. H. Freeman and Co.)
, D. J., [Wyatt]{}, W. F., [Caldwell]{}, N., [Conroy]{}, M. A., [Furesz]{}, G., & [Tokarz]{}, S. P. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. [R. A. Shaw, F. Hill, & D. J. Bell]{}, 249–+
, M., & [Nagai]{}, R. 1975, , 27, 533
, L., [Bellazzini]{}, M., [Bonifacio]{}, P., [et al.]{} 2007, , 464, 201
, L., [Bellazzini]{}, M., [Bonifacio]{}, P., [et al.]{} 2005, , 441, 141
, L., [Bellazzini]{}, M., [Ferraro]{}, F. R., & [Pancino]{}, E. 2003, , 597, L25
, R. R., [Carlin]{}, J. L., [Frinchaboy]{}, P. M., [et al.]{} 2006, , 650, L51
, R. R., [Majewski]{}, S. R., & [Johnston]{}, K. V. 2008, , 679, 346
, R. R., [Majewski]{}, S. R., [Zaggia]{}, S., [et al.]{} 2006, , 649, 201
, J. A., [Monet]{}, D. G., [Levine]{}, S. E., [et al.]{} 2004, , 127, 3034
—. 2008, , 136, 895
, H. J., [Yanny]{}, B., [Rockosi]{}, C., [et al.]{} 2002, , 569, 245
, R. P., & [Merrifield]{}, M. R. 1998, , 297, 943
, J., [Belokurov]{}, V., [Evans]{}, N. W., [et al.]{} 2010, , 408, L26
, C., & [Meylan]{}, G. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 50, Structure and Dynamics of Globular Clusters, ed. [S. G. Djorgovski & G. Meylan]{}, 357
, C., [Piatek]{}, S., & [Olszewski]{}, E. W. 2010, , 139, 839
, M. J., & [Brunthaler]{}, A. 2004, , 616, 872
, M. J., [Menten]{}, K. M., [Zheng]{}, X. W., [et al.]{} 2009, , 700, 137
, A. C., [Reyl[é]{}]{}, C., [Derri[è]{}re]{}, S., & [Picaud]{}, S. 2003, , 409, 523
, L., [Bonifacio]{}, P., [Buonanno]{}, R., [et al.]{} 2007, , 465, 815
, R. P. 2007, , 171, 146
, D. J., [Finkbeiner]{}, D. P., & [Davis]{}, M. 1998, , 500, 525
, F. H., [Kapteyn]{}, J. C., [van Rhijn]{}, P. J., [Joyner]{}, M. C., & [Richmond]{}, M. L. 1930, [Mount Wilson catalogue of photographic magnitudes in selected areas 1-139]{} (Carnegie institution of Washington)
, L., & [Zinn]{}, R. 1978, , 225, 357
, B., [Ivezi[ć]{}]{}, [Ž]{}. [Grammer]{}, S. H., [et al.]{} 2010, , 708, 717
, M., [Venn]{}, K. A., [Tolstoy]{}, E., [et al.]{} 2003, , 125, 684
, M. D., [C[ô]{}t[é]{}]{}, P., & [Sargent]{}, W. L. W. 2001, , 548, 592
, M. H., [Dotter]{}, A., [Majewski]{}, S. R., [et al.]{} 2007, , 667, L57
, M. H., [et al.]{} 2011, , accepted (ArXiv:1108.6276)
, T. A., & [McWilliam]{}, A. 2002, ArXiv:0205411
, Y., [Honma]{}, M., & [Omodaka]{}, T. 2009, , 61, 227
, S. T., [Majewski]{}, S. R., [Mu[ñ]{}oz]{}, R. R., [et al.]{} 2007, , 663, 960
, E., [Hill]{}, V., & [Tosi]{}, M. 2009, , 47, 371
, E., [Venn]{}, K. A., [Shetrone]{}, M., [et al.]{} 2003, , 125, 707
, J., & [Davis]{}, M. 1979, , 84, 1511
, F. 2007, , 474, 653
, K. A., [Irwin]{}, M., [Shetrone]{}, M. D., [et al.]{} 2004, , 128, 1177
, S. S., [Mateo]{}, M., [Olszewski]{}, E. W., & [Keane]{}, M. J. 1995, , 109, 151
, G., [Faber]{}, S. M., [Gonzalez]{}, J. J., & [Burstein]{}, D. 1994, , 94, 687
, B., [Newberg]{}, H. J., [Grebel]{}, E. K., [et al.]{} 2003, , 588, 824
, B., [Rockosi]{}, C., [Newberg]{}, H. J., [et al.]{} 2009, , 137, 4377
, B., [Newberg]{}, H. J., [Johnson]{}, J. A., [et al.]{} 2009, , 700, 1282
, F., [Zhu]{}, Z., & [Kong]{}, D. 2008, , 8, 714
, S., [Bonifacio]{}, P., [Bellazzini]{}, M., [et al.]{} 2004, Memorie della Societa Astronomica Italiana Supplementi, 5, 291
[^1]: Though we note that there is now evidence for extended tidal debris populations around the Carina [@mmz+06; @mmj08] and Leo I [@smm+07] dSphs. Also, some debate still exists over whether the HI Magellanic Stream derives from tidal stripping of Small or Large Magellanic Cloud gas versus from ram pressure stripping.
[^2]: Further complications have arisen due to an apparent bifurcation of the leading stream [@bze+06]; the model was not designed to address this issue. Several attempts to explain this detail invoke overlapping debris from multiple orbital wraps (@fbe+06; though @ynj+09 find similar stellar populations in both arms, likely ruling out this scenario) or a disk-galaxy progenitor for Sgr (@pbe+10; but cf. @lkm+10). The highly-elliptical shape of the Sgr dwarf has recently been reproduced by [@lkm+10], who model Sgr as a disk galaxy embedded in an extended dark halo. Tidal stirring transforms the initially disk dwarf into an extended elliptical shape over two pericentric passages; the rotation of the progenitor may also explain the bifurcation of the Sgr leading arm.
[^3]: Throughout this paper, when we refer to Galactic Cartesian ($X,Y,Z)_{\rm GC}$ coordinates, we are specifically referring to a right-handed Cartesian frame centered on the Galactic center, with $X_{\rm GC}$ positive in the direction from the Sun to the Galactic center, $Y_{\rm GC}$ in the direction of the Sun’s motion through the Galaxy, and $Z_{\rm GC}$ upward out of the plane. Assuming the Sun is at $R_0 = 8.0$ kpc from the Galactic center, this places the Sun at $(X,Y,Z)_{\rm
GC}$=(-8.0,0,0) kpc. The corresponding velocity components will be denoted ($U,V,W)_{\rm GC}$, where the “GC” denotes velocities relative to the Galactic rest frame.
[^4]: $\Lambda_{\sun}$ was defined by @msw+03 as longitude in the Sgr debris plane as seen from the Sun; $\Lambda_{\sun}=0^{\circ}$ at the present Sgr position, and increases along the trailing tail.
[^5]: The WIYN Observatory is a joint facility of the University of Wisconsin-Madison, Indiana University, Yale University, and the National Optical Astronomy Observatory.
[^6]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.
[^7]: Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. MMT telescope time was granted by NOAO (proposal ID 2008B-0448), through the Telescope System Instrumentation Program (TSIP). TSIP is funded by the NSF.
[^8]: <http://iparrizar.mnstate.edu/~juan/research/ESPECROAD/index.php>
[^9]: From the Geneva Radial-Velocity Standard Stars at <http://obswww.unige.ch/~udry/std/>
[^10]: $V_{\rm GSR} \equiv V_{\rm
helio} + 9.0 \cos b \cos l + 232.0 \cos b \sin l + 7.0 \sin b$, where $V_{\rm helio}$ is the measured heliocentric radial velocity. This calculation assumes a circular velocity of 220 km s$^{-1}$ at the solar circle, and solar peculiar motion of $(U_0, V_0, W_0) = (9.0,
12.0, 7.0)$ km s$^{-1}$.
[^11]: Model query available at http://model.obs-besancon.fr/.
[^12]: Note that we are not suggesting that Sgr has stars 15.8 Gyr old – this isochrone is simply meant as a guide to show that these stars could plausibly be BHB stars associated with Sgr. A blue horizontal branch is only seen in the oldest (log(age) > 10.15) of the @ggo+04 isochrones at this metallicity.
[^13]: The halo triaxiality is an added complication that has little bearing on the present discussion.
|
---
abstract: 'This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation (“the cell problem”), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.'
address:
- |
Department of Mathematics\
University of Arizona\
Tucson, AZ, 85721, USA
- |
Department of Mathematics and Program in Applied Mathematics\
University of Arizona\
Tucson, AZ, 85721, USA
author:
- Jeremiah Birrell
- Jan Wehr
title: A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics
---
Introduction and background
===========================
This paper studies the small mass limit of a general class of Langevin equations. Langevin dynamics is defined in terms of canonical variables—positions and momenta—by adding damping and (Itô) noise terms to Hamiltonian equations. In the limit when the mass, or masses, of the system’s particles, go to zero, the momenta homogenize, and one obtains a limiting equation for the position variables only. This is a great simplification which often allows one to see the nature of the dynamics more clearly. If the damping matrix of the original system depends on its state, a [*noise-induced drift*]{} arises in the limit. We analyze and interpret this term from several points of view. The paper consists of four parts. The first part contains general background on stochastic differential equations. In the second part, the small-mass limit of Langevin equations is studied using a multiscale expansion. This method requires making additional assumptions, but it leads to correct results in all cases in which rigorous proofs are known. The third part presents a new rigorous result about homogenization of a general class of singularly perturbed SDEs. The final part applies this result to prove a theorem about the homogenization of a large class of Langevin systems.
Stochastic differential equations
---------------------------------
Let us start from a general background on Langevin equations. The material presented here is not new, and its various versions can be found in many textbooks, see for example [@khasminskii2011stochastic]. We do not strive for complete precision or a listing of all necessary assumptions in our discussions here. The aim of the first two sections is to motivate and facilitate reading the remainder of the paper. Detailed technical considerations will be reserved for Sections 3 and 4, where we present our new results.
Consider the stochastic differential equation $$\begin{aligned}
\label{SDE}
dy_t = b(y_t)\,dt + \sigma(y_t)\,dW_t.\end{aligned}$$ The process $y_t$ takes values in $\mathbb{R}^m$, $b$ is a vector field in $\mathbb{R}^m$, $W$ is an $n$-dimensional Wiener process and $\sigma$ is an $m \times n$-matrix-valued function. Define an $m \times m$ matrix $\Sigma$ by $\Sigma = \sigma\sigma^T$. The equation [Eq.(\[SDE\])]{} defines a Markov process with the infinitesimal operator $$\begin{aligned}
\label{generator}
(Lf)(y) = {1 \over 2}\Sigma_{ij}\partial_i\partial_j f + b_i\partial_i f \end{aligned}$$ where we are writing $\partial_i$ for $\partial_{y_i}$ and suppressing the dependence of $\Sigma$, $b_i$ and $f$ on $y$ from the notation. Summation over repeating indices is implied.
We assume that this process has a unique stationary probability measure with a $C^2$-density $h(y)$. Under this assumption $h$ satisfies the equation $$\begin{aligned}
\label{stationaryFP}
L^*h = 0\end{aligned}$$ where $L^*$ denotes the formal adjoint of $L$, $$\begin{aligned}
\label{adjoint_generator}
L^*f = {1 \over 2}\partial_i\partial_j\left(\Sigma_{ij}f\right) - \partial_i\left(b_i f\right).\end{aligned}$$ That is, we have $$\begin{aligned}
\label{stationaryFP_explicit}
\partial_i\left({1 \over 2}\partial_j\left(\Sigma_{ij}h\right) - b_ih\right) = 0.\end{aligned}$$
Consider the special case when $h$ solves the equation $$\begin{aligned}
\label{inner}
{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) - b_ih = 0 .\end{aligned}$$ In this case the operator $L$ is symmetric on the space $L^2\left(\mathbb{R}^m, h\right)\equiv L^2_h$ of square-integrable functions with the weight $h$, as the following calculation shows. Using product formula, we have $$\int\left(Lf\right)gh = \int fL^*\left(gh\right) = \int f\partial_i\left({1 \over 2}\partial_j\left(\Sigma_{ij}gh\right) - b_igh\right).$$ The expression in parentheses equals $${1 \over 2}\partial_j g \left(\Sigma_{ij}h\right) + g{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) - gb_ih = {1 \over 2}\partial_j g \left(\Sigma_{ij}h\right)$$ by [Eq.(\[inner\])]{}. Applying product formula again, we obtain $$\int\left(Lf\right)gh = \int f\left({1 \over 2}\Sigma_{ij}\left(\partial_i\partial_jg\right)h + {1 \over 2}\partial_i\left(\Sigma_{ij}h\right)\partial_jg\right)$$ which, by another application of [Eq.(\[inner\])]{}, equals $$\int f\left({1 \over 2}\Sigma_{ij}\partial_i\partial_jg - b_j\partial_jg\right)h = \int f\left(Lg\right)h.$$ Here is a more complete discussion:
Detailed balance condition and symmetry of the infinitesimal operator
---------------------------------------------------------------------
We have $$\begin{aligned}
\int\left(Lf\right)gh &= \int\left(\frac{1}{2}\Sigma_{ij}\partial_i\partial_jf + b_i\partial_if\right)gh \\
&=-\frac{1}{2}\int\partial_if\partial_j\left(\Sigma_{ij}gh\right) + \int\left(\partial_if\right)b_igh \notag\\
&= -\frac{1}{2}\int\partial_if\left[\partial_j\left(\Sigma_{ij}h\right)g + \Sigma_{ij}h\partial_jg\right] + \partial\left(\partial_if\right)b_igh \notag\\
&=\int\partial_if\left[-\frac{1}{2}\partial_j\left(\Sigma_{ij}h\right) + b_ih\right]g - \frac{1}{2}\int \Sigma_{ij}\partial_if\partial_jgh. \notag\end{aligned}$$ Interchanging the roles of $f$ and $g$ and canceling the term symmetric in $f$ and $g$, we obtain $$\int\left(Lf\right)gh - \int f\left(Lg\right)h = \int\left[\left(\partial_if\right)g - \left(\partial_ig\right)f\right]\left(-{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) + b_ih\right).$$ If $h$ is a solution to the equation $$-{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) + b_ih = 0$$ then the above expression is zero, showing that the operator $L$ is symmetric on the space $L^2_h$. Conversely, for this symmetry to hold, the $\mathbb{R}^m$-valued function $-{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) + b_ih$ has to be orthogonal to all elements of the space $L^2$ (of functions with values in $\mathbb{R}^m$) of the form $\left(\partial_if\right)g - \left(\partial_ig\right)f$. It is not hard to prove that every $C^1$ function with this property must vanish, and thus, that $\frac{1}{2}\partial_j\left(\Sigma_{ij}h\right) - b_ih = 0$. Here is a sketch of a proof: suppose $\phi$ is $C^1$ and orthogonal to all such functions. That is, for every $f$ and $g$, $$\int \left[\phi_i \left(\partial_i f\right)g - \phi_i\left(\partial_i g\right)f\right] = 0.$$ Integrating the first term by parts we obtain $$\int\left[-\left(\partial_i\phi_i\right)g - 2\phi_i\partial_i g\right]f = 0.$$ Since this holds for all $f$, it follows that $$-\left(\partial_i\phi_i\right)g - 2\phi_i\partial_i g = 0$$ and thus also $$\int\left[-\left(\partial_i\phi_i\right)g - 2\phi_i\partial_i g\right] = 0.$$ Integrating the second term by parts, we get $$\int\left(\partial_i\phi_i\right)g = 0$$ and, since this is true for every $g$, it follows that $\partial_i\phi_i$ vanishes. We thus have, for every $g$ $$\phi_i \partial_ig = 0$$ and this implies that $\phi$ vanishes. In summary:
[**Proposition:**]{} If the density $h$ of the stationary probability measure is $C^2$, then $h$ satisfies the stationary Fokker-Planck equation $$\partial_i\left[-{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) + b_ih\right] = 0.$$ The stronger statement $$-{1 \over 2}\partial_j\left(\Sigma_{ij}h\right) + b_ih = 0$$ is equivalent to symmetry of the operator $L$ on the space $L^2_h$.
We are now going to relate the above symmetry statement to the detailed balance property of the stationary dynamics. First, it is clearly equivalent to the analogous property for the backward Kolmogorov semigroup: $$\int\left(P_tf\right)gh = \int f\left(P_tg\right)h$$ since $P_t = \exp\left(tL\right)$. Now, $\left(P_tf\right)(x)$ is the expected value of $f(x_t)$ for the process, starting at $x$ at time 0. In particular, for $f = \delta_y$, we obtain $P_tf(x) = p_t(x,y)$—the density of the transition probability from $x$ to $y$ in time $t$. Using the above symmetry of $P_t$ with $f = \delta_y$ and $g = \delta_x$, we obtain the detailed balance condition: $$h(x) p_t(x,y) = h(y) p_t(y, x)$$ which, conversely, implies the symmetry statement for arbitrary $f$ and $g$.
The case of a linear drift and constant noise
---------------------------------------------
When both $b(y)$ and $\sigma(y)$ are constant or depend linearly on $y$, [Eq.(\[SDE\])]{} can be solved explicitly [@arnold] and an explicit formula for its stationary distribution can be found, when it exists. We consider the special case $b(y) = - \gamma y$ and $\sigma(y) \equiv \sigma$, where $\gamma$ and $\sigma$ are constant matrices and the eigenvalues of $\gamma$ have positive real parts. The stationary Fokker-Planck equation, [Eq.(\[stationaryFP\_explicit\])]{}, reads $$\nabla \cdot \left({1 \over 2}\Sigma\nabla h + (\gamma y)h\right) = 0$$ where $\Sigma = \sigma\sigma^T$. It has a Gaussian solution $$h(y) = \left(2\pi\right)^{-{m \over 2}}\left(\det M\right)^{-{1 \over 2}}\exp\left(-{1 \over 2}\left(M^{-1}y, y\right)\right)$$ with the covariance matrix $M$ which is the unique solution of the Lyapunov equation $$\gamma M + M\gamma^T = \Sigma$$ and can be written as (see, for example, [@ortega2013matrix]) $$M = \int_0^{\infty}\exp\left(-t\gamma\right)\Sigma\exp\left(-t\gamma^T\right)dt.$$ This result can be verified by a direct calculation. We emphasize that it holds without assuming the detailed balance condition. The latter condition is satisfied if and only if the above $h$ solves the equation $${1 \over 2}\Sigma\nabla h + (\gamma y)h = 0$$ which is equivalent to $M = {1 \over 2}\gamma^{-1}\Sigma$ or, in terms of the coefficients of the system, to $$\begin{aligned}
\label{condition_forDB}
\Sigma\gamma^T = \gamma \Sigma\end{aligned}$$ To see the physical significance of this condition, let us go back to the general case and write (adapting the discussion in [@Zwanzig] to our notation) $$\gamma = {1 \over 2}\Sigma M^{-1} - i\Omega.$$ $\Omega$ represents the “oscillatory degrees of freedom” of the diffusive system. The above calculations show that the detailed balance condition is equivalent to $\Omega = 0$, in agreement with the physical intuition that there are no macroscopic currents in the stationary state.
Small mass limit—a perturbative approach {#sec:perturb}
========================================
We are now going to apply the general facts about Langevin equations to a model of a mechanical system, interacting with a noisy environment. The dynamical variables of this system are positions and momenta, and, in general, the Langevin equations which describe its time evolution, are not linear. However, when investigating the small mass limit of the system by a perturbative method, we will encounter equations closely related to those studied above. This will be explained later, when we interpret the limiting equations.
Consider a mechanical system with the Hamiltonian $\mathcal{H}(q, p)$ where $q, p \in \mathbb{R}^n$. We want to study a small mass limit of this system, coupled to a damping force and the noise. Therefore, we introduce the variable $z = {p \over \sqrt{m}}$ and assume the Hamiltonian can be written $\mathcal{H}(q,p) =H(q,z)$ where $H$ is independent of $m$. We thus have $$\begin{aligned}
\label{Langevin}
dq_t &= {1 \over \sqrt{m}}\nabla_zH(q_t, z_t)\,dt \\
dz_t &= -{1 \over \sqrt{m}}\nabla_qH(q_t, z_t)\,dt - {1 \over m}\gamma(q_t)\nabla_zH(q_t, z_t)\,dt + {1 \over \sqrt{m}}\sigma(q_t)\,dW_t .\notag\end{aligned}$$ $\gamma$ is $n \times n$-matrix-valued, $\sigma$ is $n \times k$-matrix-valued and $W$ is a $k$-dimensional Wiener process. We emphasize that $\sigma$ does not play here the same role that it played in our discussion of the general Langevin equation, since the noise term enters only the equation for $dz_t$. The number $k$ of the components of the driving noise does not have to be related to the dimension of the system in any particular way. The corresponding backward Kolmogorov equation for a function $\rho(q, z, t)$ is $$\partial_t \rho = L\rho$$ where the differential operator $L$ equals $$L = {1 \over m}L_1 + {1 \over \sqrt{m}}L_2$$ with $$\begin{aligned}
L_1 &= {1 \over 2}\Sigma\nabla_z \cdot \nabla_z - \gamma \nabla_zH \nabla_z \\
L_2 &= \nabla_zH \cdot \nabla_q - \nabla_qH \cdot \nabla_z \notag\end{aligned}$$ where $\Sigma(q) = \sigma(q)\sigma(q)^T$. We represent the solution of the Kolmogorov equation as a formal series $$\rho = \rho_0 + \sqrt{m}\rho_1 + m\rho_2 + \dots$$ Equating the expressions, proportional to $m^{-1}$, $m^{-{1 \over 2}}$ and $m^0$, we obtain the equations: $$\begin{aligned}
L_1 \rho_0 &= 0, \\
L_1 \rho_1 &= -L_2\rho_0, \notag\\
\partial_t \rho_0 &= L_1\rho_2 + L_2\rho_1. \notag\end{aligned}$$ To satisfy the first equation it is sufficient to choose $\rho_0$ which does not depend on $z$: $$\rho_0 = \rho_0(q,t).$$ If we now search for $\rho_1$ which is linear in $z$, the second equation simplifies to $$\gamma \nabla_z H \cdot \nabla_z \rho_1 = \nabla_z H \cdot \nabla_q \rho_0$$ which has a solution $$\rho_1(q,z) = \left(\gamma^{-1}\right)^T\nabla_q \rho_0\cdot z = \nabla_q\rho_0 \cdot \gamma^{-1}z.$$ Writing the third equation as $$\partial_t \rho_0 - L_2\rho_1 = L_1\rho_2$$ and applying the identity $$Ran L_1 = \left(Ker L_1^*\right)^{\perp}$$ to the space $L^2$ with respect to the $z$ variable, we see that $\partial_t \rho_0 - L_2\rho_1 = L_1\rho_2$ must be orthogonal in this space to any function $h$ in the null space of $L_1^*$. We have $$\begin{aligned}
\label{nullspace}
L_1^*h = \nabla_z\cdot\left({1 \over 2}\Sigma\nabla_zh + \left(\gamma\nabla_zH\right)h\right)\end{aligned}$$ where $\Sigma = \sigma\sigma^T$.
It is impossible to continue the analysis without further, simplifying assumptions. We are first going to study the case of a general $H$, assuming a form of the detailed balance condition in the variable $z$, at fixed $q$.
[**Assumption 1:**]{} for every $q$ there exists a nonnegative solution of the equation $$\begin{aligned}
\label{conditionalDB}
{1 \over 2}\Sigma\nabla_zh + \left(\gamma\nabla_zH\right)h = 0 \end{aligned}$$ of finite $L^1(dz)$-norm. We can thus choose $$\int h(q,z)\,dz = 1.$$ We will say in this case that the system satisfies the [*conditional detailed balance property*]{} in the variable $z$. Since $\rho_0$ does not depend on $z$, the orthogonality condition can be written as $$\partial_t\rho_0 = \int L_2\rho_1(q,z)h(q,z)\,dz.$$ We have the following explicit formula for $L_2\rho_1$ (summation over repeated indices is implied): $$\begin{aligned}
L_2\rho_1=& \partial_{z_i}H\left(\partial_{q_i}\partial_{q_j}\rho_0\right)\left(\gamma^{-1}\right)_{jk}z_k + \partial_{z_i}H\left(\partial_{q_j}\rho_0\right)\partial_{q_i}\left(\left(\gamma^{-1}\right)_{jk}\right)z_k\\
& - \partial_{q_i}H\left(\partial_{q_j}\rho_0\right)\left(\gamma^{-1}\right)_{ji}. \notag\end{aligned}$$ To integrate it against $h(q,z)$, we will use the following consequence of [Eq.(\[conditionalDB\])]{} $$\begin{aligned}
\label{averaging}
&\int\left(\partial_{z_i}H\right)z_kh(q,z)\,dz = -{1 \over 2}\int\left(\gamma^{-1}\Sigma\nabla_zh\right)_iz_k\,dz\\
=& -\int\left(\gamma^{-1}\Sigma\right)_{ij}\left(\partial_{z_j}h\right)z_k\,dz= -{1 \over 2}\left(\gamma^{-1}\Sigma\right)_{ij}\int\left(\partial_{z_j}h\right)z_k\,dz \notag\\
=& {1 \over 2}\left(\gamma^{-1}\Sigma\right)_{ij}\int h\delta_{jk}\,dz = {1 \over 2}\left(\gamma^{-1}\Sigma\right)_{ik}.\notag\end{aligned}$$ The orthogonality condition is thus $$\begin{aligned}
\partial_t \rho_0 =& -\left(\gamma^{-1}\right)_{ji}\left<\partial_{q_i}H\right>\partial_{q_j}\rho_0 + {1 \over 2}\left(\gamma^{-1}\Sigma\right)_{ik}\partial_{q_i}\left(\left(\gamma^{-1}\right)_{jk}\right)\partial_{q_j}\rho_0\\
&+ {1 \over 2}\left(\gamma^{-1}\Sigma\right)_{ik}\left(\gamma^{-1}\right)_{jk}\left(\partial_{q_i}\partial_{q_j}\rho_0\right).\notag\end{aligned}$$ In this formula, which is more general than the detailed-balance case of the rigorous result of [@Hottovy], $\left<-\right>$ denotes the average (i.e. the integral over $z$ with the density $h(q,z)$). This notation is used only in the term in which the average has not been calculated explicitly. Passing from the Kolmogorov equation to the corresponding SDE, we obtain the effective Langevin equation in the $m \to 0$ limit: $$\begin{aligned}
\label{effective}
dq_t = -\gamma(q_t)^{-1}\left(\left<\nabla_qH\right>(q_t) + S(q_t)\right)\,dt + \gamma^{-1}(q_t)\sigma(q_t)\, dW_t\end{aligned}$$ where the components of the noise-induced drift, $S(q)$, are given by $$S_i(q) = {1 \over 2}\left(\gamma^{-1}\Sigma\right)_{jk}\partial_{q_j}\left(\left(\gamma^{-1}\right)_{ik}\right)$$ and we have used $$\gamma^{-1}\sigma\left(\gamma^{-1}\sigma\right)^T = \gamma^{-1}\Sigma\left(\gamma^{-1}\right)^T.$$ We are now going to interpret the limiting equation [Eq.(\[effective\])]{}, using the stationary probability measure $h(q,z)\,dz$, as follows: from the original equations for $q_t$ and $z_t$ we obtain $$dq_t = -\gamma(q_t)^{-1}\nabla_qH\,dt + \gamma(q_t)^{-1}\sigma(q_t)\,dW_t -\sqrt{m}\gamma(q_t)^{-1}\,dz_t.$$ Integrating the last term by parts, we obtain $$\sqrt{m}\left(\gamma^{-1}(q_t)\right)_{ij}\,dz_t^j = d\left(\sqrt{m}\left(\gamma_{ij}^{-1}(q_t)\right)z_t^j\right)-\sqrt{m}\,d\left(\left(\gamma^{-1}\right)_{ij}\right)z_t^j.$$ We leave the first term out, since, under fairly general natural assumptions, it is of order $m^{1 \over 2}$ [@Hottovy]. The second term equals $$-\partial_{q_k}\left(\left(\gamma^{-1}\right)_{ij}\right)\left(\partial_{z_k}H\right)z_j\,dt.$$ We substitute this into the equation for $dq_t$ and average, multiplying by $h(q,z)$ and integrating over $z$. The calculation is as in [Eq.(\[averaging\])]{} and the result is thus the same as the equation obtained by the multiscale expansion [Eq.(\[effective\])]{}. This provides the following heuristic physical interpretation of the perturbative result: the smaller $m$ is, the faster the variation of $z$ becomes, and in the limit $m \to 0$, $z$ homogenizes instantaneously, with $q$ changing only infinitesimally.
Let us now discuss conditions, under which one may expect our conditional detailed balance assumption to hold. As seen above, at fixed $q$ this assumption is equivalent to existence of a non-negative, integrable solution of the equation $$\begin{aligned}
\label{condDB}
{1 \over 2}\Sigma \nabla_zh + \gamma\left(\nabla_zH\right)h = 0.\end{aligned}$$ This equation can be rewritten as $${\nabla_zh \over h} = -2\Sigma^{-1}\gamma\nabla_zH.$$ The left-hand side equals $\nabla_z \log h$. Letting $B = -2\Sigma^{-1}\gamma$ to simplify notation, we see that a necessary condition for existence of a solution is that $B\nabla_zH$ be a gradient. This requires $$\partial_{z_k}\left(b_{ij}\partial_{z_j}H\right) = \partial_{z_i}\left(b_{kj}\partial_{z_j}H\right)$$ for all $i, k$, where $b_{ij}$ are matrix elements of $B$. Introducing the matrix $R = \left(r_{ij}\right)$ of second derivatives of $H$, $$r_{ij} = \partial_{z_i}\partial_{z_j}H$$ we see that solvability of [Eq.(\[condDB\])]{} is equivalent to symmetry of the product $BR$: $$BR = RB^T.$$ For the system to satisfy the conditional detailed balance property, this relation has to be satisfied for all $q$ and $z$. When $H$ is a quadratic function of $z$, the matrix $R$ is constant. Even though in this case we will derive the limiting equation withouth assuming conditional detailed balance, let us remark that the above approach provides a method of determining when that condition holds, different from that used earlier. Namely, let $$H(q,z) = V(q) + {1 \over 2}Q(q)z\cdot z$$ where $Q(q)$ is a symmetric matrix. We then have $R = Q$ and the solvability condition becomes $$BQ = QB^T.$$ In a still more special—but the most fequently considered—case when $Q$ is a multiple of identity, this reduces to $$B = B^T$$ which is easily seen to be equivalent to the relation $$\gamma \Sigma = \Sigma\gamma^T.$$ We have derived this condition earlier by a different argument [Eq.(\[condition\_forDB\])]{}.
If $\gamma$ is symmetric, this becomes the commutation relation $$\gamma \Sigma = \Sigma\gamma.$$ Note that if $\gamma \Sigma = \Sigma\gamma^T$, the solution of the Lyapunov equation $$J\gamma^T + \gamma J = \Sigma$$ is given by $J = {1 \over 2}\gamma^{-1}\Sigma$. In this the case the linear Langevin equation in the $z$ variable, whose conditional equilibrium at fixed value of $q$ we are studying, has no “oscillatory degrees of freedom”, as discussed earlier (see also [@Zwanzig]).
In the case when $H$ is not a quadratic function of $z$, the matrix $BR(q,z)$ has to be symmetric for all $q$ and $z$, which means satisfying a continuum of conditions for every fixed $q$. It is interesting to ask whether there exist physically natural examples in which this happens, without each $B(q)$ being a multiple of identity. We are not going to pursue this question here.
In the case when $B(q)$ is a multiple of identity, we can write $$A = 2\beta(q)^{-1}\gamma$$ with $\beta(q)^{-1} = k_BT(q)$ and call the scalar function $T(q)$ [*generalized temperature*]{}. The limiting Kolmogorov equation reads then $$\partial_t \rho_0 = -\left(\gamma^{-1}\right)_{ji}\left<\partial_{q_i}H\right>\partial_{q_j}\rho_0 +k_BT\partial_{q_k}\left(\gamma^{-1}\right)_{jk}\partial_{q_j}\rho_0+ k_BT\left(\gamma^{-1}\right)_{ij}\left(\partial_{q_i}\partial_{q_j}\rho_0\right)$$ and the components of the noise-induced drift are thus $$S_j(q) = k_BT\partial_{q_k}\left(\gamma^{-1}\right)_{jk}.$$ The above applies in particular in the one-dimensional case, in which $\sigma(q)^2$ and $\gamma(q)$ are scalars and hence one is always an ($q$-dependent) multiple of the other: $$\begin{aligned}
k_BT(q) = {\sigma(q)^2 \over 2\gamma(q)}.\end{aligned}$$ The limiting Langevin equation is in this case $$dq_t = - {\left<\nabla_qH\right> \over \gamma}\,dt - {1 \over 2}{\nabla_q \gamma \over \gamma^3}\sigma^2\,dt + {\sigma \over \gamma}\,dW_t.$$ For a Hamiltonian equal to a sum of potential and quadratic kinetic energy, $H = V(q) + {z^2 \over 2}$, the first term equals $F \over \gamma$, where $F = -\nabla_qV\,dt$, in agreement with earlier results.
The second situation, in which the perturbative treatment of the original system can be carried out explicitly is the general quadratic kinetic energy case.
[**Assumption 2**]{}: $H = V(q) + {z^2 \over 2}$
If we follow the singular perturbation method used above, we again need to find the integral [Eq.(\[averaging\])]{}, where $\partial_{z_i}H = z_i$. In this case we know the solution of $L_1^*h = 0$ explicitly: $$h(q,z) = \left(2\pi\right)^{-{n \over 2}}\left(\det M\right)^{-{1 \over 2}}\exp\left(-{1 \over 2}M^{-1}z\cdot z\right)$$ so the integral in [Eq.(\[averaging\])]{} is the mean of $z_iz_k$ in the Gaussian distribution with the covariance $M = \left(m_{ik}\right)$, that is, $m_{ik}$. The second-order term in the Kolmogorov equation is thus $m_{ik}\left(\gamma^{-1}\right)_{jk}\partial_{q_i}\partial_{q_j}\rho_0$. The corresponding Langevin equation, which has been derived rigorously in [@Hottovy] is in this case $$\begin{aligned}
dq_t = -\gamma^{-1}(q_t)\nabla_q V(q_t)\,dt + S(q_t)\,dt + \gamma^{-1}(q_t)\sigma(q_t)\,dW_t.\end{aligned}$$ The homogenization heuristics proposed under Assumption 1 applies here as well: the limiting Langevin equation can be interpreted as a result of averaging over the conditional stationary distribution of the $z$ variable. A rigorous result, corroborating this picture has recently been proven in [@clt].
A rigorous homogenization theorem
=================================
We now develop a framework for the homogenization of Langevin equations that is able to make many of the heuristic results from the previous two sections rigorous. Our results will concern Hamiltonians of the form $$\begin{aligned}
\label{H_form}
H(t,x)=K(t,q,p-\psi(t,q))+V(t,q)\end{aligned}$$ where $x=(q,p)\in\mathbb{R}^n\times \mathbb{R}^n$, $K = K(t,q,z)$ and $V= V(t,q)$ are $C^2$, $\mathbb{R}$-valued functions, $K$ is non-negative, and $\psi$ is a $C^2$, $\mathbb{R}^n$-valued function. The splitting of $H$ into $K$ and $V$ does not have to correspond physically to any notion of kinetic and potential energy, although we will use those terms for convenience. The splitting is not unique; it will be constrained further as we continue. We now define the family of scaled Hamiltonians, parameterized by $\epsilon>0$ (generalizing the above mass parameter): $$\begin{aligned}
H^\epsilon(t,q,p)\equiv K^\epsilon(t,q,p)+V(t,q)\equiv K(t,q,(p-\psi(t,q))/\sqrt{\epsilon})+V(t,q).\end{aligned}$$
Consider the following family of SDEs:\
$$\begin{aligned}
dq^\epsilon_t=&\nabla_p H^\epsilon(t,x^\epsilon_t)dt,\label{Hamiltonian_SDE_q}\\
d p^\epsilon_t=&(-\gamma(t,x^\epsilon_t)\nabla_p H^\epsilon(t,x^\epsilon_t)-\nabla_q H^\epsilon(t,x^\epsilon_t)+F(t,x^\epsilon_t))dt+\sigma(t,x^\epsilon_t) dW_t,\label{Hamiltonian_SDE_p}\end{aligned}$$ where $\gamma:[0,\infty)\times\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n\times n}$ and $\sigma:[0,\infty)\times\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n\times k}$ are continuous, $\gamma$ is positive definite, and $W_t$ is a $\mathbb{R}^k$-valued Brownian motion on a filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$ satisfying the usual conditions [@karatzas2014brownian].
Our objective in this section is to develop a method for investigating the behavior of $x^\epsilon_t$ in the limit $\epsilon\rightarrow 0^+$; more precisely, we wish to prove the existence of a limiting “position" process $q_t$ and derive a homogenized SDE that it satisfies. In fact, the method we develop is applicable to a more general class SDEs that share certain properties with [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{}. In the following subsection, we discuss some prior results concerning [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{}. This will help motivate the assumptions made in the development of our general homogenization method, starting in subsection \[sec:gen\_homog\].
Summary of prior results
------------------------
Let $x_t^\epsilon$ be a family of solutions to the SDE [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{} with initial condition $x_0^\epsilon=(q_0^\epsilon,p_0^\epsilon)$. We assume that a solution exists for all $t\geq 0$ (i.e. there are no explosions). See Appendix B in [@BirrellHomogenization] for assumptions that guarantee this. Under Assumptions 1-3 in [@BirrellHomogenization] (repeated as Assumptions \[assump1\]-\[assump3\] in \[app:assump\], we showed that for any $T>0$, $p>0$, $0<\beta<p/2$ we have $$\begin{aligned}
\label{results_summary}
\sup_{t\in[0,T]}E\left[\|p_t^\epsilon-\psi(t,q^\epsilon_t)\|^p\right]=O(\epsilon^{p/2}) \text{ and } E\left[\sup_{t\in[0,T]}\|p_t^\epsilon-\psi(t,q^\epsilon_t)\|^p\right]=O(\epsilon^{\beta})\end{aligned}$$ as $\epsilon\rightarrow 0^+$ i.e. the point $(p,q)$ is attracted to the surface defined by $p=\psi(t,q)$.
Adding Assumption 4 (Assumption [Eq.(\[assump4\])]{} in the appendix) we also showed that $$\begin{aligned}
\label{q_eq}
d(q_t^\epsilon)^i=&(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)(-\partial_t\psi_j(t,q_t^\epsilon)-\partial_{q^j}V(t,q_t^\epsilon)+F_j(t,x^\epsilon_t))dt\\
&+(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)\sigma_{j\rho}(t,x_t^\epsilon)dW^\rho_t-(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)\partial_{q^j}K(t,q_t^\epsilon,z_t^\epsilon)dt\notag\\
&+(z_t^\epsilon)_j\partial_{q^l}(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)\partial_{z_l}K(t,q_t^\epsilon,z_t^\epsilon)dt- d((\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)(u^\epsilon_t)_j)\notag\\
&+(u_t^\epsilon)_j\partial_t(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)dt,\notag\end{aligned}$$ where $u_t^\epsilon\equiv p_t^\epsilon-\psi(t,q^\epsilon_t)$, $z_t^\epsilon\equiv u_t^\epsilon/\sqrt{\epsilon}$, and $$\begin{aligned}
\label{tilde_gamma_def}
\tilde\gamma_{ik}(t,q)\equiv\gamma_{ik}(t,q) +\partial_{q^k}\psi_i(t,q)-\partial_{q^i}\psi_k(t,q).\end{aligned}$$ We define the components of $\tilde\gamma^{-1}$ such that $$\begin{aligned}
\label{tilde_gamma_inv_def}
(\tilde\gamma^{-1})^{ij}\tilde\gamma_{jk}=\delta^i_k,\end{aligned}$$
and for any $v_i$ we define $(\tilde\gamma^{-1}v)^i=(\tilde\gamma^{-1})^{ij}v_j$.
Under the additional Assumptions 5-7 in [@BirrellHomogenization], which include further restrictions on the form of the Hamiltonian, we were then able to show that $q_t^\epsilon$ converges in an $L^p$-norm as $\epsilon\rightarrow 0^+$ to the solution of a lower dimensional SDE, $$\begin{aligned}
\label{limit_eq}
dq_t=&\tilde \gamma^{-1}(t,q_t)(-\partial_t\psi(t,q_t)-\nabla_{q}V(t,q_t)+F(t,q_t,\psi(t,q_t)))dt+S(t,q_t)dt\notag\\
&+\tilde\gamma^{-1}(t,q_t)\sigma(t,q_t,\psi(t,q_t)) dW_t.\end{aligned}$$ The [*noise-induced drift*]{} term, $S(t,q)$, that arises in the limit is the term of greatest interest here. Its form is given in Eq. (3.26) in [@BirrellHomogenization].
The homogenization technique used in [@BirrellHomogenization] to arrive at [Eq.(\[limit\_eq\])]{} relies heavily on the specific structural assumptions on the form of the Hamiltonian. Those assumptions cover a wide variety of important systems, such as a particle in an electromagnetic field, and motion on a Riemannian manifold, but it is desirable to search for a more generally applicable homogenization method. In this paper, we develop a significantly more general technique, adapted from the methods presented in [@pavliotis2008multiscale], that is capable of homogenizing terms of the form $G(t,q_t^\epsilon,(p_t^\epsilon-\psi(t,q_t^\epsilon))/\sqrt{\epsilon})dt$ for a general class of SDEs that satisfy the property [Eq.(\[results\_summary\])]{}, as well as prove convergence of $q_t^\epsilon$ to the solution of a limiting, homogenized SDE. In particular, it will be capable of homogenizing $q_t^\epsilon$ from the Hamiltonian system [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{} under less restrictive assumptions on the form of the Hamiltonian, than those made in [@BirrellHomogenization]. We emphasize that the convergence statements are proven in the strong sense, see Section \[sec:gen\_homog\].
General homogenization framework {#sec:gen_homog}
--------------------------------
Here we describe our homogenization technique in a more general context than the Hamiltonian setting from the previous section. This method is related to the cell problem method from [@pavliotis2008multiscale], our proof applies to a larger class of SDEs and demonstrates $L^p$-convergence rather that weak convergence.
We will denote an element of $\mathbb{R}^{n}\times\mathbb{R}^m$ by $x=(q,p)$, where we no longer require the $q$ and $p$ degrees of freedom to have the same dimensionality, though we still employ the convention of writing $q$ indices with superscripts and $p$ indices with subscripts. We let $W_t$ be an $\mathbb{R}^k$-valued Wiener process, $\psi:[0,\infty)\times\mathbb{R}^n\rightarrow\mathbb{R}^m$ be $C^2$ and $G_1,F_1:[0,\infty)\times\mathbb{R}^{n+m}\times\mathbb{R}^m\rightarrow\mathbb{R}^n$, $G_2,F_2:[0,\infty)\times\mathbb{R}^{n+m}\times\mathbb{R}^m\rightarrow\mathbb{R}^m$, $\sigma_1:[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n\times k}$, and $\sigma_2:[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{m\times k}$ be continuous. With these definitions, we consider the following family of SDEs, depending on a parameter $\epsilon>0$: $$\begin{aligned}
dq^\epsilon_t=&\left(\frac{1}{\sqrt{\epsilon}}G_1(t,x_t^\epsilon,z_t^\epsilon)+F_1(t,x_t^\epsilon,z_t^\epsilon)\right)dt+\sigma_1(t,x_t^\epsilon)dW_t,\label{gen_SDE1}\\
d p^\epsilon_t=&\left(\frac{1}{\sqrt{\epsilon}}G_2(t,x^\epsilon_t,z_t^\epsilon)+F_2(t,x^\epsilon_t,z_t^\epsilon)\right)dt+\sigma_2(t,x^\epsilon_t) dW_t,\label{gen_SDE2}\end{aligned}$$ where we define $z_t^\epsilon=(p_t^\epsilon-\psi(t,q_t^\epsilon))/\sqrt{\epsilon}$. We will assume, in analogy with [Eq.(\[results\_summary\])]{}, that:
\[homog\_assump1\] For any $T>0$, $p>0$, $0<\beta<p/2$ we have $$\begin{aligned}
\sup_{t\in[0,T]}E\left[\|p_t^\epsilon-\psi(t,q^\epsilon_t)\|^p\right]=O(\epsilon^{p/2}) \text{ and } E\left[\sup_{t\in[0,T]}\|p_t^\epsilon-\psi(t,q^\epsilon_t)\|^p\right]=O(\epsilon^{\beta})\end{aligned}$$ as $\epsilon\rightarrow 0^+$.
In words, we assume that the $p$ degrees of freedom are attracted to the values defined by $p=\psi(t,q)$. This is an appropriate setting to expect some form of homogenization, as it suggests that the dynamics in the limit $\epsilon\rightarrow 0^+$ can be characterized by fewer degrees of freedom—the $q$-variables.
### Homogenization of integral processes
In this section we derive a method capable of homogenizing processes of the form $$\label{M_t_def}
M^\epsilon_t\equiv \int_0^tG(s,x_s^\epsilon,z_s^\epsilon)ds$$ in the limit $\epsilon\rightarrow 0^+$. More specifically, our aim is to find conditions under which there exists some function, $S(t,q)$, such that $$\begin{aligned}
\label{homog_goal}
\int_0^tG(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^tS(s,q_s^\epsilon)ds\rightarrow 0\end{aligned}$$ in some norm, as $\epsilon\rightarrow 0^+$ , i.e. only the $q$-degrees of freedom are needed to characterize $M_t^\epsilon$ in the limit. We will call a family of processes, $S(t,q_t^\epsilon)dt$, that satisfies such a limit, a homogenization of $G(t,x_t^\epsilon,z_t^\epsilon)dt$. The technique we develop will also be useful for proving existence of a limiting process $q_s$ (i.e. $q_s^\epsilon\rightarrow q_s$), and showing that $$\begin{aligned}
\int_0^tG(s,x_s^\epsilon,z_s^\epsilon)ds\rightarrow \int_0^tS(s,q_s)ds.\end{aligned}$$ as $\epsilon\rightarrow 0^+$. We will consider this second question in Section \[sec:limit\_eq\]. Here, our focus is on [Eq.(\[homog\_goal\])]{}.
As a starting point, let $\chi(t,x,z):[0,\infty)\times\mathbb{R}^{n+m}\times\mathbb{R}^m\rightarrow\mathbb{R}$ be $C^{1,2}$, where $C^{1,2}$ is defined as follows:
- If $\sigma_1\neq 0$ then we take this to mean $\chi$ is $C^1$ and, for each $t$, $\chi(t,x,z)$ is $C^2$ in $(x,z)$ with second derivatives continuous jointly in all variables.
- If $\sigma_1=0$ then we take this to mean $\chi$ is $C^1$ and, for each $t,q$, $\chi(t,q,p,z)$ is $C^2$ in $(p,z)$ with second derivatives continuous jointly in all variables.
Eventually, we will need to carefully choose $\chi$ so that we achieve our aim, but for now we simply use Itô’s formula to compute $\chi(t,x_t^\epsilon,z_t^\epsilon)$. We defined $C^{1,2}$ precisely so that Itô’s formula is justified. For this computation, we will define $\chi^\epsilon(t,x)=\chi(t,x,(p-\psi(t,q))/\sqrt{\epsilon})$, and $$\begin{aligned}
\label{sigma_defs}
&\Sigma_{11}^{ij}= \sum_\rho (\sigma_1)^i_\rho(\sigma_1)^j_\rho,\hspace{1mm} (\Sigma_{12})^i_j= \sum_\rho (\sigma_1)^i_\rho(\sigma_2)_{j\rho}, \hspace{1mm}(\Sigma_{22})_{ij}= \sum_\rho (\sigma_2)_{i\rho}(\sigma_2)_{j\rho}.\end{aligned}$$ Itô’s formula gives $$\begin{aligned}
&\chi(t,x_t^\epsilon,z_t^\epsilon)=\chi(0,x_0^\epsilon,z_0^\epsilon)+\int_0^t\partial_s \chi(s,x_s^\epsilon,z_s^\epsilon)ds+\int_0^t\nabla_q\chi^\epsilon(s,x_s^\epsilon)\cdot dq_s^\epsilon\\
&+\int_0^t\nabla_p\chi^\epsilon(s,x_s^\epsilon)\cdot dp_s^\epsilon+\frac{1}{2}\int_0^t\partial_{q^i}\partial_{q^j}\chi^\epsilon(s,x_s^\epsilon) \Sigma_{11}^{ij}(s,x_s^\epsilon)ds\notag\\
&+\frac{1}{2}\int_0^t\partial_{q^i}\partial_{p_j}\chi^\epsilon(s,x_s^\epsilon) (\Sigma_{12})^i_j(s,x_s^\epsilon)ds+\frac{1}{2}\int_0^t\partial_{p_i}\partial_{q^j}\chi^\epsilon(s,x_s^\epsilon) (\Sigma_{12})_i^j(s,x_s^\epsilon)ds\notag\\
&+\frac{1}{2}\int_0^t\partial_{p_i}\partial_{p_j}\chi^\epsilon(s,x_s^\epsilon) (\Sigma_{22})_{ij}(s,x_s^\epsilon)ds.\notag\end{aligned}$$ Note that if $\sigma_1=0$ then only the second derivatives that we have assumed exist are involved in this computation.
We can compute these terms as follows: $$\begin{aligned}
\partial_{q^i}\chi^\epsilon(t,x)=&(\partial_{q^i}\chi)(t,x,z)-\epsilon^{-1/2}\partial_{q^i}\psi_k(t,q)(\partial_{z_k}\chi)(t,x,z),\\
\partial_{p_i}\chi^\epsilon(t,x)=&(\partial_{p_i}\chi)(t,x,z)+\epsilon^{-1/2}(\partial_{z_i}\chi)(t,x,z),\\
\partial_{q^i}\partial_{q^j}\chi^\epsilon(t,x)=&(\partial_{q^i}\partial_{q^j}\chi)(t,x,z)+\epsilon^{-1/2}\left(-\partial_{q^j}\psi_k(t,q)(\partial_{q^i}\partial_{z_k}\chi)(t,x,z)\right.\\
&\left.-\partial_{q^i}\partial_{q^j}\psi_k(t,q)(\partial_{z_k}\chi)(t,x,z)-\partial_{q^i}\psi_k(t,q)(\partial_{q^j}\partial_{z_k}\chi)(t,x,z)\right)\notag\\
&+\epsilon^{-1}\partial_{q^i}\psi_k(t,q)\partial_{q^j}\psi_l(t,q)(\partial_{z_k}\partial_{z_l}\chi)(t,x,z).\notag\\
\partial_{p_i}\partial_{p_j}\chi^\epsilon(t,x)=&(\partial_{p_i}\partial_{p_j}\chi)(t,x,z)+\epsilon^{-1/2}\left((\partial_{z_j}\partial_{p_i}\chi)(t,x,z)\right.\\
&\left.+(\partial_{p_j}\partial_{z_i}\chi)(t,x,z)\right)+\epsilon^{-1}(\partial_{z_i}\partial_{z_j}\chi)(t,x,z),\notag\\
\partial_{q^i}\partial_{p_j}\chi^\epsilon(t,x)=&(\partial_{q^i}\partial_{p_j}\chi)(t,x,z)+\epsilon^{-1/2}\left((\partial_{q^i}\partial_{z_j}\chi)(t,x,z)\right.\\
&\left.-\partial_{q^i}\psi_k(t,q)(\partial_{p_j}\partial_{z_k}\chi)(t,x,z)\right)\notag\\
&-\epsilon^{-1}\partial_{q^i}\psi_k(t,q)(\partial_{z_j}\partial_{z_k}\chi)(t,x,z),\notag\end{aligned}$$ where $z$ is evaluated at $z(t,x,\epsilon)=(p-\psi(t,q))/\sqrt{\epsilon}$ in each of the above formulae.
Using these expressions, together with the SDE [Eq.(\[gen\_SDE1\])]{}-[Eq.(\[gen\_SDE2\])]{} we find $$\begin{aligned}
&\chi(t,x_t^\epsilon,z_t^\epsilon)=\chi(0,x_0^\epsilon,z_0^\epsilon)+\int_0^t\partial_s \chi(s,x_s^\epsilon,z_s^\epsilon)ds\\
&+\int_0^t\left((\partial_{q^i}\chi)(s,x_s^\epsilon,z_s^\epsilon)-\epsilon^{-1/2}\partial_{q^i}\psi_k(s,q_s^\epsilon)(\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right) \notag\\
&\hspace{10mm}\times\left[\left(\frac{1}{\sqrt{\epsilon}}G_1(s,x_s^\epsilon,z_s^\epsilon)+F_1(s,x_s^\epsilon,z_s^\epsilon)\right)ds+\sigma_1(s,x_s^\epsilon)dW_s\right]^i\notag\\
&+\int_0^t\left((\partial_{p_i}\chi)(s,x_s^\epsilon,z_s^\epsilon)+\epsilon^{-1/2}(\partial_{z_i}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right) \notag\\
&\hspace{10mm}\times\left [\left(\frac{1}{\sqrt{\epsilon}}G_2(s,x^\epsilon_s,z_s^\epsilon)+F_2(s,x^\epsilon_s,z_s^\epsilon)\right)ds+\sigma_2(s,x^\epsilon_s) dW_s\right]_i\notag\\
&+\frac{1}{2}\int_0^t\Sigma_{11}^{ij}(s,x_s^\epsilon)\bigg[(\partial_{q^i}\partial_{q^j}\chi)(s,x_s^\epsilon,z_s^\epsilon)+\epsilon^{-1/2}\left(-\partial_{q^j}\psi_k(s,q_s^\epsilon)(\partial_{q^i}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right.\notag\\
&\hspace{15mm}\left.-\partial_{q^i}\partial_{q^j}\psi_k(s,q_s^\epsilon)(\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)-\partial_{q^i}\psi_k(s,q_s^\epsilon)(\partial_{q^j}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right)\notag\\
&\hspace{15mm}+\epsilon^{-1}\partial_{q^i}\psi_k(s,q_s^\epsilon)\partial_{q^j}\psi_l(s,q_s^\epsilon)(\partial_{z_k}\partial_{z_l}\chi)(s,x_s^\epsilon,z_s^\epsilon)\bigg] ds\notag\\
&+\int_0^t(\Sigma_{12})^i_j(s,x_s^\epsilon)\bigg[(\partial_{q^i}\partial_{p_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)+\epsilon^{-1/2}\left((\partial_{q^i}\partial_{z_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right.\notag\\
&\hspace{10mm}\left.-\partial_{q^i}\psi_k(t,q)(\partial_{p_j}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right)-\epsilon^{-1}\partial_{q^i}\psi_k(s,q_s^\epsilon)(\partial_{z_j}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\bigg]ds\notag\\
&+\frac{1}{2}\int_0^t (\Sigma_{22})_{ij}(s,x_s^\epsilon)\bigg[(\partial_{p_i}\partial_{p_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)+\epsilon^{-1/2}\left((\partial_{z_j}\partial_{p_i}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right.\notag\\
&\hspace{15mm}\left.+(\partial_{p_j}\partial_{z_i}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right)+\epsilon^{-1}(\partial_{z_i}\partial_{z_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)\bigg]ds.\notag\end{aligned}$$ Multiplying by $\epsilon$ and collecting powers, we arrive at $$\begin{aligned}
\label{homog_eq}
&\int_0^t(L\chi)(s,x_s^\epsilon,z_s^\epsilon)ds=\epsilon^{1/2} (R_1^\epsilon)_t+\epsilon\left(\chi(t,x_t^\epsilon,z_t^\epsilon)-\chi(0,x_0^\epsilon,z_0^\epsilon)+ (R^\epsilon_2)_t\right),\end{aligned}$$ where we define $$\begin{aligned}
\label{L_def}
(L\chi)(t,x,z)=&\bigg(\frac{1}{2}\Sigma_{11}^{ij}(t,x)\partial_{q^i}\psi_k(t,q)\partial_{q^j}\psi_l(t,q)\\
&\hspace{5mm}-(\Sigma_{12})^i_l(t,x)\partial_{q^i}\psi_k(t,q)+\frac{1}{2} (\Sigma_{22})_{kl}(t,x)\bigg)(\partial_{z_k}\partial_{z_l}\chi)(t,x,z)\notag\\
&+\left((G_2)_k(t,x,z)-\partial_{q^i}\psi_k(t,q)G^i_1(t,x,z)\right)(\partial_{z_k}\chi)(t,x,z),\notag\end{aligned}$$ $$\begin{aligned}
\label{R1_def}
&(R_1^\epsilon)_t\\
=&-\int_0^t(\partial_{q^i}\chi)(s,x_s^\epsilon,z_s^\epsilon)G^i_1(s,x_s^\epsilon,z_s^\epsilon)ds\notag\\
&+\int_0^t\partial_{q^i}\psi_k(s,q_s^\epsilon)(\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\left[F_1(s,x_s^\epsilon,z_s^\epsilon)ds+\sigma_1(s,x_s^\epsilon)dW_s\right]^i\notag\\
&-\int_0^t(\partial_{z_i}\chi)(s,x_s^\epsilon,z_s^\epsilon) \left [F_2(s,x^\epsilon_s,z_s^\epsilon)ds+\sigma_2(s,x^\epsilon_s) dW_s\right]_i\notag\\
&-\int_0^t(\partial_{p_i}\chi)(s,x_s^\epsilon,z_s^\epsilon)(G_2)_i(s,x^\epsilon_s,z_s^\epsilon)ds\notag\\
&-\frac{1}{2}\int_0^t\Sigma_{11}^{ij}(s,x_s^\epsilon)\left(-\partial_{q^j}\psi_k(s,q_s^\epsilon)(\partial_{q^i}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right.\notag\\
&\hspace{8.5mm}\left.-\partial_{q^i}\partial_{q^j}\psi_k(s,q_s^\epsilon)(\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)-\partial_{q^i}\psi_k(s,q_s^\epsilon)(\partial_{q^j}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right)ds\notag\\
&-\int_0^t(\Sigma_{12})^i_j(s,x_s^\epsilon)\left((\partial_{q^i}\partial_{z_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)-\partial_{q^i}\psi_k(t,q)(\partial_{p_j}\partial_{z_k}\chi)(s,x_s^\epsilon,z_s^\epsilon)\right)ds\notag\\
&-\int_0^t (\Sigma_{22})_{ij}(s,x_s^\epsilon)(\partial_{z_j}\partial_{p_i}\chi)(s,x_s^\epsilon,z_s^\epsilon)ds,\notag\end{aligned}$$ and $$\begin{aligned}
\label{R2_def}
& (R^\epsilon_2)_t\\
=&-\int_0^t\partial_s \chi(s,x_s^\epsilon,z_s^\epsilon)ds\notag\\
&-\int_0^t(\partial_{q^i}\chi)(s,x_s^\epsilon,z_s^\epsilon)\left[F_1(s,x_s^\epsilon,z_s^\epsilon)ds+\sigma_1(s,x_s^\epsilon)dW_s\right]^i\notag\\
&-\int_0^t(\partial_{p_i}\chi)(s,x_s^\epsilon,z_s^\epsilon) \left [F_2(s,x^\epsilon_s,z_s^\epsilon)ds+\sigma_2(s,x^\epsilon_s) dW_s\right]_i\notag\\
&-\frac{1}{2}\int_0^t\Sigma_{11}^{ij}(s,x_s^\epsilon)(\partial_{q^i}\partial_{q^j}\chi)(s,x_s^\epsilon,z_s^\epsilon)ds\notag\\
&-\int_0^t(\Sigma_{12})^i_j(s,x_s^\epsilon)(\partial_{q^i}\partial_{p_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)ds\notag\\
&-\frac{1}{2}\int_0^t (\Sigma_{22})_{ij}(s,x_s^\epsilon)(\partial_{p_i}\partial_{p_j}\chi)(s,x_s^\epsilon,z_s^\epsilon)ds.\notag\end{aligned}$$
First, think of simply homogenizing [Eq.(\[M\_t\_def\])]{} to a quantity of the form $\int_0^t\tilde G(s,x_s^\epsilon)ds$. Suppose we have a candidate for $\tilde G$. If we can find a $C^{1,2}$ solution, $\chi$, to the PDE $$\begin{aligned}
(L\chi)(t,x,z)=G(t,x,z)-\tilde G(t,x)\end{aligned}$$ then substituting this into [Eq.(\[homog\_eq\])]{} gives $$\begin{aligned}
\label{homog_eq2}
&\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,x_s^\epsilon)ds\\
=&\epsilon^{1/2}(R_1^\epsilon)_t+\epsilon\left(\chi(t,x_t^\epsilon,z_t^\epsilon)-\chi(0,x_0^\epsilon,z_0^\epsilon)+ (R^\epsilon_2)_t\right).\notag\end{aligned}$$ Given sufficient growth bounds for $\chi$ and its derivatives, one anticipates that the right hand side of [Eq.(\[homog\_eq2\])]{} vanishes in the limit. If in addition, $\tilde G$ is Lipschitz in $p$, uniformly in $(t,q)$, then, based on Assumption \[homog\_assump1\], one expects $$\begin{aligned}
\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds- \int_0^t\tilde G(s,q_s^\epsilon,\psi(s,q_s^\epsilon))ds\rightarrow 0\end{aligned}$$ as $\epsilon\rightarrow 0^+$.
We make this informal discussion precise in Theorem \[homog\_thm\], below. For this, we will need the following assumptions:
\[homog\_assump2\] For all $T>0$, the following quantities are polynomially bounded in $z$, with the bounds uniform on $[0,T]\times \mathbb{R}^{n+m}$:\
$G_1$, $F_1$, $G_2$, $F_2$, $\sigma_{1}$, $\sigma_{2}$, $\partial_{q^i}\psi$, $\partial_{q^i}\partial_{q^j}\psi$. If $\sigma_1=0$ then we can remove the requirement on $\partial_{q^i}\partial_{q^j}\psi$.
Recall that an $\mathbb{R}^l$-valued function, $\phi(t,x,z)$, is called [*polynomially bounded*]{} in $z$, uniformly on $[0,T]\times \mathbb{R}^{n+m}$ if there exists $q,C>0$ such that $$\|\phi(t,x,z)\|\leq C(1+\|z\|^q)$$ for all $(t,x,z)\in[0,T]\times \mathbb{R}^{n+m}\times\mathbb{R}^m$. In particular, if $\phi$ is independent of $z$, this just means it is bounded on $[0,T]\times \mathbb{R}^{n+m}$. Applying this to $\psi$, we note that Assumption \[homog\_assump2\] implies $\psi$ is Lipschitz in $q$, uniformly in $t\in[0,T]$.
\[homog\_assump3\] Given a continuous $G:[0,\infty)\times\mathbb{R}^{n+m}\times\mathbb{R}^m\rightarrow\mathbb{R}$, assume that there exists a $C^{1,2}$ function $\chi:[0,\infty)\times\mathbb{R}^{n+m}\times\mathbb{R}^m\rightarrow\mathbb{R}$ and a continuous function $\tilde G(t,x):[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}$ that together satisfy the PDE $$\begin{aligned}
\label{chi_eq}
(L\chi)(t,x,z)= G(t,x,z)-\tilde G(t,x),\end{aligned}$$ where the differential operator, $L$, is defined in [Eq.(\[L\_def\])]{}.
Assume that, for a given $T>0$, $\tilde G$ is Lipschitz in $p$, uniformly for $(t,q)\in[0,T]\times\mathbb{R}^n$. Also suppose that $\chi$, its first derivatives, and the second derivatives $\partial_{q^i}\partial_{q^j}\chi$, $\partial_{q^i}\partial_{p_j}\chi$, $\partial_{q^i}\partial_{z_j}\chi$, $\partial_{p_i}\partial_{p_j}\chi$, and $\partial_{p_i}\partial_{z_j}\chi$ are polynomially bounded in $z$, uniformly for $(t,x)\in[0,T]\times\mathbb{R}^{n+m}$. If $\sigma_1=0$ then the only second derivatives that we require to be polynomially bounded are $\partial_{p_i}\partial_{p_j}\chi$ and $\partial_{p_i}\partial_{z_j}\chi$.
\[homog\_thm\] Fix $T>0$. Let Assumptions \[homog\_assump1\]-\[homog\_assump3\] hold and $x_t^\epsilon=(q_t^\epsilon,p_t^\epsilon)$ satisfy the SDE [Eq.(\[gen\_SDE1\])]{}-[Eq.(\[gen\_SDE2\])]{}. Then for any $p>0$ we have $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,x_s^\epsilon)ds\right|^p\right]=O(\epsilon^{p/2}).\end{aligned}$$ and $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,q_s^\epsilon,\psi(s,q_s^\epsilon))ds\right|^p\right]=O(\epsilon^{p/2})\end{aligned}$$ as $\epsilon\rightarrow 0^+$.
Fix $T>0$. First let $p\geq 2$. [Eq.(\[homog\_eq2\])]{} gives $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,x_s^\epsilon)ds\right|^p\right]\\
\leq &3^{p-1}\left( \epsilon^{p/2}E\left[\sup_{t\in[0,T]}|(R_1^\epsilon)_t|^p\right]+2^p\epsilon^pE\left[\sup_{t\in[0,T]}|\chi(t,x_t^\epsilon,z_t^\epsilon)|^p\right]\right.\notag\\
&\left.\hspace{1cm}+\epsilon^p E\left[\sup_{t\in[0,T]}|(R^\epsilon_2)_t|^p\right]\right).\notag\end{aligned}$$
From [Eq.(\[R1\_def\])]{} and [Eq.(\[R2\_def\])]{} we see that $R_1^\epsilon$ and $R_2^\epsilon$ have the forms $$\begin{aligned}
(R_i^\epsilon)_t=\int_0^t V_i(s,x_s^\epsilon,z_s^\epsilon) ds+\int_0^t Q_{ij}(s,x_s^\epsilon,z_s^\epsilon) dW^j_s,\end{aligned}$$ where $V_i$ and $Q_{ij}$ are linear combinations of products of (components of) one or more terms from the following list:\
$G_1$, $F_1$, $G_2$, $F_2$, $\sigma_1$, $\sigma_2$, $\partial_{q^i}\psi$, $\partial_{q^i}\partial_{q^j}\psi$, $\partial_t\chi$, $\partial_{q^i}\chi$, $\partial_{z_i}\chi$, $\partial_{p_i}\chi$, $\partial_{q^i}\partial_{q^j}\chi$, $\partial_{q^i}\partial_{p_j}\chi$, $\partial_{q^i}\partial_{z_j}\chi$, $\partial_{p_i}\partial_{p_j}\chi$, $\partial_{p_i}\partial_{z_j}\chi$. Also note that if $\sigma_1=0$ then the only second derivatives terms that are involved are $\partial_{p_i}\partial_{p_j}\chi$ and $\partial_{p_i}\partial_{z_j}\chi$.
By assumption, these are all polynomially bounded in $z$, uniformly on $[0,T]\times \mathbb{R}^{n+m}$, as is $\chi$. Therefore, letting $\tilde C$ denote a constant that potentially varies line to line, there exists $r>0$ such that $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,x_s^\epsilon)ds\right|^p\right]\\
\leq &\tilde C \epsilon^{p/2}\left(E\left[\left(\int_0^T |V_1(s,x_s^\epsilon,z_s^\epsilon)| ds\right)^p\right]+E\left[\sup_{t\in[0,T]}\left|\int_0^t Q_{1j}(s,x_s^\epsilon,z_s^\epsilon) dW^j_s\right|^p\right]\right)\notag\\
&+\tilde C\epsilon^p\left(E\left[\left(\int_0^T |V_2(s,x_s^\epsilon,z_s^\epsilon)| ds\right)^p\right]+E\left[\sup_{t\in[0,T]}\left|\int_0^t Q_{2j}(s,x_s^\epsilon,z_s^\epsilon) dW^j_s\right|^p\right]\right.\notag\\
&\left.\hspace{15mm}+1+E\left[\sup_{t\in[0,T]}\|z_t^\epsilon\|^{rp}\right]\right).\notag\end{aligned}$$ Hölder’s inequality and polynomial boundedness yields $$\begin{aligned}
E\left[\left(\int_0^T |V_i(s,x_s^\epsilon,z_s^\epsilon)| ds\right)^p\right]\leq& T^{p-1} E\left[ \int_0^T |V_i(s,x_s^\epsilon,z_s^\epsilon)|^p ds\right]\\
\leq & \tilde C T^{p}\left( 1+\sup_{t\in[0,T]}E\left[\|z_t^\epsilon\|^{rp} \right]\right).\notag\end{aligned}$$ Applying the Burkholder-Davis-Gundy inequality to the terms involving $Q_{ij}$, (as found in, for example, Theorem 3.28 in [@karatzas2014brownian]), and then Hölder’s inequality, we obtain $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\left|\int_0^t Q_{ij}(s,x_s^\epsilon,z_s^\epsilon) dW^j_s\right|^p\right]\\
\leq &\tilde CE\left[\left( \int_0^T\| Q_i(s,x_s^\epsilon,z_s^\epsilon)\|^2ds\right)^{p/2}\right]\notag\\
\leq &\tilde C T^{p/2-1}E\left[ \int_0^T\| Q_i(s,x_s^\epsilon,z_s^\epsilon)\|^pds\right]\notag\\
\leq &\tilde C T^{p/2}\left( 1+\sup_{t\in[0,T]}E[\|z_t^\epsilon\|^{rp}]\right).\notag\end{aligned}$$ Combining these bounds, and using Assumption \[homog\_assump1\], we find $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,x_s^\epsilon)ds\right|^p\right]\\
\leq &\tilde C \epsilon^{p/2}\left(1+\sup_{t\in[0,T]}E[\|z_t^\epsilon\|^{rp}]\right)\notag\\
&+\tilde C\epsilon^p\left(1+\sup_{t\in[0,T]}E[\|z_t^\epsilon\|^{rp}]+E\left[\sup_{t\in[0,T]}\|z_t^\epsilon\|^{rp}\right]\right)\notag\\
\leq &\tilde C \epsilon^{p/2}(1+O(1))+\tilde C\epsilon^p\left(1+O(1)+O(\epsilon^{-\delta})\right)\notag\end{aligned}$$ for any $\delta>0$. Letting $\delta=p/2$ we find $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,x_s^\epsilon)ds\right|^p\right]=O(\epsilon^{p/2}).\end{aligned}$$
Now use Hölder’s inequality, the uniform Lipschitz property of $\tilde G$, and Assumption \[homog\_assump1\] again to compute $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\left|\int_0^t G(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t \tilde G(s,q_s^\epsilon,\psi(s,q_s^\epsilon))ds\right|^p\right]\\
\leq& O(\epsilon^{p/2})+\tilde C E\left[\sup_{t\in[0,T]}\left|\int_0^t \tilde G(s,x_s^\epsilon)-\tilde G(s,q_s^\epsilon,\psi(s,q_s^\epsilon))ds\right|^p\right]\notag\\
\leq& O(\epsilon^{p/2})+ \tilde CT^{p-1}E\left[\int_0^T |\tilde G(s,x_s^\epsilon)-\tilde G(s,q_s^\epsilon,\psi(s,q_s^\epsilon))|^p ds\right]\notag\\
\leq& O(\epsilon^{p/2})+\tilde C T^{p}\sup_{t\in[0,T]}E\left[ \|p_t^\epsilon-\psi(t,q_t^\epsilon)\|^p \right]\notag\\
=&O(\epsilon^{p/2}).\notag\end{aligned}$$ This proves the claim for $p\geq 2$. The result for arbitrary $p>0$ then follows from an application of Hölder’s inequality.
### Formal derivation of $\tilde G$ {#sec:formal_G_tilde}
Formally applying the Fredholm alternative to [Eq.(\[chi\_eq\])]{} motivates the form that $\tilde G$ must have in order for $\chi$ and its derivatives to possess the growth bounds required by Theorem \[homog\_thm\]. The formal calculation is simple enough that we repeat it here:\
Let $L^*$ be the formal adjoint to $L$ and suppose we have a solution, $h(t,x,z)$, to $$\label{h_eq}
L^*h=0, \hspace{2mm} \int h(t,x,z)dz=1.$$ If $\chi$ and its derivatives grow slowly enough and $h$ and its derivatives decay quickly enough, then $\int hL\chi dz$ will exist, the boundary terms from integration by parts will vanish at infinity, and we find $$\begin{aligned}
0=\int (L^*h)\chi dz=\int h L(\chi) dz=\int h (G-\tilde G)dz=\int h Gdz-\tilde G.\end{aligned}$$ Therefore we must have $$\begin{aligned}
\tilde G(t,x)=\int h(t,x,z)G(t,x,z)dz.\end{aligned}$$ In essence, the homogenized quantity is obtained by averaging over $h$, the instantaneous equilibrium distribution for the fast variables, $z$. This corroborates the heuristic discussion in Section \[sec:perturb\].
### Limiting equation {#sec:limit_eq}
We now apply the above framework to prove existence of a limiting process $q_t^\epsilon\rightarrow q_s$ and deriving an SDE satisfied by $q_s$. Specifically, we have:
\[conv\_thm\] Let $T>0$, $p\geq 2$, $0<\beta\leq p/2$, $x_t^\epsilon=(q_t^\epsilon,p_t^\epsilon)$ satisfy the SDE [Eq.(\[gen\_SDE1\])]{}-[Eq.(\[gen\_SDE2\])]{}, suppose Assumptions \[homog\_assump1\]-\[homog\_assump3\] hold, and that the SDE for $q_t^\epsilon$, [Eq.(\[gen\_SDE1\])]{}, can be rewritten in the form $$\begin{aligned}
\label{con_thm_eq}
q_t^\epsilon=q_0^\epsilon+\int_0^t\tilde F(s,x_s^\epsilon)ds+\int_0^tG(s,x_s^\epsilon,z_s^\epsilon)ds+\int_0^t\tilde\sigma(s,x_s^\epsilon)dW_s+ R^\epsilon_t\end{aligned}$$ where the components of $G$ have the properties described in Assumption \[homog\_assump3\], $\tilde F(t,x):[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}^n$, $\tilde \sigma(t,x):[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n\times k}$ are continuous, Lipschitz in $x$, uniformly in $t\in[0,T]$, and $R_t^\epsilon$ are continuous semimartingales that satisfy $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\| R_t^\epsilon\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+.\end{aligned}$$ Suppose $\tilde G$ (from Assumption \[homog\_assump3\]) is Lipschitz in $x$, uniformly in $t\in[0,T]$, and we have initial conditions $E[\|q^\epsilon_0\|^p]<\infty$, $E[\|q_0\|^p]<\infty$, and\
$E[\|q_0^\epsilon-q_0\|^p]=O(\epsilon^{p/2})$. Then $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon-q_t\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+\end{aligned}$$ where $q_t$ satisfies the SDE $$\begin{aligned}
\label{gen_limit_eq}
q_t=q_0+&\int_0^t\tilde F(s,q_s,\psi(s,q_s))ds+\int_0^t\tilde G(s,q_s,\psi(s,q_s))ds\\
&+\int_0^t\tilde\sigma(s,q_s,\psi(s,q_s))dW_s.\notag\end{aligned}$$
We will prove this theorem by verifying all the hypotheses of Lemma \[conv\_lemma\]. Define $$\begin{aligned}
\tilde R_t^\epsilon=R_t^\epsilon+\int_0^tG(s,x_s^\epsilon,z_s^\epsilon)ds-\int_0^t\tilde G(s,x_s^\epsilon)ds.\end{aligned}$$ Then $$\begin{aligned}
q_t^\epsilon=q_0^\epsilon+\int_0^t\tilde F(s,x_s^\epsilon)ds+\int_0^t\tilde G(s,x_s^\epsilon)ds+\int_0^t\tilde\sigma(s,x_s^\epsilon)dW_s+ \tilde R^\epsilon_t\end{aligned}$$ where $\tilde F+\tilde G$ and $\tilde \sigma$ are Lipschitz in $x$, uniformly for $t\in[0,T]$ and $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|\tilde R_t^\epsilon\|^p\right]=O(\epsilon^\beta)\end{aligned}$$ by Theorem \[homog\_thm\]. $E[\|q_0^\epsilon-q_0\|^p]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+$ by assumption and $$\sup_{t\in[0,T]}E[\|p_t^\epsilon-\psi(t,q_t^\epsilon)\|^p]=O(\epsilon^{p/2})\text{ as }\epsilon\rightarrow 0^+$$ by Assumption \[homog\_assump1\]. Note that the assumptions also imply that a solution $q_t$ to [Eq.(\[gen\_limit\_eq\])]{} exists for all $t\geq 0$ [@khasminskii2011stochastic].
For any $\epsilon>0$, using the Burkholder-Davis-Gundy inequality and Hölder’s inequality we obtain the bound $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\|q_t^\epsilon\|^p\right]\\
\leq&4^{p-1}\bigg(E\left[\|q_0^\epsilon\|^p\right]+\epsilon^{-p/2}E\left[\left(\int_0^T \|G_1(s,x_s^\epsilon,z_s^\epsilon)\|ds\right)^p\right]\notag\\
&+E\left[\left(\int_0^T \|F_1(s,x_s^\epsilon,z_s^\epsilon)\|ds\right)^p\right]+E\left[\sup_{t\in[0,T]}\left\|\int_0^t\sigma_1(s,x_s^\epsilon)dW_s\right\|^p\right]\bigg)\notag\\
\leq&4^{p-1}\bigg(E\left[\|q_0^\epsilon\|^p\right]+\epsilon^{-p/2}T^{p-1}\int_0^T E\left[\|G_1(s,x_s^\epsilon,z_s^\epsilon)\|^p\right]ds\notag\\
&+T^{p-1}\int_0^T E\left[\|F_1(s,x_s^\epsilon,z_s^\epsilon)\|^p\right]ds+\tilde C T^{p/2-1}\int_0^TE\left[ \|\sigma_1(s,x_s^\epsilon)\|^p_F\right]ds\bigg)\notag.\end{aligned}$$ Polynomial boundedness (see Assumption \[homog\_assump2\]) gives $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\|q_t^\epsilon\|^p\right]\leq4^{p-1}\bigg(E\left[\|q_0^\epsilon\|^p\right]+\tilde C \int_0^TE\left[ (1+\|z_s^\epsilon\|^q)^p\right]ds\bigg),\end{aligned}$$ where we absorbed all factors of $T$ and $\epsilon$ into the constant $\tilde C$. Using Assumption \[homog\_assump1\] then gives $$\begin{aligned}
&E\left[\sup_{t\in[0,T]}\|q_t^\epsilon\|^p\right]<\infty\end{aligned}$$ for all $\epsilon$ sufficiently small.
Finally, for $n>0$ define the stopping time $\tau_n=\inf\{t\geq 0:\|q_t\|\geq n\}$. Then for $0\leq t\leq T$ the Lipschitz properties together with the Burkholder-Davis-Gundy and Hölder’s inequalities imply $$\begin{aligned}
&E\left[\sup_{s\in[0,t]}\|q^{\tau_n}_s\|^p\right]\\
\leq&3^{p-1}\bigg(E[ \|q_0\|^p]+E\left[\left(\int_0^{t\wedge\tau_n} \|(\tilde F+\tilde G)(s,q^{\tau_n}_s,\psi(s,q^{\tau_n}_s))\|ds\right)^p\right]\notag\\
&\hspace{.75cm}+E\left[\sup_{t\in[0,t]}\left\|\int_0^{t\wedge\tau_n}\tilde\sigma(s,q_s^{\tau_n},\psi(s,q^{\tau_n}_s))dW_s\right\|^p\right]\bigg)\notag\\
\leq&3^{p-1}E[ \|q_0\|^p]+\tilde C\int_0^tE\left[\|q^{\tau_n}_s\|^p\right]ds\\
&+\tilde C\int_0^t\|(\tilde F+\tilde G)(s,0,\psi(s,0))\|^p+\|\tilde\sigma(s,0,\psi(s,0))\|^p_Fds\notag\\
\leq&\tilde C\left(1+ \int_0^tE\left[\sup_{r\in[0,s]}\|q^{\tau_n}_r\|^p\right]ds\right),\end{aligned}$$ where $\tilde C$ changes line to line, and is independent of $t$.
The definition of $\tau_n$, together with $E[\|q_0\|^p]<\infty$, implies that $$\sup_{s\geq 0}E\left[\sup_{r\in[0,s]}\|q^{\tau_n}_r\|^p\right]<\infty.$$ Therefore we can apply Gronwall’s inequality to get $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q^{\tau_n}_t\|^p\right]\leq \tilde Ce^{\tilde C T},\end{aligned}$$ where the constant $\tilde C$ is independent of $n$. Hence, the monotone convergence theorem yields $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t\|^p\right]\leq \tilde Ce^{\tilde C T}<\infty.\end{aligned}$$
This completes the verification that the hypotheses of Lemma \[conv\_lemma\] hold, allowing us to conclude that $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon-q_t\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+.\end{aligned}$$
Homogenization of Hamiltonian systems
=====================================
In this final section, we apply the above framework to our original Hamiltonian system, [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{} (in particular, $m=n$ in this section) in order to prove the existence of a limiting process $q_t^\epsilon\to q_t$ and derive a homogenized SDE for $q_t$. Specifically, in Sections \[sec:h\_explicit\_sol\] and \[sec:chi\_explicit\_sol\] we will study a class of Hamiltonian systems for which the PDEs [Eq.(\[h\_eq\])]{} for $h$ and [Eq.(\[chi\_eq\])]{} for $\chi$ that are needed to derive the limiting equation are explicitly solvable and the required bounds can be verified by elementary means.
The SDE [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{} can be rewritten in the general form [Eq.(\[gen\_SDE1\])]{}-[Eq.(\[gen\_SDE2\])]{}: $$\begin{aligned}
dq^\epsilon_t=&\frac{1}{\sqrt{\epsilon}}\nabla_z K(t,q^\epsilon_t,z_t^\epsilon)dt,\label{q_Hamil_eq2}\\
d p^\epsilon_t=&\left(-\frac{1}{\sqrt{\epsilon}}\left(\gamma_l(t,x^\epsilon_t)- \nabla_q\psi_l(t,q_t^\epsilon)\right)\partial_{z_l}K(t,q_t^\epsilon,z_t^\epsilon)-\nabla_q K(t,q^\epsilon_t,z_t^\epsilon)\right.\label{p_Hamil_eq2}\\
&\hspace{5mm}-\nabla_q V(t,q^\epsilon_t)+F(t,x^\epsilon_t)\bigg)dt+\sigma(t,x^\epsilon_t) dW_t,\notag\end{aligned}$$ where $\gamma_l$ denotes the vector obtained by taking the $l$th column of $\gamma$. Specifically, $$\begin{aligned}
&F_1=0, \hspace{2mm}\sigma_1=0, \hspace{2mm} \sigma_2=\sigma, \hspace{2mm} G_1(t,x,z)=\nabla_z K(t,q,z),\\
&F_2(t,x,z)=-\nabla_q K(t,q,z)-\nabla_q V(t,q)+F(t,x),\\
& G_2(t,x,z)=-\left(\gamma_l(t,x)- \nabla_q\psi_l(t,q)\right)\partial_{z_l}K(t,q,z).\end{aligned}$$ In particular, $\sigma_1=0$, so below we use the definition of $C^{1,2}$ applicable to this case.
The operator $L$, [Eq.(\[L\_def\])]{}, and its formal adjoint have the following form: $$\begin{aligned}
(L\chi)(t,x,z)=&\frac{1}{2} \Sigma_{kl}(t,x)(\partial_{z_k}\partial_{z_l}\chi)(t,x,z)\label{Hamil_L}\\
&-\tilde\gamma_{kl}(t,x)\partial_{z_l}K(t,q,z)(\partial_{z_k}\chi)(t,x,z),\notag\\
(L^*h)(t,x,z)=&\partial_{z_k}\bigg(\frac{1}{2} \Sigma_{kl}(t,x)\partial_{z_l}h(t,x,z)\label{Hamil_L_star}\\
&+\tilde\gamma_{kl}(t,x)\partial_{z_l}K(t,q,z)h(t,x,z)\bigg),\notag\end{aligned}$$ where $$\label{hamil_Sigma_def}
\Sigma_{ij}=\sum_{\rho}\sigma_{i\rho}\sigma_{j\rho}$$ and $\tilde\gamma$ was defined in [Eq.(\[tilde\_gamma\_def\])]{}. Here $\sigma$ and $\Sigma$ denote what were $\sigma_2$ and $\Sigma_{22}$ respectively in [Eq.(\[sigma\_defs\])]{} and $\sigma_1=0$. In particular, the indices on $\Sigma$ have the meaning $\Sigma_{ij}\equiv (\Sigma_{22})_{ij}$.
