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abstract: 'Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension $d_c>1$. We here identify an *antiunitary* operator that ensures the spectral symmetry of a two-dimensional HOT insulator and the existence of cornered localized states ($d_c=2$) at precise zero energy. Such an antiunitary symmetry allows us to construct a generalized HOT insulator that continues to host corner modes even in the presence of a *weak* anomalous Hall insulator and a spin-orbital density wave orderings, and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$ superconductor and the corner localized Majorana zero modes survive even in the presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also serve as the building blocks of three-dimensional second-order Weyl semimetals, supporting one-dimensional hinge modes.'
author:
- Bitan Roy
title: '**Antiunitary symmetry protected higher order topological phases**'
---
*Introduction*. The hallmark of topological phases of matter is the presence of gapless modes at the boundary, protected by the nontrivial bulk topological invariant. Traditionally, a $d$-dimensional bulk topological phase (insulating or gapless) hosts boundary modes that are localized on $d-1$ dimensional surfaces, characterized by codimension $d_c=1$ [@hasan-kane-review2010; @qi-zhang-review2011; @bernevig2006; @fu-kane2007; @volovik-book; @chiu-review2016; @armitage-review2018]. Nevertheless, the family of topological phases of matter nowadays includes its higher order cousins, and an $n$th-order topological phase features boundary modes of codimension $d_c=n>1$ [@benalcazar2017], such as the corner (with $d_c=d$) and hinge (with $d_c=d-1$) states of topological insulators (electrical and thermal) and semimetals [@benalcazar2017; @schindler2018; @serra-garcia2018; @noh2018; @peterson; @imhof2018; @song2017; @langbehn2017; @franca2018; @schindler-sciadv2018; @ezawa2018; @hsu2018; @lin-hughes-DSM; @wang1-2018; @yan2018; @calugaru2019; @okuma2018; @tanay2019; @sigrist2019; @benalcazar-prb2017; @matsugatani2018; @khalaf2018; @Vliu2018; @seradjeh2018; @ahn2018; @Klinovaja2019; @Klinovaja2019arXiv; @kaisun2019arXiv; @ghorashi-HOTSC; @vanmiert2018; @wang-arxiv2018; @trifunovic2019]. In this language, the traditional topological phases are first order. While the bulk topological invariant assures the existence of boundary modes, often (if not always) the localized topological modes get pinned at precise zero energy due to the *spectral symmetry*, which we exploit here to propose the most general setup for a two-dimensional higher-order topological (HOT) insulator, characterized by a quantized quadrupolar moment $Q_{xy}=0.5$ and supports four corner localized zero-energy modes. The central results are summarized in the phase diagram, shown in Fig. \[Fig:PD\_bandstructure\].
![Phase diagram of a time-reversal and $C_4$ symmetry breaking Dirac insulator, represented by the Hamiltonian operator $H^{\rm gen}_{\rm HOT}=\hat{h}_0+ \hat{h}_1 + \hat{h}_{\rm p}$ for $t=t_0=m=1$ \[Eqs. (\[Eq:HOTImodel\]) and (\[Eq:genHOTIpert\])\]. For small $\Delta_1$ and $\Delta_2$, the system supports four zero-energy corner modes (Fig. \[Fig:GHOTI\_CornerModes\]), protected by an antiunitary operator ($A$) and representing a generalized higher order topological insulator (GHOTI). For charged fermions GHOTI is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. The bulk band gap closes either at the $\Gamma$ point (solid line) or along the $\Gamma-{\rm M}$ line (dashed line) (Fig. \[Fig:bandstructure\]) beyond which the system becomes a trivial or normal insulator (NI), with $Q_{xy}=0$ for charged fermions. The phase diagram possesses a *reflection* symmetry about $(\Delta_1,\Delta_2)=(0,0)$, where the bands recover two-fold degeneracy \[see Fig. \[Fig:bandstructure\] (left column)\], and the system describes a regular HOTI (red dot). The phase boundaries do not depend on $\Delta$ ($C_4$ symmetry breaking mass). ](PhaseDiagram_Color.pdf){height="6.5cm" width="7.25cm"}
\[Fig:PD\_bandstructure\]
{width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"}
\[Fig:bandstructure\]
The HOT phases can be constructed (at least, in principle) by systematically reducing the dimensionality of the boundary modes at the cost of some discrete crystalline and fundamental (such as time-reversal) symmetries in the bulk of the system. For example, a two-dimensional HOT insulator, supporting four corner localized zero-energy modes ($d=0$, $d_c=2$), can be realized in the presence of a four-fold ($C_4$) and time-reversal (${\mathcal T}$) symmetry breaking perturbation that acts as a mass for two one-dimensional counter propagating helical edge modes ($d=1$, $d_c=1$) of a first-order topological insulator. The corresponding effective single-particle Hamiltonian can be decomposed as $\hat{h}^{\rm 2D}_{\rm HOT}=\hat{h}_{0}+ \hat{h}_{1}$, with $$\begin{aligned}
~\label{Eq:HOTImodel}
\hat{h}_{0} &=& t \sum^2_{j=1} \sin(k_j a) \Gamma_j + \big[ m + t_0 \sum^{2}_{j=1} \left( \cos (k_j a)-1 \right) \big] \Gamma_3, \nonumber \\
\hat{h}_{1} &=& \Delta \left[ \cos(k_x a) - \cos(k_y a) \right] \Gamma_4,\end{aligned}$$ where $\Gamma_j$’s are mutually anticommuting four-component Hermitian matrices, satisfying $\{ \Gamma_j, \Gamma_k \}=2 \delta_{jk}$ for $j,k=1, \cdots, 5$, $a$ is the lattice spacing (set to be unity) and ${\bf k}$ is spatial momenta. For $0<m/t_0<2$, $\hat{h}_{0}$ describes a first-order topological insulator. But, depending on the spinor basis and the corresponding representation of $\Gamma$ matrices (about which more in a moment), this phase represents a quantum spin-Hall insulator (QSHI) or a topological $p$-wave pairing. On the other hand, $\hat{h}_1$ lacks both $C_4$ and ${\mathcal T}$ symmetries. It (1) acts as a mass for the edge modes, since $\{ \hat{h}_1, \hat{h}_0 \}=0$, and (2) changes sign under the $C_4$ rotation, thus assuming the profile of a domain wall. Then a generalized Jackiw-Rebbi index theorem [@Jackiw-Rebbi], guarantees the existence of four corner localized zero energy modes, with $d_c=2$. We then realize a second-order topological insulator. Respectively for charged and neutral fermions, $\hat{h}_1$ represents either a spin-orbital density wave ordering and a $d$-wave pairing. In the latter case, the resulting phase stands as HOT $p+id$ pairing [@wang1-2018].
We here seek to answer the following question. *What is the most general form of the Hamiltonian operator that supports topologically protected corner modes at precise zero energy and describes a two-dimensional HOT insulator?* We note that the corner modes are pinned at zero energy due to the spectral symmetry of $\hat{h}^{\rm 2D}_{\rm HOT}$, generated by a unitary ($U$) as well as an antiunitary ($A$) operators, such that $\{ \hat{h}^{\rm 2D}_{\rm HOT}, U \}=0=\{ \hat{h}^{\rm 2D}_{\rm HOT}, A \}$. Since the maximal number of mutually anticommuting four-component $\Gamma$ matrices is *five* and only four of them appear in $\hat{h}^{\rm 2D}_{\rm HOT}$, one is always guaranteed to find $U=\Gamma_5$. On the other hand, the existence of $A$ can be assured in the following way. Note all representations of mutually anticommuting four-component Hermitian $\Gamma$ matrices are *unitarily equivalent*. Hence, without any loss of generality, we commit to a representation in which $\Gamma_1$ and $\Gamma_2$ ($\Gamma_3$ and $\Gamma_4$) are purely real (imaginary) [@okubo; @herbut-lu; @roy-herbut-halfvortex]. Then $A=K$, where $K$ is the complex conjugation [@antiunitary-comment]. Identification of the antiunitary operator $A$ allows us to construct the most general form of the Hamitonian operator $\hat{h}^{\rm gen}_{\rm HOT}=\hat{h}^{\rm 2D}_{\rm HOT} + \hat{h}_p$, such that $\{ \hat{h}^{\rm gen}_{\rm HOT}, A \}=0$ (with real $\Delta_1$ and $\Delta_2$), where $$~\label{Eq:genHOTIpert}
\hat{h}_{\rm p}= \Delta_1 \; (i \Gamma_1 \Gamma_2) + \Delta_2 \; (i \Gamma_3 \Gamma_4)
\equiv \Delta_1 \; \Gamma_{12} + \Delta_2 \; \Gamma_{34}.$$ For *small* $\Delta_1$ and $\Delta_2$, the system continues to support four zero-energy corner modes \[see Fig. \[Fig:GHOTI\_CornerModes\]\] and a quantized quadrupolar moment $Q_{xy}=0.5$ (modulo 1). The resulting phase then describes a two-dimensional *generalized* higher order topological insulator (GHOTI). However, for sufficiently large $\Delta_1$ or $\Delta_2$, the system enters into a trivial or normal insulating phase, where $Q_{xy}=0$ (modulo 1), following a band gap closing (see Fig. \[Fig:bandstructure\]). These findings are summarized in Fig. \[Fig:PD\_bandstructure\]. The physical meanings of $\Delta_1$ and $\Delta_2$ are of course representation dependent [@antiunitary-wavefunction].
{width="0.19\linewidth"}{width="0.19\linewidth"}{width="0.19\linewidth"}{width="0.19\linewidth"}{width="0.19\linewidth"}
\[Fig:GHOTI\_CornerModes\]
*Charged fermions*. We first focus on charged fermions and introduce a four-component spinor $\Psi^\top_{\bf k}= \big( c^{\bf k}_{A,\uparrow}, c^{\bf k}_{B,\uparrow}, c^{\bf k}_{A,\downarrow}, c^{\bf k}_{B,\downarrow} \big)$, where $c^{\bf k}_{X,\sigma}$ is the fermion annihilation operator on sublattice/orbital $X=A,B$ with spin projection $\sigma=\uparrow, \downarrow$ and momenta ${\bf k}$. Then $\hat{h}_0$ describes a QSHI (for $0<m/t_0<2$), when the $\Gamma$ matrices are $\Gamma_1=\sigma_3 \tau_1$, $\Gamma_2=\sigma_0 \tau_2$, $\Gamma_3=\sigma_0 \tau_3$, $\Gamma_4=\sigma_1 \tau_1$ and $\Gamma_5=\sigma_2 \tau_1$. The Pauli matrices ${\boldsymbol \sigma} ({\boldsymbol \tau})$ operate on the spin (sublattice/orbital) degrees of freedom. In this representation $A=\Gamma_1 K$, and $\Delta_1 \; (\Delta_2)$ corresponds to anomalous charge Hall (spin and orbital density-wave) order.
Note that $\hat{h}_0$ preserves both time-reversal (${\mathcal T}$) and parity (${\mathcal P}$) or inversion symmetries. Under the reversal of time ${\bf k} \to -{\bf k}$ and $\Psi_{\bf k} \to \sigma_2 \tau_0 \Psi_{-{\bf k}}$. Hence, ${\mathcal T}= \Gamma_1 \Gamma_4 K$ and ${\mathcal T}^2=-1$. Under the spatial inversion ${\bf r} \to -{\bf r}$ and $\Psi_{\bf k} \to \sigma_0 \tau_3 \Psi_{-{\bf k}}$, yielding ${\mathcal P}=\Gamma_3$. By contrast, $\hat{h}_1$ breaks ${\mathcal T}$, ${\mathcal P}$ as well as discrete $C_4$ rotation about the $z$-axis ($\hat{C}^z_4$), under which $(k_x,k_y) \to (k_y,-k_x)$ and $\hat{C}^z_4=\exp[i \frac{\pi}{4} \sigma_3 \tau_3] \equiv \exp[i \frac{\pi}{4} \Gamma_{12}]$. Nonetheless, one can define a ‘pseudo’ time-reversal operator ${\mathcal T}_{\rm ps}=i \sigma_2 \tau_3 K = \Gamma_2 \Gamma_5 K$, under which ${\bf r} \to -{\bf r}$ as well, such that $\left[ \hat{h}^{\rm 2D}_{\rm HOT}, {\mathcal T}_{\rm ps} \right]=0$ and ${\mathcal T}^2_{\rm ps}=-1$. Consequently, the valence and conduction bands of a HOT insulator ($\Delta_1=0=\Delta_2$) possess two-fold degeneracy \[see Fig. \[Fig:bandstructure\] (first column)\].
Once we turn on $\hat{h}_{\rm p}$ \[see Eq. (\[Eq:genHOTIpert\])\], the bands loose the two-fold degeneracy (see Fig. \[Fig:bandstructure\]). Note that under ${\mathcal T}$, ${\mathcal P}$ and ${\mathcal T}_{\rm ps}$, the term proportional to $\Delta_1$ ($\Delta_2$) is odd (even), even (odd) and odd (odd). Therefore, it is *impossible* to find an antiunitary operator that commutes with $\hat{h}^{\rm gen}_{\rm HOT}$ and squares to $-1$. As a result, the energy spectra of $\hat{h}^{\rm gen}_{\rm HOT}$ only contains non-degenerate bands. Still $\{ \hat{h}^{\rm gen}_{\rm HOT}, A \}=0$, assuring the spectral symmetry among the bands about the zero energy. It is worth pointing out that $\hat{h}^{\rm gen}_{\rm HOT}$ is *algebraically* similar to the *generalized* Jackiw-Rossi Hamiltonian, yielding zero-energy modes bound to the core of a vortex in $d=2$ [@herbut-lu; @roy-herbut-halfvortex; @Jackiw-Rossi; @chamon-GJR; @Roy-Goswami-GJR].
Next, we assess the stability of the HOT insulator in the presence of two perturbations, $\Delta_1$ and $\Delta_2$. As shown in Fig. \[Fig:bandstructure\] (second column) that despite loosing the two-fold degeneracy, the bands are still gapped for small $\Delta_1$ and/or $\Delta_2$. But, at an intermediate $\Delta_1$ or $\Delta_2$ the band gap closes either at the $\Gamma$ point (top row) or along the $\Gamma-{\rm M}$ line (bottom row) of the Brillouin zone \[see Fig. \[Fig:bandstructure\] (third column)\]. The line of the band gap closing at the $\Gamma$ point is given by $\Delta_1 =[m^2 + \Delta^2_2]^{1/2}$ (see the solid line in Fig. \[Fig:PD\_bandstructure\]). On the other hand, the gap closing along the $\Gamma-{\rm M}$ line takes place at momenta ${\bf k}=(\pm,\pm)k_\ast$ and the corresponding phase boundary (the dashed line in Fig. \[Fig:PD\_bandstructure\]) is determined by $\Delta_2 =[\Delta^2_1 + 2 t^2_0 \; \sin^2 (k_\ast)]^{1/2}$, where $k_\ast=\cos^{-1}\left( \frac{m-2t_0}{2 t_0}\right)$. At the gap closing points, the system is described in terms of linearly dispersing massless two-component Weyl fermions at low energies. For stronger $\Delta_1$ or $\Delta_2$, the system reenters into an insulating (but trivial) phase (see the fourth column of Fig. \[Fig:bandstructure\]). Note that the phase boundaries between GHOTI and the trivial insulator do not depend on $\Delta$, as $\hat{h}_1$ vanishes at the $\Gamma$ point and along the $\Gamma - {\rm M}$ line.
We now anchor the topological nature of these insulators, separated by a band gap closing. To this end, we numerically diagonalize the effective tight-binding model, namely $\hat{h}^{\rm gen}_{\rm HOT}$, on a square lattice of linear dimension $L$ and with an open boundary in each direction for various choices of $\Delta_1$ and $\Delta_2$. The results are shown in Fig. \[Fig:GHOTI\_CornerModes\]. For $\Delta_1=0=\Delta_2$, the system supports four near (due to a finite system size) zero energy states that are highly localized near the corner of the system, yielding a conventional HOT insulator \[see Fig. \[Fig:GHOTI\_CornerModes\](a)\].
An HOT insulator can be identified from the quantized quadrupolar moment $Q_{xy}=1/2$ (modulo 1) [@multipole1; @multipole2; @agarwala]. In order to compute $Q_{xy}$, we first evaluate $$n={\rm Re} \left[ -\frac{i}{2 \pi} {\rm Tr} \left( \ln \left\{ U^\dagger \exp \left[ 2 \pi i \sum_{\bf r} \hat{q}_{xy} ({\bf r}) \right] U \right\} \right) \right],$$ where $\hat{q}_{xy} ({\bf r})= x y \; \hat{n}({\bf r})/L^2$ and $\hat{n}({\bf r})$ is the number operator at ${\bf r}=(x,y)$, and $U$ is constructed by columnwise arranging the eigenvectors for the negative energy states. The quadrupolar moment is defined as $Q_{xy}=n-n_{\rm al}$, where $n_{\rm al}=(1/2) \; \sum_{\bf r} x y /L^2$ represents $n$ in the atomic limit and at half filling. Indeed for a HOT insulator, we find $Q_{xy}=0.5$ (within numerical accuracy). While a quantized quadrupolar moment is solely supported by the $C_4$ symmetry breaking Dirac mass ($\hat{h}_1$), the antiunitary operator ($A$) allows us to construct GHOTI.
For finite but small $\Delta_1$ and/ or $\Delta_2$, the system continues to support four corner localized zero-energy modes, and describes a GHOTI \[see Figs. \[Fig:GHOTI\_CornerModes\](b)-\[Fig:GHOTI\_CornerModes\](e)\], with $Q_{xy}=0.5$. However, with increasing $\Delta_1$ or $\Delta_2$, they gradually loose support at the corners. But, the system still continues to describe a GHOTI up to critical values of $\Delta_1$ and $\Delta_2$. Finally, beyond the band gap closing the system enters into a trivial insulating phase, where $Q_{xy}=0$. Hence, $\hat{h}^{\rm gen}_{\rm HOT}$ describes a HOT phase for small $\Delta_1$ or $\Delta_2$.
Before leaving the territory of charged fermions, we demonstrate the applicability of the above construction of GHOTI in the context of the original model of the two-dimensional HOT insulator introduced in Ref. [@benalcazar2017], the Belancazar-Bernevig-Hughes (BBH) model. The corresponding Hamiltonian operator reads $\hat{h}^{\rm BBH}_{\rm HOT}=\hat{h}^\prime_1 + \hat{h}^\prime_2$, with $$\hat{h}^\prime_{j}= \lambda_1 \sin(k_j a) \; \gamma_j + \left[ \beta + \lambda_2 \cos(k_j a) \right] \; \gamma_{2+j},$$ for $j=1,2$, where $\gamma_j$’s are mutually anticommuting four-component Hermitian matrices, satisfying $\{ \gamma_j, \gamma_k \}=2\delta_{jk}$. Notice $\hat{h}^\prime_j$ describe Su-Schrieffer-Heeger (SSH) chain in the $x$ and $y$ direction, respectively for $j=1$ and $2$. Specifically for $|\beta/\lambda_2|<1$, each SSH chain supports two endpoint zero energy modes [@SSH-original]. Decoupled $x$ and $y$ SSH chains respectively support a string of such endpoint zero modes along the $y$ and $x$ direction. However, the BBH model supports zero-energy modes only at the four corners, where both SSH chains place endpoint zero modes, yielding a second order topological insulator. This is so, since $\hat{h}^\prime_1$ acts as mass for the zero modes of $\hat{h}^\prime_2$ and vice versa as $\{ \hat{h}^\prime_1, \hat{h}^\prime_2 \}=0$. Notice $\hat{h}^{\rm BBH}_{\rm HOT}$ assumes the form of $\hat{h}^{\rm 2D}_{\rm HOT}$ \[see Eq. (\[Eq:HOTImodel\])\], with $\Gamma_1=\gamma_1$, $\Gamma_2=\gamma_2$, $\Gamma_3=\gamma_+$, $\Gamma_4=\gamma_-$, where $\gamma_\pm =\gamma_3 \pm \gamma_4$, and $t=\lambda_1$, $m=\beta+2t_0$, $t_0=\Delta=\lambda_2/2$. Therefore, our discussion on the GHOTI is equally germane to the BBH model. Without exploiting this correspondence, we can choose (without loss of generality) $\gamma_{1,2}$ ($\gamma_{3,4}$) to be purely real (imaginary), and construct GHOTI from the BBH model, respecting the spectral symmetry generated by $A=K$ and described by the Hamiltonian $\hat{h}^{\rm BBH}_{\rm HOT} + i \Delta_1 \gamma_{1} \gamma_{2} + i \Delta_2 \gamma_{3} \gamma_{4}$.
*HOT pairing*. As a penultimate topic, we focus on two-dimensional HOT superconductor, for which the Nambu spinor is $\Psi^\top_{{\bf k}}= (c_{{\bf k}, \uparrow}, c_{{\bf k}, \downarrow}, c^\ast_{-{\bf k}, \downarrow}, -c^\ast_{-{\bf k}, \uparrow})$ and $c^\ast_{{\bf k},\sigma} (c_{{\bf k},\sigma})$ is the creation (annihilation) operator for the fermionic quasiparticles with momenta ${\bf k}$ and spin projection $\sigma=\uparrow, \downarrow$. The $\Gamma$ matrices are $\Gamma_1=\eta_1 \sigma_1$, $\Gamma_2=\eta_1 \sigma_2$, $\Gamma_3=\eta_3 \sigma_0$, $\Gamma_4=\eta_2 \sigma_0$ and $\Gamma_5=\eta_1 \sigma_3$. The Pauli matrices ${\boldsymbol \eta}$ operate on the Nambu or particle-hole index. The parameter $t$ ($\Delta$) from Eq. (\[Eq:HOTImodel\]) represents the amplitude of the $p$ ($d$)-wave pairing, and the term proportional to $t_0$ yields a Fermi surface for $0<m/t_0<2$. Under that circumstance, a weak coupling $p+id$ pairing takes place around the Fermi surface and we realize a second-order topological superconductor, supporting four corner localized Mojorana zero modes [@wang1-2018]. It is worth noting that a mixed parity time-reversal odd $p+is$ pairing, by contrast, only supports gapped Majorana fermions [@goswami-roy-axion].
In the above-mentioned representation, $\Delta_1$ denotes the Zeeman coupling, while $\Delta_2$ corresponds to the amplitude of spin-singlet (real) $s$-wave pairing. Hence, our discussion on GHOTI suggests that a two-dimensional HOT pairing can be realized in the form of $p+s+id$ pairing even in the presence of (sufficiently weak) Zeeman coupling, at least when the amplitude of the $s$-wave pairing is small enough. Therefore, a quantum phase transition between HOT and a trivial paired state can be triggered by tuning the Zeeman coupling between the quasiparticles and external magnetic field.
{width="0.07\linewidth"}{width="0.45\linewidth"}{width="0.45\linewidth"}
\[Fig:HOTWeyl\]
Note that when a $d$-wave pairing sets in, it also causes a lattice distortion or electronic nematicity in the system that in turn induces a (small) $s$-wave pairing [@roy-ghorashi-foster-nevidomskyy]. Nonetheless, the amplitude of the $s$-wave pairing can be amplified and the system can also be tuned through the HOT-trivial pairing critical point by applying an external uniaxial strain along the $\langle 11 \rangle$ directions, for example.
*Three dimensions*. Using two-dimensional GHOTI as the building blocks, one can construct three-dimensional HOT phases, by stacking them along the $k_z$ direction in the momentum space. This is accomplished by replacing the term proportional to $\Gamma_3$ in Eq. (\[Eq:HOTImodel\]) by $$\Gamma_3 \left[ t_z \cos(k_z a) + m + t_0 \; \left\{ \cos(k_x a)+\cos(k_y a)-2 \right\} \; \right]. \nonumber$$ For example, when $\Delta_1=\Delta_2$, the system describes a second order Weyl semimetal (since all bands are non-degenerate) with two Weyl nodes at $(0,0,\pm k^\ast_z)$, where $k^\ast_z=\cos^{-1}(|m|/t_z)$ for $t_z>|m|$ and $m/t_0<1$. It supports localized one-dimensional *hinge* modes for $|k_z| < k^\ast_z$ \[see Fig. \[Fig:HOTWeyl\](a)\]. However, the corner localization of the hinge modes decreases monotonically as one approaches the Weyl nodes from the center of the Brillouin zone ($k_z=0$), similar to the situation with the Fermi arcs of a first-order Weyl semimetal (WSM) [@roy-Fermiarc; @arc-hinge]. Within this range of $k_z$, the quadrupolar moment is quantized to $0.5$, but vanishes for $|k_z|>k^\ast_z$ \[see Fig. \[Fig:HOTWeyl\](b)\]. By contrast, for $\Delta_2=0$, four Weyl nodes appear at $(0,0,\pm k^\alpha_z)$, where $k^\alpha_z=\cos^{-1}([m+\alpha \Delta_1]/t_z)$ for $\alpha=\pm$. Four pairs of Weyl nodes can be found at $(\pm k_0, \pm k_0, \pm k^0_z)$ when $\Delta_1=0$, where $k_0=\sin^{-1}(\Delta_2/[\sqrt{2} t_0])$ and $k^0_z=\cos^{-1}([m-2t_0-2t_0 \cos(k_0)]/t_z)$. A complete analysis of three-dimensional second-order Weyl semimetals in the $(\Delta_1, \Delta_2)$ plane is left for a future investigation. It should be noted that so far only second-order Dirac semimetals (supporting linearly touching Kramers *degenerate* valence and conduction bands) have been discussed in the literature [@lin-hughes-DSM; @wang1-2018; @calugaru2019], whereas we here demonstrate that it is conceivable to realize its Weyl counterparts (yielding linear touching between Kramers *non-degenerate* bands), protected by an *antiunitary symmetry*.
*Summary and discussions*. To summarize, we identify an antiunitary operator ($A$) that assures the spectral symmetry of a two-dimensional HOT insulator \[see Eq. (\[Eq:HOTImodel\])\] and pins four corner modes at precise zero energy. Such an antiunitary symmetry allows us to construct a GHOTI for charged as well as neutral fermions, in terms of two additional perturbations \[see Eq. (\[Eq:genHOTIpert\])\], that continues to support corner localized zero-energy mode (see Figs. \[Fig:PD\_bandstructure\] and \[Fig:GHOTI\_CornerModes\]), at least when they are small. In particular, our findings suggest that the corner localized Majorana zero modes of a HOT $p+id$ superconductor survive even in the presence of a weak Zeeman coupling and a parasitic or strain engineered $s$-wave pairing. Concomitantly, a transition between a HOT to trivial paired state can be triggered by tuning the strength of the external magnetic field or uniaxial strain, which can be instrumental for topological quantum computing based on Majorana fermions. The proposed anitiunitary symmetry protected corner and hinge modes can also be observed in highly tunable metamaterials, such as electrical circuits [@junkai].
*Acknowledgments*. The author thanks Vladimir Juriči' c, Soumya Bera, and Junkai Dong for discussions. B.R. was partially supported by the start-up grant from Lehigh University.
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A first-order WSM can be constructed by stacking 2D quantum anomalous Hall insulators (QAHIs), occupying the $xy$ plane in the momentum space (along $k_z$, for example) between two Weyl nodes. The locus of the zero-energy modes associated with the 1D edge modes of each layer of QAHI constitutes the Fermi arc [@roy-Fermiarc]. By contrast, a second-order WSM is constructed by stacking 2D GHOTI (supporting only pointlike corner modes) yielding 1D hinge states, as shown in Fig. \[Fig:HOTWeyl\](a).
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|
---
abstract: '[ We show that the ultraviolet absorption features, newly discovered in HST spectra, are consistent with being formed in a layer that extends a few kpc above the disk of the Milky Way. In this interface between the disk and the Galactic corona, high-metallicity gas ejected from the disk by supernova feedback can mix efficiently with the virial-temperature coronal material. The mixing process triggers the cooling of the lower corona down to temperatures encompassing the characteristic range of the observed absorption features, producing a net supernova-driven gas accretion onto the disk at a rate of a few ${\,{{\rm M}_\odot\,\rm yr}^{-1}}$. We speculate that this mechanism explains how the hot-mode of cosmological accretion feeds star formation in galactic disks. ]{}'
author:
- Filippo Fraternali
- Antonino Marasco
- Federico Marinacci
- James Binney
title: 'Ionized absorbers as evidence for supernova-driven cooling of the lower Galactic corona'
---
Introduction
============
How the Milky Way and other disk galaxies acquire gas from their surrounding environment is a long-standing problem in galaxy evolution. Several pieces of evidence, for example studies of the star formation in the Solar neighborhood and chemical evolution models [e.g., @Chiappini+97], show that gas accretion is needed to maintain star formation. Since their discovery, high-velocity clouds (HVCs) have been regarded as the main source of neutral ([[H]{}[I]{}]{}) gas accretion in the Galaxy [e.g., @Oort66]. However, the recent determination of the distances to the main complexes [e.g., @Wakker+08] has led to estimates of the accretion rate from HVCs that are at least an order of magnitude less than what is required [@Putman+12]. The same is true for HVCs found in external galaxies [@Fraternali09]. Mergers of gas rich satellites also appear to fall short in providing the required amount of gas accretion [@Sancisi+08].
During the last years, the possibility that gas accretion manifests itself mostly as ionized gas has attracted a growing interest. Ultraviolet spectra against bright AGNs obtained with the Space Telescope Imaging Spectrograph (STIS) and the Cosmic Origins Spectrograph (COS) on board of the Hubble Space Telescope (HST) have revealed a number of ionized species at velocities incompatible with those of the Galactic disk material [e.g., @Shull+09]. The ions include [[C]{}[II]{}]{}, [[C]{}[III]{}]{}, [[C]{}[IV]{}]{}, [[Si]{}[II]{}]{}, [[Si]{}[III]{}]{}, and [[Si]{}[IV]{}]{} and cover a range of temperatures (assuming collisional ionization equilibrium, CIE) from a few times $10^4{\,{\rm K}}$ to $2\times10^5{\,{\rm K}}$. They fill about 70-90% of the sky and they often have neutral gas associated with the absorption [@Collins+09; @Lehner+12]. In a few cases they have been identified in the spectra of halo stars, showing that at least a subsample of them lie at distances within about $10 {\,{\rm kpc}}$ of the Galactic plane and ruling out formation in the distant corona or in the Local Group medium.
From a theoretical point of view, the accretion of fresh gas into galaxy disks is a non-trivial problem that has not yet reached a definitive solution. The scheme that has recently enjoyed most success is that of the so-called cold flows , where baryons reach the center of a potential well without going through a phase of virialization. Cosmological simulations suggest that cold flows are the main channel for gas accretion in the early Universe, but they should be replaced by a hot accretion mode for a galaxy like the Milky Way at $z\lesssim1$ [e.g., @Keres+09]. Thus, the problem of how to cool the hot gas contained in an extended virial-temperature corona and transfer it to the disk to feed star formation still remains. The formation of gas clumps in the corona via thermal instabilities has been explored [e.g., @Kaufmann+06], but it has been ruled out both on theoretical [e.g., @Binney+09; @Hobbs+12] and observational [e.g., @Pisano+07] grounds. Using hydrodynamical simulations, @Marinacci+10a showed that gas at the typical temperature of the Galactic corona can cool efficiently if it is mixed with high-metallicity, cooler disk material. This mixing is enhanced by the onset of a galactic fountain circulation [e.g. @Bregman80]. @Marasco+12 [hereafter MFB12] used the fountain model of , together with the results of Marinacci et al.’s simulations, to reproduce the kinematics of the [[H]{}[I]{}]{} in the halo of the Milky Way. In this Letter, we show that the MFB12 model also explains the observational properties of the ionized absorbers detected by @Lehner+12.
The model
=========
We use the supernova-driven accretion model presented in MFB12, where cloud particles are ejected from the disk and move through the halo region interacting with the pre-existing Galactic corona. The details of this interaction are derived from the hydrodynamical simulations presented in @Marinacci+10a [@Marinacci+11 this latter hereafter M11]. In these simulations, a high-metallicity fountain cloud moves through a hot, low-metallicity plasma, representing the Galactic corona. Ram-pressure stripping and the onset of the Kelvin-Helmholtz instability produce a turbulent wake in which coronal gas is entrained and mixes efficiently with the disk material. The resulting medium at intermediate temperatures and metallicities, trailing the cloud front, can further cool down to recombination temperatures and produce a net transfer of mass from the hot coronal phase to the cold phase. This condensed coronal material is eventually transferred to the disk where it can feed star formation (see Fig. \[scheme\]). This mass transfer, together with a corresponding exchange of momentum can significantly alter the clouds’ trajectories, producing a detectable kinematic signature. MFB12 used this model to fit the kinematics of the Galactic [[H]{}[I]{}]{} halo in the LAB 21-cm Survey [@Kalberla+05] and they determined that the mixing of disk and coronal gas triggers condensation and subsequent accretion of the latter at a rate of $\sim 2 {\,{{\rm M}_\odot\,\rm yr}^{-1}}$. Here, we use the same model to predict the properties of the material at intermediate temperatures in the wakes of the fountain clouds and compare them with those of the absorption features detected by @Lehner+12. We do not perform a new fit but simply used the best-fit parameters that reproduced the kinematics of the [[H]{}[I]{}]{} halo.
From the simulations of M11 we selected only the gas in the temperature range $4.3\!<\!{\log}(T\!/\!{\,{\rm K}})\!<\!5.3$. This range is representative for the species [[Si]{}[III]{}]{}, [[Si]{}[IV]{}]{}, [[C]{}[II]{}]{}, [[C]{}[III]{}]{} and [[C]{}[IV]{}]{} assuming CIE . The gas selected with this temperature cut - hereafter the ‘warm’ gas - is located in the wake of the cloud, typically within 2 kpc of the cold front. We studied the evolution of this warm gas with time, in particular its mass ratio and velocity difference with respect to the [[H]{}[I]{}]{} (assumed to be at $\log(T\!/\!{\,{\rm K}})<4.3$) and its velocity dispersion. We found that the warm gas lags $10-20{\,{\rm km\,s}^{-1}}$ behind the cold [[H]{}[I]{}]{} front and develops a turbulent motion with a velocity dispersion of $\sim30{\,{\rm km\,s}^{-1}}$, which dominates over the thermal broadening. The details of this analysis are given in a parallel paper (Marasco et al., MNRAS, submitted). The dynamical model of MFB12 gives us the column-density of neutral gas as a function of position in the sky and line-of-sight radial velocity. From this, using the above analysis, we can predict the column-density of the warm gas at every location in the position-velocity space, which can then be compared with the HST absorption data. The photo-ionizing flux from the disk largely dominates the extragalactic contribution [@Shull+09]. Thus photo-ionization is probably non-negligible in the early stages of the clouds’ trajectories: MFB12 found that the best fit to the [[H]{}[I]{}]{} data is obtained if the particles are ionized, and therefore are not visible in [[H]{}[I]{}]{}, for $30\%$ of the ascending part of their trajectory. These ionized outflows are included in our model.
Results
=======
{width="\textwidth"}
Fig. \[allsky\] gives an all-sky view of the velocity field predicted by our model compared to the velocity centroids of the absorption lines detected by @Lehner+12. Where there are multiple detections along a single line of sight, we have plotted the average velocity. Twelve absorption systems out of 84 are considered to be related to the Magellanic Cloud/Stream and are not taken into account in our analysis. The top panel of Fig. \[allsky\] shows the median velocity field predicted by the model, while in the bottom panel velocities are extracted randomly in cells of $2.5{^{\circ}}\times2.5{^{\circ}}$ to give a measure of the turbulent motions in the wakes. In both cases, we excluded warm gas at velocities $|{v_{\rm LOS}}|\!<\!90{\,{\rm km\,s}^{-1}}$, as was done in the observations. The detections are not distributed isotropically in the sky: the targeted background sources are located mostly at positive latitudes, and no targets are present for $|b|<15{^{\circ}}$. Globally, the median velocity field predicts the correct dichotomy between detections at positive and negative velocities, indicating that the absorbing material is consistent with being part of a slowly rotating medium similar to that produced by the interaction between the galactic fountain and the corona [see also @Fox+06; @Shull+09]. However, the data show large fluctuations around the predicted median value. The random velocity field (bottom panel) shows that fluctuations of similar amplitude are present also in our model. They are caused by the large velocity dispersion ($\sigma\!=\!30{\,{\rm km\,s}^{-1}}$) of the warm material in the turbulent wakes of the fountain clouds.
![ Longitude-velocity diagrams in four different latitude bins (bottom left corners) showing the confidence contours of our model of supernova-driven corona cooling. The circles show the location in longitude and velocity of the HST absorbers detected by @Lehner+12. The empty circles are detections considered related to the Magellanic Clouds/Stream, whose [[H]{}[I]{}]{} emission is shown as the black thick contours. Five detections at very high-velocities, four of which associated to the Magellanic Stream, are not shown in this plot. []{data-label="lv"}](lv.jpg){width="60.00000%"}
Fig.\[lv\] shows the longitude-velocity distribution of the warm gas in our model in four different latitude bins. The absorption features of @Lehner+12 are over-plotted on these diagrams as filled and empty (if associated to the Magellanic Clouds/Stream) points. The contours enclose $68\%$, $95\%$ and $99.7\%$ of the total flux present in the model, and are proportional to the probability of finding a detection at a given position and velocity if the background sources were isotropically distributed in the sky. Even though this is not the case, the absorbers follow a trend very similar to that predicted. We infer a fraction of detections that is consistent with our model by comparing the predicted and the observed distributions using a Kolmogorov-Smirnov test. The details are discussed in Marasco et al. (MNRAS, submitted), where it is shown that the positions and velocities of $94^{+6}_{-3}\%$ of the UV absorbers are reproduced by our model. Thus, a large fraction of these ion absorbers are likely to be produced in the wakes of fountain clouds. These clouds would appear in [[H]{}[I]{}]{} as intermediate-velocity clouds (IVCs) (MFB12) but the ionized gas in the wake may have HVC-like velocities because it has a different kinematics and a much larger turbulence with respect to the [[H]{}[I]{}]{}. Interestingly, the fraction of ion absorbers reproduced by our model does not vary significantly if the detections that overlap with the classical [[H]{}[I]{}]{} HVCs are excluded from the calculation.
@Shull+09 obtained an average column density for the high-velocity [[Si]{}[III]{}]{} absorption lines of ${\left< \log N_{\rm Si\,III} \right>}=13.42\pm0.21$, with velocity widths ranging from $40$ to $100{\,{\rm km\,s}^{-1}}$. We compared this value with the prediction of our model by integrating line profiles of the warm gas over $70{\,{\rm km\,s}^{-1}}$ around the velocity centroid. To convert this value into a [[Si]{}[III]{}]{} column density, we further assumed Solar abundance ratios and used the average metallicity of the warm gas in the simulations ($\log[Z/Z_{\odot}]\!=\!-0.24$) with the maximum [[Si]{}[III]{}]{} fraction in CIE . We found a [[Si]{}[III]{}]{} column density, averaged over all the absorbers, of ${\left< \log N_{\rm Si\,III} \right>}=13.44\pm0.36$. Hence, the density of gas at intermediate temperatures produced in our model by mixing the fountain material with the hot Galactic corona is in remarkable agreement with the observations. Note that these column densities are derived assuming CIE, and this agreement may show that the gas is not too far from it. However, a doubling or tripling of the ion density by photo-ionization would be still compatible within the errors. We stress again that our model has not been fitted to the absorption data, the only fit that has been performed is on the [[H]{}[I]{}]{} component, which is related to the ionized material only by the cloud-corona interaction and the underlying physics of the turbulent mixing. Thus, this comparison strongly supports the validity of our approach.
A further comparison between our model and observations comes from counting the number of fountain-cloud wakes intercepted along the lines of sight. In our model, the Galactic halo is populated by $\sim10^4$ fountain clouds, given the total [[H]{}[I]{}]{} halo mass ($\sim3\times10^8{\,{\rm M}_\odot}$; MFB12) and the mass of a typical (intermediate-velocity) cloud [few $\times10^4{\,{\rm M}_\odot}$; @vanWoerden+04]. We considered that these $10^4$ clouds are distributed around the Galactic plane with the density profile obtained by MFB12 for the [[H]{}[I]{}]{} halo. We assumed that each cloud has an associated wake, for which the volume occupied by the warm gas is derived from the M11 simulations. With the above information we estimated that an observer placed at the position of the Sun should intercept an average of $0.5-1.0$ wakes per line of sight, which nicely compares with the average number of $\sim0.7$ detections per line of sight in the dataset of @Lehner+12.
@Sembach+03 and @Savage+03 provide locations, velocities and column-densities for high-ionization ([[O]{}[VI]{}]{}) absorbers surrounding the Milky Way. Marasco et al. (MNRAS, submitted) show that more than half of these absorbers are compatible with the same model of supernova-driven accretion presented here. @Shull+09 provide velocity ranges for Si ions ([[Si]{}[II]{}]{}, [[Si]{}[III]{}]{}, [[Si]{}[IV]{}]{}) found in STIS and FUSE spectra of 37 AGNs. We used the central velocities of these ranges and found that only 33.4 % of these features are compatible with our model. This percentage becomes 63.2 % when the dataset of @Shull+09 is considered together with that of @Lehner+12. This result is surprising as it seems to show the two dataset are not fully compatible. It appears that most of the difference is due to a number of absorbers at high negative velocities in the region $20<l<120$ in the @Shull+09 sample. Although this requires further investigation, it is possible that these absorbers lie in a large ionised envelope surrounding the Magellanic Stream [@BlandHawthorn+07].
Conclusions
===========
We have shown that the newly discovered UV absorption features in the Galactic halo, observed in HST spectra of bright AGNs and some halo stars, are largely consistent with the predictions of supernova-driven cooling of the Galactic corona. To make this comparison, we extended the supernova-driven accretion model of MFB12 by including the intermediate-temperature gas generated by the interaction between the galactic fountain clouds and the corona. We did not perform a new fit but used the best-fit parameters that match the [[H]{}[I]{}]{} kinematics in the LAB survey. Our model is able to reproduce the positions and velocities of the vast majority of the detected absorbers. Moreover, the model predicts the correct mean column density and the number of intervening absorbers along the line of sight.
Our findings support the idea that the vast majority of the absorbers are produced in a turbulent multi-phase layer a few kpc thick surrounding the Galactic plane. This layer is created by supernova feedback, which produces a galactic-fountain cycle that fosters the interaction between the high-metallicity disk material and the corona. The mixing of the two media lowers the temperature and increases the metallicity of the coronal gas, thus reducing dramatically its cooling time. As a consequence, part of the lower corona cools to lower and lower temperatures encompassing the range characteristic of the species considered in this work. When recombination occurs the gas is potentially visible in [[H]{}[I]{}]{}, but being buried within the fountain cycle it cannot be directly detected. This is the reason why cold gas accretion has escaped detection for so long. The only way to unveil its presence is via the effects that it has on the kinematics of the [[H]{}[I]{}]{} halo . This process produces a net accretion of fresh gas from the lower corona onto the disk at a rate of a few ${\,{{\rm M}_\odot\,\rm yr}^{-1}}$.
In Fig. \[accretionMW\] we compare the accretion rates onto the Galaxy that one can infer from the currently available gas sources. The classical HVCs contribute only $0.08{\,{{\rm M}_\odot\,\rm yr}^{-1}}$, and only half of this gas is in the neutral phase [@Putman+12]. This value is uncertain and somewhat debated but still of the same order of magnitude as the previously often quoted value of $0.1-0.2 {\,{{\rm M}_\odot\,\rm yr}^{-1}}$ [@Wakker+07]. Direct accretion of cold gas from satellites is only visible in the Magellanic Stream, which is a sporadic event and it is very unlikely to reach the Galactic disk before being ablated and thermalized. If the Magellanic Stream survives the journey, it will merge with the Galactic ISM on a timescale likely larger than $1{\,{\rm Gyr}}$ [@Putman+12]. Given the [[H]{}[I]{}]{} mass of the Stream, $\sim1.2\times10^8{\,{\rm M}_\odot}$ [@Bruns+05], this gives an upper limit to the accretion rate of $\sim0.16{\,{{\rm M}_\odot\,\rm yr}^{-1}}$. The galactic fountain instead produces an accretion of cold pristine gas onto the disk at a rate of $\sim2{\,{{\rm M}_\odot\,\rm yr}^{-1}}$ (MFB12). This value is remarkably similar to the current SFR of the Galaxy, which lies in the range $1-3{\,{{\rm M}_\odot\,\rm yr}^{-1}}$ . In addition, from the present analysis, we infer a further accretion of $\sim1{\,{{\rm M}_\odot\,\rm yr}^{-1}}$ of gas in the ionized phase, but it is unclear whether this gas can take part in the star formation process or not. Note that this value agrees with the estimates of both @Lehner+12 and @Shull+09. All the above estimates are corrected for He abundance. Clearly, the coronal material harvested by the fountain cycle provides the necessary gas supply for star formation to proceed.
![ Comparison between different sources of gas accretion for the Milky Way. The estimate for the fountain-cycle (supernova-driven accretion) comes from fitting the kinematics of the [[H]{}[I]{}]{} halo (MFB12) and it is confirmed by the ionized absorption features studied in this work.[]{data-label="accretionMW"}](accretionMW.jpg){width="\textwidth"}
Current cosmological simulations suggest that galaxies above a virial-mass threshold of a few times $10^{11} {\,{\rm M}_\odot}$ should acquire gas via hot-mode accretion, which mainly feeds the hot corona rather than the star formation in the disk. The galaxy’s central black hole accretes gas at a rate that rises steeply with the corona’s central density, and energy released by this accretion largely offsets the corona’s radiative losses, which are dominated by its dense centre . The Milky Way and similar star-forming galaxies became massive enough to enter this regime of black-hole stabilization at $z{\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$}}1$ and consequently, in simulations, their star formation rates (SFRs) have declined since then by roughly an order of magnitude. In contrast, the star formation histories of the Milky-Way and nearby galaxies of similar masses appear to have declined much more slowly . Supernova-driven accretion provides an explanation for this apparent contradiction, as the presence of an active star-forming disk allows the Galaxy to continuously cool and collect fresh gas from its corona at significant distance from the centre despite episodic re-heating by Sgr A\*. We speculate that supernova-driven gas accretion has been the way in which the cosmological hot-mode accretion has fed the star formation in the Milky Way and similar disk galaxies after the initial cold-mode phase. The ability of the MFB12 model to account so nicely for the HST absorption-line data suggests that this picture is correct.
The same mechanism also explains the dichotomy between the SFRs of blue-cloud and red-sequence galaxies of similar masses. If star formation were sustained by spontaneous cooling of cosmological coronae, it would be difficult to understand why galaxies residing in similar potential wells - and therefore realistically surrounded by similar coronae [@Crain+10] - can have completely different current SFRs. The puzzle remains even if star formation were fed by the tail of the cold-mode accretion as suggested by @Fernandez+12 among others. By contrast, in our scheme it all comes down to the presence or not of a (star-forming) disk of cold gas. The disk effectively acts as a [*refrigerator*]{} that cools and carves out the corona from below. As long as a galaxy is able to retain its gaseous disk it can hope to harvest more cold material from the corona, but when the disk is gone it becomes irreversibly “red and dead”.
[We thank the referee Mike Shull for a constructive report. FF, AM, and FM thank support from PRIN-MIUR 2008SPTACC. FM also acknowledges support from the collaborative research centre “The Miky Way system” (SFB 881) of the DFG. ]{}
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|
---
abstract: 'Out of thousands of names to choose from, picking the right one for your child is a daunting task. In this work, our objective is to help parentsmaking an informed decision while choosing a name for their baby. We follow arecommender system approach and combine, in an ensemble, the individualrankings produced by simple collaborative filtering algorithms in order to produce a personalized list of names that meets the individual parents’ taste.Our experiments were conducted using real-world data collected from the query logs of *nameling* ([nameling.net](nameling.net)), an online portal for searching and exploring names, which corresponds to the dataset released in the context of the ECML PKDD Discover Challenge 2013. Our approach is intuitive, easy to implement, and features fast training and prediction steps.'
author:
- 'Bernat Coma-Puig[^1]'
- 'Ernesto Diaz-Aviles and Wolfgang Nejdl'
bibliography:
- 'biblio.bib'
title: |
Collaborative Filtering Ensemble for\
Personalized Name Recommendation
---
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Asmelash Teka and Rakshit Gautam for their valuable feedback. This work is funded, in part, by the L3S IAI research grant for the *FizzStream!* Project. Bernat Coma-Puig is sponsored by the EuRopean Community Action Scheme for the Mobility of University Students (ERASMUS).
[^1]: Work done at the L3S Research Center as part of the ERASMUS exchangeprogram while a student at Universitat Politècnica de Catalunya – BarcelonaTech (UPC) $<$`bernat.coma@est.fib.upc.edu`$>$.
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abstract: 'We introduce algebraic sets in the products of the complex projective spaces for the mixed states in multipartite quantum systems, which are independent of their eigenvalues and only measure the “position” of their eigenvectors, as their non-local invariants (ie., remaining invariant after local unitary transformation). These invariants are naturally arised from the physical consideration of checking multipartite mixed states by measuring them with the multipartite separable pure states. The algebraic sets have to be the sum of linear subspaces if the multipartite mixed state is separable, and thus we give a new criterion of separability. A continuous family of 4-party mixed states, whose members are separable for any $2:2$ cut and entangled for any $1:3$ cut (thus bound entanglement if 4 parties are isolated), is constructed and studied by our invariants and separability criterion. Examples of LOCC-incomparable entangled tripartite pure states are also given to show that it is hopeless to characterize the entanglement proporties of multipartite pure states by the eigenvalue spectra of their partial traces. We also prove that at least $n^2+n-1$ terms of separable pure states, which are orthogonal in some sence, are needed to write a generic pure state in $H_A^{n^2} \otimes H_B^{n^2} \otimes H_C^{n^2}$ as a linear combination of them.'
author:
- |
Hao Chen\
Department of Mathematics\
Zhongshan University\
Guangzhou,Guangdong 510275\
People’s Republic of China
date: 'July,2001'
title: 'Quantum entanglement without eigenvalue spectra: multipartite case'
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Entanglement, first noted by Einstein, Podolsky and Rosen \[1\] and Schrodinger \[2\], is an essential feature of quantum mechanics. Recently it has been realized that entanglement is a useful resource for various kinds of quantum information processing, including quantum state teleportation (\[3\]), cryptographic key distribution (\[3\],\[4\]), quantum computation (\[5\]), etc., see \[6\] and \[7\].\
Entanglement of bipartite quantum systems, ie., entanglement of pure and mixed states in $H=H_A^m \otimes H_B^n$, has been a matter of intensive research, see \[7\] for a survey. It has been realized that the entanglement of pure tripartite quantum states is not a trivial extension of the entanglement of bipartite systems (\[8\],\[9\]). Recently Bennett etc., \[10\] studied the exact and asymptotic entanglement measure of multipartite pure states, which showed essential difference to that of bipartite pure states. On the other hand Acin, etc., \[11\] proved a generalization of Schmidt decomposition for pure triqubit states, which seems impossible to be generalized to arbitrary multipartite case (see Theorem 5 in this paper). Basically, the understanding of multipartite quantum entanglement for both pure and mixed states, is much less advanced.\
From the point view of quantum entanglement, states in multipartite quantum systems $H=H_{A_1}^{m_1} \otimes \cdots \otimes H_{A_n}^{m_n}$ are completely equivalent if they can be transformed by local unitary transfomations (ie., $U_{A_1} \otimes \cdots \otimes U_{A_n}$, where $U_{A_i}$ is unitary transformation of $H_{A_i}$). The property of states being separable or entangled is clearly preserved under local unitary transformations. Any good measure of multipartite entanglement must be invariant under local unitary transformations (\[12\]). It is obvious that eigenvalues of $tr_{A_{i_1}...A_{i_k}}(\rho)$, for any indices $i_1,...,i_k \in \{1,...,n\}$ and any state $\rho$ in $H$, are invariant under local unitary transformations. Many known invariants \[12\] and entanglement monotone (see \[13\]) are more or less related to eigenvalue spectra of states or their partial traces.\
It is clear that any separability criterion for bipartite mixed states , such as Peres PPT criterion \[14\] and Horodecki range criterion \[7\], can be applied to multipartite mixed states for their separability under various cuts. For example, from Peres PPT criterion, a separable multipartite mixed state necessarily have all its partial transpositions positive. In \[15\], Horodeckis proved a separability criterion for multipartite mixed states by the using of linear maps. Based on range criterion, a systematic way to construct both bipartite and multipartite PPT mixed states (thus bound entanglement) from unextendible product bases was eastablished in \[16\]. Classification of triqubit mixed states inspired by Acin, etc., \[11\] was studied in \[17\].\
In our previous work \[18\] we introduced algebraic sets in complex projective spaces for states in bipartite quantum systems, which are independent of the eigenvalues of the states and only measure the “position” of eigenvectors, as their non-local invariants. These invariants are naturally from the physical consideration to measure the bipartite mixed states by separable pure states. An “eigenvalue-free” separability criterion is also proved, which asserts that the algebraic sets have to be the sum of linear subspaces if the state is separable. This revealed that a quite large part of quantum entanglement is independent of eigenvalue spectra and only dependes on eigenvectors.\
The algebraic set invariants and the separability criterion based on these invariants in \[18\] can be extended naturally to multipartite mixed states. For any multipartite mixed state, we measure it by multipartite separable pure states and consider the “degenerating locus”. From this motivation we introduce algebraic sets (ie., zero locus of several homogeneous multi-variable polynomials, see \[19\]) in the products of complex projective spaces for the multipartite mixed states, which are independent of the eigenvalues of the mixed states and only measure the “geometric position” of eigenvectors. These algebraic sets are invariants of the mixed states under local unitary transformations, and thus many numerical algebraic-geometric invariants (such as dimensions, number of irreducible components) and Hermitian differential geometric invariants (with the product Fubini-Study metric on the products of complex projective spaces, such as volumes, curvatures) of these algebraic sets are automatically invariant under local unitary transformations. In this way many candidates for good entanglement measure or potentially entanglement monotone independent of eigenvalues of mixed states are offerd. Another important aspect is that these algebraic sets can be easily calculated. Based on these algebraic sets we prove a new separability criterion (independent of eigenvalues) which asserts that the algebraic sets have to be the sum of the products of linear subspaces if the multipartite mixed state is separable. For any entangled mixed state violating our criterion, the mixed states with the same eigenvectors and arbitrary eigenvalues are also entangled, ie., our criterion always detects continuous family of entangled multipartite mixed states.\
Based on our new separabilty criterion, a continuous family of 4 quibit mixed state is constructed as a generalizationof Smolin’s mixed state in \[20\], each mixed state in this family is separable under any $2:2$ cut and entangled under any $1:3$ cut, thus they are bound entanglement if 4 parties are isolated (Example 1). Since our invariants can be computed easily and can be used to distinguish inequivalent mixed states under local unitary transformations, it is proved that the “generic” members of this continuous family of mixed states are inequivalent under local unitary transformations, thus these 4 qubit mixed states are continuous many distinct bound entangled mixed states. Examples of LOCC-incomparable enatngled tripartite pure states are also constructed to show it is hopeless to characterize the entanglement properties of multipartite pure states by the eigenvalue vectors of their partial traces, actually the eigenvalue vectors of partial traces of the tripartite pure states in example 2 are constant. At last we proved that at least $n^2+n-1$ pure separable states, which are orthogonal in some sence, are needed to write any generic pure state in $H_A^{n^2} \otimes H_B^{n^2} \otimes H_C^{n^2}$ as a linear combination of them. This is a remarkable difference to Schmidt decompositions of bipartite pure states.\
For the algebraic geometry used in this paper, we refer to the nice book \[19\].\
We introduce the algebraic sets of the mixed states and prove the results for tripartite case. Then the multipartite case is similar and we just generalize directly.\
Let $H=H_{A}^m \otimes H_{B}^n \otimes H_{C}^l$ and the standard orthogonal base is $|ijk>$, where, $i=1,...,m$,$j=1,...,n$ and $k=1,...,l$, and $\rho$ is a mixed state on $H$. We represent the matrix of $\rho$ in the base $\{|111>,...|11l>,...,|mn1>,...,|mnl>\}$ as $\rho=(\rho_{ij,i'j'})_{1 \leq i ,i' \leq m, 1 \leq j ,j' \leq n}$, and $\rho_{ij,i'j'}$ is a $l \times l$ matrix. Consider $H$ as a bipartite system as $H=(H_{A}^m \otimes H_{B}^n) \otimes H_{C}^l$, then we have $V_{AB}^k(\rho)=\{(r_{11},...,r_{mn}) \in C^{mn}:rank( \Sigma r_{ij}r_{i'j'}^{*} \rho_{ij,i'j'})\leq k \}$ defined as in \[18\]. This set is actually the “degenerating locus” of the Hermitian bilinear form $<\phi_{12}|\rho|\phi_{12}>$ on $H_C^l$ for the given pure state $\phi_{12} =\Sigma_{i,j}^{m,n} r_{ij} |ij> \in P(H_A^m \otimes H_B^n)$. When the finer cut A:B:C is considered, it is natural to take $\phi_{12}$ as a separable pure state $\phi_{12}=\phi_1 \otimes \phi_2$, ie., there exist $\phi_1=\Sigma_i r_i^1 |i> \in P(H_A^m),\phi_2=\Sigma_j r_j^2 |j> \in P(H_B^n)$ such that $r_{ij}=r_i^1r_j^2$. In this way the tripartite mixed state $\rho$ is measured by tripartite separable pure states $\phi_1 \otimes \phi_2 \otimes \phi_3$. Thus it is natural we define $V_{A:B}^k(\rho)$ as follows. It is the “degenerating locus” of the bilinear form $<\phi_1 \otimes \phi_2 |\rho|\phi_1 \otimes \phi_2>$ on $H_C^l$.\
[**Definition 1.**]{}
*Let $\phi:CP^{m-1} \times CP^{n-1} \rightarrow CP^{mn-1}$ be the mapping defined by\
$$\begin{array}{ccccccccc}
\phi(r_1^1,...r_m^1,r_1^2,...,r_n^2)=(r_1^1r_1^2,...,r_i^1r_j^2,...r_m^1r_n^2)
\end{array}
(1)$$*
(ie., $r_{ij}=r_i^1 r_j^2$ is introduced.)
Then $V_{A:B}^k (\rho)$ is defined as the preimage $\phi^{-1}(V_{AB}^k(\rho))$.
\
Similarly $V_{B:C}^k(\rho),V_{A:C}^k(\rho)$ can be defined. In the following statement we just state the result for $V_{A:B}^k(\rho)$. The conclusion holds similarly for other $V's$.\
From this definition and Theorem 2 in \[18\] we immediately have the following result.\
[**Theorem 1.**]{} [*$V_{A:B}^k(\rho)$ is an algebraic set in $CP^{m-1} \times CP^{n-1}$.*]{}\
[**Theorem 2.**]{}[*Let $T=U_{A} \otimes U_{B} \otimes U_{C}$, where $U_{A},U_{B}$ and $U_{C}$ are unitary transformations of $H_A^m, H_B^n, H_C^l$,be a local unitary transformations of $H$. Then $V_{A:B}^k(T(\rho))=U_{A}^{-1} \times U_{B}^{-1}(V_{A:B}^k(\rho))$.*]{}\
[**Proof.**]{} Let $U_{A}=(u_{ij}^{A})_{1 \leq i \leq m,1 \leq j \leq m}$, $U_{B}=(u_{ij}^{B})_{1 \leq i \leq n, 1 \leq j \leq n}$ and $U_{C}=(u_{ij}^{C})_{1 \leq i \leq l, 1 \leq j \leq l}$, be the matrix in the standard orthogonal bases.\
Recall the proof of Theorem 1 in \[18\], we have $V_{AB}^k(T(\rho))=(U_{A} \times U_{B})^{-1}(V_{AB}^k(\rho))$ under the coordinate change\
$$\begin{array}{cccccccc}
r_{kw}'=\Sigma_{ij} r_{ij} u_{ik}^{A} u_{jw}^{B}\\
=\Sigma_{ij}r_i^1 r_j^2 u_{ik}^{A} u_{jw}^{B}\\
=\Sigma_{ij} (r_{i}^1 u_{ik}^A) (r_{j}^2 u_{jw}^B)\\
=(\Sigma_i r_i^1 u_{ik}^A)(\Sigma_j r_j^2 u_{jw}^B)
\end{array}
(2)$$
for $k=1,...,m,w=1,...,n$. Thus our conclusion follows from the definition.\
Since $U_{A}^{-1} \times U_{B}^{-1}$ certainly preserves the (product) Fubini-Study metric of $CP^{m-1} \times CP^{n-1}$, we know that all metric properties of $V_{A:B}^k(\rho)$ are preserved when the local unitary transformations are applied to the mixed state $\rho$.\
In the following statement we give a separability criterion of the mixed state $\rho$ under the cut A:B:C. The term “a linear subspace of $CP^{m-1} \times CP^{n-1}$” means the product of a linear subspace in $CP^{m-1}$ and a linear subspace in $CP^{n-1}$.\
[**Theorem 3.**]{}[*If $\rho$ is a separable mixed state on $H=H_{A}^m \otimes H_{B}^{n} \otimes H_{C}^l$ under the cut A:B:C, $V_{A:B}^k(\rho)$ is a linear subset of $CP^{m-1} \times CP^{n-1}$, ie., it is the sum of the linear subspaces.*]{}\
[**Proof.**]{} We first consider the separability of $\rho$ under the cut AB:C,ie., $\rho= \Sigma_{f=1}^g p_f P_{a_f \otimes b_f}$, where $a_f \in H_{A}^m \otimes H_{B}^n$ and $b_f \in H_{C}^l$ for $f=1,...,g$. Consider the separability of $\rho$ under the cut A:B:C, we have $a_f=a_f' \otimes a_f''$ , $a_f' \in H_{A}^m, a_f'' \in H_{B}^n$. Let $a_f=(a_f^1,...,a_f^{mn}), a_f'=(a_f'^1,...,a_f'^m)$ and $a_f''(a_f''^1,...,a_f''^n)$ be the coordinate forms with the standard orthogonal basis $\{|ij>\}$, $\{|i>\}$ and $\{|j>\}$ respectively, we have that $a_f^{ij}=a_f'^i a_f''^j$. Recall the proof of Theorem 3 in \[18\], the diagonal entries of $G$ in the proof of Theorem 3 in \[18\] are\
$$\begin{array}{cccccccc}
\Sigma_{ij}r_{ij} a_f^{ij}=\\
\Sigma_{ij} r_i^1 a_f'^i r_j^2 a_f''^j=\\
(\Sigma_i r_i^1 a_f'^i)(\Sigma_j r_j^1 a_f''^j)
\end{array}
(3)$$
Thus as argued in the proof of Theorem 3 of \[18\], $V_{A:B}^k(\rho)$ has to be the zero locus of the multiplications of the linear forms in (3). The conclusion is proved.\
For the mixed state $\rho$ in the multipartite system $H=H_{A_1}^{m_1} \otimes \cdots \otimes H_{A_k}^{m_k}$, we want to study the entanglement under the cut $ A_{i_1}:A_{i_2}:...:A_{i_l}:(A_{j_1}...A_{j_{k-l}})$, where $\{i_1,...,i_l\} \cup \{j_1,...j_{k-l}\}=\{1,...k\}$. We can define the set $V_{A_{i_1}:...:A_{i_l}}^k(\rho)$ similarly. We have the following results.\
[**Theorem 1’.**]{} [*$V_{A_{i_1}:...:A_{i_l}}^k(\rho)$ is an algebraic set in in $CP^{m_{i_1}-1} \times CP^{m_{i_l}-1}$.*]{}\
[**Theorem 2’.**]{}[*Let $T=U_{A_{i_1}} \otimes \cdots \otimes U_{A_{i_l}} \otimes U_{j_1...j_{k-l}}$, where $U_{A_{i_1}},...,U_{A_{i_l}}, U_{j_1...j_{k-l}}$ are unitary transformations of $H_{A_{i_1}},...,H_{A_{i_l}}$, be a local unitary transformations of $H$. Then $V_{A_{i_1}:...:A_{i_l}}^k(T(\rho))=U_{A_{i_1}}^{-1} \times \cdots \times U_{A_{i_l}}^{-1}(V_{A_{i_1}:...:A_{i_l}}^k(\rho))$.*]{}\
[**Theorem 3’.**]{}[*If $\rho$ is a separable mixed state on $H=H_{A_1}^{m_1} \otimes \cdots \otimes H_{A_k}^{m_k}$ under the cut $ A_{i_1}:A_{i_2}:...:A_{i_l}:(A_{j_1}...A_{j_{k-l}})$, $V_{A_{i_1}:...:A_{i_l}}^k(\rho)$ is a linear subset of $CP^{m_{i_1}-1} \times ... \times CP^{m_{i_l}-1}$,ie., it is the sum of the linear subspaces.*]{}\
We now give and study some examples of mixed states based on our above results.\
The following example, which is a continuous family (depending on 4 parameters) of mixed state in the four-party quantum system $H_A^2 \otimes H_B^2 \otimes H_C^2 \otimes H_D^2$ and separable for any $2:2$ cut but entangled for any $1:3$ cut, can be thought as a generalization of Smolin’s mixed state in \[20\].\
[**Example 1.**]{} Let $H=H_{A}^2 \otimes H_{B}^2 \otimes H_{C}^2 \otimes H_{D}^2$ and $h_1,h_2,h_3,h_4$ (understood as row vectors)are 4 mutually orthogonal unit vectors in $C^4$. Consider the $16 \times 4$ matrix $T$ with 16 rows as\
$T=(a_1h_1^{\tau},0,0,a_2 h_2^{\tau},0, a_3 h_3^{\tau},a_4 h_4^{\tau},0,0, a_5 h_3^{\tau}, a_6 h_4^{\tau},0, a_7 h_1^{\tau},0,0,a_8 h_2^{\tau})^{\tau}$. Let\
$\phi'_1,\phi'_2,\phi'_3,\phi'_4$ be 4 vectors in $H$ whose expansions with the base $|0000>,|0001>,|0010>,|0011>,|0100>,|0101>,|0110>,|0111>,|1000>$,\
$|1001>,|1010>,|1011>,|1100>,|1101>,|1110>,|1111> $ are exactly the 4 columns of the matrix $T$ and $\phi_1,\phi_2,\phi_3,\phi_4$ are the normalized unit vectors of $\phi'_1,\phi'_2, \phi'_3, \phi'_4$. Let $\rho=\frac{1}{4}(P_{\phi_1} +P_{\phi_2} +P_{\phi_3} +P_{\phi_4})$.\
It is easy to check that when $h_1=(1,1,0,0),h_2=(1,-1,0,0), h_3=(0,0,1,1), h_4=(0,0,1,-1)$ and $a_1=a_2=a_3=a_4=1$. It is just the Smolin’s mixed state in \[20\].\
Now we prove that $\rho$ is invariant under the partial transposes of the cuts AB:CD,AC:BD,AD:BC.\
Let the “representation” matrix $T=(b_{ijkl})_{i=0,1,j=0,1,k=0,1,l=0,1}$ is the matrix with columns corresponding the expansions of $\phi_1,\phi_2,\phi_3,\phi_4$.Then we can consider that $T=(T_1,T_2,T_3,T_4)^{\tau}$ is blocked matrix of size $4 \times 1$ with each block $T_{ij}=(b_{kl })_{k=0,1,l=0,1}$ a $4 \times 4$ matrix,where $ij=00,01,10,11$. Because $h_1,h_2,h_3,h_4$ are mutually orthogonal unit vectors we can easily check that $T_{ij} (T_{i'j'}^{*})^{\tau}=T_{i'j'} (T_{ij}^{*})^{\tau}$ Thus it is invariant when the partial transpose of the cut AB:CD is applied.\
With the same methods we can check that $\rho$ is invariant when the partial transposes of the cuts AC:BD, AD:BC are applied. Hence $\rho$ is PPT under the cuts AB:CD, AC:BD,AD:BC. Thus from a result in \[21\] we know $\rho$ is separable under these cuts AB:CD, AC:BD,AD:BC.\
Now we want to prove $\rho$ is entangled under the cut A:BCD by computing $V_{BCD}^1(\rho)$. From the arguments in \[18\] and this paper, we can check that $V_{BCD}^1(\rho)$ is the locus of the condition: $a_1 h_1 r_{000} + a_2 h_2 r_{011} +a_3 h_3 r_{101} +a_4 h_4 r_{110}$ and $a_7 h_1 r_{100} + a_8 h_2 r_{111} +a_5 h_3 r_{001} +a_6 h_4 r_{010}$ are linear dependent. This is equivalent to the condition that the matrix (6) is of rank 1.\
$$\left(
\begin{array}{cccccc}
a_7 r_{100} & a_8 r_{111} & a_5 r_{001} & a_6 r_{010}\\
a_1 r_{000} & a_2 r_{011} & a_3 r_{101} & a_4 r_{110}
\end{array}
\right)
(6)$$
From \[19\] pp. 25-26 we can check that $V_{BCD}^1(\rho)$ is exactly the famous Segre variety in algebraic geometry. It is irreducible and thus cannot be linear. From Theorem 3 in \[18\], $\rho$ is entangled under the cut A:BCD. Similarly we can prove that $\rho$ is entangled under the cuts B:ACD, C:ABD, D:ABC.\
Now we compute $V_{A:B}^3(\rho)$. From the arguments in \[18\] and Definition 1 , it is just the locus of the condition that the vectors $h_1(a_1 r_0^1 r_0^2 +a_7 r_1^1 r_1^2)$, $h_3 (a_3 r_0^1 r_1^2 +a_5 r_1^1 r_0^2)$, $h_4(a_4 r_0^1 r_1^2 +a_6 r_1^1 r_0^2)$, $h_2 (a_2 r_0^1 r_0^2 +a_8 r_1^1 r_1^2)$ are linear dependent. Since $h_1,h_2,h_3,h_4$ are mutually orthogonal unit vectors,we have\
$$\begin{array}{cccccccccc}
V_{A:B}^3(\rho)=\{(r_0^1,r_1^1,r_0^2,r_1^2) \in CP^1 \times CP^1:\\
(a_1 r_0^1 r_0^2 +a_7 r_1^1 r_1^2)(a_3 r_0^1 r_1^2 +a_5 r_1^1 r_0^2)(a_4 r_0^1 r_1^2 +a_6 r_1^1 r_0^2)(a_2 r_0^1 r_0^2 +a_8 r_1^1 r_1^2)=0\}
\end{array}
(7)$$
From Theorem 3 we know that $\rho$ is entangled for the cut A:B:CD, A:C:BD and A:D:BC for generic parameters, since (for example) $a_1r_0^1r_0^2+a_7r_1^1r_1^2$ cannot be factorized to 2 linear forms for generic $a_1$ and $a_7$. This provides another proof the mixed state is entangled if the 4 parties are isolated.\
Let $\lambda_1=-a_1/a_7,\lambda_2=-a_3/a_5, \lambda_3=-a_4/a_6, \lambda_4=-a_2/a_8$ and consider the family of the mixed states $\{\rho_{\lambda_{1,2,3,4}}\}$, we want to prove the following statement.\
[**Theorem 4.**]{} [*The generic memebers in this continuous family of mixed states are inequivalent under the local operations on $H=H_{A}^2 \otimes H_{B}^2 \otimes H_{C}^2 \otimes H_{D}^2$.*]{}\
[**Proof.**]{} From the above computation, $V_{A:B}^3(\rho_{\lambda_{1,2,3,4}})$ is the union of the following 4 algbraic varieties in $CP^1 \times CP^1$.\
$$\begin{array}{ccccccccccc}
V_1=\{(r_0^1,r_1^1,r_0^2,r_1^2) \in CP^1 \times CP^1:r_0^1 r_0^2 - \lambda_1 r_1^1 r_1^2=0\}\\
V_2=\{(r_0^1,r_1^1,r_0^2,r_1^2) \in CP^1 \times CP^1:r_0^1 r_1^2 - \lambda_2 r_1^1 r_0^2=0\}\\
V_3=\{(r_0^1,r_1^1,r_0^2,r_1^2) \in CP^1 \times CP^1:r_0^1 r_1^2 - \lambda_3 r_1^1 r_0^2=0\}\\
V_4=\{(r_0^1,r_1^1,r_0^2,r_1^2) \in CP^1 \times CP^1:r_0^1 r_0^2 - \lambda_4 r_1^1 r_1^2=0\}
\end{array}
(8)$$
From Theorem 2, if $\rho_{\lambda_{1,2,3,4}}$ and $\rho_{\lambda'_{1,2,3,4}}$ are equivalent by a local operation, there must exist 2 fractional linear transformations $T_1, T_2$ of $CP^1$ such that $T=T_1 \times T_2$ (acting on $CP^1 \times CP^1$) transforms the 4 varieties $V_1,V_2,V_3,V_4$ of $\rho_{\lambda_{1,2,3,4}}$ to the 4 varieties $V'_1,V'_2,V'_3,V'_4$ of $\rho_{\lambda'_{1,2,3,4}}$,ie., $T(V_i)=V'_j$.\
Introduce the inhomogeneous coordinates $x_1=r_0^1/r_1^1,x_2=r_0^2/h_1^2$. Let $T_1(x_1)=(ax_1+b)/(cx_1+d)$. Suppose $T(V_i)=V'_i, i=1,2,3,4$. Then we have $ab \lambda_1= cd \lambda'_1 \lambda'_2$ and $ab \lambda_4=cd \lambda'_3 \lambda'_4$. Hence $\lambda_1 \lambda'_3 \lambda'_4 =\lambda'_1 \lambda'_2 \lambda_4$. This means that there are some algebraic relations of parameters if the $T$ exists. Similarly we can get the same conclusion for the other possibilities $T(V_i)=V'_j$. This implies that there are some algebraic relations of parameters $\lambda_{1,2,3,4}$ and $\lambda'_{1,2,3,4}$ if $\rho_{\lambda_{1,2,3,4}}$ and $\rho_{\lambda'_{1,2,3,4}}$ are equivalent by a local operation. Hence our conclusion follows immediately.\
In \[22\] Nielsen gave a beautiful necessary and sufficient condition for the pure state $|\psi>$ can be transformed to the pure state $|\phi>$ in bipartite quantum systems by local operations and classical communications (LOCC) based on the majorization between the eigenvalue vectors of the partial traces of $|\psi>$ and $|\phi>$. In \[10\] an example was given, from which we know that Nielsen’s criterion cannot be generalized to multipartite case, [**3EPR**]{} and [**2GHZ**]{} are understood as pure states in a $4 \times 4 \times 4$ quantum system, they have the same eigenvalue vectors when traced over any subsystem. However it is proved that they are LOCC-incomparable in \[10\]\
In the following example 2, a continuous family $\{\phi\}_{\eta_1,\eta_2,\eta_3}$ of pure states in tripartite quantum system $H_{A_1}^3 \otimes H_{A_2}^3 \otimes H_{A_3}^3$ is given, the eigenvalue vectors of $tr_{A_i}(|\phi_{\eta_1,\eta_2,\eta_3}><|\phi_{\eta_1,\eta_2,\eta_3}>), tr_{A_iA_j}(|\phi_{\eta_1,\eta_2,\eta_3}><\phi_{\eta_1,\eta_2,\eta_3}|)$ are independent of parameters $\eta_1,\eta_2,\eta_3$. However the “generic” pure states in this family are entangled and LOCC-incomparable. This gives stronger evidence that it is hopeless to characterize the entanglement properties of pure states in multipartite quantum systems by only using the eigenvalue spetra of their partial traces.\
[**Example 2**]{} Let $H=H_{A_1}^3 \otimes H_{A_2}^3 \otimes H_{A_3}^3$ be a tripartite quantum system and the following 3 unit vectors are in $H_{A_1}^3 \otimes H_{A_2}^3$.\
$$\begin{array}{cccccccc}
|v_1>=\frac{1}{\sqrt{3}}(e^{i\eta_1}|11>+|22>+|33>)\\
|v_2>=\frac{1}{\sqrt{3}}(e^{i\eta_2}|12>+|23>+|31>)\\
|v_3>=\frac{1}{\sqrt{3}}(e^{i\eta_3}|13>+|21>+|32>)
\end{array}
(9)$$
,where $\eta_1,\eta_2,\eta_3$ are $3$ real parameters. Let $|\phi_{\eta_1,\eta_2,\eta_3}>=\frac{1}{\sqrt{3}}(|v_1> \otimes |1>+|v_2>\otimes |2>+|v_3> \otimes |3>)$. This is a continuous family of pure states in $H$ parameterized by three real parameters. It is clear that $tr_{A_3}=\frac{1}{3}(|v_1><v_1|+|v_2><v_2|+|v_3><v_3|)$ is a rank 3 mixed state in $H_{A_1}^3 \otimes H_{A_2}^3$. Set $g(\eta_1,\eta_2,\eta_3)=\frac{e^{i\eta_1}+e^{i\eta_2}+e^{i\eta_3}}{e^{i(\eta_1+\eta_2+\eta_3)/3}}$, $|\phi_{\eta_1,\eta_2,\eta_3}>$ and $|\phi_{\eta'_1,\eta'_2,\eta'_3}>$ are not equivalent under local unitary transformations if $k(g(\eta_1,\eta_2,\eta_3)) \neq k(g(\eta'_1,\eta'_2,\eta'_3))$, where $k(x)=\frac{x^3(x^3+216)^3}{(-x^3+27)^3}$ is the moduli function of elliptic curves, since their corresponding traces over $A_3$ are not equivalent under local unitary transformations of $H_{A_1}^3 \otimes H_{A_2}^3$ from Theorem 5 of \[18\]. Hence the “generic” members of this family of pure states in tripartite quantum system $H$ are enatngled and LOCC-incomparable from Theorem 1 in \[10\].\
On the other hand it is easy to calculate that all nonzero eigenvalues of $tr_{A_3}(|\phi_{\eta_1,\eta_2,\eta_3}><\phi_{\eta_1,\eta_2,\eta_3}|), tr_{A_1A_3}(|\phi_{\eta_1,\eta_2,\eta_3}><\phi_{\eta_1,\eta_2,\eta_3}|),tr_{A_2A_3}(|\phi_{\eta_1,\eta_2,\eta_3}><\phi_{\eta_1,\eta_2,\eta_3}|)$ are $\frac{1}{3}$. Thus all nonzero eigenvalues of $tr_{A_i}(|\phi_{\eta_1,\eta_2,\eta_3}><|\phi_{\eta_1,\eta_2,\eta_3}>), tr_{A_iA_j}(|\phi_{\eta_1,\eta_2,\eta_3}><\phi_{\eta_1,\eta_2,\eta_3}|)$ are constant $\frac{1}{3}$. Thus the “generic” members of this family of pure states have the same eigenvalue spectra but are entangled and LOCC-incomparable.\
For any pure state in a bipartite quantum system $H=H_A^m \otimes H_B^n$ , it can be written as a linear combination of at most $min\{m,n\}$ 2-way orthogonal separable pure states (\[10\]) from Schmidt decomposition. For multipartite pure states, there is no direct generaliztion of Schmidt decomposition, and those multipartite pure states with a m-way orthogonal decompositions can be distilled to cat states (see \[10\]). From the results in \[11\], it is known that we need at least 5 terms of “orthogonal” separable pure states to write a generic pure state in $H_A^2 \otimes H_B^2 \otimes H_C^2$ as a linear combination of them. This phenomenon is a remarkable difference between bipartite pure state entanglement and multipartite pure state entanglement. In the following statement it is showed what happens for generic pure states in $H_A^{n^2} \otimes H_B^{n^2} \otimes H_C^{n^2}$.\
[**Theorem 5.**]{}[*For a generic pure state $|\psi>=\Sigma_{i=1}^{n^2} \lambda_i |\psi_i^{12}> \otimes |\psi_i^3>$ in a tripartite quantum system $H_A^{n^2} \otimes H_B^{n^2} \otimes H_C^{n^2}$, where $|\psi_1^3>,...,|\psi_{n^2}^3>$ are mutually orthogonal unit vectors in $H_C^{n^2}$ and $|\psi_1^{12}>,...,|\psi_{n^2}^{12}>$ are pure states in $H_A^{n^2} \otimes H_B^{n^2}$, then there exists one of $|\psi_1^{12}>,...,|\psi_{n^2}^{12}>$ with Schmidt rank at least $n$.*]{}\
[**Proof.**]{} It is clear that $tr_C(|\psi><\psi|)=\Sigma_{i=1}^{n^2} |\lambda_i|^2 |\psi_i^{12}><\psi_i^{12}|$ is a generic rank $n^2$ mixed state in $H_A^{n^2} \otimes H_B^{n^2}$. From Theorem 4 in \[18\], at least one of $|\psi_1^{12}>,...,|\psi_{n^2}^{12}>$ has Schmidt rank (as a pure state in $H_A^{n^2} \otimes H_B^{n^2}$) at least $n$. Thus our conclusion is proved.\
In conclusion we revealed that there is a large part of multipartite quantum entanglement, which is independent of eigenvalues and only depending on the eigenvectors of the multipartite mixed states. We introduced algebraic set invariants for measuring this part of multipartite quantum entanglement, which actually are the [*degenerating locus*]{} of the Hermitian bilinear forms arising from the measurement of the mixed states by multipartite seaprable pure states. . Based on these algebraic set invariants, a new [*eigenvalue-free*]{} separability criterion has been proved. Examples to show why entanglement of tripartite pure states cannot be characterized by the spectra of their partial traces from the point of view of algebraic set invariants and separability criterion have been constructed. Generalized Smolin mixed states were introduced and studied from our invariants and separability criterion, served as examples of continuous many distinct bound entanglement in 4 qubits. We also have proved a conclusion showing the decomposing a generic tripartite pure state as “orthogonal” separable pure states is quite different to the Schmidt decomposition of bipartite pure states.\
The author acknowledges the support from NNSF China, Information Science Division, grant 69972049.\
e-mail: chenhao1964cn@yahoo.com.cn\
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|
---
author:
- Miao He
- 'Zi-liang Wang'
- 'Jian-bo Deng [^1]'
- Hua Chen
bibliography:
- 'reference.bib'
title: 'Quantum Cosmology via Quantization of Point-Like Lagrangian'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore Trailer
Introduction {#intro}
============
The motive of studying a quantum cosmological theory mainly emerges from two parts. First of all, classical gravity theory fails in precisely describing topics like the very early universe. To understand those tiny scale objects with huge energy, a successful quantum gravity theory is requisite and then we can apply it into the cosmological case. However since no such theory is available till now, one considerable effort is to view the quantum cosmology as an effective theory which can be approximately obtained by modifying the classical theory.
In classical field theory, the Lagrangian density is a function of one or more fields and their derivatives. Then one gets the action $$\label{eq:action}
S=\int L\,dt=\int\mathcal{L}(\phi,\partial_\mu\phi)\,d^4x \qquad.$$ According to the principle of least action, one can get corresponding equations like Maxwell equations, Klein-Gordon equation, Dirac equation etc. Similarly, in the general relativity, the Einstein-Hillbert actions discribed as $$\label{eq:E-H action}
S_{EH}=\int\mathcal{L}\,d^4x=\int \sqrt{-g}R\,d^4x \qquad.$$ which can get the Einstein field equation. This Lagrangian formulation leads to get Hamiltionian formulation, through Legendre transformation, which would make the transition to quantum mechanics easier. An attempt following this idea is called the minisuperspace quantization which is first purposed by DeWitt [@dewitt_quantum_1967]. At that time Wheeler had introduced the idea of superspace, the space of all three-geometries as the arena in which geometrodynamics develops, a particular four-geometries being represented by a trajectory in this space. Misner had just finished applying the Hamiltonian formulation of gravity, developed in the late 1950s and early 1960s, to cosmological models with an eye towards quantization of these cosmologies as model theories of general relativity [@misner_quantum_1969], which is another motive for studying quantum cosmology. He invented “minisuperspace” and “minisuperspace quantization” or “quantum cosmology”, to describe the evolution of cosmological spacetimes as trajectories in the finite-dimensional sector of superspace related to the finite number of parameters, which describe $t=$const slices of the models, and the quantum version of such models, respectively.
For the classical cosmology, its dynamic equation is the Friedman equation which is derived from Einstein field equation. Under standard model, for the quantum cosmology, a quantum gravity theory might be an effective approach. Cosmological minisuperspaces and their quantum versions were extensively studied in the early 1970s, but interest in them waned after about 1975 and little new work was done until Hawking revived the field in the 1980s [@hawking_boundary_1982; @hartle_wave_1983], emphasizing path-integral approaches. This started a lively resurgence of interest in minisuperspace quantization till now. For a quick review of the minisuperspace quantization see Halliwell [@halliwell1; @halliwell2; @halliwell3].
In this paper, we would like to introduce a new technique in acquiring the quantum system of cosmology via the point-like Lagrangian under a certain gravitational model. According the cosmological principle, if we use the spacial symmetry of our univese first, a point-like Langrangian can be got as bellow $$\label{eq:point-like lag}
L= \int \mathcal{L}\,d\Omega=\int \sqrt{-g}R\,d\Omega \qquad,$$ where $g$ is the determinant of Friedmann-Robertson-Walker(FRW) metric, $R$ is the scalar curvature and $d\Omega$ represents the spacial volume element. The Ensitein-Hillbert action can be written as $$\label{eq:point-like lag}
S = \int L\,dt \qquad,$$ here $t$ is the cosmological time, which can be treated as a classical mechanics action. Then quantize this model through the classical canonical quantization. It is much easier than the normal process but reaches the similar result. We will then apply this technique into a more complicated case, which contains a cosmological constant, and even more general, for the $f(R)$ gravity. We will give the exact solution to the quantum universe with a cosmological constant and discuss its meaning.
The Friedmann Equations
=======================
In the normal sense, we get the cosmological model under a certain kind of gravity by applying the cosmological principle into the gravitational field equation which is obtained from variation of the action. That is to say, considering a action with matter field $$\label{eq:action_matter}
S=\int\sqrt{-g}R\,d^4x+S_M \qquad,$$ by varying the action eq. with respect to $g_{\mu\nu}$, one can get the Enistein Field Equation. After setting the metric in the equation to the FRW metric $$\label{eq:frw-metric}
ds^2 = -dt^2 +a^2(t) \left(\frac{dr^2}{1-\kappa r^2}
+r^2(d\theta^2+\sin^2\theta d\varphi^2)\right) \qquad,$$ and the energy-momentum tensor with a perfect fluid $$T^{\mu\nu}=(-g)^{-1/2}\frac{\delta S_M}{\delta g_{\mu\nu}}=(p+\rho)U^{\mu} U^{\nu}+pg^{\mu\nu}\qquad,$$ it would end up with the Friedman Equations.
Surprisingly, one can apply the cosmology principle directly into the action before variation. This modifies the Lagrangian to a classical point-like one with respect to the scale factor. By the variation of the point-like action, it will give exactly the same equations as those in the normal process. Such method has been extensively used in studying scalar field cosmology , non-minimally coupled cosmology [@PhysRevD.62.043506; @sanyal_general_2003], scalar-tensor theory [@PhysRevD.52.3288], multiple scalar fields [@10.4137/GBI.S4273], vector field [@0264-9381-27-13-135019], Fermion field [@0264-9381-25-22-225006], $f(T)$ gravity [@wei_noether_2012], $f(R)$ gravity [@capozziello_fr_2008], high order gravity theory [@capozziello_general_2000], Gauss–Bonnet gravity [@sanyal_general_2011] and so on. Without loss of generality, we get the action as below by setting $\kappa=0$, $$\label{eq:action_matter-2}
S=\int-6a\dot{a}^2dt +S_M \qquad,$$ the variation with respect to $a$ and take note of $\delta S_M/\delta a=\delta S_M/\delta g_{\mu\nu}\cdot\delta g_{\mu\nu}/\delta a=6p/a$, one can get one of the fridemann equation $$\label{eq:friedmann_equation_1}
2\frac{\ddot{a}}{a}+\frac{\dot{a}^2}{a}=-p \qquad.$$ By combining the conservation equation of perfect fluid $\dot{\rho}=-3(p+\rho)\dot{a}/a$, all Friedmann equations can be derived from this point-like Lagrangian.
Moreover, this point-like Lagrangian may lead us to quantiza the cosmology, one would get an easier wave equation and its analytic solutions. In addition, through the classical canonical quantization, the evolution of the cosmology which can be discribed by the scale factor, may be different from the classical cosmology as it has the quantum effects. Next, we will apply this method into the quantum cosmology.
Point-Like Quantum Cosmology
============================
For simplicity, let us restrict our discussion to the flat FRW universe($\kappa=0$) in the case of without considering the matter field. Recall the point-like Lagrangian for a flat FRW universe under Einstein gravity, its effective part is $$\label{eq:point-R-lagrangian}
L = -6a\dot{a}^2 \qquad.$$ Its canonical momentum is $$\label{eq:point-R-momentum-a}
\pi_a = \frac{\partial L}{\partial \dot{a}} =
-12a\dot{a} \qquad,$$ from which we can get its Hamiltonian $$\label{eq:point-R-hamiltonian}
H = \pi_a\dot{a} -L = -6a\dot{a}^2 =
-\frac{\pi_a^2}{24a} \qquad.$$ Following the process of standard canonical quantization, according to the Weyl rule [@weyl1950theory], we can get the Hamiltonian operator $$\label{eq:point-R-hamiltonian operator}
\hat{H}=-\frac{1}{24}\frac{1}{4}(\hat{\pi}_{a}^2\frac{1}{a}
+2\hat{\pi}_{a}\frac{1}{a}\hat{\pi}_{a}+\frac{1}{a}\hat{\pi}_{a}^2)
\qquad,$$ then we should replace the canonical momentum $\pi_a$ by the operator of momentum $-i\partial_a$ and get the wave equation of the scale factor $a$ $$\label{eq:point-R-schrodinger}
i\frac{\partial\psi}{\partial t}=\frac{1}{24}(\frac{1}{a}\frac{\partial^2
\psi}{\partial{a^2}}-\frac{1}{a^2}\frac{\partial\psi}{\partial{a}}+
\frac{1}{2a^3}\psi) \qquad.$$ Since the scale factor $a$ is a non-negative real number in cosmology, this wavefunction should be a function defined only on the right half of the real line with its value being a complex number at any given time.
Assuming $\psi(a,t)$ takes the form of $e^{-i\epsilon t}\phi(\epsilon;a)$ and pluging it into eq., we get the eigen equation $$\label{eq:point-R-energy}
\frac{1}{24}[\frac{1}{a}\phi''(a)-\frac{1}{a^2}\phi'(a)+\frac{1}{2a^3}
\phi(a)]=\epsilon\phi(a) \quad,\quad a\geqslant
0 \qquad.$$ The solution of this equation depends on whether the eigenvalue is positive or not.
When $\epsilon>0$ we can first rescale the variable $\xi=(24\epsilon)^{1/3}
a$ to drop the parameters in eq. and modify it to $$\label{eq:point-R-energy-modified-pos}
\xi^2\phi''-\xi\phi'+(\frac{1}{2}-\xi^3)\phi=0 \quad,\quad \xi\geqslant 0
\qquad.$$ Through a variable substitution of $z=2\xi^{3/2}/3$ and introducing a new function $u(z)$ with the relationship $\phi(z)=\xi\,u(z)$, we get $$\label{eq:modified-bessel}
u''(z)+\frac{1}{z}u'(z)-(1+\frac{(\sqrt{2}/3)^2}{z^2})u(z)=0 \qquad,$$ which is a modified Bessel equation whose solutions can be described by a linear combination of the first modified Bessel function $I_{\sqrt{2}/3}(z)$ and the second modified Bessel function $K_{\sqrt{2}/3}(z)$.
However $I_{\sqrt{2}/3}(z)$ increases in exponential form when $z$ goes to infinity, which shows great divergent trend and can not be accepted as a reasonable wavefunction. Therefore, recovering all the transformations we made, we get the eigen function with a given eigenvalue described by eq. of $$\label{eq:point-R-wave-pos}
\phi^+(\epsilon;a)=(24\epsilon)^{\frac{1}{3}}aK_{\sqrt2/3}(\frac{4\sqrt{6
\epsilon}}{3}a^{\frac{3}{2}}) \qquad.$$
The asymptotic expansion of the second modified Bessel function with huge arguments can be described as $$\label{eq:modified-bessel-2-asymp-huge}
K_\nu(z) \sim \sqrt{\frac{\pi}{2z}}\,e^{-z} \qquad,$$ which indicates our wavefunction $\phi^+$ behaves like $$\label{eq:point-R-wave-pos-asymp-huge}
\phi^+(\epsilon;a)\sim a^{\frac{1}{4}}e^{-4\sqrt{6\epsilon}a^{3/2}/3}
\qquad,$$ and will decay to zero at an extremely quick rate with $a$ getting huge enough. This means $\phi^+$ is normalizable and is suitable for being a wavefunction.
For tiny arguments, the second modified Bessel function has the following asymptotic form, $$\label{eq:modified-bessel-2-asymp-tiny}
K_\nu(z) \sim \frac{\Gamma(\nu)}{2} \left(\frac{2}{z}\right)^\nu
\qquad.$$ Applying this into the wavefunction eq., one can find when $a\rightarrow 0$, the wavefunction $$\label{eq:point-R-wave-pos-asymp-tiny}
\phi^+(\epsilon;a)\sim a^{(2-\sqrt2)/2}$$ goes zero which can avoid the possibility of a cosmological singularity.
When $\epsilon<0$, in order to keep $\xi$ being positive, we should rescale the variable in form of $\xi=(-24\epsilon)^{1/3}a$. Now equation eq. becomes $$\label{eq:point-R-energy-modified-neg}
\xi^2\phi''-\xi\phi'+(\frac{1}{2}+\xi^3)\phi=0 \quad,\quad \xi\geqslant 0
\qquad.$$ Again we apply the substitution $z=2\xi^{3/2}/3$ and let $\phi=\xi\,u$, then we end up with a Bessel equation $$\label{eq:bessel}
u''(z)+\frac{1}{z}u'(z)+(1-\frac{(\sqrt{2}/3)^2}{z^2})u(z)=0 \qquad.$$ Normally we choose the Bessel function of first kind $J_{\sqrt{2}/3}(z)$ as one of the basis for its solution space. The other base can have different choice among which the most convenient should be the Bessel function of second kind $Y_{\sqrt{2}/3}(z)$ also known as the Weber function or the Neumann function.
So for every distinct eigenvalue, the quantum system has the degenerate degree of two with its independence basis chosen as $$\begin{aligned}
\label{eq:point-R-wave-neg-1}
\phi^-_1(\epsilon\,;a) &= &(-24\epsilon)^{\frac{1}{3}}a\,
J_{\frac{\sqrt2}{3}}(\frac{4\sqrt{-6\epsilon}}{3}\,a^{\frac{3}{2}})
\qquad,\\
\label{eq:point-R-wave-neg-2}
\phi^-_2(\epsilon\,;a)&= &(-24\epsilon)^{\frac{1}{3}}a\,
Y_{\frac{\sqrt2}{3}}(\frac{4\sqrt{-6\epsilon}}{3}\,a^{\frac{3}{2}})
\qquad.
\end{aligned}$$
For great arguments, the Bessel functions behave like the following $$\begin{aligned}
\label{eq:bessel-1-asymp-huge}
J_\nu(z) &\sim &\sqrt{\frac{2}{\pi z}} \left(\cos\left(z
-\frac{\nu\pi}{2} -\frac{\pi}{4}\right)
+O(z^{-1})\right) \qquad, \\
\label{eq:bessel-2-asymp-huge}
Y_\nu(z) &\sim &\sqrt{\frac{2}{\pi z}} \left(\sin\left(z
-\frac{\nu\pi}{2} -\frac{\pi}{4}\right)
+O(z^{-1})\right) \qquad.
\end{aligned}$$ Therefore, when $a$ goes to infinity, the wavefunctions have asymptotic expansions $$\begin{aligned}
\label{eq:point-R-wave-neg-1-asymp-huge}
\phi^-_1(\epsilon\,;a) &\sim &a^{\frac{1}{4}}
\cos\left(\frac{4\sqrt{-6\epsilon}}{3}\,
a^{\frac{3}{2}}
-\frac{2\sqrt2+3}{12}\pi\right)
+O(a^{-\frac{5}{4}}) \qquad,\\
\label{eq:point-R-wave-neg-2-asymp-huge}
\phi^-_2(\epsilon\,;a) &\sim &a^{\frac{1}{4}}
\sin\left(\frac{4\sqrt{-6\epsilon}}{3}\,
a^{\frac{3}{2}}
-\frac{2\sqrt2+3}{12}\pi\right)
+O(a^{-\frac{5}{4}}) \qquad,
\end{aligned}$$ that somehow behaves like sine-cosine with a raise to the power of $1/4$. thus $\phi^-_1,\phi^-_2$ are divergent when $z$ goes to infinity, which cannot be accepted as a reasonable wavefunction.
As shown from our discussion, only $\phi^+$ can act as the wavefuntion and is capable of avoiding the cosmological singularity. Its eigenvalue $
\epsilon$ would be a continue spectrum which can has value from zero to infinity. Normally, for any initial states $\sum_{\epsilon}c_{\epsilon}|
\epsilon\rangle$(here $|\epsilon\rangle$ is the nomalized eigen state). the expectation value of $a^2$ evolves like $$\label{eq:scale-factor-evolve}
\overline{a^2}(t)=\sum_{\epsilon\neq\epsilon'}c_\epsilon c_{\epsilon'}
\langle\epsilon|a^2|\epsilon'\rangle cos((\epsilon-\epsilon')t+\theta_
{\epsilon,\epsilon'})+\sum_
{\epsilon}|c_\epsilon|^2\langle\epsilon|a^2|\epsilon\rangle \qquad.$$ Therefore, the evolution of $a^2$ can be any functions with specific initial state, including the inflation of the early universe. However, many theory consider the cosmology constant is essential, like $\Lambda$CDM model and Quantum Field Theory. So we would like to consider a quantum cosmological model under some more generalized gravitational models like gravity with a cosmological constant.
Quantum Cosmology with A Cosmological Constant
==============================================
The point-like Lagrangian for flat FRW universe under the gravitational model with a cosmological constant has the form $$\label{eq:point-lambda-lagrangian}
L = -6a\dot{a}^2 +\Lambda a^3 \qquad.$$ The additional term of $\Lambda a^3$ does not contain the derivative of $a$, thus will not change the form of canonical momentum we get in eq.. So the Hamiltonian in this case is $$\label{eq:point-lambda-hamiltonian}
H = -\frac{\pi_a^2}{24a} -\Lambda a^3 \qquad.$$ The same quantization method we can get the wave equation $$\label{eq:point-lambda-schrodinger}
i\frac{\partial\psi}{\partial t}=\frac{1}{24}(\frac{1}{a}\frac
{\partial^2\psi}{\partial{a^2}}-\frac{1}{a^2}\frac{\partial\psi}
{\partial{a}}+\frac{1}{2a^3}\psi)-\Lambda a^3\psi \qquad.$$ Its eigen equation is given by $$\label{eq:point-lambda-energy}
\phi''-\frac{1}{a}\phi'+(\frac{1}{2a^2}-24\Lambda a^4-24\epsilon
a)\phi=0\qquad.$$ Next we will consider the asymptotic behavior of the equation.
When $a\rightarrow 0$, the wave equation can be described as $$\label{eq:point-lambda-energy-zero}
\phi''-\frac{1}{a}\phi'+\frac{1}{2a^2}\phi=0\qquad,$$ obviously, $$\label{eq:point-lambda-wave-zero}
\phi\sim a^{1+\sqrt{2}/2},a^{1-\sqrt{2}/2}\qquad.$$
When $a\rightarrow\infty$, the equation behaves like $$\label{eq:point-lambda-energy-infinity}
\phi''-(24\Lambda a^3+24\epsilon a)\phi=0\qquad,$$ it has a special solution with the form like $$\label{eq:point-lambda-wave-infinity}
\phi\sim e^{\nu a^3}\qquad.$$
Generally, assuming $\phi(a)$ takes the forms like $$\label{eq:point-lambda-wave-form-1}
\phi(a)=a^{1+\sqrt{2}/2}\cdot e^{\nu a^3}\cdot u(a)\qquad,$$ or $$\label{eq:point-lambda-wave-form-2}
\phi(a)=a^{1-\sqrt{2}/2}\cdot e^{\nu a^3}\cdot u(a)\qquad.$$ Applying eq. into the eigen equation eq. and setting $9\nu^2=24
\Lambda$, one would get the equation for $u(a)$ as bellow $$\label{eq:point-lambda-energy-modified}
au''(a)+(6\nu a^{3}+1+\sqrt{2})u'(a)+[(9+3\sqrt{2})\nu-24
\epsilon]a^2u(a)=0\qquad.$$ With substitution $z=-2\nu a^3$, rearrange eq. $$\label{eq:kummer-equation}
zu''(z)+(\frac{3+\sqrt{2}}{3}-z)u'(z)-(\frac{3+\sqrt{2}}{6}-\frac{4
\epsilon}{3\nu})u(z)=0\qquad,$$ which is a confluent hypergeometric equation that is also known as Kummer’s equation. It has a solution described by Kummer’s function defined as $$\label{eq:kummer-function}
F(\alpha,\gamma,z) = \sum_{n=0}^\infty
\frac{\alpha^{(n)}}{\gamma^{(n)}} \frac{z^n}{n!} \qquad,$$ where in our case $\alpha=(3+\sqrt2)/6-4\epsilon/3\nu$ and $\gamma=(3+\sqrt2)/3$. Here the symbol $x^{(n)}$($x=\alpha,\gamma$) stands for a rising factorial defined as $$\begin{aligned}
x^{(0)} &= &1 \qquad,\\
x^{(n)} &= &x(x+1) \cdots (x+n-1) ,\quad n\geqslant 1 \qquad.
\end{aligned}$$ Since $\gamma$ is not a integer, another solution independent with eq. can be simply introduced as $$u(z) = z^{1-\gamma} F(\alpha-\gamma+1,2-\gamma,z) \qquad.$$
Recovering from the substitution we made and considering $\nu$ can be either positive or negative, it seems that we will get four independent eigen functions for any given eigenvalue $\epsilon$ $$\begin{aligned}
\label{eq:point-lambda-wave-1}
\phi^{\pm}_1(\epsilon\,;a) &= &a^{1+\sqrt2/2}e^{\pm\nu a^3}
F(\frac{3+\sqrt2}{6}\mp \frac{4\epsilon}
{3\nu},\frac{3+\sqrt2}{3},\mp2\nu a^3)
\qquad,\\
\label{eq:point-lambda-wave-2}
\phi^{\pm}_2(\epsilon\,;a) &= &a^{1-\sqrt2/2}e^{\pm\nu a^3}F(\frac{3-
\sqrt2}{6}\mp \frac{4\epsilon}{3\nu},
\frac{3-\sqrt2}{3},\mp2\nu a^3)
\qquad,
\end{aligned}$$ where $\nu=2\sqrt{6\Lambda}/3$. However Kummer’s function obeys the property of Kummer’s transformation $$\label{eq:kummer-transformation}
F(\alpha,\gamma,z) = e^zF(\gamma-\alpha,\gamma,-z) \qquad.$$ So in fact one can check that $\phi^+_{1,2}=\phi^-_{1,2}$, and there are only two independent solutions for each $\epsilon$. In addtion, when apply eq. into eq. we get the same two solutions except a constant coefficient according to Kummer transfermation.
For great arguments, Kummer’s function can be approximately expanded as $$\label{eq:kummer-equation-asymp-huge}
F(\alpha,\gamma,z) \sim \Gamma(\gamma)
\left(\frac{e^zz^{\alpha-\gamma}}{\Gamma(\alpha)}
+\frac{(-z)^{-\alpha}}{\Gamma(\gamma-\alpha)}\right) \qquad.$$ Applying it into eq. and eq., we get the asymptotic expansions of these wavefunctions when $a$ goes to infinity, $$\begin{aligned}
\label{eq:point-lambda-wave-1-asymp-huge}
\phi_1 &\sim &\Theta(\frac{3+\sqrt2}{6},-\nu) +\Theta(\frac{3+
\sqrt2}{6},\nu) \qquad,\\
\label{eq:point-lambda-wave-2-asymp-huge}
\phi_2 &\sim &\Theta(\frac{3-\sqrt2}{6},-\nu) +\Theta(\frac{3-
\sqrt2}{6},\nu) \qquad,
\end{aligned}$$ where $$\Theta(\xi,\nu) = \Gamma(2\xi) \frac{e^{\nu
a^3} a^{(-1/2+4\epsilon/\nu)}}{(2\nu)^{(\xi -4\epsilon/3\nu)}
\Gamma(\xi+4\epsilon/3\nu)} \qquad.$$ Since $\nu=2\sqrt{6\Lambda}/3$, the sign of $\Lambda$ will decide whether $\nu$ is real, and therefore decide how the asymptotic expansions behave.
When $\Lambda<0$, $\nu=i(2\sqrt{-6\Lambda}/3)$ is an imaginary number and makes the modulus of both $\exp(\pm\nu a^3)$ and $a^{\pm 4\epsilon/\nu}$ being unit for any real $a$. So the asymptotic expansions can be simplified to $$\phi_{1,2} \sim O(a^{-1/2}) \qquad.$$ This descending with the order of minus one half is too slow, thus neither of the two eigen functions is normalizable.
When $\Lambda>0$, with $\nu$ being a real number, the behavior of $\Theta$ is completely determined by the exponential term $\exp(\pm\nu a^3)$. Fortunately, we know that $\Gamma(z)$ explodes at non-positive integer points, thus a carefully selected eigenvalue $\epsilon$ can make the exploded term $\Theta(\xi,\nu)$ in the expansions vanish.
For $\phi_1$ it requires $$\label{eq:point-lambda-wave-1-energy}
\epsilon^{(1)}_n = -\frac{(6n+3+\sqrt2)}{2}\sqrt{\frac{\Lambda}{6}} \quad,\quad
n=0,1,\dots \qquad.$$ And we have the fact that for $F(\alpha,\gamma+1,z)$ whose $\alpha$ is a non-positive integer, it can be described by Laguerre function $$\label{eq:laguerre}
L_n^{(\gamma)}(z) := {n+\gamma \choose n} F(-n,\gamma+1,z) \qquad.$$ So the eigen states $|n^{(1)}\rangle$ for eigenvalue $\epsilon^{(1)}_n$ after normalization is written as $$\label{eq:point-lambda-state-1}
\langle a|n^{(1)}\rangle = c^{(1)}_n a^{1+\sqrt2/2} e^{-\nu a^3}
L^{(\frac{\sqrt2}{3})}_n(2\nu a^3) \qquad,$$ where $c^{(1)}_n$ is the normalization factor.
Same discussion also holds for solution $\phi_2$, giving its eigenvalue $$\label{eq:point-lambda-wave-2-energy}
\epsilon^{(2)}_n = -\frac{(6n+3-\sqrt2)}{2}\sqrt{\frac{\Lambda}{6}} \quad,
\quad
n=0,1,\dots \qquad,$$ and the normalized eigen state $$\label{eq:point-lambda-state-2}
\langle a|n^{(2)}\rangle = c^{(2)}_n a^{1-\sqrt2/2} e^{-\nu a^3}
L^{(-\frac{\sqrt2}{3})}_n(2\nu a^3) \qquad,$$ where $c^{(2)}_n$ is its normalization factor.
From the definition of Kummer’s function eq. we can see that for tiny arguments, $$\begin{aligned}
\label{eq:point-lambda-wave-1-asymp-tiny}
\phi_1(\epsilon\,;a) &\sim & a^{1+\sqrt2/2} (1-\nu a^3)(1
+2\nu \frac{\alpha}{\gamma}a^3)
\qquad,\\
\label{eq:point-lambda-wave-2-asymp-tiny}
\phi_2(\epsilon\,;a) &\sim & a^{1-\sqrt2/2} (1-\nu a^3)(1
+\nu\frac{\alpha-\gamma+1}{2-
\gamma}a^3) \qquad.
\end{aligned}$$ Therefore when $a$ goes to zero, $\phi_1$ and $\phi_2$ are all vanish, hence the cosmological singularity is naturally avoided. Thus we can choose the two sets of eigen states to be the basis of the quantum system: $\epsilon^{(1)}_n$, $|n^{(1)}\rangle$ and $\epsilon^{(2)}_n$, $|n^{(2)}\rangle$.
For any real number $\alpha$, the first two Laguerre polynomials are $$\begin{aligned}
\label{eq:laguerre-zero}
L^{(\gamma)}_0 &= &1 \qquad, \\
\label{eq:laguerre-one}
L^{(\gamma)}_1 &= &1 +\gamma -x \qquad.
\end{aligned}$$ So the first set of eigen states with eigenvalue of the highest two are $$\begin{aligned}
\label{eq:point-lambda-wave-2-highest}
\langle a|0^{(1)}\rangle &= &c^{(1)}_0 a^{1+\sqrt2/2} e^{-\nu a^3}
\qquad, \\
\label{eq:point-lambda-wave-1-highest}
\langle a|1^{(1)}\rangle &= &c^{(1)}_1 a^{1+\sqrt2/2} \left(\frac{3+
\sqrt2}
{3}-2\nu a^3\right)
e^{-\nu a^3} \qquad.
\end{aligned}$$ If a certain state is a combination of only these two states $|\psi\rangle=\alpha|0^{(1)}\rangle+\beta|1^{(1)}\rangle$ at initial ($|\alpha|^2+|\beta|^2=1$), it will evolve with respect to the cosmological time $t$ as $$\label{eq:point-lambda-wave-highest-comb}
|\psi,t\rangle = \alpha|0^{(1)}\rangle e^{-i\epsilon^{(1)}_0 t}
+\beta|1^{(1)}\rangle e^{-i\epsilon^{(1)}_1 t} \qquad.$$ The evolution of the average measurement $\overline{a^2}$ can be calculated out: $$\begin{aligned}
\overline{a^2}(t)= & \langle\psi,t|a^2|\psi,t\rangle \\
= & |\alpha|^2\langle0|a^2|0\rangle
+\alpha^*\beta \langle0|a^2|1\rangle e^{i(\epsilon^{(1)}_0-
\epsilon^{(1)}_1)t}
+|\beta|^2\langle1|a^2|1\rangle
+\alpha\beta^* \langle1|a^2|0\rangle e^{-i(\epsilon^{(1)}_0-
\epsilon^{(1)}_1)t} \\
= & 2 \Re(\alpha^*\beta e^{i(\epsilon^{(1)}_0-\epsilon^{(1)}_1)t})
\langle0|a^2|1\rangle
+(|\alpha|^2\langle0|a^2|0\rangle +|\beta|^2\langle1|a^2|1\rangle) \\
= &2c_{0,1}|\alpha||\beta|
\cos\left(\frac{\sqrt{6\Lambda}}{2}\,t -\theta\right)
+(c_{0,0}|\alpha|^2 +c_{1,1}|\beta|^2) &\qquad.
\end{aligned}$$ where $c_{n,m}=\langle n^{(1)}|a^2|m^{(1)}\rangle$, $\theta=\operatorname{Arg}(\alpha)-\operatorname{Arg}(\beta)$. Fortunately, a certain state which is a combination of $|n^{(2)}\rangle$ can get the similar result as $(\epsilon_{n}^{(2)}-\epsilon_{m}^{(2)})$ is also always some integer times of $\sqrt{6\Lambda}/2$ according to eq..
Therefore it is reasonable to assuming that our universe is in a state of combination of $|n^{(1)}\rangle$ or $|n^{(2)}\rangle$. This solution suggests a pulsing universe with a characteristic time of $4\pi\sqrt{1/6\Lambda}$. Actually although we do not know which state the universe is at a certain time, we can prove that it always rebounds with the same characteristic time no matter how the initial coefficients for the state of each energy level are given. It is clear that the evolution of $\overline{a^2}$ is always described by $$\label{eq:point-lambda-average-a-normal}
\overline{a^2}(t) = \sum_{n,m\geqslant0} \eta_{nm}
\cos((\varepsilon_n -\varepsilon_m)t +\theta_{nm})
\qquad.$$ We can see $\varepsilon_n-\varepsilon_m$ is always some integer times of $\sqrt{6\Lambda}/2$ which ensures that $\overline{a}(t)$ has a period of $4\pi\sqrt{1/6\Lambda}$.
Considering the universe as we observed is experiencing an accelerating expansion now, it is reasonable to assume it is still in the first quarter of the period, implying the cosmological constant $\Lambda$ should be no bigger than $\pi^2/6T_0^2$ where $T_0$ stands for the cosmological time till now.
Another interesting thing is that, observing the combination of solutions can provide a square wave, this quantum system must have some special states that may let $\overline{a^2}$ rise as fast as possible at some certain time $t_0$. For the simplest case, considering a state composed by eigen states with real coefficients $|\psi,t\rangle=\sum_n\tau_ne^{-i\varepsilon_n t}|n\rangle$, if its scale factor evolves like $$\label{eq:square}
\overline{a^2}(t) = A \sum_{k=0}^{N} \frac{1}{2k-1}
\cos\left(\frac{\sqrt{6\Lambda}}{2}(2k-1)\,t\right) +C \qquad,$$ then choose only the coefficients of the first $2N-1$ states to be non-zero and real, we know they will satisfy the polynomial system $$\label{eq:polynomial-coefficient}
\sum_{n=0}^{2N-l-1} \tau_n\tau_{n+l}c_{n,n+l} = \frac{A}{4}
\left(\frac{1-(-1)^l}{l}\right) \quad,\quad 1\leqslant l\leqslant
2N-1 \qquad,$$
There are in total $2N$ coefficients needed to be fixed. The normalization condition together with eq. exactly give the same number of equations from which the coefficients can be solved. Therefore we can satisfy eq. for any $N$ as large as we wish. That gives the possibility of an expansion of $a$ at any velocity, which may generate an inflation with the speed even faster than exponential level as normal understanding.
Quantum Cosmology of $f(R)$ Gravity
===================================
For more general cases, we consider the quantum model of a flat FRW universe under $f(R)$ gravity. More detailed discussion on the point-like model of $f(R)$ universe can be found in the works of Capozziello [@capozziello_fr_2008]. The point-like Lagrangian with no term of matter will be like $$\label{eq:point-fR-lagrangian}
L = (f-f_RR)a^3 -6f_{RR}\dot{R}a^2\dot{a}
-6f_Ra\dot{a}^2 \qquad,$$ and the canonical momentums for $a$ and $R$ are respectively $$\begin{aligned}
\label{eq:point-fR-momentum-a}
\pi_a &= &-6f_{RR}\dot{R}a^2 -12f_Ra\dot{a} \qquad,\\
\label{eq:point-fR-momentum-R}
\pi_R &= &-6f_{RR}a^2\dot{a} \qquad.
\end{aligned}$$ So the canonical energy is $$\label{eq:point-fR-hamiltonian}
E_{L} = -(f-f_RR)a^3 -6f_{RR}\dot{R}a^2\dot{a}^2
-6f_Ra\dot{a}^2 \qquad.$$
From eq. we can directly read that $$\dot{a} = -\frac{\pi_R}{6f_{RR}a^2} \qquad,$$ Plug it into eq. and get $$6f_{RR}\dot{R}a^2 = -\pi_a +\frac{2}{a}\frac{f_R}{f_{RR}}\pi_R \qquad.$$ Applying them to eq., the Hamiltonian for the system becomes $$\begin{aligned}
H &= -(f-f_RR)a^3 +\frac{\pi_R}{6f_{RR}a^2} \left(-\pi_a
+\frac{2}{a}\frac{f_R}{f_{RR}}\pi_R\right) -6f_Ra
\left(-\frac{\pi_R}{6f_{RR}a^2}\right)^2 \\
&= -(f-f_RR)a^3
-\frac{1}{6a^2}\frac{1}{f_{RR}}\pi_R\pi_a
+\frac{1}{6a^3}\frac{f_R}{f_{RR}^2}\pi_R^2 \qquad.
\end{aligned}$$ So the wave equation that describes this quantum system is $$\begin{aligned}
\label{eq:point-fR-schrodinger}
i\frac{\partial}{\partial t}\Psi = -\frac{1}{6a^3}\frac{f_R}{f_{RR}^2}\frac{\partial^2}{\partial R^2}\Psi+\frac{1}{6a^2}\frac{1}{f_{RR}}\frac{\partial^2}{\partial a \partial R}\Psi+\frac{1}{24a^3}\left(\frac{6f_Rf_{RRR}}{f_{RR}^3}-\frac{7}{f_{RR}}\right)\frac{\partial}{\partial R}\Psi-
\frac{1}{12a^2}\\
\frac{f_{RRR}}{f_{RR}^2}\frac{\partial}{\partial a}\Psi
-\left[\frac{1}{24a^3}\left(\frac{6f_Rf_{RRR}^2}{f_{RR}^4}-\frac{5f_{RRR}}{f_{RR}^2}-\frac{2f_Rf_{RRRR}}{f_{RR}^3}\right)+a^3(f-f_RR)\right]\Psi \qquad.
\end{aligned}$$ $\Psi(t,a,R)$ is a function of cosmological time $t$, scale factor $a$ and Ricci scalar $R$.
The equation relies on the form of $f(R)$ to be exactly solved. However as a linear partial differential equation, its coefficients of all the second order terms satisfy the fact that $\Delta=1/(144a^4f_{RR}^2)$ is positive on the whole $a-R$ plane. Therefore the eigen equation of the operator $\hat{H}$ is a hyperbolic equation and can be transformed into a wave equation.
We need to point out that $a$ and $R$ have been separated via Palatini formalism. In this case, their relation is linked by one of the equations of motion rather than a given definition. So after the quantization, this relation must have degenerated to be statistically satisfied. That means even a flat universe of small scale or a huge scale universe with large curvature which are not normally allowed in the classical case will also have contribution to the possibility.
Conclusion And Discussions
==========================
The purpose of this paper is to introduce a new approach to inquire the minisuperspace model without seeking the Wheeler-DeWitt equation for a certain gravitational theory.
The technique is to apply the cosmological principle directly to the action of a gravitational system before variation, and reform the Lagrangian of geometry to a classical point-like one. It is obvious that such a process of taking the metric of a cosmological model which is truncated by an enormous degree of imposed symmetry and simply plugging it into a quantization procedure should not give an answer that is in any way an exact solution. However, strange enough, we have seen that the variation of this point-like Lagrangian gives the right equation of motion (the Friedmann equation) to describe the universe.
By quantizing this semi-classical system described by the point-like Lagrangian, we represent a quantum system that is very similar to the minisuperspace from reducing the superspace where the Wheel-DeWitt equation is defined on. The only difference is, in our situation, for solving a semi-classical Schrödinger equation we need the concept of the eigenvalue $\epsilon$ of the Hamiltonian of the system which does not exist in the classical minisuperspace theory.
It is very natural to apply our technique beyond the Einstein gravity to the gravitational model with cosmological constant and more general $f(R)$ gravity with the help of Palatini formalism and respectively get their quantum cosmological model. This especially opens the gate for considering quantum cosmology of $f(R)$ gravity.
As the second aspect of our work shown in this paper, we give the exact solutions of the quantum systems we get under Einstein gravity with and without cosmological constant. We find that the existence of a tiny positive cosmological constant is reasonable
We prove that all possible states in such a legal quantum cosmological model predict pulsing universe with the same period of a cosmological characteristic time that is inversely proportional to the square root of the cosmological constant. Considering the enormous amount of time the universe has existed, the cosmological constant must be extremely tiny.
Moreover, we show that this quantum system contains states that allow expansion at any speed as fast as possible, which could probably provide a motivation for inflation.
Acknowledgments
===============
We would like to thank the National Natural Science Foundation of China (Grant No.11571342) for supporting us on this work.
[^1]: *Corresponding author:* dengjb@lzu.edu.cn
|
---
abstract: 'We prove that every faithfully flat Hopf-Galois object is a quantum torsor in the sense of Grunspan.'
address: |
Mathematisches Institut der Universität München\
Theresienstr. 39\
80333 München\
Germany\
email: schauen@mathematik.uni-muenchen.de
author:
- Peter Schauenburg
title: 'Quantum torsors and Hopf-Galois objects'
---
Introduction
============
The main result of this short note is to complete the comparison between the notion of a quantum torsor recently introduced by Grunspan [@Gru:QT], and the older notion of a Hopf-Galois object.
An $H$-Galois object for a $k$-Hopf algebra $H$ is a right $H$-comodule algebra $A$ whose coinvariant subalgebra is the base ring $k$ and for which the canonical map $$\beta:=\left( A{\otimes}A\xrightarrow{A{\otimes}\rho}A{\otimes}A{\otimes}H\xrightarrow{\nabla{\otimes}H}A{\otimes}H\right)$$ is a bijection (where $\nabla$ is the multiplication map of $A$, and $\rho\colon A\rightarrow A{\otimes}H$ is the coaction of $H$ on $A$). The notion appears in this generality in [@KreTak:HAGEA]; we refer to Montgomery’s book [@Mon:HAAR] for background. If one specializes $A$ and $H$ to be affine commutative algebras, then they correspond to an affine scheme and an affine group scheme, respectively, and the definition recovers the definition of a $G$-torsor with structure group $G=\operatorname{Spec}(H)$, in other words the affine algebraic version of a principal fiber bundle.
In Grunspan’s definition a quantum torsor is an algebra $T$ equipped with certain structure maps $\mu\colon T\rightarrow T{\otimes}T^{{\operatorname{op}}}{\otimes}T$ and $\theta\colon T\rightarrow T$ which are required to fulfill a set of axioms that we shall recall below. The definition is also inspired by results in classical algebraic geometry, going back to work of Baer [@Bae:ES]; we refer to [@Gru:QT] for more literature. Notably, if we again specify $T$ to be an affine commutative algebra, then the definition (which now does not need the map $\theta$) is known to characterize torsors, without requiring any prior specification of a structure group; in fact two structure groups can be constructed from the torsor rather than having to be given in advance. In addition to being group-free, this characterization has advantages when additional structures, notably Poisson structures, come into play: In the latter situation one cannot expect the canonical map $\beta$ in the definition of a Hopf-Galois extension to be maps of Poisson algebras, while the structure maps of a torsor are; thus the definition of a Poisson torsor becomes more natural when given in the group-free form.
Generalizing the results on commutative torsors, Grunspan shows that any torsor $T$ in the sense of his definition has the structure of an $L$-$H$-bi-Galois extension for two naturally constructed Hopf algebras $L=H_l(T)$ and $H=H_r(T)$. Thus, as in the commutative case, a torsor is a quantum group-free way to define a quantum principal homogeneous space (with trivial base), with quantum structure group(s) that can be constructed afterwards.
The following natural question is left open (or rather, asked explicitly) in [@Gru:QT]: Are there Hopf-Galois objects that do not arise from quantum torsors? Or, on the contrary, does every Hopf-Galois object have a quantum torsor structure?
We shall prove the latter (under the mild assumptions that Hopf algebras should have bijective antipodes, and Hopf-Galois objects should be faithfully flat). Thus Grunspan’s quantum torsors are seen to be an equivalent characterization of Hopf-(bi)-Galois objects, without reference to the Hopf algebras involved, parallel to the commutative case. On the other hand, the group $\operatorname{Tor}(H)$ of quantum torsors associated to a Hopf algebra $H$ in [@Gru:QT] coincides with the group $\operatorname{BiGal}(H,H)$ of $H$-$H$-bi-Galois objects introduced in [@Sch:HBE].
Notations
=========
Throughout the paper, we work over a commutative base ring $k$.
We denote multiplication in an algebra $A$ by $\nabla=\nabla_A$, and comultiplication in a coalgebra $C$ by $\Delta=\Delta_C$; we will write $\Delta(c)=:c{{}_{(1)}}{\otimes}c{{}_{(1)}}$. We will write $\rho\colon V\to V{\otimes}C$ for the structure map of a right $C$-comodule $V$, and $\rho(v)=:v{{}_{(0)}}{\otimes}v{{}_{(1)}}$.
Let $H$ be a $k$-(faithfully) flat $k$-Hopf algebra, with antipode $S$. A right $H$-comodule algebra $T$ is an algebra $T$ which is a right $H$-comodule whose structure map $\rho\colon T\rightarrow T{\otimes}H$ is an algebra map. We say that $T$ is an $H$-Galois extension of its coinvariant subalgebra ${{T}^{\operatorname{co}H}}:=\{t\in T|\rho(t)=t{\otimes}1\}$ if the canonical map $\beta\colon T{\mathrel{\mathop{\otimes}_{{{T}^{\operatorname{co}H}}}}}T\to T{\otimes}H$ given by $\beta(x{\otimes}y)=xy{{}_{(0)}}{\otimes}y{{}_{(1)}}$ is a bijection. We will call an $H$-Galois extension $T$ whose coinvariant subalgebra is the base ring an $H$-Galois object for short. In most of this paper we will be interested in faithfully flat (i.e. faithfully flat as $k$-module) $H$-Galois objects. For an $H$-Galois object $T$, we define $\gamma\colon H\rightarrow T{\otimes}T$ by $\gamma(h):=\beta{^{-1}}(1{\otimes}h)$, and write $\gamma(h)=:h{{}^{[1]}}{\otimes}h{{}^{[2]}}$. The following facts on $\gamma$ can be found in [@Sch:RTHGE]: For all $x\in T$, $g,h\in H$ we have $$\begin{aligned}
x{{}_{(0)}}x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}}&=1{\otimes}x\label{isinv}\\
h{{}^{[1]}}h{{}^{[2]}}&={\varepsilon}(h)\cdot 1\label{nablagamma}\\
h{{}^{[1]}}{\otimes}h{{}^{[2]}}{{}_{(0)}}{\otimes}h{{}^{[2]}}{{}_{(1)}}&=h{{}_{(1)}}{{}^{[1]}}{\otimes}h{{}_{(1)}}{{}^{[2]}}{\otimes}h{{}_{(2)}}\label{colinright}\\
h{{}^{[1]}}{{}_{(0)}}{\otimes}h{{}^{[2]}}{\otimes}h{{}^{[1]}}{{}_{(1)}}&=h{{}_{(2)}}{{}^{[1]}}{\otimes}h{{}_{(2)}}{{}^{[2]}}{\otimes}S(h{{}_{(1)}})\label{colinleft}\\
(gh){{}^{[1]}}{\otimes}(gh){{}^{[2]}}&=h{{}^{[1]}}g{{}^{[1]}}{\otimes}g{{}^{[2]}}h{{}^{[2]}}\label{gammanabla}\\
1{{}^{[1]}}{\otimes}1{{}^{[2]}}&=1{\otimes}1\label{gammaeta}\end{aligned}$$ In particular, the last two equations say that $\gamma\colon H\rightarrow T^{{\operatorname{op}}}{\otimes}T$ is an algebra map.
We now recall Grunspan’s definition of a quantum torsor [@Gru:QT]: A quantum torsor $(T,\nabla,1,\mu,\theta)$ consists of a faithfully flat $k$-algebra $(T,\nabla,1)$, an algebra map $\mu\colon T\to T{\otimes}T^{{\operatorname{op}}}{\otimes}T$, and an algebra automorphism $\theta\colon T\rightarrow T$ satisfying, for all $x\in T$: $$\begin{aligned}
(T{\otimes}\nabla)\mu(x)&=x{\otimes}1\label{torsor.1}\\
(\nabla{\otimes}T)\mu(x)&=1{\otimes}x\label{torsor.2}\\
(T{\otimes}T^{{\operatorname{op}}}{\otimes}\mu)\mu&=(\mu{\otimes}T^{{\operatorname{op}}}{\otimes}T)\mu\label{torsor.3}\\
(T{\otimes}T^{{\operatorname{op}}}{\otimes}\theta{\otimes}T^{{\operatorname{op}}}{\otimes}T)(\mu{\otimes}T^{{\operatorname{op}}}{\otimes}T)\mu
&=(T{\otimes}\mu^{{\operatorname{op}}}{\otimes}T)\mu\label{torsor.4}\\
(\theta{\otimes}\theta{\otimes}\theta)\mu&=\mu\theta\label{torsor.5},\end{aligned}$$ where $\mu^{{\operatorname{op}}}\colon T{{\operatorname{op}}}\to T^{{\operatorname{op}}}{\otimes}T{\otimes}T^{{\operatorname{op}}}$ is defined by $\mu^{{\operatorname{op}}}=\tau_{(13)}\mu$, and $\tau_{(13)}$ exchanges the first and last tensor factor in $T{\otimes}T{\otimes}T$. We will also write $T$ or $(T,\mu,\theta)$ for $(T,\nabla,1,\mu,\theta)$, if the structure maps, or at least the algebra structure maps, are clear from the context. What we have defined above is what is called a $k$-torsor in [@Gru:QT], where more generally the notion of an $A$-torsor is defined for every $k$-algebra $A$. However, after extending scalars from $k$ to $A$, the notion of an $A$-torsor is covered by the above definition, which is therefore sufficient for our purposes.
The main result
===============
We shall show that every faithfully flat $H$-Galois object $T$ is a quantum torsor. To prepare, we shall show that certain elements in $T{\otimes}T$ and $T{\otimes}T{\otimes}T$ which shall occur in our calculations can be written with the righmost tensor factors taken to be scalars, or equivalently $H$-coinvariant elements:
Let $T$ be a faithfully flat $H$-Galois object. Then $$\label{skalar1}
S(x{{}_{(1)}}){{}^{[1]}}{\otimes}x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}}\in T{\otimes}k\subset T{\otimes}T$$ for all $x\in T$, and $$\label{skalar2}
h{{}_{(1)}}{{}^{[1]}}{\otimes}S(h{{}_{(2)}}){{}^{[1]}}{\otimes}h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(2)}}){{}^{[2]}}\in T{\otimes}T{\otimes}k\subset T{\otimes}T{\otimes}T$$ for all $h\in H$.
For $x\in T$ we have $$\begin{aligned}
S(x{{}_{(1)}}){{}^{[1]}}&{\otimes}\rho(x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}})\\
&=S(x{{}_{(2)}}){{}^{[1]}}{\otimes}x{{}_{(0)}}S(x{{}_{(2)}}){{}^{[2]}}{{}_{(0)}}{\otimes}x{{}_{(1)}}S(x{{}_{(2)}}){{}^{[2]}}{{}_{(1)}}\\
&{\overset{\text{\eqref{colinright}}}=}S(x{{}_{(2)}}){{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(0)}}S(x{{}_{(2)}}){{}_{(1)}}{{}^{[2]}}{\otimes}x{{}_{(1)}}S(x{{}_{(2)}}){{}_{(2)}}\\
&=S(x{{}_{(3)}}){{}^{[1]}}{\otimes}x{{}_{(0)}}S(x{{}_{(3)}}){{}^{[2]}}{\otimes}x{{}_{(1)}}S(x{{}_{(2)}})\\
&=S(x{{}_{(1)}}){{}^{[1]}}{\otimes}x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}}{\otimes}1
\end{aligned}$$ in $T{\otimes}T{\otimes}H$. Since ${{T}^{\operatorname{co}H}}=k$ and $T$ is flat over $k$, this proves the first claim. Similarly, for $h\in H$ we have $$\begin{aligned}
h{{}_{(1)}}{{}^{[1]}}&{\otimes}S(h{{}_{(2)}}){{}^{[1]}}{\otimes}\rho(h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(2)}}){{}^{[2]}})\\
&{\overset{\text{\eqref{colinright}}}=}h{{}_{(1)}}{{}^{[1]}}{\otimes}S(h{{}_{(3)}}){{}_{(1)}}{{}^{[1]}}{\otimes}h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(3)}}){{}_{(1)}}{{}^{[2]}}{\otimes}h{{}_{(2)}}S(h{{}_{(3)}}){{}_{(2)}}\\
&=h{{}_{(1)}}{{}^{[1]}}{\otimes}S(h{{}_{(4)}}){{}^{[1]}}{\otimes}h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(4)}}){{}^{[2]}}{\otimes}h{{}_{(2)}}S(h{{}_{(3)}})\\
&=h{{}_{(1)}}{{}^{[1]}}{\otimes}S(h{{}_{(2)}}){{}^{[1]}}{\otimes}h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(2)}}){{}^{[2]}}{\otimes}1,
\end{aligned}$$ proving the second claim, again by flatness of $T$.
Abusing Sweedler notation, the Lemma says that the “elements” $x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}}$ and $h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(2)}}){{}^{[2]}}$ are scalars. We will use this by moving these elements around freely in any $k$-multilinear expression in calculations below, sometimes indicating our plans by putting parentheses around the “scalar” before moving it.
Let $T$ be a faithfully flat $H$-Galois object, where $H$ is a Hopf algebra with bijective antipode. Then $(T,\mu,\theta)$ is a quantum torsor, with $$\begin{gathered}
\mu(x)=(T{\otimes}\gamma)\rho(x)=x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}}\\
\theta(x)=(x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}})S(x{{}_{(1)}}){{}^{[1]}}=S(x{{}_{(1)}}){{}^{[1]}}(x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}})
\end{gathered}$$
For all calculations, we let $x,y\in T$ and $h\in H$.
Since $\rho$ and $\gamma$ are algebra maps, so is $\mu$. We have $$(T{\otimes}\nabla)\mu(x)=x{{}_{(0)}}{\otimes}\nabla\gamma(x{{}_{(1)}})=x{{}_{(0)}}{\otimes}{\varepsilon}(x{{}_{(1)}})1=x{\otimes}1$$ by , and $(\nabla{\otimes}T)\mu(x)=x{{}_{(0)}}x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}}=1{\otimes}x$ by . Next $$\begin{aligned}
(T{\otimes}T^{{\operatorname{op}}}{\otimes}\mu)\mu(x)
&=x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}\mu(x{{}_{(1)}}{{}^{[2]}})\\
&=x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}}{{}_{(0)}}{\otimes}\gamma(x{{}_{(1)}}{{}^{[2]}}{{}_{(1)}})\\
&{\overset{\text{\eqref{colinright}}}=}x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}}{\otimes}\gamma(x{{}_{(2)}})\\
&=\mu(x{{}_{(0)}}){\otimes}\gamma(x{{}_{(1)}})\\
&=(\mu{\otimes}T^{{\operatorname{op}}}{\otimes}T)\mu(x)
\end{aligned}$$ proves . It is clear that $\theta(1)=1$. For $x,y\in T$ we have $$\begin{aligned}
\theta(xy)&=x{{}_{(0)}}y{{}_{(0)}}S(x{{}_{(1)}}y{{}_{(1)}}){{}^{[2]}}S(x{{}_{(1)}}y{{}_{(1)}}){{}^{[1]}}\\
&=x{{}_{(0)}}y{{}_{(0)}}(S(y{{}_{(1)}})S(x{{}_{(1)}})){{}^{[2]}}(S(y{{}_{(1)}})S(x{{}_{(1)}})){{}^{[1]}}\\
&{\overset{\text{\eqref{gammanabla}}}=}x{{}_{(0)}}(y{{}_{(0)}}S(y{{}_{(1)}}){{}^{[2]}}) S(x{{}_{(1)}}){{}^{[2]}}S(x{{}_{(1)}}){{}^{[1]}}S(y{{}_{(1)}}){{}^{[1]}}\\
&{\overset{\text{\eqref{skalar1}}}=}x{{}_{(0)}} S(x{{}_{(1)}}){{}^{[2]}}S(x{{}_{(1)}}){{}^{[1]}}(y{{}_{(0)}}S(y{{}_{(1)}}){{}^{[2]}})S(y{{}_{(1)}}){{}^{[1]}}\\
&=\theta(x)\theta(y),
\end{aligned}$$ so $\theta$ is an algebra map.
For $h\in H$ we have $$\label{thetari}
h{{}^{[1]}}{\otimes}\theta(h{{}^{[2]}})=S(h){{}^{[2]}}{\otimes}S(h){{}^{[1]}}$$ by the calculation $$\begin{aligned}
h{{}^{[1]}}{\otimes}\theta(h{{}^{[2]}})&=h{{}^{[1]}}{\otimes}h{{}^{[2]}}{{}_{(0)}}S(h{{}^{[2]}}{{}_{(1)}}){{}^{[2]}}S(h{{}^{[2]}}{{}_{(1)}}){{}^{[1]}}\\
&{\overset{\text{\eqref{colinright}}}=}h{{}_{(1)}}{{}^{[1]}}{\otimes}(h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(2)}}){{}^{[2]}}) S(h{{}_{(2)}}){{}^{[1]}}\\
&{\overset{\text{\eqref{skalar2}}}=}h{{}_{(1)}}{{}^{[1]}}(h{{}_{(1)}}{{}^{[2]}}S(h{{}_{(2)}}){{}^{[2]}}){\otimes}S(h{{}_{(2)}}){{}^{[2]}}\\
&{\overset{\text{\eqref{nablagamma}}}=}S(h){{}^{[2]}}{\otimes}S(h){{}^{[1]}}.
\end{aligned}$$ We conclude that $$(T{\otimes}T^{{\operatorname{op}}}{\otimes}\theta)\mu(x)
=x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}\theta(x{{}_{(1)}}{{}^{[2]}})\\
{\overset{\text{\eqref{thetari}}}=}x{{}_{(0)}}{\otimes}S(x{{}_{(1)}}){{}^{[2]}}{\otimes}S(x{{}_{(1)}}){{}^{[1]}},$$ hence $$\begin{aligned}
(T{\otimes}T^{{\operatorname{op}}}{\otimes}\theta{\otimes}T^{{\operatorname{op}}}{\otimes}T)&(\mu{\otimes}T^{{\operatorname{op}}}{\otimes}T)\mu(x)\\
&=(T{\otimes}T^{{\operatorname{op}}}{\otimes}\theta)\mu(x{{}_{(0)}}){\otimes}\gamma(x{{}_{(1)}})\\
&=x{{}_{(0)}}{\otimes}S(x{{}_{(1)}}){{}^{[2]}}{\otimes}S(x{{}_{(1)}}){{}^{[1]}}{\otimes}\gamma(x{{}_{(2)}}),
\end{aligned}$$ and on the other hand $$\begin{aligned}
(T{\otimes}\mu^{{\operatorname{op}}}{\otimes}T)\mu(x)
&=x{{}_{(0)}}{\otimes}\mu^{{\operatorname{op}}}(x{{}_{(1)}}{{}^{[1]}}){\otimes}x{{}_{(1)}}{{}^{[2]}}\\
&=x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{{}_{(1)}}{{}^{[2]}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[2]}}\\
&{\overset{\text{\eqref{colinleft}}}=}x{{}_{(0)}}{\otimes}S(x{{}_{(1)}}){{}^{[2]}}{\otimes}S(x{{}_{(1)}}){{}^{[1]}}{\otimes}x{{}_{(2)}}{{}^{[1]}}{\otimes}x{{}_{(2)}}{{}^{[2]}},
\end{aligned}$$ proving . To prove we first check $$\label{thetacol}
\rho\theta(x)=\theta(x{{}_{(0)}}){\otimes}S^2(x{{}_{(1)}}),$$ by the calculation $$\begin{aligned}
\rho\theta(x)&{\overset{\text{\eqref{skalar1}}}=}(x{{}_{(0)}} S(x{{}_{(1)}}){{}^{[2]}})\rho(S(x{{}_{(1)}}){{}^{[1]}})\\
&{\overset{\text{\eqref{colinleft}}}=}x{{}_{(0)}} S(x{{}_{(1)}}){{}_{(2)}}{{}^{[2]}}S(x{{}_{(1)}}){{}_{(2)}}{{}^{[1]}}{\otimes}S(S(x{{}_{(1)}}){{}_{(1)}})\\
&=x{{}_{(0)}}S(x{{}_{(1)}}){{}^{[2]}}S(x{{}_{(1)}}){{}^{[1]}}{\otimes}S^2(x{{}_{(2)}})\\
&=\theta(x{{}_{(0)}}){\otimes}S^2(x{{}_{(2)}}).
\end{aligned}$$ Using this, we find $$\begin{aligned}
(\theta{\otimes}\theta{\otimes}\theta)\mu(x)
&=\theta(x{{}_{(0)}}){\otimes}\theta(x{{}_{(1)}}{{}^{[1]}}){\otimes}\theta(x{{}_{(1)}}{{}^{[2]}})\\
&{\overset{\text{\eqref{thetari}}}=}\theta(x{{}_{(0)}}){\otimes}\theta(S(x{{}_{(1)}}){{}^{[2]}}){\otimes}S(x{{}_{(1)}}){{}^{[1]}}\\
&{\overset{\text{\eqref{thetari}}}=}\theta(x{{}_{(0)}}){\otimes}S^2(x{{}_{(1)}}){{}^{[1]}}{\otimes}S^2(x{{}_{(1)}}){{}^{[2]}}\\
&=\theta(x{{}_{(0)}}){\otimes}\gamma(S^2(x{{}_{(1)}}))\\
&{\overset{\text{\eqref{thetacol}}}=}\theta(x){{}_{(0)}}{\otimes}\gamma(\theta(x){{}_{(1)}})=\mu\theta(x).
\end{aligned}$$ It remains to check that $\theta$ is a bijection. Now we have seen that $\theta$ is an algebra map, and colinear, provided that the codomain copy of $T$ is endowed with the comodule structure restricted along the Hopf algebra automorphism $S^2$ of $H$. Of course $T$ with this new comodule algebra structure is also $H$-Galois. It is known [@Sch:PHSAHA Rem.3.11.(1)] that every comodule algebra homomorphism between nonzero $H$-Galois objects is a bijection.
Obviously, if we drop the requirement that $\theta$ be bijective from the definition of a quantum torsor, we can do without bijectivity of the antipode of $H$ in the proof. More precisely, the proof shows that $\theta$ is bijective if and only if $S$ is.
By the results of Grunspan, any quantum torsor $T$ has associated to it two Hopf algebras $H_l(T)$ and $H_r(T)$, which make it into an $H_l(T)$-$H_r(T)$-bi-Galois object in the sense of [@Sch:HBE]. That is, $T$ is a right $H_r(T)$-Galois object in the sense recalled above, and at the same time a left $H_l(T)$-Galois object (i.e.the same as a right Galois object, with sides switched in the definition), in such a way that the two comodule structures involved make it into an $H_l(T)$-$H_r(T)$-bicomodule. Together with these constructions, [\[mainthm.nme\] \[mainthm\]]{} shows that the notions of a quantum torsor and of a Hopf-bi-Galois extension are equivalent, provided that we complete the picture by proving the following:
1. Let $T$ be a faithfully flat $H$-Galois object, and consider the torsor associated to it as in [\[mainthm.nme\] \[mainthm\]]{}. Then $H_r(T)\cong H$, and $H_l(T)\cong L(T,H)$, where the latter is the Hopf algebra making $T$ an $L(T,H)$-$H$-bi-Galois object, see [@Sch:HBE].
2. Let $T$ be a quantum torsor. Then the quantum torsor associated as in [\[mainthm.nme\] \[mainthm\]]{} to the $H_r(T)$-Galois object $T$ coincides with $T$.
By the results in [@Sch:HBE], each of the two one-sided Hopf-Galois structures in an $L$-$H$-bi-Galois object determines the other (along with the other Hopf algebra). Thus to prove (1), it suffices to check that $L(T,H)\cong H_l(T)$, and the isomorphism is compatible with the left coactions. Now let $\xi\in T{\otimes}T^{{\operatorname{op}}}$. We write formally $\xi=x{\otimes}y$ even though we do not assume $\xi$ to be a decomposable tensor. According to the definition of $H_l(T)\subset T{\otimes}T^{{\operatorname{op}}}$ in [@Gru:QT], we have $$\begin{aligned}
\xi\in H_l(T)&
\Leftrightarrow
(T{\otimes}T^{{\operatorname{op}}}{\otimes}T{\otimes}\theta)\mu(x){\otimes}y=x{\otimes}\mu^{{\operatorname{op}}}(y)\\
&\Leftrightarrow
x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}\theta(x{{}_{(1)}}{{}^{[2]}}){\otimes}y=x{\otimes}\mu^{{\operatorname{op}}}(y)\\
&{\overset{\text{\eqref{thetari}}}\Leftrightarrow}x{{}_{(0)}}{\otimes}S(x{{}_{(1)}}){{}^{[2]}}{\otimes}S(x{{}_{(1)}}){{}^{[1]}}{\otimes}y=x{\otimes}y{{}_{(1)}}{{}^{[2]}}{\otimes}y{{}_{(1)}}{{}^{[1]}}{\otimes}y{{}_{(0)}}\\
&\Leftrightarrow x{{}_{(0)}}{\otimes}S(x{{}_{(1)}}){\otimes}y=x{\otimes}y{{}_{(1)}}{\otimes}y{{}_{(0)}}\\
&\Leftrightarrow \xi\in{{(T{\otimes}T)}^{\operatorname{co}H}}
\end{aligned}$$ where in the last step $T{\otimes}T$ is endowed with the codiagonal comodule structure, and we have used a version of [@Sch:PHSAHA Lem.3.1]. By the definition of $L(T,H)$ in [@Sch:HBE], this shows $L(T,H)=H_\ell(T)$ as algebras. A look at the respective definitions of comultiplication in $L(T,H)$ and $H_\ell(T)$ and of their coactions on $T$ shows that these also agree.
To show (2), we use the following results on $H_r(T)$ from [@Gru:QT]: $H_r(T)$ is some subalgebra of $T^{{\operatorname{op}}}{\otimes}T$, the right $H_r(T)$-comodule algebra structure of $T$ maps $x\in T$ to $x{{}_{(0)}}{\otimes}x{{}_{(1)}}:=\mu(x)\in T{\otimes}H_r(T)\subset T{\otimes}T^{{\operatorname{op}}}{\otimes}T$, and $T$ is in fact $H_r(T)$-Galois, that is, the canonical map $\beta\colon T{\otimes}T\to T{\otimes}H$ is bijective. Now the torsor structure $(T,\mu',\theta')$ induced on $T$ by its Hopf-Galois structure as in [\[mainthm.nme\] \[mainthm\]]{} satisfies $\mu'(x)=x{{}_{(0)}}{\otimes}x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}}$. To check $\mu=\mu'$, we apply $\beta$ to the two right tensor factors. Writing $\mu(x):=x{{}^{(1)}}{\otimes}x{{}^{(2)}}{\otimes}x{{}^{(3)}}$, we have $$\begin{aligned}
(T{\otimes}\beta)\mu(x)&=x{{}^{(1)}}{\otimes}\beta(x{{}^{(2)}}{\otimes}x{{}^{(3)}})\\
&=x{{}^{(1)}}{\otimes}x{{}^{(2)}}x{{}^{(3)}}{{}_{(0)}}{\otimes}x{{}^{(3)}}{{}_{(1)}}\\
&=x{{}^{(1)}}{\otimes}x{{}^{(2)}}x{{}^{(3)}}{{}^{(1)}}{\otimes}x{{}^{(3)}}{{}^{(2)}}{\otimes}x{{}^{(3)}}{{}^{(3)}}\\
&{\overset{\text{\eqref{torsor.3}}}=}x{{}^{(1)}}{{}^{(1)}}{\otimes}x{{}^{(1)}}{{}^{(2)}}x{{}^{(1)}}{{}^{(3)}}{\otimes}x{{}^{(2)}}{\otimes}x{{}^{(3)}}\\
&{\overset{\text{\eqref{torsor.1}}}=}x{{}^{(1)}}{\otimes}1{\otimes}x{{}^{(2)}}{\otimes}x{{}^{(3)}}\\
&=x{{}_{(0)}}{\otimes}1{\otimes}x{{}^{(1)}}\\
&=x{{}_{(0)}}{\otimes}\beta(x{{}_{(1)}}{{}^{[1]}}{\otimes}x{{}_{(1)}}{{}^{[2]}})\\
&=(T{\otimes}\beta)\mu'(x)
\end{aligned}$$ Since $\theta$ is determined by $\mu$, we are done.
As a result of the Proposition, the construction $L(T,H)$ for a Hopf-Galois object $T$ coincides with the construction of $H_l(T)$ as in [@Gru:QT] for the quantum torsor associated to the Hopf-Galois object $T$ as in [\[mainthm.nme\] \[mainthm\]]{}. Finally
The group $\operatorname{Tor}(H)$ of isomorphism classes of quantum torsors $T$ equipped with specified isomorphisms $H\cong H_l(T)\cong H_r(T)$ was observed by Grunspan to be a subgroup of the group $\operatorname{BiGal}(H)$ of $H$-$H$-bi-Galois objects defined in [@Sch:HBE]. We see that the two groups in fact coincide.
Ribbon transformations and the Miyashita-Ulbrich action
=======================================================
The proof we gave for [\[mainthm.nme\] \[mainthm\]]{} is rather direct. One can shorten it slightly, and perhaps provide some partial explanation for the behavior of the $\theta$ map by using the Miyashita-Ulbrich action [@Ulb:GNKR; @DoiTak:HGEAMUAAA] and the notion of a ribbon transformation of monoidal functors introduced by Sommerhäuser [@Som:RTITD]. To discuss this, we assume again that $H$ has bijective antipode.
Recall that a right-right Yetter-Drinfeld module $V\in{\mathcal{YD}^{H}_{H}}$ is a right $H$-module (with action denoted ${\leftharpoonup}$) and $H$-comodule such that $$v{{}_{(0)}}{\leftharpoonup}h{{}_{(1)}}{\otimes}v{{}_{(1)}}h{{}_{(2)}}=(v{\leftharpoonup}h{{}_{(2)}}){{}_{(0)}}{\otimes}h{{}_{(1)}}(v{\leftharpoonup}h{{}_{(2)}}){{}_{(1)}},$$ or equivalently $\rho(v{\leftharpoonup}h)=v{{}_{(0)}}{\leftharpoonup}h{{}_{(2)}}{\otimes}S(h{{}_{(1)}})v{{}_{(1)}}h{{}_{(2)}}$ holds for all $v\in V$. The category ${\mathcal{YD}^{H}_{H}}$ is a braided monoidal category. The tensor product of Yetter-Drinfeld modules is their tensor product over $k$ with the (co)diagonal action and coaction, the braiding $\sigma$ is given by $$\sigma_{VW}\colon V{\otimes}W\ni v{\otimes}w\mapsto w{{}_{(0)}}{\otimes}v{\leftharpoonup}w{{}_{(1)}}\in W{\otimes}V$$ for $V,W\in{\mathcal{YD}^{H}_{H}}$, its inverse by $\sigma_{VW}{^{-1}}(w{\otimes}v)=v{\leftharpoonup}S{^{-1}}(w{{}_{(1)}}){\otimes}w{{}_{(0)}}$.
Let $T$ be a faithfully flat $H$-Galois object. The Miyashita-Ulbrich action of $H$ on $T$ is defined by $x{\leftharpoonup}h:=h{{}^{[1]}}xh{{}^{[2]}}$ for $x\in T$ and $h\in H$. It is proved in [@Ulb:GNKR; @DoiTak:HGEAMUAAA] (without the terminology) that $T$ with its $H$-comodule structure and the Miyashita-Ulbrich action is a Yetter-Drinfeld module algebra, that is, an algebra in ${\mathcal{YD}^{H}_{H}}$. This means that it is a module algebra (it is a comodule algebra to begin with), and a Yetter-Drinfeld module. Moreover, $T$ is commutative in the braided monoidal category ${\mathcal{YD}^{H}_{H}}$, which means that we have $\nabla\sigma_{TT}=\nabla$, that is $xy=y{{}_{(0)}}(x{\leftharpoonup}y{{}_{(1)}})$ for all $x,y\in T$.
An endofunctor $F$ of ${\mathcal{YD}^{H}_{H}}$ is defined by letting $F(V)$ be the $k$-module $V$, equipped with the new right coaction $v\mapsto v{{}_{(0)}}{\otimes}S^{-2}(v{{}_{(1)}})$ and right action $v{\otimes}h\mapsto v{\leftharpoonup}S^{2}(h)$. The functor $F$ preserves the tensor product as well as the braiding of ${\mathcal{YD}^{H}_{H}}$.
According to Sommerhäuser, a ribbon transformation $\theta\colon{\mathit{Id}}\rightarrow F$ is a natural transformation such that $\theta_V{\otimes}\theta_W=\theta_{V{\otimes}W}\sigma_{WV}\sigma_{VW}$ holds for all $V,W\in{\mathcal{YD}^{H}_{H}}$ (moreover, we should have $\theta_k={\mathit{id}}_k$). The example of a ribbon transformation we will use is essentially in [@Som:RTITD], up to a switch of sides. It generalizes the map $\theta$ in the proof of [\[mainthm.nme\] \[mainthm\]]{}, and is defined by $\theta_V(v)=v{{}_{(0)}}{\leftharpoonup}S(v{{}_{(1)}})$ for $V\in{\mathcal{YD}^{H}_{H}}$ and $v\in V$. This is surely natural, and also a morphism in ${\mathcal{YD}^{H}_{H}}$, that is, $H$-linear and $H$-colinear according to the formulas $$\begin{aligned}
\rho\theta_V(v)&=\theta_V(v{{}_{(0)}}){\otimes}S^2(v{{}_{(1)}})&
\theta_V(v){\leftharpoonup}h&=\theta_V(v{\leftharpoonup}S^{-2}(h)),
\end{aligned}$$ the first of which was used in our proof of [\[mainthm.nme\] \[mainthm\]]{}; we’ll omit the proofs. Since for all $v\in V\in{\mathcal{YD}^{H}_{H}}$ and $w\in W\in{\mathcal{YD}^{H}_{H}}$ we find $$\begin{aligned}
\theta_{W{\otimes}V}\sigma(v{\otimes}w)
&=\sigma(v{\otimes}w){{}_{(0)}}{\leftharpoonup}S(\sigma(v{\otimes}w))\\
&=\sigma((v{\otimes}w){{}_{(0)}}){\leftharpoonup}S((v{\otimes}w){{}_{(1)}})\\
&=(w{{}_{(0)}}{\otimes}v{{}_{(0)}}{\leftharpoonup}w{{}_{(1)}}){\leftharpoonup}S(v{{}_{(1)}}w{{}_{(2)}})\\
&=w{{}_{(0)}}{\leftharpoonup}S(v{{}_{(2)}}w{{}_{(3)}}){\otimes}v{{}_{(0)}}{\leftharpoonup}w{{}_{(1)}}S(v{{}_{(1)}}w{{}_{(2)}})\\
&=w{{}_{(0)}}{\leftharpoonup}S(v{{}_{(2)}}w{{}_{(1)}}){\otimes}v{{}_{(0)}}{\leftharpoonup}S(v{{}_{(1)}})\\
&=\theta(w){\leftharpoonup}S(v{{}_{(1)}}){\otimes}\theta(v{{}_{(0)}})\\
&=\theta(w{\leftharpoonup}S{^{-1}}(v{{}_{(1)}})){\otimes}\theta(v{{}_{(0)}})\\
&=(\theta_W{\otimes}\theta_V)\sigma{^{-1}}(v{\otimes}w),
\end{aligned}$$ $\theta$ is a ribbon transformation.
Given the results on the ribbon transformation $\theta$ (which we could have taken by side-switching from [@Som:RTITD]), it is almost obvious that $\theta_T$ is an algebra map: $\theta_T\nabla=\nabla\theta_{T{\otimes}T}
=\nabla(\theta_T{\otimes}\theta_T)\sigma^{-2}
=\nabla\sigma^{-2}(\theta_T{\otimes}\theta_T)=\nabla(\theta_T{\otimes}\theta_T),$ using naturality of $\theta$, the ribbon property, naturality of $\sigma$, and braided commutativity of $T$.
There is also a formula for the inverse of $\theta$ in [@Som:RTITD], namely $\theta{^{-1}}(v)=v{{}_{(0)}}{\leftharpoonup}S^{-2}(v{{}_{(1)}})$. We compute for completeness: $$\theta\theta{^{-1}}(v)=\theta(v{{}_{(0)}}{\leftharpoonup}S^{-2}(v{{}_{(1)}}))
=\theta(v{{}_{(0)}}){\leftharpoonup}v{{}_{(1)}}
=v{{}_{(0)}}{\leftharpoonup}S(v{{}_{(1)}})v{{}_{(2)}}=v$$ and $$\theta{^{-1}}\theta(v)=\theta{^{-1}}(\theta(v){{}_{(0)}}){\otimes}S^{-2}(\theta(v){{}_{(1)}})
=\theta(v{{}_{(0)}}){\leftharpoonup}v{{}_{(1)}}=v{{}_{(0)}}{\leftharpoonup}S(v{{}_{(1)}})v{{}_{(2)}}.$$
Our final shortcut is not dependent on any results on ribbon transformations or Miyashita-Ulbrich actions, but rather on bijectivity of the antipode, and its consequence that $\theta$ is bijective. The morphism $\mu\colon T\rightarrow T{\otimes}T^{{\operatorname{op}}}{\otimes}T$ constructed for [\[mainthm.nme\] \[mainthm\]]{} depends only on the $H$-comodule algebra structure of $H$, but does not contain $H$, so that it surely does not change if we replace the $H$-comodule structure by the $H$-comodule structure induced along $S^2$. But since $\theta\colon T\rightarrow T$ is colinear between these two comodule structures, and an algebra isomorphism, it follows that $\theta$ also preserves $\mu$, that is, axiom holds.
[10]{}
Zur [E]{}inführung des [S]{}charbegriffs. (1929), 199–207.
opf-[Galois]{} extensions of algebras, the [Miyashita]{}-[Ulbrich]{} action, and [Azumaya]{} algebras. (1989), 488–516.
Quantum torsors. (math.QA/0204280).
Hopf algebras and [G]{}alois extensions of an algebra. (1981), 675–692.
, vol. 82 of [*CBMS Regional Conference Series in Mathematics*]{}. AMS, Providence, Rhode Island, 1993.
Hopf [Bigalois]{} extensions. (1996), 3797–3825.
Principal homogeneous spaces for arbitrary [H]{}opf algebras. (1990), 167–195.
Representation theory of [Hopf-Galois]{} extensions. (1990), 196–231.
Ribbon transformations, integrals, and triangular decompositions. (gk-mp-9707/52).
Galoiserweiterungen von nicht-kommutativen [Ringen]{}. (1982), 655–672.
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---
abstract: 'This work investigates the rank properties of $A^+(B_n)$, the additive semigroup reduct of affine near-semiring over Brandt semigroup $B_n$. In this connection, this work reports the ranks $r_1$, $r_2$, $r_3$ and $r_5$ of $A^+(B_n)$ and identifies a lower bound for the upper rank $r_4(A^+(B_n))$. While this lower bound is found to be the $r_4(A^+(B_n))$ for $n \ge 6$, in other cases where $2 \le n \le 5$, the upper rank of $A^+(B_n)$ is still open for investigation.'
address: 'Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India'
author:
- 'Jitender Kumar and K. V. Krishna'
title: 'The Ranks of the Additive Semigroup Reduct of Affine Near-Semiring over Brandt Semigroup'
---
Introduction {#introduction .unnumbered}
============
Since the work of Marczewski in [@a.mar66], many authors have studied the rank properties in the context of general algebras. The concept of rank for general algebras is equivalent to the concept of dimension in linear algebra. The dimension of a vector space is the maximum cardinality of an independent subset, or equivalently, it is the minimum cardinality of a generating set of the vector space. A subset $U$ of a semigroup $\G$ is said to be *independent* if every element of $U$ is not in the subsemigroup generated by the remaining elements of $U$, i.e. $$\forall a \in U, \; a \notin \langle U \setminus \{a\} \rangle .$$ This definition of independence is equivalent to the usual definition of independence in linear algebra. It can be observed that the minimum size of a generating set need not be equal to the maximum size of an independent set in a semigroup. Accordingly, Howie and Ribeiro have considered the following possible definitions of ranks for a semigroup $\G$ (cf. [@a.hw99; @a.hw00]).
1. $r_1(\G) = \max\{k : \forall U \subseteq \G$ with $|U| = k, U$ is independent}.
2. $r_2(\G) = \min\{|U| : U \subseteq \G, \langle U\rangle = \G\}$.
3. $r_3(\G) = \max\{|U| : U \subseteq \G, \langle U\rangle = \G, U$ is independent}.
4. $r_4(\G) = \max\{|U| : U \subseteq \G, U$ is independent}.
5. $r_5(\G) = \min\{k : \forall U \subseteq \G$ with $|U| = k, \langle U\rangle = \G\}$.
For a finite semigroup $\G$, it can be observed that $$r_1(\G) \le r_2(\G) \le r_3(\G) \le r_4(\G) \le r_5(\G).$$ Thus, $r_1(\G), r_2(\G), r_3(\G), r_4(\G)$ and $r_5(\G)$ are, respectively, known as *small rank*, *lower rank*, *intermediate rank*, *upper rank* and *large rank* of $\G$.
While all these five ranks coincide for certain semigroups, there exist semigroups for which all these ranks are distinct. For instance, all these five ranks are equal to $|\G|$ for a finite left or right zero semigroup $\G$. For $n > 2$, Howie et al. have determined all these five ranks for Brandt semigroup $B_n$ (see Definition \[d.bs\]) through the papers [@a.howie87; @a.hw99; @a.hw00] and observed that all these five ranks are different from each other.
The ranks of rectangular bands and monogenic semigroups were also established in [@a.hw99; @a.hw00]. The lower rank of completely 0-simple semigroups was obtained by Ruškuc [@a.ruskuc94]. The intermediate rank of $S_n$, the symmetric group of degree $n$, is determined to be $n-1$ by Whiston [@a.whiston00]. All independent generating sets of size $n-1$ in $S_n$ were investigated in [@a.cam02]. In [@jdm02], Mitchell studied the rank properties of various groups, semigroups and semilattices. The rank properties of certain semigroups of order preserving transformations have been investigated in [@a.howie92] and further extended to orientation-preserving transformations in [@a.ping11].
In this work, we investigate all the five ranks of the additive semigroup reduct of $A^+(B_n)$ – the affine near-semiring over Brandt semigroup $B_n$. This semigroup is indeed the semigroup generated by affine maps over $B_n$. The remaining paper has been organized into six sections. Section 1 provides a necessary background material for the subsequent four main sections which are devoted for all the five ranks of $A^+(B_n)$. We conclude the paper in Section 6.
Preliminaries
=============
In this section, we provide a necessary background material and fix our notation. For more details one may refer to [@a.jk13].
An algebraic structure $(S, +, \cdot)$ is said to be a *near-semiring* if
1. $(S, +)$ is a semigroup,
2. $(S, \cdot)$ is a semigroup, and
3. $a(b + c) = ab + ac$, for all $a,b,c \in S$.
In this work, unless it is required, algebraic structures (such as semigroups, groups, near-semirings) will simply be referred by their underlying sets without explicit mention of their operations. Further, we write an argument of a function on its left, e.g. $xf$ is the value of a function $f$ at an argument $x$.
Let $(\Gamma, +)$ be a semigroup and $M(\Gamma)$ be the set of all mappings on $\Gamma$. The algebraic structure $(M(\G), +, \circ)$ is a near-semiring, where $+$ is point-wise addition and $\circ$ is composition of mappings, i.e., for $\gamma \in \Gamma$ and $f,g \in M(\Gamma)$, $$\g(f + g)= \g f + \g g \;\;\;\; \text{and}\;\;\;\; \g(f \circ g) = (\g
f)g.$$ Also, certain subsets of $M(\G)$ are near-semirings. For instance, the set $M_c(\Gamma)$ of all constant mappings on $\Gamma$ is a near-semiring with respect to the above operations so that $M_c(\Gamma)$ is a subnear-semiring of $M(\G)$.
Now, we recall the notion of affine near-semirings from [@kvk05a]. Let $(\G, +)$ be a semigroup. An element $f \in M(\G)$ is said to be an *affine map* if $f = g + h$, for some $g \in
End(\G)$, the set of all endomorphisms over $\G$, and $h \in M_c(\G)$. This sum is said to be an *affine decomposition* of $f$. The set of all affine mappings over $\G$, denoted by $\text{Aff}(\G)$, need not be a subnear-semiring of $M(\G)$. The *affine near-semiring*, denoted by $A^+(\G)$, is the subnear-semiring generated by $\text{Aff}(\G)$ in $M(\G)$. Indeed, the subsemigroup of $(M(\G), +)$ generated by $\text{Aff}(\G)$ equals $(A^+(\G), +)$ (cf. [@kvk05b Corollary 1]). If $(\G, +)$ is commutative, then $\text{Aff}(\G)$ is a subnear-semiring of $M(\G)$ so that $\text{Aff}(\G) = A^+(\G)$.
\[d.bs\] For any integer $n \geq 1$, let $[n] = \{1,2,\ldots,n\}$. The semigroup $(B_n, +)$, where $B_n = ([n]\times[n])\cup \{\vartheta\}$ and the operation $+$ is given by $$(i,j) + (k,l) =
\left\{\begin{array}{cl}
(i,l) & \text {if $j = k$;} \\
\vartheta & \text {if $j \neq k $}
\end{array}\right.$$ and, for all $\alpha \in B_n$, $\alpha + \vartheta = \vartheta + \alpha =
\vartheta$, is known as *Brandt semigroup*. Note that $\vartheta$ is the (two sided) zero element in $B_n$.
In [@a.jk13], Jitender and Krishna have studied the structure of (both additive and multiplicative) semigroup reducts of the near-semiring $A^+(B_n)$ via Green’s relations. We now recall the results on $A^+(B_n)$ which are useful in the present work. The following concept plays a vital role in the study of $A^+(B_n)$.
Let $(\Gamma, +)$ be a semigroup with zero element $\vartheta$. For $f
\in M(\Gamma)$, the *support of $f$*, denoted by supp$(f)$, is defined by the set $${\rm supp}(f) = \{\alpha \in \Gamma \;|\; \alpha f \neq \vartheta\}.$$ A function $f \in M(\Gamma)$ is said to be of *k-support* if the cardinality of supp$(f)$ is $k$, i.e. $|{\rm supp}(f)| = k$. If $k =
|\Gamma|$ (or $k = 1$), then $f$ is said to be of *full support* (or *singleton support*, respectively). For $X \subseteq M(\Gamma)$, we write $X_k$ to denote the set of all mappings of $k$-support in $X$, i.e. $$X_k = \{ f \in X \mid f \; \text{is of $k$-support}\; \}.$$
\[r.ksupport-sum\] For $f \in M(\G)_k$ and $g \in M(\G)$, we have $|{\rm supp}(f+g)| \leq k$ and $|{\rm supp}(g + f)| \leq k$.
For ease of reference, we continue to use the following notations for the elements of $M(B_n)$, as given in [@a.jk13].
1. For $c \in B_n$, the constant map that sends all the elements of $B_n$ to $c$ is denoted by $\xi_c$. The set of all constant maps over $B_n$ is denoted by $\mathcal{C}_{B_n}$.
2. For $k, l, p, q \in [n]$, the singleton support map that send $(k, l)$ to $(p, q)$ is denoted by ${^{(k, l)}\!\zeta_{(p, q)}}$.
3. For $p, q \in [n]$, the $n$-support map which sends $(i, p)$ (where $1 \le i \le n$) to $(i\sigma, q)$ using a permutation $\sigma \in S_n$ is denoted by $(p, q; \sigma)$.
Note that $A^+(B_1) = \{(1, 1; id)\} \cup \mathcal{C}_{B_n}$, where $id$ is the identity permutation on $[n]$. For $n \ge 2$, the elements of $A^+(B_n)$ are given by the following theorem.
\[t.class.a+bn\] For $n \geq 2$, $A^+(B_n)$ precisely contains $(n! + 1)n^2 + n^4 + 1$ elements with the following breakup.
1. All the $n^2 + 1$ constant maps.
2. All the $n^4$ singleton support maps.
3. The remaining $(n!)n^2$ elements are the $n$-support maps of the form $(p, q; \sigma)$, where $p, q \in [n]$ and $\sigma \in S_n$.
As shown in the Remark \[r.ad.a+bn\], except singleton support maps, all other elements of $A^+(B_n)$ are indeed affine maps over $B_n$. We require the following proposition.
\[sn-iso-autbn\] The assignment $\sigma \mapsto \phi_\sigma: S_n \rightarrow Aut(B_n)$ is an isomorphism, where the mapping $\phi_\sigma: B_n \rightarrow B_n$ is given by, $\forall i, j \in [n]$, $$(i, j)\phi_\sigma = (i\sigma, j\sigma)\; \mbox{ and }\; \vt\phi_\sigma = \vt.$$
\[r.ad.a+bn\] For $k, l, p, q \in [n]$ and $\sigma \in S_n$, we have the following.
1. Since $\xi_{(p, p)} + \xi_{(p, q)}$ and $\xi_\vt + \xi_{(1,1)}$, respectively, are affine decompositions of $\xi_{(p, q)}$ and $\xi_{\vt}$, all constant maps are affine maps.
2. The $n$-support map $(p, q; \sigma)$ is an affine map. An affine decomposition for $(p, q; \sigma)$ is $\phi_\sigma + \xi_{(p\sigma, q)}$, where $\phi_\sigma$ is as per Proposition \[sn-iso-autbn\]. Furthermore, for $f \in Aut(B_n)$, $f + \xi_{(r, s)}$ is an $n$-support map represented by $(r\rho^{-1}, s; \rho)$, where $f = \phi_\rho$.
3. Every singleton support map can be written as sum of a constant map and an $n$-support affine map so that it is an element of $A^+(B_n)$. For instance, ${^{(k, l)}\!\zeta_{(p, q)}}$ $= \xi_{(p, q)} + g$, where $g = (l, q; \rho)$ such that $k\rho = q$.
In what follows, $A^+(B_n)$ denotes the additive semigroup reduct $(A^+(B_n), +)$ of the affine near-semiring $(A^+(B_n), +, \circ)$. We now present a necessary result on the Green’s relations $\mathcal{R}$ and $\mathcal{L}$ of the additive semigroup $A^+(B_n)$.
\[t.gr-rl\] For $1 \le i \le 2$, let $\pi_i : [n] \times [n] \rightarrow [n]$ be the $i$th projection map. That is, $(p, q)\pi_1 = p$ and $(p, q)\pi_2 = q$, for all $(p, q) \in [n]
\times [n]$.
1. For $f,g \in A^+(B_n) \setminus \{\xi_\vt\}$, $f \mathcal{R} g$ if and only if ${\rm supp}(f) = {\rm supp}(g)$ and $\alpha f\pi_1 = \alpha
g\pi_1$, for all $\alpha \in {\rm supp}(f)$.
2. For $f,g \in \mathcal{C}_{B_n} \setminus \{\xi_\vt\}$, $f \mathcal{L} g$ if and only if $\alpha f\pi_2 = \alpha
g\pi_2$, for all $\alpha \in B_n$.
3. The number of $\mathcal{R}$-classes containing $n$-support elements in $A^+(B_n)$ is $(n!)n$.
The ranks $r_1$ and $r_2$ of $A^+(B_n)$
=======================================
It can be easily observed that $A^+(B_1)$ is an independent set and none of its proper subsets generates $A^+(B_1)$. Hence, for $1 \le i \le 5$, we have $$r_i(A^+(B_1)) = |A^+(B_1)| = 3.$$ In the rest of the paper we shall investigate the ranks of $A^+(B_n)$, for $n > 1$.
In this section, after quickly ascertaining the small rank $r_1$ of $A^+(B_n)$, we will obtain its lower rank $r_2$. The small rank of $A^+(B_n)$ comes as a consequence of the following result due to Howie and Ribeiro.
\[r1-gm\] Let $\G$ be a finite semigroup, with $|\G| \ge 2$. If $\G$ is not a band, then $r_1(\G) = 1$.
Owing to the fact that $A^+(B_n)$ (for $n \ge 2$) have some non idempotent elements, it is not a band. For instance, the constant maps $\xi_{(p, q)}$ with $p \ne q$ in $A^+(B_n)$ are not idempotent. Hence, we have the following corollary of Theorem \[r1-gm\].
For $n \ge 2$, $r_1(A^+(B_n)) = 1$.
Now, in the remaining section, we construct a generating set of the minimum cardinality of $A^+(B_n)$ and obtain its lower rank in Theorem \[r2-a+bn\]. Consider the subsets $$\mathcal{S} = \{ \xi_{(i, i + 1)} \mid i \in [n - 1] \} \cup \{ \xi_{(n, 1)} \}$$ and $$\mathcal{T} = \{g + h \mid g \in Aut(B_n), \; h \in \mathcal{S} \}$$ of $A^+(B_n)$. We develop a proof of Theorem \[r2-a+bn\] through a sequence of lemmas by showing that the set $\mathcal{S} \cup \mathcal{T}$ serves our purpose.
\[gen-cm-a+bn\] For $n \ge 2$, $\langle\mathcal{S}\rangle = \mathcal{C}_{B_n}$.
Let $f \in \mathcal{C}_{B_n}$; then, either $f = \xi_\vt$ or $f = \xi_{(i, j)}$. If $f = \xi_\vt$, then, for $p \in [n-1]$, write $\xi_\vt = \xi_{(p, p + 1)} + \xi_{(p, p + 1)}$ so that $f \in \langle \mathcal{S} \rangle$. If $f = \xi_{(i, j)}$, then, for $i < j$, we have $$\xi_{(i, j)} = \xi_{(i, i + 1)} + \xi_{(i + 1, i + 2)} + \cdots + \xi_{(j - 1, j)},$$ and, for $i \ge j$, $$\xi_{(i, j)} = \xi_{(i, i + 1)} + \xi_{(i + 1, i + 2)} + \cdots + \xi_{(n - 1, n)} + \xi_{(n, 1)} + \xi_{(1, 2)}+ \cdots +\xi_{(j-1, j)}$$ so that $f \in \langle \mathcal{S}\rangle$.
\[gen-a+bn\] If $X \subset A^+(B_n)$ such that $\langle X \rangle = \mathcal{C}_{B_n}$, then $\langle X \cup \mathcal{T} \rangle = A^+(B_n)$.
Since the constant elements of $A^+(B_n)$ are generated by $X$, in view of Theorem \[t.class.a+bn\], it is sufficient to prove that the $n$-support and singleton support elements are generated by $X \cup \mathcal{T}$. However, since every singleton support map is a sum of a constant map and an $n$-support map (cf. Remark \[r.ad.a+bn\](3)), we will now observe that $X \cup \mathcal{T}$ generates the $n$-support maps of $A^+(B_n)$.
Let $f \in A^+(B_n)_n$. By Remark \[r.ad.a+bn\](2), $f = g + \xi_c$ for some $g \in Aut(B_n)$ and $c \in B_n \setminus \{\vt\}$. By Lemma \[gen-cm-a+bn\], write $\xi_c = \displaystyle \sum_{i = 1}^{k}f_i$, for some $f_i$’s from $\mathcal{S}$ so that $$f = g + \displaystyle \sum_{i = 1}^{k}f_i = g + f_1 + \displaystyle \sum_{i = 2}^{k}f_i.$$ Note that $g + f_1 \in \mathcal{T}$ and each $f_i$ is a sum of elements of $X$. Hence, $f \in \langle X \cup \mathcal{T} \rangle$.
\[gen-nsm-a+bn\] For $g_1, g_2, g_3 \in Aut(B_n)$ with $g_2 \ne g_3$ and $p, q, s, t \in [n]$ such that $p \ne s$, we have the following.
1. If $f_1 = g_1 + \xi_{(p, q)}$ and $f_1' = g_1 + \xi_{(s, t)}$, then $f_1 \ne f_1'$.
2. If $f_2 = g_2 + \xi_{(p, q)}$ and $f_3 = g_3 + \xi_{(p, q)}$, then $f_2 \ne f_3$.
Hence, $|\mathcal{T}| = n(n!)$.
As per Proposition \[sn-iso-autbn\], for $1 \le i \le 3$, let $g_i = \phi_{\sigma_i}$ so that $f_1, f_1', f_2, f_3$ are $n$-support maps represented by $(p \sigma_1^{-1}, q; \sigma_1)$, $(s \sigma_1^{-1}, t; \sigma_1)$, $(p \sigma_2^{-1}, q; \sigma_2)$ and $(p \sigma_3^{-1}, q; \sigma_3)$, respectively (cf. Remark \[r.ad.a+bn\](2)).
\(1) Since $p \ne s$ and $\sigma_1$ is a permutation on $[n]$, $f_1$ and $f_1'$ have different support so that $f_1 \ne f_1'$.
\(2) If $p \sigma_2^{-1} \ne p \sigma_3^{-1}$, then we are done. Otherwise, since $\sigma_2 \ne \sigma_3$, there exists $i_0 \in [n]$ such that $i_0 \sigma_2 \ne i_0 \sigma_3$. Now, $$(i_0, p \sigma_2^{-1})f_2 = (i_0 \sigma_2, q) \ne (i_0 \sigma_3, q) = (i_0, p \sigma_3^{-1})f_3$$so that $f_2 \ne f_3$.
Now, since $|Aut(B_n)| = n!$ (cf. Proposition \[sn-iso-autbn\]) and there are $n$ elements in $\mathcal{S}$, we have $|\mathcal{T}| = n(n!)$.
\[l.fs-sum\] For $1 \le i \le k$, let $f, f_i \in A^+(B_n)$ such that $f = \displaystyle{\sum_{i = 1}^{k}}f_i$.
1. If $f \in A^+(B_n)_{n^2 + 1}$, then $f_1 \in R_f$ and $f_k \in L_f$.
2. If $f \in A^+(B_n)_{n}$, then $f_1 \in R_f$.
\(1) If $f \in A^+(B_n)_{n^2 + 1}$, then $f_i \in A^+(B_n)_{n^2 + 1}$, for all $i$ (cf. Remark \[r.ksupport-sum\]). Let $f = \xi_{(p, q)}$ and $f_i = \xi_{(p_i, q_i)}$ so that $$\xi_{(p, q)} = \displaystyle{\sum_{i = 1}^{k}}\xi_{(p_i, q_i)}.$$ Then clearly, for $1 \le i \le k - 1$, $q_i = p_{i + 1}$ and $p_1 = p$, $q_k = q$. Hence, by Theorem \[t.gr-rl\], $f_1 \in R_f$ and $f_k \in L_f$.\
(2) If $f \in A^+(B_n)_{n}$, then $|{\rm supp}(f_i)| \ge n$, for all $i$ (cf. Remark \[r.ksupport-sum\]). Then, for each $i$, $|{\rm supp}(f_i)| = n$ or $n^2 + 1$ (cf. Theorem \[t.class.a+bn\]). Note that, there exists $j$ ($1 \le j \le k$) such that $|{\rm supp}(f_j)| = n$; otherwise, $f$ will be a constant map. If $j \ge 2$, then, by [@a.jk13 Proposition 2.9], $|{\rm supp}(f_{j-1} + f_{j})| \le 1$ so that $|{\rm supp}(f)| \le 1$; a contradiction. Thus, we have $j = 1$ and, for all $i > 1$, $|{\rm supp}(f_i)| = n^2 + 1$.
Let $(k, q; \sigma)$ and $(k', p; \tau)$ be the representations of $f$ and $f_1$, respectively, and $\displaystyle \sum_{i = 2}^{k} f_i = \xi_{(r, s)}$ for some $r, s \in [n]$. Then, note that $p = r$, $k' = k$, $\tau = \sigma$ and $s = q$. Thus, $f_1 = (k, r; \sigma)$ and $\displaystyle \sum_{i = 2}^{k} f_i = \xi_{(r, q)}$ for some $r \in [n]$. Hence, by Theorem \[t.gr-rl\], $f_1 \in R_f$.
\[gen-set-prop\] Every generating set of $A^+(B_n)$ contains at least
1. $n$ elements of full support, and
2. $n(n!)$ elements of $n$-support.
Let $V$ be a generating set of $A^+(B_n)$. For $f \in A^+(B_n)$, write $f = \displaystyle \sum_{i = 1}^{k}f_i$, for some $f_i \in V$.
\(1) If $f \in \mathcal{S}$, then by Lemma \[l.fs-sum\](1), $f_1 \in R_f$ so that $R_f \cap V \ne \vn$. Further, one can observe that if $g, h \in \mathcal{S}$ with $g \ne h$, then $R_g \cap R_h = \vn$ (cf. Theorem \[t.gr-rl\](1)). Hence, since $\mathcal{S}$ has $n$ full support elements, $V$ will have at least $n$ full support elements.
\(2) If $f \in \mathcal{T}$, then by Lemma \[l.fs-sum\](2), we have $R_f \cap V \ne \vn$. For $g, h \in \mathcal{T}$ with $g \ne h$, we show that $R_g \cap R_h = \vn$ so that $|V| \ge |\mathcal{T}| = (n!)n$. In view of Remark \[r.ad.a+bn\](2), by considering $n \mod n = n$, write $g = (r\sigma^{-1}, r + 1 \mod n; \sigma)$ and $h = (s\rho^{-1}, s + 1 \mod n; \rho)$. Since $g \ne h$, either $\sigma \ne \rho$ or $r \ne s$. Hence, by Theorem \[t.gr-rl\](1), $R_g \ne R_h$ in either case.
In view of lemmas \[gen-cm-a+bn\] and \[gen-a+bn\], we have the following corollary of Lemma \[gen-set-prop\].
\[sut-min-car\] $|\mathcal{S} \cup \mathcal{T}|$ is the minimum such that $\langle \mathcal{S} \cup \mathcal{T} \rangle = A^+(B_n)$.
Combining the results from Lemma \[gen-cm-a+bn\] through Corollary \[sut-min-car\], we have the following main theorem of the section.
\[r2-a+bn\] For $n \ge 2$, $r_2(A^+(B_n)) = n(n! + 1)$.
Intermediate rank
=================
In this section, after ascertaining certain relevant properties of independent generating sets of $A^+(B_n)$, we obtain its intermediate rank.
\[max-igs\] Let $U$ be an independent generating set of $A^+(B_n)$; then
1. $A^+(B_n)_1 \cap U = \vn$,
2. $|A^+(B_n)_n \cap U| = n(n!)$,
3. $n \le |A^+(B_n)_{n^2 + 1} \cap U| \le 2n - 2$.
Hence, $|U| \le n(n!) + 2n -2$.
$\;$
1. Let $f =\; {^{(k, l)}\!\zeta_{(p, q)}} \in A^+(B_n)_1 \cap U$. Since $U$ is a generating set, we have $\xi_{(p, r)} \in \langle U \setminus \{f\}\rangle$ and, for $k \sigma = r$, $(l, q; \sigma) \in \langle U \setminus \{f\} \rangle$ (cf. Remark \[r.ksupport-sum\]). Note that $f = \xi_{(p, r)} + (l, q; \sigma)$ so that $f \in \langle U \setminus \{f\} \rangle$; a contradiction to $U$ is an independent set.
2. By Lemma \[gen-set-prop\](2), $|A^+(B_n)_n \cap U| \ge n(n!)$. Since $A^+(B_n)_n$ contains only $n(n!)$ $\mathcal{R}$-classes (cf. Theorem \[t.gr-rl\](3)), if $U$ contain more than $n(n!)$ elements of $n$-support, then there exist distinct $f,g \in A^+(B_n)_n \cap U$ such that $f \mathcal{R} g$. By Theorem \[t.gr-rl\], $f = (k, p; \sigma)$ and $g = (k, p'; \sigma)$, for some $\sigma \in S_n$. Note that, $g = f + \xi_{(p, p')}$, where $\xi_{(p, p')} \in \langle U \setminus \{g\}\rangle$, so that $ g \in \langle U \setminus \{g\} \rangle$; a contradiction to independence of $U$. Hence, there are exactly $n(n!)$ elements of $n$-support in $U$.
3. By Lemma \[gen-set-prop\](1), $|A^+(B_n)_{n^2 + 1} \cap U| \ge n$. Further, using [@a.hw99 Theorem 3.1], one can observe that $U$ contains at most $2n-2$ full support maps.
For $n \ge 2$, $r_3(A^+(B_n)) = n(n!) + 2n - 2$.
First note that the set $$\mathcal{S'} = \{\xi_{(1, i)} \mid 2 \le i \le n \} \cup \{\xi_{(j, 1)} \mid 2 \le j \le n \}$$ generates all constant maps in $A^+(B_n)$. For instance, $\xi_\vt = \xi_{(1, 2)} + \xi_{(1, 2)}$ and $\xi_{(1, 1)} = \xi_{(1, 2)} + \xi_{(2, 1)}$. Now, for $p, q \in [n]$, clearly $\xi_{(1, p)}$ and $\xi_{(q, 1)} \in \langle \mathcal{S'} \rangle$. Further, since $\xi_{(p, q)} = \xi_{(p, 1)} + \xi_{(1, q)}$, we have $\xi_{(p, q)} \in \langle \mathcal{S'} \rangle$.
Hence, by Lemma \[gen-a+bn\], the set $V = \mathcal{S'} \cup \mathcal{T}$ is a generating set of $A^+(B_n)$. We show that $V$ is also an independent set. Since $|V| = n(n!) + 2n - 2$, by Lemma \[max-igs\], the theorem follows.
*$V$ is an independent set*: For $f \in V$, suppose $f = \displaystyle\sum_{i =1}^k f_i$ with $f_i \in V \setminus \{f\}$. Then, by Lemma \[l.fs-sum\], $f_1 \in R_f$. If $f \in \mathcal{S'} \cup \mathcal{T}$, in the following, we observe that $f = f_1$; which is a contradiction so that $f \notin \langle V \setminus \{f\} \rangle$.
If $f = \xi_{(q, 1)} \in \mathcal{S'}$ (for some $2 \le q \le n$), then $f_1 = \xi_{(q, l)}$, for some $l \in [n]$ (cf. Theorem \[t.gr-rl\](1)). Hence, $f_1 = f$. For $2 \le p \le n$, if $f = \xi_{(1, p)}$, the argument is similar.
If $f = (k, p; \sigma) \in \mathcal{T}$ (for some $k, p \in [n], \sigma \in S_n$), again by Theorem \[t.gr-rl\](1), $f_1 = (k, s; \sigma)$ for some $s \in [n]$. Consequently, $f = f_1$ (cf. construction of $\mathcal{T}$).
Upper rank
==========
It is always difficult to identify the upper rank of a semigroup and we observe that $A^+(B_n)$ is also not an exception. In order to investigate the upper rank $r_4(A^+(B_n))$, in this section, first we obtain a lower bound for the upper rank and eventually we prove that this lower bound is indeed the $r_4(A^+(B_n))$, for $n \ge 6$. We also report an independent set of 14 elements in $A^+(B_2)$.
\[lb-r4a+bn\] For $n \ge 2$, $ I = A^+(B_n)_n \cup \{\xi_{(i, i)} : i \in [n]\}$ is an independent set in $A^+(B_n)$. Hence, by Theorem \[t.class.a+bn\], $r_4(A^+(B_n)) \ge (n!)n^2 + n$.
For $f \in I$, suppose $f = \displaystyle{\sum_{j = 1}^{k}f_j}$, for $f_j \in I$. We prove that $f_1 = f$ so that $f \notin \langle I \setminus \{f\}\rangle$. Let $f = \xi_{(i, i)}$; then clearly $f_j = f$ for all $j$. We may now suppose $f \in A^+(B_n)_n$ and $(k, p; \sigma)$ be the representation of $f$. By Lemma \[l.fs-sum\](2), $f_1 \in R_f$ and $\displaystyle{\sum_{i = 2}^{k}f_i} = \xi_{(s, p)}$ for some $s \in [n]$. Note that $f_1 = (k, s; \sigma)$ (cf. Theorem \[t.gr-rl\]). If $s \ne p$, then $\xi_{(s, p)} \notin \langle I\rangle$; a contradiction. Hence, $s = p$ so that $f_1 = f$.
\[r.mi-ns\] $A^+(B_n)_n$ is an independent subset of size $(n!)n^2$ in $A^+(B_n)$.
\[l.ind-ss\] Let $Q$ be an independent subset of $B_n$ and $$Q' = \left.\left \{ {^{(k, l)}\!\zeta_{\alpha}} \right| k, l \in [n] \mbox{ and
} \alpha \in Q \right \};$$ then $Q'$ is an independent subset of $A^+(B_n)$.
For ${^{(k, l)}\!\zeta_{\alpha}} \in Q'$, $k_j, l_j \in [n]$ and $\alpha_j \in Q$ suppose $${^{(k, l)}\!\zeta_{\alpha}} = \sum_{j = 1}^{k}\; {^{(k_j, l_j)}\!\zeta_{\alpha_j}}.$$ Clearly $k_j = k$, $l_j = l$ for all $j$, and $\alpha = \displaystyle\sum_{j=1}^k\alpha_j$. Since $Q$ is independent, we have $\alpha = \alpha_i$ for some $i$ ($1 \le i \le k$). Consequently, ${^{(k, l)}\!\zeta_{\alpha}} \notin \langle Q'
\setminus \{{^{(k, l)}\!\zeta_{\alpha}}\}\rangle$ so that $Q'$ is an independent set.
\[r.mi-fs\] Since $B_n$ is isomorphic to the semigroup $\mathcal{C}_{B_n}$, by [@a.hw99 Theorem 3.3], we have $r_4(\mathcal{C}_{B_n}) = \left\lfloor n^2 /4 \right\rfloor \ + n$.
In view of Remark \[r.mi-fs\], we have the following corollary of Lemma \[l.ind-ss\].
\[r.mi-ss\] For $n \ge 2$, the maximum size of an independent subset in $A^+(B_n)_1$ is $n^2(\left\lfloor n^2 /4 \right\rfloor \ + n )$.
\[r.ins-fs\] For $f_i \in A^+(B_n)$, if $\displaystyle\sum_{i=1}^rf_i + \xi_{(p, p)} + \sum_{i = r+1}^s f_i$ is nonzero, then the sum equals $\displaystyle\sum_{i=1}^sf_i$.
For $n =2$, we provide a better lower bound in the following theorem.
\[lb-r4a+b2\] $r_4(A^+(B_2)) \ge 14$.
We claim that the 14-element set $$P = \left.\left \{ {^{(k, l)}\!\zeta_{\alpha}} \right| k, l \in [2] \mbox{ and } \alpha \in Q \right \} \cup \{\xi_{(1, 1)}, \xi_{(2, 2)}\},$$ where $Q = \{(1, 1), (1, 2), (2, 2)\}$, is an independent subset of the semigroup $A^+(B_2)$.
For $f \in P$, suppose $f = \displaystyle{\sum_{j = 1}^{k}f_j}$, for $f_j \in P$. If $f = \xi_{(i, i)}$, then $f_j = f$ for all $j$ so that $f \notin \langle P \setminus \{f\}\rangle$. Otherwise, $f$ = ${^{(k, l)}\!\zeta_{\alpha}}$ for $\alpha \in Q$. By Remark \[r.ins-fs\], the sum for $f$ can be reduced to a sum with only the singleton support elements of $P$. Hence, from the proof of Lemma \[l.ind-ss\], $P$ is independent.
\[r4-a+bn\] For $n \ge 6$, $r_4(A^+(B_n)) = (n!)n^2 + n$.
For $n \ge 6$, if an independent subset $K$ of $A^+(B_n)$ contains a single support map or a full support map of the form $\xi_{(p, q)}$, for $p \ne q$, then $|K| < (n!)n^2 + n$. Hence, the result follows by Theorem \[lb-r4a+bn\].
Let $K$ be an independent subset of $A^+(B_n)$. By Corollary \[r.mi-ns\], Remark \[r.mi-fs\] and Corollary \[r.mi-ss\], we have $|K| \le \kappa$, where $$\kappa = (n!)n^2 + \left\lfloor n^2 /4 \right\rfloor \ + n + n^2(\left\lfloor n^2 /4 \right\rfloor \ + n ).$$ In the following, we observe that, out of $\kappa$ (the maximum possible number) elements, at least $(n-1)!(n-1)$ elements will not be in $K$. Hence, since $n \ge 6$, $$|K| \le \kappa - (n-1)!(n-1) < (n!)n^2 + n.$$
*Case 1: $\xi_{(p, q)} \in K$ with $p \ne q$*. For each $\sigma \in S_n$ and $l \in [n]$, since $$(l, q; \sigma) = (l, p; \sigma) + \xi_{(p, q)},$$ the independent set $K$ cannot contain $(l, q; \sigma)$ and $(l, p; \sigma)$ together. Thus, out of $\kappa$ elements, at least $(n!)n$ elements will not be in $K$.
*Case 2: ${^{(r, s)}\!\zeta_{(p, q)}} \in K$*. For each $t \in [n]$ and $\sigma, \rho \in S_n$ such that $r \sigma = p$ and $r \rho = t$, since $${^{(r, s)}\!\zeta_{(p, q)}} = (s, t; \sigma) + (s, q; \rho),$$ the independent set $K$ cannot contain $(s, t; \sigma)$ and $(s, q; \rho)$ together.
[*Subcase 2.1*]{}
: $p = q$. Except at $t = q$, for all other choices, none of the first terms is equal to any of the second terms in the sums $(s, t; \sigma) + (s, q; \rho)$. Thus, out of $\kappa$ elements, at least $(n-1)!(n-1)$ elements (either first terms or second terms in the sums) will not be in $K$.
[*Subcase 2.2*]{}
: $p \ne q$. In the similar lines of *Subcase 2.1*, at least $(n-1)!(n-2)$ elements will not be in $K$ for the choices of $t \in [n] \setminus \{p, q\}$. If $t \in \{p, q\}$, the set of second terms of the sums for $t = p$ is equal to the set of first terms of the sums for $t = q$, which is of size $(n-1)!$. Thus, for $t \in \{p, q\}$, at least $(n-1)!$ elements will not be in $K$. Hence, a total of at least $(n-1)!(n-1)$ elements will not be in $K$.
Large Rank
==========
In this section, we obtain the large rank of $A^+(B_n)$. An element $a$ of a semigroup $(\G, +)$ is said to be *indecomposable* if there do not exist $b , c \in \G \setminus \{a\}$ such that $a = b + c$. The following key result by Howie and Ribeiro is useful to find the large rank of a finite semigroup.
\[r5-lsgp\] Let $\G$ be a finite semigroup and $V$ be a largest proper subsemigroup of $\G$; then $r_5(\G) = |V|+1.$ Hence, $r_5(\G) = |\G|$ if and only if $\G$ contains an indecomposable element.
Since $\xi_{(1, 2)}$ is an indecomposable element in $A^+(B_2)$, we have $$r_5(A^+(B_2)) = |A^+(B_2)| = 29.$$
However, as shown in the following proposition, there is no indecomposable element in $A^+(B_n)$, for $n \ge 3$.
For $n \ge 3$, all the elements of $A^+(B_n)$ are decomposable.
Refereing to Theorem \[t.class.a+bn\], we give a decomposition of each element $f \in A^+(B_n)$ in the following cases.
1. $f$ is the zero element: $\xi_\vt = \xi_{(p, q)} + \xi_{(r, s)}$, for $q \ne r$.
2. $f$ is a full or singleton support element: Let ${\rm Im}(f)\setminus \{\vt\} = \{(p, q)\}$. We have $f = g + h$, where $g, h \in A^+(B_n)$ such that ${\rm supp}(f) = {\rm supp}(g) = {\rm supp}(h)$ and ${\rm Im}(g)\setminus \{\vt\} = \{(p, r)\}$, ${\rm Im}(h)\setminus \{\vt\} = \{(r, q)\}$, for some $r \ne p, q$.
3. $f$ is an $n$-support map: Let $f = (k, p; \sigma)$. Note that $f = (k, q; \sigma) + \xi_{(q, p)}$, for $q \ne p$.
In order to find the large rank of $A^+(B_n)$, we adopt the technique that is used to find the large rank of Brandt semigroups in [@a.jk13-3]. The technique, as stated in Lemma \[gm-lsgp\], relies on the concept of prime subsets of semigroups. A nonempty subset $U$ of a semigroup $(\G, +)$ is said to be *prime* if, $\forall a, b \in \G$, $$a + b \in U \Longrightarrow a \in U \vee b \in U.$$
\[gm-lsgp\] Let $V$ be a smallest and proper prime subset of a finite semigroup $\G$; then $\G \setminus V$ is a largest proper subsemigroup of $\G$.
Using Lemma \[gm-lsgp\] and Theorem \[r5-lsgp\], now we obtain the large rank of $A^+(B_n)$ in the following theorem.
\[lr-a+bn+\] For $n \geq 2$, $r_5(A^+(B_n)) = (n!)n^2+n^2+n^4-n+3.$
We show that the set $V = \{\xi_{(n, k)} \mid 1 \le k \le n - 1\}$ is a smallest prime subset of $A^+(B_n)$. Since, $|V| = n-1$, the result follows from Theorem \[t.class.a+bn\].
*$V$ is a prime subset*: For $\xi_{(i, j)}, \xi_{(l, k)} \in A^+(B_n)$, if $\xi_{(i, j)} + \xi_{(l, k)} \in V$, then $i = n$, $j = l$ and $1 \le k \le n-1$. If $l = n$, then clearly $\xi_{(l, k)} \in V$; otherwise, $\xi_{(i, j)} \in V$.
*$V$ is a smallest prime subset*: Let $U$ be a prime subset of $A^+(B_n)$ such that $|U| < |V|$. If $U \subset V$, then let $\xi_{(n, q)} \in V \setminus U$. Now, for $\xi_{(n, p)} \in U$ and for all $i \in [n]$, clearly we have $$\xi_{(n, p)} = \xi_{(n, i)} + \xi_{(i, p)}.$$ Note that, for $i = q$, neither $\xi_{(n, i)}$ nor $\xi_{(i, p)}$ is in $U$; a contradiction to $U$ is a prime set.
Otherwise, we have $U \not \subset V$. Let $f \in U \setminus V$; then, $f$ can be (i) $\xi_\vt$, (ii) $\xi_{(n, n)}$, (iii) $\xi_{(p, q)}$, for some $p \in [n-1]$, $q \in [n]$, (iv) an $n$-support map, or (v) a singleton support map. In all the five cases we observe that $|U| \ge n-1$, which is a contradiction to the choice of $U$.
- $f = \xi_\vt$: For each $i \in [n]$, since $\xi_{(i, 1)} + \xi_{(2, i)} = \xi_{\vt}$, there are at least $n$ elements in $U$.
- $f = \xi_{(n, n)}$: For each $i \in [n-1]$, since $\xi_{(n, i)}+ \xi_{(i, n)} = \xi_{(n, n)}$, there are at least $n-1$ elements in $U$.
- $f = \xi_{(p, q)}$, for some $p \in [n-1]$, $q \in [n]$: First note that, for each $i \in [n]$, we have $\xi_{(p, i)} + \xi_{(i, q)} = \xi_{(p, q)}$. If $p = q$, the argument is similar to above (ii). Otherwise, corresponding to $n-2$ different choices of $i \ne p, q$, there are at least $n - 2$ elements in $U$. Now, including $f$, we have $|U| \ge n-1$.
- $f$ is an $n$-support map: Let $f = (k, p; \sigma)$. For $q \in [n] \setminus \{p\}$, since $(k, p; \sigma) = (k, q; \sigma) + \xi_{(q, p)}$, there are at least $n-1$ elements in $U$.
- $f$ is a singleton support map: Let $f$ = ${^{(k, l)}\!\zeta_{(p, q)}}$. For $s \in [n] \setminus \{p, q\}$, since $${^{(k, l)}\!\zeta_{(p, q)}} =\; {^{(k, l)}\!\zeta_{(p, s)}} +\; {^{(k, l)}\!\zeta_{(s, q)}},$$there are at least $n-2$ elements in $U$. Since $f \in U$, we have $|U| \ge n-1$.
Conclusion
==========
In this work, we have investigated the ranks of $A^+(B_n)$, the additive semigroup reduct of the affine near-semiring over Brandt semigroup. Using the structural properties of $A^+(B_n)$ given in [@a.jk13], we obtained the small, lower, intermediate and large ranks of $A^+(B_n)$, for all $n \ge 1$. The upper rank $r_4(A^+(B_n))$ was found for the semigroups with $n \ge 6$. For $2 \le n \le 5$, through an explicit construction of an independent set, we reported a lower bound for $r_4(A^+(B_n))$. While 14 is the lower bound for the case $n = 2$, it is $(n!)n^2 + n$ for the other cases. We conjecture that these lower bounds are indeed the upper ranks of the respective cases. In the similar lines of this work, one could also investigate on the rank properties of the multiplicative semigroup reduct of the affine near-semiring over Brandt semigroup.
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---
abstract: 'Deformable image registration and regression are important tasks in medical image analysis. However, they are computationally expensive, especially when analyzing large-scale datasets that contain thousands of images. Hence, cluster computing is typically used, making the approaches dependent on such computational infrastructure. Even larger computational resources are required as study sizes increase. This limits the use of deformable image registration and regression for clinical applications and as component algorithms for other image analysis approaches. We therefore propose using a fast predictive approach to perform image registrations. In particular, we employ these fast registration predictions to *approximate* a simplified geodesic regression model to capture longitudinal brain changes. The resulting method is orders of magnitude faster than the standard optimization-based regression model and hence facilitates large-scale analysis on a single graphics processing unit (GPU). We evaluate our results on 3D brain magnetic resonance images (MRI) from the ADNI datasets.'
address:
- 'Department of Computer Science, University of North Carolina at Chapel Hill, USA'
- 'Biomedical Research Imaging Center, University of North Carolina at Chapel Hill, USA'
- 'Imaging Genetics Center, University of Southern California, USA'
- 'Department of Radiology, University of Pennsylvania, USA'
- 'Department of Computer Science, University of Salzburg, Austria'
author:
- Zhipeng Ding
- Greg Fleishman
- Xiao Yang
- Paul Thompson
- Roland Kwitt
- Marc Niethammer
- 'The Alzheimer’s Disease Neuroimaging Initiative'
bibliography:
- 'allBibShort.bib'
title: Fast Predictive Simple Geodesic Regression
---
Fast prediction, image regression, ADNI dataset, longitudinal data
Introduction
============
Longitudinal image data provides us with a wealth of information to study aging processes, brain development and disease progression. Such studies, for example ADNI [@jack2015magnetic] and the Rotterdam study [@ikram2015rotterdam], involve analyzing thousands of images. In fact, even larger studies will be available in the near future. For example, the UK Biobank [@biobankWebsite] targets on the order of 100,000 images once completed. With the number of images increasing, large-scale image analysis typically resorts to using compute clusters for parallel processing. While this is, in principle, a viable solution, increasingly larger compute clusters will become necessary for such studies. Alternatively, more efficient algorithms can reduce computational requirements, which then facilitates computations on individual computers or much smaller compute clusters, interactive (e.g., clinical) applications, efficient algorithm development, and use of these efficient algorithms as components in more sophisticated analysis approaches (which may use them as part of iterative processes).
1ex Image registration is a key task in medical image analysis to study deformations between images. Building on image registration approaches, image regression models [@ref:georegression; @hong2012simple; @hong2012metamorphic; @ref:nikhil; @fletcher2013geodesic; @hong2014time; @singh2014splines; @hong2014geodesic; @singh2015splines; @hong2016parametric] have been developed to analyze deformation trends in longitudinal imaging studies. One such approach is geodesic regression (GR) [@ref:georegression; @ref:nikhil; @fletcher2013geodesic] which (for images) build on the large displacement diffeomorphic metric mapping model (LDDMM) [@ref:lddmm]. In general, GR generalizes linear regression to Riemannian manifolds. When applied to longitudinal image data, it can compactly express spatial image transformations over time. However, the solution to the underlying optimization problem is computationally expensive. Hence, a simplified, approximate, GR approach has been proposed [@ref:hong] (SGR) to decouple the computation of the regression geodesic into pairwise image registrations. However, even such a simplified GR approach would require months of computation time on a single graphics processing unit (GPU) to process thousands of 3D image registrations for large-scale imaging studies such as ADNI [@jack2015magnetic]. The primary reason computational bottleneck for SGR are the optimization required to compute pair-wise registrations.
{width="\textwidth"}
1ex Recently, efficient approaches have been proposed for deformable image registration [@dinggang; @miao2016real; @nonrigid; @ref:yang2016; @quicksilver; @zhang2017frequency]. In particular, for LDDMM, which is the basis of GR approaches for images, registrations can be dramatically sped up, by either working with finite-dimensional Lie algebras [@zhang2015finite] and frequency diffeomorphisms [@zhang2017frequency], or by fast predictive image registration (FPIR) [@ref:yang2016; @quicksilver]. FPIR predicts the initial conditions (specifically, the initial momentum) of LDDMM, which fully characterize the geodesic and the spatial transformation using a *learned* a patch-based deep regression model. Because numerical optimization of standard LDDMM registration is replaced by a [*single*]{} prediction step, followed by optional correction steps [@quicksilver], FPIR is dramatically faster than optimization-based LDDMM without compromising registration accuracy, as measured on several registration benchmarks [@klein2009].
1ex Besides FPIR, other predictive image registration approaches have been proposed. Dosovitskiy et al. [@flownet2015] use a convolutional neural network (CNN) to directly predict optical flow. Liu et al. [@ref:liu] use an encoder-decoder network to synthesize video frames. Schuster et al. [@schuster2016optical] investigate strategies to improve optical flow prediction via a CNN. Cao et al. [@dinggang] use a sampling strategy and CNN regression to directly learn the mapping from moving and target image pairs to the final deformation field. Miao et al. [@miao2016real] use CNN regression for 2D/3D rigid registration. Sokooti et al.[@nonrigid] use CNNs to directly predict a 3D displacement vector field from input image pairs. An unsupervised approach for image registration was proposed by de Vos et al. [@dlmia2017_unsupervised]; here, the loss function is the image similarity measure between images themselves and a deformation is parameterized via a spatial transformer (which essentially amounts to a parameterized model of deformation in image registration) which generates the sought-for displacement vector field. In [@hong2017fast], Hong et al. employ a low-dimensional band-limited representation of velocity fields in Fourier space [@zhang2015finite] to speed up SGR [@ref:hong] for population-based image analysis.
1ex In this work, we will build on FPIR, as it is a desirable approach for brain image registration for the following reasons: *First*, FPIR predicts the initial momentum of LDDMM and therefore inherits the theoretical properties of LDDMM. Consequently, FPIR results in diffeomorphic transformations, even though predictions are computed in a patch-by-patch manner; this can not be guaranteed by most other prediction methods. *Second*, patch-wise prediction allows for training of the prediction models based on a very small number of images, containing a large number of patches. *Third*, by using a patch-wise approach, even high-resolution image volumes can be processed without running into memory issues on a GPU. *Fourth*, none of the existing predictive methods address longitudinal data. However, as both FPIR and SGR are based on LDDMM, they naturally integrate and hence result in our proposed [*fast predictive simple geodesic regression (FPSGR)*]{} approach.
1ex Our *contributions* can be summarized as follows:\
Predictive geodesic regression
: We use a fast predictive registration approach for image geodesic regression. Different to [@quicksilver], we specifically validate that our approach can indeed capture the frequently subtle deformation trends of *longitudinal* image data.
Large-scale dataset capability
: Our predictive regression approach facilitates large-scale image regression within a short amount of time on a single GPU, instead of requiring months of computation time for standard optimization-based methods on a single computer, or on a compute cluster.
Accuracy
: We assess the accuracy of FPSGR by (1) studying linear models of atrophy scores (which are derived from the nonlinear SGR model) over time, as well as (2) correlations between atrophy scores and various diagnostic groups.
Validation
: We demonstrate the performance of FPSGR by analyzing $>6000$ images of the `ADNI-1` / `ADNI-2` datasets. For comparison, we also perform SGR using numerical optimization for the registrations, again on the complete `ADNI-1` / `ADNI-2` datasets.
1ex This work is an extension of a recent conference paper [@ding2017]. In particular, all our experiments are now in 3D. We also added significantly more results to further explore the behavior of FPSGR in comparison to optimization-based SGR. In particular, we added (a) an example to visualize the performance of regression models and associated quantitative comparisons (Sec. \[sec:regression\_results\]); (b) an analysis of local atrophy score correlated with clinical variables (Sec. \[sec:atrophy\]); (c) correlations within diagnostic groups (Sec. \[sec:atrophy\]); (d) a comparison with pairwise registration (Sec. \[sec:sgr\_justification\]); (e) and experiments on extrapolation on unseen data (Sec. \[sec:forecast\], Sec. \[sec:jacobian\]). 1ex **Organization.** The remainder of this article is organized as follows: Sec. \[sec:method\] describes FPSGR, Sec. \[sec:experimental\_setup\] discusses the experimental setup and the training of the prediction models. In Sec. \[sec:discussion\_of\_experimental\_results\], we present experimental results for 3D MR brain images. The paper concludes with a summary and an outlook on future work.
Fast predictive simple geodesic regression {#sec:method}
==========================================
Our fast predictive simple geodesic regression approach is a combination of two methods: *First*, fast predictive image registration (FPIR) and, *second*, integration of FPIR with simple geodesic regression (SGR). Both FPIR and SGR are based on the shooting formulation of LDDMM [@ref:nikhil]; Fig. \[fig:1\] illustrates our overall approach. The individual components are described in the following.
LDDMM
-----
Shooting-based LDDMM and geodesic regression minimize $$\begin{aligned}
\label{eq:gr}
E(I_0, m_0) = \frac{1}{2}\langle m_0, Km_0\rangle + \frac{1}{\sigma^2}\sum_i d^2(I(t_i), Y^i), \\
\notag
s.t. \quad m_t + \text{ad}^*_vm = 0, I_t + \nabla I^Tv = 0, m - Lv = 0,\end{aligned}$$ where $I_0$ is the initial image (known for image-to-image registration and to be determined for geodesic regression), $m_0$ is the initial momentum, $K$ is a smoothing operator that connects velocity $v$ and momentum $m$ as $v = Km$ and $m = Lv$ with $K = L^{-1}$, $\sigma > 0$ is a weight, $Y^i$ is the measured image at time $t_i$ (there will be only one such image for image-to-image registration at $t=1$), and $d^2(I_1, I_2)$ denotes the image similarity measure between $I_1$ and $I_2$ (for example $L_2$ or geodesic distance); $\text{ad}^*$ is the dual of the negative Jacobi-Lie bracket of vector fields: $\text{ad}_vw = -[v, w] = Dvw - Dwv$ and $D$ denotes the Jacobian. The deformation of the source image $I_0\circ\Phi^{-1}$ can be computed by solving $\Phi^{-1}_t + D\Phi^{-1}v=0,~\Phi^{-1}(0)=\text{id}$, where $\text{id}$ denotes the identity map.
FPIR
----
Fast predictive image registration [@ref:yang2016; @quicksilver] aims at predicting the initial momentum, $m_0$, between a source and a target image patch-by-patch. Specifically, we use a deep encoder-decoder network to predict the patch-wise momentum. As shown in Fig. \[fig:1\], in 3D the inputs are two layers of $15 \times 15 \times 15$ image patches ($15 \times 15$ in 2D), where the two layers are from the source and target images respectively. Two patches are taken at the same position by two parallel encoders, which learn features independently. The output is the predicted initial momentum in the $x$, $y$ and $z$ directions (obtained by numerical optimization on the training samples). Basically, the network is split into an encoder and a decoder part. An *encoder* consists of 2 blocks of three 3 $\times$ 3 $\times$ 3 convolutional layers with PReLU activations, followed by another 2 $\times$ 2 $\times$ 2 convolution+PReLU with a stride of two, serving as a “pooling” operation. The number of features in the first convolutional layer is 64 and increases to 128 in the second. In the *decoder*, three parallel decoders share the same input generated from the encoder. Each decoder is the inverse of the encoder except for using 3D transposed convolution layers with a stride of two to perform “unpooling”, and no non-linearity at the end. To speed up computations, we use patch pruning (i.e., for brain imaging, e.g., patches outside the brain are not predicted as the momentum is expected to be zero there) and a large pixel stride (e.g., 14 for $15 \times 15 \times 15$ patches) for the sliding window of the predicted patches.
{width="\textwidth"}
Correction network
------------------
We follow [@quicksilver] and use a two-step approach to improve overall prediction accuracy. An additional correction step, i.e., a [*correction network*]{}, corrects the prediction of the initial prediction network. Fig. \[fig:corr\] illustrates this two-step approach graphically. The correction network has the same structure as the prediction network. Only the inputs and outputs differ. For the prediction network, the inputs are the original moving image and the original target image; output is the predicted initial momentum. For the correction network, the inputs are the original moving image and the warped target image; the output is the momentum difference.
[ |c|c|c|c|c|c|c|c|c|]{}\
*Data Percentile* & 0.3% & 5% & 25% & 50% & 75% & 95% & 99.7%\
Longitudinal Training& **0.0156** & **0.0407** & **0.0761** & **0.1098** & **0.1559** & **0.2681** & **0.3238**\
Cross-sectional Training& 0.0544 & 0.1424 & 0.2641 & 0.3723 & 0.5067 & 0.7502 & 0.8425\
\
*Data Percentile* & 0.3% & 5% & 25% & 50% & 75% & 95% & 99.7%\
Longitudinal Training& 0.1694 & 0.4802 & 1.0765 & 1.7649 & 2.7630 & 4.8060 & 5.6826\
Cross-sectional Training& **0.1123** & **0.3024** & **0.5863** & **0.8737** & **1.2743** & **2.2659** & **2.7836**\
SGR
---
Determining the initial image, $I_0$, and the initial momentum, $m_0$, of Eq. is computationally costly. However, in simple geodesic regression, the initial image is fixed to the *first* image of a subject’s longitudinal image set (left-most part of Fig. \[fig:1\]). Furthermore, the similarity measure $d(\cdot,\cdot)$ is chosen as the geodesic distance between images and [*approximated*]{} so that the geodesic regression problem can be solved by computing pair-wise image registrations with respect to the first image. Specifically, we define the quadratic distance $d^2$ between two images $A$ and $B$ as $$\begin{aligned}
& d^2(A, B) = \frac{1}{2} \int_0^1\lVert v^*\rVert^2_L dt, \\
\notag
& \text{where} \, v^* = \operatorname*{arg\,min}_v \frac{1}{2} \int_0^1 \lVert v\rVert^2_L dt + \frac{1}{\sigma^2} \lVert Q(1) - B\rVert_2^2, \\
\notag
& \text{s.t.} \quad Q_t + \nabla Q^Tv = 0, \text{and}~Q(0) = A\enspace.
\label{eq:geo_dist}\end{aligned}$$ Assume we have an image $I(t_0)$ at time $t_0$ as well as two images $A(t_i)$ and $B(t_i)$. Further, assume that the spatial transformation $\Phi_A$ maps $A(t_i)$ to $I(t_0)$ and $\Phi_B$ maps $B(t_i)$ to $I_0$. Then $A(t_i)=I(t_0)\circ\Phi_A^{-1}$ and $B(t_i)=I(t_0)\circ\Phi_B^{-1}$. Furthermore, assume that $\Phi$ maps $A(t_i)$ to $B(t_i)$, i.e., $B(t_i) = A(t_i)\circ\Phi^{-1}$. Then $\Phi=\Phi_B\circ\Phi_A^{-1}$. Assuming that the geodesic between $I(t_0)$ and $A(t_i)$ is parameterized by the initial velocity $v^A$ and between $I(t_0)$ and $B(t_i)$ by the initial velocity $v^B$ and that we travel between $I(t_0)$ and $A(t_i)$ in time $t_i-t_0$ (and similarly for $B(t_i)$) we can rewrite the map between $A(t_i)$ and $B(t_i)$ based on the exponential map as $$\Phi = \mathrm{Exp_{Id}}((t_i-t_0)v^B)\circ \mathrm{Exp_{Id}}(-(t_i-t_0)v^A),$$ which can be approximated to first order as $$\Phi \approx \mathrm{Exp_{Id}}((t_i-t_0)(v^B-v^A)).$$ Hence, the squared geodesic distance between the two images can be approximated as $$d^2(A(t_i),B(t_i))\approx \frac{1}{2}(t_i-t_0)^2\langle K(m^B-m^A),m^B-m^A\rangle,$$ where $v^A = K m^A$ and $v^B = K m^B$. Hence, Eq. becomes $$\begin{aligned}
& E(\overline{I}, \overline{m}) = \frac{1}{2}\langle \overline{m}, K\overline{m}\rangle \notag\\
& + \frac{1}{2\sigma^2}\sum_i (t_i-t_0)^2 \langle K(\overline{m}-m_i),\overline{m}-m_i \rangle,
\label{eq:approx}\end{aligned}$$ where $\overline{m}$ is the sought-for initial momentum of the regression geodesic and $m_i$ are the initial momenta corresponding to the geodesic connecting $\overline{I}$ (the starting image of the geodesic) and the measurements $Y_i$ in time $t_i-t_0$. Differentiating Eq. w.r.t. $\overline{m}$ results in $$\begin{aligned}
\nabla_{\overline{m}} E = K [\overline{m} + \frac{1}{\sigma^2} \sum_i (t_i - t_0)^2(\overline{m}-m_i)] \overset{!}{=} 0.\end{aligned}$$ Thus, $$\overline{m} = \frac{\sum_i (t_i - t_0)^2m_i}{\sigma^2 + \sum_i (t_i - t_0)^2}.\label{eq:gr_m0}$$ In practice, $\sigma^2$ is very small and can thus be omitted. Furthermore, $m_i$ is obtained by either registering $\overline{I}$ to $Y^i$ in unit time or, as in our FPSGR approach, by predicting the momenta $m_i$ via FPIR, denoted as $\widetilde{m}_i$. As Equation \[eq:gr\_m0\] was derived assuming that images are transformed into each other in time $t_i-t_0$ instead of unit time, the obtained unit-time predicted momenta $\widetilde{m}_i$ correspond in fact to the approximation $\widetilde{m}_i\approx (t_i-t_0)m_i$. Finally, we obtain the approximated optimal $\overline{m}$ of the energy functional in Eq. , for a fixed $\overline{I}=I_0$ as $$\begin{aligned}
\qquad \qquad \overline{m} \approx \frac{\sum_i (t_i-t_0)\widetilde{m}_i}{\sum_i (t_i-t_0)^2}\label{eq:sgr}.\end{aligned}$$
Setup / Training {#sec:experimental_setup}
================
All our experiments use 3D images from the `ADNI` dataset[^1] which consists of 6471 3D MR brain images of size $220 \times 220 \times 220$ voxels. In particular, `ADNI-1` contains 3479 images from 833 subjects and `ADNI-2` contains 2992 images from 823 subjects. Images belong to various types of diagnostic categories which we will discuss later. 1ex We perform the following two types of studies:
Registration
: We assess our hypothesis that training FPIR on longitudinal data for longitudinal registrations is preferred over training using cross-sectional data. Vice versa, training FPIR on cross-sectional data for cross-sectional registrations is preferred over training using longitudinal data. Comparisons are with respect to registration results obtained by numerical optimization (i.e., LDDMM).
Regression
: As for regression, we compare linear models fitted to atrophy scores over time, where scores are either obtained from FPSGR or optimization-based SGR. Additionally, we study correlations between atrophy scores and diagnostic groups. Our hypothesis is that FPSGR is accurate enough to achieve comparable performance to optimization-based SGR, at much lower computational cost, in both situations.
Training of the prediction models
---------------------------------
We use a randomly selected set of 120 patients’ MRI images from `ADNI` for training the prediction models and to test the performance of FPIR. We use all of the `ADNI` data for our regression experiments.\
**Training for registration.** We randomly selected 120 subjects from `ADNI-1` and registered their baseline images to their 24 month follow-up images. We used the first 100 subjects for training and the remaining 20 subjects for testing. For *longitudinal training*, we registered the baseline image of a subject to the subject’s 24-month image. For *cross-sectional training*, we registered a subject’s baseline image to another subject’s 24-month image. To assess the performance of prediction models trained on these two types of paired data, we (1) perform the same type of registrations on the held-out 20 subjects and (2) compare the 2-norm of the deformation error computed from the output of the prediction models with respect to the result obtained by numerical optimization of LDDMM[^2] (which serves as the “ground-truth”). Table \[tab:1\] shows the results which confirm our hypothesis that training the prediction model with longitudinal registration cases is preferred for longitudinal registration over training with cross-sectional data. The deformation error is very small for longitudinal training / testing which provides strong evidence that the predictive method exhibits performance comparable to the (costly) optimization-based LDDMM. Another interpretation of these results is, that it is beneficial to train a prediction model with deformations that are to be *expected*, i.e., relatively small deformations for longitudinal registrations and larger deformations for cross-sectional registrations. As we are interested in longitudinal registrations for the `ADNI` data, we only train our 3D models using longitudinal registrations in the following.
1ex **Training for regression.** The `ADNI-1` dataset contains 228 normal controls, 257 subjects with mild cognitive impairment (MCI), 149 with late mild cognitive impairment (LMCI), as well as 199 subjects suffering from Alzheimer’s disease (AD). We randomly picked roughly 1/6 of patients from each diagnostic category to form a set of 139 subjects for training in `ADNI-1`, i.e., 38 normal controls, 43 MCI, 25 LMCI, as well as 33 AD subjects; this results in 139 subjects overall. The baseline images of each subject were registered to *all* the later time-points within the same subject. To maintain the diagnostic ratio, we picked (out of all registrations) 45 registrations from the normal group, 50 registrations from the MCI group, 30 registrations from the LMCI group, and 40 registrations from the AD group, resulting in 165 longitudinal registration cases for training. 0.5ex The same strategy was applied to `ADNI-2`. In detail, `ADNI-2` contains 200 normal controls, 111 subjects with significant memory complaint (SMC), 182 subjects with early mild cognitive impairment (EMCI), 175 with late mild cognitive impairment (LMCI), and 155 subjects with Alzheimer’s disease (AD). We picked 150 subjects and 140 longitudinal registrations, consisting of 35 registrations from the control group, 20 registrations from the SMC group, 30 registrations from the EMCI group, 30 registrations from the LMCI group, and 25 registrations from the AD group. Note that there are fewer registrations than subjects (140 *vs.* 150) in this setup, as our priority is to maintain the overall diagnostic ratio.
0.5ex For both, `ADNI-1` and `ADNI-2`, the remaining 5/6 of the data is used for testing. We trained four prediction models and their four corresponding correction models, leading to eight prediction models in total, listed in Table \[table:predictionmodels\]. We also note that the training sets within `ADNI-1` and `ADNI-2`, resp., were not overlapping.
[max width=0.5]{}
[ |c|p[0.7cm]{}|p[0.7cm]{}|p[0.7cm]{}|p[0.7cm]{}|p[0.7cm]{}|p[0.7cm]{}|p[0.7cm]{}|]{}\
& 6mo & 12mo & 18mo & 24mo & 36mo & 48mo\
NC& 182& 172& 8& 151& 128& 38\
MCI$^{*}$& 274 & 221 & 165 & 122 & 80 & 11\
AD& 153& 173& 66& 163& 69& 20\
**Total**& 609& 566& 239& 436& 277& 69\
& 6mo & 12mo & 18mo & 24mo & 36mo & 48mo\
NC& 182& 168& 9& 144& 119& 33\
MCI$^{*}$& 272& 224& 169& 124& 70& 10\
AD& 152& 168& 64& 160& 67& 22\
**Total**& 606& 560& 242& 428& 256& 65\
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[ |c|p[0.9cm]{}|p[0.9cm]{}|p[0.9cm]{}|p[0.9cm]{}|p[0.9cm]{}|]{}\
& 3mo & 6mo & 12mo & 24mo & 36mo\
NC$^*$& 173 & 141 & 153 & 119 & 3\
MCI$^*$& 256& 232& 207& 142& 4\
AD& 93& 95& 105& 66& 1\
**Total**& 522&468& 465& 327& 8\
& 3mo & 6mo & 12mo & 24mo & 36mo\
NC$^*$& 172& 142& 159& 122& 3\
MCI$^*$& 257& 230& 202& 149& 4\
AD& 94& 98& 101& 52& 1\
**Total** & 523& 470& 462& 323& 8\
Parameter selection
-------------------
We use the regularization kernel $$K = L^{-1} = (-a \nabla^2 - b\nabla(\nabla \cdot) + c)^{-2}$$ with $[a, b, c]$ set to $[1, 0, 0.1]$. The parameter $\sigma$, from equation , is set to $0.1$. We train our network (using `ADAM` [@adam]) over 10 epochs with a learning rate of $0.0001$.
Efficiency
----------
Once trained, the prediction models allow fast computations of registrations. We use a TITAN X (Pascal) GPU and `PyTorch`[^3] for our implementation of FPIR. For the 3D `ADNI-1` dataset ($220 \times 220 \times 220$ MR images), FPSGR took about one day to predict 2646 pairwise registrations (i.e., 25 \[s\]/prediction) and to compute the regression result. Optimization-based LDDMM[^4] would require $\approx$ 40 days of runtime. Runtime for FPIR on `ADNI-2` is identical to `ADNI-1` as the images have the same spatial dimension.
1ex Compared to the-state-of-art fast geodesic regression model [@hong2017fast], FPSGR is also at least twice as fast. The model in [@hong2017fast] achieves $\approx 16$ times speed-up compared with SGR [@ref:hong] for the same setting (parallel computing with the same number of cores). In our case, we achieve a more than 40 times speed-up compared with SGR for the same setting (a single GPU).
Experimental results for 3D ADNI data {#sec:discussion_of_experimental_results}
=====================================
![Region of Interest (ROI) significantly associated with atrophy in AD used to compute atrophy scores.[]{data-label="fig:ROI"}](image/ROI_refine.png){width="50.00000%"}
For our experiments, we created 10 different (dataset, registration approach) combinations, each combination specifically designed to assess certain properties of our proposed strategy. These combinations are described next.
- All subjects from the `ADNI-1` dataset in combination with optimization-based LDDMM.
- Two subgroups of `ADNI-1` (i.e., different training data portions) in combination with FPSGR *without* a correction network.
- The same two subgroups as in 2), but in combination with FPSGR *with* a correction network.
- The same five groups of 1-3, but for `ADNI-2`.
Our general hypothesis is that the prediction models (for `ADNI-1/2`) show similar performance to optimization-based LDDMM and that using the correction network for the predictions improves results. To assess differences, we compare differences in deformations. Specifically, for every deformation produced by the different approaches, we compute its Jacobian determinant (JD). The JDs are then warped to a common coordinate system for the entire `ADNI` dataset using existing non-linear deformations from [@GMF_ISBI_matching; @GMF_ISBI_optimization]. Each such spatially normalized JD is then averaged within a region where the rate of atrophy is significantly associated with Alzheimer’s disease (AD), i.e., within a *statistical region of interest (stat-ROI)* (see Fig. \[fig:ROI\]). Specifically, we quantify atrophy as $$\left(1 - \frac{1}{|\omega|} \int_{\omega} \text{det}(D \phi (x))~dx\right) \times 100$$ where $\text{det}(\cdot)$ denotes the determinant and $|\cdot|$ the cardinality/size of a set; $\omega$ is an area in the temporal lobes which was determined in prior studies [@GMF_ISBI_matching; @GMF_ISBI_optimization] to be significantly associated with accelerated atrophy in Alzheimer’s disease. The resulting scalar value is an estimate of the relative volume change experienced by that region between the baseline and a follow-up image. Hence, its sign is positive when the region has lost volume over time and is negative if the region has gained volume over time.
We limited our experiments to the applications in [@Xue_ADNI1; @Xue_ADNI2], wherein nonlinear registration/regression is used to quantify atrophy within regions known to be associated to varying degrees with AD ($2$), mild cognitive impairment (MCI) ($1$) (including LMCI[^5]), and normal ageing (NC: normal control) ($0$) in an elderly population. These are the diagnostic groups for `ADNI-1`. For `ADNI-2`, there are also 3 diagnostic categories[^6]: normal ageing ($0$) (including SMC), mild cognitive impairment (including EMCI and LMCI) ($1$), and AD ($2$).
1ex Specifically, we investigate the following *six* questions:
- Can the prediction models for regression qualitatively capture similar trends to the regression model obtained by numerical optimization? (Sec. \[sec:regression\_results\])
- Are atrophy measurements derived from FPSGR biased to overestimate or underestimate volume changes? (Sec. \[sec:bias\])
- Are FPSGR atrophy measurements consistent with those derived from deformations via numerical optimization (LDDMM) which produced the training dataset? (Sec. \[sec:atrophy\])
- Are regression results more stable and hence capture trends better than pairwise registrations? (Sec. \[sec:sgr\_justification\])
- Is the predictive power of the regression models strong enough to forecast deformations for unseen future timepoints (Sec. \[sec:forecast\])
- Do the prediction results capture expected trends in deformation? (Sec. \[sec:jacobian\])
If these experiments resolve favorably, then the substantially improved computational efficiency of FPSGR justifies its use for large-scale imaging studies. Tables \[tab:dist1\] and \[tab:dist2\] show the distributions of the prediction cases per time-point and the diagnostic groups in `ADNI-1` and `ADNI-2`, respectively.
Regression results {#sec:regression_results}
------------------
{width="\textwidth"}
Table \[tab:1\] indicates that FPIR can predict deformation fields similar to the ones obtained using optimization-based LDDMM, even for the subtle changes seen in longitudinal imaging data. However, it remains to be seen how a predictive model performs for image regression. Fig. \[fig:example\] shows an exemplary regression result. In this specific case, large changes can be observed around the ventricles. To illustrate differences between the methods, Fig. \[fig:example\] shows regression results based on optimization-based LDDMM, for FPSGR *without* a correction network, and for FPSGR *with* a correction network. All three methods successfully capture the expanding ventricles and generally capture the image changes. Both FPSGR methods show results that are highly similar to SGR using optimization-based LDDMM. Hence, FPSGR is useful for longitudinal image regression. To further quantify the regression accuracy, we compute the overlay error between measured images and the images on the geodesic as $$\begin{aligned}
\label{eq:overlay}
E_{overlay} (I_0 \circ \Phi^{-1}_{t_i}, Y_{i}) = \frac{1}{|\Omega|} \lVert I_0 \circ \Phi^{-1}_{t_i} - Y_{i}\rVert_{L_1}\end{aligned}$$ where $\Omega$ is the brain area, $I_0 \circ \Phi^{-1}_{t_i}$ is the regressed image at time $t_i$ and $Y_{i}$ is the measured image at time $t_i$. Table \[tab:overlay\] shows the overlay error for the population of 100 subjects which includes all diagnostic groups in `ADNI-1`. Both FPSGR methods obtain results comparable with optimization-based LDDMM. This justifies the use of the proposed methods. The correction network generally increases the prediction accuracy over using the prediction network only.
[max width=]{}
------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --
`Measured Images` $I_{6mo}$ $I_{12mo}$ $I_{18mo}$ $I_{24mo}$ $I_{36mo}$ $I_{48mo}$
Original 0.0770 $\pm$ 0.0212 0.0764 $\pm$ 0.0207 0.0890 $\pm$ 0.0220 0.0810 $\pm$ 0.0223 0.0899 $\pm$ 0.0341 0.0940 $\pm$ 0.0415
LDDMM 0.0750 $\pm$ 0.0194 0.0686 $\pm$ 0.0176 0.0734 $\pm$ 0.0190 0.0609 $\pm$ 0.0168 0.0628 $\pm$ 0.0177 0.0663 $\pm$ 0.0221
Pred-1 0.0754 $\pm$ 0.0213 0.0694 $\pm$ 0.0182 0.0742 $\pm$ 0.0195 0.0621 $\pm$ 0.0188 0.0654 $\pm$ 0.0184 0.0698 $\pm$ 0.0238
Pred+Corr-1 0.0754 $\pm$ 0.0211 0.0691 $\pm$ 0.0182 0.0734 $\pm$ 0.0192 0.0615 $\pm$ 0.0166 0.0642 $\pm$ 0.0188 0.0688 $\pm$ 0.0235
------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --
Bias {#sec:bias}
----
[max width=0.47]{}
Slope Intercept \#data
-- ------------- -------------------------- ---------------------------- --------
LDDMM-1 \[0.62, **0.70**, 0.78\] \[-0.25,**-0.08**, 0.09\]
Pred-1 \[0.37, **0.44**, 0.50\] \[-0.21, **-0.08**, 0.05\]
Pred+Corr-1 \[0.61, **0.68**, 0.75\] \[-0.15, **-0.01**, 0.13\]
LDDMM-2 \[0.57, **0.66**, 0.75\] \[-0.21, **-0.04**, 0.14\]
Pred-2 \[0.43, **0.50**, 0.57\] \[-0.16, **-0.02**, 0.11\]
Pred+Corr-2 \[0.51, **0.58**, 0.65\] \[-0.12, **0.01**, 0.15\]
LDDMM-1 \[0.72, **0.94**, 1.16\] \[-0.45, **-0.03**, 0.39\]
Pred-1 \[0.39, **0.58**, 0.78\] \[-0.43, **-0.05**, 0.33\]
Pred+Corr-1 \[0.71, **0.90**, 1.10\] \[-0.40, **-0.01**, 0.37\]
LDDMM-2 \[0.88, **1.19**, 1.50\] \[-0.65, **-0.05**, 0.55\]
Pred-2 \[0.72, **0.99**, 1.26\] \[-0.68, **-0.16**, 0.36\]
Pred+Corr-2 \[0.80, **1.07**, 1.34\] \[-0.66, **-0.14**, 0.38\]
LDDMM-1 \[0.97, **1.17**, 1.38\] \[-0.28, **0.05**, 0.39\]
Pred-1 \[0.65, **0.80**, 0.96\] \[-0.29, **-0.03**, 0.22\]
Pred+Corr-1 \[0.92, **1.09**, 1.26\] \[-0.14, **0.14**, 0.42\]
LDDMM-2 \[0.83, **1.00**, 1.17\] \[-0.21, **0.06**, 0.33\]
Pred-2 \[0.69, **0.82**, 0.96\] \[-0.20, **0.02**, 0.24\]
Pred+Corr-2 \[0.77, **0.90**, 1.04\] \[-0.15, **0.07**, 0.29\]
LDDMM-1 \[0.48, **0.72**, 0.96\] \[-0.85, **-0.42**, 0.01\]
Pred-1 \[0.26, **0.44**, 0.62\] \[-0.61, **-0.29**, 0.03\]
Pred+Corr-1 \[0.51, **0.68**, 0.86\] \[-0.52, **-0.20**, 0.13\]
LDDMM-2 \[0.54, **0.79**, 1.03\] \[-0.79, **-0.36**, 0.07\]
Pred-2 \[0.40, **0.61**, 0.83\] \[-0.62, **-0.24**, 0.14\]
Pred+Corr-2 \[0.49, **0.70**, 0.91\] \[-0.59, **-0.21**, 0.17\]
LDDMM-1 \[1.94, **2.10**, 2.27\] \[-0.28, **0.02**, 0.31\]
Pred-1 \[1.28, **1.40**, 1.53\] \[-0.24, **-0.02**, 0.20\]
Pred+Corr-1 \[1.70, **1.84**, 1.98\] \[-0.17, **0.08**, 0.33\]
LDDMM-2 \[1.75, **1.92**, 2.09\] \[-0.16, **0.14**, 0.44\]
Pred-2 \[1.42, **1.56**, 1.70\] \[-0.11, **0.14**, 0.39\]
Pred+Corr-2 \[1.49, **1.64**, 1.78\] \[-0.08, **0.17**, 0.43\]
LDDMM-1 \[1.97, **2.33**, 2.69\] \[-0.17, **0.27**, 0.70\]
Pred-1 \[1.23, **1.50**, 1.77\] \[-0.13, **0.21**, 0.54\]
Pred+Corr-1 \[1.74, **2.05**, 2.35\] \[-0.04, **0.33**, 0.70\]
LDDMM-2 \[1.92, **2.28**, 2.65\] \[-0.20, **0.24**, 0.68\]
Pred-2 \[1.56, **1.85**, 2.15\] \[-0.13, **0.22**, 0.57\]
Pred+Corr-2 \[1.65, **1.95**, 2.24\] \[-0.10, **0.25**, 0.60\]
Slope Intercept
LDDMM-1 \[0.55, **0.65**, 0.75\] \[-0.08, **0.03**, 0.13\]
Pred-1 \[0.41, **0.48**, 0.55\] \[-0.03, **0.04**, 0.12\]
Pred+Corr-1 \[0.50, **0.57**, 0.65\] \[-0.04, **0.05**, 0.13\]
LDDMM-2 \[0.51, **0.62**, 0.72\] \[-0.10, **0.01**, 0.12\]
Pred-2 \[0.47, **0.55**, 0.62\] \[-0.03, **0.05**, 0.13\]
Pred+Corr-2 \[0.35, **0.44**, 0.52\] \[-0.09, **-0.00**, 0.08\]
LDDMM-1 \[0.56, **0.79**, 1.02\] \[-0.22, **0.01**, 0.25\]
Pred-1 \[0.53, **0.68**, 0.82\] \[-0.14, **0.01**, 0.16\]
Pred+Corr-1 \[0.63, **0.80**, 0.97\] \[-0.16, **0.02**, 0.19\]
LDDMM-2 \[0.62, **0.90**, 1.18\] \[-0.32, **-0.02**, 0.28\]
Pred-2 \[0.58, **0.77**, 0.97\] \[-0.19, **0.01**, 0.22\]
Pred+Corr-2 \[0.46, **0.68**, 0.91\] \[-0.25, **-0.02**, 0.22\]
LDDMM-1 \[0.71, **0.83**, 0.94\] \[-0.13, **-0.00**, 0.12\]
Pred-1 \[0.53, **0.61**, 0.68\] \[-0.06, **0.02**, 0.10\]
Pred+Corr-1 \[0.64, **0.73**, 0.82\] \[-0.08, **0.02**, 0.11\]
LDDMM-2 \[0.71, **0.82**, 0.92\] \[-0.14, **-0.02**, 0.09\]
Pred-2 \[0.58, **0.66**, 0.73\] \[-0.05, **0.03**, 0.12\]
Pred+Corr-2 \[0.50, **0.59**, 0.67\] \[-0.12, **-0.02**, 0.07\]
LDDMM-1 \[0.03, **0.39**, 0.74\] \[-0.38, **0.05**, 0.47\]
Pred-1 \[0.05, **0.29**, 0.52\] \[-0.24, **0.05**, 0.33\]
Pred+Corr-1 \[0.08, **0.36**, 0.64\] \[-0.28, **0.05**, 0.38\]
LDDMM-2 \[0.14, **0.40**, 0.67\] \[-0.28, **0.04**, 0.35\]
Pred-2 \[0.24, **0.42**, 0.61\] \[-0.17, **0.05**, 0.28\]
Pred+Corr-2 \[0.05, **0.26**, 0.48\] \[-0.22, **0.03**, 0.29\]
LDDMM-1 \[1.65, **1.95**, 2.25\] \[-0.21, **0.13**, 0.47\]
Pred-1 \[1.09, **1.27**, 1.46\] \[-0.12, **0.09**, 0.30\]
Pred+Corr-1 \[1.39, **1.62**, 1.85\] \[-0.15, **0.11**, 0.37\]
LDDMM-2 \[1.59, **1.91**, 2.23\] \[-0.16, **0.19**, 0.53\]
Pred-2 \[1.15, **1.35**, 1.56\] \[-0.09, **0.14**, 0.36\]
Pred+Corr-2 \[1.20, **1.45**, 1.69\] \[-0.13, **0.14**, 0.41\]
LDDMM-1 \[2.49, **2.76**, 3.04\] \[-0.15, **0.07**, 0.30\]
Pred-1 \[1.74, **1.90**, 2.07\] \[-0.09, **0.04**, 0.18\]
Pred+Corr-1 \[2.14, **2.34**, 2.54\] \[-0.09, **0.08**, 0.24\]
LDDMM-2 \[2.72, **2.99**, 3.27\] \[-0.15, **0.07**, 0.29\]
Pred-2 \[1.97, **2.14**, 2.31\] \[-0.07, **0.07**, 0.21\]
Pred+Corr-2 \[2.16, **2.36**, 2.56\] \[-0.15, **0.02**, 0.18\]
: Slope and intercept values for simple linear regression of volume change over time. Our notation for *slope* and *intercept* indicate \[lower bound of 95% C.I., **point estimate**, upper bound of 95% C.I.\]. The interval of intercept estimates all contain zero. The slope changes between the different diagnostic groups. The \#data column lists the number of data points analyzed.[]{data-label="tab:slope_and_intercept"}
Estimates of atrophy are susceptible to bias [@Yushkevich_Bias]. To quantitatively assess this potential bias, we separately considered different diagnostic groups. Specifically, we considered six diagnostic change groups in our experiments: (1) NC for all time points (NC-NC), (2) starting with NC and changing to MCI or AD at a later time point (NC-MCI), (3) MCI for all time points (MCI-MCI), (4) starting with MCI and reversing to NC at later time points (MCI-NC), (5) starting with MCI and changing to AD at later time points (MCI-AD), and (6) AD for all the time points (AD-AD)[^7]. In particular, we follow [@Xue_ADNI1] and fit a straight line (i.e., linear regression) through all atrophy measurements over time, conditioned on each diagnostic change category. The intercept term is an estimate of the atrophy one would measure when registering two scans acquired on the same day; hence it should be near zero and its 95% confidence interval should contain zero. Quantitatively, Table \[tab:slope\_and\_intercept\] lists the slopes, intercepts, and 95% confidence intervals for all ten groups of `ADNI-1` and `ADNI-2`, respectively. LDDMM-1 and LDDMM-2 denote the optimization-based results split into the same testing groups used for Pred-1 and Pred-2 to allow for a direct comparison. All of the results show intercepts that are near zero relative to the range of changes observed and all prediction intercept confidence intervals contain zero. For all diagnostic change groups the prediction and prediction+correction models exhibit more stable results than the optimization-based LDDMM method as indicated by the tighter confidence intervals. Furthermore, all slopes are positive, indicating average volume loss over time. This is consistent with expectations for an aging and neuro-degenerative population. The slopes capture increasing atrophy with disease severity. In `ADNI-1`/`ADNI-2`, we expect $\text{Slope}_{\text{NC-NC}} < \text{Slope}_{\text{MCI-NC}} < \text{Slope}_{\text{NC-MCI}} < \text{Slope}_{\text{MCI-AD}} < \text{Slope}_{\text{AD-AD}}$ and all six experimental groups (i.e. LDDMM-1, Pred-1, Pred+Corr-1, LDDMM-2, Pred-2, and Pred+Corr-2) are generally consistent with this expectation. Exceptions happen in `ADNI-2` for the NC-MCI and MCI-NC cases. As the number of subjects involved is relatively small, i.e., fewer than 20, compared with the other cases (roughly 100), one may speculate that this observation is caused by the limited number of data points for NC-MCI and MCI-NC as shown in the \#data column of Table \[tab:slope\_and\_intercept\]. However, the behavior within each starting diagnostic category, is consistent, i.e., for NC $\text{Slope}_{\text{NC-NC}} < \text{Slope}_{\text{NC-MCI}}$ and for MCI $\text{Slope}_{\text{MCI-NC}} < \text{Slope}_{\text{MCI-MCI}} < \text{Slope}_{\text{MCI-AD}}$. Hence, all six groups’ slope results in `ADNI-1/ADNI-2` are generally consistent with our expectation (and also consistent with results in [@Xue_ADNI1]). The slope estimated from the prediction+correction results is larger than the slope estimated from the prediction model results and closer to the slope obtained from the optimization-based LDDMM results. This indicates that the correction network can improve prediction accuracy. Fig. \[fig:linear\] shows linear regression results for the estimated atrophy scores in `ADNI-1/2` for the Pred+Corr-1 model. Both the data points themselves (i.e., the atrophy scores), as well as kernel density estimates for the linear trends for each subject are shown. These results are consistent with the results of Table \[tab:slope\_and\_intercept\] discussed above. We conclude that (1) neither LDDMM optimization nor FPSGR produced deformations with significant bias to overestimate or underestimate volume change; (2) a linear model of atrophy scores generated by FPSGR can capture intrinsic volume change (i.e., slope) among different diagnostic change groups. Note that our LDDMM optimization results and the prediction results show the same trends. Further, they are directly comparable as the results are based on the same test images (also for the atrophy measurements).
{width="90.00000%"}
{width="90.00000%"}
Atrophy {#sec:atrophy}
-------
[max width=0.46]{}
MMSE $p$-value DX $p$-value \#data
-- ------------- --------- ----------- -------- ----------- --------
LDDMM-1 -0.4957 5.17e-39 0.5140 2.66e-42
Pred-1 -0.4642 8.09e-34 0.4754 1.30e-35
Pred+Corr-1 -0.5104 1.22e-41 0.5259 1.53e-44
LDDMM-2 -0.4667 4.17e-34 0.4814 1.75e-36
Pred-2 -0.4711 8.48e-35 0.4849 4.58e-37
Pred+Corr-2 -0.4734 3.54e-35 0.4890 9.67e-38
LDDMM-1 -0.5749 5.23e-51 0.5313 1.81e-42
Pred-1 -0.5328 9.46e-43 0.4898 1.97e-35
Pred+Corr-1 -0.5799 4.39e-52 0.5406 3.44e-44
LDDMM-2 -0.5301 6.81e-42 0.5055 1.17e-37
Pred-2 -0.5351 9.79e-43 0.5120 1.11e-38
Pred+Corr-2 -0.5374 3.73e-43 0.5155 2.89e-39
LDDMM-1 -0.4939 4.86e-16 0.4776 5.76e-15
Pred-1 -0.4659 3.18e-14 0.4313 3.37e-12
Pred+Corr-1 -0.4924 6.16e-16 0.4643 3.98e-14
LDDMM-2 -0.4385 9.50e-13 0.4000 1.12e-10
Pred-2 -0.4389 9.06e-13 0.3818 8.80e-10
Pred+Corr-2 -0.4384 9.75e-13 0.3790 1.19e-9
LDDMM-1 -0.6064 5.01e-45 0.5978 1.69e-43
Pred-1 -0.5664 2.83e-38 0.5607 2.18e-37
Pred+Corr-1 -0.6001 6.55e-44 0.5943 6.82e-43
LDDMM-2 -0.5822 4.11e-40 0.5534 1.24e-35
Pred-2 -0.5911 1.41e-41 0.5714 2.26e-38
Pred+Corr-2 -0.5898 2.28e-41 0.5709 2.65e-38
LDDMM-1 -0.5142 4.29e-20 0.5300 1.81e-21
Pred-1 -0.4731 7.38e-17 0.4926 2.42e-18
Pred+Corr-1 -0.5069 1.71e-19 0.5296 1.99e-21
LDDMM-2 -0.4334 3.79e-13 0.4815 2.93e-16
Pred-2 -0.4425 1.07e-13 0.4894 7.99e-17
Pred+Corr-2 -0.4393 1.67e-13 0.4863 1.34e-16
LDDMM-1 -0.7456 2.01e-13 0.6635 5.20e-10
Pred-1 -0.7294 1.18e-12 0.6458 2.08e-9
Pred+Corr-1 -0.7443 2.30e-13 0.6575 8.43e-10
LDDMM-2 -0.6889 2.25e-10 0.5927 1.98e-7
Pred-2 -0.6995 9.08e-11 0.6048 9.49e-8
Pred+Corr-2 -0.7005 8.31e-11 0.6067 8.49e-8
MMSE $p$-value DX $p$-value \#data
LDDMM-1 N/A N/A 0.4254 2.34e-24
Pred-1 N/A N/A 0.4142 4.72e-23
Pred+Corr-1 N/A N/A 0.4353 1.52e-25
LDDMM-2 N/A N/A 0.4409 2.77e-26
Pred-2 N/A N/A 0.4280 1.05e-24
Pred+Corr-2 N/A N/A 0.4445 9.64e-27
LDDMM-1 -0.4989 8.01e-31 0.4688 6.09e-27
Pred-1 -0.4768 6.22e-28 0.4625 3.47e-26
Pred+Corr-1 -0.5128 9.64e-33 0.4846 6.19e-29
LDDMM-2 -0.5072 4.29e-32 0.4883 1.58e-29
Pred-2 -0.4718 2.02e-27 0.4742 9.96e-28
Pred+Corr-2 -0.5066 5.25e-32 0.4913 6.33e-30
LDDMM-1 -0.4756 1.43e-27 0.4859 7.22e-29
Pred-1 -0.4530 7.32e-25 0.4771 9.39e-28
Pred+Corr-1 -0.4908 1.67e-29 0.5064 1.37e-31
LDDMM-2 -0.4937 1.07e-29 0.5026 7.05e-31
Pred-2 -0.4626 7.94e-26 0.4913 2.21e-29
Pred+Corr-2 -0.4987 2.35e-30 0.5149 1.44e-32
LDDMM-1 -0.4120 9.53e-15 0.4476 2.06e-17
Pred-1 -0.3670 8.51e-12 0.4331 2.71e-16
Pred+Corr-1 -0.4109 1.15e-14 0.4632 1.09e-18
LDDMM-2 -0.4095 2.09e-14 0.4375 1.93e-16
Pred-2 -0.3411 3.46e-10 0.3940 2.29e-13
Pred+Corr-2 -0.3943 2.20e-13 0.4336 3.79e-16
LDDMM-1 -0.2474 0.55 0.2869 0.49
Pred-1 -0.2474 0.55 0.2869 0.49
Pred+Corr-1 -0.2474 0.55 0.2869 0.49
LDDMM-2 0.0935 0.83 0.1695 0.69
Pred-2 0.0935 0.83 0.1695 0.69
Pred+Corr-2 0.0935 0.83 0.1695 0.69
: FPSGR-derived correlations with clinical variables, compared to correlations with clinical variables for SGR using optimization-based LDDMM. The \#data column lists the number of data points analyzed. indicates that FPSGR using the prediction+correction network shows the strongest correlations; indicates that FPSGR using the prediction network alone shows the strongest correlations; indicates that LDDMM SGR shows the strongest correlations. The MMSE column lists correlations between atrophy scores and the mini-mental state exam scores; the DX column lists correlations between atrophy score and diagnostic category. Finally, the $p$-value column(s) list the $p$-values for the null-hypothesis that there is no correlation. Benjamini-Hochberg procedure was employed to reduce the false discovery rate and highlight indicates statistically significant. FPSGR using the prediction+correction network generally improves performance over using the prediction network alone and frequently even performs slightly better than the SGR results obtained by optimization-based LDDMM.[]{data-label="tab:correlation_with_clinical_variables"}
[max width=0.47]{}
[ |c|c|c|c|]{}\
MMSE & LDDMM & Pred & Pred+Corr\
LDDMM& N/A& 0.1507& 0.5361\
Pred& 0.1507& N/A& 0.0183\
Pred+Corr& 0.5361& 0.0183& N/A\
\
MMSE & LDDMM & Pred & Pred+Corr\
LDDMM& N/A& 0.0005484& 0.09469173\
Pred& 0.9994516& N/A& 0.9999718\
Pred+Corr& 0.0530827& 0.0000282& N/A\
\
DX & LDDMM & Pred & Pred+Corr\
LDDMM& N/A& 0.1963& 0.2356\
Pred& 0.1963& N/A& 0.3208\
Pred+Corr& 0.2356& 0.3208& N/A\
\
DX & LDDMM & Pred & Pred+Corr\
LDDMM& N/A& 0.0010944& 0.9813582\
Pred& 0.9989056& N/A& 0.9999869\
Pred+Corr& 0.0186418& 0.0000131& N/A\
{width="80.00000%"}
{width="92.00000%"}
{width="\textwidth"}
Atrophy estimates have also been shown to correlate with clinical variables [@GMF_ISBI_matching]. To quantify this effect, we computed the Spearman rank-order correlation[^8] between our atrophy estimates and the diagnostic groups (NC = 0, MCI = 1, AD = 2), and also between our atrophy estimates and the scores of the mini-mental state exam (MMSE). We applied the Benjamini-Hochberg procedure [@FDR] for all the correlation results in this paper to reduce the false discovery rate for multiple comparisons. The overall false discovery rate was set to be 0.01, which resulted in an effective significance level of $\alpha \approx$ 0.0093. Detailed results can be found in Table \[tab:correlation\_with\_clinical\_variables\] and Fig. \[fig:correlations\_with\_clinical\_variables\], respectively. In detail, for `ADNI-1/2`, we randomly selected 200[^9] cases from each diagnostic category at each month and calculated the Spearman rank-order correlation. Fig. \[fig:correlations\_with\_clinical\_variables\] shows the results for 50 repetitions. We observe median correlations for all four prediction models in the range of $-0.36$ to $-0.75$ for MMSE and $0.36$ to $0.65$ for diagnostic category. The correlations for all four prediction+correction models were in the range of $-0.40$ to $-0.75$ for MMSE and $0.36$ to $0.65$ for diagnostic category. Previous studies reported Pearson correlations between comparable atrophy estimates and clinical variables as high as $-0.7$ for MMSE and $0.5$ for diagnostic category for 100 subjects[@GMF_ISBI_matching; @GMF_ISBI_optimization].Our two optimization-based LDDMM results achieve median correlations ranging from $-0.40$ to $-0.76$ for MMSE and $0.40$ to $0.66$ for diagnostic category, which is very similar to the predction+correlation models. In general, the correction+prediction FPSGR models outperform the models using only the prediction network. Further, using the correction network, FPSGR achieved comparable and sometimes even slightly better performance compared to the optimization-based LDDMM SGR method, see Table \[tab:correlation\_with\_clinical\_variables\] for additional quantitative results. Specifically, FPSGR using the prediction+correction network performs best in 8 out of 18 comparisons for MMSE and in 12 out of 20 comparisons for diagnostic group. In the cases where FPSGR with prediction+correction network did not perform best its difference to the best method was generally very small. In general FPSGR using the correction network performs better than FPSGR without the correction network. To check for statistical differences in the performance of FPSGR, we use a paired t-test. Table \[tab:mmse\_dx\_t\_test\] shows the resulting p-values for the three methods: optimization-based SGR (i.e., LDDMM), FPSGR without correction network (i.e., Pred) and FPSGR with correction network (i.e., Pred+Corr). In both correlation with MMSE and DX, FPSGR with correction network shows significantly better performance than LDDMM and FPSGR without correction network, which justifies the use of the FPSGR method. In summary, FPSGR captures correlations between atrophy and clinical measures well.
To further explore the correlations of atrophy with MMSE scores, we visualize them separated by diagnostic group where diagnosis did not change (i.e., NC-NC, MCI-MCI, AD-AD) in Fig. \[fig:dx\]. For the `ADNI-1` dataset, we observe (as expected) very low correlations for the normal diagnostic group (with no clear trend), and much stronger correlations for the MCI and AD groups. MCI and AD also exhibit increasingly stronger correlations with time. In case of `ADNI-2`, the MCI group shows modest correlations, which remain consistent across time. Correlations are relatively low for the normal groups. The AD groups show increasingly strong correlations over time. In contrast to `ADNI-1`, `ADNI-2` focuses mainly on earlier stages of the diagnostic groups [@Xue_ADNI2]. Hence, the deformations in `ADNI-2` are generally smaller than in `ADNI-1`. This may explain why the NC and MCI diagnostic groups show consistent correlation values over time (instead of stronger correlations as for AD in `ADNI-2` or the MCI and AD groups in `ADNI-1`).
[max width=0.47]{}
Slope Intercept
-- ----------------- --------------------------- ----------------------------
SGR Pred-1 \[0.37, **0.44**, 0.50\] \[-0.21, , 0.05\]
Pairwise Pred-1 \[0.44, **0.52**, 0.60\]
SGR Pred-2 \[0.43, **0.50**, 0.57\] \[-0.16, , 0.11\]
Pairwise Pred-2 \[0.48, **0.57**, 0.65\]
SGR Pred-1 \[0.39, **0.58**, 0.78\] \[-0.43, , 0.33\]
Pairwise Pred-1 \[0.39, **0.63**, 0.87\] \[-0.63, **-0.16**, 0.30\]
SGR Pred-2 \[0.72, **0.99**, 1.26\] \[-0.68, **-0.16**, 0.36\]
Pairwise Pred-2 \[0.65, **0.96**, 1.27\] \[-0.69, , 0.50\]
SGR Pred-1 \[0.65, **0.80**, 0.96\] \[-0.29, , 0.22\]
Pairwise Pred-1 \[0.69, **0.86**, 1.03\] \[-0.43, **-0.15**, 0.12\]
SGR Pred-2 \[0.69, **0.82**, 0.96\] \[-0.20, , 0.24\]
Pairwise Pred-2 \[0.70, **0.85**, 1.01\] \[-0.29, **-0.04**, 0.21\]
SGR Pred-1 \[0.26, **0.44**, 0.62\] \[-0.61, , 0.03\]
Pairwise Pred-1 \[0.21, **0.45**, 0.68\] \[-0.74, **-0.31**, 0.12\]
SGR Pred-2 \[0.40, **0.61**, 0.83\] \[-0.62, **-0.24**, 0.14\]
Pairwise Pred-2 \[0.29, **0.56**, 0.83\] \[-0.61, , 0.34\]
SGR Pred-1 \[1.28, **1.40**, 1.53\] \[-0.24, , 0.20\]
Pairwise Pred-1 \[1.28, **1.42**, 1.56\] \[-0.31, **-0.06**, 0.19\]
SGR Pred-2 \[1.42, **1.56**, 1.70\] \[-0.11, **0.14**, 0.39\]
Pairwise Pred-2 \[1.44, **1.60**, 1.75\] \[-0.22, , 0.33\]
SGR Pred-1 \[1.23, **1.50**, 1.77\] \[-0.13, **0.21**, 0.54\]
Pairwise Pred-1 \[1.25, **1.55**, 1.85\] \[-0.23, , 0.49\]
SGR Pred-2 \[1.56, **1.85**, 2.15\] \[-0.13, , 0.57\]
Pairwise Pred-2 \[1.53, **1.85**, 2.16\] \[-0.15, **0.23**, 0.60\]
Slope Intercept
SGR Pred-1 \[0.41, **0.48**, 0.55\] \[-0.03, , 0.12\]
Pairwise Pred-1 \[0.25, **0.33**, 0.41\]
SGR Pred-2 \[0.47, **0.55**, 0.62\] \[-0.03, , 0.13\]
Pairwise Pred-2 \[0.26, **0.35**, 0.44\]
SGR Pred-1 \[0.53, **0.68**, 0.82\] \[-0.14, , 0.16\]
Pairwise Pred-1 \[0.37, **0.57**, 0.77\] \[-0.06, **0.14**, 0.33\]
SGR Pred-2 \[0.58, **0.77**, 0.97\] \[-0.19, , 0.22\]
Pairwise Pred-2 \[0.42, **0.65**, 0.88\] \[-0.07, **0.18**, 0.42\]
SGR Pred-1 \[0.53, **0.61**, 0.68\] \[-0.06, , 0.10\]
Pairwise Pred-1 \[0.43, **0.52**, 0.61\]
SGR Pred-2 \[0.58, **0.66**, 0.73\] \[-0.05, , 0.12\]
Pairwise Pred-2 \[0.45, **0.54**, 0.63\]
SGR Pred-1 \[0.05, **0.29**, 0.52\] \[-0.24, , 0.33\]
Pairwise Pred-1 \[-0.10, **0.17**, 0.45\] \[-0.12, **0.21**, 0.53\]
SGR Pred-2 \[0.24, **0.42**, 0.61\] \[-0.17, , 0.28\]
Pairwise Pred-2 \[0.03, **0.26**, 0.49\]
SGR Pred-1 \[1.09, **1.27**, 1.46\] \[-0.12, , 0.30\]
Pairwise Pred-1 \[0.88, **1.10**, 1.32\]
SGR Pred-2 \[1.15, **1.35**, 1.56\] \[-0.09, , 0.36\]
Pairwise Pred-2 \[0.89, **1.13**, 1.37\]
SGR Pred-1 \[1.74, **1.90**, 2.07\] \[-0.09, , 0.18\]
Pairwise Pred-1 \[1.57, **1.77**, 1.96\]
SGR Pred-2 \[1.97, **2.14**, 2.31\] \[-0.07, , 0.21\]
Pairwise Pred-2 \[1.79, **1.99**, 2.19\]
: SGR prediction model compared with a pairwise prediction model. Slope and intercept values for simple linear regression of volume change over time. The notation for slope and intercept columns indicates \[Lower bound of 95% C.I., **point estimate**, Upper bound of 95% C.I.\]. indicates that the intercept is closer to zero (also, zero is within the 95% confidence interval) for SGR prediction model; indicates that the intercept is closer to zero for pairwise prediction model; indicates that the point estimate is either biased to overestimate or underestimate volume change. The SGR prediction model performs better than the pairwise prediction model.[]{data-label="tab:sgr_versus_pairwise_slope_and_intercept"}
[max width=0.47]{}
MMSE $p$-value DX $p$-value \#data
-- ----------------- --------- ----------- -------- ----------- --------
SGR Pred-1 -0.4642 8.09e-34 0.4754 1.30e-35
Pairwise Pred-1 -0.3138 2.31e-15 0.3369 1.32e-17
SGR Pred-2 -0.4711 8.48e-35 0.4849 4.58e-37
Pairwise Pred-2 -0.3431 3.51e-18 0.3680 7.24e-21
SGR Pred-1 -0.5328 9.46e-43 0.4898 1.97e-35
Pairwise Pred-1 -0.4393 4.67e-28 0.3996 4.51e-23
SGR Pred-2 -0.5351 9.79e-43 0.5120 1.11e-38
Pairwise Pred-2 -0.4465 9.61e-29 0.4154 1.00e-24
SGR Pred-1 -0.4659 3.18e-14 0.4313 3.37e-12
Pairwise Pred-1 -0.4164 2.12e-11 0.3882 5.56e-10
SGR Pred-2 -0.4389 9.06e-13 0.3818 8.80e-10
Pairwise Pred-2 -0.4078 4.52e-11 0.3356 9.38e-8
SGR Pred-1 -0.5664 2.83e-38 0.5607 2.18e-37
Pairwise Pred-1 -0.5805 1.51e-40 0.5791 2.55e-40
SGR Pred-2 -0.5911 1.41e-41 0.5714 2.26e-38
Pairwise Pred-2 -0.5927 7.34e-42 0.5811 6.26e-40
SGR Pred-1 -0.4731 7.38e-17 0.4926 2.42e-18
Pairwise Pred-1 -0.4470 5.20e-15 0.4798 2.36e-17
SGR Pred-2 -0.4425 1.07e-13 0.4894 7.99e-17
Pairwise Pred-2 -0.4538 2.08e-14 0.4990 1.59e-17
SGR Pred-1 -0.7294 1.18e-12 0.6458 2.08e-9
Pairwise Pred-1 -0.7100 8.43e-12 0.6168 1.67e-8
SGR Pred-2 -0.6995 9.08e-11 0.6048 9.49e-8
Pairwise Pred-2 -0.6709 9.65e-10 0.5924 2.01e-7
MMSE p-value DX p-value \#data
SGR Pred-1 N/A N/A 0.4142 4.72e-23
Pairwise Pred-1 N/A N/A 0.1744 6.17e-5
SGR Pred-2 N/A N/A 0.4280 1.05e-24
Pairwise Pred-2 N/A N/A 0.1503 5.64e-4
SGR Pred-1 -0.4768 6.22e-28 0.4625 3.47e-26
Pairwise Pred-1 -0.3378 5.93e-14 0.2633 7.29e-9
SGR Pred-2 -0.4718 2.02e-27 0.4742 9.96e-28
Pairwise Pred-2 -0.3312 1.70e-13 0.2849 3.14e-10
SGR Pred-1 -0.4530 7.32e-25 0.4771 9.39e-28
Pairwise Pred-1 -0.4305 2.34e-22 0.4472 3.40e-24
SGR Pred-2 -0.4626 7.94e-26 0.4913 2.21e-29
Pairwise Pred-2 -0.4223 2.30e-21 0.4374 5.72e-23
SGR Pred-1 -0.3670 8.51e-12 0.4331 2.71e-16
Pairwise Pred-1 -0.3772 3.06e-12 0.4515 9.99e-18
SGR Pred-2 -0.3411 3.46e-10 0.3940 2.29e-13
Pairwise Pred-2 -0.3517 8.89e-11 0.4239 1.99e-15
SGR Pred-1 -0.2474 0.55 0.4536 0.26
Pairwise Pred-1 -0.1650 0.70 0.2869 0.49
SGR Pred-2 0.0935 0.83 0.1695 0.69
Pairwise Pred-2 0.0935 0.83 0.2608 0.53
: SGR prediction model compared with pairwise prediction model. Results show correlations with clinical variables. The \#data column lists the number of data points analyzed. indicates a stronger correlation for the SGR prediction method; indicates a stronger correlation for the pairwise model. The $p$-value column lists $p$-values for the null-hypothesis that there is no correlation. The Benjamini-Hochberg procedure was employed to reduce the false discovery rate (FDR). The highlight indicates statistically significant results after correction for multiple comparisons. In general, SGR prediction performs better than pairwise prediction demonstrating that regression stabilizes the correlation results. `ADNI-2` 36mo only has 8 data points and the $p$-value is greater than $0.1$, thus we ignore this month in our comparison.[]{data-label="tab:sgr_versus_pairwise_correlations"}
[max width=0.47]{}
Shapiro-Wilk normality test Wilcoxon signed-rank test
------ ----------------------------- ---------------------------
MMSE 0.01943 0.0005226
DX 0.03286 0.0005083
: $p$-values for a Shapiro-Wilk normality test and Wilcoxon signed-rank test on MMSE and DX correlations between the SGR prediction model and the pairwise prediction model. The null-hypothesis for the Shapiro-Wilk normality test is that the difference of two methods is normally distributed (at a significance level of 5%). The null-hypothesis for the Wilcoxon signed-rank test is that the pairwise prediction method is statistically better than the SGR prediction method (at a significance level of 5%).\[tab:sgr\_pairwise\_w\_test\]
To address the question how stat-ROI specific measures behave over time, we explore how atrophy *locally* (i.e., voxel-by-voxel) correlates with MMSE. The local atrophy is defined as $$\left(1 - \text{det}(D \phi (x) ) \right) \times 100\enspace.$$ I.e., each voxel in a stat-ROI has an associated atrophy score. Fig. \[fig:hist\] shows kernel density estimates of the highest 10% local correlations in a violin plot. For the `ADNI-1` MCI and AD groups, a clear shift toward stronger correlations can be observed over time, consistent with the boxplots of Fig. \[fig:dx\]. This indicates the progression of the disease. Correlations for the normal groups in `ADNI 1/2` are mostly centered around a modest correlation (as expected). In `ADNI-2`, only the AD diagnostic group shows a shift towards stronger correlations over time. All the other diagnostic groups show a relatively consistent distribution over time. This is also consistent with Fig. \[fig:dx\].
Justification of SGR {#sec:sgr_justification}
--------------------
For simple geodesic regression to be a useful model it should outperform pairwise image registration. The main conceptual difference is that the regression model will recover an *average trend* based on multiple image time-points, i.e., the resulting regression geodesic will be a compromise between all the measurements. In contrast, for pairwise image registration (which can be seen as a trivial case of geodesic regression with two images only) the deformation will in general be able to match the target image well. However, just as in linear regression, this may accentuate the effects of noise. In both setups, images can be interpolated or extrapolated based on the estimated geodesic.
0.5ex Tables \[tab:sgr\_versus\_pairwise\_slope\_and\_intercept\] and \[tab:sgr\_versus\_pairwise\_correlations\] justify the use of SGR. Specifically, Table \[tab:sgr\_versus\_pairwise\_slope\_and\_intercept\] shows linear regression results of atrophy measures over time as obtained via SGR (i.e., using an SGR fit over all time-points followed by atrophy computations based on the deformations of the regression geodesic) compared with atrophy measures obtained by pairwise registration. For both the `ADNI-1` and the `ADNI-2` datasets, SGR outperforms the pairwise registration approach in two aspects: (1) the estimated intercept of SGR is generally closer to zero than for the pairwise method and the intercept 95% confidence interval is narrower; (2) 11 out of 24 of the 95% confidence intervals of the pairwise methods show bias to either overestimate or underestimate volume change, while none of the SGR results show such significant bias. Table \[tab:sgr\_versus\_pairwise\_correlations\] compares the correlations between atrophy and clinical measures (MMSE and diagnostic category) of SGR and the pairwise approach. SGR performs better than the pairwise approach in 13 out of 18 cases for MMSE and in 15 out of 20 cases for the diagnostic category. Furthermore, when the pairwise method is better than SGR, the difference is much smaller compared to the differences observed for the cases where SGR is better than the pairwise method. Also note that the pairwise method shows better performance in later months compared to earlier months. This could, for example, be because the deformations are larger for later time-points and hence the registration result becomes more stable, or because SGR is also heavily influenced by the last time-point. To address the above observation, we used a Shapiro-Wilk normality test and a Wilcoxon signed-rank test. From Table \[tab:sgr\_pairwise\_w\_test\] we see that we can reject the null-hypothesis of normality and hence, a paired $t$-test is not appropriate. As an alternative, we conducted a Wilcoxon signed-rank test to compare the SGR prediction model and the pairwise prediction model. Table \[tab:sgr\_pairwise\_w\_test\] shows that the SGR prediction model is statistically significantly better than the pairwise prediction model. Based on the above points, we conclude that SGR is more stable over time than the pairwise method and in general also results in stronger correlations.
Forecasting {#sec:forecast}
-----------
[max width=0.47]{}
MMSE p-value DX $p$-value \#data
-- -- ------------- --------- ---------- -------- ----------- --------
LDDMM-1 -0.5242 1.34e-13 0.5157 3.85e-13
Pred-1 -0.4727 5.16e-11 0.4816 1.98e-11
Pred+Corr-1 -0.5193 2.48e-13 0.5240 1.38e-13
LDDMM-2 -0.4501 2.32e-10 0.4761 1.43e-11
Pred-2 -0.4527 1.77e-10 0.4620 6.63e-11
Pred+Corr-2 -0.4582 9.97e-11 0.4652 4.73e-11
LDDMM-1 -0.4607 1.60e-10 0.4507 4.37e-10
Pred-1 -0.4132 1.45e-8 0.4364 1.75e-9
Pred+Corr-1 -0.4615 1.47e-10 0.4667 8.52e-11
LDDMM-2 -0.3662 3.18e-7 0.4233 2.15e-9
Pred-2 -0.3793 1.09e-7 0.4273 1.46e-9
Pred+Corr-2 -0.3793 1.09e-7 0.4259 1.67e-9
LDDMM-1 -0.3986 1.40e-6 0.4108 6.17e-7
Pred-1 -0.3495 2.84e-5 0.4018 1.13e-6
Pred+Corr-1 -0.3946 1.83e-6 0.4211 2.98e-7
LDDMM-2 -0.3293 4.65e-5 0.3622 6.53e-6
Pred-2 -0.3199 7.81e-5 0.3629 6.25e-6
Pred+Corr-2 -0.3187 8.35e-5 0.3609 7.12e-6
: Correlations of forecasting results. The \#data column lists the number of data points analyzed. indicates that FPSGR using the prediction+correction network shows the strongest correlations; indicates that FPSGR using the prediction network alone shows the strongest correlations; indicates that LDDMM SGR shows the strongest correlations. The Benjamini-Hochberg procedure was employed to reduce the false discovery rate (FDR). The highlight indicates statistically significant results after correction for multiple comparisons.[]{data-label="tab:forecast_leave_out"}
[max width=0.47]{}
MMSE p-value DX p-value \#data
-- -- ------------- --------- ---------- -------- ---------- --------
LDDMM-1 -0.5142 4.29e-20 0.5300 1.81e-21
Pred-1 -0.4731 7.38e-17 0.4926 2.42e-18
Pred+Corr-1 -0.5069 1.71e-19 0.5296 1.99e-21
Pred-1 -0.4583 1.09e-15 0.4825 1.93e-17
Pred+Corr-1 -0.4708 1.42e-16 0.4980 1.21e-18
Pred-1 -0.4923 3.43e-18 0.5104 1.21e-19
Pred+Corr-1 -0.5097 1.37e-19 0.5375 5.47e-22
LDDMM-2 -0.4334 3.79e-13 0.4815 2.93e-16
Pred-2 -0.4425 1.07e-13 0.4894 7.99e-17
Pred+Corr-2 -0.4393 1.67e-13 0.4863 1.34e-16
Pred-2 -0.4078 1.36e-11 0.4398 1.95e-13
Pred+Corr-2 -0.4005 3.34e-11 0.4301 7.40e-13
Pred-2 -0.4202 2.75e-12 0.4635 6.27e-15
Pred+Corr-2 -0.4164 4.51e-12 0.4582 1.38e-14
LDDMM-1 -0.7456 2.01e-13 0.6635 5.20e-10
Pred-1 -0.7294 1.18e-12 0.6458 2.08e-9
Pred+Corr-1 -0.7443 2.30e-13 0.6575 8.43e-10
Pred-1 -0.6332 5.29e-9 0.6165 1.70e-8
Pred+Corr-1 -0.6541 1.10e-9 0.6317 5.86e-9
Pred-1 -0.6446 2.27e-9 0.6478 1.78e-9
Pred+Corr-1 -0.6668 3.98e-10 0.6800 1.31e-10
LDDMM-2 -0.6889 2.25e-10 0.5927 1.98e-7
Pred-2 -0.6995 9.08e-11 0.6048 9.49e-8
Pred+Corr-2 -0.7005 8.31e-11 0.6067 8.49e-8
Pred-2 -0.6528 3.79e-9 0.5568 1.46e-6
Pred+Corr-2 -0.6403 9.25e-9 0.5460 2.55e-6
Pred-2 -0.6334 1.49e-8 0.5970 1.53e-7
Pred+Corr-2 -0.6307 1.79e-8 0.5973 1.50e-7
: Forecast results compared with real data results. The \#data column lists the number of data points analyzed. The Benjamini-Hochberg procedure was employed to reduce the false discovery rate (FDR). highlight indicates statistically significant results after corrections for multiple comparisons. Forecast results are calculated by using SGR excluding 36mo and 48mo data points and then predicting 36mo and 48mo correlations. Results are compared based on the same dataset except for two invalid data points for the 36mo data.[]{data-label="tab:forecast_versus_real"}
{width="\textwidth"}
Another interesting question for SGR and geodesic regression in general is the suitability of the model for the data. To address this question, we evaluate if SGR can *forecast* unseen future time-points. Specifically we consider this question in two different scenarios:
- [**Extrapolate-clinical:**]{} Can we extrapolate the SGR results into the future (to time-points that do not exist in the ADNI image dataset, but for the clinical data) while still obtaining strong correlations.
- [**Extrapolate-image:**]{} How well can correlations between atrophy and clinical measures be predicted for time-points when we do or do not use image data at that very time-point. We artificially leave out image measurements so that we can compare prediction results to results when we have the image measurement.
For both scenarios we use two different forecasting approaches. In the first approach ([**Forecast**]{}) we simply compute SGR results with the available image time-points and then extrapolate using the resulting regression geodesic to the desired time-point in the future. In the second approach ([**Replace**]{}), we artificially impute the missing image time-points by simply replacing them by the image at the closest measured time-point. For example, if we have images at 6, 12, and 18 month, but we want to forecast at 24 month, we use the 18 month image as the imputed 24 month image and then perform SGR on the 6, 12, 18, and the imputed 24 month images. We then obtain the deformation at 24 months from the SGR result.
1ex [**ad Q1.**]{} Table \[tab:forecast\_leave\_out\] shows correlations between atrophy and the clinical measures for the **Forecast** results for 60 month, 72 month and 84 month.The resulting correlations of atrophy with diagnostic category are all above 0.3 (or below -0.3). Furthermore, the **Forecast** correlations show a downward trend with respect to time, which means that the prediction of “far-away“ points is not as accurate as for the “near” future. On the other hand, SGR using the 6 month to 48 month time points results in correlation around -0.5 for MMSE and 0.5 for DX on average. Hence the correlation with the diagnostic category is consistent for that of 60 months. In other words, using 6 month to 48 month data, our prediction model can predict accurately up to 60 month. Our prediction+correction network performs as well as and even slightly better than SGR using optimization-based LDDMM. Fig. \[fig:JD\] shows that these forecasting results capture the trends of the changes in the temporal lobes near the hippocampus and changes in the ventricles.
1ex [**ad Q2.**]{} Table \[tab:forecast\_versus\_real\] and Fig. \[fig:forecast\] show **Forecast** and **Replace** results for correlations between atrophy and clinical measures in comparison to using all images. Specifically, for the **Forecast** and **Replace** results we did not use the available images at 36 and 48 month so we could compare against the results obtained when using these images. If FPSGR is a good model, it should results in correlation results as close to the correlation results using all images as possible. The **Forecast** correlations are only slightly weaker (0.02 to 0.05 lower) than the original correlations using all images illustrating that FPSGR can approximately forecast future changes.
The overall correlations in Table \[tab:forecast\_versus\_real\] show that the **Replace** group performs better than the **Forecast** group. In particular, we are also interested in the prediction of MCI converters, namely, MCI to NC, MCI to MCI, and MCI to AD. The boxplots in Fig. \[fig:forecast\] show the correlations for such predictions. The **Replace** group in Fig. \[fig:forecast\] show relatively worse correlation performance than the **Forecast** group in ADNI-1 Pred-1 and consistent performance in ADNI-1 Pred-2. Hence SGR on a longitudinal image data can achieves good forecasting result for MCI converters. Thus, both [**Extrapolate-clinical**]{} and [**Extrapolate-image**]{} experiments justify the use of FPSGR in predicting near future longitudinal trends especially for MCI converters.
Jacobian Determinant (JD) {#sec:jacobian}
-------------------------
![Average Jacobian determinant over time and diagnostic category for ADNI-1 Prediction-1 and ADNI-1 Prediction+Correction-1 (experiments in ADNI-2 show similar results). A value $<$ 1 means shrinkage and value $>1$ means expansion. The 60 month - 84 month results contained in the purple rectangle are forecasts using the data from 6 month - 48 month. Results show consistent volume loss over time near the temporal lobes and expansion over time near the ventricles/cerebrospinal fluid.[]{data-label="fig:JD"}](image/jd_1_new_2.png){width="50.00000%"}
The average JD images qualitatively agree with prior results [@Xue_ADNI1; @Xue_ADNI2]: severity of volume change increases with severity of diagnosis and time. Change is most substantial in the temporal lobes near the hippocampus (see Fig. \[fig:JD\]). In Fig. \[fig:JD\], 6 month to 48 month are existing data points; 60 month to 84 month are forecast results. Blue indicates volume loss. Red indicates expansion. Results are consistent with expectations: volume loss increases with time and severity of diagnosis in temporal lobes; volume expansion increases with respect to time and severity of diagnosis around the ventricles / cerebrospinal fluid. The forecast results capture visually sensible volume loss or expansion over time, qualitatively illustrating the performance of our method.
Conclusion and future work {#sec:conclusion}
==========================
In this work, we proposed a fast approach for geodesic regression (FPSGR) to study longitudinal image data. FPSGR incorporates the recently proposed FPIR [@ref:yang2016; @quicksilver] into the SGR [@ref:hong] framework, thus leading to a computationally efficient solution to geodesic regression. Since FPSGR replaces the computationally intensive intermediate step of computing pairwise initial momenta via a deep-learning prediction method, it is orders of magnitude faster than existing approaches [@ref:hong; @hong2017fast], without compromising accuracy. Consequently, FPSGR facilitates the analysis of large-scale imaging studies. Experiments on the `ADNI-1/ADNI-2` datasets demonstrate that FPSGR captures expected atrophy trends of normal aging, MCI and AD. It further (1) exhibits negligible bias towards volume changes within stat-ROIs, (2) shows high correlations with clinical variables (MMSE and diagnosis) and (3) produces consistent forecasting results on unseen data.
In future work, it will be be interesting to explore FPSGR for the task of *classifying* stable Mild Cognitive Impairment (sMCI) and progressive Mild Cognitive Impairment (pMCI). Currently, FPSGR only shows modest accuracy for distinguishing these types within MCI. Extending our approach to time-warped geodesic regression models [@hong2014time] might improve the accuracy in this context. Furthermore, end-to-end prediction of averaged initial momenta would be an interesting future direction, as this would allow *learning* representations that characterize the geodesic path among multiple time-series images, not only based on averages of momenta for two images as in FPIR [@ref:yang2016; @quicksilver].
Acknowledgements {#acknowledgements .unnumbered}
================
Research reported in this publication was supported by the National Institutes of Health (NIH) and the National Science Foundation (NSF) under award numbers NIH R01AR072013, NSF ECCS-1148870, and EECS-1711776. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH or the NSF. We also thank Nvidia for the donation of a TitanX GPU.
Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health ([www.fnih.org](www.fnih.org)). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.
References {#references .unnumbered}
==========
[^1]: Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database ([adni.loni.usc.edu](adni.loni.usc.edu)). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD).
[^2]: LDDMM results are generated using a vector momentum formulation: <https://bitbucket.org/scicompanat/vectormomentum>
[^3]: <http://pytorch.org>
[^4]: Here, we used 300 fixed iterations for each registration. 300 iterations can guarantee almost all the results converge. Note that the optimization-based LDDMM also uses a GPU implementation.
[^5]: We combine MCI and LMCI mainly because (a) the diagnostic changes available on the IDA website (<https://ida.loni.usc.edu/login.jsp>) only provide these three diagnostic groups; (b) to be consistent with the experiments conducted by Hua et al. [@Xue_ADNI1], where only Normal, MCI and AD were used as labels to classify `ADNI-1`. Hereafter, in all discussions of `ADNI-1`, MCI is a combination of MCI and LMCI of `ADNI-1`
[^6]: Similar to `ADNI-1`, a detailed diagnosis for `ADNI-2` is only available for the baseline images; MR images at later time points are only labeled as NC, MCI, and AD. Thus, we combine SMC and NC, as well as EMCI and LMCI to be consistent with the diagnostic changes in the *ADNI Diagnosis Summary* available on the IDA website. Hereafter, in all discussions of `ADNI-2`, NC includes NC and SMC and MCI includes EMCI and LMCI.
[^7]: In `ADNI-1/ADNI-2`, there are two patients who show a reversion from AD to MCI. We omitted these cases in our experiment because the number of such cases is too small.
[^8]: We used Spearman rank-order correlation instead of Pearson correlation, because the diagnostic groups imply an ordering only.
[^9]: In `ADNI-1` 48 month, the number was 60 because there was not enough data; `ADNI-2` 36 month was omitted due to lack of data.
|
---
abstract: 'Some linguistic constraints cannot be effectively resolved during parsing at the location in which they are most naturally introduced. This paper shows how constraints can be propagated in a memoizing parser (such as a chart parser) in much the same way that variable bindings are, providing a general treatment of constraint coroutining in memoization. Prolog code for a simple application of our technique to Bouma and van Noord’s (1994) categorial grammar analysis of Dutch is provided.'
author:
-
nocite:
- '[@DR88]'
- '[@DorreDyana]'
- '[@MJ93]'
- '[@BoumaNoord94a]'
- '[@ParsingAsDeduction]'
- '[@v:2]'
- '[@ts:oldt]'
- '[@wa:mem]'
- '[@Shieber85]'
title: Memoization of Coroutined Constraints
---
=cmtt10 at 9pt
\#1[$\rm #1$]{} \#1[$\underline{\rm #1}$]{} \#1[\#1.]{} \#1\#2[\#1 $\leftarrow$ \#2.]{} \#1[to 1.75cm[\#1]{} $\;$]{} \#1\#2\#3\#4
\#1\#2\#3\#4
\#1\#2\#3\#4
\#1\#2
\[Theorem\][Lemma]{} \[Theorem\][Definition]{} \[Theorem\][Algorithm]{}
Introduction
============
As the examples discussed below show, some linguistic constraints cannot be effectively resolved during parsing at the location in which they are most naturally introduced. In a backtracking parser, a natural way of dealing with such constraints is to coroutine them with the other parsing processes, reducing them only when the parse tree is sufficiently instantiated so that they can be deterministically resolved. Such parsers are particularly easy to implement in extended versions of Prolog (such as PrologII, SICStus Prolog and Eclipse) which have such coroutining facilities built-in. Like all backtracking parsers, they can exhibit non-termination and exponential parse times in situations where memoizing parsers (such as chart parsers) can terminate in polynomial time. Unfortunately, the coroutining approach, which requires that constraints share variables in order to communicate, seems to be incompatible with standard memoization techniques, which require systematic variable-renaming (i.e., copying) in order to avoid spurious variable binding.
For generality, conciseness and precision, we formalize our approach to memoization and constraints within Höhfeld and Smolka’s (1988) general theory of Constraint Logic Programming (CLP), but we discuss how our method can be applied to more standard chart parsing as well. This paper extends our previous work reported in Dörre (1993) and Johnson (1993) by generalizing those methods to arbitrary constraint systems (including feature-structure constraints), even though for reasons of space such systems are not discussed here.
Lexical rules in Categorial Grammar
===================================
This section reviews Bouma and van Noord’s (1994) (BN henceforth) constraint-based categorial grammar analysis of modification in Dutch, which we use as our primary example in this paper. However, the memoizing CLP interpreter presented below has also been applied to GB and HPSG parsing, both of which benefit from constraint coroutining in parsing.
BN can explain a number of puzzling scope phenomena by proposing that heads (specifically, verbs) subcategorize for adjuncts as well as arguments (rather than allowing adjuncts to subcategorize for the arguments they modify, as is standard in Categorial Grammar). For example, the first reading of the Dutch sentence
[by1 ()]{}
Frits Marie
‘Fritz deliberately seems to avoid Marie’ ‘Fritz seems to deliberately avoid Marie’
is obtained by the analysis depicted in Figure \[fig:cgder\]. The other reading of this sentence is produced by a derivation in which the adjunct addition rule ‘$A$’ adds an adjunct to [*lijkt te*]{}, and applies vacuously to [*ontwijken*]{}.
$${{{\displaystyle\strut {{{{{\displaystyle\strut }\atop}\atop{\displaystyle\strut
\rm Frits}}\over{\displaystyle\strut {\mbox{\footnotesize NP}}_1}}} \;\;
{{{\displaystyle\strut {{{{{\displaystyle\strut }\atop}\atop{\displaystyle\strut
\rm opzettelijk}}\over{\displaystyle\strut {\mbox{\footnotesize ADV}}}}} \;\;
{{{\displaystyle\strut {{{{{\displaystyle\strut }\atop}\atop{\displaystyle\strut
\rm Marie}}\over{\displaystyle\strut {\mbox{\footnotesize NP}}_2}}} \;\;
{{{\displaystyle\strut {{{\displaystyle\strut {{{\displaystyle\strut {{{{{\displaystyle\strut }\atop}\atop{\displaystyle\strut
\mbox{lijkt te}}}\over{\displaystyle\strut {\mbox{\footnotesize VP}}_1 / {\mbox{\footnotesize VP}}_1 }}}}\over{\displaystyle\strut {\mbox{\footnotesize VP}}_1 / {\mbox{\footnotesize VP}}_1 }}}
A}\over{\displaystyle\strut ({\mbox{\footnotesize VP}}_1 \backslash {\mbox{\footnotesize ADV}} \backslash {\mbox{\footnotesize NP}}_2) /
\Box ({\mbox{\footnotesize VP}}_1 \backslash {\mbox{\footnotesize ADV}} \backslash {\mbox{\footnotesize NP}}_2) }}} D
\;\;
{{{{\displaystyle\strut {{{{{\displaystyle\strut }\atop}\atop{\displaystyle\strut
\rm ontwijken}}\over{\displaystyle\strut {\mbox{\footnotesize VP}}_1 \backslash {\mbox{\footnotesize NP}}_2 }}}}\over{\displaystyle\strut \Box({\mbox{\footnotesize VP}}_1 \backslash {\mbox{\footnotesize ADV}} \backslash {\mbox{\footnotesize NP}}_2)}}}}
\rlap{$A$}}\over{\displaystyle\strut {\mbox{\footnotesize VP}}_1 \backslash {\mbox{\footnotesize ADV}} \backslash {\mbox{\footnotesize NP}}_2 }}}}\over{\displaystyle\strut {\mbox{\footnotesize VP}}_1 \backslash {\mbox{\footnotesize ADV}} }}}}\over{\displaystyle\strut {\mbox{\footnotesize VP}}_1 }}}}\over{\displaystyle\strut {\mbox{\footnotesize S}} }}}$$
It is easy to formalize this kind of grammar in pure Prolog. In order to simplify the presentation of the proof procedure interpreter below, we write clauses as ‘$H$ `::-` $B$’ where $H$ is an atom (the head) and $B$ is a list of atoms (the negative literals).
The atom $\verb+x(+{\it Cat, Left, Right}\verb+)+$ is true iff the substring between the two string positions [*Left*]{} and [*Right*]{} can be analyzed as belonging to category [*Cat*]{}. (As is standard, we use suffixes of the input string for string positions).
The modal operator ‘$\Box$’ is used to diacritically mark untensed verbs (e.g., [*ontwijken*]{}), and prevent them from combining with their arguments. Thus untensed verbs must combine with other verbs which subcategorize for them (e.g., [*lijkt te*]{}), forcing all verbs to appear in a ‘verb cluster’ at the end of a clause.
For simplicity we have not provided a semantics here, but it is easy to add a ‘semantic interpretation’ as a fourth argument in the usual manner. The forward and backward application rules are specified as clauses of `x`/3. Note that the application rules are left-recursive, so a top-down parser will in general fail to terminate with such a grammar. :- op(990, xfx, ::- ). :- op(400, yfx, `\` ). :- op(300, fy, \# ). x(X, Left, Right) ::- \[ x(X/Y, Left, Mid), x(Y, Mid, Right) \]. x(X, Left, Right) ::- \[ x(Y, Left, Mid), x(X`\`Y, Mid, Right) \]. x(X, \[Word|Words\], Words) ::- \[ lex(Word, X) \]. Lexical entries are formalized using a two place relation $\verb+lex(+{\it Word, Cat}\verb+)+$, which is true if [*Cat*]{} is a category that the lexicon assigns to [*Word*]{}. lex(’Frits’, np) ::- \[\]. lex(’Marie’, np) ::- \[\]. lex(opzettelijk, adv) ::- \[\]. lex(ontwijken, \#X ) ::- \[ add\_adjuncts(s`\`np`\`np, X ) \]. lex(lijkt\_te, X / \#Y ) ::- \[ add\_adjuncts((s`\`np)/(s`\`np), X0), division(X0, X/Y ) \]. The `add_adjuncts`/2 and `division`/2 predicates formalize the lexical rules ‘$A$’ (which adds adjuncts to verbs) and ‘$D$’ (the division rule). add\_adjuncts(s, s) ::- \[\]. add\_adjuncts(X, Y`\`adv) ::- \[ add\_adjuncts(X, Y) \]. add\_adjuncts(X`\`A, Y`\`A) ::- \[ add\_adjuncts(X, Y) \]. add\_adjuncts(X/A, Y/A) ::- \[ add\_adjuncts(X, Y) \].
division(X, X) ::- \[\]. division(X0/Y0, (X`\`Z)/(Y`\`Z)) ::- \[ division(X0/Y0, X/Y) \]. Note that the definitions of `add_adjuncts`/2 and `division`/2 are recursive, and have an infinite number of solutions when only their first arguments are instantiated. This is necessary because the number of adjuncts that can be associated with any given verb is unbounded. Thus it is infeasible to enumerate all of the categories that could be associated with a verb when it is retrieved from the lexicon, so following BN, we treat the predicates `add_adjuncts`/2 and `division`/2 as coroutined constraints which are only resolved when their second arguments become sufficiently instantiated.
As noted above, this kind of constraint coroutining is built-in to a number of Prolog implementations. Unfortunately, the left recursion inherent in the combinatory rules mentioned earlier dooms any standard backtracking top-down parser to non-termination, no matter how coroutining is applied to the lexical constraints. As is well-known, memoizing parsers do not suffer from this deficiency, and we present a memoizing interpreter below which does terminate.
The Lemma Table proof procedure
===============================
This section presents a coroutining, memoizing CLP proof procedure. The basic intuition behind our approach is quite natural in a CLP setting like the one of Höhfeld and Smolka, which we sketch now.
A program is a set of definite clauses of the form $$p(X) {\leftarrow}q_1(X_1)\wedge \ldots\wedge q_n(X_n)\wedge \phi$$ where the $X_i$ are vectors of variables, $p(X)$ and $q_i(X_i)$ are relational atoms and $\phi$ is a basic constraint coming from a [ *basic constraint language*]{} ${{\cal C}}$. $\phi$ will typically refer to some (or all) of the variables mentioned. The language of basic constraints is closed under conjunction and comes with (computable) notions of consistency (of a constraint) and entailment ($\phi_1\models_{{{\cal C}}}\phi_2$) which have to be invariant under variable renaming.[^1] Given a program $P$ and a goal $G$, which is a conjunction of relational atoms and constraints, a $P$-answer of $G$ is defined as [*a consistent basic constraint $\phi$ such that $\phi\rightarrow G$ is valid in every model of $P$*]{}. SLD-resolution is generalized in this setting by performing resolution only on relational atoms and simplifying (conjunctions of) basic constraints thus collected in the goal list. When finally only a consistent basic constraint remains, this is an answer constraint $\phi$. Observe that this use of basic constraints generalizes the use of substitutions in ordinary logic programming and the (simplification of a) conjunction of constraints generalizes unification. Actually, pure Prolog can be viewed as a syntactically sugared variant of such a CLP language with equality constraints as basic constraints, where a standard Prolog clause $$\rm p({\mit T}) {\leftarrow}q_1({\mit T_1}), \ldots, q_{\mit n}({\mit T_n})$$ is seen as an abbreviation for a clause in which the equality constraints have been made explicit by means of new variables and new equalities $$\begin{array}{rcl}
\rm p(X) & {\leftarrow}& X = {\mit T}, X_1 = {\mit T_1}, \ldots, X_{\mit n} = {\mit
T_n}, \\
\rm & & q_1(X_1), \ldots, q_{\mit n}(X_{\mit n}).
\end{array}$$ Here the ${\rm X}_i$ are vectors of variables and the $T_i$ are vectors of terms.
Now consider a standard memoizing proof procedure such as Earley Deduction (Pereira and Warren 1983) or the memoizing procedures described by Tamaki and Sato (1986), Vieille (1989) or Warren (1992) from this perspective. Each memoized goal is associated with a set of bindings for its arguments; so in CLP terms each memoized goal is a conjunction of a single relational atom and zero or more equality constraints. A completed (i.e., atomic) clause $\rm p({\mit T})$ with an instantiated argument $T$ abbreviates the non-atomic clause $\rm p(X) {\leftarrow}X = {\mit T}$, where the equality constraint makes the instantiation specific. Such equality constraints are ‘inherited’ via resolution by any clause that resolves with the completed clause.
In the CLP perspective, variable-binding or equality constraints have no special status; informally, all constraints can be treated in the same way that pure Prolog treats equality constraints. This is the central insight behind the Lemma Table proof procedure: general constraints are permitted to propagate into and out of subcomputations in the same way that Earley Deduction propagates variable bindings. Thus the Lemma Table proof procedure generalizes Earley Deduction in the following ways:
1. Memoized goals are in general conjunctions of relational atoms and constraints. This allows constraints to be passed [*into*]{} a memoized subcomputation.
We do not use this capability in the categorial grammar example (except to pass in variable bindings), but it is important in GB and HPSG parsing applications. For example, memoized goals in our GB parser consist of conjunctions of and ECP constraints. Because the phrase-structure rules freely permit empty categories every string has infinitely many well-formed analyses that satisfy the constraints, but the conjoined ECP constraint rules out all but a very few of these empty nodes.
2. Completed clauses can contain arbitrary negative literals (rather than just equality constraints, as in Earley Deduction). This allows constraints to be passed [*out of*]{} a memoized subcomputation.
In the categorial grammar example, the `add_adjuncts`/2 and `division`/2 associated with a lexical entry cannot be finitely resolved, as noted above, so e.g., a clause
x(\#X, \[ontwijken\], \[\]) ::- \[ add\_adjuncts(s`\`np`\`np, X ) \]. is classified as a completed clause; the `add_adjuncts`/2 constraint in its body is inherited by any clause which uses this lemma.
3. Subgoals can be selected in any order (Earley Deduction always selects goals in left-to-right order). This allows constraint coroutining [*within*]{} a memoized subcomputation.
In the categorial grammar example, a category becomes more instantiated when it combines with arguments, allowing eventually the `add_adjuncts`/2 and `division`/2 to be deterministically resolved. Thus we use the flexibility in the selection of goals to run constraints whenever their arguments are sufficiently instantiated, and delay them otherwise.
4. Memoization can be selectively applied (Earley Deduction memoizes every computational step). This can significantly improve overall efficiency.
In the categorial grammar example only `x`/3 goals are memoized (and thus only these goals incur the cost of table management).
The ‘abstraction’ step, which is used in most memoizing systems (including complex feature grammar chart parsers where it is somewhat confusingly called ‘restriction’, as in Shieber 1985), receives an elegant treatment in a CLP approach; an ‘abstracted’ goal is merely one in which not all of the equality constraints associated with the variables appearing in the goal are selected with that goal.[^2]
For example, because of the backward application rule and the left-to-right evaluation our parser uses, eventually it will search at every left string position for an uninstantiated category (the variable `Y` in the clause), we might as well abstract all memoized goals of the form $\verb+x(+\it C, L, R\verb+)+$ to $\verb+x(_+\it , L, \verb+_)+$, i.e., goals in which the category and right string position are uninstantiated. Making the equality constraints explicit, we see that the abstracted goal is obtained by merely selecting the underlined subset of these below: $$\rm \underline{\verb+x(+ X_1, X_2, X_3\verb+)+}, X_1 = {\mit C},
\underline{X_2 = {\mit L}}, X_3 = {\mit R}.$$ While our formal presentation does not discuss abstraction (since it can be implemented in terms of constraint selection as just described), because our implementation uses the underlying Prolog’s unification mechanism to solve equality constraints over terms, it provides an explicit abstraction operation.
Now we turn to the specification of the algorithm itself, beginning with the basic computational entities it uses.
A (generalized) [*goal*]{} is a multiset of relational atoms and constraints. A (generalized) [*clause*]{} ${{H_0 \leftarrow B_0}}$ is an ordered pair of generalized goals, where $H_0$ contains at least one relational atom. A relational interpretation ${{\cal A}}$ (see Höhfeld and Smolka 1988 for definition) satisfies a goal $G$ iff ${{\cal A}}$ satisfies each element of $G$, and it satisfies a clause ${{H_0 \leftarrow B_0}}$ iff either ${{\cal A}}$ fails to satisfy some element of $B_0$ or ${{\cal A}}$ satisfies each element of $H_0$.
This generalizes the standard notion of clause by allowing the head $H_0$ to consist of more than one atom. The head $H_0$ is interpreted conjunctively; i.e., if each element of $B_0$ is true, then so is each element of $H_0$. The standard definition of resolution extends unproblematically to such clauses.
We say that a clause [*${c}_0 ={{H_0 \leftarrow B_0}}$ resolves with a clause ${c}_1 ={{H_1 \leftarrow B_1}}$ on a non-empty set of literals $C\subseteq B_0$*]{} iff there is a variant ${{c}_1}'$ of ${c}_1$ of the form ${{C \leftarrow {B_1}'}}$ such that $V({c}_0) \cap V({B_1}') \subseteq V(C)$ (i.e., the variables common to ${c}_0$ and ${B_1}'$ also appear in C, so there is no accidental variable sharing).
If ${c}_0$ resolves with ${c}_1$ on $C$, then the clause ${{H_0 \leftarrow (B_0 - C) \cup {B_1}'}}$ is called a [*resolvent of ${c}_0$ with ${c}_1$ on $C$*]{}.
Now we define items, which are the basic computational units that appear on the agenda and in the lemma tables, which record memoized subcomputations.
An [*item*]{} is a pair $\langle t, {c} \rangle$ where ${c}$ is a clause and $t$ is a tag, i.e., one of ${\mbox{\sf program}}$, ${\mbox{\sf solution}}$ or ${\mbox{\sf table}}(B)$ for some goal $B$. A [*lemma table for a goal $G$*]{} is a pair $\langle G, L_G \rangle$ where $L_G$ is a finite list of items.
The algorithm manipulates a set $T$ of lemma tables which has the property that the first components of any two distinct members of $T$ are distinct. This justifies speaking of the (unique) lemma table in $T$ for a goal $G$.
Tags are associated with clauses by a user-specified control rule, as described below. The tag associated with a clause in an item identifies the operation that should be performed on that clause. The ${\mbox{\sf solution}}$ tag labels ‘completed’ clauses, the ${\mbox{\sf program}}$ tag directs the proof procedure to perform a non-memoizing resolution of one of the clause’s negative literals with program clauses (the particular negative literal is chosen by a user-specified selection rule, as in standard SLD resolution), and the ${\mbox{\sf table}}(B)$ tag indicates that a subcomputation with root goal $B$ (which is always a subset of the clause’s negative literals) should be started.
A [*control rule*]{} is a function from clauses ${{G \leftarrow B}}$ to one of ${\mbox{\sf program}}$, ${\mbox{\sf solution}}$ or ${\mbox{\sf table}}(C)$ for some goal $C \subseteq B$. A [*selection rule*]{} is a function from clauses ${{G \leftarrow B}}$ where $B$ contains at least one relational atom to relational atoms $a$, where $a$ appears in $B$.
Because ${\mbox{\sf program}}$ steps do not require memoization and given the constraints on the control rule just mentioned, the list $L_G$ associated with a lemma table $\langle G, L_G \rangle$ will only contain items of the form $\langle t, {{G \leftarrow B}} \rangle$ where $t$ is either ${\mbox{\sf solution}}$ or ${\mbox{\sf table}}(C)$ for some goal $C \subseteq B$.
To [*add an item an item $e = \langle t, {{H \leftarrow B}} \rangle$ to its table*]{} means to replace the table $\langle H, L \rangle$ in $T$ with $\langle H, [e | L] \rangle$.
The formal description of the Lemma Table proof procedure is given in Figure \[fig:ltpp\]. We prove the soundness and completeness of the proof procedure in Dörre and Johnson (in preparation). In fact, soundness is easy to show, since all of the operations are resolution steps. Completeness follows from the fact that Lemma Table proofs can be ‘unfolded’ into standard SLD search trees (this unfolding is well-founded because the first step of every ${\mbox{\sf table}}$-initiated subcomputation is required to be a ${\mbox{\sf program}}$ resolution), so completeness follows from Höhfeld and Smolka’s completeness theorem for SLD resolution in CLP.
A worked example
================
Returning to the categorial grammar example above, the control rule and selection rule are specified by the Prolog code below, which can be informally described as follows. All `x`/3 literals are classified as ‘memo’ literals, and `add_adjuncts`/2 and `division`/2 whose second arguments are not sufficiently instantiated are classified as ‘delay’ literals. If the clause contains a memo literal $G$, then the control rule returns ${\mbox{\sf table}}{([G])}$. Otherwise, if the clause contains any non-delay literals, then the control rule returns ${\mbox{\sf program}}$ and the selection rule chooses the left-most such literal. If none of the above apply, the control rule returns ${\mbox{\sf solution}}$. To simplify the interpreter code, the Prolog code for the selection rule and ${\mbox{\sf table}}{(G)}$ output of the control rule also return the remaining literals along with chosen goal. :- ensure\_loaded(library(lists)). :- op(990, fx, \[delay, memo\]).
delay division(\_, X/Y) :- var(X), var(Y). delay add\_adjuncts(\_, X/Y) :- var(X), var(Y).
memo x(\_,\_,\_).
control(Gs0, Control) :- memo(G), select(G, Gs0, Gs) -> Control = table(\[G\], Gs) ; member(G, Gs0), `\`+ delay(G) -> Control = program ; Control = solution.
selection(Gs0, G, Gs) :- select(G1, Gs0, Gs1), `\`+ delay(G1) -> G = G1, Gs = Gs1. Because we do not represent variable binding as explicit constraints, we cannot implement ‘abstraction’ by means of the control rule and require an explicit abstraction operation. The abstraction operation here unbinds the first and third arguments of `x`/3 goals, as discussed above. abstraction(\[x(\_,Left,\_)\], \[x(\_,Left,\_)\]).
[‘\_= ]{}
Figure \[fig:trace\] depicts the proof of a parse of the verb cluster in (1). Item 1 is generated by the initial goal; its sole negative literal is selected for program resolution, producing items 2–4 corresponding to three program clauses for `x`/3. Because items 2 and 3 contain ‘memo’ literals, the control rule tags them ${\mbox{\sf table}}$; there already is a table for a variant of these goals (after abstraction). Item 4 is tagged ${\mbox{\sf program}}$ because it contains a negative literal that is not ‘memo’ or ‘delay’; the resolution of this literal with the program clauses for `lex`/3 produces item 5 containing the constraint literals associated with [*lijkt te*]{}. Both of these are classified as ‘delay’ literals, so item 5 is tagged ${\mbox{\sf solution}}$, and both are ‘inherited’ when item 5 resolves with the ${\mbox{\sf table}}$-tagged items 2 and 3, producing items 6 (corresponding to a right application analysis with [*lijkt te*]{} as functor) and item 19 (corresponding to a left application analysis with [*ontwijken*]{} as functor) respectively. Item 6 is tagged ${\mbox{\sf table}}$, since it contains a `x`/3 literal; because this goal’s second argument (i.e., the left string position) differs from that of the goal associated with table 0, a new table (table 1) is constructed, with item 7 as its first item.
The three program clauses for `x`/3 are used to resolve the selected literal in item 7, just as in item 1, yielding items 8–10. The `lex`/3 literal in item 10 is resolved with the appropriate program clause, producing item 11. Just as in item 5, the second argument of the single literal in item 11 is not sufficiently instantiated, so item 11 is tagged ${\mbox{\sf solution}}$, and the unresolved literal is ‘inherited’ by item 12. Item 12 contains the partially resolved analysis of the verb complex. Items 13–16 analyze the empty string; notice that there are no ${\mbox{\sf solution}}$ items for table 2. Items 17–19 represent partial alternative analyses of the verb cluster where the two verbs combine using other rules than forward application; again, these yield no ${\mbox{\sf solution}}$ items, so item 12 is the sole analysis of the verb cluster.
A simple interpreter
====================
This section describes an implementation of the Lemma Table proof procedure in Prolog, designed for simplicity rather than efficiency. Tables are stored in the Prolog database, and no explicit agenda is used. The dynamic predicate $\verb+goal_table(+G, I\verb+)+$ records the initial goals $G$ for each table subcomputation and that table’s identifying index $I$ (a number assigned to each table when it is created). The dynamic predicate $\verb+table_solution(+I, S\verb+)+$ records all of the ${\mbox{\sf solution}}$ items generated for table $I$ so far, and $\verb+table_parent(+I, T\verb+)+$ records the ${\mbox{\sf table}}$ items $T$, called ‘parent items’ below, which are ‘waiting’ for additional ${\mbox{\sf solution}}$ items from table $I$.
The ‘top level’ goal is $\verb+prove(+G,Cs\verb+)+$, where $G$ is a single atom (the goal to be proven), and $Cs$ is a list of (unresolved) solution constraints (different solutions are enumerated through backtracking). `prove`/2 starts by retracting the tables associated with previous computations, asserting the table entry associated with the initial goal, and then calls `take_action`/2 to perform a program resolution on the initial goal. After all succeeding steps are complete, `prove`/2 returns the solutions associated with table 0. prove(Goal, \_Constraints) :- retractall(goal\_table(\_,\_)), retractall(table\_solution(\_,\_)), retractall(table\_parent(\_, \_)), retractall(counter(\_)), assert(goal\_table(\[Goal\], 0)), take\_action(program, \[Goal\]::-\[Goal\], 0), fail. prove(Goal, Constraints) :- table\_solution(0, \[Goal\]::-Constraints). The predicate $\verb+take_action(+L, C, I\verb+)+$ processes items. $L$ is the item’s label, $C$ its clause and $I$ is the index of the table it belongs to. The first clause calls `complete`/2 to resolve the ${\mbox{\sf solution}}$ clause with any parent items the table may have, and the third clause constructs a parent item term (which encodes both the clause, the tabled goal, and the index of the table the item belongs to) and calls `insert_into_table`/2 to insert it into the appropriate table. take\_action(solution, Clause, Index) :- assert(table\_solution(Index, Clause)), findall(P, table\_parent(Index, P), ParentItems), member(ParentItem, ParentItems), complete(ParentItem, Clause). take\_action(program, Head::-Goal, Index) :- selection(Goal, Selected, Body1), Selected ::- Body0, append(Body0, Body1, Body), control(Body, Action), take\_action(Action, Head::-Body, Index). take\_action(table(Goal,Other), Head::-\_Body, Index) :- insert\_into\_table(Goal, tableItem(Head, Goal, Other, Index)). `complete`/2 takes an item labeled ${\mbox{\sf table}}$ and a clause, resolves the head of the clause with the item, and calls `control`/2 and `take_action`/3 to process the resulting item. complete(tableItem(Head, Goal, Body1, Index), Goal::-Body0) :- append(Body0, Body1, Body), control(Body, Action), take\_action(Action, Head::-Body, Index). The first clause `insert_into_table`/2 checks to see if a table for the goal to be tabled has already been constructed (`numbervars`/3 is used to ground a copy of the term). If an appropriate table does not exist, the second clause calls `create_table`/3 to construct one. insert\_into\_table(Goal, ParentItem) :- copy\_term(Goal, GoalCopy), numbervars(GoalCopy, 0, \_), goal\_table(GoalCopy, Index), !, assert(table\_parent(Index, ParentItem)), findall(Sol, table\_solution(Index, Sol), Solutions), !, member(Solution, Solutions), complete(ParentItem, Solution). insert\_into\_table(Goal0, ParentItem) :- abstraction(Goal0, Goal), !, create\_table(Goal, ParentItem, Index), take\_action(program, Goal::-Goal, Index). `create_table`/3 performs the necessary database manipulations to construct a new table for the goal, assigning a new index for the table, and adding appropriate entries to the indices.
create\_table(Goal, ParentItem, Index) :- (retract(counter(Index0)) -> true ; Index0=0), Index is Index0+1, assert(counter(Index)), assert(goal\_table(Goal, Index)), assert(table\_parent(Index, ParentItem)).
Conclusion
==========
This paper has presented a general framework which allows both constraint coroutining and memoization. To achieve maximum generality we stated the Lemma Table proof procedure in Höhfeld and Smolka’s (1988) CLP framework, but the basic idea—that arbitrary constraints can be allowed to propagate in essentially the same way that variable bindings do—can be applied in most approaches to complex feature based parsing. For example, the technique can be used in chart parsing: in such a system an edge consists not only of a dotted rule and associated variable bindings (i.e., instantiated feature terms), but also contains zero or more as yet unresolved constraints that are propagated (and simplified if sufficiently instantiated) during application of the fundamental rule.
At a more abstract level, the identical propagation of both variable bindings and more general constraints leads us to question whether there is any principled difference between them. While still preliminary, our research suggests that it is often possible to reexpress complex feature based grammars more succinctly by using more general constraints.
[AI]{}
G. Bouma and G. van Noord. Constraint-Based Categorial Grammar. In [*Proceedings of the 32nd Annual Meeting of the ACL, New Mexico State University*]{}, Las Cruces, New Mexico, 1994.
B. Carpenter. The Logic of Typed Feature Structures. Cambridge Tracts in Theoretical Computer Science 32. Cambridge University Press. 1992.
J. Dörre. Generalizing [E]{}arley deduction for constraint-based grammars. In J. D[ö]{}rre (ed.), [*Computational Aspects of Constraint-Based Linguistic Description I, DYANA-2 deliverable R1.2.A*]{}. ESPRIT, Basic Research Project 6852, July 1993.
J. Dörre and M. Johnson. Memoization and coroutined constraints. ms. Institut für maschinelle Sprachverarbeitung, Universität Stuttgart.
M. Höhfeld and G. Smolka. Definite Relations over Constraint Languages. LILOG Report 53, IWBS, IBM Deutschland, Postfach 80 08 80, 7000 Stuttgart 80, W. Germany, October 1988. (available on-line by anonymous ftp from /duck.dfki.uni–sb.de:/pub/papers)
M. Johnson. Memoization in Constraint Logic Programming. Presented at [*First Workshop on Principles and Practice of Constraint Programming, April 28–30 1993, Newport, Rhode Island.*]{}
F. C. Pereira and D. H. Warren. Parsing as Deduction. In [*Proceedings of the 21st Annual Meeting of the ACL, Massachusetts Institute of Technology*]{}, pp. 137–144, Cambridge, Mass., 1983.
S. M. Shieber. Using Restriction to Extend Parsing Algorithms for Complex-Feature-Based Formalisms. In [*Proceedings of the 23rd Annual Meeting of the Association for Computational Linguistics*]{}, pp. 145–152, 1985.
Tamaki, H. and T. Sato. “OLDT resolution with tabulation”, in [*Proceedings of Third International Conference on Logic Programming*]{}, Springer-Verlag, Berlin, pages 84–98. 1986.
Vieille, L. “Recursive query processing: the power of logic”, Theoretical Computer Science 69, pages 1–53. 1989.
Warren, D. S. “Memoing for logic programs”, in [*Communications of the ACM*]{} 35:3, pages 94–111. 1992.
[^1]: This essentially means that basic constraints can be recast as first-order predicates.
[^2]: After this paper was accepted, we discovered that a more general formulation of abstraction is required for systems using a hierarchy of types, such as typed feature structure constraints (Carpenter 1992). In applications of the Lemma Table Proof Procedure to such systems it may be desirable to abstract from a ‘strong’ type constraint in the body of a clause to a logically ‘weaker’ type constraint in the memoized goal. Such a form of abstraction cannot be implemented using the selection rule alone.
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---
abstract: 'Planets open gaps in discs. Gap opening is typically modeled by considering the planetary Lindblad torque which repels disc gas away from the planet’s orbit. But gaps also clear because the planet consumes local material. We present a simple, easy-to-use, analytic framework for calculating how gaps deplete and how the disc’s structure as a whole changes by the combined action of Lindblad repulsion and planetary consumption. The final mass to which a gap-embedded gas giant grows is derived in tandem. The analytics are tested against 1D numerical experiments and calibrated using published multi-dimensional simulations. In viscous alpha discs, the planet, while clearing a gap, initially accretes practically all of the gas that tries to diffuse past, rapidly achieving super-Jupiter if not brown dwarf status. By contrast, in inviscid discs—that may still accrete onto their central stars by, say, magnetized winds—planets open deep, repulsion-dominated gaps. Then only a small fraction of the disc accretion flow is diverted onto the planet, which grows to a fraction of a Jupiter mass. Transitional disc cavities might be cleared by families of such low-mass objects opening inviscid, repulsion-dominated, overlapping gaps which allow most of the outer disc gas to flow unimpeded onto host stars.'
author:
- |
M. M. Rosenthal$^{1\href{https://orcid.org/0000-0003-3938-3099}{\includegraphics[scale=0.4]{orcid.pdf}}}$[^1], E. I. Chiang$^{2,3\href{https://orcid.org/0000-0002-6246-2310}{\includegraphics[scale=0.4]{orcid.pdf}}}$, S. Ginzburg$^{2}$, R. A. Murray-Clay$^{1\href{https://orcid.org/0000-0001-5061-0462}{\includegraphics[scale=0.4]{orcid.pdf}}}$\
$^{1}$ Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\
$^{2}$ Department of Astronomy, University of California at Berkeley, CA 94720, USA\
$^{3}$ Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA
bibliography:
- 'refs.bib'
date: Released
title: How Consumption and Repulsion Set Planetary Gap Depths and the Final Masses of Gas Giants
---
\[firstpage\]
accretion, accretion discs – planets and satellites: formation – planets and satellites: gaseous planets – planets and satellites: physical evolution – planet-disc interactions – protoplanetary discs
Introduction {#sec:intro}
============
Annular gaps in protoplanetary discs are often attributed to embedded planets. The interpretation stems from the theory of satellite-disc interactions that successfully predicted the existence of shepherd moons in planetary rings (e.g., @gt_1982). Satellites in rings, and by analogy planets in discs, repel material away from their orbits as the waves they excite at Lindblad resonances dissipate and impart angular momentum to the ambient medium (see also @gr_2001; @gs_2018). The repulsive, gap-forming planetary Lindblad torque competes against the disc’s viscous torque which diffuses material back into the gap.
Most studies of protoplanetary disc gaps concentrate exclusively on the Lindblad and viscous torques (e.g., @fsc_2014; @kmt_2015; @zzh_2018) and neglect how gaps can also deepen because embedded planets consume local disc gas. Exceptions include, e.g., @znh_2011, @dk_2015 [@dk_2017], and @mfv_2019, whose numerical simulations of planet-disc interactions allow for both planetary accretion and planetary torques. Our aim here is to give an elementary and analytic accounting of both effects: to understand, for planets on fixed circular orbits, how Lindblad repulsion and planetary consumption combine to set gap depths. This is a two-way feedback problem—planetary accretion affects the gas density inside the gap, but the density inside the gap determines the rate of planetary accretion (@gc_2019a, [-@gc_2019b]). Accordingly we will calculate how gas giants grow in tandem with their deepening gaps. Much of our analytic framework is the same as that of @tt_2016 and @tmt_2020, who used it to study nascent planets in viscous discs; we will explore both viscous and inviscid discs. The problem of planetary accretion within disc gaps is also a global one insofar as a planet can accrete gas that is brought to it from afar, from regions outside the gap. Thus we will stage our calculations within circumstellar discs that transport mass across decades in radius. This opens up another form of feedback: in feeding the planet, the disc can have its entire surface density profile changed (e.g., @ld_2006; @znh_2011; @o_2016).
Our work is organized as follows. In section \[sec:consumption\] we describe how Lindblad repulsion and planetary accretion of disc gas (“consumption”) work together to determine gap depths and surface density profiles of viscous circumstellar accretion discs. Our largely analytic considerations are supplemented with simple numerical experiments modeling planet-disc interactions and disc evolution in 1D (orbital radius). In section \[sec:consumption\] we fix, for simplicity, the planet mass; in section \[sec:mass\], we allow the planet mass to grow freely and solve the full two-way feedback problem. In section \[sec:windy\], motivated by recent theoretical and observational developments, we consider discs that transport their mass not by viscous diffusion but rather by angular momentum losses from magnetized winds. For such inviscid, wind-driven discs, accretion is not diffusive but purely advective, and embedded planets carve out especially deep gaps in the absence of viscous backflow. We summarize and discuss the implications of our findings on gas giant masses and disc structure, including the structure of transitional discs, in section \[sec:sum\].
A simplified study such as ours will not capture important (and sometimes poorly understood) effects, among them planetary migration (e.g., @kn_2012; @dhm_2014; @dk_2015 [@dk_2017]; @fc_2017; @kts_2018; @mnp_2020), eccentricity evolution (both of the planet and the disc; e.g., @pnm_2001; @gs_2003; @kd_2006; @dc_2015; @mfv_2019), and the 3D dynamics of circumplanetary discs (e.g., @fzc_2019). Our goal is not so much to be realistic but to acquire some intuition about the interplay of Lindblad repulsion and planetary accretion, and to provide a baseline understanding that can guide the development and interpretation of more sophisticated models. Where possible, we place our results in context with state-of-the-art numerical experiments in the literature (see in particular section \[sec:sum\]).
Viscous discs: Surface Density Profiles at Fixed Planet Mass {#sec:consumption}
============================================================
We study how the surface densities of viscous accretion discs are shaped by repulsive planetary Lindblad torques in addition to planetary accretion of disc gas (“consumption”). Section \[sec:oom\] contains analytic considerations which are tested numerically in section \[sec:num\]. In these sections, while we allow the disc surface density to deplete by consumption, we do not simultaneously allow the planet’s mass to increase. This fixing of the planet’s mass is done for simplicity, to see how the planet affects the disc but not vice versa. In section \[sec:mass\], we free up the planet’s mass and allow two-way feedback between planet and disc.
Order-of-magnitude scalings {#sec:oom}
---------------------------
Consider an accreting planet embedded in a viscous disc. From Figure \[fig:sig\_cartoon\] we identify three disc surface densities: $\Sigma_{\rm p}$ at the orbital radius of the planet ($r = r_{\rm p}$), $\Sigma_+$ exterior to the planet, and $\Sigma_-$ interior to the planet. The planet depresses the local surface density because it is both consuming disc gas and repelling disc gas away by Lindblad torques. Our goal is to estimate the depth of the planet’s gap in relation to the inner and outer discs: $\Sigma_{\rm p}/\Sigma_-$ and $\Sigma_{\rm p}/\Sigma_+$. We assume a steady state where the disc has viscously relaxed: given a viscosity $\nu$, the system age $t$ is at least as long as the diffusion time $r^2/\nu$ across the disc. In addition, $t$ is at most the planet growth timescale $M_{\rm p}/\dot{M}_{\rm p}$, so that we may consider the planet mass fixed at any given moment.
Mass flows steadily inward at rate $\dot{M}_+$ from the outer disc. Part of this flow is accreted by the planet at rate $\dot{M}_{\rm p}$, with the rest feeding the inner disc which accretes onto the star at rate $\dot{M}_-$. Dropping numerical pre-factors (these will be restored in later sections), we have $$\begin{aligned}
\dot{M}_+ &= \dot{M}_- + \dot{M}_{\rm p} \nonumber \\
\Sigma_+ \nu &\sim \Sigma_- \nu + \dot{M}_{\rm p} \nonumber \\
&\sim \Sigma_- \nu + A \Sigma_{\rm p} \,. \label{eqn:mass}\end{aligned}$$ There are a number of assumptions embedded in these order-of-magnitude statements. For $\dot{M}_+$ and $\dot{M}_-$ we have substituted standard steady-state expressions for a disc of shear viscosity $\nu$ (e.g., @fkr_2002), valid asymptotically at locations far from any mass sink ($|r-r_{\rm p}| \gtrsim r_{\rm p}$). At the same time, the locations we are considering in the outer and inner discs are not so far from the planet that we need to account for spatial variations in $\nu$, which may change by order-unity factors over length scale $r$.
For the planet’s accretion rate, we have assumed in (\[eqn:mass\]) that it scales linearly with the local surface density $\Sigma_{\rm p}$ with proportionality constant $A$: $$\label{eqn:Adef}
\dot{M}_{\rm p} = A \Sigma_{\rm p} \,.$$ This assumption is satisfied, e.g., by a planet accreting at the Bondi rate (e.g., @fkr_2002): $$\begin{aligned}
\label{eq:m_dot_bondi_simp}
\dot{M}_{\rm p, Bondi} &\sim \rho_{\rm p} c_{\rm s} R_{\rm B}^2 \nonumber \\
&\sim \frac{\Sigma_{\rm p}}{H} c_{\rm s} \left( \frac{GM_{\rm p}}{c_{\rm s}^2} \right)^2 \end{aligned}$$ where $\rho_{\rm p}$ is the disc midplane mass density near the planet, $c_{\rm s}$ is the disc sound speed, $R_{\rm B} = GM_{\rm p}/c_{\rm s}^2$ is the Bondi radius, $H = c_{\rm s}/\Omega$ is the disc scale height, $\Omega$ is the orbital angular frequency, and $G$ is the gravitational constant. Then $$\label{eqn:A}
A_{\rm Bondi} \sim \frac{m^2}{h^4} \Omega r^2$$ where $m \equiv M_{\rm p}/M_\star$ is the planet-to-star mass ratio, and $h\equiv H/r$ is the disc aspect ratio. Ginzburg & Chiang ([-@gc_2019a], their section 1.1) discusses how Bondi accretion may be valid for “sub-thermal” planets whose masses are less than $$\label{eqn:thermalmass_0}
M_{\rm thermal} \sim h^3 M_\star$$ the mass for which the Bondi radius $R_{\rm B}$, the Hill radius $R_{\rm H} \sim m^{1/3} r$, and the disc scale height $H$ are all equal. A sub-thermal planet has $R_{\rm B} < R_{\rm H} < H$—its gravitational sphere of influence has radius $R_{\rm B}$, set by gravity and thermal pressure—and should accrete at the Bondi rate, isotropically from the all-surrounding disc (@gc_2019a; see also fig. 1 of @tt_2016 for evidence supporting the Bondi $m^2$ scaling, taken from the 3D simulations of @dkh_2003). For a super-thermal planet having $M>M_{\rm thermal}$, the hierarchy of length scales reverses so that $R_{\rm B} > R_{\rm H} > H$—the planet’s sphere of influence, now set by gravitational tides at radius $R_{\rm H}$, “pops out” of the disc—and arguably the planet accretes in a more 2D fashion, presenting a cross-section of order $R_{\rm H} H$ to disc gas that shears by at a velocity $\Omega R_{\rm H}$. The corresponding “Hill rate” for consumption is then $$\label{eqn:hillsimple}
\dot{M}_{\rm p,Hill} \sim \rho_{\rm p} \times R_{\rm H}H \times \Omega R_{\rm H} \sim \Sigma_{\rm p} R_{\rm H}^2 \Omega$$ whence $$A_{\rm Hill} \sim m^{2/3} \Omega r^2 \,.$$ A Hill-based scaling for consumption is commonly used in 2D disc-planet hydrodynamical simulations (e.g., @znh_2011 [@dk_2015; @dk_2017; @mfv_2019]). We have assumed in writing the above that the planet masses are large enough for accretion to be hydrodynamically-limited as opposed to cooling-limited (@gc_2019a, cf. their fig. 1).
In this paper we will calculate the growth of planets from sub-thermal to super-thermal masses, so will have occasion to use both $A_{\rm Bondi}$ and $A_{\rm Hill}$. We recognize that the 2D picture motivating our Hill scaling may not be correct; in 3D, meridional flows from gap walls can feed the planet along its poles [@smc_2014; @msc_2014; @fc_2016]. Relatedly, the disc density scales with height $z$ above the midplane as $\exp[-z^2/(2H^2)]$ (for an isothermal atmosphere), which implies that a considerable fraction of the disc mass resides between $|z|= H$ and $2H$; accordingly, the planet does not pop out of the disc until it is strongly super-thermal, i.e., until $m$ is a large multiple of $h^3$ (cf. equation \[eqn:thermalmass\_0\]). An isotropic version of super-thermal accretion controlled by the Hill sphere gives $\dot{M}_{\rm p,Hill,iso} \sim \rho_{\rm p} \times R_{\rm H}^2 \times \Omega R_{\rm H}$ or $A_{\rm Hill,iso} \sim m \Omega r^2 / h$. Yet another prescription for accretion is given by @tw_2002: $A_{\rm TW} \sim m^{4/3} \Omega r^2 / h^2$, an empirical relation based on their 2D numerical simulations (see also @tt_2016). To the extent that these alternative scalings increase with $m$ more steeply than our nominal $A_{\rm Hill} \propto m^{2/3}$, whatever final super-thermal planet masses we derive should be lower limits (see sections \[sec:proc\] and \[sec:sum\]).
![Sketch of the disc surface density and accretion flow in the vicinity of a planet. The planet is located at orbital radius $r_\mathrm{p}$, inside a gap having surface density $\Sigma_\mathrm{p}$. At $r > r_{\rm p}$, the disc surface density is $\Sigma_+$ and mass accretes inward at rate $\dot{M}_+$. Downstream of the planet, at $r < r_{\rm p}$, the corresponding surface density and accretion rate are $\Sigma_-$ and $\dot{M}_-$, respectively. The difference $\dot{M}_+ - \dot{M}_-$ is the accretion rate onto the planet $\dot{M}_{\rm p}$.[]{data-label="fig:sig_cartoon"}](Fig1_newA.pdf){width="\linewidth"}
Momentum conservation provides another relation between the surface densities. It is easiest to write down downstream of the planet in the accretion flow (in the inner disc), as the flow of momentum upstream (in the outer disc) is complicated by the mass sink presented by the planet. In the inner disc there are no sinks of mass or momentum, only a steady transmission of mass inward and angular momentum outward (assuming, as we do throughout this paper, a non-migrating planet; see section \[sec:sum\] for pointers to the migrating case). The rate at which angular momentum is carried viscously outward by the inner disc equals the viscous transport rate local to the planet, plus the repulsive Lindblad torque exerted by the planet on the disc:[^2] $$\begin{aligned}
\Sigma_- \nu \Omega r^2 \sim \Sigma_{\rm p} \nu \Omega r^2 + B \Sigma_{\rm p} \Omega r^2
\label{eqn:momentum}\end{aligned}$$ with $$\label{eqn:B}
B \sim \frac{m^2}{h^3} \Omega r^2$$ given by the standard @gt_1980 linear Lindblad torque, integrating the effects of all Lindblad resonances up to the torque cutoff. A similar statement to (\[eqn:momentum\]), dropping the viscous term local to the planet, was made by @fsc_2014. Given $B$, (\[eqn:momentum\]) can be solved for the gap contrast with the inner disc: $$\begin{aligned}
\frac{\Sigma_{\rm p}}{\Sigma_-} \sim \frac{1}{1 + B/\nu}
\label{eqn:oom1}\end{aligned}$$ (see also @dm_2013; @kmt_2015; @gs_2018). Combining mass conservation (\[eqn:mass\]) with momentum conservation (\[eqn:momentum\]) yields the gap contrast with the outer disc: $$\begin{aligned}
\frac{\Sigma_{\rm p}}{\Sigma_+} \sim \frac{1}{1 + (A + B)/\nu} \,.
\label{eqn:oom2}\end{aligned}$$ An equivalent equation is derived by @tt_2016 [their appendix B] and @tmt_2020 [their equation 26]. Equations (\[eqn:oom1\]) and (\[eqn:oom2\]) inform us that planetary consumption ($A \neq 0$) leads to asymmetric gap contrasts: a deeper gap relative to the outer disc than to the inner disc. The outer gap contrast is the more important insofar as the outer disc controls surface densities everywhere downstream; in other words, $\Sigma_+$ is the independent variable while $\Sigma_{\rm p}$ and $\Sigma_-$ are dependent variables. Equation (\[eqn:oom2\]) states that, given $\Sigma_+$, the effects of accretion ($A$) and repulsion ($B$) in setting the gap depth $\Sigma_{\rm p}$ are additive (not multiplicative). If $A > B$, then consumption dominates.
For $A=A_{\rm Bondi}$ and $B$ given by (\[eqn:B\]), $$\label{eqn:ab}
A_{\rm Bondi}/B \sim 1/h > 1$$ and consumption dominates repulsion in setting the gap depth, independent of planet mass in the sub-thermal regime. On the other hand, for $A=A_{\rm Hill}$, $$\label{eqn:ab_Hill}
A_{\rm Hill}/B \sim m^{-4/3} h^3 $$ which says that for super-thermal planets that are massive enough, repulsion dominates consumption ($A_{\rm Hill}/B < 1$).
We may also solve for the relative accretion rates: $$\begin{aligned}
\frac{\dot{M}_{\rm p}}{\dot{M}_+} &\sim \frac{A\Sigma_{\rm p}}{\Sigma_+\nu} \sim \frac{A/\nu}{1+(A+B)/\nu} \label{eq:m_dot_ratio} \\
\frac{\dot{M}_-}{\dot{M}_+} &\sim \frac{\Sigma_-}{\Sigma_+} \sim \frac{1+B/\nu}{1+(A+B)/\nu} \,. \label{eqn:mdot-0}\end{aligned}$$ A couple example limiting cases of (\[eq:m\_dot\_ratio\]) and (\[eqn:mdot-0\]) are as follows. If we take $A/B = A_{\rm Bondi}/B \sim 1/h > 1$ and further assume that $B/\nu > 1$ so that the inner gap contrast is significant (equation \[eqn:oom1\]), we find $$\begin{aligned}
\frac{\dot{M}_{\rm p}}{\dot{M}_+} &\sim 1 - B/A_{\rm Bondi} \sim 1 - h \label{eqn:mdotp}\\
\frac{\dot{M}_-}{\dot{M}_+} &\sim B/A_{\rm Bondi} \sim h \label{eqn:mdot-}\end{aligned}$$ which says that the planet consumes nearly all of the mass supplied to it by the outer disc, leaving behind a fraction $h$ to feed the inner disc. If instead we take $A/B = A_{\rm Hill}/B$ and further assume $B > A_{\rm Hill} > \nu$ (repulsion-limited and deep gap), then $$\begin{aligned}
\frac{\dot{M}_{\rm p}}{\dot{M}_+} &\sim A_{\rm Hill}/B \sim m^{-4/3} h^3 < 1 \label{eqn:mdotp_Hill}\\
\frac{\dot{M}_-}{\dot{M}_+} &\sim 1 - A_{\rm Hill}/B \sim 1 - m^{-4/3} h^3 \label{eqn:mdot-_Hill}\end{aligned}$$ and the planet diverts only a small fraction, $A_{\rm Hill}/B$, of the disc accretion flow onto itself.
The order-of-magnitude considerations presented here are firmed up in subsequent sections, including in Appendix \[sec:analy\_ss\], where we derive in greater analytic detail the surface density profile and mass accretion rates, drawing from @ld_2006.
Numerical simulations {#sec:num}
---------------------
### Procedure {#sec:proc}
We solve numerically for the 1D evolution of a viscously shearing disc [e.g., @fkr_2002] with a planetary mass sink. The governing equation for the surface density $\Sigma(r,t)$ in cylindrical radius $r$ and time $t$ reads $$\begin{aligned}
\pd{\Sigma}t=\frac{3}{r}\pd{}r\left[r^{1/2}\pd{}r\left(r^{1/2}\nu\Sigma\right)\right] - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p})
\label{eq:dsig_dt}\end{aligned}$$ where $\delta$ is the Dirac delta function and $r_{\rm p}$ is the radial position of the planet (held fixed). For the viscosity $\nu$ we employ the @ss_alpha $\alpha$-prescription: $$\label{eq:ss_alpha}
\nu = \alpha c_{\rm s}^2/\Omega = \alpha h^2 \Omega r^2$$ where $\Omega$ is the Keplerian orbital frequency around a $1 \, M_\odot$ star, $c_{\rm s} = \sqrt{k_{\rm B}T/\overline{m}}$, the disc temperature is $T = 200 \, {\rm K} (r/{\rm au})^{-1/2}$, $k_{\rm B}$ is Boltzmann’s constant, $\overline{m} = 2m_{\rm H}$ is the mean molecular mass, $m_{\rm H}$ is the mass of the hydrogen atom, and $$h \equiv H/r = c_{\rm s}/(\Omega r) \simeq 0.054 \left( \frac{r}{{\rm 10 \, au}} \right)^{1/4} \,.$$ We fix $\alpha = 10^{-3}$ for the results in this section. Given these inputs, $\nu = \nu(r) \propto r^1$.
Apart from the mass sink, equation (\[eq:dsig\_dt\]), which combines the 1D mass and momentum equations, is identical to the diffusion equation governing an isolated viscous disc as derived by @lp_1974. What is missing is an explicit accounting for the repulsive Lindblad torque exerted by the planet. Many studies include the planetary torque by introducing, into the momentum equation, a term for the torque per unit radius that scales as ${\rm sgn}(x)/x^4$, where $x \equiv r - r_{\rm p}$ (e.g., @lp_1986; @ld_2006). Compared against 2D hydrodynamical simulations, this $1/x^4$ prescription has been shown in 1D studies to reproduce the azimuthally averaged surface density profiles of repulsive gaps near their peripheries (at $x \gtrsim 4 H$) but not near gap centers (at $x \lesssim 4 H$; @fsc_2014, their section 4.3). In particular the 1D torque density prescription, which assumes angular momentum is deposited locally and neglects wave propagation, fails to recover the flat bottoms of gaps and the surface densities there (cf. @gs_2018 who use the @gr_2001 wave steepening theory to lift these assumptions). This shortcoming of the $1/x^4$ prescription means that it cannot be used to compute the planetary accretion rate $\dot{M}_{\rm p}$, which depends on knowing the gas density in the planet’s immediate vicinity.
What we do instead to include the repulsive Lindblad torque when calculating planetary accretion is as follows. Within the radial grid cell at $r = r_{\rm p}$ of width $\Delta r_{\rm p}$, the surface density is reduced after every timestep $\Delta t$ according to $$\begin{aligned}
\label{eqn:euler}
\Sigma (r_{\rm p}, t+\Delta t) = \Sigma (r_{\rm p}, t) - \frac{\dot{M}_{\rm p} (t) \Delta t}{2 \pi r_{\rm p} \Delta r_{\rm p}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, ({\rm simulation})\end{aligned}$$ where the label “simulation” reminds us that this equation applies to the numerical simulation only and should not be used outside of that context. It is in evaluating $\dot{M}_{\rm p}$ that we include, in a “sub-grid” manner, the repulsive Lindblad gap: $$\begin{aligned}
\label{eqn:subgrid}
\dot{M}_{\rm p}(t) &= A \times \frac{\Sigma (r_{\rm p},t)}{1+B/\nu} \,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,({\rm simulation}).\end{aligned}$$ What equation (\[eqn:subgrid\]) says is that the disc surface density the planet actually “sees” when consuming local gas is lower than the numerically computed “grid-level” surface density $\Sigma (r_{\rm p},t)$—lower by the Lindblad reduction factor $1/(1+B/\nu)$ (equation \[eqn:oom1\]). In other words, repulsion is encoded/enforced at a sub-grid level. We stress that equation (\[eqn:subgrid\]) is used only in our numerical simulation to capture repulsion and should not be used outside of it; contrast (\[eqn:subgrid\]) with, e.g., (\[eq:m\_dot\_ratio\]), and note that $\Sigma(r_{\rm p},t)$ is notation specific to the simulation and should not be confused with $\Sigma_{\rm p}$, the actual surface density at the planet’s position.
Our numerical procedure captures the gap depth but not the gap width, as the sub-grid modification is restricted (for simplicity) to the grid cell containing the planet. We consider this crude scheme acceptable insofar as we are more interested in the gross magnitudes for $\Sigma_{\rm p}/\Sigma_+$ and $\Sigma_{\rm p}/\Sigma_-$ and less interested in the precise surface density gradients. An untested assumption underlying our numerical procedure—and in our steady-state analytics—is that material flows radially through the gap at whatever velocity $u_r$ is needed to maintain continuity, i.e., to enforce $\dot{M}_- = \dot{M}_+ - \dot{M}_{\rm p} = -2\pi \Sigma_{\rm p} r_{\rm p} u_r$ (where $u_r < 0$ for accretion toward the star). We cannot test this assumption as we do not resolve the flow dynamics inside the gap. We will call out this assumption in the results to follow (sections \[sec:nc10au\] and \[sec:numwind\]). Also, as a reminder, we note that while the surface density changes as a result of consumption, in this subsection we fix $M_{\rm p}$, i.e., we do not update $M_{\rm p}$ using $\dot{M}_{\rm p}$ (this assumption is relaxed in section \[sec:mass\]).
In evaluating the consumption and repulsion coefficients $A$ and $B$, we make choices similar to those in our earlier order-of-magnitude analysis (section \[sec:oom\]), except that now we include numerical pre-factors for greater precision: $$\begin{aligned}
A_{\rm Bondi} &=
0.5 \, \Omega r^2 \frac{m^2}{h^4} && {\rm for} \, \operatorname{sub-thermal}\,\, m \leq 3 h^3 \label{eqn:bondi}\\
A_{\rm Hill} &= 2.2 \, \Omega r^2 m^{2/3} && {\rm for} \, \operatorname{super-thermal}\,\, m > 3 h^3 \label{eqn:hill}\\
B &= 0.04 \, \Omega r^2 \frac{m^2}{h^3}
\label{eqn:kanagawa}\end{aligned}$$ where all quantities are evaluated at $r_{\rm p}$. The pre-factor of $0.5$ in equation (\[eqn:bondi\]) is calibrated using 3D simulation results for $\dot{M}_{\rm p}$ from D’Angelo et al. ([-@dkh_2003]; these are re-printed in fig. 1 of @tt_2016). The coefficient of 2.2 in equation (\[eqn:hill\]) follows from requiring that (\[eqn:bondi\]) match (\[eqn:hill\]) at the thermal mass $$\label{eqn:thermalmass}
M_{\rm thermal} \equiv 3 h^3 M_\star \simeq 0.5 \left( \frac{h}{0.054} \right)^3 M_{\rm J}$$ defined by equating $H$ with $R_{\rm H} = (m/3)^{1/3} r$, with $M_{\rm J}$ the mass of Jupiter. Equation (\[eqn:kanagawa\]) is taken from the numerical 2D simulations of @kmt_2015 [see also @dm_2013 and @d_2015 who report similar results].
Note further that the expressions we used in section \[sec:oom\] for the steady disc accretion rates $\dot{M}_+$ and $\dot{M}_-$ should be amended with the numerical pre-factor $3\pi$, i.e., $\dot{M}_+ = 3\pi \Sigma_+ \nu$ and similarly for $\dot{M}_-$ (e.g., @fkr_2002). This correction is already embedded in the diffusion equation (\[eq:dsig\_dt\]). Including this pre-factor in equation (\[eqn:mass\]) implies that $A$ should be replaced with $A/(3\pi)$ in equations (\[eqn:oom2\])–(\[eqn:mdot-\_Hill\]). Putting it all together, we have $$\begin{aligned}
\label{eq:A_B_visc}
\frac{A_{\rm Bondi}}{3 \pi B} \simeq \frac{1.3}{h} > 1\end{aligned}$$ implying that consumption always dominates for sub-thermal masses. Furthermore, $$\begin{aligned}
\label{eq:A_B_visc_Hill}
\frac{A_{\rm Hill}}{3 \pi B} \simeq 5.5 m^{-4/3} h^3 \simeq 1.0 \left( \frac{m}{5 \times 10^{-3}} \right)^{-4/3} \left( \frac{h}{0.054} \right)^3\end{aligned}$$ implying that repulsion dominates for super-thermal masses exceeding a “repulsion mass”
$$\begin{aligned}
\label{eqn:repulsionmass}
M_{\rm repulsion,visc} &\simeq 3.6 h^{9/4} M_\star \nonumber \\
& \simeq 5.3 M_{\rm J} \left( \frac{h}{0.054} \right)^{9/4} \simeq 5.3 M_{\rm J} \left( \frac{r}{10 \, {\rm au} } \right)^{9/16} \nonumber \\
&\simeq 1.2 h^{-3/4} M_{\rm thermal} \simeq 11 \left( \frac{0.054}{h} \right)^{3/4} M_{\rm thermal} \,.\end{aligned}$$
For $M<M_{\rm repulsion,visc}$, consumption dominates and the planet accretes nearly all the disc gas that tries to diffuse past; for $M > M_{\rm repulsion,visc}$, repulsion dominates and the planet’s accretion rate falls below the disc accretion rate. The above expression for $M_{\rm repulsion, visc}$ depends on our assumption that planetary accretion follows our Hill scaling $A_{\rm Hill} \propto m^{2/3}$ for super-thermal masses. As discussed in section \[sec:oom\], this assumption might not be correct. If instead of $A_{\rm Hill}$ we use $A_{\rm TW} = 0.29 \,\Omega r^2 \,m^{4/3}/h^2$ as found from the 2D numerical simulations of @tw_2002, we would find $A_{\rm TW}/(3\pi B) \simeq 4 \,( M_\mathrm{p}/M_\mathrm{J} )^{-2/3} ( h/0.054)$, in which case the mass above which repulsion dominates would change to $M_{\rm repulsion,visc,TW} \simeq 9 \, M_{\rm J} \,[r / (10 \, {\rm au})]^{3/8}$. This is nearly twice the value of $M_{\rm repulsion,visc}$ given by (\[eqn:repulsionmass\]), and would imply a more extended consumption-dominated growth phase. Insofar as our nominal model adopts $A_{\rm Hill}$ which leads to a more limited consumption-dominated growth phase, the planet masses we compute for our viscous disc model are lower limits.
So far we have described how we compute the mass sink term, which includes the sub-grid Lindblad torque, in equation (\[eq:dsig\_dt\]). The remaining diffusive term is solved in a standard way. We first change variables to $z \equiv r^{1/2} \nu \Sigma$ and $y \equiv 2 r^{1/2}$ so that the diffusive portion of equation reads $$\begin{aligned}
\label{eq:kep_simp}
\pd{z}t &= \frac{12 \nu}{y^2} \pd{^2 z}{y^2}\end{aligned}$$ with non-constant diffusion coefficient $12 \nu/y^2$. We solve equation as an initial value problem using an implicit scheme (e.g., @ptv_2007). Our computation grid extends from an inner boundary of $r_\mathrm{in} = 0.01 \, \mathrm{au}$ to an outer boundary of $r_\mathrm{out} = 500 \, \mathrm{au}$, and is divided into 300 cells that are uniform in $\Delta y$. We fix the timestep $\Delta t = 10^{-4} t_{\nu,{\rm p}}$, where $t_{\nu,{\rm p}} \equiv r_{\rm p}^2 / \nu (r_{\rm p}) \simeq 1.7$ Myr is the viscous diffusion timescale at the planet’s orbital radius of $r_{\rm p} = 10$ au (where $h \simeq 0.054$). Recognizing that our transformed variable $z$ is proportional to the viscous torque $2 \pi \nu \Sigma r^3 d \Omega / dr \propto r^{1/2} \nu \Sigma$, we use a torque-free inner boundary condition, $z (r_\mathrm{in}) = 0$, as would be the case if the disc were truncated by a co-rotating stellar magnetosphere (shearless boundary layer). At the outer boundary we assume the torque gradient $\partial z / \partial r \,(r_{\rm out}) = 0$. Neither boundary condition is critical as we are interested in the flow near the planet, away from either boundary.
The surface density of the disc is initialized with the similarity solution for an isolated viscous accretion disc with $\nu \propto r^1$ (@lp_1974; @hcg_1998): $$\begin{aligned}
\label{eqn:sim}
\Sigma(r,0) = \frac{M_{\rm disc}}{2 \pi r_1^2} \frac{r_1}{r} e^{-r/r_1} \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,
({\rm simulation})\end{aligned}$$ where $M_{\rm disc} = 15.5 \, M_{\rm J} \simeq 0.015 \, M_\odot$ is the initial mass of the disc and $r_1 = 30$ au is a characteristic disc radius (where the diffusion time is $r_1^2/\nu \simeq 5$ Myr). We consider two fixed planet masses, $M_{\rm p} = 0.3 \,M_{\rm J} < M_{\rm thermal}$ and $M_{\rm p} = 10 \,M_{\rm J} > M_{\rm thermal}$. Planet masses that freely grow are modeled in section \[sec:mass\].