Computing the noise induced drift {#sec:h_explicit_sol}
---------------------------------
In general, an explicit solution to $L^*h=0$ is not available, and so the homogenized equation can only be defined implicitly, as in Theorem \[conv\_thm\]. However, there are certain classes of systems where we can explicitly derive the form of the additional vector field, $\tilde G$, appearing in the homogenized equation. In [@BirrellHomogenization], one such class was studied by a different method. Here, we explore the case where the noise and dissipation satisfy the fluctuation dissipation relation pointwise for a time and state dependent generalized temperature $T(t,q)$, $$\begin{aligned}
\label{fluc_dis}
\Sigma_{ij}(t,q)=2k_BT(t,q) \gamma_{ij}(t,q).\end{aligned}$$ where $\Sigma$ was defined in [Eq.(\[hamil\_Sigma\_def\])]{}. We will make Assumptions \[assump1\]-\[assump4\], but make no further constraints on the form of the Hamiltonian here.
As can be verified by a direct calculation, under the assumption [Eq.(\[fluc\_dis\])]{}, the adjoint equation [Eq.(\[Hamil\_L\_star\])]{} is solved by $$\begin{aligned}
\label{h_formula}
h(t,q,z)=\frac{1}{Z(t,q)} \exp[-\beta(t,q)K(t,q,z)],\end{aligned}$$ where we define $\beta(t,q)=1/(k_BT(t,q))$ and $Z$, the “partition function", is chosen so that $\int h dz=1$. Note that Assumption \[assump3\] ensures such a normalization exists. We also point out that in this case, the antisymmetric part of $\tilde \gamma$ does not contribute to the right hand side of [Eq.(\[Hamil\_L\_star\])]{}.
An interesting point to note is that when the antisymmetric part of $\tilde\gamma$ vanishes (physically, for $K$ quadratic in $z$ this means a vanishing magnetic field), the vector field that we are taking the divergence of in [Eq.(\[Hamil\_L\_star\])]{} vanishes identically. When $\tilde\gamma$ has a non-vanishing antisymmetric part, only once we take the divergence does the expression in [Eq.(\[Hamil\_L\_star\])]{} vanish.
From [Eq.(\[q\_eq\])]{}, we see that the terms that require homogenization are $$\begin{aligned}
\label{G_gen_def}
G(t,q_t^\epsilon,z_t^\epsilon)=&-(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)\partial_{q^j}K(t,q_t^\epsilon,z_t^\epsilon)dt\\
&+(z_t^\epsilon)_j\partial_{q^l}(\tilde\gamma^{-1})^{ij}(t,q_t^\epsilon)\partial_{z_l}K(t,q_t^\epsilon,z_t^\epsilon)dt.\notag\end{aligned}$$ Using [Eq.(\[h\_formula\])]{}, the formal calculation of Section \[sec:formal\_G\_tilde\] gives $$\begin{aligned}
\tilde G(t,q)= -(\tilde\gamma^{-1})^{ij}(t,q)\langle\partial_{q^j}K(t,q,z)\rangle+\frac{\partial_{q^l}(\tilde\gamma^{-1})^{il}(t,q)}{\beta(t,q)},\end{aligned}$$ where we define $$\begin{aligned}
\label{h_avg_def}
\langle\partial_{q^j}K(t,q,z)\rangle=\frac{1}{Z(t,q)} \int \partial_{q^j}K(t,q,z)\exp[-\beta(t,q)K(t,q,z)]dz.\end{aligned}$$
Of course, this calculation is only formal. In the next section, we study a particular case where everything can be made rigorous.
Rigorous Homogenization of a class of Hamiltonian systems {#sec:chi_explicit_sol}
----------------------------------------------------------
In this section we explore a class of Hamiltonian systems for which Assumption \[homog\_assump3\] can be rigorously verified via an explicit solution to the PDE for $\chi$. We will work with Hamiltonian systems that satisfy Assumptions \[assump1\]-\[assump5\], \[assump7\]. In particular, we are restricting to the class of Hamiltonians with $$\begin{aligned}
\label{K_tilde_def}
K(t,q,z)=\tilde K(t,q,A^{ij}(t,q)z_iz_j),\end{aligned}$$ where $A(t,q)$ is valued in the space of positive definite $n \times n$-matrices. We will write $\tilde K\equiv \tilde K(t,q,\zeta)$ and $\tilde K^\prime\equiv \partial_{\zeta}\tilde K$.
We will also need the following relations between $\Sigma$, $\gamma$, and $A$ to hold:
\[proportionality\_assump\] $\sigma$ is independent of $p$ and $$\begin{aligned}
\Sigma(t,q)=b_1(t,q)A^{-1}(t,q) , \hspace{2mm} \gamma(t,q)=b_2(t,q)A^{-1}(t,q)\end{aligned}$$ where, for every $T>0$, the $b_i$ are bounded, $C^2$ functions that have positive lower bounds and bounded first derivatives, both on $[0,T]\times\mathbb{R}^n$.
Note that these relations imply a fluctuation-dissipation relation with a time and state dependent generalized temperature $T=\frac{b_1}{2k_Bb_2}$.
In [@BirrellHomogenization], we showed that Assumptions \[assump1\]-\[assump5\] imply: $$\begin{aligned}
\label{q_eq2}
d(q_t^\epsilon)^i=&\tilde F^i(t,x)dt+\tilde\sigma^i_{\rho}(t,x_t^\epsilon)dW^\rho_t+G^i(t,x_t^\epsilon,z_t^\epsilon)dt+d(R^\epsilon_t)^i,\notag\end{aligned}$$ where $$\begin{aligned}
\tilde F^i(t,x)=&(\tilde\gamma^{-1})^{ij}(t,q)(-\partial_t\psi_j(t,q)-\partial_{q^j}V(t,q)+F_j(t,x))+S^i(t,q),\\
G^i(t,q,z)=&-(\tilde\gamma^{-1})^{ij}(t,q)(\partial_{q^j}\tilde K)(t,q,A^{ij}(t,q)z_iz_j),\\
\tilde \sigma^i_\rho(t,x)=& (\tilde\gamma^{-1})^{ij}(t,q)\sigma_{j\rho}(t,x),\\
S^i(t,q)=& k_BT(t,q) \left(\partial_{q^j}(\tilde\gamma^{-1})^{ij}(t,q)-\frac{1}{2}(\tilde\gamma^{-1})^{ik}(t,q)A^{-1}_{jl}(t,q)\partial_{q^k} A^{jl}(t,q)\right),\label{S_def}\end{aligned}$$ and $R_t^\epsilon$ is a family of continuous semimartingales. $S(t,q)$ is called the [*noise induced drift*]{} (see Eq. 3.26 in [@BirrellHomogenization]).
Note, that with $K(t,q,z)$ defined by [Eq.(\[K\_tilde\_def\])]{}, the first term in [Eq.(\[G\_gen\_def\])]{} consists of two contributions—one coming from the $q$-dependence of $\tilde{K}$ and one coming from the $q$-dependence of $A$. The $G$ defined here comprises only the first contribution. The method of [@BirrellHomogenization] is able to homogenize the second term in [Eq.(\[G\_gen\_def\])]{}, as well the first contribution of the first term, leading to the noise induced drift, S, but fails when $\tilde K$ depends explicitly on $q$. However, under certain circumstances, the method developed in Section \[sec:gen\_homog\] is succeeds in homogenizing the system when $\tilde K$ has $q$ dependence, as we now show.
We will need one final assumption:
\[K\_poly\_bound\_assump\] For every $T>0$:
1. There exists $\zeta_0>0$ and $C>0$ such that $\tilde K^\prime(t,q,\zeta)\geq C$ for all $(t,q,\zeta)\in[0,T]\times\mathbb{R}^n\times[\zeta_0,\infty)$.
2. $\tilde K(t,q,\zeta)$, $\partial_t\partial_{q^i}\tilde K(t,q,\zeta)$, $\partial_{q^i}\partial_{\zeta}\tilde K(t,q,\zeta)$, and $\partial_{q^i}\partial_{q^j}\tilde K(t,q,\zeta)$ are polynomially bounded in $\zeta$, uniformly in $(t,q)\in[0,T]\times\mathbb{R}^n$.
We are now prepared to prove the following homogenization result:
\[hamil\_conv\_thm\] Let $x_t^\epsilon=(q_t^\epsilon,p_t^\epsilon)$ satisfy the Hamiltonian SDE [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{} and suppose Assumptions \[assump1\]-\[assump5\], \[assump7\], \[proportionality\_assump\], and \[K\_poly\_bound\_assump\] hold. Let $p\geq 2$ and suppose we have initial conditions that satisfy $E[\|q^\epsilon_0\|^p]<\infty$, $E[\|q_0\|^p]<\infty$, and $E[\|q_0^\epsilon-q_0\|^p]=O(\epsilon^{p/2})$. Then for any $T>0$, $0<\beta<p/2$ we have $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon-q_t\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+\end{aligned}$$ where $q_t$ is the solution to the SDE $$\begin{aligned}
\label{limit_eq}
dq_t^i=&(\tilde\gamma^{-1})^{ij}(t,q_t)(-\partial_t\psi_j(t,q_t)-\partial_{q^j}V(t,q_t)+F_j(t,q_t,\psi(t,q_t)))dt\\
&+S^i(t,q_t)dt+\tilde G^i(t,q_t)dt+(\tilde\gamma^{-1})^{ij}(t,q_t)\sigma_{j\rho}(t,q_t)dW_t^\rho\notag\end{aligned}$$ with initial condition $q_0$. See [Eq.(\[tilde\_gamma\_def\])]{}, [Eq.(\[S\_def\])]{}, and [Eq.(\[hamil\_tilde\_G\_def\])]{} for the definitions of $\tilde\gamma$, $S$, and $\tilde G$, respectively.
From [@BirrellHomogenization], Assumptions \[assump1\]-\[assump5\], \[assump7\] imply:
1. $q_t^\epsilon$ satisifies an equation of the form [Eq.(\[con\_thm\_eq\])]{}, where, for every $T>0$, $\tilde F$, $\tilde\sigma$ are bounded, continuous, and Lipschitz in $x$, on $[0,T]\times\mathbb{R}^{2n}$, with Lipschitz constant uniform in $t$.
2. Assumptions \[homog\_assump1\] holds.
3. For any $p>0$, $T>0$, $0<\beta<p/2$ we have $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|R_t^\epsilon\|^p\right]=O(\epsilon^\beta) \text{ as } \epsilon \rightarrow 0^+.\end{aligned}$$
Combined with polynomial boundedness of $\tilde K$ (Assumption \[K\_poly\_bound\_assump\]) we see that Assumption \[homog\_assump2\] also holds. Therefore, to apply Theorem \[conv\_thm\], we have to verify Assumption \[homog\_assump3\] and that the $\tilde G$ referenced therein is Lipschitz in $x$, uniformly in $t\in [0,T]$.
From Section \[sec:formal\_G\_tilde\], we expect that $$\begin{aligned}
\label{hamil_tilde_G_def}
\tilde G^i(t,q)=-(\tilde\gamma^{-1})^{ij}(t,q)\langle\partial_{q^j}\tilde K(t,q,A^{ij}(t,q)z_iz_j)\rangle\end{aligned}$$ where, similarly to [Eq.(\[h\_avg\_def\])]{}, $$\begin{aligned}
\langle\partial_{q^j}\tilde K(t,q,\|z\|_A^2)\rangle=\frac{1}{Z(t,q)} \int\partial_{q^j}\tilde K(t,q,\|z\|^2_A)\exp[-\beta(t,q)\tilde K(t,q,\|z\|^2_A)]dz.\end{aligned}$$ Here we use the shorthand $\|z\|^2_A\equiv A^{ij}(t,q)z_iz_j$ when the implied values of $t,q$ are apparent from the context.
Using our assumptions, along with several applications of the DCT, one can see that $\tilde G$ is $C^{1}$ and, for every $T>0$, is bounded with bounded first derivatives on $[0,T]\times\mathbb{R}^n$. In particular, it is Lipschitz in $q$, uniformly in $t\in[0,T]$.
We now turn to solving the equation $$\begin{aligned}
\label{L_pde}
L\chi=G-\tilde G.\end{aligned}$$ Since $G-\tilde G$ is independent of $p$ and depends on $z$ only through $\|z\|^2_A$, we look for $\chi$ with the same behavior. Using the ansatz $\chi(t,q,z)=\tilde\chi(t,q,\|z\|^2_A)$, and defining $G^i(t,q,\zeta)=-(\tilde\gamma^{-1})^{ij}(t,q)(\partial_{q^j}\tilde K)(t,q,\zeta)$, leads (on account of the antisymmetry of the matrix $\tilde{\gamma} - \gamma$) to the ODE in the variable $\zeta$: $$\begin{aligned}
&\zeta\tilde\chi^{\prime\prime}(t,q,\zeta) +\left(\frac{n}{2}- \beta(t,q) \zeta \tilde K^\prime(t,q,\zeta)\right) \tilde\chi^\prime(t,q,\zeta)\\
=&\frac{1}{2 b_1(t,q)}(G(t,q,\zeta)-\tilde G(t,q)).\notag\end{aligned}$$
This has the solution $$\begin{aligned}
\label{tilde_chi_def}
&\tilde\chi(t,q,\zeta)=\frac{1}{2b_1(t,q)}\int_0^\zeta \zeta_1^{-n/2}\exp[\beta(t,q) \tilde K(t,q,\zeta_1)]\\
&\times \int_0^{\zeta_1} \zeta_2^{(n-2)/2}\exp[-\beta(t,q) \tilde K(t,q,\zeta_2)]\left(G(t,q,\zeta_2)-\tilde G(t,q)\right)d\zeta_2d\zeta_1\notag\end{aligned}$$
Therefore $\chi(t,q,z)\equiv\tilde\chi(t,q,\|z\|_A^2)$ solves the PDE [Eq.(\[L\_pde\])]{}. One can show that it is is $C^{1,2}$ and that $\chi$ and its first derivatives are polynomially bounded in $z$, uniformly for $(t,q)\in[0,T]\times\mathbb{R}^{n}$. As a representative example, in \[app:poly\_bound\] we outline the proof that $\tilde\chi(t,q,\zeta)$ is polynomially bounded in $\zeta$, uniformly in $(t,q)\in[0,T]\times\mathbb{R}^n$. The remainder of the computations are similar and we leave them to the reader.
$\chi$ is independent of $p$, so $\partial_{p_i}\partial_{p_j}\chi=0$ and $\partial_{p_i}\partial_{z_j}\chi=0$. Therefore, this completes the verification of Assumption \[homog\_assump3\] and we are justified in using Theorem \[conv\_thm\] to conclude $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon-q_t\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+,\end{aligned}$$ where $q_t$ satisfies the SDE $$\begin{aligned}
\label{gen_limit_eq}
q_t=q_0+&\int_0^t\tilde F(s,q_s,\psi(s,q_s))ds+\int_0^t\tilde G(s,q_s)ds+\int_0^t\tilde\sigma(s,q_s)dW_s\notag \end{aligned}$$ as claimed.
Lastly, we give an example of a general class of Hamiltonians that satisfy the hypotheses of Theorem \[hamil\_conv\_thm\]. The proof of this corollary is straighforward, so we leave it to the reader.
Consider the class of Hamiltonians of the form $$\begin{aligned}
H(t,q,p)=\sum_{l=k_1}^{k_2} d_l(t,q) \left[A^{ij}(t,q) (p-\psi(t,q))_i(p-\psi(t,q))_j\right]^l+V(t,q)\end{aligned}$$ where $1\leq k_1\leq k_2$ are integers and the following properties hold on $[0,T]\times\mathbb{R}^{n}$ for every $T>0$:
1. $V$ is $C^2$ and $\nabla_q V$ is bounded and Lipschitz in $q$, uniformly in $t\in[0,T]$.
2. $\psi$ is $C^3$ and $\partial_t\psi$, $\partial_{q^i}\psi$, $\partial_t\partial_{q^i}\psi$, $\partial_{q^i}\partial_{q^j}\psi$, $\partial_t\partial_{q^j}\partial_{q^i}\psi$, and $\partial_{q^l}\partial_{q^j}\partial_{q^i}\psi$ are bounded.
3. $d_l$ are $C^2$, non-negative, bounded, and have bounded first and second derivatives.
4. $d_{k_1}$ and $d_{k_2}$ are uniformly bounded below by a positive constant.
5. $A$ is $C^2$, positive-definite, and $A$, $\partial_t A$, $\partial_{q_i} A$, $\partial_t \partial_{q^i}A$, and $\partial_{q^i}\partial_{q^j} A$ are bounded.
6. The eigenvalues of $A$ are uniformly bounded below by a positive constant.
Also suppose that
1. $\sigma$ is independent of $p$ and $$\begin{aligned}
\Sigma(t,q)=b_1(t,q)A^{-1}(t,q) , \hspace{2mm} \gamma(t,q)=b_2(t,q)A^{-1}(t,q)\end{aligned}$$ where, for every $T>0$, the $b_i$ are bounded, $C^2$ functions with positive lower bounds and bounded first derivatives.
2. $\gamma$ is $C^2$, is independent of $p$, and $\partial_t\gamma$, $\partial_{q^i} \gamma$, $\partial_t\partial_{q^j}\gamma$, $\partial_{q^i}\partial_{q^j}\gamma$ are bounded on $[0,T]\times\mathbb{R}^{n}$.
3. The eigenvalues of $\gamma$ are bounded below by some $\lambda>0$.
4. $\gamma$, $F$, and $\sigma$ are bounded.
5. $F$ and $\sigma$ are Lipschitz in $x$ uniformly in $t\in[0,T]$.
6. There exists $C>0$ such that the (random) initial conditions satisfy $K^\epsilon(0,x^\epsilon_0)\leq C$ for all $\epsilon>0$ and all $\omega\in\Omega$.
7. There is a $p\geq 2$ such that $$\begin{aligned}
E[\|q^\epsilon_0\|^p]<\infty, \hspace{2mm} E[\|q_0\|^p]<\infty,\text{ and } E[\|q_0^\epsilon-q_0\|^p]=O(\epsilon^{p/2}).\end{aligned}$$
Then all the hypotheses of Theorem \[hamil\_conv\_thm\] hold, in particular Assumptions \[assump1\]-\[assump5\], \[assump7\], \[proportionality\_assump\], and \[K\_poly\_bound\_assump\] hold, and hence, for any $\beta \in \left(0, {p \over 2}\right)$, $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon-q_t\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+,\end{aligned}$$ where $x_t^\epsilon=(q_t^\epsilon,p_t^\epsilon)$ satisfy the Hamiltonian SDE [Eq.(\[Hamiltonian\_SDE\_q\])]{}-[Eq.(\[Hamiltonian\_SDE\_p\])]{} and $q_t$ satisfies the homogenized SDE, [Eq.(\[limit\_eq\])]{}.
Material from [@BirrellHomogenization] {#app:assump}
======================================
In this appendix, we collect several useful assumptions and results from [@BirrellHomogenization]. The assumptions listed here are [*not*]{} used in the entirety of this current work. When they are needed for a particular result we explicitly references them.
\[assump1\] We assume that the Hamiltonian has the form given in [Eq.(\[H\_form\])]{} where $K$ and $\psi$ are $C^2$ and $K$ is non-negative. For every $T>0$, we assume the following bounds hold on $[0,T]\times\mathbb{R}^{2n}$:
1. There exist $C>0$ and $M>0$ such that $$\begin{aligned}
\label{K_assump1}
\max\{|\partial_t K(t,q,z)|,\|\nabla_qK(t,q,z)\|\}\leq M+CK(t,q,z).\end{aligned}$$
2. There exist $c>0$ and $M\geq 0$ such that $$\begin{aligned}
\label{K_assump2}
\|\nabla_z K(t,q,z)\|^2+M\geq c K(t,q,z).\end{aligned}$$
3. For every $\delta>0$ there exists an $M>0$ such that $$\begin{aligned}
\label{K_assump3}
\max\left\{\|\nabla_z K(t,q,z)\|,\left(\sum_{ij}|\partial_{z_i}\partial_{z_j}K(t,q,z)|^2\right)^{1/2}\right\}\leq M+\delta K(t,q,z).\end{aligned}$$
\[assump2\] For every $T>0$, we assume that the following hold uniformly on $[0,T]\times\mathbb{R}^n$:
1. $V$ is $C^2$ and $\nabla_q V$ is bounded
2. $\gamma$ is symmetric with eigenvalues bounded below by some $\lambda>0$.
3. $\gamma$, $F$, $\partial_t\psi$, and $\sigma$ are bounded.
4. There exists $C>0$ such that the (random) initial conditions satisfy $K^\epsilon(0,x^\epsilon_0)\leq C$ for all $\epsilon>0$ and all $\omega\in\Omega$.
\[assump3\] We assume that for every $T>0$ there exists $c>0$, $\eta>0$ such that $$\begin{aligned}
K(t,q,z)\geq c\|z\|^{2\eta}\end{aligned}$$ on $[0,T]\times\mathbb{R}^{2n}$.
\[assump4\] We assume that $\gamma$ is $C^1$ and is independent of $p$.
\[assump5\] We assume that $K$ has the form $$\begin{aligned}
K(t,q,z)=\tilde K(t,q,A^{ij}(t,q)z_iz_j)\end{aligned}$$ where $\tilde K(t,q,\zeta)$ is $C^2$ and non-negative on $[0,\infty)\times\mathbb{R}^n\times[0,\infty)$ and $A(t,q)$ is a $C^2$ function whose values are symmetric $n \times n$-matrices. We also assume that for every $T>0$, the eigenvalues of $A$ are bounded above and below by some constants $C>0$ and $c>0$ respectively, uniformly on $[0,T]\times\mathbb{R}^n$.
We will write $\tilde K^\prime$ for $\partial_\zeta \tilde K$ and will use the abbreviation $\|z\|_A^2$ for $A^{ij}(t,q)z_iz_j$ when the implied values of $t$ and $q$ are apparent from the context.
\[assump7\] We assume that, for every $T>0$, $\nabla_q V$, $F$, and $\sigma$ are Lipschitz in $x$ uniformly in $t\in[0,T]$. We also assume that $A$ and $\gamma$ are $C^2$, $\psi$ is $C^3$, and $\partial_t\psi$, $\partial_{q^i}\psi$, $\partial_{q^i}\partial_{q_j}\psi$, $\partial_t\partial_{q^i}\psi$, $\partial_t\partial_{q^j}\partial_{q^i}\psi$, $\partial_{q^l}\partial_{q^j}\partial_{q^i}\psi$, $\partial_t\gamma$, $\partial_{q^i} \gamma$, $\partial_t\partial_{q^j}\gamma$, $\partial_{q^i}\partial_{q^j}\gamma$, $\partial_t A$, $\partial_{q^i} A$, $\partial_t \partial_{q^i}A$, and $\partial_{q^i}\partial_{q^j} A$ are bounded on $[0,T]\times\mathbb{R}^{2n}$ for every $T>0$.
Note that, combined with Assumptions \[assump1\]-\[assump4\], this implies $\tilde\gamma$, $\tilde\gamma^{-1}$, $\partial_t\tilde\gamma^{-1}$, $\partial_{q^i}\tilde\gamma^{-1}$, $\partial_t\partial_{q^j}\tilde\gamma^{-1}$, and $\partial_{q^i}\partial_{q^j}\tilde\gamma^{-1}$ are bounded on compact $t$-intervals.
\[q\_bounded\_lemma\] Under Assumptions \[assump1\] and \[assump2\], for any $T>0$, $p>0$ we have $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon\|^p\right]<\infty.\end{aligned}$$
\[p\_decay\_lemma\] Under Assumptions \[assump1\]-\[assump3\], for any $T>0$, $p>0$ we have $$\begin{aligned}
\sup_{t\in[0,T]}E[\|p_t^\epsilon-\psi(t,q_t^\epsilon)\|^{p}]=O(\epsilon^{p/2}) \text{ as }\epsilon\rightarrow 0^+\end{aligned}$$ and for any $p>0$, $T>0$, $0<\beta<p/2$ we have $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|p_t^\epsilon-\psi(t,q_t^\epsilon)\|^{p}\right]=O(\epsilon^\beta) \text{ as }\epsilon\rightarrow 0^+.\end{aligned}$$
The following is a slight variant of the result from [@BirrellHomogenization], but the proof is identical.
\[conv\_lemma\] Let $T>0$ and suppose we have continuous functions $\tilde F(t,x):[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}^n$, $\tilde \sigma(t,x):[0,\infty)\times\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{n\times k}$, and $\psi:[0,\infty)\times\mathbb{R}^n\rightarrow\mathbb{R}^m$ that are Lipschitz in $x$, uniformly in $t\in[0,T]$.
Let $W_t$ be a $k$-dimensional Wiener process, $p\geq 2$ and $\beta>0$ and suppose that we have continuous semimartingales $q_t$ and, for each $0<\epsilon\leq\epsilon_0$, $\tilde R_t^\epsilon$, $x_t^\epsilon=(q_t^\epsilon,p_t^\epsilon)$ that satisfy the following properties:
1. $q_t^\epsilon=q_0^\epsilon+\int_0^t\tilde F(s,x_s^\epsilon)ds+\int_0^t\tilde\sigma(s,x_s^\epsilon)dW_s+\tilde R^\epsilon_t$.\[q\_eps\_eq\_assump\]
2. $q_t=q_0+\int_0^t\tilde F(s,q_s,\psi(s,q_s))ds+\int_0^t\tilde\sigma(s,q_s^\epsilon,\psi(s,q_s))dW_s$.\[q\_eq\_assump\]
3. $E[\|q_0^\epsilon-q_0\|^p]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+$. \[ics\_assump\]
4. $E\left[\sup_{t\in[0,T]}\|\tilde R_t^\epsilon\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+$.\[R\_decay\_assump\]
5. $\sup_{t\in[0,T]}E[\|p_t^\epsilon-\psi(t,q_t^\epsilon)\|^p]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+$.\[p\_decay\_assump\]
6. $E\left[\sup_{t\in[0,T]}\|q_t^\epsilon\|^p\right]<\infty$ for all $\epsilon>0$ sufficiently small.\[q\_eps\_integrable\_assump\]
7. $E\left[\sup_{t\in[0,T]}\|q_t\|^p\right]<\infty$.\[q\_integrable\_assump\]
Then $$\begin{aligned}
E\left[\sup_{t\in[0,T]}\|q_t^\epsilon-q_t\|^p\right]=O(\epsilon^\beta)\text{ as }\epsilon\rightarrow 0^+.\end{aligned}$$
Polynomial boundedness of $\tilde\chi$ {#app:poly_bound}
======================================
Changing variables, $\tilde\chi$ can be expressed as $$\begin{aligned}
\tilde\chi(t,q,\zeta)=&\frac{1}{2b_1(t,q)}\zeta\int_0^1\exp[\beta(t,q) \tilde K(t,q,s\zeta)]\\
&\times \int_0^{1} r^{(m-2)/2}\exp[-\beta(t,q) \tilde K(t,q,rs\zeta)]\left(G(t,q,rs\zeta)-\tilde G(t,q)\right)dr ds.\notag\end{aligned}$$ Applying the DCT to this expression several times, one can prove that $\tilde\chi$ is $C^{1,2}$. Using the fact that $\tilde K$ and $\partial_{q^i}\tilde K$ are polynomially bounded in $\zeta$, uniformly in $(t,q)\in[0,T]\times\mathbb{R}^n$, one can see that $\tilde\chi(t,q,\zeta)$ is bounded on $[0,T]\times\mathbb{R}^n\times[0,\zeta_0]$ for any $\zeta_0>0$. From Assumption \[K\_poly\_bound\_assump\], there exists $\zeta_0$ and $C>0$ such that $\tilde K^\prime(t,q,\zeta)\geq C$ for all $(t,q,\zeta)\in[0,T]\times\mathbb{R}^n\times[\zeta_0,\infty)$.
By combining [Eq.(\[tilde\_chi\_def\])]{} with [Eq.(\[hamil\_tilde\_G\_def\])]{}, one finds that for $\zeta\geq \zeta_0$, $\tilde \chi$ can alternatively be written as $$\begin{aligned}
\tilde\chi(t,q,\zeta)=&\tilde\chi(t,q,\zeta_0)+\frac{1}{2b_1(t,q)}\int_{\zeta_0}^\zeta \zeta_1^{-m/2}\exp[\beta(t,q) \tilde K(t,q,\zeta_1)]\\
&\times \int_{\zeta_1}^\infty \exp[-\beta(t,q)\tilde K(t,q,\zeta_2)]\zeta_2^{(m-2)/2}(\tilde G(t,q)-G(t,q,\zeta_2))d\zeta_2 d\zeta_1\notag.\end{aligned}$$ Therefore, if we can show that the second term has the polynomial boundedness property then so does $\tilde\chi$, and hence $\chi$.
Letting $\tilde C$ denote a constant that potentially changes in each line and choosing $\zeta_0$ as in Assumption \[K\_poly\_bound\_assump\], we have $$\begin{aligned}
&\|\frac{1}{2b_1(t,q)}\int_{\zeta_0}^\zeta \zeta_1^{-m/2}\exp[\beta(t,q) \tilde K(t,q,\zeta_1)]\\
&\times \int_{\zeta_1}^\infty \exp[-\beta(t,q)\tilde K(t,q,\zeta_2)]\zeta_2^{(m-2)/2}(\tilde G(t,q)-G(t,q,\zeta_2))d\zeta_2 d\zeta_1\|\notag\\
\leq &\tilde C\int_{\zeta_0}^\zeta \zeta_1^{-m/2}\exp[\beta(t,q) \tilde K(t,q,\zeta_1)]\notag\\
&\times \int_{\zeta_1}^\infty \exp[-\beta(t,q)\tilde K(t,q,\zeta_2)]\zeta_2^{(m-2)/2+q} d\zeta_2 d\zeta_1\notag\\
\leq &\tilde C \zeta_0^{-m/2}\int_{\zeta_0}^\zeta \exp[\beta(t,q) \tilde K(t,q,\zeta_1)]\notag\\
&\times \int_{\zeta_1}^\infty \exp[-\beta(t,q)\tilde K(t,q,\zeta_2)] \tilde K(t,q,\zeta_2)^{((m-2)/2+q)/\eta} \tilde K^\prime(t,q,\zeta_2) d\zeta_2 d\zeta_1\notag\\
=&\tilde C \zeta_0^{-m/2}\int_{\zeta_0}^\zeta \exp[\beta(t,q) \tilde K(t,q,\zeta_1)]\notag\\
&\hspace{1.5cm}\times\int_{\tilde K(t,q,\zeta_1)}^\infty \exp[-\beta(t,q)u]u^{((m-2)/2+q)/\eta} dud\zeta_1\notag\end{aligned}$$ for some $q>0$. To obtain the first inequality, we use polynomial boundedness of $\partial_{q^i}\tilde K$. For the second, we used Assumption [Eq.(\[assump3\])]{} together with the fact that $\tilde K^\prime\geq C>0$ on $[0,T]\times\mathbb{R}^n\times[\zeta_0,\infty)$.
Therefore we obtain $$\begin{aligned}
\|\tilde\chi(t,q,\zeta)\|\leq\tilde C\left(1+\int_{\zeta_0}^\zeta P(\tilde K(t,q,\zeta_1))d\zeta_1\right)\end{aligned}$$ for some polynomial $P(x)$ with positive coefficients that are independant of $t$ and $q$. Polynomial boundedness of $\tilde K$ then implies $$\begin{aligned}
\|\tilde\chi(t,q,\zeta)\|\leq\tilde C\left(1+ \int_{\zeta_0}^\zeta Q(\zeta_1)d\zeta_1\right)\end{aligned}$$ for some polynomial $Q(\zeta)$. This proves the desired polynomial boundedness property for $\tilde\chi$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
J.W. was partially supported by NSF grants DMS 131271 and DMS 1615045.\
|
---
abstract: 'View-invariant object recognition is a challenging problem, which has attracted much attention among the psychology, neuroscience, and computer vision communities. Humans are notoriously good at it, even if some variations are presumably more difficult to handle than others (e.g. 3D rotations). Humans are thought to solve the problem through hierarchical processing along the ventral stream, which progressively extracts more and more invariant visual features. This feed-forward architecture has inspired a new generation of bio-inspired computer vision systems called deep convolutional neural networks (DCNN), which are currently the best algorithms for object recognition in natural images. Here, for the first time, we systematically compared human feed-forward vision and DCNNs at view-invariant object recognition using the same images and controlling for both the kinds of transformation (position, scale, rotation in plane, and rotation in depth) as well as their magnitude, which we call “variation level”. We used four object categories: cars, ships, motorcycles, and animals. All 2D images were rendered from 3D computer models. In total, 89 human subjects participated in 10 experiments in which they had to discriminate between two or four categories after rapid presentation with backward masking. We also tested two recent DCNNs (proposed respectively by Hinton’s group and Zisserman’s group) on the same tasks. We found that humans and DCNNs largely agreed on the relative difficulties of each kind of variation: rotation in depth is by far the hardest transformation to handle, followed by scale, then rotation in plane, and finally position (much easier). This suggests that humans recognize objects mainly through 2D template matching, rather than by constructing 3D object models, and that DCNNs are not too unreasonable models of human feed-forward vision. In addition, our results show that the variation levels in rotation in depth and scale strongly modulate both humans’ and DCNNs’ recognition performances. We thus argue that these variations should be controlled in the image datasets used in vision research.'
author:
- 'Saeed Reza Kheradpisheh$ ^{1}$'
- 'Masoud Ghodrati$ ^{2} $'
- 'Mohammad Ganjtabesh$ ^{1}$'
- 'Timothée Masquelier$ ^{3,4,5,6,}$[^1]'
bibliography:
- 'references.bib'
title: Humans and deep networks largely agree on which kinds of variation make object recognition harder
---
Introduction
============
As our viewpoint relative to an object changes, the retinal representation of the object tremendously varies across different dimensions. Yet our perception of objects is largely stable. How humans and monkeys can achieve this remarkable performance has been a major focus of research in visual neuroscience [@dicarlo2012does]. Neural recordings showed that some variations are treated by early visual cortices, e.g., through phase- and contrast-invariant properties of neurons as well as increasing receptive field sizes along the visual hierarchy [@riesenhuber1999hierarchical; @finn2007emergence]. Position and scale invariance also exist in the responses of neurons in area V4 [@rust2010selectivity], but these invariances considerably increase as visual information propagates to neurons in inferior temporal (IT) cortex [@rust2010selectivity; @zoccolan2005multiple; @zoccolan2007trade; @brincat2004underlying; @hung2005fast], where responses are highly consistent when an identical object varies across different dimensions [@murty2015dynamics; @cadieu2013neural; @yamins2013hierarchical; @cadieu2014deep]. In addition, IT cortex is the only area in the ventral stream which encodes three-dimensional transformations through view specific [@logothetis1994view; @logothetis1995shape] and view invariant [@perrett1991viewer; @booth1998view] responses.
Inspired by these findings, several early computational models [@Fukushima1980; @LeCun1998; @Riesenhuber1999; @Serre2007.PAMI; @Masquelier2007; @Lee2009] were proposed. These models mimic feed-forward processing in the ventral visual stream as it is believed that the first feed-forward flow of information, $\sim150$ [*ms*]{} post-stimulus onset, is usually sufficient for object recognition [@thorpe1996speed; @liu2009timing; @freiwald2010functional; @anselmi2014unsupervised; @hung2005fast]. However, the performance of these models in object recognition was significantly poor comparing to that of humans in the presence of large viewpoint variations [@ghodrati2014feedforward; @pinto2011comparing; @Pinto2008].
The second generation of these feed-forward models are called deep convolutional neural networks (DCNNs). DCNNs involve many layers (say 8 and above) and millions of free parameters, usually tuned through extensive supervised learning. These networks have achieved outstanding accuracy on object and scene categorization on highly challenging image databases [@lecun2015deep; @krizhevsky2012imagenet; @zhou2014learning]. Moreover, it has been shown that DCNNs can tolerate a high degree of variations in object images and even achieve close-to-human performance [@khaligh2014deep; @kheradpisheh2015deep; @cadieu2014deep]. However, despite extensive research, it is still unclear how different types of variations in object images are treated by DCNNs. These networks are position-invariant by design (thanks to weight sharing), but other sorts of invariances must be acquired through training, and the resulting invariances have not been systematically quantified.
In humans, early behavioral studies [@bricolo1993rotation; @dill1997translation] showed that we can robustly recognize objects despite considerable changes in scale, position, and illumination; however, the accuracy drops if the objects are rotated in depth. Yet these studies used somewhat simple stimuli (respectively paperclips and combinations of geons). It remains largely unclear how different kinds of variation on more realistic object images, individually and combined with each other, affect the performance of humans, and if they affect the performance of DCNNs similarly.
Here, we address these questions through a set of behavioral and computational experiments in human subjects and DCNNs to test their ability in categorizing object images that were transformed across different dimensions. We generated naturalistic object images of four categories: cars, ships, motorcycles, and animals. Each object carefully varied across either one dimension or a combination of dimensions, among scale, position, in-depth and in-plane rotations. All 2D images were rendered from 3D planes. The effects of variations across single dimension and compound dimensions on recognition performance of humans and two powerful DCNNs [@krizhevsky2012imagenet; @simonyan2014very] were compared in a systematic way, using the same images.
Our results indicate that human subjects can tolerate a high degree of variation with remarkably high accuracy and very short response time. The accuracy and reaction time were, however, significantly dependent on the type of object variation, with rotation in-depth as the most difficult dimension. This finding does not argue in favor of three-dimensional object representation theories, but suggests that object recognition is mainly based on two-dimensional template matching. Interestingly, the results of deep neural networks were highly correlated with those of humans as they could well mimic human behavior when facing variations across different dimensions. This suggests that humans have difficulty to handle those variations that are also computationally more complicated to overcome. More specifically, variations in some dimensions, such as in-depth rotation and scale, that change the amount or the content of input visual information, make the object recognition more difficult for both humans and deep networks.
Material and methods
====================
Image generation
----------------
We generated object images of four different categories: car, animal, ship, and motorcycle. Object images varied across four dimensions: scale, position (horizontal and vertical), in-plane and in-depth rotations. Depending on the type of experiment, the number of dimensions that the objects varied across were determined (see following sections). All two-dimensional object images were rendered from three-dimensional models. There were on average 16 different three-dimensional example models per object category (car: 16, animal: 15, ship: 18, and motorcycle: 16).
To generate a two-dimensional object image, first, a set of random values were sampled from uniform distributions. Each value determined the degree of variation across one dimension (e.g., size). These values were then simultaneously applied to a three-dimensional object model. This resulted in a three-dimensional object images that has varied across different dimensions. Finally, a two-dimensional image was generated by taking a snapshot from the transformed three-dimensional model. Object images were generated with four levels of difficulty by carefully controlling the amplitude of variations across four levels, from no variation (level 0, where changes in all dimensions were very small: $\Delta_{Sc} = \pm 1\%$, $\Delta_{Po}=\pm 1\%$, $\Delta_{RD} = \pm 1^{\circ}$, and $\Delta_{RP}=\pm 1^{\circ}$; each subscript refers to a dimension: Sc: Scale, Po: Position, RD: in-depth rotation, RP: in-plane rotation; and $\Delta$ is the amplitude of variations) to high variation (level 3: $\Delta_{Sc} = \pm 60\%$, $\Delta_{Po}=\pm 60\%$, $\Delta_{RD} = \pm 90^{\circ}$, and $\Delta_{RP}=\pm 90^{\circ}$). To control the degree of variation in each level, we limited the range of random sampling to a specific upper and lower bounds.
Figure \[figure\_1\] shows several sample images and the range of variations across four levels. The size of two-dimensional images was $400\times 300$ pixels (width$ \times $height). All images were initially generated on uniform gray background. Moreover, identical object images on natural backgrounds were generated for some experiments. This was done by superimposing object images on randomly selected natural backgrounds from a large pool. Our natural image database contained 3,907 images which consisted of a wide variety of indoor, outdoor, man-made, and natural scenes.
### Different image databases
To test humans and DCNNs in invariant object recognition tasks, we generated three different image databases:
- **All-dimension:** In this database, objects varied across all dimensions, as described earlier (i.e., scale, position, in-depth and in-plane rotations). Object images were generated across four levels in terms of variation amplitude (Level 0-3, Figure \[figure\_1\].A).
- **Three-dimension:** The image generation procedure in this database was similar to all-dimension database, but object images varied across a combination of three dimensions only, while the 4th dimension was fixed. For example, objects’ size were fixed across all variation levels while other dimensions varied (i.e., position, in-depth, and in-plane rotations). This provided us with four databases: 1) $\Delta_{Sc} = 0$: object’s size were fixed to a reference size across variation levels; 2) $\Delta_{Po}=0$: objects’ position were fixed at the center of the image; 3) $\Delta_{RD} = 0$: objects were not rotated in depth (three-dimensional transformation) across variation levels; 4) $\Delta_{RP}=0$: objects were not rotated in plane (see Figure \[figure\_1\].A).
- **One-dimension:** Object images in this database varied across only one dimension (e.g., size), meaning that the variations across other dimensions were fixed to reference values. Thus, we generated four databases: 1) $\Delta_{Sc}$: only the scale of objects varied; 2) $\Delta_{Po}$: only the position of objects across vertical and horizontal axes varied; 3) $\Delta_{RP}$: objects were only rotated in plane; 4) $\Delta_{RD}$: objects were only rotated in depth. Figure \[figure\_1\].B shows several sample images from these databases.
Human psychophysical experiments
--------------------------------
We evaluated the performance of human subjects in invariant object recognition through different experiments and using different image databases. In total, the data of 89 subjects (aged between 23-31, mean = 22, 39 female and 50 male) were recorded. Subjects had normal or corrected-to-normal vision. In most experiments, data of 16-20 sessions were recorded (two experiments were ran for 5 sessions); therefore, some subjects completed all experiments and others only participated in some experiments. All subjects voluntarily participated to the experiments and gave their written consent prior to participation. Our research adhered to the tenets of the Declaration of Helsinki and all experimental procedures were approved by the ethic committee of the University of Tehran.
Images were presented on a 17" CRT monitor (LG T710BH CRT; refresh rate 80Hz, resolution 1280$ \times $1024 pixels) connected to a PC equipped with an NVIDIA GeForce GTX 650 graphic card. We used MATLAB (www.mathworks.com) with psychophysics toolbox [@brainard1997psychophysics; @pelli1997videotoolbox] (http://psychtoolbox.org) to present images. Subjects had a viewing distance of 60 [*cm*]{} and each image covered $\sim 10 \times 11$ degrees of visual angle.
Details of each experiment are explained in the following sections. Generally, we used rapid image presentation paradigm with mask to only account for the feed-forward processing in the ventral visual pathway (see Figure \[figure\_1\]). Each trial was started with a black fixation cross, presented at the center of the screen for 500 [*ms*]{}, followed by a randomly selected image from the database that was presented for either one or two frames depending on the experiment type (see the description for each experiment). Afterwards, a gray blank screen was presented as inter-stimulus interval (ISI). Finally, a $1/f$ noise mask image was presented. The timing of the image presentation, ISI, and mask depended on the experiment type (see the following sections). Subjects’ task was to categorize the presented object images. They were instructed to respond as fast and accurate as they could by pressing a key on computer keyboard (each key was labeled with a category name). The next trial was started after the key press and there was a random time delay before the start of the next trial. We recorded both subjects’ reaction times (reported in Supplementary Information) and accuracies. Each experiment was divided into a number of blocks and subjects could rest between blocks. Subjects performed some practice trials prior to the main experiment. Images in the practice trials were not presented in the main experiments. In the next sections, we describe the details of each experiment.
### Rapid invariant object categorization {#Multi-class-psycho}
In these experiments, subjects categorized rapidly presented images from four object categories (see Figure \[figure\_1\]). Each trial started with a fixation cross presented at the center of the screen for 500 [*ms*]{}. An image was then randomly selected from the pool and was presented for 25 [*ms*]{} (2 frames of 80 Hz monitor) followed by a gray blank screen for 25 [*ms*]{} (ISI). Immediately after the blank screen, a $1/f$ noise mask image was presented for 100 [*ms*]{}. Subjects were asked to rapidly and accurately press one of the four keys, labeled on keyboard, to indicate which object category was presented. The next trial started after a key press with a random time delay ($2 \pm 0.5$ second). This experiment was performed in two types that are explained as following:
- **Using the all-dimension database:** We used all-dimension database where object images varied across all dimensions (i.e., scale, position, in-depth and in-plane rotations). In each session, subjects were presented with 320 images: 4 categories $\times$ 4 levels $\times$ 20 images per category. Images were presented into two blocks of 160 images. For each background condition (i.e., uniform and natural backgrounds), we recorded the data of 16 different sessions.
- **Using three-dimension databases:** In this experiment, we used the three-dimension databases. Using these databases, we could measure how excluding variations across one dimension can affect human performance in invariant object recognition: if the fixed dimension is more difficult than the others, subjects would be able to categorize the object more accurately and within shorter time than in the case where the fixed dimension is much easier. We presented subjects with 960 images: 4 categories $\times$ 4 levels $\times$ 4 conditions ($\Delta_{Sc} = 0$, $\Delta_{Po} = 0$, $\Delta_{RP} = 0$, and $\Delta_{RD} = 0$) $\times$ 15 images per category. Images were presented in four blocks of 240 images. Note that we inter-mixed images of all conditions in each session; therefore, subjects were unaware of the type of variations. We recorded the data of 20 sessions for the case of objects on natural backgrounds and 17 sessions for objects on uniform background.
To have more accurate reaction times, we also performed two-category (car versus animal) rapid invariant object categorization tasks with similar experimental settings. The details of these two-category experiments and their results are presented in Supplementary Information.
### Ultra-rapid invariant object categorization {#Ultra-Rapid}
To assess whether the experimental design (presentation time and variation conditions) could affect our results and interpretations, we ran two ultra-rapid invariant object categorization tasks, using [*three-dimension*]{} and one-dimension databases. In each trial, we presented a fixation cross for 500 [*ms*]{}. Then, an image was randomly selected from the pool and presented to the subject for 12.5 [*ms*]{} (1 frame at 80 Hz monitor). The image was then followed by a blank screen for 12.5 [*ms*]{}. Finally, a noise mask was presented for 200 [*ms*]{}. Subjects had to accurately and rapidly press one of the four keys, labeled on the keyboard, to declare their responses. The next trial started after a key press with a random time delay ($2 \pm 0.5$ second). As mentioned above, this experiment was performed in two types that are explained as following:
- **Using three-dimension database:** We recorded the data of five human subjects. Object images were selected from natural background three-dimension database. Images are identical to those of three-dimension rapid presentation experiment described in previous section. But, here, images were presented for 12.5 [*ms*]{} followed by 12.5 [*ms*]{} blank and then 200 [*ms*]{} noise mask.
- **Using one-dimension database:** In this experiment, we used natural background one-dimension database to evaluate the effect of variations across individual dimensions on human performance. Subjects were presented with 960 images: 4 categories $\times$ 4 levels $\times$ 4 conditions ($\Delta_{Sc}$,$\Delta_{Po}$,$\Delta_{RP}$,$\Delta_{RD}$) $\times$ 15 images per category. The experiment was divided into four blocks of 240 images. We collected the data of five sessions. Note that we only used objects on natural backgrounds because this task was easier compared to previous experiments; therefore, categorizing objects on uniform background would be very easy. For the same reason, we did not used the one-dimension database in the rapid task.
Behavioral data analysis
------------------------
We calculated the accuracy of subjects in each experiment as the ratio of correct responses (i.e., Accuracy % = 100 $\times$ Number of correct trials / Total number of trials). The accuracies of all subjects were calculated and the average and standard deviation were reported. We also calculated confusion matrices for different conditions of rapid invariant object categorization experiments, which are presented in Supplementary Information. A confusion matrix allowed us to determine which categories were more miscategorized and how categorization errors were distributed across different categories. To calculate the human confusion matrix for each variation condition, we averaged the confusion matrices of all human subjects.
We also analyzed subjects’ reaction times in different experiments which are provided in Supplementary Information. In the two-category experiment, first, we removed reaction times longer than 1200 [*ms*]{} (only 7.8% of reaction times were removed across all experiments and subjects). We then compared the reaction times in different experimental conditions. The reported results are the mean and standard deviation reaction times. In four-category experiments, we removed reaction times longer than 1500 [*ms*]{} because in these tasks it could take longer time to press a key (only 8.7% of reaction times were removed across all experiments and subjects). Although the reaction times in four-category experiments might be a bit unreliable as subjects had to select one key out of four keys, they provided us with clues about the effect of variations across different dimensions on humans’ response time.
Deep convolutional neural networks (DCNNs)
------------------------------------------
DCNNs are a combination of deep learning [@schmidhuber2015deep] and convolutional neural networks [@LeCun1998]. DCNNs use a hierarchy of several consecutive feature detector layers. The complexity of features increases along the hierarchy. Neurons/units in higher convolutional layers are selective to complex objects or object parts. Convolution is the main process in each layer that is generally followed by complementary operations such as pooling and output normalization. Recent deep networks, which have exploited supervised gradient descend based learning algorithms, have achieved remarkable performances in recognizing extensively large and difficult object databases such as Imagenet [@lecun2015deep; @schmidhuber2015deep]. Here, we evaluated the performance of two of the most powerful DCNNs proposed by Krizhevsky et al. [@krizhevsky2012imagenet] and Simonyan and Zisserman [@simonyan2014very] in invariant object recognition. More explanations about these networks are provided as following:
- **Krizhevsky et al. (2012):** This model achieved an impressive performance in categorizing object images for Imagenet database and significantly defeated other competitors in the ILSVRC-2012 competition [@krizhevsky2012imagenet]. Briefly, the model contains five convolutional (feature detector) and three fully connected (classification) layers. The model uses Rectified Linear Units (ReLUs) as the activation function of neurons. This significantly sped up the learning phase. The max-pooling operation is performed in the first, second, and fifth convolutional layers. The model is trained using a stochastic gradient descent algorithm. This network has about 60 millions free parameters. To avoid overfitting during the learning procedure, some data augmentation techniques (enlarging the training set) and the dropout technique (in the first two fully-connected layers) were applied. Here, we used the pre-trained version of this model (on the Imagenet database) which is publicly available at <http://caffe.berkeleyvision.org> by Jia et. al [@jia2014caffe].
- **Very Deep (2014):**An important aspect of DCNNs is the number of internal layers, which influences their final performance. Simonyan and Zisserman [@simonyan2014very] studied the impact of the network depth by implementing deep convolutional networks with 11, 13, 16, and 19 layers. For this purpose, they used very small convolution filters in all layers, and steadily increased the depth of the network by adding more convolutional layers. Their results showed that the recognition accuracy increases by adding more layers and the 19-layer model significantly outperformed other DCNNs. Here, we used the 19-layer model which is freely available at <http://www.robots.ox.ac.uk/~vgg/research/very_deep/>.
Evaluation of DCNNs
-------------------
We evaluated the categorization accuracy of deep networks on natural background three- and one-dimension tasks. To this end, we first randomly selected 600 images from each object category, variation level, and variation condition (three- or one-dimension). Hence, we used 8 different image databases (4 variation levels $\times$ 2 variation conditions), each of which consisted of 2500 images (4 categories $\times$ 600 images). To compute the accuracy of each DCNN for given variation condition and level, we randomly selected two subsets of 1200 training (300 images per category) and 600 testing images (150 images per category) from the corresponding image database. We then fed the DCNN with the training and testing images and calculated the corresponding feature vectors of the last convolutional layer. Afterwards, we used these feature vectors to train the classifier and compute the categorization accuracy. Here we used a linear SVM classifier (libSVM implementation [@CC01a], [www.csie.ntu.edu.tw/\~cjlin/libsvm](www.csie.ntu.edu.tw/~cjlin/libsvm)) with optimized regularization parameters. This procedure was repeated for 15 times (with different randomly selected training and testing sets) and the average and standard deviation of the accuracy were computed. This procedure was done for both DCNNs and all variation conditions and levels.
To visualize the similarity between DCNNs and humans, we performed a Multidimensional Scaling (MDS) analysis across the variation levels of the three-dimension task. For each human subject or DCNN, we put together its accuracies over different variation conditions in a vector. Then we plotted the 2D MDS map based on the cosine similarities (distances) between these vectors.
![**Accuracy of subjects in rapid invariant object categorization task.** A. The accuracy of subjects in categorization of four object categories, when objects had uniform backgrounds. The dark, blue curve shows the accuracy when objects varied in all dimensions and the light, blue curve demonstrates the accuracy when objects varied in three dimensions. Error bars are the standard deviation (STD). P values, depicted at the top of curves, show whether the accuracy between all- and three-dimension experiment are significantly different (Wilcoxon rank sum test). Color-coded matrices, at the right, show whether changes in accuracy across levels statistically significant (Wilcoxon rank sum test; each matrix corresponds to one curve; see color of the frame). B. Categorization accuracy when objects had natural backgrounds.[]{data-label="figure_2"}](Figure2)
Results
=======
We ran different experiments in which subjects and DCNNs categorized object images varied across several dimensions (i.e., scale, position, in-plane and in-depth rotations, background). We measured the accuracies and reaction times of human subjects in different rapid and ultra-rapid invariant object categorization tasks, and the effect of variations across different dimensions on human performance was evaluated. The human accuracy was then compared with the accuracy of two well-known deep networks [@krizhevsky2012imagenet; @simonyan2014very] performing the same tasks as humans. We first report human results in different experiments and then compare them with the results of deep networks.
Human performance is dependent on the type of object variation
--------------------------------------------------------------
In these experiments, subjects were asked to accurately and quickly categorize rapidly presented object images of four categories (animal, car, motorcycle, and ship) appeared in uniform and natural backgrounds (see section \[Multi-class-psycho\]).
Figures \[figure\_2\].A and \[figure\_2\].B provide the average accuracy of subjects over different variation levels in all- and three-dimension conditions while objects had uniform and natural backgrounds, respectively. Figure \[figure\_2\].A shows that there is a small and negligible difference between the categorization accuracies in all- and three-dimension conditions with objects on uniform background. Also, for both experimental conditions, the categorization errors significantly increased at high variation levels (see the color-coded matrices in the right side of Figure \[figure\_2\].A). Despite the small, but significant, accuracy drop, this data shows that humans can robustly categorize object images when they have uniform background even at the highest variation levels (average accuracy above 90%). In addition, the reaction times in all- and three-dimension experiments were not significantly different (Figure S9.A).
Conversely, in the case of objects on natural backgrounds (Figure \[figure\_2\].B), the categorization accuracies in both experimental conditions substantially decreased as the variation level was increased (see the color-coded matrices in the right side of Figure \[figure\_2\].B; Wilcoxon rank sum test), pointing out the difficulty of invariant object recognition in clutter. Moreover, in contrast to the uniform background experiments, there is a large significant difference between the accuracies in all- and three-dimension experiments (see p values depicted at the top of Figure \[figure\_2\].B; Wilcoxon rank sum test). Overall, it is evident that excluding one dimension can considerably reduce the difficulty of the task. A similar trend can be seen in the reaction times (see Figure S9.B), where the reaction times in both conditions significantly increased as the variation level increased.
We then broke the trials into different conditions and calculated the mean accuracy in each condition (i.e., $ \Delta_{Sc}=0 $, $ \Delta_{Po}=0 $, $ \Delta_{RP}=0 $, $ \Delta_{RD}=0 $ ). Figure \[figure\_3\].A demonstrates the accuracies in all and three-dimension conditions, for the case of objects on uniform background. As seen, there is a small difference in the accuracies of different conditions at low and intermediate variation levels (level 0-2). However, at the highest variation level, the accuracy in $ \Delta_{RD}=0 $ (red curve) is significantly higher than the other conditions, suggesting that excluding in-depth rotation made the task very easy despite variations across other dimensions. Note that in $ \Delta_{RD}=0 $ the accuracy curve is virtually flat across levels with average of $\sim$95%. Interestingly, the accuracies were not significantly different between all-dimension experiment and $ \Delta_{Po}=0 $, $ \Delta_{Sc}=0 $, and $ \Delta_{RP}=0 $. This confirms that much of the task difficulty arises from in-depth rotation, although other dimensions have some weaker effects (e.g., scale, and rotation in-plane). This is also reflected in the bar plot in Figure \[figure\_3\].A as the absolute accuracy drop in $ \Delta_{RD}=0 $ is lower than 5%, while it is above 10% in $ \Delta_{Po}=0 $. It is also clear that humans had the maximum errors in $ \Delta_{Po}=0 $ condition, suggesting that removing position variation did not considerably affect the task difficulty (i.e., position is the easiest dimension).
The reaction times were compatible with the accuracy results (see Figure S10.A), where at the highest variation level, the human reaction times in $ \Delta_{Sc}=0 $, $ \Delta_{Po}=0 $, and $ \Delta_{RP}=0 $ significantly increased, while it did not significantly change in $ \Delta_{RD}=0 $. In other words, when objects were not rotated in-depth, humans could quickly and accurately categorize them.
In a separate experiment, subjects performed similar task while objects had natural backgrounds. Results show that there were small differences between the accuracies in all-dimension and three-dimension conditions at the first two variation levels (Figure \[figure\_3\].B). This suggests that human subjects could easily categorize object images on natural backgrounds while objects had small and intermediate degree of variations. However, accuracies became significantly different as the variation level increased (e.g., levels 2 and 3 see color-coded matrices in Figure \[figure\_3\].B). As shown in Figure \[figure\_3\].B, there is about 20% accuracy difference between $ \Delta_{RD}=0 $ and all-dimension condition at the most difficult level, confirming that the rotation in depth is a very difficult dimension. The bar plot in Figure \[figure\_3\].B, shows that the highest accuracy drop, between levels 0 and 3, belonged to $ \Delta_{Po}=0 $ and all-dimension conditions while the lowest drop was observed in $\Delta_{RD}=0$. In addition, the accuracies in $\Delta_{Sc}=0 $ and $\Delta_{RP}=0 $ fall somewhere between $\Delta_{Po}=0 $ and $\Delta_{RD}=0$, indicating that scale variations and in-plane rotation imposed more difficulty than variations in position; however, they were easier than rotation in depth. This is also evident in the accuracy drop.
Different objects have different three-dimensional properties; so, the categorization performance might be affected by these properties. In this case, one object category might bias the performance of humans in different variation conditions. To address this question, we broke the trials into different categories and calculated the accuracies (Figure S5) and reaction times (Figures S10.B and S11.B) for all variation and background conditions. The results indicated that although the categorization accuracy and reaction time may differ between categories, the order of the difficulty of different variation conditions are consistent in all categories. That is in-depth rotation and position transformation are respectively the most difficult and easy variations to process. We also calculated the confusion matrix of humans for each variation condition and level, to have a closer look at error rate and miscategorization across categories. The confusion matrices for uniform and natural background experiments are presented in Figure S6.
Analyses so far have provided information about the dependence of human accuracy and reaction time on the variations across different dimensions. However, one may ask how these results can be influenced by low-level image properties such as luminance and contrast. To address this, we computed the correlation between low-level image statistics (contrast and luminance) and the performance of human subjects. The results show that neither luminance (Figure S7) nor contrast (Figure S8) could explain human accuracy and reaction time in our invariant object recognition tasks.
We also performed similar two-category rapid tasks and their results are provided int Supplementary Information(Figure S1-S4). Interestingly, the results of two-category experiments are consistent with the four-category tasks, indicating that our results are robust to number of categories.
Human performance is independent of experimental setup
------------------------------------------------------
Although the effect of variations across different dimensions of an object on subjects’ performance was quite robust, we designed two other experiments to investigate how decreasing the presentation time would affect our results. Therefore, we reduced the image presentation time and the following blank screen from 25 [*ms*]{} to 12.5 [*ms*]{} (ultra-rapid object presentation). We also increased the time of the subsequent noise mask from 100 [*ms*]{} to 200 [*ms*]{}. In the first experiment, we repeated the natural background three-dimension categorization task with the ultra-rapid settings. We did not run uniform background condition because our results showed that this task would be easy and some ceiling effects may mask differences between conditions. For the second experiment, we studied the effect of each individual dimension (e.g., scale only) on the accuracy and reaction time of subjects. In the following, we report the results of these two experiments.
### Shorter presentation time does not affect human performance
Figure \[figure\_4\].A illustrates the results of the ultra-rapid object categorization task in three-dimension conditions with objects on natural backgrounds. Comparing the results in rapid (see Figure \[figure\_3\].B) and ultra-rapid experiments (see Figure \[figure\_4\].A, the left plot) indicates that there is no considerable difference between the accuracies in these two experiments. This shows the ability of human visual system to extract sufficient information for invariant object recognition even under ultra rapid presentation.
Similar to the rapid experiment, subjects had the highest categorization accuracy in $\Delta_{RD}=0$ condition, even at the most difficult level, with significant difference to other conditions (see the middle plot in Figure \[figure\_4\].A). However, there is a significant difference in accuracies ($\sim 10\%$) between $\Delta_{Sc}=0$ and $\Delta_{RP}=0$. In other words, tolerating scale variation seems to be more difficult than in-plane rotation in ultra-rapid presentation task. It suggests that it is easier to recognized a rotated object in plane than a small object. Comparing the accuracies in level 3 indicates that $\Delta_{RD}=0$ and $\Delta_{Sc}=0$ were the easiest tasks while $\Delta_{Po}=0$ and $\Delta_{RP}=0$ were the most difficult ones. Moreover, although there was no significant difference in reaction times of different conditions (Figure S12.A), subjects had shorter reaction times in $\Delta_{RD}=0$ at level 3 while the reaction times were longer in $\Delta_{Po}=0$ at this level.
Overall, the results of ultra-rapid experiment showed that different time setting did not change our initial results about the effect of variations across different dimensions, despite imposing higher task difficulty.
### Humans have a consistent behavior in the one-dimension experiment
In all experiments so far, object images varied across more than one dimension. In this experiment, we evaluated the performance of human subjects in ultra-rapid object categorization task while objects varied across a single dimension. Object images were presented on natural backgrounds. Figure \[figure\_4\].B illustrates that the accuracies were higher in $\Delta_{RP}$ and $\Delta_{Po}$ than in $\Delta_{RD}$ and $\Delta_{Sc}$ conditions. Hence, similar to results shown in Figure \[figure\_4\].A for three-dimension experiments, variations across position and in-plane rotation were easier to tolerate than in scale and in-depth rotation (again the most difficult). Subjects also had the highest accuracy drop between levels 0 and 3 in $\Delta_{RD}$ and $\Delta_{Sc}$ conditions while the accuracy drop in $\Delta_{RP}$ was significantly lower (bar plots in Figure \[figure\_4\].B).
The reaction times in different conditions are shown in Figure S12.B. Although the differences were not statistically significant, the absolute increase in reaction time in $\Delta_{Sc}$ and $\Delta_{RD}$ was higher than the other conditions, confirming that these variations needed more processing time (note that the results are average of five subjects, and increasing the number of subjects might lead to significant differences).
. \[figure\_6\]
DCNNs perform similarly to humans in different experiments
----------------------------------------------------------
We examined the performance of two powerful DCNNs on our three- and one-dimension databases with objects on natural backgrounds. We did not use gray background because it would be too easy. The first DCNN was the 8-layer network, introduced by Krizhevsky et al. [@krizhevsky2012imagenet], and the second was a 19-layer network, also known as Very Deep model, proposed by Simonyan and Zisserman [@simonyan2014very]. These networks achieved great performance on Imagenet as one of the most challenging current images databases.
Figures \[figure\_5\].A-D compares the accuracies of DCNNs with humans (for both rapid and ultra-rapid experiments) on different conditions of three-dimension database (i.e., $\Delta_{RP}=0$, $\Delta_{Po}=0$, $\Delta_{Sc}=0$, $\Delta_{RD}=0$). Interestingly, the overall trend in accuracies of DCNNs were very similar to humans in different variation conditions of both rapid and ultra-rapid experiments. However, DCNNs outperformed humans in different tasks. Despite significantly higher accuracies of both DCNNs compared to humans, DCNNs accuracies were significantly correlated with those of humans for both rapid (Figure \[figure\_5\].G-H) and ultra-rapid (Figure \[figure\_5\].I-J) experiments. In other words, deep networks can resemble human object recognition behavior in the face of different types of variation. Hence, if a variation is more difficult (easy) for humans, it is also more difficult (easy) for DCNNs.
We also compared the accuracy of DCNNs in different experimental conditions (Figure \[figure\_5\].E-F). Figure \[figure\_5\].E shows that the Krizhevsky network could easily tolerate variations in the first two levels (levels 0 and 1). However, the performance decreased at higher variation levels (levels 2 and 3). At the most difficult level (level 3), the accuracy of DCNNs were highest in $\Delta_{RD}=0$ while this significantly dropped to lowest accuracy in $\Delta_{Po}=0$. Also, accuracies were higher in $\Delta_{Sc}=0$ than $\Delta_{RP}=0$. Similar result was observed for Very Deep model with slightly higher accuracy (Figure \[figure\_5\].F).
We performed a MDS analysis to visualize the similarity between the accuracy patterns of DCNNs and all human subjects across variation levels (see Materials and methods). For this analysis, we used the rapid categorization data only (20 subjects), and not the ultra rapid one (5 subjects only, which is not sufficient for MDS). Figure \[figure\_6\] shows that the similarity between DCNNs and humans is high at the first two variation levels. In other words, there is no difference between humans and DCNNs in low variation levels and DCNNs treat different variations as humans. However, the distances between DCNNs and human subjects increased at the third level and became greater at the last level. This points to the fact that as the level of variation increases the task becomes difficult for both humans and DCNNs and the difference between them increases. Although DCNNs get further away from humans, it is not much greater than human inter-subject distances. Hence, it can be said that even in higher variation levels DCNNs perform similarly to humans. Moreover, the Very Deep network is closer to humans than the Krizhevsky model. This might be the result of exploiting more layers in Very Deep network which helps it to act more like humans.
To compare DCNNs with humans in the one-dimension experiment, we also evaluated the performance of DCNNs using one-dimension database with natural backgrounds (Figure \[figure\_7\]). Figure \[figure\_7\].A-D illustrates that DCNNs outperformed humans across all conditions and levels. The accuracy of DCNNs was about 100% at all levels. Despite this difference, we observed a significant correlation between the accuracies of DCNNs and humans (Figure \[figure\_7\].E-F), meaning that when a condition was difficult for humans it was also difficult for models.
To see how the accuracy of DCNNs depends on the dimension of variation, we re-plotted the accuracies of the models in different conditions (Figure \[figure\_7\].G-H). It is evident that both DCNNs performed perfectly in $\Delta_{Po}$, which is possibly inherent by their network design (the weight sharing mechanism in DCNNs [@kheradpisheh2015bio]), while they achieved relatively lower accuracies in $\Delta_{Sc}$ and $\Delta_{RD}$. Interestingly, these results are compatible with humans’ accuracy over different variation conditions of one-dimension psychophysics experiment (Figure \[figure\_4\]), where the accuracies of $\Delta_{Po}$ and $\Delta_{RP}$ were high and almost flat across the levels and the accuracies of $\Delta_{Sc}$ and $\Delta_{RD}$ were lower and significantly dropped in the highest variation level. It can be interpreted that, those variations which change the amount or the content of input visual information, such scaling and in-depth rotation, are much harder to handle (for both humans and DCNNs) than other types of variation such as position transformation and in-plane rotation.
Discussion
==========
Although it is well known that the human visual system can invariantly represent and recognize various objects, the underlying mechanisms are still mysterious. Most studies have used object images with very limited variations in different dimensions, presumably to decrease experiment and analysis complexity. Some studies investigated the effect of a few variations on neural and behavioral responses (e.g., scale and position [@brincat2004underlying; @hung2005fast; @zoccolan2007trade; @rust2010selectivity]). It was shown that different variations are differently treated trough the ventral visual pathway, for example responses to some variation (e.g., position) emerge before others (e.g., scale) [@isik2014dynamics]. However, there is not any data addressing this for other variations. Yet depending on the type of variation, the visual system may use different sources of information to handle rapid object recognition. Therefore, the responses to each variation, separately, or in different combinations can provide valuable insight about how the visual system performs invariant object recognition. Because DCNNs claim to be bio-inspired, it is also relevant to check if their performance, when facing these transformations, correlates with that of humans.
Here, we performed several behavioral experiments to study the processing of different object variation dimensions through the visual system in terms of reaction time and categorization accuracy. To this end, we generated a series of image databases consisting of different object categories which varied in different combinations of four major variation dimensions: position, scale, in-depth and in-plane rotations. These databases were divided into three major groups:1) objects that varied in all four dimensions; 2) object that varied in combination of three dimensions (all possible combinations); 3) objects that varied only in a single dimension. In addition, each database has two background conditions: uniform gray and natural. Hence, our image database has several advantages for studying the invariant object recognition. First, it contains a large number of object images, changing across different types of variation such as geometric dimensions, object instance, and background. Second, we had a precise control over the amount of variations in each dimension which let us generate images with different degrees of complexity/difficulty. Therefore, it enabled us to scrutinize the behavior of humans, while the complexity of object variations gradually increases. Third, by eliminating dependencies between objects and backgrounds, we were able to study invariance, independent of contextual effects.
Different combinations of object variations allowed us to investigate the role of each variation and combination in the task complexity and human performance. Interestingly, although different variations were linearly combined, the effects on reaction time and accuracy were not modulated in that way, suggesting that some dimensions substantially increased the task difficulty. The overall impression of our experimental results indicate that humans responded differently to different combination of variations, some variations imposed more difficulty and required more processing time. Also, reaction times and categorization accuracies indicated that natural backgrounds significantly affects invariant object recognition.
Results showed that 3D object rotation is the most difficult variation either in combination with others or by itself. In case of three-dimension experiments, subjects had high categorization accuracy when object were not rotated in-depth, while their accuracy significantly dropped in other three-dimension conditions. The situation was a similar for the reaction times: when the in-depth rotation was fixed across levels, the reaction time was shorter than the other conditions which objects were allowed to rotate in-depth. Although we expected that rotation in plane might be more difficult than scale, our results suggest the opposite. Possibly, changing the scale of the object might change the amount of information conveyed through the visual system which would affect the processing time and accuracy. Besides, the accuracy was very low when the objects were located on the center of the image but varied in other dimensions, while the accuracy was higher when we changed the object position and fixed any other dimensions. This suggests that subjects were better able to tolerate variations in objects’ position.
Moreover, we investigated whether these effects are related to low-level image features such as contrast and luminance. The results showed that the correlation between these features and reaction time and accuracy is very low and insignificant across all levels, type of variations, and objects. This suggests that although different variations affect the contrast and luminance, such low-level features have little effect on reaction time and accuracy.
We also performed ultra-rapid object categorization for the three-dimension databases with natural backgrounds, to see if our results depend on presentation condition or not. Moreover, to independently check the role of each individual dimension, we ran a one-dimension experiment in which objects were allowed to vary in only one dimension. These experiments confirmed the results of our previous experiments. Besides, we evaluated two powerful DCNNs over three- and one-dimension databases which surprisingly achieved similar results to those of humans. It suggests that humans have more difficulty for those variations which are computationally more difficult.
In addition to object transformations, background variation can also affect the categorization accuracy and time. Here, we observed that using natural images as object backgrounds seriously reduced the categorization accuracy and concurrently increased the reaction time. Importantly the backgrounds we used were quite irrelevant. We removed object-background dependency, to purely study the impacts of background on invariant object recognition. However, object-background dependency can be studied in future to investigate how contextual relevance between the target object and surrounding environment would affect the process of invariant object recognition (e.g., [@remy2013object; @bar2004visual; @harel2014task]).
Another limitation of our work, is that we did not assess the question of the extent to which previous experience is required for invariant recognition. Here, presumably both humans and DCNNs (through training) had extensive experience of the four classes we used (car, animal, ship, motorcycle) at different positions, scales, and with different viewing angles, and it is likely that this helped to develop the invariant responses. Importantly, studies have shown that difficult variations (rotation in depth) are solved in the brain later in development compared to easier ones [@nishimura2014size], suggesting that the brain needs more training to solve complex variations. It would be interesting to perform similar experiments as here with subjects of different ages, to unravel how invariance to different variations evolve through the development.
During the last decades, models have attained some scale and position invariant. However, attempts for building a model invariant to 3D variations has been marginally successful. In particular, recently developed deep neural networks has shown merits in tolerating 2D and 3D variations [@kheradpisheh2015deep; @cadieu2014deep; @ghodrati2014feedforward]. Certainly, comparing the responses of such models with humans (either behavioral or neural data) can give a better insight about their performance and structural characteristics. Hence, we did the same experiments on two of the best deep networks to see whether they treat different variations as humans do. It was previously shown that these networks can tolerate variations in similar order of the human feed-forward vision [@kheradpisheh2015deep; @cadieu2014deep]. Our results indicate that as humans they also have more difficulties with in-depth rotation and scale variation.
However, the human visual system extensively exploits feedback and recurrent information to refine and disambiguate the visual representation. Also, our vision is continues. Hence, the human visual system would have higher accuracies if it was allowed to use feedback information and continuous visual input. But deep networks lack such mechanism which could help them to increase their invariance and recognition ability. The future advances in deep networks should put more focus on feedback and continuous vision.
Finally, our results show that variation levels strongly modulate both humans’ and DCNNs’ recognition performances, especially for rotation in depth and scale. Therefore these variations should be controlled in all the image datasets used in vision research. Failure to do so may lead to noisy results, or even misleading ones. For example a category may appear easier to recognize than another one only because its variation levels happen to be small in a given dataset. We thus think that our methodology and image databases could be considered as benchmarks for investigating the power of any computational model in tolerating different object variations. Such results could then be compared with biological data (electrophysiology, fMRI, MEG, EEG) in terms of performance, but also representational dissimilarity [@Kriegeskorte2008]. It would help computational modelers to systematically evaluate their models in fully controlled invariant object recognition tasks, and could help them to improve the variation tolerance in their models, and to make them more human-like.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the High Performance Computing Center at Department of Computer Science of University of Tehran, for letting us perform our heavy and time-consuming calculations on their computing cluster.
[^1]: Corresponding author.\
Email addresses:\
kheradpisheh@ut.ac.ir (SRK),\
masoud.ghodrati@monash.edu (MGh)\
mgtabesh@ut.ac.ir (MG),\
timothee.masquelier@alum.mit.edu (TM).
|
---
abstract: 'We investigate the emission of single photons from CdSe/CdS dot-in-rods which are optically trapped in the focus of a deep parabolic mirror. Thanks to this mirror, we are able to image almost the full 4$\pi$ emission pattern of nanometer-sized elementary dipoles and verify the alignment of the rods within the optical trap. From the motional dynamics of the emitters in the trap we infer that the single-photon emission occurs from clusters comprising several emitters. We demonstrate the optical trapping of rod-shaped quantum emitters in a configuration suitable for efficiently coupling an ensemble of linear dipoles with the electromagnetic field in free space.'
author:
- Vsevolod Salakhutdinov
- Markus Sondermann
- Luigi Carbone
- Elisabeth Giacobino
- Alberto Bramati
- Gerd Leuchs
title: 'Single photons emitted by nano-crystals optically trapped in a deep parabolic mirror'
---
#### Introduction
Nano-scale light sources are used in many areas of research and applications. Examples range from the use of fluorescent nano-beads in microscopy to the role of various nanometer-sized solid-state systems as sources of single photons in future quantum technology applications. One approach to efficiently collect the emitted photons is to place the source at the focus of a parabolic mirror (PM) spanning a huge fraction of the full solid angle. Such a mirror has been used to study a laser cooled atomic ion held in a radio-frequency trap in vacuum [@maiwald2012]. More recently, the optical trapping of CdSe/CdS dot-in-rod particles in a deep PM in air has been demonstrated [@Salakhutdinov2016], and lately also the fabrication of a microscopic diamond PM around a nitrogen vacancy centre [@wan2018].
CdSe/CdS dot-in-rod (DR) particles have the remarkable properties of being single-photon emitters at room temperature as well as emitting light with a high degree of linear polarization [@Pisanello2010]. The latter property stems from the fact that their emission is dominated by a linear-dipole transition [@vezzoli2015]. In order to maximize the collection efficiency when using a PM, the axis of the linear dipole has to be aligned with the mirror’s optical axis [@Lindlein2007; @sondermann2015]. Once this is achieved, a PM with the geometry used in Refs. [@Lindlein2007; @sondermann2015] as well as here collimates 94% of the emission of a linear dipole, exceeding the capabilities of a realistic lens-based set-up by a factor of two. Moreover, the same requirement has to be fulfilled when reversing this scenario, i.e. when focusing a linear-dipole mode with the purpose of exciting a single quantum emitter with high efficiency [@Lindlein2007]. Any tilt of the quantization axis off the PM’s axis will lead to a reduced field amplitude along the transition dipole moment of the target.
For CdSe/CdS DR particles, the dipole-moment of the excitonic transition is found to be aligned along the c-axis of the rod [@vezzoli2015; @efros1996; @mathieuThesis]. Hence, the rod has to be aligned along the optical axis of the PM for maximizing the collection efficiency. This should occur in a natural way in an optical trap that is built by focusing a radially polarized beam as in Ref. [@Salakhutdinov2016]: The electric field in the focus of the PM is purely longitudinal, i.e. parallel to the PM axis. Since the polarizability of a rod is largest along its symmetry axis [@Stickler2016], the DR should thus be aligned along the optical axis of the PM. However, when trapping the particle under ambient conditions, this alignment is counteracted by random collisions with air molecules which exert a random torque on the DR (see, e.g. Refs. [@Hoang2016; @Kuhn2017]). Thus, the torque induced by the trapping laser onto the rod has to be larger than this random torque in order to achieve a good alignment of the rods.
The purpose of this work is two-fold: One aspect is checking the orientation of the DRs with respect to the optical axis of the PM. This is done by analyzing the spatially resolved pattern of the photons emitted by CdSe/CdS DR-particles. Since the rods’ alignment is governed by their geometry and independent of the small CdSe core, our results should be applicable for many types of rod-shaped quantum emitters. In this sense, the CdSe/CdS DR-particles can be considered as a proof model. We again emphasize that a favorable alignment of a rod-shaped quantum-system is a mandatory prerequisite for efficient photon collection as well as for efficient excitation.
The other aspect treated here is specific to the CdSe/CdS DR-particles. We will show that clusters of DRs can act as a source of single photons. As a matter of fact, it will turn out that indeed all single-photon sources observed in our experiment are constituted by clusters of emitters. The emission of single photons from clusters of DRs demonstrated here is in good agreement with recent observations of the same phenomenon for CdSe/CdS core-shell dots [@lv2018].
#### Experimental procedure
The CdS/CdSe DR-particles investigated here consist of a spherical CdSe core with $2.7\:\text{nm}$ diameter. The core is embedded in a CdS rod with a nominal length of $35\:\text{nm}$ and diameter of $7\:\text{nm}$. Fluorescent photons are emitted at wavelengths about $\lambda_\text{DR}=605\:\text{nm}$. The DRs are trapped at ambient conditions in air at the focus of a deep parabolic mirror in an optical dipole-trap (see Ref. [@Salakhutdinov2016] and Fig. \[fig:setup\] in appendix for details). The electric field at wavelength $\lambda_\text{trap}=1064\:\text{nm}$ in the focal region is polarized along the optical axis of the PM, which is here defined as the $z$-direction.
At the beginning of each experiment the power of the trapping laser is set to $360\:\text{mW}$ and the pulsed excitation beam is switched off. Successful trapping of DR particles is witnessed by the onset of a fluorescence signal and from scattered trap light. The fluorescence is detected with a pair of avalanche photo-diodes (APDs). At this stage, fluorescence photons stem from two-photon excitation at $\lambda_\text{trap}$ [@Salakhutdinov2016]. To determine the intensity auto-correlation function $g^{(2)}(t=0)$, the power of the trapping laser is reduced such that the count rate of photons on the APDs is hidden in the background noise. Then, more efficient pulsed excitation at $\lambda_\text{exc}=405\:\text{nm}$ is switched on and the value of $g^{(2)}(t=0)$ is measured.
In case that $g^{(2)}(0)<0.5$ is observed, i.e. the signature of single-photon emission, the experiment is continued as follows: Images of the spatial intensity distributions of fluorescence photons as found in the output aperture of the PM (see example in Fig. \[fig:fluorescence\]) are acquired with an electron-multiplying charge-coupled device (EMCCD) camera at several trapping beam powers. For each power setting we have averaged over several images, with an exposure time of $2\:\text{s}$ per image. Vertically and horizontally polarized components are recorded simultaneously on different parts of the camera chip.
Next, the trapping beam power is set again to $360\:\text{mW}$ and the motional dynamics of the particles in the trap is analyzed. The light scattered and Doppler-shifted by the particle interferes with the trapping light [@mestres2015]. The corresponding interference signal is measured with a balanced detector (cf. Ref. [@Salakhutdinov2016]). From the time series of this signal we compute the power-spectral density which contains several Lorentzian-shaped features, each of which corresponds to the motion of the particle in the trap along a certain direction. The width $\Gamma_i$ ($i=x,y,z$) of each of these peaks is proportional to the ratio $r_i/m$, where $r_i$ is the radius of the trapped object and $m$ its mass [@neukirch2015r]. Since the particles used here are non-spherical, we interpret $r_i$ as the radius of a disc that has the same area as the particle’s surface when viewed from the direction of motion (see sketch in Fig. \[fig:Pmin\]b). With $m$ being proportional to the volume of the trapped object, $\Gamma_i$ shrinks with increasing object size. Fitting a Lorentzian function to the power spectral density yields the axial damping rate $\Gamma_z$ of the particles’ motion [@Salakhutdinov2016]. The motion in radial direction is damped such strong that a corresponding spectral feature cannot be discerned.
Finally, the trapping beam power is gradually decreased to determine the power $P_\text{min}$ at which the particle is lost from the trap. The loss of the trapped particles was monitored via the signal of the balanced detector.
#### Single-photon emission from clusters
![\[fig:Pmin\] (a) Normalized second-order intensity correlation $g^{(2)}(0)$ of fluorescence photons emitted by DRs against power $P_{min}$ at which the particles escaped the trap. Error bars show the Poissonian uncertainty. Green circles denote experiments in which the spatial intensity pattern revealed azimuthal symmetry and an obvious alignment of the rods to the axis of the PM. A single such data point with $P_{min}=22\:\text{mW}$ and $g^{(2)}(0)=0.19$ is off scale. Orange squares correspond to cases where the emission pattern shows a clear azimuthal asymmetry. Black triangles mark experiments with an inconclusive outcome. The inset shows a magnified view for low power levels. (b) Damping rate of axial motion $\Gamma_z$ of trapped nano-particles against $P_\text{min}$. Error bars denote the 95% confidence interval of the fit from which $\Gamma_z$ was obtained. The dashed line denotes the function $\Gamma_z\propto
P_\text{min}^{0.48\pm0.03}$, with the power scaling obtained from a fit. The inset is a pictorial view of a cluster of rods aligned along the optical axis of the PM ($z$-axis). The yellow disc with radius $r_z$ denotes the effective area of the cluster relevant for damping the motion along the $z$-axis. ](DR_Pmin_combined_withSketch.pdf){width="8cm"}
In our experiments, we observed $g^{(2)}(0)<0.5$ for 33 different loads of the trap, see Fig. \[fig:Pmin\](a). The probability to trap a sample with this characteristic is approximately 10%. The observed values $0.15\le g^{(2)}(0)\le 0.44$ agree well with the ones observed for single DRs with comparable geometry in literature [@Vezzoli2013; @manceau2014]. Therefore, one could conclude that the trapped objects are indeed single DRs. But this conclusion is contradicted by the observation that all trapped particles, which appear as single-photon emitters, are lost at powers in the range $0.5\:\text{mW}<P_\text{min}<22\:\text{mW}$, with an average value of $2.5\pm2.1\:\text{mW}$, see histogram in Fig. \[fig:sizeHist\] in the appendix. As derived there, the power at which the trapping potential for a single DR equals $k_\text{B}T$ at ambient temperature is $P_\text{min}\approx41\:\text{mW}$. Therefore, a single DR that thermalizes to room temperature should be lost at about this power value. The volume and hence the polarizability of a trapped cluster consisting of several rods scales with the number of rods. The magnitude of $P_\text{min}$ consequently reduces by the same factor. Thus, one can infer that the smallest cluster displayed in Fig. \[fig:Pmin\] must contain at least four DRs and that the average number of rods in a cluster is $16\pm14$.
It is also evident from the same figure that two branches in the scattered data points are apparent. In one of these regions, which extends towards power values $P_\text{min}\lesssim1.5\:\text{mW}$, the data points are densely packed. We will refer to these samples as large clusters. At $P_\text{min}>1.5\:\text{mW}$ the data points are less dense. These samples will be called small clusters. The above threshold value of $P_\text{min}$ corresponds to approximately 27 rods. Further below we show that these two branches are linked to qualitative differences in the spatial intensity pattern of the emitted photons, influenced by the geometry of the clusters and their relative alignment to the PM axis. Independent of these differences, each branch has the property that $g^{(2)}(0)$ tends to increase with decreasing $P_\text{min}$ or increasing cluster size, respectively.
The trapping of clusters is furthermore confirmed by analyzing the damping rate of the particles’ motion inside the trap. Figure \[fig:Pmin\](b) shows the axial damping rate $\Gamma_z$ as a function of $P_\text{min}$. One observes a clear correlation of the two quantities with $\Gamma_z\propto \sqrt{P_\text{min}}$. Since $\Gamma_z$ increases with a decreasing size of the trapped object, we conclude that the variation in $P_\text{min}$ is due to the variation of the size and, therefore, to the polarizability of the trapped object. Furthermore, all observed damping rates are smaller than the one estimated for a single DR ($\Gamma_{1\text{DR}}/2\pi\ge
2\:\text{MHz}$, see appendix). Hence, the trapped emitters of different size must be clusters consisting of differing numbers of DRs.
Surprisingly, the trapping of DRs in clusters occurs despite the use of toluene as working solvent that should suppress the formation of clusters in the aerosol sprayed in the trapping region. Possibly, clusters might form inside the optical trap as soon as several single DRs diffuse in the focal region of the PM.
The observation of pronounced antibunching in the emission from a cluster of sources proves the emission of single photons. This is not to be expected at first sight, however, but in good agreement with findings in other experiments. In Ref. [@goodfellow2016] the quenching of the emission of CdSe/CdS core-shell quantum-dots by Förster energy-transfer to a $\text{MoSe}_2$ mono layer was demonstrated, which has earlier been also observed within assemblies of CdSe dots [@kagan1996; @crooker2002]. In principle, such an exchange could also occur between several DRs in a cluster provided that the spectra of excitonic transitions of the DRs constituting the cluster overlap. Recently, single-photon emission from clusters of CdSe/CdS core-shell dots has been attributed to Auger annihilation of excitons in neighboring nano-crystals [@lv2018]. For DRs trapped sufficiently close to each other the same mechanism might be responsible for single-photon emission.
#### Analysis of the fluorescence pattern
We now investigate the spatial intensity pattern of the photons emitted by the DRs. Since the polarizability of a rod is larger along its symmetry axis than perpendicular to it, one expects that a single rod or a small enough cluster of parallel rods is oriented along the longitudinal electric field in the focus of the PM. Under this assumption and with the dipole moment of the exciton aligned along the rod, the intensity distribution of the photons collimated by the PM has to be symmetric under rotation around the optical axis of the PM. In the majority of loads (26 out of 33) the measured distribution of the total intensity exhibits this symmetry.
![\[fig:fluorescence\] (a) Example of a spatial intensity pattern with rotational symmetry as observed in the aperture of the PM. The corresponding sample is the one with $P_\text{min}=2.9\:\text{mW}$ and $g^{(2)}(0)=0.31$ in Fig. \[fig:Pmin\]. From left to right the panels show the intensity when projected onto a vertical state of polarization, the projection onto a horizontal state of polarization, and the sum of the two, respectively. Dashed lines indicate the orientation of extinction lines as a guide to the eye. (b) Fraction of linear-dipole radiation in the fluorescence of small clusters of DRs as a function of trapping beam power. Values obtained in the same experimental run are connected by lines. The data points correspond to the ones depicted by circles in Figs. \[fig:Pmin\]. ](fluorescence_analysis.pdf){width="8cm"}
While a rotationally symmetric intensity pattern could in principle also be observed for an isotropic emitter, more insight can be gained from the polarization resolved emission pattern. For a linear dipole aligned parallel to the axis of the PM, the collimated emission is radially polarized [@Lindlein2007]. After projection onto a linear state of polarization, the intensity distribution of a radially polarized state must exhibit extinction lines perpendicular to the projected polarization. This property is illustrated in the example displayed in Fig. \[fig:fluorescence\](a). The observed extinction is not perfect, which is compatible with the existence of circular-dipole components in the DR emission.
The clusters which emit photons with a rotationally symmetric spatial distribution and which exhibit polarization patterns as described above are analyzed in further detail: We compute the azimuthal average of the total intensity distribution, resulting in a radial intensity profile [@maiwald2012]. Fitting a sum of the intensity distributions of linear and circular dipoles [@Sondermann2008] to these profiles reveals the *measured* fraction of linear-dipole radiation (see App. \[sec:dipole\]). This measured fraction will be lower than the one predicted by the properties of the excitonic transitions of the emitters due to the residual thermal vibration of the rods. The latter results in randomly fluctuating tilts of the rod off the PM’s axis. With tilts into all directions being equally likely, a linear dipole appears not as a pure linear dipole but is masked by a circular-dipole component in the long-time average. However, the thermal vibrations are counteracted by the longitudinal electric field of the trap. This field exerts a torque onto non-aligned rods and re-orients them along the PM axis. Therefore, a stronger suppression of random tilts should occur at larger field amplitudes, i.e. at larger trapping beam powers. One thus expects to observe a larger linear-dipole fraction at larger trapping beam powers.
Indeed, 16 samples exhibit such a behavior, as is evident from Fig. \[fig:fluorescence\](b). This confirms the alignment of the rods along the PM axis on average. Besides one exception, all samples (marked with circles in Figs. \[fig:Pmin\]) showing axial alignment were lost from the trap at $P_\text{min}>1.5\:\text{mW}$. These samples also constitute the group with the largest damping rates. Therefore, we conclude that the samples lost at $P_\text{min}>1.5\:\text{mW}$ are small clusters of DRs that are aligned along the axial trapping field. It should be noted that there is no correlation between the linear-dipole fraction measured at large trapping beam power and the power $P_\text{min}$ at which samples are lost from the trap. Alignment along the field of an optical trapping beam was previously observed also for silicon nanorods [@Kuhn2017] and nano-diamonds [@Hoang2016; @Geiselmann2013] as well as for nanorods trapped in a liquid [@ruijgrok2011; @head2012].
Most of the samples lost at low powers show no uniquely classifiable behavior in terms of emission pattern. This hints at emission from a large cluster without a pronounced structural order. Some of these large clusters show an emission pattern without rotational symmetry, i.e. with the dipole axis tilted off the PM’s optical axis. We conjecture that these cases correspond to emission from DRs that are attached to a large main cluster under a non-zero angle. However, once released from the trap the clusters are inevitably lost, hindering a posteriori investigations of the clusters’ structure.
A tempting explanation why the majority of aligned samples is found among the small clusters can be constructed from a geometric argument as follows: If the rods are aligned parallel to each other in closest packing, the width of the cluster equals the length of a single rod for a number of about 20 rods, resulting in an object of approximately equal size along all spatial dimensions and leading to the loss of pronounced directionality. In the experiments the apparent change from the alignment of clusters along the PM axis to no preferential alignment occurs at a number of about 27 rods. This is in reasonable agreement with the number inferred just above. But the fact that some of the large clusters exhibit radiation patterns lacking rotational symmetry hints at a more complicated scenario.
Finally, it would be interesting to compare the measured count rates of photons emitted by the DRs to the ones expected for our set-up, in particular for the samples exhibiting alignment. However, a reliable quantitative comparison is hindered by the fact that the investigated samples exhibit blinking, as detailed in App. \[sec:countRates\].
#### Concluding discussion
We have analyzed the fluorescence photons emitted by CdSe/CdS DRs which are optically trapped at the focus of a deep PM. The observation of single-photon emission from clusters of sources will be advantageous for applications, since clusters can be kept in the trap at trapping beam powers that are lower than necessary for single emitters. Moreover, for low trapping beam powers the excitation of the DRs via two-photon absorption [@Salakhutdinov2016] can be strongly suppressed, which promises a better timing accuracy for the emission of photons in a pulsed excitation scheme.
On the one hand, it could be possibly argued that the trapping of clusters of DRs, i.e. the fact that so far not a single isolated DR was trapped, might be detrimental to the efficient coupling of light to these emitters. On the other hand, this appears as a favorable condition to possibly promote the enhancement of the coupling due to collective effects when trapping a cluster of quantum targets. Also in cavity quantum-electrodynamics collective effects have been used advantageously [@thompson1992; @lambrecht1996; @brennecke2007; @colombe2007; @herskind2009]. There, large fractions of the ensemble can be hosted in the cavity mode [@brennecke2007; @colombe2007]. When coupling light and ensembles in free-space by tight focusing, the ensemble has to be located within the focal spot, which is of dimensions of the order of half a wavelength under optimum conditions [@gonoskov2012; @alber2017]. This is very difficult to achieve with ions due to Coulomb repulsion. Therefore, locating small clusters of optically trapped solid-state quantum targets in the focus of a linear-dipole wave is a feasible path towards combining tight focusing and ensemble physics.
The authors thank M.Manceau for fruitful discussions. G.L. acknowledges financial support from the European Research Council (ERC) via the Advanced Grant ’PACART’.
[39]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase 10.1103/PhysRevA.86.043431) [****, ()](\doibase
10.1364/optica.3.001181) [****, ()](\doibase 10.1021/acs.nanolett.7b04684) [****, ()](\doibase
10.1063/1.3291849) [****, ()](\doibase 10.1021/acsnano.5b01354) [****, ()](\doibase
10.1134/S1054660X07070055) in @noop [**]{}, , (, ) pp. , [****, ()](\doibase 10.1103/PhysRevB.54.4843) **, [Ph.D. thesis](https://scholar.google.com/scholar_lookup?hl=en&publication_year=2014&author=M.+Manceau&title=Single+CdSe%2FCdS+dot-in-rods+fluorescence+properties), () [****, ()](\doibase
10.1103/physreva.94.033818) [****, ()](\doibase 10.1103/physrevlett.117.123604) [****, ()](\doibase
10.1364/optica.4.000356) [****, ()](\doibase 10.1038/s41467-018-03971-w) [****, ()](\doibase
10.1063/1.4933180) [****, ()](\doibase 10.1080/00107514.2014.969492) [****, ()](\doibase 10.1016/j.optcom.2013.03.020) [****, ()](\doibase 10.1103/PhysRevB.90.035311) [****, ()](\doibase 10.1063/1.4939845) [****, ()](\doibase
10.1103/PhysRevLett.76.1517) [****, ()](\doibase 10.1103/PhysRevLett.89.186802) [ ()](https://arxiv.org/abs/0811.2098), [****, ()](\doibase
10.1038/nnano.2012.259) [****, ()](\doibase 10.1103/PhysRevLett.107.037401) [****, ()](\doibase
10.1039/C2NR30515A) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevA.86.053836) [****, ()](\doibase
10.1186/s41476-017-0043-y), [****, ()](\doibase 10.1121/1.1909020) [****, ()](\doibase 10.1007/s00348-002-0456-1) [****, ()](\doibase
10.1021/nl0717661) [****, ()](\doibase
10.1103/PhysRevLett.102.136801) [****, ()](\doibase
10.1002/adma.201203171) [****, ()](\doibase
10.1038/nature10569) [****, ()](\doibase 10.1103/PhysRevLett.113.251801) [****, ()](\doibase
10.1103/PhysRevA.95.061801)
Experimental set-up {#app:setup}
===================
Figure \[fig:setup\] shows a schematic of the experimental set-up. Nano-particles are trapped at ambient conditions in air at the focus of a deep parabolic mirror (PM). The PM has a focal length of $2.1\:\text{mm}$ and an aperture radius of $10\:\text{mm}$, which corresponds to a half-opening angle of $134^\circ$. A bore hole with $1.5\:\text{mm}$ diameter at the vertex of the PM allows for the delivery and the optical excitation of the nanoparticles. Based on these geometrical boundary conditions the collection efficiency for photons in a linear-dipole mode (circular-dipole mode) cannot exceed $94\%$ ($76\%$), where the quantization axis is defined to coincide with the optical axis of the PM. The reflectivity of the PM was determined to be $72\%$.
The trapping beam is radially polarized and has the intensity pattern of a Laguerre-Gaussian mode of first radial order (doughnut mode). The electric field at wavelength $\lambda_\text{trap}=1064\:\text{nm}$ in the focal region is polarized along the optical axis of the PM, which is here defined as the $z$-direction.
![\[fig:setup\] Scheme of the experimental set-up. CdSe/CdS dot-in-rod (DR) particles are trapped in an optical tweezers by focusing a radially polarized infrared laser beam (doughnut beam) with a parabolic mirror (PM). The DRs are optically excited by a blue pulsed laser (pump). Fluorescent photons are collimated by the PM and either detected by two avalanche photo-diodes (APDs), which are separated by a beam splitter (BS), or an electron-multiplying charge-coupled device (EMCCD) camera. A polarizing beam splitter (PBS) in front of the EMCCD camera projects the photons onto horizontally and vertically polarized states measured by different sections of the camera chip. The motion of the DRs is characterized by detection of the light of the trap laser that is scattered by the DRs and interferes with trap light back-reflected by the PM. This signal is balanced with light that is tapped off the trap laser beam and adjusted in amplitude by a neutral-density filter (NDF). ](setup.pdf){width="8cm"}
The CdS/CdSe dot-in-rod (DR) particles are surrounded by alkyl chains stored in toluene, hindering the clustering of particles. We use a home-built, toluene-resistant nebulizer for delivering DR particles to the trapping region through an opening at the vertex of the PM. The nebulizer comprises a piezo-electric transducer that is driven with an alternating voltage oscillating at about $360 - 390\:\text{kHz}$. For these frequencies one expects an aerosol of toluene droplets with a mean diameter of $6\:\mu\text{m}$ [@Lang1962; @Barreras2002]. The nebulizer is filled with toluene containing DR particles in concentrations ranging from $6\times10^{4}$ to $6\times10^{8}\,\text{DRs}/\text{l}$. We thus expect less than $10^{-4}$ DRs in a single droplet of about 100fl volume on average.
Light scattered and Doppler-shifted by the particle interferes with the trapping light [@mestres2015]. The corresponding interference signal is measured with a balanced detector (cf. Fig. \[fig:setup\] and Ref. [@Salakhutdinov2016]). The balancing is achieved with light that is tapped off the trap laser beam and adjusted in amplitude by a neutral-density filter (NDF).
Trapped DR particles are optically excited at a wavelength $\lambda_\text{exc}=405\:\text{nm}$ with pulses of $82\:\text{ns}$ duration and $2\:\mu\text{W}$ average power. The pulse repetition rate is $10^6\:\text{s}^{-1}$. The excitation pulse power is chosen such that the DRs are approximately saturated, i.e. on average one exciton is created per excitation pulse [@Vezzoli2013]. The DRs emit photons at wavelengths about $\lambda_\text{DR}=605\:\text{nm}$, which are picked off the excitation and trapping beam path by a reflective short-pass filter that attenuates light at $\lambda_\text{trap}$ in reflection. The fluorescence photons are analyzed by either a pair of APDs with $69\%$ quantum efficiency for determining the second-order intensity correlation function $g^{(2)}(t)$ or by an EMCCD camera for measuring the spatial intensity distribution as found in the output aperture of the PM. This aperture is imaged onto the camera by a demagnifying telescope. Vertically and horizontally polarized components are split by a polarizing beam splitter (PBS) and recorded simultaneously on different parts of the camera chip. The APDs as well as the camera are equipped with band-pass filters which reject light at $\lambda_\text{exc}$ and at $\lambda_\text{trap}$. Not accounting for the reflectivity of the PM as well as the APDs’ quantum efficiency, the transmission of the set-up was measured to be $83\%$.
Minimum trapping beam power {#app:trap}
===========================
The depth of the trapping potential of a dielectric particle in the Rayleigh approximation is given by $U_0=\alpha/2\cdot E_\text{max}^2$, where $\alpha$ is the polarizability of the particle and $E_\text{max}$ is the maximum amplitude of the electric field in the focal region. $E_\text{max}^2$ is proportional to the power of the focused trapping beam. The derivation of $E_\text{max}$ for our particular set-up can be found in Ref. [@Salakhutdinov2016]. Taking the polarizability $\alpha$ to be the one of a CdS rod aligned to the electric field, one has [@Kuhn2017] $$\label{eq:alpha}
\alpha=\epsilon_0 V_\text{rod}(n_\text{CdS}^2-1)$$ with $n_\text{CdS}=2.34$ the refractive index of CdS at $\lambda_\text{trap}$ and $V_\text{rod}$ the volume of the rod. Setting $U_0=k_\text{B}T$ with $T=296\:\text{K}$ yields $P_\text{min}=41\:\text{mW}$.
![\[fig:sizeHist\] (a) Distribution of minimum trapping beam powers $P_\text{min}$ at which samples escaped the trap. (b) Distribution of damping rates $\Gamma_z$ of the axial motion of the samples trapped in the experiment. The histograms have been computed for the data shown in Fig. \[fig:Pmin\] in the main text. ](Pmin_gamma_stats.pdf)
The distribution of powers $P_\text{min}$ at which samples with $g^{(2)}(0)<0.5$ escaped the trap is shown in Fig. \[fig:sizeHist\]. The average value (standard deviation) is $2.5(2.1)\:\text{mW}$. This corresponds to an average number of $16(14)$ rods in a trapped cluster.
Damping of axial motion
=======================
For a spherical particle in air, the damping of its oscillatory motion is given by [@neukirch2015r] $$\label{eq:damping}
\Gamma=\frac{6\pi\eta r}{m}
\cdot\frac{0.619}{0.619+\text{Kn}}(1+c_\text{K})\quad ,$$ with $\eta$ the viscosity of air, $r$ the sphere’s radius and $m$ its mass. $\text{Kn}=\Lambda/r$ is the Knudsen number with $\Lambda$ the mean free-path length of an air molecule and $c_\text{K}=0.31\text{Kn}/(0.785 + 1.152\text{Kn} + \text{Kn}^2)$.
Here, we interpret $r$ as the radius of the area of the particle’s surface as seen along the direction of motion. Hence, for a rod aligned parallel to the optical axis of the PM $r$ is identified with the radius of the rod. Due to the attached alkyl chains, which have a length of about $1.6\:\text{nm}$, the *effective* radius of the rod is increased, resulting in $\Gamma_\text{1DR}/2\pi\approx
2\:\text{MHz}$. Furthermore, due to residual vibration the trapped particles’ effective radius is enlarged when averaging over the time constants of translational motion. The distribution of damping rates observed in the experiments for samples with $g^{(2)}(0)<0.5$ is shown in Fig. \[fig:sizeHist\]. The average value (standard deviation) of $\Gamma_z/2\pi$ is $0.62(0.29)\:\text{MHz}$.
Dipole radiation patterns {#sec:dipole}
=========================
The analysis of the spatial intensity pattern of the photons emitted by the DRs is based on determining the relative fraction of linear- and circular-dipole radiation. We choose the quantization axis to coincide with the optical axis of the PM. Then, upon collimation by the PM the intensity distribution of a linear dipole reads [@Lindlein2007; @Sondermann2008] $$I_\pi(R)=I_{0,\pi}\cdot\frac{R^2}{(R^2/4+1)^4}$$ with $I_{0,\pi}$ a proportionality constant and $R$ the radial distance to the optical axis of the PM in units of the mirror’s focal length. The corresponding expression for a circular dipole reads $$I_\sigma(R)=I_{0,\sigma}\cdot\frac{R^4/16 + 1}{(R^2/4+1)^4}
\quad .$$ The sum of these two distributions is fit to radial intensity profiles which are obtained by computing the azimuthal average of the images acquired with the EMCCD camera. The resulting fraction of linear-dipole radiation is then given by $$A_\pi=\frac{I_{0,\pi}}{I_{0,\pi}+I_{0,\sigma}} \quad .$$
Photon count rates {#sec:countRates}
==================
In what follows, we first derive the photon count rate expected for the CdSe/CdS DRs and then compare to the ones observed in the experiment. One has to account for several characteristics of the investigated emitters. The number of electron-hole pairs created upon optical excitation with power $P$ is given by $P/P_\text{sat}$, where $P_\text{sat}$ is the saturation power defined as the power at which one exciton-hole pair is created on average [@Vezzoli2013]. The number of emitted photons is proportional to $(1-\exp(-P/P_\text{sat}))$. We have determined $P_\text{sat}$ from saturation measurements [@Vezzoli2013] using large ensembles of rods, i.e. clusters with intensity correlation $g^{(2)}(0)\approx 1$. The resulting value is $P_\text{sat}=2.63\pm0.43\:\mu\text{W}$. The used excitation power in all our experiments was $P=2\:\mu\text{W}$. Furthermore, the DRs have a non-unit quantum yield. The latter was previously determined to be about $Q_\text{DR}=0.7$ [@carbone2007].
In the experiment the photons are collected upon pulsed excitation from the auxiliary laser at a rate $\gamma_\text{exc}=10^6/\text{s}$ and at strongly reduced trapping beam power. The latter condition results in an ineffective suppression of rod vibrations, such that the *measured* radiation pattern contains a fraction of linear-dipole radiation of $A_\pi=0.31\pm0.03$ (cf. Fig. \[fig:fluorescence\] in main text). The collection efficiency of the used PM itself is $94\%$ for a linear dipole and $76\%$ for circular dipole radiation.
Taking into account the APD’s quantum efficiency $Q_\text{APD}$, the reflectivity $R_\text{PM}$ of the PM, and the transmission $T$ through the setup (all specified in Sec. \[app:setup\]), the expected photon count rate is $$\gamma_\text{exc}\cdot Q_\text{APD} \cdot T \cdot R_\text{PM}\cdot [1-\exp(-P/P_\text{sat})]
\cdot Q_\text{DR} \cdot [0.94\times A_\pi +0.76 \times (1-A_\pi)] =
(125\pm 14)\times 10^3\,\text{s}^{-1}\quad .$$
![\[fig:counts\] Photon count rates of trapped clusters as a function of power $P_\text{min}$ at which samples escaped the trap. The count rates correspond to the average rate of a grey state obtained by fitting a gaussian distribution to count rate histograms. The errorbars represent the rms width of the gaussian distribution. The meaning of different symbols is the same as in Fig. \[fig:Pmin\] in the main text. ](count_rates_vs_Pmin.pdf)
All samples investigated in the experiment show a blinking behavior which is well-known in literature. Therefore, for the analysis of the photon detection events we apply established methods accounting for this effect [@spinicelli2009; @pisanello2013; @manceau2014]. The time-resolved count rate of detected photons is obtained by binning the time-tagged detection events into a continuous series of intervals with a constant width of $500\,\mu\text{s}$. From the resulting time series the distribution of count rates is computed. For some samples, the time series is dominated by intermittent bursts from a low light level. The corresponding histograms show an exponentially decaying distribution of count rates. For 22 samples the distribution of count rates exhibits a pronounced local maximum at nonzero detection rates. The average count rates corresponding to this local maximum are reported in Fig. \[fig:counts\]. The observed rates are considerably below the expected rate.
However, the observed count rates are strongly fluctuating from sample to sample. The size of the cluster, i.e. the value of $P_\text{min}$ has no obvious influence onto the rate of detected photons. One might argue that there is a slight tendency of increasing count rate for increasing $P_\text{min}$ evident from Fig. \[fig:counts\] for small clusters with $P_\text{min}>1.5\,\text{mW}$, i.e. the aligned samples. But the corresponding count rates are of comparable magnitude to the rates observed for the large samples. Furthermore, the photon emission properties of all samples are investigated at low trapping beam powers, at which the alignment of the small clusters is weakest anyhow. There is also no evident correlation of count rate and the value of $g^{(2)}(0)$ (data not shown).
These observations indicate that the photon count rates are related to the photo-physics of the DRs and not the properties of the setup or the geomnetry of the trapped clusters. Furthermore, the blinking behaviour is typically characterized by distributions exhibiting two pronounced maxima at nonzero count rates (cf. e.g. Refs. [@spinicelli2009; @manceau2014]), where the larger rate corresponds to a so-called bright-state and the lower one to a so-called grey state which is related to the presence of excess charges [@galland2011]. None of the samples investigated here exhibits two pronounced states. Due to the low photon count rates we associate the rates given in Fig. \[fig:counts\] with a grey state. In addition, the emission from a grey state is typically a factor 2 to 6 weaker than from a bright state [@spinicelli2009; @manceau2014]. When applying this correction factor, many samples would exhibit an extrapolated bright state with count rates in reasonable agreement with the expected one. We attribute the dominance of emission from a grey state to particulary strong charging of the optically trapped samples. A potential origin of the charging is the process of spraying the aerosol containing the nano-particles into the trapping region. For example, also silica nano-spheres are typically charged after spraying [@moore2014; @frimmer2017].
|
---
abstract: 'Intuitively, if two strings $S_1$ and $S_2$ are sufficiently similar and we already have an FM-index for $S_1$ then, by storing a little extra information, we should be able to reuse parts of that index in an FM-index for $S_2$. We formalize this intuition and show that it can lead to significant space savings in practice, as well as to some interesting theoretical problems.'
bibliography:
- 'relative.bib'
title: 'Relative FM-indexes'
---
Introduction {#sec:introduction}
============
FM-indexes [@FM05] are core components in most modern DNA aligners (e.g., [@LTPS09; @LD09; @LYLLYKW09]) and have thus played an important role in the genomics revolution. Medical researchers are now producing databases of hundreds or even thousands of human genomes, so bioinformatics researchers are working to improve FM-indexes’ compression of sets of nearly duplicate strings. As far as we know, however, the solutions proposed so far (e.g., [@FGHP14; @MNSV10]) index the concatenation of the genomes, so we can search the whole database easily but searching only in one specified genome is more difficult. In this paper we consider how to index each of the genomes individually while still using reasonable space and query time.
Our intuition is that if two strings $S_1$ and $S_2$ are sufficiently similar and we already have an FM-index for $S_1$ then, by storing a little extra information, we should be able to reuse parts of that index in an FM-index for $S_2$. More specifically, it seems $S_1$’s and $S_2$’s Burrows-Wheelers Transforms [@BW94] (BWTs) should also be fairly similar. Since BWTs are the main component of FM-indexes, it is natural to try to take advantage of such similarity to build an index for $S_2$ that “reuses” information already available in $S_1$’s FM-index.
Among the many possible similarities one can find and exploit in the BWTs, in this paper we consider the longest common subsequence (LCS). The BWT sorts the characters of a string into the lexicographic order of the suffixes following those characters. For example, if $$S_1 = \mathsf{AAGTTGAGAGTGAGT},\qquad
S_2 = \mathsf{AGAGAGTCGAAGTT};$$ then $${\ensuremath{\mathsf{BWT}}}(S_1) = \mathsf{TGGGATTAAAAGTGG},\qquad
{\ensuremath{\mathsf{BWT}}}(S_2) = \mathsf{TGGGATCAAAATGG};$$ whose LCS [TGGGATAAAATGG]{} is nearly as long as either BWT. Note that in this example ${\ensuremath{\mathsf{LCS}}}(S_1,S_2)=\mathsf{AGAGAGTGAGT}$ is shorter than ${\ensuremath{\mathsf{LCS}}}({\ensuremath{\mathsf{BWT}}}(S_1),{\ensuremath{\mathsf{BWT}}}(S_2))$.
We introduce the concept of [*BW-distance*]{} ${\ensuremath{\mathsf{BWD}}}(S_1, S_2)$ between $S_1$ and $S_2$ defined as $|S_1| + |S_2| - 2 |{\ensuremath{\mathsf{LCS}}}({\ensuremath{\mathsf{BWT}}}(S_1),{\ensuremath{\mathsf{BWT}}}(S_2))|$. Note that this coincides with the edit distance between ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ when only insertions and deletions are allowed. We prove that, if we are willing to tolerate a slight increase in query times, we can build an index for $S_2$ using an unmodified FM-index for $S_1$ and additional data structures whose total space in words is asymptotically bounded by ${\ensuremath{\mathsf{BWD}}}(S_1, S_2)$ (Theorem \[theo:counting\]).
This first result is the starting point for our investigation as it generates many challenging issues. First, since we are interested in indexing whole genomes, we observe that finding the LCS of strings whose length is of the order of billions is outside the capabilities of most computers. Thus, in Section \[sec:simon&jouni\] we show how to approximate the LCS of two BWTs, using combinatorial properties of the BWT to align the sequences. In the same section we also discuss and test several practical alternatives for building the index for $S_2$ given the one for $S_1$ and we analyze their time/space trade-offs.
If one needs an index not only for counting queries but also for locating and extracting, we must enrich it with suffix array (SA) samples. Such samples usually take significantly less space than the main index. However, we may still want to take advantage of the similarities between $S_1$ and $S_2$ to “reuse” SA samples from $S_1$ for $S_2$’s index. In Section \[sec:djamal&giovanni\] we show that this is indeed possible if, instead of considering the LCS between the BWTs, we use a common subsequence with the additional constraint of being [*BWT-invariant*]{} (Theorem \[theo:locating\]). This result motivates the problem of finding the longest BWT-invariant subsequence, which unfortunately turns out to be NP-hard (Theorem \[theo:np\]). We therefore devise a heuristic to find a “long” BWT-invariant subsequence in ${\ensuremath{\mathcal{O}\!\left( {|S_1|\log|S_1|} \right)}}$ time.
We have tested our approach in practice by building an FM-index for the genome of a Han Chinese individual, “reusing” an FM-index of the human reference genome. The Han genome is about 3.0 billion base pairs, the reference is about 3.1 billion base pairs and we found a common subsequence of about 2.9 billion base pairs. A standard implementation of a stand-alone FM-index for the Han genome takes 628 MB or 1090 MB, depending on encoding, while our index uses only 256 MB or 288 MB on top of the index for the reference. On the other hand, queries to our index take about 9.5 or 4.5 times longer. Since our index is compressed relative to the underlying index for the reference, we call it a relative FM-index.
Review of the FM-index structure {#sec:review}
================================
The core component of an FM-index for a string $S [1..n]$ is a data structure supporting rank queries on the Burrows-Wheeler Transform ${\ensuremath{\mathsf{BWT}}}(S)$ of $S$. This transform permutes the characters in $S$ such that $S [i]$ comes before $S [j]$ in ${\ensuremath{\mathsf{BWT}}}(S)$ if $S [i + 1..n]$ is lexicographically less than $S [j + 1..n]$.
If the lexicographic range of suffixes of $S$ starting with $\beta$ is $[i..j]$, then the range of suffixes starting with $a \beta$ is $$\begin{gathered}
\left[ {\ensuremath{\mathsf{BWT}}}(S).{\ensuremath{\mathsf{rank}}}_a (i - 1) + 1 + \sum_{a' \prec a} S.{\ensuremath{\mathsf{rank}}}_{a'} (n).. \right.\\[-2ex]
\vspace{-1ex} \left. {\ensuremath{\mathsf{BWT}}}(S).{\ensuremath{\mathsf{rank}}}_a (j) + \sum_{a' \prec a} S.{\ensuremath{\mathsf{rank}}}_{a'} (n) \right]\end{gathered}$$ It follows that, if we have precomputed an array storing $\sum_{a' \prec a}
S.{\ensuremath{\mathsf{rank}}}_{a'} (n)$ for each distinct character $a$ (i.e., the number of characters in $S$ less than $a$), then we can find the range of suffixes starting with a pattern $P [1..m]$ — and, thus, count its occurrences — using ${\ensuremath{\mathcal{O}\!\left( {m} \right)}}$ rank queries.
If the position of $S [i]$ in ${\ensuremath{\mathsf{BWT}}}(S)$ is $j$, then the position of $S
[i - 1]$ is $${\ensuremath{\mathsf{BWT}}}(S).{\ensuremath{\mathsf{rank}}}_{S [i]} (j) + \sum_{a \prec S [i]} {\ensuremath{\mathsf{BWT}}}(S).{\ensuremath{\mathsf{rank}}}_{a} (n)\,.$$ It follows that, if we have also precomputed a dictionary storing the position of every $r$th character of $S$ in ${\ensuremath{\mathsf{BWT}}}(S)$ with its position in $S$ as satellite information, then we can find a character’s position in $S$ from its position in ${\ensuremath{\mathsf{BWT}}}(S)$ using ${\ensuremath{\mathcal{O}\!\left( {r} \right)}}$ rank and membership queries. Therefore, once we know the lexicographic range of suffixes starting with $P$, we can locate each of its occurrences using ${\ensuremath{\mathcal{O}\!\left( {r} \right)}}$ rank queries.
Finally, if we have also precomputed an array storing the position of every $r$th character of $S$ in ${\ensuremath{\mathsf{BWT}}}(S)$, in order of appearance in $S$, then given $i$ and $j$, we can extract $S [i..j]$ using ${\ensuremath{\mathcal{O}\!\left( {r + j - i} \right)}}$ rank queries.
BW-distance and relative FM-indices {#sec:BWdiv}
===================================
Given two strings $S_1 [1..n_1]$ and $S_2 [1..n_2]$ we define the [*BW-distance*]{} ${\ensuremath{\mathsf{BWD}}}(S_1, S_2)$ between $S_1$ and $S_2$ as $$\label{eq:BWDdef}
{\ensuremath{\mathsf{BWD}}}(S_1, S_2) = n_1 + n_2 - 2 |{\ensuremath{\mathsf{LCS}}}({\ensuremath{\mathsf{BWT}}}(S_1),{\ensuremath{\mathsf{BWT}}}(S_2))|.$$ Note that the BW-distance is nothing but the edit distance between ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ when only insertions and deletions are allowed [@Myers86] (also known as the shortest edit script or indel distance), and is thus at most twice their normal edit distance. We now show how to support counting queries on $S_2$ using an FM-index for $S_1$ and some auxiliary data structures taking ${\ensuremath{\mathcal{O}\!\left( {{\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ words of space. Specifically, we consider how we can support rank queries on ${\ensuremath{\mathsf{BWT}}}(S_2)$ and partial-sum queries on the distinct characters’ frequencies.
Let $C$ denote a LCS of ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ with $|C|=m$. Let $C=c_1
\cdots c_m$, and for $i=1,\ldots,m$, let $\alpha_i$ (resp. $\beta_i$) be the position of $c_i$ in ${\ensuremath{\mathsf{BWT}}}(S_1)$ (resp. ${\ensuremath{\mathsf{BWT}}}(S_s)$) with $\alpha_1 < \cdots
< \alpha_{m}$ (resp. $\beta_1 < \cdots < \beta_{m})$. Define
- bitvector $B_1 [1..n_1]$ with 0s in positions $\alpha_1, \ldots,
\alpha_m$,
- bitvector $B_2 [1..n_2]$ with 0s in positions of $\beta_1,
\ldots, \beta_m$,
- subsequence $D_1$ of ${\ensuremath{\mathsf{BWT}}}(S_1)$ marked by 1s in $B_1$; $D_1$ is the complement of $C$ in ${\ensuremath{\mathsf{BWT}}}(S_1)$,
- subsequence $D_2$ of ${\ensuremath{\mathsf{BWT}}}(S_2)$ marked by 1s in $B_2$; $D_2$ is the complement of $C$ in ${\ensuremath{\mathsf{BWT}}}(S_2)$.
We claim that if we can support fast [$\mathsf{rank}$]{} queries on ${\ensuremath{\mathsf{BWT}}}(S_1)$, $B_1$, $B_2$, $D_1$ and $D_2$ and fast ${\ensuremath{\mathsf{select}}}_0$ queries on $B_1$, then we can support fast [$\mathsf{rank}$]{} queries on ${\ensuremath{\mathsf{BWT}}}(S_2)$. To see why, notice that $$\begin{aligned}
{\ensuremath{\mathsf{BWT}}}(S_2).{\ensuremath{\mathsf{rank}}}_X (i)
& = C.{\ensuremath{\mathsf{rank}}}_X (B_2.{\ensuremath{\mathsf{rank}}}_0 (i))\\
& \quad + D_2.{\ensuremath{\mathsf{rank}}}_X (B_2.{\ensuremath{\mathsf{rank}}}_1 (i))\end{aligned}$$ and, by the same reasoning, $$\begin{aligned}
C.{\ensuremath{\mathsf{rank}}}_X (j)
& = {\ensuremath{\mathsf{BWT}}}(S_1).{\ensuremath{\mathsf{rank}}}_X (B_1.{\ensuremath{\mathsf{select}}}_0 (j))\\
& \quad - D_1.{\ensuremath{\mathsf{rank}}}_X (B_1.{\ensuremath{\mathsf{rank}}}_1 (B_1.{\ensuremath{\mathsf{select}}}_0 (j)))\,.\end{aligned}$$ Therefore, $$\begin{aligned}
{\ensuremath{\mathsf{BWT}}}(S_2).{\ensuremath{\mathsf{rank}}}_X (i)
& = {\ensuremath{\mathsf{BWT}}}(S_1).{\ensuremath{\mathsf{rank}}}_X (k)\\
& \quad - D_1.{\ensuremath{\mathsf{rank}}}_X (B_1.{\ensuremath{\mathsf{rank}}}_1 (k))\\
& \quad + D_2.{\ensuremath{\mathsf{rank}}}_X (B_2.{\ensuremath{\mathsf{rank}}}_1 (i))\end{aligned}$$ where $k = B_1.{\ensuremath{\mathsf{select}}}_0 (B_2.{\ensuremath{\mathsf{rank}}}_0 (i))$.
For example, for the strings $$S_1 = \mathsf{AAGTTGAGAGTGAGT},\qquad
S_2 = \mathsf{AGAGAGTCGAAGTT};$$ it is $${\ensuremath{\mathsf{BWT}}}(S_1) = \mathsf{TGGGATTAAAAGTGG},\qquad
{\ensuremath{\mathsf{BWT}}}(S_2) = \mathsf{TGGGATCAAAATGG};$$ and ${\ensuremath{\mathsf{LCS}}}({\ensuremath{\mathsf{BWT}}}(S_1),{\ensuremath{\mathsf{BWT}}}(S_2)) = \mathsf{TCTCGTAAAAGG}$. Hence $$\begin{aligned}
B_1 & = 0001000000000111 & D_1 &= \mathsf{GTGC}\\
B_2 & = 010000000001010 & D_2 & = \mathsf{GCC}.\end{aligned}$$ Suppose we want to compute ${\ensuremath{\mathsf{BWT}}}(S_2).{\ensuremath{\mathsf{rank}}}_\mathsf{C}$. It is $B_1.{\ensuremath{\mathsf{select}}}_0 (B_2.{\ensuremath{\mathsf{rank}}}_0 (13)) = 12$, so $$\begin{aligned}
{\ensuremath{\mathsf{BWT}}}(S_2).{\ensuremath{\mathsf{rank}}}_\mathsf{C} (13)
& = {\ensuremath{\mathsf{BWT}}}(S_1).{\ensuremath{\mathsf{rank}}}_\mathsf{C} (12)\; -\; D_1.{\ensuremath{\mathsf{rank}}}_\mathsf{C} (B_1.{\ensuremath{\mathsf{rank}}}_1 (12))
\\
& \quad + D_2.{\ensuremath{\mathsf{rank}}}_\mathsf{C} (B_2.{\ensuremath{\mathsf{rank}}}_1 (13)) \;=\; 3.\end{aligned}$$
Observing that the number of 1s in $B_1$ and $B_2$ is ${\ensuremath{\mathcal{O}\!\left( {\max (n_1, n_2) -
\ell} \right)}} = {\ensuremath{\mathcal{O}\!\left( {{\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$, we can store data structures for $B_1$, $B_2$, $D_1$ and $D_2$ in ${\ensuremath{\mathcal{O}\!\left( {{\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ space such that the desired [$\mathsf{rank}$]{}/[$\mathsf{select}$]{} queries take ${\ensuremath{\mathcal{O}\!\left( {\log {\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ time.
The only other component required for an FM-index for $S_2$ for counting, is a data structure for computing $\sum_{a' \prec a} S_2.{\ensuremath{\mathsf{rank}}}_{a'} (n)$ for each distinct character $a$. Notice that ${\ensuremath{\mathsf{BWD}}}(S_1, S_2)$ is at least the number of distinct characters whose frequencies in $S_1$ and $S_2$ differ. It follows that in ${\ensuremath{\mathcal{O}\!\left( {{\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ space we can store
- a ${\ensuremath{\mathcal{O}\!\left( {\log {\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$-time predecessor data structure storing those distinct characters,
- an array storing $\sum_{a' \prec a} S_2.{\ensuremath{\mathsf{rank}}}_{a'} (n_2)$ for each such distinct character $a$.
For any distinct character $b$, we can find the preceding distinct character $a$ whose frequencies in $S_1$ and $S_2$ differ and compute $$\sum_{a' \prec b} S_2.{\ensuremath{\mathsf{rank}}}_{a'} (n_2) = \sum_{a' \prec b} S_1.{\ensuremath{\mathsf{rank}}}_{a'} (n_1) - \sum_{a' \prec a} S_1.{\ensuremath{\mathsf{rank}}}_{a'} (n_1) + \sum_{a' \prec a} S_2.{\ensuremath{\mathsf{rank}}}_{a'} (n_2)$$ using ${\ensuremath{\mathcal{O}\!\left( {\log {\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ time. Summing up:
\[theo:counting\] If we already have an FM-index for $S_1$, we can store a relative FM-index for $S_2$ using ${\ensuremath{\mathcal{O}\!\left( {{\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ words of extra space. Counting queries on the relative FM-index take time an ${\ensuremath{\mathcal{O}\!\left( {\log {\ensuremath{\mathsf{BWD}}}(S_1, S_2)} \right)}}$ factor larger than on $S_1$.
In Section \[sec:djamal&giovanni\] we show how to build a relative FMindex supporting also locating and extracting.
A practical implementation {#sec:simon&jouni}
--------------------------
A longest common sequence of ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ can be computed in ${\ensuremath{\mathcal{O}\!\left( {n_1 n_2 /w} \right)}}$ time, where $w$ is the word size [@Myers99]. Since we are mainly interested in strings with a small BW-distance, a better alternative could be the algorithms whose running times are bounded by the number of differences between the input sequences (see eg [@LandauVN86; @Myers86]). Unfortunately none of these algorithms is really practical when working with such very large files as the complete genomes we considered in our tests. Hence, to make the construction of a relative FM-index practical, we approximate the LCS of the two Burrows-Wheeler transforms, using the combinatorial properties of the BWT to align the sequences.
Let $S_{1}$ be a random string of length $n$ over alphabet $\Sigma$ of size $\sigma$, and let string $S_{2}$ differ from it by $s$ insertions, deletions, and substitutions. In the expected case, the edit operations move $O(s
\log_{\sigma} n)$ suffixes in lexicographic order, and change the preceding characters for $O(s)$ suffixes [@MNSV10]. If we remove the characters corresponding to those suffixes from ${\ensuremath{\mathsf{BWT}}}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}(S_{2})$, we have a common subsequence of length $n - O(s \log_{\sigma} n)$ in the expected case.
Assume that we have partitioned the BWTs according to the first $k$ characters of the suffixes, for $k \ge 0$. For all $x \in \Sigma^{k}$, let ${\ensuremath{\mathsf{BWT}}}_{x}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}_{x}(S_{2})$ be the substrings of the BWTs corresponding to the suffixes starting with $x$. If we remove the suffixes affected by the edit operations, as well as the suffixes where string $x$ covers an edit, we have a common subsequence ${\ensuremath{\mathsf{BWT}}}_{x}'$ of ${\ensuremath{\mathsf{BWT}}}_{x}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}_{x}(S_{2})$. If we concatenate the sequences ${\ensuremath{\mathsf{BWT}}}_{x}'$ for all $x$, we get a common subsequence of ${\ensuremath{\mathsf{BWT}}}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}(S_{2})$ of length $n - O(s (k + \log_{\sigma} n))$ in the expected case. This suggests that we can find a long common subsequence of ${\ensuremath{\mathsf{BWT}}}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}(S_{2})$ by partitioning the BWTs, finding an LCS for each partition, and concatenating the results.
In practice, we partition the BWTs by variable-length strings. We use backward searching on the BWTs to traverse the suffix trees of $S_{1}$ and $S_{2}$, selecting a partition when either the length of ${\ensuremath{\mathsf{BWT}}}_{x}(S_{1})$ or ${\ensuremath{\mathsf{BWT}}}_{x}(S_{2})$ is at most $1024$, or the length of the pattern $x$ reaches $32$. For each partition, we use the greedy LCS algorithm [@Myers86] to find the longest common subsequence of that partition. To avoid hard cases, we stop the greedy algorithm if it would need diagonals beyond $\pm 50000$, and match only the most common characters for that partition. We also predict in advance the common cases where this happens (the difference of the lengths of ${\ensuremath{\mathsf{BWT}}}_{x}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}_{x}(S_{2})$ is over $50000$, or $x = N^{32}$ for DNA sequences), and match the most common characters in that partition directly.
We implemented the counting structure of the relative FM-index using the SDSL library [@Gog2014b], and compared its performance to a regular FM-index. To encode the BWTs and sequences $D_{1}$ and $D_{2}$, we used Huffman-shaped wavelet trees with either plain or entropy-compressed (RRR) [@Raman2007] bitvectors. We chose entropy-compressed bitvectors for marking the positions of the LCS in ${\ensuremath{\mathsf{BWT}}}(S_{1})$ and ${\ensuremath{\mathsf{BWT}}}(S_{2})$.
The implementation was written in C++ and compiled on g++ version 4.7.3. We used a system with 32 gigabyes of memory and two quad-core 2.53 GHz Intel Xeon E5540 processors, running Ubuntu 12.04 with Linux kernel 3.2.0. Only one CPU core was used in the experiments.
For our experiments, we used the 1000 Genomes Project assembly of the human reference genome as the reference sequence $S_{1}$.[^1] As sequence $S_{2}$, we used the genome of a Han Chinese individual from the YanHuang project.[^2] The lengths of the sequences were 3.10 billion bases and 3.00 billion bases, respectively, and our algorithm found a common subsequence of 2.93 billion bases. As our pattern set, we used 10 million reads of length 56. Almost 4.20 million reads had exact matches in sequence $S_{2}$, with a total of 99.7 million occurrences. The results of the experiments can be seen in Table \[table:experiments\].
[ccccccccccccc]{} & & & & & & & &\
Bitvector & & Time & Space & & Time & Size & & Time & Size & & Time & Size\
Plain & & 762s & 9124MB & & 146s & 1090MB & & 1392s & 288MB & & 954% & 26%\
RRR & & 6022s & 7823MB & & 667s & 628MB & & 3022s & 256MB & & 453% & 41%\
With plain bitvectors in the wavelet tree, the relative FM-index was 9.5 times slower than a regular FM-index, while requiring a quarter of the space. With entropy-compressed bitvectors, the relative index was 4.5 times slower and required 41% of the space. Comparing the relative FM-index using plain bitvectors to the regular index using entropy-compressed bitvectors, we see that the relative index is 2.1 times slower, while taking 46% of the space.
Bitvectors $B_{1}$ and $B_{2}$ took 70% to 80% of the total size of the relative index. We tried to encode them as sparse bitvectors [@Okanohara2007], but the result was slightly larger and clearly slower than with entropy-compressed bitvectors. By our estimates, run-length encoded bitvectors would have taken slightly more space than sparse vectors. Hybrid bitvectors using different encodings for different parts of the bitvector [@Kaerkkaeinen2014] could improve compression, but the existing implementation does not work with vectors longer than $2^{31}$ bits.
Relative FM-indices supporting locating and extracting {#sec:djamal&giovanni}
======================================================
As mentioned in Section \[sec:review\], an FM-index for $S_1$ usually has an SA sample that takes an only slightly sublinear number of bits. This sample has two parts: the first consists of a bitvector $R$ with 1s marking the positions in ${\ensuremath{\mathsf{BWT}}}(S_1)$ of every $r$th character in $S_1$, and an array $A$ storing a mapping from the ranks of those characters’ positions in ${\ensuremath{\mathsf{BWT}}}(S_1)$ to their positions in $S_1$; the second is an array storing a mapping from the ranks of those characters’ positions in $S$ to their positions in ${\ensuremath{\mathsf{BWT}}}(S_1)$. With these, given the position of a sampled character in ${\ensuremath{\mathsf{BWT}}}(S_1)$, we can find its position in $S_1$, and vice versa.
These parts are used for locating and extracting queries, respectively, and the worst-case query times are proportional to $r$. On the other hand, the size of the sample in words is proportional to the length of $S$ divided by $r$. For details on how the sample works, we direct the reader to the full description of FM-indexes [@FM05]. We note only that if we sample irregularly, then the worst-case query times for locating and extracting are proportional to the maximum distance in $S$ between two consecutive sampled characters. We leave consideration of extracting for the full version of the paper — it is nearly symmetric to locating — so we do not discuss the second part of the sample here.
Let $G = S_1[i_1]\;\cdots,\;S_1[i_\ell]$ denote a length-$\ell$ common subsequence of $S_1$ and $S_2$ (not their BWTs). That is, we have $i_1< \;\cdots\; < i_\ell$ and there exists $j_1 <
\;\cdots\; < j_\ell$ such that $$S_1[i_1]=S_2[j_1],\;\ldots,\;S_1[i_\ell]=S_2[j_\ell].$$ Since there is a one-to-one correspondence between a text and its BWT, we can define the indexes $v_1, \ldots, v_\ell$ (resp. $w_1,\ldots,w_\ell$) such that for $k=1,\ldots,\ell$, ${\ensuremath{\mathsf{BWT}}}(S_1)[v_k]$ is the character corresponding to $S_1[i_k]$ (resp. ${\ensuremath{\mathsf{BWT}}}(S_2)[w_k]$ is the character corresponding to $S_2[j_k]$). We say that the common subsequence $G$ is [*BWT-invariant*]{} if there exists a permutation $\pi: \{1,\ldots,\ell\} \rightarrow
\{1,\ldots,\ell\}$ such that we have simultaneously $$\label{eq:bwtinv}
v_{\pi(1)} < v_{\pi(2)} < \cdots < v_{\pi(\ell)}, \quad\mbox{and}\quad
w_{\pi(1)} < w_{\pi(2)} < \cdots < w_{\pi(\ell)}.$$ In other words, when we go from the texts to the BWTs the elements of $G$ are permuted in the same way in $S_1$ and $S_2$.
An immediate consequence of is that the sequence $$G' = {\ensuremath{\mathsf{BWT}}}(S_1)[v_{\pi(1)}]\,{\ensuremath{\mathsf{BWT}}}(S_1)[v_{\pi(2)}]\,\cdots\,{\ensuremath{\mathsf{BWT}}}(S_1)[v_{\pi(\ell)}]$$ is a common subsequence of ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$. We can therefore generalize and define $${\ensuremath{\mathsf{BWD}}}_G(S_1, S_2) = \max(n_1,n_2) - |G|$$ and repeat the construction of Theorem \[theo:counting\] with ${\ensuremath{\mathsf{BWD}}}$ replaced by ${\ensuremath{\mathsf{BWD}}}_G$. However, since $G$ is BWT-invariant it is now possible to reuse the the SA samples from $S_1$ relative to positions in $G$ for the string $S_2$ provided that we have
- bitvector $M_1 [1..n_1]$ with 0s in positions $i_1, \ldots, i_\ell$, supporting fast [$\mathsf{rank}$]{} queries,
- bitvector $M_2 [1..n_2]$ with 0s in positions of $j_1, \ldots,
j_\ell$, supporting fast ${\ensuremath{\mathsf{select}}}_0$ queries;
proof idea in the appendix, complete proof in the full paper. Summing up, we have:
\[theo:locating\] For any BWT-invariant subsequence $G$, if we already have an FM-index for $S_1$, then we can store ${\ensuremath{\mathcal{O}\!\left( {{\ensuremath{\mathsf{BWD}}}_G (S_1, S_2)} \right)}}$ extra space such that the time bounds for locating and extracting queries on $S_2$ are an ${\ensuremath{\mathcal{O}\!\left( {\log
{\ensuremath{\mathsf{BWD}}}_G (S_1, S_2)} \right)}}$ factor larger than on $S_1$.
In view of the above theorem, it is certainly desirable to find the longest common subsequence of $S_1$ and $S_2$ which is BWT-invariant. Unfortunately, this problem is NP-hard as shown by the following result.
\[theo:np\] It is NP-complete to determine whether there is an LCS of $S_1$ and $S_2$ which is BWT-invariant, even when the strings are over a ternary alphabet.
Clearly we can check in polynomial time whether a given subsequence of $S_1$ and $S_2$ has this property, so the problem is in NP. To show that it is NP-complete, we reduce from the NP-complete problem of permutation pattern matching [@BBL98], for which we are given two permutations $\pi_1$ and $\pi_2$ over $n$ and $m \leq n$ elements, respectively, and asked to determine whether there is a subsequence of $\pi_1$ of length $m$ such that the relative order of the elements in that subsequence is the same as the relative order of the elements in $\pi_2$. For example, if $\pi_1 = 6, 3,
2, 1, 4, 5$ and $\pi_2 = 4, 2, 1, 3$, then $6, 2, 1, 4$ is such a subsequence. Specifically, we set $$\begin{aligned}
S_1 & = \mathsf{A B^{\pi_1 [1]} A B^{\pi_1 [2]} \cdots A B^{\pi_1 [n]}}\\
S_2 & = \mathsf{A C^{\pi_2 [1]} A C^{\pi_2 [2]} \cdots A C^{\pi_2 [m]}}\,,\end{aligned}$$ so the unique LCS of $S_1$ and $S_2$ is $\mathsf{A}^m$. For our example, $$\begin{aligned}
S_1 & = \mathsf{A B^6 A B^3 A B^2 A B A B^5} = \mathsf{A B B B B B B A B B B A B B A B A B B B B B}\\
S_2 & = \mathsf{A C^4 A C^2 A C A C^3} = \mathsf{A C C C C A C C A C A C C C}\,.\end{aligned}$$ The BWT sorts the $m$ copies of [A]{} in $S_2$ according to $\pi_2$ and sorts any subsequence of $m$ copies of [A]{} in $S_1$ according to the corresponding subsequence of $\pi_1$. Therefore, there is an LCS of $S_1$ and $S_2$ such that the relative order of its characters is ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ is the same, if and only if there is a subsequence of $\pi_1$ of length $m$ such that the relative order of the elements in that subsequence is the same as the relative order of the elements in $\pi_2$.
In view of the above result, for large inputs we cannot expect to find the longest possible BWT-invariant subsequence, so, as for the LCS, we have devised the following fast heuristic for computing a “long” BWT-invariant subsequence.
We first compute the suffix array $SA_{12}$ for the concatenation $S_1\#S_2$ and we use it to define the array $A$ of size $n_1 \times 2$ as follows
- $A[i][1] = j\quad$ iff $S_1[i] = S_2[j]$ and suffix $S_2[j+1,n_2]$ immediately follows suffix $S_1[i+1,n_1]$ in $SA_{12}$. If no such $j$ exists $A[i][1]$ is undefined.
- $A[i][2] = j\quad$ iff $S_1[i] = S_2[j]$ and suffix $S_2[j+1,n_2]$ is the lexicographically largest suffix of $S_2$ preceding suffix $S_1[i+1,n_1]$ in $SA_{12}$. In no such $j$ exists $A[i][2]$ is undefined.
Next, we compute the longest subsequence $1 \leq i_1 < i_2 < \cdots < i_\ell
\leq n_1$ such that there exist $b_1, \ldots, b_\ell$, with $b_k \in \{1,2\}$ and the sequence $$A[i_1][b_1] < A[i_2][b_2] < \cdots < A[i_\ell][b_\ell]$$ is the longest possible (every $A[i_k][b_k]$ must be defined). The values $i_1, \ldots, i_\ell$ and $b_1, \ldots, b_\ell$ can be computed in ${\ensuremath{\mathcal{O}\!\left( {n_1\log n_1} \right)}}$ time using a straightforward modification of the dynamic programming algorithm for the longest increasing subsequence. Setting, for $k=1,\ldots,\ell$, $j_k = A[i_k][b_k]$ we get that $$G \;= \; S_1[i_1] S_1[i_2] \cdots S_1[i_\ell] \; = \; S_2[j_1] S_2[j_2] \cdots S_2[j_\ell]$$ is a common subsequence of $S_1$ and $S_2$.
\[lemma:bwtinv\] The subsequence $G$ is BWT-invariant.
Let $v_1, \ldots, v_\ell$ (resp. $w_1,\ldots,w_\ell$) such that for $k=1,\ldots,\ell$, ${\ensuremath{\mathsf{BWT}}}(S_1)[v_k]$ is the character corresponding to $S_1[i_k]$ (resp. ${\ensuremath{\mathsf{BWT}}}(S_2)[w_k]$ corresponds to $S_2[j_k]$). It suffices to prove that for any pair $h,k$, with $1 \leq h,k \leq \ell$, the inequality $v_h < v_k$ implies $w_h < w_k$. Let $\prec$ denote the lexicographic order. By construction, and by the properties of the BWT, we have $v_h < v_k$ iff the suffix $S_1[i_h+1,n_1] \prec S_1[i_k+1,n_1]$ and we must prove that this implies $S_2[j_h+1,n_2] \prec S_2[j_k+1,n_2]$.
Since $j_h = A[i_h][b_h]$ and $j_k = A[i_k][b_k]$, the proof follows considering the four possible cases: $b_h =1,2$ and $b_k=1,2$. We consider the case $b_h=1$, $b_k=2$ leaving the others to the reader. If $j_h =
A[i_h][1]$ and $j_k = A[i_k][2]$ then $S_2[j_h+1,n_2]$ immediately follows $S_1[i_h+1,n_1]$ in $SA_{12}$. At same time $S_2[j_k+1,n_2]$ precedes $S_1[i_h+1,n_1]$ but there are no other suffixes from $S_2$ between them. Since $j_h \neq j_k$ the only possible ordering of the suffixes in $SA_{12}$ is $$S_1[i_h+1,n_1] \;\prec\; S_2[j_h+1,n_2] \;\prec\; S_2[j_k+1,n_2] \;\prec\; S_1[i_k+1,n_1]$$ implying $S_2[j_h+1,n_2] \prec S_2[j_k+1,n_2]$ as claimed.
To evaluate whether the subsequence $G$ derived from the above procedure is still able to capture the similarity between $S_1$ and $S_2$, we have compared the length of $G$ with the [$\mathsf{LCS}$]{} length for pairs of [*S.cerevisiae*]{} genomes from the Saccharomyces Genome Resequencing Project.[^3] In particular we compared the 273614N sequence with sequences 322134S, 378604X, BC187, and DBVPG1106. For each sequence we report in Table \[table:G\] the ratio between the length of $G$ and ${\ensuremath{\mathsf{LCS}}}({\ensuremath{\mathsf{BWT}}}(S_1),{\ensuremath{\mathsf{BWT}}}(S_2))$ and the length of sequence 273614N (roughly 11.9 MB). We see that in all cases more than 85% of BWT positions are in $G$ which roughly indicates that more than 85% of the SA samples from 273614N could be reused as SA samples for the other sequences.
---------------------------------------------- --------- --------- -------- -----------
322134S 378604X BC187 DBVPG1106
\[.5ex\]
\[-1.5ex\] $|{\ensuremath{\mathsf{LCS}}}|/n$ 0.9341 0.9669 0.9521 0.9590
\[.5ex\] $|G|/n$ 0.8694 0.8655 0.8798 0.8800
\[.5ex\]
---------------------------------------------- --------- --------- -------- -----------
: Comparison between $|G|$ and $|{\ensuremath{\mathsf{LCS}}}|$. The normalizing factor $n$ is the length of sequence 273614N.\[table:G\]
Conclusions {#sec:concl}
===========
In this paper we have considered the problem of building an index for a string $S_2$ given an FM-index for a similar string $S_1$. We have shown how to build such a “relative” index using space bounded by the BW-distance between $S_1$ and $S_2$. The BW-distance is simply the edit distance between ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ when only insertions and deletions are allowed. We have also introduced the notion of BWT-invariant subsequence and shown that it can be used to determine a set of $S_1$ suffix array samples that can be easily “reused” for an index for $S_2$.
We have tested our approach by building a relative index for a Han Chinese individual with respect to an FM-index of the human reference genome. We leave as a future work the development of these ideas and the complete implementation of a relative FM-index supporting locating and extracting. We also leave as future work proving bounds on the BW-distance and the length of the longest BWT-invariant subsequence in terms of the edit distance of the strings.
Appendix: Reusing an SA Sample {#sec:sample .unnumbered}
==============================
Consider the example strings $S_1$, $S_2$ given in the introduction. The characters of ${\ensuremath{\mathsf{BWT}}}(S_1) [1..16]$ and ${\ensuremath{\mathsf{BWT}}}(S_2) [1..15]$ are mapped to their positions by the BWT from $$\begin{aligned}
& S_1 [16, 2, 6, 8, 13, 1, 12, 3, 7, 9, 14, 10, 15, 5, 11, 4]\\
& S_2 [15, 7, 2, 5, 12, 1, 11, 8, 3, 6, 13, 9, 14, 4, 10]\end{aligned}$$ respectively. (Notice the lists of indices are just the SAs of $S_1\$$ and $S_2\$$ with each value decremented.) Therefore, if $r = 3$ then $$R = 1000110010010001, \qquad
A [1..6] = [16, 13, 1, 7, 10, 4]\,.$$ Comparing $R$ and $B_1 = 0001000000000111$ we see that the sampled characters ${\ensuremath{\mathsf{BWT}}}(S_1) [1, 5, 6, 9, 12]$ that are in $C$, are $C$’s 1st, 4th, 5th, 8th and 11th characters. From $B_2 = 010000000001010$ we see that the 1st, 4th, 5th, 8th and 11th characters in $C$ in ${\ensuremath{\mathsf{BWT}}}(S_2)$ are ${\ensuremath{\mathsf{BWT}}}(S_2) [1, 5, 6, 9, 13]$, which are mapped to their positions by the BWT from $S_2 [15, 12, 1, 3, 14]$.
The relative order $5, 3, 1, 2, 4$ of the positions $15, 12, 1, 3, 14$ in $S_2$ of these characters, is [*almost*]{} the same as the relative order $5, 4, 1, 2. 3$ of the positions $16, 13, 1, 7, 10$ in $S_1$ of the sampled characters in ${\ensuremath{\mathsf{BWT}}}(S_1)$ that are in $C$, which seems promising. What if we choose $C$ and its occurrences in ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ such that the relative order in $S_1$ of all ${\ensuremath{\mathsf{BWT}}}(S_1)$’s characters that are in $C$, is the same as the relative order in $S_2$ of all ${\ensuremath{\mathsf{BWT}}}(S_2)$’s characters that are in $C$?
For example, we can choose instead $$\begin{aligned}
C' & = \mathsf{TCTCGTAAAGG}\\
B_1' & = 0001000001010101 & B_2' & = 010000010001010\\
D_1' & = \mathsf{GAGTC} & D_2' & = \mathsf{GACC}\end{aligned}$$ even though $C'$ is not then an LCS of ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ and, thus, our data structures for supporting [$\mathsf{rank}$]{} in ${\ensuremath{\mathsf{BWT}}}(S_2)$ are slightly larger. With these choices, the characters in ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$ that are in $C'$, are mapped to their positions by the BWT from $$S_1 [16, 2, 6, 13, 1, 12, 3, 7, 14, 15, 11],\qquad
S_2 [15, 2, 5, 12, 1, 11, 3, 6, 13, 14, 10]$$ and the relative order $11, 2, 4, 8, 1, 7, 3, 5, 9, 10, 6$ of the indices in those two lists is the same, as desired.
Suppose we store yet another pair of bitvectors $$M_1 = 0001100111000000,\qquad
M_2 = 000100111000000$$ with 1s marking the positions in $S_1$ and $S_2$ of characters that are not mapped into $C'$ in ${\ensuremath{\mathsf{BWT}}}(S_1)$ and ${\ensuremath{\mathsf{BWT}}}(S_2)$. We claim that if we can support fast [$\mathsf{rank}$]{} queries on $B_2'$, $R$ and $M_1$, fast access to $A$ and fast ${\ensuremath{\mathsf{select}}}_0$ queries on $B_1'$ and $M_2$, then we can support fast access to a (possibly irregular) sample SA sample for $S_2$ with as many sampled characters as there are in $C'$ in ${\ensuremath{\mathsf{BWT}}}(S_1)$. More specifically, if ${\ensuremath{\mathsf{BWT}}}(S_2) [i]$ is in $C'$ and $R [B_1'.{\ensuremath{\mathsf{select}}}_0 (B_2'.{\ensuremath{\mathsf{rank}}}_0 (i))] = 1$ — meaning the corresponding character in $C'$ in ${\ensuremath{\mathsf{BWT}}}(S_1)$ is sampled — then ${\ensuremath{\mathsf{BWT}}}(S_2) [i]$ is mapped to its position by the BWT from $$S_2 \left[ \rule{0ex}{4ex}
M_2.{\ensuremath{\mathsf{select}}}_0 \left( \rule{0ex}{3.5ex}
M_1.{\ensuremath{\mathsf{rank}}}_0 \left( \rule{0ex}{3ex}
A \left[ \rule{0ex}{2.5ex}
R.{\ensuremath{\mathsf{rank}}}_1 \left( \rule{0ex}{2ex}
B_1'.{\ensuremath{\mathsf{select}}}_0 \left(
B_2'.{\ensuremath{\mathsf{rank}}}_0 (i)
\right)
\right)
\right]
\right)
\right)
\right]\,.$$ We leave a detailed explanation to the full version of this paper. We note, however, that this approach works for any sample rate $r$, and even if the SA sample for $S_1$ is irregular itself.
In our example, since ${\ensuremath{\mathsf{BWT}}}(S_2) [10]$ is in $C'$, $B_1'.{\ensuremath{\mathsf{select}}}_0
(B_2'.{\ensuremath{\mathsf{rank}}}_0 (10)) = 9$ and $R [9] = 1$, we know ${\ensuremath{\mathsf{BWT}}}(S_2) [10]$ is mapped to its position by the BWT from position $M_2.{\ensuremath{\mathsf{select}}}_0 \left(
M_1.{\ensuremath{\mathsf{rank}}}_0 \left( \rule{0ex}{2ex}
A [R.{\ensuremath{\mathsf{rank}}}_1 (9)]
\right)
\right)
= 6$ in $S_2$.
[^1]: GRCh37, <ftp://ftp-trace.ncbi.nih.gov/1000genomes/ftp/technical/reference/>
[^2]: <ftp://public.genomics.org.cn/BGI/yanhuang/fa/>
[^3]: https://www.sanger.ac.uk/research/ projects/genomeinformatics/sgrp.html
|
---
abstract: |
**Background**: Unsupervised machine learners have been increasingly applied to software defect prediction. It is an approach that may be valuable for software practitioners because it reduces the need for labeled training data.\
**Objective**: Investigate the use and performance of unsupervised learning techniques in software defect prediction.\
**Method**: We conducted a systematic literature review that identified 49 studies containing 2456 individual experimental results, which satisfied our inclusion criteria published between January 2000 and March 2018. In order to compare prediction performance across these studies in a consistent way, we (re-)computed the confusion matrices and employed the Matthews Correlation Coefficient (MCC) as our main performance measure.\
**Results**: Our meta-analysis shows that unsupervised models are comparable with supervised models for both within-project and cross-project prediction. Among the 14 families of unsupervised model, Fuzzy CMeans (FCM) and Fuzzy SOMs (FSOMs) perform best. In addition, where we were able to check, we found that almost 11% (262/2456) of published results (contained in 16 papers) were internally inconsistent and a further 33% (823/2456) provided insufficient details for us to check.\
**Conclusion**: Although many factors impact the performance of a classifier, e.g., dataset characteristics, broadly speaking, unsupervised classifiers do not seem to perform worse than the supervised classifiers in our review. However, we note a worrying prevalence of (i) demonstrably erroneous experimental results, (ii) undemanding benchmarks and (iii) incomplete reporting. We therefore encourage researchers to be comprehensive in their reporting.
address:
-
-
-
-
author:
- Ning Li
- Martin Shepperd
- Yuchen Guo
title: A Systematic Review of Unsupervised Learning Techniques for Software Defect Prediction
---
Unsupervised learning; Software defect prediction; Machine learning; Systematic review; Meta-analysis.
Introduction {#sec:Intro}
============
Various software defect prediction models have been proposed to improve the quality of software over the past few decades [@fenton1999critique]. An increasingly popular approach is to use machine learning [@catal2009systematic; @hall2012systematic]. These approaches can be divided into supervised methods where the training data requires labels, typically faulty or not, and unsupervised methods where the data do not need to be labelled. Supervised prediction models predominate. However, in practice it is often difficult to collect defect classification labels to train a Supervised Defect Prediction (SDP) model [@nam2015clami]. As a consequence in recent years, Unsupervised Defect Prediction (UnSDP) models have begun to attract attention.
The main aim of our systematic review is to provide software practitioners and researchers with guidance for software defect prediction, particularly regarding whether the use of unsupervised prediction models is a viable option. We analyse 49 UnSDP primary studies that satisfy our inclusion criteria. From these primary studies, we investigate which unsupervised learning algorithms were deployed and the relative predictive performance of supervised and unsupervised models.
Our systematic review makes the following contributions:
1. Identification of a set of 49 primary studies related to UnSDP published between January 2000 and March 2018. These cover a wide range (14) of unsupervised prediction technique families including six different cluster labelling techniques.
2. Data extraction and regularisation from the 49 studies satisfying our inclusion criteria. We obtain 2456 individual experimental results from these studies.
3. A meta-analysis to compare the relative performance of unsupervised and supervised learners from two perspectives (i) the specific learning algorithm and (ii) the dataset. Based on the analysis results we make suggestions for machine learning-based, software defect prediction for practitioners.
4. A bibliometric analysis that considers research and publishing trends, along with the quality of reported experimental results.
The remainder of this paper is organised as follows: Section \[sec:Review\] describes our systematic review methodology including the research questions, study selection criteria, data extraction and synthesis (process, prediction performance measures and raw data summary). Next, Section \[sec:Bibmetrics\] presents current research trends and quality issues: both incomplete and inconsistent reporting of results. Section \[sec:Perform\] shows our meta-analysis results. Section \[sec:Threats\] discusses the threats to validity, followed by conclusions and discussion of actionable results in Section \[sec:Conc\].
Review Methodology {#sec:Review}
==================
Our systematic review follows the guidelines of a systematic review approach for software engineering presented in [@Kitc15]. First, we clarify our research questions, then search and identify relevant primary studies, next we synthesise results from the selected primary studies, finally we explore and answer our research questions through meta-analysis.
Research Questions
------------------
- RQ1: What are the publication trends in unsupervised software defect prediction research?
- RQ2: What is the quality of the experimental reporting (completeness and consistency)?
- RQ3: What kinds of unsupervised learning research experiments are conducted?
- RQ4: What is the difference between unsupervised and supervised software defect predictive performance?
- RQ5: Which unsupervised prediction models or model families perform better?
- RQ6: What is the impact of dataset characteristics on predictive performance?
The first three research questions will principally be of interest to software engineering researchers. The remaining three questions will be of interest to both practitioners and researchers.
Search Process
--------------
We used five search engines to include the papers published between January 2000 and March 2018. The start date was aligned with the search period of the widely cited systematic review by Hall et al. [@hall2012systematic]. We undertook our search on 7th March, 2018. The search engines include the ISI Web of Science, ACM Digital Library, IEEE Xplore, ScienceDirect and SpringerLink. Although there are small variants in the five search engines, our key search string is:
("fault prediction" OR "defect prediction"
OR "bug prediction" OR "error prediction")
AND ("unsupervised" OR "unlabel*"
OR "cluster*")
AND ("software")
Table \[tbl:review\] presents the results of our paper search and selection process. This results in 49 papers selected from an initial 1360 papers. We list these 49 primary papers at the end of this paper where P$i$ denotes the i^th^ primary study. Note that experimental results in [@yan2016self] and [@boucher2016using] are included in [P @yan2017automated] and [P @boucher2018software] respectively, thus we only took [P @yan2017automated] and [P @boucher2018software] as our primary studies.
\[tbl:review\]
Search and selection process \# of added / excluded papers \# of remaining papers
--------------------------------------------------------------------------------------------------------------------- ------------------------------- ------------------------
Search of five academic search engines +1360 1360
Exclude duplicate studies -346 1014
Exclude irrelevant studies based on title and abstract -943 71
Forward and backward chain using Google Scholar +88 159
Exclude papers that use defect labels for training data or unsupervised techniques are used only for pre-processing -96 63
Exclude papers without real-world data experiments -6 57
Exclude papers with insufficiently detailed or duplicated results -8 49
: Systematic review paper search and selection process
Inclusion Criteria
------------------
When determining whether a paper or experimental result should be included or not, the following criteria were applied.
1. Written in English.
2. Full content must be available.
3. Published between January 2000 - March 2018.
4. Uses real data (not simulations).
5. Applies at least one unsupervised method to software defect-prone module prediction.
6. Includes new software defect prediction experiments. We ignore re-analysis of previously published experiments.
7. Includes the best prediction performance results per dataset and learner (since some experiments have a primary focus on some other aspects of machine learning e.g., feature selection, parameter tuning).
8. Reports the results in sufficient detail to enable meta-analysis.
Data Extraction and Synthesis {#subsec:dataprocess}
-----------------------------
We extracted data related to our research questions from each of the 49 papers, and organised the qualitative and quantitative data into a raw data file (see our Mendeley dataset [@Li2020Data]) . Each paper contains from 1 to 751 ( median = 12) experimental results, yielding a total of 2456 individual results, which involve 128 distinct software project defect datasets (from NASA, ISM, AEEEM, PROMISE, etc.) and 25 prediction model families (14 unsupervised learners and 11 supervised learners).
### Data Extraction and Synthesis Process
From each paper we extracted the following:
- Title
- Year
- Journal/conference
- ‘Predatory’ publisher? (Y $ | $ N)
- Count of results reported in paper
- Count of inconsistent results reported in paper
- Parameter tuning in SDP? (Yes $ | $ Default $ | $ ?)
- SDP references[^1] (SDPRefs$\_ $OrigResults $ | $ SDPRefs $ | $ SDPNoRefs $ | $ OnlyUnSDP)
Then, from within each paper, we extracted for each experimental result:
- Prediction method name (e.g., DTJ48)
- Project name trained on (e.g., PC4)
- Project name tested on (e.g., PC4)
- Prediction type (within-project $ | $ cross-project)
- No. of input metrics (count $|$ NA)
- Dataset family (e.g., NASA)
- Dateset fault rate (%)
- Was cross validation used? (Y $|$ N $ | $ ?)
- Was error checking possible? (Y $ | $ N)
- Inconsistent results? (Y $ | $ N $ | $ ?)
- Error reason description (text)
- Learning type (Supervised $|$ Unsupervised)
- Clustering method? (Y $|$ N $|$ NA)
- Machine learning family (e.g., Un-NN)
- Machine learning technique (e.g., KM)
- Prediction results (including TP, TN, FP, FN, etc.)
For full details refer to the review protocol at our Mendeley dataset [@Li2020Data]. To ensure data quality, we undertook pre-processing (synthesising across studies) including name unification (project name, method name, response variable name, etc.), confusion matrix (re-)computation[^2](see Table \[tbl:matrix\]) and data quality checking with R scripts. We describe the details of confusion matrix re-computation and data quality checks in Section \[subsec:quality\].
### Prediction Performance Measures {#subsec:perfMetric}
For data classification, the confusion matrix (Table \[tbl:matrix\]) is the fundamental descriptor from which the majority of performance indicators may be derived. Although ideally all primary studies would report consistent performance indicators, in practice, a wide range of indicators are used such as accuracy, precision, recall, the F-measure[^3], the G-measure and so forth. Consequently, we reconstruct the confusion matrix wherever possible. Unfortunately, there remain about 33% (823/2456) of the experimental results for which this was not possible, due to incomplete reporting as discussed in Section \[subsec:LowQual\].
The performance indicators used by our set of 49 primary studies are summarised below. To further complicate matters, note that different studies may use different names for the same measure [@shepperd2014researcher; @bowes2014dconfusion].
\[tbl:matrix\]
Observed defective Observed defect free
------------- -------------------- ----------------------
Predicted True Positive False Positive
defective (TP) (FP)
Predicted False Negative True Negative
defect free (FN) (TN)
: Confusion Matrix
1. FPR (False Positive Rate): $ \frac{FP}{TN+FP} $
2. FNR (False Negative Rate): $ \frac{FN}{TP+FN} $
3. ER (Error Rate): $ \frac{FP+FN}{TP+TN+FP+FN} $
4. Recall: $ \frac{TP}{TP+FN} $
5. Accuracy: $ \frac{TP+TN}{TP+TN+FP+FN} $
6. Precision: $ \frac{TP}{TP+FP} $
7. F1: $ \frac{2 \times Recall \times Precision}{Recall+Precision} $
8. MCC: $ \frac{TP \times TN - FP \times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}} $ [@baldi2000assessing]
9. AUC: Area Under the Curve (AUC) ROC chart [@fawcett2006introduction].
10. Popt/ACC: Effort-aware prediction performance [@kamei2013large].
11. G-Mean:$\sqrt{ \frac{TP}{TP+FN} * \frac{TN}{TN+FP}}$ ([P @boucher2018software]).
12. G-Measure: $ \frac{2 \times Recall \times TNR}{Recall+TNR} $ . ([P @fan2017utility])
13. Balance: $ 1- \frac{\sqrt{(0-FPR)^{2} + (1-Recall)^{2} }}{\sqrt{2}} $. ([P @singh2014efficient])
14. Purity: Percentage of the most-dominated category (fault prone or not fault prone) in the cluster. [P @zhong2004analyzing]
15. MAE: Mean absolute error between prediction and observation ([P @chug2013software]).
16. MeanAIC: The mean of Akaike’s Information Criterion (AIC) ([P @iwata2012clustering]).
Despite being widely used, F1 and AUC are known to be potentially problematic [@Hand09; @Powe11; @Flac15]. F1 focuses on positive classes and ignores negative classes. This is acceptable for information retrieval type problems (since the number of say irrelevant documents, correctly not retrieved can be essentially unbounded). However, this is not the case for defect prediction because the precision of negative classes (non-defective) is also of concern to software developers. Knowing that a component is non-defective is important. Thus the application domain of defect prediction is unsuitable for F1.
The AUC metric compares algorithms without classification threshold. So it is suitable to compare classifier *families* in a theoretical sense but not in a practical sense. The reason being, in practice a deployed classifier must have a particular classification threshold. So unless a classifier strictly dominates another (i.e., for all possible threshold values) we cannot use AUC to prefer one classifier to another.
In contrast, the Matthews Correlation Coefficient (MCC) [@baldi2000assessing] is based on all four quadrants of the confusion matrix, which gives a better summary of the performance of classification algorithms. It is also known by statisticians as the $\phi$ coefficient, originally proposed by Karl Pearson. MCC is easier to interpret as correlation coefficient since it takes a value in the interval \[-1, 1\], with 1 showing a perfect classifier, -1 showing a perverse classifier, and 0 showing that the prediction is uncorrelated with the ground truth. A more detailed comparison between MCC, F1 and AUC can be found in our another work [@song2018comprehensive] (see Section 3.1 Classification Performance Measures). Therefore, we prefer to use MCC to assess classification performance in this paper. Note, however, that MCC is undefined if any of the quantities TP + FN, TP + FP, TN + FP, or TN + FN are zero [@baldi2000assessing].
### Data Summary {#subsec:rawdata}
From the 49 studies we obtain 2456 individual experimental results. Table \[tbl:statClass\] presents a summary of the categorical attributes. Over half of the results are from supervised learners which are generally deployed as comparators to the unsupervised methods. Within-project (as opposed to cross-project) classification is the dominant approach.
The remaining categories relate to primary study quality which we explore in Section \[subsec:quality\].
Attribute Count
----------------------- -------------------------------------
Learning Type Unsupervised: 947; Supervised: 1509
Prediction type WithinPrj: 1840; CrossPrj: 616
Cross validation Yes: 2210; Unknown: 246
‘Predatory’ publisher Yes: 280; No: 2176
Inconsistent data Yes: 262; No: 1371; Unknown: 823
: Summary of key categorical attributes for individual results[]{data-label="tbl:statClass"}
Likewise, Table \[tbl:statNumber\] summarises the numerical attributes where \#NA is the number of unavailable results. From our data extraction and synthesis, 262 inconsistent results were identified (see Section \[subsec:quality\]). Therefore, Table \[tbl:statNumber\] only lists a summary of the valid — in the sense of being internally consistent — 2194 experimental results after removing 262 inconsistent ones. Quite striking is the diversity of values, e.g., the fault rate $d$ ranges from 0.4% to over 93%. The AUC ranges from 0.31 to 0.948 (recall that any value below 0.5 suggests a classifier that is predicting worse than by chance). Note that \#NA of MCC is larger than 823 unchecked results, which is caused by zero in the denominator of its definition (see Section \[subsec:perfMetric\]). F1 is similarly impacted. For the fault rate, \#NA is due to unreported values in private (RSDIMU, CSAS, MIS and Embedded used in 20 experiments) or modified public datasets (KC3\_314 and JM1\_8916 are public NASA data, but are inconsistent with the original datasets and are used in a further 4 experiments). However, the overall unavailability of fault rate data is low, considering the total number of datasets and fortunately has little effect on our meta-analysis.
Statistic Fault rate ($d$) MCC F1 AUC Popt
----------- ------------------ -------- ------- ------- -------
Min 0.004 -0.524 0.016 0.310 0.276
Q1 0.122 0.245 0.381 0.610 0.583
Median 0.204 0.384 0.531 0.670 0.726
Mean 0.237 0.360 0.519 0.674 0.697
Q3 0.310 0.494 0.654 0.750 0.815
Max 0.936 0.931 0.978 0.948 0.948
\#NA 24 952 945 1734 1843
: Summary of 2194 consistent experimental results (including 823 unchecked results)[]{data-label="tbl:statNumber"}
Quality Factors {#subsec:quality}
---------------
In order to understand the quality of the different studies within our analysis, we consider two classes of explanatory factors. External factors are publication venue and type, whereas internal factors address experimental design and consistency of results.
First we explore publication venue. Our search involves a wide range of journal or conference publications. Note that the number of experimental results in these primary studies varies greatly from 1 to 751 thus some papers have more propensity for error. We classify the 49 selected papers into two groups according to whether they are found in the so-called ‘predatory’ publisher lists [@perlin2017predatory]. The list aims to identify standalone publishers with questionable approaches to peer review rigour.
We used three lists: for publisher[^4], for standalone journals[^5] and for conferences[^6]. Note that these lists have been questioned [@Berg15] so we recognise the classification is imperfect.
Second, we consider internal experiment quality. For this we check for the use of cross-validation [@kohavi1995study], as an indicator for experimental quality. More importantly, we check the internal consistency of results. Hence we recompute the confusion matrix, or construct it if not provided. This is possible for any one of the 10 combinations of performance measure reporting listed below, i.e., we solve for X. From this we can compute MCC.
For brevity we just present the details for Cases 9 and 10 (N.B. Cases 1 to 8 can be found in [@bowes2014dconfusion]). Here, $d$ is the proportion of defective modules, $a$ is accuracy, $r$ is recall and $p$ is precision. Other definitions for FPR, FNR, ER, Accuracy, etc. are given in Section \[subsec:perfMetric\]. Note that we use other measures in preference to $d$, since real $d$ might be slightly divergent from reported $d$ in the original study due to pre-processing or cross validation and differences between folds. Therefore, we only use $d$ when we cannot recompute the confusion matrix for Cases 1-4. Furthermore, we normalize TP, TN, FP and FN with $ TP+TN+FP+FN=1 $ when preprocessing raw data.
1. FPR, FNR, ER
2. FPR, FNR, Accuracy
3. FPR, Recall, Precision
4. Recall, Precision, Accuracy
5. Defect Percentage(d), FPR, FNR
6. Defect Percentage(d), FPR, ER
7. Defect Percentage(d), FPR, Recall
8. Defect Percentage(d), Recall, Precision
9. Defect Percentage(d), Accuracy, F1:\
$ TP = \frac{(1-a)*F1}{2*(1-F1)} $, $ TN = a -TP $, $ FN = d -TP $, $ FP = 1-TP-TN-FN $ .