At every timestep, we first advance $\Sigma(r,t)\rightarrow \Sigma(r,t+\Delta t)$ for all $r$ according to (\[eq:kep\_simp\]) using the implicit solver, and then we advance $\Sigma(r_{\rm p},t)\rightarrow \Sigma(r_{\rm p},t+\Delta t)$ using (\[eqn:euler\]) and (\[eqn:subgrid\]). This procedure is repeated until the disc is evolved for several $t_{\nu,{\rm p}}$, long enough for the disc near the planet to achieve a quasi-steady state.
![How the surface density profile of a viscous disc responds to a planet that both consumes disc gas, and repels gas away by Lindblad torques. Surface densities are calculated from our 1D numerical simulation of a planet of fixed mass, either $M_{\rm p} = 0.3 \,M_{\rm J}$ (top panel) or $M_{\rm p} = 10\, M_{\rm J}$ (bottom panel), at $t = 3 t_{\nu,\mathrm{p}}$ when the disc near the planet at $r_{\rm p} = 10$ au has viscously relaxed. When computing the planetary accretion rate $\dot{M}_{\rm p}$, the gap is modeled as a single cell whose “true” surface density equals the grid-level $\Sigma$ lowered by a factor of $(1+B/\nu) \simeq B/\nu$; plotted here are the true sub-grid values $\Sigma_{\rm p}$. Accordingly, the planet’s gap is not spatially resolved and its width should not be taken literally from this figure. Red double-tipped arrows have lengths equal to their associated variables in dex, and demonstrate good agreement between numerics and analytics. The planet of mass $M_{\rm p} = 0.3 \,M_{\rm J}$, accreting at the Bondi rate, creates an asymmetric gap, with the inner disc surface density $\Sigma_-$ lower than the outer $\Sigma_+$ by $A_{\rm Bondi}/(3\pi B) > 1$; conditions are always consumption-dominated for Bondi accretion and $B$ as given by (\[eqn:kanagawa\]). The planet of mass $M_{\rm p} = 10\, M_{\rm J}$, accreting at the Hill rate, creates a symmetric gap where $\Sigma_-/\Sigma_+ \sim 1$; conditions here are repulsion-dominated as $M_{\rm p} > M_{\rm repulsion,visc}$ (equation \[eqn:repulsionmass\]).[]{data-label="fig:sigma_detailed"}](Fig2_newA.pdf){width="\linewidth"}
### Results {#sec:res222}
Figure \[fig:sigma\_detailed\] shows, for $M_{\rm p} = \{0.3, 10\} M_{\rm J}$, the numerically computed surface density profiles $\Sigma(r)$ at $t = 3 t_{\nu,{\rm p}} \simeq 5$ Myr. Overlaid for comparison is our numerical solution without a planet, which we have verified matches the analytic time-dependent similarity solution of @lp_1974. For the case with a planet, rather than plot at face value the numerically computed (grid-level) $\Sigma (r_{\rm p},t)$, we plot that value multiplied by the sub-grid reduction factor $1/(1+B/\nu)$—this is the “true” value for $\Sigma_{\rm p}$ that incorporates the repulsive Lindblad torque. Since this sub-grid correction factor is applied to only a single grid point, we cannot resolve gap widths; our focus instead is on the gross gap contrasts $\Sigma_{\rm p}/\Sigma_+$ and $\Sigma_{\rm p}/\Sigma_-$.
The surface density profiles shown in Figure \[fig:sigma\_detailed\] conform to the analytic considerations of section \[sec:oom\]. For $M_{\rm p} = 0.3 \, M_{\rm J}$ (top panel), conditions are consumption-limited: $\Sigma_+/\Sigma_{\rm p} \sim A_{\rm Bondi}/(3\pi\nu)$ (equation \[eqn:oom2\] in the limit $A_{\rm Bondi}/(3\pi) > B > \nu$) and the surface density of the entire interior disc is depressed relative to the same disc without a planet by a factor of $\Sigma_+ / \Sigma_- \sim \dot{M}_+ / \dot{M}_- \simeq A_{\rm Bondi}/(3\pi B)$ (equations \[eqn:mdot-0\] and \[eqn:mdot-\]). By comparison, for $M_{\rm p} = 10 \, M_{\rm J}$ (bottom panel), the gap is more nearly symmetric, $\Sigma_+/\Sigma_- \sim 1$ (equations \[eqn:mdot-0\] and \[eqn:mdot-\_Hill\]), and deep and repulsion-dominated, $\Sigma_+/\Sigma_{\rm p} \sim B/\nu$ (equation \[eqn:oom2\] in the limit $B > A_{\rm Hill}/(3\pi) > \nu$).
So long as consumption is stronger than repulsion in the sense that $A/(3\pi) > B$—a condition that we have shown always obtains for sub-thermal masses accreting at the Bondi rate, and for sufficiently low-mass super-thermal masses accreting at the Hill rate ($M < M_{\rm repulsion,visc}$)—repulsion does not much affect the gap surface density $\Sigma_{\rm p}$. Figure \[fig:m\_final\_visc\] demonstrates that different choices for the repulsion coefficient $B = \{10^{-2}, 10^{-3}, 10^{-4}\}\times A_{\rm Bondi}$ all yield practically the same $\Sigma_{\rm p}$ (when corrected to the true sub-grid value) relative to $\Sigma_+$. What repulsion, in combination with consumption, affects instead is how much gas leaks past the planet into the inner disc: the three different values for $B$ in Figure \[fig:m\_final\_visc\] yield three inner disc surface densities that, from equation (\[eqn:mdot-0\]), scale as $\Sigma_-/\Sigma_+ \simeq (1+B/\nu)/[1 + A_{\rm Bondi}/(3\pi \nu)]$. This factor scales as $3\pi B/A_{\rm Bondi}$ when $A_{\rm Bondi}/(3\pi) > B > \nu$ (dot-dashed and dotted lines), and as $1/[1 + A_{\rm Bondi}/(3\pi\nu)]$ when $B < \nu$ (solid line; in this limit repulsion has no effect).
![Same as Figure \[fig:sigma\_detailed\] for the case $M_{\rm p} = 0.3 M_{\rm J}$, but for different choices of $B$ scaled to $A_{\rm Bondi}$. As long as $A_{\rm Bondi}/(3 \pi) > B$, the planet’s gap is consumption-dominated and its surface density $\Sigma_\mathrm{p}$ is independent of the repulsion coefficient $B$. The depression of the inner disc relative to the outer disc is, however, sensitive to $B$ for $B > \nu$; $\Sigma_-/\Sigma_+ \simeq (1+B/\nu)/[A_{\rm Bondi}/(3\pi\nu)]$. []{data-label="fig:m_final_visc"}](Fig3_newA.pdf){width="\linewidth"}
Viscous discs: Gas Giant Growth {#sec:mass}
===============================
Numerical calculation at $r_{\rm p} = 10$ au {#sec:nc10au}
--------------------------------------------
We now relax the assumption that the planet mass remains fixed, and at every timestep update $M_{\rm p}$ according to $\dot{M}_{\rm p}$ computed using equation (\[eqn:subgrid\]). Our numerical procedure is unchanged from section \[sec:num\] except that we initialize the planet mass at $M_{\rm p}(0) = 0.1 \, M_{\rm J}$ and allow it to grow. For our nominal disc parameters ($\alpha = 10^{-3}$, $h = 0.054$ at $r_{\rm p} = 10$ au), a starting planet mass of $0.1 M_{\rm J}$ ($m \simeq 0.95 \times 10^{-4}$) implies that, initially, $A = A_{\rm Bondi}$, $A_{\rm Bondi}/(3\pi B) \simeq 1.3/h \simeq 24$ (a consumption-dominated gap), $A_{\rm Bondi}/(3\pi \nu) \simeq 19$ (a strong outer gap contrast), and $B/\nu \simeq 0.79$ (a weak inner gap contrast).
Figure \[fig:sigma\_mdot\_M0\_10\_MJ\] shows two snapshots in time of $\Sigma(r)$ and the disc mass flow rate $\dot{M}_{\rm disc}(r) = -2 \pi \Sigma r u_r$, where $$\begin{aligned}
u_r = -\frac{3}{\Sigma r^{1/2}} \frac{\partial}{\partial r} \left(\nu \Sigma r^{1/2} \right)\end{aligned}$$ is the gas radial velocity (e.g., @fkr_2002) evaluated numerically from our solution for $\Sigma$ (omitting the single-point discontinuity at $r = r_{\rm p}$). Note that $\dot{M}_{\rm disc} > 0$ indicates inward mass transport, toward the star. The planet accretes predominantly from the outer disc, notwithstanding a small contribution from the inner disc before the disc has viscously relaxed; this early-time contribution can be seen at $t = 0.3\,t_{\nu,{\rm p}}$ when $\dot{M}_{\rm disc} < 0$ from $r \sim 3$ au to the planet’s orbit. The behaviour of $\dot{M}_{\rm disc}$ at $r \sim 100$ au is characteristic of a viscous disc near its turn-around “transition radius” (@lp_1974; @hcg_1998), outside of which the disc has not yet viscously relaxed; this outermost disc behaviour is not caused by the planet.
Embedded in Figure \[fig:sigma\_mdot\_M0\_10\_MJ\] is our assumption, first mentioned in section \[sec:proc\], that the disc flow inside the gap maintains continuity. At $t = 3 t_{\nu,{\rm p}}$, $\dot{M}_{\rm disc} (r > r_{\rm p})$ is, to within a factor of 2, the same as $\dot{M}_{\rm disc} (r < r_{\rm p})$. Because the gap surface density $\Sigma_{\rm p}$ at this time is about 4 orders of magnitude smaller than the surface densities $\Sigma_+$ and $\Sigma_-$ outside the gap, the radial velocity $|u_r|$ within the gap must be 4 orders of magnitude larger than the radial velocities outside, to maintain the near-constancy of $\dot{M}_{\rm disc}$ across $r_{\rm p}$. Since the radial accretion velocities away from the gap are of order $r/t_\nu \sim \nu/r \sim \alpha h c_{\rm s} \sim 2$ cm/s, we must have $|u_r| \sim 0.2$ km/s within the gap. How such a radial velocity is achieved is not specified by our model, which does not resolve the gap spatially.
Figure \[fig:Mp\_vs\_t\_visc\] displays the planet’s mass as a function of time. We identify a consumption-dominated phase during which the planet grows from 0.1 to $5 \,M_{\rm J}$ ($M < M_{\rm repulsion,visc}$; equation \[eqn:repulsionmass\]) and a slower repulsion-dominated phase between 5 and $8 \,M_{\rm J}$ ($M > M_{\rm repulsion,visc}$). During the first phase, accretion starts at the Bondi rate and switches to the Hill rate once $M_{\rm p} > M_{\rm thermal} \simeq 0.5 \,M_{\rm J}$ (equations \[eqn:thermalmass\] and \[eqn:bondi\]–\[eqn:hill\]). A consumption-dominated ($A/(3\pi) > B$) and deep ($A/(3\pi) > \nu$) gap implies from (\[eq:m\_dot\_ratio\]) that $\dot{M}_{\rm p} \simeq \dot{M}_+$, i.e., the planet’s accretion rate is about as large as it can be. During the final repulsion-limited phase, when $M_{\rm p} > 5 \,M_{\rm J}$ and $A_{\rm Hill}/(3\pi) > B$, consumption slows and the planet undergoes a last near-doubling in mass as the remainder of the disc diffuses away, onto the star.
Analytic estimates of the final planet mass
-------------------------------------------
We can compare our numerical result for the final mass at $r_{\rm p} = 10$ au to the following analytic estimates, derived by neglecting the initial short-lived Bondi accretion phase and assuming that at all times the planet accretes at the Hill rate ($A=A_{\rm Hill}$) and has a large inner gap contrast ($B >\nu$): $$\begin{aligned}
\dot{m} &= \frac{A_{\rm Hill} \Sigma_{\rm p}}{M_\star} \nonumber \\
&= \frac{A_{\rm Hill}}{M_\star} \frac{\Sigma_+ \nu}{A_{\rm Hill}/(3\pi) + B}\end{aligned}$$ where we have used (\[eqn:oom2\]). At small orbital distances, final planet masses exceed $M_{\rm repulsion,visc}$ and so their final growth phase is repulsion-limited: $$\begin{aligned}
\label{eqn:viscmdot}
\dot{m} & = \frac{A_{\rm Hill}}{B} \frac{\Sigma_+ \nu}{M_\star} \nonumber \\
& = 55 \alpha h^5 m^{-4/3} \frac{\Sigma_+ r_{\rm p}^2}{M_\star} \Omega \,.\end{aligned}$$ We approximate $\Sigma_+$ using the similarity solution for an isolated viscous disc with no planet and $\nu \propto r^1$: $$\begin{aligned}
\label{eqn:simtime}
\Sigma_+ \sim \frac{M_{\rm disc}}{2\pi r_1^2} \left(\frac{r_1}{r_{\rm p}}\right) T^{-3/2} e^{ -(r_{\rm p}/r_1)/T}\end{aligned}$$ where $T \equiv 1 + t/t_1$, $t_1 \equiv r_1^2/[3 \nu(r_1)]$, and $M_{\rm disc}$ is the initial disc mass (@lp_1974; @hcg_1998). Integrating equation (\[eqn:viscmdot\]) from $t=0$ to $t$ gives $$\begin{aligned}
\label{eqn:fullsolvisc}
m(t) \sim & \left(\frac{385}{18\sqrt{\pi}} \frac{M_{\rm disc}}{M_\star} \frac{h^5}{h_1^2} \frac{r_1}{r_{\rm p}} \right)^{3/7} \nonumber \\ &\times \left[\text{Erf}\left(\sqrt{\frac{r_{\rm p}}{r_1}}\right)-\text{Erf}\left(\sqrt{\frac{r_{\rm p} t_1}{r_1 (t + t_1)}}\right)\right]^{3/7} (\operatorname{repulsion-limited})\end{aligned}$$ where $h_1$ is the disc aspect ratio at $r_1$. As $t \rightarrow \infty$, equation (\[eqn:fullsolvisc\]) simplifies to $$\begin{aligned}
\label{eqn:asymsolvisc}
m_\mathrm{final,visc} \sim& \left[\frac{385}{18\sqrt{\pi}} \frac{M_{\rm disc}}{M_\star} \frac{h^5}{h_1^2} \frac{r_1}{r_{\rm p}}
\text{Erf}\left(\sqrt{\frac{r_{\rm p}}{r_1}}\right)
\right]^{3/7} (\operatorname{repulsion-limited})
\end{aligned}$$ which further simplifies in the limit $r_{\rm p} \ll r_1$ (away from the initial disc outer edge) to $$\begin{aligned}
\label{eqn:solviscsimple}
M_{\mathrm{final,visc}} &\sim 10 \, M_\mathrm{J} \, \left( \frac{M_{\rm disc}}{15.5 \,M_{\rm J}} \right)^{3/7} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{9/28} \,\, (\operatorname{repulsion-limited})\end{aligned}$$ for our fiducial parameters. Note that $M_{\rm final,visc}$ in these limits is independent of $\alpha$ and $r_1$. Equation (\[eqn:solviscsimple\]) may be reproduced to order-of-magnitude by multiplying $\dot{m}$ (evaluated at $t = t_1$) by $t_1$. In Figure \[fig:Mp\_vs\_t\_visc\] we plot equation (\[eqn:asymsolvisc\]) as the uppermost horizontal dashed line, labeled $M_{\rm final,visc}$.
At the largest orbital distances, conditions tend to remain consumption-limited as $M_{\rm p}$ stays below $M_{\rm repulsion,visc}$. Then the planet accretes nearly all of the disc gas that tries to diffuse past the planet—and diffusion can be in the outward direction ($\dot{M}_{\rm disc} < 0$) if the planet is located near or beyond the disc’s turn-around radius. Accordingly we estimate the planet mass as $$\begin{aligned}
M_{\rm p}(t) \sim \int^t_0 |\dot{M}_{\rm disc}| dt \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\operatorname{consumption-limited}) \end{aligned}$$ where $\dot{M}_{\rm disc}$ is approximated by the no-planet similarity solution (equation 35 of @hcg_1998). For $r_{\rm p}\leq r_1/2$, $$\begin{aligned}
\label{eqn:consume1}
M_{\rm p}(t) &\sim M_{\rm disc} \left( e^{-r_{\rm p}/r_1} - \frac{e^{-(r_{\rm p}/r_1)/T}}{\sqrt{T}} \right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\operatorname{consumption-limited})\end{aligned}$$ and for $r_{\rm p}>r_1/2$, $$\begin{aligned}
\label{eqn:consume2}
M_{\rm p}(t) \sim M_{\rm disc} \left( \sqrt{\frac{2r_1}{r_{\rm p}}} e^{-1/2} - e^{-r_{\rm p}/r_1} - \frac{e^{-(r_{\rm p}/r_1)/T}}{\sqrt{T}} \right) \nonumber \\
(\operatorname{consumption-limited}).\end{aligned}$$ We will make use of equations (\[eqn:fullsolvisc\]), (\[eqn:consume1\]), and (\[eqn:consume2\]) in section \[sec:sum\] when we discuss, in the context of observations, how the final planet mass depends on disc mass and orbital distance.
![Snapshots of the surface density profile $\Sigma(r)$ and disc accretion rate $\dot{M}_{\rm disc}(r) = -2 \pi \Sigma u_r r$ ($> 0$ for accretion toward the star) for a planet embedded at $r_{\rm p} = 10$ au in a viscous $\alpha = 10^{-3}$ disc. The planet mass is allowed to freely grow starting from $M_{\rm p}(0) = 0.1 M_{\rm J}$. At $t = 0.3 t_{\nu,{\rm p}}$, the planet resides in a consumption-dominated, asymmetric gap (top panel, dashed curve) and accretes from regions both exterior and interior to its orbit which have not yet viscously relaxed (bottom panel, dashed curve). At the later time $t = 3 t_{\nu,{\rm p}}$, the planet has grown sufficiently (see also Figure \[fig:Mp\_vs\_t\_visc\]) that its gap is now repulsion-dominated and more symmetric (top panel, solid curve); the planet now accretes only from the outer disc, reducing the flow of mass into the inner disc by less than a factor of 2 (bottom panel, solid curve. At this time we have multiplied $\dot{M}_{\rm disc}$ by a factor of 5 for easier viewing).[]{data-label="fig:sigma_mdot_M0_10_MJ"}](Fig4_newA_v3.pdf){width="\linewidth"}
![Accretion history of a planet of initial mass $M_{\rm p}(0) = 0.1 M_{\rm J}$ embedded at $r_{\rm p} = 10$ au (where $h = 0.054$) in a viscous disc of initial mass $M_{\rm disc} = 15.5 M_{\rm J}$. Transitions from Bondi accretion to Hill accretion ($M_{\rm thermal}$, equations \[eqn:bondi\]–\[eqn:hill\] and \[eqn:thermalmass\]), and from consumption to repulsion-dominated gaps ($M_{\rm repulsion,visc}$, equation \[eqn:repulsionmass\]), are indicated. An analytic estimate of the final planet mass is plotted as $M_{\rm final,visc}$ (equation \[eqn:asymsolvisc\]), computed assuming repulsion-dominated conditions (at $r_{\rm p} = 10$ au for this disc mass, conditions are actually intermediate between the repulsion and consumption limits, and so plotting equation \[eqn:consume1\] which assumes consumption-dominated conditions would give a similar result as equation \[eqn:asymsolvisc\]; see also Figure \[fig:m\_final\_r\_visc\]).[]{data-label="fig:Mp_vs_t_visc"}](Fig5_loglog_3lines.pdf){width="\linewidth"}
Planets in Inviscid Wind-Driven Discs {#sec:windy}
=====================================
Motivated by recent ALMA observations that point to little or no turbulence in protoplanetary discs (e.g., @p_etal_2016; @f_etal_2017), and by theoretical work arguing that discs are, for the most part, laminar because they are too cold and dusty to support magnetorotational turbulence (e.g., @g_1996; @pc_2011; @b_2011), we here turn away from the $\alpha$-based picture of turbulent and diffusive discs, and consider instead inviscid (zero viscosity) discs that accrete by virtue of magnetized winds (e.g., @b_etal_2016; @b_2016). We review how wind-driven discs work in section \[sec:wind\] and how planets open repulsive gaps in inviscid discs in section \[sec:inv\]. We then study how repulsion combines with consumption to set gap depths and planetary accretion rates, analytically in section \[sec:invoom\] and numerically in section \[sec:numwind\].
Wind-driven accretion discs {#sec:wind}
---------------------------
Inviscid, wind-driven accretion discs do not behave diffusively. Instead they are governed by simple advection: at every radius $r$, material moves inward with a vertically-averaged radial speed $u_r$ because it has lost angular momentum to a magnetized wind. The mass carried away by the wind itself is small compared to the mass advected inward through the disc (see Appendix \[sec:winds\], in particular the discussion below equation \[eqn:lambda\]). Then from continuity, including our planetary mass sink, $$\label{eqn:cont2}
\frac{\partial \Sigma}{\partial t} = \frac{1}{ r} \frac{\partial}{\partial r} \left( \Sigma r u_r \right) - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p}) \,.$$ In Appendix \[sec:winds\] we show how a wind-driven disc inspired by @b_etal_2016 and @b_2016 can have $u_r$ approximately constant ($<0$ for accretion). We utilize here, for simplicity, a constant $u_r \equiv c < 0$ model: $$\label{eqn:cont3}
\frac{\partial \Sigma}{\partial t} = \frac{c}{r} \frac{\partial}{\partial r} (\Sigma r) - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p}) \,.$$
It is instructive to examine the solution to (\[eqn:cont3\]) when $\dot{M}_{\rm p}=0$. The no-planet solution is separable: $$\label{eqn:sol}
\Sigma(r,t) = f(r) g(t) = \frac{M_{\rm disc}}{2\pi (ct_{\rm adv})^2} \frac{|c|t_{\rm adv}}{r} e^{-r/(|c|t_{\rm adv})} e^{-t/t_{\rm adv}}$$ for constants $M_{\rm disc}$ (the initial disc mass) and $t_{\rm adv}$, which we interpret as a disc radial advection time or drain-out time. For $c = -4$ cm/s (a value we relate to magnetic field parameters in Appendix \[sec:winds\]) and $t_{\rm adv} = 3$ Myr, the characteristic disc size is $|c|t_{\rm adv} \simeq 25$ AU, which seems reasonable. Equation (\[eqn:sol\]) resembles the @lp_1974 solution for a viscous disc which gives, for $\nu \propto r^1$, a surface density profile that scales as $r^{-1} \exp(-r/r_1)$ at fixed $t$ (equation \[eqn:simtime\]). This spatial resemblance is not surprising, as our viscous disc happens also to have an accretion velocity that is constant with radius: $|u_r| \sim r/t_\nu \sim \nu/r =$ constant. However, the solutions differ in their time behaviours; at fixed $r$, the wind-driven surface density decays exponentially as $\exp(-t/t_{\rm adv})$, whereas our viscous disc decays as a power law $t^{-3/2}$ (within viscously relaxed regions at small radii; @lp_1974 [@hcg_1998]). Viscous discs evolve more slowly because they conserve their total angular momentum; they can only drain away on the inside by redistributing their angular momentum to the outside in a kind of zero-sum game. Wind-driven discs are not so constrained; they lose their angular momentum wholesale to a wind, and so can dissipate more quickly.
We emphasize that $u_r = c$ is a vertically averaged, mass weighted, radial accretion velocity. In simulations by @bs_2013 of discs whose magneto-thermal winds are anchored at their electrically conductive surfaces, accretion actually occurs in a vertically thin, rarefied layer several scale heights above the midplane. The radial accretion velocity in this high-altitude layer is fast, on the order of the sound speed $c_{\rm s}$. The bulk of the mass of the disc, below this layer, is inert (see fig. 10 of @bs_2013). It is with this static and inviscid gas, extending from the midplane to a couple scale heights above and below, that the planet interacts, as we now describe.
Repulsion in inviscid discs {#sec:inv}
---------------------------
Without viscosity, disc gas in the vicinity of the planet depletes indefinitely, as it is repelled by the planetary Lindblad torque but cannot diffuse back. Under these conditions, @gc_2019a derived how the gas density at the center of the planet’s gap scales with elapsed time $t$, for a given planet-to-star mass ratio $m = M_{\rm p}/M_\star$ and disc aspect ratio $h = H/r$ (see the inviscid branch of their equation 17, and also their appendix): $$\begin{aligned}
\label{eqn:inv}
\frac{\Sigma_{\rm p}}{\Sigma_-} \sim h^{549/49} m^{-4} (\Omega t)^{-39/49} \equiv \widetilde{B}_{\rm inv}^{-1}\end{aligned}$$ where $\Omega$ is the orbital frequency of the planet, $\Sigma_{\rm p}$ is the surface density within the gap, and $\Sigma_-$ is the surface density downstream of the planet in the accretion flow (see Figure \[fig:sig\_cartoon\]). By construction, $t$ is the time over which the planet’s mass is close to its given value $m$ (say within a factor of 2). In practice, for inviscid discs where gaps deepen dramatically with increasing planet mass, the mass doubling time of a planet lengthens with each doubling, so $t$ is of order the system age.
Equation (\[eqn:inv\]) does not apply when $\widetilde{B}_{\rm inv} < 1$, i.e., when a repulsive gap has not yet been opened because not enough time has elapsed for a given planet mass. To account for this possibility, we generalize (\[eqn:inv\]) using $$\label{eqn:invgen}
\frac{\Sigma_{\rm p}}{\Sigma_-} \sim \frac{1}{1+\widetilde{B}_{\rm inv}}$$ by analogy with equation (\[eqn:oom1\]) for the viscous case. Note that $\widetilde{B}_{\rm inv}$ is dimensionless while its viscous counterpart $B$ has dimensions of viscosity.
Consumption and repulsion combined {#sec:invoom}
----------------------------------
We now assemble the physical ingredients laid out in sections \[sec:wind\] and \[sec:inv\] into a sketch of how consumption and repulsion combine in an inviscid, wind-driven disc. Following by analogy our analysis in section \[sec:oom\] for a viscous disc, we first write down mass conservation (see equation \[eqn:mass\] and Figure \[fig:sig\_cartoon\]): $$\begin{aligned}
\label{eqn:massinv}
\dot{M}_+ &= \dot{M}_- + \dot{M}_{\rm p} \nonumber \\
2\pi \Sigma_+ r |c| &= 2\pi \Sigma_- r |c| + \dot{M}_{\rm p} \nonumber \\
&= 2\pi \Sigma_- r |c| + A \Sigma_{\rm p} \end{aligned}$$ where in lieu of the viscosity we now have $r |c|$. After replacing $\Sigma_-$ in (\[eqn:massinv\]) using our momentum relation (\[eqn:invgen\]), we have $$\begin{aligned}
\Sigma_+ \sim (1+\widetilde{B}_{\rm inv})\Sigma_{\rm p} + \frac{A}{2\pi r|c|} \Sigma_{\rm p}\end{aligned}$$ which implies the outer gap contrast $$\begin{aligned}
\label{eqn:outer2}
\frac{\Sigma_{\rm p}}{\Sigma_+} \sim \frac{1}{1+A/(2\pi r|c|) + \widetilde{B}_{\rm inv}} \,.\end{aligned}$$ As in the viscous case (equation \[eqn:oom2\]), we see here that consumption ($A/(2\pi r|c|)$) and repulsion ($\widetilde{B}_{\rm inv}$) add. Taking $A$ to be the Bondi value (equation \[eqn:bondi\]) gives the ratio $$\begin{aligned}
\label{eqn:sensitive}
\frac{A_{\rm Bondi}/(2\pi r|c|)}{\widetilde{B}_{\rm inv}} \sim & \,\frac{0.5}{2\pi} \frac{h^{353/49}}{m^{2} (\Omega t)^{39/49}} \frac{\Omega r_{\rm p}}{|c|} \nonumber \\
\sim & \,0.04 \left( \frac{M_{\rm p}}{0.1 \, M_{\rm J}} \right)^{-2} \left( \frac{t}{3 \, {\rm Myr}} \right)^{-39/49} \times \nonumber \\
& \left( \frac{|c|}{4 \, {\rm cm/s}} \right)^{-1} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{489/196}\end{aligned}$$ which informs us that repulsion dominates consumption ($\widetilde{B}_{\rm inv} > A_{\rm Bondi}/(2\pi r|c|)$) when $$\begin{aligned}
M_{\rm p} > M_{\rm repulsion,inv} \sim & \, 0.02 \,M_{\rm J} \left( \frac{t}{3 \, {\rm Myr}} \right)^{-39/98} \times \nonumber \\
& \left( \frac{|c|}{4 \, {\rm cm/s}} \right)^{-1/2} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{489/392} \,.\end{aligned}$$ That repulsion dominates consumption even for small masses is in contrast to the viscous case (see equation \[eqn:repulsionmass\] for $M_{\rm repulsion,visc}$). Repulsion-dominated gaps are symmetric between the inner and outer discs (equations \[eqn:invgen\] and \[eqn:outer2\]): $$\begin{aligned}
\Sigma_{\rm p}/\Sigma_- & \sim \Sigma_{\rm p}/\Sigma_+ \sim 1/(1+\widetilde{B}_{\rm inv}) \nonumber \\
& \sim 2 \times 10^{-3} \left( \frac{h}{0.054} \right)^{549/49} \left( \frac{10^{-4}}{m} \right)^4 \left( \frac{3 \, {\rm Myr}}{t} \right)^{39/49} \label{eqn:symmetric}\end{aligned}$$ where for the last equality we have assumed that the gaps are deep ($\widetilde{B}_{\rm inv} > 1$). Under these conditions, we may estimate a final accreted planet mass by time-integrating $$\begin{aligned}
\dot{M}_{\rm p} &= A_{\rm Bondi} \Sigma_{\rm p} \nonumber \\
&\sim A_{\rm Bondi} \frac{\Sigma_+}{\widetilde{B}_{\rm inv}} \nonumber \\
& \sim \frac{A_{\rm Bondi}}{\widetilde{B}_{\rm inv}} \frac{M_{\rm disc}}{2\pi (ct_{\rm adv})^2} \frac{|c|t_{\rm adv}}{r_{\rm p}} e^{-r_{\rm p}/(|c|t_{\rm adv})} e^{-t/t_{\rm adv}} \label{eqn:int}\end{aligned}$$ from $t=0$ to $\infty$, where for $\Sigma_+$ we have employed the no-planet solution (\[eqn:sol\]). This last approximation is analogous to the one we made in (\[eqn:simtime\]) for a viscous disc. Equation (\[eqn:int\]) integrates to yield $$\begin{aligned}
\label{eqn:m_final_wind_full}
M_\mathrm{final,inv} & \sim \left[\frac{1.5}{2\pi} \, \Gamma\left(\frac{10}{49}\right) \left(\frac{M_{\rm disc}}{M_\star} \right) \left(\frac{r_{\rm p}}{ |c|t_{\rm adv}} \right) \right. \nonumber \\ & \times \left. h^{353/49}(\Omega t_{\rm adv})^{10/49} e^{-r_{\rm p}/(|c|t_{\rm adv})} \right]^{1/3} M_\star \nonumber \\
& \sim 0.3 \, M_{\rm J} \left( \frac{M_{\rm disc}}{15.5 \,M_{\rm J}} \right)^{1/3} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{163/196} e^{-r_{\rm p}/(3|c|t_{\rm adv})}\nonumber \\
& (\operatorname{repulsion-limited})\end{aligned}$$ where $\Gamma$ is the gamma function, and the numerical evaluation uses our fiducial parameters including $|c| = 4 $ cm/s, $t_{\rm adv} = 3$ Myr, and $M_\star = 1 M_\odot$. Our estimated final mass of $0.3 \, M_{\rm J}$ at $r_{\rm p} = 10$ au remains smaller than $M_{\rm thermal} \simeq 0.5 M_{\rm J}$ and so our use of $A_{\rm Bondi}$ is self-consistent.
Our expression (\[eqn:m\_final\_wind\_full\]) for $M_{\rm final,inv}$ resembles equation (19) of @gc_2019a; ours is an improvement as we have accounted explicitly for the transport properties of the disc through the radial velocity $c$ (see the discussion of transport-limited accretion in their section 4.1).
Numerical simulations {#sec:numwind}
---------------------
We test the ideas in section \[sec:invoom\] by numerically solving the continuity equation (\[eqn:cont3\]) and the momentum equation (\[eqn:invgen\]). To model the planetary mass sink in equation (\[eqn:cont3\]), we utilize the same sub-grid procedure of section \[sec:num\], replacing equation (\[eqn:subgrid\]) with $$\begin{aligned}
\label{eqn:m_dot_win}
\dot{M}_{\rm p}(t) = A \times \frac{\Sigma(r_{\rm p},t)}{1+ \widetilde{B}_{\rm inv}} \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\,
({\rm simulation})\end{aligned}$$ where $\Sigma(r_{\rm p},t)$ is the grid-level surface density in the bin containing the planet, and $A$ and $\widetilde{B}_{\rm inv}$ are given by equations (\[eqn:bondi\])–(\[eqn:hill\]) and (\[eqn:inv\]), respectively. The initial mass of the planet is set to $M_{\rm p}(0) = 0.1 \, M_{\rm J}$ (we will see that using smaller initial masses hardly changes the outcome). We solve the advective portion of equation (\[eqn:cont3\]) with a first-order upwind scheme (e.g., @ptv_2007) applied to a grid that extends from $r_{\rm in} = 0.01$ au to $r_{\rm out} = 500$ au across 300 cells uniformly spaced in $\log r$. We fix $c = -4$ cm/s and initialize the grid using (\[eqn:sol\]), with $t_{\rm adv} = 3$ Myr and $M_{\rm disc} = 15.5 M_{\rm J} = 0.015 M_\odot$, the same value chosen for our viscous disc calculations. Our timestep is set to $\Delta t = 0.2 \Delta r_{\mathrm{min}}/|c|$, where $\Delta r_\mathrm{min} = 3 \times 10^{-3} \, \mathrm{au}$ is our smallest bin width. Other disc properties such as $h(r)$ and $\Omega(r)$ are the same as before. For the outer boundary condition we impose a ghost cell just outside $r_{\rm out}$ where the surface density is fixed at 0.
Figure \[fig:m\_final\_wind\_tdisc\_a\] (the inviscid counterpart to Figure \[fig:sigma\_detailed\]) shows $\Sigma(r)$ at $t = t_{\rm adv}$ when $M_{\rm p}$ has grown to $0.3 \, M_{\rm J}$, illustrating many of the features anticipated from our analytic treatment. Without a planet, the surface density profile follows $r^{-1} \exp[-r/(|c|t_{\rm adv})]$ as expected from equation (\[eqn:sol\]). With a planet, a gap is created that is nearly symmetric between the inner and outer discs, and whose depth is dominated by Lindblad repulsion (enforced by our sub-grid scheme), not consumption (equation \[eqn:symmetric\]). The inviscid gap is deep (scaling as $m^{-4}$; @gs_2018; @gc_2019a; see also @d_2020). Figure \[fig:m\_final\_wind\_tdisc\_b\] (analogous to Figure \[fig:sigma\_mdot\_M0\_10\_MJ\]) provides snapshots of $\Sigma(r)$ and $\dot{M}_{\rm disc}(r)$ taken at different times, and Figure \[fig:sigma\_mdot\_M0\_wind\] (analogous to Figure \[fig:Mp\_vs\_t\_visc\]) plots $M_{\rm p}(t)$. Unlike in a viscous disc, our example planet in an inviscid disc does not consume most of the disc mass exterior to its orbit; the disc accretion rate profile $\dot{M}_{\rm disc}(r)$ is not much affected by the planet except during an initial transient phase at $t < t_{\rm adv}$. We see a need for a high radial accretion velocity $|u_r|$ within the gap (see also section \[sec:nc10au\]): to ensure that $\dot{M}_{\rm disc}$ grades smoothly across the gap as shown in Figure \[fig:m\_final\_wind\_tdisc\_b\], $|u_r|$ must increase in proportion to the gap contrast $\Sigma/\Sigma_{\rm p}$. Inviscid gap contrasts are on the order of $10^5$, and so $|u_r| \sim 10^5 |c| \sim 4$ km/s, comparable to the orbital velocity. Note that simulations of planets in inviscid discs have not reproduced the deep gaps expected from our analytics, finding gap contrasts only up to a factor of $\sim$10 (e.g., @fc_2017 [@mnp_2019; @mnp_2020]). On the one hand the simulations are of limited duration and so their gaps may not have fully developed; on the other hand, the simulations allow for orbital migration and hydrodynamical instabilities, effects which may prevent gaps from becoming too deep in reality.
That the disc accretion flow proceeds largely unimpeded from outside to inside the planet’s orbit is a consequence of the gap being repulsion-dominated (equation \[eqn:mdot-0\], with $\nu$ replaced by $r|c|$). The planet diverts such a small fraction of the disc flow that it grows from $0.1 \, M_{\rm J}$ to only $0.3 \, M_{\rm J}$; most of the original $15.5 \, M_{\rm J}$ contained in the disc drains onto the star. Figure \[fig:sigma\_mdot\_M0\_wind\] also shows that reducing the initial seed mass to $M_{\rm p}(0) = 0.01 M_{\rm J}$ hardly affect the final mass.
![How the surface density profile of an inviscid disc responds to a planet that consumes disc gas and repels gas away by Lindblad torques. The planet, located at $r_{\rm p} = 10$ au, freely accretes starting from a seed mass of $0.1 M_{\rm J}$; the $\Sigma$ profile shown here is taken at a time $t = t_{\rm adv} = 3$ Myr, when the planet has grown to $\sim$$0.3 M_{\rm J}$ (see also Figure \[fig:sigma\_mdot\_M0\_wind\]). As is the case throughout this paper, the planet’s gap is not spatially resolved, but is modeled as a single cell. The “true” surface density inside this cell equals the grid-level $\Sigma$ lowered by a factor of $\widetilde{B}_{\rm inv}$, whose magnitude is given by the red double-tipped arrow. The gap is repulsion and not consumption dominated ($\widetilde{B}_{\rm inv} > A/(2\pi r|c|)$, equation \[eqn:sensitive\]); as such, the gap is symmetric in the sense that the surface density contrast with the outer disc is practically the same as with the inner disc. This figure is the inviscid counterpart to Figure \[fig:sigma\_detailed\] which was made for a viscous disc.[]{data-label="fig:m_final_wind_tdisc_a"}](Fig6_newA.pdf){width="\linewidth"}
![Snapshots of the surface density profile $\Sigma(r)$ and disc accretion rate $\dot{M}_{\rm disc}(r) = -2 \pi \Sigma u_r r$ ($> 0$ for accretion toward the star) for a planet embedded in an inviscid, wind-driven disc. The planet mass is allowed to freely grow starting from $M_{\rm p}(0) = 0.1 \,M_{\rm J}$; the masses corresponding to the plotted times are $0.27\, M_{\rm J}$ ($t = 0.1\, t_{\rm adv} = 0.3$ Myr) and $0.34 \,M_{\rm J}$ ($t = 3 \,t_{\rm adv} = 9$ Myr; see also Figure \[fig:sigma\_mdot\_M0\_wind\]). At $t=3\,t_{\rm adv}$, the disc has relaxed into a quasi-steady state in the presence of the planetary mass sink, and $\dot{M}_{\rm disc}(r)$ looks essentially the same as it would without the planet; the accretion rate onto the planet is negligible compared to the disc accretion rate—the gap is repulsion-dominated—and so the disc is not materially affected. Even at $t = 0.1\,t_{\rm adv}$, the interior surface density $\Sigma_-$ and $\dot{M}_{\rm disc}$ depress by only $\sim$15% because of consumption.[]{data-label="fig:m_final_wind_tdisc_b"}](Fig7_newA.pdf){width="\linewidth"}
![Mass evolution of a planet embedded at $r_{\rm p} = 10$ au in an inviscid but still accreting disc of initial mass $M_{\rm disc} = 15.5 \,M_{\rm J}$. Within $\sim$1 disc advection time $t_{\rm adv}$, the planet, whose gap is repulsion-dominated ($\widetilde{B}_{\rm inv} > A_{\rm Bondi}/(2\pi r_{\rm p}|c|)$), grows to a mass of $\sim$$0.35 \, M_{\rm J}$. The final planet mass varies by only $\sim$10% when the initial seed mass $M_{\rm p}(0)$ varies by a factor of 10. This figure is the inviscid counterpart to Figure \[fig:Mp\_vs\_t\_visc\] which was made for a viscous disc.[]{data-label="fig:sigma_mdot_M0_wind"}](Fig8_newA.pdf){width="\linewidth"}
Summary and Discussion {#sec:sum}
======================
Planets open gaps in circumstellar discs in two ways: by repelling material away via Lindblad torques, and by consuming local disc gas. Measured relative to the disc outside the planet’s orbit, the two effects are additive: both repulsion and consumption add to deepen the planet’s gap relative to the outer disc (see equation \[eqn:oom2\] or \[eqn:outer2\]). Relative to the inner disc, downstream of the mass sink presented by the planet, the gap surface density contrast is set by repulsion only (see equation \[eqn:oom1\] or \[eqn:invgen\]).
Many planet formation studies (e.g., @tt_2016; @lee_2019) take the planet’s hydrodynamically-limited accretion rate $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc})$, where $\dot{M}_{\rm hydro}$ is the planetary accretion rate computed according to the hydrodynamics of flows in the immediate vicinity of the planet, and $\dot{M}_{\rm disc}$ is the local disc accretion rate (the mass crossing the planet’s orbital radius, per time). Prescribing the planet’s accretion rate in this way is equivalent to comparing consumption, as measured by the “consumption coefficient” $A \equiv \dot{M}_{\rm p}/\Sigma_{\rm p}$, where $\Sigma_{\rm p}$ is the surface density inside the gap, and repulsion, as measured by the “repulsion coefficient” $B \equiv T/(\Sigma_{\rm p} \Omega r^2)$, where $T$ is the repulsive planetary torque and $\Omega r^2$ is the angular momentum per unit mass (see also @tt_2016 and @tmt_2020 who use the same framework). Under consumption-limited conditions ($A/(3\pi) > B$), the planet’s accretion rate saturates to nearly the disc’s accretion rate: $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc}) = \dot{M}_{\rm disc}$. Otherwise, under repulsion-limited conditions ($A/(3\pi) < B$), $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc}) = \dot{M}_{\rm hydro}$.
Final planet masses {#sec:final}
-------------------
In conventional viscous discs with large enough $\alpha$-diffusivities[^3] and our assumed parameters, planets begin their growth under consumption-dominated conditions and possibly continue their growth under repulsion-dominated conditions, arriving at final masses well in excess of a Jupiter. We show in Figure \[fig:m\_final\_r\_visc\] the final mass of a planet embedded in an $\alpha = 10^{-3}$ disc, as a function of the planet’s orbital distance $r_{\rm p}$, computed using our numerical code of sections \[sec:consumption\]–\[sec:mass\]. Final planet masses increase gradually from $4 \, M_{\rm J}$ at 1 au, to $8\,M_{\rm J}$ at 30 au, in a disc of initial mass $M_{\rm disc} = 15.5 M_{\rm J}= 0.015 M_\odot $. In a disc $5\times$ more massive, the corresponding range of planet masses is 9–$20 \, M_{\rm J}$. The final masses are not sensitive to $\alpha$ insofar as $\alpha$ controls only the timescale over which the disc evolves (modulo disc dispersal by some other means, e.g., photoevaporation; see @tmt_2020). Final masses do depend on the initial mass of the disc, scaling as $M_{\rm disc}^{3/7}$ under repulsion-dominated conditions (equation \[eqn:asymsolvisc\]) and $M_{\rm disc}^1$ under consumption-dominated conditions (equation \[eqn:consume1\] or \[eqn:consume2\]). The trend of final planet mass with distance shown in Figure \[fig:m\_final\_r\_visc\] follows, for the most part, the trend predicted for repulsion-limited conditions, except at large $r_{\rm p}$ where consumption dominates. The final mass profiles in Figure \[fig:m\_final\_r\_visc\] recall those of the super-Jupiters in the HR 8799 system; the four planets, located between 15 and 70 AU of their host star, have practically the same mass, about 6–7 $M_{\rm J}$ (@wgd_2018).
![Final planet masses grown from viscous discs having $\alpha =
10^{-3}$ and varying total mass (top vs. bottom panels). Planet masses are initialized at $0.1 \,M_{\rm J}$ and grown using the 1D numerical code of section \[sec:mass\], which utilizes the repulsive gap contrast of Kanagawa et al. ([-@kmt_2015]; see also @dm_2013 and @fsc_2014) and gas accretion that switches from Bondi to Hill at the thermal mass. Points are plotted at $t = 50\,t_1 = 85$ Myr, where $t_{\rm 1} = r_1^2/[3\nu(r_1)]$ is the viscous diffusion time at $r_1 = 30$ au. Analytic curves are given by equation (\[eqn:fullsolvisc\]) for the repulsion limit (dashed blue), and equations (\[eqn:consume1\])–(\[eqn:consume2\]) for the consumption limit (dotted orange), also evaluated at $t = 50 t_1$. At most orbital distances, planet mass growth is limited by repulsion-dominated gaps; only at the largest distances, where the disc aspect ratio is large, are gaps relatively harder to open and conditions remain consumption-limited. The analytics, which are derived assuming the planet mass is small compared to the disc mass, are a better guide for the more massive disc in the bottom panel.[]{data-label="fig:m_final_r_visc"}](Fig9_two_panel_50_ts_v2.pdf){width="\linewidth"}
Initially and everywhere in a viscous disc, a planet, despite opening a gap, consumes practically all of the disc gas that tries to diffuse past its orbit (equation \[eq:m\_dot\_ratio\] with $A/(3\pi) > B > \nu$, where $\nu$ is the disc viscosity). This consumption-limited behaviour persists up to a repulsion mass $M_{\rm repulsion,visc}
\simeq 5 \, M_{\rm J} \,[r_{\rm p}/(10 \, {\rm au})]^{9/16}$ (equation \[eqn:repulsionmass\]), above which repulsion dominates. The repulsion mass is not the thermal mass $M_{\rm thermal}$ (equation \[eqn:thermalmass\]), but exceeds it by a factor of $\sim$$h^{-3/4}$, where $h$ is the disc aspect ratio. Growth continues more slowly at $M_{\rm p} > M_{\rm repulsion,visc}$, with the planet mass increasing beyond $M_{\rm repulsion,visc}$ by up to a factor of $\sim$4 for our parameter choices.
Equation (\[eqn:fullsolvisc\]) gives an approximate analytic expression for the planet mass vs. time during this final repulsion-limited stage. It predicts that planet masses are of order $10 \,M_{\rm J}$ by the time the disc dissipates. This result is derived by assuming the planet accretes at a rate that scales as $A_{\rm Hill} = 2.2 \, \Omega r^2 \, m^{2/3}$, where $m$ is the planet-to-star mass ratio; this prescription is commonly adopted by hydrodynamical simulations of planet-disc interactions, and might be appropriate for super-thermal masses. If instead of $A_{\rm Hill}$ we use the empirical formula $A_{\rm TW} = 0.29 \, \Omega r^2 m^{4/3}/h^2$ drawn from 2D numerical simulations by @tw_2002, then the mass above which repulsion dominates changes to $M_{\rm repulsion,visc,TW} \simeq 9 \, M_{\rm J} \,[r / (10 \, {\rm au})]^{3/8}$, nearly twice the value of $M_{\rm repulsion,visc}$ derived using the Hill scaling. Using $A_{\rm TW}$ leads to a more extended consumption-dominated growth phase, and final planet masses larger by order-unity factors compared to those of the solid curves in Figure \[fig:m\_final\_r\_visc\]. Overall, it appears that in viscous discs, planets accrete a not-small fraction of the disc mass, which can be many tens of Jupiter masses (@tab_2017, their fig. 10; see also @pmp_2019). This is in agreement with @tmt_2020, who limit giant planet growth by incorporating photoevaporative mass loss from the disc.
In inviscid discs, conditions tend to be repulsion-dominated even at low planet masses. Without viscosity or turbulent transport to compete against, planetary Lindblad torques carve deep gaps that are repulsion-dominated even for sub-thermal planets accreting at the Bondi rate (equation \[eqn:sensitive\]). Repulsion-dominated gaps are symmetric in the sense that gap contrasts between the outer and inner discs are the same; accordingly, disc accretion rates are nearly continuous across the gap (e.g., Figure \[fig:m\_final\_wind\_tdisc\_b\]), which means that most of the disc mass is not diverted onto the planet (in the language of @tt_2016, $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc}) = \dot{M}_{\rm hydro}$). Maintaining the disc accretion rate across a gap demands that the radial accretion velocity within the gap be as large as the gap is deep. Whether such fast inflows are possible, and whether inviscid gaps can be as deep as expected from our analytics (cf. numerical simulations that find only shallow gaps; @fc_2017 [@mnp_2019; @mnp_2020]), are unresolved issues.
Figure \[fig:m\_final\_r\_wind\], analogous to Figure \[fig:m\_final\_r\_visc\], shows that final planet masses in our model inviscid discs range between $\sim$0.05 and $1\, M_{\rm J}$, more than an order-of-magnitude smaller than their viscous disc counterparts. For the most part, the masses computed for inviscid discs using our numerical 1D code are well reproduced by equation (\[eqn:m\_final\_wind\_full\]), derived in the repulsion limit. This formula, which predicts that final planet masses scale as $M_{\rm disc}^{1/3}$ and $r_{\rm p}^{163/196} \simeq r_{\rm p}^{0.83}$, is similar to that derived by Ginzburg & Chiang ([-@gc_2019a], their equation 19),[^4] and improves upon it by accounting for the structure and transport properties of the parent disc—specifically how the disc may accrete by shedding angular momentum through a magnetized surface wind (e.g., @b_2016).
Orbital migration in viscous discs has been shown in numerical simulations to enhance $\dot{M}_{\rm p}$ relative to the migration-free case [e.g., @dk_2017]. Including migration would only amplify our finding that final planet masses in viscous discs are large, approaching if not well within the regime of brown dwarfs. Accretion rates should also increase for planets migrating in inviscid, wind-driven discs;in 3D, strongly sub-thermal planets have been shown to migrate inward (@mnp_2020). We may need such enhancements in $\dot{M}_{\rm p}$ to explain, within an inviscid scenario, giant planets like our own Jupiter, i.e., to bring planet masses up to $1 M_{\rm J}$ at distances of 1–10 au (Figure \[fig:m\_final\_r\_wind\]). On the other hand, sub-Jupiter masses, down to $\sim$0.1 $M_{\rm J}$ in many cases, are inferred from ALMA observations of disc gaps (@zzh_2018), and suggest that planets there are strongly repelling inviscid gas.
The asymmetric gap we computed for the viscous disc model in Figure \[fig:sigma\_detailed\] suggests a strong, mostly one-sided migration torque forcing the planet inward. However, this is misleading because our numerical procedure does not spatially resolve the gap, whose true radial width lies between $H$ (the pressure scale height) and $r_{\rm p}$ (@gs_2018). Most of the migration torque is exerted by disc gas on the bottoms of gaps, displaced radially from the planet by $\sim$$\pm H$, and here the actual surface density gradients, and of course the surface density itself, are small (see also @kts_2018).
![Final planet masses in an inviscid, wind-driven disc of varying mass (top vs. bottom panels). Planet masses are initialized at $0.01 M_{\rm J}$ and grown using the 1D numerical code of section \[sec:numwind\], which uses the time-dependent gap contrast of @gc_2019a to model repulsion, in a purely advective disc whose height-averaged radial accretion velocity is $c = -4$ cm/s and exponential drain-out time is $t_{\rm adv} = 3$ Myr. Points are plotted after $5 t_{\rm adv} = 15$ Myr. They mostly respect equation (\[eqn:m\_final\_wind\_full\]), which gives final planet masses grown in repulsion-limited and deep ($\widetilde{B}_{\rm inv}$ > 1) gaps (dashed curve not including the drop-off at the largest distances). At $r_{\rm p} \sim 100$ au, the disc has such low density that the planet’s initial growth timescale $M_\mathrm{p}/\dot{M}_\mathrm{p}$ is comparable to $t_\mathrm{adv}$; here there are not many doublings before the disc drains away. In this regime the planet does not open a substantial gap ($\widetilde{B}_\mathrm{inv} < 1$) and its final mass can be estimated analytically by integrating $\dot{M}_\mathrm{p} = A_\mathrm{Bondi} \Sigma$ with $\Sigma$ given by the no-planet solution (\[eqn:sol\]); the dashed curve is the minimum of the resulting expression (not shown) and . []{data-label="fig:m_final_r_wind"}](Fig10_newA.pdf){width="\linewidth"}
Transitional discs {#sec:transitional}
------------------
We have shown how a planet accreting from its parent disc can change the disc’s entire complexion. This make-over is most evident for a planet that siphons away most of the disc’s accretion flow—as it can in a viscously diffusing disc—carving out a consumption-limited gap that divides a gas-rich outer disc with surface density $\Sigma_+$ from a gas-poor inner one with surface density $\Sigma_-$. Transitional discs have just such an outer/inner structure (e.g., @emj_2014; @dvh_2017), suggesting that they represent viscous discs whose inner regions are cleared by accreting planets (with dust filtration at the outer gap edge, and grain growth in the inner disc, enhancing the surface density contrast in dust over gas; @drz_2012; @znd_2012).
In a viscous disc, a single accreting planet suffices to deplete the entire disc interior to its orbit. The 2D single-planet simulations of @znh_2011 bear this out; they find an outer vs. inner disc contrast of $\Sigma_+/\Sigma_- \sim 10$ for a $1 \, M_{\rm J}$ planet that accretes at the Hill rate from a disc of $h \simeq 0.05$ (their fig. 1, model P1A1). This numerical result agrees with our analytic theory, which predicts according to equations (\[eqn:mdot-0\]) and (\[eq:A\_B\_visc\_Hill\]) that $$\begin{aligned}
\frac{\Sigma_+}{\Sigma_-} \simeq \frac{A_{\rm Hill}}{3\pi B} \simeq 7 \left( \frac{M_{\rm p}}{M_{\rm J}} \right)^{-4/3} \left( \frac{h}{0.05} \right)^3\end{aligned}$$ for a consumption-dominated and deep gap with $A_{\rm Hill}/(3\pi) > B > \nu$, where $\nu$ is the disc viscosity. In steady state, $\Sigma_+/\Sigma_- = \dot{M}_+/\dot{M}_-$, the ratio of outer-to-inner disc accretion rates. A value of $\dot{M}_+/\dot{M}_- \sim 10$, as we have found for the above parameters, accords with the observation that the median accretion rate for stars hosting transitional discs is lower than that of stars hosting non-transitional discs by a factor of $\sim$10 (@nsm_2007; @kwm_2013). However, the corresponding factor-of-10 reduction in $\Sigma$ seems too small to match observed gas depletions in transitional disc cavities. In the disc studied in CO by @dvh_2017, the gas surface density declines by $\sim$$10^3$ from $r = 70$ au to 15 au. As recognized by Zhu et al. ([-@znh_2011]; see also @o_2016), it is a challenge to simultaneously explain how disc inner cavities can be strongly depleted in density while their central stars continue to accrete at near-normal rates. This challenge seems more easily met in the repulsion limit, where deep gaps are carved by planets which alter the disc accretion flow only modestly—assuming radial accretion velocities within the gap are large enough to maintain mass transport rates across it.
The repulsion limit is attained in viscous discs by planets having $M > M_{\rm repulsion,visc} \simeq 5.3 \, M_{\rm J}\, [r/(10\,{\rm au})]^{9/16}$, or in inviscid discs by planets having $M > M_{\rm repulsion,inv} \simeq 0.02 \,M_{\rm J}
[r/(10 \,{\rm au})]^{489/392}$. In both cases, multiple planets with adjoining gaps would be required to evacuate transition disc cavities spanning decades in radius—more planets in a viscous scenario where each gap has a radial width closer to $H$, and fewer in an inviscid scenario where each gap is of order $r_{\rm p} > H$ wide (@gs_2018; note that widths are not captured by our single-grid-point treatment of gaps). The inviscid picture requires only super-Earth masses and appeals more, insofar as observations seem to have already ruled out transitional discs containing families of super-Jupiters as required in the viscous scenario. Inviscid discs can still accrete, either by virtue of magnetized winds (@b_2016; @wg_2017), or by the repulsive torques of their embedded planets (@gr_2001; @sg_2004; @fc_2017).
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Xuening Bai, Jeffrey Fung, Willy Kley, Eve Lee, James Owen, and Hidekazu Tanaka for helpful exchanges. An anonymous referee provided an encouraging report. EC acknowledges NASA grants 80NSSC19K0506 and NNX15AD95G/NEXSS. SG is supported by the Heising-Simons Foundation through a 51 Pegasi b Fellowship. MMR and RMC acknowledge support from NSF CAREER grant number AST-1555385.
Data availability {#data-availability .unnumbered}
=================
The code underlying this article will be shared on reasonable request to the corresponding author.
Analytic Steady-State Solution for $\Sigma$ and $\dot{M}$ For Viscous Disc with Planet {#sec:analy_ss}
=======================================================================================
In this appendix we provide an analytic expression for the surface density profile of a disc with an embedded planet. Our derivation here is more careful than our order-of-magnitude sketch in section \[sec:oom\], and similar to that presented in Lubow & D’Angelo ([-@ld_2006], their section 2.4), with a couple of differences: we reduce the surface density at the planet’s location by a factor $1 + B/\nu$ to account for repulsive Lindblad torques (see section \[sec:oom\]), and we express our solution in terms of the surface density at infinity as opposed to the surface density at the planet’s location.
Using the same notation as in section \[sec:oom\], and neglecting for the moment the Lindblad torque, the equations of mass and angular momentum conservation with a mass sink at $r = r_\mathrm{p}$ read $$\begin{aligned}
\frac{1}{r}\frac{d\left( \mu u_r r / \nu\right)}{dr} & =-\frac{\dot{M}_\mathrm{p}}{2\pi r}\delta\left(r-r_\mathrm{p}\right) \label{eq:mass_cons_ss}\\
\frac{r^{2}\Omega \mu u_r}{\nu} & \label{eq:ang_cons_ss} =-\frac{d}{dr}\left(3\mu\Omega r^{2}\right)\end{aligned}$$ where $u_r$ is the radial velocity and $\mu \equiv \Sigma \nu$. Equation indicates that the mass flow rate $\dot{M}_+ = -2\pi \mu_+ u_r r/\nu$ is spatially constant in regions exterior to the planet’s orbit (the outer disc), and likewise for $\dot{M}_-$ in regions interior to the planet’s orbit (the inner disc): $$\begin{aligned}
\label{eq:m_dot_cons}
\dot{M}_- = \dot{M}_+ - A \frac{\mu_\mathrm{p}}{\nu_\mathrm{p}}\end{aligned}$$ where we have used $\dot{M}_{\rm p} = A \mu_{\rm p}/\nu_{\rm p}$ and $\nu_\mathrm{p} \equiv \nu \left( r_\mathrm{p} \right)$. Since $\dot{M}_-$ and $\dot{M}_+$ are constants, equation can be solved to yield $$\begin{aligned}
3 \pi \mu_\pm(r) = \dot{M_\pm} + \frac{C_\pm}{\sqrt{r}}\end{aligned}$$ where $C_\pm$ are integration constants. For the inner disc we use the boundary condition $\mu_-(r_\star) = 0$, whence $$\begin{aligned}
\label{eq:mu_sol}
3\pi\mu_{-}\left(r\right)=\dot{M}_{-}\left(1-\sqrt{\frac{r_\star}{r}}\right) \,.\end{aligned}$$ Following our treatment in the main text, we encode the planetary gap caused by Lindblad torques at a sub-grid level, i.e., we force the surface density at the planet’s location to be depleted relative to the surface density just interior to the planet according to $$\begin{aligned}
\label{eq:inner_mu_p}
\mu_{\rm p} = \mu_-(r_{\rm p}) \left( 1 + B/\nu_{\rm p} \right)^{-1}\end{aligned}$$ where subscript $p$ denotes the planet’s location. For the outer disc, we fix the surface density at infinity, $\mu(\infty) = \mu_\infty$, so that $$\dot{M}_+ = 3 \pi \mu_\infty \,.$$ Then from equations (\[eq:m\_dot\_cons\]), (\[eq:mu\_sol\]), and (\[eq:inner\_mu\_p\]) we have $$\begin{aligned}
\label{eq:ss_mup}
\frac{\mu_\mathrm{p}}{\mu_\infty}=\frac{1}{\frac{1}{1-\sqrt{r_\star/r_\mathrm{p}}} + \left( \frac{A}{3 \pi} + \frac{B}{1-\sqrt{r_\star/r_\mathrm{p}}}\right)/\nu_\mathrm{p}}\end{aligned}$$ which can be compared to equation (\[eqn:oom2\]). We may also solve for $$\begin{aligned}
\frac{\dot{M}_\mathrm{p}}{\dot{M}_+} &= \frac{A/(3\pi \nu_\mathrm{p})}{\frac{1}{1-\sqrt{r_\star/r_\mathrm{p}}} + \left(\frac{A}{3 \pi} + \frac{B}{1-\sqrt{r_\star/r_\mathrm{p}}}\right)/\nu_\mathrm{p}}\\
\frac{\dot{M}_-}{\dot{M}_+} &= \frac{1 + B/\nu_\mathrm{p}}{1 + \left[ \frac{A}{3 \pi} \left(1-\sqrt{\frac{r_\star}{r_\mathrm{p}}}\right) + B \right]/\nu_\mathrm{p}}\end{aligned}$$ which can be compared to equations (\[eq:m\_dot\_ratio\]) and (\[eqn:mdot-0\]). Finally, stitching the outer disc solution to the inner disc solution implies $\mu_+(r_{\rm p}) = \mu_-(r_{\rm p}) = \mu_{\rm p} (1+B/\nu_{\rm p})$ and $$\begin{aligned}
\label{eq:plus}
3 \pi \mu_+(r) = 3\pi \mu_\infty \left[1 - \sqrt{\frac{r_\mathrm{p}}{r}} \left(1-\frac{\mu_\mathrm{p}\left( 1 + B/\nu_\mathrm{p} \right)}{\mu_\infty} \right) \right] \,.\end{aligned}$$ The equations above mirror the results in section \[sec:oom\], with the addition of a factor of $3 \pi$ (see section \[sec:num\]) and the factor of $1-\sqrt{r_\star/r_\mathrm{p}}$ which accounts for the star’s ability to divert material from the planet.
In Figure \[fig:sig\_analy\_sol\] we plot equations (\[eq:mu\_sol\]), (\[eq:inner\_mu\_p\]) and (\[eq:plus\]), adopting parameters as close as possible to those used in the top panel of Figure \[fig:sigma\_detailed\] so that we may compare the numerical result there to the analytic result here (see caption to Figure \[fig:sig\_analy\_sol\] for details).
![Analytic solution (black solid curve) for the surface density profile of a viscous disc perturbed by a planet, as given by equations (\[eq:mu\_sol\]), (\[eq:inner\_mu\_p\]) and (\[eq:plus\]), using parameters as close as possible to those used in the top panel of Figure \[fig:sigma\_detailed\], whose numerical result is overlaid here for comparison (orange dashed curve). For our analytic unperturbed “no planet” disc (blue dashed curve) we use a power law of slope -1 and normalization at 1 au equal to the corresponding “no planet” curve in Figure \[fig:sigma\_detailed\]. The $A$ and $B$ coefficients are taken from equations and for $M_\mathrm{p} = 0.3 \,M_\mathrm{J}$. The differences between the analytic and numeric curves mainly arise from the behaviour of the outermost disc near the turn-around “transition radius” (@lp_1974; @hcg_1998). This transition radius, which varies with time, does not appear in our steady-state solution.[]{data-label="fig:sig_analy_sol"}](ss_cons_sol_v4.pdf){width="\linewidth"}
Magnetized winds and disc accretion {#sec:winds}
===================================
We motivate here our simple, constant accretion velocity model for a wind-driven disc using the numerical simulations of Bai and collaborators. From continuity (equations 1, 6, and 9 of @b_2016), $$\begin{aligned}
\label{continuity}
\frac{\partial \Sigma}{\partial t} &= +\frac{1}{2\pi r} \frac{\partial
\dot{M}_{\rm disc}} {\partial r} - \frac{1}{2\pi r} \frac{\partial
\dot{M}_{\rm wind}}{\partial r} \nonumber \\
&= + \frac{1}{2\pi r}
\frac{\partial}{\partial r} \left[ 2(\lambda-1) r \frac{\partial \dot{M}_{\rm wind}}{\partial r} \right]
- \frac{1}{2\pi r} \frac{\partial
\dot{M}_{\rm wind}}{\partial r} \end{aligned}$$ where $$\begin{aligned}
\dot{M}_{\rm wind} (r) = \int^r \frac{\partial \dot{M}_{\rm
wind}}{\partial r} dr\end{aligned}$$ is the cumulative rate at which mass is carried to infinity by the wind (integrated over the disc within $r$). From equation (20) of @b_etal_2016, $$\label{eqn:rhov}
\frac{\partial \dot{M}_{\rm wind}}{\partial r} = 2\pi r \rho_0 u_{\rm p0}$$ where $\rho_0$ and $u_{\rm p0}$ are the volumetric mass density and poloidal velocity of the wind where it is launched, near the disc surface. All quantities subscripted with 0 are evaluated at the wind base $(r_0,z_0)$.
The disc accretion rate $$\label{eqn:usual}
\dot{M}_{\rm disc} \equiv -2\pi \Sigma r u_r = 2(\lambda-1) r \frac{\partial \dot{M}_{\rm wind}}{\partial r}$$ for surface density $\Sigma$ and radial velocity $u_r$ is identical in definition to the variable $\dot{M}_{\rm disc}$ used throughout our paper. Unlike $\dot{M}_{\rm wind}$, $\dot{M}_{\rm disc}$ is not a cumulative quantity, but measures the mass crossing a circle of radius $r$ per unit time, and uses a sign convention such that $\dot{M}_{\rm disc} > 0$ for $u_r <0$.
Disc accretion by a wind hinges on the “magnetic lever arm” $$\lambda = (r_{\rm A}/r_0)^2 \label{eqn:lambda}$$ where $r_{\rm A}$ is the Alfvén radius for the wind streamline running through $r_0$. A lever arm $\lambda > 1$ enables $\dot{M}_{\rm disc} > 0$ by having the wind carry away more specific angular momentum than the Keplerian disc has at $r_0$. The fiducial wind model of Bai ([-@b_2016], their fig. 2) has $(\lambda-1)$ ranging from $\sim$30 at $r = 0.3$ AU to $\sim$2 at 30 AU; therefore the first term in (\[continuity\]) dominates the second term by a factor of order $2(\lambda-1) \sim 4$–60. Only the first term is modeled in our paper.
@b_2016 and the magnetized disc wind literature dating back to @bp_1982 parameterize the wind mass-loss rate in terms of the dimensionless mass loading parameter $$\mu = \frac{\omega r_0}{B_{\rm p0}} \times k = \frac{\omega
r_0}{B_{\rm p0}} \times \frac{4\pi \rho u_{\rm p}}{B_{\rm p}}$$ where $k$ is the ratio of poloidal mass flux to poloidal field strength $B_{\rm p}$ ($k$ is constant along a magnetic field line), and $\omega$ is the angular velocity of a field line, approximately equal to the Keplerian frequency $\Omega_{\rm K}$ at $r_0$. Note that $\mu$ (not to be confused with $\mu$ in Appendix \[sec:analy\_ss\]) varies with $r$ from field line to field line. Evaluating $\mu$ at the wind base, we rewrite (\[eqn:rhov\]) as $$\frac{\partial \dot{M}_{\rm wind}}{\partial r} = \frac{\mu B_{\rm p0}^2}{2\omega}$$ (@b_etal_2016, equation 21). Now parameterize $B_{\rm p0}$ in terms of the midplane plasma beta: $$\beta_0 = \frac{8\pi}{B_{\rm p0}^2} \frac{\Sigma k_{\rm B} T}{\sqrt{2\pi} \overline{m} H }$$ where $k_{\rm B}$ is Boltzmann’s constant, $T$ is the disc temperature, $H = c_{\rm s}/\Omega_{\rm K}$ is the disc scale height, $c_{\rm s} = \sqrt{k_{\rm B}T/\overline{m}}$ is the gas sound speed, and $\overline{m}$ is the mean molecular weight. Then $$\label{eight}
\frac{\partial \dot{M}_{\rm wind}}{\partial r} = \frac{\sqrt{8\pi}k_{\rm B}}{\beta_0 \overline{m}} \frac{T}{H} \frac{\mu \Sigma}{\omega} = \frac{\sqrt{8\pi k_{\rm B}}}{\sqrt{\overline{m}}\beta_0} T^{1/2} \mu \Sigma \,.$$ Combine (\[eight\]) and (\[eqn:usual\]) to find $$\label{eqn:end}
u_r = -\sqrt{\frac{8}{\pi}} \frac{\sqrt{k_{\rm B}}}{\sqrt{\overline{m}}\beta_0} T^{1/2} \mu (\lambda-1) \sim -\frac{\mu(\lambda-1)}{\beta_0} c_{\rm s} \,.$$ In the fiducial model of Bai ([-@b_2016], see their fig. 2), $\mu$ increases from $\sim$0.06 at $r = 0.3$ AU to $\sim$4 at 30 AU, and $(\lambda-1)$ decreases from $\sim$30 to $\sim$2 over the same range; therefore the product $\mu(\lambda-1)$ increases from $\sim$2 to $\sim$8, scaling roughly as $r^{0.3}$. Their model temperature scales as $T \propto r^{-1/2}$; therefore the combination $T^{1/2}\mu (\lambda-1)$ is nearly constant with $r$. Assuming it to be constant implies from (\[eqn:end\]) that $u_r$ is similarly constant (cf. @kdk_2020), if $\beta_0$ is constant: $$u_r \sim -4 \left( \frac{10^5}{\beta_0} \right) {\rm cm}/{\rm s} \,.$$ Taking $\beta_0$ to be a strict constant corresponds to a model intermediate between the conserved-flux model of @b_2016 (dashed line in the right panel of their fig. 5) and their flux-proportional-to-mass model (solid line). Using their initial fiducial $\beta_0 = 10^5$ implies the disc at $r = 30$ AU drains out in $r/|u_r| \simeq 3$ Myr.
\[lastpage\]
[^1]: E-mail: mmrosent@ucsc.edu
[^2]: The planet excites waves in the inner disc which carry negative angular momentum inward. This is equivalent to transmitting positive angular momentum outward.
[^3]: If the Shakura-Sunyaev $\alpha \lesssim 10^{-4}$, discs respond to planetary torques as if they were inviscid (@gc_2019a, their fig. 1).
[^4]: Our final planet masses are a factor of $\sim$3 lower than theirs, a consequence largely of their choice for $h$ which is 50% larger.
|
LPTENS-05/13\
[**Dual Form of the Paperclip Model**]{}
[**Sergei L. Lukyanov**]{}$^{1,2}$ and [**Alexander B. Zamolodchikov**]{}$^{1,2,3}$
${}^{1}$ NHETC, Department of Physics and Astronomy\
Rutgers University\
Piscataway, NJ 08855-0849, USA\
${}^{2}$ L.D. Landau Institute for Theoretical Physics\
Chernogolovka, 142432, Russia
and
${}^{3}$ Chaire Internationale de Recherche Blaise Pascal\
Laboratoire de Physique Th${\acute {\rm e}}$orique de l’Ecole Normale Sup${\acute {\rm e}}$rieure\
24 rue Lhomond, Paris Cedex 05, France\
**Abstract**
------------------------------------------------------------------------
\
[October 2005]{}
Introduction
============
In this work we describe the dual form of the “paperclip model” of boundary interaction in 2D Quantum Field Theory. The paperclip model was introduced in Ref.[@LVZ], where its basic properties are discussed. The model involves two-component Bose field ${\bf X}(\sigma,\tau)=\big(X(\sigma,\tau)
, Y(\sigma,\tau)\big)$ which lives on a semi-infinite cylinder with Cartesian coordinates $(\sigma,\tau)$, $\sigma\geq 0,\ \tau\equiv\tau+1/T$[^1]. In the bulk, i.e. at $\sigma>0$, the dynamics is described by the free-field action: \[baction\] [A]{}\_[bulk]{}\[ X,Y \]= [14]{} \_0\^[1/T]{}\_[0]{}\^ . The interaction takes place at the boundary at $\sigma=0$, due to a non-linear boundary constraint: the boundary values ${\bf X}_B = (X_B, Y_B) \equiv {\bf X}|_{\sigma=0}$ of the field ${\bf
X}$ are restricted to the “paperclip curve” \[bconstraint\] r( ) - ( )= 0 , |Y\_[B]{}| a , Here $a,\ b,$ and $r$ are real positive parameters, the first two being related as follows[^2], \[salskj\] a\^2-b\^2= .
As usual, the non-linear constraint requires renormalization, but one can check (up to two loops) that the RG transformation affects the curve only through renormalization of the parameter $r$, which “flows” according to the equation \[rgflow\] [E\_\*E]{} = 4b\^2 (1 - r\^2) r\^[4b\^2]{}, where $E$ is the RG energy, and $E_{*}$ is the integration constant of the RG equation, which sets up the “physical scale” in the model. As in [@LVZ], in what follows we always take the scale $E$ proportional to the temperature $ T$, namely \[kappadef\] E = 2T.
It is useful to introduce also the external field ${\bf h} = (h_x,h_y)$ which couples to the boundary values ${\bf
X}_B$, i.e. to add the boundary term \[askj\] [A]{}\_[h]{}\[X\_B,Y\_B\]=\_0\^[1/T]{} ( h\_x X\_B+ h\_y Y\_B ) to the action . Then, the first object of interest is the partition function, \[fint\] Z\_0 = [ D]{} X [ D]{} Y \^[-[A]{}\_[bulk]{}\[X,Y\]- [A]{}\_[h]{}\[X\_B,Y\_B\] ]{} , where the functional integration is over all fields ${\bf X}(\sigma,
\tau)$ obeying the boundary constraint .
General definition of the model involves additional parameter, the topological angle $\theta$. Topologically, the paperclip curve is a circle, hence the configuration space for the field ${\bf X}(\sigma,\tau)$ consists of sectors characterized by an integer $w$, the number of times the boundary value ${\bf X}_B$ winds around the paperclip curve when one goes around the boundary at $\sigma=0$. The contributions from the topological sectors can be weighted with the factors ${\mbox{e}}^{{\rm i}w\theta}$. Thus, in general \[topsum\] Z\_= \_[w=-]{}\^\^[[i]{} w]{} Z\^[(w)]{} , where $Z^{(w)}$ is the functional integral taken over the fields from the sector $w$ only. Physics of the model, in particular its infrared (i.e. low temperature) behavior, depends on $\theta$ in a significant way (see [@LVZ; @LTZ] for details).
The ultraviolet (high-$T$) limit of the paperclip model is understood in terms of the conformally invariant “hairpin model” of boundary interaction [@LVZ]. In this limit the parameter $r$ tends to zero, and the paperclip curve becomes a composition of two “hairpins”, as shown in Fig.1.
![The paperclip formed by junction of two hairpins.[]{data-label="fig-ampla"}](Fig2.eps){width="10cm"}
The hairpin model is defined by replacing the paperclip constraint by the non-compact “hairpin” curve ${\textstyle {r\over 2}}\,\exp\big(\pm {\textstyle{X_B\over 2\,b}}\big)=
\cos\big( {\textstyle{Y_B\over 2\,a}}\big)$. We refer to this model as the “left” or the “right” hairpin, depending on the sign in the exponential; the two models are related by simple field transformation $(X,Y)\to (-X,Y)$). Note that the left hairpin corresponds to the right part in Fig.2, and vice versa (just like the way a human brain is wired to the rest of the body). The left (right) hairpin model is conformally invariant with the linear dilaton $D({\bf X})=
{X\over 2b}$ $\big( D({\bf X})=-{X\over 2b}\big)$. More details on the hairpin model can be found in [@LVZ] and in Section\[sectwo\] below.
The paperclip model has many features in common with the so-called “sausage” sigma model studied in Ref.[@falz]. The UV splitting of the paperclip into the hairpins is analogous to the UV splitting of the sausage into two semi-infinite “cigars” (see [@falz]). Like the sausage sigma model, the paperclip model seems to be integrable at two values of the topological angle, $\theta=0$ and $\theta=\pi$. The sausage model is known to admit a dual description, where the non-trivial metric is replaced by certain potential (“tachion”) term in the action [@fatea]. This analogy is one of the reasons to expect that a similar dual representation exists for the paperclip model. The aim of this work is to introduce the dual representations of both the hairpin model and the paperclip model.
In this paper we argue that the paperclip model is equivalent (or “dual”) to another model of boundary interaction. The dual model also involves a two-component Bose field $(X(\sigma,\tau),{\tilde
Y}(\sigma,\tau))$ (where ${\tilde Y}$ is interpreted as the T-dual[^3] of $Y$) on the semi-infinite cylinder, which has free-field dynamics in the bulk, and obeys [*no constraint*]{} at the boundary $\sigma=0$; instead it interacts with an additional boundary degree of freedom. It is best to discuss the dual model in terms of its Hamiltonian representation, with the cyclic coordinate $\tau\equiv
\tau+1/T$ taken as the Euclidean (or Matsubara) time. In this picture the partition function admits the dual representation as the trace \[tracce\] Z\_ = [Tr]{}\_ , taken over the space ${\tilde{\cal H}} = {\cal
H}_{X,{\tilde Y}}\otimes {\mathbb C}^2$, where ${\cal H}_{X,{\tilde
Y}}$ is the space of states of the two-component boson $\big(X(\sigma), {\tilde Y}(\sigma)\big)$ on the half-line $\sigma
\geq 0$ (with no constraint at $\sigma=0$) and ${\mathbb C}^2$ is the two-dimensional space representing the new boundary degree of freedom. The dual Hamiltonian in consists of the bulk and the boundary parts, ${\hat H} =
{\hat H}_{\rm bulk} + {\hat H}_{\rm boundary}$. The bulk part is just the free-field Hamiltonian \[hbulk\] [H]{}\_[bulk]{} =[14]{} \_[0]{}\^ , where $(\Pi_{X}, \Pi_{\tilde Y})$ are momenta conjugated to the field operators $(X, {\tilde
Y})$ [^4] acting in ${\cal H}_{X,{\tilde Y}}$ (${\hat H}_{\rm bulk}$ acts as identity in the ${\mathbb C}^2$ component of ${\tilde{\cal H}}$). The boundary term describes coupling of the boundary values $(X_B,
{\tilde Y}_B)\equiv
(X,{\tilde Y})|_{\sigma=0}$ of the fields to the additional boundary degree of freedom represented by ${\mathbb C}^2$ ($\sigma_{\pm}$ and $\sigma_3$ are the Pauli matrices acting in ${\mathbb C}^2$), \[hbdry\] [H]{}\_[boundary]{} = h\_xX\_B + ah\_y\_3 + [ V]{} , where[^5]: \[shasyaasd\] [V]{}=\_[B]{} , with $\mu_B$ related to the scale $E_*$ in as \[alskjs\] \_[B]{}= . In Eqs.,, $a,\, b$ and $(h_x, h_y)$ are the same as in ,.
Dual form of the hairpin \[sectwo\]
===================================
In this section we describe the dual form of the conformal hairpin model. To be definite, throughout this section we concentrate attention on the [*left*]{} hairpin model; the right hairpin is obtained by reflection $X \to -X$. The left hairpin boundary constraint has the form \[hairpin\] ()= ( ) , |Y\_[B]{}| a . Here $r_{*}$ relates to the energy scale in a simple way \[rstar\] [E\_\*2T]{}=4b\^2 (r\_[\*]{})\^[4 b\^2]{} .
The boundary state of the (left) hairpin model was described in [@LVZ], \[hairstate\] B\_ | = \_[**P**]{}\^2[**P**]{} B\_([**P**]{}) I\_[ **P**]{} | , where $\langle\, I_{\bf P}\, |$ are the Ishibashi states associated with the $W$-algebra of the hairpin model (see [@LVZ] and Section\[sectwoone\] below), ${\bf P}$ is the zero-mode momentum of the free boson ${\bf X}$, and the amplitude $B_{\supset}({\bf
P})$ has the following explicit form in terms of the components $(P,Q)$ of the vector ${\bf P}$ \[bright\] B\_(P,Q) = [g]{}\_D\^2 r\_\*\^[-2[i]{} b P]{} [2b (2bP) (1+ [P2b]{})([12]{} - a Q+ b P) ([12]{} + a Q + bP )]{} , where ${\rm g}_D=2^{-1/4}$. The amplitude $B_{\supset}(P,Q)$ coincides with the partition function of the hairpin model on the semi-infinite cylinder of circumference $1/T$: \[zetb\] Z\_(h\_x,h\_y) = B\_(P,Q), where the dependence of $Z_{\supset}$ on the parameters $h_x,\,h_y$ is brought about through the linear boundary term , and $P,Q$ in the right hand side are related to the these parameters as follows \[pxi\] P= [ h\_xT]{} , Q= [ h\_yT]{} .
$W$-algebra and dual potential \[sectwoone\]
--------------------------------------------
As was explained in [@LVZ], the hairpin boundary condition is conformally invariant (with linear dilaton $X/b$), and moreover it has extended conformal symmetry with respect to certain $W$-algebra. Here we use the notation $W_{\supset}$ for the $W$-algebra of the left hairpin model. It is generated by a set of local currents $\{\,W_s (z), s=2,\,3,\,4\ldots\,\}$ (built from derivatives of the free fields $X,Y$) which commute with two “screening operators”[^6], \[screenings\] \_zw W\_s (z) \^[bX a[Y]{}]{}(w,[|w]{}) =0 . Then, it is almost trivial observation that this $W$-symmetry is present in any model of boundary interaction which has no boundary constraint, but instead whose action has an additional boundary potential term \[bpotential\] [A]{}\_[BP]{} = \_[0]{}\^[1/T]{} . Here $S_{\pm}$ may be either c-numbers, or more generally any non-trivial boundary degrees of freedom whose own dynamics is “topologically invariant”, i.e. invariant with respect to any diffeomorphism $\tau \to f(\tau)$ of the boundary. Since such boundary interaction has the hairpin $W$-algebra as its symmetry, it seems natural to expect that under appropriate choice of the boundary degree of freedom ${S}_{\pm}$ the model with the boundary potential is equivalent (or, in modern speak, “dual”) to the left hairpin model.
Because of the essentially quantum nature of the boundary degree of freedom $S_{\pm}$ (see below), it will be convenient to discuss in terms of the Hamiltonian representation of the model, as was mentioned in Introduction. The partition function of the left hairpin model is written as the trace ${\rm Tr}\big[\,{\mbox{e}}^{-{{\hat H}\over T}}\,\big]$, where ${\hat H}$ is the sum ${\hat H}_{\rm bulk} + {\hat V}_{\supset}$ of the bulk free-field Hamiltonian , and the boundary term \[hairdualv\] [V]{}\_ = [S]{}\_[+]{} \^[b X\_B+[i]{}a [Y]{}\_B]{}+ [S]{}\_[-]{} \^[b X\_B-[i]{} a [Y]{}\_B]{} ,corresponding to the term in the action. Here ${\mathbb S}_{\pm}$ are the operators associated with the boundary degree of freedom $S_{\pm}(\tau)$ in . The latter operators must commute with the field operators, i.e. $[\, {\mathbb S}_{\pm}\, ,\, {\bf X}(\sigma)]=
[\, {\mathbb S}_{\pm}\,,\, {\bf\Pi}(\sigma)\,] =0$ at any $\sigma$, lest the $W$-symmetry of the theory be violated, but the commutation relations among the boundary observables ${\mathbb S}_{\pm}$ themselves are not fixed [*a priori*]{}. Our goal here is to identify the algebra of these operators in the dual hairpin model, as well as its representation $\rho$.
Some relations can be inferred from the following simple argument. The boundary potential term vanishes in the limit $X \to -\infty$. In the absence of this term both fields $X$ and ${\tilde Y}$ would obey the von Neumann (free) boundary conditions. Equivalently, in the absence of the potential term the field $Y$ would obey the Dirichlet (fixed) boundary condition, i.e. the boundary values $(X_B,Y_B)$ would lay on a straight brane parallel to the $X$-axis. Note that in the same limit $X\to-\infty$ the right hairpin curve becomes a composition of two parallel branes, \[braneass\] Y\_Ba [as]{} X\_B- , separated by the distance $2\pi a$. In the full hairpin curve, these two “legs” are bridged at the right, allowing for a passage from the upper leg to the lower leg and vice versa. In the dual representation of the hairpin model such passages should be attributed to two terms in the boundary potential . The boundary vertex operators ${\mbox{e}}^{\pm {\rm i}a {\tilde Y}_B}(\tau)$ create jumps in the boundary value of $Y$ exactly of the desired magnitude, $Y_B (\tau+0) -
Y_{B}(\tau-0) = \pm 2\pi a$. The fact that the hairpin has only two “legs” clearly suggests that the operators ${\mathbb S}_{\pm}$ in must obey the “fermionic” relations[^7] \[fermionic\] [S]{}\_[+]{}\^2=[S]{}\_[-]{}\^2=0 .
So far in the discussion of the dual hairpin we have ignored the coupling to the external field $(h_x,h_y)$. Adding the term $h_x X_B$ to the dual hairpin Hamiltonian is straightforward, but in the description in terms of the T-dual field ${\tilde Y}$ the term $h_y Y_B$ would be non-local. One can circumvent this difficulty by introducing the “fermion number” operator ${\mathbb
N}$, associated with the boundary degrees of freedom ${\mathbb
S}_{\pm}$, which satisfies with them the following commutation relations: \[fnumber\] \[ [N]{} , [S]{}\_[+]{} \]=2 [S]{}\_[+]{} , =-2 [S]{}\_[-]{} . Then one can check that the sum $Y_B - \pi a\,{\mathbb N}$ commutes with the Hamiltonian . Therefore, in the presence of the external field ${\bf h}$, expected form of the full Hamiltonian of the dual hairpin model is \[fullhairdual\] [H]{}= [H]{}\_0 + [V]{}\_, where ${\hat V}_{\supset}$ is the boundary potential operator , and \[hnot\] [H]{}\_[0]{} = [H]{}\_[bulk]{} + h\_xX\_B + ah\_y [N]{}, with the operators ${\mathbb N}$ and ${\mathbb S}_{\pm}$ in satisfying the relations and .
Singularities of the hairpin boundary amplitude
-----------------------------------------------
Looking at the right hairpin amplitude as a function of complex variables $P$ and $Q$, one observes two sets of singularities. First one is a sequence of poles at $2{\mbox{i}}b\,P =0,\, -1,\, -2,\ldots$ due to the first gamma-factor in the numerator in . These poles admit straightforward interpretation in terms of potential divergences of the functional integral , with the non-compact boundary constraint , at the infinite end of the hairpin $X \to -\infty$ (see Ref.[@LVZ]). The second set is a string of poles at \[poles\] P\_k = 2bk, k= 1, 2,3…due to the second gamma-factor in . Note that in the weak-coupling limit of the hairpin model, i.e. at $b\to\infty$, these poles depart to infinity. Clearly, in terms of the hairpin functional integral these singularities represent non-perturbative effects. Instead, the poles become most visible at small $b$, which is the weak coupling domain in the dual representation of the hairpin model, and indeed they admit simple interpretation in terms of the dual hairpin model. Since the boundary potential vanishes in the limit $X\to
-\infty$, there is a potential divergence of the functional integral associated with the dual hairpin[^8]. Following [@Goulian], one can first integrate out the constant mode of the field $X$. This integration produces poles in $P$ exactly at the points whose residues \[resus\] R\_k = [Res]{}\_[ P= 2[i]{} bk]{} (k=0,1,2…) are expressed through the integrals of the $2k$-point correlation functions \[intcorr\] R\_k =(2T)\^[nk\^2]{} [T]{}\_[2k]{}\_[1]{} [V]{}\_ (\_[2k]{})\_(\_1) \_[0]{} , where the $\tau$-ordering symbol ${\cal T}$ signifies that the integration is performed over the domain $1/T\geq\tau_{2k}\geq\cdots \geq \tau_{1}
\geq 0$. In ${\hat V}_{\supset} (\tau) =
{\mbox{e}}^{\tau{\hat H}_0} \,{\hat V}_{\supset}\, {\mbox{e}}^{-\tau{\hat
H}_0}$ is the unperturbed Matsubara operator associated with the boundary potential , and $\langle\langle\, \cdots
\, \rangle\rangle_{0}
\equiv {\rm Tr}\big[\,\cdots\,{\mbox{e}}^{-{{{\hat H}_0}\over T}}\, \big]$. Since the unperturbed Hamiltonian ${\hat H}_0$ involves no interaction between the boundary variables ${\mathbb S}_{\pm}$ and the fields $X,\,{\tilde Y}$, the expectation value in factorizes in terms of these two parts of the system. In view of , it can be written in the form \[formfactor\] R\_k =&&\_[\_[i]{}=]{} [Tr]{}\_\
&& [T]{}\_[2k]{} \_[1]{} [V]{}\^[\_[2k]{}]{}\_[B]{}(\_[2k]{})[ V]{}\^[\_[1]{}]{}\_[B]{}(\_[1]{}) \_[0]{} ,which involves the traces over the space of states $\rho$ of the boundary degrees of freedom ${\mathbb S}_{\pm}$, as well as the free-field expectation values $\langle\,\cdots\,\rangle_{0}$ of the boundary values ${\cal V}^{\pm}_B (\tau)$ of the vertex operators . Thanks to the relations , there are only two nonvanishing contributions to the sum in , with $(\epsilon_1\,
\cdots\,\epsilon_{2k}) = (+ - + - \cdots + -)$ and $(\epsilon_1\,\cdots\,\epsilon_{2k}) = (- + - + \cdots- +)$. With this observation, and using explicit form of the free-field correlators in , this expression can be brought to the form: \[factorform\] R\_[0]{} &=& [[g]{}\_[D]{}\^2]{} [Tr]{}\_ ,\
R\_[k]{} &=& [[g]{}\_[D]{}\^2]{} (2T)\^[-k]{}F\_kG\_k (k=1, 2,…) , where \[xitraces\] F\_k = \^[[i]{}a Q]{} [Tr]{}\_= \^[-[i]{}a Q]{}\_ , and $G_k$ are given by the $2k$-fold integrals \[gk\] &&G\_k=\_[0]{}\^[2]{}[u\_k]{}\_[0]{}\^[u\_k ]{}[v\_k]{} \_[0]{}\^[v\_k ]{}[u\_[k-1]{}]{}\_[0]{}\^[v\_[2]{} ]{}[u\_1]{}\_[0]{}\^[u\_1 ]{}[v\_1]{}\
&& \_[j>i]{}\^k \_[ji]{}\^k\^[-4b\^2-1]{}\
&& \_[j> i]{}\^k\^[-4b\^2-1]{} 2 . The overall factor ${{\rm g}_{D}^2\over 2\pi}$ in appears because involves unnormalized free-field correlation functions; here ${\rm g}_D = 2^{-1/4}$ is well known “boundary entropy” factor [@affleck] associated with the Dirichlet and von-Neumann boundary conditions for a free boson[^9].
Obviously, the way they are written above, the integrals diverge for all positive $b^2$. As is common in conformal perturbation theory, we assume here a version of “analytic regularization”, where the expressions are understood as analytic continuations of these integrals from the domain $\Re e\,b^2 <0$[^10]. This procedure is performed in Appendix. Remarkably, the integrals are evaluated in a closed form (the calculations are presented in Appendix), \[gkanswer\] G\_k= [(2)\^[k+1]{} (-4 b\^2)\^[-k]{} (1-4 b\^2 k)k! ( [12]{}-2b\^2 k-a Q )([12]{}- 2b\^2 k+a Q )]{} (k1) . Note that the $Q$-dependence of is exactly as expected from , and moreover coincide with the residues of , at the points provided \[trzero\] [Tr]{}\_=2 (a Q) , and \[trk\] [Tr]{}\_= [Tr]{}\_= ()\^k ( k=1,2… ) .
The equations and are sufficient to identify the representation $\rho$ of the algebra , of the boundary degrees of freedom. From one finds \[dimrho\] [dim]{}() =2 , and \[trn\] [N]{}\^2 = [I]{} , [Tr]{}\_= 0 . Next, the two-dimensional representation of the algebra must satisfy additional relation \[anticomm\] { [S]{}\_[+]{} , [S]{}\_[-]{}}= [I]{} . Indeed, $\rho$ is necessarily an irreducible representation of the algebra. As follows from Eqs.,, the anticommutator $\{\, {\mathbb S}_{+}\, ,\, {\mathbb S}_{-}\}$ commutes with ${\mathbb S}_{\pm}$ and ${\mathbb N}$, therefore it should be a constant in $\rho$. With Eq. this implies the condition .
There is a unique (up to equivalence) two-dimensional representation of ,, which satisfies : \[sigmarep\] : [S]{}\_[+]{}= \_+ , \_[-]{}= \_- , =\_3 , where $\sigma_a$ are conventional Pauli matrices. Thus we identify the boundary degree of freedom of the dual hairpin model with the spin $s=1/2$. Note that two eigenvalues of $\sigma_3$ are associated with two legs of the hairpin. Note also that according to the operators ${\mathbb S}_{\pm}$ have dimension $[\,{energy}\, ]^{1\over
2}$, as required by the balance of dimensions in Eq. (in view of the vertex operators ${\mbox{e}}^{bX_B\pm {\rm i}a{\tilde Y}_B}$ have dimensions $[\, {energy}\,]^{1\over
2}$).
Dual to the paperclip model
===========================
The dual representation for the paperclip model can be identified using similar line of arguments. The idea of the paperclip being the composition of the left and right hairpins makes it natural to look for the Hamiltonian of the dual paperclip model in the form \[dualclip\] [H]{} = [H]{}\_[bulk]{}+h\_x X\_B+ a h\_y [N]{} + [V]{} , where ${\hat H}_{\rm bulk}$ is the same free-field bulk Hamiltonian , and \[dualclipv\] [V]{} &= & . Here ${\mathbb A}_{\pm}$, ${\mathbb B}_{\pm}$ and ${\mathbb N}$ are operators representing boundary degrees of freedom, which commute with the field operators ${\bf X}(\sigma), \ {\bf
\Pi}(\sigma)$. Note that we have explicitly put the factor $\propto \sqrt{E_*}$ in , so that the operators ${\mathbb A}_{\pm}$, ${\mathbb B}_{\pm}$ are dimensionless. Obviously, the first two terms in are associated with the left hairpin component of the paperclip, i.e. the operators ${\mathbb A}_{\pm}$ play the same role as ${\mathbb S}_{\pm}/\sqrt{E_{*}\over 2\pi}$ in . The last two terms in are associated in a similar way with the dual form of the right hairpin. This correspondence suggests that the operators ${\mathbb B}_{\pm}$, as well as ${\mathbb A}_{\pm}$, satisfy the fermionic relations analogous to \[ffermionic\] [A]{}\_\^2 =0 , [B]{}\_\^2 =0 . In addition, they have to satisfy the commutation relations with ${\mathbb N}$ \[ffnumber\] \[ [N]{} , [A]{}\_\] = 2[A]{}\_ , \[ [N]{} , [B]{}\_\] = 2[B]{}\_ , and the anticommutation relations \[anticomma\] { [A]{}\_[+]{} , [A]{}\_[-]{}} = { [B]{}\_[+]{} , [ B]{}\_[-]{} } = [I]{} .
Intuitively, these relations can be advocated by the same arguments that were considered in the previous section. The paperclip curve in Fig.1 can be regarded as the combination of two nearly straight parallel D-branes connected to each other by the left and the right hairpin curves. Then, the presence of the vertex operators ${\mbox{e}}^{bX_B \pm {\rm i}a{\tilde Y}_B}$ in is the way how the dual representation reflects the possibility of passages from one straight brane to another via the connection at the right, i.e. at sufficiently large positive $X_B$. Likewise, the operators ${\mbox{e}}^{-b
X_B \pm {\rm i}a {\tilde Y}_B}$, which become significant at large negative $X_B$, describe the transitions between the nearly parallel branes via the connection at the left. This picture suggests that the space of states associated with the boundary degrees of freedom in , i.e. the supporting space of the representation $\rho$ of the above algebra, is two-dimensional, with two basic vectors (the eigenvectors of ${\mathbb N}$) corresponding to the two constituent straight branes. Then, Eq. simply express the statement that there is s single copy of the paperclip curve (as is specified by the $Y_B$ bound in Eq.). Also, since $\rho$ is irreducible, and since in the limit $E_{*}\to 0$ we have to recover , the relations follow.
It is easy to check that any two-dimensional representation $\rho$ of the algebra , and is equivalent to the following one: \[rhoo\] : \_ = \^[ [2]{}]{} \_ , [B]{}\_ = \^[ [2]{}]{} \_ , where $\theta$ is an arbitrary complex parameter. It is possible to show that this parameter must be real and, moreover, it coincides with the $\theta$-angle of the paperclip model.
The easiest way to verify this identification of $\theta$ is to analyze specific logarithmic divergences generated by the interaction . The divergences appear due to the singular term in the Operator Product Expansions \[ope\] \^[bX\_Ba[Y]{}\_B]{}() \^[-bX\_B a [ Y]{}\_B]{}(’) = [1]{} + [regular terms]{} , and it is easy to check that in view of they can be absorbed by local boundary counterterm \[counterterm\] -2E\_\*() (/E\_[\*]{}) , where $\Lambda$ is the UV cut-off. Exactly the same counterterm, with $\theta$ being the topological angle, is required in the original formulation of the paperclip model, where its role is to compensate for “small instanton” divergences (see [@LVZ] for details).
The above observation suggests that the instanton contributions of the original paperclip model are reproduced by certain terms of conformal perturbation theory of the dual model ,. To be sure, the conformal perturbation theory, understood as a regular expansion in powers of $E_{*}$, can not be literally valid for this model. The model has many properties in common with the boundary sinh-Gordon model, and in view of the analysis in [@bsinh], one expects rather complicated structure of the UV expansion in . For example, in the case $h_x=0$ and at $T\gg E_{*}$ the partition function $Z(h_x,h_y\,|\, T)|_{h_x=0}$ of the model is expected to develop a large logarithmic term, \[lskjsal\] Z(0,h\_y | T) = G( Q | ) ( ) + F( Q | ) , which derives from large $\big(\sim b^{-1}\log({ 1\over \kappa}) \big)
$ fluctuations of the zero mode of the field $X$. The coefficient $G$ depends on $Q={\mbox{i}}\, { h_y
\over T}$ and \[kshsakhs\] = [ [E\_[\*]{}2T]{}]{} . Up to an overall factor $\kappa^{2\kappa\cos(\theta)}$ (whose origin could be traced down to the singular term in Eq.), it admits a small-$\kappa$ expansion in double series in powers of $\kappa$ and $\kappa^{1\over 2{b^2}}$ (the term $F$ has a similar expansion). The term $\sim\kappa$ of this expansion is given by the integral \[cpt\] G( Q | ) &=& [[[g]{}\_D\^2]{}]{} [Tr]{}\_, where ${\hat V}_{\supset}$ and ${\hat V}_{\subset}$ correspond to the first two terms and the second two terms on Eq., respectively. Putting in explicit correlation functions of the exponential fields in and evaluating the trace, one can bring this expression to a more explicit form \[cptt\] &&G( Q | ) = [[g]{}\_D\^2 ]{} 2(a Q) ( 1 + D(Q) ()+ … ) , where the factor $\cos(\theta)$ in the second term appears as the result of evaluation of the trace using Eqs., and $D(Q)$ is the integral \[alss\] D(Q) =[12]{} \_[0]{}\^[2]{}[u\_2]{}\_[0]{}\^[u\_2]{}[u]{}\_1+[counterterm]{} , where $u_1$ and $u_2$ differ from $\tau_1$ and $\tau_2$ in by a factor of $ 2\pi T$. The integral logarithmically diverges when $u_1 \to u_2$, but its divergent part cancels with the counterterm , and the $Q$-dependent finite part is evaluated explicitly, in terms of Euler’s function $\psi(x)={{\rm d}\over {\rm d} x}\, \log\Gamma(x)$. As the result, the expansion of $G(Q\, |\,\kappa)$ has the form with \[cpttt\] D(Q) = 2 () - (-a Q)- (+a Q) . With this, the coefficient in front of $\cos(\theta)$ in exactly matches one-instanton contribution to the paperclip partition function (see Eq.(110) of [@LVZ]).
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to Vladimir V. Bazhanov, Vladimir A. Fateev and Alexei B. Zamolodchikov for discussions and interest to this work.
The research is supported in part by DOE grant $\#$DE-FG02-96 ER 40959. ABZ gratefully acknowledges kind hospitality of Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, and generous support from Foundation de l’Ecole Normale Sup${\acute {\rm e}}$rieure, via chaire Internationale de Recherche Blaise Pascal.
Appendix: Calculation of the integral $G_k$
============================================
Contour integral representation of $G_k$
----------------------------------------
Up to overall factor, the integrand in coincides with the free-field expectation value of product of $2k$ chiral vertex operators \[asisal\] V\_= \^[2b X\_R2[i]{}a Y\_R]{} , where ${\bf X}_R=(X_R,\, Y_R)$ in the exponent stands for the holomorphic part of the Bose field ${\bf X}(\sigma,\tau)=
{\bf X}_R(\tau+{\mbox{i}}\sigma)+{\bf X}_L(\tau-{\mbox{i}}\sigma)$. As usual, the free-field expectation values are fully determined by the two-point functions, \[twopoint\] X\_R () X\_R(’)\_[0]{} = Y\_R () Y\_R(’)\_[0]{} = - .
It is useful to introduce the following set of integrated products \[jahsh\] J\_[p]{}(\_1 \_p )= \_[\_0]{}\^[\_0+1/T ]{}\_p\_[\_0]{}\^[\_[p]{}]{} \_[p-1]{}\_[\_0]{}\^[\_[2]{}]{}\_[1]{}V\_[\_p]{}(\_p)V\_[\_1]{}(\_[1]{}) , where $(\epsilon_1 \cdots\epsilon_p )$ is a set of signs, $\tau_0$ is fixed real parameter, and integration is along the real axis. Clearly, the integral is certain linear combinations of expectation values of $J_{2k}( \epsilon_1
\cdots\epsilon_{2k})$ with $(\epsilon_1 \cdots\epsilon_{2k} )$ being one of two alternating sign sequences $(+-+\cdots+-)$ and $(-+-\cdots-+)$.
As the first step in calculation we transform into contour integrals. To do this we introduce the “screening operators”, the integrals of the chiral vertex operators: \[salksas\] x\_=[1q-q\^[-1]{}]{}\_[\_0]{}\^[\_0+1/T]{} \^[2b X\_R2[i]{}a Y\_R]{} , where[^11] \[ajhsgx\] q=\^[ [i]{}n]{} . Here and bellow in this Appendix we use the notation from Ref.[@LVZ]: \[ssjsl\] n4 b\^2 . The following relations, \[klash\] &&(q-q\^[-1]{}) x\_+J\_[2k-1]{}(--)= J\_[2k]{}(+-)-q\^[k]{} J\_[2k]{}(-+)\
&&(q-q\^[-1]{}) x\_-J\_[2k-1]{}(--)=0\
&&(q-q\^[-1]{}) x\_+J\_[2k]{}(-+)=J\_[2k+1]{}(++)\
&&(q-q\^[-1]{}) x\_-J\_[2k]{}(-+)=q\^k J\_[2k+1]{}(--) , can be easily established by rearranging the integration domains. Additional set of relations is obtained from these by replacing $x_{\pm} \to x_{\mp}$ and simultaneously changing all signs, $J_{p}(\epsilon_1\cdots \epsilon_p)\to
J_{p}(-\epsilon_1\cdots -\epsilon_p)$. Recursively applying , one can prove that \[hsgahg\] &&J\_[2k]{}(+-)= (q-q\^[-1]{})\^k [(-1)\^k q\^[-[k(k+1)2]{}]{}\_q!]{}( q\^k (x\_-x\_+)\^k+(x\_+x\_-)\^k )\
&&J\_[2k]{}(-+)= (q-q\^[-1]{})\^k [(-1)\^k q\^[-[k(k+1)2]{}]{}\_q!]{}( q\^k (x\_+x\_-)\^k+(x\_-x\_+)\^k ) ,\
where the standard notations, \[skasjh\] \[k\]\_q!=\[1\]\_q\[2\]\_q…\[k\]\_q , \_q=[q\^k-q\^[-k]{}q-q\^[-1]{}]{} , are used.
To make the next step more transparent we change to the new coordinate \[slkssk\] z=\^[2T ]{} . The screening charges become contour integrals in the variable $z$. More precisely, the action of the operators $x_{\pm}$ on any state created by a set of local insertions in the $z$-plane is written as the integral \[akkjs\] x\_ (…) = [1q-q\^[-1]{}]{} \_[C]{} dz V\_(z) (… ) , where the contour $C$ starts from the point $\zeta_0 = {\mbox{e}}^{2\pi{\rm
i}T \tau_0 }$, goes around all the insertions in the counterclockwise direction, and then returns back to $\zeta_0$, as is shown in Fig.\[fig-analita1\]. The integrand in should be understood in terms of free-field operator product expansions. In general, the contour is not closed since operator product in the integrand is multivalued function of $z$.
![The integration contour in Eq..[]{data-label="fig-analita1"}](contour13.eps){width="4cm"}
Using the integral $G_k$ can be rewritten in the form: \[saaaks\] &&G\_k=2 (-1)\^k q\^[ k\^22]{} , or more explicitly \[sssks\] &&G\_k=[2\^[1-k]{} \^k \^[-[[i]{}2]{} nk\^2]{} \_[j=1]{}\^k(n j)]{}\_[C\_k]{}\_k\_[S\_k]{}z\_k…\_[C\_1]{}\_1 \_[S\_1]{}z\_1\
&& , where the integration is over a set of contours $C_j$, $S_j$ $(j=1,2,\, \ldots\, k)$ each starting at $\zeta_0$ and returning to the same point after going around the point $0$, and arranged so that $C_k$ lays entirely (except for the point $\zeta_0$ itself) inside $S_k$, $S_{k}$ lays inside $C_{k-1}$, $C_{k-1}$ lays inside $S_{k-1}$, etc., as is depicted in Fig.\[fig-count\].
![The integration contours in Eq..[]{data-label="fig-count"}](contour11.eps){width="6cm"}
The expectation values in the integrand in involves a certain vertex operator, \[vlambda\] V\_ =(- X\_R+ Y\_R ) , with $\lambda_{\pm}$ to be specified below, inserted at the origin $z=0$, so that \[kasjshs\] && V\_-(\_k) V\_+(z\_[k]{})…V\_+(z\_1) V\_(0) =\^[-k]{} \^[-[[i]{}2]{} n k\^2]{} \_[j=1]{}\^[k]{} z\_j\^[n\_+]{} \_j\^[n\_-]{}\
&& \_[j>i]{} \_[j>i]{}\^[-n-1]{} \_[j=1]{}\^k(\_j-z\_j)\^[-n-1]{} . We assume that the brunches of the power functions in are chosen to give real positive values at \[slksjsasl\] 0<z\_1<\_1…<z\_k<\_k . In Eqs., the following notations are used \[alksjalksj\] n\_+=[kn-12]{}+a Q+ , n\_-=[kn-12]{}-a Q , where $\varepsilon$ is a complex parameter, which is assumed to be small, and eventually will be sent to zero.
Combinatorics of the contour integrals
--------------------------------------
Now let us introduce another set of integrated products of vertex operators, \[ajahsh\] I\_[p]{}(\_1 \_p )=\_[\_0]{} \^[0 ]{}z\_p \_[\_0]{}\^[z\_p]{}z\_[p-1]{}\_[\_0]{}\^[z\_[2]{}]{}z\_[1]{} V\_[\_1]{}(z\_1)V\_[\_p]{}(z\_[p]{})V\_(0) . Here $V_{\lambda}$ is the vertex operator . Although the final result of the calculations below does not depend on a choice of the point $\zeta_0$, for convenience we choose it to be real and negative, and we assume that all the integrations in are along the real axis. The operator products in are multivalued functions of the integration variables, and we assume the same choice of the branch as in . Note that the integrals are similar but different from , the main difference being in the form of integration contours.
The monodromies of the operator products in are determined by symbolic relations \[jhsagshg\] [A]{}\_C\[ V\_(z)V\_() \]=q\^[2\_]{} V\_(z)V\_() , and, in particular, \[kajsk\] [A]{}\_C\[ V\_[+]{}(z)V\_[-]{}() \]=q\^[-2]{} V\_[+]{}(z)V\_[-]{}() , where the symbol ${\cal A}_C[\ldots]$ denotes analytic continuation in the variable $z$ along the contour $C$ shown on Fig.\[fig-analit\].
![The contour of analytic continuation in Eq..[]{data-label="fig-analit"}](analit2.eps){width="4cm"}
Using these relations one can derive the following identities: \[aslskjsl\] x\_[+]{} I\_[2k]{}(+…-)&=&-q\^[\_+-k]{} \[\_+\]\_qI\_[2k+1]{}(+…+)\
x\_[+]{} I\_[2k]{}(-…+)&=& -q\^[\_+-k]{} \[\_+-k\]\_q I\_[2k+1]{}(+…+)\
x\_[+]{} I\_[2k-1]{}(+…+)&=& 0\
x\_[+]{} I\_[2k-1]{}(-…-)&=& -q\^[\_+-k]{} \[\_+-k\]\_q I\_[2k]{}(+…-) -\
&&q\^[\_+-k]{} \[\_+\]\_q I\_[2k]{}(-…+) ,as well as similar identities obtained from these by simultaneous change of the signs: $\{x_{\pm}\to x_{\mp},\ \lambda_{\pm}\to
\lambda_{\mp},\ I_{p}(\epsilon_1\cdots \epsilon_p)\to
I_{p}(-\epsilon_1\cdots -\epsilon_p)\}$. Here the action of the screening charges $x_{\pm}$ is defined by Eq.. The recursion allows one to express the ordered integrals in terms of the screening charges. In particular, one finds \[skasjhsj\] I\_[2k]{}(-…+)=[1c\_k \[\_+\]\_q]{}( \[\_+-k\]\_q (x\_-x\_+)\^k+\[\_+\]\_q (x\_+x\_-)\^k ) V\_() , where \[alksjska\] c\_k=(-1)\^kq\^[k(\_++\_–k)]{} [\[k\]\_q! \[\_++\_–1\]\_q!\_q!]{} .
Now assume that $\lambda_{\pm}$ are given by , and consider the limit $\varepsilon\to 0$. It is easy to check that at $\varepsilon=0$ the expression becomes invariant with respect to simultaneous rescaling of all the integration variables, hence the integral develops logarithmic divergence at $z_i \to 0$. Thus, as the function of $\varepsilon$, $I_{2k}(-+ \cdots +)$ is expected to have a simple pole at $\varepsilon=0$. The appearance of the pole is very explicit in , since the coefficient $c_k$ vanishes at $\varepsilon=0$, \[alsas\] c\_k [2 q-q\^[-1]{}]{} (-1)\^k \[k\]\_q!\[k-1\]\_[q]{}! . The representation also makes it easy to isolate the residue at this pole, \[alksjl\] &&\_[0]{} I\_[2k]{}(-…+)= [(-1)\^k (q-q\^[-1]{})2 \[k\]\_q!\[k-1\]\_q! ((a Q+[kn2]{}))]{}\
&& ( ([(a Q-[kn2]{}]{})(x\_-x\_+)\^k+([(a Q+[kn2]{}]{})(x\_+x\_-)\^k ) V\_() .Comparing with we observe that the desired integrals $G_k$ can be expressed through such residues, \[laskjss\] G\_k&=& 2\^[k+1]{} \^k \^[[[i]{}2]{} nk\^2]{}\_[j=1]{}\^[k-1]{}(n j) \
&& ((a Q+ ))\_[0]{} I\_[2k]{}(-…+) |\_[=0]{} . Furthermore, it is easy to see that \[kajshks\] && \_[0]{} I\_[2k]{}(-…+) |\_[=0]{} =\
&& \_[0]{}\^1z\_[2k-1]{}…\_[0]{}\^[z\_3]{}[z]{}\_2 \^[z\_[2]{}]{}\_0[z]{}\_[1]{} V\_-(1) V\_+(z\_[2k-1]{})…V\_+(z\_[1]{}) V\_(0) ,and hence \[sssksa\] &&G\_k=[ 2\^[k+1]{} k]{}\_[j=1]{}\^[k-1]{}(n j) ((a Q+ ))\
&&\_[0]{}\^[1]{}z\_k\_[0]{}\^[z\_k]{}\_[k-1]{}…\_[0]{}\^[\_[1]{}]{} z\_1\_[j=1]{}\^[k-1]{} \_[j>i]{}(\_j-\_i)\
&&\_[j=1]{}\^[k]{} \_[j>i]{}(z\_j-z\_i)\_[j,i]{}|z\_i-\_j|\^[-n-1]{} .
Final step of the calculation
-----------------------------
Integrations over the variables $z_1,\,\ldots\, z_k$ in can be eliminated by using the following identity[^12]: \[aslsas\] && \_[\_[k-1]{}]{}\^[\_k]{} z\_k \_[\_[k-1]{}]{}\^[\_[k-2]{}]{}z\_[k-1]{}…\_[\_[0]{}]{}\^[\_[1]{}]{}z\_[1]{} \_[j>i]{} (z\_j-z\_i)\
&& \_[j=0]{}\^[k]{} \_[i=1]{}\^k |\_j-z\_i|\^[\_j]{}= [\_[j=0]{}\^[k]{}(1+\_j)(k+1+\_[j=0]{}\^k \_j )]{} \_[j>i]{} (\_j-\_i)\^[\_i+ \_j+1]{} ,where $\zeta_k>\zeta_{k-1}>\ldots>\zeta_0$ is an ordered set of real numbers. In the case under consideration $\zeta_0=0,\ \zeta_{k}=1$ and \[slsakjlk\] \_0=[kn-12]{}+a Q , \_[j]{}=-n-1 (j=1,… k) . Thus \[asksjsal\] G\_k= [2\^[k+1]{} \^2 D\_[k-1]{}k ([1-kn2]{}-a Q) ([1-kn2]{}+a Q)]{}\^k(-n) \_[j=1]{}\^[k-1]{}(nj) , where $D_{k-1}$ is the Selberg integral [@Selberg; @Dot]: \[salkks\] &&D\_[k-1]{} =\_[0]{}\^[1]{}\_[k-1]{} \_[0]{}\^[\_[k-1]{}]{}\_[k-2]{}…\_[0]{}\^[\_[2]{}]{}\_[1]{} \_[j=1]{}\^[k-1]{} \_j\^[(k-1)n-1]{}(1-\_j)\^[-2n]{}\
&& \_[j>i]{}(\_j-\_i)\^[-2n]{}= [ \^[k-1]{} (-n)\^[-k]{} (1-kn)(k-1)! \^k(-n)]{} [1\_[j=1]{}\^[k-1]{}(n j )]{} . Combining Eqs. and one arrives to.
[99]{}
S.L. Lukyanov, E.S. Vitchev and A.B. Zamolodchikov, Nucl. Phys. [**B683**]{}, 423 (2004).
S.L. Lukyanov, A.M. Tsvelik and A.B. Zamolodchikov, Nucl. Phys. [**B719**]{}, 103 (2005).
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V.A. Fateev, Phys. Lett. [**B357**]{}, 397 (1995).
A.M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, Cambridge, UK: Univ. Press (1995).
M. Goulian and M. Li, Phys. Rev. Lett. [**66**]{}, 2051 (1991).
I. Affleck and A. Ludwig, Phys. Rev. Lett. [**67**]{}, 161 (1991).
P. Fendley, H. Saleur and N.P. Warner, Nucl. Phys. [**B430**]{}, 577 (1994).
Al.B. Zamolodchikov, unpublished notes (2001).
R.B. Zhang, J. Phys. A: Math. Gen. [**27**]{}, 817 (1994).
P. Baseilhac and V. A. Fateev, Nucl. Phys. [**B532**]{}, 567 (1998).
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[^1]: Euclidean formulation of the theory is implied. Due to the compactification $\tau\equiv
\tau+1/T$ it is equivalent to the Matsubara representation of the $1+1$ dimensional theory at thermodynamic equilibrium at the temperature $T$.
[^2]: The parameter $n$ used in [@LVZ] is related to $a$ and $b$ as $$a={\textstyle{\sqrt {n+2}\over 2}}\, ,\ \ \ \
\ \ b={\textstyle {\sqrt {n}\over 2}}\ .$$
[^3]: The T-dual of the free massless field is defined as usual, through the relations: $\partial_{\tau}{\tilde Y} = {\mbox{i}}\, \partial_{\sigma} Y$ and $
\partial_{\sigma}{\tilde Y} = -{\mbox{i}}\, \partial_{\tau} Y$. \[dual\]
[^4]: Normalization is such that, for instance, $[\, X (\sigma)\, ,\, \Pi_X(\sigma')
\,]=
2\pi{\mbox{i}}\, \delta(\sigma-\sigma')\ .$
[^5]: We assume canonical conformal normalization of the boundary vertex operators (see e.g. Eq.) involved in this interaction term.
[^6]: Note that we write the screening operators in terms of the T-dual field ${\tilde Y}$. This is done in preparation to the discussion of the dual hairpin below. At this point the distinction makes no difference in the definition of the $W_{\supset}$-algebra, since the equation is sensitive to the holomorphic parts of the fields $X$ and $Y$ only.
[^7]: It is important at this point that our definition of the hairpin (and of the paperclip) model involves [*uncompactified*]{} field $Y$. Also, the bound $|Y_B| < \pi a$ in (and in ) is essential. Without the bound, Eq. (as well as Eq.) would define a series of disconnected curves, $Y \to Y+ 4\pi a\, {\mathbb Z}$ copies of the original hairpin (or paperclip). Although the models of boundary interaction which involve more then one copy also deserve attention, they are different from the hairpin (paperclip) model as defined in [@LVZ], and we do not address them here.
[^8]: It is not difficult to write down the full functional integral for the dual hairpin model, which involves integration over the fields $X$ and ${\tilde Y}$, as well as the integral over the boundary spin ${\bf S} =
(S_{+},S_{-}, S_{3})$ with the Wess-Zumino term (see, e.g., Ref.[@Polyakov]). Such expression is not very useful for our analysis, except for the observation that the only terms in the full action which involve the constant mode of $X$ are the boundary potential term , and the term ${\tilde {\mathscr A}}_{{\bf h}} = \int_{0}^{1/T}{\mbox{d}}\tau\,\big[\,
h_x X_B(\tau) + \pi a h_y\,S_3
(\tau)\, \big]$ responsible for the coupling to the external field $(h_x,h_y)$.
[^9]: The $g$-factor of uncompactified boson $X$ with the von-Neumann boundary condition diverges. Formally, it involves the factor ${{\rm g_D}\over 2\pi}\ {\rm d} X_0$, where $X_0$ is the zero mode of $X$ [@saleur]. In the integration over $ X_0$ is already performed – this integration was the origin of the poles . Additional factor ${\rm g}_D$ comes from the unperturbed partition function of ${\tilde Y}$. Since it is the T-duality transform of $Y$, its partition function equals to $\int {\mbox{d}}{\tilde Q} \ \langle \,B_N\,|\, {\tilde Q}\,\rangle = {\rm g}_D$, where $\langle\, B_N\, |$ is the boundary state associated with the von Neumann boundary condition for the field ${\tilde Y}$.
[^10]: More generally, these divergences have to be canceled by adding a counterterm $M\,{\mbox{e}}^{2bX_B}$ (with the cutoff-dependent coefficient $M$) to the Hamiltonian .
[^11]: As it is known, the screening operators $x_{\pm}$, together with the two zero mode operators, $\int_{\tau_0}^{\tau_0+1/T} {{\mbox{d}}\tau}\,
\big(\, a\partial_\tau X_R\pm {\mbox{i}}\, b\partial_\tau Y_R\, \big)$, form the Chevelley basis of the Borel subalgebra of quantum superalgebra $U_{q,q}(sl(2|1))$ [@Zhang].
[^12]: This identity can be interpreted as the Dotsenko-Fateev representation [@Dot] of the simple conformal block of Virasoro algebra with the central charge $c=-2$. A similar identity was used in Ref.[@fata].
|
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abstract: 'The NEMO collaboration is presently mounting the NEMO-3 detector in the Fréjus Underground Laboratory. This detector, which will be completed by the end of the year 2000, is devoted to the search of neutrinoless double beta decay with various isotopes. Much attention has been focused on $^{100}$Mo and $^{82}$Se with their large $Q_{\beta\beta}$-values. The detector is based on the direct detection of the two electrons by a tracking device and on the measurement of their energies by a calorimeter. The aim of the experiment is to have a sensitivity for the effective neutrino mass on the order of 0.1 eV. The status and the expected performance of the NEMO-3 detector for both internal and external background rejections and for signal detection are presented.'
author:
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NEMO Collaboration\
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Contributed Paper for the XIX International Conference on Neutrino Physics and Astrophysics, Neutrino 2000, Sudbury, Canada, June 16-21, 2000, presented by X. Sarazin\
and for the XXX International Conference on High Energy Physics,\
ICHEP2000, Osaka, Japan, July 27 - August 2, 2000, presented by D. Lalanne.
title: 'Status Report on the double-$\beta$ decay experiment NEMO-3'
---
Introduction
============
Several strong indications in favor of neutrino masses and mixing have been observed in atmospheric and solar neutrinos. However, direct detection of neutrino masses has not been measured. The most stringent upper limit obtained by tritium beta decay is $m_{\nu}<2.8$ eV (95% C.L.) [@mainz]. Another fundamental question of neutrino physics is the nature of massive neutrinos. Are massive neutrinos Dirac particles or neutral Majorana particles having all lepton numbers equal to zero? The neutrinoless double beta decay $\beta\beta(0\nu)$ which is a process beyond the electroweak Standard Model, is the only way to prove the existence of Majorana neutrinos. In some phenomenologically viable neutrino scenarios, the effective Majorana neutrino mass $\langle m_{\nu} \rangle$ can be 0.1 eV (in a three-neutrino scenario with two mass-degenerate neutrinos) or even as large as 1 eV (in a four-neutrino scenario which accomodates all the oscillation measurements) [@bilenki].
To date, the most stringent limit on the $\beta\beta(0\nu)$ half-life is obtained in the $^{76}$Ge Heidelberg-Moscow experiment [@laura]: $$T_{1/2}^{0\nu} > 1.6 \; 10^{25} \mbox{ yr (90\% C.L.)}$$ From this limit, an upper limit on $\langle m_{\nu} \rangle$ can be inferred with the relation: $$(T_{1/2}^{0\nu})^{-1} = \left(\frac{\langle m_{\nu} \rangle}{m_e}\right)^2 \times |M_{0\nu}|^2 \times F_{0\nu}$$ where $M_{0\nu}$ is the nuclear matrix element of the relevant isotope and $F_{0\nu}$ is the phase-space factor.\
Calculations of $M_{0\nu}$ have unfortunately large theoretical uncertainties. Depending on the calculation of $M_{0\nu}$, one obtains limits on $\langle m_{\nu} \rangle$ ranging from 0.4 eV to 1 eV [@laura]. The limit $\langle m_{\nu} \rangle <$ 1 eV is obtained by using calculations performed in the framework of the Shell Model [@caurier]. $F_{0\nu}$ is analytically calculable and is proportional to $Q_{\beta\beta}^5$ ($Q_{\beta\beta}=2040 keV$ for $^{76}$Ge) . Therefor to improve the sensitivity of a double-$\beta$ decay experiment, an isotope with a larger $Q_{\beta\beta}$ seems to be preferable in order to get a larger $F_{0\nu}$, but also to reduce the background in the search for a $\beta\beta0\nu$ signal.
The aim of the NEMO-3 detector, which will operate in the Fréjus Underground Laboratory, referred to as the Laboratoire Souterrain de Modane (LSM), is to search for $\beta\beta(0\nu)$ with various isotopes with large Q$_{\beta\beta}$ values. The detector is able to accomodate at least 10 kg of double beta decay isotopes. Much attention has been focused on $^{100}$Mo ($Q_{\beta\beta}$ = 3034 keV), $^{82}$Se ($Q_{\beta\beta}$ = 2995 keV) and $^{116}$Cd ($Q_{\beta\beta}$ = 2802 keV).
The NEMO-3 detector
===================
The NEMO-3 experiment is based on the direct detection of the two electrons by a tracking device and on the measurement of their energies by a calorimeter. The NEMO-3 detector, as shown in the Figure \[fig:nemo3\], is similar in function to the earlier prototype NEMO-2 [@nemo2].\
The detector is cylindrical in design and divided into 20 equals sectors. Thin ($\sim 50 \mu m$) source foils are fixed vertically between two concentric cylindrical tracking volumes composed of open octagonal drift cells, 270 cm long, operating in Geiger mode. In order to minimize multiple scattering effects, the tracking volume is filled with a mixture of helium gas and 4% ethyl alcohol. The wire chamber provides three-dimensional tracking. The tracking volume is covered with calorimeters made of large blocks of plastic scintillators coupled to very low radioactivity 3” and 5” PMTs. The finished detector contains 6180 drift-Geiger cells and 1940 scintillators.
A solenoid surrounding the detector produces a magnetic field of 30 Gauss in order to recognize ($e^+e^-$) pair production events in the source foils. An external shield, in the form of 20 cm thick low radioactivity iron, covers the detector to reduce $\gamma$-rays and thermal neutron fluxes. Outside of this shield, an additional shield is added to thermalize fast neutrons.
Current Status of Construction
==============================
The construction of the 20 sectors of the NEMO-3 detector has been completed. Currently, 12 sectors are in the Underground Laboratory and 6 of them are equipped with their source foils and mounted on the detector frame. The detector will be completed by the end of the year 2000.
The energy resolution of each scintillator block has been measured with a 1 MeV electron spectrometer during the construction of the calorimeter. The energy resolution is $\sigma(E)/E=5.6\%$ at 1 MeV which is lower than the energy resolution of 7% at 1 MeV specified in the detector’s proposal.
The double-$\beta$ decay isotopes which are being mounted in the detector are the following: 7 kg of $^{100}$Mo (corresponding to 12 sectors), 1 kg of $^{82}$Se (2.3 sectors), 0.6 kg of $^{116}$Cd (1 sector), 0.7 kg of $^{130}$Te (1.8 sectors), 50 g of $^{150}$Nd, 16 g of $^{96}$Zr and 8 g of $^{48}$Ca. Also, 2.7 sectors are devoted to external background measurements: 1 sector is equipped with an ultra-pure copper foil and 1.7 sectors with 0.9 kg of $^{nat}$TeO$_2$. To date, $^{82}$Se, $^{116}$Cd, $^{nat}$TeO$_2$ and the copper foils are mounted. The choice of Cu and $^{nat}$TeO$_2$ is explained below. We are now starting to mount the $^{100}$Mo sources.
Three sectors installed on the detector frame have been succesfully running since the end of April 2000 (without a magnetic field and an external shield). The NEMO collaboration has decided to start operating with these 3 sectors in order to test the tracking and calorimeter parts of the detector. The wire chamber and the PMTs coupled to the scintillators are running well and only 0.3% of Geiger cells are out-of-order. Geiger $\beta$ tracks obtained with these 3 sectors and with the finalized NEMO-3 trigger and acquisition system, are shown in Figure \[fig:event1\] and \[fig:event2\].
Expected background
===================
There are three origins of expected background which can occur in this search for a $\beta\beta0\nu$ signal around 3 MeV. The first background comes from the beta decays of $^{214}$Bi (Q$_{\beta} =$ 3.2 MeV) and $^{208}$Tl (Q$_{\beta} =$ 5.0 MeV) which are present in the source, from the Uranium and Thorium decay chains. They can mimic $\beta\beta$ events by $\beta$ emission followed by M$\ddot{o}$ller effect or by a $\beta-\gamma$ cascade followed by a Compton interaction. Thus, the experiment requires ultra-pure enriched $\beta\beta$ isotopes. A second origin of $\beta\beta0\nu$ background is due to high energy gamma rays ($>$ 2.6 MeV) interacting with the source foil. Their origin is from neutron captures occuring inside the detector. The interactions of these gammas in the foil can lead to 2 electrons by $e^+e^-$ pair creation, double Compton scattering or Compton followed by M$\ddot{o}$ller scattering. Finally, given the energy resolution, the ultimate background is the tail of the $\beta\beta2\nu$ decay distribution. It defines the half-life limits to which the $\beta\beta0\nu$ can be studied.
Radiopurity of the sources in $^{214}$Bi and $^{208}$Tl
-------------------------------------------------------
### $^{100}$Mo source
Maximum levels of $^{214}$Bi and $^{208}$Tl contamination in the source have been calculated to insure that $\beta\beta2\nu$ is the limiting background. These limits are $^{214}$Bi $<$ 0.3 mBq/kg and $^{208}$Tl $<$ 0.02 mBq/kg. These activities in $^{214}$Bi and $^{208}$Tl correspond to a level of $2 \; 10^{-11}$ g/g in $^{238}$U and $10^{-11}$ g/g in $^{232}$Th respectively when we assume the natural radioactive families of $^{238}$U and $^{232}$Th are in equilibrium. To reach these specifications, two methods have been developed to purify the enriched Molybdenum isotope.
The first method developed by ITEP (Moscow, Russia), is a purification by local melting of solid Mo with an electron beam and drawing a monocrystal from the liquid portion. One gets an ultra-pure $^{100}$Mo monocrystal. The crystal is then rolled into a metallic foil for use in the detector. Much attention has been focused on this rolling process. To date 0.5 kg of foil has been produced and no contaminant activity have been measured with HP-Ge in the LSM.
The second purification method is a chemical process done at INEEL (Idaho, USA) which leaves the Mo in a powder form that is then used to produce foils with a binding paste and mylar strips which have been etched with an ion beam and a chemical process. To date 3 kg of $^{100}$Mo have been purified and 2 kg more are being processed and will be ready towards the end of September 2000. No activity has been observed in the purified $^{100}$Mo after 1 month of HP-Ge measurements in the LSM and the most stringent limits obtained for radiopurities are $^{214}$Bi $<$ 0.2 mBq/kg and $^{208}$Tl $<$ 0.05 mBq/kg. The radiopurity in $^{214}$Bi is already better than the design specifications. The task of measuring the required limits for $^{208}$Tl is beyond the practical measuring limits of the HP-Ge detectors in the LSM. However, the chemical extraction factors defined as the ratio of contamination before and after purification were measured with a $^{nat}$Mo sample. Applying the $^{208}$Tl extraction factor to the $^{208}$Tl activity measured in the $^{100}$Mo before purification, one obtains after purification an expected level in $^{208}$Tl of 0.01 mBq/kg which is again lower than the design specifications.
### $^{82}$Se source
Some low activities in $^{214}$Bi and $^{208}$Tl have been measured in the 1 kg $^{82}$Se source foils with HP-Ge studies. The activities are 1.2 $\pm$ 0.5 mBq/kg in $^{214}$Bi and 0.4 $\pm$ 0.1 mBq/kg in $^{208}$Tl. This corresponds to an expected background of 0.2 events/yr/kg from $^{214}$Bi and 1 event/yr/kg from $^{208}$Tl.\
The same contamination had been measured with $^{82}$Se foils used in the NEMO-2 prototype and contaminants were found to be concentrated in small “hot-spots” and rejected in the analysis thanks to the tracking device [@selenium_nemo2]. We believe that the contamination in these $^{82}$Se foils is identical and will be suppressed by software analysis.
External background from neutrons and $\gamma$-rays
---------------------------------------------------
The effect of neutrons and $\gamma$-rays on the background in the $\beta\beta0\nu$ energy region was studied for 10,700 hours of live time with the NEMO-2 prototype [@neutrons]. Various shields and measuremements with a neutron source were used to identify the different components.
This study has shown that there is no contribution from thermal neutrons which are stopped in a few centimeters of the iron shielding but that the dominating background is due to fast neutrons ($>$ 1 MeV) from the laboratory. Fast neutrons going through the iron shielding, are thermalized in the plastic scintillators and then captured in copper, iron or hydrogen inside the detector. To compare the data and Monte Carlo calculations, a study required the development of an interface between GEANT/MICAP and a new library for $\gamma$-ray emission after capture or inelastic scattering of neutrons. Good agreement was obtained between the experiments and simulations.
It was demonstrated with the neutron simulations for NEMO-3 that an appropriate neutron shield (like paraffin) and a 30 Gauss magnetic shield will make the neutron background negligible [@neutrons].
Radiopurity of the detector
---------------------------
Additionally, the components of the detector have to be ultra-pure in $^{214}$Bi, $^{208}$Tl and $^{40}$K to have a low background in the $\beta\beta2\nu$ energy spectrum. This is required to not only measure the $\beta\beta2\nu$ period with high accuracy but also to see any distortions in the $\beta\beta2\nu$ spectrum due to Majoron emission. Finally, the high radiopurity is required so that we can measure the $e\gamma$ and $e\gamma\gamma$ events which identify the Tl activity in the source.
The activities of all materials used in the detector were measured with HP-Ge detectors in the LSM or at the CENBG laboratory in Bordeaux (France). This exhausting examination of samples, corresponding to about 1000 measurements, reasulted in the rejection of numerous glues, plastics, and metals. Activities in $^{214}$Bi, $^{208}$Tl and $^{40}$K, of the main components of the detector are listed in Table \[tab:activities\].
As expected, the radioactive contamination in the detector is dominated by the low radioactivity glass in the PMTs. The activity of these PMTs are three orders of magnitude below standard PMT levels. With a total activity of 300 Bq for $^{214}$Bi and 18 Bq for $^{208}$Tl in the 600 kg of PMTs, the expected signal-to-background ratio ($S/B$) in the integrated $\beta\beta2\nu$ energy spectrum is $S/B \sim 400$ from $^{214}$Bi and $S/B \sim 900$ from $^{208}$Tl with 7 kg of $^{100}$Mo (T$_{1/2}(\beta\beta2\nu) = 0.95 \; 10^{19}$y). This ratio becomes about 10 times smaller with $^{82}$Se since its $\beta\beta2\nu$ half-life is about 10 times larger (T$_{1/2}(\beta\beta2\nu) = 0.8 \; 10^{20}$y).
Activities of all other components are under our measurement sensitivity and negligeable compare to the PMTs.
-------------- ------------- ---------------- -------------- ---------------- ---------------
Total Activity (in Bq)
Components Weight (kg) $^{40}$K $^{214}$Bi $^{208}$Tl $^{60}$Co
PMTs 600 830 300 18
scintil. 5,000 $<$100 $<$0.7 $<$0.3 1.8 $\pm$ 0.4
copper 25,000 $<$125 $<$25 $<$10 $<$6
petals iron 10,000 $<$50 $<$6 $<$8 17 $\pm$ 4
$\mu$ metal 2,000 $<$17 $<$2 2.0 $\pm$ 0.7 4.3 $\pm$ 0.7
wires 1.7 $<$8.10$^{-3}$ $<$10$^{-3}$ $<$6.10$^{-4}$ 10$^{-2}$
shield. iron 180,000 $<$3000 $<$300 $<$300 300 $\pm$ 100
-------------- ------------- ---------------- -------------- ---------------- ---------------
: Total activities (in Bq) for the main components of the NEMO-3 detector, measured with HP-Ge detectors in the Fréjus Underground Laboratory.[]{data-label="tab:activities"}
$^{nat}$TeO$_2$ and Copper foils to measure external background
---------------------------------------------------------------
Foils of $^{nat}$TeO$_2$ inserted into the NEMO-3 detector allow one to measure the external background for $^{100}$Mo. The effective $Z$ of these foils is nearly the same as that of molybdenum foils. This is useful because the external $\gamma$-ray background can give rise to pair production, double Compton scattering, or Compton-M$\ddot{o}$ller which are all proportional to $Z^2$. Thus, the background for $^{100}$Mo and $^{nat}$TeO$_2$ foils should give rise to similar event rates. However, $^{nat}$TeO$_2$, which is 34.5% $^{130}$TeO$_2$, produces no $\beta\beta$ pairs in the energy region above the $Q_{\beta\beta}$-value of $^{130}$TeO$_2$ (2.53 MeV), so a background subtraction is possible for $^{100}$Mo foils given the spectrum of $^{nat}$TeO$_2$. The copper foils provide a similar study for a smaller value of $Z$.
Number of background events in the $\beta\beta0\nu$ energy region
-----------------------------------------------------------------
The expected numbers of background events, in the energy range 2.8 to 3.2 MeV around the $\beta\beta0\nu$ signal peak are summarized in Table \[tab:expect\_bkg\] for $^{100}$Mo and $^{82}$Se.
------------------- -------------- --------------
$^{100}$Mo $^{82}$Se
events/yr/kg events/yr/kg
$^{214}$Bi $<$ 0.03 negl.
$^{208}$Tl $<$ 0.04 negl.
$\beta\beta2\nu$ 0.11 0.01
External neutrons $<$ 0.01 $<$ 0.01
TOTAL $<$ 0.18 0.01
------------------- -------------- --------------
: Expected number of background events, in the energy window 2.8 to 3.2 MeV, per year per kg. For $^{82}$Se, it is believed that the background from $^{214}$Bi and $^{208}$Tl will be limited to “hot-spots” (see text).[]{data-label="tab:expect_bkg"}
Expected sensitivity of NEMO-3
==============================
The sensitivity that the NEMO-3 detector will reached after 5 years of data collection, has been calculated with 7 kg of $^{100}$Mo and 1 kg of $^{82}$Se. After 5 years, in the energy window 2.8 to 3.2 MeV, a total of 6 background events are expected with 7 kg of $^{100}$Mo and no background events are expected with 1 kg of $^{82}$Se. The $\beta\beta0\nu$ detection efficiency in the same energy window, 2.8 to 3.2 MeV, is $\epsilon(\beta\beta0\nu) = 14\%$. The expected sensitivities are summarized in Table \[tab:sensitivity\].
7 kg $^{100}$Mo 1 kg $^{82}$Se
--------------------------- ------------------------------ -------------------------------
Number of events 6 background events expected 0 background events expected
in the energy window 6 events observed 0 events observed
2.8 to 3.2 MeV 5 $\beta\beta0\nu$ excluded 2.5 $\beta\beta0\nu$ excluded
$T_{1/2}^{0\nu}$ $>$ 4. 10$^{24}$ yr $>$ 1.5 10$^{24}$ yr
$\langle m_{\nu} \rangle$ $<$ 0.25 - 0.7 eV $<$ 0.6 - 1.2 eV
: Expected sensitivity (90% C.L.) for NEMO-3 after 5 years of data with 7 kg of $^{100}$Mo and 1 kg of $^{82}$Se (the number of events are given in the energy window 2.8 to 3.2 MeV around the $\beta\beta0\nu$ signal peak).[]{data-label="tab:sensitivity"}
[References]{} Ch. Weinheimer et al., Phys. Lett. B, 460 (1999) 219-226 S.M. Bilenki et al., Phys. Lett. B, 465 (1999) 193-202 L. Baudis et al., Phys. Rev. Lett., 83 (1999) 41-44 E. Caurier et al., Phys. Rev. Lett., 77 (1996) 1954-1957 R. Arnold et al., Nucl. Inst. Meth., A 354, (1995) 338-351 R. Arnold et al., Nucl. Phys., A 636, (1998) 209-223 Ch. Marquet et al., to be published
|
---
bibliography:
- 'bibliography.bib'
date: 2019
title: 'Cylindrical vector beam generator using a two-element interferometer'
---
### Authors: {#authors .unnumbered}
Job Mendoza-Hernández, Manuel F. Ferrer-Garcia, Jorge Arturo Rojas-Santana, and Dorilian Lopez-Mago$^{*}$
### Addresses: {#addresses .unnumbered}
Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L., México, 64849.\
$^\ast$ dlopezmago@tec.mx
abstract
========
We realize a robust and compact cylindrical vector beam generator which consists of a simple two-element interferometer composed of a beam displacer and a cube beamsplitter. The interferometer operates on the higher-order Poincaré sphere transforming a homogeneously polarized vortex into a cylindrical vector (CV) beam. We experimentally demonstrate the transformation of a single vortex beam into all the well-known CV beams and show the operations on the higher-order Poincaré sphere according to the control parameters. Our method offers an alternative to the Pancharatnam-Berry phase optical elements and has the potential to be implemented as a monolithic device.
Introduction
============
Optical vector beams are structured fields which posses a space-dependent polarization distribution [@Brown:10]. Two particular examples are the radially and azimuthally polarized beams, which are commonly called cylindrical vector (CV) beams [@zhan_cylindrical_2009; @Chen2018]. These are characterized by a donut shape intensity containing a central singularity surrounded by an azimuthally varying pattern of linear polarizations. The family of CV beams can be conveniently described as a coherent superposition of optical vortices with orthogonal polarizations and opposite helicities [@Galvez:12]. The corresponding Jones vector of the transverse electric field can be written as $$\mathbf{E}(\mathbf{r})= \frac{1}{\sqrt{2}}\left[ \mathrm{LG}_{-m}(\mathbf{r})\, \mathbf{\hat{c}}_{R} + \exp(i\beta)\, \mathrm{LG}_{m}(\mathbf{r}) \, \mathbf{\hat{c}}_{L}\right].
\label{EQ:CV}$$ In the above equation, we have considered a monochromatic light beam, with wavelength $\lambda=2\pi/k$ and $\mathbf{r}=x\mathbf{\hat{x}}+y\mathbf{\hat{y}}$, propagating along the $z$-axis. The unit vectors $\mathbf{\hat{c}}_{R}$ and $\mathbf{\hat{c}}_{L}$ are the right- and left-handed unit polarization vectors, respectively. The function $\mathrm{LG}_{m}(\mathbf{r})$ represents an optical vortex whose complex amplitude is given by the single-ringed Laguerre-Gaussian (LG) beams. The LG beams belong to the class of helical modes with azimuthal dependence $\exp(im\phi)$, where the integer index $m$, also known as the topological charge, characterizes the helical wavefront of the beam. This helical wavefront indicates that the LG beams carry orbital angular momentum (OAM). In fact, for the LG beams, the amount of OAM per unit power and unit length is equal to $m$. For the particular case $m=1$, the phase difference $\beta$ in Eq. (\[EQ:CV\]) can be adjusted to generate radial, azimuthal or hybrid modes[@Wang:hybrid2010]. Higher-order polarization singularities can be realized with $m>1$ [@Otte:17; @Freund].
The CV beams have been extensively studied, in particular, due to their tight focusing properties which present a strong longitudinal component and a smaller focal spot compared to a focused Gaussian beam [@pu_tight_2010; @porfirev_polarization_2016]. They have been applied in multiple areas such as microscopy [@biss_dark-field_2006; @hell_breaking_1994], optical manipulation [@grier_revolution_2003; @man_optical_2018; @turpin_optical_2013; @weng_creation_2014], material processing [@kraus_microdrilling_2010; @hnatovsky_role_2013; @hamazaki_optical-vortex_2010; @hnatovsky_polarization-dependent_2012], and telecommunications [@gibson_free-space_2004; @bozinovic_terabit-scale_2013].
The generation of CV beams can be performed using active or passive methods[@zhan_cylindrical_2009]. In the active generation method the beams are obtained directly from the light source using a specially design intracavity resonator [@kozawa_generation_2005; @ahmed_multilayer_2007]. The passive methods in free space are based on the modification of spatially-homogeneous polarized beams with devices that variant the polarization distribution. All the interferometric methods, where the wavefront is modified using spatial light modulators, belong to the passive methods [@rosales-guzman_simultaneous_2017; @wang_generation_2007]. Other interferometric configurations have been explored which include compact and robust designs [@Chen2011; @wang_generation_2007; @li_efficient_2014; @liu_compact_2019]. Furthermore, single-element vector beams generators have been introduced such as metasurfaces and Pancharatnam-Berry phase optical elements [@Capasso; @Cardano:12], which allow us to create complex polarization patterns based on a specific input beam. Nevertheless, these elements are not so common and their purchase or fabrication is not accessible for some laboratories.
The CV beams can be used as an extended basis to describe light beams with space-dependent polarization states, which are visualized in a higher-order Poincaré sphere (HOPS) [@milione_higher-order_2011; @PhysRevLett.108.190401; @Holleczek2011]. In this new representation, the family of CV beams is described in a hybrid spatial-polarization basis, which is formed by the product of orthogonal spatial modes and circular polarization basis, i.e. the basis elements are $\{ \mathrm{LG}_{m} , \mathrm{LG}_{-m} \} \otimes \{ \mathbf{\hat{c}}_{R}, \mathbf{\hat{c}}_{L}\}$. This representation permits to easily visualize the effect of phase retarders and polarizers on the polarization pattern. Further details of this formalism can be found in Millione *et al.* [@milione_higher-order_2011] and Holleczek *et al.* [@Holleczek2011].
Due to the properties and applications of the CV beams, it is desirable to find alternatives to generate them. In this work, we introduce a compact and simple interferometric method to transform scalar vortex beams into CV beams using optical elements easily found in an optics laboratory. The main component of our method is a two-element interferometer composed of a beam displacer and a cube beam splitter with its semi-reflecting layer placed parallel to the optical axis of the system. We introduce the theoretical description for the CV beam generator and explain the effect of the control parameters on the higher-order Poincaré sphere. An experiment is performed to demonstrate our proposal by realizing all well-known CV beams.
A two-element interferometer as a vector beam generator. {#sec:CVBgen}
========================================================
Our CV beam generator scheme is presented in Fig. \[fig1\]. It is mainly composed of a beam displacer (BD) and a cube beam splitter (CBS). Therefore, we call this device a *two-element interferometer*. It is important to notice that the BD is not equivalent to a polarizing beam splitter. The BD separates the **s**- and **p**-polarization components of the input beam *without* reflecting the **s** component. On the contrary, a polarizing beam splitter reflects the **s** component changing its helicity and hence its spatial mode. A requirement for our idea to work, it is that both beams arriving at the BS have the same spatial mode (except for an overall phase and amplitude factor). Therefore, the BD separates the beam according to our requirements. In addition, the output beams are parallel to the input beam, which facilitates alignment and reduces the number of optical elements. As explained later, a vector beam is already realized after the BS, however, in order to generate the polarization structure described by Eq. (\[EQ:CV\]), we use a quarter-wave plate (QWP) following the CBS to change the Cartesian polarization basis into a circular polarization basis. The input to the device is a vortex beam described as $$\mathbf{U_{in}}= \mathrm{LG}_{m} \, \mathbf{\hat{e}} = \mathrm{LG}_{m} (\cos\alpha \mathbf{\hat{x}}+ \exp{(i\theta)} \sin \alpha \mathbf{\hat{y}}),
\label{eq1}$$ where $\mathbf{\hat{x}},\mathbf{\hat{y}}$ are unit polarization vectors directed along the $x$ and $y$ directions, respectively. The unit vector $\mathbf{\hat{e}}= \cos\alpha \mathbf{\hat{x}}+ \exp{(i\theta)} \sin \alpha \mathbf{\hat{y}}$ is a generic elliptical polarization state with inclination angle $\alpha$ and constant phase difference $\theta$. The complex amplitude $\mathrm{LG}_{m}$ is given by $$\mathrm{LG}_{m}= C_m (r/w_0)^{|m|} \exp(-r^2/w_0^2)\exp(i m\phi),
\label{eq2}$$ where we have used a cylindrical coordinate system $(x,y)=(r\cos\phi,r\sin\phi)$, $w_0$ is the beam waist at the plane $z=0$ and $C_m$ is a normalization constant, such that the beam has unit power, i.e. $\int \int |\mathrm{LG}_{m}|^{2} \mathrm{d}x \mathrm{d}y=1$. The optical vortex given by Eq. can be generated in multiple ways [@Yao:11]. In our case, taking advantage of a recent experiment, we opted to generate the vortex beam using a spatial light modulator (SLM) [@Arrizon:05]. Nevertheless, the vortex beam could be generated using a spiral phase plate or by transforming a Hermite-Gaussian mode using a pair of cylindrical lenses [@Neil2000]. The vortex beam passes through a combination of a quarter-wave plate and a half-wave plate (not shown in Fig. \[fig1\]) in order to generate the elliptic polarization of Eq. (\[eq1\]).
![Top view of the experimental scheme to generate CV beams. The input beam shown in the inset corresponds to an LG beam with homogeneous polarization. The LG beam passes through our two-element interferometer composed of a beam displacer (BD) and a cube beamsplitter (BS), Transverse lines and dots are used to indicate the horizontal and vertical polarization directions, respectively. The quarter-wave plate (QWP) is used to change the polarization basis. The output $\mathbf{U_{3}}$ is a CV beam as represented by the inset.[]{data-label="fig1"}](figure1.pdf){width="12"}
The vortex beam $\mathbf{U_{in}}$ enters the BD and is spatially separated in two orthogonally-polarized optical vortices $\mathbf{U_{1}}$ and $\mathbf{U_{2}}$ with the same topological charge and helicity. The outputs $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ after the BD can be written as $$\begin{aligned}
\mathbf{U_{1}}=& \cos\alpha \, \mathrm{LG}_{m} \, \mathbf{\hat{x}},\\
\mathbf{U_{2}}=& \exp(i\theta)\sin \alpha \, \mathrm{LG}_{m} \, \mathbf{\hat{y}}.\end{aligned}$$ In the above equation, we have neglected a phase difference between $\mathbf{U_{1}}$ and $\mathbf{U_{2}}$ that might arise due to the different optical paths between the ordinary ($\mathbf{U_{2}}$) and extraordinary modes ($\mathbf{U_{1}}$) of the BD. However, this phase difference can be compensated through phase $\theta$, which can be introduced using a quarter-wave plate. Nevertheless, in the experimental results that follow, we did not need to compensate for that phase difference.
After the BD, $\mathbf{U_1}$ and $\mathbf{U_2}$ are incident onto the CBS whose semi-reflecting layer is placed parallel to the propagation direction of the beams (refer to Fig. \[fig1\] to see the ray trajectories inside the CBS). This configuration was inspired by the single-element interferometer proposed by Ferrari *et al.* [@ferrari_single-element_2007]. The CBS divides each input into two rays, one transmitted and one reflected. The transmitted parts of the beams are replicas of the input beams multiplied by a factor $1/\sqrt{2}$. The reflected parts, in addition to the multiplicative factor $1/\sqrt{2}$, gain a phase-shift of $\pi/2$ and their OAM helicity (the sign of the topological charge $m$) is inverted. Notice that the trajectory of the reflected parts resembles the action of a Dove prism. It has been shown that an optical vortex inverts its OAM after traversing a Dove prism, as described by Gonzalez *et al.* [@Gonzalez2006]. Therefore, for the reflected parts we make the transformation $m \rightarrow -m$.
The output fields $\mathbf{U'_3}$ and $\mathbf{U'_4}$ after the CBS are given by the superposition of the transmitted and reflected parts as $$\begin{aligned}
\mathbf{U'_3}=& \frac{1}{\sqrt{2}}\left[ i\, \cos \alpha\,\, \mathrm{LG}_{-m} \, \mathbf{\hat{x}} \, + \sin \alpha \exp(i\theta) \, \mathrm{LG}_{m} \, \mathbf{\hat{y}} \right], \\
\mathbf{U'_4}=& \frac{1}{\sqrt{2}} \left[\cos \alpha\, \mathrm{LG}_{m} \, \mathbf{\hat{x}} \, + i \sin \alpha \exp(i\theta) \, \mathrm{LG}_{-m} \, \mathbf{\hat{y}}\right].\end{aligned}$$
Finally, in order to change the polarization basis from linear to circular, we use a quarter-wave plate whose fast axis is placed at 45 degrees with respect to the $x$-direction. The resulting beams are $$\begin{aligned}
\mathbf{U_3}=& \frac{i}{\sqrt{2}}\left[\, \cos \alpha \, \textrm{LG}_{-m} \, \mathbf{\hat{c}}_{R} \, - \sin \alpha \exp(i\theta) \, \textrm{LG}_{m} \, \mathbf{\hat{c}}_{L}\right], \label{eq:out3} \\
\mathbf{U_4}=& \frac{1}{\sqrt{2}} \left[ \cos \alpha\, \textrm{LG}_{m} \, \mathbf{\hat{c}}_{R} \, + \sin \alpha \exp(i\theta) \, \textrm{LG}_{-m} \, \mathbf{\hat{c}}_{L}\right], \label{eq:out4}\end{aligned}$$ where $\mathbf{\hat{c}}_{R}= \mathbf{\hat{x}}-i \,
\mathbf{\hat{y}}$ and $\mathbf{\hat{c}}_{L}= \mathbf{\hat{x}}+i \,\mathbf{\hat{y}}$ are the right and left handed polarization vectors, respectively. It must be noticed that the spatial distribution of the superposition can be modified only by varying the input beam polarization state. For example, the case with $\alpha=-\pi/4$ in Eq. (\[eq:out3\]) reproduces the polarization distribution given by Eq. (\[EQ:CV\]), which corresponds to the family of CV beams.
Experimental implementation
===========================
The CV beam generator is constructed using a HeNe laser (Thorlabs HNL020LB) centered at a wavelength of $632.8$ nm. The single-ringed LG beams are generated applying a similar technique as the one used by Arrizon *et al.* [@Arrizon:05] by using an amplitude-only SLM (HOLOEYE LC2002). The laser beam is spatially filtered, expanded and collimated before impinging the SLM. It is then passed through a combination of a 4f system with an aperture to recover the LG mode from the first diffraction order. Subsequently, the LG beam passes through the quarter- and half-wave plates to define the polarization state according to Eq. (\[eq1\]).
The resulting LG beam goes through the BD (ThorLabs BD40) which provides a 4 mm spatial displacement between the two output beams. The output beams have a beam diameter of about 1 mm, and hence, the beam displacer separates the beam without overlapping them.
As mentioned in the previous section, both output beams are parallel to the input beam and have orthogonal polarizations. They enter the CBS which is aligned according to the scheme of Fig. \[fig1\]. The CBS is a 50:50 non-polarizing cube beamsplitter (Thorlabs BS013).
Since the polarization distribution is constructed by the superposition of a reflected and transmitted beam, it must be noticed that the alignment of the CBS is crucial to ensure it. Following this condition, a three-axis mount has been used for the CBS in order to obtain a correct control over the superposition. After interfering at the beam splitter, the output beams are directed to a quarter-wave plate. The inhomogeneous polarization distribution is reconstructed by using traditional Stokes polarimetry $$\begin{aligned}
S_0(x,y) &=& I_H (x,y)+ I_V (x,y) = I(x,y), \\
S_1(x,y) &=& I_H (x,y) - I_V (x,y),\\
S_2(x,y) &=& I_D (x,y)- I_A(x,y), \\
S_3(x,y) &=& I_R(x,y)-I_L(x,y),\end{aligned}$$ where $I(x,y)$ is the total transverse intensity of the beam and the subscripts $H,V,D,A,R,L$ are the horizontal, vertical, diagonal at $45^{\circ}$, diagonal at $135^{\circ}$, right and circular intensities, respectively. The intensity pattern of each output beam was captured using a CCD camera (Thorlabs DCU224M).
We compute the polarization ellipses at each spatial point of the transverse plane using the following equations [@Chipmanbook]: $$\begin{aligned}
\Psi = \frac{1}{2} \arctan \left( \frac{S_{2}}{S_{1}}\right),\label{eq:psi}\\
\varepsilon = \frac{|S_{3}|}{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}} + \sqrt{S_{1}^{2}+S_{2}^{2}}},\label{eq:ellip}\end{aligned}$$ where $\Psi$ is the orientation of the major axis of the ellipse and $\varepsilon$ is the ellipticity. The handedness of the polarization state is determined by the sign of $S_3$. To illustrate the polarization distribution we consider a subset of equally spaced points on the transverse plane.
Results and discussion
======================
We focus our experimental measurements on the output beam $\mathbf{U_3}$ given by Eq. (\[eq:out3\]). Notice that the output beam $\mathbf{U_4}$ provides similar polarization patterns. In fact, $\mathbf{U_4}$ is given by $\mathbf{U_3}$ with the transformations $m\rightarrow -m$ and $\theta \rightarrow \theta +\pi$. Figure \[fig3:OR1\] shows our first experimental results. We realized the typical CV beams consisting of a radial and an azimuthal polarization pattern. Figure \[fig3:OR1\](a) shows the reconstructed polarization distribution along with the measured Stokes parameters for the azimuthally polarized mode. The intensity distribution $S_0$ is used as the intensity background for the polarization distribution, where the brighter regions represent larger intensities. The polarization ellipses drawn on top of the intensity are computed with Eqs. and employing the measured Stokes parameters shown next to the polarization distribution. The Stokes images are accompanied by theoretical simulations. The colormap for the Stokes images $S_1$, $S_2$ and $S_3$ represents positive values with the brighter regions and negative values with the darker regions.
![Experimentally generated cylindrical vector beams. The results show the polarization distribution and their corresponding Stokes parameters. (a) Azimuthally-polarized CV beam produced with $m=-1$, $\alpha=\pi/4$, and $\theta=0$ in Eq. (\[eq:out3\]), (b) Radially-polarized CV beam $(m=-1, \, \alpha=-\pi/4, \, \theta=0 )$. Theoretical (up) and experimental (down) Stokes images.[]{data-label="fig3:OR1"}](figure2.pdf){width="12"}
Similarly, Fig. \[fig3:OR1\](b) shows the results for the radially-polarized CV mode. In both results, the Stokes image $S_0$ shows the typical donut-shaped profile. However, the donut shape is not quite circular and presents astigmatism due to the CBS. In addition, the measured Stokes parameters show some background noise attributed to the waveplates and the protective IR window of the CCD camera. Furthermore, there is a small circular polarization component shown in the $S_3$ image, which should be ideally zero. These small deviations from the expected linear states on both cases are attributed to the angular dependence of the CBS, which introduces a small phase-shift that depends on the incidence angle [@Pezzaniti1994]. Nevertheless, we consider that the results show good agreement with the theoretical images and the polarization distribution clearly outlines the concentric rings and the radial lines characteristic of the azimuthally and the radially polarized modes, respectively.
Figure \[fig4:comparative\] shows further examples of the possible polarization distributions that can be generated with our method. For visualization purposes, we show the polarization ellipses without the background intensity. We generate the most common CV beams in terms of the topological charge and polarization state of the input field $\mathbf{U_{in}}$. Radial and azimuthal directions are both achieved when the input LG mode carries a topological charge of $m=-1$ and its polarization is in the diagonal basis, while the hybrid modes are obtained when $m=1$. Circular polarization in the input beam generates equally weighted superposition of the previous states, obtaining spiral polarization distributions when $m=-1$, and the hybrid modes when $m=1$.
![Experimental measurement of the output polarization patterns according to the polarization (top row) and topological charge (left column) of the input beam $\mathbf{U_{in}}$.[]{data-label="fig4:comparative"}](figure3.pdf){width="12"}
Finally, our proposal has the versatility of modifying the topological charge of the initial beam to obtain high-order polarization distributions[@Otte:17; @Freund]. We obtained the polarization distributions, called flower and spider webs, as shown in Fig. \[fig5:high\], where an LG beam with topological charge $m =\pm 2$ and antidiagonal polarization in the input field $\mathbf{U_{in}}$ is used. A polarization distribution of a flower is shown in Fig. \[fig5:high\](a) and the spider web polarization distribution is shown in Fig. \[fig5:high\](b).
![Experimentally generated higher-order polarization singularities. (a) a two-fold vectorial flower and (b) a six-fold vectorial spider web. Theoretical (up) and experimental (down) Stokes images.[]{data-label="fig5:high"}](figure4.pdf){width="12"}
Transformations in the higher-order Poincaré sphere
===================================================
As shown in the above sections, the output polarization distribution is completely described by the polarization state and topological charge of the input beam. We can visualize the transformation of the input LG mode using the HOPS representation. A new set of Stokes parameters in the higher-order Poincaré sphere in terms of the input polarization parameters $(\alpha,\theta)$ are defined as [@milione_higher-order_2011] $$\begin{aligned}
S'_1&=& \sin(2\alpha) \cos(\theta), \\
S'_2&=& \sin(2\alpha) \sin(\theta), \\
S'_3&=& \cos^2(\alpha)- \sin^2(\alpha).\end{aligned}$$
Figure \[fig2:trans\] illustrates the input polarization state on the Poincaré sphere and its respective output polarization state on the corresponding HOPS, which depends on the topological charge of the input beam. From the new set of Stokes parameters, it is noticeable that a variation on the inclination angle $\alpha$ of the polarization state at the input beam represents a translation along a meridian. Meanwhile, a variation on the phase difference between the Cartesian components $\theta$ is mapped as a translation along the latitude. Both effects are illustrated in Figure \[fig2:trans\] using blue and red arrows, respectively. Since the orthonormal basis of the HOPS depends on the sign of the topological charge, we obtain two independent spheres: one that includes the spirally polarized beams (in which the radial and azimuthal polarization directions are contained) when $m=-1$ and another one with the hybrid polarized beams when $m=1$.
![Visual representation of the input and output polarization states and their transformations on the higher-order Poincaré sphere. Input and their respective output polarizations states are indicated by different color markers. The arrows indicate the increasing direction of the parameters in the intervals $\alpha=[0,2\pi]$ (red arrows) and $\theta=[-\pi/2,\pi/2]$ (blue arrows). (a) Input polarization state on the Poincaré sphere (PS). The corresponding output polarization distributions are represented on the higher-order Poincaré sphere (HPS) for (b) $m=1$ and (c) $m=-1$.[]{data-label="fig2:trans"}](figure5.pdf){width="\linewidth"}
Conclusions
===========
In this work, we have demonstrated the feasibility of a robust and compact cylindrical vector beam generator which consists of a simple two-element interferometer composed of a beam displacer and a cube beamsplitter. The interferometer operates on the higher-order Poincaré sphere transforming a homogeneously polarized vortex into a CV beam. We experimentally demonstrated the transformation of a single vortex beam into all the well-known CV beams and higher-order polarization singularities, and showed the operations on the higher-order Poincaré sphere according to the control parameters. Our method offers an alternative to the Pancharatnam-Berry phase optical elements and has the potential to be implemented in a monolithic device. Furthermore, since our method only employs refractive elements with a high-damage threshold, it can be used to create high-power vector beams.
Funding {#funding .unnumbered}
=======
Consejo Nacional de Ciencia y Tecnología (CONACYT) (Grants: 257517, 280181, 293471, 295239, APN2016-3140).
Acknowledgments {#acknowledgments .unnumbered}
===============
JMH acknowledges partial support from CONACyT, México. JMH thanks Israel Melendez Montoya for his help with the polarimeter.
|
---
author:
- 'Sara Beery, Grant Van Horn, and Pietro Perona'
bibliography:
- 'bibliography.bib'
title: Recognition in Terra Incognita
---
|
---
abstract: 'We sharpen an estimate for the growth rate of preimages of a point under a transitive piecewise monotone interval map. Then we apply our estimate to study the continuity of the operator which assigns to such a map its constant slope model.'
author:
- Michal Malek
- Samuel Roth
title: Constant Slope Models and Perturbation
---
Introduction
============
Motivation {#motivation .unnumbered}
----------
\
Over 50 years ago, W. Parry showed that each continuous topologically transitive piecewise monotone interval map is conjugate by an increasing homeomorphism to a map with constant slope. Recently, Ll. Alsedà and M. Misiurewicz pointed out that this constant slope model is unique, and thus it makes sense to study the operator $\Phi$ which assigns to a map its constant slope model. They showed that $\Phi$ is not continuous, essentially because $C^0$ perturbation of the map can lead to a jump in topological entropy. Nevertheless, they conjectured that within the space of transitive maps of a fixed modality, $\Phi$ is continuous at each point of continuity of the topological entropy [@AM page 13]. This paper confirms that conjecture.
Definitions {#definitions .unnumbered}
-----------
\
We study maps $f:[0,1]\to[0,1]$ which are continuous and *piecewise monotone*, i.e. with a finite set of *critical points*: $$\operatorname{Crit}(f)=\{0,1\} \cup \{x \,|\, f \text{ is not monotone on any neighborhood of }x\}.$$ The *modality* of $f$ is the cardinality of $\operatorname{Crit}(f)\cap(0,1)$. We are interested in the spaces
- ${\mathcal{T}}$ - the space of topologically transitive piecewise monotone maps,
- ${\mathcal{T}}_m$ - the subspace of transitive maps of a fixed modality $m\in\mathbb{N}$, and
- $\mathcal{H}^+$ - the space of monotone increasing homeomorphisms of $[0,1]$ with itself.
Each of these spaces is contained in $\mathcal{C}^0$, the space of all continuous functions from $[0,1]$ to itself. Moreover, we equip our spaces with the topology of uniform convergence given by the usual $C^0$ metric $$d(f,g)=\max_{x\in[0,1]} |f(x)-g(x)|.$$ A map $f\in{\mathcal{T}}$ has *constant slope* $\lambda$ if $|f'(x)|=\lambda$ for $x\notin\operatorname{Crit}(f)$. We say that $\tilde{f}$ is a *constant slope model* for $f$ if $\tilde{f}$ has some constant slope $\lambda$ and there is a homeomorphism $\psi\in\mathcal{H}^+$ such that $f=\psi\circ\tilde{f}\circ\psi^{-1}$. It is known [@AM Theorem 8.2 and Corollary 1] that each map $f\in{\mathcal{T}}$ has a unique constant slope model, the conjugating homeomorphism is likewise unique[^1], and the constant slope is the exponential of the topological entropy of $f$. Thus, we are interested in two operators and one real-valued function on the space ${\mathcal{T}}$, namely $$\begin{aligned}
{2}
\Phi&:{\mathcal{T}}\to{\mathcal{T}}, &\qquad \Phi(f)&=\text{the constant slope model for }f,\\
\Psi&:{\mathcal{T}}\to\mathcal{H}^+, & \Psi(f)&=\text{the conjugating homeomorphism, and}\\
h&:{\mathcal{T}}\to\mathbb{R}, & h(f)&=\text{the topological entropy of }f.\end{aligned}$$ Since conjugacy preserves modality, $\Phi$ preserves the spaces ${\mathcal{T}}_m$. We denote the restrictions of our operators to these spaces by $\Phi_m:{\mathcal{T}}_m \to{\mathcal{T}}_m$, $\Psi_m:{\mathcal{T}}_m \to\mathcal{H}^+$.
The role of these operators is summarized in the following commutative diagram. $$\begin{CD}
[0,1] @>\Phi(f)>> [0,1]\\
@V{\Psi(f)}VV @VV{\Psi(f)}V\\
[0,1] @>>f> [0,1]
\end{CD}$$
Results {#results .unnumbered}
-------
\
We start with a theorem concerning the growth rate of the number of iterated preimages of an arbitrary point $x\in[0,1]$ under a transitive, piecewise monotone map $f$. It is already known that the exponential growth rate of the sequence $\left(\#f^{-n}(x)\right)$ gives the entropy of $f$, [@MR Theorem 1.2]. We show that the “subexponential part” of this sequence does not converge to zero.
\[th:preimages\] Let $f\in{\mathcal{T}}$ and fix $x\in[0,1]$. Then $$\limsup_{n\to\infty} \frac{\# f^{-n}(x)}{e^{n h(f)}} > 0.$$
Our second result allows us to verify that a family of homeomorphisms in $\mathcal{H}^+$ is an equicontinuous family.
\[th:equicontinuous\] If $K$ is a compact subset of ${\mathcal{T}}_m$, then $\Psi_m(K)$ is an equicontinuous family.
In the spirit of “dynamical topology,”[^2] we may also state this result in purely topological terms. In light of the Arzela-Ascoli theorem, this says that the $\Psi_m$ image of a compact set is precompact, i.e., has a compact closure in $\mathcal{C}^0$.
These two theorems allow us to prove the main result of our paper, namely,
\[th:main\] If a sequence of maps $g_n\in{\mathcal{T}}_m$ converges uniformly to $f\in{\mathcal{T}}_m$ and if $h(g_n)\to h(f)$, then the constant slope models $\Phi(g_n)$ converge uniformly to $\Phi(f)$.
As a corollary, we get a positive answer for the conjecture of Alseda and Misiurewicz,
The operator $\Phi_m$ is continuous at each continuity point of $h|_{{\mathcal{T}}_m}$. In particular, $\Phi_m$ is continuous for $m\leq 4$.
Sharpness of the Results {#sharpness-of-the-results .unnumbered}
------------------------
\
We remark that all of the hypotheses in Theorem \[th:main\] are essential. This is illustrated by the following two examples.
*Perturbation with a jump in modality.*\
For each value $0\leq t\leq\frac{1}{4}$, put $a=\frac12- t$, $b=\frac12+ t$, $\lambda=3+2t$, and let $\tilde{g}_t$ be the (unique) map with constant slope $\lambda$, 10 critical points $0=c_0<c_1<\cdots<c_9=1$, and critical values $g(c_0)=a$, $g(c_2)=a+t^2$, $g(c_3)=0$, $g(c_4)=b+t^2$, $g(c_5)=a-t^2$, $g(c_6)=1$, $g(c_7)=b-t^2$, and $g(c_9)=b$. For $t>0$ let $\psi_t$ be the “connect-the-dots” map with dots at $(0,0)$, $(a,t)$, $(b,1-t)$, and $(1,1)$. Then put $g_t=\psi_t\circ\tilde{g}_t\circ\psi^{-1}_t$. Finally, let $f$ be the full 3-horseshoe, i.e. the “connect-the-dots” map with dots $(0,0)$, $(\frac13,1)$, $(\frac23,0)$, and $(1,1)$. As $t\to0$ we have uniform convergence of modality-8 maps to a modality-2 map $g_t \rightrightarrows f$ and convergence of entropy $h(g_t)=\log(3+2t) \to h(f)$, but the constant slope models converge to the “wrong” limit $\Phi(g_t)=\tilde{g}_t \rightrightarrows \tilde{g}_0 \neq f=\Phi(f)$. We omit the proofs of transitivity (when $t>0$) and uniform convergence – these proofs are tedious but routine calculations, since all maps involved are piecewise affine.\
[Y[0.2]{} Y[0.2]{} Y[0.2]{} Y[0.2]{} Y[0.2]{}]{} $g_t$ & $f$ & $\psi_t$ & $\tilde{g}_t$ & $\tilde{g}_0${width="2.7cm"} & {width="2.7cm"} & {width="2.7cm"} & {width="2.7cm"} & {width="2.7cm"}
\
Another perspective: when the map $\Phi$ carries the arc $(g_t)_{t\in[0,\frac14]}$ (writing $g_0=f$ for the endpoint) from ${\mathcal{T}}$ to the space of constant slope models $\Phi({\mathcal{T}})$, it “breaks off” the endpoint. Notice that the point $\tilde{g}_0$ is not even in $\Phi({\mathcal{T}})$ because it is not transitive.
(0,0) (75,53)[$\Phi$]{} (28,4)[${\mathcal{T}}$]{} (115,4)[$\Phi({\mathcal{T}})$]{} (47,26)[$f$]{} (130,23)[$\Phi(f)$]{} (117,48)[$\tilde{g}_0$]{}
*Perturbation with a jump in entropy.*\
Let $f:[0,72]\to[0,72]$ be the “connect-the-dots” map with dots $(0,32)$, $(20,52)$, $(24,60)$, $(25,58)$, $(32,72)$, $(52,32)$, $(58,20)$, $(60,24)$, $(72,0)$. The map has modality $5$. It was introduced in [@M2] as a point of discontinuity of $h:{\mathcal{T}}_5\to\mathbb{R}$. Numerical calculations give $h(f)\approx \log 1.81299$, while [@M2] shows rigorously that $h(f)<\log 2$. On the other hand, $f$ has a 2-cycle consisting of critical points $24\mapsto 60\mapsto 24$. Form $g_t$ by perturbing $f$ on the $t$-neighborhoods of those critical points, increasing the slope from 2 to 3 as in the figure below. As $t\to 0$ we have uniform convergence $g_t \rightrightarrows f$ within the space ${\mathcal{T}}_5$ (the proof of transitivity is omitted). On the other hand, $h(g_t)\geq\log2$ because $g_t^2$ has a 4-horseshoe. Thus, we may be sure that $\Phi(g_t)\not\rightrightarrows\Phi(f)$, because the slopes do not converge to the slope of $\Phi(f)$, see [@M2 Lemma 8.3].
[Y[.25]{}Y[.05]{}Y[.25]{}Y[.05]{}Y[.25]{}]{} A 2-cycle of critical points && Perturbation near the 2-cycle && The 2nd iterates near\
one point of the 2-cycle {width="3.3cm"} && {width="3.3cm"} && {width="3.3cm"}
------------------------------------------------------------------------
$f$ &&
------------------------------------------------------------------------
$f$ $g_t$ &&
------------------------------------------------------------------------
$f$ $g^2_t$
Outline of the Paper {#outline-of-the-paper .unnumbered}
--------------------
\
Section \[sec:preimages\] counts preimages to prove Theorem \[th:preimages\]. The central observation is that the strongly connected components of the Hofbauer diagram corresponding to a map $f\in{\mathcal{T}}$ are known to be positive recurrent.
Section \[sec:equicontinuous\] establishes our equicontinuity result, Theorem \[th:equicontinuous\]. The proof requires us to upgrade several facts about mixing piecewise monotone maps to use with perturbation.
Sections \[sec:flatspots\] and \[sec:noflatspots\] examine more closely what happens when we have a convergent sequence $g_n \rightrightarrows f$ in ${\mathcal{T}}_m$. We establish several properties of each (subsequential) limit $\psi$ of the corresponding homeomorphisms $\psi_n=\Psi(g_n)$.
Section \[sec:wrapup\] applies these properties to complete the proof of Theorem \[th:main\].
Section \[sec:further\] leaves the reader with two open problems for further research.
Counting Preimages {#sec:preimages}
==================
Definitions {#definitions-1 .unnumbered}
-----------
\
The *Markov shift* associated to a countable (possibly finite) directed graph $\mathcal{G}$ with vertex set $\mathcal{V}$ is the set of all biinfinite paths on $\mathcal{G}$, $$\Sigma_{\mathcal{G}}=\left\{v\in\mathcal{V}^\mathbb{Z} \mid v_n \to v_{n+1} \text{ in }\mathcal{G}\text{ for all }n\in\mathbb{Z}\right\},$$ together with the shift map $(\sigma v)_n=v_{n+1}$. Its *irreducible Markov subshifts* are the Markov shifts associated with the maximal strongly connected subgraphs of $\mathcal{G}$, where *strongly connected* means that for each pair $(v,w)$ of vertices there is a path from $v$ to $w$. $\Sigma_{\mathcal{G}}$ becomes a measurable space when we equip it with the Borel sigma algebra, where the topology is induced from the product topology on $\mathcal{V}^\mathbb{Z}$. In the absence of compactness, the entropy of the shift is defined (following Gurevich [@G]) simply as the supremum of metric entropies $$\begin{aligned}
h(\Sigma_{\mathcal{G}}) &=
\sup \{ h_\mu(\sigma) \,|\, \mu \text{ is a $\sigma$-invariant Borel probability measure on $\Sigma_{\mathcal{G}}$} \} \\
&= \sup \{ h_\mu(\sigma) \,|\, \mu \text{ is an ergodic $\sigma$-invariant Borel probability measure on $\Sigma_{\mathcal{G}}$} \}\end{aligned}$$ Given a strongly connected graph $\mathcal{G}$, the Markov shift $\Sigma_{\mathcal{G}}$ is called *positive recurrent* if $\mathcal{G}$ has “enough” loops, so that if we fix a vertex $v$ and let $l_n$ count the number of length $n$ loops in $\mathcal{G}$ which start and end at $v$, we require that $\limsup_{n\to\infty} l_n e^{-nh(\Sigma_{\mathcal{G}})} > 0$. The strong connectedness of $\mathcal{G}$ guarantees that this property does not depend on the choice of the vertex $v$ (see, eg., [@VJ]).
Two shift spaces associated with an interval map {#two-shift-spaces-associated-with-an-interval-map .unnumbered}
------------------------------------------------
\
Let $f:[0,1]\to[0,1]$ be a transitive, piecewise monotone interval map. In particular, this implies that $f$ is surjective, piecewise strictly monotone, and has positive topological entropy. We follow the work of J. Buzzi and consider two shift spaces associated to $f$.
The first is a shift space usually called the *symbolic dynamics of $f$*. It is a subshift in the alphabet $\mathcal{A}$ whose letters are the maximal open intervals on which $f$ is monotone. It is given by $$\Sigma=\{A\in\mathcal{A}^\mathbb{Z} \mid \forall{n\in\mathbb{Z}} \forall{k\geq0}\,\, A_n \cap f^{-1}(A_{n+1}) \cap \cdots \cap f^{-k}(A_{n+k})\neq\emptyset \}$$ together with the shift map $\sigma$. This shift space has the advantage that its alphabet is finite, but the disadvantage that it need not be a Markov shift.
The second is a Buzzi’s variant of the Hofbauer shift. It is the Markov shift ${\hat{\Sigma}}= \Sigma_{\mathcal{D}}$ associated to a certain directed graph $\mathcal{D}$ called the *complete Markov diagram* of $\Sigma$. In the language of our original interval map, the definitions read as follows. A *word* is a finite concatenation of letters from the alphabet $\mathcal{A}$. The set of points which are “just finishing the itinerary” given by a word is called the *follower set* of the word, $${\operatorname{Fol}(A_{-m}\cdots A_0)} := f^m\left(A_{-m} \cap f^{-1}(A_{-m+1}) \cap \cdots \cap f^{-m}(A_0)\right).$$ By convention, the follower set of the empty word is the whole space $[0,1]$. A word is *forbidden* if its follower set is empty. A *constraint word* is a word $A_{-m}\cdots A_0$, $m\geq0$ whose follower set changes if we cross off the left-hand letter, i.e., such that $$\emptyset \neq {\operatorname{Fol}(A_{-m}\cdots A_0)} \subsetneq {\operatorname{Fol}(A_{-m+1} \cdots A_0)}$$ The collection of all constraint words will be denoted $\mathcal{C}$. Each nonforbidden word may be shortened to a constraint word by crossing off letters on the left, leaving behind the *minimal* suffix with the same follower set. This motivates the definition $$\begin{gathered}
\min(A_{-m}\cdots A_0)=A_{-k}\cdots A_0 \text{ if and only if} \\
k\leq m \text{ and } {\operatorname{Fol}( A_{-m}\cdots A_0)} = \cdots = {\operatorname{Fol}( A_{-k}\cdots A_0)} \subsetneq {\operatorname{Fol}( A_{-k+1}\cdots A_0)}.\end{gathered}$$
Finally we define the complete Markov diagram $\mathcal{D}$ as the directed graph with vertex set $\mathcal{C}$ and all arrows of the form $
\alpha \to \min(\alpha A),
$ where $\alpha\in\mathcal{C}$, $A\in\mathcal{A}$, and $\alpha A$ is not forbidden.[^3]
Properties of these shift spaces {#properties-of-these-shift-spaces .unnumbered}
--------------------------------
\
Now we can begin to exploit the connections between our transitive piecewise monotone map $f$, its symbolic dynamics $\Sigma$, and the Markov shift ${\hat{\Sigma}}$ associated with its complete Markov diagram $\mathcal{D}$. We start by gathering together four known results:
- $h(f)=h(\Sigma)$.
- $h(\Sigma)=h({\hat{\Sigma}})$.
- ${\hat{\Sigma}}$ contains only finitely many positive-entropy irreducible Markov subshifts, and all of them are (strongly) positive recurrent.
- The entropy of ${\hat{\Sigma}}$ is the supremum of the entropies of its irreducible Markov subshifts.
The first result follows from Misiurewicz and Szlenk’s characterization of the entropy of $f$ in terms of lap numbers, and noting that $f^n$ has as many laps as the number of length $n$ words appearing in the language of $\Sigma$ [@MS]. The next two results are Buzzi’s, and are based on the fact that $\Sigma$, although not of finite type, is still a subshift of quasi-finite type [@B Theorem 3 and Lemma 7]. The fourth result follows because each ergodic invariant probability measure on a Markov shift is necessarily concentrated on one of its irreducible subshifts.
If we combine all four results, we may derive immediately
\[prop:subgraph\] There exists a strongly connected subgraph $\mathcal{D}_0\subseteq\mathcal{D}$ whose associated Markov shift is positive recurrent and has the same entropy as $f$.
Follower Sets and Loops {#follower-sets-and-loops .unnumbered}
-----------------------
\
Several properties of follower sets follow immediately from the definitions. For example, we can observe immediately that
- ${\operatorname{Fol}(A_{-m}\cdots A_0)} \subseteq A_0$.
If we apply $n$ times the identity $A\cap f^{-1}(B)=f|_A^{-1}(B)$ to the definition of a follower set, we obtain ${\operatorname{Fol}(A_{-m}\cdots A_0)} = f^{m}\left[ f|_{A_{-m}}^{-1} f|_{A_{-m+1}}^{-1} \cdots f|_{A_{-1}}^{-1} A_0 \right]$. Since the monotone preimage of a connected set is connected, we can see that
- Each follower set is an interval.
Now consider the meaning of an arrow $\alpha\to\beta$ in $\mathcal{D}$. Write $\alpha=A_{-m}\cdots A_0$ and $\beta=\min(A_{-m}\cdots A_0 B)$. The definition of $\min(\cdot)$ gives ${\operatorname{Fol}(\beta)}={\operatorname{Fol}(A_{-m}\cdots A_0 B)} = f^{m+1}\left[ A_{-m}\cap \cdots \cap f^{-m}(A_0) \cap f^{-m-1}(B) \right]$. On the other hand, the image of the follower set of $\alpha$ is $f({\operatorname{Fol}(\alpha)})=f^{m+1}\left[ A_{-m}\cap \cdots \cap f^{-m}(A_0) \right]$, which proves the implication
- If there is an arrow $\alpha\to\beta$, then $f({\operatorname{Fol}(\alpha)}) \supseteq {\operatorname{Fol}(\beta)}$.
We also need a disjointness result. Fix a constraint word $\alpha\in\mathcal{C}$. If we form two more constraint words $\beta=\min(\alpha B)$ and $\gamma=\min(\alpha C)$ with $B\neq C$, then the disjointness of $B$ and $C$ gives disjointness of follower sets:
- If there are arrows $\alpha\to\beta$ and $\alpha\to\gamma$, $\beta\neq\gamma$, then ${\operatorname{Fol}(\beta)}\cap{\operatorname{Fol}(\gamma)}=\emptyset$.
If we combine all of these observations, we are ready to prove
\[lem:fol-loops\] To each length-$n$ loop from a vertex $\alpha$ to itself in the complete Markov diagram, there corresponds a subinterval of ${\operatorname{Fol}(\alpha)}$ which is mapped homeomorphically by $f^n$ onto ${\operatorname{Fol}(\alpha)}$. Moreover, the subintervals corresponding to distinct length-$n$ loops are pairwise disjoint.
Let $\alpha=\alpha_0\to\alpha_1\to\cdots\to\alpha_n=\alpha$ be a loop. Put $I={\operatorname{Fol}(\alpha_0)}\cap f^{-1}{\operatorname{Fol}(\alpha_1)}\cap \cdots \cap f^{-n}{\operatorname{Fol}(\alpha_n)}$. Applying $n$ times the identity $A\cap f^{-1}(B)=f|_A^{-1}(B)$, this becomes $$I=f|_{{\operatorname{Fol}(\alpha_0)}}^{-1} f|_{{\operatorname{Fol}(\alpha_1)}}^{-1} \cdots f|_{{\operatorname{Fol}(\alpha_{n-1})}}^{-1} {\operatorname{Fol}(\alpha_n)}.$$ Since the image of each follower set in the loop contains the next one, and since $f$ restricted to each follower set is monotone, we see that $f^n$ maps the interval $I$ monotonically and surjectively (i.e., homeomorphically) onto ${\operatorname{Fol}(\alpha_n)}$.
Now let $\alpha=\beta_0\to\beta_1\to\cdots\to\beta_{n-1}\to\beta_n=\alpha$ be another length-$n$ loop which starts and ends at the same vertex $\alpha$, and let $J$ be the corresponding subinterval. Let $k$ be the minimum index so that $\alpha_k\neq\beta_k$. By hypothesis, $1\leq k\leq n-1$. Thus the two arrows $\alpha_{k-1}\to\alpha_k$ and $\beta_{k-1}\to\beta_k$ originate from the same vertex, so that ${\operatorname{Fol}(\alpha_k)}\cap{\operatorname{Fol}(\beta_k)}=\emptyset$. But $f^k(I)\subseteq{\operatorname{Fol}(\alpha_k)}$ and $f^k(J)\subseteq{\operatorname{Fol}(\beta_k)}$. Therefore $I\cap J=\emptyset$ also.
Counting Preimages {#counting-preimages .unnumbered}
------------------
\
We now have all the tools we need to prove Theorem \[th:preimages\], which we restate here for the reader’s convenience.
Let $f\in{\mathcal{T}}$ and fix $x\in[0,1]$. Then $$\limsup_{n\to\infty} \frac{\# f^{-n}(x)}{e^{n h(f)}} > 0.$$
Let $\mathcal{D}$ be the complete Markov diagram of $f$ and $\mathcal{D}_0$ the subgraph promised by Proposition \[prop:subgraph\]. Fix a vertex $\alpha=A_{-m}\cdots A_0 \in\mathcal{D}_0$. Under a transitive piecewise monotone interval map, every point has a dense set of preimages [@Ru see Theorem 2.19 and Proposition 2.34]. Therefore, we can find a natural number $n_0$ such that $f^{-n_0}(x)\cap\langle A_{-m}\cdots \underline{A_0}\rangle\neq\emptyset$. Let $l_n$ be the number of loops of length $n$ in $\mathcal{D}_0$ which start and end at $\alpha$. By Lemma \[lem:fol-loops\] we get $\#f^{-n-n_0}(x)\geq l_n$, so using positive recurrence we get $$\limsup_{n\to\infty} \frac{\# f^{-n}(x)}{e^{nh(f)}} \geq
\limsup_{n\to\infty} \frac{l_{n-n_0}}{e^{nh(\Sigma_{\mathcal{D}_0})}} > 0. \qedhere$$
Equicontinuity {#sec:equicontinuous}
==============
To prove our equicontinuity result, we need to understand the behavior of critical points under perturbation in the space ${\mathcal{T}}_m$. We will use the following notation for the $\zeta$-neighborhood of a point $f$ in ${\mathcal{T}}_m$, $$N_\zeta(f):=\{g\in{\mathcal{T}}_m \mid d(f,g)<\zeta\}.$$
The following lemma records the simple observation that critical points vary continuously under perturbation in ${\mathcal{T}}_m$.
\[lem:perturbcrit\] *“Critical points vary continuously”*\
Fix $f\in{\mathcal{T}}_m$ and $\rho>0$. Then there exists $\zeta>0$ such that if $g\in N_\zeta(f)$, then there is a bijection $\operatorname{Crit}(f)\to\operatorname{Crit}(g)$, $c\mapsto c'$, such that $0'=0$, $1'=1$, and $|c-c'|<\rho$ for all $c\in\operatorname{Crit}(f)$.
Choose $\rho'<\rho$ less than half the distance between any two adjacent points in $\operatorname{Crit}(f)$. Now choose $\zeta>0$ sufficiently small so that $\zeta<\frac12|f(c)-f(w)|$ for $c\in \operatorname{Crit}(f)\setminus\{0,1\}$, $w=c\pm\rho'$.
Suppose $g\in N_\zeta(f)$. We construct the bijection $\operatorname{Crit}(f)\to\operatorname{Crit}(g)$. Put $0'=0$ and $1'=1$. Now let $c\in\operatorname{Crit}(f)\setminus\{0,1\}$. By the choice of $\zeta$, the values $g(c-\rho')$, $g(c+\rho')$ both lie on the same side of $g(c)$. Therefore $g$ has a critical point $c'$ satisfying $|c-c'|<\rho'<\rho$. The mapping so defined is injective by the choice of $\rho'$. Since $f,g$ have the same modality, it is also surjective.
Now we give a known property of interval maps, as well as an upgraded version for use with perturbation. They concern the accessibility of endpoints of the interval from the interior of the interval.
\[lem:onto\] *[@M2 Lemma 15] “Accessibility of endpoints”*\
For all $f\in{\mathcal{T}}_m$, $f^2((0,1))=[0,1].$
\[lem:ontopert\] *“Equi-accessibility of endpoints”*\
For all $f\in{\mathcal{T}}_m$ there exist $\rho,\zeta>0$ such that for all $g\in N_\zeta(f)$, $g^2([\rho,1-\rho])=[0,1].$
We use the notation of “$\rho$-neighborhoods” for points $a$ and sets $A$, $$\begin{gathered}
N_\rho(a):=\{x\in[0,1] \mid |x-a|<\rho\} \\
N_\rho(A):=\{x\in[0,1] \mid |x-a|<\rho \text{ for some } a\in A\}.\end{gathered}$$ Choose $\rho>0$ small enough that $$\begin{aligned}
N_\rho(a)\cap N_\rho(\operatorname{Crit}(f)\setminus\{a\})&=\emptyset, \text{ and}\\
N_\rho(f(N_\rho(a))) \cap N_\rho(\operatorname{Crit}(f)\setminus\{f(a)\})&=\emptyset, \text{ for }a=0,1.\end{aligned}$$ Find $\zeta<\rho$ corresponding to $f$ and $\rho$ in Lemma \[lem:perturbcrit\]. Fix $g\in N_\zeta(f)$. Then for both $a=0,1$,
1. \[it:one\] The only critical point of $g$ in $N_\rho(a)$ is $a$, and
2. \[it:two\] $g$ has at most one critical point in $g(N_\rho(a))$ (i.e. $f(a)'$).
Lemma \[lem:onto\] applied to $g$ gives a pair of points $x,y\not\in\{0,1\}$ with $$x\mapsto g(x)\mapsto g^2(x)=0 \quad \text{and} \quad y\mapsto g(y)\mapsto g^2(y)=1.$$ Any point which maps into an endpoint of the interval $[0,1]$ must be critical. Therefore $g(x),g(y)\in\operatorname{Crit}(g)$. We claim that at least one of the points $x,g(x)$ belongs to $[\rho,1-\rho]$. Indeed, if $g(x)$ does not, then \[it:one\] implies $g(x)\in\{0,1\}$. Then $x\in\operatorname{Crit}(g)$, and \[it:one\] implies $x\in[\rho,1-\rho]$. The same argument shows that at least one of the points $y,g(y)$ belongs to $[\rho,1-\rho]$.
We finish the proof in cases.\
*Case 1:* $x,y\in[\rho,1-\rho]$. Then $g^2([\rho,1-\rho])=[0,1]$.\
*Case 2:* $g(x),g(y)\in[\rho,1-\rho]$. Then $g([\rho,1-\rho])=[0,1]$, so also $g^2([\rho,1-\rho])=[0,1]$.\
*Case 3:* $x,g(y)\in[\rho,1-\rho]$, $g(x),y\not\in[\rho,1-\rho]$. Then \[it:one\] implies $g(x)\in\{0,1\}$.
1. *Case 3a:* $g(x)=0$. Then $g([\rho,1-\rho])=[0,1]$, so also $g^2([\rho,1-\rho])=[0,1]$.
2. *Case 3b:* $g(x)=1$. By the intermediate value theorem there is a point $z\in(x,1)$ with $g(z)=x$. Since $1,z$ both map to critical points, \[it:two\] implies $z\not\in N_\rho(1)$. So $g^2([\rho,1-\rho])$ contains both $0=g^2(x)$ and $1=g^2(z)$.
*Case 4:* $x,g(y)\notin[\rho,1-\rho]$, $g(x),y\in[\rho,1-\rho]$. Analogous to Case 3.
The last ingredient we need is a kind of uniform locally eventually onto property. Unfortunately, it holds only in the subspace ${\mathcal{M}}_m\subset {\mathcal{T}}_m$ of maps which are topologically weak mixing. We remind the reader that $f$ is called (topologically) *weak mixing* if for any nonempty open sets $A,B,C,D$ there exists $n\geq0$ such that $f^n(A)\cap B\neq\emptyset$ and $f^n(C)\cap D\neq\emptyset$ simultaneously. The next two lemmas say that a piecewise monotone weak mixing interval map $f$ is uniformly locally eventually onto, and the nearby maps in ${\mathcal{T}}_m$ are even equi-uniformly locally eventually onto. We can formulate analogous results when $f$ is transitive but not weak mixing, but the statements and proofs are technically involved and offer little insight, and are therefore deferred to the appendix.
\[lem:unileo\] *[@Ru Lemma 2.28] “Uniformly locally eventually onto”*\
For all $f\in{\mathcal{M}}_m$ and all $\epsilon>0$ there exists $k\in\mathbb{N}$ such that for all $x,y\in[0,1]$, $$y-x>\epsilon \implies f^k([x,y])=[0,1].$$
\[lem:equiunileo\] *“Equi-uniformly locally eventually onto.”*\
For all $f\in{\mathcal{M}}_m$ and all $\epsilon>0$ there exist $k\in\mathbb{N}$ and $\eta>0$ such that for all $g\in N_\eta(f)$ and all $x,y\in[0,1]$, $$y-x>\epsilon \implies g^{k+2}([x,y])=[0,1].$$
Let $f,\epsilon$ be given. Choose $k$ as in Lemma \[lem:unileo\] and $\rho,\zeta$ as in Lemma \[lem:ontopert\]. Finally, choose $\eta<\zeta$ small enough that $d(f,g)<\eta$ implies $d(f^k,g^k)<\rho$. For each $g\in N_\eta(f)$ and $x,y\in[0,1]$ with $y-x>\epsilon$, we get $$g^{k+2}([x,y]) = g^2(g^k([x,y])) \supseteq g^2([\rho,1-\rho]) = [0,1].$$ The containment follows because $f^k([x,y])=[0,1]$ and $g^k$ is $\rho$-close to $f^k$. The final equality is just Lemma \[lem:ontopert\].
Now we are ready to formulate and prove our equicontinuity result.
If $K$ is a compact subset of ${\mathcal{T}}_m$, then $\Psi_m(K)$ is an equicontinuous family.
Topological entropy on the space of piecewise monotone maps of modality $m$ is not continuous, but jumps of entropy are bounded in the sense that $\limsup_{g\rightrightarrows f} h(g) \leq \max\{ h(f), \log 2 \}$, see [@M Theorem 1]. Thus each point $f\in K$ has a neighborhood on which $h$ is bounded above. By compactness, this implies that $h$ is bounded above on $K$, so put $\lambda = \sup \{ \exp h(f) \mid f\in K\}$. This $\lambda$ is a common Lipschitz constant for the constant slope models of all the maps in $K$.[^4]
Assume now that $K\subset \mathcal{M}_m$. The general case $K\subset \mathcal{T}_m$ is addressed in the appendix.
Fix $\epsilon>0$. The neighborhoods around each $f\in K$ guaranteed by Lemma \[lem:equiunileo\] form an open cover of $K$. Pass to a finite subcover and let $k_0$ be the maximum of the corresponding values of $k$. Fix $g\in K$. It belongs to one of those neighborhoods, so choosing $x,y\in[0,1]$ with $y-x>\epsilon$ we have $$g^{k_0+2}([x,y])=[0,1].$$ Write $\tilde{g}=\Phi_m(g)$ for the constant slope model and $\psi=\Psi_m(g)$ for the conjugating homeomorphism. Now we pass through the conjugacy $g=\psi \circ \tilde{g} \circ \psi^{-1}$ to obtain $$\tilde{g}^{k_0+2}([\psi^{-1}x, \psi^{-1}y])=[0,1].$$ But $\tilde{g}$ has Lipschitz constant $\lambda$, so an interval which it stretches to length $1$ in $k_0+2$ steps must have length at least $\lambda^{-k_0-2}$. Writing $\delta=\lambda^{-k_0-2}$ we have proved that $$\forall \epsilon>0\,\, \exists \delta>0\,\, \forall \psi\in\Psi_m(K)\,\, \forall x,y\in[0,1]\, :\, y-x>\epsilon \implies \psi^{-1} y - \psi^{-1} x > \delta.$$ But this says exactly that the family $\Psi_m(K)$ is equicontinuous.
\[cor:equicontinuous\] Suppose $g_n \rightrightarrows f$ in ${\mathcal{T}}_m$. Then the sequence of homeomorphisms $\psi_n=\Psi_m(g_n)$ is equicontinuous.
The set $K=\{g_1, g_2, \ldots\} \cup \{ f \}$ is compact.
Flat Spots {#sec:flatspots}
==========
Consider now the situation of a uniformly convergent sequence $g_n \rightrightarrows f$ in the space ${\mathcal{T}}_m$. We investigate the implications of this convergence on the corresponding constant slope models and conjugating homeomorphisms, as denoted in the following diagrams: $$\label{cd}
\begin{CD}
[0,1] @>\Phi(g_n)>> [0,1]\\
@V{\psi_n}VV @VV{\psi_n}V\\
[0,1] @>>g_n> [0,1]
\end{CD}
\qquad\qquad
\begin{CD}
[0,1] @>\Phi(f)>> [0,1]\\
@V{\phi}VV @VV{\phi}V\\
[0,1] @>>f> [0,1]
\end{CD}
\qquad\qquad
\arraycolsep=1.4pt
\begin{array}{rl}
\lambda_n&=\exp h(g_n) \\[1em]
\lambda&=\exp h(f)
\end{array}$$
In this section we consider what happens if the homeomorphisms $\psi_n=\Psi(g_n)$ in converge uniformly to a map $\psi$. A priori, $\psi$ is weakly monotone, but need not be a homeomorphism. It may have *flat points:* $$\operatorname{Flat}(\psi):=\{x \,|\, \psi \text{ is constant on some neighborhood of } x\}$$ A *flat value* is any element of the set $\psi(\operatorname{Flat}\psi)$.
The remaining properties of $\psi$ can be more or less guessed if we imagine that the maps $\Phi(g_n)$ are converging uniformly to some constant slope extension of $f$ with $\psi$ giving the semiconjugacy. In reality, we are able to prove the following results.
\[lem:limpoints\] Suppose in that $g_n\rightrightarrows f$ in ${\mathcal{T}}_m$ and $\psi_n\rightrightarrows \psi$. If a sequence of points $y_{n}\rightarrow y$, then all limit points of the sequence $(\psi^{-1}_{n}(y_{n}))$ are in $\psi^{-1}(y)$.
Choose $\epsilon>0$ arbitrarily and let $x_{n_i} \rightarrow x'$, where $x_{n_{i}} = \psi^{-1}_{n_{i}}(y_{n_{i}})$. Then for sufficiently large $i$ we have\
$|y - y_{n_{i}}| < \epsilon$ by convergence $y_{n_{i}} \rightarrow y$;\
$|\psi(x_{n_{i}}) - y_{n_{i}}| < \epsilon$ by uniform convergence $\psi_{n_{i}} \rightrightarrows \psi$ and $y_{n_{i}} = \psi_{n_{i}}(x_{n_{i}})$;\
$|\psi(x_{n_{i}}) - \psi(x')| < \epsilon$ by convergence $x_{n_{i}} \rightarrow x'$ and continuity of $\psi$.\
Combining these three inequalities $| y - \psi(x')| < 3 \epsilon$. Since $\epsilon$ was arbitrary, this shows $y = \psi(x')$, i.e., $x' \in \psi^{-1}(y)$.
\[prop:rectangles\] *“Growth of rectangles”*\
Suppose in that $g_n\rightrightarrows f$ in ${\mathcal{T}}_m$, $\psi_n \rightrightarrows \psi$, and $\lambda_n\rightarrow\lambda'$. If $y<y'$ are two points such that $[y,y'] \cap \operatorname{Crit}(f)=\emptyset$ and none of $y,y',f(y),f(y')$ are flat values of $\psi$, then $$|\psi^{-1}\circ f (y')-\psi^{-1}\circ f(y)| = \lambda' \cdot |\psi^{-1}(y')-\psi^{-1}(y)|.$$
The meaning of the proposition is visualized in Figure \[fig:rect\], which shows the graph of a purported limit map $\psi$. We imagine $f$ acting on the vertical axis. The dashed gray lines represent critical points of $f$. The meaning of the lemma, then, is that the rectangle determined by $f(y),f(y')$ is $\lambda'$ times wider than the rectangle determined by $y,y'$.
Graph of $\psi$\
![Growth of Rectangles[]{data-label="fig:rect"}](rectangles.jpg "fig:"){width="5cm"}
(0,0) (-151,58)[$y$]{} (-152,67)[$y'$]{} (-160,83)[$f(y)$]{} (-162,107)[$f(y')$]{}
Using Lemma \[lem:limpoints\] and then (\[cd\]) we have $$\begin{aligned}
|\psi^{-1}\circ f(y')-\psi^{-1}\circ f(y)| &=& \lim_{n\rightarrow\infty} |\psi^{-1}_{n}\circ g_{n}(y') - \psi^{-1}_{n}\circ g_{n}(y)| \\
&=& \lim_{n\rightarrow\infty} |\Phi(g_n) \circ \psi^{-1}_{n}(y') - \Phi(g_n) \circ \psi^{-1}_{n}(y)|\\\end{aligned}$$ Now since $\Phi(g_n)$ are constant slope maps and since by Lemma \[lem:perturbcrit\] for sufficiently large $n$ both $\psi^{-1}_{n}(y'),$ $\psi^{-1}_{n}(y)$ lie in the same lap of $\Phi(g_n)$, $$\begin{aligned}
|\psi^{-1}\circ f(y')-\psi^{-1}\circ f(y)| &=& \lim_{n\rightarrow\infty} \lambda_{n} \cdot |\psi^{-1}_{n}(y') - \psi^{-1}_{n}(y)|.\end{aligned}$$ We finish by using Lemma \[lem:limpoints\] again $$\begin{aligned}
|\psi^{-1}\circ f(y')-\psi^{-1}\circ f(y)| &=& \lambda' \cdot |\psi^{-1}(y') - \psi^{-1}(y)|.\end{aligned}$$
By letting $y,y'$ approach a point $b$ (possibly a flat value) from opposite sides, we prove
\[prop:flat\] *“Growth of flat spots”*\
Suppose in that $g_n\rightrightarrows f$ in ${\mathcal{T}}_m$, $\psi_n \rightrightarrows \psi$, and $\lambda_n\rightarrow\lambda'$.
1. \[it:noncritical\] If $b\notin\operatorname{Crit}(f)$, then $\operatorname{len}(\psi^{-1}(f(b)))=\lambda'\cdot\operatorname{len}(\psi^{-1}(b))$.
2. \[it:endpoint\] If $b\in\{0,1\}$, then $\operatorname{len}(\psi^{-1}(f(b)))\geq\lambda'\cdot\operatorname{len}(\psi^{-1}(b))$.
\(a) Let $(y_{i})$ be an increasing sequence and let $(y'_{i})$ be a decreasing sequence both with limit equal to $b$, moreover such that $[y_{1},y'_{1}] \cap \operatorname{Crit}(f) = \emptyset$ and none of $y,y',f(y),f(y')$ are flat values of $\psi$. Then $f(y_{i})$, $f(y'_{i})$ converge to $f(b)$ from opposite sides. Now $$\operatorname{len}\psi^{-1}(b) = \lim_{i\rightarrow\infty} |\psi^{-1}(y'_{i}) - \psi^{-1}(y_{i})|.$$ Using the previous proposition we arrive at $$\operatorname{len}\psi^{-1}(f(b)) = \lim_{i\rightarrow\infty} |\psi^{-1}(f(y'_{i})) - \psi^{-1}(f(y_{i}))|
= \lambda' \cdot \lim_{i\rightarrow\infty} |\psi^{-1}(y'_{i}) - \psi^{-1}(y_{i})| = \lambda' \cdot \operatorname{len}\psi^{-1}(b).$$
\(b) We proof the assertion for the case $b=0$. Using Lemma \[lem:limpoints\] all limit points of the sequence $(\psi^{-1}_{n}(g_n(0)))$ belong to the compact set $\psi^{-1}(f(0))$. So replacing $(\psi_{n})$ by a subsequence and taking into account $\psi_n \rightrightarrows \psi$ we may assume that $\psi^{-1}_{n}(g_{n}(0)) \rightarrow x \in \psi^{-1}(f(0))$. If $0<y$, $(0,y] \cap \operatorname{Crit}(f)=\emptyset$, and $\{y,f(y)\} \cap \psi(\operatorname{Flat}\psi) = \emptyset$, then $$\begin{aligned}
|\psi^{-1}\circ f(y) - x | &=& \lim_{n\rightarrow\infty} |\psi^{-1}_{n}\circ g_{n}(y) - \psi^{-1}_n\circ g_{n}(0)| \label{eq:prop-flat-a}\\
&=& \lim_{n\rightarrow\infty} |\Phi(g_n)\circ\psi^{-1}_{n}(y)-\Phi(g_n)\circ\psi^{-1}_{n}(0)|\label{eq:prop-flat-b}\\
&=& \lim_{n\rightarrow\infty} \lambda_{n} \cdot |\psi^{-1}_{n}(y) - \psi^{-1}_{n}(0)|\label{eq:prop-flat-c}\\
&=& \lambda' \cdot |\psi^{-1}(y) - 0|,\label{eq:prop-flat-d}\end{aligned}$$ see Figure \[fig:rect2\]. Equation (\[eq:prop-flat-a\]) holds by Lemma \[lem:limpoints\], (\[eq:prop-flat-b\]) holds by (\[cd\]), (\[eq:prop-flat-c\]) holds since all $\Phi(g_n)$ have constant slope, and again (\[eq:prop-flat-d\]) holds by Lemma \[lem:limpoints\].
Graph of $\psi$\
![Growth of Flat Spots[]{data-label="fig:rect2"}](rectangles2.jpg "fig:"){width="5cm"}
(0,0) (-150,0)[$0$]{} (-150,23)[$y$]{} (-160,91)[$f(0)$]{} (-160,109)[$f(y)$]{} (-57,-4)[$x$]{}
Now if $y_{i}$ converges to zero, $0<y_{i}$, $(0,y_{i}] \cap \operatorname{Crit}(f)=\emptyset$, and $\{y_{i},f(y_{i})\} \cap \psi(\operatorname{Flat}\psi) = \emptyset$ for all $i$, then $f(y_{i})$ converges to $f(0)$ from one side. Therefore $$\operatorname{len}(\psi^{-1}(0)) = \lim_{i\rightarrow\infty} |\psi^{-1}(y_{i})|$$ and $$\operatorname{len}(\psi^{-1}(f(0))) \geq \lim_{i\rightarrow\infty} |\psi^{-1}(f(y_{i})) - x| =
\lim_{i\rightarrow\infty} \lambda' |\psi^{-1}(y_{i}) - 0| = \lambda' \cdot \operatorname{len}(\psi^{-1}(0)).$$
\[rem:c\] One might also consider the growth of flat spots of $\psi$ at critical points of $f$. Unfortunately, because of the folding, the strongest result we could prove was
1. If $b\in\operatorname{Crit}(f)$, then $\operatorname{len}(\psi^{-1}(f(b)))\geq\frac{\lambda'}{2}\cdot\operatorname{len}(\psi^{-1}(b))$.
This result has no value for us, since $\lambda'/2$ may be less than $1$.
No Flat Spots {#sec:noflatspots}
=============
\[prop:homeomorphism\] Suppose in that $g_n\rightrightarrows f$ in ${\mathcal{T}}_m$, $\psi_n\rightrightarrows \psi$, and $h(g_n)\to h(f)$. Then $\psi$ is a homeomorphism.
Here is the idea of the argument. We want to show that $\psi$ has no flat spots. Suppose to the contrary that $\psi$ collapses an interval of length $l>0$ to a single point $b$. As long as we avoid critical points, we can use Proposition \[prop:flat\] to produce other flat spots. The natural choice is to work forward along the orbit $b, f(b), \ldots$ until we find a flat spot of length $\lambda^n l > 1$, as in Table \[tab:flat1\]. Unfortunately, this argument, even using Remark \[rem:c\], does not rule out flat spots on periodic orbits containing critical points. So instead we work backward along an orbit, producing flat spots at all points along a whole tree of preimages, as in Table \[tab:flat2\].
[Y[.125]{} Y[0.50]{} Y[0.25]{}]{} & Flat spot & Length & $b$ & $l$ & $f(b)$ & $\lambda l$ & $f^2(b)$ & $\lambda^2 l$ & $\vdots$ & $\vdots$
[Y[0.67]{} Y[0.33]{}]{} Flat spots & Length $b$ & $l$ Each point of $f^{-1}(b)$ & $\lambda^{-1}l$ Each point of $f^{-2}(b)$ & $\lambda^{-2}l$ $\vdots$ & $\vdots$
Suppose to the contrary that $\operatorname{Flat}(\psi)\neq\emptyset$. Choose $b\in\psi(\operatorname{Flat}\psi)$ and suppose for now that $b\notin\{0,1\}$. Let $\Gamma = (V,E)$ be the smallest directed graph such that $b \in V$ and if $v \in V$, $w\in[0,1]\setminus\operatorname{Crit}(f)$, and $f(w)=v$, then $w \in V$ and $(w,v) \in E$. For any $v \in V$ write $|v|=\operatorname{len}\psi^{-1}(v)$ for the length of the flat spot. By Proposition \[prop:flat\], $$\label{eq:prop-noflat-a}
\text{if } (w, v) \in E, \text{ then } |w| = \lambda^{-1} |v|.$$ By supposition $|b|>0$ and consequently $|v|>0$ for all $v \in V$. From (\[eq:prop-noflat-a\]) we get that $\Gamma$ contains no loops (because $\lambda=\exp h(f)>1$), i.e. $\Gamma$ is a tree.
Unfortunately, we do not know if all preimages of $b$ are contained in this tree; we still need to avoid critical points. Since $f$ is transitive and piecewise-monotone, each point of $(0,1)$ has non-critical preimages. It follows that there is a backward orbit $$\dots \mapsto b_{-3} \mapsto b_{-2} \mapsto b_{-1} \mapsto b_{0}$$ with $(b_{-n-1},b_{-n}) \in E$ for all $n \in \mathbb{N}$. In other words, the set $B=\{b_{-1},b_{-2},\ldots\}$ contains no critical points of $f$. Choose $n_0\geq1$ such that $$\label{eq:prop-noflat-b}
\begin{gathered}
\text{for each }c\in\operatorname{Crit}(f)\text{ whose forward orbit enters }B,\\
\text{ if the first entrance is at }b_{-m}\text{, then }m<n_0.
\end{gathered}$$ We will prove that $$\label{eq:prop-noflat-c}
b_{-n_0}\text{ is not in the forward orbit of any critical point.}$$
For suppose to the contrary that $f^t(c)=b_{-n_0}$, $c\in\operatorname{Crit}(f)$. Choose $r<t$ maximal such that $f^r(c)\in\operatorname{Crit}(f)$. Choose $s>r$ minimal such that $f^s(c)\in B$. By , $f^s(c)=b_{-m}$ with $m<n_0$. $$c \mapsto \cdots \mapsto
\underbrace{f^r(c)}_{\mathclap{\substack{\text{last critical}\\\text{point before $b_{-n_0}$}}}}
\mapsto
\overbrace{
\cdots \mapsto
\underbrace{f^s(c)=b_{-m}}_{\mathclap{\substack{\text{first entrance to $B$}\\\text{after $f^r(c)$}}}}
\mapsto \cdots
}^{\substack{\text{No critical points}\\\text{by maximality of $r$}}}
\mapsto
\overbrace{
f^t(c)=b_{-n_0} \mapsto \cdots \mapsto b_{-m}
}^{\substack{\text{No critical points}\\\text{by construction of $B$}}}$$ This gives us a periodic point $b_{-m}\in V$ with no critical points in its periodic orbit. This contradicts the fact that $\Gamma$ contains no loops, proving .
Now we add together the lengths of the flat spots at all preimages of $b_{-n_0}$. By we have $f^{-i}(b_{-n_0})\subset V$ for all $i\in\mathbb{N}$. By the absence of loops in $\Gamma$ we have $f^{-i}(b_{-n_0})\cap f^{-j}(b_{-n_0}) = \emptyset$ for $i\neq j$. When we apply and Theorem \[th:preimages\] we get $$\sum_{v\in V} |v| \geq |b_{-n_0}| \sum_{i\in \mathbb{N}} \lambda^{-i} \# f^{-i}(b_{-n_0}) = \infty.$$ This is our contradiction, since the flat spots of $\psi$ are pairwise disjoint intervals contained in $[0,1]$.
Finally, consider the case when $b \in \{0,1\}$. If $f(b)$ belongs to $(0,1)$, then by Proposition \[prop:flat\] \[it:endpoint\] we may replace $b$ with $f(b)$ and proceed as before. If both $b, f(b) \in \{0,1\}$ but $f^2(b)$ belongs to $(0,1)$, then we use two applications of Proposition \[prop:flat\] \[it:endpoint\] to replace $b$ with $f^2(b)$ and proceed as before.
If all three points $b, f(b), f^2(b) \in \{0,1\}$, then two of these three points must coincide, which is impossible since by Proposition \[prop:flat\] \[it:endpoint\] we have $\operatorname{len}(\psi^{-1}(b)) < \operatorname{len}(\psi^{-1}(f(b))) < \operatorname{len}(\psi^{-1}(f^{2}(b)))$.
Concluding Arguments {#sec:wrapup}
====================
\[lem:joint-cont\] Composition $(f_{1},f_{2}) \mapsto f_{1}\circ f_{2}$ is jointly continuous as a map $\mathcal{C}^0\times\mathcal{C}^0 \rightarrow \mathcal{C}^0$. Inversion $\psi \mapsto \psi^{-1}$ is continuous as a map $\mathcal{H}^+ \to \mathcal{H}^+$.
We could not find the first statement anywhere, so we prove it here. Let $f_{1},f_{2}$ be given. Fix $\epsilon > 0$. Uniform continuity of $f_{1}$ give us $\eta$ such that if $|x-x'| < \eta$ then $|f_{1}(x)-f_{1}(x)'|<\frac{1}{2}\epsilon$. If $f'_{1},f'_{2} \in \mathcal{C}^0$ are such that $d(f_{1},f'_{1}) < \frac12\epsilon$ and $d(f_{2},f'_{2}) < \eta$ then $$|f'_{1}\circ f'_{2}(x) - f_{1}\circ f_{2}(x)| \leq
|f'_{1}\circ f'_{2}(x) - f_{1}\circ f'_{2}(x)| + |f_{1}\circ f'_{2}(x) - f_{1}\circ f_{2}(x)| <
\frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$ Therefore $d(f'_{1}\circ f'_{2}, f_{1}\circ f_{2}) < \epsilon$.
The second statement is similar. A proof appears in [@KMS Lemma 3.1 (c)]
\[lem:checkCS\] If $f$ is piecewise monotone, $\psi\in\mathcal{H}^+$, $\lambda>1$, and $$\label{checkCS} |\psi^{-1}\circ f(y') - \psi^{-1}\circ f(y)| = \lambda \cdot |\psi^{-1} y' - \psi^{-1} y|$$ holds for all $y<y'$ such that $[y,y']\cap\operatorname{Crit}(f)=\emptyset$, then $\psi^{-1}\circ f\circ\psi$ has constant slope $\lambda$.
This is a simplified version of [@ALM Lemma 4.6.4], but we include the proof for the reader’s convenience. Write $\tilde{f}=\psi^{-1}\circ f\circ\psi$. Two points $x<x'$ belong to the interior of a lap of monotonicity of $\tilde{f}$ if and only if the corresponding points $y=\psi(x), y'=\psi(x')$ satisfy $[y,y']\cap\operatorname{Crit}(f)=\emptyset$, and in this case Equation reduces to $$|\tilde{f}(x')-\tilde{f}(x)| = \lambda \cdot |x'-x|,$$ which says exactly that $\tilde{f}$ has constant slope $\lambda$.
If a sequence of maps $g_n\in{\mathcal{T}}_m$ converges uniformly to $f\in{\mathcal{T}}_m$ and if $h(g_n)\to h(f)$, then the constant slope models $\Phi(g_n)$ converge uniformly to $\Phi(f)$.
We need to show uniform convergence $\Phi(g_n) \rightrightarrows \Phi(f)$, i.e. $$\psi^{-1}_n \circ g_n \circ \psi_n \rightrightarrows \phi^{-1} \circ f \circ \phi,$$ where $\psi_n=\Psi(g_n)$, $\phi=\Psi(f)$ as in . We already have $g_n\rightrightarrows f$ by hypothesis, so by Lemma \[lem:joint-cont\] it is enough to show uniform convergence $\psi_n \rightrightarrows \phi$.
By Theorem \[th:equicontinuous\], $\{\psi_n\}$ is an equicontinuous family. Let $\psi$ be any subsequential limit $\psi_{n_i} \rightrightarrows \psi$. By Proposition \[prop:homeomorphism\], $\psi$ is a homeomorphism, so we may consider the map $\psi^{-1} \circ f \circ \psi$. By Proposition \[prop:rectangles\] and Lemma \[lem:checkCS\] it has constant slope $\lambda=\exp h(f)=\lim_{n\to\infty} \exp h(g_n)$. By the uniqueness of constant slope models, $\psi^{-1}\circ f \circ \psi = \phi^{-1}\circ f \circ \phi$. By the uniqueness of the conjugating homeomorphism, $\psi=\phi$. Since this is true for every subsequential limit, it follows from equicontinuity that we have uniform convergence of the whole sequence $\psi_n \rightrightarrows \phi$, as desired.
Open Questions {#sec:further}
==============
There remains still the interesting question of what happens when we have convergence of maps to a limit $f$ in ${\mathcal{T}}_m$ without convergence of entropy. What other maps besides $\Phi(f)$ can the constant slope models converge to?
Characterize the set of all limit points of $\Phi(g)$ as $g\rightrightarrows f$ in ${\mathcal{T}}_m$. Is each limit point a constant slope extension of $f$? What other necessary or sufficient conditions are there? What slopes are possible?
We can also ask about stronger versions of continuity for the operator $\Phi_m$.
Is $\Phi_m$ locally Lipschitz or Holder continuous at continuity points of the entropy? For $m\leq4$ can we get also global Lipschitz or Holder continuity?
Appendix {#appendix .unnumbered}
========
The goal of this appendix is to finish the proof of Theorem \[th:equicontinuous\] by developing analogs of Lemmas \[lem:unileo\] and \[lem:equiunileo\] for maps which are transitive but not weak mixing. In essence, this is nothing more than a long technical obstacle, because for piecewise monotone interval maps, the various topological notions of a system’s indecomposability are very closely related. In particular, it is known that
- The notions of topological weak mixing, topological strong mixing, and topological exactness (the locally eventually onto property) coincide.
- If $f$ is transitive but not weak mixing, then it has a unique fixed point $e$, it interchanges $[0,e]$ with $[e,1]$, and both of the maps $f^2|_{[0,e]}$, $f^2|_{[e,1]}$ are weak mixing.
A nice exposition of these results can be found in Ruette’s textbook [@Ru Proposition 2.34 and Theorem 2.19]. We will use these results freely throughout this section.
Fix $f\in{\mathcal{T}}_m\setminus{\mathcal{M}}_m$ with unique fixed point $e$ and fix $\rho>0$. Then there exists $\zeta>0$ such that if $g\in N_\zeta(f)$, then $g$ has a unique fixed point $e'$ and $|e-e'|<\rho$.
The proof is similar to the proof of Lemma \[lem:perturbcrit\], and is left as an exercise for the reader.
If $f\in{\mathcal{T}}_m$ has a unique fixed point $e$, then $f([0,e])\supseteq [e,1]$ and $f^2([0,e)) \supseteq [0,e]$.
Notice that a map whose graph intersects the diagonal only once must lie above the diagonal to the left of this fixed point and below the diagonal to the right. Together with surjectivity, this implies that $f$ maps some point less than $e$ to $1$ and some point greater than $e$ to $0$. This gives the first result $f([0,e])\supseteq [e,1]$.
This also implies the containment $f^2([0,e))\supseteq [0,e)$. Now if $f$ is weak mixing, then $f^2$ is also weak mixing (iteration preserves the weak mixing property), hence $[0,e)$ is not invariant under $f^2$ and so the containment is strict $f^2([0,e))\supsetneq [0,e)$, yielding the second result.
Now suppose that $f\in{\mathcal{T}}_m\setminus{\mathcal{M}}_m$. Then the second iterate restricted to $[0,e]$ is locally eventually onto, so there is a minimal natural number $N$ with $f^{2N}([0,e))=[0,e]$. Surjectivity and the intermediate value theorem give us immediately the following containments $$[0,e) \subseteq f^2([0,e)) \subseteq f^4([0,e)) \subseteq \cdots,$$ and as soon as one of these containments is an equality, then all the following containments must be equalities also. Since $[0,e)$ omits only one point, we get $N\leq 1$
Fix $f\in{\mathcal{T}}_m\setminus{\mathcal{M}}_m$ with unique fixed point $e$. Then there exist $\rho,\zeta>0$ such that if $g\in N_\zeta(f)$, then $g^4([\rho,e-\rho])\cup g^5([\rho,e-\rho]) = [0,1]$.
We choose $\rho$ as in the proof of Lemma \[lem:ontopert\], but with the following additional requirement. Since $e$ is fixed and not critical, we may choose $\rho$ so that $$N_\rho(f^i(N_\rho(e))) \cap N_\rho(\operatorname{Crit}f)=\emptyset \text{ for }i=0,1,2,3.$$ Now choose $\zeta$ answering to $\rho$ in Lemmas \[lem:perturbcrit\] and \[lem:perturbcrit\]$'$, and satisfying $\zeta<\rho$ and $$d(f,g)<\zeta \implies d(f^i,g^i)<\rho \text{ for }i=0,1,2,3.$$ Now fix $g\in N_\zeta(f)$. Our choice of $\rho$ and $\zeta$ gives $$g^i(N_\rho(e)) \cap \operatorname{Crit}(g)=\emptyset \text{ for }i=0,1,2,3.$$ If a point $c$ is critical for $g^2$, then either $c$ or $g(c)$ is critical for $g$. Therefore $$\label{nocrit}
N_\rho(e) \cap \operatorname{Crit}(g^2) = \emptyset \text{ and } g^2(N_\rho(e)) \cap \operatorname{Crit}(g^2) = \emptyset.$$ By Lemma \[lem:perturbcrit\]$'$, $g$ has a unique fixed point $e'$ and $e'\in N_\rho(e)$. To finish the proof it suffices to show the following three containments:
1. \[it:step1\] $g^2([\rho,e-\rho]) \supseteq [0,g^2(e-\rho)]$,
2. \[it:step2\] $g^2([0,g^2(e-\rho)]) \supseteq [0,e']$, and
3. \[it:step3\] $g([0,e']) \supseteq [e',1]$,
By Lemma \[lem:ontopert\], there is $x\in[\rho,1-\rho]$ with $g^2(x)=0$. Then $x\in\operatorname{Crit}(g^2)$, so by , $x\notin N_\rho(e)$. Since $f^2([e,1])=[e,1]$ and $d(g^2,f^2)<\rho<e$, $x\notin [e,1]$ either. Therefore $x\in[\rho,e-\rho]$. This proves \[it:step1\].
By Lemma \[lem:onto\]$'$ applied to $g$ we can find $y<e'$ with $g^2(y)=\max g^2|_{[0,e']}\geq e'$. Since this is a local maximum we have $y\in\operatorname{Crit}(g^2)$. By neither $x$ nor $y$ belongs to $g^2(N_\rho(e))\supset (g^2(e-\rho),e')$. We have shown $0\leq x,y \leq g^2(e-\rho)$, which proves \[it:step2\].
Lemma \[lem:onto\]$'$ applied to $g$ gives \[it:step3\].
For all $f\in{\mathcal{T}}_m$ and all $\epsilon>0$ there exists $k\in\mathbb{N}$ such that for all $x,y\in[0,1]$, $$y-x>\epsilon \implies \exists i\in\{0,1\} : f^{2k+i}([x,y]) = [0,e].$$
Let $e$ be the unique fixed point of $f$. We may apply Lemma \[lem:unileo\] to $f^2$ on each side of the phase space $[0,e]$, $[e,1]$ separately. Fixing $\epsilon>0$ we find a natural number $k$ so that if $y-x>\epsilon/2$ and $x,y\in[0,e]$, then $f^{2k}([x,y])=[0,e]$, whereas if $y-x>\epsilon/2$ and $x,y\in[e,1]$, then $f^{2k}([x,y])=[e,1]$. But $f([e,1])=[0,e]$. Now if $y-x>\epsilon$, then either $x,y$ are both on the same side of $e$, or one of $e-x$, $y-e$ is greater than $\epsilon/2$. In either case the result follows.
For all $f\in{\mathcal{T}}_m\setminus{\mathcal{M}}_m$ and all $\epsilon>0$ there exist $k\in\mathbb{N}$ and $\eta>0$ such that for all $g\in N_\eta(f)$ and all $x,y\in[0,1]$, $$y-x>\epsilon \implies \exists i\in\{0,1\} : g^{2k+i+4}([x,y])\cup g^{2k+i+5}([x,y]) =[0,1].$$
Let $f,\epsilon$ be given. Let $e$ be the unique fixed point of $f$. Choose $k$ as in Lemma \[lem:unileo\]$'$ and $\rho,\zeta$ be as in Lemma \[lem:ontopert\]$'$. Choose $\eta<\zeta$ small enough that $d(f,g)<\eta$ implies $d(f^{2k+i},g^{2k+i})<\rho$ for $i=0,1$. Fix $g\in N_\eta(f)$ and $x,y\in [0,1]$ with $y-x>\epsilon$. Find $i\in\{0,1\}$ with $f^{2k+i}([x,y])=[0,e]$. Then $$g^4(g^{2k+i}([x,y])) \cup g^5(g^{2k+i}([x,y])) \supseteq g^4([\rho,e-\rho]) \cup g^5([\rho,e-\rho]) = [0,1].$$ This completes the proof.
Finally, we show how to complete the proof of Theorem \[th:equicontinuous\] in the general case $K\subset {\mathcal{T}}_m$, allowing for maps which are transitive but not weak mixing.
Fix $\epsilon>0$. The neighborhoods around each $f\in K$ guaranteed by Lemmas \[lem:equiunileo\] and \[lem:equiunileo\]$'$ form an open cover of $K$. Pass to a finite subcover and let $k_0$ be the maximum of the corresponding values of $k$. Let $g\in K$. It belongs to one of those neighborhoods, so choosing $x,y\in[0,1]$ with $y-x>\epsilon$ we have $$\exists k\leq k_0\,\, \exists i\in\{0,1\} : g^{k+2}([x,y])=[0,1] \text{ or } g^{2k+i+4}([x,y])\cup g^{2k+i+5}([x,y])=[0,1].$$ In either case, further applications of the map $g$ to both sides of the equation yields $$g^{2k_0+5}([x,y]) \cup g^{2k_0+6}([x,y]) = [0,1].$$ Write $\tilde{g}=\Phi_m(g)$ for the constant slope model and $\psi=\Psi_m(g)$ for the conjugating homeomorphism. Now we pass through the conjugacy $g=\psi \circ \tilde{g} \circ \psi^{-1}$ to obtain $$\tilde{g}^{2k_0+5}([\psi^{-1}x, \psi^{-1}y]) \cup \tilde{g}^{2k_0+6}([\psi^{-1}x, \psi^{-1}y])=[0,1].$$ Thus at least one of the two intervals in the union above has length $\geq\frac12$. But $\tilde{g}$ has Lipschitz constant $\lambda$, so an interval which it stretches to length $\frac12$ in at most $k_0+6$ steps must have length at least $\frac12\lambda^{-k_0-6}$. Writing $\delta=\frac12\lambda^{-k_0-6}$ we have proved that $$\forall \epsilon>0\,\, \exists \delta>0\,\, \forall \psi\in\Psi_m(K)\,\, \forall x,y\in[0,1]\, :\, y-x>\epsilon \implies \psi^{-1} y - \psi^{-1} x > \delta.$$ But this says exactly that the family $\Psi_m(K)$ is equicontinuous.
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[^1]: This is not explicitly stated in [@AM], but is an easy corollary of [@AM Theorem 8.2]. If we have two conjugating homeomorphisms $\tilde{f}=\psi^{-1}\circ f \circ \psi = \phi^{-1} \circ f \circ \phi$, then $\psi^{-1}\circ\phi\circ\tilde{f}\circ\phi^{-1}\circ\psi=\tilde{f}$, so that by [@AM Theorem 8.2], $\psi^{-1}\circ\phi={\textnormal{id}}$, that is, $\psi=\phi$.
[^2]: The term dynamical topology was coined in [@KMS]. Simply put, it means we investigate topological properties of spaces of maps and operators on those spaces which are defined in terms of their dynamical properties.
[^3]: In Hofbauer’s original work the vertices are the follower sets, rather than the constraint words. We follow Buzzi’s approach simply because his work contains the theorems we needed.
[^4]: Incidentally, this shows that $\Phi_m(K)$ is also an equicontinuous family. We remark, however, that the set of inverse homeomorphisms $\{\psi^{-1} | \psi\in\Psi(K)\}$ is not necessarily an equicontinuous family. That is why we defined the operator $\Psi$ in the direction we did, imagining a map’s constant slope model as an extension rather than as a factor.
|
---
abstract: 'We present numerical simulations of the spectral evolution and emission of [radio components in relativistic jets. We compute jet models by means of a relativistic hydrodynamics code. We have developed an algorithm (SPEV) for the transport of a population of non-thermal electrons including radiative losses. For large values of the ratio of gas pressure to magnetic field energy density, ${\alpha_{_{\rm B}}}\sim 6\times 10^4$, quiescent jet models show substantial spectral evolution, with observational consequences only above radio frequencies. Larger values of the magnetic field (${\alpha_{_{\rm B}}}\sim 6\times 10^2$), such that synchrotron losses are moderately important at radio frequencies, present a larger ratio of shocked-to-unshocked regions brightness than the models without radiative losses, despite the fact that they correspond to the same underlying hydrodynamic structure. We also show that jets with a positive photon spectral index result if the lower limit $\gamma_{\rm min}$ of the non-thermal particle energy distribution is large enough. A temporary increase of the Lorentz factor at the jet inlet produces a traveling perturbation that appears in the synthetic maps as a superluminal component. We show that trailing components can be originated not only in pressure matched jets, but also in over-pressured ones, where the existence of recollimation shocks does not allow for a direct identification of such features as Kelvin-Helmholtz modes, and its observational imprint depends on the observing frequency. If the magnetic field is large (${\alpha_{_{\rm B}}}\sim 6\times 10^2$), the spectral index in the rarefaction trailing the traveling perturbation does not change much with respect to the same model without any hydrodynamic perturbation. If the synchrotron losses are considered the spectral index displays a smaller value than in the corresponding region of the quiescent jet model.]{}'
author:
- 'P. Mimica, M.-A. Aloy, I. Agudo, J. M. Martí, J. L. Gómez'
- 'J. A. Miralles'
title: 'Spectral evolution of superluminal components in parsec-scale jets'
---
Introduction
============
Relativistic jets are routinely observed emerging from active galactic nuclei and microquasars, and presumably they are behind the phenomenology detected in gamma-ray bursts. It is a broadly recognized fact that the observed VLBI radio-maps of parsec-scale jets are not a direct map of the physical state (density, pressure, velocity, magnetic field) of the emitting plasma. The emission structure is greatly modified by the fact that a distant (Earth) observer detects the radiation emitted from a jet which moves at relativistic speed and forms a certain angle with respect to the line of sight. Time delays between different emitting regions, Doppler boosting and light aberration shape decisively the observed aspect of every time-dependent process in the jet. The observed patterns are also influenced by the travel path of the emitted radiation towards the observer since Faraday rotation and, most importantly, opacity modulate total intensity and polarization radio maps. Finally, there are other effects that can be very important for shaping VLBI observations which do not unambiguously depend on the hydrodynamic jet structure, namely, radiative losses, particle acceleration at shocks, pair formation, etc. In this work we try to account for some of these elements by means of numerical simulations.
The basis for currently accepted interpretation of the phenomenology of relativistic jets was set by [@BK79] and [@Koenigl81]. A number of analytic works have settled the basic understanding that accounts for the non-thermal synchrotron and inverse Compton emission of extragalactic jets (e.g., [@Marscher80]), as well as the spectral evolution of superluminal components in parsec-scale jets (e.g., [@BM76; @HAA85; @MGT92; @MG85]). Assuming kinematic jet models, the numerical implementation of these analytic results enables one to extensively test the most critical theoretical assumptions by comparison with the observed phenomenology [both for steady (e.g., [@DM88; @HAA89a; @HAA91; @GAM93; @GAM94; @Gomezetal94]) and unsteady jets (e.g., [@Jones88])]{}. Basically, the aforementioned numerical implementation consists on integrating the synchrotron transfer equations assuming that radiation originated from an idealized jet model and accounting for all the effects mentioned in the previous paragraph.
The advent of multidimensional relativistic (magneto-)hydrodynamic numerical codes has allowed to replace the previously used kinematic, steady jet models by multidimensional, time-dependent hydrodynamic models of thermal plasmas (for a review see, e.g., [@Gomez02]). The works of [@GMMIM95; @GMMIA97], (hereafter G95 and G97, respectively) [@DHO96] or [@KF96] compute the synchrotron emission of relativistic hydrodynamic jet models with suitable algorithms that account for a number of relativistic effects (e.g., Doppler boosting, light aberration, time delays, etc.). Their models assume that there exists a proportionality between the number and the energy density of the hydrodynamic (thermal) plasma and the corresponding number and energy density of the emitting population of non-thermal or supra-thermal particles. These authors assumed that the magnetic field was dynamically negligible, that the emitted radiation had no back-reaction on the dynamics, and that that synchrotron losses were negligible. All these assumptions are very reasonable for VLBI jets at radio observing frequencies if the jet magnetic field is sufficiently weak. Consistent with their assumptions, the former papers included neither a consistent spectral evolution of the non-thermal particle (NTP) population, nor the proper particle and energy transport along the jet.
[The]{} spectral evolution of NTPs and its transport in classical jets and radiogalaxies have been carried out by [@JRE99], [@Miconoetal99] and [@TJR01]. In these works a coupled evolution of a non-relativistic plasma along with a population of NTPs has been used to asses either the signatures of diffusive shock acceleration in radio galaxies [@JRE99; @TJR01] or the observational imprint of the non-linear saturation of Kelvin-Helmholtz (KH) modes developed by a perturbed beam [@Miconoetal99]. [@CM03] have also developed a scheme to perform multidimensional Newtonian magneto-hydrodynamical simulations coupled with stochastic differential equations adapted to test particle acceleration and transport in kilo-parsec scale jets. Dealing with the spectral evolution of NTPs is relevant in view of the multiband observations of extragalactic jets where, a significant aging of the emitting particles seems to be present at optical to X-ray frequencies (M87, [@HB97; @Marshalletal02]; Cen A, [@KFJM01]).
This paper builds upon the lines opened by G95 and G97. G95 concentrated on the emission properties from steady relativistic jets, focusing on the role played by the external medium in determining the jet opening angle and presence of standing shocks. G97 used a similar numerical procedure to study the ejection, structure, and evolution of superluminal components through variations in the ejection velocity at the jet inlet. [@Agudoetal01] discussed in detail how a single hydrodynamic perturbation triggers pinch body modes in a relativistic, axisymmetric beam which result in observable superluminal features trailing the main superluminal component. Finally, [@AMGAMI03] extended the work of [@Agudoetal01] to three-dimensional, helically perturbed beams. Here, we combine multidimensional relativistic models of compact jets with a new algorithm to compute the spectral evolution of supra-thermal particles evolving in its bosom, i.e., including their radiative losses, and their relevance for the emission and the spectral study of relativistic jets.
[This work is composed of two parts. In the first part, we present a new numerical scheme to evolve populations of relativistic electrons in relativistic hydrodynamical flows including radiative losses (§ \[sec:SPEV\]). For the purpose of calibration the new method, our work is]{} based upon the same axisymmetric, relativistic, hydrodynamic jet models as employed in G97. Using the same jet parameters allows us to quantify the relevance of including radiative losses and, along the way, to compare the emission properties of parsec-scale jets computed according to two different methods: (1) the new method presented in this paper and (2) the method presented in G95 and G97, to which we will refer, for simplicity, as [*Adiabatic Method*]{} (AM). [In the second part of the paper, we apply the new method to quantify the relevance of radiative losses in the evolution of both quiescent and dynamical jet models.]{} We will show (§ \[sec:radio\]) the regimes in which both approaches yield similar synthetic total intensity radio maps and when synchrotron losses modify substantially the results. We also show which are the key parameters to trigger a substantial NTP aging and, therefore, to significantly change the appearance of the radio maps corresponding to the same underlying, quiescent jet models. The spectral evolution of a hydrodynamic perturbation [travelling]{} downstream the jet, will be discussed in Sect. \[sec:spec\_evol\_component\]. Finally, we [discuss our main results and conclusions]{} in Sect. \[sec:conclusions\].
Hydrodynamic models {#sec:jets}
===================
[cccc]{} model & $P_b/P_a$ & $b_b$\[G\] & ${\alpha_{_{\rm B}}}$\
PM-S & 1.0 & 0.002 & $6\times 10^6$\
PM-L & 1.0 & 0.02 & $6\times 10^4$\
PM-H & 1.0 & 0.20 & $6\times 10^2$\
OP-L & 1.5 & 0.03 & $6\times 10^4$\
OP-H & 1.5 & 0.30 & $6\times 10^2$\
Two quiescent, relativistic, axisymmetric jet models constitute our basic hydrodynamic set up [(see Tab. \[tab:models\])]{}. They correspond to the same pressure-matched (PM), and over-pressured (OP) models of G97. The models were computed in cylindrical symmetry with the code RGENESIS [@MAMB04]. The computational domain spans $(10R_b\times 200R_b)$ in the $(r \times z)$-plane ($R_b$ is the beam cross-sectional radius at the injection position). A uniform resolution of 8 numerical cells/$R_b$ is used. The code module that integrates the relativistic hydrodynamics equations is a conservative, Eulerian implementation of a Godunov-type scheme with high-order spatial and temporal accuracy (based on the GENESIS code; [@AIMM99; @API99]). We follow the same nomenclature as G97 where quantities affected by subscripts [*a*]{}, [*b*]{} and [*p*]{} refer to variables of the atmosphere, of the beam at the injection nozzle and of the perturbation (§ \[sec:perturb\]), respectively. The jet material is represented by a diffuse ($\rho_b / \rho_a=10^{-3}$; $\rho$ being the rest-mass density), relativistic (Lorentz factor $\Gamma_b=4$) ideal gas of adiabatic exponent $4/3$, with a Mach number $M_b =1.69$. At the injection position, model PM has a pressure $P_b = P_a$, while model OP has $P_b = 1.5P_a$. Pressure in the atmosphere decays with distance $z$ according to $P(z) = P_a/[1 + (z/z_c)^{1.5}]^{1.53}$, where $z_c = 60R_b$. With such an atmospheric profile both jet models display a paraboloid shape, which introduces a small, distance-dependent, jet opening angle which is compatible with observations of parse-scale jets. At a distance of $200R_b$, the opening angles for the models PM and OP are $0.29^\circ$ and $0.43^\circ$, respectively. Pressure equilibrium in the atmosphere is ensured by including adequate counter-balancing, numerical source terms. However, despite the fact that the initial model is very close to equilibrium, small numerical imbalance of forces triggers a transient evolution that decays into a final quasi-steady state after roughly $2-5$ longitudinal grid light-crossing times. [We]{} treat these quiescent states as initial models. Model PM yields an [ adiabatically-expanding, smooth beam]{}. Model OP develops a collection of cross shocks in the beam, [whose spacing increases with the distance from the jet basis]{}.
Injection and Evolution of Hydrodynamic Perturbations {#sec:perturb}
-----------------------------------------------------
Variations in the injection velocity (Lorentz factor) have been suggested as a way to generate internal shocks in relativistic jets [@Rees78]. We set up a traveling perturbation in the jet as a sudden increase of the Lorentz factor at the jet nozzle (from $\Gamma_b=4$ to $\Gamma_p=10$) for a short period of time ($0.75R_b/c$; $c$ being the light speed). Since the injected perturbation is the same as in G97, its evolution is identical to the one these authors showed and, thus, we provide a brief overview here. The perturbation develops two Riemann fans emerging from its leading and rear edges (see, e.g., [@MAMB05; @MAM07]). In front of the perturbation a shock-contact discontinuity-shock structure (${\cal SCS}$) forms, while the rear edge is trailed by a rarefaction-contact discontinuity-rarefaction (${\cal RCR}$) fan. In the leading shocked region the beam expands radially owed to the pressure increase with respect to the atmosphere. In the trailing rarefied volume the beam shrinks radially on account of the smaller pressure in the beam than in the external medium. This excites the generation of pinch body modes in the beam that seem to trail the main hydrodynamic perturbation as pointed out by [@Agudoetal01]. Also the component itself splits in, at least, two parts when the forward moving rarefaction leaving the rear edge of the component merges with the reverse shock traveling backwards (in the component rest frame) that leaves from the forward edge of the hydrodynamic perturbation ([as in]{} [@AMGAMI03]).
SPEV: A new algorithm to follow non-thermal particle evolution {#sec:SPEV}
==============================================================
The [*sp*]{}ectral [*ev*]{}olution (SPEV) routines are a set of methods developed to follow the evolution of NTPs in the phase space. [Here]{} we assume that the radiative losses at radio frequencies are negligible with respect to the total thermal energy of the jet at every point in the jet. Thus, radiation back reaction onto the hydrodynamic evolution is neglected. Certainly, such an [ *ansatz*]{} is invalid at shorter wavelengths (optical, X-rays), where radiative losses shape the observed spectra (see, e.g., [@MAMB05] for X-ray-synchrotron blazar models that include the radiation back-reaction onto the component dynamics).
The [7-]{}dimensional space formed by the particle momenta, particle positions and time is split into two parts. For the spatial part of the phase space, we assume that NTPs do not diffuse in the hydrodynamic (thermal) plasma. Thereby, the spatial evolution of the NTPs is governed by the velocity field of the underlying fluid, and it implies that the NTP comoving frame is the same as the thermal fluid comoving frame. Assuming a negligible diffusion of NTPs is a sound approximation in most parts of our hydrodynamic models since the electron diffusion lengths are much smaller than the dynamical lengths in smooth flows (see, e.g., [@TJR01; @Miniati01]). Obviously, the assumption is not fulfilled wherever diffusive acceleration of NTPs takes place (e.g., at shocks or at the jet lateral boundaries). Nevertheless, there exists a strong mismatch between the scales relevant to dynamical and diffusive transport processes for NTPs of relevance to synchrotron radio-to-X-ray emissions within relativistic jets. The mismatch ensures that even in macroscopic, non-smooth regions such as the cross shocks in the beam of model OP, the assumption we have made suffices to provide a good qualitative description of the NTP population dynamics.
Consistent with the hydrodynamic discretization, we assume that the velocity field is uniform inside each numerical cell (equal to the average of the velocity inside such cell). In practice, a number of Lagrangian [*particles*]{} are introduced through the jet nozzle, each evolving the same NTP distribution but being spatially transported according to the local fluid conditions. We emphasize that these Lagrangian particles are used here for the solely purpose of representing the spatial evolution (i.e., the trajectories) of ensembles of NTPs. We integrate the trajectories of such particles using a conventional time-explicit, adaptive-step-size, fourth order Runge-Kutta (RK) integrator.
Particle Evolution in the momentum space
----------------------------------------
In order to derive the equations governing the time evolution of charged NTPs in the momentum space we follow closely the approach of [@MRL93] (see also [@Webb85], or [@Kirk94]). We start by considering the Boltzmann equation that obeys the ensemble averaged distribution function $f$ of the NTPs, each with a rest-mass $m_0$, $$p^\beta\left(\frac{\partial f}{\partial x^\beta}-
\Gamma^\alpha_{\beta\gamma}p^\gamma\frac{\partial f} {\partial p^\alpha}\right)
=\left(\frac{df}{d\tau}\right)_{coll},
\label{eq:Boltzman}$$ where $f$ is a function of the coordinates $x^\alpha$ and the components of the particle 4-momentum $p^\alpha$ with respect to the coordinate basis ${\bf e}_{(\alpha)}$. The $\Gamma^\alpha_{\beta\gamma}$ are the usual Christoffel symbols and the right hand side represents the collision term, $\tau$ being the particle proper time.
[Equation \[eq:Boltzman\]]{} can be written in terms of the particle 4-momentum components with respect to the comoving or matter frame instead of the components with respect to the coordinate basis. The comoving tetrad ${\bf e}_{(a)}$ ($a=0, 1, 2, 3$), is formed by four vectors, one of which (${\bf e}_{(0)}$) is the four velocity of the matter and the following orthonormality relation is fulfilled $$\begin{aligned}
{\bf e}_{(a)}\cdot{\bf e}_{(b)}=\eta_{ab},\end{aligned}$$ where $\eta_{ab}$ is the Minkowski metric ($\eta_{00}=-1$). We explicitly point out that the components of tensor quantities with respect to the coordinate and tetrad basis are annotated with Greek and Latin indices, respectively. The transformation between the basis ${\bf e}_{(\alpha)}$ and ${\bf e}_{(a)}$ is given by the matrix $e^\alpha_a$ and its inverse matrix $e'^a_\alpha$, $${\bf e}_{(a)}=e^\alpha_a {\bf e}_{(\alpha)}, \;\; {\bf e}_{(\alpha)}=e'^a_\alpha
{\bf e}_{(a)}$$
In terms of the comoving basis, the Boltzmann equation is $$p^b\left(e^\beta_b\frac{\partial f}{\partial x^\beta}-
\Gamma^a_{bc}p^c\frac{\partial f} {\partial p^a}\right)
=\left(\frac{\delta f}{\delta\tau}\right)_{\rm coll}.
\label{eq:Boltzmann}$$ [The connection coefficients in the tetrad frame $\Gamma^a_{bc}$ obey the following relations]{} $$\Gamma^a_{bc}=e^\beta_b e'^a_\alpha e^\alpha_{c;\beta}=e^\beta_b
e'^a_\alpha\left(e^\alpha_{c,\beta}+\Gamma^\alpha_{\beta\gamma}
e^\gamma_c\right),$$ where the comma [and the semicolon stand for partial and covariant derivatives, respectively]{}.
We introduce the two first moments of the distribution function by the equations $$n^a=\int d\Omega \frac{\rm p^2}{p^0}p^af,
\label{eq:n^a}$$ $$t^{ab}=\int d\Omega \frac{\rm p^2}{p^0}p^ap^bf,$$ where p$^2=(p^0)^2-m_0^2c^2$ is the square of the NTP three-momentum measured by the comoving observer. The solid-angle ($\Omega$) integrations are performed over all particle momentum directions. [The number of NTPs per unit volume with modulus of their three-momentum between p and p+$d{\rm p}$ for an observer comoving with the matter is $n^0({\rm p})d{\rm p}$]{}. Further integration of the above moments $n^a$ and $t^{ab}$ over p, $\int_0^\infty d{\rm p}$, [gives]{} the hydrodynamical moments.
In order to obtain the continuity equation for NTPs, we multiply the Boltzmann equation (\[eq:Boltzmann\]) by $({\rm p}^2 / p^0)$, and integrate over $\Omega$ to yield (for the details see [App. A of]{} [@Webb85]),
$$e^{\alpha}_a \frac{\partial n^a}{\partial x^\alpha} +
e^{\alpha}_{a; \alpha} n^a -
\frac{\partial}{\partial {\rm p}}
\left( \frac{p^0}{\rm p} \Gamma^0_{ab} t^{ab} \right) =
\int d\Omega\, \frac{{\rm
p}^2}{p^0} \left( \frac{\delta f}{\delta \tau}\right)_{\rm coll}\, .
\label{eq:cont}$$
The next step is to formulate the continuity equation in the diffusion approximation. Such approximation implies that the scattering of NTPs by hydromagnetic turbulence results in a quasi isotropic distribution function in the scattering (comoving) frame. Thus, it is assumed that the distribution function of the NTPs can be expressed as the sum of two terms, $f=f^{(0)}+f^{(1)}{\bf \Omega}$, where $f^{(0)}\gg f^{(1)}$ and ${\bf \Omega}$ is the unit vector in the direction of the momentum of the particle. With such an assumption, we obtain that $$n^0 \simeq 4\pi {\rm p}^2 f^{(0)} \gg n^i\, , \:\,\,\, t^{ij} \simeq \frac{{\rm
p}^2}{p^0}\frac{\delta^{ij}}{3} n^0\, , \:\,\,\, i,j=1,2,3\, ,
\label{eq:diffusion-approx1}$$ which also leads to $$t^{00}\simeq p^0 n^0\gg t^{0i}=t^{i0}.
\label{eq:diffusion-approx2}$$
Plugging the approximations (\[eq:diffusion-approx1\]) and (\[eq:diffusion-approx2\]) into Eq. (\[eq:cont\]) and neglecting the terms coming from the anisotropy of the distribution function, i.e., the terms arising from $f^{(1)}$, we obtain $$e^{\alpha}_0 \frac{\partial n^0}{\partial x^\alpha} +
e^{\alpha}_{0; \alpha} n^0 -
\frac{\partial}{\partial {\rm p}}
\left( \sum_{i=1}^3 \Gamma^i_{0i} {\rm p} \frac{n^0}{3} \right) =
\int d\Omega\, \frac{{\rm
p}^2}{p^0} \left( \frac{\delta f}{\delta \tau}\right)_{\rm coll}\, .
\label{eq:cont-diff_approx}$$
Equation (\[eq:cont-diff\_approx\]) is valid for any general metric $g_{\mu \nu}$. However, in the present work we are only interested in obtaining the transport equation for NTPs in the special relativistic regime. To restrict Eq. (\[eq:cont-diff\_approx\]) to such a regime we take a flat metric, $g_{\mu\nu} = \eta_{\mu \nu}$. [ Thereby,]{} the tetrad and the coordinate frame basis are related by a simple Lorentz transformation, i.e., $$\begin{aligned}
e^{0}_0 &=& \Gamma,\, \\
e^{0}_i &=& e^{i}_0 = \Gamma v^i, \, \\
e^{i}_j &=& \delta_{ij} + (\Gamma -1) \frac{v^i v^j}{v^2} \:\:\: (i,j=1,2,3),
\label{eq:lorentz_transformation}\end{aligned}$$ where $v^i$ $(i=1,2,3)$ are the spatial components of the velocity of matter, which is equal to the hydrodynamical velocity of the NTPs, since we make the assumption that NTPs do not diffuse in the hydrodynamic plasma. The hydrodynamical Lorentz factor of the plasma is denoted by $\Gamma=1/\sqrt{(1-v^iv_i)}$. With this transformation we obtain $$\begin{aligned}
e^{\alpha}_{0; \alpha} &=& e^{\alpha}_{0, \alpha} =
\frac{\partial \Gamma}{\partial t} + \frac{\partial \Gamma
v^i}{\partial x^i} = \Theta
\label{eq:lorentz_transformation2} \\
\sum_{i=1}^3 \Gamma^i_{0i} &=&
\frac{\partial \Gamma}{\partial t} + \frac{\partial \Gamma
v^i}{\partial x^i} = \Theta \, ,
\label{eq:lorentz_transformation3}\end{aligned}$$ $\Theta$ being the expansion of the underlying thermal fluid, which is related to [$\rho$]{} by $$\begin{aligned}
\Theta = -\frac{D \ln{\rho}}{D\tau}\, .
\label{eq:expansion}\end{aligned}$$
Plugging Eqs. (\[eq:lorentz\_transformation2\])-(\[eq:expansion\]) into Eq. (\[eq:cont-diff\_approx\]) and using the definition of the Lagrangian derivative with respect to the proper time of the comoving observer $$\begin{aligned}
\frac{D}{D\tau} = \Gamma \left(
\frac{\partial}{\partial t} + v^i \frac{\partial}{\partial x^i}
\right)\, ,\end{aligned}$$ yields $$\frac{D n^0}{D \tau} + \Theta n^0 -
\frac{\partial}{\partial {\rm p}}
\left( \frac{n^0}{3} {\rm p} \Theta \right) =
\int d\Omega\, \frac{{\rm
p}^2}{p^0} \left( \frac{\delta f}{\delta \tau}\right)_{\rm coll}\, ,
\label{eq:cont-diff_approx_SR_0}$$ which can be cast in the form $$\frac{D \ln{n^0}}{D \tau} -
\frac{{\rm p}}{3} \Theta \frac{\partial
\ln{n^0}}{\partial {\rm p}}
+ \frac{2}{3}\Theta
= \frac{1}{n^0}
\int d\Omega\, \frac{{\rm
p}^2}{p^0} \left( \frac{\delta f}{\delta \tau}\right)_{\rm coll}\, .
\label{eq:cont-diff_approx_SR_1}$$
The collision term contains the interaction between NTPs and matter, radiative losses due to synchrotron processes, etc. Let us consider first the interaction with matter. In this case, the collisions can be assumed to be isotropic in the comoving frame and elastic. In such a case, and consistently to the previous approximation, the collision term in Eq. (\[eq:cont-diff\_approx\_SR\_1\]) vanishes and we can find a solution for the homogeneous differential equation by considering $$\begin{aligned}
\frac{D \ln{n^0}}{D \tau} -
\frac{{\rm p}}{3} \Theta \frac{\partial
\ln{n^0}}{\partial {\rm p}} \, ,\end{aligned}$$ as the derivative of $\ln{n^0}$ along the following curve in the plane ($\tau$,p), parametrized by $\sigma$ $$\begin{aligned}
\frac{d \tau}{d\sigma} & = &1 \\
\frac{d {\rm p}}{d \sigma} &=& -\frac{{\rm p}}{3} \Theta\, ,\end{aligned}$$ i.e., we may write Eq. (\[eq:cont-diff\_approx\_SR\_1\]) as $$\begin{aligned}
\frac{d \ln{n^0}}{d\sigma} & =& -\frac{2}{3} \Theta\, .
\label{eq:deriv2}\end{aligned}$$ The solution of Eq. (\[eq:deriv2\]), is $$\begin{aligned}
n^0(\tau(\sigma), p(\sigma)) = n^0(\tau(\sigma_0), p(\sigma_0))
\left( \frac{\rho(\tau(\sigma))}{\rho(\tau(\sigma_0))} \right)^{\frac{2}{3}}\, ,
\label{eq:n^0_homogeneous}\end{aligned}$$ where $\sigma_0$ corresponds to some initial value of the parameter $\sigma$. Equation (\[eq:n\^0\_homogeneous\]) expresses the fact that the variation of the number of NTPs per unit of volume along a certain curve (parametrized by $\sigma$) is directly related with the variation of the rest-mass density of the thermal plasma between the initial and final points of such a curve.
We now turn back to Eq. (\[eq:cont-diff\_approx\_SR\_1\]) and derive the form of the collisions term in the case that the only relevant radiative losses are due to synchrotron processes. In such a case we have [e.g., @RL79] $$\left( \frac{d{\rm p}}{d\tau} \right)_{\rm syn} =
- \frac{4 \sigma_{\rm T} {\rm p}^2 U_{\rm B}}{3 m_{\rm e}^2 c^2}
:= {\cal B} ({\rm p},\tau)\, ,
\label{eq:dpdt_syn}$$ where $\sigma_{\rm T}$ is the Thompson cross section, $m_{\rm e}$ is the electron rest-mass, $U_B={\bf b}^2/8\pi$ is the magnetic energy density, and we have assumed that the electrons are ultrarelativistic, $p \approx \gamma m_{\rm e}$, $\gamma$ [being]{} the electron Lorentz factor (not to be confused with the plasma Lorentz factor). [If $\Theta=0$, Eq. (\[eq:cont-diff\_approx\_SR\_1\]) reads in the comoving frame]{} $$\left( \frac{D \ln{n^0}}{D\tau} \right)_{\rm syn} = \frac{1}{n^0}
\int d\Omega\, \frac{{\rm
p}^2}{p^0} \left( \frac{\delta f}{\delta \tau}\right)_{\rm coll}
\label{eq:dndtau_syn}$$ and, on the other hand, the particle number conservation yields $$\begin{aligned}
\left( \frac{Dn^0}{D\tau} \right)_{\rm syn} = -
\frac{\partial }{\partial {\rm p}} \left( n^0 {\cal B}\right)\, ,
$$ or, equivalently, $$\left( \frac{D \ln{n^0}}{D\tau} \right)_{\rm syn} =
- \frac{\partial {\cal B}}{\partial {\rm p}}
- {\cal B} \frac{\partial \ln{n^0}}{\partial {\rm p}}\, .
\label{eq:dndtau_syn2}$$
Taking into account Eq. (\[eq:dndtau\_syn\]), we may plug Eq. (\[eq:dndtau\_syn2\]) into Eq. (\[eq:cont-diff\_approx\_SR\_1\]) to account for the combined effects of synchrotron losses and adiabatic expansion/compression of the fluid $$\frac{D \ln{n^0}}{D \tau} +
\left( -\frac{{\rm p}}{3} \Theta +
{\cal B} \right) \frac{\partial \ln{n^0}}{\partial {\rm p}}
= - \frac{2}{3}\Theta -
\frac{\partial {\cal B}}{\partial {\rm p}}\, .
\label{eq:adiabatic_syn}$$ The formal solution of Eq. (\[eq:adiabatic\_syn\]), can be found following the same procedure we used above for the homogeneous continuity equation. In this case we interpret $$\begin{aligned}
\frac{D \ln{n^0}}{D \tau} +
\left( -\frac{{\rm p}}{3} \Theta +
{\cal B} \right) \frac{\partial \ln{n^0}}{\partial {\rm
p}}\, ,\end{aligned}$$ as the derivative of $\ln{n^0}$ along the curve $$\begin{aligned}
\frac{d \tau}{d\sigma} & = &1 \nonumber \\
\frac{d {\rm p}}{d \sigma} &=& -\frac{{\rm p}}{3} \Theta +
{\cal B}\, ,
\label{eq:dpdsigma}\end{aligned}$$ which yields, on the one hand, $$\begin{aligned}
\frac{d {\rm p}}{d \tau} &=& -\frac{{\rm p}}{3} \Theta +
{\cal B}({\rm p},\tau)\, ,
\label{eq:dpdt}\end{aligned}$$ and, on the other hand, $$\begin{aligned}
n^0(\tau(\sigma), p(\sigma)) & =& n^0(\tau(\sigma_0), p(\sigma_0))
\left(
\frac{\rho(\tau(\sigma))}{\rho(\tau(\sigma_0))}\right)^{\frac{2}{3}}
\nonumber \\
& & \times \exp{\left(-\int_{\sigma_0}^{\sigma} d\sigma' \frac{\partial
{\cal B}({\rm p},\tau)}{\partial {\rm p}}(\sigma')\right)}\, .
\label{eq:n^0}\end{aligned}$$ Equation (\[eq:dpdt\]) shows the evolution of the particle momentum in time, while Eq. (\[eq:n\^0\]) is only a formal solution since the exact dependence of ${\rm p}(\sigma)$, necessary to perform the integration, is only known through the differential equation (\[eq:dpdsigma\]). The first term on the right hand side of Eq. (\[eq:dpdt\]) accounts for the change of momentum due to the adiabatic expansion or compression of the fluid in which NTPs are embedded. The time dependence of ${\cal B}$ is fixed by the hydrodynamic properties of the thermal fluid and by the comoving magnetic field ${\bf b}$, assumed to be provided by hydrodynamic simulations and models of the ${\bf b}$-field (which is not directly simulated), respectively.
In order to speed up the numerical evaluation of Eqs. (\[eq:dpdt\]) and (\[eq:n\^0\]), we assume that both, the fluid expansion and the synchrotron losses (or, equivalently, [$U_{\rm B}$]{}) are constant within an small interval of proper time around $\tau(\sigma_0)$. Thus, we can write Eq. (\[eq:dpdsigma\]) as $$\begin{aligned}
\frac{d {\rm p}}{d \sigma} &=& k_{\rm a} {\rm p} - k_{\rm s} {\rm
p}^2\, ,
\label{eq:dpdsigma2}\end{aligned}$$ $k_{\rm a}$ and $k_{\rm s}$ being both constants, such that the following relations hold $$\begin{aligned}
\frac{\rho(\tau(\sigma))}{\rho(\tau(\sigma_0))} &=& {\rm e}{^{3k_{\rm
a}\Delta\sigma}} \\ \label{eq:ka_ks}
{\cal B}({\rm p}(\sigma),\tau(\sigma)) &=& -k_{\rm s}{\rm p}^2\, ,\end{aligned}$$ with $\Delta\sigma=\sigma - \sigma_0$. Equation (\[eq:dpdsigma2\]) has the following analytic solution $${\rm p}(\sigma) = {\rm p}_0 \frac{k_{\rm a} {\rm e}^{k_{\rm a}\Delta\sigma}}{k_{\rm a} + {\rm p}_0k_{\rm s} \left( {\rm e}^{k_{\rm
a}\Delta\sigma} - 1\right)}\, ,
\label{eq:p(sigma)}$$ where ${\rm p}_0 := {\rm p}(\sigma_0)$. Upon substitution of the relations (\[eq:ka\_ks\]) and (\[eq:p(sigma)\]) in Eq. (\[eq:n\^0\]) we obtain $$\begin{aligned}
n^0(\tau(\sigma),{\rm p}(\sigma)) &=& n^0(\tau(\sigma_0),
p(\sigma_0)) \times \nonumber \\
& & \left[ {\rm e}{^{k_{\rm a}\Delta\sigma}}
\left( 1 + {\rm p}_0 \frac{k_{\rm s}}{k_{\rm a}} \left({\rm e}^{k_{\rm a}\Delta\sigma} -1\right) \right) \right]^2\, .
\label{eq:n^0(sigma)}\end{aligned}$$
This equation is approximately valid in the neighborhood of $\tau(\sigma_0)$ or if the fluid expansion and magnetic field energy are both constant in a certain interval $\Delta\sigma$. Indeed, such an assumption is adequate for our purposes, since the hydrodynamic evolution is performed numerically as a succession of finite, but small, time steps. Within each hydrodynamic time step the physical variables inside of each numerical cell do not change much and, thus, the magnetic field energy and the fluid expansion are roughly constant. Alternatively, one might not assume anything about $\Theta$ or $U_{\rm B}$ and solve the system of integro-differential equations (\[eq:dpdsigma\]) and (\[eq:n\^0\]). However, such a procedure is much more computationally demanding than obtaining the evolution of ${\rm p}$ and $n^0$ from, respectively, Eqs. (\[eq:p(sigma)\]) and (\[eq:n\^0(sigma)\]). Furthermore, since the magnetic field is assumed in this work, i.e., not consistently computed, a numerical solution of the aforementioned equations does not yield a true improvement of the accuracy.
For completeness, [as]{} in the diffusion approximation $n^0=4\pi{\rm p}^2f^{(0)}$, we can specify the evolution equation for the isotropic part of the distribution function of the NTPs $$\begin{aligned}
f^{(0)}(\tau(\sigma),{\rm p}(\sigma)) &=& f^{(0)}_0
\left( 1 + {\rm p}_0 \frac{k_{\rm s}}{k_{\rm a}} \left({\rm
e}^{k_{\rm a}\Delta \sigma} -1\right) \right)^4 ,
\label{eq:f(sigma)}\end{aligned}$$ where $f^{(0)}_0=f^{(0)}(\tau(\sigma_0), p(\sigma_0))$.
Finally, we define the number density of NTPs within a certain momentum interval $[{\rm p}_a(\tau(\sigma)), {\rm p}_b(\tau(\sigma))]$ $$\begin{aligned}
{\cal N}(\tau(\sigma);{\rm p}_a,{\rm p}_b) &:=& \int_{{\rm p}_a(\tau(\sigma))}^{{\rm p}_b(\tau(\sigma))}
d{\rm p}\, n^0(\tau(\sigma_0), p(\sigma_0))\, ,
\label{eq:n_def}\end{aligned}$$ whose evolution equation can be easily obtained from Eqs. (\[eq:p(sigma)\]) and (\[eq:n\^0(sigma)\]) and reads $$\begin{aligned}
{\cal N}(\tau(\sigma);{\rm p}_a,{\rm p}_b) &=&
{\rm e}^{3k_{\rm a}\Delta\sigma} {\cal N}(\tau(\sigma_0);{\rm p}_a,{\rm p}_b)\, .
\label{eq:n(tau)}\end{aligned}$$
Equation(\[eq:n(tau)\]) shows that the time evolution of the number density of NTPs in a time-evolving momentum interval, depends only on the adiabatic changes of the NTPs in such momentum interval, but not on the synchrotron losses [(Eq. (\[eq:n(tau)\]) is independent of $k_{\rm s}$).]{}
Discretization in momentum space {#sec:discretization}
--------------------------------
In the following we normalize $p$ to $m_{\rm e} c$, which allows us to express our results in terms of the particle Lorentz factor $\gamma$ instead of $p$. In order to make Eqs. (\[eq:p(sigma)\]) and (\[eq:n\^0(sigma)\]) amenable to numerical treatment, we discretize the momentum space in $N_{\rm b}$ bins, each momentum bin $i$ having a lower bound $\gamma_i$. In the present applications we use $N_{\rm b}
= 32$ [(see App. \[sec:totalflux\]). As in, e.g., [@JRE99], [@Miniati01], or [@JK05],]{} we initially distribute $\gamma_i$ logarithmically, i.e. $$\begin{aligned}
\gamma_i(\tau_0) = \gamma_{\rm min}\left( \frac{\gamma_{\rm max}}{\gamma_{\rm min}}
\right)^{(i-1)/(N_{\rm b}-1)}\, ,\end{aligned}$$ $\gamma_{\rm min}$ and $\gamma_{\rm max}$ being the minimum and maximum Lorentz factors of the considered distribution, respectively.
On the other hand, the time dimension is also discretized in time steps. We call $\tau^n$ the interval of proper time elapsed since the beginning of our simulation, and denote $\Delta \tau =
\tau^{n+1}-\tau^n$.
Our numerical method follows the time evolution of NTPs in the momentum space employing a Lagrangian approach. We track both the evolution of the $N_{\rm b}$ interface values $n^0_i$ (from Eq. (\[eq:n\^0(sigma)\]), where we take $\tau=\sigma$ and $\sigma_0=\tau(\sigma_0):=\tau^n$), $$\begin{aligned}
n^0_i(\tau^{n+1})&:=&n^0(\tau^{n+1}, \gamma_i(\tau^{n+1})) = n^0(\tau^n, \gamma_i(\tau^n)) \times \nonumber \\
& & \left[ {\rm e}{^{k_{\rm a}\Delta\tau}}
\left( 1 + \gamma_i(\tau^n) \frac{k_{\rm s}}{k_{\rm a}} \left({\rm
e}^{k_{\rm a}\Delta \tau} -1\right) \right) \right]^2\, ,
\label{eq:n_i^0}\end{aligned}$$ as well as the $N_{\rm b}$ bin integrated values $${\cal N}_i(\tau) :=\int_{\gamma_i(\tau)}^{\gamma_{i+1}(\tau)} d\gamma\, n^0(\tau, \gamma)\, .
\label{eq:n_i(tau)}$$
The time evolution of the $N_{\rm b} + 1$ interface values $\gamma_i(\tau)$ is governed by Eq. (\[eq:p(sigma)\]).
For the purpose of efficiently computing the synchrotron emissivity (see § \[sec:synchrotron\]), inside of each Lorentz factor bin $i$, we assume that, at any time, the number of NTPs per unit of energy and unit of volume $n^0_i(\tau,\gamma)$ ($\gamma_i(\tau) \le \gamma <
\gamma_{i+1}(\tau)$) follows a power-law and, therefore, the whole momentum distribution of NTPs consists of a piecewise power-law and, $$\begin{aligned}
n^0(\tau,\gamma) = n^0_i(\tau) \left(
\frac{\gamma}{\gamma_i}\right)^{-q_i(\tau)},\: i=1, \dots, N_{\rm
b},
\label{eq:n^0_powerlaw}\end{aligned}$$ where $n^0_i(\tau)$ is the number of particles with $\gamma=\gamma_i$ at the proper time $\tau$, and $q_i(\tau)$ is the power-law index of the distribution at the $i$-Lorentz factor interval. The values of $q_i(\tau)$ are computed numerically in every time step plugging Eq. (\[eq:n\^0\_powerlaw\]) into Eq. (\[eq:n\_i(tau)\]) and solving iteratively the corresponding equation (which also involves knowing the interface values -Eq. (\[eq:n\_i\^0\])- and justifies why we need to follow the evolution of two sets of variables [per]{} bin).
The approach defined up to here has the advantage that, at every time level $\tau^n$, the momentum-space evolution and the physical space trajectory of the NTPs are decoupled during the corresponding time step $\Delta \tau$. The hydrodynamic evolution of the thermal plasma provides the values of $k_{\rm a}$ and $k_{\rm s}$ at the beginning of the time step ($\tau=\tau^n$), and once these values are known, it is possible to compute the momentum distribution of NTPs at time $\tau^{n+1}$. Thereby, it is possible to perform separately the trajectory integration of the NTPs once, and to evolve NTPs in the phase space afterward, as many times and with as many initial particle distributions as desired (viz., during a post-processing phase).
Normalization and initialization of the NTP distribution
--------------------------------------------------------
Our models are set up such that we initially inject through the jet nozzle NTPs with a momentum distribution function which follows a single power-law, i.e., $q_i=q_1, \forall i$. Therefore, the initial number and energy density in the interval $\gamma_{\rm min} \leq
\gamma \leq \gamma_{\rm max}$ read $$\begin{aligned}
{\cal N} & =& \frac{n^0_1}{q_1-1}\gamma_{\rm min}
\left[ 1-\left(\frac{\gamma_{\rm max}}{\gamma_{\rm min}}\right)^{1-q_1}
\right] \label{eq:N} ,
\\
{\cal U} &= & \frac{n^0_1}{q_1-2} \gamma_{\rm min}^2 m_{\rm e} c^2
\left[1-\left(\frac{\gamma_{\rm max}}{\gamma_{\rm
min}}\right)^{2-q_1}\right]\, .
\label{eq:E}\end{aligned}$$
Consistent with our assumptions about the relation between the thermal and non-thermal populations we assume that ${\cal N} = c_{_{\cal N}}
\rho/m_{\rm e}$ and ${\cal U} = c_{_{\cal U}} P$, where $c_{_{\cal
N}}$ and $c_{_{\cal U}}$ are constants, while $P$ and $\rho$ stand for the pressure and rest-mass density of the background fluid, respectively. Such proportionalities along with Eqs. (\[eq:N\]) and (\[eq:E\]) yield (G95) $$\gamma_{\rm min} =
\frac{c_{_{\cal U}}}{c_{_{\cal N}}}\frac{q_1-2}{q_1-1}\frac{P}{\rho
c^2}\frac{1-(\gamma_{\rm
max}/\gamma_{\rm min})^{1-q_1}}{1-(\gamma_{\rm max}/\gamma_{\rm min})^{2-q_1}},
\label{eq:pmin}$$ and we can use either Eq. (\[eq:N\]) or Eq. (\[eq:E\]) to compute $n^0_1$ if the ratio $C_\gamma:=\gamma_{\rm max}/ \gamma_{\rm min}$ is fixed. Thus, the initial distribution of particles can be determined from pressure and rest-mass density at the jet nozzle, simply by specifying $c_{_{\cal N}}$ and $c_{_{\cal U}}$.
A key difference between SPEV and AM methods is that in SPEV the dimensionless proportionality parameters $c_{_{\cal N}}$ and $c_{_{\cal U}}$ are only specified at the jet injection nozzle. In the SPEV method, the subsequent time evolution of the NTP momentum distribution, namely, the spectral shape (piecewise power-law) and the limits of the distribution $\gamma_{\rm min}$ and $\gamma_{\rm max}$ as it evolves in the physical space is computed according to Eq. (\[eq:p(sigma)\]). AM ignores synchrotron loses, which yields a fixed power-law index for the whole distribution of Lorentz factors of the NTPs. The remaining two parameters needed to specify the distribution function, $\gamma_{\rm min}$ and $\gamma_{\rm max}$ are computed from the local values of the hydrodynamic variables. On the one hand, $\gamma_{\rm min}$ follows from Eq. (\[eq:pmin\]) and $\gamma_{\rm max}$ is obtained from the fact that, $C_\gamma$ is strictly constant in time if the evolution is adiabatic. Also, in contrast to SPEV, it is necessary to assume a value of $C_\gamma$ everywhere in the simulated region and not only at the injection region.
Synchrotron radiation and synthetic radio maps {#sec:synchrotron}
==============================================
The synchrotron emissivity, at a time $\tau$, of an ensemble of NTPs advected by a thermal plasma element, can be cast in the following general form (valid both for ordered and random magnetic fields; see [@Mimica04]) $$\begin{aligned}
j(\tau, \nu) = \frac{\sqrt{3}e^3 b_{\perp}}{4\pi m_{\rm e} c^2}
\sum_{i=1}^{N_{\rm b}}
\int_{\gamma_i(\tau)}^{\gamma_{i+1}(\tau)} {{\mathrm d}}\gamma n^0(\tau, \gamma) g \left( \frac{\nu}{\nu_{\perp}
\gamma^2} \right) \, ,
\label{eq:j(nu)} $$ where $(g(x), b_\perp,\nu_\perp ) = (R(x), |{\bf b}|, \nu_0)$ if ${\bf
b}$ is randomly oriented, or $(g(x), b_\perp,\nu_\perp ) = (F(x),
|{\bf b}|\sin\alpha, \nu_0\sin\alpha)$ in case ${\bf b}$ is ordered. $\alpha$ is the angle the comoving magnetic field forms with the line of sight, and $\nu_0=3e|{\bf b}|/4\pi m_{\rm e}c$. In the previous expressions, $F$ is the first synchrotron function $$F(x) = x\int_x^\infty d\xi K_{5/3}(\xi),
\label{eq:F(x)}$$ $K_{5/3}$ being the modified Bessel function of index $5/3$, and $R$ is defined as $$R(x) :=\frac{1}{2}\int_0^\pi {{\mathrm d}}\alpha \sin^2\alpha\ \
F\left(\frac{x}{\sin\alpha}\right)\, .
\label{eq:R(x)}$$
The synchrotron self-absorption process is also included in our algorithm. Thus, we need to compute the synchrotron absorption coefficient, at a time $\tau$, of an ensemble of NTPs advected by a thermal plasma element, which can be cast in the following general form $$\begin{aligned}
\kappa(\tau, \nu) &=& \frac{\sqrt{3}e^3 b_{\perp}}{8\pi m_{\rm e}^2
c^2 \nu^2} \times \label{eq:kappa(nu)}\\[2mm]
& & \sum_{i=1}^{N_{\rm b}}
\int_{\gamma_i(\tau)}^{\gamma_{i+1}(\tau)} {{\mathrm d}}\gamma
\left[-\gamma^2\frac{{{\mathrm d}}}{{{\mathrm d}}\gamma} \left(\frac{n^0(\tau,
\gamma)}{\gamma^2} \right)
\right] g \left( \frac{\nu}{\nu_{\perp}
\gamma^2} \right) \, ,\nonumber\end{aligned}$$ In order to perform the integrals of Eq. (\[eq:j(nu)\]) and (\[eq:kappa(nu)\]), it is necessary to make some assumption about the internal distribution of NTPs within each Lorentz factor bin $i$. As explained in Sect. \[sec:discretization\], we choose to assume that NTPs distribute as power-law (Eq. \[eq:n\^0\_powerlaw\]) inside of each bin. This choice agrees with the common assumptions made in the literature and is also supported by theoretical arguments and observations of discrete radio sources (e.g., [@Pacholczyk70], chapt. 6; [@Koenigl81]), and by numerical simulations (e.g., [@AGKG01]). Furthermore, it allows us to build a very efficient and robust method for computing the local synchrotron emissivity and the local absorption coefficient. It consist of tabulating the functions $F(x)$ and $R(x)$, and then tabulating integrals over power-law distributions of particles. Proceeding in this way is $\sim 100$ times faster than computing Eqs. (\[eq:j(nu)\])-(\[eq:R(x)\]) by direct numerical integration.
The synchrotron coefficients (Eqs. \[eq:j(nu)\] and \[eq:kappa(nu)\]) of steady jet models result from the time evolution of the Lagrangian particles injected at the jet nozzle and spatially transported along the whole jet (the larger the number of Lagrangian particles, the better coverage of the whole jet). In our simulations around $N_{\rm steady}=32$ of such Lagrangian particles (i.e., about 4 particles per numerical cell at the injection nozzle) are sufficient to properly cover a quiescent jet. If the jet is not steady, e.g., because a hydrodynamic perturbation is injected, we need to follow many more Lagrangian particles. It becomes necessary to have particles everywhere the quiescent jet is perturbed. For the models in this paper, this means to inject new Lagrangian particles through the jet nozzle at all time steps after a hydrodynamic perturbation is set in. The distribution function of the NTPs injected with the perturbation is the same as that of the particles injected in the quiescent jet. This is justified since the perturbation only changes the bulk Lorentz factor, but not the pressure, or the density of the fluid. In the simulations where we have injected a hydrodynamic perturbation this implies that we must follow the evolution of more than $N_{\rm steady} \times N_{\rm timesteps} \gsim 10^5$ Lagrangian particles. This makes our SPEV simulations effectively four-dimensional (two spatial, one momentum and a -huge- number of Lagrangian particles dimension). Therefore, the spatial resolution that we may afford results severely limited.
The synchrotron coefficients depend on the magnetic field strength and orientation as well as on the spectral energy distribution $n^0(\tau,\gamma)$. In our models the magnetic field is [ dynamically negligible, thus we set it]{} up [*ad hoc*]{}. We choose that [$U_{\rm B}$]{} remains a fixed fraction of the particle energy density and that the field is randomly oriented.
Synthetic radio maps are build by integrating the transfer equations for synchrotron radiation along rays parallel to the line, accounting for the appropriate relativistic effects (time dilation, Doppler boosting, etc.). The technical details relevant for this procedure can be found in Appendix \[appendix:imaging\].
Radio emission {#sec:radio}
==============
The goals of this section are twofold. First, we validate the new algorithm comparing the synthetic radio maps obtained with SPEV without accounting for synchrotron losses with the [ones obtained]{} using AM. For this purpose, we will employ the SPEV method to evolve NTPs but taking $k_{\rm s}=0$ in Eq. (\[eq:dpdsigma2\]). We will refer to this method of evaluating the evolution of NTPs as SPEV-NL. Second, we will show the differences induced by accounting for synchrotron losses in the evolution of NTPs.
Calibration of the method {#sec:calibration1}
-------------------------
In order to properly compare SPEV-NL and AM results we set up the same spectral parameters at the jet nozzle for both: $q_1= 2.2$, $\gamma_{\rm min}=330$, $C_\gamma=10^3$ and $\rho_a =
2\times10^{-21}\,$gcm$^{-3}$. We produce all our images for a canonical viewing angle of $10^\circ$ and assuming that $R_b=0.1\,$pc. The comoving magnetic field strength is ${\rm b}_b:=\sqrt{{\bf
b}^2}=0.02\,$G [(model PM-L-NL) and 0.03G (model OP-L-NL)]{}.[^1] For the set of reference parameters we have considered, the synthetic radio maps of the quiescent jets produced with SPEV-NL yield very small differences with respect those computed with AM (Fig. \[fig:fiduc-emiss\]). Indeed, the overall agreement between both methods in the predicted quiescent radio maps is remarkably good, particularly, if we consider the fact that SPEV is a Lagrangian method while AM is Eulerian.
Looking at the synthetic radio maps of model [OP-L-NL]{} (Fig. \[fig:fiduc-emiss\]), we observe, as in G97, a regular pattern of knots of high emission, associated with the increased specific internal energy and rest-mass density of internal oblique shocks produced by the initial overpressure in this model. The intensity of the knots decreases along the jet due to the expansion resulting from the gradient in external pressure. Some authors (e.g., @DM88; G95; G97; @Marscheretal08) propose that the VLBI cores may actually correspond to the first of such recollimation shocks. Since, for the parameters we use, the source absorption for frequencies above 1GHz is negligible, the jet core reflects the ad hoc jet inlet in the [PM-L-NL]{} model, while we shall associate it with the first recollimation shock for model [OP-L-NL]{}. The rest of the knots are standing features in the radio maps for which, there exists robust observational confirmation [@Gomez02; @Gomez05].
Since the synchrotron losses affect more the higher energy part of the distribution of NTPs than the lower one, we have also validated our code by considering the dependence of the results with the limit $\gamma_{\rm max}$ and checked them against the theoretical expectations (e.g., [@Pacholczyk70]). For this we reduce the value of $\gamma_{\rm max}$ keeping all other parameters fixed and equal to those of the [PM-L]{} model. Since the value of $\gamma_{\rm max}$ is set by the ratio $C_\gamma$, in order to study the dependence of the results with $\gamma_{\rm max}$, we have computed a set of models combining three different values of $C_\gamma=\{10^3,10^2,10\}$ and ${\rm b}_b=0.02$G. Additionally, to highlight the effect of the radiative losses, we have performed the same simulations (varying $C_\gamma$) for a larger value of the beam magnetic field, [equal to that of the model PM-H]{}.
For [model PM-L]{} (Fig. \[fig:fiduc\_gmax\], left panel), radiative losses are negligible, and the reduction in $C_\gamma$ (i.e., in $\gamma_{\rm max}$), does not change appreciably the radio maps at radio observing frequencies. Indeed, except the model with the lowest $C_\gamma$ (corresponding to $\gamma_{\rm max}=2200$) beyond $160\,R_b$, all the models stay above the 100% efficient radiation limit along the whole jet.
The models with larger magnetic field ${\rm b}_b=0.2\,$G (Fig. \[fig:fiduc\_gmax\], right panel), undergo a much faster evolution. The emissivity along the jet axis drops very quickly and at $z=150R_b$, [it]{} is five orders of magnitude smaller than [for the PM-L model]{}. After a very short distance ($\simeq
1\,R_b$), synchrotron losses bring $\gamma_{\rm max}$ of all three models to a common value which is independent of the initial one (note that the variation of $\gamma_{\rm max}$ with distance is indistinguishable for the three models except in the zoom displayed in the inset of the top right panel of Fig. \[fig:fiduc\_gmax\]). The reason for this degenerate evolution resides in the relatively large magnetic field strength [(see [@Pacholczyk70], Eq. 6.20)]{}. Thus, our method is able to reproduce the common evolution of models with different values of $\gamma_{\rm max}$ and a relatively large magnetic field.
On the relevance of synchrotron losses {#sec:synlos}
--------------------------------------
Having verified that our method (SPEV-NL) compares adequately to the AM, we now turn to the specific role that synchrotron losses play in the evolution of NTPs. For that, we compare in Fig. \[fig:fiduc\_distrib\] the spectral properties of NTPs in quiescent jet models using both SPEV-NL and SPEV methods. It is obvious that the highest energy particles of the distribution cool down rather quickly (see the fast decay of the dashed black curves in the upper panels of Fig. \[fig:fiduc\_distrib\]) even for the small value of ${\rm b}_b$ considered here. Most of the spectral evolution triggered by synchrotron cooling at high values of $\gamma$ happens in the first $25R_b-50R_b$. After that location, the ratio $C_\gamma$ is much smaller than at the injection nozzle ($C_\gamma\lsim 50$), and the evolution of the NTP population is dominated by the adiabatic cooling/compression downstream the jet. In contrast, the upper limit of the SPEV-NL distribution (dashed red curves in the upper panels of Fig. \[fig:fiduc\_distrib\]) only changes by a factor of 2 along the whole jet length. Theoretically, it is well understood that it is possible to undergo a substantial spectral evolution (triggered by synchrotron losses) and, simultaneously, not to have any manifestation of such evolution at radio frequencies [e.g., @Pacholczyk70]. The substantial decrease of $\gamma_{\rm max}$ triggered by the radiative losses, does not affect much the value of the integral that has to be performed over $\gamma$ in order to compute the emissivity in Eq. (\[eq:j(nu)\]), since most of the emitted power at radio-frequencies happens relatively close to $\gamma_{\rm min}$, where synchrotron losses are negligible. Certainly, at higher observing frequencies this is not the case, and the emissivity substantially drops because of the fact that both, the synchrotron losses (Eq. \[eq:dpdt\_syn\]) and the frequency at which the spectral maximum emission is reached depend on the square of the non-thermal electron energy (and on the magnetic field strength).
We define the spectral index between two radio frequencies as $$\begin{aligned}
\label{eq:alpha_ij}
\alpha_{ij} = \frac{\log{(S_i/S_j)}}{\log{(\nu_i/\nu_j)}},\end{aligned}$$ where $S_i$ and $S_j$ are the flux densities at the frequencies $\nu_i$ and $\nu_j$, respectively. Since we compute synthetic radio maps at three different radio frequencies ($\nu_1=15\,$GHz, $\nu_2=22\,$GHz, and $\nu_3=43\,$GHz), we may define three different spectral indices. For convenience, in the following, we consider the spectral index $\alpha_{13}$ between 15GHz and 43GHz. Furthermore, we may compute $\alpha_{13}$ for both [*convolved*]{} or [ *unconvolved*]{} flux densities. The unconvolved flux density is directly obtained from the simulations and has an extremely good spatial resolution, viz. the unconvolved radio images have a resolution comparable to that of the hydrodynamic data. The convolved flux densities result from the convolution with a circular Gaussian beam of the unconvolved data. The FWHM of the Gaussian beam is proportional to the observing wavelength. This convolution is necessary to degrade the resolution of our models down to limits comparable with typical VLBI observing resolution. We note that in order to compute the spectral index for convolved flux densities, we have to employ the same FWHM convolution kernel for the data at the two frequencies under consideration. Thus, to compute $\alpha_{13}$ for convolved data, we employ the same Gaussian beam with a FWHM 6.45$R_b$ for both flux densities at 15GHz and 43GHz.
Our models are computed for an electron spectral index $q=q_1=2.2$. We verify that, at large distances to the jet nozzle, unconvolved models (Fig. \[fig:thick\_axis\] upper panels) tend to reach the expected value $\alpha=(1-q)/2=-0.6$ for an optically thin source. This asymptotic value is reached smoothly in case of the [PM-L and PM-H models]{} and it is modulated by the presence of inhomogeneities (recollimation shocks) in the beam of models [OP-L and OP-H]{}.
Close to the jet nozzle, our unconvolved models display flat or even inverted ($\alpha_{13}>0$) spectra (Fig. \[fig:thick\_axis\]), in spite of the fact that the jets are optically thin throughout their whole volume. The occurrence of flat or inverted spectrum depends on the magnetic field strength and differs for OP and PM models. [As shown in Fig. \[fig:thick\_axis\], the PM-L]{} model shows an inverted spectrum for $z\lsim2.5R_b$, [while]{} the [OP-L]{} model displays a pattern of alternated inverted and normal ($\alpha_{13}<0$) spectra for $z \lsim 12.5R_b$. The spectral inversion in the [ OP-L]{} model happens where standing features (associated to recollimation shocks in the beam) are seen in the jet.
If synchrotron losses are not included, the spectral behavior of models [PM-L and OP-L]{} remains unchanged, because in such a case loses are negligible. However, if for the models [PM-H and OP-H]{} the losses are not accounted for (which is, obviously, a wrong assumption), the jet displays an inverted spectrum up to distances $z
\sim 30R_b$.
The behavior of the spectral index exhibited by our models close to the jet nozzle contrasts with the theoretical expectations for an inhomogeneous, optically thin jet with a negative electron spectral index, for which the jet inhomogeneity is predicted to steepen the spectrum [e.g., @Marscher80; @Koenigl81]. To explain this discrepancy we argue that the analytic predictions are based on the assumption that the limits of the energy distribution of the NTPs safely yield that the contribution of the synchrotron functions (Eqs. \[eq:F(x)\] and \[eq:R(x)\]) to the synchrotron coefficients (Eqs. \[eq:j(nu)\] and \[eq:kappa(nu)\]) is proportional to some power of the frequency and of the NTP’s Lorentz factor. This situation does not happen if the lower limit of the distribution $n^0(\gamma)$, $\gamma_{\rm min}$, is (roughly) larger than the value $\gamma_{_{\rm
M}}$ at which the synchrotron function $R(x_{\rm low})$ (Eq. \[eq:R(x)\]) reaches its maximum, where $x_{\rm low}=\nu_{\rm
low} / \nu_0 \gamma^2$, and $\nu_{\rm low}$ is the smallest observing frequency in the comoving frame. Since the function $R(x)$ has a maximum for $x\simeq 0.28$, one finds that the condition to have an inverted spectrum is $\gamma_{\rm min} \gsim \gamma_{_{\rm
M}}\simeq 1.9 \times (\nu_{\rm low}/ \nu_0)^{1/2} {\cal
D}^{-1/2}$, where ${\cal D}:=1/\Gamma (1 - v\cos{\theta})$ is the Doppler factor. Since, in our case, $\nu_{\rm low}=15$GHz, we may also write $$\gamma_{\rm min} \gsim 113 \left( \frac{\nu_{\rm low}}{\rm 15\,GHz}
\right)^{1/2} \left(\frac{b}{\rm 1\,G}\right)^{-1/2} {\cal
D}^{-1/2}.
\label{eq:gamma_min2}$$ Figure \[fig:boundary\_effects\] shows how this boundary effect substantially modifies the emissivity at 15GHz and 43GHz for the model [PM-L]{}. At the injection nozzle (Fig. \[fig:boundary\_effects\] upper panel) the lower limit of the integral in Eq. (\[eq:j(nu)\]) is set by $\gamma_{\rm min}$ and not by the lower limit of $R_\nu(x)$. However, downstream the jet (Fig. \[fig:boundary\_effects\] lower panel) the situation reverses and the fast decay of $R_\nu(x)$ for $\gamma<300$ sets the lower limit of the emissivity integral. [Thus,]{} close to the nozzle, the value of the area below the $n^0(\gamma)R_{43}(x)$ curve, which is proportional to the emissivity at 43GHz, is larger than that below the curve $n^0(\gamma)R_{15}(x)$. Hence, there is an emissivity excess at 43GHz compared to that at 15GHz. As a result, the $\alpha_{13}$ becomes positive close to the jet nozzle. Far away from the nozzle the emissivity at 15GHz almost doubles that at 43GHz, explaining why values of $\alpha_{13}< 0$ are reached [asymptotically]{}.
The convolved models display some traces of the behavior shown for the uncovolved ones. For example, OP models display a flat or inverted spectrum very close to the jet nozzle (Fig. \[fig:thick\_axis\_conv\] right panels). This is not the case for [PM-L model]{} (Fig. \[fig:thick\_axis\_conv\]). Since the resolution of the convolved data is much poorer than that of the unconvolved one, $\alpha_{13}$ exhibits a quasi monotonically decreasing profile from the jet nozzle (where $-0.1\lsim \alpha_{13}\lsim 0$). The coarse resolution of the convolved data also blurs any signature in the spectral index associated to the existence of cross shocks in the beam of OP models. Furthermore, the decay with distance of the spectral index is shallower for the convolved flux data than for the unconvolved one. [Hence,]{} the theoretical value $\alpha_{13}=-0.6$, which is expected for an optically thin synchrotron source, is reached nowhere in the jet models [PM-L and OP-L]{} (Fig. \[fig:thick\_axis\_conv\]).
As expected, at frequencies below a few hundred MHz, the jet is strongly self-absorbed everywhere (Fig. \[fig:spec-axis\]). Close to the jet nozzle, there is not a clear turnover frequency between the self-absorbed part of the spectrum and the optically thin one. Instead, we observe a smooth transition between both regimes. Far from the nozzle, the self-absorption turnover is much more peaked. It is known [@TK07] that in contrast with a distribution of NTP that follows a power-law extending to $\gamma_{\rm min}\gsim 1$, if the power-law is restricted to a relatively large, but not unrealistic $\gamma_{\rm min}$, or if the electron distribution [was]{} monoenergetic, the intensity can be flat over nearly two decades in frequency (which implies that the energy flux grows linearly over the same frequency range). Our [PM-L]{} models have $\gamma_{\rm
min}=330$ at the injection nozzle and reduce it to $\gamma_{\rm
min}\simeq 200$ at $z=200R_b$ because of the adiabatic expansion of the jet (Fig. \[fig:fiduc\_distrib\] upper left). As we have argued in § \[sec:radio\] close to the jet nozzle, $\gamma_{\rm
min}\gsim\gamma_{_{\rm M}}$, which means that $\gamma_{\rm min}$ is sufficiently large to be in the range where a smooth turnover transition is expected, in agreement with [@TK07]. Far away from the nozzle, since $\gamma_{\rm min}$ decreases, we recover the more standard situation in which an obvious turnover frequency can be identified.
[Provided]{} that close to the nozzle our PM (also OP) models are weakly self-absorbed ([at 15GHz, the solid black line in Fig. \[fig:spec-axis\] has not reached the power-law regime yet]{}), one may question whether the spectral inversion we have found is not also the result of opacity effects. We have dismissed such a possibility by running models with the SPEV method [including no]{} absorption.
### Dependence with the magnetic field strength {#sec:magfield}
In order to [study the effect of]{} intense synchrotron losses we [consider models PM-H and OP-H]{} (Figs. \[fig:highB-emiss\] and \[fig:high-B\]). Very close to the injection nozzle ($Z\sim
50\,R_b$) the line denoting the evolution of $\gamma_{\rm max}^{\rm
SPEV}$ crosses the line corresponding to a 10% synchrotron efficiency limit (lower blue thick line; Fig. \[fig:high-B\]) and most of the synchrotron emissivity falls outside of the observational frequency. [Because of a]{} stronger magnetic field [than in models PM-L and OP-L]{}, more energy is lost close to the jet nozzle than far from it and, thus, SPEV-radio-maps look much shorter than SPEV-NL-radio-maps (Fig. \[fig:highB-emiss\]). An alternative way to see such an effect is through the rapid decay of $\gamma_{\rm
max}^{\rm SPEV}$ in the first $10R_b$ in Fig. \[fig:high-B\], right panel. Afterwards, the adiabatic changes dominate the NTP evolution. The initial period of fast evolution is even shorter if a larger magnetic field [were to be]{} considered.
The intensity contrast between shocked and unshocked jet regions of [model OP-H (Fig. \[fig:high-B\]) is larger than that of model OP-H-NL]{}. Indeed, the [OP-H]{} model appears as a discontinuous jet (Fig. \[fig:highB-emiss\]) because of the slightly larger intensity increase than in the [OP-H-NL]{} model when the NTP distribution passes through cross-shocks and the much more pronounced intensity decrease at rarefactions. We note that, although the adiabatic evolution is followed with the same algorithm in SPEV and SPEV-NL, the radiative losses change substantially the NTP distribution that it is injected through the nozzle after very short distances. The consequence being that the NTP distribution $n^0(\tau,\gamma)$ that faces shocks and rarefactions downstream the nozzle is rather different when using SPEV or SPEV-NL method and, therefore, the relative intensity of shocked and unshocked regions is also different depending on whether synchrotron losses are included or not in the calculation.
The outlined differences between [models OP-H and OP-H-NL (with shocks in the beam)]{}, have to be interpreted with caution since none of the [methods accounts]{} for the injection of high-energy particles at shocks. But independent of this, we expect that if the magnetic field is sufficiently large, the SPEV method will yield a rather fast evolution of such particles and, thereby, a faster decay of the intensity downstream the shock.
The most relevant difference between the upper and lower panels of Fig. \[fig:highB-emiss\], is the increased brightness of the jet close to the injection nozzle and the steeper fading of the jet when energy losses are included. This fact poses the paradox that the method that accounts for radiative losses (SPEV) yields brighter standing features close to the injection nozzle (far from the nozzle the situation reverses and the SPEV-NL model is brighter than SPEV one). In order to disentangle this apparent contradiction, we shall consider that the plasma is compressed at standing shocks, which yields a growth of the magnetic field energy density (proportional to the pressure in our case), and triggers a faster cooling of the high-energy particles. Since the SPEV method conserves the number density of NTPs (Eq. \[eq:N\]), due to the synchrotron losses, high-energy particles reduce their energy and accumulate into an interval of Lorentz factor which is smaller than in the case of SPEV-NL models. As in such reduced Lorentz factor interval NTPs radiate more efficiently at the considered radio-frequencies, the emissivity of SPEV models at strong compressions (like, e.g., the considered cross shocks) becomes larger than that corresponding to models which do not include synchrotron losses. It is important to note that this situation happens in our models rather close to the jet nozzle. The reason being that after the NTPs have suffered a substantial synchrotron cooling, the evolution of the NTP distribution is dominated by the adiabatic terms of Eq. \[eq:dpdsigma2\]. In such a regime, reached by our models at a certain distance from the jet nozzle, the evolution of SPEV-NL and SPEV models is qualitatively similar. Considering the different qualitative evolution of the NTP distribution close to the nozzle and far from it, we refer to such epochs as [*losses-dominated*]{} and [*adiabatic*]{} regimes, respectively. These terms agree with the commonly used in the literature to refer to similar phenomenologies (e.g., @MG85).
For [PM-H and OP-H models]{}, the spectral behavior is dominated by the change of slope of the NTP Lorentz factor distribution beyond the synchrotron cooling break at $\gamma=\gamma_{\rm
br}$. Theoretically, an optically thin inhomogeneous jet shall display a spectral index $\alpha=(q+1)/2$, if the radiation in the observational band is dominated by the electrons with Lorentz factors $\gamma\gsim\gamma_{\rm br}$, or $\alpha\simeq-2.7$ if the emission is dominated by electrons with Lorentz factors close to $\gamma_{\rm
max}$ [@Koenigl81][^2]. Figure \[fig:thick\_axis\] (lower panels) shows that asymptotically (viz., at large $z$) unconvolved models reach values $\alpha_{13}\lsim -2.5$, implying that the highest energy electrons with $\gamma\sim\gamma_{\rm max}$ are the most efficiently radiating at the considered observing frequencies. The value of $\gamma_{\rm max}$ differs significantly when synchrotron loses are not included. This fact explains the inversion of the spectrum along the whole jet if synchrotron losses were not included [(PM-H-NL and OP-H-NL models)]{}. Thereby, synchrotron losses tend to produce a “normal” spectrum ($\alpha_{ij}<0$) if the magnetic field is large.
Regarding the convolved data, we note that models with a higher magnetic field display the same qualitative phenomenology discussed in § \[sec:radio\]. In this case, the theoretical value $\alpha\simeq-2.7$ is not reached neither by the [PM-H]{} ($\alpha_{13,min}^{\rm PM-H}=-1.4$) nor by the [OP-H]{} ($\alpha_{13,min}^{\rm OP-H}=-1.1$) model (Fig. \[fig:thick\_axis\_conv\] lower panels).
Infrared to X-rays emission {#sec:infrared_to_X}
===========================
We have computed the spectral properties of some of our quiescent jet models above radio frequencies. [We note, that we have not included any particle acceleration process at shocks in the SPEV method, thus,]{} the spectrum beyond infrared frequencies has to be taken carefully. If any shock acceleration mechanism were included, a larger contribution of the shocked regions will be present. In addition, the inverse Compton process, may shape the emission at such high energies, and such cooling process is presently not included in SPEV.
[The results for models PM-S and PM-L (Tab. \[tab:models\]), which have no or extremely weak shocks are displayed in Fig. \[fig:spec-axis\], where we show]{} the spectral energy distribution at selected distances from the nozzle for points located along the jet axis. [The small magnetic field of model PM-S (Fig. \[fig:spec-axis\] dashed lines) minimizes the energy losses, but also the observed flux in the optical or X-ray band, rendering observable at such wavelengths the hydrodynamic jet models considered here (if the jet is sufficiently close). In the case of model PM-L,]{} right at the nozzle ($z=0$ in Fig. \[fig:spec-axis\]), the energy flux cut-off is located at $\simeq 10^{18}\,$Hz. This means [that, we]{} could observe the jet core in the soft X-ray band, if the source was sufficiently close. However, the core size at such frequencies is very small [(as it is expected; see]{} e.g., @MG85). This is reproduced in our models since at such a short distance as $z=5R_b$, the jet can barely be observed in the Near Ultraviolet or, perhaps in the optical band (Fig. \[fig:spec-axis\], red solid line), but there is virtually no flux in the X-ray band because of the fast NTP cooling for the considered magnetic field energy density at the jet nozzle. In the Near Infrared range, the jet could perhaps be observable up to distances of $10R_b-15R_b$. [A]{} larger magnetic field drives [a]{} faster cooling, rendering undetectable the jet even at infrared frequencies. This phenomenology has been invoked to explain the relative paucity of optical jets with respect to radio jets. However, there are a number of authors which claim that a large proportion of jets generate significant levels of both optical and, even, X-ray emission (e.g., [@Perlmanetal06]). Our results shall not be taken in support of any of the two thesis since energy losses depend also on the magnetic field strength (Eq. \[eq:dpdt\_syn\]), which we fix [*ad hoc*]{}.
Evolution of a superluminal component {#sec:spec_evol_component}
=====================================
In this section we discuss the time dependent observed emission once a hydrodynamic perturbation is injected into the jet (see Sec. \[sec:perturb\]). Following the convention of G97, [we call [*components*]{}]{} to local increases of the specific intensity in a radio map, while we use [*perturbation*]{} to denominate the variation of the hydrodynamic conditions injected through the jet nozzle. In order to magnify the effect of synchrotron losses in our models, we [discuss models PM-H and OP-H]{} in Sec. \[sec:losses\], and we also look for the differences between the PM and OP models.[^3] While the standing shocks of the beam of model OP-H are very weak, the shocks developed by the hydrodynamic perturbation are rather strong. Since we have not included in our method the acceleration of NTPs at relativistic shocks, computing the synchrotron emission at frequencies above radio may yield inconsistent results. Therefore, we only analyze the spectral properties of the emission in radio bands. Finally we show spacetime plots of hydrodynamic and emission properties along the jet axis in Sec. \[sec:spacetime\_analysis\].
On the relevance of synchrotron losses {#sec:losses}
--------------------------------------
The magnetic field energy density is set [*ad hoc*]{} in our models (Sect. \[sec:synchrotron\]), and we can change it freely if the resulting magnetic field does not become dynamically relevant.
[For the sake of a better illustration of the effect of the synchrotron losses on the morphologies displayed in the radio maps, we have computed models PM-H and OP-H (Fig. \[fig:PMOP-highB-snap\]), and PM-H-NL and OP-H-NL (Fig. \[fig:PMOP-highB-nl\])]{}. A noticeable general characteristic of SPEV-NL models is that all the features identifiable in the radio maps are more elongated (along the jet axial direction) than in the case that synchrotron losses are included. The reason is that without synchrotron losses, the beam of the jet is brighter at longer distances. Thus, in the unconvolved data, the parts [located downstream the jet weight more in the convolution beam]{} than in the case where synchrotron losses are included, biasing the isocontours of flux density along the axial, downstream jet direction. For the same reason, the models which include synchrotron losses display a more knotty morphology than those which do not include them, both in the unconvolved and in the convolved data. This feature is more important in case of [OP-H and OP-H-NL models]{} (compare, e.g., panels two, three and six -from top- of Figs. \[fig:PMOP-highB-snap\] and \[fig:PMOP-highB-nl\]) than in case of [PM-H and PM-H-NL models]{}.
The [main component undergoes losses-dominated (first) and adiabatic (later) regimes as quiescent jet models do]{}. In the losses-dominated regime (upper two panels of Fig. \[fig:PMOP-highB-snap\]), SPEV models exhibit a brighter component than SPEV-NL models. Later, in the adiabatic regime, SPEV models display a dimmer component than SPEV-NL ones. As we argued in Sect. \[sec:magfield\], the conservation of the NTPs number density explains this phenomenology. The main component clearly splits into two parts when synchrotron losses are included in model OP-H (Fig. \[fig:PMOP-highB-snap\] panels 2 and 3 from top; see also the movie “PMOP-highB.mpg” in the online material). The component splitting is not so apparent in model [OP-H-NL]{}, although it also takes place farther away from the nozzle than in the model including losses (Fig. \[fig:PMOP-highB-nl\], third panel from top). The splitting of the main component happens during the losses-dominated regime and the rear part of the component is brighter than the forward one if losses are included, otherwise, the forward part of the component is brighter than the rear one. However, the fact that the component is seen as a double peaked structure is not the direct result of the splitting of the hydrodynamic perturbation in two parts (§ \[sec:perturb\]), because the projected separation of these two hydrodynamic features is smaller than the convolution beam, even at 43GHz. Instead, this results from the interaction of the hydrodynamic perturbation with the cross shocks in the beam of model OP. Because of the small viewing angle, the increased emission triggered in the component when it crosses over a recollimation shock is seen by the observer to arrive simultaneously with the radiation emitted when the hydrodynamic perturbation was crossing over the preceding (upstream) cross shock.
Figure \[fig:PM-highB-spectral-snap\] shows the evolution of the component at 15Ghz (left panels) and 22GHz (right panels) for the [PM-H model]{}. The convolution beam depends linearly on the wavelength of observation, thereby, it is larger at smaller frequencies. Except for the obvious disparity of resolutions the evolution of the main component along the pressure matched jet at 15GHz, 22GHz and 43GHz [does]{} not display large differences. The main component appears as a moving bright spot at all three frequencies (upper three panels of Fig. \[fig:PMOP-highB-snap\] [*left*]{} and Fig. \[fig:PM-highB-spectral-snap\]).
We have also checked that the profile outlined above does not depend on including synchrotron losses either. However, the smaller the magnetic field, the larger the increase in the spectral index behind the intensity maxima associated to the main component (i.e., associated with the rarefaction trailing the main hydrodynamic perturbation). The time evolution of the prototype spectral profile of a hydrodynamic perturbation injected at the nozzle is characterized by a substantial steepening of the spectrum behind the intensity maxima (Figs. \[fig:PM-highB-spec\]c and \[fig:PM-fiduc-spec\]c,d) compared to the quiescent jet model. This behavior of the spectral index has also been found in previous theoretical papers, and it is attributed to the fact that the NTP distribution evolves on timescales smaller than the light crossing time of the source (e.g., [@CG99]).
Comparing Figs. \[fig:PM-highB-spec\]e and \[fig:PM-fiduc-spec\]e, it is remarkable that trailing components pop up precisely to the left (i.e., behind) of the local relative spectral index minimum (at $Z\sim 18R_b$ in Fig. \[fig:PM-fiduc-spec\]e and $Z\sim 20R_b$ in Fig. \[fig:PM-highB-spec\]e) that follows the local relative maximum of the spectral index reached in the wake of the main perturbation. Furthermore, we notice that the intensity relative to the background jet of the trailing components identifiable at 43GHz, depends on the strength of the initial magnetic field, in spite of the fact that in our models the magnetic field is dynamically negligible.[^4] At higher magnetic field strength the intensity of the trailing components is lowered and, some of them are hardly visible (e.g., the leading trailing at $\sim 25R_b$ is evident in Fig. \[fig:PM-fiduc-spec\]f, while it is difficult to identify in Fig. \[fig:PM-highB-spec\]f). [Thereby,]{} the observational imprint of trailing components is frequency dependent.
The evolution of the perturbation in model [OP-L]{} displays a slightly different profile at 43GHz than in model [PM-L. The main component splits into two sub-components at the highest observing frequency (Fig. \[fig:OP-fiduc-spec\]b)]{}. At 15GHz and 22GHz, the profile of the perturbation is qualitatively the same as for the [PM-L]{} model. [The]{} spectral index displays a behavior very similar to that of the [PM-L]{} model. However, the evolution after the passage of the main component in model [ OP-L]{} (Fig. \[fig:OP-fiduc-spec\]d – \[fig:OP-fiduc-spec\]f) is different from that of model [PM-L]{}. The number of bright spots popping up in the wake of the main perturbation is smaller and they are brighter (in relation to the quiescent jet) in the [OP-L]{} model than in the [PM-L]{} one. [Identifying]{} these features as trailing components ([ Sect. \[sec:spacetime\_analysis\]]{}), we realize that they do not only appear at 43GHz, but also at 22GHz, and one may guess them even at 15GHz.
Spacetime analysis {#sec:spacetime_analysis}
------------------
In order to relate the hydrodynamic evolution with the features observed in the synthetic radio maps, we have built up several space-time diagrams of the evolution of the component as seen by a distant observer. In Fig. \[fig:PMOP-sptm\] we plot the difference in intensity, averaged over the beam cross-section, between the perturbed and quiescent models. This difference accounts for the net effects that the passage of the hydrodynamic perturbation triggers on the quiescent jet. The trajectory of the main component is seen as a bright (yellow) region close to the top of each plot.
Its superluminal motion is apparent when the slope of the trajectory is compared to that of the dashed line, which denotes the slope corresponding to the speed of light. Below the main component, the dark (blue) region is associated to the reduced intensity that the rarefaction trailing the hydrodynamic perturbation leaves.
As in G97, while in models PM-L and PM-H the main component and the reduced intensity region trailing it are continuous in the space-time diagrams (Fig. \[fig:PMOP-sptm\] upper panels), in OP-L and OP-H models they flash intermittently as they cross over standing cross shocks of the beam (larger intensity – Fig. \[fig:PMOP-sptm\] lower left panel) and then expand in the rarefactions that follow such standing shocks (smaller intensity). The interaction of the perturbation with the standing shocks of the quiescent OP model results in a displacement of the position of the shocks also noticed in G97. The temporarily dragging of standing components, is clearly visible in the lower left panel of Fig. \[fig:PMOP-sptm\]. The second (from the left) of the well identified bright spots, oscillates with an amplitude of $\sim 1.4
R_b$ in $\sim 10\,$months. The trend being to increase both the oscillation period and the amplitude with the distance to the jet nozzle.
Besides
the main component, we observe several trailing components [@Agudoetal01], identified in Fig. \[fig:PMOP-sptm\] by “threads” with an intensity larger than in the quiescent model, which emerge from the wake of the main component. In Fig. \[fig:PMOP-sptm\] we also overplot (black dots) the world-lines of a number of bright features observed in the convolved 43GHz-radio images resulting from the difference between the hydrodynamic models with and without an injected perturbation. These world-lines show only those local intensity maxima which could be unambiguously tracked in [convolved radio maps]{}. Except for the bright features closer to the jet nozzle, the world-lines match fairly well the unconvolved trails of high intensity. The latest three trailing components of Fig. \[fig:PMOP-sptm\] (upper left panel) do actually recede[^5] in the convolved 43GHz maps as much as $0.5R_b$ for 1 to 4 moths, soon after they are identified (i.e., at an apparent speed $\sim 0.5c - 0.9c$).
[Like]{} in the case of PM models, in the wake of the main component of model OP a number of bright spots seem to emerge with increasingly larger apparent velocities as they pop up far away from the jet nozzle. However, looking at the locations from where these components seem to emerge, we notice that they are in clear association with the locus of the standing shocks of the OP models. Such an association is even more evident when we look at the world-lines of the brighter features trailing the main component as they are localized in the 43GHz radio maps. [The]{} physical origin of these trailing features differs from that of the trailing components seen in PM models. [There]{} trailing components are local increments of the pressure and of the rest-mass density of the flow produced by the [ *linear*]{} growth of KH modes in the beam, generated by the passage of the main hydrodynamic perturbation. In the beam of OP models, intrinsically [*non-linear*]{} standing shocks are present. Nonetheless, the interaction of a non-linear hydrodynamic perturbation with non-linear cross shocks yields an observational trace which resembles much that of a trailing component. Thereby we keep calling such features trailing components, following [@Agudoetal01].
If the jet is not pressure matched, all the KH modes excited in the beam are blended with standing knots. Indeed, we realize that close to the jet nozzle, the locus of the first two bright spots is almost standing and, at larger distances, the subsequent knots show a clear increment of its pattern speed. The fist two trailing components are, actually, the traces of standing shocks which are dragged along with the main perturbation and oscillate around their equilibrium positions. The remaining trailing components move much faster and they can probably be due to the pattern motion of KH modes in the OP beam.
Comparing the traces left by the passage of the main hydrodynamic perturbation in the PM and OP models (Fig. \[fig:PMOP-sptm\]), it turns out that the signatures of such perturbation are much cleaner and numerous in [PM than in OP models]{}. The number of trailing components is smaller in [OP than in PM models]{}, and their world-lines are more oscillatory than in the latter case. [For a lager magnetic field (models PM-H and OP-H; Fig. \[fig:PMOP-sptm\] right panels)]{} NTPs cool faster and radiate more energy, [and thus,]{} one can basically see only features happening close to the jet nozzle. The unconvolved data for both PM and OP models, and independently of the magnetic field strength, is compatible with not having any time lag between the high and low frequency radiation emitted by the main component, i.e., the radiation at all three frequencies is co-spatial (Fig. \[fig:PMOP-sptm-traj-nconv\]). However, the convolved data display a number of positive and negative time lags which result from the difference in the size of the convolution beam at every frequency. In case of the PM models, there is a trend of the 43GHz maximum emission to lie behind the corresponding maxima at 22GHz and 15GHz (Fig. \[fig:PMOP-sptm-traj-conv\] upper panels). Thereby, the low energy radiation from the main component is seen first, and later an observer detects radiation at higher frequencies. Nevertheless, considering that the resolution of the convolved data is worse at smaller frequencies, the emission from the component is consistent with having no time-lags between low and high frequency emission. This trend is independent of the magnetic field strength, but it is more obvious for the model [PM-H model]{} (note the large separation between the different symbols beyond $Z_{\rm obs} \sim 15
R_b$ in the Fig. \[fig:PMOP-sptm-traj-conv\] upper right panel). Therefore, any positive or negative time lag of radiation at different frequencies measured from convolved data has to be taken with care.
For OP-L models, positive and negative time lags between the high and low energy radiation are observed along the $z$-axis (Fig. \[fig:PMOP-sptm-traj-conv\] lower left panel). Such time lags are smaller than for the [PM-H]{} model and, indeed, the data are compatible with no-time lags at all. For [OP-H]{}, in most cases, the high-frequency emission dots lie in front of the lower frequency ones (Fig. \[fig:PMOP-sptm-traj-conv\] lower right panel). But still, considering the difference in linear resolution for the location of the maxima, the radiation at different frequencies is almost co-spatial.
Trailing components can only be tracked at 43GHz close to the jet nozzle. Only after a certain distance, it is possible to see them at 22GHz and even at 15GHz (see the last two trailing world lines in each panel of Fig. \[fig:PMOP-sptm-traj-conv\]). The world lines of trailing components at 22GHz and, particularly, at 15GHz, undergo substantial velocity changes. During some time intervals the convolved data shows receding trailing components at such frequencies. In the OP models, there are no clear trends, independent of the magnetic field strength, since it is very difficult to locate any local maxima at 15GHz, and the 22GHz data lie almost on top of the 43GHz points. We note that there is a mismatch between the data points at different frequencies in the [OP-H model]{} at the first two recollimation shocks (vertical threads at $z_{\rm obs}\sim 4R_b$ and $7R_b$). It is produced because there is a rather small relative difference in the emissivity of the perturbed and the quiescent jet models at 43GHz until $z_{\rm obs} \lsim 10R_b$. In such conditions, the algorithm to detect local maxima in the space-time diagrams yields oscillatory results. A large mismatch between the world-lines of the peak intensity of trailing components at different frequencies also happens in other trailing features (e.g., the fourth and fifth threads in Fig. \[fig:PMOP-sptm-traj-conv\] lower right panel). This mismatch does not exist in the corresponding unconvolved data (Fig. \[fig:PMOP-sptm-traj-nconv\]) and, hence, we conclude it is an artifact of the finite size of the convolution beam at the observing frequencies.
Discussion and conclusions {#sec:conclusions}
==========================
We have presented a new method (SPEV) to compute the evolution of NTPs coupled to relativistic plasmas under the assumption that these NTPs do not diffuse across the underlying hydrodynamic fluid. NTPs change their energy because of the variable hydrodynamic conditions in the flow and because of their synchrotron losses in an assumed background magnetic field. The inclusion of synchrotron losses and a transport algorithm for NTPs are major steps forward with respect to previous approaches we have followed. The new method has been validated with another preexisting algorithm suited for the same purpose, but without including synchrotron losses and transport of NTPs (AM algorithm). The validation process shows that the SPEV method reproduces the same qualitative phenomenology as outlined in the previous works of our group (G95, G97). The power of the new method in its whole blossom shows up when synchrotron cooling dominates the NTP evolution.
#### Quiescent jet models:
When synchrotron losses are considered, the resulting phenomenology can be split into two regimes: losses-dominated and adiabatic regime (following the convention of [@MG85]). In the losses-dominated regime, the knots displayed in the radio maps, which are close to the jet nozzle, are brighter than in models which do not include synchrotron cooling at the considered frequencies. [Indeed, quiescent jet models including radiative losses are more knotty than those models which do not include them. These features result]{} from the conservation of the number density of NTPs. Since the same number of particles per unit of volume that initially extends from $\gamma_{\rm min}(t=0)$ to a certain upper limit $\gamma_{\rm max}(t=0)$ is confined into a narrower Lorentz factor interval, wherein more NTPs are efficiently emitting in the considered observational radio bands. In the adiabatic regime (reached relatively far away from the jet nozzle), the spectral changes, that the NTP population experiences as it is advected downstream the jet, of models with and without losses is qualitatively similar, since most of the high-energy NTPs (which evolve faster) have cooled down to energies where losses are negligible. The beam of the jet in the adiabatic regime is dimmer at radio-frequencies than in models where synchrotron losses are not included. Our method lacks of a suitable scheme to account for diffusive shock acceleration of NTPs. However, all shocks existing in the quiescent jet models are rather weak and, for practical purposes they can be considered as compressions in the flow, where an enhanced emission is obtained due to the local increase of density and of pressure.
[One of the main results of this work is that for the same background hydrodynamic jet model, dynamically negligible magnetic fields of different strengths yield substantially different observed morphologies. This introduces a new source of degeneracy (in addition to relativistic effects, such as, time delay, aberration, etc.) when inferring physical parameters out of observations of radio jets. For example, the difference in the observational properties of models OP-L and OP-H (Sect. \[sec:synlos\]) shows, that increasing the magnetic field strength by a factor of 10 triggers a much faster cooling of the NTPs, resulting in a much shorter losses-dominated regime and shorter jets, despite magnetic field remaining dynamically unimportant.]{} Furthermore, jet models with such a large magnetic field display a larger flux density contrast between shocked and unshocked jet regions. The reason being that after the losses-dominated regime, $\gamma_{\rm max}$ is reduced so much that most of the NTP population is inefficiently radiating at the considered radio wavelengths and, only when the non-thermal electrons are compressed at cross shocks of the beam, they partly reenter into the efficiently radiating regime at the considered frequencies.
#### Spectral inversion:
[In this paper we suggest that an inverted spectrum may also result if the lower limit of the NTP distribution $\gamma_{\rm min}$ is larger than the value of $\gamma_{_{\rm M}}$ for which the synchrotron function $R(x)$ reaches its maximum (Eq. \[eq:gamma\_min2\]), in agreement with the theoretical predictions of [@TK07].]{} Evidences for flat, optically thin radio spectra in several active galactic nuclei have been shown by, e.g., [@HAA89b; @Melrose96], and [@Wangetal97]. These authors consider different kinds of Fermi-like acceleration schemes to be responsible for the hardness of the electron energy spectra. [@SP08] show that stochastic interactions of radiating ultrarelativistic electrons with turbulence characterized by a power-law spectrum naturally result in a very hard (actually inverted) electron energy distribution which yields a synchrotron emissivity at low frequencies with an spectral index $\simeq 1/3$. Alternatively, Birk, Crusius-Wätzel & Lesch (2001) argue that optically thin synchrotron emission due to hard electron spectra produced in magnetic reconnection regions may explain the origin of flat or even inverted spectrum radio sources. [In contrast to our findings, these authors, explain the spectral inversion in some sources as a result of a flatter electron energy distribution. Observationally, it could be possible to discriminate between both possibilities by looking at the high-energy spectrum of the source. If there are external seed photons (e.g., from the AGN), which were Compton up-scattered by the non-thermal electrons of the jet, the spectral index at high energies could discriminate between the alternative explanations for the optically-thin inverted spectra at radio frequencies.]{}
Since $\gamma_{\rm min}$ is fixed in our model through Eq. (\[eq:pmin\]) and it is not derived from first principles, one may question whether the value we obtain for $\gamma_{\rm min}$ could be too large and, therefore, the spectral inversion we are explaining on the basis of taking $\gamma_{\rm min}\gsim\gamma_{_{\rm M}}$ is unlikely to happen in nature. This would be the case if the jet was composed by and electron-positron plasma, in which case $\gamma_{\rm
min}\simeq 1$ (e.g.,@Marscheretal07). For plasmas made out of electrons and protons, [@Wardle77] obtained that for synchrotron sources with a brightness temperature $\simeq 10^{12}\,$K and $q=2$, $\gamma_{\rm min}\gsim 161$ in order to account for the low degree of depolarization in parsec-scale emission regions. More recently, [@Blundelletal06] inferred $\gamma_{\rm min}\sim 10^4$ at the hot-spots of 6C0905+3955 (see also @TK07, and references therein). Thus, the exact value of $\gamma_{\rm min}$ is probably source dependent, and our minimum Lorentz factor threshold ($\gamma_{\rm min}\simeq 330$) can be well accounted by present day theory and observations if the jet is not a pure electron-positron plasma.
#### Radio components:
We have applied the SPEV method to calculate the spectral evolution of superluminal components in relativistic, parsec-scale jets. These components are set up as hydrodynamic perturbations [at the jet nozzle]{}. For a small value of the magnetic field (the same as in G97), synchrotron losses are negligible and we recover the phenomenology shown by G97 and [@Agudoetal01].
The main component is characterized by a hardening of the spectrum. Pressure matched models yield a generic spectral profile of the component, which is rather independent of synchrotron losses. The hydrodynamic perturbation looks in the radio maps like a burst at every radio-frequency and, just behind it, there is a decrease of the flux density. The shape of the burst is asymmetric in the axial jet direction, being brighter upstream than downstream. The shape of the burst is also frequency dependent because the convolution beam grows linearly with the observing wavelength (at lower frequencies the component is more symmetric in the axial jet direction). This triggers a decrease of the spectral index in the forward region of the main component, until it reaches a minimum (which precedes the intensity maxima at the highest observing frequency).
When radiative losses are important, a number of differences can be observed:
1. Main component splitting in OP-H model: The main component splits in the radio-maps much more clearly than in OP-L model (Sect. \[sec:losses\]), and the splitting takes place farther away from the nozzle in the latter than in the former case. The rear part of the component is brighter than the forward one if losses are included. The spectral index profile is unaffected by the apparent splitting of the component. We conclude that the apparent splitting of the main component is an artifact of the sampling of the results in the observer frame. It is necessary to perform a finer time sampling of the radio jet than the $\sim 3.5\,$months we have considered in the radio maps, in which case, the main component exhibits an intermittent variation of its flux density (see on line material). If observations do not have the sufficient time resolution, there is another hint that can help to disentangle whether the splitting is apparent or real. In a true splitting of the component, each part may show a different spectral aging due to their different hydrodynamic evolutions.
2. Radio features: Main and trailing components display a less elongated aspect in radio maps. The reason is that without losses, the beam itself is brighter at longer distances. Thus, in the unconvolved data, the parts located downstream the jet weight more in the convolution beam than in the case where synchrotron losses are included. For the same reason, models which include synchrotron losses display a more knotty morphology. In the losses-dominated regime, SPEV models exhibit a brighter main superluminal component than SPEV-NL models. This behavior reverses in the adiabatic regime. Also, the ratio between the peak specific intensity of a trailing component to the specific intensity of the region of the beam immediately behind it, is larger than if losses are not included. The conservation of the NTPs number density explains this phenomenology (Sect. \[sec:magfield\]).
3. Spectral properties: Behind the main component, the spectral index returns almost monotonically to its unperturbed value. In contrast, when losses are negligible, there is a softening of the spectrum, just behind the main component (where the spectral index reaches a maximum).
#### Time lags:
[In this paper we explicitly show,]{} that the convolved data has to be interpreted [ carefully. During most of the time the main component is observable, the radiation emitted by the component at low energy (15GHz) arrives to the observer before that at high energy (43GHz). Indeed, for models PM-H and OP-H, a substantial mismatch between the world-lines of the peak intensity of trailing components at different frequencies is possible. This mismatch is an artifact due to the finite size of the convolution beam at the observing wavelengths.In contrast]{}, the unconvolved data is consistent with a simultaneous emission of radiation at the three frequencies under consideration. This behavior matches our expectations, since the interval of observing wavelengths is too narrow to display a substantial [frequency dependent]{} separation of the regions of maximum emission.
#### On the nature of trailing components:
The journey of the main component downstream the jet generates a number of frequency dependent bright spots which pop up in its wake. [They differentiate themselves from the main component because (1) they do not emerge from the jet core, (2) they posses substantially smaller (sometimes subluminal or even, receding) speeds [@Agudoetal01] and, as we demonstrate here, (3) they do not exhibit an obvious change in the spectral index with respect to the quiescent jet model, but (4) their observational imprint is frequency dependent (they are clearly visible at the highest radio-observing frequencies, but at 22GHz and, particularly, at 15GHz they are wiped out by the large convolution beams at this wavelengths).]{} In pressure matched jet models, [trailing components result from the linear growth of KH modes in the beam, after the passage of the main hydrodynamic perturbation [@Agudoetal01]. Here, we also consider overpressured jet models, where the situation is qualitatively different from pressure matched ones,]{} since the beam of such models develops standing shocks ([*non-linear*]{} structures). Nonetheless, the interaction of a non-linear hydrodynamic perturbation with non-linear cross shocks yields an observational trace which resembles that of a trailing component. Therefore, sticking to the definition of [@Agudoetal01], we also call trailing components to the bright spots following the main component in overpressured models, although the dynamical origin of such components differs. [In this sense, every bright spot that results from the interaction between a strong hydrodynamic perturbation with a relativistic beam, which moves slower than the main component and is not ejected from the jet core shall be considered as a trailing component.]{} We shall add an obvious cautionary note: striving for the knowledge of the jet parameters, on the basis of a fit of the intensity variations behind a main perturbation to a number or KH modes, requires that the jet is pressure matched [(if the jet is not pressure matched, all the KH modes excited in the beam are blended with standing knots and the predicted jet parameters might be inaccurate). Furthermore,]{} it is necessary that the linear resolution of the convolved (observational) data was rather good. We have tested that the unconvolved results are roughly recovered if the FWHM of the beam at 43GHz is smaller than $0.25 R_b$. Insufficient linear resolution biases the observed features in hardly predictable ways, rendering inadequate the identification of features in the radio maps with hydrodynamic structures.
In the future we plan to apply the SPEV method to perform additional parametric studies of relativistic parsec scale jets. Among the parameters which can be interesting to look at, we give preference to the electron spectral index. Also, the SPEV algorithm can be coupled to relativistic magnetohydrodynamic codes. This will drop any assumption about the topology and strength of the magnetic field in the jet, and it will enable us to perform also parametric studies of polarization of the jet emission and of superluminal components.
A. Imaging algorithm {#appendix:imaging}
====================
Equations given in Secs. \[sec:SPEV\] and \[sec:synchrotron\] are, in principle, sufficient to compute the synchrotron emissivity at any position in space and at any instant of time in the observer’s frame, either using SPEV or AM methods, accounting for the appropriate transformations from the frame comoving with the fluid (where the emissivity \[Eq. \[eq:j(nu)\]\], absorption coefficient \[Eq. \[eq:kappa(nu)\]\], number density of NTPs \[Eq. \[eq:n\^a\]\], etc. are computed).[The purpose of this Appendix is to explain the algorithm used to produce synthetic radio maps from discrete spatial and temporal elements.]{}
Geometry and arrival time
-------------------------
While [in our simulations]{} the hydrodynamic state of the fluid is axisymmetric regardless of the jet viewing angle, the observed emission is, in general, not axisymmetric. We introduce the azimuthal angle $\phi$ (measured in the $xy$-plane from the $x$-axis) and define the laboratory frame [(attached to the center of the AGN)]{} 3D Cartesian coordinate system [$(x, y, z) := (R\cos\phi, R\sin\phi, Z)$, where the $z$-axis coincides with the jet axis.]{} We denote the jet viewing angle by $\theta$, and choose the following observer coordinate system (rotated with respect to the 3D Cartesian system by an angle $\theta$ around the $y$-axis) $$\label{eq:3Dobs}
(x_{\rm obs},y_{\rm obs},z_{\rm obs}) := (x\cos\theta + z\sin\theta, y, -x\sin\theta + z\cos\theta)\,$$ in which the observer is located along the $z_{\rm obs}$ axis, far from the jet. For a given elapsed simulation time $T$ in the jet frame the time of observation $t_{\rm obs}$ is defined as $$\label{eq:tobs}
t_{\rm obs} := T - z_{\rm obs}/c$$ The task of the imaging algorithm is to produce image in the $(x_{\rm
obs}, y_{\rm obs})$ plane for a fixed arrival time $t_{\rm obs}$ (note that the image will be symmetric with respect to the $x_{\rm
obs}$-axis if the magnetic field is completely random). From Eqs. \[eq:3Dobs\] - \[eq:tobs\] it is clear that we need to have information about states of the jet at multiple instants of laboratory frame time in order to correctly compute the contribution at a single $t_{\rm obs}$. In a numerical hydrodynamic simulation we only have a finite number of discrete iterations, but each iteration has an associated time step $\Delta T$. In order to correctly take this into account, in the following we assume that the time instant $t_{\rm
obs}$ has a finite duration $\Delta T$ as well, and all radiation arriving between $t_{\rm obs} - \Delta T/2$ and $t_{\rm obs} + \Delta
T/2$ is arriving precisely at $t_{\rm obs}$.
Particle images
---------------
Owing to the axisymmetric nature of the problem, we only follow the Lagrangian particle motion and evolution in two dimensions (see Sec. \[sec:SPEV\]). However, for the purposes of imaging, a three-dimensional particle distribution needs to be created. We assume that each particle which is injected at the jet nozzle has a radius $\Delta r := R_{\rm b} / (2N_{\rm p})$, where $R_{\rm b}$ is the beam radius and $N_{\rm p}$ number of particles per beam radius. That means that a particle in two dimensions correspond to a revolution annulus in the $(x, y, z)$ coordinate system[^6]. In principle, by knowing the particle position $(R_{\rm p}, Z_{\rm p})$ in the 2D grid we could compute from Eqs. \[eq:3Dobs\] - \[eq:tobs\] all combinations of $(x, y, z)$ and, hence, all combinations $x_{\rm
obs}$, $y_{\rm obs}$ and $t_{\rm obs}$ to which the particle annulus corresponds for a fixed $T$. In practice, we approximate every annulus by a series of cubes which are distributed along a circle with radius $R_{\rm p}$, whose center is in $(0, 0, Z_{\rm p})$. The number of cubes, evenly distributed in the azimuthal direction, necessary for an optimal volume coverage of the annulus depends on the relation between $R_{\rm p}$ and the particle radius $\Delta r$ (see next subsection). By virtue of the symmetry of the jet, as seen by the observer, with respect to the $x_{\rm obs}$-axis, we only need to compute the contribution from one half-annulus, i.e. for those cubes where $y = y_{\rm obs} \geq 0$.
[We]{} assume that both the emissivity and the absorption coefficient are homogeneous within each cube. Thus, knowing the particle velocity and the azimuthal angle of a given cube, we can transform its emissivity and its absorption into the observer frame.
### Approximation of annuli by cubes
Given a particle with radius $\Delta r$ and cylindrical coordinates $(R_{\rm p}, Z_{\rm p})$, we approximate the corresponding half-revolution annulus by cubes evenly tessellating a circumference centered at $(0, 0, Z_{\rm p})$. The angular separation between the cubes is defined as $\Delta\phi = \min{(\pi, 2 \Delta r / R_{\rm
p})}$. [Thus,]{} there are $N_\phi = \pi/\Delta \phi =
\max{(1, R_{\rm p} \pi / (2\Delta r))}$ cubes in a half-annulus [of]{} volume $V_{\rm cubes}=N_\phi (2\Delta r)^3 = 4\pi R_{\rm
p}(\Delta r)^2$ whereas the true volume of the half-annulus is $$V_{_{\rm HA}} = \left\{
\begin{array}{rl}
[(R_{\rm p} + \Delta r)^2 - (R_{\rm p} - \Delta r)^2]\pi(2\Delta r)/2
= 4\pi R_{\rm p}(\Delta r)^2 & \rm{if\ } R_{\rm p}>\Delta r\\[4mm]
(R_{\rm p} + \Delta r)^2\pi(2\Delta r)/2 = (R_{\rm p} + \Delta
r)^2\pi\Delta r & \rm{if\ } R_{\rm p}\leq \Delta r
\end{array}
\right.$$ [In the limit of $R_{\rm p} > \Delta r$,]{} the total volume of the cubes is approximating that of the half-annulus. For $R_{\rm
p}<\Delta r$, $N_\phi$ is reset to $1$ and the volume for which $y>0$ is always $4(\Delta r)^3$, which is close to the average over all possible values $R_{\rm p}\le \Delta r$ of the true volume $7\pi(\Delta r)^3/6$.
Radiative transfer
------------------
[To compute an image we subdivide]{} the $(x_{\rm obs}, y_{\rm obs})$ plane into rectangular pixels, and compute the contributions to each pixel by checking which particle cube[^7] intersects which pixel at the right observation time. The ratio of the area of intersection to the pixel area, gives a “weight” of the contribution of a particular cube to the intensity [of the pixel]{}. For a given $T$, the value of $z_{\rm obs}$ for each [particle]{} gives the distance from the observer, so that we create a “line-of-sight” (LoS) for each pixel and sort along this line all contributing particles according to [$z_{\rm obs}$]{} (note that these contributions generally come from different instants of the laboratory frame time $T$). Since in every pixel we sum up the contributions spanning the observer time range $[t_{\rm obs}-\Delta T/2, t_{\rm
obs}+\Delta T/2]$, the intersections of every LoS with particle cubes are segments, not points (which would be the case if in every pixel we would only consider the instantaneous contributions at $t_{\rm obs}$). After all the contributions (i.e., intersection segments) to a pixel have been accounted for, we solve the standard radiative transfer equation to evaluate the final pixel intensity. The above procedure can be performed simultaneously for a number of different values of $t_{\rm obs}$, so that a “movie” in the observer frame can be created. In order to transform the intensity detected in a pixel into a flux we need to multiply by the pixel area.
Tests of the method
-------------------
In order to validate our imaging algorithm we have developed two tests which are based upon the idea that, increasing the number of Lagrangian particles, both the volume filling factor[^8] and the total detected flux should converge. [We first show the convergence of the volume filling method. Then we show that the images and the total flux of the quiescent PM-L and OP-L models converge with increasing number of particle families.]{}
### Volume filling
We have created a toy model consisting of a cylindrical [*jet*]{} with uniform velocity parallel to the jet axis, and with a length equal to the particle size $\Delta r$. The half-volume of such jet is $V_{\rm
j,1/2}=\pi R_{\rm b}^2\Delta r$. We inject $N_{\rm p}$ particles in the jet evenly distributed across the jet radius (i.e., $\Delta
r=R_{\rm b}/(2N_{\rm p})$, or $R_{\rm b}=(2i+1) \Delta r$, $i=0,
\dots, N_{\rm p} - 1$). If particles do not overlap, the volume filling factor is $$\frac{\sum_{i=0}^{N_{\rm p}-1} V_{{\rm cubes},i}}{V_{\rm j,1/2}} =
\frac{\sum_{i=0}^{N_{\rm p}-1} 4\pi(2i+1) (\Delta r)^3}{\pi R_{\rm
b}^2 \Delta r} =
1 - \frac{1}{N_{\rm p}}\, .
\label{eq:Vfilling}$$
Since we have a finite number of particles, the jet volume is only partially patched by the volume occupied by such Lagrangian particles, [i.e., the volume filling factor is smaller than one]{}. [ Increasing]{} the number of particles [brings it closer]{} to one. To test the volume filling method, we produce an “image” of the jet at an observer time $t_{\rm obs}=0$ with a $90^\circ$ viewing angle, accumulating in each pixel the contributions corresponding to a laboratory frame time interval $\Delta T=2 R_{\rm b}/c$. However, instead of summing up the emissivity, we add up the length of the intersection of each particle’s volume with each pixel in the $(x_{\rm
obs}, y_{\rm obs})$ plane (as described above). The idea behind the substitution of the emissivity by the intersection length is that, at $90^\circ$ the intersection length and the intersection volume of the particles are proportional and, thus, measuring lengths or volumes is equivalent.
Since we accumulate in every pixel all contributions in the range $[-\Delta T/2, \Delta T/2]$, the intersection length with each particle equals the size of the particle perpendicular to the LoS ($2\Delta r$). [Hence,]{} the value accumulated in a pixel ${\cal
P}:=(x_{\rm obs}, y_{\rm obs})$, namely $L_{\rm px}$, [is]{} $$L_{\rm px} = \sum_i \frac{A_i}{A_{\rm px}} 2\Delta r\, ,
\label{eq:Lpx}$$ where $A_i$ and $A_{\rm px}$ are the area of intersection of a particle with a pixel and the pixel area, respectively. The sum in Eq. (\[eq:Lpx\]) extends over all particles that are intersected by the line of sight that departs from ${\cal P}$. In the limit $\Delta r
\rightarrow 0$ (equivalently, $N_{\rm p} \rightarrow \infty$) $A_i
\rightarrow 4(\Delta r)^2$. On the other hand, the number of particles intersected by the LoS departing from ${\cal P}$ and having a cross sectional area $A_{\rm px}$ is $N_{\rm px}= A_{\rm px}/(2\Delta
r)^2$. Therefore, we have $$\lim_{\Delta r \rightarrow 0} L_{\rm px} = \lim_{\Delta r
\rightarrow 0} 8 (\Delta r)^3 \frac{N_{\rm px}}{A_{\rm px}} = 2
\sqrt{(R_{\rm b}^2 - y_{\rm obs}^2)}\, .
\label{eq:Lpx2}$$ Equation \[eq:Lpx2\] simply expresses that, in the limit $N_{\rm p}
\rightarrow \infty$, the length measured in the pixel ${\cal P}$ should tend to the length of the chord determined by the intersection of the jet body with the line of sight from ${\cal P}$. [ Figure \[fig:circle\]a shows that for $N_p\geq 16$ the results converge very rapidly]{} to the analytic expectation (thick black line).
### Total flux {#sec:totalflux}
To test the convergence of the imaging algorithm we have produced images of quiescent [PM-L]{} and [OP-L]{} models with varying $N_p$. [The total number of particles in the grid grows as $N_{\rm p}^2$, it is thus important to minimize the number of particle families for numerical purposes]{}. On Fig. \[fig:circle\]b we show the total image flux at $15$GHz for [PM-L]{} and [OP-L]{} models as a function of $N_{\rm p}$. The values are normalized to the flux of the model with the largest number of injected particles per beam radius ($N_{\rm p}=40$), which we consider the reference value. This test is important because the total flux represents a global value of every model, since it is computed by summing up the individual fluxes arriving to each pixel in the detector, and multiplying by the corresponding pixel area. Remarkably, for $N_{\rm p}\geq 16$ the flux does not deviate more than $5\%$ form the reference value. Thus, any model with $N_{\rm p}\geq 16$ has sufficiently converged to an appropriate total flux. This has motivated our choice to work with $N_{\rm p}=32$ in the current paper, [since it yields an optimal trade-off between numerical accuracy and computational cost]{}. Figures \[fig:PM-conv\] and \[fig:OP-conv\] show images corresponding to the convergence tests for models [PM-L]{} and [OP-L]{}, respectively.
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[^1]: With such values of ${\rm b}_b$ the magnetic field is dynamically negligible.
[^2]: [çWe obtain this value from]{} the expression $\alpha_{s3}=(m+2-n)/m$ of [@Koenigl81] with $m=1.15$ and $n=0$. The values of $m$ and $n$ are computed from the decay with the distance to the jet nozzle of the magnetic field strength [ $|{\bf b}|\propto z^{-m}$]{} and of the number density of NTPs per unit of energy [ $n^0\propto z^{-n}$]{}, respectively.
[^3]: [In the online material we provide a movie (“PMOP-fiduc.mpg”) where the evolution of the total intensity at 43GHz is displayed for models PM-L and OP-L.]{}
[^4]: According to [@MAM07], the boundary separating magnetic fields dynamically relevant from those in which the magnetic field is dynamically negligible is around $U_{\rm b}\simeq 0.03P$. In our case, even for the model with the largest comoving magnetic field, we have $U_{\rm b}=0.01P$.
[^5]: Trailing components are pattern motions in the jet beam.
[^6]: Note that also annuli are generated from the rotation of 2D cylindrical numerical cells around the jet axis and, thereby we can apply the same imaging procedure when we use the cell-based algorithm AM.
[^7]: One might also use spheres instead of cubes, but we use cubes to avoid dealing with trigonometric functions and square roots when checking for the intersection between rectangular pixels and particles.
[^8]: We define the volume filling factor as the fraction of the jet volume occupied by our finite size Lagrangian particles.
|
---
abstract: 'We analyze online collective evaluation processes through positive and negative votes in various social media. We find two modes of collective evaluations that stem from the existence of filter bubbles. Above a threshold of collective attention, negativity grows faster with positivity, as a sign of the burst of a filter bubble when information reaches beyond the local social context of a user. We analyze how collectively evaluated content can reach large social contexts and create polarization, showing that emotions expressed through text play a key role in collective evaluation processes.'
author:
- 'Adiya Abisheva, David Garcia, Frank Schweitzer'
bibliography:
- 'websci2016.bib'
title: |
When the Filter Bubble Bursts:\
Collective Evaluation Dynamics in Online Communities
---
Introduction
============
#### When the filter bubble bursts
Rebecca Black, an amateur teenage singer, posted a music video on [`YouTube`]{} on February 10, 2011. The song originally circulated mostly among the Facebook friends of its 13-year old singer and was loved and positively commented. Rebecca Black’s song received the “all the usual friends things” [@Larsen2011] and was enough to please her, but it suddenly went viral *in the wrong direction*. From initial 4,000 views on [`YouTube`]{} her song skyrocketed to 13 Million views. This sudden popularity brought mostly negative attention, up to the point of becoming officially the most disliked [`YouTube`]{} video, and by June 15, 2011 the song received 3.2 Million dislikes in [`YouTube`]{} against less than half a million likes. From *local* fame her song soared to the heights of *global* shame.
The anecdotal example of Rebecca Black’s song is paradigmatic of some aspects of the collective dynamics of evaluations in online media. A video can become relatively popular within a small community and receive initial positive evaluations, but when larger audiences are reached, negativity rises faster than in early moments. Figure \[fig:example\] shows this phenomenon through an example of the relative daily volume of likes and dislikes of a [`YouTube`]{} video. Initially, the video is positively evaluated, but the volume of likes decreases quickly. While initial dislikes also decrease, they start rising after the fourth day, reaching a peak at the ninth day.
![**Example of evaluation dynamics in Youtube.** Normalized daily volume of likes and dislikes for a video in our [`YouTube`]{} dataset. Likes appear soon after the video is uploaded, while dislikes tend to appear later.[]{data-label="fig:example"}](Example-small.pdf){width="68.00000%"}
The early viewers of a [`YouTube`]{} video are prone to like it, either due to a social connection with the uploader, or given the similarity of the video with their past liked content. This is a consequence of the purpose of social filtering mechanisms and recommender systems, which is to personalize content selection such that users find content that they consider relevant and of good quality. In contrast, the video can also spread through other media towards more general users, and eventually reach a global audience with users more critical or negative towards the video. Beyond [`YouTube`]{} videos, this phenomenon can be seen as another aspect of the filter bubble [@Pariser2011]: The reinforcement of opinions caused by filtering mechanisms creates an initial pocket of positivity, but *when the filter bubble bursts*, collective negativity can backlash.
Our study sets out to understand collective evaluation processes in various social media through likes and dislikes, as manifestations of opinions towards the evaluated content. We test the duality of collective evaluations in the local versus global behavior illustrated above, looking for the existence of a threshold of positivity after which negative evaluations rise faster and polarization emerges.
#### Emotions in polarization
Technological filters are not the only factor that shapes collective evaluations; emotions influence cognitive information processing, shaping opinions and attitudes towards online content. A number of studies in social psychology show how emotions influence individual evaluations, judgements, and opinions [@Kuhne2012; @Gonzalez-Bailon2010; @Gorn2011], based on the theory of *core affect* [@Russell1999]. Within this theoretical framework, emotions are composed of two dimensions: i) *valence*, which characterizes the feeling of pleasure or displeasure, and ii) *arousal*, which encompasses a feeling of activation or deactivation, and quantifies mobilization and energy [@Russell1999]. Additional dimensions can improve the representation of emotional experience, such as potency or surprise [@Fontaine2007], but their consistent inclusion in psychological research about opinions is still to be explored.
Research in psychology on the role of emotions in evaluations show that arousal can lead to extreme reactions and polarized responses [@Reisenzein1983]. The theory of misattribution explains this effect [@Zillmann1971; @Reisenzein1983] as a transfer of residual emotions between events that intensifies the reaction to the second event. For instance, men in a state of high emotional arousal (for example from physical exercises) give more extreme ratings of attractiveness to women in comparison to the situation in which raters are in a calm emotional state. Similarly, valence can be misattributed and bias evaluations [@Schwarz1988], in particular when individuals inspecting their current feelings, which might be caused by an incidental source rather than the evaluated content.