10. Defect Percentage(d), Precision, F1:\
$ TP = \frac{p*d*F1}{2*p-F1} $, $ FN = d -TP $, $ FP = TP *(1-p)/p$,\
$ TN = 1 - TP-FN-FP $.
As part of our re-computation, we also check the correctness and consistency of the experimental results. If the reported or recomputed result satisfies one of the following six inconsistency rules, we label it as problematic data and remove it from our data analysis. In total, we removed 262 results (Rule 1: 171; Rule 2: 7; Rule 3: 60; Rule 4: 3; Rule 5: 19; Rule 6: 2) out of 2456 experimental results.
Rule 1. The recomputed performance values (including TP, F1, MCC, d, etc.) are out of their correct ranges (for example: $TP \notin [0,1]$, $ F1 \notin [0,1] $, $MCC \notin [-1,1] $, $d \notin [0,1] $ ).
Rule 2. The recomputed $ d $ is 0 (we treat 0 as problematic data since all experiments for defect prediction should include defective modules).
Rule 3. Compare the recomputed $d$ with the original reported defect percentage, we treat them as problematic data if the difference is greater than 0.1 (i.e., we allow for some rounding errors).
Rule 4. Compare our recomputed results with the original ones where available. If the difference of a measure is greater than the rounding error range, we label them as problematic data. Here, the rounding error intervals are computed by adding $\pm0.05$ to the original data. Note, compared to the rounding error tolerance of 0.01 used by [@bowes2014dconfusion], we apply a wider range of 0.05.
Rule 5. Special values from Cases 1–10. Here, we use Case1 as example. For a given FPR, FNR, ER in Case1, the confusion matrix is recomputed. We derive the formulae: $ d = \frac{ER-FPR}{FNR-FPR} $, $ FN= d*FNR $, $ FP= (1-d)*FNR$, $TP=d-FN $, $TN=1-TP-FP-FN $. If the numerator is not zero and the denominator is zero, then it is a problematic result. In this case, if $ER \neq FPR $ and $FNR=FPR$ (e.g., FNR=0.24, FPR=0.24, ER=0.13), it will be labeled as problematic or inconsistent. A complete explanation for all of 10 cases can be found in our Mendeley dataset (see Section \[subsec:dataprocess\]).
Rule 6. Obvious problematic experimental data is reported by the primary study. For example, the confusion matrix in [P @sandhu2010k] is inconsistent with their dataset. In [P @sandhu2010k], their Table X shows that there are 134 bugs (BUGs=true) in their JEdit dataset, however the following clustering prediction result Table XI and Table XII show that there are only 61 bugs in that dataset (Table XIII: TP=46, FN=15).
Bibliometric analysis {#sec:Bibmetrics}
=====================
RQ1: What are the publication trends in unsupervised software defect prediction research? {#subsec:PubTrends}
-----------------------------------------------------------------------------------------
Recall that our search has identified 49 primary studies that have applied unsupervised learning to the task of software defect prediction. These are published over the period of 2000-2018[^7] and are summarised in Table \[Tab:PapersByYear\] and also by Figure \[Fig:PubTrends\]. From this we can see that conference papers tend to dominate and that there has been pronounced growth in the overall number of papers published in recent years as indicated by the smoother in Figure \[Fig:PubTrends\].
Conf Journal Total
------- ------ --------- -------
2000 2 0 2
2004 1 1 2
2006 1 0 1
2008 2 0 2
2009 3 0 3
2010 2 2 4
2011 0 1 1
2012 2 1 3
2013 3 2 5
2014 2 3 5
2015 2 3 5
2016 4 1 5
2017 7 2 9
2018 0 2 2
Total 31 18 49
: Research papers published by type and year[]{data-label="Tab:PapersByYear"}
![Unsupervised software defect prediction papers (2000-2018). The broad blue line shows total publications using a loess smoother and 95% confidence limits where these are non-negative, i.e., 2005-onwards. N.B. 2018 is incomplete.[]{data-label="Fig:PubTrends"}](PubTrends.png){width=".9\linewidth"}
RQ2: What is the quality of the experimental reporting (completeness and consistency)? {#Subsec:quality}
--------------------------------------------------------------------------------------
The 49 primary studies contain a total of 2456 individual experimental results. This ranges from 1 to 751 results with the median number being 12 results contained in a single paper. Problematic results are analyzed in Section \[Subsec:quality\], so here we just provide an overall summary of the remaining 2194 non-erroneous results in Table \[Tab:ResErrCt\].
Result Count % of total
---------------------------- ------- ------------
Inconsistent results 262 10.67%
Non-error results 2194 89.33%
(Non-error) Cannot check 823 33.51%
(Non-error) Can check - ok 1371 55.82%
Total 2456 100%
: Counts and proportions of experimental results
\[Tab:ResErrCt\]
### ‘Predatory’ publishing
Next we consider the prevalence of papers published in so-called ‘predatory’ journals and conferences. Table \[Tab:PredPubByYear\] reveals that 18 out of the 49 papers (i.e., approximately 37%) were from venues considered to be ‘predatory’ publishers. This was a higher proportion than we had expected. This phenomenon appears to have started in the late 2000s and is possibly declining since 2014 (see also Figure \[Fig:PubQualityTrends\]). This could be a reaction to the adverse publicity such publishers are gaining.
No Yes Sum
------ ---- ----- -----
2000 2 0 2
2004 2 0 2
2006 0 1 1
2008 1 1 2
2009 1 2 3
2010 0 4 4
2011 1 0 1
2012 3 0 3
2013 1 4 5
2014 1 4 5
2015 4 1 5
2016 5 0 5
2017 8 1 9
2018 2 0 2
Sum 31 18 49
: Papers published by year and by venue, i.e., ‘predatory’ or ‘non-predatory’ publisher[]{data-label="Tab:PredPubByYear"}
![Unsupervised software defect prediction papers (2000-2018) by venue. The broad pink line shows ‘predatory’ publications using a loess smoother. N.B. 2018 is incomplete.[]{data-label="Fig:PubQualityTrends"}](PubQualityTrends.png){width=".9\linewidth"}
### Problems with Incomplete Reporting {#subsec:LowQual}
Next we consider *potential* issues which may arise from incomplete reporting of experimental design and results. First, we examine whether a study explicitly reports if a cross-validation procedure was used or not. We were surprised to find that 25 out of 49 studies did *not* report this information. This is important because although unsupervised learners need not initially be trained with data that have class labels (i.e., defective or not), the learner must still be configured to classify and hence be evaluated on unseen data. Of course some, or even all, of the other studies may have applied cross-validation techniques, however this *ought* to be confirmed by the authors in the first instance.
In terms of results, as described in Section \[subsec:quality\] we endeavoured to apply basic consistency checks to the raw results. Unfortunately this was not possible for over 30% of the results (823/2456) listed in Table \[Tab:ResErrCt\]. Given, as we discuss in the following section, that we uncovered 262 (262/1633 $ \approx $ 16%) instances of problematic data in the results we are able to check, this is a little disturbing.
### Errors and inconsistent results {#subsec:Inconsistent}
As previously indicated, we were able to apply consistency checking to approximately 66% (1633 out of 2456) of all the results in our systematic review. Figure \[Fig:ErrsPerPaper\] shows the distribution of errors (or inconsistencies) per paper. There are 14 papers that are excluded due to incomplete reporting. Note also that in order to improve the visual qualities of the histogram, the largest bin is 10+ and it contains a single extreme outlier of 165 inconsistencies and two studies including 19 and 30 inconsistencies respectively.
![Histogram of discovered errors (inconsistencies) per paper[]{data-label="Fig:ErrsPerPaper"}](ErrsPerPaper.png){width="\linewidth"}
Next, we consider whether there is any relationship between publication venue and study errors. For example, one might expect that ‘non-predatory’ publishers deploy more rigorous peer review, or that ‘predatory’ publishers are less diligent in this regard. To validate such an expectation, we examine the odds ratio [@grissom2012effect] which is defined as:
$$OR = \frac{Odds_{pc}}{Odds_{notpc}} = \frac{P(T|pc)/P(not T|pc)}{P(T|not pc)/P(not T|not pc)}$$
where $Odds_{pc}$ is the odds that a member of Population $pc$ will fall into Category $T$, similarly $Odds_{notpc}$ is the odds that a member not from Population $pc$ will fall into Category $T$.
When calculating the odds ratio (from Table \[Tab:ErrorsPredPub\]) the likelihood of containing an error is $ 1.21 $(with 95% confidence limits $[0.31, 4.73]$) between ‘non-predatory’ and ‘predatory’ publishers. This is close to unity and so suggests little effect at the study level. However, a similar analysis for individual experimental results (from Table \[Tab:ErrorsDataNum\]) reveals a different picture of $0.02$ (with 95% confidence limits $[0.02, 0.04]$). This suggests individual results from a ‘non-predatory’ paper clearly are less likely to contain errors than a ‘predatory’ paper, however, this is skewed by a small number of papers that contain very high numbers of results (and inconsistencies).
Predatory? Error Ok Unchecked
------------- ------- ---- -----------
No 10 11 10
‘Predatory’ 6 8 4
: Paper contains consistency errors and ‘predatory’ publisher[]{data-label="Tab:ErrorsPredPub"}
Predatory? Error Ok Unchecked
------------- ------- ------ -----------
No 88 1307 781
‘Predatory’ 174 64 42
: Predatory publisher and Error Data[]{data-label="Tab:ErrorsDataNum"}
### Use of comparator benchmarks {#subsubsec:benchmarks}
When assessing prediction methods such as UnSDP it is usual to provide some kind of comparator or benchmark. This is particularly helpful when a new or improved learning method is proposed. However, there is a risk that most of the research ‘energy’ is invested in the new method, as opposed to the benchmark. As Michie et al. [@Mich94] remarked 25 years ago, we need to be vigilant about the problems of comparing ‘pet’ algorithms in which researchers are expert with others with which they are less familiar. They also note the more general danger of selecting comparator benchmarks that are not state of the art. How does this apply to our systematic review? It potentially raises a bias most obviously with RQ4 that compares UnSDP and SDP predictive performance.
**SDPRef** **Description** **Study Count**
----------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------
SDPRefs + OrigResults Supervised methods refer to some literature, and they use the results reported in those literature 4
SDPRefs Supervised methods refer to some literature, but do not use the reported results 16
SDPNoRefs Supervised methods used but no reference to literature (5 out of 8 papers used very common supervised learners, e.g., Naïve Bayes, Random Forest, Logistic Regression) 8
Only UnSDP No supervised comparator models 21
Total 49
: Use of Supervised Learner Benchmarks and the Literature (by paper)
\[Tab:UseOfBenchmarks\]
Unfortunately we cannot know researchers’ intentions and making judgements as to when a method is a ‘pet’ requires a great deal of subjectivity. However, we can at least examine how papers use the literature with regard to SDPs. Table \[Tab:UseOfBenchmarks\] shows that of the 28 papers that use SDPs as a comparator for their UnSDPs, 71% (20/28) cite related articles, although only four use previous prediction results. In contrast 8/28 papers use supervised methods without citation suggesting that common methods such as logistic regression are seen as basic default methods that do not warrant discussion or tuning (see Section \[subsec:finemcc\] for an analysis of hyper-parameter tuning).
RQ3: What kinds of unsupervised learning research experiments are conducted?
----------------------------------------------------------------------------
For a more detailed analysis of unsupervised learning techniques, we categorise them into clustering and non-clustering techniques as presented in Table \[tbl:UnSvSDP\]. We further divide these unsupervised prediction techniques into 14 families and 21 sub-families.
\[tbl:UnSvSDP\]
**Clustering?** **MethodFamily** **Sub. Abbr** **SubFamily Approaches** **Related Study**
----------------- -------------------------------------------------------------------------------------------------------------------------------------------------- --------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
KM K-means: Each data point is assigned into the cluster according to the distance with the centroid of a cluster, which is the mean of all points within the cluster. The initial number of clusters is assigned manually.
XM X-means: An extended K-Means which tries to automatically determine the number of clusters. [P @catal2009software],[P @catal2010metrics], [P @park2014software]
KMD K-medoids: A clustering algorithm related to the k-means, while using an actual point in the cluster as centroid to clustering. [P @b2011application], [P @zhang2016cross]
**Hierarchical (HC)**: Grouping data into a hierarchy of clusters. HC Hierarchical clustering: Can be either agglomerative (bottom-up) or divisive (top-down). [P @chug2013software], [P @gupta2013estimating]
**Density (Density)**: Discover non-spherical clusters by finding core objects with dense neighborhoods. DBC Density based clustering: Connects core objects and their neighborhoods to form dense regions as clusters. [P @bishnu2012software],[P @chug2013software], [P @sandhu2010density]
NGas [Neural-Gas clustering: A competitive learning technique with SoftMax learning rule.]{}
SOMs Self-Organizing Maps: Produces a low-dimensional representation of the input space. [P @iwata2013error; @abaei2013fault; @marian2015software; @czibula2016novel; @iwata2012clustering], [P @abaei2015empirical], [P @boucher2017predicting],[P @boucher2018software]
FCM Fuzzy C-Means: Assigns membership to each data point corresponding to each cluster center on the basis of distance between the center and itself.
FSubC Fuzzy subtractive clustering: Uses fuzzy inferences based on rules generated by subtractive clustering. [P @yuan2000application],[P @sidhu2010subtractive]
FSOMs Fuzzy Self-Organizing Maps: Combines SOMs with the concept of fuzziness in fuzzy clustering. [P @czibula2016novel]
[**Spectral Clustering (SC)**: Make use of the eigenvalues of the similarity matrix to reduce dimensionality before clustering.]{} SC Partitions a dataset based on the connectivity between its nodes in a graph with spectral clustering. [P @zhang2016cross],[P @chang2017novel]
**Expectation maximization clustering (EM)**: Get probabilities of cluster memberships based on probability distributions. EM EM clustering: Finds the maximum likelihood by iteratively alternating between expectation step and maximization step. [P @coelho2014applying],[P @yang2006software], [P @park2014software],[P @guo2000software], [P @chang2017novel]
**Optimization (Opt)**: Taking clustering as an optimization problem, such as apply Particle Swarm Optimization to solve. PSO Particle Swarm Optimization clustering: Find the optimal clusters by iteratively improving a candidate solution with regard to a given measure. [P @coelho2014applying]
**Affinity Propagation (AP)**: Finds exemplars, members of the input set that are representative of clusters. AP Affinity Propagation clustering: Obtains the data similarities, and iteratively exchanges the real-valued messages between data points to clustering. [P @yang2008software],[P @yang2008Metrics]
CLA Clustering and labelling: Compute instances violation degree based on the relation between metric values and corresponding median values, then clustering by the violation or consistency information.
CLAMI Based on CLA, add two extra steps: Metric selection and instance selection based on violation degree. [P @yang2016defect],[P @n2015clami], [P @yan2017automated]
ACL Average Clustering and labelling: Clustering by metrics of instances violation score (MIVS), which is computed based on the relation between each metric value and their average metric value. [P @yang2016defect]
**Clustering Ensemble (CLEM)**: Ensemble multiple clustering solutions. CLEM Clustering Ensemble: Uses multiple algorithms to clustering, then combines the different clustering solutions and produces a single cluster. [P @coelho2014applying]
**Threshold (THD)$^{1} $**: Threshold for selected metric. It might be determined by experiences, rules, calculations, or machine learning, etc. THD Threshold: Use some metric thresholds to classify instances directly. e.g., given a metric threshold vector, if any metric of an instance exceeds the corresponding threshold, it will be defect-prone. [P @c2009clustering],[P @zhong2004analyzing], [P @abaei2013fault],[P @n2015clami], [P @catal2013fault],[P @abaei2015empirical], [P @boucher2018software]
**Expert (EXP)**: Determine by experience. EXP Expert determine the labels of each instance directly by their experience. [P @n2015clami]
**Ranking (MR)**: Metric Ranking MR Metric Ranking: For some change metrics in just-in-time prediction, using 1/Metric to rank instances in ascending order. [P @yang2016effort],[P @fu2017revisiting], [P @liu2017code; @yan2017file; @fan2017utility; @huang2017supervised], [P @chen2018multi]
For ‘Threshold’, if the threshold is calculated with defective labels in some method, we will mark it as supervised method, e.g. STHD-ROC in [P @boucher2018software] .
Clustering-based prediction approaches dominate. They typically include two phases: clustering and labelling. Firstly, all instances in a dataset are clustered into different groups, then each group is labeled as defective or not. In addition, there are also a few non-clustering unsupervised prediction techniques, such as using thresholds that label instances directly.
Labelling is a necessary step in unsupervised clustering defect prediction. Table \[tbl:labelling\] summarises all labelling approaches located by our review. This shows considerable diversity. Generic thresholds, as opposed to Pareto or distribution methods, are the most popular. However, practitioners have to consider what is the most suitable labelling technique when employing some particular learners, such as Clustering and Labelling (CLA, see Table \[tbl:UnSvSDP\]).
\[tbl:labelling\]
**Approaches** **Description** **Paper**
--------------------- ---------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------
Expert Decided cluster label by expert [P @zhong2004analyzing],[P @z2004unsupervised], [P @n2015clami]
Threshold Decided cluster label by particular metric threshold values [P @c2009clustering], [P @abaei2015empirical]
Distribution According to the number of nodes in a cluster, scattered (smaller) is defect-prone; concentrative (larger) is defect-free. [P @yang2008software],[P @park2014software],[P @guo2000software], [P @yuan2000application],[P @yang2008Metrics]
Majority vote Majority voting by the most three similar clustering centres. [P @abaei2015increasing]
Top half Half of the top clusters are defect-prone, e.g., rank clusters according to violation degree in descending order. [P @zhang2016cross],[P @n2015clami],[P @yan2017file], [P @yan2017automated]
Supervised learning Use supervised models to predict cluster labels. [P @boucher2018software]
‘Expert’ and ‘Threshold’ are used for cluster label decision in Table \[tbl:labelling\], and for instance label decision in Table \[tbl:UnSvSDP\].
Analysis of Unsupervised Defect Prediction Performance {#sec:Perform}
======================================================
This section addresses research questions RQ4–RQ6. In RQ4, we conduct a meta-analysis by vote-counting. This is not an ideal approach to meta-analysis, but necessary in order to avoid losing important primary studies which employ a range of response measures (performance measures) and experimental designs. Meta-analysis by vote-counting is a simple, though less powerful, procedure than directly using effect size, to compare the performance of two groups. It is only recommended when there are problems with a direct approach [@Bush09]. Our meta-analysis is hindered in that some studies only report AUC. So even though the experiments are informative, they would not contribute to the meta-analysis if we choose F1 to make comparisons. Essentially vote-counting does not directly use effect size estimates from each primary study. Although at its simplest it proceeds by comparing the number of ‘positive’ studies with ‘negative’ studies[^8], more sophisticated vote-counting methods are available (see Bushman and Wang [@Bush09] for an overview). We use the Hedges and Olkins method for unequal sample sizes [@Hedges1985statistical pp. 47–74] in order to compare UnSDP and SDP.
$ Study $ $ Expt^{1} $ $ Prj^{1} $
-- ------------------------------------------------------------- -------------- ----------------- -----
49 2456 128
be counted if some result is consistent or incomplete$ ^2 $ 47 2194 125
not be counted as long as a result is inconsistent $ ^2 $ $ 33^3 $ 2194 89
$ 26^{4} $ expts: 2052 - 110 expts: 2128
22 1178 85
5 351 16
22 426 85
5 107 16
: The volume of data involved in analysis[]{data-label="Tab:NumAnalysis"}
Expt = count of experimental results; Prj = count of software projects.
One study could contain both inconsistent and consistent results. When we use “be counted if some result is consistent or incomplete”, then Expt will be increased by 1, while when using “not be counted as long as a result is inconsistent”, Expt will not be increased. Similarly with Prj.
There are 16 studies that include inconsistent experiments according to Figure \[Fig:ErrsPerPaper\], thus only 33 studies could be used.
In total, there are 28 studies that include both UnSDP and SDP model, but all of the SDP experiment results in these two studies are inconsistent, so we removed these two papers.
We also carry out a finer grained meta-analysis. Table \[Tab:NumAnalysis\] describes the number of studies, experimental results and projects (or datasets) in our meta-analysis. Note that in the RQ4 vote-counting comparison, 26 studies including 2052 results are involved when we take study as a voting unit, and 110 projects including 2128 results are employed when project is taken as the voting unit.
RQ4: What is the difference between unsupervised and supervised software defect predictive performance? {#subsec:UnSDPvSDP}
-------------------------------------------------------------------------------------------------------
In this section, we compare UnSDP and SDP models at a coarse (Section \[subsec:vote-counting\]) and finer (Section \[subsec:finemcc\]) level respectively. All 1306 supervised experimental results are taken from our 26 primary studies which include both unsupervised and supervised models.
### Comparison by vote-counting with multiple measures {#subsec:vote-counting}
We carry out vote-counting comparison in this section. Note the possibility that some conclusions might be unstable depending on whether making the voting unit a project or study (paper). For example, this has been a factor for studies [P @yang2016effort], [P @fu2017revisiting]. So we carry out the vote-counting analysis from both perspectives: (i) by study (each study can vote once) and (ii) dataset (a vote per project). Based on the method of Hedges and Olkin [@Hedges1985statistical], we carry out the vote-counting as follows.
1. Identify voting units which must include *both* unsupervised and supervised prediction models to enable comparability. Voting unit type can be study or project. Depending on the choice of voting unit, all eligible studies or projects are identified. Each eligible study or project will vote once to determine whether UnSDP is better than SDP.
2. Determine the sample size (number of experiments) for each paper or project, and calculate mean, standard deviation of response variable for each group (UnSDP and SDP). The priority of response variable is: MCC, F1, AUC, Popt, Accuracy, ER, Recall, Precision, GMean, ACC, GMeasure, Balance, Purity, MAE and MeanAIC. For example, if MCC and F1 are available in primary study $ P1 $, MCC will be used to vote for $ P1 $. This priority is determined by the number of all available results for each response variable among 2194 consistent experiment results.
3. Determine $X$ for each voting unit, where $ X $ is the sign of ES (Effect Size based on the mean difference between two groups). If $ \bar{Y}_{i}^{U} - \bar{Y}_{i}^{S} > 0 $, $ X =1$ otherwise, $ X=0 $, where $ \bar{Y}_{i}^{U}$ refers to the mean of response variable for unsupervised models, and $\bar{Y}_{i}^{S} $ is the mean for supervised models.
4. Construct the log likelihood function, and obtain the maximum likelihood estimator $ \hat{\theta}$ ($ \theta $ refers to ES).
$$\label{logfunc}
\centering
\small
\begin{split}
& \displaystyle{\sum_{i=1}^{k}} \{X_{i} log[1-\Phi (-\sqrt{\bar{{n}_{i}}}\theta)] + (1-X_{i}) log \Phi (-\sqrt{\bar{{n}_{i}}}\theta) \}
\end{split}$$
where $ k $ is the number of eligible vote entities, $ \Phi $ is the cumulative distribution function, $ \bar{{n}_{i}}$ is average sample size, $\bar{{n}_{i}} = (n1 \times n2)/(n1+n2)$, $n1$ and $n2$ are the sample sizes of unsupervised and supervised experiments respectively.
5. Compute the 95% confidence interval for $\hat{\theta}$: \[$\theta_{L}$, $\theta_{U}$\] .
$\theta_{L} = \hat{\theta} - C_{\alpha/2} \sqrt{Var(\hat{\theta})}$
$\theta_{U} = \hat{\theta} + C_{\alpha/2} \sqrt{Var(\hat{\theta})}$
where, $Var(\hat{\theta}) $ is the variance of $ \hat{\theta} $. $ C_{\alpha/2} $ is the two-tailed critical value of the standard normal distribution. In this paper, we use $ C_{\alpha/2}=1.96 $ (central area=0.95, $Z_{\alpha} $=1.96). The variance computation is:
$ Var(\hat{\theta}) = \{\displaystyle{\sum_{i=1}^{k}} (
D_{i}^{1} + D_{i}^{2} - (D_{i}^{1})^2 \frac{(1-2p_{i})}{p_{i}(1-p_{i})}) \}^{-1}$
where,
$p_{i} = p(\delta,\bar{{n}_{i}}) = 1-\Phi (-\sqrt{\bar{{n}_{i}}}\delta)) $,
$D_{i}^{1}= \frac{\partial{p(\delta,\bar{{n}_{i}})}}{\partial{\delta}}
= \sqrt{\frac{\bar{{n}_{i}}}{2\pi}} exp(-\frac{1}{2}\bar{{n}_{i}} \delta^2 )$,
$D_{i}^{2}= \frac{\partial^2{p({\delta},\bar{{n}_{i}})}}{\partial{\delta}^2}
= -{\frac{\bar{{n}_{i}}*{\delta}}{\sqrt{2\pi}}} exp(-\frac{1}{2}\bar{{n}_{i}} \delta^2 )$.
$\delta $ is the population standardized mean difference, and it is equal to the estimated $\hat{\theta}$ in Step 4.
In terms of voting unit, from our meta-analysis there are 26 studies (papers) or 110 projects that conduct both unsupervised and supervised learning for defect prediction. If we carry out vote-counting for all combinations of Prediction Type, Predatory and Cross validation, unfortunately there are only sufficient results for 8 conditions, these are listed in bold in Table \[tbl:votecounting\]. This is because the number of eligible study or project results is only zero or one for the other 8 combinations, e.g., if we use project as the voting unit there are no results for cross-project prediction from ‘predatory’ publishers that explicitly use cross-validation. Another point worth mentioning is that the sum of studies or projects in Table \[tbl:votecounting\] or Figure \[fig:vote\] is not necessarily 26 or 110. For example: one study may include both within-project and cross-project prediction, it would be counted more than once.
--------- ----------- ------ ------- --------
Voting Pred Pred Cross Count
Unit Type Pub? Val?
Study WithinPrj Yes Yes 1
Study WithinPrj Yes ? 1
Study WithinPrj No Yes **19**
Study WithinPrj No ? **4**
Study CrossPrj Yes Yes 0
Study CrossPrj Yes ? 0
Study CrossPrj No Yes **4**
Study CrossPrj No ? 1
Project WithinPrj Yes Yes **7**
Project WithinPrj Yes ? 1
Project WithinPrj No Yes **94**
Project WithinPrj No ? **7**
Project CrossPrj Yes Yes 0
Project CrossPrj Yes ? 0
Project CrossPrj No Yes **38**
Project CrossPrj No ? **10**
--------- ----------- ------ ------- --------
: Valid combinations for vote-counting[]{data-label="tbl:votecounting"}
Figures \[fig:vote-paper\] and \[fig:vote-prj\] show the results of voting by study (paper) and by (software) project respectively. Here, positive values of ES mean UnSDP models perform better than SDP models and negative values the reverse. The vertical dashed line indicates no difference ($ES=0$). The length of each horizontal line shows the 95% Confidence Intervals (CI) and the rectangle the estimate of the effect from the relevant set of results. Note that some of the CI lines (e.g., the second case in Figure \[fig:vote-paper\]) end with an arrow, meaning the lower or upper bound of CI exceeds the bound of the legend $[-0.6 ,0.6]$.
Figure \[fig:vote-paper\] shows the results of voting by 26 studies including 2052 experiments. For within-project prediction, although the sign of effect sizes is negative, both of their 95% CI contain zero, which indicates that it is possible that neither performs better than the other. In other words, UnSDP appears comparable with SDP. The same explanation also can be applied in Cross-project prediction as the 95% CI goes across zero.
Figure \[fig:vote-prj\] shows the results of voting by 110 projects including 2128 experiments. These results are broadly similar to voting by paper in that the 95% CIs straddle zero for all five cases. So this suggests that for both within-project and cross-project prediction UnSDP has potential, since it has data collection advantages over SDP, specifically the need for labelled training data is reduced or removed.
Note that the CIs are widest when there are only a few data points. It would also seem that the results are most positive to UnSDP when it is unclear that a cross-validation has been employed. This suggests a certain degree of caution is warranted before claims of superiority are made.
Overall, the results of vote-counting show that UnSDP models are comparable with SDP models, which could make practitioners and researchers more confident to use UnSDP models. However differences in performance are reduced to single votes. So next we focus on the more informative performance indicator MCC, despite the fact that this reduces the amount of data available.
### Comparison of UnSDP and SDP using MCC {#subsec:finemcc}
{width="95.00000%"}
The previous vote-counting comparison does not consider the impact of different prediction performance measures, dataset, etc. For subsequent analysis we restrict our data to all available, non-predatory results for which MCC is available. This yields 1178 observations to compare UnSDP and SDP. N.B. the MCC results are all recomputed from the reported raw results from the primary studies.
The side by side boxplots in Figure \[fig:wp-mcc\] and Figure \[fig:cp-mcc\] illustrate within-project and cross-project defect prediction performance respectively. For more details on the prediction model family refer to Table \[tbl:UnSvSDP\] (for unsupervised learners) and Table \[tbl:SvSDP\] (for supervised learners). Although sample size varies considerably among these learners (from 1 to 127), they still illustrate some overall patterns. The boxplot notches indicate the 95% confidence intervals of the median. For smaller samples these are wider. Although not a formal test, there is evidence that medians differ if the notches do not overlap [@chambers1983graphical].
From Figure \[fig:wp-mcc\], we observe that unsupervised models for within-project prediction show greater variance in prediction performance than the supervised ones. Although it is hard to obtain a definitive overall conclusion which is better, these results are consistent with our vote-counting results, namely the evidence does not differentiate their relative performances to any substantial degree.
In more detail, and restricting our comments to sample sizes $ > 10$, and therefore more reliable (from Figure \[fig:wp-mcc\]) we note the following.
1. Un\_Fuzzy (the blue line in Figure \[fig:wp-mcc\]) outperforms other SDP and UnSDP model families in that the median of Un\_Fuzzy is higher than that of the other families of learners. However, the CI is wide so it overlaps with other models at least in some contexts. Among the supervised approaches Bayes appears to be the best model which has also been noted by previous studies [@hall2012systematic].
2. Traditional unsupervised KPart (partition-based clustering including the widely used KMeans and its variants) family did not perform very well. The median of KPart (the red line in Figure \[fig:wp-mcc\], 0.33) is lower than the median of all SDP models(0.41). This suggests that researchers should be cautious in using KMeans as a benchmark with which to compare their new UnSDP.
3. Un\_MR (metric or 1/metric ranking) performs least well. However, this is based on only a small subset of the data, since only 39 out of the 234 Un\_MR experimental results use MCC with the majority only reporting Popt or ACC.
![Defect prediction performance (MCC) in cross-project prediction. N.B. the numbers indicate the count of experiments (sample size).[]{data-label="fig:cp-mcc"}](cp-mcc){width="40.00000%"}
Figure \[fig:cp-mcc\] shows the performance of cross-project prediction models. Un\_Fuzzy again seems to have the best performance. Presently there are quite limited UnSDP results so further experimentation would be welcome.
We also observe that an unsupervised spectrum clustering models approach (Un\_SC family) was proposed in [P @zhang2016cross], and their model was evaluated with 26 different datasets. It outperformed most of the SDP and other UnSDP models in within-project prediction, and outperformed all others UnSDP and SDP in cross-project prediction. But unfortunately they only reported AUC, so their model could not be included in Figure \[fig:wp-mcc\] and Figure \[fig:cp-mcc\]. This is somewhat frustrating as more complete reporting (of the confusion matrix) would enable integration of their results into meta-analyses.
In summary, UnSDP models are comparable with SDP models both in within-project and cross-project prediction. Compared with most of the SDP models, Un\_Fuzzy, Un\_NN and Un\_SC are potentially the strongest UnSDP approaches. However, we also checked all 28 primary studies that include SDP models to see whether any parameter tuning was used. Surprisingly, we found only 3 studies state clearly that tuned SDP models are used in their comparisons, 6 studies use default parameter and 19 studies provide no information about tuning. Therefore, most of SDP models used in comparison might not be best models. We discuss this potential source of bias in the threats to validity (Section \[sec:Threats\]).
### Comparison of UnSDP and SDP using Popt for JiT {#subsec:finepopt}
Due to incomplete reporting, there are 823 unchecked experimental results that could not be compared with MCC in Section \[subsec:finemcc\]. Among these results, AUC and Popt are dominant reported performance measures. Because effort-aware just-in-time (JiT) defect-prone commit prediction differentiates with defect-prone module prediction, effort consideration is its important characteristic. To present a comprehensive comparison of UnSDP and SDP, we carry out the meta-analysis with Popt for JiT prediction in this section. To solid our meta-analysis results, we also compare the unchecked UnSDP and SDP with AUC in Section \[sec:Threats\].
Figure \[fig:popt\] presents Popt comparison for JiT prediction. It illustrates that SDP model GA (multi-objective optimization based on genetic algorithm) performs better than the only UnSDP model family UN\_MR. However, GA needs to cost more time than UN\_MR for model construction analyzed in [P @chen2018multi]. From Figure \[fig:popt\], we can found that UN\_MR is still a good choice for practitioners since it has better prediction performance and less time. In summary, for JiT defect prediction, the comparisons with Popt indicates UnSDP models are comparable with SDP models both in within-project and cross-project prediction.
RQ5:Which unsupervised prediction models or model families perform best? {#subsec:UnSDP}
------------------------------------------------------------------------
To investigate which models are better among UnSDP models, we compare them with more detailed subfamily categories. Table \[tbl:UnSDPFamily\] lists the mean value of MCC, the corresponding 95% CI and the number of experiments for all available unsupervised models in within-project defect prediction. Here, we only list the models in which experimental sample size is greater than 1.
\[tbl:UnSDPFamily\]
MTinyFamily MCC \#Exp %95 CI
---- ---------------- --------- ------- -------------------------
1 **FCM** 0.5972 11 \[ 0.4357 , 0.7586 \]
2 **FSOMs** 0.5646 5 \[ 0.3309 , 0.7984 \]
3 EXP 0.5600 6 \[ 0.3757 , 0.7442 \]
4 KMD 0.4970 9 \[ 0.3276 , 0.6665 \]
5 NGas 0.3949 10 \[ 0.1936 , 0.5962 \]
6 SC 0.3799 6 \[ 0.2814 , 0.4784 \]
7 THD 0.3740 28 \[ 0.3200 , 0.4279 \]
8 ACL 0.3549 16 \[ 0.2925 , 0.4172 \]
9 CLA 0.3489 23 \[ 0.3014 , 0.3965 \]
10 SOMs 0.3485 117 \[ 0.3144 , 0.3826 \]
11 CLAMI 0.3003 52 \[ 0.2648 , 0.3358 \]
12 KM 0.2967 86 \[ 0.2529 , 0.3406 \]
13 EM 0.2790 3 \[ 0.0165 , 0.5414 \]
14 CLAMIPlus $^1$ 0.2139 13 \[ 0.1358 , 0.2920\]
15 MR-OneWay $^2$ -0.0368 16 \[ -0.1003 , 0.0268 \]
16 MR -0.1316 23 \[ -0.1952 , -0.0681 \]
: UnSDP families MCC performance (within-project). Since there is only one experiment in the family of FSubC and AP, %95 CI could not be calculated and we removed these two ones from 426 experiments.
CLAMIPlus[P @yan2017automated] is a variant of CLAMI.
MR-OneWay[P @fu2017revisiting] is an improved method based on MR. The same name is also used in Table \[tbl:UnSDPJitMethod\].
From these results, we can see that FCM and FSOMs learners perform best. Although EXP also performs competitively, we do not recommend it since EXP requires software modules to be classified by an expert manually. Hence we do not consider it to be an effective UnSDP approach.
For JiT defect prediction, we conduct the comparison based on 197 high quality Popt values including 12 UnSDP methods based on MR (metric ranking). Here, we named these methods with MR-$v$ and $v$ is the metric used to rank, e.g. MR-RFC. Also if time-wise cross validation is used in one method, we name it MR-$v$\_TIME, otherwise it means normal n-fold cross validation is used. Table \[tbl:UnSDPJitMethod\] shows MR-CCUM performs best among those unsupervised JiT within-project prediction methods. MR-LT and MR-AGE are followed by MR-CCUM. For 97 cross-project JiT prediction experiments, only MR-CCUM, MR-LT and MR-AGE are included. MR-CCUM is also the best method among them.
\[tbl:UnSDPJitMethod\]
MethodName Popt \#Exp %95CI
---- ----------------- -------- ------- -----------------------
1 MR-CCUM $^1$ 0.8930 5 \[ 0.8311 , 0.9549 \]
2 MR-CCUM\_TIME 0.8648 6 \[ 0.8217 , 0.908 \]
3 MR-LT$^2$ 0.7486 24 \[ 0.7247 , 0.7726 \]
4 MR-AGE$^3$ 0.7403 18 \[ 0.7057 , 0.775 \]
5 MR-LT\_TIME 0.7120 6 \[ 0.6784 , 0.7456 \]
6 MR-NF $^4$ 0.7065 6 \[ 0.6382 , 0.7748 \]
7 MR-AGE\_TIME 0.7017 6 \[ 0.6461 , 0.7572 \]
8 MR-OneWay 0.6933 6 \[ 0.6379 , 0.7487 \]
9 MR-NUC $^5$ 0.6795 6 \[ 0.5863 , 0.7727 \]
10 MR-OneWay\_TIME 0.6750 4 \[ 0.5923 , 0.7577 \]
11 MR-RFC $^6$ 0.6317 10 \[ 0.5659 , 0.6975 \]
12 MR-AMC $^7$ 0.6285 10 \[ 0.5612 , 0.6958 \]
: UnSDP methods performance for JiT prediction (within-project)
CCUM : Code churn based unsupervised model. [P @yan2017automated]
LT: Lines of code in a file before the current change. [P @yang2016effort]
AGE: The average time interval (in days) between the last and the change over the files that are touched.[P @yang2016effort]
NF: Number of files touched by the current change. [P @yang2016effort]
NUC: Number of unique changes to the modified files. [P @yang2016effort]
RFC: Response of a Class. [P @yan2017file]
AMC: Average Method Complexity.[P @yan2017file]
RQ6: What is the impact of dataset characteristics on predictive performance? {#subsec:Dataset}
-----------------------------------------------------------------------------
Studies ([P @yang2016effort] and [P @fu2017revisiting]) reveal inconsistent results between predicting defect-prone modules by each project and by all projects as a whole in a just-in-time context. Stimulated by these findings, we also undertake a comparison between UnSDP and SDP models based on individual projects.
Before the comparison of RQ6, we removed two kinds of unsuitable data from Table \[tbl:UnSDPFamily\]: (i) EXP related data since expert-based approach is more subjective. (ii) Just-in-time datasets(including BUG, POS, MOZ, PLA, JDT, COL) due to their prominent characteristics compared with defect-prone module prediction dataset.
Figure \[fig:relation\] presents the comparison between the *best* UnSDP and SDP models by project. In this figure, horizontal axis label stands for each project and its fault rate. For example, Ant (0.24) denotes the project Ant (Ant1.3, Ant 1.4, etc. are different versions which we unify as Ant). The fault rate 0.24 is average value across all versions. Figure \[fig:relation\] shows that there are similar performance change tendencies for UnSDP and SDP over the different projects. The two groups both have higher prediction performance on some projects (such as Argouml, AR5, Weka, etc.) and lower performance on other projects (such as KC2, ZXing, etc.). This suggests dataset characteristics have a non-trivial moderating effect on prediction performance. We analysed the largest two abnormal gaps Xerces and Xalan (Diff $ > $ 0.2) between UnSDP and SDP in Figure \[fig:relation\]. According to the original primary study [P @yang2016defect], the gaps of Xerces and Xalan seem to be caused by their dataset characteristics.
{width="98.00000%"}
We investigated one dataset characteristic, namely imbalance (fault rate), but the one-way ANOVA analysis for fault rate and max MCC shows that it is a very weak explanatory factor for MCC (one-way ANOVA F = 0.66, p-value = 0.6261). In summary, we consider the dataset characteristics have an obvious impact on predictive performance. However, it is unclear which kinds of characteristics are important. We consider this would be well worth exploring in the future.
Threats to Validity {#sec:Threats}
===================
Threats to *internal validity* relate to our ability to reconstruct a confusion matrix for each experimental result. Ideally, we would recompute the confusion matrix for all 2456 individual experimental results, however, there are 823 ($~33\%$) results that could not be checked due to incomplete reporting. Where we could check we found 262 results are problematic, but it may well be that there is an interaction between non-reporting and error-proneness so simply extrapolating from our error-rate findings may not be safe.
Another threat is our implementation of re-computation with R. Where possible we compared our partial analysis with the java tool DConfusion [@bowes2014dconfusion]. Our results are consistent with DConfusion.
The third threat is that we only carry out the fine level comparison between UnSDP and SDP with MCC. Although MCC is the best choice for our meta-analysis, to strengthen our analysis, we also conduct an extra analysis with high quality 458 AUC results (Predatory = No and AUC is not NA) . Our comparison results also indicate that Un\_Fuzzy is the most potential approach for within-project prediction. However, the number of experiments is relatively smaller, and more evaluations of Un\_Fuzzy would be recommended. The family of Un\_SC reported only with AUC performs best in cross-project prediction. The analysis results could be found in our Mendeley dataset.
Further, we could only check for consistency errors so it is quite possible that the *actual* rate of experimental analysis errors is greater than we were able to detect. We have to remark that an overall (knowable) error rate of 11% across all publication venues does not engender confidence. We hope a move towards more open science [@Muna17] will assist in this regard.
Threats to *external validity* concern the selected experiments and the extent to which the experiments we have located generalise. By undertaking an explicitly systematic approach [@Kitc15] to this review we hope to have included all relevant studies. These have used a wide range of datasets but we cannot be certain how representative these might be of all possible software defect prediction scenarios.
Another difficulty with the comparison of unsupervised and supervised classification is that the majority of papers appear to focus principally upon novel or innovative uses of UnSDP and that the comparator supervised approaches serve the role of very basic benchmarks. Consequently it is not always obvious that state of the art supervised algorithms are deployed, nor that much effort has gone into hyper-parameter tuning. For instance, only 3/28 papers explicitly state they have tuned the SDP models and a further 6/28 indicated defaults were chosen. Thus there is a danger we are sometimes comparing state of the art UnSDP with off-the-shelf SDP.
A final danger is researcher bias and the tendency to confirm what we already believe to be true. This is an additional reason why we have made our materials and research processes public$^1$.
Summary and actionable findings {#sec:Conc}
===============================
UnSDP techniques are attracting more and more researchers and practitioners since labelling information for the training data is not a prerequisite. We reviewed 49 unsupervised software defect prediction primary studies published between January 2000 to March 2018, which includes 2456 independent experimental results. These 2456 independent results involved 128 software projects (from NASA, PROMISE, ISM, AEEEM, etc.) and 25 prediction model families (172 models). All the UnSDP models and labelling techniques for clusters are listed in Table \[tbl:UnSvSDP\] and Table \[tbl:labelling\]. These tables also provide researchers with an indication of the diversity of unsupervised learning techniques used for software defect prediction.
We have carried out two kinds of analysis.
Firstly, we conducted a bibliometric analysis. We found a growth in research activity in recent years. We also have found some problems with experimental data quality. We were surprised to find that 25 out of 49 studies did *not* explicitly report whether a cross-validation procedure was used or not, and there were 14 papers (out of 49) that could not be checked for the existence of problematic data due to incomplete reporting. Therefore, the quality and completeness of reporting should be paid more attention when publishing papers. Where we were able to assess consistency of results, we found something of the order of 11% of all results were demonstrably in error due to inconsistencies. This was an issue in both ‘predatory’ and non-predatory publication venues.
Secondly, we undertook a meta-analysis concerning the performance of UnSDP models. To compare these results in a reliable way, we recomputed the confusion matrices of the primary studies to obtain consistent performance measures. In our vote-counting analysis, MCC is the main performance measure and others (such as AUC, F1, Popt, etc) are only used when MCC is unavailable. We found, that UnSDP models are comparable with SDP models both for within-project and cross-project prediction. This indicates UnSDP models may not be as problematic as might be supposed, so we suggest they might be considered in most situations when labelled training data is scarce. Among the different UnSDP learners, Un\_Fuzzy and Un\_SC appear to have most potential. Meanwhile, for effort-aware JiT defect prediction, the state-of-the-art complex SDP model GA (multi-objective optimization based on genetic algorithms) performs better than the simple sorting UnSDP model (UN\_MR). However, it less clear whether it has better performance when effort is ignored. Overall, we consider that UN\_MR remains a good choice for practitioners since it has comparable prediction performance to GA model and requires less time.
We also note that when clustering-based UnSDP approaches are applied, it is a simple and effective way to use the node distribution in clusters (see Table \[tbl:labelling\]) to label each cluster as defective or not according to the Pareto principle (sometimes referred to as the 80:20 rule).
Finally, we found dataset characteristics can have a clear impact on predictive performance, no matter whether UnSDP or SDP models are used. Therefore it will be worth exploring in more depth which of these dataset characteristics are most important. Likewise, the exploration of the interaction between learner and dataset could be fruitful, as it would appear unlikely that a single learning algorithm, be it supervised or unsupervised, will always be optimal.
Primary Studies {#sec:paper .unnumbered}
===============
Search completed 7th March, 2018.
bibitem\[1\]
@filesw auxout
biblabel\#1[\[P\#1\]]{}
[10]{} C. Catal, U. Sevim, B. Diri, Software fault prediction of unlabeled program modules, in: Proceedings of the world congress on engineering, Vol. 1, 2009, pp. 1–6.
C. Catal, U. Sevim, B. Diri, Clustering and metrics thresholds based software fault prediction of unlabeled program modules, in: Information Technology: New Generations, 2009. ITNG’09. Sixth International Conference on, IEEE, 2009, pp. 199–204.
S. Zhong, T. M. Khoshgoftaar, N. Seliya, Analyzing software measurement data with clustering techniques, IEEE Intelligent Systems 19 (2) (2004) 20–27.
S. Zhong, T. M. Khoshgoftaar, N. Seliya, Unsupervised learning for expert-based software quality estimation., in: HASE, 2004, pp. 149–155.
P. S. Bishnu, V. Bhattacherjee, Software fault prediction using quad tree-based k-means clustering algorithm, IEEE Transactions on knowledge and data engineering 24 (6) (2012) 1146–1150.
P. S. Bishnu, V. Bhattacherjee, Application of k-medoids with kd-tree for software fault prediction, ACM SIGSOFT Software Engineering Notes 36 (2) (2011) 1–6.
S. Singh, R. Singla, Comparative performance of fault-prone prediction classes with k-means clustering and mlp, in: Proceedings of the Second International Conference on Information and Communication Technology for Competitive Strategies, ACM, 2016, p. 65.
S. Varade, M. Ingle, Hyper-quad-tree based k-means clustering algorithm for fault prediction, International Journal of Computer Applications 76 (5).
R. A. Coelho, F. dos RN Guimar[ã]{}es, A. A. Esmin, Applying swarm ensemble clustering technique for fault prediction using software metrics, in: Machine Learning and Applications (ICMLA), 2014 13th International Conference on, IEEE, 2014, pp. 356–361.
A. Chug, S. Dhall, Software defect prediction using supervised learning algorithm and unsupervised learning algorithm, in: Confluence 2013: The Next Generation Information Technology Summit (4th International Conference), IET, 2013, pp. 173–179.
D. Gupta, V. K. Goyal, H. Mittal, Estimating of software quality with clustering techniques, in: Advanced Computing and Communication Technologies (ACCT), 2013 Third International Conference on, IEEE, 2013, pp. 20–27.
B. Yang, Q. Yin, S. Xu, P. Guo, Software quality prediction using affinity propagation algorithm, in: Neural Networks, 2008. IJCNN 2008.(IEEE World Congress on Computational Intelligence). IEEE International Joint Conference on, IEEE, 2008, pp. 1891–1896.
B. Yang, X. Zheng, P. Guo, Software metrics data clustering for quality prediction, Computational Intelligence (2006) 959–964.
D. Kaur, A comparative study of various distance measures for software fault prediction, arXiv preprint arXiv:1411.7474.
P. S. Sandhu, J. Singh, V. Gupta, M. Kaur, S. Manhas, R. Sidhu, A k-means based clustering approach for finding faulty modules in open source software systems, World academy of science, Engineering and technology 72 (2010) 654–658.
P. Singh, S. Verma, An efficient software fault prediction model using cluster based classification, International Journal of Applied Information Systems (IJAIS) 7 (3) (2014) 35–41.
D. Kaur, A. Kaur, S. Gulati, M. Aggarwal, A clustering algorithm for software fault prediction, in: Computer and Communication Technology (ICCCT), 2010 International Conference on, IEEE, 2010, pp. 603–607.
G. Abaei, A. Selamat, Increasing the accuracy of software fault prediction using majority ranking fuzzy clustering, in: Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing, Springer, 2015, pp. 179–193.
C. Catal, U. Sevim, B. Diri, Metrics-driven software quality prediction without prior fault data, in: Electronic Engineering and Computing Technology, Springer, 2010, pp. 189–199.
M. Park, E. Hong, Software fault prediction model using clustering algorithms determining the number of clusters automatically, International Journal of Software Engineering & Its Applications 8 (7) (2014) 199–204.
P. Guo, M. R. Lyu, Software quality prediction using mixture models with em algorithm, in: Quality Software, 2000. Proceedings. First Asia-Pacific Conference on, IEEE, 2000, pp. 69–78.
P. S. Sandhu, M. Kaur, A. Kaur, A density based clustering approach for early detection of fault prone modules, in: Electronics and Information Engineering (ICEIE), 2010 International Conference On, Vol. 2, IEEE, 2010, pp. 525–530.
X. Yuan, T. M. Khoshgoftaar, E. B. Allen, K. Ganesan, An application of fuzzy clustering to software quality prediction, in: Application-Specific Systems and Software Engineering Technology, 2000. Proceedings. 3rd IEEE Symposium on, IEEE, 2000, pp. 85–90.
R. S. Sidhu, S. Khullar, P. S. Sandhu, R. Bedi, K. Kaur, A subtractive clustering based approach for early prediction of fault proneness in software modules, World Academy of Science, Engineering and Technology 4 (7) 1165–1169.
F. Zhang, Q. Zheng, Y. Zou, A. E. Hassan, Cross-project defect prediction using a connectivity-based unsupervised classifier, in: Proceedings of the 38th International Conference on Software Engineering, ACM, 2016, pp. 309–320.
J. Yang, H. Qian, Defect prediction on unlabeled datasets by using unsupervised clustering, in: High Performance Computing and Communications; IEEE 14th International Conference on Smart City; IEEE 2nd International Conference on Data Science and Systems (HPCC/SmartCity/DSS), 2016 IEEE 18th International Conference on, IEEE, 2016, pp. 465–472.
B. Yang, X. Chen, S. Xu, P. Guo, Software metrics analysis with genetic algorithm and affinity propagation clustering, in: DMIN, 2008, pp. 590–596.
K. Iwata, T. Nakashima, Y. Anan, N. Ishii, Error prediction methods for embedded software development using hybrid models of self-organizing maps and multiple regression analyses, in: Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing 2013, Springer, 2013, pp. 185–200.
G. Abaei, Z. Rezaei, A. Selamat, Fault prediction by utilizing self-organizing map and threshold, in: Control System, Computing and Engineering (ICCSCE), 2013 IEEE International Conference on, IEEE, 2013, pp. 465–470.
Z. Marian, G. Czibula, G. Czibula, S. Sotoc, Software defect detection using self-organizing maps, Studia Universitatis Babes-Bolyai, Informatica 60 (2) (2015) 55–69.
I.-G. Czibula, G. Czibula, Z. Marian, V.-S. Ionescu, A novel approach using fuzzy self-organizing maps for detecting software faults, Studies in Informatics and Control 25 (2) (2016) 207–216.
K. Iwata, T. Nakashima, Y. Anan, N. Ishii, Clustering and analyzing embedded software development projects data using self-organizing maps, Software Engineering Research, Management and Applications 2012 (2012) 47–59.
S. Aleem, L. Capretz, F. Ahmed, Benchmarking machine learning techniques for software defect detection, Int. J. Softw. Eng. Appl 6 (3).
J. Nam, S. Kim, Clami: Defect prediction on unlabeled datasets, in: Automated Software Engineering (ASE), 2015 30th IEEE/ACM International Conference on, IEEE, 2015, pp. 452–463.
C. Catal, B. Diri, A fault detection strategy for software projects, Tehni[č]{}ki vjesnik 20 (1) (2013) 1–7.
Y. Yang, Y. Zhou, J. Liu, Y. Zhao, H. Lu, L. Xu, B. Xu, H. Leung, Effort-aware just-in-time defect prediction: simple unsupervised models could be better than supervised models, in: Proceedings of the 2016 24th ACM SIGSOFT International Symposium on Foundations of Software Engineering, ACM, 2016, pp. 157–168.
W. Fu, T. Menzies, Revisiting unsupervised learning for defect prediction, in: Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering, ACM, 2017, pp. 72–83.
G. Abaei, A. Selamat, H. Fujita, An empirical study based on semi-supervised hybrid self-organizing map for software fault prediction, Knowledge-Based Systems 74 (2015) 28–39.
J. Liu, Y. Zhou, Y. Yang, H. Lu, B. Xu, Code churn: A neglected metric in effort-aware just-in-time defect prediction, in: Empirical Software Engineering and Measurement (ESEM), 2017 ACM/IEEE International Symposium on, IEEE, 2017, pp. 11–19.
M. Yan, Y. Fang, D. Lo, X. Xia, X. Zhang, File-level defect prediction: Unsupervised vs. supervised models, in: Empirical Software Engineering and Measurement (ESEM), 2017 ACM/IEEE International Symposium on, IEEE, 2017, pp. 344–353.
Y. Fan, C. Lv, X. Zhang, G. Zhou, Y. Zhou, The utility challenge of privacy-preserving data-sharing in cross-company defect prediction: An empirical study of the cliff&morph algorithm, in: Software Maintenance and Evolution (ICSME), 2017 IEEE International Conference on, IEEE, 2017, pp. 80–90.
Q. Huang, X. Xia, D. Lo, Supervised vs unsupervised models: A holistic look at effort-aware just-in-time defect prediction, in: Software Maintenance and Evolution (ICSME), 2017 IEEE International Conference on, IEEE, 2017, pp. 159–170.
S. Singh, R. Singla, Classification of defective modules using object-oriented metrics, International Journal of Intelligent Systems Technologies and Applications 16 (1) (2017) 1–13.
R. Chang, X. Shen, B. Wang, Q. Xu, A novel method for software defect prediction in the context of big data, in: Big Data Analysis (ICBDA), 2017 IEEE 2nd International Conference on, IEEE, 2017, pp. 100–104.
A. Boucher, M. Badri, Predicting fault-prone classes in object-oriented software: An adaptation of an unsupervised hybrid som algorithm, in: Software Quality, Reliability and Security (QRS), 2017 IEEE International Conference on, IEEE, 2017, pp. 306–317.
M. Yan, X. Zhang, C. Liu, L. Xu, M. Yang, D. Yang, Automated change-prone class prediction on unlabeled dataset using unsupervised method, Information and Software Technology 92 (2017) 1–16.
A. Boucher, M. Badri, Software metrics thresholds calculation techniques to predict fault-proneness: An empirical comparison, Information and Software Technology 96 (2018) 38–67.
R. Sasidharan, P. Sriram, Hyper-quadtree-based k-means algorithm for software fault prediction (2014) 107–118.
X. Chen, Y. Zhao, Q. Wang, Z. Yuan, Multi: Multi-objective effort-aware just-in-time software defect prediction, Information and Software Technology 93 (2018) 1–13.
Supervised defect prediction procedure
======================================
For performance comparison, we conclude supervised defect prediction models used in our review studies as Table \[tbl:SvSDP\].
\[tbl:SvSDP\]
[|p[3.7cm]{}|p[1.6cm]{}|p[6.0cm]{}|p[5cm]{}|]{} **Family** & **Sub Abbr** &**SubFamily Approaches** & **Related Study**\
& NB & Naïve Bayes &[P @c2009clustering], [P @zhong2004analyzing], [P @bishnu2012software], [P @b2011application], [P @singh2014efficient], [P @abaei2015increasing], [P @zhang2016cross], [P @aleem2015benchmarking], [P @abaei2015empirical], [P @fan2017utility], [P @boucher2017predicting], [P @boucher2018software]\
& TDT & Treedisc Decision Tree &[P @zhong2004analyzing]\
& DT &Decision tree &[P @czibula2016novel],[P @aleem2015benchmarking],[P @boucher2018software]\
&DTJ48 &J48 C4.5 &[P @zhong2004analyzing],[P @yang2016defect],[P @aleem2015benchmarking]\
&LMT& Logistic Model Tree &[P @zhang2016cross]\
&RF &Random Forest&[P @abaei2015increasing],[P @zhang2016cross],[P @aleem2015benchmarking],[P @fu2017revisiting], [P @abaei2015empirical],[P @boucher2017predicting]\
& SVM & Support Vector Machine &[P @z2004unsupervised],[P @czibula2016novel],[P @aleem2015benchmarking],[P @yan2017file],[P @yan2017automated],[P @fan2017utility],[P @boucher2018software]\
&SMO &Sequential Minimal Optimization &[P @zhong2004analyzing]\
& LR & Linear Regression &[P @zhong2004analyzing],[P @zhang2016cross],[P @yang2016defect],[P @czibula2016novel],[P @n2015clami],[P @yan2017file],[P @huang2017supervised]\
&BLR& Binary Logistic Regression &[P @czibula2016novel]\
&MLR& Multiple Linear Regression &[P @iwata2013error],[P @czibula2016novel]\
&kNN& k nearest neighbor &[P @zhong2004analyzing], [P @aleem2015benchmarking],[P @fu2017revisiting]\
&LWLS &Locally weighted learning & [P @zhong2004analyzing]\
&CBR& Case-based reasoning &[P @zhong2004analyzing]\
& ONER & One Rule algorithm &[P @zhong2004analyzing]\
&RDR &Ripple down rule algorithm &[P @zhong2004analyzing],[P @chen2018multi]\
&RBM &Rule-based modeling &[P @zhong2004analyzing]\
&JRIP &Repeated incremental pruning& [P @zhong2004analyzing]\
&DTABLE &Decision table &[P @zhong2004analyzing],[P @boucher2018software]\
&ADT &Alternating decision table &[P @zhong2004analyzing]\
&PART &Rules from partial decision trees &[P @zhong2004analyzing]\
& ANN & Artificial neural networks &[P @zhong2004analyzing],[P @czibula2016novel],[P @iwata2012clustering],[P @abaei2015empirical],[P @fan2017utility],[P @boucher2017predicting],[P @boucher2018software]\
&CCN &Cascade Correlation Networks & [P @czibula2016novel]\
&GMDHN & GMDH Network & [P @czibula2016novel]\
&RBFN &Radial basis function network &[P @aleem2015benchmarking], [P @yan2017file]\
&MLP &Multilayer perceptron&[P @singh2016comparative],[P @aleem2015benchmarking],[P @yan2017file]\
& LBOOST & LogitBoost classifier &[P @zhong2004analyzing], [P @yan2017file],[P @singh2017classification]\
&ABOOST &AdaBoost classifier &[P @zhong2004analyzing],[P @aleem2015benchmarking],[P @chen2018multi]\
&BAG &Bagging classifier &[P @zhong2004analyzing],[P @aleem2015benchmarking]\
& GA & Genetic algorithms (evolutionary algorithms) &[P @zhong2004analyzing],[P @czibula2016novel],[P @chen2018multi]\
& EALR & Effort aware&[P @yang2016effort],[P @fu2017revisiting],[P @liu2017code], [P @huang2017supervised],[P @chen2018multi]\
&TLEL &Two-layer ensemble learning &[P @liu2017code]\
& RSET & Rough sets-based classifier &[P @zhong2004analyzing]\
&MCOST & MetaCost classifier &[P @zhong2004analyzing]\
&GeneEP &Gene Expression Programming& [P @czibula2016novel]\
&HPIPES &Hyperpipes algorithm &[P @zhong2004analyzing]\
&LDA &Linear Discriminant Analysis&[P @bishnu2012software],[P @b2011application]\
&[SOMs-MLR]{}&Hybrid: SOMs and Multiple Logic Regression &[P @iwata2013error]\
&STHD &Threshold determined by supervised method &[P @boucher2018software]\
Acknowledgements {#sec:Ack .unnumbered}
================
We would like to thank the editors and the anonymous reviewers for their insightful comments and suggestions. We also wish to acknowledge the use of the DConfusion tool developed by David Bowes and David Gray whilst at the University of Hertfordshire. This work was supported by the National Key Basic Research Program of China \[2018YFB1004401\]; the National Natural Science Foundation of China \[61972317, 61402370\].
References {#references .unnumbered}
==========
bibitem\[1\]
@filesw auxout
biblabel\#1[\[\#1\]]{}
[10]{} N. E. Fenton, M. Neil, A critique of software defect prediction models, IEEE Transactions on software engineering 25 (5) (1999) 675–689.
C. Catal, B. Diri, A systematic review of software fault prediction studies, Expert systems with applications 36 (4) (2009) 7346–7354.
T. Hall, S. Beecham, D. Bowes, D. Gray, S. Counsell, A systematic literature review on fault prediction performance in software engineering, IEEE Transactions on Software Engineering 38 (6) (2012) 1276–1304.
J. Nam, S. Kim, Clami: Defect prediction on unlabeled datasets (t), in: Automated Software Engineering (ASE), 2015 30th IEEE/ACM International Conference on, IEEE, 2015, pp. 452–463.
B. Kitchenham, D. Budgen, P. Brereton, Evidence-based Software Engineering and Systematic Reviews, CRC Press, 2015.
M. Yan, M. Yang, C. Liu, X. Zhang, Self-learning change-prone class prediction., in: SEKE, 2016, pp. 134–140.
A. Boucher, M. Badri, Using software metrics thresholds to predict fault-prone classes in object-oriented software, in: 2016 4th Intl Conf on Applied Computing and Information Technology/3rd Intl Conf on Computational Science/Intelligence and Applied Informatics/1st Intl Conf on Big Data, Cloud Computing, Data Science & Engineering (ACIT-CSII-BCD), IEEE, 2016, pp. 169–176.
N. Li, M. Shepperd, Y. Guo, A Systematic Review of Unsupervised Defect Prediction Dataset, Mendeley Data,v1. <http://dx.doi.org/10.17632/h24ctmyx73.1>
D. Powers, Evaluation: from precision, recall and [F]{}-measure to [ROC]{}, informedness, markedness and correlation, Journal of Machine Learning Technologies 2 (1) (2011) 37–63.
M. Shepperd, D. Bowes, T. Hall, Researcher bias: The use of machine learning in software defect prediction, IEEE Transactions on Software Engineering 40 (6) (2014) 603–616.
D. Bowes, T. Hall, D. Gray, [DC]{}onfusion: a technique to allow cross study performance evaluation of fault prediction studies, Automated Software Engineering 21 (2) (2014) 287–313.
P. Baldi, S. Brunak, Y. Chauvin, C. A. Andersen, H. Nielsen, Assessing the accuracy of prediction algorithms for classification: an overview, Bioinformatics 16 (5) (2000) 412–424.
T. Fawcett, An introduction to roc analysis, Pattern recognition letters 27 (8) (2006) 861–874.
Y. Kamei, E. Shihab, B. Adams, A. E. Hassan, A. Mockus, A. Sinha, N. Ubayashi, A large-scale empirical study of just-in-time quality assurance, IEEE Transactions on Software Engineering 39 (6) (2013) 757–773.
D. Hand, Measuring classifier performance: a coherent alternative to the area under the [ROC]{} curve, Machine Learning 77 (1) (2009) 103–123.
P. Flach, M. Kull, Precision-recall-gain curves: [PR]{} analysis done right, in: Advances in Neural Information Processing Systems, 2015, pp. 838–846.
Q. Song, Y. Guo, M. Shepperd, A comprehensive investigation of the role of imbalanced learning for software defect prediction, IEEE Transactions on Software Engineering 45 (12) (2018) 1253–1269.
M. S. Perlin, T. Imasato, D. Borenstein, Is predatory publishing a real threat? evidence from a large database study, Scientometrics (2017) 1–19.
M. Berger, J. Cirasella, Beyond [B]{}eall’s list: [B]{}etter understanding predatory publishers, College & Research Libraries News 76 (3) (2015) 132–135.
R. Kohavi, et al., A study of cross-validation and bootstrap for accuracy estimation and model selection, in: Ijcai, Vol. 14, Montreal, Canada, 1995, pp. 1137–1145.
R. J. Grissom, J. J. Kim, Effect sizes for research: Univariate and multivariate applications, Routledge, 2012.
D. Michie, D. Spiegelhalter, C. Taylor, Machine learning, neural and statistical classification, Ellis Horwood Series in Artifical Intelligence, Ellis Horwood, Chichester, Sussex, UK, 1994.
B. Bushman, M. Wang, Vote-counting procedures in meta-analysis, Russell Sage Foundation, New York, NY, US, 2009, pp. 207–220.
F. Mosteller, G. Colditz, Understanding research synthesis (meta-analysis), Annual Review of Public Health 17 (1) (1996) 1–23.
L. V. Hedges, I. Olkin, Statistical methods for meta-analysis, Academic Press, 1985.
J. M. Chambers, Graphical methods for data analysis, Wadsworth International Group, 1983.
M. Munaf[ò]{}, B. Nosek, D. Bishop, K. Button, C. Chambers, N. du Sert, U. Simonsohn, E. Wagenmakers, J. Ware, J. Ioannidis, A manifesto for reproducible science, Nature Human Behaviour 1 (1) (2017) 0021.
[^1]: Here we seek to understand the researchers’ approach to SDP benchmark through their citations of related work and published results.
[^2]: We parenthesise the ‘re’ of computation to convey that for some papers we are constructing the confusion matrix *ab initio*. In other situations there is already an explicit matrix but we re-construct it from other data provided, although potentially by different means from that presented in the original paper. For brevity, in the remainder of our paper we simply state ‘re-computation’.
[^3]: The F-measure that researchers use is the balanced F score and is often referred to as F1. Other weightings are possible but in practice seldom used [@Powe11].
[^4]: https://beallslist.weebly.com/
[^5]: https://beallslist.weebly.com/standalone-journals.html
[^6]: https://libguides.caltech.edu/c.php?g=512665&p=3503029
[^7]: Note that year 2018 is incomplete.
[^8]: Despite the recent emergence of software defect studies that only vote count studies where there is a statistically ‘significant’ effect sometimes referred to as a win-tie-loss procedure, this is in error and studies should always be included [@Most96; @Bush09].
|
---
abstract: 'Originally proposed by Read [@Read89:62] and Jain [@Jain89:199], the so-called “composite-fermion” is a phenomenological attachment of two infinitely thin local flux quanta seen as nonlocal vortices to two-dimensional (2D) electrons embedded in a strong orthogonal magnetic field. In this letter, it is described as a highly-nonlinear and coherent mean-field quantum process of the soliton type by use of a 2D stationary Schr[ö]{}dinger-Poisson differential model with only two Coulomb-interacting electrons. At filling factor $\nu=\frac{1}{3}$ of the lowest Landau level, it agrees with both the exact two-electron antisymmetric Schr[ö]{}dinger wave function and Laughlin’s Jastrow-type guess for the fractional quantum Hall effect, hence providing this later with a tentative physical justification based on first principles.'
author:
- Gilbert Reinisch
- Vidar Gudmundsson
- Andrei Manolescu
title: Coherent Nonlinear Quantum Model for Composite Fermions
---
Perhaps the most spectacular physical concept introduced in the description of Fractional Quantum Hall Effect (FQHE) is Composite Fermion (CF). It consists in an intricate mixture of $N_e$ electrons and vortices in a two-dimensional (2D) electron gas orthogonal to a (strong) magnetic field such that the lowest Landau level (LLL) is only partially occupied. Actually, the CF concept provides an intuitive phenomenological way of looking at electron-electron correlations as a part of sophistiscated many-particle quantum effects where charged electrons do avoid each other by correlating their relative motion in the energetically most advantageous fashion conditioned by the magnetic field. Therefore it is picturesquely assumed that each electron lies at the center of a vortex whose trough represents the outward displacement of all fellow electrons and, hence, accounts for actual decrease of their mutual repulsion [@Read89:62; @Stormer99:875]. Or equivalently, in the simplest case of $N_e=2$ electrons considered in the present letter, that two flux quanta $\Phi_0=hc/e$ are [*“attached”*]{} to each electron, turning the pair into a LLL of two CFs with a $6\Phi_0$ resulting flux [@Jain89:199]. The corresponding Aharonov-Bohm quantum phase shift equals $2\pi$. In addition to the $\pi$ phase shift of core electrons, it agrees with the requirements of the Laughlin correlations expressed by the Jastrow polynomial of degree $3$ and corresponding to the LLL filling factor $\nu = \frac{1}{3}$ [@Laughlin83:1395; @Laughlin83:3383]. Laughlin’s guessed wavefunction for odd polynomial degree was soon regarded as a Bose condensate [@Girvin87:1252; @Zhang89:62; @Rajaraman96:793] whereas for even degree, it was considered as a mathematical artefact describing a Hall metal that consists of a well defined Fermi surface at a vanishing magnetic field generated by a Chern-Simons gauge transformation of the state at exactly $\nu = \frac{1}{2}$ [@Halperin93:47; @Rezayi94:72].
Although they provide a simple appealing single-particle illustration of Laughlin correlations, the physical origin of the CF auxiliary field fluxes remains unclear. In particular, the way they are fixed to particles is not explained. Hence tentative theories avoiding the CF concept like e.g. a recent topological formulation of FQHE [@Jacak12:26]. In the present letter, we show how a strongly-nonlinear mean-field quantum model provides an alternative Hamiltonian physical description, based on first principles, of the debated CF quasiparticle.
Consider the 2D electron pair confined in the $x-y$ plane under the action of the orthogonal magnetic field $ {\bold B} $. It is situated at $z_{1,2}=x_{1,2}+iy_{1,2}$. Adopt the usual center-of-mass ${\bar z}=(z_1+z_2)/2$ and internal coordinate $z =(z_1-z_2)/\sqrt{2}$ separation and select odd-$m$ angular momenta $m\hbar$ in order to comply with the antisymmetry of the two-electron wavefunction under electron interchange. The corresponding internal motion radial eigenstate $\Psi_m(x,y)=u_m(r)e^{im\phi}$ with $z =x +iy =re^{i\phi}$ is defined in units of length and energy by the cyclotron length $\lambda_c=\sqrt{\hbar / (M\omega_c)}$ and by the Larmor energy $\hbar \omega_L=\frac{1}{2} \hbar \omega_c = \hbar e B/(2Mc)$ where $M$ denotes the effective mass of the electron which may incorporate many-body effects. The eigenstate $u_m$ is given by [@Laughlin83:3383]: $$\label{eq-SchroeRBL}
\Bigl[\nabla_X^2 + E_m +m
-\frac{m^2}{X^2} -\frac{X^2}{4} - \frac{K}{X} \Bigr] u_m =0.$$ The radial part of the 2D Laplacian operator is $\nabla_X^2=d^2/dX^2 + X^{-1}(d/dX)$, the energy eigenvalue is $E_m$ and $X=r/\lambda_c$. The dimensionless parameter $$\label{eq-K}
K=\sqrt{2}\,\,\frac{ e^2/(\epsilon \lambda_c )}{\hbar\omega_c}
,$$ where $\epsilon $ is the dielectric constant of the semiconductor host, compares the Coulomb interaction between the two particles with the cyclotron energy. Obviously, $K=0$ corresponds to the free-particle case. Actually, the internal motion could be approximated by the 2D free-particle harmonic oscillator eigenstate $|m\rangle$ as long as $K\leq \sqrt{2}$, i.e. $B \leq 6$ T in GaAs [@Laughlin83:3383]. However, in FQHE experimental conditions, the magnetic field is much higher. In [@Du93:70], the energy gaps of FQHE states related to samples A and B at filling factors $p/(2p\pm1)$ between $\nu=\frac{1}{4}$ and $\nu=\frac{1}{2}$ are shown to increase linearly with the deviation of $B$ from the respective characteristic values $B_{\frac{1}{2}}^A=9.25$ T, $B_{\frac{1}{4}}^A=18.50$ T and $B_{\frac{1}{2}}^B=19$ T (where the superscripts refer to the samples). The corresponding slopes respectively yield the direct measures $M_A=0.63$, $M_A=0.93$ and $M_B=0.92$ of the effective electron mass in units of the electron mass $m_e$. Indeed, since these masses scale like $\lambda_c^{-1}$ and hence like $\sqrt{B}$ for they are determined by electron-electron interaction, we have $0.63/\sqrt{9.25}=0.207\approx 0.93/\sqrt{18.50}=0.216 \approx 0.92/\sqrt{19}=0.211$. Therefore, introducing the parameter $\kappa$ that accounts for the above experimental results, we have [@Du93:70]: $$\label{eq-MvsSQRTofB}
\frac{M}{m_e} \sim \kappa\sqrt{B}\quad ;\quad 0.207\leq \kappa\leq 0.216,$$ where $B$ is given in Tesla.
![Upper plot: the “trajectory” $C_{|1\rangle}(0)$ vs $[du_{|1\rangle}/dX]_0$ for increasing $K$ values whithin the interval $[ 0,15 ]$ as indicated by the arrows. It is defined by the initial conditions of (\[eq-SchroeRBL\]), or equivalently of (\[eq-nlSchroeLAUGH\]-\[eq-CspLAUGH\]). Lower plot: the corresponding “trajectory” defined by the initial conditions of the SP differential system (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]). The circle indicates the $\nu=\frac{1}{3}$ FQHE solution defined by (\[eq-Kvalue\]). In both plots, the $K=0$ free-electron case is defined by the upper left point $[du/dX]_0 = 0 $ and $C(0)=2$.[]{data-label="fig1"}](Fig-1.eps){width="48.00000%"}
Equation (\[eq-SchroeRBL\]) is linear and hence dispersive in its free-particle angular-momentum eigenspace. Its stationary solutions are expected to spread out over more and more eigenstates $|m\rangle$ when the perturbation defined by $ K\neq 0$ grows. This is best illustrated by Fig. \[fig1\] (upper plot). Starting at $K= 0$ (no interaction) from $m=1$ lowest-energy and most stable free-electron vortex state $|1\rangle$ defined by (\[eq-SchroeRBL\]), it implicitely displays in terms of increasing $K$ the “trajectory” corresponding to the solution of (\[eq-SchroeRBL\]) in its initial-condition phase space. Let us rewrite (\[eq-SchroeRBL\]) under the form of the following equivalent differential system: $$\label{eq-nlSchroeLAUGH}
\Bigl[\nabla_X^2 + C_{|1\rangle} -\frac{1}{X^2} -\frac{X^2}{4} \Bigr] u_{|1\rangle}=0,$$ $$\label{eq-nlPoisLAUGH}
\nabla_X^2 C_{|1\rangle} = K \delta(X),$$ with $$\label{eq-CspLAUGH}
C_{|1\rangle}(X)=\mu_{|1\rangle} -{\cal W}_{|1\rangle}(X),$$ where $\delta(X)$ is the Dirac function, the radial Laplacian $\nabla_X^2$ is 3D in (\[eq-nlPoisLAUGH\]) while it remains 2D in (\[eq-nlSchroeLAUGH\]) (this point will be discussed further below), the eigenvalue $\mu_{|1\rangle}$ stands for $E _{|1\rangle}+1$ due to the Larmor rotation at $m=1$ and ${\cal W}_{|1\rangle}=K/X$ describes the particle-particle interaction potential defined by the last term in (\[eq-SchroeRBL\]). Then the initial-condition phase space becomes $C_{|1\rangle}(0)$ vs $[du_{|1\rangle}/dX]_0$. Indeed, we let $u(0)=[dC/dX]_0=0$ due to, respectively, complete depletion in the vortex trough and radial symmetry (no cusp). There are clearly two discontinuities in Fig. \[fig1\] (upper plot) which describe the “jumps” of the initial $m=1$ solution to higher orbital momenta when $K$ grows. In particular, there is a phase transition (infinite slope) at $[du_{|1\rangle}/dX]_{0}\sim 0.85$ and $C_{|1)}(0)\sim -7$. Now compare with Fig. \[fig1\] (lower plot). It displays the trajectory of the nonlinear Schr[ö]{}dinger-Poisson (SP) solution which is defined from (\[eq-nlSchroeLAUGH\]-\[eq-CspLAUGH\]) by adding the mean-field source term $u_{|1)}^2$ to Poisson equation (\[eq-nlPoisLAUGH\]), namely: $$\label{eq-nlPois}
\nabla^2 C_{|1)} = u_{|1)}^2,$$ (we emphasize the eigenstate’s nonlinear nature imposed by Eq. (\[eq-nlPois\]) by using parentheses instead of kets).
{width="80.00000%"}
The spectral coherence of the new solution —i.e. the invariance of its angular momentum with respect to the increase of $K$— is obvious: instead of discontinuously spreading out in the momenta space like in Fig. \[fig1\] (upper plot), the SP solution starts spiraling down while keeping its $m=1$ initial value [@Reinisch12:902]. No phase transition towards higher angular momenta occurs for $0\leq K \leq 15$ (e.g. in Fig. \[fig1\], lower plot, at $[du_{|1)}/dX]_{0}\sim 1.75$ and $C_{|1)}(0)\sim 1.3$).
This phenomenon resembles the well-known soliton coherence in hydrodynamics due to the cancellation of the dispersive effects by nonlinearity. The mathematical tool that explains the stability of the resulting solitary wave is the so-called nonlinear spectral transform. It provides a theoretical link between linear spectral —and nonlinear dynamical and/or structural properties of the wave [@Remoissenet99]. This is what our nonlinear transformation from (\[eq-nlPoisLAUGH\]) to (\[eq-nlPois\]) is doing. It introduces an explicit ab-initio nonlinearity that cancels the angular-momentum dispersion displayed by Fig. \[fig1\] (incidently, this transformation is usually done the opposite way in classical soliton physics: one starts with the “real” nonlinear wave equation and ends up with its formal “spectral transformed” linear counterpart [@Remoissenet99]). As shown by Fig. \[fig2\]a, the resulting nonlinear eigenstate $|1)$ defined by the SP differential system (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]) yields an average spatial extension which fits with the prediction of the linear equation (\[eq-SchroeRBL\]) provided $K$ is chosen in the following FQHE experimental range defined by (\[eq-K\]-\[eq-MvsSQRTofB\]): $$\label{eq-Kvalue}
11.07\leq K=
\frac{4\pi^{3/2}M c^2}{\epsilon \Phi_0^{3/2}\sqrt{B}}=\kappa\frac{4\pi^{3/2}
m_e c^2}{\epsilon \Phi_0^{3/2}}\leq 11.56.$$ This range of relevant $K$ values is indicated by the circle in Fig. \[fig1\] (lower plot) and by the two vertical marks in Figs \[fig2\]a,b,d. Most important for the aim of the present letter, the nonlinear transformation from (\[eq-nlPoisLAUGH\]) to (\[eq-nlPois\]) yields the expected CF properties about gap stability and flux quantization (see Fig. \[fig2\]b and 2d, respectively) while the spatial extension of the nonlinear eigenstate $|1)$ also agrees with Laughlin’s $\nu=\frac{1}{3}$ two-electron normalized wavefunction whose modulus $|\Psi_3|$ is derived from [@Chakraborty95]: $$\label{eq-jastrow}
\Psi_3\propto (z_1-z_2)^{\displaystyle 3}\, e^{\displaystyle -\frac{(|z_1|^2 + |z_2|^2)}{4\lambda_c}}
\propto X^{\displaystyle 3} e^{\displaystyle 3i\phi}e^{\displaystyle -\frac{X^2}{4}},$$ up to the unimportant factor $\exp[\frac{1}{2}({\bar z}/\lambda_c)^2]$ related to the external degree of freedom. Figure \[fig2\]c indeed displays the FQHE states defined by (\[eq-Kvalue\]) (broken-line profiles in Fig. \[fig3\]), namely $|1)$ (nonlinear: bold green) and $|1\rangle$ (linear: bold blue), as compared to normalized $|\Psi_3|$ (thin red).
The amplitude of the wavefunction defined by Eqs (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]) reads $\Psi_{|1)}=u_{|1)}\sqrt{M \omega_L/(\pi\hbar {\cal N}_{|1)})}$ where [@Reinisch12:902] $$\label{eq-normeDEi}
{\cal N}_{|1)}=\int_0^{\infty} u_{|1)}^2 XdX,$$ in order to achieve normalization according to $\int_0^{\infty}|| \Psi_{|1)}(x,y)||^2 dx dy = 1$. Norm (\[eq-normeDEi\]) is the nonlinear order parameter of our SP model [@Reinisch12:902]. As it grows, the amplitude and width of $u_{|1)}$ increases while its corresponding normalized profile $u_{|1)}/\sqrt{{\cal N}_{|1)}}$ spreads out: see Fig. \[fig3\]. Comparing the (last) interaction term of the bracket in (\[eq-SchroeRBL\]) with the asymptotic solution $$\label{eq-PoissonASYMPT}
\lim_{X\rightarrow \infty}{\cal W}_{|1)}(X)={\cal N}_{|1)} G(X)$$ of (\[eq-CspLAUGH\]-\[eq-nlPois\]) where the 3D Green function defined by (\[eq-nlPoisLAUGH\]) is $G(X)=X^{-1}$, we obtain: $$\label{eq-Nvalue}
{\cal N}_{|1)}=K.$$ Equation (\[eq-Nvalue\]) operates the link between the two-particle linear description (\[eq-SchroeRBL\]) and the mean-field single-(quasi)particle provided by (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]). In [@Du93:70] the filling factor $\nu=\frac{1}{3}$ lies at the intersection of two slopes concerning sample A. Taking their average, we obtain $K \sim 11.23$ which will be our reference value in interval (\[eq-Kvalue\]). Note that it largely exceeds $K=\sqrt{2}$ in [@Laughlin83:3383]. Similarly, ${\cal N}_{|1)}=11.23$ in accordance with (\[eq-Nvalue\]) yields a nonlinearity: e.g. compare it with $2.53$ in quantum-dot helium [@Reinisch12:902]. This is quite spectacularly illustrated by Fig. \[fig3\] where the linear solution $|1\rangle$ of (\[eq-SchroeRBL\]) —or equivalently of system (\[eq-nlSchroeLAUGH\]-\[eq-CspLAUGH\])— is compared with solution $|1)$ of the nonlinear SP system (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]) for increasing $K$ values. The profiles defined by $\nu=\frac{1}{3}$ FQHE value $K={\cal N}_{|1)} =11.23$ are displayed in broken lines: obviously they are strongly modified with respect to the free-particle ones (in dotted lines).
Let us now be specific about some technical points used in order to obtain the above results. In FQHE, the zeros of the Jastrow-type many-electron wave function proposed by Laughlin for odd polynomial degree look like 2D charges which repel each other by logarithmic interaction, yielding a negative value for the energy $E$ per electron [@Chakraborty95]. Consequently we solve the Poisson equation (\[eq-nlPois\]) in 2D, which indeed ensures that the corresponding Green function becomes $G(X)=-\log(X)$. Therefore we obtain the “fully 2D” self-consistent SP differential system (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]) whose eigensolution $u_{|1)}(X)$ is defined by \[eq-Kvalue\], (\[eq-normeDEi\]) and (\[eq-Nvalue\]).
The radius of SP’s nonlinear state $|1)$ displayed by Fig. \[fig2\]a: $$\label{eq-Xbarre}
{\bar X}=\sqrt{\langle {\bar z}^2 \rangle}_{|1)}=
\frac{1}{\sqrt{2}}[\langle X^2\rangle_{|1)}+\langle X\rangle^2_{|1)}]^{\frac{1}{2}},$$ is obtained from ${\bar z}=\frac{1}{2} (z_1+z_2)$ by quantum-averaging $X$ and $X^2$ in the state $|1)$ (hence the subscripts) since the two electrons located at $z_1$ and $z_2$ are both in the same state $|1)$[@Griffiths05]. On the other hand, $|z_1-z_2|$ is the diameter of the two-electron orbit defined by the linear internal degree of freedom $|1\rangle$ when assuming that the external degree of freedom ${\bar z}$ is frozen in its ground state. Therefore its radius: $$\label{eq-Ybarre}
{\bar Y}=\frac{1}{2} \langle |z_1-z_2| \rangle_{|1\rangle}= \frac{1}{\sqrt{2}}\langle X \rangle_{|1\rangle}$$ (broken line in Fig. \[fig2\]a ) can indeed be compared with (\[eq-Xbarre\]) (continuous line). Like already emphasized, these two radii coincide at the experimental range (\[eq-Kvalue\]) displayed by the couple of adjacent vertical marks in Fig. \[fig2\]a.
![Left: The broadening and flatening of the normalized linear profile $-u_{|1\rangle}(X)/\sqrt{{\cal N}_{|1\rangle}}$ corresponding to eigenstate $|1\rangle$ of (\[eq-nlSchroeLAUGH\])-(\[eq-CspLAUGH\]) (in bold blue and multiplied by $-1$ for the sake of clarity), compared with the normalized nonlinear SP wave function $u_{|1)}(X)/\sqrt{{\cal N}_{|1)}}$ of (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]) (in bold red) when ${\cal N}_{|1)}=K$ increases by integer steps from $0$ (free-electron solution: dotted profiles) to $15$. The SP solution $u_{|1)}(X)$ is displayed in green and increases with $K$. The CF profiles defined by ${\cal N}_{|1)}= K=11.23$ are displayed in broken lines. []{data-label="fig3"}](Fig-3.eps){width="48.00000%"}
The nonlinear energy eigenvalue $\mu_{|1)}$ solution of (\[eq-nlSchroeLAUGH\]), (\[eq-CspLAUGH\]) and (\[eq-nlPois\]) (continuous line in Fig. \[fig2\]b ) is obtained from (\[eq-CspLAUGH\]) by use of the 2D Green function $G(X)=-\log(X)$ either from the initial condition $C_0=C_{|1)}(0)$ with ${\cal W}_{|1)}(0)=\int_0^{\infty} G(X) u_{|1)}^2(X) dX$ in agreement with (\[eq-nlPois\]); or at the boundary $C_{|1)}(X\rightarrow \infty) $ by use of (\[eq-PoissonASYMPT\]). The equivalence of these two definitions constitutes a test for the relevance of our numerical code: they fit within a relative error of $10^{-7}$. On tne other hand, the energy eigenvalue $\mu_{|1\rangle}$ corresponding to the solution $u_{|1\rangle}$ of the linear differential system (\[eq-nlSchroeLAUGH\]-\[eq-CspLAUGH\]) can also be obtained from (\[eq-CspLAUGH\]) at the two above limits by simply using the explicit 2D definition ${\cal W}_{|1\rangle}(X)=-K \log(X)$. It is displayed in Fig. \[fig2\]b by the broken line. The negative energy gap $\Delta=\mu_{|1)}-\mu_{|1\rangle}$ yields the stability of $|1)$ when compared with $|1\rangle$. Though small, it is clearly visible. We obtain at the experimental FQHE value $K=11.23$: $\mu_{|1\rangle}=-11.0807$ $\hbar\omega_L$ $=-0.6977$ $e^2/\epsilon \lambda_c$ and $\mu_{|1)}=-11.7164$ $\hbar\omega_L$ $=-0.7377$ $e^2/\epsilon \lambda_c$ (cf. (\[eq-K\])). Therefore $\Delta=-0.04$ $e^2/\epsilon \lambda_c$, to be compared with some experimental value $\Delta\sim -0.1$ $e^2/\epsilon \lambda_c$ [@Du93:70]. Moreover, the nonlinear eigenenergy per particle in our $N_e=2$ system is $E_2=\frac{1}{2}\mu_{|1)}=-0.37$ $e^2/\epsilon \lambda_c$. In the $20\leq N_e\leq 144$ interacting electron case in the disk geometry with the filling factor $\nu=\frac{1}{3}$, $E_2\sim -0.39$ $e^2/\epsilon \lambda_c$ by extrapolation to $N_e=2$ of the Monte Carlo evaluation of the ground-state energy per particle [@Morf86:33]. On the other hand, $E\sim -0.41$ $e^2/\epsilon \lambda_c$ per particle is almost insensitive to the system size for $4\leq N_e\leq 6$ [@Yoshioka83:50]. Therefore our $E_2=-0.37$ $e^2/\epsilon \lambda_c$ seems quite acceptable in this context.
Figure \[fig2\]d displays the fundamental property of the present model, namely, its $6\Phi_0$ flux quantization for the LLL filling factor $\nu=\frac{1}{3}$ in the experimental range defined by \[eq-Kvalue\]. Indeed we have respectively from (\[eq-Xbarre\]) and (\[eq-Ybarre\]): $$\label{eq-fluxNL}
\frac{\Phi_{|1)}}{\Phi_0}=\frac{\pi B \lambda_c^2}{\Phi_0} \,{\bar X}^2=
\frac{1}{2}{\bar X}^2=6,$$ $$\label{eq-fluxL}
\frac{\Phi_{|1\rangle}}{\Phi_0}=\frac{\pi B \lambda_c^2}{\Phi_0}\, {\bar Y}^2=
\frac{1}{4} \langle X \rangle_{|1\rangle}^2=6,$$ between the two vertical marks. The (blue) horizontal broken line at $\Phi/\Phi_0=6$ refers to Laughlin’s $\nu=\frac{1}{3}$ normalized wavefunction ansatz (\[eq-jastrow\]) which does not depend on $K$. Its flux is indeed $6\Phi_0$ [@Laughlin87]).
In conclusion, we stress the self-consistency of our FQHE nonlinear model. The spectral coherence of the mean-field SP mode $|1)$ slightly lowers the energy per electron with respect to that obtained from Schr[ö]{}dinger’s two-electron internal mode $|1\rangle$. This gap makes the nonlinear mode $|1)$ energetically favourable for the fulfilment of the flux quantization condition (\[eq-fluxNL\]) than the linear SP mode $|1\rangle$ for (\[eq-fluxL\]). This property might be considered as the attachment of two flux quanta per electron and the subsequent transformation of this later into a nonlinear soliton-like CF whose 3rd remaining flux quantum makes it behave as a mere quasiparticle in LLL Quantum Hall Effect.
GR gratefully acknowledges Lagrange lab’s hospitality at the Observatoire de la Cote d’Azur (Nice, France). The authors acknowledge financial support from the Icelandic Research and Instruments Funds, the Research Fund of the University of Iceland.
[19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , ****, ().
, ** (, ).
, ****, ().
, ** (, ).
, ** (, ).
, ** (, ).
, ****, ().
, , , ****, ().
|
---
abstract: |
We describe $D=4$ twistorial membrane in terms of two twistorial three–dimensional world–volume fields. We start with the $D$–dimensional $p$–brane generalizations of two phase space string formulations: the one with $p+1$ vectorial fourmomenta, and the second with tensorial momenta of $(p+1)$-th rank. Further we consider tensionful membrane case in $D=4$. By using the membrane generalization of Cartan–Penrose formula we express the fourmomenta by spinorial fields and obtain the intermediate spinor–space-time formulation. Further by expressing the world–volume dreibein and the membrane space-time coordinate fields in terms of two twistor fields one obtains the purely twistorial formulation. It appears that the action is generated by a geometric three–form on two–twistor space. Finally we comment on higher–dimensional ($D>4$) twistorial $p$–brane models and their superextensions.
PACS numbers: 11.25.-w, 11.10.Ef, 11.30.Pb
author:
- 'Sergey Fedoruk,${}^1$ Jerzy Lukierski${}^2$[^1]'
title: 'TWO–TWISTOR DESCRIPTION OF MEMBRANE'
---
Introduction
============
Since long time the idea of twistor space (see e. g. [@PenMac]) as describing the basic geometric arena for the physical phenomena is tested in various ways. In particular it is known that massless relativistic particles can be equivalently described by free one–twistor particle model (see e. g. [@Fer; @Shir; @BC; @STV]); further it can be shown that massive relativistic particles with spin require in twistorial approach the two–twistor space [@Hug; @Bet; @FedZim; @BetAz; @Az; @FFLM][^2]. Further recently there was derived the twistor space action for general Yang–Mills (Euclidean) gauge theory [@Mas1; @Mas2] providing a breakthrough in extending the twistor construction to non–self–dual field theories. Important contribution to the twistor programme is also provided by the proof that large class of perturbative amplitudes in $N=4$ $D=4$ supersymmetric YM theory and conformal supergravity can be derived by using tensionless superstring moving in supertwistor space. In such approach (see e. g. [@Wit; @BerWit]) the twistorial classical string is described by two–dimensional $CP(3|4)$ $\sigma$–model with (twisted) $N=2$ world sheet SUSY. The correspondence with the space–time picture was derived [@Wit] on the level of quantized supertwistor string, providing twistor superstring field theory. By assuming the topological CS action for the (Euclidean) twistor superstring field one can reduce all the superstring excitations to the massless spectrum describing SUSY YM and supergravity theories. On the other hand recently in $D=4$ the close link between the tensionful string model (Nambu-Goto action) and twistor geometry has been proposed on the classical level, with target space–time string coordinates composite in terms of twistor string fields [@FL; @Tay]. It appears that in such a model the twistorial target space in bosonic $D=4$ string model consists of a pair of twistor string coordinates.
Recently in $M$–theory the (super)strings have lost their privileged role as the candidates for the Theory of Everything, and higher–dimensional $p$–branes ($p>1$), in particular membranes, were considered. In this paper we shall show that analogously to the result presented in [@FL] one can introduce in two–twistor target space the purely twistorial membrane action. In the intermediate spinor–space-time formulation with fundamental Lorentz spinors and space–time coordinates we shall get the membrane action which can be also linked with the $p$–brane description in the framework of spinorial harmonics ([@BZ-p]; see also [@BPSTV; @BSV]). It should be added that the presence of cosmological term in Polyakov type action for membrane permits to obtain the purely twistorial model without the use of gauge fixing procedure, which was a necessary step in string case [@FL].
For simplicity we shall describe our composite membrane models in detail in $D=4$, and without supersymmetric extension. Taking however into consideration that the multidimensional and supersymmetric extensions of twistors applied to twistorial string formulations has been already considered (see for example [@Wit; @BerWit; @BAM; @Uv]), we believe that this paper can be also useful e. g. for the consideration of the composite “$M$–theoretic” $D=11$ supermembrane.
In order to achieve our goal firstly in Sect. 2 we shall show how to extend from string to $p$–brane the two known phase space formulations of bosonic string theory: [ the one using vectorial momenta fields [@Sieg2; @Pav-PL] and the second using tensorial momenta fields (see e. g. [@GG; @AurS; @GZ; @Nieto])]{}. These two formulations are based on the following two Liouville $(p+1)$–forms which define the $p$–brane momenta fields:
a)
: [Vectorial momenta]{} model: $$\label{Th-p-int}
\Theta^{(1)} = p_\mu \,d X^\mu \quad \rightarrow \quad \Theta^{(p+1)} = P_\mu \wedge d X^\mu$$ where $P_\mu$ is the following $p$–form $$\label{P-form-int}
P_\mu =P_\mu^{\,m}\epsilon_{mn_1\ldots n_p} d \xi^{n_1} \ldots d \xi^{n_p} \,.$$ It is easy to see that if $p=1$ we obtain the Siegel formula [@Sieg2] for the string momenta one form $$\label{P-st-int}
P_\mu =P_\mu^{\,m} d \xi^{n} \epsilon_{mn} \,.$$
b)
: Tensorial momenta model: $$\label{tTh-p-int}
\Theta^{(1)}\! = p_\mu \,d X^\mu \, \rightarrow \, {\tilde\Theta}{}^{(p+1)}\! = P_{\mu_1
\ldots \,\mu_{p+1}} d X^{\mu_1} \wedge \ldots \wedge d X^{\mu_{p+1}}\,.$$ If $p=1$ one obtains the tensorial string momenta $P_{\mu\nu}=-P_{\nu\mu}$ ([@GG; @GZ]; for arbitrary $p$ see [@Nieto]).
In Sect. 2 we shall consider $p$–branes with arbitrary $p$ and in arbitrary space–time dimension $D$. We shall show that both phase space formulations using the momenta fields (\[P-form-int\]) or (\[tTh-p-int\]) are equivalent to the $p$–brane Dirac–Nambu–Goto action [@Dirac; @Nambu] as well as to the $\sigma$–model action ($p$–brane extension [of Howe–Tucker–Polyakov action for strings and membranes [@HT; @Pol; @Sug])]{}. Further, in Sect. 3 we consider for the case $p=2$ and $D=4$ (fourdimensional membrane) the intermediate spinor–space-time models for both phase space formulations. It appears that in such a model the spinors are constrained.[^3] In Sect. 4 we shall consider purely twistorial action for $D=4$ membrane. We show that in this action the Lagrangian density is described by a canonical twistorial three–form. In Sect. 5 we present an outlook: we comment on the description of twistorial membranes in higher dimensions, their supersymmetric extensions and present the remarks about the purely twistorial $p$–branes ($p>2$).
Two phase space formulations of the tensionful $p$–brane in $D$–dimensional space–time
======================================================================================
The tensionful $p$–brane propagating in flat Minkowski space is described by the nonlinear Dirac–Nambu–Goto action [^4] $$\label{action-NG}
S=-T\int d\,^{p+1}\xi\, \sqrt{- g}$$ where $\xi^m=(\tau,\sigma^1,\ldots,\sigma^p)$ are the world–volume coordinates, $$\label{g-det}
g\equiv \det (g_{mn})$$ and $$\label{ind-metr}
g_{mn}= \partial_m X^\mu\partial_n X_\mu$$ is the induced metric on the ($p+1$)–dimensional $p$–brane volume, $T$ is the $p$–brane tension. In the case $p=1$ the action (\[action-NG\]) is the Nambu–Goto action for string [@Nambu] whereas if $p=2$ the action (\[action-NG\]) is the Dirac action for relativistic membrane [@Dirac].
Because the twistor coordinates replace the standard phase space variables, the transition to twistorial formulation should be imposed on the Hamiltonian–like formulations. In the case of tensionful $p$–branes (\[action-NG\]) there are known two Hamiltonian descriptions.
Phase space formulation with vectorial momenta
----------------------------------------------
Let us use firstly the vectorial momenta defined by the formula (\[P-form-int\]). The corresponding action of the tensionful $p$–brane looks as follows [@Pav-PL] $$\begin{aligned}
S= \int d^{p+1}\xi & \!\!\Big[ & \!\! P_\mu^{\,m}\partial_m X^\mu + {\textstyle
\frac{1}{2T}}(-h)^{-1/2}h_{mn}P_\mu^{\,m} P^{\,\mu\,n} + \nonumber\\
&& + {\textstyle \frac{T}{2}} (p-1) (-h)^{1/2} \Big]\,. \label{act-s}\end{aligned}$$ We note that last ‘cosmological’ term in the action is absent if $p=1$ (string case).
Let us write down the equation of motion obtained after varying the world–volume metric $h_{mn}$. Using $\delta h=hh^{mn}\delta h_{mn}= -hh_{mn}\delta h^{mn}$ one gets $$\label{3}
P^{\,m}_\mu P^{\,\mu\,n}-{\textstyle \frac{1}{2}} h^{mn} \Big[ \, h_{kl}P^{\,k}_\mu P^{\,l \mu}
+ (p-1)h T^2 \Big]=0\,.$$
Further expressing $P^{\,m}_\mu$ in the action (\[act-s\]) by its equation of motion ($h^{mk}h_{kn}=\delta^{m}_{n}$) $$\label{2}
P^{\,m}_\mu= -T(-h)^{1/2}h^{mn}\partial_n X_\mu$$ one obtains the $\sigma$–model action for the $p$–brane [ of the Howe–Tucker–Polyakov type]{} [^5] $$\label{action-2}
S=-{\textstyle \frac{T}{2}}\int d^{p+1}\xi\, (-h)^{1/2} \Big[\, h^{mn}\partial_m X^\mu\partial_n X_\mu
-(p-1) \,\Big]$$ where the variables $h^{mn}$ can be treated as independent ones. The equations of motions for $h^{mn}$ give $$\partial_m X^\mu\partial_n X_\mu - {\textstyle\frac{1}{2}}h_{mn} \Big[\, h^{kl}\partial_k X^\mu\partial_l X_\mu
-(p-1) \,\Big]=0$$ and lead to (using $h^{mn}h_{mn}=p+1$) $$\label{h-g}
h_{mn}= g_{mn}$$ where $g_{mn}$ is the induced metric (\[ind-metr\]) on the $p$–brane volume.[^6] After substitution of (\[h-g\]) in (\[action-2\]) we obtain after simple algebraic calculation the action (\[action-NG\]).
It can be shown that from the action (\[act-s\]) one can derive the $p+1$ Virasoro constraints, generating the world–volume diffeomorphisms. We shall divide world–volume indices $m,n = 0,1,\ldots ,p$ into one with zero value and remaining $\underline{m},\underline{n} = 1,\ldots ,p$, that is $m =(0, \underline{m})$. The equations (\[2\]) lead to expression of the ‘auxiliary space momenta’ $$\label{sp-mom}
P^{\,\underline{m}}_\mu= - h_{0\underline{n}} \,
\underline{h}^{\underline{n}\,\underline{m}} \, P_\mu -T(-h)^{1/2}\,
\underline{h}^{\underline{m}\,\underline{n}} \,\partial_{\underline{n}} X_\mu$$ where $P_\mu= P^{\,0}_\mu$ is true momentum and $(p\times p)$ matrix $$\label{h-und}
\underline{h}^{\underline{m}\,\underline{n}} = h^{\underline{m}\,\underline{n}} \,-\,
\frac{h^{0 \underline{m}}\,h^{0\underline{n}}}{h^{00}}$$ is the inverse matrix for $(p\times p)$ space part of the world volume metric ($\underline{h}^{\underline{m}\,\underline{n}} \, h_{\underline{n}\,\underline{k}} =
\delta^{\underline{m}}_{\underline{k}}$). We note that $$\label{h-tozh}
\frac{1}{h^{00}} = h\, \det (\underline{h}^{\underline{m}\,\underline{n}}) \,, \qquad
\frac{h^{0 \underline{m}}}{h^{00}} = - h_{0\,\underline{n}}
\,\underline{h}^{\underline{n}\,\underline{m}} \,.$$ Inserting (\[sp-mom\]) in the the action (\[act-s\]) we obtain the first order Lagrangian
$$\label{L-ap1}
{\cal L} = P_\mu \dot X^\mu - {\textstyle\frac{\sqrt{-h}}{2T}} \, \det
(\underline{h}^{\underline{m}\,\underline{n}}) \,\Big( P_\mu P^\mu \Big) \,-\,
{\textstyle\frac{T\sqrt{-h}}{2}}\, \underline{h}^{\underline{m}\,\underline{n}} \, \Big(
\partial_{\underline{m}} X^\mu\partial_{\underline{n}} X_\mu \Big) \,- \,
h_{0\,\underline{n}} \,\underline{h}^{\underline{n}\,\underline{m}} \, \Big(
P_\mu\partial_{\underline{m}} X^\mu \Big) \,+\, {\textstyle\frac{T}{2}} \, (p-1)\sqrt{-h}
\,. \nonumber$$
Using the equality $$\underline{h}^{\underline{m}_1\,\underline{n}_1} = {\textstyle\frac{1}{(p-1)!}}\,\det
(\underline{h}^{\underline{m}\,\underline{n}})\,\epsilon^{\underline{m}_1 \underline{m}_2
\ldots \underline{m}_p}\,\epsilon^{\underline{n}_1 \underline{n}_2\ldots \underline{n}_p}\,
h_{\underline{m}_2\,\underline{n}_2} \ldots h_{\underline{m}_p\,\underline{n}_p}$$ and the equations of motions (\[h-g\]) we obtain that third term in Lagrangian (\[L-ap1\]) takes the form $$\label{term3-ap}
-\, {\textstyle\frac{T\sqrt{-h}}{2}}\, \underline{h}^{\underline{m}\,\underline{n}} \,
\Big( \partial_{\underline{m}} X^\mu\partial_{\underline{n}} X_\mu \Big) = -\,
{\textstyle\frac{T\sqrt{-h}}{2}}\,\, p\, \det
(\underline{h}^{\underline{m}\,\underline{n}}) \, \det (g_{\underline{m}\,\underline{n}})
\,.$$ If the equations (\[h-g\]) are valid we get $$\label{det-det}
\det (\underline{h}^{\underline{m}\,\underline{n}}) \, \det (g_{\underline{m}\,\underline{n}}) =1$$ and the last term in the Lagrangian (\[L-ap1\]) can be written as follows $$\label{term5-ap}
{\textstyle\frac{T}{2}} \, (p-1)\sqrt{-h} = {\textstyle\frac{T}{2}} \, (p-1)\sqrt{-h} \,
\det (\underline{h}^{\underline{m}\,\underline{n}}) \, \det
(g_{\underline{m}\,\underline{n}}) \,.$$ Inserting the expressions (\[term3-ap\]), (\[term5-ap\]) in the Lagrangian (\[L-ap1\]) we obtain the following standard form of the Lagrangian density in first order formalizm $$\label{L-ap}
{\cal L} = P_\mu \dot X^\mu - {\textstyle\frac{\sqrt{-h}}{2}} \, \det
(\underline{h}^{\underline{m}\,\underline{n}}) \,\Big[ {\textstyle\frac{1}{T}}\, P_\mu
P^\mu \,+\, T\, \det (g_{\underline{m}\,\underline{n}}) \Big] \,- \, h_{0\,\underline{n}}
\,\underline{h}^{\underline{n}\,\underline{m}} \, \Big( P_\mu\partial_{\underline{m}} X^\mu
\Big) \,.$$
We see that the formula (\[L-ap\]) describes the set of $p+1$ Virasoro constraints $$\begin{aligned}
H_0 &\equiv& {\textstyle\frac{1}{T}}\, P_\mu P^\mu \,+\, T\,
\det (g_{\underline{m}\,\underline{n}}) \approx 0 \,, \label{cons-0}\\
&& \nonumber \\
H_{\underline{m}} &\equiv& P_\mu\partial_{\underline{m}} X^\mu \approx 0 \label{cons-n}\end{aligned}$$ with the Lagrange multipliers which are some nonlinear functions of the world–volume metric $h_{{m}\,{n}}$.
Let us finally deduce from the relations (\[3\]) the $p$–brane mass–shell condition. Multiplying (\[3\]) by $h_{mn}$ we get for $p>1$ $$\label{mass-p}
h_{mn}P^{\,m}_\mu P^{\,\mu\,n} = -(p+1)\,h \,T^2$$ or $$\label{mass-Lag}
{\textstyle\frac{1}{2T}}(-h)^{-1/2}h_{mn}P_\mu^{\,m} P^{\,\mu\,n} = {\textstyle\frac{1}{2}} (p+1) (-h)^{1/2} T\,.$$ In string case ($p=1$) the contraction of l. h. s. of (\[3\]) with $h_{mn}$ is identically vanishing and in any space–time dimension the string mass condition (\[mass-p\]) is absent.
Phase space formulation with tensorial momenta
----------------------------------------------
Other phase space formulation of the $p$–brane (\[action-NG\]) is the model with tensorial momenta. It is obtained by the use of the Liouville $(p+1)$–form (\[tTh-p-int\]). Such a formulation is directly related with the interpretation of $p$–branes as describing the dynamical $(p+1)$–dimensional world volume elements described by the following $(p+1)$–forms [^7]
$$\label{els}
d S^{\mu_1 \ldots \,\mu_{p+1}} = d X^{\mu_1} \wedge \ldots \wedge d
X^{\mu_{p+1}} =
\partial_{m_1} X^{\mu_1} \ldots \partial_{m_{p+1}} X^{\mu_{p+1}} \epsilon^{m_1 \ldots \, m_{p+1}} d^{p+1}\xi \,.$$
The $p$–brane action with tensorial momenta looks as follows [@Nieto] $$\label{tens-4}
S = {\textstyle \frac{2}{\sqrt{(p+1)!}}} \int d^{p+1}\xi \, \Bigg[ \, P_{\mu_1 \ldots
\,\mu_{p+1}}\, \Pi^{\mu_1 \ldots \,\mu_{p+1}} - \Lambda\left( P^{\mu_1 \ldots \,\mu_{p+1}}
P_{\mu_1 \ldots \,\mu_{p+1}} + {\textstyle\frac{T^2}{4}}\right) \Bigg]$$
where $$\label{Pi}
\Pi^{\mu_1 \ldots \,\mu_{p+1}} \equiv \epsilon^{m_1 \ldots \, m_{p+1}}
\partial_{m_1} X^{\mu_1} \ldots \partial_{m_{p+1}} X^{\mu_{p+1}} \,.$$ Expressing $P_{\mu_1 \ldots \,\mu_{p+1}}$ by its equation of motion, we get $$\label{p-mn}
P^{\mu_1 \ldots \,\mu_{p+1}}={\textstyle\frac{1}{2\Lambda}}\,\Pi^{\mu_1 \ldots \,\mu_{p+1}}\,.$$ After substituting (\[p-mn\]) in the action (\[tens-4\]) we obtain the $2(p+1)$-th order action $$\label{tens-5}
S={\textstyle\frac{1}{2\sqrt{(p+1)!}}}\int d^{p+1}\xi \left[ \, \Lambda^{-1} \Pi^{\mu_1
\ldots \,\mu_{p+1}} \Pi_{\mu_1 \ldots \,\mu_{p+1}} - \Lambda T^2 \, \right]\,.$$ Eliminating the auxiliary field $\Lambda$ we obtain $$\label{action-NG-Pi}
S=-{\textstyle\frac{T}{\sqrt{(p+1)!}}}\int d\,^{p+1}\xi\, \sqrt{- \Pi^{\mu_1 \ldots
\,\mu_{p+1}} \Pi_{\mu_1 \ldots \,\mu_{p+1}}} \,.$$ But the determinant of the matrix (\[ind-metr\]) is given by the formula $$\begin{aligned}
\det (g_{mn}) &=& {\textstyle\frac{1}{(p+1)!}}\, \epsilon^{m_1 \ldots \,
m_{p+1}}\epsilon^{n_1 \ldots \, n_{p+1}} g_{m_1 n_1} \ldots g_{m_{p+1} n_{p+1}} \nonumber\\
&=& {\textstyle\frac{1}{(p+1)!}}\, \Pi^{\mu_1 \ldots \,\mu_{p+1}} \Pi_{\mu_1 \ldots
\,\mu_{p+1}} \,. \label{Pi-det}\end{aligned}$$ We see that the action (\[action-NG-Pi\]) is classically equivalent to the action (\[action-NG\]).
The formula (\[tens-5\]) is very useful if we wish to consider for any $p$ the tensionless limit $T \rightarrow 0$. We obtain the formula [@Schild; @Lind; @Nieto] $$\label{tensionless}
S_{T=0}={\textstyle\frac{1}{2\sqrt{(p+1)!}}}\int d^{p+1}\xi \,
{\textstyle\frac{1}{\Lambda}}\, \Pi^{\mu_1 \ldots \,\mu_{p+1}} \Pi_{\mu_1 \ldots
\,\mu_{p+1}}\,.$$ The formula (\[tensionless\]) describes the $p$–brane counterpart of Brink–Schwarz action for massless particle [@BrSch].
Tensionful $D=4$ membrane ($p=2$) in intermediate spinor–space-time formulations
================================================================================
Formulation with vectorial momenta
----------------------------------
In order to obtain from the action (\[act-s\]) the intermediate spinor–space-time action we should eliminate the fourmomenta $P_\mu^m$ by means of the the membrane generalization of the Cartan–Penrose formula expressing fourmomenta as spinorial bilinears. On $d=3$ curved world volume it has the form [^8] $$\label{P-res}
P_{\alpha\dot\alpha}^{\,m}=e\,
\tilde{\lambda}_{\dot\alpha}\rho^m \lambda_{\alpha} =e
e^m_a\tilde{\lambda}_{\dot\alpha}^i(\rho^a)_i{}^j\lambda_{\alpha j}$$ where $\lambda_{\alpha i}$ ($i=1,2$) are two $D=4$ commuting Weyl spinors, $(\rho^a)_i{}^j$ are $2\times 2$ Dirac matrices in three–dimensional Minkowski space–time ($\{\rho^a ,
\rho^b \} =2 \eta^{ab}$) and $\rho^m = e^m_a \rho^a$. After using (\[P-res\]) the second term in the action (\[act-s\]) takes the form [^9] $$\label{2-term}
{\textstyle \frac{1}{2T}}(-h)^{-1/2}h_{mn}P_\mu^{\,m} P^{\,n \mu}={\textstyle
\frac{3}{4T}}\,e\, (\lambda^{\alpha i}\lambda_{\alpha i})
(\tilde{\lambda}_{\dot\alpha}^j\tilde{\lambda}^{\dot\alpha}_j)$$ where we used ${\rm Tr}(\rho^m \rho^n) =2h^{mn}$.
Let us recall the condition (\[mass-Lag\]) which is valid for $p>1$ i. e. also for membrane. In order to get consistency of (\[mass-Lag\]) and (\[2-term\]) we should introduce the following constraint on spinors $\lambda$ $$\label{la-mass}
A \equiv (\lambda \lambda)(\tilde{\lambda} \tilde{\lambda}) - 2\, T^2 =0$$ (we use notations $(\lambda \lambda) \equiv(\lambda^{\alpha i}
\lambda_{\alpha i})$, $(\tilde{\lambda} \tilde{\lambda})
\equiv(\tilde{\lambda}_{\dot\alpha}^i\tilde{\lambda}^{\dot\alpha}_i)$; note that $\tilde{\lambda}_{\dot\alpha}^i\tilde{\lambda}^{\dot\alpha}_i
= \bar{\lambda}_{\dot\alpha}^i\bar{\lambda}^{\dot\alpha}_i$). Putting (\[P-res\]) and (\[2-term\]) in (\[act-s\]) and imposing via Lagrange multiplier the constraint (\[la-mass\]) we obtain the action $$\label{act-mix-mem}
S= \int d^3\xi \, \Bigg[ e\left( \tilde{\lambda}_{\dot\alpha} \rho^m\!\lambda_{\alpha }\,
\partial_m X^{\dot\alpha\alpha}+ 2\, T \right) + \Lambda A \Bigg]$$ which provides the intermediate spinor–space-time formulation of the membrane. Let us observe that the action (\[act-mix-mem\]) is invariant under the following Abelian local gauge transformation $$\label{phase-tr}
\lambda^\prime_{\alpha i} = e^{i\gamma}\lambda_{\alpha i}$$ with real local parameter $\gamma(\xi)$. By fixing the gauge (\[phase-tr\]) we can replace one real constraint (\[la-mass\]) by the following pair of constraints [^10] $$\label{la-mass-2}
(\lambda \lambda)=(\tilde{\lambda} \tilde{\lambda}) = \sqrt{2}\, T \,.$$
Formulation with tensorial momenta
----------------------------------
Let us consider the general action which has the following form $$\label{act-sim}
S= \int d^{3}\xi \, \Big( e e^m_a Q_m^a + 2eT \Big)$$ where $Q_m^a=Q_m^a(X,\lambda)$ do not depend on $e_m^a$. Using the relations $$e=-{\textstyle\frac{1}{3!}}\epsilon^{mnk}\epsilon_{abc} e_m^a e_n^b e_k^c\,, \qquad
ee^m_a=-{\textstyle\frac{1}{2}}\epsilon^{mnk}\epsilon_{abc} e_n^b e_k^c$$ we obtain the following equation of motion for $e^m_a$ $$\label{eq-e}
e_m^a = -{\textstyle\frac{1}{T}}\, Q_m^a \,.$$ Subsequently the action (\[act-sim\]) takes the following classically equivalent form $$\label{act-sim-fin}
S= - {\textstyle\frac{1}{6T^2}} \int d^{3}\xi \, \epsilon_{abc} \epsilon^{mnk} Q_m^a Q_n^b Q_k^c\,.$$ Choosing in the action (\[act-sim\]) $$\label{Q-lX}
Q_m^a = (\tilde{\lambda}_{\dot\alpha}
\rho^a \lambda_{\alpha })\,
\partial_m X^{\dot\alpha\alpha}$$ and after supplementing the constraint (\[la-mass\]) one gets our membrane action (\[act-mix-mem\]). Inserting the formula (\[eq-e\]) we obtain [^11] $$\label{act-mix-tens}
S= {\textstyle\frac{2}{\sqrt{6}}} \int d^{3}\xi \Big( P_{\alpha\dot\alpha, \beta\dot\beta,
\gamma\dot\gamma} \epsilon^{mnk} \partial_m X^{\dot\alpha\alpha} \partial_n
X^{\dot\beta\beta} \partial_k X^{\dot\gamma\gamma} + \Lambda A \Big)$$ where tensorial momenta are composites in term of fundamental spinors [^12] $$\begin{aligned}
\label{P-tens-l}
P_{\mu\nu\lambda} &=& P_{\alpha\dot\alpha, \beta\dot\beta, \gamma\dot\gamma} \\
&=& - {\textstyle\frac{1}{2\sqrt{6}\, T^2}} \, \epsilon_{abc}
(\tilde{\lambda}_{\dot\alpha} \rho^a \lambda_{\alpha }) (\tilde{\lambda}_{\dot\beta} \rho^b
\lambda_{\beta }) (\tilde{\lambda}_{\dot\gamma} \rho^c \lambda_{\gamma }) \nonumber\,.\end{aligned}$$
We see that the intermediate spinor–space-time action (\[act-mix-tens\]) with composite tensorial momenta is obtained after the elimination of dreibein variables $e^a_m$. Taking into account the relation $$\label{l-vec}
(\tilde{\lambda}_{\dot\alpha}\rho^a \lambda_{\alpha }) (\tilde{\lambda}^{\dot\alpha} \rho_b
\lambda^{\alpha }) = \delta^a_b\, T^2\,,$$ following from (\[la-mass\]), and $\epsilon^{abc}\epsilon_{abc}=-3!$ we can show easily that the tensor (\[P-tens-l\]) satisfies the membrane mass shell condition (compare with the constraint in the action (\[tens-4\])) $$\label{P-sq}
P^{\mu\nu\lambda}P_{\mu\nu\lambda} = P^{\alpha\dot\alpha, \beta\dot\beta, \gamma\dot\gamma}
P_{\alpha\dot\alpha, \beta\dot\beta, \gamma\dot\gamma} = - {\textstyle\frac{T^2}{4}} \,.$$ We see that in the action (\[act-mix-tens\]) the condition (\[P-sq\]) follows from the constraint (\[la-mass\]).
Purely twistorial formulation of the membrane ($p=2$) in $D=4$ space–time
=========================================================================
Further we introduce second half of twistor coordinates $\mu^{\dot\alpha}_i$, $\bar{\mu}^{\alpha i}$ by postulating the Penrose incidence relations generalized for $D=4$ membrane fields $$\label{Pen-inc}
\mu^{\dot\alpha}_i=X^{\dot\alpha\alpha}\lambda_{\alpha i} \,,\qquad \tilde{\mu}^{\alpha
i}=\tilde\lambda^i_{\dot\alpha} X^{\dot\alpha\alpha}\,.$$ We shall rewrite the action (\[act-mix-mem\]) by taking into account the relations (\[Pen-inc\]). Using the relations (\[P-res\]), (\[Pen-inc\]) we obtain $$\begin{aligned}
\label{PX-lm}
P_{\alpha\dot\alpha}^{\,m}\partial_m X^{\dot\alpha\alpha} &=& e\,
\tilde{\lambda}_{\dot\alpha}\rho^m\!\lambda_{\alpha}\, \partial_m X^{\dot\alpha\alpha} \\
&=& {\textstyle \frac{1}{2}}\,e \, e_a^m \left(\tilde{\lambda}_{\dot\alpha}\rho^a\partial_m
\mu^{\dot\alpha} -\tilde{\mu}^\alpha\rho^a\partial_m\lambda_\alpha \right) + {\it c.c.}
\nonumber\end{aligned}$$ If we introduce the four–component twistors ($A=1,\cdots,4$) $$\label{ZbZ}
Z_{Ai}=(\lambda_{\alpha i}, \mu_i^{\dot\alpha}), \qquad \tilde
Z^{Ai}=(\tilde\mu^{\alpha i}, -\tilde\lambda_{\dot\alpha}^{i} ),$$ the relations (\[PX-lm\]) takes the form $$\begin{aligned}
\label{PX-ZbZ}
P_{\alpha\dot\alpha}^{\,m}\partial_m X^{\dot\alpha\alpha} &=& e\,
\tilde{\lambda}_{\dot\alpha}\rho^m\!\lambda_{\alpha}\, \partial_m X^{\dot\alpha\alpha} \\
&=& {\textstyle \frac{1}{2}}\, e\, e_a^m \left(\partial_m\tilde Z^{A}\rho^a Z_{A} - \tilde
Z^{A}\rho^a \partial_m Z_{A} \right)\,. \nonumber\end{aligned}$$
Incidence relations (\[Pen-inc\]) with real space–time membrane position field $X^{\dot\alpha\alpha}$ imply that the twistor field variables satisfy the constraints $$\label{V}
V_i{}^j \equiv \lambda_{\alpha i}\tilde{\mu}^{\alpha j} - \mu^{\dot\alpha}_i
\tilde{\lambda}_{\dot\alpha}^j \approx 0$$ which can be rewritten equivalently $$\label{V-Z}
V_i{}^j = Z_{Ai}\tilde Z^{Aj}\approx 0\,.$$ We obtain the following membrane action (\[act-mix-mem\]) in twistor formulation with dreibein $$\begin{aligned}
S= \int d^{3}\xi \!&\! \Bigg[ \!&\! {\textstyle \frac{1}{2}}\, e\, e_a^m
\left(\partial_m\tilde
Z^{A}\rho^a Z_{A} - \tilde Z^{A}\rho^a \partial_m Z_{A} \right) + \nonumber\\
&& + \, 2e\, T + \Lambda A +\Lambda_j{}^i V_i{}^j \Bigg] \label{action-tw-mem}\end{aligned}$$ where $\Lambda$ and $\Lambda_i{}^j$ are the Lagrange multipliers. If we define the asymptotic twistors [@PenMac; @Hug] $$\label{I}
I^{AB}=\left(
\begin{array}{cc}
\epsilon^{\alpha\beta} & 0 \\
0 & 0
\end{array}
\right)\,, \qquad I_{AB}=\left(
\begin{array}{cc}
0 & 0 \\
0 & \epsilon^{\dot\alpha\dot\beta}
\end{array}
\right)$$ one can introduce the following notation $$\begin{aligned}
(\lambda\lambda) \equiv & \lambda^{\alpha i} \lambda_{\alpha i} = \epsilon^{ij} I^{AB}
Z_{Ai}Z_{Bj} &\equiv (ZZ)\,, \label{ll-ZZ1}\\
(\tilde\lambda \tilde\lambda) \equiv & \tilde\lambda_{\dot\alpha}^{i}
\tilde\lambda^{\dot\alpha}_i = \epsilon_{ij} I_{AB} \tilde Z^{Ai}\tilde Z^{Bj} &\equiv
(\tilde Z\tilde Z) \label{ll-ZZ2}\end{aligned}$$ and write down the fourlinear constraint (\[la-mass\]) in the following twistorial form: $$\label{Z-mass}
A \equiv (Z Z)(\tilde Z\tilde Z) - 2T^2 =0\,.$$
We shall eliminate the dreibein $e^a_m$ by employing the formula (\[eq-e\]). The action (\[action-tw-mem\]) correspond to the choice $$\label{Q-Z}
Q_m^a = {\textstyle \frac{1}{2}}\,
\left(\partial_m\tilde Z^{A}\rho^a
Z_{A} - \tilde Z^{A}\rho^a \partial_m Z_{A} \right) \,.$$ One gets the final action depending only on two twistorial fields $Z_{Ai}(\tau, \sigma^1,
\sigma^2)$ and suitably rescaled (in comparison with (\[action-tw-mem\])) the Lagrange multipliers $\Lambda$, $\Lambda_j{}^i$:
$$\begin{aligned}
S \!\!\!&=& \!\!\!-{\textstyle\frac{1}{48 T^2}}\, \int d^{3}\xi \, \Bigg[ \epsilon_{abc}
\epsilon^{mnk} \left(\partial_m\tilde Z^{A}\rho^a Z_{A} \!-\! \tilde Z^{A}\rho^a \partial_m
Z_{A} \right)\! \left(\partial_n\tilde Z^{B}\rho^b Z_{B} \!-\! \tilde Z^{B}\rho^b
\partial_n Z_{B} \right)\! \left(\partial_k\tilde Z^{C}\rho^c
Z_{C} \!-\! \tilde Z^{C}\rho^c \partial_k Z_{C} \right) + \nonumber\\
&& \qquad\qquad\qquad \qquad\qquad + \, \Lambda A + \Lambda_j{}^i V_i{}^j \Bigg] \,.
\label{act-pure-tw}\end{aligned}$$
The model (\[act-pure-tw\]) describes the $D=4$ membrane in purely twistorial formulation. Introducing three one–forms with world-volume–vectorial index $$\label{1-forms}
\Theta_{\!(1)}^{\,a} \equiv d\tilde Z^{A}\rho^a
Z_{A} \!-\! \tilde Z^{A}\rho^a d Z_{A}$$ one can obtain the action (\[act-pure-tw\]) as induced on the membrane world volume by the following three–form $$\label{3-form}
\Theta_{\!(3)} = \epsilon_{abc}\, \Theta_{\!(1)}^{\,a} \wedge \Theta_{\!(1)}^{\,b} \wedge \Theta_{\!(1)}^{\,c}\,.$$
Outlook
=======
In this paper we presented the new description of the twistorial membrane in $D=4$ space–time. We would like now to comment on two generalizations:
i)
: to $p$–branes with $p>2$ in arbitrary $D$–dimensional ($D>p+1$) space–time
ii)
: to super–$p$–branes in higher dimensions.
The basic relation in the construction of twistorial formulation of the $p$–branes in dimension $D$ is a suitable generalization of Cartan–Penrose formula (\[P-res\]). For arbitrary $p$ and arbitrary $D$ such a formula looks as follows: $$\label{P-res-D}
P_{\mu}^{\,m}= e e^m_a\,\tilde{\lambda}^{\hat\alpha i}(\rho^a)_i{}^j\lambda_{\hat\beta j}\,
(\gamma_\mu)_{\hat\alpha}{}^{\hat\beta}$$ where $\gamma_\mu$ are the Dirac matrices in $D$–dimensional space–time and $\rho^a$ are Dirac matrices in $d=p+1$ dimensional space (tangent space to curved world volume geometry of the $p$–brane). Thus the spinor $\lambda_{\hat\alpha i}$ has two indices: it is spinor in $D$ dimensions with the components described by index $\hat\alpha$ and as well spinor in $d$ dimensions with index $i$; we employ also the spinor $\tilde{\lambda}^{\hat\alpha i}$ which is a Dirac–conjugated spinor with respect to both indices: $\tilde{\lambda}^{\hat\alpha i} =(\lambda_{\hat\beta j})^+
(\rho^0)^{ji}(\gamma^0)^{\hat\beta\hat\alpha}$. For general $D$ and $p$ we get that $\hat\alpha=1,\ldots ,2^{[\frac{D}{2}]}$ and $i =1,\ldots ,2^{[\frac{p+1}{2}]}$. Thus in order to obtain the composite $p$–brane momenta we must use at least $2^{[\frac{p+1}{2}]}$ twistors. For definite dimensions $p$ and $D$ we can further decrease the number of the elementary spinor components $\lambda_{\hat\alpha i}$ by imposing consistently Majorana-, Weyl- or Majorana–Weyl conditions.
Inserting (\[P-res-D\]) in the action (\[act-s\]) we shall obtain the intermediate spinor–space-time formulation. Due to the mass–shell for the vectorial momenta (see (\[mass-p\])) the elementary spinors $\lambda_{\hat\alpha i}$ will be constrained (compare with (\[la-mass\]) and (\[l-vec\])). In order to get the purely twistorial formulation of $p$–branes one has to introduce $D$–dimensional incidence relation (\[Pen-inc\]) which provides the doubling of spinor components and lifts the Lorentz spinors to twistors. It should be stress however that in general case the $D$–dimensional incidence relation will introduce extended $D$–dimensional space–time. Only suitable use of the additional spinor structures (e. g. quaternionic in $D=6$) and imposition of the algebraic constraints in twistor space (e. g. selecting only null twistors lying on null hyperplanes) permits to obtain the incidence relations just with the Minkowski space–time coordinates.
The extension of the twistorial formalism for bosonic $p$–branes to super–$p$–branes requires the introduction of $p$–brane supertwistors. The techniques of supersymmetrization of various twistorial $p$–brane models were already studied (see e. g. [@BAM; @Uv]). It should be also mentioned that our construction can be linked to the analysis based on the use of Lorentz harmonics [@BZ-p] as well as with the formalism using $d=11$ BPS preons [@BAIL; @BAPV] described by generalized $OSp(1|64)$ supertwistor fields.
S.F. would like to thank Institute for Theoretical Physics, Wroc[ł]{}aw University for kind hospitality and a very friendly creative atmosphere. He would like to thank Bogoliubov–Infeld program for financial support. The work of S.F. was partially supported also by the RFBR grant 06-02-16684 and the grant INTAS-05-7928.
[99]{}
R. Penrose and M.A.H. MacCallum, Phys. Reports [**C6**]{}, 241 (1972);\
R. Penrose, in “Quantum Gravity”, ed. C.J. Isham, R. Penrose and D.W. Sciama, p. 268, Oxford Univ. Press, 1975.
A. Ferber, Nucl. Phys. [**B132**]{}, 55 (1978).
T. Shirafuji, Prog. Theor. Phys. [**70**]{}, 55 (1983).
N. Bengtsson and M. Cederwall, Nucl. Phys. [**B302**]{}, 81 (1988).
D. Sorokin, V. Tkach and D.V. Volkov, Mod. Phys. Lett. [**A4**]{}, 901 (1989).
L.P. Hughston, *Twistors and Particles*, Lecture Notes in Physics [**97**]{}, Springer Verlag, Berlin (1979)
A. Bette, J. Math. Phys. [**25**]{}, 2456 (1984).
S. Fedoruk and V.G. Zima, J. Kharkov Univ. [**585**]{}, 39 (2003) .
A. Bette, J.A. de Azcárraga, J. Lukierski and C. Miquel-Espanya, Phys. Lett. [**B595**]{}, 491-497 (2004) .
J.A. de Azcárraga, S. Fedoruk, A. Frydryszak, J. Lukierski and C. Miquel-Espanya, [**D73**]{}, 105011 (2006) .
S. Fedoruk, A. Frydryszak, J. Lukierski and C. Miquel-Espanya, Int. J. Mod. Phys. [**A21**]{}, 4137 (2006) .
I. Bars and M. Picon, Phys. Rev. [**D73**]{}, 064002 (2006) ; [**D73**]{}, 064033 (2006) .
L.J. Mason, JHEP [**0510**]{}, 009 (2005) .
R. Boels, L. Mason and D. Skinner, JHEP [**0702**]{}, 014 (2007) ; Phys. Lett. [**B648**]{}, 90 (2007) .
E. Witten, Commun. Math. Phys. [**252**]{}, 189 (2004) .
N. Berkovits and E. Witten, JHEP [**0408**]{}, 009 (2004) .
S. Fedoruk and J. Lukierski, Phys. Rev. [**D75**]{}, 026004 (2007) .
T.R. Taylor, [*Non-commutative Field Theory with Twistor Coordinates*]{}, [arXiv: 0704.2071 \[hep-th\]]{}.
I.A. Bandos and A.A. Zheltukhin, Class. Quant. Grav. [**12**]{}, 609 (1995) .
I. Bandos, P. Pasti, D. Sorokin, M. Tonin and D. Volkov, Nucl. Phys. [**B446**]{}, 79 (1995) .
I. Bandos, D. Sorokin and D. Volkov, Phys. Lett. [**B352**]{}, 269 (1995) .
I.A. Bandos, J.A. de Azcárraga and C. Miquel-Espanya, JHEP [**0607**]{}, 005 (2006) .
D.V. Uvarov, Class. Quant. Grav. [**23**]{}, 2711 (2006) ; [*Gauge symmetries of strings in supertwistor space*]{}, ; [*Supertwistor formulation for higher dimensional superstrings*]{}, .
W. Siegel, Nucl. Phys. [**B263**]{}, 93 (1985).
M. Pavsic, Phys. Lett. [**B197**]{}, 327 (1987).
M. Gurses and F. Gursey, Phys. Rev. [**D11**]{}, 967 (1975).
A. Aurilia and E. Spallucci, Class. Quant. Grav. [**10**]{}, 1217 (1993) .
O.E. Gusev and A.A. Zheltukhin, JETP Lett. [**64**]{}, 487 (1996).
J.A. Nieto, Mod. Phys. Lett. [**A16**]{}, 2567 (2001) .
P.A.M. Dirac, Proc. Roy. Soc. Lond. [**A268**]{}, 57 (1962).
Y. Nambu, Lectures at the Copenhagen Summer Symposium (1970);\
T. Goto, Prog. Theor. Phys. [**46**]{}, 1560 (1971).
P.S. Howe and R.W. Tucker, J. Phys. [**A10**]{}, L155 (1977).
A.M. Polyakov, Phys. Lett. [**B103**]{}, 207 (1981).
A. Sugamoto, Nucl. Phys. [**B215**]{}, 381 (1983).
L. Brink, P. Di Vecchia and P.S. Howe, Phys. Lett. [**B65**]{}, 471 (1976); Nucl. Phys. [**B118**]{}, 76 (1977).
V.A. Soroka, D.P. Sorokin, V.V. Tkach and D.V. Volkov, JETP Lett. [**52**]{}, 526 (1990), Int. Journ. Mod. Phys. [**A7**]{}, 5977 (1992).
A. Schild, Phys. Rev. [**D16**]{}, 1722 (1977).
A. Karlhede and U. Lindstrom, Class. Quant. Grav. [**3**]{}, L73 (1986);\
J. Isberg, U. Lindstrom and B. Sundborg, Phys. Lett. [**B293**]{}, 321 (1992) .
L. Brink and J.H. Schwarz, Phys. Lett. [**B100**]{}, 310 (1981).
I.A. Bandos, J.A. de Azcárraga, J.M. Izquierdo and J. Lukierski, Phys. Rev. Lett. [**86**]{}, 4451 (2001) .
I.A. Bandos, J.A. de Azcárraga, M. Picon and O. Varela, Phys. Rev. [**D69**]{}, 085007 (2004) .
[^1]: Supported by KBN grant 1 P03B 01828
[^2]: One should add that unconventional application of one–twistor geometry to massive particles has been proposed in [@Bars].
[^3]: We propose alternative way of generating constraints in comparison with the framework of Lorentz harmonic approach [@BZ-p; @BPSTV; @BSV].
[^4]: The indices $m,n=0,1,\ldots,p$ are vector world-sheet indices; $\mu,\nu=0,1,\ldots, D-1$ is vector space–time ones. We use the following flat metrics: $\eta^{ab}=(-,+,\ldots,+)$, $\eta^{\mu\nu}=(-,+,\ldots,+)$.
[^5]: The string actions as $d=2$ world sheet gravity interacting with string coordinate fields were originally proposed in [@BrDes], where as well the description of spinning string using interacting $d=2$ supergravity is presented.
[^6]: The relation (\[h-g\]) is unique for $p\neq 1$; for $p= 1$ one can introduce in (\[h-g\]) an arbitrary local scaling factor.
[^7]: Total antisymmetric tensors $\epsilon^{m_1 \ldots \, m_{p+1}}$, $\epsilon_{m_1 \ldots \, m_{p+1}}$ and $\epsilon^{a_1
\ldots \, a_{p+1}}$, $\epsilon_{a_1 \ldots \, a_{p+1}}$ have the components $\epsilon^{01
\ldots \, p}=1$, $\epsilon_{0 1 \ldots \, p}=-1$.
[^8]: $h_{mn}=e_m^a e_{n a}$ is a world–volume metric, $e_m^a$ is the dreibein, $e_m^a e^m_b = \delta^a_b$, $e=\det(e_m^a)=\sqrt{-h}$. The indices $a,b=0,1,2$, $m,n=0,1,2$ are $d=3$ vector indices; the indices $i,j=1,2$ are $d=3$ Dirac spinor indices. We use bar for complex conjugate quantities, $\bar{\lambda}_{\dot\alpha}^i =(\overline{\lambda_{\alpha i}})$, and tilde for Dirac–conjugated $d=2$ spinors, $\tilde{\lambda}_{\dot\alpha}^i =
\bar{\lambda}_{\dot\alpha}^j(\rho^0)_j{}^i$.
[^9]: The matrix of charge conjugation is the $2\times 2$ skew-symmetric tensor $\epsilon^{ij}$. For definiteness, we take $\epsilon^{12}=1$. The rules of lifting and lowering the indices are following: $a^i=\epsilon^{ij}a_j$, $a_i=\epsilon_{ij}a^j$ where $\epsilon^{ij}\epsilon_{jk}=\delta^i_k$. Also, we use $D=4$ Penrose spinor–vector conventions [@PenMac] in which $P_\mu P^\mu =
P_{\alpha\dot\alpha}P^{\dot\alpha\alpha}$, $P_\mu X^\mu =
P_{\alpha\dot\alpha}X^{\dot\alpha\alpha}$ ($P_{\alpha\dot\alpha} = \frac{1}{\sqrt{2}}P_\mu
\sigma^\mu_{\alpha\dot\alpha}$ etc.)
[^10]: Compare with the string case considered in [@FL].
[^11]: The Lagrange multiplier in (\[act-mix-tens\]) is obtained from the one in (\[act-mix-mem\]) by the rescaling $\Lambda \rightarrow \frac{2}{\sqrt{6}}\,\Lambda$.
[^12]: The coefficient in (\[P-tens-l\]) are chosen in consistency with the general $p$–brane formula (\[tens-4\]).
|
---
author:
- 'P. Zacharias [^1]'
- 'H. Peter'
- 'S. Bingert'
title: Ejection of cool plasma into the hot corona
---
[The corona is highly dynamic and shows transient events on various scales in space and time. Most of these features are related to changes in the magnetic field structure or impulsive heating caused by the conversion of magnetic to thermal energy. ]{}[We investigate the processes that lead to the formation, ejection and fall of a confined plasma ejection that was observed in a numerical experiment of the solar corona. By quantifying physical parameters such as mass, velocity, and orientation of the plasma ejection relative to the magnetic field, we provide a description of the nature of this particular plasma ejection. ]{}[The time-dependent three-dimensional magnetohydrodynamic (3D MHD) equations are solved in a box extending from the chromosphere, which serves as a reservoir for mass and energy, to the lower corona. The plasma is heated by currents that are induced through field line braiding as a consequence of photospheric motions included in the model. Spectra of optically thin emission lines in the extreme ultraviolet range are synthesized, and magnetic field lines are traced over time. We determine the trajectory of the plasma ejection and identify anomalies in the profiles of the plasma parameters. ]{}[Following strong heating just above the chromosphere, the pressure rapidly increases, leading to a hydrodynamic explosion above the upper chromosphere in the low transition region. The explosion drives the plasma, which needs to follow the magnetic field lines. The ejection is then moving more or less ballistically along the loop-like field lines and eventually drops down onto the surface of the Sun. The speed of the ejection is in the range of the sound speed, well below the Alfvén velocity. ]{}[The plasma ejection observed in a numerical experiment of the solar corona is basically a hydrodynamic phenomenon, whereas the rise of the heating rate is of magnetic nature. The granular motions in the photosphere lead (by chance) to a strong braiding of the magnetic field lines at the location of the explosion that in turn is causing strong currents which are dissipated. Future studies need to determine if this process is a ubiquitous phenomenon on the Sun on small scales. Data from the Atmospheric Imaging Assembly on the Solar Dynamics Observatory (AIA/SDO) might provide the relevant information. ]{}
Introduction
============
The solar corona is highly structured and very dynamic. This ranges from large disruptive events such as flares and coronal mass ejections (CMEs) down to small-scale explosive events in the transition region between the chromosphere and the corona. Because the coronal dynamics are a direct signature of the processes governing the energy balance in the upper atmosphere, i.e., coronal heating, radiative losses and heat conduction, the investigation of coronal transients on all temporal and spatial scales is a key for also understanding the nature of the corona of the Sun and other stars. We present results from a three-dimensional magnetohydrodynamic (3D MHD) numerical experiment of the solar corona in which an explosion-like event drives a confined plasma ejection ballistically through the corona along the magnetic field lines.
Imaging instruments provide information on the temporal and spatial variation of the coronal emission. Especially during transient events, their interpretation is often non-trivial and might be attributed to either waves or mass flows. For example, early on, intensity variations were associated with the evaporation and condensation of plasma and the motion of the condensation region [@antiochos+sturrock:1978]. Outward-propagating features of fan-like structures have often been interpreted as (slow) magneto-acoustic waves, [e.g., @demoortel:2002a].
The investigation of cool plasma of several 10.000K as seen in the Ly-$\alpha$ line above the limb indicates that small brightenings are moving along the magnetic field lines. These have been analyzed in depth by [@Schrijver:2001] using data from TRACE [Transition Region and Coronal Explorer; @1999SoPh..187..229H]. The movie provided with the online version of [@Schrijver:2001] clearly shows three types of bright cool structures moving in the corona: (1) features appearing up in the corona that are falling down, (2) brightenings moving upward that subsequently disappear, and (3) small brightenings initially moving upward before they fall back again. These types of brightenings have been interpreted in different ways.
Type (1) features can be interpreted as condensations in the corona caused by a loss of thermal equilibrium. These features have been investigated in depth by [@mueller:2003; @mueller:2004] through 1D loop models and compared to observations [@mueller:2005]. These models show that if the heating is concentrated in the lower atmosphere and is not sufficient to balance the radiative cooling at the loop apex, a runaway process will be initiated, because the cooler plasma can cool more efficiently and through this rapidly forms a condensation. Eventually, this condensation slides down the field lines with speeds of up to 100km/s. [@degroof:2004] observed that features moving downward are indeed plasma flows rather than magneto-acoustic waves.
The second type of cool brightenings that move upward and subsequently disappear might be associated with the high-speed transition region line upflows as discussed by [@mcintosh+depontieu:2009]. The authors conclude that these upflows might be visible as propagating disturbances in the extreme ultraviolet. They may or may not be the small-scale relatives of explosive events and spicules. [@mcintosh+depontieu:2009] and [@depontieu:2009] found these upflows through a blueward asymmetry and line broadening of transition region and coronal lines in active regions that have been reported before for transition lines in the quiet Sun network [@peter:2000; @peter:2001]. The cool brightenings eventually heat up and disappear from the observations of the cool plasma (as in Ly-$\alpha$), leaving a footprint in the asymmetry of lines formed at higher coronal temperatures above $10^6$K [@hara+al:2008; @peter:2010].
The type (3) cool brightenings that move up and fall back might be interpreted as plasma that is propelled upward, follows the magnetic field lines and then falls back onto the Sun without being heated up to coronal temperatures as the type (2) brightenings are. We found evidence for this type (3) process in our numerical experiments of the solar corona. The driving of the magnetic field in the photosphere through horizontal convective (granular) motions results in a high energy input over a short time caused by Ohmic dissipation in the transition region. An explosion event of this kind would be pushing cool material upward, which then follows the magnetic field lines up to the apex and slides down on the other side of the magnetic loop.
The high energy deposition that eventually leads to the plasma ejection comes about self-consistently when (by chance) the granular motions in the photosphere lead to a magnetic configuration with a high rate of Ohmic dissipation in a small region in the simulation box.
By analyzing the data of the numerical experiment, we will investigate the formation, rise and fall of the plasma ejection in the 3D MHD model and quantify its physical parameters, such as mass, velocity and orientation relative to the magnetic field. In addition, we synthesize emission lines that are observable with current instrumentation.
We emphasize that this study is not investigating spicule-like phenomena. Related high-speed upflows that have recently been observed from the chromosphere through the transition region to the corona [@depontieu+al:2011] most likely play a significant role for the supply of hot plasma into the corona. Likewise, observations have shown strong upflows at the edges of active regions [e.g., @sakao+al:2007; @Doschek+al:2008]. Observational evidence exists that these upflows appear as asymmetries in the line profiles [e.g., @depontieu:2009; @mcintosh+depontieu:2009] or as second components in the spectral profiles [e.g., @hara+al:2008; @peter:2010]. However, these high-speed upflow structures seem to be relatively stable. Probably, the transient plasma ejection we discuss here is closer related to miniature coronal mass ejections (mini-CMEs) as described by, e.g., [@Innes+al:2009; @Innes+al:2010].
We briefly explain the setup of the numerical expriment in [Sect.\[model\]]{}. A qualitative description of the observed phenomenon is provided in Sect.\[phenomenology\], and the onset of the eruption is investigated in detail in Sect.\[onset\]. The dynamic evolution of the ejection along its trajectory is discussed in Sect.\[evolution\], and in Sect.\[aia\_section\] we comment on the visibility of this phenomenon with respect to the Atmospheric Imaging Assembly onboard the Solar Dynamics Observatory (AIA/SDO). Finally, the results are summarized and discussed in Sect.\[discussion\].
{width="\textwidth"}
Setup of the numerical experiment {#model}
=================================
We analyze an ejection of cool material into the corona in a 3D MHD simulation of the solar corona above a small active region. The setup of the 3D model is identical to the one described in [@bingert:2010]. We used the Pencil code [@brandenburg+dobler:2002] to solve the 3D MHD equations. The heat conduction term parallel to the magnetic field and the radiative losses were included in the energy equation to obtain a realistic description of the coronal pressure and thus of the emerging synthesized emission from the transition region and corona. The corona is heated through Ohmic dissipation of currents that are induced by the photospheric footpoint motions shuffling around the magnetic field. The computational domain encompasses 50Mm ${\times}$ 50Mm horizontally and 30Mm vertically. The initial magnetic field at the lower boundary is taken from an observed active region magnetogram. After approximately 30min of simulation time, the solution is independent of the initial condition, and the magnetic field in the corona is computed self-consistently based on the MHD equations. After this time period, the corona has reached a dynamic equilibrium, where the overall appearance and spatially averaged quantities do not change significantly, while the situation is still dynamic with strong flows and changes in the plasma properties (cf. movie attached to [Fig.\[fig.movie\]]{}). For more details, we refer the reader to [@bingert:2010].
The model does not contain the proper physics of the chromosphere, especially (non-LTE) radiative transfer is not included in the calculations. However, because the phenomenon we describe emerges from *above* the chromosphere, this is not expected to be a severe issue. At the top of the chromosphere, the energy input is typically $10^{-5}\,\mbox{Wm}^{-2}$, while for the plasma ejection described here, the heating rate just *above* the chromosphere rises to values a factor of 100 higher than that just before the ejection. Therefore, we do not expect the inclusion of non-LTE radiative transfer to have a major effect here. Additional energy transfer, e.g., through radiation, would certainly slightly mitigate the effect, but would not fully compensate the strong increase in heating rate (see also [Sect.\[energy\]]{}).
From the results of the 3D MHD model, we calculated the emission in the transition region and corona. Emission line spectra of various transition region and coronal lines are synthesized using the CHIANTI atomic data base [@dere+al:1997; @landi+al:2006]. This procedure follows [@zacharias:2009; @zacharias:2010.doppler]. The O (1032Å) line was chosen to represent cool transition region material, and the Mg (625Å) line to represent hot coronal plasma (see Table\[lines\]). The emission of these lines is roughly comparable to the 304Å and 171Å bands observed with AIA/SDO (cf. Sect.\[aia\_section\]), which were derived using the temperature response functions [@Boerner+al:2011] as provided by the instrument team in SolarSoft.[^2] The plasma is assumed to be optically thin and in a state of ionization equilibrium (Peter et al., [-@peter+al:2006]). The constant elemental abundances are taken from Mazotta et al. ([-@mazzotta+al:1998]).
The specific choice of the O and Mg lines is not crucial. The temporal evolution of the ejection looks quite similar in C and Ne, respectively. The synthesized Mg images are similar to those synthesized for the 195[Å]{} channel of AIA, the Ne images are similar to those of the 171[Å]{} AIA band. In this study, the difference between these four coronal images is not significant, and therefore the validity of the images displayed in Figs.\[emiss\], \[aia\] and \[fig.movie\] does not depend on the choice of the respective line. Also, the small differences in appearance between images of C, O and AIA304[Å]{} are not significant for the illustrative purpose of these figures. Actually, O is more similar to the AIA304[Å]{} band than C. The AIA304[Å]{} band has some contribution from high temperatures, which originates from hot emission lines in the bandpass. Likewise, O shows some contribution from high temperatures that is caused by di-electronic recombination. C, being a Li-like ion, also shows this effect, but only to some extent. The quantitative analysis in Sects.\[onset\] and \[evolution\] is not influenced by the choice of the lines or bands, of course.
----------- ------------ -----------------------------------
line wavelength formation temperature
\[Å\] $\log ( T_{\rm } / [\mbox{K}] ) $
O 1032 5.5
Mg 625 6.0
AIA / He 304 4.9
AIA / Fe 171 5.9
----------- ------------ -----------------------------------
: Emission lines and AIA/SDO bands synthesized from 3D MHD model.\[lines\]
[*Note 1:*]{} For O and Mg, the line formation temperature is given, i.e., the temperature where $G(T)\propto(n_{\rm ion}/n_{\rm el})~T^{-1/2}\,\exp{\big[{-}h\nu / (k_B T)\big]}$ reaches its maximum value (assuming ionization equilibrium).\
[*Note 2:*]{} For the AIA bands, the main contributing lines are given along with the center wavelength and the formation temperature of the plasma dominating the emission in the respective band.
Phenomenology
=============
We are focusing on a time sequence of about 70 minutes duration in the numerical experiment. All points of time discussed here refer to time $t{=}0$ at the beginning of this sequence. The actual simulation was started long before $t{=}0$, so that any impact of the initial condition is excluded during the investigated time sequence.
The ejection lifts off at about $t{=}40$min and lasts for about 11min. It is observed in the form of a “bubble” of plasma that rises and falls back down again along a curved trajectory. Snapshots of the emissivities integrated along the horizontal $y$-direction of the simulation box are presented in Fig.\[emiss\] for different time steps of the simulation run. In addition, the corresponding Ohmic heating rates ($\propto j^2$, the current density squared) are shown. For details on the displayed images and a movie of the temporal evolution, see Appendix \[appendix\] and Eqs.\[E:jnorm\] and \[E:jfluct\]).
Before the plasma eruption, an increased heating rate is observed just above the chromosphere in the region outlined by the red box and arrow $a$ in Fig.\[emiss\] at $t{=}38$min (best seen in the movie with Fig.\[fig.movie\], indicated by an arrow). At $t{=}41.5$min, the plasma ejection lifts off (Fig. \[emiss\], second row), and the cool ejection is clearly visible in the O line formed at transition region temperatures of about $3{\cdot}10^5$ K (arrow $b$). The cool ejection, however, is hardly visible in the Mg coronal line formed at approximately $10^6$ K (arrow $c$). In this line, the plasma ejection merely appears as a darkish shade because of its low temperature compared to the surroundings. At $t{=}47$min, the ejection reaches its maximum height and starts descending. When falling down, the ejection compresses the magnetic field at its front. As a consequence, currents increase in front of the ejection (arrow $f$), plasma is heated, and the front is brightening (arrows $d,e$; again, best visible in the movie). The magnetic field compression caused by the ejection produces a heating front that is traveling ahead of the actual plasma ejection. Thus, the plasma is heated at low heights just above the chromosphere and consequently shines bright in the coronal line (arrow $h$). At this location and time ($t{=}52$min), the emission in the transition region line is not yet brightening (arrow $g$), but it does when the ejection actually hits the chromosphere (as can be seen in the movie). There is a time delay between the brightening of the coronal line and the transition region line (at the location of arrows $g$ and $h$) of about 1min. In the movie, some of the plasma can also be seen to bounce up again in the rear part of the ejection because of the increased pressure when the ejection hits the chromosphere. This observation is similar to the bouncing of condensations falling down on the solar surface, as was found by [@mueller:2003; @mueller:2004]. A few minutes later, the plasma ejection has completely disappeared (Fig.\[emiss\], bottom row).
![Coronal box as seen in O (1032 Å) synthesized from 3D MHD model (emission increases from blue over violet to red). Overplotted are the magnetic field lines that cover the trajectory of the ejection (snapshot at 45min). Visualization using VAPOR [@clyne+rast:2005; @clyne+al:2007]. []{data-label="vapor"}](fig2.ps){width="1.0\columnwidth"}
The correlation of the plasma ejection and the magnetic field is depicted in Fig.\[vapor\], which provides a 3D view of the computational domain as it would be seen in the emission of the O line at time $t{=}45$min, i.e., close (in time) to the middle row of Fig.\[emiss\].
The ejection, shown together with magnetic field lines in the vicinity of its trajectory in Fig.\[vapor\], has almost reached its maximum altitude and is moving nearly parallel to the magnetic field structure (cf.Sect.\[mag.interaction\]). As can also be seen in Fig.\[vapor\], the magnetic field is expanding with height, which causes the varying diameter of the ejection (cf. Fig.\[emiss\] and movie provided in Appendix\[appendix\]). Fig.\[vapor\] also shows that the magnetic field is not helical.
Onset of the plasma ejection {#onset}
============================
In this section, the processes that trigger and drive the plasma ejection will be investigated. To study the onset of the plasma eruption in terms of time and space, peculiarities in the plasma properties must be looked for, mainly with regard to the current density, which is an indicator for plasma heating that can cause an eruption (Sect.\[formation1\]).
When looking at the computational domain from the side, the ejection becomes clearly visible after it has emerged above the ambient transition region. This is the case at a height of approximately 10.5Mm, between $t{=}39$min and 40min. The shape of the ejection appears tailed when ascending or descending and ellipsoidal when it is close to its maximum height. The ejection returns to the initial height at $t{=}51.5$min on the other side of the loop-shaped magnetic field lines. Subsequently, it slowly dissolves into the bright plasma and has completely disappeared after $t{=}56$ min.
To determine the trajectory of the ejection, the plasma that is part of the ejection must be defined. It proved to be useful to consider ejected plasma in the volume where the emissivity of O is observed to exceed $10^{-7}$ W m$^{-3}$. Thus, the surface of the ejection can be defined and its center of gravity localized. Tracing the center of gravity as a function of time allows us to determine the trajectory of the ejection. This procedure works best for the time periods after the ejection has detached from the ambient transition region, during its rising phase and before it touches the ambient transition region again when it falls back onto the solar surface. The analysis of the plasma parameters along this trajectory is presented in Sect.\[formation2\].
{width="\linewidth"}
Heating rate and pressure in the formation region {#formation1}
-------------------------------------------------
Fig.\[blob\_start\] visualizes the evolution of both the heating rate and the pressure in a small volume indicated by the red box in Fig.\[emiss\] (the volume extension is identical for both the $x$ and the $y$ direction) chosen to capture the formation site of the ejection. Before the eruption, the Ohminc heating rate, $H_{\rm{Ohm}}{=}\mu_0{\eta}j^2$, continuously increases by approximately two orders of magnitude within 20 minutes (Fig.\[blob\_start\], left) causing a temperature increase in the formation region. At the same time, the density increases through chromospheric evaporation. These occurances lead to a considerable rise in pressure (Fig.\[fig3\]) that eventually results in the observed explosion at $t{\approx}39$min.
The rapid rise of the Ohmic dissipation, and thus the pressure, is limited to the small volume indicated by the red box in Fig.\[emiss\] extending 2Mm${\times}$2Mm horizontally and 1Mm vertically). In general, the heating of the corona in the applied model is caused by the braiding of magnetic field lines [@Parker:1972; @Parker:1988]. Numerical 3D MHD models for this process have been developed by [@Gudiksen+Nordlund:2002; @Gudiksen+Nordlund:2005a; @Gudiksen+Nordlund:2005b] for the first time. In these models, horizontal motions in the photosphere shuffle around the magnetic field which results in the braiding of magnetic field lines. Thus, currents are induced that are subsequently dissipated in the upper atmosphere.
The photospheric driving by the horizontal motions in our 3D MHD model [see @bingert:2010] leads by chance, not by prescription, to a strong increase of the heating rate at the spot that will later become the origin of the plasma ejection. This rapid increase of the pressure is comparable to an explosion that subsequently drives plasma away from the explosion site.
According to the realistic driver used to describe the photospheric motions, it appears plausible that a strong increase of the current density just above the chromosphere can also happen on the Sun, e.g., in the periphery of an active region or pore.
![Propagation of the pressure disturbance in the early phase of the ejection. During this interval, the ejection still rises almost vertically. The different curves show the (peak) pressure as a function of height in a vertical column indicated by the dotted lines in Fig.\[emiss\] (bottom right panel). As time progresses, the location of the peak pressure moves upward. The vertical extension of the red boxes in Fig.\[emiss\] is indicated by the vertical dashed-dotted lines. []{data-label="fig3"}](fig4.ps){width="\columnwidth"}
Pressure along the trajectory of the ejection {#formation2}
---------------------------------------------
The trajectory of the ejection must be defined before we investigate the ejection properties along the trajectory. We determined the volume of the ejected plasma as described above (i.e., where the emissivity of O exceeds $10^{-7}$ W m$^{-3}$) and calculated its center of mass for each snapshot of the numerical model every 30s. Thus, the path of the ejection can be tracked from $t{=}40$min to $51.5$min, at which time the entire plasma ejection is visible and distinguishable from the background. For time steps prior to $t{=}40$min, the trajectory is obtained by a linear downward extrapolation (basically vertical).
For the early onset of the ejection before $t{=}39$min, the vertical column containing the formation site of the ejection was examined in detail just below the spot where the ejection becomes visible. This column has a quadratic cross section (2Mm$\times$2Mm) and is indicated in Fig. \[emiss\] (bottom right) by the vertical dotted lines. The (peak) pressure in the vertical column is shown as a function of height for different time steps in Fig.\[fig3\]. The dashed-dotted vertical lines in Fig.\[fig3\] indicate the vertical extension of the red box shown in Fig\[emiss\].
At $t{=}32.5$min, a small kink appears in the pressure profile at a height of approximately 5Mm, indicating a pressure anomaly. Remaining more or less at the same location, it develops into a local pressure peak within a few minutes. This marks the phase of temperature increase and evaporation of the plasma in response to the increased heating rate. It is followed by the explosion, and, as a result, the pressure peak starts moving upward, as can be seen on snapshots at times $t{=}37$min to 39min. The part of the pressure disturbance moving downward is effectively dissipated by the dense plasma of the low atmospheric layers and is therefore not visible.
Typically, a disturbance that travels upward into regions of lower density steepens and eventually forms a shock. However, dissipation can counteract this tendency to form a shock, and thus, the shape of the perturbation will be invariant on its way upward in the simulation box. This phenomenon of a soliton-like structure is indeed what is observed here. The pressure disturbance appears in the form of a soliton-like structure that keeps its amplitude and half width, leaves the top of the red box at $t{=}38.5$min and continues to travel upward.
The subsequent evolution of the pressure of the ejection on its way upward is shown in Fig.\[fig4\]. As the maximum pressure of the ejection is found roughly at its center of gravity, these values match the ones shown in Fig.\[fig3\] (for better comparability, the actual height in the simulation box is shown rather than the arc length along the trajectory). Thus, the lowermost location of the ejection as shown in Fig.\[fig4\] at $t{=}38.5$min exactly corresponds to the uppermost location of the ejection shown in Fig.\[fig3\], which also occurs at $t{=}38.5$min. This shows that the plasma ejection continues to travel upward, still in a soliton-like fashion, but now the dispersion slowly becomes more important: The shape flattens as the ejection expands in the course of the flight.
![Similar to Fig.\[fig3\], but for the peak pressure of the ejection along its trajectory (shown as a dashed line in the lower panels of Fig.\[emiss\], see also red line in Fig.\[fieldlines\] for a 3D impression). For better comparison with Fig.\[fig3\], the pressure is plotted as a function of height, viz. altitude, along the trajectory (and not as a function of arc length along its path). The first time step (39min, leftmost peak) corresponds to the last time step shown in Fig.\[fig3\]. []{data-label="fig4"}](fig5.ps){width="\columnwidth"}
Energy considerations for the onset of the ejection {#energy}
---------------------------------------------------
In the previous section (\[formation2\]) we outlined a scenario in which the increasing heating rate leads to an increase in pressure. The resulting pressure gradient drives the ejection — similar to the situation of an explosion. We will confirm this scenario by investigating the energy budget.
[Fig.\[fig3\]]{} shows that the pressure $p$ increases by 0.3Nm$^{-2}$ within $\tau{=}5$min (the pressure starts increasing at $t{=}32.5$min and reaches its peak at about 37.5min). For an ideal monoatomic gas, the internal energy density $e$ is given through $e{=}3/2\,p$. Thus, the rate of change of the internal energy density is about $e/\tau=1.5$mWm$^{-3}$. This nicely corresponds to the Ohmic heating rate $H_{\rm{Ohm}}$ of 1.5 - 2mWm$^{-3}$ during that time (see [Fig.\[blob\_start\]]{}, left panel). Thus, the pressure increase is indeed caused by the increased Ohmic heating rate of the plasma. The heating rate of some 2mWm$^{-3}$ is about a factor of 100 higher compared to normal regions at or just above the top of the chromosphere.
The acceleration during the early phase of the ejection can be derived from [Fig.\[fig3\]]{}. At times $t=(38,38.5,39)$min the ejection reaches heights of about $z=(5.4,5.7,6.7)$Mm when traced by the peak pressure.
From this, one can calculate the velocity to be about $v\approx25$kms$^{-1}$ and the acceleration to be $a\approx800$kms$^{-2}$. The density in this region is $\rho\approx5\cdot 10^{-12}$kgm$^{-3}$. Thus, the rate of change of the kinetic energy density of the ejection is about $\partial_t(\frac{1}{2}{\rho}v^2)={\rho}va\approx0.1$mWm$^{-3}$. This number corresponds well to the amount of work done by the pressure gradient as shown in the right panel of [Fig.\[blob\_start\]]{} and gives a clear indication that the work done by the pressure gradient is pushing the plasma upward.
We can therefore conclude that the ejection is driven by the pressure gradient. The middle panel of [Fig.\[blob\_start\]]{} shows the work performed on the plasma by the magnetic field, or more precisely by the Lorentz force. This is about a factor of 10 weaker than the work performed by the pressure gradient, and it would not be strong enough to drive the ejection. Therefore, the Lorentz force does not play a significant role for the onset of the plasma ejection. When the ejection leaves the source region just before time $t=40$min, $v{\cdot}(j{\times}B)$ becomes negative, i.e., the plasma performs work on the magnetic field by pushing aside the magnetic field lines.
Finally, we note that the kinetic energy of the plasma ejection is about $10^{18}$J or $10^{25}$erg (assuming $10^9$kg for the average mass and 50kms$^{-1}$ for the average velocity of the ejection). This is about a factor of ten higher than the typical energy content of a nanoflare [e.g., @Parker:1988].
Dynamics and evolution of the plasma ejection {#evolution}
=============================================
Mass balance and chromospheric evaporation {#massbalance}
------------------------------------------
After defining the volume of the plasma ejection, the total mass of the plasma ejection can be tracked. After a total of about $0.5{\cdot}10^9$kg has been ejected following the initial explosion, the gas bubble acquires additional mass of about $10^9$kg on its way along the magnetic field (see Fig.\[mass\]). The total mass of the coronal part of the computational box, i.e., the mass of the plasma at temperatures above $10^6$K, is found to be about $5{\cdot}10^9$kg. Therefore, the cool plasma ejected up into the corona is a significant fraction of the total coronal mass. Still, it is only a tiny fraction of the entire mass in the computational domain owing to the high density of the photospheric and chromospheric plasma.
The (linear) increase of the mass of the ejection that flies through the corona as depicted in Fig.\[mass\] cannot simply be accounted for by an accumulation of plasma in the corona before the lift-off of the ejection. Before and during the ejection, the (curved) tube along the trajectory of the ejection is filled by evaporated chromospheric material. The evaporation is caused by the increased heating rate before and during the plasma ejection. Thus, without the occurence of the eruptive ejection it is likely that a coronal loop (with increased density) would have formed through the long-known process of chromospheric evaporation following increased heating. After the explosion-type event, the ejection collects all material, which prevents the coronal loop from forming. The reason for this is that it sweeps away the evaporated material and transports it back to the chromosphere on the other side, where it is thermalized once the ejection hits the surface again.
![Temporal evolution of the mass of the plasma ejection. \[mass\] ](fig6.ps){width="0.55\columnwidth"}
Dynamics
--------
The vertical velocity $v_z$ of the plasma ejection is shown as a function of time in the left panel of Fig.\[vel\_beta\]. The dependency is characterized by the vertical component of the velocity of the center of gravity of the ejection. During the upward moving phase, the center of gravity is decelerated nearly constantly, until its initial speed of $v_z{\approx}75$km/s at $t{=}40$min has decreased to zero at $t{=}46$min, reaching the apex of the trajectory at a height of 21.5 Mm. The ejection then continues (roughly) along the magnetic field lines and falls down with a roughly constant acceleration, reaching about 55km/s at $t{=}51$min, just before it hits the ambient transition region.
![[*[Left:]{}*]{} Temporal evolution of the vertical velocity; overplotted in gray are linear fits in the time intervals $t{=}40-45$min and $t{=}46.5-50.5$min. [*[Right:]{}*]{} Temporal evolution of kinetic plasma-$\beta$ at the center of gravity of the plasma ejection (see Eq.\[beta\] in Sect.\[mag.interaction\]).[]{data-label="vel_beta"}](fig7.ps){width="\columnwidth"}
The deceleration and acceleration of the ejection are derived from a linear fit to the velocities in the time-periods $t{=}40$min to 45min and $t{=}46.5$min to 50.5min, respectively, during the rising and descending phases (Fig.\[vel\_beta\], left panel, gray lines). The corresponding numbers are listed in Table \[accel\].
After the initial explosion-like acceleration, the ejection basically follows a ballistic flight. During the rising phase, the front of the ejection is stronger decelerated than the center of gravity (${\approx}255$m/s$^2$ for $t$=39.5-44 min, not shown in Fig.\[vel\_beta\]) because the ejection collects the material from the ambient corona at its front. Consequently, the shape of the ejection turns from tailed to ellipsoidal.
During the descending phase, the acceleration of the ejection is well below the solar surface gravitational acceleration (Table \[accel\]). This is consistent with the mass balance as discussed in [Sect.\[massbalance\]]{}. After the chromospheric evaporation, while the ejection collects the material along the trajectory, part of its momentum goes into the acceleration of the collected material, which is initially at rest. Consequently, the ejection is not accelerated as fast as a free falling body on its way down along the magnetic field lines.
phase \[min\] direction acceleration \[m/s$^2$\]
--------------- ----------- --------------------------
40.0 - 45.0 upward 231
46.5 - 50.5 downward 157
: Acceleration of plasma ejection (center of gravity).\[accel\]
Note: Derived from linear fits to the vertical velocity shown in Fig.\[vel\_beta\].
Interaction with the magnetic field {#mag.interaction}
-----------------------------------
The ejection roughly follows the magnetic field lines. However, because of its high momentum, it is capable to distort the coronal magnetic field (cf.Fig.\[fieldlines\]). This is evident when investigating the kinetic plasma-$\beta$ term, which can be defined as the ratio of the kinetic energy density $\rho v^2/2$, and the magnetic energy density $B^2/2\mu_0$, i.e., $$\label{beta}
\beta_{\rm{kin}} = \mu_0 \frac{\rho v^2}{B^2} ~ .
$$ Here, $\mu_0$ is the magnetic vacuum permeability, $\rho$ and $v$ are the mass density and velocity of the plasma ejection (at its center of gravity), and $B$ is the magnetic field strength.
In Fig. \[vel\_beta\] (right panel) $\beta_{\rm{kin}}$ at the center of gravity of the ejection is shown as a function of time. Evidently, $\beta_{\rm{kin}}$ is approximately unity and can therefore displace the magnetic field, i.e., induce electric currents. These currents are visible as a tail behind the ejection in the current maps (right panels of Fig.\[emiss\] and lower left panel in the movie attached to Fig.\[fig.movie\]). This means that the plasma ejection bends the magnetic field.