Further theoretical explanations for the role of emotions in evaluations pose the reduction of cognitive complexity induced by emotional states, which bias the formulation of evaluations towards fast rather than informed responses. The theory of affect priming explains this through the attribution of an individual’s mood to similarly valenced signals in memory, which helps reducing the effort of evaluation tasks [@Isen1978]. Empirical evidence shows that the subjective experience of arousal motivates evaluation on the extremes [@Paulhus1994]. For example, the ratings of famous figures by students are found to be more polarized right before taking an exam, in comparison to weeks before or after. This kind of reactions are especially salient when arousal is experienced along with negative valence (such as the stress before an exam), and thus we can expect the expression of negative and aroused emotions to motivate more polarized collective evaluations in social media. The digital traces of collective evaluations allow us to analyze further the role of emotions in online evaluation processes.
#### Contributions of this article
In this work we analyze collective evaluations across different social media to reveal statistical regularities related to information filters and emotions. First, we test if the distributions of likes and dislikes of evaluated content shows signs of the existence of multiplicative growth processes of social interaction. Second, we test if the relationship between likes and dislikes is non-linear with a division in to two different modes, corresponding to local and global collective evaluations. Third, we test if the emotions expressed in the evaluated content lead to global and polarized collective responses. Our work provides insights into the properties of collective evaluations and tests established psychology theories on the role of emotions in opinion formation.
Background
==========
#### Collective dynamics
In the last years, lots of research focused on the topic of online communities, i.e. large groups of individuals that interact through an online medium. Collective phenomena such as dynamics of trends [@Wu2007; @Wang2012], or viral marketing [@Leskovec2007] can be assessed with data from online communities. Examples of studies on online user behaviour are understanding dynamics of replying activity and website engagement of users [@Rowe2014], buyer activity in online shopping websites [@Lee2015] and communication dynamics in forums [@Kan2011]. Another example is research on social influence, which was shown to exist in [`YouTube`]{} [@Crane2008], in [`Facebook`]{} [@Onnela2010], and in [`Twitter`]{}[@Varol2014]. Furthermore, social influence on popularity of `Facebook` applications has been shown to arise from a mixture of local and global signals [@Onnela2010]. While the former notion indicates how friends and local community influence an individual’s behaviour, the latter suggests the effect of the aggregate popularity of products or behaviours on an individual. Additionally, previous results for popularity distributions show that the amount of votes for `Digg` stories [@Mieghem2011b] and tweets in trending topics [@Asur2011] follow log-normal distributions that are explained by social coupling.
#### Collective evaluations
Online voting dynamics and dynamics of human appraisal were studied in a number of previous research. Studies on collective evaluations mostly interrelates and finds explanations in research on collective popularity of the online content, with the assumption that more likes lead to item’s popularity. Despite the differences in the ways of measuring popularity, as a number of views in [`YouTube`]{} [@Szabo2010], or as a number of likes and dislikes in [`Reddit`]{} [@Mieghem2011a], or as a number of votes in `Digg` [@Mieghem2011b] or as a time span of trending topics in [`Twitter`]{} [@Asur2011], these measures showed the existence of statistical regularities of content popularity, and fit to the log-normal distribution.
Studies on collective dynamics of negative evaluations are scarcer, but some recent works illustrate that social influence effects are present in movie ratings from `imdb.com` [@Lorenz2009], and that controversiality expressed through movie ratings evolves with time [@Amendola2015]. Additionally, herding effects have been observed in random manipulations of votes in [`Reddit`]{} [@Weninger2015], which shows that the way users vote depends on the votes of other users. Further research on [`Reddit`]{} [@Mieghem2011a] showed a non trivial dependency between likes and dislikes at the collective level, in line with the questions we address in this article.
#### Online polarization
The proliferation of online participatory media, such as social network sites, blogs and online fora, increases users engagement in discussions on political and societal issues, which in its turn may - under certain conditions - split individuals apart in their opinion space. Opinion polarization is characterzied by a division of the population into a small number of fractions with high internal consensus and sharp disagreement between them [@Flache2011]. Agent-based models [@Lorenz2007; @Mas2013] and experimental studies [@Van2009] explain some aspects of opinion formation and its role in consensus and polarization.
Based on data from digital traces, previous research investigated polarization from the network perspective in political blogs [@Adamic2005], in follower and mention links in [`Twitter`]{} [@Conover2011a], and in Swiss politicians profiles [@Garcia2015a], as well as in a non-political domains like friendship networks [@Guerra2013], and cultural expression [@Garcia2013]. Additionally, exprerimental evidence shows that group processes like polarization function differently in computer-mediated communication than in a face-to-face interaction [@Taylor2002], for example as the relative annonymity of online media dampens inhibiting effects like the spiral of silence [@Noelle1993].
#### Online emotions
Emotional expression through online text has been analyzed in earlier research on data from MySpace [@Thelwall2010b], Yahoo answers [@Kucuktunc2012], IRC channels [@Garas2012], Wikipedia [@Iosub2014], BBC, Digg, YouTube and Twitter [@Thelwall2012]. Availability of large-scale quantitative datasets allows us to understand emotions and their role in various domains. Studies in the field of subjective well-being leverage extensively on quantifying emotions through text. For instance, subjective well-being is manifested in [`Facebook`]{} status updates [@Wang2014], and shows a pattern of assortativity in social networks [@Quercia2012] in relation to feelings of loneliness [@Burke2010]. This is in a close relation to the quantification of mood in [`Twitter`]{} which has been used to validate theories of periodic mood oscillations [@Golder2011].
[`Twitter`]{} mood measured in terms of valence and arousal reveals aspects of the relation between mood states and online interaction and participation [@DeChoudhury2012], and the psycholinguistic analysis of emotions reveal the traces of mental health issues [@DeChoudhury2014]. Furthermore, segregation patterns in geographical space [@Lin2014] and gender-based patterns [@Kivran2012; @Thelwall2010b] can be partially attributed to differences in emotional expression. In online interaction, for example in real-time chat conversations [@Garas2012] and product reviews [@Garcia2011], emotions are not a phenomenon characteristic to just an individual, but exhibit collective properties [@Schweitzer2010].
Lastly, information-centric role of online emotions has been studied through blogs [@Miller2011], in [`Twitter`]{} [@Pfitzner2012], and in `Yahoo` answers [@Kucuktunc2012]. Emotions are the building blocks for a creation of social network structures [@West2014; @Tan2011] through empathy [@Kim2012] that lead to correlations between emotional expression and popularity [@Kivran2011; @Tchokni2014]. Negative emotional posts were shown to be drivers of communication among users and responsible for extension of the lifetime of online discussions in forums [@Chmiel2011b]. Furthermore, the digital traces of emotions synchronize with political outcomes [@Gonzalez-Bailon2010], which goes inline with the findings that political discussions are emotionally charged [@Hoang2013], in particular during election periods [@Schweitzer2012].
Data and Methods
================
Data on collective evaluations
------------------------------
#### Datasets
The data used in this research is the result of our crawl of four publicly accessible online communities.
[`YouTube`]{} (<http://www.youtube.com/>) is a video sharing website on which registered users can upload and view videos, as well as post comments and rate videos with likes and dislikes. Our crawlwas launched in June 2011 to daily collect a combination of top videos in various categories and to iteratively explore the channels of general users [@Abisheva2014], including 6.3 Million videos by February 2015.
[`Reddit`]{} (<http://www.reddit.com>) is a message board in which registered users submit posts with links and text, and vote up and down for posts to appear on a frontpage. Conversations between users appear in one of the many thematic boards, called subreddits, covering diverse topics from politics to science fiction and adult content. From 2012 to 2014 our daily [`Reddit`]{}crawlcollected 338,000 submissions from 1,972 subreddits. While the user interface of [`Reddit`]{} provides fuzzed amounts of votes, it is possible to construct the total amount of up and downvotes to a submission based on the JSON fields of reddit score and like ratio. This way, we count with the text and the final amount of up and downvotes for each submission in our dataset.
[`Imgur`]{} (<http://www.imgur.org/>) is an image hosting and sharing website where registered users upload, rate, and discuss uploaded images. Image sharing traffic of [`Imgur`]{} has a large presence in [`Reddit`]{} such that every 6th successful [`Reddit`]{} post has a link to an image on [`Imgur`]{} [@Olson2015]. Our daily crawlcollected 200,000 images and their user activity statistics between December 2015 and January 2016.
Finally, [`Urban Dictionary`]{} (<http://www.urbandictionary.com/>) is an online crowdsourced platform consisting of non-standard lexicon of slang words and idioms. Registered users can submit new terms and provide definitions, and all users of the website, registered and anonymous, can vote up and down for the best definitions. Between April and May 2013 our python-based crawl collected 220,000 definitions and their votes.
All platforms provide functionality for users to evaluate uploaded content positively and negatively by clicking an upvote/[`like`]{} or downvote/[`dislike`]{}button respectively. For simplicity, from now on we refer to evaluated videos, submissions, images and definitions as *items* and we denote as likes and dislikes to positive and negative evaluations, including up and down votes respectively.
#### Sentiment Analysis
To quantify emotional expression, we applied sentiment analysis to headers or titles of each item, leaving for a future research the analysis of longer descriptions, transcripts, and comments. We applied sentiment analysis techniques to video descriptions in [`YouTube`]{}, image titles in [`Imgur`]{}, submission headers in [`Reddit`]{} and term definitions in [`Urban Dictionary`]{}. Headers and titles are a good proxy of the emotional tone of a discussion, in line with earlier research on forum-like conversations [@Gonzalez-Bailon2010].
We measured emotional content of items by applying two complementary sentiment analysis methods. First, we apply a lexicon of affective norms of valence **V**, arousal [**A**]{} and dominance [**D**]{} of nearly 14,000 English words [@Warriner2013]. In line with previous findings [@Warriner2013], the scores of valence and dominance in our dataset are highly correlated, in comparison with the weaker correlation between valence and arousal as explained more in detail in the Results section. This motivates our focus to only valence and arousal as suggested by the theory of core affect.
Second, we apply the SentiStrength classifier [@Thelwall2010a; @Thelwall2012] a state-of-the-art lexicon-based method [@Kucuktunc2012; @Abbasi2014] that has been used in earlier research on the online data from MySpace [@Thelwall2010b], Yahoo! [@Kucuktunc2012], IRC channels [@Garas2012], BBC, Digg, YouTube [@Thelwall2012], Twitter [@Thelwall2012; @Pfitzner2012] and Wikipedia [@Iosub2014]. The core of the SentiStrength method is to predict the sentiment of a text, based upon the occurrences of the words from a lexical corpora, which contains the set of terms with known sentiment of a text. The classifier incorporates various rules, which strengthen or weaken sentiments of the lexicon words detected in the short text. Among the rules are syntactic rules, e.g. exclamation marks and punctuation, language modifiers and intensifiers, such as negation and booster words, and spelling correction rules. The final sentiment score is composed of a positive [**P**]{} and a negative [**N**]{} score for each text as two discrete values in the range of $[+1,+5]$ and $[-5,-1]$ respectively. In our analysis, we normalize all emotions variables to $[0..1]$ mapping [**P**]{}from $[+1,+5]$ to $[0,1]$ and reversing and rescaling [**N**]{} from $[-1,-5]$ to $[0,1]$.
To ensure a valid measurement of sentiment and collective evaluations, we apply two filters to our datasets. First, since both sentiment analysis techniques are designed only for English texts, we apply language classification [@Nakatani2010] and filter out all non-English texts. Second, we remove all items with less than a like and a dislike, and that existed for less than a year in all platforms, to ensure that positive and negative evaluations are stable. Detailed statistics on the number of posts in each dataset are shown in Table \[tab:stats-num-posts\], showing that they are still sufficient for large scale analyses. We will make these datasets available for research purposes.
Statistical analysis methods
----------------------------
#### Distribution fits
We apply a Maximum Likelihood criterion to fit the distributions of likes and dislikes [@Alstott2013], to confirm early findings of the fits of the popularity distribution to the log-normal distribution [@Mieghem2011b; @Asur2011]. We use the `powerlaw` python package to fit four statistical distributions related to complex growth phenomena [@Mitzenmacher2004]: power law, log-normal, truncated power law and exponential distributions. We compare the likelihood of each distribution using the log-likelihood ratio $R=ln(L_1/L_2)$ between the two candidate distributions and its significance value $p$. Positive ratios indicate evidence for the first distribution, and negative ratios for the second one. Instead of testing the hypothesis of the data following a certain distribution, this comparative test answers the question of which parametric distribution provides the best fit available, following the principle of Maximum Likelihood estimation [@Alstott2013]. To finally assess the quality of the best fit, we measure the Kolmogorov-Smirnov distance between the best fitting distribution and the emprical data.
#### Dual regime analysis
We test the existence of a dual local versus global regime in collective evaluations by analyzing the non-linear properties of the relationship between the amounts of likes and dislikes for each item.
We use an extension of a traditional linear modelling, multivariate adaptive regression splines (MARS) [@Friedman1991; @Friedman1993] implemented in the **** programming language package *earth*. MARS fits a continuous piecewise regression function with *knots* that join locally linear pieces. In our analysis, we are interested to test a dual pattern in the relationship between the number of likes L and the number of dislikes D, therefore we set the number of knots to one and fit a model of the form $$\text{D(L)} = I + \alpha_1 * max(0,\text{L}-L_c) + \alpha_2 *
max(0,L_c-\text{L})$$ The values of likes above $L_c$ the values of dislikes above D($L_c$) correspond to observations in the global regime, after the bubble bursts, and the values in which any is below map to the local regime.
To evaluate the quality of the MARS model, we compare it to the Ordinary Least Squares (OLS) regression using the Generalized Cross-Validation prediction error (GCV) defined as $$GCV = \frac{RSS}{N*(1-\frac{\text{ENP}}{N})^2}$$ where $N$ is the number of observations, $RSS$ is the residual sum of squares, and $ENP$ is the effective number of parameters to avoid overfitting [@Friedman1993]. We use the implementation provided by the package *boot* in [****]{} as well as the coefficient of determination $R^2$ of both OLS and MARS fits.
#### Emotion and polarization analysis
Having identified the two regimes and their thresholds in the relationship between the number of dislikes and the number of likes, we can mark items either in the global or the local regime as a binary class. We test how emotions influence the chances of items reaching the global regime through two logistic regression models, one for each sentiment analysis technique. Similarly, we combine the values of likes and dislikes through their geometric mean to measure polarization, as manifested by simultaneous large amounts of positive and negative evaluations. We regress this measure of polarization through two linear models depending on the emotions expressed on the items.
Prior to modelling, we examine the normalized emotional dimensions for multicollinearity by computing the Spearman’s rank correlation coefficients, to avoid singularities. We assess the quality of fits in comparison to null models, by measuring the $\chi^2$ statistic of model likelihood ratio tests implemented in the *lmtest* **** package.
Results
=======
Stylized facts of evaluation distributions
------------------------------------------
Figure \[fig:04-ccdfFitsLNLD\] shows the probability density functions of the distributions of the amount of likes and dislikes for items in each of the four datasets. To understand the process that generates these distributions, we fit a set of parametric distributions that provide insights into how likes and dislikes are given to items. Following the categorization of [@Mitzenmacher2004], generative mechanisms produce stylized size distributions that can be traced back to the properties of growth processes. If the appearance of likes and dislikes follows an uncorrelated process and new evaluations are independent of previous ones, likes and dislikes should follow *exponential* distributions. On the other hand, the presence of likes and dislikes can motivate further evaluations through social effects, creating multiplicative growth (also known as preferential attachment in the context of networks). In the presence of multiplicative growth, if items have similar lifespans, likes and dislikes follow *log-normal* distributions. On the other hand if multiplicative growth is combined with heterogeneous lifespans, likes and dislikes follow a *power law* distribution. This power law can be corrected by adding an exponential cutoff if finite size effects limit the growth of likes and dislikes, a case in which the distributions would be better fitted by a *truncated power law*.
![**Probability density function of collective evaluations**. Probability density function of the number of likes (top) and the number of dislikes (bottom) with exponential binning and fits to log-normal distribution $ln\mathcal{N}(\mu,\sigma)$ (red dashed lines). For all datasets, the results of the log-likelihood pairwise comparisons of the four distributions (see text) identified the log-normal distribution as the best fit.[]{data-label="fig:04-ccdfFitsLNLD"}](PDFs.pdf){width="95.00000%"}
For all datasets, the results of pairwise comparisons of the four proposed distributions identified the *log-normal* distribution as the best fit, with significant and positive log-likelihood ratios as shown in Table \[tab:03-ln-ks\] along with the best fitting parameter estimates. The dashed lines in Figure \[fig:04-ccdfFitsLNLD\] show the fitted distributions, revealing the quality of the fit. The cases of [`YouTube`]{} and [`Urban Dictionary`]{} provide very good fits with extremely low Kolmogorov-Smirnov $D$ statistics. The fits are not so good at the tails of [`Reddit`]{} and [`Imgur`]{}, but the the Kolmogorov-Smirnov $D$ statistic provide good values below 0.05 and the *log-normal* distribution clearly outperforms all others. The worst fit is for the number of likes in [`Imgur`]{}, for which Figure \[fig:04-ccdfFitsLNLD\] suggests a bimodal pattern. Identifying the possible mechanisms that can produce such bimodality goes beyond the scope of this research. We can conclude that the amounts of likes and dislikes display a general heavy tailed behavior of *log-normal* distributions, lending evidence for the production of evaluations following socially coupled growth processes with homogeneous life spans.
The dual pattern of collective evaluations
------------------------------------------
We explore the existence of a dual relationship between likes and dislikes through non-linear MARS fits, testing if the relationship can be divided in a local and a global regime. We restrict the number of model terms to have a single knot, measuring if a dual model outperforms a linear pattern. Figure \[fig:03-2DhistLDplot\] shows the results of MARS fits between the logarithms of likes and dislikes. Vertical and horizontal lines mark the likes cutoff value $L_c$ and its corresponding value of dislikes in the fit D($L_c$). These cutoff values divide the system in a local versus a global regime, with the fitted functions of the form $D \propto L^{\lambda}$ and $D \propto L^{\gamma}$ respectively.
In all datasets, the exponent of the global regime is larger than exponent of the local one, for example in [`YouTube`]{} $\gamma = 0.93 > \lambda = 0.29$. While both exponents are below $1$ and indicate sublinear scaling, the much higher value of the second one shows that, beyond a threshold value of likes, the dislikes given to items grow faster than below the threshold as a sign of the burst of a filter bubble. The presence of scaling in [`Reddit`]{} votes was previously reported in a smaller data subsample [@Mieghem2011a], concluding the existence of superlinear scaling of dislikes with likes. Our analysis shows that the relationship between likes and dislikes in [`Reddit`]{} is better approximated by a dual regime model, in line with the results of the other three datasets.
We evaluate the goodness of the dual model against a single regime model in Table \[tab:03-r2-lm-mars\]. The dual model outperforms in $R^2$ and GCV to the single regime model, lending strong evidence to the existence of two regimes. We further tested if additional knots could improve the fits, and found that a dual regime is the optimal model for [`Urban Dictionary`]{}, [`YouTube`]{}, and [`Reddit`]{}, and only a 4 knot model could improve the [`Imgur`]{} fit by a marginal GCV of less than 0.01.
![**Relationship between the number of dislikes and likes**. Two-dimensional joint distributions with 50 bins, bin colors indicate the count of observations within the bin. Purple and red lines show the local and global regimes of the non-linear relationship between the number of dislikes and the number of likes. Threshold estimates are located at $L_c$, estimated as $L_{c} = 155$ in [`Urban Dictionary`]{}; $L_{c} = 131$ in [`YouTube`]{}; $L_{c} = 7$ in [`Reddit`]{}; and $L_{c} =
27$ in [`Imgur`]{}.](LDs.pdf){width="95.00000%"}
\[fig:03-2DhistLDplot\]
Emotions in the global regime
-----------------------------
Figure \[fig:07-emoscorrplot\] illustrates the rank correlations between emotional dimensions. In all datasets valence and dominance are *highly correlated* with $\rho \geqslant 0.7^{***}$, and therefore we discard the dominance variable from regression analysis as it is difficult to distinguish from valence. Valence and positivity [**P**]{} have a minor positive significant correlation $\rho \in [0.2,0.3]$, and valence and negativity [**N**]{} have a slightly negative correlation $\rho \approx -0.3$, illustrating the relation of emotion variables accross both valence/arousal and positive/negative models.
We fit two regression models in which the probability of the event of an item reaching the global regime $G$ depends on the emotions expressed in the evaluated item. The first model uses [**V**]{} and [**A**]{} as explanatory variables, and focuses on the role of emotions as quantified through their pleasant/unpleasant and active/calm dimensions. The second model takes [**P**]{}and [**N**]{} as predictors, and measures significance of positive and negative sentiments in bringing an item to global regime. Table \[tab:07-lr-pn-pol\] reports the results of logistic regression of the form $logit(G) \sim V + A$ and $logit(G) \sim P + N$ respectively. The role of arousal is heterogeneous, having a significant positive effect in [`Urban Dictionary`]{} and [`Imgur`]{}, but a weak negative effect in [`YouTube`]{} and a non-significant one in [`Reddit`]{}. The effect of valence is also mixed, in [`Urban Dictionary`]{} and [`YouTube`]{} the chances of reaching the global regime grow with valence, while in [`Reddit`]{} and [`Imgur`]{} is the opposite case. The second model sheds more light to this: the pattern is the same for positive sentiment, but negative sentiment increases the chance of reaching the global regime in all datasets but [`Reddit`]{}, where the effect is not significant.
\[tab:07-lr-pn-pol\]
Analysis of emotions in polarization
------------------------------------
Since the distributions of likes and dislikes are approximately log-normal, we can treat the logarithms of likes $ln(L)$ and dislikes $ln(D)$ as centrally distributed around their means ${\left\langle ln(D) \right\rangle}$ and ${\left\langle ln(L) \right\rangle}$. We standardize the logarithmic counts $ln(D)$ and $ln(L)$ as: $$Z_L = \frac{ln(L) - {\left\langle ln(L) \right\rangle}}{sd(ln(L))} \quad Z_D = \frac{ln(D) -
{\left\langle ln(D) \right\rangle}}{sd(ln(D))}$$ where $sd(ln(L))$ and $sd(ln(D))$ are the standard deviations. Then, we compute a measure of polarization as the geometric mean of both values: $Pol=\sqrt{Z_L*Z_D}$. This measure captures the principle that polarization is high under simultaneous large amounts of positive and negative evaluations, and that polarization is low when only one of the values is dominant.
To understand which kind of emotional content creates polarization, we fit two regression models as in the previous section, one of polarization as a function of valence and arousal in the evaluated item, and another as a function of positive and negative sentiment scores. The results of the fits are shown in Table \[table:LikesModels\]. In line with the theory that links arousal to more extreme opinions, we find a general pattern in three datasets where arousal leads to higher levels of polarization. While there is no significant effect in [`Reddit`]{}, all the other datasets show that items that contain words that transmit higher arousal also create a stronger polarized response.
This also manifests in the model using positive and negative scores, where negative content predicts higher polarization in the same three cases as for arousal. The results of these two metrics are consistent with the hypothesis that the expression of activating and negative feelings, such as anger or outrage, tend to create more polarized responses, in line with the theoretical argument that poses emotions as mechanisms to speed up evaluation processes at the expense of more extreme reactions.
Valence in evaluated items creates different responses. Two communities, [`Imgur`]{} and [`Reddit`]{}, show a negative relation of polarization with valence and positive sentiment. The other two, [`Urban Dictionary`]{} and [`YouTube`]{}, show the opposite, where polarization increases with valence. This suggests a context dependent interpretation of positive expression, which does not necessarily motivate positive empathy but can also fuel polarized responses. The positive and negative scores model works better than the valence and arousal model in all cases but [`Reddit`]{}, where the valence and arousal model was more explanatory for polarization, as evidenced by $\chi^2$ tests comparing both models.
Discussion
==========
Our study of emotions focuses on understanding the role of emotions expressed in the text of items with relation to the chances that the items reach the global regime and produce polarized evaluations. While we used two established and validated sentiment analysis methods based on metrics from psychology, future advanced techniques can reveal new patterns and potentially falsify the conclusions of our analysis with current techniques. Furthermore, deeper analyses on individual data can correlate the expression of individual emotions in the comments of a user and the evaluations given by the user, bridging closer this way the measurement of emotional states and evaluations and providing a better understanding of interpersonal emotions.
Following an observational approach to collective evaluations has the advantage of having high ecological validity, but lacks the level of control that can be induced in experimental scenarios. We can deduce insights on the factual properties of collective evaluations, such as the dual regime between likes and dislikes, but testing the conditions that produce them requires a controlled set up. Our motivation and explanation for the dual regime stems from the phenomenon of filter bubbles [@Pariser2011], but to fully understand how these filters affect our behavior we need to experiment on how individual evaluations respond to filtering mechanisms. While these experiments can be carried out it in typical psychological settings and surveys, large platforms like [`Facebook`]{} can also experiment with the behavior of their users in this respect (under the appropriate ethical considerations). A complete understanding of online evaluations can only be achieved when our results are complemented by experimental approaches.
The use of observational data has the advantage of taking a *natural exposure* approach: we analyze the evaluations of what people actually see, rather than the *forced exposure* to content in experiments [@McPhee1963]. In contrast, using digital traces of evaluations contains a selection bias by which some users might be responsible for much larger amounts of likes and dislikes than other users. While this selection bias is natural at the collective level, inferring conclusions about the behavior of individuals needs to consider corrections and use richer datasets [@Cuddeback2004], or apply agent-based modelling approaches to connect the micro and macro levels [@Schweitzer2010].
We explain the dual pattern between likes and dislikes as the result of filter bubbles, but other possible explanations might also be plausible. Some unkown deleting mechanism might downsample videos with a lot of dislikes in the local regime, or some external factor like audience size might explain the values of the thresholds. The results of our statistical analyses of distributions of likes and dislikes fit to hypothetical mechanisms of multiplicative growth, in line with previous findings on popularity metrics rather than evaluations [@Mieghem2011b; @Asur2011]. Our in depth statistics also provide a clear view on the limits of our results, for example in the worse fits of log-normal distributions in [`Imgur`]{}. Future research can conjecture on the possible alternative explanations of our findings, in particular with respect to which filtering mechanisms are in place. Our results do not allow us to distinguish social filtering, based on friends and follower links, from recommender systems, which are based on previous evaluations of a user. Further research with information on individual behavior can shed light on these different processes, for example measuring evaluation tendencies to content produced by friends versus strangers, or across assortative and disassortative links with respect to opinions.
Our analysis of the relation between likes and dislikes is based on the amounts given to items after a long time has passed. This way, we evaluate items after they do not attract lots of attention and their counts are stable. In a figurative way, we study the *fossils* of broken filter bubbles, but we do not study them in a live setting. To fully understand the dynamics of collective evaluations, we need data with temporal resolution on the counts of likes and dislikes. In general, such data is not publicly available on the sites, which requires a much more powerful crawling approach to monitor items on a frequent basis, or access to proprietary data.
Conclusions
===========
Our analysis of collective evaluations across various online media shows statistical regularities in the distributions of evaluations and their relationships. Our contribution is threefold: First we report that the distributions of the amounts of likes and dislikes per item are well fitted by log-normal distributions, a result that gives insights into the properties of the process that creates evaluations. Second, we test the existence of a dual pattern in the relation between likes and dislikes, finding robust evidence of the existence of a local and a global regime that is consistent with our hypotheses about the burst of filter bubbles. Third, we found evidence for the role of emotions in the creation of polarization and the access to the global regime, lending support for psychology theories about the role of affect, in particular arousal, in the polarization of opinions.
Our results have implications for the design of online platforms and filtering mechanisms. Recommender systems and filtering mechanisms allow users to discover content of relevance and quality, but can have unintended consequences in the large scale. Our results suggest that the increasing polarization levels of discussions might be created by these filtering mechanisms, and that users are at risk of receiving a negative backlash to their content when it goes beyond their local social context. Such abrupt behavior with respect to negative evaluations can have important consequences to user motivation and engagement, which might only be visible on the long run.
Our findings shed light on fundamental polarization processes, in particular with respect to the role of emotions. Increasing levels of polarization pose a risk of social conflict and hinder collaboration and common goods, but a healthy society needs certain level of disagreement to be able to deliberate, discuss, and take decisions about important topics. Calibrating the design of web and social media offers this way the chance to find a balance between stagnation and polarization, leading to productive interaction in our current online society.
Acknowledgments:
================
This research was funded by the Swiss National Science Foundation (CR21I1\_146499/1).
|
---
abstract: 'We consider the problem of determining the minimal time for which an energy supply source should operate in order to supply a system with a desired amount of energy in finite time.'
address: |
Department of Mathematics and Natural Sciences, American College of Greece\
Aghia Paraskevi, Athens 15342, Greece
author:
- Andreas Boukas
title: Minimal Operation Time of Energy Devices
---
Introduction
============
While boiling water or any other liquid, most of us have noticed that the heater can be switched off at an intuitively chosen time and the liquid will still reach its boiling point no long after the heater is switched off. A natural question arises: how can switch-off time be chosen in an optimal way so that electrical energy will not be wasted? In other words what is the earliest time at which the heater can be turned off while still reaching the liquid’s boiling point in a finite time? A general formulation of the problem is as follows: Let D be a device that supplies energy to a system S through a supply line. Let $E^{\prime}(t)$ be the energy supply rate. We assume that D can be switched on and off and that it continues to supply energy, at a decreasing rate, for some time after it has been switched off. Question: What is the minimum switch-off time of D (corresponding to the minimal operational time of D) in order to transfer to S a total amount of energy $Q$ (where $Q>0$ is given)? The device $D$ can also be viewed as a control mechanism for bringing the system $S$ from an energy level $E_1$ to a higher energy level $E_2$ in finite time while operating for the minimum time possible. Clearly, the solution of this problem can have a lot of applications , both civilian (e.g energy conservation) and military (e.g minimizing detection risk).
Examples
========
EXAMPLE 1. Exponential Model.
We consider a simple example of an energy device supplying energy to a system at time $t\geq0$ at a rate (which in what follows we consider to include the rate at which the transferred energy is possibly radiating from the system and/or the supply line) given by
$$E^{\prime}(t)=
\left\{
\begin{array}{ll}
e^{a\,t}-1&\mbox{if $0 \leq t \leq t_0$}\\
e^{a\,t_0}-1&\mbox{if $t_0 \leq t \leq t_1$}\\
\frac{e^{a\,t_0}-1}{1-e^{-b\,T}}\,\left(e^{-b\,(t-t_1)}-e^{-b\,T}\right)&\mbox{if $t \geq t_1$}
\end{array}
\right.$$
where $a$ and $b$ are positive real numbers characteristic of the source but also depending on the environment, $t_0$ is the time at which the energy supply rate is at its peak, and $t_1$ is the switch-off time. We assume that the optimal switch-off time is after the rate of energy supply has been stabilized i.e that $\hat{ t_1}\geq t_0$. We also assume that after switching-off at time $t_1\geq t_0$, the source stops transferring energy to the device at time $t_1+T$, where $T>0$ is independent of $t_1$. Let $Q>0$ be the amount of energy that we wish to transfer to the system. We assume that the transfer of this amount of energy will occur at some time $t_2 \geq t_1\geq t_0$. The energy $Q$ could be, for example, the energy required to bring a liquid substance to its boiling temperature, or the energy required for complete phase transition. In the latter case $Q=m\,L_v$, where $m$ is the mass of the liquid and $L_v$ is its latent heat of vaporization (cf. \[2\]). We require that
$$\int_0^{t_2}\,E^{\prime}(s)\,ds =Q$$
which implies that
$$\int_0^{t_0}\,E^{\prime}(s)\,ds +\int_{t_0}^{t_1}\,E^{\prime}(s)\,ds+ \int_{t_1}^{t_2}\,E^{\prime}(s)\,ds =Q$$
or, by the definition of $E^{\prime}(s)$,
$$\int_0^{t_0}\, \left( e^{a\,s}-1 \right) \,ds +\int_{t_0}^{t_1}\, \left( e^{a\,t_0}-1 \right) \,ds+ \int_{t_1}^{t_2}\, \frac{e^{a\,t_0}-1}{1-e^{-b\,T}} \,\left(e^{-b\,(s-t_1)} -e^{-b\,T}\right) \,ds =Q$$
which implies that
$$\left[ \frac{e^{a\,s}}{a}-s\right]^{s=t_0}_{s=0} + \left( e^{a\,t_0}-1 \right)\,(t_1-t_0)+ \frac{e^{a\,t_0}-1}{1-e^{-b\,T}}\,\left[-\frac{e^{-b\,(s-t_1)} }{b} -e^{-b\,T}\,s\right]_{s=t_1}^{s=t_2} =Q$$
from which letting
$$\begin{aligned}
y&:=&t_2-t_1\\
L_0&:=&\frac{1}{e^{a\,t_0}-1 }\,\left(Q+\frac{1}{a}-\frac{ e^{a\,t_0}}{a}+e^{a\,t_0}\,t_0+\frac{ 1-e^{a\,t_0}}{1-e^{-b\,T} }\,\frac{1}{b}\right)\\
L_1&:=&\frac{1}{b(1-e^{-b\,T}) }\\
L_2&:=&\frac{1}{1-e^{-b\,T}}\end{aligned}$$
we obtain
$$t_1= L_0+L_1\,e^{-b\,y}+L_2\,y.\label{ee1}$$
Notice that if $y=0$ then $t_1=L_0+L_1$ is the time the device supplies the desired energy level $Q$ without being switched-off. The maximum value of $y$ is $y=y_{max}=(t_1+T)-t_1=T$. The optimal switch-off time $\hat{t_1}$ is therefore determined from (\[ee1\]) by letting $y=T$ and it is given by
$$\hat{t_1}=L_0+L_1\,e^{-b\,T}+L_2\,T.$$
Energy level $Q$ is reached at time
$$t_2=\hat{t_1}+T=L_0+L_1\,e^{-b\,T}+(L_2+1)\,T.$$
EXAMPLE 2. Linear Model.
A simplified version of the previous model is obtained by assuming that
$$E^{\prime}(t)=
\left\{
\begin{array}{ll}
\frac{a}{t_0}\,t&\mbox{if $0 \leq t \leq t_0$}\\
a&\mbox{if $t_0 \leq t \leq t_1$}\\
-\frac{a}{T}\,(t-t_1)+a&\mbox{if $t \geq t_1$}
\end{array}
\right.$$
where $t_0>0$, $a>0$, and $T$ is as in Example 1 above. In this case
$$\int_0^{t_2}\,E^{\prime}(s)\,ds =Q$$
implies that
$$\int_0^{t_0}\,E^{\prime}(s)\,ds +\int_{t_0}^{t_1}\,E^{\prime}(s)\,ds+ \int_{t_1}^{t_2}\,E^{\prime}(s)\,ds =Q$$
or, by the definition of $E^{\prime}(s)$,
$$\int_0^{t_0}\,\frac{a}{t_0}\,s \,ds +\int_{t_0}^{t_1}\, a\,ds+\int_{t_1}^{t_2}\, \left( -\frac{a}{T}\, (t-t_1) +a \right) \,ds =Q$$
which implies that
$$\left[ \frac{ a}{t_0 }\,\frac{s^2}{2}\right]^{s=t_0}_{s=0} + a\,(t_1-t_0)+\left[-\frac{a}{2\,T}\,(s-t_1)^2+a\,s\right]_{s=t_1}^{s=t_2} =Q$$
and letting $y=t_2-t_1$ we obtain
$$t_1=\frac{1}{2\,T}\,y^2-y+\frac{Q}{a}+\frac{t_0}{2}.$$
As in Example 1, substituting $y$ by $ y_{max}=T$ we obtain
$$\hat{t_1}=-\frac{1}{2}\,T+\frac{Q}{a}+\frac{t_0}{2}$$
which is bigger or equal to $t_0$ if and only if $\frac{Q}{a} -\frac{1}{2}\,T-\frac{t_0}{2}\geq0$. Energy level $Q$ is reached at time
$$t_2=\hat{t_1}+T=\frac{t_0}{2}+\frac{Q}{a} +\frac{1}{2}\,T.$$
General Description of the Optimal Switch-Off Time
==================================================
The examples treated in detail in the previous section suggest the following general theorems.
Let, for each $t_1\geq t_0$
$$E_{t_1}^{\prime}(t)=
\left\{
\begin{array}{ll}
f(t)&\mbox{if $0 \leq t \leq t_0$}\\
f(t_0)&\mbox{if $t_0 \leq t \leq t_1$}\\
g(t)&\mbox{if $t \geq t_1$}
\end{array}
\right.$$
where $f$ is continuous and increasing with $f(0)=0$, and $g$ is continuous and decreasing with $g(t_1)=f(t_0)$. Let $F$ and $G$ denote the anti-derivatives of $f$ and $g$ respectively. If there exists $T>0$ such that $g(t_1+T)=0$ for all $t_1\geq t_0$, then the optimal switch-off time $\hat{t_1}\geq t_0$ is the solution of
$$F(t_0)+f(t_0)\,(\hat{t_1}-t_0)+G( \hat{t_1}+T )-G(\hat{t_1})=Q.$$
The condition
$$\int_0^{t_2}\,E_{t_1}^{\prime}(s)\,ds =Q$$
implies that
$$\left(F(t_0)-F(0)\right)+f(t_0)\,(t_1-t_0)+\left(G(t_2)-G(t_1)\right)=Q$$
which by $F(0)=0$ and the fact that for the optimal $\hat{t_1}$, $t_2=\hat{t_1}+T$ implies that
$$F(t_0)+f(t_0)\,(\hat{t_1}-t_0)+G( \hat{t_1}+T )-G(\hat{t_1})=Q.$$
We can generalize the above theorem to the case of a heat source that supplies energy at a possibly strictly increasing (or even arbitrary continuous) rate as follows.
Let, for each $t_1\geq t_0$, the energy function $E_{t_1}(t)$ be an increasing continuously differentiable function of $t$. The optimal switch-off time $\hat{t_1}$ and the associated time $t_2$ at which the energy level reaches $Q$ are determined by the system of equations
$$E_{\hat{t_1}}(t_2)=Q,\,\,\,\,\, E_{\hat{t_1}}^{\prime}(t_2)=0.$$
It is clear that the optimal switch-off time $\hat{t_1}$ makes full use of the energy source in the sense that the desired energy amount $Q$ is supplied at the moment when the source dies out i.e when $E_{\hat{t_1}}^{\prime}(t_2)=0$. For such a $t_2$,
$$\int_0^{t_2}\,E_{ \hat{t_1} }^{\prime}(s)\,ds =Q$$
implies
$$\int_0^{\hat{t_1}}\,E_{ \hat{t_1} }^{\prime}(s)\,ds+ \int_{\hat{t_1} }^{t_2}\,E_{\hat{t_1} }^{\prime}(s)\,ds =Q$$
i.e $E_{ \hat{t_1} }( \hat{t_1})- E_{ \hat{t_1} }( 0 ) +E_{\hat{t_1} }( t_2)-E_{ \hat{t_1} }( \hat{t_1}) =Q$ and since $E_{\hat{t_1}}(0)=0 $, we obtain $E_{\hat{t_1}}(t_2)=Q$.
Noisy Supply Line
=================
The energy supply rate functions $\phi_{ t_1}(t):=E^{\prime}(t)$ used in Examples 1 and 2 above, can be viewed on each of the intervals $[0,t_0]$, $[t_0,t_1]$ and $[t_1,t_2]$, as solutions of ordinary differential equations of the form
$$d\phi_{ t_1}(t)=(c_1\,\phi_{ t_1}(t)+c_2)\,dt\label{n1}$$
for appropriate constants $c_1,c_2$ (depending on the interval) and initial condition $\phi_{ t_1}(0)=0$, with the different branches of $\phi_{ t_1}(t)$ tied up at $t_0$ and $t_1$. It could be the case however, that the supply line connecting the energy source $D$ with the system $S$ is affected by noise, appearing in (\[n1\]) in the form of additive noise
$$d\phi_{ t_1}(t)=(c_1\,\phi_{ t_1}(t)+c_2)\,dt+ (c_3\,\phi_{ t_1}(t)+c_4) \,dB(t)\label{n2}$$
or equivalently
$$\phi_{ t_1}(t)=\int_0^t\,\left(c_1\,\phi_{ t_1}(s)+c_2\right)\,ds+\int_0^t\,\left(c_3\,\phi_{ t_1}(s)+c_4\right)\,dB(s)\label{n3}$$
where $c_1,c_2,c_3,c_4 $ are constants, $B(s)$ is one dimensional Brownian motion and the stochastic integral on the right hand side of (\[n3\]) is in the sense of Itô (cf. \[1\]). In that case $\phi_{ t_1}(t)$ is actually a stochastic process $\phi_{ t_1}(t,\omega)$ and the problem of finding the optimal switch-off time $\hat{t_1}$ now amounts to finding the first $t_1$ for which
$$\int_0^{t_2}\,\mu_{t_1}(s) \,ds =Q$$
for some finite $t_2\geq t_1$, where $\mu_{t_1}(t)$ denotes the mathematical expectation of $\phi_{ t_1}(t,\omega)$ . Since the mathematical expectation of an Itô stochastic integral with respect to Brownian motion is equal to zero, (\[n3\]) implies upon taking the expectation of both sides that
$$\mu_{ t_1}(t)=\int_0^t \,\left(c_1\,\mu_{ t_1}(s)+c_2\right)\,ds$$
which can be solved explicitly and yields the mean energy supply rate
$$\mu_{ t_1}(t)=\frac{c_2}{c_1}\,\left( e^{c_1\,t}-1 \right)$$
where on each interval the constants $c_1$ and $c_2$ are determined by using the initial and tying up conditions. Thus we are reduced to the deterministic problem considered in the examples of Section 2, but this time for the mean energy supply function. The method extends directly to the case when equations (\[n1\]) and (\[n2\]) are replaced by the more general equations
$$d\phi_{ t_1}(t)=f(t,\phi_{ t_1}(t))\,dt$$
and
$$d\phi_{ t_1}(t)= f(t,\phi_{ t_1}(t)) \,dt+ g(t,\phi_{ t_1}(t))\,dB(t)$$
respectively.
[9]{}
Oksendal, B., *Stochastic Differential Equations*, 2nd edition, Springer-Verlag 1989.
Faughn J.S., Serway R. A., Vuille C., Bennett C. A., *College Physics*, 7th edition, Thomson Brooks/Cole 2006.
|
Introduction {#sectionI}
============
Recently Andersson [@andersson] discovered that gravitational radiation tends to destabilize the [*r-*]{}modes of rotating stars. Friedman and Morsink [@friedman-morsink] then showed that this instability is generic, in the sense that gravitational radiation tends to make [*all r-*]{}modes in [*all*]{} rotating stars unstable. Lindblom, Owen, and Morsink [@lindblom-owen-morsink] have recently evaluated the timescales associated with the growth of this instability. Gravitational radiation couples to these modes through the current multipoles rather than the more typical mass multipole moments. This coupling is stronger than anyone anticipated for these modes, and is so strong in fact that the viscous forces present in hot young neutron stars are not sufficient to suppress the gravitational radiation driven instability. Gravitational radiation is expected therefore to carry away most of the angular momentum of hot young neutron stars. These results have now been verified by Andersson, Kokkotas, and Schutz [@andersson-et-al].
In this paper we study the gravitational waveforms that are produced as the [*r-*]{}mode instability grows and radiates away the bulk of the angular momentum of a hot young rapidly rotating neutron star. The properties of the [*r-*]{}modes and the instability associated with them are reviewed in Sec. \[sectionII\]. The equations that describe (approximately) the evolution of the [*r-*]{}modes as they grow and spin down a rapidly rotating neutron star are derived in Sec. \[sectionIII\]. These equations are solved numerically and the results are also presented in Sec. \[sectionIII\]. The gravitational waveforms associated with the [*r*]{}-mode instability are evaluated in Sec. \[sectionIV\]. General analytical and detailed numerical expressions for these waveforms are presented. In Sec. \[sectionV\] we evaluate the detectability of this type of gravitational wave signal by the laser interferometer gravitational wave detectors such as LIGO [@science], VIRGO [@virgo], and GEO [@geo600]. We consider the detectability of signals produced by single nearby sources, and also the detectability of a stochastic background of sources from throughout the universe. Finally, in Sec. \[secVI\] we discuss the prospects for gravitational-wave astronomy opened up by the $r$-modes.
The [*r*]{}-Mode Instability {#sectionII}
============================
The [*r-*]{}modes of rotating Newtonian stars are generally defined to be solutions of the perturbed fluid equations having (Eulerian) velocity perturbations of the form
= R f(r/R) \^[B]{}\_[lm]{} e\^[it]{},\[2.1\]
where $R$ and $\Omega$ are the radius and angular velocity of the unperturbed star, $f(r/R)$ is an arbitrary dimensionless function, and $\vec{Y}^{B}_{l\,m}$ is the magnetic type vector spherical harmonic defined by
\^[B]{}\_[lm]{}= \[l(l+1)\]\^[-1/2]{}r (r Y\_[lm]{}). \[2.2\]
For barotropic stellar models, of primary concern to us here, the Euler equation determines the form of these modes: The radial dependence $f(r/R)$ is determined to be $f(r/R)=\alpha
(r/R)^l$, where $\alpha$ is an arbitrary constant [@provost]. These modes exist with velocity perturbations as given by Eq. (\[2.1\]) if and only if $l=m$ [@provost]. Also, the frequencies of these modes are given by [@papaloizou-pringle]
= - [(l-1)(l+2)l+1]{}.\[2.3\]
These modes represent large scale oscillating currents that move (approximately) along the equipotential surfaces of the rotating star. The restoring force for these oscillations is the Coriolis force; hence the frequencies of these modes are low compared to the usual $f$ and $p-$modes in slowly rotating stars. These expressions for $\delta \vec{v}$ and $\omega$ are the lowest order terms in an expansion in terms of the angular velocity $\Omega$. The exact expressions contain additional terms of order $\Omega^3$. There may exist other modes of rotating barotropic stellar models with properties similar to these classical [*r-*]{}modes; however, our discussion here is limited to the properties of these classical [*r-*]{}modes.
The density perturbation associated with the [*r-*]{}modes can be deduced by evaluating the inner product of $\vec{v}$ (the unperturbed fluid velocity) with the perturbed Euler equation, and the equation for the perturbed gravitational potential [@ipser-lind]:
&&= R\^2\^2\
&&Y\_[l+1l]{} e\^[it]{}.\[2.4\]
The quantity $\delta\Psi$ is proportional to the perturbed gravitational potential $\delta\Phi$, and is the solution to the ordinary differential equation
&&+ [2r]{} [d(r)dr]{} +(r)\
&&= - [8G l2l+1]{} ([rR]{})\^[l+1]{},\[2.5\]
which satisfies appropriate asymptotic boundary conditions. We note that $\delta \rho$ is proportional to $\Omega^2$ and hence is small (i.e., higher order in $\Omega$) compared to $\delta\vec{v}$ in slowly rotating stars. We also note that $\delta\rho$ is proportional to $Y_{l+1\,l}$—having spherical harmonic index one order in $l$ higher than that of the velocity perturbation. Equation (\[2.4\]) is the complete expression for the density perturbation to order $\Omega^2$. The exact expression for $\delta\rho$ includes additional terms of order $\Omega^4$.
The [*r-*]{}modes evolve with time dependence $e^{i\omega t - t/\tau}$ as a consequence of ordinary hydrodynamics and the influence of the various dissipative processes. The real part of the frequency of these modes, $\omega$, is given in Eq. (\[2.3\]), while the imaginary part $1/\tau$ is determined by the effects of gravitational radiation, viscosity, etc. The simplest way to evaluate $1/\tau$ is to compute the time derivative of the energy $\tilde{E}$ of the mode (as measured in the rotating frame). $\tilde{E}$ can be expressed as a real quadratic functional of the fluid perturbations:
=[12]{}d\^3x. \[2.6\]
Thus the time derivative of $\tilde{E}$ is related to the imaginary part of the frequency $1/\tau$ by
= -[2]{}. \[2.7\]
Since the specific expressions for the time derivative of $\tilde{E}$ due to the influences of gravitational radiation [@thorne] and viscosity [@ipser-lindblom91] are well known, Eq. (\[2.7\]) may be used to evaluate the imaginary part of the frequency.
It is convenient to decompose $1/\tau$:
= [1\_[GR]{}()]{} + [1\_[S]{}()]{} + [1\_[B]{}()]{}, \[2.8\]
where ${1/ \tau_{\scriptscriptstyle GR}}$, ${1/
\tau_{\scriptscriptstyle S}}$, and ${1/ \tau_{\scriptscriptstyle B}}$ are the contributions due to gravitational radiation emission, shear viscosity and bulk viscosity respectively. Expressions for these individual contributions for the $r-$modes are given by [@lindblom-owen-morsink]:
=-[32G\^[2l+2]{}c\^[2l+3]{}]{}&&[ (l-1)\^[2l]{}\^2]{} ([l+2l+1]{})\^[2l+2]{}\
&&\_0\^Rr\^[2l+2]{} dr,\[2.9\]
= (l-1)(2l+1) \_0\^R r\^[2l]{} dr ( \_0\^R r\^[2l+2]{} dr)\^[-1]{}. \[2.10\]
and
||\^2d\^3x(\_0\^R r\^[2l+2]{}dr)\^[-1]{},\[2.11\]
where $\delta\rho$ is given in Eq. (\[2.4\]). We note that the expression for $1/\tau_{\scriptscriptstyle B}$ in Eq. (\[2.11\]) is only approximate. The exact expression should contain the Lagrangian density perturbation $\Delta\rho$ in place of the Eulerian perturbation $\delta\rho$. The bulk viscosity (see Eq. \[2.13\]) is a very strong function of the temperature, being proportional to $T^6$. Thus, the result of any error that might occur in our approximation for $1/\tau_{\scriptscriptstyle B}$ is simply to shift slightly the temperature needed to achieve a given viscosity timescale. Numerical estimates show that changing this quantity by even a factor of one hundred (as suggested by Ref. [@andersson-et-al]), does not substantially affect the important physical quantities computed here (i.e. the spindown rate or the final angular velocity of the star).
We have evaluated these expressions for the imaginary parts of the frequency for a “typical” neutron star model with a polytropic equation of state: $p=k\rho^2$, with $k$ chosen so that a 1.4$M_\odot$ model has the radius $12.53$km. We use the usual expressions for the viscosity of hot neutron star matter [@visrefs]:
=347\^[9/4]{}T\^[-2]{},\[2.12\]
=6.010\^[-59]{} ([l+12]{})\^2\^2T\^6,\[2.13\]
where all quantities are expressed in cgs units. We have evaluated the expressions, Eqs. (\[2.9\])–(\[2.11\]), for the dissipative timescales with fiducial values of the angular velocity $\Omega=\sqrt{\pi G \bar{\rho}}$ and temperature $T=10^9$K. These fiducial timescales ${\tilde{\tau}_{\scriptscriptstyle GR}}$, ${\tilde{\tau}_{\scriptscriptstyle V}}$, and ${\tilde{\tau}_{\scriptscriptstyle B}}$ are given in Table \[table1\] for the [*r-*]{}modes with $2\leq l \leq 6$. It will be useful in the following to define a timescale associated with the viscous dissipation $1/\tau_{\scriptscriptstyle
V}=1/\tau_{\scriptscriptstyle S} +1/\tau_{\scriptscriptstyle B}$. The viscous timescale $\tau_{\scriptscriptstyle V}$ and the gravitational timescale $\tau_{\scriptscriptstyle GR}$ can be expressed then in terms of the fiducial timescales in a way that makes their temperature and angular velocity dependences explicit:
= [1\_[S]{}]{} ([10\^9 [K]{}T]{})\^2 + [1\_[B]{}]{} ([T10\^9 [K]{}]{})\^6 ([\^2G |]{}), \[2.14\]
= [1\_[GR]{}]{} ([\^2G |]{})\^[l+1]{}. \[2.15\]
$\qquad l\qquad $ $\qquad \tilde{\tau}_{\scriptscriptstyle GR}\qquad$ $\qquad\tilde{\tau}_{\scriptscriptstyle S}\qquad$ $\qquad\tilde{\tau}_{\scriptscriptstyle B}\qquad$
------------------- ----------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
2 $-3.26\times 10^0$ $2.52\times 10^8$ $6.99\times 10^8$
3 $-3.11\times 10^1$ $1.44\times 10^8$ $5.13\times 10^8$
4 $-2.85\times 10^2$ $1.07\times 10^8$ $4.01\times 10^8$
5 $-2.37\times 10^3$ $8.79\times 10^7$ $3.26\times 10^8$
6 $-1.82\times 10^4$ $7.58\times 10^7$ $2.74\times 10^8$
: Gravitational radiation and viscous timescales (in seconds) are presented for $T=10^9\,$K and $\Omega=\sqrt{\pi G \bar{\rho}}$.
\[table1\]
Evolution of the [*r-*]{}Modes {#sectionIII}
==============================
To determine the gravitational waveform that will result from the instability in the [*r-*]{}modes, we must estimate how the neutron star evolves as the instability grows and radiates the angular momentum of the star away to infinity. Initially the mode will be a small perturbation that is described adequately by the linear analysis that we have described above. However, as the mode grows, non-linear hydrodynamic effects become important and eventually dominate the dynamics. At the present time we do not have available the tools to follow exactly this non-linear phase of the evolution. Instead, we propose a simple approximation that includes (we believe) the basic features of the exact evolution.
We treat the star as a simple system having only two degrees of freedom: the uniformly rotating equilibrium state parameterized by its angular velocity $\Omega$, and the [*r-*]{}mode parameterized by its amplitude $\alpha$. The total angular momentum $J$ of this simple model of the star is given by,
J = I+ J\_c,\[3.1\]
where $I$ is the moment of inertia of the equilibrium state of the star, and $J_c$ is the canonical angular momentum of the [*r-*]{}mode.
In this simple model of the star the angular momentum $J$ is a function of the two parameters that characterize the state of the system: $J=J(\Omega,\alpha)$. We can determine this functional relationship approximately as follows. The canonical angular momentum of an [*r-*]{}mode can be expressed in terms of the velocity perturbation $\delta\vec{v}$ by [@friedman-schutz],
J\_c = -[l2 (+l)]{} \^ d\^3x. \[3.2\]
For the $l=2$ [*r-*]{}mode of primary interest to us here this expression reduces (at lowest order in $\Omega$) to
J\_c=-\^2MR\^2,\[3.3\]
where $\tilde{J}$ is defined by
=[1MR\^4]{}\_0\^R r\^[6]{}dr.\[3.4\]
For the polytropic models studied in detail here the dimensionless constant $\tilde{J}=1.635\times 10^{-2}$. The moment of inertia $I$ can also be conveniently expressed as
I = MR\^2,\[3.5\]
where $\tilde{I}$ is given by
= [83 MR\^2]{}\_0\^Rr\^4dr.\[3.6\]
For the polytropic models considered here $\tilde{I}= 0.261$. Thus, our simple model of the angular momentum of the perturbed star is
J(,)= ( -\^2)MR\^2. \[3.7\]
The perturbed star loses angular momentum primarily through the emission of gravitational radiation. Thus, we compute the evolution of $J(\Omega,\alpha)$ by using the standard multipole expression for angular momentum loss. The $l=2$ [*r-*]{}mode is the primary source of gravitational radiation in our simple model of this system, and this mode loses angular momentum primarily through the $l=m=2$ current multipole. Thus the angular momentum of the star evolves as
= - [c\^316G]{} ([43]{})\^5 (S\_[22]{})\^2.\[3.8\]
The $l=m=2$ current multipole moment $S_{22}$ for this system is given by
S\_[22]{} = R\^3 .\[3.9\]
Combining Eq. (\[3.8\]) for the angular momentum evolution of the star with Eqs. (\[2.9\]), and (\[3.7\]), we obtain one equation for the evolution of the parameters $\Omega$ and $\alpha$ that determine the state of the star:
(-\^2)[ddt]{} -3=[3\^2\_[GR]{}]{}. \[3.10\]
During the early part of the evolution of the star, the perturbation analysis of the [*r-*]{}modes described earlier applies. In addition to radiating angular momentum from the star via gravitational radiation, the mode will also lose energy via gravitational radiation and neutrino emission (from the bulk viscosity) and also deposit energy into the thermal state of the star due to shear viscosity. It is most convenient to obtain the equation for the energy balance during this part of the evolution in terms of the energy $\tilde{E}$ of the mode as defined in Eq. (\[2.6\]). For the $l=2$ [*r-*]{}mode $\tilde{E}$ is given by
= \^2\^2 MR\^2. \[3.11\]
The time derivative of $\tilde{E}$ is precisely the quantity that was used to determine the imaginary part of the frequency of the mode in Eq. (\[2.7\]):
=-2( [1\_[GR]{}]{}+ [1\_[V]{}]{}) .\[3.12\]
Equation (\[3.12\]) together with (\[3.11\]) therefore provides a second equation for determining the evolution of the parameters $\Omega$ and $\alpha$ that specify the state of the star:
+ =-( [1\_[GR]{}]{}+ [1\_[V]{}]{}) .\[3.13\]
Equations (\[3.10\]) and (\[3.13\]) can be combined then to determine the evolution of $\Omega$ and $\alpha$ during the portion of the evolution in which the perturbation remains small:
=-[2\_[V]{}]{} [\^2 Q1+\^2Q]{} ,\[3.14\]
=-[\_[GR]{}]{} -[\_[V]{}]{}[1-\^2Q1+\^2Q]{} .\[3.15\]
The equation of state dependent parameter $Q$ that appears in Eqs. (\[3.14\]) and (\[3.15\]) is defined by $Q=3\tilde{J}/2\tilde{I}$. For the polytropic model considered in detail here $Q=9.40\times10^{-2}$. We note that during the initial linear evolution phase the angular velocity of the star $\Omega$ is nearly constant, evolving according to Eq. (\[3.14\]) on the viscous dissipation timescale. During this phase the amplitude of the mode $\alpha$ grows exponentially on a timescale that is comparable to the gravitational radiation timescale.
After a short time (about $500\,$s in our numerical solutions) the amplitude becomes so large that non-linear effects can no longer be ignored. We have not yet developed the tools needed to follow the evolution exactly during this non-linear phase. However, we do have some intuition about the non-linear hydrodynamical evolution of gravitationally driven instabilities in rotating stars. This intuition comes from the studies of the effects of gravitational radiation reaction on the evolution of the ellipsoidal models [@detweiler-lindblom; @lai-shapiro]. In that case the unstable mode grows exponentially until its amplitude is of order unity. At that point a kind of non-linear saturation occurs, and the growth of the mode stops. The excess angular momentum of the star is radiated away and the star evolves toward a new lower angular momentum equilibrium state. We expect a similar situation to pertain in the evolution of the [*r-*]{}modes. Thus, we expect non-linear effects will saturate and halt the further growth of the mode when the amplitude of the mode becomes of order unity. Thus, when the amplitude $\alpha$ grows to the value
\^2=, \[3.16\]
(where $\kappa$ is a constant of order unity) we stop evolving the star using Eqs. (\[3.14\]) and (\[3.15\]). Instead we set $d\alpha/dt=0$ during the saturated non-linear phase of the evolution, while continuing to evolve the angular velocity $\Omega$ by Eq. (\[3.10\]) as angular momentum is radiated away to infinity by gravitational radiation. During this phase, then the angular velocity evolves by
= [2\_[GR]{}]{} [Q1-Q]{}. \[3.17\]
The [*r-*]{}mode will evolve during the saturated non-linear phase of its evolution approximately according to Eqs. (\[3.16\]) and (\[3.17\]). During this phase the star will lose most of its angular momentum, and spin down to a state having an angular velocity that is much smaller than $\Omega_{\scriptstyle K}\approx
{\scriptstyle \frac {2}{3}}\sqrt{\pi G \bar{\rho}}$. The star will eventually (in about 1 year in our numerical solutions) evolve to a point where the angular velocity and temperature become sufficiently low that the [*r-*]{}mode is no longer unstable. The end of the evolution is characterized by a phase in which the viscous forces and gravitational radiation damp out the energy remaining in the mode and move the star slowly to its final equilibrium configuration. During this final phase, the mode is again of small amplitude and so the linear approximation is adequate to describe the evolution. We monitor the quantity on the right side of Eq. (\[3.15\]) throughout the non-linear evolution phase. When it becomes negative we change the evolution equations again, from Eqs. (\[3.16\]) and (\[3.17\]) back to the linear equations Eqs. (\[3.14\]) and (\[3.15\]).
In summary then, we model the evolution of the [*r-*]{}mode as having three distinct phases: (i) The hot young neutron star is born rapidly rotating with a small initial excitation in the $l=2$ [*r-*]{}mode. This mode initially grows exponentially according to Eqs. (\[3.14\]) and (\[3.15\]). (ii) The amplitude of the mode saturates due to non-linear hydrodynamic effects at a value of order unity. The bulk of the angular momentum of the star is radiated away by gravitational radiation during this phase according to Eqs. (\[3.16\]) and (\[3.17\]). (iii) The final phase of the evolution begins when the right side of Eq. (\[3.15\]) becomes negative so that the mode begins to be damped out. During the final phase the star evolves again according to Eqs. (\[3.14\]) and (\[3.15\]).
In order to complete our model for the evolution of the [*r-*]{}modes we must specify how the temperature of the star evolves with time. We do this by adopting one of the standard descriptions of the cooling of hot young neutron stars. These stars are expected to cool primarily due to the emission of neutrinos via a modified URCA process. The temperature during this phase falls quickly by a simple power law cooling formula [@shapiro-teukolsky]:
=\^[-1/6]{},\[3.18\]
where $T_i$ is the initial temperature of the neutron star, and $\tau_c$ is a parameter that characterizes the cooling rate. For the modified URCA process $\tau_c\approx 1$y. A typical value for the initial temperature is $T_i\approx 10^{11}$K. Equation (\[3.18\]) can now be inserted into Eqs. (\[3.14\])–(\[3.17\]) to provide explicit differential equations for the time evolution of the angular velocity of the star and the amplitude of the mode. These equations can be solved numerically in a straightforward manner.
Figs. \[fig1\] and \[fig2\] illustrate the solutions to these equations. The dashed curves in Figs. \[fig1\] and \[fig2\] show the critical angular velocity $\Omega_c$, defined by $1/\tau(\Omega_c)=0$, above which the [*r-*]{}modes are unstable. Figure \[fig1\] shows the evolution ($\Omega$ plotted versus $T$) of the angular velocity of the star for $\kappa=1.0$ and a range of values of the initial value of the parameter $\alpha$. In these simulations we have assumed that the initial angular velocity of the star is $\Omega=\Omega_{\scriptstyle K}$. This figure illustrates that the final non-linear part of the evolution is remarkably insensitive to the initial size of the perturbation.
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Figure \[fig2\] illustrates the dependence of the evolution on the parameter $\kappa$ by showing several evolutions with initial values of $\alpha=10^{-6}$. The parameter $\kappa$ measures the degree of saturation that occurs in the non-linear spindown phase of the star. In our numerical studies we examine the limited range $0.25\leq\kappa\leq 2$. If $\kappa$ is taken to be too small, the mode simply does not grow to the point that non-linear effects can stop its growth. Conversely if $\kappa$ is taken too large, then our simple evolution equations based in part on the linear perturbation theory become singular, e.g. Eq. (\[3.17\]). The equations break down because the (negative) canonical angular momentum of the mode equals the (positive) angular momentum of the equilibrium configuration, and therefore Eq. (\[3.1\]) yields the unphysical result that the star has no net angular momentum.
We artificially stop all of our evolution curves when the temperature of the star falls to $10^9\,$K. Below this temperature we expect superfluidity and perhaps other non-perfect fluid effects to make our simple simulation highly inaccurate [@lindblom-mendell]. Figure \[fig2\] illustrates that the gravitational radiation instability in the [*r-*]{}modes is nevertheless effective in radiating away most of the angular momentum of the star before the star cools to the point that superfluidity or other effects are expected to become important. Figure \[fig2\] shows that the amount of angular momentum lost in this process is remarkably insensitive to the value of $\kappa$. Thus, the final upper limit on the angular velocity of the star is (fortunately) fairly insensitive to our assumption about the exact nature of the non-linear portion of the star’s evolution.
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In this simple model of the evolution of the unstable star, we have ignored the effect that viscous heating might have on the cooling rate of the star. If there were too much viscous heating, then the cooling formula given in Eq. (\[3.18\]) would not be correct. We have evaluated the importance of this re-heating effect by comparing the rate at which thermal energy is being radiated away from the star by neutrinos according to Eq. (\[3.18\]) with the rate that viscous dissipation deposits thermal energy into the star. Neutrino cooling removes energy from the thermal state of the star at the rate [@shapiro-teukolsky]
= 7.410\^[39]{}([T10\^9[K]{}]{})\^8 [ergs/s]{}.\[3.19\]
Thermal energy is generated by shear viscosity as the star evolves, but not by bulk viscosity. Bulk viscosity radiates away its excess energy directly by neutrinos without significantly interacting with the thermal energy contained in the star. Thus, energy is transfered from the canonical energy of the [*r-*]{}mode to the thermal energy of the star by the formula
=[2\^2\^2 MR\^2 \_[S]{}]{}. \[3.20\]
Figure \[fig3\] compares the values of $dU/dt$ and $dE_c/dt$ for our numerical evolution with $\kappa=1.0$ and initial $\alpha=10^{-6}$. We find that viscosity does not deposit energy into the thermal state of the star at a significant rate until the temperature of the system falls to about $10^9\,$K. At this point the star has already lost most of its angular momentum to gravitational radiation, and other dissipative effects (such as those associated with superfluidity) which are not modeled here will start to play a significant role [@lindblom-mendell]. Thus, we are justified in ignoring the effects of viscous re-heating on the thermal evolution of the star during the early part of its evolution modeled here.
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The modified URCA process that determines the thermal evolution used in our evolutions is the standard mechanism by which neutron stars are expected to cool down to about $10^9\,$K. Other less standard mechanisms have also been proposed which could significantly speed up the cooling [@shapiro-teukolsky]. These mechanisms include neutrino emission processes that require the presence of exotic species (such as quarks or pions) as free particles in the cores of these stars. We have ignored these possibilities in the evolutions described above. If these particles do exist in the cores of neutron stars, we expect that they will only be present in a small volume of material at the centers of these stars. The cores of these stars may well cool rapidly, but the outer layers where the [*r-*]{}mode is large will continue to cool at the rate given in Eq. (\[3.18\]) until thermal conduction can move energy from the outer layers back into the core.
To estimate what effect a somewhat more rapid cooling might have on the evolution of the [*r-*]{}modes, we have artificially varied the value of the parameter $\tau_c$ that determines the cooling rate in Eq. (\[3.18\]). Figure \[fig4\] shows the results of the evolution of an [*r-*]{}mode with initial $\alpha=10^{-6}$ and $\kappa=1.0$ for several values of $\tau_c$. We see that while the details of the evolution are somewhat effected, the total amount of angular momentum radiated away by gravitational radiation, and the final angular velocity of the star are fairly insensitive to the rate at which the star cools to $10^9\,$K.
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Gravitational waveforms {#sectionIV}
=======================
As the [*r-*]{}mode grows and evolves it emits gravitational radiation. In this section we calculate the waveforms for the gravitational wave strain $h(t)$ and its Fourier transform $\tilde{h}(f)$ that are produced by this [*r-*]{}mode instability. These are the quantities that can be measured by the gravitational wave detectors now under construction (LIGO, VIRGO, GEO, etc.). During the non-linear saturation phase of the [*r-*]{}mode evolution, gravitational radiation controls the dynamics. In this case $\tilde{h}(f)$ turns out to be independent of the details of the evolution, and thus can be determined quite generally. An evolutionary model [*is*]{} needed, however, to determine $df/dt$, the time dependence of the various quantities, and the initial and final frequencies of the saturation phase. In this section we present the general model independent derivation of $\tilde{h}(f)$ for the non-linear saturated phase of the evolution. We also give expressions for the gravitational wave strain that apply to the early phases of the evolution using the simple model discussed in Sec. \[sectionIII\].
The frequency-domain gravitational waveform
\[4.1\] [h]{}(f) \_[-]{}\^ e\^[2i f t]{}h(t) dt
is determined completely by the assumption that the angular momentum radiated as gravitational waves comes directly from the angular momentum of the star. This assumption is expected to be satisfied during the non-linear saturated phase of the evolution, but not during the early evolution when the mode is growing exponentially. This derivation is based on Blandford’s analysis (as discussed in [@300yrs]) of white dwarf collapse to a neutron star which is halted by centrifugal forces (see also [@schutz89; @thorneIAU]). Such a star can only collapse to a neutron star by shedding its excess angular momentum through gravitational waves. In that situation as in the non-linear saturation phase of the [*r-*]{}mode evolution, gravitational radiation determines the rate at which angular momentum leaves the system, and this in turn determines the rate at which the frequency of the radiation evolves with time.
In the stationary phase approximation (which is always valid for a secular instability) the gravitational wave strain $h(t)$ is related to its Fourier transform $\tilde{h}(f)$ by
|h(t)|\^2=|h(f)|\^2 |[dfdt]{}|. \[4.2\]
Throughout this discussion we treat $h(t)$ as a complex quantity with purely positive frequency. For the $l=2$ mode of primary importance here, the mode frequency is $\omega = {4\over 3}
\Omega$, or $f = {2\over 3\pi} \Omega $, where $f$ is the frequency of the emitted gravitational waves measured in Hz. Assuming the star is uniformly rotating, its angular momentum is $J = I \Omega$, where $I$ is the star’s moment of inertia. The moment of inertia is fairly independent of angular velocity (especially at small angular velocities where most of the detectable signal from these sources is likely to originate) and is also fairly independent of the amplitude of the excited $r$-mode (see Eq. \[3.7\]). Thus, $I$ is reasonably well approximated by its non-rotating value. Thus
= [32]{} I.\[4.3\]
The rate at which angular momentum is radiated away by a source is related to the gravitational wave amplitude by the expression [@thorne]
= 4D\^2 [m]{}[1]{} \^2 h\_+\^2 +h\_\^2\[4.4\]
where $h_+$ and $h_\times$ are the amplitudes of the two polarizations of gravitational waves, $D$ is the distance to the source, and $\langle\ldots\rangle$ denotes an average over the orientation of the source and its location on the observers sky. Using $dt/df = dJ/df (dJ/dt)^{-1}$ and combining Eqs. (\[4.2\])–(\[4.4\]) we obtain
|h\_+(f)|\^2 + |h\_(f)|\^2= [3I4D\^2 f]{}. \[4.5\]
The measured value of $|{\tilde h}(f)|^2$ depends on the orientation of the source and its location on the detector’s sky. Averaged over these angles, its value is given by
|h(f)|\^2= |h\_+(f)|\^2 + |h\_(f)|\^2. \[4.6\]
We are actually interested in the average over source locations in three-dimensional space, not just the two angles on the sky. The spatial average weights more strongly those orientations that yield stronger signals, effectively increasing $\langle|\tilde{h}(f)|^2\rangle$ by about ${3\over2}$. Combining these results then, the average value of $\tilde{h}$ produced by our fiducial [*r-*]{}mode source (with $M=1.4M_\odot$, $D=20$Mpc, $R=12.5$km) is
(f)= 5.710\^[-25]{} [Hz]{}\^[-1]{}.\[4.7\]
Note that this expression does not depend (in the frequency domain) on the details of the evolutionary model apart from the upper and lower frequency cutoffs. This expression, Eq. (\[4.7\]), only depends on the assumption that the frequency of the mode evolves as angular momentum is radiated by the star according to Eq. (\[4.3\]). We expect this to be satisfied during the non-linear saturated phase of the [*r-*]{}mode evolution, but probably not during the early linear growth phase of the mode.
To obtain the complete waveforms for a particular evolutionary model, we start with the usual expression for the gravitational field in terms of its multipoles [@thorne]. We average this expression over angles in the manner described above to obtain
h(t)= .\[4.8\]
For the simple two-parameter evolution of the [*r-*]{}mode instability described in Sec. \[sectionIII\], this expression can be simplified using Eq. (\[3.9\]) to
h(t)= 4.410\^[-24]{} ()\^3 ([20[Mpc]{}D]{}).\[4.9\]
This simple evolutionary model also gives a simple formula for the frequency evolution. During the non-linear saturation phase of the evolution Eq. (\[3.17\]) can be written as
\[4.10\] [dfdt]{}-1.8([f1]{})\^7[Hz/s]{},
where we have assumed that $\kappa Q\ll 1$. The time for the gravitational wave frequency to evolve to its minimum value $f_{\min}$ is obtained by integrating Eq. (\[4.10\]):
\[4.11\] t ([120[Hz]{}f\_]{})\^6 [y]{}.
Analogous model dependent expressions can also be derived for the early linear phase of the evolution. During this period the amplitude $\alpha$ grows exponentially on the gravitational timescale according to Eq. (\[3.14\]), while the frequency of the mode changes extremely slowly according to Eq. (\[3.15\]). Solving these equations approximately gives
-[2.7 \^2t]{} ([f1 [k Hz]{}]{})\^3, \[4.12\]
Using Eq. (\[4.9\]) this implies that $\tilde{h}$ during the linear growth phase is given approximately by
\[4.13\] (f)4.710\^[-25]{} ([f1[k Hz]{}]{})\^[3/2]{} [Hz]{}\^[-1]{}. where $t \equiv t(f)$ is obtained from Eq. (\[4.12\]). The factor $f^{3/2}$ that appears in Eq. (\[4.13\]) is essentially constant, being given by the initial mode frequency as determined by the initial angular velocity of the star. Since this factor is essentially constant during the linear evolution phase, this implies that $\tilde{h}(f)$ grows as $\sqrt{t}$. The duration of the initial linear growth phase can be estimated from the solution for the amplitude $\alpha$:
\[4.14\] t = 1.510\^3 ([1[k Hz]{}f]{})\^6 [s]{},
where $\alpha_o$ is the initial size of the perturbation. During this interval, the mode frequency decreases by an amount of order $0.1 \kappa\, {\rm Hz}$. This is the width of the initial spikes shown in Figs. 7 and 8. The maximum amplitude achieved by $\tilde{h}$ can also be determined from Eqs. (\[4.13\]) and (\[4.14\]):
()1.810\^[-23]{} &&([1[k Hz]{}f]{})\^[9/2]{}\
&&\^[1/2]{} [Hz]{}\^[-1]{}. \[4.15\]
This expression for $\max({\tilde{h}})$ is fairly insensitive to the duration of the growth phase, and as well to the exact point at which the transition to the non-linear saturation phase occurs. The value of $\max(\tilde{h})$ [*is*]{} rather sensitive to our expression for $df/dt$ however. If $df/dt$ were to differ during the linear growth phase from the expression used here by any small effect (such as non-linear modifications of the frequency of the mode) then the the resulting change on $\max(\tilde{h})$ could be large. The analytical expressions given here, Eqs. (\[4.13\])–(\[4.15\]), do however accurately represent (to within a few percent) the exact numerical solutions to the equations for our simple model in this region.
Figure \[fig5\] illustrates our full numerical solutions for the time dependence of the gravitational wave amplitude $h(t)$ for several values of the parameters $\alpha$ with $\kappa=1$. Figure \[fig6\] illustrates the time dependence of $h(t)$ for various values of $\kappa$ with $\alpha = 10^{-6}$. All of these curves represent the gravitational radiation emitted by a neutron star initially spinning with angular velocity $\Omega=\Omega_K={\scriptstyle {2\over
3}}\sqrt{\pi G \bar{\rho}}$. In this section and the next we terminate the evolution once the star has cooled to $10^9$K. Below this temperature the evolution will be significantly affected by mechanisms not dealt with in this paper, such as the superfluid transition, the re-heating of the star by viscous dissipation in the mode, and dissipation mechanisms (e.g. plate tectonics) associated with the rapidly forming crust.
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Figs. \[fig7\] and \[fig8\] illustrate the frequency dependence of $\tilde{h}(f)$ for a $1.4M_\odot$ neutron star located at $20\,$Mpc using the same values of $\alpha$ and $\kappa$ used in Figs. \[fig5\] and \[fig6\]. Figure \[fig7\] illustrates that $\tilde{h}(f)$ is remarkably insensitive to the initial size of the perturbation $\alpha$. The sharp vertical spikes appearing at the high-frequency ends of these curves are due to the extremely monochromatic gravitational waves emitted during the linear growth phase. The structure of this spike in our model is accurately described by Eqs. (\[4.13\])–(\[4.15\]), but it is not clear that this spike is a robust feature of our model. During the phase of the evolution that produces the spike, the amplitude of the mode is quite small except for a period of about one minute. Thus the total amount of radiated energy and angular momentum contained in this spike is quite small. The spike is not likely to play an important role in the detection of these sources even if it is a real feature of the $r$-mode instability.
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Detectability {#sectionV}
=============
In this section we discuss the detectability of the gravitational radiation emitted by young neutron stars spinning down due to the $r$-mode instability. This radiation might be detected as strong sources from single spindown events or as a stochastic background made up of many weaker sources.
Single Sources
--------------
First we estimate the signal-to-noise ratio $S/N$ for a single source located at a distance $D$, chosen to be large enough so that there is likely to be a reasonable event rate (say a few events per year). This distance $D$ must be large enough then to include several thousand galaxies (assuming that the observed supernova rate is comparable to the neutron star formation rate). Thus we take this fiducial distance to have the value $D=20\,$Mpc [@thorne78], the approximate distance to the Virgo cluster of galaxies. We estimate the optimal value of $S/N$ that could be obtained by matched filtering. Because matched filtering is probably not feasible for these sources, this estimate provides only an upper limit of what might be achieved. We briefly discuss two more realistic search strategies based on barycentered Fourier transforms of the data [@bccs].
Using matched filtering, the power signal-to-noise ratio $(S/N)^2$ of the detection is given by
([SN]{})\^2 = 2\_0\^ [ [|[h]{}(f)|\^2 ]{} ]{} df \[5.1\]
where $S_h(f)$ is the power spectral density of the detector strain noise. The constant in front of the integral in Eq. (\[4.5\]) is 2 (instead of 4 as in Ref. [@cutler-flanagan]) because our $h$ is complex (with purely positive frequency). Equation (\[5.1\]) can also be written
\[5.2\] ([SN]{})\^2 =2([h\_ch\_[rms]{}]{})\^2.
Here the rms strain noise $h_{\rm rms}$ in the detector is given by
\[5.3\] h\_[rms]{} ,
where $S_h(f)$ is the power spectral density of the noise, and the characteristic amplitude $h_c$ of the signal is defined by [@300yrs]
\[5.4\] h\_c h.
The quantity multiplying $h$ on the right side of Eq. (\[5.4\]) is generally interpreted as the number of cycles radiated while the frequency changes by an amount of order $f$. This interpretation is correct as long as the frequency evolution is very smooth.
However, our evolutionary model of the frequency evolution contains a discontinuity as the mode stops linearly evolving and saturates. Consequently the actual number of cycles spent near the initial frequency is far fewer than indicated by Eq. (\[5.4\]). The quantity $h_c$ is a useful estimator of the effective filtered amplitude of a signal because the [*integral*]{} of $(h_c/h_{\rm
rms})^2$ always gives the optimal $(S/N)^2$. Therefore a spike in $h_c$ must be interpreted with some caution—the peak value of the spike is useless unless one also knows the bandwidth of the spike.
In Figure \[fig9\] we plot $h_c$ versus frequency, superimposed on $h_{\rm rms}$ for three LIGO interferometer configurations. In the saturation phase (i.e., not including the spike) $h_c$ is well approximated by
h\_c 5.710\^[-22]{}([f1]{})\^[1/2]{}.
We plot $h_{\rm rms}$ for the LIGO “first interferometers” [@science] (which we abbreviate LIGO I), the “enhanced interferometers” [@r-d] (LIGO II), and the “advanced interferometers” [@science] (LIGO III). The noise power spectral density for LIGO I is well approximated by the analytical fit [@thorne-fit]
\[5.5\] S\_h(f)=[S\_o3]{}
where $S_o=4.4\times 10^{-46}$ Hz${}^{-1}$ and $f_o=175\,$Hz. For LIGO II we construct the approximation
\[5.6\] S\_h(f)=[S\_o11]{}{2([f\_0f]{})\^[9/2]{} +[92]{}}
where $S_o=8.0\times 10^{-48}$ Hz${}^{-1}$ and $f_o=112$Hz. For LIGO III the noise spectral density is well approximated by [@hughes]
\[adv\] S\_h(f)=[S\_o5]{}{([f\_0f]{})\^4 +2}
where $S_o=2.3\times 10^{-48}$ Hz${}^{-1}$ and $f_o=76\,$Hz.
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Most of the contribution to $S/N$ in Eq. (\[5.2\]) comes from the saturation phase of the evolution which is largely model independent as discussed earlier. Given a detector noise curve, $S/N$ is thus independent of most details of the waveform except the final frequency of the neutron star. We therefore have
\[5.7\] ([SN]{})\^2 = [9I10D\^2]{} \_[f\_]{}\^[f\_]{} [dffS\_h(f)]{}.
The minimum frequency $f_{\min}$ reached by the $r$-mode evolution is about $120$ Hz. At frequencies slightly larger than $f_{\min}$, the LIGO II noise is dominated by photon shot noise. If we ignore the other noise components, $S_h(f)$ becomes
S\_h(f) = 2.6 10\^[-52]{} f\^2. \[5.8\]
For $f_{\max}\gg f_{\min}$ the integral is dominated by the lower cutoff. Thus, the LIGO II $S/N$ is given approximately by
8.8 ( [I]{})\^[1/2]{} ([D]{})\^[-1]{} ([[f\_]{}]{})\^[-1]{}. \[5.9\]
Including the other components of the noise decreases $S/N$ somewhat. For the numerical evolutions of our simple model with $\kappa=1.0$ which terminate at $10^9\,$K, we find we find $S/N =
1.2$, $7.6$, and $10.6$ for LIGO I, II, and III respectively at $D=20$ Mpc. These last three results scale with $I$ and $D$ just as in Eq. (\[5.9\]), but the dependence on $f_{\min}$ is now more complicated. The contribution to $S/N$ from the high frequency spike in our model is $0.6$ for LIGO II, and about 0.1 for LIGO I. While the height of the spike in $h_c$ may not be a robust feature of our simple model, the contribution that this spike makes to the overall $S/N$ probably is. These numbers indicate that the gravitational radiation from the [*r-*]{}mode instability is a probable source for LIGO II if some near-optimal data analysis strategy can be developed. It appears unlikely that the radiation from the high frequency spike will be detectable.
Given the recent discovery of the ultrafast young pulsar PSR J0537–6910 in the Large Magellanic Cloud [@lmc], it is interesting to examine the effect on detectability of a relatively high superfluid transition temperature. If the initial period of PSR J0537–6910 was about 7 ms (as extrapolated from the braking indices of typical young pulsars), that could indicate a superfluid transition temperature of about $2\times10^9\,$K and final gravitational-wave frequency of about 200 Hz. Cutting off our simulations at this point in the evolution, we obtain for LIGO II the $S/N$ of about 5.
It is clear from Fig. \[fig9\] that the first-generation LIGO and VIRGO detectors will not see $r$-mode events from the Virgo cluster. Their sensitivity is a factor of about 8 worse than the enhanced detectors. We have also considered the possibility that GEO [@geo600] might detect these sources by using narrow-banding, where it can improve its sensitivity in a restricted frequency range at the expense of worse sensitivity elsewhere. However, for the kind of broad-spectrum signal produced by the $r$-mode instability, narrow-banding is in fact neutral: the gain of signal-to-noise ratio in the selected band is just compensated by the loss over the rest of the spectrum. So GEO is not likely to see these signals either. Nor will advanced resonant detectors: at their frequencies and in their relatively narrow bandwidths, there is just not enough power in these signals if the sources are in the Virgo cluster. For example, the proposed GRAIL detector [@grail] operating at its quantum-limited sensitivity ($S_h = 1.6\times 10^{-48}\rm\;Hz^{-1}$) between 500 and 700 Hz would have $S/N \approx 1$ for a source at the distance of the Virgo cluster.
Matched filtering using the year-long waveform templates that would be needed to describe these sources completely could yield the signal-to-noise ratios quoted above. However, this is not a practical strategy for this type of signal due to the prohibitively large number of templates that would be needed to parameterize our ignorance of these sources and the resulting high computational cost of filtering the data with these templates. Other strategies equivalent to combining the results of shorter template searches might well be computationally feasible, although they would obtain less than the optimal $S/N$.
The barycentered fast Fourier transform (FFT) technique that has been designed to search for nearly periodic signals [@bccs] might well provide one such method. Figure \[fig10\] shows the spindown age
\_[sd]{}= -f[dtdf]{}
for a neutron star spinning down due to the $r$-mode instability. This quantity provides a reasonably good estimate of the time spent by the evolving star in the saturated non-linear phase, but it is not a good estimate of the amount of time spent during the linear growth phase (for the reasons outlined above). During the saturated phase $\tau_{\rm sd}$ is given approximately by
\_[sd]{} ([1f]{} )\^6 .
The signal becomes quite monochromatic after about one day, in the sense that the spindown timescale is long compared to the inevitable daily modulation of the signal due to the motion of the Earth. Thus the search techniques for periodic sources should work well. In many cases the supernova will be observed, yielding the location of the source and allowing a search over spindown parameters only, a [*directed spindown search.*]{}
The most straightforward way to conduct a directed spindown search is to search over generic spindown parameters as discussed by Brady and Creighton [@brady-creighton]. This involves re-sampling the data so as to render sinusoidal a signal with arbitrary (smooth) frequency evolution, then Fourier transforming it. The signal frequency evolution (which determines the re-sampling) is modeled as
\[5.13\] f(t)=f\_0\_[n=0]{}\^N [1n!]{}([t\_[n]{}]{})\^n,
where $f_0$ is the frequency at the beginning of the FFT, $t$ is the time measured from the beginning of the FFT, and $\tau_n$ are expansion parameters with $\tau_1=\tau_{\rm sd}$. The number $N$ of spindown parameters needed is set by the requirement that the frequency drift due to the next term in the series (\[5.13\]) be less than one frequency bin of the FFT (which is in turn determined by the integration time $\tau_{\rm int}$). This implies (assuming $\tau_n\approx\tau_{\rm sd}$) that
\_[int]{}\^[N+1]{} \_[sd]{}\^N.\[5.14\]
One must choose a set of points in the spindown parameter space for which to perform the re-sampling and FFT. Too few points and one misses signals by re-sampling at the wrong rate; too many and the computational cost of performing all the FFTs becomes prohibitive if data analysis is to be performed “on-line,” i.e. keep up with data acquisition. The points in parameter space are chosen using a metric which relates distance in parameter space to loss of $S/N$ [@bccs]. The integration time, which is set so as to optimize the sensitivity of an on-line search for the (fixed) computational power available, is far shorter than a year even for a teraflop computer. Therefore such a search would achieve at best a fraction of the optimal $S/N$ [@brady-creighton]. It is possible that a hierarchical version of this strategy could be developed, in which the best candidates from a year of shorter FFTs are somehow combined to give an improved confidence level. Developing such a strategy would require extensive further investigation.
One way to increase the sensitivity of a directed spindown search would be to constrain the spindown parameter space by taking advantage of whatever information we have about the source from physical models. In practice we do not have and probably will never have completely reliable models for this type of system. The phenomenological model presented here has many physical assumptions that may prove to be inadequate. For instance, in the saturation phase it is highly unlikely that the star can be represented simply as a uniformly rotating equilibrium configuration plus a linear perturbation. When the mode reaches saturation, the mean velocity perturbation is comparable to the rotation rate of the star. Thus in the saturation phase, the velocity field may develop complicated nonlinear structures (such as the cyclones on Jupiter) that produce gravitational radiation involving many different multipoles. Long-lived non-axisymmetric structures requiring higher-order modes are seen in the density perturbations of simulations of star formation when there is enough angular momentum [@tohline]. Although the physics is very different in such simulations, a priori we see no reason to believe that the velocity analogues of these structures are not formed by the $r$-mode instability.
However, not all of the physics affects the waveforms. Some kind of phenomenological model of the signal (as opposed to the star) could be enough to substantially reduce the volume of spindown parameter space to be searched. For a reasonable model of the neutron star, all of the terms in the expansion for $df/dt$ might be determined from a relatively small number of phenomenological parameters. Although quite crude, the model of [*r-*]{}mode instability gravitational waveforms provided here has several features which should be fairly robust: the form of $\tilde{h}(f)$ during the spindown phase, the approximate frequency range of the expected radiation, the approximate timescale for the spindown to occur, etc. Presumably these robust features can be used to reduce considerably the volume of general spindown parameter space which need be searched.
0.3cm
Stochastic Background
---------------------
We now consider the gravitational-wave stochastic background generated by spin-down radiation from neutron star formation throughout the universe. A stochastic background is detected by looking for correlations in the response of two or more detectors. The sensitivity of the network to the background drops rapidly for gravitational wave frequencies much higher than the inverse light-travel time between detectors. For the present application, the important networks are therefore the VIRGO-GEO pair (high frequency cut-off $f_{\rm cut}\approx 400$ Hz) and the Washington-Louisiana pair of LIGO detectors ($f_{\rm cut} \approx 100$ Hz). The gravitational radiation generated by the [*r-*]{}mode spin-down process has significant power at these relatively low-frequencies, $f
\approx 100-400$ Hz. In addition the cosmological redshift (with $z \approx
1-4$) of these sources has the beneficial effect of shifting much of the radiation into the detectable band.
Neutron stars have presumably been formed since the beginning of star formation. If each neutron star formation event converts a reasonable fraction of a solar mass into gravitational radiation via the $r$-mode instability, then the sum of this radiation constitutes a random background that may be detectable by LIGO III. We now make a rough estimate of the spectrum and detectability of this background radiation. A more detailed analysis is being carried out by Vecchio and Cutler [@vecchio-cutler].
The spectrum of the gravitational wave background is typically represented by the following dimensionless quantity:
\_[gw]{}(f) , \[e:om\_f\]
where $\rho_{\rm gw}$ is the energy density in gravitational waves, and $\rho_c=3c^2H_0^2/3\pi G \approx 1.6\times 10^{-8}
h^2_{100}$ erg/cm${}^3$ is the critical energy density just needed to close the universe. ($H_0$ is the Hubble constant and $h_{100}$ is $H_0$ divided by 100 km/s/Mpc.) The signal-to-noise with which this background can be detected in a correlation experiment between two detectors (here assumed to have uncorrelated noise) is given by [@flanagan; @allen]
\[snsb\] ([SN]{})\^2 = [9H\_0\^4 T50\^4]{} \_0\^df [\^2(f)\^2\_[gw]{}(f) f\^6 [\^1S]{}\_h(f) [\^2S]{}\_h(f)]{}.
where $T$ is the integration time, $^1S_h(f)$ and $^2S_h(f)$ are the noise spectral densities of the two detectors, and $\gamma(f)$ is the (dimensionless) overlap reduction function, which accounts for the fact that the detectors will typically have different locations and orientations.
We can get a rough estimate of ${\bf \Omega}_{\rm
gw}(f)$ due to the stochastic background of gravitational radiation from the [*r-*]{}mode instability as follows. A $1.4 M_\odot$ neutron star rotating with Keplerian angular velocity $\Omega_K \approx
{\scriptstyle {2\over 3}} \sqrt{\pi G\bar{\rho}}$ has rotational kinetic energy $E_{\rm K}\approx 0.025 M_\odot c^2$, two-thirds of which is radiated as gravitational waves. For now, assume that all neutron stars are born with angular velocity $\Omega_K$. For a single supernova occurring at $z=0$, the spin-down radiation has spectrum $d
E_{\rm gw}/df \approx {4\over 3}E_{\rm K}\, f/f_{\max}^2$, where $f_{\max} \approx 2\Omega_K/ 3\pi \approx 1400$ Hz.
We are chiefly interested in ${\bf \Omega}_{\rm gw}(f)$ for $f <
f_{\min} \approx 120$ Hz, since that is the range where the pair of LIGO II detectors will have their best sensitivity. Let $n(z)\,dV\,dz$ be the number of supernovae occurring within co-moving volume $dV$ and redshift interval $dz$. In the frequency range of interest then the spectrum of gravitational radiation in the universe today is given by
f = [ 4E\_[K]{}3f\_\^2]{} \_[z\_(f)]{}\^[z\_(f)]{}[n(z) [f’f’dz 1+z]{}]{}, \[e:drhodf\]
where $f'\Delta f'= (1+z)^2 f \Delta f$ are the values of the frequencies as emitted by the source, $z_{\min}(f) \equiv {\rm
max}\{0,f_{\min}/f-1\}$, $z_{\max}(f)\equiv
\min\{z_*,f_{\max}/f-1\}$, and $z_{*}$ corresponds to the maximum redshift where there was significant star formation. The factor $1+z$ in the denominator in Eq. (\[e:drhodf\]) accounts for the redshift in energy of the gravitational radiation.
To evaluate the integral in Eq. (\[e:drhodf\]), we must make some assumption about the rate of Types Ib, Ic, and II supernovae (which are the ones that leave behind neutron stars). The combined rate in our galaxy at present is roughly one per 100 years. If this rate had been constant, the Galaxy would today contain about $10^8$ neutron stars. However at earlier times, for $z$ between 1 and 4, the rate (with respect to proper time) was significantly higher. A reasonable estimate is that our Galaxy contains $3\times 10^8$ neutron stars today. Let $n(z)\Delta z$ be the density of neutron star births (per unit comoving volume) between redshifts $z$ and $z + \Delta z$. As a rough, first-cut at this problem, we model $n(z)$ as constant $n(z)\equiv n_o$ for $0 < z < z_{*}$, where $z_{*} \approx 4$, and take $n(z) = 0$ for $z > z_{*}$. In this case the integral in Eq. (\[e:drhodf\]) can be performed to obtain:
\[rspec\] [[****]{}\_[gw]{}(f)Af\^2]{} = {
[ll]{} 0, &,\
(z\_\* + 1)\^2 - ([f\_/ f]{})\^2 , &,\
z\_\*(z\_\* + 2), &,\
(f\_/f)\^2 - 1, &,\
0, &,\
.
where $A \equiv { 2n_o E_{\rm K}/(3 \rho_c f_{\max}^2)}$, $f_1 \equiv f_{\min}/(1+z_*)$ and $f_2 \equiv f_{\max}/(1+z_*)$.
We can estimate the value of the constant $n_o$ as follows. We assume that the number of neutron stars in a given location is roughly proportional to the luminosity of the visible matter at that location. The total luminosity of our Galaxy is $1.4 \times 10^{10} L_\odot$ [@binney], while the number of neutron stars in the Galaxy is about $3\times 10^8$. Thus the neutron star mass to total luminosity of matter ratio is about $0.03 M_\odot/L_\odot$. The mean luminosity of the universe is $1.0\times 10^8 h_{100} L_\odot/{\rm
Mpc}^3$ [@peebles]. Thus, the mean mass density of neutron stars is about $\rho_{\rm ns}\approx3 \times 10^{6} h_{100} M_\odot/{\rm
Mpc}^3\approx 1.1\times 10^{-5} h_{100}^{-1}\,\rho_{\rm c}$. The current density of neutron stars is related to the constant $n_o$ by $\rho_{\rm ns}\approx 1.4 M_\odot z_*n_o \approx 5.6 M_\odot
n_o$. Thus ${2\over 3} n_o E_{\rm K}/\rho_{\rm c} \approx 3.3\times
10^{-8}h_{100}^{-1}$. For LIGO, the most important range in Eq. (\[rspec\]) is $f_1 < f < f_{\min}$; we can re-write the result for this portion in the more useful form
\_[gw]{}(f) 2.410\^[-10]{} h\_[100]{}\^[-1]{} . \[e:omgwf2\]
Evaluating at $f=50$ Hz, with $h_{100}=0.7$ we find ${\bf \Omega}_{\rm
gw}(f=50\,{\rm Hz}) \approx 1.1 \times 10^{-9}$.
Using the above spectrum Eq. (\[rspec\]), we evaluate the $S/N$ using Eq. (\[snsb\]). For an integration time of $T=10^7$s we find $S/N = 0.0022$, $0.34$, and $2.6$ for LIGO I, II, and III respectively. Since there has been some discussion of building a second kilometer-size interferometer in Europe, we also consider the sensitivity to the $r$-mode background of this detector paired with VIRGO. We assume that the detectors will be located less than about $
300$ km apart and have the same orientation. (To model this, we simply set $\gamma(f) = 1$ in the integral Eq. \[snsb\]). We find $S/N = 5.6$ assuming both these detectors have LIGO II sensitivity, $S/N = 0.9$ assuming one detector has LIGO I sensitivity while the other has LIGO III sensitivity, and $S/N = 23$ assuming both have LIGO III sensitivity. Thus we see that detection of the $r$-mode background will have to wait either for development of “advanced” interferometers or for the construction of two nearby detectors with “enhanced” sensitivity. Two nearby “advanced” interferometers could see quite a strong signal. All the above results on correlation measurements of the stochastic background assume that magnets will be eliminated from the LIGO design. With the current design long-range correlated magnetic fields from Schuman resonances and lightning strikes will mimic a stochastic background with ${\bf \Omega}_{\rm
gw}$ approximately $10^{-7}$ to $10^{-9}$ [@allen-romano].
These calculations assumed that all neutron stars are born with spins near their maximal value $\Omega_K$. It should be clear, however, that these results for the $S/N$ achievable by the LIGO pair depend only on the stars being born with spins greater than about $300$ Hz. Of course, it could well be that some fraction $F$ of neutron stars are born with rapid spins, while $(1-F)$ are born slowly spinning. The values of $S/N$ in this case could be estimated from those given above by multiplying by $F$. (See Spruit & Phinney [@phinney] for a recent argument that most neutron stars should be born with very slow rotation rates.)
It has previously been suggested that there could be a detectable gravitational wave background produced by supernova events [@blair; @ferrari]. The stochastic background due gravitational radiation from the $r$-modes differs from that previously envisioned in two important respects. Previously it was assumed that the radiation would be emitted in short bursts, forming a random but not continuous background. For the $r$-mode background, the long duration of the emission guarantees that it will be a continuous hum rather than an occasional pop. Also, the spectrum from spindown radiation extends to lower frequencies than had previously been expected from supernova events.
Discussion {#secVI}
==========
The discovery of a strong source of gravitational waves that is ubiquitous and is associated with such interesting objects as supernovae and neutron stars inevitably opens up a rich prospect for obtaining astronomical information from gravitational wave observations. We shall discuss here some of the more obvious prospects.
[*Background radiation from $r$-modes.*]{} Pleasantly, the background requires no detailed modeling of the signal in order to detect it. However, detection of background radiation from $r$-modes will probably have to wait for the development of “advanced” interferometers (or the construction of two nearby “enhanced” interferometers), even if we assume that a large fraction of neutron stars are born rapidly rotating. For nearby detectors with LIGO III sensitivity, $S/N$ is high enough that one could experimentally measure the spectrum ${\bf \Omega}_{gw}(f)$ with reasonable accuracy. This might provide very interesting cosmological information. For instance, imagine that most neutron stars are born rapidly rotating. (This could be verified, at least for neutron stars born today, by direct LIGO detections of nearby supernovae.) The background spectrum between $25$ and $50$Hz would then give us direct information about the star formation rate in the early universe. And its spectrum between 50 and perhaps 300 Hz would tell us about the distribution of initial rotation speeds of neutron stars.
[*Individual $r$-mode events associated with known supernovae.*]{} Observations of individual spindown events will be more difficult to achieve but can be very rewarding. The easiest case is if the supernova that leads to the neutron star is seen optically. This gives some hint of when the $r$-mode radiation should be looked for, but more importantly it gives a position. That reduces the difficulty of extracting the signal from the detector data stream. Detection of the radiation will return the amplitude of the signal, its polarization, the final spin of the star, and the values of the parameters of the waveform. Assuming that the three large detectors (the two LIGO installations and VIRGO) all detect the signal with $S/N\approx 8$, the effective combined $S/N$ will be $8\sqrt{3}\approx
14$. Values of the various parameters will then typically be determined to 10-20% accuracy. The polarization measures the orientation of the spin axis of the neutron star, which will be difficult to relate to any other observable. So it may not be of much interest. But the power spectrum of the radiation measures, as we have seen, the loss of rotational energy of the star, so it is proportional to $I/D^2$. If the host galaxy’s distance can be determined to better than the accuracy of the gravitational wave measurement, which seems likely, then this will provide a direct measure of the moment of inertia of a neutron star.
There will likely be several detections per year, which will shed light on a number of uncertainties. Even if the parameters are only the Taylor expansion coefficients for the frequency, they will constrain models of the $r$-mode spindown. We can expect to get some information about cooling rates, viscosity, crust formation, the equation of state of neutron matter, and the onset of superfluidity (or some combination of these). We also expect variability from event to event, due to different initial conditions after gravitational collapse, such as differential rotation or even the mass of the neutron star. Significant differential rotation might affect the final spin rate of neutron stars; hence any variability in the final spin speed might shed light on these initial conditions.
If we find we can detect this radiation with confidence, then the absence of it after a supernova could be a hint that a black hole has formed instead of a neutron star. More rarely it might indicate that the neutron star remained bound in a binary system, and the orbital modulation of the signal made it impossible to find.
Some supernovae will be especially nearby, and some neutron stars will be oriented more favorably, so there might be a handful of events with single-detector $S/N\approx 15-20$. These will provide particularly good constraints on the moment of inertia, superfluidity, viscosity, etc. For these events it may also be possible to trace the waveform back to its initial stages, and thereby to measure the initial spin of the collapsed star. In some cases, a star might be formed with more mass that it can support after spinning down, and so the $r$-mode spindown would lead to and be interrupted by collapse to a black hole. This would probably happen relatively soon after the star is formed; but for strong events, it might be possible to detect this break.
[*Using $r$-mode events as supernova detectors.*]{} Perhaps the most exciting use of $r$-mode observations would be to identify hidden or unnoticed supernovae. If it proves possible to create search strategies that are efficient enough to detect $r$-mode spindown even without prior positional information from an optical observation, then the gravitational wave detectors will become supernova monitors for the Virgo cluster. Perhaps as many as half of the supernovae in Virgo go unnoticed, hidden in thick dust clouds. LIGO and VIRGO would not give optical observers advance notice of the supernova: they will identify a neutron star only a year or so after it was formed. But they may be able to locate the position of the event with an accuracy of better than one arc-second.
This great precision is achieved from the modulation of the signal produced by the motion of the earth [@bfs]. The angular accuracy is similar to that achieved for pulsar observations in the radio. Fundamentally it is the diffraction limit of a gravitational wave telescope with the diameter of the earth’s orbit, because the detector acts like a synthesis array as it builds up signal along its orbit around the Sun. The ratio of the gravitational wavelength of about 1000 km to the diameter of 2 AU is about $3\times10^{-6}$ radians. This angular accuracy improves with $S/N$ as well, but it can be degraded by uncertainties in the spindown parameters. This assumes, as we have in this paper, that neutron-star cooling takes a year or more. If the alternative cooling scenarios are correct and the star cools in a few days, then the angular accuracy of observations will be very poor.
For an event at $20$ Mpc, arc-second angular resolution corresponds to distance resolution of about $60$ pc. Observations would therefore not only tell us in which galaxy the event occurred, but even in which molecular cloud. Detailed follow-up searches will then be possible for the expanding nebula, starting perhaps one year after the event.
[*Other implications for gravitational radiation from supernovae.*]{} The $r$-mode instability also has implications for our expectations of other kinds of gravitational radiation from supernova events. Although detecting supernovae was the goal of initial bar detector development, it has not been possible before now to provide reliable predictions of radiation from gravitational collapse. The $r$-mode instability is a reliable prediction, but only of radiation long after the collapse event. It is clear that a collapse that produces a rapidly rotating star will be more likely to radiate strongly, especially if it can reach the dynamical bar-mode instability that is seen in the lower-density star-formation simulations. But although rapid rotation is in some sense natural in gravitational collapse, the fact that young pulsars like the Crab are slow rotators seemed to indicate that neutron stars do not form with fast spins. Now the $r$-mode instability has provided an explanation for the slow spins of young pulsars; there is no longer any observational restriction on the initial spins of neutron stars.
It therefore seems to us much more likely than before that the gravitational collapse event can also be a strong source of gravitational waves. The following scenario seems plausible in at least the extreme cases where rotation completely dominates the last stages of collapse. The collapsed object has so much spin that it forms a bar shape on a dynamical timescale. This radiates away angular momentum in gravitational waves until the star is finally able to adopt a stable axisymmetric shape. The strong gravitational radiation ceases, to be replaced by the developing $r$-mode radiation. The first burst would be detectable by LIGO II at the distance of the Virgo cluster, and a network of such detectors could give a rough position, which would allow notification of optical astronomers of the event and multi-wavelength follow-up observations. In the gravitational wave data stream, intensive searching for the $r$-mode radiation would follow.
The unexpected strength of the $r$-mode gravitational wave instability in young neutron stars has therefore completely changed the prospects for detection of gravitational waves from supernova events. Not only can we look forward to regular detections of the spindown radiation when enhanced detectors begin operating, but we also now have more reason to expect strong bursts from the collapse events themselves. Particularly exciting is the prospect for using networks of LIGO-type detectors as supernova monitors, able to pinpoint the positions of hidden supernovae with arc-second precision. In order to achieve these goals, however, much work needs yet to be done to eliminate the uncertainties in our models of young neutron stars and to develop effective ways of searching for signals of this kind.
We thank Patrick Brady, Jolien Creighton, Teviet Creighton, Scott Hughes, Sterl Phinney, Joseph Romano, and Kip Thorne for helpful discussions. This research was supported by NSF grants AST-9417371 and PHY-9796079, by the NSF graduate program, and by NASA grant NAG5-4093.
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---
abstract: 'We use the quantum corrected chargino production cross section, including information on the beam polarization and chargino helicities, to estimate the precision achievable at a 1 TeV, 500 ${\mathrm fb}^{-1}$ Linear Collider on the determination of fundamental supersymmetric parameters from measurements of light chargino mass and production cross sections. We show that to get meaningful results higher order corrections should be included.'
author:
- |
H. BAER${}^1$, M.A. DÍAZ${}^2$[^1], M.A. RIVERA${}^2$, AND D.A. ROSS${}^3$\
\
[*${}^1$Dept. of Physics, Florida State University, Tallahassee FL 32306, USA*]{}\
[*${}^2$D. de Física, U. Católica de Chile, Av. V. Mackenna 4860, Santiago, Chile*]{}\
[*${}^3$Dept. of Physics, U. of Southampton, Southampton SO17 1BJ, U.K.*]{}
title: 'CHARGINO PAIR PRODUCTION AT ONE-LOOP'
---
It has been established that it will be possible to make precision measurements of supersymmetric observables at a future Linear Collider [@LC]. Therefore, it is imperative to match the experimental precision with higher order theoretical calculations [@Review]. Chargino production at an LC, $$e^+(p_2) + e^-(p_1) \quad \longrightarrow \quad
\tilde\chi^+_b(k_2) + \tilde\chi^-_a(k_1)$$ is a well studied laboratory for the determination of the parameters of the MSSM. The total unpolarized cross section was first calculated at one-loop in the approximation where only quarks and squarks were included [@qsq], while in the same approximation, the cross section with polarized beams was found in [@DKR2]. These early works showed the importance of the quark-squark loops. The first complete one-loop calculation of the polarized production cross section is in [@BH], where the importance of box graphs was demonstrated. This was followed by the first calculation of polarized cross sections which keeps the full information on the helicity state of the charginos and includes self energies, triangles and boxes, excluding only the QED corrections [@prototype; @DRhelicity]. Here the cross sections are given in terms of helicity amplitudes, $$\frac{d\sigma(\alpha,\lambda_2,\lambda_1)}{d\cos\theta}
\ = \ \frac{\lambda^{1/2}(s,\ma^2,\mb^2)}{128 \, \pi \, s}
\left| {\cal A}^\alpha_{\lambda_2,\lambda_1} \right|^2,$$ where $\lambda_1$ and $\lambda_2$ are the helicities of the $\chi^-$ and $\chi^+$ respectively, and $\alpha$ is the electron polarization. The renormalized helicity amplitudes are given in terms of generalized $Q$-charges, [*e.g.*]{}, $$\begin{aligned}
{\cal A}^L_{+-}&=&
-{\cal Q}^L_{31}(1+v)\,(1+\cos\theta)
-{\cal Q}^L_{32}\,s\,v\,(1+v)\,\sin^2\theta
\nonumber\\ &&
-{\cal Q}^L_{41}(1-v)\,(1+\cos\theta)
-{\cal Q}^L_{42}\,s\,v\,(1-v)\,\sin^2\theta
\nonumber\\ &&
-4{\cal Q}^L_{52}\sqrt{s}\,\sqrt{1 - v^2}\,(1+\cos\theta)\end{aligned}$$ where $v$ is the velocity of the chargino and $\theta$ is the scattering angle. All quantum corrections are included in the $Q$-charges.
Here we are interested in the determination of MSSM parameters from chargino observables hypothetically taken at an LC with 1 TeV center of mass energy and $500\, {\mathrm fb}^{-1}$ integrated luminosity. To this end we use a $\chi^2$ analysis whose input is the Supergravity benchmark model B [@benchmark] with $m_0=100$ GeV, $M_{1/2}=250$ GeV, $\tan\beta=10$, and $\mu>0$. In this model, the lightest chargino has a mass $m_{\chi^+_1}=181$ GeV and it is pair produced at the LC with an unpolarized cross section $\sigma=420$ fb, expecting of the order of $10^5$ events. Other observables we include in our $\chi^2$ are the left cross sections $\sigma_L^{++}=15$ fb, $\sigma_L^{+-}=250$ fb, and $\sigma_L^{-+}=140$ fb, and the right cross sections $\sigma_R^{++}=0.05$ fb, $\sigma_R^{+-}=0.44$ fb, and $\sigma_R^{-+}=0.29$ fb. We do not include observables from the heavy chargino. For the cross sections we consider only the statistical error ($N\pm\sqrt{N}$), and for the chargino mass we take a 1% error.
.5cm
The results of the $\chi^2$ analysis are summarized in Fig. \[chi\_elipse\], where regions of normalized $\chi^2<1$ are depicted (inner ellipses). For comparison we also show the regions where $\chi^2<4$ (central ellipses) and $\chi^2<10$ (outer ellipses). The center of each ellipse indicates the output of the $\chi^2$ analysis which should coincide with the input, and the width of each ellipse indicates the error in the determination of the corresponding parameter. Filled circles in two of the quadrants correspond to the tree level prediction, with a mediocre normalized $\chi^2=65$. These results are summarized in Table \[tab:cases\].
parameter input output error percent
-------------------- ------- -------- ------- --------- --
$M_2 $ 193.6 192 3 2
$\mu $ 328.2 326 9 3
$\tan\beta $ 10.0 10 3 30
$m_{\tilde\nu_e} $ 188.0 185 6 3
$\cos\phi_L $ 0.452 0.46 0.01 2
$\cos\phi_R $ 0.273 0.28 0.01 4
$m_{\chi^+_2} $ 365.0 364 8 2
$m_{\tilde t_1} $ 392.0 450 110 24
\[tab:cases\]
: Parameter determination from light chargino pair production at one-loop. All dimensionful parameters are expressed in GeV.
From the parameters in the chargino mass matrix at tree level, the gaugino mass $M_2$ and the higgsino mass $\mu$ can be determined within 2-3%, but $\tan\beta$ is determined within $30\%$ due to a weaker dependence of the observables on this parameter. The tree level mass matrix is diagonalized with two rotation matrices defined by the angles $\phi_L$ and $\phi_R$ and their cosine can be found with a 2-4% error. The heavy chargino and the sneutrino masses (the former intervenes at tree level into the production cross section with left handed polarized electrons) can be found with a 2-3% error. Last but not least, we can see in Fig. \[chi\_elipse\] the determination of the stop quark mass $m_{\tilde t_1}$, which is a pure one-loop effect: it is found to be $m_{\tilde t_1}=450\pm110$ GeV, while the input from benchmark B is $m_{\tilde t_1}=392$ GeV. This is very interesting since it indicates that precision measurements in the chargino sector may shed light into the squark sector, in the same way that the SM precision measurements gave us information on the top quark mass (and after its discovery, on the Higgs mass).
In summary, we have shown that in order to determine the underlying parameters of the MSSM in the chargino sector, it is necessary to work with the one-loop corrected cross sections and masses. In addition, precision measurements in the chargino sector can give us information on the squark masses.
[99]{} American Linear Collider Working Group (T. Abe et al.), hep-ex/0106055, hep-ex/0106056, hep-ex/0106057, hep-ex/0106058; ECFA/DESY LC Physics Working Group (J.A. Aguilar-Saavedra et al.), hep-ph/0106315; ACFA Linear Collider Working Group (Koh Abe et al.), hep-ph/0109166. For a review see, [*e.g.*]{}, W. Majerotto, hep-ph/0209137. M.A. Díaz, S.F. King, and D.A. Ross, [*Nucl. Phys. B*]{} [**529**]{}, 23 (1998); S. Kiyoura, M.M. Nojiri, D.M. Pierce, Y. Yamada, [*Phys. Rev. D*]{} [**58**]{}, 075002 (1998). M.A. Díaz, S.F. King, D.A. Ross, [*Phys. Rev. D*]{} [**64**]{}, 017701 (2001). T. Blank, W. Hollik, hep-ph/0011092. M.A. Díaz, and D.A. Ross, [*JHEP*]{} [**0106**]{}:001 (2001). M.A. Díaz, D.A. Ross, hep-ph/0205257. M. Battaglia, ., hep-ph/0112013.
[^1]: Speaker. Presented at Workshop on Physics and Experiments with Future Electron-Positron Linear Colliders August 26-30, 2002, Jeju Island, Korea.
|
---
abstract: 'We present a new video compression framework (ViSTRA2) which exploits adaptation of spatial resolution and effective bit depth, down-sampling these parameters at the encoder based on perceptual criteria, and up-sampling at the decoder using a deep convolution neural network. ViSTRA2 has been integrated with the reference software of both the HEVC (HM 16.20) and VVC (VTM 4.01), and evaluated under the Joint Video Exploration Team Common Test Conditions using the Random Access configuration. Our results show consistent and significant compression gains against HM and VVC based on Bj[ø]{}negaard Delta measurements, with average BD-rate savings of 12.6% (PSNR) and 19.5% (VMAF) over HM and 5.5% (PSNR) and 8.6% (VMAF) over VTM.'
author:
- 'Fan Zhang, , Mariana Afonso, , and David R. Bull, [^1] [^2] [^3]'
bibliography:
- 'IEEEabrv.bib'
- 'MyRef.bib'
title: 'ViSTRA2: Video Coding using Spatial Resolution and Effective Bit Depth Adaptation'
---
Spatial resolution adaptation, effective bit depth adaptation, video compression, machine learning based compression, HEVC, VVC
Introduction {#sec:intro}
============
The development of video technology for film, television, terrestrial and satellite transmission, surveillance, and especially Internet video, demands much higher bandwidth than ever before to support new services and ever increasing numbers of users consuming ever more content. Although communication techniques have improved significantly recently due to the advances in network and physical layers, the bitrate available to an individual user at the application layer is still limited, due to the increasing amounts of video data (with higher quality and more immersive formats) consumed everyday. Video compression offers the solution to this problem, but equally presents major challenges in how we achieved yet higher coding gains.
Since the first video coding standard, H.120 [@r:h120], was introduced in 1984, standardisation of video formats and coding algorithms have played an important role in the application and success of video technology [@b:Bull]. In the last three decades, video compression standards have been improved significantly, with each new standard providing approximately twice the compression efficiency compared to its predecessor. The latest effort, Versatile Video Coding (VVC) [@s:VVC1], initiated in 2018, is targeting 30-50% overall bitrate savings over the current standard, High Efficiency Video Coding (HEVC), through integrating more sophisticated tools.
Inspired by recent advances in artificial intelligence, machine learning-based methods have seen increasing utility in video compression algorithms both for end to end compression and to enhance conventional coding tools [@j:Liu3; @j:Ma1; @j:ZhangYun]. Although some of these works, particularly those using Convolutional Neural Networks (CNNs), have reported evident coding gains, very few of them have been adopted by standards due to their high complexity and unconventional architectures needed for the CNN models [@s:JVET-Jnotes].
Alongside conventional coding tools, spatial and temporal resolution adaptation have also been employed in video compression to improve coding efficiency. This type of approach has previously been adopted for relatively low bitrate scenarios [@c:Nguyen; @j:Georgis], as the quality of up-sampled video content can be inconsistent at higher bitrates due to the blurring or aliasing artefacts introduced by resolution adaptation. It is also noted that, in these approaches, only spatial and temporal (frame rate) resolution adaptations have been exploited for video compression and there is little reported work on bit depth adaptation.
Based on our previous work on spatial resolution (SR) and effective bit depth (EBD) [@j:Zhang9; @c:Zhang19; @c:Zhang23; @c:Zhang29; @c:Zhang26; @p:Zhang] adaptations, and perceptual quality assessment [@c:Zhang20], a new video compression framework, ViSTRA2, is proposed in this paper. It dynamically down-samples the spatial resolution and bit depth of the input video at the encoder, and reconstructs its original resolution and bit depth during decoding. To improve the final reconstruction quality, a deep CNN is employed for both SR and EBD up-sampling, trained on a large, multi-resolution video database for HEVC and VVC compression at various quantisation levels. The proposed approach has been integrated with the reference test models of HEVC (HM 16.20) and VVC (VTM 4.01), and evaluated on the Joint Video Exploration Team (JVET) Common Test Conditions (CTC). The results exhibit significant coding gains over both HEVC (HM) and VVC (VTM), with average BD-rate gains (based on PSNR) of 12.6% and 5.5% respectively.
The main contributions of this paper are summarised as follows:
1. Integration of both spatial resolution (SR) and effective bit depth (EBD) adaptation into one coding framework.
2. The employment of a single deep CNN for both SR and EBD up-sampling (using different model parameters for various scenarios);
3. Robust spatial resolution adaptation decisions based on a bespoke SR-dependent quality metric.
4. Demonstration that using deep networks in an end to end system offers flexibility in the distribution of complexity across that system.
Comparing to this paper, our previous contributions [@c:Zhang19; @c:Zhang23; @c:Zhang29; @c:Zhang26] focus solely on the adaptation of spatial resolution or effective bit depth. In addition, the employed CNN architecture in this paper has been enhanced compared to that in [@j:Zhang9; @c:Zhang19; @c:Zhang23]. Furthermore, an extended training database has been employed here to achieve improved reconstruction results. Comprehensive evaluation results are presented for ViSTRA2, against both HEVC and VVC using various quality assessment methods. An early version of ViSTRA2 was contributed by the University of Bristol (JVET-J0031) to the JVET “Call for Proposals” [@s:Zhang1] for Versatile Video Coding (VVC).
The remainder of this paper is organised as follows. Section \[sec:background\] reviews the state-of-the-art of video coding algorithms, specifically covering the approaches based on resolution adaptation and deep learning. In Section \[sec:background\], the proposed coding framework is described in detail alongside its important components. The performance of the presented work is fully evaluated in Section \[sec:results\], while Section \[sec:conclusion\] provides the conclusions and outlines future work.
Background {#sec:background}
==========
This section is divided into three subsections. The first overviews the development of video compression standards and the recent advances on royalty-free coding formats. Research work on machine learning based video coding is then briefly reviewed, followed by a summary of compression algorithms employing resolution adaptation.
Video Coding Standards: A Brief Overview
----------------------------------------
Since the early 1980s, video compression has been subject to global standardisation. Each standard defines the bitstream format and the decoder structure, alongside the associated test model (reference encoder), which generates standard-compliant streams and provides benchmark coding performance.
One of the most successful examples, H.264/AVC (Advanced Video Coding) [@r:h264] was jointly delivered in 2004 by ITU-T (VCEG) and ISO/IEC (MPEG). It remains the most widely adopted coding standard for Internet streaming, HDTV and Blu-ray players, although its successor H.265/HEVC [@r:HEVC], developed in 2013, offers nearly double the coding efficiency. More recently, aiming to achieve even higher compression performance (30%-50%) than HEVC, a new coding standard, Versatile Video Coding (VVC) [@s:VVC1] is under development, offering better support for immersive video including high dynamic range and 360.
In order to provide open source and royalty-free solutions for media delivery, the Alliance for Open Media (AOM) [@w:AOM] was formed in 2015 by a consortium of companies. Its first generation codec, AV1 (AOMedia Video 1) [@w:AV1], was released in 2018, and is considered one of the primary competitors of MPEG standards. Other notable coding standards also include Essential Video Coding/MPEG-5 [@r:EVC], Microsoft WMV (Windows Media Video) [@w:WMV9], BBC (British Broadcasting Corporation) Dirac [@w:Dirac], and AVS standards [@w:AVS].
Machine Learning based Video Compression
----------------------------------------
With recent advances in computational equipment and graphics processing (GPU) devices, deep neural networks, especially convolutional neural networks (CNNs), now offer tractable solutions to many image processing problems. They are being increasingly applied in image and video compression to enhance various coding tools including intra prediction [@c:Laude; @j:Yeh1; @j:Li4; @c:Pfaff], motion estimation [@j:Liu2; @c:ZhangHan; @j:Zhao1; @j:Yan], transforms [@c:Liu1], quantisation [@c:Alam], entropy coding [@c:Song; @c:Puri] and loop filtering [@c:Kuanar; @c:Park; @j:Yang4; @c:Yang]. Detailed reviews on machine learning based compression can be found in [@j:Liu3; @j:Ma1; @j:ZhangYun]. Among the responses to the Call for Proposals for VVC [@s:beyondHEVC], there were five proposals containing coding tools based on neural networks, but few of these have been adopted by VVC [@s:VVC1], though its development is still ongoing. This is mainly due to their relatively high computational complexity, especially when CNN-based calculation has to be executed at the decoder.
Resolution Adaptation for Compression
-------------------------------------
In video compression, resolution adaptation based approaches are utilised to trade off the relationship between quantisation and (spatial and temporal) resolutions. These methods encode a lower resolution version of a video, and reconstruct its original resolution during decoding. This process can be applied at different coding stages – for each macroblock or Coding Tree Unit (CTU) [@b:Uslubas; @c:Nguyen; @j:Li], frame [@j:Shen; @c:Zhang19], group of pictures [@j:Wang4; @j:Zhang9], or sequence [@j:Georgis; @j:Dong2]. The reconstruction quality is highly content dependent and relies on the up-sampling methods employed. This is why this type of approach was mainly applicable in low bitrate cases [@j:Shen; @j:Georgis; @c:Nguyen] or for intra coding [@c:Zhang19], when simple interpolation filters were employed. Inspired by the recent development of (especially CNN-based) super-resolution algorithms [@c:Ledig; @j:Dong; @c:Kim; @c:Lim], it is now possible to extend these approaches to higher bitrate ranges with improved reconstruction quality.
Due to the content-dependent nature of these adaptation approaches, it is also important to characterise the relationship between spatial resolution, quantisation and visual quality. For this purpose, subjective video databases [@c:Zhang21; @j:Bampis], were developed to investigate perceptual quality across a range of spatial resolutions, quantisation levels and up-sampling methods. Based on these databases, objective quality metrics have been designed for generic [@j:Bampis; @w:VMAF] or bespoke [@c:Zhang20] applications, which can facilitate quantisation resolution optimisation during resolution adaptation inside the coding loop.
Proposed Algorithm {#sec:algorithm}
==================
{width="1.02\linewidth"}
The proposed ViSTRA2 coding framework is illustrated in Fig. \[fig:workflow\]. At the encoder, the spatial resolution (SR) is firstly determined by a Quantisation-Resolution Optimisation (QRO) module according to the input content and the initial base quantisation parameter ($\mathrm{QP}_\mathrm{base}$) values configured in the host encoder. The original video frames are then spatially down-sampled (by 2 in this case[^4]) if SR adaptation is enabled, by applying an anti-aliasing modified separable Lanczos3 filter [@b:Turkowski] with a kernel width of 12. In order to signal the SR adaptation decision to the decoder, a flag bit is inserted into the bitsteam as side information.
Secondly, the effective bit depth (EBD) (for both luma and chroma channels) is down-sampled by 1 bit prior to encoding through bitwise right shift. Here EBD is defined as the actual bit depth used to represent the video content, which is different from the coding bit depth (CBD) defined as the pixel bit depth, e.g. *InternalBitDepth* in HEVC HM and VVC VTM reference encoders. Throughout the coding process, EBD is lower than or equal to CBD in the proposed coding workflow, while CBD remains constant. It is noted that there is no optimisation module for EBD adaptation here. This is because, through observing the coding results on the training data and the preliminary results generated in [@c:Zhang26], in most cases, CNN-based EBD adaptation can provide improved (or equivalent) compression performance. Therefore, EBD adaptation is always enabled in the coding framework of ViSTRA2, and does not require any side information indicating the decoder the EBD changes.
During encoding, to obtain similar bitrate ranges and facilitate comparison with anchor codecs (with the same $\mathrm{QP}_\mathrm{base}$ values), a fixed quantisation parameter (QP) offset is applied on $\mathrm{QP}_\mathrm{base}$. This offset equals -6 when only EBD adaptation is enabled, and becomes -12 if both SR and EBD adaptations are applied. These two values were empirically obtained through the observation on the coding statistics of training sequences [@c:Zhang19; @c:Zhang26].
At the decoder, based on the value of the flag bit, decoded SR and EBD down-sampled video frames are up-sampled to their original SR and EBD using a deep CNN. If SR adaptation is enabled, the spatial resolution of the decoded video frames is firstly up-sampled by 2 using a nearest neighbour filter before CNN reconstruction[^5]. The network architecture and training process alongside the detail description of the QRO module are presented below.
The Employed CNN Architecture
-----------------------------
{width="1\linewidth"}
The architecture of the convolutional neural network employed to reconstruct full spatial resolution and effective bit depth is shown in Fig. \[fig:network\]. The input of the CNN is a 96$\times$96 compressed RGB colour image block with reduced EBD (and re-sampled SR if SR adaptation is enabled), while the output is expected to be the corresponding original image block with the same size. This architecture is modified based on the generator (SRResNet) of SRGAN [@c:Ledig], which was developed for uncompressed image super-resolution. It starts from an initial convolutional layer with a Parametric Rectified Linear Unit (PReLU) as the activation function, and ends with another single convolutional layer with a Tanh activation. Before the final output, a skip connection is employed between the input of the CNN and the output the last convolutional layer. Between these two convolutional layers, there are $N$ identical residual blocks, one of each consisting of two convolutional layers and one PReLU in between. In each residual block, a skip connection is applied between the input of first convolutional layer and the output of the second. Another skip connection is also employed between the output of the first convolutional layer and the output of all residual blocks. In all convolutional layers, the stride value is 1 and the kernel size equals 3$\times$3. The number of feature maps is 64 for most convolutional layers except the last one (3 feature maps there).
Comparing to the original architecture of SRResNet [@c:Ledig], two modifications were applied:
- Batch normalisation (BN) layers have not been used here, as they were found to decrease image feature variability and influence overall performance [@c:Lim].
- The loss function employed for training the network is $\ell$1 loss rather than $\ell$2 in [@c:Ledig]. This is based on recent work on CNN super-resolution [@c:Johnson], which claims improved reconstruction quality achieved from this change.
Network Training and Evaluation {#sec:training}
-------------------------------
The CNN employed for EBD and SR-EBD (if SR adaptation is enabled) up-sampling was trained using 432 video clips at various spatial resolutions (108 source sequences $\times$ 4 spatial resolutions), including 3840$\times$2160, 1920$\times$1080, 960$\times$540, 480$\times$270, each of them having 64 frames with a CBD (coding bit depth) of 10 bits. All clips were collected from publicly available databases, including BVI-HFR [@j:Zhang8], BVI-Texture [@c:Zhang10], Netflix Chimera [@w:NetflixChimera] and Harmonic 4K [@w:Harmonic]. Their EBD (by 1 bit) and SR-EBD down-sampled (SR adaptation ratio of 2) versions were compressed using both HEVC HM 16.20 and VVC VTM 4.01 encoders for five different initial base QP values 22, 27, 32, 37 and 42 (QP offsets of -6 or -12 was applied during encoding for EBD or SR-EBD version). The same coding configuration was used as the Random Access (RA) mode (Main10 profile) in the Joint Video Exploration Team (JVET) Common Test Conditions (CTC), with a fixed intra period of 64.
This results in 20 groups of reconstructed videos (2 codecs $\times$ 2 adaptation versions $\times$ 5 QP groups), each group containing 432 reconstructed sequences for a specific codec (HM or VTM), adaptation version (EBD or SR-EBD) and QP group (22-42). For each SR-EBD down-sampled and compressed video, its frames were spatially up-sampled to their original spatial resolution using a nearest neighbour filter (see footnote \[fn:nn\]). The video frames of all reconstructed sequences in each group and their original counterparts, were randomly selected and split into 96$\times$96 image blocks, which were then converted to the RGB colour space. Block rotation has also been employed for data augmentation to further enhance model generalisation. For each training group, the total number of training image block pairs is approximately 100,000, much more than that of our previous training sets in [@j:Zhang9; @c:Zhang26].
The training process was conducted in the Tensorflow environment [@w:Tensorflow], using the following parameters: batch size of 16, learning rate of 0.0001, Adam optimisation [@c:Kingma], weight decay of 0.1 and a total number of 200 training epochs. The number of residual blocks ($N$) is fixed as 16, which is the same as that used in [@c:Ledig]. This generates 20 different CNN models ($\mathrm{model}_{c,v,q}$), each for one group. Here $c$ stands for the used codec (HM or VTM), $v$ represents adaptation versions (EBD or SR-EBD), and $q$ is denoted to the QP group. For a specific codec and adaptation version, the CNN model used in evaluation depends on the initial (before applying the offset) base QP ($\mathrm{QP_{base}}$):
$$\left\{
\begin{array}{l r }
\mathrm{model}_{c,v,22}, \ \text{if} & \mathrm{QP_{base}}\leq 24.5 \\
\mathrm{model}_{c,v,27}, \ \text{if} & 24.5<\mathrm{QP_{base}}\leq 29.5 \\
\mathrm{model}_{c,v,32}, \ \text{if} & 29.5<\mathrm{QP_{base}}\leq 34.5 \\
\mathrm{model}_{c,v,37}, \ \text{if} & 34.5<\mathrm{QP_{base}}\leq 39.5 \\
\mathrm{model}_{c,v,42}, \ \text{if} & \mathrm{QP_{base}}\geq 39.5
\end{array}
\right.
\label{eq:models}$$
When these CNN models were employed for evaluation, each EBD or SR-EBD down-sampled frame (pre-CNN SR up-sampling has been applied here if SR adaptation is enabled) is firstly segmented into 96$\times$96 overlapping blocks with an overlap size of 4 pixels as CNN input (after converting to the RGB space). The output full SR and EBD blocks were then aggregated back following the same segmentation pattern to form the final reconstructed frame.
Quantisation-Resolution Optimisation
------------------------------------
Spatial resolution adaptation does not always lead to coding improvement, being highly dependent on the original resolutions, spatio-temporal characteristics and the host codec. Our previous work [@j:Zhang9] employed four video features to predict QP thresholds, beyond which encoding lower resolution content produces higher compression efficiency. In ViSTRA2, an improved machine learning based approach has been developed to make decisions on resolution adaptation, based on a spatial resolution dependent quality metric, SRQM [@c:Zhang20], temporal information (TI) and initial base quantisation parameter.
SRQM is an efficient, full reference objective video quality metric, which characterises the relationship between variations in spatial resolution and visual quality. It employs wavelet decomposition, subband combination with perceptually inspired weights, and spatio-temporal pooling to estimate the relative quality between the frames of a high resolution reference video, and one that has been spatially adapted through a combination of down- and up-sampling. In this work, SRQM is applied between the uncompressed, original resolution video frames and their re-sampled versions (using Lanczos3 filter for both down- and up- sampling). The temporal information (TI) here is defined as the average absolute frame difference between luma pixels in the current frame and its neighbouring ones (one frame before and another after if available).
Both SRQM and TI values were computed for all frames in the 432 uncompressed sequences in the training dataset at various resolutions from 3840$\times$2160 to 480$\times$270. This database does not include any test sequences used in Section \[sec:results\], and was used for training the CNN in Section \[sec:training\]. The average SRQM and TI values for each sequence, together with tested initial base QP values (22, 27, 32, 37 and 42), were employed as input features to train a fully connected, shadow neural network, which contains a hidden layer and an output layer. The target binary output of the network was generated through comparing the rate-distortion performance (PSNR on luma values was used here as a quality metric) of the original anchor codecs (HM and VTM) and that of ViSTRA2 (with the trained CNN models) for all five tested initial base QPs on these training sequences. For a training sequence and $\mathrm{QP_{base}}$, if the rate-PSNR point of ViSTRA2 codec is above the rate-PSNR curve of the corresponding anchor codec, the binary decision (for this sequence and QP) is defined as 1 – the adaptation should be applied. Otherwise, it is configured to 0. This results in 432$\times$5 binary decisions for each codec. The network was trained offline using the Matlab function *train*, and two sets of network model parameters were then obtained for HEVC HM 16.20 and VVC VTM 4.01.
When these models were employed in ViSTRA2 to predict whether spatial resolution should be enabled, the average feature values used for the QRO are obtained for a number of frames – one GOP for Random Access mode in HEVC HM and VVC VTM – in order to maintain the same latency. The decision is applied to the frames assessed. Due to the possibility of content variations (although not very common in the JVET SDR test set), the QRO module may lead to frequent resolution changes during encoding. This should not cause any problem if resolution changing is supported at the frame or GOP level in the encoding loop. It is noted that although an Ad-Hoc Group was established by JVET to investigate resolution adaptivity [@s:JVET-O0008], this feature had not been formally adopted by VVC when this paper was written. Video sequence segmentation is therefore executed in the current version of ViSTRA2, when different resolutions have to be applied during encoding. If a video sequence has to be segmented into more than one segments, their bitstreams are generated sequentially by the host encoder. In order to avoid frequent sequence segmentation, each video segment is constrained here to be longer than 1 second. A flag bit is inserted within the header of the bitstream for each video segment to indicate the decoder resolution change. This solution will be modified when resolution adaptivity feature is integrated into standardised coding algorithms.
Results and Discussion {#sec:results}
======================
Metric
------------------- ------------ ----------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------
Codec
Class-Sequence H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs
A-Campfire -28.0% +2.0% -22.0% -16.7% -35.5% -19.9% -28.9% -19.6% -24.2% -25.5% -42.2% -41.2%
A-FoodMarket4 -12.2% -3.5% -7.9% -7.2% -18.0% -11.8% -14.4% -12.5% -12.2% -12.2% -24.5% -20.9%
A-Tango2 -11.9% -6.8% -8.6% -9.7% -17.4% -15.2% -11.9% -10.0% -12.5% -15.4% -22.3% -20.7%
A-CatRobot1 -4.4% -0.3% -11.9% -12.3% -15.1% -13.1% -6.3% -3.7% -18.4% -22.2% -23.1% -23.7%
A-DaylightRoad2 +1.1% +2.4% -10.6% -14.3% -11.2% -13.7% -0.5% -2.1% -17.5% -25.3% -21.7% -26.5%
A-ParkRunning3 -26.5% -17.4% -18.5% -13.7% -31.2% -25.2% -29.7% -27.2% -21.7% -20.4% -36.7% -34.9%
**Class A** **-13.7%** **-4.0%** **-13.2%** **-12.3%** **-21.4%** **-16.5%** **-15.3%** **-12.5%** **-17.8%** **-20.2%** **-28.4%** **-28.0%**
B-BQTerrace -0.3% -0.2% -11.0% -9.8% -11.0% -9.8% -1.1% -2.8% -21.7% -28.2% -21.7% -28.2%
B-BasketballDrive -1.3% -1.7% -11.0% -10.5% -10.0% -10.5% -2.9% -4.9% -11.7% -14.2% -12.6% -14.2%
B-Cactus -0.4% -0.3% -9.5% -9.8% -9.5% -9.8% -3.7% -4.9% -15.2% -18.6% -16.8% -18.6%
B-MarketPlace -0.7% +2.3% -4.8% -4.5% -6.7% -4.7% -3.3% -0.8% -12.4% -13.7% -18.0% -15.9%
B-RitualDance -1.8% +1.2% -8.0% -7.0% -9.8% -7.2% -2.0% -1.2% -12.7% -13.6% -16.1% -14.7%
**Class B** **-0.9%** **+0.3%** **-8.9%** **-8.3%** **-9.4%** **-8.4%** **-2.6%** **-2.9%** **-14.7%** **-17.7%** **-17.0%** **-18.3%**
C-BQMall +0.1% +1.6% -9.5% -8.7% -9.5% -8.7% -1.0% -0.5% -13.1% -13.3% -13.1% -13.3%
C-BasketballDrill -2.1% +1.2% -11.9% -8.8% -11.9% -8.8% -5.5% -4.4% -15.6% -13.8% -15.6% -13.8%
C-PartyScene +0.0% +1.6% -8.0% -7.4% -8.0% -7.4% -1.1% -1.1% -11.2% -12.4% -11.2% -12.4%
C-RaceHorses -3.5% -2.0% -10.5% -8.4% -10.5% -8.4% -4.6% -5.4% -12.7% -13.4% -12.7% -13.4%
**Class C** **-1.4%** **+0.6%** **-10.0%** **-8.3%** **-10.0%** **-8.3%** **-3.0%** **-2.9%** **-13.1%** **-13.2%** **-13.1%** **-13.2%**
D-BQSquare +0.8% +2.1% -15.6% -15.8% -15.6% -15.8% -1.3% -2.3% -18.1% -22.7% -18.1% -22.7%
D-BasketballPass -3.2% -0.8% -11.7% -9.8% -11.7% -9.8% -4.7% -4.3% -13.1% -12.3% -13.1% -12.3%
D-BlowingBubbles -0.4% +1.3% -8.0% -7.1% -8.0% -7.1% -2.4% -1.3% -12.5% -12.0% -12.5% -12.0%
D-RaceHorses -3.0% -1.2% -11.2% -9.3% -11.2% -9.3% -5.2% -5.4% -14.0% -14.2% -14.0% -14.2%
**Class D** **-1.5%** **+0.3%** **-11.6%** **-10.5%** **-11.6%** **-10.5%** **-3.4%** **-3.3%** **-14.4%** **-15.3%** **-14.4%** **-15.3%**
**Overall** **-5.2%** **-1.0%** **-11.1%** **-10.0%** **-13.8%** **-11.4%** **-6.9%** **-6.0%** **-15.3%** **-17.0%** **-19.3%** **-19.7%**
\[tab:results1\]
Metric
------------------- ----------- ----------- ----------- ----------- ------------ ----------- ------------ ----------- ----------- ------------ ------------ ------------
Codec
Class-Sequence H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs H-QPs L-QPs
A-Campfire -20.2% -9.0% -11.4% -11.3% -30.3% -16.8% -24.6% -21.3% -16.7% -20.9% -38.3% -36.5%
A-FoodMarket4 -3.9% +2.0% -2.5% -0.6% -7.3% -1.9% -11.0% -6.2% -5.6% -3.4% -14.7% -9.6%
A-Tango2 -4.6% +0.1% -3.0% -1.9% -7.9% -3.9% -9.4% -5.1% -5.5% -5.2% -12.3% -8.4%
A-CatRobot1 +4.0% +4.7% -5.3% -3.6% -4.8% -3.3% +0.4% -0.0% -7.3% -7.9% -9.0% -8.0%
A-DaylightRoad2 +1.9% +2.9% -4.1% -5.8% -5.0% -5.8% -0.3% -1.5% -7.1% -11.1% -9.1% -11.1%
A-ParkRunning3 -19.1% -17.0% -15.7% -17.0% -24.0% -25.0% -22.1% -19.7% -16.7% -16.3% -26.6% -25.7%
**Class A** **-7.0%** **-2.7%** **-7.0%** **-6.7%** **-13.2%** **-9.5%** **-11.2%** **-9.0%** **-9.8%** **-10.8%** **-18.3%** **-16.5%**
B-BQTerrace +1.6% +4.5% -1.7% +0.1% -1.7% +0.1% +0.3% +0.8% -1.5% -3.8% -1.5% -3.8%
B-BasketballDrive -0.8% +0.3% -4.0% -3.9% -4.0% -3.9% -2.6% -3.5% -3.4% -5.1% -3.4% -5.1%
B-Cactus -1.5% -0.4% -4.4% -4.0% -4.4% -4.0% -3.7% -4.6% -7.3% -8.7% -7.3% -8.7%
B-MarketPlace +7.1% +7.3% +4.4% +4.2% +4.4% +4.2% +5.3% +4.1% -0.7% -1.8% -0.7% -1.8%
B-RitualDance +0.4% +2.0% -3.2% -2.2% -3.2% -2.2% -1.3% -1.9% -6.7% -6.7% -6.7% -6.7%
**Class B** **+1.4%** **+2.7%** **-1.8%** **-1.2%** **-1.8%** **-1.2%** **-0.4%** **-1.0%** **-3.9%** **-5.2%** **-3.9%** **-5.2%**
C-BQMall +3.6% +4.9% -2.2% -2.1% -2.2% -2.1% +2.3% +2.1% -2.5% -2.8% -2.5% -2.8%
C-BasketballDrill +3.3% +7.2% -1.8% +0.7% -1.8% +0.7% -0.7% +0.3% -3.3% -2.7% -3.3% -2.7%
C-PartyScene +0.9% +2.6% -4.3% -3.2% -4.3% -3.2% -0.3% +0.4% -4.8% -3.7% -4.8% -3.7%
C-RaceHorses -1.4% -0.6% -4.0% -3.3% -4.0% -3.3% -3.4% -3.9% -6.3% -6.9% -6.3% -6.9%
**Class C** **+1.6%** **+3.5%** **-3.1%** **-2.0%** **-3.1%** **-2.0%** **-0.5%** **-0.2%** **-4.2%** **-4.0%** **-4.2%** **-4.0%**
D-BQSquare +5.7% +11.1% -5.7% -1.8% -5.7% -1.8% +3.9% +12.1% -1.8% +6.4% -1.8% +6.4%
D-BasketballPass +0.1% +1.9% -5.6% -4.6% -5.6% -4.6% -0.6% -0.6% -6.2% -5.4% -6.2% -5.4%
D-BlowingBubbles +1.1% +2.6% -3.5% -2.6% -3.5% -2.6% -0.5% +0.0% -5.4% -4.0% -5.4% -4.0%
D-RaceHorses -1.9% -0.5% -6.5% -5.6% -6.5% -5.6% -5.9% -6.0% -10.3% -10.4% -10.3% -10.4%
**Class D** **+1.3%** **+3.8%** **-5.3%** **-3.7%** **-5.3%** **-3.7%** **-0.8%** **+1.4%** **-5.9%** **-3.4%** **-5.9%** **-3.4%**
**Overall** **-1.2%** **+1.4%** **-4.5%** **-3.6%** **-6.4%** **-4.5%** **-3.9%** **-2.9%** **-6.3%** **-6.3%** **-9.0%** **-8.2%**
\[tab:results2\]
The ViSTRA2 coding framework has been integrated into the HEVC (HM 16.20) and VVC (VTM 4.01) reference software, and has been fully tested under JVET CTC [@s:JVETCTC] using the Random Access configuration (Main10 profile). In order to evaluate the proposed approach across a wider bitrate range, the initial base QPs tested include 22, 27, 32, 37 and 42. The SDR (standard dynamic range) video classes A1, A2, B, C and D from JVET CTC were employed here as test content, none of which were utilised in the training of CNN and QRO modules in Section \[sec:algorithm\].
The rate quality performance of the integrated ViSTRA2 has been compared to the corresponding reference codecs. Bj[ø]{}ntegaard delta measurements (BD) [@r:Bjontegaard] results were generated for both low QP (22, 27, 32 and 37) and high QP (27, 32, 37 and 42) values over all frames using two quality metrics (on luma values only), Peak Signal-to-Noise Ratio (PSNR) and Video Multimethod Assessment Fusion (VMAF, version 0.6.1) [@w:VMAF]. PSNR is the most commonly used image quality metric for evaluating video compression performance, while the recently developed VMAF is a machine learning based assessment method, combining the Visual Information Fidelity measures (VIF) [@j:Sheikh], Detail Loss Metric (DLM) [@j:Li2] and temporal frame difference [@w:VMAF] together using a Support Vector Machine (SVM) regressor [@j:Cortes]. As the content generated by ViSTRA2 contains both compression distortions and resolution re-sampled artefacts, VMAF, which is one of few quality metrics being trained and evaluated on compressed and resolution adapted content, is expected to provide more reliable assessment results (better correlation with subjective opinions) than PSNR.
In order to further benchmark the contribution of CNN reconstruction, results through up-sampling using Lanczos3 [@b:Turkowski] (for SR) and simple bitwise left shifting (for EBD) were also produced (denoted ViSTRA2-w/o CNN), based on the same QRO decisions (as CNN-based ViSTRA2). Moreover, the results generated by CNN-based EBD adaptation only (denoted EBDA-CNN) were presented as well, providing another benchmark for the full ViSTRA2.
Finally, the computational complexity of ViSTRA2 was calculated and normalised to the corresponding anchor codecs. The encoding process was executed on a shared cluster, BlueCrystal Phase3 [@w:BC3] based in the University of Bristol, in which each CPU node has 16 $\times$2.6 GHz SandyBridge cores and 64GB RAM. The decoding was run on the GPU nodes of the BlueCrystal Phase 4 [@w:BC4], each of which has 14 core 2.4 GHz Intel E5-2680 v4 (Broadwell) CPUs, 138GB of RAM and two NVIDIA P100 graphic cards.
Compression results
-------------------
Tables \[tab:results1\] and \[tab:results2\] summarise the compression results of ViSTRA2 for JVET CTC test sequences when integrated with HM 16.20 and VTM 4.01, using PSNR and VMAF as quality metrics. The rate-quality curves for selected sequences are also shown in Fig. \[fig:curves\]. It can be observed that ViSTRA2 provides consistent bitrate savings for both host codecs based on PSNR, with average BD-rate values of -12.6% for HM and -5.5% for VTM. The coding gains are greater when perceptual quality metric VMAF is employed for quality assessment, and the BD-rate savings (based on VMAF) are -19.5% and -8.6% for HM and VTM respectively. The coding improvement from ViSTRA2 for VVC VTM is relatively lower (although consistent) than that for HEVC HM. This is likely to be due to the significant enhancements already achieved by VTM over HM.
Comparing to ViSTRA2-w/o CNN, ViSTRA2 is superior in terms of overall coding efficiency for all test sequences at all resolutions. This improvement is even higher when VMAF is used to calculate the rate quality performance. It is also noted that the overall compression performance of EBDA-CNN (without SR adaptation) is worse than that of the full ViSTRA2, especially for UHD content, although the former has already been improved from our previous work [@c:Zhang26] due to the much larger training set used. Comparing to the compression results presented in [@j:Zhang9; @s:Zhang1], where the improvement is only evident on high resolution and low QP cases, ViSTRA2 offers enhanced coding performance across a wider QP and resolution range. This is due to the more sophisticated network architecture and more diverse training data employed.
Finally, it can also be observed that ViSTRA2 performs best on UHD sequences (Class A), where both SR and EBD adaptation were enabled for most tested QP values. For lower resolutions, SR adaptation was less frequently activated, and the coding performance of ViSTRA2-CNN is therefore very close (or identical) to that of EBDA-CNN. Between the two tested QP ranges, ViSTRA2 offers an improved performance for higher QP cases than in low QP scenarios. This becomes more evident on UHD content (Class A), when spatial resolution adaptation was more frequently employed.
{width="1.1\linewidth"}
\(a) Campfile-PSNR
{width="1.1\linewidth"}
\(b) DaylightRoad2-PSNR
{width="1.1\linewidth"}
\(c) RitualDance-PSNR
{width="1.1\linewidth"}
\(d) Cactus-PSNR
{width="1.1\linewidth"}
\(e) RaceHorses-PSNR
{width="1.1\linewidth"}
\(f) BasketballPass-PSNR
{width="1.1\linewidth"}
\(g) Campfile-VMAF
{width="1.1\linewidth"}
\(h) DaylightRoad2-VMAF
{width="1.1\linewidth"}
\(i) RitualDance-VMAF
{width="1.1\linewidth"}
\(j) Cactus-VMAF
{width="1.1\linewidth"}
\(k) RaceHorses-VMAF
{width="1.1\linewidth"}
\(l) BasketballPass-VMAF
Complexity figures
------------------
--------- --------- --------- --------- ---------
Encoder Decoder Encoder Decoder
Class A 57% 2,168% 91% 1,842%
Class B 78% 4,153% 102% 3,690%
Class C 102% 9,273% 99% 7,679%
Class D 102% 12,785% 103% 11,662%
Average 80% 6,421% 98% 5,625%
--------- --------- --------- --------- ---------
: Computational Complexity of ViSTRA2.
\[tab:complexity\]
The complexity figures of ViSTRA2 are presented in Table \[tab:complexity\]. It is important to note the machine learning architecture employed in ViSTRA2 takes account of the simple filter-based re-sampling process employed in the encoder. This means that, although we benefit from the reconstruction power of a deep network, the encoding process complexity remains similar to the reference encoder. In fact, the average encoding time of ViSTRA2 is shorter (if SR adaptation is enabled) or equivalent to that of the corresponding anchor codecs. This is due to the encoder processing down-sampled frames, but also because we do not employ a neural network at the encoder. During decoding process, the use of CNN-based up-sampling has significantly increased the execution time, on average 64 times that of HM and 56 times that of VTM.
Conclusion {#sec:conclusion}
==========
A new video coding framework (ViSTRA2) has been presented using CNN-based spatial resolution and effective bit depth re-sampling. This approach adaptively reduces the spatial resolution and effective bit depth of input video content for encoding, and employs a deep CNN at the decoder to reconstruct its original format. ViSTRA2 has been integrated with HEVC HM 16.20 and VVC VTM 4.01 reference software, showing consistent coding gains for test sequences at various resolutions based on different quality metrics. It has further shown that, when deep learning methods are used in an end to end compression system, there is flexibility in the distribution of complexity increases and these are not necessarily located at the encoder. Future work should focus on the complexity optimisation of the CNN to enable more efficient decoding process.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors acknowledge funding from the UK Engineering and Physical Sciences Research Council (EPSRC, project No. EP/M000885/1), and the NVIDIA GPU Seeding Grants. The authors would also like to thank Mr Di Ma for his contribution in developing the large video database for training the employed CNN.
[Fan Zhang]{} (M’12) received the B.Sc. and M.Sc. degrees from Shanghai Jiao Tong University (2005 and 2008 respectively), and his Ph.D from the University of Bristol (2012). He is currently working as a Senior Research Associate in the Visual Information Laboratory, Department of Electrical and Electronic Engineering, University of Bristol, on projects related to video compression and machine learning. His research interests include perceptual video compression, video quality assessment and machine learning based video coding.
[Mariana Afonso]{} received the B.S./M.S. degree in Electrical and Computers Engineering from the University of Porto, Portugal, in 2015 and a PhD in Electronic and Electrical Engineering from the University of Bristol, UK, in 2019. During her PhD, she was a part of the PROVISION ITN European Commission’s FP7 Project, a network of leading academic and industrial organizations in Europe, working on the perceptual video coding. She also completed a secondment at Netflix, USA, in 2017. She is currently a Research Scientist in the Video Algorithms team at Netflix. Her research interests include video compression, video quality assessment, and machine learning.
[David R. Bull]{} (M’94-SM’07-F’12) received the B.Sc. degree from the University of Exeter, Exeter, U.K., in 1980; the M.Sc. degree from University of Manchester, Manchester, U.K., in 1983; and the Ph.D. degree from the University of Cardiff, Cardiff, U.K., in 1988.
Dr Bull has previously been a Systems Engineer with Rolls Royce, Bristol, U.K. and a Lecturer at the University of Wales, Cardiff, U.K. He joined the University of Bristol in 1993 and is currently its Chair of Signal Processing and Director of its Bristol Vision Institute. In 2001, he co-founded a university spin-off company, ProVision Communication Technologies Ltd., specializing in wireless video technology. He has authored over 450 papers on the topics of image and video communications and analysis for wireless, Internet and broadcast applications, together with numerous patents, several of which have been exploited commercially. He has received two IEE Premium awards for his work. He is the author of three books, and has delivered numerous invited/keynote lectures and tutorials. Dr. Bull is a fellow of the Institution of Engineering and Technology.
[^1]: Manuscript drafted at September 2019.
[^2]: Fan Zhang, Mariana Afonso, and David R. Bull are with the Visual Information Laboratory, University of Bristol, Bristol, UK.
[^3]: E-mail: fan.zhang@bristol.ac.uk, mariana.afonso@bristol.ac.uk, and dave.bull@bristol.ac.uk
[^4]: Greater improvement may be achieved by applying multiple re-sampling ratios as in [@c:Zhang23]. This will however requires more sophisticated CNN training process and accurate QRO prediction. In this paper, as effective bit depth adaptation has also been employed in the coding workflow, a single spatial resolution re-sampling ratio is used for algorithm simplification.
[^5]: It is noted that, in our previous work [@j:Zhang9; @c:Zhang23; @s:Zhang1], pre-CNN up-sampling was conducted using a Lanczos3 filter. However we have found that by using a nearest neighhour filter here the overall reconstruction performance can be slightly improved.\[fn:nn\]
|
---
abstract: 'A simple model consisting of three distinct dimer sublattices is proposed to describe the magnetism of NH$_4$CuCl$_3$. It explains the occurrence of magnetization plateaus only at 1/4 and 3/4 of the saturation magnetization. The field dependence of the excitation modes observed by ESR measurements is also explained by the model. The model predicts that the magnetization plateaus should disappear under high pressure.'
author:
- Masashige Matsumoto
date: 'September 12, 2003'
title: 'Microscopic model for the magnetization plateaus in NH$_4$CuCl$_3$'
---
[[NH$_4$CuCl$_3$]{}]{}, [[TlCuCl$_3$]{}]{}, and [[KCuCl$_3$]{}]{} are isostructual quantum spin systems consisting of two-leg ladders separated by NH$_4^+$, Tl$^+$, and K$^+$ ions. The two-leg ladders are composed of Cu$^{2+}$ ions with spin $S=1/2$ which interact antiferromagnetically (AF) through the Cl$^-$ ions. These compounds can be considered as coupled two-leg $S=1/2$ Heisenberg AF spin ladders, and show various types of magnetization curves. Shiramura [*et al.*]{} found that [[NH$_4$CuCl$_3$]{}]{} has magnetization plateaus at 1/4 and 3/4 of the saturation moment, [@Shiramura-1998] while [[TlCuCl$_3$]{}]{} and [[KCuCl$_3$]{}]{} have no plateaus in their magnetization curves. [@Shiramura-1997] For [[NH$_4$CuCl$_3$]{}]{}, Kurniawan [*et al.*]{} performed ESR experiments which revealed that [[NH$_4$CuCl$_3$]{}]{} has a finite excitation gap in the plateau regions. [@Kurniawan-1] By studying specific heat, they found AF order already at zero field and suggested a phase diagram in finite fields containing three magnetically ordered phases. [@Kurniawan-2]
The origin of magnetization plateaus has attracted much interest recently. For weakly interacting dimer systems, the ground state is a spin singlet liquid with only short-range correlations between spins, and the triplet excitations require a finite excitation energy. An excited gas of triplet magnons has two characteristic energies. [@Rice] Exchange interactions cause the transfer of excited triplets between neighboring dimers leading to a form of kinetic energy. They also lead to a short range repulsion between triplets in addition to the hard core repulsion forbidding double occupancy of a dimer. The triplet excitations are split in an external magnetic field, and the energy of the lowest component can be driven through zero. When the kinetic energy is dominant, which is realized in [[TlCuCl$_3$]{}]{} and [[KCuCl$_3$]{}]{}, we have no plateaus in the magnetization curve. [@Shiramura-1997] In these case, antiferromagnetic order perpendicular to the field appears simultaneously with magnetization parallel to the field, creating [*field-induced magnetic order*]{}, [@Tanaka-2001] which can be interpreted as a condensation of the lowest lying magnon mode. [@Kato; @Oosawa-2002; @Cavadini-1999; @Cavadini-2001; @Cavadini-2002; @Rueegg-2002; @Rueegg-2003; @Nikuni; @Matsumoto] In contrast to these case, when the interaction energy dominates, magnetization plateaus can appear accompanied by a magnetic superlattice. [@Oshikawa] Such plateaus have been observed at a rational fraction of the saturation moment. [@Shiramura-1998; @Narumi] In SrCu$_2$(BO$_3$)$_2$, the frustrated form of the dimer lattice leads to a narrow bandwidth for triplet excitations and magnetization plateaus associated with a magnetic superlattice of localized triplets. [@Miyahara; @Kodama]
An unexpected feature of [[NH$_4$CuCl$_3$]{}]{} is the absence of a magnetization plateau at a value 1/2 of the saturation magnetization although this would correspond to the simplest superlattice. In this letter, an explanation of the appearance of only the values 1/4 and 3/4 is proposed. Recently, Oosawa [*et al*]{}. performed an inelastic neutron scattering experiment at zero field in ND$_4$CuCl$_3$, and found two almost non-dispersive excitation branches at 1.8 meV and 3 meV. [@Oosawa-2003] The feature of these excitation gaps is also reported in specific heat measurements above the transition temperature. [@Kurniawan-2] Since the system is already ordered at zero field, we can expect another low lying gapless branch which is hard to resolve. Very recently, Shimaoka [*et al.*]{} found by using NMR that 1/4 of the dimers are in a triplet configuration below 8.5K in both the ordered phase and the 1/4 plateau magnetic field regions ($H<6$T), [@Shimaoka] indicating that the symmetry is lowered already above the magnetic transition temperature, and that three weakly interacting dimer sublattices are preformed already above the transition temperature. These results are completely different from the SrCu$_2$(BO$_3$)$_2$ case where a phase transition breaking translational symmetry takes place in 1/8 plateau region. Motivated by these results, we propose a model composed of three distinct dimer sublattices to account for the magnetization plateaus of [[NH$_4$CuCl$_3$]{}]{}. In addition, we will discuss the consequences of increased interladder interactions, e.g. due to application of an external pressure, which can lead to the suppression of the plateaus.
![ Hamiltonian of our model consisting of three distinct dimers, A, B, and C. Circles represent $S=1/2$ spin. Intradimer interactions are expressed by solid lines whose width represents the strength of the interaction. []{data-label="fig:Hamiltonian"}](fig1.eps){width="7cm"}
The NH$_4^+$ ion is not spherical and is much larger than Tl$^+$ and K$^+$ ions. The NH$_4$ molecules rotate at high temperatures, and they freeze at low temperatures. [@Rueegg-private] In fact, elastic constant shows anomaly at 70K, [@Schmidt] and the NMR line splits below this temperature. [@Shimaoka] We speculate that the inclusion of the NH$_4^+$ ions between the ladders gives rise to a lattice distortion at low temperatures, modulating the exchange couplings. It is possible, because the angle of Cu-Cl-Cu pathway for the intradimer interaction is close to 90$^\circ$ (close to ferromagnetic exchange interaction), and a slight change of the angle may give drastic change in the intradimer interaction. There are various ways to distribute the three distinct dimer sublattices. We show a possible simple model in Fig. \[fig:Hamiltonian\] with three distinct ladders consisting of the A, B, and C dimers. The left and right side of each dimer is equivalent in this model. Two-leg ladders run along the $a$ axis. $J_i$ $(i=a,b,c)$ is an intradimer interaction for A, B, and C dimers, respectively. We assume $J_a \sim 0.3$ meV, $J_b\sim 1.8$ meV, and $J_c\sim 3$ meV to reproduce the magnon excitation modes. $J_i'$ $(i=a,b,c)$ is an interdimer interaction along the ladder. $J_{ij}$ $(i,j=a,b,c)$ is an interdimer interaction between $i$ and $j$ ladders. These path ways between $S=1/2$ spins are extracted from the crystal structure of [[NH$_4$CuCl$_3$]{}]{}. In our model, we neglect other interactions, because they are expected to be small. The spin structure in the ordered phases of [[NH$_4$CuCl$_3$]{}]{} has not yet been determined, so we assume that its pattern is similar to field-induced staggered order of [[TlCuCl$_3$]{}]{} and [[KCuCl$_3$]{}]{}. Due to the modulation of the exchange couplings, the unit cell is larger than that of [[TlCuCl$_3$]{}]{} and [[KCuCl$_3$]{}]{}, with A and C dimers and two B dimers in the unit cell. In the limit A, B, and C dimers are identical, the present model reduces to the models for [[TlCuCl$_3$]{}]{} and [[KCuCl$_3$]{}]{} in our previous work. [@Matsumoto]
Taking the $z$ axis parallel to the external magnetic field, we introduce the following variational wave function at the $i$th dimer: [@Oosawa-2002-2] $$\psi_i = c_{s i} |S\rangle
+ c_{\uparrow i} e^{-i\chi_i} |\uparrow\uparrow\rangle
+ c_{\downarrow i} e^{i\chi_i} |\downarrow\downarrow\rangle.
\label{eqn:Hamiltonian}$$ Here, $|S\rangle$ is the singlet wave function. $c_{s i}$, $c_{\uparrow i}$, and $c_{\downarrow i}$ are coefficients expressed as $c_{s i}=\cos{\theta_i}$, $c_{\uparrow i}=\sin{\theta_i}\cos{\phi_i}$, and $c_{\downarrow i}=-\sin{\theta_i}\sin{\phi_i}$. Since the left and right side of the dimer is equivalent in our model, the $|\uparrow\downarrow\rangle+|\uparrow\downarrow\rangle$ triplet component does not appear in the wavefunction. The expectation value of the spin operator of each site of a dimer is given by $$\begin{aligned}
\langle S_x \rangle_r &=& -\langle S_x \rangle_l =
\frac{1}{2\sqrt{2}}\sin{2\theta_i}(\cos{\phi_i}+\sin{\phi_i})\cos{\chi_i}, \cr
\langle S_y \rangle_r &=& - \langle S_y \rangle_l =
\frac{1}{2\sqrt{2}}\sin{2\theta_i}(\cos{\phi_i}+\sin{\phi_i})\sin{\chi_i}, \cr
\langle S_z \rangle_r &=& \langle S_z \rangle_l =
\frac{1}{2}\sin^2{\theta_i}\cos{2\phi_i}.\end{aligned}$$ Here, $\langle\cdots\rangle_{r(l)}$ represents the expectation value on the right (left) side of the dimer. The perpendicular ($x$ and $y$) component is staggered (i.e. spins are aligned oppositely on $l$ and $r$ sites). The parameter $\chi_i$ governs the rotation around the $z$-axis, and we can determine only the relative phase, $\chi_a=\chi_c=\chi_b+\pi$, such that the spin configuration gains the AF interladder interaction energy. The angles, $\theta_i$ and $\phi_i$ $(i=a,b,c)$, are variational parameters to be determined minimizing the expectation value of the Hamiltonian. This variational method is identical to that used in our previous paper where we introduced unitary transformations and minimized the $c$ number term of the transformed Hamiltonian. [@Matsumoto]
![ Magnetization curves at $T=0$K. (a) Normalized staggered moment, $M_{xy}$, per volume for A, B, and C dimers. Since there are two B dimers in the unit cell, the staggered magnetization for the B dimer in the phase II is about twice as large as for the A and C dimers in the phase I and III, respectively. (b) Uniform magnetization, $M_z$, at the A, B, and C dimer sites, not normalized by the number of dimers. The total magnetization is $A/4$+$B/2$+$C/4$, and it increases by 1/4, 1/2, and 1/4 upon passing through phases I, II, and III, respectively. Parameters were chosen to reproduce the experimental magnetization curves in unit of meV: $J_{a}=0.3$, $J_{b}=1.75$, $J_{c}=2.95$, $J_{a}'=0.25$, $J_{b}'=0.2$, $J_{c}'=0.35$, $J_{ac}=0.25$, $J_{bb}=0.1$, $J_{ab}=0.25$, $J_{bc}=0.25$. []{data-label="fig:p=1"}](fig2a.eps "fig:"){width="6.5cm"} ![ Magnetization curves at $T=0$K. (a) Normalized staggered moment, $M_{xy}$, per volume for A, B, and C dimers. Since there are two B dimers in the unit cell, the staggered magnetization for the B dimer in the phase II is about twice as large as for the A and C dimers in the phase I and III, respectively. (b) Uniform magnetization, $M_z$, at the A, B, and C dimer sites, not normalized by the number of dimers. The total magnetization is $A/4$+$B/2$+$C/4$, and it increases by 1/4, 1/2, and 1/4 upon passing through phases I, II, and III, respectively. Parameters were chosen to reproduce the experimental magnetization curves in unit of meV: $J_{a}=0.3$, $J_{b}=1.75$, $J_{c}=2.95$, $J_{a}'=0.25$, $J_{b}'=0.2$, $J_{c}'=0.35$, $J_{ac}=0.25$, $J_{bb}=0.1$, $J_{ab}=0.25$, $J_{bc}=0.25$. []{data-label="fig:p=1"}](fig2b.eps "fig:"){width="6.5cm"}
Figure \[fig:p=1\] shows the magnetization curves obtained by the variational wavefunction. There are three ordered phases, I, II, and III, which are driven by the A, B, and C dimers, respectively. At the A site, there is a finite staggered moment ($M_{xy}$) already at zero field, which induces staggered moment at B and C sites. As the field increases, this staggered moment first develops and then decreases. At $H_{c1}$, which is a saturation field for the A dimer, the staggered moments disappear and the A dimer is fully polarized by the field. Since there is one A dimer in the unit cell as in Fig. \[fig:Hamiltonian\], we have a magnetization plateau at 1/4 for $H>H_{c1}$. Above $H_{c2}$, which is a critical field for the B dimer, a staggered moment develops at the B site, inducing staggered moments at A and C sites. Although this field region is above the saturation field for the A dimer, the interaction between the A and B ladders induces a staggered moment at the A site. Accordingly, the uniform magnetization ($M_z$) at the A site decreases somewhat in the phase II as we can see in Fig. \[fig:p=1\](b). Above $H_{c3}$, which is a saturation field for the B dimer, both B and A dimers are fully polarized. Since there are two B dimers in the unit cell, we have a magnetization plateau at 3/4 for $H>H_{c3}$. Above $H_{c4}$ (critical field for the C dimer), a staggered moment develops at the C site, inducing the staggered moment at A and B sites. Above $H_s$ (saturation field for the C dimer), the all dimers are fully polarized. Thus, the magnetization plateaus at 1/4 and 3/4 of the saturation moment can be understood as successive quantum phase transitions driven by magnetic field, which occur in interacting three distinct dimer sublattices.
Specific heat and high-field magnetization measurements found three magnetically ordered phases on the $H-T$ phase diagram. [@Kurniawan-2] It is interesting to compare it with Fig. \[fig:p=1\](a). Since we expect a larger staggered magnetic moment leads to a higher transition temperature, we can account for the resemblance between the $H-T$ phase diagram and $M_{xy}(H)$ curves.
![ Schematic picture to understand the field dependence of excitation modes detected by ESR measurements. [@Kurniawan-1] $\omega_1$, $\omega_3$, and $\omega_6$ are excitation modes from triplet to singlet, while $\omega_2$ and $\omega_4$ are excitation modes from singlet to triplet. $W_{\rm B}$ and $W_{\rm C}$ represent bandwidth of triplet magnon excitations on the B and C dimer sublattices, respectively, which are related to the width of the ordered phases II and III. []{data-label="fig:esr"}](fig3.eps){width="6.5cm"}
In our model, the excitation branches at 1.8 meV and 3 meV found by neutron scattering measurements are triplet excitations. Therefore, each of them should split into three branches when we apply external magnetic field. It can explain the origins of the four low lying excitation modes, $\omega_i~(i=1\sim 4)$ (see Fig. \[fig:esr\]), detected by ESR measurements. [@Kurniawan-1] In the 1/4-plateau region, a $\omega_1$-mode increases linearly with the field, while the $\omega_2$-mode decreases and goes soft at $H_{c2}$. In this field region, each A dimer is fully polarized by the field, and is in the lowest lying triplet $|\uparrow\uparrow\rangle$, while the configuration of B dimer is dominated by the singlet component. Therefore, the $\omega_1$-mode can be identified with an excitation from a triplet to singlet on the A dimer, and the $\omega_2$-mode with an excitation from the singlet to the lowest lying triplet on the B dimer. Thus, an excitation energy gap opens up in the 1/4-plateau region. In the 3/4-plateau region, the energy of the $\omega_3$-mode increases with the field, while $\omega_4$-mode decreases and goes soft at $H_{c4}$. Similarly to the 1/4-plateau region, the $\omega_3$-mode can be identified with an excitation from the lowest triplet to the singlet on the B dimer, and $\omega_4$ with an excitation from singlet to the lowest triplet on the C dimer. $\omega_6$ in Fig. \[fig:esr\] is an excitation from the lowest triplet to the singlet on the C dimer, which is not detected by ESR measurements. There is no singlet-triplet excitation gap on the A dimer, since an AF order appears on the A dimer already at zero field.
![ Pressure dependence of critical fields and saturation field. Interladder interactions ($J_{ab}$, $J_{ac}$, $J_{bb}$, $J_{bc}$) are increased by a factor of $p$ which increases linearly with pressure: $p=\alpha(P-P_0)+1$. Here, $P$ is pressure, $P_0$ is atmosphere pressure, and $\alpha$ is a constant. $p_{c1}=2.0$ ($p_{c2}=2.3$) corresponds to a critical pressure on which the 1/4 (3/4) plateau vanishes. []{data-label="fig:hc"}](fig4.eps){width="6.5cm"}
Consider now what we can predict from our model. The critical fields ($H_{c1}$, $H_{c2}$, $H_{c3}$, $H_{c4}$) are functions of interdimer interactions which we can control by pressure. If we assume that interladder interactions increase linearly with pressure, while the intraladder interactions are constant, we obtain the pressure dependence of the critical fields shown in Fig. \[fig:hc\]. As pressure increases, the three distinct dimer sublattices couple more strongly, and a stronger field is required to saturate spins. Thus the plateau onset fields, $H_{c1}$, $H_{c3}$, and $H_s$, increase with pressure, since they correspond to saturation fields for A, B, and C dimers, respectively. The plateau end fields, $H_{c2}$ and $H_{c4}$, are critical fields where the lowest triplet component on the B and C dimers are driven to zero energy. Consider first the case of the B-sublattice. The increase with pressure of the intrasublattice, $J_{bb}$, broadens the triplet magnon bandwidth for the B dimer and lowers $H_{c2}$. The coupling through the C-sublattice $J_{bc}$ acts similarly but the coupling to the already polarized A-sublattice acts oppositely to increase $H_{c2}$, since the spins on the A sites are oriented parallel to the field. The net result however is a decrease in $H_{c2}(p)$ as in Fig. \[fig:hc\]. This decrease leads to the disappearance of the plateau between $H_{c1}$ and $H_{c2}$ at a critical pressure. In Fig. \[fig:p=pc1\], we show the magnetizations $M_{xy}(H)$ and $M_z(H)$ at this quantum critical point. The case of the 3/4-plateau is slightly different, since $H_{c4}(p)$ increases with the pressure. This occurs because there is negligible direct intrasublattice coupling for C dimers and both the A and B sublattices are now polarized parallel to $H$. None the less the 3/4-plateau width also narrows with increasing $p$ and eventually disappears at a higher critical pressure above which all the ordered phases I, II, and III merge into a single phase.
![ Magnetization curves under pressure $p=p_{c1}$. (a) Staggered moment, (b) uniform magnetization. Interladder interactions ($J_{ac}$, $J_{bb}$, $J_{ab}$, $J_{bc}$) are simply increased by a factor of $p_{c1}$. The other couplings are the same as in Fig. \[fig:p=1\]. []{data-label="fig:p=pc1"}](fig5a.eps "fig:"){width="3.3cm"} ![ Magnetization curves under pressure $p=p_{c1}$. (a) Staggered moment, (b) uniform magnetization. Interladder interactions ($J_{ac}$, $J_{bb}$, $J_{ab}$, $J_{bc}$) are simply increased by a factor of $p_{c1}$. The other couplings are the same as in Fig. \[fig:p=1\]. []{data-label="fig:p=pc1"}](fig5b.eps "fig:"){width="3.3cm"}
In summary, we have proposed a model consisting of three distinct dimer sublattices in order to account for the magnetization plateaus that appear in [[NH$_4$CuCl$_3$]{}]{}. There are various way to distribute the three distinct dimer sublattices. In this paper, we proposed a simple model in which the dimer sublattices distribute on the $b-c$ plane as illustrated in Fig. \[fig:Hamiltonian\]. In this case, triplet magnon excitations propagate predominantly along the ladder direction. Below $H_{c2}$, the spin structure of our model is characterized by a wave vector (0,$\pi$,$\pi$) or (0,0,2$\pi$). An alternate possibility is to distribute the three distinct dimer sublattices along the ladders ($a$ axis), which is proposed by studying NMR. [@Shimaoka] Our theory is applicable also to this case. The distribution pattern of the three distinct dimer sublattices is not essential to our main results. The important assumption of our model is that there are three weakly interacting distinct dimer sublattices whose intradimer interactions are characterized by the values 0.3 meV, 1.8 meV, and 3 meV observed by the neutron experiments. [@Oosawa-2003] The volume fraction of the dimers should be 1/4, 1/2, and 1/4, respectively. Note this is consistent with the fact that the intensity of the magnon branch at 1.8 meV is about twice as large as the branch at 3 meV, [@Oosawa-2003] and it explains why the magnetization plateaus take place only at 1/4 and 3/4. We made a prediction that the magnetization plateaus disappear by applying pressure, which is supported by a recent magnetization measurement under high pressure. [@Tanaka-2003]
The author expresses his sincere thanks to T. M. Rice for valuable discussions and critical reading of the manuscript. He is very grateful to F. Mila, C. Rüegg, and H. Tanaka for fruitful discussions. He also would like to thank H. Kusunose, B. Lüthi, B. Normand, A. Oosawa, Y. Shimaoka, and M. Sigrist for many useful discussions. This work is supported by Japan Society for the Promotion of Science (JSPS) and the MaNEP program of the Swiss National Fund.
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---
abstract: 'We show that the two-weight estimate for the dyadic square function proved by Lacey–Li in [@Lacey-Li] is sharp.'
author:
- Spyridon Kakaroumpas
title: 'A note on a two-weight estimate for the dyadic square function'
---
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|
---
abstract: 'This paper explores object detection in the small data regime, where only a limited number of annotated bounding boxes are available due to data rarity and annotation expense. This is a common challenge today with machine learning being applied to many new tasks where obtaining training data is more challenging, in medical images with rare diseases that doctors sometimes only see once in their life-time. In this work we explore this problem from a generative modeling perspective by learning to generate new images with associated bounding boxes, and using these for training an object detector. We show that simply training previously proposed generative models does not yield satisfactory performance due to them optimizing for image realism rather than object detection accuracy. To this end we develop a new model with a novel unrolling mechanism that jointly optimizes the generative model and a detector such that the generated images improve the performance of the detector. We show this method outperforms the state of the art on two challenging datasets, disease detection and small data pedestrian detection, improving the average precision on NIH Chest X-ray by a relative 20% and localization accuracy by a relative 50%.'
author:
- |
Lanlan Liu$^{1,2}$[^1] Michael Muelly$^{2}$ Jia Deng$^{3}$ Tomas Pfister$^{2}$[^2] Jia Li$^{4}$\
$^{1}$University of Michigan, Ann Arbor $^{2}$Google Cloud AI $^{3}$Princeton University $^{4}$Stanford University\
[ tpfister@google.com]{}
bibliography:
- 'bboxgan.bib'
title: 'Generative Modeling for Small-Data Object Detection'
---
Introduction
============
Generative Adversarial Networks (GANs) [@goodfellow2014generative] have recently advanced significantly, with the latest models [@anonymous2019large; @karras2018style] being able to generate high quality photo-realistic images that are almost indistinguishable from real images. A natural question that has recently started being explored [@li2017perceptual; @ouyang2018pedestrian; @shrivastava2017learning] is whether these generated images are useful in some other ways; for example, could they be useful training data for downstream tasks?
One common computer vision task that could benefit from generated data is object detection [@lin2018focal; @renNIPS15fasterrcnn] which currently requires a large amount of training data to obtain good performance. But for many object detection tasks, large datasets are difficult to obtain due to rare objects and difficulties in obtaining object location annotations. One common example is with medical images – disease detection has very little labeled object bounding box data because the diseases by nature are rare, and annotations can only be done by professionals, and thus are costly. Solving such rare data object detection problems is valuable: for example, for disease localization, a good disease detector can help provide assistance to radiologists to accelerate the analysis process and reduce the chance of missing tumors, or even provide a medical report directly if a radiologist is not available.
In this paper we explore using generative models to improve the performance in small-data object detection. Directly applying existing generative models is problematic. First, previous work on object insertion for generative models often needs segmentation masks, which are often not available in disease detection tasks. Second, GANs are designed to produce realistic images (indistinguishable from real images), but realism does not guarantee that it can help with the downstream object detection task. In particular, there is no direct feedback from the detector to the generator; which means the generator cannot be trained explicitly to improve the detector.
![ DetectorGAN generates object-inserted images as synthesized data to improve the detection performance. DetectorGAN integrates a detector into the generator-discriminator loop. []{data-label="fig:overview"}](overview_5.pdf){width="1\linewidth"}
To address this, we propose a new DetectorGAN model (shown in Fig. \[fig:overview\]) that connects the detector and the GAN together. This joint model integrates a detector into the generator-discriminator pipeline and trains the generator to explicitly improve the detection performance.
DetectorGAN has two branches after the generator: one with discriminators to improve realism and interpretability of the generated images, and another with a detector to give feedback on how well the generated images improve the detector. We jointly optimize the adversarial losses and detection losses. To generate images that are beneficial for the detector, the loss formulation is non-trivial. One difficulty is that our goal is for the generated images to improve the detector performance of real images, but the generator cannot receive gradients from the detection loss on real images because the real images are not generated. To address this, the proposed method bridges this link between the generator and the detection loss on real images by unrolling one forward-backward pass of the detector training.
We demonstrate the effectiveness of using DetectorGAN to improve small-data object detection in two datasets for disease detection and pedestrian detection. The detector-integrated GAN model achieves state of the art performance on the NIH chest X-ray disease localization task, benefiting from the additional generated training data. In particular, DetectorGAN improves the Average Precision of the nodule detector by a relative 20% by adding 1000 synthetic images, and outperform the state of the art on localization accuracy by a relative 50%. We also show that the proposed framework significantly improves the quality of the generated images: a radiologist prefers generated images by DetectorGAN over alternative methods in 96% of cases. The detector model can be integrated into almost any existing GAN models to force them to generate images that are both realistic and useful for downstream tasks. We give the pedestrian detection task and the associated PS-GAN [@ouyang2018pedestrian] as an example, demonstrating a significant quantitative and qualitative improvement in the generated images.
Our contributions are:
1. To the best of our knowledge, this work is first to integrate a detector into the GAN pipeline so that the detector gives direct feedback to the generator to help generate images that are beneficial for detection.
2. We propose a novel unrolling method to bridge the gap between the generator and the detection performance on real images.
3. The proposed model outperforms GAN baselines on two challenging tasks including disease detection and pedestrian detection, and achieves the state-of-the-art performance on NIH chest X-ray disease localization.
4. We are the first few works to explore GANs with downstream vision tasks such as small-data object detection.
Related Work
============
#### Image-to-image Translation.
Based on a conditional version of Generative Adversarial Networks (GANs) [@goodfellow2014generative], Isola et al [@isola2017image] pioneered the general image-to-image translation task. Afterwards multiple other works have also exploited pixel-level reconstruction constraints to transfer between source and target domains [@zhang2017stackgan; @wang2018pix2pixHD]. These image-to-image translation frameworks are powerful, but require training data with paired source/target images, which are often difficult to obtain. Unpaired image-to-image translation frameworks [@zhu2017unpaired; @liu2017unsupervised; @shrivastava2017learning; @DBLP:conf/icml/KimCKLK17; @lee2018unsupervised] remove this requirement of paired-image supervision; in CycleGAN [@zhu2017unpaired] this is achieved by enforcing a bi-directional prediction between source and target. The proposed DetectorGAN falls in the category of unpaired image-to-image translation frameworks. Its novelty is that it integrates a detector into GAN to generate images as training data for object detection.
#### Object Insertion with GANs.
The idea of manipulating images by GANs has been explored recently [@lee2018context; @hong2018learning; @chien2017detecting; @lin2018st; @ouyang2018pedestrian; @lee2019inserting; @lin2018st]. These works use generative models to edit objects in the scene. In contrast, (1) our method doesn’t require any segmentation information; and (2) our goal is to gain quantitative improvement on object detection task while prior works focus on qualitative improvement such as realism.
#### Integration of GANs and Classifiers.
Beyond the basic idea of using adversarial losses to generate realistic images, some GAN models integrate auxiliary classifiers into the generative model pipeline, such as Auxiliary Classifier GAN (ACGAN) and related works [@odena2017conditional; @bazrafkan2018versatile; @dash2017tac; @bousmalis2017unsupervised; @hoffman2018cycada]. At a first glance, these models bear some similarity with our integration with detector. However, we differ from them both conceptually and technically. Conceptually, these methods only improve the realism of the generated images and have no intention to improve the integrated classifier; in contrast, the purpose of our integration is to improve the detection performance. Technically, our loss formulation is different: ACGAN minimizes classification losses only on synthetic images and has no guarantee for improving performance on real images, whereas ours optimizes losses on both synthetic and real by adding unrolling step. Nevertheless, we construct a baseline with ACGAN-like losses, which only minimizing detection losses on synthetic images, and show that our proposed method outperforms it.
#### Data Augmentation for Object Detection
There are some works using data augmentation to improve object detection. A-Fast-RCNN [@WangCVPR17afrcnn] uses adversarial learning to generate hard data augmentation transformations, specifically for occlusions and deformations. It differs from the method in this paper in two major ways: (1) It is not a GAN model – it does not generate images but instead adds adversarial data augmentation into the detector network. In contrast, our model has a discriminator and detector that work together to generate synthetic images. (2) Its goal is to ‘learn an object detector that is invariant to occlusions and deformations’. In contrast, this paper focuses on generating synthetic data for the problem setting where the amount of training data is limited.
Perceptual GAN [@li2017perceptual] generates synthetic images to improve the detection. However, it is designed specifically for small-sized object detection by super-resolving the small-sized objects into better representations. Their method does not generalize to general object detection.
Concurrent unpublished work PS-GAN [@ouyang2018pedestrian] is most closely related: synthetic images are generated to improve pedestrian detection. They generate synthetic images using a traditional generator-discriminator architecture. In contrast, we add a detector in the generator-discriminator loop and have direct feedback from the detector to the generator.
DetectorGAN
===========
Our DetectorGAN method generates synthetic training images to directly improve the detection performance. It has three components: a generator, (multiple) discriminators, and a detector. The detector gives feedback to the generator about whether the generated images are improving the detection performance. The discriminators improve the realism and interpretablity of the generated images; that is, the discriminators help to produce realistic and understandable synthetic images.
Model Architecture
------------------
We implement our architecture based on CycleGAN [@zhu2017unpaired]. The generator in DetectorGAN generates synthetic labelled (object-inserted) images that are fed into two branches later: the discriminator branch and the detector branch. We consider clean images without objects belong to domain X, and labelled images with objects belong to domain Y.
#### Generators.
We use a ResNet generator with 9 blocks as our generators $G_X$ and $G_Y$ following [@zhu2017unpaired; @he2016deep]. The forward generator $G_X$ takes two inputs: one is a real clean image, which is used as the background image to insert objects. The other one is a mask where the pixels inside the bounding box of the object to insert are filled with ones while the rest are zeros. The output of the generator is a synthetic image with the input background and an object inserted at the marked location. Inversely, the backward generator $G_Y$ takes a real labelled image and a mask showing the object location, and outputs an image with the indicated object removed.
Plausible inserting locations of objects are difficult to obtain. In this paper, for the NIH disease task, we obtain these locations by pre-processing and random sampling. In theory, the location could be in any position in the lung area, but since in practice we do not have segmentation mask for the lung area, we first match each clean image to the most similar labelled image with bounding box and then randomly shift the location around to get the sampled ground-truth box location. For the pedestrian detection task, we follow the setup in the previous work [@ouyang2018pedestrian]. It is notable that the selection of mask locations does not change our method – as an alternative one could use trainable methods to predict plausible locations.
#### Discriminators.
Our method contains two global discriminators $DIS_{globalX}$ and $DIS_{globalY}$ as in Cycle-GAN[@zhu2017unpaired], and a local discriminator $DIS_{localX}$ for local area realism [@lee2019inserting; @li2017generative]. The global discriminator $DIS_{globalX}$ and the local discriminator $DIS_{localX}$ discriminates between real labelled images and synthetic labelled images (generated by $G_X$), globally on the whole images or locally on the bounding box crops. $DIS_{globalY}$ discriminates what $G_Y$ generates (synthetic clean images by removing objects from real labelled images) and real clean images. $DIS_{localY}$ is not needed because conceptually we do not care much about the local realism after removing an object. We use $70\times70$ PatchGAN following [@johnson2016perceptual; @isola2017image; @zhu2017unpaired] for all of our discriminators.
#### Detector.
The detector $DET$ takes both real and synthetic labelled images with objects as input and outputs bounding boxes. In our implementation we use the RetinaNet detector [@lin2018focal]. But we are not only limited to RetinaNet: as long as the detector is trainable, we can integrate it into the loop.
Train Generator with Detection Losses
-------------------------------------
The objective of the generator $G_X$ is to generate images with objects inserted that are both realistic and beneficial to improve object detection performance. One of our main contributions is that we propose a way to backpropagate the gradients derived from detection losses back to the generator to help the generator to generate images that can better help improve the detector. In other words, the detection losses give the generator feedback to generate useful images for the detector.
We note the detection loss (regression and classification losses) as $L_d(\cdot)$, where $\cdot$ is a labelled image, either real or synthetic. The detection loss on real images and synthetic images are: $$\begin{split}
L_{Det}^{real}(DET) = E_{y \sim p_{data}(Y)}[L_{d}(DET(y))] \\
\label{eqn:loss_detb}
\end{split}$$
$$\begin{split}
L_{Det}^{syn}(G_X, DET) = E_{x \sim p_{data}(X)}[L_{d}(DET(G_X(x)))] \\
\label{eqn:loss_deta}
\end{split}$$
#### Unroll to Optimize Detection Loss on Real Images.
![ The illustration for Eqn. \[eqn:loss\_unroll\] – unrolling one forward-backward pass for training $DET$ to bridge the link between $G_X$ and $L_{Det}^{real}$ (detection loss on real images). Detection loss on real images has no direct link to the generator $G_X$. Last step of training old $DET$ (noted as $DET'$ in the figure, refers to same $DET$ module but in the previous training step) is unrolled as in the dotted rectangle. The red arrow represents the fact that there is a differentiable link between $G_X$ and $L_{Det}^{real}$ after the unrolling. []{data-label="fig:method"}](method_2.pdf){width="1\linewidth"}
Intuitively, given a real image $y$, the goal of $G_X$ is to use generated images to help minimize the detection loss on real images. That is, $G_X$ should be trained to minimize the loss $L_{Det}^{real}$ in Eqn. \[eqn:loss\_detb\]. However, there is no $G_X$ involved at the first glance – the loss $L_{Det}^{real}$ does not depend on the weights of the $G_X$ so $G_X$ cannot be trained. But we observe that even though there is no direct link in one forward-backward loop from $G_X$ to real images, the detector is trained by synthetic images generated by $G_X$ in the previous step. We propose to bridge the link between $G_X$ and the real image detection loss $L_{Det}^{real}$ by unrolling a single forward-backward pass of the detector as shown in Eqn. \[eqn:loss\_unroll\]. A visualization of this unrolling process is shown in Fig. \[fig:method\]. This allows us to train $G_X$ with respect to the loss $L_{Det}^{real}$.
$$\begin{split}
\tilde{L}_{Det}^{real}(G_X, DET) = E_{y \sim p_{data}(Y)}[L_{d}(DET(y))] \\
\text{where weights of $DET$, $W_{DET}$, is updated with} \\
\frac{\partial (L_{Det}^{real}(DET) + L_{Det}^{syn}(G_X, DET)))} {\partial W_{DET}} \\
\label{eqn:loss_unroll}
\end{split}$$
Specifically, we train the weights $DET$ with synthetic images and real images for one iteration and obtain the gradients on $DET$. These gradients are linked to the generated synthetic images and thus to the weights in the generator $G_X$. Then we use the updated $DET$ to get the $L_{Det}^{real}$ loss and gradients. In this way, we obtain a link from $G_X$ to $DET$ and then to $L_{Det}^{real}$.
Intuitively, this Eqn. \[eqn:loss\_unroll\] can be seen as a simple estimation of how the change in $G_X$ will change detection performance on real images in Eqn. \[eqn:loss\_detb\].
#### Detection Loss on Synthetic Images.
The generator aims to make the synthetic images helpful for the detector. It maximizes the detection loss on synthetic images (Eqn. \[eqn:loss\_deta\]) to generate images that the detector has not seen before and cannot predict well. In this case the generated images can help improve the performance.
One might think the generator should instead minimize the detection loss on synthetic images. This shares some similar ideas with ACGAN-like losses, where the auxiliary classification loss on synthetic images is minimized to improve realism. But for our goal to improve the detection performance on real images, minimizing detection losses on synthetic images may not help, or may even hurt the detection performance on real images. The intuition behind this is that synthetic objects may distract away from the optimization goal of the detector. In our experiments, we show that minimizing synthetic image losses like ACGAN harms detection performance on real images.
Training data Nodule AP Nodule Recall
----------------------------------------- ----------- ---------------
Real data only 0.124 0.184
Real + syn from ACGAN-like losses 0.154 0.607
Real + syn from CycleGAN + BboxLoss 0.196 0.541
Real + syn from DetectorGAN - unrolling 0.203 0.544
Real + syn from DetectorGAN **0.236** **0.649**
$T_{IOU}$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Avg
------------------------------------------------ ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
Wang et al. [@wang2017chestx] 0.14 0.05 0.04 0.01 0.01 0.01 0.00 0.04
Zhe et al. [@li2017thoracic] **0.40** 0.29 0.17 0.11 0.07 0.03 0.01 0.15
RetinaNet: real 0.15 0.15 0.15 0.08 0.08 0.00 0.00 0.09
RetinaNet: real + syn from CycleGAN + BboxLoss 0.31 0.31 0.23 0.23 0.00 0.00 0.00 0.15
RetinaNet: real + syn from DetectorGAN 0.31 **0.31** **0.31** **0.23** **0.23** **0.15** **0.08** **0.23**
Overall Losses and Training
---------------------------
#### Overall Losses.
The objective of the generator $G_X$ is to generate images with an object inserted at the indicated location in background images. The generated images should be both realistic and beneficial to improve object detection performance. In other words, the DetectorGAN model should generate images that: (1) can help to train a better detector; (2) have an object inserted; and (3) are indistinguishable from real images, globally and locally.
We have introduced losses to help the detector above. For inserting an object, we use an L1 loss to minimize the loss between the synthetic object crop and the real object crop (refered as BboxLoss):
To generate realistic images, we have adversarial losses for the global discriminators and the local discriminator. For $DIS_{globalX}$, the adversarial loss is $L_{GAN}(G_X, DIS_{globalX})$ as in Eqn. \[eqn:loss\_globalx\]. $L_{GAN}(G_Y, DIS_{globalY})$ and $L_{GAN}(G_X, DIS_{localX})$ are similar.
$$\begin{split}
L_{GAN}(G_X, DIS_{globalX}) = E_{y \sim p_{data}(Y)}[\text{log} DIS_{globalX}(y)] \\
+ E_{x \sim p_{data}(X)}[\text{log} (1 - DIS_{globalX}(G_X(x)))]
\label{eqn:loss_globalx}
\end{split}$$
In addition, we use cycle consistency losses and identity losses to help preserve information from the whole image.
Here $G_X$ and $G_Y$ aim to fool the discriminators while the discriminators aim to discriminate between fake and real images. The generator and discriminators thus optimize $\min_{G_X, G_Y} \max_{DIS_{globalX}, DIS_{localX}, DIS_{globalY}}$ $[L_{GAN} (G_X, DIS_{globalX}) + L_{GAN} (G_Y, DIS_{globalY}) + L_{GAN} (G_X, DIS_{localX})]$.
We update the weights of the detector $DET$ by minimizing the detection losses for both real images and synthetic images: it minimizes Eqn. \[eqn:loss\_detb\] and Eqn. \[eqn:loss\_deta\].
#### Training.
In summary, when updating the discriminators, the goal is to maximize the discriminator losses on generated images and minimize the losses on real nodule images. When updating the detector, the goal is to minimize the detection losses for both real and generated nodule images. When updating the generator, the goal is to: (1) minimize the discriminator losses on generated images; (2) minimize detection loss on real object images, (3) maximize detection loss on generated images.
We use a history of synthetic images [@shrivastava2017learning], and for faster convergence we pretrain the discriminator-generator pair and the detector separately and then train them jointly. When we have a labelled image without bounding box annotations, we still update the discriminator $DIS_{globalX}$ to improve global realism.
{width="1\linewidth"}
{width="1\linewidth"}
$T_{IOBB}$ 0.1 0.25 0.5 0.75 Avg
------------------------------------------------ ---------- ---------- ---------- ---------- ----------
Wang et al. [@wang2017chestx] 0.15 0.05 0.00 0.00 0.04
Zhe et al. [@li2017thoracic] **0.40** 0.25 0.11 0.07 0.18
RetinaNet: real 0.15 0.15 0.08 0.08 0.09
RetinaNet: real + syn from CycleGAN + BboxLoss 0.31 0.31 0.00 0.00 0.12
RetinaNet: real + syn from DetectorGAN 0.31 **0.31** **0.23** **0.23** **0.22**
--------------------- -------- -------- -- ---------- ---------- -- -------- -------
Object Whole Object Whole Object Whole
CycleGAN + BboxLoss 20 4 1.31 1.18 0.68 0.53
DetectorGAN **80** **96** **2.69** **3.88** 0.85 0.73
--------------------- -------- -------- -- ---------- ---------- -- -------- -------
{width="1\linewidth"}
Experiments
===========
In this section we demonstrate the effectiveness of DetectorGAN on two tasks: nodule detection task with the NIH Chest X-ray dataset and pedestrian detection with the Cityscapes dataset. We obtain significant improvements over baselines and achieve state of the art results on the nodule detection task.
Disease Localization
--------------------
### Dataset
We use the NIH Chest X-ray dataset [@wang2017chestx] and focus on the nodule detection task. The NIH Chest X-ray dataset contains 112,120 X-ray images – 60,412 clean images and 51,708 disease images, 880 of which have bounding boxes. For the nodule class, there are 6,323 nodule images, 78 of which have bounding boxes.
#### Improved and Extended Annotations.
The bounding box annotation for this dataset is however not satisfying due to the following issues: (1) In the original paper and previous work [@wang2017chestx; @li2017thoracic], there is no standard train/test/validation split. (2) The bounding box annotations are not complete; that is, for each image there is only at most one bounding box for each class annotated, while there are actually many nodules present in the image. (3) Even with a standard train/test/validation split, the test and validation sets are too small to obtain stable and meaningful results.
To address these problems, we make the following efforts to make the disease detection task more standard and easy to conduct research on: (1) Generating a no-patient-overlap train/test/validation split with 0.7/0.2/0.1 portion of the data, yielding 57/13/9 images with 57/13/9 object instances. (2) Asking radiologists to re-annotate the current validation and test images using additional images from labeled images in the test/validation sets. These efforts result in 36 images and 80 images in validation and test sets accordingly, with 159 and 309 object instances. These splits and extended annotations will be published online to facilitate future research into this topic. We did not re-annotate or expand the training set as we want to demonstrate the effectiveness of the proposed method in learning small-data object detection tasks.
We refer to the 9/13 validation/test settings as “old annotations” and the 36/80 validation/test settings as the “new annotations”. We obtain the detection AP on the “new annotations” and localization accuracy on the “old annotations” for fair comparison with previously published results.
#### Baselines and Previous Work.
The baselines are: training with only real images, with additional synthetic images generated from CycleGAN and BboxLoss, and with additional synthetic images generated from ACGAN-like losses. The ACGAN-like losses refers to that in addition to discriminator losses, we also minimize the detection loss on synthetic images, similar to what ACGAN does for a classifier. We compare these methods on the new high quality annotations. In addition, we compare to two previously published best-performing works [@wang2017chestx; @li2017thoracic] using their evaluation split and their annotations (the “old annotations”).
#### Evaluation Metrics.
We use the standard object detection metric, average precision (AP), as the evaluation measure for the detection task. For comparisons to previous work, we also use their metric: localization accuracy, which is defined as the percentage of images that obtain correct predictions. An image is considered having correct predictions if the intersection over union ($IOU$) ratio between the predicted regions (can be non-rectangle) and the ground truth box is above threshold $T_{IOU}$. Another metric that is used by these works is to replace the $IOU$ with intersection over bounding boxes $IOBB$. However, we encourage researchers to use the proposed new annotations and evaluation metric in the future for standard comparisons.
### Quantitative Comparison
#### New Annotation with Average Precision.
In Table \[tlb:ap\], we compare the results of using only real data, using synthetic data from the proposed method as well as from other baseline GAN models. We observe that DetectorGAN significantly improves the average precision. Compared to training on real data only, the AP nearly doubles from 0.124 to 0.236, and recall over triples from 0.184 to 0.649. Compared to ACGAN-like losses and CycleGAN + BboxLoss, we obtain relatively 50% and 20% improvement.
We notice that ACGAN-like losses performs more poorly than using discriminator losses only, even though it has an additional loss to improve the detection performance on synthetic images. One explanation is that the generator and the detector learn only to detect synthetic objects, which is different from the goal of detecting real objects, leading to poor performance.
To further demonstrate the benefits of using the unrolling step to bridge the gap between the generator and the detection performance on real images, we also experiment with a ‘DetectorGAN - unrolling’ network without unrolling. We observe a significant boost for adding the unrolling step, from 0.203 to 0.236 AP.
#### Old Annotation with Localization Accuracy.
For comparison with previous work, we evaluate detection results using the localization accuracy metric with different $IOU$ and $IOBB$ thresholds. Results are shown in Table \[tlb:acc\_u\] and Table \[tlb:acc\_bb\]. We significantly outperform competing methods by relative 50% and 22%.
### Qualitative Analysis
#### Generated Image Quality.
We show DetectorGAN’s generated images, along with CycleGAN-generated images in Fig. \[fig:img\_quality\]. We observe that images are much better in terms of realism and blend-in.
#### Detected Nodules.
We show that the detector helps to detect undetectable nodules in Fig. \[fig:nodule\_detected\]. We observe that every nodule captured by the baseline (trained on real images only) is also captured by the model trained using synthetic images. Meanwhile, adding synthetic images helps capture more nodules that baseline cannot capture. Moreover, the box locations are generally more accurate.
### User Study
We also conduct user study with a radiologist to evaluate the quality of the generated images. We ask the radiologists to rate the realism of the inserted nodule and the global image on a Likert scale (scale 1–5, with 5 indicating highest quality). As shown in Table \[tab:userstudy\], the images from DetectorGAN are better than those from CycleGAN + BboxLoss in 96% of cases, with generated objects (nodules) better in 80% of cases. Moreover, the average Likert scores are significantly higher: 2.69 vs 1.31 for the objects, and 3.88 vs 1.18 for the whole image, demonstrating the benefits of our method.
Data Real +DetectorGAN +PS-GAN +pix2pix
------ ------- -------------- --------- ----------
AP 0.593 **0.613** 0.602 0.574
: Pedestrian detection AP trained with real data, synthetic data generated by PS-GAN, pix2pix and DetectorGAN.
\[tlb:ped\_num\]
Pedestrian Detection
--------------------
As a demonstration of the applicability of DetectorGAN to other datasets and problems, we apply it to pedestrian detection with a different base architecture. We follow PS-GAN [@ouyang2018pedestrian] to synthesize images with pedestrians inserted and improve pedestrian detection. We demonstrate a quantitative and qualitative improvement in the generated images by adding the detector into the loop.
#### Dataset.
We use the Cityscapes dataset, which contains 5,000 urban scene images with high-quality annotations. We follow the instructions in the PS-GAN paper to filter images with small or occluded pedestrians obtain about 2,000 images with about 9,000 labeled instances.
#### Baseline and Architecture.
We use PS-GAN[@ouyang2018pedestrian] as the backbone architecture and add the detector into the model. The PS-GAN uses the standard pix2pix framework with local discriminators. This also shows that the DetectorGAN idea is versatile — it can be integrated with different GAN models. We fine-tune the model from the pretrained PS-GAN model.
#### Quantitative Results.
Table \[tlb:ped\_num\] shows that we improve the detection performance for pedestrian detection as well. We observe that DetectorGAN further improves the performance over PS-GAN.
All models here are trained using the same setting. The real-images-only baseline performance is slightly different from what is reported in the PS-GAN paper because we do not have access to the exact details of the detector setting used in the PS-GAN paper.
#### Qualitative Results.
Qualitative results are shown in Fig. \[fig:ped\_img\]. We observe that DetectorGAN can generate qualitatively better images with less artifacts.
Conclusion
==========
In this work we explored the object detection problem in the small data regime from a generative modeling perspective by learning to generate new images with associated bounding boxes. We have shown that simply training an existing generative model does not yield satisfactory performance due to it optimizing for image realism instead of object detection accuracy. To this end we developed a new model with a novel unrolling step that jointly optimizes a generative model and a detector such that the generated images improve the performance of the detector. We show that this method significantly outperforms the state of the art on two challenging datasets.
[^1]: This work was conducted when Lanlan Liu was an intern at Google.
[^2]: Corresponding author.
|
---
abstract: 'In the quantum version of prisoners’ dilemma, each prisoner is equipped with a single qubit that the interrogator can entangle. We enlarge the available Hilbert space by introducing a third qubit that the interrogator can entangle with the other two. We discuss an enhanced interrogation technique based on tripartite entanglement and analyze Nash equilibria. We show that for tripartite entanglement approaching a W-state, there exist Nash equilibria that coincide with the Pareto optimal choice where both prisoners cooperate. Upon continuous variation between a W-state and a pure bipartite entangled state, the game is shown to have a surprisingly rich structure. The role of bipartite and tripartite entanglement is explored to explain that structure.'
author:
- George Siopsis
- Radhakrishnan Balu
- Neal Solmeyer
title: 'Quantum Prisoners’ Dilemma under Enhanced Interrogation'
---
Quantum games as a field received a lot of attention from the early works of Meyer [@Meyer1999], and has grown steadily ever since. Connections exist between quantum games and various other fields, such as Bell non-locality [@Brunner2013] and quantum logic [@Piotrowski2003], to name a few. Various aspects of quantum games, including the role of entanglement and multiple player extensions, explored by different authors can be found in references [@Flitney2005; @Iqbal2002; @Hayden2002]. A solution to the quantum prisoners’ dilemma in which players have a Nash equilibrium that is Pareto optimal ignited interest in quantum games [@Eisert1999]. However, this initial formulation drew criticism because it dramatically restricted the strategy space of the players and did not persist under maximal entanglement if arbitrary quantum strategies were allowed [@Benjamin2001].
In this work, we enlarge the Hilbert space in a minimal way in order to arrive at a Nash equilibrium (NE) that is Pareto-optimal with maximal entanglement. In addition to the two player qubits, we consider a resource qubit that the referee, or interrogator, has access to. In the three-qubit Hilbert space, one can introduce tripartite entanglement which will lead to a much richer Nash equilibrium structure. Interestingly, for tripartite entanglement close to maximum (approaching a W-state), we obtain Nash equilibria that coincide with Pareto optimal points. The bipartite entanglement between the players’ qubits partially explains the structure. In addition, it is shown that there is no NE for a GHZ state, which has a fundamentally different type of entanglement [@bibdur].
We briefly review the standard formulation of the prisoners’ dilemma. It is a game of two players, Alice and Bob, who must decide independently whether they defect (strategy $D$) or cooperate (strategy $C$). Each player receives a payoff and chooses a strategy that maximizes it. For concreteness, we shall use Table \[table:1\] to determine the payoff [@bibu].
C D
--- ---------------------- ----------------------
C (11,9) (1,10)
D (10,1) (6,6)
: \[table:1\]Prisoners’ payoff matrix. In each pair, the first (second) entry is payoff for Alice (Bob).
The best strategy is $CC$ (both prisoners cooperate), which is the Pareto optimal choice. However, by making unilateral decisions, they choose $DD$, which is the Nash equilibrium.
The classical game outlined above has been quantized as follows. Suppose that Alice and Bob are in possession of one qubit each with the state $|0\rangle$ ($|1\rangle$) corresponding to the choice $C$ ($D$). Thus, the set of four classical possibilities $\{ CC, CD, DC, DD \}$ corresponds to the basis $\{ |00\rangle, |01\rangle, |10\rangle, |11\rangle \}$. A general quantum state is a linear superposition of the four basis vectors, \[eqpsi\]|= \_[x=0]{}\^3 a\_x |xwith $x$ written in binary notation. The payoff for Alice is then \[eqpay\] \$\_A = \_[x=0]{}\^[3]{} \$\_[A,x]{} |a\_x|\^2 where $\$_{A,x}$ are the classical payoff values for Alice given in Table \[table:1\], and similarly for Bob.
![\[fig:1\] Payoffs for $A$ (upper line) and $B$ (lower line) *vs*. $\delta$ for $\gamma = \pi/2$. The strategies correspond to (C,C), or $U_A=U_B = I$.](maximalEntanglement.jpg)
Alice and Bob start with qubits in the state $|0\rangle$. The interrogator then entangles them by applying the unitary \[eq1\] J() = e\^[i \_X\_X]{} = + i \_X\_X Then the prisoners choose strategies represented by applying unitaries $U_A$ and $U_B$, respectively, to their qubits. We adopt the use of quaternionic strategy choices [@Landsburg], i.e., the Pauli matrices and the identity matrix, so that there are 4 strategy choices, $\{I, \sigma_X, \sigma_Y, \sigma_Z\}$. This choice of a strategy set ensures that the analysis of the behavior of the game captures the behavior as if the players had complete freedom to play any quantum strategy.
Finally, the interrogator reverses the setup, by applying the unitary $J^\dagger (\gamma) = J(-\gamma)$ to the prisoners’ qubits. The circuit is shown below: @C=1em @R=.7em [ & & & &&\
& & & &&\
]{} \[eq:circ0\]The final state is of the form for which payoffs $\$_{A,B}$ are computed using . This results in new Nash equilibria which are distinct from the classical one. They reduce to the classical case if $\gamma = 0$ (no entanglement). Nash equilibria exist for degree of entanglement $\gamma$ below a certain value ($\gamma < 1.15$). In the case of maximal entanglement, no Nash equilibria exist [@bibu].
Our enlarged Hilbert space consists of a resource qubit $Q$, as well qubits $A$ and $B$ that belong to the two prisoners, respectively. The prisoners’ payoffs are given in Table \[table:1\]. Each qubit is in the state $|0\rangle$ initially. The interrogator starts by entangling $A$ and $B$ using the unitary , as before. Subsequently, he uses $A$ as control to apply the rotation $e^{i\delta \sigma_X}$ on qubit $Q$, with a control parameter $\delta$ that can vary. The state of the system $QAB$ becomes |(,)= |000+ i (|0+ i|1)|11Then he uses $B$ and $Q$ as controls to flip $A$. The state becomes \[eq6\] |(,)= |000+ i |011-|101Notice that, if $\tan\frac{\gamma}{2} = \sqrt{2}$, and $\delta = \frac{\pi}{4}$, it is maximally entangled. We obtain \[eq6W\] |= ( |000+ i |011- |101) This state is the tripartite $W$-state [@bibdur], \[eqW\] |W= ( |001+ |010+ |100) up to single-qubit transformations (application of $\sigma_X$ on the third qubit (prisoner B), and phase changes).
If $\delta =0$, this is equivalent to the setup considered above.
Then the prisoners apply strategies $U_A$ and $U_{B}$, respectively. After they are done strategizing, the referee reverses the setup of the enhanced interrogation.
The circuit is shown below: @C=.2em @R=.7em [ &&& & && & &&&&\
&& & && && & &&&\
&& && & & && &&&\
]{} \[eq:circ1\]For $\gamma > 1.15$, there are no NE with the circuit . In fig. \[fig:1\], we show the results for the modified circuit, for maximal entanglement, i.e. $\gamma = \frac{\pi}{2}$. The payoff for both players at the NE is plotted as a function of the control parameter, $\delta$. For $\delta < 0.41$, there are no NE. But for $\delta > 0.56$, they exist. In fact, they coincide with the Pareto optimal choice where both prisoners confess! This equilibrium is formed by both players playing the identity matrix, i.e., $I \otimes I$ .
For $\delta \in (0.41,0.56)$, a different NE exists that is not the Pareto optimal choice. This equilibrium is formed by players $A$ and $B$ playing the strategy choices $\sigma_Z \otimes \sigma_X$.
In fig. \[fig:2\], we plot the results as a function of both bipartite entanglement, and the control parameter $\delta$. For $\delta = 0$, the curve reproduces the results that have been previously found for the prisioners’ dilemma with partial entanglement. As $\delta$ increases, the payoff begins to lower generally, and the region near maximal entanglement where there is no NE shrinks. Where there are multiple NE, we plot the payoff for the NE with the highest payoff. Qualitatively, there is no NE when the control and entanglement parameters fall within an ellipse centered on $\delta = 0, \gamma = \pi/2$ with radii $r_\gamma = 0.42$ and $r_\delta = 0.41$. In fact, this is the only region of the parameter space that has no NE.
In addition, the Pareto optimal solutions are shown to form a plateau centered on $\delta = \pi/2, \gamma = \pi/2$. Numerically, fitting the data to an ellipse, we find that when the control and entanglement parameters fall within an ellipse centered on $\delta = \pi/2, \gamma = \pi/2$, with radii $r_\gamma = 0.90$ and $r_\delta = 1.00$, the payoff at the NE is equal to the Pareto-optimal choices. Outside of that region, there is a discontinuous drop of the payoff at NE that continuously deforms into the NE that exists at $\delta = 0$ in the standard quantum prisoners’ dilemma with partial entanglement.
The various types of equilibria that occur show a surprisingly complicated phase-diagram-like structure. This structure is shown in fig. \[fig:3\]. The strategy choices of the NE in the Pareto optimal plateau are $I \otimes I$ (type $F$ in fig. \[fig:3\]). There are two types of equilibria that exist with partial entanglement: $\sigma_Z \otimes \sigma_X$ (type $G$) and $\sigma_X \otimes \sigma_Z$ (type $H$). Type $B$ occurs everywhere except for two elliptical regions: a region centered on $\gamma = \pi/2, \delta = \pi/2$ with radii $r_\gamma = 0.75$ and $r_\delta = 0.60$, and a region centered on $\gamma = \pi/2, \delta = 0$ with radii $r_\gamma = 0.42$ and $r_\delta = 0.41$. It should be noted that the so-called elliptical regions, in reality have a slightly different curvature than ellipses, but their theoretical and analytic description remains elusive.
The type $H$ only occurs when $\gamma$ is less than a curved boundary value near $\gamma \sim \frac{5 \pi}{16}$.There are regions which only have types $F$ or $G$, and also regions which overlap with types $\{F,G \}$, or $\{G,H\}$, or $\{F,G,H\}$. Type $G$ and $H$ equilibria give the same payoff for the players, while type $F$ is different. Along the line of $\gamma = 0$, there are two additional equilibria not shown on the graph. They are given by $\sigma_Z \otimes \sigma_Z$ and $\sigma_X \otimes \sigma_X$, and have the same payoff as types $G$ and $H$.
![\[fig:2\] Payoffs for $A$ are shown for all values of $\delta$ and $\gamma$. The payoffs for $B$ are qualitatively the same, with different values, owing to the asymmetry of the payoff matrix.](partialEntanglement.jpg)
![\[fig:3\] The NE in different regions of the plot. The various NE span different parameter regions of the plot, there is an ellipse-like region centered around $\gamma = \pi/2, \delta = 0$ that has no NE, and an elipse-like region centered around $\gamma = \pi/2, \delta = \pi/2$ which has a Pareto optimal NE.](types.jpg)
In order to gain insight into the role of entanglement in the NE, the three bipartite entanglements are computed for the qubits just before the players apply their strategy choices. A partial trace is taken of one qubit to obtain the density matrix of the other two, then the concurrence of the resulting density matrix is computed. The three values of concurrence, i.e., entanglement between A and B, between Q and B, and between Q and A, are computed for all values of $\delta$ and $\gamma$, and the results are shown in fig. \[fig:conc\].
The red surface of fig. \[fig:conc\] represents the bipartite entanglement between A and B showing that it does not remain maximal for $\gamma = \frac{\pi}{2}$ as $\delta$ increases. The region with no NE follows a contour of 0.91 concurrence between qubits A and B suggesting that the absence of a NE can be explained by the bipartite entanglement between A and B being larger than a threshold value, as is the case in the original quantum prisoners’ dilemma.
![\[fig:5\] Payoffs for $A$ (upper line) and $B$ (lower line) *vs*. $t$, where $\gamma = t \tan^{-1} \sqrt{2} + (1-t) \frac{\pi}{2}$, $\delta = t \frac{\pi}{4}$, and $\eta = (1-t) \frac{\pi}{2}$, interpolating between the GHZ-type state ($t=0$) and the $W$-type state ($t=1$).](WGHZ.jpg)
The region where there is no NE for type G in fig. \[fig:3\] and the plateau of Pareto optimal solutions appear as if they might follow contours of the concurrence between Q and B, yet upon close analysis, this is not found to be the case, suggesting that their explanation requires an analysis of more than just the bipartite entanglement.
As noted above, for certain values of $\gamma$ and $\delta$, NE coincide with the Pareto optimal choice. This range of parameters includes the maximally entangled state (with $\tan \frac{\gamma}{2} = \sqrt{2}$, $\delta = \frac{\pi}{4}$), which can be transformed to the tripartite $W$-state with single-qubit transformations. Then the question arises whether the other independent tripartite maximally entangled state (GHZ state), | = ( |000+ |111) has a similar effect on NE. To answer this question, we draw the circuit:
@C=.2em @R=.7em [ &&& & & && & & &&&&\
&& & && & & && & &&&\
&& && && && && &&&\
]{} \[eq:circW\]
In the circuit above, there is an additional step the interrogator performs before $A$ and $B$ get a chance to play. It depends on an additional parameter $\eta$. The interrogator uses qubit $A$ as control to apply the rotation $e^{i\eta \sigma_X}$ on the resource qubit $Q$. If $\eta = 0$, this reduces to the previous setup. In general, the state is transformed to \[eq6a\] && |000+ i |011\
&&+ i |111-|101By varying the parameters, we can interpolate between the $W$-type state (with $\tan\frac{\gamma}{2} = \sqrt{2}, \delta = \frac{\pi}{4}, \eta = 0$), and the GHZ-type state (with $\gamma = \frac{\pi}{2}, \delta =0, \eta = \frac{\pi}{2}$), \[eq6GHZ\] |’ = ( |000+ i |111) Notice that with the choice of parameters that yield the GHZ-type state , the circuit reduces to @C=.5em @R=.7em [ &&& && &&&&\
&& & & & & &&&\
&& && && &&&\
]{} \[eq:circGHZ\]Therefore, the state of the resource qubit $Q$ does not affect the prisoners. Without a resource qubit, it is known that the maximally entangled state does not have NE. Consequently, the GHZ-type state yields no NE. An interpolation between the $W$-type state and the GHZ-type state is shown in fig. \[fig:5\]. For $t = 0$, we obtain a GHZ state which has no NE, while for $t=1$, we have a W-state with a NE that coincides with the Pareto optimal choice. As we increase $t$ from $t=0$, we move away from the GHZ state and we see no NE up to $t=0.4$. There is no smooth transition of NE as we increase $t$ further. Near $t=1$ (W-states), we recover the NE found earlier (fig. \[fig:1\]).
The quantum prisoners’ dilemma has shed light on many aspects of quantum games including the existence of new NE not readily available in the classical games, and the absence of NE for maximally entangled inputs. The addition of entanglement to a third qubit controlled by the referee is seen to result in a rich structure that can be either beneficial, or detrimental to the players in terms of their payoff. If the bipartite entanglement between the players is near maximal, there is no NE, but by preparing the player’s qubits near a W-state, the referee can steer the game so that it has a Pareto optimal NE. Interestingly, not all tripartite entangled states are equivalent, as the GHZ state gives no NE, unlike the W-state which yields NE that coincide with the Pareto optimal choice.
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S. C. Benjamin and P. M. Hayden, *“Comment on ‘Quantum Games and Quantum Strategies’,"* Phys. Rev. Lett. **87**, 069801 (2001).
W. Dür, G. Vidal, and J. I. Cirac, *“Three qubits can be entangled in two inequivalent ways,"* Phys. Rev. A **62**, 062314 (2000).
N. Solmeyer, R. Dixon, and R. Balu, *“Characterizing the Nash equilibria of a three-player Bayesian quantum game,"* Quantum Inf. Process. **16**, 146 (2017).
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|
---
abstract: 'Deep learning usually achieves the best results with complete supervision. In the case of semantic segmentation, this means that large amounts of pixelwise annotations are required to learn accurate models. In this paper, we show that we can obtain state-of-the-art results using a semi-supervised approach, specifically a self-training paradigm. We first train a teacher model on labeled data, and then generate pseudo labels on a large set of unlabeled data. Our robust training framework can digest human-annotated and pseudo labels jointly and achieve top performances on Cityscapes, CamVid and KITTI datasets while requiring significantly less supervision. We also demonstrate the effectiveness of self-training on a challenging cross-domain generalization task, outperforming conventional finetuning method by a large margin. Lastly, to alleviate the computational burden caused by the large amount of pseudo labels, we propose a fast training schedule to accelerate the training of segmentation models by up to 2x without performance degradation.'
author:
- |
Yi Zhu, Zhongyue Zhang, Chongruo Wu[^1], Zhi Zhang, Tong He,\
Hang Zhang, R. Manmatha, Mu Li, Alexander Smola
bibliography:
- 'egbib.bib'
title: |
Improving Semantic Segmentation\
via Self-Training
---
Introduction {#sec:introduction}
============
Semantic segmentation is a fundamental computer vision task whose goal is to predict semantic labels for each pixel. Great progress has been made in the last few years in part thanks to the collection of large and rich datasets with high quality human-annotated labels. However, pixel-by-pixel annotation is prohibitively expensive, e.g., labeling all pixels in one Cityscapes image takes more than an hour [@Cordts2016Cityscapes]. Thus until now, we only have semantic segmentation datasets consisting of thousands or tens of thousands annotated images [@Cordts2016Cityscapes; @Neuhold2017mapillaryVista; @BDD100K], which are orders of magnitude smaller than datasets in other domains [@Sun2017JFT300M; @Mahajan2018WeakSupLimit].
Given the fact that learning with annotated samples alone is neither scalable nor generalizable, there is a surge of interest in using unlabeled data by semi-supervised learning. [@Souly2017SemiGAN; @Hung2018Adversarial] adopt the concept of adversarial learning to improve a segmentation model. Self-training [@Luc2017futureSeg; @Zou2018DAClassBalance; @Li_2019_bidirection; @Zou_2019_CRST; @Lian_2019_Pyramid] often uses a teacher model to generate extra annotations from unlabeled images. Recently, [@Mustikovela2016labelPropagation; @Budvytis2017augmentation] use temporal consistency constraints to propagate ground truth labels to unlabeled video frames. However, these models often beat a competing baseline in their own settings but have difficulty achieving state-of-the-art semantic segmentation performance on widely adopted benchmark datasets.
![Overview of our self-training framework. (a) Train a teacher model on labeled data. (b) Generate pseudo labels on a large set of unlabeled data. (c) Train a student model using the combination of both real and pseudo labels[]{data-label="fig:self_training"}](overview.png){width="1.0\linewidth"}
In this paper, we would like to revisit semi-supervised learning in semantic segmentation, particularly on driving scene segmentation. Driving scene segmentation suffers from insufficient training labels but it has access to unlimited unlabeled images collected by running vehicles. Semi-supervised learning is thus perfectly suited for such a task. In addition, motivated by an open problem in semantic segmentation, i.e., a segmentation model trained on one driving dataset may not generalize to another due to domain gap, we also evaluate our model on a challenging cross-domain generalization task (e.g., from Cityscapes to Mapillary) with a small number of annotations in the target domain.
The method we used is based on the self-training framework [@Xie2019NoisyStudent], which is illustrated in Fig. \[fig:self\_training\]. We first train a *teacher model* on a small set of labeled data, which we refer as “real labels”. We then generate “pseudo labels” by using this teacher model to predict on a large set of unlabeled data. In the end, a *student model* is trained using the combination of both real labels and pseudo labels. If the noise of the pseudo labels can be properly handled, the student model often outperforms the teacher model. And in particular, when the pseudo labels come from a different domain, the student model should generalize better than the teacher model on this new domain.
As we can generate unlimited pseudo labels, it significantly increases the training computational cost. A key reason segmentation models are hard to train is because of the high resolution input images. We propose a schedule that reduces the image resolution during training from time to time. A small image resolution reduces both the computation cost per image and also allows a large batch for a better system efficiency. We carefully design the schedule so that it will not impact the final performance given a fixed number of images processed.
Extensive experimental results demonstrate the effectiveness of our approach on three driving scene segmentation datasets [@Cordts2016Cityscapes; @Brostow2008camvid; @Geiger2012CVPR]. Take Cityscapes [@Cordts2016Cityscapes] as an example, we achieve an mIoU of $82.7\%$ on its test set using only fine annotations. We outperform all prior arts that use the same training data, and several methods [@Chen2018SearchSegDPC; @Liu2019autodeeplab; @Bulo2018inplaceABN; @Zhu2019VPLR] trained using additional labeled data.
Our contributions are summarized below:
- We introduce a self-training framework for semantic segmentation, which is able to properly handle the noisy pixel-level pseudo labels.
- We propose a training schedule that adjusts the image resolution during the training to speedup the performance without losing model accuracy.
- We extensively evaluate the proposed method to show that it achieves state-of-the-art performance on Cityscapes, CamVid and KITTI datasets, while using significantly less annotations.
- We demonstrate the effectiveness of self-training on a challenging cross-domain generalization task with different semantic categories.
Related Work {#sec:related_work}
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**Semantic segmentation.** Starting from the seminal work of FCN [@Long2015FCN], semantic segmentation has made significant progress [@Zhao2017pspnet; @Chen2018deeplabv3plus; @Bulo2018inplaceABN; @zhangli_dgcn; @Cheng2019DeepLabPano] with respect to model development. Recent work has focused more on exploiting object context by using attention [@Yuan2019OCNetv2; @Fu2018DANet; @Zhang_encnet_CVPR18; @Fu2019ACNet; @zhao2018psanet; @Zhang2019ACFNet], designing more efficient networks [@Huang2019CCNet; @Poudel2019FastSCNN; @Yu_BiSeNet_2018eccv; @Li2019DFANet; @wu2019fastfcn] and performing neural architecture search [@Liu2019autodeeplab; @Chen2018SearchSegDPC; @Nekrasov_2019_fastNASSeg; @Zhang_2019_CNAS], etc.
Our work is different as we introduce a self-training framework for semantic segmentation to explore the benefit of using unlabeled data. We demonstrate that our method is orthogonal to model development. We can improve a number of widely adopted models irrespective of the network architectures.
**Semi-supervised learning.** Recently we have witnessed the power of semi-supervised learning in the image classification domain. By using a large amount of unlabeled data, [@Yalniz2019BillionSemi; @Xie2019NoisyStudent] are able to achieve state-of-the-art performance on ImageNet. There are also numerous papers on semi-supervised semantic segmentation, such as using adversarial learning [@Souly2017SemiGAN; @Hung2018Adversarial], self-training [@Luc2017futureSeg; @Zou2018DAClassBalance; @Li_2019_bidirection; @Zou_2019_CRST; @Lian_2019_Pyramid], consistency regularization [@Mittal2019HighLow; @French2019Perturbations], knowledge distillation [@Xie2018TeacherStudent; @Liu2019KDSegmentation], video label propagation [@Mustikovela2016labelPropagation; @Budvytis2017augmentation], etc. However, these models often beat a competing baseline in their own settings but have difficulty achieving state-of-the-art semantic segmentation performance on widely adopted benchmark datasets.
In this paper, we revisit semi-supervised learning for semantic segmentation. Our work is different from the previous self-training literature [@Luc2017futureSeg; @Zou2018DAClassBalance; @Li_2019_bidirection; @Zou_2019_CRST; @Lian_2019_Pyramid] because our problem settings are different, and no direct comparisons can be made. Besides, we adopt the teacher-student paradigm without using additional regularization as in [@Zou2018DAClassBalance; @Zou_2019_CRST; @Lian_2019_Pyramid]. Our work also differs from knowledge distillation [@Xie2018TeacherStudent; @Liu2019KDSegmentation] since our goal is to get state-of-the-art performance by using unlabeled data, not distilling a light-weight student model for fast semantic segmentation.
**Domain adaptation.** Our cross-domain generalization task is related to unsupervised domain adaptation (UDA). The goal of UDA is to learn a domain invariant feature representation to address the domain gap/shift problem. Numerous approaches have been proposed in this field, such as the dominating adversarial learning pipeline [@Hoffman2018CyCADA; @Tsai_adaptseg_2018; @Vu_2019_CVPR; @Luo_2019_SAIB; @zhang2019category; @Yang2020LabelRecons], conservative loss [@Zhu_2018_ECCV], texture/structure invariant [@Chen_2019_geometric; @chang2020texture], consistency regularization [@Chen_2019_CrDoCo; @Lee_2019_SWD; @Yue_2019_DRPC], etc.
Our work differs from conventional UDA in multiple ways. First, our settings are different. Most UDA literature adopt a synthetic-to-real setting (e.g., GTA5/SYNTHIA to Cityscapes), while we consider a real-to-real setting (e.g., Cityscapes to Mapillary/BDD100K). Second, our framework is more generic and simpler. We do not use adversarial learning or specially designed loss functions to reduce the domain gap, and yet achieve decent generalization performance. Third, we show promising results on a challenging but practical task where the target domain has new classes and we have a few labeled samples. Most UDA literature do not consider this scenario. They often assume there is no labeled data in the target domain and only report performance when the source and target domain have the same number of classes.
Methodology {#sec:methodology}
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In this section, we introduce our self-training method for semantic segmentation. We first describe the employed teacher-student framework in Sec. \[subsec:self\_training\]. With the large amount of pseudo labels, we propose a centroid data sampling technique in Sec. \[subsec:centroid\_sampling\] to combat the class imbalance problem and the noisy label problem. Following this, we design a fast training schedule in Sec. \[subsec:fast\_training\] to handle the large expanded training set. This approach speeds up the model training by up to 2x without performance degradation. Finally, we demonstrate in Sec. \[subsec:cross\_domain\] that our self-training method can also greatly improve the performance on a challenging cross-domain generalization task with only a few labels.
Self-Training using Unlabeled Data {#subsec:self_training}
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While semi-supervised learning has been widely studied for semantic segmentation [@Luc2017futureSeg; @Zou2018DAClassBalance; @Mittal2019HighLow; @French2019Perturbations; @Xie2018TeacherStudent; @Liu2019KDSegmentation], it is difficult to beat human supervised counterparts [@Cheng2019DeepLabPano; @Zhu2019VPLR; @Li2019GALDNet; @Yuan2019OCNetv2]. In this section, we introduce a teacher-student framework to perform self-training on semantic segmentation. Our goal is to use a small set of labeled data and a large quantity of unlabeled data to improve both the accuracy and robustness of semantic segmentation models. In this way, the human labeling effort can be largely reduced.
We present an overview of our self-training framework in Fig. \[fig:self\_training\]. Given a small quantity of labeled training samples (an image and a human-annotated segmentation mask), we first train a teacher model with standard cross-entropy loss. We adopt a number of training techniques specially designed for semantic segmentation, to make the teacher model as good as possible. We will discuss the training details in the following sections.
We then use the teacher model to generate pseudo labels on a large number of unlabeled images. The better the teacher model is, the higher the quality of the generated pseudo labels can be. As can be seen in Fig. \[fig:self\_training\], our teacher-generated pseudo labels have a good quality that are close to human annotations. More visualizations of pseudo labels can be found in the Appendix.
Finally, we train a student model using both human-annotated labels (real labels) and teacher-generated labels (pseudo labels). Note that the teacher-student framework is widely studied in the literature of distillation, however, it has been reported in [@He2019BagTricks] to have the limitation that teacher and student are expected to have a similar architecture to work well. We would like to point out that with our approach, the student model may have a different network architecture which does not have to be the same as that of the teacher’s. In our experiments, we use a single teacher but train several student models with various backbones and network architectures. Our self-training framework consistently improves all of them, which demonstrate its generalizability.
Fighting Class Imbalance with Centroid Sampling {#subsec:centroid_sampling}
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The biggest challenge in self-training is how to deal with the noise in pseudo labels [@Yalniz2019BillionSemi; @Xie2019NoisyStudent; @Zou2018DAClassBalance; @Li_2019_bidirection; @Zou_2019_CRST; @Lian_2019_Pyramid]. With the increased number of training samples, there is a high chance that noisy samples mislead and confuse the model during training. In addition, semantic segmentation is a dense prediction problem and each erroneous pixel prediction is a noisy sample. That explains why previous self-training papers [@Mustikovela2016labelPropagation; @Luc2017futureSeg] do not observe significant improvement even though they generate a massive amount of pseudo labels from unlabeled images.
There are several widely adopted approaches to control the usage of pseudo samples, such as (1) lowering the ratio of pseudo labels in each mini-batch (or in each training epoch); (2) selecting pseudo labels with high confidence; (3) setting lower weights in computing the loss for pseudo labels. However, these methods do not account for the problem of class imbalance. We argue that this problem will become more severe in the self-training paradigm because samples of each class are amplified by a biased teacher model with the increasing number of unlabeled training samples. For example, in the original Cityscapes training set, the “road” class has 360 times more pixels than the “motorcycle” class. In the expanded training set with pseudo labels, this ratio is on the scale of thousands. Hence, a better way to control the usage of pseudo labels is essential to make self-training work in the semantic segmentation domain.
We introduce a centroid sampling strategy similar to [@Bulo2018inplaceABN; @Zhu2019VPLR] but with several major differences. The idea is to make sure that our model can see instances from all classes in the expanded noisy set within each epoch, even for the underrepresented classes. To be specific, we first record the centroid of areas containing the class of interest before training. A centroid is the arithmetic mean position of all the points within an object. For example, the centroid of a car instance will be a point roughly in the middle of the car. Then during training, we can query training samples using such class-level information, i.e., crop an image patch around the centroid. This centroid sampling strategy has two advantages. First, we can train a semantic segmentation model without worrying about the problem of class imbalance, no matter how large the noisy set is. We can choose to uniformly query training mini-batches using the centroids so that the model can see instances from all classes. Second, it does not break the underlying data distribution because we only crop the image around the centroids. The ratio of pixel count among all classes will remain approximately the same.
We now illustrate the differences between our centroid sampling technique and [@Zhu2019VPLR]. First, [@Zhu2019VPLR] fixes the number of iterations within each epoch, which means the model can only see a small fraction of all the pseudo labels. This is not suitable for the self-training paradigm, because we can easily generate a huge number of pseudo labels. We relax this constraint to flexibly adjust the ratio between real and pseudo labels in each epoch. In this way, we can fully understand the effect brought by the pseudo labels, and determine the appropriate amount of pseudo labels to train a good student model. Second, [@Zhu2019VPLR] only selects underrepresented classes such as “fence”, “rider”, “train” to augment the original dataset. We instead use all the classes because we believe it is better to train the model following the real-world data distribution. It is not necessary to force class “road” and class “motorcycle” to have the same amount of pixels.
Fast Training Schedule for Large Expanded Dataset {#subsec:fast_training}
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Once we have the pseudo labels and a principled way to control the noisiness, it is time to kickoff training. However, semantic segmentation consumes lots of computing resources. Due to the large crop size (e.g., $800 \times 800$), we can only use a small batch size (2 or 4) depending on the network architecture to fit into the GPU memory. Hence, even training on a medium-scale dataset like Cityscapes with only 3K training samples using a high-end 8-GPU machine, takes days to finish. Now if we increase the size of dataset to 10 times more, the training will take weeks to complete, which leads to long research cycles. This is part of the reason why self-training hasn’t been investigated thoroughly in the field of semantic segmentation.
Given that the slowness is caused by using a small batch size, can we reduce the crop size during training to increase the batch size? Several researchers [@Li2019GALDNet; @Zhao2017pspnet; @Chen2018deeplabv3plus] have done this experiment to accelerate their model training, and they came to the conclusion that reducing crop size hurts the results. Thus, trading crop size for batch size is not worth it. We agree with the conclusion because semantic segmentation is a per-pixel dense prediction problem and a small crop size will lead to the problem of losing global context and detailed boundary information, which is of course harmful. However, what if we can design a training schedule that iterates between a small crop size and a larger crop size, so that the training time can be reduced without losing segmentation accuracy?
![Overview of our proposed fast training schedules. x-axis is the epoch number and y-axis is the crop size. See texts in Section \[subsec:fast\_training\] for more details[]{data-label="fig:fast_training"}](schedule_2.pdf)
In this work, inspired by the coarse-to-fine concept in computer vision, we design several training schedules to speed up the experiments. Our goal is to avoid the speed and accuracy trade-off, and achieve faster training without losing accuracy. As shown in Fig. \[fig:fast\_training\], we introduce 4 learning schedules, namely coarse2fine, fine2coarse, coarse2fine+ and fine2coarse+. To be specific, (1) coarse2fine means we first use small crop sizes such as 400 in the early epochs, then change to 480, 560, 640, 720 and eventually 800 for the rest of the training. Each crop size stays constant for several epochs, e.g., we change the crop size every 30 epochs. (2) fine2coarse means we first use a large crop size such as 800 in the early epochs, then change to 720, 640, 560, 480 and 400 as we progress to the end. (3) coarse2fine+ means we iterate the crop size every epoch to maximize the scale variation during model learning. For example, we use crop size 400 for epoch 0, 480 for epoch 1, 560 for epoch 2, and so on. And similarly, (4) fine2coarse+ will be the reverse process of (3) where we start with a larger (finer) crop. We will show the performance of each learning schedule in Section \[sec:experiments\], and demonstrate that the coarse2fine+ schedule is the best candidate. It is able to speed up the training by up to 2x with the same segmentation accuracy.
We would like to point out that our fast training schedule is a general technique. It is not only suitable for our self-training framework with large expanded training set, but also applicable to standard semantic segmentation training on large-scale datasets such as Mapillary [@Neuhold2017mapillaryVista] and BDD100K [@BDD100K].
Cross-Domain Generalization with New Categories {#subsec:cross_domain}
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Using the self-training framework, we can improve the accuracy of semantic segmentation on the current dataset by leveraging a large set of external unlabeled data. However, a segmentation model trained on one dataset may not generalize to another. For instance, two driving datasets collected in different locations may be significantly different in terms of traffic, lighting and viewpoint. On the other hand, in practice it is a meaningful task as people usually train model on one mature dataset with abundant labels, and expect to test on another new dataset with a small number of annotations. Note that the target dataset could have new categories which makes the task even more difficult, as the model needs to learn knowledge from new scenarios with just a few labels.
A conventional approach would be training a model on the source dataset, and finetuning it on the new dataset assuming the two datasets share similar distribution. However, this approach requires a large amount of annotations from the target dataset to achieve good performance. Here we introduce the improvement with our self-training framework for such a challenging setting.
We first train a good model on the source domain, using both real labels from the source domain and pseudo labels from the target domain. In this manner, our model learns the prior knowledge of the data distribution in the target domain, even though information about the new semantic categories is not provided. Then we finetune this model on the small set of labeled samples from the target domain, so that it can quickly adapt to the new domain. In our experiments, we have shown that our self-training framework can handle the problem of cross-domain generalization with new categories better than conventional finetuning approach. Even when there are just 10 annotations per category, our method is able to achieve decent segmentation accuracy.
Experiments {#sec:experiments}
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In this section, we describe the implementation details of our framework. Then we report the performance on three widely adopted driving scene segmentation datasets, Cityscapes [@Cordts2016Cityscapes], CamVid [@Brostow2008camvid] and KITTI [@Geiger2012CVPR]. We perform all the ablation studies on Cityscapes because it is the most benchmarked dataset. In the end, we evaluate our model on a cross-domain generalization task from Cityscapes to Mapillary. For all the datasets, we use the standard mean Intersection over Union (mIoU) metric to report segmentation accuracy.
Datasets {#subsec:datasets}
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**Cityscapes** contains 5K high quality annotated images, splitting into $2975$ training, $500$ validation, and $1525$ test images. The dataset defines $19$ semantic labels and a background class. There are also $20$K coarsely annotated images, but we ignore their labels in our self-training framework. **KITTI** has the same data format and metrics with Cityscapes, but with varying image resolution. The dataset consists of 200 training and 200 test images, without an official validation set. **CamVid** defines $32$ semantic labels, however, most literature only focuses on 11 of them. It includes 701 densely annotated images, splitting into 367 training, 101 validation and 233 test images. **Mapillary Vista** is a recent large-scale benchmark with global reach and includes more varieties. The dataset has 18K training, 2K validation and 5K test images. We use its research edition which contains 66 classes. We will refer to it as Mapillary in the paper.
Implementation Details {#subsec:implementation}
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We employ the SGD optimizer for all the experiments. We set the initial learning rate to $0.02$ for training from scratch and $0.002$ for finetuning. We use a polynomial learning rate policy [@Liu2017parsenet], where the initial learning rate is multiplied by $(1 - \frac{\text{epoch}}{\text{max}\_\text{epoch}})^\text{power}$ with a power of $0.9$. Momentum and weight decay are set to $0.9$ and $0.0001$ respectively. Synchronized batch normalization [@Zhao2017pspnet] is used with a default batch size of 16. Using our fast training schedule, the batch size can increase to $64$ when the crop size is smaller. The number of training epochs is set to $180$ for both Cityscapes and Mapillary, $80$ for CamVid and $50$ for KITTI. The crop size is set to $800$ for both Cityscapes and Mapillary, $640$ for CamVid and $368$ for KITTI due to different image resolutions. For data augmentation, we perform random spatial scaling (from 0.5 to 2.0), horizontal flipping, Gaussian blur and color jittering (0.1) during training. We adopt DeepLabV3+ [@Chen2018deeplabv3plus] as our network architecture, and use ResNeXt50 [@Xie2017ResNeXt] as the backbone for the ablation studies, and WideResNet38 [@Wu2016WideOrDeep] for the final test-submissions. We adopt the OHEM loss following [@Wu2016OHEMSeg; @Yuan2019OCNetv2]. For all the ablation experiments, we run the same training recipe five times and report the average mIoU.
Cityscapes {#subsec:cityscapes}
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We first perform ablation studies on the validation set of Cityscapes to justify our framework design and then report our performance on its test set.
**Self-training baseline** Here we establish the baseline for all of our experiments that follow. As shown in the top of Table \[table:baseline\], our baseline model (DeepLabV3+ with ResNeXt50 backbone using OHEM loss) trained on the 3K Cityscapes fine annotations has an mIoU of $78.1\%$ [@Zhu2019VPLR]. We adopt this model as the teacher to start self-training.
Next, we use this model to generate pseudo labels on the Cityscapes coarse images and Mapillary images. Note that the images in both the Cityscapes coarse dataset and Mapillary dataset are annotated but we ignore the labels and treat them as unlabeled data. Since both new datasets have 20K pseudo labels, we randomly pick half the training samples (1.5K) from the Cityscapes fine annotations, and another half (1.5K) from the centroids of generated pseudo labels for a fair comparison. It has been shown that longer training is beneficial for dense prediction tasks such as semantic segmentation because it can refine the boundaries [@Poudel2019FastSCNN; @Cheng2019DeepLabPano; @Li2019GALDNet]. In this way, the total number of training samples (3K) within an epoch remains the same with our baseline.
As shown in the bottom of Table \[table:baseline\], with the pseudo labels generated on the Cityscapes coarse images, it brings $0.9\%$ mIoU improvement ($78.1\% \shortrightarrow 79.0\%$). Together with the pseudo labels generated on the Mapillary images, we obtain an $1.2\%$ mIoU improvement over the baseline ($78.1\% \shortrightarrow 79.3\%$). Our preliminary results suggest the potential of self-training for semantic segmentation. We can improve a strong baseline without using extra labeled data. Since adding pseudo labels from both Citysapes coarse and Mapillary gives the best result, we will use them for the rest of our experiments, which means the amount of our pseudo labels is 40K. Again for clarity, we will term the 3K Cityscapes fine annotations as our real labels, and the 40K teacher-generated labels as pseudo labels afterwards.
**Ratio of pseudo labels to real labels** In the previous experiment, we only pick 1.5K samples randomly from the pool of 40K pseudo labels, thus we may miss most of our teacher-generated data and cannot fully explore the potential of self-training. Hence, we would like to increase the ratio between pseudo labels and real labels from 1:1 to 3:1, 5:1 and 7:1. Specifically, for each ratio setting we still randomly pick 1.5K real labels and pick 1.5K, 4.5K, 7.5K and 10.5K samples from the pool of pseudo labels. Under each setting, we also compare training with and without the centroid sampling. The experimental results are reported in Table \[table:ratio\]a. To cancel out the effect from longer training, we design another group of baseline experiments by duplicating the real labels. For example, a ratio of 3:1 between pseudo and real labels equals 6K training samples, whose baseline counterpart is obtained by duplicating the Cityscapes training set twice. The experimental results are reported in Table \[table:ratio\]b.
As we can see in Table \[table:ratio\]a, increasing the ratio indeed improves the segmentation accuracy, from $79.3\%$ to $80.0\%$. We also show that centroid sampling is essential to achieving good results. Without it, our model trained on the same amount of pseudo labels can only reach an mIoU of $78.9\%$, which is $1.1\%$ worse than using centroid sampling. Furthermore, in the situation that pseudo labels dominate the training set (e.g., 10.5K samples), centroid sampling does a good job in controlling label noisiness, otherwise the performance starts to drop ($78.9\% \shortrightarrow 78.7\%$). One observation we want to point out is, our final performance of $80.0$ is even better than a model pre-trained on Mapillary labeled data and finetuned on both Cityscapes fine and coarse annotations [@Zhu2019VPLR], using a total of 43K real labels. Our network architecture and training hyperparameters are the same as [@Zhu2019VPLR], which implies that the improvement is from self-training. This result is inspiring because it indicates the effectiveness of the self-training paradigm for semantic segmentation. We may not need a ultra large-scale labeled dataset to achieve good performance. Especially for autonomous driving, we have unlimited videos recorded during driving but not the resources to label them. With this self-training technique, we may generalize the model to various cities or situations with images alone.
As shown in Table \[table:ratio\]b, increasing the number of iterations by duplication is helpful, but not as good as using self-training ($79.1\%$ vs. $80.0\%$). In addition, the performance from duplication saturates at a certain number but self-training with pseudo labels continue to improve, which shows its potential to scale. Since a ratio of 7:1 between pseudo and real labels gives the best result, we will use this setting for the rest of our experiments unless otherwise stated.
**Generalizing to other students** Self-training is model-agnostic. It is a way to increase the number of training samples, and improve the accuracy and robustness of model itself. Here we would like to show that the pseudo labels generated by our teacher model (DeepLabV3+ with ResNeXt50 backbone), can improve the performance of (1) a heavier model (DeepLabV3+ with WideResNet38 backbone [@Wu2016WideOrDeep]); (2) a fast model (FastSCNN [@Poudel2019FastSCNN]) and (3) another widely adopted segmentation model (PSPNet with ResNet101 backbone [@Zhao2017pspnet]).
As shown in Table \[table:otherstudents\], our self-training method can improve the student model irrespective of the backbones and network architectures, which demonstrates its great generalization capability. We want to emphasize again that for all three students, our results are not only better than their comparing baseline, but also outperforms the models pre-trained on Mapillary labeled data. In addition, our trained FastSCNN model achieves an mIoU score of $72.5\%$ on the Cityscapes validation set, with only $1.1$M parameters. We believe this is a strong baseline for real-time semantic segmentation as compared to recent literature [@Nekrasov_2019_fastNASSeg; @Li2019DFANet; @Yu_BiSeNet_2018eccv].
**Fast training schedules** The surge in amount of training samples requires a faster training schedule. We introduce four of them: coarse2fine, fine2coarse, coarse2fine+ and fine2coarse+. As shown in Table \[table:fast\_main\], our baseline uses a crop size of 800 throughout the training, and achieves $80.0\%$ mIoU. Next we simply switch to our proposed fast learning schedule, and obtain slightly worse performance with 1.8x speed up. However, our goal is to avoid the speed and accuracy trade-off. We find that a good initialization is important for semantic segmentation. Hence, we propose to warm-up the crop size in the early epochs. We use a large crop size of 800 in the first 20 epochs, and then switch to the fast training schedules. We can see that by using coarse2fine+ with crop size warm-up, our fast training schedule is able to match the performance of baseline with 1.7x speed up. Note that, the speed up is model dependent. We can enjoy 2x speed up when training with a heavier model using the WideResNet38 backbone.
**Comparison to state-of-the-art** We compare our self-training method to recent literature on the test set of Cityscapes. For the test submission, we train our model using the best recipe suggested above, with several modifications. We use WideResNet38 [@Wu2016WideOrDeep] as the backbone and adopt a standard multi-scale strategy following [@Zhao2017pspnet; @Chen2018deeplabv3plus] to perform inference on multi-scaled (0.5, 1.0 and 2.0), left-right flipped and overlapping-tiled images. As we can see in Table \[table:cs\_sota\], our self-training method achieves an mIoU of $82.7\%$ using only the Cityscapes fine annotations, outperforming all prior methods that use the same training data. In addition, we even outperform some recent approaches using external labeled data, such as InPlaceABN [@Bulo2018inplaceABN], Auto-DeepLab-L [@Liu2019autodeeplab], SSMA [@Valada2019SSMA] and DPC [@Chen2018SearchSegDPC], etc.
In order to show the contribution from self-training alone, we perform a fair comparison in the bottom of Table \[table:cs\_sota\]. We design three training settings with the same network architecture, (a) Baseline: we only use the fine annotations; (b) Mapillary pre-trained: we use Mapillary labeled data to pre-train the model and then finetune it on Cityscapes; and (c) Self-training: our proposed method using only the fine annotations from Cityscapes and pseudo labels generated from Mapillary. As we can see, our self-training approach has the same segmentation accuracy as the model pre-trained using Mapillary labeled data ($82.7\%$). To clarify, we cannot do joint training when using Mapillary labeled data because Mapillary has different classes. We also show several visual examples in Fig. \[fig:visualization\], and demonstrate that self-training can handle class confusion better than the baseline model. In conclusion, we demonstrate the effectiveness of our proposed self-training framework for semantic segmentation, achieving state-of-the-art performance while requiring significantly less supervision.
Lastly, we also experimented with several modifications, for instance comparison between using hard (a one-hot distribution) or soft (a continuous distribution) labels, single-loop or multi-loop teacher-student, joint training or pre-training followed by finetuning. In terms of labels, our experiments show that using hard labels in general performs better than using soft labels. This intuitively makes sense because dense prediction problems favor hard labels [@He2019BagTricks]. In terms of teacher-student iteration, we do not observe improvement using more loops of self-training for semantic segmentation, i.e., putting back the student as teacher and train another student model. In terms of training, we try a conventional setting [@Yalniz2019BillionSemi] to first pre-train a model on pseudo labels and then finetune it on real labels, but we observe worse performance than joint training with all the labels. Detailed comparisons on these ablation studies can be found in the Appendix.
Finetuning on Other Datasets {#subsec:kitti_camvid}
----------------------------
In this section, we would like to show that our student model serves as a good initialization for finetuning on other driving scene segmentation datasets, such as CamVid and KITTI. To be specific, we take our well-trained student model (DeepLabV3+ with WideResNet38 backbone), and finetune it on target datasets to report the performance.
For CamVid, we only report single-scale evaluation scores on its test set for fair comparison to previous approaches. As can be seen in Table \[table:camvid\], we achieve state-of-the-art performance, an mIoU of $81.6\%$. We want to point out that the previous best method [@Zhu2019VPLR] uses a model pre-trained on both Cityscapes and Mapillary labeled data as initialization to finetune on CamVid. We outperform it by $1.8\%$. This result is encouraging because generalizing to other datasets is usually more meaningful [@He2019MomentumContrast]. In addition, it supports our claim that a model learned using the self-training framework is more robust.
We also achieve promising results on the KITTI leaderboard, with an mIoU of $71.41\%$. Given limited space, per-class results and comparisons can be found in the Appendix.
Cross-Domain Generalization with New Categories {#subsec:mapillary}
-----------------------------------------------
Here, we first describe the task of cross-domain generalization, and then show the robustness of our self-training method. We set Cityscapes dataset as the source domain, and Mapillary dataset as the target domain. We choose Mapillary due to its large size and variability. In addition, Mapillary has many more semantic categories than Cityscapes, 66 compared to 19. Hence, most classes are considered new, never seen by the model trained on the source domain, which makes the problem very challenging. We randomly select 1/10, 1/5 and 1/2 of the original Mapillary training set to form different evaluation scenarios.
We compare two methods, (1) conventional finetuning approach: a model trained on Cityscapes fine annotations, and then finetuned on the given Mapillary labeled samples; (2) our self-training approach: a model trained on both fine annotations from Cityscapes and pseudo labels from Mapillary, and then finetuned on the given Mapillary labeled samples. For simplicity, we do not use OHEM loss and other non-default training options, to better demonstrate the contributions from model initialization. Our goal is to see whether self-training can help improve a model’s generalization capability across domains.
We can see the results in Fig. \[fig:few\_shot\]. First, the model trained using our self-training framework achieves the best performance across all scenarios compared to the finetuning approach. This indicates that the self-training method can largely reduce the human labeling effort when generalizing to new domains. Second, in terms of using the full training set, our approach also performs better. This maybe due to the fact that self-training serves as effective data augmentation and provides a better model initialization. Third, given fewer and fewer training samples, our method starts to reveal its robustness. For example, using only 1/10 of the training set, we are able to achieve $37.8\%$ mIoU, which even outperforms the finetuning model using 1/5 of the data. We can see that the gap between the finetuning model and our self-training model becomes larger.
We would like to push this task even further to the few-shot case, where we only have $10$ training samples per class. As can be seen in Fig. \[fig:few\_shot\], our method is still able to achieve decent performance of $33.3\%$ mIoU. We significantly outperform the finetuning approach, an improvement of $7.3\%$ ($26.0\% \shortrightarrow 33.3\%$). In addition, our performance using 10 samples per class is even better than the finetuning approach using 1/10 of the full training set (30 samples per class). Hence, the results suggest that our self-training technique can generalize well to various locations with just images and a few annotations from the new scene. Another cross-domain generalization experiment, from Cityscapes to BDD100K, can be found in the Appendix.
![Visual comparisons on Cityscapes. We demonstrate that self-training can effectively handle class confusion, such as between tree and vegetation (row 1), road and sidewalk (row 2), wall and fence (row 3).[]{data-label="fig:visualization"}](visualization_v2.png){width="1.0\linewidth"}
Conclusion {#sec:conclusion}
==========
In this work, we introduce a self-training framework for driving scene semantic segmentation. The self-training method can leverage a large number of unlabeled data to improve both accuracy and robustness of the segmentation model. Together with our proposed training techniques, we achieve state-of-the-art performance on three driving scene benchmark datasets, Cityscapes, CamVid and KITTI, while requiring significantly less supervision. We also demonstrate that our self-training method works well on a challenging cross-domain generalization task. Even with 10 labeled samples per class, we show that our model is able to achieve decent segmentation accuracy generalizing from Cityscapes to Mapillary. Lastly, we propose a fast training schedule, which is a general technique to speed up model learning by 2x without performance degradation.
Appendix {#appendix .unnumbered}
========
We will provide more details about the results in the main paper, and show more visualizations. To be specific, we first discuss three alternative design choices of our self-training framework in Appendix. \[sec:ablation\]. Then we show visualizations of the pseudo labels, successful predictions and failure cases in Appendix. \[sec:visualizations\]. Furthermore, we present two experiments on using a better teacher and cross-domain generalization in Appendix. \[sec:generalization\]. In the end, we provide more details of our KITTI results in Appendix. \[sec:kitti\].
Ablation Studies {#sec:ablation}
================
We have indicated at the end of Sec. 4.3 of the main paper that we could only briefly describe the ablation studies due to space limitations. Here, we discuss more details of the ablation studies. The baseline is a DeepLabV3+ model with ResNeXt50 backbone, the same as in the main paper. The ratio between pseudo labels to real labels defaults to 7:1. We still run the same experiment five times and report the average mIoU.
Hard vs soft labels {#subsec:hard}
-------------------
The generated labels could be either hard or soft. Hard means it is a one-hot distribution. We can think of them as ground truth labels and use standard cross-entropy loss to start training. Soft means it is a continuous distribution (i.e., each label has an associated probability). We need to save the probabilities from the teacher’s predictions as supervision and use sparse cross-entropy loss to train the student model.
Table \[table:soft\_hard\] shows that the model does better with hard labels than soft labels. This observation agrees with [@He2019BagTricks] that dense prediction problems favor hard labels. A potential explanation is that soft labels may cause ambiguity around object boundaries, which is harmful for semantic segmentation. This also indicates the difference between image classification and semantic segmentation, because soft labels are usually preferred in the image classification domain [@Yalniz2019BillionSemi; @Xie2019NoisyStudent].
Joint learning vs pre-training {#subsec:joint}
------------------------------
Once we have a large set of pseudo labels, we train our student model using a joint learning process with both human-annotated real labels and teacher-generated pseudo labels. Here, we would like to compare it with a conventional alternative following [@Yalniz2019BillionSemi]: pre-training on the large set of pseudo labels first and then finetuning on the real labels.
Table \[table:joint\] shows that joint learning outperforms the conventional pipeline using pre-training and then finetuning. A potential explanation is that joint training with both real and pseudo labels serves as an effective data augmentation to regularize the model learning.
Single-loop or multi-loop {#subsec:loop}
-------------------------
Teacher-student learning could be iterative which means we can use the student as teacher, generate more accurate pseudo labels and then retrain another student model. Here, we use more loops of self-training to see if helps semantic segmentation.
As seen in Table \[table:loop\], using a single-loop of teacher-student training is able to achieve promising results ($80.0\%$). 2-loop obtains slightly worse results ($79.9\%$), and 3-loop is slightly better ($80.2\%$). In terms of a good trade-off between accuracy and resources, we only perform a single iteration of teacher-student for all experiments.
Model-dependent speed up {#subsec:speed}
------------------------
Recall from Sec. 3.3 in the main paper, that the speed up using our proposed fast training schedule is model-dependent. Here, we show the detailed speed up information for various models.
As seen in Table \[table:fast\_appendix\], larger models tend to benefit more from the fast training schedule. For example, we achieve 2x speed up when training on a DeepLabV3+ model with WideResNet38 backbone. This is because when the model is bigger, the time spent on network computation dominates the training time. If we reduce the crop size, we save a lot of computation.
![Visualizations of our teacher-generated pseudo labels. Left: on Cityscapes coarse images. We can see our pseudo labels have higher quality than human-labeled coarse annotations based on polygon. Black regions in coarse annotations represent background class. Right: on Mapillary images. Our teacher model is able to provide reasonable segmentation predictions despite the large domain gap.[]{data-label="fig:pseudo_labels"}](pseudo_labels_small.png){width="1.0\linewidth"}
Visualizations {#sec:visualizations}
==============
Here, we first show several visualizations of our teacher-generated pseudo labels in Appendix. \[subsec:pseudo\]. Then we provide some successful predictions and failure cases on Cityscapes dataset in Appendix. \[subsec:cityscapes\_appendix\].
Visualizations of pseudo labels {#subsec:pseudo}
-------------------------------
As mentioned in Sec. 3.1 of the main paper, we show more visualizations of pseudo labels here in Fig. \[fig:pseudo\_labels\].
First, we look at our generated pseudo labels on Cityscapes coarse images (left column Fig. \[fig:pseudo\_labels\]). Compared to the coarse labels annotated by humans (i.e., polygons), our pseudo labels have higher quality, such as sharper boundaries, correct predictions, etc. For example on row 3, our pseudo labels successfully capture several persons on the right of the sidewalk, while the coarse annotations completely ignore them.
Then, we show our generated pseudo labels on Mapillary images. The Mapillary dataset is collected worldwide, and includes different seasons, time of the day, traffic etc. Our teacher model is able to provide reasonable predictions on these challenging situations, such as uphill road (row 1), cloudy weather (row 2) and other countries (row 3 and 4).
Despite some erroneous predictions, the quality of our teacher-generated pseudo labels are in general good. This is part of the reason our self-training method works, because the student model won’t learn well if the pseudo labels contain too much noise.
![Successful predictions on Cityscapes dataset. From left to right: image, ground truth, our prediction and their differences. We can see that our model is able to assign correct semantic labels to each pixel except the object boundaries due to challenges such as annotation ambiguity. Black regions in ground truth represent background class[]{data-label="fig:cs_good"}](cs_good_small.png){width="1.0\linewidth"}
![Visualizations of failure cases. Top: five common scenarios of class confusion. From rows (a) to (e), our model has difficulty in segmenting: (a) terrain and vegetation, (b) person and rider, (c) wall and fence, (d) car and truck, (e) road and sidewalk. Bottom: challenging situations, (f) objects overlapping, (g) reflection in the mirror, (h) objects far away with strong illumination, and (i) annotation ambiguity. []{data-label="fig:cs_fail"}](cs_fail_v2.png){width="1.0\linewidth"}
Visualization on Cityscapes {#subsec:cityscapes_appendix}
---------------------------
We first show several visualizations of our successful predictions in Fig. \[fig:cs\_good\]. We see that our model is able to handle small objects, crowded scenes, complicated lighting, etc. Our predictions are accurate and sharp, as demonstrated by the difference images in the rightmost column. We can assign correct semantic labels to each pixel except the object boundaries due to challenges such as annotation ambiguity.
Then we show visualizations of the failure cases of our model on Cityscapes dataset in Fig. \[fig:cs\_fail\]. We show five common scenarios of class confusion. From rows (a) to (e), our model has difficulty in segmenting: (a) terrain and vegetation, (b) person and rider, (c) wall and fence, (d) car and truck, (e) road and sidewalk. Some of these situations are very challenging. For example, the wall in row (c) has holes, which by definition should be a fence.
In addition, we show four even challenging scenarios from rows (f) to (i). In Fig. \[fig:cs\_fail\] (f), the bicycle is overlapping with a car. It is hard to correctly tell them apart from such a long distance. In Fig. \[fig:cs\_fail\] (g), our model predicts a reflection in the mirror as a person. This is an interesting result because our prediction should be considered correct in terms of appearance without reasoning about context. In Fig. \[fig:cs\_fail\] (h), it is very hard to tell whether it is a car or bus when the object is far away, especially when there is strong illumination. In Fig. \[fig:cs\_fail\] (i), this is an ambiguous situation because the handbag is neither a person or car. It should be a background class.
\[table:better\_teacher\]
Generalization {#sec:generalization}
==============
Does a better teacher help? {#subsec:better_teacher}
---------------------------
In the main paper, we always use a DeepLabV3+ model with ResNeXt50 backbone as the teacher model, given its good accuracy and speed trade-off. However, a straightforward question arises, does a better teacher help?
Here, we use a better model (DeepLabV3+ network with WideResNet38 backbone) as our teacher, we call it teacher B. Its performance on the validation set of Cityscapes is $80.5\%$, higher than $78.1\%$ of the old teacher which we call teacher A. We will use teacher B to generate pseudo labels and compare to Table 3 in the main paper to answer the question.
As can be seen in Table \[table:better\_teacher\], a better teacher indeed helps. The performance of all the student models improve. However, the improvements range from 0.2 to 0.5, not significant compared to the performance gap between teacher A and teacher B ($78.1\%$ vs $80.5\%$).
Cross-domain generalization: from Cityscapes to BDD100K {#subsec:cross}
-------------------------------------------------------
As mentioned in Sec. 3.4 of the main paper, generalizing a trained model to other domains (i.e, datasets or locations) given limited supervision is one of the motivations and contributions of our work.
We have done an experiment, generalizing from a model trained on Cityscapes to Mapillary, in Sec. 4.5 of the main paper. We demonstrate that our model can generalize to other domains with new categories, given a few labeled data. Here, we perform another experiment, generalizing from a model trained on Cityscapes to BDD100K [@BDD100K]. Despite the fact that BDD100K has the same number of classes (19) as Cityscapes, this is a challenging situation because the data of BDD100K is collected in United States, which has a big domain gap compared to the data from Cityscapes.
As seen in Table \[table:cross\], our self-training method outperforms the conventional finetuning approach by a large margin. We want to emphasize that even using 10 samples per class from BDD100K (i.e., a total of 200 training samples), our model can achieve an mIoU of $58.9\%$, close to the finetuning approach using the full dataset ($62.3\%$ by using 7K training samples). This result strongly indicates the effectiveness of our method’s generalization capability.
![Successful predictions on KITTI dataset. From left to right: image, our prediction and the difference between our prediction and ground truth. Note that we do not have the ground truth for the test set. The difference images are provided by the KITTI official evaluation server. Black regions represent background class[]{data-label="fig:kitti_good"}](kiiti_good_small.png){width="1.0\linewidth"}
KITTI Results {#sec:kitti}
=============
Recall from Sec. 4.4 in the main paper, that we also achieved promising results on the KITTI leaderboard, with an mIoU of $71.41\%$ ranking 2nd. Here, we report the detailed results on the test set in Table \[table:kitti\].
We would like to emphasize that our model is only pre-trained on Cityscapes fine annotations (about 3K samples), while other approaches [@meletis2018AHiSS; @Ivan_ladderLDN_iccvw2017; @Bulo2018inplaceABN; @Zhu2019VPLR] use external training data, such as Mapillary Vista dataset and Cityscapes coarse annotations (for a total of about 43K samples). VPLR [@Zhu2019VPLR], the top performer, also uses Cityscapes video data to help regularize the model training. For fair comparison in terms of training data usage, our method uses the same training data with SegStereo [@yang2018segstereo] but outperforms it by $12.3\%$.
Qualitatively, we show several visualizations of our successful predictions in Fig. \[fig:kitti\_good\]. These images are of the test set and the difference images are provided by KITTI official evaluation server.
[^1]:
|
---
abstract: 'The efficiency of time dependent density matrix renormalization group methods is intrinsically connected with the rate of entanglement growth. We introduce a new measure of entanglement in the space of operators and show, for transverse Ising spin $1/2$ chain, that the simulation of observables, contrary to simulation of typical pure quantum states, [*is*]{} efficient for initial local operators. For initial operators with a finite index in Majorana representation, the operator space entanglement entropy saturates with time to a level which is calculated analytically, while for initial operators with infinite index the growth of operator space entanglement entropy is shown to be logarithmic.'
author:
- Tomaž Prosen
- Iztok Pižorn
title: Operator space entanglement entropy in transverse Ising chain
---
Introduction {#sect:introduction}
============
The entanglement is an intrinsic property of composite quantum systems and represents a cornerstone in quantum information theory [@nielsenbook]. It is important to understand the role of quantum entanglement in classical manipulation of quantum objects, and to quantify the degree of entanglement. Although the question of quantification is not clearly resolved, the quantum information theory offers several measures [@eisertJMO; @plenio0504163] of entanglement. Quantum information theory also gave a new birth or fresh interpretation of a class of methods for numerical simulation of many-body quantum systems which, due to the exponential growth of Hilbert space, cannot be manipulated using exact diagonalization. The methods originally known as density matrix renormalization group (DMRG) [@whitePRL69] deploy the fact that many degrees of freedom are redundant in quantum state description; the system is therefore adequately described by taking into account maximally entangled components only. Thus, sufficiently slow growth of the entanglement is of crucial importance. DMRG enjoyed remarkable success in determining the ground state properties of large one-dimensional quantum models, for which the degree of entanglement scales at most logarithmically with size [@latorre; @jinJSP116; @keatingCMP252; @holzhey; @osborne; @osterloh], however its [*time-dependent*]{} version (tDMRG) [@vidalPRL91; @whitePRL93] is often plagued by abundance of entanglement with time evolution [@prosenPRE75]. For efficient classical simulation of many body quantum dynamics using tDMRG it is required that the computational costs grow polynomially in time meaning that the degree of entanglement of any quantum object which can be represented as an element of a scalable tensor product Hilbert space (either a pure state, or a mixed state/operator, etc) must grow no faster than logarithmically. It was recently shown that this is generically not the case for [*quantum chaotic*]{} Ising spin chain in a tilted magnetic field where the entanglement entropies grow linearly and hence the computation costs increase exponentially in time [@prosenPRE75].
In this paper we shall consider the model of quantum Ising chain in transverse magnetic field which is integrable and an explicit analytical solution exists. Calabrese and Cardy [@calabreseJSM4] have shown numerically that the growth of entanglement entropy is [*linear*]{} for evolution of pure initial states which are ground states of quenched hamiltonians; see also Ref.[@chiaraJSM6]. However, from efficiency of tDMRG for time evolution of local operators [@prosenPRE75] one may conclude that entanglement entropy computed in the space of operators grows only logarithmically. Here we address this problem theoretically using the idea [@prosenPRE60] of re-formulating the Heisenberg evolution in an algebra of operators in terms of a Schr" odinger evolution generated by a different - adjoint hamiltonian acting on the Hilbert space of operator algebra. We show that operator space entanglement entropy saturates in time for initial local operators of finite index (precise definitions follow) and explicitly compute the saturation values in the critical case. Further, for initial local operators of infinite index we give accurate numerical evidence and a theoretical hint that the growth is logarithmic (in thermodynamic limit) with prefactor $1/6$ in critical, or $1/3$ in non-critical case.
We note that, to best of our knowledge, the entanglement in operator space is a new concept which has not yet been considered theoretically, and it is clearly not equivalent to the conventional concept of entanglement of density operators as discussed in sect.\[sect:conclusions\]. Yet, it is the one which we expect to be more closely related to computational complexity of time-evolution in infinite interacting quantum systems.
Fermion representation of dynamics in operator space
====================================================
The dynamics of a transverse Ising chain of length $2L$ is described in terms of canonical Pauli matrices $\sigma^\alpha_j$ for sites $j\in {\mathbb{Z}}_{2L}\equiv\{ -L+1,\ldots,0,1,\ldots,L\}$ and the hamiltonian $$H = \sum_{j=-L+1}^{L-1} \sigma_j^{\rm z} \sigma_{j+1}^{\rm z} +
h \sum_{j=-L+1}^{L} \sigma_j^{\rm x}
\label{eq:H}$$ with open boundary conditions, which can be diagonalized by means of Jordan-Wigner transformation and introduction of Majorana fermion operators [@brandtjacoby; @prosenPRE60], $X_n = \big(\prod_{j<n}\sigma_j^{\rm z}\big) \sigma_n^{\rm x}$ and $Y_n = \big(\prod_{j<n}\sigma_j^{\rm z}\big) \sigma_n^{\rm y}$ fulfilling the anti-commutation relations $\{X_i,X_j\}=\{Y_i,Y_j\}=2\delta_{ij}$ and $\{X_i,Y_j\}=0$. Heisenberg equations of motion ${ {\rm d} }A/{ {\rm d} }t = { {\rm i} }[H,A]$ for Majorana operators can be written: $${\dot X_n} =
2(Y_{n-1}\!-\! h Y_n ),
\qquad
{\dot Y_n} = -2(X_{n+1}\!-\! h X_n).
\label{eq:dXndt}$$ An operator corresponding to an arbitrary physical observable can be written as a superposition of products of Majorana operators $X_j$, $Y_j$, namely $P_{\mathbf{n},\mathbf{n'}} = X_{-L+1}^{n_{-L+1}} Y_{-L+1}^{n'_{-L+1}}\cdots X_L^{n_{L}} Y_L^{n'_{L}}$ with powers $n_j,n'_j \in \{0,1\}$. A set of $4^{2L}$ operators $\{ {{\vert P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp} \}$ spans an orthonormal basis of a Hilbert space, namely the matrix alebra $\AA={\mathbb{C}}^{2^{2L}\times 2^{2L}}$, with an inner product ${{}_\sharp\langle A \vert B \rangle_\sharp}= 2^{-2L} \tr A^\dagger B$. $\AA$ can be conveniently interpreted as a Fock space of $2L$ [*adjoint fermions*]{} (we shall call them a-fermions) with pseudo-spin (distinguishing between Majorana $X_j$ and $Y_j$ operator). An arbitrary operator $A$ is then a-fermion state ${{\vert A \rangle}_\sharp} = \sum_{\mathbf{n},\mathbf{n'}} a_{\mathbf{n},\mathbf{n'}}
{{\vert P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp}$. A-fermi operators over $\AA$, ${ {\hat c} }_{j\uparrow},{ {\hat c} }_{j\downarrow}$, can be introduced by $$\begin{aligned}
{ {\hat c} }_{j\uparrow} {{\vert P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp} &=&
n_j {{\vert X_j P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp} \\
{ {\hat c} }_{j\downarrow} {{\vert P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp} &=&
n'_j {{\vert Y_j P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp} $$ satisfying canonical anti-commutation relations.
The index of an operator $P_{\mathbf{n},\mathbf{n'}}$ is defined as $I_{\mathbf{n},\mathbf{n'}} = \sum_{j}(n_j+n'_j)$ and for index-one operators eq. (\[eq:dXndt\]) rewrites: $$\begin{aligned}
\frac{{ {\rm d} }}{{ {\rm d} }t} {{\vert X_n \rangle}_\sharp} &=&
2({ {\hat c} }_{n-1\downarrow}^\dagger - h { {\hat c} }_{n\downarrow}^\dagger)
{ {\hat c} }_{n\uparrow} {{\vert X_n \rangle}_\sharp}\\
\frac{{ {\rm d} }}{{ {\rm d} }t} {{\vert Y_n \rangle}_\sharp} &=&
-2({ {\hat c} }_{n+1\uparrow}^\dagger - h { {\hat c} }_{n\uparrow}^\dagger)
{ {\hat c} }_{n\downarrow} {{\vert Y_n \rangle}_\sharp}
\label{eq:dXndYn2}\end{aligned}$$ which can be interpreted as a Schr" odinger equation $({ {\rm d} }/{ {\rm d} }t) {{\vert A \rangle}_\sharp} =
-{ {\rm i} }\hat{\mathcal{H}}{{\vert A \rangle}_\sharp}$ for the adjoint hamiltonian $$\hat{\mathcal{H}} =
2 {\rm i}\!\!\sum_{n\in{\mathbb{Z}}_{2L}}\!\!\!\Big[
({ {\hat c} }_{n-1\downarrow}^\dagger { {\hat c} }_{n\uparrow} - { {\hat c} }_{n+1\uparrow}^\dagger { {\hat c} }_{n\downarrow})
+ h ( { {\hat c} }_{n\uparrow}^\dagger { {\hat c} }_{n\downarrow} - { {\hat c} }_{n\downarrow}^\dagger c_{n\uparrow})
\Big].
\label{eq:hamiltonianHcal}$$ Since the adjoint time-evolution is a [*homomorphism*]{} the Schr" odinger equation extends to arbitrary element of the operator algebra ${{\vert A \rangle}_\sharp}\in\AA$. Note that the number of a-fermions, $\hat{\mathcal{N}}=\sum_{n,s} { {\hat c} }^\dagger_{n,s} { {\hat c} }_{n,s}$ is conserved, unlike the number of ordinary Majorana fermions.
Operator space entanglement entropy
===================================
It is clear that classical simulability of quantum states is quantified by the entanglement entropy of half-half (or worst case) bipartition of the lattice. However, for simulability of quantum observables (or density operators of mixed states), the decisive quantity is an analog of entanglement entropy defined for an arbitrary element of operator algebra $\AA \ni {{\vert A \rangle}_\sharp} = \sum_{\mathbf{n},\mathbf{n'}} a_{\mathbf{n},\mathbf{n'}}
{{\vert P_{\mathbf{n},\mathbf{n'}} \rangle}_\sharp}$, with the adjoint reduced density matrix $$\begin{aligned}
&& R_{(n_{-\!L\!+\!1},n'_{-\!L\!+\!1},\ldots n_0,n'_0),(m_{-\!L\!+\!1},m'_{-\!L\!+\!1},\ldots m_0,m'_0)} = \nonumber \\
&& \sum_{n_1,n'_1,\ldots n_L,n'_L} a_{(n_{-\!L\!+\!1},\ldots n_0,n_1,\ldots n_L),(n'_{-\!L\!+\!1},\ldots n'_0,n'_1,\ldots n'_L)} \times \nonumber \\
&& a_{(m_{-\!L\!+\!1},\ldots m_0,n_1,\ldots n_L),(m'_{-\!L\!+\!1},\ldots m'_0,n'_1,\ldots n'_L)}^*,\end{aligned}$$ namely $$S = -\tr R \ln R.
\label{eq:SreducedR}$$ For a spin 1/2 chain it is perhaps more natural to use a set of $4^{2L}$ Pauli operators $Q_{s_{-\!L\!+\!1},\ldots s_L} = \sigma_{-\!L\!+\!1}^{s_{-\!L\!+\!1}}\cdots \sigma^{s_L}_L$, where $s_j\in\{0,{\rm x},{\rm y},{\rm z}\}$, $\sigma^0\equiv \one$, as a physical basis of operator algebra $\AA$, and define bi-partition and entanglement entropy with respect to $Q_{{{\mathbf s}}}$. However, it is easy to show that the result is identical to (\[eq:SreducedR\]) since the transformation between the bases $\{ P_{{{\mathbf n}},{{\mathbf n}}'}\}$ and $\{ Q_{{{\mathbf s}}} \}$ is a simple [*permutation*]{} of multiindices $({{\mathbf n}},{{\mathbf n}}')\leftrightarrow {{\mathbf s}}$ (with multiplications by $\pm 1$), and even though it is [*non-local*]{} it maps first $L$ a-fermions to only first $L$ spins and vice versa.
Let us now try to compute [*time-dependence*]{} of operator space entanglement entropy $S(t)$ for some simple initial operators $A$. For convenience, we introduce staggered a-fermi operators ${ {\hat w} }_j$, $j\in {\mathbb{Z}}_{4L}=\{ -2L+1,\ldots,0,1,\ldots,2L\}$, such that ${ {\hat w} }_{2n-1} \equiv { {\hat c} }_{n\uparrow}$ and ${ {\hat w} }_{2n} \equiv { {\hat c} }_{n\downarrow}$. Any operator acting solely in a space of first $L$ a-fermions (or first $L$ spins) can be expressed in terms of $2L$ anti-comuting operators ${ {\hat w} }_j$ with $j \in {\mathbb{Z}}_{2L}^- \equiv \{-2L+1, \ldots, -1, 0\}$. We follow Ref.[@jinJSP116] and express $2^{2L}$ eigenvalues of adjoint reduced density matrix $R$, as $\rho_{{{\mathbf n}}} = \prod_j\left( n_j \gamma_j + (1\!-\!n_j)(1\!-\!\gamma_j)\right)$, $n_j\in\{0,1\}$ where $\gamma_j$ are eigenvalues of time-dependent $2L\times 2L$ correlation matrix $$\Gamma_{mn}(t) = {{}_\sharp{\langle A \vert}} { {\hat w} }_m^\dagger(t) { {\hat w} }_n(t) {{\vert A \rangle}_\sharp},
\quad m,n \in {\mathbb{Z}}_{2L}^-$$ Then, the entanglement entropy (\[eq:SreducedR\]) simply reads $$S(t) = \sum_j e( \gamma_{j} ),\; e(x) \equiv -x\ln x - (1\!-\!x) \ln(1\!-\!x).
\label{eq:S}$$ This procedure (see [@calabreseJSM4] for details) results in an efficient numerical method which essentially only requires diagonalization of $2L$ dimensional matrix $\Gamma$ for the solution of a quantum problem on $2^{4L}$ dimensional Hilbert space.
The time-dependent a-fermi operators ${ {\hat w} }_m(t)$ are obtained from [*linear*]{} Heisenberg type equations ${\dot { {\hat w} }_m} = -{ {\rm i} }[{ {\hat w} }_m, \mathcal{H}]$, namely, ${\dot { {\hat w} }_{2j}}=2({ {\hat w} }_{2j+1}-h{ {\hat w} }_{2j-1})$ and ${\dot { {\hat w} }_{2j-1}}=2(-{ {\hat w} }_{2j-2}+h{ {\hat w} }_{2j})$. The solution of such Heisenberg equations, written as $ {\dot { {\hat w} }_m} = -{ {\rm i} }\sum_n G_{mn} { {\hat w} }_n$, is obtained by diagonalizing a $2L\times 2L$ matrix $G = V \cdot \Lambda \cdot V^{\dagger}$ which yields $${ {\hat w} }_m(t) = \sum_n \Big( \sum_k V_{mk} { {\rm e} }^{-{ {\rm i} }t \Lambda_k} V_{nk}^* \Big) { {\hat w} }_n.$$
However, in the ‘critical case’ $h=1$, the time-evolution of ${ {\hat w} }_m(t)$ can be solved exactly and some analytical solutions can be given. Namely in such a case the sets of Heisenberg eqs. are identical and are solved via discrete sine-transform with $V_{mk}={ {\rm i} }^{m}\sqrt{\frac{2}{4L+1}}\sin\big[\frac{(m\!+\!2L)k\pi}{4L+1}\big]$, $${ {\hat w} }_m(t) = \sum_n \Big[
\sum_{k=0}^{4L} V_{mk} { {\rm e} }^{{ {\rm i} }4 \cos(\frac{k\pi}{4L+1}) t} V_{nk}^{*}
\Big] { {\hat w} }_n.
\label{eq:discretefoursolution}$$
In the following we shall be interested in the results in the [*thermodynamic limit*]{} (TL), $L\to\infty$. The infinite sum over $k$ in (\[eq:discretefoursolution\]) is transformed onto an integral which yields ${ {\hat w} }_{m}(t) = \sum_{n\in{\mathbb{Z}}}\Phi_{nm}(4t) {\hat w_n}$ in terms of Bessel functions $\Phi_{ab}(x) \equiv J_{a-b}(x)$. The correlation matrix elements are therefore (using ${\hat n_b} = {{ {\hat w} }_b}^\dagger {{ {\hat w} }_b}$) $$\Gamma_{mn}(t) = \sum_{b\in {\mathbb{Z}}}\Phi_{bm}(4t)\Phi_{bn}(4t)
{{}_\sharp{\langle A \vert}}{\hat n_b}{{\vert A \rangle}_\sharp}.$$ We also assume that the initial operator $A$ is [*local*]{}, i.e. a product of [*finite*]{} number of Pauli matrices $\sigma^\alpha_j$. This implies that: (i) either $A$ has a [*finite*]{} index, i.e. ${{\langle { {\hat n} }_b \rangle_\sharp}}\equiv {{}_\sharp{\langle A \vert}}\hat{n}_b{{\vert A \rangle}_\sharp} = 0$, for $|b| > b_0$ for some $b_0 \in {\mathbb{Z}}^+$, or (ii) $A$ has an infinite index and ${{\langle { {\hat n} }_b \rangle_\sharp}}=1$ for $b < -b_0$ and ${{\langle { {\hat n} }_b \rangle_\sharp}}=0$ for $b > b_0$ (such as e.g. $A=\sigma_1^{\rm x}$). Then as $L\to \infty$, an arbitrary large fixed finite piece of correlation matrix can be asymptotically written as $$\Gamma_{mn}(t) = \sum_{b\in {\mathbb{Z}}}
J_{b-m}(4t)J_{b-n}(4t)
{{\langle \hat n_b \rangle_\sharp}}.
\label{eq:approximationgamma}$$ Note that $\Gamma_{mn}$ has effectively finite rank $\sim x=4t$, namely $\Gamma_{m,n} \sim \delta _{mn}$, for $-m,-n > x$. For brevity we shall be omitting the argument of Bessel functions always equal to $x=4t$.
Initial operators of finite index
---------------------------------
First, we focus on the case (i) of finite index initial operators $A$. It was conjectured in [@prosenPRE75] that in such cases the entanglement entropy in thermodynamic limit saturates in time. Using the a-fermion algebra we are now able to calculate the exact saturation value since RHS in (\[eq:approximationgamma\]) is a finite sum. Consider $\Gamma_{mn}$ as a real matrix over ${\mathbb{R}}^{\infty}$ with canonical basis $\{{{\vert m \rangle}},m\in{\mathbb{Z}}^-\}$ and write a set of [*non-orthogonal*]{} vectors $\{{{\vert \psi_\alpha \rangle}}\}$, namely ${\langle m \vert \psi_\alpha \rangle} = (-1)^{\alpha} J_{\alpha-m}(4t) = {\langle \psi_\alpha \vert m \rangle}$. Let us write initial operator of finite index ($K$) as $A = O_{j_1}\cdots O_{j_K}$ where $O_{2j-1}\equiv X_j$ and $O_{2j}\equiv Y_j$. Then we have $\Gamma_{mn} = J_{j_1-m}J_{j_1-n} + \cdots + J_{j_K-m}J_{j_K-n}$, or $$\Gamma_{mn}(t) =
{\langle m \vert \psi_{j_1} \rangle}{\langle \psi_{j_1} \vert n \rangle} +
\cdots +
{\langle m \vert \psi_{j_K} \rangle}{\langle \psi_{j_K} \vert n \rangle}.
\label{eq:operatorfinitegamma}$$ This means that the rank of $\Gamma_{mn}$ is bounded by $K$, in fact it is $K$, and its non-trivial eigenspaces are spanned by $\{{{\vert \psi_{j_k} \rangle}},1\le k \le K\}$. Let $\{{{\vert \phi_k \rangle}},1\le k \le K\}$ be an orthonormalized set obtained from $\{{{\vert \psi_{j_k} \rangle}},1\le k \le K\}$ by a standard Gramm-Schmidt procedure, for which the only input is the set of scalar products ${\langle \psi_\alpha \vert \psi_\beta \rangle} =
\sum_{k\in {\mathbb{Z}}^-} J_{k-\alpha}J_{k-\beta}$ which can be in TL evaluated analytically for [*any*]{} $t$ in terms of finite sums and approach the following long time asymptotics ${\langle \psi_\alpha \vert \psi_\beta \rangle}\vert_{t=\infty} = (1/2)\delta_{\alpha\beta} - \sin[\pi(\alpha-\beta)/2]/[\pi(\alpha-\beta)]$. The non-vanishing part of the spectrum $\{\gamma_j\}$ of the correlation matrix (\[eq:operatorfinitegamma\]) entering eq. (\[eq:S\]) is thus given by the eigenvalues of the following $K\times K$ matrix $$\tilde{\Gamma}_{kl} =
{\langle \phi_k \vert \psi_{j_1} \rangle}{\langle \psi_{j_1} \vert \phi_{l} \rangle} +
\cdots +
{\langle \phi_k \vert \psi_{j_K} \rangle}{\langle \psi_{j_K} \vert \phi_{l} \rangle}.
\label{eq:gammatilde}$$ Thus, eq. (\[eq:gammatilde\]) is our main result for the case of finite index initial operators.
![Entanglement entropy for finite-index operators ${{\vert X_1 \rangle}_\sharp}$ (black) and ${{\vert X_1 Y_1 \rangle}_\sharp}={ {\rm i} }{{\vert \sigma^{\rm z}_1 \rangle}_\sharp}$ (gray) compared to theoretic saturation value for $t\to\infty$ and $h=1$ (thick lines). Three different values of magnetic field are considered: $h=1$ (solid curve), $h=0.5$ (dotted), $h=2$ (dashed), $h=3$ (dash-dotted). We set $2L=200$, such that no finite size effect were noticable. []{data-label="fig:figA"}](figA.eps){width="\columnwidth"}
For illustration, let us calculate the asymptotic value $S(t\to\infty)$ for the simplest two cases: (a) $A=O_j$, e.g. $A=\cdots \sigma^{\rm z}_{-2}\sigma^{\rm z}_{-1}\sigma^{\rm x}_0$, and (b) $A= O_j O_{j+1}$, e.g. $A=\sigma^{\rm z}_1$. In case (a), $K=1$, the result is $\gamma_1 = {\langle \psi_j \vert \psi_j \rangle} $ with $\gamma_1\vert_{t=\infty} = \frac{1}{2}$ which gives the entanglement entropy of $S(\infty) = \ln 2$. In case (b), $K=2$, we have $\gamma_{1,2}=\frac{1}{2}[{\langle \psi_j \vert \psi_j \rangle}+{\langle \psi_{j+1} \vert \psi_{j+1} \rangle} \pm \sqrt{ ({\langle \psi_j \vert \psi_j \rangle}+{\langle \psi_{j+1} \vert \psi_{j+1} \rangle})^2+4{\langle \psi_{j} \vert \psi_{j+1} \rangle}^2}]
$ with $\gamma_{1,2}\vert_{t=\infty} = \frac{1}{2}\pm \frac{1}{\pi}$ and $S(\infty) = 2 \ln(\gamma_1^{-\gamma_1}\gamma_2^{-\gamma_2})$. Both results agree excellently with numerical solutions of Heisenberg eqs. for ${ {\hat w} }_j(t)$ shown in Fig. \[fig:figA\], for the case $h=1$, whereas saturation is observed for any $h$.
Initial operators of infinite index
-----------------------------------
Second, we consider the case (ii) of infinite index initial operator $A$. In thermodynamic limit, local spin operators such as $\sigma_1^{\rm x}$ are products of infinite number of Majorana operators $X_n$, $Y_n$, in particular ${{\vert \sigma_1^x \rangle}_\sharp}={{\vert \cdots X_{-1} Y_{-1} X_0 Y_0 X_1 \rangle}_\sharp}$ and previous disciussion does not apply. As conjectured in [@prosenPRE75] time complexity for such initial operators only grows polynomially in time which corresponds to logarithmic growth of the entanglement entropy.
![Entanglement entropy for infinite-index operators ${{\vert F \rangle}_\sharp}$ (black) and ${{\vert F Y_1 \rangle}_\sharp}={{\vert \sigma_1^{\rm y} \rangle}_\sharp}$ (gray) and different magnetic fields (same styling as in Fig. \[fig:figA\]) as they appear in the legend from top to bottom. Thick dashed lines correspond to $(1/3)\ln t$ and $(1/6)\ln t$. []{data-label="fig:figB"}](figB.eps){width="\columnwidth"}
Let us define an infinite index operator $F = \cdots X_{-1}Y_{-1}X_0 Y_0$ which corresponds to a half-filled Fermi sea ${{\vert F \rangle}_\sharp}$ of a-fermions. Any operator of interest here can be written as $A = F B$ where $B$ is a finite-index operator, again ${{\vert A \rangle}_\sharp}$ can be intepreted as a Fermi sea with a finite number of particle and hole excitations. Figure \[fig:figB\] shows results for $S(t)$ (\[eq:S\]) based on numerical solution of Heisenberg equations for ${ {\hat w} }_j(t)$ for $2L$ up to $800$ such that no finite size effects are noticable in the figure. For any initial operator of the form $A = F B $, we observe a clean asymptotic logarithmic growth $$S(t) \asymp c \ln t + c',
\textrm{ where }
c = \left\{ \begin{array}{ll}
1/6 & \textrm{ if } |h| = 1; \\
1/3 & \textrm{ if } |h| \neq 1,
\end{array}
\right.
\label{eq:Sc}$$ and $c'$ is a constant which, for given $h$, only depends on the choice of finite index operator $B$. Note an intriguing similarity with the size scaling of entanglement entropy of the ground state of (\[eq:H\]) [@latorre; @calabreseJSM2004]. Analytical explanation of this interesting phenomenon may be non-trivial, so in the following we limit ourselves to the case of critical field $h=1$ and consider only the simplest initial operator of infinite index, namely $A=F$ where the problem can be connected to the theory of block Toeplitz determinants. In order to avoid negative indices, we re-define the correlation matrix as $\Gamma'_{mn} \equiv
\Gamma_{-m,-n}$, so from eq. (\[eq:approximationgamma\]) follows $$\Gamma'_{mn} = (-1)^{m+n}\sum_{b=0}^\infty J_{b-m}J_{b-n},
\quad
m,n=0,1,\ldots
\label{eq:gammaZZ}$$ It should be noted that correlation matrix can be factorized $\Gamma'_{mn} = \sum_{l=0}^\infty \Psi_{m l} \Psi_{l n}$ as a square of a matrix $\Psi_{mn} = (-1)^n J_{m-n}$. Note that $\Psi_{mn}$ is a real symmetric infinite block Toeplitz matrix $$\Psi =
\begin{pmatrix} \Pi_0 & \Pi_1 & \ddots \\
\Pi_{-1} & \Pi_0 & \ddots \\
\ddots & \ddots & \ddots
\end{pmatrix},\quad
\Pi_l \equiv \begin{pmatrix}
J_{2l} & J_{2l+1} \\
-J_{2l-1} & -J_{2l}
\end{pmatrix}.$$ Following Ref. [@jinJSP116] we express the time-dependent entanglement entropy (\[eq:S\]) in terms of a formula involving Block Toeplitz determinant $$S = \frac{1}{2\pi { {\rm i} }} \int_{\Xi} e(\lambda^2) \big[
\frac{{ {\rm d} }}{{ {\rm d} }\lambda}\ln \det(\lambda\one - \Psi)
\big] { {\rm d} }\lambda
\label{eq:residua}$$ where $\Xi$ is a closed curve in complex plane enclosing unit disk, avoiding point $1$ and interval $[-1,0]$. Note that eigenvalues of infinite dimensional matrix $\Psi$ come in pairs $\lambda,-\lambda$ with accumulation points $\pm 1$. For any $\epsilon > 0$ there is only a finite number, $N_\epsilon(t) \sim t$, of eigenvalues of $\Psi$ which are not in $\epsilon$ vicinity of $\pm 1$. However, we find numerically that most of these eigenvalues cluster around $0$, and only $\sim \ln t$ of them lie outside $\epsilon$ vicinity of $0$ which are the only eigenvalues contributing to entanglement entropy result (\[eq:Sc\]).
At present state of the theory of block Toeplitz determinants - in connection to the theory of integrable Fredholm operators and the Riemann-Hilbert problem [@deiftAMST] - the formula (\[eq:residua\]) can be explicitly evaluated (see e.g. Ref.[@itsJPA38]) provided the [*matrix symbol*]{} $\Phi(z) = \lambda \one - \sum_{k\in{\mathbb{Z}}} \Pi_k z^k$, admits explicit Wiener-Hopf factorizatons $\Phi(z) = U^+(z) U^-(z)=V^-(z) V^+(z)$ where the matrix functions $U^{\pm}(z),V^{\pm}(z)$ are analytic inside($+$)/outsize($-$) the unit circle. Even though the matrix symbol has an appealing explicit form $$\Phi(z) = \left( \begin{array}{cc}
\lambda - f \! \bar{f} + g \bar{g} & f \bar{g}/z - g \bar{f} \\
z g \bar{f} - f \bar{g} & \lambda + f \! \bar{f} - g \bar{g}
\end{array} \right)$$ where $f=f(z),\bar{f}=f(z^{-1}),g=g(z),\bar{g}=g(z^{-1})$ and $f(z) \equiv {\rm cosh}(2t \sqrt{z})$ and $g(z) \equiv {\rm sinh}(2t \sqrt{z})/\sqrt{z}$ are [*entire*]{} analytic functions, its Wiener-Hopf factorizaton is at present unknown and poses a challenging problem.
Conclusions {#sect:conclusions}
===========
We have studied complexity of time evolution of initial local operators under dynamics given by the transverse Ising chain. Such complexity can be characterized in terms of entanglement entropy of operators treated as elements of a product Hilbert space corresponding to a bi-partition of a chain and is directly related to time efficiency of simulation methods such as tDMRG. Note that operator space entanglement entropy, of say a density operator, is [*not*]{} simply related to a traditional notion of entanglement of the corresponding mixed state. For example, consider a macroscopic convex combination of $~2^L$ product states. This corresponds to a [*non-entangled*]{} mixed state but has a macroscopic operator space entanglement entropy $\sim L$ and hence it is difficult to simulate classically. Thus it seems that the traditional concept of [*state entanglement*]{} is not sufficient to characterize classical complexity of quantum operators. In this paper we have shown, in parts analytically and numerically, that operator space entanglement entropy of transverse Ising model grows at most logarithmically for initial operators which are local products of Pauli matrices. This result has to be contrasted with a a linear growth of entanglement entropy for time evolution of pure states [@calabreseJSM4]. Explanation of deeper physical reasons for this dramatic effect is needed.
Stimulating discussions with J. Eisert and M. Žnidarič and support by the grants P1-0044 and J1-7347 of Slovenian Research Agency are gratefully acknowledged.
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abstract: 'We present Hitomi observations of N132D, a young, X-ray bright, O-rich core-collapse supernova remnant in the Large Magellanic Cloud (LMC). Despite a very short observation of only 3.7 ks, the Soft X-ray Spectrometer (SXS) easily detects the line complexes of highly ionized S K and Fe K with 16–17 counts in each. The Fe feature is measured for the first time at high spectral resolution. Based on the plausible assumption that the Fe K emission is dominated by He-like ions, we find that the material responsible for this Fe emission is highly redshifted at $\sim$800 km s$^{-1}$ compared to the local LMC interstellar medium (ISM), with a 90% credible interval of 50–1500 km s$^{-1}$ if a weakly informative prior is placed on possible line broadening. This indicates (1) that the Fe emission arises from the supernova ejecta, and (2) that these ejecta are highly asymmetric, since no blue-shifted component is found. The S K velocity is consistent with the local LMC ISM, and is likely from swept-up ISM material. These results are consistent with spatial mapping that shows the He-like Fe concentrated in the interior of the remnant and the S tracing the outer shell. The results also show that even with a very small number of counts, direct velocity measurements from Doppler-shifted lines detected in extended objects like supernova remnants are now possible. Thanks to the very low SXS background of $\sim$1 event per spectral resolution element per 100 ks, such results are obtainable during short pointed or slew observations with similar instruments. This highlights the power of high-spectral-resolution imaging observations, and demonstrates the new window that has been opened with Hitomi and will be greatly widened with future missions such as the X-ray Astronomy Recovery Mission (XARM) and Athena.'
author:
- 'Hitomi Collaboration, Felix <span style="font-variant:small-caps;">Aharonian</span>, Hiroki <span style="font-variant:small-caps;">Akamatsu</span>, Fumie <span style="font-variant:small-caps;">Akimoto</span>, Steven W. <span style="font-variant:small-caps;">Allen</span>, Lorella <span style="font-variant:small-caps;">Angelini</span>, Marc <span style="font-variant:small-caps;">Audard</span>, Hisamitsu <span style="font-variant:small-caps;">Awaki</span>, Magnus <span style="font-variant:small-caps;">Axelsson</span>, Aya <span style="font-variant:small-caps;">Bamba</span>, Marshall W. <span style="font-variant:small-caps;">Bautz</span>, Roger <span style="font-variant:small-caps;">Blandford</span>, Laura W. <span style="font-variant:small-caps;">Brenneman</span>, Gregory V. <span style="font-variant:small-caps;">Brown</span>, Esra <span style="font-variant:small-caps;">Bulbul</span>, Edward M. <span style="font-variant:small-caps;">Cackett</span>, Maria <span style="font-variant:small-caps;">Chernyakova</span>, Meng P. <span style="font-variant:small-caps;">Chiao</span>, Paolo S. <span style="font-variant:small-caps;">Coppi</span>, Elisa <span style="font-variant:small-caps;">Costantini</span>, Jelle <span style="font-variant:small-caps;">de Plaa</span>, Cor P. <span style="font-variant:small-caps;">de Vries</span>, Jan-Willem <span style="font-variant:small-caps;">den Herder</span>, Chris <span style="font-variant:small-caps;">Done</span>, Tadayasu <span style="font-variant:small-caps;">Dotani</span>, Ken <span style="font-variant:small-caps;">Ebisawa</span>, Megan E. <span style="font-variant:small-caps;">Eckart</span>, Teruaki <span style="font-variant:small-caps;">Enoto</span>, Yuichiro <span style="font-variant:small-caps;">Ezoe</span>, Andrew C. <span style="font-variant:small-caps;">Fabian</span>, Carlo <span style="font-variant:small-caps;">Ferrigno</span>, Adam R. <span style="font-variant:small-caps;">Foster</span>, Ryuichi <span style="font-variant:small-caps;">Fujimoto</span>, Yasushi <span style="font-variant:small-caps;">Fukazawa</span>, Akihiro <span style="font-variant:small-caps;">Furuzawa</span>, Massimiliano <span style="font-variant:small-caps;">Galeazzi</span>, Luigi C. <span style="font-variant:small-caps;">Gallo</span>, Poshak <span style="font-variant:small-caps;">Gandhi</span>, Margherita <span style="font-variant:small-caps;">Giustini</span>, Andrea <span style="font-variant:small-caps;">Goldwurm</span>, Liyi <span style="font-variant:small-caps;">Gu</span>, Matteo <span style="font-variant:small-caps;">Guainazzi</span>, Yoshito <span style="font-variant:small-caps;">Haba</span>, Kouichi <span style="font-variant:small-caps;">Hagino</span>, Kenji <span style="font-variant:small-caps;">Hamaguchi</span>, Ilana M. <span style="font-variant:small-caps;">Harrus</span>, Isamu <span style="font-variant:small-caps;">Hatsukade</span>, Katsuhiro <span style="font-variant:small-caps;">Hayashi</span>, Takayuki <span style="font-variant:small-caps;">Hayashi</span>, Kiyoshi <span style="font-variant:small-caps;">Hayashida</span>, Junko S. <span style="font-variant:small-caps;">Hiraga</span>, Ann <span style="font-variant:small-caps;">Hornschemeier</span>, Akio <span style="font-variant:small-caps;">Hoshino</span>, John P. <span style="font-variant:small-caps;">Hughes</span>, Yuto <span style="font-variant:small-caps;">Ichinohe</span>, Ryo <span style="font-variant:small-caps;">Iizuka</span>, Hajime <span style="font-variant:small-caps;">Inoue</span>, Yoshiyuki <span style="font-variant:small-caps;">Inoue</span>, Manabu <span style="font-variant:small-caps;">Ishida</span>, Kumi <span style="font-variant:small-caps;">Ishikawa</span>, Yoshitaka <span style="font-variant:small-caps;">Ishisaki</span>, Masachika <span style="font-variant:small-caps;">Iwai</span>, Jelle <span style="font-variant:small-caps;">Kaastra</span>, Tim <span style="font-variant:small-caps;">Kallman</span>, Tsuneyoshi <span style="font-variant:small-caps;">Kamae</span>, Jun <span style="font-variant:small-caps;">Kataoka</span>, Satoru <span style="font-variant:small-caps;">Katsuda</span>, Nobuyuki <span style="font-variant:small-caps;">Kawai</span>, Richard L. <span style="font-variant:small-caps;">Kelley</span>, Caroline A. <span style="font-variant:small-caps;">Kilbourne</span>, Takao <span style="font-variant:small-caps;">Kitaguchi</span>, Shunji <span style="font-variant:small-caps;">Kitamoto</span>, Tetsu <span style="font-variant:small-caps;">Kitayama</span>, Takayoshi <span style="font-variant:small-caps;">Kohmura</span>, Motohide <span style="font-variant:small-caps;">Kokubun</span>, Katsuji <span style="font-variant:small-caps;">Koyama</span>, Shu <span style="font-variant:small-caps;">Koyama</span>, Peter <span style="font-variant:small-caps;">Kretschmar</span>, Hans A. <span style="font-variant:small-caps;">Krimm</span>, Aya <span style="font-variant:small-caps;">Kubota</span>, Hideyo <span style="font-variant:small-caps;">Kunieda</span>, Philippe <span style="font-variant:small-caps;">Laurent</span>, Shiu-Hang <span style="font-variant:small-caps;">Lee</span>, Maurice A. <span style="font-variant:small-caps;">Leutenegger</span>, Olivier <span style="font-variant:small-caps;">Limousin</span>, Michael <span style="font-variant:small-caps;">Loewenstein</span>, Knox S. <span style="font-variant:small-caps;">Long</span>, David <span style="font-variant:small-caps;">Lumb</span>, Greg <span style="font-variant:small-caps;">Madejski</span>, Yoshitomo <span style="font-variant:small-caps;">Maeda</span>, Daniel <span style="font-variant:small-caps;">Maier</span>, Kazuo <span style="font-variant:small-caps;">Makishima</span>, Maxim <span style="font-variant:small-caps;">Markevitch</span>, Hironori <span style="font-variant:small-caps;">Matsumoto</span>, Kyoko <span style="font-variant:small-caps;">Matsushita</span>, Dan <span style="font-variant:small-caps;">McCammon</span>, Brian R. <span style="font-variant:small-caps;">McNamara</span>, Missagh <span style="font-variant:small-caps;">Mehdipour</span>, Eric D. <span style="font-variant:small-caps;">Miller</span>, Jon M. <span style="font-variant:small-caps;">Miller</span>, Shin <span style="font-variant:small-caps;">Mineshige</span>, Kazuhisa <span style="font-variant:small-caps;">Mitsuda</span>, Ikuyuki <span style="font-variant:small-caps;">Mitsuishi</span>, Takuya <span style="font-variant:small-caps;">Miyazawa</span>, Tsunefumi <span style="font-variant:small-caps;">Mizuno</span>, Hideyuki <span style="font-variant:small-caps;">Mori</span>, Koji <span style="font-variant:small-caps;">Mori</span>, Koji <span style="font-variant:small-caps;">Mukai</span>, Hiroshi <span style="font-variant:small-caps;">Murakami</span>, Richard F. <span style="font-variant:small-caps;">Mushotzky</span>, Takao <span style="font-variant:small-caps;">Nakagawa</span>, Hiroshi <span style="font-variant:small-caps;">Nakajima</span>, Takeshi <span style="font-variant:small-caps;">Nakamori</span>, Shinya <span style="font-variant:small-caps;">Nakashima</span>, Kazuhiro <span style="font-variant:small-caps;">Nakazawa</span>, Kumiko K. <span style="font-variant:small-caps;">Nobukawa</span>, Masayoshi <span style="font-variant:small-caps;">Nobukawa</span>, Hirofumi <span style="font-variant:small-caps;">Noda</span>, Hirokazu <span style="font-variant:small-caps;">Odaka</span>, Takaya <span style="font-variant:small-caps;">Ohashi</span>, Masanori <span style="font-variant:small-caps;">Ohno</span>, Takashi <span style="font-variant:small-caps;">Okajima</span>, Naomi <span style="font-variant:small-caps;">Ota</span>, Masanobu <span style="font-variant:small-caps;">Ozaki</span>, Frits <span style="font-variant:small-caps;">Paerels</span>, Stéphane <span style="font-variant:small-caps;">Paltani</span>, Robert <span style="font-variant:small-caps;">Petre</span>, Ciro <span style="font-variant:small-caps;">Pinto</span>, Frederick S. <span style="font-variant:small-caps;">Porter</span>, Katja <span style="font-variant:small-caps;">Pottschmidt</span>, Christopher S. <span style="font-variant:small-caps;">Reynolds</span>, Samar <span style="font-variant:small-caps;">Safi-Harb</span>, Shinya <span style="font-variant:small-caps;">Saito</span>, Kazuhiro <span style="font-variant:small-caps;">Sakai</span>, Toru <span style="font-variant:small-caps;">Sasaki</span>, Goro <span style="font-variant:small-caps;">Sato</span>, Kosuke <span style="font-variant:small-caps;">Sato</span>, Rie <span style="font-variant:small-caps;">Sato</span>, Toshiki <span style="font-variant:small-caps;">Sato</span>, Makoto <span style="font-variant:small-caps;">Sawada</span>, Norbert <span style="font-variant:small-caps;">Schartel</span>, Peter J. <span style="font-variant:small-caps;">Serlemtsos</span>, Hiromi <span style="font-variant:small-caps;">Seta</span>, Megumi <span style="font-variant:small-caps;">Shidatsu</span>, Aurora <span style="font-variant:small-caps;">Simionescu</span>, Randall K. <span style="font-variant:small-caps;">Smith</span>, Yang <span style="font-variant:small-caps;">Soong</span>, [Ł]{}ukasz <span style="font-variant:small-caps;">Stawarz</span>, Yasuharu <span style="font-variant:small-caps;">Sugawara</span>, Satoshi <span style="font-variant:small-caps;">Sugita</span>, Andrew <span style="font-variant:small-caps;">Szymkowiak</span>, Hiroyasu <span style="font-variant:small-caps;">Tajima</span>, Hiromitsu <span style="font-variant:small-caps;">Takahashi</span>, Tadayuki <span style="font-variant:small-caps;">Takahashi</span>, Shin’ichiro <span style="font-variant:small-caps;">Takeda</span>, Yoh <span style="font-variant:small-caps;">Takei</span>, Toru <span style="font-variant:small-caps;">Tamagawa</span>, Takayuki <span style="font-variant:small-caps;">Tamura</span>, Takaaki <span style="font-variant:small-caps;">Tanaka</span>, Yasuo <span style="font-variant:small-caps;">Tanaka</span>, Yasuyuki T. <span style="font-variant:small-caps;">Tanaka</span>, Makoto S. <span style="font-variant:small-caps;">Tashiro</span>, Yuzuru <span style="font-variant:small-caps;">Tawara</span>, Yukikatsu <span style="font-variant:small-caps;">Terada</span>, Yuichi <span style="font-variant:small-caps;">Terashima</span>, Francesco <span style="font-variant:small-caps;">Tombesi</span>, Hiroshi <span style="font-variant:small-caps;">Tomida</span>, Yohko <span style="font-variant:small-caps;">Tsuboi</span>, Masahiro <span style="font-variant:small-caps;">Tsujimoto</span>, Hiroshi <span style="font-variant:small-caps;">Tsunemi</span>, Takeshi Go <span style="font-variant:small-caps;">Tsuru</span>, Hiroyuki <span style="font-variant:small-caps;">Uchida</span>, Hideki <span style="font-variant:small-caps;">Uchiyama</span>, Yasunobu <span style="font-variant:small-caps;">Uchiyama</span>, Shutaro <span style="font-variant:small-caps;">Ueda</span>, Yoshihiro <span style="font-variant:small-caps;">Ueda</span>, Shin’ichiro <span style="font-variant:small-caps;">Uno</span>, C. Megan <span style="font-variant:small-caps;">Urry</span>, Eugenio <span style="font-variant:small-caps;">Ursino</span>, Shin <span style="font-variant:small-caps;">Watanabe</span>, Norbert <span style="font-variant:small-caps;">Werner</span>, Dan R. <span style="font-variant:small-caps;">Wilkins</span>, Brian J. <span style="font-variant:small-caps;">Williams</span>, Shinya <span style="font-variant:small-caps;">Yamada</span>, Hiroya <span style="font-variant:small-caps;">Yamaguchi</span>, Kazutaka <span style="font-variant:small-caps;">Yamaoka</span>, Noriko Y. <span style="font-variant:small-caps;">Yamasaki</span>, Makoto <span style="font-variant:small-caps;">Yamauchi</span>, Shigeo <span style="font-variant:small-caps;">Yamauchi</span>, Tahir <span style="font-variant:small-caps;">Yaqoob</span>, Yoichi <span style="font-variant:small-caps;">Yatsu</span>, Daisuke <span style="font-variant:small-caps;">Yonetoku</span>, Irina <span style="font-variant:small-caps;">Zhuravleva</span>, Abderahmen <span style="font-variant:small-caps;">Zoghbi</span>,'
title: 'Hitomi Observations of the LMC SNR N132D: Highly Redshifted X-ray Emission from Iron Ejecta [^1]'
---
Introduction {#sect:intro}
============
As the main drivers for matter and energy in the Universe, supernova remnants (SNRs) are excellent laboratories for studying nucleosynthesis yields and for probing the supernova (SN) engine and dynamics. Core-collapse SNRs, in particular, address fundamental questions related to the debated explosion mechanism and the aftermath of exploding a massive star.
The mechanism of core-collapse supernova explosions has been one of the central mysteries in stellar astrophysics. While one-dimensional simulations have failed to explode a star, only very recently, successful explosions of massive stars have been achieved in three-dimensional simulations invoking convection or standing accretion shock instabilities (SASI; see [@Janka2016] for a recent review). The ejecta composition and dynamics as a function of the progenitor star’s mass and environment have formed another puzzle, with predictions largely relying on the assumption of spherically symmetric models and with yields that vary depending on metallicity, mass loss, explosion energy, and other assumptions (e.g., [@Nomoto2006; @WoosleyHeger2007]).
Significant progress has been made to answer these central questions, thanks to high-resolution imaging and spectroscopic mapping of ejecta (in space and velocity) in core-collapse SNRs, including the oxygen-rich, very young and bright Cassiopeia A SNR in our Galaxy [@HwangLaming2012; @Grefenstette2014] and more evolved SNRs with ejecta signatures such as the O-rich Galactic SNRs G292.2$+$1.8 [@Park2007; @Kamitsukasa2014] and Puppis A [@Hwang2008; @Katsuda2013], and the ejecta-dominated SNR W49B ([@Lopez2013a], ). While such observations have opened a new window to understanding the physics and aftermath of core-collapse explosions, several complications remain in interpreting the observations. First, resolving ejecta from the shocked interstellar medium (ISM) requires fine spectral resolution of extended objects in the X-ray. Second, there is a strong dependence of the elemental distribution and plasma state on both the evolutionary stage of the SNR and on the surrounding environment shaped by the exploded progenitor star. Mixed-morphology SNRs, expanding into an inhomogeneous medium and often interacting with molecular clouds, need the additional treatment of over-ionized (recombining) plasma, as opposed to under-ionized (ionizing) plasma in the younger remnants or SNRs expanding into a low-density and homogeneous medium (e.g., [@Ozawa2009; @Uchida2015]). The advent of high-spectral-resolution imaging detectors such as the Soft X-ray Spectrometer (SXS) aboard Hitomi has promised to revolutionize our three-dimensional mapping of ejecta dynamics and composition, while spectroscopically differentiating between shocked ejecta and the shocked circumstellar/interstellar environment shaped by the progenitor star [@Takahashi16; @Hughes2014].
A natural early target for Hitomi was N132D, the X-ray brightest SNR in the LMC, with an age estimated to be $\sim$2500yr [@VogtDopita2011]. High-velocity ejecta were first detected and studied in optical wavelengths in N132D [@Danziger76; @Sutherland95a; @Morse95; @Morse96]. Optical/UV spectra from the Hubble Space Telescope show strong emission of C/Ne-burning elements (i.e., C, O, Ne, Mg), but little emission from O-burning elements (i.e., Si, S), leading to an interpretation of a Type Ib core-collapse supernova origin for this SNR [@Blair00].
In the X-ray band, the Einstein Observatory made the first observation of N132D, revealing its clear shell-like morphology [@Mathewson83] which has been interpreted as arising from the SN blast wave expanding within a cavity produced by the progenitor star’s H <span style="font-variant:small-caps;">ii</span> region [@Hughes1987]. Einstein also performed the first high-resolution spectral observations with the Focal Plane Crystal Spectrometer (FPCS), clearly seeing strong oxygen and other emission lines and obtaining the first measurements of line flux ratios and constraints on the temperature and ionization state [@Hwang93]. The following ASCA observations revealed that elemental abundances of the entire SNR are consistent with the mean LMC values. This suggests that the X-ray-emitting plasma is dominated by the swept-up ISM [@Hughes98]. Beppo-SAX detected Fe K line emission arising from a hot plasma [@Favata97]. High-resolution X-ray images from XMM-Newton and Chandra have shown that the Fe K-emitting material is concentrated in the interior of the SNR, contrasting with the material emitting at softer energies of O, Ne, Mg, Si, S, and Fe L [@Behar01; @Canizares01; @Borkowski07; @XiaoChen08; @Plucinsky2016]. X-ray emission from O-rich ejecta knots has also been discovered with Chandra, showing a spatial correlation with the optical O emission [@Borkowski07]. The centroid and intensity of the Fe K line emission measured with Suzaku support the core-collapse origin [@Yamaguchi14b]. Very recently, a combined NuSTAR and Suzaku analysis revealed that the hot, Fe K-emitting plasma is in a recombining state with a large relaxation timescale of $\sim$10$^{12}$cm$^{-3}$s, implying that the plasma underwent rapid cooling in the very beginning of its life [@Bamba17].
N132D is the brightest among all known SNRs in GeV and TeV bands [@Ackermann16; @HESS2015]. The spectral energy distribution from radio to gamma-rays including synchrotron X-rays detected with NuSTAR suggests that the gamma-ray emission originates from hadronic processes [@Bamba17]. The total proton energy required to explain the spectral energy distribution was derived to be $\sim$10$^{50}$erg, showing that N132D is an efficient particle accelerator.
We here present Hitomi observations of N132D. These commissioning phase observations were expected to explore the emission line structure of the remnant with exquisite spectral resolution, unprecedented for an extended object at the energy of Fe K ($\sim$6.7 keV). Unfortunately, due to poor satellite attitude control during the majority of the observation (see section \[sect:obs\] for details), only a short exposure was obtained with the Hitomi/SXS microcalorimeter. Nevertheless, owing to the excellent spectral resolution and gain accuracy of the SXS, we detect spectral features of strong emission from S, Ar, and Fe, allowing us to investigate the bulk velocity of the shocked material in this SNR using the Doppler shift of these emission lines. We demonstrate the superior capability of high-resolution spectrometers particularly for low-statistics data, which provide positive prospects for future observations of distant or faint objects with future X-ray microcalorimeter missions, like the X-ray Astronomy Recovery Mission (XARM), Athena [@AthenaWP2013], and Lynx[^2]. We also present the analysis of Soft X-ray Imager (SXI) data, simultaneously obtained from this observation but with longer exposure (and hence higher statistics) owing to its wide field of view (FoV).
This paper is organized as follows. In section \[sect:obs\], we describe the details of the Hitomi observations. We present spectral analysis of the SXS and SXI in sections \[sect:sxs\_analysis\] and \[sect:sxi\_analysis\], respectively. We discuss the results in section \[sect:disc\] and summarize in section \[sect:conc\]. Throughout the paper, we assume 50 kpc for the distance to the LMC [@Westerlund90], and $v_{\rm helio,LMC} = 275 \pm 4$ km s$^{-1}$ as the heliocentric velocity of the LMC ISM immediately surrounding N132D [@VogtDopita2011]. Heliocentric velocities noted by $v_{\rm helio}$ have been corrected to the Solar System barycentric standard of rest. The errors quoted in the text and table represent the 90% confidence level, and the error bars given in the spectra represent 68% confidence.
Observations and Data Reduction {#sect:obs}
===============================
The Hitomi X-ray Observatory was launched in February 2016 and tragically lost at the end of March [@Takahashi16]. During the month of operation, the SXS successfully demonstrated its in-orbit performance by achieving an unprecedented spectral resolution ($\Delta E \approx 5$eV) across a broad energy (2–12keV) for extended sources [@Kelley16; @Porter16]. This led to accurate determination of the turbulent velocity of hot plasma in the Perseus Cluster by measuring the line width of the Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ fine structure [@Hitomi16].
After the Perseus observations, Hitomi aimed at the SNR N132D for performance verification of the SXS and SXI using another line-rich source. The other detectors, the Hard X-ray Imager (HXI) and Soft Gamma-ray Detector (SGD), were not yet turned on. Unfortunately, the satellite attitude control system lost control about 30 minutes after the observation started due to problems in the star tracker system, as illustrated in figure \[fig:att\]. Because of this, the SNR drifted out of the $3\arcmin\times3\arcmin$ SXS FoV and remained out of view for the remainder of the observation. Thanks to its larger FoV, the SXI was able to observe the source during the entire observation.
(100,80mm)[fig1a.eps]{} (60mm,80mm)[fig1b.eps]{}
\[fig:att\]
As this observation took place during the commissioning phase, several instrument settings were non-standard compared to expected science operation. First, the SXS gate valve was in the closed configuration to reduce the chance of molecular contamination from spacecraft out-gassing. The gate valve had a $\sim$260 $\mu$m thick Be window to allow observations while closed, but this absorbed almost all X-rays below $\sim$2 keV and reduced the effective area by $\sim$50% at higher energies [@Eckart2016]. Thus we limit our SXS analysis to the 2–10 keV regime. Second, while the SXS was close to thermal equilibrium at this point in the commissioning phase [@Fujimoto2016; @Noda2016], no on-orbit, full-array energy scale (or gain) calibration had been performed with the filter-wheel calibration sources. The Modulated X-ray Source (MXS; de Vries et al. 2017) was also not available for contemporaneous gain measurement. A dedicated calibration pixel that was outside of the aperture and continuously illuminated by a collimated $^{55}$Fe source served as the only contemporaneous energy-scale reference, and the time-dependent scaling required to correct its gain was applied to each pixel in the array [@Porter16b]. It was well known prior to launch that the time-dependent gain-correction function for this calibration pixel generally would not adequately correct the energy scale of the array pixels. In particular, the relationship between changes on the calibration pixel and on the array was not fixed, but rather depended on the temperatures of various shields and interfaces in the SXS dewar. Therefore, although the relative drift rates across the array were characterized during a later calibration with the filter-wheel $^{55}$Fe source (Eckart et al. in prep.), changes in SXS cryocooler settings between the N132D observation and that calibration limit the usefulness of that characterization.
In fact, the measured relative gain drift predicts a much larger energy-scale offset between the final two pointings of the Perseus Cluster than was actually observed. Using source-free SXS observations taken during the period with the same cryocooler settings as the N132D observation (2016 March 7–-15) in order to circumvent this limitation, we measured the center of the Mn K$\alpha$ instrumental line (Kilbourne et al. in prep.), and conclude that the SXS energy scale is shifted by at most $+1\pm0.5$ eV at 5.9 keV (Eckart et al. in prep.). There are no sufficiently strong low-energy lines in the same data set, but extrapolating from Perseus Cluster observations, we estimate a gain shift of $-2\pm1$ eV at 2 keV (Hitomi Collaboration, in prep. \[Perseus cluster atomic data paper\]). In the filter-wheel $^{55}$Fe data set, errors in the position of the Mn K$\beta$ line ranged from $-$0.6 to $+$0.2 eV across the array. Since this line is at 6.5 keV, less than 1 keV from the Mn K$\alpha$ reference line, gain errors at other energies further from the reference may be substantial. This is especially true in science data, for which drift of the energy scale can only be corrected via the data from the calibration pixel. To be conservative, we use a systematic gain error of $\pm2$ eV at all energies in the analysis below.
We analyzed the cleaned event data of the final pipeline processing (Hitomi software version 6, CALDB version 7) with the standard screening for both SXS and SXI [@Angelini16], with one exception. To maximize the good SXS observing time, we relaxed the requirement that eliminates data when the aimpoint is further than 1$.\!\!^\prime$5 from the target position. Using a maximum angular offset of 2$.\!\!^\prime$2 ensures that at least 50% of the SNR is still in the FoV, and it increases the good SXS exposure time from 2,610 s to 3,737 s (by 43%) and the total SXS counts in the 2–10 keV band from 198 to 233 (by 18%). Relaxing this criterion increased the counts in the Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ band (defined in section \[subsect:sxs\_fe\_analysis\]) from 16 to 17, and in the S<span style="font-variant:small-caps;">xv</span> He$\alpha$ band (defined in section \[subsect:sxs\_s\_analysis\]) from 13 to 16. As we show in section \[sect:sxs\_analysis\], with the very low SXS background and very high spectral resolution, this small number of counts is sufficient to derive interesting constraints for the line centers. Some of the additional broad-band counts are from the region outside the N132D emission peak, as shown in figure \[fig:sxscounts\], so they are likely background counts. The extra counts in the lines are consistent with locations in the remnant, also shown in figure \[fig:sxscounts\]; in particular, to the extent that we can infer locations from the $\sim$1 Hitomi PSF, the S counts are found largely in the rim of the remnant, while the additional Fe K count (and all the Fe K counts) are concentrated in the remnant center, consistent with what is seen with XMM-Newton [@Behar01].
(8cm,8cm)[fig2.eps]{}
We constructed an SXS source spectrum by extracting only GRADE Hp (high-resolution primary) events from the entire SXS field of view of OBSID 100041010, and created the redistribution matrix file (RMF) with `sxsrmf`, using the medium size option. The ancillary response file (ARF) was generated with `aharfgen`, using a high-resolution Chandra image as input to the ray-tracing. A non-X-ray background (NXB) spectrum with the same sampling of magnetic cut-off rigidity as the observation and with identical filtering as the source data (except for Earth elevation criteria) was extracted from the SXS archive NXB event file using `sxsnxbgen`. In the 2–10 keV band, we expect $23.2 \pm 0.6$ NXB counts, about 10% of the observed count rate, and corresponding to $\sim$0.4 counts per spectral resolution element per 100 ks. In the narrow bands used for the analysis that follows, the NXB count rate is less than 5% of the observed rate as the SXS NXB is almost featureless and nearly constant over the energy range (Eckart et al. in prep.).
For the SXI, both OBSIDs 100041010 and 100041020 were used, although for the former we enforced the requirement that the aimpoint be within 1$.\!\!^\prime$5 of the target to eliminate complications in constructing a response for a source moving across the FoV. For OBSID 100041020, we used only times when the attitude was stable, although the source was not at the expected aimpoint and was partially obstructed by the chip gaps (see figure \[fig:att\]). The final good exposure time for the SXI was 35.4 ks.
An SXI spectrum was extracted from a 2$.\!^\prime$5 radius circle with center (RA,Dec) = (,). The NXB spectrum was produced with `sxinxbgen`, using the entire SXI FoV excluding the source in order to increase the statistics. To properly scale the NXB normalization between the full FoV and source region, the instrumental lines of Au L$\alpha$ and L$\beta$ were used, producing a scaling factor of 0.0070. RMF and ARF files were generated with and `sxirmf` and `aharfgen`, respectively.
SXS Spectral Analysis {#sect:sxs_analysis}
=====================
With only 233 counts, the SXS spectrum is dominated by Poisson low-count statistics. In addition, with the SXS gate valve closed, the bright emission lines of C, O, Ne, and Mg below 2 keV are not observable. However, three emission features are easily seen in the full-band spectrum shown in figure \[fig:sxsfullspec\], the He$\alpha$ transition features of He-like S ($\sim$2.45 keV), Ar ($\sim$3.1 keV), and Fe ($\sim$6.7 keV). These lines are clearly detected in previous observations dating back to BeppoSAX [@Favata97], although the combination of an extended source and lower sensitivity at these energies complicates their measurement by X-ray grating instruments like Chandra/HETGS and XMM-Newton/RGS. From narrow bands centered on each expected line centroid, the total number of counts and estimated NXB counts are 16 total ($0.30\pm0.07$ NXB) counts for S<span style="font-variant:small-caps;">xv</span> He$\alpha$; 14 total ($0.28\pm0.06$ NXB) counts for Ar<span style="font-variant:small-caps;">xvii</span> He$\alpha$; and 17 total ($0.8\pm0.1$ NXB) counts for Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$. The signal-to-noise of these features and the underlying continuum is insufficient to obtain useful constraints on the metal abundance, temperature, or velocity broadening of the emitting plasma. However, as we show below, given a reasonable spectral model from other sources, the exquisite spectral resolution of SXS allows us to measure the line centers and thus the average line-of-sight Doppler velocity of two of these components, S and Fe.
All spectral fitting described below was performed with XSPEC v12.9.1d [@Arnaud1996], using atomic and non-equilibrium ionization (NEI) emissivity data from AtomDB v3.0.8 [@Foster2012], and abundance ratios from @AndersGrevesse1989. In each restricted fitting region, we allowed only the line-of-sight velocity and normalization of the appropriate thermal component (described below) to vary in the initial fit. While we include the cosmic X-ray background (CXB), it is negligible; a reasonable model for the 2–10 keV contribution of the CXB power law component with $\Gamma = 1.4$, $S$(2–10 keV) = $5.4\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$ arcmin$^{-2}$ (e.g., [@Ueda1999; @Bautzetal2009]) predicts a mean of 1.5 CXB counts across the entire band and less than 0.1 CXB counts in any of the narrow spectral analysis bands. This is less than 1% of the detected counts. Galactic foreground emission is negligible above 2 keV toward this direction ($l =
280\degree$, $b = -32.\!\!\degree8$).
(8cm,8cm)[fig3.eps]{}
Iron Region Spectral Analysis {#subsect:sxs_fe_analysis}
-----------------------------
Fe K emission in N132D has been explored previously [@Favata97; @Behar01; @XiaoChen08; @Yamaguchi14b], with the conclusion that this feature is dominated by Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ emission. The XMM-Newton/EPIC observations are successfully fit above 2.5 keV with a two-temperature-component model with $kT =$ 0.89 and 6.2 keV [@Behar01]. The cooler component produces the strong soft emission lines seen with XMM-Newton/RGS, and the hotter component explains the Fe K emission. In particular, @Behar01 emphasize the lack of a temperature component at $\sim$1.5 keV to explain the lack of observed L-shell emission from Li-, Be-, and B-like Fe in the XMM-Newton spectrum. A recent study using 240 ks of Suzaku data combined with a 60 ks NuSTAR observation [@Bamba17] has produced a two-component broad-band spectral model of N132D with a similar cool temperature ($kT \approx$ 0.8 keV) but that interprets the Fe K emission arising primarily from an over-ionized, recombining plasma component with $kT_{\rm e} = 1.5$ keV, $kT_{\rm init} > 20$ keV, and relaxation timescale $n_e
t~\approx~10^{12}$ s cm$^{-3}$. Crucially, the Suzaku data show a clear detection of H-like Fe Ly$\alpha$ emission, indicating that an under-ionized (ionizing) plasma is unlikely to contribute significantly to the emission at these energies, and thus much of the otherwise unresolved Fe K emission is likely due to He-like Fe rather than lower ionization states.
These previous observations provide confidence that we know where the line centroid should be for the Fe K complex, and can cleanly measure the line-of-sight velocity. However, we emphasize that this is one possible interpretation of a plasma with strong Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ and measurable Fe<span style="font-variant:small-caps;">xxvi</span> Ly$\alpha$ emission. A more complicated temperature structure, such as from multiple unassociated, spatially unresolved components, could produce a very different complex of lines in this spectral region. We address this possibility further in section \[sect:disc\]. To ease comparison to current work, we adopt the model from @Bamba17 as a baseline model, shown in figure \[fig:model\_spec\] and table \[tab:sxs\_params\].
(8cm,8cm)[fig4.eps]{}
\[tab:sxs\_params\]
[lcccc]{} Model Parameter & &\
& no broadening & with broadening$\dagger$ & no broadening & with broadening$\dagger$\
\
$kT$ (keV) &\
$Z_{\rm Si}$ (solar) &\
$Z_{\rm S}$ (solar) &\
$Z_{\rm Fe}$ (solar) &\
$v_{\rm helio}$ (km s$^{-1}$) & & 210$^{+370}_{-380}$ & 520$^{+770}_{-620}$\
$\sigma$ (km s$^{-1}$) & & 0 & 520$^{+780}_{-340}$\
flux, 2–10 keV& & 5.6$^{+2.9}_{-1.9}$ & 5.5$^{+3.1}_{-1.8}$\
flux, fitting band& & 1.3$^{+0.4}_{-0.2}$ & 1.3$^{+0.4}_{-0.2}$\
\
$kT$ (keV) &\
$kT_{\rm init}$ (keV) &\
$n_et$ ($10^{12}$ s cm$^{-3}$) &\
$Z_{\rm Si}$ (solar) &\
$Z_{\rm S}$ (solar) &\
$Z_{\rm Fe}$ (solar) &\
$v_{\rm helio}$ (km s$^{-1}$) & 1440$^{+100}_{-1000}$ & 1140$^{+640}_{-810}$ & 1440 & 1140\
$\sigma$ (km s$^{-1}$) & 0 & 510$^{+1060}_{-330}$ & 0 & 510\
flux, 2–10 keV& 9.5$^{+4.5}_{-3.0}$ & 9.7$^{+4.2}_{-3.2}$ & 6.1 & 6.2\
flux, fitting band& 0.48$^{+0.25}_{-0.16}$ & 0.49$^{+0.24}_{-0.16}$ & 0.34 & 0.34\
\
$\Gamma$ &\
flux, 2–10 keV&\
flux, fitting band& &\
spectral fitting band & &\
C-stat / d.o.f. & 107.9 / 696 & 106.5 / 695 & 61.0 / 157 & 59.1 / 156\
goodness-of-fit (KS)& 24% & 20% & 62% & 31%\
goodness-of-fit (CvM)& 35% & 21% & 62% & 46%\
The Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ complex, shown in figure \[fig:fe\_spec\], was fit within the energy range 6.45–6.80 keV. This range includes sufficient width to constrain the continuum and measure velocity shifts up to $\sim$7000 km s$^{-1}$, but avoids contamination from a possible 6.4 keV Fe K line and any H-like Fe features. It is clear from figure \[fig:model\_spec\] that in this very clean fitting region the model is dominated by emission from the recombining plasma component by at least a factor of 100 over the cooler collisional ionization equilibrium (CIE) component. Therefore, while we included the entire model with all components for the Fe region fit, we only allowed parameters related to the NEI component to vary. To allow for differences in the observed flux due to the smaller SXS FoV and attitude drift, we fixed the ratio of the CIE to NEI component normalizations to that derived by @Bamba17, and allowed the NEI flux to vary along with the line-of-sight velocity. The CIE component was modeled by a variable-abundance `vapec` model in XSPEC, while the NEI component was modeled by a variable-abundance recombining plasma model, `vrnei`. We included a single Gaussian broadening parameter to allow for thermal and turbulent broadening as well as unresolved bulk motion.
(75mm,80mm)[fig5a.eps]{} (75mm,80mm)[fig5b.eps]{}
\[fig:s\_spec\]
Parameter estimation was performed in two ways. First, maximum likelihood estimation was done by minimizing the fit statistic, using `cstat` in XSPEC, a modified @Cash1979 statistic. With the broadening width fixed at zero, this fitting revealed a highly non-monotonic parameter space for the velocity (see figure \[fig:fe\_post\_nobroad\]), likely due to the combination of low-count Poisson statistics in the data and discrete spectral features in the model. The best-fit velocity of $v_{\rm helio} =
1440$ km s$^{-1}$ is significantly larger than the value of the local LMC ISM surrounding N132D, $v_{\rm helio,LMC} = 275\pm4$ km s$^{-1}$ [@VogtDopita2011]. Allowing a free broadening width eliminated this non-monotonicity (see figure \[fig:fe\_post\_broad\]), resulting in a best-fit $v_{\rm helio} = 1140$ km s$^{-1}$ and broadening of $\sigma =
510$ km s$^{-1}$.
Second, to fully explore parameter space, we performed Markov chain Monte Carlo (MCMC) simulations within XSPEC using Bayesian inference. These simulations were run with and without velocity broadening, using both a flat (uniform) prior distribution and a Gaussian prior distribution for the broadening width. The width of the Gaussian prior distribution was chosen to reflect current upper limits on the velocity broadening. In particular, observations with CCD-based X-ray observatories such as Suzaku (e.g. [@Bamba17]) have not found measurable broadening. The typical spectral resolution of such instruments near 6 keV is $\sim$150–180 eV FWHM, depending on the epoch of observation, with a typical 1-$\sigma$ calibration uncertainty of 5%[^3]. This calibration uncertainty can be thought of as an upper limit on the detectable line broadening velocity. Since the broadening is a convolution, this extra velocity component adds in quadrature with the instrumental width. We find that a 5% increase on the 150–180 eV FWHM instrumental width is equivalent to an extra broadening component with FWHM of 48–58 eV, or $\sigma$ = 900–1100 km s$^{-1}$ in the center of our fitting band. We therefore adopted 1000 km s$^{-1}$ as a natural 1-$\sigma$ width to use for the Gaussian prior distribution. We performed MCMC simulations using both the flat, uninformative prior and the weakly informative Gaussian prior.
The MCMC results are consistent with the local `cstat` minima in velocity parameter space for fits with and without broadening, as shown by the MCMC posterior probability distributions in figures \[fig:fe\_post\_nobroad\] and \[fig:fe\_post\_broad\]. In particular, the complicated velocity posterior distribution shows up clearly in the MCMC runs without broadening, but with the most likely value (highest mode) near $v_{\rm helio} = 800$ km s$^{-1}$ instead of 1400 km s$^{-1}$ as found in the `cstat` minimization. The MCMC chain steps shown in figure \[fig:fe\_post\_nobroad\] (right) indicate that the simulation is well-behaved and samples the posterior distribution adequately despite the multimodal structure. The runs with broadening result in Gaussian posterior distributions with peak near 1000 km s$^{-1}$. Using either a Gaussian or Cauchy form for the chain proposal distribution produced the same results.
(80mm,80mm)[fig6a.eps]{} (80mm,80mm)[fig6b.eps]{}
(80mm,80mm)[fig7a.eps]{} (80mm,80mm)[fig7b.eps]{}
We used these posterior distributions to obtain central credible intervals on $v_{\rm helio}$. For the fit with no broadening, a single interval is uninformative due to the complicated structure. We obtain a 68% credible interval of 730–1460 km s$^{-1}$, 90% interval of 440–1540 km s$^{-1}$, and 95% interval of 160–1620 km s$^{-1}$. A line-of-sight velocity consistent with $v_{\rm helio,LMC}$ is ruled out at 93% confidence under this model. With broadening, a single credible interval is sufficient to characterize the Gaussian-shaped distribution, and we find 90% credible intervals of 330–1780 km s$^{-1}$ for broadening with a Gaussian prior distribution, and 0–2090 km s$^{-1}$ for a flat prior. The conservative gain uncertainty of $\pm$2 eV (see section \[sect:obs\]) produces a systematic uncertainty of $\pm$90 km s$^{-1}$, well within the statistical uncertainty. It is apparent that imposing an flat, uninformative prior on the broadening width distribution allows unrealistic values exceeding $\sigma = 3000$ km s$^{-1}$ with a broad tail to very high values. This greatly exceeds the thermal width of an Fe emission feature at 2 keV ($\sigma \sim 50$ km s$^{-1}$), and requires either extreme turbulence or very large bulk motions. If we adopt the results with the Gaussian prior, which has sufficient width to allow a blueshifted and redshifted component separated by up to $\sim$2000 km s$^{-1}$, a mean line-of-sight velocity consistent with $v_{\rm helio,LMC}$ is ruled out at 91% confidence under this model. The model parameters are listed in table \[tab:sxs\_params\].
The measured photon flux in the fitting band, $4.6^{+2.3}_{-1.4}\times10^{-5}$ ph cm$^{-2}$ s$^{-1}$, is more than a factor of two higher than previous estimates of the Fe K$\alpha$ line flux, e.g. $1.83 \pm 0.17 \times10^{-5}$ ph cm$^{-2}$ s$^{-1}$ ([@Yamaguchi14b]; errors are 90%). This is likely due to a combination of the Hitomi attitude uncertainty and the use of a broad-band X-ray image to produce the response files. While much of this broad-band X-ray emission is found in a shell with diameter $\sim\,$2, the Fe K$\alpha$ emission appears centrally concentrated (e.g., [@Behar01]). Using the more spatially extended broad-band image produces a lower response as some of the PSF-broadened flux falls outside of the $3\arcmin\times3\arcmin$ SXS FOV, thereby increasing the inferred model flux for a given count rate. Our inclusion of data with large pointing offset of up to 2.2$\arcmin$ and the large attitude drift undoubtedly exacerbate this effect. For this reason, the flux calibration is so uncertain that a flat, uninformative prior is a good representation of our knowledge of the SXS effective area for this observation.
Once the minimum fit statistic and parameter distribution function were determined, we explored the effects of adjusting other `vrnei` parameters within a reasonable range of uncertainty. In addition, we ran fits testing plasma models with higher over-ionization (setting $n_et$ to a small value), under-ionization (an ionizing plasma, setting $kT_{\rm init}
< kT$), and collisional ionization equilibrium (CIE, setting $kT_{\rm init}
= kT$). The fit statistic was consistent in all cases, indicating that we cannot distinguish between various ionization states with the Hitomi/SXS data alone. In all cases, neither the best-fit velocity nor its posterior distribution from the MCMC analysis changed appreciably, indicating that our results are insensitive to the exact emission model used so long as it is not highly complex.
Neither the XSPEC `cstat` statistic nor the MCMC analysis provides an estimate of the goodness of fit. We used two tests available in XSPEC, Kolmogorov-Smirnov (KS) and Cramer-von Mises (CvM), both of which treat the observed and model spectra as empirical distribution functions and compute a statistical difference between the two. Drawing parameter values for velocity, normalization, and broadening width from the full posterior distributions, we performed 1000 simulations of the observed 3.7 ksec spectrum for the fits with and without broadening. These simulated spectra were then fit with the model, and the resulting KS and CvM test statistics were compared with the values from the original fits. For the fit without broadening, 24% of the realizations produced a smaller KS statistic than the best fit, and 35% produced a smaller CvM statistic. For the fits with broadening, the fractions were 20% for KS and 21% for CvM. We can only say that our best-fit models are not statistically inconsistent with the data.
Since this asymmetric velocity structure is unexpected, we constrained a potential blue-shifted emission feature by adding a second `vrnei` component with identical model parameters. The velocity of the first component was fixed to the best-fit value of 1140 km s$^{-1}$, while that of the new component was fixed to $-590$ km s$^{-1}$, to force symmetry about $v_{\rm helio,LMC}$. The `vrnei` normalizations, initially equal, were allowed to vary independently. We find that a blue-shifted feature is allowed at up to 30% of the flux of the redshifted component, with a similar fit statistic and goodness-of-fit measure. Varying the blueshift within a reasonable range did not improve the fit or change the upper limit to its flux. The best-fit broadening width ($\sigma \sim$ 500 km s$^{-1}$ or FWHM $\sim$ 1200 km s$^{-1}$) allows some blueshifted component, but the emission-weighted mean velocity is not centered on the LMC velocity. We conclude that the bulk of the He-like-iron-bearing material is receding asymmetrically, at a velocity $\sim$800 km s$^{-1}$ with respect to the swept-up ISM surrounding N132D.
Sulfur Region Spectral Analysis {#subsect:sxs_s_analysis}
-------------------------------
(80mm,80mm)[fig8a.eps]{} (80mm,80mm)[fig8b.eps]{}
Spectral fitting of the S<span style="font-variant:small-caps;">xv</span> He$\alpha$ line proceeded in a similar manner to the Fe K region. We restricted the energy range to 2.40–2.48 keV, leaving 16 total counts of which $0.30\pm0.07$ ($\sim$2%) are estimated to be from the NXB. Consistent with other recent work, we interpret the S<span style="font-variant:small-caps;">xv</span> He$\alpha$ emission to arise predominantly from a CIE plasma with $kT \sim$ 1 keV [@Behar01; @Borkowski07; @XiaoChen08]. In our baseline model, the CIE component dominates the NEI emission by a factor of $\sim$5–10 in this region. Thus we allowed some small contamination from the high-redshift NEI emission by freezing the velocity and broadening of the `vrnei` component to the best-fit values, and fixed the ratio of the `vapec` to `vrnei` normalizations to that found by @Bamba17. Only the velocity and normalization of the CIE `vapec` component were allowed to vary in the initial fit, but as with the Fe fit, we included broadening with similar priors to explore the effect on the derived velocity. The S region spectrum and model are shown in figure \[fig:s\_spec\], posterior probability distributions are shown in figure \[fig:s\_post\], and best-fit parameters are given in table \[tab:sxs\_params\].
Using the `cstat` maximum likelihood estimator, we obtain a best-fit line-of-sight velocity of $v_{\rm helio} = 210$ km s$^{-1}$ with broadening fixed at zero. Allowing a single broadening component results in $v_{\rm
helio} = 520$ km s$^{-1}$ with $\sigma = 520$ km s$^{-1}$. As with the Fe fitting, the posterior distributions in figure \[fig:s\_post\] are considerably wider when broadening is included, with 90% credible intervals on $v_{\rm helio}$ of $-$170 to $+$580 km s$^{-1}$ with no broadening and $-$100 to $+$1290 km s$^{-1}$ with a Gaussian prior on broadening with $\sigma =
1000$ km s$^{-1}$. Unlike for Fe, the velocity of the S component is completely unconstrained with a flat broadening prior. Our adopted SXS gain uncertainty of $\pm$2 eV (245 km s$^{-1}$; see section \[sect:obs\]) is again well within this statistical uncertainty, which itself is consistent with the local LMC velocity of 275 km s$^{-1}$.
We performed additional spectral fitting, allowing $kT$ of the CIE component and $kT$, $kT_{\rm init}$, $n_et$, and $\sigma$ of the recombining plasma component to vary over a broad range as in the Fe region fitting described in the previous section. The best-fit velocity and credible intervals did not change. We performed the same goodness-of-fit tests to the S region fits as the Fe region fits, finding that 30–60% of the simulated datasets produced a smaller test statistic. The model is thus consistent with the data, and we conclude that the He-like-sulfur-bearing gas is consistent with being at rest relative to the local LMC ISM, if we assume that line broadening is small.
Argon Region Spectral Analysis {#subsect:sxs_ar_analysis}
------------------------------
Spectral fitting of the Ar<span style="font-variant:small-caps;">xvii</span> He$\alpha$ line is complicated by both the low number of total counts (14) and the estimated contributions from both CIE and NEI components. In fact, the Ar abundance is not constrained in either component, leading to a degeneracy between the normalization and abundance in each component and further difficulty fitting different velocities. As a simple test, we fixed the `vapec` and `vrnei` normalizations to the @Bamba17 values, fixed the Ar abundance to solar for both components, and fit a single line-of-sight velocity and normalization. The best-fit velocity is $v_{\rm helio} = 2400$ km s$^{-1}$, with a 90% credible interval of 570–5900 km s$^{-1}$. This is consistent with both velocity ranges of Fe<span style="font-variant:small-caps;">xxv</span> and S<span style="font-variant:small-caps;">xv</span>. If the velocities are tied at the offset to the best-fit values so that $v_{\texttt{vrnei}} =
v_{\texttt{vapec}} + 1200$ km s$^{-1}$, the fit statistic is only slightly worse (`cstat` = 81.2 vs. 80.8), and the best-fit values are $v_{\rm
helio} = 1800$ km s$^{-1}$ for the `vrnei` component and 600 km s$^{-1}$ for the `vapec`, with similar uncertainties. Given the uncertainties in the model, we can only conclude that the Ar<span style="font-variant:small-caps;">xvii</span> fit is consistent with the Fe and S line results.
SXI Spectral Analysis {#sect:sxi_analysis}
=====================
For the following analysis, the same version of XSPEC, AtomDB, NEI emissivity data, and abundance tables were used as in the analysis of the SXS spectrum (see section \[sect:sxs\_analysis\]). The NXB-subtracted spectrum is shown in figure \[fig:sxi\_spec\]. In the N132D observation, the event and split thresholds are 600 eV and 30 eV, respectively. Since charge from a detected X-rays may be split among multiple CCD pixels, the quantum efficiency (QE) can be affected by split events well above the event threshold. Given the limited amount of calibration information available in these early observations, we conservatively exclude the energy band below 2 keV in this study.
We detect emission lines at $2.456\pm0.010$ keV and $6.68\pm0.04$ keV, which correspond to the same He$\alpha$ lines of S and Fe detected in SXS, respectively. The SXI is affected by light leak when the satellite is in daylight, which can result in an observed line center shift (Nakajima et al. in prep.). We investigated the line center shift in the N132D data, and confirmed that daylight illumination of the spacecraft has no effect. The S<span style="font-variant:small-caps;">xv</span> He$\alpha$ line center is fully consistent with the centroid of the line complex measured with SXS (see figure \[fig:fe\_spec\]). The Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ line center is marginally consistent with SXS within the uncertainty (see figure \[fig:s\_spec\]), and likely includes some unresolved contribution from Fe<span style="font-variant:small-caps;">xxvi</span> Ly$\alpha$ at $\sim$7 keV (see figure \[fig:model\_spec\]).
Following the SXS analysis, we adopted a spectral model with two thin-thermal plasmas, a low-temperature `vapec` and high-temperature `vrnei`. From the model of @Bamba17, we also include a 6.4 keV neutral Fe K line, a non-thermal component, and the CXB. In the SXI analysis, the normalizations of the two plasmas are set to be free and all the other thermal parameters are fixed to those of @Bamba17. The normalization of the Fe <span style="font-variant:small-caps;">i</span> K line was tied to that of the `vrnei` component using the ratio of normalizations from @Bamba17. A power-law model was added for the possible non-thermal component, with both photon-index $\Gamma$ and normalization allowed to vary. For the CXB, another power-law model with fixed parameters of $\Gamma=1.4$ and surface brightness $5.4\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$ arcmin$^{-2}$ in the 2–10 keV band was used [@Ueda1999; @Bautzetal2009]. This CXB intensity is expected from observations with previous X-ray imaging instruments with similar PSF, and thus similar confusion limits. Since we are in the high-counts regime with at least 30 counts per spectral bin in the total (unsubtracted) source spectrum and high statistics in the NXB spectrum, we expect the background-subtracted spectral bins to be Gaussian distributed and use $\chi^2$ minimization. We obtain $\chi^2 / $d.o.f.$=234/243$ and an acceptable fit at the 90% confidence level. The best-fit model with individual components is shown in figure \[fig:sxi\_spec\_fit\]. To check for potential bias in the use of $\chi^2$ statistics, we perform the fit again excluding the poorest statistical region above 9 keV, and obtain similar results.
(7cm,8cm)[fig9.eps]{}
(7cm,8cm)[fig10.eps]{}
The lower and higher temperature plasmas produce the majority of the He$\alpha$ lines of S and Fe, respectively, consistent with the result of the previous study (see also figure \[fig:model\_spec\]). The best-fit `vapec` normalization was $0.92\pm0.03$ of the value from @Bamba17, while the `vrnei` was $0.86\pm0.10$ of their best-fit value. The model fitting results in a non-thermal component with flux $1.3\pm1.1\times10^{-13}$ erg cm$^{-2}$ s$^{-1}$ in the 2–10 keV band. If we assume this non-thermal component exists, the fit constrains the photon index to be $\Gamma<3.0$. With the SXI data in hand, we are unable to say conclusively that the non-thermal component is required, only that it is consistent with the observed spectrum.
Discussion {#sect:disc}
==========
We have revealed a significant redshift of the emission lines of He-like Fe, constraining the line-of-sight velocity to be $\sim$1100 km s$^{-1}$, or $\sim$800 km s$^{-1}$ faster than the local LMC ISM. The emission of S<span style="font-variant:small-caps;">xv</span> He$\alpha$, on the other hand, shows a velocity consistent with the radial velocity of the LMC ISM, albeit with large uncertainty, especially when broadening is included in the model. These results suggest different origins of the Fe and S emission: the former is dominated by the fast-moving ejecta and the latter by the swept-up ISM. This interpretation is consistent with the previous work by XMM-Newton, which revealed that the Fe emission has a centrally-filled morphology and the S emission is found along the outer shell [@Behar01].
This interpretation hinges on our assumed underlying emission model. Previous results from XMM-Newton [@Behar01] and the detection of an Fe<span style="font-variant:small-caps;">xxvi</span> Ly$\alpha$ line in the Suzaku spectrum [@Bamba17] suggest minor contamination from lower-energy, lower-ionization states of Fe. It is possible that the H-like Fe emission arises from a much hotter plasma that does not produce He-like emission, and the Fe K complex in question is produced by lower-temperature plasma unresolved by both the Suzaku and Hitomi PSF. Although L-shell lines of lower-ionization Fe were not detected by [@Behar01], it is further possible that the L-shell energy band is dominated by the low temperature swept-up ISM component, hindering detection of faint ejecta lines. We are unable to conclusively demonstrate the validity of our assumptions with existing X-ray data, and we stress that the discussion that follows assumes the Fe K emission is dominated by He-like Fe.
The best-fit broadening widths for both Fe K and S K, $\sigma \sim 500$ km s$^{-1}$, greatly exceed thermal broadening at these temperatures. It is unclear whether the constraints on broadening are physical or somehow related to the combination of low statistics and complicated line structure in the thermal model. The addition of broadening simplifies the posterior velocity distribution without greatly changing the 90% credible interval, and we can speculate that if this line broadening is physical, there could be Fe K-emitting material at a range of velocities due to bulk motion, including very high ones. Much better statistics at similar spectral resolution are required to further understand the velocity structure in both Fe K and S K.
These Fe-rich ejecta display very different line-of-sight velocity structure compared to the O-rich ejecta explored in detail in the optical. The O-rich ejecta traced by \[O <span style="font-variant:small-caps;">iii</span>\] $\lambda 5007$ emission have an average blueshifted velocity of $\sim\,-500$ km s$^{-1}$ with respect to the local LMC when an elliptical shell model is fit in projected space and velocity [@Morse95]. @VogtDopita2011 confirm this systematic offset, but point out that the complicated spatial structure of the ejecta heavily biases the average velocity of the emission as different clumps interact with the reverse shock at different times. The ring structure of the O-rich ejecta first suggested by @Lasker1980 and confirmed in several successive studies is possibly accompanied by a polar jet associated with a “run-away” knot and the enhanced X-ray emission along the southwestern shell [@VogtDopita2011]. It is tempting to speculate that the Fe emission is associated with such a jet, but a more significant detection at higher spatial resolution is required.
The lack of blue-shifted emission indicates a highly asymmetric distribution of the Fe-rich ejecta. Such asymmetry is seen morphologically in the ejecta of other core-collapse SNRs, such as Cas A [@Grefenstette2017], G292.2$+$1.8 [@Bhalerao2015], and W49B [@Lopez2013a], and in the more evolved SNRs dominated by shocked ISM/CSM but with Fe knots such as Puppis A [@Hwang2008; @Katsuda2008; @Katsuda2013]. Notably, the Fe ejecta in these remnants are not always centrally concentrated, as would be expected in a typical core-collapse explosion. In Cas A, the mismatch between the shocked Fe ejecta and more concentrated, redshifted $^{44}$Ti has been interpreted in light of the SN explosion mechanism involving instabilities such as SASI [@Grefenstette2017]. N132D is more evolved than Cas A and perhaps better compared to W49B, with which it is comparable in age. The X-ray morphology of N132D is more symmetric than W49B, and relatively symmetric among core-collapse SNRs in general [@Lopez11], despite the obvious differences between the bright southern shell and the blown-out northeastern region. This symmetry could indicate a projection effect and an axis of symmetry along the line-of-sight. If N132D were observed perpendicular to the direction it is, it might appear more highly asymmetric, like W49B.
The origin of the over-ionized plasma is not completely clear. Interestingly, both N132D and W49B show evidence for overionization of the Fe ejecta [@Ozawa2009; @Bamba17], suggesting a possible connection between asymmetric ejecta distribution and overionization. In addition, recombining plasma is observed in several mixed-morphology SNRs that are interacting with molecular clouds (e.g., [@Yamaguchi2009; @Uchida2015]); although the mechanism responsible for the peculiar plasma conditions in these remnants is still unclear, a possible connection is the inhomogeneous medium into which the SNR is expanding. In N132D, the entire southern half of the remnant is surrounded in projection by molecular gas, with Mopra 22-m telescope CO data showing that the outer shell is sweeping through the cloud [@Banas1997; @Sano2015]. This molecular gas distribution combined with the X-ray emission morphology showing a brighter shell impinging on the cloud in the south suggest that the shock is slowing here due to the cloud, while the fainter shell blowing out toward the north and northeast suggests that the shock is expanding faster here. The detection of both GeV emission [@Ackermann16] and neutral Fe K [@Bamba17] from N132D further suggest that accelerated protons are interacting with the nearby molecular cloud [@Bamba17]. It is likely that N132D is expanding into a highly inhomogeneous medium.
In W49B, the recombining plasma is detected on the west side of the remnant whereas the molecular cloud is to the east, suggesting that the dominant cooling mechanism producing the over-ionized plasma is rapid expansion of the inner ejecta [@Miceli2010; @Lopez2013a]. A similar density gradient is apparent in N132D, however due to the insufficient spatial resolution of either Suzaku or Hitomi we are unable to identify exactly where the recombining plasma is located. Comparing the Fe K map from XMM-Newton (figure 4a of [@Behar01]) with the molecular gas map (figure 1b of [@Sano2015]), we see that the Fe K peak is not in the center of the remnant nor toward the blown-out low-density northeast region, but offset closer to the bright southeastern shell. Since the recombining Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ is the brightest feature seen in this spectral region, this hints that the over-ionized plasma is located near the molecular cloud. With the data currently in hand, and with likely projection effects along the line-of-sight, a firm conclusion is not possible.
It was unfortunate that the first microcalorimeter observation of a thermally dominated SNR was not fully performed due to an attitude control problem. However, this short-exposure observation of N132D demonstrates the power of high-spectral-resolution detectors by detecting clear emission features with extremely low photon counts—a similarly short CCD observation would not have detected these features, let alone placed interesting constraints on the velocity. The very low SXS background of $\sim$1 event per spectral resolution element per 100 ks is also vital for this result, and it opens the possibility of using slew observations for similar science with similar future instruments. For N132D, revealing the Fe ejecta line-of-sight velocity structure, along with its detailed spatial distribution and proper motion, is a vital step to determine its three-dimensional velocity. Future observations with the X-ray Astronomy Recovery Mission (XARM) microcalorimeter, identical in performance to that on Hitomi, will be sufficient to spatially resolve the remnant into two regions and explore in detail the line-of-sight velocity and ionization state for each element. Observations with Athena [@AthenaWP2013] will also be crucial to more accurately constrain the kinematics and ionization state of this SNR.
Conclusions {#sect:conc}
===========
In this paper, we have presented observations of the LMC SNR N132D taken with Hitomi. Using only a short, 3.7 ks observation with the SXS, we detect emission lines of Fe<span style="font-variant:small-caps;">xxv</span> and S<span style="font-variant:small-caps;">xv</span> He$\alpha$ with only 17 and 16 counts, respectively. Assuming a plausible emission model and prior on the velocity broadening, the Fe line shows a redshift of 800 km s$^{-1}$ (50–1500 km s$^{-1}$ 90% credible interval) compared to the local LMC ISM, indicating that it likely arises from highly asymmetric ejecta. The S line is consistent with the local LMC standard of rest, shifted by $-$65 km s$^{-1}$ ($-$450 to $+$435 km s$^{-1}$ 90% credible interval) assuming no broadening, and likely arises from the swept-up ISM. Longer SXI observations produce results consistent with a recent combined Suzaku+NuSTAR spectral analysis, including a recombining thermal plasma component responsible for the Fe<span style="font-variant:small-caps;">xxv</span> He$\alpha$ emission and constraints on a non-thermal component that dominates at high energies [@Bamba17]. In addition to this first result on SNRs with a microcalorimeter, the observations highlight the power of high-spectral-resolution X-ray imaging instruments in even short exposures.
E. Miller and H. Yamaguchi led this study and wrote the final manuscript along with S. Katsuda, K. Nobukawa, M. Nobukawa, S. Safi-Harb, and M. Sawada. E. Miller, T. Sato, M. Sawada, and H. Yamaguchi performed the SXS data reduction and analysis. K. Nobukawa and M. Nobukawa performed the SXI data reduction and analysis. C. Kilbourne contributed estimates and discussion of the SXS gain uncertainty. A. Bamba contributed the detailed spectral model used for both the SXS and SXI analysis. M. Sawada contributed to optimizing the SXS data screening. K. Mori contributed analysis of the SXI light leak. L. Gallo, J. Hughes, R. Mushotzky, C. Reynolds, T. Sato, M. Tsujimoto, and B. Williams contributed valuable comments on the manuscript. The science goals of Hitomi were discussed and developed over more than 10 years by the ASTRO-H Science Working Group (SWG), all members of which are authors of this manuscript. All the instruments were prepared by joint efforts of the team. The manuscript was subject to an internal collaboration-wide review process. All authors reviewed and approved the final version of the manuscript.
We thank the support from the JSPS Core-to-Core Program. We acknowledge all the JAXA members who have contributed to the ASTRO-H (Hitomi) project. All U.S. members gratefully acknowledge support through the NASA Science Mission Directorate. Stanford and SLAC members acknowledge support via DoE contract to SLAC National Accelerator Laboratory DE-AC3-76SF00515. Part of this work was performed under the auspices of the U.S. DoE by LLNL under Contract DE-AC52-07NA27344. Support from the European Space Agency is gratefully acknowledged. French members acknowledge support from CNES, the Centre National d’Études Spatiales. SRON is supported by NWO, the Netherlands Organization for Scientific Research. Swiss team acknowledges support of the Swiss Secretariat for Education, Research and Innovation (SERI). The Canadian Space Agency is acknowledged for the support of Canadian members. We acknowledge support from JSPS/MEXT KAKENHI grant numbers 15H00773, 15H00785, 15H02090, 15H03639, 15H05438, 15K05107, 15K17610, 15K17657, 16H00949, 16H06342, 16K05295, 16K05300, 16K13787, 16K17672, 16K17673, 21659292, 23340055, 23340071, 23540280, 24105007, 24540232, 25105516, 25109004, 25247028, 25287042, 25400236, 25800119, 26109506, 26220703, 26400228, 26610047, 26800102, JP15H02070, JP15H03641, JP15H03642, JP15H03642, JP15H06896, JP16H03983, JP16K05296, JP16K05309, JP16K17667, and JP16K05296. The following NASA grants are acknowledged: NNX15AC76G, NNX15AE16G, NNX15AK71G, NNX15AU54G, NNX15AW94G, and NNG15PP48P to Eureka Scientific. H. Akamatsu acknowledges support of NWO via Veni grant. C. Done acknowledges STFC funding under grant ST/L00075X/1. A. Fabian and C. Pinto acknowledge ERC Advanced Grant 340442. P. Gandhi acknowledges JAXA International Top Young Fellowship and UK Science and Technology Funding Council (STFC) grant ST/J003697/2. Y. Ichinohe, K. Nobukawa, T. Sato, and H. Seta are supported by the Research Fellow of JSPS for Young Scientists. N. Kawai is supported by the Grant-in-Aid for Scientific Research on Innovative Areas “New Developments in Astrophysics Through Multi-Messenger Observations of Gravitational Wave Sources”. S. Kitamoto is partially supported by the MEXT Supported Program for the Strategic Research Foundation at Private Universities, 2014-2018. B. McNamara and S. Safi-Harb acknowledge support from NSERC. T. Dotani, T. Takahashi, T. Tamagawa, M. Tsujimoto and Y. Uchiyama acknowledge support from the Grant-in-Aid for Scientific Research on Innovative Areas “Nuclear Matter in Neutron Stars Investigated by Experiments and Astronomical Observations”. N. Werner is supported by the Lendület LP2016-11 grant from the Hungarian Academy of Sciences. D. Wilkins is supported by NASA through Einstein Fellowship grant number PF6-170160, awarded by the Chandra X-ray Center, operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.
We thank contributions by many companies, including in particular, NEC, Mitsubishi Heavy Industries, Sumitomo Heavy Industries, and Japan Aviation Electronics Industry. Finally, we acknowledge strong support from the following engineers. JAXA/ISAS: Chris Baluta, Nobutaka Bando, Atsushi Harayama, Kazuyuki Hirose, Kosei Ishimura, Naoko Iwata, Taro Kawano, Shigeo Kawasaki, Kenji Minesugi, Chikara Natsukari, Hiroyuki Ogawa, Mina Ogawa, Masayuki Ohta, Tsuyoshi Okazaki, Shin-ichiro Sakai, Yasuko Shibano, Maki Shida, Takanobu Shimada, Atsushi Wada, Takahiro Yamada; JAXA/TKSC: Atsushi Okamoto, Yoichi Sato, Keisuke Shinozaki, Hiroyuki Sugita; Chubu U: Yoshiharu Namba; Ehime U: Keiji Ogi; Kochi U of Technology: Tatsuro Kosaka; Miyazaki U: Yusuke Nishioka; Nagoya U: Housei Nagano; NASA/GSFC: Thomas Bialas, Kevin Boyce, Edgar Canavan, Michael DiPirro, Mark Kimball, Candace Masters, Daniel Mcguinness, Joseph Miko, Theodore Muench, James Pontius, Peter Shirron, Cynthia Simmons, Gary Sneiderman, Tomomi Watanabe; ADNET Systems: Michael Witthoeft, Kristin Rutkowski, Robert S. Hill, Joseph Eggen; Wyle Information Systems: Andrew Sargent, Michael Dutka; Noqsi Aerospace Ltd: John Doty; Stanford U/KIPAC: Makoto Asai, Kirk Gilmore; ESA (Netherlands): Chris Jewell; SRON: Daniel Haas, Martin Frericks, Philippe Laubert, Paul Lowes; U of Geneva: Philipp Azzarello; CSA: Alex Koujelev, Franco Moroso.
We finally acknowledge helpful comments from Mikio Morii on the statistical analysis, and valuable comments from the anonymous referee that greatly improved the manuscript.
[61]{} natexlab\#1[\#1]{}
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[^1]: The corresponding authors are Eric D. Miller, Hiroya Yamaguchi, Kumiko Nobukawa, Makoto Sawada, Masayoshi Nobukawa, Satoru Katsuda, and Hideyuki Mori.
[^2]: https://wwwastro.msfc.nasa.gov/lynx
[^3]: See Table 3.2 and Figure 7.11 of the Suzaku Technical Description, ftp://legacy.gsfc.nasa.gov/suzaku/nra\_info/suzaku\_td\_xisfinal.pdf.
|
---
abstract: 'After over a decade of experiments generating and studying the physics of quantized vortices in atomic gas Bose-Einstein condensates, research is beginning to focus on the roles of vortices in quantum turbulence, as well as other measures of quantum turbulence in atomic condensates. Such research directions have the potential to uncover new insights into quantum turbulence, vortices and superfluidity, and also explore the similarities and differences between quantum and classical turbulence in entirely new settings. Here we present a critical assessment of theoretical and experimental studies in this emerging field of quantum turbulence in atomic condensates.'
author:
- 'Angela C. White'
- 'Brian P. Anderson'
- 'Vanderlei S. Bagnato'
title: Vortices and turbulence in trapped atomic condensates
---
Introduction
============
[S]{}ince Onsager’s groundbreaking theoretical work linking turbulence and point vortex dynamics in a two-dimensional (2D) fluid [@Onsager1949], it has been hoped that the simple nature of quantum vortices in superfluids will aid in understanding the nature of turbulence. After many years of research with superfluid helium systems, the field of quantum turbulence (QT) is now well established, and has led to numerous new insights and developments regarding QT and the universality of turbulence [@QVDSF]. The discovery of links between classical turbulence and QT remains a strong motivating factor for QT research, particularly in the emerging field of QT studies with Bose-Einstein condensates (BECs). BECs present a new platform for QT studies due to their compressibility, weak interatomic interactions, and availability of new experimental methods for probing and studying superfluid flow [@Pet2008]. The relationship between QT and vortex dynamics in these systems is consequently an inherently interesting new research topic as well.
Classical turbulence is composed of eddies of continuous vorticity and size and it is necessary to solve the Navier-Stokes equation to mathematically describe viscous fluid dynamics [@Davidson2004]. For turbulent fluid flow, which consists of scale-invariant flow dynamics across a wide range of length scales, this procedure becomes difficult to tackle from first principles. In comparison, QT is comprised of vortices of less complexity, each with a localized and well-defined vortex core structure and quantized circulation. Superfluid flow is inviscid and vortices cannot decay by viscous diffusion of vorticity: a quantized vortex cannot simply “spin down” and dissipate energy via viscosity in the same way a classical vortex can. Incompressible kinetic energy is instead diffused through emission of sound waves and then dissipated due to the presence of a thermal cloud in BECs or the normal fluid component in superfluid He.
Despite the differences arising from the nature of vortices, classical and quantum turbulence share profound similarities that underscore the universality of turbulence. We briefly illustrate this idea with the structure of kinetic energy spectra in three-dimensional (3D) turbulence. At locations in the fluid far from vortex cores, and for length scales greater than the average inter-vortex spacing, vortex core structure is unimportant and quantized vortex lines are analogous to vortex filament lines of an Euler fluid. Under these conditions, classical and quantum turbulence are known to possess similar macroscopic and statistical properties. The most striking similarity is the existence of the same Kolmogorov spectrum in the inertial range for 3D turbulence. For non-equilibrium steady-state forced turbulence, at length scales larger than the distance given by the average inter-vortex separation and smaller than the scale corresponding to energy injection, the incompressible kinetic energy spectrum scales with wavenumber $k$ as $E(k)\sim k^{-5/3}$ [@Maurer1998]. This scaling is thought to arise from a Richardson cascade process as large vortices break up into smaller and smaller vortices until at very small scales energy is dissipated. The same classical Kolmogorov scaling has been verified numerically and experimentally in turbulent superfluid $^{4}$He and $^{3}$He-B [@Maurer1998; @Nore1997; @Stalp1999; @Araki2002; @Bradleyqc; @Salort2010PF; @Baggaley2011PRE; @Baggaley2011PRB] and has also been established numerically as a feature of trapped [@Kobayashi2007; @Kobayashi2008] and homogeneous [@Kobayashi2005; @Sasa2011] atomic condensates. There is thus reason to believe that a quantitative understanding of aspects of turbulence in one system, even a quantum fluid, may aid in the general understanding of the subject.
Among the central issues in the development of an understanding of QT is therefore understanding how quantized vortices move about in a plane in 2D QT, or how they bend, interact, and create complex tangles in 3D QT. The relationship to vortex dynamics and statistical measures of turbulence, such as energy and velocity spectra, is also an ongoing and active research topic. While the physics of such processes and the links between the various aspects of the problems may remain intricate, the reduction of vorticity to well-defined cores enables new approaches to studying QT generally unattainable in classical turbulence. This article is aimed at presenting a brief overview of theoretical, numerical, and experimental progress — as well as potential capabilities — for achieving a deeper understanding of QT by tackling these central issues. We present atomic BECs as a new and distinct tool that will contribute towards an extended understanding of QT and vortex dynamics through both theory and experiment. Firstly we discuss the unique aspects of turbulence in BECs in comparison to superfluid $^{4}$He and $^{3}$He-B. We examine the recent theoretical and experimental advances in the BEC turbulence field and detail the new regimes of turbulence made accessible in BECs. Finally we overview experimental measures of turbulence and future directions for the field, focusing on progress in both 3D and 2D QT.
Atomic BECs: beyond superfluid helium
=====================================
Studies of quantum turbulence have historically been restricted to superfluid $^{4}$He and $^{3}$He-B. These systems have a huge range of accessible length scales and as a result, turbulent vortex tangles can consist of hundreds of thousands of vortices. The vortex tangles are well separated with typical inter-vortex separation distances of $l\sim10^{-4}$m. Vortices have small vortex core diameters, $\xi\sim10^{-10}$m for superfluid $^4$He, and consequently the turbulent tangles in $^{4}$He and $^{3}$He-B are characterised by a large ratio of inter-vortex spacing to vortex core radius, on order of $10^{5}-10^{6}$. The intrinsic superfluid parameters such as atom-atom interaction strength are fixed, and the superfluids are homogeneous and of constant density (i.e., incompressible). Controlling single-vortex dynamics in a turbulent $^{4}$He and $^{3}$He-B superfluid is exceptionally challenging, and probing the behaviour of turbulence at small scales remains an open problem from an experimental standpoint. Finally, due to the strong interactions between atoms in liquid and superfluid helium, simulations of wave function dynamics are generally only qualitatively accurate.
In contrast, trapped atomic condensates are rarely homogeneous and have a smaller range of accessible length scales over which vortex dynamics can be probed. With the exception of shallow traps and nominally hard-wall confining potentials where the condensate is to a good approximation homogeneous, for realistic trapped systems the condensate density is non-uniform, with the non-uniformity arising as a consequence of the form of the trapping potential and the compressibility of BEC systems. It is this compressibility that also lends significant new approaches to the study of QT. Also because of this, manipulating the trapping potential allows one to manipulate the condensate homogeneity throughout the trapping region. The typical vortex core diameter in atomic BECs is much larger than that in superfluid He, being on order of the coherence or healing length, $\sim0.5\mu$m. Turbulent BECs consist of small numbers of vortices (generally no more than $\sim 100$) that are less sparsely separated (ratio of inter-vortex spacing to vortex core radius on the order of 10). From a theoretical standpoint, the models describing vortex tangles in superfluids are amenable to numerical simulation, and for BECs are well established and quantitatively accurate at both zero and non-zero temperatures [@FTProukakis; @FTBlakie].
BECs also permit adjustment of intrinsic atomic properties that lead to macroscopic flexibility not readily achieved in other superfluid systems. The strength of atom-atom interactions can be controlled and even driven from attractive to repulsive by tuning an external magnetic field around a Feshbach resonance [@RevModPhys.82.1225]. Individual vortex dynamics and position can be well controlled [@DavisPRA2009; @Samson2012], and combined with a wide variety of imaging techniques opens up the field to investigations of few-vortex systems and the relationship of chaotic vortex dynamics to turbulent flow. The tunable features of trapped atomic condensates further extends to their dimensionality. Highly oblate condensates, with vortex dynamics well into the 2D regime can be routinely created. This makes BECs the first system for which 2D QT is experimentally accessible [@Samson2012; @Neely2010phd; @Neely2012; @Wilson2013]. We note that for such studies, quasi-2D BECs that have chemical potential less than the most strongly confining dimension’s mode spacing are [*not*]{} needed. Rather, highly oblate but nevertheless 3D BECs may suppress superfluid flow along the tight-confining direction enough that 2D superfluid vortex dynamics and 2D turbulence are obtained [@RooneyPRA2011]. We will henceforth refer to such BECs as 2D, and the associated conditions, where vortex dynamics occur within a plane, as being 2D. In addition to creating conditions for studying both 2D turbulence and 3D turbulence, BEC trapping parameters are extremely flexible, opening up the possibility of investigating transitions between 2D and 3D QT.
Theoretical formalism
---------------------
The dynamics of quantized vortices in zero-temperature BECs with scalar order parameter are described by evolution of the Gross-Pitaevskii equation, $$i \hbar\frac{\partial \psi\left({\bf{r}},t\right)}{\partial t} = \left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V\left({\bf{r}},t\right) + U_{0} |\psi\left({\bf{r}},t\right)|^{2}\right]\psi\left({\bf{r}},t\right)\,.$$ Here $\psi$ is a classical mean-field wave function representing the condensate trapped by a potential $V\left({\bf{r}},t\right)$. $U_{0}=4\pi\hbar^{2}a/m$ describes interactions between bosons of mass $m$ in the condensate, where $a$ is the $s$-wave scattering length, and $\hbar$ is the reduced Plank’s constant [@Pet2008]. For simplicity, we consider BECs in cylindrically symmetric harmonic trapping potentials. When the axial trapping frequency is much greater than the radial trapping frequency, $\omega_{z}\gg\omega_{r}$, the condensate will be highly oblate, enabling studies of 2D QT and vortex dynamics limited to the radial plane. For $\omega_z \sim \omega_r$, 3D QT may be explored. The dynamics of condensates at non-zero temperatures can be accurately described by applying classical field methods [@FTBlakie]. For a description of non-zero-temperature models that could be applied to superfluid turbulence, we refer the interested reader to the reviews [@FTProukakis; @FTBlakie].
### Compressible and Incompressible kinetic energy
Decomposing the condensate kinetic energy into compressible and incompressible parts is a useful technique frequently applied to analyse how kinetic energy due to vortex lines and sound is distributed over length scales [@Nore1997]. This is performed by defining a density weighted velocity field ${\bf{\Upsilon}}=\sqrt{n}\textbf{v}$, in terms of the superfluid velocity, $\textbf{v}=\frac{\hbar}{m}\nabla\theta$, where $n$ and $\theta$ are the position-dependent condensate density and phase profiles respectively. Utilizing the fundamental theorem of vector calculus, we can write ${\bf{\Upsilon}} = {\bf{\Upsilon}}^{i}+{\bf{\Upsilon}}^{c}$, where ${\bf{\Upsilon}}^{i}$ is an incompressible (i.e., divergence-free) component satisfying $\nabla\cdot{\bf{\Upsilon}}^{i}=0$, and ${\bf{\Upsilon}}^{c}$ is a compressible (i.e., irrotational) component for which $\nabla\times{\bf{\Upsilon}}^{c}=0$. The compressible and incompressible kinetic energy spectra $E^{i,c}_{\mathrm{kin}}(k)$ are defined by $$E^{i,c}_{\mathrm{kin}}=\frac{1}{2}\int \text{d}\textbf{r}|{\bf{\Upsilon}}^{i,c}|^{2}=\int_{0}^{\infty}\text{d}kE^{i,c}_{\mathrm{kin}}(k),$$ where $k=|\vec{k}|$ is a wavenumber and $E^{i}_{\mathrm{kin}}$ and $E^{c}_{\mathrm{kin}}$ are the total incompressible and compressible kinetic energies in the system (per unit mass). The flow of incompressible kinetic energy across wavenumbers and its relationship to vortex dynamics is one of the central and critically important and challenging issues in classical turbulence as well as QT. An understanding of spectra in 2D and 3D QT thus relates directly to the dynamics and distribution of quantized vortices. Considerable effort has been recently concentrated on constructing the incompressible kinetic energy spectrum and also the angle-averaged momentum and velocity distribution in order to determine their scaling in relation to the distribution of vortices in 2D [@Nowak2011; @Nowak2012; @Schole2012; @Bradley2011; @Kusumura2012] and 3D QT in condensates [@Nowak2011].
### Quantum Pressure Energy
Contributions to the quantum pressure energy, $$E_{\mathrm{qp}}=\frac{1}{2}\int\text{d}\textbf{r}|\nabla\sqrt{n}|^{2}\,,$$ arise only where the condensate density varies sharply, such as at vortex cores, dark solitons, and other density discontinuities. Under conditions where solitons and other features of the compressible component are negligible or damped, this quantity may provide a useful theoretical measure of the total vortex number in 2D condensates. For 3D condensates, quantum pressure energy scales with the vortex line length. While evaluating incompressible, compressible and quantum pressure energies and spectra is useful from a theoretical point of view, it remains an open problem how to measure these individual components experimentally. However, velocity correlations and the angle-averaged momentum spectrum are quantities potentially experimentally accessible in atomic BECs and we outline some measurement prospects later in the article.
Vortex generation
=================
A plethora of experimental methods are available to induce vortices in BECs [@Anderson2010]. We highlight a non-exhaustive selection of techniques that allow the preparation of well defined initial states from which to investigate the evolution or decay of turbulence. Deterministic induction of vortices into the condensate at precisely defined positions can be achieved by the controlled methods of phase imprinting, where the phase profile of the condensate is engineered [@Ketterle2002; @Shibayama2011], or by the coherent transfer of orbital angular momentum to a condensate by a two-photon stimulated raman process [@Andersen2006; @Wright2008; @Wright2009; @Leslie2009]. The creation of non-equilibrium vortex states with arbitrary winding has been beautifully demonstrated by the transfer of orbital angular momentum from a holographically produced light beam [@Brachmann2011]. Applying this technique has the potential to create arbitrarily complex initial vortex distributions and even vortex knots in BECs [@Proment2012; @Dennis2010]. Alternatively, vortex knots could be imprinted in BECs applying the techniques demonstrated in classical fluids, through the acceleration of shaped hydrofoils [@Kleckner2013], made by a 3D laser structure or a shaped nano-tube in a condensate.
Laser stirring can also be applied to create vortices in atomic BECs in a deterministic manner [@Neely2010]. The distribution of vortices throughout a condensate is known to depend on the path of the laser stirrer [@White2012]. For a 2D condensate, this means the path the laser stirrer takes through the condensate can create clusters of like-signed vortices as shown in Fig. 1(A), or a more random distribution of vortices of differing sign. For a 3D condensate, this implies that a stirring path might be optimised to generate well-distributed vortex configurations or more polarised tangles, where the vortex distribution is aligned preferentially along a particular direction. Reeves et al.[@Reeves2012] have shown that the potential strength of a laser stirrer and its speed can also be chosen such that the laser stirrer sheds single dipoles, clusters, or oblique solitons in a trapped 2D BEC. These ideas should also extend to stirring or flow past an obstacle in a 3D condensate.
In addition to laser stirring, combined rotation and precession around three cartesian axes has been shown to create isotropic vortex tangles in 3D condensates [@Kobayashi2008]. Reducing the rotation and precession of the condensate to only two directions decreases the degree of isotropy of the resulting vortex tangle and could also be applied to create polarised distributions of vortices [@Kobayashi2007]. The methods of laser stirring, phase imprinting, transfer of orbital angular momentum and combined rotation and precession which have been highlighted here, have the additional advantage of producing only small amounts of phononic excitations, where the acoustic energy density is much less than the incompressible energy density; see [@Reeves2012] for further discussion of this relationship.
Chaos and few-vortex dynamics
=============================
The ability to create well-defined initial distributions of vortices [@Samson2012] opens up the possibility of directly probing systems of few vortices and their resulting dynamics. Previous experiments have observed the precession of single filled vortices in trapped condensates [@And2000.PRL85.2857], and the motion of vortex dipoles deterministically created prior to expansion of a sequence of BECs [@Neely2010]. Recent experimental advances have demonstrated measurements of few-vortex dynamics within a single BEC [@Freilich2010; @Navarro2013]. This method is accomplished by allowing only a small fraction of the atoms in a BEC to be imaged after expansion, and sequential images from multiple expansion steps allows determination of vortex dynamics. This technique could be incorporated into future measurements of chaotic vortex dynamics. With the future development of real-time in situ imaging of vortex dynamics it may become possible to directly observe vortex dynamics, including vortex-vortex annihilation and reconnection events directly within BECs. The motion of four point vortices in a plane can be chaotic [@Aref1980; @Aref1982; @Aref1988] and similar chaotic dynamics are expected to be observable in oblate trapped BECs. The number of vortices that determines the crossover from chaotic to turbulent vortex dynamics and the role of chaos in turbulence [@Are1983.ARFM15.345] remain open questions that atomic BECs may be able to address.
Three-dimensional turbulence
============================
Non-equilibrium steady-state turbulence
---------------------------------------
One of the defining features of continuously forced turbulence in a bulk 3D fluid is the existence of a direct Kolmogorov cascade corresponding to the conserved flow of energy from the forcing scale to smaller length scales. Although the direct Kolmogorov energy cascade has been established numerically for trapped atomic condensates [@Kobayashi2007; @Kobayashi2008], it is yet to be confirmed experimentally. Indeed for a realistic tangle of vortices, where the BEC diameter is typically less than $\sim100\mu$m, the small range of length scales present may prohibit its detection. On the other hand, this implies trapped condensates may be a suitable system in which to determine the lower bound on the system size and vortex line density for which Kolmogorov scaling may be observed. Although the $k^{-5/3}$ scaling of the incompressible energy spectrum is established in superfluid He systems and is thought to arise due to a Richardson cascade process [@Maurer1998; @Nore1997; @Stalp1999; @Araki2002; @Salort2010PF; @Baggaley2011PRE; @Baggaley2012PRL], a corresponding break up of large eddies into smaller and smaller eddies has not been theoretically or experimentally verified. If a Richardson cascade process is present in QT, it may be directly observable in trapped atomic condensates as detection schemes that experimentally image vortices directly are feasible. We outline such techniques near the end of this article. Tracking the length of individual vortices is also possible numerically, and may help answer questions about the relationship of an energy cascade to a Richardson cascade. Establishing the existence or absence of a Richardson cascade process coupled to a direct Kolmogorov cascade is perhaps one of the most exciting prospects in future experimental and numerical studies of turbulent vortex tangles in trapped atomic BECs.
Small-scale dynamics and dissipation
------------------------------------
On length scales smaller than the typical inter-vortex spacing, quantum turbulence is quite different from classical turbulence. Characteristics that are dependent on the vortex core structure and microscopic nature of the fluid are expected to be unique for quantum fluids. One such property is the velocity statistics of turbulent quantum fluids. The velocity components of trapped and homogeneous turbulent BECs in 2D and 3D, as well as turbulent superfluid He, follow power-law like behaviour [@Salort2010PF; @Paoletti2008PRL; @White2010; @Adachi2011], in stark contrast to gaussian velocity statistics of turbulent classical fluids [@Vincent1991; @Noullez1997; @Gotoh2002]. This difference in the velocity statistics of quantum fluids arises from the singular nature of quantized vorticity and the $1/r$ velocity field of a quantum vortex [@White2010; @Min1996]. Measurement techniques in superfluid ${}^{4}$He can also probe the ‘quasi-classical’ limit, that is they do not measure length-scales smaller than the average vortex separation distance. In this limit, velocity statistics lead to a gaussian distribution [@Baggaley2011PRE; @Salort2012]. This is another demonstration of features of classical turbulence emerging for quantum turbulent systems, at large scales, where the microscopic vortex core structure is not probed.
In 3D turbulence, the direct Kolmogorov cascade is coupled to dissipation of incompressible kinetic energy at small scales. In superfluid He, at zero temperature this sink of incompressible kinetic energy is facilitated by a Kelvin wave cascade process at scales smaller than the average inter-vortex separation distance. The Kelvin-wave cascade process occurs when the helical perturbations on vortex lines cascade to shorter and shorter wavelengths, until at small scales phonons are excited, dissipating the kinetic energy of vortices [@VinenPRL2003]. For zero temperature condensates, such a Kelvin wave cascade process could also provide a sink of incompressible kinetic energy, in particular as Kelvin waves can be generated by vortex reconnection events [@KivotidesPRL] that are more frequent due to the smaller inter-vortex spacing in 3D condensates. As the collision of finite amplitude sound waves, or rarefaction waves with vortex lines can also generate Kelvin waves [@BerloffPRA2004], they may feature prominently in tangles generated by methods that induce a significant quantity of sound. Whether the Kelvin wave cascade process is driven by local or non-local transfer of energy remains an open and contentious question [@KozikPRL2004; @LauriePRB2010; @LebedevJLTP2010; @KozikJLTP2010; @LebedevJTLTPreply; @LNLTP10; @BouePRB2011; @Sonin2012]. However, Gross-Pitaevskii equation simulations for Kelvin waves in a homogeneous system find scaling consistent with that predicted for a non-local energy cascade caused by 4-wave interactions [@KrstulovicPRE2012]. Numerical simulations of few-vortex systems in 3D trapped condensates at non-zero temperature have shown Kelvin waves to be important in facilitating the decay of vortices by their movement out of the trap [@RooneyPRA2011]. This work also showed that Kelvin mode excitations can be effectively frozen out by altering the condensate dimensionality through flattening the condensate, or increasing its confinement along the vortex line direction [@RooneyPRA2011].
In addition to the dissipation of incompressible kinetic energy through a Kelvin wave cascade process, for finite size trapped condensates, the dissipation of incompressible kinetic energy by sound generated through reconnection events [@Leadbeater01; @Zuccher12] will also be important. It has been suggested that the break up of vortex loops into smaller vortex rings could form a self-generating reconnection process to smaller and smaller scales until they are dissipated by self-annihilation and sound waves [@BayerPRB2011; @Simula2011]. Whether sound produced by a Kelvin wave cascade process, or from reconnection events, is the dominant process dissipating incompressible kinetic energy in zero temperature trapped atomic condensates, and how this scales with condensate size, vortex line density, and the polarity of vortex tangles, remain open questions to be addressed.
Decay of turbulence
-------------------
In addition to forced 3D turbulence, it is also possible to investigate decaying QT in trapped 3D BECs. The decay of turbulence is expected to be a complex process facilitated by vortex-vortex reconnections with the rate of decay of vortex line density also influenced by the condensate homogeneity and temperature. A numerical study of the decay of a vortex tangle induced from straight vortices imprinted along each cartesian direction employed a phenomenological model of temperature and found vortex-line length to decay faster with increased dissipation, qualitatively modelling higher temperatures [@White2010]. No distinction between scalings of vortex line-length decay predicted from ‘Vinen’ or ‘ultra-quantum’ models of turbulence or ‘quasi-classical’ models of turbulence describing decaying turbulence in superfluid He was observed. ‘Ultra-quantum’ models of turbulence describe random distributions of vortices in superfluid He turbulence, with vortex decay facilitated by a Kelvin-wave cascade process corresponding to a decay of vortex line-density $L$ as $L\sim t^{-1}$ [@Golovqcvsuq; @Baggaleyqcvsuq]. For turbulence in superfluid He where a Kolmogorov cascade features, vortices are locally aligned [@Volovik03], and the decay of line density is observed to scale as $L\sim t^{-3/2}$ [@Bradleyqc; @Golovqcvsuq; @Baggaleyqcvsuq]. A unique scaling of the decay of vortex line-length, dictated by the length-scale at which turbulence was forced, may feature in the decay of turbulent tangles in trapped condensates. As the number of vortices in a turbulent BEC is much less than in turbulent superfluid He, yet vortex tangles are typically more dense, the processes dictating vortex decay are expected to be strongly influenced by the density of vortices and trapping inhomogeneity.
![A). Density profile of a 2D condensate where a cluster of like-signed vortices has been induced by stirring with a laser paddle as in [@White2012]. Vortices of positive and negative winding are denoted by magenta $+$ and cyan $-$ symbols respectively. B). 3D decaying turbulence: distribution of vortex lines (magenta) introduced by phase imprinting a lattice of $17$ straight vortices in a harmonically trapped condensate [@White2010]. The condensate edge defined by a density that is 25% of the peak density is depicted in cyan. ](Figure1.pdf){width="0.98\linewidth"}
Experiments on three-dimensional QT
-----------------------------------
Between the years 1999 and 2009, numerous aspects of the physics and characteristics of quantized vortices were studied in experiments with superfluid atomic gases [@Anderson2010]. These investigations included methods of vortex generation, imaging, and manipulation; properties of highly rotating condensates; vortex interactions, structure, stability; and relationships to superfluidity. However, few references to turbulence can be found in this body of work, and appear primarily as brief qualitative statements given to refer to disordered distributions of vortices. Such observations were made in the contexts of rotating single-component [@Che2000.PRL85.2223; @Ram2001.PRL87.210402] and two-component [@Sch2004.PRL93.210403] BECs, and with fast laser sweeps through a BEC [@Ino2001.PRL87.080402].
In arguably the first experiment principally aimed at studying a turbulent tangle of vortices in a 3D BEC, a non-uniform distribution of vortices was created in a cigar-shaped condensate by exciting a surface mode instability induced by an applied external oscillatory potential [@Henn09a; @Hennpra09]. Varying the strength and duration of the perturbing potential increased the number of vortices nucleated, enabling the observation of few-vortex configurations [@threevortex] through to a tangled distribution of vortices. In the regime of many vortices, it is difficult to identify individual vortex cores, particularly with detection of an expanded BEC’s column density distribution. In these experiments on 3D QT, the turbulence was in a regime with few enough vortices that absorption images of expanded condensates showed the presence of vortex lines of varying orientations. A tangled vortex configuration was then inferred from the absorption images. On release of the trap, the condensate aspect ratio (defined by the measured ratio of BEC width to height) was conserved and self-similar expansion was observed [@Henn09a]. This behaviour is in stark contrast to the typical inversion of the aspect ratio observed on expansion of a vortex-free cigar shaped condensate and is thought to be an indication of a tangle of vorticity throughout the condensate. This conclusion is supported by theoretical investigations that studied the influence on the expansion of a cigar-shaped cloud arising from the presence of randomly distributed vorticity [@Caracanhas2013] and distributions of vorticity with preferred directions in a hydrodynamic framework [@Caracanhas2012]. However, full numerical simulations going beyond the hydrodynamic limit and including quantum effects are still required. Measuring the condensate momentum distribution could potentially also provide useful information about the vortex distribution throughout the condensate [@Nowak2011]. Measurements towards this aim are underway at the Unversidade de São Paulo.
Two-dimensional turbulence
==========================
The existence of the direct Kolmogorov cascade for 3D superfluid turbulence suggests that large scale features of 2D turbulent classical flow [@Kraichnan1980; @Kellay2002; @Bofetta2012] may also crossover to 2D turbulence in quantum fluids. One of the characteristic features of classical 2D turbulence is the possible appearance of an inverse energy cascade (IEC) in which energy flow across length scales may occur in a direction opposite that of 3D turbulence; that is, with kinetic energy injected at small length scales, energy flows towards larger length scales in an inertial range free of forcing and dissipation mechanisms [@Kra1967.PF10.1417]. This inertial range, with $E_{\mathrm{kin}}^i(k) \propto k^{-5/3}$, corresponds to merging and growth of vortices; the IEC and vortex growth can continue until the largest wavelength modes of the system contain significant energy in the form of a vortex (or vortex dipole) on the scale of the size of the system. The IEC is enabled by the 2D nature of the system, where local vorticity must be normal to the 2D plane. This dimensional restriction gives rise to a conservation law for enstrophy (net squared vorticity) not present in 3D turbulence. As conserved energy and enstrophy flux cannot simultaneously occur in two dimensions, 2D turbulence can simultaneously display a flux of energy towards length scales larger than that of forcing (the IEC), and flux of enstrophy towards smaller length scales (a direct enstrophy cascade). The enstrophy cascade exhibits $E_{\mathrm{kin}}^i(k) \propto k^{-3}$ scaling over wavenumbers larger than that of the forcing scale.
Whereas the classical IEC of a 2D fluid is associated with the growth of patches of vorticity due to vortex merging and pushing energy towards larger length scales, the enstrophy cascade is associated with the stretching of patches of vorticity. Experiments on forced and decaying 2D turbulence have shown that an IEC that corresponds to vortex merging can appear with an enstrophy cascade [@Rutgers1998a], or may be observed without a simultaneous enstrophy cascade [@Som1986.JFM170.139; @Par1997.PRL79.4162]. Similarly, evidence for enstrophy cascades with approximate $k^{-3}$ dependence in the energy spectrum, and corresponding observations of vortex stretching, may appear without an IEC particularly in decaying 2D turbulence [@PhysRevLett.74.3975; @Mar1998.PRL80.3964]. The type of forcing used in experiments may thus have a significant effect on the characteristics of observed spectra and flow dynamics.
These cascades are only beginning to be explored in 2D QT. Among the most significant research topics within this field are the determination of (i) conditions under which an IEC and an enstrophy cascade can be found in forced or decaying 2D QT, either together or individually; (ii) superfluid dynamics that accompany either cascade process, in particular the clustering of quantized vortices accompanying an IEC, which is the process most readily envisioned to correspond to vortex growth in 2D QT, and the appearance of large-scale flow that may be produced by energy flux into large scales; and (iii) vortex dynamics that may provide insight into the mechanisms underlying turbulent dynamics, such as vortex-antivortex annihilation and few-vortex dynamics. The remainder of this section is devoted to addressing aspects of these open problems.
In 2D QT enstrophy is proportional to the number of vortices [@Bradley2011; @Numasato2010]. In decaying 2D QT, enstrophy is rarely conserved throughout the entire system, as vortex-antivortex annihilation events occur [@Numasato2010; @Numasato2010jltp] and serve as a dissipation mechanism. However, annihilation does not automatically preclude the appearance of an IEC in a forced system, and evidence for IECs have been observed in numerical simulations. These will be discussed below. For decaying 2D QT, existence of IECs remains an open question. To complicate matters further, in smaller systems, vortex dynamics are strongly influenced by the condensate confining geometry. For harmonically trapped BECs, the energy of a single vortex can be lost to the thermal cloud as the vortex precesses on a radial trajectory from the condensate centre towards the BEC boundary [@Freilich2010; @Rosenbusch2002].
Another effect potentially prohibitive to establishing IECs in BECs is the generation of large amounts of phononic excitations. Sound can be generated from vortex-dipole recombination events, the movement of vortices, and in particular by the method applied to induce vortices into the condensates. Sound may also increase the frequency of vortex-antivortex annihilation events. Furthermore, in consideration of Onsager’s arguments [@Onsager1949], there may be a minimum local number density of vortices or a forcing rate necessary to generate an IEC process. These issues imply the presence of IEC processes in atomic BECs is no more clear cut than it is in classical turbulence, and considerable theoretical and experimental studies have been concentrated on determining conditions for its existence. It is furthermore necessary to establish the dominant parameters that dictate different regimes of vortex dynamics for continuously forced and decaying 2D QT, so that these various regimes can be associated with features of energy spectra, forcing, and dissipation.
To date, most of the work on compressible 2D QT has been numerical in nature. We turn our attention to summarizing some of the findings that are among the most relevant for understanding vortex dynamics and energy cascades. Decaying homogeneous 2D QT generated by random phase initial conditions was observed to exhibit a direct energy cascade with $k^{-5/3}$ scaling [@Numasato2010]. In a study of vortex clustering, statistical measures of clustering were applied to quantify the distribution of vortices nucleated from a moving laser stirrer in a harmonically trapped condensate. This study found no increase in clustering of vortices and only a significant degree of clustering was observed when the clustering was forced by the path of the laser stirrer [@White2012]. In a different study using an annular trap, signs of weak clustering were observed, along with energy spectra consistent with (but not directly confirming) an IEC; this study was made in conjunction with experimental observations, and will be discussed at the end of this section [@Neely2012]. Reeves et al.[@Reeves2012] looked at a laser stirrer nucleating turbulent vortices and exciting acoustic energy into the BEC in order to distinguish regimes of vortex and acoustic turbulence. It was found that in a particular regime of stirring where clustering of vortices of the same sign occurred, Kolmogorov $k^{-5/3}$ scaling was observed, but became more transient as the fraction of vortices that were clustered decreased.
Another recent 2D QT study of a homogeneous BEC simulated superfluid flow past past four static stirrers, and varied the degree of damping in the system. Regimes of vortex clustering and energy spectra were then determined. For one range of damping, incompressible kinetic energy spectra approximately proportional to $k^{-5/3}$ for length scales larger than that of the forcing scale were found [@Reeves2013PRL]. Along with this spectral power law, a flux of incompressible kinetic energy into the longest wavelength modes of the system was measured. The growth of vortex clusters was simultaneously observed: localized regions of vortices of the same sign were observed to grow in circulation quanta, and in size by a factor of approximately 5, both in time and distance away from the forcing region. Together, these three observations show that IECs can be supported in 2D QT and that there is a correspondence between an IEC and the growth of patches of vorticity coinciding with vortex clusters, as in 2D classical turbulence. Nevertheless, there still remain numerous open issues regarding vortex clustering and IECs that must be resolved. These include a broad discovery of forcing and dissipation parameters that lead to the development of an IEC and vortex clustering, the roles and effects of boundary conditions and system geometry, the possibility of an enstrophy cascade developing, and the conditions under which a large region of vorticity on the scale of the system size might develop.
Finally, an analytical approach to understanding the spectra of 2D QT for a compressible superfluid was pursued in [@Bradley2011]. In this work, the shape of vortex cores in a BEC was used to analytically derive an expression for incompressible kinetic energy spectra of various vortex configurations in a homogeneous BEC. Intriguingly, for small enough length scales, the spectrum is proportional to $k^{-3}$, although does this not correspond to an enstrophy cascade. A particular type of vortex clustering was shown to lead to a $k^{-5/3}$ distribution in the spectrum for large enough length scales. This paper proposed a scenario of forced 2D QT in which an IEC might appear over length scales larger than an assumed forcing scale at the lower end of the $k^{-3}$ range, although there is no enstrophy cascade in the proposed picture. By considering the fraction of vortices involved in clusters, numerical results of forced turbulence correspond surprisingly well to the analytical predictions [@Reeves2013PRL]. An analytical derivation of the Kolmogorov constant for forced 2D QT in a BEC was also proposed, suggesting that new approaches to 2D QT could eventually lead to new insights on 2D turbulence.
From the body of recent theoretical work performed to date, we can conclude that the IEC, vortex clustering, and system-scale vortex growth may appear in 2D QT, but understanding the full range of conditions for the appearance of these features will require much further exploration. Simulations have shown that the manner in which vortices, as well as sound, are forced into the BEC can play a large role in the spectra observed, and this is one of the more complex aspects of the 2D QT field. In the above summaries, we have focused on investigations that center on IECs and inertial regimes in energy spectra, but there are indeed other important aspects of 2D QT in BECs. In particular, discussions of non-thermal fixed points lead to new means of understanding and discussing quantum turbulence [@Nowak2011; @Nowak2012; @Schole2012], and studies of vortex dynamics and correlations may provide further means of analyzing regimes of QT [@Kusumura2012].
Experiments on two-dimensional QT
---------------------------------
As with 3D QT in BECs, there are still few experiments on 2D QT in trapped atomic condensates, and there is much to learn. The first 2D QT experiment by Neely et al.[@Neely2012] observed the formation and evolution of disordered distributions of vortices forced into a highly oblate annular BEC by stirring with a laser beam, as shown in Fig. \[fig2DQTexpt\](B). The technique excited very little acoustic energy, and injected on the order of 20 vortices, which may well be enough to display the physics of turbulence in such a small system. Subsequent to stirring, the system evolved into a persistent current as vortex number decayed; while this observation corresponds to the growth of energy in large-scale flow given small-scale forcing, the extent to which this observation directly relates to the vortex and spectral dynamics described above is currently unclear. Numerical simulations of the process showed the presence of pairs of like-signed vortices, an incompressible energy spectrum proportional to $k^{-5/3}$ over a range of length scales larger than the forcing scale, and conservation of enstrophy over the time period associated with the growth of the $k^{-5/3}$ spectrum. While these observations are consistent with an IEC, the energy flux has not been determined, and the definite existence of an IEC in these simulations remains an open issue.
Other methods have also been experimentally explored for producing 2D QT in highly oblate BECs [@Wilson2013], as shown in Fig. \[fig2DQTexpt\](C)-(F). These include modulating the strength of the trapping potential, suddenly applying then removing a repulsive laser potential to a localized region in the BEC, modulating the intensity of a localized repulsive laser potential, and spinning a slightly elliptical highly oblate trapping potential within the radial plane. Within the parameter ranges explored, all of these methods have been experimentally found to induce large numbers of vortices in harmonic and annular traps, and are candidates for further 2D QT studies. Unfortunately for the study of vortex turbulence, these methods also generally excite acoustic or breathing modes of the BEC, and may render 2D QT studies difficult without further modification. A deeper understanding the physical origins of vortex generation in each case is also needed.
. 2D QT can be studied in a highly oblate BEC, as shown in an in-situ image of the BEC taken along the tightly confining axis and a corresponding in-situ image taken along the radial plane (inset). (B) In an annular BEC (created by directing a repulsive focused laser beam through the BEC along the tight trapping direction) excited by small-scale stirring using a repulsive laser potential, vortices were observed after the repulsive central laser barrier was removed prior to BEC expansion. Note that the circulation of vortices can not be determined in this expansion method. Taken from the experiment of [@Neely2012]. The remaining images of expanded BECs show 2D QT generation due to (C) modulation of the harmonic potential in a trap with a central focused repulsive laser potential, followed by 50 ms of hold time, (D) laser beam modulation (image taken 100 ms after removing beam), (E) sudden localized repulsive laser beam application and removal followed by 125 ms of hold time, and (F) trap spinning followed by 2.5 s of hold time. In all cases, the BEC is expanded prior to imaging. See Ref. [@Wilson2013] for details on these images and experimental methods.](Figure2.pdf){width="0.98\linewidth"}
Quantum Turbulence in BECs: future prospects
============================================
Central experimental challenges
-------------------------------
Here we outline some of the main experimental challenges that must be overcome to explore aspects of QT in BECs. Ideally, an experiment would be able to watch all vortices in real time, examine their dynamics and how they interact with the other vortices of a BEC, determine the circulations of all vortices, and measure corresponding energy spectra. This is an exceptionally challenging task! But there are likely to be realistic methods for approaching at least some of these goals. In 2D QT, methods for in situ imaging of vortices in the plane of a highly oblate BEC are currently being explored, and these show promise for use in multiple imaging methods that will allow for real-time probes of 2D QT vortex dynamics. The 3D case is much more demanding.
Atom interferometry presents an available method for circulation measurement [@Ino2001.PRL87.080402; @Mat1999.PRL83.2498; @Che2001.PRA64.031601], and in principle may be used even in three dimensions. In practice, whether in two or three dimensions, the basic application of atom interferometry in these cases would involve the interference of a turbulent BEC with a reference BEC. The reference BEC could be obtained either from coherently splitting a vortex-free condensate prior to driving one of the components into a turbulent state, or by creating two fully independent BECs of the same species. Trapping two BECs simultaneously is challenging; driving one and not the other into a turbulent state adds another layer of challenge that may be achievable with laser stirring beams, for example. Matter wave interference that is suitable for resolving circulations of a high-vorticity BEC is another open experimental challenge. Add to this the vortex tangles of 3D turbulence, and the experimental challenges are formidable, but not out of the question.
Another experimental goal involves the generation of QT with minimal acoustic excitation so that forced QT can be reliably studied. Forcing techniques are currently one of the main topics of numerical investigation. As mentioned, a major experimental advance will occur with techniques that permit measurement of the kinetic energy spectrum of a BEC, although the correspondence of the total kinetic energy spectrum with the incompressible component is an open question. Finally, on-demand vortex generation and manipulation techniques will allow for studies of vortex dynamics and interactions, as mentioned earlier.
Conclusions
===========
To summarise, atomic BECs are a highly tuneable system that hold much promise for the development of theoretical and experimental insights into some of the unanswered questions surrounding the theory of quantum turbulence. This holds particularly true regarding aspects of compressibility and QT, and the nature of 2D QT, both of which have not been explored prior to recent work with BECs. With BECs, emerging methods for controlled vortex creation may soon allow deterministic preparation of initial states necessary for investigations into the chaotic dynamics of few-vortex systems. Routine creation of highly oblate condensates provides the first system in which 2D quantum turbulence can be experimentally explored. Atomic gas superfluids also provide numerous other opportunities not mentioned in this paper, such as possibilities to investigate QT in spinor systems, or in degenerate Fermi gases.
Atomic BECs are also currently the most accessible system in which to study the small scale properties of turbulent vortex flow in three dimensions. Theoretical investigations have begun to build up a picture of vortex dynamics and the processes contributing to the forcing and decay of turbulence at small scales and there is great scope for experimental verification in atomic BECs. At large scales, features of classical turbulence is an emergent feature of quantum turbulence, suggesting that research with BEC systems may provide insight into some of the outstanding questions of turbulence.
BPA acknowledges funding from the US National Science Foundation, grant PHY-1205713. ACW acknowledges funding from EPSRC grant No. EP/H027777/1.
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---
abstract: 'Recent observations have revealed the presence of small fibres or sub-filaments within larger filaments. We present a numerical fragmentation study of fibrous filaments investigating the link between cores and sub-filaments using hydrodynamical simulations performed with the moving-mesh code <span style="font-variant:small-caps;">Arepo</span>. Our study suggests that cores form in two environments: (i) as isolated cores, or small chains of cores, on a single sub-filament, or (ii) as an ensemble of cores located at the junction of sub-filaments. We term these *isolated* and *hub* cores respectively. We show that these core populations are statistically different from each other. Hub cores have a greater mean mass than isolated cores, and the mass distribution of hub cores is significantly wider than isolated cores. This fragmentation is reminiscent of parsec-scale hub-filament systems, showing that the combination of turbulence and gravity leads to similar fragmentation signatures on multiple scales, even within filaments. Moreover, the fact that fragmentation proceeds through sub-filaments suggests that there exists no characteristic fragmentation length-scale between cores. This is in opposition to earlier theoretical works studying fibre-less filaments which suggest a strong tendency towards the formation of quasi-periodically spaced cores, but in better agreement with observations. We also show tentative signs that global collapse of filaments preferentially form cores at both filament ends, which are more massive and dense than other cores.'
author:
- |
S. D. Clarke$^{1}$[^1], G. M. Williams$^{2}$ and S. Walch$^{1,3}$.\
$^{1}$I. Physikalisches Institut, Universit[ä]{}t zu K[ö]{}ln, Z[ü]{}lpicher Str. 77, D-50937 K[ö]{}ln, Germany\
$^{2}$Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire,\
College Lane, Hatfield, Al10 9AB, UK\
$^{3}$Cologne Centre for Data and Simulation Science, University of Cologne, Cologne, Germany [^2]
bibliography:
- 'ref.bib'
title: 'The hierarchical fragmentation of filaments and the role of sub-filaments'
---
\[firstpage\]
ISM: clouds - ISM: kinematics and dynamics - ISM: structure - stars: formation
Introduction {#SEC:INTRO}
============
Observations using the Herschel Space Observatory show that filaments act as an intermediate step in the star formation process, linking the gas on the molecular cloud scale with the gas in cores [@And10; @Arz11; @Sch12; @Arz13; @Kon15; @Mar16]. The current filament paradigm argues that molecular clouds first form a complex network of filamentary structures which then proceed to fragment into dense cores [@And14]. It is therefore imperative to better understand the fragmentation process of filaments if one wishes to understand how mass accumulates in cores.
Theories suggest that isothermal filaments should fragment when their line-mass is close to the critical line-mass given by: $$\mu_{_{\rm CRIT}} = \frac{2 c_s^2}{G},$$ where $c_s$ is the isothermal sound speed and $G$ is the gravitational constant [@Ost64; @Lar85; @InuMiy92]. Models of equilibrium filaments show that there exists a fastest growing wavelength density perturbation, suggesting that a filament should fragment into a chain of quasi-periodically spaced cores [@InuMiy92; @InuMiy97; @FisMar12]. The spacing of these cores is related to the width of the fragmenting filaments, around 4 times the diameter. The non-equilibrium model presented by @Cla16, which includes the effects of accretion, shows a more complicated dispersion relation linking perturbation wavelength and growth rate. Yet a fastest growing mode remains which produces quasi-periodically spaced cores when realistic initial density perturbations are used.
{width="0.95\linewidth"}
Studies investigating the substructure of filaments present two challenges to this picture. First, quasi-periodically spaced cores are relatively uncommon [@And14]. Second, cores are not the only substructure. Filaments appear to fragment into smaller filaments termed fibres if observed in position-position-velocity (PPV) space or sub-filaments if observed in position-position (PP) or position-position-position (PPP) space [@Hac13; @TafHac15; @Feh16; @Dha18; @Suri19]. We use this distinction between fibre and sub-filament throughout the paper but note that other authors in the literature differ in their definitions.
Recent simulations agree with observations and show that filaments harbour numerous smaller filaments, though it is currently unclear if these sub-filaments form in situ within the parent filament or form separately and are gathered together into a large scale filament [@Smi16; @Cla17]. In both scenarios it is the sub-filaments which then go on to fragment into cores. Further, @Cla18 show that the accretion-driven turbulence, which causes the in situ fragmentation of a filament into sub-filaments, leads to the appearance of fibres in synthetic C$^{18}$O observations. While the fibres identified in PPV space are not always directly related to the sub-filaments in PPP space, they propose that as both structures are formed due to the internal turbulence of a filament, a filament which contains fibres also contains sub-filaments. The fact that fibres and sub-filaments are not identical structures is corroborated by cloud-scale simulations presented in @Zam17.
In this paper, we present a numerical study of the fragmentation of fibrous filaments focusing on the link between sub-filaments and cores. While the simulations are carried out in 3D we perform this study in 2D, i.e. on column density maps, to more closely resemble observational fragmentation studies. From these simulations we aim to produce a generalised picture of the hierarchical fragmentation of filaments and the role that sub-filaments play in it. In section \[SEC:SIM\] we detail the simulations used. In section \[SEC:TECH\] we present the fragmentation analysis techniques used. In section \[SEC:IDEN\] we describe how we identify and locate cores and how we determine the spines of the sub-filaments and the main filament. In section \[SEC:RES\] we present the results of the analysis and discuss the link between sub-filaments and core properties, the presence of characteristic fragmentation length-scales, and fragmentation signatures of end-dominated collapse. In section \[SEC:CON\] we conclude.
Simulations {#SEC:SIM}
===========
The simulations used for this study are the same as those originally presented in @Cla18, all of which were shown to contain fibres using synthetic C$^{18}$O observations. We summarise the pertinent details here.
The simulations were performed using the moving-mesh code <span style="font-variant:small-caps;">Arepo</span> [@Spr10]. The code solves the three-dimensional hydrodynamic equations including self-gravity with time-dependent coupled chemistry and thermodynamics. The current simulations are purely hydrodynamical, the inclusion of a magnetic field is part of a future study as it has been shown to affect fragmentation [@Nak93].
The computational domain is a slightly flattened box defined by $|x| \leq 3.0$ pc, $|y| \leq 3.0$ pc and $|z| \leq 2.5$ pc. The boundary conditions are periodic with respect to the hydrodynamics but isolated for self-gravity.
{width="0.95\linewidth"}
The initial set-up is an idealised cylindrically symmetric colliding flow. We note that larger-scale influences and non-cylindrical accretion will not be captured by these simulations. Rather the purpose of the simulations is to investigate the basic underlying physics of filament fragmentation without these added complexities. In addition, it allows the simulations to attain the high resolution (see below) necessary to tackle this problem.
The set-up’s colliding flow meets at the $z$-axis with an initial velocity of 0.75 km/s and an initial density profile given by $\rho_o / r$, where $\rho_o$ = 15 M${_{_{\odot}}}$ pc$^{-3}$ and $r$ is the cylindrical radius. This results in a mass accretion rate from the colliding flow of $\sim$ 70 M${_{_{\odot}}}$ Myr$^{-1}$ pc$^{-1}$ towards the $z$-axis. Turbulence is seeded in the inflowing gas with a thermal mix of compressive and solenoidal modes, and a mean velocity dispersion of 1 km/s. It is not driven and allowed to decay. Ten simulations are performed with different turbulent random seeds, labelled <span style="font-variant:small-caps;">SIM01</span> to <span style="font-variant:small-caps;">SIM10</span>. Simulation <span style="font-variant:small-caps;">SIM01</span> is used throughout as an example.
The set-up quickly forms a dense filament at the $z$-axis. Accretion from the colliding flows drives internal turbulence and the filaments fragment to form numerous cores, see figure \[fig::cores\]. Mesh cell refinement is used to ensure that the Truelove criterion is always met [@Tru97], i.e. we ensure that the local Jeans mass is resolved by at least 8 cell radii. This requirement leads to a resolution of $\sim 3 \times 10^{-3} - 3 \times 10^{-4}$ pc in the dense gas of the filament, $\rho > 10^{-21} \rm{g \,cm^{-3}}$. Sink particles are inserted at $n \sim 10^7$ cm$^{-3}$ to allow the simulations to proceed after the initial collapse. Here the simulations are stopped when $\sim$ 5 - 10 $\%$ of the mass of the filament is in sinks. At this point the simulations are analysed. This occurs about 0.5 - 0.6 Myr after the start of the simulation and the filaments have reached a mass between $\sim$ 150 M${_{_{\odot}}}$and $\sim$ 250 M${_{_{\odot}}}$. Before analysis, the <span style="font-variant:small-caps;">Arepo</span> mesh is mapped to a fixed Cartesian grid with resolution of 0.01 pc. We note that our simulations do not include any protostellar feedback effects which may have an effect on core properties and fragmentation; this will be investigated in a forthcoming paper (Clarke et al. in prep.).
FragMent: a tool to study filament fragmentation {#SEC:TECH}
================================================
<span style="font-variant:small-caps;">FragMent</span> is an open-source <span style="font-variant:small-caps;">Python/C</span> library[^3] presented in @Cla19 which includes a number of fragmentation analysis tools. Here, we use the nearest neighbour separation distribution, the minimum spanning tree edge length distribution, and the two-point correlation function to detect the presence of characteristic fragmentation length-scales. We also make use of the null hypothesis tests included in the package to test the statistical significance of the results.
A modification has been made to the two-point correlation function included in <span style="font-variant:small-caps;">FragMent</span>. We follow @Gra06 by estimating the error of the two-point correlation function due to Poisson noise with the relation: $$\sigma_w (r) = \sqrt{\frac{1 + |w(r)|}{DD(r)}},$$ where $w(r)$ is the two-point correlation function evaluated at separation $r$ and $DD(r)$ is the distribution of separation distances of the complete graph constructed from the core locations. This enables us to quantify the significance of any characteristic fragmentation length-scale detected with the two-point correlation function.
Several modifications have been made to the null hypothesis test functions which result in more than an order of magnitude speed up. These modifications are included in the latest version of <span style="font-variant:small-caps;">FragMent</span>.
{width="0.95\linewidth"}
Structure identification {#SEC:IDEN}
========================
Core identification {#SEC:CORE}
-------------------
Cores are identified using the dendrogram python package <span style="font-variant:small-caps;">astrodendro</span>[^4]. The parameters used for the dendrogram are: <span style="font-variant:small-caps;">min$\_$value</span> = $2 \times 10^{22} \; \rm{cm^{-2}}$, <span style="font-variant:small-caps;">min$\_$delta</span> = $10^{22} \; \rm{cm^{-2}}$, <span style="font-variant:small-caps;">min$\_$npix</span> = 9. Due to the high value of <span style="font-variant:small-caps;">min$\_$value</span> breaking the filament into numerous high column-density regions, the resulting leaves are relatively insensitive to the choice of <span style="font-variant:small-caps;">min$\_$delta</span>. The value of the parameter <span style="font-variant:small-caps;">min$\_$npix</span> leads to a minimum effective diameter of a leaf of 0.03 pc; this helps to ensure that cores are not overly segmented. We identify the leaves of the dendrogram as cores and take their column density weighted centre as the core location. Figure \[fig::cores\] shows an example of this process, the red contours show the outline of the cores and the black dots the core locations. Over the total of 10 simulations, 116 cores are identified, giving an average of 11.6 cores per filament. The minimum number of cores detected in a filament is 9, and the maximum is 16.
Spine identification {#SEC:SPINE}
--------------------
As noted in @Cla18, the simulated filaments contain numerous, smaller sub-filaments. @Cla18 detect these in position-position-position (PPP) space, but they are also apparent in the column density maps of these filaments (PP space). This makes the identification of a spine more complicated. Here we use two different techniques: one to determine the spines of each sub-filament and one to determine the spine of the whole filament.
### Sub-filaments {#SSEC:SUBFIL}
To find the spines of the sub-filaments seen in column density we used the tool <span style="font-variant:small-caps;">DisPerSe</span> [@Sou11]. <span style="font-variant:small-caps;">DisPerSe</span> has been used to identify filaments in numerous observations using column density and line emission maps and has been shown to be sensitive to substructures [e.g. @Suri19]. We run <span style="font-variant:small-caps;">DisPerSe</span> using a persistence value of $2 \times 10^{21} \; \rm{cm^{-2}}$, and a threshold of $2 \times 10^{21} \; \rm{cm^{-2}}$. The persistence and threshold are relatively low as we wish to detect sub-structures, not just the prominent main spine. Before analysis we use the inbuilt <span style="font-variant:small-caps;">DisPerSe</span> function to smooth the spines with a smoothing length of 5 pixels. We also use the <span style="font-variant:small-caps;">assemble</span> option to join spines which meet but are at an angle to each other; we use a value of 70 degrees so as to join as many spines as possible together. Finally, we exclude all spines which consist of fewer than 10 pixels as they are mostly artefacts.
Figure \[fig::sub-fils\] shows an example of the sub-filaments found using this technique. Here 16 sub-filaments are found. One can see that they relate to column density ridges, and that the high column density region at $x \; \sim \; 0.3$ pc shows a high number of sub-filaments due to its complexity.
Over all 10 simulations, 148 sub-filaments are found. The minimum number of sub-filaments identified in a simulation is 10 and the maximum is 19. @Cla18 analyse these simulations using synthetic C$^{18}$O observations, though at a slightly earlier time ($\sim 0.1$ Myr earlier), and find that each filament contained on average 22 fibres in PPV space. When identifying sub-filaments in PPP space, they find on average 26 per filament. It is clear that the level of substructure and complexity which is readily identifiable lessens as one uses a reduced amount of information with respect to the underlying 3D density field.
{width="0.95\linewidth"}
### Main filament {#SSEC:MAINFIL}
To apply the tools in <span style="font-variant:small-caps;">FragMent</span> one needs the spine of the entire filament, hereafter main filament, rather than the individual spines of the column density sub-filaments picked out by <span style="font-variant:small-caps;">DisPerSe</span>. Here we detail a method similar to that described in @Sch14. Each step can be seen in figure \[fig::Main\_spine\].
The first step is to convolve the column density map with a Gaussian beam (figure \[fig::Main\_spine\], top left panel), this is to dilute the substructure present so that it does not affect the spine. However, if the beam size is too large the resulting spine is often shifted from that which one would identify by eye. We find that a standard deviation of between 3 and 5 pixels is sufficient to hide substructure without distorting the spine. We use a standard deviation of 4 pixels for all 10 simulations.
A column density threshold is applied to the convolved column density map to identify the entirety of the filament (figure \[fig::Main\_spine\], top right panel). This produces a binary image, i.e. one consisting of 1s and 0s, which shows the filament. If the column density threshold is set too low, features seen in the turbulent accretion flow may be detected. Set too high and the filament is broken into small pieces. A value of $4 \times 10^{21}$ cm$^{-2}$ is used here, for all simulations, as it is a good compromise for these simulated filaments.
To produce the skeleton of this binary image we use the medial axis transformation included in the <span style="font-variant:small-caps;">Python</span> package <span style="font-variant:small-caps;">scikit-image</span> [@scikit]. The median axis is defined as the set of points each having more than one closest point on the image boundary. It is therefore able to reduce an image to a one pixel wide skeleton while preserving the general morphology.
As seen in the bottom left panel of figure \[fig::Main\_spine\], the result from the medial axis transformation is not the spine of the filament; it contains unwanted side-branches and in some cases there exist small gaps in the skeleton. These side-branches must be removed and the gaps must be filled manually. The result is a continuous, single, one pixel wide skeleton, which traces the spine of the main filament, seen as the black line in the bottom right panel of figure \[fig::Main\_spine\]. However, this process may result in areas where the spine does not pass through clear high column density ridges. Therefore, small modifications are made manually to force this to be the case; this is seen as the red line in the bottom right panel of figure \[fig::Main\_spine\]. This is the main filament spine used.
Results {#SEC:RES}
=======
Link between cores and sub-filaments {#SSEC:SUBCORE}
------------------------------------
![A KDE showing the distribution of sub-filament surface density, $S_{\rm Nsub}$, at the 116 core locations across all 10 simulations. The small vertical lines show each individual data point. Cores in the red shaded region, $0.9 \leq S_{\rm Nsub} \leq 1.1$, are termed *isolated* cores and those in the green shaded region, $S_{\rm Nsub} \geq 1.9$, are termed *hub* cores. The distribution is normalised such that the integral is equal to 1. The KDE bandwidth is 0.18.[]{data-label="fig::SFSDvsCORE"}](./SD.png){width="0.95\linewidth"}
It is clear that the cores lie on, or very close to, the spine of the main filaments (see figures \[fig::cores\] and \[fig::Main\_spine\]). However, it is unclear how the cores are related to the sub-filaments identified in the column-density map. In similar simulations to those analysed here, @Cla17 note that a number of the elongated structures in PPP space overlap and merge to form hub-like structures in which cores form. In observations studying fibres in PPV space, @TafHac15 propose a scenario of filament fragmentation termed ‘*fray and fragment*’ which claims that filaments fragment into numerous fibres which then proceed to fragment into cores independently of each other. This scenario may also occur in the sub-filaments identified here in PP space. Here we attempt to characterise the link between cores and sub-filaments.
We construct a dimensionless measure of the surface density of sub-filaments across the column density map. This is done by considering the minimum distance between each sub-filament and a pixel. The sub-filament surface density $S_{\rm Nsub}$ for pixel $j$ is given by the equation:
$$S_{\rm{Nsub},j} = \sum_i^N e^{-r_{min,ij}^2 / 2 \sigma^2_{\rm Nsub}} \; ;
\label{eq::surface_density}$$
where $r_{min,ij}$ is the minimum distance between the pixel co-ordinates, ($x_j$,$y_j$), and one of the spine pixels of sub-filament $i$, $\sigma_{\rm Nsub}$ is the bandwidth of the kernel, and $N$ is the number of sub-filaments. The idea for this quantity is that each sub-filament’s contribution to a pixel’s surface density is weighted by a Gaussian kernel of the minimum distance between the pixel and the sub-filament. As the number of sub-filaments increases, or the sub-filaments become closer to the pixel, the measure $S_{\rm Nsub}$ increases; it thus acts similar to a surface density. As only the minimum distance to each sub-filament is used, this measure is not affected by how many sub-filament spine pixels are nearby but only how many different sub-filament spine pixels are. The term is dimensionless and the exact value is not important (it may range from 0 to $N$), but the relative values across the map are important. We take $\sigma_{\rm Nsub}$ to be 4 pixels here. The exact value chosen is arbitrary. We take $\sigma_{\rm Nsub} = 4$ which results in a kernel with a full width half maximum of $\sim$ 0.1 pc as our column density map resolution is 0.01 pc.
[@\*7l@]{} Core population & Number of cores & Median & Interquartile range & Mean & Standard deviation & Range\
All cores & 116 & 3.3 & 3.8 & 4.7 & 4.4 & 0.6 - 24.9\
Isolated cores ($0.9 \leq S_{\rm Nsub} \leq 1.1$) & 38 & 2.6 & 2.4 & 2.9 & 1.8 & 0.7 - 8.5\
Hub cores ($S_{\rm Nsub} \geq 1.9$) & 47 & 4.5 & 5.4 & 6.2 & 5.6 & 0.6 - 24.9\
Intermediate cores ($1.1 \leq S_{\rm Nsub} \leq 1.9$) & 30 & 2.9 & 3.6 & 4.2 & 3.3 & 0.6 - 12.5\
Figure \[fig::sub\_surface\] shows the surface density of sub-filaments for <span style="font-variant:small-caps;">SIM01</span>. The majority of the filament shows a value close to 1, denoting a close presence of only one sub-filament. There are bright spots with a surface density of 2, showing areas where two sub-filaments meet. The complex region located at $x \; \sim \; 0.3$ pc shows a surface density as high as 4.
![A probability density function of the core masses considering all 116 cores from the 10 simulations (black), the isolated core population (blue) and the hub core population (red). The small vertical lines show each individual data point. The distribution is normalised such that the integral is equal to 1. The KDE bandwidth is 0.08 for the all core data set, 0.10 for the isolated core data set, and 0.16 for the hub core data set.[]{data-label="fig::CMF"}](./Core_mass.png){width="0.95\linewidth"}
Figure \[fig::SFSDvsCORE\] shows a Kernel Density Estimator (KDE) of the surface density of sub-filaments at the core locations determined in section \[SEC:CORE\]. The bandwidths of all KDEs are calculated using Scott’s rule [@Scott]. There is a large grouping of core locations where the surface density is close to 1 ($\sim 33\%$ of all cores have surface densities between 0.9 and 1.1), showing that they lie on or very close to a single sub-filament. This is consistent with the idea of the fray and fragment model, that each sub-filament fragments into a core, or cores, independently of each other. We call these cores *isolated cores*. However, there are a significant number of cores where the surface density is 1.9 or greater ($\sim 41\%$ of all cores) showing that there exist hub-sub-filament systems within the main filament, as proposed by @Cla17. We call these cores *hub cores*. There also exists some cores which lie in between these extremes ($\sim 26\%$ of all cores) which we call *intermediate cores.* These are typically cores on the edge of hubs.
One may ask if the cores formed in individual sub-filaments differ from those formed at the junctions of multiple sub-filaments. We first focus on their mass. Figure \[fig::CMF\] shows a KDE of the core mass distribution considering all 116 cores. There is no significant difference between the cores found in the 10 simulations. The distribution appears close to a log-normal, though with a flat top. However, we do not expect to fully represent the observed core mass function due to the idealised nature of the setup.
Figure \[fig::CMF\] also shows the core mass distributions for the isolated and the hub cores. It is clear that there is a difference in the core mass distribution between these populations; isolated cores tend to lower masses and have a narrower distribution. A two sample Kolmogorov-Smirnov test, a null hypothesis test where the null is that the two samples come from the same underlying distribution, returns a large distance statistic, 0.39, with the corresponding p-value of 0.005. We can thus confidently reject the null hypothesis and assert that the two populations have a significantly different core mass distribution. We also perform a two-sided Mann-Whitney U test [@Mann47]. The Mann-Whitney U test is a non-parametric test and its null hypothesis is typically stated as neither distribution having stochastic dominance over the other. This can formally be expressed as the probability of a variable drawn from distribution X having a greater value than a variable drawn from distribution Y being equal to the reverse, i.e. $P(X>Y) = P(Y>X)$. Thus the test is sensitive to differences in location, width and form between the two tested distributions, similar to the KS test. Using the Mann-Whitney U test we are able to reject the null hypothesis at high confidence, a $p$-values of 0.0036, and can therefore state that hub cores have a greater mass than isolated cores on average. Table \[tab::core\_mass\] summaries the properties of these core mass distributions.
![A probability density function of the core surface density for all pixels in all 10 simulations belonging to isolated or hub regions. Isolated sub-filaments typically host no core or one core, while hubs of sub-filaments usually host at least one core. The distributions are normalised such that their integral are equal to 1. The bandwidth of both KDEs is 0.027.[]{data-label="fig::cd_hist"}](./CD_kde.png){width="0.95\linewidth"}
Junctions of sub-filaments may also harbour a higher density of cores than single sub-filaments. We thus compare the sub-filament surface density to the core surface density. The core surface density is constructed by convolving the core locations with a two-dimensional Gaussian kernel with a standard deviation of 4 pixels, the same standard deviation as that used for the sub-filament surface density. We divide the map into hub sub-filament regions (the sub-filament surface density is greater than 1.9), and isolated sub-filament regions (the sub-filament surface density lies between 0.9 and 1.1). Figure \[fig::cd\_hist\] shows a KDE of the core surface density in isolated and hub regions, constructed using only map pixels with a core surface density above 0.1, or around 8 pixels away from a core location. Isolated sub-filament regions are skewed towards low core surface densities, peaking at 0.1, and also showing a smaller peak at 1. However, hub sub-filament regions have their maximum likelihood at a core density of around 1. Above a core surface density of 1, the hub sub-filament regions show a clear excess of pixels over the isolated sub-filament regions. Thus, hub sub-filaments show a more clustered form of core formation than that seen in isolated sub-filaments. A two sample Kolmogorov-Smirnov test returns a distance statistic of 0.22 and a corresponding p-value of $10^{-69}$, therefore this difference is statistically significant.
We are thus able to build a coherent scenario of how sub-filaments impact core formation in filaments. Filaments first fragment into sub-filaments due to their internal turbulence, which is both inherited from the large scale flows (i.e. the gas which initially formed the filament) and maintained and driven by sustained accretion. Regions where numerous sub-filaments meet and interact form. We term these hub-sub-filament systems. Such hub-sub-filament systems can be seen in Orion using N$_2$H$^+$ [@Hac18]. Isolated sub-filaments fragment into a single core, or a small chain of cores, independent of each other, while hub sub-filaments lead to a small ensemble of cores to form in the hub. Cores formed in such hubs do not only appear more clustered but also more massive than those that form on isolated sub-filaments. This is analogous to core formation in much larger hub-filament systems [@Mye09; @Per13; @Per14; @Wil18].
{width="0.95\linewidth"}
Fragmentation spacing {#SSEC:FRAGSPACE}
---------------------
While sub-filaments are intimately linked to core formation in fibrous filaments, there may exist imprints of the fragmentation process of the main filament which is apparent in the core spacing. The widths of the filaments is between 0.1-0.2 pc and are in agreement with the observational results of @Arz11 and @Arz19. According to equilibrium models we may expect a characteristic core spacing of 0.4-0.8 pc. However, as shown by @Cla16 and @Cla17, the characteristic fragmentation length-scale is unconnected to the filament width for non-equilibrium filaments. In this section we use the fragmentation analysis tool package <span style="font-variant:small-caps;">FragMent</span> to detect the presence of characteristic fragmentation length-scales.
Before applying the <span style="font-variant:small-caps;">FragMent</span> tool one must use the spines of the main filaments to straighten the filaments. As detailed in @Cla19, the process of straightening a filament is required to reduce the dimensionality of the problem and remove the complexity of a filament’s curvature. The function used to straighten the filaments is <span style="font-variant:small-caps;">Straighten$\_$filament$\_$weight</span>, included in the <span style="font-variant:small-caps;">FragMent</span> package. The parameters used are: <span style="font-variant:small-caps;">n$\_$pix</span>=90, <span style="font-variant:small-caps;">max$\_$dist</span>=30, <span style="font-variant:small-caps;">order</span>=10 and <span style="font-variant:small-caps;">h$\_$length</span>=0.5. The function to map the core positions to the straightened filaments is <span style="font-variant:small-caps;">Map$\_$cores</span>, also included in <span style="font-variant:small-caps;">FragMent</span>, and the value of the parameter <span style="font-variant:small-caps;">order</span> is 10.
Figure \[fig::straight\_example\] shows the filament from <span style="font-variant:small-caps;">SIM01</span> which has been straightened and the cores, which have been identified using dendrograms, mapped onto this straightened filament. Here one can see that the cores now lie on, or close to, the longitudinal axis of the filament and their positions can be used to investigate the presence of characteristic fragmentation length-scales.
{width="0.48\linewidth"} {width="0.48\linewidth"}
[@\*6l@]{} Nearest neighbour separations & (section \[SSSEC::NNS\]) & & &\
<span style="font-variant:small-caps;">SIM</span> & Number of cores & Median \[pc\] & Mean \[pc\] & Minimum $p$-values & Rejection rate\
01 & 9 & 0.12 (0.14) & 0.28 (0.28) & 0.236 (MS) & 0/4\
02 & 12 & 0.12 (0.14) & 0.16 (0.12) & 0.126 (AD) & 0/4\
03 & 12 & 0.13 (0.18) & 0.21 (0.17) & 0.387 (MI) & 0/4\
04 & 11 & 0.14 (0.18) & 0.18 (0.10) & 0.315 (MI) & 0/4\
05 & 12 & 0.12 (0.14) & 0.16 (0.08) & 0.229 (MI) & 0/4\
06 & 13 & 0.13 (0.08) & 0.14 (0.06) & 0.014 (AD) & 1/4\
07 & 16 & 0.11 (0.05) & 0.14 (0.09) & 0.034 (AD) & 1/4\
08 & 9 & 0.25 (0.29) & 0.33 (0.20) & 0.147 (KS) & 0/4\
09 & 13 & 0.10 (0.10) & 0.16 (0.13) & 0.088 (KS) & 0/4\
10 & 9 & 0.21 (0.05) & 0.23 (0.13) & 0.292 (KS) & 0/4\
All simulations & 116 & 0.12 (0.14) & 0.19 (0.15) & 0.065 (AD) & 0/4\
For the null hypothesis tests included in <span style="font-variant:small-caps;">FragMent</span> one needs a boundary box within which one places randomly-placed cores, defined by the minimum and maximum radius and length. These are different for each simulation. The minimum and maximum values for the length are taken from the spine. The minimum and maximum values for the radius are taken as the minimum and maximum radial locations of the cores, multiplied by 1.1. Typically the spine is roughly 3-4 pc long and the radial width is $<$ 0.1 pc, resulting in a box with an aspect ratio of $\gtrsim$ 30-40; sufficiently high to confidently investigate the mainly longitudinal spacings.
We use all four null hypothesis tests included in <span style="font-variant:small-caps;">FragMent</span>, the Kolmogorov-Smirnov and Anderson-Darling tests, and the two variants of the average-width test where either the mean and standard deviation or the median and interquartile range of the data distribution is compared to the average and width measurements resulting from from the null hypothesis. All null hypothesis tests are run using 100,000 realisations and with a minimum separation of 0.06 pc. A value of 0.06 pc is taken as it corresponds to two times the minimum effective core diameter from the dendrogram parameter, <span style="font-variant:small-caps;">min$\_$npix</span> = 9. The exact value is arbitrary, with the only constraint being that it is smaller than the smallest separation, though it should be informed by the beam size (in real observations) and the minimum allowed size used by the core finding algorithm. As a consequence of this minimum separation value, when constructing the KDEs in the following two subsections we use a reflective boundary condition at 0.06 pc to avoid the presence of artificial peaks.
The separation statistics and the results of the null hypothesis tests resulting from the following sections are summarised in tables \[tab::NNS\] and \[tab::MST\]. We take 0.05 as the threshold $p$-value for rejecting the null hypothesis.
### Nearest neighbour separation {#SSSEC::NNS}
{width="0.48\linewidth"} {width="0.48\linewidth"}
[@\*6l@]{} Minimum spanning tree edge lengths & (section \[SSSEC::MST\]) & & &\
<span style="font-variant:small-caps;">SIM</span> & Number of cores & Median \[pc\] & Mean \[pc\] & Minimum $p$-values & Rejection rate\
01 & 9 & 0.22 (0.68) & 0.41 (0.35) & 0.007 (MI) & 1/4\
02 & 12 & 0.20 (0.25) & 0.25 (0.16) & 0.490 (MI) & 0/4\
03 & 12 & 0.16 (0.30) & 0.27 (0.20) & 0.285 (MI) & 0/4\
04 & 11 & 0.26 (0.24) & 0.28 (0.19) & 0.359 (MI) & 0/4\
05 & 12 & 0.21 (0.16) & 0.23 (0.17) & 0.692 (MS) & 0/4\
06 & 13 & 0.17 (0.20) & 0.27 (0.22) & 0.226 (AD) & 0/4\
07 & 16 & 0.12 (0.22) & 0.21 (0.15) & 0.138 (MI) & 0/4\
08 & 9 & 0.41 (0.36) & 0.42 (0.22) & 0.242 (MI) & 0/4\
09 & 13 & 0.18 (0.23) & 0.27 (0.23) & 0.419 (MS) & 0/4\
10 & 9 & 0.26 (0.39) & 0.40 (0.27) & 0.399 (MI) & 0/4\
All simulations & 116 & 0.21 (0.25) & 0.29 (0.23) & 0.292 (AD) & 0/4\
Figure \[fig::NNS\] shows the nearest neighbour separation distribution for <span style="font-variant:small-caps;">SIM01</span>, and the separation distribution taken from combining the results from all 10 simulations. <span style="font-variant:small-caps;">SIM01</span> contains 9 cores and the total sample contains 116. For <span style="font-variant:small-caps;">SIM01</span>, the distribution is dominated by a peak at small separations, close to the separation limit of 0.06 pc, but has a secondary peak at higher separations, $\sim 0.8$ pc. The total distribution is similar but the peak at small separations is considerably stronger and exhibits a tail that extends to higher separations. The small separation peak shows the prevalence of core/clump fragmentation within the filament. The width of both distributions is comparable to the average separation (see table \[tab::NNS\]) suggesting a lack of a characteristic fragmentation length-scale.
For <span style="font-variant:small-caps;">SIM01</span>, all four null hypothesis tests return $p$-values greater than 0.05; therefore, we can not reject the null hypothesis that the cores are randomly placed. We repeat this analysis for the other 9 simulations and find that in only two of the simulations the null can be rejected (<span style="font-variant:small-caps;">SIM06</span> and <span style="font-variant:small-caps;">SIM07</span>). However, these can only be rejected using one of the tests, the Anderson-Darling test. The other $p$-values for these simulations range from 0.06 to 0.98. We therefore consider it only a tentative rejection. Thus, studying each filament separately leads to no statistically significant and robust fragmentation spacing being detected using the nearest neighbour approach. This may be surprising considering the peaked distributions seen in figure \[fig::NNS\], but the null hypothesis distribution is itself peaked. This is due to the fact one has a fixed length filament with a certain number of cores lying in it, and so introduces a length-scale to the problem. This highlights the need for the null hypothesis test and not to rely on a peaked distribution. The results for each simulation are summarised in table \[tab::NNS\].
We therefore test the significance of the total separation distribution. As each filament has a unique boundary box due to their varying lengths and the radial distances of the cores, sampling the separation distribution resulting from the null hypothesis is slightly more involved. For each simulation we randomly place the same number of cores as detected in the data in that simulation’s boundary box. We then produce the nearest neighbour separation distribution resulting from these randomly placed cores. We do this for each of the 10 simulations and combine the distributions to produce the total nearest neighbour separation distribution from the 116 randomly placed cores. This is treated as 1 realisation of the null hypothesis’ distribution and is repeated 100,000 times to produce a well-sampled null hypothesis distribution. None of the four resulting $p$-values are below 0.05. Despite the large sample size the null can not be rejected and the core fragmentation is indistinguishable from randomly placed cores. Note that by combining the results from all 10 simulations we make the implicit assumption that all 10 of the filaments share a common characteristic fragmentation length-scale. This is not necessarily the case and complicates the combination of data from multiple filaments. We thus emphasise that the most important quantity is the number of cores per filament, as pointed out in @Cla19.
### Minimum spanning tree {#SSSEC::MST}
Figure \[fig::MST\] shows the minimum spanning tree edge length distribution for <span style="font-variant:small-caps;">SIM01</span>, and the distribution taken from combining the results from all 10 simulations. The distribution resulting from <span style="font-variant:small-caps;">SIM01</span> looks similar to the results from the nearest neighbour separation method, a dominant peak at $\sim 0.1$ pc with a small secondary peak at $\sim 0.8$ pc. The same is true for the total distribution; however, the shoulder feature at $\sim$ 0.6 pc suggests that the distribution is bimodal, a narrow distribution around $\sim$ 0.2 pc and a wider one around 0.6 pc. The width of both distributions are comparable to their average values, making a characteristic fragmentation length-scale unlikely.
For <span style="font-variant:small-caps;">SIM01</span> the average-width null hypothesis test returns $p$-values of 0.007 using the median-interquartile range, but the other three tests return $p$-values much greater than 0.05. We therefore consider this a tentative rejection. When this analysis is repeated for the other nine simulations we find that the null can not be rejected in any of the simulations. This is similar to the result from the nearest neighbour approach. The results for each simulation are summarised in table \[tab::MST\].
For the total distribution, all four null hypothesis tests return $p$-values greater than 0.05 and we can not reject the null hypothesis.
### Two-point correlation function {#SSSEC:2point}
![The two-point correlation function resulting from <span style="font-variant:small-caps;">SIM01</span>. The blue shaded region shows the 1$\sigma$ errors as each point due to Poisson noise. The horizontal dashed line at $y=0$ is to help guide the reader.[]{data-label="fig::2point"}](./2point.png){width="0.95\linewidth"}
Figure \[fig::2point\] shows the two-point correlation function for <span style="font-variant:small-caps;">SIM01</span>. The shaded area denotes the 1$\sigma$ error as described in section \[SEC:TECH\]. There exists an extended feature above zero between 0.1 and 0.4 pc, which correspond to the short spacings seen in the minimum spanning tree and nearest neighbour methods. However, it is clear that this feature is approximately a 2$\sigma$ feature and so is only marginally significant. The next peaks occur at approximately 0.9 and 1.2 pc. The first peak is similarly significant, $\sim 2\sigma$, while the second is considerably more, lying around $5\sigma$ above zero. The peak at $\sim$0.9 pc corresponds to the larger scale separations seen in the minimum spanning tree and nearest neighbour methods. However, the peak at $\sim$1.2 pc does not. @Cla19 show that two-tier fragmentation produces peaks in the two-point correlation function which correspond to superpositions of the underlying characteristic fragmentation length-scales, i.e. if there are two length-scales at $x_1$ and $x_2$, the two-point correlation function may present peaks at $x_2 - x_1$ and at $x_2 + x_1$. As the difference between the peaks at $\sim$ 0.9 pc and 1.2 pc, 0.3 pc, corresponds to the feature between 0.1 and 0.4 pc, we suggest that this is tentative evidence of two-tier fragmentation. However, it is unclear why the signal from the superposition is much stronger than the signal at the two underlying characteristic fragmentation length-scales, or why, at the expected location of the $x_2 - x_1$ superposition, there is a deficit of separations (though statistically insignificant due to the error size). As noted by @Cla19, signatures of two-tier fragmentation in the two-point correlation function are complex and typically larger number statistics are needed.
As there is no way to combine the results of two-point correlation functions from multiple different data sets we present the individual two-point correlation function plots from the other 9 simulations in appendix \[app::2point\]. In general, there is no strong signature in any of the two-point correlation functions and no coherent picture can be formed. This is similar to the results from using the nearest neighbour separation and minimum spanning tree methods.
### Lack of a characteristic fragmentation length-scale {#SSSEC:LACK}
There exists no strong signature for the existence of a characteristic fragmentation length-scale in any of the 10 simulations, or the combination of all 10 simulations. The results from the nearest neighbour and minimum spanning tree methods are unable to pass the null hypothesis test, and the results from the two-point correlation function are tentative at best. While this could be due to the small number of cores, @Cla19 show that only a few cores are typically needed to detect single-tier fragmentation, $N\sim10$; a condition which is satisfied here. There may exist an underlying two-tier, or more complex, fragmentation pattern but we are unable to detect it as $N>20$ is typically needed. We therefore can not rule out the null hypothesis, that there exists no characteristic fragmentation length-scale in fibrous filaments. This could be due to the fact that the fragmentation into cores proceeds via sub-filaments, rather than directly from the main filament. @Cla17 show that the formation of sub-filaments and hubs is linked to the turbulent gas motions within the filament, and so has no preferred length-scale. This intermediate fragmentation step erases the expected characteristic fragmentation length-scales that has been seen in previous works of fibre-less filaments [@InuMiy92; @InuMiy97; @FisMar12; @Cla16; @Cla17]. We thus suggest that filaments containing fibres/sub-filaments should lack characteristic fragmentation length-scales, while those without such substructure are more likely to exhibit a fragmentation length-scale due to the dominance of gravity.
This result may appear in contradiction to recent observational work [@Jac10; @Bus13; @Lu14; @Beu15; @Hen16; @Tei16; @Kai17; @Lad20; @Zha20]. These works used a variety of methods to investigate the presence of a characteristic fragmentation length-scale; some used null hypothesis tests and others did not, some relied on clustering in the core separations and others used more advanced techniques such as the two-point correlation function and the N$^{th}$ nearest neighbour method. As shown here, and in more detail in @Cla19, a null hypothesis test is necessary whichever method one uses; clustering or peaks in the separation distribution does not allow one to draw strong conclusions. Moreover, @Cla19 show that certain techniques such as the N$^{th}$ nearest neighbour method are fairly insensitive to the presence of characteristics length-scales and may even produce spurious signatures of such a length-scale when the distribution of cores is random. They also show that the construction of the random core positions for the two-point correlation function must be handled carefully if one wishes to investigate filament fragmentation, i.e. the random cores should be placed on or very close to the filament spine rather than the entire map. Thus, in light of this recent work one should be cautious about the presence of characteristic fragmentation length-scales when these techniques have been used. A re-examining of the data with the techniques presented here and in @Cla19 is encouraged.
The lack of a characteristic fragmentation length-scale does not change when we consider the simulations from a different viewing angle (see appendix \[app::proj\]).
End-dominated collapse {#SSEC:END}
----------------------
![The cumulative probability function of the distance between end cores and the spine beginning and end. The solid blue line shows the distribution derived from the simulations, the dashed red line shows the distribution assuming the null hypothesis.[]{data-label="fig::end_dist"}](./EndDomSep.png){width="0.98\linewidth"}
While there may not exist a characteristic fragmentation length-scale due to filament fragmentation, there may exist another signature of core formation in filaments. Due to non-linear terms in the gravitational acceleration becoming important at the ends of a filament, global collapse proceeds via an *end-dominated* mode [@Bas83]. This means that the ends of the filament are accelerated to greater speeds than the filament interior, sweeping up gas and becoming dense [see @Cla15 and references therein for more details on this phenomenon]. Thus, dense and massive cores should preferentially form close to the filament ends. Observations support that this end-dominated collapse may occur in isolated filaments and induce star formation [@Zer13; @Beu15; @Kai16; @Dew19; @Yuan20; @Bha20; @Liu20].
The ends of the filaments are defined as the two longitudinal parts at either end of the filament where its column density drops to the background value. This is non-obvious in the case of a non-ideal filament without a sharp boundary at either end. Here we use the first and last pixel of the spine to denote the ends of the filament as the main spine is defined by a column density cut, as seen in section \[SSEC:MAINFIL\].
We first investigate if cores do preferentially form at the ends of filaments. We do this by measuring the distance between the start of the spine and the first core, and the distance between the last core and the end of the spine. We call these cores the *end* cores, and the remainder of the cores *interior* cores. We perform a null hypothesis test where the null hypothesis is that cores are placed randomly and thus there is no preferential formation of cores near the filament ends. We use the same 100,000 random realisations as those used in the preceding section.
Figure \[fig::end\_dist\] shows the cumulative probability function for the simulations and resulting from the null hypothesis test. The two distributions have very similar medians, 0.15 and 0.17 pc for the distribution resulting from the data and the null hypothesis, respectively. However, the null hypothesis distribution extends to considerably larger separations while the distribution from the data is steeper. This is evident in the larger interquartile range of the null hypothesis distribution, 0.26 pc, compared to the data distribution, 0.09 pc. We apply a two sample Kolmogorov-Smirnov test and Anderson-Darling test to evaluate if the distribution from the data is distinct from that resulting from the randomly placed cores. For the KS test, the test statistic is large, 0.29, with a corresponding $p$-value of 0.06, and the AD test also returns a large test statistic, 2.40, and a $p$-value of 0.03. We thus tentatively reject the null hypothesis.
We next ask if these end cores are more massive than the interior cores. The left panel of figure \[fig::end\_clump\] shows kernel density estimators of the mass distribution of end cores and interior cores. One can see that the mass distribution for end cores is shifted to higher masses compared to the interior cores. The mean and standard deviation of the mass distributions are: 6.4 M${_{_{\odot}}}$ and 5.8 M${_{_{\odot}}}$ for the end cores, and 4.3 M${_{_{\odot}}}$ and 3.9 M${_{_{\odot}}}$ for the interior cores. A KS test returns a test statistic of 0.29 with a corresponding $p$-value of 0.10, and the AD test returns a test statistic of 1.27 with a $p$-value of 0.10. The Mann-Whitney U test returns similar $p$-value of 0.08. We are therefore unable to reject the null hypothesis that the masses of end cores and interior cores are sampled from the same underlying distribution, but the test statistics are relatively large and more simulations could help reject the null.
We also compare the peak column density of end cores to that of interior cores. The right panel of figure \[fig::end\_clump\] shows kernel density estimators of the peak column density distribution of end cores and interior cores. One can see that the distribution for end cores is peaked at slightly higher column densities compared to the bimodal interior core distribution. The mean and standard deviation of the column density distributions are: 3.5 $\times 10^{23}$ cm$^{-2}$ and 3.4 $\times 10^{23}$ cm$^{-2}$ for the end cores, and 2.1 $\times 10^{23}$ cm$^{-2}$ and 2.6 $\times 10^{23}$ cm$^{-2}$ for the interior cores. A KS test returns a test statistic of 0.36 with a corresponding $p$-value of 0.02, and the AD test returns a test statistic of 2.39 with a $p$-value of 0.03. The Mann-Whitney U test also returns a similarly small $p$-value of 0.04. We therefore reject the null hypothesis and can say that the peak column density of end cores is statistically higher than those of interior cores.
We conclude that there is a tentative signature of end-dominated collapse, leading to the formation of cores close to the filament ends which are slightly more massive and dense than cores located in the filament’s interior. An increase in number statistics would likely increase this signature due to the large test statistic values returned by the KS and AD tests.
{width="0.48\linewidth"} {width="0.48\linewidth"}
Conclusions {#SEC:CON}
===========
Our numerical study presents an intriguing scenario of how sub-filaments impact fragmentation. First, filaments fragment into sub-filaments due to their internal turbulence, likely driven by accretion from the surrounding medium. Regions where a number of sub-filaments join and meet form; we term these regions hub-sub-filament systems due to their similarity to the parsec-scale hub-filament systems commonly seen in molecular clouds [e.g. SDC13 and MonR2, @Wil18; @Trev19]. These hubs fragment into small ensembles of clustered cores. Away from these hubs, sub-filaments fragment into isolated single cores, or a small chain of cores. Cores formed in the hubs are on average more massive than those formed in the isolated sub-filaments, and also show a wider mass distribution. This is also reminiscent of the parsec-scale hub-filament systems, leading to the conclusion that the combination of turbulence and gravity leads to similar patterns of fragmentation on multiple scales.
The fact that filaments first fragment into sub-filaments which then proceed to form cores erases any evidence of the expected characteristic fragmentation length-scale of filament fragmentation. This fits well with observations which have been unable to find robust evidence for quasi-periodically spaced cores, and is a significant departure from previous fragmentation models. We thus expect that filaments containing sub-filaments to show no clear sign of characteristic fragmentation length-scales, while those filaments without sub-filaments are more likely to.
End-dominated collapse leads to the preferential formation of cores close to a filament’s ends. These cores are slightly more massive and dense than cores located in the interior due to gravitational focusing. However, this signature is weak and requires good number statistics for a robust detection.
As this work used an idealised cylindrical colliding flow set-up (e.g. without magnetic fields) to investigate the basic underlying physics of filament fragmentation, large-scale molecular cloud simulations would be a valuable follow up, allowing the addition of environmental complexities. Such a study (Ganguly et al. in prep.) will be carried out within the SILCC-zoom project [@Wal15; @Sei17], which simulates the formation of molecular clouds from the galactic, multi-phase interstellar medium.
Acknowledgments {#SEC:ACK}
===============
We thank the referee for their comments which have helped to improve the quality of the paper. SDC and SW acknowledges support from the ERC starting grant No. 679852 ‘RADFEEDBACK’. GMW acknowledges support from the UK’s Science and Technology Facilities Council under grant number ST/R000905/1. SW thank the DFG for funding through the Collaborative Research Center (SFB956) on the ‘Conditions and impact of star formation’ (sub-project C5).
Data availability {#SEC:ACK}
=================
The data underlying this article will be shared on reasonable request to the corresponding author. The principle analysis tool of this work, <span style="font-variant:small-caps;">FragMent</span>, is made freely available at https://github.com/SeamusClarke/FragMent.
The two-point correlation function results for <span style="font-variant:small-caps;">SIM02</span> - <span style="font-variant:small-caps;">SIM010</span> {#app::2point}
==========================================================================================================================================================
The individual two-point correlation functions resulting from the nine other simulations, <span style="font-variant:small-caps;">SIM02</span> - <span style="font-variant:small-caps;">SIM10</span>, are shown in figure \[fig::2pointappendix\]. The results from each are summarised in table \[tab::2point\]. There is no clear sign of any characteristic fragmentation length-scale.
{width="0.9\linewidth"}
[@\*3l@]{} & Two-point correlation function & (section \[SSSEC:2point\])\
<span style="font-variant:small-caps;">SIM</span> & Inference & Comment\
01 & Tentative two-tier fragmentation & Possible features at $\sim$0.2 and $\sim$0.9 pc which produce a superposition at $\sim$1.2 pc.\
02 & Tentative single-tier fragmentation & Feature at $\sim$0.6 pc with possible harmonic at $\sim$1.2 pc.\
03 & No clear signal & Weak feature at $\sim$0.2 pc, no harmonics. Peak at 1.5 pc also apparent.\
04 & Tentative single-tier fragmentation & Feature at $\sim$0.65 pc and harmonic at $\sim$1.4 pc.\
05 & Single-tier fragmentation & While individually each peak is weak the harmonics are clear at $\sim$0.25, 0.5, 0.75, 1.0, 1.25 pc.\
06 & No clear signal & Extended feature between 0.8 and 1.2 pc.\
07 & No clear signal & Nearly zero everywhere.\
08 & Tentative two-tier fragmentation & Possible features at $\sim$0.2 and 0.7 pc with weak superpositions at $\sim$0.5 and 0.9 pc.\
09 & Tentative single-tier fragmentation & Possible feature and harmonics at $\sim$0.3, 0.6, 0.9 and 1.3 pc.\
10 & No clear signal & Close to zeros everywhere with no clear harmonics.\
{width="0.98\linewidth"}
![The core mass distribution using the cores identified with the main viewing angle (black, solid line), and the appendix viewing angle (blue, dashed line). The small vertical lines show each individual data point.[]{data-label="appfig::coremass"}](./Appendix_plots/Core_mass_compare.png){width="0.98\linewidth"}
The effect of viewing angle {#app::proj}
===========================
As this work has been preformed on column density maps from 3D simulations created by considering a viewing angle aligned with the $z$-axis, we check the effect of projection by considering a viewing angle aligned with the $y$-axis. We call the viewing angle (or projection) aligned with the $z$-axis the *main* viewing angle (or projection) as it is discussed in the main body of the paper, and that aligned with the $y$-axis, the *appendix* viewing angle (or projection).
Core and spine identification {#app::iden}
-----------------------------
Figure \[appfig::spine\] shows the column density plot of <span style="font-variant:small-caps;">SIM01</span> from the appendix viewing angle. Overlaid in red are the cores and in white is the main filament spine; both are found using the same techniques and parameters as in sections \[SEC:CORE\] and \[SSEC:MAINFIL\], respectively. From all ten simulations, 105 cores are identified, only slightly fewer than that observed with the main viewing angle. Figure \[appfig::coremass\] shows the core mass distributions from the two viewing angles, it is clear there are few differences. A two sample KS test returns a small test statistic of 0.08 and a $p$-value of 0.84. Thus, the core masses from the two viewing angles are statistically indistinguishable.
Characteristic fragmentation length-scale {#app::frag}
-----------------------------------------
### Nearest neighbour results {#app::NNS}
Figure \[appfig::NNS\] shows the nearest neighbour separation distribution from <span style="font-variant:small-caps;">SIM01</span> for the cores identified with the appendix viewing angle, and also the separation distribution when combining results from all 10 simulations. The results are summarised in table \[apptab::NNS\]. As with the main viewing angle, the results from the nearest neighbour show no strong evidence of a characteristic fragmentation length-scale, and the null hypothesis can not be rejected strongly.
### Minimum spanning tree results {#app::MST}
Figure \[appfig::MST\] shows the minimum spanning tree edge length distribution from <span style="font-variant:small-caps;">SIM01</span> for the cores identified with the appendix viewing angle, and also the edge length distribution when combining results from all 10 simulations. The results are summarised in table \[apptab::MST\]. As with the main viewing angle, the results from the minimum spanning tree show no strong evidence of a characteristic fragmentation length-scale, and the null hypothesis can not be rejected strongly.
### Two-point correlation function {#appB::2point}
Figure \[appfig::2point\] shows the two-point correlation function for <span style="font-variant:small-caps;">SIM01</span> for the cores identified with the appendix viewing angle. Here there are strong signs of two-tier fragmentation, a small-scale peak at $\sim$0.2 pc, and a large scale one at $\sim$1.1 pc, with superpositions at $\sim$0.9 pc and $\sim$1.3 pc. This bimodality is seen in the nearest neighbour and minimum spanning tree results but neither could robustly reject the null hypothesis.
Figure \[appfig::all2point\] shows the two-point correlations functions for <span style="font-variant:small-caps;">SIM02</span> to <span style="font-variant:small-caps;">SIM10</span>. There is no strong evidence for any characteristic length-scale in these nine simulations, other than in <span style="font-variant:small-caps;">SIM04</span> which suggests a possible signal at $\sim$0.6 pc and a harmonic at $\sim$1.2 pc. This is similar to the results presented in the main body of the paper, section \[SSSEC:2point\].
{width="0.48\linewidth"} {width="0.48\linewidth"}
[@\*6l@]{} Nearest neighbour separations & (section \[app::NNS\]) & & &\
<span style="font-variant:small-caps;">SIM</span> & Number of cores & Median \[pc\] & Mean \[pc\] & Minimum $p$-values & Rejection rate\
01 & 8 & 0.14 (0.77) & 0.40 (0.37) & 0.005 (MI) & 2/4\
02 & 12 & 0.11 (0.12) & 0.15 (0.10) & 0.427 (AD) & 0/4\
03 & 11 & 0.13 (0.08) & 0.21 (0.21) & 0.190 (MS) & 0/4\
04 & 10 & 0.13 (0.07) & 0.15 (0.05) & 0.368 (KS) & 0/4\
05 & 9 & 0.15 (0.14) & 0.25 (0.21) & 0.453 (MS) & 0/4\
06 & 15 & 0.09 (0.06) & 0.11 (0.05) & 0.002 (AD) & 1/4\
07 & 10 & 0.19 (0.09) & 0.20 (0.13) & 0.711 (MI) & 0/4\
08 & 8 & 0.27 (0.40) & 0.37 (0.24) & 0.105 (MI) & 0/4\
09 & 15 & 0.10 (0.08) & 0.16 (0.18) & 0.042 (MS) & 1/4\
10 & 7 & 0.17 (0.07) & 0.22 (0.19) & 0.115 (KS) & 0/4\
All simulations & 105 & 0.14 (0.12) & 0.21 (0.20) & 0.133 (AD) & 0/4\
{width="0.48\linewidth"} {width="0.48\linewidth"}
[@\*6l@]{} Minimum spanning tree edge lengths & (section \[app::MST\]) & & &\
<span style="font-variant:small-caps;">SIM</span> & Number of cores & Median \[pc\] & Mean \[pc\] & Minimum $p$-values & Rejection rate\
01 & 8 & 0.16 (0.75) & 0.44 (0.38) & 0.001 (MI) & 1/4\
02 & 12 & 0.20 (0.25) & 0.25 (0.17) & 0.610 (MS) & 0/4\
03 & 11 & 0.17 (0.40) & 0.28 (0.23) & 0.044 (MI) & 1/4\
04 & 10 & 0.24 (0.18) & 0.28 (0.18) & 0.886 (AD) & 0/4\
05 & 9 & 0.26 (0.40) & 0.38 (0.30) & 0.322 (MI) & 0/4\
06 & 15 & 0.14 (0.12) & 0.22 (0.23) & 0.082 (MI) & 0/4\
07 & 10 & 0.20 (0.35) & 0.32 (0.23) & 0.317 (MI) & 0/4\
08 & 8 & 0.56 (0.37) & 0.46 (0.22) & 0.056 (MI) & 0/4\
09 & 15 & 0.15 (0.21) & 0.23 (0.19) & 0.451 (MI) & 0/4\
10 & 7 & 0.16 (0.76) & 0.44 (0.38) & 0.001 (MI) & 1/4\
All simulations & 105 & 0.20 (0.35) & 0.32 (0.28) & 0.005 (MS) & 2/4\
![The two-point correlation function resulting from <span style="font-variant:small-caps;">SIM01</span>. The blue shaded region shows the 1-sigma errors as each point due to Poisson noise. The horizontal dashed line at $y=0$ is to help guide the reader.[]{data-label="appfig::2point"}](./Appendix_plots/2point1.png){width="0.95\linewidth"}
{width="0.9\linewidth"}
\[lastpage\]
[^1]: E-mail: clarke@ph1.uni-koeln.de
[^2]: www.cds.uni-koeln.de
[^3]: https://github.com/SeamusClarke/FragMent
[^4]: http://www.dendrograms.org/
|
---
abstract: 'We introduce a semi-supervised learning estimator which tends to the first kernel principal component as the number of labeled points vanishes. Our approach is based on the notion of optimal target vector, which is defined as follows. Given an input data-set of ${\bf x}$ values, the optimal target vector $\mathbf{y}$ is such that treating it as the target and using kernel ridge regression to model the dependency of $y$ on ${\bf x}$, the training error achieves its minimum value. For an unlabeled data set, the first kernel principal component is the optimal vector. In the case one is given a partially labeled data set, still one may look for the optimal target vector minimizing the training error. We use this new estimator in two directions. As a substitute of kernel principal component analysis, in the case one has some labeled data, to produce dimensionality reduction. Second, to develop a semi-supervised regression and classification algorithm for transductive inference. We show application of the proposed method in both directions.'
author:
- 'Leonardo Angelini, Daniele Marinazzo, Mario Pellicoro and Sebastiano Stramaglia'
title: ' Semi-supervised learning by search of optimal target vector'
---
Introduction {#intro}
============
The problem of effectively combining [*unlabeled*]{} data with [*labeled*]{} data, semi-supervised learning, is of central importance in machine learning; see, for example, [@zu; @zhu; @ch] and references therein. Semi-supervised learning methods usually assume that adjacent points and/or points in the same structure (group, cluster) should have similar labels; one may assume that data are situated on a low dimensional manifold which can be approximated by a weighted discrete graph whose vertices are identified with the empirical (labeled and unlabeled) data points. This can be seen as a form of regularization [@smo]. A common feature of these methods, see also [@arg], is that, as the number of labeled points vanishes, the solution tends to the constant vector. An interesting survey on semi-supervised learning literature may be found on the web [@zz]. Improving regression with unlabeled data is the problem considered in [@zhou], where co-training is achieved using k-NN regressors. A statistical physics approach, based on the Potts model, is described in [@getz]. An issue closely related to semi-supervised learning is active-learning: some attempts to combine active learning and semi-supervised learning has been made [@zzz].
The purpose of this work is to introduce a semi-supervised learning estimator which, as the number of labeled points vanishes, tends to the first kernel principal component [@kpca]; when a suitable number of labeled points is available, it may be used for transductive inference [@vapnik]. Our approach is based on the following fact. Given an unlabeled data set, its first kernel principal component is such that, treating it as target vector, supervised kernel ridge regression provides the minimum training error. Now, suppose that you are given a partially labeled data set: still one may look for the target vector minimizing the training error. This optimal target vector may be seen as the generalization of the first kernel principal component to the semi-supervised case.
The paper is organized as follows. In the next Section we describe our approach, while in Section 3 the experiments we performed are described. Some conclusions are drawn in Section 4.
Methods
=======
Kernel ridge regression
-----------------------
We briefly recall the properties of kernel ridge regression (KRR), while referring the reader to [@st] for further technical details. Let us consider a set of $\ell$ independent, identically distributed data $S=\{ ({\bf x}_i, y_i) \}_{i=1}^\ell$, where ${\bf x}_i$ is the $n$-dimensional vector of input variables and $y_i$ is the scalar output variable. Data are drawn from an unknown probability distribution; we assume that both ${\bf x}$ and $y$ have been centered, i.e. they have been linearly transformed to have zero mean. The regularized linear predictor is $y={\bf w}\cdot{\bf x}$, where ${\bf
w}$ minimizes the following functional: $$\label{lagrangian-function-without-b}
L(\mathbf{w}) = \sum_{i=1}^\ell \left ( y_i - {\bf w}\cdot {\bf x}_i \right )^2 +
\lambda || {\mathbf w} ||^2.$$ Here $|| \mathbf{w} || = \sqrt{\bf{w}\cdot\bf{w}} $ and $\lambda >0$ is the regularization parameter. For $\lambda =0$, predictor (\[lagrangian-function-without-b\]) is invariant when new variables, statistically independent of input and target variables, are added to the set of input variables (IIV property, [@as]). One may show that this invariance property holds, for (\[lagrangian-function-without-b\]), also at finite $\lambda
> 0$.
KRR is the [*kernel*]{} version of the previous predictor. Calling $\bf{y}$ = $(y_1, y_2,
..., y_\ell)^\top$ the vector formed by the $\ell$ values of the output variable and $K(\cdot,\cdot)$ being a positive definite symmetric function, the predictor has the following form: $$\label{notlinear}
y = f({\bf x}) = \sum_{i=1}^\ell c_i K({\bf x}_i,{\bf x}),$$ where coefficients $\{c_i\}$ are given by $$\label{w2}
\bf{c} = \left (\bf{K} + \lambda \bf{I} \right)^{-1} \bf{y},$$ $\bf{K}$ being the $\ell \times \ell$ matrix with elements $K({\bf x}_i,{\bf x}_j)$. Equation (\[notlinear\]) may be seen to correspond to a linear predictor in the feature space $ \Phi({\bf x}) = ( \sqrt{\alpha_1}\psi_1 ({\bf x}), \sqrt{\alpha_2}\psi_2 ({\bf
x}), ...,\sqrt{\alpha_N} \psi_N ({\bf x}), ... ), $ where $\alpha_i$ and $\psi_i$ are the eigenvalues and eigenfunctions of the integral operator with kernel $K$. One may show [@prep] that, for KRR predictors with nonlinear kernels, the IIV property does not generically hold, even for those kernels, discussed in [@as], for which the property holds at $\lambda=0$. Regularization breaks the IIV invariance in those cases.
Due to (\[notlinear\]) and (\[w2\]), the predicted output vector $\bf{\bar{y}}$, in correspondence of the [*true*]{} target vector $\bf{y}$, is given by $\bf{\bar{y}}$$=\mathbf{G}\bf{y}$, where the symmetric matrix $\mathbf{G}$ is given by $$\label{G}
\mathbf{G}=\mathbf{K}\left(\mathbf{K}+\lambda \mathbf{I}\right)^{-1}.$$ Note that matrix $\mathbf{G}$ depends only on the distribution of $\{\mathbf{x}\}$ values: $\mathbf{G}$ embodies information about the structures present in $\{\mathbf{x}\}$ data set. Indeed, for $i\ne j$, the matrix element $G_{ij}$ quantifies how much the target value of the $j-th$ point influences the estimate of the target of point $i$. Let us now consider the leave-one-out scheme; let data point $i$ be removed from the data set and the model be trained using the remaining $\ell -1$ points. We denote $\tilde{y}_i$ the target value thus predicted, in correspondence of $\bf{x_i}$. It is well known [@st] that the leave-one-out-error $\tilde{y}_i -y_i$ and the training error obtained using the whole data set $\bar{y}_i -y_i$ satisfy: $$\label{loo} \tilde{y}_i -y_i={\bar{y}_i -y_i \over 1-G_{ii}}.$$ This formula shows that the closer $G_{ii}$ to one, the farther the leave-one-out predicted value from those obtained using also point $i$ in the training stage. Consider a point $i$ in a dense region of the feature space: one may expect that removing this point from the data-set would not change much the estimate since it can be well predicted on the basis of values of neighboring points. Therefore points in low density regions of the feature space are characterized by diagonal values $G_{ii}$ close to one, while $G_{ii}$ is close to zero for points $\mathbf{x_i}$ in dense regions: the diagonal elements of $\mathbf{G}$ thus convey information about the structure of points in the feature space. It is worth stressing that, given a kernel function, the corresponding features $\psi_\gamma ({\bf x})$ are not centered in general. One can show [@kpca] that centering the features ($\psi_\gamma \to \psi_\gamma -\langle \psi_\gamma\rangle$, for all $\gamma$) amounts to perform the following transformation on the kernel matrix: $$\mathbf{K }\to \mathbf{\tilde{K}}=\mathbf{K}-\mathbf{I}_\ell \mathbf{K}-\mathbf{K}\mathbf{I}_\ell +\mathbf{I}_\ell \mathbf{K} \mathbf{I}_\ell,$$ where $\left( I_\ell\right)_{ij}=1/\ell$, and to work with the centered kernel $\mathbf{\tilde{K}}$. In the following we will assume that the kernel matrix $\mathbf{K}$ has been centered.
Optimal target vector
---------------------
The training error of the KRR model is proportional to $(\bf{y}-\mathbf{G}\bf{y})^\top(\bf{y}-\mathbf{G}\bf{y})=\bf{y}^\top \mathbf{H}\bf{y},$ where $\mathbf{H}= \mathbf{I}-2\mathbf{G}+\mathbf{G}\mathbf{G}$ is a symmetric and positive matrix. In the unsupervised case the data set is made of $\mathbf{x}$ points, $\{ {\bf x}_i\}_{i=1}^\ell$, the target function $\mathbf{y}$ is missing. However we may pose the following question: what is the vector $\mathbf{y}\in \mathbf{R}^\ell$ such that treating it as the target vector leads to the best fit, i.e. the minimum training error $\bf{y}^\top \mathbf{H}\bf{y}$? We expect that this [*optimal*]{} target vector would bring information about the structures present in the data. To avoid the trivial solution $\mathbf{y}=\mathbf{0}$, we constrain the target vector to have unit norm, $\mathbf{y}^\top\mathbf{y}=1$; it follows that the optimal vector is the normalized eigenvector of $\mathbf{H}$ with the smallest eigenvalue. On the other hand, matrix $\mathbf{H}$ is a function of matrix $\mathbf{K}$: hence it has the same eigenvectors of $\mathbf{K}$ while the corresponding eigenvalues $\mu_H$ and $\mu_K$ are related by the following monotonically decreasing correspondence: $$\mu_H=\left(1-{\mu_K\over \mu_K+\lambda}\right)^2.$$ Therefore, independently of $\lambda$, the smallest eigenvalue of $\mathbf{H}$ corresponds to the largest eigenvalue of $\mathbf{K}$, and the optimal vector coincides with the first kernel principal component. To conclude this subsection, we have shown that the method in \[10\] may be motivated also as the search for the optimal target vector.
The notion of optimal target vector has been introduced in [@ang], where a kernel method for dichotomic clustering has been proposed, consisting in finding the ground state of a class of Ising models.
Semi-supervised learning
------------------------
Now we consider the case that we are given a set $S=\{ {\bf x}_i \}_{i=1}^\ell$ of data points with unknown targets $\{t_i\}_{i=1}^\ell$, and a set $S'=\{ ({\bf x}_j, u_j)
\}_{j=\ell+1}^{N}$, where $N=\ell +m$, of input-output data. Without loss of generality we assume that the labeled points belong to two classes, and take $u_j\in
\{-1/\sqrt{N},+1/\sqrt{N}\}$ for all $j$’s. The $N$ dimensional full vector of targets $\mathbf{y}$ is obtained appending $\{t\}$ (unknown) and $\{u\}$ (known) values: $$\mathbf{y}=(\mathbf{t}^\top \mathbf{u}^\top)^\top.$$ Keeping the kernel and $\lambda$ fixed, we look for the unit norm target vector $\mathbf{y}$ minimizing the training error $\mathbf{y}^\top \mathbf{H} \mathbf{y}$. The $N\times N$ matrix $\mathbf{H}$ has the block structure $$\mathbf{H} = \left( \begin{array}{cc}
\mathbf{H_0} & \mathbf{H_1} \\
\mathbf{H_1^\top}& \mathbf{H_2}
\end{array}\right),$$ where $\mathbf{H_0}$ is an $\ell \times \ell$ matrix. Neglecting a constant term, the optimal vector is determined by the vector $\mathbf{t}$ minimizing $$\mathcal{E}(\bf{t})=\mathbf{t}^\top \mathbf{H_0} \mathbf{t} +2\mathbf{t}^\top
\mathbf{H_1} \mathbf{u} \label{eeee}$$ under the constraint $|| \mathbf{t} ||^2=1-|| \mathbf{u} ||^2$. The first term of $\mathcal{E}$ favors projections of the $\ell$ points with great variance, whereas the second term measures their consistency with labeled points. Let us denote $\{\Psi_{\alpha'}\}$ and $\{\mu_{\alpha'}\}$ the eigenvectors and eigenvalues of $\mathbf{H_0}$, sorted into increasing $\mu_{\alpha'}$. We express $\mathbf{t}=\sum_{\alpha' =1}^\ell \xi_{\alpha'} \Psi_{\alpha'}$. The coefficients $\xi_{\alpha'}$ for the minimum are given by $$\xi_{\alpha'} ={f_{\alpha'} \over \mu -\mu_{\alpha'}},$$ where $f_{\alpha'}= \Psi_{\alpha'}^\top\mathbf{H_1} \mathbf{u}$, and $\mu$ is a Lagrange multiplier which must to be tuned to satisfy: $$\label{csi} g(\mu)= \sum_{\alpha' =1}^\ell
\left({f_{\alpha'} \over \mu -\mu_{\alpha'}}\right)^2=1-|| \mathbf{u} ||^2.$$
Equation (\[csi\]) has always at least one solution with $\mu < \mu_1$, see figure 1, and usually this is the one minimizing $\mathcal{E}$. However all the solutions of (\[csi\]) must be compared according to their [*energies*]{} $\mathcal{E}$; those corresponding to the lowest $\mathcal{E}$, $\bf{y^\star}$, is then selected. Clearly as $m\to 0$ one recovers the first eigenvector of $\mathbf{H_0}$, i.e. the first kernel principal component: $\bf{y^\star}$ thus constitutes a generalization of the latter to the semi-supervised case. To construct the other generalized kernel principal components, we make the following transformation on matrix $\mathbf{H}$: $$\mathbf{\tilde{H}}=\mathbf{H}-\mathbf{P^\star}\mathbf{H}
-\mathbf{H}\mathbf{P^\star}+\mathbf{P^\star}\mathbf{H}\mathbf{P^\star},$$ where $\mathbf{P^\star}=\bf{y^\star}\bf{y^\star}^\top$ is the projector on the linear subspace spanned by $\bf{y^\star}$. The symmetric matrix $\mathbf{\tilde{H}}$ has the lowest eigenvalue equal to zero and corresponding to eigenvector $\bf{y^\star}$. The system of eigenvectors of $\mathbf{\tilde{H}}$ constitutes a generalization of kernel principal components to the semi-supervised case.
Experiments
===========
Generalizing kernel principal components
----------------------------------------
Now we present some simulations of the proposed method, focusing on the dimensionality reduction issue and comparing with fully unsupervised kernel principal component analysis. We consider three well known data sets: IRIS (100 points in a four-dimensional space, second and third classes, versicolor and virginica); colon cancer data set of [@alon], consisting in 40 tumor and 22 normal colon tissues samples, each sample being described by the $100$ most discriminant genes; the leukemia data set of [@golub], consisting of samples of tissues of bone marrow samples, $47$ affected by acute myeloid leukemia (AML) and $25$ by acute lymphoblastic leukemia (ALL), each sample being described by the $500$ most discriminant genes. The following question is addressed: is $\bf{y^\star}$ more correlated to the true labels than the fully unsupervised first kernel principal component? Here we restrict our analysis to the linear kernel.
We start with IRIS and proceed as follows. We randomly select $m=4$ points and, treating them as labeled, we find the system of eigenvectors of $\mathbf{\tilde{H}}$. Then we evaluate the linear correlation $R$ between the eigenvectors and the true labels of the whole data-set. The distributions of $R$ for the four eigenvectors are depicted in figure 2. We observe that in most cases the vector $\bf{y^\star}$ is more correlated with the true classes than the fully unsupervised principal component: the one-dimensional projection of data onto $\bf{y^\star}$ is more informative than the first principal component. However there are situations where use of labeled points leads to poor results; a typical example is depicted in figure 3. In figure 4 a situation is depicted where knowledge of labeled points leads to a relevant improvement.
In general, we denote $f$ the fraction of instances such that $\bf{y^\star}$ is more correlated to the true labels than the first principal component. In figure 5 we depict $f$ as a function of $\bar{m}=m/N$ for the three data sets here considered. At $\bar{m}=0.16$ $f$ is already nearly one. The semi-supervised method here proposed outperforms principal components almost always for large $\bar{m}$.
Transductive inference
----------------------
In this subsection we demonstrate the effectiveness of the proposed approach for estimating the values of a function at a set of test points, given a set of input-output data points, without estimating (as an intermediate step) the regression function.
The boston data set is a well-known problem where one is required to estimate house prices according to various statistics based on $13$ locational, economic and structural features from data collected by U.S. Census Service in the Boston Massachusetts area. For $\ell =5,10,15,20,25$, we partition the data-set of $N=506$ observations randomly 100 times into a training set of $N-\ell$ observations and a testing set of $\ell$ observations. We use a Gaussian kernel with $\sigma =1$ and set $\lambda =1$; results are stable against variations of these parameters. In Table 1 we report the mean squared error (MSE) on the test set averaged over the 100 runs, for each value of $\ell$, we obtain using the optimal target vector $\bf{y^\star}$. In Table 1 we also report the MSE obtained using the classical KRR in the two step procedure: (i) estimation of the regression function using the training data-set (ii) calculation of the regression function at points of interest (test data-set). The improvement achieved using the optimal target approach, over classical KRR, is clear.
$\ell$ OT KRR
-------- -------- --------
5 2.3790 3.6312
10 2.7938 4.0111
15 2.9460 4.1057
20 3.1024 4.1802
25 3.1569 4.1653
: \[tab:table1\]The mean square error on the Boston data set obtained using the optimal target (OT) approach and the classical kernel ridge regression (KRR) method. The size of the test set is $\ell$.
We also consider five well known data sets of pattern recognition from UCI database: we evaluate the optimal target vector, points are then attributed to classes according to the sign of $\bf{y^\star}$. We compare with the transductive linear discrimination (TLD) approach developed in [@trans]; the performance of a classifier is measured by its average error over 100 partitions of the data-sets into training and testing sets. We use the linear kernel with $\lambda =1$, however the results are stable to variations of $\lambda$. Obviously, our approach and TLD are applied to the same partitions of data-sets, so that the comparison is meaningful. The results are shown in Table 2: our approach outperforms TLD.
TLD OT
--------------- ------ -------
Diabetes 23.3 11.98
Titanic 22.4 6.52
Breast Cancer 25.7 16.7
Heart 15.7 3.3
Thyroid 4.0 4.0
: \[tab:table2\]The percentage test error of transductive linear discrimination and optimal target approach, on five datasets from UCI database.
It is worth stressing that our results are obtained without a fine-tuning of parameters. In particular,note that our definition of optimal target vector fixes the relative importance of the two terms in equation (\[eeee\]).
Conclusions {#conc}
===========
We have presented a new approach to semi-supervised learning based on the notion of optimal target vector, the target vector such that KRR provides the minimum training error over all the possible target vectors. The proposed algorithm is characterized by the fact that the first kernel principal component is recovered as the cardinality of labeled points vanishes; hence it may be seen as a semi-supervised generalization of Kernel Principal Components Analysis. The effectiveness of the proposed approach for transductive inference has also been demonstrated.
0.4 cm
[**Acknoledgements.**]{} The authors thank Olivier Chapelle for a valuable correspondence on the subject of this paper. Discussions on semi-supervised learning with Eytan Domany and Noam Shental are warmly acknowledged.
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|
---
abstract: 'We report the selective stabilization of chiral rotational direction of bacterial vortices, from turbulent bacterial suspension, in achiral circular microwells sealed by an oil-water interface. This broken-symmetry, originating from the intrinsic chirality of bacterial swimming near hydrodynamically different top and bottom surfaces, generates a chiral edge current of bacteria at lateral boundary and grows stronger as bacterial density increases. We demonstrate that chiral edge current favors co-rotational configurations of interacting vortices, enhancing their ordering. The interplay between the intrinsic chirality of bacteria and the geometric properties of the boundary is a key-feature for the pairing order transition of active turbulence.'
author:
- Kazusa Beppu
- Ziane Izri
- Tasuku Sato
- Yoko Yamanishi
- Yutaka Sumino
- 'Yusuke T. Maeda'
- Kazusa Beppu
- Ziane Izri
- Tasuku Sato
- Yoko Yamanishi
- Yutaka Sumino
- 'Yusuke T. Maeda'
title: Edge Current and Pairing Order Transition in Chiral Bacterial Vortex
---
Turbulent flows offer a rich variety of structures at large length scales and are usually obtained by driving flows out of equilibrium[@zhang] while overcoming viscous dampening. A peculiar class of out-of-equilibrium fluids, stimulated from the lower scales, also present turbulence-like structures called active turbulence [@ramaswamy; @marchetti]. For example, a dense bacterial suspension is driven out of equilibrium by the autonomous motion of self-propelled bacteria suspended therein[@yeomans; @frey; @kokot; @Li]. The alignment between bacteria shapes the active turbulence patterns of collective swimming into vortices of similar size[@wioland1; @wioland2; @beppu; @hamby; @clement; @nishiguchi2]. However, this vortical order decays over distance, making it a long-standing issue for the development of ordered dynamics at larger scales. Hence, a growing attention is paid to novel strategies to control active turbulence with simple geometric design.
![**Chiral bacterial vortex.** (a) Experimental setup: a dense bacterial suspension confined in hydrophilic-treated PDMS microwells and sealed under oil/water interface stabilized with a surfactant. (b) Ensemble of chiral bacterial vortices, in microwells with radius (top row) and (bottom row). Color map codes for the direction of the velocity field. Scale bar, . Schematic illustration of a bacterial vortex in a single microwell. “+” defines the positive sign of CCW rotation. CCW and CW occurrences are displayed in blue and red arrows, respectively.[]{data-label="fig1"}](figure1_9.jpg)
Chirality, i.e. the non-equivalence of opposite handedness, is ubiquitous across scales[@bahr], and is commonly involved in active systems[@glotzer; @lowen; @levis; @lenz], either biological, such as bacteria[@whitesides; @haoran; @howard; @petroff], cytoskeletons and molecular motors[@tee; @frey2; @kim], or non-biological, consisting of self-propelled colloids[@jiang; @bechinger1; @irvine]. One of the effects of chirality is the non-equivalence of clockwise (CW) and counter-clockwise (CCW) rotations. As for bacteria, broken mirror-symmetry in flagellar rotation (CCW rotation around the tail-to-head direction during swimming) results in the opposite rotation of the cell body, which generates a net torque onto the solid surface the bacterium swims over, and in turn bends its trajectory circularly[@whitesides]. Despite such intrinsic chirality in individual motion, active turbulence reported in the past showed CW and CCW global rotational directions have equal probability, indicating that mirror symmetry was recovered at the collective level[@wioland2; @beppu; @nishiguchi2; @hamby]. Can microscopic chirality of bacterial motion be transferred into the macroscopic order of collective swimming? Such question is a great challenge that would provide both fundamental understanding of active turbulence and technical applications for controlled material transport[@fabrizio; @aronson].
In this Letter, we report the chiral collective swimming of a dense bacterial suspension confined in an asymmetric (different top and bottom interfaces) but achiral (perfectly circular lateral interface) hydrodynamic boundary. Non-equivalence between CW and CCW collective swimming is enhanced as the bacterial density increases, and the selected CCW rotation with respect to the bottom-top direction, later referred to as “top view”, induces persistent edge current mostly observed on bacteria swimming near the lateral boundaries. Such edge current can alter the geometric constraints ruling the self-organization of bacterial vortices by suppressing the anti-rotational mode. The extended geometric rule, which is in excellent agreement with experiment, brings new understanding of chiral active matter in order to organize larger scale flow.
![**Chiral edge current.** (a) Color map of the orientation of collective motion in a chiral bacterial vortex ($R = \SI{50}{\micro\meter}$). The edge, defined as the area within from the lateral boundary, is separated from the rest of the microwell (the bulk) by a dashed white line. Scale bar, . (b) Normalized azimuthal velocity $v_{\theta}$ in microwells of various sizes. The blue circles indicate the edge current, and the green squares indicate the motion in bulk. $v_{\theta}$ is averaged over 10 s and plotted with error bars representing standard deviation.[]{data-label="fig2"}](figure2_8.jpg)
A bacterial suspension of *Escherichia coli* was confined in microwells with the depth of , made of poly-dimethyl siloxane (PDMS) rendered hydrophilic with a polyethylene glycol coating, and sealed with an oil/water interface stabilized with a surfactant (Fig. \[fig1\](a))[@izri]. Top/bottom hydrodynamic asymmetry (later referred to as “asymmetric conditions”) consists in the solid bottom interface of the microwell and its top fluidic interface (Fig. S1 in [@supplement]). Bacteria in the suspension collectively move following the circular boundary of the microwell, but show a surprisingly selective CCW rotational direction (top view). The orientation map $\theta_i$ of the velocity field $\bm{v}(\bm{r}_i)$ obtained from particle image velocimetry (PIV) shows vortical structure maintained across the microwell of $R = \SI{20}{\micro\meter}$ ( Fig. \[fig1\](b)) . CCW-biased vortex (called “chiral bacterial vortex” hereafter) occurs without built-in chirality of the confinement, e.g. ratchets[@fabrizio; @aronson; @leonardo]. CCW rotation is strongly favored at 95% ($N = 145$, Fig. S3) probability in chiral bacterial vortex, while bacterial vortex rotates at equal probability in CCW or CW direction in water-in-oil droplets between *symmetric solid (top)/solid (bottom)* interfaces (symmetric conditions, Fig. S2). Flow reversal in vortex was not observed during our observations (Fig. S3) which emphasizes that chiral bacterial vortex is much more stable than the bacterial vortices in droplets[@hamby](Fig. S3). Chiral vortical structure is also observed in larger microwells ($R = \SI{35}{\micro\meter}$) but only within a distance of away from the circular boundary. This chiral motion with a coherent orientation near the lateral boundary, called edge current, is known to be a key feature in chiral many-body systems[@jiang; @bechinger1; @irvine]. We in turn examined the size-dependence of chiral vortices and the edge current in order to investigate the mechanism of such stability and selectivity in rotational direction. We define the tangential vector $\bm{t}(\theta_i)$ in CCW direction along the circular boundary and the azimuthal velocity $v_{\theta}(\bm{r}_i)=\bm{v}(\bm{r}_i)\cdot \bm{t}_i$. The orientation of the edge current along the boundary wall is analyzed by $\langle v_{\theta}(\bm{r}_i)/ |\bm{v}| \rangle$ where $\langle \cdot \rangle$ denotes the average over all possible site $i$ (Fig. \[fig2\](a)). Surprisingly, this edge current was maintained even in very large microwells ($R\geq$), the size of which is much larger than the critical size of a stable bacterial vortex in the bulk ($\approx\SI{35}{\micro\meter}$) (Fig. \[fig2\](b)).
This persistence of edge current motivates us to further investigate its physical origin, by analyzing the interplay between the boundary and the intrinsic chirality of bacteria, which is the only element having chirality in the present system. With respect to the tail-to-head direction, the flagella of the bacterium rotate in the CCW direction. Torque balance then imposes a CW rotation of the body. Those two opposite rotations result in opposite frictions against the bottom interface, which ultimately converts into a CCW rotation (top view) of bacteria swimming near the top interface (CW rotation near bottom interface) (Fig. \[fig3\](a))[@whitesides]. Because of this CCW bias of their trajectories beneath the top, bacteria that collide with the lateral boundaries align with it and swim in a CCW rotation direction (CW rotation direction on the bottom). However, this effect alone does not determine net chirality because bacteria swim in opposite directions on the top and bottom interfaces of the microwell. In order to find the origin of net chirality in edge current, we recorded the trajectories of individual bacteria in dilute ($0.2\% v/v$) suspensions, such that interactions between bacteria do not affect their swimming. Fig. \[fig3\](b) presents the probability distribution function of the azimuthal velocities of individual bacteria, $P(v_{\theta})$, in a microwell ($R = \SI{35}{\micro\meter}$). Individual bacterial motion beneath the top fluidic interface shows visible nonequivalence between CW and CCW swimming, as more than half ($71\%$) of the tracked bacteria swim in CCW direction. By contrast, $58\%$ of individual bacteria onto the bottom solid interface swim in CW rotational direction, indicating weaker chiral bias. To compare those swimming chiralities, we define the chirality index $CI(v_{\theta})$ as antisymmetric part of $P(v_{\theta})$, i.e. $$CI(v_{\theta}) = P(v_{\theta}) - P(-v_{\theta}).$$ Positive chirality index means CCW rotation is dominant, and the larger the absolute value of the index, the more biased the rotational direction. For bacteria swimming near the top interface, chirality index is positive, while it is negative near the bottom interface (Fig. \[fig3\](c)). This tells that bacterial motion is CCW-biased near the top interface, and CW near the bottom interface. In addition, near the top interface, $|CI(v_{\theta})|>0.05$ whereas near the bottom interface $|CI(v_{\theta})| < 0.02$. This indicates that bacterial swimming is more biased near the top (fluidic) interface than near the bottom (solid) interface. This difference in the amplitude of the bias at each interface is responsible for the chiral rotation of the whole bacterial vortex, and therefore is at the origin of chiral edge current (Fig. \[fig3\](d)).
![**Bacterial swimming in dilute ($0.2\% v/v$) suspension** (a) Schematic illustrations of chiral bacterial swimming near the top fluidic interface (left) and near the bottom solid interface (right). Near the top interface, individual bacterial swimming is CCW-biased (blue curved arrow) while near the bottom interface it is CW-biased (see Fig. S5). (b) Histograms of azimuthal velocity and their average (vertical dashed line) of single bacteria swimming near top interface (left, schematic illustration in insert, $N=5004$, average ) and bottom interface (right, schematic illustration in insert, $N=3702$, average ). Proportions of CCW and CW occurrences are displayed in blue and red at the top of each plot. (c) Corresponding chirality indices plotted against azimuthal velocity, near the top (blue) and bottom (red) interfaces. (d) Chiral bias of swimming near the top and bottom interfaces have opposite signs but different amplitudes, under asymmetric conditions.[]{data-label="fig3"}](figure3_7.jpg)
![**Chiral bacterial vortex is collective effect.** A dense ($20\% v/v$) bacterial suspension is confined under asymmetric conditions. Represented chirality indices against azimuthal velocity of (a) individual fluorescent bacteria (schematic illustration in insert) observed under confocal microscopy near the top (left) and bottom (right) interface, (b) fluid flow with tracer particles also observed under confocal microscopy, and (c) collective bacterial motion observed under bright field. Two microwell sizes were considered: $R = \SI{35}{\micro\meter}$ (small blue circles) and $R = \SI{50}{\micro\meter}$. In the larger microwells were considered two regions: the boundary layer within from the lateral boundary (larger blue diamonds), and the bulk that is more than away from the lateral wall (green disks). Sample size and average azimuthal velocity of each case can be found in Fig. S6.[]{data-label="fig4"}](figure4_7.jpg)
As the density of self-propelled particles increases, their mutual interactions affect more significantly their global motion, which ultimately gives rise to collective behavior. As implied already, the rotational biases of handedness of individual bacteria ($71\%$ in top and $58\%$ in bottom) are too weak to account for the 95%-selective chirality in the bacterial vortex (Fig. \[fig1\](c)). To resolve this gap, we therefore characterized the structure of chiral bacterial vortices in much denser ($20\% v/v$) suspensions. Fig. \[fig4\](a) presents chirality index of individual bacteria near the top and bottom interfaces in a microwell ($R = \SI{35}{\micro\meter}$). Interestingly, both interfaces present dominantly CCW rotational direction, although the top interface is more strongly biased. This indicates that the effect of interactions between bacteria forces the rotational direction to be the same across the microwell. Intriguingly, chirality index of individual motion becomes larger as the bacterial density increases. Trajectories of tracer particles were also predominantly CCW, indicating that fluid flow also has CCW handedness. Moreover, chirality index of the fluid velocity (Fig. \[fig4\](b)) and the collective velocity (Fig. \[fig4\](c)) are comparable to that of individual motion. This suggests that a slight chiral bias in individual bacteria is amplified by collective bacterial interactions, which turns into a global vortex with a unidirectional rotation.
When we analyzed the vortex flow near the lateral boundary of larger microwells ($R = \SI{50}{\micro\meter}$), individual motion at high density has a large positive chirality index, similarly to smaller microwells. However, near the center of larger microwells, chirality is null for individual and collective bacterial motion as well as for the fluid flow (Fig. \[fig4\](a) to (c)). The necessity to be close to the lateral boundaries to observe coherent chiral swimming emphasizes the importance of spatial constraint to amplify the rotational bias and turn it into chiral edge current.
![**Edge current favors co-rotational vortex pairing** (a) Schematic illustration and definition of relevant geometric parameters. (b) Illustration of the co-rotational vortex pairing (FMV pattern, left) and the anti-rotational pairing (AFMV pattern, right) with edge current. The edge current deviates the orientation angle of bacteria $\theta$ around the *tip*. In FMV pattern, bi-particle alignment and edge current deviate bacteria in the same direction, but in AFMV pattern, those effects compete. (c) Vorticity map of bacterial vortex pairs at various values of $\Delta$ with $R = \SI{19}{\micro\meter}$. (d) Order parameter $\Phi_{FMV}$ of FMV pairs of interacting bacterial vortices, against $\Delta/R$. Under no edge current (full grey circles), FMV to AFMV transition occurs at $\Delta/R\simeq\sqrt{2}$, while under CCW edge current (inverted black triangles) it occurs at a larger value of $\Delta/R\simeq1.9$.[]{data-label="fig5"}](figure5_8.jpg)
The edge current plays a crucial role in the ordering of interacting bacterial vortices. When two bacterial vortices interact with one another via near-field interaction, they have either the same rotational direction (defined as ferromagnetic vortices, FMV) or opposite rotational directions (anti-ferromagnetic vortices, AFMV) in a geometry-dependent manner[@wioland2; @beppu; @nishiguchi2]. To reveal how a chiral edge current affects the ordering of interacting bacterial vortices, we construct a theoretical model of interacting bacterial vortices in doublets of overlapping identical circular boundaries (Fig. \[fig5\](a))[@beppu].
Two identical overlapping circular microwells, with a radius $R$ and an inter-center $\Delta$, offer the means for a systematic investigation of the pairing order transition[@beppu]. The ratio $\Delta/R = 2\cos\Psi$ is an important geometric parameter characterizing the pairing order transition from FMV to AFMV: if $\Delta$/R is small enough, the two vortices align in co-rotational direction and pair into a FMV pattern. Transition from FMV to AFMV occurs at a predicted value $\Delta_c/R = 2 \cos \SI{45}{\degree} = \sqrt{2}$ because that is the only configuration at which the two pairing patterns are equiprobable[@beppu]. How does edge current affect the previously established design principle?
To answer this question, the orientation dynamics of bacteria with the heading angle $\theta$ is considered at the vicinity of the sharp areas of the boundaries (“*tip*”). Given the CCW fluid flow near boundary reorients the bacteria at the *tip*, the effective torque for reorientation $\bm{\tau_e} = - \partial U_e/\partial \theta$ has geometry-dependent potential $U_e = -2 \gamma_e \sin\theta\cos\Psi$ with $\gamma_e$ representing the ratio of fluid flow to bacterial swimming ($\gamma_e > 0$). This reorientation maintains the CCW rotation of the edge current along boundary in both FMV pairing (Fig. \[fig5\](b), left) and AFMV pairing (Fig. \[fig5\](b), right). In addition, the result of the bi-particular collision near the *tip* between bacteria coming from different circular parts[@beppu] also affects the vortex pairing. Bacterial collision is ruled by a polar alignment as a source of geometry-dependence[@vicsek; @supplement]. By considering the most probable configurations (general case detailed in [@supplement]), the orientation near the *tip* is decided by the potential of FMV pairing $U_p^{FMV} = 2\gamma_p\sin\Psi$ at $\theta = 0$ (Fig. \[fig5\](b), left) or of AFMV pairing $U_p^{AFMV} = 2\gamma_p\cos\Psi$ at $\theta = \pi/2$ (Fig. \[fig5\](b), right), with the strength of the polar alignment $\gamma_p>0$. The respective sums of the potentials coincide at the transition point, i.e. $U_p^{FMV} + U_e|_{\theta=0} = U_p^{AFMV} + U_e|_{\theta=\pi/2}$, which leads to $$\gamma_p \sin\Psi_c = (\gamma_p- \gamma_e) \cos\Psi_c,$$ where $\gamma_p- \gamma_e$ indicates the suppression of AFMV pairing by chiral edge current. Hence, $\Delta_c/R=2\cos\Psi_c$ at which FMV and AFMV pairings occur at equal probability is $$\label{chiral_geometric rule}
\frac{\Delta_c}{R} = \frac{2}{\sqrt{1+(1-\gamma_e/\gamma_p)^2}}\simeq \sqrt{2} \Bigl( 1 + \frac{\gamma_e}{2\gamma_p}\Bigr).$$ FMV pattern is stabilized in $\Delta/R \geq \sqrt{2}$ and the relative strength of the edge current $\gamma_e/\gamma_p$ determines the shift of the transition point.
To test those chirality effects, we examined the pairing of doublets of chiral vortices with $R = \SI{19}{\micro\meter}$ within the range of $0 \leq \Delta < 2R = \SI{38}{\micro\meter}$. FMV pattern of chiral vortices is dominant in $0 \leq \Delta/R \leq 1.9$ and exhibits CCW rotation (Fig. \[fig5\](c)). The pairing order transition is also analyzed by using the order parameter of FMV pairing $\Phi_{FMV}$ [@supplement]. $\Phi_{FMV}$, which reaches 1 for FMV while it goes down to 0 for AFMV, is defined as $\vert \langle \bm{p}_i \cdot \bm{u}_i \rangle \vert$ with the orientation of velocity $\bm{p}_i$ measured experimentally at site $i$ and the expected orientation of FMV pattern $\bm{u}_i$ calculated numerically at corresponding site. Under asymmetric condition, $\Phi_{FMV}$ shows a transition from $1$ to $0$ at $\Delta/R\simeq1.9$, while it occurs at $\Delta/R\simeq1.4$ under symmetric condition (absence of chiral edge current) (Fig. \[fig5\](d)). FMV pairing pattern appears to be favored in the presence of edge current. Moreover, according to Eq. with the experimental values $\gamma_p=0.5$ (obtained from independent experiment, Fig. S7) and $\Delta_c/R\simeq1.9$, the coefficient of edge current is $\gamma_e=0.3$ that is reasonably comparable to the ratio of fluid flow to bacterial swimming velocity (Figs.\[fig4\](a) and 4(b)). The excellent agreement with experiment means that chiral edge current affects the pairing order transition.
In conclusion, we revealed that confining a dense bacterial suspension in microwells with asymmetric top/bottom interfaces but achiral circular lateral boundaries stabilizes a chiral vortex. Hydrodynamically different top/bottom interfaces give then rise to a subtle broken symmetry in bacterial swimming, that is amplified by collective bacterial motion and becomes an edge current persistent over larger length scale. We showed that it is possible to generate and control the break of mirror-symmetry without using built-in chirality, unlike most current experimental setup, e.g. with built-in chiral ratchet-shape[@fabrizio; @aronson; @leonardo]. Moreover, in present chiral bacterial vortex, rigid rod much longer than bacterial body can consistently rotate in CCW direction over multiple rounds at (Fig. S8). Thus, asymmetric hydrodynamic boundary offers simple and fast material transport, even without built-in chirality. Finally, the edge current favors co-rotational FMV pairing pattern of doublets of identical vortices. Even in triplets of identical overlapping circular microwells, the pairing order transition from FMV to (frustrated) AFMV patterns is also shifted to higher values (from $1.7$ to $1.9$) (Fig. S9), suggesting that chiral bacterial vortex has fewer limitation from geometric frustration. Although beyond the scope of this study, understanding how the nature of an interface affects the amplitude of the chiral flow it generates could give the means of a finer tuning of the global rotation of confined bacterial vortices.
The finding of edge current in chiral bacterial vortex opens new directions for the tailoring of collective motion. Such asymmetric hydrodynamic boundaries are also involved in flocking and lane formation of active droplets[@stone]. The edge current observed in active droplets with similar collective effect may advance the generic understanding of chiral collective behavior[@glotzer; @tjhung]. Furthermore, stabilized co-rotational pairing order is a key to clarify how broad class of active matter amplifies microscopic chirality. As such constrained pairing order is also relevant to chiral spinners[@tierno], controlling chirality-induced order with simple geometric rule would verify the validity of geometric approach to chiral many-body systems.
This work was supported by Grant-in-Aid for Scientific Research on Innovative Areas (JP18H05427 and JP19H05403) and Grant-in-Aid for Scientific Research (B) JP17KT0025 from MEXT.
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Supplemental information for\
Edge Current and Pairing Order Transition in Chiral Bacterial Vortex {#supplemental-information-for-edge-current-and-pairing-order-transition-in-chiral-bacterial-vortex .unnumbered}
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Materials and Methods
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Device microfabrication
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The device used in this experiment is a flow cell that contains an array of microwells of various radii $R$ ($\SI{20}{\micro\meter} \leq R \leq \SI{150}{\micro\meter}$) and three types of shape: single circular microwells, pairs and triplets of overlapping identical circular microwells. Their depth is in all the experiments[@beppu]. The flow cell is made of a cover glass slide (Matsunami, S1127, ) and a cover slip (Matsunami, C218181, ) separated by a double-sided adhesive tape (NICHI-BAN, NW-10, ).
The microwells are fabricated using standard soft lithography techniques. Briefly, PDMS polymer and curing agent (Dow Corning, 98-0898, Sylgard184) at 90-to-10 mass ratio was spin-coated at 1000 rpm for 30 seconds to reach a thickness of , on a mold made of a photoresist (SU-8 3025, Microchem) pattern (thickness ) cured and developed through conventional photo-lithography on a silicon wafer (Matsuzaki, Ltd., $\phi$2-inch wafer). After a curing at for an hour, the PDMS film is cut around a single array of microwells and peeled off to be bonded on its unpatterned surface to the glass cover slide that has been exposed to air plasma beforehand (, corona discharge gun, Shinko Denso).
This assemblage is then once again exposed to air plasma (, corona discharge gun), covered with a solution of polyethylene glycol-poly-L-lysine (PEG-PLL, Nanocs, PG2k-PLY), and left to rest for 30 minutes. Ungrafted PEG-PLL is then washed away with deionized water. PDMS microwells and glass cover slide are now treated hydrophilic, and PEG-PLL coating prevents non-specific adhesion of bacteria. Finally, the flow cell is completed with the glass cover slip (untreated) being attached to the glass cover slide with the adhesive spacer placed over the unpatterned areas of the PDMS sheet. It is used immediately after fabrication (Fig. \[fig.s1\] Step 1).
![**Schematic illustration of device fabrication.** Step 1: device construction. A thin PDMS layer patterned with microwells is transferred between the two glass slides of a flow cell and coated with PEG-PLL to avoid unspecific adhesion of bacteria. Step 2: bacterial suspension is injected from one side of the flow cell. Step 3: excess bacterial suspension is flushed out by a mixture of oil and surfactant. This mixture seals the microwells on their top side. Step 4: device is sealed at both ends with epoxy glue.[]{data-label="fig.s1"}](figureS1_3.jpg)
Filling protocol
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Once a flow cell is ready, it is filled with of a bacterial suspension (two volume fractions were available: dense at $20\% v/v$, dilute at $0.2\% v/v$). The PEG-PLL coating of the PDMS allows the bacterial suspension to reach the inside of the microwell and fill them properly. Once the flow cell is completely filled, oil (light mineral oil, Sigma Aldrich) with surfactant (SPAN80, Nacalai) at $2 wt\%$ is injected from the same side, while excess bacterial suspension over the microwells is flushed out and absorbed with filter paper from the other side of the flow cell. This seals the microwells under an oil/water interface. To suppress unwanted flow in the flow cell, both of its ends are sealed with epoxy glue (Huntsmann, Ltd.). The array of microwells is then ready to be observed under microscope (Fig. \[fig.s1\] Steps 2-4)[@izri].
Persistently straight-swimming bacterial strain
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We used bacterial strain *Escherichia coli* RP4979 that lacks tumbling ability. These bacteria swim smoothly without tumbling and show persistent straight motion in the bulk of a fluid. Highly motile bacteria were obtained by inoculating a single bacterial colony into of LB medium (NaCl , yeast extract , tryptone , pH7.2) and incubating it overnight at . The next day, of this overnight culture solution is transferred to of T-Broth (NaCl , trypton , pH7.2), and the inoculated cultures are incubated at and agitated at 150 rpm for about 6 hours. After reaching an optical density of $0.4$, the culture medium is centrifuged at 3000 rpm at room temperature for 10 minutes to concentrate the bacterial suspension density to $20-25\% v/v$.
The straight swimming of bacteria was measured in a bulk fluid, and the fluctuation of the heading angle $\theta(t)$ was analyzed. Single bacteria are considered as self-propelled points particle, with a position $\bm{r}(t) = (\bm{x}(t), \bm{y}(t))$ and an orientation $\bm{d}(t)$(Fig.\[figs\_diffusion\](a)). For two-dimensional coordinates, the orientation of single bacteria is expressed by the unit vector along the long-axis of bacteria $\bm{d} = \bm{d}(\theta(t)) = (\cos\theta(t) , \sin\theta(t))$. Because the mean-square angle displacement (MSD) is given by $\langle \bigl[\bm{d}(\theta(t)) - \bm{d}(\theta(0))\bigr]^2 \rangle_t = 2(1-\exp[-D t])$ with time interval $\delta t$ and the angular diffusion coefficient $D$ that reflects the fluctuation of bacterial orientation at single cell level[@jain](Fig.\[figs\_diffusion\](b)). The obtained coefficient $D$ is 0.12 rad$^2$/sec, indicating that RP4979 bacteria persistently swim in one direction at a low density in bulk.
![**Straight swimming of single bacteria in bulk.** (a) Typical trajectory of swimming single bacteria in bulk. Scale bar is . (b) Mean square displacement of heading angle of single bacteria is plotted with time. The slope of this MSD curve is fitted with $2(1-\exp[-Dt])$, where $D$ reflects orientation fluctuation of smoothly swimming bacteria[@jain].[]{data-label="figs_diffusion"}](figureS_fluctuation.jpg)
Bacterial density measurement and image velocimetry
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To measure bacterial density, we used a mixture - 99-to-1 ratio - of two genetically modified bacteria (strain RP4979) that constitutively express fluorescent protein (either YFP or dTomato). That fluorescent labeling allows the quantitative recording of the trajectories of individual bacterial swimming in either dilute (Fig. 3 in main text) or dense suspensions (Fig. 4(a) in main text). By tracking dTomato-expressing bacteria, we can record the individual trajectories of bacteria inside collective vortical motion.
In the analysis of single bacteria and tracer particles in a suspension, they were tracked by means of a plugin of Particle Tracker 2D/3D in the Fiji(ImageJ) software. Bright-field optical imaging and video-microscopy were performed by using an inverted microscope (IX73, Olympus) with a CCD camera (DMK23G445, Imaging Source) that enables us to record bacterial collective motion at 30 frames per second. The velocity field of bacterial collective motion $\bm{v}(r,t)$ was analyzed by Particle Image Velocimetry(PIV) with Wiener filter method using PIVlab based on MATLAB software, and its grid size was $\times$. Acquired velocity fields were further smoothed by averaging over 1 sec.
In addition, polystyrene tracer particles with red fluorescence (Molecular probes) were used to track the flow field (Fig. 4(b) in main text). The tracer particles were dispersed at a low density of $0.026\% v/v$, where individual particles could be tracked inside the bacterial suspension. Recording of the trajectories of red-labeled bacteria and red tracer particles was done with a confocal microscope (IX73 inverted microscope from Olympus, and confocal scanning unit CSU-X1 from Yokogawa Electric Cor. Ltd., iXon-Ultra EM-CCD camera from Andor Technologies) under red fluorescence channel. All the recordings were done at 30 frames per second.
Experimental details
====================
Preponderance and stability of CCW bacterial vortex
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![**Preponderance and stability of chiral bacterial vortex.** (a) A droplet of a dense bacterial suspension confined in a microwell ($R = \SI{20}{\micro\meter}$) between *asymmetric fluidic (top) and solid (bottom)* interfaces. (b) Control experiment for the effect of top/bottom interfaces on the directionality of the bacterial vortex. A droplets ($\SI{15}{\micro\meter}\leq R\leq\SI{35}{\micro\meter}$) of an emulsion of dense bacterial suspension in oil between *symmetric solid (top)/solid (bottom)* interfaces. (c) Distributions of vorticity averaged over 10 s and their respective average (vertical dashed line) under asymmetric interfaces ($N = 145$, average ) in a microwell ($R = \SI{20}{\micro\meter}$). Proportions of CCW and CW occurrences are displayed in blue and red at the top of each plot. (d) The distributions of vorticity and their respective average (vertical dashed line) under symmetric solid/solid interfaces ($N = 126$, average ). Under symmetric interfaces, CW and CCW rotations are observed with the same frequency. (e) Persistent dynamics of vorticity. The vorticity is always positive without sign change, meaning CCW rotation is stable over the range of our measurement (30 sec). (f) Dynamics of vorticity change under solid / solid symmetric interfaces show the frequent change of vorticity sign, indicating that switching between CW and CCW rotations occurs.[]{data-label="fig.s2"}](figureS_vorticity.jpg)
The chiral bias of bacterial swimming can be affected by the nature of the top and bottom interfaces, i.e. whether the two interfaces are hydrodynamically equivalent or not as seen in main text. We compare this effect on chiral bacterial vortices (dense suspension) between asymmetric (Fig. \[fig.s2\](a)) and symmetric top/bottom interfaces (Fig. \[fig.s2\](b)). Fig. \[fig.s2\] (c) shows the distribution of vorticity of an ensemble of chiral bacterial vortices. CCW rotation is favored between *asymmetric fluidic (top)/solid (bottom)* interfaces (later referred to as asymmetric conditions) and the probability of CCW rotation is 95%. We next tested the vortex rotation of bacterial collective motion under symmetric top/bottom interface. We confined dense bacterial suspension in water-in-oil droplets between *symmetric solid (top)/solid (bottom)* interfaces (symmetric conditions). By examining the vorticity of bacterial collective motion an emulsion of $\SI{15}{\micro\meter}\leq R\leq\SI{35}{\micro\meter}$, a vortex rotated at equal probability in CCW (52%) or CW direction (48%) (Fig. \[fig.s2\](d)). Thus, the selective chirality is observed only in the asymmetric top/bottom interfaces. Moreover, such prepondence in chirality can be involved in the stability of rotational direction of vortical flow in dynamics. we analyzed the dynamics of vorticity for 30 sec in order to examine the flow reversal in chiral bacterial vortex. For chiral bacterial vortex under asymmetric condition, the sign reversal was not observed in this typical observation time (Fig. \[fig.s2\](e)). In contrast, for achiral bacterial vortex under symmetric condition, the vorticity shows periodical change with sign reversal as reported in previous study[@hamby] (Fig. \[fig.s2\](f)). Thus, chiral bacterial vortex is highly selective (95% in CCW rotation) and stable (without flow reversal for few tens of sec).
Edge current is not observed in a microwell with symmetric top/bottom condition
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In Figure 2 in main text, we showed the edge current in CCW rotation in chiral bacterial vortex. This edge current is also observed in larger microwells but only within a distance of away from the circular boundary. To test whether this edge current is unique to the spatial confinement between *asymmetric fluidic (top)/solid (bottom)* interfaces, we also examined the bacterial vortex in the circular microwell between *symmetric solid (top)/solid (bottom)* interfaces (Fig. \[fig.edge\](a)).
The solid interface was made of PEG-coated PDMS and the dense bacterial suspension (20% volume fraction) was confined in microwells. We define the tangential vector $\bm{t}(\theta_i)$ in CCW direction along the circular boundary and the azimuthal velocity $v_{\theta}(\bm{r}_i) = \bm{v}(\bm{r}_i)\cdot \bm{t}_i$. The edge, defined as the area within from the lateral boundary, is separated from the rest of the microwell (the bulk). The orientation of the edge current along the boundary wall is analyzed by $\langle v_{\theta}(\bm{r}_i)/ |\bm{v}| \rangle$ where $\langle \cdot \rangle$ denotes the average over all possible site $i$ and observation time of 10 sec (Fig. 2(a) in main text). However, the stable edge current in symmetric conditions is observed only in smaller microwells with $R\leq$(red open triangle: CW direction, blue open circle: CCW direction, in Fig. \[fig.edge\](b)). The threshold value is comparable to the critical size of a stable bacterial vortex in the bulk (green filled square in Fig. \[fig.edge\](b)), and this characteristic length is much shorter than the one for chiral edge current ($\approx \SI{100}{\micro\meter}$, see Fig 2(b) in main text). In addition, collective motion near boundary has no preference of either CW or CCW directions in symmetric conditions. Thus, the collective motion in symmetric conditions does not have persistent edge current in CCW rotation, indicating that chiral edge current is uniquely stabilized in asymmetric top/bottom conditions.
![**Weak edge current of bacteria in a microwell with symmetric condition.** (a) Schematic illustration of a dense bacterial suspension confined in a microwell between *symmetric solid (top)/solid (bottom)* interfaces. (b) The edge current in symmetric PDMS microwell. The edge, defined as the area within from the lateral boundary, is separated from the rest of the microwell (the bulk) as same as Figure 2 in main text. Normalized azimuthal velocity of the flow in microwells of various sizes. The blue circles (red circles) indicate the edge current in CCW direction (in CW direction), and the green squares indicate the vortical collective motion in bulk. Normalized azimuthal velocities averaged over 10 s are plotted with error bars that present standard deviations of their time series.[]{data-label="fig.edge"}](figureS_edge.jpg)
Trajectory of single bacteria near solid interface
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Bacteria of the RP4979 strain swim persistently straight in the bulk of their environment due to their inability to tumble. However, in the vicinity of a solid interface, the torque generated by flagella rotation induces a CW (while looking from above, the bacteria being on top of the interface) deviation of the bacterial swimming that eventually leads to a circular trajectory. RP 4979 bacteria, in a dilute suspension, swimming on top of PDMS interface in the absence of lateral confinement show, as predicted, circular trajectories with a CW rotation direction (Fig. \[fig.s4\](a)(d)(g)). Such circular trajectory reflects the hydrodynamic interaction between the chiral rotation of flagella and boundaries.
A bacterial suspension was confined in PDMS microwells and sealed with an oil/water interface, which means that two interfaces were present in interaction with bacterial swimming. We thus tracked the swimming of individual bacteria in a dilute suspension, near the top oil/water interface, and the bottom PDMS interface of circular microwells (Fig. 3 in main text). Fig. \[fig.s4\](b)(e)(h) show the trajectory of bacteria that swim below the top oil/water interface, with top view. Their trajectories are curved towards the CCW direction. In addition, Fig. \[fig.s4\](c)(f)(i) shows the swimming trajectories near the bottom solid PDMS interface.
Furthermore, we also analyzed the bacterial swimming, fluid flow, and collective motion (analyzed by PIV) at a dense ($20\% v/v$) bacterial suspension confined under asymmetric conditions. Figs. \[fig.s5\](a) and (b) show the distribution of average azimuthal velocity $P(v_{\theta})$ at higher density. The small fraction of bacteria expressing dTomato (red fluorescent protein) was mixed at 1% in dense bacterial suspension expressing YFP (yellow fluorescent protein) and the trajectory of individual bacteria was recorded by confocal microscopy. Fig. \[fig.s5\](a) (or \[fig.s5\](b)) is the histogram of the azimuthal velocity of bacteria in top fluidic interface (or bottom solid interface). In Figs. 4(a) and 4(b) in main text, the chirality index $CI(v_{\theta}) = P(v_{\theta}) - P(-v_{\theta})$ is presented by using those data.
Figs. \[fig.s5\](c) and (d) show the distribution of average azimuthal velocity $P(v_{\theta})$ of tracer particle (, red fluorescence) at higher density. The trajectories of particles were recorded at the top (\[fig.s5\](c)) or the middle (\[fig.s5\](d)) in the microwell by using confocal microscopy.
Finally, Figs. \[fig.s5\](e) is the distribution function of average azimuthal velocity $P(v_{\theta})$ from PIV analysis at higher density. One can find large fraction of bacteria (82-95%) shows CCW rotation near boundary.
![**Intrinsic bacterial chiral swimming near interfaces.** (a-c) Schematic illustrations of (a) CW rotation (top view) of bacterial swimming on a solid interface with open lateral boundaries, (b) CCW rotation of bacterial swimming below oil/water interface, with lateral solid PDMS wall, and (c) CW rotation of bacterial swimming on the solid PDMS interface, with lateral solid PDMS wall. Scale bar in (a) is , and scale bars in (b) and (c) are both . (d-f) Representative trajectories of individual bacterial swimming in corresponding conditions ((d): open lateral boundaries, (e) top oil/water interface, (f) bottom solid PDMS interface), in dilute bacterial suspensions. The start points of each trajectory are fixed at the center of the plot, and their end points are indicated by a black cross. Colors code for different trajectories. (g-i) Histogram of the mean angular velocities of the detected trajectories in the corresponding cases ((g): open lateral boundaries, (h) top oil/water interface, (i) bottom solid PDMS interface). Proportions of rotation directions are given in blue for CCW and red for CW.[]{data-label="fig.s4"}](figureS4_3.jpg)
![**Histograms of azimuthal velocities in dense bacterial suspension.** (a and b) individual bacteria swimming near (a) the top oil/water interface and (b) the bottom PDMS interface, (c and d) fluid flow with tracer particles near (c) the top oil/water interface and (d) the middle of PDMS microwell, and (e) PIV velocity field of collective bacterial motion observed under bright field. The blue bars correspond to the probability of CCW rotation while the red bars to CW rotation. We also analyzed collective motion in two different sizes of microwells: $R = \SI{35}{\micro\meter}$ (small blue circle) and $R=\SI{50}{\micro\meter}$. In the larger microwells, we considered two regions: the boundary layer within from the lateral boundary (larger blue diamond), and the bulk that is more than away from the lateral wall (green filled square).[]{data-label="fig.s5"}](figureS5_5.jpg)
solid interface (Fig.\[fig.s4\](a)) Top fluidic interface (Fig.\[fig.s4\](b)) Bottom solid interface (Fig.\[fig.s4\](c))
-------------------------- ------------------------------------- ------------------------------------------- --------------------------------------------
spatial constraint Boundary-free Circular microwell Circular microwell
sample size, N N = 264 N = 1469 N = 876
average angular velocity
: Sample sizes and average velocities of histograms presented in Fig. \[fig.s4\] (PTV (bacteria)).
\[table.s1\]
[|c|c|c|c|ll]{} & **$R$= , boundary** & **$R$= , boundary** & **$R$ = , bulk** &\
PTV (bacteria) TOP & $N = 491$, $\langle v_{\theta}\rangle = \SI{4.24}{\micro\meter}$/s & $N = 784$, $\langle v_{\theta}\rangle = \SI{3.44}{\micro\meter}$/s & $N = 1129$, $\langle v_{\theta}\rangle = \SI{0.369}{\micro\meter}$/s &\
PTV (bacteria) BOTTOM & $N = 619$, $\langle v_{\theta}\rangle = \SI{2.35}{\micro\meter}$/s & $N = 1022$, $\langle v_{\theta}\rangle = \SI{2.16}{\micro\meter}$/s & $N = 1654$, $\langle v_{\theta}\rangle = \SI{-0.130}{\micro\meter}$/s &\
PTV (tracer particles) TOP & $N = 442$, $\langle v_{\theta}\rangle = \SI{2.86}{\micro\meter}$/s & $N = 406$, $\langle v_{\theta}\rangle = \SI{2.06}{\micro\meter}$/s & $N = 628$, $\langle v_{\theta}\rangle = \SI{0.557}{\micro\meter}$/s &\
PTV (tracer particles) MIDDLE & $N = 574$, $\langle v_{\theta}\rangle = \SI{3.12}{\micro\meter}$/s & $N = 736$, $\langle v_{\theta}\rangle = \SI{2.59}{\micro\meter}$/s & $N = 1379$, $\langle v_{\theta}\rangle = \SI{0.541}{\micro\meter}$/s &\
PIV & $N = 88536$, $\langle v_{\theta}\rangle = \SI{1.71}{\micro\meter}$/s & $N = 131240$, $\langle v_{\theta}\rangle = \SI{1.28}{\micro\meter}$/s & $N = 231200$, $\langle v_{\theta}\rangle = \SI{-0.048}{\micro\meter}$/s &\
\[table.s2\]
The distribution of orientation angle in bacterial collective motion
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To analyze the fluctuation of the heading angle of bacteria $\theta(t)$ at a higher density, we measured the trajectory of single bacteria inside dense bacterial suspension confined in a microwell of overlapping two circles. The swimming bacteria show angular fluctuation $D$ as shown in Fig.\[figs\_parameter\], but such fluctuation should be changed in the presence of orientation alignment due to collective motion. The strength of orientation interaction, which is defined as $\gamma_p$, is an important parameter for their collective motion. Indeed, the variance of the distribution function of heading angle $\sigma_{\theta}^2$ is closely related to both the coefficient of polar orientation interaction $\gamma_p$ and the fluctuation of bacterial heading angle $D$ as $\sigma_{\theta}^2=\frac{D}{2\gamma_p\sin\Psi}$ for FMV pattern ($\sigma_{\theta}^2=\frac{D}{2\gamma_p \cos\Psi}$ for AFMV pattern) because the polar orientation reorients the direction of bacterial swimming and in turn suppresses the angular fluctuation. The ratio between the angular fluctuation $D=0.12$ and the coefficient of polar alignment $\gamma_p$ is approximately equal to the variance of the distribution at the given geometric parameter $\Psi$. From the data for the straight swimming of bacteria in a bulk fluid (Fig.\[figs\_parameter\]), the angular diffusion coefficient $D$ is 0.12 rad$^2$/sec. In addition, the variance of angle distribution has 0.21. By using these values, we can estimate $\gamma_p$=$\frac{D}{2\sigma_{\theta}^2\sin\Psi}$=0.47. This value is used to examine the effect of chiral edge current for the pairing transition of FMV and AFMV, later.
![**Orientation distribution of bacteria at high density.** A bacterial population was enclosed in a microwell with the boundary shape of two identical overlapping circles ($\Delta/R$ = 1.58). The group of bacteria exhibited collective motion in an FMV pattern, and we observed single bacteria labeled with fluorescent dTomato protein that swim away from the tip and the distribution of its heading angle was measured. The variance of this probability distribution is used to estimate the coefficient $\gamma_p$ of the polar alignment as $\sigma^2=\frac{D}{2\gamma \sin \Psi}$.[]{data-label="figs_parameter"}](figureS_parameter.jpg)
Active stirring of a micron-sized object in chiral bacterial vortex
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![**Rotation of rod-shaped object.** (a) Movement of a rigid rod in chiral bacterial vortex showed in (left) snapshots over $\SI{6}{\second}$. Red point indicates the head-end of the rod. Scale bar, . (b) Corresponding orientation angle increases linearly with time, exhibiting a constant CCW angular velocity (for comparison, dashed line has a slope of ). (c) Snapshots under bright field over of the random rotation and translation of a rigid rod in a dense bacterial suspension under quasi-two-dimensional confinement. Rod of interest is colored yellow in the image, and its head is arbitrarily defined by a red dot. Scale bar, . (d) Time evolution of the orientation angle of the marked rod. Blue color represents the CCW rotation and red color, the CW rotation. Both rotation directions have comparable angular velocities (black dashed line at ). Scale bar, . (e) Velocity field (black arrows) and vorticity map (red to blue colormap) obtained from the PIV analysis of the turbulent flow in a dense ($20\% v/v$) bacterial suspension under quasi-two-dimensional confinement between PDMS sheet (top) and a glass slide (bottom). Blue stands for CCW rotation and red stands for CW rotation. (f) Power spectrum of the previous velocity field. Peak is at which corresponds to a wavelength of . (g) Corresponding histogram of vorticity (Average vorticity ). Proportions of rotation directions are given in blue for CCW and red for CW.[]{data-label="fig.s3"}](figureS3_6.jpg)
Collective motion of suspended self-propelled particles enhances transport properties in active fluids, which has been used conjunctly with built-in geometry to direct the motion of objects larger than the suspended particles (e.g. gear-shape[@fabrizio; @aronson] or ratchet-shape[@leonardo]). To further illustrate the transport properties chiral bacterial vortices, we confined a rigid rod ( length and thickness, made of SU-8 photoresist) that is much longer than bacterial body ( length and thickness). The rod consistently rotates in CCW direction over multiple rounds at , i.e. 6 times faster than previously known ratcheted gears in a bacterial suspension (Fig. \[fig.s3\]). As one would expect it, disordered vortices seen in quasi two-dimensional channel are also able to rotate rigid rods, larger than bacteria (Fig. \[fig.s3\](c)). However, the direction of rod rotation switches stochastically between CW and CCW and the absolute value of the angular velocity remains constant ($\simeq\SI{0.7}{\radian\per\second}$) (Fig. \[fig.s3\](d)). At high concentration, RP4979 strain bacteria present a turbulent behavior characterized by a large number of dynamic and intermingled vortices under quasi-two-dimensional confinement (Fig. \[fig.s3\](e)). The PIV analysis of their collective motion reveals a widely distributed power spectrum with a peak at (Fig. \[fig.s3\](f)). This suggests that the vortices observed in this turbulent active flow have a size of typically or more. The PIV analysis of their collective motion also shows that the distribution of vorticity is even (Fig. \[fig.s3\](f)), which indicates that CW and CCW behaviors are identical under symmetric and quasi-two dimensional confinement. This equiprobability of CW and CCW leads to the stochastic change of rod rotation in two-dimensional chamber (Fig. \[fig.s3\](d)). Therefore, chiral bacterial vortex in present setup offers simple and fast material transport, even without built-in chirality such as gear-shape.
FMV pairing order in triplet of circular microwells
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To determine the strength of the ordering induced by chiral bacterial vortices, we tested in against the rotational frustration observed in triplets of identical overlapping circular microwells. Here again, the microwell radius is noted $R$, and the center-to-center distance is noted $\Delta$ and is the same for all the pairs of microwells in the triplet. We used microwells with $R$ = and $0 \leq \Delta/R \leq 1.98$ (Figs. \[fig.s6\](a) and \[fig.s6\](b)). When we confine a dense($20\% v/v$) bacterial suspension in such microwells, under symmetric conditions (top and bottom interfaces are in PDMS), geometric frustration is responsible for a shift of the FMV-AFMV (frustrated order) transition point $\Delta/R$ from 1.3-1.4 (observed in doublets of overlapping microwells under symmetric conditions, Fig. \[fig.s6\](a) top) to 1.6-1.7 (Fig. \[fig.s6\](b) top). Under asymmetric conditions (top interface is in oil/water and bottom interface is in PDMS), the chiral edge current of the interacting vortices further shifts that transition point to 1.8-1.9, similarly to what is observed in doublets of microwells (Figs. \[fig.s6\](a) bottom and \[fig.s6\](b) bottom). This result indicates that the chirality of bacterial vortices can generate larger unidirectional flow and overcome geometric frustration of interacting triplet vortices.
![**Edge current favors co-rotational vortex pairing**. Vortex pairing is affected by geometric frustration and chiral edge current. Dense bacterial suspension confined in multiple circular identical overlapping microwells ($R = \SI{19}{\micro\meter}$) can present various paring ((a) left and (b) left). In doublets of microwells ((a) top right and bottom right) no frustration is present and the effect of chirality shifts the transition point between FMV and AFMV pairing towards higher values of $\Delta/R$. In triplets of microwells (b), in the absence of chirality ((b) top right), FMV pairing is favored by geometric frustration and transits to AFMV pairing (frustrated order) at increased value of $\Delta/R$. When chirality is added to geometric frustration ((b) bottom right), FMV pairing is further favored, and the value of the transition point is further increased. The transition point from FMV to AFMV is $\Delta_c/R=1.9$ in both doublet and triplet circle microwells, which indicates that chiral edge current induces larger shift than the effect of frustration in triplet circle microwell.[]{data-label="fig.s6"}](figureS6_5.jpg)
Order parameter of FMV pattern
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To analyze the ordered pattern of ferromagnetic vortex (FMV) pattern of bacteria, we used the order parameter $\Phi_{FMV}$ according to our previous study[@beppu]. Before giving the definition of $\Phi_{FMV}$, we need to derive analytical form of the angular velocity of interacting vortices inside a overlapping circular microwell. For this aim, we firstly find the analytical form for angular velocity of single vortex, ${v}_{\theta}(r)$ inside a circle of the radius $R$. We assume the boundary condition at $r=R$ is ${v}_{\theta}(R)$=0, and ${v}_{\theta}(r)$ is proportional to $r$. The spatial distribution of vorticity inside the circle is given by $$\label{angvel1}
\omega(r) = \begin{cases}
\omega & (0 \leq r \leq R-s) \\
- \omega \Bigl[1- \frac{(R-s)^2}{R^2} \Bigr] & (R-s \leq r \leq R)
\end{cases}$$ where $R-s$ is the position with maximum angular velocity and the size $s$ is estimated from experimental data. By solving , one can express the orthoradial velocity in a circular microwell $\bm{v}(r,\theta)$=$v_{\theta}(r)\bm{t}(\theta)$ as $$\label{angvel2}
\bm{v}(r,\theta) = \begin{cases}
\frac{\omega}{2} \Bigl[1- \frac{(R-s)^2}{R^2} \Bigr]r \bm{t}(\theta)
& (0 \leq r \leq R-s) \\
\frac{\omega}{2} \Bigl(1- \frac{s}{R}\Bigr)^2\frac{R^2-r^2}{r} \bm{t}(\theta) & (R-s \leq r \leq R) \\
0 & (r>R)
\end{cases}$$ where $\bm{t}(\theta)=(-\sin \theta, \cos \theta)$ is the unit orthoradial vector at the angular position $\theta$. In the following section, this analytic formulation is used to define the order parameter of FMV pattern.
Here we show the derivation of order parameter of ferromagnetic vortices (FMV) pattern used in Fig. 5(d). This order parameter reports the correlation of orientation field of velocity between the experimentally observed vortex pairing and numerically calculated FMV. For the numerical calculation of FMV pattern, the vortex confined in circular boundary is firstly considered: for each circle composing the doublet microwell, we set an index $j$, 1 stands for the left side and 2 for the right side. We define two sets of polar coordinates $(r_j, \theta_j)$; one for left circle is $(r_1, \theta_1)$ and the other for right circle is $(r_2, \theta_2)$. The origin of $j$ polar coordinates is set at the center of $j$ circle. We consider $\bm{t}_j(\theta_j)$ the orthoradial unit vector at the angular position $\theta_j$ centered on the center of the circle $j$ for $0\leq r_j \leq R$. In particular, we have $\bm{v}_j$$(r_j,\theta_j)=v_{\theta}(r_j)\bm{t}_j(\theta_j)$ where $v_{\theta}(r_j)$ is given by .
![**Orientation fields of FMV pattern.** (a) Orientation field of FMV pattern obtained from numerical method using Eq.. (b) Typical orientation field of interacting vortices in FMV pattern obtained in experiment.[]{data-label="fig.Order"}](figureS_FMV.jpg)
We then construct velocity field for vortices showing FMV pattern in the doublet microwell. In addition to the boundary condition of a doublet circle that is characterized by $R$ and $\Delta$, the polar coordinates $(r, \theta)$ defines the internal space. The origin of polar coordinates is placed at the centroid of the doublet shape and the velocity field, $\bm{v}$$(r, \theta)$, is in turn considered as the superposition of two vortices in $j$=1 and 2 circles. Because two vortices in FMV pattern show same angular velocities of $\bm{t}_1(\theta)=\bm{t}_2(\theta)$, we can describe the velocity field as $$\label{angvel3}
\bm{v}(r,\theta) = \sum_{j} \bm{v}_j(r_j, \theta_j) = \sum_{j} v_{\theta}(r_j) \bm{t}_j(\theta_j) .$$
The orientation field of an FMV pattern lies on the unit vector $\bm{u}(r,\theta)$ such that $$\label{angvel4}
\bm{u}(r,\theta) =\frac{\bm{v}(r,\theta)}{|\bm{v}(r,\theta)|} .$$
By using the inner product of expected orientation map $\bm{u}(r,\theta)$ and the one measured experimentally $\bm{p}(r,\theta)$, the order parameter $\Phi_{FMV}$ is then defined as $$\Phi_{FMV}=\vert \langle \bm{p}(r,\theta)\cdot \bm{u}(r,\theta) \rangle \vert$$ where $\langle \cdot \rangle$ denotes the ensemble average over all sites in a doublet microwell. One can estimate the deviation of experimentally obtained velocity field from ideal FMV pattern because $\Phi_{FMV} = 1$ if it matches the FMV pattern, and $\Phi_{FMV} = 0$ if it becomes an AFMV pattern or a disordered turbulence.
Theoretical analysis
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Geometric rule of bacterial vortex transition with no edge current (without chirality)
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![**Boundary condition in theoretical model.** (a) The shape of the boundary conditions used in the theoretical model. Geometric parameters are shown in the figure. (b) The direction of movement and flow of particles along the wall near the tip. The horizontal right direction is defined as angle $\theta=0$, and the counterclockwise direction is a positive angle direction. The counterclockwise tangential direction of the left circular microwell (Red arrow) is an angle $\pi/2-\Psi$, and the clockwise tangential angle of the right circular microwell (Blue arrow) is $\pi/2+\Psi$.[]{data-label="fig.s7"}](figureS7_1.jpg)
To explain the transition of ferromagnetic vortex (FMV) pattern and anti-ferromagnetic vortex (AFMV) patterns under geometric constraints, we construct theoretical model of orientational dynamics by considering the motion of self-propelled particles in a fluid. Bacteria is considered as a self-propelled point particle, with a position $\bm{r}_m(t) = (x_m(t), y_m(t))$ and an orientation $\bm{d}_m(t)$. For two-dimensional coordinates, the orientation of the particles is also expressed with the unit vector along the long-axis of bacteria $\bm{d}_m = \bm{d}(\theta_{m}) = (\cos\theta_m, \sin\theta_m)$. We impose the confinement of a circular microwell on the fluid and bacteria as shown in Fig.\[fig.s7\](a). The bacteria swim along the boundary wall at low noise limit and their heading angle $\theta_m$ is parallel to the tangential direction of boundary (Fig.\[fig.s7\](b)). The geometry of boundary shape is a pair of overlapping circular microwells with geometric parameter $\Delta/R$. The “tip” of this doublet is a point where two bacterial vortices intersect and bacteria from different circular parts of the microwell collide. The parameters $\Delta$ and $R$ are also rewritten with the elevation angle $\Psi$ as $\cos\Psi = \frac{\Delta}{2R}$ .
The bacteria swim in their surrounding fluid and, when their density increases, their mutual interaction increases the alignment of their orientations. The evolution of $\bm{r}_m(t)$ is given by: $$\dot{\bm{r}}_m(t) = v_0 \bm{d}(\theta_m)$$ where particles move at a constant speed $v_0$.
We next consider that the orientation angle of the bacteria $\theta(t)$ is determined by to three distinct effects: mutual alignment of bacteria due to collision, hydrodynamic processes and random rotational diffusion. Swimming bacteria exert a force on the fluid and in turn induce fluid advection $\bm{v}(\bm{r},t)$$ \propto \bm{p}$, where $\bm{p}$ is a local polar vector defined by $$\label{polar}
\bm{p}(\bm{r}) = \langle\bm{d}(\theta_m)\rangle_{\bm{r}}.$$ Collective motion occurs at a higher density and it generates the active fluid flow as $\bm{v}=V_0 \bm{p}$, with $V_0<v_0$. Such hydrodynamic flow rotates bacteria through velocity gradient effect.
The orientational dynamics of local alignment and rotation of bacteria by flow is $$\label{orientation_all}
\dot{\theta}_m = - \gamma_p \underbrace{\sum_{\vert \bm{r}_{mn} \vert < \epsilon} \sin(\theta_m - \theta_n)}_{\rm polar \: alignment} - \gamma_n \underbrace{\sum_{\vert \bm{r}_{mn} \vert < \epsilon} \sin2(\theta_m - \theta_n)}_{\rm nematic \: alignment} + \gamma_a \underbrace{[\bm{d}_m \times (\bm{d}_m \cdot \nabla )\bm{v}]\cdot \bm{e}_z}_{\rm flow-induced \: alignment} + \eta_m(t),$$ where the first and the second terms govern the polar[@beppu][@wioland1] and nematic[@Li] alignments of bacteria due to mutual collision, and the third term shows rotation by velocity gradient of fluid flow that generates the torque on bacteria[@hamby][@Doi]. The forth term $\eta_m(t)$ represents the random fluctuation of the rotational direction, that satisfies $\langle \eta_m(t) \rangle$=0, $\langle \eta_m(t) \eta_n(t') \rangle$=$2D \delta_{mn}\delta(t-t')$ where $\delta_{mn}$ and $\delta(t)$ are Kronecker’s delta symbol and Dirac’s delta function, respectively. $D$ is the amplitude of the random noise that affects the orientation of bacteria.
As shown in Fig. 5 in main text, the geometrical feature of the boundary shape induces the transition from FMV to AFMV. However, it is not clear if this geometry-induced transition occurs due to the polar interaction (first term in Eq.(\[orientation\_all\])) or the nematic interaction (second term in Eq.(\[orientation\_all\])) depending on the orientation of the bacteria, or the flow-induced rotation change in fluid advection (third term in Eq.(\[orientation\_all\])).
What we need to consider is the interaction of bacteria from left or right circles in a doublet microwell defined by geometric constant $\Psi$. In the tip where the two circles intersect, bacteria swimming in the left and right wells interact and become oriented. If bacteria swim in counterclockwise direction along the wall in the left microwell, their heading angle is $\pi/2 - \Psi$ at the tip, while bacteria swimming clockwise in the right microwell have an angle $\pi/2 + \Psi$ at the tip (Fig.\[fig.s8\](a)).
In the following, a theoretical analysis is performed on how each orientational dynamics is affected by the geometric shape of the boundary. We assume the bacteria flowing from the left well to be aligned initially in CCW rotation, and hence the fluid flow follows the same rotation direction. In the case of AFMV (FMV) pattern, we consider the right well has CW (CCW) bacteria motion and flow rotation. To have the insight of the coupling, we assume it to be weak. In this way, we estimate the interaction between the vortices residing in different microwell by comparison with the swimming manner of bacteria and fluid flow of independent ones. In addition, we assume the fluid flow of interacting vortices to be the linear superposition of the independent ones. In this way, we can consider how each term in Eq. affects the orientation of bacteria. We first consider the effects of polar alignment due to mutual collision (including near field hydrodynamic interaction) and show that polar interaction can account for both (1) the preference of AFMV and FMV depending on geometrical parameter $\Psi$ and (2) the transition point consistent with experimental result. Because the effect of nematic alignment due to mutual collision cannot explain experimental results, the geometric dependence in the pairing order transition is decided by polar alignment. We then consider the flow-induced alignment of bacteria near the tip in order to explain the effect of edge current with rotational chirality. Finally, by extending the model of polar alignment with this chiral edge current, we propose geometric rule to account for both the selection of FMV pattern and the shift of the pairing order transition.
### The effect of polar alignment explains geometry-dependent FMV-AFMV transition
At first, we consider the effect of polar alignment described by
$$\label{orientation_polar}
\dot{\theta}_m = - \gamma_p \sum_{\vert \bm{r}_{mn} \vert < \epsilon} \sin(\theta_m - \theta_n) + \eta_m(t).$$
where $\bm{r}_{mn}=\bm{x}_m-\bm{x}_n$ and $\epsilon$ is the effective radius of particle interaction. Polar interaction is involved in collective swimming in a highly dense suspension of bacteria, and a condition where the bacteria are oriented along the long axis with distinguishing the head and tail is favorable. Suppose the high-density of bacteria, the distribution of bacterial particles is homogeneous in space, and the density fluctuation is negligible.
The aim of this theoretical analysis is to clarify whether polar interaction can explain the transition between FMV and AFMV patterns depending on the boundary shape $\Delta/R=2\cos\Psi$, and whether it also agrees with the transition point seen in experiment. Therefore, what we should consider is the collective motion under the spatial constraint where bacteria move along the walls of the left or right circles in a doublet microwell defined by geometric constant $\Psi$, and then collide mutually at the tip.
We first consider the AFMV pattern at which two bacterial vortices have opposite rotational direction. If bacteria swim in counterclockwise (CCW) along the wall in the left microwell, its heading angle is $\pi/2 - \Psi$ at the tip, while bacteria swimming clockwise (CW) in the right microwell have the angle $\pi/2 + \Psi$ at the tip (Fig.\[fig.s8\](b)). By taking mean field approximation[@beppu][@vicsek][@peruani], the orientational dynamics Eq. is rewritten by $$\label{orientation_polar2}
\dot{\theta}_m = - \gamma_p \bigl(\sin(\theta_m - \pi/2 + \Psi) + \sin (\theta_m - \pi/2 - \Psi)\bigr) + \eta_m(t).$$
![**Polar alignment at the tip.** (a) Diagram of bacterial polar interaction showing anti-rotational vortex pairing at the tip. Bacteria moving along the left wall collide with bacteria moving from right circle at the tip. This pattern of polar interaction corresponds to AFMV pattern. (b) Diagram of bacterial polar interaction for co-rotational vortex pairing at the tip. The impact angle of the bacteria changes, and collective motion is in turn induced in the x-axis direction. This pattern eventually leads to FMV pattern.[]{data-label="fig.s8"}](figureS8_3.jpg)
Then the dynamics of bacterial orientation in AFMV order is reduced to $$\dot{\theta}_m(t) = 2 \gamma_p \cos \theta_m \cos \Psi+ \eta_m(t).$$ Once the orientational dynamics is obtained, one can derive the Fokker-Planck equation for the probability distribution of particle orientation, $P_{AFMV}(\theta,t;\Psi)$, in AFMV pattern as $$\label{polarFP_afmv}
\frac{\partial P_{AFMV}}{\partial t} = D \frac{\partial^2 P_{AFMV}}{\partial \theta^2} - 2\gamma_p \frac{\partial}{\partial \theta} \Bigl[(\cos\theta \cos \Psi) P_{AFMV} \Bigr].$$
As for FMV pattern, the bacteria swimming in CCW rotation from the left circle has an angle $\pi/2 - \Psi$ at the tip while the bacteria entering the right circle has the heading angle of $- \pi/2 + \Psi$ at the tip (Fig.\[fig.s8\](b)). To construct the FMV pattern from Eq., we take the same mean-field approximation and give the orientational dynamics of FMV pattern at the tip $$\label{orientation_polar3}
\dot{\theta}_m = - \gamma_p \bigl(\sin(\theta_m - \pi/2 + \Psi) + \sin (\theta_m + \pi/2 - \Psi)\bigr) + \eta_m(t).$$ Then the dynamics of bacterial orientation in FMV order is reduced to $$\dot{\theta}_m(t) = - 2 \gamma_p \sin \theta_m \sin \Psi+ \eta_m(t).$$
The Fokker-Planck equation for the probability distribution of particle orientation, $P_{FMV}(\theta,t;\Psi)$, in FMV pattern is derived as $$\label{polarFP_fmv}
\frac{\partial P_{FMV}}{\partial t} = D \frac{\partial^2 P_{FMV}}{\partial \theta^2} + 2\gamma_p \frac{\partial}{\partial \theta} \Bigl[(\sin\theta \sin \Psi) P_{FMV} \Bigr],$$
Because we focus on the static pattern of confined bacterial vortices, the left-hand side of both Eq. and Eq. is set at $\frac{\partial P}{\partial t} = 0$. By solving Eq. and Eq. , one can find the probability distribution $P_{AFMV}(\theta;\Psi)$ and $P_{FMV}(\theta;\Psi)$ at the steady state, $$\label{soln_afmv}
P_{AFMV}(\theta;\Psi)=\frac{\exp \bigl(\frac{2\gamma_p}{D}\sin \theta \cos \Psi \bigr)}{2 \pi I_0\bigl(\frac{\gamma_p}{D}\cos \Psi \bigr)}.$$ and $$\label{soln_fmv}
P_{FMV}(\theta;\Psi)=\frac{\exp \bigl(\frac{2\gamma_p}{D}\cos \theta \sin \Psi \bigr)}{2 \pi I_0\bigl(\frac{\gamma_p}{D}\sin \Psi \bigr)}.$$ where $I_0(\cdot)$ is first order Bessel function. In particular, AFMV pattern pointing in the vertical direction at $\theta = \pi/2 $ and the probability of FMV pattern pointing in the horizontal direction at $\theta = 0$ are both maximal. The unique solution $P_{FMV}(\theta = 0; \Psi_c) = P_{AFMV}(\theta = \pi/2; \Psi_c)$, meaning that AFMV and FMV patterns occur at equal probability, is at the transition point $\Psi_c$. By comparing Eqs. and , the unique transition point is obtained as $$\label{geometricrule}
\sin \Psi_c = \cos \Psi_c,$$ which sets the transition at $\Psi_c = \pi/4$. Thus in the collective motion of self-propelled particles without chiral edge current, the geometric condition for the transition from FMV to AFMV is $$\label{geometricrule2}
\Delta_c/R = 2 \cos \Psi_c = \sqrt{2}.$$. In addition, at $\Psi \leq \pi/4$, the probability of FMV pattern $P^{FMV}(\theta = 0; \Psi)$ is always larger than that of AFMV pattern $P^{AFMV}(\theta = \pi/2; \Psi)$, indicating that FMV pattern is favored at $\Delta/R\leq \sqrt{2}$. This geometric dependence is in excellent agreement with experimental result (Fig. 5(c) in main text and Fig.\[fig.s6\](a)), $\Delta_c/R \approx 1.3-1.4$.
This analysis suggests that polar alignment is considered to be the primary effect that can explain both the pattern formation of FMV and AFMV and its transition at $\Delta_c/R=1.3-1.4$ observed in the experiment.
$ $
### Nematic alignment cannot explain geometry dependent FMV-AFMV transition
We next consider the effect of nematic alignment due to mutual collisions $$\label{orientation_nematic}
\dot{\theta}_m = - \gamma_n \sum_{\vert \bm{r}_{mn} \vert < \epsilon} \sin2(\theta_m - \theta_n) + \eta_m(t).$$
We assume the left well has CCW chirality; bacteria swim in CCW direction, and fluid flows in CCW direction as well. Bacteria from left well are considered to have the orientation angle $\theta=\pi/2-\Psi$ at the tip. In the case of AFMV pattern, we assume bacteria from CW right well have the orientation angle $\theta=\pi/2+\Psi$ By taking mean field approximation of particle orientation, the dynamics of $\theta$ at time $t$ is given by $$\label{orientation_nematic-AFMV}
\dot{\theta}_m = - \frac{ \gamma_n }{2} \left\{\sin2(\theta_m - (\pi/2-\Psi)) + \sin2(\theta_m - (\pi/2+\Psi))\right\} + \eta_m(t)=\cos 2\Psi \sin 2\theta_m + \eta_m(t)$$ where $\gamma_n$ is the coefficient of nematic alignment.
In the case of FMV pattern, CCW right well has the orientation angle $\theta=-\pi/2+\Psi$. In this case, the dynamics of $\theta$ at time $t$ is also given by $$\label{orientation_nematic-FMV}
\dot{\theta}_m = - \frac{\gamma_n}{2} \left\{\sin2(\theta_m - (\pi/2-\Psi)) + \sin2(\theta_m - (-\pi/2+\Psi))\right\} + \eta_m(t)=\cos 2\Psi \sin 2\theta_m + \eta_m(t).$$
Both AFMV and FMV follow the same dynamics of $\theta_m$ by comparing Eqns. and , and one can find the interaction term is identical in $P_{AFMV}$ and $P_{FMV}$. FMV and AFMV patterns can be obtained with the same probability for all geometric parameters under nematic alignment of bacteria. In other words, the nematic interaction at the tip do not make any preference between AFMV and FMV. This analytical result is not consistent with the experimental results for geometrical transitions of FMV-AFMV patterns, suggesting that nematic orientation is not needed in orientational dynamics for our experimental observation.
$ $
### Chiral edge current as advection-induced alignment
![**The alignment due to velocity gradient.** (a) A particle with orientation angle $\theta_m$ is affected by the velocity gradient of the fluid. Velocity of fluid is denoted by $\bm{v}=v_f \Theta(x)\bm{e}_y$. The calculation leads to the alignment of bacteria with the newly imposed flow direction, $\pi/2$. (b) When the imposed flow direction is given by $\theta_f$, bacteria is aligned with the direction of $\theta_f$.[]{data-label="fig.fa"}](flow-align.jpg)
Bacteria experience torque due to velocity gradient, as given by $$\label{orientation_adv}
\dot{\theta}_m=\gamma_a [\bm{d}_m \times (\bm{d}_m \cdot \nabla )\bm{v}]\cdot \bm{e}_z+ \eta_m(t).$$ Assuming step-like velocity field proportional to the step function $\Theta(x)$, we have $\bm{v}=v_f \Theta(x)\bm{e}_y$ as shown in Fig.\[fig.fa\]. The bacteria heading to this step experiences torque, which can be calculated as $$\label{orientation_rot}
[\bm{d}_m \times (\bm{d}_m \cdot \nabla )\bm{v}]\cdot \bm{e}_z=v_f\delta(x)(\cos^2 \theta)$$ Bacteria whose orientation angle $\theta_m$ is heading to the step of velocity fields has normal velocity $v_0 \cos\theta$. Thus, the total change of the orientation $\Delta \theta$ due to the step-like flow fields can be obtained by $$\label{orientation_change}
\Delta \theta = \int dt \dot{\theta}= \frac{\gamma_a v_f }{v_0}\int_{-\infty}^{\infty} \cos \theta(x) dx = -\frac{\gamma_a v_f }{v_0} \sin\bigl(\theta_m-\frac{\pi}{2}\bigr).$$ Thus, dynamics of the bacterial orientation under the effect of step-like flow field can be described as $$\label{orientation-flow}
\dot{\theta}_m=-\gamma_e \sin\bigl(\theta_m-\frac{\pi}{2}\bigr)+ \eta_m(t),$$ where $\gamma_e = \frac{ \gamma_a v_f }{v_0 \Delta t}$. When the direction of flow $\theta_f$ is arbitrary set to $\bm{v}=v_f (\cos \theta_f \bm{e}_x+\sin \theta_f \bm{e}_y)$, the same argument applies and the dynamics of the bacteria orientation is denoted as $$\label{orientation_change2}
\dot{\theta}_m=- \gamma_e \sin(\theta_m-\theta_f)+ \eta_m(t).$$ where $\gamma_e$ is the coefficient for the alignment along the boundary edge ($\gamma_e\geq0$).
![**The edge current of bacteria and fluid streaming near boundary.** Diagram of interaction and superposition of bacteria and vortical flow explaining edge current in CCW direction. (left) Bacteria moving along the left wall collide with the vortical flow on the right microwell. (right) bacteria along the wall from the right collide with the vortical flow on the left microwell. Collective motion can be described with the sum of these two polar alignment of bacteria and vortical flow in co-rotational vortex pair. []{data-label="fig.s9"}](figureS9_3.jpg)
As we noted earlier, collective motion occurs at a higher density and it generates the active fluid flow along the polarized direction as $\bm{v}=V_0 \bm{p}$, and we assume the flow-like vortical rotation can be present at the steady state, with velocity $\bm{v}(\bm{r},t)$. At the tip, bacteria collide with the advection flowing at velocity $\bm{v}(\bm{r},t)$ as shown in Fig.\[fig.s7\]. The direction of the flow advection is also directed to the tangential direction of the boundary wall, which is considered to be a stable vortical flow.
In Figure 4 in main text, we found that chiral bacterial vortices show strong CCW bias in both bacterial swimming and fluid flow at boundary region. This fact allows one to propose the mathematical description of edge current of bacteria as follows: we consider the orientational dynamics under fluid advection for two interacting vortices with rotational flow in CCW direction. Since the bacterial population is constrained by a quasi-two dimensional space, the hydrodynamic flow is considered to be two-dimensional. The fluid flow in the microwell can be described as a superposition of the vortical flows present in the left and right circles of the doublet. In addition, we assume that bacteria do not collide with each other while they are aligned with the advective flow driven by collective motion that appears in a neighboring microwell. Then, it can be approximated that bacterial and flow alignment is limited to two combinations: the first is the alignment of bacteria moving in the left microwell along the vortical flow in the right microwell, and the second is the alignment case for bacteria moving in the right microwell with the vortical flow present in the left microwell (Fig. \[fig.s9\]).
As for edge current in CCW direction (Fig. \[fig.s7\](c)), the probability distribution functions $P_{L}(\theta,t;\Psi)$ and $P_{R}(\theta,t;\Psi)$ representing the angular distribution of bacteria in left and right are obtained, respectively (Fig. \[fig.s9\](a)). If the bacteria at the left side have the heading angle $\pi/2 - \Psi$ while the bacteria at the right side have the angle $-\pi/2 + \Psi$, the orientation probability distribution of bacteria can be obtained based on the equation describing orientational dynamics. The Fokker-Planck equations for probability of heading angle are $$\label{advFP_left_fmv}
\frac{\partial P_{L}}{\partial t} = D \frac{\partial^2 P_{L}}{\partial \theta^2} + \gamma_e \frac{\partial}{\partial \theta} \Bigl[\cos(\theta - \Psi)P_{L} \Bigr],$$ and $$\label{advFP_right_fmv}
\frac{\partial P_{R}}{\partial t} = D \frac{\partial^2 P_{R}}{\partial \theta^2} + \gamma_e \frac{\partial}{\partial \theta} \Bigl[\cos(\theta + \Psi)P_{R} \Bigr].$$
At this time, the bacteria swimming from the left microwell have an angle $\pi/2 - \Psi$, and the orientation changes due to the vortical flow of the right microwell. The vortical flow at the tip is directed along $- \pi/2 + \Psi$. Thereby, the change in orientation is given by $- \gamma_e \cos (\theta - \Psi)$. On the other hand, the bacteria entering the right microwell have an angle $- \pi/2 + \Psi$ at the tip, but in this case, the orientation is changed by being released from the flow at the angle $\pi/2 - \Psi$. Thus, the change in orientation is represented by $- \gamma_e \cos (\theta + \Psi)$. Instead of finding the general solutions of Eqs. (\[advFP\_left\_fmv\]) and (\[advFP\_right\_fmv\]), we focus on the symmetry of probability distribution: because the shape of the boundary condition is highly symmetric at the tip, one can expect only negligible difference between $P_L(\theta,t)$ and $P_R(\theta,t)$ at this point. This geometric argument allows one to define the probability distribution of orientation for edge current by $P_{edge}=(P_{L}+P_{R})/2$. Hence, Eqs. and can be reduced to
$$\begin{aligned}
\label{advFP_fmv}
\frac{\partial P_{edge}}{\partial t} &=& D \frac{\partial^2 P_{edge}}{\partial \theta^2} + \frac{\partial}{\partial \theta} \Bigl(\gamma_e \Bigl[\cos(\theta + \Psi)+\cos(\theta - \Psi)\Bigr]P_{edge}\Bigr) \nonumber \\
&=& D \frac{\partial^2 P_{edge}}{\partial \theta^2} + \frac{\partial}{\partial \theta} \Bigl(\Bigl[2\gamma_e \cos\theta \cos\Psi \Bigr]P_{edge}\Bigr),\end{aligned}$$
where $P_{edge}(\theta,t;\Psi)$ is the probability distribution of particles at the tip with a heading $\theta$ at the time $t$ under CCW edge current.
Eq.(\[advFP\_fmv\]) tells that the reorientation by the edge current has geometric dependence $\cos\Psi$. In addition, it tends to trap bacterial particle nearby boundary and rotate bacteria in CW direction (due to negative sign in right hand side). Due to such reorientation, bacteria keep swimming nearby boundary edge in CCW direction. In next section, in order to account for how chiral bacterial vortex favors FMV order and move the transition point, we consider the effect of particle reorientation due to this edge current in addition to polar alignment.
Geometric rule of bacterial vortex transition with chiral edge current
----------------------------------------------------------------------
In this section, we theoretically explain that interacting chiral vortices favors FMV pattern in a wide range of geometric condition, by adding the effect of chiral edge current in CCW direction in the orientational dynamics. From the above analysis, one can propose the orientational dynamics of bacterial particle swimming close to boundary under the edge current in CCW direction and polar alignment at the tip position. Because the direction of chiral symmetry breaking originates from bacterial swimming onto surface, the direction of edge current should be continuously formed in the CCW direction regardless of whether the vortex pairing pattern is FMV or AFMV. When CCW-chiral collective motion remains near the wall, the heading angle tends to be downward, resulting in an edge current that continues along the wall as an FMV pattern. By taking mean-field approximation, the change of bacterial orientation under edge current with CCW rotation is described by $\dot{\theta} = - \gamma_e ( \cos(\theta_m - \Psi) + \cos(\theta_m + \Psi)) = - 2 \gamma_e \cos\theta_m\cos\Psi$ that corresponds to the second term in right-hand side in Fokker-Planck equation Eq.. In addition, as shown in Eq., it is effective to make the polar interaction $\sum_{\vert \bm{r}_{mn} \vert < \epsilon} \sin(\theta_m - \theta_n) $ the primary effect in the orientational dynamics. We thus start the orientational dynamics with edge current in CCW chirality by taking minimal Vicsek-style model : $$\dot{\theta}_m = - \gamma_p \sum_{\vert \bm{r}_{mn} \vert < \epsilon} \sin(\theta_m - \theta_n) - 2 \gamma_e \cos\theta_m\cos\Psi + \eta_m(t).$$
Then, one can write the Fokker-Planck equations for AFMV or FMV pattern with chiral edge current as follows:
for AFMV order with edge current: $$\begin{aligned}
& &\frac{\partial P_{AFMV}}{\partial t}= D \frac{\partial^2 P_{AFMV}}{\partial \theta^2} - \frac{\partial}{\partial \theta} \Bigl(\bigl[2\gamma_p \cos \theta \cos \Psi - 2\gamma_e \cos \theta \cos \Psi \bigr] P_{AFMV}\Bigr),\end{aligned}$$ and for FMV order with edge current: $$\begin{aligned}
& &\frac{\partial P_{FMV}}{\partial t}= D \frac{\partial^2 P_{FMV}}{\partial \theta^2} + \frac{\partial}{\partial \theta} \Bigl(\bigl[2\gamma_p \sin \theta \sin \Psi + 2 \gamma_e \cos \theta \cos \Psi \bigr] P_{FMV}\Bigr).\end{aligned}$$ As is apparent from these equations, the distribution function of the bacterial collective motion is affected by edge current and determined by the ratio of its rotational speed to the rotational diffusion constant $D$, and deviates the geometric rule of $\Delta_c/R = \sqrt{2}$ that characterizes the FMV-AFMV transition in the absence of chiral edge current. The probability of realizing the FMV and AFMV patterns is obtained by $$P_{AFMV}(\theta;\Psi)=A(\Psi) \exp \biggl[\frac{2}{D}((\gamma_p- \gamma_e)\sin\theta \cos\Psi)\biggr].$$ and $$P_{FMV}(\theta;\Psi)=B(\Psi) \exp \biggl[\frac{2}{D}(\gamma_p \cos\theta\sin\Psi + \gamma_e \sin\theta\cos\Psi)\biggr]$$ where $A(\Psi)$ and $B(\Psi)$ are normalization factor with $\Psi$. Suppose that $\theta = 0$ is assigned to $P_{FMV}(\theta; \Psi) $, and $\theta = \pi/2$ is assigned to $P_{AFMV}(\theta; \Psi) $, we can rewrite $$P_{AFMV}(\theta=\pi/2;\Psi)=A(\Psi)\exp\biggl[\frac{2}{D}\bigl((\gamma_p - \gamma_e)\cos\Psi\bigr)\biggr]$$ and $$P_{FMV}(\theta=0;\Psi)=B(\Psi)\exp\biggl[\frac{2}{D}(\gamma_p \sin\Psi)\biggr].$$
$A(\Psi)$ and $B(\Psi)$ are not equal unless $ \gamma_e = 0$, Then, the transition point with $P_{FMV}(\theta = 0; \Psi) = P_{AFMV}(\theta = \pi/2; \Psi)$ is given by $$\label{transition_chiral}
\log\biggl[\frac{A(\Psi)}{B(\Psi)}\biggr] + \frac{2(\gamma_p -\gamma_e)}{D} \cos\Psi = \frac{2\gamma_p}{D}\sin\Psi .$$
![**Phase diagram of vortex patterns.** (a) Illustration of the chiral bias of vortex pairing by edge current. In the configuration $\Psi =\pi/4$, that is $\Delta_c/R =2\cos(\pi/4)=\sqrt{2}$, CW and CCW rotations are equiprobable in both overlapping microwells in the absence of chirality effect (left). However, since the edge current creates a flow around the tip, the FMV pattern with CCW rotation is maintained, and the transition to the AFMV pattern shifts to the point of $\Delta_c/R\geq\sqrt{2}$ (right). (b) Phase diagram of FMV, chirality-induced FMV and AFMV is shown with geometric parameter $\Delta/R$ and the coefficient of chiral edge current $\gamma_e$. The dotted line indicates the transition point of chiral bacterial vortices and the horizontal broken line is the original transition point $\Delta/R=\sqrt{2}$ without chiral edge current.[]{data-label="fig.s10"}](figureS10_2.jpg)
By calculating $\Psi$ satisfying the Eq. (\[transition\_chiral\]) numerically, the transition point to AFMV pattern can be obtained. FIG. \[fig.s10\] is the phase diagram showing the transition point $\Delta_c/R$ with the chiral edge current. If bacterial vortex does not have such edge current, the transition point is $\Delta_c/R=\sqrt{2}$. In other word, the chiral edge current in bacterial vortex shifts the transition from FMV to AFMV at a point deviating from $\sqrt{2}$. If the transition from FMV pattern occurs at $\Delta_c/R > \sqrt{2}$ at non-zero $\gamma_e$, that is classified as the chirality-induced FMV.
Interestingly, one can find that the shift of transition point increases in proportion to the coefficient of chiral edge current $\gamma_e$ in the range where chirality is small. In order to understand why such a linear relation holds, Eq. (\[transition\_chiral\]) was solved and the approximate solution of the transition point from chiral-FMV to AFMV was determined. To analyze the shift of transition point, we suppose the geometric parameter $\Psi$ that is close to $\cos\Psi=\sin\Psi=1/\sqrt{2}$ because this condition allows one to get $A \simeq B$ and then the first term in Eq. can approximate $\log\bigl[\frac{A(\Psi)}{B(\Psi)}\bigr] \simeq 0$. By solving Eq.(\[transition\_chiral\]) with $\sin^2\Psi + \cos^2\Psi =1$, the transition point $\Delta_c/R$ is $$\label{chiral_rule}
\frac{\Delta_c}{R} = \frac{2}{\sqrt{1 + \bigl(1 - \frac{ \gamma_e }{\gamma_p}\bigr)^2}}$$ where $\Delta_c/R$ is in the range of $0 \leq \Delta/R \leq 2$ by definition. When Eq. (\[chiral\_rule\]) is linearized for $ \gamma_e /\gamma_p\ll1$, the transition point is rewritten as $$\label{chiral_rule2}
\frac{\Delta_c}{R} \approx \sqrt{2} \Bigl(1 + \frac{ \gamma_e }{2\gamma_p} \Bigr).$$ The obtained geometric relation Eq.(\[chiral\_rule2\]) means that the term related to chirality is added as a linear sum to the original expression of $\Delta_c/R=\sqrt{2}$ obtained when there is no chiral edge current. The shift of transition point is determined by the ratio between the effect of polar alignment $\gamma_p$ and the effect of edge current $ \gamma_e $. In addition, the transition point of chirality-induced FMV to AFMV is always larger than $\sqrt{2}$, suggesting that the edge current extends FMV order to a broader range of geometric conditions, in agreement with our experiments.
[9]{} R. Jain and K.L. Sebastian, Journal of Chemical Physics **146**, 214102 (2017). M. Doi and S. F. Edwards, Theory of Polymer Dynamics (Oxford University Press.) p. 293 (1986). F. Peruani, A. Deutsch, M. Bär. Euro. Phys. J. 157, 111-122 (2008).
|
---
author:
- |
Tomek Bartoszynski[^1]\
Department of Mathematics,\
Boise State University,\
Boise, Idaho 83725
- |
Winfried Just[^2]\
Department of Mathematics,\
Ohio University,\
Athens, Ohio 45701\
- |
Marion Scheepers[^3]\
Department of Mathematics,\
Boise State University,\
Boise, Idaho 83725
title: 'Covering games and the Banach-Mazur game: $k$-tactics.'
---
mssymb
\[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Claim]{}
Introduction
============
Let $J$ be a free ideal of subsets of a given set. By $\JA$ we denote the $\sigma$-ideal generated by $J$ ($\JA$ could turn out to be the power set of $\cup J$). Two concrete examples of ideals motivated much of our work. The one is ${J_{{\Bbb R}}}$, the ideal of nowhere dense subsets of the real line ${\Bbb R}$. In this case $\JAR$ is the ideal of meager sets of reals. The other is $[\kappa]^{<\lambda}$ where $\omega=cof(\lambda)\leq\lambda\leq\kappa$ are cardinal numbers.
We are interested in games of the following type: Player ONE plays a set $O_n \in \JA$ during inning $n$, to which TWO responds with a set $T_n \in J$. ONE is required to play an increasing sequence of sets; TWO’s objective is to cover $\bigcup_{n \in \omega} O_n$ with $\bigcup_{n \in \omega } T_n$. As long as TWO remembers the complete history of the game, this task is trivial. However, it often happens that TWO needs to know only the last $k$ moves of the opponent in order to win. A strategy that accomplishes this is called a [*winning $k$-tactic*]{}.
We consider four such games, $MG({\cal A},J)$, $\mg$, the “monotonic game”, $\smg$, the “strongly monotonic game”, and $\vsg$, the “very strong game”. The study of these games was initiated in [@S1], and motivated by Telgarský’s conjecture that for every $k > 0$ there exists a topological space $(X,\tau )$ such that TWO has a winning $k+1$-tactic but no winning $k$-tactic in the Banach-Mazur game on $(X,\tau )$ (see section 4.4 for more information). However, we find the games considered here of interest independent of the original motivation. The game $\mg$ was introduced in [@S1], as was the game $\smg$; the games $MG({\cal A},J)$ and $\vsg$ appear here for the first time.
In sections 2 and 3, we introduce and discuss pseudo Lusin sets, the irredundancy property and the coherent decomposition property of ideals. These properties, together with the $\omega$-path partition relation, are the main tools for constructing winning $k$-tactics in our games. These combinatorial properties of ideals are very likely of independent interest - they have already appeared in the literature in various guises.
In section 4 we apply the results of sections 2 and 3 to give various conditions sufficient for the existence of winning $k$-tactics for TWO in the games mentioned above. Not surprisingly, as the game becomes more favorable for TWO, weaker conditions suffice. Among other things, our results show that in the Banach-Mazur game on the space that inspired the invention of meager-nowhere dense games, TWO has a winning 2-tactic.
The appendix is devoted to a proof of an unpublished consistency result of Stevo Todorcevic, which we use in section 4.
Our notation is mostly standard. One important exception may be that we use the symbol $\subset$ exclusively to mean “is a proper subset of". Where we otherwise deviate from standard notation or terminology we explicitly alert the reader. For convenience we also assume the consistency of traditional (Zermelo-Fraenkel) set theory. All statements we make about the consistency of various mathematical assertions must be understood as consistency which can be proven by means of that theory. The reader might find having a copy of [@S1] and [@S2] handy when reading parts of this paper a bit more comfortable than otherwise.
We are grateful to Stevo Todorčević for sharing with us his insights about the matters we study here, and for his kind permission to present in this paper some of his answers to our questions.
The irredundancy property.
==========================
For a partially ordered set $(P,<)$ which has no maximum element we let $$add(P,<)$$ be the least cardinal number, $\lambda$, for which there is a collection of cardinality $\lambda$ of elements of $P$ which do not have an upper bound in $P$. This cardinal number is said to be the [*additivity*]{} of $(P,<)$. Note that $add(P,<)$ is either $2$, or else it is infinite. In the latter case $(P,<)$ is said to be [*directed*]{}. We attend exclusively to directed partially ordered sets in this paper. Isbell [@I] and some earlier authors also refer to the additivity of a partially ordered set as its [*lower character*]{}; they denote it by $\ell(P,<)$.
A free ideal $J$ on a set $S$ is partially ordered by $\subset$. The partially ordered set $(J,\subset)$ is directed. When $add(J,\subset)=\aleph_{0}$, the symbol $\langle J
\rangle$ denotes the $\sigma$-completion of $J$ (i.e., the smallest collection which contains each union of countably many sets from $J$). We say that $J$ is a $\sigma$-[*complete ideal*]{} if $J=\langle J\rangle$.
The other important example for our study is the set $^{\omega}\omega$ of sequences of nonnegative integers; we use ${\frak c}$ to denote the cardinality of this set. We say $g$ [*eventually dominates*]{} $f$ and write $f\ll g$ if: $\lim_{n\rightarrow\infty} (g(n)-f(n))=\infty$. It is customary to denote $add(^{\omega}\omega, \ll)$ by ${\frak b}$.
A well known theorem of Miller ([@M], p. 94, Theorem 1.2) states that $$add(\langle J_{{\Bbb R}} \rangle,\subset) \leq add(^{\omega}\omega,\ll)
(={\frak b}).$$
Again, for an arbitrary partially ordered set $(P,<)$ the symbol $$cof(P,<)$$ denotes the least cardinal number, $\kappa$, for which there is a collection $X$ of cardinality $\kappa$ of elements of $P$ such that: for each $p\in P$ there is an $x\in X$ such that $p\leq x$. This cardinal number is said to be the [*cofinality*]{} of $(P,<)$. Some authors (see e.g. [@I], p. 397) also call this cardinal number the [*upper character*]{} of $(P,<)$ and denote it by $u(P,<)$. It is customary to denote $cof(^{\omega}\omega, \ll)$ by ${\frak d}$.
A theorem of Fremlin ([@F], Proposition 13(b)) states that $$({\frak
d}=) cof(^{\omega}\omega, \ll)\leq cof(\langle J_{{\Bbb R}}
\rangle,\subset).$$
Let $(P,<)$ be a directed partially ordered set. The [*bursting number*]{} of $(P,<)$ ([@I], p. 401) is the smallest cardinal number which exceeds the cardinality of each of the bounded subsets of $(P,<)$. This cardinal number is denoted by $burst(P,<)$. More important is the [ *principal bursting number*]{} of $(P,<)$, denoted $bu(P,<)$ and define as $$bu(P,<)=\min\{burst(Q,<): Q\mbox{ is a cofinal subset of }P\}$$ (following [@I], p. 409). It is always the case that $add(P,<)\leq bu(P,<)$.
A directed partially ordered set $(P,<)$ has the [*irredundancy property*]{} if: $$bu(P,<)= add(P,<).$$
The cofinal subfamily ${\cal A}$ of $(P,<)$ is said to be [*irredundant*]{} if $burst({\cal A},<)\leq add(P,<)$.
Not all $\sigma$-complete ideals have the irredundancy property. Here is an ad hoc example. Let $S_{1}$ and $S_{2}$ be disjoint sets such that $S_{i}$ has cardinality $\aleph_{i}$ for each $i$. Define an ideal $J$ on the union of these sets by admitting a set $Y$ into $J$ if: $Y\cap S_{1}$ is countable and $Y\cap S_{2}$ has cardinality less than $\aleph_{2}$. Then $add(J,\subset)=\aleph_{1}$ and $cof(J,\subset)=\aleph_{2}$. No cofinal family of $J$ is irredundant.
A refined version of the classical notion of a Lusin set is instrumental in verifying the presence of the irredundancy property in many directed partially ordered sets. Since what we’ll define is not exactly the same as the classical notion, we call our “Lusin sets" [ *pseudo Lusin sets*]{} (more about this after the definition). Let $\kappa$ and $\lambda$ be infinite cardinal numbers. Let $(P,<)$ be a directed partially ordered set.
A subset $L$ of $P$ is a $(\kappa,\lambda)$ pseudo Lusin set if:
1. [$\lambda$ is the cardinality of $L$ and]{}
2. [for each $x\in P$ the cardinality of the set $\{y\in L:y\leq x\}$ is less than $\kappa$.]{}
$(\kappa,\lambda)$ pseudo Lusin sets are interesting only when $\kappa\leq\lambda$. If a directed partially ordered set $(P,<)$ has a $(\kappa,\lambda)$ pseudo Lusin set, then $add(P,<)\leq\kappa$ and $\lambda\leq cof(P,<)$. Moreover, every partially ordered set has an $(add(P,<),add(P,<))$ pseudo Lusin set. Thus, if $add(P,<)=cof(P,<)$, then these are the only types of pseudo Lusin sets in $(P,<)$.
Let $J$ be a free ideal on a set $S$. The uniformity number of $J$, written $unif(J)$, is the minimal cardinal $\kappa$ such that there is a subset of $S$ which is of cardinality $\kappa$, which is not an element of $J$.
Consider the partially ordered set $(\langle
J_{{\Bbb R}}\rangle,\subset)$. If $L\subset {{\Bbb R}}$ is a Lusin set in the classical sense (i.e., $L$ is uncountable and every meager set meets $L$ in only countably many points), then $\{\{x\}:x\in L\}$ is an $(\omega_1,\mid
L\mid)$ pseudo Lusin set. There will be pseudo Lusin sets even when there are no (classical) Lusin sets: If $unif(\langle J_{{\Bbb R}}\rangle)>add(\langle
J_{{\Bbb R}}\rangle,\subset)$ then every set of real numbers of cardinality $\aleph_1$ is meager, whence there is no Lusin set in the classical sense. Now let $\{M_{\alpha}:\alpha<add(\langle
J_{{\Bbb R}}\rangle,\subset)\}$ be a family of meager sets such that
1. [ $M_{\alpha}\subset M_{\beta}$ whenever $\alpha<\beta<add(\langle
J_{{\Bbb R}}\rangle,\subset)$ and]{}
2. [$\cup_{\alpha<add(\langle
J_{{\Bbb R}}\rangle,\subset)}M_{\alpha}$ is not meager.]{}
Then the set $L=\{M_{\alpha}:\alpha<add(\langle
J_{{\Bbb R}}\rangle,\subset)\}$ is a $(add(\langle
J_{{\Bbb R}}\rangle,\subset),add(\langle
J_{{\Bbb R}}\rangle,\subset))$ pseudo Lusin set.
It is also well known that these hypotheses on the ideal of meager subsets of the real line are consistent. For example, it is consistent that the real line is a union of $\aleph_1$ meager sets and that each set of real numbers of cardinality less than $\aleph_2$ is meager (see e.g. [@M], §6).
The reader should also compare our notion of a $(\kappa,\lambda)$ - pseudo Lusin set with Cichon’s notion of a $(\kappa,\lambda)$ - Lusin set (see [@Ci]).
The connection between the irredundancy property and the existence of certain pseudo Lusin sets is given by the following proposition. The argument in its proof is well known in the special case when $P$ is the collection of countable subsets of some infinite set, ordered by set inclusion (see the proof of 4.4 on p. 409 of [@I]).
\[refinelusin\]Let $(P,<)$ be a directed partially ordered set. Then the following statements are equivalent:
1. [There is an $(add(P,<),cof(P,<))$ pseudo Lusin set for $(P,<)$,]{}
2. [$(P,<)$ has the irredundancy property,]{}
3. [There is a cofinal $(add(P,<),cof(P,<))$ pseudo Lusin set for $(P,<)$,]{}
Proof.
: [That 1. implies 2:\
Let be such a pseudo Lusin set and let $\{a_{\xi}:\xi<cof(P,<)\}$ be a cofinal subfamily of $P$. For each $\xi<cof(P,<)$ choose $z_{\xi}\in P$ such that $x_{\xi},a_{\xi}\leq z_{\xi}$. Put ${\cal
A}=\{z_{\xi}:\xi<cof(P,<)\}$. Then ${\cal A}$ is an irredundant cofinal family.\
That 2. implies 3:\
Let ${\cal A}$ be an irredundant cofinal family. We may assume that the cardinality of this family is $cof(P,<)$. Then ${\cal A}$ is an example of a cofinal $(add(P,<),cof(P,<))$ pseudo Lusin set.\
It is clear that 3. implies 1. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
\[cor:cardinalityideals\] Let $\kappa>\lambda\geq\aleph_0$ be cardinals, $\lambda$ regular. If $cof([\kappa]^{<\lambda},\subset) = \kappa$, then $([\kappa]^{<\lambda},\subset)$ has the irredundancy property.
Proof.
: [Let $\{S_{\alpha}:\alpha<\kappa\}$ be a pairwise disjoint subcollection from $[\kappa]^{<\lambda}$. Then this family is a $(\lambda,\kappa)$ pseudo Lusin set for this ideal. Applying the cofinality hypothesis we conclude that this ideal has the irredundancy property. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
The ideal of finite subsets of an infinite set has the irredundancy property; the set of one-element subsets of such an infinite set forms an appropriate pseudo Lusin set for this ideal.
\[equivalence\] Let $\kappa>\lambda$ be an uncountable cardinal numbers, $\lambda$ regular. Then the following statements are equivalent:
1. [ The ideal $([\kappa]^{<\lambda},\subset)$ has cofinality $\kappa$.]{}
2. There is a free ideal $J$ such that:
1. [$add(J,\subset)=\lambda$,]{}
2. [$cof(J,\subset)=\kappa$ and]{}
3. [$(J,\subset)$ has the irredundancy property.]{}
Proof.
: [The proof of [*1*]{}$\Rightarrow$ [*2*]{} is trivial. We show that [*2*]{} implies [*1*]{}. Let $J$ be a free ideal on the set $S$ such that $cof(J,\subset)=\kappa$ and $add(J,\subset)=\lambda$, and $(J,\subset)$ has the irredundancy property. Let $L\subset J$ be an $(\lambda,\kappa)$ pseudo Lusin set for $J$. Also let ${\cal C}\subset J$ be a cofinal family of cardinality $\kappa$. For each $X\in{\cal C}$ define: $S_X=\{Y\in L:Y\subseteq X\}$. Then the collection ${\cal B}=\{S_X: X\in {\cal C}\}$ is cofinal in $([L]^{<\lambda},\subset)$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
The following examples play an important role in our game-theoretic applications.
[**[Example 1:]{} The ideal of countable subsets of an infinite set.**]{}
Let $\kappa$ be an uncountable cardinal number. Then $add([\kappa]^{\leq\aleph_{0}},\subset)=\aleph_{1}$ and $bu([\kappa]^{\leq\aleph_{0}},\subset)\geq\aleph_{1}.$ For uncountable cardinal numbers $\kappa$ it is always the case that $\kappa\leq cof([\kappa]^{\leq\aleph_{0}}, \subset)$. A set of the form $\{\{\alpha_{\xi}\}:\xi<\kappa\}$ (where this enumeration is bijective and $\lambda\leq\kappa$) is an $(\omega_{1},\kappa)$ pseudo Lusin set for $[\kappa]^{\leq\aleph_{0}}$. The only difficult cases to decide whether or not the irredundancy property is present are those where $\kappa< cof([\kappa]^{\leq\aleph_{0}}, \subset)$; this occurs for example when $\kappa$ has countable cofinality. It turns out that for these the irredundancy property is not decidable by the axioms of traditional set theory:
1. In [@To3], Todorčević shows that if for each uncountable cardinal $\lambda$ of countable cofinality the assertions
1. [$cof([\lambda]^{\leq\aleph_{0}},\subset)=\lambda^{+}$ and]{}
2. [$\square_{\lambda}$]{}
are true, then for each uncountable cardinal number $\kappa$ there is a cofinal family ${\cal K}\subset [\kappa]^{\aleph_{0}}$ such that $|\{A\cap
X:X\in{\cal K}\}|\leq\aleph_{0}$ for any countable subset $A$ of $\kappa$. Such a family ${\cal K}$ is an example of an $(add([\kappa]^{\leq\aleph_{0}},\subset),
cof([\kappa]^{\leq\aleph_{0}},\subset))$ pseudo Lusin set for $([\kappa]^{\leq\aleph_{0}},\subset)$. These particular examples of pseudo Lusin sets are called [*cofinal Kurepa families*]{}. Thus it is true in the constructible universe, ${\bold L}$ that $([\kappa]^{<\aleph_0},\subset)$ has the irredundancy property for each infinite $\kappa$.
2. One might ask if any hypotheses beyond $ZFC$ are necessary to obtain the conclusion that $([\kappa]^{\leq\aleph_{0}},\subset)$ has the irredundancy property. Todorčević has shown in [@To2] that for an infinite cardinal number $\kappa$ the following statements are equivalent:
1. [$bu([\kappa]^{\leq\aleph_{0}},\subset)=\aleph_{1}$.]{}
2. [$([\kappa]^{\leq\aleph_{0}},\subset)$ has the irredundancy property.]{}
He also noted (p. 843 of [@To4]) that the version $$(\aleph_{\omega+1},\aleph_{\omega})\rightarrow(\omega_{1},\omega)$$ of Chang’s Conjecture implies that $\aleph_{1}<bu([\aleph_{\omega}]^{\leq\aleph_{0}}, \subset)$ (and thus this ideal does not have the irredundancy property). Now [@L-M-S] established the consistency of the above version of Chang’s Conjecture modulo the consistency of the existence of a fairly large cardinal.
3. [This takes care of uncountable cardinals of countable cofinality. What is the situation for those of uncountable cofinality? It is clear that $([\kappa]^{\leq\aleph_0},\subset)$ has the irredundancy property if $\kappa$ is $\aleph_n$ for some finite $n$ or if, for some $m<\omega$, $\kappa$ is the $m$-th successor of a singular strong limit cardinal of uncountable cofinality. In fact, the axiomatic system of traditional set theory has to be strengthened fairly dramatically before one could create circumstances where there is a cardinal number of uncountable cofinality which is strictly less than the cofinality of its ideal of countable sets; it follows from Lemma 4.10 of [@J-M-P-S] that if there is a cardinal number of uncountable cofinality which is smaller than the cofinality of its ideal of countable sets, then there is an inner model with many measurable cardinal numbers. ]{}
Information about the ideal of countable subsets of some infinite set can be used to gain information about some other ideals, using the notion of a locally small family.
A family ${\cal F}$ of subsets of a set $S$ is [*locally small*]{} if:$$|\{Y\in {\cal F}:Y\subseteq X\}|\leq \aleph_{0}$$for each $X$ in ${\cal
F}$.
If the ideal of countable subsets of an infinite set has an irredundant cofinal family then that cofinal family is ipso facto locally small. If there is an $(\omega_{1},cof(J,\subset))$ pseudo Lusin set for the $\sigma$-complete free ideal $J$ on the set $S$, then $J$ contains a locally small cofinal family.
[**[Example 2:]{} The ideal of meager subsets of the real line**]{}
Assume that $add(\langle J_{{\Bbb R}}\rangle,\subset)=cof(\langle
J_{{\Bbb R}}\rangle,\subset)$ (This equation is for example implied by Martin’s Axiom). Then $\langle J_{{\Bbb R}} \rangle$ has the irredundancy property. In this case one may insure that the cofinal family which witnesses the irredundancy is a well-ordered chain of meager sets. By the results cited from [@M] and [@F], the hypothesis implies that ${\frak
b}={\frak d}$. It is well known that the reverse implication is not provable.
Irredundancy does not require having a well ordered cofinal chain of meager sets. For let an initial ordinal be given. According to a theorem of Kunen ([@K], p. 906, Theorem 3.18) it is consistent that the cardinality of the real line is regular and larger than that initial ordinal, and at the same time there is an $(\omega_{1},{\frak c})$ pseudo Lusin set. It follows that $\langle
J_{{\Bbb R}}\rangle$ has a locally small cofinal family of cardinality ${\frak c}$. In particular, $\langle J_{{\Bbb R}}\rangle$ has the irredundancy property. If the continuum is larger than $\aleph_1$ it also follows that this ideal has no cofinal well-ordered chain.
Stevo Todorčević has informed us that it is also consistent, modulo the consistency of a form of Chang’s Conjecture that $\langle J_{{{\Bbb R}}}\rangle$ does not have the irredundancy property. Actually, something apparently weaker than that form of Chang’s Conjecture is used: we present this result of Todorčević’s in Theorem \[consistency1\], which he kindly permitted us to include in this paper.
\[consistency1\] If “[*ZFC+$MA_{\aleph_1}$+ there is no Kurepa family in $[\aleph_{\omega}]^{\aleph_0}$ of cardinality larger than $\aleph_{\omega}$*]{}" is a consistent theory, then so is the theory “[*ZFC + $bu(\langle{J_{{\Bbb R}}}\rangle , \subset )>add(\langle{J_{{\Bbb R}}}\rangle, \subset)
= \aleph_1$*]{}".
Proof
: Let ${\bold P}$ be the set of finite functions with domain a subset of $\aleph_{\omega}$ and range a subset of $\omega$ (in other words, ${\bold P}$ is the standard set for adding $\aleph_\omega$ Cohen reals). For $p$ and $q$ in ${\bold P}$ we write $p<q$ if $q\subset p$. For $D$ a countable subset of $\aleph_\omega$ we write ${\bold P}(D)$ for the set of elements of ${\bold P}$ whose domains are subsets of $D$.
Suppose we have a sequence $\{N_{\xi}:\xi<\theta\}$ ($\theta>\aleph_{\omega}$) of ${\bold P}$-names for meager sets of reals. Let $D_{\xi}\in[\aleph_\omega]^{\aleph_0}$ be the support of $N_\xi$ i.e., $N_\xi\in {\bold V}^{{\bold P}(D_{\xi})}$. By the hypothesis of the theorem and by Theorem 1 of [@To3] there is an uncountable set $A\subset\theta$ such that $D=\cup_{\xi\in A}D_{\xi}$ is countable. Thus, $N_{\xi}\in{\bold
V}^{{\bold P}(D)}$ for each $\xi\in A$. Since ${\bold P}(D)$ is essentially the poset for adding [*one*]{} Cohen real and since $MA_{\aleph_1}$ holds, ${\bold
V}^{{\bold P}(D)}\models``\cup_{\xi\in A}N_{\xi} \mbox{ is
meager}"$ (because ${\bold V}^{{\bold P}(D)}\models``MA(\sigma-\mbox{centered})"$). ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
The hypothesis of Theorem \[consistency1\] is consistent modulo the consistency of the relevant form of Chang’s Conjecture, because that form of the conjecture is preserved by c.c.c. generic extensions.
The coherent decomposition property
===================================
Let $J$ be a free ideal on a set $S$ and let $\langle J\rangle$ be its $\sigma$-completion. Let ${\cal A}$ be a subcollection of $\langle
J\rangle$.
1. ${\cal A}$ has a coherent decomposition if there is for each $A\in {\cal A}$ a sequence $(A^n:n<\omega)$ such that:
1. [$A^n\in J$ for each $n$,]{}
2. [$A^n\subseteq A^m$ whenever $n<m<\omega$, and]{}
3. [For all $A$ and $B$ in ${\cal A}$ such that $A\subset B$, there is an $m$ such that $A^n\subseteq B^n$ whenever $n\geq m$.]{}
The collection $\{(A^n:n<\omega): A\in {\cal A}\}$ is said to be a coherent decomposition for ${\cal A}$.
2. [The ideal $J$ has the coherent decomposition property if some cofinal subset of $\langle J\rangle$ has a coherent decomposition.]{}
It is worth mentioning that if $J$ has the coherent decomposition property and if $\langle J\rangle$ has a cofinal chain, than the family $\langle J\rangle$ itself has a coherent decomposition. We now explore the coherent decomposition property for our examples.
[**[Example 1:]{} (continued)**]{}
\[locsmallth\] Let ${\cal A}$ be a locally small family of countable sets such that $({\cal A},\subset)$ is a well-founded partially ordered set. Then ${\cal A}$ has a coherent decomposition.
Proof.
: Let $\Phi:{\cal A}\rightarrow\alpha$ be a function to an ordinal $\alpha$ such that $\Phi(A)<\Phi(B)$ for all $A\subset
B$ in ${\cal A}$ (i.e., a rank function). Since ${\cal A}$ is locally small we may assume that $\alpha$ is $\omega_{1}$.
For $A$ in ${\cal A}$ with $\Phi(A)=0$, choose a sequence $(A^{n}:n<\omega)$ of finite subsets of $A$ such that $A=\cup_{n<\omega}A^{n}$ and $A^{n}\subseteq A^{n+1}$ for all n.
Let $0<\beta<\omega_{1}$ be given and assume that we have already assigned to each $A$ in ${\cal A}$ for which $\Phi(A)<\beta$, a sequence $(A^{n}:n<\omega)$ in compliance with [*1*]{} and [*2*]{}. Now Let $B$ be an element of ${\cal A}$ such that $\Phi(B)=\beta$. Write $F(B)=\{A\in {\cal
A}: A\subset B\}.$
To begin, arbitrarily choose a sequence $(S_{n}:n<\omega)$ of finite sets such that $B=\cup_{n<\omega}S_{n}$. For each $A\in F(B)$, define $g_{A}:\omega\rightarrow\omega$ such that for each $n<\omega$, $$g_{A}(n)=min\{k<\omega:A^{n}\subseteq S_{0}\cup\dots\cup
S_{k}\}.$$ Then $\{g_{A}:A\in F(B)\}$ is countable since ${\cal A}$ is locally small. Let $f\in$ be a strictly increasing function such that $g_{A}\ll f$ for each $A$ in $F(B)$. Define: $$B^{n}=S_{0}\cup\dots\cup S_{f(n)}$$ for each n. Then $(B^{n}:n<\omega)$ is as required. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[localsmallcor\]Let $J$ be a free ideal on a set $S$ and let ${\cal A}$ be a locally small family of sets in $\langle J\rangle$ such that $({\cal A},\subset)$ is a well-founded partially ordered set. Then ${\cal A}$ has a coherent decomposition.
Proof.
: For each $B$ in ${\cal A}$, let $(S_{n}(B):n<\omega)$ be a sequence from $J$ such that $B=\cup_{n<\omega}S_{n}(B)$. Also write $\Gamma(B)=\{A\in{\cal A}:A\subseteq
B\}$. Then is a well-founded, locally small collection of countable subsets of ${\cal A}$. Choose, by Theorem \[locsmallth\], for each $A\in {\cal A}$ a sequence $(\Gamma(A)^{n}:n<\omega)$ of finite subsets of $\Gamma(A)$ such that:
1. [$\Gamma(A)=\cup_{n<\omega}\Gamma(A)^{n}$ where $\Gamma(A)^{n}\subseteq
\Gamma(A)^{n+1}$ for each n, and]{}
2. [for all $A$ and $B$ in ${\cal A}$ with $A\subset B$ there exists an $m$ such that: $$\Gamma(A)^{n}\subseteq \Gamma(B)^{n}$$ for all $n\geq m$.]{}
For each $A$ in ${\cal A}$ and each $n<\omega$ define: $$A^{n}=\cup\{S_{j}(B):j\leq n \mbox{ and } B\in\Gamma(A)^{n}\}.$$ Then the sequences $(A^{n}:n<\omega)$ are as required. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
If $([\kappa]^{\leq \aleph_0},\subset)$ has the irredundancy property, then it has the coherent decomposition property.
Proof
: [An irredundant cofinal family is necessarily locally small. We may thin out any cofinal family to a well-founded cofinal family. Now apply Theorem \[locsmallth\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
[**[Example 2:]{} (continued)**]{}
We show that the ideal of meager sets of the real line has the coherent decomposition property, and also that it has a second combinatorial property which plays an important role in our game-theoretic applications. It is convenient, for this section, to work with the set $^{\omega}2$, with the usual Tychonoff product topology ($2=\{0,1\}$ is taken to have the discrete topology) in place of ${\Bbb R}$. For a subset $S$ of the domain of a function $g$, the symbol $g\lceil_{S}$ denotes the restriction of $g$ to the set $S$. For $s$ an element of $^{<\omega}2$, the symbol $[s]$ denotes the set of all those $x$ in $^{\omega}2$ for which $x\lceil_{length(s)}=s$. Subsets of $^{\omega}2$ of the form $[s]$ where $s$ ranges over $^{<\omega}2$, form a base for the topology of $^{\omega}2$. Let $f\in\mbox{$^{\omega}\omega$}$ be a strictly increasing sequence and let $x$ be an element of $^{\omega}2$. Define:
$$B_{x,f}=\{z\in\mbox{$^{\omega}2$}:\forall^{\infty}_n(z\lceil_{[f(n),f(n+1))}\not=
x\lceil_{[f(n),f(n+1))})\}.$$
Now also fix an $n\in \omega$ and define
$$B^n_{x,f}= \{z\in \mbox{$^{\omega}2$}:(\forall k\geq
n)(z\lceil_{[f(k),f(k+1))}\not=x\lceil_{[f(k),f(k+1))})\}.$$ Then $B^m_{x,f}\subseteq B^n_{x,f}$ whenever $m<n<\omega$; also, $B_{x,f}=\cup_{n<\omega}B^n_{x,f}$.
\[crucial\] For $x,y\in\mbox{$^{\omega}2$}$ and strictly increasing $f,g\in \mbox{$^{\omega}\omega$}$, the following assertions are equivalent:
1. [$B_{x,f}\subset B_{y,g}$.]{}
2. 1. [$B_{x,f}\not=B_{y,g}$ and]{}
2. [$(\forall^{\infty}_n) (\exists k) (g(n)\leq f(k)<f(k+1)\leq
g(n+1)\mbox{ and }
x\lceil_{[f(k),f(k+1))}=y\lceil_{[f(k),f(k+1))})$]{}
Proof.
: That [*1*]{} implies [*2*]{} requires some thought:\
If [*1*]{} holds, then (a) of [*2*]{} holds. Assume the negation of [*2*]{}(b). It reads: $$(\exists^{\infty}_n)(\forall k)(\neg(g(n)\leq f(k)<f(k+1)\leq
g(n+1)) \mbox{ or
}\neg(x\lceil_{[f(k),f(k+1))}=y\lceil_{[f(k),f(k+1))}))$$
Put $S=\{n<\omega:(\forall
k)(\neg([f(k),f(k+1)]\subseteq[g(n),g(n+1)]) \mbox{ or }\neg(
x\lceil_{[f(k),f(k+1))}= y\lceil_{[f(k),f(k+1))})\}$. Our hypothesis is that $S$ is an infinite set.
Consider an $n$ in $S$. For each $k$, there are the following possibilities:
1. [$\neg([f(k),f(k+1)]\subseteq[g(n),g(n+1)]$]{}
2. [$[f(k),f(k+1)]\subseteq[g(n),g(n+1)]$, but $x\lceil_{[f(k),f(k+1))}\not= y\lceil_{[f(k),f(k+1))}$.]{}
Put $S_n=\{k:\mbox{{\em 2} holds for }k\}$. We consider two cases.\
[**Case 1:**]{} There are infinitely many $n$ for which $S_n$ is nonempty.\
Choose an infinite sequence $(n_1,n_2,n_3,\dots)$ from $S$ such that:
1. [$S_{n_m}\neq\emptyset$,]{}
2. [$n_{m+1}>g(n_m+1)$, and]{}
3. [$(\exists k)(g(n_m+1)<f(k)<g(n_{m+1}))$, for each $m$, and]{}
4. [$f(1)<g(n_1)$.]{}
This is possible because $f$ and $g$ are increasing, and $S$ is infinite. Put $T=\cup_{j=1}^{\infty}[g(n_j),g(n_j+1))$. Define $z$, an element of $^{\omega}2$, so that $z\lceil_{T}=y\lceil_{T}$ and $z(n)=1-x(n)$ for each $n\in\omega\backslash T$. Then $z\in B_{x,f}$ while $z\not\in
B_{y,g}$. Thus [*1*]{} fails in this case.\
[**Case 2**]{}: There are only finitely many $n\in S$ for which $S_n$ is nonempty.\
We may assume that $S_n=\emptyset$ for each $n\in S$. Consider $n\in S$. We then have that for each $k\in \omega$, $[f(k),f(k+1))\not\subseteq[g(n),g(n+1))$. We distinguish between two possibilities:
1. [$(\exists k)(g(n)\leq f(k)<g(n+1))$ or]{}
2. [$(\forall k)(f(k)\not\in[g(n),g(n+1))$]{}
[**Case 2 (A):**]{} Possibility [*1*]{} occurs for infinitely many $n\in S$:\
Choose $n_1<n_2<n_3<\dots$ from $S$ such that
- [$2\cdot n_j\leq n_{j+1}$ for each $j$,]{}
- [for each $j$ there is a $k$ such that $g(n_j+1)<f(k)<g(n_{j+1})$,]{}
- [for each $j$ there is a $k$ such that $f(k)\in[g(n_j),g(n_{j+1}))$, and]{}
- [$f(1)<g(n_1)$.]{}
Put $T=\cup_{j=1}^{\infty}[g(n_j),g(n_j+1))$ and define $z$ so that $z\lceil_T=y\lceil_T$, and $z(n)=1-x(n)$ for each $n\in\omega\backslash T$. From the hypothesis of Case 2(A) it follows that $z\in B_{x,f}$, but $z\not\in B_{y,g}$. Thus, [*1*]{} of the Proposition fails also in this case.\
[**Case 2 (B):**]{} Possibility [*1*]{} occurs for only finitely many $n\in S$:\
We may assume that possibility [*2*]{} occurs for each $n\in
S$. Choose $k_1<k_2<k_3<\dots$ such that for each $j$ there is an $n\in S$ with $[g(n),g(n+1))\subset[f(k_j),f(k_j+1))$. For each $j$ choose $n_j\in S$ such that $[g(n_j),g(n_j+1))\subset
[f(k_j),f(k_j+1))$. As before define $T=\cup_{j=1}^{\infty}[g(n_j),g(n_j+1))$. Finally, define $z$ so that $z\lceil_T=y\lceil_T$ and $z(n)=1-x(n)$ for each $n\in\omega\backslash T$. Then $z\in B_{x,f}$ and $z\not\in B_{y,g}$, showing that [*1*]{} of the Proposition fails also in this case.
This completes the proof of the Proposition. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[equality\]Let $f$ and $g$ be strictly increasing elements of $^{\omega}\omega$ for which there is some $k<\omega$ such that $g(n+k)=f(n)$ for all but finitely many $n$. If $B_{x,f}\subseteq
B_{y,g}$, then $B_{x,f}=B_{y,g}$.
Proof.
: Assume that $B_{x,f}\neq B_{y,g}$ and suppose that $B_{y,g}\not\subseteq B_{x,f}$. We show that $B_{x,f}\not\subseteq B_{y,g}$. Let $z$ be an element of $B_{y,g}\backslash B_{x,f}$. Fix $N$ such that
1. [$z\lceil_{[g(n+k),g(n+k+1))}\neq y\lceil_{[g(n+k),g(n+k+1))}$ and]{}
2. [$f(n)=g(n+k)$]{}
for each $n\geq N$.
Since $z$ is not an element of $B_{x,f}$, there are infinitely many $n\geq N$ for which $z\lceil_{[f(n),f(n+1))}=x\lceil_{f(n),f(n+1))}$. Consequently the set $S=\{n\geq N:x\lceil_{[f(n),f(n+1))}\neq
y\lceil_{[f(n),f(n+1))}\}$ is infinite. Now define $t$ such that $t\lceil_{[f(n),f(n+1))}=y\lceil_{[f(n),f(n+1))}$ for each $n\in S$, and $t(m)=1-x(m)$ for each $m\in\omega\backslash(\cup_{n\in
S}[f(n),f(n+1)))$. Then $t$ is in $B_{x,f}$ but not in $B_{y,g}$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
Under the hypothesis of Lemma \[equality\], $x(n)=y(n)$ for all but finitely many $n$.
\[orderpres\] Let $x,y$ be elements of $^{\omega}2$ and let $f,g$ be increasing elements of $^{\omega}\omega$. Of the following two assertions, [*1*]{} implies [*2*]{}.
1. [$B_{x,f}\subset B_{y,g}$.]{}
2. [$f\ll g$.]{}
Proof.
: [ Assume that $B_{x,f}\subset
B_{y,g}$. Fix, by Proposition \[crucial\], an $N$ such that\
$(\forall n\geq N) (\exists k) ([f(k),f(k+1)]\subseteq[g(n),g(n+1)]$ and $x\lceil_{[f(k),f(k+1))}=y\lceil_{[f(k),f(k+1))}).$\
For each $n\geq N$ choose $k_n$ such that $[f(k_n),f(k_n+1)]\subseteq [g(n),g(n+1)]$. It follows that $k_n+1\leq
k_{n+1}$ for each $n\geq N$ (since $f$ and $g$ are increasing).\
[**Claim:**]{} $[f(k_n),f(k_n+1)]\subset [g(n),g(n+1)]$ for infinitely many n.\
[**Proof of the claim:**]{} For otherwise, fix $M\geq N$ such that $[f(k_n),f(k_{n+1}]=[g(n),g(n+1)]$ for each $n\geq M$. Then we have $k_{n+1}=k_n+1$ for each $n\geq M$. It follows that $g(n)=f(n+(k_M-M))$ for all $n\geq M$. Then Lemma \[equality\] implies that $B_{x,f}=B_{y,g}$, contrary to the fact that $B_{x,f}$ is a proper subset of $B_{y,g}$. This completes the proof of the claim. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$\
Thus, there are infinitely many $n$ for which $k_{n+1}>k_n+1$. Let $m>1$ be given, and fix $L\geq M$ such that $|\{n<L:k_{n+1}>k_n+1\}|\geq
k_1+m$. Then $k_n>(n+m)$ for each $n\geq L$; we have $$f(n+1)<f(n+m)\leq f(k_n)<g(n+1)$$ for each $n\geq L$. In particular, $m\leq g(n+1)-f(n+1)$ for each $n\geq L$. This completes the proof that $f\ll g$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
\[coherencyprop\] Let $x$ and $y$ be elements of $^{\omega}2$ and let $f$ and $g$ be increasing elements of $^{\omega}\omega$. If $B_{x,f}\subset B_{y,g}$, then there is an $m<\omega$ such that $B^n_{x,f}\subseteq B^n_{y,g}$ whenever $n\geq m$.
Proof
: From our hypotheses and Proposition \[crucial\] there is an $m$ such that for each $n\geq m$ there is a $k$ such that $[f(k),f(k+1))\subseteq[g(n),g(n+1))$ and $x\lceil_{[f(k),f(k+1))}=y\lceil_{[f(k),f(k+1))}$. By Proposition \[orderpres\] there is an $M>m$ such that $f(j)\leq
g(j)$ for each $j\geq M$. We show that $B^n_{x,f}\subseteq B^n_{y,g}$ for each $n\geq M$.
Let $z$ be an element of $B^n_{x,f}$. Then $z\lceil_{[f(j),f(j+1))}\neq x\lceil_{[f(j),f(j+1))}$ for each $j\geq
n$. But consider any $j\geq $n. Then there is a $k$ such that $[f(k),f(k+1))\subset [g(j),g(j+1))$; $k\geq j$ for any such $k$, by the choice of $M$. It follows that $z\lceil_{[g(j),g(j+1))}\neq
y\lceil_{[g(j),g(j+1))}$. Thus, $z$ is also an element of $B^n_{y,g}$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[cofinalprop\] For each $X\in \langle
J_{{\Bbb R}}\rangle$ there are an $x$ in $^{\omega}2$ and an increasing $f$ in $^{\omega}\omega$ such that $X\subset B_{x,f}$.
Proof.
: Let $X$ be a meager set. We may assume that $X=\cup_{n=0}^{\infty}X_n$ where $X_n\subseteq X_{n+1}$ and $X_n$ is closed, nowhere dense for each $n$. Fix a well-ordering of $^{<\omega}2$, and define $(s_n:n<\omega)$ and $f$ in $^{\omega}\omega$ as follows:\
Take $s_0=\emptyset$ and $f(0)=0$. Assume that $s_1,s_2,\dots,s_n$ and $f(1),\dots,f(n)$ have been defined so that:
1. [$s_1$ is the first element of $^{<\omega}2$ such that $[s_1]\cap
X_1=\emptyset$ and $f(1)=length(s_1)$,]{}
2. [$s_{j+1}$ is the first element of $^{<\omega}2$ such that $[t^{\frown}s_{j+1}]\cap X_{j}=\emptyset$ for each $t$ in $^{\leq
f(j)}2$, and $f(j+1)=\sum_{i=0}^{j+1}length(s_i)$ for each $j<n$.]{}
Then let $s_{n+1}$ be the first element of $^{<\omega}2$ such that $[t^{\frown}s_{n+1}]\cap X_n=\emptyset$ for each $t$ in $^{\leq
f(n)}2$; put $f(n+1)=f(n)+length(s_{n+1})$.
Finally, set $x=s_1^{\frown}s_2^{\frown}s_3^{\frown}\dots$.\
[**Claim:**]{} $X\subseteq B_{x,f}$.\
For suppose that $z$ is not an element of $B_{x,f}$. Then there are infinitely many $n$ for which $z\lceil_{[f(n),f(n+1))}=x\lceil_{[f(n),f(n+1))}$; in other words, there are infinitely many $n$ for which $z\lceil_{[f(n),f(n+1))}=s_{n+1}$. Now fix an $m$. Choose an $n>m$ such that $z\lceil_{[f(n),f(n+1))}=s_{n+1}$. From the choice of $s_{n+1}$ it follows that $[z\lceil_{f(n+1)}]\cap X_m=\emptyset$; in particular, $z\not\in X_m$. Consequently, $z$ is not an element of $X$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[meagernessprop\] Each $B^n_{x,f}$ is in $J_{{\Bbb R}}$.
Proof.
: [ Consider an $s$ from $^{\omega}2$ for which $[s]\cap B^n_{x,f}\neq\emptyset$. Choose $m$ such that $f(m)>length(s)$ and $m>n$. Then choose $t$ from $^{<\omega}2$ such that $length(s^{\frown}t)\geq f(m+1)$ and $s^{\frown}t\lceil_{[f(m),f(m+1))}= x\lceil_{[f(m),f(m+1))}$. Then $[s^{\frown}t]\cap B^n_{x,f}=\emptyset$. It follows that $B^n_{x,f}$ is nowhere dense. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
Consequently, $B_{x,f}$ is a meager set for each $x$ in $^{\omega}2$ and for each increasing $f$ from $^{\omega}\omega$.
\[important\] $\langle J_{{\Bbb R}}\rangle$ has a cofinal family which embeds in $(^{\omega}\omega,\ll)$ and which has the coherent decomposition property.
Proof.
: [ By Propositions \[meagernessprop\] and \[cofinalprop\] the family of sets of the form $B_{x,f}$ where $f$ is an increasing element of $^{\omega}\omega$ and $x$ is an element of $^{\omega}2$, is a cofinal family of meager sets. By Proposition \[coherencyprop\], this family has the coherent decomposition property. Also, the mapping which assigns $f$ to $B_{x,f}$ is, according to Proposition \[orderpres\], an order preserving mapping. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
[**[Example 3:]{} Cardinals of countable cofinality**]{}
Here is a result which is quite analogous to Theorem \[locsmallth\].
\[unctble,countblecof\] Let $\lambda$ be an uncountable cardinal number which has countable cofinality. Let $\lambda_{0}<\lambda_{1}<\dots$ be a sequence of infinite regular cardinal numbers which converges to $\lambda$. Let $({\cal A},\subset)$ be a well-founded family of sets, each of cardinality $\lambda$, such that $$|\{Y\in{\cal A}:Y\subseteq X\}|\leq\lambda$$ for each $X$ in ${\cal A}$. Then ${\cal A}$ has the coherent decomposition property. In particular:\
There exists for each $A\in{\cal A}$ a sequence $(A^{n}:n<\omega)$ such that:
1. [$|A^{n}|\leq\lambda_{n}$ for all $n$,]{}
2. [$A^{n}\subseteq A^{n+1}$ for all n,]{}
3. [$A=\cup_{n=0}^{\infty}A^{n}$ and]{}
4. [if $A\subset B$, then there is an $m<\omega$ such that $A^{n}\subseteq
B^{n}$ for all $n\geq m$.]{}
Proof.
: Let $\Phi:{\cal A}\rightarrow\lambda^{+}$ be a rank function. For all $A$ in ${\cal A}$ with $\Phi(A)=0$, choose $(A^{n}:n<\omega)$ arbitrary, subject only to [*1, 2*]{} and [*3*]{}.
Let $0<\gamma<\lambda^{+}$ be given and assume that $(A^{n}:n<\omega)$ has been assigned to each $A$ from ${\cal A}$ for which $\Phi(A)<\gamma$, in such a way that [*1, 2, 3*]{} and [*4*]{} are satisfied. Consider $B$ in ${\cal A}$ with $\Phi(B)=\gamma$. Write $F(B)$ for $\{A\in{\cal A}:A\subseteq
B\}$ and write $F(B)=\cup_{n=0}^{\infty}F_{n}(B)$ where
1. [$F_{0}(B)\subseteq F_{1}(B)\subseteq\dots$, and]{}
2. [$|F_{n}(B)|\leq\lambda_{n}$ for all $n$.]{}
Also let $B=\cup_{n=0}^{\infty}X_{n}$ where $X_{0}\subseteq
X_{1}\subseteq\dots$ and $X_{n}\leq \lambda_{n}$ for all $n$. Finally put $B^{n}=(\cup\{A^{n}:A\in F_n(B)\})\cup X_{n}$ for each $n$. Then $(B^{n}:n<\omega)$ is as required. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[cor:countablecof\] Let $\lambda$ be a cardinal number of countable cofinality. If $([\kappa]^{\leq\lambda},\subset)$ has the irredundancy property then it has the coherent decomposition property.
Applications
============
The $\omega-path$ partition relation is the one other combinatorial ingredient in our technique for constructing winning $k$-tactics, or for defeating a given $k$-tactic for TWO. For a positive integer $n$, infinite cardinal number $\lambda$ and a partially ordered set $(P,<)$, the symbol $$(P,<)\rightarrow(\omega-path)^{n}_{\lambda/<\omega}$$ means that for every function $F:[P]^{n}\rightarrow\lambda$ there is an increasing $\omega$-sequence $$p_{1}<p_{2}<\dots<p_{m}<\dots$$ such that the set $\{F(\{p_{j+1},\dots,p_{j+n}\}):j<\omega\}$ is finite. The negation of this assertion is denoted by the symbol $$(P,<)\not\rightarrow(\omega-path)^{n}_{\lambda/<\omega}.$$ This partition relation has been studied in [@S2]. The reader should consult this reference about the various facts concerning the $\omega$-path relation which are used in the sequel.
The game $MG({\cal A},J)$
-------------------------
For a free ideal $J$ on an infinite set $S$ and for a family ${\cal
A}$ in $\langle J\rangle$ with the property that for each $X\in{\cal A}$ there is a $Y\in{\cal A}$ such that $X\subset Y$, the game $MG({\cal A},J)$ is defined so that an $\omega$-sequence $(O_{1},T_{1},\dots,O_{n},T_{n},\dots)$ is a play if for each $n$,
1. [$O_{n}\in {\cal A}$ is player ONE’s move in inning $n$,]{}
2. [$T_{n}\in J$ is player TWO’s move in inning $n$, and]{}
3. [$O_{n}\subset O_{n+1}$.]{}
Player TWO wins this play if $\cup_{n=1}^{\infty}O_{n}\subseteq\cup_{n=1}^{\infty}T_{n}.$
\[mgth\] Let $J$ be a free ideal on a set $S$. If ${\cal A}$ is a family of sets in $\langle J\rangle$ such that:
1. [for each $X\in{\cal A}$ there is a $Y\in{\cal A}$ such that $X\subset Y$,]{}
2. [$({\cal A},\subset)\not\rightarrow(\omega-path)^{k}_{\omega/<\omega}$ for some $k\geq 2$, and]{}
3. [${\cal A}$ has a coherent decomposition]{}
then TWO has a winning $k$-tactic in $MG({\cal A},J)$.
Proof.
: Choose a function $F:[{\cal
A}]^{k}\rightarrow\omega$ which witnesses hypothesis [*2*]{}. Also associate with each $A$ in ${\cal A}$ a sequence $(A^{n}:n<\omega)$ such that hypothesis [*3*]{} is satisfied.
Define a $k$-tactic, $\Upsilon$ for TWO as follows. Let $(X_{1},\dots,X_{j})$ be given such that $j\leq k$, $X_{1}\subset\dots\subset X_{j}$ and $X_{i}\in{\cal A}$ for $i\leq j$.
1. [If $j<k$: Then put $\Upsilon(X_{1},\dots,X_{j}) =
X^{1}_{1}\cup\dots\cup X^{1}_{j}.$]{}
2. If $j=k$: Let $m$ be such that
- [$m\geq F(\{X_{1},\dots,X_{k}\})$ and]{}
- [$X^{n}_{1}\subseteq\dots\subseteq X^{n}_{k}$ for all $n\geq m$.]{}
Put $\Upsilon(X_{1},\dots,X_{k})=X^{m}_{1}\cup\dots\cup X^{m}_{k}$.
Then $\Upsilon$ is a winning $k$-tactic for TWO. For let $(O_{1},T_{1},\dots,O_{n},T_{n},\dots)$ be a play of $MG({\cal A},J)$ where:
- [$T_{j}=\Upsilon(O_{1},\dots,O_{j})$ for each $j\leq k$]{}
- [$T_{n+k}=\Upsilon(O_{n+1},\dots,O_{n+k})$ for each $n<\omega$.]{}
For each $t\geq 1$ let $m_{t}$ be the number associated with $(O_{t},\dots,O_{t+k-1})$ in part 2 of the definition of $\Upsilon.$ By the properties of $F$, the set $\{m_{t}:t=1,2,3,\dots\}$ is infinite. Thus choose $t_{1}<t_{2}<\dots$ such that $m_{j}<m_{t_{r}}$ for all $j<t_{r}$. It follows from the criteria used in the choices of the numbers $m_{t}$ that $$O^{m_{t_{r}}}_{1}\subseteq\dots\subseteq O^{m_{t_{r}}}_{m_{t_{r}}}$$ for all $r$. But $O^{m_{t_{r}}}_{m_{t_{r}}}\subseteq
T_{m_{t_{r}}}$ for all $r$, according to the definition of $\Upsilon$. It follows that $\cup_{n=1}^{\infty}O_{n}\subseteq\cup_{n=1}^{\infty}T_{n}.$ ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[cor:mgreals\] There is a cofinal family ${\cal
A}\subset\langle J_{{{\Bbb R}}}\rangle$ such that TWO has a winning $2$-tactic in $MG({\cal A},J_{{{\Bbb R}}})$.
Proof.
: [Let ${\cal A}$ be the family of meager sets provided by Theorem \[important\]. Thus, there is an order preserving function from $({\cal A},\subset)$ to $(^{\omega}\omega,\ll)$. But then $({\cal
A},\subset)\not\rightarrow(\omega-path)^{2}_{\omega/<\omega}$ holds, since $(^{\omega}\omega,\ll)\not\rightarrow(\omega-path)^{2}_{\omega/<\omega}$ holds. By Theorem \[important\] the family ${\cal A}$ also satisfies the third hypothesis of Theorem \[mgth\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
\[cor;localsmall\] Let $J$ be a free ideal on an infinite set. If ${\cal A}$ is a family of sets in $\langle J\rangle$ such that:
1. [${\cal A}$ is locally small,]{}
2. [for each $X\in{\cal A}$ there is a $Y\in{\cal A}$ such that $X\subset Y$, and]{}
3. [$({\cal A},\subset)$ is well-founded,]{}
then TWO has a winning $2$-tactic in $MG({\cal A},J)$.
Proof.
: [The proof is analogous to that of Corollary \[cor:mgreals\]; now we refer to the proof of Theorem \[locsmallth\], we observe that $\omega_{1}\leq{\frak b}$, and invoke Theorem \[mgth\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
\[cor:countablecof2\] Let $\lambda\leq\kappa$ be infinite cardinal numbers such that:
1. [$\lambda$ has countable cofinality,]{}
2. [$\lambda^{+}\not\rightarrow(\omega-path)^{2}_{\omega/<\omega}$, and]{}
3. [$[\kappa]^{\leq\lambda}$ has the irredundancy property.]{}
Then there is a cofinal family ${\cal A}\subset[\kappa]^{\lambda}$ such that TWO has a winning 2-tactic in $MG({\cal A},[\kappa]^{<\lambda})$.
Proof.
: [Let ${\cal A}$ be a well-founded cofinal family in $[\kappa]^{\lambda}$ which is irredundant. Since there is a rank-function from ${\cal A}$ to $\lambda^{+}$ it follows from hypothesis [*2*]{} that $({\cal A},\subset)\not\rightarrow(\omega-path)^{2}_{\omega/
<\omega}$. From Corollary \[cor:countablecof\] it follows that ${\cal A}$ also satisfies the third hypothesis of Theorem \[mgth\]. By that theorem TWO then has a winning $2$-tactic in the game $MG({\cal
A},[\kappa]^{<\lambda})$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
Theorem \[slightprogress\] shows that under certain circumstances there is for each $n$ a free ideal $J_n$ and a cofinal family ${\cal A}_n\subset \langle J_n\rangle$ such that TWO does not have a winning $n$-tactic, but does have a winning $n+1$-tactic in $MG({\cal A}_n,J_n)$. We think that Theorem \[slightprogress\] indicates some relevance of the games as considered here for Telgarsky’s Conjecture (see 3.4).
\[slightprogress\]Let $\lambda$ be an infinite cardinal number. If there is a linearly ordered set $(L,<)$ such that:
1. [$cof(L,<)>\omega$,]{}
2. [$(L,<)\rightarrow (\omega-path)^2_{\lambda/<\omega}$, but]{}
3. [$(L,<)\not\rightarrow(\omega-path)^3_{\lambda/<\omega}$,]{}
then there is for each $n$ a free ideal $J_n$ and a cofinal family ${\cal
A}_n\subset \langle J_n\rangle$ such that TWO does not have a winning $n$-tactic, but does have a winning $n+1$-tactic in $MG({\cal A}_n,J_n)$.
Proof.
: Let $\lambda$ and $(L,<)$ be as in the hypotheses. It follows from Propositions 3 and 4 of [@S2] that there is for each integer $m>1$ a linearly ordered set $(L_n,<_n)$ such that $\omega<cof(L_n,<_n)$ and: $$(L_n,<_n)\rightarrow(\omega-path)^{n}_{\lambda/<\omega}$$ but $$(L_n,<_n)\not\rightarrow(\omega-path)^{n+1}_{\lambda/<\omega}$$ Let $n>1$ and $(L_n,<_n)$ be fixed for the rest of the proof. We may assume that the underlying set, $L_n$, is disjoint from ${\EuScript
P}({\EuScript P}(\lambda))\cup{\EuScript P}(\lambda)\cup\lambda$.\
Define a free ideal $J_n$ as follows: The underlying set on which $J_n$ lives, say $S_n$, is $[\lambda]^{<\aleph_0}\cup L_n$. For each $\alpha\in\lambda$ let $X_{\alpha}$ be the set $\{Z\in
[\lambda]^{<\aleph_0}: \alpha\not\in Z\}$. Let ${\cal T}$ be $\{X_{\alpha}:\alpha\in\lambda\}$. Put a subset $X$ of $S_n$ in $J_n$ if:
> $X\cap[\lambda]^{<\aleph_0}$ is a subset of a union of finitely many elements of ${\cal T}$, and $X\cap L_n$ is bounded above.
Then the cofinality of $\langle J_n\rangle$ is $cof(L_n,<_n)$. Define ${\cal A}_n$ so that $X\in {\cal A}_n$ if: $$X\cap L_n=\{t\in L_n:t<z\} \mbox{for some $z\in L_n$.}$$ Then ${\cal A}_n$ is cofinal in $\langle J_n\rangle$.\
[**Claim 1:**]{} TWO does not have a winning $n$-tactic in $MG({\cal
A}_n,J_n)$.\
For let $\Phi$ be an $n$-tactic of TWO. For $x\in L_n$ put $V_x=[\lambda]^{<\aleph_0}\cup\{y\in L_n:y<_nx\}$. Define a partition $\Psi:[L_n]^{n}\rightarrow[\lambda]^{<\aleph_0}$ so that $$(\Phi(V_{x_1}) \cup \Phi(V_{x_1},V_{x_2})\cup \dots \cup
\Phi(V_{x_1},\dots,V_{x_n}))\cap
[\lambda]^{<\aleph_0}$$ is a subset of $\cup\{X_{\alpha}:
\alpha\in\Psi(\{x_{1},\dots,x_{n}\})\}$.
By $(1)$ we obtain an $\omega$-path $x_1<_nx_2<_n\ldots<_nx_k<_n\ldots$ and a finite set $F\subset\lambda$ such that $\Psi(x_{j+1},\ldots,x_{j+n})
\subseteq F$ for all $j$. For each $m$ we define: $O_m=[\lambda]^{<\aleph_0}\cup V_{x_m}$. Letting $(O_1,T_1,\ldots,O_k,T_k,\ldots)$ be the corresponding $\Phi$-play, we find that TWO has lost this play since $[\lambda]^{<\aleph_0}\cap(\cup_{m=1}^{\infty}T_m)\subseteq \cup_{\alpha\in
F} X_{\alpha}\not=[\lambda]^{<\aleph_0}.$
It follows that TWO does not have a winning $n$-tactic.\
[**Claim 2:**]{} TWO has a winning $n+1$-tactic in $MG({\cal
A}_n,J_n)$.\
First observe that $\cup_{\alpha\in F}X_{\alpha}=[\lambda]^{<\aleph_0}$ whenever $F$ is an infinite subset of $\lambda$.
Here is a definition of an $n+1$-tactic for TWO in this game: Let $\{t_{\alpha}:\alpha<\lambda\}$ enumerate $[\lambda]^{<\aleph_0}$ bijectively. Let $\Phi:[L_n]^{n+1}\rightarrow\lambda$ be a coloring which witnesses that $(L_n,<_n)\not\rightarrow(\omega-path)^{n+1}_{\lambda/<\omega}$. For each $X$ in ${\cal A}_n$ let $\phi_X$ be that element of $L_n$ for which $X\cap L_n=\{t\in L_n:t<\phi_X\}$.
For $U_1\subset\ldots\subset U_{n+1}$ elements of ${\cal A}_n$, observe that $\phi_{U_1}\leq\ldots\leq\phi_{U_{n+1}}$. For $X\subset Y$ sets in ${\cal
A}_n$ such that $X\cap[\lambda]^{<\aleph_0}\neq Y\cap[\lambda]^{<\aleph_0}$ we set $\Psi(X,Y)=\min\{\alpha:t_{\alpha}\in Y\backslash X\}$.
Let $U_1\subset\ldots\subset U_{n+1}\in {\cal A}_n$ be given. We define:\
1. [$G(U_1,\ldots,U_j)=\emptyset$ when $j<n+1$,]{}
2. [$G(U_1,\ldots,U_{n+1})=X_{\alpha}\cup (L_n\cap U_{n+1})$ when $\phi_{U_1}<\ldots<\phi_{U_{n+1}}$, and $\Phi(\{\phi_{U_1},\ldots,\phi_{U_{n+1}}\}) = \alpha$,]{}
3. [$G(U_1,\ldots,U_{n+1})=X_{\alpha}\cup (L_n\cap U_{n+1})$ where $\alpha$ is minimal such that $t_{\alpha}\in U_{i+1}\backslash U_i$ for some $i\leq n$, otherwise.]{}
We show that $G$ is a winning $n+1$-tactic for TWO. Thus, let $$(O_1,T_1,\ldots, O_m,T_m,\ldots)$$ be a $G$-play of the game. For typographical convenience we define:
1. [$x_i=\phi_{O_i}$ for each $i$, and]{}
2. [$\alpha_i=\Psi(O_i,O_{i+1})$ for each $i$ for which this is defined.]{}
There are two cases to consider.\
CASE 1: $\{i:x_i=x_{i+1}\}$ is finite.\
Choose $m$ such that $x_i<x_{i+1}$ for all $i\geq m$. Then the set $$\{\Phi(\{x_{m+k+1}, \ldots, x_{m+k+n+1}\}): k=1, 2, \ldots\}$$ is an infinite subset of $\lambda$ and it follows from 2. in the definition of $G$ that this play is won by TWO.\
CASE 2: $\{i:x_i=x_{i+1}\}$ is infinite.\
Then the set $\{i:\Psi(O_i,O_{i+1})\mbox{ is defined}\}$ is infinite. But then it follows from 3. in the definition of $G$ that TWO wins this play. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
The hypotheses of Theorem \[slightprogress\] are realized under any of the following axiomatic circumstances (one uses Corollary 27 and Proposition 29 of [@S2]):
1. [$2^{<{\frak c}}={\frak c}+EH$,]{}
2. [${\frak c}<2^{\aleph_1}$, i.e., the negation of $LH$ (Lusin’s second Continuum Hypothesis)]{}
3. [There is an infinite regular cardinal number $\kappa$ such that $2^{\kappa}=\kappa^+$.]{}
For the case when $\lambda=\omega$, the example constructed in the proof of Theorem \[slightprogress\] shows that hypothesis 2 of Theorem \[mgth\] is to some extent necessary. This is because:
1. [${\cal A_n}$ has the coherent decomposition property: For choose $\alpha_1<\alpha_2<\dots<\alpha_n<\dots$ from $\omega$, and set $T_m=X_{\alpha_1}\cup\dots\cup X_{\alpha_m}$ for each $m$. Then $[\omega]^{<\aleph_0}=\cup_{m=1}^{\infty}X_{\alpha_m}$, and $X_{\alpha_j}\subseteq X_{\alpha_i}$ for $j<i$. For $A\in {\cal A_n}$ we put $A_m=(A\cap T_m)\cup(A\cap L_n)$.]{}
2. [$({\cal
A}_m,\subset)\rightarrow(\omega-path)^m_{\omega/<\omega}$, but]{}
3. [$({\cal
A}_m,\subset)\not\rightarrow(\omega-path)^{m+1}_{\omega/<\omega}$.]{}
At this point it is an open problem whether the hypotheses (and for that matter the conclusion) of Theorem \[slightprogress\] are satisfied simply in the theory ZFC (see Problem 9 of [@S2]).
The game MG(J)
--------------
$MG(J)$ denotes the version of $MG({\cal A},J)$ where $\langle
J\rangle = {\cal A}$. In Problem 1 of [@S1] it was asked whether there is for each $k$ a free ideal $J_k$ such that TWO does not have a winning $k$-tactic in $MG(J_k)$, but does have a winning $k+1$-tactic in $MG(J_k)$. This problem is still open. In [@S1], Corollary 10, it was proven that TWO does not have a winning $2$-tactic in the game $MG(J_{{\Bbb R}})$, but that TWO has a winning $3$-tactic in $MG(J_{{\Bbb R}})$ if for example the Continuum Hypothesis is assumed. We now extend these results in two directions.
1. [In Problem 3 of that paper it was asked if player TWO has a winning $3$-tactic if instead of the Continuum Hypothesis one uses the theory $ZFC+MA+EH+\neg CH$, which is explained below. We now show that the answer is affirmative.]{}
2. [We identify circumstances under which TWO does not have a winning $k$-tactic in $MG(J_{{\Bbb R}})$ for any $k$; combining this with a consistency result of Todorcevic (given in the appendix), it follows that it is also consistent that there is no $k$ for which TWO has a winning $k$-tactic in $MG(J_{{\Bbb R}})$.]{}
It follows that the existence of a winning $k$-tactic for TWO in $MG(J_{{\Bbb R}})$ is not decided by the axioms of traditional set theory. One might now wonder if it is consistent that for example TWO does not have a winning $3$-tactic in $MG(J_{{\Bbb R}})$, but does have a winning $4$-tactic? This is not possible since a theorem of [@S3] implies that either TWO has a winning $3$-tactic, or else there is no $k$ such that TWO has a winning $k$-tactic in $MG(J_{{\Bbb R}})$.
Let $EH$ (which abbreviates [*Embedding Hypothesis*]{}) denote the statement: $$\mbox{{\em every linearly ordered set of cardinality $\leq {\frak c}$
embeds in }$(^{\omega}\omega,\ll)$.}$$ The hypothesis $EH$ is a consequence of the Continuum Hypothesis. Laver has proven ([@L]) that the theory $ZFC+EH+\neg CH$ is consistent, and Woodin ([@W], pp. 31-47), extending this, has proven the consistency of the theory $ZFC+MA+EH+\neg CH$. This theory implies that $2^{<\frak c}={\frak c}+EH$, which in turn is strong enough to prove that the partition relation $$({\cal
P}({\frak c}),\subset)\not\rightarrow(\omega-\mbox{path})^{3}_{\omega
/<\omega}$$ holds (see [@S2], top of p. 60). Thus we have:
\[problem3\] The theory $``ZFC+\neg CH+$ TWO has a winning 3-tactic in $MG(J_{{{\Bbb R}}})$" is consistent.
Proof.
: Consider any model of $ZFC+EH+\neg
CH+2^{<\frak c}={\frak c}$ in which $\langle J_{{\Bbb R}}\rangle$ has a cofinal chain. Let ${\EuScript C}$ denote this cofinal chain. By Theorem \[important\] we may assume that this cofinal chain has a coherent decomposition and that it satisfies the partition relation $({\EuScript C},\subset)\not\rightarrow(\omega-path)^2_{\omega/<\omega}.$ Since we also have $({\cal
P}({\frak c}),\subset)\not\rightarrow(\omega-\mbox{path})^{3}_{\omega
/<\omega}$ it follows that:
1. [$(\langle J_{{\Bbb R}}\rangle,\subset)\not \rightarrow
(\omega-path)^3_{\omega/<\omega}$, and]{}
2. [The family $\langle J_{{\Bbb R}}\rangle$ has a coherent decomposition.]{}
Theorem \[mgth\] implies that TWO has a winning $3$-tactic in $MG(\langle J_{{\Bbb R}}\rangle, J_{{\Bbb R}})$. This completes the proof of the proposition. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
Indeed, our proof of Proposition \[problem3\] shows more generally that if $J$ is a free ideal on a set of cardinality at most ${\frak c}$, and if $\JA$ has a cofinal chain and the coherent decomposition property, then the theory $ZFC+EH+2^{<{\frak c}}={\frak
c}$ proves that TWO has a winning $3$-tactic in $MG(J)$ . This generalizes Theorem 8(a) of [@S1].
Next we give hypotheses under which there is no $k$ for which TWO has a winning $k$-tactic in $MG(J_{{{\Bbb R}}})$. In the appendix we give a proof that these hypotheses are consistent with $ZFC$. This consistency result is due to Todorcevic.
\[noktactic\] Assume that $cof(J_{{{\Bbb R}}},\subset)=\lambda$ and that the partition relation $({\EuScript
P}({\frak c}),\subset)\rightarrow(\omega-path)^{3}_{\lambda/<\omega}$ holds. Then there is no $k$ for which TWO has a winning $k$-tactic in $MG(J_{{\Bbb R}})$.
Proof.
: Let $k$ as well as a $k$-tactic $F$ for $TWO$ be given. Let $X$ be a nowhere dense subset of cardinality ${\frak c}$ of ${{\Bbb R}}\backslash{{\Bbb Q}}$. Let ${\cal
A}=\{A_{\alpha}:\alpha<\lambda\}$ be a bijectively enumerated cofinal subfamily of $J_{{\Bbb R}}$.
Define a partition $\Phi:[{\EuScript
P}(X)]^{k}\rightarrow\lambda$ so that $$\Phi(\{X_{1},\cdots,X_{k}\})=\beta$$ where $\beta$ is minimal such that $$F({{\Bbb Q}}\cup X_{1})\cup\cdots\cup
F({{\Bbb Q}}\cup X_{1},\cdots,{{\Bbb Q}}\cup X_{k})\subset
A_{\beta}.$$
Since $({\EuScript
P}({\frak c}),\subset)\rightarrow(\omega-path)^{3}_{\lambda/<\omega}$, it follows that $({\EuScript
P}({\frak c}),\subset)\rightarrow(\omega-path)^{k}_{\lambda/<\omega}$ ( see [@S2], Proposition 36). Accordingly, choose a finite set $G\subset
\lambda$ and an increasing $\omega$-sequence $X_{1}\subset X_{2}\subset\cdots$ of subsets of $X$ such that $\Phi(\{X_{j+1},\cdots,X_{j+k}\})\in
G$ for all $j$. Put $O_{n}=X_{n}\cup{{\Bbb Q}}$ for all $n$. Let $B$ be the nowhere dense set $\cup\{A_{\alpha}:\alpha\in G\}$. Also define $T_{j}=F(O_{1}\cdots,O_{j})$ for $j\leq k$, and $T_{j+k}=F(O_{j+1},\cdots,O_{j+k})$ for all $j$. Then $$(O_{1},T_{1},O_{2},T_{2},\cdots)$$ is an $F$-play of $MG(J_{{\Bbb R}})$ for which ${{\Bbb Q}}\subset\cup_{n=1}^{\infty}O_{n}$ and $\cup_{n=1}^{\infty}T_{n}\subseteq B$. Since $B$ is nowhere dense, ${{\Bbb Q}}\backslash B\neq \emptyset$. It follows that TWO has lost this play. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
We now consider games of the form $MG([\kappa]^{<\lambda})$. In Proposition 15 of [@S1] it was shown that if TWO has a winning $k$-tactic in this game for some $k$, then TWO in fact has a winning $3$-tactic. It is not known if $``3"$ is optimal (this is Problem $7$ of [@S1]). It also follows from [@S1], Proposition $5$, that if $\lambda\rightarrow(\omega-path)^{2}_{\omega/<\omega}$, then TWO does not have a winning $k$-tactic in this game for any $k$. We now present slightly sharper results.
\[th:countablecof\] Let $\lambda$ be an uncountable cardinal number of countable cofinality. Let $k>1$ be an integer. The following statements are equivalent:
1. [Player TWO has a winning $k$-tactic in the game $MG([\lambda^{+}]^{<\lambda})$.]{}
2. [$([\lambda^+]^{\leq\lambda},\subset)\not\rightarrow
(\omega-path)^k_{\omega/<\omega}$.]{}
3. [$\lambda^{+}\not\rightarrow(\omega-path)^{2}_{\omega/<\omega}$ and $({\cal
P}(\lambda),\subset)\not\rightarrow(\omega-path)^{k}_{\omega/<\omega}$.]{}
Proof.
: By Theorem 1 and Proposition 15 of [@S1] we may assume that $k\in\{2,3\}$. Let $\lambda_1<\ldots<\lambda_n<\ldots$ be a sequence of cardinal numbers converging to $\lambda$.\
$1.\Rightarrow 2.$\
Let $F$ be a winning $k$-tactic for TWO in $MG([\lambda^+]^{<\lambda})$. Put $S=\lambda^+\backslash \lambda$. Define a coloring $\Phi:[[S]^{\leq\lambda}]^k\rightarrow\omega$ so that $$\Phi(X_1,\dots,X_k)=\min\{n:|F(\lambda\cup X_1,\dots,\lambda\cup
X_k|\leq \lambda_n\}.$$ Since $F$ is a winning $k$-tactic for TWO, $\Phi$ is a coloring which witnesses the partition relation in $2$.
$2.\Rightarrow 1.$\
According to Corollary \[cor:countablecof\], $([\lambda^+]^{<\lambda}, \subset)$ has the coherent decomposition property. Since $[\lambda^+]^{\leq\lambda}$ has a cofinal chain it follows that this family of sets itself has a coherent decomposition. The partition property in $2$ implies that the family $[\lambda^+]^{\leq\lambda}$ satisfies the hypotheses of Theorem \[mgth\]; thus TWO has a winning $k$-tactic in $MG([\lambda^+]^{<\lambda})$.
The equivalence of 2. and 3. is also easy to establish. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
Let $\lambda$ be an uncountable cardinal number of countable cofinality. Assume $ZFC+EH+\lambda<{\frak c}+{\frak c}=2^{<{\frak c}}$. Then TWO has a winning $2$-tactic in $MG([\lambda^{+}]^{<\lambda})$.
Proof.
: [The hypothesis $EH+{\frak c}=2^{<{\frak c}}$ implies that both $\lambda^{+}$ and $({\EuScript P}(\lambda),\subset)$ embed in $(^{\omega}\omega,\ll)$ for any $\lambda<{\frak c}$. It then follows from Corollary 13 of [@S2] that the partition relations in $3.$ of Theorem \[th:countablecof\] hold for $k=2$ for each $\lambda<{\frak c}$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
The game SMG(J)
---------------
For a free ideal $J$ on an infinite set $S$, the game $SMG(J)$ (read “strongly monotonic game on J") is defined so that an $\omega$-sequence $(O_{1},T_{1},\dots,O_{n},T_{n},\dots)$ is a play if for each $n$,
1. [$O_{n}\in\langle J\rangle$ is player ONE’s move in inning $n$,]{}
2. [$T_{n}\in J$ is player TWO’s move in inning $n$, and]{}
3. [$O_{n}\cup T_{n}\subseteq O_{n+1}$.]{}
Player TWO wins this play if $\cup_{n=1}^{\infty}O_{n}=\cup_{n=1}^{\infty}T_{n}.$
Throughout this section we assume that $\langle J\rangle$ is a proper ideal on $S$.
\[smgth\] Let $J\subset{\EuScript P}(S)$ be a free ideal and let ${\cal A}$ be a cofinal subfamily of $\langle J\rangle$ such that:
1. [TWO has a winning $k$-tactic in $MG({\cal A},J)$,]{}
2. there are functions $\Phi_1:\langle J\rangle\rightarrow J$ and $\Phi_2:\langle J\rangle\rightarrow{\cal A}$ such that:
1. [$A\subset \Phi_2(A)$ for each $A\in \langle J\rangle$, and]{}
2. [$\Phi_2(A)\subset\Phi_2(B)$ whenever $A\cup\Phi_1(A)\subseteq
B\in \langle J\rangle$.]{}
Then TWO has a winning $2$-tactic in $SMG(J)$.
Proof
: Let ${\cal A}$, $\Phi_1$ and $\Phi_2$ be as in the hypotheses. For each $A$ in $\langle J\rangle$ define $(A_1,\dots,A_k)$ so that $A_1=\Phi_2(A)$ and $A_{j+1}=\Phi_2(A_j)$ for each $j<k$. Also define: $\Psi(A)=\Phi_1(A)\cup\Phi_1(A_1)\cup\dots\cup \Phi_1(A_k)$.
Let $F$ be a winning $k$-tactic for TWO in $MG({\cal A},J)$. Define a $k$-tactic, $G$, for TWO as follows. Let $A\subset B$ be given.\
CASE 1: $G(A)=F(A_1)\cup\dots\cup F(A_1,\dots,A_k)\cup\Psi(A)$.\
CASE 2: If $A_k\subset B_1$, we let $G(A,B)$ be the set $$F(A_2,\dots,A_k,B_1)\cup
F(A_3,\dots,A_k,B_1,B_2)\cup\dots\cup F(B_1,\dots,B_k)\cup\Psi_1(B).$$
CASE 3: Otherwise we put $G(A,B)=G(B).$\
Then $G$ is a winning $2$-tactic for TWO in $SMG(J)$. For let $$(O_{1},T_{1},\dots,O_{n},T_{n},\dots)$$ be a play of $SMG(J)$ during which TWO followed the $2$-tactic $G$. For each $j$ we put $M_j^1=\Phi_2(O_j),\dots,M_j^k=\Phi_2(M_j^{k-1})$. An inductive computation shows that
- [$(M_1^1,M_1^2,\dots,M_1^k,M_2^1,M_2^2,\dots,M_2^k,\dots)$ is a sequence of legal moves for $ONE$ in the game $MG({\cal A},J)$, and that]{}
- 1. [$F(M_1^1)\cup\dots\cup F(M_1^1,\dots,M_1^k)\subseteq T_1$, and]{}
2. [$F(M_j^1,\dots,M_j^k)\cup F(M_j^2,\dots,M_j^k,M_{j+1}^1)\cup \dots
\cup F(M_j^k,M_{j+1}^1,\dots,M_{j+1}^{k-1})\\ \subseteq T_{j+1}$ for each $j$.]{}
Since $F$ is a winning $k$-tactic for TWO in the game $MG({\cal A},J)$, and since $\cup_{n=1}^{\infty}O_n\subseteq
\cup_{n=1}^{\infty}M_n^1$, TWO won the given play of $SMG(J)$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
The next corollary solves Problems 10 and 11 of [@S1].
\[corsmg\] Player $TWO$ has a winning 2-tactic in the game $SMG(J_{{{\Bbb R}}})$.
Proof.
: Fix, by Corollary \[cor:mgreals\], a cofinal family ${\cal A}\subset \langle J_{{\Bbb R}}\rangle$ such that TWO has a winning $2$-tactic in $MG({\cal A},J_{{\Bbb R}})$.
We define $\Phi_1:\langle J_{{\Bbb R}}\rangle\rightarrow J_{{\Bbb R}}$ and $\Phi_2:\langle J_{{\Bbb R}}\rangle\rightarrow {\cal A}$ as follows:\
Fix $X\in\langle J_{{\Bbb R}}\rangle$, and choose a sequence $(X_0,X_1,\dots, X_n,\dots)$ such that:
1. [$X_0=X$,]{}
2. [$X_{n+1}\in{\cal A}$ and ${\Bbb N}\cdot X_n\subseteq X_{n+1}$]{}
for each $n$. Put $\Phi_2(X)=\cup_{n=1}^{\infty}X_n.$
Fix $X\in\langle J_{{\Bbb R}}\rangle$ and let $\Phi_1(X)$ be a nowhere dense set for which $\Phi_2(X)\subset {\Bbb N}\cdot\Phi_1(X)$.
Then ${\cal A}$, $\Phi_1$ and $\Phi_2$ are as required by Theorem \[smgth\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[Th21\] For each of the ideals $J_n$ constructed in the proof of Theorem \[slightprogress\], TWO has a winning $2$-tactic in $SMG(J_n)$.
Proof.
: [Let ${\cal A}_n$ be as in the proof of Theorem \[slightprogress\]. For each $X\in\langle J_n\rangle$ we let $\Phi_2(X)$ be an element of ${\cal A}_n$ which contains it, and we let $\Phi_1(X)=\{a_X\}$ where $a_X\in L_n\backslash\Phi_2(X)$. Then ${\cal A}_n$, $\Phi_1$ and $\Phi_2$ are as required by Theorem \[smgth\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
Before giving another application of Theorem \[smgth\] we give an example of free ideals $J$ which show that TWO does not always have a winning $k$-tactic in the game $SMG(J)$ for some $k$. These examples are also relevant to the material of the next section. The symbol $M(\omega,2)$ denotes the smallest ordinal $\alpha$ for which the partition relation $\alpha\rightarrow(\omega-path)^{2}_{\omega/<\omega}$ holds. $M(\omega,2)$ is a regular cardinal less than or equal to ${\frak c}^{+}$. It in fact satisfies the partition relation $M(\omega,2)\rightarrow(\omega-path)^{n}_{\omega/<\omega}$ for all $n$. Let $\kappa$ be an initial ordinal number. It is consistent that $M(\omega,2)$ is equal to $\aleph_{2}$ while ${\frak c}$ is larger than $\kappa$ (this is yet another result of Todorcevic).
\[counterexample\]Let $\lambda$ be a cardinal number of countable cofinality and let $\kappa$ be a cardinal number larger than $\lambda$. If $M(\omega,2) \leq \lambda^+$, then there is no $k$ such that player TWO has a winning $k$-tactic in $SMG([\kappa]^{<\lambda})$.
Proof.
: Let $F$ be a $k$-tactic for TWO.
Player ONE’s counter-strategy will be to play judiciously chosen subsets from $\kappa$. We first single out those sets from which ONE will make moves.
Choose sets $S_0\subset S_1\subset\dots\subset S_{\alpha}\subset
\dots\in[\kappa]^{\lambda}$ for $\alpha<\lambda^+$ such that:
1. [$\lambda\subset S_0$,]{}
2. [$\cup\{F(S_{i_{1}},\dots,S_{i_j}):j\leq k,\
i_1<\dots<i_j<\alpha\}\subset S_{\alpha}$ for each $0<\alpha<\lambda^+$.]{}
Now let $\lambda_{1}<\lambda_{2}<\dots<\lambda$ be an increasing sequence of regular cardinal numbers converging to $\lambda$. Define a function $
\Gamma:[\lambda^+]^{k}\rightarrow\omega
$ so that $$\Gamma(\xi_{1},\dots,\xi_{k})=\min\{m:|F(S_{\xi_{1}},\dots,S_{\xi_{k}})|
\leq\lambda_{m}\}.$$
Then, on account of the relation $M(\omega,2)\leq\lambda^{+}$, choose an $m<
\omega$ and a sequence $\alpha_{k+1}<\dots<\alpha_{k+m}<\dots$ from $\lambda^+$ such that $\Gamma(\alpha_{j+1},\dots,\alpha_{j+k})\leq m$ for all $j$.
Consider the sequence $$(S_{\alpha_{1}}, F(S_{\alpha_{1}}),\dots,S_{\alpha_{k}},
F(S_{\alpha_{1}},\dots,S_{\alpha_{k}}),\dots,S_{\alpha_{k+m}},
F(S_{\alpha_{1+m}},\dots,S_{\alpha_{k+m}}),\dots).$$ It is a play of the game $SMG([\kappa]^{<\lambda})$ during which TWO used the $k$-tactic $F$. To see that TWO lost this play, let $T_{k}$ denote TWO’s $k$-th move. The choice of the sequence $\alpha_{k+1}<\dots$ implies that $\cup_{n=1}^{\infty}T_{n}$ has cardinality less than $\lambda$. The union of the sets played by ONE has cardinality $\lambda$; TWO didn’t catch up with ONE. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[cor:countablecof4\] For $\omega=cof(\lambda)\leq\lambda<\kappa$ cardinal numbers with $cof([\kappa]^{\leq\lambda},\subset)=\kappa$, the following statements are equivalent:
1. [TWO has a winning $2$-tactic in $SMG([\kappa]^{<\lambda})$.]{}
2. [$\lambda^+\not\rightarrow(\omega-path)^2_{\omega/<\omega}$.]{}
Proof.
: It follows from Theorem \[counterexample\] that $1.$ implies $2.$\
That $2.$ implies $1.$:\
By the cofinality hypothesis and by $2.$ we find, according to Corollary \[cor:countablecof2\], a well-founded cofinal family ${\cal A}$ such that TWO has a winning $2$-tactic in $MG({\cal
A},[\kappa]^{<\lambda})$. We may assume that there is an enumeration $\{A_{\alpha}:\alpha<\kappa\}$ of ${\cal A}$ for which $\alpha\in
A_{\alpha}$ for each $\alpha$. Define $\Phi_1$ and $\Phi_2$ as follows:\
For $X\in[\kappa]^{\leq\lambda}$ define a sequence $(X_0,\dots,X_m,\dots)$ such that:
1. [$X_0=X$, and]{}
2. [$X_{n+1}=\cup_{\alpha\in X_n}A_{\alpha}$]{}
for each $n$.
Choose $\Phi_2(X)\in{\cal A}$ such that $\cup_{n<\omega}X_n\subseteq\Phi_2(X)$.\
Pick $z_X\in(\kappa\backslash\Phi_2(X))$ and pick $\rho_X$ minimal such that $\rho_X\not\in\Phi_2(X)$, and $\Phi_2(X)\subset A_{\rho_X}$. Put $\Phi_1(X)=\{z_X, \rho_X\}$.\
Then ${\cal A}$, $\Phi_1$ and $\Phi_2$ are as required by Theorem \[smgth\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
This result will be discussed at greater length after Theorem \[3tacticth\].
We finally mention that it is still unknown whether there is for each $m$ a free ideal $J_m$ such that TWO does not have a winning $m$-tactic, but does have a winning $m+1$-tactic in $SMG(J_m)$. This is Problem 9 of [@S1]. In this connection it is worth noting the following relationship between winning $k$-tactics in $MG(J)$ and winning $m$-tactics in $SMG(J)$. The proof uses ideas as in the proof of Theorem \[smgth\].
If TWO has a winning $k$-tactic in $MG(J)$, then TWO has a winning $2$-tactic in $SMG(J)$.
The game VSG(J)
---------------
For a free ideal $J$ on an infinite set $S$, the game $VSG(J)$ (read “ very strong game on J") is defined so that an $\omega$-sequence $(O_{1},(T_{1},S_{1}),\dots,O_{n},(T_{n},S_{n}),\dots)$ is a play if for each $n$,
1. [$O_{n}\in\langle J\rangle$ is player ONE’s move in inning $n$,]{}
2. [$(T_{n},S_{n})\in J\times\langle J\rangle$ is player TWO’s move in inning $n$, and]{}
3. [$O_{n}\cup T_{n}\cup S_{n}\subseteq O_{n+1}$.]{}
Player TWO wins this play if $\cup_{n=1}^{\infty}O_{n}=\cup_{n=1}^{\infty}T_{n}.$
We assume for this section that $\langle J\rangle$ is also a proper ideal on $S$. Given a cofinal family ${\cal A}\subset\langle
J\rangle$, we may assume whenever convenient that ONE is playing from ${\cal A}$ in the game $VSG(J)$. It is clear that if TWO has a winning $k$-tactic in $SMG(J)$, then TWO has a winning $k$-tactic is $VSG(J)$. The converse is not so clear.
Let $J$ be a free ideal on a set $S$ and let $k$ be a positive integer. Is it true that if TWO has a winning $k$-tactic in $VSG(J)$, then TWO has a winning $k$-tactic in $SMG(J)$?
In the next theorem we find a partial converse.
\[SMG,VSG\] Let $J$ be a free ideal on a set $S$ and let $k$ be a positive integer. If $add(\langle J\rangle,\subset) = cof(\langle
J\rangle,\subset)$, then the following statements are equivalent:
1. [TWO has a winning $2$-tactic in $SMG(J)$.]{}
2. [TWO has a winning $k$-tactic in $SMG(J)$.]{}
3. [TWO has a winning $k$-tactic in $VSG(J)$.]{}
Proof
: That $1.$ and $2.$ are equivalent: This is Theorem 19 of [@S1].\
That $2.$ implies $3.$: Let $F$ be a winning $k$-tactic for TWO in $SMG(J)$. Define $G$ so that $$G(A_1,\dots,A_j)=(F(A_1,\dots,A_j),A_j\cup F(A_1,\dots,A_j))$$ for $j\leq k$. Then $G$ is a winning $k$-tactic for TWO in $VSG(J)$.\
That $3.$ implies $2.$: Let $G$ be a winning $k$-tactic for TWO in $VSG(J)$. Then choose a sequence $(M_\xi:\xi<cof(\langle J\rangle,\subset))$ such that:
1. [$M_\xi\subset M_\nu$ for $\xi<\nu<cof(\langle J\rangle,\subset)$ and]{}
2. [$\{M_\xi:\xi<cof(\langle J\rangle,\subset)\}$ is cofinal in $\langle
J\rangle$.]{}
Now $cof(\langle J\rangle,\subset)$ is a regular uncountable cardinal number. We may thus further assume that the sequence $(M_\xi:\xi<cof(\langle
J\rangle,\subset))$ has been chosen such that if $(U,T)=G(M_{\xi_1},\dots,
M_{\xi_j})$, then $U\cup T\subset M_{\eta}$ for all $\xi_j<\eta< cof(\langle
J\rangle,\subset)$.
For each $X\in \langle J\rangle$ define $\alpha(X)=\min\{\xi:X\subset
M_\xi\}$. For each $\xi$ choose $z_\xi\in S\backslash M_\xi$. We now define a $k$-tactic, $F$, for TWO in $SMG(J)$.
Let $X_1\subset\dots\subset X_j\in\langle J\rangle$ for a $j\leq k$ be given.\
CASE 1:$\alpha(X_1)<\dots<\alpha(X_j)$. Let $(U,T)=G(M_{\alpha(X_1)},\dots,
M_{\alpha(X_j)})$ and define $F(X_1,\dots,
X_j)=U\cup\{z_{\alpha(X_j)+1}\}$.\
CASE 2: Otherwise, set $F(X_1,\dots, X_j)=\{z_{\alpha(X_j)+1}\}$. Then $F$ is a winning $k$-tactic for TWO in $SMG(J)$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
There is the following analogue of Theorem \[smgth\] for the very strong game:
\[mgtovsg\]Let $J$ be a free ideal on a set $S$. If there is a cofinal family ${\cal A}\subset\langle J\rangle$ such that TWO has a winning $k$-tactic in $MG({\cal A},J)$, then TWO has a winning $2$-tactic in $VSG(J)$.
Proof.
: Let ${\cal A}\subset\langle J\rangle$ be a cofinal family such that TWO has a winning $k$-tactic in $MG({\cal A},J)$. We will define a winning 2-tactic for TWO for the game $VSG(J)$. To this end, choose a winning $k$-tactic, $F$, for TWO for the game $MG({\cal A},J)$. For each $X\in\langle J\rangle$ choose a set $A_1(X)\subset\dots\subset A_k(X)$ from ${\cal A}$ such that $X\subset A_1(X)$, and choose $\Psi(X)$ from ${\cal A}$ such that $A_k(X)\subset\Psi(X)$.
Let $X\subset Y$ be sets from $\langle J\rangle$.\
CASE 1: $G(X)=(F(A_1(X))\cup\dots\cup
F(A_1(X),\dots,A_k(X)),\Psi(X))$.\
CASE 2: Define $G(X,Y)$ so that:
1. [ $G(X,Y)= (F(A_2(X),\dots,A_k(X),A_1(Y) \cup\dots\cup F(A_1(Y),\dots,A_k(Y)), \Psi(Y))$ if $\Psi(X)\subset Y$, and]{}
2. [$G(X,Y)= G(Y)$ otherwise.]{}
Then $G$ is a winning $2$-tactic for TWO in $VSG(J)$. For let $(O_{1},(T_{1},S_{1}),O_{2},(T_{2},S_{2}),\dots)$ be a play of $VSG(J)$ such that $(T_1,S_1)=G(O_1)$ and $(T_{n+1},S_{n+1})=G(O_n,O_{n+1})$ for all $n$. Then $S_n=\Psi(O_n)$ and $A_k(O_n)\subset A_1(O_{n+1})$ for each $n$. An inductive computation, using this information, shows that TWO won this play of $VSG(J)$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
Combining Theorem \[SMG,VSG\] and Theorem \[counterexample\] we see that TWO does not always have a winning $k$-tactic in games of the form $VSG(J)$. Combining Theorem \[SMG,VSG\] and Corollary \[cor:countablecof4\] we obtain another game-theoretic characterization of the partition relation $\lambda^{+}\rightarrow(\omega-path)^{2}_{\omega/<\omega}$ when $\lambda$ is an uncountable cardinal of countable cofinality.
Analogous to the case of the ideal of countable subsets of an infinite set, there is for each uncountable cardinal number $\lambda$ which is of countable cofinality, a proper class of cardinals $\kappa$ for which the ideal $[\kappa]^{\leq\lambda}$ has the irredundancy property. It is also a consequence of $MA+{\frak c}>\lambda$ that the partition relation $\lambda^{+} \not \rightarrow(\omega-path)^{2}_{\omega/<\omega}$ holds. Accordingly it is consistent that there is a proper class of cardinals $\kappa$ such that TWO has a winning $2$-tactic in the game $VSG([\kappa]^{\leq\lambda})$. The following problem (to be compared with the upcoming Conjecture \[conj:countable-finite\]) is open.
Let $\lambda$ be an uncountable cardinal of countable cofinality. Is it true that if TWO has a winning $2$-tactic in the game $VSG([\lambda^{+}]^{<\lambda})$, then TWO has a winning $2$-tactic in $VSG([\kappa]^{<\lambda})$ for all $\kappa>\lambda$?
Our next theorem (Theorem \[3tacticth\]) applies to abstract free ideals whose $\sigma$-completions have small principal bursting number. It is not clear to us whether “3" occurring in Theorem \[3tacticth\] is optimal. One of its applications is that ZFC+GCH implies that TWO has a winning 3-tactic in $VSG([\kappa]^{<\aleph_0})$ for all $\kappa$. It is [*very*]{} likely that the “3” appearing in this application is not optimal, as will be discussed later.
\[3tacticth\]Let $J$ be a free ideal such that:
1. [$bu(\langle J\rangle,\subset) = \aleph_{2}$,]{}
2. [$add(\langle J\rangle,\subset) = \aleph_{1}$,]{}
3. [$cof(\langle J\rangle,\subset) = \lambda$ and]{}
4. [$[\lambda]^{<\aleph_0}$ has the coherent decomposition property.]{}
Then player TWO has a winning $3$-tactic in $VSG(J)$.
Proof.
: Let $J$ be a free ideal (on a set $S$) as in the hypotheses. Let ${\cal A}$ be a well-founded cofinal family of cardinality $\lambda$, such that $|\{B\in{\cal
A}:B\subseteq A\}|\leq\aleph_1$ for each $A\in{\cal A}$.
For each $A\in{\cal A}$ fix $\nu_A\leq\omega_{1}$ and a bijective enumeration $\{J_{\xi}(A):\xi < \nu_A\}$ of the set $\{X\in
{\cal A}:X\subseteq A\}$.
Choose a sequence $(C_{\xi}:\xi<\omega_{1})$ from $\langle J\rangle$ such that:
1. [$C_{\xi}\subset C_{\nu}$ for $\xi < \nu$ and]{}
2. [$\cup_{\xi<\omega_{1}}C_{\xi}\not\in \langle J\rangle$.]{}
For $A\in{\cal A}$ define $\xi_A=\min\{\xi<\omega_1:C_{\xi}\not\subseteq A\}$.
For $A\subset B$ elements from ${\cal A}$, define a set $\tau(A,B)$ such that $(S_{1},\dots,S_{n})$ is in $\tau(A,B)$ if:
1. [$2\leq n <\omega$,]{}
2. [$S_{1}=B$ and $S_{2}=A$,]{}
3. [$S_{j+1}\in\{J_{\xi}(S_{j}):\xi<\nu_{S_j}\mbox{ and }
C_{\xi}\subset S_{j-1}\}$ for $2\leq j< n$.]{}
For $(S_{1},\dots,S_{n})$ and $(T_{1},\dots,T_{m})$ in $\tau(A,B)$ define $(S_{1},\dots,S_{n}) < (T_{1},\dots,T_{m})$ if $n<m$ and $(S_{1},\dots,S_{n})=(T_{1},\dots,T_{n})$. Then $(\tau(A,B),<)$ is a tree. Each branch of this tree is finite since $({\cal A},\subset)$ is well-founded. Indeed, $\tau(A,B)$ is a countable set.
Define $F(A,B)$ to be the set of $X\in {\cal A}$ such that $X\in
\{S_{1},\dots,S_{m}\}$ for some $(S_{1},\dots,S_{m})\in \tau(A,B)$. Then $F(A,B)$ is a countable set. Notice that if $C\subset A\subset B$ are elements of ${\cal A}$ such that $C\in
\{J_{\xi}(A):\xi\leq\nu_A \mbox{ and } C_{\xi}\subset B\}$, then $F(C,A)\subset F(A,B)$.
Let ${\cal B}\subset [{\cal A}]^{\aleph_0}$ be cofinal, well-founded and with the coherent decomposition property. For each $B\in{\cal B}$ choose a decomposition $B=\cup_{n=1}^{\infty}B^n$ where each $B^n$ is finite, and these decompositions satisfy the coherent decomposition requirement. By Proposition 15 of [@S2] we also fix a function $${\cal K}:[{\cal B}]^2\rightarrow\omega$$ which witnesses that $({\cal
B},\subset)\not\rightarrow(\omega-path)^2_{\omega/<\omega}$.
Define $\Phi_1:[{\cal A}]^2\rightarrow{\cal B}$ such that
$$(\cup\{F(X,Y):(\exists(S_1,\dots,S_n)\in\tau(A,B))(X\subset Y \mbox{
and }X,Y\in\{S_1,\dots,S_n\})\})$$
is a subset of $\Phi_1(A,B).$ Also define $\Phi_2:[{\cal
A}]^2\rightarrow{\cal A}$ such that
$$C_{\xi}\cup C_{\xi_B}\cup(\cup\Phi_1(A,B))\subset\Phi_2(A,B)$$
where now $A=J_{\xi}(B)$.
Note that if $A, \ B,$ and $C$ are elements of ${\cal A}$ such that $A\subset B\subset \Phi_2(A,B)\subset C$, then $\Phi_1(A,B)\subset\Phi_1(B,C)$.
Finally, choose for each $A\in {\cal A}$ a $\Phi_3(A)\in{\cal A}$ such that $A\cup C_{\xi_A}\subseteq \Phi_3(A)$.
Choose for each $A\in {\cal A}$ a sequence of sets $A^{0}\subseteq\dots,A^{n}\subseteq\dots$ such that each $A^{i}$ is in $J$ and $A=\cup_{n=0}^{\infty}A^{n}$.
We now define a $3$-tactic for TWO: First note that for the very strong game we may make the harmless assumption that player ONE’s moves are all from the cofinal family ${\cal A}$. Let $A\subset
B\subset C$ be sets from ${\cal A}$. Here are player TWO’s responses ${\cal F}(A)$, ${\cal F}(A,B)$ and ${\cal F}(A,B,C)$:\
Case 1: ${\cal F}(A)=(\emptyset,\Phi_3(A))$\
Case 2: ${\cal F}(A,B)=(\emptyset,\Phi_2(A,B))$\
Case 3: ${\cal F}(A,B,C)=(D,\Phi_2(B,C))$\
if $\Phi_2(A,B)\subseteq C$, where $D=C^m_1\cup\dots\cup C^m_r$ is given by: $m\geq {\cal K}(\{\Phi_1(A,B),\Phi_1(B,C)\})$ is minimal such that $(\Phi_1(A,B))^n\subseteq(\Phi_1(B,C))^n$ for all $n\geq m$, and $(\Phi_1(B,C))^m=\{C_1,\dots,C_r\}$.\
Case 4: In all other cases define ${\cal F}(A,B,C)={\cal
F}(B,C)$.\
To see that ${\cal F}$ is a winning $3$-tactic for TWO, consider a play $$(O_{1},(T_{1},S_{1}),O_{2},(T_{2},S_{2}),\dots)$$ of $VSG(J)$ for which
1. [$(T_{1},S_{1})={\cal F}(O_{1})$,]{}
2. [$(T_{2},S_{2})={\cal F}(O_{1},O_{2})$ and]{}
3. [$(T_{n+3},S_{n+3})={\cal F}(O_{n+1},O_{n+2},O_{n+3})$]{}
for all $n$.
Then $T_1=T_2=\emptyset$, $S_1=\Phi_3(O_1)$, $S_2=\Phi_2(O_1,O_2)$ and $S_{n+1}=\Phi_2(O_n,O_{n+1})$ for all $n\geq 2$. From the fact that $O_n\supseteq S_{n-1}$ for all $n\geq 2$ it follows that $$O_1\subset O_2\subset\Phi_2(O_1,O_2)\subseteq O_3\subset
\Phi_2(O_2,O_3)\subseteq O_4\subset\dots,$$ whence $\Phi_1(O_1,O_2)\subset\Phi_1(O_2,O_3)\subset\Phi_1(O_3,O_4)\subset\dots$. For each $k$ let $m_k$ denote the minimal integer such that
1. [${\cal K}(\{\Phi_1(O_k,O_{k+1}),\Phi_1(O_{k+1},O_{k+2})\})\leq
m_k$ and]{}
2. [$(\Phi_1(O_k,O_{k+1}))^n\subseteq(\Phi(O_{k+1},O_{k+2}))^n$ for all $n\geq m_k$.]{}
From the properties of ${\cal K}$ it follows that there are infinitely many $k$ such that $m_j<m_k$ for each $j<k$. Fix $i$, and fix the smallest $j\geq i$ such that $O_i\in\Phi_1(O_j,O_{j+1})$. Then let $t$ be minimal such that $O_i\in(\Phi_1(O_j,O_{j+1}))^t$. Then for each $k$ such that $m_{\ell}<m_k$ for each $\ell<k$, and $t<m_k$, $O^{m_k}_i\subseteq T_k$. It follows that $O_i\subseteq\cup_{n=1}^{\infty}T_n$. From this it follows that TWO won this ${\cal F}$-play of $VSG(J)$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[vsg:countablefinite\] For every infinite cardinal number $\kappa$, TWO has a winning $3$-tactic in $VSG([\kappa]^{<\aleph_0})$.
The results of Corollaries \[cor:countablecof4\] and \[vsg:countablefinite\] should be compared with those of Koszmider [@Ko] for the game $MG([\kappa]^{<\aleph_{0}})$. In Corollary \[cor:countablecof4\] we show that there is a proper class of $\kappa$ such that TWO has a winning $2$-tactic in $SMG([\kappa]^{<\aleph_0})$, and thus in $VSG([\kappa]^{<\aleph_0})$. This class includes $\aleph_n$ for all $n<\omega$. In [@Ko] it is proven that TWO has a winning 2-tactic in $MG([\aleph_{n}]^{<\aleph_{0}})$ for all $n\in \omega$ ([@Ko], Theorem 18). Under the additional set theoretic assumption that both $\square_{\lambda}$ holds and $\lambda^{\aleph_{0}}=\lambda^{+}$ for all uncountable cardinal numbers $\lambda$ which are of countable cofinality, Koszmider further proves that player TWO has a winning 2-tactic in $MG([\kappa]^{<\aleph_{0}})$ for all $\kappa$ ([@Ko], Theorem 19). In light of these results it is consistent that TWO has a winning 2-tactic in the game $SMG([\kappa]^{<\aleph_{0}})$ and thus in the game $VSG([\kappa]^{<\aleph_0})$ for all $\kappa$.
All this evidence leads us to believe that one could prove (without recourse to additional set theoretic hypotheses) that player TWO has a winning 2-tactic in the game $VSG([\kappa]^{<\aleph_{0}})$ for all infinite $\kappa$. We suspect even more: that TWO has a winning 2-tactic in $SMG([\kappa]^{<\aleph_0})$ for all $\kappa$. We state this formally as a conjecture:
\[conj:countable-finite\] Player TWO has a winning 2-tactic in the game $SMG([\kappa]^{<\aleph_{0}})$ for each infinite cardinal number $\kappa$.
One can modify the proof of Theorem \[3tacticth\] to obtain the following result:
\[3tacticthgeneralized\]Let $J$ be a free ideal on a set $S$ such that
1. [$bu(\langle J\rangle,\subset)=\aleph_n$ for some finite $n$,]{}
2. [there is an $(\omega_k,\omega_k)$-pseudo Lusin set in $(\langle
J\rangle,\subset)$ for each $k\in\{1,\ldots,n\}$,]{}
3. [$cof(\langle J\rangle,\subset)=\lambda)$, and]{}
4. [$([\lambda]^{<\aleph_0},\subset)$ has the coherent decomposition property.]{}
Then player TWO has a winning $n+1$-tactic in $VSG(J)$.
We now give an example which shows, assuming the Continuum Hypothesis, that the hypothesis that $add(\langle J\rangle,\subset)=
\aleph_1$ of Theorem \[3tacticth\] is necessary (see Corollary \[cor:counterexample\]).
Let $\omega_{\alpha}$ be the initial ordinal corresponding with ${\frak c}$. Then there is a free ideal $J\subset{\EuScript P}(\omega_{\alpha+1})$ such that $cof(\langle
J\rangle,\subset)=\aleph_{\alpha+1}$ and there is no positive integer $k$ for which TWO has a winning $k$-tactic in $VSG(J)$.
Proof
: Define $J\subset{\EuScript
P}(\omega_{\alpha+1})$ such that $X\in J$ if, and only if, $|X|\leq\aleph_{\alpha}$ and $X\cap\omega$ is finite. Then $cof(\langle J\rangle,\subset)=add(\langle J\rangle,\subset)=
\omega_{\alpha+1}.$ By Theorem \[SMG,VSG\] it suffices to show that TWO doesn’t have a winning $2$-tactic in $SMG(J)$.\
Let $F$ be a $2$-tactic for TWO in $SMG(J)$. For $\omega<\eta<\omega_{\alpha+1}$ put $\phi(\eta)=\sup(\eta\cup F(\eta)).$ Let $C\subseteq\omega_{\alpha+1}\backslash(\omega+1)$ be a closed unbounded set such that $\phi(\gamma)<\beta$ whenever $\gamma<\beta$ are in $C$.\
For each $\eta\in C$ define $\phi_{\eta}:C\backslash(\eta+1)\rightarrow \omega_{\alpha+1}$ so that $\phi_{\eta}(\beta)=\sup(\beta\cup F(\eta,\beta))$ for all $\beta$. Then choose a closed, unbounded set $C_{\eta}\subseteq C\backslash(\alpha+1)$ such that $\phi_{\eta}(\beta)<\gamma$ whenever $\beta<\gamma$ are in $C_{\eta}$.
Let $D$ be the diagonal intersection of $(C_{\eta}:\eta\in C)$; i.e., $D=\{\xi\in C:\xi\in\cap\{C_{\eta}:\eta<\xi \mbox{ and }\eta\in C\}$. Then $D$ is an unbounded subset of $\omega_{\alpha+1}$. Now observe that if $\eta_1<\eta_2<\eta_3$ are elements of $D$, then
1. [$\eta_2\in C_{\eta_1}$,]{}
2. [$\eta_3\in C_{\eta_1}\cap C_{\eta_2}$, and thus]{}
3. [$F(\eta_1)\subseteq\eta_2$ and $F(\eta_1,\eta_2)\subseteq
\eta_3$.]{}
Define $\Phi:[D]^{2}\rightarrow\omega$ so that $$\Phi(\eta,\beta)=\max(\omega\cap(F(\eta)\cup F(\eta,\beta))).$$ By the Erdös-Rado theorem we obtain an $n<\omega$ and an uncountable $X\subset D$ such that $\Phi(\eta,\beta)=n$ for all $\eta<\beta\in X$. Pick $\eta_1<\eta_2<\dots<\eta_m<\dots$ from $X$ and put $O_n=\eta_n$ for each $n$. Put $T_1=F(O_1)$ and $T_{n+1}=F(O_n,O_{n+1})$ for each $n$.
Then $(O_1,T_1,\dots,O_n,T_n,\dots)$ is an $F$-play of $SMG(J)$ which is lost by TWO. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[cor:counterexample\] Assume the Continuum Hypothesis. Then there is a free ideal $J\subset {\EuScript P}(\omega_2)$ such that $cof(\langle J\rangle,\subset)=\aleph_2$, and there is no positive integer $k$ for which $TWO$ has a winning $k$-tactic in $VSG(J)$.
We don’t know if there is for each $m$ a free ideal $J_m$ such that TWO does not have a winning $m$-tactic, but does have a winning $m+1$-tactic in $VSG(J_m)$.
Is there for each $m$ a free ideal $J_m$ such that TWO does not have a winning $m$-tactic, but does have a winning $m+1$-tactic in $VSG(J_m)$?
The Banach-Mazur game and an example of Debs
--------------------------------------------
The Banach-Mazur game is defined as follows for a topological space $(X,\tau)$. Players ONE and TWO alternately choose nonempty open subsets from $X$; in the n-th inning player ONE first chooses $O_{n}$ and TWO responds with $T_{n}$. An inning is played for each positive integer. The sets chosen by the players must satisfy the rule $$O_{n+1}\subseteq T_{n}\subseteq O_{n}$$ for all n. Player TWO wins the play $$(O_{1},T_{1},\dots,O_{n},T_{n},\dots)$$ if the intersection of these sets is nonempty; otherwise player ONE wins. Following Galvin and Telgarsky [@G-T], we denote this game by $BM(X,\tau)$. In the early 1980’s Debs [@D] solved Problem 3 of [@F-K] by giving examples of topological spaces $(X,\tau)$ for which player TWO has a winning strategy in the game $BM(X,\tau)$, but no winning $1$-tactic. In all but one of Debs’ examples it was known (in $ZFC$) that TWO has a winning $2$-tactic. We show here that also for the remaining example player TWO has a winning $2$-tactic (Corollary \[vstobm\]). This was previously known under the assumption of some additional hypotheses.
This result eliminates this example as a candidate for providing evidence (consistent, modulo $ZFC$) towards the following conjecture of Telgarsky:
For each positive integer $k$ there is a topological space $(X_{k},\tau_{k})$ such that TWO does not have a winning $k$-tactic, but does have a winning $k+1$-tactic in the game $BM(X_{k},\tau_{k})$.
The following unpublished result of Galvin is the only theorem known to us which gives general conditions under which TWO has a winning $2$-tactic if TWO has a winning strategy in the Banach-Mazur game:
Let $(X,\tau)$ be a topological space for which TWO has a winning strategy in the Banach-Mazur game. If this space has a pseudo base ${\EuScript P}$ with the property that
- [$|\{V\in{\EuScript P}:B\subseteq V\}|<s(B)$ for each $B$ in ${\EuScript P}$,]{}
then TWO has a winning $2$-tactic.
Here the cardinal number $s(B)$ is defined to be the minimal $\kappa$ such that $B$ does not contain a collection of $\kappa$ pairwise disjoint nonempty open subsets; it is said to be the [ *Souslin number*]{} of $B$.
This subsection is organised as follows. We first prove a theorem concerning $k$-tactics in the Banach-Mazur game which is analogous to Theorem 5 of [@G-T]. It provides an equivalent formulation of Telgarsky’s conjecture which allows player TWO slightly more information: TWO may also remember the inning number. After this we give our result on Debs’ example.
### Markov $k$-tactics.
Whereas a $k$-tactic for player TWO remembers at most the latest $k$ moves of the opponent, a strategy for TWO which remembers in addition to this information also the number of the inning in progress is called a [ *Markov $k$-tactic*]{}. This choice of terminology is by analogy with the terminology “tactic" (used by Choquet [@C], p. 116, Definition 7.11 for what we call a $1$-tactic) and “Markov strategy" (used by Galvin and Telgarsky [@G-T], p. 52 for what we call a Markov $1$-tactic). A k-tactic is the special case of a Markov k-tactic where the inning number is ignored by the player.
Note that if $(X,\tau)$ has a dense set of isolated points then player TWO has a winning $1$-tactic in $BM(X,\tau)$. Thus we may assume that if at all possible, player ONE will avoid playing an open set which contains an isolated point. From the point of view of $k$-tactics for TWO we may therefore restrict our attention to topological spaces without isolated points. By the following proposition we may further restrict our attention to topological spaces in which each nonempty open set contains infinitely many pairwise disjoint open subsets.
\[filterdecomp\]Let $(X,\tau)$ be a topological space with no infinite set of pairwise disjoint open subsets. Then there is a positive integer $n$ such that: $$\tau\backslash\{\emptyset\}=\tau_{1}\cup\dots\cup\tau_{n}$$ where each $\tau_{i}$ has the finite intersection property.
Proof.
: Claim 1: There is a positive integer n such that every collection of pairwise disjoint nonempty open subsets is of cardinality $\leq n$.
Proof of Claim 1: This is a well known fact: see e.g. [@C-N], Lemma 2.10, p. 31. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
Now let $n$ be the minimal positive integer satisfying Claim 1. Let ${\cal U}=\{U_{1},\dots,U_{n}\}$ be a collection of pairwise disjoint nonempty open subsets of the space. Then ${\cal U}$ is a maximal pairwise disjoint family.
For $1\leq i\leq n$, let $\tau_{i}$ be a maximal family of nonempty open sets such that:
1. [$U_{i}\in\tau_{i}$,]{}
2. [any two elements of $\tau_{i}$ have nonempty intersection.]{}
Claim 2: $\tau\backslash\{\emptyset\}=\tau_{1}\cup\dots\cup\tau_{n}$
Proof of Claim 2: Assume the contrary and let $Y$ be a nonempty open set which is in none of the $\tau_{i}$. Then we find for each $i$ an $X_{i}$ in $\tau_{i}$ which is disjoint from $Y$ (by maximality of each $\tau_{i}$). We may assume that $X_{i}\subseteq U_{i}$ for each $i$. But then $\{X_{1},\dots,X_{n},Y\}$ is a collection of $n+1$ pairwise disjoint nonempty open subsets of $(X,\tau)$, contradicting the choice of $n$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
Each $\tau_{i}$ has the finite intersection property. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[finitaryth\]Let $(X,\tau)$ be a topological space for which:
1. [Player TWO has a winning strategy in the game $BM(X,\tau)$ and]{}
2. [every collection of pairwise disjoint open subsets is finite.]{}
Then TWO has a winning $1$-tactic in $BM(X,\tau)$.
Proof.
: Write, by Proposition \[filterdecomp\], $$\tau\backslash\{\emptyset\}=\tau_{1}\cup\dots\cup\tau_{n}$$ where each $\tau_{i}$ has the finite intersection property, and $n$ is minimal. Choose a pairwise disjoint collection $\{U_{1},\dots,U_{n}\}$ such that $U_{j}\in \tau_{j}$ for each $j$.
Claim 3: For each $j$, if $S_{1}\supseteq S_{2}\supseteq\dots$ is a denumerable chain from $\tau_{j}$, then $\cap_{n=1}^{\infty}S_{n}\neq
\emptyset$.
Proof of Claim 3: Assume the contrary, and fix $j$ and a chain $S_{1}\supseteq
S_{2}\supseteq\dots$ in $\tau_{j}$ such that $\cap_{n=1}^{\infty}S_{n}=
\emptyset$. We may assume that $S_{n+1}\subset S_{n}\subset U_{j}$ for all $n$.
Let $F$ be a winning perfect information strategy for TWO in $BM(X,\tau)$. Consider the play $$(O_{1},T_{1},\dots,O_{m},T_{m},\dots)$$ which is defined so that:
1. [$O_{1}=S_{1}$,]{}
2. [$T_{m}=F(O_{1},\dots,O_{m})$ for all $m$ and]{}
3. [$O_{m+1}=T_{m}\cap S_{m+1}$.]{}
Note that since each $S_{m}$ is a subset of $U_{j}$, each response by player TWO using $F$ is a member of $\tau_{j}$, whence each $O_{m}$ is a legal move by ONE. We now get the contradiction that TWO lost this play despite the fact that TWO was playing according to a winning strategy. This completes the proof of Claim 3. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
We now define a winning $1$-tactic, $G$, for TWO. Let $U$ be a nonempty open subset of $X$. Choose the minimal $j$ such that $U_{j}\cap U\neq\emptyset$ and put $G(U)=U_{j}\cap U$. Claim 3 implies that this is a winning $1$-tactic for TWO. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
\[reductionth\]Let $k$ be a positive integer. If player TWO has a winning Markov $k$-tactic in the Banach Mazur game on some topological space, then TWO has a winning $k$-tactic in the Banach-Mazur game on that space.
Proof.
: Let $k$ be a positive integer and let $(X,\tau)$ be a topological space such that TWO has a winning Markov $k$-tactic in the game $BM(X,\tau)$. We may assume that this space has no isolated points. By Proposition \[finitaryth\] we may also assume that every nonempty open subset of $X$ contains infinitely many pairwise disjoint open subsets (player ONE may safely avoid playing open subsets not having this property). By Theorem 5 of [@G-T] we may assume that $k>1$.
Let $F$ be a winning Markov $k$-tactic for TWO. For each nonempty open set $U$, let $\{J_{m}(U):0<m<\omega\}$ bijectively enumerate a collection of infinitely many pairwise disjoint nonempty open subsets of $U$.
Define a $k$-tactic $G$ for TWO as follows. Let $U_{1}\supseteq\dots\supseteq
U_{j}$ be nonempty open sets, where $1\leq j\leq k$.
Case 1: $j=1$: Put $G(U_{1})=F(J_{2}(U_{1}),1)$.
Case 2: $j>1$ and $U_{i+1}\subseteq J_{l+i+1}(U_{i})$ for $1\leq i<j$, for some $l$. Put $G(U_{1},\dots,U_{j})=F(J_{l+2}(U_{1}),\dots,J_{l+j+1}(U_{j}),l+j)$.
Case 3: In all other cases define $G(U_{1},\dots,U_{j})=G(U_{j})$.
To see that $G$ is a winning $k$-tactic for TWO, consider a play $$(O_{1},T_{1},\dots,O_{m},T_{m},\dots)$$ such that
- [$T_{j}=G(O_{1},\dots,O_{j})$ for $j\leq k$ and]{}
- [$T_{n+k}=G(O_{n+1},\dots,O_{n+k})$ for all $n$.]{}
From the definition of $G$ and the rules of the Banach-Mazur game it follows that $T_{1}$ is defined by Case 1 and $T_{m}$ for $m>1$ by Case 2. In particular, writing $S_{n}$ for $J_{n+1}(O_{n})$ we find that:
1. [$T_{j}=F(S_{1},\dots,S_{j},j)$ for $j\leq k$ and]{}
2. [$T_{n+k}=F(S_{n+1},\dots,S_{n+k},n+k)$]{}
for all n. Indeed, $$O_{1}\supseteq S_{1}\supseteq T_{1}\supseteq O_{2}\supseteq S_{2}\supseteq\dots$$
Since $F$ is a winning Markov $k$-tactic, it follows that $\cap_{n=1}^{\infty}O_{n}\neq\emptyset$. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
### Debs’ example
Let $\sigma$ be the topology of the real line whose elements are of the form $U\backslash M$ where $U$ is open and $M$ is meager in the usual topology. The symbol $BM({{\Bbb R}},\sigma)$ denotes the Banach-Mazur game, played on the topological space $({{\Bbb R}},\sigma)$. It is known that TWO has a winning strategy but does not have a winning $1$-tactic in $BM({{\Bbb R}},\sigma)$.
\[vstobm\] Player TWO has a winning 2-tactic in the game $BM({{\Bbb R}},\sigma)$.
Proof.
: [Theorem 22 of [@S1] and Corollary \[corsmg\]. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
Appendix: Consistency of the hypotheses of Theorem \[noktactic\].
=================================================================
We start with a ground model $V$ and let ${\bold
P} \in V$ be a forcing notion of cardinality $\leq {\frak c}$. For a cardinal $\kappa$, denote by ${{\bold P}_{\kappa}}$ the product of $\kappa$ copies of $\bold P$ taken side-by-side with countable supports.
\[lemma:consistency2\] Let $\lambda$ be an uncountable cardinal. Suppose:
1. $\kappa\geq\kappa_1\geq\kappa_2\geq\kappa_3\geq\omega_2$ are cardinal numbers such that
- [$\kappa$ is a regular cardinal,]{}
- [$\kappa\rightarrow(\kappa_1)^2_{\lambda}$,]{}
- [$\kappa_1\rightarrow(\kappa_2)^3_{{\frak c}}$,]{}
- [$\kappa_2\rightarrow(\kappa_3)^2_{\lambda}$ and]{}
- [$\kappa_3\rightarrow(\omega_2)^3_{{\frak c}}$.]{}
2. [Forcing with ${\bold P}$ adds a real to the ground model.]{}
Then ${\frak c}{\rightarrow (\omega - path )^2_{\lambda /<\omega }}$ holds in the forcing extension ${V^{{{\bold P}_{\kappa}}}}$.
Proof.
: Let $\lambda, \kappa, \kappa_1,
\dots , \kappa_3, \bold P$ be as in the assumptions. Our argument closely follows section 2 of [@To1].
For sets $A,B$ the symbol $A/B$ denotes $\{ {\{ \alpha , \beta\} }: \ \alpha \in A ,
\ \beta \in B , \ \alpha < \beta \}$.
Note that ${V^{{{\bold P}_{\kappa}}}}$ satisfies ${\frak c}= \kappa$; we prove that $\kappa {\rightarrow (\omega - path )^2_{\lambda /<\omega }}$ holds in ${V^{{{\bold P}_{\kappa}}}}$.
Let $[\kappa ]^2 = \bigcup_{i < \lambda} {\dot K}_i$ be a given partition in ${V^{{{\bold P}_{\kappa}}}}$. Let $\dot U$ be a ${{\bold P}_{\kappa}}$-name for a member of $[\kappa ]^\kappa$. Pick $A \in [\kappa ]^\kappa$ and for each $\alpha \in A$, a $q_\alpha \in {{\bold P}_{\kappa}}$ such that $q_\alpha {\mathrel{\|}\joinrel\mathrel{-}}\alpha \in {\dot U}$ and such that the $q_\alpha$’s form a $\Delta$-system. Define $H: [A]^2 \rightarrow
(\lambda+1)$ so that $H(\{ \alpha, \beta \}) = i$ if $i$ is the minimal $j$ such that $p {\mathrel{\|}\joinrel\mathrel{-}}\{ \alpha , \beta \} \in {\dot
K}_j$ for some $p \leq q_\alpha,
q_\beta$ if such $j$ exists (i.e., if $q_\alpha$ and $q_\beta$ are compatible), and $H(\{ \alpha , \beta \} ) = \lambda$ if $q_\alpha$ is incompatible with $q_\beta$.
By our choice of $\kappa$, the partition relation $\kappa \rightarrow (\kappa_1 )^2_\lambda$ holds. Therefore, choose $A_1 \subset [A]^{\kappa_1}$ and $i \leq \lambda$ such that $H''[A_1]^2 = \{ i \}$. Since ${{\bold P}_{\kappa}}$ satisfies the ${\frak c}^+$-c.c., we have $i < \lambda$.
Let $\langle {p_{\alpha , \beta}}: \ \{ \alpha , \beta \} \in [A_1]^2 \rangle$ be a fixed sequence of conditions such that ${p_{\alpha , \beta}}\leq q_\alpha , q_\beta$ and ${p_{\alpha , \beta}}{\mathrel{\|}\joinrel\mathrel{-}}{\{ \alpha , \beta\} }\in {{\dot K}_i}$. For $\alpha < \beta < \gamma$ in $A_1$ we define $H_0 (\{ \alpha , \beta , \gamma \} )$ to be a pair $(c,d)$, where $c$ codes ${p_{\alpha , \beta}}$ and ${p_{\alpha , \gamma}}$ as structures as well as relations between the ordinals of $dom ( {p_{\alpha , \beta}})$ and $dom ({p_{\alpha , \gamma}})$, and $d$ does the same for ${p_{\alpha , \gamma}}$ and ${p_{\beta , \gamma}}$. Since there are only ${\frak c}$ such pairs, and since $\kappa_1 \rightarrow (\kappa_2 )^3_{{\frak c}}$ holds, choose $A_2 \in [A_1]^{\kappa_2}$ and $(c,d)$ such that $H_0''[A_2]^3 = \{ (c,d) \}$. For convenience, assume that $A_2$ has order type $\kappa_2$. It follows that for each $\alpha \in A_2$ the sequence $\langle {p_{\alpha , \beta}}: \ \beta \in A_2 \backslash
(\alpha + 1) \rangle$ forms a $\Delta$-system with root $p^0_\alpha$ ($\leq q_\alpha$), and that for each $\gamma \in A_2$ the sequence $\langle {p_{\beta , \gamma}}: \ \beta \in A_2 \cap \gamma \rangle$ forms a $\Delta$-system with root $p^1_\gamma$ ($\leq q_\gamma$). Moreover, the $p_\alpha^0$’s and $p^1_\gamma$’s form $\Delta$-systems with roots $p^0$ and $p^1$ respectively. To see the latter, note that we may shrink $A_2$ to a cofinal subset $A_3$ so that the relevant $p^0_\alpha$’s and $p^1_\alpha$’s do in fact form a $\Delta$-system. Now consider $\alpha , \beta , \gamma \in A_3$, and $\alpha ' , \beta '
\in A_2$. Comparing $H_0(\{ \alpha , \beta , \gamma \})$, $H_0( \{ \alpha , \beta ' , \gamma \} )$ and $H_0 ( \{\alpha ' ,
\beta ' , \gamma \} )$, one sees that the sequence $\langle p_\alpha^0 : \ \alpha \in A_2 \rangle$ forms a $\Delta$-system. A similar argument works for the $p^1_\gamma$’s.
Also, $p^0$ is compatible with $p^1$. We call $\langle {p_{\alpha , \beta}}: \ {\{ \alpha , \beta\} }\in B/B \rangle$ a [*double $\Delta$-system*]{} with root $p^0 \cup p^1$.
There is no reason why for a given $\alpha$ the conditions $p^0_\alpha$ and $p^1_\alpha$ should be compatible: if these were always compatible, our argument would yield a consistency proof of ${\frak c}\rightarrow (\omega )^2_\lambda$, which is false in $ZFC$.
We now save as much of the compatibility between $p_\alpha^0$ and $p_\alpha^1$ as is needed for the consistency proof of ${\frak c}{\rightarrow (\omega - path )^2_{\lambda /<\omega }}$. Thin out $A_2$ to a cofinal subset $A_3$ such that $dom(p^0_\alpha \cup p^1_\alpha )
\cap dom(p^0_\beta \cup p^1_\beta) = dom (p^0 \cup p^1)$ for all ${\{ \alpha , \beta\} }\in A_3 / A_3$. Then in particular $p^1_\alpha$ and $p^0_\beta$ are compatible for ${\{ \alpha , \beta\} }\in A_3/A_3$.
Now repeat the reasoning above with $A_2$ in place of $A$, $\kappa_2$ in place of $\kappa$, $\kappa_3$ in place of $\kappa_1$, and $\omega_2$ in place of $\kappa_2$. Also, $p^1_\alpha$ will now play the role of $q_\alpha$, and $p^0_\beta $ the role of $q_\beta$ for $\{ \alpha , \beta \} \in
A_3 / A_3$. We get a set $A_4 \subset A_3$ of order type $\omega_2$ and some $j < \lambda$ (which may be different from $i$), conditions ${{\bar p}_{\alpha , \beta }}$ for ${\{ \alpha , \beta\} }\in A_4 / A_4$ that form a double $\Delta$-system with root ${{\bar p}^0}\cup {{\bar p}^1}$, and we get roots ${{{\bar p}^0}_\alpha}$ and ${{{\bar p}^1}_\gamma}$ as before. Now ${{\bar p}_{\alpha , \beta }}{\mathrel{\|}\joinrel\mathrel{-}}{\{ \alpha , \beta\} }\in {\dot K}_j$ for ${\{ \alpha , \beta\} }\in A_4 / A_4$.
Our choice of ${{\bar p}_{\alpha , \beta }}$ at the beginning of the second run of the argument insures that ${{{\bar p}^0}_\alpha}\leq p_\alpha^1$ and ${{{\bar p}^1}_\gamma}\leq p^0_\gamma$, and hence ${{\bar p}^0}\leq p^1$ and ${{\bar p}^1}\leq p^0$.
Now let $\bold G$ be a generic subset of ${{\bold P}_{\kappa}}$. Define:
${\dot X}= \{ \alpha \in A_4 : \ p^0_\alpha \in {\bold G}\}$,
${\dot Y}= \{ \alpha \in A_4 : \ p^1_\alpha \in {\bold G} \}$,
${\dot W}= \{ \alpha \in A_4 : \ {{{\bar p}^0}_\alpha}\in {\bold G} \}$,
${\dot Z}= \{ \alpha \in A_4 : \ {\bar p}^1_\alpha \in {\bold G} \}$.
Then ${\dot Z}\subset {\dot X}$ and ${\dot W}\subset {\dot Y}$, and all four sets are cofinal in $A_4$.
Now ${\bar p}^0 \cup {\bar p}^1$ forces the following facts:
\(1) $\exists \delta_1 \in \omega_2 \forall \alpha \in {\dot X}\backslash \delta_1
\ \{ \beta \in {\dot W}: \ {\{ \alpha , \beta\} }\in {{\dot K}_i}\} $ is cofinal in $A_4$, and
\(2) $\exists \delta_2 \in \omega_2 \forall \alpha \in {\dot Y}\backslash
\delta_2 \ \{ \beta \in {\dot Z}: \ {\{ \alpha , \beta\} }\in {\dot K}_j \}$ is cofinal in $A_4$.
The combination of (1) and (2) suffices to construct in ${V^{{{\bold P}_{\kappa}}}}$ an $\omega$-path of the given partition that uses only colors $i$ and $j$:
Let $\delta = max \{ \delta_1 , \delta_2 \}$. Inductively define an increasing sequence $\langle x_n: \ n \in \omega \rangle$ of ordinals such that $x_{2k} \in Z$ (and hence in $X$), $x_{2k+1} \in W$, and $\{ x_{2k}, x_{2k+1} \} \in {{\dot K}_i}$ (by (1)); $\{ x_{2k+1}, x_{2k+2} \} \in {\dot K}_j$ (by (2)).
It remains to prove (1) and (2). We shall prove (1) only; the proof of (2) is similar, and is a special case of [@To1], section 2, property (1)\].