Since the non-ideal MHD equations are solved in the numerical experiment, i.e., diffusivity is taken into account, the ejection will start moving through the magnetic field lines. The characteristic time for resistive diffusion is $\tau = \ell^2/\eta$, where $\ell$ is the length scale over which the ejection moves across the magnetic field, and $\eta$ is the magnetic diffusivity. Consequently, $\ell=(\eta\,\tau)^{1/2}$. Thus, for a typical flight time $\tau=5$ min and a magnetic diffusivity $\eta = 10^{10}$ m$^2$/s used in the numerical experiment, our estimate implies that the trajectory of the plasma ejection can be shifted perpendicular to the direction of the magnetic field by $\ell \approx 1\dots2$ Mm. This number is consistent with the offset of the ejection trajectory perpendicular to the magnetic field lines (cf. Fig.\[fieldlines\]).
![Magnetic field configuration at timestep $t{=}15$min (well before the plasma ejection lifts off) together with trajectory of the plasma ejection (center-of-gravity) overplotted as a red line. The map at the bottom shows the vertical magnetic field at the bottom boundary of the simulation box scaled from $-$2000G (blue) to $+$2000G (red).[]{data-label="fieldlines"}](fig8.ps){width="\columnwidth"}
Visibility in AIA/SDO {#aia_section}
=====================
As outlined in the introduction, cool structures are ubiquitously present in the hot corona. So far, we have focused on emission in O and Mg to represent the hot and cool plasma. The Atmospheric Imaging Assembly [AIA; @Lemen+al:2011] on the Solar Dynamics Observatory (SDO) is an appropriate instrument to observe ejections as those investigated in the present paper. The channels centered at 304Å and 171Å are dominated by lines of He and Fe, respectively. Their temperature contribution functions peak at ${\log}T\,[{\rm{K}}]=4.9$ and 5.9, i.e., not very far from the O, and close to the Mg formation temperature (cf. Table \[lines\]). Consequently, the images synthesized for the AIA channels displayed in Fig.\[aia\] do not show significant differences from those in O and Mg presented in Figs.\[emiss\] and \[fig.movie\].
To synthesize maps as AIA would observe the plasma ejection, we used the temperature response functions for the AIA channels as provided in SolarSoft.[^3] These are based on synthesized emission line spectra folded with the wavelength response of the filters. Applying the temperature response function at each grid point, multiplying by the density squared and then integrating along the line of sight provides the count rate to be expected in the respective band, i.e., the digital number DN/s/pixel. In Fig.\[aia\] these maps are displayed for horizontal integration (along $y$) for the 304Å band that shows the cool plasma at transition region temperatures and for the 171Å band that displays the hotter plasma at coronal temperatures. The images are degraded to match the spatial sampling of AIA (0.6arcsec/pixel corresponding to 435km/pixel). The dynamic range of the AIA synthetic images in Fig.\[aia\] is 1000 (304Å) and 300 (171Å), which is consistent with the estimates for the AIA dynamic range as provided in the AIA instrument paper [@Lemen+al:2011]. Also, the absolute values of the count rates of approximately 100DN/s/pixel are consistent with real solar observations of AIA.
{width="\textwidth"}
The plasma ejection is hardly visible in the 171Å band (right panel of Fig.\[aia\]), it basically appears as a dark bubble with a bright ring. The reason for this is that the ejection is cool in the middle with the temperature rising to its circumference. However, we cannot really expect to see this ejection in the 171Å channel in real observations. In the present case, the line-of-sight integration is over 50Mm only, i.e., the length of the computational box. On the real Sun, structures over a far greater length would contribute along the line of sight. Thus, the weak signature of the plasma ejection can be expected to be drowned in the background. If the density of the ejection is sufficiently high and if the temperature is sufficiently low, it might significantly absorb light at the wavelengths of the 171Å channel through free-bound absorption [@anzer+heinzel:2005]. However, even then, ample structures in the foreground might contribute to the emission along the line of sight on the real Sun and cloud the visibility of the ejection in the 171Å band and the other coronal channels of AIA.
The situation is different for the 304Å channel. This band mainly displays cooler material at less than 100.000K and shows a clear signature of the plasma ejection during its flight. To our knowledge, no reports have been published so far on similar structures on ballistic orbits through the corona. However, it must be noted that SDO is fully operational since last summer only. We therefore propose to look for these structures in the AIA 304Å data to confirm the counterparts of the (even cooler) structures seen in Ly-$\alpha$ by [@Schrijver:2001], which were discussed in the introduction, and which are consistent with the plasma ejection described here. When analyzing the AIA 304Å data one will have to struggle with the confusion of structures through the integration along the line of sight. Therefore, we cannot estimate yet if the mechanism proposed in this paper will be applicable for a wider range of processes seen in the 304Å AIA channel. However, we hope that the very high temporal resolution of AIA will allow us to see these cool ejecta flying in the background through the corona.
Conclusions {#discussion}
===========
We have analyzed a numerical experiment that describes the solar atmosphere above a small active region in which — without external triggering — a bubble of dense cool plasma is accelerated upward into the corona, moves along the magnetic field lines, and eventually falls back down onto the surface at a different place. The ejection is a direct consequence of the increased heating just above the chromosphere (and in the low transition region) for a short period of time. This can be considered an explosion, where magnetic energy is transformed effectively to thermal energy. As a consequence of the explosion, a pressure perturbation travels in a ballistic fashion away from the heating site along the magnetic field lines.
While the heating rate in our model is based on magnetic processes (i.e., braiding of field lines and Ohmic dissipation of the induced currents), the transient event itself is basically a hydrodynamic phenomenon. The situation here is different from reconnection events, where plasmoids are driven away from the reconnection site by direct magnetic forcing. For the case described in this study, the increased heating leads to an explosion-type event, and the pressure gradient pushes the material up into the corona.
The increased heating rate along the magnetic field lines that eventually leads to the ejection also causes (a more gradual) evaporation of chromospheric material before and during the event. Thus, the bubble that flies through the corona and collects mass on its way contains plasma evaporated from both footpoints of its trajectory.
The ejection has sufficient kinetic energy to deform the magnetic field on its flight through the corona. This leads to current sheets in front of the ejection that propagate faster than the ejection and lead to heating (and evaporation) of plasma ahead of the ejection.
Because the ejection consists of cool plasma, it cannot be expected to be observed in coronal images of the Sun that display plasma at temperatures around and above $10^6$K, and which are available through the 171Å channel of SDO/AIA. The ejection might be visible in data that image lower temperature plasma, such as the 304Å AIA channel. However, more detailed studies will be needed to ascertain if these events are visible considering the confusion of structures along the line of sight.
The plasma ejection observed in our numerical experiment is consistent with the type (3) brightenings as seen in the TRACE Ly-$\alpha$ channel by [@Schrijver:2001] and discussed in the introduction. Even though the plasma seen in Ly-$\alpha$ is most probably cooler than the ejection described in this work, our proposal of an ejection driven by an explosion-type event might be a good explanation for structures flying on ballistic orbits through the corona. Further comparisons between numerical experiments and observations will have to show to what extent this is an ubiquitous mechanism for transient events seen through cool plasma in the hot corona.
[ The work of PZ and SB was supported by the Deutsche Forschungsgemeinschaft (DFG). We thank the anonymous referee for comments that helped improve the manuscript. ]{}
Temporal evolution of the plasma ejection {#appendix}
=========================================
{width="\textwidth"}
The temporal evolution of the plasma ejection is best displayed by a movie showing the emission as well as the heating rate integrated through the computational box. To mimic the situation at the limb, a horizontal line of sight is chosen, e.g., the $y$-direction. The trajectory of the ejection is tilted by approximately 20$^\circ$ with respect to the $x$-direction (cf. Fig.\[fieldlines\]), thus, these projections nicely show the ejection flying through the corona.
The intensities $I(x,z;t) = \int_y \varepsilon(x,y,z;t) ~{\rm{d}}y$, are displayed in the top row of Fig.\[fig.movie\], where $\varepsilon$ is the emissivity at a given grid point $x,y,z$ and time $t$. To avoid Moiré patterns when displaying the images, the values for density and temperature were interpolated on a grid with higher spatial resolution before the emissivities were calculated using CHIANTI (see also Sect.\[phenomenology\]). This procedure is similar to that of [@peter+al:2004; @peter+al:2006]. The intensities in Fig.\[fig.movie\] are displayed on a logarithmic scale with dynamic ranges of 3000 (O) and 50 (Mg) to achieve a better contrast.
The bottom panels in Fig.\[fig.movie\] and the movie show the current $j$, a measure of the Ohmic heating rate (${\propto}j^2$), in two different normalizations. The lower left panel shows $j^2$ normalized by the average trend with height, i.e., the line-of-sight-integrated currents ($\int_y\dots{\rm{d}}y$) averaged along the $x$-direction and in time, $\langle\dots\rangle_{\displaystyle{x,t}}$, $$\label{E:jnorm}
{j^2_N} ~ (x,z;t) ~ = ~ \frac{\int_{y} j^2 (x,y,z;t)~{\rm{d}}y} {\langle ~ \int_{y} j^2 (x,y,z;t)~{\rm{d}}y ~ \rangle_{\displaystyle x,t} } ~.$$ This normalization is necessary to see structures in the heating rate (${\propto}j^2$), which when averaged horizontally, drops roughly exponentially with a scale height of about 5Mm [@bingert:2010].
To emphasize the temporal variability, the heating rate was normalized at each grid point in the 3D computational domain by the temporal average, $\langle\dots\rangle_{\displaystyle{t}}$, at the respective grid point. This normalized quantity is then integrated along the line of sight, i.e., $$\label{E:jfluct}
{j^2_F} ~ (x,z;t) ~ = ~ \int_{y} \frac{ j^2 (x,y,z;t) } {\langle ~ j^2 (x,y,z;t) ~ \rangle_{\displaystyle t} } ~{\rm{d}}y ~.$$ The term ${j^2_F}$ basically shows fluctuations along the line of sight, which is why it is indexed $F$. In the lower right panel of Fig.\[fig.movie\], this quantity is plotted using a rainbow color table, where violet represents low values of ${j^2_F}$ and red represents high values.
A potential magnetic field is used as an upper boundary condition [*above*]{} the computational domain in the 3D MHD model. Of course, the magnetic field inside the box is computed self-consistently based on the MHD equations. As a result, increased currents are observed near the top boundary. These currents have no impact on the magnetic structure and plasma heating in the main part of the box. They are small compared to those at lower heights, which are responsible for heating the coronal plasma (note that the average currents decrease exponentially with height).
[^1]: *Present address:* International Space Science Institute, Hallerstrasse 6, CH-3012 Bern, Switzerland
[^2]: http://www.lmsal.com/solarsoft/
[^3]: http://www.lmsal.com/solarsoft/
|
---
abstract: 'Automatic post-editing (APE) seeks to automatically refine the output of a black-box machine translation (MT) system through human post-edits. APE systems are usually trained by complementing human post-edited data with large, artificial data generated through back-translations, a time-consuming process often no easier than training a MT system from scratch. In this paper, we propose an alternative where we fine-tune pre-trained BERT models on both the encoder and decoder of an APE system, exploring several parameter sharing strategies. By only training on a dataset of 23K sentences for 3 hours on a single GPU we obtain results that are competitive with systems that were trained on 5M artificial sentences. When we add this artificial data, our method obtains state-of-the-art results.'
author:
- |
Gonçalo M. Correia\
Instituto de Telecomunicações\
Lisbon, Portugal\
`goncalo.correia@lx.it.pt` André F. T. Martins\
Instituto de Telecomunicações & Unbabel\
Lisbon, Portugal\
`andre.martins@unbabel.com`
bibliography:
- 'acl2019.bib'
title: |
A Simple and Effective Approach to Automatic Post-Editing\
with Transfer Learning
---
Introduction
============
The goal of [**automatic post-editing**]{} [APE; @simard2007rule] is to automatically correct the mistakes produced by a black-box machine translation (MT) system. APE is particularly appealing for rapidly customizing MT, avoiding to train new systems from scratch. Interfaces where human translators can post-edit and improve the quality of MT sentences [@Alabau2014; @Federico2014; @Denkowski2015; @Hokamp2018] are a common data source for APE models, since they provide [**triplets**]{} of [*source*]{} sentences ([src]{}), [*machine translation*]{} outputs ([mt]{}), and [*human post-edits*]{} ([pe]{}).
Unfortunately, human post-edits are typically scarce. Existing APE systems circumvent this by generating [**artificial triplets**]{} [@junczys2016log; @negri2018escape]. However, this requires access to a high quality MT system, similar to (or better than) the one used in the black-box MT itself. This spoils the motivation of APE as an alternative to large-scale MT training in the first place: the time to train MT systems in order to extract these artificial triplets, combined with the time to train an APE system on the resulting large dataset, may well exceed the time to train a MT system from scratch.
Meanwhile, there have been many successes of [**transfer learning**]{} for NLP: models such as CoVe [@mccann2017learned], ELMo [@peters2018deep], OpenAI GPT [@radford2018improving], ULMFiT [@howard2018universal], and BERT [@devlin2018bert] obtain powerful representations by training large-scale language models and use them to improve performance in many sentence-level and word-level tasks. However, a language generation task such as APE presents additional challenges.
In this paper, we build upon the successes above and show that [**transfer learning is an effective and time-efficient strategy for APE**]{}, using a pre-trained BERT model. This is an appealing strategy in practice: while large language models like BERT are expensive to train, this step is only done once and covers many languages, reducing engineering efforts substantially. This is in contrast with the computational and time resources that creating artificial triplets for APE needs—these triplets need to be created separately for every language pair that one wishes to train an APE system for.
Current APE systems struggle to overcome the MT baseline without additional data. This baseline corresponds to leaving the MT uncorrected (“do-nothing” baseline).[^1] With only the small shared task dataset (23K triplets), our proposed strategy outperforms this baseline by $-$4.9 TER and $+$7.4 BLEU in the English-German WMT 2018 APE shared task, with 3 hours of training on a single GPU. Adding the artificial eSCAPE dataset [@negri2018escape] leads to a performance of 17.15 TER, a new state of the art.
Our main contributions are the following:
We combine the ability of BERT to handle [**sentence pair inputs**]{} together with its pre-trained multilingual model, to use both the [src]{} and [mt]{} in a [**cross-lingual encoder**]{}, that takes a multilingual sentence pair as input.
We show how pre-trained BERT models can also be used and fine-tuned as the [**decoder**]{} in a language generation task.
We make a thorough empirical evaluation of different ways of coupling BERT models in an APE system, comparing different options of parameter sharing, initialization, and fine-tuning.
Automatic Post-Editing with BERT {#sec:ape_bert}
================================
Automatic Post-Editing
----------------------
APE [@simard2007rule] is inspired by human post-editing, in which a translator corrects mistakes made by an MT system. APE systems are trained from triplets ([src]{}, [mt]{}, [pe]{}), containing respectively the source sentence, the machine-translated sentence, and its post-edited version.
#### Artificial triplets.
Since there is little data available (e.g WMT 2018 APE shared task has 23K triplets), most research has focused on creating artificial triplets to achieve the scale that is needed for powerful sequence-to-sequence models to outperform the MT baseline, either from “round-trip” translations [@junczys2016log] or starting from parallel data, as in the eSCAPE corpus of , which contains 8M synthetic triplets.
#### Dual-Source Transformer.
The current state of the art in APE uses a Transformer [@vaswani2017attention] with [**two encoders**]{}, for the [src]{} and [mt]{}, and [**one decoder**]{}, for [pe]{} [@junczys2018ms; @tebbifakhr2018multi]. When concatenating human post-edited data and artificial triplets, these systems greatly improve the MT baseline. However, little successes are known using the shared task training data only.
By contrast, with transfer learning, our work outperforms this baseline considerably, even without any auxiliary synthetic dataset; and, as shown in §\[sec:experiments\], it achieves state-of-the-art results by combining it with the aforementioned artificial datasets.
![**Dual-Source BERT.** Dashed lines show shared parameters in our best configuration.[]{data-label="fig:transformer_diagram"}](diagram_8-dashed.pdf){width="\columnwidth"}
BERT as a Cross-Lingual Encoder
-------------------------------
Our transfer learning approach is based on the Bidirectional Encoder Representations from Transformers [BERT; @devlin2018bert]. This model obtains deep bidirectional representations by training a Transformer [@vaswani2017attention] with a large-scale dataset in a masked language modeling task where the objective is to predict missing words in a sentence. We use the BERT~BASE~ model, which is composed of $L=12$ self-attention layers, hidden size $H=768$, $A=12$ attention heads, and feed-forward inner layer size $F=3072$. In addition to the word and learned position embeddings, BERT also has [**segment embeddings**]{} to differentiate between a segment A and a segment B—this is useful for tasks such as natural language inference, which involve two sentences. In the case of APE, there is also a pair of input sentences ([src]{}, [mt]{}) which are in different languages. Since one of the released BERT models was jointly pre-trained on 104 languages,[^2] we use this multilingual BERT pre-trained model to encode the bilingual input pair of APE.
Therefore, the whole encoder of our APE model is the multilingual BERT: we encode both [src]{} and [mt]{} in the same encoder and use the segment embeddings to differentiate between languages (Figure \[fig:transformer\_diagram\]). We reset positional embeddings when the [mt]{} starts, since it is not a continuation of [src]{}.
BERT as a Decoder {#sec:ape_bert_decoder}
-----------------
Prior work has incorporated pre-trained models in [*encoders*]{}, but not as [**decoders**]{} of sequence-to-sequence models. Doing so requires a strategy for generating fluently from the pre-trained model. Note that the bidirectionality of BERT is lost, since the model cannot look at words that have not been generated yet, and it is an open question how to learn decoder-specific blocks (e.g. context attention), which are absent in the pre-trained model.
One of our key contributions is to use BERT in the decoder by experimenting different strategies for initializing and sharing the self and context attention layers and the positionwise feed-forward layers. We tie together the encoder and decoder embeddings weights (word, position, and segment) along with the decoder output layer (transpose of the word embedding layer). We use the same segment embedding for the target sentence ([pe]{}) and the second sentence in the encoder ([mt]{}) since they are in the same language. The full architecture is shown in Figure \[fig:transformer\_diagram\]. We experiment with the following strategies for coupling BERT pre-trained models in the decoder:
**Transformer.** A Transformer decoder as described in without any shared parameters, with the BERT~BASE~ dimensions and randomly initialized weights.
**Pre-trained BERT.** This initializes the decoder with the pre-trained BERT model. The only component initialized randomly is the context attention (CA) layer, which is absent in BERT. Unlike in the original BERT model—which only encodes sentences—a mask in the self-attention is required to prevent the model from looking to subsequent tokens in the target sentence.
**BERT initialized context attention.** Instead of a random initialization, we initialize the context attention layers with the weights of the corresponding BERT self-attention layers.
**Shared self-attention**. Instead of just having the same initialization, the self-attentions (SA) in the encoder and decoder are tied during training.
**Context attention shared with self-attention.** We take a step further and *tie* the context attention and self attention weights—making all the attention transformation matrices (self and context) in the encoder and decoder tied.
**Shared feed-forward.** We tie the feed-forward weights (FF) between the encoder and decoder.
Experiments {#sec:experiments}
===========
We now describe our experimental results. Our models were implemented on a fork of OpenNMT-py [@klein2017opennmt] using a Pytorch [@paszke2017automatic] re-implementation of BERT.[^3] Our model’s implementation is publicly available.[^4]
#### Datasets.
We use the data from the WMT 2018 APE shared task [@Chatterjee2018] (English-German SMT), which consists of 23,000 triplets for training, 1,000 for validation, and 2,000 for testing. In some of our experiments, we also use the eSCAPE corpus [@negri2018escape], which comprises about 8M sentences; when doing so, we oversample 35x the shared task data to cover $10\%$ of the final training data. We segment words with WordPiece [@wu2016google], with the same vocabulary used in the Multilingual BERT. At training time, we discard triplets with 200+ tokens in the combination of [src]{} and [mt]{} or 100+ tokens in [pe]{}. For evaluation, we use TER [@snover2006study] and tokenized BLEU [@papineni2002bleu].
TER$\downarrow$ BLEU$\uparrow$
------------------------ ----------------- ----------------
Transformer decoder 20.33 69.31
Pre-trained BERT 20.83 69.11
CA $\leftarrow$ SA 18.91 71.81
[ ]{} [ ]{} Encoder SA **18.44** **72.25**
[ ]{} [ ]{} SA 18.75 71.83
[ ]{} [ ]{} Encoder FF 19.04 71.53
: Ablation study of decoder configurations, by gradually having more shared parameters between the encoder and decoder (trained without synthetic data). $\leftrightarrow$ denotes parameter tying and $\leftarrow$ an initialization. []{data-label="tab:ablation_smt"}
----------------------------------------- ------------ ----------------- ---------------- ----------------- ---------------- ----------------- ----------------
Model Train Size TER$\downarrow$ BLEU$\uparrow$ TER$\downarrow$ BLEU$\uparrow$ TER$\downarrow$ BLEU$\uparrow$
MT baseline (Uncorrected) 24.76 62.11 24.48 62.49 24.24 62.99
22.89 — 23.08 65.57 — —
18.92 70.86 19.49 69.72 — —
$\times 4$ 18.86 71.04 19.03 70.46 — —
— — — — 18.62 71.04
17.81 72.79 18.10 71.72 — —
$\times 4$ 17.34 73.43 17.47 72.84 18.00 72.52
Dual-Source Transformer$^\dagger$ 27.80 60.76 27.73 59.78 28.00 59.98
BERT Enc.+Transformer Dec. (*Ours*) 20.23 68.98 21.02 67.47 20.93 67.60
BERT Enc.+BERT Dec. (*Ours*) 18.88 71.61 19.03 70.66 19.34 70.41
BERT Enc.+BERT Dec. $\times 4$ (*Ours*) **18.05** **72.39** **18.07** **71.90** **18.91** **70.94**
BERT Enc.+BERT Dec. (*Ours*) 16.91 74.29 17.26 73.42 17.71 72.74
BERT Enc.+BERT Dec. $\times 4$ (*Ours*) **16.49** **74.98** **16.83** **73.94** **17.15** **73.60**
----------------------------------------- ------------ ----------------- ---------------- ----------------- ---------------- ----------------- ----------------
#### Training Details.
We use Adam [@kingma2014adam] with a triangular learning rate schedule that increases linearly during the first 5,000 steps until $5\times 10^{-5}$ and has a linear decay afterwards. When using BERT components, we use a $\ell_2$ weight decay of $0.01$. We apply dropout [@srivastava2014dropout] with $p_{drop}=0.1$ to all layers and use label smoothing with $\epsilon=0.1$ [@pereyra2017regularizing]. For the small data experiments, we use a batch size of 1024 tokens and save checkpoints every 1,000 steps; when using the eSCAPE corpus, we increase this to 2048 tokens and 10,000 steps. The checkpoints are created with the exponential moving average strategy of with a decay of $10^{-4}$. At test time, we select the model with best TER on the development set, and apply beam search with a beam size of 8 and average length penalty.
#### Initialization and Parameter Sharing.
Table \[tab:ablation\_smt\] compares the different decoder strategies described in §\[sec:ape\_bert\_decoder\] on the WMT 2018 validation set. The best results were achieved by sharing the self-attention between encoder and decoder, and by initializing (but not sharing) the context attention with the same weights as the self-attention. Regarding the self-attention sharing, we hypothesize that its benefits are due to both encoder and decoder sharing a common language in their input (in the [mt]{} and [pe]{} sentence, respectively). Future work will investigate if this is still beneficial when the source and target languages are less similar. On the other hand, the initialization of the context attention with BERT’s self-attention weights is essential to reap the benefits of BERT representations in the decoder—without it, using BERT decreases performance when compared to a regular transformer decoder. This might be due to the fact that context attention and self-attention share the same neural block architecture (multi-head attention) and thus the context attention benefits from the pre-trained BERT’s better weight initialization. No benefit was observed from sharing the feed-forward weights.
#### Final Results.
Finally, Table \[tab:results\_smt\] shows our results on the WMT 2016–18 test sets. The model named *BERT Enc.+BERT Dec.* corresponds to the best setting found in Table \[tab:ablation\_smt\], while *BERT Enc.+Transformer Dec.* only uses BERT in the encoder. We show results for single models and ensembles of 4 independent models.
Using the small shared task dataset only (23K triplets), our single *BERT Enc.+BERT Dec.* model surpasses the MT baseline by a large margin ($-$4.90 TER in test 2018). The only system we are aware to beat the MT baseline with only the shared task data is @berard2017lig, which we also outperform ($-$4.05 TER in test 2017). With only about 3 GPU-hours and on a much smaller dataset, our model reaches a performance that is comparable to an ensemble of the best WMT 2018 system with an artificial dataset of 5M triplets ($+$0.02 TER in test 2016), which is much more expensive to train. With 4$\times$ ensembling, we get competitive results with systems trained on 8M triplets.
When adding the eSCAPE corpus (8M triplets), performance surpasses the state of the art in all test sets. By ensembling, we improve even further, achieving a final 17.15 TER score in test 2018 ($-$0.85 TER than the previous state of the art).
Related Work {#sec:related}
============
In their Dual-Source Transformer model, also found gains by tying together encoder parameters, and the embeddings of both encoders and decoder. Our work confirms this but shows further gains by using segment embeddings and more careful sharing and initialization strategies. explore parameter sharing between Transformer layers. However, they focus on sharing decoder parameters in a *one-to-many* multilingual MT system. In our work, we share parameters between the encoder and the decoder.
As stated in §\[sec:experiments\], @berard2017lig also showed improved results over the MT baseline, using exclusively the shared task data. Their system outputs edit operations that decide whether to insert, keep or delete tokens from the machine translated sentence. Instead of relying on edit operations, our approach mitigates the small amount of data with transfer learning through BERT.
Our work makes use of the recent advances in transfer learning for NLP [@peters2018deep; @howard2018universal; @radford2018improving; @devlin2018bert]. Pre-training these large language models has largely improved the state of the art of the GLUE benchmark [@wang2018glue]. Particularly, our work uses the BERT pre-trained model and makes use of the representations obtained not only in the encoder but also on the decoder in a language generation task.
More closely related to our work, pre-trained a BERT-like language model using parallel data, which they used to initialize the encoder and decoder for supervised and unsupervised MT systems. They also used segment embeddings (along with word and position embeddings) to differentiate between a pair of sentences in different languages. However, this is only used in one of the pre-training phases of the language model (translation language modelling) and not in the downstream task. In our work, we use segment embeddings during the downstream task itself, which is a perfect fit to the APE task.
used our model on the harder English-German NMT subtask to obtain better TER performance than previous state of the art. To obtain this result, the transfer learning capabilities of BERT were not enough and further engineering effort was required. Particularly, a conservativeness factor was added during beam decoding to constrain the changes the APE system can make to the [mt]{} output. Furthermore, the authors used a data weighting method to augment the importance of data samples that have lower TER. By doing this, data samples that required less post-editing effort are assigned higher weights during the training loop. Since the NMT system does very few errors on this domain this data weighting is important for the APE model to learn to do fewer corrections to the [mt]{} output. However, their approach required the creation of an artificial dataset to obtain a performance that improved the MT baseline. We leave it for future work to investigate better methods to obtain results that improve the baseline using only real post-edited data in these smaller APE datasets.
Conclusion and Future Work
==========================
We proposed a transfer learning approach to APE using BERT pre-trained models and careful parameter sharing. We explored various ways for coupling BERT in the decoder for language generation. We found it beneficial to initialize the context attention of the decoder with BERT’s self-attention and to tie together the parameters of the self-attention layers between the encoder and decoder. Using a small dataset, our results are competitive with systems trained on a large amount of artificial data, with much faster training. By adding artificial data, we obtain a new state of the art in APE.
In future work, we would like to do an extensive analysis on the capabilities of BERT and transfer learning in general for different domains and language pairs in APE.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the European Research Council (ERC StG DeepSPIN 758969), and by the Fundação para a Ciência e Tecnologia through contracts UID/EEA/50008/2019 and CMUPERI/TIC/0046/2014 (GoLocal). We thank the anonymous reviewers for their feedback.
[^1]: If an APE system has worse performance than this baseline, it is pointless to use it.
[^2]: <https://github.com/google-research/bert/blob/master/multilingual.md>
[^3]: <https://github.com/huggingface/pytorch-pretrained-BERT>
[^4]: <https://github.com/deep-spin/OpenNMT-APE>
|
---
abstract: 'We present new photometric and spectroscopic observations of an unusual luminous blue variable (LBV) in NGC 3432, covering three major outbursts in October 2008, April 2009 and November 2009. Previously, this star experienced an outburst also in 2000 (known as SN 2000ch). During outbursts the star reached an absolute magnitude between -12.1 and -12.8. Its spectrum showed H, He I and Fe II lines with P-Cygni profiles during and soon after the eruptive phases, while only intermediate-width lines in pure emission (including He II $\lambda$ 4686) were visible during quiescence. The fast-evolving light curve soon after the outbursts, the quasi-modulated light curve, the peak magnitude and the overall spectral properties are consistent with multiple episodes of variability of an extremely active LBV. However, the widths of the spectral lines indicate unusually high wind velocities (1500-2800 km s$^{-1}$), similar to those observed in Wolf-Rayet stars. Although modulated light curves are typical of LBVs during the S-Dor variability phase, the luminous maxima and the high frequency of outbursts are unexpected in S-Dor variables. Such extreme variability may be associated with repeated ejection episodes during a giant eruption of an LBV. Alternatively, it may be indicative of a high level of instability shortly preceding the core-collapse or due to interaction with a massive, binary companion. In this context, the variable in NGC 3432 shares some similarities with the famous stellar system HD 5980 in the Small Magellanic Cloud, which includes an erupting LBV and an early Wolf-Rayet star.'
author:
- 'A. Pastorello$^{1}$[^1], M. T. Botticella$^{1}$, C. Trundle$^{1}$, S. Taubenberger$^{2}$, S. Mattila$^{3,4}$,'
- 'E. Kankare$^{5,3}$, N. Elias-Rosa$^{6}$, S. Benetti$^{7}$, G. Duszanowicz$^{8}$, L. Hermansson$^{9}$,'
- 'J. E. Beckman$^{10,11}$, F. Bufano$^{7}$, M. Fraser$^{1}$, A. Harutyunyan$^{12}$, H. Navasardyan$^{7}$,'
- |
S. J. Smartt$^{1}$, S. D. van Dyk$^{6}$, J. S. Vink$^{13}$, R. M. Wagner$^{14}$\
$^{1}$Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom\
$^{2}$ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85741 Garching bei München, Germany\
$^{3}$ Tuorla Observatory, Department of Physics & Astronomy, University of Turku, Väisäläntie 20, FI-21500 Piikkiö, Finland\
$^{4}$ Stockholm Observatory, Stockholm University, AlbaNova University Center, SE-10691, Stockholm, Sweden\
$^{5}$ Nordic Optical Telescope, Apartado 474, E-38700 Santa Cruz de La Palma, Spain\
$^{6}$ Spitzer Science Center, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA\
$^{7}$ INAF Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy\
$^{8}$ Moonbase Observatory, Otto Bondes vag 43, SE-18462 Akersberga, Sweden\
$^{9}$ Sandvretens Observatory, Linnégatan 5A, SE-75332 Uppsala, Sweden\
$^{10}$ Instituto de Astrofísica de Canarias C/ Vía Láctea s/n, E-38205, La Laguna, S/C de Tenerife, Spain\
$^{11}$ Consejo Superior de Investigaciones Cientícas of Spain, C/ Serrano 117, E-28006, Madrid, Spain\
$^{12}$ Fundación Galileo Galilei - INAF, Telescopio Nazionale Galileo, Rambla José Ana Fernández Pérez 7, E-38712 Breña Baja, Tenerife, Spain\
$^{13}$ Armagh Observatory, College Hill, Armagh BT61 9DG, United Kingdom\
$^{14}$ Large Binocular Telescope Observatory, 933 North Cherry Avenue, Tucson, AZ 85721, USA\
date: 'Accepted .... Received .....; in original form .....'
title: Multiple Major Outbursts from a Restless Luminous Blue Variable in NGC 3432
---
\[firstpage\]
stars: variables: other - supernovae: general - supernovae: individual: 2000ch - galaxies: individual: NGC 3432.
Introduction
============
Luminous Blue Variable (LBV) stars are debated subjects of modern astrophysics, because they signify a key stage in the fate of the most massive stars. Such stars may experience an unstable post-main-sequence LBV stage before becoming Wolf-Rayet (WR) stars. LBVs lose most of the hydrogen (H) envelope rapidly [typically 10$^{-5}$ M$_\odot$ yr$^{-1}$, up to 10$^{-4}$ M$_\odot$ yr$^{-1}$, see e.g. @hum94], forming massive circumstellar nebulae like the spectacular [*Homunculus*]{} surrounding $\eta$-Car. Later massive stars eventually explode as type Ib/c supernovae (SNe) [see the case of SN 2001am, @sci08]. However, there is increasing evidence that even LBVs can [*directly*]{} explode producing core-collapse SNe [@kot06; @gal07; @gal08; @smi07; @smi08; @tru08; @agn08].
An attempt to characterize LBVs was done by @hum94. However, LBVs do not form a homogeneous group of stars, since they span quite a large range of magnitudes and variability types. Some of them show an S-Dor variability type. During this phase, the star expands and contracts with some regularity over timescales of years, varying its apparent temperature (and, as a consequence, spectral type). Its bolometric luminosity is generally not thought to change significantly [but see @cla09; @gro09] and the upper limit to the luminosity/mass ratio for static stellar atmospheres (the so-called “Eddington limit”) is not violated. With S-Dor, other classical examples of this variability type are AG Car in the Galaxy and R 127 in the Large Magellanic Cloud (LMC). As mentioned by @vge01, the S-Dor variability has typical periods of few years ([*short S-Dor phase*]{}) or decades ([*long S-Dor phase*]{}). Other LBVs occasionally experience giant eruptions, during which they increase their luminosity and can reach $M \approx$ -14 (4-6 mag brighter than their typical quiescence magnitudes), temporarily exceeding the Eddington Limit. The most popular giant eruption of an LBV in our Galaxy is that of $\eta$-Car during the 19th century.
Spectra of LBVs’ giant eruptions are characterized by incipient narrow (with full-width-at-half-maximum velocity, $v_{FWHM}$, lower than 1000 km s$^{-1}$) H lines in emission, resembling those of type IIn SNe. For this reason and their relatively high peak luminosities, they are occasionally misclassified as real SN explosions. However, since the stars survive the eruptions, they are labelled as [*SN impostors*]{} [@van00]. Well-studied cases are SN 1954J [also known as V12 in NGC 2403, @tam68; @smi01; @van05], SN 1997bs [@van00] and SN 2002kg [@wei05; @mau06; @mau08][^2]. The first registered outburst of the variable star in NGC 3432 discussed in this paper was designated as SN 2000ch [@pap00; @wag04]. Hereafter, this transient will be dubbed as NGC 3432 2000-OT. Contrary to genuine SNe IIn, SN impostors are less luminous and their spectra do not show clear evidence of broad lines ($>$ 3000-4000 km s$^{-1}$) produced by SN ejecta. SN impostors will hereafter be referred to without the misleading “SN” label.
In this paper we will analyse the October 2008, April 2009 and November 2009 outbursts of the same variable in NGC 3432 that was responsible for NGC 3432 2000-OT [see the wide discussion in @wag04]. We attempt to reconstruct the variability history of this star and derive information on its nature. The paper is organized as follows: in Sect. 2 we introduce the variable in NGC 3432 and describe its environment. In Sect. 3 and Sect. 4 photometric and spectroscopic evolutions are presented, while in Sect. 5 we analyse the evolution of the spectral energy distribution (SED) of the variable. A discussion on the nature of the star follows in Sect. 6.
The luminous variable in NGC 3432 and its environment
=====================================================
![SDSS colour image ([*http://cas.sdss.org/astrodr6/en/tools/chart/*]{}) of the interacting galaxy system Arp 206. The main galaxy is NGC 3432, while the dwarf companion UGC 5983 is located south-west of it. The field of view is about 11$\arcmin \times $11$\arcmin$. North is up, East is to the left.[]{data-label="fig0"}](fig1.ps){width="84mm"}
NGC 3432 is a well-studied irregular galaxy [see e.g. @ho97] interacting with a low-surface brightness dwarf companion (UGC 5983), which is located South-West of the main galaxy (Figure \[fig0\]). This interacting system is included in Arp’s catalogue [@arp66] with the label Arp 206. @eng97 noted the presence of an extended arm of the galaxy in the North-East direction, which ends with a luminous clump in which the variable star lies. The disturbed morphology of NGC 3432 and the presence of an extended North-East region detached from the main body of the galaxy likely result from the tidal interaction of NGC 3432 with UGC 5983 [see also @swa02; @vdk07]. @lee09 determined for NGC 3432 a star formation rate (SFR) of 0.49 M$_\odot$ yr$^{-1}$ from UV data, and of 0.32 M$_\odot$ yr$^{-1}$ from H$\alpha$ luminosity. Interaction between galaxies is well known to trigger star formation, and the relatively low SFR observed in NGC 3432 may appear as an inconsistency. However, a starburst episode may occur with a significant delay (10$^8$-10$^9$ yrs) with respect to the major tidal distortions [@mih94]. According to @eng97 Arp 206 is consistent with this scenario, and would currently be in a pre-starburst phase.
Measuring the intensity of the \[O II\] $\lambda\lambda$ 7320,7330 doublet and the intensity ratio of strong \[O II\] $\lambda$ 3727 and \[O III\] $\lambda\lambda$ 4959,5007 lines in H II regions, @pyl07 estimated an average O abundance of 12 log (O/H) = 8.3 $\pm$ 0.1 dex for NGC 3432 which is comparable to the metallicity of the LMC [$\sim$8.35 dex, @hun07]. However, the metallicity in the region of the variable star might be significantly different from this average value.
The distance of the galaxy system is computed through the recessional velocity corrected for Local Group infall into Virgo[^3]. The Hyperleda database[^4] gives $v_{Vir}$ = 779 km s$^{-1}$, providing a distance of 10.8 $\pm$ 1.2 Mpc [distance modulus $\mu$ = 30.17 $\pm$ 0.56 mag, adopting $H_0$ = 72 $\pm$ 8 km s$^{-1}$ Mpc$^{-1}$, @fre01].
@sch98 quote a Galactic reddening of $E(B-V)$= 0.013 in the direction of NGC 3432, while the peripheral position of the objects in the host galaxy suggests a modest internal reddening. This is also indicated from the lack of narrow interstellar absorption lines in the spectra of the variable star (see Sect. \[sp\]). Therefore, hereafter we will adopt a total reddening of $E(B-V)$= 0.013 mag, in agreement with @wag04.
The discovery of a variable star in NGC 3432 on 2000 May 3 UT was reported by @pap00 at the position $\alpha = 10^{h}52^{m}41\fs40$ and $\delta = +36\degr40\arcmin08\farcs5$ (equinox J2000.0), which is 123$\arcsec$ East and 180$\arcsec$ North of the nucleus of the host galaxy. The transient had an unfiltered peak magnitude of $m$ = 17.4. A KAIT image was obtained 4 days prior to this on 2000 April 29 showing nothing brighter than $m$ = 19 at the position of the variable [@wag04b]. The low luminosity and the spectra dominated by narrow Balmer lines were consistent with a SN impostor [e.g. @van00]. Wagner et al. presented optical plus near-infrared light curves of NGC 3432 2000-OT covering a period of $\sim$3 months, and noted a somewhat erratic variability over very short timescales (of the order of weeks). Additional sparse archive observations led these authors to conclude that no major outburst was previously observed, and the likely quiescent magnitude of the LBV was around $R$ = 19.4 $\pm$ 0.4 mag. They also registered a deep magnitude minimum ($R \sim$ 20.8) soon after the luminosity peak, and interpreted it as the result of dust formation, occultation or eclipse phase. We will show in Sect. \[ph\] that the star has been detected (occasionally) at a magnitude even fainter than $R \approx$ 21 in subsequent follow-up observations.
![Comparison between an unfiltered KAIT image of NGC 3432 2000-OT (2000ch) obtained on May 4, 2000 and a R-band image of the 2008 transient obtained at TNG on 25 October, 2008. The position of the two transients is coincident, within the errors. This allows us to conclude that the two outbursts were produced by the same star.[]{data-label="fig1"}](fig2.ps){width="82mm"}
![Unfiltered images of NGC 3432 obtained with a 32-cm reflector in Akersberga (Sweden). The 4 images show the evolution of the variable star, whose position is marked with a circle, on 2006 November 15 (top-left), 2007 March 17 (top-right), 2008 February 25 (bottom-left) and 2008 October 7 (bottom-right). The transient is clearly visible in the March 2007 and October 2008 images, while in the other two images it is below the detection limit. North is up, East is to the left.[]{data-label="fig2"}](fig3.ps){width="82mm"}
Another transient was announced on 2008 October 7.12 UT by [@dus08] at a position $\alpha = 10^{h}52^{m}41\fs33$, $\delta = +36\degr40\arcmin08\farcs9$ (equinox J2000.0), very close (but apparently not coincident) to that of the 2000 outburst (Figure \[fig1\]). The new transient (labelled as NGC 3432 2008-OT in this paper) had an apparent peak magnitude ($R$ = 17.5) similar to that of 2000-OT. Its faint absolute magnitude ($M_R$ $\approx$ -12.7) was again consistent with that of a SN impostor.
In order to prove that the position of 2008-OT was coincident with that of 2000-OT, relative astrometric calibration was performed using as template a good-quality image of the 2008 transient and as target image the discovery frame of the 2000 transient obtained with the 0.76-m KAIT telescope (Filippenko et al. 2001). In order to align the two images, we identified 15 sources in common to both images and measured their centroid positions. This allowed us to derive a geometric transformation for the two images (implying a shift, a scaling to a common pixel scale and a rotation) using the IRAF package GEOMAP. To employ a general non-linear transformation, we selected a 2nd-order polynomial model that computes the geometric alignment function. The images were finally registered to the target image using GEOTRAN. The accuracy of the alignment is 0.145 px in the X-axis and 0.154 px in Y. Accounting for the pixel scale of the target image, the rms errors of the transformation in the two axes are 116 mas and 123 mas. We measured the position of the transients in the two images with aperture photometry and found in the transformed images a discrepancy in the position of the two transients of $\Delta x$ = 0.039 px (31 mas) and $\Delta y$ = 0.013 px (10 mas), which are well below the uncertainty of the transformation. We are therefore confident that the two transients were two major outbursts of the same stellar source (Figure \[fig1\]).
{width="154mm"}
More precise coordinates than those previously reported for the NGC 3432 variable, as determined from Sloan Digital Sky Survey (SDSS) images, are $\alpha = 10^{h}52^{m}41\fs256$ and $\delta = +36\degr40\arcmin08\farcs$95 (equinox J2000.0). Unfiltered amateur images showing the region of the variable star at 4 epochs during the period 2006-2008 are shown in Figure \[fig2\]. Finally, two further outbursts of the variable in NGC 3432 were registered in April 2009 (hereafter labelled as 2009-OT1) and November 2009 (2009-OT2), during routine monitoring observations of the star.
Photometry {#ph}
==========
After the discovery by @dus08, we started a systematic monitoring of the variable in the optical bands. At the same time, we collected a number of pre-discovery images of NGC 3432 from amateur astronomers and professional telescopes available through science data archives. The calibration of the optical photometry obtained with standard Johnson-Bessell filters has been performed using the local standard star magnitudes presented in @wag04 for the B, V and R bands, while the U and I band data have been calibrated independently. The calibration of the Sloan filter photometry was performed using the SDSS photometry catalogue[^5].
{width="100mm"}
Johnson-Bessell and Sloan magnitudes (or magnitude limits) of the variable in NGC 3432 are reported in Appendix (Tables \[tab1\] and \[tab2\]). The light curves, including the photometry of @wag04 and spanning a period of almost 16 years, are shown in Figure \[fig3\]. The right-hand panels show in detail the evolution of the variable during the 4 major outbursts: from top to bottom 2009-OT2, 2009-OT1, 2008-OT and 2000-OT. As already mentioned, the outbursts showed comparable peak magnitudes ($R \approx$ 17.5-18.5 mag) and a number of subsequent luminosity fluctuations, although the post-peak evolution is somewhat different in the 4 cases. This is clearly visible when the R band absolute light curves of the 4 outbursts are plotted together (Figure \[fig4\]). While 2000-OT showed a fast post-peak decline, with a deep minimum reached after a few days, 2008-OT and 2009-OT1 reached the minimum 2-3 months after maximum. We can summarize the photometric evolution of the 4 outbursts as follows:
- [**2000-OT.**]{} During this outburst, the star reached a peak magnitude of $R$ = 17.4 ($M_R \approx$ -12.8) on May 3rd, 2000. The maximum was followed by a steep decline of 3.5 mag in about a week (but an observation obtained 2 days after maximum already showed the transient being 2.9 mag fainter than at peak). A sequence of secondary peaks was also observed (at magnitude around 18.6), with a temporal distance of about 12-15 days between each other [@wag04 and Figure \[fig3\], bottom-right panel].
- [**2008-OT.**]{} During this eruptive episode, the variable arrived at a maximum magnitude of $R$ = 17.5 ($M_R \approx$ -12.7, October 7th, 2008), and experienced a rebrightening ($R \approx$ 18) at $\sim$18 days after the main maximum. The rebrightening was followed by a large luminosity drop (more than 3 mag). Although the light curve is poorly sampled in this period, the star probably remained fainter than $R \sim$ 20.5 ($V \sim$ 21.5) for about 1 month. Then its luminosity increased again, reaching a peak of $R \sim$ 18.6 at about 70 days after the main outburst. Finally the luminosity slowly declined and leveled off at $R \approx$ 19.5 (see Figure \[fig3\], right, second panel from bottom).
- [**2009-OT1.**]{} Surprisingly, about 200 days after the previous outburst the variable brightened again. In this episode the peak magnitude (on April 22nd, 2009) was around $R \sim$ 17.9 ($M_R \sim$ -12.3). The primary peak was followed by a number of magnitude oscillations that occurred on shorter time-scales than those observed in 2008-OT. They were followed by a dramatic drop of $\Delta R$ $\geq$ 3.3 mag from the peak to the minimum (Figure \[fig4\]).
- [**2009-OT2.**]{} A further rebrightening of the variable in NGC 3432 was finally registered in November 2009, again [**$\sim$**]{}200-210 days after the previous episode. This might be either a coincidence in the luminosity fluctuations or, alternatively, might suggest some periodicity in the major episodes of stellar variability. The consequences of this latter scenario will be discussed in Sect. \[prog\]. Interestingly, the first maximum of 2009-OT2 reached $M_R \sim$ -11.7, but was followed after 8 days by a second peak that was even slightly brighter ($M_R \sim$ -12.1, see Figure \[fig4\]). After the second peak, the luminosity of the star dropped down by more than 2 mag in $\sim$1 month.
![Comparison of the R-band absolute light curve of the variable star in NGC 3432 (period october 2008 - June 2009) with those of some well-studied transient events: 1997bs [@van00], 2009ip [@smi09b], the 2004 outburst preceding SN 2006jc [UGC 4904 OT 2004-1, @pasto07a], 2008S [@bot08] and M85 OT 2006-1 [@kul07]. SDSS r-band magnitudes of the variable in NGC 3432 were rescaled to Johnson-Bessell magnitudes by applying a zero-point shift of $\Delta R$ = -0.177 mag (see caption in Figure \[fig3\]).[]{data-label="fig5"}](fig6.ps){width="87mm"}
The comprehensive light curve (period 1994-2009, see Figure \[fig3\], main panel) shows a rather erratic evolution. Luminosity peaks occur on unusually short timescales and the range of spanned magnitudes ($\Delta m$ $\sim$ 4 mag, Figure \[fig4\]) are atypical for an S-Dor-like event. Together with the main peaks we also note a few sparse detections at magnitudes 18.5-19.5, and some minima (e.g. at JDs 2449475 and 2453081). Such optical deficits can be possibly due to post outburst dust formation episodes [as proposed by @wag04]. The implications of this unusual photometric evolution in constraining the nature of the variable star will be discussed in Sect. 5.
In Figure \[fig5\] we compare the R band absolute light curve of the variable in NGC 3432 during the period from October 2008 to June 2009 with those of the SN impostors 1997bs [@van00] and 2009ip [@smi09b], the 2004 luminous eruption in UGC 4904 that heralded SN 2006jc [hereafter UGC 4904 OT 2004-1, @pasto07a] and two additional under-luminous transients whose nature (core-collapse SN, LBV outburst, peculiar luminous nova) is still debated: M85 OT 2006-1 and 2008S [@kul07; @rau07; @pasto07b; @ofek08; @pri08a; @tho08; @wan08; @bot08; @smi09]. The peak magnitudes of the NGC 3432 variable (in the range -12.1 to -12.8) are much fainter than those of 1997bs ($M_R \approx$ -15.3) and 2009ip ($M_R \approx$ -14.6), marginally fainter than those of 2008S and UGC 4904 OT 2004-1 ($M_R \approx$ -13.7). However, they are comparable with the maximum magnitude of M85 OT 2006-1 [@kul07], although the two transients have different photometric evolution. M85 OT 2006-1, indeed, presents a slowly evolving light curve with a sort of plateau, while the variable in NGC 3432 has a fast evolution, characterized by steep post-peak luminosity declines and subsequent rebrightenings similar to those observed in 2009ip [@smi09b] and other SN impostors [@mau06].
In Figure \[fig6\] long-term light curves of the NGC 3432 variable and a few well monitored LBVs and SN imposters are shown: $\eta$ Car[^6], S-Dor and HD 5980[^7], 1997bs [@van00], 2002kg [@wei05; @mau06; @mau08], 2009ip [@smi09b], UGC 4904 OT 2004-1 [@pasto07a]. Remarkably, even though some modulation in the light curve of the variable star in NGC 3432 may suggest an S-Dor-type variability, its overall characteristics are reminiscent of those of major outbursts. Some similarity can be found, indeed, with the years-long outburst of $\eta$ Car during the 19th century, and -more marginally- with the eruption in 1993-1994 of the stellar system HD 5980 [see @koe04 and references therein]. We will better address the latter similarity in forthcoming sections.
{width="126mm"}
Spectroscopy {#sp}
============
Date JD Instrument Grism/grating Exposure time (s) Range (Å) Resolution (Å)
----------- ------------ ---------------- --------------- ------------------- ---------------------- ---------------- -- --
2008Oct14 2454753.75 WHT+ISIS R300B,R158R 2$\times$2400 3250-5410,5140-10150 5.4,6.3
2008Oct17 2454756.71 CAHA2.2m+CAFOS b200 2727.3 3400-8770 10.5
2008Oct17 2454756.74 TNG+LRS LRB 2700 3700-7990 15
2008Dec21 2454821.75 NOT+ALFOSC gm4 3600 3300-9050 18
2008Dec22 2454822.63 TNG+LRS LRR 3600 5050-9800 13
2009Jan22 2454853.64 TNG+LRS LRB 3600 3560-7920 12
2009Feb22 2454884.56 NOT+ALFOSC gm4 2$\times$2700 3400-8100 18
2009Mar22 2454912.52 TNG+LRS LRB 2$\times$2700 3370-7930 15
2009Apr24 2454945.60 TNG+LRS LRB 3600 3260-7940 11
2009May12 2454964.49 TNG+LRS LRB 2700 3280-7940 11
2009May19 2454971.41 TNG+LRS LRB 2700 3400-7930 11
2009Nov27 2455162.67 CAHA2.2m+CAFOS b200 2700 3500-8730 12
Spectra of the luminous variable in NGC 3432 were obtained after the October 2008 outburst using the 4.2-m William Herschel Telescope (WHT) equipped with ISIS, the 3.58-m Telescopio Nazionale Galileo (TNG) with LRS, the 2.56-m Nordic Optical Telescope (NOT) with ALFOSC and the 2.2-m Telescope of the Calar Alto Observatory plus CAFOS. Information on the spectra obtained during the period from October 2008 to November 2009 is reported in Table \[tab3\]. The best signal-to-noise (S/N) spectra of the star witnessing the evolution of the 2008-OT and 2009-OT1 outbursts, but also the subsequent quiescent phases, are shown in Figure \[fig7\] together with spectra obtained in 2000/2001 (2000-OT) from @wag04. The spectra of the variable obtained soon after 2008-OT are characterized by prominent H lines in emission (see line identifications in Figure \[fig8\]). He I lines are also clearly visible with $v_{FWHM} \approx$ 2000 km s$^{-1}$, which is only slightly lower than that of H$\alpha$ ($v_{FWHM} \approx$ 2300 km s$^{-1}$). The spectra of the 2008 episode display significant differences compared with those of 2000-OT described in @wag04 [and shown atop of Figure \[fig7\]]. In the 2000 event the He I lines were weaker and the FWHM velocities measured for H$\alpha$ were slightly lower, being in the range 1600-1800 km s$^{-1}$. Such differences in line velocities may suggest a more powerful outburst (but we cannot confirm it on the basis of our photometric data) or less envelope mass ejection in the 2008 outburst. However, the integrated flux of H$\alpha$ obtained averaging the flux measured from the three available spectra of October 2008 is about 4 $\times$ 10$^{-14}$ erg cm$^{-2}$ s$^{-1}$, which is a factor of two more than the flux reported by @wag04 for the 2000 event, implying that the 2008 event had an H$\alpha$ luminosity twice that of the 2000 event. Adopting the same relation between the H$\alpha$ luminosity and the radius (R) of the emitting region and the same assumptions as Wagner et al., we obtain $R(H\alpha)$ = 0.25 pc [from the H$\alpha$ luminosity computed from the 2000 May 14 spectrum, $R(H\alpha)$ = 0.2 pc was determined by @wag04]. Remarkably, the peak of the H$\alpha$ line is asymmetric and slightly blue-shifted. This phenomenon is visible also in the late-time spectra of some core-collapse SNe and is expected when newly-formed dust extinguishes the light coming from the receding hemisphere [see e.g. SN 1999em, @elm03].
{width="165mm"}
![Line identification in the spectra obtained on December 21, 2008, and January 22, 2009. The apparent narrow P-Cygni feature visible at the position of He I $\lambda$5876 in the 2009 January 22 spectrum is in reality a spike due to a poorly removed hot pixel. The insert shows in detail the complex structure of the He II $\lambda$4686 line in the January 22 spectrum. []{data-label="fig8"}](fig9.ps){width="90mm"}
High S/N spectra were then obtained on December 21-22, 2008, showing a remarkable evolution. The continuum is very blue, and prominent P-Cygni profiles are now visible. It is worth noting that P-Cygni profiles were marginally detected (especially in H$\beta$) also in the 2000-OT spectrum of May 6, 2000 [@wag04]. Together with H and He I, Fe II lines of the multiplet 42 and Ca II H$\&$K[^8] are clearly detected (see Figures \[fig7\] and \[fig8\]). Now H$\alpha$ does not show a P-Cygni profile and its $v_{FWHM}$ is about 2000 km s$^{-1}$, consistent with the velocity deduced from the position of the minimum of H$\beta$. Other Balmer lines show significantly lower P-Cygni velocities (about 1600 km s$^{-1}$), very close to those of He I lines ($\sim$1500-1550 km s$^{-1}$). Finally, unblended Ca II $\lambda$ 3934 and Fe II $\lambda$ 5169 P-Cygni lines have absorption components which are blue-shifted by about 1250 km s$^{-1}$. Comprehensive line identification is shown in Figure \[fig8\].
A further spectrum of the variable was obtained on January 22, 2009. Within one month, the spectrum of the variable has evolved significantly. P-Cygni profiles are no longer visible, He I and Fe II lines are marginally detected, while a prominent He II $\lambda$ 4686 line is now visible in emission (see insert in Figure \[fig8\]). The same line was visible also in the second spectrum of 2000-OT (May 31, 2000) presented by @wag04 and, marginally, also in the January 19, 2001, spectrum. The presence of this line would indicate high temperatures of the emitting material. He II lines in emission are common features in He-rich, hot Wolf-Rayet stars and early O-type supergiants, and are only rarely detected in LBVs during quiescence [e.g. in AG Car, @sta86]. Finally, two spectra were collected on 2009 February 22 and March 22, and the object did not show any further evolution. We noted only a weakening of the He II line. The line widths are very similar to those measured in the January spectrum, viz. $v_{FWHM} \approx$ 2750 km s$^{-1}$.
The spectra obtained during the 2009-OT1 event (April-May 2009, see Figure \[fig9\]) show prominent P-Cygni lines of H, He I and Fe II, and share striking similarities with the December 2008 spectra, although the continuum is now significantly bluer. This is probably due to the fact that these spectra were obtained closer to the epoch of the outburst onset. From May 12th to May 19th the continuum temperature decreases significantly and the spectral lines show more prominent P-Cygni absorption components. The H and He I lines in the first spectrum of Figure \[fig9\] (April 24, 2009) show two-component absorption profiles. We identify a lower-velocity trough (labelled with [*a*]{} in Figure \[fig9\]) which is blue-shifted by about $v_a \approx$ 3000 km s$^{-1}$, and another absorption component (labelled [*b*]{}) with a core velocity $v_b$ of about 5300 km s$^{-1}$. The broad wings of the absorption feature extend up to a velocity of $v_{edge} \approx$ 9000 km s$^{-1}$. As H$\beta$ is not saturated, $v_{edge}$ is not a perfect indicator for the terminal wind velocity ($v_\infty$). @pri90 found that $v_{edge}$ overestimates $v_\infty$ by 20-25$\%$ ($v_\infty \approx$ 0.8 $v_{edge}$), and we can therefore more accurately estimate the terminal velocity of H$\beta$ as 7200 km s$^{-1}$.
This peculiar line profile is remarkably similar to those observed by [@tru08] in SN 2005gj. The similarity of this absorption line splitting in the SN spectra with those of well known LBVs [@sta01; @sta03] led [@tru08] to conclude that the progenitor of SN 2005gj was likely an LBV that experienced a mass-loss episode a few decades before the SN explosion. In the spectrum of 2009-OT1 we measure velocities which are more than one order of magnitude higher that those reported by [@tru08], very different to those of ordinary LBV winds. The higher velocity trough (b) is only marginally detectable in the subsequent spectrum of 2009-OT1 (May 12, 2009), while the lower velocity one is still visible and has $v_a$ $\approx$ 2000-2200 km s$^{-1}$. The whole absorption feature extends to blue wavelengths corresponding to $v_{edge} \approx$ 6000-7000 km s$^{-1}$ ($v_\infty \approx$ 4800-5600 km s$^{-1}$). The last spectrum of the variable in Figure \[fig9\] (May 19, 2009) shows P-Cygni lines with a single absorption component, though with a slightly asymmetric profile. The minimum of the absorption indicates $v_a \sim$ 2000 km s$^{-1}$, but with the blue wing extending to $v_{edge} \approx$ 3000-3500 km s$^{-1}$ ($v_\infty \approx$ 2400-2800 km s$^{-1}$).
![Spectroscopic evolution of the April-May 2009 outburst (2009-OT). A zoom-in of the H$\beta$ region of the April 24 spectrum is shown in the insert. Individual features mentioned in the main text are labelled. []{data-label="fig9"}](fig10.ps){width="90mm"}
The last spectrum of the variable in NGC 3432 (see Table \[tab3\]) was obtained at the time of the 2009-OT2 outburst (November 26, 2009), but it is not shown in Figure \[fig7\] because of its poor S/N. It showed a blue continuum and narrow H lines, with $v_{FWHM}$ of about 1300-1400 km s$^{-1}$, still consistent with the lower line velocities displayed by this variable during all earlier outbursts.
@vink02 suggested that oscillations in the wind velocity (and mass-loss rate) similar to those we observe in the spectra of the variable in NGC 3432 can be associated with changes in the efficiency of line driving, as a result of variations in the Fe ionization of the wind [bi-stability in the wind velocity in early-type stars, see @lam95; @vink99]. We note that also @koe06 invoke the bi-stability mechanism to explain the changes in wind velocity of HD 5980 (see Sect. \[prog\]). Despite the high degree of variability, the wind velocities of the variable in NGC 3432 remain remarkably high both during outbursts and in quiescence, making an ordinary LBV scenario for this object rather problematic.
Comparison with Luminous Stars and Stellar Outbursts
----------------------------------------------------
{width="81mm"} {width="81mm"} {width="81mm"} {width="81mm"}
During the LBV phase, massive stars vary their apparent temperatures and, as a consequence, spectral types. The high degree of variability of LBVs is generally evident in their spectra. During the quiescent phase LBVs tend to have blue spectra often similar to those of B supergiants. However, a number of LBVs such as AG Car and HD 269582 [@sta86] show Opfe/WN9 spectra during quiescence. In this phase, these objects have also strong He II in emission, that disappears during outburst when the spectra display prominent P-Cygni profiles. Nevertheless, the remarkably high velocities measured in the spectra of the variable in NGC 3432 both during outburst and in quiescence are unusual.
In order to better constrain the nature of this variable, we compare its spectra with those of well-known classes of luminous stars or other types of transients. In Figure \[fig10\] (panel [**A**]{}) a spectrum of the variable star during the quiescence after 2008-OT is shown along with those of a number of Wolf-Rayet (of WN type) stars. While early WN (WNE) stars show relatively broad emission lines with a FWHM which is quite consistent with that observed in the spectrum of the variable in NGC 3432, their spectra are characterized by prominent He II emission lines, which dominate over the H features. (Unblended He II line at 4686Å is visible only in the 2009 January 22 spectrum, when the star was quiescent, and its intensity was much lower than that observed in spectra of WNE stars). Late WN (WNL) stars have instead more intense H lines, but very narrow. In addition, lines such as N III, N IV, C III, C IV which are prominent in WN stars, are not visible in the spectra of the variable [see @ham95a; @cro95; @cro95b; @cro95c; @nota96 for a detailed information on the sub-classification of WN stars]. The spectrum of the variable in NGC 3432 shares some similarity with those of other luminous hot stellar types, such as Be, Oe stars and O-type supergiants. Although Be stars show conspicuous H$\alpha$ lines in emission, sometimes the line profiles are double-peaked, suggesting the presence of a disk [^9].
Finally, Oe and O-supergiant stars still show prominent H lines, and He II features are weak (usually weaker than He I lines). This is the case of N11-020, classified as O5 I(n)fp by @evan06 [see Figure \[fig10\], bottom of panel [**A**]{}], whose spectral lines are relatively broad, but still too narrow in comparison with those of the variable. In addition, lines of other ions visible in all stellar types described above (N II, N III, C III, C IV, Si IV) [*are not visible*]{} in the spectra collected so far for the luminous star in NGC 3432.
In Figure \[fig10\] (panel [**B**]{}) two spectra of the variable in NGC 3432 are compared with those of the SN impostors 1997bs [@van00], 2009ip (Padova-Asiago Supernova Archive), 2003gm [@mau06] and 2002kg [@mau06]. It is worth noting that the spectral lines of the NGC 3432 variable are broader than those of the other SN impostors. H$\alpha$ is indeed dominated by an intermediate-width component ($v_{FWHM} \approx$ 2300 km s$^{-1}$) and weak, marginally detectable broader wings.
The other SN impostors shown in Figure \[fig10\] have FWHM velocities of H$\alpha$ that typically are $\leq$ 1000 km s$^{-1}$, similar to that measured for 2001ac [$v_{FWHM} \approx$ 900 km s$^{-1}$, @mat01]. Only 2009ip competes with the variable in NGC 3432 in line widths [see also the discussion in @smi09b; @fol10], although its spectrum does not show He II $\lambda$4686. The spectrum of 1997bs presented by @van00 and shown in Figure \[fig10\] (panel [**B**]{}) has a two-component emission profile, with a narrow component ($v_{FWHM} \approx$ 500 km s$^{-1}$) atop a broader one, with $v_{FWHM} \approx$ 1500 km s$^{-1}$. In this context, the spectral lines of the major outbursts of the variable in NGC 3432 are broader by a factor of 1.5-3, with FWHM velocities which are close to those observed in the intermediate components of some interacting SNe (see Figure \[fig10\], panel [**D**]{}). Nevertheless, the most important spectral features observed in the variable in NGC 3432 (due to H, He I and Fe II transitions) are commonly observed also in the spectra of SN impostors [see e.g. @mau06]. The only exception is the He II $\lambda$ 4686 line visible in the quiescent spectrum of the variable, which is rarely observed in spectra of erupting LBVs.
We compare in Figure \[fig10\] (panel [**C**]{}) the 2008 October 14 spectrum of the NGC 3432 variable with those of a sample of enigmatic transients whose overall similarity was first discussed by @pri08b and @tho08. The nature of this group of transients, which includes SN 2008S [@pri08a; @bot08; @smi08], NGC 300 2008-OT [@bond09; @ber09] and M85 OT 2006-1 [@kul07], is still debated. According to @smi08, @bond09 and @ber09 these transients are likely SN impostors caused by the luminous eruptions of massive (10-20 M$_\odot$) stars. While the progenitors of SN 2008S and NGC 300-OT were identified as dust-enshrouded massive stars in archive [*Spitzer*]{} images [of about 10M$_\odot$, according to @pri08a; @tho08; @bot08], the star producing M85 OT 2006-1 was never detected, and the outburst was associated to the merging of low-mass stars by @kul07, @rau07 and @ofek08. @pasto07b questioned both the mass limit derived for the progenitor of M85 OT 2006-1 and the nature of the outburst itself, proposing that it was the underluminous core collapse of a $\sim$8M$_\odot$ star. Finally, @bot08 discussed the possibility that SN 2008S and probably also the other transients of this group were electron-capture SNe from super-AGB stars [see also @wan08; @pumo09]. H, He I and Fe II lines are detected both in the spectra of the luminous star in NGC 3432 and in objects of the family. However, the latter objects show a prominent \[Ca II\] feature at about 7300Å (not visible in our spectra of the NGC 3432 variable), narrower spectral lines and a much slower photometric evolution.
Our first spectrum after the October 2008 outburst is finally compared in Figure \[fig10\] (panel [**D**]{}) with those of three type IIn SNe [1995G, 1995N, 1988Z; see @pasto02; @pasto05; @tura93]. The spectra of SNe 1995G and 1995N were obtained more than one year after their explosions, while that of SN 1988Z is probably earlier. The FWHM of the lines in the spectrum of the variable in NGC 3432 is similar to that of the intermediate-velocity line components of the three type IIn SNe [see @fra02; @pasto02; @zam05; @tura93; @are99]. In interacting SNe these line components are thought to originate from the shocked CSM.
All these comparisons indicate that the spectra alone do not allow us to discriminate between the different types of explosions. Additional information provided by the photometry (specifically the low peak luminosity, the fast luminosity evolution of the outburst, the variability history of the star) makes us confident in identifying the 2000/2008/2009 transients in NGC 3432 as the most recent episodes of a long series of major mass-loss events in this exceptional, restless variable.
{width="170mm"}
Evolution of the Line Profiles {#prof}
------------------------------
A view of the spectral sequence in Figure \[fig7\] suggests that two types of spectra were observed during follow-up observations of the variable in NGC 3432 (October 2008 to June 2009).
1. A few spectra (marked with 1, 8, 9, 13, 14, 15 in Figure \[fig11\]) show a blue continuum with prominent P-Cygni lines of H, He I and (although weak) Fe II. On average, the FWHM velocity of the emission features is around 1600-2200 km s$^{-1}$. In these spectra there is no clear evidence for the presence of He II lines. The reason why the spectral energy distribution (SED) peaks at the blue wavelengths during outbursts will be addressed more in detail in Section \[sed\].
2. Other spectra (e.g. 2, 3, 4, 10, 11, 12) show a much redder continuum and lines of H and He I in pure emission. The FWHM velocities are significantly higher (2400-2800 km s$^{-1}$) [^10]. In these spectra, the He II $\lambda$4686 feature is unequivocally detected.
In order to understand the reasons of the rapid variability of the spectra, we have visualized in Figure \[fig11\] the evolution of the profile of the most important H lines, and correlated the spectra with the phases of the three outbursts. In particular, for each spectrum in Figure \[fig11\] (left panels) we have assigned an identification number. The phases of the light curves reported in the right panels of Figure \[fig11\] are relative to the epoch of the outburst peaks. We have marked the epochs of the individual spectra with vertical dashed lines. We note that the low S/N spectrum marked with 1 (May 6, 2000), which shows evidence of H$\beta$ with P-Cygni profile, was captured when the object was in the deep, post-maximum decline (R $>$ 20.5) but still very close in time ($\sim$3 days) to the epoch of the outburst. Other spectra obtained soon after the outburst episodes show an excess at blue wavelengths and clear P-Cygni profiles. On the other hand, spectra with pure emission lines, including He II 4868Å, were mostly obtained in periods of relative quiescence (R between 19 and 20). In addition, counter intuitively, we found that $v_{FWHM}$ of the line components in emission is lower immediately after outbursts than when the star is quiescent.
In other words, close to the outburst epochs, the spectra of the variable in NGC 3432 are reminiscent to those of regular erupting LBVs, while in quiescence the spectrum acquires some resemblance with those of young WRs. As mentioned above, although spectra of some LBVs may show He II lines during quiescence [e.g. AG Car, @sta86], the velocity of the material ejected by the variable in NGC 3432 (as deduced from spectroscopy) is too high for a single LBV eruptor. As an alternative, supported by marginal evidence of modulation in the light curve, one may propose a multiple-system scenario. During outburst the total flux is dominated by the LBV eruptor, while in quiescence we start to see a WR companion. This scenario, similar to that proposed for the famous stellar system HD 5980 [see e.g. @koe04 and references therein], is widely discussed in Section \[prog\].
Spectral Energy Distribution {#sed}
============================
{width="132mm"}
The evolution of the SED for the variable in NGC 3432 is shown in Figure \[fig12\]. The four panels show the SED at 3 epochs representative of crucial phases in the evolution of each major outburst: 2000-OT (top-left panel), 2008-OT (top-right), 2009-OT1 (bottom-left) and 2009-OT2 (bottom-right). We account only for epochs in which multi-band observations of the source are available and, typically, close to a light curve maximum, a minimum and in quiescence.
As mentioned in Section \[ph\], the light curve peak and the subsequent minimum after 2000-OT are extremely sharp, and multi-band observations are available only from day +11 [@wag04], when the transient is in the post-minimum rise. The SEDs at this and subsequent epochs (phases $\sim$ +18 and +33; Figure \[fig12\], top-left panel) are extremely blue, with a strong U-band flux contribution (the reddening-corrected U-B ranges from -0.6 to -0.8 during this period).
The evolution of the SED after the 2008-OT eruptive episode is shown in the top-right panel. Again, multi-band observations of the luminosity peak are missing. However, at day +10 from the main outburst (in coincidence with a sharp minimum) the source shows a redder SED ($V-I \approx$ 1), with a clear R-band excess probably due to the flux contribution of H$\alpha$. The SED is then computed at +75 days when it becomes again much bluer ($U-B \approx$ -0.6; $V-I \approx$ 0.5), which is probably the result of a secondary eruptive event[^11]. Finally, at +107 days from the main event, the absolute magnitude of the object is about $M_R \approx$ -10.7, close to that expected during the quiescence. In this phase, there is a huge R-band (H$\alpha$) excess and the colour is still very red ($V-I \approx$ 1).
In Figure \[fig12\] (bottom-left panel, upper insert) we report the SED of the star at $\sim$9 days before the peak of the 2009-OT1 outburst, when the luminosity of the object is rising after a deep minimum, and is close to the magnitude at the quiescence. The SED is quite similar to that computed on 2008 October 17. The central insert shows the SED around maximum light, which is extremely blue ($U-B \approx$ -1, $V-I \approx$ 0.4). In the subsequent epoch (+27 days) the SED is computed during a sharp luminosity minimum, but still with the object being in a very active, eruptive phase. Consequently, despite the faint absolute magnitude ($M_R \approx$ -10.3), the variable still showed a moderately blue SED ($U-B \approx$ -0.5, $V-I \approx$ 0.4).
The last panel in Figure \[fig12\] (bottom-right) shows the SED evolution before (-33 days, in a deep magnitude minimum, with $M_R \approx$ -9), during and after (+114 days) the 2009-OT2 outburst. The evolution of the SED is similar to that shown for 2009-OT1 (bottom-left panel). At day -33 there is still a huge R-band excess and the SED is very red ($V-I \approx$ 0.8), then it becomes blue during outburst ($U-B \approx$ -0.9, $V-I \approx$ 0.45) and finally redder again in quiescence ($V-I \approx$ 1 at day +114). Unfortunately, the U, J and H band limits reported at phase $\sim$ +114 days are not stringent.
The observed SED is dominated by the U-band emission at all epochs that are relatively close in time to an eruptive episode. This is clearly visible around and soon after the 2009 episodes (2009 April 22-24 and November 27-28), but also on 2008 December 21 when possibly a secondary eruptive episode occurred after 2008-OT. In our attempts to fit the SED around the time of the outbursts with a single blackbody we obtain reasonable fits to the fluxes in the B, V, R, I bands with temperatures of 8500-9500 K, whilst we fail to simultaneously fit the U band flux, which shows a clear excess. At these epochs, the spectra show prominent and narrow P-Cygni features, and there is no evidence for the presence of He II $\lambda$4686. After the outbursts, the SED evolution suggests a decline of the temperatures [in agreement with @wag04 who fitted the optical SED of 2000 May 14, i.e. +11 days from the outburst, with a $T$ = 7800 K blackbody].
In quiescence or in pre-outburst phases (e.g. 2008 October 17, 2009 January 22, April 14, October 26) the SED indicates somewhat cooler temperatures, and a clear excess in the R-band flux can be noted. At these phases, the spectral lines are broader and in pure emission, He II $\lambda$4686 becomes strong and H$\alpha$ contributes significantly to the R-band flux.
The SED of an LBV is expected to shift from the UV to the optical during outbursts. This appears to be in contradiction with what we observe for the variable in NGC 3432 which, in outburst, moves its SED to bluer wavelengths (i.e. higher temperatures). On the other hand, during quiescence, the variable in NGC 3432 shows moderate temperatures [$V-I \sim$ 1 would imply $T_{eff} \approx$ 5300 K, @dri00] that are typical of yellow (G-type) supergiants, and are inconsistent with early spectral types that LBVs are expected to show during quiescence [e.g. @wol97].
The large U-band emission displayed by the variable in NGC 3432 during and soon after the eruptive events might provide key information. Lacking UV observations, we can only speculate on the nature of the U-band excess. A plausible scenario is that the observed SED is in reality due to two variable contributions: a very hot component which peaks in the UV domain associated with the outburst, plus a warm component that peaks in the optical and declines in temperature with time after the outburst, possibly due to the hypergiant star. A detailed study on the UV variability of the source appears to be crucial to understand the nature of the luminous star in NGC 3432. Spectral modelling of the continuum and the strong stellar outflow would also give estimates for $T_{eff}$ and $log L$ of the star during its transitional phases, to determine if the observables are consistent with our ideas of LBVs expanding and cooling during outbursts.
Nevertheless, a further stellar component is still necessary to account for the presence of the He II $\lambda$ 4686 line in the spectra of the variable and the high-velocity wind observed during quiescence. These observables would be more consistent with a scenario involving also a hot Wolf-Rayet star (see Sect. \[prog\]).
The nature of the variable in NGC 3432 {#prog}
======================================
The luminous variable in NGC 3432 is one of the most intriguing stars discovered in the Local Universe. It experienced at least 4 luminous outbursts within a decade, but a number of minor rebrightenings were also observed (e.g. Figure \[fig6\]). Since the 3 most recent outbursts occurred with intervals of about 200-210 days, one may suggest that subsequent eruptive events occur with some periodicity, although the modulation of the light curve is not regular enough to support this claim. In quiescence the object has an unusually high intrinsic luminosity ($M_R$ $\approx$ -10.8), consistent with that of a luminous LBV. Also, to our knowledge, its spectra are unprecedented. Prominent H Balmer lines, dominating over other spectral lines (e.g. He I, Fe II), are typical of LBVs. However, the velocity of the ejected material, 1500-2500 km s$^{-1}$, is high (by a factor $\sim$ 5-10) for erupting LBVs, and are more consistent with velocities expected in winds of Wolf-Rayet stars [or in some very hot stars, see @cas87 that are not expected to produce prominent H lines like those observed in the spectra of the NGC 3432 variable]. In addition, the presence of He II during the quiescent phase is indicative of high gas temperatures, which is unusual in LBVs, and is commonly observed in Wolf-Rayet stars [including Ofpe/WN9 transition types, see @nota96; @mor96]. However, in all these stars the He II lines (in particular that at 4686Å) are expected to dominate over all the other lines. This is not the case for the variable in NGC 3432, in which He II $\lambda$4686 is always less prominent than H$\beta$. A relatively weak He II $\lambda$4686 line was occasionally observed in quiescent LBVs [@sta86], but always at much lower velocities than those observed in the spectra of the variable in NGC 3432.
In addition, the asymmetric profile (double-peaked, see insert in Figure \[fig8\]) of the He II $\lambda$4686 line is puzzling. It is currently unclear if this peculiar profile is due to a blend with other spectral lines or if it is an intrinsic asymmetry of the He II line. A similar profile in the He II $\lambda$4686 feature was observed also in the spectra of the luminous OB supergiant S18 in the Small Magellanic Cloud (SMC), classified as a B\[e\] type [@nota96]. In that case, the line asymmetry and its variability with time were interpreted as a signature of accretion onto a hot companion inside a complex, two-component circumstellar wind [a hot, high-velocity polar wind plus a cooler, dense equatorial disk, see @shor87; @zick89]. However, S18 has a moderate photometric variability[^12], it is slightly fainter [$M_{bol} \approx$ -9.2 to -9.4, @zick89] and the spectral lines are narrower than those observed in the spectra of the variable in NGC 3432. Nevertheless, despite the evident differences between the variable in NGC 3432 and S18 in SMC, a scenario involving a companion star and a complex circumstellar medium (CSM) might explain both the quasi-modulated variability in the light curves and the variability of the He II $\lambda$4686 line.
The large photometric variability of the luminous star in NGC 3432 is another characteristic that is difficult to explain. A variability of 4-5 mag is not unusual for an LBV outburst. However, the frequency of these episodes is puzzling. According to @hum94, different types of variability, in terms of magnitude amplitudes and time scales, are observed in LBVs:
1. [*giant eruptions*]{}, like that of $\eta$ Car during the 1840s, are rare and expected to occur in intervals of hundreds to thousands of years. During these outbursts the star increases its magnitude by more than 2 mag and ejects a significant amount of its envelope. Erratic individual brightenings have short life, although the phase of intense activity may last even a few years. This phase of major activity is then followed by a long period of quiescence.
2. [*Eruptions*]{} are much more frequent, being usually observed over time scales of decades. In these episodes the star brightens by less than 2 mag, although the bolometric luminosity remains almost constant, and the eruptive phase may last for months to years, followed by a luminosity minimum of similar duration. AG Car experienced this kind of variability during the 1980s [@bat; @sta01].
3. [*Smaller oscillations*]{} ($\Delta m$ $\leq$ 0.5) or even [*micro-variations*]{} ($\Delta m$ $\leq$ 0.1) are also frequently observed in LBVs on timescales from weeks to several months.
It is evident that the variable in NGC 3432 has unique characteristics, since it does not match any of these variability scenarios. Although modulated light curves (e.g. Figure \[fig6\]) are typical of LBVs during the S-Dor phase, the fact that the variable in NGC 3432 experienced at least 4 major outbursts in 9 years plus a few further minor episodes in a very short period of the stellar evolution (around 15 yrs) is unexpected and probably cannot be connected to an S-Dor type of variability. @smi01 noted some similarity with the erratic fluctuations of the light curve of V12 in NGC 2403 during the period 1949-1954 [@tam68; @hum99], and suggested that these oscillations may be typical of LBVs prior to and during a giant eruption.
Another possibility is that we are observing a very massive star in the latest stages of its life, which experiences a major phase of instability. Frequent ejections of H-rich shells through a pulsational pair instability mechanism, followed by their collisions, have been proposed to explain the properties of the over-luminous type IIn SN 2006gy [@woo07]. A similar scenario has been proposed for another type IIn event, SN 1994W [@des08][^13]. In both cases, the shell-shell interaction was able to fully explain the observed properties of these events, even without the need to invoke an additional contribution from radioactive $^{56}$Ni [e.g. @smi07].
As mentioned above, the spectral lines identified in 2008-OT are consistent with those of LBVs. The spectra of the variable in NGC 3432 are indeed reminiscent to those of LBV NGC 2363-V1 presented in @pet06. The spectral lines and their evident P-Cygni profiles are remarkably similar in these two objects, but the derived velocities are about one order of magnitude lower in the LBV NGC 2363-V1. LBVs in eruption may have significantly higher velocities, but usually without reaching the extreme values registered in the cases of the major outbursts of the variable in NGC 3432 (see Figure \[fig10\] panel [**B**]{} and discussion in Sect. \[sp\]).
The recurrent mass ejections observed in the variable in NGC 3432 are then expected to produce an increasingly He-rich CSM, stripping the star of its massive envelope. The above scenario, as suggested by @woo07, may also explain the sequence of events preceding SN 2006jc and the formation of its dense, He-rich CSM [@nak06; @pasto07a]. Remarkably, the FWHM velocity of the H and He I lines in recent spectra of the variable in NGC 3432 ($\sim$2300 km s$^{-1}$) is very close to that reported by @pasto07a for the He I circumstellar lines of SN 2006jc (about 2200 km s$^{-1}$). This wind velocity (as mentioned above) is significantly higher than that expected for eruptions of LBVs, and more consistent with that of a Wolf-Rayet wind. However, H is still the most prominent spectral feature. The N and He II lines which characterize spectra of young Wolf-Rayet stars [of WN type, @cro95 see also Sect. \[sp\]] are relatively weak (He II $\lambda$4686) or not detected. In addition, the progenitor stars of the variable in NGC 3432 and SN 2006jc were different. In the case of SN 2006jc, a young Wolf-Rayet star[^14] was proposed as progenitor [@pasto07a; @fol07], while the variable in NGC 3432 is significantly less evolved, since the spectra are still dominated by H lines. This would suggest that it is in a transitional phase of its evolution, possibly approaching the Wolf-Rayet stage. In the light of all of this, the frequent outbursts of the NGC 3432 variable star may possibly herald a similar fate as the precursor of SN 2006jc and an imminent SN explosion.
As an alternative to the pulsational instability, these oscillations in the light curve may indicate another plausible scenario, in which the NGC 3432 variable is a member of a close binary stellar system [like S18 in SMC, @zick89] in a complex circumstellar environment. Binarity could enhance the LBV instability. In that case, bipolar structures can be expected in the ejected material. This might be verified through high-resolution spectroscopic observations or with (spectro)polarimetry, both of them unfortunately still out of reach for the variable in NGC 3432.
Another object which the luminous star in NGC 3432 may share some similarity with is the stellar system HD 5980 in SMC [see @koe04 for a review]. HD 5980 is one of the best studied stellar objects, and the brightest source [with integrated magnitude $M_V$ = -7.3, @bp91] in the SMC. However, there is substantial evidence that it is not a single exceptionally massive star, but is a triple system comprising of a massive erupting LBV/WR (usually labelled as [*star A*]{}), an early Wolf-Rayet ([*star B*]{}) and, possibly, an O-type companion[^15] ([*star C*]{}). The three stars have, in quiescence, individual absolute magnitudes of about $M_V \sim$ -6 [@foe08]. [*Star A*]{} [with a mass of about 50 M$_\odot$, @nie97] and [*star B*]{} ($M \approx$ 28 M$_\odot$)[^16] constitute a rather close, eclipsing binary system with a period of about 19 days and a rather eccentric orbit ($e$ = 0.3). During past decades HD 5980 gradually changed its absolute magnitude and its spectroscopic properties. This was because [*star A*]{} entered a very active phase of variability, during which it changed its spectrum from that of a Wolf-Rayet star [of WN4-type, in early 1980s, @nie97] to that of an LBV [at the time of the eruption, in 1993-1994, see @koe96 and references therein] and back to a WN type in more recent years. During the late 1970s to the early 1980s, the spectrum of HD 5980 showed strong, high-velocity ($v_\infty \approx$ 2000-3000 km s$^{-1}$) He II lines which are traditionally seen in WN-type Wolf-Rayet stars. However, at the time of the outburst of [*star A*]{}, HD 5980 increased its luminosity by about 3 mag, and the spectrum started to show prominent and narrower lines of H and He I [@bat94; @bar95; @koe95], that are typical of LBV eruptions. At the time of the major eruption, the minimum wind velocities measured from the spectra of HD 5980 were about 500 km s$^{-1}$, but they rapidly increased in the subsequent few months to about 1000-1300 km s$^{-1}$ [@koe98]. It is generally assumed that in the pre-outburst phase the spectrum of HD 5980 was dominated by the Wolf-Rayet features attributed to [*star B*]{}, while during outburst the spectrum was dominated by the lower-velocity LBV-like features of [*star A*]{}. However, it is still puzzling how [*star A*]{} could be a hot WN star before (and after) its eruption as an LBV. The pieces of evidence [**1)**]{} that the spectra of HD 5980 show strong H lines (with line velocities that are, however, too high for an LBV) and [**2)**]{} that only one eruption has been registered so far, suggest that [*star A*]{} is not a Wolf-Rayet star in the classical sense, but it is in a sort of pre-LBV evolutionary phase [@foe08]. It is interesting to note that also a few other luminous, H-rich Wolf-Rayet stars are observed in the SMC [@foe03].