Assume that ${\bar p}^0 \cup {\bar p}^1$ does not force (1). Then we can find a condition ${\bar p}^2 \leq {\bar p}^0 \cup {\bar p}^1$ and a ${{\bold P}_{\kappa}}$-name $\dot D \in [{\dot X}]^{\omega_2}$ and for each $\beta \in \dot D$ a $\gamma_\beta \in A_4 \backslash (\beta + 1)$ such that ${\bar p}^2 {\mathrel{\|}\joinrel\mathrel{-}}\{ \beta_ , \delta \}
\notin {{\dot K}_i}$ whenever $\delta \in \dot W \backslash \gamma_\beta$.
Working in $V$, we pick $B \in [A_4]^{\omega_2}$ such that for each $\beta \in B$ we find $r_\beta \leq p^0_\beta \cup {\bar p}^2$ such that $r_\beta {\mathrel{\|}\joinrel\mathrel{-}}\beta \in \dot D$, and $r_\beta$ decides the value of $\gamma_\beta$. We may assume that the $r_\beta$’s form a $\Delta$-system with root $\leq {\bar p}^2 \leq {\bar p}^0 \cup {\bar p}^1$, and that $\gamma_\beta < \delta$ for all $\{\beta , \delta\} \in B/B$. Since $\langle {p_{\alpha , \beta}}: \ \beta \in B \backslash (\alpha + 1) \rangle$ forms a $\Delta$-system, we may also assume that $dom(r_\beta) \cap dom(p_{\beta , \delta}\backslash p^0_{\beta} )
= \emptyset $ for all $ \delta > \gamma_\beta$ in $A_4$.
Pick $\delta \in A_2$ such that $B \cap \delta$ is uncountable and $dom({\bar p}^0_\delta ) \cap dom ({\bar p}^2) = dom ({\bar p}^0)$. Since $\langle p_{\beta , \delta} : \ \beta \in B \cap \delta \rangle$ forms a $\Delta$-system with root $p^1_\delta$ and since $dom({\bar p}^0_\delta )$ is countable, we have $dom(p_{\beta , \delta} \backslash p^1_\delta) \cap dom ({\bar p}^0_\delta )
\not= \emptyset$ for only countably many $\beta \in B \cap \delta$. So pick a $\beta \in B \cap \delta$ such that $dom(p_{\beta , \delta} \backslash p^1_\delta) \cap dom ({\bar p}^0_\delta )
= \emptyset$.
Define $r \in {{\bold P}_{\kappa}}$ as follows:
$dom (r) = dom (r_\beta ) \cup dom ({\bar p}^0_\delta ) \cup
dom (p_{\beta , \delta } \backslash p^1_\delta )$,
$r| dom (r_\beta \cup {\bar p}^0_\delta ) = r_\beta \cup {\bar p}^0_\delta$,
and
$r(\xi ) = p_{\beta , \delta } (\xi ) $ for $\xi \in
dom (p_{\beta , \delta } \backslash dom (r_\beta \cup {\bar p}^0_\delta ))$.
Then $r$ is a well-defined condition with the property that $r \leq r_\beta , {\bar p}^0_\delta$ and $p_{\beta , \delta}$. So $r$ forces that $\{\beta , \delta\} \in X/W$ and that $\{ \beta , \delta \} \in K_i$, which is a contradiction. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$
If ${{\bold P}_{\kappa}}$ is as in the assumptions of Lemma \[lemma:consistency2\], then ${{\bold P}_{\kappa}}$ is a ${\frak c}^+$-c.c. poset. If GCH holds in the ground model and $\lambda = \omega_1$, then our proof works if $\kappa \geq \aleph_8$. One can obtain the consistency of ${\frak c}\rightarrow (\omega -path)^2_{\omega_1/<\omega}$ with a smaller size of the continuum, but this is not essential for our purposes. Todorcevic has for example shown that, adjoining at least $\omega_2$ Cohen reals to a model of the Continuum Hypothesis, produces a model in which $\omega_2\rightarrow(\mbox{$\omega$-path})^2_{\omega/<3}$.\
We have actually proved something apparently stronger than ${\frak c}{\rightarrow (\omega - path )^2_{\lambda /<\omega }}$ in ${V^{{{\bold P}_{\kappa}}}}$, namely a relation denoted by ${\frak c}\rightarrow (\omega - path)^2_{\lambda/<3}$.
We do not know an answer to the following two problems concerning the $\omega$-path partition relation:
Is it for each integer $k>2$ consistent, for some infinite cardinal numbers $\kappa$ and $\lambda$, that $\kappa\not\rightarrow(\omega\mbox{-path})^{2}_{\lambda/<k}$, but $\kappa\rightarrow(\omega\mbox{-path})^{2}_{\lambda/<k+1}$?
Is it consistent, for some infinite cardinal numbers $\kappa$ and $\lambda$, that for each $k<\omega$, $\kappa\not\rightarrow(\omega\mbox{-path})^{2}_{\lambda/<k}$, but $\kappa\rightarrow(\omega\mbox{-path})^{2}_{\lambda/<\omega}$?
\[consistency2\] If $ZFC$ is a consistent theory, then so is the theory $ZFC \ + \ cof(\langle{J_{{\Bbb R}}}\rangle ,
\subset ) = \aleph_1 \ + \ {\frak c} {\rightarrow (\omega - path )^2_{\omega_1/<\omega }}$.
Proof
: [ Theorem \[consistency2\] is an immediate consequence of Lemma \[lemma:consistency2\]: It is well known that if CH holds in the ground model, and $\bold P$ is e.g. Sacks or Prikry-Silver forcing, then (b) and (c) of the lemma hold for every $\kappa$. It is also known that adding any number of Sacks or Prikry-Silver reals side-by-side with countable supports to a model of CH, one obtains a model where the collection of meager sets whose Borel codes are from the ground model, is a cofinal subfamily of ${J_{{\Bbb R}}}$ (see [@M]). Since $| ^{\omega}\omega\cap
V| = \aleph_1$, we get $cof(\langle{J_{{\Bbb R}}}\rangle , \subset ) =
\aleph_1$ in the forcing extension. ${\vrule width 6pt height 6pt depth 0pt \vspace{0.1in}}$]{}
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[^1]: Supported by Idaho State Board of Education grant 92-096
[^2]: Supported by NSF grant DMS-9016021 and Research Challenge Grant RC 89-64 from Ohio University
[^3]: Supported by Idaho State Board of Education grant 91-093.
|
---
abstract: 'We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global regularity for solutions when dissipation is slightly supercritical. Both results utilize a nonlocal maximum principle.'
address: 'Dept. of Mathematics, University of Wisconsin, Madison, USA'
author:
- Tam Do
bibliography:
- 'refs.bib'
title: On a 1D transport equation with nonlocal velocity and supercritical dissipation
---
Active scalar ,Supercritical dissipation ,Nonlocal velocity ,Modulus of continuity
Introduction
============
We will consider solutions to the following initial value problem $$\begin{aligned}
\label{eq:PDE}
\theta_t = (H\theta)\theta_x-(-\Delta)^\alpha \theta \\
\theta(x,0) = \theta_0(x) \nonumber\end{aligned}$$ for $(x,t)\in {\ensuremath{\mathbb{R}}\xspace}\times [0,\infty)$ and $0\le \alpha <1/2$. Here, $H$ is the Hilbert transform $$H\theta(x)=P.V. \frac{1}{\pi}\int \frac{\theta(y)}{x-y}\, dy$$ The equation has an $L^\infty$ maximum principle which makes the range $0\le \alpha<1/2$ supercritical. The equation can be thought of as a 1D model for the 2D surface quasi-geostrophic (SQG) equation $$\begin{aligned}
\theta_t &=& u\cdot \nabla \theta-(-\Delta)^\alpha \theta \quad\mbox{on}\,\,\, {\ensuremath{\mathbb{R}}\xspace}^2\times[0,\infty) \\
u &=& \nabla^\perp \Delta^{-1/2} \theta.\end{aligned}$$ In addition, $\eqref{eq:PDE}$ also has similarities with the Birkhoff-Rott equations for the evolution of vortex sheets (see [@Blow2] for more references). For $\alpha \ge 1/2$, the problem $\eqref{eq:PDE}$ is globally well-posed for initial data in $H^{3/2-2\alpha}$ and locally well-posed for $0\le \alpha <1/2$. [@Dong1]. For $0\le \alpha <1/4$, finite time blow up has been shown to be possible [@Blow1; @Blow2].
In the range $1/4 \le \alpha <1/2$, it is unknown whether there is finite time blow-up or global regularity. One can think of the term $(-\Delta)^\alpha\theta$ as smoothing while the nonlinear term $(H\theta)\theta_x$ as introducing singular behavior. In the supercritical range, scaling favors the nonlinear term and standard methods for proving regularity are not sufficient. For $1/4\le \alpha < 1/2$, if one were to prove the more interesting result of global regularity, one would need to discover some mechanism of nonlinear depletion present in the equation.
In this paper, we will show that two results that are true for the SQG equation also hold for this 1D model. First, we will show eventual regularization for dissipation in the supercritical regime $0< \alpha <1/2$ with non-negative initial data. Second, we will prove global regularity for the slightly supercritical version of this equation. For the SQG equation, the arguments rely on dissipation in the “perpendicular” direction [@EvReg] or incompressibility of the fluid velocity [@Dabk; @Luis], which are both absent in our 1D setup. In our results, we will need to carefully use the structure of the nonlinearity as well as the exact formula for the dissipation term. The new ingredients in our proof are Lemma 2.7 and a part of section 3.3.
Eventual Regularization
=======================
In this section, we will closely follow the arguments of Kiselev [@EvReg]. We will work with solutions to the dissipative regularization of $\eqref{eq:PDE}$: $$\begin{aligned}
\label{eq:reg}
\theta_t = (H\theta)\theta_x-(-\Delta)^{\alpha} \theta+\epsilon \Delta \theta \\
\theta(x,0)=\theta_0(x). \nonumber\end{aligned}$$ The solutions of $\eqref{eq:reg}$ will be smooth and we will estimate the Holder norms of these solutions uniformly in $\epsilon>0$. The limit obtained by letting $\epsilon\to 0$ will yield a candidate for a weak solution $\theta(x,t)\in C_w([0,\infty);L^2) \cap L^2([0,\infty); H^{1/2})\cap L^\infty([0,\infty); L^2)$. However, this regularity appears to be insufficient to conclude that $\theta $ solves $\eqref{eq:PDE}$ in the standard weak sense. The equation is not conservative. On the other hand, the limit will inherit our estimates on the regularization. By having control of high enough Holder norms, the following theorem allows us to conclude smoothness :
Let $\theta$ be a solution of $\eqref{eq:reg}$ with non-negative initial data. Let $\beta>1-2\alpha$ and let $0<t_0<t<\infty$. If $\theta\in L^\infty([t_0,t]; C^\beta({\ensuremath{\mathbb{R}}\xspace}))$ then $\theta\in C^\infty((t_0,t]\times {\ensuremath{\mathbb{R}}\xspace})$ with bounds independent of $\epsilon$.
The proof of Theorem 2.1 is analogous to the proof of Theorem 3.1 in Constantin and Wu [@Constantin] where they showed a similar result is true for the SQG equation. Their argument for SQG uses Besov space techniques and does not rely heavily on incompressibility, the key difference between $\eqref{eq:PDE}$ and SQG. Since we have non-negative initial data, solutions of $\eqref{eq:PDE}$ are bounded in $L^2$ without the need of incompressibility [@Blow2], a condition necessary to show the analogous result for SQG. Also, $\eqref{eq:PDE}$ posess the same scaling as SQG and the Hilbert transform is bounded on the Holder and Besov spaces, like the Riesz transforms, which is needed in the proof.
The main result of this section is the following theorem.
\[reg\] Let $\theta(x,t)$ be the limit obtained in $ \eqref{eq:reg}$ by letting $\epsilon\to 0$ with $\theta_0\in L^\infty\cap H^{3/2-2\alpha}$ and non-negative. Then there exist times $0<T_1(\alpha, \theta_0)\le T_2(\alpha, \theta_0)$ such that $\theta$ is smooth for $0<t<T_1$ and $t>T_2$ ($\theta$ will be a classical solution of $\eqref{eq:PDE}$ for these times).
[**Remark.**]{} For $T_1\le t\le T_2$, it is unclear in what sense the $\theta$ above is a solution of $\eqref{eq:PDE}$. The theorem follows from uniform in $\epsilon$ estimates for (2) and such estimates can be regarded as the main result of this section.
To control Holder norms, we will show that a certain family of moduli of continuity is eventually preserved under the evolution.
A function $\omega(\xi): (0,\infty)\mapsto (0,\infty)$ is a modulus of continuity if $\omega$ is increasing, continuous on $(0,\infty)$, concave, and piecewise $C^2$ with one-sided derivatives defined at every point in $[0,\infty)$. A function $f(x)$ obeys $\omega$ if $|f(x)-f(y)|<\omega(|x-y|)$ for all $x\ne y$.
To prove that solutions preserve a modulus of continuity we state the following lemma, which describes the scenario in which the modulus is broken.
\[break\] Let $\theta(x,t)$ be a solution of $\eqref{eq:PDE}$. Suppose that $\omega(\xi,t)$ is continuous on $(0,\infty)\times [0,T]$, piecewise $C^1$ in the time variable (with one-sided derivatives defined at all points) for each fixed $\xi>0$, and that for each fixed $t\ge 0$, $\omega(\xi,t)$ is a modulus of continuity. Assume in addition that for each $t\ge 0$, either $\omega(0+,t)>0$, or $\partial_\xi \omega(0+,t)=\infty$, or $\partial_{\xi\xi}^2\omega(0+,t)=-\infty$, and that $\omega(0+,t)$, $\partial_\xi \omega(0+,t)$ are continuous in $t$ with values in ${\ensuremath{\mathbb{R}}\xspace}\cup \infty$. Let the initial data $\theta_0(x)$ obey $\omega(\xi,0)$. Suppose that for some $t>0$ the solution $\theta(x,t)$ no longer obeys $\omega(\xi,t)$. Then there exist $t_1>0$ and $x,y\in {\ensuremath{\mathbb{R}}\xspace}$, $x\ne y$ such that for all $t<t_1$, $\theta(x,t)$ obeys $\omega(\xi,t)$ while $$\theta(x,t_1)-\theta(y,t_1)= \omega(|x-y|,t_1).$$
The proof of preceding lemma can be found in [@EvReg] for the periodic case. Decay results for solutions from [@Dong1] allow the lemma to be extended to the non-periodic setting [@Du]. We will use the same moduli of continuity as in [@EvReg]: $$\omega(\xi,\xi_0) = \left\lbrace \begin{array}{ll} \beta H\delta^{-\beta}\xi_0^{\beta-1}\xi+(1-\beta)H\delta^{-\beta}\xi_0^\beta, & 0<\xi<\xi_0 \\ H(\xi/\delta)^\beta, & \xi_0 \le \xi\le \delta \\
H, & \xi>\delta \end{array}\right.$$ where $\beta>1-2\alpha$. Observe that if $2\|\theta_0\|_{L^\infty} \le \omega(0,\delta)=(1-\beta) H$, then $\theta_0$ obeys $\omega(\xi,\delta)$. Thus, for every bounded initial data, we can find a modulus that is obeyed.
It is known that for $0<\alpha<1/2$, $$(-\Delta)^\alpha \theta(x) = P.V.\int_{-\infty}^\infty \frac{\theta(x)-\theta(x+y)}{|y|^{1+2\alpha}}\, dy,$$ see [@Cordoba] for a proof. We will need the following estimate of the dissipation terms:
\[dis\] (Dissipation Estimate) Let $\xi=|x-y|$. Then $$-(-\Delta)^\alpha \theta(x,t) + (-\Delta)^\alpha \theta(y,t) \le D_\alpha(\xi, t)$$ where $$D_\alpha(\xi,t) = c_\alpha \left(\int_0^{\frac{\xi}{2}} \frac{\omega(\xi+2\eta,t)+\omega(\xi-2\eta,t)-2\omega(\xi,t)}{\eta^{1+2\alpha}}\, d\eta + \int_{\frac{\xi}{2}}^\infty \frac{\omega(\xi+2\eta, t)-\omega(2\eta -\xi,t)-2\omega(\xi,t)}{\eta^{1+2\alpha}}\, d\eta\right).$$
See [@EvReg] for the proof. Theorem \[reg\] is an easy consequence of the following lemma, which we will prove later.
\[lem\] Assume that $\theta_0(x)$ of $\eqref{eq:PDE}$ obeys $\omega(\xi,\delta)$. Then there exist positive constants $C_{1,2}= C_{1,2}(\alpha,\beta)$ such that if $\xi_0(t)$ is a solution of $$\label{eq:ODE}
\frac{d\xi_0}{dt}=-C_2 \xi_0^{1-2\alpha}, \quad \xi_0(0)=\delta,$$ and $H\le C_1 \delta^{1-2\alpha}$, then the solution $\theta(x,t)$ obeys $\omega(\xi,\xi_0(t))$ for all $t$ such that $\xi_0(t)\ge 0$.
[**Proof of Theorem \[reg\]**]{} The solution $\xi_0(t)$ of $\eqref{eq:ODE}$ becomes zero and stays zero in finite time. Then eventually, the solution $\theta(x,t)$ obeys $\omega(\xi,0)$ and we can uniformly bound its $C^\beta$ norm, $\beta>1-2\alpha$. $\Box$
To prove lemma \[lem\], we will show that the breakthrough scenario described in lemma \[break\] cannot happen. Suppose there exists $t_1>0$ such that $\theta(x,t)$ obeys $\omega(\xi,t)$ for $t<\xi_0(t_1)$ and $\theta(x,t_1)-\theta(y,t_1)=\omega(\xi,\xi_0(t_1))$ where $\xi=|x-y|$. Then it is not hard to see that $\nabla\theta(x,t_1)=\partial_\xi \omega(\xi,\xi_0(t_1)) \frac{x-y}{\xi}= \nabla\theta(y,t_1)$ and $\Delta \theta(x,t_1)-\Delta\theta(y,t_1) \le 2 \partial_{\xi\xi}^2\omega(\xi,t_1)$ (details are in [@EvReg]). Also, by , lemma \[dis\],
$$\label{eq:break}
\partial_t \left. \left[ \frac{\theta(x,t)-\theta(y,t)}{\omega(\xi,\xi_0(t))}\right] \right|_{t=t_1} \le
\frac{\Omega(x,y,t_1)\partial_{\xi}\omega(\xi,\xi_0(t_1)) +d_\alpha(\xi,t_1)+2\epsilon \partial_{\xi\xi}^2 \omega(\xi,t_1) - \partial_t \omega(\xi,\xi_0(t_1))}{\omega(\xi, \xi_0(t_1))}$$
where $\Omega(x,y,t_1)=H\theta(x,t_1)-H\theta(y,t_1)$ and $$d_\alpha(\xi,t_1)= \frac{1}{2} \left(-(-\Delta)^\alpha \theta(x,t_1)+(-\Delta)^\alpha \theta(y,t_1)\right) + \frac{1}{2} D_\alpha(\xi,t_1).$$ If we can show that the numerator of the right hand side of $\eqref{eq:break}$ is negative, then the modulus of continuity must have been broken at an earlier time, a contradiction. Because of the concavity of $\omega$, $2\epsilon \partial_{\xi\xi}^2 \omega(\xi,t_1) \le 0$ and since we want our estimates to be independent of $\epsilon$, we will ignore this term.
\[key\] $$\Omega(x,y)=H\theta(x)-H\theta(y) \le C\left[ \xi^{2\alpha} \left((-\Delta)^\alpha \theta(x)-(-\Delta)^\alpha \theta(y)\right)+ \xi \int_{\xi/2}^\infty \frac{\omega(r)}{r^2}\, dr\right]$$
For simplicity of expression, we have omitted time in our expressions.
[[**Proof.**]{} ]{}
The first term on the right side will control the singular behavior of the Hilbert transforms near the kernel singularity and the second term will control the behavior away from the singularity. Where appropriate, integrals will be understood in the principal value sense.
Let ${\widetilde}{x}= \frac{x+y}{2}$. Then $$\begin{aligned}
\left|\int_{|x-z|\ge \xi} \frac{\theta(z)}{x-z}\, dz-\int_{|y-z|\ge \xi} \frac{\theta(z)}{y-z}\, dz\right| &=& \left|\int_{|x-z|\ge \xi} \frac{\theta(z)-\theta({\widetilde}{x})}{x-z}\, dz-\int_{|y-z|\ge \xi} \frac{\theta(z)-\theta({\widetilde}{x})}{y-z}\, dz \right| \\ &\le & \int_{|{\widetilde}{x}-z|\ge \xi/2} \left|\frac{1}{x-z}-\frac{1}{y-z}\right| |\theta(z)-\theta({\widetilde}{x})|\, dz \\ & \le & C\xi \int_{|{\widetilde}{x}-z|\ge \xi/2} \frac{1}{|{\widetilde}{x}-z|^2} |\theta(z)-\theta({\widetilde}{x})|\, dz \le C\xi \int_{\xi/2}^\infty \frac{\omega(r)}{r^2}\, dr\end{aligned}$$
Now, we will estimate the other part. Observe that $$\begin{aligned}
H\theta(x) -H\theta(y)= \int_{-\infty}^\infty \frac{\theta(x+z)-\theta(x)}{z}\, dz - \int_{-\infty}^\infty \frac{\theta(y+z)-\theta(y)}{z}\, dz\end{aligned}$$
Then
$$\begin{aligned}
&\, &\int_{|z|<\xi} \frac{\theta(x+z)-\theta(x)}{z}\, dz - \int_{|z|<\xi} \frac{\theta(y+z)-\theta(y)}{z}\, dz - \xi^{2\alpha}((-\Delta)^\alpha\theta(x)- (-\Delta)^{\alpha} \theta(y)) \\ &=& \int_{|z|<\xi} \frac{\theta(x+z)-\theta(x)}{z}\, dz - \int_{|z|<\xi} \frac{\theta(y+z)-\theta(y)}{z}\, dz - \xi^{2\alpha}\int_{-\infty}^\infty \frac{\theta(x)-\theta(x+z)}{|z|^{1+2\alpha}}\, dz \\ &\, & +\xi^{2\alpha}\int_{-\infty}^\infty \frac{\theta(y)-\theta(y+z)}{|z|^{1+2\alpha}}\, dz \\ &=&
\int_{|z|<\xi} \left(\theta(x+z)-\theta(y+z)+\theta(y)-\theta(x)\right) \left(\frac{1}{z}+\frac{\xi^{2\alpha}}{|z|^{1+2\alpha}}\right) \\
& \, & + \int_{|z|>\xi} \left(\theta(x+z)-\theta(y+z)+\theta(y)-\theta(x)\right) \frac{\xi^{2\alpha}}{|z|^{1+2\alpha}}\, dz \\
&=&\int_{|z|<\xi} \left(\theta(x+z)-\theta(y+z)-\omega(\xi)\right) \left(\frac{1}{z}+\frac{\xi^{2\alpha}}{|z|^{1+2\alpha}}\right)\\ &\,& +\int_{|z|>\xi} \left(\theta(x+z)-\theta(y+z)-\omega(\xi)\right) \frac{\xi^{2\alpha}}{|z|^{1+2\alpha}}\, dz \le 0\end{aligned}$$
The last inequality follows from the facts that $$\theta(x+z)-\theta(y+z)-\omega(\xi) \le 0$$ and that in our region of integration $$\frac{1}{z}+\frac{\xi^{2\alpha}}{|z|^{1+2\alpha}} \ge \frac{1}{z}+\frac{1}{|z|} \ge 0$$ Thus, we have control over the Hilbert transforms near the kernel singularity. Combining our estimates, we get the result. $\Box$
[**Proof of Lemma \[lem\]**]{}
We want to show $$\label{eq:1}
\partial_t \omega(\xi,t_1) > (H\theta(x,t_1)-H\theta(y,t_1))\partial_{\xi}\omega(\xi,\xi_0(t_1)) +d_\alpha(\xi,t_1)$$
From Lemma 3.3 of [@EvReg], we can choose the constant $C_2$ in $\eqref{eq:ODE}$ small enough so that we have $\partial_t \omega(\xi,\xi_0(t))> \frac{1}{4}D_\alpha (\xi,t)$ at $t=t_1$ ($\xi_0'(t)$ is small). By Lemma 5.3 of [@EvReg], we can replace $\ds \xi \int_{\xi/2}^\infty \frac{\omega(r)}{r^2}\, dr$ in Lemma 2.7 by $\omega(\xi,\xi_0)$. Using an argument very similar to Lemma 3.3 of [@EvReg], it can be shown that for all $0<\xi<\delta$, $$C\omega(\xi,\xi_0(t_1))\partial_\xi \omega(\xi,\xi_0(t_1)) \le - \frac{1}{4} D_\alpha (\xi,t_1).$$ where $C$ is the constant from Lemma 2.7. Now, for $0<\xi<\delta$, we have $$\begin{aligned}
C\xi^{2\alpha} \partial_{\xi}\omega(\xi,\xi_0(t_1)) &\le & C\beta H\xi^{2\alpha} \delta^{-\beta}\xi^{\beta-1} = C\beta \frac{H}{\delta^{1-2\alpha}} \left(\frac{\xi}{\delta}\right)^{2\alpha+\beta-1} \end{aligned}$$
By choosing $C_1$ in $H\le C_1 \delta^{1-2\alpha}$ small enough we can bound the expression above by $\ds \frac{1}{2}$. Then $$C\xi^{2\alpha} \partial_{\xi}\omega(\xi,\xi_0(t_1)) \left((-\Delta)^\alpha \theta(x,t_1)-(-\Delta)^\alpha \theta(y,t_1)\right) \le \frac{1}{2} \left((-\Delta)^\alpha \theta(x,t_1)-(-\Delta)^\alpha \theta(y,t_1)\right)$$ Combining these estimates with Lemma \[key\], we have $\eqref{eq:1}$. $\Box $
Well-posedness for Slightly Supercritical Hilbert Model
=======================================================
In this section, we prove global regularity for our model for which the dissipation can be supercritical by a logarithm. Specifically, we will look at solutions of the following equation $$\label{eq:super}
\theta_t= (H\theta)\theta_x-\mathscr{L}\theta , \quad \theta(x,0)=\theta_0(x)$$ for $\theta_0\in H^{3/2}({\ensuremath{\mathbb{R}}\xspace})$, where $\mathscr{L} \theta = \frac{(-\Delta)^{1/2}}{\log(1-\Delta)}\theta$ is a Fourier multiplier operator with multiplier $$P(\xi)= \frac{|\xi|}{\log(1+|\xi|^2)}.$$ For simplicity, we will only concern ourselves with a dissipative operator of this form. The results of this section can easily be generalized to other similar dissipative operators. The main result of this section is the following
[Assume that $\theta_0\in H^{3/2}({\ensuremath{\mathbb{R}}\xspace})$. Then there exists a unique smooth solution $\theta$ of $\eqref{eq:super}$.]{}
First, we have local existence of smooth solutions that we will eventually show can be extended.
[[**Proof.**]{} ]{}This result is analogous to Theorem 4.1 and Proposition 6.2 of Dong [@Dong1] where it is done for the usual fractional laplacian dissipation. The argument for the dissipation we are using is very similar. We will present the modification necessary to make their proof work. The general idea is that $\mathscr{L}$ is more dissipative then $(-\Delta)^\alpha$ for $0<\alpha<1/2$. Let $\theta$ be a solution of $\eqref{eq:super}$ and let $\theta_j=\Delta_j \theta$ be the $j$th Littlewood-Paley projection. Applying $\Delta_j$ to both sides of $\eqref{eq:super}$ we get $$\label{paley}
\partial_t \theta_j+(H\theta) (\theta_j)_x+ \mathscr{L}\theta_j = [H\theta,\Delta_j]\theta_x$$ where $[H,\Delta_j]$ is a commutator with $ [H\theta,\Delta_j]\theta_x= (H\theta)(\theta_j)_x - \Delta_j((H\theta)(\theta_x))$. By applying Plancherel and using that $\Delta_j$ localizes $\theta$ in the frequency space, $$\int_{\ensuremath{\mathbb{R}}\xspace}\theta_j \mathscr{L}\theta_j\, dx \ge 2^{2\alpha j} C \|\theta_j\|_{L^2}^2$$ for some constant $C$. Then by multiplying both sides of $\eqref{paley}$ by $\theta_j$ and integrating we get $$\frac{1}{2} \frac{d}{dt} \|\theta_j\|_{L^2}^2 + 2^{2\alpha j} C \|\theta_j\|_{L^2}^2 \le \int_{\ensuremath{\mathbb{R}}\xspace}\left([H\theta, \Delta_j]\theta_x+ H\theta_x \theta_j/2\right)\theta_j\, dx.$$ This is the same type of inequality used in Dong [@Dong1] and one can apply the methods there to arrive at the a priori bounds needed to conclude local existence as well as higher regularity despite the absence of a divergence free property
Dong also proves a Beale-Kato-Majda type blow up criterion for $\eqref{eq:PDE}$ and the result still holds true for $\eqref{eq:super}$ with our logarithmic dissipation. The contribution from the dissipation term is still non-negative, which is the only fact used about dissipation in his proof. Specifically, by Plancherel, for any regular enough function $f$, $$\int_{\ensuremath{\mathbb{R}}\xspace}f\mathscr{L}f\, dx \ge 0$$ $\Box$
Thus, if we can show $\|\nabla \theta\|_\infty$ is bounded uniformly in time, then Theorem 3.1 is proved. Having such a bound will allow us to extend local solutions indefinitely. To have a bound on $\|\nabla \theta\|_\infty$, we will show the evolution preserves a family of moduli of continuity. If a function $f\in C^2 ({\ensuremath{\mathbb{R}}\xspace})$ obeys a modulus $\omega$ satisfying $\omega'(0)<\infty$ and $\omega''(0)=-\infty$, then $\|\nabla f(x)\|_\infty< \omega'(0)$(see [@SQG]). Therefore, if $\theta$ preserves a modulus of continuity, $\|\nabla \theta\|_\infty< \omega'(0)$.
Writing $\mathscr{L}$ as dissipative nonlocal operator
------------------------------------------------------
In the proofs, it will be easier to write $\mathscr{L}$ as a nonlocal dissipative nonlocal operator, which the following version of lemmas 5.1 and 5.2 from [@Super] allows us to do.
Since are not assured positivity of the kernel $K$, by the previous lemma, we will not have the $L^\infty$ maximum principle. The following result (Lemma 5.4 from [@Super]) allows us to circumvent this.
Using the notation from Lemma 3.3, let $\varphi$ be a smooth radially decreasing function that is identically $1$ on $|y|\le \sigma$ and vanishes identically on $|y|\ge 2\sigma$. Let $$\begin{aligned}
K_1(y) &=& K(y)\varphi(y) \\
K_2(y) &=& K(y)(1-\varphi(y))\end{aligned}$$ Now, we decompose the dissipation term $\mathscr{L}$: $$\label{eq:decomp}
\mathscr{L}\theta(x)= \mathscr{L}_1\theta(x)+\mathscr{L}_2\theta(x) := \int_{\ensuremath{\mathbb{R}}\xspace}(\theta(x)-\theta(x+y))K_1(y)\, dy+ \int_{\ensuremath{\mathbb{R}}\xspace}(\theta(x)-\theta(x+y))K_2(y)\, dy.$$ Let $$m(r)=\frac{1}{C} P(r^{-1}) \varphi(r)$$ where $C$ is the constant from Lemma 3.3. Then we have the following lower bound on $\mathscr{L}_1$ that we will use extensively: $$\mathscr{L}_1\theta(x)\ge \int_{\ensuremath{\mathbb{R}}\xspace}(\theta(x)-\theta(x+y))\frac{m(|y|)}{|y|}\, dy$$ The operator $\mathscr{L}_1$ satisfies the following conditions satisfied by more general nonlocal dissipative operators
1. there exists a positive constant $C_0>0$ such that $$rm(r)\le C_0\,\,\mbox{for all}\,\, r\in (0,2\sigma)$$
for some $r_0>0$.
2. there exists some $a>0$ such that $r^{a} m(r)$ is non-increasing.
We also have the following dissipation estimate whose proof is analogous to Lemma \[dis\]. [Suppose $\theta$ obeys a modulus of continuity $\omega$. Suppose there exists $x,y$ with $|x-y|=\xi>0$ such that $\theta(x)-\theta(y)=\omega(\xi)$. Then $$\mathscr{L}_1\theta(x)-\mathscr{L}_1\theta(y) \ge \mathscr{D}(\xi)$$ where $$\begin{aligned}
\mathscr{D}(\xi) = &A& \int_0^{\xi/2} \left(2\omega(\xi)-\omega(\xi+2\eta)-\omega(\xi-2\eta)\right) \frac{m(2\eta)}{\eta}\, d\eta \\ &+& A\int_{\xi/2}^\infty \left(2\omega(\xi)-\omega(\xi+2\eta)+\omega(2\eta-\xi)\right) \frac{m(2\eta)}{\eta}\, d\eta\end{aligned}$$ and $A$ is a constant.]{}
The Moduli of Continuity
------------------------
The modulus from [@Super] will work here. Fix a small constant $\kappa >0$. For any $B\ge 1$, define $\delta(B)$ to be the solution of $$m(\delta(B)) = \frac{B}{\kappa}.$$ We can also assume that $\delta(B)\le \sigma/2$ by choosing $\kappa$ small enough. Let $\omega_B(\xi)$ be a continuous function with $\omega_B(0)=0$ and $$\begin{aligned}
\omega_B'(\xi) &=& B - \frac{B^2}{2C_a \kappa} \int_0^\xi \frac{3+ \log(\delta(B)/\eta)}{\eta m(\eta)}\, d\eta, \quad \mbox{for}\,\, 0<\xi <\delta(B), \\
\omega_B'(\xi) &=& \gamma m(2\xi), \,\,\,\qquad\qquad\qquad\qquad\qquad\qquad \mbox{for}\,\, \xi>\delta(B),\end{aligned}$$
where $C_a = (1+3a)/a^2$ and $\gamma>0$ is a constant dependent on $\kappa, A,$ and $m$. It is shown in [@Super] that $\omega_B$ is a indeed a modulus of continuity.
Now, we will show that solutions will initially obey some $\omega_B(\xi)$ for some $B$ large enough. Since evolution immediately smooths out the initial data, we can assume $\theta_0$ is a smooth as needed. By Lemma 3.4, it suffices to find $B$ such that $\omega_B(\xi)\ge \min\{\xi \|\nabla \theta_0\|_{L^\infty}, 2M\}$ for all $\xi>0$ where $M$ is from Lemma 3.4. By concavity of $\omega$, we are left to show that $\omega_B(b)\ge 2M$ where $b = 2M/\|\nabla \theta_0\|_{L^\infty}$. Choose $B$ large so $b>\delta(B)$, so $$\omega_B(b)= \omega_B(\delta(B))+\int_{\delta(B)}^b \omega_B'(\eta)\, d\eta \ge \gamma \int_{\delta(B)}^b m(2\eta)\, d\eta \to \infty$$ as $\delta(B)\to 0$. By choosing $B$ possibly even larger we can have $\omega_B(\sigma)\ge 2M\ge 2\|\theta(\cdot,t)\|_{L^\infty}$ where $\sigma$ is from our decomposition of $\mathscr{L}$ earlier. Therefore, the modulus can only be broken for $0<\xi<\sigma$ and solutions will initially obey a modulus from the family $\{\omega_B\}_{B\ge 1}$.
The moduli are preserved
------------------------
To prove a modulus of continuity is preserved, we will rule out the breakthrough scenario described in Lemma \[break\]. Let $t_1$ be the time of breakthrough. By using $\eqref{eq:super}$ and Lemma 3.5, $$\partial_t\left.(\theta(x,t)-\theta(y,t))\right|_{t=t_1} \le \left( H\theta(x,t_1)-H\theta(y,t_1)\right) \omega'(\xi) -\mathscr{D}_B(\xi) + \mathscr{L}_2\theta(y,t_1)-\mathscr{L}_2\theta(x,t_1)$$ where $\xi = |x-y|$. If the right side of the equation above is negative then the modulus was broken at an earlier time, a contradiction. In [@Super], they show $$|\mathscr{L}_2\theta(x,t)-\mathscr{L}_2\theta(y,t)|\le \frac{1}{2} \mathscr{D}_B(\xi)$$ for $0<\xi<\sigma$ so to prove Theorem 3.1, it suffices to show $$\label{eq:show}
\left( H\theta(x,t_1)-H\theta(y,t_1)\right) \omega_B'(\xi) -\frac{1}{2}\mathscr{D}_B(\xi) < 0$$ for $0<\xi<\sigma$ where $\mathscr{D}_B$ is the expression from Lemma 3.5 with $\omega_B$ being the modulus. For simplicity, we will now omit $t_1$ from our expressions involving $\theta$.
[****]{}
By a similar argument to the proof of Lemma \[key\], $$\left|\int_{|x-z|\ge 2\xi} \frac{\theta(z)}{x-z}\, dz-\int_{|y-z|\ge 2\xi} \frac{\theta(z)}{y-z}\, dz\right| \le C\xi \int_\xi^\infty \frac{\omega_B(\eta)}{\eta^2}\, d\eta.$$ Integrating by parts $$\xi \int_\xi^\infty \frac{\omega_B(\eta)}{\eta^2}\, d\eta = \omega_B(\xi)+\gamma \xi \int_\xi^\infty \frac{m(2\eta)}{\eta}\, d\eta.$$ By property (2) of $m$, $$\int_\xi^\infty \frac{m(2\eta)}{\eta}\, d\eta \le \xi^a m(2\xi)\int_\xi^\infty \frac{1}{\eta^{1+a}}\, d\eta \le \frac{m(2\xi)}{a}$$ Now, we have $$\xi \int_\xi^\infty \frac{\omega_B(\eta)}{\eta^2}\, d\eta \le \omega_B(\xi) + \frac{\gamma\xi m(2\xi)}{a}$$ For $\delta(B) \le \xi \le 2\delta(B)$, it is not hard to see that $$\frac{\gamma\xi m(2\xi)}{a} \le \omega_B(\xi),$$ the details are in [@Super]. For $\xi > 2\delta(B)$, we have $\xi -\delta(B) \ge \xi/2$ so $$\omega_B(\xi) \ge \gamma \int_{\delta(B)}^\xi m(2\eta)\, d\eta \ge \gamma m(2\xi)(\xi-\delta(B)) \ge \frac{\gamma \xi m(2\xi)}{2}.$$ Thus, we have, $$C\xi \int_\xi^\infty \frac{\omega_B(\eta)}{\eta^2}\, d\eta \le C\left(1+ \frac{2}{a}\right) \omega_B(\xi).$$ In [@Super], they prove the following estimate on the dissipation term $$-\mathscr{D}_B (\xi) \le -\frac{2-c_a}{C} \omega_B(\xi) m(2\xi)$$ where $c_a=1+(3/2)^{-a}$. Then we obtain $$\left(\int_{|x-z|\ge 2\xi} \frac{\theta(z)}{x-z}\, dz-\int_{|y-z|\ge 2\xi} \frac{\theta(z)}{y-z}\, dz\right) \omega_B'(\xi) - \frac{1}{4} \mathscr{D}_B(\xi) \le \left(C\gamma \frac{a+2}{a} - \frac{2-c_a}{4C}\right) \omega_B(\xi) m(2\xi) < 0$$ if we set $\gamma$ small enough. In other words, we have used some of the dissipation to control the modulus of the Hilbert transform away from the kernel singularity. Now, we will concern ourselves with the other part of the Hilbert transform. A novel step is that instead of using $\mathscr{D}_B$ we will use the expression for $\mathscr{L}_1\theta$ directly. We want to show $$\label{eq:show1}
\left(\int_{|x-z|\le 2\xi} \frac{\theta(z)}{x-z}\, dz-\int_{|y-z|\le 2\xi} \frac{\theta(z)}{y-z}\, dz\right) \omega_B'(\xi) - \frac{1}{4}( \mathscr{L}_1\theta(x)-\mathscr{L}_1\theta(y)) < 0.$$ After a similar manipulation as in the proof of Lemma 2.6, the left side of $\eqref{eq:show1}$ is precisely $$\begin{aligned}
&\, & \int_{|z| <2\xi} \left(\theta(y)-\theta(z+y)-\theta(x)+\theta(x+z)\right) \left[ \frac{\omega_B'(\xi)}{z} +\frac{1}{4} \frac{m(z)}{|z|}\right]\, dz \\
&\, & + \int_{|z|\ge 2\xi} \left(\theta(y)-\theta(z+y)-\theta(x)+\theta(x+z)\right) \frac{1}{4} \frac{m(z)}{|z|}\, dz\end{aligned}$$ By hypothesis, we have $\theta(y)-\theta(x)= \omega_B(\xi)$, so $$\theta(y)-\theta(z+y)-\theta(x)+\theta(x+z)= \omega_B(\xi)- \theta(z+y)+\theta(x+z)< 0.$$ If we can show $$\frac{\omega_B'(\xi)- \frac{1}{4}m(z)}{z} > 0$$ for $-2\xi<z<0$ then we are done. However, $$\omega_B'(\xi)- \frac{1}{4}m(z) = \gamma m(2\xi)-\frac{1}{4}m(z) < 0$$ from the fact that $m$ is non-increasing and choosing $\gamma$ small enough. Therefore, we have $\eqref{eq:show}$ and the case when $\xi \ge \delta(B)$ is complete.
[****]{} The argument for this case is exactly the same as in [@Super] with no modifications. Therefore, the proof of theorem 3.1 is complete. $\Box$
The author wishes to thank Prof. Alex Kiselev for the introduction into the subject and Prof. Hongjie Dong for helpful comments. The author also acknowledges the support of the NSF grant DMS 1147523 and DMS 1159133 at UW-Madison.
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---
abstract: 'We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with a given number of states that is consistent with a given finite sample of weight-labelled words or trees? We show that, over both words and trees, this decision problem is interreducible with the problem of deciding the truth of existential first-order sentences over the field of rationals—whose decidability is a longstanding open problem.'
address:
- 'University of Oxford, UK'
- 'University of Oxford, UK'
- 'University of Oxford, UK'
author:
- Stefan Kiefer
- Ines Marušić
- James Worrell
bibliography:
- 'references.bib'
title: Minimisation of Multiplicity Tree Automata
---
|
---
abstract: 'Let $G$ be a finite group. In this short note, we give a criterion of nilpotency of $G$ based on the existence of elements of certain order in each section of $G$.'
address: |
Marius Tărnăuceanu\
Faculty of Mathematics\
”Al.I. Cuza” University\
Iaşi\
Romania
author:
- Marius Tărnăuceanu
date: '2016/2017'
title: A nilpotency criterion for finite groups
---
Introduction
============
The problem of detecting structural properties of finite groups by looking at element orders has been considered in many recent papers (see e.g. [@1] and [@3]-[@6]). In the current note, we identify a new property detecting nilpotency of a finite group $G$ that uses the function $$\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$$introduced and studied in [@9]. The proof that we present is founded on the structure of minimal non-nilpotent groups (also called *Schmidt groups*) given by [@8].
It is well-known that a finite nilpotent group $G$ contains elements of order $\exp(G)$. Moreover, all sections of $G$ have this property. Under the above notation, this can be written alternatively as $$\varphi(S)\neq 0 \mbox{ for any section } S \mbox{ of } G.{\leqno}(1)$$Our main theorem shows that the converse is also true, that is we have the following nilpotency criterion.
[**Theorem 1.**]{} [*Let $G$ be a finite group. Then $G$ is nilpotent if and only if $\varphi(S)\neq 0$ for any section $S$ of $G$.*]{}
Note that (1) implies $$\varphi(S)\neq 0 \mbox{ for any subgroup } S \mbox{ of } G{\leqno}(2)$$and in particular $$\varphi(G)\neq 0.{\leqno}(3)$$We observe that the condition (3) is not sufficient to guarantee the nilpotency of $G$, as shows the elementary example $G=\mathbb{Z}_6\times S_3$; we can even construct a non-solvable group $G$ for which $\varphi(G)\neq 0$, namely $G=\mathbb{Z}_n\times H$, where $H$ is a simple group of exponent $n$. A similar thing can be said about the condition (2).
[**Example.**]{} Let $G$ be a nontrivial semidirect product of a normal subgroup isomorphic to $$E(5^3)=\langle x,y \mid x^5=y^5=[x,y]^5=1, [x,y]\in Z(E(5^3))\rangle$$by a subgroup $\langle a\rangle$ of order $3$ such that $a$ commutes with $[x,y]$. Then $G$ is a non-CLT group of order $375$, more precisely it does not have subgroups of order $75$. We infer that its subgroups are: $G$, all subgroups contained in the unique Sylow $5$-subgroup, all Sylow $3$-subgroups, and all cyclic subgroups of order $15$. Clearly, $G$ satisfies the condition (2), but it is not nilpotent.
Finally, we note that our criterion can be used to prove the non-nilpotency of a finite group by looking to its sections. In [@9] we have determined several classes of groups $G$ satisfying $\varphi(G)=0$, such as dihedral groups $D_{2n}$ with $n$ odd, non-abelian $P$-groups of order $p^{n-1}q$ [(]{}$p>2,q$ primes, $q \mid p-1$[)]{}, symmetric groups $S_n$ with $n\geq3$, and alternating groups $A_n$ with $n\geq4$. These examples together with Theorem 1 lead to the following corollary.
[**Corollary 2.**]{} [*If a finite group $G$ contains a section isomorphic to one of the above groups, then it is not nilpotent.*]{}
Proof of Theorem 1
==================
We will prove that a finite group all of whose sections $S$ satisfy $\varphi(S)\neq 0$ is nilpotent. Assume that $G$ is a counterexample of minimal order. Then $G$ is a Schmidt group since all its proper subgroups satisfy the hypothesis. By [@8] (see also [@2; @7]) it follows that $G$ is a solvable group of order $p^mq^n$ (where $p$ and $q$ are different primes) with a unique Sylow $p$-subgroup $P$ and a cyclic Sylow $q$-subgroup $Q$, and hence $G$ is a semidirect product of $P$ by $Q$. Moreover, we have:
- if $Q=\langle y\rangle$ then $y^q\in Z(G)$;
- $Z(G)=\Phi(G)=\Phi(P)\times\langle y^q\rangle$, $G'=P$, $P'=(G')'=\Phi(P)$;
- $|P/P'|=p^r$, where $r$ is the order of $p$ modulo $q$;
- if $P$ is abelian, then $P$ is an elementary abelian $p$-group of order $p^r$ and $P$ is a minimal normal subgroup of $G$;
- if $P$ is non-abelian, then $Z(P)=P'=\Phi(P)$ and $|P/Z(P)|=p^r$.
We infer that $S=G/Z(G)$ is also a Schmidt group of order $p^rq$ which can be written as semidirect product of an elementary abelian $p$-group $P_1$ of order $p^r$ by a cyclic group $Q_1$ of order $q$ (note that $S_3$ and $A_4$ are examples of such groups). Clearly, we have $$\exp(S)=pq.$$On the other hand, it is easy to see that $$L(S)=L(P_1)\cup\{Q_1^x \mid x\in S\}\cup\{S\}.$$Thus, the section $S$ does not have cyclic subgroups of order $pq$ and consequently $\varphi(S)=0$, a contradiction. This completes the proof.
------------------------------------------------------------------------
[10]{} , [Sums of element orders in finite groups]{}, [*Comm. Algebra*]{} [**37**]{} (2009), no. 9, 2978–2980. , [On finite minimal non-nilpotent groups]{}, [*Proc. Amer. Math. Soc.*]{} [**133**]{} (2015), 3455–-3462. , [Inequalities detecting structural properties of a finite group]{}, [*Comm. Algebra*]{} [**45**]{} (2017), no. 2, 677–687. , [An exact upper bound for sums of element orders in non-cyclic finite groups]{}, http://arxiv.org/abs/1610.03669. , [Finite groups determined by an inequality of the orders of their subgroups]{}, [*Bull. Belg. Math. Soc. Simon Stevin*]{} [**15**]{} (2008), 699–-704. , [An inequality detecting nilpotency of finite groups]{}, http://arxiv.org/abs/1207.1020. , [The Schmidt subgroups, its existence, and some of their applications]{}, [*Ukraini. Mat. Congr. 2001*]{}, Kiev, 2002, Section 1, 81–90. , [Groups whose all subgroups are special]{}, [*Mat. Sb.*]{} [**31**]{} (1924), 366–372. , [A generalization of the Euler’s totient function]{}, [*Asian-Eur. J. Math.*]{} [**8**]{} (2015), no. 4, article ID 1550087.
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---
abstract: 'We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive examples that confirm this conjecture. As a by-product, we also obtain a classification of all spherical Pythagorean triples, previously found by Smith.'
author:
- 'Alexander Kolpakov & Sinai Robins'
title: |
Spherical tetrahedra with rational volume,\
and spherical Pythagorean triples
---
Introduction {#section:pre}
============
A *spherical tetrahedron* $T$ can be defined as the intersection of a simplicial cone in $\mathbb R^4$ with the unit sphere $\mathbb{S}^3$ centred at the origin. In other words, $T$ has four vertices connected by spherical geodesics on $\mathbb{S}^3$ that comprise its edges, and each of its vertices is the intersection of exactly three of its spherical facets. A *spherical Coxeter tetrahedron* $T$ is a spherical tetrahedron whose six dihedral angles are of the form $\pi/n$, with $n\geq 2$.
A complete list of spherical Coxeter tetrahedra was produced by Coxeter [@Coxeter], and shows that there are eleven types of spherical Coxeter tetrahedra in $\mathbb{S}^3$. Let $S_i$, $i=1,\dots,11$, denote these spherical tetrahedra, as presented in Table \[tabular:coxeter-tetrahedra\].
In the present paper we study *rational spherical tetrahedra*, as generalisations of spherical Coxeter tetrahedra, where we now allow their dihedral angles to be arbitrary rational multiples of $\pi$. An important focus here is the determination of their volume, which is also called a solid angle in some of the literature.
The volume of a spherical Coxeter tetrahedron is easily seen to be a rational multiple of the total volume of the sphere $\mathbb{S}^3$, which is $2 \pi^2$. We describe a wide class of rational spherical tetrahedra whose volumes are rational multiples of $\pi^2$, in relation to the work of Cheeger and Simons [@CS].
[c|c|c|c]{} $i$& Symbol& Coxeter diagram& Volume\
1& $A_4$&
(0,0) – (1,0); (1,0) – (2,0); (2,0) – (3,0);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{60}$\
2& $B_4$&
(0,0) – (1,0); (1,0) – (2,0); (2,0.05) – (3,0.05); (2,-0.05) – (3,-0.05);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{192}$\
3& $D_4$&
(0,0) – (1,0); (1,0) – (2,0.5); (1,0) – (2,-0.5);
(0,0) circle(.1); (1,0) circle(.1); (2,0.5) circle(.1); (2,-0.5) circle(.1);
& $\frac{\pi^2}{96}$\
4& $H_4$&
(0,0) – (1,0); (1,0) – (2,0); (2,0.05) – (3,0.05); (2,0) – (3,0); (2,-0.05) – (3,-0.05);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{7200}$\
5& $F_4$&
(0,0) – (1,0); (1,0.05) – (2,0.05); (1,-0.05) – (2,-0.05); (2,0) – (3,0);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{576}$\
6& $A_3\times A_1$&
(0,0) – (1,0); (1,0) – (2,0);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{24}$\
7& $B_3\times A_1$&
(0,0) – (1,0); (1,0.05) – (2,0.05); (1,-0.05) – (2,-0.05);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{48}$\
8& $H_3\times A_1$&
(0,0) – (1,0); (1,0.05) – (2,0.05); (1,0) – (2,0); (1,-0.05) – (2,-0.05);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{120}$\
9& $I_2(k)\times I_2(l)$&
(0,0) – (1,0); (2,0) – (3,0);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
at (0.5, 0.3) [$k$]{}; at (2.5, 0.3) [$l$]{};
& $\frac{\pi^2}{2kl}$\
10& $I_2(k)\times A_1^{\times 2}$&
(0,0) – (1,0);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
at (0.5, 0.3) [$k$]{};
& $\frac{\pi^2}{4k}$\
11& $A_1^{\times 4}$&
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
& $\frac{\pi^2}{8}$\
In this work, an angle $\alpha$ (assumed to be a plane angle of a polygon, or a dihedral angle of a polyhedron) is called *rational* if $\alpha \in \pi\,\mathbb{Q}$. Similarly, an edge of a polygon (or an edge length of a polyhedron) of length $l$ is called *rational* if $l \in \pi \, \mathbb{Q}$. Finally, an $n$-tuple of numbers $(x_1, \dots, x_n)$ is *rational*, if $x_i \in \pi\,\mathbb{Q}$ for every $1 \leq i \leq n$.
Descending to $\mathbb{S}^2 \subset \mathbb R^3$, a [*spherical Pythagorean triple*]{} is defined to be a rational solution $(p, q, r)$ to $$\label{eq:Pythagorean-triple}
\cos p \cdot \cos q + \cos r = 0,$$ where $\pi - p$, $\pi - q$, and $\pi - r$ are the side lengths of a spherical right triangle $T$. The side lengths of a spherical triangle are subject to several additional constraints on $p$, $q$ and $r$: $$0<\, p,\,\, q,\,\, r\, < \pi, \hspace {0.25in} p + q + r < 2\pi,$$ $$p + q < r,\hspace {0.5in} p + r < q,\hspace {0.5in} q + r < p.$$ We relax the above conditions and call any solution of , with $0 < p, q, r < \pi$, a Pythagorean triple.
\[q1\] Is there any reasonably simple classification of rational Pythagorean triples, corresponding to the side lengths of a spherical triangle?
Returning to $\mathbb{S}^3$, we focus on a broader class of “Pythagorean quadruples”, that will become useful in the discussion of $\mathbb{Z}_2$-symmetric spherical tetrahedra with rational dihedral angles (or *rational tetrahedra*, for short) later on. To this end, we call $(p, q, r, s)$ a *spherical Pythagorean quadruple* if it is a solution to the equation $$\label{eq:Pythagorean-quadruple}
\cos p \cdot \cos q+ \cos \frac{r+s}{2}\cdot \cos \frac{r-s}{2} = 0.$$ Here, we shall suppose that $0 < p, q, r, s < \pi$. The corresponding spherical tetrahedron, if it exists, looks akin to the one depicted in Figure \[fig:z2-symmetric-tetrahedron\] and is called a $\mathbb{Z}_2$-symmetric (spherical) tetrahedron.
![The dihedral angles (left) and edge lengths (right) of a $\mathbb{Z}_2$-symmetric tetrahedron $T$.[]{data-label="fig:z2-symmetric-tetrahedron"}](z2-symmetric-tetrahedron.pdf)
We note that a quadruple with $r = s$ corresponds to the usual Pythagorean triple $(p, q, r)$.
\[q2\] Is there any reasonably simple classification of rational Pythagorean quadruples corresponding to the dihedral angles of a spherical tetrahedron?
We shall answer Questions \[q1\] and \[q2\] simultaneously by classifying all Pythagorean quadruples.
\[thm:Pythagorean\] There exist exactly $59$ sporadic Pythagorean quadruples, and $42$ continuous families of Pythagorean quadruples corresponding to the dihedral angles of a $\mathbb{Z}_2$-symmetric spherical tetrahedron.
The proof of Theorem \[thm:Pythagorean\] is contained in Section 2.1 for the case of sporadic instances listed in Appendix A, and in Section 2.2 for the case of continuous families listed in Appendix B. The main tool in our proof is a very basic enumeration realised by a `SageMath` script Monty [@Monty]. Thus, Theorem \[thm:Pythagorean\] extends a result (unpublished) of Smith [@Smith] that classifies rational spherical Pythagorean triples by using a beautiful geometric connection with *three-dimensional* Coxeter simplices:
\[thm:Smith\] Aside from the trivial continuous family of solutions $(\pi/2, b, \pi/2)$, $0\leq b \leq \pi/2$, there is exactly one solution $(a,b,c)$ to $\cos a \cos b = \cos c$ with $0 \leq a, b, c \leq \pi/2$ being rational multiples of $\pi$, namely $(\pi/4, \pi/4, \pi/3)$.
In terms of equation , the non-trivial triple in the above theorem is $(\pi/4, \pi/4, 2\pi/3)$. All rational Pythagorean triples found by Smith belong to the continuous families of quadruples described in Section \[section:spherical-triples\].
The following statement is an observation which had its origin in the list of rational spherical Pythagorean quadruples, and which is of interest in the context of [@Felikson1; @Felikson2].
\[thm:rational-not-coxeter-tetrahedron\] There exists a rational tetrahedron in $\mathbb{S}^3$ whose volume has a value in $\pi^2\,\mathbb{Q}$, and which is not decomposable into any finite number of spherical Coxeter tetrahedra.
Thus, we can show that the property of “being rational” for a spherical tetrahedron is very far from “being Coxeter”, even if its volume is a rational multiple of $\pi^2$, which is always true for Coxeter tetrahedra in $\mathbb{S}^3$. Here we recall that $\mathbb{S}^3$ has volume $2\pi^2$ in its natural metric of constant sectional curvature $+1$, and that every Coxeter polyhedron in $\mathbb{S}^3$ is a tetrahedron, which generates a finite discrete reflection group by reflection in its faces.
The first open problem that Theorem \[thm:rational-not-coxeter-tetrahedron\] vaguely relates to is Schläfli’s Conjecture:
(Schläfli) Let $T$ be an orthoscheme in $\mathbb{S}^3$ with rational dihedral angles. Then the volume of $T$ takes values in $\pi^2\, \mathbb{Q}$ if and only if $T$ is a Coxeter orthoscheme.
The above statement can be generalised for spherical simplices of dimension $\geq 4$, and this is how it actually appears in Scläfli’s original work [@Schlafli p. 267, Formeln (4)–(5)]. Here, an orthoscheme is a tetrahedron with three mutually orthogonal faces that do not share a common vertex. However, the tetrahedron mentioned in Theorem \[thm:rational-not-coxeter-tetrahedron\] is not an orthoscheme.
Another related open problem is the following question posed in [@CS] by Cheeger and Simons, and known as the Rational Simplex Conjecture:
*Is it true that the volume of a rational spherical tetrahedron always takes values in $\pi^2\,\mathbb{Q}$?*
The putative answer would be negative for “virtually all” rational simplices. Our results only show that the Rational Simplex Conjecture may hold for a tetrahedron which is geometrically “far enough” from a Coxeter tetrahedron, and thus one may still expect many “positive examples”. Finally, we can produce many pairs of non-isometric rational tetrahedra with equal volumes and Dehn invariants. In view of Hilbert’s 3^rd^ problem, it would be natural to ask if our examples are scissors congruent.
**Acknowledgements**\
[The authors gratefully acknowledge the support they received from Brown University and the Institute for Computational and Experimental Research in Mathematics - ICERM (Providence, USA), where most of this work has been conceived and carried out, and especially thank the ICERM program “Point Configurations in Geometry, Physics and Computer Science” for hospitality and excellent working atmosphere during their stay. A.K. has been supported by the Swiss National Science Foundation - SNSF project no. PP00P2-170560, S.R. has been supported by the São Paulo Research Foundation - FAPESP project no. 15/10323-7. The authors are grateful to Don Zagier (MPIM, Germany and ICTP, Italy), John Parker (Durham University, UK), Richard Schwartz (Brown University, USA) and Ruth Kellerhals (Université de Fribourg, Switzerland) for stimulating discussions. They also thank the anonymous referees for their valuable comments and suggestions. ]{}
Pythagorean quadruples {#section:spherical-triples}
======================
Let a spherical tetrahedron $T$ be defined as an intersection of a simplicial cone $C$ in $\mathbb{R}^4$ centred at the origin with the unit sphere $\mathbb{S}^3 = \{ \mathbf{v} = (x_1, x_2, x_3, x_4) \in \mathbb{R}^4\, |\, \| \mathbf{v} \| = 1 \}$. We suppose that the dihedral angles of $C$ belong to the interval $(0, \pi)$.
The dihedral angles of $T$ are equal to the corresponding dihedral angles between the three-dimensional faces of its defining cone $C$ measured at its two-dimensional faces. The edge lengths of $T$ correspond to the plane angles in the two-dimensional proper sub-cones measured at the origin.
The polar dual $T^*$ of a spherical tetrahedron $T$, defined by a cone $C$, is the intersection of the dual cone $C^*$ with $\mathbb{S}^3$.
We recall that a spherical tetrahedron is called $\mathbb{Z}_2$-symmetric, if it admits such a distribution of dihedral angles values as shown in Figure \[fig:z2-symmetric-tetrahedron\]. A Pythagorean quadruple of dihedral angles $(p, q, r, s)$ of a $\mathbb{Z}_2$-symmetric spherical tetrahedron is a solution to equation .
Then, by cosidering polar duals, one can deduce from Proposition 6 of [@KMP] the following:
\[prop:volume-angles\] If $p$, $q$, $r$ and $s$ are the dihedral angles of a $\mathbb{Z}_2$-symmetric spherical tetrahedron $T$, for which equation holds, then the volume of $T$ can be expressed as $$\label{eq:volume-angles}
\mathrm{Vol}\, T = \frac{1}{2}\left( \frac{r\,(2\pi-r)}{2} + p^2 + q^2 + \frac{s\,(2\pi-s)}{2} - \pi^2 \right).$$
Thus, once the dihedral angles of a tetrahedron $T$ as above are rational, then its volume has a value in $\pi^2\,\mathbb{Q}$. It also follows from [@KMP Proposition 6] (and the discussion preceding it), that a rational $\mathbb{Z}_2$-symmetric tetrahedron has rational edge lengths. Namely, the following holds.
\[prop:edge-lengths\] If $(p, q, r, s)$ is the quadruple of dihedral angles of a $\mathbb{Z}_2$-symmetric spherical tetrahedron $T$, for which equation holds, then the lengths of its respective edges, as depicted in Figure \[fig:z2-symmetric-tetrahedron\] are given by the quadruple $(\ell_p, \ell_q, \ell_r, \ell_s) = (p, q, \pi - r, \pi - s)$.
Once we have $r = s$ for a spherical $\mathbb{Z}_2$-symmetric tetrahedron $T$, we get a triple $(p, q, r)$, which corresponds in this case to a symmetric spherical tetrahedron, rather than to a triangle. However, $(p,q,r)$ is a Pythagorean triple in the sense of our initial definition. Indeed, for each vertex $v$ of $T$ in this case, its link $\mathrm{Lk}_v$ is a rational spherical triangle with plane angles $p$, $q$, and $r$. Its dual $\mathrm{Lk}^*_v$ is a spherical triangle with edge length $\pi - p$, $\pi - q$, $\pi - r$, while $p$, $q$, and $r$ satisfy equation .
A Pythagorean quadruple $(p, q, r, s)$ represents the dihedral angles of a $\mathbb{Z}_2$-symmetric spherical tetrahedron $T$, if and only if the associated Gram matrix $$\label{eq:Gram-matrix}
G = G(T) := \left( \begin{array}{cccc}
1& -\cos r& -\cos p& -\cos q\\
-\cos r& 1& -\cos q& -\cos p\\
-\cos p& -\cos q& 1& -\cos s\\
-\cos q& -\cos p& -\cos s& 1
\end{array} \right)$$ is positive definite [@Luo Lemma 1.2].
Thus, once we have a rational solution $(p, q, r, s)$ to , then we only need to check if the Gram matrix $G(T)$ given by is positive definite. If it is indeed the case, then we obtain a rational spherical tetrahedron $T$ such that $\mathrm{Vol}\,T \in \pi^2\, \mathbb{Q}$.
First of all, finding a solution to equation is equivalent to finding a solution to the equation $$\label{eq:quad-angles-1}
\cos(a) + \cos(b) + \cos(c) + \cos(d) = 0,$$ while the correspondence between two sets of solutions is given by $$\label{eq:quad-angles-2}
p = \frac{a+b}{2}, \,\, q = \frac{a-b}{2}, \,\, r = c,\,\, s = d.$$
We shall search for all possible solutions to – , such that $0 < p, q, r, s < \pi$, and $r \geq s$. The former condition is necessary for the dihedral angles of a spherical tetrahedron $T$, and the latter can be assumed since $r$ and $s$, as drawn in Figure \[fig:z2-symmetric-tetrahedron\], can be interchanged by an isometry of $\mathbb{S}^3$ without interchanging $p$ and $q$.
If $(a, b, c, d)$ is a rational quadruple, then turns out to be a trigonometric Diophantine equation which has been studied by Conway and Jones in [@CJ]. All of its solutions such that $0 < a$, $b$, $c$, $d < \frac{\pi}{2}$ are listed in [@CJ Theorem 7]. For convenience, its statement is reproduced below, although using a slightly different notation.
\[thm:cj\] Suppose that we have at most four rational multiples of $\pi$ lying strictly between $0$ and $\pi/2$ for which some rational linear combination $S$ of their cosines is rational, but no proper subset has this property. Then $S$ is proportional to one of the following list:
- $\cos \frac{\pi}{3} - \cos \frac{\pi}{3}$ ($=0$),
- $-\cos t + \cos \left( t + \frac{\pi}{3} \right) + \cos \left( t - \frac{\pi}{3} \right)$ ($=0$),
- $\cos \frac{\pi}{5} - \cos \frac{2\pi}{5} - \cos \frac{\pi}{3}$ ($=0$),
- $\cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} - \cos \frac{\pi}{3}$ ($=0$),
- $ \cos \frac{\pi}{5} - \cos \frac{\pi}{15} + \cos \frac{4\pi}{15} - \cos \frac{\pi}{3}$ ($=0$),
- $-\cos \frac{2\pi}{5} + \cos \frac{2\pi}{15} - \cos \frac{7\pi}{15} - \cos \frac{\pi}{3}$ ($=0$),
- $\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} - \cos \frac{\pi}{21} + \cos \frac{8\pi}{21}$ $\left( =\frac{1}{2} \right)$,
- $\cos \frac{\pi}{7} - \cos \frac{2\pi}{7} + \cos\frac{2\pi}{21} - \cos \frac{5\pi}{21}$ $\left( =\frac{1}{2} \right)$,
- $- \cos \frac{2\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{4\pi}{21} + \cos \frac{10 \pi}{21}$ $\left( = \frac{1}{2} \right)$,
- $- \cos \frac{\pi}{15} + \cos \frac{2\pi}{15} + \cos \frac{4\pi}{15} - \cos \frac{7\pi}{15}$ $\left( = \frac{1}{2} \right)$.
According to Theorem \[thm:cj\] there is a single continuous family of linear combinations of cosines, depending on a real-valued parameter $t$, which, for every instance of $t \in \pi \mathbb{Q}$, provides a rational solution to . The remaining linear combinations we call sporadic, in order to distinguish them from continuous families. Also, our methods to handle sporadic solutions to and their continuous families will be slightly different, since the former require more computations to be performed (first, numerically, and then exactly by verifying the respective minimal polynomials), while the latter need more symbolic algebra and the use of `SymPy` [@SymPy].
Rational spherical tetrahedra: $59$ sporadic instances {#section:sporadic}
------------------------------------------------------
Let the rational length of a quadruple $(a, b, c, d)$ giving rise to the trigonometric sum $S = \cos a + \cos b + \cos c + \cos d$ in be defined as the maximal length of its sub-sum $S'$, such that $S' \in \mathbb{Q}$, but for any proper sub-sum $S''$ of $S'$ it still holds that $S'' \notin \mathbb{Q}$.
Then, we can already notice that there is no solution to of rational length $4$. Indeed, each linear combination of rational length $4$ would yield an expression $S$ equal to the right-hand side of items $7$, $8$, $9$, or $10$ in Theorem \[thm:cj\], up to a sign. None of those sums evaluates to $0$.
The sporadic solutions to mentioned in items $4$, $5$, and $6$ of Theorem \[thm:cj\] have rational length $3$. The one mentioned in item 3 has rational length $2$. Finally, only those solutions where each cosine term of $S$ above is a rational number have rational length $1$. The latter is possible only if $a, b, c, d \in \{ 0, \pi/3, \pi/2 \}$, given that $0 \leq a$, $b$, $c$, $d \leq \frac{\pi}{2}$ .
However, Theorem \[thm:cj\] provides only the sub-sums realising the rational length of $S$, and says nothing about the remaining part of the sum, which may have itself various rational length (e.g. if $S$ has rational length $2$ realised by a sub-sum $S'$, then $S - S'$ may have rational length $1$ or $2$).
We shall need a wider range of dihedral angles represented by the Pythagorean quadruple $(a, b, c, d)$, namely $0 < a$, $b$, $c$, $d < \pi$. Thus, for each dihedral angle in each entry on the list of Theorem \[thm:cj\], we also consider its complement to $\pi$ and $2\pi$, respectively. However, we always keep in mind that any angle in the interval $(0,\pi)$, as above, can be brought to an angle in $(0, \pi/2)$ in such a way that we do not create any new sums as compared to Theorem \[thm:cj\], since the only difference will be some cosines in $S$ changing their signs.
Moreover, if we assume that $a$, $b$, $c$, $d \in (0, \pi)$ instead of $(0, \pi/2)$, we need to consider one more continuous family in addition to the ones already mentioned in Theorem \[thm:cj\]. Namely, we need to consider $\cos \alpha + \cos \beta = 0$, with $\alpha = t$, $\beta = \pi - t$, and $t \in (0, \pi)$, as well as all possible complements of $\alpha$ and $\beta$ to $\pi$ and $2\pi$.
In order to simplify our search algorithm (at the cost of making it overall less efficient), we shall for each rational length of $S$ look at the set of possible denominators of the angles involved in $S'$ realising said length, and at the set of denominators realising any possible rational length of $S - S'$. Then we shall obtain a list of possible denominators $\delta_a$, $\delta_b$, $\delta_c$, $\delta_d$ that $a = \frac{\nu_a}{\delta_a} \pi$, $b = \frac{\nu_b}{\delta_b} \pi$, $c = \frac{\nu_c}{\delta_c} \pi$, $d = \frac{\nu_d}{\delta_d} \pi$ may have, and choose their numerators $\nu_a$, $\nu_b$, $\nu_c$, $\nu_d$ so that $0 < a$, $b$, $c$, $d < \pi$. If any number of the form $\frac{\nu}{\delta} \pi$ equals $0$, then we assume $\delta = \infty$. Such an approach is still practically reasonable, and takes about $90$ minutes in total to run in `SageMath` [@Sage] on a MacPro 2.3 GHz Intel Core i5 Processor with 8 Mb RAM. An observation from Galois theory implies that if $S = \cos a + \cos b + \cos c + \cos d$ has rational length $1$, then the list of possible denominators of angles in $S$ is $L_0= \{ 1, 2, 3, \infty \}$.
If $S$ has rational length $2$ realised by a sub-sum $S'$, then the list of possible denominators in $S'$ is $L_1 = \{ 3, 5 \}$ as indicated by item $3$ of Theorem \[thm:cj\], while the denominators in $S - S'$ can belong either to $L_0$ or to $L_1$.
If $S$ has rational length $3$ realised by a sub-sum $S'$, then the denominators of angles in $S'$ belong either to the list $L_2 = \{ 3, 7 \}$, or to $L_3 = \{ 3, 5, 15 \}$, as indicated by items $4$, $5$ and $6$ of Theorem \[thm:cj\], while the denominators of the remaining term $S - S'$ belong to $L_0$.
In Monty [@Monty] we use a brute-force search over the set of all dihedral angles with denominators from the union of the above mentioned lists $L_i$, $i \in \{0, 1, 2, 3\}$. This does not result in an à priori efficient search, however turns out to be sufficient to find all sporadic solutions to and, subsequently, to .
Each time a “numerical” zero is obtained in Monty’s search, i.e. the condition $|S| < 10^{-8}$ is satisfied (which is a very generous margin for a numerical zero, since Monty’s machine precision is $10^{-16}$), the minimal polynomial for $S$ is computed. Since $S$ is an algebraic integer, this test is sufficient to verify that $S = 0$.
In each of the cases above, we check if the resulting dihedral angles $p$, $q$, $r$, $s$ of a “candidate” tetrahedron $T$ belong to the interior of the interval $(0, \pi)$, and whether the corresponding Gram matrix $G = G(T)$ is positive definite. The former condition guarantees that the first two corner minors $G_1 = 1$ and $G_2 = \sin^2 r$ of $G$, respectively of rank $1$ and $2$, are positive, and we need specifically to check only $G_3$ and $G_4 = \det G$. In Monty’s search, $G_i$ is considered positive if $G_i > 10^{-8}$, which is again a generous numerical margin to decide if a number is positive. In order to verify that no possible solution is left out, we check if $G_i$ within the $10^{-8}$-neighbourhood of $0$ is actually $0$, by using minimal polynomials. Otherwise, $G_i < - 10^{-8}$, and is indeed negative.
Finally, Monty finds $172$ sporadic solutions. Since the dihedral angles of all the listed tetrahedra satisfy equation , their volumes are rational multiples of $\pi^2$ by Proposition \[prop:volume-angles\].
There are, however, some of the sporadic solutions which belong by chance to one of the $42$ continuous families described in the next section. For brevity, we exclude them from our final list, and only $59$ genuinely sporadic solutions are presented in Appendix A.
Rational spherical tetrahedra: $42$ continuous families {#section:families}
-------------------------------------------------------
By using a method analogous to the above, we find $34$ one-parameter continuous families, and $8$ two-parameter continuous families of rational spherical tetrahedra whose volumes take values in $\pi^2 \mathbb{Q}$. Those families are listed in Appendix B. When dealing with symbolic computations in Monty [@Monty], we employ `SymPy` [@SymPy] in order to simplify expressions and check whether $S = 0$, rather than using the minimal polynomial test.
In the case of continuous families, we have only two types of sub-sums $S'$ appearing in $S$, which depend on a parameter:
- either a sub-sum of the form indicated in item 2 of Theorem \[thm:cj\],
- or a sub-sum of the form $S'(t) = \cos(t) - \cos(t) = \cos(t) + \cos(\pi - t)$.
In the former case three of the angles $a$, $b$, $c$ and $d$ is belong to the list $L_0 = \{ \pi/3 - t, \pi/3 + t, 2\pi/3 - t, 2\pi/3 + t, \pi - t, t, \pi + t, 5 \pi/3 - t, 5\pi/3 + t \}$, with $t \in (0, \pi/6)$, and the remaining one belongs to $L_1 = \{ \pi/2, 3\pi/2 \}$. In the latter case, one pair of angles from $a$, $b$, $c$ and $d$ equals $\{ t, \pi - t \}$, with $t \in (0, \pi)$, and the remaining pair equals $\{ s, \pi - s \}$, with $s \in (0, \pi)$.
In case (i), we choose to produce graphs of the minors $G_3$ and $G_4 = \det G$ of the Gram matrix $G$ of each candidate tetrahedron, in order to check their positivity. The ones that appear positive on the whole interval $(0, \pi/6)$ indeed turn to $0$ only at the ends, or only one of the ends of the interval $(0, \pi/6)$. Then we check that those which appear negative on the interval $(0, \pi/6)$ do not turn positive near the end-points $0$ and $\pi/6$, but at worst become equal to $0$ at one or both of them. In order to verify all the above mentioned inequalities we use interval arithmetic implemented in `SageMath`, and for the equalities we use minimal polynomials, as before.
In case (ii), we know that the tetrahedron $T^\ast$ with Coxeter diagram $A^{\times 4}_1$ belongs to any possible continuous family. The tetrahedron $T^\ast$ has all right angles, and thus the minors $G_3(\pi/2, \pi/2)$ and $G_4(\pi/2, \pi/2)$ have to be positive for any family containing geometrically realisable tetrahedra. This filter leaves us with only few possible families, for which $G_3(s, t)$ and $G_4(s, t)$ have very simple form, amenable to elementary analysis for determining their positivity domains.
Finally, case (i) produces $34$ continuous families of tetrahedra depending on a single parameter, and case (ii) produces $8$ continuous families of tetrahedra depending on two parameters. All of them are listed in Appendix B, together with the domains of admissible parameter values, and the corresponding volume formulas.
Splitting rational polytopes into Coxeter tetrahedra
====================================================
Below we give a proof of Theorem \[thm:rational-not-coxeter-tetrahedron\]. We begin by considering more closely one of the many Pythagorean quadruples of Theorem \[thm:Pythagorean\], namely $$(p, q, r, s) = \left(\frac{5}{18}\,\pi, \frac{2}{9}\,\pi, \frac{13}{18}\,\pi, \frac{11}{18}\,\pi \right),$$ which corresponds to item $11$ in Appendix B with parameter $t = \frac{\pi}{18}$.
The corresponding $\mathbb{Z}_2$-symmetric rational tetrahedron $T$ has edge lengths $$(\ell_p, \ell_q, \ell_r, \ell_s) = \left(\frac{5}{18}\,\pi, \frac{2}{9}\,\pi, \frac{5}{18}\,\pi, \frac{7}{18}\,\pi\right),$$ and volume $\mathrm{vol}\, T = \pi^2/162$.
We shall prove that $T$ cannot be decomposed into any finite number of spherical Coxeter tetrahedra $S_i$, $i=1,\dots,11$, c.f. Table \[tabular:coxeter-tetrahedra\].
Suppose that it were indeed the case: then the vertex links of $T$ would be decomposed into a finite number of vertex links of Coxeter tetrahedra. The latter correspond to any of the Coxeter spherical triangles $\Delta_{2,2,n}$, $n\geq 2$, $\Delta_{2,3,3}$, $\Delta_{2,3,4}$ or $\Delta_{2,3,5}$.
Let us consider one of the vertices $v$ of $T$ whose link $\mathrm{Lk}_v$ is a spherical triangle $\tau$ with angles $\alpha = \frac{5\pi}{18}$, $\beta = \frac{2\pi}{9}$ and $\gamma = \frac{11\pi}{18}$. The side lengths of this triangle opposite to the above mentioned angles are denoted by $\ell_\alpha$, $\ell_\beta$ and $\ell_\gamma$, respectively. The spherical law of cosines [@Ratcliffe Theorem 2.5.3] grants that $\frac{\pi}{6} < \ell_\alpha, \ell_\beta, \ell_\gamma < \frac{\pi}{2}$. We can thus position $\tau$ on the sphere $\mathbb{S}^2 = \{ (x,y,z)\in \mathbb{R}^2 | x^2 + y^2 + z^2 = 1 \}$ so that one of its vertices has coordinates $(1,0,0)$, and its adjacent vertex has coordinates $(\cos \ell_\gamma, \sin \ell_\gamma, 0)$, while the third one is in the intersection of the positive orthant $\{ (x,y,z)\in \mathbb{R}^2 | x,\, y,\, z \geq 0 \}$ with $\mathbb{S}^2$. Then we can verify that all the vertices of $\tau$ lie in the circle of radius $\frac{\pi}{4}$ centred at $p = \left(\cos\frac{4 \pi}{25}, \sin\frac{4 \pi}{25}, 0\right)$, c.f. Monty [@Monty].
Thus, $\mathrm{diam}\,\mathrm{Lk}_v < \frac{\pi}{2}$, and none of the triangles $\Delta_{2,2,n}$ is a part of the decomposition of $\mathrm{Lk}_v$. The remaining cases are limited to a decomposition into $k\geq 0$ triangles of type $\Delta_{2,3,3}$, $l\geq 0$ triangles of type $\Delta_{2,3,4}$, and $m\geq 0$ triangles of type $\Delta_{2,3,5}$. Then the obvious sum of areas equality holds: $$k\, \mathrm{Area}\,\Delta_{2,3,3} + l\, \mathrm{Area}\,\Delta_{2,3,4} + m\, \mathrm{Area}\, \Delta_{2,3,5} = \mathrm{Area}\, \mathrm{Lk}_v,$$ which can be simplified down to $$10 k + 5 l + 2 m = \frac{20}{3}$$ by using the angle excess formula for the area of a spherical triangle [@Ratcliffe Theorem 2.5.5]. The latter never holds with $k,l,m \in \mathbb{Z}$.
Another spherical rational tetrahedron $T^\prime$ with volume $\pi^2/162$ is given by the Coxeter diagram
(0,0) – (1,0); (2,0) – (3,0);
(0,0) circle(.1); (1,0) circle(.1); (2,0) circle(.1); (3,0) circle(.1);
at (0.5, 0.3) [$9$]{}; at (2.5, 0.3) [$9$]{};
Both $T$ and $T^\prime$ have equal volumes and equal Dehn invariants: the former is by construction, and the latter follows from the fact that their dihedral angles are rational multiples of $\pi$, which implies that their Dehn invariants vanish.
Are the tetrahedra $T$ and $T^\prime$, as above, scissors congruent?
Rational Lambert cubes
======================
A Lambert cube $L := L(a, b, c)$ is depicted in Figure \[fig:Lambert-cube\]. It is realisable as a spherical polytope $L \subset \mathbb{S}^3$, if $ \pi/2 < \alpha, \beta, \gamma < \pi$, c.f. [@Diaz]. All other dihedral angles of $L$, apart from the *essential* ones $a$, $b$ and $c$, are always equal to $\pi/2$.
![The Lambert cube $L(a, b, c)$ with essential angles marked[]{data-label="fig:Lambert-cube"}](Lambert-cube.pdf)
The following fact stated as Proposition 4 in [@DM] holds for the volume function $\mathrm{Vol}\, L$, which allows us to seek rational Lambert cubes i.e. $L = L(a, b, c)$ with $a, b, c \in \pi\, \mathbb{Q}$, having rational volume $\mathrm{Vol}\, L \in \pi^2\, \mathbb{Q}$.
\[prop:volume-Lambert-cube\] Suppose that the essential angles of a spherical Lambert cube $L = L(a, b, c)$ satisfy the relation $\cos^2 a + \cos^2 b + \cos^2 c = 1$. Then $$\mathrm{Vol}\, L = \frac{1}{4}\, \left(\frac{\pi^2}{2} - (\pi-a)^2 - (\pi-b)^2 - (\pi-c)^2\right).$$
By using Monty [@Monty] we find, in a way analogous to the discussion in Sections \[section:sporadic\] – \[section:families\], that there are only two sporadic rational Lambert cubes satisfying the conditions of Proposition \[prop:volume-Lambert-cube\]. No continuous families are present in this case, as follows from Theorem \[thm:cj\].
Namely, only the following two Lambert cubes come out of our analysis: $L_1 = L(\frac{3\pi}{4}, \frac{2\pi}{3}, \frac{2\pi}{3})$ and $L_2 = L(\frac{2\pi}{3}, \frac{3\pi}{5}, \frac{4\pi}{5})$. By applying Proposition \[prop:volume-Lambert-cube\], we obtain that $\mathrm{Vol}\, L_1 = 31/576\, \pi^2$ and $\mathrm{Vol}\, L_2 = 17/360\, \pi^2$.
It is easy to produce a pair of spherical rational simplices $T_1$ and $T_2$ such that the respective $L_i$ and $T_i$, $i=1, 2$, have equal volumes and equal Dehn invariants. Let $T_1$ be given by the quadruple $(\frac{\pi}{2}, \frac{\pi}{2}, \frac{\pi}{2}, \frac{31\pi}{144})$, and let $T_2$ be given by $( \frac{\pi}{2}, \frac{\pi}{2}, \frac{\pi}{2}, \frac{17\pi}{90})$. Both $T_i$’s belong to the family $I_2(k)\times A_1^{\times 2}$ in Table \[tabular:coxeter-tetrahedra\], if we allow $k$ to take rational values.
Are the tetrahedron $T_1$ (resp. $T_2$) and the cube $L_1$ (resp. $L_2$), as above, scissors congruent?
By [@Felikson2], we have that $L_1$ is the only spherical Lambert cube that can be represented as a union of mutually isometric Coxeter tetrahedra.
Is the Lambert cube $L_2$ decomposable into any finite number of Coxeter tetrahedra?
Higher-dimensional aspects
==========================
As in the proof of Theorem \[thm:rational-not-coxeter-tetrahedron\], suppose that a rational $n$-dimensional, $n\geq 3$, spherical simplex $T \subset \mathbb{S}^n$ is given. Then the fact that $T$ splits into a finite number of Coxeter simplices (identified facet to facet in order to form the initial simplex $T$) will imply that all the vertex links $\mathrm{Lk}_{v_i}$, $i = 1, \dots, n+1$, of $T$ can be decomposed into a finite number of co-dimension one Coxeter simplices $T^i_j$, $j = 1, \dots, n_i$. If one of the vertex links in $T$ does *not* have this property, then neither has $T$.
Let us now suppose that the three-dimensional rational tetrahedron $T^{(3)}_1 = T$ from Theorem \[thm:rational-not-coxeter-tetrahedron\] is a vertex link of a four-dimensional rational spherical simplex $T^{(4)}_1 \subset \mathbb{S}^4$. Then we obviously have an example of a four-dimensional simplex that does not split into any finite number of Coxeter simplices. More generally, if a rational simplex $T^{(n)}_1 \subset \mathbb{S}^n$ that is *not* decomposable into Coxeter pieces can be realised as a vertex link of a rational simplex $T^{(n+1)}_1 \subset \mathbb{S}^{(n+1)}$, then $T^{(n+1)}_1$ gives us a rational simplex with the analogous property in a higher dimension.
Constructing such a family of rational spherical simplices $T^{(n)}_1$, $n\geq 3$, starting from $T^{(3)}_1$ is simple: let $G_3 = G(T^{(3)}_1)$ be the Gram matrix of $T^{(3)}_1$, and then let $T^{(n)}_1$, $n\geq 3$, be the spherical simplex with the block-diagonal Gram matrix $$G_n := \left( \begin{array}{cccc}
G_3& \mathbf{0}& \mathbf{0}& \mathbf{0}\\
\mathbf{0}& 1& 0& 0\\
\mathbf{0}& 0& \ddots& 0\\
\mathbf{0}& 0& 0& 1
\end{array} \right).$$
The volume of $T^{(n)}_1$, $n\geq 4$, equals $\mathrm{Vol}\, T^{(n)}_1 = \mathrm{Vol}\, T^{(3)}_1 \cdot \frac{\mathrm{Vol}\, \mathbb{S}^{n}}{2^{n-3}\, \mathrm{Vol}\, \mathbb{S}^3}$, which is a rational multiple of $\mathrm{Vol}\,\mathbb{S}^n$ once $T^{(3)}_1$ has rational volume.
If we apply the above construction to the tetrahedron $T^{(3)}_2 = T^{\prime}$, then we obtain a family of Coxeter tetrahedra $T^{(n)}_2$, each generating the respective finite reflection group $I_2(9)\times I_2(9)\times (A_1)^{n-3}$, for $n\geq 3$. The volumes and Dehn invariants of each pair $T^{(n)}_1$ and $T^{(n)}_2$, $n\geq 3$, are equal, although the former is not decomposable into any finite number of Coxeter tetrahedra, and the latter is a Coxeter tetrahedron itself.
Are $T^{(n)}_1$ and $T^{(n)}_2$, $n \geq 3$, scissors congruent?
It is also worth mentioning that if there exists a tetrahedron $T\subset \mathbb{S}^3$ with “rational” dihedral angles, but “irrational” volume, i.e. a counterexample to the initial conjecture by Cheeger and Simons [@CS], then using the above construction we can also produce a counterexample in every dimension $n \geq 3$.
[00]{} J. Cheeger, J. Simons: “Differential characters and geometric invariants,” in: Geometry and Topology. Lecture Notes in Mathematics **1167**. Springer: Berlin – Heidelberg (1985) J.H. Conway, A.J. Jones: “Trigonometric diophantine equations,” Acta Arithm. **30**, 229 – 240 (1976) H.S.M. Coxeter: “Discrete groups generated by reflections,” Ann. Math. **35**, 588 – 621 (1934) D.A. Derevnin, A.D. Mednykh: “The volume of the Lambert cube in spherical space”, Math. Notes **86** (1–2), 176 – 186 (2009) R. Díaz: “A characterization of Gram matrices of polytopes,” Discrete Comput. Geom. **21** (4), 581 – 601 (1999) A. Felikson: “Spherical simplices generating discrete reflection groups,” Sb. Math. **195** (4), 585 – 598 (2004) A. Felikson: “Lambert cubes generating discrete reflection groups,” Math. Notes **75** (1–2), 250 – 258 (2004) A. Kolpakov, A. Mednykh, M. Pashkevich: “Volume formula for a $\mathbb{Z}_2$-symmetric spherical tetrahedron through its edge lengths,” Ark. Mat. **51** (1), 99 – 123 (2013) A. Kolpakov, S. Robins: Ancillary files available at Harvard Dataverse, <https://doi.org/10.7910/DVN/LJBNGO> F. Luo: “On a problem of Fenchel,” Geom. Dedicata **64** (3), 277 – 282 (1997) A. Meurer A, et al.: “SymPy: symbolic computing in Python,” Peer J. Computer Science 3:e103 (2017) <https://doi.org/10.7717/peerj-cs.103> J.G. Ratcliffe: “Foundations of hyperbolic manifolds”, Graduate Texts in Mathematics **149**. New–York: Springer, 2013. SageMath: Sage Mathematics Software System (Version 8.4), <http://www.sagemath.org> L. Schläfli: “Lehre von den sphärischen Kontinuen,” in: Gesammelte Mathematische Abhandlungen, Band 1. Birkhäuser: Basel (1950) W.D. Smith: “Pythagorean triples, rational angles, and space-filling simplices”, manuscript available at <https://rangevoting.org/WarrenSmithPages/homepage/works.html>
[l@[1.5in]{}l]{}
---------------------------------------
*Alexander Kolpakov\
*Institut de Mathématiques\
*Université de Neuchâtel\
*Suisse/Switzerland\
*kolpakov.alexander(at)gmail.com*****
---------------------------------------
&
-----------------------------------------
*Sinai Robins\
*Departamento de ciência da computação\
*Instituto de Matemática e Estatistica\
*Universidade de São Paulo\
*Brasil/Brazil\
*sinai.robins(at)gmail.com******
-----------------------------------------
Appendix A
==========
----- ---------------------------- ------------------------------------ ----------------
no. $(p, q, r, s)$ $(\ell_p, \ell_q, \ell_r, \ell_s)$ $\mathrm{Vol}$
1 (2/3, 1/3, 3/5, 1/5) (2/3, 1/3, 2/5, 4/5) 7/90
2 (25/42, 11/42, 4/7, 2/7) (25/42, 11/42, 3/7, 5/7) 67/1764
3 (2/5, 4/15, 3/5, 8/15) (2/5, 4/15, 2/5, 7/15) 19/900
4 (2/5, 1/5, 2/3, 1/2) (2/5, 1/5, 1/3, 1/2) 7/720
5 (6/7, 2/7, 1/3, 2/7) (6/7, 2/7, 2/3, 5/7) 299/1764
6 (19/30, 17/30, 11/15, 1/3) (19/30, 17/30, 4/15, 2/3) 209/900
7 (2/3, 2/3, 4/5, 2/5) (2/3, 2/3, 1/5, 3/5) 31/90
8 (6/7, 5/7, 5/7, 2/3) (6/7, 5/7, 2/7, 1/3) 1013/1764
9 (13/30, 11/30, 11/15, 1/3) (13/30, 11/30, 4/15, 2/3) 29/900
10 (7/20, 3/20, 2/3, 3/5) (7/20, 3/20, 1/3, 2/5) 17/3600
11 (4/5, 3/5, 2/3, 1/2) (4/5, 3/5, 1/3, 1/2) 59/144
12 (23/30, 11/30, 7/15, 1/3) (23/30, 11/30, 8/15, 2/3) 161/900
13 (5/7, 1/7, 1/3, 2/7) (5/7, 1/7, 2/3, 5/7) 47/1764
14 (17/30, 11/30, 2/3, 4/15) (17/30, 11/30, 1/3, 11/15) 59/900
15 (2/3, 1/5, 2/5, 1/3) (2/3, 1/5, 3/5, 2/3) 37/900
16 (13/30, 7/30, 3/5, 1/2) (13/30, 7/30, 2/5, 1/2) 67/3600
17 (5/7, 3/7, 4/7, 1/3) (5/7, 3/7, 3/7, 2/3) 335/1764
18 (1/5, 2/15, 4/5, 11/15) (1/5, 2/15, 1/5, 4/15) 1/900
19 (31/42, 25/42, 5/7, 3/7) (31/42, 25/42, 2/7, 4/7) 613/1764
20 (11/15, 3/5, 3/5, 8/15) (11/15, 3/5, 2/5, 7/15) 319/900
21 (23/30, 13/30, 1/2, 2/5) (23/30, 13/30, 1/2, 3/5) 847/3600
22 (17/42, 11/42, 5/7, 3/7) (17/42, 11/42, 2/7, 4/7) 25/1764
23 (17/30, 7/30, 1/2, 2/5) (17/30, 7/30, 1/2, 3/5) 127/3600
24 (23/30, 19/30, 2/3, 8/15) (23/30, 19/30, 1/3, 7/15) 371/900
25 (1/3, 1/3, 4/5, 2/5) (1/3, 1/3, 1/5, 3/5) 1/90
26 (4/7, 2/7, 4/7, 1/3) (4/7, 2/7, 3/7, 2/3) 83/1764
27 (3/5, 3/5, 2/3, 2/5) (3/5, 3/5, 1/3, 3/5) 109/450
28 (1/3, 1/5, 2/3, 3/5) (1/3, 1/5, 1/3, 2/5) 7/900
29 (11/30, 7/30, 2/3, 8/15) (11/30, 7/30, 1/3, 7/15) 11/900
30 (3/5, 2/5, 3/5, 1/3) (3/5, 2/5, 2/5, 2/3) 49/450
----- ---------------------------- ------------------------------------ ----------------
: Sporadic spherical $\mathbb{Z}_2$-symmetric tetrahedra: dihedral angles have the form $(p \pi, q \pi, r \pi, s \pi)$, side lengths have the form $(\ell_p \pi, \ell_q \pi, \ell_r \pi, \ell_s \pi)$, and volumes are $v \pi^2$, with $p, q, r, s, \ell_p, \ell_q, \ell_r, \ell_s, v \in \mathbb{Q}$[]{data-label="table:A1"}
---- --------------------------- ---------------------------- -----------
31 (13/15, 4/5, 4/5, 11/15) (13/15, 4/5, 1/5, 4/15) 601/900
32 (5/7, 4/7, 2/3, 3/7) (5/7, 4/7, 1/3, 4/7) 545/1764
33 (3/5, 4/15, 7/15, 2/5) (3/5, 4/15, 8/15, 3/5) 49/900
34 (23/30, 17/30, 3/5, 1/2) (23/30, 17/30, 2/5, 1/2) 1267/3600
35 (2/5, 2/5, 2/3, 2/5) (2/5, 2/5, 1/3, 3/5) 19/450
36 (17/20, 7/20, 2/5, 1/3) (17/20, 7/20, 3/5, 2/3) 797/3600
37 (4/5, 2/5, 1/2, 1/3) (4/5, 2/5, 1/2, 2/3) 163/720
38 (3/7, 2/7, 2/3, 3/7) (3/7, 2/7, 1/3, 4/7) 41/1764
39 (13/15, 1/5, 4/15, 1/5) (13/15, 1/5, 11/15, 4/5) 91/900
40 (19/30, 7/30, 7/15, 1/3) (19/30, 7/30, 8/15, 2/3) 41/900
41 (2/3, 2/5, 2/3, 1/5) (2/3, 2/5, 1/3, 4/5) 103/900
42 (3/5, 1/5, 1/2, 1/3) (3/5, 1/5, 1/2, 2/3) 19/720
43 (3/5, 1/3, 2/3, 1/5) (3/5, 1/3, 1/3, 4/5) 43/900
44 (4/5, 1/3, 2/5, 1/3) (4/5, 1/3, 3/5, 2/3) 157/900
45 (19/30, 13/30, 2/3, 4/15) (19/30, 13/30, 1/3, 11/15) 119/900
46 (4/5, 1/5, 1/3, 1/5) (4/5, 1/5, 2/3, 4/5) 31/450
47 (17/20, 13/20, 2/3, 3/5) (17/20, 13/20, 1/3, 2/5) 1817/3600
48 (4/5, 2/3, 4/5, 1/2) (4/5, 2/3, 1/5, 1/2) 1691/3600
49 (4/5, 2/15, 4/15, 1/5) (4/5, 2/15, 11/15, 4/5) 31/900
50 (2/5, 1/3, 4/5, 1/3) (2/5, 1/3, 1/5, 2/3) 13/900
51 (1/5, 1/5, 4/5, 2/3) (1/5, 1/5, 1/5, 1/3) 1/450
52 (2/7, 1/7, 5/7, 2/3) (2/7, 1/7, 2/7, 1/3) 5/1764
53 (11/15, 2/5, 7/15, 2/5) (11/15, 2/5, 8/15, 3/5) 169/900
54 (31/42, 17/42, 4/7, 2/7) (31/42, 17/42, 3/7, 5/7) 319/1764
55 (4/5, 4/5, 4/5, 2/3) (4/5, 4/5, 1/5, 1/3) 271/450
56 (4/5, 2/3, 2/3, 3/5) (4/5, 2/3, 1/3, 2/5) 427/900
57 (11/15, 2/3, 11/15, 1/2) (11/15, 2/3, 4/15, 1/2) 493/1200
58 (2/3, 3/5, 4/5, 1/3) (2/3, 3/5, 1/5, 2/3) 253/900
59 (13/20, 3/20, 2/5, 1/3) (13/20, 3/20, 3/5, 2/3) 77/3600
---- --------------------------- ---------------------------- -----------
: Sporadic spherical $\mathbb{Z}_2$-symmetric tetrahedra (cont.)[]{data-label="table:A2"}
Appendix B
==========
----- --------------------------------------------------------------------- ------------------------------------------------------------------ ----------------------- ------------------------------------
no. $(p, q, r, s)$ $(\ell_p, \ell_q, \ell_r, \ell_s)$ domain $\mathrm{Vol}$
1 $(1/2 \pi + t, 1/2 \pi, 1/2 \pi, 1/2 \pi )$ $(t + \pi/2, \pi/2, $ $ $(2 t + \pi )^2/8$
\pi/2, \pi/2 )$
2 $(3/4 \pi - 1/2 t, 1/4 \pi - 1/2 t, 1/3 \pi - t, 1/3 \pi + t )$ $(-t/2 + 3 \pi/4, -t/2 + \pi/4, -t + 2 \pi/3, t + 2 \pi/3 )$ $ $ $-t^2/4 -
\pi t/2 + 13 \pi^2/144$
3 $(1/2 \pi + t, 1/2 \pi, 1/3 \pi + t, 2/3 \pi - t )$ $(t + \pi/2, $ $ $\pi (6 t + \pi )/9$
\pi/2, t + \pi/3, -t + 2 \pi/3 )$
4 $(1/2 \pi, 1/6 \pi + t, 2/3 \pi - t, 1/3 \pi + t )$ $( \pi/2, t + $ $ $ \pi t/3$
\pi/6, -t + 2 \pi/3, t + \pi/3 )$
5 $(2/3 \pi - t, 1/3 \pi, 1/3 \pi + t, 1/2 \pi )$ $(-t + 2 \pi/3, $ $ $t^2/4 - \pi t/3 + 5 \pi^2/48$
\pi/3, \pi/2, -t + 2 \pi/3 )$
6 $(1/2 \pi, 1/2 \pi - t, 1/3 \pi + t, 2/3 \pi - t )$ $( \pi/2, -t + $ $ $ \pi (-3 t + \pi )/9$
\pi/2, t + \pi/3, -t + 2 \pi/3 )$
7 $(1/3 \pi + t, 1/3 \pi, 1/2 \pi, 2/3 \pi - t )$ $(t + \pi/3, $ $ $t^2/4 + \pi t/6 + \pi^2/48$
\pi/3, t + \pi/3, \pi/2 )$
8 $(2/3 \pi, 1/3 \pi - t, 1/2 \pi, 1/3 \pi - t )$ $(2 \pi/3, -t + $ $ $t^2/4 - 2 \pi t/3 + 5 \pi^2/48$
\pi/3, t + 2 \pi/3, \pi/2 )$
9 $(2/3 \pi, 1/3 \pi + t, 1/3 \pi + t, 1/2 \pi )$ $(2 \pi/3, t + $ $ $t^2/4 + 2 \pi t/3 + 5 \pi^2/48$
\pi/3, \pi/2, -t + 2 \pi/3 )$
10 $(1/2 \pi, 1/2 \pi - t, 1/3 \pi - t, 2/3 \pi + t )$ $( \pi/2, -t $ $ $ \pi (-6 t + \pi )/9$
+ \pi/2, -t + \pi/3, t + 2 \pi/3 )$
11 $(1/4 \pi + 1/2 t, 1/4 \pi - 1/2 t, 2/3 \pi + t, 2/3 \pi - t )$ $(t/2 + \pi/4, -t/2 + \pi/4, t + \pi/3, -t + \pi/3 )$ $ $ $-t^2/4 +
\pi^2/144$
12 $(1/2 \pi + t, 1/2 \pi, 1/3 \pi - t, 2/3 \pi + t )$ $(t + \pi/2, $0 \leq t \leq \pi/6$ $ \pi (3 t + \pi )/9$
\pi/2, -t + \pi/3, t + 2 \pi/3 )$
13 $(1/2 \pi, 1/6 \pi + t, 1/2 \pi, 1/2 \pi )$ $( \pi/2, t + \pi/6, for no. 1 – no. 34 $(6 t + \pi )^2/72$
\pi/2, \pi/2 )$
14 $(1/2 \pi, 1/2 \pi - t, 1/2 \pi, 1/2 \pi )$ $( \pi/2, -t + \pi/2, $ $ $(2 t - \pi )^2/8$
\pi/2, \pi/2 )$
15 $(1/3 \pi, 1/3 \pi - t, 2/3 \pi + t, 1/2 \pi )$ $( \pi/3, -t + $ $ $t^2/4 - \pi t/6 + \pi^2/48$
\pi/3, \pi/2, -t + \pi/3 )$
16 $(3/4 \pi + 1/2 t, 1/4 \pi + 1/2 t, 1/3 \pi - t, 1/3 \pi + t )$ $(t/2 + 3 \pi/4, t/2 + \pi/4, -t + 2 \pi/3, t + 2 \pi/3 )$ $ $ $-t^2/4 +
\pi t/2 + 13 \pi^2/144$
17 $(3/4 \pi + 1/2 t, 1/4 \pi + 1/2 t, 1/3 \pi + t, 1/3 \pi - t )$ $(t/2 + 3 \pi/4, t/2 + \pi/4, t + 2 \pi/3, -t + 2 \pi/3 )$ $ $ $-t^2/4 +
\pi t/2 + 13 \pi^2/144$
18 $(1/4 \pi + 1/2 t, 1/4 \pi - 1/2 t, 2/3 \pi - t, 2/3 \pi + t )$ $(t/2 + \pi/4, -t/2 + \pi/4, -t + \pi/3, t + \pi/3 )$ $ $ $-t^2/4 +
\pi^2/144$
19 $(2/3 \pi + t, 1/3 \pi, 1/3 \pi - t, 1/2 \pi )$ $(t + 2 \pi/3, $ $ $t^2/4 + \pi t/3 + 5 \pi^2/48$
\pi/3, \pi/2, t + 2 \pi/3 )$
20 $(1/2 \pi, 1/6 \pi - t, 1/2 \pi, 1/2 \pi )$ $( \pi/2, -t + \pi/6, $ $ $(6 t - \pi )^2/72$
\pi/2, \pi/2 )$
21 $(2/3 \pi, 1/3 \pi + t, 1/2 \pi, 1/3 \pi + t )$ $(2 \pi/3, t + $ $ $t^2/4 + 2 \pi t/3 + 5 \pi^2/48$
\pi/3, -t + 2 \pi/3, \pi/2 )$
22 $(3/4 \pi - 1/2 t, 1/4 \pi - 1/2 t, 1/3 \pi + t, 1/3 \pi - t )$ $(-t/2 + 3 \pi/4, -t/2 + \pi/4, t + 2 \pi/3, -t + 2 \pi/3 )$ $ $ $-t^2/4 -
\pi t/2 + 13 \pi^2/144$
23 $(1/2 \pi, 1/2 \pi - t, 2/3 \pi - t, 1/3 \pi + t )$ $( \pi/2, -t $ $ $ \pi (-3 t + \pi )/9$
+ \pi/2, -t + 2 \pi/3, t + \pi/3 )$
24 $(1/2 \pi, 1/6 \pi + t, 1/3 \pi + t, 2/3 \pi - t )$ $( \pi/2, t + $ $ $ \pi t/3$
\pi/6, t + \pi/3, -t + 2 \pi/3 )$
25 $(1/2 \pi + t, 1/2 \pi, 2/3 \pi + t, 1/3 \pi - t )$ $(t + \pi/2, $ $ $ \pi (3 t + \pi )/9$
\pi/2, t + 2 \pi/3, -t + \pi/3 )$
----- --------------------------------------------------------------------- ------------------------------------------------------------------ ----------------------- ------------------------------------
: Continuous families of $\mathbb{Z}_2$-symmetric spherical tetrahedra[]{data-label="table:B1"}
---- --------------------------------------------------------------------- --------------------------------------------------------------- ---------- -----------------------------------------
26 $(3/4 \pi + 1/2 t, 3/4 \pi - 1/2 t, 2/3 \pi - t, 2/3 \pi + t )$ $(t/2 + 3 \pi/4, -t/2 + 3 \pi/4, -t + \pi/3, t + \pi/3 )$ $ $ $-t^2/4 +
73 \pi^2/144$
27 $(1/3 \pi + t, 1/3 \pi, 2/3 \pi - t, 1/2 \pi )$ $(t + \pi/3, $ $ $t^2/4 + \pi t/6 + \pi^2/48$
\pi/3, \pi/2, t + \pi/3 )$
28 $(2/3 \pi - t, 1/3 \pi, 1/2 \pi, 1/3 \pi + t )$ $(-t + 2 \pi/3, $ $ $t^2/4 - \pi t/3 + 5 \pi^2/48$
\pi/3, -t + 2 \pi/3, \pi/2 )$
29 $(1/3 \pi, 1/3 \pi - t, 1/2 \pi, 2/3 \pi + t )$ $( \pi/3, -t + $ $ $t^2/4 - \pi t/6 + \pi^2/48$
\pi/3, -t + \pi/3, \pi/2 )$
30 $(1/2 \pi, 1/2 \pi - t, 2/3 \pi + t, 1/3 \pi - t )$ $( \pi/2, -t $ $ $ \pi (-6 t + \pi )/9$
+ \pi/2, t + 2 \pi/3, -t + \pi/3 )$
31 $(2/3 \pi + t, 1/3 \pi, 1/2 \pi, 1/3 \pi - t )$ $(t + 2 \pi/3, $ $ $t^2/4 + \pi t/3 + 5 \pi^2/48$
\pi/3, t + 2 \pi/3, \pi/2 )$
32 $(2/3 \pi, 1/3 \pi - t, 1/3 \pi - t, 1/2 \pi )$ $(2 \pi/3, -t + $ $ $t^2/4 - 2 \pi t/3 + 5 \pi^2/48$
\pi/3, \pi/2, t + 2 \pi/3 )$
33 $(1/2 \pi + t, 1/2 \pi, 2/3 \pi - t, 1/3 \pi + t )$ $(t + \pi/2, $ $ $ \pi (6 t + \pi )/9$
\pi/2, -t + 2 \pi/3, t + \pi/3 )$
34 $(3/4 \pi + 1/2 t, 3/4 \pi - 1/2 t, 2/3 \pi + t, 2/3 \pi - t )$ $(t/2 + 3 \pi/4, -t/2 + 3 \pi/4, t + \pi/3, -t + \pi/3 )$ $ $ $-t^2/4 +
73 \pi^2/144$
35 $(1/2 \pi, 1/2 \pi - u, \pi - t, t)$ $( \pi/2, -u + \pi/2, -t + Domain A $-t^2/2 + \pi t/2 + u^2/2 - \pi u/2$
\pi, t)$
36 $(1/2 \pi, 1/2 \pi - u, t, \pi - t)$ $( \pi/2, -u + \pi/2, t, -t Domain A $-t^2/2 + \pi t/2 + u^2/2 - \pi u/2$
+ \pi)$
37 $(1/2 \pi + u, 1/2 \pi, \pi - t, t)$ $(u + \pi/2, \pi/2, -t + Domain A $-t^2/2 + \pi t/2 + u^2/2 + \pi u/2$
\pi, t$)
38 $(1/2 \pi + u, 1/2 \pi, t, \pi - t)$ $(u + \pi/2, \pi/2, t, -t + Domain A $ -t^2/2 + \pi t/2 + u^2/2 + \pi u/2$
\pi)$
39 $(1/2 \pi, 1/2 \pi - t, \pi - u, u)$ $( \pi/2, -t + \pi/2, -u + Domain B $ t^2/2 - \pi t/2 - u^2/2 + \pi u/2$
\pi, u)$
40 $(1/2 \pi, 1/2 \pi - t, u, \pi - u)$ $( \pi/2, -t + \pi/2, u, -u Domain B $ t^2/2 - \pi t/2 - u^2/2 + \pi u/2$
+ \pi)$
41 $(1/2 \pi + t, 1/2 \pi, \pi - u, u)$ $(t + \pi/2, \pi/2, -u + Domain B $t^2/2 + \pi t/2 - u^2/2 + \pi u/2$
\pi, u)$
42 $(1/2 \pi + t, 1/2 \pi, u, \pi - u)$ $(t + \pi/2, \pi/2, u, -u + Domain B $t^2/2 + \pi t/2 - u^2/2 + \pi u/2$
\pi)$
---- --------------------------------------------------------------------- --------------------------------------------------------------- ---------- -----------------------------------------
: Continuous families of $\mathbb{Z}_2$-symmetric spherical tetrahedra (cont.)[]{data-label="table:B2"}
-------------------------------- --------------------------------
Domain A: Domain B:
$0 \leq u \leq \frac{\pi}{2}$, $0 \leq u \leq \pi$,
$0 \leq t \leq \pi$, $0 \leq t \leq \frac{\pi}{2}$,
$t \geq u$. $t \leq u$.
-------------------------------- --------------------------------
|
---
abstract: 'We impose perfect fluid concept along with slow expansion approximation to derive new solutions which, considering non-static spherically symmetric metrics, can be treated as Black Holes. We will refer to these solutions as Quasi Black Holes. Mathematical and physical features such as Killing vectors, singularities, and mass have been studied. Their horizons and thermodynamic properties have also been investigated. In addition, relationship with other related works (including mcVittie’s) are described.'
address: |
$1$ Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran,\
$^2$ Physics Department, Shahid Beheshti University, Evin, Tehran 19839, Iran.
author:
- 'H. Moradpour$^1$[^1] and N. Riazi$^2$[^2]'
title: Spherically symmetric solutions in a FRW background
---
Introduction \[Introduction\]
=============================
The Universe expansion can be modeled by the so called FRW metric $$\begin{aligned}
\label{FRW}
ds^2=-dt^2+a(t)^2[\frac{dr^2}{(1-kr^2)}+r^2d\theta^2+r^2sin(\theta)^2d\phi^2],\end{aligned}$$ where $k=0, +1, -1$ are curvature scalars which represent the flat, closed and open universes, respectively. The WMAP data confirms a flat ($k=0$) universe [@Roos]. $a(t)$ is the scale factor and for a background which is filled by a perfect fluid with equation of state $p=\omega \rho$, there are three classes of expanding solutions. These three solutions are $$\begin{aligned}
\label{Scale factor1}
a(t)=a_0 t^{\frac{2}{3(\omega + 1)}}\end{aligned}$$ for $ \omega\neq 0$ when $-1<\omega$ and , $$\begin{aligned}
\label{Scale factor2}
a(t)=a_0 e^{Ht}\end{aligned}$$ for $\omega=-1$ (dark energy), and for the Phantom regime ($\omega<-1$) is $$\begin{aligned}
\label{Scale factor3}
a(t)=a_0(t_0-t)^{\frac{2}{3(\omega + 1)}},\end{aligned}$$ where $t_0$ is the big rip singularity time and will be available, if the universe is in the phantom regime.
In Eq. (\[Scale factor2\]), $H(\equiv\frac{\dot{a}(t)}{a(t)})$ is the Hubble parameter and the current estimates are $H=73^{+4}_{-3}kms^{-1}Mpc^{-1}$ [@Roos].
Note that, at the end of the Phantom regime, everything will decompose into its fundamental constituents [@Mukh]. In addition, this spacetime can be classified as a subgroup of the Godel-type spacetime with $\sigma=m=0$ and $k^{\prime}=1$ [@godel].
A signal which was emitted at the time $t_0$ by a co-moving source and absorbed by a co-moving observer at a later time $t$ is affected by a redshift ($z$) as $$\begin{aligned}
1+z=\frac{a(t)}{a(t_0)}.\end{aligned}$$ The apparent horizon as a marginally trapped surface, is defined as [@SWR] $$\begin{aligned}
\label{aph1}
g^{\mu \nu}\partial_{\mu}\xi \partial_{\nu}\xi=0,\end{aligned}$$ which for the physical radius of $\xi=a(t)r$, the solution will be: $$\begin{aligned}
\label{aph2}
\xi=\frac{1}{\sqrt{H^2+\frac{k}{a(t)^2}}}.\end{aligned}$$ The surface gravity of the apparent horizon can be evaluated by: $$\begin{aligned}
\label{sg1}
\kappa=\frac{1}{2\sqrt{-h}}\partial_a(\sqrt{-h}h^{ab}\partial_b
\xi).\end{aligned}$$
Where the two dimensional induced metric is $h_{ab}=diag(-1,\frac{a(t)}{(1-kr^2)})$. It was shown that the first law of thermodynamics is satisfied on the apparent horizon [@S0; @S1; @S2; @S3]. The special case of $\omega=-1$ is called the dark energy, and by a suitable change of variables one can rewrite this case in the static form [@Poisson]: $$\begin{aligned}
\label{static dark}
ds^2=-(1-H^2r^2)dt^2+\frac{dr^2}{(1-H^2r^2)}+r^2d\Omega^2.\end{aligned}$$ This metric belongs to a more general class of spherically symmetric, static metrics. For these class of spherically symmetric static metrics, the line element can be written in the form of: $$\begin{aligned}
\label{SSM}
ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2,\end{aligned}$$ where the general form of $f(r)$ is: $$\begin{aligned}
f(r)=1-2\frac{m}{r}+\frac{Q^2}{r^2}-H^2r^2.\end{aligned}$$ In the above expression, $m$ and $Q$ represent mass and charge, respectively. For this metric, one can evaluate redshift: $$\begin{aligned}
1+z=(\frac{1-2\frac{m}{r}+\frac{Q^2}{r^2}-H^2r^2}
{1-2\frac{m}{r_0}+\frac{Q^2}{r_0^2}-H^2r_0^2})^\frac{1}{2}.\end{aligned}$$ Where, $r_0$ and $r$ are radial coordinates at the emission and the absorption points. For the horizons, the radius and the surface gravity can be found using equations $$\begin{aligned}
\label{SG}
g_{tt}&=&f(r)=0 \longrightarrow r_h \\ \nonumber \kappa
&=&\frac{f^\prime(r)}{2}|_{r_h},\end{aligned}$$ where $(^\prime)$ denotes derivative with respect to the coordinate $r$ [@Poisson]. From the thermodynamic laws of Black Holes (BHs) we know $$\begin{aligned}
\label{Temp2}
T=\frac{\kappa}{2\pi},\end{aligned}$$ which $T$ is the temperature on the horizon [@Poisson]. Validity of the first law of the thermodynamics on the static horizons for the static spherically symmetric spacetime has been shown [@Cai1; @Padm1].
The BHs with the FRW dynamic background has motivated many investigations. The first approach, which is named Swiss Cheese, includes efforts in order to find the effects of the expansion of the Universe on the gravitational field of the stars [@P1], introduced originally by Einstein and Straus $(1945)$ [@ES]. In these models, authors tried to join the Schwarzschild metric to the FRW metric by satisfying the junction conditions on the boundary, which is an expanding timelike hypersurface. The inner spacetime is described by the Schwarzschild metric, while the FRW metric explains the outer spacetime. These models don’t contain dynamical BHs, Because the inner spacetime is in the Schwarzschild coordinate, hence, is static [@saida]. In addition, the Swiss Cheese models can be classified as a subclass of inhomogeneous Lemabitre-Tolman-Bondi models [@MD1; @CLure].
Looking for dynamical BHs, some authors used the conformal transformation of the Schwarzschild BH, where the conformal factor is the scale factor of the famous FRW model. Originally, Thakurta $(1981)$ have used this technique and obtained a dynamical version of the Schwarzschild BH [@Thak]. Since the Thakurta spacetime is a conformal transformation of the Schwarzschild metric, it is now accepted that its redshift radii points to the co-moving radii of the event horizon of BH [@MD1; @MR; @RMS]. By considering asymptotic behavior of the gravitational lagrangian (Ricci scalar), one can classify the Thakurta BH and its extension to the charged BH into the same class of solutions [@MR; @RMS]. The Thakurta spacetime sustains an inward flow, which leads to an increase in the mass of BH [@MR; @RMS; @Gao1]. This ingoing flow comes from the back-reaction effect and can be neglected in a low density background [@Gao1]. In fact, for the low density background, the mass will be decreased in the Phantom regime [@Bab]. Also, the radius of event horizon increases with the scale factor when its temperature decreases by the inverse of scale factor [@MR; @RMS].
Using the Eddington-Finkelstein form of the Schwarzschild metric and the conformal transformation, Sultana and Dyer $(2005)$ have constructed their metric and studied its properties [@SD]. In addition, unlike the Thakurta spacetime, the curvature scalars do not diverge at the redshift singularity radii (event horizon) of the Sultana and Dyer spacetimes. Since the Sultana and Dyer spacetimes is conformal transformation of the Schwarzschild metric, it is now accepted that the Sultana and Dyer spacetimes include dynamic BHs [@MD1]. Various examples can be found in [@MD1; @MD2; @FJ; @MN]. Among these conformal BHs, only the solutions by M$^{\textmd{c}}$Clure et al. and Thakurta can satisfy the energy conditions [@RMS; @MD1]. Static charged BHs which are confined into the FRW spacetime and the dynamic, charged BHs were studied in [@O1; @O2; @O3; @O4; @O5; @O6; @O7; @O8]. The Brane solutions can be found in [@BS1; @BS2; @BS3].
In another approach, mcVittie found new solutions including contracting BHs in the coordinates co-moving with the universe’s expansion [@mcvittie]. Its generalization to the arbitrary dimensions and to the charged BHs can be found in [@Gao0; @Gao]. In these solutions, it is easy to check that the curvature scalars diverge at the redshift singularities. In this approach, authors have used the isotropic form of the FRW metric along as the perfect fluid concept and could find their solutions which can contain BHs [@Far]. The mass and the charge of their BHs seem to be decreased with the scale factor. Also, it seems that the redshift singularities does not point to a dynamic event horizon [@nol1; @nol2; @SUS; @fri]. Unlike the Swiss Cheese models, the energy conditions are violated by these solutions [@MD1]. These solutions can be considered as Models for cosmological inhomogeneities [@CLure].
This paper is organized as follows: in the next section, we consider the conformal transformation of a non-static spherically symmetric metric, where conformal factor has only time dependency. In addition, we derive the general possible form of metric by using perfect fluid concept. In section $3$, slow time varying approximation is used in order to find the physical meaning of the parameters of metric. In continue, the mcVittie like solution and its thermodynamic properties are addressed. In section $4$, we generalized our debates to the charged spacetime, when the effects of the dark energy are considerable. In section $5$, we summarize and conclude the results.
Metric, general properties and basic assumptions
================================================
Let us begin with this metric: $$\begin{aligned}
ds^2=a(\tau)^2[-f(\tau,r)d\tau^2+\frac{dr^2}{(1-kr^2)f(\tau,r)}+r^2d\theta^2+r^2sin(\theta)^2d\phi^2].\end{aligned}$$ Where $a(\tau)$ is the arbitrary function of time coordinate $\tau$. This metric has three Killing vectors $$\begin{aligned}
\label{symm}
\partial_{\phi},\ \ \sin\phi \ \partial_{\theta}+\cot\theta \ \cos\phi \
\partial_{\phi}\ \ \textmd{and}\ \ \cos\phi \ \partial_\theta - \cot\theta \ \sin\phi \ \partial_\phi .\end{aligned}$$ Now, if we define new time coordinate as $$\begin{aligned}
\tau \rightarrow t=\int a(\tau)d\tau,\end{aligned}$$ we will get $$\begin{aligned}
\label{Metric1}
ds^2=-f(t,r)dt^2+a(t)^2[\frac{dr^2}{(1-kr^2)f(t,r)}+r^2d\theta^2+r^2sin(\theta)^2d\phi^2],\end{aligned}$$ which possesses symmetries like as Eq. (\[symm\]). From now, it is assumed that $a(t)$ is the cosmic scale factor similar to the FRW’s. For $f(t,r)=1$, Eq. (\[Metric1\]) is reduced to the FRW metric (\[FRW\]). Also, conformal BHs can be achieved by choosing $f(t,r)=f(r)$ where, the general form of $f(r)$ is [@MR]: $$\begin{aligned}
f(r)=1-\frac{2m}{r}+\frac{Q^2}{r^2}-\frac{\Lambda r^2}{3}.\end{aligned}$$ Therefore, conformal BHs can be classified as a special subclass of metric (\[Metric1\]). $n_{\alpha}=\delta^r_{\alpha}$ is normal to the hypersurface $r=const$ and yields $$\begin{aligned}
\label{nhs}
n_{\alpha}n^{\alpha}=g^{rr}=\frac{(1-kr^2)f(t,r)}{a(t)^2},\end{aligned}$$ which is timelike when $(1-kr^2)f(t,r)<0$, null for $(1-kr^2)f(t,r)=0$ and spacelike if we have $(1-kr^2)f(t,r)>0$. For an emitted signal at the coordinates $t_0$ and $r_0$, when it is absorbed at coordinates $t$ and $r$ simple calculations lead to $$\begin{aligned}
\label{redshift}
1+z=\frac{\lambda}{\lambda_0}=\frac{a(t)}{a(t_0)}(\frac{f(t,r)}{f(t_0,r_0)})^{\frac{1}{2}},\end{aligned}$$ as induced redshift due to the universe expansion and factor $f(t,r)$. Redshift will diverge when $f(t_0 , r_0)$ goes to zero or $1+z\longrightarrow \infty$. This divergence as the signal of singularity is independent of the curvature scalar ($k$), unlike the Mcvittie’s solution and its various generalizations [@Gao0; @Gao], which shows that our solutions are compatible with the FRW background. As a desired expectation, it is obvious that the FRW result is covered when $f(t_0 , r_0)=f(t,r)=1$. The only non-diagonal term of the Einstein tensor is $$\begin{aligned}
\label{off diagonal}
G^{t
r}=-\frac{1-kr^2}{f(t,r)a(t)^3r}(a(t)\dot{f}(t,r)-{f}^\prime(t,r)\dot{a}(t)r),\end{aligned}$$ which $(\dot{})$ and $(^\prime)$ are derivatives with respect to time and radius, respectively. Using $\frac{\partial f}{\partial
t}=\dot{a}\frac{\partial f}{\partial a}$, one gets $$\begin{aligned}
\label{off diagonal2}
G^{t
r}=-\frac{(1-kr^2)\dot{a}(t)}{f(a(t),r)a(t)^3r}(a(t)\tilde{f}(a(t),r)-{f}^\prime(a(t),r)r),\end{aligned}$$ where $\tilde{f}(a(t),r)=\frac{\partial f}{\partial a}$. In order to get perfect fluid solutions, we impose condition $G^{tr}=0$ and reach to $$\begin{aligned}
\label{f}
f(t,r)=f(a(t)r)=\sum_n b_n (a(t)r)^n.\end{aligned}$$ Although Eq. (\[f\]) includes numerous terms, but the slow expansion approximation helps us to attribute physical meaning to the certain coefficients $b_n$. Since $G_{tr}=0$, we should stress that here that there is no redial flow and thus, the backreaction effect is zero [@RMS; @Gao1], which means that there is no energy accretion in these solutions [@fj]. Finally and briefly, we see that the perfect fluid concept is in line with the no energy accretion condition. The only answer which is independent of the rate of expansion can be obtained by condition $b_n=\delta_{n0}$ which is yielding the FRW solution.
mcVittie like solution in the FRW background
============================================
The mcVittie’s solution in the flat FRW background can be written as [@MD1] $$\begin{aligned}
\label{mc1}
ds^2=-(\frac{1-\frac{M}{2a(t)\tilde{r}}}{1+\frac{M}{2a(t)\tilde{r}}})^2dt^2+
a(t)^2 (1+\frac{M}{2a(t)\tilde{r}})^4[d\tilde{r}^2+
\tilde{r}^2d\Omega^2].\end{aligned}$$ This metric possess symmetries same as metric (\[Metric1\]). $\tilde{r}$ is isotropic radius defined by: $$\begin{aligned}
\label{r1}
r=\tilde{r}(1+\frac{M}{2 \tilde{r}})^2.\end{aligned}$$ There is a redshift singularity at radii $\tilde{r}_h=\frac{M}{2a(t)}$ which yields the radius $r_h=\frac{M}{2a(t)}(1+a(t))^2$ [@Fara]. In addition, $\tilde{r}_h$ is a spacelike hypersurface, and can not point to an event horizon [@fj].
Consider $f(a(t)r)=1-\frac{2b_{-1}}{a(t)r}$. This assumption satisfies condition (\[f\]) and leads to $$\begin{aligned}
\label{SCHWM}
ds^2=-(1-\frac{2b_{-1}}{a(t)r})dt^2+a(t)^2
[\frac{dr^2}{(1-kr^2)(1-\frac{2b_{-1}}{a(t)r})}+r^2d\Omega^2].\end{aligned}$$ For $b_{-1}\neq0$, this metric will converge to the FRW metric when $r\longrightarrow \infty$. The Schwarzschild metric is obtainable by putting $a(t)=1$, $b_{-1}=M$ and $k=0$. Metric suffers from three singularities at $a(t)=0$ (big bang), $r=0$ and $$\begin{aligned}
\label{SCHR}
f(a(t)r)=0\Rightarrow a(t)r_h=2b_{-1}.\end{aligned}$$ Third singularity exists if $b_{-1}>0$. In this manner, Eq. (\[redshift\]) will diverge at $r_0=r_h$. In addition and in contrast to the Gao’s solutions, the radii of the redshift singularity $(r_h)$ in our solutions is independent of the background curvature $(k)$, while for the flat case our radius is compatible with the previous works [@mcvittie; @Gao; @MD1]. Also, metric changes its sign at $r=r_h$ just the same as the schwarzschild spacetime. In addition, curvature scalars diverge at this radius as well as the mcVittie spacetime. Accordingly, this singularity point to a naked singularity which can be considered as alternatives for BHs [@virb; @virb1]. In continue, we will point to the some physical and mathematical properties of this singularity which has the same behaviors as event horizon if one considers slow expansion approximation. The surface area integration at this radius leads to $$\begin{aligned}
\label{surface area1}
A=\int\sqrt{\sigma}d\theta d\phi=4\pi r^2_h a(t)^2=16\pi
(b_{-1})^{2}.\end{aligned}$$ The main questions that arise here are: what is the nature of $b_{-1}$? and can we better clarify the meaning of $r_h$? For these purposes, we consider the slow expansion approximation $(a(t)\approx
c)$, define new coordinate $\eta=cr$ and get $$\begin{aligned}
\label{apmetric}
ds^2\approx-(1-\frac{2b_{-1}}{\eta})dt^2+\frac{d\eta^2}{(1-k^{\prime}\eta^2)
(1-\frac{2b_{-1}}{\eta})}+\eta^2d\theta^2+\eta^2sin(\theta)^2d\phi^2,\end{aligned}$$ where $k^{\prime}=\frac{k}{c^2}$. In these new coordinates, $(t,\eta,\theta,\phi)$, and from Eq. (\[nhs\]) it is apparent that for $b_{-1}>0$, hypersurface with equation $\eta=\eta_h=2b_{-1}$ is a null hypersurface. When our approximation is broken, then $\eta_h$ may not be actually a null hypersurface, despite its resemblance to that. We call this null hypersurface a quasi event horizon which is signalling us an object like a BH and we refer to that as a quasi BH. From now, we assume $b_{-1}>0$, the reason of this option will be more clear later, when we debate mass. Therefore by the slow expansion approximation, $r_h$ ($=\frac{2b_{-1}}{c}$) plays the role of the co-moving radius of event horizon and it is decreased with time. In order to find an answer to the first question about the physical meaning of $b_{-1}$, we use Komar mass: $$\begin{aligned}
\label{mass1}
M=\frac{1}{4\pi}\int_S n^{\alpha} \sigma_{\beta}
\triangledown_\alpha \xi^\beta_t dA,\end{aligned}$$ where $\xi^\beta_t$ is the timelike Killing vector of spacetime. Since the Komar mass is only definable for the stationary and asymptotically flat spacetimes [@W1], one should consider the flat case ($k=0$) and then by bearing the spirit of the stationary limit in mind (the slow expansion approximation) tries to evaluate Eq. (\[mass1\]).
Consider $n_{\alpha}=\sqrt{1-\frac{2b_{-1}}{a(t)r}}\delta^t_\alpha$ and $\sigma_\beta=\frac{a(t)}{\sqrt{1-\frac{2b_{-1}}{a(t)r}}}\delta^r_\beta$ as the unit timelike and unit spacelike four-vectors, respectively. Now using Eq. (\[mass1\]) and bearing the spirit of the slow expansion approximation in mind, one gets $$\begin{aligned}
\label{mass}
M=\frac{1}{4\pi}\int_S n^{\alpha} \sigma_{\beta}
\Gamma^{\beta}_{\alpha t}dA=b_{-1},\end{aligned}$$ which is compatible with the no energy accretion condition ($G_{tr}=0$). In addition, we will find the same result as Eq. (\[mass\]), if we considered the flat case ($k=0$) of metric (\[apmetric\]) and use $n_{\alpha}=\sqrt{1-\frac{2b_{-1}}{\eta}}\delta^t_\alpha$ and $\sigma_\beta=\frac{1}{\sqrt{1-\frac{2b_{-1}}{\eta}}}\delta^\eta_\beta$. Since the integrand is independent of the scale factor ($a(t)$), the slow expansion approximation does not change the result of integral. But, the accessibility of the slow expansion approximation is necessary if one wants to evaluate the Komar mass for dynamical spacetimes [@W1]. Indeed, this situation is the same as what we have in the quasi-equilibrium thermodynamical systems, where the accessibility of the quasi-equilibrium condition lets us use the equilibrium formulation for the vast thermodynamical systems [@callen]. It is obvious that for avoiding negative mass, we should have $b_{-1}>0$. Relation to the Komar mass of the mcVittie’s solution can be written as [@MD1; @Gao] $$\begin{aligned}
\label{komar}
M_{mcVittie}=\frac{M}{a(t)}.\end{aligned}$$ In addition, some studies show that the Komar mass is just a metric parameter in the mcVittie spacetime [@nol1; @nol2; @fj]. Indeed, Hawking-Hayward quasi-local mass satisfies $\dot{M}=0$, which is compatible with $G_{r t}=0$ and indicates that there is no redial flow and thus the backreaction effect, in the mcVittie’s solution [@RMS; @Gao1; @Bab; @fj]. In order to clarify the mass notion in the mcVittie spacetime, we consider the slow expansion approximation of the mcVittie spacetime which yields $$\begin{aligned}
\label{mse}
ds^2 \approx -(\frac{1-\frac{M}{2\eta}}{1+\frac{M}{2\eta}})^2dt^2+
(1+\frac{M}{2\eta})^4[d\eta^2+ \eta^2d\Omega^2].\end{aligned}$$ This metric is signalling us that the $M$ may play the role of the mass in the mcVittie spacetime. In addition, by defining new radii $R$ as $$\begin{aligned}
R(t,r)=a(t)\tilde{r}(1+\frac{M}{2\tilde{r}})^2,\end{aligned}$$ one can rewrite the mcVittie spacetime in the form of $$\begin{aligned}
\label{mn}
ds^2=-(1-\frac{2M}{R}-H^2R^2)dt^2-\frac{2HR}{\sqrt{1-\frac{2M}{R}}}dtdR+
\frac{dR^2}{1-\frac{2M}{R}}+R^2d\Omega^2,\end{aligned}$$ where $H=\frac{\dot{a}}{a}$ [@ff]. This form of the mcVittie spacetime indicates these facts that the Komar mass is a metric parameter and $M$ is the physical mass in this spacetime [@ff]. Finally, we see that the results of the slow expansion approximation (Eq. (\[mse\])) and Eq. (\[mn\]) are in line with the result of the study of the Hawking-Hayward quasi-local mass in the mcVittie spacetime [@nol1; @nol2; @fj; @ff]. For the flat case ($k=0$) of our spacetime (Eq. (\[SCHWM\])), by considering Eq. (\[komar\]) and following the slow expansion approximation, we reach at $$\begin{aligned}
\label{nm1}
ds^2\approx-(1-\frac{2M}{\eta})dt^2+\frac{d\eta^2}{(1-\frac{2M}{\eta})}
+\eta^2d\theta^2+\eta^2sin(\theta)^2d\phi^2.\end{aligned}$$ Also, if we define new radius $R$ as $$\begin{aligned}
r=\frac{R}{a}(1+\frac{M}{2R})^2,\end{aligned}$$ we obtain $$\begin{aligned}
\label{nm}
ds^2&=&
-(\frac{(1-\frac{M}{2R})^2}{(1+\frac{M}{2R})^2}-\frac{R^2H^2(1+\frac{M}{2R})^6}
{(1-\frac{M}{2R})^2})dt^2-
\frac{2RH(1+\frac{M}{2R})^5}{(1-\frac{M}{2R})}dtdR\\ \nonumber
&+&(1+\frac{M}{2R})^4[dR^2+R^2d\Omega^2].\end{aligned}$$ Both of the equations (\[nm1\]) and (\[nm\]) as well as the no energy accretion condition suggest that, unlike the mcVittie’s spacetime, the Komar mass may play the role of the mass in our solution. From Eq. (\[nm\]) it is apparent that $R=\frac{M}{2}$ points to the spacelike hypersurface where, in the metric (\[mn\]), $R=2M$ points to the null hypersurface. In the next subsection and when we debate thermodynamics, we will derive the same result for the mass notion in our spacetime.Only in the $a(t)=1$ limit (the Schwarzschild limit), Eqs. (\[nm\]) and (\[mc1\]) will be compatible which shows that our spacetime is different with the mcVittie’s. Let us note that the obtained metric (Eq. (\[nm\])) is consistent with Eq. (\[mn\]), provided we take $M=0$ (the FRW limit).
Horizons, energy and thermodynamics {#horizons-energy-and-thermodynamics .unnumbered}
-----------------------------------
There is an apparent horizon in accordance with the FRW background which can be evaluated from Eq. (\[aph1\]): $$\begin{aligned}
(1-kr_{ap}^2)(1-\frac{2M}{a(t)r_{ap}})^2-r_{ap}^2\dot{a}(t)^2=0.\end{aligned}$$ This equation covers the FRW results in the limit of $M\longrightarrow0$ ( see Eq. (\[aph2\])). In addition, one can get the Schwarzschild radius by considering $\dot{a}(t)=0$, which supports our previous definition for $b_{-1}$. Calculations for the flat case yield four solutions. The only solution which is in full agreement with the limiting situation of the FRW metric (in the limit of zero $M$) is $$\begin{aligned}
\label{hcr}
r_{ap}=\frac{1+\sqrt{1-8HM}}{2\dot{a}}.\end{aligned}$$ Therefore, the physical radius of apparent horizon $(\xi_{ap}=a(t)r_{ap})$ is $$\begin{aligned}
\xi_{ap}=\frac{1+\sqrt{1-8HM}}{2H},\end{aligned}$$ which is similar to the conformal BHs [@RMS]. It is obvious that in the limit of $M\longrightarrow0$, the radius for the apparent horizon of the flat FRW is recovered. For the surface gravity of apparent horizon, one can use Eq. (\[sg1\]) and gets: $$\begin{aligned}
\label{fsg}
\kappa=\frac{\kappa_{FRW}}{(1-\frac{2M}{a(t)r_{ap}})^2}+\frac{M}{a(t)^2}
[\frac{1}{r_{ap}^2}+\frac{1}{(1-\frac{2M}{a(t)r_{ap}})^2}(\ddot{a}(t)+
2\frac{\dot{a}(t)^2}{a(t)})],\end{aligned}$$ where $h^{ab}=diag(-\frac{1}{1-\frac{2M}{a(t)r}},\frac{1-\frac{2M}{a(t)r}}{a(t)^2})$, $r_{ap}$ is the apparent horizon co-moving radius (\[hcr\]) and $\kappa_{FRW}$ is the surface gravity of the flat FRW manifold $$\begin{aligned}
\kappa_{FRW}=-\frac{\dot{a}(t)^2+a(t)\ddot{a}(t)}{2a(t)\dot{a}(t)}.\end{aligned}$$
The schwarzschild limit ($\kappa=\frac{1}{4M}$) is obtainable by inserting $a(t)=1$ in Eq. (\[fsg\]). In the limiting case $M\longrightarrow0$, Eq. (\[fsg\]) is reduced to the surface gravity of the flat FRW spacetime, as a desired result. The Misner-Sharp mass inside radius $\xi$ for this spherically symmetric spacetime is defined as [@Ms]: $$\begin{aligned}
\label{MS}
M_{MS}=\frac{\xi}{2}(1-h^{ab}\partial_a \xi \partial_b \xi).\end{aligned}$$ Because this definition does not yield true results in some theories such as the Brans-Dicke and scalar-tensor gravities, we are pointing to the Gong-Wang definition of mass [@Gong]: $$\begin{aligned}
\label{GW}
M_{GW}=\frac{\xi}{2}(1+h^{ab}\partial_a \xi \partial_b \xi).\end{aligned}$$ It is apparent that, for the apparent horizon, Eqs. (\[MS\]) and (\[GW\]) yield the same result as $M_{GW}=M_{MS}=\frac{\xi_{ap}}{2}$. In the limit of $M\longrightarrow0$, the FRW’s results are recovered and we reach to $M_{GW}=M_{MS}=\rho V$ as a desired result [@Cai1]. Using Eqs. (\[MS\]) and (\[GW\]) and taking the slow expansion approximation into account, we reach to $M_{GW}=M_{MS}\simeq M$ as the mass of quasi BH. Also, this result supports our previous guess about the Komar mass as the physical mass in our solution, and is in line with the result of Eqs. (\[nm1\]) and (\[nm\]). For the Mcvittie metric, Eqs. (\[MS\]) and (\[GW\]) yield $M_{GW}=M_{MS}\simeq\frac{M}{4}$ as the confined mass to radius $\xi_h=a(t)\tilde{r}_h=\frac{M}{2}$. Also, Eqs. (\[mass\]), (\[MS\]) and (\[GW\]) leads to the same result in the Schwarzschild’s limit ($\mathcal{M}=M_{GW}=M_{MS}=M$). For the flat background, using metric (\[apmetric\]), Eq. (\[SG\]) and inserting results into Eq. (\[Temp2\]), one gets $$\begin{aligned}
\label{Temp1}
T\simeq\frac{1}{8 \pi M},\end{aligned}$$ for the temperature on the surface of quasi horizon. The same calculations yield similar results, as the temperature on the horizon of the Mcvittie’s solution. For the conformal Schwarzschild BH, the same analysis leads to $$\begin{aligned}
T\simeq\frac{1}{8 \pi a(t)M},\end{aligned}$$ which shows that the $a(t)M$ plays the role of mass, and is compatible with the energy accretion in the conformal BHs [@RMS; @Gao1; @Bab; @Fara]. Again, we see that the temperature analysis can support our expectation from $M$ as the physical mass in our solutions. For the area of quasi horizon, we have $$\begin{aligned}
\label{surface area}
A=\int\sqrt{\sigma}d\theta d\phi=4\pi a(t)^2 r_h^2=16\pi M^{2}.\end{aligned}$$ In the mcVittie spacetime, this integral leads to $A=16 \pi M^2$. In order to vindicate our approximation, we consider $S=\frac{A}{4}$ for the entropy of quasi BH. In continue and from Eq. (\[Temp1\]), we get $$\begin{aligned}
TdS\simeq dM=dE.\end{aligned}$$
Whereas, we reach to $TdS\simeq dM\neq dE $ for the mcVittie spacetime. In the coordinates $(t,\eta,\theta,\phi)$, we should remind that, unlike the mcVittie spacetime, $E=M_{GW}=M_{MS}\simeq
M$ is valid for quasi BH and the work term can be neglected as the result of slow expansion approximation ($dW\sim 0$) [@Fara]. Finally and unlike the mcVittie’s horizon, we see that $TdS\simeq
dE$ is valid on the quasi event horizon. This result points us to this fact that the first law of the BH thermodynamics on quasi event horizon will be satisfied if we use either the Gong-Wang or the Misner-Sharp definitions for the energy of quasi BH. $TdS\simeq dE$ is valid for the conformal Schwarzschild BH, too [@Fara]. For the flat background, we see that the surface area at redshift singularity in our spacetime is equal to the mcVittie metric which is equal to the Schwarzschild metric. In continue and by bearing the slow expansion approximation in mind, we saw that the temperature on quasi horizon is like the Schwarzschild spacetime [@RMS]. In addition, we saw that the quality of the validity of the first law of the BH thermodynamics on quasi event horizon is like the conformal Schwarzschild BH’s and differs from the mcVittie’s solution.
In another approach and for the mcVittie spacetime, if we use the Hawking-Hayward definition of mass as the total confined energy to the hypersurface $\tilde{r}=\frac{M}{2a(t)}$, we reach to $$\begin{aligned}
\label{Tempp}
TdS\simeq dM=dE,\end{aligned}$$ where we have considered the slow expansion approximation. In addition, Eq. (\[Tempp\]) will be not valid, if one uses the Komar mass (\[komar\]). Finally, we saw that the first law of thermodynamics will be approximately valid in the mcVittie’s solution, if one uses the Hawking-Hayward definition of energy. Also, none of the Komar, Misner-Sharp and Gong-Wang masses can not satisfy the first law of thermodynamics on the mcVittie’s horizon.
Other Possibilities
===================
According to what we have said, it is obvious that there are two other meaningful sentences in expansion (\[f\]). The first term is due to $n=-2$ and points to the charge, where the second term comes from $n=2$ and it is related to the cosmological constant. Therefore, the more general form of $f(t,r)$ can be written as: $$\begin{aligned}
\label{tf}
f(t,r)=1-\frac{2M}{a(t)r}+\frac{Q^2}{(a(t)r)^2}-\frac{1}{3}\Lambda
(a(t)r)^2,\end{aligned}$$ where we have considered the slow expansion approximation and used these definitions $b_{-2}\equiv Q^2$ and $b_2\equiv-\frac{1}{3}\Lambda$. Imaginary charge $(b_{-2}<0)$ and the anti De-Sitter $(\Lambda<0)$ solutions are allowed by this scheme, but these possibilities are removed by the other parts of physics. Consider Eq. (\[tf\]) when $\Lambda=0$, there are two horizons located at $r_+=\frac{M+\sqrt{M^2-Q^2}}{a(t)}$ and $r_-=\frac{M-\sqrt{M^2-Q^2}}{a(t)}$. These radiuses are same as the Gao’s flat case [@Gao]. In the low expansion regime ($a(t)\sim
c$), these radiuses point to the event and the Coushy horizons, as the Riessner-Nordstorm metric [@Poisson]. Hence, we refer to them as quasi event and quasi Coushy horizons. The case with $Q=0$, $M=0$ and $\Lambda>0$ has attractive properties. Because in the low expansion regime $(a(t)\simeq c)$, one can rewrite this case as $$\begin{aligned}
ds^2\approx -(1-\frac{\Lambda}{3}\eta^2)dt^2+\frac{d\eta^2}
{(1-\frac{\Lambda}{3}\eta^2)}+\eta^2 d\Omega^2.\end{aligned}$$ This is nothing but the De-Sitter spacetime with cosmological constant $\Lambda$, which points to the current acceleration era.
Horizons and temperature {#horizons-and-temperature .unnumbered}
------------------------
Different $f(t,r)$ yield apparent horizons with different locations, and one can use Eqs. (\[aph1\]) and (\[sg1\]) in order to find the location and the temperature of apparent horizon. For every $f(t,r)$, using the slow expansion regime, we get: $$\begin{aligned}
\label{Metricf}
ds^2\approx -f(\eta)dt^2+\frac{d\eta^2}{f(\eta)}+\eta^2 d\Omega^2.\end{aligned}$$ Now, the location of horizons and their surface gravity can be evaluated by using Eq. (\[SG\]). Their temperature is approximately equal to Eq. (\[Temp2\]), or briefly: $$\begin{aligned}
T_i \simeq \frac{f^{\prime}(\eta)}{4 \pi}|_{\eta_{hi}},\end{aligned}$$ where $(^\prime)$ is derivative with respect to radii $\eta$ and $\eta_{hi}$ is the radii of i$^{\textmd{th}}$ horizon.
Conclusions \[Conclusions\]
===========================
We considered the conformal form of the special group of the non-static spherically symmetric metrics, where it was assumed that the time dependence of the conformal factor is like as the FRW’s. We saw that the conformal BHs can be classified as a special subgroup of these metrics. In order to derive the new solutions of the Einstein equations, we have imposed perfect fluid concept and used slow expansion approximation which helps us to clarify the physical meaning of the parameters of metric. Since the Einstein tensor is diagonal, there is no energy accretion and thus the backreaction effect is zero. This imply that the energy (mass) should be constant in our solutions. These new solutions have similarities with earlier metrics that have been presented by others [@mcvittie; @Gao; @Gao0]. A related metric which is similar to the special class of our solutions was introduced by mcVittie [@mcvittie; @Gao]. These similarities are explicit in the flat case (temperature and entropy at the redshift singularity), but the differences will be more clear in the non-flat case ($k\neq0$), and we pointed to the one of them, when we debate the redshift. In addition and in the flat case, we tried to clear the some of differences between our solution and the mcVittie’s. We did it by pointing to the behavior of the redshift singularity in the various coordinates, the mass notion, and thermodynamics. Meanwhile, when our slow expansion approximation is broken then there is no horizon for our solutions. Indeed, these objects can be classified as naked singularities which can be considered as alternatives for BHs [@virb; @virb1].
For the our solutions and similar with earlier works [@mcvittie; @Gao; @Gao0], the co-moving radiuses of the redshift singularities are decreased by the expansion of universe. Also, unlike the previous works [@mcvittie; @Gao; @Gao0], the redshift singularities in our solutions are independent of the background curvature. By considering the slow expansion approximation, we were able to find out BH’s like behavior of these singularities. We pointed to these objects and their surfaces as quasi BHs and the quasi horizons, respectively. In continue, we introduced the apparent horizon for our spacetime which should be evaluated by considering the FRW background.
In order to compare the mcVittie’s solution with our mcVittie’s like solution, we have used the three existing definitions of mass including the Komar mass, the Misner-Sharp mass ($M_{MS}$) and the Gong-Wang mass ($M_{GW}$). We saw that the notion of the Komar mass of quasi BH differs from the mcVittie’s solution. Also, in our spacetime, we showed that the $M_{MS}$ and $M_{GW}$ masses yield the same result on the apparent horizon and cover the FRW’s result in the limiting situations. In addition, using the slow expansion approximation, we evaluated $M_{MS}$ and $M_{GW}$ on the quasi event horizon of our mcVittie’s like solution, which leads to the same result as the Komar mass. In addition, we should express that, the same as the mcVittie spacetime, the energy conditions are not satisfied near the quasi horizon.
In addition, we have proved that, unlike the mcVittie’s solution, the first law of thermodynamics may be satisfied on the quasi event horizon of our mcVittie’s like solution, if we use the Komar mass or either $M_{MS}$ or $M_{GW}$ as the confined mass and consider the slow expansion approximation. This result is consistent with previous studies about the conformal BHs [@Fara], which shows that the thermodynamics of our solutions is similar to the conformal BHs’. In order to clarify the mass notion, we think that the full analysis of the Hawking-Hayward mass for our solution is needed, which is out of the scope of this letter and should be considered as another work, but our resolution makes this feeling that the predictions by either the slow expansion approximation or using the suitable coordinates for describing the metric for mass, may be in line with the Hawking-Hayward definition of energy, and have reasonable accordance with the Komar, $M_{MS}$ and $M_{GW}$ masses of our solutions. Indeed, this final remark can be supported by the thermodynamics considerations and the no energy accretion condition ($G_{tr}=0$). Moreover, we think that, in dynamic spacetimes, the thermodynamic considerations along as the slow time varying approximation can help us to get the reasonable assumptions for energy and thus mass. Finally, we saw that the first law of thermodynamics will be approximately valid in the mcVittie’s and our solutions if we use the Hawking-Hayward definition of the mass in the mcVittie spacetime and the Komar mass as the physical mass in our solution, respectively. In continue, the more general solutions such as the charged quasi BHs and the some of their properties have been addressed.
Results obtained in this paper may help achieving a better understanding of black holes in a dynamical background. From a phenomenological point of view, this issue is important since after all, any local astrophysical object lives in an expanding cosmological background. Finally, we tried to explore the concepts of mass, entropy and temperature in a dynamic spacetime.
Acknowledgments
===============
We are grateful to referee for appreciable comments which led to sensible improvements in this manuscript. This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM).
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[^1]: h.moradpour@riaam.ac.ir
[^2]: n$\_$riazi@sbu.ac.ir
|
---
abstract: 'In this article we define the $-_+$-construction and the $-^+$-construction, that was crucial in the theory of canonical induction formulas (see [@Boltje1998b]), in the setting of biset functors, thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors. Additionally, this provides a unified approach to the study of a class of functors including the Burnside ring, the monomial Burnside ring and global representation ring.'
author:
- |
Robert Boltje\
Department of Mathematics\
University of California\
Santa Cruz, CA 95064\
U.S.A.\
boltje@math.ucsc.edu
- |
Gerardo Raggi-Cárdenas\
Centro de Ciencias Matemáticas\
UNAM\
Morelia, Mich\
MEXICO\
raggi@matmor.unam.mx
- |
Luis Valero-Elizondo\
Facultad de Cs Fis-Mat\
Universidad Michoacana\
Morelia, Mich\
MEXICO\
valero@fismat.umich.mx
date: 'May 23, 2018'
title: 'The $-_+$ and $-^+$ Constructions for Biset Functors [^1] [^2] '
---
Introduction
============
This paper can be considered as the first step to extend the framework of canonical induction formulas, introduced by the first author in [@Boltje1998b], to the setting of biset functors, a notion introduced by Bouc, see [@Bouc2010a]. The most basic example of a canonical induction formula is the one for the character ring $R(H)$ of a finite group, which one can regard as a canonical section of $a_H\colon R(H)\to {R^{\mathrm{ab}}}_+(H)$ of the natural map $b_H\colon {R^{\mathrm{ab}}}_+(H)\to R(H)$. Here, ${R^{\mathrm{ab}}}_+(H)$ is the free abelian group on $H$-conjugacy classes $[K,\varphi]_H$ of pairs $(K,\varphi)$, where $K{\leqslant}H$ and $\varphi$ is a linear character of $K$, i.e., a character of degree $1$, and $b_H([K,\varphi]_H)={\mathrm{ind}}_K^H(\varphi)$. In [@Boltje1998b], both $R$ and ${R^{\mathrm{ab}}}_+$ were considered as Mackey functors, where $H$ runs through all subgroups of a fixed finite group $G$. The maps $b_H$, $H{\leqslant}G$, commute with conjugations, restrictions, and inductions, while the maps $a_H$, $H{\leqslant}G$, commute only with conjugations and restrictions. The groups ${R^{\mathrm{ab}}}_+(H)$, $H{\leqslant}G$, were constructed via the $-_+$-construction from the groups ${R^{\mathrm{ab}}}(H)$, $H{\leqslant}G$. Here ${R^{\mathrm{ab}}}(H)$ denotes the subgroups of $R(H)$ generated generated by linear characters of $H$. These groups allow conjugation and restriction maps, but no induction maps. Canonical induction formulas exist for a variety of representation rings: The character ring, the Brauer character ring, the group of projective representations, the trivial source ring, and the linear source ring, see [@Boltje1998b]. There were two major unsatisfactory aspects to these constructions: The first is the restriction to subgroups of a fixed finite group $G$, the second one is the disregard for other natural operations, in particular [*inflations*]{}. Both these aspects were a consequence of using the framework of Mackey functors, the main functorial setup for representation rings available at that time. Bouc’s theory of biset functors removes both of these restrictions and gives additional freedom of choosing certain of the operations of restriction, induction, inflation, deflation, conjugations, isomorphism, or even more general sets of operators.
The aim of this paper is to introduce the $-_+$ and $-^+$-constructions within the framework of biset functors. The $-^+$-construction is an auxiliary construction which is crucial in the proofs and the understanding of the $-_+$-construction. This should also be of interest independent of the theory of canonical induction formulas, since the $_+$ -construction yields various important biset functors (see Example \[ex Burnside examples\]: The Burnside functor is the $-_+$-construction of the constant biset functor with values ${\mathbb{Z}}$. For an abelian group $A$, the monomial (also called $A$-fibered) Burnside ring is the $-_+$-construction applied to the biset functor mapping a finite group $G$ to the free ${\mathbb{Z}}$-module with basis ${\mathrm{Hom}}(G,A)$, see [@Dress], [@Barker], or [@BoltjeCoskun] for instance. This was used in the known examples of canonical induction formulas for the representation rings mentioned above, where $A$ is a subgroup of the unit group of an appropriate field. More recently, the second and the third author introduced the notion of the [*global representation ring*]{}, see [@RaggiValero2015], a combination of the Burnside ring and the character ring. Again, this construction turns out to be the $-_+$-construction applied to the character ring functor. Proving statements in general about the $-_+$-construction thus has applications for a variety of interesting examples, and unifies previous proofs for these examples. In the case of the first example, the Burnside ring $B(G)$ is the $-_+$-construction of the constant functor ${\mathbb{Z}}$ and the $-^+$-constructions of the constant functor ${\mathbb{Z}}$ yields its ghost ring $(\prod_{H{\leqslant}G}{\mathbb{Z}})^G$, where the exponent $G$ denotes taking $G$-fixed points with respect to permuting the components according to the conjugation action of $G$ on its subgroups. One of the main tools to study the Burnside ring is the [*mark homomorphism*]{} $B(G)\to(\prod_{H{\leqslant}G}{\mathbb{Z}})^G$ introduced by Burnside. This feature generalizes to our set-up.
We start our axiomatic setup with the choice of a family ${\mathcal{G}}$ of finite groups and for every $G,H\in{\mathcal{G}}$, a choice of a set ${\mathcal{S}}(G,H)$ of subgroups of $G\times H$, satisfying axioms that lead to a category ${\mathcal{D}}$, a subcategory of Bouc’s biset category ${\mathcal{C}}$. This general setup accommodates for instance the representation theory of the symmetric groups, where only symmetric groups on finite sets, their Young subgroups, and only restrictions and inductions between them are of interest. In Section 2 we recall basic definitions and facts on bisets and biset functors. Section 3 defines how to construct from $({\mathcal{G}},{\mathcal{S}})$ associated pairs $({\mathcal{G}},{\mathcal{S}}_+)$ and $({\mathcal{G}},{\mathcal{S}}^+)$, which lead to subcategories ${\mathcal{D}}_+$ and ${\mathcal{D}}^+$ of ${\mathcal{C}}$. Sections 4 and 5 describe the construction of the biset functor $F_+$ on ${\mathcal{D}}_+$ and the biset functor $F^+$ on ${\mathcal{D}}^+$ associated to a biset functor $F$ on ${\mathcal{D}}$. In Section 6 we define the mark morphism $F_+\to F^+$ as a natural transformation. We prove that, under certain conditions on the base ring, the mark morphism is injective, or even bijective. Section 7 deals with the situation that $F$ has a multiplicative structure, more precisely, that $F$ is a Green biset functor. We show that then also $F_+$ and $F^+$ inherit Green biset functor structures and that the mark morphism is multiplicative. We also show how the species of $F_+(G)$ are determined by the species of $F(H)$, for $H{\leqslant}G$ with $H\in{\mathcal{G}}$. Finally, in Section 9 we prove adjunction properties of the functor $F\mapsto F_+$ similar to the properties proven in [@Boltje1998b].
Bisets and Biset Functors
=========================
Throughout this section, let $R$ denote a commutative ring (associative with $1$). We recall the notions of bisets and biset functors from [@Bouc2010a].
[*$(G,H)$-bisets and $B(G,H)$.*]{} For finite groups $G$ and $H$, a [*$(G,H)$-biset*]{} is a finite set $U$ equipped with a left $G$-action and a right $H$-action which commute: $g(uh)=(gu)h$ for all $g\in G$, $h\in H$, $u\in U$. The $(G,H)$-bisets form a category, whose morphisms are the functions that are $G$-equivariant and $H$-equivariant. Denote by $B(G,H)$ the Grothendieck group of $(G,H)$-bisets with respect to coproducts (disjoint unions). Identifying $(G,H)$-bisets with left $(G\times H)$-sets via $(g,h)u=guh^{-1}$ for $g\in G$, $h\in H$, and $u\in U$, the abelian group $B(G,H)$ is free with [*standard basis elements*]{} $[G\times H/D]$, where $D{\leqslant}G\times H$ runs through a set of representatives of the conjugacy classes of subgroups of $G\times H$. Here, for a $(G,H)$-biset $U$, $[U]$ denotes the associated element in the Grothendieck group.
If also $K$ is a finite group, if $U$ is a $(G,H)$-biset, and $V$ is an $(H,K)$-biset, then $U\times V$ is an $H$-set via $h(u,v):=(uh^{-1},hv)$, for $u\in U$, $v\in V$ and $h\in H$. The $H$-orbit of $(u,v)$ is denoted by $[u,_H v]$ (or just $[u,v]$ if there is no risk of confusion) and the set of $H$-orbits is denoted by $U\times_H V$. The latter is naturally a $(G,K)$-biset via $g[u, v]k:=[gu, vk]$ for $(u,v)\in U\times V$ and $(g,k)\in G\times K$. This construction induces a bilinear map $$\label{eqn ten of bisets}
-{\mathop{\cdot}\limits_{H}}-\colon B(G,H)\times B(H,K)\to B(G,K)\,.$$
[*The biset category ${\mathcal{C}}$ and the biset functor category ${\mathcal{F}}_{{\mathcal{D}},R}$.*]{}Let ${\mathcal{C}}$ denote the following category. Its objects are the finite groups, and for finite groups $G$ and $H$, one sets ${\mathrm{Hom}}_{{\mathcal{C}}}(H,G):=B(G,H)$. The identity morphism of $G$ is $[G]$, where $G$ is viewed as $(G,G)$-biset by left and right multiplication. For finite groups $G,H,K$, the composition in ${\mathcal{C}}$ is defined by the map in (\[eqn ten of bisets\]).
For any subcategory ${\mathcal{D}}$ of ${\mathcal{C}}$, a [*biset functor on ${\mathcal{D}}$ over $R$*]{} is an additive functor ${\mathcal{F}}\colon{\mathcal{D}}\to {\llap{\phantom{|}}_{R}{\mathsf}{Mod}}$. The biset functors on ${\mathcal{D}}$ over $R$ form an abelian category ${\mathcal{F}}_{{\mathcal{D}},R}$ whose morphisms are the natural transformations between biset functors.
[*Subgroups of $G\times H$.*]{}Let $G$ and $H$ be finite groups and let $D{\leqslant}G\times H$. We write $p_1\colon G\times H\to G$ and $p_2\colon G\times H\to H$ for the natural projection maps. Moreover, we set $$k_1(D):=\{g\in G\mid (g,1)\in D\}\quad\text{and}\quad k_2(D):=\{h\in H\mid (1,h)\in D\}\,.$$ Note that the projection maps $p_i$ induce isomorphisms $D/(k_1(D)\times k_2(D)){\buildrel\sim\over\to}p_i(D)/k_i(D)$, for $i=1,2$. The resulting canonical isomorphism $\eta_D\colon p_2(D)/k_2(D){\buildrel\sim\over\to}p_1(D)/k_1(D)$ is characterized by $\eta(hk_2(D))=gk_2(D)$ if and only if $(g,h)\in D$, for $g\in p_1(D)$ and $h\in p_2(D)$.
\[noth \* product\] [*The $*$-product.*]{}For subgroups $D{\leqslant}G\times H$ and $E{\leqslant}H\times K$, one sets $$D*E:=\{(g,k)\in G\times K\mid \exists h\in H \text{ with } (g,h)\in D \text{ and } (h,k)\in E\}{\leqslant}G\times K\,.$$ Note that this product is associative. For a subgroup $H_1{\leqslant}H$, one defines $D*H_1{\leqslant}G$ in a similar way, by identifying $H$ with $H\times \{1\}$ and $G$ with $G\times \{1\}$. Note also that if $U$ is a $(G,H)$-biset, $V$ is an $(H,K)$-biset, and $(u,v)\in U\times V$, then $$\label{eqn stab *}
(G\times K)_{[u,_H v]} = (G\times H)_u * (H\times K)_v\,.$$
The following theorem gives an explicit formula for the product in (\[eqn ten of bisets\]) in terms of standard basis elements, see [@Bouc2010a Lemma 2.3.24].
\[thm Mackey formula\]([@Bouc2010a Lemma 2.3.24]) Assume that $G$, $H$, $K$ are finite groups and $D{\leqslant}G\times H$, $E{\leqslant}H\times K$ are subgroups. Then, $$\left[\frac{G\times H}{D}\right]{\mathop{\cdot}\limits_{H}}\left[\frac{H\times K}{E}\right] = \coprod_{t\in [p_2(D)\backslash H/p_1(E)]}
\left[\frac{G\times K}{D*{\setbox0=\hbox{$E$} \setbox1=\vbox to
\ht0{}\,\box1^{(t,1)}\!E}}\right] \ \in B(G,K)\,.$$
[*Elementary biset operations.*]{} Assume that $G$ is a finite group. For a subgroup $H{\leqslant}G$ one sets $${\mathrm{res}}^G_H:=[{\llap{\phantom{|}}_{H}{G}}_G]=\left[\frac{H\times G}{\Delta(H)}\right]\in B(H,G)\quad\text{and}\quad
{\mathrm{ind}}_H^G:=[{\llap{\phantom{|}}_{G}{G}}_H]=\left[\frac{G\times H}{\Delta(H)}\right]\in B(G,H)\,,$$ where $G$ is viewed as a $(G,H)$-biset and as an $(H,G)$-biset with via left and right multiplication and $\Delta(H):=\{(h,h)\mid h\in H\}$. For a normal subgroup $N\trianglelefteq G$ one sets $$\begin{gathered}
{\mathrm{inf}}^G_{G/N}:=[{\llap{\phantom{|}}_{G}{(G/N)}}_{G/N}]=\left[\frac{G\times G/N}{\{(g,gN)\mid g\in G\}}\right]\in B(G,G/N)\quad\text{and}\\
{\mathrm{def}}^G_{G/N}:=[{\llap{\phantom{|}}_{G/N}{(G/N)}}_G]=\left[\frac{G/N\times G}{\{(gN,g)\mid g\in G\}}\right]\in B(G/N,G)\,,\end{gathered}$$ where $G/N$ is viewed as a $(G,G/N)$-biset and as a $(G/N,G)$-biset via the natural epimorphism $G\to G/N$ and left and right multiplication. Finally, if $\alpha\colon G{\buildrel\sim\over\to}G'$ is a group isomorphism, we set ${\mathrm{iso}}_\alpha:=[G']\in B(G',G)$, where $G'$ is considered as $(G',G)$-biset via $g'xg:=g'x\alpha(g)$ for $g',x\in G'$ and $g\in G$. The five elements defined above are referred to as [*restriction*]{}, [*induction*]{}, [*inflation*]{}, [*deflation*]{}, and [*isogation*]{}. When $F$ is a biset functor then these elements induce maps between the respective evaluations of $F$.
For arbitrary finite groups $G$ and $H$, one has a canonical decomposition of a standard basis element $\left[\frac{G\times H}{D}\right]$ of $B(G,H)$ into five elementary bisets:
\[thm elementary decomposition\]([@Bouc2010a Lemma 2.3.26]) Let $G$ and $H$ be finite groups and let $D{\leqslant}G\times H$. Then, in the category ${\mathcal{C}}$, the morphism $\left[\frac{G\times H}{D}\right]$ can be written as the following composition of elementary biset operations: $${\mathrm{ind}}_{p_1(D)}^G\circ{\mathrm{inf}}_{p_1(D)/k_1(D)}^{p_1(D)}\circ{\mathrm{iso}}_{\eta_D}\circ{\mathrm{def}}^{p_2(D)}_{p_2(D)/k_2(D)}\circ{\mathrm{res}}^{H}_{p_2(D)}\,.$$
The $-^+$ and $-_+$ Constructions on Subcategories ${\mathcal{D}}$ of ${\mathcal{C}}$ {#sec D_+}
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For a finite group $G$ we denote by $\Sigma(G)$ the set of all subgroups of $G$.
\[noth GS\] [*The data $({\mathcal{G}},{\mathcal{S}})$.*]{}(a) Throughout this section we consider a class ${\mathcal{G}}$ of finite groups and a family ${\mathcal{S}}=({\mathcal{S}}(G,H))_{G,H\in{\mathcal{G}}}$, with ${\mathcal{S}}(G,H)\subseteq\Sigma(G\times H)$ for $G,H\in{\mathcal{G}}$. We will assume throughout, that ${\mathcal{G}}$ and ${\mathcal{S}}$ satisfy the following axioms:
\(i) For all $G\in{\mathcal{G}}$ one has $\Delta(G)\in{\mathcal{S}}(G,G)$.
\(ii) For all $G,H\in{\mathcal{G}}$ the set ${\mathcal{S}}(G,H)$ is closed under $G\times H$-conjugation.
\(iii) For all $G,H,K\in{\mathcal{G}}$ and all $D\in{\mathcal{S}}(G,H)$ and $E\in{\mathcal{S}}(H,K)$ one has $D*E\in{\mathcal{S}}(G,K)$.
For $G\in{\mathcal{G}}$ we will denote by $\Sigma_{\mathcal{G}}(G)$ the set of all subgroups $H$ of $G$ with $H\in{\mathcal{G}}$.
\(b) In the sequel we will also consider additional properties of $({\mathcal{G}},{\mathcal{S}})$ that we will require as necessary. They are
\(iv) For all $G,H\in {\mathcal{G}}$, all $D\in{\mathcal{S}}(G,H)$ and all $K\in\Sigma_{{\mathcal{G}}}(H)$, one has $D*K\in{\mathcal{G}}$ and $D*\Delta(K)\in{\mathcal{S}}(D*K,K)$. Note that $D*\Delta(K)=D\cap (G\times K)$ and that $p_1(D*\Delta(K))=D*K$, but in general $p_2(D*\Delta(K))=p_2(D)\cap K$ can be a proper subgroup of $K$.
\(v) For all $G\in{\mathcal{G}}$ and all $H\in\Sigma_{{\mathcal{G}}}(G)$, one has $\Delta(H)\in{\mathcal{S}}(G,H)$.
\(vi) For all $G\in{\mathcal{G}}$ and all $H\in\Sigma_{{\mathcal{G}}}(G)$, one has $\Delta(H)\in{\mathcal{S}}(H,G)$.
\(vii) For all $G,H\in{\mathcal{G}}$ and all $D\in{\mathcal{S}}(G,H)$ one has $p_2(D)\in{\mathcal{G}}$, and for all $K\in\Sigma_{{\mathcal{G}}}(p_2(D))$ one has $D*K\in{\mathcal{G}}$ and $D*\Delta(K)\in{\mathcal{S}}(D*K,K)$. Note that $D*\Delta(K)=D\cap (G\times K)$, $p_1(D*\Delta(K))=D*K$ and $p_2(D*\Delta(K))=K$. Note also that this condition is symmetric (cf. Proposition \[prop D\^+\]).
\(c) Assume that $({\mathcal{G}},{\mathcal{S}})$ satisfies the additional axiom (iv) or (vii). If $G\in{\mathcal{G}}$, $H\in \Sigma_{{\mathcal{G}}}(G)$, and $g\in G$, then also ${\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H}\in{\mathcal{G}}$ and ${\setbox0=\hbox{$\Delta(H)$} \setbox1=\vbox to
\ht0{}\,\box1^{(g,1)}\!\Delta(H)}\in{\mathcal{S}}({\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H},H)$. In fact, ${\setbox0=\hbox{$\Delta(G)$} \setbox1=\vbox to
\ht0{}\,\box1^{(g,1)}\!\Delta(G)}\in{\mathcal{S}}(G,G)$ by (i) and (ii), and by (iv) or (vii) we obtain ${\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H}={\setbox0=\hbox{$\Delta(G)$} \setbox1=\vbox to
\ht0{}\,\box1^{(g,1)}\!\Delta(G)}*H\in{\mathcal{G}}$ and ${\setbox0=\hbox{$\Delta(H)$} \setbox1=\vbox to
\ht0{}\,\box1^{(g,1)}\!\Delta(H)}={\setbox0=\hbox{$\Delta(G)$} \setbox1=\vbox to
\ht0{}\,\box1^{(g,1)}\!\Delta(G)}*\Delta(H) \in{\mathcal{S}}({\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H},H)$.
\[noth D(G,S)\] [*The category ${\mathcal{D}}={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$.*]{}Given $({\mathcal{G}},{\mathcal{S}})$ as in \[noth GS\](a), we define the subcategory ${\mathcal{D}}={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$ of the biset category ${\mathcal{C}}$ by ${\mathrm{Ob}}({\mathcal{D}})={\mathcal{G}}$ and, for $G,H\in{\mathcal{G}}$, $${\mathrm{Hom}}_{{\mathcal{D}}}(H,G) := \big\langle\left[\frac{G\times H}{D}\right] \mid D\in{\mathcal{S}}(G,H)\big\rangle_{{\mathbb{Z}}} \subseteq B(G,H)\,.$$ Axioms (i)—(iii) in \[noth GS\](a) and Theorem \[thm Mackey formula\] imply immediately that this is in fact a subcategory of ${\mathcal{C}}$. Note that Axiom \[noth GS\](v) (resp. (vi)), if valid, ensures that the category ${\mathcal{D}}$ contains all possible inductions (resp. restrictions).
\[def D\_+\] Let $({\mathcal{G}},{\mathcal{S}})$ be as in \[noth GS\](a). For $G,H\in{\mathcal{G}}$ we define ${\mathcal{S}}_+(G,H)$ as the set of all subgroups $D{\leqslant}G\times H$ such that $${\mathcal{S}}_+(G,H):=\{ D{\leqslant}G\times H\mid p_1(D)\in{\mathcal{G}}\text{ and }D\in{\mathcal{S}}(p_1(D),H)\}\,.$$ The following proposition shows that if $({\mathcal{G}},{\mathcal{S}})$ satisfies also Axiom (iv) then $({\mathcal{G}},{\mathcal{S}}_+)$ satisfies again the axioms (i)–(iii) in \[noth GS\](a), so that we obtain a category ${\mathcal{D}}_+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}_+)$ by \[noth D(G,S)\].
Note that each $D\in{\mathcal{S}}_+(G,H)$ can be written as $\Delta(p_1(D))*D$ with $\Delta(p_1(D)){\leqslant}G\times p_1(D)$ and $D\in{\mathcal{S}}(p_1(D),H)$. Thus, $\left[\frac{G\times H}{D}\right] = {\mathrm{ind}}_{p_1(D)}^G \circ \left[\frac{p_1(D)\times H}{D}\right]$ with $\left[\frac{p_1(D)\times H}{D}\right]\in{\mathrm{Hom}}_{{\mathcal{D}}}(H,p_1(D))$.
\[prop D\_+\] Assume that $({\mathcal{G}},{\mathcal{S}})$ is as in \[noth GS\](a), satisfying Axioms (i)–(iii) and additionally Axiom (iv).
For $G, H\in{\mathcal{G}}$ and $D{\leqslant}G\times H$, one has $D\in{\mathcal{S}}_+(G,H)$ if and only if, for all $K\in\Sigma_{{\mathcal{G}}}(H)$, one has $D*K\in{\mathcal{G}}$ and $D*\Delta(K)\in{\mathcal{S}}(D*K,K)$. In particular, by Axiom (iv), ${\mathcal{S}}(G,H)\subseteq{\mathcal{S}}_+(G,H)$ for all $G,H\in{\mathcal{G}}$.
$({\mathcal{G}},{\mathcal{S}}_+)$ satisfies the axioms (i)–(v) in \[noth GS\].
One has ${\mathcal{D}}\subseteq {\mathcal{D}}_+$, with equality if and only if $({\mathcal{G}},{\mathcal{S}})$ satisfies Axiom (v). In particular, by Part (b), $({\mathcal{D}}_+)_+={\mathcal{D}}_+$, and ${\mathcal{D}}_+$ is the smallest ${\mathbb{Z}}-linear$ subcategory of ${\mathcal{C}}$ containing ${\mathcal{D}}$ and all inductions ${\mathrm{ind}}_{H}^G$ with $G\in{\mathcal{G}}$ and $H\in\Sigma_{{\mathcal{G}}}(G)$.
Let $G,H\in{\mathcal{G}}$ and $D{\leqslant}G\times H$ with $p_1(D)=G$. Then $D\in{\mathcal{S}}(G,H)$ if and only if $D\in{\mathcal{S}}_+(G,H)$. In particular, ${\mathcal{D}}_+$ contains a given restriction, inflation, or deflation if and only if ${\mathcal{D}}$ does.
\(a) Let $G,H\in{\mathcal{G}}$ and $D{\leqslant}G\times H$. First assume $D\in{\mathcal{S}}_+(G,H)$ and $K\in\Sigma_{{\mathcal{G}}}(H)$. Then $p_1(D)\in{\mathcal{G}}$ and $D\in{\mathcal{S}}(p_1(D),H)$ by the definition of ${\mathcal{S}}_+(G,H)$. Applying Axiom (iv) to $D\in{\mathcal{S}}(p_1(D),H)$ and $K$, we obtain $D*K\in{\mathcal{G}}$ and $D*\Delta(K)\in{\mathcal{S}}(D*K,K)$. For the converse, apply the condition to $K=H$ and note that $D*H=p_1(D)$ and $D*\Delta(H)=D$.
\(b) Clearly, $({\mathcal{G}},{\mathcal{S}}_+)$ satisfies Axiom (i).
To see that it satisfies Axiom (ii), let $D\in{\mathcal{S}}_+(G,H)$ and $(a,b)\in G\times H$. We have $p_1({\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,b)}\!D})={\setbox0=\hbox{$p_1(D)$} \setbox1=\vbox to
\ht0{}\,\box1^{a}\!p_1(D)}\in{\mathcal{G}}$ and ${\setbox0=\hbox{$\Delta(p_1(D))$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,1)}\!\Delta(p_1(D))}\in {\mathcal{S}}({\setbox0=\hbox{$p_1(D)$} \setbox1=\vbox to
\ht0{}\,\box1^{a}\!p_1(D)},p_1(D))$ by \[noth GS\](c). Thus, ${\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,1)}\!D}={\setbox0=\hbox{$\Delta(p_1(D))$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,1)}\!\Delta(p_1(D))}*D\in{\mathcal{S}}({\setbox0=\hbox{$p_1(D)$} \setbox1=\vbox to
\ht0{}\,\box1^{a}\!p_1(D)},H)$ by Axiom (iii), and therefore, by Axiom (ii), ${\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,b)}\!D}={\setbox0=\hbox{$($} \setbox1=\vbox to
\ht0{}\,\box1^{(1,b)}\!(}{\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,1)}\!D}) \in{\mathcal{S}}({\setbox0=\hbox{$p_1(D)$} \setbox1=\vbox to
\ht0{}\,\box1^{a}\!p_1(D)},H) = {\mathcal{S}}(p_1({\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{(a,b)}\!D}),H)$. This shows that $({\mathcal{G}},{\mathcal{S}}_+)$ satisfies Axiom (ii).
Next we show that $({\mathcal{G}},{\mathcal{S}}_+)$ satisfies Axiom (iii). Let $G,H,K\in{\mathcal{G}}$ and $D\in{\mathcal{S}}_+(G,H)$ and $E\in{\mathcal{S}}_+(H,K)$. We will use the characterization from Part (a). Let $L\in\Sigma_{{\mathcal{G}}}(K)$. Since $(D*E)*L= D*(E*L)$, we see that $E*L\in\Sigma_{\mathcal{G}}(H)$ and then $(D*E)*L\in\Sigma_{\mathcal{G}}(G)$. Next we show that $(D*E)*\Delta(L)\in {\mathcal{S}}(D*E*L,L)$. We have $(D*E)*\Delta(L) = (D*\Delta(E*L))*(E*\Delta(L))$. By Part (a) we have $E*\Delta(L)\in {\mathcal{S}}(E*L,L)$ and also $D*\Delta(E*L)\in{\mathcal{S}}(D*E*L,E*L)$. Now Axiom (iii) for $({\mathcal{G}},{\mathcal{S}})$ implies that $(D*E)*\Delta(L)\in{\mathcal{S}}(D*E*L,L)$.
Axiom (iv) for $({\mathcal{G}},{\mathcal{S}}_+)$ follows immediately from Part (a) and Axiom (iv) for $({\mathcal{G}},{\mathcal{S}})$.
Axiom (v) clearly holds for $({\mathcal{G}},{\mathcal{S}}_+)$ by the definition of ${\mathcal{S}}_+$, since $\Delta(H)\in{\mathcal{S}}(H,H)$, for all $H\in{\mathcal{G}}$.
\(c) Since ${\mathcal{S}}(G,H)\subseteq{\mathcal{S}}_+(G,H)$ by Part (a), we have ${\mathcal{D}}\subseteq{\mathcal{D}}_+$. If ${\mathcal{D}}_+={\mathcal{D}}$, then ${\mathcal{S}}={\mathcal{S}}_+$ and, by Part (b), $({\mathcal{G}},{\mathcal{S}}_+)$ satisfies Axiom (v), so also $({\mathcal{G}},{\mathcal{S}})$ does. Conversely, assume $({\mathcal{G}},{\mathcal{S}})$ satisfies Axiom (v), and let $D\in{\mathcal{S}}_+(G,H)$, for some $G,H\in{\mathcal{G}}$. Then $p_1(D)\in{\mathcal{G}}$ and $D\in{\mathcal{S}}(p_1(D),H)$. Moreover $\Delta(p_1(D))\in{\mathcal{S}}(G,p_1(D))$ by Axiom (v) for $({\mathcal{G}},{\mathcal{S}})$. By Axiom (iii) for $({\mathcal{G}},{\mathcal{S}})$, also $D=\Delta(p_1(D))*D\in {\mathcal{S}}(G,H)$. The last statement in (c) follows immediately from the paragraph following Definition \[def D\_+\].
\(d) This follows immediately from the definition of ${\mathcal{S}}_+$.
\[def D\^+\] Let $({\mathcal{G}},{\mathcal{S}})$ be as in \[noth GS\](a). For $G,H\in{\mathcal{G}}$ we define $${\mathcal{S}}^+(G,H):=\{D{\leqslant}G\times H\mid \text{$p_1(D)\in {\mathcal{G}}$, $p_2(D)\in{\mathcal{G}}$, and $D\in{\mathcal{S}}(p_1(D),p_2(D))$}\}\,.$$ The following proposition shows that if $({\mathcal{G}},{\mathcal{S}})$ also satisfies Axiom (vii) then $({\mathcal{G}},{\mathcal{S}}^+)$ satisfies again Axioms (i)–(iii) in \[noth GS\](a), so that we obtain a category ${\mathcal{D}}^+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}^+)$ by \[noth D(G,S)\].
Note that each $D\in{\mathcal{S}}^+(G,H)$ can be written as $\Delta(p_1(D))*D*\Delta(p_2(D))$ with $\Delta(p_1(D)){\leqslant}G\times p_1(D)$, $D\in{\mathcal{S}}(p_1(D),p_2(D))$, and $\Delta(p_2(D))\in{\mathcal{S}}(p_2(D),H)$. Thus, $\left[\frac{G\times H}{D}\right] = {\mathrm{ind}}_{p_1(D)}^G \circ \left[\frac{p_1(D)\times p_2(D)}{D}\right]\circ{\mathrm{res}}^H_{p_2(D)}$ with $\left[\frac{p_1(D)\times p_2(D)}{D}\right]\in{\mathrm{Hom}}_{{\mathcal{D}}}(p_2(D),p_1(D))$.
\[prop D\^+\] Assume that $({\mathcal{G}},{\mathcal{S}})$ is as in \[noth GS\](a), satisfying Axioms (i)–(iii) and additionally Axiom (vii).
For $G,H\in G$ and $D{\leqslant}G\times H$, the following are equivalent:
$D\in{\mathcal{S}}^+(G,H)$.
$p_2(D)\in{\mathcal{G}}$ and for all $L\in\Sigma_{\mathcal{G}}(p_2(D))$ one has $D*L\in{\mathcal{G}}$ and $D*\Delta(L)\in{\mathcal{S}}(L*D,L)$.
$p_1(D)\in{\mathcal{G}}$ and for all $K\in\Sigma_{\mathcal{G}}(p_1(D))$ one has $K*D\in{\mathcal{G}}$ and $\Delta(K)*D\in{\mathcal{S}}(K,K*D)$.
In particular, ${\mathcal{S}}(G,H)\subseteq{\mathcal{S}}^+(G,H)$ for all $G,H\in{\mathcal{G}}$.
$({\mathcal{G}},{\mathcal{S}}^+)$ satisfies Axioms (i)–(iii), (v), (vi), and (vii).
One has ${\mathcal{D}}\subseteq{\mathcal{D}}^+$, with equality if and only if $({\mathcal{G}},{\mathcal{S}})$ satisfies Axioms (v) and (vi). In particular, by Part (b), $({\mathcal{D}}^+)^+={\mathcal{D}}^+$ and ${\mathcal{D}}^+$ is the smallest ${\mathbb{Z}}$-linear subcategory of ${\mathcal{C}}$ containing ${\mathcal{D}}$ and all inductions ${\mathrm{ind}}_H^G$ and restrictions ${\mathrm{res}}^G_H$ with $G\in{\mathcal{G}}$ and $H\in\Sigma_{\mathcal{G}}(G)$.
Let $G,H\in{\mathcal{G}}$ and $D{\leqslant}G\times H$ with $p_1(D)=G$ and $p_2(D)=H$. Then $D\in{\mathcal{S}}(G,H)$ if and only if $D\in{\mathcal{S}}^+(G,H)$. In particular, ${\mathcal{D}}^+$ contains a given inflation or deflation if and only if ${\mathcal{D}}$ does.
\(a) Let $D{\leqslant}G\times H$. First assume that $D\in{\mathcal{S}}^+(G,H)$. Then, by definition, $p_2(D)\in{\mathcal{G}}$. Further assume that $L\in\Sigma_{\mathcal{G}}(p_2(D))$. Then, by Axiom (vii) for $({\mathcal{G}},{\mathcal{S}})$ and $D$, we have $D*L\in{\mathcal{G}}$ and $D*\Delta(L)\in{\mathcal{S}}(D*L,L)$. Thus (ii) holds. Conversely, assume that $D$ satisfies the condition in (ii). Then $p_2(D)\in{\mathcal{G}}$. Applying the condition in (ii) to $L:=p_2(D)$, we obtain $p_1(D)=D*p_2(D)\in{\mathcal{G}}$ and $D=D*\Delta(p_2(D))\in{\mathcal{S}}(p_1(D),p_2(D))$. Thus, (i) holds. Since the definition of ${\mathcal{S}}^+$ is symmetric, the equivalence of (ii) and (iii) is proved in the same way. The last statement in (a) follows immediately from this characterization and from $({\mathcal{G}},{\mathcal{S}})$ satisfying Axiom (vii).
\(b) Axioms (i)–(iii) and (v) follow from the same arguments as in the proof of \[prop D\_+\]. Axiom (vi) follows immediately from the definition of ${\mathcal{S}}^+$, and Axiom (vii) for $({\mathcal{G}},{\mathcal{S}}^+)$ follows from Part (a) and Axiom (vii) for $({\mathcal{G}},{\mathcal{S}})$.
\(c) This is proved in a similar way as Part (c) of Proposition \[prop D\_+\] using the decomposition of $\left[\frac{G\times H}{D}\right]$ in the paragraph following Definition \[def D\^+\].
\(d) This follows immediately from the definition of ${\mathcal{S}}^+(G,H)$.
We will need the following lemma in the subsequent example. Its proof is an easy verification.
\[lem conditions and \*\] Let $G,H,K$ be finite groups and let $D{\leqslant}G\times H$ and $E{\leqslant}H\times K$.
Let $i\in\{1,2\}$. If $k_i(D)=\{1\}$ and $k_i(E)=\{1\}$ then $k_i(D*E)=\{1\}$.
If $p_1(D)=G$ and $p_1(E)=H$ then $p_1(D*E)=G$. If $p_2(D)=H$ and $p_2(E)=K$ then $p_2(D*E)=K$.
\[ex D\_+ and D\^+\] We say that a subgroup $D$ of $G\times H$ satisfies condition $k_1$ (resp. $k_2$, resp. $p_1$, resp. $p_2$) if $k_1(D)=\{1\}$ (resp. $k_2(D)=\{1\}$, resp. $p_1(D)=G$, resp. $p_2(D)=H$). For any subset $C$ of $\{k_1,k_2,p_1,p_2\}$, let ${\mathcal{S}}_C(G,H)$ denote the set of subgroups $D$ of $G\times H$ which satisfy all the conditions in $C$.
Clearly, for any class ${\mathcal{G}}$ of finite groups and any choice of $C$, $({\mathcal{G}},{\mathcal{S}}_C)$ satisfies Axioms (i) and (ii) in \[noth GS\](a). Moreover, by Lemma \[lem conditions and \*\], $({\mathcal{G}},{\mathcal{S}}_C)$ also satisfies Axiom (iii) of \[noth GS\](a). Thus, by \[noth D(G,S)\], we obtain a subcategory ${\mathcal{D}}_C({\mathcal{G}}):={\mathcal{D}}({\mathcal{G}},{\mathcal{S}}_C)$ of ${\mathcal{C}}$.
Assume from now on that ${\mathcal{G}}$ is the class of all finite groups and write ${\mathcal{D}}_C$ for ${\mathcal{D}}_C({\mathcal{G}})$. It is easy to verify, using Theorems \[thm Mackey formula\] and \[thm elementary decomposition\], that the following holds: ${\mathcal{D}}_C$ is the subcategory of ${\mathcal{C}}$ generated by all isogations together with all the elementary operations contained in $E\subseteq\{{\mathrm{res}},{\mathrm{ind}},{\mathrm{inf}},{\mathrm{def}}\}$, where $E$ contains ${\mathrm{res}}$ (resp. ${\mathrm{ind}}$, resp. ${\mathrm{inf}}$, resp. ${\mathrm{def}}$) if and only if $C$ does [*not*]{} contain $p_2$ (resp. $p_1$, resp. $k_1$, resp. $k_2$). It is also an easy verification that $({\mathcal{G}},{\mathcal{S}}_C)$ satisfies Axioms (iv) and (vii) of \[noth GS\](a). Therefore, by Proposition \[prop D\_+\](c), we obtain $({\mathcal{D}}_C)_+={\mathcal{D}}_{C\smallsetminus\{p_1\}}$, or equivalently that $({\mathcal{D}}_C)_+$ is the subcategory of ${\mathcal{C}}$ generated by all the elementary operations that are already contained in ${\mathcal{D}}$ together with all inductions. Similarly, by Proposition \[prop D\^+\](c), we obtain $({\mathcal{D}}_C)^+={\mathcal{D}}_{C\smallsetminus\{p_1,p_2\}}$, or equivalently that $({\mathcal{D}}_C)^+$ is generated by all the elementary operations that are already contained in ${\mathcal{D}}$ together with all inductions and all restrictions.
\[ex D\_+ and D\^+ local\] Let $G$ be a finite group and let ${\mathcal{G}}_G$ denote the set of all subgroups of $G$. Replacing all isogations with just conjugation isomorphisms induced by elements in $G$ and using a similar notation as in Example \[ex D\_+ and D\^+\], we obtain three subcategories ${\mathcal{D}}_{G,C}$ of ${\mathcal{C}}$ for the three choices of subsets $C=\{k_1,k_2,p_1,p_2\}$, $C=\{k_1,k_2, p_1\}$, and $C=\{k_1,k_2\}$, whose objects are the elements of ${\mathcal{G}}_G$ and whose morphisms are generated by all conjugations (resp. all conjugations and restrictions, resp. all conjugations, restrictions and inductions). For a commutative ring $R$, the functor categories ${\mathcal{F}}_{{\mathcal{D}}_{G,C},R}$ for these three choices are then very closely related to the conjugation functors (resp. restriction functors, resp. Mackey functors) on $G$ over $R$, as defined in [@Boltje1998b Def. 1.1]. In fact, the categories ${\mathcal{F}}_{{\mathcal{D}}_{G,C},R}$ are precisely the subcategories of the conjugation functors, restriction functors and Mackey functors in [@Boltje1998b], satisfying the additional axiom that for $H{\leqslant}G$ and $g\in C_G(H)$, the conjugation map induced by $g$ on the evaluation at $H$ is the identity. We will denote these categories by ${\mathrm{Con}_R^f}(G)$, ${\mathrm{Res}_R^f}(G)$, and ${\mathrm{Mack}_R^f}(G)$, respectively. See [@Bouc2015] for a more detailed discussion on [*fused*]{} Mackey functors versus Mackey functors that elaborates on this aspect. We are grateful to Serge Bouc for pointing out this difference.
The Functor $-_+\colon {\mathcal{F}}_{{\mathcal{D}},R}\to{\mathcal{F}}_{{\mathcal{D}}_+,R}$ {#sec F_+}
===========================================================================================
Throughout this section, let $R$ denote a commutative ring and let $({\mathcal{G}},{\mathcal{S}})$ be as in \[noth GS\](a), satisfying Axioms (i)–(iv). Let ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$ (cf. \[noth D(G,S)\]) and let ${\mathcal{S}}_+$ and ${\mathcal{D}}_+$ be defined as in Definition \[def D\_+\]. The goal of this section is the construction of a functor $-_+\colon {\mathcal{F}}_{{\mathcal{D}},R}\to{\mathcal{F}}_{{\mathcal{D}}_+,R}$ that generalizes the construction in [@Boltje1998b Section 2] in the situation of Example \[ex D\_+ and D\^+ local\].
\[not exponential\] For a finite group $H$, an $H$-set $X$ and an element $x\in X$, we denote by $H_x:=\{h\in H\mid hx=x\}$ the stabilizer of $x$ in $H$. If also $G$ is a finite group and $U$ is a $(G,H)$-biset, then, for $u\in U$ and $K{\leqslant}H$, we set $${\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{u}\!K}:=\{g\in G\mid \exists k\in K\colon gu=uk\}\,.$$ It is easy to verify that $$(G\times K)_u = (G\times H)_u*\Delta(K) = (G\times H)_u\cap(G\times K)\,,\quad {\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{u}\!K}=p_1((G\times K)_u) = (G\times H)_u*K\,,$$ and that, for an $H$-set $X$ and $x\in X$, one has $$G_{[u, x]}={\setbox0=\hbox{$H_x$} \setbox1=\vbox to
\ht0{}\,\box1^{u}\!H_x} = (G\times H)_u*H_x\,.$$
[*The category $\Gamma_F(G)$.*]{} Let $G\in{\mathcal{G}}$, let $X$ be a finite $G$-set such that $G_x\in{\mathcal{G}}$ for all $x\in X$, and let $F\in{\mathcal{F}}_{{\mathcal{D}},R}$. A [*section of $F$ over $X$*]{} is a function $s\colon X\to \bigoplus_{x\in X} F(G_x)$ such that $s(x)\in F(G_x)$ for all $x\in X$. These sections form an $R$-module via point-wise constructions. The group $G$ acts $R$-linearly on the set of these sections by $(g\cdot s)(x):={\setbox0=\hbox{$s(g^{-1}x$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!s(g^{-1}x})$, where ${\setbox0=\hbox{$m$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!m}:=F({\mathrm{iso}}_{c_g})(m)$, for $g\in G$ and $m\in F(G_{g^{-1}x})$, and $c_g\colon G_{g^{-1}x}\to G_x$ is the conjugation isomorphism mapping $h\in G_{g^{-1}x}$ to $ghg^{-1}\in G_x$. A section $s$ of $G$ over $F$ is called [*$G$-equivariant*]{} if $g\cdot s=s$ for all $g\in G$.
For $G\in {\mathcal{G}}$ and $F$ as above, we denote by $\Gamma_F(G)$ the category whose objects are the pairs $(X,s)$, where $X$ is a finite $G$-set such that $G_x\in {\mathcal{G}}$, for all $x\in X$, and $s$ is a $G$-equivariant section of $F$ over $X$. A morphism $\alpha\colon (X,s)\to (Y,t)$ in $\Gamma_F(G)$ is a morphism of $G$-sets $\alpha\colon X\to Y$ such that for all $x\in X$, one has $G_{\alpha(x)}=G_x$ and $t(\alpha(x))=s(x)$.
[*The functor $\Gamma_F(U)\colon\Gamma_F(H)\to\Gamma_F(G)$.*]{}Let $G,H\in{\mathcal{G}}$, let $F\in{\mathcal{F}}_{{\mathcal{D}},R}$, and let $U$ be a $(G,H)$-biset with $(G\times H)_u\in {\mathcal{S}}_+(G,H)$ for all $u\in U$. We define a functor $$\Gamma_F(U)\colon\Gamma_F(H)\to\Gamma_{F}(G)\,,\quad (X,s)\mapsto(U\times_H X,U(s))\,,$$ where $$\label{eqn U(s)}
U(s)([u, x]):= F\left(\left[\frac{G_{[u,x]}\times H_x}{(G\times H_x)_u}\right]\right)(s(x))\,.$$ For a morphism $\alpha\colon (X,s)\to(Y,t)$ in $\Gamma_F(H)$ we set $\Gamma_F(\alpha):=U\times_H\alpha$. For the rest of this subsection we show that these definitions are well-defined and yield a functor.
\(a) First note that $H_x\in{\mathcal{G}}$, for all $x\in X$, by the definition of $\Gamma_F(H)$ and that $G_{[u,x]}=(G\times H)_u * H_x\in{\mathcal{G}}$, for all $u\in U$ and $x\in X$, by \[not exponential\] and Proposition \[prop D\_+\](a). Moreover, $(G\times H_x)_u= (G\times H)_u*\Delta(H_x)\in{\mathcal{S}}((G\times H)_u*H_x,H_x) = {\mathcal{S}}(G_{[u,x]},H_x)$, again by \[not exponential\] and Proposition \[prop D\_+\](a). Therefore, $F$ can be applied to the class of the biset in (\[eqn U(s)\]).
\(b) Next we show that the definition in (\[eqn U(s)\]) does not change, when $(u,x)$ is replaced with $(uh^{-1},hx)$ for some $h\in H$. To see this, note that $$\label{eqn comp}
F\left(\left[\frac{G_{[uh^{-1}, hx]}\times H_{hx}}{(G\times H_{hx})_{uh^{-1}}}\right]\right)(s(hx)) =
F\left(\left[\frac{G_{[u, x]}\times H_{hx}}{(G\times H_x)_u*\Delta_{c_{h^{-1}}}(H_{hx})}\right]\right)(s(hx))\,,$$ where $\Delta_{c_{h^{-1}}}(H_{hx}):=\{(h^{-1}h'h,h')\mid h'\in H_{hx}\}$, since $(G\times H_{hx})_{uh^{-1}} =
(G\times H_x)_u*\Delta_{c_{h^{-1}}}(H_{hx})$, as a quick computation shows. Further, $$\frac{G_{[u, x]}\times H_{hx}}{(G\times H_x)_u*\Delta_{c_{h^{-1}}}(H_{hx})} \cong
\frac{G_{[u, x]}\times H_x}{(G\times H_x)_u} \times_{H_x} \frac{H_x\times H_{hx}}{\Delta_{c_{h^{-1}}}(H_{hx})}\,,$$ so that, using the functor property of $F$, we may continue (\[eqn comp\]) with $$\cdots = \left(F\left(\left[\frac{G_{[u, x]}\times H_x}{(G\times H_x)_u}\right]\right)\circ
F({\mathrm{iso}}_{c_{h^{-1}}})\right)(s(hx))
= F\left(\left[\frac{G_{[u, x]}\times H_x}{(G\times H_x)_u}\right]\right)({\setbox0=\hbox{$s(hx)$} \setbox1=\vbox to
\ht0{}\,\box1^{h^{-1}}\!s(hx)})\,,$$ and arrive at the same expression, since $h\cdot s=s$.
\(c) Next we show that the section $U(s)$ is $G$-equivariant. For $g\in G$ we have $$\begin{aligned}
(g\cdot U(s))[u, x] & = {\setbox0=\hbox{$\bigl(U(s)([g^{-1}u,x])\bigr)$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!\bigl(U(s)([g^{-1}u,x])\bigr)} \\
& = F\left(\left[\frac{G_{[u, x]} \times G_{[g^{-1}u, x]}}{\Delta_{c_g}(G_{[g^{-1}u, x]})}\times_{G_{[g^{-1}u, x]}}
\frac{G_{[g^{-1}u, x]}\times H_x}{(G\times H_x)_{g^{-1}u}}\right]\right)(s(x)) \\
& = F\left(\left[\frac{G_{[u, x]}\times H_x}{\Delta_{c_g}(G_{[g^{-1}u, x]})*(G\times H_x)_{g^{-1}u}}\right]\right)(s(x))\\
& = F\left(\left[\frac{G_{[u, x]}\times H_x}{(G\times H_x)_u}\right]\right)(s(x))\,,\end{aligned}$$ since $\Delta_{c_g}(G_{[g^{-1}u, x]})*(G\times H_x)_{g^{-1}u} = (G\times H_x)_u$.
\(d) Next let $(X,s)$ and $(Y,t)$ be objects in $\Gamma_F(H)$ and let $\alpha\colon X\to Y$ be a morphism of $H$-sets such that, for all $x\in X$, one has $H_{\alpha(x)}=H_x$ and $s(x)=t(\alpha(x))$. We need to show that $U\times_H \alpha$ is a morphism in $\Gamma_F(G)$ between $(U\times_H X,U(s))$ and $(U\times_H Y, U(t))$. Since $H_{\alpha(x)}=H_x$ we also have $G_{[u, x]}=G_{[u, \alpha(x)]}$ for all $u$ in $U$ and all $x\in X$ (see \[not exponential\]). Moreover, for all $u\in U$ and $x\in X$, we have $$\begin{aligned}
U(t)([u, \alpha(x)])
& = F\left(\left[\frac{G_{[u,\alpha(x)]}\times H_{\alpha(x)}}{(G\times H_{\alpha(x)})_u}\right]\right)
\bigl(t(\alpha(x))\bigr)\\
& = F\left(\left[\frac{G_{[u,x]}\times H_x}{(G\times H_x)_u}\right]\right) \bigl(s(x)\bigr) = U(s)([u, x])\,,\end{aligned}$$ so that $U\times_H\alpha$ is a morphism in $\Gamma_F(G)$.
\(e) Since $U\times_H -$ preserves compositions and the identity, we have verified that the above definitions yield a well-defined functor $\Gamma_F(U)\colon \Gamma_F(H)\to\Gamma_F(G)$.
\[prop functorial in U\] Let $G$, $H$, and $K$ be finite groups in ${\mathcal{G}}$, let $U$ be a finite $(G,H)$-biset such that $(G\times H)_u\in{\mathcal{S}}_+(G,H)$ for all $u\in U$, and let $V$ be a finite $(H,K)$-biset such that $(H\times K)_v\in{\mathcal{S}}_+(H,K)$ for all $v\in V$. Then $(G\times K)_{[u,v]}\in {\mathcal{S}}_+(G,K)$ for all $u\in U$ and $v\in V$. Moreover, the two functors $\Gamma_F(U\times _HV)$ and $\Gamma_F(U)\circ\Gamma_F(V)$ from $\Gamma_F(K)$ to $\Gamma_F(G)$ are naturally isomorphic.
The first statement follows immediately from Equation (\[eqn stab \*\]) and Axiom (iii) in \[noth GS\](a), which holds for $({\mathcal{G}}, {\mathcal{S}}_+)$ by Proposition \[prop D\_+\](b).
The two functors applied to an object $(X,s)$ of $\Gamma_F(K)$ yield the objects $$((U\times_H V)\times_K X, (U\times_H V)(s)) \text{ and }(U\times_H(V\times_K X),U(V(s)))$$ of $\Gamma_F(G)$. Let $$\alpha\colon (U\times_H V)\times_K X{\buildrel\sim\over\to}U\times_H (V\times_K X)\,,\quad [[u,_H v],_Kx]\mapsto [u,_H[v,_K x]]\,,$$ for $(u,v,x)\in U\times V\times X$, be the usual natural isomorphism of $G$-sets. In order to show that $\alpha$ is a morphism between $\Gamma_F(U\times_H V)(X,s)$ and $(\Gamma_F(U)\circ\Gamma_F(V))(X,s)$, we still need to show that $G_{[[u,_H v],_Kx]}=G_{[u,_H[v,_K x]]}$ and that $((U\times_H V)(s))([[u,_H v],_Kx]) = U(V(s))([u,_H[v,_K x]])$, for all $(u,v,x)\in U\times V\times X$. The first statement is immediate, since $\alpha$ is an isomorphism of $G$-sets. For the second statement, we evaluate both sides. The left hand side equals $$\label{eqn evaluation}
F\left(\left[\frac{G_{[[u,_H v],_K x]}\times K_x}{(G\times K_x)_{[u,_H v]}}\right]\right) (s(x))\,,$$ and the right hand side equals $$\begin{aligned}
&\quad F\left(\left[\frac{G_{[u,_H[v,_K x]]}\times H_{[v,_K x]}}{(G\times H_{[v,_K x]})_u}\right]\right)(V(s)(x)) \\
& = \left(F\left(\left[\frac{G_{[u,_H[v,_K x]]}\times H_{[v,_K x]}}{(G\times H_{[v,_K x]})_u}\right]\right) \circ
F\left(\left[\frac{H_{[v,_K x]}\times K_x}{(H\times K_x)_v}\right]\right)\right)(s(x)) \\
& = F\left(\left[\frac{G_{[u,_H[v,_K x]]}\times H_{[v,_K x]}}{(G\times H_{[v,_K x]})_u}\times_{H_{[v,_K x]}}
\frac{H_{[v,_K x]}\times K_x}{(H\times K_x)_v}\right]\right)(s(x))\,.\end{aligned}$$ A quick computation shows that $p_1((H\times K_x)_v) = H_{[v,_K x]}$. Thus, Theorem \[thm Mackey formula\] implies that the last expression is equal to $$F\left(\left[\frac{G_{[u,_H[v,_K x]]}\times K_x}{(G\times H_{[v,_K x]})_u*(H\times K_x)_v}\right]\right)(s(x))\,,$$ which coincides with the element in (\[eqn evaluation\]), since $G_{[u,_H[v,_K x]]}=G_{[[u,_H v],_Kx]}$ and $(G\times H_{[v,_K x]})_u*(H\times K_x)_v = (G\times K_x)_{[u,_H v]}$. In fact, the first equation was established before and the second equation is an easy verification.
\[def 2 operations\] For $G\in{\mathcal{G}}$ and $F\in{\mathcal{F}}_{{\mathcal{D}},R}$, we define the following two operations in the category $\Gamma_F(G)$.
\(a) Given two arbitrary objects $(X,s)$ and $(Y,t)$ in $\Gamma_F(G)$, their [*coproduct*]{} $(X,s)\coprod(Y,t)$ is defined as $(X\coprod Y, s\coprod t)$, where $X\coprod Y$ is the disjoint union of $X$ and $Y$ and $s\coprod t$ is the section on $X\coprod Y$ which is defined on $X$ as $s$ and on $Y$ as $t$. This construction together with the obvious inclusions from $X$ and $Y$ to $X\coprod Y$ is also a categorial coproduct in $\Gamma_F(G)$.
\(b) Given two objects $(X,s)$ and $(X,t)$ with the same underlying $G$-set $X$, we have an object $(X,s+t)$, where $s+t$ is the pointwise sum of the two sections $s$ and $t$.
\[def F\_+\] Let $G\in{\mathcal{G}}$ and $F\in{\mathcal{F}}_{{\mathcal{D}},R}$. We define the abelian group $F_+(G)$ as the free abelian group on the set of isomorphism classes $\{X,s\}$ of objects $(X,s)$ of $\Gamma_F(G)$ modulo the subgroup generated by all elements of the form $$\label{eqn relations}
\{X\coprod Y, s\coprod t\}-\{X,s\}-\{Y,t\} \quad\text{and}\quad \{X,s+u\}-\{X,s\}-\{X,u\}\,,$$ where $(X,s),(X,u),(Y,t)$ are objects of $\Gamma_F(G)$. We will denote the coset of $\{X,s\}$ in $F_+(G)$ by $[X,s]$.
Note that the above free abelian group is an $R$-module, via $r\{X,s\}:=\{X,rs\}$, using the point-wise $R$-module structure of sections on a fixed $G$-set $X$, and note that the subgroup generated by the elements in (\[eqn relations\]) is an $R$-submodule. Thus, $F_+(G)$ has a natural $R$-module structure.
Note also that by the first type of relations in (\[eqn relations\]), the classes of the elements $\{G/H,s\}$, where $H\in\Sigma_{{\mathcal{G}}}(G)$, and where $s$ is a $G$-equivariant section of $F$ over the $G$-set $G/H$, form a generating set of the abelian group $F_+(G)$.
For every element $a\in F(H)$, there exists a unique $G$-equivariant section $s_a$ of $F$ over $G/H$ with $s(H)=a$. We abbreviate the class of $\{G/H,s_a\}$ by $[H,a]_G\in F_+(G)$.
\[thm F\_+\] Let $R$ be a commutative ring, let $({\mathcal{G}},{\mathcal{S}})$ be a as in \[noth GS\](a) satisfying Axioms (i)–(iv), set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$ and ${\mathcal{D}}_+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$, and let $F\in{\mathcal{F}}_{{\mathcal{D}},R}$ be a biset functor on ${\mathcal{D}}$ over $R$.
Mapping a finite group $G$ to the $R$-module $F_+(G)$ and an element $[U]\in B(G,H)$, where $U$ is a finite $(G,H)$-biset with point stabilizers in ${\mathcal{S}}_+(G,H)$, to the $R$-linear map $F_+([U])\colon F_+(H)\to F_+(G)$, induced by the functor $\Gamma_F(U)\colon \Gamma_F(H)\to\Gamma_F(G)$, yields a biset functor $F_+\in{\mathcal{F}}_{{\mathcal{D}}_+,R}$.
The association $F\mapsto F_+$ defines an $R$-linear functor $-_+\colon {\mathcal{F}}_{{\mathcal{D}},R}\to{\mathcal{F}}_{{\mathcal{D}}_+,R}$.
For each $G\in{\mathcal{G}}$, one has an $R$-module isomorphism $$F_+(G)\cong \left(\bigoplus_{H\in\Sigma_{{\mathcal{G}}}(G)} F(H)\right)_G,$$ where the above direct sum $M:=\bigoplus_{H\in\Sigma_{{\mathcal{G}}}(G)} F(H)$ is an $RG$-module with $g\in G$ mapping $a\in F(H)$ to ${\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!a}\in F({\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H})$, and where $M_G$ denotes the $G$-cofixed points of $M$, i.e., the $R$-module $M/IM$, where $I$ is the augmentation ideal of $RG$. The isomorphism associates the class of the element $a\in F(H)$, for $H\in\Sigma_{\mathcal{G}}(G)$, to $[H,a]_G\in F_+(G)$.
\(a) It is straightforward to show that if $U$ and $V$ are isomorphic $(G,H)$-bisets with stabilizers in ${\mathcal{S}}_+(G,H)$, then the functors $\Gamma_F(U)$ and $\Gamma_F(V)$ are naturally isomorphic. Thus, every $(G,H)$-biset $U$ induces a well-defined group homomorphism $f_{[U]}$ between the free abelian groups associated to $\Gamma_F(H)$ and $\Gamma_F(G)$. Moreover, $f_{[U]}$ is an $R$-module homomorphism and maps elements of the type in (\[eqn relations\]) to elements of the same type. Thus $f_{[U]}$ induces an $R$-module homomorphism $F_+([U])\colon F_+(H)\to F_+(G)$. It is also easy to see that $f_{[U\coprod V]}= f_{[U]}+f_{[V]}$. Thus, there is a unique group homomorphism $F_+\colon B(G,H)\to {\mathrm{Hom}}_R(F_+(H),F_+(G))$ with $F_+([U])$ as defined above for every $(G,H)$-biset $U$. Moreover, Proposition \[prop functorial in U\] implies that $F_+$ is in fact a functor in ${\mathcal{F}}_{{\mathcal{D}}_+,R}$.
\(b) If $\varphi\colon F\to F'$ is a morphism between objects $F,F'\in{\mathcal{F}}_{{\mathcal{D}},R}$, then, for every $G\in{\mathcal{G}}$, one obtains an induced functor $\Gamma_F(G)\to\Gamma_{F'}(G)$, which maps an object $(X,s)$ to the object $(X,\phi(s))$, where $(\phi(s))(x):=\phi_{G_x}(s(x))$. This functor induces an $R$-linear map from $F_+(G)$ to $F'_+(G)$. Moreover, it is straightforward to verify that this construction respects compositions, sends the identity morphism to the identity morphism, and is $R$-linear.
\(c) For every object $(X,s)$ of $\Gamma_F(G)$, we can define an element $\phi_G(X,s)\in (\bigoplus_{H\in\Sigma_{\mathcal{G}}(G)} F(H))_G$ as the class of $\sum_{x\in [G\backslash X]} s(x)$, with $s(x)\in F(G_x)$, where $[G\backslash X]$ denotes a set of representatives of the $G$-orbits of $X$. Isomorphic objects lead to the same elements and the elements in (\[eqn relations\]) are mapped to $0$. Altogether we obtain an $R$-linear map $\phi_G\colon F_+(G)\to (\bigoplus_{H\in\Sigma_{\mathcal{G}}(G)} F(H))_G$, which maps $[H,a]_G$ to the class of the element $a\in F(H)$. Conversely, for every $H\in\Sigma_{\mathcal{G}}(G)$, one has a map $F(H)\to F_+(G)$, $a\mapsto[H,a]_G$. Since $[{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H},{\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!a}]_G=[H,a]_G$, for all $g\in G$, one obtains a well-defined map in the opposite direction which maps the class of $a\in F(H)$ to $[H,a]_G$. Thus, the two constructed maps are inverse to each other.
One could have defined $F_+(G)$ directly as $(\bigoplus_{H\in\Sigma_{{\mathcal{G}}}(G)} F(H))_G$. But then a proof that $F\mapsto F_+$ yields a functor from ${\mathcal{F}}_{{\mathcal{D}},R}$ to ${\mathcal{F}}_{{\mathcal{D}}_+,R}$ would have been longer, even more technical, and uglier. The reason to introduce the category $\Gamma_F(G)$ was to give a more conceptual and also shorter proof.
Note that $\bigoplus_{H\in \Sigma_{\mathcal{G}}(G)} F(H) = F(G)\oplus (\bigoplus_{G\neq H\in\Sigma_{\mathcal{G}}(G)} F(H))$ is a decomposition into $RG$-submodules and that $G$ acts trivially on $F(G)$. Thus, with the identification in Theorem \[thm F\_+\](c), we obtain a decomposition $$\label{eqn F_+ decomp}
F_+(G) = F(G) \oplus F_+^<(G)\,,$$ where $F_+^<(G):=(\bigoplus_{G\neq H\in\Sigma_{\mathcal{G}}(G)} F(H))_G$. We denote the corresponding projection by $$\label{eqn pi}
\pi_G\colon F_+(G)\to F(G)\,.$$ Note that for $H\in\Sigma_{{\mathcal{G}}}(G)$ and $a\in F(H)$ one has $\pi_G([H,a]_G)=0$ if $H\neq G$ and $\pi_G([H,a]_G)=a$ if $H=G$.
For all practical purposes it is easier to use the version $F_+(G)=(\bigoplus_{H\in\Sigma_{{\mathcal{G}}}(G)} F(H))_G$ and its generating elements $[H,a]_G$. Using the isomorphism between the two versions we can translate the general biset operation as follows: Assume that $G,H\in{\mathcal{G}}$, $U$ is a $(G,H)$-biset with point stabilizers in ${\mathcal{S}}_+(G,H)$, that $K\in\Sigma_{{\mathcal{G}}}(H)$ and $a\in F(K)$. Then $$F_+([U])([K,a]_H) = \sum_{[u, hK]\in[G\backslash(U\times_H (H/K))]}\bigl[ G_{[u,hK]},
F\left(\left[\frac{G_{[u, hK]}\times {\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!K}}{(G\times {\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!K})_u}\right]\right) ({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!a})\bigr]_G\,,$$ where $[G\backslash(U\times_H (H/K))]$ denotes a set of representatives of the $G$-orbits of $U\times_H (H/K)$. In particular, if $D\in{\mathcal{S}}_+(G,H)$, then $$\label{eqn explicit F_+}
F_+\bigl(\bigl[\frac{G\times H}{D}\bigr]\bigr)([K,a]_H) =
\sum_{h\in [p_2(D)\backslash H/K]} \bigl[D*{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!K},
F\bigl(\bigl[\frac{D*{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!K}\times {\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!K}}{D*\Delta({\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!K})}\bigr]\bigr) ({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!a}) \bigr]_G \,,$$ where $[p_2(D)\backslash H/K]$ denotes a set of representatives of the double cosets of $G$ with respect to $p_2(D)$ and $K$. This formula specializes in the case of elementary bisets as follows:
- Assume that $G,G'\in{\mathcal{G}}$, $\alpha\colon G{\buildrel\sim\over\to}G'$ is an isomorphism with $\Delta_\alpha(G)\in{\mathcal{S}}_+(G',G)$, $K\in\Sigma_{\mathcal{G}}(G)$ and $a\in F(K)$. Then $$F_+({\mathrm{iso}}_\alpha)([K,a]_G) = [\alpha(K),F({\mathrm{iso}}_\alpha)(a)]_{G'}\,.$$
- Assume that $G\in{\mathcal{G}}$, $H\in\Sigma_{{\mathcal{G}}}(G)$ with $\Delta(H)\in{\mathcal{S}}_+(H,G)$, $K\in\Sigma_{{\mathcal{G}}}(G)$, and that $a\in F(K)$. Then $$F_+({\mathrm{res}}^G_H)([K,a]_G) = \sum_{g\in[H\backslash G/K]} [H\cap{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!K},F({\mathrm{res}}^{{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!K}}_{H\cap{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!K}})({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!a})]_H\,.$$
- Assume that $G\in{\mathcal{G}}$, $H\in\Sigma_{{\mathcal{G}}}(G)$ with $\Delta(H)\in{\mathcal{S}}_+(G,H)$, $K\in\Sigma_{{\mathcal{G}}}(H)$, and that $a\in F(K)$. Then $$F_+({\mathrm{ind}}_H^G)([K,a]_H) = [K,a]_G\,.$$ Note that even if ${\mathcal{D}}$ contains inductions, the induction operations for $F_+$ are independent of the induction operations for $F$.
- Assume that $G\in{\mathcal{G}}$, $N\trianglelefteq G$ with $G/N\in{\mathcal{G}}$ and $\{(g,gN)\mid g\in G\}\in{\mathcal{S}}_+(G,G/N)$, $N\trianglelefteq K{\leqslant}G$ with $K/N\in{\mathcal{G}}$, and $a\in F(K/N)$. Then $$F_+({\mathrm{inf}}_{G/N}^G)([K/N,a]_{G/N}) = [K,F({\mathrm{inf}}_{K/N}^K)(a)]_G\,.$$
- Assume that $G\in{\mathcal{G}}$, $N\trianglelefteq G$ with $G/N\in{\mathcal{G}}$ and $\{(gN,g)\mid g\in G\}\in{\mathcal{S}}_+(G/N,G)$, that $K\in\Sigma_{\mathcal{G}}(G)$ and $a\in F(K)$. Then $$F_+({\mathrm{def}}^G_{G/N})([K,a]_G) = \left[KN/N,F\left(\left[\frac{(KN/N)\times K}{\{(kN,k)\mid k\in K\}}\right]\right)(a)\right]_{G/N}\,.$$ If also $K/(K\cap N)\in{\mathcal{G}}$ and ${\mathrm{iso}}_\alpha$ and ${\mathrm{def}}^K_{K/(K\cap N)}$ are in ${\mathcal{D}}$, where $\alpha\colon K/K\cap N{\buildrel\sim\over\to}KN/N$ is the canonical isomorphism induced by the inclusion $K{\leqslant}KN$, then we can write the above as $$\bigl[KN/N, \bigl(F({\mathrm{iso}}_\alpha)F({\mathrm{def}}^K_{K/(K\cap N)})\bigr)(a)\bigr]_{G/N}\,.$$
\[ex Burnside examples\] Assume that ${\mathcal{G}}$ is the class of all finite groups and consider ${\mathcal{S}}(G,H):={\mathcal{S}}_{\{p_1\}}(G,H)=(\{D{\leqslant}G\times H\mid p_1(D)=G\}$, for $G,H\in{\mathcal{G}}$. Then ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}) = {\mathcal{D}}_{\{p_1\}}$, cf. Example \[ex D\_+ and D\^+\]. Then ${\mathcal{S}}_+(G,H)$ consists of all subgroups of $G\times H$ and ${\mathcal{D}}_+={\mathcal{C}}$.
\(a) Consider the constant functor $F\in{\mathcal{F}}_{{\mathcal{D}},{\mathbb{Z}}}$ associating to each $G\in{\mathcal{G}}$ the abelian group ${\mathbb{Z}}$ and to each $[(G\times H)/D]$ (with $G,H\in{\mathcal{G}}$ and $D{\leqslant}G\times H$ with $p_1(D)=G$) the identity map from $F(H)$ to $F(G)$. This is indeed a functor by Theorem \[thm Mackey formula\]. Then $F_+$ identifies to the Burnside ring functor $B$ by mapping $[H,a]_G\in F_+(G)$ to $a[G/H]\in B(G)$, for $H{\leqslant}G$ and $a\in{\mathbb{Z}}$.
\(b) Let $A$ be an abelian group and define $F(G)$ as the free ${\mathbb{Z}}$-module with basis ${\mathrm{Hom}}(G,A)$. Then $F\in{\mathcal{F}}_{{\mathcal{D}},{\mathbb{Z}}}$, if endowed with the obvious isogation, restriction, inflation and deflation operations. The resulting biset functor $F_+$ is isomorphic to the $A$-fibered Burnside ring functor $B^A$, see [@Dress], [@Barker], or [@BoltjeCoskun]. In particular, if $A={\mathbb{C}}^\times$ is the unit group of the field of complex numbers, we obtain $F(G)={R^{\mathrm{ab}}}(G)$ and $F_+(G)={R^{\mathrm{ab}}}_+(G)$ for any finite group $G$, cf. [@Boltje1998b].
\(c) If $F=R$ denotes the character ring biset functor on ${\mathcal{D}}$ over ${\mathbb{Z}}$, then $F_+$ is the global representation ring functor introduced in [@RaggiValero2015].
The Functor $-^+\colon {\mathcal{F}}_{{\mathcal{D}},R}\to{\mathcal{F}}_{{\mathcal{D}}^+,R}$ {#sec F^+}
===========================================================================================
Throughout this section, let $R$ be a commutative ring and let $({\mathcal{G}},{\mathcal{S}})$ be as in \[noth GS\](a) satisfying the axioms (i)-(iii) and (vii) in \[noth GS\]. Let ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$ (cf. \[noth D(G,S)\]) and let ${\mathcal{S}}^+$ and ${\mathcal{D}}^+$ be defined as in Definition \[def D\^+\]. We additionally assume that for all $G,H\in{\mathcal{G}}$ and $D\in{\mathcal{S}}^+(G,H)$ one has $k_2(D)=\{1\}$, i.e., that $(G\times H)/D$ is free as $H$-set. This is equivalent to assuming the same for all $D\in{\mathcal{S}}(G,H)$. In fact, ${\mathcal{S}}\subseteq{\mathcal{S}}^+$, by Proposition \[prop D\^+\](a), and if $D\in{\mathcal{S}}^+(G,H)$, then $D\in{\mathcal{S}}(p_1(D),p_2(D))$, by definition. Thus, ${\mathcal{D}}$ and ${\mathcal{D}}^+$ don’t have deflations.
The goal of this section is to construct a functor $-^+\colon{\mathcal{F}}_{{\mathcal{D}},R}\to{\mathcal{F}}_{{\mathcal{D}}^+,R}$ that generalizes the construction in [@Boltje1998b Section 2] in the situation of Example \[ex D\_+ and D\^+ local\], up to the modification of working with [*fused*]{} Mackey functors and [*fused*]{} conjugation functors, cf. [@Bouc2015].
\[not K\^u\] Similar to the notation in \[not exponential\], one defines for a finite $(G,H)$-biset $U$, $u\in U$, and $K{\leqslant}G$, $$K^u:=\{h\in G\mid \exists k\in K\colon (k,h)\in (K\times H)_u\}{\leqslant}H\,.$$ Note that $$(K\times H)_u = \Delta(K)*(G\times H)_u\quad\text{and}\quad K^u=K*(G\times H)_u\,.$$
\[def F\^+\] For $G\in{\mathcal{G}}$ we define $$F^+(G):=\left(\prod_{H\in\Sigma_{{\mathcal{G}}}(G)} F(H)\right)^G\,,$$ where the exponent $G$ denotes taking $G$-fixed points. That is, a tuple of elements $a_H\in F(H)$, for $H\in\Sigma_{\mathcal{G}}(G)$, belongs to $F^+(G)$ if and only if ${\setbox0=\hbox{$a_H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!a_H}=a_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{g}\!H}}$ for all $H\in\Sigma_{{\mathcal{G}}}(G)$ and all $g\in G$. Note that if ${\tilde{\Sigma}}_{\mathcal{G}}(G)\subseteq\Sigma_{\mathcal{G}}(G)$ is a set of representatives of the conjugacy classes of $\Sigma_{\mathcal{G}}(H)$ then projection onto the components indexed by ${\tilde{\Sigma}}_{{\mathcal{G}}}(G)$ defines an $R$-module isomorphism $$\label{eqn F_+ projection iso}
F^+(G){\buildrel\sim\over\to}\prod_{H\in{\tilde{\Sigma}}_{\mathcal{G}}(G)} F(H)^{N_G(H)}\,.$$
For any $G,H\in{\mathcal{G}}$ and any $(G,H)$-biset $U$ such that $(G\times H)_u\in{\mathcal{S}}^+(G,H)$ for all $u\in U$, we define $$\begin{aligned}
\label{eqn F^+(U)}
\notag F^+([U])=F^+(U)\colon F^+(H) & \to F^+(G)\,,\\
(a_L)_{L\in\Sigma_{{\mathcal{G}}}(H)} & \mapsto \left(\sum_{\substack{u\in[U/H]\\ K{\leqslant}p_1((G\times H)_u)}}
F\left(\left[\frac{K\times K^u}{(K\times H)_u}\right]\right)(a_{K^u})\right)_{K\in\Sigma_{{\mathcal{G}}}(G)}\,,\end{aligned}$$ where $[U/H]$ denotes a set of representatives of the $H$-orbits of $U$. The above expression is well-defined, since $K\in{\mathcal{G}}$ implies $K^u=K*(G\times H)_u\in{\mathcal{G}}$ and $(K\times H)_u= \Delta(K)*(G\times H)_u\in{\mathcal{S}}(K,K^u)$, by \[not K\^u\] and Proposition \[prop D\^+\](a) (i)$\Rightarrow$(iii). Thus, $F$ can be applied to the class of the $(K,K^u)$-biset $(K\times K^u)/(K\times H)_u$. Finally, the expression on the right hand side in (\[eqn F\^+(U)\]) does not depend on the choice of representatives $[U/H]$ and neither on the choice of $U$ in its isomorphism class. Clearly, $F^+([U])$ is also an $R$-module homomorphism. Since the sum in (\[eqn F\^+(U)\]) is additive in $[U]$, we also obtain an induced group homomorphism $$F^+\colon B(G,H)\to {\mathrm{Hom}}_R(F^+(H),F^+(G))\,.$$
\[thm F\^+ is functor\] Let $({\mathcal{G}},{\mathcal{S}})$ be as introduced at the beginning of the section, satisfying Axioms (i)–(iii) and (vii) of \[noth GS\] and assume that $k_2(D)=1$ for every $G,H\in\cal G$ and $D\in{\mathcal{S}}(G,H)$. Set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$ and ${\mathcal{D}}^+={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}^+)$. Then the constructions in Definition \[def F\^+\] define an $R$-linear functor $-^+\colon{\mathcal{F}}_{{\mathcal{D}},R}\to{\mathcal{F}}_{{\mathcal{D}}^+,R}$.
We first show that for all finite groups $G,H,K\in{\mathcal{G}}$, all finite $(G,H)$-bisets $U$ with stabilizers in ${\mathcal{S}}^+(G,H)$ and all finite $(H,K)$-bisets $V$ with stabilizers in ${\mathcal{S}}^+(H,K)$, one has $$\label{eqn F^+ functor}
F^+(U\times_H V) = F^+(U)\circ F^+(V)\colon F^+(K)\to F^+(G)\,.$$ Let $c=(c_N)_{N\in\Sigma_{\mathcal{G}}(K)}\in F^+(K)$. Then for every $L\in\Sigma_{\mathcal{G}}(G)$, the $L$-component of $F^+(U\times_H V)(c)$ equals $$\label{eqn lhs}
\sum_{\substack{[u,v]\in [(U\times_H V)/K]\\ L\in p_1((G\times K)_{[u, v]})}}
F\left(\left[\frac{L\times L^{[u,v]}}{(L\times K)_{[u, v]}}\right]\right) (c_{L^{[u,v]}}) \,.$$ Next we compute the right hand side of (\[eqn F\^+ functor\]) evaluated at $c$. Setting $b=(b_M)_{M\in\Sigma_{\mathcal{G}}(H)}:=F^+(V)(c)$ and $a=(a_L)_{L\in\Sigma_{{\mathcal{G}}}(G)}:= F^+(U)(b)$ we have, for every $M\in\Sigma_{\mathcal{G}}(H)$, $$b_M= \sum_{\substack{v\in [V/K]\\ M{\leqslant}p_1((H\times K)_v)}} F\left(\left[\frac{M\times M^v}{(M\times K)_v}\right]\right)(c_{M^v})\,,$$ and for every $L\in\Sigma_{\mathcal{G}}(G)$, $$\begin{aligned}
a_L & = F^+(U)(b)= \sum_{\substack{u\in [U/H]\\ L{\leqslant}p_1((G\times H)_u)}}
F\left(\left[\frac{L\times L^u}{(L\times H)_u}\right]\right)(b_{L^u})\\\
& = \sum_{\substack{u\in [U/H] \\ L{\leqslant}p_1((G\times H)_u)}} \sum_{\substack{v\in [V/K] \\ L^u{\leqslant}p_1((H\times K)_v)}}
F\left(\left[\frac{L\times L^u}{(L\times H)_u}\times_{L^u}\frac{L^u\times (L^u)^v}{(L^u\times K)_v}\right]\right) (c_{(L^u)^v})\,.\end{aligned}$$ Since $U$ is right-free, for any sets of representatives $[U/H]$ and $[V/K]$, the set $\{[u,v]\mid (u,v)\in [U/H]\times [V/K]\}$ is a set of representatives of $(U\times_H V)/K$. Moreover, by Lemma \[lem double condition\](ii), the sum in (\[eqn lhs\]) and the double sum in the last formula for $a_L$ run over the the same indexing set. Finally, we have $(L^u)^v=L^{[u,v]}$ by Lemma \[lem double condition\](i) and $(L\times H)_u*(L^u\times K)_v = (L\times K)_{[u,v]}$, see Equation (\[eqn stab \*\]). This finishes the proof of the equality in (\[eqn F\^+ functor\]). That $F^+$ preserves identity morphisms and is $R$-linear follows immediately from the definitions.
The proof of the following lemma is an easy exercise left to the reader.
\[lem double condition\] Let $U$ be a finite $(G,H)$-biset, let $V$ be a finite $(H,K)$-biset, let $(u,v)\in U\times V$ and let $L{\leqslant}G$.
One has $(L^u)^v=L^{[u,_Hv]}$.
Assume that $U$ is right-free. Then $L{\leqslant}p_1((G\times K)_{[u,v]})$ if and only if $L{\leqslant}p_1((G\times H)_u)$ and $L^u{\leqslant}p_1((H\times K)_v)$.
The Mark Morphism {#sec mark morphism}
=================
Let $({\mathcal{G}},{\mathcal{S}})$ be as in \[noth GS\] and assume that $({\mathcal{G}},{\mathcal{S}})$ satisfies axioms (i)–(iv), (vi), and (vii) in \[noth GS\] and the condition $k_2$ in Example \[ex D\_+ and D\^+\]. Set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$, ${\mathcal{D}}_+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}_+)$ and ${\mathcal{D}}^+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}^+)$. Then Propositions \[prop D\_+\](c) and \[prop D\^+\](c) imply that ${\mathcal{D}}_+={\mathcal{D}}^+$. We denote this category by ${\mathcal{E}}$. Let $F\in {\mathcal{F}}_{{\mathcal{D}},R}$ and $G\in {\mathcal{G}}$. We define an $R$-linear map $m_{F,G}:=(\pi_H\circ F_+({\mathrm{res}}^G_H))_{H\in\Sigma_{\mathcal{G}}(G)} : F_+(G)\longrightarrow F^+(G)$, using the projection map $\pi_H\colon F_+(H)\to F(H)$ from (\[eqn pi\]). Thus, $m_{F,G}$ maps the class $[X,s]\in F_+(G)$ of an object $(X,s)$ in $\Gamma_F(G)$ to the element $(a_L)_{L\in\Sigma_{{\mathcal{G}}}(G)}\in F^+(G)$ with $$a_L := \sum_{\substack{x\in X \\ L\leq G_x}}F({\mathrm{res}}^{G_x}_L)(s(x))\,.$$ Note that for the definition of $F^+(G)$ and the map $m_{F,G}$ one does not need axiom (vii) in \[noth GS\] and the condition $k_2$ in Example \[ex D\_+ and D\^+\]. But it is necessary in the next theorem. In general, the map $m_{F,G}$ does not commute with deflations.
\[thm mark diagram\] Assume that $({\mathcal{G}},{\mathcal{S}})$ satisfies axioms (i)–(iv), (vi), and (vii) in \[noth GS\] and the condition $k_2$ in Example \[ex D\_+ and D\^+\]. Let $G,H\in{\mathcal{G}}$ and let $U$ be a finite $(G,H)$-biset with stabilizers in ${\mathcal{S}}_+(G,H)={\mathcal{S}}^+(G,H)$. Then, for any $F\in{\mathcal{F}}_{{\mathcal{D}},R}$, the diagram $$\begin{CD}
F_+(H) @>m_{F,H}>> F^+(H)\\
@VF_+([U])VV @VVF^+([U])V\\
F_+(G) @>m_{F,G}>> F^+(G)
\end{CD}$$ commutes. In particular, the $R$-linear maps $m_{F,G}$, $G\in{\mathcal{G}}$, define a morphism $m_F\colon F_+\to F^+$ in ${\mathcal{F}}_{{\mathcal{E}},R}$ which we will call the [*mark morphism*]{} associated with $F$.
Let $(X,s)$ be in $\Gamma_F(H)$, let $[X,s]\in F_+(H)$ be the associated class, and let $L\in\Sigma_{\mathcal{G}}(G)$. Then the $L$-component of $(F^+([U])(m_{F,H}([X,s])))$ is equal to $$\begin{aligned}
& \sum_{\substack{u\in [U/H] \\L\leq p_1((G\times H)_u)}}
F\left(\left[\frac{L\times L^u}{(L\times H)_u}\right]\right)
\Bigl(\sum_{\substack{x\in X\\L^{u}\leq H_x}}
F({\mathrm{res}}^{H_x}_{L^u})(s(x))\Bigr)\\
= & \sum_{\substack{u\in [U/H] \\x\in X\\L\leq p_1((G\times H)_u)\\L^{u}\leq H_x}}
F\left(\left[\frac{L\times L^u}{(L\times H)_u}\times_{L^u}
\frac{L^u\times H_x}{\Delta(L^u)}\right]\right)(s(x))\,.\end{aligned}$$ Moreover, the $L$-component of $m_{F,G}(F_+([U])([X,s]))=m_{F,G}([U\times_H X, U(s)])$ is equal to $$\begin{aligned}
& \sum_{\substack{[u,x]\in U\times_H X \\L\leq G_{[u,x]}}} F({\mathrm{res}}^{G_{[u,x]}}_L)(U(s)([u,x]))\\
= & \sum_{\substack{[u,x]\in U\times_H X \\L\leq G_{[u,x]}}}
F\left(\left[\frac{L\times G_{[u,x]}}{\Delta(L)}
\times_{G_{[u,x]}} \frac{G_{[u,x]}\times H_x}{(G\times H_x)_u}\right]
\right)(s(x))\\
= & \sum_{\substack{[u,x]\in U\times_H X \\L\leq G_{[u,x]}}}
F\left(\left[ \frac{L\times H_x}{\Delta(L)* (G\times H_x)_u}\right]
\right)(s(x))\,.\end{aligned}$$ Since $U$ is right-free, the map $[U/H]\times X\to U\times_H X$, $(u,x)\mapsto [u,x]$, is bijective. Moreover, Lemma \[lem double condition\](b) applied to the case $K=\{1\}$ shows that the conditions for the subgroup $L$ are equivalent in both sums. Thus, there is a bijection between the indexing sets of both sums. Finally, notice that for all $L\in\Sigma_{\mathcal{G}}(G)$, \[not exponential\] and \[not K\^u\] imply that $\Delta(L)*(G\times H_x)_u =
(L\times H_x)_u = (L\times H)_u * \Delta(L^u)$. This completes the proof.
The following definition gives a map that is close to an inverse to the mark morphism.
\[def sigma\] Assume that $({\mathcal{G}},{\mathcal{S}})$ satisfies axioms (i)–(iv), (vi) in \[noth GS\] and set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$. Let $F\in{\mathcal{F}}_{{\mathcal{D}},R}$ and $G\in\Sigma_{\mathcal{G}}(G)$. We define a map $n_{F,G}:F^+(G)\longrightarrow F_+(G)$ by $$n_{F,G}((a_H)_{H\in \Sigma_{{\mathcal{G}}}(G)}) = \sum_{\substack{L, K\in \Sigma_{{\mathcal{G}}}(G)}}
|L| \mu(L,K) [L,{\mathrm{res}}^K_L(a_K)]\,.$$ Here, $\mu$ denotes the Möbius function of the poset $\Sigma_{\mathcal{G}}(G)$, ordered by inclusion.
Let $({\mathcal{G}},{\mathcal{S}})$, ${\mathcal{D}}$, $F$, and $G$ be as in Definition \[def sigma\]. Then $n_{F,G}\circ m_{F,G}=|G|\cdot{\mathrm{id}}_{F_+(G)}$ and $m_{F,G}\circ n_{F,G} = |G|\cdot{\mathrm{id}}_{F^+(G)}$.
Let $(X,s)\in\Gamma_F(G)$. We have $$\begin{aligned}
(n_{F,G}\circ m_{F,G})([X,s])
= & \sum_{\substack{L, K\in\Sigma_{{\mathcal{G}}}(G) }} | L | \mu(L,K) [L, F({\mathrm{res}}^K_L)(m_{F,G}([X,s])_K)]_G\\
= & \sum_{\substack{L,K\in\Sigma_{\mathcal{G}}(G) }}
\sum_{\substack{x\in X \\ K\leq G_x }} | L | \mu(L,K) [L, F({\mathrm{res}}^K_L \mathop{\cdot}\limits_K{\mathrm{res}}^{G_x}_K)(s(x))]_G\\
= & \sum_{\substack{L\in\Sigma_{\mathcal{G}}(G) }}
\sum_{\substack{x\in X}} | L |
\sum_{\substack{K\in\Sigma_{{\mathcal{G}}}(G)\\ L\leq K\leq G_x }} \mu(L,K) [L, F({\mathrm{res}}^{G_x}_L)(s(x))]_G\,,\end{aligned}$$ where $m_{F,G}([X,s])_K$ denotes the $K$-component of $m_{F,G}([X,s])$. By the standard properties of $\mu$, the expression $$\sum_{\substack{K\in\Sigma_{\mathcal{G}}(G)\\ L\leq K\leq G_x }} \mu(L,K)$$ equals 1 if $L=G_x$, and 0 otherwise. Therefore, the last expression equals
$$\sum_{\substack{L\leq G }} \sum_{\substack{x\in X \\ L = G_x}} | L | [L, s(x)]_G
= \sum_{\substack{x\in X}} | G_x | [G_x, s(x)]_G
= \sum_{\substack{x\in [G\backslash X]}} | G | [G_x, s(x)]_G = | G | [X,s]\,.$$
The second equality is a similar adaptation of the proof of [@Boltje1998b Proposition 2.4].
\[cor m iso\] Let $({\mathcal{G}},{\mathcal{S}})$, ${\mathcal{D}}$, $F$, and $G$ be as in Definition \[def sigma\]. If $|G|$ is invertible in $R$ then $m_{F,G}$ and $|G|^{-1}n_{F,G}$ are mutually inverse isomorphisms.
Let $({\mathcal{G}},{\mathcal{S}})$, ${\mathcal{D}}$, $F$, and $G$ be as in Definition \[def sigma\]. If $F_+(G)$ has trivial $|G|$-torsion then $m_{F,G}$ is injective.
Multiplicative Structures
=========================
For any group homomorphism $\varphi\colon H\to K$ we denote by ${\llap{\phantom{|}}_{H^\varphi}{K}}_K$ the $(H,K)$-biset $K$ on which $K$ acts by multiplication and $H$ acts via $\varphi$ and multiplication. Similarly, we define the $(K,H)$-biset ${\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}$. Note that ${\llap{\phantom{|}}_{H^\varphi}{K}}_K\cong (H\times K)/{D^{\varphi}}$ as $(H,K)$-bisets, where ${D^{\varphi}}=\{(h,\varphi(h))\mid h\in H\}$, and that ${\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}\cong (K\times H)/{{\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!D}}$, where ${{\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!D}}:=\{(\varphi(h),h)\mid h\in H\}$.
Green biset functors were defined by Bouc in [@Bouc2010a §8.5]. An alternative definition and proof for their equivalence was given by Romero in [@RomeroThesis Definición 3.2.7, Lema 4.2.3]. In our context, Romero’s definition amounts to the following definition.
\[def Green biset functor\] Let $({\mathcal{G}}, {\mathcal{S}})$ satisfy axioms (i)–(iii) in \[noth GS\] and let ${\mathcal{D}}= {\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$.
\(a) A [*Green biset functor*]{} on ${\mathcal{D}}$ over $R$ is an object $F\in {\mathcal{F}}_{{\mathcal{D}},R}$ together with the datum of an $R$-algebra structure on each $F(G)$, $G\in{\mathcal{G}}$, such that the following axioms are satisfied for all $H,K\in{\mathcal{G}}$ and all group homomorphisms $\varphi\colon H\to K$:
\(i) If ${D^{\varphi}}\in S(H,K)$ then the map $F([{\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}])\colon F(K)\to F(H)$ is a morphism of $R$-algebras.
\(ii) If ${D^{\varphi}}\in S(H,K)$ and ${{\setbox0=\hbox{$D$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!D}}\in S(K,H)$ then, for all $a\in F(H)$ and $b\in F(K)$, one has $$F([{\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}])(a)\cdot b = F([{\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}])(a\cdot F([{\llap{\phantom{|}}_{H^\varphi}{K}}_K])(b))$$ and $$b\cdot F([{\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}])(a) = F([{\llap{\phantom{|}}_{K}{K}}_{{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{\varphi}\!H}}])
(F([{\llap{\phantom{|}}_{H^\varphi}{K}}_K])(b)\cdot a)\,.$$
\(b) Let $F_1,F_2\in{\mathcal{F}}_{{\mathcal{D}},R}$ be Green biset functors on ${\mathcal{D}}$ over $R$. A morphism of Green biset functors between $F_1$ and $F_2$ is a morphism $\eta\colon F_1\to F_2$ in the category ${\mathcal{F}}_{{\mathcal{D}},R}$ with the additional property that $\eta_G\colon F_1(G)\to F_2(G)$ is an $R$-algebra homomorphism, for all $G\in{\mathcal{G}}$. We denote the category of Green biset functors on ${\mathcal{D}}$ over $R$ by ${\mathcal{F}}^\mu_{{\mathcal{D}},R}$.
The following theorem shows that the constructions $F_+$ and $F^+$ also work for Green biset functors. Interestingly, to define the multiplication on $F_+$ one needs restriction maps. The mark morphism is then multiplicative.
\[thm Green +\] Let $({\mathcal{G}},{\mathcal{S}})$ satisfy the axioms (i)–(iii) in \[noth GS\] and set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$. Further, let $F\in{\mathcal{F}}^\mu_{{\mathcal{D}},R}$.
Assume that $({\mathcal{G}},{\mathcal{S}})$ also satisfies axioms (iv) and (vi) in \[noth GS\] and set ${\mathcal{D}}_+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}_+)$. Then $F_+\in{\mathcal{F}}^\mu_{{\mathcal{D}}_+,R}$ and one obtains a functor $-_+\colon {\mathcal{F}}^\mu_{{\mathcal{D}},R}\to{\mathcal{F}}^\mu_{{\mathcal{D}}_+,R}$.
Assume that $({\mathcal{G}},{\mathcal{S}})$ also satisfies axiom (vii) in \[noth GS\] and condition $k_2$ in Example \[ex D\_+ and D\^+\], and set ${\mathcal{D}}^+:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}}^+)$. Then $F^+\in{\mathcal{F}}^\mu_{{\mathcal{D}}^+,R}$ and one obtains a functor $-^+\colon {\mathcal{F}}^\mu_{{\mathcal{D}},R}\to{\mathcal{F}}^\mu_{{\mathcal{D}}^+,R}$.
If $({\mathcal{G}},{\mathcal{S}})$ also satisfies axioms (iv) and (vi) in \[noth GS\] then the mark morphism $m_{F,G}\colon F_+(G)\to F^+(G)$ is an $R$-algebra homomorphism for all $G\in{\mathcal{G}}$. In particular, if $({\mathcal{G}},{\mathcal{S}})$ additionally satisfies axiom (vii) in \[noth GS\] and condition $k_2$ in Example \[ex D\_+ and D\^+\], we obtain a morphism $m_F\colon F_+\to F^+$ in the category of Green biset functors, ${\mathcal{F}}^\mu_{{\mathcal{E}},R}$, where ${\mathcal{E}}={\mathcal{D}}^+={\mathcal{D}}_+$.
The proof of the previous theorem is straightforward but involves a large number of verifications. We only point out the multiplicative structures on $F_+(G)$ and $F^+(G)$ and leave the verifications to the reader. In $F_+(G)$, the product is given by $[X,s]\cdot [Y,t]:=[X\times Y, s\times t]$, where $(X,s),(Y,t)\in\Gamma_F(G)$ and $s\times t\colon X\times Y\to \coprod_{(x,y)\in X\times Y} F(G_{(x,y)})$ maps $(x,y)$ to $F({\mathrm{res}}_{G_{(x,y)}}^{G_x})(s(x)) \cdot F({\mathrm{res}}_{G_{(x,y)}}^{G_y})(t(y))$. This translates to the product $$[H,a]_G\cdot[K,b]_G:= \sum_{g\in H\backslash G/K} [H\cap{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!K}, F({\mathrm{res}}^H_{H\cap{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!K}})(a)\cdot
F({\mathrm{res}}^{{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!K}}_{H\cap{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!K}})({\setbox0=\hbox{$b$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!b})]_G\,,$$ for $H,K\in\Sigma_{\mathcal{G}}(G)$ and $a\in F(H)$, $b\in F(K)$. In $F^+(G)$ we define the product coordinate-wise. In Part (c), note that the maps $F_+({\mathrm{res}}^G_H)$ and $\pi_H\colon F_+(H)\to F(H)$ are $R$-algebra homomorphisms, for all $H\in\Sigma_{\mathcal{G}}(G)$.
\[ex multiplicative local\] (a) Let $G$ be a finite group. Note that the categories ${\mathrm{Con}_R^f}(G)$, ${\mathrm{Res}_R^f}(G)$, ${\mathrm{Mack}_R^f}(G)$, considered in Example \[ex D\_+ and D\^+ local\], have Green biset functor versions ${\mathrm{Con}_R^{f,\mu}}(G)$, ${\mathrm{Res}_R^{f,\mu}}(G)$, ${\mathrm{Mack}_R^{f,\mu}}(G)$ which are [*fused*]{} versions of the $R$-algebra conjugation, restriction and Mackey functors considered in [@Boltje1998b Section 1]. With this notation, we have again functors $-_+\colon {\mathrm{Res}_R^{f,\mu}}(G)\to{\mathrm{Mack}_R^{f,\mu}}(G)$ and $-^+\colon {\mathrm{Con}_R^{f,\mu}}(G)\to{\mathrm{Mack}_R^{f,\mu}}(G)$ that are restrictions of the functors $-_+$ and $-^+$ given in [@Boltje1998b Section 2].
\(b) The multiplicative structures on the Burnside ring, $A$-fibered Burnside ring, and the global representation ring (see Example \[ex Burnside examples\]) coincide with the multiplication on $F_+(G)$ defined above.
Next we turn our attention to [*species*]{} of $F_+(G)$, with the goal to describe them in terms of species of $F$. For our purposes, a [*species*]{} of a ring $\Lambda$ is a ring homomorphism $\sigma\colon \Lambda\to{\mathbb{C}}$. We denote the set of species of $\Lambda$ by ${\mathrm{Sp}}(\Lambda)$. For the remainder of this section we assume that $R={\mathbb{Z}}$, that $({\mathcal{G}},{\mathcal{S}})$ satisfies axioms (i)–(iv) and (vi) in \[noth GS\], and that $F\in{\mathcal{F}}_{{\mathcal{D}}}^\mu:={\mathcal{F}}_{{\mathcal{D}},{\mathbb{Z}}}^\mu$ is a Green biset functor, where ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$. For $G\in{\mathcal{G}}$ we set $${\mathrm{Sp}}_+(F,G):= \{(H,\tau)\mid H\in\Sigma_{\mathcal{G}}(G), \tau\in{\mathrm{Sp}}(F(H))\}$$ and $${\widetilde{{\mathrm{Sp}}}}_+(F,G):=\{(H,{\tilde{\tau}})\mid H\in\Sigma_{{\mathcal{G}}}(G), {\tilde{\tau}}\in{\mathrm{Sp}}(F(H)^{N_G(H)})\}\,,$$ where $F(H)^{N_G(H)}$ denotes the $N_G(H)$-fixed points of $F(H)$, a subring of $F(H)$. Note that $G$ acts on ${\mathrm{Sp}}_+(F,G)$ from the right: For $x\in G$ we define $(H,\tau)^x:=(H^x,\tau^x)$, where $\tau^x:=\tau\circ F(c_x)\colon F(H^x)\to{\mathbb{C}}$. Similarly, $G$ acts on ${\widetilde{{\mathrm{Sp}}}}_+(F,G)$ and the map $$\label{eqn rho def}
{\mathrm{Sp}}_+(F,G)\to{\widetilde{{\mathrm{Sp}}}}_+(F,G)\,,\quad (H,\tau)\mapsto (H,\tau|_{F(H)^{N_G(H)}})\,,$$ is $G$-equivariant. For every $(H,{\tilde{\tau}})\in{\widetilde{{\mathrm{Sp}}}}_+(F,G)$ we defined the map $$\sigma_{(H,{\tilde{\tau}})}\colon F_+(G)\to{\mathbb{C}}\,,\quad \omega\mapsto{\tilde{\tau}}\bigl( m_{F,G}(\omega)_H\bigr)\,,$$ where $m_{F,G}(\omega)_H$ denotes the $H$-component of $m_{F,G}(\omega)\in(\prod_{H\in\Sigma_{\mathcal{G}}(G)}F(H))^G$. Note that $m_{F,G}(\omega)_H\in F(H)^{N_G(H)}$. Since $m_{F,G}$ is a ring homomorphism (see Theorem \[thm Green +\](c)), we have $\sigma_{(H,{\tilde{\tau}})}\in{\mathrm{Sp}}(F_+(G))$. It is easy to verify that $\sigma_{(H,{\tilde{\tau}})}=\sigma_{(H,{\tilde{\tau}})^x}$ for all $x\in G$. Thus, we obtain a map $${\tilde{\sigma}}_{F,G}\colon {\widetilde{{\mathrm{Sp}}}}_+(F,G)/G\to {\mathrm{Sp}}(F_+(G))\,, \quad [H,{\tilde{\tau}}]\mapsto {\tilde{\sigma}}_{(H,{\tilde{\tau}})}\,,$$ and a commutative diagram $$\label{diag rho sigma}
\parbox{7cm}{
\begin{diagram}
{\mathrm{Sp}}_+(F,G)/G & \movearrow(10,0){\Ear[30]{\sigma_{F,G}}} & \movevertex(20,0){{\mathrm{Sp}}(F_+(G))} &&
\Sar{\rho_{F,G}} & & \movearrow(-10,0){\neaR{{\tilde{\sigma}}_{F,G}}} &&
{\widetilde{{\mathrm{Sp}}}}_+(F,G)/G & & &&
\end{diagram}}$$ where $\sigma_{F,G}:={\tilde{\sigma}}_{F,G}\circ\rho_{F,G}$ and $\rho_{F,G}$ denotes the map induced by the map in (\[eqn rho def\]). For $(H,\tau)\in{\mathrm{Sp}}_+(F,G)$ we set $\sigma_{(H,\tau)}:=\sigma_{(H,{\tilde{\tau}})}$, where ${\tilde{\tau}}:=\tau|_{F(H)^{N_G(H)}}$.
Let $({\mathcal{G}},{\mathcal{S}})$ satisfy (i)–(iv) and (vi) in \[noth GS\], set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$, let $F\in{\mathcal{F}}_{{\mathcal{D}}}^\mu$, and let $G\in{\mathcal{G}}$. Then the map $\sigma_{F,G}$ in the commutative diagram (\[diag rho sigma\]) is injective and the map ${\tilde{\sigma}}_{F,G}$ is surjective. If moreover $F(H)$ is commutative for all $H\in\Sigma_{{\mathcal{G}}}(G)$ then all three maps $\sigma_{F,G}$, ${\tilde{\sigma}}_{F,G}$ and $\rho_{F,G}$ are bijective.
We first prove that $\sigma_{F,G}$ is injective. Let $(H,\tau),(K,\rho)\in {\mathrm{Sp}}_+(F,G)$ with $\sigma_{(H,\tau)}=\sigma_{(K,\rho)}$. We will show that $(H,\tau)$ and $(K,\rho)$ are $G$-conjugate. Since $\sigma_{(K,\rho)}([H,1]_G) = \sigma_{(H,\tau)}([H,1]_G) = [N_G(H):H]\neq 0\in {\mathbb{C}}$, we have $K{\leqslant}{\setbox0=\hbox{$H$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!H}$ for some $x\in G$. Similarly, $H{\leqslant}{\setbox0=\hbox{$K$} \setbox1=\vbox to
\ht0{}\,\box1^{y}\!K}$ for some $y\in G$. Thus, $K$ and $H$ are conjugate, and we may assume that they are equal. For every $a\in F(H)^{N_G(H)}$, we have $[N_G(H):H] \tau(a) = s_{(H,\tau)}([H,a]_G) = s_{(K,\rho)}([H,a]_G) = [N_G(H):H] \rho(a)$, which shows that $\tau$ and $\rho$ coincide on $F(H)^{N_G(H)}$. Furthermore, for every $b\in F(H)$, the element $a:=\sum_{x\in [N_G(H)/H]} {\setbox0=\hbox{$b$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!b}$ lies in $F(H)^{N_G(H)}$, so that $\sum_{x\in[N_G(H)/H]}\tau^x(b) = \tau(a) = \rho(a) = \sum_{x\in [N_G(H)/H]} \rho^x(b)$. This implies $\sum_{x\in [N_G(H)/H]}\tau^x=\sum_{x\in [N_G(H)/H]}\rho^x$. Since ring homomorphisms from $F(H)$ to ${\mathbb{C}}$ are ${\mathbb{C}}$-linearly independent in the ${\mathbb{C}}$-vector space of all functions from $F(H)$ to ${\mathbb{C}}$, there exists $x\in N_G(H)$ with $\tau=\rho^x$, and the proof of the injectivity of $\sigma_{F,G}$ is complete.
Next we prove that ${\tilde{\sigma}}_{F,G}$ is surjective. Let $\sigma\in{\mathrm{Sp}}(F_+(G))$ and let $\sigma_{\mathbb{C}}\colon {\mathbb{C}}\otimes F_+(G)\to{\mathbb{C}}$ be its extension to a ${\mathbb{C}}$-algebra homomorphism. By the arguments in the proof of Lemma 7.5(ii) in [@Boltje1998b], one has a canonical ${\mathbb{C}}$-algebra isomorphism ${\mathbb{C}}\otimes F_+(G){\buildrel\sim\over\to}({{\setbox0=\hbox{$F$} \setbox1=\vbox to
\ht0{}\,\box1^{{\mathbb{C}}}\!F}})_+(G)$, where ${{\setbox0=\hbox{$F$} \setbox1=\vbox to
\ht0{}\,\box1^{{\mathbb{C}}}\!F}}\in{\mathcal{F}}_{{\mathcal{D}},{\mathbb{C}}}^\mu$ denotes the obvious scalar extension of $F$ from ${\mathbb{Z}}$ to ${\mathbb{C}}$. We write again $\sigma_{\mathbb{C}}\colon ({{\setbox0=\hbox{$F$} \setbox1=\vbox to
\ht0{}\,\box1^{{\mathbb{C}}}\!F}})_+(G)\to{\mathbb{C}}$ for the corresponding ${\mathbb{C}}$-algebra homomorphism. Next we consider the commutative diagram
(-70,0)[F\_+(G)]{} & (-65,0) & (-60,0)[F\^+(G)]{} & & (-30,0) & & (30,0)[\_[H\_(G)]{} F(H)\^[N\_G(H)]{}]{} && (-70,0) & & & & & & (40,0) && (-70,0)[([[0= 1=to 01\^F]{}]{})\_+(G)]{} & (-65,0) & (-60,0)[([[0= 1=to 01\^F]{}]{})\^+(G)]{} & (-55,0) & (-20,0)[ \_[H\_(G)]{}(F(H))\^[N\_G(H)]{}]{} & (10,0) & (50,0)[\_[H\_(G)]{}F(H)\^[N\_G(H)]{}]{} &&
where ${\tilde{\Sigma}}_{\mathcal{G}}(G)\subseteq\Sigma_{\mathcal{G}}(G)$ denotes a set of representatives of the conjugacy classes of $\Sigma_{\mathcal{G}}(G)$ and all morphisms other than $m_{F,G}$ and $m_{{{\setbox0=\hbox{$F$} \setbox1=\vbox to
\ht0{}\,\box1^{{\mathbb{C}}}\!F}},G}$ are the obvious natural maps, cf. (\[eqn F\_+ projection iso\]). Note that the right hand map of the bottom row is a ${\mathbb{C}}$-algebra isomorphism, since ${\mathbb{C}}$ is flat as ${\mathbb{Z}}$-module (see the proof of Lemma 7.5(i) in [@Boltje1998b]). Note also that $m_{{{\setbox0=\hbox{$F$} \setbox1=\vbox to
\ht0{}\,\box1^{{\mathbb{C}}}\!F}},G}$ is a ${\mathbb{C}}$-algebra isomorphism by Theorem \[thm Green +\](c) and Corollary \[cor m iso\]. Thus, all maps in the bottom row are ${\mathbb{C}}$-algebra isomorphisms and it follows that $\sigma_{\mathbb{C}}\colon ({{\setbox0=\hbox{$F$} \setbox1=\vbox to
\ht0{}\,\box1^{{\mathbb{C}}}\!F}})_+(G)\to {\mathbb{C}}$ must come via composition with these isomorphisms from a ${\mathbb{C}}$-algebra homomorphism ${\mathbb{C}}\otimes F(H)^{N_G(H)}\to {\mathbb{C}}$, for some $H\in{\tilde{\Sigma}}_{{\mathcal{G}}}(G)$. The restriction of this homomorphism to $F(H)^{N_G(H)}$ yields a species ${\tilde{\tau}}\in{\mathrm{Sp}}(F(H)^{N_G(H)})$ with $\sigma=\sigma_{(H,{\tilde{\tau}})}$. Thus, ${\tilde{\sigma}}_{F,G}$ is surjective.
Assume for the rest of the proof that $F(H)$ is commutative for all $H\in\Sigma_{\mathcal{G}}(G)$. By Theorem 1.8.1 and the first part of the proof of Theorem 2.9.1 in [@Benson1984], every ${\mathbb{C}}$-algebra homomorphism ${\tilde{\tau}}\colon ({\mathbb{C}}\otimes F(H))^{N_G(H)}\to{\mathbb{C}}$ extends to a ${\mathbb{C}}$-algebra homomorphism $\tau\colon {\mathbb{C}}\otimes F(H)\to{\mathbb{C}}$. Since $({\mathbb{C}}\otimes F(H))^{N_G(H)}\cong {\mathbb{C}}\otimes F(H)^{N_G(H)}$ as ${\mathbb{C}}$-algebras (with the same argument as above), this implies that the map in (\[eqn rho def\]) and therefore also the map $\rho$ in Diagram (\[diag rho sigma\]) is surjective. Since $\sigma_{F,G}$ is injective, also $\rho_{F,G}$ is injective. Thus, $\rho$ is bijective. Since $\sigma_{F,G}$ is injective, ${\tilde{\sigma}}_{F,G}$ is surjective and $\rho_{F,G}$ is bijective, all three maps must be bijective. This completes the proof of the Theorem.
Adjointness
===========
\[not adj D\] Unless otherwise stated, throughout this section let $R$ denote a commutative ring. Let $({\mathcal{G}},{\mathcal{S}})$ satisfy axioms (i) – (iv), and set ${\mathcal{D}}:={\mathcal{C}}({\mathcal{G}},{\mathcal{S}})$. For $H,K\in{\mathcal{G}}$ set ${\mathcal{S}}_-(H,K) = \{ D \in {\mathcal{S}}(H,K) \mid p_1(D) = H \}$.
The proof of the following lemma is straightforward and is left to the reader.
Let $({\mathcal{G}},{\mathcal{S}})$ be as in \[not adj D\]. Then $({\mathcal{G}},{\mathcal{S}}_-)$ also satisfy axioms (i)–(iv) and $({\mathcal{S}}_-)_+ = {\mathcal{S}}_+$. We can thus define ${\mathcal{D}}_- = {\mathcal{C}}({\mathcal{G}},{\mathcal{S}}_-)$ and obtain ${\mathcal{D}}_-\subseteq{\mathcal{D}}\subseteq{\mathcal{D}}_+$. For any $F\in{\mathcal{F}}_{{\mathcal{D}}_-,R}$ one has a morphism $\eta_F\colon F\to {\mathrm{Res}}^{{\mathcal{D}}_+}_{{\mathcal{D}}_-}(F_+)$ in ${\mathcal{F}}_{{\mathcal{D}}_-,R}$ given by $\eta_{F,G}(a):=[G,a]_G$ for any $G\in{\mathcal{G}}$ and $a\in F(G)$.
\[thm adjunction\] Let $({\mathcal{G}},{\mathcal{S}})$ be as in \[not adj D\]. Then the functor $-_+\colon {\mathcal{F}}_{{\mathcal{D}}_-,R}\to {\mathcal{F}}_{{\mathcal{D}}_+,R}$ is left adjoint to the restriction functor ${\mathrm{Res}}^{{\mathcal{D}}_+}_{{\mathcal{D}}_-}\colon {\mathcal{F}}_{{\mathcal{D}}_+,R}\to{\mathcal{F}}_{{\mathcal{D}}_-,R}$. More precisely, for any $F\in{\mathcal{F}}_{{\mathcal{D}}_-,R}$ and $M\in{\mathcal{F}}_{{\mathcal{D}}_+,R}$, the map $\varphi\mapsto \varphi\circ\eta_F$ defines an $R$-linear isomorphism $${\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_+,R}}(F_+,M){\buildrel\sim\over\to}{\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_-,R}}(F,{\mathrm{Res}}^{{\mathcal{D}}_+}_{{\mathcal{D}}_-}(M))\,.$$
We will show that for any $\psi\in{\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_-,R}}(F,{\mathrm{Res}}^{{\mathcal{D}}_+}_{{\mathcal{D}}_-}(M))$ there exists a unique $\varphi\in {\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_+,R}}(F_+,M)$ with $\psi=\varphi\circ\eta_F$. We show the uniqueness first. Let $G\in{\mathcal{G}}$. Then every element in $F_+(G)$ can be written as an $R$-linear combination of elements of the form $[H,a]_G=F_+({\mathrm{ind}}_H^G)([H,a]_H)$ with $a\in F(H)$ and $H\in\Sigma_{{\mathcal{G}}}(G)$. Since $[H,a]_H$ is in the image of $\eta_{F,H}$ and $\varphi$ commutes with inductions, $\varphi$ is uniquely determined by the condition $\psi=\varphi\circ\eta_F$. Next we show the existence of $\varphi$. For $G\in{\mathcal{G}}$ and an object $(X,s)$ in $\Gamma_F(G)$, we define $\varphi_G([X,s]):=\sum_{x\in [G\backslash X]} M({\mathrm{ind}}^G_{G_x}) (\psi_{G_x}(s(x)))$. Note that this yields a well-defined map $\varphi_G\colon F_+(G)\to M(G)$. In fact, the above sum does not depend on the choice of $[G\backslash X]$, and by the definition of $F_+(G)$ we only have to check that the relations in Definition \[def 2 operations\] are respected, which is an easy verification. Note also that, for $H\in\Sigma_{\mathcal{G}}(G)$ and $a\in F(H)$, we have $\varphi_G([H,a]_G)=M({\mathrm{ind}}_H^G)(\psi_H(a))$. Choosing $H=G$, this shows that $\varphi\circ\eta_F=\psi$. Next we show that $\varphi\in{\mathcal{F}}_{{\mathcal{D}}_+,R}$. So let $G,H\in{\mathcal{G}}$, $D\in{\mathcal{S}}_+(G,H)$ and set $U:=(G\times H)/D$. Let $L\in\Sigma_{\mathcal{G}}(H)$ and $a\in F(L)$. On the one hand we have $$\begin{aligned}
M([U])(\varphi_H([L,a]_H)) & = & M([U])\left({\mathrm{ind}}^H_{L} (\psi_{L}(a)) \right)\\
& = & M\left(\left[ \frac{G\times H}{D} \times_H \frac{H\times L}{\Delta(L)} \right] \right) (\psi_L(a))\\
& = & \sum_{x\in [p_2(D)\backslash H / L]} M\left( \left[\frac{G\times L}{D*{\setbox0=\hbox{$\Delta(L)$} \setbox1=\vbox to
\ht0{}\,\box1^{(x,1)}\!\Delta(L)}} \right]\right)(\psi_L(a))\end{aligned}$$ On the other hand, using (\[eqn explicit F\_+\]), we have $$F_+([U])([L,a]_H) = \sum_{x\in[p_2(D)\backslash H/L]}
\left[ D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}, F\left(\left[\frac{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}\times {{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}{D*\Delta({{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})}\right]\right)({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{h}\!a})\right]_G$$ and therefore $$\begin{aligned}
\varphi_G\bigl(F_+([U])([L,a]_H)\bigr) & = \sum_{x\in[p_2(D)\backslash H/L]}
M({\mathrm{ind}}_{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}^G)\bigl(\psi_{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}\bigl(F\left(\left[\frac{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}\times{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}{D*\Delta({{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})}\right]\right)({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!a})\bigr)\bigr)\\
& = \sum_{x\in[p_2(D)\backslash H/L]}
M({\mathrm{ind}}_{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}^G)\bigl( M\left(\left[\frac{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}\times{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}{D*\Delta({{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})}\right]\right) (\psi_{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!a}))\bigr)\\
& = \sum_{x\in[p_2(D)\backslash H/L]}
M\left(\left[\frac{G\times D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}{\Delta(D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})}\mathop{\cdot}\limits_{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}
\frac{D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}\times{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}{D*\Delta({{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})}\right]\right)(\psi_{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!a}))\\
& = \sum_{x\in[p_2(D)\backslash H/L]}
M\left(\left[\frac{G\times {{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}{\Delta(D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})*D*\Delta({{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})}\right]\right)(\psi_{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!a}))\,.\end{aligned}$$ Since $\psi_{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$a$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!a}) = M(c_x)(\psi_L(a))$ and $\Delta(D*{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})*D*\Delta({{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}})*{\setbox0=\hbox{$\Delta(L)$} \setbox1=\vbox to
\ht0{}\,\box1^{(x,1)}\!\Delta(L)} = D*{\setbox0=\hbox{$\Delta(L)$} \setbox1=\vbox to
\ht0{}\,\box1^{(x,1)}\!\Delta(L)}$, we obtain $M([U])(\varphi_H([L,a]_H)) = \varphi_G(F_+([U])([L,a]_H))$. Thus, $\varphi\in
{\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_+,R}}(F_+,M)$, and the proof is complete.
If $F\in{\mathcal{F}}_{{\mathcal{D}}_-,R}^\mu$ is a Green biset functor then the natural transformation $\eta_F$ is multiplicative, i.e., a morphism in ${\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_-,R}^\mu}(F,{\mathrm{Res}}^{{\mathcal{D}}^+}_{{\mathcal{D}}_-}(F^+))$. Theorem \[thm adjunction\] has the following multiplicative version.
\[thm adjunction mu\] Assume that $({\mathcal{G}},{\mathcal{S}})$ satisfies Axioms (i)–(iv) and Axiom (vi) in \[noth GS\]. Then the functor $-_+\colon {\mathcal{F}}^\mu_{{\mathcal{D}}_-,R}\to {\mathcal{F}}^\mu_{{\mathcal{D}}_+,R}$ is left adjoint to the restriction functor ${\mathrm{Res}}^{{\mathcal{D}}_+}_{{\mathcal{D}}_-}\colon {\mathcal{F}}^\mu_{{\mathcal{D}}_+,R}\to{\mathcal{F}}^\mu_{{\mathcal{D}}_-,R}$. As in Theorem \[thm adjunction\], the adjunction bijection is given by composition with $\eta_F$.
Let $F\in{\mathcal{F}}^\mu_{{\mathcal{D}}_-,R}$ and $M\in{\mathcal{F}}^\mu_{{\mathcal{D}}_+,R}$ and let $\psi\in{\mathrm{Hom}}_{{\mathcal{F}}^\mu_{{\mathcal{D}}_-,R}}(F,M)$. We define $\varphi\in{\mathrm{Hom}}_{{\mathcal{F}}_{{\mathcal{D}}_+,R}}(F_+,M)$ as in the proof of Theorem \[thm adjunction\]. It suffices to show that, for any $G\in{\mathcal{G}}$, the map $\varphi_G\colon F_+(G)\to M(G)$ is multiplicative. Let $K,L\in\Sigma_{{\mathcal{G}}}(G)$, $a\in F(K)$, and $b\in F(L)$. On the one hand we have $$\begin{aligned}
& \ \ \varphi_G([K,a]_G\cdot[L,b]_G) \\
= & \sum_{\substack{x\in [ K\backslash G/L ]}} \varphi_G([K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}, F({\mathrm{res}}^{K}_{K\cap {{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})(a) \cdot
{\mathrm{res}}^{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$b$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!b})]_G) \\
= & \sum_{\substack{x\in [ K\backslash G/L ]}} M({\mathrm{ind}}^G_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})
\left(\psi_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}( F({\mathrm{res}}^{K}_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})(a)) \cdot
\psi_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}( F({\mathrm{res}}^{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}_{K\cap {{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})({\setbox0=\hbox{$b$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!b}))\right)\,.
\end{aligned}$$ On the other hand $$\begin{aligned}
& \ \ \varphi_G([K,a]_G)\cdot \varphi_G([L,b]_G) = M({\mathrm{ind}}^G_K) (\psi_K(a)) \cdot M({\mathrm{ind}}^G_L) (\psi_L(b))\\
= & \ \ M({\mathrm{ind}}^G_K)\left(\psi_K(a)\cdot M({\mathrm{res}}^G_K \mathop{\cdot}\limits_G {\mathrm{ind}}^G_L)(\psi_L(b)) \right)\\
= & \sum_{\substack{x\in [ K\backslash G/L ]}} M({\mathrm{ind}}^G_{K})
\left(\psi_{K}(a)\cdot M({\mathrm{ind}}^K_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}) \bigl( M({\mathrm{res}}^{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})
(\psi_{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$b$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!b})) \bigr) \right) \\
= & \sum_{\substack{x\in [ K\backslash G/L ]}} M({\mathrm{ind}}^G_{K})
\bigl(M({\mathrm{ind}}^K_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})\bigl( M({\mathrm{res}}^K_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}) ( \psi_K(a))\cdot
M({\mathrm{res}}^{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}_{K\cap{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}})(\psi_{{{\setbox0=\hbox{$L$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!L}}}({\setbox0=\hbox{$b$} \setbox1=\vbox to
\ht0{}\,\box1^{x}\!b}))\bigr)\bigr)\,.
\end{aligned}$$ Since induction is transitive and $\psi$ commutes with restrictions (note that ${\mathcal{D}}_-$ contains all possible restrictions), the two expressions coincide.
[XXXX]{}
Fibred permutation sets and the idempotents and units of monomial Burnside rings. [*J. Algebra*]{} [**281**]{} (2004), 535–566. Modular representation theory: new trends and methods. Lecture Notes in Mathematics, 1081. Springer-Verlag, Berlin, 1984. A general theory of canonical induction formulae. [*J. Algebra*]{} [**206**]{} (1998), 293–343. Fibered biset functors. arXiv:1612.01117. Biset functors for finite groups. Lecture Notes in Mathematics, 1990. Springer-Verlag, Berlin, 2010 Fused Mackey functors. [*Geom. Dedicata*]{} [**176**]{} (2015), 225–240. The ring of monomial representations I. Structure theory. [*J. Algebra*]{} [**18**]{} (1971) 137–157. Global representation rings. [*J. Algebra*]{} [**441**]{} (2015), 426–440. Funtores de Mackey. PhD Thesis, UNAM Morelia, 2011.
[^1]: MR Subject Classification 19A22, 20C15, 20C20
[^2]: Partially supported by Projects 2015 CN-15-43 UC MEXUS-CONACYT Collaborative Research Grants “Representation rings of finite groups”, PAPIIT IN101416 Anillo global de representaciones y funtores asociados, and by the CIC at the UMSNH
|
---
abstract: 'We present the results of a search for the most luminous star-forming galaxies at redshifts $z\approx 6$ based on CFHT Legacy Survey data. We identify a sample of 40 Lyman break galaxies brighter than magnitude $z''=25.3$ across an area of almost 4 square degrees. Sensitive spectroscopic observations of seven galaxies provide redshifts for four, of which only two have moderate to strong [Ly$\alpha$]{} emission lines. All four have clear continuum breaks in their spectra. Approximately half of the Lyman break galaxies are spatially resolved in 0.7 arcsec seeing images, indicating larger sizes than lower luminosity galaxies discovered with the [*Hubble Space Telescope*]{}, possibly due to on-going mergers. The stacked optical and infrared photometry is consistent with a galaxy model with stellar mass $\sim 10^{10}\, {\rm M}_\odot$. There is strong evidence for substantial dust reddening with a best-fit $A_V=0.75$ and $A_V>0.48$ at $2\sigma$ confidence, in contrast to the typical dust-free galaxies of lower luminosity at this epoch. The spatial extent and spectral energy distribution suggest that the most luminous $z\approx 6$ galaxies are undergoing merger-induced starbursts. The luminosity function of $z=5.9$ star-forming galaxies is derived. This agrees well with previous work and shows strong evidence for an exponential decline at the bright end, indicating that the feedback processes which govern the shape of the bright end are occurring effectively at this epoch.'
author:
- 'Chris J. Willott, Ross J. McLure, Pascale Hibon, Richard Bielby, Henry J. McCracken, Jean-Paul Kneib, Olivier Ilbert, David G. Bonfield, Victoria A. Bruce, and Matt J. Jarvis,'
title: An exponential decline at the bright end of the $z=6$ galaxy luminosity function
---
Introduction
============
The light from distant galaxies brings evidence of the conditions and physical processes at play in the early Universe. By studying the changes in galaxy properties over cosmic time we obtain a deeper understanding of how our Universe evolved. At 1 billion years after the Big Bang, galaxies were typically smaller, less luminous and less dusty than today (Bouwens et al. 2006). Such observations need to be explained by cosmological simulations which account for the hierarchical merging of dark matter halos and the gas accretion and cooling inside them necessary to form stars (e.g. Finlator et al. 2011).
One of the key measurements of the evolving Universe is the galaxy luminosity function. This function is related to the star formation rate occurring at an epoch and how the star formation is distributed across the galaxy population. Ultraviolet luminosity is well correlated with the formation rate of young, hot stars, with the caveat that dust extinction, common in starbursts, reduces the observed ultraviolet flux. Comparison of the observed luminosity function with that predicted by models constrains the important physical processes occurring. For example, the faint end slope of the galaxy luminosity function at redshifts up to at least $z=6$ is flatter than the dark matter halo mass function (Bouwens et al. 2007), which could be explained by feedback from supernovae winds (Cole 1991) or from photoevaporation and heating during reionization (Barkana & Loeb 2001). At the bright end, the galaxy luminosity declines much more sharply than the halo mass function. This is usually ascribed to AGN feedback and inefficient gas cooling in high mass halos (e.g. Benson et al. 2003).
Surveys for UV-continuum-selected galaxies at $z {\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}}6$ have been largely focused on due to the small sizes of the galaxies and low background in space. Hundreds of such galaxies have been identified in deep ACS imaging of the GOODS, Hubble Ultra Deep (HUDF) and parallel fields (Dickinson et al. 2004; Bunker et al. 2004; Yan & Windhorst 2004; Bouwens et al. 2006). The extremely deep imaging over the very small area of the HUDF identifies galaxies with absolute magnitude at 1350Å, $M_{1350}=-18$, corresponding to a star formation rate of only $1\, $M$_\odot \,{\rm yr}^{-1}$ (Kennicut 1998). These surveys have led to a good description of the $z=6$ luminosity function over the range $-21<M_{1350}<-18$ (Bouwens et al. 2007, 2008; Su et al. 2011). The luminosity function can be fit by a Schechter function with characteristic break luminosity of $M_{1350}=-20.2$. However, the few sources detected brighter than the break leave the exact nature of the high-luminosity decline uncertain.
Several surveys have been carried out from the ground over wider areas to find the rarer, more luminous $z\approx 6$ galaxies. McLure et al. (2006, 2009) and Curtis-Lake et al. (2012) report on $5<z<6.5$ galaxies found within the 0.8 square degree Subaru/[*XMM-Newton*]{} Deep Survey (optical) and UKIDSS Ultra Deep Survey (near-IR). This SXDS/UDS work provides tighter constraints on the luminosity function at $-22<M_{1350}<-21$ than from GOODS (McLure et al. 2009). Shimasaku et al. (2005), Nagao et al. (2007) and Jiang et al. (2011) have discovered many $z\approx 6$ galaxies in the 0.25 square degree Subaru Deep Field (SDF) which provides good luminosity function constraints for $-21<M_{1350}<-20$. Together these works show that the galaxy luminosity function at $z=6$ can be fit by a Schechter function, the same parameterization that successfully fits at lower redshift. The surveys discussed above only contain enough volume to discover galaxies with a space density of ${\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}}10^{-6}\, {\rm Mpc}^{-1}$. They show there is a steepening of the luminosity function, but do not confirm whether the cut-off is exponential or less steep than this.
At even higher redshifts ($z\sim 7 -8$) there have been mixed results from ground-based surveys to determine the bright end of the luminosity function. Castellano et al. (2010) found a decrease in the space density from $z=6$ to $z=6.8$ of a factor of 3.5. Capak et al. (2011) found three UV-bright $z\sim 7$ galaxy candidates in the 2 square degree COSMOS field. Although there is some evidence that these galaxies are low-$z$ interlopers, if they are truly at $z\approx 7$ then the galaxy luminosity function does not decline precipitously as a Schechter function, but rather as a power-law. Luminous $z\approx 8$ LBG candidates have recently been reported by Yan et al. (2011), also suggesting a bright end decline less steep than a Schechter function. Although both the above studies could be affected by low-$z$ contamination, it is important to determine how well the $z=6$ luminosity function is fit by a Schechter function, because there is little cosmic time available between $z=6$ and $z=7$ for significant evolution of the shape of the luminosity function. Capak et al. noted that their result could be explained if AGN feedback is less effective at early times due to the time required for supermassive black holes to build up their mass via Eddington-limited accretion and mergers.
The [*Deep*]{} component of the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) provides the deepest optical data covering several square degrees and the largest area survey capable of finding $z \approx 6$ galaxies based on their rest-frame UV continuum. At a total of nearly 4 square degrees the volume probed is approximately 40 times that of the GOODS survey and five times the previous largest area studied, the SXDS/UDS. We present the results of a search for $z\approx 6$ galaxies in the CFHTLS. Section \[imaging\] describes the optical and near-IR data used and how the galaxies were selected. Section \[spectra\] gives details of spectroscopic followup of a subset of the galaxies. Section \[sizes\] considers the physical sizes of these galaxies. In Section \[stacked\] we stack together the optical and IR images for the galaxy sample to demonstrate their high-$z$ nature and determine the typical galaxy SED. Section \[lumfun\] presents the resulting luminosity function and our conclusions are drawn in Section \[conc\].
All optical and near-IR magnitudes in this paper are on the AB system. Cosmological parameters of $H_0=70~ {\rm km~s^{-1}~Mpc^{-1}}$, $\Omega_{\mathrm M}=0.30$ and $\Omega_\Lambda=0.7$ are assumed throughout. These parameters have been adopted to ease comparison with previous work, even though they are slightly different from the best fit in Jarosik et al. (2011).
Imaging and sample selection {#imaging}
============================
Imaging observations
--------------------
The imaging data used to select high-redshift galaxies come primarily from the 3.6m Canada-France-Hawaii Telescope. Optical observations with MegaCam in the $u^*g'r'i'z'$ filters are from CFHTLS Deep which covered four $\approx 1$ square degree fields with typical total integration time of 75ks in $u^*$, 85ks in $g'$, 145ks in $r'$, 230ks in $i'$ and 175ks in $z'$. The seeing in the final stacks at $i'$ and $z'$ range from 0.66 to 0.76 arcsec. The data used here are from the 6th data release, T0006, which contains all the data acquired over the five years of the project.
These optical data are complemented by near-IR data from the WIRCam Deep Survey (WIRDS; Bielby et al. 2012). WIRDS used the WIRCam near-IR imager at the CFHT to cover 2.4 square degrees of the CFHTLS Deep reaching typical 50% completeness depth of AB magnitude 24.5 in the $JHK_s$ filters. A few high-redshift galaxy candidates not in the regions covered by WIRDS had $J$ band photometry obtained from the Gemini-North Telescope using GNIRS and from the ESO New Technology Telescope using SOFI. More recently, near-IR data for some of these regions has become available from the ESO VISTA telescope. The D1 and D2 fields are fully covered by the first public data releases of the VIDEO survey (Jarvis et al. 2012) and UltraVISTA survey (McCracken et al. 2012), respectively. These data reach about a magnitude deeper than WIRDS, so all sources in D1 and D2 have $YJHK$ photometry in this paper from the VISTA surveys instead of the original WIRDS data. Note that the VIDEO data used has been corrected for the non-optimal sky subtraction detailed in the release notes. Small parts of the D2 and D3 field are covered by the Multi-Cycle Treasury program CANDELS (Grogin et al. 2011; Koekemoer et al. 2011).
Photometry was carried out in dual-image mode using the Sextractor source extractor software (Bertin & Arnouts 1996). The $z'$ band was used as the detection band because it provides the highest signal-to-noise (S/N) for $z\approx 6$ galaxies. 2 arcsec diameter photometric apertures were used. Aperture magnitudes were corrected to total magnitudes assuming that the objects are spatially unresolved. This gives a lower limit on the flux for spatially extended objects.
Sample Selection
----------------
There are two methods for selecting high-redshift galaxies from optical/IR broad-band imaging, Lyman break and photometric redshifts. The Lyman break technique adopts hard color cuts in one or more colors possibly including non-detections in certain filters, whereas photometric redshifts use all available filters to derive a redshift probability distribution. Photometric redshifts are most suitable when the targets will be detected in many filters. However they reduce to a Lyman break-type selection if only a few filters are deep enough to constrain the relevant objects. In this work we use the Lyman break technique for homogeneous selection because of the variable near-IR data quality in the different Deep fields.
We set a magnitude limit of $z'<25.3$ to ensure that the objects are real and not too faint for good photometry. Over most of the survey area this limit corresponds to a $7\sigma$ detection in our 2 arcsec apertures. It will be shown in Section \[evidence\] that for the fields with the deepest near-IR data (D1 and D2) all 39 $z'$ selected objects (galaxies and brown dwarfs) have a counterpart at $i'$ and/or $J$, showing the rate of spurious $z'$ band detections in the sample is very low. The high z’ threshold ($\approx 7 \sigma$) minimizes the problem of photometric scatter of objects into our sample with true colors different from the selection criteria. The luminosity function has been well studied by others at magnitudes fainter than this limit. The primary selection criterion is color $i'-z'>2$ which corresponds to the break across the [Ly$\alpha$]{} line. This criterion is somewhat stricter than other studies (e.g. Bouwens et al. 2006; Jiang et al. 2011) but ensures that contamination from low redshift galaxies and brown dwarfs is kept to a minimum. Two of our objects lie in slightly less deep than average regions at $i'$ band and are undetected at $i'$ with measured limits of $i'-z'>1.97$ and $i'-z'>1.99$. These are included in the sample because there is a high likelihood they would have $i'-z'>2$ if deeper $i'$ data were available.
Most Lyman break surveys adopt two colour criteria and therefore define a box in two-dimensional color-color space. We adopt a similar, but slightly different, method. Instead of a hard cut-off in $z'-J$ color we consider all the properties of every source in the $i'-z'>2$ selection region and determine whether it is most likely a high-redshift galaxy or something else. This is important here because the CFHTLS Deep Fields contain varying amounts of extra multi-wavelength data which can be used as an additional constraint.
The initial automated search routine revealed 136 possible candidates brighter than the magnitude limit and having $i'-z'>2$ in the $\sim$ 4 square degrees CFHTLS Deep. The images were inspected by eye. Sometimes more detailed manual photometry for objects in locations with varying background was performed. Non-detections in the CFHTLS $u^*g'r'$ filters were also required for good LBG candidates. We checked that this criterion would not eliminate true LBGs using the model galaxy simulations to be described in Section \[completeness\] that account for the observed variation of galaxy and intergalactic medium (IGM) properties. It was found that only 0.02% of galaxies at $5.7<z<6.0$ have colors $r'-z'<3.5$ and hence could potentially be detected at $r'$ band (typical $2\sigma$ limit is $r'\sim 27.5$ to $28$) for the brightest galaxies that have $z'
\approx 24.5$. The most common problems leading to rejection from the sample were due to structured background near bright stars or close to the edges of the fields where the effective exposure time is lower. A total of 69 candidates were removed from the list in this process. This left 67 true astronomical sources with $i'-z'>2$.
For each of these, the available data was studied to determine its nature. The primary filter for this process is $J$ because brown dwarfs are known to have much higher $z'-J$ colors than high-redshift galaxies. 7 of the 67 sources do not have $J$ band coverage. The next most important is $Ks$ band because reddened galaxies at $z\approx 1$ would be expected to be bright at $Ks$. In addition, three of the fields have at least partial IRAC coverage available from SWIRE (Lonsdale et al. 2003), S-COSMOS (Sanders et al. 2007) and AEGIS (Barmby et al. 2008). The COSMOS field also has deep Subaru imaging with broad-band depth similar to CFHTLS, additional medium-band filters (Taniguchi et al. 2007) and HST F814W imaging (Scoville et al. 2007) which is useful for compact sources.
[l l c c c r r c]{} WMH 1 & 02:24:13.79 $-$04:56:41.4 & $27.18\pm0.26$ & $25.15\pm0.10$ & $> 25.00 $ & $2.02 $ & $< 0.15$ & 1.36\
WMH 2 & 02:24:15.10 $-$04:20:47.0 & $27.29\pm0.29$ & $25.14\pm0.10$ & $24.58\pm0.24$ & $2.14 $ & $0.56 $ & 0.92\
WMH 3 & 02:24:51.14 $-$04:03:29.4 & $27.42\pm0.32$ & $24.78\pm0.07$ & $24.57\pm0.24$ & $2.63 $ & $0.21 $ & 0.94\
WMH 4 & 02:25:19.64 $-$04:28:06.8 & $27.47\pm0.42$ & $25.21\pm0.13$ & $> 25.30 $ & $2.26 $ & $< -0.09$ & 0.88\
WMH 5 & 02:26:27.03 $-$04:52:38.3 & $27.47\pm0.34$ & $24.54\pm0.06$ & $24.17\pm0.17$ & $2.92 $ & $0.37 $ & 1.11\
WMH 6 & 02:27:18.77 $-$04:50:08.4 & $26.97\pm0.22$ & $24.85\pm0.08$ & $25.16\pm0.39$ & $2.11 $ & $-0.31 $ & 1.94\
WMH 7 & 02:27:29.03 $-$04:33:04.3 & $27.69\pm0.48$ & $25.26\pm0.11$ & $24.87\pm0.31$ & $2.42 $ & $0.39 $ & 1.45\
WMH 8 & 02:27:46.20 $-$04:30:32.2 & $27.46\pm0.34$ & $25.16\pm0.10$ & $24.35\pm0.20$ & $2.29 $ & $0.81 $ & 1.73\
WMH 9 & 09:58:45.49 +02:23:24.8 & $27.67\pm0.49$ & $25.27\pm0.12$ & $24.84\pm0.30$ & $2.39 $ & $0.43 $ & 1.21\
WMH 10 & 09:58:59.84 +01:59:48.8 & $27.42\pm0.37$ & $25.24\pm0.12$ & $25.16\pm0.39$ & $2.18 $ & $0.08 $ & 0.84\
WMH 11 & 09:59:44.49 +02:09:36.7 & $27.29\pm0.33$ & $25.06\pm0.10$ & $24.90\pm0.32$ & $2.23 $ & $0.16 $ & 1.02\
WMH 12 & 09:59:52.74 +02:25:53.2 & $27.34\pm0.41$ & $25.17\pm0.13$ & $25.03\pm0.35$ & $2.17 $ & $0.13 $ & 0.99\
WMH 13 & 09:59:56.54 +02:12:27.1 & $26.78\pm0.21$ & $24.78\pm0.08$ & $24.53\pm0.23$ & $2.00 $ & $0.25 $ & 0.86\
WMH 14 & 10:00:19.93 +02:25:36.8 & $26.98\pm0.26$ & $24.88\pm0.09$ & $24.83\pm0.07$ & $2.09 $ & $0.05 $ & 0.74\
WMH 15 & 10:00:26.37 +02:13:46.8 & $> 27.35 $ & $24.99\pm0.10$ & $24.68\pm0.07$ & $> 2.35$ & $0.31 $ & 1.24\
WMH 16 & 10:00:30.58 +02:19:35.1 & $27.56\pm0.42$ & $25.27\pm0.12$ & $24.51\pm0.07$ & $2.28 $ & $0.76 $ & 1.11\
WMH 17 & 10:00:50.10 +02:13:03.3 & $27.37\pm0.38$ & $25.17\pm0.12$ & $24.60\pm0.25$ & $2.20 $ & $0.57 $ & 1.05\
WMH 18 & 10:00:59.82 +01:57:26.6 & $> 27.27 $ & $25.24\pm0.15$ & $24.63\pm0.26$ & $> 2.02$ & $0.61 $ & 1.10\
WMH 19 & 10:01:18.99 +02:11:11.6 & $27.64\pm0.46$ & $25.15\pm0.11$ & $> 25.20 $ & $2.48 $ & $< -0.05$ & 1.29\
WMH 20 & 10:01:21.54 +02:12:22.5 & $26.83\pm0.22$ & $24.65\pm0.07$ & $24.14\pm0.17$ & $2.18 $ & $0.51 $ & 0.86\
WMH 21 & 10:01:21.94 +02:19:37.6 & $27.20\pm0.30$ & $25.12\pm0.10$ & $24.15\pm0.17$ & $2.07 $ & $0.97 $ & 1.25\
WMH 22 & 10:01:38.37 +01:43:48.1 & $27.43\pm0.45$ & $25.28\pm0.13$ & $> 25.20 $ & $2.14 $ & $< 0.08$ & 0.81\
WMH 23 & 10:01:38.71 +02:28:20.0 & $> 27.16 $ & $25.05\pm0.12$ & $24.31\pm0.19$ & $> 2.10$ & $0.74 $ & 1.32\
WMH 24 & 14:16:20.38 +52:13:23.4 & $> 27.07 $ & $24.72\pm0.07$ & — & $> 2.34$ & — & 1.03\
WMH 25 & 14:17:47.97 +52:38:09.4 & $> 27.28 $ & $25.04\pm0.10$ & $> 25.19 $ & $> 2.23$ & $< -0.15$ & 1.53\
WMH 26 & 14:18:10.47 +52:19:43.4 & $27.79\pm0.45$ & $25.13\pm0.10$ & — & $2.65 $ & — & 1.24\
WMH 27 & 14:18:40.04 +52:35:08.2 & $27.20\pm0.32$ & $25.09\pm0.10$ & $> 24.61 $ & $2.10 $ & $< 0.48$ & 1.15\
WMH 28 & 14:18:52.23 +53:07:47.3 & $27.31\pm0.29$ & $25.23\pm0.11$ & $> 24.16 $ & $2.08 $ & $< 1.07$ & 1.00\
WMH 29 & 14:19:20.53 +52:52:38.7 & $26.52\pm0.14$ & $24.41\pm0.05$ & $24.42\pm0.05$ & $2.10 $ & $-0.01 $ & 1.11\
WMH 30 & 14:20:40.80 +52:52:45.3 & $27.47\pm0.34$ & $24.72\pm0.07$ & $> 24.75 $ & $2.75 $ & $< -0.03$ & 1.08\
WMH 31 & 14:21:03.74 +52:12:16.2 & $26.90\pm0.21$ & $24.87\pm0.08$ & — & $2.02 $ & — & 1.08\
WMH 32 & 22:13:42.68 $-$17:56:33.2 & $27.42\pm0.38$ & $25.01\pm0.10$ & $> 24.44 $ & $2.41 $ & $< 0.57$ & 0.77\
WMH 33 & 22:14:06.97 $-$17:37:59.7 & $> 27.21 $ & $25.24\pm0.13$ & — & $> 1.96$ & — & 0.73\
WMH 34 & 22:14:46.63 $-$17:39:22.2 & $27.31\pm0.35$ & $24.98\pm0.11$ & — & $2.33 $ & — & 1.02\
WMH 35 & 22:15:04.28 $-$17:57:20.5 & $> 27.18 $ & $25.03\pm0.12$ & $> 24.48 $ & $> 2.14$ & $< 0.55$ & 1.44\
WMH 36 & 22:15:48.91 $-$17:35:45.7 & $27.30\pm0.34$ & $25.27\pm0.14$ & $> 24.85 $ & $2.02 $ & $< 0.42$ & 1.17\
WMH 37 & 22:16:26.67 $-$18:04:45.0 & $> 27.25 $ & $25.26\pm0.13$ & $> 24.12 $ & $> 1.98$ & $< 1.14$ & 0.69\
WMH 38 & 22:16:38.04 $-$17:37:00.9 & $27.50\pm0.44$ & $25.21\pm0.14$ & $> 24.85 $ & $2.29 $ & $< 0.36$ & 0.73\
WMH 39 & 22:17:08.83 $-$17:38:43.9 & $26.86\pm0.24$ & $24.61\pm0.08$ & $24.57\pm0.26$ & $2.25 $ & $0.04 $ & 1.53\
WMH 40 & 22:17:13.55 $-$18:11:45.7 & $27.15\pm0.34$ & $25.03\pm0.10$ & — & $2.11 $ & — & 0.80
As discussed in the following section, the results of this process showed that 40 of the 67 candidates were most likely $z \approx 6$ Lyman break galaxies and the remaining 27 are most likely to be brown dwarfs.
Evidence for $z\approx 6$ galaxies {#evidence}
----------------------------------
Three-colour ($r'i'z'$) images for the the 40 candidates classified as high-redshift galaxies are shown in Figure \[fig:colorimages\]. Photometric measurements in the $i'z'J$ filters are given in Table \[tab:photom\]. In the case that an object is not detected, a magnitude limit is given. These magnitude limits are determined from the variance of photometric aperture fluxes at random locations close to the object. At $i'$ band, many sources appear to be marginally detected. Therefore $i'$ magnitudes are quoted down to magnitude errors of 0.5 magnitudes, rather than the strict $2\sigma$ limits which are often brighter. This should be kept in mind if using $i'$ magnitudes or $i'-z'$ colors from Table \[tab:photom\]. Note that six of the 40 galaxies do not have $J$ band observations. The evidence in favor of these being high-redshift galaxies will be discussed later. The final column gives the measured FWHM in the $z'$ band. Many of the galaxies appear more extended than the seeing, a fact that will be discussed later.
Figure \[fig:izjcol\] shows the distribution of points on the $i'-z'$ vs $z'-J$ color-color diagram for the 60 of 67 candidates which are covered by $J$ band observations. Almost all the sources detected at $J$ band have colors close to the expected loci of $z>5.7$ Lyman break galaxies or brown dwarfs. The derivation of the brown dwarf locus is given in Delorme et al. (2008). Close to the $i'-z'=2$ boundary the dwarfs tend to lie below the line. A similar tendency was noted for the Canada-France Brown Dwarf Survey (Reylé et al. 2010) and is at least partially due to photometric scatter of somewhat bluer dwarfs with intrinsic $i'-z'\approx 1.5$. Note that a similar effect will occur for the high-redshift galaxies. This is why we began with a relatively high $i'-z'$ cut, so that even those scattered from lower $i'-z'$ have intrinsic colors matching only brown dwarfs or high-$z$ galaxies. The brown dwarfs in the CFHTLS/WIRDS survey also have well-defined loci in the $z'-J$ vs $J-H$ and $J-H$ vs $H-Ks$ diagrams with no significant outliers.
The gray circles show colors of model galaxies at $0.5<z<3$ determined using the 2008 stellar population synthesis models of Charlot & Bruzual (priv. comm.). Passively evolving and dusty star-forming galaxies with ages ranging from a million years to the age of the universe at that redshift are included. Extreme dust extinction with optical depth up to $\tau=10$ is included, because extreme dust is required to get such large $i'-z'$ colors. Note that due to the larger wavelength difference between $z'$ and $J$ than between $i'$ and $z'$, the effect of extreme dust reddening is to push low-$z$ galaxies into the brown dwarf region of the diagram. A combination of extreme reddening plus high photometric scatter would be required to obtain the colors of our sample. The low-$z$ galaxy models closest to the location of the LBGs have $z \approx 1.2$ and $\tau \approx 10$. Such galaxies would be extremely rare and faint.
21 of the 34 LBG-classified targets with $J$ band data are detected at $J$. The other 13 have $z'-J$ limits well separated from the regions covered by brown dwarfs and reddened $0.5<z<3$ galaxies. We expect almost all the red squares in Figure \[fig:izjcol\] to be high-redshift galaxies as discussed below.
As noted in Table 1, many of the LBGs appear spatially resolved at $z'$ band. Section \[sizes\] discusses this in much more detail and it is concluded that at least half of the LBGs are spatially resolved from the ground. In addition, observations of a subset also show that they are resolved. This provides further evidence against contamination by brown dwarfs.
A final source of contamination to consider is that by quasars at $z\approx 6$ which will have similar colors. A search for such quasars going to magnitude $z'<24.5$ found none in the CFHTLS Deep survey area (Willott et al. 2010), but one at $z=6.01$ with magnitude $z'=24.4$ in the SXDS/UDS (McLure et al. 2006; Willott et al. 2009). At such faint magnitudes, the quasar luminosity function slope is likely to be fairly flat (Willott et al. 2010), so the expectation would be for $\approx 1$ more quasar in the magnitude range $24.5<z'<25.3$ across the full survey region.
Sources without $J$ photometry
------------------------------
7 of the 67 $i'-z'>2$ sample do not have $J$ photometry because they lie outside the WIRDS and VISTA surveys and their nature, high-$z$ galaxy or brown dwarf, had to be ascertained using other available information. Given the ratio of galaxies to brown dwarfs in the regions with $J$ photometry we would expect 4 of these to be galaxies and 3 to be brown dwarfs.
One object is in the D4 field and a WIRCam $Ks$ band image provided by Genevieve Soucail shows it to have $Ks=22$. This red color of $z'-Ks=2.8$ is consistent with a L dwarf.
The six remaining objects are retained in the list of likely $z
\approx 6$ galaxies. WMH24, WMH26, WMH31 and WMH34 have measured $z'$ band FWHM $> 1$ arcsec and therefore appear to be spatially resolved. As will be discussed in Section \[sizes\], only 15% of brown dwarfs are found to have such high FWHM. The probability that none of these four are brown dwarfs based on the FWHM argument is therefore 60%. WMH33 has FWHM=0.73, so is consistent with being unresolved. However it is faint at $z'$ band with $z'=25.25$ and most sources close to the detection limit are galaxies. It is not detected in a WIRCam $Ks$ band image that reaches to $Ks=22$. If it were a L dwarf it would be expected to have $Ks \approx 22$ to $22.5$, so this limit, although not definitive, suggests that it may be a galaxy. Therefore it is retained in the galaxy sample. WMH40 has FWHM=0.80 arcsec and $z'=25.05$. It is retained within the galaxy sample but we note it has almost equal probability to be a high-$z$ galaxy, or a brown dwarf with these properties. In conclusion, at least four of these six objects are likely high-$z$ galaxies. Taking into consideration there could be one $z\approx 6$ quasar in our sample, the contamination of our $z\approx 6$ galaxy sample is most likely $<10$%.
Spectroscopy {#spectra}
============
Observations
------------
The photometrically-selected high-$z$ galaxies are spread across nearly 4 square degrees. Given that the typical field-of-view of multi-object spectrographs on large telescopes is $\ll 1$ degree, there is little opportunity for a high-multiplex factor. Spectroscopic observations have been attempted for 7 of the 40 candidates. Priority was given to those which are brightest at $z'$ band to minimise the integration time required. All spectroscopy was performed using the GMOS spectrographs on the 8.2m Gemini Telescopes. The R400 gratings were used with 1 arcsec slits to give a resolution of $R=1000$. CCD pixels were binned by a factor of two in the spectral direction so that each binned pixel covers 1.34Å. WMH13 was observed in long-slit mode, WMH 29 was observed equally in long-slit and multi-object mode, and all other observations used multi-object mode. Occasionally, two $z\approx
6$ galaxies could be fit on the same mask. The masks were filled with lower redshift Lyman break galaxies and photometric redshift selected galaxies ($z>3.5$). All observations used the nod-and-shuffle mode to ensure good subtraction of the sky background spectrum.
[l c c c c c c]{} WMH 5 & 2.5 & $-22.65$ & $6.068$ & A & $13 \pm 4$ & $220 \pm 30$\
WMH 6 & 2.5 & $-21.75$ & $5.645$ & C & – & –\
WMH 13 & 2.5 & $-22.06$ & $5.983$ & A & $27 \pm 8$ & $210 \pm 50$\
WMH 15 & 4.0 & $-21.98$ & $5.847$ & C & – & –\
WMH 29 & 5.0 & $-22.27$ & $5.757$ & B & $4 \pm 2$ & unres.?\
WMH 34 & 2.3 & $-21.67$ & $5.759$ & C & – & –\
WMH 39 & 2.0 & $-22.06$ & $5.733$ & B & – & –\
Serendip & 2.5 & $-21.33$ & $5.618$ & A & $31 \pm 11$ & $240 \pm 40$
The long-slit observations were performed in queue-mode in excellent conditions in 2009. The multi-object observations were performed in classical mode on the nights of 12 December 2010, 27 and 28 June 2011 in variable conditions. Details of the spectroscopic observations and results are given in Table \[tab:spec\]. Due to the faintness of these targets, the typical continuum S/N per pixel between 8500Å and 9000Å was only $\sim 0.5$. Therefore, redshifts could only be quickly and unambiguously determined for the few sources which showed very strong [Ly$\alpha$]{} emission lines. However, even at S/N $=0.5$ per pixel, the continuum can be significantly detected over broader wavelength intervals and redshifts determined from continuum breaks (Spinrad et al. 1998).
Model fitting
-------------
To fully utilize the spectral information, a routine was developed to fit the observed spectrum to model templates. The galaxy models used are based on the observed composite $z=3$ Lyman break galaxy spectrum of Shapley et al. (2003). Two galaxy models are used: one with a strong [Ly$\alpha$]{} emission line and one with only [Ly$\alpha$]{} absorption. The fitting routine allows a combination of these models to represent a range in [Ly$\alpha$]{} strength. IGM absorption is accounted for using the model of Songaila (2004). The galaxy spectrum is multiplied by a power-law to account for the variation in observed UV continuum slope of galaxies. There are a total of four free parameters: normalization, redshift, power-law slope and [Ly$\alpha$]{} flux. The [Ly$\alpha$]{} flux was fixed at zero for those galaxies without obvious [Ly$\alpha$]{} emission in their 2D spectra to avoid over-fitting of noise peaks with the [Ly$\alpha$]{} line.
Best-fit galaxy models are determined by fitting the models to the observed spectra. The uncertainty on each spectral pixel due to the sky noise is determined from the variance of blank sky pixels in each of the masks. The best fit is determined by the lowest value of the reduced $\chi^2$. The spectra, along with best fit galaxy models, are shown in Figure \[fig:spec\]. The observed spectra, models and sky noise spectrum have been smoothed by 20 pixels (27Å) for display purposes. Those sources with measurable [Ly$\alpha$]{} emission have an inset panel in Figure \[fig:spec\] with a 220Å long segment of the 2D spectrum centred on the [Ly$\alpha$]{} emission.
Results of spectroscopy
-----------------------
Each target has a redshift [*Quality Flag*]{} in Table \[tab:spec\] which gives a measure of the confidence of the redshift determined. The categories follow those in Vanzella et al. (2009) where ‘A’ means unambiguous, ‘B’ is likely and ‘C’ is uncertain. Strong [Ly$\alpha$]{} emission was only found in two of the $z\approx 6$ galaxies, WMH5 and WMH13. These are assigned QF=A. In addition, weak, but still significant, [Ly$\alpha$]{} was observed in WMH29 which is assigned QF=B. Note that this [Ly$\alpha$]{} line at $z=5.757$ is in a region of very low sky noise. It is doubtful that such a line would have been significantly detected at slightly higher redshift. Due to the weakness of this line it cannot be determined if it is asymmetric and therefore definitely [Ly$\alpha$]{}. If it is instead [\[O[ii]{}\]]{} at $z=1.20$ then the 4000Å break would be expected at 8790Å, whereas the continuum break is observed to be at $<8500$Å. The apparent emission line at 8400Å in WMH15 in Figure \[fig:spec\] is found to be due to sky lines upon examination of the full resolution spectrum.
One of the serendipitous mask-filling galaxies with photometric redshift $z>3.5$ was also found to have strong [Ly$\alpha$]{} at $z=5.618 $, just below the redshift range of our sample. It is labeled “Serendip” and is the last object in Table \[tab:spec\] and Figure \[fig:spec\]. It has QF=A and is located at 02:26:37.02 $-04$:55:20.4. This galaxy has $i'-z'=1.1$, well below our selection criterion of $i'-z'=2.0$. The reason for this much lower color is that the strong [Ly$\alpha$]{} line is still within the $i'$ filter at this redshift. When an emission line of this strength shifts from the $i'$ band to the $z'$ band (at $z\approx 5.8$) it increases the $i'-z'$ color by 0.5.
All four of the galaxies with [Ly$\alpha$]{} emission show significant spectral breaks across the [Ly$\alpha$]{} line. All four have best fit galaxy spectra at redshifts equal to their [Ly$\alpha$]{} redshifts. This is not too surprising for those three with strong [Ly$\alpha$]{} lines, because inability to fit the line would increase the $\chi^2$. However it is encouraging for WMH 29, which has only weak [Ly$\alpha$]{}, where the continuum break contributes most of the weight in the fit. For the four galaxies without [Ly$\alpha$]{}emission, we depend upon continuum breaks to constrain the redshift. WMH39 also shows a break at $z=5.73$ and is assigned QF=‘B’ based on this break. WMH15 shows a likely continuum break between $z=5.85$ and $z=5.95$, however this break is not so clear as that in WMH39 leaving the redshift uncertain and hence QF=‘C’.
The other two galaxies, WMH6 and WMH34, show continuum flux in the $z'$ band and regions with no flux at shorter wavelength, but the exact break redshifts are not well-constrained, so are assigned QF=‘C’. These two spectra have the highest noise (along with WMH13 which has the strongest [Ly$\alpha$]{} line) which likely accounts for the lack of a clear redshift solution. The best fit redshift of $z=5.645$ for WMH6 is quite unlikely given its color of $i'-z'=2.1$ (see Section \[completeness\]). We note that in any color-selected sample there is a possibility for rare interlopers with unusual colors (e.g. Capak et al. 2011; Hayes et al. 2012) to masquerade as high-redshift galaxies. However, based on the evidence we have and the stringent color cuts, we expect there to be very few of these in our sample. Only WMH13 out of the seven color-selected spectroscopic targets has a [Ly$\alpha$]{} line with rest-frame equivalent width $>25$Ågiving a fraction of 14%. This is comparable to the fraction of $20
\pm 8$% for luminous $z\sim 6$ LBGs found by Stark et al. (2011) and lower than the $54 \pm 21$% observed by Curtis-Lake et al. (2012).
Table \[tab:spec\] includes the absolute magnitudes, $M_{1350}$ of the galaxies. These were calculated from the $z'$ band magnitudes after taking account of the observed [Ly$\alpha$]{} emission line contributions. At these redshifts, the $z'$ band contains flux at or very close to rest-frame 1350Å and hence the main uncertainty on the absolute magnitudes comes not from $k$-corrections, but the $z'$ band photometry and is $\approx 10$%. With absolute magnitudes ranging from $-21.67$ to $-22.65$, the spectroscopic targets are some of the most luminous LBGs known at $z\sim 6$. For comparison, the break in the $z\approx 6$ luminosity function is at $M_{1350}=-20.2$ (Bouwens et al. 2007). WMH5 has $M_{1350}=-22.65$, which is even more luminous than the lowest known luminosity $z\approx 6$ quasar, CFHQSJ0216-0455 ($M_{1450}=-22.21$, Willott et al. 2009). WMH5 is only 2.1 arcsec from the center of a galaxy in Figure \[fig:colorimages\]. This is an inclined disk galaxy with a photometric redshift of $z=1.01$, rest-frame $B$-band absolute magnitude of $M_B=-21.5$ and disk half-light radius $4$kpc. Using the results of studies of the Tully-Fisher and similar relations for $z=1$ disk galaxies (Dutton et al. 2011; Miller et al. 2011), we find the circular velocity for this galaxy should be $\approx
160$kms$^{-1}$. Modeling the gravitational potential as an isothermal sphere, the gravitational lensing magnification of WMH5 due to this galaxy is a factor of 1.27. Hence the intrinsic absolute magnitude of WMH5 is still extremely luminous with $M_{1350} \approx
-22.4$.
Galaxy sizes {#sizes}
============
An important cosmological observation is that galaxies are smaller at higher redshifts than similar galaxies at lower redshift (Ferguson et al. 2004; Bouwens et al. 2004). This is expected due to the evolution in the virial radii of dark matter halos, but the exact nature of the evolution also depends upon details of gas accretion, retention and star formation efficiency. The faint $z\sim6$ galaxies discovered in GOODS and HUDF typically have half-light radii of $\approx 0.1$ arcsec (equivalent to 0.6kpc; Bouwens et al. 2006) and are therefore not expected to be resolved in typical seeing-limited, ground-based observations. Table \[tab:photom\] showed that the measured $z'$ band FWHM of many CFHTLS $z\approx 6$ LBGs are larger than the seeing of $\approx 0.7$ arcsec of the images.
Four (WMH14, WMH15, WMH16 and WMH29) of the 40 LBGs (including two with spectroscopy) are located within the CANDELS survey and therefore have high-resolution ACS and WFC3 imaging available (Grogin et al. 2011; Koekemoer et al. 2011). We have retrieved these data in order to get a more detailed view of the galaxy morphologies for this small sub-sample. The WFC3 F125W and F160W images are shown in Figure \[fig:wmhgalfit\]. We have used [GALFIT]{} (Peng et al. 2010) to fit galaxy models to the main galaxy component in each filter independently. The best fit single model galaxy and the residuals after subtraction are also plotted in Figure \[fig:wmhgalfit\]. Similar models and residuals are found for both filters in all cases, providing a high degree of confidence in the model galaxy fits. Inspection of the F814W images showed that the residuals have similarly red F814W-F125W colors as the main galaxies, indicating the residuals are at $z\approx 6$ and not at lower redshift.
For WMH14 the residual emission is compact and co-incident with the main galaxy. For the other three galaxies, the residual emission is more extended and clumpy, extending at least an arcsec (equivalent to 6kpc) from the main galaxy centroid. This is suggestive of galaxy mergers and interactions being common in the most luminous galaxies at this epoch. The model galaxies have a wide range of half-light radii from only 0.3 and 0.5kpc for WMH14 and WMH29, respectively, to 1.0 and 1.5kpc for WMH15 and WMH16, respectively.
Six of the 15 galaxies in D2 are well-detected in the ACS F814W observations of the COSMOS field. They are all spatially resolved and half of them show multiple components. The F814W filter extends further redward than the CFHT $i'$ filter, so such detections in F814W are consistent with the $z\approx 6$ LBG classification. The frequent occurrence of multiple components in both the WFC3 and ACS data indicate that about half of the LBG sample have signs of interactions/mergers.
The only size data available for the full sample are the measured FWHM in the $z'$ band. Figure \[fig:size\] plots FWHM against magnitude for the 40 LBGs and 27 brown dwarfs. The brown dwarfs are included as a comparison sample, because they are expected to be almost all unresolved. Although many brown dwarfs exist in binary systems (Burgasser et al. 2007) only a small fraction would have the relevant component separations to appear as a single, resolved source in these images. The spread in observed FWHM for dwarfs is due to three factors: the small difference in quality between the four Deep fields, the variation in stellar FWHM from the field centers to the edges and photometric noise. The photometric noise is evident by the increased spread in FWHM for dwarfs at fainter magnitudes. It is for this reason that we use the dwarfs as the unresolved comparison sample to compare to the LBGs rather than stars at brighter magnitudes.
Figure \[fig:size\] also shows the 25%, 50% and 75% centiles of the measured FWHM distribution as a function of magnitude for simulated point sources inserted into the data (see Section \[completeness\]). The observed distribution for the dwarfs agrees well with the simulation. Note that the three dwarfs with measured FWHM $>1$ arcsec all have FWHM $\approx 0.6$ in $J$ band imaging with higher S/N than $z'$ band for these red sources, confirming photometric noise as the reason for a small fraction of large FWHM values. It is found that 3 out of 20 (15%) dwarfs with magnitudes in the range of LBGs are observed to have FWHM $>1$ arcsec and lie beyond the 75% simulated centile due to noise. In contrast, we find that 26 out of the 40 LBGs (63%) have FWHM $>1$ arcsec and that the same percentage lie beyond the 75% centile. Assuming that 15% of these LBGs are outliers due to noise, then there remain $\approx 50$% which are truly resolved in our ground-based imaging.
Given the uncertainty in the broadening of the FWHM values by seeing and noise, it is not possible to reliably deconvolve each value to determine the intrinsic size distribution of our LBG sample. However, a typical intrinsic half-light radius of 2kpc (0.35 arcsec) is required to observe a galaxy with FWHM 1.1 arcsec (median for our LBG sample) when the FWHM of a point source would be 0.85 arcsec (50% centile for the simulated point sources with magnitude $z'=25.0$). This contrasts with the typical half-light radius of 1kpc for $z=26$ galaxies in GOODS (Bouwens et al. 2006). The high-resolution WFC3 imaging of four LBGs showed that one is compact, one has a compact core with low surface brightness extended emission and two have relatively large half-light radii plus extended emission. This suggests that both large galaxy sizes and multiple emission components are common in our sample of the most luminous LBGs. Our finding is consistent with an extension to the brightest galaxies of the strong correlation between UV luminosity and linear size at $z=6$ presented by Dow-Hygelund et al. (2007).
Stacked galaxy properties {#stacked}
=========================
In order to check that the typical LBG properties match those of high-redshift galaxies and to get an idea of the typical galaxy spectral shape, we have stacked the photometric data at the positions of the $z \approx 6$ galaxies. After registering the images, the median flux of each pixel was determined since it provides the most robust estimator that is least affected by outliers (White et al. 2007). Because the near-IR data from VISTA has an extra filter ($Y$) and is deeper than WIRDS, we just use the 23 galaxies in D1 and D2 with VISTA data. We have checked that the stacked optical and near-IR magnitudes for D3 and D4 are consistent. Deep IRAC data is available for the 15 galaxies in D2 from the S-COSMOS survey (Sanders et al. 2007). We include these data because these wavelengths are important for constraining the stellar mass and age, but the reader should remember that the IRAC points only correspond to a subset of the objects used for the $r'i'z'YJHKs$ stacks.
The resulting median stacked images are shown in Figure \[fig:imstack\]. There are significant detections at all filters except for $r'$ band. Photometry of the ground-based data was performed to obtain 3 arcsec aperture magnitudes or limits. These are equivalent to total magnitudes, within the uncertainties. These 3 arcsec apertures are larger than the 2 arcsec used previously in this paper for individual object magnitudes (many of which had low S/N necessitating small apertures) and provide a better match to the IRAC apertures. For the IRAC data which has a broader PSF, 3.8 arcsec apertures were used and a PSF correction to total magnitudes applied. Smoothing of the images showed that for all the optical filters, there is a negative “hole” in the background covering the central five arcseconds. This is most apparent for $r'$ and $i'$ in Figure \[fig:imstack\]. This is showing us that our $z=6$ galaxy sample is incomplete. The CFHTLS Deep $r'$ and $i'$ band data is so deep that it is close to the confusion limit and we are missing $z=6$ galaxies that have foreground galaxies contaminating the photometry aperture in $r'$ and $i'$. Another consequence of this is that we are unlikely to have strongly gravitational lensed galaxies in our sample due to the lensing galaxy contaminating the source galaxy aperture. Photometric measurements were performed carefully to ensure that the appropriate background level was set. The measured magnitudes of the stacks are given below the images in Figure \[fig:imstack\]. The 2$\sigma$ limit on the $r'$ band magnitude (determined from the noise in the background pixels because this region has lower variance than the rest of the field which was populated by galaxies in the input images) gives $r'>29.8$. This is almost 5 magnitudes fainter than the $z'$ band magnitude and provides compelling evidence that these galaxies are truly at $z\approx 6$ rather than lower redshift galaxies or dwarf stars. The stack colors of $i'-z'=2.44 \pm 0.14$ and $z'-J=0.29 \pm 0.06$ are plotted in Figure \[fig:izjcol\] and are entirely consistent with a typical $z=5.9$ galaxy.
Figure \[fig:sedstack\] plots the stack fluxes as a function of wavelength. The $z'$ flux has been reduced by a factor of 1.1 to account for the typical [Ly$\alpha$]{} emission line contribution. It is apparent that the $YJHKs$ fluxes are described by a power-law with index (defined as $f_\lambda \propto \lambda^{\beta}$) redder than $\beta=-2$ (which would be flat on this plot which uses $f_\nu$). A power-law fit to the $YJHKs$ fluxes gives $\beta=-1.44 \pm 0.10$, where the $1\sigma$ uncertainty comes from 1000 bootstrap resample trials. To ensure that this result is not an artifact of the stacking method, we have also fitted the $YJHKs$ fluxes of the 23 LBGs individually. Excluding two extreme outliers which have poor fits, it is found that the mean $\beta=-1.38$ and the median $\beta=-1.02$. The standard error on the mean is $\pm 0.20$.
The value of $\beta$ found is significantly redder than the typical values previously observed for the most luminous $z\approx 6$ LBGs in deep surveys of $\beta=-1.78 \pm 0.11$ (Bouwens et al. 2012), $\beta=-2.10 \pm 0.16$ (Dunlop et al. 2012), $\beta=-2.04 \pm 0.17$ (Wilkins et al. 2011) and $\beta=-2.05 \pm 0.11$ (Finkelstein et al. 2012). Finkelstein et al. showed that although at $z\approx 6$ there is only a weak correlation between $\beta$ and UV luminosity, there is a stronger correlation between $\beta$ and stellar mass. Whilst this is not surprising, because in a UV flux-limited sample, a redder galaxy model will require a higher stellar mass, the typical value of $\beta$ for their most massive bin ($10^9$ to $10^{10}$M$_\odot$) is $-1.78$ and the typical stellar mass of our sample is $10^{10}$M$_\odot$ (see below). The simplest explanation for dust in luminous galaxies comes from combining the metallicity–dust, mass-metallicity and mass-luminosity correlations. Therefore our typical $\beta=-1.4$ for $z\approx 6$ luminous LBGs is not at odds with the work of Finkelstein et al. Our findings suggest that a substantial number of these galaxies have significant dust reddening.
Galaxy models were fitted to the stacked photometry using the photometric redshift method of McLure et al. (2011). The fitted models used Bruzual & Charlot (2003) stellar populations with the Chabrier IMF and the Calzetti et al. (2000) dust attenuation law. Various star formation histories were fitted, but none were preferred at a high significance. Therefore we quote the constant star formation rate results, since these most-luminous LBGs are unlikely to be observed in a state of rapidly declining or increasing star formation rate. Figure \[fig:sedstack\] shows the best-fit model and its parameters. The best fit redshift was found to be $z=5.95$ with a $2\sigma$ range of $5.87<z<5.97$. The best-fit metallicity is solar. The best-fit stellar mass, $M_\star$, is $1.1\times 10^{10}{\rm M}_\odot$ with a $2\sigma$ range of $ 4\times 10^{9} < M_\star < 1.9\times 10^{10}{\rm
M}_\odot$. The age and star formation rate are obviously degenerate with the star formation history and dust reddening, so no strong constraints could be placed on those parameters. As was found in the analysis of $\beta$, the UV spectral slope is fairly red and this requires significant dust reddening. The best fit has dust extinction of $A_{V}=0.75$ and the $2\sigma$ range is $0.48<A_{V}<1.48$.
Galaxy luminosity function {#lumfun}
==========================
Completeness
------------
There are a number of factors that affect the completeness of the sample, or equivalently, the effective volume of the survey. Firstly, the sample does not contain every object brighter than a certain $z'$ magnitude limit due to incompleteness in the source detection algorithm close to the limit. In addition, blending with brighter objects becomes an issue in ground-based data at these faint magnitudes. As was shown in the stacks of Section \[stacked\], there is evidence for incompleteness of the sample due to foreground contamination at $i'$ band which prevents some true high-redshift galaxies from having $i'-z'>2$ in their aperture magnitudes.
In order to determine the effective sky area surveyed, we have carried out an analysis of insertion and recovery of simulated sources into the data. As found in Section \[sizes\], about 50% of the $z=6$ LBGs are spatially resolved, whereas 50% are consistent with point sources. The simulated sources have a similar distribution of observed FWHM to the LBGs. The selection criteria for recovery are identical to those of the automated candidate selection. In addition, we check whether there is a contaminating object at $i'$ band which would cause the observed $i'-z'$ color to fall below the threshold. Figure \[fig:zcomplete\] shows the effective sky area surveyed in all four Deep fields as a function of the input magnitude of the artificial galaxies. The dashed line shows the effective area if one does not consider the foreground contamination of the $i'-z'$ color. The effective area declines slowly from $z'=23$ to $z'=24.8$ and then falls more rapidly to $z'=25.5$. Note that the effective area is still considerable for input magnitudes $25.3<z'<25.5$, so some galaxies fainter than the nominal magnitude limit will be in the sample due to photometric scatter.
The next factors to consider for completeness are the effects of the $i'-z'>2$ color criterion and the conversion from $z'$ magnitude to absolute magnitude. These are assessed by considering the range of properties expected for high-redshift galaxies, including [Ly$\alpha$]{} emission line strength, UV continuum spectral slope and IGM absorption of the UV continuum. As in Section \[spectra\] we use the $z=3$ LBG composite of Shapley et al. (2003) as the basis for the galaxy template. Absorption shortward of [Ly$\alpha$]{} due to foreground neutral hydrogen is modeled as in Willott et al. (2010) using the mean and scatter derived by Songaila (2004).
A range of UV spectral slopes is implemented, as required by observations. At $z\approx 6$ and high luminosity the reported typical value in the literature is $\beta=-2$ (Bouwens et al. 2012; Finkelstein et al. 2012). Our stacking and individual LBG fitting results showed that for even more luminous $z\approx 6$ LBGs, the typical $\beta=-1.4$. Combining our results with other work, we assume a typical value for the population of $\beta=-1.8$ and a Gaussian distribution with $1\sigma$ scatter of $0.5$. We have checked and using a mean of $\beta=-1.5$ or $\beta=-2$ makes no difference to our results.
Several spectroscopic studies have investigated the distribution of [Ly$\alpha$]{} equivalent widths at high redshifts. We combine the works of Stark et al. (2011), Curtis-Lake et al. (2012) and our own work in Section \[spectra\] to derive a [Ly$\alpha$]{} rest-frame equivalent width probability distribution for luminous $z\approx 6$ LBGs of $P(W_{{\rm
Ly}\alpha}) \propto \exp(-W_{{\rm Ly}\alpha}/25)$. This is similar to the exponential parameterization of Dijkstra & Wyithe (2012), also based on the data of Stark et al.
A sample of 500 simulated galaxies were generated with properties randomly drawn from these distributions and their $i'z'J$ colors and absolute magnitudes, $M_{1350}$, determined if located at all redshifts between $z=5.5$ and $z=6.7$. Photometric scatter in the colors is also included. The median $i'-z'$ and $z'-J$ color as a function of redshift is plotted in Figure \[fig:izjcol\]. This curve runs very close to the stacked values of $i'-z'$ and $z'-J$, suggesting the models are indeed a good representation of the data. At redshifts $5.6\leq z \leq 6$, some fraction of galaxies are too blue in $i'-z'$ to be included in the sample. In particular, we find that galaxies with strong [Ly$\alpha$]{} lines at $z<5.9$ are much less likely to be included in our sample than those with weak [Ly$\alpha$]{} lines. This could be the reason why we find a relatively low [Ly$\alpha$]{} fraction for the galaxies with best-fit redshifts $z \approx 5.8$ (c.f. Curtis-Lake et al. 2011). At $z>5.9$, where this bias does not exist, the fraction with strong [Ly$\alpha$]{} in our sample is quite high (two out of three). Figure \[fig:colcomplete\] shows the completeness as a function of redshift due to the color selection criterion.
Magnitude distributions {#magdistrib}
-----------------------
Before using these data to derive the $z=6$ galaxy luminosity function, we consider simply the observed $z'$ band magnitude distributions of the galaxies and brown dwarfs. Figure \[fig:maghist\] shows these two distributions. The dwarf magnitudes extend to $z'=22.8$ and have a fairly flat distribution with a peak at $z\approx 25.0$. This flat distribution is a consequence of galactic structure and the survey completeness as a function of magnitude. At magnitude $z'=25$, mid-L and mid-T dwarfs are 500 and 200 pc away, respectively. The CFHTLS fields are at high galactic latitude and hence the space density of brown dwarfs declines beyond the disk. The galaxy distribution is quite different and much steeper. There are no galaxies brighter than $z'=24.4$ and almost 70% of the galaxies lie in the small magnitude range of $25<z'<25.3$. The steepness of this distribution, despite the declining survey completeness at faint magnitudes, is due to the very steep bright end of the galaxy luminosity function.
The gray histogram in Figure \[fig:maghist\] shows the expected magnitude distribution for $z=6$ LBGs in our survey based on the evolving luminosity function in Section 5.3 of Bouwens et al. (2008). At $3.5 < z < 6.5$ this luminosity function is constrained by the LBG luminosity functions determined in Bouwens et al (2007) based on ACS observations in the GOODS, HUDF and HUDF-Parallel fields. The expected counts take into account the selection criteria, completeness and photometric uncertainty of our sample. It is clear from Figure \[fig:maghist\] that both the shape and overall normalization of the expected counts are close to those observed in our sample. This is remarkable given that our survey volume is 40 times greater than that used by Bouwens et al. (2007) and hence their data contained very few sources as bright as ours. The expected magnitude distribution predicts 46 LBGs in our sample, compared with 40 observed. The number expected at $z'<25$ is 13, identical to that observed.
Luminosity function derivation
------------------------------
Because our LBG sample covers only a limited range of apparent and absolute magnitudes, we cannot use it to determine the full galaxy luminosity function at $z=6$. The luminosity function at the break and at fainter magnitudes has already been well-studied from deep surveys over small sky areas (Bouwens et al. 2007). The main contribution of our study is to determine the space density of very rare, highly luminous LBGs.
Due to the fact that most CFHTLS LBGs have unknown redshifts (within the Lyman break redshift selection range) and unknown $k$-corrections due to [Ly$\alpha$]{} emission line contribution and spectral slope, we cannot map a particular apparent magnitude onto an absolute magnitude and redshift. Therefore we cannot carry out standard luminosity function derivation methods where each observed source corresponds to a point on the redshift, luminosity plane or by counting the number of galaxies in luminosity bins. Instead we use the stepwise maximum-likelihood method of Efstathiou et al. (1988) where the luminosity function is characterized by values in a number of absolute magnitude bins. We choose to use 5 regularly spaced nodes at $M_{1350}=[-22.5,-22,-21.5,-21,-20.5]$ since this covers the full extent of the absolute magnitude range of our detected LBGs and ensures that extrapolation of the luminosity function beyond this range will not bias our results.
[c c c c]{} $-22.5 $&$ 2.66\times 10^{-8} $&$ 9.08\times 10^{-9} $&$ 7.78\times 10^{-8}$\
$-22.0 $&$ 2.18\times 10^{-6} $&$ 8.70\times 10^{-7} $&$ 9.70\times 10^{-6}$\
$-21.5 $&$ 1.45\times 10^{-5} $&$ 2.88\times 10^{-6} $&$ 2.92\times 10^{-5}$\
$-21.0 $&$ 1.29\times 10^{-4} $&$ 7.06\times 10^{-5} $&$ 2.19\times 10^{-4}$\
$-20.5 $&$ 2.30\times 10^{-4} $&$ 9.34\times 10^{-5} $&$ 5.77\times 10^{-4}$
The comparison of the model with the data is performed in observed magnitude space because this is the only way of accounting for the selection effects and completeness discussed in Section \[completeness\] and the non-unique mapping of apparent to absolute magnitude. For a model luminosity function (defined as power-laws connecting and extending beyond the five nodes), the completeness information and photometric noise are used to generate a model observed $z'$ magnitude distribution, using the same method as in Section \[magdistrib\]. These model magnitudes are binned in $\delta z'=0.1$ magnitude bins, $n_{\rm mod}$, and compared to the observed $z'$ magnitude histogram, $n_{\rm obs}$, (Figure \[fig:maghist\]). The data are Poissonian and the likelihood, $\mathcal{L}$, of observing the data given the model over all $N$ magnitude bins is given by $$\mathcal{L}=\prod_{i=1}^{N} \frac{e^{-n_{{\rm mod}, i}} n_{{\rm mod}, i}^{n_{{\rm obs}, i}}}{n_{{\rm obs}, i}!}.$$ The galaxy space densities at the five nodes are the five free parameters which are adjusted using an amoeba algorithm (Press et al. 1992) to determine the maximum likelihood. The 68% range on the best fit values are determined via bootstrapping by 400 trials of Poisson perturbation of the $n_{{\rm obs}, i}$ values.
The results are given in Table \[tab:lf\] . Figure \[fig:lf\] plots the data and compares with previous results from the literature. Our best-fit values match well the Schechter parameterizations of Bouwens et al. (2007, 2008) and McLure et al. (2009) and agree with previous binned values determined by Bouwens et al. (2007) and McLure et al. (2009). Our two most luminous data points show the steep decline in space density towards high luminosity required by our data. The 68% uncertainties on our data points are fairly large due to the small sample size and the fact that sources can be redistributed to neighboring nodes in different trials. The total volume of our survey is $\approx 10^7$Mpc. Therefore we are not truly measuring a space density of $<10^{-7}$Mpc$^{-1}$ at $M_{1350}=-22.5$. The low value of this node is showing that there must be a sharp decline brighter than $M_{1350}=-22$ or else we would observe more bright LBGs. One of our spectroscopically confirmed galaxies, WMH5, has $M_{1350}=-22.65$, but most of the rest have absolute magnitudes close to $M_{1350}=-22$, in agreement with this sharp decline in number density.
Field-to-field variance
-----------------------
Part of the robustness of our results comes from the fact the CFHTLS Deep is spread over four separate 1 square degree fields. We now consider the variance in $z\approx 6$ LBG counts across the four fields. We identified 8, 15, 8 and 9 LBGs in D1, D2, D3 and D4, respectively. The two fields with the least certain identifications due to shallower $J$-band (D3 and D4) do not have an unusual number of counts, providing confidence in the LBG classification in these fields. The field with the largest deviation is D2 (COSMOS) which, with 15 LBGs, is 1.6$\sigma$ beyond the mean, assuming Poisson variance only. Another source of variance is the large-scale distribution of matter, so called cosmic variance. This could be important if the most luminous LBGs are hosted by rare, massive dark matter halos. We use the cosmic variance calculator of Trenti & Stiavelli (2008) assuming a duty cycle of 0.5 for the halos. For this space density and duty cycle the minimum dark matter halo mass hosting a galaxy is $2 \times 10^{12 }$M$_\odot$. The variance calculation shows that the cosmic variance contribution to the total variance in each field is expected to be only half the Poisson variance contribution. Hence including cosmic variance the D2 field is 1.4$\sigma$ beyond the mean and not unexpectedly overdense.
As well as chance, another factor that could be contributing to the higher counts in D2 is that the CFHTLS data is a little shallower in that field. With shallower data, there are three possible effects that alter the expected number of candidates found: (i) the larger magnitude uncertainties cause more objects to scatter in from faintward of the $z'=25.3$ magnitude limit; (ii) the larger magnitude uncertainties cause more objects to scatter in from blueward of the $i'-z'=2$ color limit; (iii) the detection completeness will be lower at the faint end. All of these effects are included for the sample as a whole in Section \[completeness\], but we do not analyze them on a field-by-field basis. Certainly, effect (i) is not significant because comparison of the CFHTLS $z'$ and Subaru $z'$ magnitudes in D2 shows that only two of the 15 have Subaru $z'>25.3$. A similar fraction is found for D1 using VIDEO $Z$ magnitudes and is expected based on Figure \[fig:zcomplete\]. We do not have similar comparison data to determine the likely strength of effects (ii) and (iii) (which affect the number of candidates in opposite directions). Therefore we ascribe the higher number of LBGs in D2 being mostly due to chance and the total of 40 in the sample is unbiased and fully accounted for by our selection function.
Discussion
----------
Our results show that there is a sharp decline in the $z\approx 6$ galaxy space density for galaxies brighter than $M_{1350}=-22$, consistent with the exponential function of Bouwens et al. (2007; 2008). The steepness of this function is much steeper than that of the dark matter halo mass function for the expected host halo masses, ${M_H}$, of these luminous LBGs ($10^{12}<{M_H}<10^{13}\,{\rm M}_\odot$; Lee et al. 2009; Trenti et al. 2010). This implies suppression of star formation in the most massive halos at high redshift. The galaxy luminosity and dark matter halo mass functions can be related using the conditional luminosity function (CLF) method (Yang et al. 2003). At low redshifts, the bright end of the galaxy luminosity function is consistent with the relationship $L \sim M_H^{0.28}$ (Vale & Ostriker 2006). The physical explanation for this is likely feedback from an AGN and/or inefficient gas cooling in high mass halos (Benson et al. 2003).
Because our best-fit luminosity function is so similar to that of Bouwens et al. (2007), we can adopt the $z=6$ CLF derivation of Trenti et al. (2010) which was fit to this function. Trenti et al. (2010) showed that by requiring only recently formed (200 Myr) halos to host LBGs, the duty cycle for high mass halos must be almost unity, unlike the values of $\approx 20$% previously estimated for typical LBGs at this epoch (Stark et al. 2007; Lee et al. 2009). Under this assumption that the duty cycle does not vary much over the luminosity range of interest, Trenti et al. (2010) showed that the high luminosity end of the CLF follows a relationship of $L \sim M_H^{0.5}$. This is somewhat steeper than is found at low redshift, but still indicates that luminosity is a slowly varying function of halo mass.
Models with star formation efficiency and duty cycle (or star formation timescale) which are independent of halo mass do not predict such a steep decline in the luminosity function at the brightest magnitudes (Stark et al. 2007; Muñoz & Loeb 2011). These works were able to fit the $z=6$ luminosity function of Bouwens et al. (2007) because those data did not provide evidence of the sharp exponential decline. Our results show that the same process that limits star formation in high mass halos at low redshift is also acting at a time just one billion years after the Big Bang.
Conclusions {#conc}
===========
We have performed the largest area survey for luminous LBGs at $z\approx 6$. These galaxies have been used to study their physical sizes and spectral energy distributions. The large sky area has enabled the most robust results on the space density decline at the bright end of the luminosity function. Below we list the specific conclusions drawn from this work.
- Spectroscopy of seven galaxies revealed redshifts for four of them. Three of these have [Ly$\alpha$]{} emission lines, one of which is extremely weak with $W_{{\rm Ly}\alpha} = 4$Å. All four galaxies show continuum redward of [Ly$\alpha$]{} which enables redshift determination from the break location.
- About half of the LBGs are resolved in the ground-based CFHT imaging observations. This is consistent with a typical intrinsic half-light radius for luminous $z\approx 6$ galaxies of 2kpc, higher than the typical 1kpc in GOODS. High resolution imaging of a subsample shows that some galaxies have a dominant component with large half-light radii and some have multiple components. A larger sample with high resolution imaging is necessary to determine if there is a relationship between interaction frequency and star-formation rate.
- The properties of the stacked photometry have been investigated. These data show a flux decrement in the observed-frame optical of nearly 100 due to the Lyman break. The $YJHK$ photometry are fit by a power-law of slope $\beta=-1.44 \pm 0.10$, much redder than for less luminous galaxies at this epoch. The stacked SED is well fit by a $z=5.95$ constant star formation model with mass $\approx
10^{10} {\rm M}_\odot$. Significant dust reddening is required ($A_V=0.75$) showing that dust is more prevalent in the most massive systems, again possibly due to these being merger-induced starbursts.
- The magnitude distributions of our LBG sample and a $i'-z'$ color-matched sample of L/T dwarfs are considered. Whilst the dwarfs have a fairly flat distribution with a peak at $z'\approx 25$, the LBGs have a distribution that rises sharply with magnitude. Our $z\approx 6$ luminosity function derivation provides clear evidence for a sharp decline in space density brighter than $M=-22$.
- The decline in space density at high luminosity is consistent with a relationship between galaxy luminosity and dark matter halo mass of $L \sim M_H^{0.5}$. This shows that the processes limiting star formation in high mass halos at low redshift are also operating effectively at a time just 1 billion years after the Big Bang.
This paper is dedicated to the memory of Steve Rawlings, who taught CJW to look for the signal in the noise. Thanks to Genevieve Soucail for providing reduced near-IR data for some of our targets. Thanks to the anonymous referee for suggestions that considerably improved the paper. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. We gratefully acknowledge use of data from the ESO Public Survey programs 179.A-2005 and 179.A-2006 with the VISTA telescope.This work uses observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil) and CONICET (Argentina). This paper uses data from Gemini programs GN-2009A-Q-2, GS-2009A-Q-3, GN-2010B-C-9 and GN-2011A-C-1.
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abstract: 'The quantum spin $1/2$ XXZ chain with anisotropy parameter $\Delta=-1/2$ possesses a dynamic supersymmetry on the lattice. This supersymmetry and a generalisation to higher spin are investigated in the case of open spin chains. A family of non-diagonal boundary interactions that are compatible with the lattice supersymmetry and depend on several parameters is constructed. The cohomology of the corresponding supercharges is explicitly computed as a function of the parameters and the length of the chain. For certain specific values of the parameters, this cohomology is shown to be non-trivial. This implies that the spin-chain ground states are supersymmetry singlets. Special scalar products involving an arbitrary number of these supersymmetry singlets for chains of different lengths are exactly computed. As a physical application, the logarithmic bipartite fidelity of the open quantum spin $1/2$ XXZ chain with $\Delta=-1/2$ and special diagonal boundary interactions is determined.'
title: '**Open spin chains with dynamic lattice supersymmetry**'
---
=1
Introduction {#sec:Introduction}
============
The spin $1/2$ XXZ Heisenberg chain is arguably one of the most important exactly solvable quantum models of interacting spins in one dimension. One the one hand, its study has inspired the development of many modern techniques of quantum integrability such as the Bethe ansatz [@bethe:31], the quantum-inverse scattering method [@korepin:93; @maillet:00_2] or the vertex-operator approach [@jimbo:94]. On the other hand, it is related to a variety of physically interesting problems, most notably the theory of quantum magnetism [@maillet:07; @schollwoeck:08].
In this article, we study the open integrable XXZ Heisenberg chain with arbitrary spin $\ell/2$ [@kirillov:87]. We focus on a particular value of the anisotropy parameter where the spin chain exhibits an additional symmetry beyond its integrability: *supersymmetry* [@witten:82]. For concreteness, let us consider the familiar case $\ell=1$. For diagonal boundary fields the spin-chain Hamiltonian of a chain of length $L$ is given by
$$H = -\frac{1}{2}\sum_{j=1}^{L-1} \left(\sigma_j^1\sigma_{j+1}^1 + \sigma_j^2\sigma_{j+1}^2+ \Delta \sigma_j^3\sigma_{j+1}^3\right)+ p \sigma_1^3 + p'\sigma_L^3
\label{eqn:GenericOpenXXZ}$$
where $\sigma^1,\sigma^2$ and $\sigma^3$ denote the Pauli matrices, $\Delta$ the anisotropy parameter, and $p,p'$ the boundary magnetic fields. Fendley and Yang [@yang:04] showed that for $$\Delta = -\frac{1}{2}, \quad p=p' = -\frac{1}{4}$$\[eqn:OpenXXZIntro\]
the Hamiltonian is supersymmetric: up to a constant shift (that we specify later) it can be written as the anticommutator of a *supercharge* and its adjoint. This supercharge is a nilpotent operator that maps states of a chain of length $L$ to states of a chain of length $L+1$: It is dynamic. Dynamic supersymmetry on the lattice has since been observed for many other spin chains [@beisert:04; @hagendorf:12; @hagendorf:13; @meidinger:14], in particular for the periodic and twisted spin $\ell/2$ XXZ Heisenberg chains with a particular value of its anisotropy parameter that depends on $\ell$ [@hagendorf:13]. In the following, we refer to this value as the supersymmetric point.
In this article, we determine a multi-parameter family of boundary magnetic fields for the open integrable XXZ Heisenberg chains with spin $\ell/2$ at the supersymmetric point that are compatible with a dynamic lattice supersymmetry. These boundary terms generically are non-diagonal and may differ at the first and last site of the chain. We achieve this through a generalisation of the supercharges found in previous works. Furthermore, we identify all values of the parameters for which the spin-chain Hamiltonians possess so-called supersymmetry singlets [@witten:82]. Supersymmetry singlets are special eigenstates of the Hamiltonian that are annihilated by both the supercharge and its adjoint. If they exist then they are automatically ground states and therefore of great physical interest. Their existence is related to the existence of a non-trivial cohomology of the supercharge. We explicitly compute this cohomology.
We use our cohomology results in order to determine sum rules for special scalar products involving the supersymmetry singlets. Specifically, let us denote by $|\psi_L\rangle$ a singlet for a spin chain of length $L$. We consider the overlaps $$\langle \psi_L|\left(|\psi_{L_1}\rangle\otimes |\psi_{L_2}\rangle\otimes \cdots \otimes |\psi_{L_m}\rangle\right), \quad L_1+L_2+\cdots+L_m = L.
\label{eqn:GeneralGSOverlap}$$ These scalar products and their scaling limit are of interest in the field of quantum quenches and quantum entanglement. In particular, the case $m=2$ is related to an entanglement measure called the *bipartite fidelity* [@dubail:11]. For one-dimensional quantum critical systems, this quantity has a large-$L$ asymptotic expansion whose first few terms have been predicted by conformal field theory (CFT) techniques [@dubail:11; @dubail:13]. These CFT predictions have been confirmed by lattice derivations at the leading order in a few cases [@dubail:13; @weston:11; @weston:12]. Here, we use the supersymmetry to show that the scalar product in can (in a suitable normalisation) be computed from the sole knowledge of a single special component of each involved singlet. For the Hamiltonian , we provide exact finite-size expressions of these components and therefore an explicit formula for the scalar products as a function of $L_1,\dots,L_m$. This allows us to exactly compute the large-$L$ expansions of the scalar products and show that they match the CFT predictions both at leading *and* subleading orders.
The layout of this article is as follows. In \[sec:SUSYSpinChains\] we review the formalism of dynamic lattice supersymmetry for open quantum spin chains. In particular, we discuss the supercharge of the open integrable XXZ Heisenberg chain with spin $\ell/2$ at its supersymmetric point and particular diagonal boundary interactions. We present a new multi-parameter deformation of this supercharge in \[sec:DeformedSUSY\]. The deformation allows us to identify a family of non-diagonal boundary interactions that is compatible with an exact lattice supersymmetry. The purpose of \[sec:E0States\] is to determine the values of the deformation parameters for which the supercharges possess supersymmetry singlets. We achieve this by exactly computing the cohomology of the supercharge. In \[sec:SP\], we analyse a number of properties of these supersymmetry singlets. In particular, we find sum rules for the scalar products . We analyse them for the supersymmetry singlets of the Hamiltonian in \[sec:SPXXZ\]. In particular, we compute their scaling limit and compare our findings to the predictions of conformal field theory. We present our conclusions in \[sec:Conclusion\].
Dynamic lattice supersymmetry {#sec:SUSYSpinChains}
=============================
In this section, we recall the concept and formalism of dynamic lattice supersymmetry for spin chains [@yang:04; @hagendorf:12; @hagendorf:13; @meidinger:14]. Furthermore, we discuss the supercharges for the open integrable XXZ spin chains with spin $\ell/2$ at their supersymmetric point that we analyse and generalise in this article.
#### Supercharge and Hamiltonian.
Throughout this article, we consider open quantum spin chains of finite length. We denote by $V^L$ the Hilbert space of a spin chain of length $L$. It is given by the $L$-fold tensor product of single-spin Hilbert spaces $V$: $$V^L = \underset{L\,\text{times}}{\underbrace{V\otimes V\otimes \cdots \otimes V}}.$$ We focus on the case $ V = \mathbb C^{\ell+1}$ where $\ell$ is an arbitrary fixed positive integer. We refer to [@meidinger:14] for a more general discussion of models where $V$ is a super vector space.
The supercharge ${\mathfrak{Q}}$ of our spin-chain models is a length-increasing operator that maps $V^L$ to $V^{L+1}$ for each $L\geqslant 1$.[^1] When acting on $V^L$, it is given by the following alternating sum: $${\mathfrak{Q}}= \sum_{j=1}^L (-1)^j {\mathfrak{q}}_j.
\label{eqn:SCSupercharge}$$ Here, the length-increasing operators ${\mathfrak{q}}_j$ are $${\mathfrak{q}}_j = \underset{j-1}{\underbrace{1\otimes \cdots \otimes 1}} \otimes {\mathfrak{q}}\otimes \underset{L-j}{\underbrace{1 \otimes \cdots \otimes 1}}$$ where ${\mathfrak{q}}:V\to V\otimes V$ denotes the so-called *local supercharge*. One checks that if ${\mathfrak{q}}$ obeys the relation $$({\mathfrak{q}}\otimes 1-1\otimes {\mathfrak{q}}){\mathfrak{q}}= 0
\label{eqn:Coassociativity}$$ then the operator ${\mathfrak{Q}}$ is nilpotent, $${\mathfrak{Q}}^2=0,
\label{eqn:QNP}$$ in the sense that the action of ${\mathfrak{Q}}^2 : V^L \to V^{L+2}$ yields zero on any element of $V^L$, for each $L\geqslant 1$. We refer to as the coassociativity property.
The canonical (complex) scalar product of the spin-chain Hilbert space allows us to define the adjoint supercharge ${\mathfrak{Q}}^\dagger$. It is a length-decreasing operator that maps $V^L$ to $V^{L-1}$ for each $L\geqslant 2$. We have $\langle \psi|({\mathfrak{Q}}^\dagger|\phi\rangle) = (\langle \phi|({\mathfrak{Q}}|\psi\rangle))^\ast$ for all $|\phi\rangle \in V^{L}, |\psi\rangle \in V^{L-1}$ and each $L\geqslant 2$. It follows from that the adjoint supercharge is also nilpotent, $$({\mathfrak{Q}}^\dagger)^2=0.
\label{eqn:QDNP}$$ This means that for each $L\geqslant 3$, the application $({\mathfrak{Q}}^\dagger)^2:V^L\to V^{L-2}$ yields zero on every element of $V^L$.
In supersymmetric quantum mechanics, the Hamiltonian is given by the anticommutator of a supercharge ${\mathfrak{Q}}$ and its adjoint ${\mathfrak{Q}}^\dagger$: $$H = {\mathfrak{Q}}{\mathfrak{Q}}^\dagger + {\mathfrak{Q}}^\dagger {\mathfrak{Q}}.
\label{eqn:DefHFromQ}$$ Using the specific supercharge we find that the Hamiltonian is the sum of a bulk part, describing nearest-neighbour interactions, and boundary terms: $$H = \sum_{i=1}^{L-1} h_{i,i+1} + (h_{\text{\rm \tiny B}})_1+(h_{\text{\rm \tiny B}})_L.
\label{eqn:Hamiltonian}$$ Here, $h_{i,i+1}$ denotes the Hamiltonian density $h: V^2\to V^2$, acting on the sites $i$ and $i+1$. In terms of the local supercharge it is given by $$h = -(1\otimes {\mathfrak{q}}^\dagger)({\mathfrak{q}}\otimes 1) - ({\mathfrak{q}}^\dagger \otimes 1)(1\otimes {\mathfrak{q}})+{\mathfrak{q}}{\mathfrak{q}}^\dagger + \frac12\left({\mathfrak{q}}^\dagger{\mathfrak{q}}\otimes 1 +1 \otimes {\mathfrak{q}}^\dagger {\mathfrak{q}}\right).
\label{eqn:HamiltonianDensity}$$ Furthermore, the boundary interaction at the first and last site of the chain is encoded in the operator $h_{\text{\rm \tiny B}}:V\to V$. In terms of the local supercharge, we find $$h_{\text{\rm \tiny B}} = \frac{1}{2}{\mathfrak{q}}^\dagger{\mathfrak{q}}.
\label{eqn:BoundaryHamiltonian}$$ In the boundary terms are the same at both ends of the spin chain. In \[sec:DeformedSUSY\], we show for a specific choice of ${\mathfrak{q}}$ that the definition supercharge can be generalised in order to incorporate unequal boundary terms at both ends without modifying the bulk part.
Unlike ${\mathfrak{Q}}$ and ${\mathfrak{Q}}^\dagger$ the Hamiltonian $H$ is not a length-changing operator. From and follow the following relations: $$H {\mathfrak{Q}}= {\mathfrak{Q}}H, \quad H{\mathfrak{Q}}^\dagger = {\mathfrak{Q}}^\dagger H.
\label{eqn:CommuteHQ}$$ Hence, both ${\mathfrak{Q}}$ and ${\mathfrak{Q}}^\dagger$ are formally symmetry operators, and therefore the Hamiltonian $H$ is said to be supersymmetric. The supersymmetry is dynamic in the sense that the Hamiltonians on the left- and right-hand side of act on the Hilbert spaces of spin chains whose lengths differ by one. This supersymmetry leads to special properties of the spectrum of $H$. The construction implies that this spectrum is real and non-negative. Furthermore, it follows from the commutation relations that the spectra for chains of different length have common eigenvalues. We discuss these properties in detail below in \[sec:Cohomology\].
#### Local supercharges for XXZ chains with arbitrary spin.
The work of this article is based on a particular local supercharge. It acts on the canonical basis states $|0\rangle,|1\rangle,\dots,|\ell\rangle$ of $V$ according to
\[eqn:TrigSupercharge\] $${\mathfrak{q}}|0\rangle = 0, \quad \text{and} \quad {\mathfrak{q}}|m\rangle = \sum_{k=0}^{m-1}a_{m,k}|k,m-k-1\rangle, \quad m=1,\dots, \ell,$$ where we abbreviated $|m_1,m_2\rangle = |m_1\rangle \otimes |m_2\rangle$. The coefficients $a_{m,k}$ are strictly positive real numbers defined for $0\leqslant k < m\leqslant \ell$. They are given by $$\label{eqn:Defamk}
a_{m,k} = \sqrt{\frac{\{m+1\}}{\{m-k\}\{k+1\}}}, \quad
\quad\{m\} = \frac{q^m-q^{-m}}{q-q^{-1}},$$ where $q$ is the root of unity $$q = e^{{\text{i}}\pi/(\ell+2)}.
\label{eqn:RootOfUnity}$$
It follows that the adjoint local supercharge ${\mathfrak{q}}^\dagger$ acts on a basis vector of $V^2$ according to $$\label{eqn:ActionAdjointTrigSC}
{\mathfrak{q}}^\dagger|m_1,m_2\rangle =
\begin{cases}
a_{m_1+m_2+1,m_1}|m_1+m_2+1\rangle, &\text{if } m_1+m_2<\ell,\\
0, &\text{if } m_1+m_2 \geqslant \ell.
\end{cases}$$
The local supercharge was introduced in [@hagendorf:13]. It is closely related to the supercharge of the so-called $M_\ell$ models of Fendley, Nienhuis and Schoutens [@fendley:03]. These models describe supersymmetric fermions on a one-dimensional lattice with an exclusion constraint that limits the length of connected fermion clusters to $\ell$. Their supercharge splits connected clusters of $m$ fermions into pairs of adjacent clusters of $k$ and $m-k-1$ fermions, for $k=0,\dots,m-1$, with an amplitude $a_{m,k}$. Locally, the $M_\ell$ models are equivalent to the spin chains considered in this article. However, the spin-chain language easily allows us to generalise the supersymmetry and add new features that are rather difficult to implement in the fermion language. One such feature is particle-hole symmetry for the fermions. In fact, the particule-hole transformation is a complicated non-local operation because of the exclusion constraint. In the spin-chain language, it translates to a simple spin-reversal symmetry. Indeed, the special values for the constants $a_{m,k}$ in lead to a Hamiltonian density that is invariant under a spin-reversal transformation as we shall see below. Another feature is the introduction of boundary interactions that break the particle-number conservation in the fermion model (without breaking the supersymmetry). In the spin-chain language, this translates to boundary conditions that break the conservation of the magnetisation. Below, we construct a new family of such boundary interactions for the spin-chain models by deforming the action of the supercharge on the boundary sites of the spin chain.
Let us now illustrate the nature of the spin-chain Hamiltonians resulting from . For $\ell=1$, defines a local supercharge for the well-known spin $1/2$ XXZ chain with anisotropy parameter $\Delta=-1/2$, found by Fendley and Yang [@yang:04]: $${\mathfrak{q}}|0\rangle = 0, \quad {\mathfrak{q}}|1\rangle = {|0\rangle}\otimes{|0\rangle}.
\label{eqn:DefQXXZ}$$ The corresponding Hamiltonian density and the boundary terms are readily evaluated from and . The full spin-chain Hamiltonian is obtained from . Identifying $$|0\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \quad
|1\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix},$$ it can conveniently be written in terms of the Pauli matrices $$\sigma^1
=\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\quad
\sigma^2
=\begin{pmatrix}
0 & -{\text{i}}\\
{\text{i}}& 0
\end{pmatrix},
\quad
\sigma^3
=\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.$$ Indeed, one finds the XXZ Hamiltonian with anisotropy parameter $\Delta=-1/2$ and diagonal boundary magnetic fields: $$\begin{aligned}
\label{eqn:XXZOpenChainND}
H = -\frac{1}{2}\sum_{j=1}^{L-1} \left(\sigma_j^1\sigma_{j+1}^1+\sigma_j^2\sigma_{j+1}^2-\frac{1}{2}\sigma_j^3\sigma_{j+1}^3\right)-\frac{\sigma_1^3+\sigma_L^3}{4}+\frac{3L-1}{4}.\end{aligned}$$ Up to a constant shift, this is the Hamiltonian that we discussed in \[sec:Introduction\]. Besides its lattice supersymmetry and its relation to the (open) $M_1$ model of supersymmetric fermions [@beccaria:05], it has a few interesting features. First, it is isospectral to another XXZ chain that describes the so-called Temperley-Lieb stochastic process [@nichols:05; @degier:05]. The connection to a stochastic process implies in particular that the spectrum of is non-negative and contains the non-degenerate ground-state eigenvalue $E=0$. In \[sec:E0States\], we use the supersymmetry in order to provide new insights into the properties of the corresponding zero-energy state. Second, we note that belongs to a family of Hamiltonians that have a spectral overlap with the Hamiltonians of the famous quantum-group invariant XXZ spin chains of Pasquier and Saleur’s [@pasquier:90]. This connection has recently been used in order to proof the reality of the spectra of the Pasquier-Saleur spin chains [@morin:16].
For arbitrary $\ell$, the Hamiltonian density that derives from was explicitly computed in [@hagendorf:13]. We write it as follows:
\[eqn:ExplicitHamDensity\] $$h = \sum_{m_1,m_2=0}^\ell
\sum_{n=-M_1}^{M_2}\beta_{m_1,m_2}^n|m_1+n,
m_2-n\rangle\langle m_1,m_2|,$$ where we abbreviate $M_1 = \min(m_1,\ell-m_2)$ and ${{M_2}} = \min(m_2,\ell-m_1)$. The coefficients have the symmetry property $\beta^n_{m_1,m_2} = \beta_{m_2,m_1}^{-n}$. For $n>0$, they are given by $$\beta^n_{m_1,m_2} = -\frac{1}{\{n\}}\sqrt{\frac{\{M_1+1\}\{M_2-n+1\}}{\{M_2+1\}\{M_1+n+1\}}}.$$ Furthermore, we have $$\beta^0_{m_1,m_2} = c_{M_1+1}+c_{M_2+1}, \quad
c_m = \sum_{k=1}^m \frac{\{k+1\}-\{k-1\}}{2\{k\}}.$$
Furtermore, the term that describes the boundary interactions is diagonal for any $\ell$: $$\label{eqn:hBND}
h_{\text{\rm \tiny B}} =\frac{1}{2}{\mathfrak{q}}^\dagger{\mathfrak{q}}= \sum_{m=1}^\ell c_m|m\rangle\langle m|.$$ For $\ell=2$, yields a Hamiltonian density which coincides, up to a simple unitary transformation, with the Hamiltonian density of the integrable spin-one XXZ chain (of Fateev and Zamolodchikov [@zamolodchikov:81]) at its supersymmetric point. Further investigations in [@hagendorf:13] support the conjecture that for arbitrary $\ell$ is, up to a simple unitary transformation, the Hamiltonian density of the quantum integrable spin $\ell/2$ XXZ chain at a particular value of its anisotropy parameter. This observation is consistent with an analysis of the Bethe-ansatz equations [@meidinger:14] but remains to be proven. A possible proof could be obtained from Mangazeev’s explicit expressions for the $R$-matrices of fused vertex models [@mangazeev:14]. We leave the details of this proof to future investigations.
Supercharges and boundary conditions {#sec:DeformedSUSY}
====================================
In this section, we generalise the supersymmetry of the open XXZ spin chain at $\Delta=-1/2$, described by the Hamiltonian , and its higher-spin analogues at their supersymmetric point. In \[sec:OneParamDef\] we find a family of local supercharges ${\mathfrak{q}}(y)$, depending non-trivially on a complex parameter $y$, that have the same Hamiltonian density as ${\mathfrak{q}}$ defined in . The resulting boundary terms however depend non-trivially on $y$ and generically are non-diagonal. Using the properties of ${\mathfrak{q}}(y)$, we generalise in \[sec:BoundaryConditions\] the action of the supercharge at the first and last site of the spin chain. The resulting supersymmetric spin-chain Hamiltonians have unequal boundary terms at both ends of the chain.
A one-parameter deformation {#sec:OneParamDef}
---------------------------
In order to construct ${\mathfrak{q}}(y)$, we need to discuss two special local supercharges: the image $\bar {\mathfrak{q}}$ of ${\mathfrak{q}}$ defined in under spin reversal and a so-called local gauge supercharge.
We start our discussion with $\bar {\mathfrak{q}}$ and some of its properties. The spin-reversal operator $R$ on $V^L,\,L\geqslant 1,$ is a linear operator defined by the following action on the canonical basis states $|m_1,m_2,\dots,m_L\rangle=|m_1\rangle \otimes |m_2\rangle \otimes \cdots\otimes |m_L\rangle$: $$R|m_1,\dots,m_L\rangle = |\ell-m_1,\dots,\ell-m_L\rangle.$$ We define $\bar {\mathfrak{q}}= R {\mathfrak{q}}R$. This operator acts on the basis vectors of $V$ according to $$\label{eqn:BarTrigSupercharge}
\bar {\mathfrak{q}}|\ell\rangle = 0,\quad \text{and} \quad \bar {\mathfrak{q}}|m\rangle = \sum_{k=m+1}^\ell a_{\ell-m,\ell-k}|k, \ell+1+m-k\rangle, \quad m=0,\dots,\ell-1.$$ It follows from $R^2=1$ that $\bar {\mathfrak{q}}$ has the coassociativity property . Furthermore, it was shown in [@hagendorf:13] that, for the specific choice of the coefficients $a_{m,k}$, the Hamiltonian densities $h$ of ${\mathfrak{q}}$ and $\bar h$ of $\bar {\mathfrak{q}}$ are equal: $$\label{eqn:EqualityHHBar}
h = \bar h.$$ For our construction of ${\mathfrak{q}}(y)$, we note furthermore that ${\mathfrak{q}}$ and $\bar {\mathfrak{q}}$ obey a certain anticommutation relation up to boundary terms. Indeed, for any $|\psi\rangle \in V$ we have the relation $$\label{eqn:AntiCommutationQQBar}
\left((-{\mathfrak{q}}\otimes 1+1\otimes {\mathfrak{q}})\bar {\mathfrak{q}}+(-\bar{\mathfrak{q}}\otimes 1+ 1\otimes \bar{\mathfrak{q}}) {\mathfrak{q}}\right)|\psi\rangle = |{\protect\raisebox{0.25ex}{$\chi$}}\rangle \otimes |\psi\rangle - |\psi\rangle \otimes |{\protect\raisebox{0.25ex}{$\chi$}}\rangle$$ where the vector $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle\in V^2$ is given by $$|{\protect\raisebox{0.25ex}{$\chi$}}\rangle = \sum_{m=0}^\ell {\protect\raisebox{0.25ex}{$\chi$}}_m |m,\ell-m\rangle, \quad {\protect\raisebox{0.25ex}{$\chi$}}_m = \frac{1}{\{m+1\}}.
\label{eqn:DefChi}$$
Next, let us recall the concept of a local gauge supercharge ${\mathfrak{q}}_\phi$ [@hagendorf:13]. It depends on a vector $|\phi\rangle \in V$ and acts on any vector $|\psi\rangle \in V$ according to $${\mathfrak{q}}_\phi|\psi\rangle = |\phi\rangle \otimes |\psi\rangle + |\psi\rangle \otimes |\phi\rangle.
\label{eqn:DefGaugeSC}$$ The operator ${\mathfrak{q}}_\phi$ does not have the coassociativity property . However, it obeys a similar relation:$$({\mathfrak{q}}_\phi\otimes 1 - 1 \otimes {\mathfrak{q}}_\phi)
{\mathfrak{q}}_\phi|\psi\rangle = |{\protect\raisebox{0.25ex}{$\chi$}}_\phi\rangle \otimes |\psi\rangle - |\psi\rangle \otimes |{\protect\raisebox{0.25ex}{$\chi$}}_\phi\rangle, \quad
|{\protect\raisebox{0.25ex}{$\chi$}}_\phi\rangle = |\phi\rangle \otimes |\phi\rangle.
\label{eqn:QuasiCoassociativityGauge}$$
We define the local supercharge ${\mathfrak{q}}(y)$ as a linear combination of ${\mathfrak{q}}$, $\bar {\mathfrak{q}}$ and a local gauge supercharge. The idea is to adjust the gauge term in such a way that ${\mathfrak{q}}(y)$ has the coassociativity property. Let us write $${\mathfrak{q}}(y) = x\left({\mathfrak{q}}+ y^{\ell+2}\bar {\mathfrak{q}}+ {\mathfrak{q}}_{\phi(y)}\right), \quad x= \frac{1}{\sqrt{1+|y|^{2(\ell+2)}}}
\label{eqn:DefLocalQy}$$ where $|\phi(y)\rangle\in V$ is the vector characterising the gauge term. Using , and one checks that ${\mathfrak{q}}(y)$ satisfies if $|\phi(y)\rangle$ solves the quadratic equation $$\left({\mathfrak{q}}+y^{\ell+2}\bar {\mathfrak{q}}\right)|\phi(y)\rangle +|\phi(y)\rangle \otimes |\phi(y)\rangle= y^{\ell+2}|{\protect\raisebox{0.25ex}{$\chi$}}\rangle.
\label{eqn:EqnForPhi}$$ This equation is solved by[^2] $$|\phi(y)\rangle = \sum_{m=0}^\ell \phi_m(y) |m\rangle, \quad \phi_m(y) = -\frac{y^{m+1}}{\sqrt{\{m+1\}}}.
\label{eqn:DefPhiGauge}$$ The equations and define the local supercharge ${\mathfrak{q}}(y)$. It has the property that ${\mathfrak{q}}(y=0)={\mathfrak{q}}$ and $\lim_{y\to \infty} {\mathfrak{q}}(y) =\bar {\mathfrak{q}}$ where the limit is taken along the real axis. The construction implies that ${\mathfrak{q}}(y)$ has a well-defined transformation property under spin reversal. Indeed, one checks that $$R {\mathfrak{q}}(y) R =\left(\frac{y}{|y|}\right)^{\ell+2}{\mathfrak{q}}(y^{-1})
\label{eqn:ROnQy}$$ for non-zero values of $y$. We note that this is compatible with the values taken at $y=0$ and for $y\to \infty$. Furthermore, this equation shows that the local supercharge ${\mathfrak{q}}(y)$ is invariant under spin reversal up to a sign if and only if $y=\pm 1$.
#### Hamiltonian density.
We will now show that the Hamiltonian density of ${\mathfrak{q}}(y)$ is independent of $y$. To this end, we use a result for local supercharges that are invariant under a parity transformation. The parity operator $P$ on $V^L,\,L\geqslant 1,$ is the linear operator defined by the following action on the canonical basis states: $$P|m_1,m_2,\dots,m_L\rangle = |m_L,m_{L-1}\dots,m_1\rangle.$$ A local supercharge ${\mathfrak{q}}$ is parity-invariant if $P{\mathfrak{q}}={\mathfrak{q}}$ on $V$. The following property of parity-invariant local supercharges was shown in [@hagendorf:13]:
\[lem:ParityProperty\] Let ${\mathfrak{q}}$ be a parity-invariant local supercharge and ${\mathfrak{q}}_\phi$ any gauge local supercharge, then the Hamiltonian densities of ${\mathfrak{q}}$ and ${\mathfrak{q}}+ {\mathfrak{q}}_\phi$ are equal.
One checks that the local supercharges ${\mathfrak{q}}$ and $\bar {\mathfrak{q}}$ defined in and , respectively, are parity-invariant. We use this observation to prove the following statement:
\[prop:HamDensity\] The Hamiltonian density $h(y)$ of the local supercharge ${\mathfrak{q}}(y)$ defined by , is independent of the parameter $y$.
It follows from \[lem:ParityProperty\] that $h(y)$ is equal to the Hamiltonian density of $x({\mathfrak{q}}+ y^{\ell+2} \bar {\mathfrak{q}})$. This Hamiltonian density is obtained from . In order to evaluate it, we note that ${\mathfrak{q}}$ and $\bar {\mathfrak{q}}$ obey the following relations
\[eqn:QQBarRelations\] $$\begin{aligned}
\bar {\mathfrak{q}}{\mathfrak{q}}^\dagger = (1\otimes {\mathfrak{q}}^\dagger)(\bar {\mathfrak{q}}\otimes 1) +({\mathfrak{q}}^\dagger\otimes 1)(1\otimes \bar {\mathfrak{q}}),\quad \bar {\mathfrak{q}}^\dagger {\mathfrak{q}}= 0,\\
{\mathfrak{q}}\bar {\mathfrak{q}}^\dagger = (1\otimes \bar {\mathfrak{q}}^\dagger)({\mathfrak{q}}\otimes 1) +(\bar {\mathfrak{q}}^\dagger\otimes 1)(1\otimes {\mathfrak{q}}),\quad {\mathfrak{q}}^\dagger \bar{\mathfrak{q}}= 0.
\end{aligned}$$
These relations follow from a straightforward calculation, using and . Combining with , we obtain $$h(y) = x^2(h + |y|^{2(\ell+2)}\bar h),$$ where $h$ and $\bar h$ denote the Hamiltonian densities of ${\mathfrak{q}}$ and $\bar {\mathfrak{q}}$, respectively. Furthermore, $\bar h= h$ as noted above. Using the value of $x$ given in , it follows that $h(y)=h$, which is independent of $y$.
We write $$\label{eqn:DefQy}
{\mathfrak{Q}}(y) = \sum_{j=1}^L (-1)^j {\mathfrak{q}}(y)_j$$ for the supercharge constructed from ${\mathfrak{q}}(y)$. The corresponding Hamiltonian $H(y) = {\mathfrak{Q}}(y){\mathfrak{Q}}(y)^\dagger+{\mathfrak{Q}}(y)^\dagger {\mathfrak{Q}}(y)$ is of the form . In the following, we will often write $H$ for $H(y=0)$.
#### Magnetisation.
We define the magnetisation operator by the following action on the basis states of $V^L$: $$M|m_1,\cdots, m_L\rangle = \left(\frac{\ell L}{2}-\sum_{i=1}^L m_i\right)|m_1,\cdots, m_L\rangle.$$ From , it follows that $$e^{{\text{i}}\theta M}{\mathfrak{q}}(y)e^{-{\text{i}}\theta M} = e^{{\text{i}}\theta (\ell+2)/2}{\mathfrak{q}}(e^{-{\text{i}}\theta}y).$$ In particular, for $y =0$ we have $e^{{\text{i}}\theta M}{\mathfrak{q}}e^{-{\text{i}}\theta M} = e^{{\text{i}}\theta (\ell+2)/2}{\mathfrak{q}}$. Using , we find that the Hamiltonian density $h$ of ${\mathfrak{q}}$ conserves the magnetisation, $[h,M]=0$. Since $h(y)=h$ for any $y$ according to \[prop:HamDensity\], we conclude that the bulk part of the Hamiltonian $H(y)$ conserves the magnetisation.
For $y=0$, the boundary interactions are given by , which obviously has the property $[h_{\text{\rm \tiny B}},M]=0$. This implies that for $y=0$, the full Hamiltonian conserves the magnetisation: $[H,M]=0$. Conversely, for generic values of $y$ this conservation law is broken by the boundary terms, which are given by $$h_{\text{\rm \tiny B}}(y) = \frac{1}{2}{\mathfrak{q}}(y)^\dagger {\mathfrak{q}}(y).
\label{eqn:BoundaryTerm}$$ We have $e^{{\text{i}}\theta M}h_{\text{\rm \tiny B}}(y)e^{-{\text{i}}\theta M} = h_{\text{\rm \tiny B}}(e^{-{\text{i}}\theta}y)$. It is possible albeit tedious to explicitly compute the matrix elements of this operator with respect to the canonical basis for arbitrary $\ell$. Their evaluation shows that the dependence on $y$ is non-trivial for both diagonal and off-diagonal matrix elements.
#### Parity and spin reversal.
The Hamiltonian density $h(y)=h$ is parity-invariant. Indeed, we have $PhP=h$ on $V^2$ as a consequence of the parity-invariance of the local supercharge ${\mathfrak{q}}$. Since the boundary interactions are the same at both ends of the chain, we conclude that for any value of $y$ the Hamiltonian of the spin chain is parity-invariant: $$PH(y)P=H(y).$$ The transformation property of the local supercharge ${\mathfrak{q}}(y)$ under spin reversal imply that the Hamiltonian density $h(y)=h$ is spin-reversal invariant: We have $RhR=h$ on $V^2$. Because of the boundary terms, this spin-reversal invariance does however not extend to the Hamiltonian of the spin chain for generic values of $y$. We have $$RH(y)R = H(y^{-1})$$ and therefore spin-reversal invariance if and only if $y = \pm 1$.
#### Example.
As an example, we discuss the case $\ell=1$. The action of ${\mathfrak{q}}(y)$ on the basis vectors $|0\rangle,|1\rangle$ is given by[^3] $$\begin{aligned}
{\mathfrak{q}}(y)|0\rangle &= x\left(-2y|00\rangle+y^3|11\rangle -y^2(|01\rangle+|10\rangle)\right),\\
{\mathfrak{q}}(y)|1\rangle &= x\left(|00\rangle-2y^2|11\rangle -y(|01\rangle+|10\rangle)\right).\end{aligned}$$ One checks the independence of the Hamiltonian density on $y$ by an explicit calculation. In order to write the boundary terms $h_{\text{\rm \tiny B}}(y)$ in a convenient way, we set $y=\rho e^{{\text{i}}\theta}, \, \rho = |y|,$ and express $h_{\text{\rm \tiny B}}(y)$ in terms of the identity matrix and the Pauli matrices. We find
\[eqn:BoundaryHamXXZ\] $$h_{\text{\rm \tiny B}}(y) = \left(\frac{1+5\rho^2+\rho^4}{4(1-\rho^2+\rho^4)}\right)\bm 1+\sum_{j=1}^3 \lambda_j\sigma^j,$$ with $$\lambda_1 = -\frac{\rho \cos \theta}{1+\rho^2},\quad \lambda_2 = -\frac{\rho \sin \theta}{1+\rho^2},\quad \lambda_3 = -\frac{1}{4}\left(\frac{1-\rho^2}{1+\rho^2}\right).$$
We conclude that the open XXZ chain at $\Delta=-1/2$ is supersymmetric for the family of boundary interactions , parametrised by $\rho \geqslant 0$ and $\theta$. For $\rho \neq 0,\infty$ the off-diagonal terms of $h_{\text{\rm \tiny B}}$ are non-zero and therefore generalise the diagonal boundary interactions of found by Fendley and Yang [@yang:04].
Boundary terms {#sec:BoundaryConditions}
--------------
In this section, we modify the action of ${\mathfrak{Q}}(y)$ defined in on the first and last site of the spin chain. This allows us to show that the lattice supersymmetry can be present for unequal boundary terms at both ends of the spin chain. Both these boundary terms depend on the parameter $y$. Furthermore, each boundary term is individually characterised by an integer label $j=0,\dots,\ell+1$.
The main ingredient of our construction are the vectors $$|\xi_k(y)\rangle = x(|\phi(y)\rangle-|\phi(q^{2(k+1)}y)\rangle), \quad k=0,\dots, \ell+1,
\label{eqn:DefXi}$$ where $x$ is defined in and $|\phi(y)\rangle$ in . Because of $q^{2(\ell+2)}=1$, we trivially have $|\xi_{\ell+1}(y)\rangle=0$. The action of ${\mathfrak{q}}(y)$ on these vectors is very simple. Indeed, using it is not difficult to show that $${\mathfrak{q}}(y)|\xi_k(y)\rangle = |\xi_k(y)\rangle\otimes |\xi_k(y)\rangle,\quad \, k=0,\dots, \ell+1.
\label{eqn:ActionQXi}$$
For any pair $0\leqslant j,k \leqslant \ell+1$, we consider an operator ${\mathfrak{Q}}_{j,k}(y)$ that acts on any $|\psi\rangle \in V^L$ according to $$\label{eqn:DefQjk}
{\mathfrak{Q}}_{j,k}(y)|\psi\rangle = |\xi_j(y)\rangle \otimes |\psi\rangle + (-1)^{L-1} |\psi\rangle \otimes |\xi_k(y)\rangle + {\mathfrak{Q}}(y)|\psi\rangle .$$ We note that ${\mathfrak{Q}}_{\ell+1,\ell+1}(y)={\mathfrak{Q}}(y)$. Using ${\mathfrak{Q}}(y)^2=0$ and , one checks that $${\mathfrak{Q}}_{j,k}(y)^2=0.$$
The corresponding Hamiltonian $H_{j,k}(y) = {\mathfrak{Q}}_{j,k}(y){\mathfrak{Q}}_{j,k}(y)^\dagger + {\mathfrak{Q}}_{j,k}(y)^\dagger {\mathfrak{Q}}_{j,k}(y)$ is readily evaluated. It is given by a sum of nearest-neighbour interactions and boundary terms that depend on $j$ and $k$: $$H_{j,k}(y) = \sum_{i=1}^{L-1} h_{i,i+1} + (h_{\text{\rm \tiny B}}^{(j)}(y))_1+ (h_{\text{\rm \tiny B}}^{(k)}(y))_L.$$ Here $h$ is the Hamiltonian density of ${\mathfrak{q}}(y)$ and the boundary terms are given by $$h_{\text{\rm \tiny B}}^{(k)}(y) = h_{\text{\rm \tiny B}}(q^{2(k+1)}y).$$
We conclude that all boundary conditions that result from the deformation of the local supercharge and a modification of the action of the supercharge on the first and last site of the spin chain are parameterised by a complex number $y$ and two integers $0\leqslant j,k\leqslant \ell+1$. However, not all choices of these parameters lead to unequal spectra. To see this, we note that the spectrum of $H_{j,k}(y)$ is the same as the spectrum of $e^{{\text{i}}\theta M} H_{j,k}(y)e^{-{\text{i}}\theta M} = H_{j,k}(e^{-{\text{i}}\theta}y)$ (for any real value of $\theta$) and $(H_{j,k}(y))^\ast = H_{J,K}(y^\ast)$ where $J=\ell-j \,(\text{mod}\, \ell+2),\,K=\ell-k \,(\text{mod}\, \ell+2)$. An appropriate choice for $\theta$ allows us to conclude that it is sufficient to restrict the parameters to real values for $y$, $j=\lfloor (\ell+1)/2\rfloor , \dots,\ell+1$ and $k=\ell+1$. For example, if $\ell=1$ there are two distinct cases $j=1,k=2$ and $j=k=2$ for any real value of $y$.
Zero-energy states {#sec:E0States}
==================
In this section, we analyse whether the family of Hamiltonians $H_{j,k}(y)$ possesses so-called supersymmetry singlets or zero-energy states. If they exist then the zero-energy states are the ground states of the Hamiltonian.
In \[sec:Cohomology\], we recall a few basic facts about the spectrum of supersymmetric Hamiltonians. Moreover, we explain the relation between zero-energy states and the so-called cohomology of the supercharge. As we shall see, the structure of the cohomology depends on whether the parameter $y$ is non-zero or zero. We compute the cohomology for $y\neq 0$ in \[sec:GenericY\] and for $y=0$ in \[sec:Y0\].
Spectrum, zero-energy states and cohomology {#sec:Cohomology}
-------------------------------------------
Let us recall the characteristics of the spectrum and the eigenstates of a supersymmetric Hamiltonian $H={\mathfrak{Q}}{\mathfrak{Q}}^\dagger + {\mathfrak{Q}}^\dagger {\mathfrak{Q}},\,{\mathfrak{Q}}^2=0$ for a generic supercharge ${\mathfrak{Q}}$ [@witten:82]. Clearly, $H$ is a Hermitian operator and therefore diagonalisable. Furthermore, its spectrum is non-negative. Indeed, the Schrödinger equation $H|\psi\rangle = E |\psi\rangle$ implies $$||{\mathfrak{Q}}|\psi\rangle||^2+||{\mathfrak{Q}}^\dagger|\psi\rangle||^2 = E ||\psi||^2.
\label{eqn:SchroedingerProjected}$$ Since the left-hand side is non-negative and the norm of an eigenvector $|\psi\rangle$ non-vanishing, we must have $E\geqslant 0$. We call the non-zero solutions of the Schrödinger equation with $E>0$ and $E=0$ positive-energy states and zero-energy states, respectively. They differ in their behaviour under the action of the supercharge and its adjoint.
#### Positive-energy states.
The eigenstates with strictly positive energy $E>0$ organise in *doublets*. They are given by a pair of non-zero vectors $|\psi\rangle, \,{\mathfrak{Q}}|\psi\rangle$ with ${\mathfrak{Q}}^\dagger|\psi\rangle = 0$. The two states in the doublet are called superpartners. Since the Hamiltonian commutes with the supercharge, the superpartners have the same eigenvalue $E$.
In our case, the supercharge increases the length of the chain. Hence, the supersymmetry leads to spectral degeneracies for chains with lengths differing by one. \[fig:SpectraXXZ\] illustrates this spectral degeneracy for the Hamiltonians $H_{j,k}(y)$ with $\ell=1$ and unequal boundary conditions at both ends of the chain.
![The spectrum of the Hamiltonian $H_{j,k}(y)$ for $\ell=1$ with boundary conditions labelled by $j=1,k=2$ for $L=3$ (left panel) and $L=4$ sites (right panel) as a function of $\rho=|y|$. The solid lines correspond to exact common eigenvalues in the two spectra.[]{data-label="fig:SpectraXXZ"}](SpectrumXXZN3 "fig:"){width=".475\textwidth"} ![The spectrum of the Hamiltonian $H_{j,k}(y)$ for $\ell=1$ with boundary conditions labelled by $j=1,k=2$ for $L=3$ (left panel) and $L=4$ sites (right panel) as a function of $\rho=|y|$. The solid lines correspond to exact common eigenvalues in the two spectra.[]{data-label="fig:SpectraXXZ"}](SpectrumXXZN4 "fig:"){width=".475\textwidth"}
#### Zero-energy states.
In the following, we focus on the solutions of the Schrödinger equation with $E=0$. Since the spectrum of the Hamiltonian is non-negative, any non-zero solution of $H|\psi\rangle = 0$ is automatically a ground state of the system. The existence of these zero-energy states is, according to , equivalent to the existence of non-zero solutions of the system of equations $$\label{eqn:ZeroEnergyStateEqns}
{\mathfrak{Q}}|\psi\rangle = 0, \quad {\mathfrak{Q}}^\dagger |\psi\rangle = 0.$$ These equations imply that the zero-energy states are *singlets* (as stated in the introduction) in the sense that no other eigenstates of the Hamiltonian can be obtained by acting on them with the supercharge or its adjoint.
The first equation of requires that a zero-energy state be in the kernel of the supercharge. We call the elements of $\ker {\mathfrak{Q}}$ *cocycles*. Since ${\mathfrak{Q}}^2=0$, the kernel contains all states that are in the image of ${\mathfrak{Q}}$. We call the elements of $\text{im}\, {\mathfrak{Q}}$ *coboundaries*. The second equation of leads to the following property of zero-energy states:
\[lem:E0NotCoboundary\] A zero-energy state is not a coboundary.
Let $|\psi\rangle$ be a zero-energy state and assume that $|\psi\rangle= {\mathfrak{Q}}|\phi\rangle$ for some vector $|\phi\rangle$. Then its square norm is $\langle \psi|\psi\rangle = \langle \psi|({\mathfrak{Q}}|\phi\rangle) = \left(\langle \phi|({\mathfrak{Q}}^\dagger |\psi\rangle)\right)^\ast = 0$ where we used . Hence, $|\psi\rangle=0$, which is a contradiction.
#### Cohomology.
\[lem:E0NotCoboundary\] suggests that we consider the kernel of the supercharge modulo its image in order to analyse the existence and the properties of the zero-energy states. We define therefore the *cohomology* of the supercharge $$\mathcal H^\bullet({\mathfrak{Q}}) = \bigoplus_{L=1}^\infty \mathcal H^L({\mathfrak{Q}}),$$ where $\mathcal H^1({\mathfrak{Q}}) = \text{ker}\{ {\mathfrak{Q}}:V^1\to V^{2}\}$ and $$\mathcal H^L({\mathfrak{Q}}) = \frac{\text{ker}\{ {\mathfrak{Q}}:V^L\to V^{L+1}\}}{\text{im}\{ {\mathfrak{Q}}:V^{L-1}\to V^{L}\}}, \quad \text{for}\quad L\geqslant 2.$$ The elements of $\mathcal H^L({\mathfrak{Q}})$ are equivalence classes or cohomology classes. Any cohomology class of $\mathcal H^L({\mathfrak{Q}})$ can be represented by a cocycle $|\psi\rangle\in V^L$, which is called *its* *representative*. Any two cocycles of $V^L$ differing by a coboundary represent the same element of $\mathcal H^L({\mathfrak{Q}})$. We denote the equivalence class of a cocycle $|\psi\rangle \in V^L$ by $[|\psi\rangle]$. Hence $[|\psi\rangle + {\mathfrak{Q}}|\phi\rangle] = [|\psi\rangle]$ for all $|\phi\rangle \in V^{L-1}$. If $\mathcal H^L({\mathfrak{Q}})=0$ for each $L\geqslant 1$ then we call the cohomology trivial. This is the case if and only if all cocycles are coboundaries.
It can be shown [@witten:82] that $\mathcal H^L({\mathfrak{Q}})$ is isomorphic to the subspace of $V^L$ that is spanned by the zero-energy states of the Hamiltonian $H$ for a chain of length $L$. Furthermore, if $|\phi\rangle \in V^L$ is a representative of a non-zero element of $\mathcal H^L({\mathfrak{Q}})$, then there is a state $|\phi'\rangle \in V^{L-1}$ such that $$|\psi\rangle = |\phi\rangle + {\mathfrak{Q}}|\phi'\rangle
\label{eqn:RepresentativeE0State}$$ is a zero-energy state [@witten:82]. In the following, we use this connection with cohomology classes and their representatives to investigate some properties of the zero-energy states of our models.
The case YNot0 {#sec:GenericY}
--------------
In this and the following subsection, we explicitly compute $\mathcal H^L({\mathfrak{Q}}_{j,k}(y))$. This computation allows us to characterise the space of zero-energy states of the Hamiltonian $H_{j,k}(y)$ as a function of the parameter $y$, the integer labels $j,k$ and the system size $L$.
Here, we consider the case $y\neq 0$. We prove the following theorem:
\[thm:CohomGenericY\] For $y\neq 0$ and each $j,k=0,\dots,\ell+1$, the cohomology $\mathcal{H}^\bullet ({\mathfrak{Q}}_{j,k}(y))$ is trivial.
This theorem implies that for $y\neq 0$ and any length of the chain $L$ the Hamiltonian $H_{j,k}(y)$ does not possess zero-energy states. Thus its spectrum is strictly positive.
The proof is based on two lemmas. The first lemma deals with a mapping $s$ that is akin to a so-called *contracting homotopy* [@loday:92].
\[lem:ContractingHomotopy\] Let ${\mathfrak{Q}}$ be an arbitrary supercharge. Suppose that for each $L\geqslant 2$ there is a mapping $s:V^L\to V^{L-1}$ such that $$s{\mathfrak{Q}}+{\mathfrak{Q}}s = 1.
\label{eqn:ContractingHomotopy}$$ Then for each $L\geqslant 2$, we have $\mathcal H^L({\mathfrak{Q}}) = 0.$
We show that any cocycle $|\psi\rangle \in V^L$ is a coboundary. Indeed, applying to $|\psi\rangle$ we obtain $$|\psi\rangle = (s{\mathfrak{Q}}+ {\mathfrak{Q}}s)|\psi\rangle = {\mathfrak{Q}}(s|\psi\rangle).$$ Hence, $\mathcal H^L({\mathfrak{Q}}) =0$.
Our aim is to construct such a mapping $s$ for the supercharge ${\mathfrak{Q}}_{j,k}(y)$. To this end, we use the vectors $|\xi_0(y)\rangle,\dots,|\xi_\ell(y)\rangle$, defined in . The second lemma needed for our proof of \[thm:CohomGenericY\] establishes that for non-vanishing $y$ these vectors span the Hilbert space $V$ of a single spin:
\[lem:BasisVectors\] For $y\neq 0$ the vectors $|\xi_0(y)\rangle,\dots,|\xi_\ell(y)\rangle$ constitute a basis of $V$.
The components of $|\xi_n(y)\rangle$ with respect to the canonical basis are given by $$\Xi_{mn} = \langle m |\xi_n(y)\rangle = x \frac{y^{m+1}}{\sqrt{\{m+1\}}}(1-q^{2(m+1)(n+1)}).$$ To prove the lemma, it is sufficient to show that the matrix $\Xi = (\Xi_{mn})_{m,n=0}^\ell$ is invertible. This is indeed the case. One checks that the entries of the inverse matrix are given by $$\left(\Xi^{-1}\right)_{mn} = - \frac{\sqrt{\{n+1\}}}{(\ell+2)xy^{n+1}}q^{-2(m+1)(n+1)}.$$ This ends the proof.
We now prove that if $y\neq 0$ then $\mathcal H^L({\mathfrak{Q}}_{j,k}(y))=0$ for each $L\geqslant 1$ and each $j,k=0,\dots,\ell+1$. For $L=1$, the proof is trivial: one readily checks that $\ker\{{\mathfrak{Q}}_{j,k}(y):V\to V^2\}=0$, using \[lem:BasisVectors\]. Hence, we focus on $L\geqslant 2$. The proof is based on the construction of a mapping $s_j$ that obeys for each $j=0,\dots,\ell+1$. We separately consider the cases $0\leqslant j\leqslant \ell$ and $j=\ell+1$.
Let us first consider $0\leqslant j\leqslant \ell$. It follows from \[lem:BasisVectors\] and standard properties of the tensor product that for $y\neq 0$, every vector $|\psi\rangle\in V^L$ can be written as $$|\psi\rangle = \sum_{m=0}^\ell |\xi_m(y)\rangle \otimes |\psi_m\rangle$$ with unique vectors $|\psi_0\rangle,\dots,|\psi_\ell\rangle \in V^{L-1}$. We define the mapping $s_j$ by $$s_j |\psi\rangle = |\psi_j\rangle.
\label{eqn:DefSj}$$ Using action of the local supercharge on the special basis vectors, it is easy to see that $(s_j {\mathfrak{Q}}(y) + {\mathfrak{Q}}(y) s_j)|\psi\rangle = -|\xi_j(y)\rangle\otimes|\psi_j\rangle = -|\xi_j(y)\rangle\otimes s_j|\psi\rangle$. We combine this identity with the definition of the supercharge and find that $$\label{eqn:CommQs}
s_j {\mathfrak{Q}}_{j,k}(y) + {\mathfrak{Q}}_{j,k}(y) s_j= 1$$ for each $j=0,\dots,\ell$ and $k=0,\dots,\ell+1$.
Second, for $j=\ell+1$, we define $$s_{\ell+1}=-\sum_{j=0}^\ell s_j.$$ Using the definition of $s_j$, it is easy to see that holds for $j=\ell+1$ and $k=0,\dots,\ell+1$, too.
In both cases, it follows from \[lem:ContractingHomotopy\] $\mathcal H^L({\mathfrak{Q}}_{j,k}(y))=0$ for any $L\geqslant 2$ and each $j,k=0,\dots,\ell+1$. This ends the proof of the theorem.
We notice that the proof only relies on the existence of a basis $|\xi_0(y)\rangle,\dots,|\xi_\ell(y)\rangle$ of $V$ with the property ${\mathfrak{q}}(y)|\xi_k(y)\rangle=|\xi_k(y)\rangle\otimes |\xi_k(y)\rangle$ for each $k=0,\dots,\ell$. This is the case for a variety of other physically-relevant spin chains. An example is the quantum spin $1/2$ XYZ chain along a special line of couplings [@hagendorf:13].
The case YIs0 {#sec:Y0}
-------------
The proof of \[thm:CohomGenericY\] is not generalisable to the cases $y=0$ and $y \to \infty$. They have to be separately treated. In this subsection, we consider the case $y = 0$. For this value of $y$, the local supercharge reduces to ${\mathfrak{q}}(y=0) = {\mathfrak{q}}$, defined in . The supercharge ${\mathfrak{Q}}_{j,k}(y=0)$ is independent of the indices $j,k$ and we simply denote it by ${\mathfrak{Q}}$. Its cohomology is non-trivial. The results presented here below can easily be modified in order to cover the case $y\to \infty$. In this case, the local supercharge reduces to ${\mathfrak{q}}(y\to \infty)=\bar {\mathfrak{q}}$, which is the image of ${\mathfrak{q}}$ under spin reversal. In fact, the representatives of $\mathcal H^L(\bar {\mathfrak{Q}})$ are simply obtained by applying the spin-reversal operator to the representatives of $\mathcal H^L({\mathfrak{Q}})$.
The main result of this subsection is the following theorem:
\[thm:OCCohom\] $\mathcal H^L(\mathfrak Q)$ is spanned by the cohomology class of the state[^4]
\[eqn:RepE0States\] $$|\underset{n\,\text{times}}{\underbrace{{\protect\raisebox{0.25ex}{$\chi$}}\cdots{\protect\raisebox{0.25ex}{$\chi$}}}}\rangle\quad \text{if}\quad L=2n,$$ and $$|0\rangle\otimes |\hspace{-.12cm}\underset{n-1\,\text{times}}{\underbrace{{\protect\raisebox{0.25ex}{$\chi$}}\cdots{\protect\raisebox{0.25ex}{$\chi$}}}}\hspace{-.12cm}\rangle \quad \text{if}\quad L=2n-1,$$
where $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle\in V^2$ is the state defined in and $n$ a positive integer.
This theorem implies that for $y=0$ the spin-chain Hamiltonian possesses a zero-energy state for each length $L$. The state is unique up to normalisation. This result is expected from the above-mentioned mapping between the spin-chain models at $y=0$ and the $M_\ell$ models of supersymmetric fermions on open intervals. For these fermion models, the dimension of the space of zero-energy states was indeed shown to be one-dimensional in previous works [@huijse:10_1; @huijse:15]. To our knowledge, the structure of the corresponding cohomology has however so far remained undetermined and unexploited. \[thm:OCCohom\] provides this structure for the spin-chain models. We use it in the forthcoming sections in order to compute certain non-trivial scalar products that involve the zero-energy states.
The proof of \[thm:OCCohom\] is based on several lemmas. They establish the existence of an explicit bijection between $\mathcal H^{L}({\mathfrak{Q}})$ and $\mathcal H^{L+2}({\mathfrak{Q}})$ for each $L\geqslant 1$. Hence, we may construct $\mathcal H^{L}({\mathfrak{Q}})$ from $\mathcal H^{1}({\mathfrak{Q}})$ and $\mathcal H^{2}({\mathfrak{Q}})$. We explicitly compute them in the following lemma:
\[lem:lowdim\] $\mathcal H^1({\mathfrak{Q}})$ and $\mathcal H^2({\mathfrak{Q}})$ are spanned by the cohomology classes of the states $|0\rangle$ and $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$, respectively.
For $L=1$, recall that $\mathcal H^1({\mathfrak{Q}}) =\ker\{{\mathfrak{Q}}:V^1\to V^2\}$. According to , the only solution to ${\mathfrak{q}}|\psi\rangle = 0$ is $|\psi\rangle = |0\rangle$, up to a factor.
For $L=2$, we consider a cocycle $|\psi\rangle \in V^2$. We write $\psi_{m,n} = \langle m,n|\psi\rangle$ for its components with respect to the canonical basis of the Hilbert space. From ${\mathfrak{Q}}|\psi\rangle = 0$, it follows that $$\psi_{m,\ell}=\psi_{\ell,m} =0 \quad \text{for } m = 1,\dots,\ell
\label{eqn:APsi1}$$ and $$a_{m,0}\psi_{m,n} = a_{n+1,0} \psi_{m-1,n+1}\quad \text{for } m = 1,\dots,\ell\text{ and } n=0,\dots,\ell-1.
\label{eqn:APsi2}$$ We distinguish three cases. *(i)* For $m+n > \ell$, we find from the relation $$\psi_{m,n} = \psi_{m+n-\ell,\ell}\prod_{j=n+1}^\ell \frac{a_{j,0}}{a_{m+n+1-j,0}}.
\label{eqn:PsiRec}$$ From , it follows that $\psi_{m,n}=0$. *(ii)* For $m+n=\ell$, the relation still holds. The explicit form of the coefficients $a_{m,k}$, defined in , leads to $$\psi_{m,\ell-m} = \psi_{0,\ell} {\protect\raisebox{0.25ex}{$\chi$}}_m$$ where ${\protect\raisebox{0.25ex}{$\chi$}}_m,\, m=0,\dots,\ell,$ are the components of the vector $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$ defined in . *(iii)* For $m+n< \ell$, we obtain from the relation $$\psi_{m,n} = \psi_{0,m+n}\prod_{j=1}^m \frac{a_{n+j,0}}{a_{j,0}} = \psi_{0,m+n} \frac{a_{m+n+1,m}}{a_{m+n+1,0}}.
\label{eqn:PsiRec2}$$
Combining *(i)*, *(ii)* and *(iii)*, we find after some algebra $$|\psi\rangle = \psi_{0,\ell}|{\protect\raisebox{0.25ex}{$\chi$}}\rangle + {\mathfrak{Q}}\left(\sum_{p=0}^{\ell-1} \frac{\psi_{0,p}}{a_{p+1,0}}|p+1\rangle\right).$$ Hence $[|\psi\rangle]=\psi_{0,\ell}[|{\protect\raisebox{0.25ex}{$\chi$}}\rangle]$ with an arbitrary coefficient $\psi_{0,\ell}$. The cohomology class of $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$ cannot be zero since this state is a linear combination of basis states of $V^2$ which are clearly not in the image of ${\mathfrak{q}}$.
Our next aim is to study $\mathcal H^L(\mathfrak Q)$ for $L\geqslant 3$. In the following technical lemma, we determine a convenient choice of their representatives.
\[lem:technical\] For each $L\geqslant3$ any element in $\mathcal H^L(\mathfrak Q)$ can be represented by a cocycle $|\psi\rangle \in V^L$ with $|\psi\rangle = \sum_{m=0}^\ell |\psi_m\rangle \otimes |m\rangle$ such that $|\psi_0\rangle = |\psi'_{\ell,0}\rangle \otimes |\ell\rangle$ for some vector $|\psi'_{\ell,0}\rangle\in V^{L-2}$.
Let us a consider the cocycle $|\psi'\rangle$ representing an element of $\mathcal H^L({\mathfrak{Q}})$. Then for any $|\phi\rangle\in V^{L-1}$ the vector $|\psi\rangle = |\psi'\rangle+{\mathfrak{Q}}|\phi\rangle$ is also a cocycle, representing the same element of $\mathcal H^L({\mathfrak{Q}})$. We write the vector $|\psi\rangle$ (and likewise $|\psi'\rangle$, $|\phi\rangle$) as a superposition $$|\psi\rangle = \sum_{m=0}^\ell |\psi_m\rangle \otimes |m\rangle, \quad |\psi_m\rangle \in V^{L-1}.
\label{eqn:decomposition}$$ This leads to $$|\psi_m\rangle = |\psi'_m\rangle+{\mathfrak{Q}}|\phi_m\rangle+(-1)^L \sum_{k=m+1}^\ell a_{k,m}|\phi_k\rangle \otimes |k-m-1\rangle.$$ In order to prove the lemma, we consider the case $m=0$: $$\label{eqn:psi0}
|\psi_0\rangle = |\psi'_0\rangle+{\mathfrak{Q}}|\phi_0\rangle+(-1)^L \sum_{k=1}^\ell a_{k,0}|\phi_k\rangle \otimes |k-1\rangle.$$ Again, we decompose the vector $|\psi'_0\rangle$ with respect to the last site, $|\psi'_0\rangle = \sum_{m=0}^\ell|\psi'_{m,0}\rangle \otimes |m\rangle$, and substitute this decomposition into . The choices $$|\phi_0\rangle = 0, \quad \text{and} \quad |\phi_k\rangle = (-1)^{L+1}a_{k,0}^{-1} |\psi'_{k-1,0}\rangle, \quad \text{for }k=1,\dots,\ell$$ lead to $|\psi_0\rangle = |\psi_{\ell,0}'\rangle\otimes |\ell\rangle$. This ends the proof.
For the next two lemmas, we introduce the operator $S$ which acts on any vector $|\psi\rangle \in V^L,\,L\geqslant 1,$ according to $$S|\psi\rangle = |\psi\rangle \otimes |{\protect\raisebox{0.25ex}{$\chi$}}\rangle.$$ One checks that it commutes with the supercharge ${\mathfrak{Q}}$, $$S {\mathfrak{Q}}= {\mathfrak{Q}}S$$ because $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$ is annihilated by the supercharge. It follows that $S$ can be extended to a mapping $S^\sharp$ defined on the cohomology [@loday:92; @masson:08]: Its action on the cohomology classes is given by $S^\sharp[|\psi\rangle] = [S|\psi\rangle]$.
\[lem:surjectivity\] For each $L\geqslant 1$ the mapping $S^\sharp:\mathcal H^{L}({\mathfrak{Q}})\to \mathcal H^{L+2}({\mathfrak{Q}})$ is surjective.
Let $|\psi\rangle$ be a cocycle representing an element of $\mathcal H^{L+2}({\mathfrak{Q}})$. We decompose it as in with respect to the last site. The equation ${\mathfrak{Q}}|\psi\rangle = 0$ leads to $${\mathfrak{Q}}|\psi_m\rangle = (-1)^{L+1}\sum_{k=m+1}^\ell a_{k,m}|\psi_k\rangle \otimes |k-(m+1)\rangle.$$ Let us consider the case $m=0$. From \[lem:technical\] we know that without loss of generality $|\psi\rangle$ can be chosen in such a way that $|\psi_0\rangle = |\psi_{\ell,0}\rangle\otimes |\ell\rangle$ for some state $|\psi_{\ell,0}\rangle\in V^{L}$. This choice leads to $${\mathfrak{Q}}|\psi_{\ell,0}\rangle \otimes |\ell\rangle +(-1)^{L+1} |\psi_{\ell,0}\rangle \otimes \sum_{k=0}^{\ell-1}a_{\ell, k}|k,\ell-k-1\rangle=(-1)^{L+1}\sum_{k=1}^\ell a_{k,0}|\psi_k\rangle \otimes |k-1\rangle.$$ Comparing both sides, we obtain $${\mathfrak{Q}}|\psi_{\ell,0}\rangle =0,\quad\text{and}\quad |\psi_k\rangle = \left(\frac{a_{\ell,\ell-k}}{a_{k,0}}\right)|\psi_{\ell,0}\rangle \otimes |\ell-k\rangle, \quad \text{for } k=1,\dots,\ell.$$ According to and , $a_{\ell,\ell-k}/a_{k,0}={\protect\raisebox{0.25ex}{$\chi$}}_k$. Hence, we find $$|\psi\rangle = |\psi_{\ell,0}\rangle\otimes |\ell, 0\rangle + |\psi_{\ell,0}\rangle \otimes \sum_{k=1}^\ell {\protect\raisebox{0.25ex}{$\chi$}}_k|\ell-k,k\rangle = S|\psi_{\ell,0}\rangle$$ with a cocycle $|\psi_{\ell,0}\rangle \in V^{L}$. For the corresponding cohomology classes we find thus $[|\psi\rangle]=S^\sharp[|\psi_{\ell,0}\rangle]$.
\[lem:injectivity\] For each $L\geqslant 1$ the mapping $S^\sharp:\mathcal H^{L}({\mathfrak{Q}})\to \mathcal H^{L+2}({\mathfrak{Q}})$ is injective.
Consider an element of $\ker\{ S^\sharp:\mathcal H^{L}({\mathfrak{Q}})\to\mathcal H^{L+2}({\mathfrak{Q}})\}$. It can be represented by a cocycle $|\psi\rangle \in V^L$ such that $S|\psi\rangle = {\mathfrak{Q}}|\phi\rangle$ for some vector $|\phi\rangle\in V^{L+1}$. As before, it is useful to decompose the state with respect to the last site: $|\phi\rangle=\sum_{m=0}^\ell |\phi_m\rangle \otimes |m\rangle$. We find $$S|\psi\rangle = |\psi\rangle \otimes |{\protect\raisebox{0.25ex}{$\chi$}}\rangle = \sum_{m=0}^\ell {\mathfrak{Q}}|\phi_m\rangle \otimes |m\rangle + (-1)^{L+1} \sum_{m=0}^\ell\sum_{k=0}^{m-1}a_{m,k}|\phi_m\rangle \otimes |k,m-k-1\rangle.$$ We select on both sides the terms corresponding to $|0\rangle$ on the last site, and find $${\protect\raisebox{0.25ex}{$\chi$}}_\ell|\psi\rangle \otimes |\ell\rangle = {\mathfrak{Q}}|\phi_0\rangle + (-1)^{L+1}\sum_{m=0}^\ell a_{m,m-1} |\phi_m\rangle \otimes |m-1\rangle.
\label{eqn:intermed}$$ Notice that the sum on the right-hand side does not contain any term proportional to $|\ell\rangle$ on the last site. ${\mathfrak{Q}}|\phi_0\rangle$ may however contain such a term. To see this, we decompose $|\phi_0\rangle = \sum_{m=0}^\ell|\phi_{m,0}\rangle\otimes |m\rangle$. We apply the supercharge to this decomposition, insert it into and obtain $${\protect\raisebox{0.25ex}{$\chi$}}_\ell|\psi\rangle \otimes |\ell\rangle= {\mathfrak{Q}}|\phi_{\ell,0}\rangle \otimes |\ell\rangle + \sum_{k=0}^{\ell-1}|\tilde \phi_k\rangle \otimes |k\rangle.$$ The states $|\tilde \phi_k\rangle$ can in principle be computed but we won’t need them. This equality implies $|\psi\rangle = {\mathfrak{Q}}|\phi_{\ell,0}\rangle$. Hence, $|\psi\rangle$ is a coboundary. We conclude that $\ker \{S^\sharp :\mathcal H^{L}({\mathfrak{Q}})\to\mathcal H^{L+2}({\mathfrak{Q}})\}=0$. This proves the claim.
We are now ready to prove the main result of this section.
From \[lem:surjectivity,lem:injectivity\] we conclude for each $L\geqslant 1$ the mapping $S^\sharp: \mathcal H^{L}({\mathfrak{Q}})\to \mathcal H^{L+2}({\mathfrak{Q}})$ is both surjective and injective. Hence $ \mathcal H^{L+2}({\mathfrak{Q}})$ is isomorphic to $\mathcal H^{L}({\mathfrak{Q}})$. By transitivity, we obtain $$\begin{aligned}
\mathcal H^{2n-1}({\mathfrak{Q}}) = (S^{\sharp})^{n-1}\mathcal H^{1}({\mathfrak{Q}}),\quad
\mathcal H^{2n}({\mathfrak{Q}}) = (S^{\sharp})^{n-1}\mathcal H^{2}({\mathfrak{Q}})
\end{aligned}$$ for each $n\geqslant 1$. $\mathcal H^{1}({\mathfrak{Q}})$ and $\mathcal H^{2}({\mathfrak{Q}})$ were obtained in \[lem:lowdim\]. This allows us to compute representatives of the elements of $\mathcal H^L({\mathfrak{Q}})$ for odd and even $L$ from the repeated action of $S$ on $|0\rangle$ and $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$, respectively, which leads to .
Components and scalar products of the zero-energy states {#sec:SP}
========================================================
In this section, we analyse the zero-energy states of the spin-chain Hamiltonians $H$ with $y=0$. Our main goal is to unveil some of their properties with the help of \[thm:OCCohom\]. In \[sec:Representations\], we discuss two representations of the zero-energy states arising from the representatives of the cohomology and their homology analogues. We deduce from these representations their magnetisation, parity and relations between certain components. In \[sec:ScalarProducts\], we introduce a family of physically-interesting scalar products that involve an arbitrary number of zero-energy states for systems of different lengths. We use the supersymmetry to show that the knowledge of a single special component of each involved state is sufficient to evaluate the scalar product.
Representation of zero-energy states {#sec:Representations}
------------------------------------
We denote by $|\psi_L\rangle$ a zero-energy state of the Hamiltonian $H$ for a chain of length $L$. For convenience, we allow $L=1$ even though $H$ is not defined for chains consisting of a single site. Furthermore, we write $$(\psi_L)_{m_1m_2\dots m_L} = \langle m_1m_2\cdots m_L|\psi_L\rangle$$ for its components with respect to the canonical basis of $V^L$.
#### Cohomology representation.
It follows from \[thm:OCCohom\] that $|\psi_L\rangle$ can be written as the sum of a multiple of a special representative of $\mathcal H^L({\mathfrak{Q}})$ and an element of the image of the supercharge. Specifically, we have
\[prop:CohomRepr\] For each $n\geqslant 1$, we have $$|\psi_{2n}\rangle = \lambda_{2n}|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle +{\mathfrak{Q}}|\phi_{2n}\rangle, \quad \lambda_{2n}=(\psi_{2n})_{0\ell\cdots 0\ell},
\label{eqn:psi2n}$$ with $|\phi_{2n}\rangle\in V^{2n-1}$ and $$|\psi_{2n-1}\rangle =\lambda_{2n-1} |0\rangle\otimes |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle +{\mathfrak{Q}}|\phi_{2n-1}\rangle, \quad \lambda_{2n-1}= (\psi_{2n-1})_{0\ell\cdots 0\ell 0},
\label{eqn:psi2n1}$$ with $|\phi_{2n-1}\rangle\in V^{2(n-1)}$.
For simplicity, we focus on the case $L=2n$. According to and \[thm:OCCohom\], there is a scalar $\lambda_{2n}$ and a state $|\phi_{2n}\rangle\in V^{2n-1}$ such that the zero-energy state can be written as $$|\psi_{2n}\rangle = \lambda_{2n} |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle +{\mathfrak{Q}}|\phi_{2n}\rangle.$$ The state $|\phi_{2n}\rangle$ cannot be obtained from $\mathcal H^{2n}({\mathfrak{Q}})$. It is fixed by the requirement that the zero-energy state be annihilated by the adjoint supercharge.
We notice however that the unknown term ${\mathfrak{Q}}|\phi_{2n}\rangle$ does not contribute to the scalar product of the zero-energy state with any vector $|\omega\rangle \in \text{ker}\{{\mathfrak{Q}}^\dagger:V^{2n}\to V^{2n-1}\}$. Indeed, we have $$\begin{aligned}
\langle \omega |\psi_{2n}\rangle &= \lambda_{2n} \langle \omega |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle+ \langle \omega |{\mathfrak{Q}}|\phi_{2n}\rangle\nonumber= \lambda_{2n} \langle \omega |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle+ \langle \phi_{2n} |{\mathfrak{Q}}^\dagger|\omega\rangle^\ast\nonumber \\
&=\lambda_{2n} \langle \omega |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle \label{eqn:SPPsi2n}.\end{aligned}$$ A simple choice for $|\omega\rangle$ is given by a canonical basis vector $ |\omega\rangle = |m_1,\dots,m_{2n}\rangle \in
V^{2n}$ that *(i)* has a non-zero scalar product with $|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle$ and *(ii)* is annihilated by the adjoint supercharge. The requirement *(i)* leads to the constraint
\[eqn:ConstraintsBasisVector\] $$m_{2i-1}+m_{2i} = \ell, \quad i=1,\dots,n,$$ because the magnetisation of the state $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$ vanishes. The requirement *(ii)* leads to the constraint $$m_{2i}+m_{2i+1} \geqslant \ell, \quad i=1,\dots,n-1,$$ which follows from the action of the adjoint local supercharge ${\mathfrak{q}}^\dagger$.
The constraints have many solutions. One solution is given by $$|\omega \rangle =|0\ell \cdots 0\ell\rangle.$$ Using , we find that $\langle \omega |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle = 1$ and thus $\lambda_{2n}= \langle \omega |\psi_{2n}\rangle = (\psi_{2n})_{0\ell\cdots 0\ell}$. This concludes the proof for $L=2n$.
The proof for $L=2n-1$ is similar and uses the vector $|\omega \rangle = |0\ell\cdots 0\ell 0\rangle$.
The representations of the zero-energy states given in and allow us to derive a number of simple properties of the zero energy states. Two immediate consequences are:
\[corr:NonZeroComps\] For each $n\geqslant 1$ the components $(\psi_{2n})_{0\ell\cdots 0\ell}$ and $(\psi_{2n-1})_{0\ell\cdots 0\ell 0}$ are non-zero.
Consider $L=2n$ and suppose that $(\psi_{2n})_{0\ell\cdots 0\ell}=0$. Then it follows from that $|\psi_{2n}\rangle \in \text{im}\,{\mathfrak{Q}}$ which contradicts \[lem:E0NotCoboundary\]. The case $L=2n-1$ is completely analogue.
\[corr:Magnetisation\] We have $M|\psi_{2n}\rangle = 0$ and $M|\psi_{2n-1}\rangle = (\ell/2)|\psi_{2n-1}\rangle$.
The $E=0$ eigenspace of $H$ is one-dimensional. Furthermore the Hamiltonian conserves the magnetisation $[H,M]=0$. Hence, we must have $M|\psi_{L}\rangle = m_L|\psi_L\rangle$ for any $L\geqslant 1$. To find $m_L$, it is sufficient to project this equality onto simple basis vectors. For $L=2n$, we find $$0 = \langle 0\ell \cdots 0\ell|M|\psi_{2n}\rangle = m_{2n} \langle 0\ell \cdots 0\ell|\psi_{2n}\rangle.$$ According to \[corr:NonZeroComps\], $\langle 0\ell \cdots 0\ell |\psi_{2n}\rangle = (\psi_{2n})_{0\ell\cdots 0\ell}$ is non-zero and therefore $m_{2n}= 0$. For $L=2n-1$ the proof is similar.
For $L=2n$, the constraints have many solutions. This allows us to relate certain components of the zero-energy state to the special component $(\psi_{2n})_{0\ell\cdots 0\ell}$.
\[prop:SpecialCompL2n\] For any weakly increasing sequence $p_1,\dots,p_n$ of integers, we have the component $$(\psi_{2n})_{p_1,\ell-p_1,\dots,p_n,\ell-p_n} =\frac{(\psi_{2n})_{0\ell\cdots 0\ell}}{\prod_{i=1}^n \{p_i+1\}}.
\label{eqn:lambda2n}$$
It is trivial to check that holds for $m_{2i-1}=p_i$ and $m_{2i}=\ell-p_i$ where $i=1,\dots,n$. Hence, using and we find $$\begin{aligned}
(\psi_{2n})_{p_1,\ell-p_1,\dots,p_n,\ell-p_n} &= (\psi_{2n})_{0\ell\cdots 0\ell}\langle p_1,\ell-p_1,\dots,p_n,\ell-p_n|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle
\nonumber
\\
&= (\psi_{2n})_{0\ell\cdots 0\ell}\prod_{i=1}^n{\protect\raisebox{0.25ex}{$\chi$}}_{p_i}=\frac{(\psi_{2n})_{0\ell\cdots 0\ell}}{\prod_{i=1}^n \{p_i+1\}},
\end{aligned}$$ which concludes the proof.
The \[prop:CohomRepr,prop:SpecialCompL2n\] imply that the zero-energy state is even under the action of the parity operator.
For any $L\geqslant 1$ we have $P|\psi_L\rangle = |\psi_L\rangle$.
The proof follows the lines of \[corr:Magnetisation\], using \[prop:SpecialCompL2n\] in the case of even $L$.
#### Homology representation.
Up to now, we have focused on the cohomology of the supercharge ${\mathfrak{Q}}$ and the resulting representations of the zero-energy states given in \[prop:CohomRepr\]. The definition $H = {\mathfrak{Q}}{\mathfrak{Q}}^\dagger + {\mathfrak{Q}}^\dagger {\mathfrak{Q}}$ of a supersymmetric Hamiltonian suggests that we could as well have considered the adjoint supercharge ${\mathfrak{Q}}^\dagger$. Its *homology* is defined by $$\mathcal H_\bullet({\mathfrak{Q}}) = \bigoplus_{L=1}^\infty \mathcal H_L({\mathfrak{Q}})$$ where $\mathcal H_1({\mathfrak{Q}}^\dagger) = V/ \text{im} \{{\mathfrak{Q}}^\dagger : V^{2} \to V\}$ and $$\mathcal H_L({\mathfrak{Q}}^\dagger) = \frac{\ker \{{\mathfrak{Q}}^\dagger : V^L \to V^{L-1}\}}{\text{im} \{{\mathfrak{Q}}^\dagger : V^{L+1} \to V^{L}\}}, \quad \text{for}\quad L\geqslant 2.$$ The existence of a non-degenerate inner product on $V^L$ implies that the cohomology and homology are isomorphic [@loday:92; @masson:08]: $$\mathcal H_L({\mathfrak{Q}}^\dagger)\simeq \mathcal H^L({\mathfrak{Q}})\quad \text{for each}\quad L\geqslant 1.$$ In particular, we have $$\dim \mathcal H_L({\mathfrak{Q}}^\dagger)=\dim \mathcal H^L({\mathfrak{Q}})=1\quad \text{for each}\quad L\geqslant 1.$$ We now determine an alternative representation of the zero-energy states using this property.
\[prop:HomRepr\] For each $n\geqslant 1$, we have $$\label{eqn:AdjointRepresentation}
|\psi_{2n}\rangle = \mu_{2n} |0\ell\cdots 0\ell\rangle + {\mathfrak{Q}}^\dagger |\tilde \phi_{2n}\rangle, \quad \mu_{2n} = \langle {\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}|\psi_{2n}\rangle,$$ with $ |\tilde \phi_{2n}\rangle\in V^{2n+1}$ and $$|\psi_{2n-1}\rangle = \mu_{2n-1} |0\ell\cdots 0\ell 0\rangle + {\mathfrak{Q}}^\dagger |\tilde \phi_{2n-1}\rangle, \quad \mu_{2n-1} = \left(\langle 0|\otimes \langle {\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}|\right)|\psi_{2n-1}\rangle,$$ with $|\tilde \phi_{2n-1}\rangle\in V^{2n}$.
We focus on the case $L=2n$. Since $\mathcal H_{2n}({\mathfrak{Q}}^\dagger)$ is one-dimensional, each of its non-zero elements can be represented by a non-zero multiple of a fixed vector in $V^{2n}$. This vector is in the kernel (but not in the image) of ${\mathfrak{Q}}^\dagger$. We claim that such a vector is given by $$|\omega\rangle = |0\ell\cdots 0\ell\rangle.$$ Using , one readily checks that $|\omega\rangle$ is annihilated by ${\mathfrak{Q}}^\dagger$. Furthermore, it cannot be in the image of ${\mathfrak{Q}}^\dagger$. Indeed, otherwise if $|\omega\rangle = {\mathfrak{Q}}^\dagger |\tilde \omega\rangle$ for some $|\tilde\omega\rangle \in V^{2n+1}$ then it follows that $(\psi_{2n})_{0\ell\cdots 0\ell}= \langle \omega|\psi_{2n}\rangle= \langle \tilde \omega|{\mathfrak{Q}}|\psi_{2n}\rangle=0$. This contradicts \[corr:NonZeroComps\].
It follows that the zero-energy states for $L=2n$ have the representation $$|\psi_{2n}\rangle = \mu_{2n} |0\ell\cdots 0\ell\rangle + {\mathfrak{Q}}^\dagger |\tilde \phi_{2n}\rangle$$ for some non-zero scalar $\mu_{2n}$ and $|\tilde \phi_{2n}\rangle\in V^{2n+1}$. The vector $|\tilde \phi_{2n}\rangle$ cannot be determined from homological arguments. The factor $\mu_{2n}$, however, can be found by computing the scalar product of the zero-energy state with suitable states $|\gamma\rangle$ that are annihilated by the supercharge ${\mathfrak{Q}}$ and have a non-zero scalar product with the representative. In the present case, we choose $|\gamma\rangle = |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle$. A short calculation, similar to , leads to $$\mu_{2n} = \langle {\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}|\psi_{2n}\rangle.$$
The argument for $L=2n-1$ is similar, with the choice ${|\gamma\rangle}={|0\rangle}\otimes{|{\protect\raisebox{0.25ex}{$\chi$}}\dots {\protect\raisebox{0.25ex}{$\chi$}}\rangle}$.
Square norm and scalar products {#sec:ScalarProducts}
-------------------------------
In this subsection, we use the \[prop:CohomRepr,prop:HomRepr\] in order to derive sum rules for the square norm of the zero-energy states and certain scalar products.
An immediate consequence of the (co)homology representations of the zero-energy states given in \[prop:CohomRepr,prop:HomRepr\] is:
For each $n\geqslant 1$, we have
\[eqn:SquareNorm\] $$||\psi_{2n}||^2 = (\psi_{2n})_{0\ell\cdots 0 \ell}\langle \psi_{2n}|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle$$ and $$||\psi_{2n-1}||^2 = (\psi_{2n-1})_{0\ell\cdots 0 \ell 0}\langle \psi_{2n-1}|\Bigl( |0\rangle\otimes |{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle\Bigr).$$
For $L=2n$, we find $$\begin{aligned}
||\psi_{2n}||^2 &= (\mu_{2n}^\ast \langle 0\ell\cdots 0\ell| + \langle \tilde{\phi}_{2n}|{\mathfrak{Q}})|\psi_{2n}\rangle = \mu_{2n}^\ast (\psi_{2n})_{0\ell\cdots 0 \ell}\\
&=(\psi_{2n})_{0\ell\cdots 0 \ell}\langle \psi_{2n}|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle \nonumber.
\end{aligned}$$ For $L=2n-1$ the proof is similar.
The expressions for the square norm motivate the analysis of a family of scalar products involving the zero-energy states of arbitrary length. To be specific, let us consider $L=2n$. We find that $$\frac{\langle \psi_{2n}|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle}{||\psi_{2n}||} = \frac{1}{(\psi_{2n})_{0\ell\cdots 0 \ell}/||\psi_{2n}||}.
\label{eqn:SimpleExample}$$ On the left-hand side of this equality, we have the projection of the normalised zero-energy state onto an $n$-fold tensor product of zero-energy states of the two-site chain. On the right-hand site, we find the inverse of a special component of the normalised zero-energy state on $2n$ sites.
It is natural to investigate if the result remains equally simple when we replace the $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle$’s by the zero-energy energy states of spins chains of generic lengths. To this end, we cut an open chain of length $L$ into $m$ subchains of lengths $L_1,\dots,L_m>0$ as is illustrated in \[fig:Subdivision\].
(0,0) – (4,0); (4,0) – (5,0); (5,0) – (8,0); in [0,2,3.5,5.5,8]{} (0,-.1) – (0,.1); (0,-.4) – (2,-.4); (2,-.4) – (3.5,-.4); (5.5,-.4) – (8,-.4); (0,.4) – (8,.4);
in [0,0.25,...,3.75]{} (+.125,0) circle (1.25pt);
in [5,5.25,...,7.75]{} (+.125,0) circle (1.25pt);
(4.55,-0.4) node [$\cdots$]{}; (4,.4) node\[above\] [$L$]{}; (1.1,-.4) node\[below\] [$L_1$]{}; (2.8,-.4) node\[below\] [$L_2$]{}; (6.8,-.4) node\[below\] [$L_m$]{};
Our aim is to compute projection of the zero-energy state of the complete chain onto the tensor product of the zero-energy states of these $m$ subchains. We consider thus the scalar product
$$Z(L_1,\dots,L_m) = \frac{\langle \psi_L|\left(\bigotimes_{j=1}^m|\psi_{L_j}\rangle\right)}{||\psi_L||\prod_{j=1}^m ||\psi_{L_j}||}, \quad\text{for} \quad L= L_1+\dots+L_m .
\label{eqn:MultipleOverlap}$$
The division by the norms of the states makes this quantity normalisation-independent. For $L=2n$ and $L_j=2$ for $j=1,\dots,n$, we recover (up to factor) the left-hand side of .
\[thm:ResultForZ\] If $L_1,\dots,L_m$ are even, then $$Z(L_1,\dots,L_m) = \frac{\prod_{j=1}^m (\psi_{L_j})_{0\ell\cdots 0\ell }/ ||\psi_{L_j}||}{ (\psi_L)_{0\ell \cdots 0\ell }/ ||\psi_L||}.
\label{eqn:Zeven}$$ If $L_k$ is odd (for some $1\leqslant k\leqslant m$) and $L_1,\dots,L_{k-1},L_{k+1},\dots,L_m$ are even, then $$Z(L_1,\dots,L_m) = \frac{(\psi_{L_k})_{0\ell 0\cdots \ell 0}/ ||\psi_{L_k}||\prod_{j=1,j\neq k}^m (\psi_{L_j})_{0\ell \cdots 0\ell }/ ||\psi_{L_j}||}{ (\psi_L)_{0\ell \cdots 0\ell 0}/ ||\psi_L||}.
\label{eqn:Zoddeven}$$ In all other cases, the scalar product vanishes.
We note that the results and are remarkably simple. The zero-energy states typically are very complicated states with many non-zero components. Nonetheless, the scalar product can be inferred from the sole knowledge of a single component of each involved (normalised) state. Furthermore, the exchange of $L_i$ and $L_j$ for $i\neq j$ leaves the result invariant even though it can completely change the subdivision as shown in \[fig:Subdivision\].
For simplicity, we focus on the proof of . Using , we find $$\langle \psi_L|\Biggl(\bigotimes_{j=1}^m|\psi_{L_j}\rangle\Biggr) = \mu_L^\ast \prod_{j=1}^m (\psi_{L_j})_{0\ell \cdots 0\ell}+ \langle \tilde{\phi}_L|{\mathfrak{Q}}\Biggl(\bigotimes_{j=1}^m|\psi_{L_j}\rangle\Biggr).$$ The second term on the right-hand side of this equation vanishes. Indeed, we obtain $${\mathfrak{Q}}\Biggl(\bigotimes_{j=1}^m|\psi_{L_j}\rangle\Biggr) = \sum_{i=1}^m (-1)^{L_1+\dots+L_{i-1}}\bigotimes_{j=1}^{i-1}|\psi_{L_j}\rangle\otimes {\mathfrak{Q}}|\psi_{L_i}\rangle\otimes \bigotimes_{j=i+1}^{m}|\psi_{L_j}\rangle$$ which is zero because ${\mathfrak{Q}}|\psi_{L_i}\rangle=0$ for $i=1,\dots,m$. Hence, we find $$\langle \psi_L|\Biggl(\bigotimes_{j=1}^m|\psi_{L_j}\rangle\Biggr) = \mu_L^\ast \prod_{j=1}^m (\psi_{L_j})_{0\ell \cdots 0\ell}.$$ Using , we write $\mu_L^\ast$ in terms of the square norm and the alternating component. Upon division by the norms of the zero-energy states, we obtain .
The proof of is similar. Eventually, the scalar product vanishes in all other cases because of the definite magnetisation of the zero-energy states.
There are many generalisations of that have equally simple expressions in terms of special components. As an example, we consider the product
$$\tilde Z(L_1,\dots, L_m) = \frac{\langle \psi_L|\left(|\psi_{L_1}\rangle\otimes \bigotimes_{j=2}^{m}(|\ell\rangle \otimes |\psi_{L_j}\rangle)\right)}{||\psi_L||\prod_{j=1}^m ||\psi_{L_j}||}$$
for $$L=L_1+\dots+L_m +m-1.$$
\[thm:ResultForZTilde\] If $L_1,\dots,L_m$ are odd then $$\tilde Z(L_1,\dots,L_m) = \frac{ (\psi_L)_{0\ell \cdots 0\ell 0}/ ||\psi_L||}{\prod_{j=1}^m (\psi_{L_j})_{0\ell\cdots 0\ell 0}/ ||\psi_{L_j}||}.
\label{eqn:ZTildeodd}$$ If $L_k$ is even (for some $1\leqslant k\leqslant m$) and $L_1,\dots,L_{k-1},L_{k+1},\dots,L_m$ are odd, then $$\tilde Z(L_1,\dots,L_m) =\frac{ (\psi_L)_{0\ell \cdots 0\ell}/ ||\psi_L||}{(\psi_{L_k})_{0\ell \cdots 0\ell}/ ||\psi_{L_k}||\prod_{j=1,j\neq k}^m (\psi_{L_j})_{0\ell \cdots 0\ell 0 }/ ||\psi_{L_j}||}.
\label{eqn:ZTildeoddeven}$$ In all other cases, the scalar product vanishes.
Let us prove . The reasoning is similar to the proof of \[thm:ResultForZ\]. We use in order to write $$\label{eqn:ZTildeIntermediate}
\langle \psi_L|\Biggl(|\psi_{L_1}\rangle\otimes \bigotimes_{j=2}^{m}(|\ell\rangle \otimes |\psi_{L_j}\rangle)\Biggr) = \left(\lambda_L^\ast {\langle 0|}\otimes
{\langle {\protect\raisebox{0.25ex}{$\chi$}}\cdots{\protect\raisebox{0.25ex}{$\chi$}}|} + \langle \phi_L|{\mathfrak{Q}}^\dagger \right)\Biggl(|\psi_{L_1}\rangle\otimes \bigotimes_{j=2}^{m}(|\ell\rangle \otimes |\psi_{L_j}\rangle)\Biggr).$$ The term involving ${\mathfrak{Q}}^\dagger$ vanishes. Indeed, one checks that if both $|\psi\rangle$ and $|\psi'\rangle$ are in the kernel of ${\mathfrak{Q}}^\dagger$ then ${\mathfrak{Q}}^\dagger(|\psi\rangle\otimes |\ell\rangle \otimes |\psi'\rangle)=0$ as a consequence of . Applying this property repeatedly, we reduce to $$\langle \psi_L|\Biggl(|\psi_{L_1}\rangle\otimes \bigotimes_{j=2}^{m}(|\ell\rangle \otimes |\psi_{L_j}\rangle)\Biggr) = \lambda_L^\ast \prod_{j=1}^m \mu_{L_j}.$$ Using , we find .
The proof of is similar, using in addition the parity invariance of the zero-energy state. In all other cases, the scalar products vanish because of the definite magnetisation of the zero-energy states.
It is possible to extend the definition of $\tilde Z$ in order to account for the case where $L_k=0$ for some $1\leqslant k\leqslant m$ and the $L_1,\dots,L_{k-1},L_{k+1},\dots,L_m$ are odd. In this case, we have $$\begin{aligned}
\tilde Z(L_1,\dots,L_k=0,\dots, L_m) &= \frac{\langle \psi_L|\left( \bigotimes_{j=1}^{k-1}( |\psi_{L_j}\rangle\otimes |\ell\rangle)\otimes\bigotimes_{j=k+1}^{m}( |\ell\rangle\otimes |\psi_{L_j}\rangle )\right)}{||\psi_L||\prod_{j=1,j\neq k}^m ||\psi_{L_j}||}
\nonumber
\\
&=\frac{ (\psi_L)_{0\ell \cdots 0\ell}/ ||\psi_L||}{\prod_{j=1,j\neq k}^m (\psi_{L_j})_{0\ell \cdots 0\ell 0 }/ ||\psi_{L_j}||}.\end{aligned}$$
The bipartite fidelity of the supersymmetric open XXZ chain {#sec:SPXXZ}
===========================================================
In this section, we discuss the scalar products found in \[thm:ResultForZ,thm:ResultForZTilde\] for $\ell=1$. In this case, the Hamiltonian is given by and describes the open XXZ spin chain with anisotropy parameter $\Delta=-1/2$ and special diagonal boundary terms. In \[sec:FiniteSize\], we provide exact finite-size expressions for certain special components of the normalised zero-energy state. We use them to exactly evaluate the scalar products $Z(L_1,\dots,L_m)$ and $\tilde Z(L_1,\dots,L_m)$. Furthermore, we evaluate the scaling limits of the scalar products. In \[sec:LBF\], we compare these scaling limits for $m=2$ to the predictions of conformal field theory.
In contrast to the previous sections, the results presented here below are non-rigorous. Many statements are based on conjectures that we inferred from the exact diagonalisation of the spin-chain Hamiltonian.
Special components and scaling behaviour {#sec:FiniteSize}
----------------------------------------
#### A finite-size conjecture.
We computed the zero-energy state $|\psi_L\rangle$ of the XXZ Hamiltonian up to $L=16$ sites using <span style="font-variant:small-caps;">Mathematica</span>. We find that the ratio of the components $(\psi_{2n-1})_{01\cdots 010}$, $(\psi_{2n})_{01\cdots 01}$ and the norm of the corresponding states can be expressed in terms of two integer sequences $A_{\text{\rm \tiny V}}(2n+1)$ and $N_8(2n)$. These two sequences enumerate $(2n+1)\times (2n+1)$ vertically-symmetric alternating sign matrices and cyclically-symmetric self-complementary plane partitions in a $2n\times 2n \times 2n$ cube, respectively [@bressoudbook]. Explicitly, they are given by $$\label{eqn:DefAVN8}
A_{\text{\rm \tiny V}}(2n+1)= \frac{1}{2^n}\prod_{k=1}^n \frac{(6k-2)!(2k-1)!}{(4k-1)!(4k-2)!}, \quad N_{8}(2n)=\prod_{k=0}^{n-1} \frac{(3k+1)(6k)!(2k)!}{(4k)!(4k+1)!}.$$ Our numerical investigation is consistent with the following conjecture:
\[conj:NormalisedComponents\] For each $n\geqslant 1$ we have $$\label{eqn:NormalisedComponents}
\frac{(\psi_{2n-1})_{01\cdots 010}}{||\psi_{2n-1}||} = \sqrt{\frac{N_8(2n)}{A_{\text{\rm \tiny V}}(2n+1)}}
, \quad \frac{(\psi_{2n})_{01\cdots 01}}{||\psi_{2n}||} = \sqrt{\frac{A_{\text{\rm \tiny V}}(2n+1)}{N_8(2n+2)}}.$$
The occurrence of sequences that enumerate alternating sign matrices and plane partitions in the ground state of an XXZ spin chain at $\Delta=-1/2$ comes by no means as a surprise (see for example [@razumov:00; @razumov:01; @batchelor:01; @degier:02; @difrancesco:06; @razumov:07]). We provide a proof of \[conj:NormalisedComponents\] along with many other combinatorial properties of the ground state $|\psi_L\rangle$ in a separate publication [@hagendorf:tbp].
#### Scaling behaviour.
The sequences $A_{\textrm{\tiny V}}(2n+1)$ and $N_{8}(2n)$ are given by ratios of products of factorials. Hence, they can be expressed in terms of Barnes’ $G$-function [@whittaker:27]. We use the well-known asymptotic expansion of this function to evaluate the components for large system sizes:
$$\begin{aligned}
\frac{(\psi_{2n-1})_{01\cdots 010}}{||\psi_{2n-1}||} & = C_1(2n)^{1/12}\left(\frac{3^{3/4}}{2}\right)^{-2n}\left(1+O(n^{-1})\right),\\\frac{(\psi_{2n})_{01\cdots 01}}{||\psi_{2n}||} &= C_2(2n)^{-1/12}\left(\frac{3^{3/4}}{2}\right)^{-2n}\left(1+O(n^{-1})\right).\end{aligned}$$
Here $C_1$ and $C_2$ are the constants $$C_1 = \frac{\sqrt{\Gamma(1/3)}}{\pi^{1/4}}, \quad C_2=\left(\frac{2}{\sqrt{3}}\right)^{3/2} \frac{\pi^{1/4}}{\sqrt{\Gamma(1/3)}}.$$ \[eqn:asymptotics\]
We use these expressions to extract the scaling behaviour of $Z(L_1,\dots,L_m)$, for $\ell=1$. It is obtained when the lengths of the subintervals $L_1,\dots,L_m$ become large in such a way that the ratios $L_i/L$ approach certain scaling variables $0<x_i<1$ for each $i=1,\dots,m$. $Z(L_1,\dots,L_m)$ is then given by an asymptotic series with respect to the system size $L$. The series coefficients are functions of $x_1,\dots,x_m$. Notice that together, \[thm:ResultForZ\] and imply that this asymptotic series is only well defined if the parity of the integers $L_1,\dots,L_m$ is fixed. There are two interesting cases, corresponding to and :
1. $L_i$ is even for each $i=1,\dots,m$. In this case the length of the chain $L$ is even. Using , we obtain $$\label{eqn:ZScalingEven}
Z(L_1,\dots,L_m) = L^{-(m-1)/12}C_2^{m-1}\prod_{i=1}^{m}x_i^{-1/12}\left(1+O(L^{-1})\right).$$
2. $L_j$ is odd for a certain $j$ and $L_i$ is even for each $i=1,\dots,j-1,j+1,\dots,m$. In this case, $L$ is odd. From , we obtain $$\label{eqn:ZScalingEvenOdd}
Z(L_1,\dots,L_m)=L^{-(m-1)/12}C_2^{m-1}x_j^{1/12}\prod_{i=1, i\neq j}^{m}x_i^{-1/12}\left(1+O(L^{-1})\right).$$
We obtain the scaling behaviour of the scalar product $\tilde Z(L_1,\dots,L_m)$ from \[thm:ResultForZTilde\] and . The resulting leading-order terms of and are up to a factor equal to and , respectively. Hence, without loss of generality we focus in the following on $Z(L_1,\dots,L_m)$.
\[eqn:ScalingLimitsZ\]
Scaling behaviour and conformal field theory {#sec:LBF}
--------------------------------------------
The power-law decay of the scalar products in as well as their algebraic dependence on the scaling coordinates $x_1,\dots,x_{m}$ suggest that they could be related to correlation functions of conformal field theory (CFT). In this section, we discuss this relation for the special case of $m=2$. Specifically, we consider the so-called *logarithmic bipartite fidelity* (LBF): $$\mathcal F(L_1,L_2) = - \ln |Z(L_1,L_2)|^2.$$ It was introduced by Dubail and Stéphan as an entanglement measure for the ground state of interacting quantum many-body systems in one dimension[@dubail:11; @dubail:13]. In particular, they predicted the leading-order terms of the asymptotic expansion of the LBF with respect to the system size $L$ for one-dimensional quantum critical systems from CFT arguments.
From and we exactly compute these leading-order terms. We express them in terms of $x=x_1$. In case *(i)*, we obtain
$$\mathcal F = \frac{1}{6}\ln L + \frac{1}{6}\ln x(1-x)-2\ln C_2+O(L^{-1}).
\label{eqn:LBFScalingEven}$$
In the case *(ii)*, we take the scaling limit with $L_1$ even and $L_2$ odd. This leads to $$\mathcal F = \frac{1}{6}\ln L + \frac{1}{6}\ln \left(\frac{x}{1-x}\right)-2\ln C_2+O(L^{-1}).
\label{eqn:LBFScalingOdd}$$ \[eqn:LBFScaling\]
The aim of this section is to show that perfectly matches the CFT predictions. To this end, we briefly discuss the relation between the scaling limit of the open XXZ chain at $\Delta=-1/2$ and superconformal CFT in \[sec:cft\]. In \[sec:lbf\] we compare our findings for the scaling limit of the spin chain’s LBF to the CFT results.
### CFT connection {#sec:cft}
It is well known that in suitable scaling limits, many properties of the XXZ spin with anisotropy parameter $-1\leqslant \Delta \leqslant 1$ are accurately described by CFT. For open chains and real diagonal boundary terms, the field theory is expected to be given by a free boson theory with central charge $c=1$ and a compactification radius that depends on the value of $\Delta$ [@affleck:90]. If $\Delta=-1/2$ then this compactification radius is fixed to a value where the field theory coincides with the first model of the so-called $\mathcal N=2$ superconformal minimal series [@waterson:86; @friedan:88; @lerche:89] (see also [@huijse:10] for a compact introduction accessible to non-experts). This implies in particular that the CFT space of states divides into the so-called Ramond and Neveu-Schwarz sectors. Each sector organises in a finite number of irreducible highest-weight representations of the so-called $\mathcal N=2$ superconformal algebra. From the representation theory of this algebra, we know that the corresponding highest-weight states are labeled by a pair $(h,\alpha)$. Here $h$ denotes the conformal weight and $\alpha$ the so-called $U(1)$ charge of the state.[^5]
In the following, we focus on the Ramond sector. For the first $\mathcal N=2$ superconformal minimal model, it contains three highest-weight representations. Two of them are singlet representations with the corresponding highest-weight states labeled by
$$\left(\frac{1}{24},\frac{1}{2\sqrt{3}}\right),\,\left(\frac{1}{24},-\frac{1}{2\sqrt{3}}\right).$$
Furthermore, there is a doublet representation with a highest-weight state labeled by $$\left(\frac{3}{8},\frac{\sqrt{3}}{2}\right)$$ \[eqn:CFTData\]
It has a superpartner with the same conformal weight but the opposite $U(1)$-charge: $(3/8,-\sqrt{3}/2)$. Notice that since we deal with a free boson theory we have $h=\alpha^2/2$ for all these states [@difrancesco:97].
#### Lattice and CFT quantities.
The XXZ Hamiltonian possesses an explicit lattice realisation of the supersymmetry. This suggests that for finite $L$ its low-energy eigenstates constitute an approximation to the Ramond sector of the CFT Hilbert space. In order to substantiate this claim, we identify the representation data with certain properties of the ground state and first excited state of the lattice Hamiltonian (see also [@degier:05] and furthermore [@huijse:11_2] for a related analysis of the $M_1$ model of supersymmetric fermions).
The conformal weight $h$ of a low-energy eigenstate of the Hamiltonian is related to the finite-size scaling of its eigenvalue. Let us denote by $E_0(L) \leqslant E_1(L) \leqslant E_2(L) \leqslant \cdots$ the eigenvalues of the Hamiltonian at size $L$. Then we have for large $L$ and small $i$ the following expansion [@cardy:86_3; @affleck:86]: $$E_i(L) = L E_{\text{bulk}}+E_{\text{bdr}}+\frac{\pi v_F}{L}\left(h_i-\frac{c}{24}\right)+O(L^{-2}),
\label{eqn:FiniteSizeScaling}$$ where the central charge takes the value $c=1$. The factor $v_F$ is the so-called Fermi velocity that can be computed by Bethe-ansatz techniques [@korepin:93]. For $\Delta=-1/2$ we have $v_F = 3\sqrt{3}/2$. Furthermore, $E_{\text{bulk}}$ and $E_{\text{bdr}}$ are non-universal constants that depend on the definition of the Hamiltonian. In our case, these constants have to be zero, $E_{\text{bulk}}=E_{\text{bdr}}=0$, because the spectrum contains the ground-state eigenvalue $E_0(L)=0$ for any $L\geqslant 2$. Furthermore, fixes the conformal weight of the zero-energy states to $h_0=1/24$. To probe the consistency of these assignments, we have numerically examined the dependence of the eigenvalue $E_1(L)$ on $L$. Its scaling behaviour matches well with $h_1=3/8$.
Furthermore, it is natural to assume that the $U(1)$-charge of an eigenstate of the Hamiltonian is related to a linear function of its magnetisation. We fix this function by comparing the $U(1)$-charges of the two Ramond ground states in and the magnetisation of the lattice ground states that we obtained in \[corr:Magnetisation\]. This suggests that the operator $$J_0 = \frac{(1-4M)}{2\sqrt{3}}.$$ measures the CFT $U(1)$-charge: we have $$J_0|\psi_{2n-1}\rangle = -\frac{1}{2\sqrt{3}}|\psi_{2n-1}\rangle \quad \text{and} \quad J_0|\psi_{2n}\rangle = \frac{1}{2\sqrt{3}}|\psi_{2n}\rangle
\label{eqn:U1ZeroEnergyStates}$$ for each $n\geqslant 1$. To see that this choice is consistent, we inspect the first excited state of the Hamiltonian. We denote this state by $|\phi_{L}\rangle$ for a system of length $L$. Our exact diagonalisation data supports the conjecture that for each $n\geqslant 1$, we have ${\mathfrak{Q}}^\dagger |\phi_{2n-1}\rangle =0$ and ${\mathfrak{Q}}|\phi_{2n-1}\rangle = |\mathcal \phi_{2n}\rangle$. Furthermore, the magnetisation of the states that we computed for small systems is compatible with the conjecture $$J_0|\phi_{2n-1}\rangle = \frac{\sqrt{3}}{2}|\phi_{2n-1}\rangle \quad \text{and} \quad J_0|\phi_{2n}\rangle = - \frac{\sqrt{3}}{2}|\phi_{2n}\rangle.$$ for each $n\geqslant 1$. This suggests that the first excited state for spin chains of odd length is a lattice approximation to the CFT highest-weight state $(3/8,\sqrt{3}/2)$.
#### Conformal weight of the cut.
In order to compare our results and to the predictions from conformal field theory, we need a last ingredient: the $U(1)$-charge of the cut. To motivate the introduction of this quantity, recall that the scalar product $Z(L_1,L_2)$ is non-zero only if the product-state $|\psi_{L_1}\rangle\otimes |\psi_{L_2}\rangle$ and $|\psi_L\rangle$ have the same magnetisation. Let us now rephrase this statement in terms of the $U(1)$-charges of the three zero-energy states $|\psi_{L_1}\rangle$, $|\psi_{L_2}\rangle$ and $|\psi_L\rangle$, which we denote by $\alpha_1,\alpha_2$ and $\alpha_3$, respectively. We find that $Z(L_1,L_2)$ is non-vanishing only if $$\alpha_\text{c} \equiv \alpha_3-\alpha_1-\alpha_2 =-\frac{1}{2\sqrt{3}}.$$ The quantity $\alpha_\text{c}$ is the $U(1)$-charge of the cut. It allows us to formally define the conformal weight of the cut $h_\text{c}=\alpha_\text{c}^2/2=1/24$. Within CFT, it is possible to identify $h_\text{c}$ with the conformal weight of a so-called boundary condition changing operator. We refer to [@dubail:13] for more details.
### Logarithmic bipartite fidelity {#sec:lbf}
#### Finite-size scaling of the LBF.
For quantum critical systems, the leading-order terms of the asymptotic expansion of the LBF with respect to the system size $L$ is given by [@dubail:13]:
$$\mathcal F = \left(\frac{c}{8}+h_{\text{c}}\right)\ln L + f(x) + g(x) L^{-1}\ln L+ O(L^{-1}).
\label{eqn:LBFCFT}$$
Here, $c$ is the central charge of the theory and $h_{\text{c}}$ the conformal weight of the cut. Furthermore, $x=x_1$ denotes the scaling variable defined above. The functions $f(x)$ and $g(x)$ depend on the conformal weights and $U(1)$-charges associated to the involved states and the cut. Their explicit general form can be found in [@dubail:13]. Here, we only consider the case of a $c=1$ free field theory, where they can be written in terms of the $U(1)$ charges alone: $$\begin{aligned}
f(x) =& \left(\frac{1}{24}\left(2x-1+\frac{2}{x}\right)+\left(1-\frac{1}{x}\right)\alpha_1^2 +(1-x)\alpha_3^2 -\frac{\alpha_\text{c}^2}{2}-\alpha_2^2-2\alpha_\text{c}\alpha_2\right)\ln (1-x)
\nonumber\\
&+ \left\{x\to 1-x; \,\alpha_1\leftrightarrow \alpha_2\right\} + C
\label{eqn:fFromCFT}\end{aligned}$$ and $$g(x) = \xi\times \frac{1}{2}\left(\alpha_3^2 - \frac{1}{12} + \left(\frac{1}{12}-\alpha_1^2\right)\frac{1}{x}+\left(\frac{1}{12}-\alpha_2^2\right)\frac{1}{1-x}\right).
\label{eqn:gFromCFT}$$\[eqn:CFTPrediction\]
Here $C$ and $\xi > 0$ are non-universal constants. In particular, $\xi$ is called the *extrapolation length* [@diehl:81]. Apart from these two constants, the expressions are universal.
#### Comparison to scaling limit.
The identification of the $U(1)$-charges of the zero-energy states and the cut allow us to finally compare our findings with the CFT prediction . First, we notice that since $c=1$ and $h_\text{c}=1/24$, the leading order term in is $\frac16 \ln L$. This is consistent with our findings in . Second, for the subleading terms we find the following results. In case *(i)* we have $$\alpha_1=\alpha_2=\alpha_3 = \frac{1}{2\sqrt{3}}, \, \alpha_\text{c} = -\frac{1}{2\sqrt{3}}, \quad f(x) =\frac{1}{6}\ln x(1-x)+C,\quad g(x) = 0.$$ In case *(ii)* we have $$\alpha_2=\alpha_3=\alpha_\text{c}=-\frac{1}{2\sqrt{3}}, \, \alpha_1=\frac{1}{2\sqrt{3}}, \quad f(x) =\frac{1}{6}\ln \left(\frac{x}{1-x}\right)+C,\quad g(x) = 0.$$ In both cases, the expressions for $f(x)$ and $g(x)$ match perfectly the scaling limit of the lattice results, provided that we set $C=-2\ln C_2$. The absence of the $O(L^{-1}\ln L)$-term is worth mentioning. Similar instances of vanishing finite-size corrections for the quantum spin $1/2$ XXZ chain at $\Delta=-1/2$ were reported in the literature [@banchi:09]. In the present case, the vanishing is a direct consequence of the supersymmetry as it fixes the $U(1)$-charges of the zero-energy states to $\pm 1/(2\sqrt{3})$.
We end our analysis with a comment on the higher-spin cases $\ell>1$. For these cases we expect the $O(L^{-1}\ln L)$ term to vanish, too. Indeed, for arbitrary $\ell$ the scaling limit of the spin chains should be described by the Ramond sector of the $\ell$-th $\mathcal N=2$ superconformal minimal model. Indeed, this conjecture was made for the related $M_\ell$ models of strongly-interacting fermions with supersymmetry [@fendley:03]. It is well-known that the Ramond ground states have the conformal weight $h=c/24$ . Furthermore, for arbitrary $c$ the scaling function $g(x)$ is given by [@dubail:13] $$g(x) = \xi\times \left(h_3 - \frac{c}{24} + \left(\frac{c}{24}-h_1\right)\frac{1}{x}+\left(\frac{c}{24}-h_2\right)\frac{1}{1-x}\right), \quad \xi > 0.$$If all of the involved states are Ramond ground states (or at least lattice approximations thereof) then this expression identically vanishes.
Conclusion {#sec:Conclusion}
==========
In this article, we have studied the dynamic lattice supersymmetry for the open XXZ Heisenberg chains at their supersymmetric point. We have determined the family ${\mathfrak{Q}}_{j,k}(y)$ of supercharges and identified a family of non-diagonal boundary interactions that are compatible with the supersymmetry. Furthermore, we have computed the cohomology of the supercharges and shown that it is non-trivial if and only if $y=0$. From this cohomology computation we have deduced the existence of a zero-energy state, unique up to normalisation, of the spin-chain Hamiltonians with $y=0$. A central result of this article is the sum rules presented in \[thm:ResultForZ,thm:ResultForZTilde\]: They reveal that a large family of scalar products involving an arbitrary number of normalised zero-energy states can be computed in terms of certain distinguished components. We emphasise that these results are solely based on the supersymmetric structure and do not make use of any integrability techniques. Eventually, we have computed the scaling behaviour of logarithmic bipartite fidelity for the open spin $1/2$ quantum XXZ chain at $\Delta=-1/2$ with special boundary magnetic fields. Its scaling behaviour matches the predictions from conformal field theory at both leading and sub-leading orders.
The results of this work have a few interesting generalisations. First, it is natural to consider closed spin chains with periodic or twisted boundary conditions. In this case, the dynamic lattice supersymmetry only exists in certain (anti-)cyclic subspaces of the spin-chain Hilbert space [@hagendorf:13]. Therefore, the computation of the cohomology of the supercharges is more challenging. Nonetheless, we expect that there is a connection between the cohomology of supercharges of the open and the closed chains, based on known examples in the mathematical literature [@loday:92]. It should allow to establish some generalisations of \[thm:ResultForZ,thm:ResultForZTilde\] to closed spin chains for the class of models discussed in this article. Second, the computation of similar scalar products in off-critical models with lattice supersymmetry, in particular their scaling limit in the vicinity of a critical point, is of potential interest. An example is the open staggered $M_1$ model [@beccaria:12]. We hope to address these problems in forthcoming publications.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work is supported by the Belgian Interuniversity Attraction Poles Program P7/18 through the network DYGEST (Dynamics, Geometry and Statistical Physics). We thank Alexi Morin-Duchesne for many interesting remarks and stimulating suggestions. Furthermore, we thank Luigi Cantini, Jérôme Dubail, Axel Marcillaud de Goursac and Anita Ponsaing for discussions.
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[^1]: In previous works, the restriction of ${\mathfrak{Q}}$ to $V^L$ was indicated by a subscript: ${\mathfrak{Q}}_L$. We omit these subscripts for the supercharges (and all other operators) to keep the notation as simple as possible. If needed, we write ${\mathfrak{Q}}:V^L\to V^{L+1}$ in order to emphasise that we consider the action of the supercharge on $V^L$.
[^2]: The solution is not unique. For instance, $\ell+1$ other solutions are given by $|\phi(q^{2(k+1)}y)\rangle$ with $k=0,\dots,\ell$.
[^3]: Here and in the following, we omit the commas whenever the sequences that label the basis vectors take specific values. For example, we write ${|01\rangle}={|0,1\rangle}={|0\rangle}\otimes {|1\rangle}$.
[^4]: We abbreviate the tensor product $|{\protect\raisebox{0.25ex}{$\chi$}}\rangle \otimes \cdots \otimes |{\protect\raisebox{0.25ex}{$\chi$}}\rangle$ by $|{\protect\raisebox{0.25ex}{$\chi$}}\cdots {\protect\raisebox{0.25ex}{$\chi$}}\rangle$ in order to simplify the notation.
[^5]: We use the common notation $h$ for conformal weights. This is not to be confused with the Hamiltonian density discussed in previous sections. Furthermore, for the $U(1)$-charges $\alpha$, we follow the conventions of [@dubail:13] that differ from the standard choice in the literature on superconformal field theory. For the first $\mathcal N=2$ superconformal minimal model that choice is $q_{\text{\tiny SCFT}}=\alpha/\sqrt{3}$.
|
---
abstract: 'In order to get insight into the connection between the vibrational dynamics and the [*atomic level*]{} Green-Kubo stress correlation function in liquids we consider this connection in a model crystal instead. Of course, vibrational dynamics in liquids and crystals are quite different and it is not expected that the results obtained on a model crystal should be valid for liquids. However, these considerations provide a benchmark to which the results of the previous molecular dynamics simulations can be compared. Thus, assuming that vibrations are plane waves, we derive analytical expressions for the atomic level stress correlation functions in the classical limit and analyze them. These results provide, in particular, a recipe for analysis of the atomic level stress correlation functions in Fourier space and extraction of the wavevector and frequency dependent information. We also evaluate the energies of the atomic level stresses. Obtained energies are significantly smaller than the energies that were obtained in MD simulations of liquids previously. This result suggests that the average energies of the atomic level stresses in liquids and glasses are largely determined by the structural disorder. We discuss this result in the context of equipartition of the atomic level stress energies. Analysis of the previously published data suggests that it is possible to speak about configurational and vibrational contributions to the average energies of the atomic level stresses in a glass state. However, this separation in a liquid state is problematic. We also consider peak broadening in the pair distribution function with increase of distance. We find that peak broadening (by $\approx 40\%$) occurs due to the transverse vibrational modes, while contribution from the longitudinal modes does not change with distance. Finally, we introduce and consider atomic level transverse current correlation function.'
author:
- 'V.A. Levashov'
title: |
Atomic Level Green-Kubo Stress Correlation Function for a Model Crystal:\
An Insight into Molecular Dynamics Results on a Model Liquid.
---
Introduction {#sec:intro}
============
In order to understand abrupt increase in viscosity of liquids approaching the glass transition, it is necessary to understand well the nature of viscosity itself. This understanding, however, is still limited [@HansenJP20061; @EvansDJ19901; @Boon19911; @Hetero20131; @Berthier2011; @Tanaka20091; @Tanaka2011E; @Levashov20111; @Levashov2013; @Iwashita2013].
Computer simulations has proved to be an important tool in addressing properties of supercooled liquids [@HansenJP20061; @EvansDJ19901; @Boon19911; @Hetero20131; @Berthier2011]. One standard approach to calculate viscosity in computer simulations is based on the Green-Kubo expression that relates viscosity to the integral of the macroscopic stress correlation function [@HansenJP20061; @EvansDJ19901; @Boon19911; @Green1954; @Kubo1957; @Helfand1960; @EvansDJ19811; @Hoheisel19881].
Properties of the stress correlation function have been extensively studied previously from [*a macroscopic*]{} perspective [@HansenJP20061; @EvansDJ19901; @Boon19911; @Hoheisel19881]. There have been significantly fewer studies that tried to address how behavior of the system at the atomic level translates into the macroscopic behavior of the stress correlation function [@Woodcock19911; @Woodcock20062; @Stassen19951; @Stassen19952; @Levashov20111; @Levashov2013; @Iwashita2013].
The situation is similar with a closely related but somewhat different approach, i.e., the approach based on considerations of the transverse current correlation function [@HansenJP20061; @EvansDJ19901; @Boon19911; @Tanaka20091; @Tanaka2011E; @Tanaka20081; @Mizuno2013; @Mountain19821]. Studies of vibrational dynamics in disordered media with the transverse current correlation function are very common and several important results were obtained with it relatively recently [@Tanaka20091; @Tanaka2011E; @Tanaka20081]. However, in all these studies the transverse current correlation function is treated as a [*macroscopic*]{} quantity. Thus the relations between the atomic level processes and the macroscopic behavior of the transverse current correlation function remain obscure [@Tanaka20091; @Tanaka2011E].
We previously studied [*atomic level structure*]{} of the macroscopic Green-Kubo stress correlation function by decomposing it into correlation functions between the atomic level stresses [@Levashov20111; @Levashov2013]. The approach represents further development of preceding works [@Hoheisel19881; @Woodcock19911; @Woodcock20062; @Stassen19951; @Stassen19952]. Our data clearly show presence of stress waves in the atomic level stress correlation function and that the stress waves contribute to viscosity [@Levashov20111; @Levashov2013]. However, it was not previously discussed [*how*]{} stress waves and their properties translate into the observed atomic level stress correlation functions. It is difficult to address this issue in liquids, even qualitatively, as vibrational and configurational dynamics in liquids are mixed [@Stillinger20131; @HeuerA20081; @KeyesT19971]. Moreover, vibrational and configurational dynamics in disordered media are puzzles by themselves [@Stillinger20131; @HeuerA20081; @Zwanzig19651; @KeyesT19971; @Taraskin2000; @Taraskin2002; @Scopigno20071; @Tanaka20081; @Mizuno2013; @Keys2011; @Iwashita2013; @Kob2006; @Frenkel1947; @Trachenko2009; @Bolmatov2012; @Bolmatov2013].
On the other hand, as it appears from the review of the previous literature, the details of the connection between vibrational dynamics and [*the atomic level*]{} stress correlation function were not addressed previously even for those systems for which it could be done relatively easily, i.e., for the crystals. Applicability of results obtained from crystal models to liquids, in general, is not expected and should be considered with caution. However, it has been demonstrated that parallels between liquid and solid states can be useful [@Frenkel1947; @Trachenko2009; @Bolmatov2012; @Bolmatov2013].
Thus, in order to gain at least some qualitative or semiquantitative insight into the connection between the vibrational dynamics of a model liquid and the atomic level stress correlation functions observed in MD simulations [@Levashov20111; @Levashov2013], we examine a crystal-like model in which vibrations are represented by plane waves. Considerations in this paper represent further developments and more detailed discussions of some ideas and a model first presented in Ref.[@Egami19821].
Another goal of this paper is to develop a framework for analysis in Fourier space of the MD data from a model liquid [@Levashov20111; @Levashov2013]. This analysis is presented in Ref.[@Levashov20141]. It relies on the results presented in this paper.
To make derivations of the expressions for the stress correlation functions clearer it is useful to address several other issues. In particular, we calculate atomic level stress energies. In this context we discuss the data from previously published MD simulations on liquids and glasses [@Egami19821; @Chen19881; @Levashov2008B; @Levashov2008E].
In the framework of the model it is easy to evaluate the peak broadening in the pair distribution function with increasing distance. Calculations show that the peak broadening (by $\approx 40\%$) occurs because of the transverse waves, while the contribution from the longitudinal waves only weakly depends on distance.
Finally we briefly discuss [*the atomic level transverse current correlation function*]{} and argue that it is possible to study its behavior in MD simulations in a way which we previously applied to the atomic level stress correlation function.
The paper is organized as follows. In section \[sec:model\] we describe the model. Section \[sec:derivations\] is focused on derivations and analysis of the obtained results. In section \[sec:discussion\] we discuss obtained results in the broader context of some results obtained previously.
The model \[sec:model\]
=======================
We consider a single component system and assume that different atoms have identical environments. In particular, we assume that every atom interacts harmonically with $N_c$ nearest neighbors. We also assume that distribution of these neighbors is spherically symmetric and that their equilibrium distance from the central atom is $a$. Finally we assume that vibrational motion in the system is described by plane waves.
Continuous spherical approximation
----------------------------------
In the following derivations we will usually perform summation for every atom $n$ over its nearest neighbors $m$. In performing these summations we will utilize a continuous spherical approximation. Thus we will change summation over $m$ into the integration over the spherical angles: $$\begin{aligned}
\sum_m f(\theta_m, \phi_m ) \rightarrow \frac{N_c}{4\pi}\int f(\theta ,\phi)\sin(\theta)d\theta d\phi
\;\;.\;\;\;\;\;\;\;\;
\label{eq;sphericalapprox01}\end{aligned}$$
Debye’s Model
-------------
In order to estimate various quantities to which many different waves contribute we will assume that different waves contribute independently. We will also utilize Debye’s model, i.e., we will change summation over different waves into the integration over the wavevector: $$\begin{aligned}
\frac{dN}{N}=\left(\frac{a}{2\pi}\right)^3\,4\pi\,q^2\,dq\;\;,\;\;\;\;Q_{max} =\left(\frac{\pi}{a}\right)\left(\frac{6}{\pi}\right)^{1/3}\;\;,\;\;\;\;\;\;\;\;
\label{eq;debye01}\end{aligned}$$ where $N$ is the total number of atoms in the system and also the total number of vibrational states for one polarization. $dN$ is the number of states in the interval $dq$, and $Q_{max}$ is the maximum value of the wavevector. Equations in (\[eq;debye01\]) are written for one particular polarization of the waves. We will assume further, as usual, that there are one longitudinal and two transverse polarizations.
The value of $Q_{max}$ and the value of the prefactor $\left(a/(2\pi)\right)^3$ in (\[eq;debye01\]) are connected by the normalization condition. In principle, one can assume different values of $Q_{max}$ for different polarizations of the waves. We will not elaborate on this issue further.
Long wavelength approximation
-----------------------------
In the following we will sometimes assume that: $$\begin{aligned}
\sin\left(\bm{q}\bm{a}_{nm}\right)\approx \left(\bm{q}\bm{a}_{nm}\right)
\;\;,\;\;\;
\cos\left(\bm{q}\bm{a}_{nm}\right)\approx 1\;\;.\;\;\;\;\;\;\;\;
\label{eq;LWapprox01}\end{aligned}$$ Equations (\[eq;LWapprox01\]) are correct if the wavelength of the wave is much larger than the interatomic distance $a\equiv |\bm{a}_{nm}|$. Usually we will give the results obtained without long wavelength approximation and then, for comparison, the results obtained with long wavelength approximation.
Derivations \[sec:derivations\]
===============================
Potential energy of an atom due to a plane wave
-----------------------------------------------
Let us assume that $\bm{r}_n^o$ is the equilibrium position of the particle $n$ and $\bm{u}_n$ is the displacement of the particle $n$ from equilibrium. Then $\bm{r}_n = \bm{r}_n^o + \bm{u}_n$, $r_{nm} = \left|\bm{r}_m - \bm{r}_n\right|$, $\bm{a}_{nm}=a\bm{\hat{a}}_{nm}=\left(\bm{r}_m^o-\bm{r}_n^o\right)$, $\bm{u}_{nm}=\bm{u}_m-\bm{u}_n$. With these notations potential energy for the nearest neighbor atoms $n$ and $m$ in the harmonic approximation is given by: $$\begin{aligned}
U_{nm}=\frac{k\left(r_{nm}-r_{nm}^o\right)^2}{2} \approx
\frac{k}{2}\left(\bm{\hat{a}}_{nm}\bm{u}_{nm}\right)^2\;\;\;\;.
\label{eq;poten3dx1}\end{aligned}$$
The solutions for particle displacements in classical harmonic crystals are plane waves. For a particular wave: $$\begin{aligned}
&&\bm{u}_n(\bm{q}) =u_q\bm{\hat{e}}_q\,Re \left\{\chi_n(\bm{q})\right\}\label{eq;poten3dx21}\;\;\;,\;\;\;\\
&&\chi_n(\bm{q}) =
\exp\left[-i\left(\omega_{\bm{q}} t - \bm{q}\bm{r}_n +\phi_{\bm{q}}\right)\right]\;\;\;,\;\;\;
\label{eq;poten3dx22}\end{aligned}$$ where $u_q$ (real scalar) is the amplitude of the wave and $\bm{\hat{e}}_q$ (real vector) is its polarization vector.
From (\[eq;poten3dx21\],\[eq;poten3dx22\]) we get: $$\begin{aligned}
\bm{u}_{nm}(\bm{q}) = u_q\bm{\hat{e}}_q
Re\left\{\chi_n(\bm{q})\left[\exp\left(i\bm{q}\bm{a}_{nm}\right) - 1\right]\right\}\;\;.
\label{eq;unm3plane1}\end{aligned}$$
It is straightforward to show from (\[eq;poten3dx1\],\[eq;unm3plane1\]) that time average of the potential energy of the atom $n$ due to a particular wave is: $$\begin{aligned}
\left<U_{n}\right>_t \approx \left(\frac{1}{2}\right)\,ku_q^2\sum_m\left(\bm{\hat{a}}_{nm}\bm{\hat{e}}_q\right)^2
\sin^2\left(\frac{\bm{q}\bm{a}_{nm}}{2}\right)\;\;,
\label{eq;poten3dx3}\end{aligned}$$ where we introduced factor $1/2$ to take into account that half of the elastic energy belongs to the atom $n$, while another half to the atom $m$.
Force on an atom and dispersion relations
-----------------------------------------
It follows from (\[eq;poten3dx1\]) that the force on the atom $n$ due to its interaction with the atom $m$ is: $$\begin{aligned}
f_{nm}^{\alpha} = -\frac{\partial U_{nm}}{\partial u_n^{\alpha}}=k(\bm{\hat{a}}_{nm}\bm{u}_{nm})\hat{a}_{nm}^{\alpha}\;\;.
\label{eq;force01}\end{aligned}$$ Using the expression (\[eq;unm3plane1\]) for $\bm{u}_{nm}(\bm{q})$ in (\[eq;force01\]) for the total force on the atom $n$ we get: $$\begin{aligned}
f_{n}^{\alpha} = && \sum_m (ku_q)\,\left(\bm{\hat{a}}_{nm}\bm{\hat{e}}_{\bm{q}}\right)\,
a_{nm}^{\alpha}\cdot\label{eq;force02}\\
&&
Re\left\{\chi_n(\bm{q})\left[\exp\left(i\bm{q}\bm{a}_{nm}\right) - 1\right]\right\}\;\;\;\;.\;\;
\nonumber\end{aligned}$$ Let us further suppose that we consider crystal lattices with central symmetry. Then for every neighbor $m$ there is another neighbor $m'$ such that $\bm{a}_{nm'}=-\bm{a}_{nm}$. This assumption should be true in the continuous spherical approximation. Taking this into account we can rewrite (\[eq;force02\]) as: $$\begin{aligned}
f_{n}^{\alpha} = && \sum_m \left(ku_q\right)\left(\bm{\hat{a}}_{nm}\bm{\hat{e}}_{\bm{q}}\right)
a_{nm}^{\alpha}\cdot \label{eq;force03}\\
&&\left[\cos\left(\bm{q}\bm{a}_{nm}\right) - 1\right]\cdot
\cos\left(\omega_{\bm{q}} t - \bm{q}\bm{r}_n +\phi_{\bm{q}}\right)\;\;\;\;.\;\;\;\;
\nonumber\end{aligned}$$ From (\[eq;poten3dx21\],\[eq;poten3dx22\],\[eq;force03\]) and Newton’s second law we get: $$\begin{aligned}
\hat{e}_{\bm{q}}^{\alpha}\omega_{L,T}^2(\bm{q})=\left(\frac{2k}{M}\right)\sum_m
\left(\bm{\hat{a}}_{nm}\bm{\hat{e}}_{\bm{q}}\right)
\sin^2\left(\frac{\bm{q}\bm{a}_{nm}}{2}\right) a_{nm}^{\alpha}
\;\;,\;\;\;\;\;\;\;
\label{eq;dispersion3D01}\end{aligned}$$ where $M$ is the particle’s mass. Indexes $L$ and $T$ label longitudinal and transverse polarizations. Multiplication of both sides of (\[eq;dispersion3D01\]) on $\hat{e}^{\alpha}_{\bm{q}}$ with the following summation over $\alpha$ leads to: $$\begin{aligned}
\omega_{L,T}^2(\bm{q})=\left(\frac{2k}{M}\right)\sum_m
\left(\bm{\hat{a}}_{nm}\bm{\hat{e}}_{\bm{q}}\right)^2
\sin^2\left(\frac{\bm{q}\bm{a}_{nm}}{2}\right)
.\;\;\;\;
\label{eq;dispersion3D02}\end{aligned}$$ Expression (\[eq;dispersion3D02\]) is very similar to expression (\[eq;poten3dx3\]). This, of course, is not an occasion as the average potential energy of a site (\[eq;poten3dx3\]) due to a wave with the amplitude $u_{\bm{q}}$ should be equal to $M\omega^2_{\bm{q}} u^2_{\bm{q}}/4$.
![Dispersion curves for the longitudinal and transverse waves in the continuous spherical approximation. Thick lines show the results obtained without the long wavelength approximation. Thin lines show the results obtained with the long wavelength approximation. Note that $\left(Q_{max}a\right) \cong 3.9$ []{data-label="fig:dispersion02"}](Fig01){width="3.3in"}
In the continuous spherical approximation (\[eq;dispersion3D02\]) should not depend on the direction of $\bm{q}$ and an analytical expression for (\[eq;dispersion3D02\]) could be obtained for the longitudinal and transverse waves. For a longitudinal wave it is sufficient to assume that $(\bm{q} \parallel \bm{\hat{z}})$ and $(\bm{\hat{e}_q} \parallel \bm{\hat{z}})$. For a transverse wave it is sufficient to assume that $(\bm{q} \parallel \bm{\hat{z}})$ and $(\bm{\hat{e}_q} \parallel \bm{\hat{x}})$. Thus we can rewrite (\[eq;dispersion3D02\]) as: $$\begin{aligned}
\omega_{L,T}^2(q)=\omega_o^2 D_{L,T}(qa)\equiv\frac{2}{4\pi}
\int f_{L,T}(\xi,\theta,\phi)d\Omega\;\;\;,\;\;\;\;
\label{eq;dispersion3D03}\end{aligned}$$ where $$\begin{aligned}
\omega_o^2 \equiv \left(\frac{k}{M}\right)N_c\;\;\;,\;\;\;\;\;\;\xi\equiv \frac{qa}{2}\;\;\;.\;\;
\label{eq;xi}\end{aligned}$$ For longitudinal and transverse waves: $$\begin{aligned}
&&f_L(\xi,\theta,\phi)=\cos^2(\theta)\,\sin^2\left(\xi\cos(\theta)\right)\;\;\;,\;\;\nonumber\\
&&f_T(qa,\theta,\phi)=\sin^2(\theta)\cos^2(\phi)\,
\sin^2\left(\xi\cos(\theta)\right)\;\;\;.\;\;\;\;\;\nonumber\end{aligned}$$ Integrations over the spherical angles using the Maple(^TM^) program [@Maple5] lead to: $$\begin{aligned}
&&D_L(qa)= \left[L_1(\xi)+L_2(\xi)\right]/\left[6\xi^3\right]\;\;\;,\;\;\;\\
&&D_T(qa)= \left[T_1(\xi)+T_2(\xi)\right]/\left[12\xi^3\right]\;\;\;,\;\;\;
\label{eq;poten3dx3sin2}\end{aligned}$$ where $$\begin{aligned}
&&L_1(\xi)=- 6\,\xi^{2}\cos(\xi)\,\sin(\xi) + 2\,\xi^{3}
- 6\,\xi\,\cos^2(\xi)\;\;\;,\nonumber\\
&&L_2(\xi) = 3\,\cos(\xi)\,\sin(\xi) + 3\,\xi\;\;\;,\nonumber\\
&&T_1(\xi)=4\xi^{3} + 6\,\xi\,\cos^2(\xi)\nonumber\\
&&T_2(\xi) = -3\,\cos(\xi)\,\sin(\xi) - 3\,\xi\;\;\;.
\label{eq;poten3dx3sin3}\end{aligned}$$ The dependencies $\sqrt{D_L(qa)}$ and $\sqrt{D_T(qa)}$ on $qa$, i.e., the dispersion relations, are plotted in Fig.\[fig:dispersion02\].
The dispersion relations for the longitudinal and transverse waves could also be calculated in the long wavelength ([*lw*]{}) approximation: $$\begin{aligned}
&&\omega_{lw}^2(L,q) = \frac{\omega_o^2}{10}(qa)^2,\;\;\;\;\;
\omega_{lw}^2(T,q) = \frac{\omega_o^2}{30}(qa)^2.\;\;\;\;\;
\label{eq;dispersionlw}\end{aligned}$$ Thus in the long wavelength approximation speeds of the longitudinal waves are $\sqrt{3}$ times larger than the speeds of the transverse waves.
Equipartition and mean square displacements
-------------------------------------------
If we will assume that equipartition holds for our spherical approximation then the average potential energy of every wave should be equal to $k_b T/2$. Thus we should have: $$\begin{aligned}
&&\frac{M\omega_{L,T}^2(q) u_{L,T}^2(q)}{4}=\frac{1}{2}\frac{k_b T}{N}\;\;,\nonumber\\
&&u_{L,T}^2(q)=2\left(\frac{k_bT}{kN_c}\right)\left(\frac{1}{D_{L,T}(qa)}\right)\frac{1}{N}\;\;,\;\;\;\;\;\;\;
\label{eq;equip3D01}\end{aligned}$$ where $u_{L,T}^2(q)$ is the average square amplitude of the longitudinal or transverse waves with the magnitude of the wave vector $q$. Thus the squares of the amplitudes are inversely proportional to the dispersion curves shown in the Fig.\[fig:dispersion02\]. Note that wave’s amplitudes diverge for small wavevectors.
Mean square displacements due to all waves
------------------------------------------
In order to find the mean square displacements due to all waves, assuming that all of them are independent, we have to take half (since ($<u_n^2>=(1/2)u_q^2$) of (\[eq;equip3D01\]) and integrate it over all $q$ using (\[eq;debye01\]).
In this way for the mean square displacements due to all longitudinal waves and both polarizations of all transverse waves we get: $$\begin{aligned}
&&<u^2(L,T)>=\left(\frac{k_bT}{kN_c}\right)\gamma(L,T)\;\;,\;\;\;
\label{eq;u2longAll01}\\
&&\gamma(L)=2.8160\;\;,\;\;\;\;\;\gamma(T)=13.3615\;\;.\;\;\;\nonumber\end{aligned}$$ Note that $<u^2(T)>$ is significantly larger than $<u^2(L)>$.
It is simpler to evaluate the values of the mean square displacements in the long wavelength approximation. In this case we get: $$\begin{aligned}
\gamma_{lw}(L)=1.9746\;\;,\;\;\;\gamma_{lw}(T)=11.8478\
\label{eq;u2longWaveGamma01}\end{aligned}$$ Note that the values of the coefficients in the long wavelength approximation are smaller than without the long wavelength approximations. This is consistent with (\[eq;equip3D01\]), as the values of the frequencies are always larger in the long wavelength approximation.
The widths of peaks in the pair distribution function
-----------------------------------------------------
Atoms located close to each other in the lattice should exhibit a certain degree of coherence in their motion. Because of this peaks in the pair distribution function at small distances should be narrower than at large distances. The dependence of the peak’s widths on distance was investigated previously using a detailed model and evolved simulations [@Chung1997; @Chung1999]. It the frame of our model we can provide a simple evaluation of the size of the effect.
The average square of the peak width in the pair distribution function is determined by [@Chung1997; @Chung1999]: $$\begin{aligned}
\left<\left(\Delta r_{nm}\right)^2\right>\cong
\left<\left(\bm{\hat{r}}_{nm}^o \bm{u}_{nm}\right)^2\right>\;\;\;.\;\;\;
\label{eq;pdf3D01}\end{aligned}$$ In (\[eq;pdf3D01\]) the notation $\left<...\right>$ is used for the time and spherical averages. Expression for $\left<\left(\Delta r_{nm}\right)^2\right>$ is completely analogous to the expression (\[eq;poten3dx1\]), but with $\bm{r}_{nm}^o$ instead of $\bm{a}_{nm}$.
In analogy with (\[eq;poten3dx1\],\[eq;poten3dx3\]) and using the expression (\[eq;equip3D01\]) for $u_{\bm{q}}^2$ we get: $$\begin{aligned}
\frac{k\left<\left(\Delta r_{nm}(\bm{q}\right)^2\right>}{2 k_b T} \cong
\frac{\sum_m\left(\bm{\hat{r}}_{nm}\bm{\hat{e}}_{\bm{q}}\right)^2
\sin^2\left(\frac{\bm{q}\bm{r}_{nm}}{2}\right)}
{\sum_m\left(\bm{\hat{a}}_{nm}\bm{\hat{e}}_{\bm{q}}\right)^2
\sin^2\left(\frac{\bm{q}\bm{a}_{nm}}{2}\right)}\;\;\;\;\;\;
\label{eq;pdf3D03}\end{aligned}$$ In order to estimate the peak width due to all waves it is necessary to integrate the numerator and denominator of (\[eq;pdf3D03\]) over the spherical angles and then their ratio over all $q$ using (\[eq;debye01\]). The results of these integrations (assuming that $N_c =1$) for all longitudinal waves and one polarization of all transverse waves are shown in Fig.\[fig:uij3D01\].
![The value of the ratio of sums in (\[eq;pdf3D03\]) integrated over all wavevectors as a function of $r$. The blue curve represents contributions from all longitudinal waves. The red curve represents contributions from one polarization of all transverse waves. []{data-label="fig:uij3D01"}](Fig02){width="3.3in"}
For large $r_{nm}$ motions of the atoms $n$ and $m$ should be uncorrelated. It is straightforward to show from (\[eq;pdf3D01\]) that if atoms $n$ and $m$ vibrate independently then $<(\Delta r_{nm})^2> = (2/3) <(u_n)^2>$. This should be the large $r_{nm}$ limit of the peak’s width. In order to get this limit from the curves in Fig.\[fig:uij3D01\] it is necessary to multiply the limiting value by 2 for the longitudinal waves (the prefactor in \[eq;pdf3D03\]) and by 4 for the transverse waves (the prefactor and two polarizations). Then the results can be compared with (\[eq;u2longAll01\]).
It is interesting that convergence to the final value for the transverse waves is slower than for the longitudinal waves. Note also that contribution to the peak width from the shear waves increases by more than twice as the distance increases. There is essentially no change in the peak’s widths with distance due to the longitudinal waves.
Atomic level stress elements
----------------------------
Similarly to the previous definitions [@Egami19821; @Chen19881; @Levashov2008B], we define the $\alpha\beta$-component of the local atomic stress element on a particle $n$ as: $$\begin{aligned}
s_n^{\alpha\beta}=\frac{1}{2}\sum_{m \neq n} f_{nm}^{\alpha} r_{nm}^{\beta}\;\;,
\label{eq;stress3elem1}\end{aligned}$$ where, $f_{nm}^{\alpha}$ is the $\alpha$-component of the force on the particle $n$ caused by the interaction with the particle $m$ and $r_{nm}^{\beta}$ is the $\beta$-component of the radius vector from the particle $n$ to the particle $m$. The sign in (\[eq;stress3elem1\]) was chosen in such a way that an atom under compression will have a negative stress/pressure.
If there are interactions between the nearest neighbors only, we can rewrite (\[eq;stress3elem1\]) using (\[eq;force01\]) as: $$\begin{aligned}
s_n^{\alpha\beta}=\frac{\left(ka\right)}{2}\sum_{m \neq n} (\bm{u}_{nm} \bm{\hat{a}}_{nm})
\hat{a}_{nm}^{\alpha}\hat{a}_{nm}^{\beta}\;\;,
\label{eq;stress3elem2}\end{aligned}$$
Using (\[eq;unm3plane1\]) in (\[eq;stress3elem2\]) for the complex stress we obtain: $$\begin{aligned}
&&s_{n}^{\alpha\beta}(\bm{q}) = \frac{\left(ka\right)}{2}u_{\bm{q}}\chi_n(\bm{q}) \cdot \nonumber\\
&&\cdot\sum_{m \neq n}\left(\bm{\hat{e}_q}\bm{\hat{a}}_{nm}\right)
\left[\exp\left(i\bm{q}\bm{a}_{nm}\right) - 1\right]
\hat{a}_{nm}^{\alpha}\hat{a}_{nm}^{\beta}\;\;\;.\;\;\;\;
\label{eq;unm3plane22}\end{aligned}$$ Let us, like in the transition from (\[eq;force02\]) to (\[eq;force03\]), again assume that we consider crystal lattices with the central symmetry. For the real part of the stress from (\[eq;unm3plane22\]) we get: $$\begin{aligned}
s_{n}^{\alpha\beta}(\bm{q}) = \frac{\left(kau_q\right)}{2}
N_c\Upsilon^{\alpha\beta}_1\left(\bm{q},\bm{\hat{e}}_q\right)
\sin\left(\omega_{\bm{q}} t - \bm{q}\bm{r}_n +\phi_{\bm{q}}\right)\;,\;\;\;\;\;\;\;\;
\label{eq;unm3plane40}\end{aligned}$$ where $$\begin{aligned}
\Upsilon^{\alpha\beta}_1\left(\bm{q},\bm{\hat{e}}_q\right)\equiv \frac{1}{N_c}
\sum_{m \neq n}\left(\bm{\hat{e}}_{\bm{q}} \bm{\hat{a}}_{nm}\right)
\sin\left(\bm{q}\bm{a}_{nm}\right)
\hat{a}_{nm}^{\alpha}\hat{a}_{nm}^{\beta}\;\;.\;\;\;\;\;\;\;\;
\label{eq;unm3plane41}\end{aligned}$$ Formulas (\[eq;unm3plane40\],\[eq;unm3plane41\]) express local atomic stress elements due to a particular wave through the parameters of the lattice and the parameters of the propagating wave.
Atomic level pressure
---------------------
In accord with [@Egami19821; @Chen19881; @Levashov2008B], we define atomic level pressure as: $$\begin{aligned}
p_n(\bm{q}) = \frac{1}{3v_o}
\left[s^{xx}_n(\bm{q})+s^{yy}_n(\bm{q})+s^{zz}_n(\bm{q})\right]
\;\;\;,\;\;\;\;
\label{eq;pressure01}\end{aligned}$$ where $v_o$ is atomic volume. Here we will assume that atomic volume is a constant approximately equal to the inverse of the number density, i.e., $v_o\approx 1/\rho_o$.
It follows from (\[eq;unm3plane40\],\[eq;unm3plane41\],\[eq;pressure01\]) that: $$\begin{aligned}
p_{n}(\bm{q}) = \frac{\left(kau_q\right)N_c}{6v_o}
\Upsilon^{p}_1\left(\bm{q},\bm{\hat{e}}_q\right)
\sin\left(\omega_{\bm{q}} t - \bm{q}\bm{r}_n +\phi_{\bm{q}}\right)
\;\;,\;\;\;\;\;\;
\label{eq;pressure02}\end{aligned}$$ where $$\begin{aligned}
\Upsilon^{p}_1\left(\bm{q},\bm{\hat{e}}_q\right)=\frac{1}{N_c}
\sum_{m \neq n}\left(\bm{\hat{e}}_{\bm{q}} \bm{\hat{a}}_{nm}\right)
\sin\left(\bm{q}\bm{a}_{nm}\right)\;\;\;.\;\;\;\;
\label{eq;pressure03}\end{aligned}$$ Summation over $m$ (spherical integration) for a longitudinal wave leads to: $$\begin{aligned}
\Upsilon^{p}_1\left(L,qa\right)=\left[\frac{\sin(2\xi)-(2\xi)\cos(2\xi)}{(2\xi)^2}\right],\;\;\;
\xi=\frac{qa}{2}\;.\;\;\;\label{eq;pressure04L11}\end{aligned}$$
It also follows from (\[eq;pressure03\]) that transverse waves do not contribute to the pressure.
Mean square of the atomic level pressure
----------------------------------------
It follows from (\[eq;pressure02\],\[eq;pressure03\],\[eq;pressure04L\]) that time averaged square of the pressure due to a longitudinal wave with the wavevector of magnitude $q$ is: $$\begin{aligned}
\left<\left[p_{n}(q)\right]^2\right> = \frac{\left(kau_q\right)^2 N_c^2}{72v_o^2}
\cdot\Upsilon^{p}_2\left(L,qa\right)\;\;\;,\;\;\;\;
\label{eq;pressure05}\end{aligned}$$ where $$\begin{aligned}
\Upsilon^{p}_2\left(L,qa\right) \equiv \left[\Upsilon^{p}_1\left(L,qa\right)\right]^2\;\;.\;\;\;\;
\label{eq;pressure04L}\end{aligned}$$ In the long wavelength approximation: $$\begin{aligned}
\Upsilon^{p}_{2,lw}\left(L,qa\right)\approx \frac{1}{9}(qa)^2\;\;\;.\;\;\;\end{aligned}$$
Expressing $u_q^2$ from (\[eq;equip3D01\]) and then using it in (\[eq;pressure05\]) leads, after integration (\[eq;debye01\]) over $q$, to: $$\begin{aligned}
\left<p_{n}^2\right> \approx
k_bT\cdot \left(\frac{ka^2N_c}{36v_o^2}\right)\cdot 0.29\;\;\;\;.
\label{eq;pressure06}\end{aligned}$$ Calculations in the long wavelength approximation lead to $\approx 1.11$ instead of $\approx 0.29$.
Atomic level pressure energy
----------------------------
In several previous publications atomic level stress energies were discussed [@Egami19821; @Chen19881; @Levashov2008B]. These quantities are of interest, in particular, because of their values in the liquid states. According to MD simulations, the stress energy for every stress component is very close to $(1/4)k_bT=(1/6)(3/2)k_b T$.
It is well known that the average potential energy of a classical $3D$ harmonic oscillator is equal to $(3/2)k_bT$. Thus the values of the atomic levels stress energies are such that it appears that the average potential energy of [*some $3D$ harmonic oscillator*]{} is equally divided between the six independent components of the atomic level stresses. Thus it is interesting to estimate the values of the local atomic stress energies in our model.
The expression for the local atomic pressure energy is [@Egami19821; @Levashov2008B]: $$\begin{aligned}
<U^p> \equiv \frac{v_o\left<p_{n}^2\right>}{2B} \;\;\;\;.\;\;\;\;\;\;
\label{eq;pressure06}\end{aligned}$$ In order to evaluate the expression we need to know the value of the bulk modulus $B$. The expressions for the elastic constants were discussed before [@Egami19821; @Levashov2008B]. The results of their evaluations are: $$\begin{aligned}
B =\frac{\varkappa}{8}\;\;,\;\;\;\;\;\;
G=\frac{\varkappa}{30}\;\;,\;\;\;\;\;\;
\varkappa = \left(ka^2\right)\frac{N_c}{v_o}\;\;,\;\;\;\;\;\;\;
\label{eq;elastC01}\end{aligned}$$ where $G$ is the shear modulus. Using the value of the bulk modulus $B$ for the average pressure stress energy we get: $$\begin{aligned}
<U^p> \approx
\left(\frac{1}{4}\right)k_bT \cdot \left(\frac{1}{7.76}\right)\;\;.
\label{eq;pressure06}\end{aligned}$$ This energy is significantly smaller than the value of the pressure energy that was obtained for liquids.
In the long wavelength approximation we get: $$\begin{aligned}
<U^p_{lw}> \cong
\left(\frac{1}{4}\right)k_bT \cdot \frac{1}{2.03}\;\;.
\label{eq;pressure07}\end{aligned}$$
Thus in the long wavelength approximation the atomic level pressure stress energy is approximately 2 times smaller than the equipartition value, in agreement with [@Egami19821]. However, without the long wavelength approximation the local atomic pressure energy is more than 7 times smaller than the equipartition value. We address these differences further in the discussion section.
Pressure-pressure correlation function
--------------------------------------
Our goal here is to address the behavior of the atomic level stress correlation function that is analogous to the function $F(t,r)$ that could be derived from the macroscopic Green-Kubo stress correlation function and that was studied by MD simulations previously [@Levashov20111; @Levashov2013]. Thus we introduce: $$\begin{aligned}
C^p(t,r)=\left(\frac{a}{2\pi}\right)^3N\int_0^{Q_{max}} C^p(t,r,q)\,4\pi q^2dq\;\;\;,\;\;\;\;
\label{eq;CPcorr01}\end{aligned}$$ where $$\begin{aligned}
C^p(t,r,q)=\left<p_{n}(t_o,q)\cdot p_{m}(t_o+t,q)\right>\;\;\;.\;\;\;\;
\label{eq;CPcorr02}\end{aligned}$$ Spherical averaging and the averaging over $t_o$ are assumed in (\[eq;CPcorr02\]).
Note that the correlation function that we introduced in (\[eq;CPcorr01\],\[eq;CPcorr02\]) is the correlation function per pair of particles and not the correlation function between “a central particle" and “the particles in the spherical annulus", as it was done in [@Levashov20111; @Levashov2013].
![Pressure-Pressure correlation function without long wavelength approximation. []{data-label="fig:pp-correlations-1"}](Fig03){width="2.4in"}
![Pressure-Pressure correlation function with long wavelength approximation. []{data-label="fig:pp-correlations-lw-1"}](Fig04){width="2.4in"}
From (\[eq;pressure02\],\[eq;pressure03\]) it follows that: $$\begin{aligned}
&&C^p(t,r,q)=
\frac{\left(kau_q\right)^2}{36v_o^2}\cdot
\Upsilon^{p}_2\left(L,qa\right)\cdot\;\;\;\;\;\;\;\;\;\\
&&\left<\sin\left[\omega_{\bm{q}} t_o - \bm{q}\bm{r}_n +\phi_{\bm{q}}\right]
\sin\left[\omega_{\bm{q}} (t_o+t) - \bm{q}\bm{r}_m +\phi_{\bm{q}}\right]\right>
\;\;\;.\;\;\;\;\nonumber
\label{eq;ppcf01}\end{aligned}$$ From representing the product of [*sines*]{} as a difference of [*cosines*]{} it follows that one of the [*cosines*]{} gives zero on averaging over $\phi_{\bm{q}}$. Thus we get: $$\begin{aligned}
&&C^p(t,r,q)=\nonumber\\
&&\frac{\left(kau_q\right)^2}{36v_o^2}\cdot
\Upsilon^{p}_2\left(L,qa\right)\cdot
\frac{1}{2}\left<\cos\left[\frac{\omega_{\bm{q}} t - \bm{q}\bm{r}_{nm}}{2}\right]\right>
\label{eq;ppcf02}\end{aligned}$$ Further we rewrite $\cos\left[\frac{\omega_{\bm{q}} t - \bm{q}\bm{r}_{nm}}{2}\right]$ as: $$\begin{aligned}
\left<
\cos\left[\frac{\omega_{\bm{q}} t}{2}\right]
\cos\left[\frac{\bm{q}\bm{r}_{nm}}{2}\right]
\right>
+
\left<
\sin\left[\frac{\omega_{\bm{q}} t}{2}\right]
\sin\left[\frac{\bm{q}\bm{r}_{nm}}{2}\right]
\right>\;\;\;\;\;\;\;
\label{eq;ppcf03}\end{aligned}$$ Spherical averaging of the second term over the directions of $\bm{r}_{nm}$ is zero. Spherical averaging in the first term gives: $$\begin{aligned}
\left<\cos\left[\frac{\bm{q}\bm{r}_{nm}}{2}\right]\right>=
2\frac{\sin\left(qr/2\right)}{(qr/2)}\;\;\;.
\label{eq;ppcf04}\end{aligned}$$ Using the expression (\[eq;equip3D01\]) for $u_q^2$ we rewrite (\[eq;ppcf02\]) as: $$\begin{aligned}
&&C^p(t,r,q)=k_bT\left(\frac{ka^2N_c}{v_o^2}\right)\cdot\frac{1}{36}\cdot\nonumber\\
&&\cdot\frac{2}{N}\left\{\frac{\Upsilon^{p}_2\left(L,qa\right)}{D_L(qa)}\right\}\cdot
\left\{\frac{\cos\left(\omega_{q} t/2\right)\sin\left(qr/2\right)}{(qr/2)}\right\}\;\;,\;\;\;\;\;
\label{eq;ppcf05}\end{aligned}$$ where, according to (\[eq;dispersion3D03\]), $\omega_q = \omega_o \sqrt{D_L(qa)}$. The product of the [*cosine*]{} and [*sine*]{} in (\[eq;ppcf05\]) could be rewritten as: $$\begin{aligned}
\frac{1}{2}\left[\sin\left(\frac{qr-\omega_{q} t}{2}\right)+\sin\left(\frac{qr+\omega_{q} t}{2}\right)\right]\;\;\;.\;\;\;\;\;\;
\label{eq;ppcf06}\end{aligned}$$ The first [*sine*]{} corresponds to a wave propagating from the central particle. This [*sine*]{} is zero when $r-(\omega_{\bm{q}}/q)t=0$. The argument of the second [*sine*]{} is always positive for positive $t$ and $r$. For positive times the contribution to the stress correlation function due to all waves from the second [*sine*]{} is much smaller than from the first [*sine*]{}. However, for negative times the second [*sine*]{} behaves like the first [*sine*]{} for positive times.
In order to find the pressure correlation function due to all waves we have to integrate (\[eq;ppcf05\]) over all $q$ using (\[eq;debye01\]). Fig.\[fig:pp-correlations-1\] shows the results for the pressure-pressure correlation function due to all waves without long wavelength approximation. Fig.\[fig:pp-correlations-lw-1\] shows the result with long wavelength approximation.
An example of the local atomic shear stress and shear stress energy
-------------------------------------------------------------------
In accord with references [@Egami19821; @Chen19881; @Levashov2008B] we define: $$\begin{aligned}
\sigma^{\epsilon}_n(\bm{q},\bm{\hat{e}}_{\bm{q}}) =
\left(\frac{\sqrt{2}}{v_o}\right)s^{xy}_n(\bm{q},\bm{\hat{e}}_{\bm{q}})\;\;\;.\;\;\;\;\;
\label{eq;Sepsilon01}\end{aligned}$$ Both longitudinal and transverse waves contribute to $\sigma^{\epsilon}_n(\bm{q},\bm{\hat{e}}_{\bm{q}})$. Their contributions depend on the magnitude and direction of $\bm{q}$ and the direction of $\bm{\hat{e}}_{\bm{q}}$. Below, for shortness, we present the formulas for the transverse waves only. The formulas for the longitudinal waves are analogous.
From (\[eq;unm3plane40\],\[eq;unm3plane41\],\[eq;Sepsilon01\],\[eq;equip3D01\]) we get: $$\begin{aligned}
&&\left<\left[\sigma^{\epsilon}_n(T,q)\right]^2\right> =
\left(k_bT\right)\left(\frac{ka^2 N_c}{2v_o^2}\right)\frac{1}{N}
\left[\frac{\Upsilon^{xy}_2\left(T,qa\right)}{D_T\left(qa\right)}\right],\;\;\;\;\;
\label{eq;Sepsilon02}\end{aligned}$$ where: $$\begin{aligned}
\Upsilon^{xy}_2\left(T,qa\right) \equiv
\left<\left[\Upsilon^{xy}_1\left(T,\bm{q},\bm{\hat{e}_q},a\right)\right]^2\right>
\label{eq;Sepsilon03}\;,\end{aligned}$$ and $\Upsilon^{xy}_1\left(T,\bm{q},\bm{\hat{e}_q},a\right)$ is given by (\[eq;unm3plane41\]). The averaging in (\[eq;Sepsilon03\]) is over all directions of $\bm{\hat{e}}_{\bm{q}}$ orthogonal to $\bm{q}$ and then over the directions of $\bm{q}$.
We were not able to produce analytical expressions for $\Upsilon^{xy}_2\left(T,qa\right)$ and $\Upsilon^{xy}_2\left(L,qa\right)$. However, we calculated them numerically [@numerically]. Fig.\[fig:with-q2-1-crop\] shows the dependencies of $$\begin{aligned}
H^p(L,qa)\equiv&&\left[\frac{\Upsilon^{p}_{2}(L,qa)}{D_L(qa)}\right](qa)^2\;\;\;,\;\;\;\;
\label{eq;Hqa031}\\
H^{xy}(L,qa)\equiv&&\left[\frac{\Upsilon^{xy}_{2}(L,qa)}{D_L(qa)}\right](qa)^2\;\;\;,\;\;\;\;
\label{eq;Hqa032}\\
H^{xy}(T,qa)\equiv&&\left[\frac{\Upsilon^{xy}_{2}(T,qa)}{D_T(qa)}\right](qa)^2\;
\label{eq;Hqa033}\end{aligned}$$ on $qa$ without long wavelength approximation. In (\[eq;Hqa031\],\[eq;Hqa032\],\[eq;Hqa033\]) we introduced the factor $(qa)^2$ assuming further integrations over $q$.
We also obtained analytical expressions for (\[eq;Sepsilon03\]) in the long wavelength approximation: $$\begin{aligned}
\frac{\Upsilon^{xy}_{2,lw}(L,qa)}{D_L(qa)} \approx \left[\frac{8}{675}\right],\;\;\;\;\;
\frac{\Upsilon^{xy}_{2,lw}(T,qa)}{D_T(qa)} \approx \left[\frac{18}{675}\right].\;\;\;\;\;
\label{eq;epsilon03}\end{aligned}$$
![Functions $H^p(L,qa)$, $H^{xy}(L,qa)$, and $H^{xy}(T,qa)$ from (\[eq;Hqa031\],\[eq;Hqa032\],\[eq;Hqa033\]). Note that $H^{xy}(L,qa)$ curve was scaled by $100$, while $H^{xy}(T,qa)$ curve was scaled by $50$. Thus the contribution of the transverse waves to the average square of the shear stress is significantly larger than the contribution from the longitudinal waves. Also note that there are two polarizations of the transverse waves, while the figure shows the contribution from one polarization only. []{data-label="fig:with-q2-1-crop"}](Fig05){width="3.3in"}
In order to evaluate mean square stresses due to all waves we have to integrate (\[eq;Sepsilon02\]), i.e., the curves in Fig.\[fig:with-q2-1-crop\], over all $q$ using (\[eq;debye01\]). The results of these integrations, expressed in terms of the average energy of the atomic level shear stresses, are: $$\begin{aligned}
&&\frac{v_o\left<\left[\sigma_{n}^{\epsilon}(L,T)\right]^2\right>}{4G} \approx
\left(\frac{1}{4}\right)k_bT \cdot \tau(L,T)\;\;\;,\;\;\;\label{eq;epsilon041}\\
&&\tau(L) \approx \frac{3.1}{135}, \;\;\;\;\;\tau(T)\approx\frac{20.6}{135}\;\;\;.\;\;\;
\nonumber\end{aligned}$$ The shear stress energy coefficient due to all waves is: $$\begin{aligned}
\tau(L) + 2\tau(T) \approx (44.3/135) \approx \frac{1}{3}\;\;\;.\;\;
\label{eq;epsilon042}\end{aligned}$$ Thus we got the result which is 3 times smaller than the equipartition result for certain MD liquids [@Chen19881; @Levashov2008B]. Also note that this result is more than two times larger than the result that was obtained for the pressure (\[eq;pressure06\]).
In the long wavelength approximation we get: $$\begin{aligned}
\tau_{lw}(L) = \frac{24}{135}, \;\;\;\;\;\;\;\;\;\tau_{lw}(T)=\frac{54}{135},\;\;\;\\
\tau_{lw}(L) + 2\tau_{lw}(T) = (132/135) \approx 1\;.\;\;\;
\label{eq;epsilon043}\end{aligned}$$ Thus in the long wavelength approximation we essentially have $(1/4)k_bT$ dependence.
Shear Stress Correlation Function
---------------------------------
In order to introduce a shear stress correlation function which is analogous to the function $F(t,r)$ in [@Levashov20111; @Levashov2013] we first introduce a correlation function due to a particular wave. For a particular transverse wave we write: $$\begin{aligned}
&&C^{\epsilon}_{\bm{q}}(T,t,r,\bm{q},\bm{\hat{e}_q})=
\left(\frac{2}{v_o^2}\right)\cdot\\
&&\left<s_{n}^{xy}(t_o,\bm{q},\bm{\hat{e}_q})
\cdot s_{m}^{xy}(t_o+t,\bm{q},\bm{\hat{e}_q})\right>=\\
&&\left(k_bT\right)\left(\frac{ka^2 N_c}{2v_o^2}\right)\frac{1}{N}
\left[\frac{\left(\Upsilon^{xy}_1\left(T,\bm{q},\bm{\hat{e}_q},a\right)\right)^2}{D_T\left(qa\right)}\right]
\cdot\\
&&\left<\sin\left[\omega_{\bm{q}} t_o - \bm{q}\bm{r}_n +\phi_{\bm{q}}\right]
\sin\left[\omega_{\bm{q}} (t_o+t) - \bm{q}\bm{r}_m +\phi_{\bm{q}}\right]\right>\;.
\;\;\;\;\;\;\;\;
\label{eq;CXYcorr02}\end{aligned}$$ Similarly to how it was done in the transition from (\[eq;ppcf01\]) to (\[eq;ppcf05\]) for the pressure-pressure correlation function we now average over the different directions of $\bm{r}_{nm}$. The difference with the pressure-pressure case is that now the prefactor depends on the direction and the polarization of the wave, while in the pressure-pressure case it depends only on the magnitude of the wavevector. This difference is, however, irrelevant for the averaging over the directions of $\bm{r}_{nm}$. Then we perform the averaging over the polarization and the direction of the wave. This averaging is identical to the averaging that was done in derivations of (\[eq;Sepsilon02\],\[eq;Sepsilon03\]). Thus we get: $$\begin{aligned}
C^{\epsilon}(T,t,r)=\left(\frac{a}{2\pi}\right)^3N\int_0^{Q_{max}} C^{\epsilon}(T,t,r,q)\,4\pi q^2dq
\;,\;\;\;\;\;\;\;
\label{eq;CXYcorr03}\end{aligned}$$ where $$\begin{aligned}
&&C^{\epsilon}(T,t,r,q)=k_bT\left(\frac{ka^2N_c}{v_o^2}\right)\cdot E(\omega_q t)\cdot\nonumber\\
&&\cdot\frac{2}{N}\left\{\frac{\Upsilon^{xy}_2\left(T,qa\right)}{D_T(qa)}\right\}\cdot
\left\{\frac{\cos\left(\omega_{q} t/2\right)\sin\left(qr/2\right)}{(qr/2)}\right\}\;\;\;.\;\;\;\;
\label{eq;CXYcorr04}\end{aligned}$$ In (\[eq;CXYcorr04\]) we introduced the function $E(\omega_q t)$ artificially. It should not be there if the waves do not decay with time or distance. However, if we want to make a comparison with the stress correlation functions calculated in MD simulations on liquids, then it is reasonable to assume that waves decay. For the sake of a qualitative comparison we assume that [@Mizuno2013]: $$\begin{aligned}
E(\omega_q, t) = \exp\left[-0.3(\omega_q/\omega_o)^2 \omega_o t\right]\;\;\;.\;\;\;
\label{eq;efun}\end{aligned}$$
Figure \[fig:sscf-xy-T-total-1-1\] shows the stress correlation function calculated numerically from (\[eq;CXYcorr03\],\[eq;CXYcorr04\]) under the assumption that the first row in (\[eq;CXYcorr04\]) is equal to 1. Panel (a) of Figure \[fig:sscf-joined-1\] again shows the stress correlation function (\[eq;CXYcorr03\],\[eq;CXYcorr04\]), but now with $E(\omega_q, t)$ given by (\[eq;efun\]). We do not show in the figure the contribution from the longitudinal waves. This contribution is qualitatively similar to the contribution from the transverse waves. However, this contribution is significantly smaller in magnitude. Also, since the speeds of the longitudinal waves are $\approx \sqrt{3}$ times larger than the speeds of the transverse waves, the diagonal lines in the contribution from the longitudinal waves have slopes which are $\approx \sqrt{3}$ times smaller than the slopes of the diagonal lines from the transverse waves.
It is interesting to compare the panel (a) of Fig.\[fig:sscf-joined-1\] with the panel (b) of Fig.4 in Ref.[@Levashov2013]. Note that in the present paper we changed the axes and now $x$-axis shows the distance, while $y$-axis shows the time. In the panel (b) of Fig.4 in Ref. [@Levashov2013] we see two waves. One wave is longitudinal and another wave is transverse. There we also see [[*pdf*]{}]{}- like contribution to the stress correlation function. In the panel (a) of Fig.\[fig:sscf-joined-1\] we see the contribution from the transverse waves only, since we did not include in it the contribution from the longitudinal waves or the [[*pdf*]{}]{}- like structure. Besides the differences mentioned above, it is clear that the contribution to the stress correlation function from the transverse waves observed in MD simulation and in the present calculations are qualitatively similar. In particular, if we consider how intensity changes with increase of time for a given distance we observe at first positive intensity and then negative intensity.
Formula (\[eq;ppcf06\]) suggests that the speed of the wave corresponds to the slope of the first boundary between the positive and negative intensities. This interpretation is different from the one adopted in Ref.[@Levashov2013]. There it was assumed that the center of the wave corresponds to the maximum of the positive intensity. With the new interpretation the speed of the longitudinal waves in the panel (b) of Fig.4 in Ref. [@Levashov2013] is $c_l \approx 7500$ (m/s) while before it was argued that it is 6000 (m/s). The new speed of the transverse waves is $c_t \approx 5000$ (m/s), while before it was argued that it is 3000 (m/s). Note that, according to the new values, $c_l/c_t \approx 1.5$, which is not quite $\sqrt{3}$.
![Contribution from one polarization of the transverse waves to the shear stress correlation function. Scaled time is $\omega_o t$. No damping of the waves is assumed.[]{data-label="fig:sscf-xy-T-total-1-1"}](Fig06){width="3.0in"}
Fourier transforms of the shear stress correlation function
-----------------------------------------------------------
The atomic level stress correlation functions, like those in (\[eq;ppcf05\],\[eq;CXYcorr04\]), could be calculated in MD simulations [@Levashov20111; @Levashov2013]. In this section we analyze what information could be obtained by performing Fourier transforms of these stress correlation functions (\[eq;ppcf05\],\[eq;CXYcorr04\]). Thus further we consider a function which is structurally similar to the stress correlation functions (\[eq;ppcf05\],\[eq;CXYcorr04\]): $$\begin{aligned}
f(t,r) \equiv \int_0^{Q_{max}} h(q,t)
\cos\left(\frac{\omega_q t}{2}\right)
\sin\left(\frac{qr}{2}\right)dq\;\;\;.\;\;\;\;\;\;
\label{eq;ftr11}\end{aligned}$$ For the shear stress correlation function due to the transverse waves (\[eq;CXYcorr04\]), for example, we have: $$\begin{aligned}
&&f(t,r) \equiv r\cdot C^{\epsilon}(T,t,r)\label{eq;ftr121}\;\;\;,\;\;\;\\
&&h(q,t)\equiv
\alpha \left(\frac{Y_2^{xy}(T,q)}{D_T(qa)}\right)\cdot E(\omega_q, t)\cdot q\;\;\;,\;\;\;
\label{eq;ftr12}\end{aligned}$$ where $\alpha$ is a numerical coefficient. Note that $f(t,r)$, as we define it, is the correlation function per pair of particles multiplied by $r$.
Further we define: $$\begin{aligned}
&&\tilde{f}(t,q) \equiv \int_0^{\infty} f(t,r)\,\sin(qr)dr \;\;\;,\;\;\;\label{eq;ftr21x}\\
&&\tilde{f}(\omega,r) \equiv \int_0^{\infty} f(t,r)\,\cos(\omega\,t)dt \;\;\;,\;\;\;\label{eq;ftr22x}\\
&&\tilde{f}(\omega,q) \equiv \int_0^{\infty}\int_0^{\infty} f(t,r)\,\cos(\omega\,t)\,\sin(qr)\,dt\,dr \;\;\;.\;\;\;\;\;\label{eq;ftr23x}\end{aligned}$$ From (\[eq;ftr21x\],\[eq;ftr11\],\[eq;ftr12\]) we get: $$\begin{aligned}
\tilde{f}(t,q) \equiv \left(\frac{\pi}{2}\right) h(2q,t)
\cos\left(\frac{\omega_{2q}t}{2}\right)\;\;\;\;,\label{eq;ftr21}\;\;\;\;\end{aligned}$$ where $\omega_{2q}^2=\omega_o^2 D_T(2qa)$. Thus $\tilde{f}(t,q) $, for every value of $q$, oscillates in time with the period determined by the dispersion relation. The decrease in the amplitude of oscillations with increase of time is determined by the damping function $E(\omega_q, t)$ (\[eq;ftr12\]). If there were no damping function the amplitude of oscillations would remain constant.
The situation with the Fourier transform of $f(t,r)$ over time is more complicated. In general, for liquids the damping function is not exponential, but a function that describes different relaxation regimes. For an exponential damping function it is possible to perform the Fourier transform of the integrand in (\[eq;ftr11\]) in a closed analytical form. However, then it is still necessary to integrate the obtained analytical expression over $q$. Here we will not consider the case with damping in more detail. If there is no damping, i.e., if $E(\omega_q, t)=1$, then from (\[eq;ftr22x\],\[eq;ftr11\],\[eq;ftr12\]) the Fourier transform of $f(t,r)$ over $t$ is: $$\begin{aligned}
\tilde{f}(\omega,r) \equiv \left(\frac{\pi}{2}\right) h(q_{2\omega})
\sin\left(\frac{q_{2\omega}r}{2}\right)\;\;\;,\;\;\;\label{eq;ftr22}\end{aligned}$$ where $(2\omega)^2=\omega_o^2D_T(q_{2\omega}a)$. Thus, in the absence of damping, the Fourier transform of $f(t,r)$ over time (\[eq;ftr22\]) should exhibit for every frequency constant amplitude oscillations with wavelength determined by the dispersion relation.
In the absence of damping, the Fourier transforms of $f(t,r)$ over $r$ and $t$ (\[eq;ftr23x\]) lead to: $$\begin{aligned}
\tilde{f}(\omega,q) \equiv \left(\frac{\pi}{2}\right)^2 h(2q)
\delta\left(\omega-\frac{\omega_{2q}}{2}\right)\;\;,\label{eq;ftr23}\;\;\;\end{aligned}$$ i.e., to the dispersion relation. It is clear from (\[eq;ftr21\]) that damping should lead to the broadening of the $\delta$-function in (\[eq;ftr23\]) .
Panel (a) of Fig.\[fig:sscf-joined-1\] shows the function $f(t,r)$ from (\[eq;ftr121\]). Shear stress correlation function, $C^{\epsilon}(T,t,r)$, was obtained from (\[eq;CXYcorr04\]) by integration over all $q$ with $E(\omega_q, t)$ given by (\[eq;efun\]). It was assumed that $\alpha = 1$. Panels (c,b,d) show the Fourier transforms (\[eq;ftr21x\],\[eq;ftr22x\],\[eq;ftr23x\]) of the function $f(t,r)$.
{width="6.0in"}
Transverse current correlation function
---------------------------------------
In agreement with the previous definitions [@HansenJP20061; @EvansDJ19901; @Boon19911; @Tanaka20091; @Tanaka20081; @Mizuno2013; @Mountain19821], we are using the following expression for the real part of the transverse current: $$\begin{aligned}
J_T(\bm{k},t) = \sum_i \bm{v}^T_i(t)\cdot
\cos\left[\bm{k}\bm{r}_i(t)\right]\;\;,
\label{eq;jt01}\end{aligned}$$ where $\bm{v}^T_i(t) = \bm{v}_i(t) - (\bm{v}_i(t)\bm{\hat{k}})$. For the contribution from a transverse wave with the wavevector $\bm{q}$ and the polarization $\bm{\hat{e}_q}$ from (\[eq;jt01\],\[eq;poten3dx21\],\[eq;poten3dx22\]) we get: $$\begin{aligned}
&&J_T(\bm{k},\bm{q},\bm{\hat{e}_q},t) = - u_q \bm{\hat{e}}_q\omega_q B_1(\bm{k},\bm{q},t)\;,\;\nonumber\\
&&B_1(\bm{k},\bm{q})\equiv\sum_i
\sin\left[\omega_q t -\bm{q}\bm{r}_i(t) +\phi_q\right]
\cos\left[\bm{k}\bm{r}_i(t)\right]\;.\;\;\;\;
\label{eq;jt02}\end{aligned}$$ Then for the correlation function due to this wave we have: $$\begin{aligned}
C_{JT}(\bm{k},\bm{q},\bm{\hat{e}_q},t) \equiv \frac{1}{N}
\left< J_T(...,t_o)J_T(...,t_o+t)\right>\;.\;\;\;
\label{eq;tccf01}\end{aligned}$$ The averaging in (\[eq;tccf01\]) is over the initial time $t_o$.
Using the same logic that was used in the derivations of (\[eq;ppcf05\],\[eq;CXYcorr04\]) from (\[eq;jt02\],\[eq;tccf01\],\[eq;equip3D01\]) we get: $$\begin{aligned}
&&C_{JT}(\bm{k},\bm{q},\bm{\hat{e}_q},t) =
\left(\frac{C_e}{N}\right)
\cos\left(\omega_q t\right)\left[X_1 + X_2\right]
\;,\;\;\label{eq;tccf011}\\
&&C_e \equiv \frac{u_q^2 \omega_q^2}{4}=\frac{k_bT}{2NM}\;\;,\;\;\;
X_{1,2} \equiv \sum_{ij}\frac{\sin\left(|\bm{k}\pm \bm{q}|r_{ij}\right)}
{|\bm{k}-\bm{q}|r_{ij}}\;.\;\;\;\;\;\;
\label{eq;tccf012}\end{aligned}$$ In derivations of (\[eq;tccf011\],\[eq;tccf012\]) there also appear two other terms which, however, vanish in the limit $N\rightarrow \infty$. After the integration over $\bm{q}$ contributions from $X_1$ and $X_2$ terms are equal to each other. Note that if $\bm{k}=0$ then the structure of (\[eq;tccf011\],\[eq;tccf012\]) is rather similar to the structures of (\[eq;ppcf05\],\[eq;CXYcorr04\]).
It follows from (\[eq;tccf011\],\[eq;tccf012\]) that it is possible to introduce and consider [*atomic level*]{} transverse current correlation function similarly to how it was done for the atomic level stress correlation function in Ref. [@Levashov20111; @Levashov2013].
Discussion \[sec:discussion\]
=============================
The primary goal of this paper is to gain an insight into the connection between the atomic level vibrational dynamics and the atomic level Green-Kubo stress correlation function. The understanding of this connection is needed to interpret the results of the previous MD simulations of a model liquid [@Levashov20111; @Levashov2013]. For this purpose, we considered a simple model in which vibrations are plane waves. Such representation of vibrations does not imply that we think that vibrations in liquids or glasses are plane waves. The situation in disordered materials is much more complex [@KeyesT19971; @Taraskin2000; @Taraskin2002; @Scopigno20071]. However, the model that we consider is solvable, and it provides the needed insight and a recipe for the analysis in Fourier space of the atomic level stress correlation functions obtained in MD simulations [@Levashov20141].
Atoms, as they move, do not decompose their motions into the orthogonal vibrational modes. Instead, they experience forces and stresses. From this perspective, the comparisons of the atomic level stress correlation functions from different liquids and temperatures may provide valuable and, probably, more physical insights into the atomic scale dynamics than the considerations of the vibrational eigenmodes.
Further we discuss why atomic level stress energies obtained in MD simulations are significantly larger than the values obtained in this paper. In the framework of the studied model the average atomic level stress energies in the long wavelength approximation and at high temperatures are similar to those in Ref.[@Egami19821].
To summarize, if the long wavelength approximation is not assumed then the pressure/shear stress energy within the model is approximately 8/3 times smaller than the value obtained from MD simulations [@Chen19881; @Levashov2008B]. If the long wavelength approximation is assumed then the pressure stress energy is 2 times smaller than the value from MD simulations, while the shear stress energy is approximately equal to the value from MD. Please, see also the text preceding to equation (\[eq;elastC01\]).
The dependencies of the atomic level stress energies on temperature, obtained in MD simulations on a liquid and its glass, can be found in Fig.3 of Ref.[@Chen19881] and in Fig.5 of Ref.[@Levashov2008B]. If $T \gtrsim 1000$ (K) the system is in a liquid state. If $T \lesssim 1000$ (K), the system is in a glass state.
Note, in the figures, that at $T=0$ (K) the atomic level stress energies have a finite value $U_o$. At $T=0$ (K) there are no vibrations in the classical systems. Thus the values of the atomic level stress energies at $T=0$ (K) are determined only by the structural disorder, which also includes variations in the coordination numbers between the different atoms, as can be seen in Fig.2 of Ref.[@Levashov2008E].
In our present calculations of the atomic level stress energies, it was assumed that every atom interacts with $N_c$ neighbors. It was also assumed that $N_c$ is the same for every atom. But in the previous calculations of the atomic level stress energies with MD simulations, no distinction was made between the atoms with different coordination numbers [@Chen19881; @Levashov2008B].
Formulas (\[eq;pressure06\],\[eq;Sepsilon02\]) show that average squares of the atomic level stresses are proportional to $N_c$, while (\[eq;elastC01\]) shows that elastic constants are also proportional to $N_c$. From this perspective, if all atoms have the same coordination, the atomic level stress energies, which are proportional to the ratio of the average squares of the stresses to the relevant elastic constants, should not exhibit dependence on $N_c$. However, in the previous MD simulations, the average squares of the stresses and the average values of the elastic constants were obtained by averaging over the atoms with different $N_c$. Thus a (large) part of the average atomic level stress energies obtained in previous MD simulations in the glass and in the liquid states might be related to the variations in the coordination numbers between the different atoms. Another possibility might be associated with inapplicability of the spherical approximation for certain coordination numbers or with inapplicability of the plane wave approximation.
Atomic level stresses were originally applied to a model of metallic glass in order to describe structural disorder [@Egami1980]. Later it was briefly discussed that it might be possible to speak about structural and vibrational contributions to the atomic level stresses [@Egami19821]. However, no systematic attempts to separate structural and vibrational contributions to the atomic level stresses were previously made. Considerations of the atomic level stress energies for the subsets of atoms with different coordination numbers should help to elucidate the roles of structural disorder and vibrational dynamics. This is in agreement with the results of [@Iwashita2013].
In the glass state, dependencies of the atomic level stress energies on temperature (obtained in MD simulations) can approximately be described by a formula: $U_o + (1/6)k_b T$. It is natural to associate $U_o$ with the structural contribution, while $(1/6)k_b T$ with the vibrational contribution. Note that the rates of increase of the atomic level stress energies in the glass obtained from MD simulations are larger than the values that we derived in this paper (see formulas (\[eq;pressure06\]) and (\[eq;epsilon041\])). Note also that, according to the present paper, the rates of increase of the pressure and shear energies should be different. However, previous MD simulations show similar rates. This can reflect the fact that vibrations in disordered media are not plane waves as we assumed here. It also can reflect the fact that structural disorder changes even in the glass state. For example, as temperature increases (still in the glass state) there is a weak change in the number of atoms with a given coordination as can be seen in Fig.2 of Ref.[@Levashov2008E]. However, this weak change can also be a consequence of the vibrational dynamics.
In the liquid state temperature dependence of the atomic level stress energies closely follows $(1/4)k_b T$. Absence of $U_o$ in the last formula suggests that in the liquid state there is no “frozen in" structural contribution. Thus Fig.3 of Ref.[@Chen19881] and Fig.5 of Ref.[@Levashov2008B] suggest that it might be possible to speak about vibrational and configurational degrees of freedom in the glass state. However, in the liquid states these degrees of freedom appear to be completely mixed.
In derivations of the $(1/4)k_b T$ law for the atomic level stress energies, it was assumed that the values of atomic level stresses can vary from $-\infty$ to $+\infty$ [@Egami19821; @Levashov2008B]. Any, in magnitude, contribution to the atomic level stress can come from the repulsive part of the potential for a given coordination number because very large forces can originate from the repulsive core. On the other hand, only a limited contribution can come from the attractive part of the potential for a given coordination number. This also can be the reason why the equipartition result does not hold for a fixed coordination number, but holds when averaging is done over all coordination numbers.
Derivation of the equipartition from the Boltzmann distribution [@Egami19821; @Levashov2008B] implies the presence of ergodicity for every central atom and its coordination shell. In the liquid state, time averaging over every atom is equal to the ensemble average. In the glass state, as the coordination numbers of many atoms are fixed on the timescale of a simulation, time average of the stress for every particular atom is not equal to the ensemble average. Thus, breakdown of equipartition of the atomic level stress energies at the glass transition might signal ergodicity breaking.
Finally, we note that the total potential energy of the system per atom follows $(3/2)k_bT$ law (see Fig.3 of Ref.[@Levashov2008E]) in the glass state, i.e., it follows the equipartition law and it grows faster than the energies of the atomic level stresses. Thus, the energies of the atomic level stresses obtained from our calculations here or in the previous MD simulations do not capture the total increase in potential energy.
In the liquid state, the total potential energy grows faster than $(3/2)k_bT$ in the range of temperatures between the glass transition temperature and the potential energy landscape crossover temperature (see Fig.3 of Ref.[@Levashov2008E]). Thus, the total potential energy again grows faster than the energy of the atomic level stresses. In this range of temperatures, the total potential energy follows the Rosenfeld-Tarazona law $U=U_g + bT^{3/5}$ [@Rosenfeld1998; @Gebremichael2005], where $U_g$ is the energy at the glass transition. The value of the coefficient $b$ is such that in the studied range of temperatures (which covers all reasonable temperatures), potential energy of a liquid is larger than $(3/2)k_bT$. Thus, the energies of the local atomic level stresses and the harmonic approximation made in their derivations do not reflect all processes that happen in the model liquid upon heating.
Acknowledgments
===============
We would like to thank T. Egami, V.N. Novikov, and K.A. Lokshin for useful discussions.
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For transverse waves it is convenient to represent $\bm{\hat{e}}_{\bm{q}}^T$ as a linear combination of vectors $\bm{\hat{e}}_{\bm{q}}^1$, $\bm{\hat{e}}_{\bm{q}}^2$ which are orthogonal to $\bm{\hat{q}}$ and to each other. If angles $\theta$ and $\phi$ define the direction of $\bm{q}$ then we can chose $\bm{\hat{e}}_{\bm{q}}^1 \equiv\left[\cos(\theta)\cos(\phi),\cos(\theta)\sin(\phi),-\sin(\theta)\right]$ and $\bm{\hat{e}}_{\bm{q}}^2 \equiv \left[ \bm{\hat{q}} \times \bm{\hat{e}}_{\bm{q}}^1\right]=
\left[-\sin(\phi),\cos(\phi),0\right]$. Thus we write $\bm{\hat{e}}_{\bm{q}}^T \equiv \bm{\hat{e}}_{\bm{q}}^1 \cos(\psi) + \bm{\hat{e}}_{\bm{q}}^2 \sin(\psi)$. In (\[eq;Sepsilon03\]), after taking the square, averaging should be done over $\psi$ and then over the directions of $\bm{\hat{q}}$. For longitudinal waves $\bm{\hat{e}}_{\bm{q}}^L || \bm{\hat{q}}$.
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|
---
abstract: 'Nonparametric estimation of a statistic, in general, and of the error rate of a classification rule, in particular, from just one available dataset through resampling is well mathematically founded in the literature using several versions of bootstrap and influence function. This article first provides a concise review of this literature to establish the theoretical framework that we use to construct, in a single coherent framework, nonparametric estimators of the AUC (a two-sample statistic) other than the error rate (a one-sample statistic). In addition, the smoothness of some of these estimators is well investigated and explained. Our experiments show that the behavior of the designed AUC estimators confirms the findings of the literature for the behavior of error rate estimators in many aspects including: the weak correlation between the bootstrap-based estimators and the true conditional AUC; and the comparable accuracy of the different versions of the bootstrap estimators in terms of the RMS with little superiority of the .632+ bootstrap estimator.'
address: 'Ph.D., ECE dep., University of Victoria, Canada; and CS dep., Helwan University, Egypt.'
author:
- 'Waleed A. Yousef'
title: 'AUC: Nonparametric Estimators and Their Smoothness'
---
Statistical Learning,Machine Learning,Bootstrap,Jackknife,Influence Function, Nonparametric Assessment,Error Rate,Receiver Operating Characteristic,ROC,Area Under the Curve,AUC.
|
---
abstract: 'In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-representations for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-representations.'
author:
- |
Tobias Kildetoft, Marco Mackaay,\
Volodymyr Mazorchuk and Jakob Zimmermann
title: |
Simple transitive $2$-representations\
of small quotients of Soergel bimodules
---
Introduction and description of the results {#s1}
===========================================
Classical representation theory makes a significant emphasis on problems and techniques related to the classification of various classes of representations. The recent “upgrade” of classical representation theory, known as [*$2$-representation theory*]{}, has its abstract origins in [@BFK; @CR; @KL; @Ro]. The first classification result in $2$-representation theory was obtained in [@CR Subsection 5.4.2] where certain “minimal” $2$-representations of the Chuang-Rouquier $2$-analogue of $U(\mathfrak{sl}_2)$ were classified.
The series [@MM1; @MM2; @MM3; @MM4; @MM5; @MM6] of papers initiated a systematic study of the so-called [*finitary*]{} $2$-categories which are natural $2$-analogues of finite dimensional algebras. The penultimate paper [@MM5] of this series proposes the notion of a [*simple transitive $2$-representation*]{} which seems to be a natural $2$-analogue for the classical notion of a simple module. Furthermore, [@MM5; @MM6] classifies simple transitive $2$-representations for a certain class of finitary $2$-categories with a weak involution which enjoy particularly nice combinatorial properties, the so-called [*(weakly) fiat*]{} $2$-categories with [*strongly regular two-sided cells*]{}. Examples of the latter $2$-categories include projective functors for finite dimensional self-injective algebras, finitary quotients of finite type $2$-Kac-Moody algebras and Soergel bimodules (over the coinvariant algebra) in type $A$. The classification results of [@MM5; @MM6] assert that, for such $2$-categories, every simple transitive $2$-representation is, in fact, equivalent to a so-called [*cell $2$-representation*]{}, that is a natural subquotient of the regular (principal) $2$-representation defined combinatorially in [@MM1; @MM2]. In [@KM1], this classification was used to describe projective functors on parabolic category $\mathcal{O}$ in type $A$.
The problem of classifying simple transitive $2$-representations was recently studied for several classes of finitary $2$-categories which are not covered by the results in [@MM5; @MM6]. In particular, in [@Zi] it is shown that cell $2$-representations exhaust the simple transitive $2$-representations for the $2$-category of Soergel bimodules in Weyl type $B_2$. In [@MZ], a similar result was proved for the $2$-categories of projective functors for the two smallest non-self-injective finite dimensional algebras. Some other related results can be found in [@GM1; @GM2; @Zh1; @Zh2].
An essential novel step in this theory was made in the recent paper [@MaMa] where a classification of the simple transitive $2$-representations was given for certain $2$-subquotient categories of Soergel bimodules in two dihedral Coxeter types. In one of the cases, it turned out that cell $2$-representations do not exhaust the simple transitive $2$-representations. A major part of [@MaMa] is devoted to an explicit construction of the remaining simple transitive $2$-representation. This construction involves a subtle interplay of various category theoretic tricks.
The present paper explores to which extent the techniques developed in [@MM5; @Zi; @MaMa] can be used to attack the problem of classification of the simple transitive $2$-representations for $2$-categories of Soergel bimodules over coinvariant algebras in the general case of finite Coxeter systems. We develop the approach and intuition described in [@Zi; @MaMa] further and single out a situation in which this approach seems to be applicable. The combinatorial structure of the $2$-category of Soergel bimodules is roughly captured by the Kazhdan-Lusztig combinatorics of the so-called [*two-sided Kazhdan-Lusztig cells*]{}. The minimal, with respect to the two-sided order, two-sided cell corresponds to the identity element. If we take this minimal two-sided cell out, in what remains there is again a unique minimal two-sided cell. This is the two-sided cell which contains all simple reflections. The main object of study in the present paper is the unique “simple” quotient of the $2$-category of Soergel bimodules in which only these two smallest two-sided cells survive. This is the $2$-category which we call the [*small quotient*]{} of the $2$-category of Soergel bimodules. Our main result is the following statement that combines the statements of Theorems \[thm1700\], \[thm49\], \[thm82\], \[thm84\], \[thm86\] and \[thm88\].
\[thmintro\] Let $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ be the small quotient of the $2$-category of Soergel bimodules over the coinvariant algebra associated to a finite Coxeter system $(W,S)$.
1. \[thmintro.1\] If $W$ has rank greater than two or is of Coxeter type $I_2(n)$, with $n=4$ or $n>1$ odd, then every simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation.
2. \[thmintro.2\] If $W$ is of Coxeter type $I_2(n)$, with $n>4$ even, then, apart from cell $2$-representations, $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ has two extra equivalence classes of simple transitive $2$-representations, see the explicit construction in Subsection \[s6.7\]. If $n\neq 12,18,30$, these are all the simple transitive $2$-representations.
We note that Coxeter type $I_2(4)$ is dealt with in [@Zi] and the result in this case is similar to Theorem \[thmintro\]. In fact, the construction in Subsection \[s6.7\] also works in type $I_2(4)$, however, it results in $2$-representations which turn out to be equivalent to cell $2$-representations. The exceptional types $I_2(12)$, $I_2(18)$ and $I_2(30)$ exhibit some strange connection, which we do not really understand, to Dynkin diagrams of type $E$ that, at the moment, does not allow us to complete the classification of simple transitive $2$-representations in these types.
Theorem \[thmintro\] is proved by a case-by-case analysis. Similarly to the general approach of [@MM5; @MM6; @Zi; @MaMa], in each case, the proof naturally splits into two major parts:
- the first part of the proof determines the non-negative integral matrices which represent the action of Soergel bimodules corresponding to simple reflections;
- to each particular case of matrices determined in the first part, the second part of the proof provides a classification of simple transitive $2$-representations for which these particular matrices are realized.
Although being technically more involved, the second part of the proof is fairly similar to the corresponding arguments in [@MM5; @MM6; @Zi; @MaMa]. For Theorem \[thmintro\], an additional essential part is the construction of the two extra simple transitive $2$-representations. This construction follows closely the approach of [@MaMa Section 5]. Apart from this, the main general difficulty in the proof of Theorem \[thmintro\] lies in the first part of the proof. Both the difficulty of this part and the approach we choose to deal with this part depends heavily on each particular Coxeter type we study.
For Coxeter groups of rank higher than two, we use a reduction argument to the rank two case. For this reduction to work, we need the statement of Theorem \[thmintro\] in Coxeter types $I_2(n)$, where $n=3,4,5$. For $n=3,5$, we may directly refer to Theorem \[thmintro\]. As already mentioned above, the case $n=4$ is treated in [@Zi]. Behind this reduction is the observation that Soergel bimodules corresponding to the longest elements in rank two parabolic subgroups of $W$ do not survive projection onto $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$.
The argument for the first part of the proof which we employ in the rank two case is based on an analysis of spectral properties of some integral matrices. We observe that Fibonacci polynomials are connected to minimal polynomials of certain integral matrices which encode the combinatorics of simple transitive $2$-representations. Using the factorization of Fibonacci polynomials over $\mathbb{Q}$, see [@Lev], and estimating the values of the maximal possible eigenvalues for the matrices which encode the action of Soergel bimodules corresponding to simple reflections in simple transitive $2$-representations, allows us to deduce that the only possibility for these matrices are the ones appearing in cell $2$-representations.
Apart from Theorem \[thmintro\], we also prove the following general result. We refer to Subsection \[s2.4\] and [@CM Subsection 3.2] for the definition of the apex and to Subsection \[s2.3\] and [@MM2 Subsection 4.2] for the definition of the abelianization $\overline{\mathbf{M}}$ of a $2$-representation $\mathbf{M}$.
\[thmmintrotwo\] Let $\mathbf{M}$ be a simple transitive $2$-representation of a fiat $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$ with apex $\mathcal{I}$. Then, for every $1$-morphism $\mathrm{F}\in \mathcal{I}$ and every object $X$ in any $\overline{\mathbf{M}}(\mathtt{i})$, the object $\mathrm{F}\,X$ is projective. Moreover, $\overline{\mathbf{M}}(\mathrm{F})$ is a projective functor.
Our proof of this statement is based crucially on the results from [@KM2]. Theorem \[thmmintrotwo\] will be very useful in further studies of simple transitive $2$-representations. It applies directly to all cases we consider and substantially simplifies some of the arguments. For example, in Coxeter type $I_2(5)$, we originally had an independent argument for a similar result which involved a very technical statement that a certain collection of twenty seven linear inequalities is equivalent to the fact that all these inequalities are, in fact, equalities. This full argument was three pages long and just covered one Coxeter type. A similar argument in other Coxeter types of the form $I_2(n)$ seemed, for a long period of time, unrealistic.
Our results form a first step towards classification of simple transitive $2$-representations of Soergel bimodules in all types. However, for the moment, the technical difficulty of solving the first part of the problem (as described above) in the general case seems too high. Already in type $B_3$ the classification of simple transitive $2$-representations is not complete. It is also of course very natural to ask what happens in positive characteristics. However, the situation there is expected to be even more complicated.
The paper is organized as follows: In Section \[s2\] we collect all necessary preliminaries on $2$-categories and $2$-representations. Section \[s3\] collects preliminaries on $2$-categories of Soergel bimodules. In Section \[s4\] we collect several general results related to classification of simple transitive $2$-representations of small quotients of Soergel bimodules. It also contains our proof of Theorem \[thmmintrotwo\], see Theorem \[thmminimal\] in Subsection \[s5.5\]. Section \[sim\] contains auxiliary results on some spectral properties of integral matrices. Coxeter type $I_2(n)$, for $n$ odd, is studied in Section \[s17\]. Coxeter type $I_2(n)$, for $n$ even, is studied in Section \[s6\]. Section \[s7\] collects our study of all non-simply laced Coxeter types of rank higher than two (all simply laced types are covered by the results of [@MM5; @MM6]). Finally, in Section \[s8\] we propose a new general construction of finitary $2$-categories. The novel component of this construction is that indecomposable $2$-morphisms are, in general, defined using decomposable functors. This allows us to give an alternative description of one interesting example of a finitary $2$-category from [@Xa Example 8].
[**Warning.**]{} All Soergel bimodules considered in this paper are over the coinvariant algebra, not the polynomial algebra. Furthermore, our setup is ungraded.
[**Acknowledgment.**]{} This research was partially supported by the Swedish Research Council (for V. M.), Knut and Alice Wallenbergs Foundation (for T. K. and V. M.) and G[ö]{}ran Gustafsson Foundation (for V. M.). We thank Vanessa Miemietz and Xiaoting Zhang for poining out a gap in one of the proofs. We thank the referee for helpful comments.
$2$-categories and $2$-representations {#s2}
======================================
Notation and conventions {#s2.1}
------------------------
We work over the field $\mathbb{C}$ of complex numbers and denote $\otimes_{\mathbb{C}}$ by $\otimes$. A module always means a [*left*]{} module. All maps are composed from right to left.
Finitary and fiat $2$-categories {#s2.2}
--------------------------------
For generalities on $2$-categories, we refer the reader to [@Le; @Mc; @Ma].
By a $2$-category we mean a category enriched over the monoidal category $\mathbf{Cat}$ of small categories. In other words, a $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$ consists of objects (which we denote by Roman lower case letters in a typewriter font), $1$-morphisms (which we denote by capital Roman letters), and $2$-morphisms (which we denote by Greek lower case letters), composition of $1$-morphisms, horizontal and vertical compositions of $2$-morphisms (denoted $\circ_0$ and $\circ_1$, respectively), identity $1$-morphisms and identity $2$-morphisms. These satisfy the obvious collection of axioms.
For a $1$-morphism $\mathrm{F}$, we denote by $\mathrm{id}_{\mathrm{F}}$ the corresponding identity $2$-morphism. We often write $\mathrm{F}(\alpha)$ for $\mathrm{id}_{\mathrm{F}}\circ_0\alpha$ and $\alpha_{\mathrm{F}}$ for $\alpha\circ_0\mathrm{id}_{\mathrm{F}}$.
We say that a $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$ is [*finitary*]{} if each category ${\sc\mbox{C}\hspace{1.0pt}}(\mathtt{i},\mathtt{j})$ is an idempotent split, additive and Krull-Schmidt $\mathbb{C}$-linear category with finitely many isomorphism classes of indecomposable objects and finite dimensional morphism spaces, moreover, all compositions are assumed to be compatible with these additional structures, see [@MM1 Subsection 2.2] for details.
If ${\sc\mbox{C}\hspace{1.0pt}}$ is a finitary $2$-category, we say that ${\sc\mbox{C}\hspace{1.0pt}}$ is [*fiat*]{} provided that it has a weak involution $\star$ together with adjunction $2$-morphisms satisfying the axioms of adjoint functors, for each pair $(\mathrm{F},\mathrm{F}^{\star})$ of $1$-morphisms, see [@MM1 Subsection 2.4] for details. Similarly, we say that ${\sc\mbox{C}\hspace{1.0pt}}$ is [*weakly fiat*]{} provided that it has a weak anti-autoequivalence $\star$ (not necessarily involutive) together with adjunction $2$-morphisms satisfying the axioms of adjoint functors, for each pair $(\mathrm{F},\mathrm{F}^{\star})$ of $1$-morphisms, see [@MM6 Subsection 2.5] for details.
$2$-representations {#s2.3}
-------------------
Let ${\sc\mbox{C}\hspace{1.0pt}}$ be a finitary $2$-category. The $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$-afmod consists of all [*finitary $2$-representations*]{} of ${\sc\mbox{C}\hspace{1.0pt}}$ as defined in [@MM3 Subsection 2.3]. Objects in ${\sc\mbox{C}\hspace{1.0pt}}$-afmod are strict functorial actions of ${\sc\mbox{C}\hspace{1.0pt}}$ on idempotent split, additive and Krull-Schmidt $\mathbb{C}$-linear categories which have finitely many isomorphism classes of indecomposable objects and finite dimensional spaces of morphisms. In ${\sc\mbox{C}\hspace{1.0pt}}$-afmod, $1$-morphisms are strong $2$-natural transformations and $2$-morphisms are modifications.
Similarly, we consider the $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$-mod consisting of all [*abelian $2$-representations*]{} of ${\sc\mbox{C}\hspace{1.0pt}}$. These are functorial actions of ${\sc\mbox{C}\hspace{1.0pt}}$ on categories equivalent to module categories over finite dimensional algebras, we again refer to [@MM3 Subsection 2.3] for details. The $2$-categories ${\sc\mbox{C}\hspace{1.0pt}}$-afmod and ${\sc\mbox{C}\hspace{1.0pt}}$-mod are connected by the diagrammatically defined [*abelianization $2$-functor*]{} $$\overline{\hspace{1mm}\cdot\hspace{1mm}}:
{\sc\mbox{C}\hspace{1.0pt}}\text{-}\mathrm{afmod}\to {\sc\mbox{C}\hspace{1.0pt}}\text{-}\mathrm{mod},$$ see [@MM2 Subsection 4.2] for details. For $\mathbf{M}\in {\sc\mbox{C}\hspace{1.0pt}}\text{-}\mathrm{afmod}$, objects in $\overline{\mathbf{M}}$ are diagrams of the form $X\longrightarrow Y$ over $\mathbf{M}(\mathtt{i})$’s and morphisms are quotients of the space of (solid) commutative squares of the form $$\xymatrix{
X\ar[rr]\ar[d]&&Y\ar[d]\ar@{.>}[dll]\\
X\ar[rr]&&Y'
}$$ modulo the subspace for which the right horizontal map factorizes via some dotted map. The action of ${\sc\mbox{C}\hspace{1.0pt}}$ is defined component-wise.
We say that two $2$-representations are [*equivalent*]{} provided that there is a strong $2$-natural transformation between them which restricts to an equivalence of categories, for each object in ${\sc\mbox{C}\hspace{1.0pt}}$.
A finitary $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$ is said to be [*transitive*]{} provided that, for any indecomposable objects $X$ and $Y$ in $\displaystyle
\coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}\mathbf{M}(\mathtt{i})$, there is a $1$-morphism $\mathrm{F}$ in ${\sc\mbox{C}\hspace{1.0pt}}$ such that the object $Y$ is isomorphic to a direct summand of the object $\mathbf{M}(\mathrm{F})\, X$. A transitive $2$-representation $\mathbf{M}$ is said to be [*simple transitive* ]{} provided that $\displaystyle \coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}\mathbf{M}(\mathtt{i})$ does not have any non-zero proper ideals which are invariant under the functorial action of ${\sc\mbox{C}\hspace{1.0pt}}$. Given a finitary $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$, the [*rank*]{} of $\mathbf{M}$ is the number of isomorphism classes of indecomposable objects in $$\coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}\mathbf{M}(\mathtt{i}).$$
For simplicity, we will often use the “module” notation $\mathrm{F}\, X$ instead of the corresponding “representation” notation $\mathbf{M}(\mathrm{F})\, X$.
Combinatorics {#s2.4}
-------------
Let ${\sc\mbox{C}\hspace{1.0pt}}$ be a finitary $2$-category and $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$ the corresponding [*multisemigroup*]{} in the sense of [@MM2 Section 3]. The objects in $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$ are isomorphism classes of indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}$. For $\mathrm{F},\mathrm{G}\in \mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$, we set $\mathrm{F}\leq_L\mathrm{G}$ provided that $\mathrm{G}$ is isomorphic to a summand of $\mathrm{H}\circ \mathrm{F}$, for some $1$-morphism $\mathrm{H}$. The relation $\leq_L$ is called the [*left*]{} (pre)-order on $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$. The [*right*]{} and [*two-sided*]{} (pre)-orders are defined similarly using multiplication on the right or on both sides. Equivalence classes with respect to these orders are called [*cells*]{} and the corresponding equivalence relations are denoted $\sim_L$, $\sim_R$ and $\sim_J$, respectively. As usual, for simplicity, we say “cells of ${\sc\mbox{C}\hspace{1.0pt}}$” instead of “cells of $\mathcal{S}[{\sc\mbox{C}\hspace{1.0pt}}]$”.
Given a two-sided cell $\mathcal{J}$ in ${\sc\mbox{C}\hspace{1.0pt}}$, we call ${\sc\mbox{C}\hspace{1.0pt}}$ [*$\mathcal{J}$-simple*]{} provided that every non-zero two-sided $2$-ideal in ${\sc\mbox{C}\hspace{1.0pt}}$ contains the identity $2$-morphisms for some (and hence for all) $1$-morphisms in $\mathcal{J}$, see [@MM2 Subsection 6.2] for details.
A two-sided cell is called [*strongly regular*]{}, see [@MM1 Subsection 4.8], provided that
- different left (resp. right) cells inside $\mathcal{J}$ are not comparable with respect to $\leq_L$ (resp. $\leq_R$);
- the intersection of any left and any right cell inside $\mathcal{J}$ consists of exactly one element.
By [@KM2 Corollary 19], the first condition is automatically satisfied for all fiat $2$-categories.
By [@CM Subsection 3.2], each simple transitive $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$ has an [*apex*]{}, which is defined as a unique two-sided cell that is maximal in the set of all two-sided cells whose elements are not annihilated by $\mathbf{M}$.
Cell $2$-representations {#s2.5}
------------------------
For every $\mathtt{i}\in {\sc\mbox{C}\hspace{1.0pt}}$, we will denote by $\mathbf{P}_{\mathtt{i}}:={\sc\mbox{C}\hspace{1.0pt}}_A(\mathtt{i},{}_-)$ the corresponding [*principal*]{} $2$-representation. For a left cell $\mathcal{L}$ in ${\sc\mbox{C}\hspace{1.0pt}}$, there is $\mathtt{i}\in {\sc\mbox{C}\hspace{1.0pt}}$ such that all $1$-morphisms in $\mathcal{L}$ start at $\mathtt{i}$. Let $\mathbf{N}_{\mathcal{L}}$ denote the $2$-subrepresentation of $\mathbf{P}_{\mathtt{i}}$ given by the additive closure of all $1$-morphisms $\mathbf{F}$ satisfying $\mathbf{F}\geq_L \mathcal{L}$. Then $\mathbf{N}_{\mathcal{L}}$ has a unique maximal ${\sc\mbox{C}\hspace{1.0pt}}$-invariant ideal and the corresponding quotient $\mathbf{C}_{\mathcal{L}}$ is called the [*cell*]{} $2$-representation associated to $\mathcal{L}$. We refer the reader to [@MM1 Section 4] and [@MM2 Subsection 6.5] for more details.
Matrices in the Grothendieck group {#s2.6}
----------------------------------
Let ${\sc\mbox{C}\hspace{1.0pt}}$ be a finitary $2$-category and $\mathbf{M}$ a finitary $2$-representation of ${\sc\mbox{C}\hspace{1.0pt}}$. Fix a complete and irredundant list of representatives in all isomorphism classes of indecomposable objects in $\displaystyle \coprod_{\mathtt{i}}\mathbf{M}(\mathtt{i})$. Then, for every $1$-morphism $\mathrm{F}$ in ${\sc\mbox{C}\hspace{1.0pt}}$, we have the corresponding matrix $\Lparen \mathrm{F}\Rparen$ which describes multiplicities in direct sum decompositions of the images of indecomposable objects under $\mathbf{M}(\mathrm{F})$.
If ${\sc\mbox{C}\hspace{1.0pt}}$ is fiat, then each $\overline{\mathbf{M}}(\mathrm{F})$ is exact and we also have the matrix $\llbracket \mathrm{F}\rrbracket$ which describes composition multiplicities of the images of simple objects under $\overline{\mathbf{M}}(\mathrm{F})$. By adjunction, the matrix $\llbracket \mathrm{F}^{\star}\rrbracket$ is transposed to the matrix $\Lparen \mathrm{F}\Rparen$.
Based modules over positively based algebras {#s2.7}
--------------------------------------------
In this subsection we recall some notation and results from [@KM2]. Let $A$ be a finite dimensional $\mathbb{C}$-algebra with a fixed basis $\{a_1,a_2,\dots,a_n\}$. Assume that $A$ is [*positively based*]{} in the sense that all structure constants $\gamma_{i,j}^s$ with respect to this basis, defined via $$a_ia_j=\sum_{s=1}^n\gamma_{i,j}^sa_s,\quad\text{ where } \gamma_{i,j}^s\in\mathbb{C},$$ are non-negative real numbers. For $s,j\in\{1,2,\dots,n\}$, we set $a_s\geq_L a_j$ provided that $\gamma_{i,j}^s\neq 0$ for some $i$. We set $a_s\sim_L a_j$ provided that $a_s\geq_L a_j$ and $a_j\geq_L a_s$ at the same time. This defines the [*left order*]{} and the corresponding [*left cells*]{}. The right and two-sided orders and the right and two-sided cells are defined similarly (using multiplication from the right or from both sides) and denoted $\geq_R$, $\geq_J$, $\sim_R$ and $\sim_J$, respectively.
For each left cell $\mathcal{L}$, we have the corresponding [*left cell*]{} $A$-module $C_{\mathcal{L}}$. The module $C_{\mathcal{L}}$ contains a subquotient $L_{\mathcal{L}}$, called the [*special subquotient*]{}, which has composition multiplicity one in $C_{\mathcal{L}}$. This subquotient is defined as the unique subquotient of $C_{\mathcal{L}}$ which contains the Perron-Frobenius eigenvector of $C_{\mathcal{L}}$ for the linear operator $\sum_i a_i$. We have $L_{\mathcal{L}}\cong L_{\mathcal{L}'}$ if $\mathcal{L}$ and $\mathcal{L}'$ belong to the same two-sided cell. The latter justifies the notation $L_{\mathcal{J}}:=L_{\mathcal{L}}$, where $\mathcal{J}$ is the two-sided cell containing $\mathcal{L}$.
An $A$-module $V$ with a fixed basis $\{v_1,v_2,\dots,v_k\}$ is called [*positively based*]{} provided that each $a_i\cdot v_j$ is a linear combination of the $v_m$’s with non-negative real coefficients. A positively based module $V$ is called [*transitive*]{} provided that, for any $v_i$ and $v_j$, there is an $a_s$ such that $v_j$ appears in $a_s\cdot v_i$ with a non-zero coefficient. For each transitive $A$-module $V$, there is a unique two-sided cell $\mathcal{J}(V)$ which is the maximum element (with respect to the two-sided order) in the set of all two-sided cells whose elements do not annihilate $V$. The two-sided cell $\mathcal{J}(V)$ is called the [*apex*]{} of $V$. The two-sided cell $\mathcal{J}(V)$ is [*idempotent*]{} in the sense that it contains some $a_i$, $a_j$ and $a_s$ (non necessarily different) such that $\gamma_{i,j}^s\neq 0$. The simple subquotient $L_{\mathcal{J}(V)}$ has composition multiplicity one in $V$ and is also called the [*special*]{} subquotient of $V$.
Decategorification {#s2.8}
------------------
For a finitary $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$, we denote by $A_{{\scc\mbox{C}\hspace{1.0pt}}}$ the complexification of the Grothendieck decategorification of ${\sc\mbox{C}\hspace{1.0pt}}$. The algebra $A_{{\scc\mbox{C}\hspace{1.0pt}}}$ is positively based with respect to the [*natural basis*]{} given by the classes of indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}$, see [@Ma Subsection 1.2] or [@KM2 Subsection 2.5] for details.
Given a $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$, the Grothendieck decategorification of $\mathbf{M}$ is, naturally, a positively based $A_{{\scc\mbox{C}\hspace{1.0pt}}}$-module. Moreover, the latter is transitive if and only if $\mathbf{M}$ is transitive. This allows us to speak about the [*apex*]{} of $\mathbf{M}$ in the obvious way, see [@CM Subsection 3.2] for details.
We denote by $\mathrm{F}_{\mathrm{tot}}$ a multiplicity free direct sum of all indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}$. Given a $2$-representation $\mathbf{M}$ of ${\sc\mbox{C}\hspace{1.0pt}}$, we set $\mathtt{M}_{\mathrm{tot}}:=\Lparen \mathrm{F}_{\mathrm{tot}}\Rparen$. Then $\mathbf{M}$ is transitive if and only if all coefficients of $\mathtt{M}_{\mathrm{tot}}$ are non-zero (and thus positive integers).
Soergel bimodules {#s3}
=================
Soergel bimodules for finite Coxeter groups {#s3.1}
-------------------------------------------
Let $(W,S)$ be a finite irreducible Coxeter system and $\varphi:W\to\mathrm{GL}(V)$ be the geometric representation of $W$ as in [@Hu Section 5.3]. Here $V$ is a real vector space. We denote by $\leq$ the Bruhat order and by $\mathbf{l}:W\to\mathbb{Z}_{\geq 0}$ the length function. For $w\in W$, we denote by $\underline{w}$ the corresponding element in the Kazhdan-Lusztig basis of $\mathbb{Z}[W]$, see [@KaLu; @So2; @EW]. Each $\underline{w}$ is a linear combination of elements in $W$ with non-negative integer coefficients.
Let $\mathtt{C}$ be the (complexified) coinvariant algebra associated to $(W,S,V)$. For $s\in S$, we denote by $\mathtt{C}^s$ the subalgebra of $s$-invariants in $\mathtt{C}$. A [*Soergel $\mathtt{C}\text{-}\mathtt{C}$–bimodule*]{} is a $\mathtt{C}\text{-}\mathtt{C}$–bimodule isomorphic to a bimodule from the additive closure of the monoidal category of $\mathtt{C}\text{-}\mathtt{C}$–bimodules generated by $\mathtt{C}\otimes_{\mathtt{C}^s}\mathtt{C}$, where $s$ runs through $S$, see [@So; @So2; @Li]. We also note that by the [*additive closure*]{} of some $X$ we mean the the full subcategory whose objects are isomorphic to finite direct sums of direct summands of $X$. Isomorphism classes of indecomposable Soergel bimodules are naturally indexed by $w\in W$, we denote by $B_w$ a fixed representative from such a class. There is an isomorphism between $\mathbb{Z}[W]$ and the Grothendieck ring of the monoidal category of Soergel bimodules for $(W,S,V)$. This isomorphism sends $\underline{w}$ to the class of $B_w$, see [@So2; @E; @EW].
Let $\mathcal{C}$ be a small category equivalent to the category $\mathtt{C}\text{-}\mathrm{mod}$. Define the $2$-category ${\sc\mbox{S}\hspace{1.0pt}}={\sc\mbox{S}\hspace{1.0pt}}(W,S,V,\mathcal{C})$ of [*Soergel bimodules*]{} associated to $(W,S,V,\mathcal{C})$ as follows:
- ${\sc\mbox{S}\hspace{1.0pt}}$ has one object $\mathtt{i}$, which we can identify with $\mathcal{C}$;
- $1$-morphisms in ${\sc\mbox{S}\hspace{1.0pt}}$ are all endofunctors of $\mathcal{C}$ which are isomorphic to endofunctors given by tensoring with Soergel $\mathtt{C}\text{-}\mathtt{C}$–bimodules;
- $2$-morphisms in ${\sc\mbox{S}\hspace{1.0pt}}$ are natural transformations of functors (these correspond to homomorphisms of Soergel $\mathtt{C}\text{-}\mathtt{C}$–bimodules).
The $2$-category ${\sc\mbox{S}\hspace{1.0pt}}$ is fiat. The algebra $A_{{\scc\mbox{S}\hspace{1.0pt}}}$ is isomorphic to $\mathbb{C}[W]$ and is positively based with respect to the Kazhdan-Lusztig basis in $\mathbb{C}[W]$. For $w\in W$, let $\theta_w$ denote a fixed representative in the isomorphism class of indecomposable $1$-morphisms in ${\sc\mbox{S}\hspace{1.0pt}}$ given by tensoring with $B_w$.
Small quotients of Soergel bimodules {#s3.2}
------------------------------------
Let $(W,S)$, $V$, $\mathcal{C}$ and ${\sc\mbox{S}\hspace{1.0pt}}$ be as above.
\[lem1\] For any $s,t\in S$, we have $\theta_s\sim_J\theta_t$.
As $(W,S)$ is irreducible, we may assume that $st\neq ts$. As $\theta_s\theta_t\cong \theta_{st}$, we have $\theta_s\leq_R\theta_{st}$. Further, as $st\neq ts$, we have $\theta_{st}\theta_s\cong \theta_{sts}\oplus \theta_s$. This implies that $\theta_s\geq_R\theta_{st}$ and hence $\theta_s\sim_R\theta_{st}$. Similarly one shows that $\theta_t\sim_L\theta_{st}$. The claim follows.
After Lemma \[lem1\], we may define the two-sided cell $\mathcal{J}$ of ${\sc\mbox{S}\hspace{1.0pt}}$ as the two-sided cell containing $\theta_s$, for all $s\in S$. As $\underline{s}$, where $s\in S$, generate $\mathbb{Z}[W]$, it follows that $\mathcal{J}$ is the unique minimal element, with respect to the two-sided order, in the set of all two-sided cells of ${\sc\mbox{S}\hspace{1.0pt}}$ different from the two-sided cell corresponding to $\theta_e$.
Let ${\sc\mbox{I}\hspace{1.0pt}}$ be the unique $2$-ideal of ${\sc\mbox{S}\hspace{1.0pt}}$ which is maximal with respect to the property that it does not contain any $\mathrm{id}_{\mathrm{F}}$, where $\mathrm{F}\in \mathcal{J}$. The $2$-category ${\sc\mbox{S}\hspace{1.0pt}}/{\sc\mbox{I}\hspace{1.0pt}}$ will be called the [*small quotient*]{} of ${\sc\mbox{S}\hspace{1.0pt}}$ and denoted by $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$. By construction, the $2$-category $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is fiat and $\mathcal{J}$-simple. It has two two-sided cells, namely, the two-sided cell corresponding to $\theta_e$ and the image of $\mathcal{J}$, which we identify with $\mathcal{J}$, abusing notation.
Generalities on simple transitive $2$-representations of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ {#s4}
==================================================================================================
Basic combinatorics of $\mathcal{J}$ {#s4.1}
------------------------------------
Here we describe the basics of the Kazhdan-Lusztig combinatorics related to the two-sided cell $\mathcal{J}$. Recall Lusztig’s $\mathbf{a}$-function $\mathbf{a}:W\to \mathbb{Z}_{\geq 0}$, which is defined in [@Lu1]. This function has, in particular, the following properties: it is constant on two-sided cells in $W$, and, moreover, we have the equality $\mathbf{a}(w)=\mathbf{l}(w)$ provided that $w$ is the longest element in some parabolic subgroup of $W$.
\[propJKL\]
1. \[propJKL.1\] The map $\mathcal{L}\mapsto \mathcal{L}\cap S$ is a bijection between the set of left cells in $\mathcal{J}$ and $S$.
2. \[propJKL.2\] For any left cell $\mathcal{L}$ in $\mathcal{J}$, the unique element in $\mathcal{L}\cap S$ is the Duflo involution in $\mathcal{L}$.
3. \[propJKL.3\] The map $\mathcal{R}\mapsto \mathcal{R}\cap S$ is a bijection between the set of right cells in $\mathcal{J}$ and $S$.
4. \[propJKL.4\] For any right cell $\mathcal{R}$ in $\mathcal{J}$, the unique element in $\mathcal{R}\cap S$ is the Duflo involution in $\mathcal{R}$.
5. \[propJKL.5\] An element $w\in W$ such that $w\neq e$ belongs to $\mathcal{J}$ if and only if $w$ has a unique reduced expression.
6. \[propJKL.6\] An element $w\in W$ belongs to $\mathcal{J}$ if and only if $\mathbf{a}(w)=1$.
If $W$ is a Weyl group, all these results are mentioned in [@Do] with references to [@Lu1; @Lu2; @Lu3]. We note the difference in both the normalizations of the Hecke algebra and the choice of the Kazhdan-Lusztig basis in this paper and in [@Do]. Below, we outline a general argument.
We start by observing that different simple reflections have different left (right) descent sets and hence are not right (left) equivalent.
Further, if $w\in W$ has more than one reduced expression, then any such reduced expression can be obtained from any other by means of the braid relations, see [@Bo §IV.1.5]. This means that, with respect to the Bruhat order, $w$ is bigger than or equal to the longest element in some parabolic subgroup of $W$ of rank two. As longest element in parabolic subgroup of $W$ of rank two are not in $\mathcal{J}$ (since the value of $\mathbf{a}$ on such elements is strictly bigger than $1=\mathbf{a}(s)$, where $s\in S$), we get $w\not\in\mathcal{J}$.
By induction with respect to the rank of $W$, one checks that any two elements of the form $xs$ and $ys$ (resp. $sx$ and $sy$) which, moreover, have unique reduced expressions, belong to the same left (resp. right) cell. Indeed, the basis of the induction is the rank two case which is standard. The induction step follows from the relation $\theta_s\theta_t\theta_s=\theta_{sts}\oplus \theta_s$ which is true for any pair $s$, $t$ of non-commuting simple reflections. This, combined with the observation in the previous paragraph, yields , , and .
That simple reflections are Duflo involutions in their cells follows directly from the definitions since the value of $\mathbf{a}$ on $\mathcal{J}$ is $1$, implying , . This completes the proof.
An important consequence of Proposition \[propJKL\] is the following: if $s$ and $t$ are different elements in $S$, then the longest element in the parabolic subgroup of $W$ generated by $s$ and $t$ has two different reduced expressions and thus does not belong to $\mathcal{J}$. We also refer the reader to [@De; @BW] for related results.
Examples and special cases {#s4.2}
--------------------------
Using Proposition \[propJKL\], one can describe elements in $\mathcal{J}$ explicitly as products of simple reflections. Here we list several examples and special cases under the following conventions:
- we provide a picture of the Coxeter diagram followed by the list of elements in $\mathcal{J}$ organized in a square table, each element is given by its unique reduced expression;
- vertices of the Coxeter diagram are numbered by positive integers and we write $i$ for the corresponding simple reflection $s_i$;
- the left cell $\mathcal{L}_i$ containing $i$ is the $i$-th column of the diagram;
- the right cell $\mathcal{R}_i$ containing $i$ is the $i$-th row of the diagram;
To verify these examples one could use the following general approach:
- check that all elements given in the example have a unique reduced expression;
- check that, making any step outside the list in the example, gives an element with more than one reduced expression;
- note that all elements in a left cell start with the same simple reflection on the right, and all elements in a right cell start with the same simple reflection on the left;
- use Proposition \[propJKL\].
We start with types $A_3$ and $D_4$. $$\xymatrix{1\ar@{-}[r]&2\ar@{-}[r]&3}\qquad \qquad
\begin{array}{c||c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3\\
\hline\hline
\mathcal{R}_1&1&12&123\\
\hline
\mathcal{R}_2&21&2&23\\
\hline
\mathcal{R}_3&321&32&3
\end{array}$$ $$\xymatrix{&4&\\1\ar@{-}[r]&2\ar@{-}[r]\ar@{-}[u]&3}\qquad \qquad
\begin{array}{c||c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\mathcal{L}_4\\
\hline\hline
\mathcal{R}_1&1&12&123&124\\
\hline
\mathcal{R}_2&21&2&23&24\\
\hline
\mathcal{R}_3&321&32&3&324\\
\hline
\mathcal{R}_4&421&42&423&4
\end{array}$$
Similarly to the above, we have the following observation which follows directly from Proposition \[propJKL\].
\[corsimplylaced\] If the Coxeter diagram of $W$ is simply laced, then the cell $\mathcal{J}$ is strongly regular. Namely, the intersection $\mathcal{L}_i\cap \mathcal{R}_j$ consists of $s_js_{i_k}\dots s_{i_2}s_{i_1}s_i$, where $j-i_k-\dots- i_1-i$ is the unique path in the diagram connecting $i$ and $j$.
Because of Corollary \[corsimplylaced\], in the simply laced case, any simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation by [@MM5 Theorem 18]. We note that the original formulation of [@MM5 Theorem 18] has an additional assumption, the so-called [*numerical condition*]{} which appeared in [@MM1 Formula (10)]. This additional assumption is rendered superfluous by [@MM6 Proposition 1].
Consequently,
**in what follows, we assume that $W$ is not simply laced.**
In type $G_2$, we have the following. $$\xymatrix{1\ar@{-}[r]^{6}&2}\qquad \qquad
\begin{array}{c||c|c}
&\mathcal{L}_1&\mathcal{L}_2\\
\hline\hline
\mathcal{R}_1&1,121,12121&12,1212\\
\hline
\mathcal{R}_2&21,2121&2,212,21212
\end{array}$$
The general type $B_n$, for $n\geq 2$, corresponds to the Coxeter diagram $$\xymatrix{1\ar@{-}[r]^{4}&2\ar@{-}[r]&3\ar@{-}[r]&\dots\ar@{-}[r]&n}$$ and is given below: $$\begin{array}{c||c|c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\dots&\mathcal{L}_n\\
\hline\hline
\mathcal{R}_1&1,121&12&123&\dots&12\cdots n \\
\hline
\mathcal{R}_2&21&2,212&23, 2123&\dots&23\cdots n,212\cdots n \\
\hline
\mathcal{R}_3&321&32,3212&3,32123&\dots&34\cdots n,3212\cdots n \\
\hline
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots \\
\hline
\mathcal{R}_n&n\cdots 21&n\cdots 32,n\cdots212&n\cdots43,n\cdots 2123&\dots&n,n\cdots212\cdots n
\end{array}$$
The remaining Weyl type $F_4$ looks as follows: $$\xymatrix{1\ar@{-}[r]&2\ar@{-}[r]^{4}&3\ar@{-}[r]&4}\qquad \qquad
\begin{array}{c||c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\mathcal{L}_4\\
\hline\hline
\mathcal{R}_1&1,12321&12,1232&123&1234\\
\hline
\mathcal{R}_2&21,2321&2,232&23&234\\
\hline
\mathcal{R}_3&321&32&3,323&34,3234\\
\hline
\mathcal{R}_4&4321&432&43,4323&4,43234\\
\end{array}$$
The exceptional Coxeter type $H_3$ is the following: $$\xymatrix{1\ar@{-}[r]^{5}&2\ar@{-}[r]&3}\qquad \qquad
\begin{array}{c||c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3\\
\hline\hline
\mathcal{R}_1&1,121&12,1212&123,12123\\
\hline
\mathcal{R}_2&21,2121&2,212&23,2123\\
\hline
\mathcal{R}_3&321,32121&32,3212&3,32123
\end{array}$$
The exceptional Coxeter type $H_4$ has the Coxeter diagram $$\xymatrix{1\ar@{-}[r]^{5}&2\ar@{-}[r]&3\ar@{-}[r]&4}$$ and the corresponding table looks as follows: $$\begin{array}{c||c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\mathcal{L}_4\\
\hline\hline
\mathcal{R}_1&1,121&12,1212&123,12123&1234,121234\\
\hline
\mathcal{R}_2&21,2121&2,212&23,2123&234,21234\\
\hline
\mathcal{R}_3&321,32121&32,3212&3,32123&34,321234\\
\hline
\mathcal{R}_4&4321,432121&432,43212&43,432123&4,4321234
\end{array}$$
The general dihedral type with Coxeter diagram $$\xymatrix{1\ar@{-}[r]^{k}&2},$$ where $k\geq 3$, looks as follows: $$\begin{array}{c||c|c}
&\mathcal{L}_1&\mathcal{L}_2\\
\hline\hline
\mathcal{R}_1&1,121,\dots,12\cdots 21&12,1212,12\cdots 12\\
\hline
\mathcal{R}_2&21,2121,\dots, 21\cdots 21&2,212,\dots,21\cdots 12
\end{array}$$ Note that the length of all elements in the latter table is strictly less than $k$. In particular, the diagonal boxes always contain $\lfloor\frac{k}{2}\rfloor$ elements while the off-diagonal boxes contain $\lfloor\frac{k}{2}\rfloor$ elements if $k$ is odd and $\lfloor\frac{k}{2}\rfloor-1$ elements if $k$ is even.
The principal element in $\mathbb{Z}[W]$ {#s4.4}
----------------------------------------
Following [@Zi], we consider the element $$\mathbf{s}:=\sum_{s\in S} \underline{s}\in \mathbb{Z}[W],$$ which we call the [*principal*]{} element. This element is the decategorification of $$\mathrm{F}_{\mathrm{pr}}:=\bigoplus_{s\in S} \theta_s.$$ The main aim of this subsection is to determine the form of the matrix $$\mathtt{M}=\mathtt{M}_{\mathrm{pr}}:=\Lparen \mathrm{F}_{\mathrm{pr}}\Rparen.$$ We start by recalling the following.
\[lem2\] Let $\mathbf{M}$ be a transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ and $s\in S$. Then there is an ordering of indecomposable objects in $\mathbf{M}(\mathtt{i})$ such that $$\Lparen \theta_s\Rparen=
\left(\begin{array}{c|c}2E&*\\\hline 0&0\end{array}\right),$$ where $E$ is the identity matrix.
Mutatis mutandis the proof of [@Zi Lemma 6.4].
\[lem3\] Let $\mathbf{M}$ be a transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$ and $P$ an indecomposable object in $\mathbf{M}(\mathtt{i})$. Then there exists a unique $s\in S$ having the property $\theta_s\, P\cong P\oplus P$.
Assume that $\theta_s\, P\not \cong P\oplus P$ for every $s\in S$. From Lemma \[lem2\] it then follows that $\mathtt{M}$ has a zero row. Therefore the same row will be zero in any power of $\mathtt{M}$. Note that any indecomposable $1$-morphism of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ which is not isomorphic to $\theta_e$ appears as a direct summand of some power of $\mathrm{F}_{\mathrm{pr}}$. If $\mathbf{M}$ had rank one, it would follow that all indecomposable $1$-morphisms of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ which are not isomorphic to $\theta_e$ annihilate $\mathbf{M}$. This contradicts our assumption that $\mathcal{J}$ is the apex of $\mathbf{M}$. Therefore $\mathbf{M}$ has rank at least two. However, in this case, the same row which is zero in all powers of $\mathtt{M}$ must have a zero entry in $\mathtt{M}_{\mathrm{tot}}$. This contradicts transitivity of $\mathbf{M}$ and hence establishes existence of $s\in S$ such that $\theta_s\, P\cong P\oplus P$.
Assume that $s$ and $t$ are two different elements in $S$ having the property that $\theta_s\, P\cong P\oplus P$ and $\theta_t\, P\cong P\oplus P$. Consider the parabolic subgroup $W'$ of $W$ generated by $s$ and $t$. The additive closure of $P$ is invariant under the action of the $2$-subcategory of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ generated by $\theta_s$ and $\theta_t$. The decategorification of the latter $2$-representation gives a $1$-dimensional $\mathbb{C}[W']$-module on which both $\underline{s}$ and $\underline{t}$ act as the scalar $2$. Thus this is the trivial module and it is not annihilated by the element $\underline{w'_0}$, where $w'_0$ is the longest element in $W'$. This contradicts $w'_0\not\in \mathcal{J}$, see Proposition \[propJKL\] and the remark after Proposition \[propJKL\]. The assertion of the lemma follows.
Write $S=\{s_1,s_2,\dots,s_n\}$. Let $\mathbf{M}$ be a transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Choose an ordering $$\label{eqo1}
P_1,P_2,\dots,P_m$$ on representatives of isomorphism classes of indecomposable projectives in $\mathbf{M}(\mathtt{i})$ such that, for all $i,j\in\{1,2,\dots,n\}$ such that $i<j$ and for all $a,b\in\{1,2,\dots,m\}$, the isomorphisms $\theta_{s_i}\, P_a\cong P_a\oplus P_a$ and $\theta_{s_j}\, P_b\cong P_b\oplus P_b$ imply $a<b$.
\[prop4\] Let $\mathbf{M}$ be a transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Then, with respect to the above ordering, we have $$\label{eqpr4-1}
\mathtt{M}=\left(
\begin{array}{c|c|c|c|c}
2E_1& B_{12} &B_{13}&\dots & B_{1n}\\\hline
B_{21}& 2E_2 &B_{23}&\dots & B_{2n}\\\hline
B_{31}& B_{32} &2E_3 &\dots & B_{3n}\\\hline
\vdots&\vdots&\vdots&\ddots&\vdots\\\hline
B_{n1}& B_{n2} & B_{n3} &\dots & 2E_n
\end{array}
\right),$$ where $E_i$, for $1\leq i\leq n$, are non-trivial identity matrices and, for $1\leq i\neq j\leq n$, the matrix $B_{ij}$ is non-zero if and only if $s_is_j\neq s_js_i$.
After Lemmata \[lem2\] and \[lem3\], it remains to show that $B_{ij}$ is non-zero if and only if $s_is_j\neq s_js_i$. If $s_is_j= s_js_i$, then $B_{ij}$ is zero as $s_is_j\not\in\mathcal{J}$ and $\mathcal{J}$ is the apex of $\mathbf{M}$.
Assume that $s_is_j\neq s_js_i$ and $B_{ij}=0$. Let as assume that $i>j$, the other case is similar. Taking into account the previous paragraph and the fact that the underlying graph of the Coxeter diagram of $W$ is a tree, we may rearrange the order of the elements in $S$ such that the matrix $\mathtt{M}$ has the following block form: $$\left(
\begin{array}{c|c}
*&*\\\hline
0&*
\end{array}
\right).$$ As any power of such a matrix will contain a non-empty zero south-west corner, we get a contradiction with the assumption that $\mathbf{M}$ is transitive. The claim follows.
Let $\{s_1,s_2,\dots,s_n\}$ be an ordering of simple reflections in $S$ such that the block $2E_i$ in corresponds to $s_i$, for all $i=1,2,\dots,n$, then we have [$$\Lparen\theta_{s_1}\Rparen=\left(
\begin{array}{c|c|c|c|c}
2E_1& B_{12} &B_{13}&\dots & B_{1n}\\\hline
0& 0 &0&\dots & 0\\\hline
0& 0 &0 &\dots & 0\\\hline
\vdots&\vdots&\vdots&\ddots&\vdots\\\hline
0& 0 & 0 &\dots & 0
\end{array}
\right),\quad
\Lparen\theta_{s_2}\Rparen=\left(
\begin{array}{c|c|c|c|c}
0& 0 &0&\dots & 0\\\hline
B_{21}& 2E_2 &B_{23}&\dots & B_{2n}\\\hline
0& 0 &0 &\dots & 0\\\hline
\vdots&\vdots&\vdots&\ddots&\vdots\\\hline
0& 0 & 0 &\dots & 0
\end{array}
\right),$$ ]{} and so on.
The special $\mathbb{C}[W]$-module for $\mathcal{J}$ {#s4.3}
----------------------------------------------------
Here we describe the special $\mathbb{C}[W]$-module, in the sense of [@KM2], see also Subsection \[s2.7\], associated to $\mathcal{J}$.
\[propspecialJ\] The special $\mathbb{C}[W]$-module associated to $\mathcal{J}$ is $V\otimes\mathbb{C}_{\mathrm{sign}}$, where $\mathbb{C}_{\mathrm{sign}}$ is the sign $\mathbb{C}[W]$-module.
If $W$ is a Weyl group, this can be derived from [@Do], if one takes into account the difference in the normalizations of the Hecke algebra and the choice of the Kazhdan-Lusztig basis.
Let $s$ and $t$ be two different simple reflections in $W$ and $W'$ the corresponding parabolic subgroup of rank two, with the longest element $w'_0$. The restriction of $V\otimes\mathbb{C}_{\mathrm{sign}}$ to $W'$ decomposes, by construction, into a direct sum of $V'\otimes\mathbb{C}'_{\mathrm{sign}}$, where $V'$ and $\mathbb{C}'_{\mathrm{sign}}$ are the geometric and the sign representations of $W'$, respectively, and a number of copies of $\mathbb{C}'_{\mathrm{sign}}$. As both $V'\otimes\mathbb{C}'_{\mathrm{sign}}$ and $\mathbb{C}'_{\mathrm{sign}}$ are annihilated by $\underline{w'_0}$, it follows that $\underline{w'_0}$ annihilates $V\otimes\mathbb{C}_{\mathrm{sign}}$.
Let $I$ be the ideal in $\mathbb{C}[W]$ with the $\mathbb{C}$-basis $\underline{w}$, where $w\not\in\mathcal{J}\cup\{e\}$. From Proposition \[propJKL\], it follows that $I$ is generated by all $\underline{w'_0}$ as in the previous paragraph. Therefore the previous paragraph implies that the vector space $V\otimes\mathbb{C}_{\mathrm{sign}}$ is, in fact, a non-zero $\mathbb{C}[W]/I$-module.
If $W$ is simply laced, then the cell $\mathbb{C}[W]$-module corresponding to any cell in $\mathcal{J}$ is simple. Hence in this case the claim of the proposition follows directly from the previous paragraph.
Now note that the dimension of $V\otimes\mathbb{C}_{\mathrm{sign}}$ equals the number of left cells in $\mathcal{J}$. From detailed lists of elements in $\mathcal{J}$ provided in Subsection \[s4.2\] we therefore have that $V\otimes\mathbb{C}_{\mathrm{sign}}$ is the only simple $\mathbb{C}[W]/I$-module of this dimension in types $B_n$ and $F_4$. This implies the claim of the proposition for these types.
For the general dihedral type, one may note that $V\otimes\mathbb{C}_{\mathrm{sign}}\cong V$ and hence the claim of the proposition follows from [@KM2].
It remains to consider the two exceptional types $H_3$ and $H_4$. We do explicit computations in both types. Note that, from the lists in Subsection \[s4.2\] we see that there are two non-isomorphic simple $\mathbb{C}[W]/I$-modules of the same dimension.
Let us start with type $H_3$. The action of $\mathbf{s}$ (cf. Subsection \[s4.4\]) on $C_{\mathcal{L}_1}$ in the basis $$1,121,21,2121,321,32121,$$ taken from the corresponding table in Subsection \[s4.2\], is given by the matrix $$\left(
\begin{array}{cccccc}
2&0&1&0&0&0\\
0&2&1&1&0&0\\
1&1&2&0&1&0\\
0&1&0&2&0&1\\
0&0&1&0&2&0\\
0&0&0&1&0&2
\end{array}
\right).$$ The eigenvalue of maximal absolute value for this matrix is $2+\frac{\sqrt{2\sqrt{5}+10}}{2}$. At the same time, the action of $\mathbf{s}$ on $V\otimes\mathbb{C}_{\mathrm{sign}}$ in the basis corresponding to $1$, $2$ and $3$, is given by the matrix $$\left(
\begin{array}{ccc}
2&-\frac{1+\sqrt{5}}{2}&0\\
-\frac{1+\sqrt{5}}{2}&2&-1\\
0&-1&2
\end{array}
\right).$$ As $2+\frac{\sqrt{2\sqrt{5}+10}}{2}$ is also an eigenvalue for this latter matrix, we obtain our statement in type $H_3$.
Type $H_4$ is treated similarly by noticing that the matrices $$\left(
\begin{array}{cccccccc}
2&0&1&0&0&0&0&0\\
0&2&1&1&0&0&0&0\\
1&1&2&0&1&0&0&0\\
0&1&0&2&0&1&0&0\\
0&0&1&0&2&0&1&0\\
0&0&0&1&0&2&0&1\\
0&0&0&0&1&0&2&0\\
0&0&0&0&0&1&0&2
\end{array}
\right) \qquad\text{ and }\qquad
\left(
\begin{array}{cccc}
2&-\frac{1+\sqrt{5}}{2}&0&0\\
-\frac{1+\sqrt{5}}{2}&2&-1&0\\
0&-1&2&-1\\
0&0&-1&2\\
\end{array}
\right)$$ which represent the action of $\mathbf{s}$ on $C_{\mathcal{L}_1}$ and $V\otimes\mathbb{C}_{\mathrm{sign}}$, respectively, have a common eigenvalue which is of maximal absolute value for both of them (this eigenvalue is approximately $3.98904$). This completes the proof.
\[remeigen\] [ In type $H_3$, we used the matrix calculator that can be found at www.mathportal.org. As the outcome is given as a real number, it is easily checked by hand. In type $H_4$ we used the matrix calculator available at www.arndt-bruenner.de and cross-checked the result on the matrix calculator available at www.bluebit.gr. ]{}
$1$-morphisms act as projective functors {#s5.5}
----------------------------------------
Our aim in this subsection is to prove the following very general result which extends and unifies [@MM5 Lemma 12], [@Zi Lemma 6.14] and [@MaMa Lemma 10].
\[thmminimal\] Let $\mathbf{M}$ be a simple transitive $2$-representation of a fiat $2$-category ${\sc\mbox{C}\hspace{1.0pt}}$. Let, further, $\mathcal{I}$ be the apex of $\mathbf{M}$ and $\mathrm{F}\in \mathcal{I}$.
1. \[thmminimal.1\] For every object $X$ in any $\overline{\mathbf{M}}(\mathtt{i})$, the object $\mathrm{F}\,X$ is projective.
2. \[thmminimal.2\] The functor $\overline{\mathbf{M}}(\mathrm{F})$ is a projective functor.
Without loss of generality we may assume that $\mathcal{I}$ is the maximum two-sided cell of ${\sc\mbox{C}\hspace{1.0pt}}$.
Denote by $Q$ the complexification of the split Grothendieck group of $$\coprod_{\mathtt{i}\in{\scc\mbox{C}\hspace{1.0pt}}}{\mathbf{M}}(\mathtt{i}).$$ Let $\mathbf{B}$ denote the distinguished basis in $Q$ given by classes of indecomposable modules. Let $\mathrm{F}_1,\mathrm{F}_2,\dots,\mathrm{F}_k$ be a complete and irredundant list of indecomposable $1$-morphisms in $\mathcal{I}$. For $i=1,2,\dots,k$, let $\mathcal{X}^{(i)}_{\bullet}$ be a minimal projective resolution of $\mathrm{F}_i\, X$. Further, for $j\geq 0$, denote by $v_j^{(i)}$ the image of $\mathcal{X}^{(i)}_{j}$ in $Q$. Note that $v_j^{(i)}$ is a linear combination of elements in $\mathbf{B}$ with non-negative integer coefficients.
As ${\sc\mbox{C}\hspace{1.0pt}}$ is assumed to be fiat, there exist $i,j\in\{1,2,\dots,k\}$ such that $\mathrm{F}_i\circ \mathrm{F}_j\neq 0$. Indeed, one can, for example, take any $i$ and let $\mathrm{F}_j$ be the Duflo involution in the left cell of $\mathrm{F}_i$, see [@MM1 Proposition 17]. Therefore we may apply [@KM2 Proposition 18], which asserts that the algebra $A_{{\scc\mbox{C}\hspace{1.0pt}}}$ has a unique idempotent $e$ of the form $$\sum_{i=1}^kc_i[\mathrm{F}_i],$$ where all $c_i\in\mathbb{R}_{>0}$ and $[\mathrm{F}_i]$ denotes the image of $\mathrm{F}_i$ in $A_{{\scc\mbox{C}\hspace{1.0pt}}}$. The vector space $Q$ is, naturally, an $A_{{\scc\mbox{C}\hspace{1.0pt}}}$-module.
Let $j\geq 0$ and set $$v(j):=\sum_{i=1}^kc_iv_j^{(i)}.$$ Applying any $1$-morphism $\mathrm{G}$ to a minimal projective resolution of some object $Y$ and taking into account that the action of $\mathrm{G}$ is exact as ${\sc\mbox{C}\hspace{1.0pt}}$ is fiat, gives a projective resolution of $\mathrm{G}\,Y$ which, a priori, does not have to be minimal. By construction and minimality of $\mathcal{X}^{(i)}_{\bullet}$, this implies that $ev(j)-v(j)$ is a linear combination of elements in $\mathbf{B}$ with non-negative real coefficients. Note that transitivity of $\mathbf{M}$ implies that, for any linear combination $z$ of elements in $\mathbf{B}$ with non-negative real coefficients, the element $ez$ is also a linear combination of elements in $\mathbf{B}$ with non-negative real coefficients, moreover, if $z\neq 0$, then $ez\neq 0$. Therefore the equality $e^2=e$ yields $ev(j)=v(j)$. Consequently, the projective resolution $\mathrm{F}_i\mathcal{X}^{(l)}_{\bullet}$ of $\mathrm{F}_i(\mathrm{F}_l\, X)$ is minimal, for all $i$ and $l$.
Now the proof is completed by standard arguments as in [@MM5 Lemma 12]. From the previous paragraph it follows that homomorphisms $\mathcal{X}^{(l)}_{1}\to \mathcal{X}^{(l)}_{0}$, for $l=1,2,\dots,k$, generate a ${\sc\mbox{C}\hspace{1.0pt}}$-invariant ideal of $\mathbf{M}$ different from $\mathbf{M}$. Because of simple transitivity of $\mathbf{M}$, this ideal must be zero. Therefore all homomorphisms $\mathcal{X}^{(l)}_{1}\to \mathcal{X}^{(l)}_{0}$ are zero which implies $\mathcal{X}^{(l)}_{0}\cong \mathrm{F}_l\, X$. Claim follows. Claim follows from claim and [@MM5 Lemma 13].
The matrix $\mathtt{M}$ is symmetric {#s4.37}
------------------------------------
Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Consider $\overline{\mathbf{M}}$ and let $P_1$, $P_2$,…, $P_k$ be a full list of pairwise non-isomorphic indecomposable projectives in $\overline{\mathbf{M}}(\mathtt{i})$. Let $L_1$, $L_2$,…, $L_k$ be their respective simple tops.
\[propn31\] For every $i\in\{1,2,\dots,k\}$, we have $\mathrm{F}_{\mathrm{pr}}\, L_i\cong P_i$.
We know from Theorem \[thmminimal\] that $\mathrm{F}_{\mathrm{pr}}\, L_i$ is projective. From the matrix $\mathtt{M}$ described in Subsection \[s4.4\], we see that there is a unique $s\in S$ such that $\theta_s\, L_i\neq 0$. Therefore $\mathrm{F}_{\mathrm{pr}}\, L_i\cong \theta_s\, L_i$. From the matrix $\llbracket \theta_s\rrbracket$, see Subsections \[s2.6\] and \[s4.4\] in combination with the remark that each $\theta_s$ is self-adjoint, we see that $\theta_s\,L_i$ has two subquotients isomorphic to $L_i$ and all other subquotients are killed by $\theta_s$. Note that $\theta_s\,L_i$ has length at least three, as each row of $\mathtt{M}$ must contain at least one non-zero block $B_{ij}$ by Proposition \[prop4\]. This means that $\theta_s\,L_i$ does contain some subquotients which are not isomorphic to $L_i$. If $L_j$ is such a subquotient, then, by adjunction, $$\mathrm{Hom}_{\overline{\mathbf{M}}(\mathtt{i})}(\theta_s\,L_i,L_j)=
\mathrm{Hom}_{\overline{\mathbf{M}}(\mathtt{i})}(L_i,\theta_s\,L_j)
=\mathrm{Hom}_{\overline{\mathbf{M}}(\mathtt{i})}(L_i,0)=0$$ and $L_j$ does not appear in the top of $\theta_s\,L_i$. Similarly, $L_j$ does not appear in the socle of $\theta_s\,L_i$ either. It follows that $\theta_s\,L_i$ can only have $L_i$ in top and socle. This means that one of the $L_i$’s is in the top of $\theta_s\,L_i$ and the other one is in the socle. In particular, $\theta_s\,L_i$ is indecomposable. Therefore $\theta_s\,L_i\cong P_i$ by the projectivity of $\theta_s\,L_i$. This completes the proof.
\[corn32\] Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Then the matrix $\mathtt{M}$ is symmetric.
By Proposition \[propn31\], the row $i$ of $\mathtt{M}$ describes composition multiplicities of simple modules in $P_i$. By definition, the column $i$ of $\mathtt{M}$ describes the direct sum decomposition of $\mathrm{F}_{\mathrm{pr}}\,P_i$. Note that $\mathrm{F}_{\mathrm{pr}}$ is exact as ${\sc\mbox{S}\hspace{1.0pt}}$ is fiat. Therefore, from Proposition \[propn31\] we have that the multiplicity of $P_j$ as a direct summand in $\mathrm{F}_{\mathrm{pr}}\,P_i$ coincides with the composition multiplicity of $L_j$ in $P_i$. The claim follows.
Some spectral properties of integral matrices {#sim}
=============================================
Setup {#sim.1}
-----
Denote by $\mathbb{M}$ the set of all matrices with non-negative integer coefficients, that is $$\mathbb{M}:=\bigcup_{k,m\in\mathbb{Z}_{>0}}\mathrm{Mat}_{k\times m}(\mathbb{Z}_{\geq 0}).$$ For each $X\in \mathbb{M}$, we can consider the symmetric matrices $XX^{\mathrm{tr}}$ and $X^{\mathrm{tr}}X$. Each of these two matrices is diagonalizable over $\mathbb{R}$, moreover, all eigenvalues are non-negative. Furthermore, these two matrices have the same spectrum (considered as a multiset) with only possible exception of the multiplicity of the eigenvalue zero. We denote by $\mathbf{m}_X$ the eigenvalue of $XX^{\mathrm{tr}}$ or, equivalently, $X^{\mathrm{tr}}X$ with the maximal absolute value. Note that $\mathbf{m}_X=0$ if and only if $X$ is the zero matrix. The aim of this section is to describe all $X\in \mathbb{M}$ such that $\mathbf{m}_X<4$.
Staircase matrices {#sim.2}
------------------
A [*staircase matrix*]{} is a matrix of the form $$\label{simeq1}
\left(
\begin{array}{cccccccc}
1&1&0&0&0&\dots&0&0 \\
0&1&1&0&0&\dots&0&0 \\
0&0&1&1&0&\dots&0&0 \\
0&0&0&1&1&\dots&0&0 \\
0&0&0&0&1&\dots&0&0 \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots \\
0&0&0&0&0&\dots&1&1\\
0&0&0&0&0&\dots&0&1
\end{array}
\right) \quad
\left(
\begin{array}{ccccccccc}
1&1&0&0&0&\dots&0&0&0 \\
0&1&1&0&0&\dots&0&0&0 \\
0&0&1&1&0&\dots&0&0&0 \\
0&0&0&1&1&\dots&0&0&0 \\
0&0&0&0&1&\dots&0&0&0 \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\
0&0&0&0&0&\dots&1&1&0\\
0&0&0&0&0&\dots&0&1&1
\end{array}
\right)$$ or its transpose. Note that each staircase matrix is of the size $k\times k$ or $k\times (k+1)$ or $(k+1)\times k$, for some positive integer $k$.
An [*extended staircase matrix*]{} is a matrix obtained from a staircase matrix by adding one additional row (or column, but not both) as shown below. In the pictures, the line separates the original staircase matrix from the added row (respectively, column). The picture below shows extended staircase matrices for the left matrix in . $$\left(
\begin{array}{c|cccccccc}
1&1&1&0&0&0&\dots&0&0 \\
0&0&1&1&0&0&\dots&0&0 \\
0&0&0&1&1&0&\dots&0&0 \\
0&0&0&0&1&1&\dots&0&0 \\
0&0&0&0&0&1&\dots&0&0 \\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots \\
0&0&0&0&0&0&\dots&1&1\\
0&0&0&0&0&0&\dots&0&1
\end{array}
\right) \quad
\left(
\begin{array}{cccccccc}
1&1&0&0&0&\dots&0&0 \\
0&1&1&0&0&\dots&0&0 \\
0&0&1&1&0&\dots&0&0 \\
0&0&0&1&1&\dots&0&0 \\
0&0&0&0&1&\dots&0&0 \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots \\
0&0&0&0&0&\dots&1&1\\
0&0&0&0&0&\dots&0&1 \\ \hline
0&0&0&0&0&\dots&0&1
\end{array}
\right)$$ The next picture shows extended staircase matrices for the right matrix in . $$\left(
\begin{array}{c|ccccccccc}
1&1&1&0&0&0&\dots&0&0&0 \\
0&0&1&1&0&0&\dots&0&0&0 \\
0&0&0&1&1&0&\dots&0&0&0 \\
0&0&0&0&1&1&\dots&0&0&0 \\
0&0&0&0&0&1&\dots&0&0&0 \\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\
0&0&0&0&0&0&\dots&1&1&0\\
0&0&0&0&0&0&\dots&0&1&1
\end{array}
\right)\quad
\left(
\begin{array}{ccccccccc|c}
1&1&0&0&0&\dots&0&0&0&0 \\
0&1&1&0&0&\dots&0&0&0&0 \\
0&0&1&1&0&\dots&0&0&0&0 \\
0&0&0&1&1&\dots&0&0&0&0 \\
0&0&0&0&1&\dots&0&0&0&0 \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots \\
0&0&0&0&0&\dots&1&1&0&0 \\
0&0&0&0&0&\dots&0&1&1&1
\end{array}
\right)$$ For the transposes of , the above pictures should also be transposed. Note that each extended staircase matrix is of the size $k\times (k+1)$ or $k\times (k+2)$ or $(k+1)\times k$ or $(k+2)\times k$, for some positive integer $k$.
The main result {#sim.3}
---------------
\[propsim\] Let $X\in\mathbb{M}$ be such that both $XX^{\mathrm{tr}}$ and $X^{\mathrm{tr}}X$ are irreducible and $\mathbf{m}_X<4$. Then, using independent permutations of rows and columns, $X$ can be reduced to a staircase matrix or an extended staircase matrix or one of the following matrices (or their transposes): $$X_1:=\left(
\begin{array}{ccc}
1&0&0\\
1&1&1\\
0&0&1
\end{array}
\right),\quad
X_2:=\left(
\begin{array}{cccc}
1&1&0&0\\
0&1&1&1\\
0&0&0&1
\end{array}
\right),\quad
X_3:=\left(
\begin{array}{cccc}
1&0&0&0\\
1&1&0&0\\
0&1&1&1\\
0&0&0&1
\end{array}
\right).$$
The statement of Proposition \[propsim\] can be interpreted in the way that there is a natural correspondence between matrices $X$ as in the proposition and simply laced Dynkin diagrams. Staircase matrices correspond to type $A$, extended staircase matrices correspond to type $D$ and the three exceptional matrices correspond to type $E$. Starting from a matrix $X$ appearing in the classification provided by Proposition \[propsim\], we can consider the matrix $$\left(\begin{array}{cc}2E&X\\X^{\mathrm{tr}}&2E\end{array}\right)$$ and pretend that it appears as $\mathtt{M}$ in some $2$-representation. If we compute the underlying algebra of that $2$-representation as in many examples later on, see e.g. Subsection \[s6.9\], we will get a doubling of a simply laced Dynkin quiver.
Proof of Proposition \[propsim\] {#sim.4}
--------------------------------
Let $X\in\mathbb{M}$ be such that both $XX^{\mathrm{tr}}$ and $X^{\mathrm{tr}}X$ are irreducible and $\mathbf{m}_X<4$.
Assume that $X$ has an entry that is greater than or equal to $2$. Then $XX^{\mathrm{tr}}$ has a diagonal entry which is greater than or equal to $4$. Let $\lambda$ be the Perron-Frobenius eigenvalue of $XX^{\mathrm{tr}}$. Then, by the Perron-Frobenius Theorem, the limit $$\lim_{i\to\infty}\frac{(XX^{\mathrm{tr}})^i}{\lambda^i}$$ exists, which means that $\lambda\geq 4$, a contradiction. Therefore $X$ is a $0$-$1$-matrix.
For a matrix $M$, a submatrix $N$ of $M$ is the matrix obtained by taking entries in the intersection of a non-empty set of rows of $M$ and a non-empty set of columns of $M$. If $N$ is a submatrix of $M$, then, clearly, $\mathbf{m}_N\leq \mathbf{m}_M$. An argument similar to the one in the previous paragraph shows that $X$ cannot have any submatrix which is equal to either of the following matrices, nor their transposes, $$\label{simeq2}
\left(\begin{array}{cc}1&1\\1&1\end{array}\right),\qquad
\left(\begin{array}{cccc}1&1&1&1\end{array}\right).$$ Taking the first matrix in into account and using the fact that both $XX^{\mathrm{tr}}$ and $X^{\mathrm{tr}}X$ are irreducible, we see that, by independent permutation of row and columns, $X$ can be reduced to the form $$\left(\begin{array}{cccccccccc}
1&\dots&1&1&0&\dots&0&0&0&\dots\\
0&\dots&0&1&1&\dots&1&1&0&\dots\\
0&\dots&0&0&0&\dots&0&1&1&\dots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots
\end{array}
\right).$$ So, from now on we may assume that $X$ is of the latter form. Taking the second matrix in into account, we see that each row and each column of $X$ contains at most three non-zero entries.
Next we claim that the total number of rows and columns in $X$ which contain three non-zero entries is at most one. For this we have to exclude, up to transposition, two types of possible submatrices in $X$. The first one is the matrix $$\left(\begin{array}{cccccccccc}
1&0&0&0&\dots&\dots&0&0&0&0\\
1&0&0&0&\dots&\dots&0&0&0&0\\
1&1&0&0&\dots&\dots&0&0&0&0\\
0&1&1&0&\dots&\dots&0&0&0&0\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\
0&0&0&0&\dots&\dots&1&1&0&0\\
0&0&0&0&\dots&\dots&0&1&1&1
\end{array}
\right).$$ For this matrix it is easy to check that the vector $(2,2,\dots,2,1,1)^{\mathrm{tr}}$ is an eigenvector of $X^{\mathrm{tr}}X$ with eigenvalue $4$, leading to a contradiction. The second one is the matrix $$\left(\begin{array}{cccccccccc}
1&1&1&0&0&\dots&0&0&0&0\\
0&0&1&1&0&\dots&0&0&0&0\\
0&0&0&1&1&\dots&0&0&0&0\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\
0&0&0&0&0&\dots&1&1&0&0\\
0&0&0&0&0&\dots&0&1&1&1
\end{array}
\right).$$ For this matrix it is easy to check that the vector $(1,1,\dots,1)^{\mathrm{tr}}$ is an eigenvector of $XX^{\mathrm{tr}}$ with eigenvalue $4$, leading again to a contradiction.
If $X$ has neither rows nor columns with three non-zero entries, then $X$ is a staircase matrix. By the above, if $X$ has a row or a column with three non-zero entries, it is unique. Now, the claim of the proposition follows from the observation that, for the following matrices $X$: $$\left(\begin{array}{cccc}1&1&0&0\\0&1&0&0\\0&1&1&0\\0&0&1&1\\0&0&0&1\\\end{array}\right),\qquad
\left(\begin{array}{ccc}1&0&0\\1&1&0\\0&1&0\\0&1&1\\0&0&1\\\end{array}\right),$$ the matrix $X^{\mathrm{tr}}X$ has $4$ as an eigenvalue.
Simple transitive $2$-representations of ${\sc\mbox{S}\hspace{1.0pt}}$ in Coxeter type $I_2(n)$ with $n$ odd {#s17}
============================================================================================================
Setup and the main result {#s17.1}
-------------------------
In this section we assume that $W$ is of Coxeter type $I_2(n)$, for some $n\geq 3$, and $S=\{s,t\}$ with the Coxeter diagram $$\xymatrix{s\ar@{-}[r]^n&t.}$$ In this case there are three two-sided cells, namely
- the two-sided cell of the identity element;
- the two-sided cell of $w_0$;
- the two-sided cell $\mathcal{J}$.
Our aim in this section is to prove the following.
\[thm1700\] For $n\geq 3$ odd, every simple transitive $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ is a cell $2$-representation.
If $\mathbf{M}$ is a simple transitive $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ whose apex is not $\mathcal{J}$, then $\mathbf{M}$ is a cell $2$-representation by the same argument as in [@MM5 Theorem 18], see also [@MaMa; @Zi] for similar arguments. Taking [@MM2 Theorem 19] into account, from now on, we may assume that $\mathbf{M}$ is a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Our goal is to prove that $\mathbf{M}$ is a cell $2$-representation.
Fibonacci polynomials in disguise {#s17.2}
---------------------------------
For $i=0,1,2,\dots$, we define, recursively, polynomials $f_i(x)\in\mathbb{Z}[x]$ as follows: $f_0(x)=0$, $f_1(x)=1$, and, for $i>1$, $$f_i(x)=
\begin{cases}
f_{i-1}(x)-f_{i-2}(x),& i\text{ is odd};\\
xf_{i-1}(x)-f_{i-2}(x),& i\text{ is even}.
\end{cases}$$ Coefficients of $f_i$ are given by Sequence A115139 in [@OEIS]. The values of $f_i$ for small $i$ are given here: $$\begin{array}{c||l|l}
i&f_i(x)&\text{ factorization over }\mathbb{Q}\\
\hline\hline
0&0\\
1&1&1\\
2&x&x\\
3&x-1&x-1\\
4&x^2-2x&x(x-2)\\
5&x^2-3x+1&x^2-3x+1\\
6&x^3-4x^2+3x&x(x-1)(x-3)\\
7&x^3-5x^2+6x-1&x^3-5x^2+6x-1\\
8&x^4-6x^3+10x^2-4x&x(x-2)(x^2-4x+2)\\
9&x^4-7x^3+15x^2-10x+1&(x-1)(x^3-6x^2+9x-1)\\
10&x^5-8x^4+21x^3-20x^2+5x&x(x^2-3x+1)(x^2-5x+5)\\
11&x^5-9x^4+28x^3-35x^2+15x-1&x^5-9x^4+28x^3-35x^2+15x-1\\
12&x^6-10x^5+36x^4-56x^3+35x^2-6x&x(x-1)(x-2)(x-3)(x^2-4x+1).
\end{array}$$
For $i=0,1,2,\dots$, we define, recursively, polynomials $g_i(x)\in\mathbb{Z}[x]$ in the following way: $g_0(x)=0$, $g_1(x)=1$, and $g_i(x)=xg_{i-1}(x)+g_{i-2}(x)$, for $i>1$. These polynomials are called [*Fibonacci polynomials*]{}, see Sequence A011973 in [@OEIS]. By comparing the two definitions, we have, $$\begin{array}{rclr}
(-1)^{\lfloor\frac{i}{2}\rfloor}f_i(-x^2)&=&g_i(x),&i\text{ is odd};\\
\frac{(-1)^{\frac{i}{2}}}{x}f_i(-x^2)&=&g_i(x),&i\text{ is even}.
\end{array}$$ Now, from [@Lev Lemma 5], it follows that, for each $i\in\{1,2,3,\dots\}$, the polynomial $f_i(x)$ has a unique irreducible (over $\mathbb{Q}$) factor, denoted $\underline{f}_i(x)$, which is not an irreducible factor of any $f_j(x)$, for $j<i$. Furthermore, $$f_i(x)=\prod_{d\vert i}\underline{f}_d(x).$$ From [@Lev Definition 1], it follows that, for $i>2$, the polynomial $\underline{f}_i(x)$ has degree $\frac{\phi(i)}{2}$, where $\phi$ is Euler’s totient function, and all roots of $\underline{f}_i(x)$ are positive real numbers less than $4$. Furthermore, the sequence of maximal roots of $\underline{f}_i(x)$ is strictly increasing and converges to $4$, when $i\to\infty$. For small $i$, the polynomials $\underline{f}_i(x)$ are given in the following table: $$\begin{array}{c||l}
i&\underline{f}_i(x)\\
\hline\hline
0&0\\
1&1\\
2&x\\
3&x-1\\
4&x-2\\
5&x^2-3x+1\\
6&x-3\\
7&x^3-5x^2+6x-1\\
8&x^2-4x+2\\
9&x^3-6x^2+9x-1\\
10&x^2-5x+5\\
11&x^5-9x^4+28x^3-35x^2+15x-1\\
12&x^2-4x+1\\
13&x^6-11x^5+45x^4-84x^3+70x^2-21x+1\\
14&x^3-7x^2+14x-7\\
15&x^4-9x^3+26x^2-24x+1.
\end{array}$$ We refer the reader to [@Lev; @WP] for further properties of the above polynomials.
Disguised Fibonacci polynomials and $2$-representations of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ {#s17.3}
----------------------------------------------------------------------------------------------------
Recall that we are in Coxeter type $I_2(n)$, for some $n\geq 3$. Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. By Subsection \[s4.4\] and Corollary \[corn32\], we have $$\mathtt{M}=\left(
\begin{array}{c|c}
2E_1&B\\\hline
B^{\mathrm{tr}}&2E_2
\end{array}
\right),\quad
\Lparen\theta_s\Rparen=\left(
\begin{array}{c|c}
2E_1&B\\\hline
0&0
\end{array}
\right),\quad
\Lparen\theta_t\Rparen=\left(
\begin{array}{c|c}
0&0\\\hline
B^{\mathrm{tr}}&2E_2
\end{array}
\right).$$
\[lem1701\] Both $BB^{\mathrm{tr}}$ and $B^{\mathrm{tr}}B$ are annihilated by $f_n(x)$.
Set $C=B^{\mathrm{tr}}$. For simplicity, for $i\geq 1$, let $s_i=stst\dots$ be the element of length $i$ and $t_i=tsts\dots$ be the element of length $i$. Using induction on $i$ and the multiplication rule $$\label{eq1705}
\theta_s\theta_{t_i}\cong
\begin{cases}
\theta_{s_2}, & i=1;\\
\theta_{s_{i+1}}\oplus \theta_{s_{i-1}}, & i>1;
\end{cases}
\qquad
\theta_t\theta_{s_i}\cong
\begin{cases}
\theta_{t_2}, & i=1;\\
\theta_{t_{i+1}}\oplus \theta_{t_{i-1}}, & i>1;
\end{cases}$$ one proves, by induction, that $$\Lparen\theta_{s_{i}}\Rparen=\left(
\begin{array}{c|c}
2f_i(BC)&f_i(BC)B\\\hline
0&0
\end{array}
\right),\quad
\Lparen\theta_{t_{i}}\Rparen=\left(
\begin{array}{c|c}
0&0\\\hline
f_i(CB)C&2f_i(CB)
\end{array}
\right),\quad$$ if $i$ is odd, and $$\Lparen\theta_{s_{i}}\Rparen=\left(
\begin{array}{c|c}
f_i(BC)&2f_i(BC)C^{-1}\\\hline
0&0
\end{array}
\right),\quad
\Lparen\theta_{t_{i}}\Rparen=\left(
\begin{array}{c|c}
0&0\\\hline
2f_i(CB)B^{-1}&f_i(CB)
\end{array}
\right),\quad$$ if $i$ is even (here $C^{-1}$ just means that the rightmost appearances of $C$ in $f_i(BC)$ should be deleted, and similarly for $B^{-1}$ with respect to $f_i(CB)$). Note that $C (BC)^l = (CB)^l C$, for any $l\in\{0,1,2,\dots\}$, and therefore $B f_i(CB) C = BC f_i(BC)$, for any $i$.
As $\mathbf{l}(w_0)=n$, the claim now follows from our assumption that $\mathbf{M}$ has apex $\mathcal{J}$ and therefore $\Lparen\theta_{w_0}\Rparen=0$.
\[cor1702\] If $n$ is odd, then $B$ is invertible.
If $n$ is odd, then $f_n(x)$ has a non-zero constant term. Therefore, from Lemma \[lem1701\] we have that both $BB^{\mathrm{tr}}$ and $B^{\mathrm{tr}}B$ are invertible and thus $B$ is invertible as well.
\[cor1705\] Let $n\in\{3,4,5,\dots\}$ and $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ in Coxeter type $I_2(n)$ with apex $\mathcal{J}$ and with the corresponding matrix $\mathtt{M}$. Let $n'\in\{3,4,5,\dots\}$ and $\mathbf{M}'$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ in Coxeter type $I_2(n')$ with apex $\mathcal{J}$ and with the corresponding matrix $\mathtt{M}'$. If $\mathtt{M}=\mathtt{M}'$, then $n=n'$.
Let $w$ denote the word $stst\dots$ of length $n$. Then $w=w_0$ in Coxeter type $I_2(n)$. From Lemma \[lem1701\], we have that $\mathbf{M}'(\theta_w)=0$. Therefore $w\not\in\mathcal{J}$ in Coxeter type $I_2(n')$ and we obtain $n'\leq n$. By symmetry, we also have $n\leq n'$ and hence $n=n'$.
The explicit form of $B$ {#s17.4}
------------------------
\[prop1704\] If $n=2k+1$, then, by independent permutations of rows and columns, the matrix $B$ can be reduced to a $k\times k$ staircase matrix.
If we can prove that, by independent permutations of rows and columns, $B$ can be reduced to a square staircase matrix, then the fact that the size of such matrix is $k\times k$ follows from Corollary \[cor1705\]. From Corollary \[cor1702\], we have that $B$ is an invertible matrix, say of size $m\times m$. Set $Q=B^{\mathrm{tr}}B$. Then, from Lemma \[lem1701\] and Subsection \[s17.2\], we have that all eigenvalues of $Q$ are positive real numbers that are strictly less than $4$.
Therefore, by Proposition \[propsim\], the matrix $B$ is either a staircase matrix or coincides with one of the matrices $X_1$ or $X_3$. If $B=X_3$, then $$Q=\left(\begin{array}{cccc}2&1&0&0\\1&2&1&1\\0&1&1&1\\0&1&1&2\end{array}\right).$$ The characteristic polynomial of $Q$ is $x^4-7x^3+14x^2-8x+1$ and is irreducible over $\mathbb{Q}$. The only odd $i$, for which $\phi(i)=8$, is $i=15$. However, we already know that $\underline{f}_{15}(x)=x^4-9x^3+26x^2-24x+1$. Therefore $\underline{f}_{i}(Q)\neq 0$ for any odd $i$. This means that such $B$ is not possible.
If $B=X_1$, we have $$Q=\left(\begin{array}{ccc}2&1&1\\1&1&1\\1&1&2\end{array}\right).$$ The characteristic polynomial of $Q$ has factorization $(x-1)(x^2-4x+1)$. This means that $\underline{f}_{12}(x)$ is a factor of any annihilating polynomial of $Q$ and thus $\underline{f}_{i}(Q)\neq 0$ for any odd $i$. The claim of the proposition follows.
Proof of Theorem \[thm1700\] {#s17.6}
----------------------------
We are now ready to prove Theorem \[thm1700\]. Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Proposition \[prop1704\] gives explicitly the matrix $\mathtt{M}$, in particular, it shows that this matrix is uniquely determined. Hence this matrix coincides with the corresponding matrix for the cell $2$-representation $\mathbf{C}_{\mathcal{L}_s}$.
Consider the abelianization $\overline{\mathbf{M}}$. Now, starting from the simple module $L_{k}$, we can use Proposition \[propn31\] to get $\theta_s\, L_k\cong P_k$. Using the explicit formulae for the actions of $\theta_s$ and $\theta_t$ together with , we see that, applying $\theta_w$, for $w\in \mathcal{L}_s$, to $L_k$, we obtain all indecomposable projective objects in $\overline{\mathbf{M}}(\mathtt{i})$.
The rest of the proof is now similar to the corresponding parts of the proofs in the literature, see [@MM5 Proposition 9], [@MZ Sections 6 and 9] or [@MaMa Subsection 4.9]. Let $\mathbf{N}$ be the additive $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ obtained by restricting the action of ${\sc\mbox{S}\hspace{1.0pt}}$ to the category of projective objects in $\overline{\mathbf{M}}(\mathtt{i})$. Then $\mathbf{N}$ is equivalent to $\mathbf{M}$, see [@MM2 Theorem 11]. There is a unique strong $2$-natural transformation from $\mathbf{P}_{\mathtt{i}}$ to $\overline{\mathbf{M}}$ sending $\mathbbm{1}_{\mathtt{i}}$ to $L_k$. It induces a strong $2$-natural transformation from $\mathbf{C}_{\mathcal{L}_s}$ to $\mathbf{N}$. Since $\mathbf{C}_{\mathcal{L}_s}(\mathtt{i})$ is simple transitive with apex $\mathcal{J}$, everything that we established earlier for $\mathbf{M}$ (in particular, about the structure of projective modules etc.) also holds for $\mathbf{C}_{\mathcal{L}_s}$. In particular, the Cartan matrices of the underlying algebras of $\mathbf{C}_{\mathcal{L}_s}(\mathtt{i})$ and $\mathbf{N}(\mathtt{i})$ agree. Therefore the above $2$-natural transformation is an equivalence between these two categories. This completes the proof.
Simple transitive $2$-representations of ${\sc\mbox{S}\hspace{1.0pt}}$ in Coxeter type $I_2(n)$ with $n$ even {#s6}
=============================================================================================================
Setup and preliminaries {#s6.1}
-----------------------
In this section we assume that $W$ is of type $I_2(n)$ with $n=2k>4$ and $S=\{s,t\}$ with the Coxeter diagram $$\xymatrix{s\ar@{-}[r]^{n}&t.}$$ Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. By Subsection \[s4.4\] and Corollary \[corn32\], we have $$\mathtt{M}=\left(
\begin{array}{c|c}
2E_1&B\\\hline
B^{\mathrm{tr}}&2E_2
\end{array}
\right),\quad
\Lparen\theta_s\Rparen=\left(
\begin{array}{c|c}
2E_1&B\\\hline
0&0
\end{array}
\right),\quad
\Lparen\theta_s\Rparen=\left(
\begin{array}{c|c}
0&0\\\hline
B^{\mathrm{tr}}&2E_2
\end{array}
\right).$$ From Lemma \[lem1701\], we have that both $BB^{\mathrm{tr}}$ and $B^{\mathrm{tr}}B$ are annihilated by $f_{n}(x)$.
The main result of the section is the following statement.
\[thm49\] Assume that $W$ is of type $I_2(n)$ with $n=2k>4$ and $n\neq 12,18,30$. Then every simple transitive $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ is equivalent to either a cell $2$-representation or one of the $2$-representations $\mathbf{N}_s^{(n)}$, $\mathbf{N}_t^{(n)}$ constructed in Subsection \[s6.7\]. All these $2$-representations are pairwise non-equivalent.
Additional simple transitive $2$-representations {#s6.7}
------------------------------------------------
In this subsection we just assume that $n=2k>4$. The left cell $\mathcal{L}_s$ of the element $\theta_s\in{\sc\mbox{S}\hspace{1.0pt}}$ consists of the elements $\theta_s$, $\theta_{ts}$, $\theta_{sts}$, $\theta_{tsts}$ and so on, with $2k-1$ elements in total. Consider the corresponding cell $2$-representation $\mathbf{C}_{\mathcal{L}_s}$. In this subsection we follow closely the approach of [@MaMa Subsection 5.8] to construct, starting from $\mathbf{C}_{\mathcal{L}_s}$, a new simple transitive $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$, which we later on will denote by $\mathbf{N}_s^{(n)}$.
Set $\mathbf{Q}:=\mathbf{C}_{\mathcal{L}_s}$ and consider $\overline{\mathbf{Q}}$. Let $P_w$, $w\in \mathtt{L}:=\{s,ts,sts,tsts,\dots\}$, be representatives of the isomorphism classes of the indecomposable projective objects in $\overline{\mathbf{Q}}(\mathtt{i})$ and $L_w$, $w\in\mathtt{L}$, be the respective simple tops. Note that $|\mathtt{L}|$ is odd. With respect to this choice of a basis, we have $$\Lparen\theta_s\Rparen=
\left(
\begin{array}{cccccc}
2&1&0&0&0&\dots\\
0&0&0&0&0&\dots\\
0&1&2&1&0&\dots\\
0&0&0&0&0&\dots\\
0&0&0&1&2&\dots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{array}
\right)\qquad
\Lparen\theta_t\Rparen=
\left(
\begin{array}{cccccc}
0&0&0&0&0&\dots\\
1&2&1&0&0&\dots\\
0&0&0&0&0&\dots\\
0&0&1&2&1&\dots\\
0&0&0&0&0&\dots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{array}
\right).$$ In particular, in this case the matrix $B$ is a $k\times (k-1)$ staircase matrix. Consequently, we have the following Loewy filtrations of indecomposable projective modules: $$\xymatrix{
L_s\ar@{-}[d]&&L_{ts}\ar@{-}[dl]\ar@{-}[dr]&&&L_{sts}\ar@{-}[dl]\ar@{-}[dr]&\\
L_{ts}\ar@{-}[d]&L_s\ar@{-}[dr]&&L_{sts}\ar@{-}[dl]&L_{ts}\ar@{-}[dr]&&L_{tsts}\ar@{-}[dl]\\
L_s&&L_{ts}&&&L_{sts}&\\
}$$ and so on (the last module in the series is uniserial of length three, just like the first one). Let $H$ denote the basic underlying algebra of $\overline{\mathbf{Q}}(\mathtt{i})$. Then the above implies that $H$ is the quotient of the path algebra of $$\xymatrix{
\mathtt{s}\ar@/^/[r]&\mathtt{ts}\ar@/^/[r]\ar@/^/[l]&\mathtt{sts}\ar@/^/[r]\ar@/^/[l]&
\mathtt{tsts}\ar@/^/[r]\ar@/^/[l]&\dots\ar@/^/[l]
}$$ modulo the relations that any path of the form $\mathtt{v}\to\mathtt{u}\to \mathtt{w}$ is zero if $v\neq w$ and all paths of the form $\mathtt{v}\to\mathtt{u}\to \mathtt{v}$ coincide. Let $f_w$, $w\in\mathtt{L}$, denote pairwise orthogonal primitive idempotents of $H$ corresponding to $P_w$.
As $\mathbf{Q}$ is simple transitive, all $\theta_w$, $w\in\mathcal{J}$, send simple objects in $\overline{\mathbf{Q}}(\mathtt{i})$ to projective objects in $\overline{\mathbf{Q}}(\mathtt{i})$ and act as projective endofunctors of $\overline{\mathbf{Q}}(\mathtt{i})$, by Theorem \[thmminimal\]. In terms of $H$, the action of $\theta_s$ is given by tensoring with the $H$-$H$–bimodule $$(Hf_{s}\otimes f_{s}H)\oplus(Hf_{sts}\otimes f_{sts}H)\oplus\dots$$ while the action of $\theta_{t}$ is, similarly, given by tensoring with the $H$-$H$–bimodule $$(Hf_{ts}\otimes f_{ts}H)\oplus(Hf_{tsts}\otimes f_{tsts}H)\oplus\dots.$$
Set $\mathbf{Q}^{(0)}:=\mathbf{Q}$ and let $\mathbf{Q}^{(1)}$ denote the $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ given by the action of ${\sc\mbox{S}\hspace{1.0pt}}$ on the category of projective objects in $\overline{\mathbf{Q}}^{(0)}(\mathtt{i})$. Recursively, for $k\geq 1$, define $\mathbf{Q}^{(k)}$ as the $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ given by the action of ${\sc\mbox{S}\hspace{1.0pt}}$ on the category of projective objects in $\overline{\mathbf{Q}}^{(k-1)}(\mathtt{i})$. For every $k\geq 0$, we have a strict $2$-natural transformation $\Lambda_k:{\mathbf{Q}}^{(k-1)}\to \mathbf{Q}^{(k)}$ which sends an object $X$ to the diagram $0\to X$ and a morphism $\alpha:X\to X'$ to the diagram $$\xymatrix{
0\ar[rr]\ar[d] && X\ar[d]^{\alpha}\\
0\ar[rr] && X'.
}$$ Clearly, each such $\Lambda_k$ is an equivalence. Denote by $\mathbf{K}$ the inductive limit of the directed system $$\label{eq5z}
\mathbf{Q}^{(0)}\overset{\Lambda_0}{\longrightarrow}
\mathbf{Q}^{(1)}\overset{\Lambda_1}{\longrightarrow}
\mathbf{Q}^{(2)}\overset{\Lambda_2}{\longrightarrow}\dots.$$ Then $\mathbf{K}$ is a $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ which is equivalent to $\mathbf{Q}$.
Let $x'$ be the element which you get by swapping $s$ and $t$ in the reduced expression for $x$.
\[lem43\] There is a strict $2$-natural transformation $\Psi:\mathbf{K}\to \mathbf{K}$ which is an equivalence and which acts on the isomorphism classes of the indecomposable projective objects by swapping each $P_x$ with $P_{x'w_0}$, for $x\in\mathtt{L}$ such that $x'w_0\neq x$, and fixing the unique $P_{x}$ for which $x'w_0=x$.
Consider the unique strict $2$-natural transformation $\Phi:\mathbf{P}_{\mathtt{i}}\to \overline{\mathbf{Q}}$ which sends $\mathbb{1}_{\mathtt{i}}$ to $L_{tw_0}$. Using Proposition \[propn31\] and the explicit formulae for the matrices representing the action of all $\theta_w$, we have $$\theta_s\, L_{tw_0}\cong P_{tw_0},\quad
\theta_{ts}\, L_{tw_0}\cong P_{stw_0},\quad
\text{ and so on}.$$ As the Cartan matrix of $H$ is invariant under swapping $P_x$ with $P_{x'w_0}$, for $x\in\mathtt{L}$ such that $x'w_0\neq x$, and fixing the unique $P_{x}$ for which $x'w_0=x$, it follows by the usual arguments, see for example [@MaMa Subsection 4.9], that the $2$-natural transformation $\Phi$ factors through $\mathbf{C}_{\mathcal{L}_s}$ and, therefore, gives a strict equivalence $\Phi^{(0)}:\mathbf{Q}^{(0)}\to \mathbf{Q}^{(1)}$. For $k\geq 0$, via abelianization, we get a strict equivalence $\Phi^{(k)}:\mathbf{Q}^{(k)}\to \mathbf{Q}^{(k+1)}$, which is compatible with . Now we can take $\Psi$ as the inductive limit of $\Phi^{(k)}$.
Consider a new finitary $2$-representation $\mathbf{K}'$ of ${\sc\mbox{S}\hspace{1.0pt}}$ defined as follows:
- Objects of $\mathbf{K}'(\mathtt{i})$ are sequences $(X_n,\alpha_n)_{n\in\mathbb{Z}}$, where $X_n$ is an object in $\mathbf{K}(\mathtt{i})$ and $\alpha_n\colon \Psi(X_n)\to X_{n+1}$ an isomorphism in $\mathbf{K}(\mathtt{i})$, for all $n\in\mathbb{Z}$.
- Morphisms in $\mathbf{K}'(\mathtt{i})$ from $(X_n,\alpha_n)_{n\in\mathbb{Z}}$ to $(Y_n,\beta_n)_{n\in\mathbb{Z}}$ are sequences of morphisms $f_n\colon X_n\to Y_n$ in $\mathbf{K}(\mathtt{i})$ such that $$\xymatrix{
\Psi(X_n)\ar[rr]^{\alpha_n}\ar[d]_{\Psi(f_n)}&&X_{n+1}\ar[d]^{f_{n+1}}\\
\Psi(Y_n)\ar[rr]^{\beta_n}&&Y_{n+1}\\
}$$ commutes for all $n\in\mathbb{Z}$.
- The action of ${\sc\mbox{S}\hspace{1.0pt}}$ on $\mathbf{K}'(\mathtt{i})$ is inherited from the action of ${\sc\mbox{S}\hspace{1.0pt}}$ on $\mathbf{K}(\mathtt{i})$ component-wise.
The construction of $\mathbf{K}'(\mathtt{i})$ from $\mathbf{K}(\mathtt{i})$ is the standard construction which turns a category with an autoequivalence (in our case $\Psi$) into an equivalent category with an automorphism, cf. [@Ke; @BL].
We have the strict $2$-natural transformation $\Pi:\mathbf{K}'\to \mathbf{K}$ given by projection onto the zero component of a sequence. This $\Pi$ is an equivalence, by construction. We also have a strict $2$-natural transformation $\Psi':\mathbf{K}'\to \mathbf{K}'$ given by shifting the entries of the sequences by one, that is, sending $(X_n,\alpha_n)_{n\in\mathbb{Z}}$ to $(X_{n+1},\alpha_{n+1})_{n\in\mathbb{Z}}$, with the similar obvious action on morphisms. Note that $\Psi':\mathbf{K}'(\mathtt{i})\to \mathbf{K}'(\mathtt{i})$ is an automorphism. The functor $\Psi'$ acts on the isomorphism classes of indecomposable objects in $\mathbf{K}'(\mathtt{i})$ in the same way as $\Psi$ does. As $\Psi^2$ is isomorphic to the identity functor on $\mathbf{K}(\mathtt{i})$, it follows, by construction, that $(\Psi')^2$ is isomorphic to the identity functor on $\mathbf{K}'(\mathtt{i})$.
Let $\mathbf{K}''$ be the $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ given by the action of ${\sc\mbox{S}\hspace{1.0pt}}$ on the category of projective objects in $\overline{\mathbf{K}'}$. We denote by $\Psi'':\mathbf{K}''\to \mathbf{K}''$ the diagrammatic extension of $\Psi'$ to $\mathbf{K}''$. Again, $\Psi'':\mathbf{K}''(\mathtt{i})\to \mathbf{K}''(\mathtt{i})$ is an automorphism and $(\Psi'')^2$ is isomorphic to the identity functor on $\mathbf{K}''(\mathtt{i})$. We need the following stronger statement.
\[lem44\] Let $\mathrm{Id}:\mathbf{K}''\to \mathbf{K}''$ denote the identity $2$-natural transformation.
1. \[lem44.1\] There is an invertible modification $\eta:\mathrm{Id}\to (\Psi'')^2$.
2. \[lem44.2\] For any $\eta$ as in , we have $\mathrm{id}_{(\Psi'')^2}\circ_0 \eta=\eta\circ_0\mathrm{id}_{(\Psi'')^2}$.
3. \[lem44.3\] For any $\eta'\in\mathrm{Hom}_{{\scc\mbox{S}\hspace{1.0pt}}\text{-}\mathrm{mod}}(\mathrm{Id},(\Psi'')^2)$, we have $\mathrm{id}_{(\Psi'')^2}\circ_0 \eta'=\eta'\circ_0\mathrm{id}_{(\Psi'')^2}$.
Let $L$ be a simple object in $\overline{\mathbf{K}'}$ corresponding to $P_1$ (thus $L$ is a simple corresponding to the Duflo involution in $\mathcal{L}_s$). Fix an isomorphism $\alpha:L\to (\Psi')^2(L)$. For $w\in \mathtt{L}$, set $\eta_{\theta_w\,L}:=\mathrm{id}_{\theta_w}\circ_0\alpha$. As $\{\theta_w\,L\,:\,w\in\mathtt{L}\}$ is a complete list of pairwise non-isomorphic indecomposable objects in $\mathbf{K}''(\mathtt{i})$, this uniquely defines a natural transformation $\eta:\mathrm{Id}\to (\Psi'')^2$.
If $\mathrm{F}$ is a $1$-morphism in ${\sc\mbox{S}\hspace{1.0pt}}$, then, for any $w\in \mathtt{L}$, we have $\mathrm{F}\circ \theta_w\cong \mathrm{F}_1\oplus \mathrm{F}_2$, where $\mathrm{F}_1\, L\in
\mathrm{add}(\{\theta_w\,L\,:\,w\in\mathtt{L}\})$ and $\mathrm{F}_2\, L=0$. From the definition in the previous paragraph we thus get $\mathrm{F}(\eta_{\theta_w\,L})=\eta_{\mathrm{F}_1\,L}$ which implies that $\eta$ is, in fact, a modification. Claim follows. Claims and are now proved similarly to [@MaMa Lemma 17(2) and (3)].
\[prop45\] There is an invertible modification $\eta:\mathrm{Id}\to (\Psi'')^2$ for which we have the equality $\mathrm{id}_{\Psi''}\circ_0 \eta=\eta\circ_0\mathrm{id}_{\Psi''}$.
Mutatis mutandis the proof of [@MaMa Proposition 18].
From now on we fix some invertible modification $\eta:\mathrm{Id}\to (\Psi'')^2$ as given by Proposition \[prop45\]. Define a small category $\mathbf{L}(\mathtt{i})$ as follows:
- objects in $\mathbf{L}(\mathtt{i})$ are all $4$-tuples $(X,Y,\alpha,\beta)$, where we have $X,Y\in \mathbf{K}''(\mathtt{i})$, while $\alpha:X\to \Psi''(Y)$ and $\beta:Y\to \Psi''(X)$ are isomorphism such that the following conditions are satisfied $$\label{eqnn9}
\left\{
\begin{array}{rcl}
\eta_{Y}^{-1}\circ_1 {\Psi''}(\alpha)\circ_1\beta&=&\mathrm{id}_Y,\\
\beta\circ_1\eta_{Y}^{-1}\circ_1 {\Psi''}(\alpha)&=&\mathrm{id}_{\Psi''(X)},\\
\eta_{X}^{-1}\circ_1{\Psi''}(\beta)\circ_1\alpha&=&\mathrm{id}_X,\\
\alpha\circ_1\eta_{X}^{-1}\circ_1{\Psi''}(\beta)&=&\mathrm{id}_{\Psi''(Y)}.
\end{array}
\right.$$
- morphisms in $\mathbf{L}(\mathtt{i})$ from $(X,Y,\alpha,\beta)$ to $(X',Y',\alpha',\beta')$ are pairs $(\zeta,\xi)$, where $\zeta:X\to X'$ and $\xi:Y\to Y'$ are morphisms in $\mathbf{K}''(\mathtt{i})$ such that the diagrams $$\xymatrix{
X\ar[rr]^{\alpha}\ar[d]_{\zeta}&&\Psi''(Y)\ar[d]^{\Psi''(\xi)}\\
X'\ar[rr]^{\alpha'}&&\Psi''(Y')\\
} \quad\text{ and }\quad
\xymatrix{
Y\ar[rr]^{\beta}\ar[d]_{\xi}&&\Psi''(X)\ar[d]^{\Psi''(\zeta)}\\
Y'\ar[rr]^{\beta'}&&\Psi''(X')\\
}$$ commute;
- the composition and identity morphisms are the obvious ones.
The category $\mathbf{L}(\mathtt{i})$ comes equipped with an action of ${\sc\mbox{S}\hspace{1.0pt}}$, defined component-wise, using the action of ${\sc\mbox{S}\hspace{1.0pt}}$ on $\mathbf{K}(\mathtt{i})$. This is well-defined as the $2$-natural transformation $\Psi$ is strict by Lemma \[lem43\] and, moreover, $\eta$ is a modification. We denote the corresponding $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ by $\mathbf{L}$.
\[lem46\] Restriction to the first component of a quadruple defines a strict $2$-natural transformation $\Upsilon:\mathbf{L}\to \mathbf{K}''$. This $\Upsilon$ is an equivalence,
Mutatis mutandis the proof of [@MaMa Lemma 19].
Define an endofunctor $\Theta$ on $\mathbf{L}(\mathtt{i})$ by sending $(X,Y,\alpha,\beta)$ to $(Y,X,\beta,\alpha)$ with the obvious action on morphisms. From all symmetries in the definition of $\mathbf{L}(\mathtt{i})$, it follows that $\Theta$ is a strict involution and it also strictly commutes with the action of ${\sc\mbox{S}\hspace{1.0pt}}$.
Next, consider the category $\mathbf{L}'(\mathtt{i})$ defined as follows:
- $\mathbf{L}'(\mathtt{i})$ has the same objects as $\mathbf{L}(\mathtt{i})$,
- morphisms in $\mathbf{L}'(\mathtt{i})$ are defined, for objects $X,Y\in\mathbf{L}(\mathtt{i})$, via $$\mathrm{Hom}_{\mathbf{L}'(\mathtt{i})}(X,Y):=
\mathrm{Hom}_{\mathbf{L}(\mathtt{i})}(X,Y)\oplus \mathrm{Hom}_{\mathbf{L}(\mathtt{i})}(X,\Theta(Y)),$$
- composition and identity morphisms in $\mathbf{L}'(\mathtt{i})$ are induced from those in $\mathbf{L}(\mathtt{i})$ in the obvious way, see [@CiMa Definition 2.3] for details.
The fact that $\Psi$ preserves the isomorphism class of the indecomposable object $P_{sts}$ implies that the endomorphism algebra of the corresponding object in $\mathbf{L}'(\mathtt{i})$ contains a copy of the group algebra of the group $\{\mathrm{Id},\Theta\}$ and hence is not local. This means that $\mathbf{L}'(\mathtt{i})$ is not idempotent split. Denote by $\mathbf{N}_s^{(n)}(\mathtt{i})$ the idempotent completion of $\mathbf{L}'(\mathtt{i})$.
\[prop47\]
1. \[prop27.1\] The category $\mathbf{N}_s^{(n)}(\mathtt{i})$ is a finitary $\mathbb{C}$-linear category.
2. \[prop27.2\] The category $\mathbf{N}_s^{(n)}(\mathtt{i})$ is equipped with an action of ${\sc\mbox{S}\hspace{1.0pt}}$ induced from that on $\mathbf{L}'(\mathtt{i})$.
3. \[prop27.3\] The obvious functor $\Xi:\mathbf{L}(\mathtt{i})\to \mathbf{N}_s^{(n)}(\mathtt{i})$ is a strict $2$-natural transformation.
The only thing which is different from [@MaMa Proposition 20] is the fact that the category $\mathbf{L}'(\mathtt{i})$ is not idempotent split. However, since we define $\mathbf{N}_s^{(n)}(\mathtt{i})$ as the idempotent completion of $\mathbf{L}'(\mathtt{i})$, it follows that $\mathbf{N}_s^{(n)}(\mathtt{i})$ is idempotent split and hence $\mathbf{N}_s^{(n)}(\mathtt{i})$ is a finitary $\mathbb{C}$-linear category. The rest is similar to [@MaMa Proposition 20].
The above gives us the $2$-representation $\mathbf{N}_s^{(n)}$ of ${\sc\mbox{S}\hspace{1.0pt}}$. From the fact that $\mathbf{C}_{\mathcal{L}_s}$ is simple transitive, it follows that $\mathbf{N}_s^{(n)}$ is simple transitive. The underlying algebra $\mathbf{N}_s^{(n)}$ is the quotient of the path algebra of the following quiver: $$\xymatrix{
&&&&\circ\ar@/^/[d]&\\
\bullet\ar@/^/[r]&\bullet\ar@/^/[r]\ar@/^/[l]&\dots\ar@/^/[r]\ar@/^/[l]&\bullet\ar@/^/[r]\ar@/^/[l]
&\bullet\ar@/^/[r]\ar@/^/[l]\ar@/^/[u]&\circ\ar@/^/[l]
}$$ where we mod out by the relations that any path of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{k}$ is zero if $i\neq k$ and all paths of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{i}$ coincide. Here the vertices $\bullet$ correspond to pairs $\{P_x,P_{x'w_0}\}$, where $x\in\mathtt{L}$ is such that $x\neq x'w_0$, while the two vertices $\circ$ correspond to the unique $x\in\mathtt{L}$ for which $x'w_0=x$.
If $k$ is odd, then the unique $x\in\mathtt{L}$ for which $x'w_0=x$ has the form $stst\dots s$. This implies that the matrix $B$ is, up to permutation of rows and columns, an extended staircase matrix of size $\frac{k+3}{2}\times \frac{k-1}{2}$. If $k$ is even, then the unique $x\in\mathtt{L}$ for which $x'w_0=x$ has the form $tst\dots s$. This implies that the matrix $B$ is, up to permutation of rows and columns, an extended staircase matrix of size $\frac{k}{2}\times \frac{k+2}{2}$. The explanation why $B$ has only one row or column with three non-zero elements is the fact that the quiver above has only one vertex which is connected to three other vertices.
We denote by $\mathbf{N}_t^{(n)}$ the $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$ constructed similarly starting from $\mathcal{L}_t$. Note that $\mathbf{C}_{\mathcal{L}_s}$ and $\mathbf{C}_{\mathcal{L}_t}$ are not equivalent as their decategorifications contain different one-dimensional simple $W$-modules and hence are not isomorphic. It is easy to check that the decategorifications of $\mathbf{N}_s^{(n)}$ and $\mathbf{N}_t^{(n)}$ also contain different one-dimensional simple $W$-modules. This implies that $\mathbf{N}_s^{(n)}$ and $\mathbf{N}_t^{(n)}$ are not equivalent. Comparing decategorifications, we, in fact, see that the $2$-representation $\mathbf{C}_{\mathcal{L}_s}$, $\mathbf{C}_{\mathcal{L}_t}$, $\mathbf{N}_s^{(n)}$ and $\mathbf{N}_t^{(n)}$ are pairwise not equivalent.
The matrix $B$ {#s6.2}
--------------
Our aim in this section is to describe the matrix $B$ from Subsection \[s6.1\].
\[prop42\] Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Assume that $n\neq 12,18,30$. Then, up to permutation of rows and columns, the matrix $B$ is a staircase matrix of size $k\times (k-1)$ or an extended staircase matrix of size $\frac{k}{2}\times \frac{k+2}{2}$ (if $k$ is even) or of size $\frac{k+3}{2}\times \frac{k-1}{2}$ (if $k$ is odd).
For example, if $n=6$, then $B$ is one of the following matrices: $$\left(\begin{array}{ccc}1&1&0\\0&1&1\end{array}\right),\quad
\left(\begin{array}{cc}1&0\\1&1\\0&1\end{array}\right),\quad
\left(\begin{array}{ccc}1&1&1\end{array}\right),\quad
\left(\begin{array}{c}1\\1\\1\end{array}\right).$$ If $n=8$, then $B$ is one of the following matrices: $$\left(\begin{array}{cccc}1&1&0&0\\0&1&1&0\\0&0&1&1\end{array}\right),\quad
\left(\begin{array}{ccc}1&0&0\\1&1&0\\0&1&1\\0&0&1\end{array}\right),\quad
\left(\begin{array}{ccc}1&1&1\\0&0&1\end{array}\right),\quad
\left(\begin{array}{cc}1&0\\1&0\\1&1\end{array}\right).$$
As $\mathbf{M}$ is simple transitive, the non-negative matrix $\mathtt{M}$ is primitive. From the computation in the proof of Lemma \[lem1701\] we thus get that both $BB^{\mathrm{tr}}$ and $B^{\mathrm{tr}}B$ must be irreducible non-negative matrices. The combination of Lemma \[lem1701\] with Subsection \[s17.2\] implies that both $BB^{\mathrm{tr}}$ and $B^{\mathrm{tr}}B$ are diagonalizable with real eigenvalues, moreover, all eigenvalues are contained in the half-open interval $[0,4)$.
From Proposition \[propsim\], we thus obtain that $B$ is either a staircase matrix or an extended staircase matrix or coincides with $X_1$ or $X_2$ or $X_3$. Note that all staircase and extended staircase matrices appear as $B$ for some type $I_2(l)$, where $l$ can be arbitrary. Therefore, thanks to Corollary \[cor1705\], to complete the proof of our proposition, it is enough to show that $B$ cannot coincide with any of $X_1,X_2$ or $X_3$.
The minimal polynomial of the matrix $X_1X_1^{\mathrm{tr}}$ is $(x-1)(x^2-4x+1)=
\underline{f}_{3}(x)\underline{f}_{12}(x)$. The arguments in the proof of Corollary \[cor1705\] imply that $B$ can be equal to $X_1$ only in the case $n=12$.
The minimal polynomial of the matrix $X_2X_2^{\mathrm{tr}}$ is $x^3-6x^2+9x-3=\underline{f}_{18}(x)$. The arguments in the proof of Corollary \[cor1705\] imply that $B$ can be equal to $X_1$ only in the case $n=18$.
The minimal polynomial of the matrix $X_3X_3^{\mathrm{tr}}$ is $x^4-7x^3+14x^2-8x+1=\underline{f}_{30}(x)$. The arguments in the proof of Corollary \[cor1705\] imply that $B$ can be equal to $X_1$ only in the case $n=30$.
As the cases $n=12,18,30$ are excluded, the claim of the proposition follows.
Proof of Theorem \[thm49\] {#s6.8}
--------------------------
Let $\mathbf{M}$ be a simple transitive $2$-representation of ${\sc\mbox{S}\hspace{1.0pt}}$. Since $n\neq 12,18,30$, we can apply Proposition \[prop42\] to get four possibilities for $B$ which are in a natural bijection with the $2$-representations $\mathbf{C}_{\mathcal{L}_s}$, $\mathbf{C}_{\mathcal{L}_t}$, $\mathbf{N}_s^{(n)}$ and $\mathbf{N}_t^{(n)}$. That in the first two cases we have that $\mathbf{M}$ is equivalent to $\mathbf{C}_{\mathcal{L}_s}$ or, respectively, $\mathbf{C}_{\mathcal{L}_t}$, is proved similarly to Subsection \[s17.6\]. That in the last two cases we have that $\mathbf{M}$ is equivalent to $\mathbf{N}_s^{(n)}$ or, respectively, $\mathbf{N}_t^{(n)}$, is proved similarly to [@MaMa Subsection 5.10].
Exceptional types $I_2(12)$, $I_2(18)$ and $I_2(30)$ {#s6.9}
----------------------------------------------------
In Coxeter type $I_2(12)$, the proof of Proposition \[prop42\] leaves the possibility of $B=X_1$. In this case $$\mathtt{M}=\left(\begin{array}{cccccc}2&0&0&1&0&0\\0&2&0&1&1&1\\0&0&2&0&0&1
\\1&1&0&2&0&0\\0&1&0&0&2&0\\0&1&1&0&0&2\end{array}\right).$$ If we assume that a simple transitive $2$-representation with such $\mathtt{M}$ exists, then the underlying algebra of this $2$-representation must be the quotient of the path algebra of the following quiver (here the number of each vertex corresponds to the numbering of columns in $\mathtt{M}$): $$\xymatrix{
&&\mathtt{5}\ar@/^/[d]&&&&\\
\mathtt{1}\ar@/^/[r]&\mathtt{4}\ar@/^/[r]\ar@/^/[l]&\mathtt{2}\ar@/^/[r]\ar@/^/[l]\ar@/^/[u]&
\mathtt{6}\ar@/^/[r]\ar@/^/[l]&\mathtt{3}\ar@/^/[l]
}$$ modulo the relations that any path of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{k}$ is zero if $i\neq k$ and all paths of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{i}$ coincide. This algebra is the quadratic dual of the preprojective algebra of the underlying Dynkin quiver of type $E_6$. This suggests a relation between type $I_2(12)$ and type $E_6$ which we do not understand. We do not know whether this hypothetical $2$-representation exists and we see no reasons why it should not exist. On the decategorified level, the corresponding representation of the group algebra, on which the elements of the Kazhdan-Lusztig basis act via the corresponding non-negative matrices certainly does exist. Because of Subsection \[s7.3\], there is also a possibility of connection between type $I_2(12)$ and type $F_4$.
Similarly, Coxeter types $I_2(18)$ and $I_2(30)$ are connected to Dynkin types $E_7$ and $E_8$, respectively. Because of Subsection \[s7.2\], Coxeter type $I_2(30)$ could also be connected to Coxeter type $H_4$.
Simple transitive $2$-representations of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ in other Coxeter types of rank higher than two {#s7}
=================================================================================================================================
Coxeter type $H_3$ {#s7.1}
------------------
In this subsection we assume that $W$ is of Coxeter type $H_3$ and $S=\{r,s,t\}$ with the Coxeter diagram $$\xymatrix{r\ar@{-}[r]^5&s\ar@{-}[r]&t.}$$
\[prop81\] Assume that $W$ is of Coxeter type $H_3$. Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Then the isomorphism classes of indecomposable objects in $\mathbf{M}(\mathtt{i})$ can be ordered such that, with respect to the ordering $r,s,t$, we have $$\mathtt{M}=\left(
\begin{array}{cccccc}
2&0&1&0&0&0\\
0&2&1&1&0&0\\
1&1&2&0&1&0\\
0&1&0&2&0&1\\
0&0&1&0&2&0\\
0&0&0&1&0&2
\end{array}
\right).$$
To prove this statement we use reduction to rank two Coxeter subgroups. Let ${\sc\mbox{C}\hspace{1.0pt}}_1$ denote the $2$-full $2$-subcategory of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ which is monoidally generated by $\theta_r$ and $\theta_s$. By construction, the $2$-category ${\sc\mbox{C}\hspace{1.0pt}}_1$ has a quotient which is biequivalent to the $2$-category $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ in Coxeter type $I_2(5)$. Let ${\sc\mbox{C}\hspace{1.0pt}}_2$ denote the $2$-full $2$-subcategory of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ which is monoidally generated by $\theta_s$ and $\theta_t$. By construction, the $2$-category ${\sc\mbox{C}\hspace{1.0pt}}_2$ has a quotient which is biequivalent to the $2$-category $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ in Coxeter type $A_2$.
We can restrict the action of ${\sc\mbox{C}\hspace{1.0pt}}_1$ to the additive closure $\mathcal{X}$, in $\mathbf{M}(\mathtt{i})$, of all indecomposable objects which are not annihilated by either $\theta_r$ or $\theta_s$. From Theorem \[thm1700\], ${\sc\mbox{C}\hspace{1.0pt}}_1$ has a unique (up to equivalence) simple transitive $2$-representation which does not annihilate any non-zero $1$-morphisms. In this $2$-representation, the actions of $\theta_r$ and $\theta_s$ are given by the following matrices: $$\label{eq81-1}
\left(
\begin{array}{cccc}
2&0&1&0\\
0&2&1&1\\
0&0&0&0\\
0&0&0&0
\end{array}
\right),\qquad
\left(
\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
1&1&2&0\\
0&1&0&2
\end{array}
\right).$$ By the combination of [@KM2 Corollary 20] and [@CM Theorem 25], the actions of $\theta_r$ and $\theta_s$ on $\mathcal{X}$ are then given by a direct sum of blocks of the form .
We can restrict the action of ${\sc\mbox{C}\hspace{1.0pt}}_2$ to the additive closure $\mathcal{Y}$, in $\mathbf{M}(\mathtt{i})$, of all indecomposable objects which are not annihilated by either $\theta_s$ or $\theta_t$. From [@MM5 Theorem 18], ${\sc\mbox{C}\hspace{1.0pt}}_2$ has a unique (up to equivalence) simple transitive $2$-representation which does not annihilate any non-zero $1$-morphisms. In this $2$-representation, the actions of $\theta_s$ and $\theta_t$ are given by the following matrices: $$\label{eq81-2}
\left(
\begin{array}{cc}
2&1\\
0&0
\end{array}
\right)\qquad
\left(
\begin{array}{cc}
0&0\\
1&2
\end{array}
\right).$$ By the combination of [@KM2 Corollary 20] and [@CM Theorem 25], the actions of $\theta_s$ and $\theta_t$ on $\mathcal{Y}$ are then given by a direct sum of blocks of the form .
We need to combine blocks of the form with blocks of the form to get an irreducible non-negative matrix for the principal element $\mathbf{s}$, of size $3m$, for some $m$. The latter restriction is due to the fact that the decategorification of $\mathbf{M}$ is a sum of $3$-dimensional simple $W$-modules. Note that allows us to “connect” (in the sense of having a non-zero element on the intersection of the corresponding row and column) one indecomposable which is not annihilated by $\theta_s$ with one indecomposable which is not annihilated by $\theta_t$. Therefore the only way is to connect each of the two indecomposables in some block of the form which are not annihilated by $\theta_s$ with an external pair of two indecomposables which are not annihilated by $\theta_t$. This gives exactly the matrices given in the formulation of the proposition (up to permutation of basis elements). The claim of the proposition follows.
\[thm82\] Assume that $W$ is of Coxeter type $H_3$. Then every simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation.
Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. We assume that the equivalence classes of indecomposable objects in $\mathbf{M}(\mathtt{i})$ are ordered such that the decategorification matrices of $\theta_r$, $\theta_s$ and $\theta_t$ are gives as in Proposition \[prop81\]. Consider $\overline{\mathbf{M}}$ and the $2$-representation $\mathbf{N}$ of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ given by restriction of the action to the category of projective objects in $\overline{\mathbf{M}}(\mathtt{i})$. Then $\mathbf{M}$ and $\mathbf{N}$ are equivalent by [@MM2 Theorem 11]. We call indecomposable projectives in $\overline{\mathbf{M}}(\mathtt{i})$, in order, $P_1,P_2,\dots,P_6$ and their corresponding simple tops $L_1,L_2,\dots,L_6$. Using the explicit matrices given by Proposition \[prop81\] and arguments similar to the one used in Subsection \[s17.6\], one shows that $$\begin{gathered}
\theta_r\,L_1\cong P_1,\quad
\theta_{sr}\,L_1\cong P_3,\quad
\theta_{rsr}\,L_1\cong P_2,\quad
\theta_{tsr}\,L_1\cong P_5,\\
\theta_{srsr}\,L_1\cong P_4,\quad
\theta_{tsrsr}\,L_1\cong P_6.\end{gathered}$$ Now the proof is completed as in Subsection \[s17.6\]. There is a unique strong $2$-natural transformation from $\mathbf{P}_{\mathtt{i}}$ to $\overline{\mathbf{M}}$ sending $\mathbbm{1}_{\mathtt{i}}$ to $L_1$. It induces a strong $2$-natural transformation from $\mathbf{C}_{\mathcal{L}_s}$ to $\mathbf{N}$. Comparing the Cartan matrices, we see that the latter $2$-natural transformation is, in fact, an equivalence. Therefore $\mathbf{C}_{\mathcal{L}_s}$ and $\mathbf{M}$ are equivalent.
\[rem82-1\] [The underlying algebra of a cell $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$ is the quotient of the path algebra of the following quiver (here the number $\mathtt{i}$ corresponds to $P_i$): $$\xymatrix{
&\mathtt{1}\ar@/^/[d]&&&\\
\mathtt{5}\ar@/^/[r]&\mathtt{3}\ar@/^/[r]\ar@/^/[l]\ar@/^/[u]&
\mathtt{2}\ar@/^/[r]\ar@/^/[l]&\mathtt{4}\ar@/^/[r]\ar@/^/[l]&\mathtt{6}\ar@/^/[l]
}$$ modulo the relations that any path of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{k}$ is zero if $i\neq k$ and all paths of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{i}$ coincide. The Loewy filtrations of the indecomposable projective modules for this algebra are: $$\xymatrix@!=0.6pc{
1\ar@{-}[d]&&2\ar@{-}[dl]\ar@{-}[dr]&&&3\ar@{-}[d]\ar@{-}[dl]\ar@{-}[dr]
&&&4\ar@{-}[dr]\ar@{-}[dl]&&5\ar@{-}[d]&6\ar@{-}[d]\\
3\ar@{-}[d]&3\ar@{-}[dr]&&4\ar@{-}[dl]&1\ar@{-}[dr]&2\ar@{-}[d]&5\ar@{-}[dl]&2\ar@{-}[dr]&&6\ar@{-}[dl]&3\ar@{-}[d]&4\ar@{-}[d]\\
1&&2&&&3&&&4&&5&6
}$$ This algebra is the quadratic dual of the preprojective algebra of the underlying Dynkin quiver of type $D_6$, cf. [@Du]. For further connections between these two Coxeter groups, see [@De] and references therein. ]{}
Coxeter type $H_4$ {#s7.2}
------------------
In this subsection we assume that $W$ is of Coxeter type $H_4$ and $S=\{r,s,t,u\}$ with the Coxeter diagram $$\xymatrix{r\ar@{-}[r]^5&s\ar@{-}[r]&t\ar@{-}[r]&u.}$$
\[prop83\] Assume that $W$ is of Coxeter type $H_4$. Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Then the isomorphism classes of indecomposable objects in $\mathbf{M}(\mathtt{i})$ can be ordered such that, with respect to the ordering $r,s,t,u$, we have $$\mathtt{M}=\left(
\begin{array}{cccccccc}
2&0&1&0&0&0&0&0\\
0&2&1&1&0&0&0&0\\
1&1&2&0&1&0&0&0\\
0&1&0&2&0&1&0&0\\
0&0&1&0&2&0&1&0\\
0&0&0&1&0&2&0&1\\
0&0&0&0&1&0&2&0\\
0&0&0&0&0&1&0&2
\end{array}
\right).$$
Mutatis mutandis the proof of Proposition \[prop81\].
\[thm84\] Assume that $W$ is of Coxeter type $H_4$. Then every simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation.
Mutatis mutandis the proof of Theorem \[thm82\].
\[rem84-1\] [The underlying algebra of a cell $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$ is the quotient of the path algebra of the following quiver (here the number $\mathtt{i}$ corresponds to $P_i$): $$\xymatrix{
&&\mathtt{1}\ar@/^/[d]&&&&\\
\mathtt{7}\ar@/^/[r]&\mathtt{5}\ar@/^/[r]\ar@/^/[l]&\mathtt{3}\ar@/^/[r]\ar@/^/[l]\ar@/^/[u]&
\mathtt{2}\ar@/^/[r]\ar@/^/[l]&\mathtt{4}\ar@/^/[r]\ar@/^/[l]&\mathtt{6}\ar@/^/[l]\ar@/^/[r]&\mathtt{8}\ar@/^/[l]
}$$ modulo the relations that any path of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{k}$ is zero if $i\neq k$ and all paths of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{i}$ coincide. This algebra is the quadratic dual of the preprojective algebra of the underlying Dynkin quiver of type $E_8$. For further connections between these two Coxeter groups, see [@De] and references therein. ]{}
Weyl type $F_4$ {#s7.3}
---------------
In this subsection we assume that $W$ is of Weyl type $F_4$ and $S=\{r,s,t,u\}$ with the Coxeter diagram $$\xymatrix{r\ar@{-}[r]&s\ar@{-}[r]^4&t\ar@{-}[r]&u.}$$
\[prop85\] Assume that $W$ is of Weyl type $F_4$. Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Then the isomorphism classes of indecomposable objects in $\mathbf{M}(\mathtt{i})$ can be ordered such that, with respect to the ordering $r,s,t,u$, we have either $$\mathtt{M}=\left(
\begin{array}{cccccc}
2&0&1&0&0&0\\
0&2&0&1&0&0\\
1&0&2&0&1&0\\
0&1&0&2&1&0\\
0&0&1&1&2&1\\
0&0&0&0&1&2
\end{array}
\right)\quad\text{ or }\quad
\mathtt{M}=\left(
\begin{array}{cccccc}
2&1&0&0&0&0\\
1&2&1&1&0&0\\
0&1&2&0&1&0\\
0&1&0&2&0&1\\
0&0&1&0&2&0\\
0&0&0&1&0&2
\end{array}
\right).$$
Mutatis mutandis the proof of Proposition \[prop81\].
\[thm86\] Assume that $W$ is of Weyl type $F_4$. Then every simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation.
Mutatis mutandis the proof of Theorem \[thm82\].
\[rem86-1\] [We have two different cell $2$-representations of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$ in Weyl type $F_4$. However, the underlying algebras of these cell $2$-representation are isomorphic. This common algebra is the quotient of the path algebra of the following quiver (here the number $\mathtt{i}$ corresponds to $P_i$ in the left matrix in Proposition \[prop85\]): $$\xymatrix{
&&\mathtt{6}\ar@/^/[d]&&&&\\
\mathtt{1}\ar@/^/[r]&\mathtt{3}\ar@/^/[r]\ar@/^/[l]&\mathtt{5}\ar@/^/[r]\ar@/^/[l]\ar@/^/[u]&
\mathtt{4}\ar@/^/[r]\ar@/^/[l]&\mathtt{2}\ar@/^/[l]
}$$ modulo the relations that any path of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{k}$ is zero if $i\neq k$ and all paths of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{i}$ coincide. This algebra is the quadratic dual of the preprojective algebra of the underlying Dynkin quiver of type $E_6$. ]{}
Coxeter type $B_n$ {#s7.4}
------------------
In this subsection we assume that $W$ is of type $B_n$ and $S=\{r,s,t,u,\dots,v\}$ with Coxeter diagram $$\xymatrix{
r\ar@{-}[r]^{4}&s\ar@{-}[r]&t\ar@{-}[r]&u\ar@{-}[r]&\dots\ar@{-}[r]&v.
}$$
\[prop87\] Assume that $W$ is of type $B_n$. Let $\mathbf{M}$ be a simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$. Then the isomorphism classes of indecomposable objects in $\mathbf{M}(\mathtt{i})$ can be ordered such that, with respect to the ordering $r,s,t,u,\dots,v$, we have either $$\mathtt{M}=\left(
\begin{array}{cccccccc}
2&0&1&0&0&\dots&0&0\\
0&2&1&0&0&\dots&0&0\\
1&1&2&1&0&\dots&0&0\\
0&0&1&2&1&\dots&0&0\\
0&0&0&1&2&\dots&0&0\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&0&0&0&\dots&2&1\\
0&0&0&0&0&\dots&1&2
\end{array}
\right)$$ or $$\mathtt{M}=\left(
\begin{array}{cccccccccccc}
2&1&1&0&0&0&0&\dots&0&0&0&0\\
1&2&0&1&0&0&0&\dots&0&0&0&0\\
1&0&2&0&1&0&0&\dots&0&0&0&0\\
0&1&0&2&0&1&0&\dots&0&0&0&0\\
0&0&1&0&2&0&1&\dots&0&0&0&0\\
0&0&0&1&0&2&0&\dots&0&0&0&0\\
0&0&0&0&1&0&2&\dots&0&0&0&0\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\
0&0&0&0&0&0&0&\dots&2&0&1&0\\
0&0&0&0&0&0&0&\dots&0&2&0&1\\
0&0&0&0&0&0&0&\dots&1&0&2&0\\
0&0&0&0&0&0&0&\dots&0&1&0&2
\end{array}
\right).$$
Mutatis mutandis the proof of Proposition \[prop81\].
\[thm88\] Assume that $W$ is of type $B_n$. Then every simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation.
Mutatis mutandis the proof of Theorem \[thm82\].
\[rem88-1\] [We have two different cell $2$-representations of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ with apex $\mathcal{J}$ in type $B_n$. For the first one (which corresponds to the first choice in Proposition \[prop87\]) the underlying algebra is the quotient of the path algebra of the following quiver: $$\xymatrix{
&\mathtt{2}\ar@/^/[d]&&&&\\
\mathtt{1}\ar@/^/[r]&\mathtt{3}\ar@/^/[r]\ar@/^/[l]\ar@/^/[u]&\mathtt{4}\ar@/^/[r]\ar@/^/[l]&
\dots\ar@/^/[r]\ar@/^/[l]&\mathtt{n+1}.\ar@/^/[l]
}$$ For the second one (which corresponds to the second choice in Proposition \[prop87\]) the underlying algebra is the quotient of the path algebra of the following quiver: $$\xymatrix{
\mathtt{2n-1}\ar@/^/[r]&
\dots\ar@/^/[r]\ar@/^/[l]&
\mathtt{5}\ar@/^/[r]\ar@/^/[l]&
\mathtt{3}\ar@/^/[r]\ar@/^/[l]&
\mathtt{1}\ar@/^/[r]\ar@/^/[l]&
\mathtt{2}\ar@/^/[r]\ar@/^/[l]&
\mathtt{4}\ar@/^/[r]\ar@/^/[l]&
\dots\ar@/^/[r]\ar@/^/[l]&
\mathtt{2n-2}.\ar@/^/[l]
}$$ In both cases, we mod out by the relations that any path of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{k}$ is zero if $i\neq k$ and all paths of the form $\mathtt{i}\to\mathtt{j}\to \mathtt{i}$ coincide. The first algebra is the quadratic dual of the preprojective algebra of the underlying Dynkin quiver of type $D_{n+1}$. The second algebra is the quadratic dual of the preprojective algebra of the underlying Dynkin quiver of type $A_{2n-1}$. ]{}
New examples of finitary $2$-categories {#s8}
=======================================
Symmetric modules {#s8.1}
-----------------
Let $\Bbbk$ be an algebraically closed field. In this subsection we assume that $\mathrm{char}(\Bbbk)\neq 2$. Let $A$ be a connected finite dimensional $\Bbbk$-algebra with a fixed automorphism $\iota:A\to A$ such that $\iota^2=\mathrm{id}_A$. For $M\in A\text{-}\mathrm{mod}$, denote by ${}^{\iota}M$ the $A$-module with the same underlying space as $M$ but with the new action $\bullet$ of $A$ twisted by $\iota$: $$a\bullet m:=\iota(a)\cdot m,\quad\text{ for all }\quad a\in A\,\,\text{ and }\,\, m\in M.$$
For a fixed $A$-module $Q$, consider the category $\mathcal{Q}:=\mathrm{add}(Q\oplus {}^{\iota}Q)$. Define the category $\hat{\mathcal{Q}}=\hat{\mathcal{Q}}(A,{\iota},Q)$ in the following way:
- the objects in $\hat{\mathcal{Q}}$ are all diagrams of the form $$\label{eq92}
\xymatrix{M\ar[rr]^{\alpha}&&{}^{\iota}M}$$ where $M\in \mathcal{Q}$ and $\alpha:M\to {}^{\iota}M$ is an isomorphism in $A\text{-}\mathrm{mod}$ such that $\alpha^2\cong \mathrm{id}_M$;
- morphisms in $\hat{\mathcal{Q}}$ are all commutative diagrams of the form $$\xymatrix{M\ar[d]_{\varphi}\ar[rr]^{\alpha}&&{}^{\iota}M\ar[d]^{\varphi}\\
N\ar[rr]^{\beta}&&{}^{\iota}N,
}$$ where $\varphi:M\to N$ is a homomorphism in $A\text{-}\mathrm{mod}$ (note that $\varphi:{}^{\iota}M\to {}^{\iota}N$ is a homomorphism in $A\text{-}\mathrm{mod}$ as well);
- identity morphisms in $\hat{\mathcal{Q}}$ are given by the identity maps;
- composition in $\hat{\mathcal{Q}}$ is induced from composition in $\mathcal{Q}$ in the obvious way.
We will call $\hat{\mathcal{Q}}$ the category of [*$\iota$-symmetric $A$-modules*]{} over $\mathcal{Q}$. Directly from the definitions it follows that $\hat{\mathcal{Q}}$ is additive, $\Bbbk$-linear and idempotent split.
We note that, if $M$ is an $A$-module such that $M\cong {}^{\iota}M$, then an isomorphism $\alpha:M\to {}^{\iota}M$ can always be chosen such that $\alpha^2=\mathrm{id}_M$. Indeed, if $\alpha:M\to {}^{\iota}M$ is any isomorphism, then $\iota^2=\mathrm{id}_A$ implies that $\alpha^2$ is an automorphism of $M$. As $\alpha^{-2}$ is invertible, there exists an automorphism $\beta$ of $M$ which is a polynomial in $\alpha^{-2}$ such that $\beta^2=\alpha^{-2}$. In particular, $\beta$ commutes with $\alpha$. Then $(\alpha\beta):M\to {}^{\iota}M$ is an isomorphism and $(\alpha\beta)^2=\mathrm{id}_M$.
\[prop91\] The category $\hat{\mathcal{Q}}$ is Krull-Schmidt and has finitely many isomorphism classes of indecomposable objects.
Let $Q_1$, $Q_2$,…, $Q_n$ be a complete list of pairwise non-isomorphic indecomposable objects in $\mathcal{Q}$. For every $Q_i$, we either have $Q_i\cong {}^{\iota}Q_i$ or we have $Q_i\cong {}^{\iota}Q_j$, for some $j\neq i$. For each $i$, set $Q_i^{(\iota)}:=Q_i\oplus {}^{\iota}Q_i$ and let $\alpha_i:Q_i^{(\iota)}\to {}^{\iota}Q_i^{(\iota)}$ be the homomorphism which swaps the components of the direct sum. Then $$\label{eqqeenn2}
\xymatrix{Q_i^{(\iota)}\ar[rr]^{\alpha_i}&&{}^{\iota}Q_i^{(\iota)}}$$ is an object in $\hat{\mathcal{Q}}$ which we denote by $(Q_i^{(\iota)},\alpha_i)$.
If $Q_i\cong {}^{\iota}Q_j$, for some $j\neq i$, then $(Q_i^{(\iota)},\alpha_i)$ is, clearly, indecomposable. Moreover, it is easy to see that $(Q_i^{(\iota)},\alpha_i)$ and $(Q_j^{(\iota)},\alpha_j)$ are isomorphic.
Assume now that $\varphi:Q_i\cong {}^{\iota}Q_i$ is an isomorphism such that $\varphi^2=\mathrm{id}_{Q_i}$. Then $\iota^2=\mathrm{id}_A$ implies that $\varphi:{}^{\iota}Q_i\cong Q_i$ is an isomorphism as well and hence the matrix $$\Phi:=\left(\begin{array}{cc}0&\varphi\\\varphi&0\end{array}\right)$$ gives rise to an endomorphism of the object $\alpha_i:Q_i^{(\iota)}\to {}^{\iota}Q_i^{(\iota)}$. Note that $\Phi^2$ is the identity on this object. Write $Q_i^{(\iota)}\cong X_i\oplus Y_i$, where $X_i$ denotes the eigenspace of $\Phi$ for the eigenvalue $1$ and $Y_i$ denotes the eigenspace of $\Phi$ for the eigenvalue $-1$. Clearly, $X_i\cong Y_i\cong Q_i$, as $A$-modules. At the same time, the objects $$\xymatrix{X_i\ar[rr]^{(\alpha_i)|_{X_i}}&&{}^{\iota}X_i}
\qquad\text{ and }\qquad
\xymatrix{Y_i\ar[rr]^{(\alpha_i)|_{Y_i}}&&{}^{\iota}Y_i}$$ are both in $\hat{\mathcal{Q}}$ and are, clearly, indecomposable.
For each $i$ such that $Q_i\cong {}^{\iota}Q_i$, we fix an isomorphism $\beta_i:Q_i\to {}^{\iota}Q_i$ such that $(\beta_i)^2=\mathrm{id}_{Q_i}$. We claim that each indecomposable object in $\hat{\mathcal{Q}}$ is isomorphic to $(Q_i^{(\iota)},\alpha_i)$, for some $i$ such that $Q_i\not\cong {}^{\iota}Q_i$, or is isomorphic to one of the objects $$\label{eqeqnn1}
\xymatrix{Q_i\ar[rr]^{\beta_i}&&{}^{\iota}Q_i}\qquad\text{ or }\qquad
\xymatrix{Q_i\ar[rr]^{-\beta_i}&&{}^{\iota}Q_i},$$ for some $i$ such that $Q_i\cong {}^{\iota}Q_i$. Indeed, consider an indecomposable object of the form . If $M$ contains, as a direct summand, some $N\cong Q_i$ such that $Q_i\not\cong {}^{\iota}Q_i$, then $N\oplus \alpha(N)$ is a direct summand of $M$ isomorphic to $(Q_i^{(\iota)},\alpha_i)$ and hence $M$ is isomorphic to the latter module. If $M$ contains, as a direct summand, some $N\cong Q_i$ such that $Q_i\cong {}^{\iota}Q_i$, then from the previous paragraph it follows that $M$ is isomorphic to $\xymatrix{Q_i\ar[rr]^{\alpha}&&{}^{\iota}Q_i}$, for some isomorphism $\alpha$ such that $\alpha^2=\mathrm{id}_{Q_i}$. We claim that each such object is isomorphic to one from the list . From the commutative diagram $$\xymatrix{
Q_i\ar[rrrr]^{\alpha}
\ar[dd]_{\left(\begin{array}{c}\mathrm{id}_{Q_i}\\\alpha\end{array}\right)}
&&&&{}^{\iota}Q_i\ar[dd]^{\left(\begin{array}{c}\mathrm{id}_{{}^{\iota}Q_i}\\\alpha\end{array}\right)}\\\\
Q_i\oplus {}^{\iota}Q_i
\ar[rrrr]^{\left(\begin{array}{cc}0&\mathrm{id}_{{}^{\iota}Q_i}\\\mathrm{id}_{Q_i}&0\end{array}\right)}
\ar[dd]_{\left(\begin{array}{cc}\mathrm{id}_{Q_i}&\alpha\end{array}\right)}
&&&&
{}^{\iota}Q_i\oplus Q_i
\ar[dd]^{\left(\begin{array}{cc}\mathrm{id}_{{}^{\iota}Q_i}&\alpha\end{array}\right)}\\\\
Q_i\ar[rrrr]^{\alpha}&&&&{}^{\iota}Q_i
}$$ and the fact that $\mathrm{char}(\Bbbk)\neq 2$, it follows that $\xymatrix{Q_i\ar[rr]^{\alpha}&&{}^{\iota}Q_i}$ is a summand of . As we already established in the previous paragraph, has two direct summands. So, it is enough to argue that the two objects in the list are not isomorphic. The latter follows easily from the fact that $\beta$, being an automorphism of an indecomposable module, has only one eigenvalue and this eigenvalue is non-zero and hence $\beta$ cannot be conjugate to $-\beta$ whose unique eigenvalue is different.
The claim of the proposition follows.
Symmetric projective bimodules {#s8.2}
------------------------------
Let $\Bbbk$ be an algebraically closed field and $A$ a finite dimensional $\Bbbk$-algebra with a fixed automorphism $\iota:A\to A$ such that $\iota^2=\mathrm{id}_A$. We extend $\iota$ to an automorphism $\underline{\iota}$ of $A\otimes_{\Bbbk}A^{\mathrm{op}}$ component-wise and have $\underline{\iota}^2=\mathrm{id}_{A\otimes_{\Bbbk}A^{\mathrm{op}}}$. As usual, we identify $A\otimes_{\Bbbk}A^{\mathrm{op}}$-modules and $A$-$A$–bimodules.
Consider the $A$-$A$–bimodule $Q:=A\oplus \left(A\otimes_{\Bbbk}A\right)$ and the corresponding categories $\mathcal{Q}$ and $\hat{\mathcal{Q}}$. Note that $Q$ is isomorphic to its twist by $\underline{\iota}$, as the latter amounts to simultaneously twisting both the left and the right actions of $A$ on $Q$ by $\iota$. The category $\hat{\mathcal{Q}}$ is additive, $\Bbbk$-linear, idempotent split, Krull-Schmidt and has finitely many isomorphism classes of indecomposable objects by Proposition \[prop91\]. The category of $A$-$A$–bimodules has the natural structure of a tensor category given by tensor product over $A$. This structure, applied component-wise, turns $\hat{\mathcal{Q}}$ into a tensor category. We denote by ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ the strictification of $\hat{\mathcal{Q}}$ as described, for example, in [@Le Subsection 2.3]. Then ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ is a strict tensor category or, equivalently, a $2$-category with one object which we call $\mathtt{i}$.
\[prop93\] The $2$-category ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ is finitary, moreover, it is weakly fiat provided that $A$ is self-injective.
The fact that ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ is finitary follows directly from the construction and Proposition \[prop91\]. If $A$ is self-injective, then the $2$-category ${\sc\mbox{C}\hspace{1.0pt}}_A$ is weakly fiat, cf. [@MM1 Subsection 7.3]. The weak anti-automorphism lifts from ${\sc\mbox{C}\hspace{1.0pt}}_A$ to ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ in the obvious way. We claim that even adjunction morphisms can be lifted from ${\sc\mbox{C}\hspace{1.0pt}}_A$ to ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$. Indeed, let $X\in\mathrm{add}(A\oplus \big(A\otimes_{\Bbbk}A\big))$ and $\varphi:X\to A$ be a homomorphism of bimodules. Then the following diagram commutes: $$\xymatrix{
X\oplus {}^{\iota}X^{\iota}\ar[dd]_{\left(\begin{array}{c}
\varphi\\\iota\circ \varphi\end{array}\right)}
\ar[rrr]^{\left(\begin{array}{cc}
0&\mathrm{Id}_X\\\mathrm{Id}_X&0
\end{array}\right)}&&&X\oplus {}^{\iota}X^{\iota}\ar[dd]^{\left(\begin{array}{c}
\varphi\\\iota\circ \varphi\end{array}\right)}\\\\
A\ar[rrr]^{\iota}&&&A
}$$ and hence the vertical arrows give $2$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$. Taking $\varphi$ to be adjunction morphisms in ${\sc\mbox{C}\hspace{1.0pt}}_A$ gives rise, in this way, to adjunction morphisms in ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$. This implies that ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ is weakly fiat.
The novel component of this example compared to various examples which can be found in [@MM1]–[@MM6], especially to the $2$-category ${\sc\mbox{C}\hspace{1.0pt}}_A$ from [@MM1 Subsection 7.3], is the fact that [*indecomposable*]{} $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ are given, in general, by [*decomposable*]{} endofunctors of $A$-mod. This, in particular, allows us to give an alternative construction for [@Xa Example 8], see the next subsection.
An example {#s8.3}
----------
Here we use Subsection \[s8.2\] to construct an example of a fiat $2$-category which has a left cell for which the Duflo involution (see [@MM1 Proposition 17]) is not self-adjoint. The example is essentially the same as [@Xa Example 8], however, it is constructed using completely different methods.
Let $A$ be the quotient of the path algebra of the quiver $$\xymatrix{\mathtt{1}\ar@/^/[rr]^{\alpha}&&\mathtt{2}\ar@/^/[ll]^{\beta}}$$ modulo the ideal generated by the relations $\alpha\beta=\beta\alpha=0$. Let $\varepsilon_{\mathtt{1}}$ be the trivial path at $\mathtt{1}$ and $\varepsilon_{\mathtt{2}}$ be the trivial path at $\mathtt{2}$. Let $\iota$ be the automorphism of $A$ given by swapping $\varepsilon_{\mathtt{1}}$ with $\varepsilon_{\mathtt{2}}$ and $\alpha$ with $\beta$. Consider the corresponding $2$-category ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$.
From Subsection \[s8.1\] it follows that indecomposable $1$-morphisms in ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ are given by $\mathrm{F}_1:=(A,\iota)$ and $\mathrm{F}_2:=(A,-\iota)$ together with $$\mathrm{G}_1:=\left(
\big(A\varepsilon_{\mathtt{1}}\otimes_{\Bbbk}\varepsilon_{\mathtt{1}}A\big)\oplus
\big(A\varepsilon_{\mathtt{2}}\otimes_{\Bbbk}\varepsilon_{\mathtt{2}}A\big),
\left(\begin{array}{cc}0&\underline{\iota}\\\underline{\iota}&0\end{array}\right)
\right),$$ $$\mathrm{G}_2:=\left(
\big(A\varepsilon_{\mathtt{1}}\otimes_{\Bbbk}\varepsilon_{\mathtt{2}}A\big)\oplus
\big(A\varepsilon_{\mathtt{2}}\otimes_{\Bbbk}\varepsilon_{\mathtt{1}}A\big),
\left(\begin{array}{cc}0&\underline{\iota}\\\underline{\iota}&0\end{array}\right)
\right).$$ It is easy to see that we have two two-sided cells, namely, $\{\mathrm{F}_1,\mathrm{F}_2\}$ and $\{\mathrm{G}_1,\mathrm{G}_2\}$. They both are, at the same time, both left and right cells. From [@MM1 Subsection 7.3] it follows that the objects $\mathrm{G}_1$ and $\mathrm{G}_2$ are, in fact, biadjoint to each other. Therefore, ${\sc\mbox{C}\hspace{1.0pt}}_{(A,\iota)}$ is a fiat $2$-category and the Duflo involution of the left cell $\{\mathrm{G}_1,\mathrm{G}_2\}$ cannot be self-adjoint as neither $\mathrm{G}_1$ nor $\mathrm{G}_2$ is.
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T. K.: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, SWEDEN. Present address: Department of Mathematics, [Å]{}rhus University, Ny Munkegade 118, 8000 Aarhus C, DENMARK, email: [kildetoftmath.au.dk]{}
M. M.: Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matem[á]{}tica, Instituto Superior T[é]{}cnico, 1049-001 Lisboa, PORTUGAL & Departamento de Matem[á]{}tica, FCT, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, PORTUGAL, email: [mmackaayualg.pt]{}
V. M.: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, SWEDEN, email: [mazormath.uu.se]{}
J. Z.: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, SWEDEN, email: [jakob.zimmermannmath.uu.se]{}
|
---
abstract: 'Predictive Feedback Control is an easy-to-implement method to stabilize unknown unstable periodic orbits in chaotic dynamical systems. Predictive Feedback Control is severely limited because asymptotic convergence speed decreases with stronger instabilities which in turn are typical for larger target periods, rendering it harder to effectively stabilize periodic orbits of large period. Here, we study stalled chaos control, where the application of control is stalled to make use of the chaotic, uncontrolled dynamics, and introduce an adaptation paradigm to overcome this limitation and speed up convergence. This modified control scheme is not only capable of stabilizing more periodic orbits than the original Predictive Feedback Control but also speeds up convergence for typical chaotic maps, as illustrated in both theory and application. The proposed adaptation scheme provides a way to tune parameters online, yielding a broadly applicable, fast chaos control that converges reliably, even for periodic orbits of large period.'
address: |
${}^{a}$Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany\
${}^b$Bernstein Center for Computational Neuroscience (BCCN), 37077 Göttingen, Germany\
${}^c$Institute for Mathematics, Georg–August–Universität Göttingen, 37073 Göttingen, Germany\
${}^d$III. Physical Institute—Biophysics, Georg–August–Universität Göttingen, 37077 Göttingen, Germany\
${}^e$Institute for Nonlinear Dynamics, Georg–August–Universität Göttingen, 37077 Göttingen, Germany
author:
- 'Christian Bick${}^{a,b,c}$'
- 'Christoph Kolodziejski${}^{a,d}$'
- 'Marc Timme${}^{a,e}$'
bibliography:
- 'ChaosControl.bib'
title: Controlling Chaos Faster
---
> Chaos control underlies a broad range of applications across physics and beyond. To successfully use chaos control schemes in applications, different robustness and convergence properties need to be considered from a practical point of view. For instance, for control to be useful in praxis, a method does not only need to guarantee convergence to the desired state, but convergence also has to be sufficiently fast.
>
> Predictive Feedback Control provides an easy-to-implement way to realize chaos control in discrete time dynamical systems (iterated maps). However, periodic orbits of larger periods are typically highly unstable, leading to slow convergence. Here, we systematically investigate a recently introduced extension of Predictive Feedback Control obtained by stalling control and complement it with an adaptation mechanism. The stalling of control, i.e., repeated transient interruption of control, takes advantage of the uncontrolled chaotic dynamics, thereby speeding up convergence. Adaptation provides a way to tune the control parameters online to values which yield optimal speed.
>
> Specifically, we show how the efficiency of stalling control depends on both the local stability properties of the periodic orbits to be stabilized and the choice of control parameters. Furthermore, we derive conditions for stabilizability of periodic orbits in systems of higher dimensions. In addition to speeding up convergence, the gradient adaptation scheme presented also further increases the overall convergence reliability. Hence, Adaptive Stalled Predictive Feedback Control yields an easy-to-implement, noninvasive, fast, and reliable chaos control method for a broad scope of applications.
Introduction
============
Typically, chaotic attractors contain infinitely many unstable periodic orbits [@Katok1995]. The goal of chaos control is to render these orbits stable. After first being introduced in the seminal work by Ott, Grebogi, and Yorke [@Ott1990] about two decades ago, it has not only been hypothesized to be a mechanism exploited in biological neural networks [@Rabinovich1998] but it has found its way into many applications [@Scholl2007; @Garfinkel1992] including chaotic lasers, stabilization of cardiac rhythms, and more recently into the control of autonomous robots [@Steingrube2010].
Predictive Feedback Control (PFC) [@DeSousaVieira1996; @Polyak2005] is well suited for applications: little to no prior knowledge about the system is required, it is non-invasive, i.e., control strength vanishes upon convergence, and it is very easy to implement due to the nature of the control transformation. In PFC, a prediction of the future state of the system together with the current state is fed back into the system as a control signal, similar to time-delayed feedback control [@Pyragas1992]. In fact, it can be viewed as a special case of a recent effort to determine all unstable periodic points of a discrete time dynamical system [@Schmelcher1997; @Schmelcher1998] which has been studied and extended [@Pingel2000; @Crofts2006; @Doyon2002; @Davidchack1999] for its original purpose.
In any real world application not only the existence of parameters that lead to stabilization, but also the speed of convergence is of importance. Speed is crucial, for example, if a robot is controlled by stabilizing periodic orbits in a chaotic attractor [@Steingrube2010], since the time it needs to react to a changing environment is bounded by the time the system needs to converge to a periodic orbit of a given period. In most of the literature, however, speed of convergence has been overlooked. Stabilizing periodic orbits of higher periods becomes quite a challenge; due to the increasing instability of the orbits, the PFC method yields only poor performance in terms of asymptotic convergence speed even when the control parameter is chosen optimally. Any method optimizing speed within the PFC framework [@Bick2010b] therefore is subject to the same limitation.
In this article we investigate Stalled Predictive Feedback Control (SPFC), a recently proposed extension of Predictive Feedback Control that can overcome this “speed limit”[@Bick2012]. Here, we derive conditions for the local stability properties of periodic orbits that imply stabilizability. Furthermore, we propose an adaptation mechanism that is capable of tuning the control parameter online to reach optimal asymptotic convergence speed within the regime of convergence. The resulting adaptive SPFC method is an easy-to-implement, non-invasive, and broadly applicable chaos control method that stabilizes even periodic orbits of large periods reliably without the need to fine-tune parameter values a priori.
This article is organized as follows. In the following section, we formally introduce the PFC method, briefly discuss its limitations and present SPFC as an alternative. The third section is dedicated to an in-depth look at the SPFC method; we identify regimes in parameter space in which stabilization is successful. In the fourth section, we apply our algorithm to “typical” maps with chaotic dynamics and calculate and compare convergence speeds. Adaptive methods for the control parameter are explored in Section \[sec:Adaptation\] before giving some concluding remarks.
Preliminaries {#sec:Prelim}
=============
Suppose $f: \Rn\to\Rn$ is a differentiable map such that the iteration given by the evolution equation $x_{k+1}=f(x_k)$ gives rise to a chaotic attractor $A\subset\Rn$ with a dense set of unstable periodic orbits. We refer to such a map as a *chaotic map*. Let $\FP(f)=\set{\xf\in\Rn}{f(\xf)=\xf}$ denote the set of fixed points of $f$ and $\id$ the identity map on $\Rn$. The main result of Schmelcher and Diakonos[@Schmelcher1998] reads as follows.
\[prop:PFC\] Suppose $\FPs(f)\subset\FP(f)$ is the set of fixed points such that both $\drv{f}{\xf}$ and $\drv{f}{\xf}-\id$ are nonsingular and diagonalizable (over $\C$). Then there exist finitely many orthogonal matrices $M_k\in O(\maxdim)$, $k=1, \ldots, K$, such that we have $$\FPs(f) = \bigcup_{k=1}^{K}\Cc(f, M_k)$$ where the sets $\Cc(f, M_k)$ are characterized by the the property that for $\xf\in\Cc(f, M_k)$ there exists $\mu\in(0,1)$ such that $\xf$ is a stable fixed point of the map $g_{\mu, 1}$ obtained by the transformation $S(\mu, M_k): f\mapsto\id+\mu M_k(f-\id)=g_{\mu, 1}.$
Predictive Feedback Control
---------------------------
This result may be cast into a control method. Let $\N$ denote the set of natural numbers. A periodic orbit of period $p\in\N$ is a fixed point of the $p$th iterate of $f$ denoted by $$f_p:=\ite{f}{p}=\underbrace{f\circ\cdots\circ f}_{p \text{ times}},$$ and therefore we use the terms fixed point and periodic orbit interchangeably depending on what is convenient in the context. Let $\Per(f) = \bigcup_{p\in\N}\FP(f_p)$ denote the set of all periodic points of $f$. Define the set of periodic orbits of minimal period $p$ as $\FP(f, p) = \set{\xf\in\FP(f_p)}{\ite{f}{q}(\xf)\neq\xf \text{ for }q<p}$. Furthermore, we define $\FPs(f, p) = \FP(f, p)\cap \FPs(f_p)$. Predictive Feedback Control is now a consequence of Proposition \[prop:PFC\] by replacing $f$ with $f_p$.
Let $p\in\N$. For every $\xf\in\FPs_g(f, p) := \FPs(f, p)\cap\left(\Cc(f_p, \id)
\cup\Cc(f_p, -\id)\right)$ there exists a $\mu\in(-1, 1)$ such that $\xf$ is a stable fixed point of the *Predictive Feedback Control* method given by the iteration $$x_{k+1} = g_{\mu, p}(x_k+1) := f_p(x_k)+\eta(x_k-f_p(x_k))$$ with $\eta = 1-\mu$ and *control perturbation* $c_{\mu, p}(x) = \eta\left(x_k-f_p(x_k)\right)$.
The elements of $\FPs_g(f, p)$ are referred to as *PFC-stabilizable* periodic orbits of period $p$. The cardinality of the set $\FPs_g(f, p)$ depends on the chaotic map $f$ and contains roughly half of the periodic orbits of a given period in two-dimensional systems [@Schmelcher1998; @Pingel2000].
Fix $\xf\in\FPs(f, p)$. Local stability of $g_{\mu, p}$ at $\xf$ is readily computed. Let $\drv{f}{x}$ denote the total derivative of $f$ at $x$ and suppose that $\lambda_j$, $j=1, \ldots, \maxdim$. are the eigenvalues of the linearization. The derivative of $g_{\mu, p}$ at $\xf$ evaluates to $\drv{g_{\mu, p}}{x} = \id+\mu(\drv{f_p}{x}-\id).$ Hence, stability is determined by the eigenvalues of $\drv{g_{\mu, p}}{\xf}$ given by $$\label{eq:StabilityPFC}
\kappa_j(\mu) = 1+\mu(\lambda_j-1)$$ for $j=1, \ldots, \maxdim$. Hence, $\xf\in\FPs_g(f, p)$ iff there exists a $\mu_0\in(-1,1)$ such that the spectral radius $\vrho(\drv{g_{\mu_0, p}}{\xf})=\max_{j=1, \ldots, N}\abs{\kappa_j(\mu_0)}$ is smaller than one. In particular, for a two-dimensional system these are the periodic orbits of saddle type[@Pingel2000] with stable direction $\lambda_1\in(-1, 1)$ and $\lambda_2<-1$. Note that optimal convergence speed is achieved for the value of $\mu$ which corresponds to the minimal spectral radius.
Speed Limit of Predictive Feedback Control
------------------------------------------
For increasing instability, however, the optimal convergence speed becomes increasingly slow [@Bick2012]. This applies in particular to periodic orbits of larger periods as the periodic orbits become increasingly unstable on average[@Cvitanovi2010] and asymptotic convergence speed decreases. Let $\card$ denote the cardinality of a set. The slowdown of PFC can be explicitly calculated by evaluating the functions
\[eq:StabPerG\] $$\begin{aligned}
\label{eq:StabPerGFirst}
\underline{\rho}_g(p) &= 1-\min_{\xf\in\FPs_g(f,p)}\vrho_{\text{min}}^{g}(\xf),\\
\rho_g(p) &= 1-\frac{1}{\card(\FPs_g(f,p))}\sum_{\xf\in \FPs_g(f,p)}\vrho_{\text{min}}^{g}(\xf),\\
\label{eq:StabPerGLast}
\overline{\rho}_g(p) &= 1-\max_{\xf\in\FPs_g(f,p)}\vrho_{\text{min}}^{g}(\xf),\end{aligned}$$
that quantify to the best, average, and worst asymptotic convergence speed for all periodic orbits of a given period respectively.
The slowdown becomes explicit in specific examples. We evaluated these functions for a map which describes the evolution of a two-dimensional neuromodule[@Pasemann2002]. Let $l_{11}=-22, l_{12}= 5.9, l_{21}=-6.6$, and $l_{22}= 0$ and define the sigmoidal function $\s(x)=(1+\exp(-x))^{-1}$. The dynamics of the neuromodule is given by the map $f:\Rr^2\to\Rr^2$ where $$\begin{gathered}
\label{eq:Example2D}
f(x_1, x_2) = (l_{11}\s(x_1) + l_{12}\s(x_2) - 3.4,\\ l_{21}\s(x_1) + l_{22}\s(x_2) +3.8).\end{gathered}$$ The values of the functions are depicted in Figure \[fig:SrPFC\]. One can clearly see that even the lower bound on asymptotic convergence speed for the PFC method, corresponding to the smallest spectral radius as determined by $1-\underline{\rho}_g$, approaches one exponentially on average for increasing periods. This scaling of convergence speed of PFC is quite typical; other maps with chaotic attractors, such as the Hénon map exhibit a similar behavior when subject to PFC [@Bick2012].
![\[fig:SrPFC\]Best, average, and worst asymptotic convergence speed decreases as the period of periodic orbits of increases. Here, bounds on the spectral radius are plotted for the two-dimensional map .](img/stalledcontrol/gCompSD.pdf)
Stalled Predictive Feedback Chaos Control
-----------------------------------------
By making use of the uncontrolled dynamics, i.e., “stalling control”, it was recently shown that this speed limit may be overcome[@Bick2012]. Stalled Predictive Feedback Control scheme is an extension of standard Predictive Feedback Control. For a map $\psi:\Rn\to\Rn$ define the “zeroth iterate” by $\ite{\psi}{0}:=\id$.
\[def:SPFC\] Suppose that the iteration of $F:\Rn\to\Rn$ defines a dynamical system. For $M_k\in\sset{\pm\id}$ and $\mu\in\R$ let $S(\mu, M_k)(F)=\id+\mu M_k(F-\id)=:G_\mu$ denote the map obtained by applying the Predictive Feedback Control transformation; cf. Proposition \[prop:PFC\]. For parameters $\prm,\prn\in\N_0=\N\cup\{0\}$ and $\mu\in\R$, the iteration of $$\label{eq:DelayedStabDef}
H_{\mu}^{(\prm,\prn)} = \ite{\left(F\right)}{\prn}\circ\ite{\left(G_\mu\right)}{\prm}$$ is referred to as Stalled Predictive Feedback Control.
The function $H_{\mu}^{(\prm,\prn)}$ defined above stalls Predictive Feedback Control in the following sense. In the PFC method, the control signal is applied at every point in time. By iterating $H_{\mu}^{(\prm,\prn)}$ we “stall” the application of the control perturbation by adding extra evaluations of the original, uncontrolled map $F$.
Henceforth, we adopt the period-dependent notation introduced above: the uncontrolled dynamics were given by iterating $f:\Rn\to\Rn$ and the PFC transformed map is denoted by $g_{\mu,p}$. Stalled Predictive Feedback Control is given by the iteration of $$\label{eq:DelayedStab}
h_{\mu, p} = h_{\mu, p}^{(\prm,\prn)} := \ite{\left(f_p\right)}{\prn}\circ\ite{\left(g_{\mu, p}\right)}{\prm},$$ where $\prm,\prn\in\N_0$ are parameters. By definition, we have $h_{\mu, p}^{(0,1)} = f_{p}$ and we recover the original PFC method for $h_{\mu, p}^{(1,0)} = g_{\mu, p}$. In general, we will omit the superscript $(\prm,\prn)$ unless the choice is important.
Stability of Stalled Predictive Feedback Chaos Control {#sect:StallCtrlProp}
======================================================
The stability of a periodic orbit in the controlled system depends on its stability properties for the uncontrolled dynamics. In this section we derive criteria for a periodic orbit to be stabilizable for Stalled Predictive Feedback Control.
Local stability of periodic orbits for $h_{\mu, p}$ {#sec:LocStability}
---------------------------------------------------
The local stability properties of $h_{\mu, p}$ can be calculated from $f_p$ and $g_{\mu, p}$. By definition we have $\FP(f_p)\subset\FP(h_{\mu, p})$. Suppose that $\xf\in\FPs(f, p)$ and the eigenvalues of $\drv{f_p}{\xf}$ are given by $\lambda_j$ where $j=1, \dotsc, \maxdim$. Note that the eigenvectors of $\drv{g_{\mu, p}}{\xf}$ and $\drv{f_p}{\xf}$ are the same. Hence, the local stability properties of $h_{\mu, p}$ are readily computed from the $\lambda_j$ and the local stability properties of the PFC transformed map $g_{\mu, p}$ as given by . The eigenvalues of the Jacobian of $h_{\mu, p}$ at $\xf$ evaluate to $$\Lambda_j = \lambda_j^{\prn}\kappa_j(\mu)^{\prm}=
\lambda_j^{\prn}\left(1+\mu(\lambda_j-1)\right)^{\prm}$$ for $j=1, \dotsc, \maxdim$. Hence, local stability at $\xf$ is given by the spectral radius $$\sr(\drv{h_{\mu, p}}{\xf}) = \max_{j=1, \dotsc, \maxdim}\abs{\Lambda_j}.$$ If all eigenvalues are of modulus smaller than one, the fixed point $\xf$ is stable for $h_{\mu, p}$. In other words, a periodic orbit $\xf\in\FPs(f, p)$ is called *SPFC-stabilizable* if there are parameters $\prm,\prn\in\N_0$ and $\mu\in(-1,1)$ such that $$\sr\big(\drv{h_{\mu, p}^{(\prm,\prn)}}{\xf}\big)<1.$$
Let $\FPs_h(f, p)$ denote the set of SPFC-stabilizable periodic orbits and, clearly, $\FPs_g(f, p)\subset\FPs_h(f, p),$ that is, every PFC-stabilizable periodic orbit is also SPFC-stabilizable.
To compare the “performance” of Stalled Predictive Feedback Control with that of original Predictive Feedback Control we have to rescale the stability properties. Since $h_{\mu, p}^{(m,n)}$ contains $n+m$ evaluations of $f_p$ we take the $(m+n)$th root to obtain functions $$\alts{l}_j(\prm,\prn,\mu) = \abs{\lambda_j^{\prn}\left(1+\mu(\lambda_j-1)\right)^{\prm}}^{\frac{1}{\prm+\prn}},$$ where $j=1, \dotsc, \maxdim$. With the parameter $\alpha=\frac{\prn}{\prm+\prn}$ we thus obtain an equivalent set of functions $$\label{eq:SingStab}
l_j(\alpha, \mu) = \abs{\lambda_j}^\alpha\abs{\left(1+\mu(\lambda_j-1)\right)}^{1-\alpha}$$ for $j=1, \dotsc, \maxdim$ which determine the local stability properties of $h_{\mu, p}$ rescaled to a single evaluation of $f_p$. Conversely, for any rational $\alpha\in [0, 1]\cap\Q$ we obtain a pair $(\prm,\prn)$. In the following, we refer to both $\alpha$ and the pair $\prm,\prn$ as *stalling parameters*, depending what is convenient in the context. When using the stalling parameter $\alpha$, we may also write $h_{\mu, p}^\alpha$.
Rescaled local stability of Stalled Predictive Feedback Control for a given periodic orbit $\xf\in\FPs(f,p)$ of period $p$ is hence determined by the *stability function* $$\label{eq:StabilityFunction}
\vrho_\xf(\alpha, \mu) = \max_{j=1,\dotsc, \maxdim}l_j(\alpha, \mu).$$ In comparison to the original Predictive Feedback Control, Stalled Predictive Feedback Control depends on two parameters: the control parameter $\mu$ and the stalling parameter $\alpha$.