In this context, the observables of the variable star in NGC 3432 appear rather similar to those of HD 5980 during the eruptive phase. This might indicate that the variable in NGC 3432 is in a similar evolutionary path as [*star A*]{}, i.e. in an initial LBV stage. This might explain the high wind velocities in an otherwise LBV-like spectrum. But it does not explain the strength and the variability of the He II line. In addition, one can speculate that some of the photometric and spectroscopic variability observed in the luminous variable in NGC 3432 can in some way be related to the close interaction with a companion WR star, in analogy to that observed in the stellar system HD 5980. Interestingly, both NGC 3432 and SMC have quite low oxygen abundances, of 12 + lg(O/H) = 8.3 (see Section 2) and 8.0 [@hun07; @tru07], respectively.
Long-term monitoring of the hyper-active variable in NGC 3432 is required to understand the configuration of the stellar system, the reasons of its peculiar variability and if the observed properties are an indication for a forthcoming SN explosion.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper is based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica); the Liverpool Telescope operated on the island of La Palma at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias; the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias; the Hale Telescope, Palomar Observatory, as part of a continuing collaboration between the California Institute of Technology, NASA/JPL, and Cornell University; the 2.2m telescope of the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC); the 1.82m telescope of INAF-Asiago Observatory.
This paper is also based in part on data obtained from the Isaac Newton Group Archive which is maintained as part of the CASU Astronomical Data Centre at the Institute of Astronomy, Cambridge; on data collected at the Kiso observatory (University of Tokyo) and obtained from the SMOKA, which is operated by the Astronomy Data Center, National Astronomical Observatory of Japan and on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS 5-26555. This manuscript made also use of SDSS data. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is [*http://www.sdss.org/*]{}. This manuscript used information contained in the Bright Supernova web pages ([*http://www.RochesterAstronomy.org/snimages*]{}, maintained by D. Bishop), as part of the Rochester Academy of Sciences and in the AAVSO International Database. We are grateful to the amateur astronomers K. Itagaki, C. Deforeit, T. Tamura, G. Youman, B. Häusler, R. Johnson and E. Prosperi for sharing their images of NGC 3432 with us, to P. Holmström for his help in the observations at the Sandvreten Observatory, and to G. Gräfener for useful discussions. We also thank S. Otero and K. Weis for giving us access to the comparison data of $\eta$ Car and 2002kg, and C. Inserra and the Padova-Asiago Supernova Group to provide us an unpublished spectrum of 2009ip.
S.B. and E.K. acknowledge partial financial support from ASI contracts ‘COFIS’. S.T. acknowledges support by the Trans-regional Collaborative Research Centre TRR33 “The Dark Universe” of the German Research Foundation (DFG). S.M. acknowledges support from the Academy of Finland (project 8120503). This work, conducted as part of the award “Understanding the lives of massive stars from birth to supernovae” (S.J. Smartt) made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme (see [*http://www.esf.org/euryi*]{}).
[99]{} Agnoletto, I. et al., 2009, ApJ, 691, 1348 Aretxaga, I., Benetti, S., Terlevich, R. J., Fabian, A. C., Cappellaro, E., Turatto, M., della Valle, M. 1999, MNRAS, 309, 343 Arp, H. C. 1966, [*Atlas of Peculiar Galaxies*]{} (Caltech: Pasadena) Bateson, F. M. 1988-2000, RASNZ Var. Star Sect. Circ. Bateson, F. M., Gilmore, A., Jones, A. 1994, IAU Circ. 6102 Baba, H. et al. 2002, ADASS XI, eds. D. A. Bohlender, D. Durand, & T. H. Handley, ASP Conference Series, Vol.281, p.298 Barbá R. H., Niemela, V. S., Baume, G., Vásquez, R. A., 1995, ApJ, 446, L23 Berger, E. et al. 2009, ApJ, 699, 1850 Blondin, S., Masters, K., Modjaz, M., Kirshner, R., Challis, P., Matheson, T., Berlind, P. 2006, CBET, 636, 1 Bond, H. E., Bedin, L. R., Bonanos, A. Z., Humphreys, R. M., Monard, B. L. A. G.; Prieto, J. L., Walter, F. M. 2009, ApJ, 695, L154 Botticella, M. T. et al . 2009, MNRAS, 398, 1041 Branch, D., Cowan, J. J. 1985, ApJ, 297L, 33 Breysacher, J., Perrier, C. 1991, in IAU Symp. 143, [*Wolf-Rayet Stars and Interrelations with Other Massive Stars in Galaxies*]{}, ed. K. van der Hucht & B. Hidayat, Kluwer Academic Publishers, Dordrecht, p.229 Cassinelli, J. P., Lamers, H. J. G. L. M. 1987, in [ *Exploration of the Universe with the IUE Satellite*]{}, ed. Y. Kondo, (Dordrecht: Reidel), p.139 Chu, Y.-H., Gruendl, R. A., Stockdale, C. J., Rupen, M. P., Cowan, J. J., Teare, S. W. 2004, AJ, 127, 2850 Chugai, N. N. et al. 2004, MNRAS, 352, 1213 Clark, J. S., Crowther, P. A., Mikles, V. J. 2009, A&A 507, 1567 Cowan, J. J., Henry, R. B. C., Branch, D., 1988, ApJ, 329, 116 Crowther, P. A., Smith, I. J., Hillier, D. J., Schmutz, W. 1995a, A&A, 293, 427 Crowther, P. A., Hillier, D. J., Smith, L. J. 1995b, A&A, 293, 403 Crowther, P. A., Hillier, D. J., Smith, L. J. 1995c, A&A, 293, 172 Davidson, K., Ebbets, D., Weigelt, G., Humphreys, R. M., Hajian, A. R., Walborn, N. R., Rosa, M. 1995, AJ, 109, 1784 Davidson, K., Humphreys, R. M. 1997, AAR&A, 35, 1 Dessart, L., Hillier, D. J., Gezari, S., Basa, S., Matheson, T. 2009, MNRAS, 349, 21 Drilling, J. S., Landolt, A. U. 2000, in [*Allen’s astrophysical quantities*]{}, 4th ed. Edited by Arthur N. Cox. ISBN: 0-387-98746-0. Publisher: New York: AIP Press; Springe r, 2000, p.381 Duszanowicz, G., Nakano, S., Itagaki, K. 2008, CBET, 1534, 1 Elmhamdi, A. et al. 2003, MNRAS, 338, 939 English, J., Irwin, J. A. 1997, AJ, 113, 2006 Evans, C. J., Lennon, D. J., Smartt, S. J., Trundle, C. 2006, A&A, 456, 623 Fesen, R. A. 1985, ApJ, 297L, 29 Filippenko, A. V., Barth, A. J., Bower, G. C., Ho, L. C., Stringfellow, G. S., Goodrich, R. W., Porter, A. C. 1995, AJ, 110, 2261 Filippenko, A. V., Li, W. D., Modjaz, M. 1999, IAU Circ. 7152, 2 Filippenko, A. V., Li, W. D., Treffers, R. R., Modjaz, M. 2001, [*Small-Telescope Astronomy on Global Scales*]{}, ASP Conf. Ser. 246, ed. W. P. Chen, C. Lemme, B. Paczyński (San Francisco: ASP), 121 Foellmi, C., Moffat, A. F. J., Guerrero, M. A. 2003, MNRAS, 338, 1025 Foellmi, C. et al. 2008, RevMexAA, 44, 3 Foley, R. J. et al., 2007, ApJ, 657, L105 Foley, R. J. et al., 2010, ApJ submitted (arXiv:1002.0635) Fransson, C. et al. 2002, ApJ, 572, 350 Freedman, W. L. et al. 2001, ApJ, 553, 47 Frew, D. J. 2004, JAD, 10, 6 Fukugita, M., Ichikawa, T., Gunn, J. E., Doi, M., Shimasaku, K., and Schneider, D. P. 1996, AJ, 111, 1748 Gal-Yam, Avishay, Leonard, D. C., Fox, D. B., Cenko, S. B. 2007, ApJ, 656, 372 Gal-Yam, Avishay, Leonard, D. C. 2009, Nature, 458, 865 Goodrich, R. W., Stringfellow, G. S., Penrod, G. D., Filippenko, A. V. 1989, ApJ, 342, 908 Groh, J. H., Hillier, D. J., Damineli, A., Whitelock, P. A., Marang, F., Rossi, C. 2009, ApJ, 698, 1698 Hamann, W.-R., Koesterke, L., Wessolowski, U. 1995a, A&A, 229, 151 Hamann, W.-R., Koesterke, L., Wessolowski, U. 1995b, A&AS, 113, 459 Harutyunyan, A., Navasardyan, H., Benetti, S., Turatto, M., Pastorello, A., Taubenberger, S. 2007, CBET 1184, 1 Ho, L. C., Filippenko, A. V., Sargent, W. L. W. 1997, ApJS, 112, 315 Humphreys, R. M., Davidson, K. 1994, PASP, 106, 1025 Humphreys, R. M., Davidson, K., Smith, N. 1999, PASP, 111, 1124 Hunter, I. et al. 2007, A&A, 466, 277 Koenigsberger, G., Guinan, E., Auer, L., Georgiev, L. 1995, ApJ, 452, 107 Koenigsberger, G., Guinan, E., Auer, L., Georgiev, L. 1996, RevMexAA, 5, 92 Koenigsberger, G., Pena, M., Schmutz, W., Ayala, S. 1998, ApJ, 499, 889 Koenigsberger, G. 2004, RevMexAA, 40, 107 Koenigsberger, G., Fullerton, A. W., Massa, D., Auer, L. H. 2006, AJ, 132, 1527 Kotak, R., Vink, J. S. 2006, A&A, 460L, 5 Kulkarni, S. R. et al. 2007, Nature, 447, 458 Lamers, H. J. G. L. M., Snow, T. P., Lindholm, D. M. 1995, ApJ, 455, 269 Lee, J. C. et al. 2009, ApJ, 706, 599 Li, W., Filippenko, A. V., Van Dyk, S. D., Hu, J., Qiu, Y., Modjaz, M., Leonard, D. C. 2002, PASP, 114, 403L Matheson, T., Calkins, M. 2001, IAU Circ. 7597, 3 Maund, J. R. et al., 2006, MNRAS, 369, 390 Maund, J. R. et al., 2008, MNRAS, 387, 1344 Mihos, J. C., Hernquist, L. 1994, ApJ, 425, L13 Morris, P. W., Eenens, P. R. J., Hanson, M. M., Conti, P. S., Blum, R. D. 1996, ApJ, 470, 597 Nakano, S., Igataki, K., Puckett, T., Gorelli, R. 2006, CBET 666, 1 Niemela, V. S., Barbá, R. H., Morrell, N. I., Corti, M. 1997, in ASP Conf. Ser. 120, [*Luminous Blue Variables: Massive Stars in Transition*]{}, ed. A. Nota & H. Lamers (S. Francisco: ASP), 222 Nota, A., Pasquali, A., Drissen, L., Leitherer, C., Robert, C., Moffat, A. F. J., Schmutz, W. 1996, ApJS, 102, 383 Ofek, E. O. et al. 2008, ApJ, 674, 447 Pastorello, A. et al. 2002, MNRAS, 333, 27 Pastorello, A., Aretxaga, I., Zampieri, L., Mucciarelli, P., Benetti, S. 2005, in [*1604-2004: Supernovae as Cosmological Lighthouses*]{}, ASP Conference Series, Vol. 342, Proceedings of the conference held 15-19 June, 2004 in Padua, Italy. Edited by M. Turatto, S. Benetti, L. Zampieri, and W. Shea. San Francisco: Astronomical Society of the Pacific, 2005., p.285 Pastorello, A. et al. 2007a, Nature 447, 829 Pastorello, A. et al. 2007b, Nature 449, 1 Pastorello, A. et al. 2008, MNRAS, 389, 955 Papenkova, M., Li W. 2000, IAU Circ. 7415, 1 Petit, V., Drissen, L., Crowther, P. A. 2006, AJ, 132, 1756 Pilyugin, L. S., Thuan, T. X. 2007, ApJ, 669, 299 Prieto, J. L. et al. 2008, ApJ, 681L, 9 Prieto, J. L., Kistler, M. D., Stanek, K. Z., Thompson, T. A., Kochanek, C. S., Beacom, J. F. 2008, Astron. Tel. 1596, 1 Prinja, R. K., Barlow, M. J., Howarth, I. D. 1990, ApJ, 361, 607 Pumo, M. L. et al. 2009, ApJ, 705, L138 Rau, A., Kulkarni, S. R., Ofek, E. O., Yan, L. 2007, ApJ, 659, 1536 Schlegel, D. J., Finkbeiner, D. P., Davis, M. 1998, ApJ, 500, 525 Schinzel, F. K., Taylor, G. B., Stockdale, C. J., Granot, J., Ramirez-Ruiz, E. 2009, ApJ, 691, 1380 Shore, S. N., Sanduleak, N., Allen, D. A. 1987, A&A, 176, 59 Smith, J.A. et al. 2002, AJ, 123, 2121 Smith, N., Humphreys, R. M., Gehrz, R. D. 2001, PASP, 113, 692 Smith, N. et al. 2007, ApJ, 666, 1116 Smith, N., Chornock, R., Li, W., Ganeshalingam, M., Silverman, J. M., Foley, R. J., Filippenko, A. V., Barth, A. J 2008, 686, 467 Smith, N. et al. 2009, ApJ, 697L, 49 Smith, N. et al. 2010, AJ, 139, 1451 Sollerman, J., Cumming, R. J., Lundqvist, P. 1998, ApJ, 493, 933 Stahl, O. 1986, A&A, 164, 321 Stahl, O., Jankovics, I., Kovács, J., Wolf, B., Schmutz, W., Kaufer, A., Rivinius, Th., Szeifert, Th. 2001, A&A, 375, 54 Stahl, O., Gäng, T., Sterken, C., Kaufer, A., Rivinius, T., Szeifert, T., Wolf, B. 2003 A&A, 400, 279 Stanishev, V. 2007, AN, 328, 948 Stockdale, C. J., Rupen, M. P., Cowan, J. J., Chu, Y.-H., Jones, S. S. 2001, AJ, 122, 283 Swaters, R. A., van Albada, T. S., van der Hulst, J. M., Sancisi, R. 2002, A&A, 390, 829 Tammann, G. A., Sandage, A., 1968, ApJ, 151, 825 Terry, J. N., Paturel, G., Ekholm, T. 2002, A&A, 393, 57 Thompson, T. A., Prieto, J. L., Stanek, K. Z., Kistler, M. D., Beacom, J. F., Kochanek, C. S. 2009, ApJ, 705, 1364 Tominaga, N. et al. 2008, ApJ, 687, 1208 Trundle, C., Dufton, P. L., Hunter, I., Evans, C. J., Lennon, D. J., Smartt, S. J., Ryans, R. S. I. 2007, A&A, 471, 625 Trundle, C., Kotak, R., Vink, J. S., Meikle, W. P. S. 2008, A&A, 483L, 47 Turatto, M., Cappellaro, E., Danziger, I. J., Benetti, S., Gouiffes, C., della Valle, M. 1993, MNRAS, 262, 128 Utrobin, V. P. 1984, Ap&SS, 98, 115 van der Kruit, P. C. 2007, A&A, 466, 883 Van Dyk, S. D., Peng, C. Y., King, J. Y., Filippenko, A. V., Treffers, R. R., Li, W., Richmond, M. W. 2000, PASP, 112, 1532 Van Dyk, S. D., Filippenko, A. V., Li, W. 2002, PASP, 114, 700 Van Dyk, S. D., Filippenko, A. V., Chornock, R., Li, W., Challis, P. M. 2005, PASP, 117, 553 van Genderen, A. M. 2001, A&A, 366, 508 van Genderen, A. M., Sterken, C. 2002, A&A, 386, 926 Vink, J. S., de Koter, A., Lamers, H. J. G. L. M. 1999, A&A, 350, 181 Vink, J, S., de Koter, A. 2002. A&A, 393, 543 Vink, J. S., Davies, B., Harries, T. J., Oudmaijer, R. D., Walborn, N. R. 2009, A&A, 505, 743 Wagner, R. M. et al. 2004a, PASP, 116, 326 Wagner, R. M., Schmidt, G. D., Smith, P., Hines, D., Starrfield, S. G. 2004b, IAU Circ. 7417, 2 Wanajo, S., Nomoto, K., Janka, H.-Th., Kitaura, F. S., Mueller, B. 2009, ApJ, 695, 208 Weis, K., Bomans, D. J. 2005, A&A, 429L, 13 Wolf, B., Kaufer, A. 1997, in [*Luminous Blue Variables: Massive Stars in Transition.*]{} ASP Conference Series; Vol. 120; 1997; ed. Antonella Nota and Henny Lamers (1997), p.26 Woosley, S. E., Blinnikov, S., Heger, A. 2007, Nature, 450, 390 Zampieri, L., Mucciarelli, P., Pastorello, A., Turatto, M., Cappellaro, E., Benetti, S. 2005, MNRAS, 364, 1419 Zickgraf, F.-J., Wolf, B., Stahl, O., Humphreys, R. M. 1989, A&A, 220, 206
Unpublished photometry of the variable in NGC 3432
==================================================
[cccccccc]{}
\
Date & avg. JD & U & B & V & R & I & Source\
1994May02 & 2449474.50 & & & & 21.09 (0.10) & & INT$^a$\
1994May02 & 2449474.51 & & & & 21.17 (0.16) & & INT$^a$\
1995Nov29 & 2450051 & & $>$20.90 & $>$20.63 & $>$20.12 &$>$20.34 & KISO$^b$\
1996Mar13 & 2450156 & & $>$20.57 & $>$20.60 & $>$20.32 &$>$19.70 & KISO$^b$\
1999Dec12 & 2451524.91 & & & & $>$19.04 & & IRO$^c$$\star$\
[**2000May03**]{} & [**2451667.7**]{} & & & & [**17.4 (0.1)**]{} & & KAIT$^d$$\star$\
2001Jan04 & 2451914.59 & & & 19.42 (0.17) & & & TT$\star$\
2004May15 & 2453140.83 & & 20.72 (0.03) & 20.43 (0.03) & 19.48 (0.02) & & HALE\
2005Mar11 & 2453440.74 & & & $>$19.54 & $>$19.03 & & GY\
2005Mar12 & 2453441.84 & & $>$20.04 & & & & GY\
2005Mar19 & 2453449.34 & & & & $>$19.27 & & MO$\star$\
2005Apr06 & 2453466.71 & & & & $>$20.40 & & GY$\star$\
2005Apr11 & 2453472.35 & & & & $>$19.28 & & MO$\star$\
2006Apr22 & 2453848.47 & & & & 20.35 (0.24) & & CD$\star$\
2006Oct28 & 2454036.60 & & & & $>$19.34 & & MO$\star$\
2006Nov15 & 2454054.69 & & & & $>$19.75 & & MO$\star$\
2006Dec18 & 2454087.57 & & & & $>$19.62 & & MO$\star$\
2007Jan23 & 2454123.58 & & & & 19.67 (0.36) & & MO$\star$\
2007Mar17 & 2454177.36 & & & & 18.45 (0.15) & & MO$\star$\
2007Nov24 & 2454428.67 & & & & $>$18.98 & & MO$\star$\
2008Jan27 & 2454493.50 & & & & $>$19.61 & & MO$\star$\
2008Feb25 & 2454522.41 & & & & $>$19.79 & & MO$\star$\
2008Mar06 & 2454532.41 & & & & 19.39 (0.40) & & MO$\star$\
2008Mar12 & 2454537.52 & & 20.10 (0.21) & 19.75 (0.12) & 18.67 (0.20) & & RJ\
2008Mar12 & 2454537.75 & & & 19.72 (0.09) & & & RJ$^e$\
2008Oct06 & 2454746.22 & & & & 17.57 & & TKC$^f$ $\star$\
[**2008Oct07**]{} & [**2454746.62**]{} & & & & [**17.52 (0.11)**]{} & & MO$\star$\
2008Oct09 & 2454749.28 & & & & 17.71 (0.14) & & KI$\star$\
2008Oct17 & 2454756.67 & & 19.98 (0.10) & 19.99 (0.11) & 18.93 (0.06) & 18.89 (0.08) & CAHA\
2008Oct17 & 2454756.71 & & 20.04 (0.28) & 19.92 (0.23) & & & TNG\
2008Oct17 & 2454756.72 & & & & 18.94 (0.12) & & TNG$\star$\
2008Oct25 & 2454764.74 & & 18.82 (0.03) & 18.59 (0.02) & 18.04 (0.02) & 17.96 (0.02) & TNG\
2008Nov09 & 2454779.69 & & 22.59 (0.46) & 21.92 (0.32) & & & LT\
2008Nov16 & 2454786.51 & & & & $>$20.35 & & SO$\star$\
2008Nov19 & 2454788.52 & & & 21.52 (0.02) & & 21.05 (0.02) & HST\
2008Nov19 & 2454789.66 & & 22.03 (0.21) & 21.67 (0.10) & & & LT\
2008Nov24 & 2454794.65 & & 22.56 (0.20) & 22.13 (0.12) & 21.11 (0.12) & 21.64 (0.19) & TNG\
2008Nov24 & 2454794.72 & & 22.55 (0.20) & 21.90 (0.10) & & & LT\
2008Dec09 & 2454809.62 & & 19.79 (0.03) & 19.65 (0.03) & 19.32 (0.03) & 19.25 (0.06) & EK\
2008Dec10 & 2454810.61 & & 19.42 (0.07) & 19.18 (0.04) & & & LT\
2008Dec17 & 2454817.67 & & 19.47 (0.02) & 19.09 (0.03) & & & LT\
2008Dec21 & 2454821.72 & & & 19.34 (0.01) & 18.79 (0.01) & & NOT\
2008Dec21 & 2454821.78 & 19.22 (0.02) & 19.77 (0.02) & & & 18.78 (0.02) & NOT\
2008Dec22 & 2454822.58 & & 19.79 (0.03) & 19.43 (0.02) & 18.84 (0.04) & 18.67 (0.02) & TNG\
2008Dec23 & 2454823.50 & & & & 18.86 (0.06) & & SO$\star$\
2008Dec23 & 2454823.54 & & 19.80 (0.23) & 19.44 (0.13) & 18.87 (0.12) & 18.69 (0.13) & SO\
2008Dec29 & 2454829.64 & & 20.39 (0.07) & 19.85 (0.04) & & & LT\
2008Dec30 & 2454830.68 & & & 19.86 (0.07) & 19.12 (0.04) & & CAHA\
2009Jan01 & 2454833.46 & & & & 19.22 (0.17) & & SO$\star$\
2009Jan02 & 2454833.53 & & $>$20.24 & 20.17 (0.17) & 19.21 (0.09) & & SO\
2009Jan05 & 2454836.69 & & 20.51 (0.05) & 20.05 (0.04) & & & LT\
2009Jan11 & 2454842.59 & & & & 19.09 (0.30) & & MO$\star$\
2009Jan13 & 2454844.63 & & 20.83 (0.21) & 20.37 (0.11) & & & LT\
2009Jan14 & 2454845.62 & & 20.90 (0.07) & 20.44 (0.05) & & & LT\
\
\
Date & JD & U & B & V & R & I & Source\
2009Jan20 & 2454851.57 & & & & 19.53 (0.06) & & TNG$\star$\
2009Jan20 & 2454852.49 & & 21.04 (0.07) & 20.63 (0.05) & & & LT\
2009Jan21 & 2454852.56 & & 21.11 (0.05) & 20.65 (0.02) & & & LT\
2009Jan22 & 2454853.67 & $>$20.28 & 21.07 (0.19) & 20.83 (0.05) & 19.51 (0.02) & 19.82 (0.03) & TNG\
2009Jan25 & 2454856.55 & & 21.11 (0.05) & 20.69 (0.04) & & & LT\
2009Feb01 & 2454863.57 & $>$20.98 & & & & & NOT\
2009Feb01 & 2454863.57 & & & & 19.49 (0.13) & & NOT$\star$\
2009Feb02 & 2454865.40 & & & & 19.51 (0.11) & & SO$\star$\
2009Feb02 & 2454865.45 & & & & 19.49 (0.10) & & SO\
2009Feb02 & 2454865.48 & & 21.05 (0.05) & 20.85 (0.05) & 19.48 (0.02) & 19.99 (0.03) & NOT\
2009Feb12 & 2454874.61 & & & 20.71 (0.03) & 19.45 (0.02) & 19.81 (0.07) & NOT\
2009Feb22 & 2454884.51 & & & & 19.56 (0.09) & & NOT\
2009Feb22 & 2454884.52 & & & & 19.55 (0.13) & & NOT$^g$$\star$\
2009Feb22 & 2454885.07 & $>$19.84 & $>$20.31 & $>$19.35 & & & UVOT$^h$\
2009Feb25 & 2454888.43 & & 21.11 (0.28) & $>$19.69 & $>$18.73 & $>$18.55 & EK$^g$\
2009Mar15 & 2454906.46 & & & & 19.91 (0.12) & & SO$\star$\
2009Mar15 & 2454906.49 & & & 20.97 (0.24) & & & SO\
2009Mar19 & 2454910.37 & & & 21.00 (0.04) & 19.98 (0.03) & 20.15 (0.10) & CAHA\
2009Mar21 & 2454912.46 & & & & 19.95 (0.13) & & TNG\
2009Apr14 & 2454935.62 & & 20.82 (0.04) & 20.59 (0.03) & 19.73 (0.03) & 20.03 (0.02) & NOT$\star$\
2009Apr15 & 2454937.49 & & & 20.06 (0.01) & 19.45 (0.01) & 19.67 (0.02) & NOT\
2009Apr22 & 2454943.58 & 17.58 (0.02) & 18.58 (0.01) & 18.45 (0.01) & 18.04 (0.01) & 18.00 (0.02) & NOT\
[**2009Apr22**]{} & [**2454944.41**]{} & & & [**18.33 (0.05)**]{} & [**17.92 (0.04)**]{} & & SO\
2009Apr24 & 2454945.55 & & 18.60 (0.01) & 18.50 (0.02) & 18.05 (0.01) & 18.01 (0.02) & TNG\
2009Apr24 & 2454946.44 & & & & 18.11 (0.11) & & BH$\star$\
2009Apr25 & 2454947.40 & & 18.94 (0.07) & 18.81 (0.06) & 18.40 (0.05) & 18.29 (0.05) & SO\
2009Apr29 & 2454951.43 & 19.24 (0.08) & 19.85 (0.03) & 19.67 (0.02) & 19.05 (0.03) & 19.13 (0.05) & CAHA\
2009May03 & 2454955.39 & & & 19.66 (0.03) & 19.07 (0.03) & 19.20 (0.06) & CAHA\
2009May05 & 2454956.55 & & & & 19.11 (0.03) & & NOT\
2009May08 & 2454959.67 & & & 18.54 (0.02) & 18.05 (0.02) & 18.13 (0.02) & P60\
2009May13 & 2454964.51 & 18.21 (0.03) & 19.00 (0.01) & 18.81 (0.01) & 18.41 (0.02) & 18.42 (0.02) & TNG\
2009May19 & 2454971.38 & 19.77 (0.05) & 20.25 (0.03) & 20.07 (0.03) & 19.51 (0.03) & 19.68 (0.04) & TNG\
2009May22 & 2454974.42 & & & & $>$18.73 & & EP\
2009May25 & 2454977.36 & & 19.91 (0.07) & 19.65 (0.05) & 19.18 (0.04) & 19.05 (0.06) & EK\
2009May28 & 2454980.35 & & & & 18.97 (0.06) & & EK\
2009Jul16 & 2455029.33 & & & & $>$18.36 & & EP\
2009Jul18 & 2455031.36 & & & $>$21.01 & $>$20.98 & & CAHA\
2009Jul22 & 2455035.35 & & & & $>$20.58 & & EK\
2009Oct19 & 2455122.69 & & & 21.17 (0.38) & 20.39 (0.12) & & EK\
2009Oct26 & 2455130.76 & & 21.61 (0.16) & 21.36 (0.07) & 20.18 (0.04) & 20.55 (0.06) & NOT\
2009Nov15 & 2455150.74 & & & & 20.00 (0.17) & & NOT\
2009Nov19 & 2455154.67 & & & 19.66 (0.03) & 19.23 (0.02) & & CAHA\
2009Nov20 & 2455155.74 & & & & 18.50 (0.02) & & NOT\
2009Nov21 & 2455156.62 & & 19.48 (0.07) & 19.29 (0.08) & 18.73 (0.05) & 18.71 (0.08) & CAHA\
2009Nov22 & 2455157.76 & & 21.26 (0.12) & 20.75 (0.11) & & & LT\
2009Nov27 & 2455162.61 & & 19.88 (0.07) & 19.65 (0.07) & 19.23 (0.03) & 19.18 (0.03) & CAHA\
[**2009Nov28**]{} & [**2455163.71**]{} & & [**18.72 (0.03)**]{} & [**18.53 (0.02)**]{} & & & LT\
2009Nov29 & 2455164.72 & & 18.83 (0.04) & 18.57 (0.03) & & & LT\
2009Nov30 & 2455165.68 & & 19.10 (0.03) & 18.93 (0.02) & & & LT\
2009Dec02 & 2455167.76 & & 19.33 (0.04) & 19.27 (0.12) & & & LT\
2010Jan19 & 2455215.61 & & & 20.93 (0.13) & 20.11 (0.15) & 19.75 (0.17) & EK\
2010Feb07 & 2455235.38 & & & & $>$20.30 & & SO$\star$\
2010Feb07 & 2455235.45 & & & & $>$20.00 & & MO$\star$\
2010Feb24 & 2455251.50 & & 21.71 (0.23) & 21.51 (0.22) & & & LT\
2010Feb24 & 2455252.47 & & 21.67 (0.31) & 21.49 (0.36) & & & LT\
2010Mar03 & 2455259.53 & & 22.42 (0.37) & 21.98 (0.32) & & & LT\
2010Mar19 & 2455274.69 & & & & 21.22 (0.19) & & NOT\
2010Mar21 & 2455277.41 & & 22.03 (0.21) & 21.92 (0.26) & & & LT$^i$\
$^a$ ING archive: [*http://casu.ast.cam.ac.uk/casuadc/archives/ingarch*]{}.\
$^b$ Subaru-Mitaka-Okayama-Kiso Archive System [@baba02].\
$^c$ Template image from the Supernova Search in Late-type Galaxies using the 0.37-m Rigel Robotic Telescope (Winer Observatory - University of Iowa; [*http://phobos.physics.uiowa.edu/images/snsearch.html*]{}).\
$^d$ Measurement from [@wag04].\
$^e$ Astrodon Generation 1 Series I Luminance filter.\
$^f$ Kryachko and Korotkiy estimated $R$ = 17.65 for 2008-OT with reference to star USNO-A2.0 1200-06602541 (USNO $R$ = 14.0), which is coincident with star D in the sequence of @wag04; $R$ = 13.924. The 2008-OT magnitude was rescaled accordingly.\
$^g$ Poor-quality night (clouds and poor seeing).\
$^h$ Additional UVOT detection limits: $UVW1 >$ 19.85, $UVW2 >$ 20.18, $UVM2 >$ 19.44.\
$^i$ Additional NIR detection limits were obntained with LT + SupIRCam: $J >$ 18.07 and $H >$ 17.03.\
\
INT = 2.54-m Isaac Newton Telescope + CCD-EEV5 (La Palma, Canary Islands, Spain);\
KISO = 1.05-m Schmidt Telescope + TC215-1K CCD (Kiso Observatory, Inst. of Astronomy, University of Tokyo, Japan);\
IRO = 0.37-m f/14 Rigel Telescope + FLI with SITe-003 CCD (Obs. Chris Anson, Winer Observatory, Sonoita, Arizona, US);\
KAIT = 0.76-m Katzman Automatic Imaging Telescope + CCD (Lick Observatory, Mt. Hamilton, California, US);\
TT = 0.4-m Meade LX200 Telescope + SBIG ST-8 CCD (Obs. T. Tamura, Kagawa, Japan);\
HALE = Hale 5.08-m Telescope + COSMIC direct imaging 2048x2048 SITe thinned CCD (Palomar Observatory, San Diego County, California, US);\
GY = Takahashi FS-128G Telescope + SBIG ST-10 Dual CCD Camera (Obs. G. Youman, Penryn, California, US);\
MO = 0.32-m f/3.1 Newton Telescope + Starlight Xpress MX716 CCD (Obs. G.D., Moonbase Observatory, Akersberga, Sweden);\
CD = 0.46-m f/2.8 Centurion Telescope + ST8E CCD camera (Obs. C. Deforeit, Caen, Normandie, France);\
RJ = 0.356-m f/10 LX200R Meade Telescope + SBIG STL-11000XM CCD camera (Obs. R. Johnson, Mantrap Lake, Minnesota, US);\
TKC = 0.3-m Ritchey-Chrétien Telescope + Apogee Alta U9000 CCD (Kazan State Univ., Astrotel Obs., Karachay-Cherkessia, Russia);\
KI = 0.60-m f/5.7 Reflector Telescope + KAF-1001E 1024$\times$1024pixel CCD (Obs. Koichi Itagaki, Yamagata, Japan);\
CAHA = 2.2-m Calar Alto Telescope + CAFOS (German-Spanish Astron. Center, Sierra de Los Filabres, Andalucía, Spain);\
TNG = 3.58-m Telescopio Nazionale Galileo + Dolores (Fundación Galileo Galilei-INAF, La Palma, Canary Islands, Spain);\
LT = 2.0-m Liverpool Telescope + RatCAM (La Palma, Canary Islands, Spain);\
SO = 0.44-m f/4.4 Newton Telescope + SBIG ST-7E Dual CCD (Obs. L.H., Sandvreten Obs., Uppsala, Sweden);\
HST = Hubble Space Telescope + WFPC camera (P.I. W. Li; proposal ID:10877);\
EK = 1.82-m Copernico Telescope + AFOSC (INAF - Osservatorio Astronomico di Asiago, Mt. Ekar, Asiago, Italy);\
NOT = 2.56m Nordic Optical Telescope + ALFOSC (La Palma, Canary Islands, Spain);\
UVOT = Swift Gamma-Ray Burst Explorer + UVOT (proposal ID 00036566001);\
BH = Meade LX200 12“ f-6.3 telescope + SBIG ST-10 Dual CCD Camera (Obs. B. Häusler, Rimpar, Germany);\
P60 = Palomar 60-inch Telescope + CCD camera (Palomar Mountain, San Diego County, California, US);\
EP = Meade LX200 12” f-6.3 telescope + ST10XME CCD (Obs. E. Prosperi, Osservatorio Remoto Skylive, Pedara, Catania, Italy).
Date JD u g r i z Source
----------- ------------ -------------- -------------- -------------- -------------- -------------- -------- --
2003Apr26 2452755.72 20.39 (0.09) 19.99 (0.04) 19.20 (0.03) 19.57 (0.08) 19.54 (0.10) SDSS
2004Mar16 2453080.89 $>$22.37 22.26 (0.29) 21.46 (0.13) 21.51 (0.17) 21.23 (0.37) SDSS
2008Nov09 2454779.69 21.87 (0.32) 20.67 (0.14) $>$21.37 $>$20.90 LT
2008Nov19 2454789.66 22.02 (0.24) 20.90 (0.10) 21.54 (0.19) $>$21.89 LT
2008Nov24 2454794.66 22.12 (0.20) 21.31 (0.11) 21.52 (0.10) LT
2008Dec10 2454810.61 18.79 (0.06) 19.17 (0.05) LT
2008Dec17 2454817.67 18.79 (0.02) 18.94 (0.03) LT
2008Dec22 2454822.55 18.99 (0.03) LT
2008Dec29 2454829.64 19.23 (0.02) 19.53 (0.03) LT
2009Jan05 2454836.60 19.31 (0.02) LT
2009Jan05 2454836.69 19.30 (0.03) 19.63 (0.03) LT
2009Jan13 2454844.63 19.43 (0.03) 19.94 (0.12) LT
2008Jan14 2454845.62 19.60 (0.03) 20.04 (0.03) LT
2008Jan20 2454851.54 19.58 (0.02) LT
2008Jan20 2454852.49 19.53 (0.05) 20.12 (0.06) LT
2008Jan21 2454852.56 19.57 (0.02) 20.18 (0.05) LT
2009Jan25 2454856.55 19.61 (0.02) 20.24 (0.04) LT
2009Jan29 2454860.51 19.62 (0.02) LT
2009Feb16 2454879.40 19.66 (0.06) LT
2009Feb16 2454884.44 19.71 (0.11) LT
2009Mar09 2454900.41 19.85 (0.09) LT
2009Mar23 2454914.38 20.08 (0.02) LT
2009Mar30 2454921.36 20.11 (0.03) LT
2009Apr06 2454928.36 20.13 (0.07) LT
2009Apr13 2454935.37 19.90 (0.03) LT
2009Apr21 2454943.35 18.24 (0.04) LT
2009Apr23 2454945.42 18.29 (0.01) LT
2009Apr28 2454950.46 19.05 (0.01) LT
2009Apr29 2454951.48 19.15 (0.02) LT
2009May06 2454957.53 18.33 (0.02) LT
2009May11 2454963.39 19.99 (0.02) LT
2009May14 2454966.39 19.79 (0.02) LT
2009May17 2454969.39 20.30 (0.04) LT
2009May20 2454972.38 19.37 (0.04) LT
2009May23 2454975.39 19.20 (0.02) LT
2009May26 2454978.39 19.35 (0.03) LT
2009May29 2454981.40 18.97 (0.03) LT
2009Jun09 2454992.43 18.95 (0.02) LT
2009Jun12 2454995.39 19.73 (0.04) LT
2009Jun14 2454997.39 20.37 (0.08) LT
2009Jun22 2455005.39 20.99 (0.05) LT
2009Jun23 2455006.43 21.11 (0.12) LT
2009Jun29 2455012.40 21.35 (0.10) LT
2009Jul08 2455021.39 21.31 (0.17) LT
2009Jul09 2455022.38 $>$21.18 LT
2009Jul12 2455025.40 21.68 (0.12) LT
2009Jul17 2455030.38 $>$21.37 LT
2009Jul18 2455031.38 $>$21.27 LT
2009Nov14 2455149.75 20.34 (0.04) LT
2009Nov22 2455157.75 19.96 (0.12) LT
2009Nov22 2455157.76 21.42 (0.19) 20.87 (0.12) 20.01 (0.08) 20.52 (0.18) 20.41 (0.34) LT
2009Nov24 2455159.75 20.82 (0.10) LT
2009Nov25 2455160.74 20.62 (0.06) LT
2009Nov26 2455161.67 20.70 (0.12) LT
2009Nov27 2455162.75 19.03 (0.02) LT
2009Nov28 2455163.71 18.42 (0.06) 18.61 (0.08) 18.32 (0.02) 18.48 (0.04) 18.59 (0.04) LT
2009Nov29 2455164.72 18.69 (0.03) 18.71 (0.09) 18.35 (0.03) 18.52 (0.04) 18.54 (0.04) LT
2009Nov30 2455165.68 18.88 (0.05) 18.96 (0.03) 18.61 (0.03) 18.90 (0.03) 18.81 (0.05) LT
2009Dec02 2455167.76 19.53 (0.09) 19.28 (0.06) 18.79 (0.05) 18.96 (0.07) 19.02 (0.13) LT
2009Dec09 2455174.61 19.93 (0.06) LT
2009Dec27 2455192.56 20.41 (0.24) LT
2010Feb24 2455251.50 $>$21.75 20.96 (0.10) 20.97 (0.17) 21.09 (0.22) LT
2010Feb24 2455252.47 $>$21.94 21.14 (0.22) 21.03 (0.16) 21.28 (0.33) LT
2010Mar04 2455259.53 $>$22.23 21.59 (0.35) 21.55 (0.37) 21.49 (0.46) LT
2010Mar21 2455277.41 $>$22.03 21.36 (0.13) 21.32 (0.32) $>$20.82 LT
SDSS = 2.5-m Sloan Digital Sky Survey Telescope + TK2048E CCDs (Apache Point Observatory, New Mexico, USA);\
LT = 2.0-m Liverpool Telescope + RatCAM (La Palma, Canary Islands, Spain).
[^1]: E-mail: [a.pastorello@qub.ac.uk]{}
[^2]: Many other transients have been proposed to be SN impostors, including SN 1999bw [@fil99; @li02], SN 2001ac [@mat01], SN 2003gm [@mau06], SN 2006fp [@blo06], SN 2007sv [@har07], SN 2009ip [@smi09b; @fol10]. In addition, the controversial SN 1961V, whose nature (SN explosion or LBV outburst) is still debated [@utr84; @fes85; @bra85; @cow88; @goo89; @fil95; @sto01; @van02; @chu04] deserves to be mentioned.
[^3]: The Local Group infall velocity here adopted is $v_{LG-infall}$ = 208 $\pm$ 9 km s$^{-1}$ [@ter02]
[^4]: http://leda.univ-lyon1.fr/
[^5]: Sloan Digital Sky Survey web site: [ *http://www.sdss.org/*]{}. The SDSS filter definition can be found in @fuk96, while the photometric system is defined in @smt02
[^6]: Historical observations of $\eta$ Car’s Great Eruption in the 1840s were kindly provided by @fre04, while the collection of the normalized V band photometric studies of the $\eta$ Car’s variability after 1952 are from S. Otero (private communication). Extensive light curves of $\eta$ Car can be found at the following URL address: [*http://varsao.com.ar/Curva$_{-}$Eta$_{-}$Carinae.htm*]{}
[^7]: Data from AAVSO International Database: [*http://www.aavso.org*]{}
[^8]: The identification of Ca II is unequivocal, being the line at $\lambda$ 3934Å unblended in the Nordic Optical Telescope spectrum of December 21st, 2008.
[^9]: Note, however, that only linear spectropolarimetry can unveil the real geometry of a stellar system. @vink09, indeed, recently questioned the disk-hypothesis for Oe stars due to the lack of polarimetric proofs.
[^10]: In the spectra of the variable obtained during the period 2000-2001, the FWHM velocities were slightly lower, between 1600 and 2100 km s$^{-1}$ in the post-outburst phases.
[^11]: There is a rebrightening in the light curve at day $\sim$60 after the peak of 2008-OT, clearly visible e.g. in Figure \[fig11\], middle-right panel.
[^12]: The variability of S18 is $<$1 mag, according to @vange02, while most of B\[e\]-type stars do not show significant magnitude oscillations.
[^13]: Data for SN 1994W were presented by [@sol98]. The SN was extensively modelled also by [@chug04] who interpreted its observed properties to result from the SN ejecta interacting with a dense and extended CS envelope ejected in a violent event of the progenitor star just $\sim$1.5 years before its explosion.
[^14]: Alternatively, [@tom08] proposed that the precursor of SN 2006jc was a more evolved WCO star.
[^15]: According to @foe08 we cannot exclude that the membership of [*star C*]{} to the stellar system HD 5980 is only a mere line-of-sight coincidence.
[^16]: We should note that @foe08 revised the classical mass estimates of @nie97 proposing significantly higher masses for both [*star A*]{} and [*star B*]{}, being 58-79 M$_\odot$ and 51-67 M$_\odot$, respectively.
|
---
abstract: 'Magnetic fields play an important role in almost all astrophysical phenomena including star formation. But due to the difficulty in analytic modeling and observation, magnetic fields are still poorly studied and numerical simulation has become a major tool. We have implemented a cosmological magnetohydrodynamics package for Enzo which is an AMR hydrodynamics code designed to simulate structure formation. We use the TVD solver developed by S. Li as the base solver. In addition, we employ the constrained transport (CT) algorithm as described by D. Balsara. For interpolation magnetic fields to fine grids we used a divergence free quadratic reconstruction, also described by Balsara. We present results from several test problems including MHD caustics, MHD pancake and galaxy cluster formation with magnetic fields. We also discuss possible applications of our AMR MHD code to first star research.'
author:
- Hao Xu
- 'David C. Collins'
- 'Michael L. Norman'
- Shengtai Li
- Hui Li
title: A cosmological AMR MHD module for Enzo
---
[ address=[Department of Physics and CASS, University of California, San Diego, San Diego, CA]{} ,altaddress=[Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM]{} ]{}
[ address=[Department of Physics and CASS, University of California, San Diego, San Diego, CA]{} ]{}
[ address=[Department of Physics and CASS, University of California, San Diego, San Diego, CA]{} ]{}
[ address=[Mathematical Modeling and Analysis, Los Alamos National Laboratory, Los Alamos, NM]{} ]{}
[ address=[Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM]{} ]{}
Introduction
============
Adaptive mesh refinement(AMR) cosmological hydrodynamics simulations play an important role in the study of structure formation of different scales from galaxy clusters to first stars in the past ten years. Its ability to achieve very high resolution in large scale simulations with relatively small computer resources has helped us to understand the first stars formed in our Universe. The possible effects of magnetic fields have been largely ignored. It is well established that magnetic fields are present on different scales, from intracluster medium to interstellar medium. The origin and evolution of these magnetic fields and their role on the structure formation are still unclear. So, it is imperative to include magnetic fields into current hydrodynamics AMR cosmology code. In this paper, we present the newly developed MHD version of the Enzo, which is wildly used in the study of first stars [@Abel02; @O'Shea05; @O'Shea07].
MHD with ENZO
=============
The MHD equations in the comoving coordinates are:
$$\begin{aligned}
{}
\frac{\partial \rho}{\partial t} + \frac{1}{a} \nabla \cdot ( \rho \bf{v}) & = & 0 \\
\frac{\partial \rho \bf{v}}{\partial t} + \frac{1}{a} \nabla \cdot (\rho \bf{v}\bf{v} + \bar{p} - \bf{B}\bf{B}) & = & - \frac{\dot{a}}{a} \rho \bf{v}- \frac{1}{a}\rho \nabla \phi \\
\frac{\partial E}{\partial t} + \frac{1}{a}\nabla \cdot [\bf{v}(\bar{p}+E)-\bf{B}(\bf{B}\cdot \bf{v})] & = & - \frac{\dot{a}}{a}(\rho v^{2}+3p+\frac{B^{2}}{2}) \nonumber \\
& & - \frac{\rho}{a} \bf{v}\cdot \nabla \phi \\
\frac{\partial \bf{B}}{\partial t} - \frac{1}{a} \nabla \times (\bf{v} \times \bf{B}) & = & -\frac{\dot{a}}{2a}\bf{B}\end{aligned}$$
with $$\begin{aligned}
E & = & \frac{1}{2} \rho v^2 + \frac{p}{\gamma-1} + \frac{1}{2}B^2 \\
\bar{p} & = & p + \frac{1}{2} B^2\end{aligned}$$ where all variables have their usual meaning, a is the expansion parameter. To track the pressure more accurately in the supersonic region, we have also implemented the modified entropy equation given in @Ryu93 and the internal energy equation given in @Bryan95 in our code.
The MHD solver used for all the test problems here is a high-order Godunov-type finite-volume numerical solver developed by S.T. Li [@Li03; @Li05]. This solver was recently successfully used to study magnetic jet problems [@Li06; @Nakamura06; @Nakamura07].
We used a constrained transport(CT) scheme flux CT [@Balsara99] to maintain divergence-free magnetic fields. For the AMR hierarchy in Enzo, we used a modified divergence-free reconstruction scheme original proposed by @Balsara01 including second order accurate divergence-free restriction and prolongation for magnetic fields. Details of the CT and AMR of magnetic fields in the MHD Enzo can be found in @Collins07.
The MHD module has been tested and shown to be compatible with other physics packages installed in Enzo, such as radiative cooling, star formation and feedback.
Tests
=====
MHD Caustic and Pancake
-----------------------
The MHD Caustic test is taken from @Li07 which generalizes the HD test of [@Ryu93]. The initial sinusoidal velocity field in the x-direction has the peak value $1/2\pi$, the initial density and pressure has been set to be uniform with $\rho=1$ and $p=10^{-10}$. Then caustics will be formed because of the compression by the velocity field. An initial uniform magnetic field of $10^{-3}$ in code units in the y direction was added to the simulation. Figure \[fig:caustic\] compares the density and $B_y$ at $t=3$ of unigrid and AMR runs. AMR run had 256 cells in the root grid and 2 level refinements by 2. The AMR solution is indistinguishable from a uniform grid solution with 1024 cells.
Pancake is another standard test problem of cosmological hydrodynamics simulation [@Ryu93]. We have run the collapse of a one-dimension pancake in a purely baryonic universe with $\Omega=1$ and $h=\frac{1}{2}$. Initially, at $a_{i}=1$, which corresponds to $z_{i}=20$ in this test, a sinusoidal velocity field with the peak value $0.65/(1+z_{i})$ in the normalized unit has been imposed in a box with the comoving size $64h^{-1}Mpc$, so that the shock forms at $z=1$. The initial baryonic density and pressure have been set to be uniform with $\rho=1$ and $p=6.2\times 10^{-8}$ in the normalized code units. We applied initial uniform magnetic fields $B_y = 2.0 \times 10^{-5}G,~B_x=B_z=0$ in the simulation. We did the calculations with unigrid with 1024 cells and AMR run with 256 cells of root grid and 2 level refinements. Figure \[fig:pancake\] shows the density and $B_y$ at $z=0$.
![Plots of density and y component of magnetic fields of MHD caustics at t=3. The initial magnetic field is $10^{-3}$.[]{data-label="fig:caustic"}](crefinedensityb3.jpg "fig:"){width=".23\textwidth"} ![Plots of density and y component of magnetic fields of MHD caustics at t=3. The initial magnetic field is $10^{-3}$.[]{data-label="fig:caustic"}](crefinebyb3.jpg "fig:"){width=".23\textwidth"}
![Plots of density and y component of magnetic fields of MHD pancake.The initial magnetic field is $2 \times 10^{-5}G.$[]{data-label="fig:pancake"}](zrefinedensityb75.jpg "fig:"){width=".23\textwidth"} ![Plots of density and y component of magnetic fields of MHD pancake.The initial magnetic field is $2 \times 10^{-5}G.$[]{data-label="fig:pancake"}](zrefinebyb75.jpg "fig:"){width=".23\textwidth"}
Cluster Formation with Magnetic Fields
--------------------------------------
Cluster formation is one of the problems most widely studied by Enzo [@Norman05]. We have done this problem and compared with results from Enzo-ppm to test our new code in large scale structure formation. The simulation is a $\Lambda$CDM model with parameters $h=0.7$, $\Omega_{m}=0.3$, $\Omega_{b}=0.026$, $\Omega_{\Lambda}=0.7$, $\sigma_{8}=0.928$. The survey volume is 256 $h^{-1}$ Mpc on a side. The simulations were computed from a $128^3$ root grid and 2 level nested static grids in the Lagrangian region where the cluster forms which gives an effective root grid resolution of $512^3$ cells (0.5 $h^{-1}$ Mpc) and dark matter particles of mass $1.49 \times 10^{10} M_\odot$. AMR is allowed only in the region where the galaxy cluster forms, with a total of 8 levels of refinement beyond the root grid, for a maximum spatial resolution of 7.8125 $h^{-1}$kpc.
We first present the results of no initial magnetic fields and compare them with the results from Enzo-ppm. The cluster parameters from the MHD code is almost identical to those from Enzo-ppm. The virial radii are 2.229Mpc from hydro and 2.226Mpc from MHD while the virial masses are $1.265 \times 10^{15} M_\odot$ for hydro and $1.260 \times 10^{15} M_\odot$ for MHD. Figure \[fig:cluster2\] compare the projections of the baryon density and temperature.
![Logarithmic projected gas density(top) and logarithmic projected X-ray weighted temperature(down) at $z=0$ of adiabatic simulations. The images cover the inner 4 $h^{-1}$Mpc of cluster centers. The left panels show results from the PPM solver and the right panels show results from the MHD solver. The units of density and temperature are $M_\odot /Mpc^3$ and Kelvin respectively.[]{data-label="fig:cluster2"}](density_temperature_bw.jpg){width=".45\textwidth"}
We have performed simulation with initial magnetic fields, $B_x=B_z=0$, $B_y = 1.0^{-9}G$ with radiative cooling, star formation and star formation feedback. Figure \[fig:cluster4\] shows the images of baryon density, temperature, magnetic energy density and Faraday rotation measurement of the cluster center. The rotation measurement is integral along the projection direction.
![Images of the baryon density (logarithmic, $M_\odot /Mpc^3$), temperature (logarithmic, Kelvin), magnetic energy density (logarithmic, $erg/cm^3$) and rotation measurement ($rad/m^2$) of the inner 1$h^{-1}$Mpc of cluster center. The initial magnetic field is $B_y = 1.0^{-9}G$ at $z=0$.[]{data-label="fig:cluster4"}](by=-7_feedback_bw.jpg){width=".45\textwidth"}
Another simulation we performed is without initial magnetic field but with the Biermann battery effect. The induction equation is modified by adding an additional battery term[@Kulsrud97]:
$$\begin{aligned}
\frac{\partial \bf{B}}{\partial t} & = & \nabla \times (\bf{v} \times \bf{B}) + \nabla \times (\frac{c \nabla p_e}{n_e e}) \\
& = & \nabla \times (\bf{v} \times \bf{B}) + \frac{c m_H}{e} \frac{1}{1+\chi} \nabla \times (\frac{p}{\rho})\end{aligned}$$
where c is speed of light, $p_e$ is the pressure of electron, $n_e$ is the electron number density, e is the electron charge, $m_H$ is the hydrogen mass and $\chi$ is the ionization fraction. We took $\chi = 1$ constant in space in our simulation. We performed this computation with radiative cooling. Figure \[fig:cluster5\] shows the projection of logarithmic baryon density and the magnetic energy density of the cluster center.
![Images of the baryon density (logarithmic, $M_\odot/Mpc^3$) and magnetic energy density (logarithmic, $erg/cm^3$) of the simulation with Biermann battery effect of the inner 1$h^{-1}$Mpc of cluster center. []{data-label="fig:cluster5"}](biermann_cooling_bw.jpg){width=".45\textwidth"}
Discussion
==========
We have introduced our MHD module for Enzo and presented some test results. With this new module, we have the ability to perform AMR MHD cosmology simulations. This module uses high accuracy TVD MHD solver and employs CT and AMR divergence-free magnetic fields reconstruction scheme to guarantee divergence-free of magnetic fields. MHD simulations using this MHD module can use all the exist physics packages in Enzo.
It is widely believed that the magnetic fields should have little effects in the formation of the first stars, since maybe there were no magnetic fields at all. But even there are no magnetic fields from the early universe, the Biermann battery effect should have generated some very small fields during the collapse to form those stars. If the first generation stars spin very fast, these small seed fields could be maintained and amplified by dynamo effect. Then the magnetic fields in the first stars may pollute the environment by their explosion and play a part in the formation of next generation stars. This would be investigated in the near future.
This research was supported by IGPP at Los Alamos National Laboratory.
[9]{}
Abel, T., Bryan, G. & Norman, M.L. 2002, Science, 295, 93
Balsara, D. 2001, Journal of Computational Physics, 174, 614
Balsara, D. & Spicer, D., 1999, J. Comput. Phys., 149, 270
Bryan, G. et al., 1995, Comp. Phys., 89, 149
Collins, D. et al. 2007, in preparation
Kulsrud, R. M. et al., 1997, ApJ, 480, 481
Li, S & Li, H. 2003, Technical Report, Los Alamos National Laboratory
Li, S., 2005 J. Comput. Phys., 203,344
Li, H., Lapenta, G., Finn J.M., Li, S. & Colgate, S. A., 2006, ApJ, 643, 92
Li, S et al.,2007, ApJS, Accepted
Norman, M.L., 2005, Proc. Int. Sch. Phys. IOS, 1
Nakamura, M., Li, H. & Li, S., 2006, ApJ, 652, 1059
Nakamura, M., Li, H. & Li, S., 2007, ApJ, 656, 721
O’Shea, B. et al., ApJ., 2005, 628, L5
O’Shea, B. & Norman, M.L., 2007, ApJ, 654,66
Ryu, D. et al.,1993, ApJ, 414, 1
Xu, H. et al. in preparation
|
---
abstract: 'We study the Gowers uniformity norms of functions over ${\mathbf{Z}}/p{\mathbf{Z}}$ which are trace functions of $\ell$-adic sheaves. On the one hand, we establish a strong inverse theorem for these functions, and on the other hand this gives many explicit examples of functions with Gowers norms of size comparable to that of “random” functions.'
address:
- |
Université Paris Sud, Laboratoire de Mathématique\
Campus d’Orsay\
91405 Orsay Cedex\
France
- |
ETH Zürich – D-MATH\
Rämistrasse 101\
8092 Zürich\
Switzerland
- 'EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland '
author:
- Étienne Fouvry
- Emmanuel Kowalski
- Philippe Michel
date: ', '
title: 'An inverse theorem for Gowers norms of trace functions over ${\mathbf{F}}_p$'
---
Introduction
============
The *Gowers uniformity norms* were introduced by Gowers in his work on Szemerédi’s theorem. As one sees from the definition (see [@tv Def. 11.2]), these norms (or a suitable power of them) have very algebraic definitions when applied to functions defined over a finite abelian group. In particular, one may consider a finite field $k$, and attempt to understand the Gowers norm of functions of algebraic nature on $k$. The most natural definition of such functions seems to be the *trace functions* of suitable sheaves, as we will recall below. Indeed, in recent works [@fkm; @fkm-counting; @fkm-primes], we have shown that such functions (in the special case $k={\mathbf{F}}_p$) can be exploited powerfully in analytic arguments of various types (amplification method for averages against Fourier coefficients of modular forms, bilinear forms for averages over primes, etc).
In this note, we consider the Gowers norms of trace functions. Maybe the most crucial issue in the study of these norms has been to understand which bounded functions have “large” Gowers norm, a suitable structural answer being known as an “inverse theorem” for these norms. As it turns out, the rigidity of the structure of trace functions, and especially Deligne’s proof of the Riemann Hypothesis, also leads to a rather precise structure theorem for Gowers norms of trace functions over ${\mathbf{F}}_p$.
Although this was not anticipated at first,[^1] it also turns out that the estimates we obtain give many simple explicit examples of functions with Gowers norms of size comparable to that of “random” functions, in a precise sense recalled below. Since this fact may be of interest to people interested in pseudorandomness measures of various sequences (see also, among others, the papers [@niederreite-rivat] of Niederreiter and Rivat, [@liu] of Liu and [@fmrs] of Fouvry, Michel, Rivat and Sárközy), we first state a concrete result which does not require any advanced algebraic-geometry language. In the statement, $\|\cdot\|_{U_d}$ is the $d$-th Gowers uniformity norm (normalized as in [@tv Def. 11.2]), the definition of which is recalled in Section \[sec-prelims\].
\[th-explicit\] For an odd prime $p$, and $x\in{\mathbf{Z}}/p{\mathbf{Z}}$, let $$\begin{aligned}
\varphi_1(x)&=\Bigl(\frac{f(x)}{p}\Bigr)\\
\varphi_2(x)&=e\Bigl(\frac{\bar{x}}{p}\Bigr)\text{ if $x\not=0$ and
}
\varphi_2(0)=0,\\
\varphi_3(x)&=\frac{S(x,1;p)}{\sqrt{p}} \text{ if $x\not=0$ and }
\varphi_3(0)=-\frac{1}{\sqrt{p}},\\
\varphi_4(x)&=\frac{1}{\sqrt{p}}\Bigl(p-|\{(u,v)\in{\mathbf{F}}_p^2\,\mid\,
v^2=u(u-1)(u-x)\}|\Bigr)\text{ if $x\notin \{0,1\}$ and }\\
\varphi_4(0)&=\varphi_4(1)=\frac{1}{\sqrt{p}},\nonumber\end{aligned}$$ where $f\in{\mathbf{Z}}[X]$ has degree $m{\geqslant}1$ and is not proportional to the square of another polynomial, $\bigl(\tfrac{\cdot}{p}\bigr)$ is the Legendre symbol, and $S(a,b;c)$ is a classical Kloosterman sum. Then for $d{\geqslant}1$, we have $$\begin{aligned}
\|\varphi_1\|_{U_d}^{2^d}&{\leqslant}(5m+10)^{(d+1)2^d}p^{-1},
\\
\|\varphi_2\|_{U_d}^{2^d}&{\leqslant}15^{(d+1)2^d}p^{-1},
\\
\|\varphi_3\|_{U_d}^{2^d}&{\leqslant}20^{(d+1)2^d}p^{-1},
\\
\|\varphi_4\|_{U_d}^{2^d}&{\leqslant}25^{(d+1)2^d}p^{-1}.\end{aligned}$$
In [@tv Ex. 11.1.17], Tao and Vu note that if $\varphi$ is a *random function*, in the sense that the values $\varphi(x)$, for $x\in{\mathbf{F}}_p$, are independent random variables (on some probability space) with $|\varphi(x)|{\leqslant}1$ for all $x$ and with expectation zero, then we have $${\mathbf{E}}(\|\varphi\|_{U_d}^{2^d})\ll p^{-1},$$ where the implied constant depends only on $d$. Thus this result gives concrete examples of functions which are as uniform as random functions (note that, by Weil’s bound for Kloosterman sums and by Hasse’s bound for the number of points on elliptic curves over finite fields, we have $|\varphi_3|$, $|\varphi_4|{\leqslant}2$). We are not aware of previous examples with this property in the literature (though the cases of $\varphi_1$ and $\varphi_2$ are accessible to techniques based only on the Weil bounds for character sums.)
We will explain the proof of this result in Section \[sec-refined\]. We now discuss the inverse theorems for general trace functions. We first recall the general setup of these functions (further examples are given in Example \[ex-simple-examples\] and in Section \[sec-examples\]).
We fix a prime $p$ and a finite field $k$ of characteristic $p$. Let $\ell\not=p$ be a prime number. For any algebraic variety $X/k$, any finite extension $k'/k$ and $x\in X(k')$, we denote by ${t_{{{{\mathcal{{F}}}},{k'}}}({x})}$ the value at $x$ of the trace function of some $\ell$-adic (constructible) sheaf ${\mathcal{{F}}}$ on $X/k$. We will write ${t_{{{{\mathcal{{F}}}},{k'}}}}$ for the function $x\mapsto
{t_{{{{\mathcal{{F}}}},{k'}}}({x})}$ defined on $X(k')$.
We will always assume that some isomorphism $\iota\,:\,
\bar{{\mathbf{Q}}}_{\ell}{\longrightarrow}{\mathbf{C}}$ has been chosen and we will allow ourselves to use it as an identification. Thus, for instance, by $|{t_{{{{\mathcal{{F}}}},{k}}}({x})}|^2$, we will mean $|\iota({t_{{{{\mathcal{{F}}}},{k}}}({x})})|^2$.
Given any finite field $k$ and any function $\varphi\,:\, k{\longrightarrow}\bar{{\mathbf{Q}}}_{\ell}$ or $\varphi\,:\, k{\longrightarrow}{\mathbf{C}}$, we denote $$U_d(\varphi)=\|\varphi\|_{U_d}^{2^d}$$ where $\|\cdot\|_{U^d}$ is the $d$-th uniformity norm, and $$U_d({\mathcal{{F}}};k)=U_d({t_{{{{\mathcal{{F}}}},{k}}}})$$ (in fact, we will call $U_d(\varphi)$ the Gowers $d$-pnorm of $\varphi$ – the ‘p’ is silent, as in “ptarmigan” or “Psmith” – to avoid confusion.)
We will work mostly with middle-extension sheaves, in the sense of [@katz-gkm], i.e., constructible sheaves ${\mathcal{{F}}}$ on ${\mathbf{A}}^1/k$ such that, for any open set $U$ on which ${\mathcal{{F}}}$ is lisse, with open immersion $j\,:\, U{\hookrightarrow}{\mathbf{A}}^1$, we have $${\mathcal{{F}}}\simeq j_*j^*{\mathcal{{F}}}.$$
Given any constructible sheaf ${\mathcal{{F}}}$, lisse on an open set $U\subset {\mathbf{A}}^1$, with $j\,:\, U{\hookrightarrow}{\mathbf{A}}^1$ the open immersion, the direct image $j_*{\mathcal{{F}}}$ is the unique middle-extension sheaf on ${\mathbf{A}}^1$ which is isomorphic to ${\mathcal{{F}}}$ on $U$. In particular ${\mathcal{{F}}}$ and $j_*{\mathcal{{F}}}$ have the same trace functions on $U$, but those may differ at the singularities ${\mathbf{A}}^1-S$. Thus the middle-extension condition can be seen as ensuring that a lisse sheaf on an open set $U$ of ${\mathbf{A}}^1$ is extended “optimally” to all of ${\mathbf{A}}^1$.
As in [@katz-esde §7], a middle-extension sheaf as above is called pointwise pure of weight $0$, if $j^*{\mathcal{{F}}}$ is pointwise pure of weight $0$ on $U$, and it is called arithmetically irreducible (resp. semisimple, resp. geometrically irreducible, geometrically semisimple) if $j^*{\mathcal{{F}}}$ corresponds to an irreducible (resp. semisimple) representation of the fundamental group $\pi_1(U,\bar{\eta})$ (resp. of the geometric fundamental group $\pi_1(U\times\bar{k},\bar{\eta})$), for some geometric generic point $\bar{\eta}$ of $U$. By the *semisimplification* of ${\mathcal{{F}}}$, we mean the middle-extension sheaf $$j_*{\mathcal{{F}}}^{ss}$$ where ${\mathcal{{F}}}^{ss}$ is the semisimplification of the restriction of ${\mathcal{{F}}}$ to $U$.
Note that $${t_{{{{\mathcal{{F}}}},{k}}}}={t_{{{{\mathcal{{F}}}^{ss}},{k}}}},$$ so that for any question involving the trace function of ${\mathcal{{F}}}$, we may assume that the sheaf is arithmetically semisimple.
We measure the complexity of a sheaf on ${\mathbf{P}}^1$ over a finite field by its *conductor*: if ${\mathcal{{F}}}$ is such a sheaf, of rank $\operatorname{rank}({\mathcal{{F}}})$ with singularities at $\operatorname{Sing}({\mathcal{{F}}})\subset
{\mathbf{P}}^1$, we define the *(analytic) conductor* of ${\mathcal{{F}}}$ to be $$\label{eq-conductor}
\operatorname{c}({\mathcal{{F}}})=\operatorname{rank}({\mathcal{{F}}})+
\sum_{x\in \operatorname{Sing}({\mathcal{{F}}})}{\max(1,\operatorname{Swan}_x({\mathcal{{F}}}))}.$$
An important subclass of sheaves is that of *tamely ramified* sheaves, which by definition are those where $\operatorname{Swan}_x({\mathcal{{F}}})=0$ for all $x$, so that only the rank and number of singularities appear as measures of complexity.
If ${\mathcal{{F}}}$ is a sheaf on $U\subset {\mathbf{A}}^1\subset {\mathbf{P}}^1$, the conductor is defined as that of the direct image to ${\mathbf{P}}^1$ (i.e., the Swan conductor at any point $x\in{\mathbf{P}}^1$ is that of the invariants under inertia at $x$ of the fiber of ${\mathcal{{F}}}$ over a generic geometric point.)
We now state a first version of our main structural result (see Theorem \[th-precise\] for a more precise form from which it will be deduced; for technical reasons, we are currently only able to treat fully the case of prime fields $k={\mathbf{Z}}/p{\mathbf{Z}}$, which is the most directly relevant to analytic number theory.)
\[th-inverse\] Let $p$ be a prime number and let $\ell\not=p$ be an auxiliary prime. Let $d{\geqslant}1$ be an integer such that $p>d$.
Let ${\mathcal{{F}}}$ be a middle-extension $\ell$-adic sheaf on ${\mathbf{A}}^1/{\mathbf{F}}_p$ which is pointwise pure of weight $0$ and arithmetically semisimple.
Then we can write $${t_{{{{\mathcal{{F}}}},{{\mathbf{F}}_p}}}}=t_1+t_2$$ where $t_1$ and $t_2$ are themselves trace functions and:
– We have $$\label{eq-bound-inverse}
U_d(t_1){\leqslant}(5\operatorname{c}({\mathcal{{F}}}))^{(d+1)2^d}p^{-1};$$
– There exists some non-negative integer $j{\leqslant}\operatorname{rank}({\mathcal{{F}}}){\leqslant}\operatorname{c}({\mathcal{{F}}})$, polynomials $P_i\in {\mathbf{F}}_p[X]$ of degree at most $d-1$ and coefficients $\beta_i$ bounded by $\operatorname{rank}({\mathcal{{F}}})$ for $1{\leqslant}i{\leqslant}j$, such that $$t_2(x)=\sum_{i=1}^{j}{\beta_i e\Bigl(\frac{P_i(x)}{p}\Bigr)}.$$
Note that the condition $p>d$ is certainly not a problem in this horizontal direction, where $p$ is the main variable and we think of having sheaves ${\mathcal{{F}}}_p$ for every $p$ with conductor uniformly bounded as $p$ varies (or even growing not too fast).
The more precise structural results will imply a particularly strong inverse theorem if ${\mathcal{{F}}}$ is assumed to be geometrically irreducible (and also imply Theorem \[th-explicit\]).
\[cor-inverse\] Let $p$, $\ell$ and ${\mathcal{{F}}}$ be as in the theorem, and assume that ${\mathcal{{F}}}$ is geometrically irreducible. For $d<p$, exactly one of the following two possibilities holds:
– There exists $P\in {\mathbf{F}}_p[X]$ of degree ${\leqslant}d-1$ and a complex number $\alpha$ of modulus $1$ such that $$\label{eq-strong-correlation}
{t_{{{{\mathcal{{F}}}},{{\mathbf{F}}_p}}}({x})}=\alpha e\Bigl(\frac{P(x)}{p}\Bigr)$$ for all $x\in {\mathbf{F}}_p$;
– Or we have $$U_d({\mathcal{{F}}};{\mathbf{F}}_p){\leqslant}(5\operatorname{c}({\mathcal{{F}}}))^{(d+1)2^d}p^{-1}.$$
Geometric irreducibility, for sheaves ${\mathcal{{F}}}$ with small conductor, is equivalent with approximate $L^2$-normality of the trace function over ${\mathbf{F}}_p$, i.e., with the condition $$\frac{1}{p}\sum_{x\in{\mathbf{F}}_p}{
|{t_{{{{\mathcal{{F}}}},{{\mathbf{F}}_p}}}({x})}|^2
}\approx 1$$ (see [@fkm-counting Lemma 3.5] for a precise statement of orthonormality for trace functions.)
The case of tamely ramified sheaves is also simpler since non-trivial Artin-Schreier sheaves are not tame. In fact, the technical difficulty in extending Theorem \[th-inverse\] to all finite fields, and the condition $d<p$, are then removed:
\[cor-tame\] Let $k$ be a finite field of characteristic $p$, let $\ell\not=p$ be given and let ${\mathcal{{F}}}$ be a tamely ramified $\ell$-adic middle-extension sheaf on ${\mathbf{A}}^1_k$ which does not geometrically contain the trivial sheaf. For $d{\geqslant}1$, we have $$U_d({\mathcal{{F}}};k){\leqslant}(5\operatorname{c}({\mathcal{{F}}}))^{(d+1)2^d}|k|^{-1}.$$
Using the triangle inequality, these results of course extend to “small” linear combinations of trace functions, which include the (centered) characteristic functions of sets with algebraic structure (e.g., values of polynomials, or definable sets in the language of rings [@cvm]). We study in [@fkm-norms] some aspects of the norms on functions over finite fields which arise naturally from this point of view.
A number of authors have proved inverse theorems for Gowers norms of functions over finite fields, the most general version being the one of Tao and Ziegler [@tao-ziegler-2] (see also, without exhaustivity, the paper [@green-tao] of Green and Tao, and the earlier paper [@tao-ziegler] of Tao and Ziegler; note also that, as shown in [@green-tao], and independently discovered by Lovett, Meshulam and Samorodnitsky, there do exist counterexamples in large characteristic to the most naive guess for an inverse theorem.) The focus in these papers is different: they consider arbitrary functions on $k^n$ as $n$ grows, and the finite field $k$ is fixed (in [@green-tao], the functions involved are themselves polynomials).
The arguments in all these works are much more delicate than the ones of the present paper, and this applies even more to the article [@green-tao-ziegler] of Green, Tao and Ziegler which establishes an inverse theorem for the Gowers norms on (in effect) ${\mathbf{Z}}/N{\mathbf{Z}}$, where the main variable is indeed $N{\rightarrow}+\infty$. There also, polynomial phases do not give the only obstruction to having small Gowers norms, and more general objects related to nilmanifolds are required.
Our proof relies instead on the formalism of algebraic geometry and on the Riemann Hypothesis over finite fields. Thus this note is another illustration of the great power of Deligne’s results, and of the interest in dealing with trace functions of arbitrary sheaves as objects of interest, and tools, in analytic number theory (a point of view which is already apparent in [@fkm; @fkm-counting; @fkm-primes]).
\[ex-simple-examples\] Possibly the simplest examples of trace functions modulo $p$ are given by $$\varphi(x)=e\Bigl(\frac{P(x)}{p}\Bigr)$$ where $P\in{\mathbf{Z}}[X]$ is a polynomial. In that case, the associated sheaf is a so-called Artin-Schreier sheaf, denoted ${\mathcal{{L}}}_{\psi(P)}$ for a suitable additive character $\psi$, and has rank $1$. It is therefore geometrically irreducible. Its conductor is $1+\deg(P)$, hence our theorem states that, for $p>\deg(P)$, we have $$\|\varphi\|_{U_d}{\leqslant}(5(1+\deg(P)))^{d+1}p^{-2^{-d}}$$ for all $d{\leqslant}\deg(P)$, where the implied constant depends only on $\deg(P)$ (on the other hand, it is easy to check that $\|\varphi\|_{U_d}=1$ if $d>\deg(P)$). The reader may check that in this special case, our proof can be expressed using only Weil’s theory of character sums in one variable.
In [@tv Ex. 11.1.12], Tao and Vu observe that one can prove elementarily the estimates $$\|\varphi\|_{U_d}{\leqslant}\Bigl(\frac{d-1}{p}\Bigr)^{2^{-\deg(P)}}$$ for $1{\leqslant}d{\leqslant}\deg(P)$, which is weaker in terms of $p$, except if $d=\deg(P)$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Thanks to R. Pink for help with questions concerning $\ell$-adic cohomology, and to J. Wolf for interesting discussions concerning Gowers norms. Thanks to F. Jouve for his remarks and comments concerning the manuscript, and thanks also to B. Green for suggesting the inclusion of a concrete statement like Theorem \[th-explicit\].
Notation {#notation .unnumbered}
--------
As usual, $|X|$ denotes the cardinality of a set, and we write $e(z)=e^{2i\pi z}$ for any $z\in{\mathbf{C}}$. We write ${\mathbf{F}}_p={\mathbf{Z}}/p{\mathbf{Z}}$.
By $f\ll g$ for $x\in X$, or $f=O(g)$ for $x\in X$, where $X$ is an arbitrary set on which $f$ is defined, we mean synonymously that there exists a constant $C{\geqslant}0$ such that $|f(x)|{\leqslant}Cg(x)$ for all $x\in
X$. The “implied constant” refers to any value of $C$ for which this holds. It may depend on the set $X$, which is usually specified explicitly, or clearly determined by the context. We write $f(x)\asymp
g(x)$ to mean $f\ll g$ and $g\ll f$.
For a constructible sheaf ${\mathcal{{F}}}$ on ${\mathbf{A}}^1/k$, and $h\in k$, we write $[+h]^*{\mathcal{{F}}}$ for the pullback of ${\mathcal{{F}}}$ under the map $x\mapsto x+h$. If ${\mathcal{{F}}}$ is a middle-extension sheaf on ${\mathbf{A}}^1/k$, we also write $\operatorname{D}({\mathcal{{F}}})$ for the middle-extension dual of ${\mathcal{{F}}}$, i.e., given a dense open set $j\,:\, U{\hookrightarrow}{\mathbf{A}}^1$ where ${\mathcal{{F}}}$ is lisse, we have $$\operatorname{D}({\mathcal{{F}}})=j_*((j^*{\mathcal{{F}}})'),$$ where the prime denotes the lisse sheaf of $U$ associated to the contragredient of the representation of the fundamental group of $U$ which corresponds to $j^*{\mathcal{{F}}}$ (see [@katz-esde 7.3.1]). If ${\mathcal{{F}}}$ is pointwise pure of weight $0$, it is known that $${t_{{{\operatorname{D}({\mathcal{{F}}})},{k'}}}({x})}=\overline{{t_{{{{\mathcal{{F}}}},{k'}}}({x})}}$$ for all finite extensions $k'/k$ and all $x\in k'$ (this property is obvious for $x\in U(k')$ and the point is that this extends to the singularities when the dual is suitably defined.) Note for instance that $\operatorname{c}(\operatorname{D}({\mathcal{{F}}}))=\operatorname{c}({\mathcal{{F}}})$.
Preliminaries {#sec-prelims}
=============
We recall the inductive definition of the Gowers norms, as in [@tv Def. 11.2] (one can sheafify it for trace functions, i.e., one can see the Gowers norm of ${t_{{{{\mathcal{{F}}}},{k}}}}$ as essentially the sum over $x\in k$ of the trace function of a suitable “Gowers sheaf” $\mathcal{U}_d({\mathcal{{F}}})$; it might in fact be interesting to search for some kind of “motivic” inverse theorem for these Gowers sheaves, but we will not pursue this point of view in this paper.)
Let $k$ be a finite field and $\varphi\,:\, k{\longrightarrow}{\mathbf{C}}$ an arbitrary function. The $U_d$-norms of $\varphi$ are defined inductively for $d{\geqslant}1$ by $$\|\varphi\|_{U_1}^2=\frac{1}{|k|^2}\sum_{h\in k}{\sum_{x\in k}{
\varphi(x+h)\overline{\varphi(x)} }},$$ and $$\|\varphi\|_{U_{d+1}}^{2^{d+1}}=
\frac{1}{|k|}\sum_{h\in k}{
\|\xi_h(\varphi)\|_{U_d}^{2^d}}$$ for $d{\geqslant}1$, where $$\xi_h(\varphi)(x)=\varphi(x+h)\overline{\varphi(x)}.$$
We wish to apply this recursive definition to trace functions. We first observe the trivial bound: recalling that $$|{t_{{{{\mathcal{{F}}}},{k}}}({x})}|{\leqslant}\operatorname{rank}({\mathcal{{F}}})$$ for all $x\in k$ if ${\mathcal{{F}}}$ is a middle-extension sheaf on ${\mathbf{A}}^1/k$ which is pointwise pure of weight $0$, we get:
\[cor-trivial\] Let ${\mathcal{{F}}}$ be pointwise pure of weight $0$ and rank $r{\geqslant}1$. Then for $d{\geqslant}1$, we have $$\label{eq-trivial-pnorm}
0{\leqslant}U_d({\mathcal{{F}}};k){\leqslant}r^{2^d}{\leqslant}\operatorname{c}({\mathcal{{F}}})^{2^d}.$$
Now we get a sheaf-theoretic interpretation of the recursive definition of Gowers norms. Given a middle-extension sheaf ${\mathcal{{F}}}$ and $h\in k$, the trace function of the constructible sheaf $$\xi_h({\mathcal{{F}}})=[+h]^*{\mathcal{{F}}}\otimes\operatorname{D}({\mathcal{{F}}})$$ coincides with $$\xi_h({t_{{{{\mathcal{{F}}}},{k}}}}),$$ “almost everywhere”. Precisely it does at any $x$ which is not a singularity of either $[+h]^*{\mathcal{{F}}}$ or $\operatorname{D}({\mathcal{{F}}})$ (this is clear if $x$ is a singularity for neither of these, and easy to see when $x$ is a singularity of one of them only). In fact, denoting by $S$ the set of singularities of ${\mathcal{{F}}}$ in ${\mathbf{A}}^1$, we have $${t_{{{\xi_h({\mathcal{{F}}})},{k}}}}=\xi_h({t_{{{{\mathcal{{F}}}},{k}}}})$$ as functions on $k$ provided $$h\notin E=\{h\in k^{\times}\,\mid\, S\cap (S-h)\not=\emptyset\},$$ and moreover $\xi_h({\mathcal{{F}}})$ is a middle-extension sheaf on ${\mathbf{A}}^1$ for $h\notin E$. Note that $$|E|{\leqslant}|S|(|S|-1){\leqslant}\operatorname{c}({\mathcal{{F}}})(\operatorname{c}({\mathcal{{F}}})-1)$$ since any $h\in E$ is a difference of two (distinct) elements of $S$, and hence we get:
\[lm-sheaf-rec\] Let $k$ be a finite field, and let ${\mathcal{{F}}}$ be an $\ell$-adic middle-extension sheaf on ${\mathbf{A}}^1/k$ which is pointwise pure of weight $0$. Denote $$\xi_h({\mathcal{{F}}})=[+h]^*{\mathcal{{F}}}\otimes\operatorname{D}({\mathcal{{F}}})$$ for given ${\mathcal{{F}}}$ and $h\in k$. Then $\xi_h({\mathcal{{F}}})$ is a constructible sheaf, which is tame if ${\mathcal{{F}}}$ is tame. Further, let $$E=\{h\in k^{\times}\,\mid\, S\cap (S-h)\not=\emptyset\},\quad\text{
where }\quad S=\{\text{singularities of }{\mathcal{{F}}}\}.$$
Then each $\xi_h({\mathcal{{F}}})$ with $h\notin E$ is a middle-extension sheaf on ${\mathbf{A}}^1/k$, pointwise pure of weight $0$, and we have $$\label{eq-induction}
U_{d+1}({\mathcal{{F}}};k)=
\frac{1}{|k|}
\sum_{h\in k-E}{U_d(\xi_h({\mathcal{{F}}});k)}+\theta
\frac{|E|\operatorname{rank}({\mathcal{{F}}})^{2^{d+1}}}{|k|}$$ for any $d{\geqslant}1$, where $|\theta|{\leqslant}1$.
Since it is clear that $\xi_h({\mathcal{{F}}})$ is indeed tame if ${\mathcal{{F}}}$ is, the only thing that remains is to justify the error term in (\[eq-induction\]). But if $\varphi={t_{{{{\mathcal{{F}}}},{k}}}}$, we have a trivial bound $$\frac{1}{|k|}U_d(\xi_h(\varphi)){\leqslant}\operatorname{rank}({\mathcal{{F}}})^{2\cdot 2^d}|k|^{-1}$$ for any $h\in k$ (similar to the previous corollary), hence the result.
We now state the essential result that allows us to get optimal bounds, which is a general version of the Riemann Hypothesis, for sums in one variable. The version we use is as follows:
\[th-rh\] Let $p$ be a prime number and ${\mathcal{{F}}}$ an $\ell$-adic middle-extension sheaf on ${\mathbf{A}}^1/k$, pointwise pure of weight $0$, such that $H^2_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})=0$. Then we have $$\Bigl|
\sum_{x\in k}{{t_{{{{\mathcal{{F}}}},{k}}}({x})}}
\Bigr|{\leqslant}2\operatorname{c}({\mathcal{{F}}})^2\sqrt{|k|}.$$
By the Grothendieck-Lefschetz trace formula and the Riemann Hypothesis, we have $$\Bigl|\sum_{x\in k}{
{t_{{{{\mathcal{{F}}}},{k}}}({x})}}\Bigr|
{\leqslant}\dim H^1_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})\sqrt{|k|}$$ since $H^0_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})=H^2_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})=0$.