Conditions for stabilizability {#sec:Stabilizability}
------------------------------
To derive conditions for SPFC-stabilizability, consider some general properties of functions of type . Fix $w\in\Cs := \pn{\C}{0}$. Let $\So:= \set{z\in\C}{\abs{z}=1} \cong
\Rr/2\pi\Z$ denote the unit circle. We will choose a realization to describe elements of $\So$ depending on what is convenient in the context. Consider the function $L_w:\R^2\to\R$ given by $$L_w(\alpha, \mu) := \abs{w}^\alpha\abs{1+\mu(w-1)}^{1-\alpha}.$$ By definition, we have $L_w(0, 0) = 1$ and in a sufficiently small open ball $V$ around $(0, 0)$ the function $L_w$ is differentiable and the derivative is bounded away from zero. Hence, in this ball the curve defined by $$V_0 := \set{(\alpha, \mu)\in V}{L_w(\alpha, \mu)=1}$$ is a one-dimensional submanifold of $\Rr^2$. If $V$ is chosen small enough, it may be written as a disjoint union $$V = V_0 \cup V_+ \cup V_-$$ where $V_+ = \set{(\alpha, \mu)\in V}{L_w(\alpha, \mu) > 1}$ and $V_- = \set{(\alpha, \mu)\in V}{L_w(\alpha, \mu) < 1}$.
The goal is to get a linearized description close to the origin. Let $\grad$ denote the gradient and $\langle\,\cdot\,,\cdot\,\rangle$ the usual Euclidean scalar product. Define the line $$\label{eq:Halfplane}
\gamma(w) = \set{x\in\Rr^2}{\langle\grad(L_w)|_{(0, 0)}, x\rangle = 0}$$ which is tangent to $V_0$ at the origin. Let $$\Hp:=\tset{x\in\Rr^2}{\langle\grad(L_w)|_{(0, 0)}, x\rangle < 0}$$ denote one of the half planes defined by the line $\gamma(w)$. Moreover, the sets $Q_j := \big(\frac{(j-1)\pi}{2}, \frac{j\pi}{2}\big)$ for $j\in\sset{1, 2, 3, 4}$ denote the open segments of $\So$ that lie in one of the four quadrants of $\Rr^2$.
\[defn:StabTuple\] Suppose that $w\in\Cs$. The connected subset $C_w := \Hp\cap\So$ is called the domain of stability of $w$. For a tuple $\tilde w=(w_1, \dotsc, w_\maxdim)\in(\Cs)^\maxdim$ define the domain of stability to be $$\label{eq:SomOfStab}
C_{\tilde w} := \bigcap_{k=1}^{\maxdim}C_{w_k}.$$ If $C_{\tilde w}\cap
\overline{\left(Q_1\cup Q_4\right)}\neq\emptyset$ then the tuple $\tilde w$ is called [stabilizable]{}.
In a sufficiently small neighborhood $U\subset V$ of the origin, the “linearized” version of $V_-$ is given by the set $\Hp\cap U$.
\[lem:TupleStab\] If the domain of stablity $C_w$ of a tuple $w=(w_1, \dotsc, w_\maxdim)\in(\Cs)^\maxdim$ is nonempty then there exist $(\mu_0, \alpha_0)$ such that $L_{w_j}(\mu_0, \alpha_0) < 1$ for all $j=1,\dotsc,\maxdim$. If the tuple $w$ is stabilizable then then $\alpha_0$ may be chosen such that $\alpha_0\geq 0$.
Suppose that $V_-$ and $\Hp$ are defined as above. Because of continuity, for every $w\in\Cs$ there exists an open ball $B_w\subset V_-\cap\Hp$ that is tangent to the origin. If a tuple $\tilde w = (w_1, \dotsc, w_\maxdim)$ has nonempty domain of stability $C_{\tilde w}$ then $$B:=\bigcap_{j=1}^\maxdim B_{w_j}\neq \emptyset.$$ By construction, any $(\mu_0, \alpha_0)\in B$ has the desired property.
If in addition $w$ is stabilizable then the intersection $B\cap \set{(x,y)\in\R^2}{x\geq 0}$ is not empty. This proves the second assertion.
The domain of stability is determined by the gradient of $L_w$ at the origin. Let $\ln$ denote the (real) natural logarithm. We have $\grad (L_w)|_{(0, 0)} = (\ln{\abs{w}}, \Re(w)-1)$. Define $$\begin{aligned}
R_1 &:= \set{z\in \C}{\Re(z) > 1},\\
R_2 &:=\set{z\in \C}{\abs{z} < 1},\\
R_3&:=\set{z\in \C}{\abs{z} > 1\text{, }\Re(z) < 1}.\end{aligned}$$ These regions are sketched in Figure \[fig:Stabilizability\](a). If $w\in R_1$ then $\norm{\grad(L_w)|_{(0, 0)}}^{-1}\cdot\grad(L_w)|_{(0, 0)}\in Q_1$ and therefore $Q_3\subset C_w$. Similarly, if $w\in R_2$ then $Q_1\subset C_w$ and if $w\in R_3$ then $Q_2\subset C_w$ (Figure \[fig:Stabilizability\](b)–(d)). For $w$ on the boundary of the $R_k$ the gradient lies on one of the coordinate axes and we obtain similar conditions.
These observations have implications for stabilizability for a tuple $(w_1, \dotsc, w_\maxdim)$: if for any fixed $k\in\sset{1,2,3}$ all $w_j\in R_k$ for $j=1, \dotsc, \maxdim$ then the tuple is stabilizable. Furthermore, if either $w_j\in R_1 \cup R_{3}$ or $w_j\in R_2 \cup R_{3}$ for all $j=1,\dotsc,\maxdim$ then the tuple is stabilizable. For any other combination the condition of stabilizability is more difficult; in two dimensions linear dependence of the gradients tells us for $(w_1, w_2)$ with $w_1 \in R_1$ and $w_2 \in R_2$ the tuple is stabilizable iff $$\label{eq:StabCond}
\ln(\abs{w_2})\Re(w_1) \neq \ln(\abs{w_1})\Re(w_2).$$ Note that this condition is satisfied for a set of full Lebesgue measure.
\[rem:Conjugate\] Note that stabilizability is not affected by taking the complex conjugate. Hence, stabilizability of a tuple of nonzero complex numbers as defined in Definition \[defn:StabTuple\] does not change when an entry of the tuple is replaced by its complex conjugate.
For $w=0$ the function $L_0$ has a discontinuity at $\alpha = 0$. In case $\alpha>0$ we have $L_0(\alpha, \mu)=0$ and for $\alpha=0$ and $\mu\in(-1,1)$ we have $L_0(0, \mu) = 1-\mu$. Therefore, define the domain of stabilizability of zero to be $C_0=\{0, \frac{\pi}{2}\}\cup Q_1\cup Q_4$. For , stabilizability of a tuple with one component equal to zero may be reduced to stabilizability of the “reduced” tuple where the zero entry is omitted.
With the notation as above, we are now able to relate these general results to the local stability properties of a given periodic orbit.
Suppose that $\xf\in\FPs(f, p)$ is a periodic orbit of $f$ and suppose that the eigenvalues of $\drv{f_p}{\xf}$ are given by $\lambda_j$ with $j=1,\dotsc,\maxdim$. The periodic orbit is called [locally stabilizable]{} if the tuple $\lambda = (\lambda_1, \dotsc,
\lambda_{\maxdim})$ is stabilizable as a tuple, as defined in Definition \[defn:StabTuple\].
This definition links the notion of stabilizability of a tuple defined above and the local dynamics close to a periodic orbit. Recall the notion of uniform hyperbolicity [@Katok1995]. Suppose that a differentiable function $f$ defines a discrete time dynamical system on $\Rn$. We call an $f$-invariant set $A\subset\Rn$ hyperbolic if for every $x\in A$ no eigenvalue of $\drv{f}{x}$ is of absolute value one.
Suppose that the chaotic map $f:\Rn\to\Rn$ gives rise to a hyperbolic attractor and for $\xf\in\FPs(f, p)$ let $\lambda =
(\lambda_1, \dotsc, \lambda_\maxdim)$ denote the eigenvalues of $\drv{f_p}{\xf}$. If $\xf$ is locally stabilizable then $\xf$ is SPFC-stabilizable. Moreover, if the domain of stability $C_\lambda$ satisfies $$\left\{\frac{\pi}{2}, \frac{3\pi}{2}\right\}\cap C_\lambda\neq\emptyset$$ then $\xf$ is PFC-stabilizable.
If a periodic orbit $\xf$ is locally stabilizable, then tuple $\lambda$ is stabilizable. Thus, according to Lemma \[lem:TupleStab\], there are parameters $(\alpha_0, \mu_0)$ such that $L_{\lambda_j}(\alpha_0, \mu_0) < 1$ for all $j=1,\dotsc,\maxdim$ simultaneously. Recall that local stability of $h_{\mu_0, p}^{\alpha_0}$ at $\xf$ is given by $l_j(\alpha, \mu)=L_{\lambda_j}(\alpha, \mu)$ according to Equation . Therefore, local stability of a periodic orbit is equivalent to the existence of parameters $(\alpha_0, \mu_0)$ with $\alpha_0\geq 0$ and $$\sr(\drv{h_{\mu_0, p}^{\alpha_0}}{\xf})<1.$$ which proves the first statement.
If $\tsset{\frac{\pi}{2}, \frac{3\pi}{2}}\cap C_\lambda\neq\emptyset$ then there exists a parameter $\mu_0$ such that $\sr(\drv{h_{\mu_0, p}^{0}}{\xf})<1$. Since Stalled Predictive Feedback Control reduces to classical Predictive Feedback Control for a stalling parameter of $\alpha=0$, the claim follows.
The conditions derived for stabilizability of tuples translate directly into conditions on the local stability properties of a periodic orbit. For dynamics in two dimensions we obtain the following immediate consequence.
Suppose that $f:\Rr^2\to\Rr^2$ is a chaotic map where all periodic orbits $\xf\in\Per(f)$ are of saddle type with eigenvalues $\lambda_1, \lambda_2$ that satisfy condition , i.e., we have $$\ln(\abs{\lambda_2})\Re(\lambda_1) \neq \ln(\abs{\lambda_1})\Re(\lambda_2).$$ Then all periodic orbits $\xf\in\Per(f)$ are SPFC-stabilizable.
Note that the number of constraints for stabilizability grows with increasing dimension of the dynamical system. In order to determine the absolute number of periodic orbits which are stabilizable for higher dimensional systems, a more detailed knowledge about the “average” local stability properties of periodic orbits is needed.
Since the system is real, complex eigenvalues of the derivative will always come in complex conjugate pairs. According to Remark \[rem:Conjugate\] above, this actually results in an effective decrease of the number of constraints.
A Geometric Interpretation
--------------------------
The local stability considerations also explain why Stalled Predictive Feedback Control increase asymptotic convergence speed[@Bick2012]. Consider a periodic orbit $\xf$ of saddle type in a two-dimensional system where contraction along the stable direction is given by $\lambda_1\in(-1,1)$ and expansion along the unstable manifold by $\lambda_2<-1$. As discussed above, these are the PFC-stabilizable periodic orbits. Suppose that $\mu_\text{opt}>0$ is the value of the control parameter for which the spectral radius of the linearization of the PFC-transformed map $g_{\mu, p}$ takes its minimum. For $\lambda_2\ll -1$ we have $\mu_\text{opt}\approx 0$ and therefore $\kappa_1(\mu_\text{opt})\approx 1$ determines the asymptotic convergence speed of the dominating direction if the periodic orbit is stabilized. Therefore the trajectory will approach the periodic orbit along the direction corresponding to $\lambda_1$; cf. Figure \[fig:geometry\]. The slowdown of Predictive Feedback Control is caused by the fact that for highly unstable periodic orbits, the trajectories converge to the originally stable manifold along which convergence is slow in the transformed system.
Stalling PFC exploits exactly this property. First, iteration of $g_{\mu, p}$ takes the trajectory closer to the stable manifold. Second, iteration of $f_p$ leads to fast convergence along the stable manifold while diverging from the stable manifold; cf. Figure \[fig:geometry\]. Thus, asymptotic convergence speed of $h_{\mu, p}$ is increased by making use of the (increasing) stability of the stable direction. For given stalling parameters $\prm,\prn$ the optimal value of the control parameter $\mu$ is close to the zero of $\kappa_2(\mu)$. For this value, convergence to the stable direction is strongest, taking full advantage of the fast convergence given by $\lambda_{1}$ along the stable manifold of the chaotic map $f$. The question of how to chose the stalling parameters $\prm,\prn$ will be addressed in the following sections.
Convergence Speed for Chaotic Maps {#sec:Example}
==================================
In the previous section we analyzed the stability properties of the SPFC method for periodic orbits in dependence of their stability properties. The improvements due to stalling can be calculated explicitly for some “typical” two and three-dimensional chaotic maps.
With $\sr_{\text{min}}^{h}(\xf) = \inf_{\mu, \alpha}\sr_{\xf}(\alpha, \mu)$ denoting the rescaled stability of the linearization for the optimal parameter values, we calculated the functions
\[eq:StabPerH\] $$\begin{aligned}
\label{eq:StabPerHFirst}
\overline{\rho}_h(p) &= 1-\min_{\xf\in\FPs_h(f,p)}\sr_{\text{min}}^{h}(\xf),\\
\rho_h(p) &= 1-\frac{1}{\card(\FPs_h(f,p))}\sum_{\xf\in\FPs_h(f,p)}\sr_{\text{min}}^{h}(\xf),\\
\label{eq:StabPerHLast}
\underline{\rho}_h(p) &= 1-\max_{\xf\in\FPs_h(f,p)}\sr_{\text{min}}^{h}(\xf)\end{aligned}$$
numerically in the same fashion as to asses the scaling of optimal asymptotic convergence speed of Stalled Predictive Feedback Control for a given chaotic map across different periods. That is, for every periodic orbit of $f$ of minimal period $p$ we calculated the spectral radius at the optimal parameter values and then took the minimum, maximum, and mean of these values. In particular, $1-\underline{\rho}_h$ is the upper limit and $1-\overline{\rho}_h$ is the lower limit for the best asymptotic convergence speed of all SPFC-stabilizable periodic orbits of a given period $p$ rescaled to one evaluation of $f_p$.
The increase of the number of stabilizable orbits for PFC and SPFC can be quantified by looking at the fractions of stabilizable periodic orbits that are given by $$\label{eq:Stabilizability}
\nu_h(p) = \frac{\card(\FPs_h(f,p))}{\card(\FP(f,p))}\text{ and }
\nu_g(p) =\frac{\card(\FPs_g(f,p))}{\card(\FP(f,p))},$$ respectively.
Stabilizability for chaotic maps
--------------------------------
Consider the two-dimensional neuromodule discussed above and let $\xf$ be some periodic orbit. The stability function describes local stability at $\xf$; cf. Figure \[fig:2DSingle\]. The region of stability in $(\alpha, \mu)$-parameter space is bounded by the lines $l_j(\alpha, \mu) = 1$ where $j=1,2$. The intersection of the half planes defined by the lines $\eqref{eq:Halfplane}$ gives the sector $C_{\lambda}$ that describes stability around $(\alpha, \mu)=0$ where $\lambda=(\lambda_1,
\lambda_2$) are the eigenvalues of $\drv{f_p}{\xf}$; cf. Section \[sect:StallCtrlProp\]. Note that for fixed $\alpha$, the range of $\mu$ which yields stability becomes smaller for larger $\alpha$.
![\[fig:ScalingStab2D\][(color online) ]{}Stalling PFC increases optimal asymptotic convergence speed for the 2D-Neuromodule . SPFC yields period-independent asymptotic convergence speed. The shading indicates that more periodic orbits can be stabilized. The fraction of stabilizable orbits is shaded in gray; dark indicates stabilizability for both with and without stalling, light indicates stabilizability for SPFC only.](img/stalledcontrol/gTheorConv-Silke2D)
To compare the scaling of the spectral radius across periods, we plotted the functions and in Figure \[fig:ScalingStab2D\]. The original PFC method exhibits asymptotic convergence speeds that approach one exponentially for increasing period. A fit of $\underline{\rho}$, corresponding to the best asymptotic convergence speed, by a function $\phi(x) = a\exp(-bx)$ yields a slope of $b=0.1334$. By contrast, stalling the control significantly improves this scaling. We obtain values close to zero for all periods $p\in\{1, \dotsc, 20\}$ and hence period-independent asymptotic convergence speed in terms of evaluations of $f_p$. A fit with an exponential function of $\underline{\rho}_h(p)$, i.e., the worst convergence speed, yields an exponent of $b=3.8112\cdot 10^{-8}$.
Qualitatively similar results are obtained for other two-dimensional chaotic maps[@Bick2012] such as the Hénon map [@Henon1976] and the Ikeda map [@Ikeda1980] (not shown).
As an example of a three-dimensional system, we analyzed a three-dimensional extension of the Hénon map [@Baier1990] given by $$\label{eq:3DHenon}
f(x_1, x_2, x_3)=\left(a-x_2^2-bx_3, x_1, x_2\right)$$ with parameters $a=1.76, b=0.1$. Stability properties of a periodic orbit of period $p=6$ are depicted in Figure \[fig:3DSingle\].
Due to additional constraints on stabilizability, the situation is different compared to the two-dimensional example above. In our example, the periodic orbits have a two-dimensional unstable manifold. If both eigenvalues corresponding to that manifold are real, the regime of stability depends on their sign and distance. If they have opposite signs, the periodic orbit cannot be stabilized, neither with nor without stalling. In case both eigenvalues have the same sign, the situation is depicted in Figure \[fig:3DSingle\]; there is a maximal value for $\alpha$ beyond which stabilization fails. For a pair of complex conjugate eigenvalues, the stability properties depend on the quotient of the real and imaginary part; cf. Figure \[fig:PFCvSPFC\]. In particular, if the imaginary part is large, optimal asymptotic convergence speed is achieved for the PFC method, i.e., for a choice of $\prn=0$.
![\[fig:ScalingStab3D\][(color online) ]{}Stalling Predictive Feedback Control yields period-independent scaling for periodic orbits of even period for the three-dimensional Hénon generalization . Effectivity of stalling for odd periods increases with increasing period. The number of stabilizable periodic orbits roughly doubles for higher periods as indicated by the shading, cf. Figure \[fig:ScalingStab2D\].](img/stalledcontrol/gTheorConv-Henon3D)
When looking at the scaling of optimal asymptotic convergence speed across periods we have to distinguish between even and odd periods (Figure \[fig:ScalingStab3D\]). For even periods, we obtain a period-invariant scaling of both the mean and the best optimal asymptotic convergence speed similar to the two-dimensional system. While the upper bound on convergence speed will also increase to one due to the existence of periodic orbits with complex conjugate pairs, it will typically stay above the best convergence speed for the original PFC method. For odd periods, the number of periodic orbits with complex conjugate pairs of eigenvalues corresponding to the unstable directions is large. Therefore, we see the same performance as for the PFC method. Interestingly, for larger odd periods $p>10$ stalling becomes more effective at increasing optimal asymptotic convergence speed, boosting the best speed close to one.
A similar scaling behavior is present in other three-dimensional examples; period-independent scaling for even periods $p$ is observed for a three-dimensional neuromodule [@Pasemann2002] (not shown).
Convergence speed in applications {#sec:Simulation}
---------------------------------
The scaling of the spectral radius indicates only the best possible asymptotic convergence speed for Stalled Predictive Feedback Control, i.e., the speed for the linearized dynamics. We ran simulations to compare the convergence speed for the full nonlinear system with the theoretical results for the linearized dynamics. In order to approximate a real-world implementation where control is turned on at a “arbitrary point in time” initial conditions were distributed randomly on the attractor according to the chaotic dynamics.
To evaluate convergence speed of Stalled Predictive Feedback Control, we compared the speed of $g_{\mu, p}=h^{0}_{\mu, p}$ with $h^{\alpha}_{\mu, p}$ for both $\alpha = 3^{-1}$ and $\alpha=(p+1)^{-1}$. In terms of the parameters $\prm,\prn$, a value of $\alpha = 3^{-1}$ corresponds to $\prm=2$, $\prn=1$ and $\alpha=(p+1)^{-1}$ to $\prm=p$, $\prn=1$. In our implementation, convergence time is the time $T$ for the dynamics to satisfy $$\label{eq:StalledConvCrit}
\norm{x_T-\psi(x_T)}\leq\theta\ind{conv},$$ where $\psi$ is one of the functions above. Convergence was only achieved if the criterion was fulfilled before a timeout of $T\ind{timeout}=3000$ iterations. The convergence times were rescaled to evaluations of $f_p$ to make them comparable. To calculate the best theoretical convergence time, we calculated the smallest spectral radius $$\underline{\rho}^{\alpha}(p) = \min_{\xf\in\FPs_h(f,p)}
\inf_{\mu}\sr\big(\drv{h^{\alpha}_{\mu, p}}{\xf}\big)$$ for all periodic orbits of a given period $p$ with variable $\mu$ while keeping the stalling parameter $\alpha(m,n)$ fixed. By assuming $\norm{\xf-x_\tau}=\norm{\xf-x_0}\big(\underline{\rho}^{\alpha}(p)\big)^{\tau}$ for the linear system we have that for an initial separation of $\norm{\xf-x_0}=d\ind{ini}$ the convergence criterion is satisfied for $$\label{eq:TheorConvSpeed}
\tau^{\alpha}(p)=\left(\ln\!\left(\frac{\theta\ind{conv}}{d\ind{ini}}\right)-
\ln\!\left(1-\underline{\rho}^{\alpha}(p)\right)\right)\ln\!\left(\underline{\rho}^{\alpha}(p)\right)^{-1}$$ Thus, $\tau^{\alpha}(p)$ is the convergence time of the linearized system for an initial condition $x_0$ with (period-independent) initial separation $d\ind{ini}$. For the simulations presented here, we chose $\theta\ind{conv}=10^{-13}$ and $d\ind{ini}=0.1$.
The results are shown in Figure \[fig:NumSpeed\]. The errorbars depict mean and standard deviation for all $500$ runs with initial conditions given by transient iteration of random length on the attractor. The value of the control parameter $\mu$ in the numerical simulations was chosen for each period to be the optimal value that yielded at least a fraction of $0.95$ of convergent initial conditions. In other words, $\mu$ was chosen to yield the optimal speed with at least $95\%$ reliability.
As predicted by the calculation of the spectral radius, stalling PFC leads to an increase in convergence speed across all periods. A scaling of convergence times (scaling is indicated by dashed lines) which is almost period-independent as observed in the theoretical calculations cannot be achieved in our simulations. This is due to several factors. First, in contrast to the linearized dynamics, the numerical simulations take the full nonlinear system into account. This includes the influence of the transient dynamics and the increasing complexity of the phase space (the number of fixed points increases with increasing period) on convergence times. Second, in the theoretical calculations we consider only the fixed point for which convergence is fastest. However, even in our simulations, stalling improves both absolute convergence times was well as their scaling across periods compared to classical PFC. Furthermore, it increases the number of periods that can be stabilized. For some periods, only Stalled Predictive Feedback Control yields convergence within a reasonable time. The scaling of the convergence speeds is independent of whether the stalling parameter is fixed or scales with $p$. However, a period-dependent stalling parameter will generally reduce the standard deviation of the different convergence times.
Relation to earlier results {#sec:Before}
---------------------------
Stalled Predictive Feedback Control as defined in Definition \[def:SPFC\] is a proper extension of the PFC method. In fact, the iteration of $h_{\mu, 1}^{(1,1)}$ has been considered before in the context of Predictive Feedback Control when trying to overcome the odd number limitation [@Schuster1997; @Morgul2006] as well as in the context of an experimental setup where measurements are time-delayed [@Claussen2004]. These studies were only concerned with whether or not fixed points can be stabilized, completely ignoring the aspect of convergence speed. Although for systems of dimension $\maxdim<3$ stalling control increases the number of fixed points that can be stabilized; even for $\maxdim=3$ there are points that can be stabilized using PFC but not using SPFC when the stalling parameter $\alpha$ is as large as in [@Schuster1997; @Morgul2006; @Claussen2004] (Figure \[fig:PFCvSPFC\]). Hence, the introduction of an arbitrary stalling parameter is the key to both maximizing the number of fixed points subject to stabilization through PFC as well as minimizing the convergence speed.
The idea of periodically turning control on and off has been mentioned before in the literature on control theory; both “act-and-wait” control [@Insperger2007] and “intermittent” control [@Gawthrop2010] are stated for linear control problems in discrete and continuous time. At the same time, for linear control problems with many control parameters, “pole placement” techniques [@Sontag1998] are used to control the eigenvalues of the linearization. By contrast, SPFC aims at stabilizing many unstable periodic orbits of a given nonlinear system maintaining the simplicity of the simple one-parameter feedback control scheme. The situation where control is turned on at an arbitrary point in time as described above is of particular interest; here, the system is likely to be far from the linear regime. As shown above, stalling PFC improves performance even in this situation.
Stalling Predictive Feedback Control is also related to a recent application of chaos control [@Steingrube2010]. Because of implementation restraints, Steingrube [et. al. effectively]{} iterated . In some sense, this is similar to iterating $h_{\mu, p}^{(p, 1)}$, but the stability analysis is not straightforward since one has to keep track of the (changing) point on the periodic orbit to be stabilized when iterating . Moreover, both this control and SPFC are related to an effort by Polyak [@Polyak2005] to introduce a generalized PFC method, which is capable of stabilizing periodic orbits with an arbitrary small perturbation. This method, however, is limited in applicability, because the control perturbation depends on predictions of the state of the system many time steps in the future.
Adaptive Control {#sec:Adaptation}
================
In the previous sections, we showed that for an optimal choice of parameters the asymptotic convergence speed of Predictive Feedback Control can be significantly increased when stalling control. This speedup is not only of theoretical nature, but also persists in an implementation with random initial conditions. Bot how does one find the set of optimal parameter values for a given chaotic map $f$? If no a priori estimates are available, adaptation methods provide a way to tune the control parameters online for optimal convergence speed.
Here, we consider the case where the stalling parameter $\alpha$ (corresponding to some choice of $m,n$) is fixed and $\mu\geq 0$ is subject to adaptation. We explore different adaptation mechanisms and propose a hybrid gradient adaptation approach that leads to fast and highly reliable adaptation across different periods for initial conditions distributed randomly on the chaotic attractor.
Simple and gradient adaptation
------------------------------
First, recall a simple adaptation scheme[@Steingrube2010]. We assume that the period $p$ is fixed within this subsection. A suitable objective function for finding a periodic point of period $p$ is given by $$G_1(x, p) = \norm{f_p(x)-x}^2$$ for some vector norm $\enorm$ on $\Rn$. For $\mu = 0$ the map $h_{0, p}$ as defined in . reduces to some iterate of $f$ and adaptation should lead to sequences $x_k\to\xf$ and $\mu_k\to\mu^*$ with $\xf\in\FP(f_p)$ and $\sr_h(\alpha, \mu^*) < 1$. The objective function above suggests a simple adaptation rule (SiA) with $$\label{eq:Adapt}
\Delta\mu_k = \nu(p) G_1(x_k, p)$$ where $\nu(p)$ is the (possibly period-dependent) adaptation parameter and dynamics of $\mu$ given by $$\label{eq:MuChange}
\mu_0=0,\quad \mu_{k+1} = \mu_k+\Delta\mu_k.$$ This adaptation rule increases the control parameter $\mu$ monotonically. Suppose that $\xf$ is a fixed point of $f$, i.e., . If we have a converging sequence $x_k\to \xf$ as $k\to\infty$ then the sequence $\Delta\mu_k$ tends to zero. In other words, adaptation stops in the vicinity of a fixed point $\xf$ of $f_p$.
For this adaptation mechanism, the quantity $\Delta\mu_k$ is extremely easy to calculate and yields decent results in applications [@Steingrube2010]. Adaptation, however, strongly depends on the choice of the adaptation parameter $\nu(p)$. If $\nu(p)$ is too small, it will take a long time to reach a regime in which convergence takes place. On the other hand, if $\nu(p)$ is too large and the interval $M$ of possible values of $\mu$ in which convergence takes place is rather narrow, it is possible that $\mu_k > \sup M$ for some $k$, even if $\mu_{l}\in M$ for some values $l<k$. Hence, it is possible for the control parameter to “jump out of” the range of stability. Also, note that by construction, this simple adaptation will not optimize for asymptotic convergence speed. For small $\nu(p)$, adaptation will stop close to the boundary of the convergent regime, leading to slow asymptotic convergence speed; cf. Figure \[fig:MuDyn\] (a).
Adaptation may be improved, if the objective function takes local stability into account. For some matrix norm $\enorm$, such an objective function is given by $$G_2(x, \mu, p)=\norm{\drv{h_{\mu, p}}{x}}$$ Since any matrix norm is an upper bound for the spectral radius $\vrho(A)$ of a matrix $A$, that is $\vrho(A)\leq\norm{A}$, minimizing the norm potentially leads to increased convergence speed[@Bick2010b]. At the same time, for a generic point on the attractor, this objective function is highly nonconvex with steep slopes (Figure \[fig:GlobalOpt\]) making straightforward minimization through, for example, gradient descent [@Fradkov1998] difficult.
![\[fig:GlobalOpt\]The objective function $G_2(x, \mu, p)$ is nonconvex for a generic point $x$ on the attractor leading to a difficult optimization problem.](img/stalledcontrol/gGlobalOptimization)
We therefore propose an adaptation rule that combines aspects of simple adaptation as reviewed above and the objective function $G_2$. Let $\partial_\mu$ denote the derivative with respect to $\mu$ and define $\Theta(x) = \tanh\left((pG_1(x, p))^{-1}\right)$. Consider the modified gradient adaptation rule (GrA) given by with $$\begin{gathered}
\label{eq:AdaptNew}
\Delta\mu_k = \lambda(p)\left(G_1(x, p)\right. \\ -\left. p\tanh\left(\Theta(x_k) \partial_\mu G_2(h_{\mu, p}(x_k), \mu, p)\right)\right).\end{gathered}$$ This adaptation rule has the following properties. Far away from a period $p$ orbit $\xf\in\FP(f_p)$, i.e. for $G_1(x, p)\gg 0$, we have $\Theta(x)\approx 0$. Therefore, adaptation is dominated by the first term and leads to adaptation as given by the simple adaptation rule to increase $\mu$ to reach a regime of convergence. On the other hand, in the vicinity of a fixed point we have $\Theta(x)\approx 1$ and $G_1(x, p)\approx 0$. Hence, adaptation occurs by bounded gradient descent and the dynamics of the control parameter $\mu$ are perpendicular to the level sets of the objective function $G_2$ towards a (local) minimum. The bound induced by the $\tanh$ prevents large fluctuations of the objective function $G_2$ from leading to a too large change of the control parameter $\mu$.
The adaptation parameter $\nu(p)$ again determines the size of the adaptation steps. In contrast to the simple adaptation method, the modified gradient adaptation adapts bidirectionally in order to minimize both objective functions $G_1$ and $G_2$ as depicted in Figure \[fig:MuDyn\](a). Clearly, the control parameter is adapted to the regime of stability of a periodic orbit by the modified gradient adaptation and $\Delta\mu_k\to 0$ as optimal asymptotic convergence speed is achieved. Statistics for a large number of initial conditions show that the population mean $\langle\mu_k\rangle$ for many runs is already close to the optimal value after only 70 iterations; cf. Figure \[fig:MuDyn\](b).
Convergence reliability
-----------------------
To assess the performance of the adaptive Stalled Predictive Feedback Chaos Control algorithm in a real-world application we performed large scale numerical simulations for the two-dimensional neuromodule . Periodic orbits were stabilized using SPFC with the incorporation of the adaptation mechanisms given by and . The scaling of the adaptation parameter was given by $\nu(p)=\frac{\nu_0}{p}$ and for every $\nu_0$ we iterated for 500 initial conditions distributed randomly on the chaotic attractor by iterating for a transient of random length. To determine reliability, i.e., the fraction of runs where the trajectory converged to a periodic orbit of the desired period, we checked the period of the limiting periodic orbit (if any) to a threshold of $\theta=10^{-6}$.
As discussed above, the adaptation parameter $\nu_0$ influences both speed and reliability. The results for period $p=5$ are plotted in Figure \[fig:SinglePer\]. We find that Gradient Adaptation not only decreases the total number of time steps needed to fulfill the convergence criterion but it also decreases the overall variation across runs (the standard deviation is depicted as an error bar). Of particular interest for applications is the range where convergence is highly reliable. In contrast to the simple adaptation scheme, for gradient adaptation the range of adaptation parameter values leading to highly reliable convergence is broadened. On the one hand, the gradient adaptation method optimizes for convergence speed, thereby increasing the chance that the convergence criterion is fulfilled before the timeout. At the same time, the bidirectional adaptation decreases the likelihood of the control parameter leaving the regime of convergence. Gradient adaptation therefore improves both overall convergence speed while reducing its variation and increasing the reliability of control.
The improvement of reliability compared to the simple adaptation scheme can be seen across all periods; cf. Figure \[fig:Reliability\]. The broad range of adaptation parameters giving highly reliable convergence allows for the choice of an adaptation parameter $\nu_0$ that will lead to reliable convergence across different periods, effectively eliminating this parameter.
Similar results are obtained for numerical simulations for other two- as well as three-dimensional chaotic maps (not shown). These include the Hénon map[@Bick2012] and a three-dimensional neuromodule[@Pasemann2002]. Convergence speed of $\mu_k$ to the optimal parameter value can be further increased by using higher order methods, such as Newton’s method (not shown). The use of higher order methods (also with respect to comparing simple and gradient adaptation) comes with a higher absolute computational cost. For any implementation the improvement always needs to be related to the effective improvement.
Discussion
==========
In this article, we studied the effect stalling has on Predictive Feedback Control. By stalling control, the inherent speed limit of standard Predictive Feedback Control may be overcome. We highlighted that only by taking all possible stalling parameters into account, the maximum number of periodic orbits can be stabilized. The conditions on stabilizability that we derived show that stabilizability is reduced to the conditions imposed by the eigenvalues corresponding to the unstable directions. Stalling is very easy to implement and, in addition to increasing convergence speed, the resulting chaos control method is capable of stabilizing more periodic orbits. Using numerical simulations we showed that in applications where chaos control is turned on at a random point in time, convergence speed is greatly improved across all periods. Although our method was stated in terms of discrete time dynamical systems, it also applies to continuous time dynamics if discretized for example through a Poincaré map.
As examples we studied “typical” low-dimensional chaotic systems. In higher dimensions, for example when studying chaotic collective effects in networks, we expect our method to behave qualitatively similar as in the three-dimensional case, although an increase in dimension of the unstable manifold of periodic orbits places additional constraints on stabilizability. A priori estimates of the local stability properties of the periodic orbits embedded in the attractor yield an estimate of how many periodic orbits can be stabilized. This limitation could be overcome by tuning the eigenvalue corresponding to some eigenvector separately. From a mathematical point of view, a different approach would be to allow the control parameter to take complex values, turning the problem into one of complex dynamics in several complex variables [@Bick2010b]. On the other hand, the local stability property conditions provide design principles for attractors to contain many unstable periodic orbits that our Stalled Predictive Feedback Control method is capable of stabilizing. These important questions, however, are beyond the scope of the current article and will have to be addressed in further research.
Conversely, the local stability properties and the narrowing of the regime of stability for the control parameter $\mu$ while $\alpha > 0$ is fixed can actually be exploited. Different local stability properties of the unstable periodic orbits allow for stabilization of a specific set of periodic orbits. Hence, through the choice of parameters, the targeted periodic orbits can become stable periodic orbits of the dynamics.
Adaptation mechanisms not only provide a way to tune the adaptation parameter to a suitable value, but they also allow for an increase in both speed and reliability. In contrast to previously proposed adaptation [@Steingrube2010; @Lehnert2011], the proposed hybrid algorithm also adapts for optimal convergence speed. A broad range of parameters allows for a period-independent choice of adaptation parameter, hence giving a chaos control method with a set of parameters for which it stabilizes many periodic points of most periods quickly and reliably. Adaptation using the objective function also prevents the system from converging to one of the periodic orbits potentially induced by stalling control. However, as our adaptation method merely serves as a proof of concept, it still leaves room for improvement. In particular, the cap of adaptation speed through the sigmoidal function is a major source of slowdown. Moreover, adaptation could be extended to the stalling parameter $\alpha$.
Since stalling PFC increases the number of evaluations of $f_p$ needed for a single iteration of $h_{p, \mu}$, it would be desirable to extend the theory to a “fractional stalling parameter,” i.e., to allow for stalling by composing with $\ite{f}{q}$ where $q<p$. With such stalling, however, one needs to track the point of the periodic orbit, as discussed in Section \[sec:Before\], rendering the theoretical analysis more subtle.
In conclusion, Stalled Predictive Feedback Control of Chaos together with a suitable adaptation scheme is a step towards a fast, reliable, easy-to-implement, and broadly applicable chaos control method. It would be interesting to see it applied in experimental setups in the future.
Acknowledgements {#acknowledgements .unnumbered}
================
CB would like to thank Laurent Bartholdi for making this project possible. This work was supported by the Federal Ministry of Education and Research (BMBF) by grant numbers 01GQ1005A and 01GQ1005B.
|
---
abstract: |
We report the discovery of a cluster of galaxies in the field of UM425, a pair of quasars separated by 6.5. Based on this finding, we revisit the long-standing question of whether this quasar pair is a binary quasar or a wide-separation lens. Previous work has shown that both quasars are at $z=1.465$ and show broad absorption lines. No evidence for a lensing galaxy has been found between the quasars, but there were two hints of a foreground cluster: diffuse X-ray emission observed with [*Chandra*]{}, and an excess of faint galaxies observed with the [*Hubble Space Telescope*]{}. Here we show, via VLT spectroscopy, that there is a spike in the redshift histogram of galaxies at $z=0.77$. We estimate the chance of finding a random velocity structure of such significance to be about 5%, and thereby interpret the diffuse X-ray emission as originating from $z=0.77$, rather than the quasar redshift. The mass of the cluster, as estimated from either the velocity dispersion of the $z=0.77$ galaxies or the X-ray luminosity of the diffuse emission, would be consistent with the theoretical mass required for gravitational lensing. The positional offset between the X-ray centroid and the expected location of the mass centroid is $\sim$40kpc, which is not too different from offsets observed in lower redshift clusters. However, UM425 would be an unusual gravitational lens, by virtue of the absence of a bright primary lensing galaxy. Unless the mass-to-light ratio of the galaxy is at least 80 times larger than usual, the lensing hypothesis requires that the galaxy group or cluster plays a uniquely important role in producing the observed deflections.
Based on observations performed with the Very Large Telescope at the European Southern Observatory, Paranal, Chile.
author:
- 'Paul J. Green,'
- 'Leopoldo Infante,'
- 'Sebastian Lopez,'
- 'Thomas L. Aldcroft,'
- 'Joshua N. Winn,'
title: |
Discovery of a Galaxy Cluster in the Foreground of the\
Wide-Separation Quasar Pair UM425
---
Introduction {#intro}
============
The probability that a distant quasar is gravitationally lensed by an intervening potential is sensitive to the volume of the universe, so lensing statistics place interesting constraints on $\Omega_{\Lambda}$ (@turner90, but see also @keeton02 and references therein). Wide separation lensed QSOs ($\Delta\theta > 5\arcsec$) probe more massive deflectors like groups or clusters of galaxies. The properties of these dark matter-dominated halos provide strong tests for the current standard theory of structure formation involving cold dark matter (CDM). Wide-separation lensed QSOs can measure the fractional matter density $\Omega_{\Lambda}$ and the rms linear density fluctuation in spheres of 8$h^{-1}$Mpc $\sigma_8$ [@lopes04]. But the detection of wide lensed QSOs has not been easy. Only 3 confirmed examples exist with $\Delta\theta > 5\arcsec$: Q0957+561 (6.3 at $z$=1.41; @walsh79), RX J0921+4529 (6.9 at $z$=1.65; @munoz01), and the record-setting quadruple-image quasar SDSSJ1004+4112 (14.6; @inada03 [@oguri04]).
Wide quasar pairs at similar redshifts are under intense study as possible wide lenses and have been hunted in large surveys like the 2dF [@miller04] and SDSS (@inada03 [@fukugita04]). The lensed quasar candidates are selected to have very similar redshifts, colors and optical/UV spectra. If no obvious lensing galaxy is seen between the quasar constituents, deciding whether such pairs are lensed can be tricky. Such a case was probed recently by @faure03 for the 5 pair LBQS1429-0053. Evidence for lensing may include (1) photometric monitoring and identification of a time delay between the quasar images, (2) shear in the images of the quasar host galaxy, (3) a weak lensing signal (correlated ellipticity) in galaxies in the field, (4) X-ray diffuse emission and/or (5) an overdensity of galaxies in the field at a redshift appropriate to a candidate lensing cluster.
Even if these wide pairs are not found to be lensed, they can be very illuminating in terms of the study of AGN physics, the way that twin studies are in medicine. For instance, to warrant detailed study of such quasars as candidate lensed quasars, the pair must have nearly identical spectra, ages and environment, yet may differ greatly in luminosity. In this paper, we study the intriguing pair UM425.
UM425
=====
[UM425]{} is a pair of quasars at redshift $z=1.465$ discovered by @meylan89 in a search for lenses among anomalously bright (presumably magnified) high redshift quasars. Separated by 6.5, the 2 brightest images have nearly identical optical/UV spectra and close velocities: $\Delta
v_{A-B}=200\pm100$[ [km s]{}\^[-1]{} km s$^{-1}~$]{} from @meylan89; and $\Delta
v_{A-B}=630\pm130$[ [km s]{}\^[-1]{} km s$^{-1}~$]{} from @michalitsianos97. Both $A$ and $B$ show strong broad absorption lines (BALs), which occur in a fraction (10 – 20%) of optically-selected QSOs [@hewett03; @reichard03b]. Colors of the two images are indistinguishable from UV through near-IR.
While UM425 has long been a strong wide lens candidate, ground-based optical imaging to $R \sim 24$ reveals no obvious deflector, arcs, or arclets [@courbin95], whereas a massive lens should be present to cause the large separation.
We proposed deep (110 ksec) Chandra observations of the pair, in part because UM425A (perhaps due to lensing) is one of the brightest known BALQSOs. X-ray studies of BALQSOs address the debate on whether the BAL phenomenon is one of orientation, or perhaps accretion rate or some other intrinsic physical property (e.g., @becker00 [@green01; @gallagher02a]). Our analysis of the [UM425]{} observation (ObsId 3013 obtained on 2001-Dec-13; see @aldcroft03, AG03 hereafter) showed that the $\sim$5000 count spectrum of UM425A (the brighter component) is well-fit with a power law (photon spectral index $\Gamma=2.0$) partially covered by a hydrogen column of $3.8\times 10^{22}$ cm$^{-2}$. This slope is typical for quasars, and the heavy intrinsic absorption is expected for a BALQSO [@green01]. Assuming the same $\Gamma$ for the much fainter (30 count) spectrum of UM425B yields an obscuring column 5 times larger. This X-ray spectral difference (and the difference in $f_X/f_{opt}$ between the two images) could be accounted for in a lens scenario by differing (or varying) absorbing columns and/or dust-to-gas ratios along the two sightlines. Indeed, analysis of the Ly$\alpha$ emission line region in archival HST STIS spectra of the two components shows - despite their widely disparate $\frac{S}{N}$ - that both the emission line and absorption profiles differ (AG03). Spectral differences of this magnitude have been noted previously in [*bona fide*]{} lensed quasars (HE 2149-2745 @burud02a, SBS 1520+530 - @burud02b).
A striking discovery in our Chandra image was significant diffuse extended emission in the direction of UM425. Such emission arises from the hot gas bound in massive galaxy clusters or groups. The initial analysis by AG03 suggested that the cluster $L_X$ (and thereby mass) was probably too low on its own to explain the wide separation of UM425 in a lens scenario. A similar analysis of the X-ray data, informed by a draft of AG03, was published in a Letter by @mathur03. They strongly advocated that the cluster was at the quasar redshift, while acknowledging that further data were necessary.
AG03 also analyzed archival HST WFPC2 and NICMOS images of the field, finding no evidence for a luminous lensing galaxy. However, a 3-$\sigma$ excess of faint galaxies in the UM425 field was seen, with plausible magnitudes for a galaxy group at a redshift ($z\sim0.6$) well-positioned to lens a $z$=1.465 quasar. The X-ray and optical evidence for a plausible lensing cluster, as well as the debate over the cluster’s redshift, is what motivated the VLT observations described here. For luminosity and distance calculations, we adopt a $H_0 = 70$ kms$^{-1}$Mpc$^{-1}$, $\Omega_{\Lambda} = 0.7$, and $\Omega_{M} = 0.3$ cosmology throughout.
VLT Observations {#vlt}
================
After pre-imaging at the ESO Very Large Telescope (VLT), we obtained optical spectra of galaxies in the [UM425]{} field in multi-object spectroscopy (MOS) mode.
Imaging
-------
We obtained pre-imaging data in Bessel $V$, $R$, and $I$ filters at the VLT across a 6.7 field at a scale of 0.25$\arcsec$/pixel amid seeing of 0.8 on UT 13 March 2004. Calibration images were also obtained. We used SExtractor software [@bertin96] to derive intrumental photometry, using objects detected in the $I$-band (which had the largest number of detected objects) as a reference catalog in [ASSOC]{} mode. We use [MAG\_AUTO]{} for $I$ band total magnitudes, and for colors we use the difference between aperture magnitudes in apertures of diameter $\sim3\times$FWHM, or about 2.4 (20kpc at $z$=0.77; see § \[spectra\]). Our images are complete to $V$=24.0, $R$=23.5, and $I$=23.5 (based on a conservative limit 1mag brighter than the turnover in the galaxy counts histogram). The wide-field $I$ band image is shown in Figure \[fig\_tile\], along with small-field insets showing the immediate vicinity of UM425 in optical and X-ray bands.
For a cluster at $z$=0.77 (see § \[spectra\]), $R-I$ approximates a restframe $U-B$ color, spanning the 4000Å Balmer break. $V-I$ spans an even larger (bluer) range, so is more sensitive to any recent or on-going star formation. Figure \[fig\_ivmi\] shows a $V-I$ vs. $I$ color-magnitude diagram. Several recent papers (e.g., @delucia04) indicate that a single-burst model provides a reasonable fit to the red sequence of high-redshift clusters. We assume a formation redshift of 3, so that galaxies at $z$=0.77 are 5Gyr old, and $M^*_V\sim$–21 (e.g., @delapparent03). Using the [*HyperZ*]{} photometric redshift code of @hyperz00, the expected colors of bright $M^*$ elliptical galaxies at $z$=0.77 are in the range $V$–$I$$\sim$2.5 – 2.9, with $21<I<22$.
No strong red sequence is visible among objects within 1 of UM425. But our CMD probes only brighter galaxies: $M_V=-20$ corresponds to an observed-frame $I$-band magnitude of about 22.2 at $z$=0.77. Many of the cluster elliptical galaxies composing the sequence are expected to lie fainter or redder than our current completeness limits (shown as dashed lines in Figure \[fig\_ivmi\]). In addition, the cluster may have a high fraction of star-forming galaxies, as borne out by our spectroscopy (§ \[spectra\]). The colors of star-forming galaxies, strongly affected by the strength of the break and [ ]{} emission lines, have much larger scatter. Blue galaxies like these are candidate progenitors of nearby present-day faint red sequence ellipticals (@kodama01 [@depropris03]). However, even when galaxies with spectroscopically-identified emission lines are excluded, substantial ($\sim$0.3mag rms) scatter is typically seen in cluster red sequence colors.
Deeper imaging in these filters could better determine the number of galaxies that are likely to be cluster members, and thereby estimate an extent and optical centroid for comparison to the X-ray centroid. If the diffuse X-ray emission originates in a cluster at the redshift of UM425, the 4000Å break is close to 1$\mu$m, so that $z$ band or near-IR imaging is required to estimate cluster membership with reasonable accuracy. Space-based imaging would allow for a variety of other important measurements (see § \[conclude\] below).
Spectroscopy {#spectra}
------------
We obtained spectra of the high-redshift galaxy candidates on the nights of 10, 12, and 20 May 2004, using the Focal Reducer Spectrograph (FORS2) at the ESO VLT (U4), using the grism GRIS\_150I with filter OG590 and a slit width of 6 pixels (0.76$\arcsec$). We effectively covered the spectral range from $\lambda 6000$Å to 1$\mu$m with a spectral scale of 6.86Å/pixel, yielding a spectral resolution of FWHM$\sim$19Å. We accumulated 3.26h of exposure time with an effective seeing of about 0.75$\arcsec$. Standard reduction (bias subtraction, correction for flat field variation, cosmic ray elimination) for both the photometry and spectroscopy data, as well as rebinning to the observed wavelength for the spectra, was performed independently using both MIDAS (SL) and IRAF routines (LI). The spectral extraction pipeline we used runs under MIDAS, and simultaneously fits the spatial profiles (with a Gaussian PSF) and the sky lines (using a Levenberg-Marquardt algorithm). Pixels are variance-weighted in the fit. Cosmic rays are assigned with infinite variances, so they do not contribute.
Radial velocities were also measured independently. Results are identical to within the errors, which are themselves conservative (typically 5$\times$ the difference in redshift estimates). In Table \[ztab\], we present the average of the two measurements, and the resulting random errors. We also tabulate (observed frame) emission line equivalent widths for [ ]{}$\lambda3727$
The redshift histogram in Figure \[fig\_zhisto\] reveals a strong cluster of redshifts near $z\sim$ 0.75. We analyze all 9 velocities near the cluster redshift (those with 0.6$<z<$0.85), using the robust biweight estimator of @beers90, which includes the velocity errors and yields a redshift (similar to a median) of 0.7686$\pm$0.003. The final flux-calibrated spectra for cluster galaxies are displayed in Figure \[fig\_vltspec\]. Half the spectra show strong emission line signatures of star formation. The widest spectroscopically confirmed cluster galaxies in the field (\#8 and 33 in Figure \[fig\_vltspec\] and Table \[ztab\]) span 2Mpc projected separation (at the cosmological angular scale of 7.4kpc/arcsec).
Redshift structures in a given patch of sky trace large scale structures in the Universe, which may be found in any direction. What are the chances that a redshift spike such as we find is unrelated to the diffuse X-ray cluster emission? We cannot directly answer this question, but we can estimate the chances of randomly finding a redshift spike like this in a similar-sized region on the sky.
Models of large scale structure, including galaxy luminosity functions and galaxy evolution in the field, in filaments, and in clusters, might allow simulations of a patch of sky and hence a test for spikes. But observations of significant samples of galaxies at these faint magnitudes are both rare and recent, so that such models are as yet poorly constrained. Instead, we turn directly to the best extant deep wide-field spectroscopic data, in particular the Great Observatories Origins Deep Survey (GOODS)-North field. There are 1813 reliable spectroscopic galaxy redshifts tabulated by @wirth04 and @cowie04 in the GOODS-North field, which spans about 17$\times$10. To test whether a redshift spike as strong as the one we find is expected in a random sky direction, we restrict their full catalog to objects with GOODS $i$ magnitude $<$23, resulting in a bright subcatalog of 630 redshifts across the GOODS spectroscopic field. Due to observing constraints, the VLT spectra we obtained were constrained to a thin strip on the sky of approximately 7.5$\times$0.6(Figure \[fig\_tile\]). We sampled the bright subcatalog in strips of this size, scanning across the full GOODS field with a range of step sizes and initial strip positions. At each position, we tested whether at least 21 redshifts existed in a thin strip. If fewer than 21 were found, we moved another step. If 21 or more were found, we randomly chose 21 within each thin strip. We accumulated a histogram of redshifts within each thin strip, using the same bin width as in our UM425 histogram ($\Delta z$=0.05) and then we counted the number of redshift spikes of 9 or more galaxies. For 17,000 such thin sample trials, we found a redshift spike of 9 or more galaxies in just 3.8% of samples. Since the median $I$ mag of our VLT redshifts is about 21, we tried using a variety of magnitude-limited subsamples from the GOODS-North sample, and found that an average fraction of 6.6% of 21-member thin subsamples had spikes of 9 or larger. Since we [*only*]{} retained thin subsamples with 21 or more redshifts, we have biased the subsamples toward significant structures already, so we consider either of our quoted fractions to be conservative. We therefore consider it very unlikely that this redshift spike is a coincidence unrelated to the diffuse X-ray cluster emission.
Cluster Properties {#cluster}
==================
Velocity Dispersion {#vdisp}
-------------------
We calculate the galaxy radial velocities as $$V_r = \frac{(1+z)^2 - 1}{(1+z)^2 + 1}~c$$ and measure their RMS dispersion about the mean. Assuming a normal distribution yields $\sigma_v = 1130$[ [km s]{}\^[-1]{} km s$^{-1}~$]{} for the 9 galaxies within $\Delta z$=0.05 of the mean. A more conservative measure might remove the farthest velocity outlier as a potential interloper, \#15 at $z$=0.7464, which yields a mean (and also median) optical cluster redshift $z$=0.7688 and $\sigma_v =
670$[ [km s]{}\^[-1]{} km s$^{-1}~$]{}. The most robust estimates of scale (similar to RMS dispersion in the Gaussian case) for velocity samples of this size are from the gapper or biweight methods [@beers90], which yield 695$\pm$300 and 552$\pm$280[ [km s]{}\^[-1]{} km s$^{-1}~$]{}, respectively. With more redshifts, these various estimates would likely converge.
For lensed images, the image separation depends only on $\sigma_v$ of the lens and the ratio of the comoving distances[^1] between the lens and the source, $D_{LS}$, and the observer and the source, $D_{OS}$. In the SIS model for the lensing mass, and using $z$=0.77, the observed image separation of 6$\arcsec$.5 implies a cluster velocity dispersion of $\sigma_v=580 {\ifmmode~{\rm km~s}^{-1}\else ~km~s$^{-1}~$\fi}$ or more. The same model produces a “minimum flux redshift” of 0.6 (AG03), only 10–15% different (in the relevant angular diameter distance) from the observed mean redshift. We thus have strong evidence for an optical cluster of sufficient mass to lens the quasar pair. But are the optical and X-ray characteristics compatible?
Diffuse X-ray Analysis {#xcluster}
----------------------
Because of the bright X-ray point source coincident with UM425A, it is difficult to estimate the number of diffuse X-ray photons detected in the vicinity. After subtracting the point source as well as possible, AG03 estimated a lower limit of 51$\pm$12 counts in the cluster, from a region where the excess above background was clearly visible. Now with corroborating optical evidence for a cluster, we have re-analyzed these data more thoroughly, to better characterize the cluster X-ray emission. The somewhat complicated procedure we describe below reflects the difficulty of studying the $\sim 200$ X-ray counts from the diffuse emission in close proximity to the $\sim 3000$ quasar counts.
First, in a 0.5-2 keV image, we fit a PSF model to [UM425A]{} yielding an estimate of 3448 total counts. From the full image, we then exclude a 7pixel (3.44)[^2] region centered on [UM425A]{}, and fill the hole at the background level[^3] using the [dmfilth]{} tool in CIAO3.1. We smooth the resulting image using [ csmooth]{} (adapted for CIAO from @ebeling05) which adapts the smoothing scale to result in $\geq$2.5$\sigma$ significance above background. The task also generates an image of the smoothing scale (kernel size map). Since the PSF depends on energy, we then use our best-fit quasar spectral model (AG03) as input to ChaRT[^4] and MARX[^5] to create a simulated PSF at the position of [UM425A]{}. We excise from this simulated quasar image the same 7pixel circular region as in the cluster image, and smooth with the original smoothing scale map. Now we have an quasar image that should be an exact model of the effect of [UM425A]{} in the excised and smoothed cluster image. We thus subtract the two images. Finally, we refill the excised hole in the cluster image with a constant level determined from the same size region reflected across the axis of cluster symmetry in that same image. The regions of [UM425B]{} and the nearby galaxy (labeled “g” in Figure 1) were also excised and refilled at the background level.[^6]
From the smoothed cluster image shown in Figure \[cluster\_psfsub\_smooth\] it is clear that the peak of diffuse emission does not coincide with the center of the outer contours (which are noticeably elliptical). This indicates that the X-ray emitting gas in the cluster is not fully relaxed, and may be composite. We determine that the cluster center, based on the outer contours, is at $11^h\,23^m\,20.5^s\,+01^{\circ}\,37^{\prime}\,46\arcsec$ (J2000). The emission peak is approximately 2.6 to the northwest of that position. Approximate major and minor axes for the cluster X-ray emission are ($a,b$)=17.7,14.3 at position angle $\theta$=40deg (counter-clockwise from North). A variety of methods (different combinations of PSF subtraction, smoothing, and the use of ellipse centroids or flux peaks) all yield positions within about 4.
To determine the total flux in the cluster we applied essentially the same steps described for preparing the smoothed image, but used raw image data with no smoothing. This gives our best estimate of the true underlying cluster emission with all contribution from UM425 (A and B) and the nearby galaxy removed. Figure \[radprof\] shows the background-subtracted radial profile, as well as the accumulations of counts above background with radius. Based on the point at which the cumulative counts profile flattens out to the background level, we estimate a total of 181$\pm$25 net cluster counts within the 32 radial bin[^7].
If we assume a Raymond-Smith plasma at $z$=0.77 with a rest-frame temperature $kT= 2$ keV and abundance 0.2 solar, the unabsorbed flux is $f = 8.2\pm
1.1 \times 10^{-15}$ ergs$^{-1}$cm$^{-2}$ (0.5-2 keV). We ignore $K$-corrections, since they are small ($<20\%$) for clusters of $T\geq2$keV [@jones98]. Based on the log$N$-log$S$ of extragalactic diffuse X-ray sources [@boschin02; @moretti04], the likelihood of finding an X-ray cluster of this brightness (or brighter) within 10 of any random point on the sky is at most about 4$\times 10^{-4}$. Thus this cluster is [*somehow*]{} associated with UM425, either as the lensing mass or perhaps (see § \[hizcluster\]) as a host cluster to two (presumably unlensed) BALQSOs.
At a redshift of 0.7685 in our adopted cosmology, this spectral model and flux correspond to an X-ray cluster luminosity of $L_X$(0.1-2.4 keV)$ = 2.24\times
10^{43}$ ergs$^{-1}$, consistent with a typical cluster. This luminosity differs from the earlier estimate of AG03 for two reasons. They measured a smaller region, resulting in a [*lower limit* ]{} on the cluster counts that was a factor 3.5 smaller than our more detailed analysis here. Furthermore, they were forced to assume a redshift, so reasonably used the minimum flux redshift of 0.6. The revised $L_X$ estimate here is another factor of 1.8 higher, because it is now based on the measured optical cluster redshift.
From @mulchaey98, who analyzed clusters and groups together, both the luminosity and the assumed $T$=2keV temperature correspond roughly to $\sigma_v$=550[ [km s]{}\^[-1]{} km s$^{-1}~$]{}.[^8] This in turn corresponds to a mass estimate sufficient to achieve the observed splitting of UM425A/B for a singular isothermal sphere with optimal placement.
Gravitational Lensing Models {#models}
============================
Even without any quantitative analysis, it is clear that UM 425 would be an unusual gravitational lens: both the angular separation and the flux ratio of the quasars are large, and no primary lensing galaxy has been detected between the two quasars in deep optical or near-infrared images. Only three out of about 80 well-established lenses have a separation greater than 6: Q 0957+564 (Walsh, Carswell, & Weymann 1979), RX J0921+4529 (Muñoz et al. 2001) and SDSS J1004+4112 (Inada et al. 2003). In those three cases, there is a central massive lensing galaxy between the quasar images, whose gravitational deflection is supplemented by a surrounding galaxy cluster. The large flux ratio between UM 425A and B ($\approx$100 at optical wavelengths, and even larger at X-ray wavelengths) would be the largest of any known lens. Large magnification ratios between the two brightest images of a quasar lens are unexpected, because they generally require the fine-tuned placement of the source quasar near a caustic of the lensing mass distribution. In this section, we use the simplest plausible lens model to illustrate these points. With a quasar redshift of 1.465, and assuming a single lens plane at a redshift of 0.77, the critical surface density for strong lensing is $\Sigma_c = 0.6$ g cm$^{-2} =
1.6\times 10^{11}$ M$_{\odot}$ arcsec$^{-2}$.
For a singular isothermal sphere (SIS: $\rho \propto r^{-2}$), the fractional cross-section for producing systems with a magnification ratio greater than $R$ is $$\frac{\sigma(>R)}{\sigma_{\rm total}} = \frac{4R}{(1+R)^2}$$ implying that only 4% of randomly placed background sources would produce an image pair with a magnification ratio greater than 100. The actual probability of finding such a system is even lower because of magnification bias; systems with large $R$ have a total magnification of only 2, the smallest possible magnification for a multiple-image system.
The parameters in the SIS model are the Einstein radius $b$ (which sets the overall mass scale), the sky coordinates of the center of mass, and the coordinates and intrinsic flux of the unlensed quasar. With only the image separation and magnification ratio as constraints, we have exactly as many parameters as constraints. Adopting a magnification ratio of 100, the Einstein radius is $b = 3\farcs 2$ and the center of mass is located only 64 mas from component B, along the A–B line. (The magnification of each image is proportional to its distance from the lens center.) The differential time delay between the quasar images is 11 years. To estimate the line-of-sight velocity dispersion $\sigma_v$, we use the standard relation $$b = 4\pi \left(\frac{\sigma_v}{c}\right)^2 \left(\frac{D_{\rm LS}}{D_{\rm S}}\right),$$ where $D_{\rm LS}$ and $D_{\rm S}$ are the angular-diameter distances between the lens and source, and between the observer and source, respectively. The result is $\sigma_v = 540$ km s$^{-1}$. If this is interpreted as a single elliptical galaxy of normal mass-to-light ratio, the corresponding luminosity predicted from the Faber-Jackson relation is $L/L_{\star} = (\sigma_v/$220 km s$^{-1})^4 = 40$, which is ruled out by the non-detection of any galaxy along the A–B line in optical and near-infrared images. Aldcroft & Green (2003) ruled out a galaxy brighter than $0.05L_{\star}$ more than 03 from B, and $0.5L_{\star}$ even if it were coincident with B.
Thus, unless the mass-to-light ratio of the galaxy is at least 80 times larger than usual, the lensing hypothesis requires that a galaxy group or cluster plays an important role, just as it does for the other 3 known large-separation lenses. There are two indications that there is indeed a sufficiently massive foreground cluster. First, there is an overdensity of optically detected galaxies at $z=0.77$, with an estimated velocity dispersion of 695 km s$^{-1}$. Second, the diffuse X-ray emission can be naturally interpreted as a $z=0.77$ cluster with velocity dispersion of 550 km s$^{-1}$. The problems are that the centroid of the optically detected galaxies is very poorly known, and the X-ray centroid seems to be closer to A than to B (2 vs. 8).
One can ask whether it is possible that there is a low-mass and hitherto-undetected lensing galaxy in an appropriate place between the quasar images, and that the foreground cluster magnifies the image separation to its observed large value. A suitably-placed elliptical galaxy with $L/L_{\star} < 0.5$ would have a velocity dispersion of $<$185 km s$^{-1}$, and would need to be supplemented by a cluster convergence of $\Sigma_c/\Sigma_0 > 0.88$. One issue is that the domination of the cluster convergence causes the differential time delay between the quasar images to be small; in this case it is only 1.3 years. Monitoring by Courbin et al. (1995) detected a flare in UM425B. The flare is consistent with a microlensing event, but if instead was intrinsic to the quasar, the lack of similar burst in UM425A yields a 3-year upper limit to the time delay which is incompatible with the prediction from the cluster convergence above. If the cluster is given a shear of magnitude 0.1 in the direction of the major axis of the X-ray isophotes, the time delay is reduced even further to 0.8 year. The least massive SIS lens galaxy that produces a time delay greater than 3 years has $\sigma=280$ km s$^{-1}$ and a Faber-Jackson luminosity of $L/L_{\star} = 2.7$, which is ruled out.
In short, the interpretation of UM 425 as a lens seems to require that at least one of the following possibilities hold: \[1\] The quasar variability observed by Courbin et al. (1995) and used to place an upper bound on the time delay does not reflect intrinsic variability (due perhaps to microlensing as stated by those authors). \[2\] The observed quasar flux ratio does not reflect the lensing magnification ratio (due perhaps to an extreme case of differential extinction or to microlensing). \[3\] The mass distribution is unprecedently “dark.” \[4\] The X-ray centroid is $\sim$6 (44 kpc at $z=0.77$) from the true center of mass.
Relevant to the last possibility, recent measurements of clusters of galaxies show that all of the following mass centroid estimators may show significant offsets from each other: \[1\] X-ray diffuse emission centroid, \[2\] optical galaxy counts centroid, \[3\] central dominant (cD) galaxy position, and \[4\] weak lensing mass reconstruction. These offsets may be attributable to significant substructure, ongoing mergers, or more generally to the effects of unsettled local dynamical activities on the intracluster gas. Chandra and XMM-Newton have shown X-ray cluster emission centroids offset by 30-70kpc from lensing centroid estimates (e.g., @belsole05 [@clowe04; @machacek02; @jeltema01]). For UM425, the cluster X-ray centroid is rather poorly constrained due to the small number of cluster counts and contamination by the bright quasar. Even so, the required dark matter centroid offset is well within the range seen elsewhere.
X-ray Cluster Distance {#hizcluster}
======================
Could the cluster be at the redshift of [UM425]{} as suggested by @mathur03? If the diffuse emission arises at the quasar redshift, the X-ray flux for a $T$=3keV model (consistent with the newly-derived $L_X$ below) would be $f = 1.1\pm 0.2 \times
10^{-14}$ ergs$^{-1}$cm$^{-2}$ (0.5-2 keV), and the luminosity would be about 1.5$\times 10^{44}$ ergs$^{-1}$. This is not at all an unusual cluster luminosity for a massive cluster, so remains an alternate possibility. The linear scale at that redshift, 8.4kpc/arcsec yields a reasonable cluster size. A bright cD galaxy should be easily detected in our $I$ band image. However, in this high-$z$ scenario, the quasars are distinct (unlensed) objects. Either or both are likely to be hosted by massive galaxies. Assuming that either of these host galaxies is a cD progenitor (or if they were to merge as a more massive cD somewhere along the line connecting them), we again are faced with significant ($\geq$20kpc), but certainly not unprecedented offsets from the X-ray cluster centroid.
With sufficient X-ray exposure, a definitive redshift can be measured using the FeK$\alpha$ line from the cluster gas even for a distant X-ray cluster (e.g., @hashimoto04 [@rosati04]). Given the low flux of the cluster in the [UM425]{} field, Chandra exposure times to achieve an X-ray redshift would be excessive (i.e., weeks), and an XMM-Newton image would suffer greatly from contamination by the bright quasar due to XMM’s broader PSF. Worse, BALQSOs themselves may show strong FeK$\alpha$ emission [@gallagher04], which would worsen these contamination issues considerably..
Despite great interest, clusters at redshifts above unity remain elusive. Quasars at moderately high redshifts to date are unfortunately not reliable signposts of massive clusters (e.g., @donahue03), although there are rare exceptions [@aneta05]. Even at low redshift, no example of a cluster with two luminous quasars has been published. If this is in part the result of a counter-conspiracy of cosmic evolutions (that quasar space density peaks near $z$$\sim$2 before massive clusters have formed) then an X-ray luminous cluster hosting two luminous BALQSOs appears indeed unlikely.
Our discovery of an optical cluster renders considerably less probable the hypothesis of Mathur & Williams (2003) is correct that the cluster is at the same redshift as UM425. If there is also a cluster at $z$=1.465, then very deep spectroscopy, or perhaps deep ($H\sim$22) multi-band near-IR imaging could provide redshifts.
The Binary Interpretation {#binarity}
=========================
Mortlock et al. (1999), building on the work of Kochanek, Falco & Muñoz (1999), argued that a high degree of quasar spectral similarity is expected in true binary quasars with sufficient frequency to explain most of those quasar pairs suspected as lenses but still lacking detection of a lens galaxy. They further suggest that binary quasars “are only observable as such in the early stages of galactic collisions, after which the quiescent supermassive black holes orbit in the merger remnant for some time.” Clearly then, true quasar pairs are of great interest for understanding the hypothesis that interactions trigger accretion events prior to a merger (AG03).
While a redshift of 0.77 for the cluster strengthens the case for the lens interpretation, we still cannot demonstrate convincingly from the existing evidence that UM425 is lensed. Therefore, genuine binarity for UM425 is not ruled out. At least two previous quasar pair studies (@peng99 for Q1634+267 and @faure03 for LBQS1429-0053) suffered a similar dilemma. While the images show strikingly similar high-S/N spectra, deep imaging revealed no signs of a lens galaxy, so both studies’ judgment weighed toward a binary interpretation. However, neither found any evidence for a foreground cluster as we have here.
Conclusion {#conclude}
==========
UM425A was selected by its anomalous brightness as a lens candidate. UM425B was found in deep followup to have a similar redshift. Since these properties were by selection, they are not by themselves convincing evidence for lensing. More intriguing is that both components show evidence for broad absorption lines, for which the [*a priori*]{} probability is 1 – 5% in optically-selected quasars [@hewett03; @reichard03b]. Additionally, X-ray and optical evidence for a massive intervening cluster presented in this paper are compelling, because at the observed significance, the joint probability in the field of such a redshift spike ($\sim$7%) or of diffuse X-ray emission (0.04%) is extremely small.
Based on its large angular separation and flux ratio, and the absence of a massive lensing galaxy, if UM425 can be confirmed as a lens it would be an especially interesting one. There may be a particularly dark (high $M/L$ ratio) galaxy, or a large offset between the cluster mass and X-ray emission centroids. Or the images may suffer extreme differential extinction or variability.
Confirmation of lensing for UM425 is most efficiently achieved with further deep, ground-based spectroscopy of the field, combined with deep, high spatial resolution imaging (e.g., using the ACS aboard [*Hubble*]{}) to (1) better characterize cluster membership via morphology, magnitude, red sequence colors and photometric redshifts (2) centroid the optical cluster galaxies and luminosity for comparison to the X-ray centroid and as input to a lensing model, and (3) study the field galaxies for evidence of weak lensing/tangential shear.
PJG and TLA gratefully acknowledge support through NASA contract NAS8-03060 (CXC). Work by J.N.W. was supported by NASA through Hubble Fellowship grant HST-HF-01180.02-A, 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. Imaging and spectroscopy presented here are based on observations made with ESO telescopes at the Paranal Observatories under program IDs 271.A-5009(A) and 073.A-0352(B). Thanks to Tim Beers for providing the [rostat]{} biweight estimator code.
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[rllccl]{} \[ztab\]
2 & 11 23 7.8 & +01 36 19.0 & 0.8646 & 60 & 7.1 [ ]{}\
4 & 11 23 8.8 & +01 36 30.1 & 0.0309 & 46 & 21 [H$\alpha$]{}\
7 & 11 23 11.1 & +01 36 28.8 & 0.4201 & 56 &\
8 & 11 23 11.1 & +01 36 46.0 & 0.7682 & 44 & 21.1 [ ]{}, 7.1 [H$\beta$]{}, 15.1 [ ]{}\
10 & 11 23 10.4 & +01 38 3.6 & 0.4114 & 82 &\
11 & 11 23 11.8 & +01 37 36.9 & 0.7611 & 96 &\
15 & 11 23 13.9 & +01 37 59.1 & 0.7463 & 35 & 135 [ ]{}, 51 [H$\beta$]{}, 78 [ ]{}\
16 & 11 23 15.2 & +01 37 43.5 & 0.7692 & 52 & 37 [ ]{}, 14 [H$\beta$]{}, 12 [ ]{}\
17 & 11 23 15.8 & +01 37 53.0 & 0.5424 & 55 &\
18 & 11 23 18.5 & +01 37 50.5 & 0.8735 & 55 & 54 [ ]{}, 17 [H$\beta$]{}, 28 [ ]{}\
23 & 11 23 21.3 & +01 37 46.0 & 0.2470 & 45 & 45 [ ]{}, 86 [H$\alpha$]{}\
25 & 11 23 21.9 & +01 38 5.7 & 0.7663 & 55 & 34 [ ]{}, 11 [H$\beta$]{}\
26 & 11 23 22.4 & +01 38 8.7 & 0.5442 & 57 & 14 [H$\beta$]{}, 22 [ ]{}\
28 & 11 23 23.9 & +01 38 0.9 & 0.7767 & 51 &\
30 & 11 23 24.5 & +01 38 21.0 & 0.7707 & 67 & 12 [ ]{},\
31 & 11 23 25.9 & +01 38 20.5 & 0.7644 & 51 & 26 [ ]{}, 10 [H$\beta$]{}, 19: [ ]{}\
33 & 11 23 26.4 & +01 38 43.7 & 0.7723 & 62 &\
35 & 11 23 28.5 & +01 38 52.8 & 0.1426 & 53 & 44 [H$\alpha$]{}\
36 & 11 23 29.7 & +01 39 4.9 & 0.8654 & 48 & 41 [ ]{}, 17 [H$\beta$]{}\
38 & 11 23 30.7 & +01 39 30.6 & 0.8661 & 58 & 61 [ ]{}, 22 [H$\beta$]{}\
39 & 11 23 30.7 & +01 39 42.2 & 0.3298 & 34 & 11 [H$\beta$]{}, 8 [ ]{}, 77 [H$\alpha$]{}\
[^1]: We calculate angular size distances in our cosmological model using the [ANGSIZ]{} code of @kayser97.
[^2]: Based on PSF modeling we expect about 200 counts from [UM425A]{} outside a 7 pixel radius.
[^3]: Using a source-free annulus from 70–90 radius, we determined the 0.5-2keV background level to be 0.0195counts/pixel (0.047counts/arcsec$^2$).
[^4]: http://asc.harvard.edu/chart/
[^5]: http://space.mit.edu/CXC/MARX/
[^6]: Due to the small size of these regions and low surface brightness of the cluster, filling with counts from the cluster region rather than background makes no difference ($<$2counts).
[^7]: Neither adequate spatial nor spectral information is available from these diffuse counts to better constrain the total flux via spectral or $\beta$-model spatial fits alà @ettori04
[^8]: A similar estimate of $\sigma_v$=600[ [km s]{}\^[-1]{} km s$^{-1}~$]{} would be found from more recent samples (e.g., the REFLEX sample; @ortiz04 after correcting for different assumed cosmologies).
|
---
abstract: |
Phase transition into the phase with Bose-Einstein (BE) condensate in the two-band Bose-Hubbard model with the particle hopping in the excited band only is investigated. Instability connected with such a transition (which appears at excitation energies $\delta<\lvert t_0' \rvert$, where $\lvert t_0' \rvert$ is the particle hopping parameter) is considered. The re-entrant behaviour of spinodales is revealed in the hard-core boson limit in the region of positive values of chemical potential. It is found that the order of the phase transition undergoes a change in this case and becomes the first one; the re-entrant transition into the normal phase does not take place in reality. First order phase transitions also exist at negative values of $\delta$ (under the condition $\delta>\delta_{\mathrm{crit}}\approx-0.12\lvert
t_0' \rvert$). At $\mu<0$ the phase transition mostly remains to be of the second order. The behaviour of the BE-condensate order parameter is analyzed, the $(\Theta,\mu)$ and $(\lvert t_0' \rvert,\mu)$ phase diagrams are built and localizations of tricritical points are established. The conditions are found at which the separation on the normal phase and the phase with the BE condensate takes place. Bose-Hubbard model, hard-core bosons, Bose-Einstein condensation, excited band 03.75.Hh, 03.75.Lm, 64.70.Tg, 71.35.Lk, 37.10.Jk, 67.85.-d
address: 'Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine'
author:
- 'I.V. Stasyuk, O.V. Velychko'
---
Introduction
============
During the recent years Bose-Hubbard model (BHM) is proved to be a valuable tool in the theory of systems of strongly correlated particles. The model achieves a wide recognition due to a successful description of thermodynamics and dynamics of ultracold Bose atoms in optical lattices where a phase transition to the phase with the Bose-Einstein (BE) condensate (so-called Mott insulator (MI) – superfluid state (SF) transition) occurs at very low temperatures. Experimental evidence of BE condensation in optical lattices was found for the first time in works [@wrk01; @wrk02] while theoretical predictions of such an effect were given earlier [@wrk03]. Starting from the 90-ies of the past century, a series of papers were devoted to the theory of this phenomenon. Among the first key articles on the subject one should mention the work [@wrk04] where BHM was studied in the mean field approximation. The calculated therein phase diagrams demonstrate that in the simplest case (i.e., hopping of Bose particles in the presence of a single-site Hubbard repulsion) the MI-SF transition is of the second order. Moreover, it is supposed that particles reside in the ground state of local potential wells in the lattice. Forthcoming theoretical investigations in this field were performed with the use of various techniques, e.g., the random phase approximation (RPA) in the Green function method [@wrk05; @wrk06], a strong-coupling perturbation theory [@wrk07; @wrk08], the dynamical mean field theory (Bose-DMFT) [@wrk11; @wrk12] as well as quantum Monte-Carlo calculations [@wrk09; @wrk10] and other numerical methods.
The Bose-Hubbard model is also intensively used for a theoretical description of a wide range of phenomena: quantum delocalization of hydrogen atoms adsorbed on the surface of transition metals [@wrk13; @wrk14], quantum diffusion of light particles on the surface or in the bulk [@wrk15; @wrk16], thermodynamics of the impurity ion intercalation into semiconductors [@wrk17; @wrk18].
In the last mentioned applications, there is usually a restriction on the position occupation number ($n_i\leqslant1$), which corresponds to the limit of an infinite Hubbard repulsion for the considered model. Such a model of hard-core ions (where particles are described by the Pauli statistics) is also known as the fundamental one for the investigation of a wide range of problems, e.g., superconductivity due to a local electron pairing [@wrk19] or ionic hopping in ionic (superionic) conductors [@wrk20; @wrk21].
The study of a quantum delocalization or diffusion reveals an important role of excited vibrational states of particles (ions) in localized (interstitial) positions with a much higher probability of ion hopping between them [@wrk15; @wrk22; @wrk23]. A similar issue of a possible BE condensation in the excited bands in optical lattices is also considered but the condition of their sufficient occupation due to the optical pumping (see, e.g.[@wrk24]) is imposed. An orbital degeneration of the excited $p$-state is accompanied by anisotropy of hopping parameters and causes the appearance of variously polarized bands in the one-particle spectrum. Such bands correspond by convention to different sorts (so-called “flavours”) of bosons and their number correlates with the lattice dimensionality. In the framework of the necessary generalization of the Bose-Hubbard model, a possibility of the MI-SF transition to the phase with BE condensate in the pumping-induced quasi-equilibrium long-living state of the system has been established [@wrk25].
In the equilibrium case, the issue of BE condensation involving the excited states in the framework of ordinary Bose-Hubbard model was not considered in practice. The exception is the system of spin-1 bosons [@wrk26; @wrk27] where a hyperfine splitting gives rise to multiplets of local states resulting in closely-spaced excited levels. As demonstrated in [@wrk28; @wrk29], the MI-SF phase transition could be of the first order when a single-site spin interaction is of the “antiferromagnetic” type. A similar change of the phase transition order also takes place for multicomponent Bose systems in the optical lattices [@wrk30].
In the present work we consider an equilibrium thermodynamics of the Bose-Hubbard model taking into account only one nondegenerated excited state on the lattice site besides the ground one. On the one hand, such a model corresponds to 1D or strongly anisotropic (quasi-1D) optical lattice, and on the other hand, it is close to a situation that is characteristic of a system of light particles adsorbed on the metal surface. For example, the excited states of hydrogen atoms on the Ni(111) surface are sufficiently distant [@wrk22] so only the lowest one could be taken into account. We shall investigate a condition of instability of a normal state of the Bose system with respect to BE condensation considering a criterion of divergence of the susceptibility ($\chi\sim\langle\langle c_l | c_p^+
\rangle\rangle_{\omega}|_{q=0,\omega=0}$) characterizing the system response with respect to the field related to a spontaneous creation or annihilation of particles. We shall also study the behaviour of the order parameter $\langle c_0
\rangle$ ($\langle c_0^+ \rangle$) as well as the grand canonical potential in the region of the MI-SF transition and shall build relevant phase diagrams. Special attention will be paid to a change of the phase transition order and localization of tricritical points at different values of excitation energy, particle hopping parameter and temperature.
We shall limit ourselves to the hard-core boson (HCB) limit where a limitation on occupation numbers is present: no more than one particle per site regardless of the state (excited or ground) occupied by it. Thus, the single-site problem is a three-level one (contrary to the two-level ordinary HCB case). For this reason, it is convenient to use the formalism of Hubbard operators [@wrk31] (standard basis operators [@wrk32]).
Two-state Bose-Hubbard model in RPA: normal phase
=================================================
The Bose-Hubbard model is used for description of the system of Bose particles which are located in a periodic field and can reside in lattice sites. Taking into account only the ground and the first excited vibrational levels in the potential well on the site, one can express the model Hamiltonian as: $$\begin{aligned}
\hat{H} &= (\varepsilon-\mu)\sum_i b_i^+ b_i
+ (\varepsilon'-\mu)\sum_i c_i^+ c_i
+ \frac{U_b}{2}\sum_i n_i^b(n_i^b-1)
+ \frac{U_c}{2}\sum_i n_i^c(n_i^c-1)
+ U_{bc}\sum_i n_i^b n_i^c
\notag\\
&\quad + \sum_{ij} t_{ij}^b b_i^+ b_j
+ \sum_{ij} t_{ij}^c c_i^+ c_j
+ \sum_{ij} t_{ij}^{bc} (b_i^+ c_j + c_i^+ b_j),
\label{eq2-01}\end{aligned}$$ where $b_i$ and $b_i^+$ ($c_i$ and $c_i^+$) are Bose operators of annihilation and creation of particles in the ground (excited) state, $\varepsilon$ and $\varepsilon'$ are respective energies of state and $\mu$ is the chemical potential of particles. Such a Hamiltonian includes the single-site Hubbard repulsions with energies $U_b$, $U_c$ and $U_{bc}$ as well as the particle hopping between ground ($t^b$), excited ($t^c$) and different ($t^{bc}$) states. Hereinafter we assume $U_b=U_c=U_{bc}$ for simplicity.
Let us define a single-site basis $|n_i^b,n_i^c\rangle\equiv|i;n_i^b,n_i^c\rangle$ (which is formed by particle occupation numbers in the ground and in the excited states, i.e., eigenvalues of operators $n_i^b=b_i^+ b_i$ and $n_i^c=c_i^+ c_i$) as well as introduce Hubbard operators (standard basis operators) $$X_i^{n,m;n',m'} \equiv |i;n,m\rangle\langle i;n',m'|.
\label{eq2-02}$$ Annihilation and creation Bose operators may be written as $$\begin{aligned}
b_i &= \sum_n\sum_m \sqrt{n+1}\, X_i^{n,m;n+1,m},
&
b_i^+ &= \sum_n\sum_m \sqrt{n+1}\, X_i^{n+1,m;n,m};
\notag\\
c_i &= \sum_n\sum_m \sqrt{m+1}\, X_i^{n,m;n,m+1},
&
c_i^+ &= \sum_n\sum_m \sqrt{m+1}\, X_i^{n,m+1;n,m}.
\label{eq2-03}\end{aligned}$$ Corresponding occupation numbers look as follows $$\begin{aligned}
n_i^b &= \sum_n\sum_m n X_i^{n,m;n,m},
&
n_i^c &= \sum_n\sum_m m X_i^{n,m;n,m},
\label{eq2-04}\end{aligned}$$ where summation indices $n,m=0,\dotsc,\infty$ in both and formulae.
In the $X$-operator representation, the single-site part of Hamiltonian can be written as $$\begin{aligned}
\hat{H}_0 &= \sum_i\sum_n\sum_m \lambda_{nm} X_i^{n,m;n,m},
\label{eq2-05}\\
\intertext{where}
\lambda_{nm} &= n(\varepsilon-\mu)+m(\varepsilon'-\mu)
+\frac{U}{2}(n+m)(n+m-1).
\label{eq2-06}\end{aligned}$$ Terms describing an inter-site transfer in Hamiltonian are transformed in a similar way.
Our primary goal is to calculate the two-time temperature boson Green’s functions $\langle\langle b | b^+ \rangle\rangle$ and $\langle\langle c | c^+ \rangle\rangle$, which describe an excitation spectrum and make it possible to investigate the conditions of the system’s instability with respect to the spontaneous symmetry breaking and the appearance of a BE condensate. As follows from definitions $$\begin{aligned}
\langle\langle b_l | b_p^+ \rangle\rangle_{\omega}
&=
\sum_{nm}\sum_{rs} \sqrt{n+1}\,\sqrt{r+1}\,
\langle\langle
X_l^{n,m;n+1,m} | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}\,,
\notag\\
\langle\langle c_l | c_p^+ \rangle\rangle_{\omega}
&=
\sum_{nm}\sum_{rs} \sqrt{m+1}\,\sqrt{s+1}\,
\langle\langle
X_l^{n,m;n,m+1} | X_p^{r,s+1;r,s}
\rangle\rangle_{\omega}\,.
\label{eq2-07}\end{aligned}$$
We will use the equation-of-motion method for the evaluation of $X$-operator Green’s functions. For the first one, from relations one could write $$\begin{aligned}
\hbar\omega
\langle\langle
X_l^{n,m;n+1,m} | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
&=
\frac{\hbar}{2\pi}
\langle
X_l^{n,m;n,m} - X_l^{n+1,m;n+1,m}
\rangle
\delta_{lp}\delta_{nr}\delta_{ms}
\notag\\
&\quad
+
\langle\langle
[X_l^{n,m;n+1,m},\hat{H}] | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}\,.
\label{eq2-08}\end{aligned}$$ Let us write the commutators $$\begin{aligned}
[X_l^{n,m;n+1,m},\hat{H}_0] &= (\lambda_{n+1,m}-\lambda_{n,m})
X_l^{n,m;n+1,m},
\label{eq2-09}\\
\refstepcounter{equation}
[X_l^{n,m;n+1,m},b_i^+] &= \delta_{li}
\sqrt{n+1} \left(X_l^{n,m;n,m}-X_l^{n+1,m;n+1,m}\right),
\tag{\theequation{}a}\label{eq2-10a}\\
[X_l^{n,m;n+1,m},b_i] &= \delta_{li}
\left(\sqrt{n+2}\, X_l^{n,m;n+2,m}
- \sqrt{n}\, X_l^{n-1,m;n+1,m}\right),
\tag{\theequation{}b}\label{eq2-10b}\\
[X_l^{n,m;n+1,m},c_i^+] &= \delta_{li}
\left(\sqrt{m}\, X_l^{n,m;n+1,m-1}
- \sqrt{m+1}\, X_l^{n,m+1;n+1,m}\right),
\tag{\theequation{}c}\label{eq2-10c}\\
[X_l^{n,m;n+1,m},c_i] &= \delta_{li}
\left(\sqrt{m+1}\, X_l^{n,m;n+1,m+1}
- \sqrt{m}\, X_l^{n,m-1;n+1,m}\right).
\tag{\theequation{}d}\label{eq2-10d}\end{aligned}$$ The latter are originated from the commutation of an initial $X$-operator with the inter-site transfer terms of the Hamiltonian, thus producing the higher-order Green’s functions $$\langle\langle
X_l^{\dotsm} b_j | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
\, ,\quad
\langle\langle
X_l^{\dotsm} b_j^+ | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
\, ,\dotsc,
\label{eq2-11}$$ where $X_l^{\dotsm}$ stands for operators on the right-hand side of expressions –.
Decoupling of functions in the random phase approximation (RPA) is performed in the following way: $$\langle\langle
X_l^{\dotsm} b_j | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
\approx
\langle
X_l^{\dotsm}
\rangle
\langle\langle
b_j | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
+
\langle
b_j
\rangle
\langle\langle
X_l^{\dotsm} | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
\, .
\label{eq2-12}$$ In the case of the normal phase (which will be studied herein) $\langle b_j \rangle=\langle b_j^+ \rangle=0$. Thus, retaining only the averages $\langle X_l^{\dotsm}\rangle$ of diagonal $X$-operators we have $$[X_l^{n,m;n+1,m},\hat{H}]
\approx
\Delta_{nm} X_l^{n,m;n+1,m}
+
\sqrt{n+1}\, Q_{nm} \sum_j t_{lj} b_j
+
\sqrt{n+1}\, Q_{nm} \sum_j t_{lj}'' c_j
\label{eq2-13}$$ and equation can be rewritten as $$\begin{gathered}
\langle\langle
X_l^{n,m;n+1,m} | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
=
\frac{\hbar}{2\pi} \delta_{lp}\delta_{nr}\delta_{ms}
\frac{Q_{nm}}{\hbar\omega-\Delta_{nm}}
\\
+
\frac{\sqrt{n+1}\, Q_{nm}}{\hbar\omega-\Delta_{nm}}
\sum_j t_{lj}
\langle\langle
b_j | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
+
\frac{\sqrt{n+1}\, Q_{nm}}{\hbar\omega-\Delta_{nm}}
\sum_j t_{lj}''
\langle\langle
c_j | X_p^{r+1,s;r,s}
\rangle\rangle_{\omega}
\,
.
\label{eq2-14}\end{gathered}$$ The following notations are introduced $$Q_{nm} = \langle X_l^{n,m;n,m} - X_l^{n+1,m;n+1,m} \rangle,
\qquad
\Delta_{nm} = \lambda_{n+1,m} - \lambda_{n,m}\,,
\label{eq2-15}$$ for the occupation difference of adjacent levels and the related transition energies when the number of Bose particles in the ground state (with the energy $\varepsilon$) on the site increases by one.
Proceeding from $X$-operators in equation to the Bose operators $b$ and $b^+$ according to definition we obtain $$\langle\langle
b_l | b_p^+
\rangle\rangle_{\omega}
=
\frac{\hbar}{2\pi} \delta_{lp} g_0(\omega)
+
g_0(\omega)
\biggl(
\sum_j t_{lj}
\langle\langle
b_j | b_p^+
\rangle\rangle_{\omega}
+
\sum_j t_{lj}''
\langle\langle
c_j | b_p^+
\rangle\rangle_{\omega}
\biggr)
,
\label{eq2-16}$$ where the function $$g_0(\omega)
=
\sum_{nm}
\frac{Q_{nm}}{\hbar\omega-\Delta_{nm}}
(n+1)
\label{eq2-17}$$ has the meaning of the unperturbed Green’s function for bosons residing in the single-site ground state.
Equations of motion for “mixed” Green’s functions $\langle\langle c | b^+ \rangle\rangle$ are obtained in the way similar to the above described scheme. Using decoupling one can write $$[X_l^{n,m;n,m+1},\hat{H}]
\approx
\Delta_{nm}' X_l^{n,m;n,m+1}
+
\sqrt{m+1}\, Q_{nm}' \sum_j t_{lj}'' b_j
+
\sqrt{m+1}\, Q_{nm}' \sum_j t_{lj}' c_j
\,,
\label{eq2-18}$$ which results in the equation $$\langle\langle
c_l | b_p^+
\rangle\rangle_{\omega}
=
g_0'(\omega)
\biggl(
\sum_j t_{lj}''
\langle\langle
b_j | b_p^+
\rangle\rangle_{\omega}
+
\sum_j t_{lj}'
\langle\langle
c_j | b_p^+
\rangle\rangle_{\omega}
\biggr)
.
\label{eq2-19}$$ Here, similarly to and $$Q_{nm}' = \langle X_l^{n,m;n,m} - X_l^{n,m+1;n,m+1} \rangle,
\quad
\Delta_{nm}' = \lambda_{n,m+1} - \lambda_{n,m}\,,
\quad
g_0'(\omega)
=
\sum_{nm}
\frac{Q_{nm}'}{\hbar\omega-\Delta_{nm}'}
(m+1),
\label{eq2-20}$$ and the function $g_0'(\omega)$ is the unperturbed Green’s function for bosons residing in the excited state.
By means of the Fourier transform $$\langle\langle
b_l | b_p^+
\rangle\rangle_{\omega}
=
\frac{1}{N}\sum_q \mathrm{e}^{\mathrm{i}\mathbf{q}(\mathbf{R}_l-\mathbf{R}_p)}
\langle\langle
b | b^+
\rangle\rangle_{q,\omega}
\,,
\label{eq2-21}$$ one can proceed to the momentum representation obtaining a system of equations $$\begin{aligned}
\langle\langle
b | b^+
\rangle\rangle_{q,\omega}
&=
\frac{\hbar}{2\pi} g_0(\omega)
+
g_0(\omega)
t_q
\langle\langle
b | b^+
\rangle\rangle_{q,\omega}
+
g_0(\omega)
t_q''
\langle\langle
c | b^+
\rangle\rangle_{q,\omega}
\,,
\notag\\
\langle\langle
c | b^+
\rangle\rangle_{q,\omega}
&=
g_0'(\omega)
t_q''
\langle\langle
b | b^+
\rangle\rangle_{q,\omega}
+
g_0'(\omega)
t_q'
\langle\langle
c | b^+
\rangle\rangle_{q,\omega}
\,,
\label{eq2-22}\end{aligned}$$ where $t_q$, $t_q'$ and $t_q''$ stand for the Fourier transforms of hopping parameters.
A pair of equations for other Green’s functions are obtained in a similar way $$\begin{aligned}
\langle\langle
b | c^+
\rangle\rangle_{q,\omega}
&=
g_0(\omega)
t_q
\langle\langle
b | c^+
\rangle\rangle_{q,\omega}
+
g_0(\omega)
t_q''
\langle\langle
c | c^+
\rangle\rangle_{q,\omega}
\,,
\notag\\
\langle\langle
c | c^+
\rangle\rangle_{q,\omega}
&=
\frac{\hbar}{2\pi} g_0'(\omega)
+
g_0'(\omega)
t_q''
\langle\langle
b | c^+
\rangle\rangle_{q,\omega}
+
g_0'(\omega)
t_q'
\langle\langle
c | c^+
\rangle\rangle_{q,\omega}
\,.
\label{eq2-23}\end{aligned}$$
Solutions of equations and are as follows: $$\begin{aligned}
\langle\langle
b | b^+
\rangle\rangle_{q,\omega}
&=
\frac{\hbar}{2\pi} \frac{1}{D_q(\omega)}
g_0(\omega)(1-g_0'(\omega)t_q')
,
\notag\\
\langle\langle
c | c^+
\rangle\rangle_{q,\omega}
&=
\frac{\hbar}{2\pi} \frac{1}{D_q(\omega)}
g_0'(\omega)(1-g_0'(\omega)t_q)
,
\notag\\
\langle\langle
c | b^+
\rangle\rangle_{q,\omega}
&=
\frac{\hbar}{2\pi} \frac{1}{D_q(\omega)}
g_0(\omega)g_0'(\omega)t_q''
=
\langle\langle
b | c^+
\rangle\rangle_{q,\omega}
\,,
\label{eq2-24}\end{aligned}$$ where $$D_q(\omega)
=
1-g_0(\omega)t_q-g_0'(\omega)t_q'+g_0(\omega)g_0'(\omega)
\left[
t_q t_q' - (t_q'')^2
\right]
.
\label{eq2-25}$$
The equation $D_q(\omega)=0$ gives the excitation spectrum which is obtained here in the RPA. On the other hand, the divergence of boson Green’s functions at the zero values of wave vector and frequency is the criterion of instability with respect to BE condensation [@wrk05; @wrk37], thus giving the following condition $$D_{q=0}(\omega=0)=0,
\label{eq2-26}$$ which can be rewritten in the explicit form $$1-g_0(\omega)t_q-g_0'(\omega)t_q'+g_0(\omega)g_0'(\omega)
\left[
t_q t_q' - (t_q'')^2
\right]
=0
,
\label{eq2-27}$$ where $$g_0(0)
=
-\sum_{nm}
\frac{Q_{nm}}{(n+m)U-\mu}
(n+1),
\quad
g_0'(0)
=
-\sum_{nm}
\frac{Q_{nm}'}{(n+m)U+\delta-\mu}
(m+1),
\label{eq2-28}$$ and $\delta=\varepsilon'-\varepsilon$ is the excitation energy.
We should point out that divergence of the $\langle\langle b | b^+ \rangle\rangle_{0,0}$ function correlates with the appearance of the BE condensate in the ground state while at the divergence of the $\langle\langle c | c^+ \rangle\rangle_{0,0}$ function, BE condensation takes place in the excited state. In general, both condensates appear simultaneously except the case $t_q''=0$ (e.g. due to symmetry reasons) where these effects become independent and only the one type of condensate arises in the instability point.
Equation , mutually relating the chemical potential, hopping parameters and temperature, allows us to construct spinodal surfaces (or lines) in the above mentioned coordinates and to find the temperature of the phase transition to the phase with BE condensate (so-called SF phase) where such a transition is of the second order. Below, this problem (especially the issue of the phase transition order) will be investigated more in detail.
NO phase instability in HCB limit
=================================
Let us consider now a simple special case of the HCB limit when occupation numbers in the $|n,m\rangle$ state are restricted by a condition $n+m\leqslant1$. In the framework of the model, it formally means $U\to\infty$.
In this case, the model becomes a three-level one with the local energies $$\lambda_{00}=0,
\qquad
\lambda_{01}=\delta-\mu,
\qquad
\lambda_{10}=-\mu
\label{eq3-01}$$ and the following transition energies $$\Delta_{00}=-\mu,
\qquad
\Delta_{00}'=\delta-\mu.
\label{eq3-02}$$ Thus, equation can be rewritten in the form $$1
-\frac{Q_{00}}{\mu}t_0
-\frac{Q_{00}'}{\mu-\delta}t_0'
+\frac{Q_{00}Q_{00}'}{\mu(\mu-\delta)}
\left[
t_0 t_0' - (t_0'')^2
\right]
=0
,
\label{eq3-03}$$ where $$Q_{00}=
\frac{1-\mathrm{e}^{\beta\mu}}
{1+\mathrm{e}^{\beta\mu}+\mathrm{e}^{\beta(\mu-\delta)}},
\qquad
Q_{00}'=
\frac{1-\mathrm{e}^{\beta(\mu-\delta)}}
{1+\mathrm{e}^{\beta\mu}+\mathrm{e}^{\beta(\mu-\delta)}}
\label{eq3-04}$$ in the zero approximation with respect to hopping.
[i]{}[0.5]{}
{width="47.00000%"}
We take into account (according to estimations made in [@wrk15; @wrk25]) that boson wave functions in adjacent potential wells overlap in greater extent in the excited states compared to the ground ones. Accordingly, we shall put here $t_0=0$. For a centrosymmetric lattice and in the case of different parity of wave functions of ground and excited states we have also $t_0''=0$. Finally, we follow a usual convention of the BH model for optical lattices taking $t_0'<0$. In this way equation can be reduced to $$\frac{{\lvertt_0'\rvert}}{\delta-\mu}
\frac{1-\mathrm{e}^{\beta(\mu-\delta)}}
{1+\mathrm{e}^{\beta\mu}+\mathrm{e}^{\beta(\mu-\delta)}}
=1
.
\label{eq3-05}$$ Its solutions determine the stability region boundaries of the normal (NO) phase. Respective lines of spinodals are numerically calculated and presented in figure \[fig00\] (here and below the energy quantities are given in units of ${\lvertt_0'\rvert}$).
As illustrated in figure \[fig00\], at $\delta<{\lvertt_0'\rvert}$ spinodals surround an asymmetric area in the $(\Theta,\mu)$ plane which is located between the points $\mu=\delta-{\lvertt_0'\rvert}$ and $\mu=0$ of the abscissa axis. In this region, the NO phase is unstable; this is connected with the appearance of BE condensate. At $\delta<{\lvertt_0'\rvert}/2$ and $\mu>0$ the backward path of spinodal is observed and a lower temperature of the NO phase instability appears, thus suggesting a possibility of the SF phase existence in the intermediate temperature range (so-called “re-entrant transition”). However, as will be shown further, in the mentioned region a real thermodynamic behaviour is even more complicated. The order of the NO-SF transition can change to the first one and the SF-phase remains stable up to the zero temperature.
Phase diagrams in MFA
=====================
For a more detailed treatment of the NO-SF transition issue, let us study the thermodynamics of the considered system in the HCB limit, thus reducing the problem to a three-state model with the Hamiltonian $$\hat{H} = \sum_{ip} \lambda_p X_i^{pp}
+ \sum_{ij} t_{ij} X_i^{10} X_j^{01}
+ \sum_{ij} t_{ij}' X_i^{20} X_j^{02}
+ \sum_{ij} t_{ij}'' (X_i^{10} X_j^{02}+X_i^{20} X_j^{01}),
\label{eq4-01}$$ where the shorthand notations are used $$| 0 \rangle \equiv | 00 \rangle,\;
| 1 \rangle \equiv | 10 \rangle,\;
| 2 \rangle \equiv | 01 \rangle;
\qquad
\lambda_0 =\lambda_{00},\;
\lambda_1 =\lambda_{10},\;
\lambda_2 =\lambda_{01}\,.
\label{eq4-02}$$
Possibility of BE condensation will be studied in the MFA. Average values of creation (annihilation) operators for Bose particles in the ground or excited local state $$\eta = \langle X_i^{10} \rangle = \langle X_i^{01} \rangle
\;(\equiv\langle b_i \rangle),
\qquad
\xi = \langle X_i^{20} \rangle = \langle X_i^{02} \rangle
\;(\equiv\langle c_i \rangle)
\label{eq4-03}$$ play the role of order parameters for the SF-phase. Hence, the mean-field Hamiltonian is as follows: $$\begin{aligned}
\hat{H}_{\mathrm{MF}} &=
-N(t_0 \eta^2 + t_0' \xi^2 + 2 t_0'' \eta\xi)
+ \sum_{ip} \lambda_p X_i^{pp}
\notag\\
&\quad
+ \sum_{i}
\left[
t_0 \eta (X_i^{10}+X_i^{01})
+ t_0' \xi (X_i^{20}+X_i^{02})
+ t_0'' \xi (X_i^{10}+X_i^{01})
+ t_0'' \eta(X_i^{20}+X_i^{02})
\right]
.
\label{eq4-04}\end{aligned}$$ Self-consistency equations for parameters $\eta$ and $\xi$ $$\eta = Z^{-1}\operatorname{Sp}[X_i^{10}\exp(-\beta\hat{H}_{\mathrm{MF}})],
\qquad
\xi = Z^{-1}\operatorname{Sp}[X_i^{20}\exp(-\beta\hat{H}_{\mathrm{MF}})]
\label{eq4-05}$$ are equivalent to the condition of minimum of the grand canonical potential $\Omega=-\Theta \ln Z$, where $Z=\operatorname{Sp}\exp(-\beta\hat{H}_{\mathrm{MF}})$.
Limiting our consideration to the case of particle hopping only through excited states ($t_0'\ne0$, $t_0=t_0''=0$) we can diagonalize Hamiltonian by a rotation transformation $$\left(
\begin{array}{l}
| 0 \rangle \\
| 1 \rangle \\
| 2 \rangle
\end{array}
\right)
=
\left(
\begin{array}{ccc}
\cos\vartheta & 0 & -\sin\vartheta \\
0 & 1 & 0 \\
\sin\vartheta & 0 & \cos\vartheta
\end{array}
\right)
\left(
\begin{array}{l}
| \tilde{0} \rangle \\
| \tilde{1} \rangle \\
| \tilde{2} \rangle
\end{array}
\right),
\label{eq4-06}$$ where $$\cos2\vartheta =
\frac{\lambda_2-\lambda_0}{\sqrt{(\lambda_2-\lambda_0)^2+4(t_0'\xi)^2}},
\qquad
\sin2\vartheta =
\frac{2{\lvertt_0'\rvert}\xi}{\sqrt{(\lambda_2-\lambda_0)^2+4(t_0'\xi)^2}}
\label{eq4-07}$$ and $\lambda_2-\lambda_0=\delta-\mu$. In terms of operators $\widetilde{X}^{rs}=|\tilde{r}\rangle\langle\tilde{s}|$ $$\hat{H}_{\mathrm{MF}} =
N{\lvertt_0'\rvert}\xi^2+\sum_{ip}\tilde{\lambda}_p\widetilde{X}_i^{pp}.
\label{eq4-08}$$ New energies of single-site states are $$\tilde{\lambda}_{0,2} =
\frac{\delta-\mu}{2}\mp
\sqrt{\left(\frac{\delta-\mu}{2}\right)^2+(t_0'\xi)^2},
\qquad
\tilde{\lambda}_1 = -\mu.
\label{eq4-09}$$ In the new basis $$X_i^{02}+X_i^{20} =
-(\widetilde{X}_i^{22}-\widetilde{X}_i^{00})\sin2\vartheta
+(\widetilde{X}_i^{20}-\widetilde{X}_i^{02})\cos2\vartheta,
\label{eq4-10}$$ which yields after averaging $$\xi =
\frac{1}{2}
\langle\widetilde{X}_i^{00}-\widetilde{X}_i^{22}\rangle
\sin2\vartheta.
\label{eq4-11}$$ Taking into account that $\langle\widetilde{X}^{pp}\rangle=Z^{-1}\exp(-\beta\tilde{\lambda}_p)$, $Z=\sum_p\exp(-\beta\tilde{\lambda}_p)$ we come to the equation for
[i]{}[0.5]{}
{width="47.00000%"}
the order parameter $\xi$: $$\xi =
\frac{1}{Z}
\,
\frac{{\lvertt_0'\rvert}\xi}{\sqrt{(\delta-\mu)^2+4(t_0'\xi)^2}}
\left(
\mathrm{e}^{-\beta\tilde{\lambda}_0}
-
\mathrm{e}^{-\beta\tilde{\lambda}_2}
\right)
.
\label{eq4-12}$$ Solution $\xi=0$ corresponds to the NO phase. A nonzero solution describing the BE condensate is obtained from the equation $$\frac{1}{Z}
\,
\frac{{\lvertt_0'\rvert}}{\sqrt{(\delta-\mu)^2+4(t_0'\xi)^2}}
\left(
\mathrm{e}^{-\beta\tilde{\lambda}_0}
-
\mathrm{e}^{-\beta\tilde{\lambda}_2}
\right)
= 1
.
\label{eq4-13}$$ In the limit $\xi\to0$ this equation determines the line where the order parameter for the SF phase tends to zero. One can readily see that it coincides with spinodal equation thus defining the line of the second order NO-SF phase transition (when just the transition of such an order takes place).
![Low-temperature behaviour of the order parameter $\xi$ for the reduced three-level (HCB) model at zero and negative excitation energies $\delta$ and various temperatures (${\lvertt_0'\rvert}=1$).[]{data-label="fig05"}](fig04 "fig:"){width="47.00000%"}![Low-temperature behaviour of the order parameter $\xi$ for the reduced three-level (HCB) model at zero and negative excitation energies $\delta$ and various temperatures (${\lvertt_0'\rvert}=1$).[]{data-label="fig05"}](fig05 "fig:"){width="47.00000%"}
Numerical solutions of equation make it possible to study the behavior of the order parameter $\xi$ depending on chemical potential $\mu$ at various temperatures as illustrated in figure \[fig01\]. In the main, at negative values of chemical potential the parameter $\xi$ changes smoothly and the phase transition to the SF phase is of the second order. But at $\mu\gtrsim0$ and low enough temperatures, the $\xi(\mu)$ dependence has an S-like bend. In this case, the first order phase transition with an abrupt change of the parameter $\xi$ takes place. This phase transition occurs at a certain value of the chemical potential which could be calculated using the Maxwell rule or considering the minimum of the grand canonical potential $\Omega(\mu)$ as a function of the chemical potential (see below). Obviously, the point of $\xi$ nullification does not anymore correspond here to the phase transition.
Similar behaviour of the parameter $\xi$ holds even at zero excitation energy ($\delta=0$) where the first order phase transition remains for nonzero temperatures whereas at $T=0$ its order changes to the second one (figure \[fig05\]). At negative values of $\delta$ (which corresponds to inversion of $\varepsilon$ and $\varepsilon'$ levels and to hopping between ground states) the second order of the transition is preserved in the low-temperature region close to $T=0$ transforming to the first order transition at the temperature increase and recovering henceforth (figure \[fig05\]).
{width="46.00000%"}{width="48.00000%"}\
{width="49.00000%"}{width="45.00000%"}\
Changes in the NO-SF phase transition order and localization of the corresponding tricritical points are depicted in figure \[fig06\], where phase diagrams are given for various values of the excitation energy $\delta$. At temperatures lower than tricritical, spinodal lines and phase transition curves come apart as one can see comparing figures \[fig00\] and \[fig06\]. At small values of $\delta$, the discrepancy is quite significant (figure \[fig07\]). In the case of $\delta<0$, two critical points appear at a certain distance; the latter tends to zero at $\delta=\delta_{\mathrm{crit}}\approx-0.12{\lvertt_0'\rvert}$ and the first order phase transitions at a further increase of $\delta$ (figure \[fig08\]) is suppressed.
Phase diagrams in the $({\lvertt_0'\rvert},\mu)$ plane at various temperatures for $\delta>0$ are depicted in figure \[fig09\] with indication of tricritical points. In distinction to the standard two-level HCB model [@wrk38] (where the SF phase transition is of the second order) the diagrams are asymmetric. In the limit $T\to0$ for $\mu>0$ the first order transition occurs at $\mu=(\sqrt{\delta}-\sqrt{{\lvertt_0'\rvert}})^2$ (see the next section) whereas for $\mu<0$ they are of the second order on the line $\mu=\delta-{\lvertt_0'\rvert}$.
Phase separation at fixed boson concentration
=============================================
Let us consider now the thermodynamics of the model at a fixed concentration of Bose particles. We will utilize a connection between the concentration and the chemical potential of bosons which can be established using its definition in such a form $$\begin{aligned}
n &\equiv \langle n_i^b + n_i^c \rangle
= \langle X_i^{11} + X_i^{22} \rangle
\label{eq5-01}\\
\intertext{or basing on the relationship}
n &= -\frac{\partial(\Omega/N)}{\partial\mu}\,.
\label{eq5-02}\end{aligned}$$
In the first case similarly to equality one can obtain a relation $$X_i^{11} + X_i^{22}
=
\widetilde{X}_i^{11}
+\widetilde{X}_i^{00} \sin^2\vartheta
+\widetilde{X}_i^{22} \cos^2\vartheta
+(\widetilde{X}_i^{02}+\widetilde{X}_i^{20}) \sin\vartheta \cos\vartheta
\label{eq5-03}$$ which results in $$\begin{aligned}
n &= \langle \widetilde{X}_i^{11} \rangle
+ \langle \widetilde{X}_i^{00} \rangle \sin^2\vartheta
+ \langle \widetilde{X}_i^{22} \rangle \cos^2\vartheta =
\notag\\
&= \frac{1}{Z}
\left\{
\mathrm{e}^{-\beta\tilde{\lambda}_1}
+
\left[
\frac{1}{2}-
\frac{\delta-\mu}{2\sqrt{(\delta-\mu)^2+4(t_0'\xi)^2}}
\right]
\mathrm{e}^{-\beta\tilde{\lambda}_0}
+
\left[
\frac{1}{2}+
\frac{\delta-\mu}{2\sqrt{(\delta-\mu)^2+4(t_0'\xi)^2}}
\right]
\mathrm{e}^{-\beta\tilde{\lambda}_2}
\right\}.
\label{eq5-04}\end{aligned}$$ In the second case, taking into account that $$\Omega / N
=
{\lvertt_0'\rvert}\xi^2 - \Theta \ln Z,
\qquad
Z
=
\mathrm{e}^{\beta\mu}
+
\mathrm{e}^{-\beta(\delta-\mu)/2}
\cosh\beta\sqrt{\bigl(\textstyle\frac{\delta-\mu}{2}\bigr)^2+(t_0'\xi)^2}
\label{eq5-05}$$ and differentiating with respect to $\mu$, one can come to the same expression as .
There are different relationships between $n$ and $\mu$ in NO and SF phases; in the last case, a nonzero value of $\xi$ (a solution of equation ) should be substituted into expression . Order parameter $\xi$ has a jump at the first order phase transition, so a stepwise change of concentration $n$ takes place. In the $n=\mathrm{const}$ regime (at the value of $n$ in the region of step) it means a phase separation into two phases with different concentrations: the NO phase ($\xi=0$ and a larger concentration of bosons) and the SF phase ($\xi\neq0$ and their smaller concentration).
![Lines of the NO-SF phase transition and the phase separation region in the $(\Theta,n)$ plane at various excitation energies $\delta$ including the case of small, zero and negative values of $\delta$ (${\lvertt_0'\rvert}=1$).[]{data-label="fig11"}](fig10 "fig:"){width="46.00000%"}![Lines of the NO-SF phase transition and the phase separation region in the $(\Theta,n)$ plane at various excitation energies $\delta$ including the case of small, zero and negative values of $\delta$ (${\lvertt_0'\rvert}=1$).[]{data-label="fig11"}](fig11 "fig:"){width="48.00000%"}
The above described situation is illustrated in figure \[fig11\], where the numerically calculated $(\Theta,n)$ phase diagrams are presented. At $\delta>0$, phase separation region spans up to tricritical temperatures. When $\delta$ goes to zero and finally reverses its sign, the shape of the separation region changes in a peculiar way moving off abscissa axis (figure \[fig11\]). Now the phase separation begins at nonzero temperatures and vanishes at $\delta<\delta_{\mathrm{crit}}$; the line of the second order phase transition remains only. At the further increase of ${\lvert\delta\rvert}$ (in the $\mu<0$ region) the $(\Theta,n)$ diagram becomes more and more symmetric, approaching by its shape the diagram known for the usual HCB model [@wrk39] (see also [@wrk40]).
![Phase diagram with the indication of possible phases (above) and lines of the NO-SF phase transition in the $({\lvertt_0'\rvert},\mu)$ plane at various temperatures $\Theta$ (energy quantities are given in units of $\delta$).[]{data-label="fig12"}](fig12a "fig:"){width="47.00000%"}![Phase diagram with the indication of possible phases (above) and lines of the NO-SF phase transition in the $({\lvertt_0'\rvert},\mu)$ plane at various temperatures $\Theta$ (energy quantities are given in units of $\delta$).[]{data-label="fig12"}](fig12b "fig:"){width="47.00000%"}
Phase diagrams in the $({\lvertt_0'\rvert},n)$ coordinates are given in figure \[fig12\] where the regions of NO, SF and separated phases are shown at various temperatures.
The case of the zero temperature can be studied more in detail in a pure analytic way. In this limit there are three branches of order parameter $\xi$ as a function of the chemical potential (see figure \[fig01\]): $$\begin{aligned}
(1) &\colon \xi=\frac{1}{2{\lvertt_0'\rvert}}
\sqrt{{\lvertt_0'\rvert}^2-(\mu-\delta)^2},
\notag\\
(2) &\colon \xi=\sqrt{\mu\delta}/{\lvertt_0'\rvert},
\notag\\
(3) &\colon \xi=0.
\label{eq5-07}\end{aligned}$$ After elimination of $\xi$ parameter, one can obtain the grand canonical potential $\Omega$ as follows: $$\begin{aligned}
(1) &\colon \Omega/N=
\frac{(\mu-\delta+{\lvertt_0'\rvert})^2}{4{\lvertt_0'\rvert}},
\notag\\
(2) &\colon \Omega/N= (\delta/{\lvertt_0'\rvert}-1)\mu,
\notag\\
(3) &\colon \Omega/N=
\begin{cases}
0,& \mu<0,\\
-\mu,& \mu>0.
\end{cases}
\label{eq5-08}\end{aligned}$$ Differentiating expressions with respect to $\mu$ we have $$\begin{aligned}
(1) &\colon n= \frac{1}{2}+\frac{\mu-\delta}{2{\lvertt_0'\rvert}},
\notag\\
(2) &\colon n= 1-\delta/{\lvertt_0'\rvert},
\notag\\
(3) &\colon n=
\begin{cases}
0,& \mu<0,\\
1,& \mu>0.
\end{cases}
\label{eq5-09}\end{aligned}$$
At the first order phase transition from the SF phase to the NO phase, the order parameter $\xi$ jumps from branch (1) to branch (3). This occurs at the $\mu=\mu^*\equiv(\sqrt{{\lvertt_0'\rvert}}-\sqrt{\delta})^2$ value given by equality of respective grand canonical potentials $\Omega_{(1)}=\Omega_{(3)}$. Then boson system separates into SF and NO phases with concentrations of bosons: $$\begin{aligned}
n_{\mathrm{SF}}&= \frac{1}{2}+\frac{\mu^*-\delta}{2{\lvertt_0'\rvert}},
\notag\\
n_{\mathrm{NO}}&= 1.
\label{eq5-10}\end{aligned}$$
Discussion and conclusions
==========================
As was shown in this work, the transition to the SF phase (the phase with BE condensate) in the Bose-Hubbard model with two local states (the ground and excited ones) on the lattice site can be of the first order in the case, when the particle hopping takes place only in the excited band. Calculations and estimates for optical lattices give evidence of significant distinction between hopping parameters $t_0$ and $t_1$ in the ground and excited bands, respectively. It follows from estimates [@wrk25] that $t_1/t_0\approx30-50$ depending on depth $V_0$ of local potential wells (one can produce effect on $V_0$ changing the intensity of laser beams which create an optical lattice). Similar results are obtained in the studies of quantum delocalization of the adsorbed hydrogen atoms. One can see from calculations [@wrk22; @wrk23] of energy spectrum of the H-atom subsystem on the nickel surface that the ground-state band has a negligible bandwidth. At the same time, for excited bands, the bandwidth varies in the range from 15 to 45 meV (depending on the excited state symmetry and on the crystallographic orientation of metal surface), being mostly of the order of half the corresponding excitation energy $\Delta\varepsilon_{\alpha}=\varepsilon_{\alpha}-\varepsilon_0$. There are, however, the cases of strong delocalization (e.g. H on the Ni(110) surface) where the excited bands overlap, and the width of the lowest one is of the same order as $\Delta\varepsilon_{\alpha}$ [@wrk22].
The values of hopping parameters greatly increase at the decrease of $V_0$; the distance between the local energy levels becomes smaller in this case (see [@wrk33; @wrk34]). It is one of the possible ways of changing the relation between the hopping parameters and excitation energy (${\lvertt_0'\rvert}$ and $\delta$ in our model). Another possibility (discussed in [@wrk35]) is connected with an essential reduction of the energy gap between local $s$- and $p$-levels due to sufficiently strong interspecies Feshbach resonance in the presence of Fermi atoms added to the Bose system in optical lattice.
Along with investigation of BE condensation in the excited band (or bands $p_x,p_y$ ($p_x,p_y,p_z$) in two- (three-) dimensional case) on condition that certain concentration of Bose-atoms has been created in the band by optical pumping [@wrk25; @wrk34], an attempt was made in [@wrk36] to study the effect of excited bands on the physics of BE condensation in the lowest ($s$-) band (when the $s$-band hopping is taken into account). The case of finite values of the one-site interaction $U$ was considered. The possibility of the re-entrant behaviour of the MI-SF transition was claimed. However, the order of phase transition was not investigated; the consideration was restricted to the case of zero temperature. As we show in this work, re-entrant type dependence on $T$ or $\mu$ takes place only for spinodals and the return to the initial MI phase from the SF phase could be possible only in the case of the second order phase transitions. In reality, the order of phase transition changes to the first order in this region. In the HCB limit (no more than one particle per lattice site), it takes place mainly at positive values of chemical potential of particles; at $\mu<0$, the transition remains, for the most part, of the second order. The region of existence of SF phase is restricted, as a whole, to the interval $-{\lvertt_0'\rvert}<\mu<{\lvertt_0'\rvert}$, while excitation energy should obey the inequality $\delta<{\lvertt_0'\rvert}$. We have constructed the corresponding phase diagrams and established localization of tricritical points, where the order of phase transition changes. The separation on SF and NO phases at the fixed particle concentration is investigated; the conditions of the appearance of phase-separated state are analyzed.
It should be mentioned that phase diagrams in figures \[fig01\]–\[fig12\] are close by their shape to the diagrams obtained in the framework of Bose-Hubbard model for Bose atoms with spin $S=1$ in optical lattices [@wrk29]. The excited levels are formed in that case by the higher spin single-site states and corresponding interactions of the “ferromagnetic” or “antiferromagnetic” type (the Hund-rule-like splitting), while the hopping parameter is taken the same for all bands. The similarity of the mentioned diagrams points out to the fact that the role of the excited states in the change of the phase transition order in going to the phase with the BE condensate is the same in both cases. Distinction, however, consists in another genesis of the single-site spectrum. In our model, in the limiting case of HCB there are no effects connected with the level splitting due to interaction; the excited single-particle states are taken by us into account instead.
The consideration developed in this work can be extended to the systems with the close or degenerate excited local levels. Generalization of the model by adding inter-site interactions is also important. It could even make it possible to take into consideration other phases (density-modulated or supersolid) besides NO and SF ones.
We finally emphasize that the hopping parameter $t_{ij}'$ in the excited band can be positive; in particular, this concerns the $p$-bands [@wrk35]. In such a situation, the condensation takes place into states with wave vector $\vec{Q}$ on the boundary of the Brillouin zone, while the order parameters $\langle c_Q
\rangle$, $\langle c_Q^+ \rangle$ describe the modulated condensate. Since $t_Q'=-t_0'$, the results obtained in this work are also valid (with ${\lvertt_Q'\rvert}$ in place of ${\lvertt_0'\rvert}$) in that case.
[99]{}
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|
---
author:
- 'M.H.M. Morais'
- 'A.C.M. Correia'
title: 'Stellar wobble caused by a nearby binary system: eccentric and inclined orbits'
---
Introduction
============
In , we studied a triple system consisting of a star perturbed by a binary system. Our aim was to derive an expression for the star’s radial velocity in order to determine whether the binary’s effect could be mistaken for that of a planet companion to the star. This question is relevant if one or even both binary components are unresolved which can happen for instance when these are faint but very common M stars. About $76\%$ of nearby main sequence stars are M-type [@Starrysky2001]. Moreover, in the solar neighborhood, over $50\%$ of G and K-stars and about $30\%$ of M-stars belong to binary or even multiple systems.
We previously showed that the binary system’s effect on the star is a sum of two periodic signals with very close frequencies. These signals, if detected, could be associated with two planets on circular orbits around the star. However, these planets would have very close orbits and we should be able to discard them as unstable systems. Moreover, we derived expressions for the frequencies and amplitudes of these signals, thus were able to identify the hidden binary’s parameters. As explained in , our results do not agree with previous work by who studied the case of an equal-mass binary and concluded that its effect is a single periodic signal that mimics a planet on an eccentric orbit.
Nevertheless, in we saw that there are realistic situations where we can identify one of the periodic signals but not the other. In this case, we may mistake the binary’s effect for that of a planet on a circular orbit around the star. However, we can still apply our theory to compute the hidden binary’s parameters and then try to detect these objects.
Our work was based on a full three-body model but assumed coplanar motion and initial circular orbits for the star and binary. However, general triple-star systems are likely to have non-coplanar motion and eccentric orbits. Therefore, in this article, we extend our study to the case of three-dimensional non-circular motion of the star and binary system. In Sect. 2, we present the model and in Sect. 3 we discuss the circumstances that lead to the binary being mistaken by one or more planets. In Sect. 4, we compare the theoretical predictions with results obtained from simulations of hypothetical triple systems, we discuss the triple system HD 188753, and we (re)analyze the exoplanets discovered within binary systems. Finally, in Sect. 5 we present our conclusions.
Modeling a star perturbed by a binary system
============================================
Equations of motion and radial velocity
---------------------------------------
We consider the framework of the general three-body problem: we assume that a star with mass $M_{\star}$ has a nearby binary system with masses $M_{1}$ and $M_{2}$ (Fig. \[fig1\]).
![A star with mass $M_{\star}$ is perturbed by a binary system with masses $M_{1}$ and $M_{2}$. The Jacobi coordinates $\vec{r}_b$ and $\vec{r}$ are, respectively, the inter-binary distance and the distance of $M_{\star}$ to the binary’s center of mass. The unperturbed Keplerian described by $\vec{r}$ is in the plane $(x',y')$. The unperturbed Keplerian described by $\vec{r}_b$ is defined with respect to the $(x',y')$ plane by the angles $\Omega$ (relative node longitude) and $i$ (relative inclination). The observer’s frame $(x,y,z)$ with $z$ being the line of sight is obtained by rotating the auxiliary frame $(x',y',z')$ around the $x$ axis by an angle $I$ (inclination of $\vec{r}$’s orbit with respect to $(x,y)$). Moreover, $S=\angle(\vec{r},\vec{r}_ {b})$, $\theta=\angle({\hat{x}},\vec{r})$, and $\theta_{b}=\angle({\hat{x}},\vec{r}_{b})$. \[fig1\]](14812FG1.eps){width="8cm"}
Following , we use Jacobi coordinates, which are the inter-binary distance, $\vec{r_b}$, and the distance, $ \vec{r} $, from the star $ M_{\star} $ to the binary’s center of mass (see Fig. \[fig1\]). We also assume that $ \rho = |\vec{r_b}| / |\vec{r}| \ll 1$.
In the auxiliary frame (Fig. 1), $\vec{r}=(x',y',z')$, where $$\begin{aligned}
\label{xyz}
x' &=& r \cos{\theta} \ , \nonumber \\
y' &=& r \sin{\theta} \ , \nonumber \\
z' &=& 0 \ ,\end{aligned}$$ and $\vec{r}_{b}=(x'_{b},y'_{b},z'_{b})$, where $$\begin{aligned}
\label{xyzb}
x'_{b} &=& r_{b} (\cos{\Omega}\cos(\theta_{b}-\Omega)-\sin{\Omega}\sin(\theta_{b}-\Omega)\cos{i}) \ , \nonumber \\
y'_{b} &=& r_{b} (\sin{\Omega}\cos(\theta_{b}-\Omega)+\cos{\Omega}\sin(\theta_{b}-\Omega)\cos{i}) \ , \nonumber \\
z'_{b} &=& r_{b} \sin(\theta_{b}-\Omega) \sin{i} \ ,\end{aligned}$$ where $\theta$ and $\theta_b$ are the orbits’ true longitudes, $\Omega$ is the relative node longitude, and $i$ is the relative inclination.
The transformation from the auxiliary frame $(x',y',z')$ to the observer’s frame $(x,y,z)$ is $$\begin{aligned}
\label{transfxyz}
\hat{x} &=& \hat{x'} \ , \\ \nonumber
\hat{y} &=& \cos{I}\,\hat{y'}-\sin{I}\,\hat{z'} \ , \\ \nonumber
\hat{z} &=& \sin{I}\,\hat{y'}+\cos{I}\,\hat{z'} \ .\end{aligned}$$
Moreover, we define the angle $S=\angle(\vec{r},\vec{r}_ {b})$ which is given by $$\label{coS0}
\cos{S} = \hat{r} \cdot \hat{r_{b}} \ ,$$ where $\hat{r}$ is the versor of $\vec{r}$ and $\hat{r_{b}}$ is the versor of $\vec{r_{b}}$.
Jacobi coordinates are useful to this problem because the unperturbed motion is a composition of two Keplerian ellipses. As seen in , including terms up to first order in $\rho$, we have $$\begin{aligned}
\ddot{\vec{r}}_b=-G\frac{M_{1}+M_{2}}{r_{b}^2} \left( \hat{r}_{b}+
\epsilon_{b} (3 \cos{S} \hat{r}-\hat{r}_{b}) \right) \ ,\end{aligned}$$ where, unless $M_{\star} \gg M_{1}+M_{2}$, $$\epsilon_{b}= \frac{ M_{\star}}{M_{1}+M_{2}} \rho^3 \ll 1$$ and $$\begin{aligned}
\label{eqr}
\ddot{\vec{r}}& = & -G\frac{ (M_{1}+M_{2} +M_{\star})}{r^2} \times \nonumber \\
& & \left[ \hat{r }+ \epsilon \left( (-3/2+15/2 \cos^2{S}) \hat{r} - 3 \cos{S} \hat{r}_{b} \right) \right] \ ,\end{aligned}$$ where $$\epsilon=\frac{M_{1} M_{2}}{( M_{1}+M_{2} )^2} \rho^2 \le \frac{\rho^2}{4} \ll 1 \ .$$ Thus we can write $\vec{r}_{b} = \vec{r}_{b0}+\epsilon_{b} \vec{r}_{b1}$ , where $$\ddot{\vec{r}}_{b0}=-G\frac{M_{1}+M_{2}}{r_{b0}^2} \hat{r}_{b0} \ ,$$ and $\vec{r} = \vec{r}_{0}+\epsilon \vec{r}_{1}$ , where $$\ddot{\vec{r}}_{0} = -G\frac{ (M_{1}+M_{2} +M_{\star})}{r_{0}^2} \hat{r}_{0} \ .$$ Therefore the 0th order solution, $\vec{r}_{b0}$, is a Keplerian ellipse with constant semi-major axis, $a_b$, and frequency $$\label{freqbin}
n_{b}=\sqrt{\frac{G ( M_{1}+M_{2} )}{a_{b}^3}} \ ,$$ and the 0th order solution, $\vec{r}_{0}$, is a Keplerian ellipse with constant semi-major axis, $a$, and frequency $$\label{freqstar}
n=\sqrt{\frac{G (M_{1}+M_{2} + M_{\star})}{a^3}} \ .$$ Furthermore, from Eqs. (\[freqbin\]) and (\[freqstar\]) we see that in general $n \ll n_{b}$ since $ a_b \ll a $ (this is true unless $M_{\star} \gg M_{1}+M_{2}$).
Now, the distance of the star to the triple system’s center of mass is $$\vec{r}_{\star} = \frac{M_{1}+M_{2}}{M_{1}+M_{2}+M_{\star}} \, \vec{r} \ ,$$ hence from Eq. (\[eqr\]) we have $$\begin{aligned}
\ddot{\vec{r}}_{\star}& = & -G \frac{M_{1}+M_{2}}{r^2} \times \nonumber \\
& & \left[ \hat{r }+ \epsilon \left( (-3/2+15/2 \cos^2{S}) \hat{r} - 3 \cos{S} \hat{r}_{b} \right) \right] \ .\end{aligned}$$ Since $r^{-2}=r_{0}^{-2} (1+O(\epsilon))$, $\hat{r}=\hat{r}_{0} (1+O(\epsilon))$ and $\hat{r}_{b}=\hat{r}_{b0} (1+O(\epsilon_{b}))$, we obtain an approximation to the previous equation that is accurate up to $O(\epsilon)$, i.e. $$\begin{aligned}
\label{eqrstar}
\ddot{\vec{r}}_{\star } &=& -G\frac{ M_{1}+M_{2}}{r^2}\hat{r} \\ \nonumber
&& -G\frac{ M_{1}+M_{2} }{r_{0}^2} \epsilon \left[
\left(-\frac{3}{2}+\frac{15}{2} \cos^2{S} \right) \hat{r}_{0} - 3 \cos{S} \hat{r}_{b0} \right] \ , \end{aligned}$$ whose solution, $\vec{r}_{\star}=\vec{r}_{\star0}+ \epsilon \vec{r}_{\star1}$, is thus a combination of a Keplerian ellipse (with frequency $n$), $\vec{r}_{\star0}$, and a perturbation term $$\label{eqrstar1}
\ddot{\vec{r}}_{\star1 } = -G\frac{ M_{1}+M_{2} }{r_{0}^2} \left[
\left(-\frac{3}{2}+\frac{15}{2} \cos^2{S} \right) \hat{r}_{0} - 3 \cos{S} \hat{r}_{b0} \right] \ .$$
The radial velocity is the projection of the star’s barycentric velocity along the line of sight (defined as the $z$-axis in Fig. \[fig1\]), i.e., $V_{r}=\dot{z}_{\star}$. Therefore, we only need to compute the $z$ component of the first order correction, $\epsilon \vec{r}_{\star 1}$ (Eq. (\[eqrstar1\])), i.e. $$\begin{aligned}
\label{eqz}
\ddot{z}_{\star} & = & -G \frac{M_{1}+M_{2}}{r_{0}^2} \epsilon \times \nonumber \\
& & \left( \left( \frac{9}{4}+\frac{15}{4} \cos(2 S) \right)
\hat{r}_{0} \cdot \hat{z} - 3 \cos{S} \hat{r}_{b0} \cdot \hat{z} \right) \ .\end{aligned}$$
Additionally, from Eqs. (\[xyz\]), (\[xyzb\]), (\[transfxyz\]), and (\[coS0\]) we have $$\begin{aligned}
\label{z0}
\hat{r}_{0} \cdot \hat{z} &=& \sin(\theta) \sin{I} \ ,\\
\label{zb0}
\hat{r}_{b0} \cdot \hat{z} &=& \sin(\theta_{b}) \sin{I}+\sin(\theta_{b}-\Omega)\times \\ \nonumber
&& \left( \sin{i}\,\cos{I}-(1-\cos{i})\cos{\Omega}\,\sin{I} \right) \, \\
\label{coS}
\cos{S} &=& \hat{r} \cdot \hat{r}_{b} \approx \hat{r}_{0} \cdot \hat{r}_{b0} \nonumber \\
&=& \cos(\theta_{b}-\theta)-(1-\cos{i})\,\sin(\theta-\Omega)\,\sin(\theta_{b}-\Omega) \ .\end{aligned}$$
Star and binary on circular orbits
----------------------------------
In this case , $r_{0}=a$, $r_{b0}=a_b$, $\theta=\lambda=n\,t+\lambda_0$, $\theta_b =\lambda_b = n_{b}\,t+\lambda_{b0}$, where $\lambda$ and $\lambda_b$ are the orbits’ mean longitudes. Thus Eq. (\[eqz\]) becomes $$\begin{aligned}
\label{eqz3Dcircular}
\ddot{z}_{\star} &=& \frac{3}{4}\,\delta\,n_{b}^2\,\sin{I}\times
\left[ 4 (1-3\,{ \cos{i}}^{2})
\sin( n\,t +\lambda_0 ) \right. \nonumber \\
&&+ 2 (1-{ \cos{i}}^{2})
\sin( n\,t +\lambda_0 -2\,\Omega ) \nonumber \\
&&- 10 (1-{\cos{i}}^{2})
\sin( 3\,n\,t +3\,\lambda_0 -2\,\Omega) \nonumber \\
&&+ 5 (1+ \cos{i})^{2}
\sin( (2\, n_b -3\,n)\,t +2\,\lambda_{b0}-3\,\lambda_0 ) \nonumber \\
&& -5 (1-{\cos{i}})^{2} )
\sin( (2\, n_b + 3\,n)\,t +2\,\lambda_{b0} +3\,\lambda_0 -4\,\Omega ) \nonumber \\
&& - (1+{\cos{i}} )^{2}
\sin( (2\, n_b -n)\,t +2\,\lambda_{b0} -\lambda_0 ) \nonumber \\
&&+ 6 ( 1-{\cos{i}}^{2} )
\sin( (2\, n_b -n)\,t +2\,\lambda_{b0} -\lambda_0 -2\,\Omega ) \nonumber \\
&& + (1- \cos{i})^{2}
\sin( (2\, n_b +n)\,t +2\,\lambda_{b0} +\lambda_0 -4\,\Omega ) \nonumber \\
&& \left. - 6 (1-{ \cos{i}}^{2} )
\sin( 2\, n_b +n)\,t +2\,\lambda_{b0} +\lambda_0 -2\, \Omega ) \right] \nonumber \\
&& + 6\,\delta\,n_{b}^2\,\cos{I}\times
\left[ 2\, \sin{i}\,\cos{i}\,
\sin ( n\,t +\lambda_0 -\Omega) \right. \nonumber \\
&& + \sin{i} (1+ \cos{i})
\sin( (2\, n_b -n)\,t +2\,\lambda_{b0} -\lambda_0 -\Omega ) \nonumber \\
&& \left. +\sin{i} (1-\cos{i})
\sin( (2\, n_b +n)\,t +2\,\lambda_{b0} +\lambda_0 -3\,\Omega) \right] \ ,\end{aligned}$$ where $$\label{delta}
\delta=\frac{M_{1} M_{2}}{8 ( M_{1}+M_{2} )^2} \left( \frac{a_b}{a} \right)^4 a_{b} \,$$ as defined in .
The right-hand side of Eq. (\[eqz3Dcircular\]) is a sum of periodic terms, hence $V_{r}=\dot{z}_{\star}$ is obtained by integrating them with respect to time. Therefore, $\dot{z}_{\star}$ is a linear combination of six periodic terms with frequencies $n$, $3\,n$, $2\,n_{b}-n$, $2\,n_{b}+n$, $2\,n_{b}-3\,n$, and $2\,n_{b}+3\,n$.
In the coplanar case ($i=0$), we recover the solution obtained in $$\begin{aligned}
\dot{z}_{\star} & = & 6 \delta \sin{I} \frac{n_{b}^2}{n} \cos(n\,t+\lambda_{0}) \nonumber \\
&& -15 \delta \sin{I} \frac{n_{b}^2}{2\,n_{b}-3\,n} \cos((2\,n_{b}-3\,n)\,t+2\,\lambda_{0}-3\,\lambda_{b0}) \nonumber \\
&& +3 \delta \sin{I} \frac{n_{b}^2}{2\,n_{b}-n} \cos((2\,n_{b}-n)\,t+2\,\lambda_{0}-\lambda_{b0}) \,\end{aligned}$$ i.e., $\dot{z}_{\star}$ is a composition of three periodic terms with frequencies $n$, $2\,n_{b}-n$, and $2\,n_{b}-3\,n$.
We note that when $i=0$ (coplanar case) the angle $I$ defines the projection of the star’s orbit along the line of sight and thus $\ddot{z}$ (and $\dot{z}$) scale with $\sin{I}$. However, when $i\neq 0$ the angle that defines the projection of the star’s orbit along the line of sight depends on $I$, $i$, and $\Omega$. When $i\neq 0$, the amplitudes associated with the frequencies $n$ and $2\,n_{b}\pm n$ indeed do not scale with $\sin{I}$ (cf. last three terms in Eq. (\[eqz3Dcircular\])). In particular, when $I=0$ (pole-on configuration) and if $i\neq 0$ we can still detect these terms (frequencies $n$ and $2\,n_{b}\pm n$), which are due to the binary’s effect on the star.
In Fig. (2), we show the normalized amplitudes ($A$) associated with the frequencies $2\,n_{b}\pm n$ and $2\,n_{b}\pm 3\,n$, as a function of $i$ for $\Omega=0$, $I=90^\circ$ (Fig. 2, top) and $\Omega=90^\circ$, $I=90^\circ$ (Fig. 2, bottom). In Fig. (3), we show the equivalent picture for $I=0$. The true amplitudes are $A\,\delta\,n_{b}$.
At $I=90^\circ$ (equator-on configuration) and small $i$, the term of frequency $2\,n_{b}-3\,n$ is dominant but as $i$ increases its amplitude decreases, while the amplitudes of the terms with frequencies $2\,n_{b}+3\,n$ and $2\,n_{b}\pm n$ increase until they are all approximately equal at $i=90^\circ$. At $I=0$ (pole-on configuration), only the terms with frequencies $n$ and $2\,n_{b}\pm n$ appear (cf. Eq. (\[eqz3Dcircular\])), hence we cannot detect the star’s motion around the binary’s center of mass but if $i\neq 0$ we can still detect the binary’s effect on the star. Moreover, the amplitudes associated with the frequencies $2\,n_{b}\pm 3\,n$ are independent of $\Omega$ at any value of $I$ while the amplitudes associated with the frequencies $2\,n_{b} \pm n$ depend on the orbits’ intersection $\Omega$ except when $I=0$ (cf. Eq. (\[eqz3Dcircular\])). The pictures for $90^\circ > i \ge 0^\circ$ (star and binary on prograde orbits) and $180^\circ \ge i >90^\circ$ (binary on a retrograde orbit) are symmetrical except that the frequency $n$ is replaced with $-n$, which corresponds to inverting the star’s (or binary’s) motion.
![Circular case with $I=90^\circ$ (equator-on configuration). Normalized amplitudes $A$ of periodic terms as function of the relative inclination $i$. These are obtained by integrating Eq. (\[eqz3Dcircular\]) with respect to time. The true amplitudes are $A\,\delta\,n_{b}$.](14812FG2up.eps "fig:"){width="6cm"} ![Circular case with $I=90^\circ$ (equator-on configuration). Normalized amplitudes $A$ of periodic terms as function of the relative inclination $i$. These are obtained by integrating Eq. (\[eqz3Dcircular\]) with respect to time. The true amplitudes are $A\,\delta\,n_{b}$.](14812FG2do.eps "fig:"){width="6cm"}
![Circular case with $I=0$ (pole-on configuration). Normalized amplitudes $A$ of periodic terms as function of the relative inclination $i$. These are obtained by integrating Eq. (\[eqz3Dcircular\]) with respect to time. The true amplitudes are $A\,\delta\,n_{b}$.](14812FG3.eps){width="6cm"}
Star and binary on eccentric orbits
-----------------------------------
We assume that the Keplerian orbit, $\vec{r}_{0}$, has a semi-major axis $a$, eccentricity $e$, and true longitude $\theta=f+\varpi$ (where $f$ is the true anomaly and $\varpi$ is the longitude of periastron) and that the Keplerian orbit, $\vec{r}_{b0}$, has semi-major axis $a_b$, eccentricity $e_b$, and true longitude $\theta_{b}=f_{b}+\varpi_{b}$ (where $f_b$ is the true anomaly and $\varpi_b$ is the longitude of periastron). Thus Eq. (\[eqz\]) becomes $$\begin{aligned}
\label{eqzeccentric}
\ddot{z}_{\star} & = & -G \frac{M_{1}+M_{2}}{a^2} \epsilon_{0} \left( \frac{a}{r_0} \right)^4 \left( \frac{r_{b0}}{a_b} \right)^2 \times \nonumber \\
& &
\left( \left( \frac{9}{4}+\frac{15}{4} \cos(2 S) \right)
\hat{r}_{0} \cdot \hat{z} - 3 \cos{S} \hat{r}_{b0} \cdot \hat{z} \right) \ ,\end{aligned}$$ where $$\epsilon_{0}=\frac{M_{1} M_{2}}{( M_{1}+M_{2} )^2} \left( \frac{a_b}{a} \right)^2 \ .$$
We can express the orbital solutions, $\vec{r}_{0}$ and $\vec{r}_{b0}$, as Fourier series of the mean anomalies $M=n\,t+M_{0}$ and $M_{b}=n_{b}\,t+M_{b0}$, respectively, also known as elliptic expansions. In particular, we obtain expressions for $(a/r_{0})^4$ and $(r_{b0}/a_{b})^2$ using elliptic expansions for $r/a$ and $r_{b}/a_{b}$ (e.g. @murray_dermott1999). We obtain similar expressions for $\hat{r}_{0} \cdot \hat{z}$ (Eq. (\[z0\])), $\hat{r}_{b0} \cdot \hat{z}$ (Eq. (\[zb0\])), and $\cos{S}$ (Eq. (\[coS\])) using elliptic expansions for $\cos(f)$, $\sin(f)$, $\cos(f_b)$, and $\sin(f_b)$ (e.g. @murray_dermott1999).
### Coplanar motion
In this case, $i=0$, thus from Eqs. (\[z0\]), (\[zb0\]), and (\[coS\]) $$\begin{aligned}
\hat{r}_{0} \cdot \hat{z} &=& \sin(\theta) \sin{I} \ , \\
\hat{r}_{b} \cdot \hat{z} &=& \sin(\theta_{b}) \sin{I} \ , \\
\cos{S} &=& \cos(\theta_{b}-\theta) \ ,\end{aligned}$$ where $\theta=f+\varpi$ and $\theta_{b}=f_{b}+\varpi_{b}$.
The next step is to replace these expressions into Eq. (\[eqzeccentric\]) using elliptic expansions for $(a/r_{0})^4$, $(r_{b0}/a_{b})^2$, $\sin(\theta)$, $\sin(\theta_{b})$, $\cos(\theta_{b}-\theta)$, and $\cos(2\,(\theta_{b}-\theta))$. We start by using elliptic expansions up to first order in the eccentricities.
As in , our aim is to obtain a solution to Eq. (\[eqzeccentric\]) that is a linear combination of periodic terms. However, when $e \ne 0$ we have a constant term of the type $e \sin(\varpi)$ appearing in $\ddot{z}_{\star}$. We can remove this secular term by rewriting Eq. (\[eqrstar1\]) as $$\label{eqrstar1new}
\ddot{\vec{r}}_{\star 1} = -G \frac{M_{1}+M_{2}}{r_{0}^2} \vec{p}+\ddot{\vec{r'}}_{\star 1} \ ,$$ where the vector $\vec{p}=(p_{x},p_{y},p_{z})$ has radius $(3/4)\,e$, and is aligned with the periapse of the unperturbed orbit, $\vec{r}_0$, i.e. $$\begin{aligned}
p_{x} &=& (3/4) e \cos(\varpi) \ , \\
p_{y} &=& (3/4) e \sin(\varpi)\cos{I} \ , \\
p_{z} &=& (3/4) e \sin(\varpi) \sin{I} \ ,\end{aligned}$$ and $$\begin{aligned}
\label{eqrstar1reduced}
\ddot{\vec{r'}}_{\star 1} &=& G \frac{M_{1}+M_{2}}{r_{0}^2} \vec{p}
-G \frac{M_{1}+M_{2}}{r_{0}^2} \times \nonumber \\
& & \left( \left( \frac{9}{4}+\frac{15}{4} \cos(2 S) \right)
\hat{r}_{0} - 3 \cos{S} \hat{r}_{b0} \right) \ .\end{aligned}$$ Now, the first term in Eq. (\[eqrstar1new\]) can be interpreted as the slow rotation (frequency $\sim\epsilon_{0}^{1/2}\,n \ll n \ll n_b$) of the unperturbed orbit $\vec{r}_0$’s periapse described by the vector $\vec{p}$. Therefore, we can concentrate in obtaining the solution to Eq. (\[eqrstar1reduced\]).
The $z$ component of the first order correction, $\epsilon \vec{r'}_{\star 1}$ given by Eq. (\[eqrstar1reduced\]), becomes $$\begin{aligned}
\label{eqz2Deccentric1}
\ddot{z}_{\star} &=& 3\,\delta\,n_{b}^{2}\sin{I} \times \left[ -2\,\sin \left( n\,t+\lambda_{0} \right) \right. \nonumber \\
&& -\sin \left( (2\,n_{b}-n)\,t +2\,\lambda_{b0}-\lambda_0 \right) \nonumber \\
&& +5\,\sin \left( (2\,n_{b}-3\,n)\,t +2\,\lambda_{b0}-3\,\lambda_{0} \right) \nonumber \\
&& -15\,e_{b}\,\sin \left((n_{b}-3\,n)\,t+\lambda_{b0}-3\,\lambda_{0}+\varpi_{b} \right) \nonumber \\
&& -e_{b}\,\sin \left((3\,n_{b}-n)\,t+3\,\lambda_{b0}-\lambda_{0} -\varpi_{b} \right) \nonumber \\
&& +5\,e_{b}\,\sin \left((3\,n_{b}-3\,n)\,t+3\,\lambda_{b0}-3\,\lambda_{0} -\varpi_{b} \right) \nonumber \\
&& +2\,e_{b}\,\sin \left((n_{b}+n)\,t+\lambda_{b0}+\lambda_{0} -\varpi_{b} \right) \nonumber \\
&& +3\,e_{b}\,\sin \left((n_{b}-n)\,t +\lambda_{b0}-\lambda_{0} +\varpi_{b} \right) \nonumber \\
&& -2\,e_{b}\,\sin \left((n_{b}-n)\,t+\lambda_{b0}-\lambda_{0}-\varpi_{b} \right) \nonumber \\
&& -6\,e\sin \left(2\,n\,t+2\,\lambda_{0} -\varpi \right) \nonumber \\
&& -e\sin \left(2\,n_{b}\,t+2\,\lambda_{b0}-\varpi \right) \nonumber \\
&& +25\,e\sin \left((2\,n_{b}-4\,n)\,t+2\,\lambda_{b0}-4\,\lambda_{0}+\varpi \right) \nonumber \\
&& -3\,e\sin \left((2\,n_{b}-2\,n)\,t+2\,\lambda_{b0}-2\,\lambda_{0}+\varpi \right) \nonumber \\
&& \left. -5\,e\sin \left((2\,n_{b}-2\,n)\,t+2\,\lambda_{b0}-2\,\lambda_{0}-\varpi \right) \right] \ ,\end{aligned}$$ where $\lambda_{0}=M_{0}+\varpi$ and $\lambda_{b0}=M_{b0}+\varpi_{b}$.
The right-hand side of Eq. (\[eqz2Deccentric1\]) is a sum of periodic terms, hence $V_{r}=\dot{z}_{\star}$ is obtained by integrating these with respect to time. Therefore, when including terms up to first order in the eccentricities, $\dot{z}_{\star}$ is a composition of 12 periodic terms with frequencies: $n$, $2\,n_{b}-n$ and $2\,n_{b}-3\,n$ (zero order); $n_{b}-3\,n$, $3\,n_{b}-n$, $3\,n_{b}-3\,n$, $n_{b}+n$ and $n_{b}-n$ (first order: $e_b$); $2\,n$, $2\,n_{b}$, $2\,n_{b}-4\,n$ and $2\,n_{b}-2\,n$ (first order: $e$). The amplitudes of the terms with frequencies $n_{b}-n$ and $2\,n_{b}-2\,n$ depend on $\varpi_b$ and $\varpi$, respectively.
We can easily extend our theory by including higher-order eccentricity terms in the elliptic expansions[^1] (i.e., Fourier series of the Keplerian orbital solutions $\vec{r}_{0}$ and $\vec{r}_{b0}$). In particular, if we include terms up to second order in the eccentricities, there are additional periodic terms with frequencies: $3\,n$, $2\,n_{b}+n$, $4\,n_{b}-n$, $4\,n_{b}-3\,n$ (second order: $e_{b}^2$); $3\,n$, $2\,n_{b}+n$, $2\,n_{b}-5\,n$ (second order: $e^2$); $3\,n_{b}-2\,n$, $n_b$, $3\,n_{b}$, $n_{b}-4\,n$, $n_{b}+2\,n$, $n_{b}-2\,n$, $3\,n_{b}-4\,n$ (second order: $e\,e_b$). Moreover, the amplitudes associated with the frequencies $2\,n_{b}-n$ and $2\,n_{b}-3\,n$ (zero order) have corrections of second order in the eccentricities.
In Fig. (4), we show the largest normalized amplitudes ($A>2$) of the periodic terms that appear up to fourth order in the eccentricities (except harmonics of $n$) as functions of $e_b$ when $e=0.1$. The expansion in $e_b$ converges rapidly thus it is not necessary to include higher order terms at least up to $e_{b}=0.4$. When $e_b$ is small, the term with frequency $2\,n_{b}-3\,n$ is dominant but when $e_{b}>0.16$, the term with frequency $n_{b}-3\,n$ has the largest amplitude. Other important terms have frequencies $2\,n_{b}-4\,n$ and $n_{b}-4\,n$.
In Fig. (5), we show the largest normalized amplitudes ($A>2$) of the periodic terms that appear up to 10th order in the eccentricities (except harmonics of $n$) as functions of $e$ when $e_{b}=0$. We include corrections in the amplitudes up to 12th order in the eccentricities. These high order expansions are necessary to obtain results valid up to $e=0.4$ since the expansion in $e$ converges slowly. When $e$ is small, the term with frequency $2\,n_{b}-3\,n$ has the largest amplitude but as $e$ increases, the dominant term becomes $2\,n_{b}-4\,n$ ($e>0.18$), $2\,n_{b}-5\,n$ ($e>0.28$), and $2\,n_{b}-6\,n$ ($e>0.36$). As $e$ increases, the number of terms with frequencies near $2\,n_{b}$ and similar amplitudes increases.
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### Non-coplanar motion
When $i\neq 0$, we must follow the same procedure described in the previous section, expressing the orbital solutions $\vec{r}_{0}$ and $\vec{r}_{b0}$ using elliptic expansions (e.g. @murray_dermott1999), but now using the general expressions for $\hat{r}_0 \cdot \hat{z}$, $\hat{r}_{b0} \cdot \hat{z}$, and $\cos{S}$ (Eqs. (\[z0\]), (\[zb0\]), and (\[coS\])).
We can show that, including terms up to first order in the eccentricities, $\ddot{z}_{\star}$ (and $\dot{z}_{\star}$) are linear combinations of 21 periodic terms with frequencies $n$, $2\,n$, $3\,n$, $4\,n$, $2\,n_{b}$, $2\,n_{b}\pm n$, $2\,n_{b}\pm 3\,n$, $n_{b}\pm n$, $n_{b} \pm 3\,n$, $3\,n_{b} \pm n$, $3\,n_{b} \pm 3\,n$, $2\,n_{b}\pm 4\,n$, and $2\,n_{b}\pm 2\,n$. We note that, except for the additional harmonics of $n$, these include the set of frequencies already present in the coplanar eccentric case plus an additional set of frequencies obtained from the previous set by replacing $n$ with $-n$, which as noted previously corresponds to inverting the star’s (or binary’s) motion.
We already described how the terms with frequencies $2\,n_{b} \pm n$, and $2\,n_{b} \pm 3\,n$ (circular case) behaved with $I$, $i$, and $\Omega$ (Figs. (2),(3) and Sect. 2.2). To understand what happens when $e$ and $e_b$ are small but non-zero, we now describe the behavior of the terms that appear to first order in the eccentricities.
When $e_{b} \ne 0$, the frequencies $n_{b}\pm 3\,n$, $3\,n_{b} \pm n$, $n_{b} \pm n$, and $3\,n_{b} \pm 3\,n$ appear. In Fig. (6), we show the normalized amplitudes ($A$) as functions of $i$ for $I=90^\circ$ (equator-on configuration), $e_{b}=0.1$, and $\Omega=90^\circ$. At $I=90^\circ$ and small $i$, the term with frequency $n_{b}-3\,n$ has maximum amplitude. As the relative inclination $i$ increases, this amplitude decreases and at $i=90^\circ$ the terms with frequencies $n_{b} \pm n$ have maximum amplitude.
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When $e \ne 0$, the frequencies $2\,n_{b}$, $2\,n_{b} \pm 2\,n$, and $2\,n_{b} \pm 4\,n$ appear. In Fig. (7), we show the normalized amplitudes ($A$) as functions of $i$ for $I=90^\circ$ (equator-on configuration), $e=0.1$, and $\Omega=90^\circ$. At $I=90^\circ$ and small $i$, the term with frequency $2\,n_{b}-4\,n$ has maximum amplitude. As the relative inclination $i$ increases, this amplitude decreases and at $i=90^\circ$ it matches the amplitudes of the terms with frequency $2\,n_{b}\pm 2\,n$.
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In Figs. (8) and (9), we show the normalized amplitude ($A$) when $I=0$, $e_{b}=0.1$, and $e=0.1$, respectively. In this configuration ($I=0$), only the frequencies $3\,n_{b} \pm n$, $n_{b} \pm n$, $2\,n_{b}$, and $2\,n_{b} \pm 2\,n$ have non-zero amplitude. As noted previously, in this case (pole-on configuration) we cannot detect the star’s motion around the binary’s center of mass but if $i\neq 0$, we can still detect the binary’s effect on the star.
{width="6cm"}
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Moreover, when $i \ne 0$ only the amplitudes associated with the frequencies $n_{b} \pm 3\,n$, $3\,n_{b} \pm n$, $n_{b} \pm n$, $2\,n_{b}$, and $2\,n_{b} \pm 2\,n$ depend on the relative node longitude, $\Omega$, since the other frequencies originate from the angles $2\,\theta_{b}\pm 3\,\theta$ and $3\,\theta$ whose amplitudes are independent of $\Omega$ (cf. third, fourth and fifth terms in Eq. (\[eqz3Dcircular\])). We can also show that the amplitudes associated with the frequencies $n_{b} \pm 3\,n$ and $n_{b} \pm n$ depend on the binary’s longitude of pericenter $\varpi_b$, while the amplitudes associated with the frequencies $2\,n_{b}$ and $2\,n_{b} \pm 2\,n$ depend on the star’s longitude of pericenter $\varpi$.
The pictures for $90^\circ > i \ge 0^\circ$ (star and binary on prograde orbits) and $180^\circ \ge i >90^\circ$ (binary on a retrograde orbit) are symmetrical except that the frequency $n$ is replaced with $-n$, which corresponds to inverting the star’s (or binary’s) motion.
Can a binary system mimic a planet?
===================================
As in , we wish to identify the circumstances in which we can we mistake the binary’s effect for that of a planet, when we do not know whether a binary is present (one or even both components may be unresolved). We have seen that the star’s radial velocity will have a slow Keplerian term, which can be associated with a mass, $M_{1}+M_{2}$, located at the binary’s center of mass. Hence, from the radial velocity we can infer the presence of a nearby “star”, $M_{1}+M_{2}$, except when $I=0$ (pole-on configuration) where we can nevertheless detect the wobble due to the binary if $i\neq 0$. Moreover, the radial velocity of the star, $M_{\star}$, that has a planet companion, $M_p$, in a circular orbit with frequency $n_p$, semi-major axis $a_p$ and inclination $I_p$, and another nearby “star”, $M_{1}+M_{2}$, is $$\label{radialvelstarpl}
V_{r}=V_{0}+\frac{M_{p} \sin{I_p}}{M_{\star}+M_{1}+M_{2}+M_{p}} n_{p} a_{p} \cos(n_{p} t+\varpi_{p}) \ ,$$ where $V_{0}$ includes the Keplerian motion around the “star” $M_{1}+M_{2}$, and we can neglect $M_{p}$ in the denominator of Eq. (\[radialvelstarpl\]) since the planet’s mass is negligible with respect to the stars.
We saw above that the radial velocity of a star perturbed by a binary system is a composition of several periodic signals. The terms that include harmonics of the frequency $n$ can in principle[^2] be associated with the star’s slow Keplerian motion around the binary’s center of mass (i.e., $V_0$). Therefore, we have $$\label{radialvelbinary}
V_{r}=V_{0}+ \sum K_{p} \cos(n_{p} t+\varpi_{p}) \ ,$$ where $$\label{amplitudek}
K_{p} \approx A \delta n_{b}= A \frac{\sqrt{G (M_{1}+M_{2})}}{8} \frac{M_{1} M_{2}}{( M_{1}+M_{2} )^2} \left( \frac{a_b}{a} \right)^4 a_{b}^{-1/2} \ ,$$ and the factor $A$ is a normalized amplitude that, as explained previously, can be obtained for each frequency $n_p$.
Each of the terms with amplitude $K_{p}$ could be identified as a planet on a circular orbit around the star with frequency $n_{p}=p\,n_{b} \pm q\,n$, where $p>0$ and $q \ge 0$ are integers. However, since we have a limit on the observations’ precision, we can sometimes detect only a few periodic terms with the largest amplitudes. On the other hand, since $n_{b} \gg n$, terms of equal $p$ have very close frequencies. Hence, if we can detect these latter terms, we associate them with planets on very close orbits that can be discarded as unstable configurations. However, since we have a limit on the observations’ resolution, we are sometimes unable to distinguish terms with very close frequencies.
The radial velocity method of extrasolar planet’s detection consists of measuring tiny Doppler shifts in the star’s spectral lines caused by the star’s motion. These Doppler shifts are collected over an observation timespan, $t_{obs}$, in order to obtain the star’s radial velocity curve. The instrument’s precision is the smallest Doppler shift that can be detected, hence the smallest detectable amplitude in the radial velocity’s Fourier transform. On the other hand, the data’s resolution is the smallest frequency difference that can be detected, i.e., $\delta f = 1/t_{obs}$. Therefore, to distinguish two terms with frequencies that differ by $\Delta\,q \cdot n$, we must have $$\label{tobs}
t_{obs}>\frac{T}{\Delta\,q} \ ,$$ where $T=2\pi/n$.
Now, if all detected frequencies are well separated (i.e., have different $p$) we may then be led to believe that the star has one or more planet companions. In that case, we apply Kepler’s third law to derive the fake planets’ semi-major axis, $a_p$, from the signal’s frequencies, $n_p$, i.e. $$\label{semajoraxis}
a_{p}=\frac{( G M_{\star})^{1/3}}{(n_{p})^{2/3}} \ ,$$ and from Eqs. (\[radialvelstarpl\]) and (\[radialvelbinary\]) we obtain the associated minimum masses, $M_{p} \sin{I_p}$, from the signal’s amplitudes $K_p$ (Eq. (\[amplitudek\])), i.e. $$\label{minmasspl}
\frac{M_{p} \sin{I_p}}{M_{\star}+M_{1}+M_{2}}=\frac{K_{p}}{a_{p} n_{p}} \ .$$ Moreover, from Eq. (\[amplitudek\]) we see that the binary’s effect is more pronounced (i.e $K_p$ is larger) for a large ratio $a_{b}/a$, small inter-binary distance $a_b$, and a massive binary of equal stars ($M_{1}=M_{2}$).
Circular coplanar orbits
------------------------
This was studied in . In the case of coplanar circular prograde orbits, we recall that the periodic terms due to the binary’s effect have frequencies $n_{p}=2\,n_{b}-n$ and $n_{p}=2\,n_{b}-3\,n$, and associated amplitudes $K_{p} \approx (3/2) n_{b} \delta \sin{I}$ and $K_{p} \approx (15/2) n_{b} \delta \sin{I}$, respectively. As noted previously, these signals may at first sight be identified as two planets orbiting the star with very close frequencies. However, if we could resolve these close frequencies (i.e., if $t_{obs}>T/2$, cf. Eq. (\[tobs\])), we should be able to discard the planet’s existence since these systems are most certainly unstable. There are nevertheless situations where we can mistake the effect of a binary system for a planet:
- We cannot resolve the two close frequencies (i.e., if $t_{obs}<T/2$, cf. Eq. (\[tobs\])), hence we detect only one signal at $\sim 2\,n_b$.
- Owing to limited instrument’s precision, we detect the signal at $2\,n_{b}-3\,n$ but not the one at $2\,n_{b}-n$ (these have amplitudes in the ratio $5/1$.)
In the latter scenario, from Eq. (\[semajoraxis\]), we obtain the associated fake planet’s semi-major axis $$a_{p}=\frac{( G M_{\star})^{1/3}}{(2\,n_{b}-3\,n)^{2/3}}
\approx \left( \frac{M_{\star}}{4 (M_{1}+M_{2})} \right)^{1/3} a_{b} \ ,$$ and from Eqs. (\[amplitudek\]) and (\[minmasspl\]), since the motion is coplanar ($I_{p}=I$), we obtain the fake planet’s mass $$\frac{M_p}{M_{\star}+M_{1}+M_{2}}=\frac{K_{p}}{a_{p} (2\,n_{b}-3\,n)}
\approx \frac{15}{32} \frac{M_{1} M_{2}}{(M_{1}+M_{2})^2} \left( \frac{a_b}{a}\right)^4 \frac{a_b}{a_p} \ .$$ In the former scenario, the signal’s amplitude can vary by $20\%$ depending on the unresolved signals’ phases. Nevertheless, these formulae are still approximately valid.
Non-circular, non-coplanar orbits
---------------------------------
The case of inclined and eccentric orbits is similar. We saw that, when including terms up to first order in the eccentricities, the star’s radial velocity is a composition of 21 periodic terms. Four terms are harmonics of the frequency $n$, hence can in principle be discarded as associated with the slow Keplerian motion around the binary’s center of mass. Four terms have frequencies close to $n_b$, nine terms have frequencies close to $2\,n_b$, and four terms have frequencies close to $3\,n_{b}$.
Comparing Figs. 2, 3, 6, 7, 8, and 9 we see that the maximum radial velocity variations associated with dominant periodic terms occur when $I=90^\circ$ (equator-on configuration) and the orbits are nearly coplanar ($i \approx 0$ or $i \approx 180^\circ$). In the case of coplanar prograde motion ($i=0$), the dominant term has frequency $2\,n_{b}-3\,n$ (for small $e$ and $e_b$; cf. Figs. 4 and 5) and frequency $n_{b}-3\,n$ (for large $e_b$; cf. Fig. 4). In particular, when the binary’s orbit is eccentric it is possible to detect signals at $2\,n_{b}-3\,n$ and $n_{b}-3\,n$, which can mimic two planets with orbital periods in the ratio 2/1. However, as $e$ increases the number of terms with frequencies close to $2\,n_{b}$ and similar amplitudes increases (Fig. 5), hence it should be easier to detect signals with very close frequencies.
Examples
========
Theory versus simulations
-------------------------
To test our model, we performed some numerical simulations of a triple system composed of main sequence stars with different spectral types G, K, and M, and masses $M_{\star}=M_{\odot}$, $M_{1}=0.70\,M_{\odot}$, and $M_{2}=0.35\,M_{\odot}$, respectively. The smaller K and M stars form a binary system with semi-major axis $a_{b}=1.1$ AU, eccentricity $e_b$, and inclination $I_b$. The G star is in a wide orbit with semi-major axis $a=10$ AU, eccentricity $e$, and inclination $I$ around the binary’s center of mass. The M star is much fainter than the K or G stars, hence it represents the unresolved component of the binary.
We numerically integrated this triple system and computed the radial velocity of the G star (the brightest body in the system) assuming an equator-on configuration ($I=90^\circ$). From this, we simulated 100 observational data points for a time span of 4000 days (about 11 years). We assumed that the data were obtained with a precision limit of $0.535$ m/s, which corresponds to $A=4.46$ (cf. Eq. \[amplitudek\]) and is about the highest precision that can presently be obtained (HARPS, @Harps2003). We then applied the traditional techniques used to search for planets.
Since $t_{obs}=4000$ days and the output step is $\Delta\,t \approx 40$ days, the largest detectable frequency is $f_{c}=2/(\Delta\,t)=1/20$ days$^{-1}$ and the frequency resolution is $\Delta\,f=1/4000$ days$^{-1}$. We present the theoretical predictions versus simulations in Table 1.
In a first step, we can only detect the large-amplitude radial velocity variations due to the star’s slow motion (with period $T=2\pi/n$) around the binary’s center of mass. Once we subtract these long-term variations we are able to detect the signal due to the binary’s motion, which could be mistaken for one or more planet companions to the star.
Example 1 is a circular coplanar system ($i=0$) already presented in , where the binary mimics a planet of approximately 20 Earth masses. Examples 2 and 3 are also circular but inclined systems with $i=30^{\circ}$ and $i=60^{\circ}$, respectively. These demonstrate that increasing the relative inclination $i$ leads to a decrease in the planet’s mass $M_p$, as we would expect from Eq. (\[eqz3Dcircular\]) since the amplitude associated with the frequency $2\,n_{b}-3\,n$ is proportional to $(1+\cos{i})^2$.
The next four examples are coplanar systems ($i=0$) with $e=0.1$ and $e_{b}=$0.1, 0.2, 0.3, and 0.4, respectively. When $e_{b}=0.1$ (Ex. 4), we detect the signals with frequencies $2\,n_{b}-3\,n$ and $n_{b}-3\,n$ (cf. Fig. 4 with $A=4.46$). When $e_{b} \ge 0.2$ (Exs. 5, 6 and 7), the signal with frequency $n_{b}-4\,n$ is above the detection limit (cf. Fig. 4 with $A=4.46$) but we cannot distinguish it from the signal with frequency $n_{b}-3\,n$ since the resolution is not high enough. To resolve a frequency difference of $n$ we would indeed require a resolution of at least $\Delta\,f=2\pi/n$, which corresponds to a time span $T=2\pi/n$, i.e., about 22 years. At present, such a large timespan is not realistic since exoplanet detections started only 15 years ago. This lack of resolution prevents the correct detection of the signal with frequency $n_{b}-3\,n$ in the 5th, 6th, and 7th examples since this is “contaminated” by the close signal with frequency $n_{b}-4\,n$.
In examples 1, 2, and 3, the binary mimics a planet while in examples 4, 5, 6, and 7, it mimics two planets with orbital periods in the ratio 2/1. In these cases, the data’s precision and observation timespan do not allow us to distinguish the binary’s effect from planet(s). We now present an example where we detect signals with very close frequencies thus can reject the planet(s) hypothesis.
Example 8 is a coplanar system ($i=0$) with $e_{b}=0.05$, and $e=0.4$. According to Fig. 5, when $e=0.4$ the terms with frequencies $2\,n_{b}-6\,n$, $2\,n_{b}-5\,n$, $2\,n_{b}-7\,n$, $2\,n_{b}-8\,n$, and $2\,n_{b}-9\,n$ are all above the detection level ($A=4.46$). However, we only detect signals at approximately $2\,n_{b}-6\,n$ and $2\,n_{b}-8\,n$ as the observation timespan, $t_{obs}\approx T/2$, is not long enough to resolve the other frequencies. These nearby unresolved frequencies prevent the correct detection of the signals at $2\,n_{b}-6\,n$ and $2\,n_{b}-8\,n$. Additional Fourier analysis of the residuals show the presence of a signal at $2\,n_{b}-10\,n$ but the fitting to the signals at $2\,n_{b}-6\,n$ and $2\,n_{b}-8\,n$ deteriorates when including this frequency. The residuals are 1.7 m/s, i.e., above the precision limit ($\sim 0.5$ m/s), which indicates the presence of unresolved / undetected frequencies.
Finally, we note what happens when $e$ is large but the observation time span is short. We performed a new analysis of example 8 assuming $t_{obs}=2000$ days and $t_{obs}=1000$ days, i.e., about $T/4$ and $T/8$, respectively. In the first case, we detected signals at 243 days ($2\,n_{b}-6\,n$) and 281 days ($2\,n_{b}-11\,n$), with amplitudes 5.0 m/s and 2.7 m/s, respectively. However, the residuals after the fit are still 1.3 m/s. In the second case, we detect only one signal at 257 days ($\sim 2\,n_b$) with an amplitude 6.7 m/s,, which corresponds to a $0.3\,M_{J}$ planet at $0.79$ AU. Since $t_{obs}\ll T$, we cannot resolve the signals at nearby frequencies although these affect the measured amplitude. The residuals after the fit are $\sim 0.4$ m/s, hence there is no hint of unresolved / undetected frequencies.
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Ex. Parameters $n_p$ $T_{p}$ (d) $K_p$ (m/s) $T_{p}$ (d) $K_p$ (m/s) $a_p$ (AU) $M_p$ $e_p$
1 $i=0^\circ$ $n$ 8069 - $8032.8\pm 2.6$ $6922.0\pm 0.3$ 9.97 $1.05\,M_{\odot}$ $0.00$
$e=e_{b}=0$ $2\,n_{b}-3\,n$ 222.6 0.961 $223.2\pm 0.5$ $0.96\pm 0.06$ 0.720 $18.7\,M_{\oplus}$ $0.00$
2 $i=30^\circ$ $n$ 8069 - $8037.6\pm 2.7$ $6916.1\pm 0.3$ 9.97 $1.05\,M_{\odot}$ $ 0.00$
$e=e_{b}=0$ $2\,n_{b}-3\,n$ 222.6 0.837 $222.9\pm 0.6$ $0.83\pm 0.06$ 0.719 $16.0\,M_{\oplus}$ $0.00$
3 $i=60^\circ$ $n$ 8069 - $8048.3\pm 2.7$ $6904.5\pm 0.3$ 9.98 $1.05\,M_{\odot}$ $0.00$
$e=e_{b}=0$ $2\,n_{b}-3\,n$ 222.6 0.541 $222.3\pm 0.9$ $0.52\pm 0.06$ 0.718 $10.16\,M_{\oplus}$ $0.00$
4 $i=0^\circ$ $n$ 8069 - $8035.1\pm 2.8$ $6958.1\pm 0.4$ 9.97 $1.05\,M_{\odot}$ $0.10$
$e=0.1$ $2\,n_{b}-3\,n$ 222.6 0.869 $222.6\pm 0.6$ $0.90\pm 0.06$ 0.719 $17.52\,M_{\oplus}$ $0.00$
$e_{b}=0.1$ $n_{b}-3\,n$ 485.5 0.585 $485.4\pm 3.7$ $0.65\pm 0.07$ 1.209 $16.01\,M_{\oplus}$ $0.00$
5 $i=0^\circ$ $n$ 8069 - $8026.8\pm 2.8$ $6959.4\pm 0.4$ 9.97 $1.05\,M_{\odot}$ $0.10$
$e=0.1$ $n_{b}-3\,n$ 485.5 1.149 $485.2\pm 1.8$ $1.27\pm 0.07$ 1.208 $31.9\,M_{\oplus}$ $0.00$
$e_{b}=0.2$ $2\,n_{b}-3\,n$ 222.6 0.804 $222.2\pm 0.6$ $0.84\pm 0.06$ 0.718 $16.2\,M_{\oplus}$ $0.00$
6 $i=0^\circ$ $n$ 8069 - $8074\pm 6$ $6993\pm 5$ 10.02 $1.06\,M_{\odot}$ $0.10$
$e=0.1$ $n_{b}-3\,n$ 485.5 1.673 $476.8\pm 1.4$ $1.53\pm 0.07$ 1.194 $38.5\,M_{\oplus}$ $0.00$
$e_{b}=0.3$ $2\,n_{b}-3\,n$ 222.6 0.701 $222.6\pm 0.8$ $0.68\pm 0.06$ 0.719 $13.36\,M_{\oplus}$ $0.00$
7 $i=0^\circ$ $n$ 8069 - $8103\pm 6$ $7020\pm 6$ 10.06 $1.07\,M_{\odot}$ $ 0.10$
$e=0.1$ $n_{b}-3\,n$ 485.5 2.135 $477.6\pm 1.1$ $1.98\pm 0.07$ 1.196 $50.1\,M_{\oplus}$ $ 0.00$
$e_{b}=0.4$ $2\,n_{b}-3\,n$ 222.6 0.569 $223.1\pm 0.9$ $0.57\pm 0.06$ 0.720 $11.2\,M_{\oplus}$ $0.00$
8 $i=0^\circ$ $n$ 8069 - $7928\pm 2$ $7550.1\pm 0.4$ 9.88 $1.05\,M_{\odot}$ $0.39$
$e=0.4$ $2\,n_{b}-6\,n$ 242.7 1.104 $241.1\pm 0.3$ $2.60\pm 0.07$ 0.758 $51.4\,M_{\oplus}$ $ 0.12$
$e_{b}=0.05$ $2\,n_{b}-8\,n$ 258.2 0.840 $253.7\pm 0.4$ $2.52 \pm 0.07$ 0.784 $51.1\,M_{\oplus}$ $0.00$
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Ex. $M_1$ ($M_{\odot}$) $M_2$ ($M_{\odot}$) $a$ (AU) $a_b$ (AU) $T$ (y) $K_p$ (m/s) $T_p$ (y) $a_p$ (AU) $M_p$ ($M_{\oplus}$)
----- --------------------- --------------------- ---------- ------------ --------- ------------- ----------- ------------ ----------------------
1 0.3 0.3 10 1.0 $10.26$ $0.541$ $0.685$ $0.777$ $16.00$
2 1.0 1.0 10 1.0 $10.26$ $0.987$ $0.375$ $0.520$ $23.90$
3 1.0 1.0 50 5.0 $114.7$ $0.447$ $4.197$ $2.60$ $23.90$
4 1.0 1.0 50 2.5 $114.7$ $0.039$ $1.482$ $1.30$ $1.494$
5 1.0 1.0 100 10.0 $324.4$ $0.312$ $11.86$ $5.20$ $23.90$
6 1.0 1.0 100 5.0 $324.4$ $0.028$ $4.197$ $2.6$ $1.494$
Scaling of the observables
--------------------------
In the previous section, we presented examples of several configurations of a triple star system where the binary is at $a=10$ AU from the target star. However, if the binary’s stars are not both faint, it may be difficult to obtain precise radial velocity measurements because the target star’s spectrum is likely to be contaminated by light received from the binary’s stars. This will cause spectral line blending and it may be very difficult to separate the target star’s motion from the binary’s motion. Therefore, a realistic scenario would be a distant binary system (e.g. 50 or 100 AU) or a close binary (e.g. 10 AU) composed of faint stars (e.g. M-type).
In Table 2, we show some examples of triple systems composed of a solar mass star and a binary system on circular coplanar orbits. In examples 1 and 2, the planets could be detected by the current most precise instruments (e.g. HARPS, @Harps2003). On the other hand, the effect of the binaries in examples 3 and 5 is slightly below the current detection limit but the long periods imply that we would need a long observation timespan to constraint the orbits. Finally, the planets mimicked by the binaries in examples 4 and 6 are not detectable by the current techniques but could be accessible to the planned exoplanet search program EXPRESS0/CODEX [@EXPRESSOCODEX2007].
Our results can be easily scaled:
- If we increase $a$ and $a_b$ by a factor $\gamma$, then the ratio $a_{b}/a$ does not change, $K_p$ decreases by $\gamma^{-1/2}$ (Eq. \[amplitudek\]), $a_p$ increases by a factor $\gamma$ (Eq. \[semajoraxis\]), and $M_p$ remains unchanged (Eq. \[minmasspl\]), while the periods increase by $\gamma^{3/2}$.
- If we decrease $M_1$ and $M_2$ by a factor $\gamma$, then $K_p$ decreases by $\gamma^{1/2}$ (Eq. \[amplitudek\]), $T_p$ increases by $\gamma^{-1/2}$, $a_p$ increases by a factor $\gamma^{-1/3}$ (Eq. \[semajoraxis\]), and $M_p$ decreases by $\gamma^{1/3}$ (Eq. \[minmasspl\]), while $T$ remains unchanged.
The triple system HD 188753
---------------------------
The system HD 188753 is composed of a star (A) and a close binary (Ba+Bb) with an orbital period of $155$ days that orbits the star (A) with a $25.7$ year period [@Konacki2005]. A hot Jupiter with a $3.35$ day period orbit around the A star was announced by @Konacki2005 but this was later challenged by @Eggenberger_etal2008 who claimed they could not find evidence for this planet. This shows the difficulty in identifying planets of stars with a nearby binary system. As noted previously, when observing the target star to obtain its spectrum we are likely to also detect light from the nearby stars. In the case of HD 188753, the contribution of Bb is modest since this star is faint but we cannot ignore the contribution of Ba, which is blended within the target star A’s spectrum [@Eggenberger_etal2008].
The parameters of this triple system are $M_{\star}=1.06\,M_{\odot}$, $M_{1}=0.96\,M_{\odot}$, $M_{2}=0.67\,M_{\odot}$, $a=12.3$ AU, $a_{b}=0.67$ AU, $e=0.5$, $e_{b}=0.1$, and $I=34^\circ$ [@Konacki2005]. We can apply our theory to estimate the effect of the binary on the star assuming $50$ data points and $t_{obs}=500$ days, which are approximately the parameters reported in @Eggenberger_etal2008. Assuming that the triple system is coplanar, we obtain radial velocity oscillations with an 85 day period (frequency $\sim 2\,n_b$) and amplitude 0.5 m/s (i.e., 5.5 times the value obtained with a circular coplanar model). This can be interpreted as a fake planet of 4 Earth masses located at 0.38 AU. As $t_{obs} \ll T=25.7$ years, we cannot resolve additional frequencies near $2\,n_{b}$. However, the predictions made here will change for different values of the angle variables (phases and relative inclination) and if we increase the observation time span. Nevertheless, the precision obtained for this system is currently only $60$ m/s [@Eggenberger_etal2008], hence this effect is not detectable.
Planets in binaries
-------------------
In , we reviewed all currently known planets of stars which are part of a moderately close binary system ($a\la 100$ AU). Our aim was to see whether these could be fake planets due to a binary composed of the companion star, $M_1$, and another unresolved star, $M_2$. We computed the parameters of the hidden binary component ($M_2$ and $a_b$) that could mimic these planets (Table 1, ) and saw that $M_2$ would either be too massive to be realistic or at least too massive to be unreported.
The results in are only valid in the context of triple star systems on circular coplanar orbits. Here, we saw that when the triple system’s orbits are eccentric, the magnitude of the binary’s effect can increase with respect to the circular case (Secs. 4.1 and 4.3). Therefore, in the eccentric case a planet can be mimicked by a less massive binary than in the circular case.
The most interesting cases in Table 1 , namely $\gamma$Cep and HD196885, are binary systems with eccentric orbits. The star $\gamma$Cep A orbits $\gamma$Cep B with $e=0.41$ and $I=119^\circ$ [@Neuhaeuser_etal_2007; @Mugrauer_etal_2008], while $t_{obs}\approx T/2$ [@Torres_2007]. The star HD196885 A orbits HD196885 B with $e=0.46$ but $I$ is unknown while $t_{obs}\approx T/4$ [@Correia_etal_2008]. However, extrapolating from the results in Sect. 4.1, if there were hidden companion stars close to $\gamma$Cep B or HD196885 B, we would expect to be able to detect, in each case, several nearby peaks and not single peaks at the planets’ orbital frequencies.
Moreover, the stars $\gamma$Cep A+B have been directly imaged and $\gamma$Cep B’s brightness is consistent with that of a single M4-type dwarf [@Neuhaeuser_etal_2007]. Additionally, we performed some 3-body fits to the data using a model with HD196885 A+B and an unresolved star nearby HD196885 B. We saw that these fits were worse (residuals at least 31 m/s and poorly constrained orbital elements) than fits using a model with the binary star system (HD196885 A+B) and a planet companion to HD196885 A (residuals of 11 m/s in agreement with @Correia_etal_2008). Therefore, the planets around $\gamma$Cep A and HD196885 A, respectively, are not likely to be due to hidden star companions to $\gamma$Cep B or HD196885 B.
Conclusion
==========
We have studied a triple system composed of a star and a binary system. The star and binary system have eccentric and inclined orbits. This is an extension of earlier work where we assumed that the star and binary system are on circular coplanar orbits .
We demonstrated that if we are unaware of the binary system’s presence (one or even both components may be unresolved for instance because they are faint M stars) we may then be led to believe that the star has one or even two planet companions. Although the radial velocity variations due to a binary are distinct from those due to planet(s), in practice, the measured effect depends on the instrument’s precision and the observation time span.
We have shown that because of the limited instrumental precision, we may only detect periodic terms with well separated frequencies hence we mistake these for planet(s). We also saw that, if the observation time span is shorter than the star’s long-period motion around the binary’s center of mass, we may not be able to resolve terms with nearby frequencies, which means that we cannot distinguish fake planet(s) from a binary.
We have also demonstrated that the binary’s effect is more likely to be mistaken for planet(s) when the radial velocity oscillations are composed of large dominant periodic terms with well separated frequencies. This is more likely to happen in the case of coplanar orbits observed equator-on. Moreover, we have seen that when the orbits are eccentric, the magnitude of the binary’s effect can increase with respect to the circular case. Nevertheless, our model is valid for any triple system’s configuration.
We have presented an example of a binary system that affects a nearby star’s motion. When the star’s long-period motion has low to medium eccentricity, the binary can mimic planet(s) of 10 to 50 Earth masses, which are near the current detection limit. However, if the long-period motion has a high eccentricity we are more likely to detect multiple signals with very close frequencies and therefore reject the planet hypothesis. An exception occurs when the long period motion has a high eccentricity but its period is poorly constrained (due to a short observational time span) in which case we may mistake the binary for a planet of about 100 Earth masses.
We showed that when the binary has an eccentric orbit it can mimic two planets with periods approximately in the ratio 2/1. Therefore we may be misled to think that we found planets in the 2/1 orbital resonance when we have an eccentric hidden binary. This is somehow analogous to the case studied by @Anglada-Escude2010 where two planets on circular orbits in the 2/1 orbital resonance can be mistaken for a single planet with an eccentric orbit. However, our scenario is probably more realistic since planets in orbital resonances are not likely to have circular orbits.
We propose that new planet detections in close binary systems, especially Earth-sized objects that are the targets of the planned search program EXPRESSO/CODEX [@EXPRESSOCODEX2007], be checked carefully because they could indeed be artifacts caused by a hidden binary. This could be done by comparing fits to the data using (1) a model composed of the binary star system and a planet with (2) those obtained for a model composed of a hierarchical triple star system. If the fits of (2) are at least as good as (1), then the hidden binary hypothesis should be considered.
We acknowledge financial support from FCT-Portugal (grant PTDC/CTE-AST/098528/2008).
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[^1]: We performed these expansions using the computer algebra software Maple.
[^2]: This is not always true since the binary’s contribution may prevent a Keplerian fit, in particular if the data set covers the long-period $T=2\pi/n$.
|
---
abstract: 'There are two natural choices for a volume form on the algebraic group $\Gl_n/\Q$: the first is the integral form (unique up to sign), the other is the product of the primitive classes in algebraic de Rham cohomology. We work out the explicit comparision factor between the two.'
author:
- Annette Huber and Wolfgang Soergel
title: 'Comparing natural volume forms on $\Gl_n$'
---
Introduction
============
Consider the group $\Gl_n$ and its Lie algebra $\gl_n$. Both are defined over $\Z$. The isomorphism $$H^i(\gl_n,\Q)\to H^i_\dR(\Gl_n,\Q)$$ ([@Ho] Lemma 4.1) can be used to define an integral structure in algebraic de Rham cohomology as the image of integral Lie algebra cohomology.
Let $$\rho_\Z^\dR\in H^{n^2}_\dR(\Gl_n,\Q)$$ be the image of a generator of $$H^{n^2}(\gl_n,\Z)=\bigwedge^{n^2}\gl_n^*$$ where $\gl_n^*$ is the $\Z$-dual of the integral Lie algebra $\gl_n$.
Note that $\rho_\Z^\dR$ is only well-defined up to sign.
Let $p_i^\dR\in H^{2i-1}_\dR(\Gl_n,\Q)$ be the primitive element normalized as suspension of the universal Chern class $c_i^\dR\in H^{2i}(B\Gl_n,\Q)$.
We call $$\omega^\dR=p_1^\dR\wedge\dots\wedge p_n^\dR\in H^{n^2}_\dR(\Gl_n,\Q)$$ the [*Borel element*]{}.
The Borel element occurs in his definition of a regulator on higher algebraic $K$-theory of number fields. In [@borel] he relates it to special values of Dedekind $\zeta$-functions of number fields, at least up to a rational factor.
The purpose of this note is to verify the following comparison result:
\[mainresult\] $$\omega^\dR=\pm \left(\prod_{j=1}^{n}(j-1)!\right)\rho_\Z^\dR$$
Our strategy is to use the comparison isomorphism between de Rham cohomology and singular cohomology, which is compatible with Leray spectral sequences, products and Chern classes. The structure of singular cohomology of $\Gl_n(\C)$ with integral coefficients is well-known and in particular the product of the primitive classes is an integral generator of $H^{n^2}_\sing(\Gl_n(\C),\Z)$. It remains to compare it with $\rho_\Z^\dR$. This is done by integrating the differential form $\rho_\Z^\dR$ over a fundamental cycle, i.e., over $\Un$.
The interest for this result comes from an ongoing joint project of the first author and G. Kings relating the unkown rational factor in Borel’s work to the Bloch-Kato conjecture for Dedekind-$\zeta$-functions.
[*Acknowledgements:*]{} We would like to thank A. Glang, S. Goette, G. Kings and M. Wendt for discussions and H. Klawitter for a numerical check in low degrees.
Singular cohomology
===================
Let $E\Gl_n$ be the simplicial scheme with $E_n\Gl_n=\Gl_n^k$ with boundary maps given by projections and degeneracies by diagonals. It carries a natural diagonal operation of $\Gl_n$. The classifying space $B\Gl_n$ is the quotient of $E\Gl_n$ by this action.
We view $\Gl_n(\C)$ etc. as topological spaces with the analytic topology. Let $\Un$ be the unitary group as real Lie group.
\[sing\]Let $c_j^\sing\in H^{2j}_\sing(B\Gl_n(\C),\Q)$ be the universal $j$-th Chern class in singular cohomology. Let $$s_j:H^{2j}_\sing(B\Gl_n(\C),\Z)\to H^{2j-1}_\sing(\Gl_n(\C),\Z)$$ be the suspension map. Let $p_j^\sing=s_j(c_j^\sing)$. Then:
1. $$H^*_\sing(B\Gl_n(\C),\Z)=\Z[c_1^\sing,c_2^\sing,\dots,c_n^\sing]$$ as graded algebras.
2. With $P_n=\bigoplus_{j=1}^n\Z p_j^\sing$ we have $$H^*_\sing(\Gl_n(\C),\Z)=\bigwedge^*_\Z P_n$$ as graded Hopf-algebras.
Let $S^i$ be the $i$-sphere. Integral cohomology of the group is computed as $$H^*(S^{2n+1}\times S^{2n-1}\times\dots\times S^1,\Z)$$ in [@Bo1] Proposition 9.1. This means it is an exterior algebra on generators $y_1,\dots,y_n$. By loc. cit. Proposition 19.1 (b) $H^*_\sing(B\Gl_n(\C),\Z)$ is a polynomial algebra on the same generators.
Let $T$ be the diagonal torus of $\Gl_n(\C)$ and $W$ the Weyl group (i.e. the symmetric group). Then we have ([@Hu] Ch. 18, Theorem 3.2) $$H^*_\sing(B\Gl_n(\C),\Z)= H^*_\sing(B T(\C),\Z)^W=\Z[c_1^\sing,\dots,c_n^\sing]$$ This implies that $y_i$ can be identified with the universal Chern class $c_i^\sing$.
This is the statement in the form usually used in algebraic topology. From the point of view of complex or algebraic geometry it would be more natural to view $c_j$ as an element of $H^{2j}_\sing(B\Gl_n(\C),(2\pi i)^j\Z)$. There is a hidden choice of $i$ or orientation on $\C$ behind the translation from one point of view to the other.
\[fund\] The product $$\omega_\sing=p_1^\sing\wedge\dots p_n^\sing$$ is a generator of $H^{n^2}_\sing(\Gl_n(\C),\Z)$. It is the dual of the fundamental class of $$[\Un]\in H_{n^2}^\sing(\Un,\Z)\isom H_{n^2}^\sing(\Gl_n(\C),\Z)$$
The first statement is contained in the proposition. $\Un\subset \Gl_n(\C)$ is a homotopy equivalence. $\Un$ is compact, orientable and connected, hence $H_{n^2}^\sing(\Un,\Z)$ is generated by the manifold $\Un$ itself ([@GH] Theorem 22.24).
De Rham cohomology
==================
Let $X$ be a smooth algebraic variety over $\Q$. Its [*algebraic de Rham cohomology*]{} is defined as $$H^j_\dR(X)=H^j(X,\Omega^*_X)$$ the hypercohomology of the algebraic de Rham complex.
Recall that there is a natural isomorphism of functors $$\sigma: H^j_\sing(X(\C),\Z)\otimes_\Z\C\to H^j_\dR(X)\otimes_\Q\C$$ It is induced by the inclusion $\Z_X\to\C_X$ of sheaves for the analytic topology on $X(\C)$ and the quasi-isomorphism $$\C_X\to \Omega^{\an,*}_X$$ with the holomorphic de Rham complex (holomorphic Poincaré Lemma) on the one hand and the comparison between algebraic and holomophic de Rham cohomology on the other hand. In particular, $\sigma$ is compatible with products.
Let $c_j^\dR\in H^{2j}_\dR(B\Gl_n)$ be the universal $j$-th Chern class in algebraic de Rham cohomology. Let $$s_j:H^{2j}_\dR(B\Gl_n)\to H^{2j-1}_\dR(\Gl_n)$$ be the suspension map. Let $p_j^\dR=s_j(c_j^\dR)$. Then:
1. $$H^*_\dR(B\Gl_n)=\Q[c_1^\dR,c_2^\dR,\dots,c_n^\dR]$$ as graded algebras.
2. With $P_n=\bigoplus_{j=1}^n\Q p_j^\dR$ we have $$H^*_\dR(\Gl_n)=\bigwedge\Q^*P_n$$ as graded Hopf-algebras.
There are different arguments for this fact. Once algebraicity of the Chern classes is known, the result follows directly from Proposition \[sing\] and the existence of the comparison isomorphism.
\[comparison\] The comparison isomorphism $\sigma$ is compatible with Chern classes. More precisely, $$\sigma((2\pi i)^jc_j^\sing)=c_j^\dR$$
Recall that the $j$-th Chern class is the $j$-th elementary symmetric polynomial in the $1$-st Chern class of diagonal torus (splitting principle). Hence it suffices to consider the case $j=1$.
For singular cohomology (or rather cohomology of sheaves on $\Gl_1(\C)=\C^*$) consider the exact sequence of sheaves $$0\to \Z\xrightarrow{2\pi i} \Oh^\an\xrightarrow{\exp} \Oh^{\an *}\to 1$$ $c_1^\sing$ is the image of the invertible function $z$ (the coordinate function of $\C^*$) under the connecting homomorphism. For algebraic or holomorphic de Rham cohomology consider the morphism of complexes $$\Oh^*[-1]\to \Omega^*\hspace{3ex}f\mapsto \frac{df}{f}$$ $c_1^\dR=\frac{dz}{z}$ is the image of the invertible function $z$ under this morphism of complexes. The two constructions are nearly (but not quite) compatible with the definition of the comparison functor $\sigma$ which asks for $\Z$ to be naturally embedded into the constant functions $\C\subset\Oh^\an$. This gives the factor $2\pi i$ as claimed.
\[comparisontop\] Let as before $\omega^\dR=p_1^\dR\wedge\dots\wedge p_n^\dR\in H^{n^2}_\dR(\Gl_n)$. Then $$\sigma ((2\pi i)^{\frac{n(n+1)}{2}}\omega^\sing)=\omega^\dR$$
By Proposition \[comparison\] and compatibility of $\sigma$ with the suspension map we have $$\sigma ((2\pi i)^jp_j^\sing)=p_j^\dR$$ Moreover, $\sigma$ is compatible with products.
\[rho\] For $i,j=1,\dots,n$ let $z_{ij}$ be the natural coordinate on $n\times n$-matrices. Recall that $\rho_\Z^\dR$ is the integral generator of $H^{n^2}(\gl_n,\Z)\subset H^{n^2}_\dR(\Gl_n)$. Then $$\rho_\Z=\frac{1}{\det^n}\bigwedge_{i,j=1}^n dz_{ij}$$
The complex $\Omega^*(\Gl_{n,\Q})$ is quasi-isomorphic to $\bigwedge^*\gl_{n,\Q}^*$ where elements in $\gl_{n,\Q}^*$ are viewed as left-invariant differential forms (see [@Ho] Lemma 4.1). The differential form in the statement is clearly $\Gl_n$-invariant. In order to check that it is an integral basis, it suffices to restrict to the tangent space of $1\in\Gl_n$. There it is the standard generator.
All computations are up to sign, hence we do not have to specify a prefered ordering of the coordinates.
A volume computation
====================
\[volumecomp\] Let $z_{ij}$ be the standard holomorphic coordinates on $\Gl_n(\C)$. Then $$\int_{\Un}\frac{1}{\det^n}\bigwedge^n_{ij=1} dz_{ij}
=\pm \prod_{\nu=0}^{n-1}\frac{(2\pi\op {i})^{\nu+1}}{\nu!}$$
Before going into the proof, we review integration of differential forms over fibres of a bundle, thereby fixing notation. Consider $p:X\to Y$ a fibre bundle with smooth compact fibres of dimension $c$ and a $C_\infty$-volume form $\omega$ on $Y$. Recall the definition of the volume form $\int_p\omega$ on $Y$: for every $y\in Y$ and tangent vectors $v_1,\dots,v_q\in T_yY$, the volume form $\omega[v_1,\dots,v_q]$ on $p^{-1}(y)$ assigns to all $x\in p^{-1}y$ and $w_1,\dots,w_c\in T_x p^{-1}(y)$ the value $$\omega[v_1,\dots,v_q](w_1,\dots,w_c)=\omega(w_1,\dots,w_c,\tilde{v}_1,\dots,\tilde{v}_q)$$ where $\tilde{v}_i$ is a preimage of $v_i$. The form $\omega[v_1,\dots,v_q]$ is independent of the choice of these $\tilde{v}_i$. Then $$\left(\int_p\omega\right)(v_1,\dots,v_q)=\int_{p^{-1}(y)}\omega[v_1,\dots,v_q]$$
Recall $$\rho_\Z=\frac{1}{\det^n}\bigwedge^n_{ij=1} dz_{ij}$$ We argue by induction on $n$. For $n=1$ we have $$\int_{S^1}\frac{dz}{z}=2\pi i$$ by Cauchy’s formula.
Suppose now the formula holds true for $n$. We abbreviate the value by $\pm C(n)$. The claim reads $$C(n+1)=\pm \frac{(2\pi i)^{n+1}}{n!}C(n)$$
We consider the diagram $$\xymatrix{
A \ar@{|->}[r] & \op{diag}(1,A)& (z_0, \ldots, z_n)\ar@{=}[d]\\
{\op{U}} (n) \ar@{^{(}->}[r] & {\op{U}} (n+1) \ar[r]\ar[d]^-p & \mathbb
C^{n +1}\owns (x_0 + {i}y_0, \ldots, x_n +
{i}y_n) \\
&S^{2 n+1} \ar@{^{(}->}[r] &\mathbb R^{2n+2}\ar[u]^-\wr \owns (x_0,y_0,=
\ldots, x_n,y_n)\ar@{|->}[u]
}$$ with the left vertical $p$ given by application to the first vector of the standard basis $ \vec{a}_0 =(1,0, \ldots, 0)^\top $ $\in \mathbb R^{2n+2}$.
We integrate $\rho_\Z$ over the fibres. The resulting form $\int_p\rho_\Z$ is $\U(n+1)$-invariant and uniquely determined by its value in $\vec{a}_0$, which we are going to compute. Let $\vec{v}_1,\dots,\vec{v}_{2n+1}\in T_{\vec{a}_0}S^{2n+1}$ be tangent vectors. Then $\rho_\Z[v_1,\dots,v_{2n+1}]$ is an $\Un$-invariant form and uniquely determined by its value in the unit matrix $E$.
We choose as basis of the tangent space of $S^{2n+1}$ in $\vec a_0$ the other vectors in the standard basis of $\R^{2n+2}$ and denote them $$\vec b_0, \vec a_1, \vec b_1, \ldots, \vec a_n,
\vec b_n .$$ Let $\rho_0$ be the unique $\U(n+1)$-equivariant form on $S^{2n+1}$ with $$\rho_0(\vec{b}_0,\vec{a}_1,\dots,\vec{b}_n)=1$$
For later use, we record that the surface of the unit ball in dimension $2n+2$ is computed by $$\tag{*}\label{vol} \int_{S^{2n+1}}\rho_0=2\frac{\pi^{n+1}}{n!}$$
We have to choose preimages in the tangent space ${ T}_E {\op{U}}(n+1)$ for these vectors. We can use arbitrary hermitian matrices $A_\nu$ for $1 \leq \nu\leq n$ und $B_\nu$ for $0 \leq \nu \leq n$ such that $$A_\nu \vec a_0 =\vec a_n\hspace{3ex}B_\nu\vec a_0 = \vec b_\nu.$$ A simple choice are the complex matrices $$\begin{aligned}
A_\nu &= E_{\nu 0} - E_{0\nu}\hspace{5ex}\nu \geq 1\\
B_\nu &={i}E_{\nu 0} + {i} E_{0\nu}\hspace{3ex}\nu \geq 1\\
B_0 &={i} E_{00}\end{aligned}$$ Here we are using the usual notation for the standard basis of the matrix ring over $\C$ but with indices starting from $0$.
It is more convenient to pass to complexified tangent spaces. ${ T}^{\mathbb C}_{\vec a_{0}} S^{2n +1}$ has the simpler basis $$\begin{aligned}
&\vec{v}_0=i\vec{b_0}\\
&\vec{v}_\nu=( \vec a_{\nu} - {i}\vec b_\nu)/2\hspace{7ex}\nu=1,\dots,n\\
&\vec{v}_{n+\nu}=( \vec a_\nu + {i} \vec b_\nu)/2\hspace{4ex}\nu=1,\dots,n\end{aligned}$$ Its lift to ${ T}^{\mathbb C}_{E} {\op{U}}(n+1)$ is given by $$\begin{aligned}
&\tilde{v}_0=-E_{00}\\
&\tilde{v}_\nu=+E_{\nu 0}\hspace{7ex}\nu=1,\dots, n\\
&\tilde{v}_{n+\nu}=-E_{0\nu}\hspace{4ex}\nu=1,\dots,n\end{aligned}$$ By evaluating in a standard basis of $ T_E^\C\op{U}(n)$ we get $$\rho_\Z[\vec{v}_0,\dots,\vec{v}_{2n}]=\pm dz_{11} \wedge \ldots \wedge dz_{nn}
=\rho_\Z^{\Un}$$ By inductive hypothesis this implies $$\left(\int_p\rho_\Z\right)(\vec{v}_0,\dots,\vec{v}_{2n})=\pm C(n)$$ We now translate back to the original basis. We easily find for $\nu\geq 1$ $$\vec{v}_\nu\wedge \vec{v}_{n+\nu}=\frac{1}{4}(\vec{a}_\nu-i\vec{b}_\nu)\wedge (\vec{a}_\nu+i\vec{b}_\nu)=\frac{i}{2}\vec{a}_\nu\wedge\vec{b}_\nu$$ and hence $$\begin{gathered}
\bigwedge_{i=0}^{2n}\vec{v}_i= \pm( \vec b_0 \wedge \vec a_1\wedge \vec b_1 \wedge \ldots
\wedge \vec a_n \wedge \vec b_\nu ) \cdot
\frac{i^{n+1}}{2^n}\Rightarrow\\
\left(\int_p\rho_\Z\right)(\vec{b}_0,\vec{a}_1,\vec{b}_1,\dots,\vec{b}_{n})=\pm C(n) i^{n+1}2^n\Rightarrow\\
\int_p\rho_\Z=\pm C(n)i^{n+1}2^n\rho_0\end{gathered}$$ Together with equation (\[vol\]) for the unit sphere this yields $$\begin{aligned}
C(n+1)&= \int_{\U(n+1)}\rho_\Z=\int_{S^{2n+1}}\left(\int_p\rho_\Z\right)\\
&= \pm C(n)i^{n+1}2^{n}\int_{S^{2n+1}}\rho_0\\
&=\pm C(n)\frac{(2\pi i)^{n+1}}{n!} \end{aligned}$$ This proves the claim.
Proof of the main result
========================
We want to compare the elements $\rho_\Z^\dR$ (see Lemma \[rho\]) and $\omega^\dR$ (see Corollary \[comparisontop\]) in $H^{n^2}_\dR(\Gl_n)$. Let $\alpha\in\Q^*$ such that $$\omega_\dR=\alpha \rho_\Z^\dR$$ By Corollary \[comparisontop\] $$\sigma (\omega^\sing)=(2\pi i)^{-\frac{n(n+1)}{2}}\alpha\rho_\Z^\dR$$ The comparison isomorphism between singular cohomology and holomorphic de Rham cohomology can be reformulated as integration. By Corollary \[fund\], Lemma \[rho\] and Proposition \[volumecomp\], this means $$(2\pi i)^{\frac{n(n+1)}{2}}\alpha^{-1}=\int_{\Un}\frac{1}{\det^n}\wedge_{i,j=1}^n dz_{ij}
=\pm \prod_{\nu=0}^{n-1}\frac{(2\pi i)^{\nu +1}}{\nu!}$$ where $z_{ij}$ are the holomorphic coordinates on the space on $n\times n$ matrices. Hence $$\alpha=\pm \prod_{\nu=0}^{n-1}\frac{1}{\nu!}$$ as claimed.
[9999]{} A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57, (1953). 115–207. A. Borel, Cohomologie de ${\rm SL}\sb{n}$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636. M. Greenberg, J.R. Harper, Algebraic topology. A first course. Mathematics Lecture Note Series, 58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. G. Hochschild, Cohomology of algebraic linear groups, Illinois J. Math. 5 (1961) 492–519. D. Husemoller, Fibre bundles. Second edition. Graduate Texts in Mathematics, No. 20. Springer-Verlag, New York-Heidelberg, 1975.
|
\
Andrew Gitlin\
March 2018\
**Abstract**
[1in]{}[1in]{} We say that a category $\mathscr{D}$ is dimension zero over a field $F$ provided that every finitely generated representation of $\mathscr{D}$ over $F$ is finite length. We show that ${\textrm{Rel}(R)}$, a category that arises naturally from a finite idempotent semiring $R$, is dimension zero over any infinite field. One special case of this result is that ${\textrm{Rel}}$, the category of finite sets with relations, is dimension zero over any infinite field.
Introduction and Preliminaries
==============================
We define a representation of a category $\mathscr{D}$ over a field $F$ to be a functor from $\mathscr{D}$ to $Vect_{F}$, the category of vector spaces over $F$. We say that a category $\mathscr{D}$ is dimension zero over a field $F$ provided that every finitely generated representation of $\mathscr{D}$ over $F$ is finite length. The purpose of this paper is to show that ${\textrm{Rel}}$, the category of finite sets with relations, is dimension zero over any infinite field. Our method of argument allows this result to be generalized to categories that we call ${\textrm{Rel}(R)}$, where $R$ is any finite idempotent semiring (Definition 1.8). Bouc and Thévenaz [@correspond] have independently shown that ${\textrm{Rel}}$, the category of finite sets with relations, is dimension zero over any field. Theorem 3.2 states that ${\textrm{Rel}(R)}$ is dimension zero over any infinite field for any finite idempotent semiring $R$.
For the rest of this paper, let $\mathscr{D}$ be a combinatorial category, i.e. a category such that ${\mathrm{Hom}}(a,b)$ is finite for all objects $a,b \in \mathscr{D}$, and let $F$ be a field. We will let $Vect_F$ denote the category of vector spaces over $F$; the objects are vector spaces over $F$ and the morphisms are linear transformations. Finally, for the rest of this paper, let $[n]$ be the set $\{1,...,n\}$ for any whole number $n$.
We will now introduce several notions in representation theory which will be important in this paper.
A of $\mathscr{D}$ over $F$ is a functor from $\mathscr{D}$ to $Vect_{F}$.
Concretely, a representation $V$ of $\mathscr{D}$ over $F$ takes every object $d \in \mathscr{D}$ to a vector space $V(d)$ over $F$, takes every morphism $g \in {\mathrm{Hom}}(d,e)$ to a linear map $V(g) \in {\mathrm{Hom}}(V(d),V(e))$ for all $d,e \in \mathscr{D}$, and satisfies the following two properties.
- $V(f \circ g) = V(f) \circ V(g)$
- $V(Id_d) = Id_{V(d)}$
Let $V$ be a representation of $\mathscr{D}$ over $F$. A of $V$ is a representation $W$ of $\mathscr{D}$ over $F$ such that $W(d)$ is a vector subspace of $V(d)$ for all $d\in\mathscr{D}$ and $W(f)$ is the restriction of $V(f)$ to $W(d)$ for all $d,d'\in\mathscr{D}$ and $f\in{\mathrm{Hom}}_{\mathscr{D}}(d,d')$.
Two particularly easy examples of a representation of $\mathscr{D}$ over $F$ are the zero representation and the trivial representation. The zero representation sends every object of $\mathscr{D}$ to $0$ and every morphism in $\mathscr{D}$ to the zero transformation. The trivial representation sends every object of $\mathscr{D}$ to $F$ and every morphism in $\mathscr{D}$ to the identity transformation.
A representation $V$ is provided that $V$ is not the zero representation and that the only subrepresentations of $V$ are the zero representation and $V$ itself.
A representation $V$ of $\mathscr{D}$ over $F$ is provided that there exist objects $d_1,...,d_i \in\mathscr{D}$ and $v_{1,1},...,v_{1,j_1} \in V(d_1),...,v_{i,1},...,v_{i,j_i} \in V(d_{i})$ such that if $W$ is a subrepresentation of $V$ and $v_{1,1},...,v_{1,j_1} \in W(d_1),...,v_{i,1},...,v_{i,j_i} \in W(d_{i})$ then $W = V$.
A representation $V$ is provided that any non-repetitive chain of subrepresentations of $V$ is finite.
An equivalent definition of finite length is that a representation $V$ is finite length provided that there exists some non-repetitive finite chain $0=W_{0}\subsetneq...\subsetneq W_{n}=V$ of subrepresentations of $V$ such that each $W_{i+1} / W_{i}$ is irreducible. When this is the case, $W_0,...,W_n$ is called a composition series for $V$ and $n$ is called the length of $V$. The Jordan-Hölder Theorem guarantees that if $W'_0,...,W'_m$ is another composition series for $V$ then $m = n$ (and thus the length of $V$ is well-defined) and in fact that the $W'_{i+1} / W'_{i}$ are a permutation of the $W_{i+1} / W_{i}$. A statement and proof of the Jordan-Hölder Theorem, in a more general setting, can be found as Theorem 2.1 in $\cite{JH}$; since $Vect_F$ is an abelian category, the category of functors from $\mathscr{D}$ to $Vect_F$ is also an abelian category and thus this theorem applies here.
A category $\mathscr{D}$ is over a field $F$ provided that every finitely generated representation of $\mathscr{D}$ over $F$ is finite length.
This paper considers a natural construction of a category from a finite idempotent semiring. We will now define what a finite idempotent semiring is and explain this construction.
A is a finite set $R$ equipped with two binary operations, denoted $+$ (addition) and $*$ (multiplication), satisfying the following axioms.
- $+$ is commutative, associative, and idempotent and there exists an additive identity $0 \in R$
- $*$ is associative and there exists a multiplicative identity $1 \in R$
- $*$ distributes over $+$
- $0*a = a*0 = 0$ for all $a \in R$
For the rest of this paper, let $R$ be a finite idempotent semiring.
We will now define a category, which we will denote $\underline{{\textrm{Rel}(R)}}$, which arises naturally from the finite idempotent semiring $R$. The objects are the whole numbers. For any whole numbers $x,y$, a morphism from $x$ to $y$ is a $x \times y$ matrix with elements of $R$ as its entries. The composition of morphisms is given by matrix multiplication.
Throughout the rest of this paper, for any whole numbers $x,y$, if $A \in {\mathrm{Hom}}_{{\textrm{Rel}(R)}}(x,y)$ and the $(i,j)$ entry of $A$ is $a_{i,j}$ for all $i \in [x],j \in [y]$, then we will let $(a_{i,j})$ denote the morphism $A$. Using this notation, we can now express the rule for composing morphisms in ${\textrm{Rel}(R)}$ more explicitly. If $x,y,z$ are whole numbers and $A = (a_{i,j}) \in {\mathrm{Hom}}(x,y)$ and $B = (b_{i,j}) \in {\mathrm{Hom}}(y,z)$, then $$B \circ A = AB = \left ( \sum_{k = 1}^y a_{i,k} * b_{k,j} \right ) \in {\mathrm{Hom}}(x,z).$$
A special case of the above discussion is when $R = \{0,1\}$ and $+$ and $*$ are given by logical OR and logical AND, respectively. In this special case, ${\textrm{Rel}(R)}$ is called the category of finite sets with relations and is denoted ${\textrm{Rel}}$.
Another example of a finite idempotent semiring is the truncated tropical semiring $R = \{0,1,...,n,\infty\}$ where $n$ is a fixed whole number. Addition $\oplus$ and multiplication $\otimes$ on $R$ are defined as follows. $$x \oplus y := \min(x,y) \indent x \otimes y :=
\left\{
\begin{array}{ll}
\min(x+y,n) & \textrm{if } x,y \neq \infty \\
\infty & \textrm{if } x = \infty \textrm{ or } y = \infty
\end{array}
\right.$$ The truncated tropical semiring is a truncated version of the tropical semiring $\mathbb{R} \cup \{\infty\}$ where the addition operation $\oplus$ is given by $x \oplus y := \min(x,y)$ and the multiplication operation $\otimes$ is given by $x \otimes y := x+y$. The reason for the truncation is that a finite idempotent semiring must be a finite set. For an introduction to the tropical semiring, see $\cite{speyer}$.
A Partial Order on ${\textrm{Rel}(R)}$
======================================
For any $a,b \in R$, we write $a \subseteq b$ and $b \supseteq a$ when $a+b = b$.
The following lemma lists some important properties of $\subseteq$.
1. $\subseteq$ is a partial order on $R$ with minimal element $0$
2. for all $a,b \in R$, we have $a \subseteq a + b$
3. for all $a,b,c \in R$, if $a,b \subseteq c$ then $a+b \subseteq c$
The proof of Lemma 2.1 is a series of routine computations and is left as an exercise for the reader.
${\textrm{Rel}(R)}$ is Dimension Zero over any Infinite Field
=============================================================
Fix $d,x,y\in\mathscr{D}$. For any $s\in{\mathrm{Hom}}(x,x)$, define a matrix $M_{s}$ with rows and columns indexed by ${\mathrm{Hom}}(d,x)$ by letting the $(f,g)$ entry be $1$ if $s\circ f=g$ and $0$ otherwise for all $f,g\in{\mathrm{Hom}}(d,x)$. We write $x\leq_{d} y$ when ${\textrm{span}}_{F}\{M_{t} : t\in{\mathrm{Hom}}(x,y,x)\}$ contains the identity matrix, where ${\mathrm{Hom}}(x,y,x)$ is defined to be $\{\lambda\in{\mathrm{Hom}}(x,x) :$ there exists $a\in{\mathrm{Hom}}(x,y), b\in{\mathrm{Hom}}(y,x)$ such that $\lambda = b\circ a\}$.
Proposition 2.5 in [@WG15] says that $\leq_{d}$ is a preorder on the objects of $\mathscr{D}$. Furthermore, Theorem 1.2 in [@WG15] uses $\leq_{d}$ to provide a criterion for determining whether or not $\mathscr{D}$ is dimension zero over $F$.
A category $\mathscr{D}$ is dimension zero over a field $F$ if and only if ${\mathrm{Hom}}_{\mathscr{D}}(a,b)$ is finite for all $a,b\in\mathscr{D}$ and for all $d\in\mathscr{D}$ there exists a finite set $Y_{d}$ of objects of $\mathscr{D}$ such that for all $x\in\mathscr{D}$ there exists some $y\in Y_{d}$ such that $x\leq_{d} y$ over $F$.
Theorem 1.2 in [@WG15] was proven in the more general setting of representations of categories over rings.
Proposition 2.1 in [@WG15] gives one useful property of $\leq_{d}$.
Let $d,x,y\in\mathscr{D}$. If ${\mathrm{Hom}}(d, x)$ is finite, then ${\textrm{span}}_{F}\{M_{t}$ : $t\in{\mathrm{Hom}}(x,y,x)\}$ contains an invertible matrix if and only if $x \leq_{d} y$.
The following proposition provides one method for proving that $x \leq_{d} y$ for some fixed $d,x,y \in \mathscr{D}$.
Let $d,x,y\in\mathscr{D}$. If ${\mathrm{Hom}}(d,x)$ is finite and there exists a partial order $\preceq$ on ${\mathrm{Hom}}(d,x)$ and a function $s: {\mathrm{Hom}}(d,x) \rightarrow {\mathrm{Hom}}(x,y,x)$ such that $s(f) \circ f = f$ and $s(f) \circ h \succeq h$ for all $f,h \in{\mathrm{Hom}}(d,x)$, then $x\leq_{d}y$ over any infinite field $A$.
[**Proof:**]{}
Extend $\preceq$ to a total order $\leqslant$ on ${\mathrm{Hom}}(d,x)$. For all $F \in {\mathrm{Hom}}(x,x)$, let the rows and columns of $M_F$ be arranged from least to greatest according to $\leqslant$. For all $f,g \in {\mathrm{Hom}}(d,x)$, let $b_{f,g}$ be the $(g,g)$ entry of $M_{s(f)}$. Note that $b_{f,f} = 1$ for all $f \in {\mathrm{Hom}}(d,x)$ since $s(f) \circ f = f$ for all $f \in {\mathrm{Hom}}(d,x)$. Also note that $b_{f,g}$ is either 1 or 0 for all $f,g \in {\mathrm{Hom}}(d,x)$.
We will first show that there exists $\{a_f \in A : f \in {\mathrm{Hom}}(d,x)\}$ such that $$\sum \limits_{f \in {\mathrm{Hom}}(d,x)} b_{f,g}a_f \neq 0$$ for all $g \in {\mathrm{Hom}}(d,x)$. Let $m = |{\mathrm{Hom}}(d,x)|$ and let ${\mathrm{Hom}}(d,x) = \{f_1,...,f_m\}$. It is enough to show that for all $n \in [m]$ there exist $a_{f_1},...,a_{f_n} \in A$ such that $$\sum \limits_{i = 1}^{n} b_{f_i,f_1}a_{f_i} \neq 0,...,\sum \limits_{i = 1}^{n} b_{f_i,f_{n}}a_{f_i} \neq 0.$$ We will use induction on $n$. If $n = 1$, then, since $b_{f_1,f_1} = 1$, we have that $a_{f_1} = 1$ is a solution to $b_{f_1,f_1}a_{f_1} \neq 0$ as desired. Suppose that $n \geq 2$ and the result holds for $n-1$. By the inductive hypothesis, there is a solution $a_{f_1},...,a_{f_{n-1}}$ to the system of equations $$\sum \limits_{i = 1}^{n-1} b_{f_i,f_1}a_{f_i} \neq 0,...,\sum \limits_{i = 1}^{n-1} b_{f_i,f_{n-1}}a_{f_i} \neq 0.$$ For all $k \in [n]$, let $S_k = - \sum_{i = 1}^{n-1} b_{f_i,f_k}a_{f_i}$. Note that $S_k \neq 0$ for all $k \in [n-1]$. Since $A$ is infinite and $\{S_k : k \in [n]\}$ is finite, there exists $a_{f_n} \in A \backslash \{S_k : k \in [n]\}$. For all $k \in [n-1]$, we have that $$\begin{array}{ll}
&\sum \limits_{i = 1}^{n} b_{f_i,f_k}a_{f_i} = b_{f_n,f_k}a_{f_n} + \sum \limits_{i = 1}^{n-1} b_{f_i,f_k}a_{f_i} = b_{f_n,f_k}a_{f_n} - S_k =
\left \{
\begin{array}{ll}
-S_k & \textrm{if } b_{f_n,f_k} = 0 \\
a_{f_n} - S_k & \textrm{if } b_{f_n,f_k} = 1
\end{array}
\neq 0.
\right.
\end{array}$$ Furthermore, noting that $b_{f_n,f_n} = 1$, we have that $$\sum \limits_{i = 1}^{n} b_{f_i,f_n}a_{f_i} = b_{f_n,f_n}a_{f_n} + \sum \limits_{i = 1}^{n-1} b_{f_i,f_n}a_{f_i} = a_{f_n} - S_n \neq 0.$$ Thus, as desired, $a_{f_1},...,a_{f_n}$ is a solution to the system of equations $\sum_{i = 1}^{n} b_{f_i,f_1}a_{f_i} \neq 0,...,\sum_{i = 1}^{n} b_{f_i,f_{n}}a_{f_i} \neq 0$.
Let $X = \sum_{f\in{\mathrm{Hom}}(d,x)} a_f M_{s(f)}$. For all $g \in {\mathrm{Hom}}(d,x)$, the $(g,g)$ entry of $X$ is $\sum_{f \in {\mathrm{Hom}}(d,x)} b_{f,g}a_f$, which is non-zero. Furthermore, since $s(f) \circ h \succeq h$ for all $f,h \in{\mathrm{Hom}}(d,x)$, $M_{s(f)}$ is upper triangular for all $f \in {\mathrm{Hom}}(d,x)$ and thus $X$ is upper triangular. Thus, since $X$ is an upper triangular matrix with all of its diagonal entries being non-zero, $X$ is invertible. Therefore, since $X\in {\textrm{span}}_A\{M_{t}$ : $t\in{\mathrm{Hom}}(x,y,x)\}$, we are done by Proposition 3.1. $\square$\
We are now ready to prove the following theorem, which is the main result of this paper.
If $R$ is a finite idempotent semiring, then ${\textrm{Rel}(R)}$ is dimension zero over any infinite field.
[**Proof:**]{}
Let $R$ be a finite idempotent semiring.
For all $x,y \in {\textrm{Rel}(R)}$, let $0_{x \times y}$ be the $x \times y$ matrix with each entry being 0. For all $x \in {\textrm{Rel}(R)}$, let $Id_x$ be the $x \times x$ matrix with each diagonal entry being 1 and every other entry being 0. For all $x,y,z \in {\textrm{Rel}(R)}$, $0_{x \times y}A = 0_{x \times z}$ for all $A \in {\mathrm{Hom}}(x,y)$ and $B0_{y \times z} = 0_{x \times z}$ for all $B \in {\mathrm{Hom}}(x,y)$. For all $x,y \in {\textrm{Rel}(R)}$, $Id_xA = A$ for all $A \in {\mathrm{Hom}}(x,y)$ and $BId_y = B$ for all $B \in {\mathrm{Hom}}(x,y)$.
Let $n = |R|$. It is enough to show $x\leq_{d}n^{d}$ for all $d,x\in {\textrm{Rel}(R)}$, since then we can apply Theorem 3.1 with $Y_d = \{n^d\}$ for all $d \in {\textrm{Rel}(R)}$. Fix $d,x \in {\textrm{Rel}(R)}$. If $x\leq n^{d}$ then the below calculation shows that $Id_{x}\in{\mathrm{Hom}}(x,n^{d},x)$ and therefore, since $M_{Id_x}$ is the identity matrix, we are done. $$\begin{bmatrix}
Id_x & 0_{x \times (n^d-x)}
\end{bmatrix}
\begin{bmatrix}
Id_x \\ 0_{(n^d-x) \times x}
\end{bmatrix}
=
Id_x$$
For the $x > n^{d}$ case, we will use Proposition 3.2.
We will first construct $\preceq$. For any $f = (a_{i,j}),g = (b_{i,j}) \in{\mathrm{Hom}}(d,x)$, we will write $f \preceq g$ if and only if $a_{p,q} \subseteq b_{p,q}$ for all $p \in [d],q \in [x]$. Since $\subseteq$ is a partial order by Lemma 2.1, $\preceq$ is also a partial order.
We will now construct $s : {\mathrm{Hom}}(d,x) \rightarrow {\mathrm{Hom}}(x,n^{d},x)$. For any $f = (a_{i,j}) \in{\mathrm{Hom}}(d,x)$, let $s(f)$ be the matrix $M=(m_{i,j}) \in {\mathrm{Hom}}(x,x)$ where $$m_{i,j}=
\left \{
\begin{tabular}{l}
$1$ if $a_{k,i} \subseteq a_{k,j}$ for all $k\in[d]$ \\
$0$ otherwise
\end{tabular}
\right.$$ for $i,j\in[x]$. The proposition below verifies that $s(f)\in{\mathrm{Hom}}(x,n^{d},x)$.
We have $s(f)\in{\mathrm{Hom}}(x,n^{d},x)$.
[**Proof:**]{}
Since there are only $n^{d}$ distinct $d$-tuples with entries from $R$, there are at most $n^{d}$ distinct columns in $f$. Note that if columns $p \in [x]$ and $q \in [x]$ of $f = (a_{i,j})$ are identical, then for all $i \in [x]$
$$m_{i,p} =
\left \{
\begin{tabular}{l}
$1$ if $a_{k,i} \subseteq a_{k,p} = a_{k,q}$ for all $k\in[d]$ \\
$0$ otherwise
\end{tabular}
\right.
= m_{i,q}$$
and thus columns $p$ and $q$ of $M$ are identical. Therefore, $M$ has at most $n^{d}$ distinct columns. Let $v$ be the number of distinct columns of $M$, noting that $v \leq n^d$. Let $D \in {\mathrm{Hom}}(x,n^d)$ be the $x \times n^{d}$ matrix which has the $v$ distinct columns of $M$ as its first $v$ columns and has every entry in its $n^d - v$ remaining columns as $0$. Let $E = (e_{i,j}) \in {\mathrm{Hom}}(n^d,x)$ be a $n^{d} \times x$ matrix where $$e_{i,j} =
\left \{
\begin{tabular}{l}
$1$ if $i \leq v$ and column $j$ of $M$ is column $i$ of $D$ \\
$0$ otherwise
\end{tabular}
\right.$$ for all $i \in [n^d],j \in [x]$. For all $i,j \in [x]$, we have $$\sum_{k \in [n^d]} d_{i,k} * e_{k,j} = \sum_{\substack{k \in [v] \\ \textrm{col. $j$ of $M$ is col. $k$ of $D$}}} d_{i,k} = \sum_{\substack{k \in [v] \\ \textrm{col. $j$ of $M$ is col. $k$ of $D$}}} m_{i,j} \stackrel{\dagger}{=} m_{i,j}.$$ The $\dagger$ step uses the idempotence of $+$. Thus $M=DE$. Thus $s(f)\in{\mathrm{Hom}}(x,n^{d},x)$ as desired. $\square$\
We are left to show that, for all $A \in{\mathrm{Hom}}(d,x)$, we have $s(A)\circ A = A$ and $s(A)\circ B \succeq B$ for all $B\in{\mathrm{Hom}}(d,x)$. Fix $A=(a_{i,j}) \in{\mathrm{Hom}}(d,x)$. Let $M=(m_{i,j})$ be $s(A)$.
We will first show that $s(A) \circ B \succeq B$ for all $B\in{\mathrm{Hom}}(d,x)$. Fix $B = (b_{i,j}) \in{\mathrm{Hom}}(d,x)$. Let $C=(c_{i,j})$ be $s(A) \circ B$, noting that $C=BM$. By definition, it is enough to show $c_{i,j} \supseteq b_{i,j}$ for all $i \in [d],j \in [x]$. For all $j\in[x]$, we have $a_{k,j} \subseteq a_{k,j}$ for all $k\in[d]$ and thus $m_{j,j}=1$. Thus, for all $i \in [d],j \in [x]$, we have $c_{i,j} = \sum_{l=1}^{x}(b_{i,l} * m_{l,j}) \supseteq b_{i,j} * m_{j,j}= b_{i,j} * 1=b_{i,j}$ as desired.
We will now show that $s(A) \circ A = A$. Let $C=(c_{i,j})$ be $s(A)\circ A$, noting that $C=AM$. We have $s(A) \circ A \succeq A$ by the previous paragraph. Therefore, it is enough to show that $s(A) \circ A \preceq A$. By definition, it is enough to show that $c_{i,j} \subseteq a_{i,j}$ for all $i \in [d],j \in [x]$. Fix $i \in [d],j \in [x]$. Fix $k\in[x]$. Note that $m_{k,j}$ is either $0$ or $1$. If $m_{k,j} = 0$ then $a_{i,k} * m_{k,j} = 0 \subseteq a_{i,j}$. If $m_{k,j} = 1$ then we have $a_{r,k} \subseteq a_{r,j}$ for all $r\in[d]$ and thus $a_{i,k} \subseteq a_{i,j}$ and thus $a_{i,k} * m_{k,j} = a_{i,k} \subseteq a_{i,j}$. Therefore $a_{i,k} * m_{k,j} = a_{i,k} \subseteq a_{i,j}$ for all $k\in[x]$, independently of $m_{k,j}$. Thus $c_{i,j} = \sum_{k=1}^{x}(a_{i,k} * m_{k,j}) \subseteq a_{i,j}$ as desired.
This completes the proof. $\square$\
Recall that if $R = \{0,1\}$ and $+$ and $*$ are given by logical OR and logical AND, respectively, then ${\textrm{Rel}(R)}$ is called the category of finite sets with relations and is denoted ${\textrm{Rel}}$. Therefore, the following corollary is a special case of Theorem 3.2.
The category of finite sets with relations, ${\textrm{Rel}}$, is dimension zero over any infinite field.
In fact, Bouc and Thévenaz [@correspond] have independently shown that ${\textrm{Rel}}$ is dimension zero over any field. Additionally, they computed the irreducible representations of ${\textrm{Rel}}$ (Theorem 17.19 in [@correspond]) and showed that the fundamental correspondence functors (Definition 4.7 in [@correspond]) appear as subfunctors of a particular functor that arises from a lattice (Theorem 14.16 in [@correspond]).
Acknowledgements
================
I would like to thank John Wiltshire-Gordon and David Speyer for working with me and for providing me with an invaluable undergraduate research experience. I would also like to thank John Wiltshire-Gordon for providing mathematical insights and motivation which pushed me towards a proof of the main result in this paper. I would like to thank the organizers of the University of Michigan REU program for giving me a wonderful opportunity to conduct mathematical research over the summer of 2015. Finally, I would like to thank Stephen DeBacker for setting me up with the REU opportunity. This research was partially supported by NSF Department of Undergraduate Education award 1347697 (REBUILD).
[2]{} Serge Bouc and Jacques Thévenaz. Correspondence functors and finiteness conditions, J. Algebra 495 (2018), 150-198. C. S. Seshadri. Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. 85 (1967), 303-336. David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag. 82 (2009), no. 3, 163-173. John D. Wiltshire-Gordon. Categories of dimension zero, 2015. [](http://arxiv.org/abs/1508.04107).
|
[**Brian Batell$^{\,(a)}$, Maxim Pospelov$^{\,(a,b)}$, Adam Ritz$^{\,(b)}$, and Yanwen Shang$^{\,(a)}$**]{}
$^{\,(a)}$[*Perimeter Institute for Theoretical Physics, Waterloo, ON, N2J 2W9, Canada*]{}
$^{\,(b)}$[*Department of Physics and Astronomy, University of Victoria,\
Victoria, BC, V8P 1A1 Canada*]{}
0.2in
**Abstract**
Secluded dark matter models, in which WIMPs annihilate first into metastable mediators, can present novel indirect detection signatures in the form of gamma rays and fluxes of charged particles arriving from directions correlated with the centers of large astrophysical bodies within the solar system, such as the Sun and larger planets. This naturally occurs if the mean free path of the mediator is in excess of the solar (or planetary) radius. We show that existing constraints from water Cerenkov detectors already provide a novel probe of the parameter space of these models, complementary to other sources, with significant scope for future improvement from high angular resolution gamma-ray telescopes such as Fermi-LAT. Fluxes of charged particles produced in mediator decays are also capable of contributing a significant solar system component to the spectrum of energetic electrons and positrons, a possibility which can be tested with the directional and timing information of PAMELA and Fermi.
1. Introduction {#introduction .unnumbered}
---------------
The search for weakly interacting massive particles (WIMPs) as a component of non-baryonic dark matter has become a focal point of modern particle physics [@review]. There are several complementary experimental and observational approaches to WIMP detection [@WIMPS]. Direct detection experiments probe the terrestrial scattering of WIMPs with nuclei and typically require low radiation environments to keep backgrounds under control. High energy colliders such as the Tevatron and the LHC offer the possibility of producing WIMPs and measuring their properties in the laboratory, provided the challenging missing energy signatures can be disentangled. Indirect searches for dark matter annihilating into gamma and cosmic rays in the galactic halo are also promising, although susceptible to various, often uncertain, astrophysical backgrounds. Finally, neutrino telescopes such as Super-Kamiokande and Ice Cube can search for indirect evidence of the annihilation of WIMPs captured in the core of the Sun and the Earth, in the form of an observable muon signature arising from neutrino charged current scattering in the detector. While the latter two examples are well-known indirect signatures for any thermal relic WIMP dark matter candidate, more generic WIMPs forming part of a larger dark sector can lead to further novel signatures. In this paper, we demonstrate that models of secluded dark matter [@PRV] present an additional observational possibility: high-energy gamma rays and charged particles arriving from a direction tightly correlated with the centers of the Sun, Earth and other planets. Such novel signatures can be effectively probed with the powerful new generation of gamma ray telescopes.
The primary feature of secluded models of dark matter [@PRV] is a two-stage dark matter annihilation process: WIMPs annihilate first into metastable mediators, which subsequently decay into Standard Model (SM) states. This breaks the more-or-less rigid link between the size of the WIMP annihilation and WIMP-nucleus scattering cross sections. It has been shown that a small mass for the mediator allows for new phenomenological possibilities in the form of enhanced WIMP annihilation at small velocities [@AFSW; @PR] that may help to explain various astrophysical anomalies, e.g. the positron excess observed by PAMELA above 10 GeV [@pamela] and perhaps the unexpectedly hard electron spectrum observed by Fermi above a few hundred GeV [@FERMI]. Furthermore, a relatively small mediator mass kinematically removes heavy SM particles from the final state [@AFSW; @PR], reconciling these effects with the absence of any enhancement in the cosmic ray anti-proton signal [@pamela2]. The lifetime of the mediator is essentially a free parameter, limited only by the Big Bang Nucleosynthesis bounds of $\tau \la 1\,$s. If this lifetime is rather long, the decay of the mediator will occur a long distance away from the point of the original WIMP annihilation. Denoting the WIMP particle $\chi$ and the mediator particle $V$, assuming $\chi\chi\to 2V$ as the main annihilation channel, and taking $m_V \ll m_\chi$, we arrive at the following estimate for the mediator travel distance: L = c \_V \_V = 310\^6 [km]{} . \[L\] Such large boosts $\gamma_V = m_\chi/m_V$ are easily achieved if the the dark matter mass is near the electroweak scale and the mediator mass is below a GeV. With regard to the annihilation of WIMPs captured within the Sun, one can see that this distance may very well exceed the solar radius ($R_\odot = 6.96\times 10^{5}$ km) in which case, unlike conventional WIMPs, most of the decay products will not be absorbed. For somewhat shorter lifetimes, the interesting possibility emerges of mediators produced by WIMP annihilation in the center of the Earth decaying directly into charged particles within neutrino telescopes. A schematic illustration of this new indirect mechanism of probing secluded WIMP models is shown in Fig. \[fig-scheme\].
A sub-GeV mass mediator will decay predominantly into light states, such as pions, muons, electrons, gammas and neutrinos. As they are produced in the decays of highly boosted mediators, these decay products will be tightly correlated with the direction to the original WIMP annihilation point, with a typical angular size $\theta_V \sim 1/\gamma_V$. While for charged particles this correlation can be reduced by the magnetic fields encountered on the way to the detector, the directionality of gammas and neutrinos is unaltered. Modern gamma ray telescopes enjoy an angular resolution much better than a degree, which may be exploited to enhance the gamma ray signal-to-background. In particular, a notable source of background to this signature is the generation of gamma rays via cosmic rays impinging on the Sun (see, [*e.g.*]{} [@Seckel; @Strong]). Such background gammas will typically display a much softer spectrum than gamma rays from mediator decays and will have no specific correlation with the solar center where most of the secluded WIMP annihilation is expected to take place. The main gain in sensitivity in detecting gammas from the decays of metastable mediators, as compared to a more conventional search for a highly-energetic neutrino signal, may come from the increase in efficiency. While the detection of multi-GeV neutrinos requires their conversion to muons, which means a loss in efficiency of around ten orders of magnitude, the efficiency of detecting gammas created outside of the solar radius can be order one.
In addition to gamma rays, many secluded WIMP models with relatively light mediators are destined to produce a significant fraction of leptons in the final state, and thus we are naturally led to the question of whether mediator decays outside the solar/planetary radii are capable of contributing significantly to the fluxes of electrons and positrons seen by PAMELA and Fermi. It is tempting to pursue the notion that these anomalies may in fact have a local origin in the solar system, powered by the annihilation of secluded dark matter trapped within solar bodies. This is an intriguing possibility, as it would offer a new dark matter interpretation of these signatures that does not rely on [*galactic*]{} WIMP annihilation, which generally requires a significant boost factor in the annihilation cross section. In this context, a less extreme hierarchy between the WIMP and mediator mass may be feasible, and this idea is akin to using local substructure to enhance the charged particle flux, albeit with a source (the Sun) which is extremely local and well-understood. These electrons and positrons, like the gamma rays discussed above, should be strongly correlated with the center of the Sun, particularly in the high energy range where the effect of the magnetic fields of the Sun and Earth is less significant. Such a hypothesis appears straightforwardly falsifiable using directional and timing data from PAMELA and Fermi. Finally, if charged particles are the primary signature, this immediately leads to a minimal gamma ray flux from the associated 3-body decays that is also directionally correlated with the center of the Sun, and thus testable by Fermi-LAT.
In this paper we analyze the feasibility of detecting electromagnetic particles, $\gamma$, $e^\pm, \cdots$, arising from the delayed decays of metastable mediators produced through WIMP annihilation in the deep interior of the Sun and other planetary bodies within the solar system. Our primary focus is on the capability of modern gamma ray telescopes to search for these local annihilation signatures and the possibility of distinguishing the signal gammas from the solar backgrounds produced by cosmic rays. We demonstrate that in certain corners of the secluded dark matter parameter space, the gamma-ray signature from the center of the Sun is potentially the most sensitive probe, superior to direct detection and other indirect signatures. This is especially the case for models in which WIMP-nucleus scattering is dominated either by spin-dependent interactions or proceeds through an inelastic transition to a nearby excited state. We also analyze the prospects for detecting gamma rays generated by dark matter annihilations inside Jupiter using atmospheric Čerenkov gamma-ray detectors, as well as the potential for observing pairs of upward going muons in neutrino telescopes generated by the decay of mediators to muons. Finally we offer some preliminary speculations concerning an interpretation of PAMELA’s rising positron fraction originating within the solar system, and discuss the obstacles to this identification as well as ‘smoking gun’ directional and temporal signatures which can confirm or rule out such an interpretation.
The rest of this paper is organized as follows. In the next section we present the WIMP trapping rates and general formulae for calculating the WIMP-powered gamma ray flux, including angular and energy distributions. Section 3 contains estimates of the gamma ray signature in two variants of secluded models, with pseudoscalar(‘axion’)- and vector-mediation. We conclude in Section 4 with a discussion of future prospects for improving sensitivity to these dark matter models via observation of gamma rays and charged particles in the solar system.
2. Capture and delayed electromagnetic decays of mediators {#capture-and-delayed-electromagnetic-decays-of-mediators .unnumbered}
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In this section we derive the general formulae for the expected gamma ray fluxes generated by metastable mediator decays. We will also consider the case of charged particle fluxes and comment on the expected effect of the magnetic fields of the Sun and Earth.
### 2.1 Solar capture and $\gamma$-flux {#solar-capture-and-gamma-flux .unnumbered}
We begin by providing the trapping efficiency of WIMPs inside the Sun, assuming elastic scattering of WIMPs on nuclei [@Gould; @Kamionkowski; @review] and normalizing unknown quantities on some fiducial values: C\_ 1.3 10\^[21]{} [s]{}\^[-1]{} ( ) \_N f\_N () S(m\_/m\_N) F\_N(m\_), \[scaling\] where the sum runs over the nuclei $N$ present in the Sun, $f_N$ denotes the fractional abundance relative to hydrogen, $\sigma^{\rm SD}_N$ $(\sigma^{\rm SI}_N)$ is the spin-dependent(spin-independent) scattering cross section and we have used the standard values $\rho_\chi = 0.3$ GeV cm$^{-3}$ and $\bar{v} = 270$ km/s for the local WIMP density and velocity dispersion. The function $S(x)=[A(x)^{3/2}/(1+A(x)^{3/2}) ]^{2/3}$, where $A(x) = (3/2)[x/(x-1)^2](v_{esc}/{\bar v})^2$, with $v_{esc} \simeq 1156$ km/s an ‘effective’ escape velocity, is a kinematic suppression factor, while $F_N(m_\chi)$ determines additional suppression from a nuclear form-factor. This approximate formula holds for both spin-independent and spin-dependent scattering, but it is important to bear in mind that coherent scattering in the former case means that heavier nuclei, such as He, O, and Fe tend to dominate the capture rate despite their reduced abundance. The lack of nuclear coherence in spin-dependent scattering means that accounting for scattering off hydrogen is generally sufficient. More general formulae may be found in Ref. [@Gould].
For most models the trapping rate in the Sun determines the overall annihilation rate, as the two processes, trapping and annihilation, are usually in dynamical equilibrium. With an $S$-wave annihilation rate of order 1 pb as dictated by the relic abundance, a per-nucleon scattering cross-section of order $10^{-48}$cm$^2$ or larger is generally sufficient for thermalization to occur over the lifetime of the Sun. Then the trapping rate $C_{\odot}$ and the probability of a single annihilation event to produce gamma quanta reaching the Earth, which we denote as $P_{\gamma}$, determines the overall flux of solar WIMP-generated gamma rays at the Earth’s location:
$$\begin{aligned}
\Phi_{\gamma\odot} = {\frac{1}{2}} \times {\frac{C_\odot P_{\gamma} }{4\pi ~ ({\rm A.U.})^2}}
= 1.8 \times 10^{-7} ~{\rm cm}^{-2}~{\rm s}^{-1} \times
{\frac{C_\odot P_{\gamma}}{10^{21}~{\rm s}^{-1}}}.
\label{flux}\end{aligned}$$
This equation is correct if the particles that produce the final photons are highly boosted along the radial direction of the Sun, and should be modified by another factor of $1/2$ if those particles travel very slowly and emit photons isotropically as the Sun is effectively opaque and photons emitted on the back side of the Sun are reabsorbed. We are only interested in the former case in the current discussion. The master formula (\[flux\]) suggests that observable gamma ray fluxes are indeed possible, provided that $P_\gamma$ is not too small. In conventional neutralino-like WIMP scenarios, this probability is in fact negligible, as the only practical way of producing gamma rays is via high-energy neutrinos interacting with the outer layer of the solar material[^1], and in this case the probability $P_\gamma$ is very small. In contrast, secluded WIMPs offer the possibility of maximizing $P_\gamma$, which arises as the product of: (i) the probability $P_{\rm out}$ that the mediator particle decays outside the solar radius; (ii) the probability ${\rm Br}_{\gamma}$ of producing a gamma quantum in the decay of the mediator; and (iii) the probability ${\rm Br}_{V}$ of producing the mediator particle in the annihilation process: P\_= gP\_[out]{} \_ \_V. \[pgamma\] Here, $g=g_\gamma g_V$ where $g_\gamma$ and $g_V$ are the multiplicities of the photon and the mediator particle $V$ being produced along the chain of reactions. If the loss of $V$ due to re-scattering inside the sun can be neglected, the mediator escape probability $P_{out}$ is well-approximated by P\_[out]{} {- R\_/(c)}, while the branching ratios ${\rm Br}_{V}$ and ${\rm Br}_{\gamma}$ and the multiplicities are model dependent. For example, ${\rm Br}_{\gamma}$ may be close to 1 in some secluded models, or more commonly may lie in the $10^{-3}-10^{-2}$ range for models where gammas are produced as radiation accompanying the decay to charged particles.
The general relations (\[scaling\]) and (\[flux\]) use elastic cross sections normalized to $10^{-42} $ cm$^2$, in conflict with the current bounds on spin-independent elastic WIMP-nucleon scattering [@SI], but well below the bounds on spin-dependent scattering [@SD]. Therefore, the best sensitivity to secluded WIMPs will occur for models where nucleon scattering is predominantly spin-dependent. Alternatively, scattering may be predominantly inelastic $\chi_1\to \chi_2$, with a small mass gap between the two WIMP states relaxing the most stringent constraints on spin-independent $\sigma_p$. We will consider examples in both these categories in Section 3.
### 2.2 Angular and spectral distributions {#angular-and-spectral-distributions .unnumbered}
We now turn to the angular and spectral distributions of gamma quanta. We will comment at the end of this section on how these distributions may be generalized for charged particles, such as $e^\pm$, for which the propagation effects of the magnetic fields of the Sun and Earth must be taken into account. For simplicity we shall assume a decay of the mediator into a pair of gammas, which are monochromatic and isotropic in the rest frame of the mediator. The Lorentz boost of the mediator then determines both the angular and energy resolution. In the absence of the boost, $\gamma \sim 1$, one would expect the Sun to acquire a constant surface brightness in gamma-rays, assuming that the decays happen not far from the solar radius, although its side would appear brighter due to simple geometrical reasons. However, in the most interesting case of large boosts, $\gamma > 10^2$, the gamma quanta in the decay products are emitted within an angle $\theta \sim 1/\gamma$ from the original direction of the mediator. For $\gamma$ as large as $10^3$, the majority of gamma rays would come from an angular spot in the sky smaller than the solar radius, and such a tight correlation is observable with modern gamma ray telescopes. Also, the gamma energy spectrum would be peaked in the direction of the solar center, although this effect is significantly smoothed out by the finite angular resolution of the detectors. In this subsection, we illustrate these angular distributions and energy spectra for the case of $m_\chi = 1$ TeV WIMPs annihilating into two metastable mediators, that further decay into $2\gamma$, so that the multiplicity factors in Eq. (\[pgamma\]) are $g_V=g_\gamma=2$ .
In the rest frame of the intermediate particle $V$, photons are emitted isotropically with a momentum distribution $f'(p')=\frac{2}
{\pi m^2_V}\delta(p'-m_V/2)$, where $p=|\vec p|$. After a Lorentz transformation, the distribution function $f(\vec p)$ measured in the observer’s frame becomes $$f(p,\alpha)=\gamma(1-\beta\cos\alpha)f'[\gamma(1-\beta\cos\alpha)p],$$ where $\beta\equiv v/c$, and $\alpha$ is the angle between the momentum of the outgoing photon and that of the particle $V$ measured in the observer’s frame. Consequently, photons are emitted with an angular distribution given by $1/[4\pi\gamma^2(1-\beta\cos\alpha)^2]$. The density of the photons arriving at the detector with an incoming angle $\theta$ whose momenta are within a small solid angle ${\textrm{d}}\Omega$ is calculated by integrating the particle density of $V$ along the line of sight weighed by an appropriate angular distribution factor: $$\label{eq:better}
\frac{{\textrm{d}}\Phi(\theta)}{{\textrm{d}}\Omega}
=\frac{\cos\theta\;\textrm{Br}_\gamma}{2\pi\tau}\int_D {\textrm{d}}\cos\alpha\;
\frac{n(l\sin\theta\csc\alpha) l \sin\theta}
{\gamma^3\sin^3\alpha(1-\beta \cos\alpha)^2}\,.$$ Here $l \approx 1\,\textrm{A.U.}$ is the Sun-detector distance and $n(r)$ is the number density of $V$ at distance $r$ from the center of the Sun. Given our assumptions, $$n(r)=\frac{C_\odot \cdot \textrm{Br}_V}{4\pi v r^2}\exp\{-r/(v\tau\gamma)\}\,,$$ where $v\simeq c(1-1/(2\gamma^2))$ is the velocity of $V$.
A few words about the range of the integral for ${\textrm{d}}\cos\alpha$, which we have denoted as $D$ above, are in order. If the Sun were transparent, $D$ would simply be the domain $[-1,\, +1]$. In the current case, however, the Sun is effectively opaque since photons traveling through the interior are instantaneously absorbed or degraded in energy through their interactions with solar material. Thus one should simply discard those photons from consideration, which leads to a lower bound for the integral at $\cos\alpha=\sqrt{1-(l\sin\theta)^2/R^2_\odot}$ whenever $l\sin\theta<R_\odot$. Consequently, $$D=
\begin{cases}
\left[\sqrt{1-(l\sin\theta)^2/R^2_\odot}\,,\;+1\right],
& \quad l\sin\theta<R_\odot\;; \\
\left[-1\,,\; +1\right],& \quad l\sin\theta\geq R_\odot\;.
\end{cases}$$ The lower integration limits are not important for $\gamma\gg 1$, as almost all the photons detected come from within a small angle $\sim 1/\gamma$ and the flux is completely negligible when $\theta\approx \theta_\odot$. We illustrate ${\textrm{d}}\Phi/{\textrm{d}}\Omega$ for a representative value of $\gamma=10^3$ in Fig. \[fig:spec\_smeared\]. Given a realistic angular resolution, it is clear that all events associated with the decays of highly boosted mediators will be concentrated in the angular bin covering the solar center.
The energy distribution of $\Phi$ follows from a similar integral. In general, we can write the full differential distribution in the form, $$\begin{split}
\label{eq:spectrum_gen}
\frac{{\textrm{d}}^2 \Phi(\theta, p)}{p^2{\textrm{d}}p \,{\textrm{d}}\Omega}
=&\frac{l\sin\theta \cos\theta \textrm{Br}_\gamma}
{\gamma\tau}\int_D {\textrm{d}}\cos\alpha \cdot
\frac{f(p,\alpha)\, n(l\sin\theta\csc\alpha)}{\sin^3\alpha}\\
=&\frac{l\sin\theta \cos\theta\; \textrm{Br}_\gamma}{\tau}
\int_D \frac{{\textrm{d}}\cos\alpha}{\sin^3\alpha} \cdot
(1-\beta \cos\alpha) f'[\gamma p(1-\beta \cos\alpha)]
n(l\sin\theta\csc\alpha)\,,
\end{split}$$ where $D$ is the same domain explained above. Specializing to the case of two-body decays of $V$ to photons, we have $$\label{eq:spectrum}
\frac{{\textrm{d}}^2 \Phi(\theta, p)}{p^2{\textrm{d}}p\, {\textrm{d}}\Omega}
=\frac{(\gamma^2-1)\sin\theta\cos\theta l\; \textrm{Br}_\gamma}
{2\pi\gamma\tau(2\gamma p_0 p-p_0^2-p^2)^{3/2}}\cdot\frac{p}{p_0}\cdot
n\left(\frac{\sqrt{\gamma^2-1}\sin\theta l p}
{\sqrt{2\gamma p_0 p-p_0^2-p^2}}\right)\,,$$ where $p_0\equiv m_V/2=m_\chi/(2 \gamma)$ is the photon energy measured in the rest frame of $V$. When $l\sin\theta\geq R_\odot$, this expression is valid as long as $$p\in \left[\;
p_0\sqrt{\frac{1-\beta}{1+\beta}}\;,\;
p_0\sqrt{\frac{1+\beta}{1-\beta}}
\;\right]\;.$$ and is understood to vanish for $p$ outside this range. For $l\sin\theta < R_\odot$, the lower limit is again modified to $p_0 \gamma^{-1}(1-\beta\sqrt{1-l^2\sin^2\theta/R^2_\odot}\,)^{-1}$ due to absorption.
The resulting photon spectrum varies dramatically within a tiny range of $\theta$ that for the Lorentz boosts considered here is typically well below the angular sensitivity of any gamma ray telescope. For a detector with angular resolution poorer than the angular size of the Sun, the integration of over ${\textrm{d}}\Omega$ produces a flat spectrum as long as $p_0/\gamma\le p\le p_0\sqrt{\frac{1+\beta}{1-\beta}}$. However, some detectors, such as Fermi, have an angular resolution smaller than the solar angular size. For demonstration purposes, we take $\gamma\sim 1000$, and average assuming a Gaussian-profile and an angular resolution $\Delta \theta$ of one-tenth the solar size. Since $\gamma$ is quite large, almost all the photons that reach the detector come from within an angle of $1/\gamma$ that is comparable to the smallest angular size the detector can resolve. Consequently, the signal that would be seen by the detector represents a (relatively) bright central spot of the Sun with an almost exactly flat spectrum. The photon flux drops rather quickly with angle. If detectable, the photon spectrum at $\theta > \Delta \theta$ is again mostly flat except for a very sharp peak near the low momentum end. We illustrate these results in Fig. \[fig:spec\_smeared\].
![\[fig:spec\_smeared\] [Angular and spectral distributions of the photon flux generated by two-photon decays of the mediators with $\gamma v\tau=0.5 R_\odot$ and $\gamma=1000$. (a) The normalized angular distribution $\frac{{\textrm{d}}\Phi(\theta)}{{\textrm{d}}\Omega}/(C_\odot \textrm{Br}_V \textrm{Br}_\gamma
\cdot 10^{-21}\textrm{cm}^{-2})$; and (b) the fully differential flux $\frac{{\textrm{d}}^2\Phi(p, \theta)}{{\textrm{d}}p\,{\textrm{d}}\Omega}/
(C_\odot \textrm{Br}_V \textrm{Br}_\gamma
m_\chi^{-1} \cdot 10^{-21}\textrm{cm}^{-2})$ in the direction $\theta=0$, averaged over a Gaussian profile that mimics an angular resolution of $\Delta\theta=0.1 \theta_\odot$; (c) The same for $\theta = 0.1 \theta\odot$.]{} ](signal){width="100.00000%"}
Up to this point we have only discussed the spectrum of photons resulting from the decays of long-lived mediators. A similar analysis for charged particles, such as electrons and positrons, is less straightforward due to the complications of the magnetic fields of the Sun and Earth, as well as the solar wind, and their effects on the propagation, energy degradation, and absorption. A proper calculation of the angular and energy distributions is beyond the scope of this work, but we wish to give a qualitative discussion of these effects, paying particular attention to the implications regarding a possible local solar system component to the PAMELA signal.
A variety of secluded models may be constructed which can lead to electrons and positrons in the final state. It is important to stress that the effects of the magnetic fields and solar atmosphere on these charged states is quite model dependent, and primarily sensitive to the lifetime and boost of the mediator produced in the WIMP annihilation as well as the production modes of the final state charged particles. If the mediators have a typical decay length on the order of the solar radius or slightly less, processes such as absorption and reflection have the potential to strongly degrade the overall signal flux, and the strong solar magnetic fields may drastically alter the trajectories of the charged particles. If the mediator escapes and decays well past the solar radius, the effects of the heliosphere and the Earth’s magnetosphere can be still be significant, especially for less energetic particles in the tens of GeV range and below (where the bulk of the anomalous PAMELA positrons reside).
We must emphasize that the main feature of this signal would be a high degree of anisotropy in the predicted flux of electrons and positrons, correlated with the center of the sun, which should make this scenario testable. An important consideration is to what degree this tight correlation may be affected or degraded through the processes discussed above. It is instructive to compare this signal with a possible pulsar component to the rising positron fraction of PAMELA [@pulsar], where one of the main signatures may again be an anisotropic signal. While the latter anisotropy may be somewhat difficult to detect, this feature in the solar component discussed here would be more pronounced due to the proximity of the Sun and the underlying production mechanism. This provides additional motivation for studies of the positional, directional, and temporal features of the existing PAMELA and Fermi data sets.
In order to go further and actually attempt to fit the PAMELA and Fermi data, a better understanding of the predicted energy spectrum would be required, which goes beyond the scope of this paper. Of course, this is also quite model dependent. For example, a simple underlying two-body decay of the mediator to $e^+e^-$ pairs will result in a very hard spectrum, which may not fit the observed spectrum. However, the spectrum can be softened through cascade decays, or the re-scattering of mediators inside the solar interior. It would also be important to understand to what extent the electron/positron energy spectrum is affected or softened by interaction with the solar atmosphere and through propagation from the sun to the earth.
### 2.3 Other local sources {#other-local-sources .unnumbered}
The Sun represents by far the largest astronomical body in the solar system, and presumably the most efficient WIMP capturing reservoir. However, the lifetime of the mediators can be such that their decays occur deep inside the solar interior and the decays outside are exponentially suppressed. Capture by planets may then be important and two cases to consider are WIMPs captured by the Earth and Jupiter. The latter option is of interest because of the possibility of very precise observations by means of atmospheric Čerenkov detectors. The flux of WIMP-generated mediators can be estimated using the basic scaling of (\[scaling\]) suitably rescaling the escape velocity, the mass, and the distance to the Earth. Using the results of [@Gould], and assuming for a moment that $V$ decays happen outside the solar interior, we find \~( )\^4 ()\^2 \~4 10\^[-9]{}, which represents a strong suppression. However, if the decay length of the mediator is less than $5\%$ of the solar radius, the gamma ray flux from Jupiter may exceed that from the Sun. In this case the use of atmospheric Čerenkov detectors such as HESS, MAGIC and VERITAS, could set additional bounds on the capture rate by Jupiter, and consequently on the WIMP-nucleus cross section.
Finally, the fluxes from the center of the Earth are difficult to present in the same compact form as (\[scaling\]), primarily because of the likely dependence on the annihilation rate. These fluxes are typically much smaller than those generated by WIMP annihilation in the solar interior except for some special cases with resonant energy loss [@Gould]. Nevertheless, the annihilation of secluded WIMPs inside the Earth does allow a probe of much shorter mediator lifetimes, and offers additional signatures in neutrino detectors. In certain models, the signature involves pair production of muons via the decays of the mediators, a possibility that can be efficiently explored with neutrino telescopes such as SuperK and IceCube.
### 2.4 Observational sensitivity {#observational-sensitivity .unnumbered}
Some of the primary $\gamma$-ray observatories, such as atmospheric Čerenkov detectors are of limited utility in this case, as they are unable to directly observe the Sun. While, as noted above, they can be used to place limits on the flux from Jupiter, a fiducial WIMP-nucleon cross section of $10^{-40}$cm$^2$ would generate a flux that is generally too low to provide competitive sensitivity from this source. Thus, we will focus on water Čerenkov detectors and space-borne $\gamma$-ray observatories such as Fermi.
The primary limit we will use here comes from the Milagro water Čerenkov detector [@milagro], which has a wide field of view and an angular resolution of 0.75$^\circ$, slightly larger than the disc of the Sun. The background arises primarily from the scattering of high-energy cosmic rays, producing $\gamma$’s which can arrive at the detector from within the Sun’s disc. Nonetheless, the limits obtained by Milagro for monochromatic sources are significant and reach up to 10 TeV. We exhibit the ensuing constraint on a contour plot of the $\gamma$-flux in Fig. \[fig-milagro\_lim\], which for characteristic scattering cross-sections, already imposes a significant constraint on the electromagnetic branching fractions. The primary flux limits obtained by Milagro are for monochromatic sources, while a monochromatic decay of the mediator as discussed above leads to a smoothed spectrum. Given that the background is a steeply-falling function of energy, while the expected signal is not, we adopt the Milagro bounds on monochromatic sources, weakening it by an order of magnitude in obtaining the bound in Fig. \[fig-milagro\_lim\].
In the near future, the Fermi-LAT detector, which is also able to observe the Sun [@lat] with an angular resolution about 10 times better than that of Milagro, should be able to improve on these limits and thus provide a very significant probe of secluded dark matter models. The energy range of Fermi-LAT is also ideally suited to probing secluded WIMP annihilation with masses of a few hundred GeV, where Milagro loses its sensitivity.
It is also possible for long-lived mediators to decay predominantly to electrons and positrons. It is tempting then to consider the implications of these decays in relation to the anomalous positrons fraction observed by PAMELA. As far as we are aware detailed timing and directional analyses have not yet been performed, leaving open the possibility of an intriguing local explanation for these anomalies within the dark matter framework. Given that the existing magnetic fields affect propagation, and the decay chains of the mediators are model-dependent, one could in principle fit the spectral shapes of the observed signals. We have obtained a rough estimate of the required integrated signal flux by using Fermi electron data with an $\sim E^{-3}$ spectrum to infer an estimate of the background and incorporating a new source with a harder power-law spectrum. Depending on the source as well as where the signal turns on (i.e. how large a component of the PAMELA signal) we find a required flux of \_[e\^+e\^-]{}(E>10 [GeV]{}) \~10\^[-4]{} - 10\^[-6]{} [cm]{}\^[-2]{} [s]{}\^[-1]{} \[interesting\] We also point out that there will be an ${\cal O}(1)$ reduction of the observed signal flux due to the fraction of the time the satellite is facing away from the sun. Given the assumption that at least part of the excess positron flux does arise from mediator decays, this leads to a [*minimal*]{} predicted gamma flux due to the accompanying final state radiation. The typical spectrum of photons resulting from this process would be $(\alpha/\pi )dE/E$, and thus the photon flux will generally be no less than 0.1% of the electron and positron flux, and one infers the following target photon flux: \^[min]{}\_(E>10 [GeV]{}) 10\^[-7]{} - 10\^[-9]{} [cm]{}\^[-2]{} [s]{}\^[-1]{}. \[interesting2\] Thus, while it is possible that the hypothesis of a local origin for the flux anomalies can be directly tested with timing and directional information, it also appears that existing EGRET data [@egret; @Strong] may already probe the larger range of this associated gamma flux, while it is feasible that Fermi-LAT will be able to cover the entire range of possible fluxes, with a potential $10^{-9} ~{\rm cm}^{-2} {\rm s}^{-1} $ level sensitivity to multi-GeV gamma rays originating from point sources within a year.
3. Secluded models vs $\gamma$-rays {#secluded-models-vs-gamma-rays .unnumbered}
-----------------------------------
Having discussed the available sensitivity to $\gamma$-rays from the Sun, in this section we will consider some secluded dark matter scenarios which would be subject to this indirect probe. Given that a large variety of model-building possibilities for WIMPs and mediators have been shown to exist [@PRV], we will simply exhibit two classes of models for which the $\gamma$-ray flux due to WIMP trapping in the Sun may, for various parameters, be the most sensitive observable and source of constraints. Our approach will be to fix the parameters such that the decay length of the mediator is sufficient to escape the Sun, and then consider whether the scattering cross-section leading to capture results in a measurable (or constrained) $\gamma$-ray flux according to the limits discussed in the preceding section.
### 3.1 Secluded WIMPs with pseudoscalar mediation {#secluded-wimps-with-pseudoscalar-mediation .unnumbered}
A fermionic dark matter candidate $\chi$ can be secluded by mediating its interaction with the SM via a pseudoscalar ‘axion’ field $a$ with $m_a<m_\ch$. We will imagine a light WIMP, $m_\chi \sim 10 \,{\rm GeV}$, as well as a very light mediator, $m_a <$ 10 MeV. The interactions comprise a series of dimension 5 operators: $$\begin{aligned}
{\cal L} &=& {\cal L}_{\rm SM} + \bar \chi (i\partial_\mu\gamma_\mu - m_\chi) \chi +{\frac{1}{2}} (\partial_\mu a)^2 -{\frac{1}{2}} m_a^2 a^2
\nonumber \\
&+& \partial_\mu a \left( {\frac{1}{f_\chi}} \bar \chi \gamma_\mu \gamma_5 \chi
+ \sum_q {\frac{1}{f_q}}\bar q \gamma_\mu \gamma_5 q + \sum_l {\frac{1}{f_l}} \bar l \gamma_\mu \gamma_5 l \right) +\frac{\al}{4\pi f_\gamma} a F_{\mu\nu}\tilde{F}^{\mu\nu}.
\label{axion}\end{aligned}$$ The coupling constants $f_i$ are above the electroweak scale but otherwise are completely arbitrary at this point. Since this is an effective field theory model, depending on the actual UV completion one can achieve the suppression of either $f_l^{-1}$ or $f_q^{-1}$ or both (see, [*e.g.*]{} [@PRV2]). The axion can be very long-lived if its mass is low enough to ensure that decays to hadrons and heavier leptons are kinematically forbidden.
We will now briefly discuss the interaction rates pertinent to the $\gamma$-ray signal. We note that axion mediation with the WIMP sector has also been explored recently in Ref. [@Nomura].
- [*Annihilation*]{}
Unless $f_\ch$ is parametrically larger than $f_f$, $\bar{\chi}{\chi} \rightarrow a a$ will be the dominant annihilation mode given $m_a \ll m_\chi$, and the thermally averaged rate is v = 2.4 10\^[ -26]{} [cm]{}\^3[s]{}\^[-1]{}, \[sigaa\] where $\beta = (1-4 m_\chi^2/s)^{1/2}$ is the velocity of the WIMPs in the c.o.m. frame. Note that the annihilation is in the $P-$wave, a consequence of identical bosons with overall even parity in the final state. At freeze-out, taking the WIMP velocity to be $\beta^2 \approx 3 T_f^2/m_\chi^2 \approx 3/20$, the relic density requires $f_\chi \simeq 120\,{\rm GeV} \times (m_\chi/10\, {\rm GeV})^{1/2}$.
- [*Pseudoscalar Decays*]{}
For axions in the MeV range, we can consider decays to photons and electrons, \_[a]{} = , \_[ae\^+ e\^-]{} = . Depending on the ratios $f_e/f_\gamma$ and $m_a/m_e$, either the photon or electron branching may dominate the total width. To maximize the photon fraction, we shall assume that $f_\gamma \sim 10$ TeV, while $f_e > {\rm few} \times 10^3$ TeV, in which case the decays are dominated by the 2-photon final states. Moreover, provided the characteristic coupling of the axions to quarks is small, $f_q > 100$ TeV, the axions will be long-lived with a decay length sufficient to escape the Sun, $$L_a= c\tau_a \gamma_a \simeq 1.2 \times 10^7 \, {\rm km} \times \left( \frac{m_\chi}{ 10\,{\rm GeV} } \right)
\left( \frac{5\,{\rm MeV}}{m_a} \right)^4 \left( \frac{f_\gamma}{10\,{\rm TeV}} \right)^2,
\label{travel}$$ where $\gamma_a \simeq m_\chi/ m_a \simeq 2\times 10^3$ for a fiducial normalization of masses and couplings. For the same normalization, one can explicitly check that the absorption cross section of energetic axions in the solar medium is too small to attenuate the flux.
- [*Scattering*]{}
The axion-like pseudoscalar mediates spin-dependent WIMP-nucleus elastic scattering, involving the effective axion-nucleon couplings ${\cal L}_{a(n,p)} = (\tilde{f}_p^{-1} \bar{p} \gamma^\mu \gamma^5 p + \tilde{f}^{-1}_n \bar{n} \gamma^\mu \gamma^5 n)\ptl_\mu a$ where $1/\tilde{f}_{(n,p)} = \sum_q \De^{(n,p)}_q/f_q$ in terms of the parameters measuring the spin-content of the nucleons, $\Delta_u^{(p)}= \Delta_d^{(n)}\simeq 0.8$, $\Delta_d^{(p)}= \Delta_u^{(n)}\simeq -0.5$, $\Delta_s^{(p)}= \Delta_s^{(n)} \simeq -0.15$. A straightforward calculation leads to the tree-level cross-section, conventionally re-expressed in terms of the ‘model independent’ WIMP-nucleon cross section, $$\begin{aligned}
\sigma_{p,n} & \equiv & \frac{ \mu^2_{p,n} }{\pi f_\ch^2 \tilde{f}_{(p,n)}^2} \times
\begin{cases}
1 \qquad\qquad\quad {\rm for} \qquad m_a^2\ll 4\mu_N^2v^2, \\
\displaystyle{\frac{16 \mu_N^4 v^4}{3 m_a^4}} \qquad {\rm for} \qquad m_a^2 \gg 4\mu_N^2 v^2,
\end{cases}\end{aligned}$$ where $\mu_N$ is the reduced mass for the WIMP-nucleus system. Note that depending on the mass of the mediator, WIMP, and type of nucleus involved in the scattering, there is either an enhancement or a suppression. For the trapping rate, we need only consider WIMP scattering with hydrogen as the scattering is spin-dependent, and in the specific case of the Sun, we must also account for the increase of the characteristic c.o.m. velocity (by a factor of 3-5) due to the WIMPs falling into the Sun’s gravitational well. The characteristic momentum transfer in this case is $|{\bf q}| \sim 2 m_N v \approx 8$ MeV, leading us to consider very light mediators in the MeV range to avoid a possible velocity suppression. A light pseudoscalar with mass in the few MeV range will mediate an enhanced interaction due to the long range force. Effectively the momentum dependence cancels and we have the usual form for the spin-dependent cross section: $$\sigma_p \simeq 2.5 \times 10^{-44}\,{\rm cm}^2 \times \left( \frac{500\,{\rm TeV}}{\widetilde{f}_p} \right)^2
\left( \frac{10 \,{\rm GeV}}{m_\chi} \right),
\label{axcross}$$ where we have used the relic abundance to relate $f_\ch$ to $m_\ch$, and we have normalized the effective nucleon couplings to be consistent with the stringent constraints arising from rare $K$ decays, as we will discuss below. We see that direct detection constraints are easily satisfied, as Eq. (\[axcross\]) displays a spin-dependent WIMP-nucleon cross section far smaller than the $\sigma_{(p,n)} < 10^{-37}$ cm$^2$ limit [@SD].
With these results in hand, we see that for light WIMPs and pseudoscalar mediators the solar $\gamma$-ray flux can be appreciable. For such light WIMPs in the GeV range, we see from Eq. (\[scaling\]) that the trapping rate can be enhanced by several orders of magnitude compared to weak scale WIMPs, but this is compensated by a generically smaller-spin dependent cross section shown in Eq. (\[axcross\]).
Interestingly such light mediators are not in conflict with astrophysical bounds when the scale of the interactions $f_q$ is below about $10^6$ GeV, because these states will then thermalize in the core of supernovae and so will not be subject to the stringent constraints from cooling. Constraints from BBN cannot rule out one additional thermalized degree of freedom, but in any case a mass in the few MeV range is sufficient to avoid these constraints entirely. Finally, as alluded to above, rare Kaon decays, in particular searches for $K^+ \rightarrow \pi^+ a$, constrain $f_q$ to be above 100 TeV [@Kaon; @PRV2; @Nomura], but pose no particular problems for the estimate of the spin-dependent scattering cross section in Eq. (\[axcross\]).
Using Eqs. (\[scaling\]), (\[flux\]), and (\[axcross\]), and assuming an ${\cal O}(1)$ branching of axions to photons and a decay length of order the solar radius as discussed above, we obtain the characteristic gamma ray flux in the secluded model with axion mediation: $$\Phi_{\gamma\odot} \sim 6 \times 10^{-8}\, {\rm cm}^{-2}\, {\rm s}^{-1} \times \left( \frac{10 \, {\rm GeV}}{m_\chi} \right)^2
\left( \frac{500 {\rm TeV}}{ \widetilde{f}_p }\right)^2.$$ This is a large flux which is already close to the range probed by satellites like EGRET (although the mass scale is below the sensitivity range of Milagro). We have focused on light WIMPs and mediators, but different parameter ranges may also yield appreciable fluxes at the expense of some fine-tuning of couplings in the quark sector. In particular, a suppression of the coupling to the top quark, with larger light quark couplings, will relax the prohibitive Kaon decay constraints and allow an enhanced capture rate for larger WIMP and mediator masses.
### 3.2 Secluded WIMPs with vector mediation {#secluded-wimps-with-vector-mediation .unnumbered}
Unlike the axion-mediated models, secluded WIMPs lying in a hidden sector with a spontaneously broken gauge symmetry do not require additional UV completion, and this sector naturally couples to the Standard Model through the kinetic mixing portal. WIMP scenarios in this framework are straightforwardly formulated [@PRV], and have been a focal point of theoretical interest in the last year due to the positron data released by PAMELA [@pamela]. A general class of WIMP models involve multi-component states $\ch$, charged under the vector mediator, and the low energy Lagrangian after symmetry breaking involving the WIMP, vector $V$, and the Higgs $h'$, takes the form, \[u1\] [L]{} &=& [L]{}\_[SM]{} + |(iD\_\_- m\_) + [L]{}\_[m]{} -14 V\_\^2 +m\_V\^2 V\_\^2 + V\_\_F\_\
&+& 12 (\_h’)\^2 - 12 m\_[h’]{}\^2( h’)\^2 + h’ V\_\^2 +If charge-conjugation symmetry is broken via ${\cal L}_{\De m}$, the Majorana components of the WIMP may be split in mass by $\De m \sim \lambda m_V/e'$, which can reduce the elastic scattering cross-section and ameliorate constraints on $\ka$ from direct detection. The remaining particle physics constraints require that $\ka$ be below a ${\rm few} \times 10^{-3}$.
The lifetime and decay channels for $V$ and $h'$ were analyzed in [@BPR]. There are two regimes in which either $V$ or $h'$ can be very long-lived. The first refers to $m_{h'} > m_V$ and $\kappa < 10^{-9}$. In this case the Higgs$'$ is short-lived, while the vector may have lifetimes in excess of a millisecond. This case is of no interest for us in this paper, because the trapping rate will scale as $\kappa^2$ and will be extremely small. The second case with long-lived particles is $m_{h'} < m_V$ and $\kappa \ga 10^{-3}$. In this case the extreme longevity of $h'$ comes from the fact that its decay may only proceed at second order in the mixing angle $\kappa$. This kinematic relation renders $h'$ extremely long-lived even for moderately small $\kappa$, while the scattering cross section and hence the trapping rate can remain large. The relevant interaction rates in this case are detailed below.
- [*Annihilation*]{}
Once trapped and accumulated in the center of the Sun, WIMP annihilation may lead to various final states: $VV$, $Vh'$, and $VVV$. The latter may only be possible when annihilation proceeds via capture into an $S=1$ WIMP-onium state [@PR], but otherwise this annihilation cross section is suppressed by an extra coupling constant. Comparison of the $VV$ and $Vh'$ final state branching is straightforward, once we fix the charge assignment for the Higgs$'$ particle and the spin for the WIMP. If we assume unit charge under for the complex Higgs$'$ field and a fermionic WIMP, then a comparison of the two final states gives: 3, where the brackets include an average over the spin orientation. The upper end of this ratio is achieved when the annihilation proceeds via the formation of WIMP-onium in which case the final state with total spin 1, decaying to $Vh'$ is three times more likely than total spin 0, that decays to $2V$. The lower end corresponds to the case when the recombination into WIMP-onium is kinematically forbidden. Notice that this ratio is not changed by Coulomb (Sommerfeld) enhancement of the cross section, as it is identical in both channels. Thus, in this model a minimum of one per every 5 annihilation events results in the production of a possibly very long-lived $h'$ particle that is boosted by $m_\chi/m_{h'}$. Note that there is no strict constraint from ensuring the correct relic abundance in this case, as we can take this as a relation which fixes the U(1)$'$ coupling $\al'$, leaving the WIMP mass as a free parameter. For the remainder of this section we will assume the $\chi\chi \rightarrow VV$ mode dominates, in which case $\alpha' \sim 0.02\times (m_\chi/500\, {\rm GeV})$ [@PRV].
- [*Higgs$'$ decays*]{}
As noted above, the regime of interest here is when $m_{h'}< m_V$ and the Higgs$'$ is long-lived, with the dominant decay $h' \to l\bar{l}$ occurring at order $\Gamma_h \sim \kappa^4 \times ({\rm loop~factor})^2$ [@BPR]. Given $m_V \gg m_{h'} \gg 2 m_f$ and a boost $\gamma_{h'} \simeq m_\ch/m_{h'}$, the $h'$ decay length is $$L_h = c\tau_{h'}\gamma_{h'} \sim 10^7\, {\rm km } \times
\begin{cases}
\displaystyle{ \left( \frac{\kappa}{5\times 10^{-4}} \right)^{-4} \left( \frac{ m_{h'} }{ 500\,{\rm MeV}} \right)^{-2}
\left( \frac{ m_{V} }{ 5\,{\rm GeV}} \right)^2 \,
\qquad {\rm for} \quad m_{h'}> 2 m_\mu}\\
\\
\displaystyle{ \left( \frac{\kappa}{ 5\times 10^{-3}} \right)^{-4} \left( \frac{ m_{h'} }{ 100\,{\rm MeV}} \right)^{-2}
\left( \frac{ m_{V} }{ 500\,{\rm MeV}} \right)^2
\, \quad {\rm for} \quad m_{h'} < 2 m_\mu},
\end{cases}
\label{twocases}$$ where $\al'$ has been chosen to fix the relic abundance, and we have assumed a WIMP mass of 500 GeV. However, the longevity of the Higgs$'$ boson does not guarantee its safe passage through the interior of the sun as there is a potential loss mechanism due to ‘inverse-Primakoff’ type conversion into $V$ on nuclei, $h'+N\to V+N$, followed by prompt $V$ decay [@BPR3]. The cross section for this process on protons can be estimated as follows, \_[abs]{} \~2’\^2 m\_V\^[-2]{} \~410\^[-38]{} [cm]{}\^2 ( )\^2 ( )\^2. \[absorption\] This ${ \cal O}(10^{-38}~{\rm cm}^2)$ scale for the cross section is sufficiently small that absorption of Higgs$'$ will be negligible, but an increase by two orders of magnitude would indeed lead to a significant loss of $h'$ in the solar interior. In particular, a choice of parameters as in the second line of (\[twocases\]) will result in attenuation of the $h'$ flux by more than an order of magnitude. Nonetheless, we see that over much of the parameter space $L_h$ can naturally be large enough for $h'$’s to escape the Sun without scattering even for relatively large values of $\ka$.
Since the dominant decays are electromagnetic, there will be significant photon production through various processes such as internal bremsstrahlung etc. However, it is of particular interest to know the branching to $\gamma$’s originating from sequential decays of pairs of neutral pions in the product of one-loop induced $h'$ decay, or through a two-loop decay directly to $\gamma\gamma$. The direct decay to $\gamma\gamma$ will have the hardest spectrum, while the $\pi_0$-mediated decay is likely to have larger photonic yield than internal bremsstrahlung. We can estimate these branchings by constructing an effective Lagrangian as follows. We first integrate out all quarks that are heavier than the vector. This leads to an effective interaction of Euler-Heisenberg type $\sim {\rm loop} \times (\kappa^2 e^4 /m_Q^4) (F_{\mu\nu})^2 (V_{\alpha\beta}^2)$, and similarly a coupling between vectors and gluons. The contribution of such heavy quarks to the relevant decays will thus be suppressed by powers of $(m_V/m_Q)^4$ which we will neglect. We are left with the light quarks $u,d,s$ as well as perhaps $c, b$ if they are lighter than the vector. The next step is to integrate out the vector from the theory at the loop level, which leads to the effective Lagrangian: $${\cal L}_{\psi} = - c_{ \psi} \sum_{f} Q_f^2 \frac{m_f}{v'} h' \overline{\psi}_f \psi_f,
\label{psi}$$ where $c_{\psi} = - 3 \kappa^2 \alpha/2 \pi$, and the sum runs over all quarks lighter than the vector. From here we can straightforwardly compute the decay $h'\rightarrow \gamma\gamma$ much as in the SM, with the result $$\Gamma_{h'\rightarrow \gamma\gamma} = \frac{\alpha' \alpha^4 \kappa^4}{64\pi^4}\frac{m_{h'}^3}{m_V^2}
\left( \sum_{f} \bigg\vert N_c Q_f^4 I\left( \frac{m_{h'}^2}{ 4 m_f^2} \right)\bigg\vert^2 \right),$$ where the sum is over all fermions lighter than the vector, and $I$ is the familiar form factor arising from the triangle diagram. The factor in the parentheses is of order one. For a heavy Higgs$'$, $m_{h'}> 2 m_\mu$ the dominant decay mode is $h'\rightarrow \mu^+\mu^-$, and the branching to a pair of photons is quite small, on the order of $10^{-4}$. However if $h'$ is lighter than the two muon threshold, $m_{h'}< 2 m_\mu$ the dominant decay mode is $h'\rightarrow e^+ e^-$ with a smaller total width. The branching to two photon pairs in this case can be sizable: $${\rm Br}_{h' \rightarrow \gamma\gamma } \simeq 10^{-2} \times \left( \frac{m_{h'}}{100\;\rm MeV}\right)^2 \qquad {\rm for} \qquad
m_{h'} < 2 m_\mu.$$
For a heavy $h'$ it is sill possible to get a large source of gammas through intermediate decays $h'\rightarrow 2 \pi^0$, followed by the $\pi^0$ fragmenting to photons. The calculation parallels the decay of a light SM Higgs to pions, as detailed in Refs. [@voloshin; @lighthiggs]. We consider the theory at even lower energies, below the charm mass, where we can write the effective Lagrangian in terms of the trace of the QCD energy momentum tensor and match on to a chiral Lagrangian. It is then straightforward to calculate the partial width for $h'\rightarrow 2 \pi^0$: $$\Gamma_{h'\rightarrow \pi^0 \pi^0}= \frac{\alpha' \alpha^2 \kappa^4 }{2^3 3^4 \pi^3 } \frac{m_{h'}^3}{m_V^2}
\left( 1- \frac{4 m_\pi^2}{m_{h'}^2} \right)^{1/2} |G(m_{h'})|^2,$$ where the function $G$ is $$G(m_{h'}) \equiv
\left\{
\left( \sum_F Q_F^2 \right)
+ \frac{m_\pi^2}{m_{h'}^2}
\left[ \left( \sum_F Q_F^2 \right)+\frac{3}{2} \frac{4z+1}{1+z} \right]
\right\},$$ with $z \equiv m_u /m_d \sim 0.56$ and the sum is over the charm and bottom quarks if they are lighter than the vector. This function is numerically ${\cal O}(1)$ for the parameters of interest here. The $2\pi^0$ partial width is somewhat smaller than the $h'\rightarrow \mu\mu$ mode, with a branching of $${\rm Br}_{h' \rightarrow \pi^0 \pi^0 } \simeq 5 \times 10^{-2} \times \left( \frac{m_{h'}}{500\;\rm MeV}\right)^2 .$$ Thus approximately five percent of all Higgs$'$ decays will result in 4 photons when $m_{h'}$ is somewhat larger than $2 m_{\pi}$. We conclude that both for a light or heavy $h'$ it is possible to obtain percent level branchings into photons.
- [*Scattering*]{}
The scattering of WIMPs with nuclei in the secluded vector model was discussed in detail in [@BPR2], and the regime of most interest here is when a small splitting $\De m$ allows for 1st-order inelastic scattering on heavier elements. The interesting feature here is that the larger gravitational potential well of the Sun boosts the c.o.m. kinetic energy of the WIMP-nucleus system relative to the case for terrestrial scattering. This leads to a window for $\De m$ which maximizes the trapping rate in the Sun, but which is outside the kinematic range for terrestrial direct detection. Given $E_{\rm kin} \mu_N,~\Delta m \mu_N \ll m_V^2$: \_[inel]{} = , which is a simple modification of scattering induced by a finite charge radius for a Dirac WIMP. However, this requires that $\De m < E_{\rm kin} \sim m_\ch^2 v^2/(2\mu_N)$. The trapping of inelastically scattering WIMPs, first studied in [@inel] was recently considered in detail in [@inel1; @inel2], and we can directly make use of their results in the present case. Indeed, the trapping in the Sun dominantly occurs through scattering on Fe nuclei, which are lighter than Ge used for example in CDMS. However, the larger velocity relevant for trapping in the Sun, means that $E_{\rm kin}$ can be larger allowing for a window in $\De m$ unconstrained by the direct detection limit. Indeed, the cross-section above can be large, \_[p]{}\^[([Fe]{})]{} \~110\^[-39]{} [cm]{}\^2 ( )\^2 ()\^4 ( ), where we have exhibited the equivalent per-nucleon cross-section for scattering off Fe nuclei, with $\al'$ traded for the WIMP mass through the relic abundance constraint and $m_\ch \gg m_N$. Direct detection constraints then arise from 2nd-order elastic scattering which for large mediator masses in the GeV range are relatively mild, restricting $\ka$ to be below 0.1 [@BPR2].
An alternative route to maximize the trapping rate while satisfying the direct detection constraints would be to introduce several (two or more) sequentially mixed $U(1)$ groups, all broken at a sub-GeV scale. This would ensure that the WIMP-nucleon cross section contains higher powers of $q^2$ and the resulting form factor suppresses coherent scattering on nuclei [@formfactor]. At the same time, due to the increased velocity inside the Sun, the degree of suppression in the capture rate will be much smaller.
Given these results, the observable $\gamma$-ray flux then follows from the $h'\rightarrow \gamma\gamma$ branching fraction ${\rm Br}_\gamma \sim 10^{-2}$ discussed above and the dominant contribution from the WIMP-iron inelastic scattering cross section $\sigma^{(Fe)} \sim 10^{-33}\,{\rm cm}^2$ to the trapping rate. This easily leads to a detectable flux of gammas for a wide range of parameters, $$\Phi_{\gamma\odot} \sim 1 \times 10^{-6}\, {\rm cm}^{-2}\, {\rm s}^{-1} \times \left(\frac{\ka}{5\times 10^{-4}}\right)^2
\left( \frac{5 \,{ \rm GeV}}{ m_V }\right)^4.
\label{result}$$ We see that this model leads to a remarkably large gamma ray flux, that seemingly would have been observed by EGRET and Fermi-LAT. Moreover, for the range $m_{h'}> 2 m_\mu$, the accompanying muon flux will generate a flux of $\nu_\mu$ well in excess of the bounds set by [*e.g.*]{} Super-Kamiokande [@SuperK]. However, it is clear that generic choices of parameters may equally well reduce the flux (\[result\]) by several orders of magnitude. A more comprehensive scan of the parameter space for this model vs the resulting flux is beyond the scope of this paper, but it clear that the gamma flux does indeed constitute a sensitive probe.
4. Discussion {#discussion .unnumbered}
-------------
Our exploratory study in this paper has suggested a number of striking indirect signatures associated with the annihilation of secluded dark matter trapped in the Sun (and planets), in the form of a high energy flux of electromagnetic particles tightly correlated with the solar center. In this section, we will finish with a number of additional remarks.
One of the issues that has become apparent is that there are relatively few competitive limits on gamma rays from the Sun, primarily because this is a difficult source to handle for many of the more sophisticated gamma-ray telescopes. However, Fermi-LAT can and has observed the Sun and its impressive angular resolution and long exposure times point to it as having the best experimental sensitivity for probing the gamma-ray signatures of secluded dark matter scenarios discussed here, with mediator lifetimes on the scale of the solar radius. Indeed, according to our analysis of model scenarios in Section 3, Fermi-LAT may well provide sensitivity to specific parts of the parameter space which is superior to all other direct or indirect probes.
Beyond direct decays to photons, secluded models also naturally produce a significant branching of mediators to high-energy electron-positron pairs, As discussed earlier, its an intriguing possibility that massive bodies in the Solar System may provide another possible source of the anomalous electron and positron fluxes in the multi-GeV range. Such scenarios could be probed via high-precision analyses of the spatial or temporal non-uniformities in these fluxes, particularly the positron fraction observed by PAMELA. Studies of this kind could significantly strengthen the parameter reach in many models of secluded dark matter. Moreover, while the requisite flux of charged particles could be achieved in many variants of secluded dark matter, they most likely will be put to the test by the upcoming Fermi gamma-ray data given the minimal flux that arises through final state radiation.
Within this general framework, we have presented two concrete models of secluded dark matter consistent with the relic abundance and direct detection constraints in which the most accessible experimental signatures are gamma rays and charged particles from the Sun. The crucial ingredient is a mediator with a lifetime long enough to escape from the Sun, but short enough on cosmological or astrophysical scales so that there exist no additional constraints arising from early cosmology beyond those already considered in the literature (see, [*e.g.*]{} [@cmb]). It is likely that other models of this type may be constructed, and indeed it would be worthwhile to explore a more generic scan to understand if the gamma-ray flux observed in these specific models is robust.
Beyond these points that were already touched upon in the text, we would also like to mention some other related issues concerning indirect detection signatures:
- Secluded mediators with long lifetimes can, depending on the particular model and mass range under consideration, greatly enhance the overall neutrino flux reaching the Earth. For example, if the mediators are heavy enough to decay to charged mesons outside the solar radius, the production of muon neutrinos is inevitable. Indeed, even if this decay occurs inside the solar radius, the neutrino yield can be significant provided the decays occur outside the dense core where all muons and pions are quickly absorbed before they are capable of producing energetic neutrinos.
- There is also the possibility of searching for the annihilation products of standard, neutralino-like WIMPs using gamma ray telescopes. If the annihilation of WIMPs in the solar interior creates large fluxes of neutrinos, the interaction of neutrinos in the upper layers of the solar atmosphere will result in the production of hadrons ($\pi_0$, $K$,...) that decay with the significant yield of $\gamma$’s. The surviving $\gamma$ fraction which escapes may again be accessible to Fermi-LAT, and can provide alternative sensitivity to the WIMP-powered neutrino flux. While ground-based neutrino telescopes are ideally suited to exploring $\nu_\mu$ neutrino fluxes, the gamma ray signature will contain information about other flavors, and be indirectly sensitive to the neutrino energy range where ground-based neutrino telescopes do not have any directional sensitivity.
- Another possibility that was not considered in this paper is WIMP annihilation to very light quasi-stable mediators, such as QCD axions. The conversion of axions into photons may occur in the magnetic field of the Sun, resulting in a $\gamma$-ray flux. This idea has some similarities with a recent proposal to observe the Sun’s transparency to gamma ray sources that may contain an axionic component [@FRT].
In conclusion, the existence of a generic dark sector with metastable states mediating the interactions between WIMP dark matter and the SM opens up new possibilities for indirect detection signatures in the solar system. While it is possible to search for charged particles produced by mediator decays, it seems the hard gamma-ray flux produced by decays outside the solar radius may be the most promising ‘smoking gun’ signature due to its tight correlation with the solar center, making it easily distinguishable from cosmic ray-induced backgrounds. As a final remark, we observe that it may be profitable to explore other galactic or extra-galactic implications of this scenario in which all stellar objects are effectively imbued with a high-energy gamma spectrum.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank M. Boezio, M. Casolino, D. Hanna and D. Hooper for helpful discussions and/or email correspondence. We also thank I. Yavin for informing us of his related work. B.B. acknowledges support in part from the DOE under contract DE-FG02-96ER40969 during the Unusual Dark Matter workshop at the University of Oregon. The work of A.R. and M.P. is supported in part by NSERC, Canada, and research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT.
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[^1]: Earlier claims of enhanced neutralino annihilation immediately outside of the solar radius [@early] were not confirmed by subsequent studies [@subsequent].
|
---
bibliography:
- 'Refs.bib'
---
[MAN/HEP/2011/16]{}
[**Non–global logs and clustering impact on jet mass with a jet veto distribution**]{}
Kamel Khelifa–Kerfa[^1]\
School of Physics & Astronomy, University of Manchester,\
Oxford Road, Manchester, M13 9PL, U.K.
[ **Abstract** ]{}
> There has recently been much interest in analytical computations of jet mass distributions with and without vetos on additional jet activity [@Ellis:2009wj; @Ellis:2010rwa; @Banfi:2010pa; @Kelley:2011tj; @Hornig:2011tg; @Li:2011hy]. An important issue affecting such calculations, particularly at next–to–leading logarithmic (NLL) accuracy, is that of non–global logarithms as well as logarithms induced by jet definition, as we pointed out in an earlier work [@Banfi:2010pa]. In this paper, we extend our previous calculations by independently deriving the full jet–radius analytical form of non–global logarithms, in the anti–$\kt$ jet algorithm. Employing the small–jet radius approximation, we also compute, at fixed–order, the effect of jet clustering on both $\CF^{2}$ and $\CF\CA$ colour channels. Our findings for the $\CF\CA$ channel confirm earlier analytical calculations of non–global logarithms in soft–collinear effective theory [@Hornig:2011tg]. Moreover, all of our results, as well as those of [@Banfi:2010pa], are compared to the output of the numerical program `EVENT2`. We find good agreement between analytical and numerical results both with and without final state clustering.
Introduction {#sec.intro}
============
Event and jet shape variables have long served as excellent tools for testing QCD and improving the understanding of its properties (for a review, see [@Dasgupta:2003iq]). Event/jet shape distributions have been used to extract some prominent parameters in QCD including the strong coupling and the quark–gluon colour ratio [@Beneke:1998ui]. Due to the fact that shape variables are, by construction, linear in momentum, they exhibit a strong sensitivity to non–perturbative (NP) effects [@Dasgupta:2003iq; @Webber:1994zd]. They have thus been exploited to gain a better analytical insight into this QCD domain [@Dasgupta:2003iq; @Beneke:2000kc]. Furthermore, jet shapes have been used not only to study the jet structure of hadronic final states, including jet multiplicities, jet rates and jet profiles (Ref. [@QCD_collider] and references therein), but also the *subjet* structure, or substructure, of the jets themselves (for a recent example, see [@Ellis:2009wj]). The latter subject has received significant attention in recent years, particularly in the area of boosted objects with the aim to separate the decay products of Beyond Standard Model (BSM) particles from QCD background at LHC (for a review, see [@Abdesselam:2010pt]).
Although shape variables are, by construction, Infrared and Collinear (IRC) safe, fixed–order perturbative (PT) calculations break down in regions of phase space where the shape variable is small. These regions correspond to gluon emissions that are soft and/or collinear to hard legs and lead to the appearance of large logs that spoil the PT expansion of the shape distribution [@QCD_collider] (and references therein). While measured shape distributions have a peak near small values of the shape variable and then go to zero, fixed–order analytical distributions diverge. To deal away with these divergences and successfully reproduce the experimentally–seen behaviour, one ought to either perform an all–orders resummation of the large logs, matched to fixed–order result, or rely on Monte Carlo event generators. We are concerned, in the present paper, with the resummation method as it paves the way for a better understanding of QCD dynamics including the process of multiple gluon radiation. The general form of resummed distributions for observables that have the property of exponentiation can be cast as [@QCD_collider] $$\label{resum_gen_struct}
\Sigma(v) = C(\as)\,\exp\left[L\,g_{1}(\as L) + g_{2}(\as L) + \as\,g_{3}(\as L)+ \cdots \right] + D(v)$$ where $L=\ln(1/v)$, $C(\as)$ is an expansion in $\as$ with constant coefficients that can be inferred from fixed–order calculations and $D(v)$ collects terms that are proportional to powers of the shape variable $v$. The function $g_{1}$ resums all the leading logs (LL) $\as^{n} L^{n+1}$, while $g_{2}$ resums the next–to–leading logs (NLL) $\as^{n} L^{n}$ and so on.
There are two types of jet shape observables [^2]: global and non–global [@Dasgupta:2001sh]. *Global* observables are shape variables that are sufficiently inclusive over the whole final state phase space. The resummation of such variables, e.g, thrust, heavy jet mass and broadening, up to NLL accuracy have long been performed [@Catani:1992ua; @Banfi:2001bz]. The resultant resummed distributions were then matched with NLO fixed–order results for a better agreement with measurements over a wide range of values of the shape variable [@Catani:1992ua; @Banfi:2010xy]. In the recent past, the NNLL $+$ NLO distribution has been obtained for energy–energy correlation [@deFlorian:2004mp], as well as NNLL $+$ NNLO [@GehrmannDeRidder:2005cm; @GehrmannDeRidder:2007hr] for the thrust distribution [@Monni:2011gb], both in $\ee$ annihilation processes in QCD. Within the framework of Soft and Collinear Effective Theory (SCET) [@Bauer:2000yr; @Bauer:2001yt], the N$^{3}$LL resummation for various event/jet variables have been performed [@Chien:2010kc; @Abbate:2010vw] and used, after matching to NNLO, for a precise determination of the coupling constant $\as$. The extracted value is consistent with the world average with significant improvements in the scale uncertainty.
At hadron colliders, what one often measures instead is jets, which only occupy patches of the phase space. The corresponding jet shape variables are thus non–inclusive, or non–global, and the resummation becomes highly non–trivial even at NLL level. Consider, for example, measuring the normalised invariant mass, $\rho$, of a subset of high–$\pt$ jets in multijet events. A veto is applied on final state soft activity to keep the jet multiplicity fixed. Jets are only defined through a jet algorithm, which generally depends on some parameters such as the jet size $R$ [@Cacciari:2005hq]. We are thus faced with a multi–scale ($\rho$, hard scale, veto, jet size) problem where potentially large logs in the ratios of these scales appear. In addition to the Sudakov leading logs, $\as^{n} \ln^{n+1}\rho$, coming from independent primary gluon emissions, there are large subleading non–global logs (NGLs) of the form $\as^{n} \ln^{n} (a/b)$, where $a$ and $b$ are two different scales, coming from secondary [^3] correlated gluon emissions.
We argued in [@Banfi:2010pa] that in the narrow well–separated jets limit, the non–global structure of the $\rho$ distribution, at hadronic colliders, becomes much like that of $\ee$ hemisphere jet mass [@Dasgupta:2001sh]. This is mainly due to the fact that non–global logs arise predominantly near the boundaries of individual jets. We had therefore considered $\ee$ dijet events where only one of the jets is measured while the other is left unmeasured. We found, in the anti–$\kt$ algorithm [@Cacciari:2008gp], NGLs in the ratio $\rho Q/2 R^{2} \Eo$ as well as $2\Eo/Q$ where $\Eo$ and $Q$ are the veto and hard scale respectively. These logs were completely missed out in [@Ellis:2009wj; @Ellis:2010rwa]. The resummation of these NGLs to all–orders had been approximated to that of the hemisphere mass [@Dasgupta:2001sh] up to terms vanishing as powers of $R$. Furthermore, we pointed out, by explicitly computing the jet mass (without jet veto) distribution under clustering, that different jet definitions differ at NLL due to clustering–induced large logs. Here we compute these logs, which we refer to as *clustering logs* (CLs), for the jet mass with a jet veto distribution.
Within the same context of $\ee$ multijet events, Kelley *et al.* [@Kelley:2011tj] (version $1$) proposed that if one measures the masses of the two highest–energy jets, instead of a single highest–energy jet as done in [@Banfi:2010pa], then the resulting distribution is free from NGLs. This is clearly not correct since the latter shape observable, which we shall refer to, following [@Kelley:2010qs], as *threshold thrust* [^4], is still non–global. To clearly see this consider, for example, the following gluonic configuration in $\ee$ dijet events at $\Or(\as^{2})$. A gluon $k_{1}$ is emitted by hard eikonal legs into the interjet energy region, $\Omega$. $k_{1}$ then emits a softer gluon $k_{2}$ into, say the quark jet region. This configuration then contributes to the quark jet mass. The corresponding virtual correction, whereby gluon $k_{2}$ is virtual, does not, however, contribute to the quark jet mass. Hence, upon adding the two contributions one is left with a real–virtual mis–cancellation resulting in logarithmic enhancement of the jet mass distribution. The latter is what we refer to as NGLs. The other, antiquark, jet receives identical enhancement. Thus the sum of the invariant masses of the two jets does indeed contain NGLs contribution. The latter is actually twice that of the single jet mass found in [@Banfi:2010pa].
Moreover, the authors of [@Kelley:2011tj] (version $1$) claimed that the anti–$\kt$ [@Cacciari:2008gp] and Cambridge–Aachen (C–A) [@Ellis:1993tq] jet algorithms only differ at NNLL for the threshold thrust [^5]. From our calculation in [@Banfi:2010pa] for the jet mass, which is not -with respect to clustering- much different from the threshold thrust, we know that the latter statement is incorrect. Nonetheless, an explicit proof will be presented below. Now, what is interesting in [@Kelley:2011tj] and triggers the current work, is that the total differential threshold thrust distribution computed in the C–A algorithm and which contains neither NGLs nor CLs contributions, seemed to somehow agree well with next–to–leading (NLO) program `EVENT2` [@Catani:1996jh].
In this paper we shall shed some light on the result of [@Kelley:2011tj] by considering the individual colour, $\CF^{2}, \CF\CA$ and $\CF\TF\nf$, contributions to the total differential distribution as well as the effect of C–A clustering. We show that at $\O(\as^{2})$ both NGLs and CLs are present and that the above agreement with `EVENT2` is, on one side merely accidental [^6], and on the other side due to the fact that the interval of the threshold thrust considered in [@Kelley:2011tj] does not correspond to the asymptotic region where large logs are expected to dominate. The current work may be regarded as an extension to [@Banfi:2010pa]. It includes: (a) computing the full $R$ dependence of the leading NGLs coefficient in the anti–$\kt$, (b) computing the small $R$ approximation of the latter as well as the leading CLs coefficient in the C–A algorithm and (c) checking our findings, as well as those of [@Banfi:2010pa], against `EVENT2`. It turns out, from the latter comparison, that the above approximation is actually valid for quite large values of $R$.
While the current paper was in preparation, a paper by Hornig *et al.* [@Hornig:2011tg] appeared in arXiv which studied NGLs in various jet algorithms, including anti–$\kt$ and C–A, within SCET. On the same day, Kelley *et al.* published version $2$ of [@Kelley:2011tj] in which they realised that this distribution is not actually free of NGLs and computed the corresponding coefficient in the anti–$\kt$ algorithm. Our findings on NGLs, which were independently derived using a different approach to both papers, confirm the results of both SCET groups. Clustering effects on primary emission sector are unique to this paper.
The organisation of this paper is as follows. In sec. \[sec.fixed\_order\_1\] we compute the full logarithmic part of the LO threshold thrust distribution. We then consider, in sec. \[sec.fixed\_order\_2\], the fixed–order NLO distribution in the eikonal limit and compute the NGLs coefficient, in both anti–$\kt$ and C–A jet algorithms. In the same section we derive an expression for the CLs’ first term as well. Note that our calculations for the C–A algorithm are performed in the small $R$ limit. Sec. \[sec.resummation\_in\_QCD\] is devoted to LL resummation of our jet shape including an exponentiation of the NGLs’ and CLs’ fixed–order terms. The latter exponentiation suffices for our purpose in this paper, which is to compare the analytical distribution with `EVENT2` at NLO. It also provides a rough estimate of the size and impact of NGLs and CLs on the total resummed distribution. In appendix \[app.tw\_in\_SCET\], the corresponding resummation in SCET [@Kelley:2011tj; @Kelley:2010qs; @Becher:2008cf] is presented. Numerical distributions of the threshold thrust obtained using the program `EVENT2` are compared against analytical results and the findings discussed in sec. \[sec.numerical\_results\]. In light of this discussion, we draw our main conclusions in sec. \[sec.conclusion\].
Fixed–order calculations: $\Or(\as)$ {#sec.fixed_order_1}
====================================
After briefly reviewing the definition of the threshold thrust observable, or simply the jet mass with a jet veto, presented in [@Kelley:2011tj; @Kelley:2010qs], a general formula for sequential recombination jet algorithms is presented. We then move on to compute the LO integrated distribution of this shape variable. At this order, all jet algorithms are identical. Note that partons (quarks and gluons) are assumed on–mass shell throughout.
Observable and jet algorithms definitions {#subsec.observable_def}
-----------------------------------------
Consider $\ee$ annihilation into multijet events. First, cluster events into jets of size (radius) $R$ with a jet algorithm. After clustering, label the momenta of the two hardest jets $p_{R}$ and $p_{L}$ and the energy of the third hardest jet $E_{3}$. The threshold thrust is then given by the sum of the two leading jets’ masses after events with $E_{3} > \Eo$ are vetoed [@Kelley:2011tj], $$\label{eq.tau_omega}
\tauo = \frac{m^{2}_{R} + m_{L}^{2}}{Q^{2}} = \frac{\rho_{R} + \rho_{L}}{4}.$$ $\rho_{R}$ and $\rho_{L}$ are the jet mass fractions for the two leading jets respectively. We have shown in [@Banfi:2010pa] that the single jet mass fraction, $\rho$, is a non–global shape variable. Thus $\tauo$ must obviously be a non–global variable too.
A general form of sequential recombination algorithms at hadron colliders is presented in [@Cacciari:2005hq]. The adopted version for $\ee$ machines may be summarised as follows [@Cacciari:2005hq]: Starting with a list of final state pseudojets with momenta $p_{i}$ [^7], energies $E_{i}$ and angles $\theta_{i}$ w.r.t. c.m frame, define the distances $$\label{jet_alg_def_theta}
d_{ij} = \min\left(E_{i}^{2 p},E_{j}^{2 p}\right)
\frac{2\left(1-\cos\theta_{ij}\right)}{R^{2}},\;\;\; d_{iB} = E_{i}^{2p},$$ where $p$ can be any (positive or negative) continuous number. At a given stage of clustering, if the smallest distance is $d_{ij}$ then $i$ and $j$ are recombined together. Otherwise if the smallest distance is $d_{iB}$ then $i$ is declared as a jet and removed from the list of pseudojets. Repeat until no pseudojets are left. The recombination scheme we adopt here is the $E$–scheme, in which pairs $(ij)$ are recombined by adding up their $4$–momenta. Two pseudojets, $i$ and $j$, are merged together if $$\label{clust_cond}
2(1-\cos\theta_{ij}) < R^{2}.$$ The anti–$\kt$, C–A and $\kt$ algorithms correspond, respectively, to $p = -1,
p= 0$ and $p=1$ in eq. . We shall only consider the first two algorithms, anti–$\kt$ and C–A in this paper. Calculations for the inclusive $\kt$ are identical to those for the C–A algorithm as shown in [@Banfi:2010pa]. With regard to notation, the jet–radius in [@Kelley:2011tj], which we shall denote $\Rs$, is given in terms of $R$ by $$\label{R_Rs_rel}
\Rs = R^{2}/4.$$ Here we work with $\Rs$ instead of $R$.
To verify that the definition is just the thrust in the threshold (dijet) limit, hence the name, we begin with the general formula of the thrust, $$\label{thrust_def}
\tau = 1- \max_{\uv} \frac{\sum_{i} |\mathbf{p}_{i}.\uv |}{\sum_{i}
|\mathbf{p}_{i}|},$$ where the sum is over all final state $3$–momenta $\mathbf{p}$ and the maximum is over directions (unit vectors) $\uv$. In the threshold limit, enforced by applying a veto on soft activity, $\ee$ annihilates into two back–to–back jets and the *thrust axis*, the maximum $\uv$, coincides with jet directions. At LO, an emission of a single gluon, $k$, that is collinear to, and hence clustered with say, $p_{R}$, produces the following contribution to the thrust $$\label{eq.T^2}
\tau \simeq \frac{E_{R} \omega}{Q}(1-\cos\theta_{k p_{R}}) + \frac{E_{L}
\omega}{Q}(1-\cos\theta_{k p_{L}}) + \frac{\omega^{2}}{Q^{2}}(1-\cos\theta_{k
p_{R}})(1-\cos\theta_{k p_{L}}),$$ where $E_{R(L)}$ is the energy of the hard leg $p_{R(L)}$, $\omega$ the gluon’s energy and we have discarded an $\Or(\tau^{2})$ term. Recalling that the first two terms in the RHS of eq. are just the mass fractions $\rho_{R}$ and $\rho_{L}$, respectively, at LO and neglecting the third term (quadratic in $\omega$) one concludes that $$\label{tau_tauo}
\tau \simeq \tauo.$$ This relation can straightforwardly be shown to hold to all–orders.
LO distribution {#sec.Fixed-PT-antikt}
---------------
In [@Banfi:2010pa] we computed the LO distribution of the jet mass fraction, $\rho$, in the small $R$ ($\Rs$) limit using the matrix–element squared in the eikonal approximation. In this section, we use the full QCD matrix–element to restore the complete $\Rs$ dependence of the singular part of the $\tauo$ distribution. The general expression for the integrated and normalised $\tauo$ distribution, or equivalently the $\tauo$ shape fraction, is given by $$\label{S1_tauo_dist}
\Sigma(\tauo,\Eo) = \int_{0}^{\tauo} \d\tauo' \int_{0}^{\Eo} \d
E_{3}\;\frac{1}{\sigma} \frac{\d^{2} \sigma}{\d\tauo' \d E_{3}},$$ where $\sigma$ is the total $\ee \rightarrow $ hadrons cross–section. The perturbative expansion of the shape fraction $\Sigma$ in terms of QCD coupling $\as$ may be cast in the form $$\label{Sigma_PT}
\Sigma = \Sigma^{(0)} + \Sigma^{(1)} + \Sigma^{(2)} + \cdots,$$ where $\Sigma^{(0)}$ refers to the Born contribution and is equal to $1$. The derivation of the first order correction, $\Sigma^{(1)}$, to the Born approximation is presented in appendix \[app.LO\_distr\]. The final result reads $$\begin{gathered}
\label{R1_full-b}
\Sigma^{(1)}(\tauo,\Eo) = \frac{\CF \as}{2\pi} \left[-2\,\ln^{2}\tauo +\left(-3
+ 4\,\ln\frac{\Rs}{1-\Rs}\right)\,\ln\tauo \right] \Theta\left(\frac{\Rs}{1+\Rs}
-\tauo\right)+ \\+ \frac{\CF \as}{2\pi} \Bigg[ - 1 + \frac{\pi^{2}}{3} -
4\,\ln\frac{\Rs}{1-\Rs} \,\ln\frac{2\Eo}{Q} + f_{\Eo}(\Rs)\Bigg],\end{gathered}$$ where we have used eq. to change the normalisation in eq. from $\sigma$ to $\sigma_{0}$. The reason for this change is that the matrix–element we have used in `EVENT2` is normalised to the Born cross–section [^8]. The only difference between the two normalisations at $\Or(\as)$ is in the one–loop constant. If we normalised to $\sigma$ we would have found $\CF(-5/2+\pi^{2}/3)$ instead of $\CF(-1+\pi^{2}/3)$. The function $f_{\Eo}(\Rs)$ is given by $$\label{f_omeg}
f_{\Eo}(\Rs) = -2\,\ln\Rs\,\ln\frac{\Rs}{1-\Rs} + 2\,\Li_{2}(\Rs) -
2\,\Li_{2}(1-\Rs) + \frac{8 \Eo}{Q}\,\ln\frac{\Rs}{1-\Rs} +
\Or\left(\frac{\Eo^{2}}{Q^{2}}\right).$$ Notice that eqs. and are identical to eqs. $(1)$ and $(2)$ of [@Kelley:2011tj] v$1$ and the sum of the $\as$ parts of eqs. $(65)$ and $(66)$ in [@Hornig:2011tg] provided that the jet radius in the latter, which we refer to as $\bar{R}$, is related to $\Rs$ by: $\tan^{2}(\bar{R}/2) = \Rs/(1-\Rs)$. It is worthwhile to note that in the limit $\Rs \rightarrow 1/2$ the $\tauo$ distribution reduces to the well known thrust distribution [@GehrmannDeRidder:2007bj] with upper limit $\tau < 1/3$. For $\Rs < 1/2$ the threshold thrust distribution includes, in addition to thrust distribution, the interjet energy flow distribution [@Oderda:1998en] too, $$\Sigma^{(1)}_{\mathrm{E\,flow}}(\Eo) = \frac{\CF \as}{2\pi} \left[-
4\,\ln\frac{\Rs}{1-\Rs} \ln\left(\frac{2\Eo}{Q}\right) +
\Or\left(\frac{\Eo}{Q}\right) \right],$$ Here the interjet region (rapidity gap), referred to in literature as $\deta$, is defined by the edges of the jets. Specifically, it is related to the jet–radius $\Rs$ by $$\label{eta-Rs}
\deta = -\ln\left(\frac{\Rs}{1-\Rs}\right).$$
The important features of the $\tauo$ distribution that are of concern to the present paper are actually contained in the second order correction term $\Sigma^{(2)}$, which we address in the next section.
Fixed–order calculations: $\Or(\as^{2})$ {#sec.fixed_order_2}
========================================
We begin this section by recalling the formula of the matrix–element squared for the $e^{+}e^{-}$ annihilation into two gluons, $\ee \longrightarrow q(p_{a})
+ \qbar(p_{b}) + g_{1}(k_{1})+ g_{2}(k_{2})$ in the eikonal approximation. Let us first define the final state partons’ $4$–momenta as $$\begin{aligned}
\label{4-momenta}
\nonumber p_{a} &=& \frac{Q}{2}(1,0,0,1) ,\\
\nonumber p_{b} &=& \frac{Q}{2}(1,0,0,-1) ,\\
\nonumber k_{1} &=& \omega_{1}(1,\sin\theta_{1}\cos\phi_{1},
\sin\theta_{1}\sin\phi_{1},\cos\theta_{1}) ,\\
k_{2} &=& \omega_{2}(1,\sin\theta_{2}\cos\phi_{2},
\sin\theta_{2}\sin\phi_{2},\cos\theta_{2}).\end{aligned}$$ where the angles $\theta_{i}$ are w.r.t. $p_{a}$ direction (which lies along the z–axis) and we assume the energies to be strongly ordered: $Q \gg \omega_{1} \gg
\omega_{2}$. This is so that one can straightforwardly extract the leading NGLs. Contributions from gluons with energies of the same order, $Q\gg \omega_{1} \sim \omega_{2}$, are subleading and hence beyond our control. The recoil effects are negligible in the former regime and are thus ignored throughout. The eikonal amplitude reads [@QCD_collider], $$\label{W_2}
S_{ab}(k_{1}, k_{2}) = \CF^{2} W_{P} + \CF\CA W_{S},$$ where $W_{P}$ and $W_{S}$ stand for primary and secondary emission amplitudes respectively. If we define the antenna function $w_{ij}(k) = 2(ij)/(ik)(kj)$ then the latter amplitudes are given by $$\label{W_P}
W_{P} = w_{ab}(k_{1}) w_{ab}(k_{2}) = \frac{16}{\omega_{1}^{2}\omega_{2}^{2}
\sin\theta_{1}^{2}\sin\theta_{2}^{2}},$$ and $$\begin{aligned}
\label{W_S}
\nonumber W_{S} &=& \frac{w_{ab}(k_{1})}{2} \left[w_{a1}(k_{2})+w_{b1}(k_{2}) -
w_{ab}(k_{2})\right],
\\
&=& \frac{8}{\omega_{1}^{2}\omega_{2}^{2} \sin\theta_{1}^{2}
\sin\theta_{2}^{2}}\left[\frac{1 - \cos\theta_{1}
\cos\theta_{2}}{1-\cos\theta_{12}} - 1\right],\label{Ws_theta}\end{aligned}$$ For completeness, the two–parton phase space is given by $$\label{PS_2}
\d\Phi_{2}(k_{1}, k_{2}) = \left[\prod_{i=1}^{2} \omega_{i}\d\omega_{i}
\frac{\sin\theta_{i}\d\theta_{i} \d\phi_{i}}{2\pi} \right]
\left(\frac{\as}{2\pi}\right)^{2},$$ It is worth noting that the primary emission, $W_{P}$, contribution to the $\tauo$ distribution is only fully accounted for by the single–gluon exponentiation in the anti–$\kt$ algorithm case. If the final state is clustered with a jet algorithm other than the latter, $W_{p}$ integration over the modified phase space, due to clustering, leads to (see below) new logarithmic terms that escape the naive single–gluon exponentiation. On the other hand, the secondary amplitude $W_{S}$ contribution is completely missing from the latter Sudakov exponentiation in both algorithms.
First we outline the full $\as^{2}$ structure of the $\tauo$ distribution up to NLL level in the anti–$\kt$ including the computation of the NGLs coefficient. After that, we investigate the effects of final state partons’ clustering on both primary and secondary emissions. The C–A algorithm is taken as a case study to illustrate the main points. Calculations where the final state is clustered with other jet algorithms should proceed in an analogous way to the C–A case.
$\tauo$ distribution in the anti–$\kt$ algorithm {#subsec.FO_anti-kt}
------------------------------------------------
The anti–$\kt$ jet algorithm works, in the soft limit, like a perfect cone. That is, a soft gluon $k_{i}$ is clustered to a hard parton $p_{j}$ if it is within an angular distance $2\sqrt{\Rs}\, ( = R)$, from the axis defined by the momentum of the latter. This feature of the algorithm greatly simplifies both fixed–order and resummation calculations. Considering all possible angular distances between $(k_{1},k_{2})$ and $(p_{a},p_{b})$ we compute below the corresponding contributions to primary and secondary pieces of the $\tauo$ distribution. Note that we use LL and NLL to refer to leading and next–to–leading logs of $\tauo$ (and not $2\Eo/Q$) in the exponent of the resummed distribution (discussed in sec. \[sec.resummation\_in\_QCD\]).
### $\CF^{2}$ term {#subsec.anti-kt_CF2}
The LL contribution to the $\tauo$ distribution comes from diagrams corresponding to two–jet final states. That is diagrams where both real gluons, $k_{1}$ and $k_{2}$, are clustered with the hard partons $p_{a}$ and $p_{b}$. Diagrams where one of the two gluons is in the interjet region, and hence not clustered with either hard parton, contribute at NLL level. Other gluonic configurations lead to contributions that are beyond our NLL control and thus not considered. The $\CF^{2}$ part of the $\Or(\as^{2})$ $\tauo$ distribution may be found by expanding the exponential of the LO result . The full expression including the running coupling at two–loop in the $\overline{\mathrm{MS}}$ will be presented in sec. \[sec.resummation\_in\_QCD\]. For the sake of comparison to the clustering case, we only report here the the LL term, which reads $$\label{FO_2_CF2}
\Sigma^{(2)}_{P}(\tauo,\Eo) = 2\,\CF^{2}
\left(\frac{\as}{2\pi}\right)^{2}\,\ln^{4}(\tauo).$$
Next we consider the derivation of the $\CF\CA$ contribution to the $\tauo$ distribution including the full jet–radius dependence.
### $\CF\CA$ term and NGLs {#subsec.anti-kt-NGL}
In the anti–$\kt$ algorithm the non–global logarithmic contribution to the $\tauo$ distribution is simply the sum of that of the single jet mass fraction, $\rho$, with a jet veto distribution studied in [@Banfi:2010pa] [^9]. This is in line with the near–edge nature of non–global enhancements. In two–jet events, the well separated [^10] jets receive the latter enhancements independently of each other.
![Schematic representation of gluonic arrangement giving rise to NGLs. We have only shown the NGLs contributions to the $p_{R}$–jet. Identical contributions apply to the $p_{L}$–jet.[]{data-label="fig.NGLs_akt"}](Figures/NGLs_2.eps){width="15cm"}
Possible final state gluonic arrangements relevant to NGLs at second order are depicted in fig. \[fig.NGLs\_akt\]. The all–orders resummed NGLs distribution may be written in the form [@Dasgupta:2001sh] $$\label{S_t_gen}
S(t) = 1+ S_{2}\,t^{2} + \cdots = 1+ \sum_{n=2}^{} S_{n}\,t^{n},$$ with $t$ being the evolution parameter defined in terms of the coupling $\as$ by $$\begin{aligned}
\label{t_param_akt}
\nonumber t &=& \frac{1}{2\pi} \int_{\kt^{\min}}^{\kt^{\max}} \frac{\d\kt}{\kt}
\;\as(\kt),\\
&=& \frac{\as}{2\pi}\; \,\ln\left(\frac{\kt^{\max}}{\kt^{\min}}\right),\end{aligned}$$ where the exact form of the upper and lower limits, $\kt^{\max}$ and $\kt^{\min}$, depend on the gluonic configuration and the second line in assumes a fixed coupling. To make contact with interjet energy flow calculations [@Dasgupta:2002bw; @Appleby:2002ke], we work in this particular section with hadronic variables $(k_{t},\eta, \phi)$ instead of $\ee$ variables $(E,\theta, \phi)$. The pseudo–rapidity $\eta$ and transverse momentum $\kt$ (both measured w.r.t. incoming beam direction) are related, respectively, to the angle and energy by [^11] $$\eta = - \ln\left(\tan\frac{\theta}{2}\right),\;\;\; E =\kt \cosh(\eta)$$ Using the secondary emissions eikonal amplitude in terms of the new variables, the NGLs coefficient $S_{2}$ reads $$\label{S2_akt_gen}
S_{2} = -4 \CF\CA
\;\int\d\Phi^{(2)}\;\left[\frac{\cosh(\eta_{1}-\eta_{2})}{\cosh(\eta_{1}-\eta_{2
}) - \cos(\phi_{1} -\phi_{2})} -1\right],$$ where the phase space measure, $\d\Phi^{(2)}$, is of the general form given in eq. with the $\kt$ integrals included in the definition of $t$ and new restrictions coming from the jet shape definition. For configuration $(a)$ in fig. \[fig.NGLs\_akt\], it reads $$\label{S2_PS_a}
\d\Phi^{(2)}_{a} = \int_{-\frac{\deta}{2}}^{\frac{\deta}{2}} \d\eta_{1}
\frac{\d\phi_{1}}{2\pi} \times 2 \int_{\frac{\deta}{2}}^{+\infty}\d\eta_{2}
\frac{\d\phi_{2}}{2\pi}\; \Theta\left( \ln\frac{\ktt}{Q\tauo} - \eta_{2}\right)
\Theta\left(\Eo - \kto\cosh(\eta_{1})\right),$$ where the interjet (gap) region, $\deta$ is given in eq. . Due to boost invariance of rapidity variables the latter region has been centred at $\eta=0$. Moreover, the factor $2$ in accounts for the $p_{L}$–jet contribution. Since neither the integrand nor the integral measure in eq. depends explicitly on the azimuthal angles ($\phi_{1}$, $\phi_{2}$), we use our freedom to set $\phi_{1} = 0$, average over $\phi_{2}$ and then perform the rapidity integration. The resultant expression for $S_{2}$ in configuration $(a)$ at the limit $\tauo \rightarrow 0$ reads, $$\label{S2_akt_a1}
S_{2,a} = -4\CF\CA \left[\frac{\pi^{2}}{12} +
\deta^{2} - \deta\,\ln\left(e^{2\deta}-1\right) -\frac{1}{2}
\Li_{2}\left(e^{-2\deta}\right) -\frac{1}{2} \Li_{2}\left(1-e^{2\deta}\right)
\right]$$ An identical expression was found for the NGLs’ coefficient in the interjet energy flow distribution [@Dasgupta:2002bw] [^12]. The fact that $S_{2,a}$ is the same for $\tauo$ and interjet energy flow distributions means that the NGLs’ coefficient only depends on the geometry of the phase space and not on the observable itself. This is of course only true in the limit where the jet shape variable goes to zero. The difference between the jet shape variables amounts only to a difference in the logarithm’s argument.
It should be understood that there are $\Theta$–function constraints on $\kto$ and $\ktt$ resulting from rapidity integrations not explicitly shown in eq. . Performing the remaining trivial $\kt$ integrals yields $$\label{t_param_a}
t_{a}^{2} = \left(\frac{\as}{2\pi}\right)^{2}\;\ln^{2}\left(\frac{2\Eo\,\Rs}{Q\tauo}\right) \;\Theta\left(\frac{2\Eo}{Q} - \frac{\tauo}{\Rs}\right),$$ where a factor of $1/2$ has been absorbed in $S_{2,a}$ .
Now consider configuration $(b)$ in fig. \[fig.NGLs\_akt\]. Adding up the corresponding virtual correction, one obtains the following phase space constraint $$\label{PS_const_b}
\Theta\left(\eta_{1} -\ln\left(\frac{\kto}{Q\tauo} \right) \right) \Theta\left(\ktt -
\frac{\Eo}{\cosh(\eta_{2})}\right).$$ The phase space measure $\d\Phi_{b}^{(2)}$ is analogous to $\d\Phi_{a}^{(2)}$ in with $1 \leftrightarrow 2$ and the two $\Theta$–functions in replaced by those in eq. . The limits on $\eta_{1}$ are then $+\infty > \eta_{1} > \max[\deta/2,\ln(\kto/Q\tauo)]$. If we impose the constraint given in eq. , i.e, $2\Eo/Q \gg
\tauo/\Rs$, then the lower limit becomes $\eta_{1} > \ln(\kto/Q\tauo)$. The NGLs coefficient $S_{2,b}$ thus reads $$\label{S2_akt_b}
S_{2,b} = -4\CF\CA \int_{\ln\frac{\kto}{Q\tauo}}^{+\infty} \d\eta_{1}
\int_{-\frac{\deta}{2}}^{\frac{\deta}{2}}\d\eta_{2}
\left[\coth(\eta_{1}-\eta_{2}) -1\right],$$ where we have averaged the eikonal amplitude $W_{s}$ over $\phi_{2}$ and moved $\kt$s’ $\Theta$–functions onto the integral of the evolution parameter $t_{b}$, which is given at $\as^{2}$ by $$\label{t_param_akt_b}
t_{b}^{2} = \left(\frac{\as}{2\pi}\right)^{2} \ln^{2}\left(\frac{2\Eo}{Q}\right)
\Theta\left(\frac{2\Eo}{Q} - \frac{\tauo}{\Rs}\right).$$ The $S_{2, b}\, t_{b}^{2}$ contribution is then beyond our NLL accuracy. In fact, $S_{2,b}$ vanishes in the limit $\tauo \rightarrow 0$ as can be seen from eq. .
The last contribution to NGLs at $\Or(\as^{2})$ comes from configuration $(c)$ in fig. \[fig.NGLs\_akt\]. Upon the addition of the virtual correction, one is left with the constraint $$\label{PS_const_c}
\Theta\left(Q\tauo - \kto e^{-\eta_{1}} \right) \Theta\left(\ktt e^{+\eta_{2}} -
Q\tauo\right).$$ The corresponding NGLs coefficient and evolution parameter read $$\begin{aligned}
\label{S2_akt_c}
S_{2,c} &=& -4\CF\CA \int_{\max\left[\ln\frac{\kto}{Q\tauo}, \frac{\deta}{2}\right]}^{+\infty}\d\eta_{1}
\int_{-\ln\frac{\ktt}{Q\tauo}}^{-\frac{\deta}{2}} \d\eta_{2}
\left[\coth(\eta_{1} -\eta_{2}) -1\right],\label{S2_b_limit}
\\
t_{c}^{2} &=& \left(\frac{\as}{2\pi}\right)^{2}
\ln^{2}\left(\tauo\,e^{\deta/2}\right),\label{t_param_akt_c}\end{aligned}$$ Since we have assumed strong ordering, $\kto \gg \ktt$, then the lower limit of $\eta_{1}$ in is $\ln(\kto/Q\tauo)$. Consequently the coefficient $S_{2,c}$ vanishes in the limit $\tauo \rightarrow0$. For this reason, this configuration will not be considered.
We conclude that in the regime $2\Eo/Q \gg \tauo/\Rs$, the only non–vanishing contribution to the NGLs comes from the phase space configuration $(a)$. Other configurations, $(b)$ and $(c)$, vanish in the limit $\tauo \rightarrow 0$. Hence $$\label{S2_akt_tot}
S_{2} = S_{2,a},\;\;\; t = t_{a}.$$ In fig. \[fig.NG\_coff\_CA\] we plot $S_{2}$ as a function of the jet–radius $\Rs$. At the asymptotic limit $\deta \rightarrow +\infty$ (or equivalently $\Rs \rightarrow 0$) $S_{2}$ saturates at $ -\CF\CA\; 2\pi^{2}/3$. This value (or rather half of it) is used as an approximation to $S_{2}$ in [@Banfi:2010pa]. From eq. , we can see that the correction to such an approximation is less than $10\%$ for jet–radii smaller than $\Rs \sim 0.28$, which is equivalent to $R \sim 1$. Furthermore, Eq. confirms the claim made in the same paper that NGLs do not get eliminated when the jet–radius approaches zero. One may naively expects that when the jet size shrinks down to $0$ ($\Rs\rightarrow 0$) there is no room for gluon $k_{2}$ to be emitted into. This means that $\tauo$ becomes inclusive and hence $S_{2}$ vanishes. To the contrary, $S_{2}$ reaches its maximum in this limit.
Few important points to note:
- If we choose to order the energy scales in the $\Theta$–functions of and the opposite way, i.e, $2\Eo/Q \ll \tauo/\Rs$ then configuration $(b)$ becomes leading, in NGLs, while the contribution from configuration $(a)$ vanishes. That is $t_{b}^{2}$ in eq. becomes $$\label{t_param_akt_b_no-ordering}
t_{b}^{2} = \left(\frac{\as}{2\pi}\right)^{2}\ \ln^{2}\left(\frac{2\Eo\Rs}{Q\tauo}\right)\,\Theta\left(\frac{\tauo}{\Rs} - \frac{2\Eo}{Q}\right).$$ and $S_{2,b} = S_{2,a}$ in eq. . We do not consider this regime here though.
- If, on the other hand, we do not restrict ourselves to any particular ordering of the scales, as it is done in Refs. [@Hornig:2011tg] and [@Kelley:2011tj], then both configurations $(a)$ and $(b)$ would contribute to the leading NGLs. Adding up $t_{a}^{2}$, in , and $t_{b}^{2}$, in , the $\Theta$–functions sum up to unity and one recovers the result reported in the above mentioned references. Notice that it is a straightforward exercise to show that eq. is equal to $f_{\mathrm{OL}} + f_{\mathrm{OR}}$ given in eq. $(28)$ of [@Hornig:2011tg] in the case where $R_{L} = R_{R} = R$ ($=\bar{R}$ given in sec. \[sec.fixed\_order\_1\]). Moreover, the coefficient $f^{\CA}_{\mathrm{NGL}}$ given in eq. $(\mathrm{B}2)$ of [@Kelley:2011tj] v$2$ is related to $S_{2,a}$ by $ f^{\CA}_{\mathrm{NGL}} = -8 \times S_{2,a}$.
- Setting the cut–off scale $\Eo \sim \tauo Q$ in $t_{a}$, eq. , and $t_{b}$, eq. , would diminish NGLs coming from both configurations $(a)$ and $(b)$ and the threshold thrust becomes essentially a global observable. This is unlike the observation made in the study of the single jet with a jet veto distribution [@Banfi:2010pa] where the above choice of $\Eo$ kills the NGLs near the measured jet but introduces other equally significant NGLs near the unmeasured jet.
In the next subsection we recompute both $\CF^{2}$ and $\CF\CA$ contributions to the $\tauo$ distribution under the C–A clustering condition. For the $\CF\CA$ term, we only focus on configuration $(a)$ and do not attempt to address the subleading contributions coming from configurations $(b)$ and $(c)$.
$\tauo$ distribution in the C–A algorithm {#sec.FO_CA}
-----------------------------------------
The definition of the C–A algorithm is given in eq. with $p = 0$. Unlike the anti–$\kt$ algorithm, which successively merges soft gluons with the nearest hard parton, the C–A algorithm proceeds by successively clustering soft gluons amongst themselves. Consequently, a soft parton may in many occasions be dragged into (away from) a jet region and hence contributing (not contributing) to the invariant mass of the latter. The jet mass, and hence $\tauo$, distribution is then modified. It is these modifications, due to soft–gluons self–clustering, that we shall address below.
Any clustering–induced contribution to the $\tauo$ distribution will only arise from phase space configurations where the two soft gluons, $k_{1}$ and $k_{2}$, are initially (that is, before applying the clustering) in different regions of phase space. Configurations where both gluons are within the same jet region, gluon $k_{1}$ is in one of the two jet regions and gluon $k_{2}$ is in the other or both gluons are within the interjet region are not altered by clustering and calculations of the corresponding contributions will yield identical results to the anti–$\kt$ algorithm. We can therefore write the $\tauo$ distribution in the C–A algorithm, at $\Or(\as^{2})$, as $$\label{Sig2_CA_akt}
\Sigma^{(2)}_{\ca}(\tauo,\Eo) = \Sigma^{(2)}_{\akt}(\tauo,\Eo) + \delta
\Sigma^{(2)}(\tauo,\Eo).$$ It is the last term in eq. that we compute in the present subsection. Starting at configurations with two gluons in two different regions, the jet algorithm either:
(A) recombines the two soft gluons into a single parent gluon if the clustering condition is satisfied. The latter parent gluon will either be in one of the two jet regions or out of both of them (and hence in the interjet region).
(B) or leaves the two gluons unclustered, if the clustering condition is not satisfied. This case is then identical to the anti–$\kt$ one but with a more restricted phase space. This restriction comes from the fact that for the two gluons to survive the clustering they need to be sufficiently far apart. Quantitatively, their angular separation should satisfy the relation $$\label{CA_12_clust_cond}
(1-\cos\theta_{12}) > 2\Rs.$$
Below, we examine the contributions from configurations (A) and (B) to the $\CF^{2}$ and $\CF\CA$ colour pieces of the $\tauo$ distributions. All calculations are performed in the small $\Rs$ approximation using the $\ee$ variables $(\omega,\theta, \phi)$.
### $\CF^{2}$ term {#subsec.CA-LL}
![A schematic representation of a three–jet final state after applying the C–A algorithm on real emission along with virtual correction diagrams. The two gluons are clustered in the E–Scheme (see sec. \[sec.fixed\_order\_1\]). Identical diagrams hold for the $p_{L}$–jet.[]{data-label="fig.CLs_CF2"}](Figures/CLs.eps){width="15cm"}
Consider the gluonic configuration in (A) where the harder gluon $k_{1}$ is in the interjet region and the softer gluon $k_{2}$ is in the $p_{R}$–jet region. We account for the $p_{L}$–jet region through multiplying the final result by a factor of two. Applying the C–A algorithm , the smallest distance is $d_{\min} = d_{12}$. Hence gluon $k_{1}$ pulls gluon $k_{2}$ out of the $p_{R}$–jet region and form a third jet, as depicted in fig. \[fig.CLs\_CF2\]. The latter is then vetoed to have energy less than $\Eo$. The corresponding clustering angular function, in the small angles limit, reads $$\begin{aligned}
\label{clust_CA_CF2_theta}
\nonumber \Theta_{\ca}(1,2) &=& \Theta(\theta_{1}^{2}- 4\Rs) \Theta(4\Rs -
\theta_{2}^{2}) \Theta(\theta_{2}^{2} - \theta_{12}^{2}),
\\
\nonumber &=& \Theta(4 \theta_{2}^{2}\cos^{2}\phi_{2} -\theta_{1}^{2})
\Theta(\theta_{1}^{2}- 4\Rs) \theta(4\Rs - \theta_{2}^{2})
\Theta\left(\theta_{2}^{2} - \frac{\Rs}{\cos^{2}\phi_{2}}\right)
\Theta\left(\cos\phi_{2} - \frac{1}{2}\right).\\\end{aligned}$$ Adding up the corresponding virtual corrections, where one or both of the gluons are virtual, one obtains the following constraint on the phase space $$\Theta(\Eo -\omega_{1} -\omega_{2}) -\Theta(\Eo -\omega_{1}) +
\Theta\left(\frac{\omega_{2}}{2Q} \theta_{2}^{2} - \tauo\right).$$ Since we are working in the strong energy–ordered regime, $\omega_{1} \gg
\omega_{2}$, only the last $\Theta$–function survives. The new contribution to the $\CF^{2}$ piece of the $\tauo$ distribution is then given by $$\begin{aligned}
\label{C2_CA}
\nonumber C_{2}^{P} t_{p}^{2} &=& 8 \int^{Q/2}_{\frac{Q\tauo}{2\Rs}}
\frac{\d\omega_{2}}{\omega_{2}}
\int^{Q/2}_{\omega_{2}}\frac{\d\omega_{1}}{\omega_{1}}
\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\frac{\d\phi_{2}}{2\pi}
\int_{2\sqrt{\Rs}}^{2\theta_{2}\cos\phi_{2}} \frac{\d\theta_{1}}{\theta_{1}}
\int_{2\sqrt{\Rs}}^{\Rs/\cos\phi_{2}} \frac{\d\theta_{2}}{\theta_{2}},
\\
&=& 0.73 \;\CF^{2}\; \left(\frac{\as}{2\pi}\right)^{2}
\ln^{2}\left(\frac{\Rs}{\tauo}\right).\end{aligned}$$ This result is identical to [^13] that found in [@Banfi:2010pa] for a single jet mass (without a jet veto) distribution. The reason for this is that the clustering requirement only affects the distribution to which the softest gluon contributes. Which is in both cases the jet mass distribution.
![A schematic representation of a two–jet final state after applying the C–A algorithm on real emission along with virtual correction diagrams. The two gluons are clustered in the E–Scheme (see sec. \[sec.fixed\_order\_1\]). Identical diagrams hold for the left ($p_{L}$–) jet.[]{data-label="fig.CLs_2"}](Figures/CLs_2.eps){width="15cm"}
The second possible configuration that corresponds to case (A) is where gluon $k_{1}$ is in, say, the $p_{R}$–jet region and the softer gluon $k_{2}$ is in the interjet region. If the two gluons are clustered, i.e, gluon $k_{1}$ pulls in gluon $k_{2}$, then upon adding real emission and virtual correction diagrams, depicted in fig. \[fig.CLs\_2\], one obtains the following phase space constraint $$\label{PS_const_CA_CF2}
-\Theta\left(\tauo -\frac{\omega_{1}}{2Q} \theta_{1}^{2}\right)
\Theta\left(\frac{\omega_{2}}{2 Q} \theta_{2}^{2} - \tauo\right) +
\Theta(\omega_{2} - \Eo),$$ where we have assumed small angles limit and employed the LL accurate approximation $$\Theta\left(\tauo - \frac{\omega_{1}}{2Q} \theta_{1}^{2} - \frac{\omega_{2}}{2Q}
\theta_{2}^{2}\right) \simeq \Theta\left(\tauo - \frac{\omega_{1}}{2Q}
\theta_{1}^{2} \right) \Theta\left(\tauo - \frac{\omega_{2}}{2Q}
\theta_{2}^{2}\right).$$ Given the fact that $\omega_{1} \gg \omega_{2}$ and $\theta_{1}$ and $\theta_{2}$ must be close to each other to be clustered, i.e, they should satisfy condition , then the first two $\Theta$–functions in eq. are substantially suppressed and one is only left with the veto on $\omega_{2}$. Applying the C–A algorithm one obtains an identical clustering function to eq. . Hence the CLs’ coefficient for this configuration is equal to $C_{2}^{P}$ given in eq. . That is $C_{2}^{P} = 0.73\,\CF^{2}$. The evolution parameter does however change. It is now given, at $\Or(\as^{2})$, by $$\label{C2_CA_b}
t_{p}^{' 2} = \left(\frac{\as}{2\pi}\right)^{2}
\ln^{2}\left(\frac{2\Eo}{Q}\right).$$ This contribution is then beyond our NLL control. Note that the CLs contribution in eq. is equal to what one would find for interjet energy flow distribution provided that the rapidity gap is defined through eq. .
Let us now turn to case (B) where the two gluons are not merged together. If gluon $k_{1}$ is in the interjet region and gluon $k_{2}$ is in one of the two jet regions then the corresponding phase space constraint reads $$\label{CF2_CA_B}
\Theta\left(\frac{2Q\tauo}{\omega_{2}} -\theta_{2}^{2}\right) \\
\Theta(\omega_{1} - \Eo) \left[1 - \Theta_{\ca}(1,2)\right].$$ The limits on $\theta_{2}$–integral are then given by: $\min(4\Rs,2Q\tauo/\omega_{2}) > \theta_{2}^{2}>0$. Imposing the constraint $2\Eo/Q \gg \tauo/\Rs$, it is straightforward to see that the above constraint yields NNLL contribution and thus beyond our control. Similarly, the configuration where gluon $k_{1}$ is in the jet region and gluon $k_{2}$ is in the interjet region yields subleading logs.
Hence the $\CF^{2}$ piece of the clustering–induced correction term $\delta\Sigma^{(2)}$, in eq. , up to NLL, reads $$\label{Sig_CA_CF2}
\delta\Sigma^{(2)}(\tauo,\Eo) = C^{P}_{2}\,t_{p}^{2},$$ Next we compute the $\CF\CA$ piece of $\delta\Sigma^{(2)}$.
### $\CF\CA$ term {#subsec.CA-NGL}
Consider the gluonic configuration $(a)$ depicted in fig. \[fig.NGLs\_akt\]. Applying the C–A clustering algorithm on the latter yields two possibilities. Namely the two gluons are either clustered or not. The former case completely cancels against virtual corrections and thus does not contribute to NGLs. It is when the two gluons survive the clustering, the latter case, that a real–virtual mismatch takes place and NGLs are induced. The corresponding evolution parameter is equal to $t$ of the anti–$\kt$ case, eq. . The clustering condition is simply one minus that in eq. . The NGLs’ coefficient can then be written, using the eikonal amplitude , as $$\label{S2_CA_theta}
S^{\ca}_{2} = S_{2} + \delta\Sigma^{(2)}_{\CF\CA}$$ where $S_{2}$ is given in eq. and $$\begin{gathered}
\delta\Sigma^{(2)}_{\CF\CA} = 8\,\CF\CA
\int_{\sqrt{\Rs}}^{2\theta_{2}\cos\phi_{2}}\frac{\d\theta_{1}}{\sin\theta_{1}}
\int_{\frac{\sqrt{Rs}}{\cos\phi_{2}}}^{2\sqrt{\Rs}}\frac{\d\theta_{2}}{
\sin\theta_{2}} \int_{-\pi/3}^{\pi/3}\frac{\d\phi_{2}}{2\pi}
\left[\frac{1-\cos\theta_{1}\cos\theta_{2}}{1-\cos\theta_{12}} -1\right] \times
\\ \times \Theta\left(\frac{\Rs}{\tauo\cos\phi_{2}} -
\frac{Q}{2\omega_{2}}\right),\end{gathered}$$ We can perform the $\theta_{1}$–integral analytically and then resort to numerical methods to evaluate the remaining $\theta_{2}$ and $\phi_{2}$ integrals. The result, in terms of the jet–radius $\Rs$, is depicted in Fig. \[fig.NG\_coff\_CA\].
![Non–global coefficient $S_{2}$ in the anti–$\kt$ and C–A algorithms.[]{data-label="fig.NG_coff_CA"}](Figures/S2_AKT-CA.eps){width="11cm"}
$- S^{\ca}_{2}$ saturates at around $0.44\times 2\pi^{2}/3\, \CF\CA\sim
2.92\, \CF\CA$, i.e, a reduction of about $55\%$ in $S_{2}$. This is due to the fact that for the two gluons to survive clustering they need to be sufficiently far apart ($\theta_{12} > R = 2\sqrt{\Rs}$). The dominant contribution to $S_{2}$ comes, however, from the region of phase space where the gluons are sufficiently close. This corresponds to the collinear region of the matrix–element; $\theta_{1} \sim \theta_{2}$. Hence the further apart the two gluons get from each other, the less (collinear) singular the matrix becomes and thus the smaller the value of NGLs coefficient.
Note that the C–A coefficient $S_{2}^{\ca} = 2\times f^{\mathrm{C/A}}_{\mathrm{OR}}$, where $f^{\mathrm{C/A}}_{\mathrm{OR}}$ is given by eq. $(38)$ in [@Hornig:2011tg], at least in the small jet–radius region. Noticeably, the two results coincide at both limits $\Rs \rightarrow 0$ and $\Rs \rightarrow 1/2$ (equivalently $R \rightarrow 0$ and $R \rightarrow \sqrt{2}$ in [@Hornig:2011tg]). In fact, the coefficient $S_{2}^{\ca}$ is valid, as we shall see in sec. \[sec.numerical\_results\], for quite large jet–radii; up to $\Rs \sim 0.3$ (equivalent to $R \sim 1$ in [@Hornig:2011tg]).
The fixed–order NLL logarithmic structure of the $\tauo$ distribution should by now be clear for both jet algorithms. In order to assess the phenomenological impact of NGLs and clustering requirement on the final cross–section, it is necessary to perform an all–orders treatment, which we do below.
Resummation of $\tauo$ distribution {#sec.resummation_in_QCD}
===================================
Resummation, which is essentially the organisation of large logs arising from soft and/or collinear radiation to all–orders, is based on the factorisation property of the pQCD matrix–element squared for multiple gluon radiation. This is only true for independent primary emissions though. Including secondary correlated emissions, the picture dramatically changes and the resummation can only be performed at some limits, eg. large–$\Nc$ limit [@Banfi:2005gj]. In the standard method [@Collins:1984kg; @Catani:1992ua; @Bonciani:2003nt], resummation is carried out in Mellin (Laplace) space instead of momentum space. Only at the end does one transform the result back to the momentum space through (inverse Mellin transform), $$\label{inv_Mell_tauo}
\Sigma_{P}(\tauo, \Eo) = \int \frac{\d\nu}{2\imath\pi\nu}\; e^{\nu\tauo}\; \int
\frac{\d\mu}{2\imath\pi\mu}\; e^{\mu\Eo}\;\widetilde{\Sigma}_{P}(\nu^{-1},
\mu^{-1}),$$ where $P$ stands for primary emission. With regard to non–global observables, the important point to notice is that the resummation of NGLs is included as a factor multiplying the single–gluon Sudakov form factor, $\Sigma_{P}$, [@Dasgupta:2001sh] $$\label{resum_tot}
\Sigma\left(\tauo, \Eo\right) = \Sigma_{P}(\tauo,\Eo)\; S\left( t\right),$$ In this section, we first consider resummation of $\tauo$ distribution in events where the final state jets are defined in the anti–$\kt$ algorithm and, second, discuss the potential changes to the resummed result when the jets are defined in the C–A algorithm instead.
Resummation with anti–$\kt$ algorithm {#subsec.resummation_QCD}
-------------------------------------
As stated in the introduction and proved in sec. \[sec.fixed\_order\_1\], the $\tauo$ observable is simply the sum of the invariant masses of the two highest–energy (or highest–$\pt$ for hadron colliders) jets. Therefore the $\tauo$ resummed Sudakov form factor is just double that computed in [@Banfi:2010pa], for a single jet mass. That is, up to NLL level we have $$\label{resum_tot_akt}
\Sigma_{P}(\tauo,\Eo) = \frac{\exp\left[-2\left(\R_{\tauo}(\tauo) + \gamma_{E}
\R'_{\tauo}(\tauo) \right)\right]}{\Gamma\left(1+2\,\R'_{\tauo}(\tauo)\right)}\;
\exp\left[ - \R_{\Eo}(\Eo)\right] .$$ The full derivation of as well as the resultant expressions of the various radiators are presented in the small jet–radius limit in Ref. [@Banfi:2010pa]. To restore the full $\Rs$ dependence we make the replacement $R^{2}/\rho \mapsto \Rs/(\tauo (1-\Rs))$ such that when expanded eq. reproduces at $\Or(\as)$ the LO distribution .