By the Euler-Poincaré formula of Grothendieck–Ogg–Shafarevich (see, e.g., [@katz-equid Ch. 14]), we also know that, under our assumptions, we have $$\begin{aligned}
\dim
H^1_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})
&=-\chi_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})\\
&=\sum_{x\in\operatorname{Sing}({\mathcal{{F}}})}{\operatorname{Swan}_x({\mathcal{{F}}})}
+\sum_{x\in \operatorname{Sing}({\mathcal{{F}}})\cap {\mathbf{A}}^1}{
(\operatorname{rank}({\mathcal{{F}}})-\dim {\mathcal{{F}}}_x)
}
-\operatorname{rank}({\mathcal{{F}}})\\
&{\leqslant}\operatorname{c}({\mathcal{{F}}})+\operatorname{rank}({\mathcal{{F}}})|\operatorname{Sing}({\mathcal{{F}}})|
\\
&{\leqslant}2\operatorname{c}({\mathcal{{F}}})^2\end{aligned}$$ hence the result.
Further examples {#sec-examples}
================
In this section, we will simply give a few examples of trace functions of various kinds. The remainder of the proof of Theorem \[th-inverse\] is found in Section \[sec-proof\], after a statement of a stronger structural result in Section \[sec-refined\].
\[ex-mixed\] If $U{\hookrightarrow}{\mathbf{A}}^1$ is a dense open subset (defined over $k$), and $f_1$ (resp. $f_2$) is a regular function $f_1\,:\, U{\longrightarrow}{\mathbf{A}}^1$ (resp. a non-zero regular function $f_2\,:\, U{\longrightarrow}{\mathbf{G}}_m$) both defined over $k$, one has the Artin-Schreier-Kummer lisse sheaf $${\mathcal{{F}}}={\mathcal{{L}}}_{\psi(f_1)}\otimes{\mathcal{{L}}}_{\chi(f_2)}$$ defined for any non-trivial additive character $\psi\,:\, k{\longrightarrow}\bar{{\mathbf{Q}}}_{\ell}^{\times}$ and multiplicative character $\chi\,:\,
k^{\times}{\longrightarrow}\bar{{\mathbf{Q}}}_{\ell}^{\times}$, which satisfy $${t_{{{{\mathcal{{F}}}},{k}}}({x})}=\psi(f_1(x))\chi(f_2(x))$$ for $x\in U(k)$. These sheaves are all of rank $1$ (in particular, they are geometrically irreducible) and pointwise pure of weight $0$. Moreover, possible geometric isomorphisms among them are well-understood (see, e.g., [@deligne Sommes Trig. (3.5.4)]): if $(g_1, g_2)$ is another pair of functions we have $${\mathcal{{L}}}_{\psi(f_1)}\otimes{\mathcal{{L}}}_{\chi(f_2)}\simeq
{\mathcal{{L}}}_{\psi(g_1)}\otimes{\mathcal{{L}}}_{\chi(g_2)}$$ if and only if: (1) $f_1-g_1$ is of the form $$f_1-g_1=h^{|k|}-h+C$$ for some regular function $h$ on $U$ and some constant $C\in\bar{k}$; (2) $f_2/g_2$ is of the form $$\frac{f_2}{g_2}=Dh^{d}$$ where $d{\geqslant}2$ is the order of the multiplicative character $\chi$, $h$ is a non-zero regular function on $U$ and $D\in\bar{k}^{\times}$.
Furthermore, the conductor of these sheaves is fairly easy to compute. The singularities are located (at most) at $x\in
{\mathbf{P}}^1-U$. For each such $x$, the Swan conductor at $x$ is determined only by $f_1$, and is bounded by the order of the pole of $f_1$ (seen as a function ${\mathbf{P}}^1{\longrightarrow}{\mathbf{P}}^1$) at $x$ (there is equality if this order is $<|k|$).
In particular, if $f_1=0$ and $f_2$ is not a $d$-th power, then ${\mathcal{{F}}}$ is tamely ramified everywhere, geometrically irreducible and non-trivial, so that Corollary \[cor-tame\] applies (over arbitrary finite fields).
\[ex-kloos\] Deligne proved that, for any $p$ and $\ell\not=p$, and any non-trivial additive character $\psi$, there exists a middle-extension sheaf ${\mathcal{{K}}}\ell$ on ${\mathbf{P}}^1/{\mathbf{F}}_p$ which is pointwise pure of weight $0$, geometrically irreducible, lisse on ${\mathbf{G}}_m$, and satisfies $${t_{{{{\mathcal{{K}}}\ell},{k}}}({a})}=-\frac{1}{\sqrt{|k|}}\sum_{x\in k^{\times}}{
\psi(ax+x^{-1})
}$$ for any finite extension $k/{\mathbf{F}}_p$ and $a\in k$. This sheaf is of rank $2$, tamely ramified at $0$ and wildly ramified at $\infty$ with Swan conductor $1$, so that $\operatorname{c}({\mathcal{{K}}}\ell)=2+1+1=4$.
The following examples are studied by Katz [@katz-esde Ex. 7.10.2]. Let $C/k$ be a smooth projective geometrically connected algebraic curve, and $$f\,:\, C{\longrightarrow}{\mathbf{P}}^1$$ a non-constant map defined over $k$ of degree $d<p$. Let $D\subset C$ be the divisor of poles of $f$. Let $Z\subset C-D$ be the set of zeros of the differential $df$, and let $S=f(Z)$ be the set of singular values of $f$. Then, denoting by $$f_0\,:\, C-D{\longrightarrow}{\mathbf{A}}^1$$ the restriction of $f$ to $C-D$, the sheaf $${\mathcal{{F}}}_f=\ker(\operatorname{Tr}\,:\, f_{0,*}\bar{{\mathbf{Q}}}_{\ell}{\longrightarrow}\bar{{\mathbf{Q}}}_{\ell})$$ is a middle-extension sheaf on ${\mathbf{A}}^1/k$, of rank $\deg(f)-1$, pointwise pure of weight $0$ and lisse on ${\mathbf{A}}^1-S$ with $${t_{{{{\mathcal{{F}}}_f},{k}}}({x})}=|\{y\in C(k)\,\mid\, f(y)=x\}|-1$$ for $x\in k-S$. This sheaf is also everywhere tamely ramified, so its conductor is $|Z|+\deg(f)-1$.
In many cases, ${\mathcal{{F}}}_f$ is also geometrically irreducible. For instance, this happens when $f$ is *supermorse*, defined to mean that $\deg(f)<p$, that all zeros of $df$ are simple, and that $f$ separates these zeros (i.e., $|S|=|Z|$).
There exists a Fourier transform on middle-extension sheaves corresponding to the Fourier transform of trace functions, which was defined by Deligne and developed especially by Laumon; precisely, consider a middle-extension sheaf ${\mathcal{{F}}}$ which is geometrically irreducible, of weight $0$, and not geometrically isomorphic to ${\mathcal{{L}}}_{\psi}$ for some additive character $\psi$. Fix a non-trivial additive character $\psi$. Then the Fourier transform ${\mathcal{{G}}}=\operatorname{FT}_{\psi}({\mathcal{{F}}})(1/2)$ satisfies $${t_{{{{\mathcal{{G}}}},{k}}}({t})}=-\frac{1}{\sqrt{|k|}}
\sum_{x\in k}{{t_{{{{\mathcal{{F}}}},{k}}}({x})}\psi(tx)}$$ for $t\in k$, and it is a middle-extension sheaf, geometrically irreducible and pointwise pure of weight $0$ (see [@katz-esde §7] for a survey and details). Moreover, one can show that the conductor of ${\mathcal{{G}}}$ is bounded polynomially in terms of the conductor of ${\mathcal{{F}}}$ (see, e.g., [@fkm Prop. 7.2], though the definition of conductor is slightly different there).
In particular, applying the Fourier transform to the previous examples, we find many examples of one-parameter families of exponential sums arising as trace functions with bounded conductor, namely $$x\mapsto -\frac{1}{\sqrt{|k|}}
\sum_{y\in (C-D)(k)}{e\Bigl(\frac{x f(y)}{p}\Bigr)}$$ for the sheaves ${\mathcal{{F}}}_f$, and $$x\mapsto -\frac{1}{\sqrt{|k|}}
\sum_{y\in k}{\chi(f_2(y))\psi(f_1(y)+xy)}$$ for Artin-Schreier-Kummer sheaves (for instance, the Kloosterman sums ${\mathcal{{K}}}\ell$ of Example \[ex-kloos\] can be seen as the Fourier transform of the Artin-Schreier sheaf ${\mathcal{{L}}}_{\psi(x^{-1})}$.)
Refined structural results {#sec-refined}
==========================
We will deduce Theorem \[th-inverse\] from the following result which gives stronger structural information concerning the Gowers pnorms of trace functions of middle-extension sheaves.
\[th-precise\] Let $p$ be a prime number and let $\ell\not=p$ be an auxiliary prime. Let $d{\geqslant}1$ be an integer such that $p>d$.
Let ${\mathcal{{F}}}$ be a middle-extension $\ell$-adic sheaf on ${\mathbf{A}}^1/{\mathbf{F}}_p$ which is pointwise pure of weight $0$ and arithmetically semisimple. Then one of the following two conditions holds:
– There exists an additive character $\psi$ of ${\mathbf{F}}_p$, possibly trivial, and a polynomial $P\in {\mathbf{F}}_p[X]$ of degree at most $d-1$ such that ${\mathcal{{F}}}$ geometrically contains the Artin-Schreier sheaf ${\mathcal{{L}}}_{\psi(P)}$;
– Or else we have $$\label{eq-bound-inverse-repeat}
U_d({\mathcal{{F}}};{\mathbf{F}}_p){\leqslant}(5\operatorname{c}({\mathcal{{F}}}))^{(d+1)2^d}p^{-1}.$$
In this section, before proving this result, we check that it implies all our previous statements.
Let ${\mathcal{{F}}}$ satisfy the assumptions of Theorem \[th-inverse\]. Since it is arithmetically semisimple, we can write $${\mathcal{{F}}}={\mathcal{{F}}}_1\oplus {\mathcal{{F}}}_2,$$ where ${\mathcal{{F}}}_2$ is the sum of all irreducible components of ${\mathcal{{F}}}$ which are geometrically isomorphic to an Artin-Schreier sheaf ${\mathcal{{L}}}_{\psi(P)}$ for some polynomial $P$ of degree ${\leqslant}d-1$. We then have $${t_{{{{\mathcal{{F}}}},{{\mathbf{F}}_p}}}}=t_1+t_2,\text{ with }
t_i={t_{{{{\mathcal{{F}}}_i},{{\mathbf{F}}_p}}}}.$$
Now we apply Theorem \[th-precise\] to ${\mathcal{{F}}}_1$ and ${\mathcal{{F}}}_2$ separately. By construction, the first part of the dichotomy can not hold for ${\mathcal{{F}}}_1$, and hence we get the desired estimate $$U_d(t_1)=U_d({\mathcal{{F}}}_1;{\mathbf{F}}_p){\leqslant}(5\operatorname{c}({\mathcal{{F}}}))^{(d+1)2^{d}}p^{-1},$$ by (\[eq-bound-inverse-repeat\]).
Now for $t_2$, we write ${\mathcal{{F}}}_2$ as a direct sum of geometrically isotypic components, which are all of the form ${\mathcal{{F}}}_{\psi(P_i)}$ for some $P_i\in {\mathbf{F}}_p[X]$ of degree ${\leqslant}d-1$. There are at most $\operatorname{rank}({\mathcal{{F}}})$ such components, and the trace function for each of them is of the form $$x\mapsto \beta_i\psi(P_i(x)),$$ where $\beta_i$ is the sum of the twisting factors $\alpha$ of all the arithmetic subsheaves of ${\mathcal{{F}}}_2$ which are geometrically isomorphic to ${\mathcal{{L}}}_{\psi(P_i)}$. Since ${\mathcal{{F}}}_2$ is pointwise pure of weight $0$, each $\alpha$ is (under $\iota$) of modulus $1$, and hence $|\beta_i|{\leqslant}\operatorname{rank}({\mathcal{{F}}})$. Thus $t_2$ is of the form claimed in Theorem \[th-inverse\].
It is also clear that Theorem \[th-precise\] implies Corollary \[cor-inverse\], since if ${\mathcal{{F}}}$ is geometrically irreducible, the only possibility for the first case of the dichotomy is that ${\mathcal{{F}}}$ be geometrically isomorphic to a sheaf ${\mathcal{{L}}}_{\psi(P)}$ with $\deg(P){\leqslant}d-1$, which immediately implies (\[eq-strong-correlation\]).
As for Corollary \[cor-tame\], it follows for $d<p$ and $k={\mathbf{F}}_p$ because if ${\mathcal{{F}}}$ is tamely ramified, the only Artin-Schreier sheaf it may geometrically contain is the trivial sheaf. The general case of Corollary \[cor-tame\] follows by inspection of the following argument (we will make remarks indicating the relevant points).
Finally, we explain how this structure result implies Theorem \[th-explicit\]; this will show that many more explicit functions with small Gowers norms can be constructed from the examples in Section \[sec-examples\].
We note first that because of Corollary \[cor-trivial\], we can assume that $d<p$ for the functions $\varphi_1$, $\varphi_2$, $\varphi_3$ and $\varphi_4$ (the respective ranks of the sheaves will be $1$, $1$, $2$, $2$), the bounds being trivial for $d{\geqslant}p$.
\(1) The function $\varphi_1$ arises as the case of Example \[ex-mixed\] for $U$ the affine line itself, $\chi$ the Legendre character modulo $p$, $f_1=0$ and $f_2=f$. The corresponding sheaf ${\mathcal{{F}}}_1$ has rank $1$ so is necessarily geometrically irreducible. It is tame and not geometrically trivial if $f$ is not proportional to the square of another polynomial, and the singularities are $\infty$ and the zeros of $f$, so the conductor of ${\mathcal{{F}}}_1$ is at most $2+m$. Hence the first alternative of Theorem \[th-precise\] can not hold (alternatively, we can apply Corollary \[cor-inverse\] here, since the sheaf is tame.)
\(2) The function $\varphi_2$ is the case of Example \[ex-mixed\] for $U={\mathbf{A}}^1-\{0\}$, $\chi=1$, $f_1(x)=x^{-1}$. The corresponding sheaf ${\mathcal{{F}}}_2$ is of rank $1$ so geometrically irreducible. It is tamely ramified at $\infty$ and wildly ramified with Swan conductor $1$ at $0$, so the conductor is $3$. The classificaition of Artin-Schreier sheaves shows that the first alternative of Theorem \[th-precise\] does not hold, so we obtain the desired bound.
\(3) The function $-\varphi_3$ is Example \[ex-kloos\], where it is explained that the conductor is $4$. The Kloosterman sheaf is geometrically irreducible (for instance, because it is the sheaf-theoretic Fourier transform of the previous sheaf ${\mathcal{{F}}}_2$, and the Fourier transform sends geometrically irreducible sheaves to irreducibles sheaves.) Since it is of rank $2$, the first alternative of Theorem \[th-precise\] is also impossible here, and we get the stated estimate.
\(4) Let $$\mathcal{E}\,:\, v^2=u(u-1)(u-x)$$ denote the Legendre family of elliptic curves over ${\mathbf{F}}_p$, viewed as an affine algebraic surface over ${\mathbf{A}}^1-\{0,1\}$ by the projection $$\pi\,:\, \begin{cases}
\mathcal{E}{\longrightarrow}{\mathbf{A}}^1-\{0,1\}\\
(u,v,x)\mapsto x.
\end{cases}$$
The function $\varphi_4$ arises from the middle-extension to ${\mathbf{P}}^1$ of the sheaf ${\mathcal{{F}}}_4=R^1\pi_!\bar{{\mathbf{Q}}}_{\ell}(1/2)$. This is a tame geometrically irreducible sheaf of rank $2$, ramified at $\{0,1,\infty\}$, so the conductor is $5$ and we obtain the result as before.
Proof of the inverse theorem {#sec-proof}
============================
We will now prove Theorem \[th-precise\]. As can be expected, the argument is by induction on $d$, and the base case $d=1$ is an easy consequence of the Riemann Hypothesis, for any finite field:
\[pr-inverse-1\] Let $k$ be a finite field and let ${\mathcal{{F}}}$ be a middle-extension sheaf on ${\mathbf{A}}^1/k$ which is pointwise pure of weight $0$. If ${\mathcal{{F}}}\otimes\bar{k}$ does not contain a trivial subsheaf, then we have $$\label{eq-d1}
U_1({\mathcal{{F}}};k){\leqslant}4\operatorname{c}({\mathcal{{F}}})^4|k|^{-1}.$$
We have by definition $$\begin{aligned}
U_1({\mathcal{{F}}};k)&
=\frac{1}{|k|^2} \sum_{(h,x)\in k^2}{
{t_{{{{\mathcal{{F}}}},{k}}}({x+h})}\overline{{t_{{{{\mathcal{{F}}}},{k}}}({x})}}}\\
&=\Bigl|\frac{1}{|k|} \sum_{x\in k}{
{t_{{{{\mathcal{{F}}}},{k}}}({x})}}\Bigr|^2,\end{aligned}$$ and hence the estimate (\[eq-d1\]) follows immediately from Theorem \[th-rh\], unless $$H^2_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})\not=0.$$
But, if ${\mathcal{{F}}}$ is lisse on the dense open subset $U$ of ${\mathbf{A}}^1$, we have $$H^2_c({\mathbf{A}}^1\times\bar{k},{\mathcal{{F}}})=
H^2_c(U\times\bar{k},{\mathcal{{F}}})=
({\mathcal{{F}}}_{\bar{\eta}})_{\pi_1(U\times\bar{k},\bar{\eta})}(-1)$$ by birational invariance and the coinvariant formula for the topmost cohomology of a lisse sheaf. Since ${\mathcal{{F}}}$ is pointwise pure of weight $0$ on $U$, it corresponds to a representation of $\pi_1(U\times\bar{k},\bar{\eta})$ which is geometrically semisimple (by results of Deligne [@weilii]), and therefore $$({\mathcal{{F}}}_{\bar{\eta}})_{\pi_1(U\times\bar{k},\bar{\eta})}\not=0$$ implies that ${\mathcal{{F}}}$ contains a trivial summand.
We will now deal with $U_d$-pnorms, $d{\geqslant}2$, using an induction on $d$ based on (\[eq-induction\]). Precisely, consider the following statement, for a given integer $d{\geqslant}1$:
**Inverse($d$)**. *For any prime $p$ with $p>d$, for any $\ell\not=p$, for any middle extension $\ell$-adic sheaf ${\mathcal{{F}}}$ on ${\mathbf{A}}^1/{\mathbf{F}}_p$, pointwise pure of weight $0$ and arithmetically semisimple, *either* there exists an additive character $\psi$ of ${\mathbf{F}}_p$, possibly trivial, and a polynomial $P\in {\mathbf{F}}_p[X]$ of degree at most $d-1$ such that ${\mathcal{{F}}}$ contains geometrically a summand isomorphic to ${\mathcal{{L}}}_{\psi(P)}$, *or else* we have $$U_d({\mathcal{{F}}};{\mathbf{F}}_p){\leqslant}(5\operatorname{c}({\mathcal{{F}}}))^{(d+1)2^d}p^{-1}.$$*
Note that Proposition \[pr-inverse-1\] implies that **Inverse($1$)** is valid, and that Theorem \[th-precise\] simply states that **Inverse($d$)** holds for all $d{\geqslant}1$. Hence we will be done by induction once we show:
\[pr-induction\] Let $d{\geqslant}1$ be such that ***Inverse($d$)*** holds. Then so does ***Inverse($d+1$)***.
For the proof of Proposition \[pr-induction\], we will use two lemmas. Before stating the first, we introduce some terminology. Given a finite field $k$ and an open dense subset $U/k$ of ${\mathbf{A}}^1/k$, a lisse sheaf ${\mathcal{{F}}}$ on $U$ is called *induced* if it is arithmetically irreducible, and the corresponding representation of $\pi_1(U,\bar{\eta})$ is isomorphic to an induced representation $\operatorname{Ind}_{H}^{\pi_1(U,\bar{\eta})}{\varrho}_0$, for some proper normal finite-index subgroup $H$ of $\pi_1(U,\bar{\eta})$ containing $\pi_1(U\times\bar{k},\bar{\eta})$ and some irreducible representation ${\varrho}_0$. We need the following corollary of elementary representation theory: if ${\mathcal{{F}}}$ is arithmetically irreducible on $U$, and is not induced, then it is geometrically isotypic.
\[lm-isotypic\] Let $k$ be a finite field of characteristic $p$, and let ${\mathcal{{F}}}$ be a middle-extension $\ell$-adic sheaf on ${\mathbf{A}}^1/k$, which is arithmetically irreducible and lisse on some dense open set $U{\hookrightarrow}{\mathbf{A}}^1$.
*(1)* Either the sheaf ${\mathcal{{F}}}$ is geometrically isotypic on $U$, or its trace function is identically zero on $U(k)$.
*(2)* Suppose that ${\mathcal{{F}}}$ is geometrically isotypic, and let ${\varrho}$ denote the geometrically irreducible representation of $\pi_1(\bar{U},\bar{\eta})$ which corresponds to the isotypic component of ${\mathcal{{F}}}$. Suppose further that, for some $h\in k$, some polynomial $P\in k[X]$ and $\ell$-adic character $\psi$, we have a geometric summand $${\mathcal{{L}}}_{\psi(P)}{\hookrightarrow}[+h]^*{\mathcal{{F}}}\otimes\operatorname{D}({\mathcal{{F}}})$$ on $U$. Then we have a geometric isomorphism $$[+h]^*{\mathcal{{F}}}\simeq {\mathcal{{F}}}\otimes{\mathcal{{L}}}_{\psi(P)}.$$
\(1) follows from the remark before the statement: if ${\mathcal{{F}}}$ is not geometrically isotypic, then it is induced so that, on $U$, the corresponding representation ${\varrho}$ is given by $${\varrho}\simeq \operatorname{Ind}_{H}^{\pi_1(U,\bar{\eta})}{\varrho}_0.$$
It is however elementary that, in this situation, the character of ${\varrho}$ is identically zero on the non-trivial cosets of $H$, and all Frobenius $\operatorname{\mathrm{Fr}}_{x,k}$ corresponding to $x\in U(k)$ have this property since we have $$H=\{g\in \pi_1(U,\bar{\eta})\,\mid\, \deg(g)\equiv 0{\,(\mathrm{mod}\,{m})}\}$$ for some $m{\geqslant}2$, where $\deg$ is the degree which gives an isomorphism $$\deg\,:\, \pi_1(U,\bar{\eta})/\pi_1(U\times\bar{k},\bar{\eta}){\longrightarrow}\hat{{\mathbf{Z}}},$$ and since $\deg(\operatorname{\mathrm{Fr}}_{x,k})=-1$ for all $x\in U(k)$.
\(2) We have a geometric isomorphism ${\mathcal{{F}}}\simeq n{\varrho}$ on $U$, for some $n{\geqslant}1$. Then the assumption gives $${\mathcal{{L}}}_{\psi(P)}{\hookrightarrow}[+h]^*{\mathcal{{F}}}\otimes\operatorname{D}({\mathcal{{F}}})\simeq
n^2 ([+h]^*{\mathcal{{{\varrho}}}}\otimes{\varrho}'),$$ on $U$, and since the right-hand side is isotypic and the left-hand side irreducible, we derive the existence of a geometric injection $${\mathcal{{L}}}_{\psi(P)}{\hookrightarrow}[+h]^*{\varrho}\otimes{\varrho}',$$ and therefore of a geometric isomorphism $$[+h]^*{\varrho}\simeq {\varrho}\otimes {\mathcal{{L}}}_{\psi(P)},$$ and hence $$[+h]^*{\mathcal{{F}}}\simeq {\mathcal{{F}}}\otimes{\mathcal{{L}}}_{\psi(P)}$$ by taking copies of this, first on $U$, and then on ${\mathbf{A}}^1$ because the sheaves involved are middle-extensions.
The next lemma gives some properties of lisse sheaves on ${\mathbf{A}}^1_{{\mathbf{F}}_p}$ which are (geometrically) “almost” invariant under some non-trivial translations. It complements certain results of [@fkm] (where the invariance under homographies in $\operatorname{PGL}_2$ acting on the projective line is a crucial issue, and where only the base field $k={\mathbf{F}}_p$ is considered.) This is also where the restriction to $k={\mathbf{F}}_p$ occurs; roughly speaking, to extend Theorem \[th-inverse\] to any finite field of characteristic $p$, we would need a similar statement as the second part of this lemma to be valid when $G$ is an arbitrary finite subgroup of $\bar{{\mathbf{F}}}_p$. However, if ${\mathcal{{F}}}$ is tame, the statement is vacuously true (with no assumption on $d$ in (2)), simply because there is no non-trivial tame sheaf which is lisse on ${\mathbf{A}}^1$.
\[lm-peel\] Let $\bar{k}$ be an algebraic closure of ${\mathbf{F}}_p$, $\ell\not=p$ an auxiliary prime. Let ${\mathcal{{F}}}$ be a lisse $\ell$-adic sheaf on ${\mathbf{A}}^1/\bar{k}$ such that ${\mathcal{{F}}}$ is irreducible and non-trivial.
*(1)* We have $\operatorname{Swan}_{\infty}({\mathcal{{F}}}){\geqslant}\operatorname{rank}({\mathcal{{F}}})$, with equality if and only if ${\mathcal{{F}}}$ is isomorphic to ${\mathcal{{L}}}_{\psi}$ for some non-trivial $\ell$-adic additive character.
*(2)* Let $d<p-1$ be given. Suppose there exists a cyclic subgroup $G\subset \bar{k}$ of order $p$ such that we have isomorphisms $$\label{eq-quasi-inv}
[+h]^*{\mathcal{{F}}}\simeq {\mathcal{{F}}}\otimes{\mathcal{{L}}}_{\psi(P_h)}$$ on ${\mathbf{A}}^1$ for all $h\in G$, where $P_h\in \bar{k}[X]$ has degree ${\leqslant}d$. Then ${\mathcal{{F}}}$ is either isomorphic to ${\mathcal{{L}}}_{\psi(Q)}$ for some non-trivial additive character $\psi$ and polynomial $Q$ of degree ${\leqslant}d+1$, or it satisfies $$\operatorname{Swan}_{\infty}({\mathcal{{F}}}){\geqslant}p+\operatorname{rank}({\mathcal{{F}}}).$$
\(1) Since the geometric fundamental group of ${\mathbf{A}}^1$ is topologically generated by the inertia subgroups and ${\mathcal{{F}}}$ is lisse on ${\mathbf{A}}^1$, we see first that ${\mathcal{{F}}}$ is irreducible as representation of the inertia group $I(\infty)$ at $\infty$.
Since ${\mathcal{{F}}}$ is lisse on ${\mathbf{A}}^1$ and non-trivial, we have $H^0_c({\mathbf{A}}^1,{\mathcal{{F}}})=H^2_c({\mathbf{A}}^1,{\mathcal{{F}}})=0$, and by the Euler-Poincaré formula, we get $$\dim H^1_c({\mathbf{A}}^1,{\mathcal{{F}}})=-\chi_c({\mathbf{A}}^1,{\mathcal{{F}}})
=\operatorname{Swan}_{\infty}({\mathcal{{F}}})-\operatorname{rank}({\mathcal{{F}}}),$$ since the Euler-Poincaré characteristic of ${\mathbf{A}}^1$ is $1$. Now, the left-hand side is a non-negative integer, and we therefore deduce $$\operatorname{Swan}_{\infty}({\mathcal{{F}}}){\geqslant}\operatorname{rank}({\mathcal{{F}}}),$$ which is the first claim.
Now suppose there is equality. Since ${\mathcal{{F}}}$ is irreducible as an $I(\infty)$ representation, it has a unique break $\lambda$ at $\infty$ such that $\operatorname{Swan}_{\infty}({\mathcal{{F}}})=\lambda\operatorname{rank}({\mathcal{{F}}})$. We therefore have equality if and only if $\lambda=1$.
We can now apply the “break-lowering lemma” in [@katz-gkm Th. 8.5.7] (it is applicable because ${\mathcal{{F}}}$ is already $I(\infty)$-irreducible). This shows that there exists a non-trivial additive $\ell$-adic character $\psi$ of $k$ such that ${\mathcal{{G}}}={\mathcal{{F}}}\otimes{\mathcal{{L}}}_{\psi(X)}$ has all breaks $<1$. But ${\mathcal{{G}}}$ is lisse on ${\mathbf{A}}^1$ and still irreducible as $I(\infty)$ representation. We claim that ${\mathcal{{G}}}$ is (geometrically) trivial. Indeed, otherwise the inequality above would be applicable to ${\mathcal{{G}}}$ and would give $$\operatorname{rank}({\mathcal{{F}}})=\operatorname{rank}({\mathcal{{G}}}){\leqslant}\operatorname{Swan}_{\infty}({\mathcal{{G}}})<\operatorname{Swan}_{\infty}({\mathcal{{F}}}),$$ which is a contradiction. Hence ${\mathcal{{G}}}$ is geometrically trivial, and we get a geometric isomorphism ${\mathcal{{F}}}\simeq
{\mathcal{{L}}}_{\bar{\psi}}$.
(An alternative proof of the equality case goes as follows: if ${\mathcal{{F}}}$ were not of this form, it would be a Fourier sheaf in the sense of [@katz-esde §7.3.5]; since it is lisse on ${\mathbf{A}}^1$ with all breaks at $\infty$ larger than $1$, denoting by ${\mathcal{{G}}}$ its Fourier transform, the latter would be lisse at $0$ by [@katz-esde Lemma 7.3.9 (3)] and we would get $$\operatorname{Swan}_{\infty}({\mathcal{{F}}})=\operatorname{rank}({\mathcal{{F}}})+\operatorname{rank}({\mathcal{{G}}})>\operatorname{rank}({\mathcal{{F}}}),$$ by [@katz-esde Lemma 7.3.9, (2)], since ${\mathcal{{G}}}$ is also irreducible by [@katz-esde Th. 7.3.8 (3)], hence has non-zero rank.)
\(2) The finite subgroup $G\subset \bar{k}$ is cyclic, hence generated by some $0\not=h\in \bar{k}$. Since $d<p-1$, we can find a polynomial $Q\in \bar{k}[X]$ of degree ${\leqslant}d+1$ such that $$Q(X+h)-Q(X)=P_h.$$
We now form the sheaf ${\mathcal{{F}}}_1={\mathcal{{F}}}\otimes
{\mathcal{{L}}}_{\psi(Q)}$. It is lisse on ${\mathbf{A}}^1$, and we have $$[+x]^*{\mathcal{{F}}}_1\simeq {\mathcal{{F}}}_1$$ for any $x\in G={\mathbf{F}}_ph$. Denoting $$\phi\,:\, {\mathbf{A}}^1{\longrightarrow}{\mathbf{A}}^1/G\simeq {\mathbf{A}}^1$$ the quotient map for the action of $G$ on ${\mathbf{A}}^1$, the fact that $G$ is cyclic of order $p$ implies that there exists a sheaf ${\mathcal{{F}}}_2$ on ${\mathbf{A}}^1/G$ such that $${\mathcal{{F}}}_1\simeq \phi^*({\mathcal{{F}}}_2).$$
We then use the invariance of Swan conductors under pushforward for virtual representations of degree $0$ (see references in [@katz-mm p. 286, line 3]), i.e., the formula $$\operatorname{Swan}_{\infty}(\phi^*{\mathcal{{F}}}_2-\operatorname{rank}({\mathcal{{F}}}_2)\bar{{\mathbf{Q}}}_{\ell})=
\operatorname{Swan}_{\infty}(\phi_*(\phi^*{\mathcal{{F}}}_2-\operatorname{rank}({\mathcal{{F}}}_2)\bar{{\mathbf{Q}}}_{\ell})),$$ where $-$ refers to the Grothendieck ring of lisse sheaves on ${\mathbf{A}}^1$. The left-hand side is equal to $$\operatorname{Swan}_{\infty}(\phi^*{\mathcal{{F}}}_2)=\operatorname{Swan}_{\infty}({\mathcal{{F}}}_1),$$ while the right-hand side is equal to $$\begin{aligned}
\operatorname{Swan}_{\infty}(\phi_*(\phi^*{\mathcal{{F}}}_2-\operatorname{rank}({\mathcal{{F}}}_2)\bar{{\mathbf{Q}}}_{\ell}))
&= \sum_{\eta\in
\hat{G}}{(\operatorname{Swan}_{\infty}({\mathcal{{F}}}_2\otimes{\mathcal{{L}}}_{\eta})
-\operatorname{rank}({\mathcal{{F}}}_2)\operatorname{Swan}_{\infty}({\mathcal{{L}}}_{\eta})
)}\nonumber\\
&= \sum_{\eta\in
\hat{G}}{(\operatorname{Swan}_{\infty}({\mathcal{{F}}}_2\otimes{\mathcal{{L}}}_{\eta})
-\operatorname{rank}({\mathcal{{F}}}_2))}+\operatorname{rank}({\mathcal{{F}}}_2),
\label{eq-induc-restr}\end{aligned}$$ where $\eta$ runs over $\ell$-adic characters of $G$, and the ${\mathcal{{L}}}_{\eta}$ are the corresponding lisse sheaves on ${\mathbf{A}}^1$.
Since ${\mathcal{{F}}}$ is irreducible, so is ${\mathcal{{F}}}_2$, and the twists ${\mathcal{{F}}}_2\otimes{\mathcal{{L}}}_{\eta}$ in the sum. If one term in this sum is zero, we get $$\operatorname{Swan}_{\infty}({\mathcal{{F}}}_2\otimes
{\mathcal{{L}}}_{\eta})=\operatorname{rank}({\mathcal{{F}}}_2\otimes{\mathcal{{L}}}_{\eta})$$ and therefore, by the equality case of (1), we have $${\mathcal{{F}}}_2\simeq {\mathcal{{L}}}_{\bar{\eta}}\otimes {\mathcal{{L}}}_{\psi'}$$ for some additive character $\psi'$. Pulling back under $\phi$, it follows that ${\mathcal{{F}}}_1$ is also an Artin-Scheier sheaf ${\mathcal{{L}}}_{\psi(aX)}$ for some $a$, and hence $${\mathcal{{F}}}\simeq {\mathcal{{L}}}_{\psi(aX)}\otimes{\mathcal{{L}}}_{\psi(-Q)}$$ in that case.
On the other hand, if none of the terms in the sum vanishes, we get $$\operatorname{Swan}_{\infty}({\mathcal{{F}}}_1)=\operatorname{Swan}_{\infty}(\phi^*{\mathcal{{F}}}_2)
{\geqslant}p+\operatorname{rank}({\mathcal{{F}}}_1).$$
In particular, by assumption, this is $>d$, and hence $$\operatorname{Swan}_{\infty}({\mathcal{{F}}})=
\operatorname{Swan}_{\infty}({\mathcal{{F}}}_1\otimes{\mathcal{{L}}}_{\psi(-Q)})=
\operatorname{Swan}_{\infty}({\mathcal{{F}}}_1) {\geqslant}p+\operatorname{rank}({\mathcal{{F}}}).$$
The final lemma gives an upper-bound for the conductor of $\xi_h({\mathcal{{F}}})$.
\[lm-cond-xih\] Let ${\mathcal{{F}}}$ be a middle-extension sheaf on ${\mathbf{A}}^1/\bar{{\mathbf{F}}}_p$ and $h\in \bar{{\mathbf{F}}}_p$ such that the set of singularities of ${\mathcal{{F}}}$ and $[+h]^*{\mathcal{{F}}}$ in ${\mathbf{A}}^1$ are distinct. Then the conductor of $\xi_h({\mathcal{{F}}})$ satisfies $$\operatorname{c}(\xi_h({\mathcal{{F}}})){\leqslant}5\operatorname{c}({\mathcal{{F}}})^2.$$
Indeed, since the singularities are disjoint, we have $$\operatorname{c}(\xi_h({\mathcal{{F}}}){\leqslant}\operatorname{rank}({\mathcal{{F}}})^2+2\operatorname{rank}({\mathcal{{F}}})\sum_{x\in
S}{\operatorname{Swan}_x({\mathcal{{F}}})} +\operatorname{Swan}_{\infty}(\xi_h({\mathcal{{F}}})).$$
But from known properties of Swan conductors [@esnault-kerz (3.2)], we have $$\operatorname{Swan}_{\infty}(\xi_h({\mathcal{{F}}})) {\leqslant}\operatorname{rank}({\mathcal{{F}}})\operatorname{Swan}_{\infty}([+h]^*{\mathcal{{F}}})+
\operatorname{rank}({\mathcal{{F}}})\operatorname{Swan}_{\infty}(\operatorname{D}({\mathcal{{F}}})) {\leqslant}2\operatorname{c}({\mathcal{{F}}})^2,$$ hence the result.
We are now able to conclude the inductive proof of the inverse theorem. The reader is encouraged to check the tame case, for an arbitrary finite field and with no assumption on $d$ compared with the characteristic $p$.
We start with the data for a case of **Inverse($d+1$)**: $p$ is a prime number $>d+1$, ${\mathcal{{F}}}$ is a middle-extension sheaf of weight $0$ on ${\mathbf{A}}^1/{\mathbf{F}}_p$ which is pointwise pure of weight $0$ and arithmetically semisimple. We will show that one of the two conditions in **Inverse($d+1$)** holds. For notational simplicity, we write $c=\operatorname{c}({\mathcal{{F}}})$ and $S=\operatorname{Sing}({\mathcal{{F}}})\cap {\mathbf{A}}^1$.
Let $U/{\mathbf{F}}_p$ be the complement of the singularities $S$ of ${\mathcal{{F}}}$ in ${\mathbf{A}}^1/{\mathbf{F}}_p$, so that ${\mathcal{{F}}}$ is lisse on $U/{\mathbf{F}}_p$. Let $${\mathcal{{F}}}=\bigoplus_{1{\leqslant}i{\leqslant}r}{{\mathcal{{F}}}_i}$$ be a decomposition of ${\mathcal{{F}}}$ into direct sum of arithmetically irreducible middle-extension sheaves. Note that $r{\leqslant}\operatorname{rank}({\mathcal{{F}}}){\leqslant}c$ and each ${\mathcal{{F}}}_i$ also has conductor ${\leqslant}c$, and hence we have $$\label{eq-bound-1}
U_{d+1}({\mathcal{{F}}};{\mathbf{F}}_p){\leqslant}c^{2^{d+1}}\sum_{i=1}^r{U_{d+1}({\mathcal{{F}}}_i;{\mathbf{F}}_p)}.$$
We now consider a fixed $i$, and the arithmetically irreducible sheaf ${\mathcal{{F}}}_i$. If ${\mathcal{{F}}}_i$ is induced, its trace function is zero on $U$, and is bounded by $\operatorname{rank}({\mathcal{{F}}}_i){\leqslant}c$ on the complement, which contains at most $c$ points, so that a trivial estimate gives $$\label{eq-induced}
U_{d+1}({\mathcal{{F}}}_i;{\mathbf{F}}_p){\leqslant}\frac{c^{1+2^{d+1}}}{p}.$$
Now we assume that ${\mathcal{{F}}}_i$ is not induced. By Lemma \[lm-sheaf-rec\], noting that the singularities of ${\mathcal{{F}}}_i$ are among those of ${\mathcal{{F}}}$, we have the inductive formula $$\label{eq-inductive}
U_{d+1}({\mathcal{{F}}}_i;{\mathbf{F}}_p)= \frac{1}{p} \sum_{h\in
{\mathbf{F}}_p-E}{U_d(\xi_h({\mathcal{{F}}}_i);{\mathbf{F}}_p)}+\theta
\frac{c^{2+2^{d+1}}}{p}$$ with $|\theta|{\leqslant}1$, where $$E=\{h\in {\mathbf{F}}_p^{\times}\,\mid\, S\cap (S-h)\not=\emptyset\}.$$
Each term in the sum can be trivially bounded by $$\label{eq-bound-2}
\frac{1}{p}U_d(\xi_h({\mathcal{{F}}}_i);{\mathbf{F}}_p){\leqslant}\operatorname{rank}(\xi_h({\mathcal{{F}}}_i))^{2^d}p^{-1}
=\operatorname{rank}({\mathcal{{F}}}_i)^{2\cdot 2^d}p^{-1}
{\leqslant}c^{2^{d+1}}p^{-1}$$ (which we can therefore use for some exceptional $h$, provided their number is not too large in terms of $c$).
Furthermore, we know that for each $h\in {\mathbf{F}}_p^{\times}-E$, the sheaf $\xi_h({\mathcal{{F}}}_i)$ is a middle-extension sheaf, lisse on $U_h=U\cap
(U-h)$ and pointwise pure of weight $0$. By Lemma \[lm-cond-xih\], its conductor is ${\leqslant}5c^2$.
We can therefore apply the induction assumption **Inverse($d$)**. We obtain the bound $$\label{eq-apply-induction}
U_{d}(\xi_h({\mathcal{{F}}}_i);{\mathbf{F}}_p){\leqslant}(5\operatorname{c}(\xi_h{\mathcal{{F}}}_i))^{(d+1)2^d}p^{-1} {\leqslant}(5c)^{(d+1)2^{d+1}}p^{-1},$$ for all those $h\in {\mathbf{F}}_p^{\times}-E$ such that there does *not* exist some $P_h\in {\mathbf{F}}_p[X]$ with $\deg(P_h){\leqslant}d-1$ with a geometric embedding $${\mathcal{{L}}}_{\psi(P)}{\hookrightarrow}\xi_h({\mathcal{{F}}}_i)=[+h]^*{\mathcal{{F}}}_i\otimes \operatorname{D}({\mathcal{{F}}}_i).$$
We denote by $F_i\subset {\mathbf{F}}_p-E$ the set of exceptional $h$ for which this last property holds (including $h=0$). By Lemma \[lm-isotypic\], (2), if $h\in F_i$, we have a geometric isomorphism $$[+h]^*{\mathcal{{F}}}_i\simeq {\mathcal{{F}}}_i\otimes {\mathcal{{L}}}_{\psi(P_h)}$$ for some polynomial $P_h$ of degree ${\leqslant}d-1$, and hence $$\begin{gathered}
F_i\subset G=\{h\in {\mathbf{F}}_p\,\mid\, [+h]^*{\mathcal{{F}}}_i\text{ is
geometrically isomorphic to } \\ {\mathcal{{F}}}_i\otimes
{\mathcal{{L}}}_{\psi(P)}\text{ for some $P$ of degree ${\leqslant}d-1$}\}.\end{gathered}$$
This subset $G$ is an additive subgroup of ${\mathbf{F}}_p$, hence either trivial or equal to ${\mathbf{F}}_p$. In the former case, we are done. Otherwise, we first note that if ${\mathcal{{F}}}_i$ has a singularity $a\in {\mathbf{A}}^1$, all elements in its orbit under the action of $G$ are also singularities, i.e., $|G|{\leqslant}c$. We can apply (\[eq-bound-2\]) for all $h\in G$, getting a contribution $$\label{eq-bound-3}
{\leqslant}|G|c^{2^{d+1}}p^{-1}{\leqslant}c^{1+2^{d+1}}p^{-1}$$ for these terms.
The other possibility is that ${\mathcal{{F}}}_i$ is lisse on ${\mathbf{A}}^1$. We can then apply Lemma \[lm-peel\], (2) (to the geometrically irreducible component of the arithmetically irreducible but non-induced sheaf ${\mathcal{{F}}}_i$)) and two possibilites arise: either ${\mathcal{{F}}}_i$ is geometrically isomorphic to a direct sum of copies of ${\mathcal{{L}}}_{\psi(Q)}$ for some polynomial $Q$ of degree ${\leqslant}d$, or otherwise we have $$c{\geqslant}\operatorname{Swan}_{\infty}({\mathcal{{F}}}_i){\geqslant}|G|=p,$$ in which case we also get the bound (\[eq-bound-3\]) for this contribution.
Combining (\[eq-bound-1\]), (\[eq-induced\]), (\[eq-bound-3\]) and the average of the inductive bounds (\[eq-apply-induction\]), we get $$U_{d+1}({\mathcal{{F}}};{\mathbf{F}}_p){\leqslant}Ap^{-1},$$ where $$A=c^{2^{d+1}}\left\{
c^{1+2^{d+1}}+c^{2+2^{d+1}}+(5c)^{(d+1)2^{d+1}}
\right\},$$ and in order to finish the induction, we must check that $A{\leqslant}(5c)^{(d+2)2^{d+1}}$, for $d{\geqslant}1$, which is easily done, e.g., using the bound $$c^{2^{d+1}}\times \Bigl\{c^{1+2^{d+1}}+c^{2+2^{d+1}}
\Bigr\} {\leqslant}2c^{(d+2)2^{d+1}}.$$
[CC]{}
Z. Chatzidakis, L. van den Dries and A. Macintyre: *Definable sets over finite fields*, J. reine angew. Math. 427 (1992), 107–135
P. Deligne: *Cohomologie étale*, S.G.A 4${{\textstyle{\frac{1}{2}}}}$, L.N.M 569, Springer Verlag (1977).
P. Deligne: *La conjecture de Weil, II*, Publ. Math. IHÉS 52 (1980), 137–252.
H. Esnault and M. Kerz: *A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)*, Acta Mathematica Vietnamica 37 (2012), 351–362; <arXiv:1208.0128v3>.
É. Fouvry, E. Kowalski and Ph. Michel: *Algebraic twists of modular forms and Hecke orbits*, preprint (2012); <arXiv:1210.0617v4>.
É. Fouvry, E. Kowalski and Ph. Michel: *Counting sheaves using spherical codes*, preprint (2012); <arXiv:1210.0851v2>.
É. Fouvry, E. Kowalski and Ph. Michel: *Algebraic trace functions over the primes*, preprint (2012); <arXiv:1211.6043v1>.
É. Fouvry, E. Kowalski and Ph. Michel: *Trace norms over finite fields*, in preparation.
É. Fouvry, P. Michel, J. Rivat and A. Sárközy, *On the pseudorandomness of the signs of Kloosterman sums*, Journal of the Australian Mathematical Society, Volume 77, December 2004, 425–436.
B.J. Green and T. Tao: *The distribution of polynomials over finite fields, with applications to the Gowers norms*, Contrib. Discr. Math. 4 (2009), 1–36; <arXiv:0711.3191>.
B.J. Green, T. Tao and T. Ziegler: *An inverse theorem for the Gowers $U^{s+1}[N]$-norm*, Annals of Math. 176 (2012), 1231–1372; <arXiv:1009.3998>.
N.M. Katz: *Gauss sums, Kloosterman sums and monodromy groups*, Annals of Math. Studies 116, Princeton Univ. Press (1988).
N.M. Katz: *Exponential sums and differential equations*, Annals of Math. Studies 124, Princeton Univ. Press (1990).
N.M. Katz: *Moments, monodromy and perversity*, Annals of Math. Studies 159, Princeton Univ. Press (2005).
N.M. Katz: *Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms*, Annals of Math. Studies 180, Princeton Univ. Press (2011).
H. Liu: *Gowers uniformity norm and pseudorandom measures of of the pseudorandom binary sequences*, International J. Number Th. 7 (2005), 1279–1302.
H. Niederreiter and J. Rivat: *On the Gowers norms of pseudorandom binary sequences*, Bull. Aust. Math. Soc. 79 (2009), 259–271.
T. Tao and V. Vu: *Additive combinatorics*, Cambridge studies adv. math. 105, Cambridge Univ. Press 2006.
T. Tao and T. Ziegler: *The inverse conjecture for the Gowers norm over finite fields via the correspondence principle*, Anal. PDE 3 (2010), 1–20; <arXiv:0810.5527>.
T. Tao and T. Ziegler: *The inverse conjecture for the Gowers norm over finite fields in low characteristics*, Annals of Combinatorics 16 (2012), 121–188; <arXiv:1101.1469>.
[^1]: We thank B. Green for pointing out that this application of our result is of interest in combinatorics.
|
---
abstract: 'We introduce a new method for constraining the redshift distribution of a set of galaxies, using weak gravitational lensing shear. Instead of using observed shears and redshifts to constrain cosmological parameters, we ask how well the shears around clusters can constrain the redshifts, assuming fixed cosmological parameters. This provides a check on photometric redshifts, independent of source spectral energy distribution properties and therefore free of confounding factors such as misidentification of spectral breaks. We find that $\sim 40$ massive ($\sigma_v=1200$ km s$^{-1}$) cluster lenses are sufficient to determine the fraction of sources in each of six coarse redshift bins to $\sim$11%, given weak (20%) priors on the masses of the highest-redshift lenses, tight (5%) priors on the masses of the lowest-redshift lenses, and only modest (20-50%) priors on calibration and evolution effects. Additional massive lenses drive down uncertainties as $N_{\rm lens}^{-{1\over 2}}$, but the improvement slows as one is forced to use lenses further down the mass function. Future large surveys contain enough clusters to reach 1% precision in the bin fractions if the tight lens mass priors can be maintained for large samples of lenses. In practice this will be difficult to achieve, but the method may be valuable as a complement to other more precise methods because it is based on different physics and therefore has different systematic errors.'
author:
- 'D. Wittman and W. A. Dawson'
title: Constraining Source Redshift Distributions with Gravitational Lensing
---
Introduction
============
Photometric redshifts are of key importance to current and future galaxy surveys. In a typical application, galaxies might be binned according to photometric redshift and then some analysis conducted on these binned source sets. Avoiding systematic error in this case requires knowledge of the true redshift distribution of each photometric redshift bin. Ideally, photometric redshift methods themselves produce reliable confidence intervals for this purpose, but external verification is important for controlling and quantifying systematic errors. For deep surveys, direct verification with a spectroscopic sample representative of the photometric sample is very difficult, and this has led to the development of other methods such as cross-correlating each photometric redshift bin with other photometric redshift bins (Schneider [[*et al.*]{}]{} 2006, Erben [[*et al.*]{}]{} 2009) or with spectroscopic redshift bins (Newman 2008, Matthews & Newman 2010). Here we present a new and independent method for reconstructing source redshift distributions, using the shear from weak gravitational lensing.
The shear induced by a given lens grows with source redshift in a well-understoood way which depends on cosmographic parameters. Other authors (Jain & Taylor 2003, Bernstein & Jain 2004) have suggested using this dependence to constrain the cosmographic parameters from the observed shear-vs-redshift relation around identified lenses. (Note that this is distinct from cosmic shear, for which the redshift dependence also involves the growth of structure.) In this paper we reverse the question and ask how well the redshift distribution of a set of galaxies can be constrained by shear measurements around lenses at a range of redshifts, if the background cosmology is already well determined (see Appendix \[sec-cosmo\] for a demonstration that cosmological uncertainties are negligible here). This may prove useful for applications which take the background cosmology as given, and it also provides an internal consistency check for surveys.
Although the shear around any given lens is rather weak and might be expected to provide very little constraint, a very large survey such as LSST (Iveziĉ [*et al.*]{} 2008) will contain billions of sources lensed by tens of thousands of galaxy clusters. Assuming the photometry is uniform over the sky, the entire dataset can be used to constrain the source redshift distribution. In this paper we use Fisher matrix formalism to estimate an upper bound on the effectiveness of this method, and we discuss challenges to implementing the method on real data. Because this method requires a very large survey, we refer to LSST throughout. We refer the reader to the LSST Science Book (LSST Science Collaborations [*et al.*]{} 2009) for more details on LSST survey parameters.
Basic idea and toy model {#sec-toy}
========================
The tangential shear $\gamma$ at a projected distance $r$ around an axisymmetric lens is: $$\gamma(r) = {4\pi G\over c^2} {D_{ls}D_l\over
D_s}[\bar{\Sigma}(<r)-\Sigma(r)]$$ where $D_{ls}$, $D_l$, and $D_s$ are respectively the angular diameter distances from lens to source, observer to lens, and observer to source, $\Sigma(r)$ is the projected surface mass density at $r$, and $\bar{\Sigma}(<r)$ is the mean projected surface mass density within $r$ (Miralda-Escudè 1991). Figure \[fig-dratio\] (top panel) shows the behavior of the distance factor ${D_{ls} D_{s}\over D_s}$ as a function of source redshift, for various fixed lens redshifts. This depends on the assumed cosmological model; we assume a WMAP7 $\Lambda$CDM cosmology of $H_0 = 70.4$ km s$^{-1}$ Mpc $^{-1}$, $\Omega_m = 0.272$, and $\Omega_\Lambda = 0.728$ throughout. It is clear that very low-redshift ($z<0.2$) source galaxies will have no measurable shear around any of the lenses in the figure, moderate-redshift ($\sim 0.5$) sources will have measurable shear only around the lower-redshift lenses, and high-redshift ($z>1$) sources will have measurable shear around all lenses. This is the conceptual basis for constraining the redshift distribution of a source set: one should be able to solve for the source redshift distribution which, when passed through the appropriate response curves, best matches the observed shears. The method requires lenses at a range of redshifts and, to obtain good constraints, some prior knowledge of the lens masses. To simplify the exposition we treat the lens propertise as known in this section, and explore requirements on prior knowledge of lens properties in the next section.
![[*Top:*]{} The distance factor ${D_{ls} D_{l}\over D_s}$ as a function of source redshift, for various fixed lens redshifts. The lens redshift for each curve can be read off at the point where the curve departs from zero. [*Bottom:*]{} as for the left panel, but plotting $\beta={D_{ls}\over D_s}$. \[fig-dratio\]](f1a.png "fig:")\
![[*Top:*]{} The distance factor ${D_{ls} D_{l}\over D_s}$ as a function of source redshift, for various fixed lens redshifts. The lens redshift for each curve can be read off at the point where the curve departs from zero. [*Bottom:*]{} as for the left panel, but plotting $\beta={D_{ls}\over D_s}$. \[fig-dratio\]](f1b.png "fig:")\
Individual galaxies as well as clusters of galaxies can serve as lenses. Typically, the shear one arcminute off axis might be of order 0.5 for clusters and 0.01 for galaxies. Galaxies have near-isothermal mass profiles (Gavazzi [*et al.*]{} 2007), in which $\Sigma(r)$ and $\bar{\Sigma}(<r)$ scale as $r^{-1}$; specifically, $\bar{\Sigma}(<r)-\Sigma(r) = {\sigma_v^2\over 2Gr}$ where $\sigma_v$ is the velocity dispersion. Because this scales as $r^{-1}$, it scales as $(\theta D_l)^{-1}$ in terms of projected [*angular*]{} distance $\theta$, and $\gamma(\theta)$ can be written $$\gamma(\theta) = {2\pi \over c^2} {D_{ls}\over D_s} {\sigma_v^2\over
\theta}$$ Isothermal lenses thus have the convenient property that $\gamma(\theta)$ can be separated into a part which depends [*only*]{} on the cosmology and a part which depends [*only*]{} on the mass or velocity dispersion of the lens. Mass profiles which are not scale-free cannot be written this way because the conversion from physical to angular scales invokes the cosmological model through $D_l$. This convenience is not required for our argument, but will substantially simplify the exposition here. Although cluster mass profiles depart from isothermality more than galaxy profiles do, for this section we will assume isothermality to simplify our argument.
We can now write a very simple model for the shear of the $i$th source galaxy due to the $j$th lens. We henceforth refer to $\sigma_v$ as $s$ to eliminate confusion with the symbol $\sigma$ which we will use later to indicate uncertainties on measurements. We also define $\beta = {D_{ls}\over D_{s}}$ (taking $\beta$ to be zero where $z_s\le
z_l$); the $\beta$ curves for various lenses are shown in the bottom panel of Figure \[fig-dratio\]. We can then write $$\gamma_{ij} = {2\pi \over c^2} \beta_{ij} {s_j^2\over \theta_{ij}}
\label{eqn-shearij}$$ where $\theta_{ij}$ is the angular separation between lens $j$ and source $i$, $\beta_{ij}$ is the distance ratio between the two, and $s_j$ is the velocity dispersion of lens $j$.
The redshift distribution of the source set can be parameterized in many ways. Here we adopt the simplest approach, uniform bins in redshift. With $N_{\rm bin}$ bins, we parametrize the distribution with $f_1,f_2,...f_{N_{\rm bin}}$, where $f_m$ is the fraction of the galaxies in the set which are actually in redshift bin $m$. The highest-redshift bin should extend to arbitrarily high redshift so that none of the measured shear can come from an unmodeled part of the distribution.[^1] Consequently, we do not require that $\sum_m f_m=1$; any difference between unity and $\sum_m f_m$ can be interpreted as due to sources at such low redshift that they are never lensed. (A derived parameter $f_0 \equiv 1-\sum_m f_m$ can be defined which encodes the fraction of sources at ultralow redshifts, but is not a parameter we solve for directly.) For sources from an unknown redshift distribution, the expected value of $\beta$ is simply $\sum_m
f_m \beta_{mj}$, where $\beta_{mj}$ is now the mean distance ratio between lens $j$ and source redshift bin $m$. Thus the expected shear for a lens-source combination is $$\langle \gamma_{ij}\rangle = {2\pi \over c^2} {s_j^2\over \theta_{ij}}\sum_m f_m \beta_{mj}
\label{eqn-shearij2}$$ In this section, we will assume that $s_j$ is perfectly known; we will consider uncertainty in lens masses in §\[sec-mass\].
The observable is the total shear[^2] on each source galaxy due to all lenses which, in the weak lensing limit, is $\gamma_i = \sum_j \gamma_{ij}$. However, we must add shears componentwise because in the multiple-deflector context there is no longer a single “tangential” component. Thus shear is usually written as a complex quantity: $$\langle\gamma_{ij}\rangle = {2\pi \over c^2} {s_j^2\over
\theta_{ij}}e^{i2\phi_{ij}} \sum_m f_m \beta_{mj}$$ where $\phi_{ij}$ is the position angle of source $i$ with respect to lens $j$, and the $i$ in the exponent represents $\sqrt{-1}$. The total expected shear on the source is then $$\langle\gamma_i\rangle = \sum_j {2\pi \over c^2} {s_j^2\over \theta_{ij}}e^{i2\phi_{ij}}
\sum_m f_m \beta_{mj}.$$
We could form a Fisher matrix with each two-component $\gamma_i$ contributing two observables; the derivatives of these observables with respect to the model parameters $f_m$ are very straightforward. However, this is computationally inefficient because most sources will be far from any modeled lens (assuming we use massive clusters rather than individual galaxies as the lenses) and thus will contribute negligible information to the Fisher matrix; and of the remaining sources, most are near only one lens so that their contribution can be captured by a single (tangential) component. Thus, at the expense of very little information, we can efficiently group sources according to their relevant lenses.
To do this, we group the observables in Equation \[eqn-shearij2\] on one side and the model parameters on the other: $$\langle\gamma_{ij}\theta_{ij}\rangle = {2\pi \over c^2} {s_j^2} \sum_m f_m \beta_{mj}.
\label{eqn-shearij3}$$ The $e^{i2\phi_{ij}}$ factor no longer appears because with only one lens involved the tangential component is unambiguous. We can collapse this down to a single observable for each lens by taking a mean of sources relevant to that lens: $$\Gamma_j \equiv \langle\gamma_{ij}\theta_{ij}\rangle_{i\in j} = {2\pi \over
c^2} {s_j^2} \sum_m f_m \beta_{mj}\label{eqn-Gammadef}$$ where ${i\in j}$ denotes averaging over all sources projected close enough to lens $j$ to add non-negligible information to the Fisher matrix. Accordingly, $${\partial \Gamma_j \over \partial f_m} = {2\pi s_j^2\over c^2}
\beta_{mj}$$
To construct the Fisher matrix elements, we also need the variance of $\Gamma_j$. Because there is essentially no uncertainty on $\theta_{ij}$, the variance of $\gamma_{ij}\theta_{ij}$ is $\sigma_{i}^2\theta_{ij}^2$ where $\sigma_{i}$ is the uncertainty on the measured shear of galaxy $i$. Furthermore, $\sigma_{i}$ is uncorrelated with position or lens, so we can approximate it as a constant $\sigma_\gamma$ and pull it out of any sums. Weighting sources by inverse variance when computing $\Gamma_j$, we then find that $$\sigma_{\Gamma_j}^{-2} = \sigma_\gamma^{-2}\sum_{i \in j} {1\over \theta_{ij}^2}\label{eqn-sigmaGamma}$$ This sum primarily involves properties of the survey rather than the lens. We can rewrite it as the integral of the areal density of the source set, $n$, over the “footprint” of the lens, that is, the area over which the lens contributes non-negligible Fisher information: $n\int \theta^{-2} dA$. (Note that $n$ here is in units of galaxies per steradian, because $\theta$ is in units of radians. We will convert to the more typically quoted units of arcmin$^{-2}$ at the end of the calculation.) We shall see that the value of this integral is only weakly related to the properties of the lens. Integrating from some $\theta_{\rm min}$ to some $\theta_{\rm max}$ yields $$\begin{aligned}
n\int \theta^{-2} dA& =& 2\pi n \int_{\theta_{\rm min}}^{\theta_{\rm max}}
\theta^{-1} d\theta\\
& = & 2\pi n \ln(\frac{\theta_{\rm max}}{\theta_{\rm min}})\label{eqn-ln}\end{aligned}$$ Because the dependence is only logarithmic, we can make a rough estimate of $\frac{\theta_{\rm max}}{\theta_{\rm min}}$ for a typical lens and apply it to all lenses without much loss of precision. Sources less than $\sim 1$[[$^\prime\,$]{}]{} from the lens center are typically not used in weak lensing analyses because of the risk of contamination by cluster members and/or errors on their shear measurements due to background gradients from cluster members, so we adopt this value of $\theta_{\rm min}$. At large enough $\theta$, shear from unrelated structures will dominate the shear from the cluster. Although information about the cluster is still there, as a practical matter the effort to recover it may not be justified given the logarithmic dependence on $\theta_{\rm max}$. The shear around a massive cluster can clearly be observed out to at least $15$[[$^\prime\,$]{}]{} from the center (e.g, Kling [[*et al.*]{}]{} 2005), so $\theta_{\rm max}=15$[[$^\prime\,$]{}]{} should be a conservative value. Thus Equation \[eqn-ln\] is approximately equal to $17n$, and only weakly dependent on the adopted values of $\theta_{\rm min}$ and $\theta_{\rm max}$. We expect the dependence on assumed shear profile to be weak as well, given that many ground-based weak-lensing cluster studies have difficulty discriminating between different forms of shear profiles (Wittman 2002). Inserting this result into Equation \[eqn-sigmaGamma\], we find that $$\sigma_{\Gamma_j}^2= \frac{\sigma_\gamma^2}{17n}.$$
Assuming a Gaussian likelihood model, we now have an $N_{\rm bin}$ by $N_{\rm bin}$ Fisher matrix: $$\begin{aligned}
\mathcal{F}_{mn} &=& \sum_j^{N_{\rm lens}} {1 \over \sigma_{\Gamma_j}^2}
{\partial \Gamma_j \over \partial f_m} {\partial \Gamma_j \over
\partial f_n}\\
&=& \left({2\pi\over c^2}\right)^2 {17n\over\sigma_\gamma^2}\sum_j^{N_{\rm lens}}
s_j^4 \beta_{mj} \beta_{nj} \label{eqn-betas}\\
& \approx & 25 (\frac{n}{\rm arcmin^{-2}}) \sum_j^{N_{\rm lens}}
(\frac{s_j}{\rm 1000\ km\ s^{-1}})^4 \beta_{mj} \beta_{nj} \end{aligned}$$ where we have adopted $\sigma_\gamma = 0.2$, which represents the irreducible “shape noise” due to the random intrinsic orientations of sources. Smaller, fainter, sources will have larger shear uncertainties due to measurement uncertainties: $\sigma_i^2 =
\sigma_\gamma^2 + \sigma_{{\rm meas},i}^2$. We account for this by defining $n$ as the [*effective*]{} source density, that is, the equivalent number of perfectly measured sources provided by the source set: ${n} = \sigma_\gamma^2 \sum {1\over \sigma_\gamma^2 +\sigma_{{\rm
meas},i}^2}$ where the sum is taken over a unit area of sky. This is the basis on which surveys or source sets should be quoted; see the explanation of Equation 14.7 in the LSST Science Book (LSST Science Collaborations [*et al.*]{} 2009) for a more detailed justification.
As a rough estimate of the potential of this method, we compute a Fisher matrix forecast for an LSST-depth ($n=40$ arcmin$^{-2}$) survey covering nineteen lenses with $\sigma_v=1200$ km s$^{-1}$ at redshifts $z=\{0.05,0.1,0.15,...,0.95\}$, which roughly corresponds to using the most massive lens in the sky at each redshift. We model six redshift bins: five covering $0<z \leq 1$ with equal spacing, plus a sixth bin containing all galaxies at $z>1$. The forecast uncertainties on $f_m$ are then $\{0.06,0.07,0.11,0.17,0.20,0.10\}$. The same Fisher information can be obtained using a larger number of less-massive lenses, but going beyond a small number of well-studied lenses introduces a new problem: lens parameters such as $\sigma_v$ will not be known precisely for all these lenses, and we must examine the effect of marginalizing over uncertainties in the lens properties.
Marginalizing over lens mass {#sec-mass}
============================
The two lens parameters in this model are redshift and mass (or equivalently, velocity dispersion). Spectroscopic redshifts are available in the literature for thousands of clusters, particularly for the most massive ones which contribute the most Fisher information. But with large imaging surveys now finding more clusters than can be followed up with spectroscopic redshift confirmation, photometric redshifts for the lenses must be considered. Photometric redshifts are much more precise for clusters than for individual galaxies, even beyond the statistical effect of averaging many member galaxies. Because cluster members are preferentially early-type galaxies with well-established spectral energy distributions, their redshifts can be established with very high confidence. Furthermore, the accuracy of cluster photometric redshifts can be easily verified by spectroscopy of a representative subsample of clusters, whereas representativeness is very difficult to achieve in spectroscopic subsamples of individual galaxies. The LSST Science Book (LSST Science Collaborations [*et al*]{} 2009) forecasts photometric redshift uncertainties for modest (10$^{14} M_\odot$) clusters as a function of redshift; their Figure 13.12 shows that the uncertainties are 0.01 or less for all $z\le 1.1$ after a single visit (and for all $z\le 1.9$ after ten years of the survey). This fixes the $\beta$ factors to within 0.01, which is far more precise than the masses will be known [*a priori*]{}. We can therefore safely ignore uncertainties in lens redshifts. In the remainder of this section, we focus on marginalizing over lens mass uncertainties.
We now regard the $s_j^2$ as nuisance parameters. We choose $s_j^2$ rather than $s_j$ simply for convenience: $s_j^2$ is conceptually convenient because it is linearly proportional to the shear, and this proportionality makes it fall out of the relevant Fisher matrix elements. From Equation \[eqn-Gammadef\] we have $${\partial \Gamma_j \over \partial (s_k^2)} = \begin{cases}
{2\pi \over c^2} \sum_m f_m \beta_{mj} & \text{if $j=k$,}\\
0 & \text{otherwise.}
\end{cases}
\label{eqn-betasum}$$ We now have a Fisher matrix which is $N_{\rm bin}+N_{\rm lens}$ elements square. The lower-right $N_{\rm lens}\times N_{\rm lens}$ block is diagonal, because ${\partial \Gamma_j \over \partial (s_k^2)}
= 0$ where $j \ne k$. These diagonal elements look like the square of the first case in Equation \[eqn-betasum\]. The upper-left $N_{\rm
bin}\times N_{\rm bin}$ block contains only elements of the form in Equation \[eqn-betas\], and the other blocks (the $N_{\rm
lens}\times N_{\rm bin}$ block in the upper right and its transpose in the lower left) contain mixed terms. Note, however, that many of the elements in these latter blocks will be zero, corresponding to lens/bin combinations for which the bin redshift is lower than the lens redshift, and thus for which $\beta_{mj}=0$. In the upper left block, no elements will vanish unless [*all*]{} lenses are at a higher redshift than the lowest-redshift bin. Note that the $f_m$ now appear in the Fisher matrix, so we must adopt fiducial values. Fisher matrix estimates should not be trusted far from the fiducial values; see Albrecht [*et al.*]{} (2006) for a clear exposition of Fisher matrix limitations. We therefore adopt a uniform fiducial distribution for now, and examine the dependence on the fiducial distribution at the end of this section.
Next, we need a prior on the lens masses. With no prior, the lowest-redshift lens would have no constraining power at all, as any shear (or lack thereof) in the lowest-redshift bin could be explained as a result of arbitrarily large (or small) lens mass. Successively higher-redshift lenses would have some constraining power, but not much. We therefore explore the importance of this prior by running Fisher matrix forecasts with the lens set of the previous section, but with priors of 1%, 2%, 5%, 10%, and 20% on the lens masses assuming the fiducial $s_j=1200$ km s$^{-1}$. We implement this by adding to the Fisher matrix the inverse variance representing a Gaussian prior before inverting the Fisher matrix to obtain the covariance matrix. To implement the 1% prior, for example, we add $(0.01(1200^2))^{-2}$ to $\mathcal{F}_{7,7},
\mathcal{F}_{8,8},...\mathcal{F}_{25,25}$, which are the nineteen elements representing the lens $s_j^2$. Figure \[fig-masspriors\] shows the results. For the low-redshift bins, obtaining constraints close to those of the previous section requires a 1-2% prior, but for higher-redshift bins the required priors are more modest, 10-20%. Priors tighter than 1% (not shown) do not yield any visible improvement in this figure.
![Constraints on the bin fractions for various lens-mass priors in the simplified physical scenario of §\[sec-toy\] with just one massive lens at each redshift. The redshift-varying prior allows for weaker lens-mass priors at higher lens redshifts where priors are both less necessary and more difficult to establish observationally; this particular form is $5\%(1+z_{\rm lens})^{2}$ which brings the prior to 20% at $z_{\rm lens}=1$. More lenses of the same mass will tighten the constraints as $N_{\rm
lens}^{-{1\over 2}}$. \[fig-masspriors\]](lensprior.png)
The most efficient observational approach may thus be to obtain tight priors on low-redshift lenses using velocity dispersions, X-ray analyses, external lensing analyses, etc, while obtaining only very rough priors on the higher-redshift lenses. We model this scenario by testing a mass prior which is 5% at $z_{\rm lens}=0$ but loosens as $(1+z_{\rm lens})^2$ (i.e., a 20% prior at $z_{\rm
lens}=1$); this is is shown as the dashed curve in Figure \[fig-masspriors\]. This yields source redshift constraints as good as those obtained with a 5-10% prior, but without much difficult high-redshift work. Indeed, optical richness alone can provide a 20-30% prior on the mass (Rykoff [[*et al.*]{}]{} 2011), thus obviating the need for followup at the high-redshift, assuming there is no redshift above which the optical richness estimator breaks down.
To a reasonable approximation, each additional lens tightens the constraints mostly on the 1-2 source redshift bins nearest the lens. Therefore, we can tweak the lens sample to obtain roughly equal constraints on all source redshift bins. For example, we see from Figure \[fig-masspriors\] that the maximum uncertainty for the efficient redshift-varying mass prior is for source redshifts 0.7-0.9, at a level of roughly twice the minimum uncertainty. Tripling the number of lenses at redshifts 0.5-0.95, for a total of 39 massive lenses, yields roughly equal constraints on most source redshift bins, averaging 0.11 (middle curve in Figure \[fig-sourcedep\]). Reaching yet lower levels of uncertainty requires more lenses overall. For example, reaching 0.01 uncertainty across the board will require $11^2$ times more lenses than the previous scenario, or about 5000 total. However, when so many lenses are considered, we cannot use only very massive lenses. Therefore, the required number of lenses will scale much more steeply than the inverse square of the desired constraint. The Fisher information corresponding to 5000 lenses with $\sigma_v =1200$ km s$^{-1}$ could be accumulated with tens of thousands of less-massive lenses, which is well within expectations for current and future large imaging surveys. Computationally, using so many lenses should be feasible because most of the tens of thousands of nuisance parameters can be marginalized over separately, reflecting the nonoverlapping nature of most lenses. However, the 1% goal will be difficult to achieve at low redshift due to the difficulty of obtaining the 5-8% lens-mass priors necessary to do so well there.
These factors provide insight into why we do not propose using individual galaxies as lenses. For every $10^{15}$ $M_\odot$ cluster, a survey would need one million $10^{12}$ $M_\odot$ galaxies to provide the same Fisher information, all other factors being equal. Although surveys such as LSST will have billions of galaxies, all other factors are not equal. The photometric redshift uncertainty on each individual galaxy is much larger than for clusters, and individual-galaxy lens photometric redshifts bring in exactly the problems this method is designed to avoid. These problems could be avoided by using only early-type galaxies with well-behaved photometric redshifts and mass priors from the fundamental plane, but then far too few lenses would be available. Furthermore, galaxy-scale lenses do not dominate their local shear in a way which allows parallelized modeling of designated patches of sky, and thus would introduce additional computational challenges.
A foreseeable application for this method is to input source sets corresponding to each of several photometric redshift bins, to determine the outlier fraction and distribution in each. Therefore, we examine the dependence on the redshift distribution of the source set, which we have assumed to be flat until now. Figure \[fig-sourcedep\] shows the results using the redshift-varying mass prior and the 39 massive lenses with three different source sets: flat (as described above, middle curve), all in the $z<0.2$ bin (lower curve), and all in the $0.8<z\le 1.0$ bin (upper curve). Characterization of outliers will be more precise for the low-$z_{\rm phot}$ source set than for the high-$z_{\rm phot}$ source set, although this is partly due to the choice of lens-mass prior which is tighter at lower redshift.
![Constraints on the bin fractions for low-redshift (lower curve), flat (middle curve), and high-redshift (upper curve) source sets. Each source set is used with the redshift-varying lens-mass prior and 39 lenses described in the text. \[fig-sourcedep\]](sourcedep.png)
Profile and calibration uncertainties\[sec-profile\]
====================================================
To this point we have assumed that prior knowledge of a cluster’s velocity dispersion[^3] translates perfectly into prior knowledge of its integrated shear statistic, given a source redshift distribution. However, the relationship between these two observables is sure to be more complicated. Clusters with the same velocity dispersion may have different profiles, resulting in different integrated shear estimates. If this is a purely stochastic effect, then the scatter in the observable-observable relation weakens the prior. But in principle, the loss of information due to this scatter can be compensated by analyzing more clusters. Conceptually more worrisome would be a redshift-dependent trend or an overall calibration error in this relationship, which would cause systematic errors.
We therefore insert nuisance parameters describing the uncertainties in calibration and redshift evolution, and explore how the results degrade as priors on these parameters are relaxed. We allow Equation \[eqn-Gammadef\] to be multiplied by a factor $a+bz_j$, where $a$ is a nuisance parameter for the calibration, $b$ is a nuisance parameter for the redshift evolution, and $z_j$ is the redshift of the $j$th lens, and repeat the analysis of the previous section with the flat source distribution and the redshift-dependent lens mass prior. Figure \[fig-evol\] shows the reconstruction precision for a range of calibration and evolution priors, with a fiducial model of $a=1$ and $b=0$. Priors of 20% on each are sufficient to preserve performance nearly identical to that of the previous section in which calibration and evolution were not factors; the bottom curve in each panel is very similar to the middle curve in Figure \[fig-sourcedep\].
![Constraints resulting from a range of calibration (top panel) and evolution (bottom panel) priors. For the top panel the evolution prior has been set to 20% in all cases, and for the bottom panel the calibration prior has been set to 20% in all cases. Priors tighter than 20% on either do not substantially improve the results: the bottom curve in each panel is very similar to the middle curve in Figure \[fig-sourcedep\], where the priors were delta functions. \[fig-evol\]](aprior.png "fig:")\
![Constraints resulting from a range of calibration (top panel) and evolution (bottom panel) priors. For the top panel the evolution prior has been set to 20% in all cases, and for the bottom panel the calibration prior has been set to 20% in all cases. Priors tighter than 20% on either do not substantially improve the results: the bottom curve in each panel is very similar to the middle curve in Figure \[fig-sourcedep\], where the priors were delta functions. \[fig-evol\]](bprior.png "fig:")
We stress that the choice of an isothermal profile as a calculation aid here is not fundamental to the method. In practice, surveys would measure the average cluster profile and construct an optimal integrated shear estimator before commencing the redshift analysis. The results of this section indicate that profile calibration and evolution uncertainties at the 20-50% level do not substantially degrade the performance of this method.
Discussion\[sec-detailedmodel\]
===============================
We have outlined a new method which uses gravitational lensing by clusters and knowledge of the cosmology to constrain source redshift distributions. Although this is an unorthodox use of the lensing-redshift information, it has some strengths and some relevant applications. The method’s strengths are:
- it is independent of other methods. It relies on no assumptions about galaxy spectral properties or biasing.
- it is applicable to any set of galaxies. One can determine the redshift distribution of a set of galaxies chosen in any number of ways, such as photometric redshift cut, pure color selection, or even X-ray or radio selection. Multiwavelength observations are not necessary as they are for photometric redshifts.
- in the context of a large imaging survey, it requires little additional data. Where lens spectroscopic redshifts are not available, photometric lens redshifts will do, and optical richness provides a sufficient lens-mass prior for the high-redshift lenses. However, pinning down the low-redshift end of the source distribution will require strong mass priors on low-redshift lenses.
Our Fisher matrix forecast involves some approximations and caveats:
- Lens morphology: we have assumed axisymmetric isothermal lenses. Although we have argued that other profiles would not yield substantially different forecasts, a precise experiment would include some sort of marginalization over profiles. This marginalization might include allowances for non-axisymmetry.
- Foreground/background structure: we have assumed that shear from foreground/background structure is negligible in the “footprint” of each lens, and totally dominant outside. In reality, unmodeled shear will be present in the footprint as well, and contribute to slightly looser constraints. However, unmodeled shear is not an irreducible source of noise. One can choose to model previously unmodeled structure and thus add information to the Fisher matrix. This entails a cost-benefit analysis rather than a hard limit.
- Magnification: the observed source population behind lenses will be slightly different than in other locations, due to lensing magnification. If unmodeled, this may result in a nontrivial systematic error. However, because the lenses are being modeled, the model magnification at any given point provides an easily available basis for correction. Deep fields which establish the source counts fainter than the main survey flux limit will be very useful for this purpose. As with the foreground/background structure, it may be possible to add information to the Fisher matrix with improved modeling of the magnification.
- Cluster contamination: the areas around clusters differ from the general area of a survey in another respect, the presence of cluster members. Care must be taken so that the source set whose shear is being measured is not contaminated by these members.
- Cosmology dependence of $\beta$: Appendix \[sec-cosmo\] shows that marginalizing over uncertainty in cosmographic parameters will not degrade the results presented here. But using the cosmology as input here implies that the redshift estimates from this method should not feed in to some other method which estimates the cosmology unless the covariances are properly treated.
- Wide source redshift bins: given a lens redshift, $\beta$ varies over the width of a source redshift bin. Accurately modeling this effect will require some assumptions about the source redshift distribution inside the bins. It may be possible to find a parameterization which reduces this difficulty.
The source redshift distributions obtained with this method will be most useful in applications where the background cosmology would have been assumed anyway, such as galaxy evolution, and especially in applications which are very sensitive to photometric redshift outliers. For example, measurements of the bright end of the galaxy luminosity function at high photometric redshift are extremely sensitive to catastrophic outliers, as a small fraction of low-redshift galaxies mistakenly put at high redshift results in many more “high-luminosity” galaxies at that redshift. Surveys can use this method to independently derive the full redshift distribution of the galaxies in each photometric redshift bin, thus characterizing the number and distribution of outliers. Indeed, the use of multiwavelength photometry to infer both source redshift [*and*]{} source spectral/photometric properties can be problematic, and this method offers a possible workaround.
Other methods such as cross-correlation with a spectroscopic sample (Newman 2008, Matthews & Newman 2010) should be able to achieve 1% accuracy without assuming a cosmology or requiring expensive lens mass priors. This method may therefore not be competitive statistically, but as it is based on different physics it may offer value in having different systematic errors. Redshift distributions derived using this method can and should be used in consistency checks. Given the best-fit cosmology from weak lensing tomography, this method should produce source redshift distributions which are consistent with those used in the weak lensing tomography. A prime application for this method may therefore be as an internal consistency check for surveys.
We thank J. A. Tyson for suggesting that we pursue the idea of using lensing to constrain source redshift distributions, and the anonymous referee for suggestions which led to improvements in the paper.
Dependence on Cosmological Parameters {#sec-cosmo}
=====================================
This appendix justifies the assumption in the main body of the paper that uncertainties in the distance ratios contribute negligible uncertainty to the parameters of interest, the fraction of sources in each redshift bin. Uncertainty in the distance ratios comes from uncertainty in cosmographic parameters, including Hubble’s constant $H_0$ and the energy densities in matter ($\Omega_m$), curvature ($\Omega_k$), and in the cosmological constant ($\Omega_\Lambda$). Of course, more general dark energy models can introduce further parameters, but the cosmological constant has a greater effect on distances than a dark energy model with equation of state parameter $w>-1$. Therefore, if the assumption is justified for a cosmological constant model, it is justified for models with $w>-1$.
Distances depend linearly on Hubble’s constant $H_0$, but $\beta =
{D_{ls}\over D_{s}}$ is a [*ratio*]{} of distances. Therefore $\beta$ does not depend on $H_0$ and uncertainty in $H_0$ is irrelevant.
The remaining parameters are more difficult to treat analytically, so we make a numerical estimate. The derivatives of $\beta$ with respect to $\Omega_m$ and $\Omega_\Lambda$ (with $\Omega_k=1-\Omega_m-\Omega_\Lambda$) depend on the lens and source redshifts, but are typically of order 0.02. The worst case is for uncertainty in $\Omega_m$ when considering high-redshift sources and lenses, where ${\partial \beta\over \partial \Omega_m} \approx 0.15$. We take a conservative approach by setting ${\partial \beta\over
\partial \Omega_m} = 0.15$ for all sources and lenses, so that the partial derivative of Equation \[eqn-Gammadef\] with respect to $\Omega_m$ is 0.15 times ${2\pi\over c^2}s_j^2$. We include $\Omega_m$ as a parameter in the Fisher matrix using these partial derivatives.
We rerun the forecast with a Gaussian prior of $\sigma=0.016$ on $\Omega_m$, which corresponds to the current WMAP7+BAO+H0 uncertainty for the “OLCDM” model in which curvature is allowed to vary.[^4] If the results were plotted on Figure \[fig-masspriors\] or Figure \[fig-sourcedep\], they would be nearly indistinguishable from the existing plots. The biggest difference would be the point in the lower right corner of Figure \[fig-sourcedep\] increasing from 0.052 to 0.055. The effect of varying $\Omega_\Lambda$ is even smaller because $\beta$ is less sensitive to it.
Albrecht, A., Bernstein, G., Cahn, R., et al. 2006, arXiv:astro-ph/0609591
Bernstein, G., & Jain, B. 2004, , 600, 17
Erben, T., Hildebrandt, H., Lerchster, M., et al. 2009, , 493, 1197
Gavazzi, R., Treu, T., Rhodes, J. D., et al. 2007, , 667, 176
Ivezic, Z., Tyson, J. A., Acosta, E., et al. 2008, arXiv:0805.2366
Jain, B., & Taylor, A. 2003, Physical Review Letters, 91, 141302
Kling, T. P., Dell’Antonio, I., Wittman, D., & Tyson, J. A. 2005, , 625, 643
LSST Science Collaborations, Abell, P. A., Allison, J., et al. 2009, arXiv:0912.0201
Matthews, D. J., & Newman, J. A. 2010, , 721, 456
Miralda-Escude, J. 1991, , 370, 1
Newman, J. A. 2008, , 684, 88
Rykoff, E. S., Koester, B. P., Rozo, E., et al. 2011, arXiv:1104.2089
Schneider, M., Knox, L., Zhan, H., & Connolly, A. 2006, , 651, 14
Wittman, D. 2002, Gravitational Lensing: An Astrophysical Tool, 608, 55
[^1]: Because $\beta$ becomes fairly flat at high redshift, a very wide high-redshift bin is no more problematic than a modest-width low-redshift bin in terms of variation of $\beta$ within each bin.
[^2]: Technically, the observable is the reduced shear $\frac{\gamma}{1-\kappa}$ where $\kappa$ is the convergence, but the argument still holds. The effect of $\kappa$ is to change the profile somewhat, which is not important for the toy model here.
[^3]: Throughout this section we refer to velocity dispersion for concreteness, but Sunyaev-Zel’dovich effect, X-ray, or optical richness measurements could equally well be used to set the prior.
[^4]: <http://lambda.gsfc.nasa.gov/product/map/dr4/params/olcdm_sz_lens_wmap7_bao_h0.cfm>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.