To account for the NGLs at all–orders in $\as$, it is necessary to consider an arbitrary ensemble of energy–ordered, soft wide–angle gluons that coherently radiate a softest gluon into the vetoed region of phase space [@Dasgupta:2001sh] [^14]. The analytical resummation of NGLs is then plague with mathematical problems coming from geometric and colour structure of the gluon ensemble. Two methods have been developed to address this issue: A numerical Monte Carlo evaluation [@Dasgupta:2001sh; @Dasgupta:2002bw] and a non–linear evolution equation that resums single logs (SL) at all–orders [@Banfi:2002hw]. Both methods are only valid in the large–$\Nc$ limit. In the latter limit and for small values of the jet–radius $\Rs$, we argued in [@Banfi:2010pa] that the form of $S(t)$ should be identical to that found in the hemisphere jet mass case [@Dasgupta:2001sh]. Since in the present paper we are not confined to the small $\Rs$ limit, we need to modify and re–run the Monte Carlo algorithm, presented in [@Dasgupta:2001sh], for medium and large values of the jet–radius should we seek to resum the $\tauo$ NGLs distribution. The latter task is, however, beyond the scope of this paper. Here, we are only aiming at comparing the analytical results with fixed–order NLO program `EVENT2`. It suffices in this case to simply exponentiate the first NGLs term in eq. , $$\label{resum_NGLs_akt}
S(t) = \exp\left(S_{2}\; t^{2}\right).$$
The distribution is of the generic form given in eq. . Explicitly, it reads $$\label{resum-form_QCD-b}
\Sigma(\tauo, \Eo) = \left(1+\sum_{k=1}^{\infty} C_{k}
\left(\frac{\as}{2\pi}\right)^{k} \right)
\exp\left[\sum_{n=1}^{\infty}\sum_{m=0}^{n+1} G_{nm}
\left(\frac{\as}{2\pi}\right)^{n} \widetilde{L}^{m} \right] +
D_{\mathrm{fin}}(\tauo),$$ where $C_{k}$ is the $k^{th}$ loop–constant, $\widetilde{L} = \ln(1/\tauo)$ and $D_{\mathrm{fin}} \equiv D$, which vanishes in the limit $\tauo \rightarrow 0$. In order to determine the coefficients $G_{nm}$ at NLO and up to NLL,we need to expand the radiators, as well as the $\Gamma$ function, in eq. up to second order in the fixed coupling $\as =
\as(Q)$. The results are presented in appendix. \[app.coeff\_in\_expansion\]. Although we have provided the NNLL coefficient, $G_{21}$ in eq. , we do not claim that it is under control. Nonetheless, it does capture all $\Rs$–dependent terms [^15]. The missing terms from $G_{21}$ include: a) coefficients of $\widetilde{L}$ which are independent of $\ln(\Rs/(1-\Rs))$ for all colour channels. These can be borrowed from thrust distribution [@Kelley:2011ng; @Monni:2011gb; @Hornig:2011iu]. b) Although $G_{21}$ has a subleading NGLs term in the $\CF\CA$ colour channel, which comes solely from the expansion of $t$ (eq. ), the full expression in this channel as well as in the $\CF\TF\nf$ channel is still missing. To properly compute the latter, one has to extend both the matrix–element and the phase space to include hard emission. Such a task will be considered elsewhere. It is worthwhile to mention that full subleading NGLs have recently been computed analytically within SCET framework for the hemisphere mass variable [@Kelley:2011ng; @Hornig:2011iu] [^16]. The two–loop constant $C_{2}$ has also been computed for the latter variable as well as the thrust [@Kelley:2011ng; @Monni:2011gb].
To make contact with SCET calculations, we provide in appendix \[app.tw\_in\_SCET\] the full formula of the Sudakov form factor for the $\tauo$ primary distribution including determination of the $G_{nm}$ coefficients in SCET.
Next we comment on the form of resummation when final state jets are defined in the C–A algorithm.
Resummation with C–A algorithm {#subsec.resum_CA}
------------------------------
With regard to primary emission piece, resumming logs induced by clustering is a cumbersome but doable task. It has been performed, for example, in [@Delenda:2006nf] for interjet energy flow distribution where final state jets are defined in the inclusive $\kt$ algorithm. The final result of the resummed radiator was written as an expansion in the jet–radius and the first four terms were determined. For secondary emissions, the resummation of NGLs has only been possible numerically and in the large–$\Nc$ limit. It has again been carried out for the above mentioned energy flow distribution in [@Appleby:2002ke]. We expect that analogous, to the interjet energy flow, analytical treatment and numerical evaluation can be achieved for the resummation of CLs and NGLs, respectively, for the $\tauo$ variable. We postpone this work to future publications.
For the sake of comparing to `EVENT2`, it is sufficient to simply exponentiate the fixed–order terms $S^{\ca}_{2}$ and $C_{2}^{P}$, just as we did with the anti–$\kt$ algorithm case. Due to the fact that logarithmic contributions induced by clustering arise mainly from soft wide–angle gluons, we expect them -clustering–induced logs- to factorise from the primary form factor at all–orders. Therefore, the resummed distribution, whereby clustering is imposed on the final state, may be written as $$\label{resum_tot_CA}
\Sigma(\tauo,\Eo) = \Sigma_{P}(\tauo, \Eo) \,S^{\ca}\left(t\right)
C^{P}\left(t_{p}\right),$$ where $S^{\ca}$ is of the form with $S_{2}$ replaced by $S_{2}^{\ca}$ and, in analogy with the NGLs factor, the CLs factor reads $$\label{resum_C_P_CA}
C^{P}\left(t_{p}\right) = \exp\left(C_{2}^{P}\,t_{p}^{2}\right),\;\;\; t_{p} =
\int^{Q/2}_{Q\tauo/2 \Rs} \frac{\d\kt}{\kt} \frac{\as(\kt)}{2\pi}.$$
In fig. \[fig.resummed\_CA\_akt\] we plot the resummed differential distributions, $\d \Sigma(\tauo,\Eo)/\d\tauo = (1/\so)\d\sigma/\d\tauo$, computed from eq. for the anti–$\kt$ algorithm and from eq for the C–A algorithm at different values of $\Rs$. The dependence on $\Eo$ has been discussed in [@Banfi:2010pa] where the all–orders NGLs resummed expression was employed. There are several points to note. Firstly the effect of NGLs is a suppression of the total cross–section relative to the primary result. This suppression is diminished by decreasing the value of $\Rs$. For example, at $\Eo=0.1 Q$ the Sudakov peak is reduced due to NGLs by about $4.02\%, 3.42\%, 2.62\%$ and $1.35\%$ for $\Rs = 0.30, 0.12, 0.04$ and $0.0025$ (equivalent to $R = 1.1, 0.7, 0.4$ and $0.1$) respectively. These values are only meant to give an idea of the effect of varying the jet–radius parameter on both NGLs and CLs corrections to the total cross–section, since we are only working with an approximation of the latter and not the full all–orders result. It has been shown in [@Appleby:2002ke], for the interjet energy distribution, that the NGLs resummed factor $S(t)$ at all–orders is much smaller (thus larger suppression of primary–only result) and of different shape, as a function of $t$, to the fixed–order exponentiated result.
Secondly the effect of clustering is reducing the phenomenological significance of NGLs. This reduction becomes larger, hence the NGLs suppression on the Sudakov peak becomes smaller, as one moves towards smaller values of $\Rs$. For the same jet veto $\Eo = 0.1 Q$, the Sudakov peak is reduced by $0.62\%$ [^17]$, 0.80\%, 0.63\%$ and $0.22\%$ for $\Rs = 0.30, 0.12, 0.04$ and $0.0025$ respectively (values are only an estimate of the impact of clustering). Comparing to the anti–$\kt$ case, we see that the effect of NGLs has been reduced by more than $70\%$ for $\Rs= 0.12, 0.04$ and $\Rs = 0.0025$. This observation suggests that instead of resumming NGLs, which is a daunting task even numerically, one should, perhaps, attempt at eliminating them at each order through requiring final state clustering and looking for the optimal value of the jet–radius, and may be the jet veto too, such that non–global corrections are wiped out. In our rough approximation, we find that NGLs are completely eliminated, leaving only the primary Sudakov form factor, at $\Rs \lesssim 3\times 10^{-5}$ (equivalent to $R \lesssim 0.01$). Although this value is very small and not of any practical significance, including the full all–orders resummed results for both NGLs and CLs might result in practically larger values. Whether this is indeed the case remains to be investigated. If it turns out that the optimal radius is relatively large, $0.04 \lesssim \Rs\; (0.4 \lesssim R)$, then final state clustering will be the key to solve the NGLs subtlety of non–global observables.
![Comparison of analytical resummed differential distribution $\d\Sigma/\d\tauo$ where: only primary term included , primary and NGLs factor included in the anti–$\kt$ algorithm and primary $+$ NGLs $+$ CLs factors included . The plots are shown for various jet–radii with a jet veto $\Eo = 0.1 Q$. The coupling is taken at the $Z$ mass to be $\as(M_{Z}) \simeq 0.118$. The plots are only meant to give a rough estimate of the effects of NGLs in non–clustered as well as clustered final states.[]{data-label="fig.resummed_CA_akt"}](Figures/Resum_diff_0.3_50.eps "fig:"){width="7.5cm"} ![Comparison of analytical resummed differential distribution $\d\Sigma/\d\tauo$ where: only primary term included , primary and NGLs factor included in the anti–$\kt$ algorithm and primary $+$ NGLs $+$ CLs factors included . The plots are shown for various jet–radii with a jet veto $\Eo = 0.1 Q$. The coupling is taken at the $Z$ mass to be $\as(M_{Z}) \simeq 0.118$. The plots are only meant to give a rough estimate of the effects of NGLs in non–clustered as well as clustered final states.[]{data-label="fig.resummed_CA_akt"}](Figures/Resum_diff_0.12_50.eps "fig:"){width="7.5cm"} ![Comparison of analytical resummed differential distribution $\d\Sigma/\d\tauo$ where: only primary term included , primary and NGLs factor included in the anti–$\kt$ algorithm and primary $+$ NGLs $+$ CLs factors included . The plots are shown for various jet–radii with a jet veto $\Eo = 0.1 Q$. The coupling is taken at the $Z$ mass to be $\as(M_{Z}) \simeq 0.118$. The plots are only meant to give a rough estimate of the effects of NGLs in non–clustered as well as clustered final states.[]{data-label="fig.resummed_CA_akt"}](Figures/Resum_diff_0.04_50.eps "fig:"){width="7.5cm"} ![Comparison of analytical resummed differential distribution $\d\Sigma/\d\tauo$ where: only primary term included , primary and NGLs factor included in the anti–$\kt$ algorithm and primary $+$ NGLs $+$ CLs factors included . The plots are shown for various jet–radii with a jet veto $\Eo = 0.1 Q$. The coupling is taken at the $Z$ mass to be $\as(M_{Z}) \simeq 0.118$. The plots are only meant to give a rough estimate of the effects of NGLs in non–clustered as well as clustered final states.[]{data-label="fig.resummed_CA_akt"}](Figures/Resum_diff_0.0025_50.eps "fig:"){width="7.5cm"}
In the next section, we compare our analytical calculations to `EVENT2`. In particular, we focus on establishing the presence of NGLs and CLs in the $\tauo$ distribution at NLO.
Numerical results {#sec.numerical_results}
=================
The $\tauo$ numerical distribution has been computed using the fixed–order NLO QCD program `EVENT2`. The program implements the Catani–Seymour subtraction formalism for NLO corrections to two– and three–jet events observables in $\ee$ annihilation. Final state partons have been clustered into jets using the `FastJet` library [@Cacciari:2011ma]. The latter provides an implementation of the longitudinally invariant $\kt$, Cambridge–Aachen (CA) and anti–$\kt$ jet finders along with many others. Cone algorithms such as SISCone [@Salam:2007xv] are also implemented as plugins for the package. It should be noted that the $\ee$ version of the aforementioned algorithms employs the following clustering condition for a pair of partons $(ij)$ $$\label{clust_cond_FJ}
1-\cos\theta_{ij} < 1-\cos(\widetilde{R}),$$ where $\widetilde{R}$ is the jet–radius parameter used in `FastJet` [^18]. Compared to eqs and , $\widetilde{R} =
\cos^{-1}(1-2\Rs)$. The exact numerical distributions $(1/\so)(\d\sigma_{e}/\d
L)$, with $L = -\widetilde{L} = \ln(\tauo)$,. for the three colour channels, $\CF^{2}, \CF\CA$ and $\CF\TF\nf$, have been obtained with $10^{11}$ events in the bin range $0>L>-14$. We have used four values for the jet–radius: $\Rs = 0.50, \Rs = 0.30, \Rs=0.12$ and $\Rs = 0.04$, with an energy veto $\Eo = 0.01\,Q$. Standard deviations on individual bins range from $10^{-4}\% $ to $ 10^{-2}\%$.
We plot the difference between the numerical and analytical distributions at both LO and NLO, $$\label{remainder}
r(L) = \frac{\d\sigma_{e}}{\so\d \,L} - \frac{\d\sigma_{r,2}}{\so\d L},$$ where $\d\sigma_{r,2}/\so\d L$ is given in eq. . Recall that at small values of the jet shape, $\tauo$, the finite remainder function $D_{\mathrm{fin}}(\tauo)$ is vanishingly small and will thus be ignored. For the case where the jet shape is global ($\Rs = 0.50$ and the threshold thrust reduces to thrust), we expect a full cancellation of singular terms and thus $r$ should be a constant line corresponding to the NNLL coefficient ($H_{21}$ in eq. ). For $\Rs < 0.5$, the jet shape is non–global and we expect $r$ to have a slope if NGLs contribution is excluded. If our analytical calculations of the NGLs’ coefficient, both for anti–$\kt$ and C–A algorithms, are correct then upon adding the latter to $H_{22}$ the slope should vanish and $r$ becomes flat signalling a complete cancellation of terms up to NLL level. Similar behaviour should be seen with the CLs’ coefficient $C_{2}^{P}$ for the C–A algorithm case. Considering figs. \[fig.LO\] - \[fig.CA\_CFTF\], we make the following observations:
- At LO, the distribution is independent of the jet definition. From eq. we have $$H_{11} = - 3 + 4 L_{\Rs} = -4, -8.51, -14.75, -20.95 ;\;\;\; \mathrm{for}\; \Rs
= 0.5, 0.3, 0.12, 0.04.$$ Compared to the numerical results shown in fig. \[fig.LO\] we see a complete agreement. The cut–off in fig. \[fig.LO\] is due to the fact that at LO $\tauo < \Rs/(1+\Rs)$ (eq. ).
![The difference between `EVENT2` and $\tauo$ LO distribution for various jet radii in both anti–$\kt$ (left) and CA (right) algorithms.[]{data-label="fig.LO"}](Figures/AKT_LO.eps "fig:"){width="7.4cm"} ![The difference between `EVENT2` and $\tauo$ LO distribution for various jet radii in both anti–$\kt$ (left) and CA (right) algorithms.[]{data-label="fig.LO"}](Figures/CA_LO.eps "fig:"){width="7.4cm"}
- For the NLO distribution in the anti–$\kt$ algorithm, fig. \[fig.AKT\_CFCA\] (left) illustrates the existence of NGLs. The flatness of the $r(L)$ curve, in fig. \[fig.AKT\_CFCA\] (right), at $L$ below about $-9$ indicates a complete cancellation up to single log level.
![The $\CF\CA$ part of the difference between `EVENT2` and (left) $\tauo$ primary (global) distribution and (right) $\tauo$ distribution including NGLs for various jet radii in anti–$\kt$ algorithm.[]{data-label="fig.AKT_CFCA"}](Figures/AKT_CFCA.eps "fig:"){width="7.4cm"} ![The $\CF\CA$ part of the difference between `EVENT2` and (left) $\tauo$ primary (global) distribution and (right) $\tauo$ distribution including NGLs for various jet radii in anti–$\kt$ algorithm.[]{data-label="fig.AKT_CFCA"}](Figures/AKT_CFCA_NGLs.eps "fig:"){width="7.4cm"}
[cc|c|c||c|c||c|c|]{} & & & &\
$\Rs$ & Jet alg & $H_{21}^{\mathrm{num}}$ & $H_{21}^{\mathrm{analyt}}$ & $H_{21}^{\mathrm{num}}$ & $H_{21}^{\mathrm{analyt}}$ & $H_{21}^{\mathrm{num}}$ & $H_{21}^{\mathrm{analyt}}$\
& & $5.20 \pm 0.14$ & $5.00$ & & $ -7.04$ & & $ -12.68 $\
& & $4.99 \pm 0.19$ & & & & &\
& &$ 13.92 \pm 0.15$ & $7.80$ & & $ -11.45$ & & $ 62.20 $\
& & $11.24 \pm 0.08$ & & & & &\
& &$ 15.75 \pm 0.12$ & $8.54$ & & $ -8.89$ & & $ 385.27 $\
& &$ 13.65 \pm 0.08$ & & & & &\
& &$ 12.86 \pm 0.19$ & $5.66$ & & $ 3.59$ & & $ 1022.43$\
& & $ 10.60 \pm 0.23$ & & & & &\
The $\CF^{2}$ and $\CF\TF\nf$ pieces are shown in fig. \[fig.AKT\_CF2-CFTF\]. In table \[tab:H21\_akt\] we provide both numerical and analytical, taken from SCET calculations , values of the NNLL coefficient, $H_{21}$, at the considered $\Rs$ values for the three colour channels. It is evident from the table that there are subleading $\Rs$–dependent NGLs for both $\CF\CA$ and $\CF\TF\nf$ channels. Such logs have been analytically computed in [@Kelley:2011ng] for the hemisphere jet mass. Our numerical results show that they are also present for finite–size jets [^19]. The primary $\CF^{2}$ channel is free from such subleading NGLs as numerical and analytical values of $H_{21}$ in the anti–$\kt$ coincide.
Notice that while the $x$–axis in all figures shown in this section corresponds to $\ln(\tauo) = \log(\tauo)$, i.e, the natural logarithm of the jet shape, that of [@Hornig:2011tg] corresponds to the logarithm of base $10$, $\log_{10}(\rho) \sim \log(\rho)/2, \; \rho \equiv \tauo$, of the jet shape. Given this, fig. \[fig.AKT\_CFCA\] above is equivalent to fig. $7$ of [@Hornig:2011tg]. Neither $\CF^{2}, \CF\TF\nf$ plots nor subleading NGLs were considered in [@Hornig:2011tg].
![The (left) $\CF^{2}$ and (right) $\CF\TF\nf$ piece of the difference between `EVENT2` and $\tauo$ distribution for various jet radii in the anti–$\kt$ algorithm.[]{data-label="fig.AKT_CF2-CFTF"}](Figures/AKT_CF2.eps "fig:"){width="7.5cm"} ![The (left) $\CF^{2}$ and (right) $\CF\TF\nf$ piece of the difference between `EVENT2` and $\tauo$ distribution for various jet radii in the anti–$\kt$ algorithm.[]{data-label="fig.AKT_CF2-CFTF"}](Figures/AKT_CFTF.eps "fig:"){width="7.5cm"}
- The asymptotic region, i.e, the region where large logs are expected to dominate over non–logarithmic contributions, corresponds to $L$ less than about $-9$ (for figs. \[fig.AKT\_CFCA\], $\CF^{2}$ piece in fig \[fig.AKT\_CF2-CFTF\] and may even be less for the $\CF\TF\nf$ piece in fig. \[fig.AKT\_CF2-CFTF\]) and seems to decrease further as $\Rs$ becomes smaller. A similar effect is seen in the thrust distribution, fig. \[fig.thrust\_NLO\], where the numerical distribution has been obtained using the full definition .
![The various colour pieces of the difference between `EVENT2` and thrust distribution using the full definition . The pQCD resummed analytical expression for thrust distribution can be found in, for example, [@QCD_collider].[]{data-label="fig.thrust_NLO"}](Figures/thrust_CF2.eps "fig:"){width="7.5cm"} ![The various colour pieces of the difference between `EVENT2` and thrust distribution using the full definition . The pQCD resummed analytical expression for thrust distribution can be found in, for example, [@QCD_collider].[]{data-label="fig.thrust_NLO"}](Figures/thrust_CFCA.eps "fig:"){width="7.5cm"} ![The various colour pieces of the difference between `EVENT2` and thrust distribution using the full definition . The pQCD resummed analytical expression for thrust distribution can be found in, for example, [@QCD_collider].[]{data-label="fig.thrust_NLO"}](Figures/thrust_CFTF.eps "fig:"){width="7.5cm"}
- Considering the clustering case with C–A algorithm, figs. \[fig.CA\_CF2\] and \[fig.CA\_CF2\_zoom\] illustrate the presence of CLs in the $\CF^{2}$ channel. Clearly, the addition of CLs makes the remainder $r$ flat in the region $L \lesssim -9$. To strengthen this observation even more, we plot in fig. \[fig.AKT-CA\] the difference between `EVENT2` distributions in anti–$\kt$ and C–A algorithms, $(\d\sigma^{\akt}_{e}/\d L -
\d\sigma^{\ca}_{e}/\d L)/\so$, for all colour pieces. The slopes for the $\CF^{2}$ and $\CF\CA$ indicate that an NLL positive $\Rs$–dependent term, and possibly an NNLL term as well, have been induced by clustering. This is confirmed in table \[tab:H21\_akt\]. Moreover, the fact that the difference between the latter distributions in the $\CF\TF\nf$ piece is non–vanishing implies an NNLL impact of clustering.
Furthermore, we note from fig. \[fig.CA\_CF2\_zoom\] that $C_{2}^{P}$ seems to slightly vary with the jet–radius parameter $\Rs$. This can be seen for large values of $\Rs$ ($\Rs = 0.3$) where our small angles approximation is not expected to apply.
![The $\CF^{2}$ part of the difference between `EVENT2` and (left) $\tauo$ primary (global) distribution and (right) $\tauo$ distribution including CLs for various jet radii in the C–A algorithm.[]{data-label="fig.CA_CF2"}](Figures/CA_CF2.eps "fig:"){width="7.5cm"} ![The $\CF^{2}$ part of the difference between `EVENT2` and (left) $\tauo$ primary (global) distribution and (right) $\tauo$ distribution including CLs for various jet radii in the C–A algorithm.[]{data-label="fig.CA_CF2"}](Figures/CA_CF2_CLs.eps "fig:"){width="7.5cm"}
![Zoomed–in plots for the $\CF^{2}$ part of the difference between `EVENT2` and analytical $\tauo$ distribution with and without CLs for (left) $\Rs=0.12$ and (right) $\Rs=0.3$ in the C–A algorithm.[]{data-label="fig.CA_CF2_zoom"}](Figures/CA_CF2_CLs_0.12.eps "fig:"){width="7.5cm"} ![Zoomed–in plots for the $\CF^{2}$ part of the difference between `EVENT2` and analytical $\tauo$ distribution with and without CLs for (left) $\Rs=0.12$ and (right) $\Rs=0.3$ in the C–A algorithm.[]{data-label="fig.CA_CF2_zoom"}](Figures/CA_CF2_CLs_0.3.eps "fig:"){width="7.5cm"}
![Plots of the three colour pieces of the difference between two `EVENT2` distributions corresponding to anti–$\kt$ and C–A algorithms for various jet radii. We only show $\CF^{2}$ and $\CF\TF\nf$ results at three values of the jet–radius due to large errors in these colour channels.[]{data-label="fig.AKT-CA"}](Figures/AKT-CA_CF2.eps "fig:"){width="7.5cm"} ![Plots of the three colour pieces of the difference between two `EVENT2` distributions corresponding to anti–$\kt$ and C–A algorithms for various jet radii. We only show $\CF^{2}$ and $\CF\TF\nf$ results at three values of the jet–radius due to large errors in these colour channels.[]{data-label="fig.AKT-CA"}](Figures/AKT-CA_CFCA.eps "fig:"){width="7.5cm"} ![Plots of the three colour pieces of the difference between two `EVENT2` distributions corresponding to anti–$\kt$ and C–A algorithms for various jet radii. We only show $\CF^{2}$ and $\CF\TF\nf$ results at three values of the jet–radius due to large errors in these colour channels.[]{data-label="fig.AKT-CA"}](Figures/AKT-CA_CFTF.eps "fig:"){width="7.5cm"}
- Similar analysis to those carried in the anti–$\kt$ algorithm apply to the $\CF\CA$ piece of the $\tauo$ distribution in the C–A algorithm. Including the NGLs makes the $r(L)$ curve looks convincingly flat in the region $L
\lesssim -9$, particularly for smaller values of $\Rs$, as shown in fig. \[fig.CA\_CFCA\]. Recall that we have used the small $\Rs$ limit in carrying out the computation of $S_{2}^{\ca}$, eq. . Our findings agree with those reported in [@Hornig:2011tg] for jet–radii up to $\Rs \sim 0.3\;(R \sim 1)$.
For completeness, the $\CF\TF\nf$ piece of the $r(L)$ in the C–A algorithm is depicted in fig. \[fig.CA\_CFTF\]. As shown in table \[tab:H21\_akt\], clustering requirement again reduces the impact of the subleading NGLs in both $\CF\CA$ and $\CF\TF\nf$ channels.
![The $\CF\CA$ part of the difference between `EVENT2` and (left) $\tauo$ primary (global) distribution and (right) $\tauo$ distribution including NGLs for various jet radii in the C–A algorithm.[]{data-label="fig.CA_CFCA"}](Figures/CA_CFCA.eps "fig:"){width="7.5cm"} ![The $\CF\CA$ part of the difference between `EVENT2` and (left) $\tauo$ primary (global) distribution and (right) $\tauo$ distribution including NGLs for various jet radii in the C–A algorithm.[]{data-label="fig.CA_CFCA"}](Figures/CA_CFCA_NGLs.eps "fig:"){width="7.5cm"}
![The $\CF\TF\nf$ piece of the difference between `EVENT2` and $\tauo$ distribution for various jet radii in $\ca$ algorithm.[]{data-label="fig.CA_CFTF"}](Figures/CA_CFTF.eps){width="7.5cm"}
In summary, we have confirmed through explicit comparison to exact numerical distributions the existence of large NGLs and large CLs for the $\tauo$ distribution at NLL and beyond. In light of these findings, the surprising cancellation between primary–only analytical distribution and `EVENT2` presented in [@Kelley:2011tj] v$1$ may be explained as follows. While each colour part ($\CF^{2}, \CF\CA$ and $\CF\TF\nf$) separately does not agree with `EVENT2`, as shown in figs. \[fig.CA\_CF2\], \[fig.CA\_CF2\_zoom\], \[fig.CA\_CFCA\] and \[fig.CA\_CFTF\], their sum seems to agree with `EVENT2` (recall that in of $[4]$ v$1$ only the sum of the three colour factors is plotted against `EVENT2`). Such an unexpected agreement can arise from the following possible sources:
- The $\ln(\tauo)$ region considered in [@Kelley:2011tj] v$1$ does not correspond to the asymptotic region where large logs are expected to dominate over non–logarithmic terms. Thus the agreement shown in plot $1$ of [@Kelley:2011tj] v$1$ does not convey any message and all one can say is that the non–logarithmic terms in the range $[-9,0]$ happen to cancel out (see fig \[fig.CA\_tot\]).
- NGLs are significantly reduced in the $\ca$ algorithm especially for a jet radius $\Rs =0.3$ (which is the one considered in [@Kelley:2011tj] v$1$), as clearly seen in figs. \[fig.NG\_coff\_CA\] and \[fig.CA\_CFCA\] (left). For smaller jet radii, there is a clear disagreement between the result of [@Kelley:2011tj] v$1$ and `EVENT2` as shown in fig. \[fig.CA\_tot\]
![Difference between the sum of the three colours and `EVENT2` for various jet radii.[]{data-label="fig.CA_tot"}](Figures/CA_NLO.eps){width="12cm"}
We have also shown that clustering the final state partons with the C–A algorithm yielded a significant reduction in NGLs impact, at NLL and beyond, albeit inducing large CLs, at NLL and beyond, in the primary emission sector.
Conclusion {#sec.conclusion}
==========
The jet mass with a jet veto, or simply the threshold thrust, is an example of a wider class of non–global observables. These have the characteristic of being sensitive to radiation into restricted regions of phase space, or sensitive to radiation into the whole phase space but differently in different regions. For such observables the universal Sudakov form factor fails to reproduce the full logarithmic structure even at NLL accuracy. New contributions that are dependent on various variables such as the jet size and jet definition appear at this logarithmic level. In this paper, we have elaborated on these very contributions for the aforementioned observable.
Considering secondary emissions, we have computed the full analytical expression of the first term, $S_{2}$, in a series of missing large logs, namely NGLs. The coefficient depends, as anticipated, on the jet size and saturates at its maximum in the limit where the latter, i.e, jet size, vanishes. This saturation value was used in [@Banfi:2010pa] as an approximation to the full value in the small $\Rs$ limit. It turns out that the approximation is valid for quite a wide range of $\Rs$. The formula for $S_{2}$ has been checked against full exact numerical result obtained by the program `EVENT2`. The difference between the analytical and numerical differential distributions was shown to be asymptotically flat signalling a complete cancellation of singular terms up to NLL level. This has all been done for final states defined in the cone–like anti–$\kt$ jet algorithm.
To illustrate the dependence of NLL on the jet definition, we have investigated the effects of applying the C–A algorithm on $\ee$ final states. The impact of soft partons clustering is two–fold. On one side, it reduces the size of NGLs through shrinking the phase space region where the latter dominantly come from. i.e, the region where the emitter and emitted soft partons are just in and just out of the jet. On the other side, it gives rise to new NLL logarithmic contributions, CLs, in the primary emission sector. In the small jet–radius limit, the corresponding coefficient at second order has been shown, through comparison to `EVENT2`, to be independent of $\Rs$.
Furthermore, our numerical analyses with `EVENT2` have shown that the asymptotic region where the said large logs, in both anti–$\kt$ and C–A jet algorithms, dominate corresponds to $L \lesssim -9$ and decreases for smaller values of the jet–radius. As a by–product, we have found that there are subleading NGLs in both $\CF\CA$ and $\CF\TF\nf$ pieces as well as subleading CLs in the $\CF^{2}$ piece of the $\tauo$ distribution. Clustering impact on NGLs has been observed to extend to NNLL level too. Regarding NGLs in $\CF\CA$ channel in both jet algorithms, our findings serve as a confirmation of the corresponding calculations performed within SCET in [@Hornig:2011tg].
Based on our rough approximation to NLL resummation, which is exponentiating the fixed–order result for both NGLs and CLs, it has been shown that it may be possible to completely eliminate the non–global correction to the primary Sudakov form factor at all–orders for events where final states clustering is applied. This elimination can be achieved by tuning the jet–radius parameter, $\Rs$, of the jet algorithm as well as the jet veto $\Eo$. If such *optimal* values of $\Rs$ and $\Eo$ are of practical significance, that is $ R \sim 0.4$ or so and $\Eo \gg \Lambda_{\mathrm{QCD}}$, then the single–gluon exponentiation should be sufficient in describing the experimental data. A concrete answer of whether such optimal values exist can only be established once an all–orders resummation of primary, NGLs and CLs is performed. We postpone this investigation to future publications.
As mentioned earlier in sec. \[sec.fixed\_order\_2\] and shown in [@Banfi:2010pa], the inclusive $\kt$ jet algorithm behaves in an identical way to C–A algorithm with regard to the threshold thrust distribution. It would be interesting to conduct similar studies for events defined in IRC cone algorithms such as the SISCone. In principle, one expects to see analogous effects not only for the threshold thrust but for all shape variables that are of non–global nature. Moreover, we reserve the extension of the findings of this paper to hadron–hadron collisions to future work. Apart from complications due to coloured initial state, we expect the gross features of this paper to apply.
Acknowledgement {#acknowledgement .unnumbered}
---------------
I am indebted to M. Dasgupta, S. Marzani and A. Banfi for collaboration on related work and helpful discussions on the current paper. I would like to thank M. Seymour for his generous help and useful feedback.
Derivation of LO distribution {#app.LO_distr}
=============================
In the present section we outline the derivation of the full logarithmic part of the LO $\tauo$ integrated distribution . For the emission of a single gluon, i.e, $\ee \rightarrow q\,\qbar\,g$, we define the kinematic variables, $x_{i} = 2p_{i}.Q/Q^{2} = 2 E_{i}/Q$ and $y_{ij} = 2p_{i}.p_{j}/Q^{2}
= 1-x_{k}$ where $i,j,k = 1(q), 2(\qbar), 3(g)$. The $\Or(\as)$ matrix–element squared can be computed by considering two Feynman graphs corresponding to real emission of the gluon $g$ off the two hard legs $q, \qbar$. Applying the appropriate QCD Feynman rules and supplementing the three–body phase space factor, the corresponding differential distribution is given by $$\label{dsig1_sig}
\frac{\d^{2}\sigma^{(1)}}{\sigma \d x_{1} \d x_{2}}= \frac{\CF \as}{2\pi}
\frac{x_{1}^{2} + x_{2}^{2}}{(1-x_{1})(1-x_{2})},$$ where $\sigma$ is the total hadronic cross–section. Up to $\Or(\as^{2})$, it is given in terms of the Born cross–section, $\so$, by the relation [@Appelquist:1973uz] $$\label{sig_had-sig_0}
\frac{\sigma}{\sigma_{0}} = 1 + \frac{\as}{2\pi} \left[\frac{3\CF}{2}\right] +
\left(\frac{\as}{2\pi}\right)^{2} K_{2} + \Or(\as^{3}).$$ with $$\label{sig_sig0_K}
K_{2} = -\CF^{2} \frac{3}{8} + \CF\CA\left(\frac{123}{8} -11\zeta_{3}\right)
+\CF\TF\nf \left(-\frac{11}{2} + 4\zeta_{3}\right).$$ The integration region, which is originally $1\geq x_{1},x_{2} \geq 0$ and $x_{1}+x_{2} \geq 1$ and which leads to divergences, gets modified by introducing the jet shape variable. For three partons in the final state, $\tauo$ is zero unless two partons are clustered together. Therefore $\tauo$ is non–vanishing only in two–jet events. For the latter events, there are six ways of ordering the energy fractions $x_{i}$ corresponding to six regions of phase space that needs to be integrated over. Due to $x_{1} \leftrightarrow
x_{2}$ symmetry of the matrix–element , one can only consider three regions and multiply the result by a factor of $2$. These regions correspond to; $x_{1}>x_{2}>x_{3}, x_{1}>x_{3}>x_{2}$ and $x_{3}>x_{1}>x_{2}$. The threshold thrust is then given by $$\begin{gathered}
\label{tauo_LO}
\tauo = (1- x_{1}) \Theta(x_{1}-x_{2},x_{2}-x_{3})\Theta\left(2\Rs -1+\cos\theta_{23} \right) +
\\
+ (1 - x_{1}) \Theta(x_{1}-x_{3},x_{3}-x_{2})\Theta\left(2\Rs-1+ \cos\theta_{23} \right) +
\\
+ (1-x_{3}) \Theta(x_{3}-x_{1}, x_{1}-x_{2}) \Theta\left(2\Rs-1+\cos\theta_{12} \right),\end{gathered}$$ where $\Theta(a-b,b-c) = \Theta(a-b) \Theta(b-c)$. To obtain the full logarithmic contribution it is sufficient to only consider regions where the gluon is the softest parton ($x_{3} = \min(x_{i})$). Other regions, last term in RHS of eq. , only contributes non–logarithmically. Adding up real and virtual contributions, in , one is only left with the virtual corrections in the range $\Theta(1- x_{1} - \tauo)$. The corresponding angular function in may be written in terms of the energy fractions as, $$\label{clust_cond_LO}
1- \cos\theta_{23} = \frac{2(1-x_{1})}{x_{2} x_{3}} \approx
\frac{2(1-x_{1})}{x_{3}},$$ where the last approximation follows from the fact that the gluon is the softest, $ x_{1}, x_{2} \gg x_{3}$. Hence the two–jet contribution to the first order shape fraction $\Sigma^{(1)}$ is given by $$\label{Sig1_2-jet}
\Sigma^{(1)}(\tauo, \Eo) = -\frac{\CF\,\as}{2\pi}
\int_{1-\Rs(1-\tauo)}^{1-\tauo}\d x_{2} \int_{1+\tauo+x_{2}}^{\frac{x_{2}
-1+\Rs(2-x_{2})}{\Rs}}\d x_{1}\,\frac{x_{1}^{2} + x_{2}^{2}}{(1-x_{1})
(1-x_{2})} \Theta\left(\frac{\Rs}{1+\Rs} - \tauo\right)$$
In case of events with three–jets in the final state, the energy of the softest jet is vetoed to be less than $\Eo$. The corresponding phase space constraint, left after real–virtual mis–cancellation, on the differential cross–section reads $$\label{E0_constr_LO}
- \Theta\left(x_{3} -\frac{2\Eo}{Q}\right) \Theta\left(\frac{1-x_{1}}{x_{3}} -
\Rs\right)\,\Theta\left(1-\Rs - \frac{1-x_{1}}{x_{3}}\right).$$ Noting that $x_{1} + x_{2} + x_{3} = 2$, one can obtain the corresponding integration limits on $x_{1}$ and $x_{2}$. Adding up the result of the latter integration with that of eq. and making use of the following dilogarithm identities [@polylogs] $$\begin{aligned}
\label{polylogs_rel}
\nonumber \Li_{2}(x) + \Li_{2}(1-x) &=& \frac{\pi^{2}}{6} - \,\ln(x)
\,\ln(1-x),\\
\Li_{2}(x) + \Li_{2}\left(\frac{1}{x}\right) &=& \frac{\pi^{2}}{3} -\frac{1}{2}
\,\ln^{2}(x).\end{aligned}$$ one obtains eq. .
$G_{nm}$ coefficients {#app.coeff_in_expansion}
=====================
The resultant coefficients from the expansion of the exponent in the resummed integrated distribution, eq. , are $$\begin{aligned}
\label{G_nm-QCD}
\nonumber G_{12} &=& -2 \CF ,
\\
\nonumber G_{11} &=& \CF \left(3 - 4 L_{\Rs}\right),
\\
\nonumber G_{10} &=& \CF\left[-4 L_{\Rs} L_{\Eo} +
\frac{\bar{f}_{0}(\Rs)}{2}\right],
\\
\nonumber G_{23} &=& \CF \left(\frac{4}{3} \TF \nf - \frac{11}{3}\CA \right),
\\
\nonumber G_{22} &=& -\frac{4 \pi^{2}}{3} \CF^{2} + \CF
\CA\left(\frac{\pi^{2}}{3} - S_{0}(\Rs) -\frac{169}{36} - \frac{22}{3}
L_{\Rs}\right) + \CF \TF\nf \left(\frac{11}{9} + \frac{8}{3} L_{\Rs}\right),
\\
\nonumber G_{21} &=& - \CF^{2}\,\frac{8 \pi^{2}}{3} L_{\Rs} - \CF\CA\bigg[2
S_{0}(\Rs) L_{\Eo} - \left(\frac{2\pi^{2}}{3} -2S_{0}(Rs) - \frac{134}{9} -
\frac{11}{3} L_{\Rs}\right) L_{\Rs}\bigg] +\\
&+& \CF\TF\nf \left(\frac{4}{3} L_{\Rs} + \frac{40}{9}\right) L_{\Rs}.\end{aligned}$$ where $L_{\Rs} = \ln(\Rs/(1-\Rs))$ and $L_{\Eo} = \ln(2\Eo/Q)$. The factor $\bar{f}_{0}(\Rs)$ only captures the first term of $f_{0}$ given in eq. . We simply replace $\bar{f}_{0} \mapsto f_{0}$ when comparing to the numerical distribution. Moreover, we have introduced, for shorthand, the function $S_{0}(\Rs)$ given by (cf. eq. ), $$\label{S_0_Rs}
S_{2} = -\CF\CA\; S_{0}(\Rs).$$ The one–loop constant is given by, eq. , $$\begin{aligned}
\label{C1_QCD}
C_{1} &=& \CF \left(-1+\frac{\pi^{2}}{3}\right),\end{aligned}$$ Expanding the total resummed distribution in eq. to $\Or(\as^{2})$ and up to NLL we have $$\begin{gathered}
\label{resum_tot_series}
\Sigma_{r,2}(\widetilde{L}) = 1 + \left(\frac{\as}{2\pi}\right)\left(H_{12}
\widetilde{L}^{2} + H_{11} \widetilde{L} + H_{10}\right) +
\left(\frac{\as}{2\pi}\right)^{2} \Big(H_{24} \widetilde{L}^{4} + H_{23}
\widetilde{L}^{3} + H_{22} \widetilde{L}^{2} + \\+ H_{21} \widetilde{L} +
H_{20}\Big),\end{gathered}$$ where (recall that $\widetilde{L} = \ln(1/\tauo) \Rightarrow \tauo =
e^{-\widetilde{L}}$) $$\begin{aligned}
\label{H_nm_coeffs}
\nonumber D_{\mathrm{fin}} (e^{-\widetilde{L}}) &=&
\left(\frac{\as}{2\pi}\right)\,d_{1}(e^{-\widetilde{L}}) +
\left(\frac{\as}{2\pi}\right)^{2}\,d_{2}(e^{-\widetilde{L}}),\\
\nonumber H_{12} &=& G_{12},\\
\nonumber H_{11} &=& G_{11},\\
\nonumber H_{10} &=& G_{10} + C_{1} + d_{1}(\tauo),\\
\nonumber H_{24} &=& \frac{1}{2} G_{12}^{2}.\\
\nonumber H_{23} &=& G_{23} + G_{12} G_{11},\\
\nonumber H_{22} &=& G_{22} + (G_{10} + C_{1}) G_{12} + \frac{1}{2}
G_{11}^{2},\\
\nonumber H_{21} &=& G_{21} + (G_{10}+ C_{1}) G_{11},\\
H_{20} &=& G_{20} + \frac{1}{2} G_{10}^{2} + C_{1} G_{10} + C_{2} +
d_{2}(\tauo). \end{aligned}$$ Differentiating w.r.t. $\widetilde{L}$, the NLO differential distribution reads $$\label{resum_expanded-sig-diff}
\frac{\d\Sigma_{r,2}}{\d \widetilde{L}} =
\frac{1}{\sigma_{0}}\frac{\d\sigma_{r,2}}{\d \widetilde{L}} =
\delta(\widetilde{L})\, D_{\delta} + \left(\frac{\as}{2\pi}\right)\,
D_{A}(\widetilde{L}) + \left(\frac{\as}{2\pi}\right)^{2}\, D_{B}(\widetilde{L}),$$ where the singular (logarithmic) terms are given by $$\begin{aligned}
\label{singular_terms_gen}
\nonumber D_{\delta} &=& 1 + \left(\frac{\as}{2\pi}\right) \left[G_{10} +
C_{1}\right] + \left(\frac{\as}{2\pi}\right)^{2} \left[G_{20} + \frac{1}{2}
G_{10}^{2} + C_{1} G_{10} + C_{2}\right],
\\
\nonumber D_{A}(\widetilde{L}) &=& 2 H_{12} \widetilde{L} + H_{11} +
\frac{\d}{\d \widetilde{L}}\, d_{1}(e^{-\widetilde{L}}),
\\
D_{B}(\widetilde{L}) &=& 4 H_{24} \widetilde{L}^{3} + 3 H_{23}
\widetilde{L}^{2} + 2 H_{22} \widetilde{L} + H_{21} + \frac{\d}{\d
\widetilde{L}}\, d_{2}(e^{-\widetilde{L}}).\end{aligned}$$
Threshold thrust distribution in SCET {#app.tw_in_SCET}
=====================================
The resummation of the threshold thrust in SCET is presented in the current section for comparison with pQCD. We shall only present the final form of the resummed result taken from Refs. [@Kelley:2011tj; @Kelley:2010qs; @Becher:2008cf]. For a full derivation and more in depth discussion one should consult the latter references. The only task we have performed here is the expansion of the full resummed distribution to $\Or(\as^{2})$.
Resummation {#subsec.resummation_SCET}
-----------
The general formula of the resummed distribution for the threshold thrust is given by [@Kelley:2011tj; @Becher:2008cf] $$\begin{gathered}
\label{resum-form_SCET-a}
\frac{\d\Sigma^{\SCET}(\tauo, R)}{\d\tauo} =
\frac{\d\sigma^{\SCET}}{\sigma_{0}\d\tauo} = \exp\left[4S(\mu_{h},\mu_{j}) +
4S(\mu_{s},\mu_{j}) - 4A_{H}(\mu_{h},\mu_{s}) + 4A_{J}(\mu_{j},\mu_{s}) \right]
\\ \times\left(\frac{\Rs}{1- \Rs}\right)^{-2A_{\Gamma}(\mu_{\omega},\mu_{s})}
\left(\frac{Q^{2}}{\mu_{h}^{2}}\right)^{-2A_{\Gamma}(\mu_{h},\mu_{j})}
H(Q^{2},\mu_{h}) \; S^{\outt}_{R}(\omega,\mu_{\omega}) \\ \times
\left[\tilde{j}\left(\ln\frac{\mu_{s}Q}{\mu_{j}^{2}} + \partial_{\eta},
\mu_{j}\right) \right]^{2} \tilde{s}^{\inn}_{\tauo}(\partial_{\eta}, \mu_{s})
\frac{1}{\tauo} \left(\frac{\tauo Q}{\mu_{s}}\right)^{\eta} \frac{e^{-\gamma_{E}
\eta}}{\Gamma(\eta)}.\end{gathered}$$ See [@Kelley:2011tj; @Becher:2008cf] for full notation. In order to compute the fixed–order expansion of up to $\Or(\as^{2})$, all scales should be set equal ($\mu_{h} = \mu_{j} = \mu_{s} = Q$). In this limit, the evolution factors $S, A_{J}$ and $A_{H}$ vanish. The differentiation w.r.t. $\eta$ is carried out using the explicit form of $\tilde{j}$ and $\tilde{s}^{\inn}_{\tauo}$. The final result of the integrated distribution may be cast in the generic form with the constants and coefficients of the logs given by $$\begin{aligned}
\label{eq.C1-C2_SCET}
C_{1} &=& \CF\left(-1+\frac{\pi^{2}}{3} \right),
\\
\nonumber C_{2} &=& \CF^{2} \left(1- \frac{3\pi^{2}}{8} +\frac{\pi^{4}}{72}
-6\zeta(3) \right) + \CF\CA \left(\frac{493}{324} + \frac{85\pi^{2}}{24}
-\frac{73\pi^{4}}{360} + \frac{283 \zeta(3)}{18} \right) +
\\
&+& \CF\TF\nf \left(\frac{7}{81} -\frac{7\pi^{2}}{6}
-\frac{22\zeta(3)}{9}\right) + C_{2}^{\inn} + C_{2}^{\outt},\end{aligned}$$ and $$\begin{aligned}
\label{G_nm_SCET}
\nonumber G_{12} &=& -2 \CF,
\\
\nonumber G_{11} &=& -\CF \left(3- 4 L_{\Rs}\right),
\\
\nonumber G_{10} &=& \CF\left(- 4 L_{\Rs} L_{\Eo} + \frac{f_{0}(\Rs)}{2}\right),
\\
\nonumber G_{23} &=& \CF \left(\frac{11}{3}\CA - \frac{4}{3} \TF \nf \right),
\\
\nonumber G_{22} &=& -\frac{4 \pi^{2}}{3} \CF^{2} + \CF
\CA\left(\frac{\pi^{2}}{3}-\frac{169}{36} - \frac{22}{3} L_{\Rs}\right) + \CF
\TF\nf \left(\frac{11}{9} + \frac{8}{3} L_{\Rs}\right),
\\
\nonumber G_{21} &=& \CF^{2}\left[-\frac{3}{4} - \pi^{2} + 4\zeta(3) +\frac{8
\pi^{2}}{3} L_{\Rs}\right] +\\\
\nonumber &+& \CF\CA\bigg[-\frac{57}{4} + 6\zeta(3) - \left(\frac{2\pi^{2}}{3}
-\frac{134}{9} -\frac{11}{3} L_{\Rs}\right) L_{\Rs}\bigg] +
\\
\nonumber &+& \CF\TF\nf\left[5 - \left(\frac{4}{3} L_{\Rs}+ \frac{40}{9}\right)
L_{\Rs}\right],\label{G21_SCET}\end{aligned}$$ $$\begin{aligned}
\label{G_nm_SCET_2}
\nonumber G_{20} &=& \CF^{2} \Bigg[-\frac{f_{0}^{2}}{8} + \left(2\pi^{2}
-16\zeta(3)\right) L_{\Rs} - \Big(\frac{11\pi^{2}}{6} + \frac{f_{0}}{2}\Big)
L^{2}_{\Rs} - L_{\Rs}^{2} \Bigg]+
\\
\nonumber &+& \CF\CA \Bigg[\frac{11\pi^{2}}{9} L_{\Rs} - \frac{11}{6} L_{\Rs}
L^{2}_{\Eo} - L_{\Eo} \left(\frac{11f_{0}}{12}
+\left[\frac{134}{9}-\frac{2\pi^{2}}{3}\right] L_{\Rs} +\frac{11}{6}
L^{2}_{\Rs}\right) \Bigg] +
\\
&+& \CF\TF\nf\Bigg[-\frac{4\pi^{2}}{9} L_{\Rs} + \frac{2}{3} L_{\Rs}
L^{2}_{\Eo} + L_{\Eo}\Bigg(\frac{f_{0}(\Rs)}{3} +\frac{40}{9} L_{\Rs} +
\frac{2}{3} L^{2}_{\Rs}\Bigg) \Bigg].\end{aligned}$$ Considering primary emission, the only missing piece in the distribution is the two–loop constants in the soft function, namely $C_{2}^{\inn}$ and $C_{2}^{\outt}$.
[^1]: `Kamel.Khelifa@hep.manchester.ac.uk`
[^2]: from the point of view of our calculations in this paper.
[^3]: These are emissions that are not radiated off primary hard legs.
[^4]: This name is more appropriate at hadron colliders where at threshold the final state jets are back–to–back and there is no beam remnant [@Kelley:2010qs].
[^5]: This claim has been removed from version $2$.
[^6]: As we shall see in sec. \[sec.numerical\_results\], while individual colour contributions do not agree with `EVENT2` their sum does, but only in the shape variable range and for the jet–radius considered in [@Kelley:2011tj]. Outside the latter range or for other smaller jet–radii they do not agree.
[^7]: $p_{i}^{\mu}$ may be the momenta of individual particles or each $p_{i}^{\mu}$ may be the total momentum of the particles whose paths are contained in a small cell of solid angle about the interaction point, as recorded in individual towers of a hadron calorimeter.
[^8]: Note that there are three sets of matrix–elements included in the program, of which only one is not normalised to the Born cross–section.
[^9]: Here we go beyond the small $\Rs$ approximation assumed in [@Banfi:2010pa].
[^10]: such that the jet–radius is much smaller than the jets’ separation; $\Rs \ll (1- \cos\theta_{ij})$, where $\theta_{ij}$ is the angle between jets $i$ and $j$.
[^11]: Otherwise, one can redefine the partons’ $4$–momenta in terms of $\eta$ and $\kt$ and use the antenna function expressions of $W_{P}$ and $W_{S}$ to rewrite them in terms of the hadronic variables.
[^12]: Our jet–radius, $\Rs$, is given in terms the parameter $c$, used in [@Dasgupta:2002bw], by the relation: $1 - c = 2\Rs$.
[^13]: It is actually twice
[^14]: In our case the vetoed region is the jet region. Due to symmetry, we can choose one jet region and multiply the final answer by a factor of two.
[^15]: as can be seen from comparison to the SCET result , which only contains the primary emission piece and is valid to NNLL.
[^16]: Recall that for the leading NGLs the corresponding coefficient for the hemisphere mass distribution corresponds to setting $\Rs = 0$ in $S_{2,a}$ .
[^17]: The discrepancy at $\Rs = 0.30$ is due to the fact that we have employed the small angles approximation in the C–A calculations.
[^18]: In `FastJet`’s manual $\widetilde{R}$ is allowed to go up to $\pi$. Since we are interested in two–jet events the jet size cannot be wider than a hemisphere. Thus we restrict $\tilde{R}$ to be less than $\pi/2$.
[^19]: as would be anticipated, since the finite–size jet mass is an extension to the hemisphere mass.
|
---
abstract: 'We report quantum and semi-classical calculations of spin current and spin-transfer torque in a free-electron Stoner model for systems where the magnetization varies continuously in one dimension. Analytic results are obtained for an infinite spin spiral and numerical results are obtained for realistic domain wall profiles. The adiabatic limit describes conduction electron spins that follow the sum of the exchange field and an effective, velocity-dependent field produced by the gradient of the magnetization in the wall. Non-adiabatic effects arise for short domain walls but their magnitude decreases exponentially as the wall width increases. Our results cast doubt on the existence of a recently proposed non-adiabatic contribution to the spin-transfer torque due to spin flip scattering.'
author:
- 'Jiang Xiao and A. Zangwill'
- 'M. D. Stiles'
title: Spin Transfer Torque for Continuously Variable Magnetization
---
Introduction
============
The use of electric current to control magnetization in nanometer-sized structures is a major theme in the maturing field of spintronics. [@Fert:2003] An outstanding example is the theoretical prediction of current-induced magnetization precession and switching in single domain multilayers [@SB] and its subsequent experimental confirmation in spin-valve nanopillars. [@Kiselev:2003] The physics of this spin-transfer effect is that a single domain ferromagnet feels a torque because it absorbs the component of an incident spin current that is polarized transverse to its magnetization. The same idea generalizes to systems with continuously non-uniform magnetization. [@Berger:1978; @Bazaliy:1998] This realization has generated a flurry of experimental [@expt] and theoretical work [@Waintal:2004; @Tatara:2004; @Zhang:2004; @Thiaville:2005; @Barnes:2005] focused on current-driven motion of domain walls in magnetic thin films.
The experiments cited just above employ Néel-type domain walls with widths $w\approx 100$ nm. This length is very large compared to the characteristic length scales of the processes that determine the local torque.[@Stiles:2002; @Zhang:2002] Therefore, it is appropriate to adopt an adiabatic approximation where the spin current is assumed to lie parallel to the local magnetization. [@Berger:1978; @Bazaliy:1998] Surprisingly, the adiabatic prediction for the current dependence of the domain wall velocity [@Berger:1986; @Tatara:2004; @Thiaville:2004] agrees very poorly with experiment. This has led theorists [@Waintal:2004; @Tatara:2004; @Zhang:2004; @Thiaville:2005; @Barnes:2005] to consider non-adiabatic effects and experimenters [@Klaui:2005; @Ravelosona:2005] to study systems with domain wall widths that are much shorter ($w\approx 10$ nm) than those studied previously.
Two groups [@Zhang:2004; @Thiaville:2004] have studied the effect on domain wall motion of a distributed spin-transfer torque represented by a sum of gradients of the local magnetization with constant coefficients. For a one-dimensional magnetization ${\bf
M}(x)$, the torque function can be written in terms of two vectors perpendicular to the magnetization $$\label{lt}
{\bf N}_{\rm st}(x) = c_1\partial_x\hat{\bf M}
+ c_2\hat{\bf M}\times\partial_x\hat{\bf M}.$$ In general, the coefficients $c_1$ and $c_2$ are functions of position. The well-established adiabatic piece of the torque is the first term in Eq. (\[lt\]) with a constant coefficient. Consistent with usage in the literature, we call all deviations from the adiabatic torque non-adiabatic. Any contributions of the second term are then called non-adiabatic. Zhang and Li [@Zhang:2004] derive a contribution along this second direction in Eq. (\[lt\]) from a consideration of magnetization relaxation due to spin-flip scattering in the context of an s-d exchange model of a ferromagnet. [@Berger:1986; @Zhang:2002] Their arguments lead them to the estimate $c_2/c_1\approx 0.01$. The authors of Ref. report that a similar value of $c_2/c_1$ produces agreement with experiment when Eq. (\[lt\]) is used in micromagnetic simulations.
In this paper, we study the applicability of Eq. (\[lt\]) to a free-electron Stoner model with one-dimensional magnetization distributions of the form $$\label{eqn:magnetization}
{\bf M}(x) = M_s{\left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta \right)}.$$ In Eq. (\[eqn:magnetization\]), the polar angle $\theta(x)$ is measured from the positive $z$ axis, $\phi(x)$ is the azimuthal angle in the $x$-$y$ plane, and $M_s$ is the saturation magnetization. We begin with the spin current and spin torque for an infinite spin spiral with constant pitch. This system turns out to be perfectly adiabatic; the torque is described by Eq. (\[lt\]) with $c_2=0$. The same is true for realistic domain walls of the sort usually encountered in experiment. Non-adiabatic effects appear only for very narrow walls. In that case, the torque is non-local and cannot be written in the form Eq. (\[lt\]). The non-adiabatic torque decreases exponentially as the wall width increases for all realistic domain wall profiles. Finally, our analysis casts doubt on the existence of a non-adiabatic contribution to the torque due to spin-flip scattering proposed recently by Zhang and Li.[@Zhang:2004].
The remainder of the paper is organized as follows. Section \[models\] describes our Stoner model and the methods we use to calculate the spin current and spin-transfer torque. Section \[spinspiral\] reports our results for an infinite spin spiral and Section \[domainwalls\] does the same for one-dimensional domain walls that connect two regions of uniform magnetization. Section \[other\] relates these calculations to previous work by others. Section \[scattering\] discusses the effects of scattering. We summarize our results and offer some conclusions in Section \[sumcon\]. Two appendices provides some technical details omitted in the main body of the paper.
Model & Methods {#models}
===============
The free electron Stoner model provides a first approximation to the electronic structure of an itinerant ferromagnet. The Hamiltonian is $$\label{eqn:Hamiltonian}
H=-{\hbar^2\over 2m}\nabla^2 - \mu \bm{\sigma} \cdot {\bf B}_{\rm ex}(x),$$ where $\bm{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ is a vector composed of the three Pauli matrices and $\mu=g\hbar e/2mc$. The magnetic field ${\bf B}_{\rm ex}(x)$ is everywhere parallel to ${\bf M}(x)$ but has a constant magnitude.[@MagSign] That magnitude is chosen so the Zeeman splitting between the majority and minority spins bands reproduces the quantum mechanical exchange energy in the limit of uniform magnetization: $$\label{eqn:exchange}
E_{\rm ex}=2\mu |{\bf B}_{\rm ex}|=\hbar^2 k_B^2/m.$$ If $E_F=\hbar^2 k_F^2/2m$ is the Fermi energy, the constant $k_B$ in Eq. (\[eqn:exchange\]) fixes the Fermi wave vectors for up and down spins, $k_F^+$ and $k_F^-$, from $$\label{eqn:kpm}
k_F^\pm = \sqrt{k_F^2\pm k_B^2}.$$
Given ${\bf B}_{\rm ex}(x)$, we use both quantum mechanics and a semi-classical approximation to calculate the spin accumulation, spin current density, and spin-transfer torque. The building blocks are the single-particle spin density ${\bf s}_\pm(x,k_x)$ and the single-particle spin current density[@Qnote] ${\bf Q}_\pm(x,k_x)$ for an up/down ($\pm$) electron with longitudinal wave vector $k_x$.
Summing over all electrons and using the relaxation time approximation gives the non-equilibrium majority and minority spin density ${\bf
s}_\pm(x)$ and spin current density ${\bf Q}_\pm(x)$ in the presence of an electric field $E\hat{\bf x}$: $$\begin{aligned}
\label{eqn:spinandcurrent}
{\bf s}_\pm(x)
&=& \int{\left[ f_\pm({\bf k}-{eE\tau\over\hbar}\bm{\hat{x}})-f_\pm({\bf k}) \right]}{\bf s}_\pm(x,k_x)d^3{\bf k} \nonumber \\
& & \\
{\bf Q}_\pm(x)
&=& \int{\left[ f_\pm({\bf k}-{eE\tau\over\hbar}\bm{\hat{x}})-f_\pm({\bf k}) \right]}{\bf Q}_\pm(x,k_x)d^3{\bf k} \nonumber\end{aligned}$$ Our use of the function $f_\pm({\bf k})=\Theta(k_F^\pm-{\left| \bf k \right|})$ implies that the distribution of electrons outside the region of inhomogeneous magnetization are characteristic of the zero-temperature bulk.[@linearBoltzmann] We shall expand this point and comment on the general correctness of (\[eqn:spinandcurrent\]) in Sec. \[scattering\]. The sum ${\bf s}(x)={\bf s}_+(x)+{\bf s}_-(x)$ is the total spin accumulation (spin density) and ${\bf Q}(x)={\bf Q}_+(x)+{\bf Q}_-(x)$ is the total spin current density. Finally, the distributed spin transfer torque is[@Stiles:2002] $$\label{eqn:spt}
{\bf N}_{\rm st}(x)=-\partial_x{\bf Q}(x).$$ The adiabatic approximation[@Berger:1978] to the spin dynamics leads to a spin current density that is proportional to the local magnetization, ${\bf Q}_{\rm ad}(x) \propto {\bf M}(x)$. This means that $$\label{eqn:adia}
{\bf N}_{\rm ad}(x) \propto \partial_x {\bf M}(x).$$ A main goal of this paper is to study the extent to which the spin-transfer torque associated with real domain wall configurations satisfies Eq. (\[eqn:adia\]).
Quantum {#sec:quantum}
-------
In light of Eq. (\[eqn:magnetization\]), the exchange magnetic field that the enters the Hamiltonian in Eq. (\[eqn:Hamiltonian\]) is $$\label{eqn:exchangeB}
{\bf B}_{\rm ex}(x) =B_{\rm ex}{\left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta \right)}.$$ For a given energy, the eigenfunctions for a conduction electron with wave-vector ${\bf k}$ take the form $\Psi_\pm({\bf r},{\bf
k})=\psi_\pm(x, k_x)e^{ik_y y+ ik_z z}$ where the spinor $\psi_\pm(x,k_x)$ satisfies $${\left[ -{d^2\over dx^2} - k_B^2{\left(
\begin{array}[c]{cc}
\cos\theta & e^{-i\phi}\sin\theta \\
e^{i\phi}\sin\theta & -\cos\theta
\end{array} \right)} \right]}\psi_\pm
= \kappa_\pm^2\psi_\pm.
\label{eqn:Schrodinger}$$ In this expression, $$\label{eqn:kdef}
\kappa_\pm^2=k_x^2\mp k_B^2,$$ and $\pm$ refers to majority/minority band electrons. The single-electron spin density and spin current density are[@Stiles:2002] $$\label{eqn:sesd}
{\bf s}_\pm(x,k_x)={\hbar \over 2}\sum_{\alpha,\beta}
\psi^\ast_{\pm,\alpha}(x,k_x)\bm{\sigma}_{\alpha,\beta}\psi_{\pm,\beta}(x,k_x)$$ and $$\label{eqn:sescd}
{\bf Q}_\pm(x,k_x)=-{\hbar^2 \over 2m}\sum_{\alpha,\beta} {\rm Im} [
\psi^\ast_{\pm,\alpha}(x,k_x)\bm{\sigma}_{\alpha,\beta}{d \over dx}
\psi_{\pm,\beta}(x,k_x)].$$
As a check, we used this formalism to calculate the equilibrium (zero applied current) spin density ${\bf s}_{\rm eq}(x)$ and equilibrium spin current density ${\bf Q}_{\rm eq}(x)$ for a magnetization distribution chosen arbitrarily except for the constraint that $|{\bf
M}(x)|$ be uniform. The densities ${\bf s}_{\rm eq}(x)$ and ${\bf
Q}_{\rm eq}(x)$ are obtained by retaining only the second term in square brackets in Eq. (\[eqn:spinandcurrent\]).[@linearBoltzmann] The lines in Fig. \[fig:equilibrium\](a) are the Cartesian components of the imposed ${\bf M}(x)$. The solid dots in Fig. \[fig:equilibrium\](a) show that the spin density ${\bf s}_{\rm eq}(x)$ is parallel to ${\bf
M}(x)$, as expected. Similarly, the electron-mediated spin-transfer torque should equal the phenomenological exchange torque density discussed by Brown.[@Brown:1962] This is confirmed by Fig. \[fig:equilibrium\](b), which shows that ${\bf N}_{\rm eq}(x)$ is proportional to ${\bf M}\times {\bf M}''(x)$ for the ${\bf M}(x)$ shown in Fig. \[fig:equilibrium\](a).
![Equilibrium (zero-current) results: (a) Cartesian components of an arbitrarily chosen magnetization ${\bf M}(x)$ (lines); Cartesian components of the calculated spin density ${\bf
s}_{\rm eq}(x)$ (solid dots); (b) Exchange torque (solid line) and calculated spin-transfer torque (solid dots). []{data-label="fig:equilibrium"}](fig1.eps){width="8cm"}
Semi-classical
--------------
A semi-classical approach to calculating the spin current density is useful for building physical intuition. Accordingly, we write an equation of motion for the spin density of every electron that contributes to the current. This idea has been used in the past, both semi-quantitatively[@Berger:1978] and qualitatively.[@Waintal:2004] Our derivation is based on the behavior of an electron with energy $E$ that moves along the $x$-axis through a uniform magnetic field ${\bf B}_{\rm ex}=B_{\rm ex}\hat{\bf
z}$. The wave function for such an electron is $$\varphi(x,E) = {\left( \begin{array}[c]{c} a~e^{ik_+ x} \\ b~e^{ik_- x} \end{array} \right)},$$ where $$\label{eqn:kapdef}
k_\pm^2=2mE/\hbar^2\pm k_B^2.$$ We compute the spin density ${\bf s}(x,E)$ and the spin current density ${\bf Q}(x,E)$ for this electron using the right sides of Eq. (\[eqn:sesd\]) and Eq. (\[eqn:sescd\]), respectively, with $\psi_\pm \to \varphi$.
It is straightforward to check that the components of these densities transverse to the magnetic field satisfy the semi-classical relations $$\label{eqn:scQs}
Q_x = s_x {\langle v \rangle}~~~~~~~~~~{\rm and}~~~~~~~~~~Q_y=s_y {\langle v \rangle},$$ where ${\langle v \rangle}$ is the velocity $$\label{eqn:vdef}
{\langle v \rangle}={\hbar\over m}{k_++k_-\over 2}={\hbar \over m}{\langle k \rangle}.$$ Moreover, the transverse components of the spin density satisfy $$\begin{aligned}
\label{eqn:perppiece}
{\langle v \rangle}{ds_x\over dx} &=& -{\hbar k_B^2\over m}s_y,{\nonumber\\}{\langle v \rangle}{ds_y\over dx} &=& -{\hbar k_B^2\over m}s_x.\end{aligned}$$ These equations are the components of the vector equation $$\label{eqn:LLGa}
{d{\bf s}\over dx} = -{k_B^2\over {\langle k \rangle}}{\bf s}\times\hat{\bf B}_{\rm ex}$$ where $\hat{\bf B}_{\rm ex}=\hat{\bf z}$.
We now make the [*ansatz*]{} that all three Cartesian components of the semi-classical majority and minority spin densities ${\bf
s}_\pm(x,k_x)$ satisfy Eq. (\[eqn:LLGa\]) when the direction of the magnetic field varies in space. Specifically, if ${\bf B}_{\rm
ex}(x)=B_{\rm ex}\hat{\bf B}_{\rm ex}(x)$, we suppose that $$\label{eqn:LLG}
{d{\bf s}_\pm(x,k_x)\over dx} = -{k_B^2\over {\langle k \rangle}}{\bf s}_\pm(x,k_x)\times\hat{\bf B}_{\rm ex}(x),$$ where $k_+$ and $k_-$ for ${\bf s}_+(x,k_x)$ are defined by Eq. (\[eqn:kapdef\]) with $E=\hbar^2(k_x^2-k^2_{\rm
B})/2m$.[@KMEvan] Similarly, $k_+$ and $k_-$ for ${\bf
s}_-(x,k_x)$ are defined by Eq. (\[eqn:kapdef\]) with $E=\hbar^2(k_x^2+k^2_{\rm B})/2m$. With suitable boundary conditions, we solve the differential equation Eq. (\[eqn:LLG\]) to determine the semi-classical, one-electron spin densities. The total, spin-resolved, spin densities follow by inserting these one-electron quantities into $$\label{scm}
{\bf s}_\pm(x)
= \int{\left[ f_\pm({\bf k}-{eE\tau\over\hbar}\bm{\hat{x}})-f_\pm({\bf k}) \right]}{\bf s}_\pm(x,k_x){k_x\over {\langle k \rangle}}d^3{\bf k}.$$ This equation differs from Eq. (\[eqn:spinandcurrent\]) by the weighting factor $k_x/{\langle k \rangle}$.[@ksign] This factor guarantees that the flux carried by each electron is proportional to its velocity (see Appendix I). This is confirmed by Fig. \[fig:LLGcheck\](b) which shows quantitative agreement between a fully quantum calculation of ${\bf s}(x)$ using Eq. (\[eqn:Schrodinger\]), Eq. (\[eqn:sesd\]) and Eq. (\[eqn:spinandcurrent\]) and a semi-classical calculation using Eq. (\[eqn:LLG\]) and Eq. (\[scm\]).
![(a) Cartesian components of an imposed magnetization ${\bf
M}(x)$ used in the other panels; (b) comparison of quantum (lines) to semi-classical (solid dots) calculations for the Cartesian components of the spin density ${\bf s}(x)$; (c) same comparison for the Cartesian components of the spin current density ${\bf Q}(x)$. []{data-label="fig:LLGcheck"}](fig2.eps){width="8cm"}
In light of the foregoing, it is reasonable to calculate the semi-classical single-electron spin current density from $$\label{eqn:scscd}
{\bf Q}_\pm(x,k_x)={\bf s}_\pm(x,k_x){\hbar k_x \over m}$$ and use the second equation in Eq. (\[eqn:spinandcurrent\]) to find ${\bf Q}_\pm(x)$. The correctness of this prescription is illustrated in Fig. \[fig:LLGcheck\](c).
Spin Spiral {#spinspiral}
===========
As a preliminary to our discussion of domain walls, it is instructive to discuss the spin density and spin current density for a spin spiral—an infinite magnetic structure where the direction of the magnetization rotates continuously as one moves along a fixed axis in space. Spin spirals occur in the ground state of some rare earth metals[@Jensen:1991] and also for the $\gamma$ phase of iron.[@Hafner:2002] Here, we focus on a spiral with uniform pitch $p$ where the magnetization rotates in the $x$-$z$ plane, [i.e.]{}, in Eq. (\[eqn:magnetization\]), $$\theta(x) = p\hspace{0.1em}x ~~~~~~~~~~{\rm and}~~~~~~~~~~\phi(x)=0.
\label{thetas}$$ A cartoon version of this ${\bf M}(x)$ is shown in Fig. \[fig:spinspiral\]. This figure also defines a [*local*]{} coordinate system that will be useful in what follows. The system $(x',y',z')$ rotates as a function of $x$ so the magnetization ${\bf
M}(x)$ always points along $+z'$.
Calvo[@Calvo:1978] solved Eq. (\[eqn:Schrodinger\]) to find the eigenstates and eigen-energies of this spin spiral. In our notation,
![Cartoon of a spin spiral where the magnetization (arrows) rotates uniformly in the $x$-$z$ plane of a fixed coordinate system where $\hat{\bf M}(x) \cdot \hat{\bf z}=\cos\theta$. The inset shows a local coordinate system where ${\bf M}(x)$ always points along the $z'$ axis. []{data-label="fig:spinspiral"}](fig3.eps){width="8cm"}
$$\varepsilon_{\pm}({\bf k})
= {\hbar^2\over 2m}{\left( k^2+{{1\over 4}}p^2 \pm\sqrt{(k_x p)^2+k_B^4 } \right)},
\label{eqn:spiralenergy}$$
and $$\psi_\pm({\bf k,r})
= e^{i{\bf k\cdot r}} e^{-i\sigma_y\theta/2} e^{-i\sigma_x\alpha/2}\eta_\pm,
\label{eqn:Neelstate}$$ where $$\label{alpha}
\sin\alpha = {k_x p\over \sqrt{(k_x p)^2+k_B^4}},$$ and $$\eta_+ = {\left( \begin{array}[c]{c} 1 \\ 0 \end{array} \right)},\hspace{20pt}
\eta_- = {\left( \begin{array}[c]{c} 0 \\ 1 \end{array} \right)}. \vspace{0.5em}$$ From these results, it is easy to compute the single-electron spin densities defined in Eq. (\[eqn:sesd\]). In the local $(x',y',z')$ frame, $$\label{eqn:spirald}
{\bf s'}_\pm(x,k_x)=\pm(0, \sin \alpha, \cos\alpha).$$
The corresponding calculation of the single-electron spin current densities Eq. (\[eqn:scscd\]) is straightforward but tedious and not very illuminating. Therefore, we pass directly to the total spin current density calculated by summing over all electrons as indicated in the second line of Eq. (\[eqn:spinandcurrent\]). Again in the local $(x',y',z')$ frame, $$\label{eqn:spiralcd}
{\bf Q'}(x)=A(p,k_{\rm B})(0,0,1).$$ where $A(p,k_{\rm B})$ is a constant. This shows that ${\bf
Q}(x)\propto {\bf M}(x)$, [*i.e.*]{}, the spin current density for a free electron spin spiral is perfectly adiabatic. Wessely [*et al.*]{}[@Nordstrom:2005] found consistent results in their density functional calculation of the steady-state spin current density associated with the helical spin density wave in erbium metal. We emphasize that Eq. (\[eqn:spiralcd\]) is independent of pitch for an infinite spin spiral. As we discuss below, a similar independence does [*not*]{} hold for domain walls. In that case, wide walls are adiabatic, but narrow ones are not.
![Electrons (wave vector $k_x$) and holes (wave vector $-k_x$) move in an effective field that is the sum of the exchange field ${\bf B}_{\rm ex}(x) \parallel \hat{\bf z}'$ and a fictitious, velocity-dependent “gradient” field induced by the spatial dependence of the exchange field. The spins align to the total effective field in an infinite spin spiral. The $x$ axis lies in $x'$-$z'$ plane. []{data-label="fig:curvature"}](fig4.eps){width="8cm"}
The semi-classical formula Eq. (\[eqn:scscd\]) provides an appealing way to understand the adiabaticity of the spin current density in the spin spiral defined by Eq. (\[thetas\]). The key point is that the angle $\alpha$ in Eq. (\[alpha\]) which fixes the direction of ${\bf
s}_\pm(x,k_x)$ in Eq. (\[eqn:spirald\]) is positive when $k_x$ is positive and negative when $k_x$ is negative (Fig. \[fig:curvature\]). Moreover, for every $k_x$ electron that contributes to the shifted Fermi surface sums in Eq. (\[eqn:spinandcurrent\]), there is a contribution from a $-k_x$ hole. Now, a hole has opposite spin density to an electron and the spin current density Eq. (\[eqn:scscd\]) contains an additional factor of $k_x$. Therefore, the two spin density vectors in Fig. \[fig:curvature\] [*subtract*]{} to give a net spin density along $\hat{\bf y}$ while the corresponding two spin current density vectors [*add*]{} to give a net spin current density along $\hat{\bf
z}'$. This occurs for all $k_x$ in the sums so ${\bf Q}(x)$ aligns exactly with the local exchange field and thus with the local magnetization.
The opposite situation occurs for fully occupied states below the Fermi energy. The spins of the forward and backward moving electrons combine to produce a net moment aligned with the exchange field, as necessary for self-consistency. Further, the spin currents, with the additional factor of $k_x$ add to give a net spin current along $\hat{\bf y}$, so that its gradient gives the correct form of the phenomenological exchange torque density.
To summarize: an electric current that passes through a spin spiral generates a spin accumulation with a component transverse to the magnetization. The spin current density possesses [*no*]{} such component due to pairwise cancellation between forward and backward moving spins of the same type (majority or minority). Moreover, since the cancellation occurs within each band, the final result is insensitive to the details of intraband scattering.
It remains only to understand the origin of the misalignment angle $\alpha$. Why does each spin not simply align itself with ${\bf
B}_{\rm ex}$? Berger[@Berger:1978] was the first to notice this fact and the physics was made particularly clear by Aharonov and Stern.[@Aharonov:1992] These authors studied the adiabatic approximation for a classical magnetic moment that moves in a slowly varying field ${\bf B}(x)$. Not obviously, the moment behaves as if were subjected to a effective magnetic field ${\bf B}_{\rm eff}(x)$ that is the sum of ${\bf B}_{\rm ex}(x)$ and a fictitious, velocity-dependent, “gradient” field ${\bf B}_{\rm g}(x)$ that points in the direction $\nabla
\hat{\bf B}(x)\times \hat{\bf B}(x)$. For our problem, $$\label{eqn:gradientfield}
{\bf B}_{\rm eff}
= {\bf B}_{\rm ex}+{\hbar^2k_x\over 2m\mu}{d\hat{\bf B}_{\rm ex}\over dx}\times\hat{\bf B}_{\rm ex}.$$ The presence of this gradient field is apparent from Eq. (\[eqn:spiralenergy\]) where the square root is proportional to $\vert {\bf B}_{\rm eff}\vert$. [*The adiabatic solution corresponds to perfect alignment of the moment with ${\bf B}_{\rm
eff}(x)$*]{}. This alignment is indicated in Eq. (\[eqn:spirald\]) and in Fig. \[fig:curvature\]. More generally, the expected motion of the magnetic moment is precession around ${\bf B}_{\rm
eff}(x)$.Nevertheless, as indicated above, the total spin current density for the spin spiral aligns with ${\bf B}_{\rm
ex}(x)$ (which is the conventional definition of adiabaticity for this quantity) when the net effect of all conduction electrons is taken into account.
Domain Walls {#domainwalls}
============
Our main interest is the spin-transfer torque associated with domain walls that connect two regions of uniform and antiparallel magnetization. A realistic wall of this kind can be described by Eq. (\[eqn:magnetization\]) with[@Hubert:1998] $$\label{eqn:asine}
\theta(x) = \pi/2 - \arcsin{\left[ \tanh{\left( x/w \right)} \right]} ,$$ The wall is Néel-type if $\phi(x)=0$ and Bloch-type if $\phi(x)=\pi/2$. We will speak of the domain wall width $w$ as “long” or “short” depending on whether $w$ is large or small compared to the characteristic length $$\label{eqn:length}
L={E_F\over E_{\rm ex}}{1\over k_F}={k_F\over k_B^2}.$$ Intuitively, the adiabatic approximation should be valid when $w \gg
L$. When applied to Eq. (\[eqn:spinandcurrent\]), the predicted adiabatic spin-transfer torque for our model is $$\label{eqn:adit}
{\bf N}_{\rm ad}(x) = -{\hbar\over 2}\eta{neE\tau\over m}\partial_x\hat{\bf M}(x),$$ where $n$ is the electron density, and $\eta$ is the polarization of the current. The calculations required to check this for long domain walls are difficult quantum mechanically (for numerical reasons) but straightforward semi-classically. At the single-electron level, adiabaticity again corresponds to alignment of the spin moment with the effective field defined in Eq. (\[eqn:gradientfield\]). The results for a typical long wall (Fig. \[fig:longwalltorque\]) demonstrate that summation over all electrons produces alignment of ${\bf Q}(x)$ with ${\bf M}(x)$ so the adiabatic formula Eq. (\[eqn:adit\]) is indeed correct in this limit.
For short walls, we have carried out calculations of ${\bf N}_{\rm st}(x)$ both quantum mechanically and semi-classically. The two methods agree very well with one another (see Fig. \[fig:LLGcheck\]) but not with the proposed form Eq. (\[lt\]). Bearing in mind that, when the magnetization changes, $\hat{\bf x}'$ points along $\partial_x {\bf
M}$ and $\hat{\bf y}$ points along ${\bf M}\times\partial_x{\bf M}$, our result for the spin-transfer torque is $$\label{eqn:Tdecompose}
{\bf N}_{\rm st}(x) = {\bf N}_{\rm ad}(x)+a(x)\hat{\bf x}' + b(x)\hat{\bf y}.$$
![Distributed spin-transfer torque for a long Nèel domain wall with $w=50$ and $L=6.25$ ($k_F=1$ and $k_B=0.4$): semi-classical calculation of ${\bf N}_{\rm st}$ (solid dots) compared to Eq. (\[eqn:adit\]) for ${\bf N}_{\rm ad}(x)$ (solid curve).[]{data-label="fig:longwalltorque"}](fig5.eps){width="8cm"}
![Distributed spin-transfer torque for a short Nèel domain wall with $w=4$ and $L=6.25$ ($k_F=1$ and $k_B=0.4$): in-plane piece $a(x)$ (heavy solid curve); out-of-plane piece $b(x)$ (dashed curve); adiabatic prediction (light solid curve); second term in Eq. (\[lt\]) scaled to match the maximum of $b(x)$ (light solid curve). []{data-label="fig:shortwalltorque"}](fig6.eps){width="8cm"}
${\bf N}_{\rm st}(x)$ differs from ${\bf N}_{\rm ad}(x)$ because gradients in the gradient field induce single electron spin moments to precess around ${\bf B}_{\rm eff}(x)$ rather to align perfectly with it. Fig. \[fig:shortwalltorque\] shows $a(x)$ and $b(x)$ as calculated for a typical short domain wall. The associated torques lie in the plane of the magnetization and perpendicular to that plane, respectively. These non-adiabatic contributions to the torque are both oscillatory functions of position that do not go immediately to zero when the the magnetization becomes uniform. In other words, $a(x)$ and $b(x)$ are generically non-local functions of the magnetization ${\bf
M}(x)$. The positive-valued function that falls to zero at the edges of the domain wall (light solid curve in Fig. \[fig:shortwalltorque\]) is the second function in Eq. (\[lt\]) with $c_2$ chosen to match $b(x)$ at their common maximum. Evidently, the proposed torque function Eq. (\[lt\]) gives at best a qualitative account of the out-of-plane non-adiabatic torque.
A convenient measure of the degree of non-adiabaticity of the spin-transfer torque is $$\label{eqn:nan}
{\cal Q}= {{\rm max}\,|\,b(x)\,| \over {\rm max}\,|\,N_{\rm ad}(x)\,|}.$$
![Non-adiabaticity in Eq. \[eqn:nan\] versus wall width scaled by the characteristic length in Eq. (\[eqn:length\]). Note the logarithmic scale.[]{data-label="fig:nonadiabaticity"}](fig7.eps){width="8cm"}
Fig. \[fig:nonadiabaticity\] plots this quantity as a function of scaled domain wall width $w/L$ on a log scale. The observed exponential decrease of the non-adiabatic torque as the wall width increases can be understood from the work of Dugaev [*et al.*]{}[@Dugaev:2002] These authors treat the gradient field in Eq. (\[eqn:gradientfield\]) as a perturbation and calculate the probability for an electron in a $(k_x \uparrow)$ state to scatter into a $(k'_x \downarrow)$ state in the Born approximation. If we choose $k_x$ and $k'_x$ as $k_F^+$ and $k_F^-$, respectively, their results imply that the probability ${\cal P}$ that a majority electron retains its spin and becomes a minority electron as it passes through a domain wall is $$\label{eqn:Dugaev}
{\cal P} \propto \exp(-\gamma w/L),$$ where $\gamma$ is a constant of order unity. This rationalizes the result plotted in Fig. \[fig:nonadiabaticity\] because the magnitude of the minority spin component determines the amplitude of the spin precession around ${\bf B}_{\rm eff}(x)$ and thus the magnitude of the non-adiabatic component of ${\bf s}$ and ${\bf Q}$ in Eq. (\[eqn:scscd\]). In fact, $N_{\rm ad} \propto 1/w$, so it is the case that
$$\label{over}
{\rm max}\,|\,b(x)\,|\propto {1\over w} \exp(-\gamma w/L).$$
The slope of the straight line in Fig. \[fig:nonadiabaticity\], [*i.e.*]{}, the value of the constant $\gamma$ in Eq. (\[eqn:Dugaev\]) depends on the sharpness of the domain wall. Using Eq. (\[eqn:asine\]) and other simple domain wall profile functions, it is not difficult to convince oneself that a suitable measure of domain wall sharpness is the maximum value of the second derivative $\theta''(x)$ for walls with the same width. The numerical results shown in Fig. \[fig:slope\] confirm this to be true. The sharper the domain wall, the less rapidly the non-adiabatic torque disappears with increasing domain wall width.
![Dependence of $\gamma$ in Eq. (\[eqn:Dugaev\]) on domain wall sharpness: Squares are calculated points. Straight line is a guide to the eye that passes through the origin.[]{data-label="fig:slope"}](fig8.eps){width="8cm"}
Relation to Other Work {#other}
======================
### Waintal & Viret
Waintal and Viret[@Waintal:2004] (WV) used a free-electron Stoner model and the Landauer-Büttiker formalism to calculate the spin transfer torque associated with a Néel wall with magnetization Eq. (\[eqn:magnetization\]) and $$\begin{aligned}
\label{WVwall}
\quad
\theta(x) &=& \left\{ \begin{array}{lr}
0, & x < -w \\
\displaystyle (\pi/2){\left( x/w + 1 \right)}, & -w\le x\le w \\
\pi, & x > w .
\end{array}\right. {\nonumber\\}\end{aligned}$$ For this wall profile (which is exactly one half-turn of a uniform spin spiral in the interval $-w\le x\le w$), WV reported oscillatory non-adiabatic contributions to the torque similar to our functions $a(x)$ and $b(x)$. This contrasts with the perfect adiabaticity we found in Sec. \[spinspiral\] for the infinite spin spiral. Moreover, the amplitude of the non-adiabatic torque reported by WV for this wall decreases only as $1/w$ rather than $(1/w)\exp(-\gamma w/L)$ as we found above.
The disparities between Ref. and the present work all arise from the unphysical nature of the domain wall Eq. (\[WVwall\]). Specifically, the divergence of $\theta''(x)$ at $x=\pm w$ locates this wall at the origin of Fig. \[fig:slope\] where $\gamma=0$. This brings their result into agreement with Eq. (\[over\]). Any rounding of the discontinuity in slope at $x=\pm w$ would yield a finite value for $\theta''(x)$ and thus a non-zero value of $\gamma$. In Appendix II, we calculate the spin-transfer torque for the wall Eq. (\[WVwall\]) using our methods. Qualitatively, the pure $1/w$ behavior of the non-adiabatic torque comes from the fact that there is a sudden jump in $\theta'(x)$ at $x=\pm w$. There is a corresponding jump in the direction of ${\bf B}_{\rm eff}(x)$ as defined by Eq. (\[eqn:gradientfield\]). Spins propagating along the the $x$-axis cannot follow this abrupt jump and thus precess around the post-jump field direction with an amplitude determined by the sine of the angle between the before-and-after field directions. The latter is proportional to the jump in $\theta'(x)$, which is $\pi/2w$ for the wall Eq. (\[WVwall\]).
### Zhang & Li
In spin spirals and long domain walls, we find that the non-equilibrium spin current is adiabatic, [*i.e.*]{}, ${\bf Q}(x)$ is aligned with ${\bf M}(x)$ \[or ${\bf B}_{\rm ex}(x)$\]. At the same time, we find in both cases that the non-equilibrium spin density ${\bf s}(x)$ is [*not*]{} aligned with the magnetization; there is component of ${\bf s}(x)$ transverse to ${\bf M}(x)$. The corresponding transverse component of the spin current density cancels between pairs of electrons moving in opposite directions. Zhang and Li[@Zhang:2004] found exactly the same form of non-equilibrium spin accumulation (called $\delta {\bf m}(x)$ by them) using a phenomenological theory. They proposed that this non-equilibrium spin density relaxes by spin-flip scattering toward alignment with the magnetization. Such a relaxation would produce a non-adiabatic torque of the form given by the second term in Eq. (\[lt\]). The correctness of this predicted non-adiabatic torque depends on the correctness of the assumed model for relaxation of transverse spin accumulation through spin flip scattering.
Zhang and Li assume a form for the rate of spin flip scattering, $\delta {\bf m}
/\tau_{\rm sf}$, that has been used successfully as a phenomenological description of [*longitudinal*]{} spin relaxation in systems with collinear magnetization. While it is plausible to extend this form, as they do, to describe [*transverse*]{} spin relaxation in non-collinear systems, our calculations indicate that it is not likely to be correct. Our reasoning is simplest to appreciate for a spin spiral with small pitch $p$. In this limit, Eqs. (\[alpha\]) and (\[eqn:spirald\]) show that the transverse component of the spin for every electron eigenstate is proportional to its velocity. This means that the majority band electrons contribute a transverse spin accumulation and an electric current that are proportional to one other. The same is true, separately, for the minority band electrons. This conclusion is independent of the details of the electron distribution. Therefore, for a fixed total current, it is impossible to relax the transverse spin accumulation without changing the longitudinal polarization of the current. No such change occurs in the model in Ref. , casting doubt on the validity of the form of the spin flip scattering assumed there.
Microscopic considerations also argue against this form of the relaxation. As we have emphasized, the natural basis for an electron spin moving though a non-collinear magnetization is not along the local exchange field ${\bf B}_{\rm ex}(x)$, but rather along a local effective field ${\bf B}_{\rm eff}(x)$, which includes the corrections due to the gradient of the magnetization \[see Eq. (\[eqn:gradientfield\])\]. Any spin that deviates from parallel or antiparallel alignment with the effective field will precess around the effective field, and on average will point parallel or antiparallel. Thus, we expect that there is [*no*]{} tendency for electron spins moving in a non-uniform magnetization to align themselves with the local exchange field ${\bf B}_{\rm ex}(x)$ by spin-flip scattering (or any other mechanism). Rather, the adiabatic solution is precisely alignment of their spins with the local effective field ${\bf B}_{\rm eff}(x)$. Without further microscopic justification, we believe that the phenomenological form of spin flip scattering assumed in Ref. should not be used in systems with non-collinear magnetizations. Hence, this analysis argues against the existence of the resulting contribution to the “non-adiabatic” torque from spin flip scattering.
Scattering
==========
We do not explicitly treat scattering in any of our calculations. However, the distribution function in Eq. (\[eqn:spinandcurrent\]), a shifted Fermi distribution, is an approximate solution of the Boltzmann equation in certain limits. First, the electric field must be small enough that the transport is in the linear regime. Then, the appropriate limits are determined by three important length scales, the Fermi wavelength, the mean free path, and the characteristic length of the structure, either the pitch of the spin spiral or the width of the domain wall. In all cases, we consider the limit in which the Fermi wavelength is short compared to the mean free path. This limit allows the description of the states of the system in terms of the eigenstates of the system in the absence of scattering. Different limits apply to the cases of domain walls and of spin spirals because the distribution functions are interpreted differently for these two structures.
We use the Boltzmann equation in two different ways. When the mean free path is much longer than the characteristic size of the structure, the distribution function describes the occupancy of the eigenstates of the entire system. This distribution function is independent of the spatial coordinate and we refer to this approach as global. In the opposite limit, the distribution function is spatially varying and describes the occupancy of eigenstates of the local Hamiltonian, which includes the exchange field and the gradient field. We refer to this approach as local, as the distribution function can vary spatially.
For spin spirals, the distribution functions are shifted Fermi functions of the eigen-energies of the spin spiral. In the limit that the pitch of the spiral is much shorter than the mean free path, the shifted distribution given in Eq. (\[eqn:spinandcurrent\]) is a solution of the global Boltzmann equation in the relaxation time approximation. The distribution function also becomes a solution in the opposite limit, where the mean free path is much shorter than the pitch of the spiral. In this limit, the Boltzmann equation is considered locally rather than globally. At each point in space the states are subject to the local exchange field, and the local gradient field. The distribution function is defined for states that are locally eigenstates of the sum of the fields. The local distribution function is given by the adiabatic evolution in the rotating reference frames of the distribution function specified in Eq. (\[eqn:spinandcurrent\]). In the limit that the pitch of the spiral goes to infinity, this distribution function locally solves the Boltzmann equation in the relaxation time approximation. Thus, for spin spirals, the distribution function given in Eq. (\[eqn:spinandcurrent\]) is a solution in the limits that the mean free path is much greater than or much less than the pitch. We speculate that the corrections in between these limits are small.
Domain walls are not uniform in the way that spin spirals are, so the distribution functions need to be given a different interpretation. For these structures, the distribution function is determined from the properties of the states in the leads. For example, in the Landauer-Büttiker approach to this problem,[@Waintal:2004] scattering is ignored in the domain wall itself and confined to the “leads” adjacent to it (these leads are assumed to be “wide” and function as electron reservoirs). An applied voltage is assumed to raise the energy of electron states in one lead relative to the other. Thus, in a formula like Eq. (\[eqn:spinandcurrent\]), the distribution function is shifted in energy rather than in velocity.
We also do not treat scattering within the domain wall explicitly, but we assume that the wall is bounded by long leads that are as “narrow” as the domain wall region and have resistances per unit length that are comparable to that of the domain wall region. Thus, the distribution of the states approaching the domain wall region is similar to the distribution of states in an extended wire, [*i.e.*]{}, to that given by Eq. (\[eqn:spinandcurrent\]). For domain walls in long wires, the distribution function for left going states is determined by the right lead and for right going states by the left lead. With this interpretation, the distribution given in Eq. (\[eqn:spinandcurrent\]) is a solution in the limit that the scattering in the domain wall is weak, that is, the domain wall is much narrower than the mean free path.
The distribution in Eq. (\[eqn:spinandcurrent\]) is also a solution in the limit that the mean free path is much shorter than the domain wall width. Since the Fermi wave length is much shorter than the mean free path, it is much less than the domain wall width. In this case, quantum mechanical reflection is negligible and the quantum mechanical states are closely related to the semiclassical trajectories. With a similar interpretation of the distribution function as was made for the spin spirals in this limit, the same conclusion holds for the domain walls.
Summary & Conclusion {#sumcon}
====================
In this paper, we analyzed spin-transfer torque in systems with continuously variable magnetization using previous results of Calvo[@Calvo:1978] for the eigenstates of an infinite spin spiral and of Aharonov and Stern[@Aharonov:1992] for the classical motion of a magnetic moment in an inhomogeneous magnetic field. Adiabatic motion of individual spins corresponds to alignment of the spin moment [*not*]{} with the exchange field (magnetization) but with an effective field that is slightly tilted away form the exchange field by an amount that depends on the spatial gradient of the magnetization. Nevertheless, when summed over all conduction electrons, the spin current density is parallel to the magnetization both for an infinite spin spiral and for domain walls that are long compared to a characteristic length $L$ that depends on the exchange energy and the Fermi energy.
Non-adiabatic corrections to the spin-transfer torque occur only for domain walls with widths $w$ that are comparable to or smaller than $L$. The non-adiabatic torque is oscillatory and non-local in space with an amplitude that decreases as $w^{-1}\exp(-\gamma w/L)$. The constant $\gamma$ is largest for walls with the sharpest magnetization gradients. This suggests that non-adiabatic torques may be important for spin textures like vortices where the magnetization varies extremely rapidly.
Using microscopic considerations, we have also argued that the role of the gradient field to tilt spins away from the exchange field casts serious doubt on a recent proposal by Zhang and Li[@Zhang:2004] that a non-negligible non-adiabatic contribution to the torque arises from relaxation of the non-equilibrium spin accumulation to the magnetization vector by spin flip scattering. We conclude that, if the second term in Eq. (\[lt\]) truly accounts for the systematics of current-driven domain wall motion, the physics that generates this term still remains to be identified.
Finally, we have carefully discussed the role of scattering in this problem with particular emphasis on the approximation used here to neglect scattering within the domain wall itself but to treat the adjacent ferromagnetic matter as bulk-like. We argue that this approximation is valid in limits that either include or bracket the most interesting experimental situations and therefore is likely to be generally useful.
Acknowledgment
==============
One of us (J.X.) is grateful for support from the Department of Energy under grant DE-FG02-04ER46170.
[**Appendix I: Semi-Classical Weighting Factor**]{}
The weighting factor $k_x/{\langle k \rangle}$ used in Eq. (\[scm\]) brings the amplitude of the dynamic (transverse) part of the semi-classical, one-electron spin density into accord with the corresponding quantum mechanical amplitude. This can be seen from a simple model problem that we solve both quantum mechanically and semi-classically. Namely, a spin initially oriented along the $+\hat{\bf x}$ direction propagates from $x=-\infty$ to $x=\infty$ through a magnetization that changes abruptly from ${\mathbf M}(x) = M(1,0,0)$ for $x<0$ to ${\mathbf M}(x) = M(0,0,1)$ for $x \ge 0$. For $x<0$, the eigenstates are $$\psi_{\uparrow}^-(x) = {1\over\sqrt{2}} {\left( \begin{array}[c]{c} 1 \\ 1 \end{array} \right)} e^{ik_{\uparrow}x},\quad
\psi_{\downarrow}^-(x) = {1\over\sqrt{2}}{\left( \begin{array}[c]{c} 1 \\ -1 \end{array} \right)} e^{ik_{\downarrow}x},$$ and for $x>0$, the eigenstates are $$\psi_{\uparrow}^+(x) = {\left( \begin{array}[c]{c} 1 \\ 0 \end{array} \right)} e^{ik_{\uparrow}x},\quad
\psi_{\downarrow}^+(x) = {\left( \begin{array}[c]{c} 0 \\ 1 \end{array} \right)} e^{ik_{\downarrow}x}.$$
If we choose the incoming state as $$\psi(x) = \psi_{\uparrow}^-(x),$$ the reflection and transmission amplitudes for spin flip $(r_{{\uparrow}{\downarrow}}, t_{{\uparrow}{\downarrow}})$ and no spin flip $(r_{{\uparrow}{\uparrow}},t_{{\uparrow}{\uparrow}})$ are determined by matching the total wave function and its derivative at $x=0$: $$\begin{aligned}
\psi_{\uparrow}^- + r_{{\uparrow}{\uparrow}} (\psi_{\uparrow}^-)^* + r_{{\uparrow}{\downarrow}} (\psi_{\downarrow}^-)^* &=& t_{{\uparrow}{\uparrow}} \psi_{\uparrow}^+ + t_{{\uparrow}{\downarrow}} \psi_{\downarrow}^+ {\nonumber\\}&& \\
k_{\uparrow}\psi_{\uparrow}^- - r_{{\uparrow}{\uparrow}} k_{\uparrow}(\psi_{\uparrow}^-)^* - r_{{\uparrow}{\downarrow}} k_{\downarrow}(\psi_{\downarrow}^-)^* &=& t_{{\uparrow}{\uparrow}} k_{\uparrow}\psi_{\uparrow}^+ + t_{{\uparrow}{\downarrow}} k_{\downarrow}\psi_{\downarrow}^+. {\nonumber\\}\end{aligned}$$ It is straightforward to confirm that these equations are solved by $$\begin{aligned}
r_{{\uparrow}{\uparrow}} = {k_{\uparrow}^2-k_{\downarrow}^2\over k_{\uparrow}^2 + 6k_{\uparrow}k_{\downarrow}+ k_{\downarrow}^2} &,&
r_{{\uparrow}{\downarrow}} = {2k_{\uparrow}(k_{\downarrow}-k_{\uparrow})\over k_{\uparrow}^2 + 6k_{\uparrow}k_{\downarrow}+ k_{\downarrow}^2} {\nonumber\\}t_{{\uparrow}{\uparrow}} = {4\sqrt{2} k_{\uparrow}k_{\downarrow}\over k_{\uparrow}^2 + 6k_{\uparrow}k_{\downarrow}+ k_{\downarrow}^2} &,&
t_{{\uparrow}{\downarrow}} = {2\sqrt{2}k_{\uparrow}(k_{\uparrow}+k_{\downarrow})\over k_{\uparrow}^2 + 6k_{\uparrow}k_{\downarrow}+ k_{\downarrow}^2}. {\nonumber\\}\end{aligned}$$ We are interested in the transmitted wave function, $$\psi_{\rm tr}(x) = t_{{\uparrow}{\uparrow}}\psi_{\uparrow}^+(x) + t_{{\uparrow}{\downarrow}}\psi_{\downarrow}^+(x)
= {\left( \begin{array}[c]{c} t_{{\uparrow}{\uparrow}}e^{ik_{\uparrow}x} \\ t_{{\uparrow}{\downarrow}} e^{ik_{\downarrow}x} \end{array} \right)},$$ which carries a spin density, $$\label{ampq}
{\mathbf s}_{\rm tr}^{\rm qm}(x) = {\hbar\over 2} \left[2t_{{\uparrow}{\uparrow}}t_{{\uparrow}{\downarrow}}\cos(\delta kx), 2t_{{\uparrow}{\uparrow}}t_{{\uparrow}{\downarrow}}\sin(\delta kx), (t_{{\uparrow}{\uparrow}}^2-t_{{\uparrow}{\downarrow}}^2)\right],$$ where $\delta k=k_{\uparrow}- k_{\downarrow}$. Notice that the oscillation is transverse to the $x\to\infty$ magnetization and of amplitude $\hbar t_{{\uparrow}{\uparrow}}t_{{\uparrow}{\downarrow}}$.
If we analyze the same problem semi-classically, a majority electron propagates freely until it reaches $x=0$. At that point, the electron feels a magnetization perpendicular to its magnetic moment and begins precession around that magnetization with unit amplitude. The associated spin density is $$\label{ampsc}
{\mathbf s}_{\rm tr}^{\rm sc}(x) = {\hbar\over 2} \left [\cos(\delta kx),\sin(\delta kx),0\right ].$$
Comparing Eq. (\[ampq\]) to Eq. (\[ampsc\]) shows that the transverse oscillation amplitudes will be equal if we multiply the semi-classical result by the weighting factor $$\label{eqn:approxa}
2t_{{\uparrow}{\uparrow}}t_{{\uparrow}{\downarrow}} =
{32 k_{\uparrow}^2 k_{\downarrow}(k_{\uparrow}+k_{\downarrow})\over (k_{\uparrow}^2 + 6k_{\uparrow}k_{\downarrow}+ k_{\downarrow}^2)^2}
\approx {2k_{\uparrow}\over k_{\uparrow}+k_{\downarrow}} = {k_x\over {\langle k \rangle}}.$$ Fig. \[fig:scfactor\] illustrates the quality of the approximation in Eq. (\[eqn:approxa\]) if we identify $k_{\uparrow}$ and $k_{\downarrow}$ with $k^+_F$ and $k^-_F$ (respectively) in Eq. (\[eqn:kpm\]). Of course, $k_x$ plays the role of $k_{\uparrow}$ in Eq. (\[scm\]).
![The semi-classical weighting factor for the spin density. Solid (dotted) curve is the expression on the left (right) side of the $\approx$ symbol in Eq. (\[eqn:approxa\]). []{data-label="fig:scfactor"}](fig9.eps){width="8cm"}
[**Appendix II: Spin Spiral Domain Wall**]{}
The semi-classical spin density associated with electron propagation through a magnetization like Eq. (\[eqn:magnetization\]) is the solution of Eq. (\[eqn:LLG\]) with suitable boundary conditions. Choosing $\phi=0$, we simplify the notation by using the prefactor $\lambda=k_B^2/{\langle k \rangle}$ and an overdot for $d/dx$ to write the components of Eq. (\[eqn:LLG\]) as $$\begin{aligned}
\dot{s}_x &=& - \lambda s_y\cos\theta \\
\dot{s}_y &=& - \lambda s_z\sin\theta + \lambda s_x\cos\theta \\
\dot{s}_z &=& \lambda s_y\sin\theta.\end{aligned}$$ In the local frame $(x',y',z')$ defined in Fig. \[fig:spinspiral\], the components of the spin density, $$\begin{aligned}
s'_x &=& s_x\cos\theta - s_z\sin\theta \\
s'_y &=& s_y \\
s'_z &=& s_x\sin\theta + s_z\cos\theta,\end{aligned}$$ satisfy $$\begin{aligned}
\dot{s}'_x &=& -\lambda s'_y - s'_z\dot{\theta} \\
\dot{s}'_y &=& \lambda s'_x \\
\dot{s}'_z &=& s'_x\dot{\theta}.\end{aligned}$$ Eliminating $s'_y$ gives $$\ddot{s}'_x + (\lambda^2+\dot{\theta}^2)s'_x + s'_z\ddot{\theta} = 0.
\label{eqn:mX}$$
The differential Eq. (\[eqn:mX\]) cannot be solved analytically for realistic domain wall profiles. However, it is easily solvable for the wall defined by Eq. (\[WVwall\]) where one-half turn of a spin spiral with pitch $p=\pi/2w$ connects two regions with uniform (but reversed) magnetization. In the limit $\pi/w\ll\lambda$ of a long wall, the components of the spin density transverse to the wall magnetization for the range $x\in[-w,w]$ are (after multiplying the weighting factor $k_x/{\langle k \rangle}$ for the semi-classical approach) $$\begin{aligned}
s'_x(x) &=& {\hbar\over 2}{k_x\over {\langle k \rangle}}{\pi \over 2w\lambda}\sin(\lambda(x\pm w)) {\nonumber\\}s'_y(x) &=& {\hbar\over 2}{k_x\over {\langle k \rangle}}{\pi \over 2w\lambda}{\left[ 1-\cos(\lambda(x\pm w)) \right]},\end{aligned}$$ where the plus (minus) refers to electrons that flow from left (right) to right (left). The associated spin current density and spin transfer torque carried by each electron follow from Eq. (\[eqn:scscd\]) and Eq. (\[eqn:spt\]), respectively. Bearing in mind that $\hat{\bf
x}'$ varies with $x$, our final result for the torque (in the local frame) generated by a single electron moving from right to left is $$\begin{aligned}
N'_x &=& {\hbar\over 2}{\hbar k_x\over m}{k_x\over {\langle k \rangle}}{\pi\over 2w}\left[1-\cos\lambda(x-a)\right]~\hat{\bf x}' {\nonumber\\}N'_y &=& {\hbar\over 2}{\hbar k_x\over m}{k_x\over {\langle k \rangle}}{\pi\over 2w}\sin\lambda(x-a)~\hat{\bf y}.\end{aligned}$$ This may be compared with the results of Ref. which pertain to the entire ensemble of conduction electrons.
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In reality, the magnetization is oppositely directed to the spin density since the g-factor for electrons is negative. For simplicity, we ignore this sign difference and assume that the magnetization is parallel to the spin density. Care is required when using spin currents to compute torques on magnetizations.
The spin current density is a tensor quantity (Ref. ). However, since the current defines the only relevant direction in space for this problem, we supress this dependence and use the components of the vector ${\bf Q}$ to denote the Cartesian components of the spin degree of freedom in the spin current density.
When $E\tau$ is sufficiently small, as we assume here, the difference in the square brackets in Eq. (\[eqn:spinandcurrent\]) is zero except in the immediate vicinity of the Fermi surface. In this limit, the integral over reciprocal space in Eq. (\[eqn:spinandcurrent\]) reduces to an integral over wave vectors restricted to the Fermi surface, reflecting the fact that non-equilibrium transport involves states near the Fermi surface. The reduction of the integration in this manner leads to the linearized Boltzmann equation. On the other hand, the equilibrium spin densities and currents, as shown in Fig. \[fig:equilibrium\], involve contributions from all occupied states, not just those near the Fermi surface.
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---
abstract: 'We prove rigidity of various types of holomorphic geometric structures on smooth complex projective varieties.'
address: |
University College Cork\
National University of Ireland
author:
- Benjamin McKay
bibliography:
- 'rational-curves-parabolic.bib'
title: Rigid geometry on projective varieties
---
Introduction
============
All manifolds and maps henceforth are assumed complex analytic, and all Lie groups, algebras, etc. are complex. We will prove a collection of global rigidity theorems for holomorphic geometric structures. As an example:
\[theorem:RigidityForQuaternionicContact\] Suppose that $M$ is a smooth connected complex projective variety of complex dimension 7, bearing a holomorphic quaternionic contact structure. Then $M=C_3/P$, where $C_3={\ensuremath{\operatorname{Sp}\left({6,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ and $P \subset C_3$ is a certain complex parabolic subgroup. The holomorphic quaternionic contact structure is the standard flat holomorphic quaternionic contact structure on $C_3/P$ (defined in section ).
The main theorem
================
Definitions required to state the main theorem
----------------------------------------------
### Definition of Cartan geometries
If $E \to M$ is a principal right $G$-bundle, we will write the right $G$-action as $r_g e = eg$, where $e\in E$ and $g\in G$.
Throughout we use the convention that principal bundles are right principal bundles.
\[def:CartanConnection\] Let $H \subset G$ be a closed subgroup of a Lie group, with Lie algebras ${\ensuremath{\mathfrak{h}}}\subset {\ensuremath{\mathfrak{g}}}$. A $G/H$-geometry, or *Cartan geometry* modelled on $G/H$, on a manifold $M$ is a choice of $C^\infty$ principal $H$-bundle $E \to M$, and smooth 1-form $\omega \in
{\ensuremath{\Omega^{1}\left({E}\right)}} \otimes {\ensuremath{\mathfrak{g}}}$ called the *Cartan connection*, which satisifies all of the following conditions:
1. $
r_h^* \omega = \operatorname{Ad}_h^{-1} \omega
$ for all $h \in H$.
2. $\omega_e : T_e E \to {\ensuremath{\mathfrak{g}}}$ is a linear isomorphism at each point $e \in E$.
3. For each $A \in {\ensuremath{\mathfrak{g}}}$, define a vector field $\vec{A}$ on $E$ by the equation $\vec{A} {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\omega = A$. Then the vector fields $\vec{A}$ for $A \in {\ensuremath{\mathfrak{h}}}$ generate the $H$-action on $E$.
Sharpe [@Sharpe:2002] gives an introduction to Cartan geometries.
The principal $H$-bundle $G \to G/H$ is a Cartan geometry, with Cartan connection $\omega=g^{-1} \, dg$ the left invariant Maurer–Cartan 1-form on $G$; this geometry is called the *model Cartan geometry*.
An *isomorphism* of $G/H$-geometries $E_0 \to M_0$ and $E_1 \to M_1$ with Cartan connections $\omega_0$ and $\omega_1$ is an $H$-equivariant diffeomorphism $F : E_0 \to E_1$ so that $F^* \omega_1 = \omega_0$.
### Definition of lift of Cartan geometries
Suppose that $H \subset H' \subset G$ are two closed subgroups, and $E \to M'$ is a $G/H'$-geometry. Let $M=E/H$. Clearly $E \to M$ is a principal $H$-bundle. We can equip $E$ with the Cartan connection of the original $G/H'$-geometry, and then clearly $E \to M$ is a $G/H$-geometry. Moreover $M \to M'$ is a fiber bundle with fiber $H'/H$. The geometry $E \to M$ is called the *$G/H$-lift* of $E \to M'$ (or simply the *lift*). Conversely, we will say that a given $G/H$-geometry *drops* to a certain $G/H'$-geometry if it is isomorphic to the lift of that $G/H'$-geometry.
A Cartan geometry which drops can be completely recovered (up to isomorphism) from anything it drops to. So dropping encapsulates the same geometry in a lower dimensional reformulation.
### Definition of generalized flag varieties
A *parabolic subgroup* $P$ of a complex semisimple Lie group $G$ is a subgroup containing a maximal solvable subgroup.
Parabolic subgroups are closed connected complex Lie subgroups.
A *generalized flag variety* is a homogeneous space $G/P$ where $G$ is a complex semisimple Lie group and $P$ is a parabolic subgroup. Every generalized flag variety is compact and connected.
### Semicanonical modules
Suppose that $G/H$ is a complex homogeneous space. An $H$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is *semicanonical* if there are integers $p \ge 0$ and $q > 0$ so that $\left(\det I\right)^{\otimes q} = \left(\det \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)\right)^{\otimes (-p)}$. An $H$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is *nontrivial* if $I \ne 0$ and $I \ne \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$.
The main theorem
----------------
In various examples, we will prove rigidity of various Cartan geometries. Among many other results, we will prove the following theorem:
Suppose that $G$ is a complex Lie group and $H \subset G$ is a maximal complex subgroup. Suppose that $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is a nontrivial semicanonical module. Suppose that $M$ is a connected smooth complex projective variety bearing a holomorphic Cartan geometry $E \to M$ modelled on $G/H$. Let $\mathcal{I} = E \times_H I \subset T^*M$. If the holomorphic subbundle $\mathcal{I}^{\perp} \subset TM$ is not everywhere bracket closed, then
1. $M=G/H$ and
2. the Cartan geometry on $M$ is the model holomorphic Cartan geometry on $G/H$ and
3. $G/H$ is a generalized flag variety.
We will prove more general theorems below, and prove a similar theorem for compact K[ä]{}hler manifolds for a different class of Cartan geometries. We will apply our theorems to prove rigidity of various types of holomorphic geometric structures.
Pfaffian systems
================
A *Pfaffian system* on a complex manifold $M$ is a holomorphic vector subbundle of the holomorphic cotangent bundle $T^* M$.
If $\mathcal{I} \subset T^*M$ is a Pfaffian system, the reader may feel more comfortable working with $\mathcal{V}=\mathcal{I}^{\perp}$, which is a holomorphic plane field (a.k.a. distribution, a.k.a. subbundle of the tangent bundle). The convenience of working with $\mathcal{I}$ rather than $\mathcal{V}$ will become clear, and will more than overcome the initial discomfort.
A Pfaffian system $\mathcal{I} \subset T^*M$ is *Frobenius* if the ideal it generates in the sheaf ${\ensuremath{\Lambda^{*} \left ( {T^*M} \right )}}$ of differential forms is $d$-closed.
Equivalently, $\mathcal{I}$ is Frobenius if $\mathcal{V}=\mathcal{I}^{\perp}$ is bracket closed. Synonyms for *Frobenius* include *integrable, completely integrable* and *involutive*.
Brackets in Cartan geometries
=============================
\[lemma:TgtBundle\] If $\pi : E \to M$ is any Cartan geometry, say with model $G/H$, then the Cartan connection of $E$ maps $$\xymatrix{0 \ar[r] & \ker \pi'(e) \ar[r] \ar[d] & T_e E \ar[r] \ar[d] & T_m M \ar[r] \ar[d] & 0 \\
0 \ar[r] & \mathfrak{h} \ar[r] & \mathfrak{g} \ar[r] &
\mathfrak{g}/\mathfrak{h} \ar[r] & 0 }$$ for any points $m \in M$ and $e \in E_m$; thus $$TM=E \times_H \left(\mathfrak{g}/\mathfrak{h}\right)
\text{ and }
T^*M=E \times_H \left(\mathfrak{g}/\mathfrak{h}\right)^*.$$ Under this identification, vector fields on $M$ are identified with $H$-equivariant functions $E \to \mathfrak{g}/\mathfrak{h}$, and sections of the cotangent bundle with $H$-equivariant functions $E \to \left(\mathfrak{g}/\mathfrak{h}\right)^*$.
If $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is an $H$-submodule, then each local holomorphic section of $E \times_H I \subset T^*M$ is identified with an $H$-equivariant holomorphic map from an open subset of $E$ to $I$.
Pseudoeffective line bundles
============================
A line bundle $L$ on a Kähler manifold is *pseudoeffective* if $c_1\left(L\right)$ can be represented by a closed positive $(1,1)$-current. (See Demailly [@Demailly:1997a] for more information.)
Zero is considered positive in this definition.
If $V$ is a holomorphic vector bundle of rank $N$ on a complex manifold $M$, let $\det V = {\ensuremath{\Lambda^{N} \left ( {V} \right )}}$.
\[lemma:Demailly\] Let $M$ be a closed Kähler manifold and $\mathcal{I} \subset T^*M$ a holomorphic Pfaffian system which is not Frobenius. Suppose that the line bundle $\det \mathcal{I}$ on $M$ is pseudoeffective. Then $\mathcal{I}$ is Frobenius.
Suppose that $\mathcal{I}$ has rank $q$. Define a line-bundle-valued differential form $\vartheta
\in {\ensuremath{\Omega^{q}\left({M}\right)}} \otimes \det\left(TM/\mathcal{I}^{\perp}\right)$ by $$\vartheta\left(v_1,v_2,\dots,v_q\right)
=
\left(v_1+I^{\perp}\right)
\wedge
\left(v_2+I^{\perp}\right)
\wedge
\dots
\wedge
\left(v_q + I^{\perp}\right).$$ By Demailly [@Demailly:2002] p. 1, Main Theorem applied to $\vartheta$, if $\det \mathcal{I}$ is pseudoeffective, then $I$ is Frobenius.
We write the canonical bundle of a complex manifold $M$ as $\kappa_M$.
A holomorphic vector bundle $\mathcal{I}$ on a complex manifold $M$ is *semicanonical* if there are integers $p \ge 0, q > 0$ so that $\left(\det \mathcal{I}\right)^{\otimes q} \otimes \kappa_M^{-\otimes p}$ is pseudoeffective.
\[proposition:ContracanonicalNotPseudoEff\] Suppose that
1. $M$ is a compact K[ähler]{} manifold and
2. $\mathcal{I} \subset T^*M$ is a Pfaffian system and
3. $\mathcal{I}$ is not Frobenius and
4. $\mathcal{I}$ is semicanonical.
Then the canonical bundle of $M$ is not pseudoeffective.
By lemma , $\det \mathcal{I}$ is not pseudoeffective. If $\kappa_M$ is pseudoeffective, then so is $\kappa_M^{\otimes p}$ for any integer $p \ge 0$. Therefore $\left(\det \mathcal{I}\right)^{\otimes q}$ is pseudoeffective for some integer $q>0$, and so $\det \mathcal{I}$ is also pseudoeffective.
Suppose that $\mathcal{I} \subset T^*M$ is a holomorphic contact structure. Then $\mathcal{I}$ is semicanonical.
Pick a local section $\vartheta$ of $I$ for which $\vartheta \wedge d \vartheta^n \ne 0$. But $\vartheta \wedge d \vartheta^n$ is a holomorphic volume form. The map $\phi : \mathcal{I} \to \kappa_M$, given on each local section by $\vartheta \mapsto \vartheta \wedge d\vartheta^n$ depends only on the value of $\vartheta$ pointwise, and scales like $\phi(f\vartheta)=f^{n+1} \phi(\vartheta)$, so $\mathcal{I}^{\otimes(n+1)} = \kappa_M$.
Suppose that $M=M_1 \times M_2$ is a product. If $\mathcal{I}_1 \subset T^*M_1$ and $\mathcal{I}_2 \subset T^*M_2$ are semicanonical, then so is $\mathcal{I}_1 \oplus \mathcal{I}_2 \subset T^*M$.
\[example:CartanTwoPlaneField\] Let $V$ be a rank 2 holomorphic subbundle $V \subset TM$ on a complex manifold $M$ with $\dim_{{\ensuremath{\mathbb{C}^{}}}} M = 5$. Say that a pair $X, Y$ of local holomorphic sections of $V$ is *nondegenerate* if the vector fields $$X, Y, [X,Y], \left[X,\left[X,Y\right]\right], \left[Y,\left[X,Y\right]\right]$$ are linearly independent at every point where $X$ and $Y$ are defined. Then $V$ is *Cartan* or *nondegenerate* if near each point of $M$ there is a nondegenerate pair of local holomorphic sections.
Given any nondegenerate pair $X$ and $Y$, let $$\xi(X,Y) =
X \wedge Y \wedge [X,Y] \wedge \left[X,\left[X,Y\right]\right] \wedge \left[Y,\left[X,Y\right]\right].$$ So $\xi$ takes a pair of sections to a section of the anticanonical bundle $\kappa_M^*$. If $X'$ and $Y'$ are any two local sections of $V$, we can write $$\begin{aligned}
X' &= a \, X + b \, Y \\
Y' &= c \, X + d \, Y\end{aligned}$$ for some holomorphic matrix valued function $$g =
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$$ on the overlap where $X', Y', X$ and $Y$ are defined. Check that $$\xi(X',Y')=\det(g)^5 \xi(X,Y).$$ Therefore if $V$ is nondegenerate, then $\det V = {\ensuremath{\Lambda^{2} \left ( {V} \right )}}$ and ${\ensuremath{\Lambda^{2} \left ( {V} \right )}}^{\otimes 5}=\kappa_M^*$. Let $\mathcal{I} = V^{\perp}$. Clearly $\det \mathcal{I} = \kappa_M \otimes \det V$, so $\left(\det \mathcal{I}\right)^{\otimes 5} = \kappa_M^{4}$, so $\mathcal{I}$ is semicanonical. By Demailly’s theorem, since $V$ is not bracket closed (i.e. $\mathcal{I}$ is not Frobenius), $\kappa_M$ is pseudoeffective.
\[example:FirstOrderNondegenerate\] A holomorphic $k$-plane field $V \subset TM$ on a complex manifold $M$ of dimension $n=\dim_{{\ensuremath{\mathbb{C}^{}}}} M = k(k+1)/2$ is *first order nondegenerate* if near each point of $M$ there are local holomorphic sections $$X_1, X_2, \dots, X_k$$ of $V$ so that $$X_1, X_2, \dots, X_k,
\left[X_1, X_2\right], \left[X_1,X_3\right], \dots, \left[X_{k-1}, X_k\right]$$ are linearly independent. Clearly any first order nondegenerate holomorphic $k$-plane field $V$ has associated Pfaffian system $\mathcal{I}=V^{\perp}$ semicanonical, with the same argument as for contact structures.
Pseudoeffectivity and Pfaffian systems in Cartan geometries
===========================================================
Clearly if $I$ is a semicanonical submodule then $\mathcal{I} = G \times_H I$ is a semicanonical vector bundle.
\[example:ContactSemicanonicalModule\] If $G/H$ has a $G$-invariant contact structure, say $G \times_H I \subset T^*(G/H)$, then $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is semicanonical.
Suppose that
1. $M$ is a compact K[ä]{}hler manifold and
2. $M$ bears a holomorphic Cartan geometry modelled on a complex homogeneous space $G/H$ and
3. $I$ is an $H$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ and
4. $I$ is semicanonical and
5. $E \times_H I \subset T^*M$ is not Frobenius.
Then $M$ does not have pseudoeffective canonical bundle.
Clear from proposition .
\[example:GeneralizedFlagVariety\] Suppose that $G/P$ is a generalized flag variety. Every $P$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is a direct sum of root spaces, since $P$ contains the Cartan subgroup of $G$. Associate to $I$ the set $S=S_I$ of roots whose root space lies in $I$. Then $S$ is a set of noncompact positive roots. For example, $\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is the direct sum of all of the root spaces of all of the noncompact positive roots. Pick any root $\alpha \in S$. Pick $\beta$ to be either a compact root or a noncompact positive root. Then either $\alpha+\beta \in S$ or $\alpha+\beta$ is not a root. Conversely, if $S$ is any set of roots with this property, let $I=I_S$ be the sum of the root spaces of the roots that lie in $S$. Then $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is a $P$-submodule. So we can draw $I$ by drawing the root lattice of $G$ and indicating somehow which roots lie in $S$.
Next we need to test when $I$ is semicanonical. Let $W_{G/P}$ be the subgroup of the Weyl group of $G$ preserving the noncompact positive roots of $G/P$. Baston and Eastwood [@Baston/Eastwood:1989] prove that we can identify the weights of $P$ with the $W_{G/P}$-invariant weights of $G$. The weight of $\det I$ is $$\sum_{\alpha \in S} \alpha.$$ In particular, the weight $\omega$ of $\det\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ as a weight of $P$ is the sum of the noncompact positive roots, say $$\omega=\sum_{\alpha \in \Delta^{\text{noncompact}}_+} \alpha$$ We can then see that $I$ is semicanonical if and only if $$\sum_{\alpha \in S} \alpha
= \frac{p}{q} \omega$$ for some rational number $0 \le \frac{p}{q} \le 1$, and nontrivial if and only if $0 < \frac{p}{q} < 1$.
\[example:GTwoModPOne\] Figure
$$\xy +(2,2)="o",0*\xybox{0;<1.5pc,0mm>:<-2.25pc,1.3pc>::,{"o"
\croplattice{-3}3{-2}2{-2.6}{2.6}{-3}3}
,"o"+(1,0)="a"*{{{\scriptscriptstyle{\bullet}}}}*+!D{}
,"o"+(1.4,0)="aL"*{\alpha}*+!D{}
,"o"+(0,0)="b"*{\bigcirc}*+!L{}
,"o"+(0,0)="c"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(0,1)="d"*{{{\scriptscriptstyle{\bullet}}}}*+!D{}
,"o"+(0,1.2)="dL"*{\beta}*+!D{}
,"o"+(1,1)="B"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(2,1)="C"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(3,1)="D"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(3,2)="E"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(-1,0)="na"*{{{\scriptscriptstyle{\times}}}}*+!D{},"o"+(0,-1)="nd"*{{{\scriptscriptstyle{\bullet}}}}*+!D{}
,"o"+(-1,-1)="nB"*{{{\scriptscriptstyle{\times}}}}*+!L{}
,"o"+(-2,-1)="nC"*{{{\scriptscriptstyle{\times}}}}*+!L{}
,"o"+(-3,-1)="nD"*{{{\scriptscriptstyle{\times}}}}*+!L{}
,"o"+(-3,-2)="nE"*{{{\scriptscriptstyle{\times}}}}*+!L{}
}
\endxy$$
shows the roots of $G_2$. The dots are the roots whose root spaces lie in ${\ensuremath{\mathfrak{p}}}$, and the crosses are the other roots. The circled dot is the origin, representing the Cartan subgroup. The compact roots lie on the line through $\beta$ and the origin. The positive noncompact roots lie in the upper half plane above this line. The sum of the noncompact positive roots is $$\omega = 10 \, \alpha + 5 \, \beta.$$
Figure shows stars ($\star$) on the noncompact positive roots $$3 \, \alpha+2 \, \beta, 2 \, \alpha + \beta, 3 \, \alpha+ \beta.$$ Let $S$ be the set of these roots, and $I=I_S$. Then $I$ is a 3-dimensional submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$.
$$\xy +(2,2)="o",0*\xybox{0;<1.5pc,0mm>:<-2.25pc,1.3pc>::,{"o"
\croplattice{-3}3{-2}2{-2.6}{2.6}{-3}3}
,"o"+(1,0)="a"*{{{\scriptscriptstyle{\bullet}}}}*+!D{}
,"o"+(1.4,0)="aL"*{\alpha}*+!D{}
,"o"+(0,0)="b"*{\bigcirc}*+!L{}
,"o"+(0,0)="c"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(0,1)="d"*{{{\scriptscriptstyle{\bullet}}}}*+!D{}
,"o"+(0,1.2)="dL"*{\beta}*+!D{}
,"o"+(1,1)="B"*{{{\scriptscriptstyle{\bullet}}}}*+!L{}
,"o"+(2,1)="C"*{\star}*+!L{}
,"o"+(3,1)="D"*{\star}*+!L{}
,"o"+(3,2)="E"*{\star}*+!L{}
,"o"+(-1,0)="na"*{{{\scriptscriptstyle{\times}}}}*+!D{},"o"+(0,-1)="nd"*{{{\scriptscriptstyle{\bullet}}}}*+!D{}
,"o"+(-1,-1)="nB"*{{{\scriptscriptstyle{\times}}}}*+!L{}
,"o"+(-2,-1)="nC"*{{{\scriptscriptstyle{\times}}}}*+!L{}
,"o"+(-3,-1)="nD"*{{{\scriptscriptstyle{\times}}}}*+!L{}
,"o"+(-3,-2)="nE"*{{{\scriptscriptstyle{\times}}}}*+!L{}
}
\endxy$$
The weight of $\det I$ is $$\begin{aligned}
\sum_{\alpha \in S} \alpha
&=
\left(3 \, \alpha+2 \, \beta\right)
+\left(2 \, \alpha + \beta\right) + \left(3 \, \alpha+ \beta\right)
\\
&= 8 \, \alpha + 4 \, \beta.\end{aligned}$$ So $I$ is semicanonical.
Inspect the root lattices of all simple Lie groups $G$ of rank 2, using the same approach as the previous example. You see that for all generalized flag varieties $G/P$ with $G$ of rank 2, all submodules $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ are semicanonical, except for a few counterexamples. These counterexamples only occur for those $G/P$ where $P=B$ is a Borel subgroup. Specifically $G/P={\ensuremath{\operatorname{SO}\left({5,{\ensuremath{\mathbb{C}^{}}}}\right)}}/B$ and $G/P=G_2/B$ have no nontrivial semicanonical modules $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ (i.e. other than $I=0$ and $I=\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$). On the other hand, $G/P=A_2/B={\ensuremath{\operatorname{SL}\left({3,{\ensuremath{\mathbb{C}^{}}}}\right)}}/B$ has precisely one nontrivial semicanonical submodule (as we will see in example ), and various nonsemicanonical submodules.
More generally, if $B \subset G$ is a Borel subgroup of a complex semisimple Lie group $G$, let $V={\ensuremath{\mathfrak{g}}}_{-\alpha}$ be the root space of any simple root $\alpha$. Let $I=V^{\perp} \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$, i.e. $I$ is the sum of the root spaces of all positive noncompact roots other than $\alpha$. So the weight of $\det I$ is the sum of all positive noncompact roots other than $\alpha$. Clearly $I$ is a $B$-submodule of $\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{b}}}\right)^*$. However, $I$ is not semicanonical unless $G={\ensuremath{\operatorname{SL}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}$. So there are some counterexamples in arbitrary rank.
\[proposition:MaximalGivesSemicanonical\] Suppose that $G$ is a a complex simple Lie group and $P \subset G$ a maximal parabolic subgroup. Then every submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is semicanonical.
The maximal semisimple subgroup $M \subset P$ from the Langlands decomposition (see Knapp [@Knapp:2002]) has Dynkin diagram given by removing the crossed (i.e. noncompact) simple roots from the Dynkin diagram of $G/P$. Since $P$ is maximal, there is one noncompact simple root, so the root lattice of $M$ spans a hyperplane in the root lattice of $G$. All weights of 1-dimensional $P$-modules lie in the line in the root lattice of $G$ perpendicular to the root lattice of $M$, by invariance under the Weyl group of $M$. So if $I$ is a $P$-submodule of $\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$, then $\det I$ has weight lying on this line. The weight $\omega$ of $\det \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is the sum of the noncompact positive roots, so is a nonzero vector on this line. Therefore $\det I$ must have weight a multiple of $\kappa$. We need to show that this multiple is not negative. This is clear because the weight is a sum of positive noncompact roots.
\[example:An\] The generalized flag variety $G/P$ with $G=A_n$ and Dynkin diagram
0;/r.20pc/: (10,10)\*+=“2”; (20,10)\*+=“3”; (30,10)\*+=“4”; (40,10)\*+[…]{}=“5”; (50,10)\*+=“6”; “2”; “3” \*\*; “3”; “4” \*\*; “4”; “5” \*\*; “5”; “6” \*\*;
represents the space of pairs $(p,L)$ where $L$ is a projective line in ${\ensuremath{\mathbb{ P}^{n}}}$ and $p \in L$ is a point of that line. Map $G/P \to {\ensuremath{\mathbb{ P}^{n}}}$ by $(p,L) \to p$. There is an obvious Frobenius Pfaffian system $\mathcal{I}_{\text{point}}$ on $G/P$ consisting of the 1-forms vanishing on the fibers of this map. Similarly there a map $G/P \to {\ensuremath{\mathbb{ P}^{n*}}}$, $(p,L) \mapsto L$, and an obvious Frobenius Pfaffian system $\mathcal{I}_{\text{line}}$ on $G/P$ consisting of the 1-forms vanishing on the fibers of this map. Let $\mathcal{I}_0 = \mathcal{I}_{\text{point}} \cap \mathcal{I}_{\text{line}}$.
As usual, $A_n$ has roots $e_i-e_j \in {\ensuremath{\mathbb{R}^{n+1}}}$ for $i \ne j$. A basis of positive simple roots is $\alpha_i = e_i - e_{i+1}$, $1 \le i \le n$. The compact roots are $e_i-e_j$ for $i,j \ge 3$ with $i \ne j$. The noncompact positive roots are $e_1-e_i$ for $i>1$ and $e_2-e_i$ for $i>2$. Write $\alpha \le \beta$ to mean that $\beta-\alpha$ is a sum of positive noncompact roots and compact roots.
For each positive root $\alpha$, let $$I_{\alpha} = \bigoplus_{\alpha \le \beta} {\ensuremath{\mathfrak{g}}}_{\beta}.$$ Note that $I_{\alpha} \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is a $P$-submodule. There are precisely 5 distinct $P$-submodules of $\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$:
1. $0$,
2. $I_{\text{point}} = I_{\alpha_1}$, $\dim_{{\ensuremath{\mathbb{C}^{}}}} I_{\text{point}} = n$,
3. $I_{\text{line}} = I_{\alpha_2}$, $\dim_{{\ensuremath{\mathbb{C}^{}}}} I_{\text{line}} = 2n-2$,
4. $I_0 = I_{\alpha_1+\alpha_2}$, $\dim_{{\ensuremath{\mathbb{C}^{}}}} I_0 = n-1$, and
5. $\left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$.
The associated vector bundles on $G/P$ are the Pfaffian systems defined above.
As above let $\omega$ be the sum of the positive noncompact roots, $$\omega = n \, \alpha_1
+ 2(n-1) \, \alpha_2
+ 2(n-2) \, \alpha_3
+
\dots
+ 2 \, \alpha_n,$$ while the weights for the various submodules are $$\begin{aligned}
\det I_{\alpha_1}
&:
n \, \alpha_1
+ (n-1) \, \alpha_2
+ (n-2) \, \alpha_3
+ \dots
+ \alpha_n, \\
\det I_{\alpha_2}
&:
(n-1) \, \alpha_1
+ 2(n-1) \, \alpha_2
+ 2(n-2) \, \alpha_3
+ \dots
+ 2 \, \alpha_n, \\
\det I_0
&:
(n-1) \, \alpha_1
+ (n-1) \, \alpha_2
+ (n-2) \, \alpha_3
+ \dots
+ \alpha_n.\end{aligned}$$ So $I_0$ is semicanonical precisely when $n=2$, while $I_{\text{point}}$ and $I_{\text{line}}$ are not semicanonical for any $n$.
Rational curves on smooth complex projective varieties
======================================================
A complex projective variety is *uniruled* if every point lies on a rational curve; see [@Kollar:1996].
\[theorem:Boucksom\] A smooth complex projective variety is uniruled just when the variety has nonpseudoeffective canonical bundle.
\[corollary:ContainsRationalCurve\] Suppose that $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is a semicanonical module. Suppose that $M$ is a smooth complex projective variety with a holomorphic Cartan geometry $E \to M$ modelled on $G/H$. If $E \times_H I$ is not Frobenius then $M$ contains a rational curve.
Dropping
========
\[Biswas, McKay [@Biswas/McKay:2010]\]\[thm:CurvesForceDescent\] Suppose that
1. $G/H$ is a complex homogeneous space,
2. $M$ is a connected compact K[ä]{}hler manifold and
3. $M$ bears a holomorphic $G/H$-geometry.
Then the geometry drops to a unique $G/H'$-geometry on a connected compact K[ä]{}hler manifold $M'$, so that
1. $H' \subset G$ is a closed complex subgroup,
2. $H'/H$ is a generalized flag variety,
3. $M \to M'$ is a holomorphic $H/H'$-bundle, and
4. the manifold $M'$ contains no rational curves.
Any other drop $M \to M''$ for which $M''$ contains no rational curves factors uniquely through holomorphic drops $M \to M' \to M''$.
\[theorem:ContracanonicalDrop\] Suppose that $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{h}}}\right)^*$ is a semicanonical module. Suppose that $M$ is a smooth connected complex projective variety with a holomorphic Cartan geometry $E \to M$ modelled on $G/H$. Suppose that $E \times_H I$ is not Frobenius.
Then the geometry drops to a unique $G/H'$-geometry on a connected smooth complex projective variety $M'$, so that
1. $H' \subset G$ is a closed complex subgroup,
2. $H'/H$ is a generalized flag variety,
3. $\dim_{{\ensuremath{\mathbb{C}^{}}}} H' > \dim_{{\ensuremath{\mathbb{C}^{}}}} H$, i.e. $\dim_{{\ensuremath{\mathbb{C}^{}}}} M' < \dim_{{\ensuremath{\mathbb{C}^{}}}} M$.
4. $M \to M'$ is a holomorphic $H/H'$-bundle, and
5. the manifold $M'$ contains no rational curves.
Any other drop $M \to M''$ for which $M''$ contains no rational curves factors uniquely through holomorphic drops $M \to M' \to M''$.
In particular, if there is no closed proper complex Lie subgroup $H' \subset G$ with $H \subset H'$ and $H'/H$ a rational homogeneous variety, then $M=G/H$ with its standard flat Cartan geometry, and $G/H$ is a rational homogeneous variety.
The manifold $M$ contains a rational curve, by corollary . By theorem , the geometry drops. If $H$ is not contained in a closed complex Lie subgroup $H' \subset G$ for which $H'/H$ is a rational homogeneous variety, then the geometry can only drop to a geometry modelled on $G/G$, a point. The original geometry on $M$ must be isomorphic to the lift of $G/G$, i.e. must be isomorphic to $G/H$.
A *parabolic geometry* is a holomorphic Cartan geometry modelled on a generalized flag variety.
Let’s develop a general criterion to ensure that a Pfaffian system $E \times_P I$ in a parabolic geometry cannot be Frobenius. There is a well known notion of regularity of parabolic geometries (see Calderbank and Diemer [@Calderbank/Diemer:2001], Čap [@Cap:2006]). Čap [@Cap:2006] (unnumbered proposition on page 9) proves that if a parabolic geometry $E \to M$ is regular at a point of $M$, and if $G \times_P I$ is not Frobenius on $G/P$, then $E \times_P I$ is also not Frobenius on $M$. We need to see when $G \times_P I$ is Frobenius. It is easy to see that if $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ is nontrivial and semicanonical, then $G \times_P I$ is not Frobenius. Therefore if $I$ is nontrivial and semicanonical, and $E \to M$ is regular at a single point of $M$, then $E \times_P I$ is not Frobenius. We will not need to make use of this regularity criterion in our examples.
Example: adjoint varieties
==========================
\[example:adjoint\] Suppose that $G$ is a complex semisimple Lie group. Pick a highest weight vector $x \in {\ensuremath{\mathfrak{g}}}$, for some choice of Cartan subalgebra of $G$ and basis of simple roots. The *adjoint variety* of $G$ is the orbit $X=G[x] \subset {\ensuremath{\mathbb{ P}^{}}}{\ensuremath{\mathfrak{g}}}$ of the line $[x]$ spanned by $x$ in ${\ensuremath{\mathfrak{g}}}$. The stabilizer of $[x]$ in $G$ is a parabolic subgroup, say $P \subset G$ and $X=G/P$. For example, if $G$ is simple, the adjoint varieties have Dynkin diagrams as in figure .
$$\begin{array}{llcl}
\textbf{Group} & \textbf{Variety} & \textbf{dim} & \textbf{Diagram} \\
A_n & {\ensuremath{\mathbb{ P}^{}}}T^*{\ensuremath{\mathbb{ P}^{n}}} & 2n-1 &
{\scriptscriptstyle{ \begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\times}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="2";
(20,0)*+{\dots}="3";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="4";
(40,0)*+{{{\scriptscriptstyle{\times}}}}="5";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"4"; "5" **\dir{-};
\end{xy} }} \\
B_n & {\ensuremath{\operatorname{Gr}_{\operatorname{null}}\left({2},{2n+1}\right)}} & 4n-5 &
{\scriptscriptstyle{ \begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\times}}}}="2";
(20,0)*+{{{\scriptscriptstyle{\bullet}}}}="3";
(20,0)*+{\dots}="4";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="5";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="6";
(50,0)*+{{{\scriptscriptstyle{\bullet}}}}="7";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"4"; "5" **\dir{-};
"5"; "6" **\dir{-};
{\ar@{=>}; "6";"7"};
\end{xy} }} \\
C_n & {\ensuremath{\mathbb{ P}^{2n-1}}} & 2n-1 &
{\scriptscriptstyle{
\begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\times}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="2";
(20,0)*+{\dots}="3";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="4";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="5";
(50,0)*+{{{\scriptscriptstyle{\bullet}}}}="6";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"4"; "5" **\dir{-};
{\ar@{=>}; "6";"5"};
\end{xy} }} \\
D_n & {\ensuremath{\operatorname{Gr}_{\operatorname{null}}\left({2},{2n}\right)}} & 4n-7 &
{\scriptscriptstyle{
\begin{xy}
0;/r.20pc/: (-20,0)*+{{{\scriptscriptstyle{\bullet}}}}="1"; (-10,0)*+{{{\scriptscriptstyle{\times}}}}="2"; (0,0)*+{\dots}="3";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="4"; (20,0)*+{{{\scriptscriptstyle{\bullet}}}}="5"; (25,9)*+{{{\scriptscriptstyle{\bullet}}}}="6";
(25,-9)*+{{{\scriptscriptstyle{\bullet}}}}="7";
"1"; "2" **\dir{-}; "2";
"3" **\dir{-}; "3"; "4" **\dir{-}; "4"; "5" **\dir{-}; "5"; "6"
**\dir{-}; "5"; "7" **\dir{-};
\end{xy}}} \\
E_6 & X^{\operatorname{ad}}_{E_6} & 21 & {\scriptscriptstyle{
\begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="2";
(20,0)*+{{{\scriptscriptstyle{\bullet}}}}="3";
(20,10)*+{{{\scriptscriptstyle{\times}}}}="4";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="5";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="6";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"3"; "5" **\dir{-};
"5"; "6" **\dir{-};
\end{xy}
}} \\
E_7 & X^{\operatorname{ad}}_{E_7} & 33 & {\scriptscriptstyle{
\begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\times}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="2";
(20,0)*+{{{\scriptscriptstyle{\bullet}}}}="3";
(20,10)*+{{{\scriptscriptstyle{\bullet}}}}="4";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="5";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="6";
(50,0)*+{{{\scriptscriptstyle{\bullet}}}}="7";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"3"; "5" **\dir{-};
"5"; "6" **\dir{-};
"6"; "7" **\dir{-};
\end{xy}
}} \\
E_8 & X^{\operatorname{ad}}_{E_8} & 57 & {\scriptscriptstyle{
\begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="2";
(20,0)*+{{{\scriptscriptstyle{\bullet}}}}="3";
(20,10)*+{{{\scriptscriptstyle{\bullet}}}}="4";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="5";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="6";
(50,0)*+{{{\scriptscriptstyle{\bullet}}}}="7";
(60,0)*+{{{\scriptscriptstyle{\times}}}}="8";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"3"; "5" **\dir{-};
"5"; "6" **\dir{-};
"6"; "7" **\dir{-};
"7"; "8" **\dir{-};
\end{xy}
}} \\
F_4 & X^{\operatorname{ad}}_{F_4} & 15 & {\scriptscriptstyle{
\xy
0;/r.10pc/: (10,0)*+{{{\scriptscriptstyle{\times}}}}="0";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(40,0)*+{>}="2";
(50,0)*+{{{\scriptscriptstyle{\bullet}}}}="3";
(70,0)*+{{{\scriptscriptstyle{\bullet}}}}="4";
"0"; "1" **\dir{-};
"1"; "3" **\dir2{-};
"3"; "4" **\dir{-};
\endxy
}} \\
G_2 & {\ensuremath{\operatorname{Gr}_{\operatorname{null}}\left({2},{\operatorname{Im} \mathbb{O}}\right)}} & 5 & {\scriptscriptstyle{
\xy
0;/r.10pc/: (30,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(40,0)*+{<}="2";
(50,0)*+{{{\scriptscriptstyle{\times}}}}="3";
"1"; "3" **\dir3{-};
\endxy
}}
\end{array}$$
If $G$ is simple, then its adjoint variety is a holomorphic contact manifold, and every homogeneous compact complex contact manifold occurs as an adjoint variety; see Landsberg [@Landsberg:2005]. There is precisely one holomorphic contact structure on any adjoint variety.
If $G$ is not simple, then up to a finite covering $G$ is a product of simple factors $$G = G_1 \times G_2 \times \dots \times G_s,$$ and correspondingly $$P = P_1 \times P_2 \times \dots \times P_s,$$ where $P_j = P \cap G_j$. For each $G_j$, we can then consider the one dimensional $P_j$-submodule $I_j \subset \left({\ensuremath{\mathfrak{g}}}_j/{\ensuremath{\mathfrak{p}}}\right)^*$ which arises from the holomorphic contact structure on $X_j=G_j/P_j$. We can then let $I=\bigoplus_j I_j$, and again $I$ is semicanonical on $X=G/P$, though not a contact structure.
Suppose that $G$ is a complex simple Lie group and that $G/P$ is an adjoint variety with holomorphic contact structure $G \times_P I$. Suppose that $E \to M$ is holomorphic parabolic geometry modelled on $G/P$, on a smooth connected complex projective variety $M$. Let $\mathcal{I} = E \times_P I \subset T^*M$. Either
1. $M$ is foliated by smooth hypersurfaces on which $\mathcal{I}=0$ or
2. $M=G/P$ with its usual adjoint variety geometry or
3. $G=A_n$, and the geometry on $M$ drops to a holomorphic projective connection on a smooth connected complex projective variety.
Either $\mathcal{I}$ is Frobenius, or the parabolic geometry drops by theorem .
The adjoint variety $X=G/P$ of $G=A_n$ is the variety of pairs of a hyperplane in ${\ensuremath{\mathbb{ P}^{n}}}$ and a point on that hyperplane. There are only two parabolic subgroups of $A_n$ containing $P$: forget the point or the hyperplane, i.e. $G/P'$ is either projective space or the dual projective space. Projective space and its dual are isomorphic, so the same parabolic geometries are modelled on either one. Suppose that $M$ is a smooth complex projective variety with a holomorphic parabolic geometry modelled on the adjoint variety of $A_n$. Then $M$ drops to a smooth complex projective variety with holomorphic projective connection.
Consider the adjoint variety $X=G/P$ of any other simple complex Lie group $G$ (i.e. $G=B_n, C_n, D_n, E_6, E_7, E_8, F_4$ or $G_2$). Then $P \subset G$ is a maximal parabolic subgroup. So there is only one regular parabolic geometry on any smooth complex projective variety modelled on that $G/P$: the model $G/P$ with its standard flat $G/P$-geometry.
We will reconsider the $A_n$-adjoint geometries in section .
Example: Cartan’s theory of 2-plane fields on 5-manifolds
=========================================================
In example , we saw that $G_2/P_1$ bears a holomorphic rank 3 Pfaffian system. We can see from the root lattice in figure that $\dim_{{\ensuremath{\mathbb{C}^{}}}} G_2/P_1 = 5$ (i.e. 5 crosses representing the 5 dimensions of ${\ensuremath{\mathfrak{g}}}_2/{\ensuremath{\mathfrak{p}}}_1$). We can also see that the rank 3 Pfaffian system is not Frobenius, because there is a pair of noncompact positive roots not among those 3 which add up to a root among those 3. The dual plane field is associated to the $P_1$-module $V=I^{\perp}$, i.e. the sum of root spaces of the two roots $-\alpha,-\alpha-\beta$. We can even see that the 2-plane field is Cartan, in the sense of example , by looking at the brackets of vector fields in ${\ensuremath{\mathfrak{g}}}_2/{\ensuremath{\mathfrak{p}}}_1$, i.e. looking at sums of the roots $-\alpha,-\alpha-\beta$. (We leave this claim to the reader to prove, since it is not essential to our arguments.)
If $\mathcal{V}$ is a Cartan 2-plane field on a 5-dimensional complex manifold $M$, then then there is a holomorphic parabolic geometry $E \to M$ modelled on $G_2/P_1$, so that $\mathcal{V} = E \times_P V \subset TM$, where $V \subset {\ensuremath{\mathfrak{g}}}_2/{\ensuremath{\mathfrak{p}}}_1$ is the $P_1$-submodule constructed in example .
The only holomorphic Cartan 2-plane field on any smooth connected complex projective variety is the standard one on $G_2/P_1$ described in example .
Suppose that $M$ is a smooth connected complex projective variety of complex dimension 5, bearing a holomorphic Cartan 2-plane field. From example , we have seen that a smooth complex projective variety with a Cartan 2-plane field must have nonpseudoeffective canonical bundle. By theorem , the variety must then be uniruled.
By Cartan’s theorem, we can assume that the Cartan 2-plane field is $E \times_P V \subset TM$. Let $I = V^{\perp}$. Since $I$ is semicanonical, and the Pfaffian system $E \times_P I$ is not Frobenius, again we see that the variety $M$ must be uniruled. By theorem , the parabolic geometry must drop to a parabolic geometry with a lower dimensional model. The parabolic geometry can only drop to a parabolic geometry modelled on a point, since $P$ is maximal, so drops just when the parabolic geometry is isomorphic to the model.
Example: 3-plane fields on 6-manifolds
======================================
A rank 3 Pfaffian system $\mathcal{I} \subset T^*M$ on a complex manifold $M$ of complex dimension 6 is *nondegenerate* if near each point of $M$ there are 3 sections of $\mathcal{I}$ with linearly independent exterior derivatives.
\[example:BryantExample\] Let $G=B_3={\ensuremath{\operatorname{PO}\left({7,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ and $G/P$ be the space of null 3-planes in ${\ensuremath{\mathbb{C}^{6}}}$ for some nondegenerate complex inner product. The Dynkin diagram of $G/P$ is
0;/r.10pc/: (0,0)\*+=“1”; (22,0)\*+=“2”; (44,0)\*+=“3”; (33,0)\*+[>]{}=“edge”; “1”; “2” \*\*; [@[=]{}; “2”;“3”]{};
. Write the simple roots of $G$ as $\alpha_1=e_1-e_2, \alpha_2=e_2-e_3, \alpha_3=e_2+e_3$ in terms of the standard basis $e_1, e_2, e_3 \in {\ensuremath{\mathbb{R}^{3}}}$. The root $\alpha_3$ will be the noncompact positive simple root. The noncompact positive roots are $$\alpha_3,
\alpha_2+\alpha_3,
\alpha_1+\alpha_2+\alpha_3,
\alpha_2 + 2 \, \alpha_3,
\alpha_1 + \alpha_2 + 2 \, \alpha_3,
\alpha_1 + 2 \, \alpha_2 + 2 \, \alpha_3.$$ Let $S$ be the set of roots $$\alpha_2 + 2 \, \alpha_3,
\alpha_1 + \alpha_2 + 2 \, \alpha_3,
\alpha_1 + 2 \, \alpha_2 + 2 \, \alpha_3$$ (i.e. the roots with $2 \, \alpha_3$ in them). Let $I=I_S \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ be the sum of the root spaces of roots in $S$; $I$ has dimension 3. We can see that $G \times_P I$ is nondegenerate in Bryant’s sense, since we can write the 3 roots in $S$ each as a sum of distinct pairs of roots not in $S$. (We again leave the reader to figure out the yoga relating exterior derivatives to root sums, since we won’t use this fact.)
\[Bryant [@Bryant:2005]\]\[theorem:Bryant\] If $\mathcal{I} \subset T^*M$ is a nondegenerate rank 3 Pfaffian system on a smooth complex projective variety $M$ with $\dim_{{\ensuremath{\mathbb{C}^{}}}} M = 6$, then there is a parabolic geometry $E \to M$ so that $\mathcal{I} = E \times_P I$.
\[theorem:BryantDrop\] Suppose that $M$ is a smooth connected complex projective variety of complex dimension 6, bearing a nondegenerate rank 3 Pfaffian system. Then $M=B_3/P$ is the model defined in example .
Suppose that $\mathcal{I} \subset T^*M$ is a nondegenerate rank 3 Pfaffian system on a 6-dimensional connected smooth complex projective variety $M$. By Bryant’s theorem, we can assume that $\mathcal{I} = E \times_P I$, for some parabolic geometry $E \to M$. Apply theorem to prove that the geometry on $M$ drops. The group $P \subset B_3$ is a maximal parabolic subgroup. Therefore $M$ must drop to a point, i.e. must be isomorphic to $B_3/P$.
Example: quaternionic contact structures {#section:QuaternionicContact}
========================================
Suppose that $M$ is a complex manifold, $\dim_{{\ensuremath{\mathbb{C}^{}}}} M = 7$ and that $\mathcal{I} \subset TM$ is a holomorphic Pfaffian system of rank 3. For any two local sections $\vartheta_0, \vartheta_1$ of $\mathcal{I}$, let $$q\left(\vartheta_0,\vartheta_1\right) =
\left.
d\vartheta_0 \wedge d\vartheta_1
\right|_{\mathcal{I}^{\perp}}.$$ It is easy to check that $q$ is a global holomorphic section of $${\ensuremath{\operatorname{Sym}^{2}\left({\mathcal{I}}\right)}}^* \otimes {\ensuremath{\Lambda^{4} \left ( {\mathcal{I}^{\perp}} \right )}}.$$ Say that $\mathcal{I}$ is *nondegenerate* if $\vartheta_0 {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}q=0$ precisely when $\vartheta_0=0$. A *quaternionic contact structure* is a nondegenerate holomorphic Pfaffian system of rank 3 on a 7-manifold.
Quaternionic contact structures are very clearly explained by Montgomery [@Montgomery:2002]. For discussion of real forms of quaternionic contact structures, see [@Biquard:2000; @Biquard:2001; @Fox:2005].
\[example:StandardQCS\] Let $X=C_3/P={\ensuremath{\operatorname{Sp}\left({6,{\ensuremath{\mathbb{C}^{}}}}\right)}}/P$ the space of subLagrangian 2-planes in ${\ensuremath{\mathbb{C}^{6}}}$, where $P$ is the stabilizer of a subLagrangian 2-plane. The Dynkin diagram of $X$ is
0;/r.10pc/: (0,0)\*+=“1”; (22,0)\*+=“2”; (44,0)\*+=“3”; (33,0)\*+[<]{}=“edge”; “1”; “2” \*\*; [@[=]{}; “2”;“3”]{};
There is a $C_3$-invariant quaternionic contact structure on $X$ defined as follows.
We can write the roots of $C_3$ as vertices and the middles of edges of an octahedron, say as $\pm e_i \pm e_j$ for $1 \le i,j \le 3$. The positive simple roots are $$\alpha_1 = e_1 - e_2, \alpha_2 = e_2 - e_3, \alpha_3 = 2 e_3.$$ The positive noncompact roots of $X$ are $$\alpha_2, \, \alpha_1+\alpha_2, \, \alpha_2+\alpha_3, \, \alpha_1 + \alpha_2 + \alpha_3, \,
2 \alpha_2 + \alpha_3, \, \alpha_1 + 2 \, \alpha_2 + \alpha_3, \,
2 \, \alpha_1 + 2 \, \alpha_2 + \alpha_3.$$ Consider the 3 roots $$2 \alpha_2 + \alpha_3 , \, \alpha_1 + 2 \, \alpha_2 + \alpha_3, \,
2 \, \alpha_1 + 2 \, \alpha_2 + \alpha_3,$$ i.e. those with $2 \alpha_2$ in them. Let $I \subset {\ensuremath{\mathfrak{p}}}$ be the sum of the root spaces of those 3 roots. Consider the Pfaffian system $\mathcal{I} = C_3 \times_P I \subset T^*\left(C_3/P\right)$. One can directly calculate using the structure equations of $C_3$ that $\mathcal{I}$ is a quaternionic contact structure. (Once more we leave this local calculation to the reader, since we don’t need this result.)
\[Montgometry [@Montgomery:2002]\] If $\mathcal{I}$ is a quaternionic contact structure on a complex manifold $M$, then there is a holomorphic parabolic geometry $E \to M$ modelled on $X=C_3/P$, so that $E \times_P I = \mathcal{I}$, for $I = \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ the semicanonical $P$-submodule defined in example .
We now prove theorem .
Suppose that $\mathcal{I}$ is a quaternionic contact structure on $M$. By Montgomery’s theorem, we can assume that $\mathcal{I} = E \times_P I$ for some holomorphic parabolic geometry $E \to M$. Apply theorem to prove that the parabolic geometry on $M$ drops. Since $P \subset C_3$ is a maximal parabolic subgroup, $M$ must drop to a point, i.e. must be isomorphic to $X$.
Example: Čap–Neusser Pfaffian systems
=====================================
\[example:CapNeusserTwo\] For $n \ge 3$, let $G=B_n={\ensuremath{\operatorname{PO}\left({2n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}$, Let $X=G/P$ be the set of all $n$-dimensional null subspaces of the standard complex linear inner product on ${\ensuremath{\mathbb{C}^{2n+1}}}$. The Dynkin diagram of $X$ is $${\scriptstyle{
\begin{xy}
0;/r.20pc/: (0,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="2";
(20,0)*+{\dots}="3";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="4";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="5";
(50,0)*+{{{\scriptscriptstyle{\times}}}}="6";
"1"; "2" **\dir{-};
"2"; "3" **\dir{-};
"3"; "4" **\dir{-};
"4"; "5" **\dir{-};
{\ar@{=>}; "5";"6"};
\end{xy}
}}$$ If we order the positive roots according to the coefficient of $\alpha_n$, there are precisely $n$ positive noncompact roots $\alpha$ with coefficient 1 and precisely $n(n-1)/2$ positive noncompact roots $\alpha$ with coefficient 2. Let $S$ be the set of noncompact positiive roots of coefficient 2. Let $I=I_S$ be the sum of the root spaces of these roots, so $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$. Then $I$ turns out to be a first order nondegenerate Pfaffian system in the sense of example . (Again we leave this statement for the reader to prove.)
\[Čap and Neusser [@Cap/Neusser:2009]\]\[theorem:CapNeusser\] Suppose that $n \ge 3$. Suppose that $M$ is a complex manifold with $\dim_{{\ensuremath{\mathbb{C}^{}}}} M = n(n+1)/2$. Suppose that $\mathcal{I} \subset T^*M$ is a holomorphic first order nondegenerate Pfaffian system. Then there is a holomorphic parabolic geometry $E \to M$ so that $E \times_P I=\mathcal{I}$, where $I$ is the semicanonical $P$-module defined in example .
\[theorem:CapNeusserDrop\] Suppose that $M$ is a smooth connected complex projective variety with $\dim_{{\ensuremath{\mathbb{C}^{}}}} M \ge 6$ bearing a holomorphic first order nondegenerate Pfaffian system. Then $M=B_n/P$ with its standard first order nondegenerate Pfaffian system as defined in example .
Suppose that $\mathcal{I}$ is a first order nondegenerate Pfaffian system on $M$. By theorem of Čap and Neusser, we can assume that $\mathcal{I} = E \times_P I$ for some holomorphic parabolic geometry $E \to M$ modelled on $B_n/P$. Apply theorem to prove that the geometry on $M$ drops. Since the model $P \subset B_n$ is a maximal parabolic subgroup, $M$ must drop to a point, i.e. must be isomorphic to $X$.
Theorem is the special case of theorem for $\dim_{{\ensuremath{\mathbb{C}^{}}}} M = 6$.
Example: parabolic geometries modelled on products
==================================================
Suppose that $E \to M$ is a holomorphic parabolic geometry on a smooth complex projective variety $M$, modelled on a generalized flag variety $G/P$. Suppose that $G$ splits into a product of simple complex Lie groups, $$G = G_1 \times G_2 \times \dots \times G_s.$$ Let $P_j = P \cap G_j$ for each $j$. Suppose that $P_j \subset G_j$ is maximal for each $j$. Suppose that each $G_j/P_j$ has a nontrivial semicanonical $P_j$-submodule $I_j \subset \left({\ensuremath{\mathfrak{g}}}_j/{\ensuremath{\mathfrak{p}}}_j\right)^*$. Let $I=I_1 \oplus I_2 \oplus \dots \oplus I_s$. Suppose that every $E \times_P I_j$ is not Frobenius. Then $M=G/P$ with its standard flat parabolic geometry.
Theorem \[theorem:ContracanonicalDrop\] ensures that the parabolic geometry drops, say to a geometry with some model $G/Q$, $P \subset Q \subset G$ on some complex manifold $M'$. Since each $P_j \subset G_j$ is maximal, the group $Q$ must be obtained by setting $Q_j=P_j$ or $Q_j=G_j$ for each value of $j$, and then $Q = Q_1 \times Q_2 \times \dots \times Q_s$. If $P_j \ne Q_j$, then $E \times_Q I_j$ is not Frobenius, since its local sections pull back to local sections of $E \times_Q I_j$. Therefore $Q=G$, and therefore $M'$ is a point, and so $M$ must be isomorphic to the model, i.e. $M=G/P$.
Example: double Legendre foliations {#section:DoubleLegendreFoliations}
===================================
We arrive at our most complicated example. We will study parabolic geometries modelled on the adjoint variety of $A_n$, but we will not obtain a complete classification.
\[example:AnAdjointAsDLF\] Consider the adjoint variety of $A_n={\ensuremath{\operatorname{SL}\left({n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}$, say $X=G/P$, $G=A_n$. Let’s first find all of the $P$-submodules $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$. Write the positive roots of $A_n$ as $\alpha_i + \alpha_{i+1} + \dots + \alpha_j$ for $i \le j$. Let $S_1$ be the set of roots $$\alpha_1, \alpha_1 + \alpha_2, \dots,
\alpha_1 + \alpha_2 + \dots + \alpha_n$$ Let $S_n$ be the set of roots $$\alpha_1 + \alpha_3 + \dots + \alpha_n,
\alpha_2 + \alpha_4 + \dots + \alpha_n,
\dots,
\alpha_n.$$ Let $I_1 \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ be the sum of all root spaces of positive noncompact roots $\alpha$ for which $\alpha \in S_1$, and similarly let $I_n \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$ be the sum of all root spaces of positive noncompact roots $\alpha$ for which $\alpha \in S_n$. It is clear that no two noncompact positive roots can add up to a root in $S_1$, and similarly for $S_n$. It turns out to follow that $G \times_P I_1$ and $G \times_P I_n$ are Frobenius. (We leave the reader to figure out the yoga relating exterior derivatives to root sums, since we will only make use of it in examples where the claims made are an elementary calculation using Cartan’s structure equations.) By a similar argument, if we let $I_{1n} = I_1 \cap I_n$, then $G \times_P I_{1n}$ is a contact structure. Indeed $I_{1n}$ corresponds to the set $S_{1n}=S_1 \cap S_n$, which is just $$\alpha_1 + \alpha_2 + \dots + \alpha_n,$$ which is a sum of noncompact positive roots $$\left(\alpha_1 + \alpha_2 + \dots + \alpha_{j-1}\right)
+
\left(\alpha_j + \alpha_{j+1} + \dots + \alpha_n\right),$$ corresponding to the exterior derivative of the contact form being a sum of the wedge products of various 2-forms. The leaves of $G \times_P I_1$ are the fibers of $G/P \to {\ensuremath{\mathbb{ P}^{n}}}$, while the leaves of $G \times_P I_n$ are the fibers of $G/P \to {\ensuremath{\mathbb{ P}^{n*}}}$. In particular, $G/P$ has two foliations (indeed fiber bundle mappings), with leaves integral manifolds of the contact structure.
\[Tabachnikov [@Tabachnikov:1993]\]A *double Legendre foliation* [@Tabachnikov:1993] of a complex manifold $M$ of complex dimension $2n-1$ is a pair $F_0,F_1 \subset TM$ of holomorphic foliations so that $F_0 \oplus F_1 \subset TM$ is a holomorphic contact structure, and the leaves of $F_0$ and of $F_1$ are Legendre submanifolds.
The $A_n$-adjoint variety has a holomorphic double Legendre foliation.
\[Tabachnikov [@Tabachnikov:1993]\]\[theorem:Tabachnikov\] Suppose that $F_0, F_1$ is a holomorphic double Legendre foliation of a complex manifold $M$ of complex dimension $2n-1$. Then there is a holomorphic parabolic geometry $E \to M$ modelled on the adjoint variety of $A_n={\ensuremath{\operatorname{SL}\left({n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ so that (in the notation of example ) $F_0 = E \times_P I_1^{\perp}$ and $F_1 = E \times_P I_n^{\perp}$.
\[example:AdjointAn\] We will construct a parabolic geometry modelled on the adjoint variety of $A_n$ by lifting a holomorphic projective connection.
Write points of ${\ensuremath{\mathbb{C}^{n+1}}}$ as columns, spanned by the standard basis $e_0, e_1, \dots, e_n$. Clearly ${\ensuremath{\mathbb{ P}^{n}}}={\ensuremath{\operatorname{PSL}\left({n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}/P_1$ where $P_1 \subset G={\ensuremath{\operatorname{PSL}\left({n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ is the subgroup of matrices of the form $$\begin{bmatrix}
p^0_0 & p^0_j \\
0 & p^i_j
\end{bmatrix},$$ $1 \le i,j \le n$. and we identify an element of ${\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}_1$ with a column in ${\ensuremath{\mathbb{C}^{n}}}$ by writing out the entries $$\begin{pmatrix}
A^1_0 \\
A^2_0 \\
\vdots \\
A^n_0
\end{pmatrix}
\in {\ensuremath{\mathbb{C}^{n}}}.$$
Let $P_n \subset G={\ensuremath{\operatorname{PSL}\left({n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ be the subgroup of matrices of the form $$\begin{bmatrix}
p^0_0 & p^0_j & p^0_n \\
p^i_0 & p^i_j & p^i_n \\
0 & 0 & p^n_n
\end{bmatrix},$$ $1 \le i,j \le n-1$. Let $P =P_1 \cap P_n \subset G={\ensuremath{\operatorname{PSL}\left({n+1,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ be the subgroup of matrices of the form $$\begin{bmatrix}
p^0_0 & p^0_j & p^0_n \\
0 & p^i_j & p^i_n \\
0 & 0 & p^n_n
\end{bmatrix},$$ $1 \le i,j \le n-1$.
Suppose that $M'$ is a complex manifold of complex dimension $n$ bearing a holomorphic projective connection, i.e. a holomorphic parabolic geometry $\pi : E \to M'$ modelled on ${\ensuremath{\mathbb{ P}^{n}}}$. We will construct the lift of $M'$ to a parabolic geometry modelled on the adjoint variety of $A_n$. Let $M={\ensuremath{\mathbb{ P}^{}}}T^*M'$, with its usual holomorphic contact structure. Write points of $M$ as $m=\left(m',H\right)$ where $m' \in M'$ and $H \subset T_{m'} M'$ is a complex hyperplane. Map $E \to M$ by $$e \in E \mapsto m=\left(m',H\right) \in M,$$ taking $H$ to be the hyperplane identified by the Cartan connection $\omega$ with the span of $e_1, e_2, \dots, e_{n-1} \in {\ensuremath{\mathbb{C}^{n}}}={\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}_1$. Because $\omega$ transforms in the adjoint $P$-representation, i.e. $$r_p^* \omega = \operatorname{Ad}(p)^{-1} \omega,$$ we can easily check that $M=E/P$. Therefore $M={\ensuremath{\mathbb{ P}^{}}}T^*M'$ is the lift of $M'$, and $E \to M$ is a parabolic geometry modelled on the adjoint variety of $A_n$.
Consider on $E$ the following two linear Pfaffian systems: let $\mathcal{I}_0 \subset T^* E$ be the system $$\omega+{\ensuremath{\mathfrak{p}}}_1 = 0,$$ and let $\mathcal{I}_1 \subset T^*E$ be the system $$\omega + {\ensuremath{\mathfrak{p}}}_n = 0.$$ The fibers of $E \to M$ are Cauchy characteristics for each of these systems, and both systems are $P$-invariant. Therefore these Pfaffian systems are pulled back from Pfaffian systems, which we denote by the same names, on $M$; see [@BCGGG:1991]. Clearly on $M$, $\mathcal{I}_0 = E \times_P I_0$ and $\mathcal{I}_1 = E \times_P I_1$. Let $F_0=\mathcal{I}_0^{\perp}$ and $F_1=\mathcal{I}_1^{\perp}$.
This holomorphic vector subbundle $F_1 \subset TM$ might not be a foliation. Clearly $F_0$ is a foliation. But $F_1$ is a foliation if and only if the projective connection on $M'$ satisfies a certain complicated condition on its curvature, ensuring the existence of a suitably large family of totally geodesic hypersurfaces, a local calculation which I leave to the reader.
On the other hand, if the projective connection on $M'$ has “enough” totally geodesic hypersurfaces, so that $F_1$ is a foliation, then each leaf of $F_1$ projects to a immersed complex hypersurface in $M'$, so that for every linear hyperplane $H \in {\ensuremath{\mathbb{ P}^{}}}T^*M'=M$, there is a unique such complex hypersurface with tangent space $H$.
Suppose that $M'$ is a complex manifold with holomorphic normal projective connection (see Kobayashi and Nagano [@Kobayashi/Nagano:1964] for the definition of normal). We leave the reader to check that a projective connection has “enough” totally geodesic hypersurfaces (i.e. one through each point with each possible tangent hyperplane, i.e. $F_1$ is a foliation) if and only if the projective connection is flat.
Suppose that $M$ is a smooth complex projective variety bearing a holomorphic double Legendre foliation. Then $M={\ensuremath{\mathbb{ P}^{}}}T^* M'$ for some smooth complex projective variety $M'$, and $M$ is the lift of a holomorphic projective connection on $M'$.
This theorem reduces the classification of double Legendre foliations on smooth complex projective varieties to that of holomorphic projective connections satisfying the required curvature condition to have “enough” totally geodesic hyperplanes.
By theorem , for any double Legendre foliation, say with contact structure $\mathcal{I} \subset T^*M$, there is a parabolic geometry $E \to M$ so that $\mathcal{I} = E \times_P I$ for some $P$-module $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}\right)^*$. Since $P \subset G$ is a maximal parabolic subgroup, by proposition , $I$ is a semicanonical $P$-module. Therefore the parabolic geometry on $M$ has a semicanonical module whose associated Pfaffian system is not Frobenius. Apply theorem to see that the geometry must drop. Unless $M$ is isomorphic to the model geometry, there is only one space $A_n/P'$ that it can drop to, since there is only one parabolic subgroup $P ' \subset A_n$ containing $P$, so $A_n/P'={\ensuremath{\mathbb{ P}^{n}}}$ a projective connection on some $M'$. (To be precise, there are actually two such subgroups, but there is only one up to outer automorphism.) So $M$ is a lift of a projective connection on $M'$. The lift of any projective connection to a parabolic geometry modelled on the adjoint variety of $A_n$ is given in detail in example , and must be $M={\ensuremath{\mathbb{ P}^{}}}T^* M'$. Since the subbundle $F_1 \subset TM$ is a foliation, the projective connection must have “enough hypersurfaces”.
Circles in parabolic geometries
===============================
So far we have one method to force dropping of Cartan geometries: semicanonical modules. Next we will describe a different method to force dropping, instead of using semicanonical modules. The method of semicanonical modules works particularly well on parabolic geometries modelled on $G/P$ with $P \subset G$ a maximal parabolic subgroup. Our new method, the method of rational circles, will work only on the opposite extreme: geometries modelled on $G/B$. All parabolic geometries, with any model $G/P$, lift to geometries modelled on $G/B$, so it is natural to focus on the $G/B$-geometries.
Suppose that $\alpha$ is a root of a complex semisimple Lie group $G$. Let ${{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}}$ be the Lie subalgebra of ${\ensuremath{\mathfrak{g}}}$ generated by the root spaces of $\alpha$ and $-\alpha$.
Any root of any complex semisimple Lie algebra is reduced, so ${{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}}$ is isomorphic to ${\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ [@Serre:2001].
Suppose that $E \to M$ is a holomorphic parabolic geometry, with Cartan connection $\omega$, modelled on a generalized flag variety $G/P$. Suppose that $\alpha$ is a positive simple root of $G/P$. Define a Pfaffian system on $E$ by the equation $$\omega = 0 \pmod{{{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}}}$$ on tangent vectors. This Pfaffian system has the fibers of $E \to E/B$ as Cauchy characteristics, and is $B$-invariant, and therefore descends to a unique Pfaffian system on $E/B$. Call the maximal integral Riemann surfaces of this system on $E/B$ *$\alpha$-circles*, or just *circles*.
There is some danger of confusion here, since $\omega$ is not actually defined on $E/B$, and since the $\alpha$-circles of $M$ are complex 1-dimensional submanifolds of $E/B$, not of $M$.
Suppose that $G/P$ is a generalized flag variety and that $G$ splits into a product $$G = G_1 \times G_2 \times \dots \times G_s$$ of simple complex Lie groups. Then the Borel subgroup $B \subset G$ has the form $$B = B_1 \times B_2 \times \dots \times B_s,$$ where $B_i = B \cap G_i$. Let $$B'_i = P_1 \times P_2 \times P_{i-1} \times B_i \times P_{i+1} \times \dots \times P_s.$$
If $\alpha$ is a positive simple root of ${\ensuremath{\mathfrak{g}}}_i$, then we can define the circles by the equation $$\omega = 0 \pmod{{{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}}}$$ on $E$ as above, but we find that in fact the fibers of $E \to E/B'_i$ are Cauchy characteristics for this linear Pfaffian system. So in fact, we can define the circles as Riemann surfaces on the various $E/B'_i$. For our purposes in this paper, this observation has no significance, but it should save computation in examples.
In the model, $G/P$, all circles are rational.
In the model $G/P$, with the standard model $G/P$-geometry, the $\alpha$-circles are precisely the orbits in $G/B$ of the connected subgroup ${\ensuremath{\operatorname{SL}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha} \subset G$ whose Lie algebra is ${{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}}$. Note that ${{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}} \cap {\ensuremath{\mathfrak{b}}}\subset {{\ensuremath{\mathfrak{sl}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}}$ is a Borel subalgebra. So the associated connected Lie subgroup ${\ensuremath{\operatorname{SL}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}_{\alpha}$ (which is either isomorphic to ${\ensuremath{\operatorname{SL}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}$ or to ${\ensuremath{\operatorname{PSL}\left({2,{\ensuremath{\mathbb{C}^{}}}}\right)}}$) must act on the orbit as a complex semisimple Lie group acting on a generalized flag variety. The orbit has one complex dimension. Therefore the orbit is a rational curve. So in the model, all circles are rational.
Take $G/P={\ensuremath{\mathbb{ P}^{2}}}$ and $G={\ensuremath{\operatorname{PSL}\left({3,{\ensuremath{\mathbb{C}^{}}}}\right)}}$. There is one noncompact positive simple root $\alpha$. Note that $G/B={\ensuremath{\mathbb{ P}^{}}}T{\ensuremath{\mathbb{ P}^{2}}}$. In fact $\alpha$-circles are precisely the lifts of complex projective lines in ${\ensuremath{\mathbb{ P}^{2}}}$ to the projectivized tangent bundle. We lift each line $L={\ensuremath{\mathbb{ P}^{1}}} \subset {\ensuremath{\mathbb{ P}^{2}}}$ by taking each point $p \in L$ to $T_p L \in {\ensuremath{\mathbb{ P}^{}}}T{\ensuremath{\mathbb{ P}^{2}}}=G/B$. Clearly ${\ensuremath{\mathbb{ P}^{}}}T{\ensuremath{\mathbb{ P}^{2}}}$ is foliated by the lifts of lines.
For each positive simple root $\alpha$ of $G/P$, the $\alpha$-circles foliate $E/B$. They are not actually defined inside $M$, although each $\alpha$-circle projects via a local biholomorphism to a Riemann surface in $M$. Therefore it is natural to picture the $\alpha$-circles as (not necessarily compact) curves in $M$.
Another description of the $\alpha$-circles: they are the leaves of the foliation $E \times_B {\ensuremath{\mathfrak{g}}}_{-\alpha} \subset T(E/B) = E \times_B \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{b}}}\right)$.
\[theorem:Brunella\] Suppose that $M$ is a compact Kähler manifold. Suppose that $F \subset TM$ is a holomorphic foliation by (not necessarily compact) curves, i.e. a rank 1 subbundle. Either (1) all of the leaves of $F$ are rational curves and $F^*$ is not pseudoeffective or (2) none of the leaves of $F$ are rational and $F^*$ is pseudoeffective.
Brunella’s theorem concerns holomorphic foliations with singularities, but we will only consider nowhere singular foliations, so case (2) above follows from Brunella’s remarks [@Brunella:2008] p. 55.
\[proposition:CircleDrops\] Suppose that $M$ is a compact K[ä]{}hler manifold bearing a holomorphic parabolic geometry modelled on a generalized flag variety $G/P$. Suppose that $B \subset P$ is a Borel subgroup.
Draw the Dynkin diagram of $P$, but then change any cross (say corresponding to some noncompact simple root $\alpha$) to a dot if the $\alpha$-circles are rational. In other words, change a cross to a dot just when the line bundle $E \times_B {\ensuremath{\mathfrak{g}}}_{\alpha}$ on $E/B$ is not pseudoeffective. Let $Q$ be the parabolic subgroup of $G$ whose Dynkin diagram we have just drawn.
Suppose $P' \subset G$ is a parabolic subgroup containing $P$. Then $M$ drops $M \to M'$ to a holomorphic parabolic geometry modelled on $G/P'$ if and only if $Q \subset P'$.
By theorem , these line bundles are pseudoeffective if and only if the $\alpha$-circles are rational.
If $M$ drops to $M \to M'$, then the $\alpha$-circles lie inside the fibers of $M \to M'$, i.e. inside generalized flag varieties $P'/P$. In these generalized flag varieties, the induced $P'/P$-geometry is the model geometry, and the $\alpha$-circles are therefore rational curves.
Conversely if the $\alpha$-circles are rational curves, theorem ensures that there is a drop $M \to M'$ so that all of the $\alpha$-circles lie in the fibers of $M \to M'$. The fibers are $P'/P$, some parabolic subgroup $P' \subset G$. For $P'/P$ to contain all of the $\alpha$-circles, $P'$ must have $\alpha$ as a compact root.
\[theorem:BorelBracketClosed\] Suppose that
1. $G$ is a complex semisimple Lie group with Borel subgroup $B \subset G$ and
2. $M$ is a compact K[ä]{}hler manifold and
3. $E \to M$ is a holomorphic parabolic geometry modelled on $G/B$.
Then either (1) this parabolic geometry drops to some lower dimensional holomorphic parabolic geometry on a compact K[ä]{}hler manifold or (2) for every $B$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{b}}}\right)^*$, the associated Pfaffian system $E \times_B I \subset T^*M$ is Frobenius.
By proposition , if the geometry does not drop, then for every positive simple root $\alpha$, the line bundle $E \times_B {\ensuremath{\mathfrak{g}}}_{\alpha}$ on $M$ is pseudoeffective. Every positive root $\alpha$ is a sum, with nonnegative integer coefficients, of positive simple roots. So for any positive root $\alpha$, not necessarily simple, the line bundle $E \times_B {\ensuremath{\mathfrak{g}}}_{\alpha}$ on $M$ is also pseudoeffective. Pick any $B$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{b}}}\right)^*$. Then $$\det I = \bigotimes_{\alpha} {\ensuremath{\mathfrak{g}}}_{\alpha},$$ where the tensor product is over positive roots $\alpha$ for which ${\ensuremath{\mathfrak{g}}}_{\alpha} \subset I$. Therefore the line bundle $
E \times_B \det I
$ on $M$ is pseudoeffective. By lemma , $E \times_B I \subset TM$ is Frobenius.
Example: second order scalar ordinary differential equations
============================================================
A path geometry is a geometric description of a system of 2nd order ordinary differential equations.
Take a 2nd order scalar order differential equation, $$\frac{d^2 y}{dx^2} = f\left(x,y,\frac{dy}{dx}\right).$$ Pick a variable $p$, and consider the associated foliation $$\begin{aligned}
dy &= p \, dx, \\
dp &= f\left(x,y,p\right) \, dx,\end{aligned}$$ whose leaves correspond to the solutions of the equation. Also consider the foliation $$\begin{aligned}
dy &= 0, \\
dx &= 0,\end{aligned}$$ whose leaves correspond to the points $(x,y)$ of the configuration space.
A *path geometry* on a complex manifold $M$ with $\dim_{{\ensuremath{\mathbb{C}^{}}}} M = 3$ is a choice of 2 nowhere tangent holomorphic foliations on $M$ by (not necessarily compact) curves, called *integral curves*, and *stalks* respectively, with both foliations being tangent to a (necessarily uniquely determined) holomorphic contact plane field.
In other words, a path geometry is a double Legendre foliation of a 3-manifold.
It turns out that near any point of $M$ there are local coordinates $x, \, y, \dot{y}$ on $M$ and there is a holomorphic function $f\left(x,y,\dot{y}\right)$ for which the integral curves are the solutions of $$dy = \dot{y} \, dx, \ d\dot{y} = f \, dx,$$ while the stalks are the solutions of $dx=dy=0$. Conversely, for any holomorphic function $f\left(x,y,\dot{y}\right)$, these two holomorphic foliations are a path geometry.
If we interchange the two foliations of a path geometry, we obtain the *dual path geometry*.
A holomorphic path geometry on a complex manifold $M$ determines and is determined by a holomorphic parabolic geometry $E \to M$ modelled on the adjoint variety $B_2/B$. The holomorphic contact structure is $E \times_P I$ for a semicanonical $P$-submodule $I \subset \left({\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{b}}}\right)^*$.
See [@Bryant/Griffiths/Hsu:1995] for a detailed exposition.
We encountered this adjoint variety in example . Suppose that $M$ is a complex manifold with path geometry, and that $E \to M$ is the induced regular parabolic geometry of Cartan’s theorem. Labelling roots and $B$-modules as in example , we can see that $E \times_B I_{12}$ is a holomorphic contact structure. This is the contact structure of the path geometry. The integral curves are circles of the root $\alpha_1$, while the stalks are the circles of the root $\alpha_2$.
Using our method of semicanonical modues, we can classify holomorphic path geometries on smooth complex projective varieties. By the method of circles, we find the complete classification on compact K[ä]{}hler manifolds.
\[theorem:PathGeometriesDrop\] Suppose that $M^3$ is a connected compact K[ä]{}hler manifold with a holomorphic path geometry. Then $M ={\ensuremath{\mathbb{ P}^{}}}TM'$, where $M'$ is a compact K[ä]{}hler surface with a holomorphic projective connection. The stalks of $M \to M'$ are either the stalks or the integral curves of $M$. The manifold $M'$ is
1. ${\ensuremath{\mathbb{ P}^{2}}}$ (and $M$ is the model $B_2/B$ with its standard flat path geometry) or
2. a complex surface with an unramified covering by the unit ball in ${\ensuremath{\mathbb{C}^{2}}}$ (and $M$ is a quotient of an open set in the model $B_2/B$, with its standard flat path geometry), or
3. a complex surface with an unramified holomorphic covering by a 2-torus (and the pullback projective connection on the torus is translation invariant).
All of these possibilities for $M'$ occur. The parabolic geometry on $M$ is the path geometry associated to the geodesic equation of the projective connection on $M'$.
We have proven in a more general setting in section that $M={\ensuremath{\mathbb{ P}^{}}}TM'$, (keeping in mind that ${\ensuremath{\mathbb{ P}^{}}}TM' = {\ensuremath{\mathbb{ P}^{}}}T^*M'$). The compact complex surfaces which bear projective connections have been classified [@Kobayashi/Ochiai:1980]: $M'={\ensuremath{\mathbb{ P}^{2}}}$ or $M'$ is an unramified ball quotient or an unramified torus quotient. The projective connections on these surfaces have also been classified [@Klingler:1998; @Klingler:2001] under the hypothesis of local flatness. In case $M'={\ensuremath{\mathbb{ P}^{2}}}$, the presence of rational curves in ${\ensuremath{\mathbb{ P}^{2}}}$ ensures, by theorem , that the holomorphic projective connection on $M'$ is flat. In case $M'$ is covered by the ball, Klinger’s arguments in [@Klingler:2001] actually go through without change, to prove local flatness. Finally, if $M'$ is covered by a torus, then any projective connection on $M'$ is translation invariant as shown in [@McKay:2008c]. A local calculation (see Bryant, Griffiths and Hsu [@Bryant/Griffiths/Hsu:1995]) shows that the lift $M$ of a projective connection on any complex surface $M'$ has parabolic geometry given by the path geometry of the geodesic equation of the projective connection.
Another perspective: either the scalar second order ordinary differential equations that comprise the parabolic geometry on $M$ are the geodesic equations of a holomorphic projective connection on a complex surface, or else they are the dual equations of such equations.
The 2nd order ODE of the path geometry, in the case when $M'={\ensuremath{\mathbb{ P}^{2}}}$ or a ball quotient, is locally equivalent to $$\frac{d^2 y}{dx^2} = 0.$$ If $M'$ is a surface covered by a torus, then the 2nd order ODE is locally equivalent to $$\frac{d^2 y}{dx^2} = p\left(\frac{dy}{dx}\right),$$ for $p$ a polynomial of degree at most 3 with constant coefficients (in linear holomorphic coordinates on the torus); see Cartan [@Cartan:1992] for proof. In either case, these equations are solvable by quadratures.
Example: third order scalar ordinary differential equations
===========================================================
Sato & Yoshikawa [@Sato/Yoshikawa:1998] have results on third order ordinary differential equations similar to Cartan’s above on second order ordinary differential equations. For any third order ordinary differential equation for one function of one variable, say $$\frac{d^3y}{dx^3}=
f\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2}\right),$$ consider the manifold $M^4$ whose coordinates are $x,y,p,q$, equipped with the exterior differential system $$dy=p \, dx, \ dp = q \, dx, \ dq = f(x,y,p,q) \, dx.$$ Sato and Yoshikawa put a parabolic geometry on $M$. Their parabolic geometry is invariant under “contact transformations”.
In this context, a contact transformation (in the sense of Lie, not the sense of contact topology) is any biholomorphism that preserves a certain complete flag of Pfaffian systems. Let $$\begin{aligned}
\vartheta_1 &= dy-p \, dx, \\
\vartheta_2 &= dp-q \, dx, \\
\vartheta_3 &= dq-f(x,y,p,q) \, dx\end{aligned}$$ and let $\mathcal{I}_j$ ($j=1,2,3$) be the Pfaffian system spanned locally by $\vartheta_1, \dots, \vartheta_j$. Then a contact transformation in Lie’s sense is a local biholomorphism preserving all of the Pfaffian systems $\mathcal{I}_j$.
The parabolic geometry of Sato and Yoshikawa is modelled on ${\ensuremath{\operatorname{Sp}\left({4,{\ensuremath{\mathbb{C}^{}}}}\right)}}/B=C_2/B=B_2/B={\ensuremath{\operatorname{PO}\left({5,{\ensuremath{\mathbb{C}^{}}}}\right)}}/B$, where $B$ is the Borel subgroup. The root lattice of $B_2/B$ is drawn in figure \[fig:latticeBTwo\]. One can see the 4 positive roots, drawn as dots. The 3 Pfaffian systems are associated to the sets of positive roots $$\begin{aligned}
& \alpha_1 \\
& \alpha_1, \alpha_1 + \alpha_2 \\
& \alpha_1, \alpha_1 + \alpha_2, \alpha_1 + 2 \, \alpha_2.\end{aligned}$$ The Dynkin diagram of the model is $
\begin{xy}
0;/r.20pc/:
(0,0)*+{{{\scriptscriptstyle{\times}}}}="1";
(5,0)*+{>}="2";
(10,0)*+{{{\scriptscriptstyle{\times}}}}="3";
"1"; "3" **\dir{=};
{\ar@{=}; "1";"3"};
\end{xy}
$. Let’s refer to a parabolic geometry with this model which arises locally from a third order ordinary differential equation, following the method of Sato and Yoshikawa, as a *third order ODE geometry*.
The fibers of the bundle map $$\begin{xy}
0;/r.20pc/: (0,10)*+{{{\scriptscriptstyle{\times}}}}="1";
(10,10)*+{}="2";
(20,10)*+{}="3";
(30,10)*+{{{\scriptscriptstyle{\times}}}}="4";
(35,10)*+{>}="5";
(40,10)*+{{{\scriptscriptstyle{\times}}}}="6";
(30,0)*+{{{\scriptscriptstyle{\bullet}}}}="7";
(35,0)*+{>}="8";
(40,0)*+{{{\scriptscriptstyle{\times}}}}="9";
"2"; "3" **\dir{-};
"4"; "6" **\dir{=};
{\ar@{->}; "2";"3"};
{\ar@{->}; "5"; "8"};
"7"; "9" **\dir{=};
\end{xy}$$ are the $e_3$-circles of the model (in the terminology of Sato & Yoshikawa) while those of $$\begin{xy}
0;/r.20pc/: (0,10)*+{{{\scriptscriptstyle{\times}}}}="1";
(10,10)*+{}="2";
(20,10)*+{}="3";
(30,10)*+{{{\scriptscriptstyle{\times}}}}="4";
(35,10)*+{>}="5";
(40,10)*+{{{\scriptscriptstyle{\times}}}}="6";
(30,0)*+{{{\scriptscriptstyle{\times}}}}="7";
(35,0)*+{>}="8";
(40,0)*+{{{\scriptscriptstyle{\bullet}}}}="9";
"2"; "3" **\dir{-};
"4"; "6" **\dir{=};
{\ar@{->}; "2";"3"};
{\ar@{->}; "5"; "8"};
"7"; "9" **\dir{=};
\end{xy}$$ are the integral curves of the model (which Sato & Yoshikawa call $e_4$-circles).
Suppose that $M'$ is a complex manifold, of complex dimension 3, with holomorphic parabolic geometry $E \to M'$ modelled on the smooth quadric hypersurface $Q^3={\ensuremath{\operatorname{PO}\left({5,{\ensuremath{\mathbb{C}^{}}}}\right)}}/P=B_2/P$. For example, a holomorphic conformal structure on $M'$ will impose such a holomorphic parabolic geometry. Conversely, every parabolic geometry modelled on $B_2/B$ imposes a holomorphic conformal structure, since the group $P$ acts in the representation ${\ensuremath{\mathfrak{g}}}/{\ensuremath{\mathfrak{p}}}$ preserving a nondegenerate quadratic cone. Let $M$ be the set of all null lines in the tangent spaces of $M'$. We can easily see that $M=E/B$, $B \subset B_2$ the Borel subgroup. Therefore $M$ is the lift of $M'$ to a $B_2/B$-geometry. Moreover, if the parabolic geometry on $M'$ is a holomorphic conformal structure, then the parabolic geometry of the lift $M$ is precisely the equation of circles in $M'$. We leave the reader to check these (purely local and elementary) assertions of parabolic geometry.
The construction of a third order ordinary differential equation out of a conformal structure has been well known since work of Wünschmann (see Chern [@Chern:1937; @Chern:1940], Dunajski & Tod [@Dunajski/Tod:2005], Frittelli, Newman & Nurowski [@Frittelli/Newman/Nurowski:2003], Sato & Yoshikawa [@Sato/Yoshikawa:1998], Silva-Ortigoza & Garc[í]{}a-God[í]{}nez [@SilvaOrtigoza/GarciaGodinez:2004], Wünschmann [@Wunschmann:1905]). Identification of the local obstruction to dropping with the Chern invariant is a long but straightforward calculation (see Sato & Yoshikawa [@Sato/Yoshikawa:1998]). Hitchin [@Hitchin:1982] pointed out that a rational curve on a surface with appropriate topological constraint on its normal bundle must lie in a moduli space of rational curves constituting the integral curves of a unique third order ordinary differential equation with vanishing Chern invariant.
\[theorem:KahlerBTwoModB\] Suppose that $M$ is a compact K[ä]{}hler 4-fold with holomorphic parabolic geometry modelled on $B_2/B$. Then the geometry on $M$ drops to a holomorphic parabolic geometry modelled on
1. the smooth quadric hypersurface $Q^3=B_2/P$ or
2. the projective space ${\ensuremath{\mathbb{ P}^{3}}}$ of null 2-planes in ${\ensuremath{\mathbb{C}^{5}}}$,
on a compact K[ä]{}hler 3-fold $M'$. In particular, if the parabolic geometry on $M$ is locally a third order ODE geometry, then this geometry is the equation of
1. circles of a holomorphic conformal structure on $M'$ or
2. circles of a holomorphic Legendre connection on $M'$
See Sato and Yoshikawa [@Sato/Yoshikawa:1998] for the definition of Legendre connection.
It is not known which compact Kähler 3-folds admit conformal geometries, or admit Legendre connections.
The model for third order ODE geometries is $B_2/B$, a quotient by a Borel subgroup, so theorem applies, ensuring that the parabolic geometry drops to one modelled on either $
\begin{xy}
0;/r.20pc/:
(0,0)*+{{{\scriptscriptstyle{\times}}}}="1";
(5,0)*+{>}="2";
(10,0)*+{{{\scriptscriptstyle{\bullet}}}}="3";
"1"; "3" **\dir{=};
{\ar@{=}; "1";"3"};
\end{xy}
$ or $
\begin{xy}
0;/r.20pc/:
(0,0)*+{{{\scriptscriptstyle{\bullet}}}}="1";
(5,0)*+{>}="2";
(10,0)*+{{{\scriptscriptstyle{\times}}}}="3";
"1"; "3" **\dir{=};
{\ar@{=}; "1";"3"};
\end{xy}
$. We call these $B_2/P_1$ and $B_2/P_2$ respectively. The variety $B_2/P_1$ is the Dynkin diagram of the model of a conformal geometry. The variety $B_2/P_2$ is the Dynkin diagram of the model of the parabolic geometry of a Legendre connection (again see Sato and Yoshikawa [@Sato/Yoshikawa:1998]). We leave the reader to check by examination of the structure equations of Sato and Yoshikawa [@Sato/Yoshikawa:1998] p. 1000 that if we take a parabolic geometry with either of these two models on some complex manifold $M'$ and lift it, say to a complex manifold $M$, then the lifted parabolic geometry on $M$ is the parabolic geometry associated by Sato and Yoshikawa to the third order ODE of the circles.
Recall that the *Lie ball* is the noncompact Hermitian symmetric space dual to the smooth quadric hypersurface.
Suppose that $M$ is a smooth complex projective 4-fold with a holomorphic third order ODE geometry. Then $M$ is the set of null lines in the tangent spaces of a 3-fold $M'$ with holomorphic conformal geometry. The 3rd order ODE geometry on $M$ is the one associated to the circles of $M'$. The smooth complex projective 3-folds $M'$ which admit holomorphic conformal structures are precisely
1. the quadric $Q^3$, with its standard flat conformal geometry,
2. 3-folds with unramified covering by an abelian 3-fold, with any translation invariant conformal geometry,
3. 3-folds covered by the Lie ball with the standard flat conformal geometry.
Apply theorem to ensure that the parabolic geometry on $M$ drops to a parabolic geometry on some 3-fold $M'$. The parabolic geometry on $M'$ could be either a conformal structure or a Legendre connection. Legendre connections admit a holomorphic contact structure, of the form $E \times_{P_2} I$, as we see from the structure equations of Sato and Yoshikawa, [@Sato/Yoshikawa:1998] p. 1000. A contact structure is semicanonical and not Frobenius. Therefore any Legendre connection on any smooth complex projective variety must be isomorphic to the model ${\ensuremath{\mathbb{ P}^{3}}}$ and this forces $M$ to be isomorphic to its model, so drops to the model $Q^3$ of conformal geometry.
Therefore we can assume that the parabolic geometry on $M$ drops to a conformal geometry on a smooth connected complex projective 3-fold $M'$. The classification of smooth connected complex projective 3-folds admitting conformal geometries is due to Jahnke and Radloff [@Jahnke/Radloff:2004].
Conclusion
==========
We have demonstrated rigidity phenomena for a large class of holomorphic geometric structures and holomorphic exterior differential systems on smooth complex projective varieties. Our motivation is the following conjecture.
Suppose that $G$ is a complex simple Lie group and $P \subset G$ a maximal parabolic subgroup. Suppose that $G/P$ is not a compact Hermitian symmetric space (or, if $G/P$ is a compact Hermitian symmetric space, then suppose that $G$ is a proper subgroup of the identity component of the biholomorphism group of $G/P$). Then up to isomorphism, the only holomorphic parabolic geometry modelled on $G/P$ on any compact K[ä]{}hler manifold is the standard flat parabolic geometry on $G/P$.
More generally, one would like to construct explicitly all of the holomorphic Cartan geometries on all compact K[ä]{}hler manifolds. The methods in this paper say nothing about the parabolic geometries modelled on compact Hermitian symmetric spaces, perhaps the most important type of parabolic geometry [@Goncharov:1987; @Klingler:2001].
It might be possible to classify the semicanonical modules of generalized flag varieties. This is a complicated combinatorial problem about root systems.
It is frustrating to have many results for smooth complex projective varieties but so few for compact K[ä]{}hler manifolds. Any Cartan 2-plane field $\mathcal{V}$ on any 5-dimensional complex manifold $M$ has a holomorphic quartic symmetric form on the 2-plane as an invariant; [@Cartan:30]. That quartic form has a discriminant, which is a holomorphic section of a positive power of the canonical bundle. If the underlying 5-fold is compact K[ä]{}hler, then the canonical bundle is not pseudoeffective, as explained above. Therefore no positive power of the canonical bundle has any nonzero sections. So the discriminant vanishes, i.e. the quartic has a multiple root at every point. The 2-plane field together with its brackets spans a 3-plane field. Similarly, Cartan defines a holomorphic quartic symmetric form on the 3-plane field, which restricts to the quartic on the two plane field. Again this quartic can’t have any nonvanishing invariants in classical invariant theory, since these all occur in positive powers of the canonical bundle. By geometric invariant theory, the projectivized zero locus of the quartic must therefore have a triple point or tacnode. One might be able to find similar information about other invariants and thereby prove vanishing of curvature to prove that $G_2/P_1$ is the only compact K[ä]{}hler 5-fold bearing a holomorphic Cartan 2-plane field. It is already known that the holomorphic Cartan 2-plane field on $G_2/P_1$ discovered by Cartan is the only holomorphic Cartan 2-plane field on $G_2/P_1$ [@Biswas/McKay:2010].
|
---
abstract: |
In 1990 J-L. Krivine introduced the notion of storage operators. They are $\l$-terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in $AF2$ type system for storage operators using Gődel translation of classical to intuitionistic logic.\
In order to modelize the control operators, J-L. Krivine has extended the system $AF2$ to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system $AF2$ can be used to find the values of classical integers.\
In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. We present also a similar result in the M. Parigot’s $\l \m$-calculus.
---
=Symbol at 16pt =Symbol at 10pt ł Ł PS. Ø v
\[section\] \[section\] \[section\] \[section\]
**Mixed Logic and Storage Operators\
**
**Karim NOUR\
LAMA - Equipe de Logique, Université de Chambéry\
73376 Le Bourget du Lac\
e-mail nour@univ-savoie.fr\
**
Introduction
============
In 1990, J.L. Krivine introduced the notion of storage operators (see \[4\]). They are closed $\l$-terms which allow, for a given data type (the type of integers, for example), to simulate in $\l$-calculus the “call by value” in a context of a “call by name” (the head reduction) and they can be used in order to modelize assignment instructions. J.L. Krivine has shown that the formula $\q x
\{ N$\*$[x] \f \neg\neg N[x] \}$ is a specification for storage operators for Church integers : where $N[x]$ is the type of integers in $AF2$ type system, and the operation $*$ is the simple Gődel translation from classical to intuitionistic logic which associates to every formula $F$ the formula $F$\* obtained by replacing in $F$ every atomic formula by its negation (see \[3\]).\
The latter result suggests many questions :
- Why do we need a Gődel translation ?
- Why do we need the type $N$\*$[x]$ which characterize a class larger than integers ?
In order to modelize the control operators, J-L. Krivine has extended the system $AF2$ to the classical logic (see \[6\]). His method is very simple : it consists of adding a new constant, denoted by $C$, with the declaration $C : \q X \{ \neg \neg X \f X \}$ which axiomatizes classical logic over intuitionistic logic. For the constant $C$, he adds a new reduction rule : $(C t t_1 ... t_n) \f (t \quad \l x (x \quad t_1 ... t_n))$ which is a particular case of a rule given by Felleisen for control operator (see \[1\]). In this system the property of the unicity of integers representation is lost, but J-L. Krivine has shown that storage operators typable in the intuitionistic system $AF2$ can be used to find the values of classical integers [^1](see \[6\]).\
The latter result suggests also many questions :
- What is the relation between classical integers and the type $N$\*$[x]$ ?
- Why do we need intuitionistic logic to modelize the assignment instruction and classical logic to modelize the control operators ?
In this paper, we present a new classical type system based on a logical system called mixed logic. This system allows essentially to distinguish between classical proofs and intuitionistic proofs. We prove that, in this system, we can characterize, by types, the storage operators and the control operators. This results give some answers to the previous questions.\
We present at the end (without proof) a similar result in the M. Parigot’s $\l
\m$-calculus.\
**Acknowledgement. We wish to thank J.L. Krivine, and C. Paulin for helpful discussions. We don’t forget the numerous corrections and suggestions from R. David and N. Bernard.**
Pure and typed $\l$-calculus
============================
- Let $t,u,u_1,...,u_n$ be $\l$-terms, the application of $t$ to $u$ is denoted by $(t)u$. In the same way we write $(t)u_1...u_n$ instead of $(...((t)u_1)...)u_n$.
- $Fv(t)$ is the set of free variables of a $\l$-term $t$.
- The $\b$-reduction (resp. $\b$-equivalence) relation is denoted by $u \f\sb{\b} v$ (resp. $u
\simeq\sb{\b} v$).
- The notation $\s(t)$ represents the result of the simultaneous substitution $\s$ to the free variables of $t$ after a suitable renaming of the bounded variables of $t$.
- We denote by $(u)^n v$ the $\l$-term $(u)...(u)v$ where $u$ occurs $n$ times, and $\sou{u}$ the sequence of $\l$-terms $u_1,...,u_n$. If $\sou{u} = u_1,...,u_n$ $n \geq 0$, we denote by $(t)\sou{u}$ the $\l$-term $(t)u_1...u_n$.
- Let us recall that a $\l$-term $t$ either has a head redex \[i.e. $t=\l x_1 ...\l
x_n (\l x u) v v_1 ... v_m$, the head redex being $(\l x u) v$\], or is in head normal form \[i.e. $t=\l x_1 ...\l x_n (x) v_1 ... v_m$\]. The notation $u \p v$ means that $v$ is obtained from $u$ by some head reductions. If $u \p v$, we denote by $h(u,v)$ the length of the head reduction between $u$ and $v$.
(see\[3\])\
1) If $u \p v$, then, for any substitution $\s$, $\s(u) \p \s(v)$, and $h(\s(u),\s(v))$=h(u,v).\
2) If $u \p v$, then, for every sequence of $\l$-terms $\sou{w}$, there is a $w$, such that $(u)\sou{w} \p w$, $(v)\sou{w} \p w$, and $h((u)\sou{w},w)=h((v)\sou{w},w)+h(u,v)$.
**Remark. Lemma 2.1 shows that to make the head reduction of $\s(u)$ (resp. of $(u)\sou{w}$) it is equivalent - same result, and same number of steps - to make some steps in the head reduction of $u$, and after make the head reduction of $\s(v)$ (resp. of $(v)\sou{w}$). $\Box$**
- The types will be formulas of second order predicate logic over a given language. The logical connectives are $\perp$ (for absurd), $\f$, and $\q$. There are individual (or first order) variables denoted by $x,y,z,...,$ and predicate (or second order) variables denoted by $X,Y,Z,....$
- We do not suppose that the language has a special constant for equality. Instead, we define the formula $u=v$ (where $u,v$ are terms) to be $\q Y(Y(u) \f Y(v))$ where $Y$ is a unary predicate variable. Such a formula will be called an equation. We denote by $a \approx
b$, if $a=b$ is a consequence of a set of equations.
- The formula $F_1 \f (F_2 \f(...\f (F_n \f G)...))$ is also denoted by $F_1,F_2,...,F_n \f G$. For every formula $A$, we denote by $\neg A$ the formula $A \f \perp$. If $\sou{v} = v_1,...,v_n$ is a sequence of variables, we denote by $\q \sou{v} A$ the formula $\q v_1...\q v_n A$.
- Let $t$ be a $\l$-term, $A$ a type, $\G = x_1 : A_1 ,..., x_n : A_n$ a context, and $E$ a set of equations. We define by means of the following rules the notion “$t$ is of type $A$ in $\G$ with respect to $E$” ; this notion is denoted by $\G\v_{AF2} t:A$ :
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- \(1) $\G\v_{AF2} x_i:A_i$ $1\leq i\leq n$.
- \(2) If $\G,x:A \v_{AF2} t:B$, then $\G\v_{AF2} \l xt:A \f B$.
- \(3) If $\G\v_{AF2} u:A \f B$, and $\G\v_{AF2} v:A$, then $\G\v_{AF2} (u)v:B$.
- \(4) If $\G\v_{AF2} t:A$, and $x$ is not free in $\G$, then $\G\v_{AF2} t:\q xA$.
- \(5) If $\G\v_{AF2} t:\q xA$, then, for every term $u$, $\G\v_{AF2} t:A[u/x]$.
- \(6) If $\G\v_{AF2} t:A$, and $X$ is not free in $\G$, then $\G\v_{AF2} t:\q XA$.
- \(7) If $\G\v_{AF2} t:\q XA$, then, for every formulas $G$, $\G\v_{AF2} t:A[G/X]$.
- \(8) If $\G\v_{AF2} t:A[u/x]$, and $u \approx v$, then $\G\v_{AF2}
t:A[v/x]$.
This typed $\l$-calculus system is called $AF2$ (for Arithmétique Fonctionnelle du second ordre).
(see \[2\]) The $AF2$ type system has the following properties :\
1) Type is preserved during reduction.\
2) Typable $\l$-terms are strongly normalizable.
We present now a syntaxical property of system $AF2$ that we will use afterwards.
(see \[8\]) If in the typing we go from $\G\v_{AF2} t:A$ to $\G\v_{AF2} t:B$, then we may assume that we begin by the $\q$-elimination rules, then by the equationnal rule, and finally by the $\q$-introduction rules.
- We define on the set of types the two binary relations $\lhd$ and $\approx$ as the least reflexive and transitive binary relations such that :
- - $\q xA \lhd A[u/x]$, if $u$ is a term of language ;
- - $\q XA \lhd A[F/X]$, if $F$ is a formula of language ;
- - $A \approx B$ if and only if $A=C[u/x]$, $B=C[v/x]$, and $u \approx v$.
Pure and typed $\l C$-calculus
==============================
The $C2$ type system
--------------------
We present in this section the J-L. Krivine’s classical type system.
- We add a constant $C$ to the pure $\l$-calculus and we denote by $\L C$ the set of new terms also called $\l C$-terms. We consider the following rules of reduction, called rules of head $C$-reduction.
- 1\) $(\l x u) t t_1 ... t_n \f (u[t / x]) t_1 ... t_n$ for every $u, t, t_1,...,t_n \in
\L C$.
- 2\) $(C) t t_1 ... t_n \f (t) \l x (x)t_1 ... t_n$ for every $ t, t_1,...,t_n
\in \L C$, $x$ being a $\l$-variable not appearing in $t_1,...,t_n$.
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- For any $\l C$-terms $t,t'$, we shall write $t \p_C t'$ if $t'$ is obtained from $t$ by applying these rules finitely many times. We say that $t'$ is obtained from $t$ by head $C$-reduction.
- A $\l C$-term $t$ is said $\b$-normal if and only if $t$ does not contain a $\b$-redex.
- A $\l C$-term $t$ is said $C$-solvable if and only if $t \p_C (f)t_1,...,t_n$ where $f$ is a variable.
It is easy to prove that : if $t \p_C t'$, then, for any substitution $\s$, $\s (t) \p_C \s
(t')$.
- We add to the $AF2$ type system the new following rule :
\(0) $\G \v C : \q X \{ \neg \neg X \f X \}$
This rule axiomatizes the classical logic over the intuitionistic logic. We call $C2$ the new type system, and we write $\G \v_{C2} t : A$ if $t$ is of type $A$ in the context $\G$.
It is clear that $\G \v_{C2} t : A$ if and only if $\G , C : \q X \{ \neg \neg X \f X \} \v_{AF2} t :
A$.
(see \[6\])\
1) If $\G \v_{C2} t:A$, and $t \f_{\b} t'$, then $\G \v_{C2} t':A$.\
2) If $\G \v_{C2} t:\perp$, and $t \p_C t'$, then $\G \v_{C2} t':\perp$.\
3) If $A$ is an atomic type, and $\G \v_{C2} t:A$, then $t$ is $C$-solvable.
The $M2$ type system
--------------------
In this section, we present the system $M2$. This system allows essentialy to distinguish between classical proofs and intuitionistic proofs\
We assume that for every integer $n$, there is a countable set of special $n$-ary second order variables denoted by $X_C,Y_C,Z_C$...., and called classical variables.\
Let $X$ be an $n$-ary predicate variable or predicate symbol. A type $A$ is said to be ending with $X$ if and only if $A$ is obtained by the following rules :
- - $X(t_1,...,t_n)$ ends with $X$;
- - If $B$ ends with $X$, then $A \f B$ ends with $X$ for every type $A$ ;
- - If $A$ ends with $X$, then $\q vA$ ends with $X$ for every variable $v$.\
A type $A$ is said to be a classical type if and only if $A$ ends with $\perp$ or a classical variable.\
We add to the $AF2$ type system the new following rules :
- (0$'$) $\G\v C : \q X_C \{ \neg \neg X_C \f X_C \}$
- (6$'$) If $\G\v t:A$, and $X_C$ has no free occurence in $\G$, then $\G\v t: \q
X_C A$.
- (7$'$) If $\G\v t: \q X_C A$, and $G$ is a classical type, then $\G\v t:A[G/ X_C]$.
We call $M2$ the new type system, and we write $\G \v_{M2} t:A$ if $t$ is of type $A$ in the context $\G$.\
We extend the definition of $\lhd$ by : $\q X_C A \lhd A[G / X_C]$ if $G$ is a classical type.
If $A$ is a classical type and $A \lhd B$ (or $A \approx B$), then $B$ is a classical type.
**Proof Easy. $\Box$**
The logical properties of $M2$
------------------------------
We denote by $LAF2$, $LC2$, and $LM2$ the underlying logic systems of respectively $AF2$, $C2$, and $M2$ type systems.\
With each classical variable $X_C$, we associate a special variable $X^{\ast}$ of $AF2$ having the same arity as $X_C$. For each formula $A$ of $LM2$, we define the formula $A$\* of $LAF2$ in the following way :
- - If $A=D(t_1,...,t_n)$ where $D$ is a predicate symbol or a predicate variable, then $A$\*=$A$ ;
- - If $A=X_C(t_1,...,t_n)$, then $A$\*$=\neg X^{\ast}(t_1,...,t_n)$ ;
- - If $A=B \f C$, then $A$\*$=B$\*$ \f C$\* ;
- - If $A=\q xB$, then $A$\*=$\q xB$\*.
- - If $A=\q XB$, then $A$\*=$\q XB$\*.
- - If $A=\q X_C B$, then $A$\*=$\q X^{\ast} B$\*.
$A$\* is called the Gődel translation of $A$.
If $G$ is a classical type of $LM2$, then $\v_{LAF2} \neg \neg G$\*$ \equi G$\*.
**Proof It is easy to prove that $\v_{LAF2} G$\*$ \f \neg \neg G$\*.\
We prove $\v_{LAF2} \neg \neg G$\*$ \f G$\* by induction on $G$.**
- - If $G = \perp$, then $G$\*=$\perp$, and $\v_{LAF2} ((\perp \f \perp) \f \perp) \f
\perp$.
- - If $G = X_C (t_1,...,t_n)$, then $G$\*=$\neg X^{\ast}(t_1,...,t_n)$, and $\v_{LAF2} \neg
\neg \neg X^{\ast}(t_1,...,t_n) \f \neg X^{\ast}(t_1,...,t_n)$.
- - If $G = A \f B$, then $B$ is a classical type and $G$\* = $A$\* $\f$ $B$\*. By the induction hypothesis, we have $\v_{LAF2} \neg \neg B$\*$ \f B$\*. Since $\v_{LAF2} \neg \neg
(A$\*$\f B$\*) $\f$ $(\neg \neg A$\*$\f \neg \neg B$\*), we check easily that $\v_{LAF2}
\neg \neg (A$\* $\f B$\*) $\f (A$\* $\f B$\*).
- - If $G = \q vG'$ where $v=x$ or $v=X$, then $G'$ is a classical type and $G$\*=$\q vG'$\*. By the induction hypothesis, we have $\v_{LAF2}
\neg \neg G'$\*$ \f G'$\*. Since $\v_{LAF2} \neg \neg \q vG'$\* $\f$ $\q v \neg \neg G'$\*, we check easily that $\v_{LAF2} \neg \neg \q vG'$\* $\f \q vG'$\*.
- - If $G = \q X_C G'$, then $G'$ is a classical type and $G$\*=$\q X^{\ast} G'$\*. By the induction hypothesis, we have $\v_{LAF2} \neg \neg G'$\*$ \f G'$\*. Since $\v_{LAF2} \neg \neg \q
X^{\ast} G'$\* $\f$ $\q X^{\ast} \neg \neg G'$\*, we check easily that $\v_{LAF2} \neg
\neg \q X^{\ast} G'$\* $\f \q X^{\ast} G'$\*. $\Box$
Let $A,G$ be formulas of $LM2$, $t$ a term, $x$ a first order variable, and $X$ a second order variable. We have :\
1) $(A[t/x])$\*$= A$\*$[t/x]$.\
2) $(A[G/X])$\*$=A$\*$[G$\*$/X]$.
**Proof By induction on $A$. $\Box$**
Let $A$ be a formula of $LM2$, $G$ a classical type, and $X_C$ a classical variable.\
$\v_{LAF2} (A[G/X_C])$\*$ \equi A$\*$[\neg G$\*$/X_C]$.
**Proof By induction on $A$.**
- - If $A = D(t_1,...,t_n)$ where $D$ is a predicate variable or a predicate symbol, then $A$\*=$A$, and $\v_{LAF2} A \equi A$.
- - If $A = X_C (t_1,...,t_n)$, then $A$\*=$\neg X^{\ast}(t_1,...,t_n)$, and, by Lemma 3.2, $\v_{LAF2} \neg \neg G$\*$ \equi G$\*.
- - If $A = B \f C$, then $A$\* = $B$\* $\f$ $C$\*. By the induction hypothesis, we have $\v_{LAF2} (B[G/X_C])$\*$ \equi B$\*$[\neg G$\*$/X_C]$ and $\v_{LAF2}
(C[G/X_C])$\*$ \equi C$\*$[\neg G$\*$/X_C]$. Therefore $\v_{LAF2} \{ (B[G/X_C])$\*$
\f (B[G/X_C])$\*$\} \equi \{ B$\*$[\neg G$\*$/X_C] \f C$\*$[\neg G$\*$/X_C] \}$.
- - If $A = \q vA'$, where $v=x$ or $v=X$, then $A$\*=$\q vA'$\*. By the induction hypothesis, we have $\v_{LAF2} (A'[G/X_C])$\*$ \equi A'$\*$[\neg G$\*$/X_C]$. Therefore $\v_{LAF2}
(\q vA'[G/X_C])$\*$ \equi \q vA'$\*$[\neg G$\*$/X_C]$.
- - If $A = \q Y_C A'$, then $A$\*=$\q Y^{\ast} A'$\*. By the induction hypothesis, we have $\v_{LAF2} (A'[G/X_C])$\* $\equi A'$\*$[\neg G$\*$/X_C]$. Therefore $\v_{LAF2}
(\q Y_C A'[G/X_C])$\*$ \equi$ $ (\q Y_C A')$\*$[\neg G$\*$/X_C]$. $\Box$
If $A_1,...,A_n \v_{LM2} A$, then $A_1$\*$,...,A_n$\* $\v_{LAF2} A$\*.
**Proof By induction on the proof of $A$ and using Lemmas 3.2, 3.3, and 3.4. $\Box$**
Let $A,A_1,...,A_n$ be formulas of $LAF2$.\
$A_1,...,A_n \v_{LM2} A$ if and only if $A_1,...,A_n \v_{LAF2} A$.
**Proof We use Theorem 3.2. $\Box$\
With each predicate variable $X$ of $C2$, we associate a classical variable $X_C$ having the same arity as $X$. For each formula $A$ of $LC2$, we define the formula $A^C$ of $M2$ in the following way :**
- - If $A=D(t_1,...,t_n)$ where $D$ is a constant symbol, then $A^C=A$ ;
- - If $A=X(t_1,...,t_n)$ where $X$ is a predicate symbol, then $A^C=X_C(t_1,...,t_n)$ ;
- - If $A=B \f C$, then $A^C=B^C \f C^C$ ;
- - If $A=\q xB$, then $A^C=\q xB^C$ ;
- - If $A=\q XB$, then $A^C=\q X_CB^C$.
$A^C$ is called the classical translation of $A$.
Let $A_1,...,A_n,A$ be formulas of $LC2$.\
$A_1,...,A_n \v_{LC2} A$ if and only if $A_1^C,...,A_n^C \v_{LM2} A^C$.
**Proof By induction on the proof of $A$. $\Box$**
Properties of $M2$ type system
==============================
By corollary 3.1, we have that a formula is provable in system $LAF2$ if and only if it is provable in system $LC2$. This resultat is not longer valid if we decorate the demonstrations by terms. We will give some conditions on the formulas in order to obtain such a result.\
We define two sets of types of $AF2$ type system : $\O^+$ (set of $\q$-positive types), and $\O^-$ (set of $\q$-negative types) in the following way :
- - If $A$ is an atomic type, then $A \in \O^+$, and $A \in \O^-$ ;
- - If $T \in \O^+$, and $T' \in \O^-$, then, $T' \f T \in \O^+$, and $T \f T' \in \O^-$ ;
- - If $T \in \O^+$, then $\q x T \in \O^+$ ;
- - If $T \in \O^-$, then $\q x T \in \O^-$ ;
- - If $T \in \O^+$, then $\q X T \in \O^+$ ;
- - If $T \in \O^-$, and $X$ has no free occurence in $T$, then $\q X T \in \O^-$.
1\) If $A \in \O^+$ (resp. $A \in \O^-$) and $A \approx B$, then $B \in \O^+$ (resp. $B \in \O^-$).\
2) If $A \in \O^-$ and $A \lhd B \f C$, then $B \in \O^+$ and $C \in \O^-$.
**Proof Easy. $\Box$**
Let $A_1,...,A_n$ be $\q$-negative types, $A$ a $\q$-positive type of $AF2$ which does not end with $\perp$, $B_1,...,B_m$ classical types, and $t$ a $\b$-normal $\l C$-term.\
If $\G = x_1:A_1,...,x_n:A_n,y_1:B_1,...,y_m:B_m \v_{M2} t:A$, then $t$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} t:A$.
**Proof We argue by induction on $t$.**
- - If $t$ is a variable, we have two cases :
- - If $t=x_i$ $1 \leq i \leq n$, this is clear.
- - If $t=y_j$ $1 \leq j \leq m$, then $A=\q \sou{v} B$ where $B_j \lhd B'_j$ and $B'_j
\approx B$. Therefore, by Lemma 3.1, $A$ is a classical type. A contradiction.
- - If $t=\l x u$, then $\G,x:E \v_{M2} u:F$, and $A=\q \sou{v}( E' \f F')$ where $E \approx
E'$, $F \approx F'$ and $\sou{v}$ does not appear in $\G$. First, by Lemma 4.1, $E \in \O^-$ and $F \in \O^+$, and then, by the induction hypothesis, $u$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n,x:E \v_{AF2} u:F$. Therefore $t$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} t:A$.
- - If $t=(x)u_1 ... u_r$ $r \geq 1$, we have two cases :
- - If $t=x_i$ $1 \leq i \leq n$, then $A_i \lhd B_1 \f C_1$, $C'_i \lhd B_{i+1} \f
C_{i+1}$ $1 \leq i \leq r-1$, $C'_r \lhd D$, $A = \q vD'$, where $C'_i \approx C_i$ $1
\leq i \leq r$, $D' \approx D$, and $\G \v_{M2} u_i:B_i$ $1 \leq i \leq r$. Since $A_i$ is a $\q$-negative types, we prove (by induction and using Lemma 4.1) that for all $1 \leq i \leq r$ $B_i$ is a $\q$-positive types. By the induction hypothesis we have $u_i$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} u_i:B_i$. Therefore $t$ is a normal $\l$-term, and $x_1:A_1,...,x_n:A_n \v_{AF2} t:A$.
- - If $t=y_j$ $1 \leq j \leq m$, then $B_j \lhd B_1 \f
C_1$, $C'_i \lhd B_{i+1} \f C_{i+1}$ $1 \leq i \leq r-1$, $C'_r \lhd D$, $A = \q vD'$, where $C'_i \approx C_i$ $1 \leq i \leq r$, $D' \approx D$, and $\G \v_{M2} u_i:B_i$ $1 \leq i
\leq r$. Therefore, by Lemma 3.1, $A$ is a classical type. A contradiction.
- - If $t=(C)uu_1 ... u_r$ $r \geq 0$, then there is a classical type $E$ such that $\G \v_{M2}
u:\neg \neg E$, $E \lhd B_1 \f C_1$, $C'_i \lhd B_{i+1} \f C_{i+1}$ $1 \leq i \leq r-1$, $C'_r \lhd
D$, $A = \q vD'$, where $C'_i \approx C_i$ $1 \leq i \leq r$, $D' \approx D$, and $\G \v_{M2}
u_i:B_i$ $1 \leq i \leq r$. Therefore, by Lemma 3.1, $A$ is a classical type. A contradiction. $\Box$
Let $A$ be a $\q$-positive type of $AF2$ and $t$ a $\b$-normal $\l C$-term.\
If $\v_{M2} t:A$, then $t$ is a normal $\l$-term, and $\v_{AF2} t:A$.
**Proof We use Theorem 4.1. $\Box$\
As for relation betwen the systems $C2$ and $M2$, we have the following result.**
Let $A_1,...,A_n,A$ be types of $C2$, and $t$ a $\l C$-term.\
$A_1,...,A_n \v_{C2} t:A$ if and only if $A_1^C,...,A_n^C \v_{M2} t:A^C$.
**Proof By induction on the typing of $t$. $\Box$**
The integers
============
- Each data type can be defined by a second order formula. For example, the type of integers is the formula : $N[x]= \q X \{ X(0), \q y(X(y) \f X(sy)) \f X(x) \}$ where $X$ is a unary predicate variable, $0$ is a constant symbol for zero, and $s$ is a unary function symbol for successor. The formula $N[x]$ means semantically that $x$ is an integer if and only if $x$ belongs to each set $X$ containing $0$ and closed under the successor function $s$.\
The $\l$-term $\so{0} = \l x \l fx$ is of type $N[0]$ and represents zero.\
The $\l$-term $\so{s} = \l n\l x\l
f(f)((n)x)f$ is of type $\q y(N[y] \f N[s(y)])$ and represents the successor function.
- A set of equations $E$ is said to be adequate with the type of integers if and only if :
- - $s(a) \not \approx 0$ ;
- - If $s(a) \approx s(b)$ , then so is $a \approx b$.
In the rest of the paper, we assume that all sets of equations are adequate with the type of integers.
- For each integer $n$, we define the Church integer $\so{n}$ by $\so{n} = \l x\l f(f)^n x$.
The integers in $AF2$
---------------------
The system $AF2$ has the property of the unicity of integers representation.
(see \[2\]) Let $n$ be an integer. If $\v_{AF2} t :N[s^n (0)]$, then $t \simeq\sb{\b}
\so{n}$.
The propositional trace $N=\q X \{ X,(X \f X) \f X \}$ of $N[x]$ also defines the integers.
(see \[2\]) If $\v_{AF2} t :N$, then, for a certain $n$, $t \simeq\sb{\b} \so{n}$.
**Remark A very important property of data type is the following (we express it for the type of integers) : in order to get a program for a function $f : N \f N$ it is sufficient to prove $\v
\q x ( N[x] \f N[f(x)] )$. For example a proof of $\v \q x ( N[x] \f N[p(x)] )$ from the equations $p(0)=0$, $p(s(x))=x$ gives a $\l$-term for the predecessor in Church intergers (see \[2\]). $\Box$**
The integers in $C2$
--------------------
The situation in system $C2$ is more complex. In fact, in this system the property of unicity of integers representation is lost and we have only one operational characterization of these integers.\
Let $n$ be an integer. A classical integer of value $n$ is a closed $\l C$-term $\th_n$ such that $\v_{C2} \th_n :N[s^n(0)]$.
(see \[6\] and \[12\]) Let $n$ be an integer, and $\th_n$ a classical integer of value $n$.
- - if $n=0$, then, for every distinct variables $x,g,y$ : $(\th_n) x g y \p_C (x) y$ ;
- - if $n \not = 0$, then there is $m \geq 1$ and a mapping $I
: \{0,...,m \} \f N$, such that for every distinct variables $x,g,x_0,x_1,...,x_m$ :
- $(\th_n) x g x_0 \p_C (g) t_1 x_{r_0}$ ;
- $(t_i) x_i \p_C (g) t_{i+1} x_{r_i}$ $1\leq i\leq m$ ;
- $(t_m) x_m \p_C (x) x_{r_m}$ ;
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0\leq i\leq m-1$.
We will generalize this result.\
Let $O$ be a particular unary predicate symbol. The typed system $C2_O$ is the typed system $C2$ where we replace the rules (2) and (7) by :
- $(2_O)$ If $\G,x:A \v_{C2_O} t:B$, $A$ and $B$ are not ending with $O$, then $\G
\v_{C2_O} \l xt:A \f B$.
- $(7_O)$ If $\G\v_{C2_O} t:\q X A$, and $G$ is not ending with $O$, then $\G \v_{C2_O}
t:A[G/X]$.
We define on the types of $C2_O$ a binary relation $\lhd_O$ as the least reflexive and transitive binary relation such that :
- $\q xA \lhd_O A[u/x]$ if $u$ is a term of language ;
- $\q XA \lhd_O A[G/X]$ if $G$ is a type which is not ending with $O$.
a\) If $\G \v_{C2_O} t:\perp$, and $t \p_C t'$, then $\G \v_{C2_O} t':\perp$.\
b) If $\G \v_{C2_O} t:A$, and $A$ is an atomic type, then $t$ is $C$-solvable.
**Proof a) It is enough to do the proof for one step of reduction. We have two cases :**
- - If $t=(\l xu)vv_1...v_m$, then $t'=(u[v/x])v_1...v_m$, $\G,x:F \v_{C2_O} u:G$, $F$ and $G$ are not ending with $O$, $G'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx \perp$, $G_j \approx G'_j$ $1 \leq j \leq m-1$, $\G \v_{C2_O} v:F$, and $\G \v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. It is easy to check that $\G \v_{C2_O} u[v/x]:G$, then $\G \v_{C2_O} t':\perp$.
- - If $t=(C)vv_1...v_m$, then $t'=(v)\l x(x)v_1...v_m$, and there is a type $A$ which is not ending with $O$ such that : $A'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx \perp$, $A \approx A'$, $G_j \approx G'_j$ $1 \leq j \leq m$, $\G \v_{C2_O}
v:\neg \neg A$, and $\G \v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. It is easy to check that $\G,x:A \v_{C2_O} (x)v_1...v_m:\perp$, but $A$ is not ending with $O$, then $\G \v_{C2_O}
\l x(x)v_1...v_m:\neg A$, and $\G \v_{C2_O} t':\perp$.
b\) Indeed, a typing of $C2_O$ may be seen as a typing of $C2$. $\Box$
a\) If $\G \v_{C2_O} t:O(a)$, and $t \p_C t'$, then $t=t'$.\
b) If $\G=y_1:A_1,...,y_n:A_n,x_1:O(a_1),...,x_m:O(a_m) \v_{C2_O} t:O(a)$, and all $A_i$ $1 \leq i\leq
n$ are not ending with $O$, then $t$ is one of $x_i$, and $a_i \approx a$ $1 \leq i \leq n$.
**Proof a) It is enough to do the proof for one step of reduction. We have two cases :**
- - If $t=(\l xu)vv_1...v_m$, then $t'=(u[v/x])v_1...v_m$, $\G,x:F \v_{C2_O} u:G$, $F$ and $G$ are not ending with $O$, $G'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx O(a)$, $G_j \approx G'_j$ $1 \leq j \leq m-1$, $\G \v_{C2_O} v:F$, and $\G
\v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. Therefore $G_j$ $1 \leq j \leq m$ is not ending with $O$, which is impossible since $G_m \approx O(a)$.
- - If $t=(C)vv_1...v_m$, then $t'=(v)\l x(x)v_1...v_m$, and there is a type $A$ which is not ending with $O$ such that : $A'\lhd_O F_1 \f G_1$, $G'_j \lhd_O F_{j+1} \f G_{j+1}$ $1 \leq j
\leq m-1$, $G_m \approx O(a)$, $A \approx A'$, $G_j \approx G'_j$ $1 \leq j \leq m$, $\G \v_{C2_O}
v:\neg \neg A$, and $\G \v_{C2_O} v_j:F_j$ $1 \leq j \leq m$. $A$ is not ending with $O$, therefore $G_j$ $1 \leq j \leq m$ is not ending with $O$, which is impossible since $G_m \approx O(a)$.
b\) By Lemma 5.1, we have $t \p_C (f)t_1...t_r$, and, by a), $t=(f)t_1...t_r$. Therefore $\G
\v_{C2_O} (f)t_1...t_r:O(a)$.
- - If $f=x_i$ $1 \leq i \leq m$, then $r=0$, $t=x_i$, and $O(a_i) \approx O(a)$, then $a_i
\approx a$.
- - If $f=y_j$ $1 \leq j \leq k$, then $A_j \lhd_O F_1 \f G_1$, $G'_k \lhd_O F_{k+1} \f
G_{k+1}$ $1 \leq k \leq r-1$, $G_r \approx O(a)$, $G_k \approx G'_k$ $1 \leq k \leq r$, and $\G\v_{C2_O} t_k:F_k$ $1 \leq k \leq r$. Since $A_j$ is not ending with $O$, then $G_k$ $1 \leq
k \leq r$ is not ending with $O$, which is impossible since $Cr \approx O(a)$. $\Box$
Let $V$ be the set of variables of $\l C$-calculus.\
Let $P$ be an infinite set of constants called stack constants [^2].\
We define a set of $\l C$-terms $\L CP$ by :
- - If $x \in V$, then $x \in \L CP$ ;
- - If $t \in \L CP$, and $x \in V$, then $\l xt \in \L CP$ ;
- - If $t \in \L CP$, and $u \in \L CP \bigcup P$, then $(t)u \in \L CP$.
In other words, $t \in \L CP$ if and only if the stack constants are in argument positions in $t$.\
Let $\s$ be a function defined on $V \bigcup P$ such that :
- - If $x \in V$, then $\s (x) \in \L CP$ ;
- - If $p \in P$, then $\s (p)=\sou{t}=t_1,...,t_n$, $n \geq 0$, $t_i \in \L CP \bigcup P$ $1 \leq i \leq n$.
We define $\s(t)$ for all $t \in \L CP$ by :
- - $\s ((u)v)=(\s (u))\s (v)$ if $v \not \in P$ ;
- - $\s (\l xu)=\l x \s (u)$ ;
- - $\s ((t)p)=(t)\sou{t}$ if $\s (p)=\sou{t}$.
$\s$ is said to be a $P$-substitution.\
We consider, on the set $\L CP$, the following rules of reduction :
- 1\) $(\l xu)tt_1...t_n \f (u[t/x])t_1...t_n$ for all $u,t \in \L CP$ and $t_1,...,t_n \in \L
CP \bigcup P$ ;
- 2\) $(C)tt_1...t_n \f (t)\l x(x)t_1...t_n$ for all $t \in \L CP$ and $t_1,...,t_n \in \L CP
\bigcup P$, and $x$ being $\l$-variable not appearing in $t_1,...,t_n$.
For any $t,t' \in \L CP$, we shall write $t \rhd_C t'$, if $t'$ is obtained from $t$ by applying these rules finitely many times.
If $t \rhd_C t'$, then $\s (t) \rhd_C \s (t')$ for all $P$-substitution $\s$.
**Proof Easy. $\Box$**
Let $t\in \L CP$ such that the stack constants of $t$ are among $p_1,...,p_m$.\
If $t \p_C t'$, and $\G=\G',p_1:O(a_1),...,p_m:O(a_m) \v_{C2_O}
t:\perp$, then $t' \in \L CP$ and $t \rhd_C t'$.
**Proof It is enough to do the proof for one step of reduction. We have two cases :**
- - If $t=(\l xu)vv_1...v_m$, then, $t'=(u[v/x])v_1...v_m$, $\G,x:F \v_{C2_O}
u:G$, $F$ and $G$ is not ending with $O$, and $\G \v_{C2_O} v:F$. Therefore $u,v \in \L CP$, and so $t' \in \L CP$ and $t \rhd_C t'$.
- - If $t=(C)vv_1...v_m$, then, $t'=(v)\l x(x)v_1...v_m$, and there is a type $A$ which is not ending with $O$ such that $\G \v_{C2_O} v:\neg \neg A$. Therefore $v \in \L CP$, and so $t' \in \L CP$ and $t \rhd_C t'$. $\Box$
Let $n$ be an integer, $\th_n$ a classical integer of value $n$, and $x,g$ two distinct variables.
- - If $n=0$, then for every stack constant $p$, we have : $(\th_n)xgp \p_C
(x)p$.
- - If $n \not = 0$, then there is $m \geq 1$, and a mapping $IÊ:Ê\{0,...,m\}\f N$, such that for all distinct stack constants $p_0,p_1,...,p_m$, we have :
- $(\th_n)xgp_0 \p_C (g)t_1 p_{r_0}$ ;
- $(t_i)p_i \p_C (g)t_{i+1}p_{r_i}$ $1 \leq i \leq m-1$ ;
- $(t_m)p_m \p_C (x)p_{r_m}$
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0 \leq i \leq m-1$.
**Proof We denote, in this proof, the term $s^i(0)$ by $i$.\
If $\v_{C2}\th_n:N[n]$, then $\v_{C2_O} \th_n: [ O(0) \f \perp ], \q y \{ [ O(y) \f \perp ]
\f [O(sy) \f \perp ] \}, O(n) \f \perp $, then $\G_1= x:O(0) \f \perp, g:\q y \{ [ O(y) \f
\perp ] \f [O(sy) \f \perp ] \}, p_0:O(n) \v_{C2_O} (\th_n)xgp_0:\perp$, therefore, by Lemma 5.1, $(\th_n)xgp_0$ is $C$-solvable, and three cases may be seen :**
- - If $(\th_n)xgp_0 \p_C (p_0)t_1...t_r$, then $r=0$, and there is a term $a$, such that $O(a) \approx \perp$. This is impossible.
- - If $(\th_n)xgp_0 \p_C (x)t_1...t_r$, then $r=1$, and $\G_1 \v_{C2_O} t_1:O(0)$. Therefore, by Lemma 5.2, $t_1=p_0$, and so $n=0$.
- - If $(\th_n)xgp_0 \p_C (g)t_1...t_r$, then $r=2$, $\G_1 \v_{C2_O} t_1:O(a) \f \perp$, $\G_1 \v_{C2_O}
t_2:O(s(a'))$, and $a \approx a'$. By Lemma 5.2, we have $t_2=p_0$, and $s(a') \approx n$, then $a
\approx n-1$. Therefore $(\th_n)xgp_0 \p_C (g)t_1p_0$, and $\G_1 \v_{C2_O} t_1:O(n-1) \f \perp$. Let $I(0)=n$.
We prove that : if $\G_i=g:\q y \{ [ O(y) \f \perp ] \f [ O(sy) \f \perp ] \} , x:O(0) \f \perp,
p_0:O(I(0)),...., p_i:O(I(i)) \v_{C2_O} (t_i)p_i:\perp$, then :\
$(t_i)p_i \p_C (g)t_{i+1}p_{r_i}$, and $\G_i \v_{C2_O} t_{i+1}:O(I(r_i)-1) \f \perp$\
or\
$(t_i)p_i \p_C (x)p_{r_i}$, and $I(r_i)=0$.\
$\G_i \v_{C2_O} (t_i)p_i:\perp$, therefore, by Lemma 5.1, $(t_i)p_i$ est $C$-solvable, and three cases may be seen :
- - If $(t_i)p_i \p_C (p_j)u_1...u_r$ $0 \leq j \leq i$, then $r=0$, and there is a term $a$, such that $O(a) \approx \perp$. This is impossible.
- - If $(t_i)p_i \p_C (x)u_1...u_r$, then $r=1$, and $\G_i \v_{C2_O} u_1:O(0)$. Therefore, by Lemma 5.2, $u_1=p_{r_i}$, and $I(r_i)=0$.
- - If $(t_i)p_i \p_C (g)u_1...u_r$, then $r=2$, $\G_i \v_{C2_O} u_1:O(a) \f \perp$, $\G_i \v_{C2_O} u_2:O(s(a'))$, and $a \approx a'$. By Lemma 5.2, we have $u_2=p_{r_i}$, and $s(a')
\approx I(r_i)$, then $a \approx I(r_i)-1$. Therefore $(t_i)p_i \p_C (g)t_{i+1} p_{r_i}$, and $\G_i \v_{C2_O} t_{i+1}:O(I(r_i)-1) \f \perp$. Let $I(i+1)=I(r_i)-1$.
This construction always terminates. Indeed, if not, the $\l C$-term $(((\th_n)\l xx)\l xx)p_0$ is not $C$-solvable. This is impossible, since $p_0:\perp \v_{C2}
(((\th_n)\l xx)\l xx)p_0:\perp$. $\Box$
Let $n$ be an integer, $\th_n$ a classical integer of value $n$, and $x,g$ two distinct variables.
- - If $n=0$, then, for every stack constant $p$, we have : $(\th_n)xgp \rhd_C
(x)p$.
- - If $n \not = 0$, then there is $m \geq 1$, and a mapping $IÊ:Ê\{0,...,m\}\f N$, such that for all distinct stack constants $p_0,p_1,...,p_m$, we have :
- $(\th_n)xgp_0 \rhd_C (g)t_1 p_{r_0}$ ;
- $(t_i)p_i \rhd_C (g)t_{i+1}p_{r_i}$ $1 \leq i \leq m-1$ ;
- $(t_m)p_m \rhd_C (x)p_{r_m}$
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0 \leq i \leq m-1$.
**Proof We use Lemma 5.4. $\Box$**
Let $n$ be an integer, and $\th_n$ a classical integer of value $n$.
- - If $n=0$, then, for every $\l C-terms$ $a,F,\sou{u}$, we have : $(\th_n)aF\sou{u} \p_C
(a)\sou{u}$.
- - If $n \not = 0$, then there is $m \geq 1$, and a mapping $IÊ:Ê\{0,...,m\}\f
N$, such that for all $\l C-terms$ $a,F,\sou{u_0},\sou{u_1},...,\sou{u_m}$, we have :
- $(\th_n)aF\sou{u_0} \p_C (g)t_1 \sou{u_{r_0}}$ ;
- $(t_i)\sou{u_i} \p_C (g)t_{i+1}\sou{u_{r_i}}$ $1 \leq i \leq m-1$ ;
- $(t_m)\sou{u_m} \p_C (a)\sou{u_{r_m}}$
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0 \leq i \leq m-1$.
**Proof We use Lemma 5.3. $\Box$**
The integers in $M2$
--------------------
According to the results of section 4, we can obtain some results concerning the integers in the system $M2$.
Let $n$ be an integer. If $\v_{M2} t :N[s^n (0)]$, then, $t \simeq\sb{\b}
\so{n}$.
**Proof We use Theorem 4.1. $\Box$\
Let $n$ be an integer. By Theorem 4.2, a classical integer of value $n$ is a closed $\l C$-term $\th_n$ such that $\v_{M2} \th_n :N^C[s^n(0)]$.**
Let $n$ be an integer, $\th_n$ a classical integer of value $n$, and $x,g$ two distinct variables.
- - If $n=0$, then, for every stack constant $p$, we have : $(\th_n)xgp \rhd_C (x)p$.
- - If $n \not = 0$, then there is $m \geq 1$, and a mapping $IÊ:Ê\{0,...,m\}\f N$, such that for all distinct stack constants $p_0,p_1,...,p_m$, we have :
- $(\th_n)xgp_0 \rhd_C (g)t_1 p_{r_0}$ ;
- $(t_i)p_i \rhd_C (g)t_{i+1}p_{r_i}$ $1 \leq i \leq m-1$ ;
- $(t_m)p_m \rhd_C (x)p_{r_m}$
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0 \leq i \leq m-1$.
**Proof We use Theorem 4.2. $\Box$**
Storage operators
=================
Storage operators for Church integers
-------------------------------------
Let $T$ be a closed $\l$-term. We say that $T$ is a storage operator for Church integers if and only if for every $n \geq 0$, there is a $\l$-term $\t_n \simeq\sb{\b} \so{n}$, such that for every $\l$-term $\th_n \simeq\sb{\b} \so{n}$, there is a substitution $\s$, such that $(T)\th_n f \p
(f)\s(\t_n)$.\
**Examples If we take :\
$T_1 = \l n((n)\d)G$ where $G = \l x\l y(x)\l z(y)(\so{s})z$ and $\d = \l f(f)\so{0}$\
$T_2 = \l n\l f(((n)f)F)\so{0}$ where $F = \l x\l y(x)(\so{s})y$,\
then it is easy to check that : for every $\th_n \simeq\sb{\b} \so{n}$, $(T_i)\th_n f \p
(f)(\so{s})^n \so{0}$ ($i=1$ or $2$) (see \[3\] and \[8\]).\
Therefore $T_1$ and $T_2$ are storage operators for Church integers. $\Box$\
It is a remarkable fact that we can give simple types to storage operators for Church integers. We first define the simple Gődel translation $F$\* of a formula $F$ : it is obtained by replacing in the formula $F$, each atomic formula $A$ by $\neg A$. For example :**
$N$\*$[x]=\q X \{\neg X(0),\q y(\neg X(y) \f \neg X(sy)) \f \neg X(x) \}$
It is well known that, if $F$ is provable in classical logic, then $F$\* is provable in intuitionistic logic.\
We can check that $\v_{AF2} T_1,T_2 : \q x \{N$\*$[x] \f\neg\neg N[x] \}$. And, in general, we have the following Theorem :
(see \[3\] and \[10\]) If $\v_{AF2} T: \q x\{N$\*$[x] \f\neg\neg N[x]\}$, then $T$ is a storage operator for Church integers.
Storage operators for classical integers
----------------------------------------
The storage operators play an important role in classical type systems. Indeed, they can be used to find the value of a classical integer.
(see \[6\] and \[7\]) If $\v_{AF2} T: \q x\{N$\*$[x] \f \neg\neg N[x]\}$, then for every $n \geq 0$, there is a $\l$-term $\t_n \simeq\sb{\b} \so{n}$, such that for every classical integer $\th_n$ of value $n$, there is a substitution $\s$, such that $(T)\th_n f \p_C (f)\s(\t_n)$.
If $\v_{AF2} T: \q x\{N$\*$[x] \f \neg\neg N[x]\}$, then for every $n \geq 0$ and for every classical integer $\th_n$ of value $n$, there is a $\l$-term $\t_n$, such that $(T)\th_n \l xx
\p_C \t_n \f\sb{\b} \so{n}$.
**Proof We use Theorem 6.2. $\Box$\
**Remark. Theorem 6.2 cannot be generalized for the system $C2$. Indeed, let $T=\l \n \l f (f) (C)(T_i)\n$ ($i=1$ or $2$).\
$\n:N$\*$[x] , f:\neg N[x] \v_{C2} (T_i)\n:\neg\neg N[x] \Longrightarrow$\
$\n:N$\*$[x] , f:\neg N[x] \v_{C2} (C)(T_i)\n: N[x] \Longrightarrow$\
$\n:N$\*$[x] , f:\neg N[x] \v_{C2} (f)(C)(T_i)\n: \perp \Longrightarrow$\
$\v_{C2} T: \q x\{N$\*$[x] \f \neg\neg N[x]\}$\
Since for every $\l C$-term $\th$, $(T)\th f \p_C (f) (C)(T_i)\th$, then it is easy to check that there is not a $\l C$-term $\t_n \simeq\sb{\b} \so{n}$ such that for every classical integer $\th_n$ of value $n$, there is a substitution $\s$, such that $(T)\th_n f \p_C
(f)\s(\t_n)$. $\Box$\
We will see that in system $M2$ we have a similar result to Theorem 6.2.\
Let $T$ be a closed $\l C$-term. We say that $T$ is a storage operator for classical integers if and only if for every $n \geq 0$, there is a $\l C$-term $\t_n \simeq\sb{\b} \so{n}$, such that for every classical integers $\th_n$ of value $n$, there is a substitution $\s$, such that $(T)\th_n
f \p_C (f)\s(\t_n)$.****
If $\v_{M2} T: \q x \{ N^C[x] \f \neg\neg N[x] \}$, then $T$ is a storage operator for classical integers.
The type system $M$ is the subsystem of $M2$ where we only have propositional variables and constants (predicate variables or predicate symbols of arity 0). So, first order variable, function symbols, and finite sets of equations are useless. The rules for typed are $0'$) 1), 2), 3), 6), $6'$), 7) and $7'$) restricted to propositional variables. With each predicate variable (resp. predicate symbol) $X$, we associate a predicate variable (resp. a predicate symbol) $X^{\di}$ of $M$ type system. For each formula $A$ of $M2$, we define the formula $A^{\di}$ of $F_C$ obtained by forgetting in $A$ the first order part. If $\G=x_1:A_1,...,x_n:A_n$ is a context of $M2$, then we denote by $\G^{\di}$ the context $x_1:A_1^{\di},...,x_n:A_n^{\di}$ of $M$. We write $\G\v_M t:A$ if $t$ is typable in $M$ of type $A$ in the context $\G$.\
We have obviously the following property : if $\G \v_{M2} t:A$, then $\G^{\di}
\v_M t:A^{\di}$.\
Theorem 6.3 is a consequence of the following Theorem.
If $\v_M T: N^C \f \neg\neg N$, then for every $n \geq 0$, there is an $m \geq 0$ and a $\l
C$-term $\t_m \simeq\sb{\b} \so{m}$, such that for every classical integer $\th_n$ of value $n$, there is a substitution $\s$, such that $(T)\th_n f \p_C (f)\s(\t_m)$.
Indeed, if $\v_{M2} T: \q x \{ N^C[x] \f \neg\neg N[x] \}$, then $\v_M T:
N^C \f \neg\neg N$. Therefore for every $n \geq 0$, there is an $m \geq
0$ and $\t_m \simeq\sb{\b} \so{m}$, such that for every classical integer $\th_n$ of value $n$, there is a substitution $\s$, such that $(T)\th_n f \p_C (f)\s(\t_m)$. We have $\v_{M2} \so{n} :
N^C[s^n(0)]$, then $ f:\neg N[s^n(0)] \v_{M2} (T) \so{n} f :\perp$, therefore $ f:\neg
N[s^n(0)]\v_{M2} (f)\so{m} :\perp$ and $\v_{M2} \so{m} : N[s^n(0)]$. Therefore $n=m$. and $T$ is a storage operator for classical integers. $\Box$\
In order to prove Theorem 6.4, we shall need some Lemmas.
If $\G,\n:N^C\v_M(\n)\sou{d}:\perp$, then $\sou{d}=a,b,d_1,...,d_r$ and there is a classical type $F$, such that : $\G,\n:N^C\v_M a:F$ ; $\G,\n:N^C\v_M b:F \f F$ ; $F \lhd E_1 \f F_1$, $F_i \lhd E_{i+1} \f F_{i+1}$ $1 \leq i \leq r-1$ ; $F_r \lhd \perp$ ; and $\G,\n:N^C\v_M c_i : E_i$ $1 \leq i \leq r$.
**Proof We use Theorem 2.2. $\Box$**
If $F$ is a classical type and $\G,x:F \v_M (x)\sou{d}:\perp$, then $\sou{d}=d_1,...,d_r$ ; $F \lhd E_1 \f F_1$ ; $F_i \lhd E_{i+1} \f F_{i+1}$ $1 \leq i \leq r-1$ ; $F_r
\lhd \perp$ ; and $\G,x:F \v_M c_i : E_i$ $1 \leq i \leq r$.
**Proof We use Theorem 2.2. $\Box$**
Let $t$ be a $\b$-normal $\l C$-term, and $A_1,...,A_n$ a sequence of classical types.\
If $x_1:A_1,...,x_n:A_n \v_M t:N$, then there is an $m \geq 0$ such that $t =
\so{m}$.
**Proof We use Theorems 4.1 and 5.2. $\Box$\
Let $\n$ and $f$ be two fixed variables.\
We denote by $x_{n,a,b,\sou{c}}$ (where $n$ is an integer, $a,b$ two $\l$-terms, and $\sou{c}$ a finite sequence of $\l$-terms) a variable which does not appear in $a,b,\sou{c}$.**
Let $n$ be an integer. There is an integer $m$ and a finite sequence of head reductions $\{ U_i
\p_C V_i \}_{1\leq i\leq r}$ such that :\
1) $U_1 = (T)\n f$ and $V_r = (f)\t_m$ where $\t_m \simeq\sb{\b}\so{m}$ ;\
2) $V_i = (\n) a b \sou{c}$ or $V_i = (x_{l,a,b,\sou{c}}) \sou{d}$ $0 \leq l \leq n-1$;\
3) If $V_i = (\n)a b \sou{c}$, then $U_{i+1} = (a)\sou{c}$ if $n=0$ and $U_{i+1} = ((b)x_{n-1,a,b,\sou{c}})\sou{c}$ if $n \neq 0$ ;\
4) If $V_i = (x_{l,a,b,\sou{c}})\sou{d}$ $0 \leq l \leq n-1$, then $U_{i+1} = (a)\sou{d}$ if $l=0$ and $U_{i+1} = ((b)x_{l-1,a,b,\sou{d}})\sou{d}$ if $l \neq 0$.
**Proof A good context $\G$ is a context of the form $\n:N^C, f:\neg N, x_{n_1,a_1,b_1,\sou{c_1}} : F_1
,..., x_{n_p,a_p,b_p,\sou{c_p}} : F_p$ where $F_i$ is a classical type, $0 \leq n_i \leq n-1$, and $1
\leq i \leq p$ .\
We will prove that there is an integer $m$ and a finite sequence of head reductions $\{
U_i \p_C V_i \}_{1\leq i\leq r}$ such that we have 1), 2), 3), 4), and there is a good context $\G$ such that $\G \v_M V_i :\perp$ $1 \leq i \leq r$.\
We have $\v_M T: N^C \f \neg\neg N$, then $\n:N^C,f:\neg N
\v_M(T)\n f:\perp$, and by Lemmas 6.1 and 6.2, $(T)\n f \p_C V_1$ where $V_1
= (f)\t$ or $V_1 = (\n)ab\sou{c}$.\
Assume that we have the head reduction $U_k \p_C V_k$ and $V_k \neq (f)\t$.**
- - If $V_k = (\n)a b \sou{c}$, then, by the induction hypothesis, there is a good context $\G$ such that $\G \v_M (\n)a b \sou{c} :\perp$. By Lemma 6.1, there is a classical type $F$, such that $\G
\v_Ma:F$ ; $\G \v_Mb:F \f F$ ; $\sou{c}=c_1,...,c_s$ ; $F \lhd E_1 \f F_1$ ; $F_i \lhd E_{i+1} \f F_{i+1}$ $1 \leq i \leq s-1$ ; $F_s \lhd \perp$ ; and $\G \v_Mc_i :
E_i$ $1 \leq i \leq s$.
- - If $n=0$, let $U_{k+1} = (a)\sou{c}$. We have $\G \v_M U_{k+1}:\perp$.
- - If $n \neq 0$, let $U_{k+1} = ((b)x_{n-1,a,b,\sou{c}})\sou{c}$. The variable $x_{n-1,a,b,\sou{c}}$ is not used before. Indeed, if it is, we check easily that the $\l C$-term $(T)\so{n} f$ is not solvable; but that is impossible because $f:\neg N \v_M(T)\so{n} f :\perp$. Therefore $\G' = \G ,x_{n-1,a,b,\sou{c}}:F$ is a good context and $\G' \v_M U_{k+1} :\perp$.
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- - If $V_k = (x_{l,a,b,\sou{c}}) \sou{d}$, then, by the induction hypothesis, there is a good context $\G$ such that $\G \v_M(x_{l,a,b,\sou{c}}) \sou{d} :\perp$. Then there is a classical type $F$ such that $x_{l,a,b,\sou{c}} : F$ is in the context $\G$. By Lemma 6.2, $\sou{c}=d_1,...,d_s$ ; $F \lhd E_1 \f F_1$ ; $F_i \lhd E_{i+1} \f F_{i+1}$ $1 \leq i
\leq s-1$ ; $F_s \lhd \perp$ ; and $\G \v_Mc_i : E_i$ $1 \leq i \leq s$.
- - If $l=0$, let $U_{k+1}=(a)\sou{c}$. We have $\G \v_M U_{k+1}:\perp$.
- - If $l \neq 0$, Let $U_{k+1} = ((b)x_{l-1,a,b,\sou{d}})\sou{d}$. The variable $x_{l-1,a,b,\sou{d}}$ is not used before. Indeed, if it is, we check that the $\l C$-term $(T)\so{n} f$ is not solvable; but this is impossible because $f:\neg N \v_M (T)\so{n} f :\perp$. Then $\G' = \G ,x_{l-1,a,b,\sou{d}}:F$ is a good context and $\G' \v_M U_{k+1} :\perp$.
Therefore there is a good context $\G'$ such that $\G' \v_M U_{k+1} :\perp$. Then, by Lemmas 6.1 and 6.2, $U_{k+1} \p_C V_{k+1}$ where $V_{k+1} = (f)\t$ or $V_{k+1} =
(\n)ab\sou{c}$ or $V_{k+1} = (x_{l,a,b,\sou{c}})\sou{d}$ $0 \leq l \leq n-1$.\
This construction always terminates. Indeed, if not, we check that the $\l C$-term $(T)\so{n} f$ is not solvable; but this is impossible because $f:\neg N \v_M (T)\so{n} f :\perp$.\
Therefore there is $r \geq 0$ and a good context $\G$ such that $\G \v_M V_{r} = (f)\t
:\perp$, and $\G \v_M \t :N$. Therefore, by Lemma 6.3, there is an $m \geq 0$ such that $\t
\simeq\sb{\b}\so{m}$. $\Box$\
Let $T$ be a $\l C$-term such that $\v_M T: N^C \f \neg\neg N$. By Theorem 6.5, there is an integer $s$ and a finite sequence of head reductions $\{ U_i \p_C V_i
\}_{1\leq i\leq r}$ such that :\
1) $U_1 = (T)\n f$ and $V_r = (f)\t_s$ where $\t_s \simeq\sb{\b}\so{s}$;\
2) $V_i = (\n) a b \sou{c}$ or $V_i = (x_{l,a,b,\sou{c}}) \sou{d}$ $0 \leq l \leq n-1$;\
3) If $V_i = (\n)a b \sou{c}$, then $U_{i+1} = (a)\sou{c}$ if $n=0$ and $U_{i+1} = ((b)x_{n-1,a,b,\sou{c}})\sou{c}$ if $n \neq 0$ ;\
4) If $V_i = (x_{l,a,b,\sou{c}})\sou{d}$ $0 \leq l \leq n-1$, then $U_{i+1} = (a)\sou{d}$ if $l=0$ and $U_{i+1} = ((b)x_{l-1,a,b,\sou{d}})\sou{d}$ if $l \neq 0$.\
Let $\th_n$ be a classical integer of value $n$, and $x,g$ two distinct variables. By Theorem 5.6 we have :\
If $n=0$, then for every stack constant $p$, we have : $(\th_n)xgp \rhd_C (x)p$.\
If $n \not = 0$, then there is $m \geq 1$, and a mapping $IÊ:Ê\{0,...,m\}\f N$, such that for all distinct stack constants $p_0,p_1,...,p_m$, we have :\
$(\th_n)xgp_0 \rhd_C (g)t_1 p_{r_0}$ ;\
$(t_i)p_i \rhd_C (g)t_{i+1}p_{r_i}$ $1 \leq i \leq m-1$ ;\
$(t_m)p_m \rhd_C (x)p_{r_m}$\
where $I(0)=n$, $I(r_m)=0$, and $I(i+1)=I(r_i)-1$ $0 \leq i \leq m-1$.\
If $n = 0$, then $(T)\th_n f \p_C (f)\t[\th_n / \n]$.
**Proof We prove by induction that for every $1 \leq i \leq r$, we have $(T)\th_n f \p_C V_i
[\th_n / \n]$.\
For $i=1$, $(T)\th_n f = \{ (T)\n f \} [\th_n / \n] = U_1 [\th_n / \n] \p_C V_1
[\th_n / \n]$.\
Assume it is true for $i$, and prove it for $i+1$.\
$(T)\th_n f \p_C V_i [\th_n / \n] = \{ (\n)ab \sou{c} \} [\th_n / \n] = \{ (\th_n)ab\sou{c} \}
[\th_n / \n] = \{ (\th_n)xgp \} [a / x , b / g , \sou{c} / p ] [\th_n / \n]$. Since $(\th_n)xgp
\p_C (x)p$, then $(T)\th_n f \p_C \{ (a)\sou{c} \} [\th_n / \n] = U_{i+1} [\th_n / \n] \p_C
V_{i+1}[\th_n / \n]$.\
So, for $i=r$, we have $(T)\th_n f \p_C V_r [\th_n / \n] = \{ (f)\t \} [\th_n / \n] = (f)\t[\th_n /
\n]$. $\Box$\
We assume now that $n \geq 1$.\
A $k-\l C$-term is a $\l C$-term of the forme $V_k[\t_1 / y_1 ] ... [\t_p / y_p ] [\th_n / \n ]$ such that :\
- $Fv(V_k) \subseteq \{ \n , f , y_1 , ... , y_p \}$\
- for every $1 \leq i \leq p$, $y_i = x_{n_i,a_i,b_i,\sou{c}_i}$ and $\t_i =t_{m_i}
[a_i / x , b_i / g , \sou{d_0} / p_0 ,...,\sou{d_{m_i-1}} / p_{m_i-1}]$ where $I(m_i)=n_i$\
- for every $0 \leq k \leq m_i-1$, there is $1 \leq l \leq r$ such that $U_l =
(a_i)\sou{d_k}$ if $I(k)=0$ and $U_r = (b_i)x_{I(k)-1,a_i,b_i,\sou{d_k}}\sou{d_k}$ if $I(k) > 0$.\
To simplify, a $k- \l C$-term is denoted by $V_k []$.**
Let $1 \leq i \leq r-1$ and $V_i []$ an $i-\l C$-term. If $(T)\th_n f \p_C V_i []$, then there is $1 \leq j \leq r$ and a $j-\l C$-term $V_j []$ such that $V_j [] \p_C V_j []$ and either $V_i [] \not = V_j []$ or $i < j$
**Proof There are only two possibilities. 1) $V_i =
(\n)ab\sou{c}$ ; 2) $V_i = (x_{\a,a,b,\sou{c}})\sou{d}$.\
We now examine each of this cases.\
1) If $V_i = (\n)ab\sou{c}$, then $V_i [] = \{ (\th_n)ab\sou{c} \}[] = \{ (\th_n)xgp_0 \} [a / x
, b / g ,\sou{c} / p_0] []$. Since $(\th_n)xgp_0 \rhd_C (g)t_1 p_{r_0}=(g)t_1 p_0$, then $V_i []
\p_C \{ (b)t_1[a / x, b / g , \sou{c} / p_0 ] \sou{c} \} [] =$\
$\{ (b) x_{n-1,a,b,\sou{c}} \sou{c} \} [t_1[a / x, b / g , \sou{c} / p_0 / x_{n-1,a,b,\sou{c}}]
[] = U_{i+1} [] \p_c V_{i+1} []$. Let $j=i+1$. We have $i < j$ and $I(1)=I(r_0)-1=I(0)-1=n-1$.\
2) If $V_i = (x_{\a,a,b,\sou{c}})\sou{d}$, then $V_i []=\{ (t_{\b}[a / x, b / g , \sou{d_0} / p_0
,...,\sou{d_{\b-1}} / p_{\b-1}])\sou{d} \} []$ where $I(\b)=\a$.\
If $I(\b)=\a \not =0$, then $U_{i+1}=(b)x_{\a-1,a,b,\sou{d}}\sou{d}=(b)x_{I(\b)-1,a,b,\sou{d}}\sou{d}$, and if $I(\b)=\a \not =0$, then $U_{i+1}= (a)\sou{d}$.\
We consider the following two cases.**
- - If $\b \leq m$, then $(t_{\b})p_{\b} \rhd_C (g)t_{{\b}+1}p_{r_{\b}}$, so that\
$V_i [] \p_C \{ (g)t_{{\b}+1}p_{r_{\b}} \}[a / x, b / g , \sou{d_0} / p_0
,...,\sou{d_{\b-1}} / p_{\b-1} , \sou{d} / p_{\b}] []$ =\
$\{ (b)t_{{\b}+1} \sou{d_{r_{\b}}} \}[a / x, b / g , \sou{d_0} / p_0 ,...,\sou{d_{\b-1}} /
p_{\b-1},\sou{d} / p_{\b}] []$.\
Since $\b \not = m$, then $I(r_\b) \not = 0$. By the hypothesis there is $1 \leq j \leq r$ such that $U_j =
(b)x_{I(r_{\b})-1,a,b,\sou{d_{r_{\b}}}}\sou{d_{r_{\b}}}$. Therefore\
$V_i [] \p_C U_j [t_{{\b}+1}[a / x, b / g , \sou{d_0} / p_0 ,...,\sou{d_{\b-1}} /
p_{\b-1},\sou{d} / p_{\b}]/ x_{I(r_{\b})-1,a,b,\sou{d_{r_{\b}}}}][] =
U_j [] \p_C V_j []$.\
If $V_i [] = V_j []$, then the head $C$-reduction $(t_{\b})p_{\b} \rhd_C
(g)t_{{\b}+1}p_{r_{\b}}$ must be an identity, in other words $(t_{\b})p_{\b} =
(g)t_{{\b}+1}p_{r_{\b}}$ and therefore $\b = r_{\b}$. And so $j = i+1 > i$.
- - If $\b = m$, then $(t_{\b})p_{\b} = (t_m)p_m \rhd_C (x)p_{r_m}$, so that\
$V_i [] \p_C \{ (x)p_{r_m} \}[a / x, b / g , \sou{d_0} / p_0 ,...,\sou{d_{m-1}} / p_{m-1}][] =
(a)t_{r_m} \}[]$.\
Since $I(r_m)=0$, then by the hypothesis there is $1 \leq j \leq r$ such that $U_j = (a)t_{r_m}$. Therefore $V_i [] \p_C U_j [] \p_C V_j []$.\
If $V_i [] = V_j []$, then the head $C$-reduction $(t_m)p_m \rhd_C (x)p_{r_m}$ must be an identity, in other words $(t_m)p_m \rhd_C (x)p_{r_m}$ and therefore $m
= r_m$. And so $j = i+1 > i$. $\Box$
There is a substitution $\s$ such that $(T)\th_n f \p_C (f)\s(\t)$.
**Proof $(T)\th_n f = \{ (T)\n f \} [\th_n / \n] = U_1 [\th_n / \n]
\p_C V_1 [\th_n / \n]$. By Lemma 6.5 we obtaine a sequence $V_{i_1} []$ , $V_{i_2} []$ , ... , $V_{i_k} []$ , ... such that $(T)\th_n f \p_C V_{i_s} []$ and if $V_{i_s} [] \not = V_{i_{s+1}} []$ then $i_s \leq i_{s+1}$. This sequence is necessarily finite, indeed $f:\neg N \v_M (T)\th_n f
:\perp$. If $V_{i_s} [] = V_{i_{s+1}} [] = ... = V_{i_{s+\a}} []$, then $i_s
< i_{s+1} < ... < i_{s+\a}$ and $\a \leq r$. Therefore there is $s$ such that $V_{i_s} = (f)\t$, then $(T)\th_n f \p_C V_{i_s} [] = \{ (f)\t \} [] = (f)\t[]$. $\Box$\
Then, by Lemma 6.4 and Corollary 6.2, $T$ is a storage operator for classical integers.**
General Theorem
---------------
In this subsection, we give (without proof) a generalization of Theorem 6.3.\
Let $T$ be a closed $\l C$-term, and $D,E$ two closed types of $AF2$ type system. We say that $T$ is a storage operator for the pair of types $(D,E)$ iff for every $\l$-term $\v_{AF2} t:D$, there is $\l$-term $\t'_t$ and $\l C$-term $\t_t$, such that $\t'_t \simeq\sb{\b} \t_t$, $\v_{AF2}
\t'_t:E$, and for every $\v_{C2} \th_t:D$, there is a substitution $\s$, such that $(T)\th_t f
\p_C (f)\s(\t_t)$.
Let $D,E$ two $\q$-positive closed types of $AF2$ type system, such that $E$ does not contain $\perp$. If $\v_{M2} T: D^C \f \neg\neg E$, then $T$ is a storage operator for the pair $(D,E)$.
Operational characterization of $\l C$-terms of type $\q X_C \{\perp \f X_C \}$ and $\q X_C \{ \neg \neg X_C \f X_C \}$
=======================================================================================================================
Let **A (for Abort) the $\l C$-term $\l x(C)\l yx$.\
**Behaviour of **A :******
(**A )$tt_1...t_n \p_C ((C)\l yt) t_1...t_n \p_C (\l yt)\l x(x)t_1...t_n \p_C t$.**
**Typing of **A :****
$x:\perp \v_{M2} \l yx:\neg \neg X_C \Longrightarrow x:\perp \v_{M2} (C)\l yx:X_C
\Longrightarrow \v_{M2}$ **A $:\q X_C \{ \perp \f X_C \}$**
If $\v_{M2} T:\q X_C \{ \perp \f X_C \}$, then for every integer $n$, and for all $\l C-terms$ $t,t_1,...,t_n$, $(T)t t_1...t_n \p_C t$.
**Proof. Let $O_1,...,O_n$ be new predicate symbols of arity 0 different from $\perp$. Let $A=O_1,...,O_n \f \perp$. If $\v_{M2} T:\q X_C \{ \perp \f X_C \}$, then $\v_{M2} T:\perp \f A$, and $\G = x:\perp,x_1:O_1,...,x_n:O_n \v_{M2} (T)xx_1...x_n:\perp$. Therefore $(T)xx_1...x_n \p_C (f)u_1...u_r$ and $\G \v_{M2} (f)u_1...u_r:\perp$.**
- - If $f=x_i$ $1 \leq i \leq n$, then $r=0$, and $O_i=\perp$. A contradiction.
- - If $f=x$, then $r=0$, and $(T)xx_1...x_n \p_C x$, therefore, for every integer $n$, and for all $\l C$-terms $t,t_1,...,t_n$, $(T)t t_1...t_n \p_C t$. $\Box$
The constant $C$ satisfies the following relations :\
$(C)t t_1...t_n \p_C (t)U$ and\
$(U)y \p_C (y)t_1...t_n$ where $y$ is a new variable.\
Let $C'=\l x(C)\l d(x)\l y(x)\l z(d)y$.\
$x:\neg \neg X_C,y:X_C,z:X_C,d:\neg X_C \v_{M2} (d)y:\perp \Longrightarrow$\
$x:\neg \neg X_C,y:X_C,d:\neg X_C \v_{M2} (x)\l z(d)y:\perp \Longrightarrow$\
$x:\neg \neg X_C,d:\neg X_C \v_{M2} (x)\l y(x)\l z(d)y:\perp \Longrightarrow$\
$x:\neg \neg X_C \v_{M2} (C)\l d(x)\l y(x)\l z(d)y:X_C \Longrightarrow$\
$\v_{M2} C': \q X_C \{\neg \neg X_C \f X_C \}$.\
The $\l C$-term $C'$ satisfies the following relations :\
$(C')t t_1...t_n \p_C (t)U$,\
$(U)y \p_C (t)V$, and\
$(V)z \p_C (y)t_1...t_n$ where $y,z$ are new variables.\
In general, we have the following characterization.
If $\v_{M2} T: \q X_C \{\neg \neg X_C \f X_C \}$, then there is an integer $m$, such that, for every integer $n$, and for all $\l C$-terms $t,t_1,...,t_n$ :
- $(T)t t_1...t_n \p_C (t)V_1$,
- $(V_i)y_i \p_C (t)V_{i+1}$ $1 \leq i\leq m-1$, and
- $(V_m)y_m \p_C (y_i)t_1...t_n$ where $y_1,...,y_m$ are new variables.
**Proof Let $O$ be a new predicate symbol of arity 0 different from $\perp$. We define as in section 3, the system $M2_O$. And we check easily that this system has the same results as Lemmas 5.1, 5.2, 5.3 and 5.4.\
Let $p$ be a stack constant and $A=O \f \perp$. If $\v_{M2} T: \q X_C \{\neg \neg X_C \f X_C \}$, then $\v_{M2_O} T: \neg \neg A \f A$, and $\G=x:\neg \neg A, p:O
\v_{M2_O} (T)xp:\perp$. Therefore $(T)xp \p_C (f)u_1...u_r$, and $\G \v_{M2_O}
(f)u_1...u_r:\perp$.**
- - If $f=p$, then $r=0$, and $O=\perp$. A contradiction.
- - If $f=x$, then, $(T)xp \rhd_C (x)U_1$, and $\G \v_{M2_O} U_1:\neg A$.
We prove (by induction) that if $\G,y_1:A,...,y_{i-1}:A \v_{M2_O} U_i:\neg A$, then \[$(U_i)y_i \rhd_C (x)U_{i+1}$, and $\G,y_1:A,...,y_i:A \v_{M2_O} U_{i+1}:\neg
A$\] or \[$(U_i)y_i \rhd_C (y_j)p$ $1 \leq j \leq i$\].\
The sequence $(U_i)_{i \geq 0}$ is not infinite. Indeed, if it is, the $\l C$-term $((T)\l x(x)z)p$ is not $C$-solvable; but this is impossible, because $z:A,p:O \v_{M2} ((T)\l x(x)z)p:\perp$.\
To obtain the Theorem, we replace the constant $p$ by the sequence $\sou{t}=t_1,...,t_n$ and we put $V_i = U_i [\sou{t} / p]$. $\Box$
The $\l\m$-calculus
===================
In this section, we give a similar version to Theorem 6.3 in the M. Parigot’s $\l \m$-calculus.
Pure and typed $\l\m$-calculus
------------------------------
$\l\m$-calculus has two distinct alphabets of variables : the set of $\l$-variables $x,y,z,...$, and the set of $\m$-variables $\a,\b,\g$,.... Terms are defined by the following grammar :
$t$ $:=$ $x$ $\mid$ $\l xt$ $\mid$ $(t)t$ $\mid$ $\m\a[\b]t$
Terms of $\l\m$-calculus are called $\l\m$-terms.\
The reduction relation of $\l\m$-calculus is induced by fives different notions of reduction :\
**The computation rules**
- ($C_1$) $(\l xu)v \f u[v/x]$
- ($C_2$) $(\m\a u)v \f \m\a u[v/$\*$\a]$
- where $u[\sou{v}/$\*$\a]$ is obtained from $u$ by replacing inductively each subterm of the form $[\a]w$ by $[\a](w)\sou{v}$.
**The simplification rules**
- ($S_1$) $[\a]\m\b u \f u[\a/\b]$
- ($S_2$) $\m\a [\a]u \f u$, if $\a$ has no free occurence in $u$
- ($S_3$) $\m\a u \f \l x \m\a u[x/$\*$\a]$, if $u$ contains a subterm of the form $[\a]\l yw$.
(see \[18\]) In $\l\m$-calculus, reduction is confluent.
The notation $u \p_{\m} v$ means that $v$ is obtained from $u$ by some head reductions.\
The head equivalence relation is denoted by : $u \sim_{\m} v$ if and only if there is a $w$, such that $u \p_{\m} w$ and $v \p_{\m} w$.\
Proofs are written in a natural deduction system with several conclusions, presented with sequents. One deals with sequents such that :\
- Formulas to the left of $\v$ are labelled with $\l$-variables ;\
- Formulas to the right of $\v$ are labelled with $\m$-variables, except one formula which is labelled with a $\l\m$-term ;\
- Distinct formulas never have the same label.\
The right and the left parts of the sequents are considered as sets and therefore contraction of formulas is done implicitly.\
Let $t$ be a $\l\m$-term, $A$ a type, $\G = x_1:A_1,...,x_n:A_n$, and $\D = \a_1:B_1,...,\a_m:B_m$. We define by means of the following rules the notion “$t$ is of type $A$ in $\G$ and $\D$”. This notion is denoted by $\G\v_{FD2} t:A,\D$.
- The rules (1),...,(8) of $AF2$ type system.
- \(9) If $\G\v_{FD2} t:A,\b:B,\D$, then $\G\v_{FD2}\m\b [\a]t:B,\a:A,\D$.
Weakenings are included in the rules (2) and (9).\
As in typed $\l$-calculus on can define $\neg A$ as $ \f \perp$ and use the previous rules with the following special interpretation of naming for $\perp$ : for $\a$ a $\m$-variable, $\a :
\perp$ is not mentioned.\
**Example Let **C =$\l x \m \a [\ph](x)\l y \m \b [\a] y$.\
$x:\neg \neg X,y:X \v_{FD2} y:X \Longrightarrow$\
$x:\neg \neg X,y:X \v_{FD2} \m \b [\a]y:\perp,\a:X \Longrightarrow$\
$x:\neg \neg X \v_{FD2} \l y \m \b [\a]y:\neg X,\a:X \Longrightarrow$\
$x:\neg \neg X \v_{FD2} \m \a [\ph] (x) \l y \b [\a]y: X \Longrightarrow$\
$\v_{FD2}$**C $: \q X \{\neg \neg X \f X \}$.******
(see \[18\] and \[20\]) The $FD2$ type system has the following properties :\
1) Type is preserved during reduction.\
2) Typable $\l\m$-terms are strongly normalizable.
Classical integers
------------------
Let $n$ be an integer. A classical integer of value $n$ is a closed $\l\m$-term $\th_n$ such that $\v_{FD2} \th_n :N[s^n(0)]$.\
Let $x$ and $f$ fixed variables, and $N_{x,f}$ be the set of $\l\m$-terms defined by the following grammar :
$u$ $:=$ $x$ $\mid$ $(f)u$ $\mid$ $\m\a[\b]x$ $\mid$ $\m\a[\b]u$
We define, for each $u \in N\sb{x,f}$ the set $rep(u)$, which is intuitively the set of integers potentially repesented by $u$ :
- - $rep(x) = \{ 0 \}$
- - $rep((f)u) = \{ n+1$ if $n \in rep(u) \}$
- - $rep(\m\a[\b]u)= \bigcap rep(v)$ for each subterm $[\a]v$ of $[\b]u$
The following Theorem characterizes the normal forms of classical integers.
(see \[19\]) The normal classical integers of value $n$ are exactly the $\l\m$-terms of the form $\l$x$\l$fu with u$\in$ $N_{x,f}$ without free $\m$-variable and such that rep(u)=$\{n\}$.
General Theorem
---------------
In order to define, in this framework, the equivalent of system $M2$, the demonstration of $\neg \neg A \f A$ should not be allowed for all formulas A, and thus we should prevent the occurrence of some formulas on the right. Thus the following definition.\
Let $t$ be a $\l\m$-term, $A$ a type, $\G = x_1:A_1,...,x_n:A_n$, and $\D = \a_1:B_1,...,\a_m:B_m$ where $B_i$ $1 \leq i \leq m$ is a classical type. We define by means of the following rules the notion “$t$ is of type $A$ in $\G$ and $\D$”, this notion is denoted by $\G\v_{M2} t:A,\D$.
- The rules of $DL2$ type system.
- (6$'$) If $\G\v t:A, \D$, and $X_C$ has no free occurence in $\G$, then $\G\v t: \q
X_C A, \D$.
- (7$'$) If $\G\v t: \q X_C A, \D$, and $G$ is a classical type, then $\G\v t:A[G/ X_C], \D$.
Let $T$ be a closed $\l\m$-term. We say that $T$ is a storage operator for classical integers if and only if for every $n \geq 0$, there is $\l \m$-term $\t_n \simeq\sb{\b} \so{n}$, such that for every classical integers $\th_n$ of value $n$, there is a substitution $\s$, such that $(T)\th_n f \sim_{\m} \m \a [\a] (f)\s(\t_n)$.
If $\v_{M2} T: \q x \{ N^C[x] \f \neg\neg N[x] \}$, then $T$ is a storage operator for classical integers.
[99]{}
M. Felleisein [*The Calculi of $\l_v -CS$ conversion: a syntactic theory of control and state in imperative higher order programming.*]{}\
Ph. D. dissertation, Indiana University, 1987.
J.L. Krivine [*Lambda-calcul, types et modèles*]{}\
Masson, Paris 1990.
J.L. Krivine [*Opérateurs de mise en mémoire et traduction de Gődel*]{}\
Archiv for Mathematical Logic 30, 1990, pp. 241-267.
J.L. Krivine [*Lambda-calcul, évaluation paresseuse et mise en mémoire*]{}\
Thearetical Informatics and Applications. Vol. 25,1 p. 67-84 , 1991.
J.L. Krivine [*Mise en mémoire (preuve générale)*]{}\
Manuscript, 1993.
J.L. Krivine [*Classical logic, storage operators and 2nd order lambda-calculus*]{}\
Ann. Pure and Applied Logic 68 (1994) p. 53-78.
J.L. Krivine [*A general storage theorem for integers in call-by-name $\l$-calculus*]{}\
Th. Comp. Sc. (to appear).
K. Nour [*Opérateurs de mise en mémoire en lambda-calcul pur et typé*]{}\
Thèse de Doctorat, Université de Chambéry, 1993.
K. Nour and R. David [*Storage operators and directed $\l$-calculus*]{}\
Journal of symbolic logic, vol 60, n 4, p. 1054-1086, 1995.
K. Nour [*Une preuve syntaxique d’un Théorème de J.L. Krivine sur les opérateurs de mise en mémoire*]{}\
C.R. Acad. Sci Paris, t. 318, Série I, p. 201-204, 1994.
K. Nour [*Opérateurs de mise en mémoire et types $\q$-positifs*]{}\
Thearetical Informatics and Applications (to appear).
K. Nour [*Entiers intuitionnistes et entiers classiques en $\l C$-calcul*]{}\
Thearetical Informatics and Applications, vol 29, n 4, p. 293-313, 1995.
K. Nour [*Quelques résultats sur le $\l C$-calcul*]{}\
C.R. Acad. Sci Paris, t. 320, Série I, p. 259-262, 1995.
K. Nour [*A general type for storage operators*]{}\
Mathematical Logic Quarterly, 41 p. 505-514, 1995.
K. Nour [*La valeur d’un entier classique en $\l\m$-calcul*]{}\
Submitted to Archive for Mathematical Logic.
K. Nour [*Caractérisation opérationnelle des entiers classiques en $\l C$-calcul*]{}\
C.R. Acad. Sci Paris, t. 320, Série I, p. 1431-1434, 1995.
M. Parigot [*Free deduction : an analyse of computations in classical logic*]{}\
Proc. Russian Conference on Logic Programming, St Petersburg (Russia), 1991, Springer LNCS 592, pp. 361-380.
M. Parigot [*$\l\m$-calculus : an algorithm interpretation of classical natural deduction*]{}\
Proc. International Conference on Logic Programming and Automated Reasoning, St Petersburg (Russia), 1992, Springer LNCS 624, pp. 190-201.
M. Parigot [*Classical proofs as programs*]{}\
To appear in Proc. 3rd Krut Gődel Colloquium KGC’93, Springer Lectures Notes in Computer Science.
M. Parigot [*Strong normalization for second order classical deduction*]{}\
To appear in Proc.LICS 1993.
[^1]: The idea of using storage operators in classical logic is due to M. Parigot (see \[19\])
[^2]: The notion of stack constants taken from a manuscript of J-L. Krivine
|
---
abstract: 'Random inductor-capacitor (LC) networks can exhibit percolative superconductor-insulator transitions (SITs). We use a simple and efficient algorithm to compute the dynamical conductivity $\sigma(\omega,p)$ of one type of LC network on large ($4000 \times 4000$) square lattices, where $\delta=p-p_c$ is the tuning parameter for the SIT. We confirm that the conductivity obeys a scaling form, so that the characteristic frequency scales as $\Omega \propto \abs{\delta}^{\nu z}$ with $\nu z \approx 1.91$, the superfluid stiffness scales as $\Upsilon \propto \abs{\delta}^t$ with $t \approx 1.3$, and the electric susceptibility scales as $\chi_E \propto \abs{\delta}^{-s}$ with $s = 2\nu z - t \approx 2.52$. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of $\Im\sigma(\omega)$ at small $\omega$ changes across the SIT. Most importantly, we find that right at the SIT $\Re\sigma(\omega) \propto \omega^{t/\nu z-1} \propto \omega^{-0.32}$, so that the conductivity *diverges* in the DC limit, in contrast with most other classical and quantum models of SITs.'
author:
- Yen Lee Loh
- Rajesh Dhakal
- 'John F. Neis'
- 'Evan M. Moen'
date: 'This version started 2012-6-20; touched 2013-6-26; compiled '
title: 'Divergence of Dynamical Conductivity at Certain Percolative Superconductor-Insulator Transitions'
---
Materials are classified as superconductors, metals, or insulators based on the way they respond to an electric field. A thin film of a superconducting material can be turned into an insulator by increasing the disorder, decreasing the thickness[@haviland1989], applying a parallel[@adams2004] or perpendicular magnetic field[@hebard1990], or changing the gate voltage[@bollinger2011; @lee2011]. As a quantum phase transition occurring at zero temperature, this superconductor-insulator transition (SIT) has attracted much interest. Early work focused on the most easily measurable quantity, the DC conductivity.[@goldmanMarkovicPhysicsToday1998; @gantmakher2010; @haviland1989; @hebard1990; @shahar1992; @adams2004; @steiner2005; @stewart2007] Recently, due to the availability of local scanning probes, attention has turned to the tunneling behavior [@sacepe2008; @sacepe2010; @sacepe2011; @mondal2011; @bouadim2011]. In the case of the disorder-tuned SIT[@bouadim2011], it has become clear that the SIT is ultimately due to a bosonic mechanism [@fisher1990] rather than a fermionic one [@finkelstein1994]. The last step towards a full characterization of the SIT is to develop an understanding of the behavior of the AC conductivity. This is a powerful probe of fluctuations on both long and short length and time scales, and it is of great interest especially as recent technological developments begin to open up more windows of the electromagnetic spectrum for measurement[@hetel2007; @liu2011; @hofmann2010; @valdesaguilar2010].
![ Dissipative conductivity $\Re\sigma(\omega)$ for the $L_{ij} C_{i}$ model. (Top) Right at the SIT ($p=p_c=0.5$), $\Re\sigma$ diverges as $\omega^{-0.32}$ at small $\omega$, indicated by a downward-sloping straight line on the log-log plot. In the superconducting state ($p>p_c$), $\Re\sigma$ obeys an $\omega^1$ power law. The delta function $\delta(\omega)$ is not shown. (Bottom) In the insulating state ($p<p_c$), $\Re\sigma$ is exponentially small ($e^{-\Omega(p)/\omega}$), such that a log plot of $\Re\sigma(\omega)$ vs $1/\omega$ shows straight lines. \[LijCiReSigmaDivergence\] ](LijCiReSigmaLogLogPlot "fig:"){width="0.99\columnwidth"} ![ Dissipative conductivity $\Re\sigma(\omega)$ for the $L_{ij} C_{i}$ model. (Top) Right at the SIT ($p=p_c=0.5$), $\Re\sigma$ diverges as $\omega^{-0.32}$ at small $\omega$, indicated by a downward-sloping straight line on the log-log plot. In the superconducting state ($p>p_c$), $\Re\sigma$ obeys an $\omega^1$ power law. The delta function $\delta(\omega)$ is not shown. (Bottom) In the insulating state ($p<p_c$), $\Re\sigma$ is exponentially small ($e^{-\Omega(p)/\omega}$), such that a log plot of $\Re\sigma(\omega)$ vs $1/\omega$ shows straight lines. \[LijCiReSigmaDivergence\] ](LijCiReSigmaLogPlot1 "fig:"){width="0.99\columnwidth"}\
{width="0.95\columnwidth"} {width="0.95\columnwidth"}
One of the most important questions concerns the AC conductivity in the “collisionless DC” limit [@damle1997; @sachdev_qpt] [^1], $\sigma^* = \lim_{\omega\rightarrow 0} \lim_{T\rightarrow 0} \sigma(\omega,T)$. This has been the subject of a large body of work, including analytical arguments involving charge-vortex duality arguments, quantum Monte Carlo calculations in various representations, and experiments [@fisher1990; @cha1991; @sorensen1992; @runge1992; @makivic1993; @batrouni1993; @cha1994; @smakov2005; @linSorensen2011; @sachdev_qpt; @haviland1989; @bollinger2011]. It is often claimed that at the SIT $\sigma^*$ is finite and takes a universal value of the order of $\sigma_Q = 4e^2/h$, but there are large discrepancies between the “universal” values from various studies, and it is not clear whether there really is a universal value.
In this Letter we study the limit of a coarse-grained superconductor-insulator composite (e.g., millimeter-sized superconducting particles deposited on an insulating substrate). In this situation quantum phase fluctuations are negligible and the SIT is governed by classical percolation. We find that for one of the simplest inductor-capacitor network models, $\sigma(\omega)$ diverges as $\omega\rightarrow 0$, so that $\sigma^*$ is infinite! See Fig. \[LijCiReSigmaDivergence\]. This is a surprising and important result, especially in the light of prior work on quantum models as well as classical models [@stroud1994]. We also elucidate the structure of $\sigma(\omega)$ in this model, addressing the static limit, characteristic frequency, reactive response, and low-frequency contributions from rare regions in the insulator and from Goldstone modes in the superconductor. Our predictions illustrate the range of behavior that can be obtained from classical percolation. This providing a baseline that will very useful for comparing with quantum Monte Carlo results and with upcoming experiments, thus separating the effects of quenched disorder from those of quantum phase fluctuations.
[**Model:**]{} In this Letter we focus on what we call the $L_{ij} C_{i}$ model. This model contains capacitances-to-ground $C_i = C_0$ on every site $i$ and inductances $L_{ij} = L_0$ along bonds $ij$ with probability $p$. Formally, the Lagrangian may be written as $$\begin{aligned}
\calL &= \sum_i \half C_i \dot{\theta}_i{}^2
- \sum_{\mean{ij}} \half L_{ij} {}^{-1} (\theta_i - \theta_j + A_{ij})^2
\label{Lagrangian}
\end{aligned}$$ where $\theta_i$ are electromagnetic phase variables such that $V_i = \dot{\theta}_i$ and $A_{ij}$ is the external vector potential integrated along bond $ij$. Note that this LC model is almost the same as a Josephson junction array, except that it lacks Coulomb blockade effects resulting from charge quantization. The dynamical electromagnetic response $\Upsilon=-dj/dA$, conductivity $\sigma=dj/dE$, and electric susceptibility $\chi_E=dP/dE$ can be defined in the usual way for a 2D system.
[**Methods:**]{} Previous authors have attacked similar problems numerically using a matrix formalism[@jonckheere1998], transfer matrix methods[@derrida1982; @herrmann1984; @zhangStroud1995], and the Frank-Lobb bond propagation algorithm[@frank1988; @zengStroud1989]. In this study we employ a variant of the equation-of-motion method[@williams1985], which is much simpler, more general, and more efficient. Our approach is the theoretical analogue of Fourier transform nuclear magnetic resonance (FT-NMR) spectroscopy, where the frequency response is inferred from the free induction decay signal – the impulse response in the time domain. We apply a transient uniform electric field $E_x(t) = \delta(t)$, evolve currents and voltages according to the dynamical Kirchhoff equations, record the uniform component of the current $I_x(t)$, and extract the dissipative conductivity $\Re \sigma(\omega)$ using a fast Fourier transform. The discretization error in the time evolution enters entirely in the form of systematic phase error, which we eliminate by a suitable transformation of the frequency variable. This procedure can be shown to be formally equivalent to Chebyshev methods such as the kernel polynomial method [@wangChebyshev1994; @silver1994; @silver1996]. The only source of error is the finite duration of the simulation, which leads to a finite frequency resolution. We use a Kaiser window function that gives $\Re\sigma(\omega)$ with sidelobe amplitude below $\Delta \sigma \approx 10^{-8}$ and main lobe width $\Delta \omega \approx \frac{15\omega_\text{max}}{M}$ (where $M$ is the number of timesteps). This prevents exponentially small tails in the spectrum from being contaminated by spectral leakage. We compute $\Im\sigma(\omega)$ using a Kramers-Kronig transformation. We estimate the superfluid stiffness $\Upsilon(\omega)$ from the weight in the lowest-frequency bin, and the electric susceptibility from $\chi_E = \int_0^{\infty} d\omega~ 2 \Re \sigma(\omega) / \omega^2$. Detailed algorithms will be published elsewhere.
We simulated $4000\times 4000$ lattices for 40000 timesteps (i.e., 40000 Chebyshev moments), giving a resolution of $\Delta\omega \approx 0.0015$ after windowing and rebinning. We quote angular frequency $\omega$ in units of $1/\sqrt{L_0C_0}$ and 2D conductivity (sheet conductance) $\sigma$ in units of $\sqrt{C_0/L_0}$.
\
{width="0.95\columnwidth"} {width="0.95\columnwidth"}
[**Results:**]{} Color plots of $\Re\sigma(\omega,p)$ and $\Im\sigma(\omega,p)$ are shown in Fig. \[ArrayPlots\]. The data contain a wealth of interesting information that we list below.
[**(a) Superfluid stiffness:**]{} On the superconducting side of the SIT ($p>p_c$), the superfluid stiffness scales as $\Upsilon(p) \sim \delta^t$ where $\delta=p-p_c$ and $t\approx 1.30$, as shown in Fig. \[LCStaticScaling\]. This agrees with results for resistor networks [@lobb1984; @hong1984].
[**(b) Characteristic frequency:**]{} For 2D percolation the correlation length diverges as $\xi \sim \abs{\delta}^{-\nu}$ with $\nu=4/3$ [@denNijs1979]. This suggests that there is a characteristic (angular) frequency $\Omega \sim \xi^{-z} \sim \abs{\delta}^{\nu z}$, where $z$ is a dynamical critical exponent. Indeed, Fig. \[ArrayPlots\] suggests that most of the spectral weight in $\Re\sigma(\omega)$ occurs above a frequency $\Omega \sim \delta^{\nu z}$, forming a shape analogous to a “quantum critical” fan.
[**(c) Divergent conductivity at SIT:**]{} At the SIT ($p=p_c$), $\Re\sigma(\omega)$ does not tend to a finite limit as in many other systems, but instead it diverges! This is illustrated in Fig. \[LijCiReSigmaDivergence\]. We find a very good fit to a power law of the form $\Re\sigma(\omega) \approx \omega^{-a}$ where $a\approx 0.32(1)$.
[**(d) Scaling collapse:**]{} Based on observations (a) and (b), we postulate the scaling form $
\sigma(\omega,\delta) = \omega^{t/\nu z - 1} f( \omega^{-1/\nu z} \delta )
$. This mandates that $\sigma(\omega) \propto \omega^{t/\nu z-1}$ at the SIT. Comparing with (c), we see that we must have $t/\nu z - 1 = -a$, so that $\nu z = t/(1-a) \approx 1.91$. Indeed, we find that both $\Re\sigma$ and $\Im\sigma$ collapse onto single curves for $\nu z\approx 1.91$, as shown in Fig. \[Collapse\]. The details of $f(x)$ will be reported elsewhere.
[**(e) Electric susceptibility:**]{} On the insulating side of the SIT ($p<p_c$), the scaling form dictates that the electric susceptibility must scale as $\chi_E \sim \abs{\delta}^{-s}$ with $s = 2\nu z - t \approx 2.52$. Indeed, Fig. \[LCStaticScaling\] shows a good fit to this power law.
[**(f) Low-frequency dissipation:**]{} The characteristic frequency $\Omega$ does *not* correspond to a hard gap in the spectrum. In the insulating state, large rare regions contribute exponentially small weight to $\Re\sigma$ all the way down to zero frequency, as shown in the bottom panel of Fig. \[LijCiReSigmaDivergence\]. This is a ramification of Griffiths-McCoy-Wu physics[@mccoy1968; @griffiths1969; @vojta2010] in systems with quenched disorder. The superconducting state has linear low-frequency dissipation $\Re\sigma \sim \omega$, as shown in the top panel of Fig. \[LijCiReSigmaDivergence\]. We believe this is due to the excitation of acoustic “transmission-line” modes that is permitted in the presence of disorder. (It is also reminiscent of Ref. .)
[**(g) Reactive conductivity:**]{} The reactive part of the dynamical conductivity, $\Im \sigma(\omega)$, is shown in Fig. \[ArrayPlots\] and implicitly in Fig. \[Collapse\]. In the insulating state, the low-frequency response is capacitive ($\Im\sigma(\omega) < 0$). In the superconducting state, it is inductive ($\Im\sigma(\omega) > 0$). This suggests that the sign of $\Im \sigma(\omega)$ may be used in experiments as a criterion to distinguish between insulating and superconducting states.
[**Importance of on-site capacitances:**]{} In previous studies of the dynamical conductivity of classical systems near percolation, the insulating (capacitive) elements were placed along bonds, in series with the superconducting (inductive) elements [@stroud1994; @jonckheere1998]. We have studied an $LC$ model of this type, which we call the $L_{ij} C_{ij}$ model. We find that for that model $\nu z=t \approx 1.3$, so that $\sigma^*$ is finite and $\chi_E$ scales differently. In addition, $\Re\sigma$ has exponentially small low-frequency dissipation in the superconducting state, and there are various peculiarities due to self-duality [@straley1976]. Details will be published elsewhere. Ultimately, we feel that the $L_{ij} C_{i}$ model is likely to be more generic than the $L_{ij} C_{ij}$ model for various reasons. Furthermore, as mentioned earlier, the $L_{ij} C_{i}$ model is the limiting case of a Josephson junction array when the charging energy is negligible.
[**Conclusions:**]{} We have studied the percolative superconductor-insulator transition in two-dimensional classical LC networks, in particular, the $L_{ij} C_{i}$ model. We used an efficient algorithm to compute $\sigma(\omega,p)$ on large lattices ($4000 \times 4000$ sites). We find the critical exponents $t \approx 1.30$ (in agreement with results on resistor networks), $\nu z \approx 1.91$, and $s = 2\nu z - t \approx 2.52$. We have extracted the complex-valued scaling function. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of $\Im\sigma(\omega)$ at small $\omega$ changes across the SIT. Most surprisingly, right at the SIT, $\Re\sigma(\omega)$ diverges as $\omega\rightarrow 0$.
As remarked in the introduction, most studies on quantum models and classical models report finite values of $\sigma^*$ at the SIT. In this light, it is extremely surprising that $\sigma^*$ is divergent for a simple classical model! Our results form an important baseline to which to compare simulations of more complicated models such as XY, Bose-Hubbard, and Fermi-Hubbard models, thus allowing one to separate the effects of quenched disorder, quantum phase fluctuations (Coulomb blockade physics), and pairbreaking physics.
We gratefully acknowledge Mason Swanson and Mohit Randeria for useful discussions.
[^1]: In general one must be careful to distinguish between $\sigma^*$ and the true DC conductivity $\sigma^{DC} = \lim_{T\rightarrow 0} \lim_{\omega\rightarrow 0} \sigma(\omega,T)$.
|
---
abstract: 'Let $A$ be a C\*-algebra and $I$ a closed two-sided ideal of $A$. We use the Hilbert C\*-modules picture of the Cuntz semigroup to investigate the relations between the Cuntz semigroups of $I$, $A$ and $A/I$. We obtain a relation on two elements of the Cuntz semigroup of $A$ that characterizes when they are equal in the Cuntz semigroup of $A/I$. As a corollary, we show that the Cuntz semigroup functor is exact. Replacing the Cuntz equivalence relation of Hilbert modules by their isomorphism, we obtain a generalization of Kasparov’s Stabilization theorem.'
address:
- 'Alin Ciuperca: Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada.'
- 'Leonel Robert: Fields Institute, 222 College St., Toronto, ON, M5T 3J1, Canada.'
- 'Luis Santiago: Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada.'
author:
- Alin Ciuperca
- Leonel Robert
- Luis Santiago
title: Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem
---
[^1]
[^2]
Introduction
============
In recent years the Cuntz semigroup has emerged as a powerful invariant in the classification of C\*-algebras, simple and nonsimple (e.g., [@bpt], [@ciuperca-elliott], [@rordam], [@toms]). In [@toms] Andrew Toms provides examples of simple $AH$ C\*-algebras that cannot be distinguished by their standard Elliott invariant ($K$-theory and traces) but that have different Cuntz semigroups. The first author and G. A. Elliott show in [@ciuperca-elliott] that in the nonsimple case, the Cuntz semigroup is a classifying invariant for all AI C\*-algebras (their approach relies on Thomsen’s classification of AI C\*-algebras; see [@thomsen]).
Here we define the Cuntz semigroups, stabilized and unstabilized, in terms of countably generated Hilbert C\*-modules over the algebra, following the approach introduced by K. Coward, G. Elliott and C. Ivanescu in [@kgc]. This construction of the Cuntz semigroup is analogous to the description of $K_0$ in terms of finitely generated projective modules, and is based on an appropriate translation of the notion of Cuntz equivalence of positive elements to the context of Hilbert C\*-modules. Our investigation is initially motivated by the following question: is the Cuntz semigroup of a quotient of a C\*-algebra implicitly determined by the Cuntz semigroup of the algebra? We deduce a satisfactory answer from the inequality in Theorem \[formula\] below, of interest in its own right.
Given a countably generated right Hilbert C\*-module $M$ over $A$, let us denote by $[M]$ the element that it defines in $Cu_s(A)$, the stabilized Cuntz semigroup of $A$. We denote by $Cu(A)$ the subsemigroup of $Cu_s(A)$ consisting of the elements $[M]$ that satisfy $M\subseteq A^n$ for some $n$. This last semigroup can also be described in terms of positive elements of $A$ (and $M_n(A)$), and is often denoted by $W(A)$.
Let $I$ be a $\sigma$-unital ideal of $A$. Then $MI$ is a countably generated right Hilbert C\*-module over $I$. We will see that $[MI]$ only depends on the equivalence class of $M$. Therefore we write $[M]I:=[MI]$.
\[formula\] Let $I$ be a $\sigma$-unital, closed, two-sided ideal of the C\*-algebra $A$ and let $\pi\colon A\to A/I$ denote the quotient homomorphism. Let $M$ and $N$ be countably generated right Hilbert C\*-modules over $A$. Then $Cu_s(\pi)([M])\leq Cu_s(\pi)([N])$ if and only if $$[M]+[N]I\leq [N]+[M]I.$$
It follows from Theorem \[formula\] that $Cu_s(\pi)([M])=Cu_s(\pi)([N])$ if and only if $[M]+[N]I=[N]+[M]I$. Adding $[l_2(I)]$ on both sides and using Kasparov’s stabilization theorem we get that $$\label{equivalence}
Cu_s(\pi)([M])=Cu_s(\pi)([N])\Longleftrightarrow [M]+[l_2(I)]=[N]+[l_2(I)].$$ We will show that the map $Cu_s(\pi)\colon Cu_s(A)\to Cu_s(A/I)$ is surjective. We conclude that the restriction of $Cu_s(\pi)$ to $Cu_s(A)+[l_2(I)]$ is an isomorphism onto $Cu_s(A/I)$.
In the case of the unstabilized Cuntz semigroups, the semigroup $Cu(A/I)$ is obtained as the quotient of $Cu(A)$ by the equivalence relation: $[M]\sim_I [N]$ if $[M]\leq [N]+[C_1]$ and $[N]\leq [M]+[C_2]$ for some $C_1$ and $C_2$, Hilbert C\*-modules over $I$. Here the assumption that the ideal $I$ is $\sigma$-unital is not needed. This result, which we prove, was first obtained by Francesc Perera in an unpublished work. It can also be deduced from [@kirchberg-rordam Lemma 4.12].
A suitable notion of exactness of sequences of ordered semigroups can be defined such that the isomorphism $Cu_s(\pi)$ between $Cu_s(A)+[l_2(I)]$ and $Cu_s(A/I)$ implies the exactness in the middle of the sequence $$0 \longrightarrow Cu_s(I) \stackrel{Cu_s(\iota)}{\longrightarrow} Cu_s(A)
\stackrel{Cu_s(\pi)}{\longrightarrow} Cu_s(A/I) \longrightarrow 0.$$ In Theorem \[cuntzsexact\] we will show that this is a short exact sequence of ordered semigroups, with splittings of the maps $Cu_s(\iota)$ and $Cu_s(\pi)$.
We can express (\[equivalence\]) more directly as follows: $M/MI$ and $N/NI$ are Cuntz equivalent as $A/I$-Hilbert C\*-modules if and only if $M\oplus l_2(I)$ and $N\oplus l_2(I)$ are also Cuntz equivalent. In Section \[proofofkasparov\] we obtain an improvement of this result, with isomorphism of Hilbert C\*-modules instead of Cuntz equivalence. We prove the following theorem.
\[kasparov-lift\] Let $A$ be a C\*-algebra and $I$ a $\sigma$-unital, closed, two-sided ideal of $A$. Let $M$ and $N$ be countably generated right Hilbert C\*-modules over $A$ and suppose that $\phi\colon M/MI\to N/NI$ is an isomorphism of $A/I$-Hilbert C\*-modules. Then there is $\Phi\colon M\oplus l_2(I)\to N\oplus l_2(I)$, isomorphism of Hilbert C\*-modules, that induces $\phi$ after passing to the quotient.
Taking $I=A$ we get Kasparov’s Stabilization Theorem ([@kasparov Theorem 2.1]). The module $M\oplus l_2(I)/MI\oplus l_2(I)$ is canonically isomorphic to $M/MI$. It is using this identification–applied also to $N$–that $\Phi$ induces $\phi$. Theorem \[kasparov-lift\] is proved by an adaptation of the proof given by Mingo and Phillips in [@mingo-phillips] of Kasparov’s Stabilization Theorem.
In the last two sections we apply Theorem \[kasparov-lift\] in the context of multiplier algebras and we prove an equivariant version of Theorem \[kasparov-lift\] assuming that the group is compact.
Preliminaries on Hilbert C\*-modules
====================================
Let $M$ and $N$ be right Hilbert C\*-modules over a C\*-algebra $A$. We shall denote by $K(M,N)$ the norm closure of the space spanned by the $A$-module maps $\theta_{u,v}\colon M\to N$, $\theta_{u,v}(x) :=v\langle u,x\rangle$, $u\in M$, $v\in N$. We shall denote by $B(M,N)$ the space of adjointable operators from $M$ to $N$. If $T\in B(M,N)$, $\ker T$ and $\operatorname{im}T$ will denote the kernel and the image of $T$ respectively. When $M=N$ the spaces $K(M,N)$ and $B(M,N)$ are C\*-algebras that we shall denote by $K(M)$ and $B(M)$ respectively. The elements of $B(M,N)$ will often be referred to simply as operators, while the elements of $K(M,N)$ will be called compact operators. Sometimes we will drop the prefix C\* and refer to Hilbert C\*-modules as Hilbert modules. The C\*-algebra will always act on the right of the Hilbert C\*-modules.
Given a Hilbert C\*-module $M$, the Hilbert C\*-module $l_2(M)$ is defined as the sequences $(m_i)_{i\in {\mathbb{N}}}$, $m_i\in M$, with the property that $\sum_i\langle m_i,m_i\rangle$ converges in norm. This module is endowed with the inner product $\langle (m_i^1),(m_i^2)\rangle :=\sum_i \langle m_i^1,m_i^2\rangle$.
Let $I$ be a closed two-sided ideal of $A$. By $MI$ we denote the span of the elements of the form $m \cdot i$, with $m \in M$, $i \in I$. This set is a closed submodule of $M$ (by Cohen’s Theorem, [@lance]) consisting of all vectors $z$ of $M$ for which $\langle z, z \rangle \in I$. The quotient $M/MI$ is a right $A/I$-Hilbert C\*-module module with inner product $\langle x + MI, y + MI \rangle := \langle x, y \rangle + I$.
The submodule $MI$ is invariant by any operator $T\in B(M)$. More generally, if $T\in B(M,N)$, then $T(MI)\subseteq NI$. In this way every operator $T$ induces an operator $\tilde\pi(T)\in B(M/MI,N/NI)$.
We say that a Hilbert C\*-module $M$ is countably generated if there is a countable set $\{v_i\}_{i=1}^\infty\subset M$ with dense span in $M$. We will make use of the following two theorems on countably generated Hilbert modules.
\[hilbert-tietze\] (Noncommutative Tietze extension Theorem for Hilbert C\*-modules.) Let $M$ and $N$ be countably generated Hilbert C\*-modules and $\phi\in B(M/MI,N/NI)$. Then there is $\Phi\in B(M,N)$ that induces $\phi$ in the quotient.
Let $H=M\oplus N$. We have $H/HI\simeq M/MI\oplus N/NI$. Using this isomorphism, we define $\psi\colon H/HI\to H/HI$, adjointable operator, by the matrix $$\psi :=
\begin{pmatrix}
0 & \phi^{*}\\
\phi & 0
\end{pmatrix}.$$ The homomorphism $\tilde\pi\colon B(H)\to B(H/HI)$ maps $\theta_{u,v}$ to $\theta_{\pi(u),\pi(v)}$ (here $\pi\colon H\to H/HI$ is the quotient map). Thus, $K(H)$ is mapped surjectively onto $K(H/HI)$ by $\tilde\pi$. Since $H$ is countably generated, $K(H)$ is $\sigma$-unital. Thus, by the noncommutative Tiezte extension Theorem ([@wegge-olsen Theorem 2.3.9]), $\tilde\pi$ is also surjective. Let $\Psi\in B(H)$ be a selfadjoint preimage of $\psi$ given by the matrix $$\Psi =
\begin{pmatrix}
A & \Phi^*\\
\Phi & B
\end{pmatrix}.$$ Then the operator $\Phi$ is a lift of $\phi$.
The following theorem is due to Michael Frank ([@frank Theorem 4.1]).
\[frank\] Let $M$ and $N$ be Hilbert C\*-modules, $M$ countably generated. Let $T\colon M\to N$ be a module morphism that is bounded and bounded from below (but not necessarily adjointable). Then $M$ is isomorphic to $\operatorname{im}T$ as Hilbert C\*-modules.
Cuntz semigroups {#cuntzsemigroups}
================
Let us briefly review the construction of the Cuntz semigroups, stabilized and unstabilized, of a C\*-algebra $A$, in terms of countably generated Hilbert C\*-modules over $A$. We refer to [@kgc] for further details.
Let $M$ be a Hilbert C\*-module over $A$. A submodule $F$ of $M$ is said to be compactly contained in $M$ if there is $T\in K(M)^+$ such that $T$ restricted to $F$ is the identity of $F$. In this case we write $F{\subseteq\!\subseteq}M$. Given two Hilbert C\*-modules $M$ and $N$ we say that $M$ is Cuntz smaller than $N$, denoted by $M\preceq N$, if for all $F$, $F{\subseteq\!\subseteq}M$, there is $F'$, $F'{\subseteq\!\subseteq}N$, isomorphic to $F$. This relation defines a preorder relation on the isomorphism classes of Hilbert modules over $A$. Let us say that $M$ is Cuntz equivalent to $N$ if $M\preceq N$ and $N \preceq M$. Let $[M]$ denote the equivalence class of all the modules Cuntz equivalent to a given module $M$. Then the relation $[M]\leq [N]$ if $M\preceq N$ defines an order on the Cuntz equivalence classes of right Hilbert modules over $A$.
Following [@kgc], the stabilized Cuntz semigroup is defined as the ordered set of Cuntz equivalence classes of countably generated Hilbert modules over $A$ endowed with the addition law $[M]+[N]:=[M\oplus N]$. We denote this ordered semigroup by $Cu_s(A)$. It is shown in [@kgc] that $Cu_s(A)=W(A\otimes K)$, where $W(A)$ is the Cuntz semigroup of $A$ defined in terms of positive elements of $\cup_{n=1}^\infty M_n(A)$. The unstabilized Cuntz semigroup of $A$, denoted by $Cu(A)$, is defined as the subsemigroup of $Cu_s(A)$ formed by the Cuntz equivalence classes $[M]$ of $A$-Hilbert modules $M$ such that $M\subseteq A^n$ for some $n\geq 1$. It is shown in [@kgc] that this ordered semigroup coincides with $W(A)$. Furthermore, we can define functors $Cu_s(\cdot)$ and $Cu(\cdot)$ from the category of C\*-algebras to the category of ordered semigroups. By choosing a suitable subcategory of the category of ordered semigroups, Coward, Elliott and Ivanescu were able to show in [@kgc] that the functor $Cu_s(\cdot)$ is continuous with respect to inductive limits.
Let $I$ be a $\sigma$-unital closed two-sided ideal of $A$. If $M$ is a countably generated Hilbert module over $A$ then $MI$ is also countably generated. Let us see that if $[M]\leq [N]$ then $[MI]\leq [NI]$. Suppose that $F{\subseteq\!\subseteq}MI$. Then there is $F'{\subseteq\!\subseteq}N$ isomorphic to it. Since $F$ and $F'$ are isomorphic and $FI=F$, we must have $F'I=F'$. So $F'\subseteq NI$. Hence $[F]=[F']\leq [NI]$. Taking supremum over $F{\subseteq\!\subseteq}MI$ we get that $[MI]\leq [NI]$. In particular, if $M$ and $M'$ are Cuntz equivalent then $MI$ and $M'I$ are also Cuntz equivalent. This justifies writing $[MI]:=[M]I$. We have seen already that the map $[M]\mapsto [M]I$ is order preserving. Since $(M\oplus N)I=MI\oplus NI$, it is also additive. Notice that $M=MI$ (i.e., $M$ is a Hilbert $I$-module) if and only if $[M]I=[M]$. If $M\subseteq A^n$ then $MI\subseteq A^n$, so the map $[M]\mapsto [M]I$ sends elements in $Cu(A)$ to elements in $Cu(A)$.
Let $\iota\colon I\to A$ and $\pi\colon A\to A/I$ denote the inclusion and quotient homomorphisms. The morphisms of ordered semigroups $Cu_s(\iota)$ and $Cu_s(\pi)$ are given by $$\begin{aligned}
Cu_s(\iota)([H]) &:=[H\otimes_\iota A]=[H],\\
Cu_s(\pi)([M]) &:=[M\otimes_\pi A/I]=[M/MI].\end{aligned}$$ The restrictions of $Cu_s(\iota)$ and $Cu_s(\pi)$ to $Cu(I)$ and $Cu(A)$ respectively, give $Cu(\iota)$ and $Cu(\pi)$.
*Proof of Theorem \[formula\].* The hypothesis $Cu_s(\pi)([M])=Cu_s(\pi)([N])$ says that $M/MI$ and $N/NI$ are Cuntz equivalent $A/I$-Hilbert C\*-modules. We will first show that if $M/MI$ is isomorphic to a submodule of $N/NI$ then we have $[M]+[N]I\leq [N]+[M]I$.
Let $\phi\colon M/MI\to N'/N'I$ be an isomorphism of $M/MI$ into $N'/N'I$, a submodule of $N/NI$. Let $C\colon M/MI\to M/MI$ be an arbitrary compact operator with dense range. This operator exists because $M$ is countably generated. Then $\phi'=\phi \circ C$ is compact and satisfies that $\operatorname{im}\phi'^*\phi'$ is dense in $M/MI$. Since $\phi'$ is compact, it is also a compact operator after composing it with the inclusion of $N'/N'I$ into $N/NI$. Let us consider $\phi'$ as a compact operator having codomain $N/NI$. Let $T\colon M\to N$ be a compact operator that lifts $\phi'$. We have a commutative diagram $$\begin{aligned}
\xymatrix{
M\ar[r]^T\ar[d] & N\ar[d]\\
M/MI\ar[r]^{\phi'} & N/NI.
}\end{aligned}$$ Since $\operatorname{im}\phi'^*\phi'$ is dense in $M/MI$, we have that $\operatorname{im}(T^*T)+MI$ is dense in $M$. Let $D_1\colon M\to M$ be positive and with $\operatorname{im}D_1$ dense in $MI$. The operator $D_1$ exists because $MI$ is countably generated (here we use that $I$ is $\sigma$-unital). Then $T^*T+D_1$ has dense range in $M$, that is, it is strictly positive. Let $\{F_n\}_{n=1}^\infty$ be an increasing sequence of submodules of $M$ such that $T^*T+D_1$ is bounded from below on $F_n$ and $\cup_n F_n$ is dense in $M$ (e.g., $F_n=\operatorname{im}\phi_n(T^*T+D_1)$, where $\phi_n\in C_0({\mathbb{R}}^+)$ has compact support and $\phi_n(t)\uparrow 1$). Let $G$ be compactly contained in $NI$. We claim that $F_n\oplus G$ is isomorphic to a submodule on $N\oplus MI$. By Theorem \[frank\], in order to prove this it is enough to find an operator (not necessarily adjointable) $\Phi\colon M\oplus NI\to N\oplus MI$ that is bounded from below when restricted to $F_n\oplus G$. Let us take $$\Phi :=
\begin{pmatrix}
T & -\iota_{NI,N} \\
D_1 & T^*
\end{pmatrix},$$ where $\iota_{NI,I}$ is the inclusion map of $NI$ in $N$. In order to show that $\Phi$ is bounded from below it is enough to show that $\Phi'\Phi$ is bounded from below, where $\Phi'$ is some bounded–possibly nonadjointable–operator. Let us choose $\Phi'\colon N\oplus MI\to M\oplus NI$ as follows: $$\Phi' :=
\begin{pmatrix}
T^* & \iota_{MI,M} \\
-D_2 & T
\end{pmatrix}.$$ where $D_2\colon N\to N$ has image in $NI$ and is bounded from below on $G$. Then $\Phi'\Phi$ has the form $$\Phi'\Phi=
\begin{pmatrix}
T^*T+D_1 & 0 \\
* & TT^*+D_2
\end{pmatrix}.$$ To show that the restriction of $\Phi'\Phi$ to $F_n\oplus G$ is bounded from below it is enough to show that the operators on the main diagonal are bounded from below (because the upper right corner is 0). This is true by our choice of $F_n$ and $D_2$. So $F_n\oplus G$ is isomorphic to a submodule of $N\oplus MI$. Taking supremum over $F_n$ and $G$ we get that $[M]+ [NI]\leq [N]+[MI]$.
Now suppose that $M/MI\preceq N/NI$. Let $F{\subseteq\!\subseteq}M$. Then $F/FI{\subseteq\!\subseteq}M/MI$, so $F/FI$ is isomorphic to a submodule of $N/NI$. It follows that $[F]+[N]I\leq [N]+[F]I$. Taking supremum over all $F$, $F{\subseteq\!\subseteq}M$, we get $[M]+ [NI]\leq [N]+[MI]$.
Let $I$ and $J$ be $\sigma$-unital ideals. Suppose that $[M/M(I\cap J)]=[N/N(I\cap J)]$. Then $$[M]I+N[J]=[M](I+J)+[N](I\cap J)=[M](I\cap J)+[N](I+J).$$
We have $[M]I+[N]J=[M(I+J)]I+[NJ]=[M(I+J)]+[NJ]I=[M](I+J)+[N](I\cap J)$.
\[cuntzquotient\] The map $Cu_s(\pi)$ restricted to $Cu_s(A)+[l_2(I)]$ is an isomorphism onto $Cu_s(A/I)$.
As remarked in the introduction, it follows from Theorem \[formula\] and Kasparov’s Stabilization Theorem that the map $Cu_s(\pi)$ is injective on $Cu_s(A)+[l_2(I)]$. $Cu_s(\pi)$ is surjective, since every $A/I$-Hilbert module can be embedded in $l_2(A/I)$, and then have its preimage taken by the quotient map $l_2(A)\to l_2(A/I)$. $Cu_s(\pi)$ is also surjective restricted to $Cu_s(A)+[l_2(I)]$, since adding $[l_2(I)]$ does not change the image in $Cu_s(A/I)$. Hence, $Cu_s(\pi)$ sends $Cu_s(A)+[l_2(I)]$ isomorphically onto $Cu_s(A/I)$.
The description of $Cu_s(A/I)$ obtained in Corollary \[cuntzquotient\] assumes that the ideal $I$ is $\sigma$-unital. It is possible to obtain $Cu(A/I)$ as a quotient of $Cu(A)$ by a suitable equivalence relation without assuming that $I$ is $\sigma$-unital. Since $Cu_s(A)\simeq Cu(A\otimes K)$, this result can also be applied to the stabilized Cuntz semigroup.
Recall that $Cu(A)\simeq W(A)$, the latter semigroup defined as equivalence classes of positive elements on $\cup_n M_n(A)$ (see [@rordam]). Given $[a],[b]\in W(A)$ let us say that $[a]\leq _I [b]$ if there are $c\in M_n(I)^+$ for some $n$ such that $[a]\leq [b]+[c]$. We say that $[a]\!\!\sim_I [b]$ if $[a]\leq_I [b]$ and $[b]\leq_I [a]$.
The semigroups $W(A)/\!\!\sim_I$ and $W(A/I)$ are isomorphic.
Let $\pi\colon A\to A/I$ be as before, the quotient homomorphism. Let us show that the map $W(\pi)([a])=\pi([a])$ induces an isomorphism after passing to the quotient by $\sim_I$. Since $\pi$ is surjective, $W(\pi)$ is also surjective. It only rests to show that $W(\pi)([a])\leq W(\pi)([b])$ if and only if $[a]\leq_I [b]$.
Let $a$ and $b$ be positive elements in $M_{n}(A)$, such that $\pi(a) \preceq \pi(b)$. For all $k$, there is $d_k \in M_n(A/I)$ such that $\|\pi(a) - d_k\pi(b)d_k^*\| \leq 1/k$. By [@kirchberg-rordam2 Lemma 2.2], there is $d_k' \in M_n(A/I)$ such that $(\pi(a) - 1/k))_+ =
d_k'\pi(b)d_k'^*$. Let $f_k\in M_n(A)$ be such that $\pi(f_k) = d_k'$. We have $$(a - 1/k)_{+} = f_kbf_k^* + i_k \leq f_kbf_k^* + i_k^+,$$ for some $i_k^+\in M_n(I)^+$. We get that $[(a - 1/k)_{+}] \leq
[b] + [i_k^+]$. Let $i \in M_n(I)^+$ be an element such that $[i]$ majorizes the sequence $[i_k^+]$ (for example, $i = \sum i_k/(2^k\|i_k^+\|)$. Taking supremum over $k$ in $[(a - 1/k)_+] \leq [b] + [i]$ we get $[a]\leq [b] + [i]$. Hence $[a] \leq_I [b]$.
Exactness of the Cuntz semigroup functor
========================================
Given $S$ and $T$ ordered, abelian semigroups, and $\phi\colon S \to T$ an order preserving semigroup map, let us define $\mathrm{Ker}(\phi)$ and $\mathrm{Im}(\phi)$ as follows: $$\begin{aligned}
\mathrm{Ker}(\phi) & := \{\, (s_{1},s_{2})\in S \times S \mid \phi(s_{1}) \leq
\phi(s_{2}) \,\},\\
\mathrm{Im}(\phi) & := \{\, (t_{1}, t_{2}) \in T \times T \mid \exists\, s_{1}, s_{2} \in S,\, t_{1}
\leq \phi(s_{2}) + t_{2} \,\}.\end{aligned}$$ We denote by $\operatorname{im}\phi$ and $\ker \phi$ the image and the kernel of $\phi$ (i.e., the elements mapped to 0), in the standard sense.
By a short exact sequence of ordered semigroups we mean one which is exact with respect to the two notions of image and kernel defined above.
\[cuntzsexact\] Let $I$ be a $\sigma$-unital ideal of $A$. The short exact sequence $$0 \longrightarrow I \stackrel{\iota}{\longrightarrow} A
\stackrel{\pi}{\longrightarrow} A/I \longrightarrow 0,$$ induces split, short exact sequences of ordered abelian semigroups $$\begin{aligned}
0 \longrightarrow Cu_s(I) \stackrel{r}{\leftrightarrows} Cu_s(A)
\stackrel{q}\leftrightarrows Cu_s(A/I) \longrightarrow 0,\label{shortstable}\\
0 \longrightarrow Cu(I) \stackrel{r}{\leftrightarrows} Cu(A)
\longrightarrow Cu(A/I) \longrightarrow 0.\label{shortunstable} \end{aligned}$$ These sequences are also exact in the standard sense.
The maps $r$ and $q$ are defined as follows: $r([H]):=[HI]$ and $q([M]):=[M']+[l_2(I)]$, where $[M']$ is such that $Cu_s(\pi)([M'])=[M]$.
We have already shown in Corollary \[cuntzquotient\] that the maps $Cu_s(\pi)$ and $Cu(\pi)$ are surjective. The maps $Cu_s(\iota)$ and $Cu(\iota)$ are injective, since if $M$ is Cuntz smaller than $N$ as $I$-modules, then the same holds when they are regarded as $A$-modules.
Let us prove the exactness of the sequence and note that the same proof works also for the sequence . Exactness at $Cu_s(I)$ and $Cu_s(A/I)$ is easily verified. To check the exactness in the middle of the sequence it suffices to prove that $\mathrm{Ker}(Cu_s(\pi))\subseteq\mathrm{Im}(Cu_s(\iota))$, the other inclusion being obvious. The pair $([M],[N])$ belongs to $\mathrm{Ker}(Cu_s(\pi))$ precisely when $Cu_s(\pi)([M])=Cu_s(\pi)([N])$, and this is equivalent by Theorem \[formula\] with the fact that $[M]+[N]I=[N]+[M]I$. This shows that $([M],[N])\in\mathrm{Im}(Cu_s(\iota))$, and hence $\mathrm{Ker}(Cu_s(\pi))\subseteq\mathrm{Im}(Cu_s(\iota))$.
It only remains to show that the maps $q$ and $r$ define splittings of $Cu_s(\pi)$ and $Cu_s(\iota)$ respectively. We have already observed that $Cu_s(\pi)$ restricted to $Cu_s(A)+[l_2(I)]$ is an isomorphism of ordered semigroups. Its inverse is $q$. We have also noted that $M=MI$ (i.e., $M$ is a Hilbert $I$-module) if and only if $[M]I=[M]$, which shows that $r$ is a splitting of $Cu_s(\iota)$.
The restriction of $r$ to $Cu(A)$ is a splitting of $Cu(\iota)$.
*Remarks.* The map $r$ does not preserve the way below relation of elements in $Cu_s(A)$ (for the definition of this relation, see [@kgc]). So, it is not a morphism in the category of ordered semigroups defined by Coward, Elliott and Ivanescu. However, $r$ does preserve directed suprema.
Let $\{[H_i]\}_{i=1}^\infty$ be an increasing sequence in $Cu_s(A)$ with supremum $[H]$. Then $[H]I=\sup_i ([H_i]I)$.
It will be enough to show that $[H]I\leq \sup_i([H_i]I)$, the other inequality being obvious. Let $F$ be a compactly contained submodule of $HI$. Then $F$ is compactly contained in $H$, hence we conclude that $[F]\leq[H_i]$ for some $i$ (see [@kgc Theorem 1]). This implies that $[F]\leq[H_i]I$, so $[F]\leq \sup_i ([H_i]I)$. Taking supremum over $F$, we get that $[H]I\leq \sup_i([H_i]I)$.
Proof of Theorem \[kasparov-lift\]. {#proofofkasparov}
===================================
By Theorem \[hilbert-tietze\], there is an operator $T\in B(M,N)$ that lifts $\phi$. The following diagram commutes: $$\xymatrix{
M\ar[r]^T\ar[d] & N\ar[d]\\
M/MI\ar[r]^\phi & N/NI.
}$$ The operator $T$ in general will not be an isomorphism. However, by the commutativity of this diagram, and the fact that $\phi^*=\phi^{-1}$, we do have that $$N=\operatorname{im}T+NI\, \hbox{ and }\,M=\operatorname{im}T^* + MI.$$ We now follow the ideas of Mingo and Phillips’s proof of the Stabilization Theorem ([@mingo-phillips Theorem 1.4]) to find $\widetilde T\colon M\oplus l_2(I)\to N\oplus l_2(I)$ such that $\widetilde T$ and $\widetilde T^*$ have dense range. The desired isomorphism $\Phi$ will be obtained by the polar decomposition of $\widetilde T$.
Since $I$ is $\sigma$-unital, the modules $MI$ and $NI$ are countably generated. Let $\{\eta_k\}$ and $\{\zeta_k\}$ be infinite sequences of generators of $MI$ and $NI$ respectively, such that each generator appears infinitely often. Let us define operators $\phi_1\colon l_2(I)\to N$, $\phi_2\colon l_2(I)\to l_2(I)$, and $\phi_3\colon l_2(I)\to M$ by the formulas $$\begin{aligned}
\phi_1((x_k)) := \sum_k \frac{1}{2^k}\eta_k x_k,\quad
\phi_2((x_k)) := (\frac{1}{4^k} x_k ),\quad
\phi_3((x_k)) := \sum_k \frac{1}{2^k} \zeta_k x_k.\end{aligned}$$ Let $\widetilde T\colon M\oplus l_2(I)\to N\oplus l_2(I)$ be defined by the matrix $$\widetilde T :=
\begin{pmatrix}
T & \phi_1\\
\phi_3^* & \phi_2
\end{pmatrix}.$$ Notice that $\widetilde T$ is still a lift of $\phi$. We have $\widetilde T(0,y)=(\phi_1y,\phi_2 y)$, for $y\in l_2(I)$. It is argued in the proof of [@mingo-phillips Theorem 1.4], that this set is dense in $NI\oplus N$. Thus $NI\oplus N\subseteq \overline{\operatorname{im}\widetilde T}$. Also, $\widetilde T(x,0)= (Tx,0) +(0,\phi_3^*y)$. So, $\operatorname{im}T\oplus 0\subseteq \overline{\operatorname{im}\widetilde T}$. We conclude that $\operatorname{im}\widetilde T$ is dense in $M_2\oplus l_2(I)$. In the same way we show that $\widetilde T^*$ has dense range. Thus, the operator $\widetilde T$ admits a polar decomposition of the form $\widetilde T=\Phi(\widetilde T^*\widetilde T)^{1/2}$, with $\Phi$ an isomorphism (see Proposition 15.3.7 [@wegge-olsen]). Passing to the quotients $M/MI$ and $N/NI$, the operator $\widetilde T^*\widetilde T$ induces the identity. So $\Phi$ lifts $\phi$.
Multiplier algebras
===================
Let $A$ be a $\sigma$-unital algebra and $I$ a $\sigma$-unital closed two-sided ideal of $A$. In this section we use Theorem \[kasparov-lift\] to explore the relationship between the multiplier algebras $M(A)$ and $M(A/I)$.
We shall consider $A$ and $I$ as countably generated right Hilbert modules over $A$. We shall identify the algebra $K(A)$ with $A$, and the algebra $B(A)$ with $M(A)$. All throughout this section we make the following two assumptions:
\(1) the ideal $I$ is stable,
\(2) $A\simeq A\oplus I$ as $A$-Hilbert modules.
Let us denote by $s\colon M(A)\to M(I)$ the map given by restriction of the multipliers of $A$ to the invariant submodule $I$. Let $\tilde\pi\colon M(A)\to M(A/I)$ be the extension of the quotient map $\pi\colon A\to A/I$ by strict continuity. Recall that, by the noncommutative Tietze extension theorem, $\tilde\pi$ is surjective.
Recall the fact that for $p,q\in M(A)\otimes M_n({\mathbb{C}})$ projections, the modules $pA^n$ and $qA^n$ are isomorphic if and only if $p$ and $q$ are Murray-von Neumann equivalent. The following lemma gives an alternative way of expressing conditions (1) and (2) above.
\[piprojections\] The following propositions are equivalent.
\(i) The ideal $I$ is a direct summand of $A$ as a right $A$-Hilbert module.
\(ii) There is a projection $P_I\in M(A)$ such that $P_IA\subseteq I$ and $s(P_I)$ is Murray-von Neumann equivalent to the unit of $M(I)$.
Any two projections of $M(A)$ that satisfy (ii) are Murray-von Neumann equivalent in $M(A)$.
Suppose we have (i). Let $A=I_1\oplus N_1$, with $I_1\simeq I$ as right Hilbert $A$-modules. Let $P_I\in M(A)$ be the projection onto $I_1$. Since $I_1$ is an $I$-module, $I_1I=I_1$, hence $P_IA=P_II=I_1\subseteq I$. Since the $I$-module $P_II$ is isomorphic to $I$, it follows that $P_I$, as an $I$ multiplier, is Murray von Neumann equivalent to the identity of $M(I)$.
Suppose we have (ii). The $I$-modules $I$ and $P_II$ are isomorphic. Hence, they are isomorphic as $A$-modules. Since $P_IA\subseteq I$, we have $P_II=P_IA$. Hence, $P_IA$ is a direct summand of $A$ isomorphic to $I$.
If $P_I^{(1)}$ and $P_I^{(2)}$ satisfy (ii) then $P_I^{(1)}A=P_I^{(1)}I\simeq I\simeq P_I^{(2)}I=P_I^{(2)}A$. Thus, $P_I^{(1)}$ and $P_I^{(2)}$ are Murray-von Neumann equivalent.
\[unitary-projections\] Suppose $A$ and $I$ satisfy conditions (1) and (2) above. The following propositions are true.
\(i) Every unitary of $M(A/I)$ lifts to a unitary of $M(A)$.
\(ii) If $p$ and $q$ are projections in $M(A)$ such that $\tilde\pi(p)$ and $\tilde\pi(q)$ are Murray-von Neumann equivalent in $M(A/I)$, then $p\oplus P_I$ and $q\oplus P_I$ are Murray-von Neumann equivalent in $M_2(M(A))$.
\(iii) For every projection $p_0\in M(A/I)$ there is $p\in M(A)$ such that $\tilde\pi (p)$ is Murray-von Neumann equivalent to $p_0$.
\(i) Let $\Phi\colon A\to A\oplus I$ be an $A$-module isomorphism. This map induces an isomorphism $\phi\colon A/I\to (A\oplus I)/(A\oplus I)I$, and composing with the canonical identification of $(A\oplus I)/(A\oplus I)I$ and $A/I$, we get a unitary $\phi'\colon A/I\to A/I$. By Theorem \[kasparov-lift\], we can lift this unitary to a unitary $\Phi'\colon A\oplus I\to A\oplus I$. Now the map $\Phi_0=(\Phi')^{-1}\Phi$ is an isomorphism of the Hilbert modules $A$ and $A\oplus I$ that induces the identity in the quotient.
Let $u\in M(A/I)$ be unitary. By Theorem \[kasparov-lift\], there is a unitary $U\colon A\oplus I\to A\oplus I$ that lifts $u$. Then $\Phi_0^*U\Phi_0\in M(A)$ is a unitary that lifts $u$.
\(ii) Since the $A/I$-modules $\pi(p)A/I$ and $\pi(q)A/I$ are isomorphic, we get $pA\oplus I\simeq qA\oplus I$. We have $P_IA\simeq I$. Hence, $pA\oplus P_IA\simeq qA\oplus P_IA$. So $p\oplus P_I$ is Murray-von Neumann equivalent to $q\oplus P_I$.
\(iii) The $A$-modules $\pi^{-1}(p_0A/I)\oplus \pi^{-1}((1-p_0)A/I)$ and $A$ are isomorphic in the quotient (to $A/I$). Thus $\pi^{-1}(p_0A/I)\oplus \pi^{-1}((1-p_0)A/I)\oplus I\simeq A\oplus I\simeq A$. So $\pi^{-1}(p_0A/I)$ is a direct summand of $A$. Let $p\in M(A)$ be such that $pA\simeq \pi^{-1}(p_0A/I)$. Then $\pi(p)A/I\simeq p_0A/I$, so $\pi(p)$ is Murray-von Neumann equivalent to $p_0$.
*Remarks.* If $A$ and $I$ satisfy conditions (1) and (2), then $M_n(A)$ and $M_n(I)$ satisfy them as well. So Proposition \[unitary-projections\] applies to the pair $(M_n(A)$, $M_n(I))$. If $A$ is stable then $A\oplus I\simeq A$ (by the Stabilization Theorem), and $I$ is stable. So (1) and (2) are verified in this case too. More generally, suppose there is $B$ stable such that $I\subseteq A\subseteq B$, and $I$ is an ideal of $B$. Then there is $P_I\in M(B)$ that satisfies (ii) of Lemma \[piprojections\]. The restriction of $P_I$ to $A$ is in $M(A)$ and satisfies (ii) of Lemma \[piprojections\]. Hence, in this case the pair $A$, $I$ satisfies (1) and (2).
Proposition \[unitary-projections\] has K-theoretical implications. Part (1), applied to the algebras $M_n(A)$, implies that the map $K_1(M(A))\to K_1(M(A/I))$ is surjective. Parts (ii) and (iii) imply that the map $K_0(M(A))\to K_0(M(A/I))$ is an isomorphism. We can improve these results as follows.
Let $B$ be a unital C\*-algebra. Let $A\otimes B$ be the minimal tensor product of $A$ and $B$. Given $H$ and $E$, Hilbert modules over $A$ and $B$ respectively, let us denote by $H\otimes E$ the external tensor product of $H$ and $E$ (see [@lance]). This is an $A\otimes B$ Hilbert module. Given $A$-Hilbert modules $H_1$ and $H_2$, $B(H_1,H_2)\otimes B$ denotes the norm closed subspace of $B(H_1\otimes B,H_2\otimes B)$ generated by operators of the form $T\otimes b$, with $T\in B(H_1,H_2)$ and $b\in B$. Note that the composition of operators in $B(H_1\otimes B,H_2\otimes B)$ with operators in $B(H_2\otimes B,H_3\otimes B)$ results in operators in $B(H_1\otimes B,H_3\otimes B)$.
Let $M(A,I)$ be the kernel of $\tilde\pi\colon M(A)\to M(A/I)$. We have $M(A,I)=\{\, x\in M(A)\mid xa,ax\in I \hbox { for all } a\in A \,\}$.
\[tensorB\] Let $B$ be a unital C\*-algebra and $A$ and $I$ as before. Let $p\in M(A,I)\otimes B$ be a projection and $P_I'=P_I\otimes 1$, with $P_I$ as in Lemma \[piprojections\] (ii). Then $p\oplus P_I'$ is Murray-von Neumann equivalent to $0\oplus P_I'$.
The multiplier projection $p$ is an operator from $A\otimes B$ to $A\otimes B$ with range contained in $I\otimes B$. Let $\widetilde p\in B(A,I)\otimes B$ denote the adjointable operator obtained by simply restricting the codomain of $p$ to $I\otimes B$. Let $\widetilde P_I'\in B(A,I)\otimes B$ be the corresponding operator for $P_I'$. Notice that $\widetilde p^*\widetilde p=p$, $\widetilde p\widetilde p^*=s(p)\in M(I)\otimes B$, and similarly for $\widetilde P_I'$. By [@wegge-olsen Lemma 16.2], there is $V\in M_2(M(I)\otimes B)$, partial isometry, such that $V^*V=s(p)\oplus s(P_I')$ and $VV^*=0\oplus s(P_I')$. Let $W$ be defined as $$W :=
\begin{pmatrix}
0 & 0\\
0 & (\widetilde P_I')^*
\end{pmatrix}
V
\begin{pmatrix}
\widetilde p & 0\\
0 & \widetilde P_I'
\end{pmatrix}.$$ Then $W^*W=p\oplus P_I'$, $WW^*=0\oplus P_I'$, and $W\in M(A,I)\otimes B$.
We have $$\begin{aligned}
K_i(M(A,I)) &=0, \\
K_i(M(A)) &\simeq K_i(M(A/I)), \\
K_{i}(M(A,I)/I) &\simeq K_{1-i}(I), \end{aligned}$$ for $i=0,1$.
From Proposition \[tensorB\] we deduce that $K_0(M(A,I)$. Taking $B=C({\mathbb{T}})$, we get $K_1(M(A,I))=0$. Now by the six term exact sequence associated to the extension $M(A,I)\to M(A)\to M(A/I)$, we have $K_i(M(A))\simeq K_i(M(A/I))$, $i=0,1$. Looking at the extension $I\to M(A,I)\to M(A,I)/I$, we get that $K_i(I)=K_{1-i}(M(A,I)/I)$, $i=0,1$.
*Question.* If $I=A$ then $A$ is stable, so the unitary group of $M(A)$ is contractible by the Kuiper-Mingo Theorem (see [@wegge-olsen Theorem 16.8]). Is the unitary group of $M(A,I)\,\widetilde{\,}$ contractible in the norm or strict topologies?
Equivariant version of Theorem \[kasparov-lift\]
================================================
Let $G$ be a locally compact (Hausdorff) group acting on the C\*-algebra $A$. A $G-A$ Hilbert C\*-module, or simply a $G-A$-module, is a right Hilbert C\*-module endowed with a continuous action of $G$ such that $$\begin{aligned}
\langle g\cdot v,g\cdot w\rangle &=g(\langle v,w\rangle),\\
g\cdot (va) &=(g\cdot v)(g(a)), \hbox{ for all }g\in G,\, v\in M, \hbox{ and }a\in A.\end{aligned}$$ An operator between $G-A$-modules is equivariant if $T(g\cdot v)=gTv$. The action of $g$ on $T$ is defined as $(g\cdot T)(v)=gT(g^{-1}v)$. $T$ is $G$-continuous if the map $g\mapsto g\cdot T$ is continuous in the norm of operators.
Given a $G-A$ module $M$ we denote by $L_2(G,M)$ the Hilbert C\*-module $L_2(G)\otimes M$, where $L_2(G)$ is the left regular representation of $G$. The action of $G$ on $L_2(G,M)$ is defined as $g\cdot (\lambda\otimes m)=(g\cdot \lambda \otimes g\cdot m)$. The $G-A$-module $L_2(G,M)$ can also be viewed as the completion of $C_c(G,M)$–the $M$-valued continuous functions on $G$ with compact support–with respect to the $A$-valued inner product $\langle h_1,h_2\rangle=\int \langle h_1(g),h_2(g)\rangle \,dg$.
Let $I$ be a $\sigma$-unital, closed, two-sided ideal of $A$ that is invariant by the action of $G$. Then we can define a quotient action of $G$ on $A/I$. More generally, given a $G-A$-module $M$, we can define a natural (quotient) structure of $G-A/I$-module on $M/MI$.
We now state an equivariant version of Theorem \[kasparov-lift\] for compact groups ([@kasparov Theorem 2.1] and [@mingo-phillips Theorem 2.5] in the case $I=A$).
Suppose that the group $G$ is compact. Let $I$ be a $\sigma$-unital, invariant, closed, two-sided ideal of $A$. Let $M$ and $N$ be countably generated $G-A$ modules. Let $\phi\colon M/MI\to N/NI$ be an equivariant isomorphism. Then there is $\Phi\colon M\oplus L_2(G,l_2(I))\to N\oplus L_2(G,l_2(I))$, equivariant isomorphism, that induces $\phi$ in the quotient.
The proof is an adaptation of the proof of Theorem \[kasparov-lift\]. The equivariant isomorphism $\phi\colon M/MI\to N/NI$ can be lifted to an equivariant operator $T\colon M\to N$ by first lifting it to an arbitrary operator $T'$, and then averaging over the group: $Tx=\int (g\cdot T')x \,dg$. (This integration is possible because for all $x\in M$, the function $(g\cdot T')x$ is continuous in $G$.)
Next we construct the operator $\widetilde T$, this time making sure it is equivariant. For this we need to replace the sequences of vectors $\{\eta_k\}$, $\{\zeta_k\}$, generators of $MI$ and $NI$, by equivariant operators $\eta_k\colon L_2(G,I)\to M$, $\zeta_k\colon L_2(G,I)\to N$, such that $\sum \operatorname{im}\eta_k$ is dense in $MI$ and $\sum \operatorname{im}\zeta_k$ is dense in $NI$. This is guaranteed by the following lemma.
Suppose that $H$ is a countably generated $G-A$ module. Let $I$ be as before. Then there is a sequence $\eta_k\colon L_2(G,I)\to H$ of $G$-continuous maps such that $\sum \operatorname{im}(\eta_k)$ is dense in $HI$. If $G$ is compact these maps can be chosen equivariant.
Before proving the lemma, let us proceed with the proof of the theorem. We define the maps $\phi_1$ and $\phi_3$ replacing the vectors $\eta_k$ and $\zeta_k$ for the operators obtained using the lemma. The definition of the map $\phi_2$ is unchanged. The resulting operator $\widetilde T$ is equivariant. Following the same argument of Mingo and Phillips, $\widetilde T$ and $\widetilde T^*$ have dense range. Since the unitary part of an equivariant operator is also equivariant, we get the equivariant isomorphism $\Phi$ by polar decomposition of $\widetilde T$.
Let us prove the lemma. First suppose that $G$ is only locally compact. It is enough to find a $G$-continuous operator from $l_2(L_2(G,I))$ to $H$ with range dense in $HI$. Let $C_1\colon l_2(L_2(G,I))\to HI$ be a $G$-continuous, surjective operator. Its existence is guaranteed by the Stabilization Theorem. Let $C_2\in K(HI)$ with dense range. Then $C_2C_1\colon l_2(L_2(G,I))\to HI$ has dense range, and since it is compact, it is still an adjointable operator after composing it with the inclusion of $HI$ in $H$. If $G$ is compact we need to choose $C_1$ and $C_2$ equivariant. $C_1$ exists by the Stabilization Theorem. We take $C_2=\int (g\cdot C_2')dg$, with $C_2'\in K(HI)^+$ strictly positive. Then $C_2$ is also strictly positive, thus of dense range.
*Remark.* In the case that $G$ is locally compact, Kasparov ([@kasparov]), and Mingo and Phillips ([@mingo-phillips]), obtain a $G$-continuous isomorphism of $M\oplus L_2(G,l_2(A))$ and $L_2(G,l_2(A))$. Thus, it would be desirable to have a $G$-continuous version of Theorem \[kasparov-lift\]. It is possible to obtain a $G$-continuous lift $T$ of $\phi$. Furthermore, the construction of the operator $\widetilde T$ can be carried through. However, the proof breaks down at the last step, since the unitary part of a $G$-continuous operator need not be $G$-continuous.
[99]{} N. P. Brown, F. Perera, A. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C\*-algebras. To appear in J. Reine Agnew. Math.
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Michael Frank, Geometrical aspects of Hilbert C\*-modules, Positivity 3 (1999), 215–243.
G. G. Kasparov, Hilbert C\*-modules: theorems of Stinespring and Voiculescu. J. Operator Theory 4 (1980), no. 1, 133–150.
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E. Kirchberg, M. Rørdam, Infinite non-simple C\*-algebras: absorbing the Cuntz algebra $O_\infty$, Advances in Math 167, No. 2 (2002), 195–264.
E. C. Lance, Hilbert C\*-Modules: A Toolkit for Operator Algebraists. Cambridge, England: Cambridge University Press, 1995.
J. A. Mingo, K-Theory and Multipliers of Stable C\*-Algebras, Transactions of the American Mathematical Society, Vol. 299, No. 1 (1987), 397–411.
J. A. Mingo, W. J. Phillips, Equivariant triviality theorems for Hilbert C\*-modules. Proc. Amer. Math. Soc. 91 (1984), 225–230.
G. K. Pedersen, SAW\*-algebras and corona C\*-algebras, contributions to noncommutative topology. J. Operator Theory 15 (1986), no. 1, 15–32.
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[^1]: 2000 Mathematics Subject Classification: Primary 46L08, Secondary 46L35.
[^2]: Leonel Robert was supported by NSERC
|
---
author:
- 'Sa’ar Zehavi[ [^1] ]{}'
- 'Ivo Fagundes David de Oliveira[ [^2] ]{}'
bibliography:
- 'documento.bib'
title: 'Not Conway’s 99-Graph Problem'
---
Abstract
========
Conway’s 99-graph problem is the second problem amongst the five 1000\$ 2017 open problems set [@CON]. Four out of the five remain unsolved to this day, including the 99-graph problem. In this paper we quote Conway’s definition of the problem and give an alternative interpretation of it, which we humorously name “not Conway’s 99-graph problem". We solve the alternative interpretation completely.
Introduction
============
Conway’s 99-graph problem [@CON] asks the following: “Is there a graph with 99 vertices in which every edge (i.e. pair of joined vertices) belongs to a unique triangle and every none-edge (pair of unjoined vertices) to a unique quadrilateral?".
Put formally, is there a simple, undirected graph $G=(V,E)$, such that the following holds?
- ($\bar{P}1$): $|V| = 99$
- ($\bar{P}2$): $\forall \{x,y\}\in E$, there exists a unique $z\in V$, such that\
$\{x,z\},\{y,z\}\in E$.
- ($\bar{P}3$): $\forall \{x,y\}\in E^c$, there exists unique pair $z,w\in V$, such that\
$\{x,z\},\{y,z\},\{x,w\},\{y,w\}\in E$.
It can be proven that a graph satisfying ($\bar{P}2-3$) is, in fact, regular. Whether a regular graph, satisfying ($\bar{P}1-3$) exists, is a known open problem. Such a graph is also called strongly regular. Strongly regular graphs are a parametric family of graphs. A counting argument shows that a strongly regular graph satisfying ($\bar{P}1-3$) has vertex degree of 14.
An alternative interpretation of Conway’s 99-graph problem is the following, which we humorously name “not Conway’s 99-graph problem":
Is there a simple, undirected graph $G=(V,E)$, such that the following holds?
- (P1): $|V| = 99$
- (P2): $\forall e\in E$, there exists a unique triangle $\Delta\in G$ such that $e\in \Delta$.
- (P3): $\forall e\in E^c$, there exists a unique quadrilateral $\Box\in G^c$ such that $e\in \Box$.
We prove the following:
\[theorem:conway\] Assume $G=(V,E)$, is a simple graph satisfying (P2) and (P3), then $|V| = 5$ or $|V| = 3$. I.e. $G$ is either a triangle or two triangles intersecting in a single vertex.
The previous theorem, in particular shows that there does not exist a $G$ satisfying (P1-3) simultaneously. In the next section we supply an overview of the proof of our theorem.
Overview
========
For simplicity, due to our use of illustrations, let us name the edges “blue edges", and the none-edges “red edges". Assuming $G$ satisfies (P2-3), we know that every blue edge belongs to a unique blue triangle, and every red edge belongs to a unique red quadrilateral. Our main 2 lemmas are the following.
\[lemma:one\] If $G=(V,E)$ satisfies (P2-3), then $G$ has no red triangles.
If $G=(V,E)$ satisfies (P2-3), then $G$ has no blue quadrilaterals.
Assuming $G$ is a graph with $n$ vertices, satisfying (P2-3), the previous lemmas imply that the graph induced by the red edges is triangle free and hence has at most $n^2/4$ edges, and that the graph induced by the blue edges is quadrilateral free, and hence has at most $(1 + \sqrt{2n - 3})n/4$ edges. The red and blue edges, together, form the complete graph, and hence the following inequality must hold: $n^2/4 + (1 + \sqrt{2n - 3})n/4 \ge {n\choose 2}$. The previous implies that $n<9$.
\[theorem:main\] If $G$ satisfies (P2-3), then $|V|<9$.
Theorem \[theorem:main\] implies the inexistence of a graph satisfying (P1-3), concluding with a *negative* answer to the “not 99-graph problem". Theorem \[theorem:main\], together with extensive search (for graphs with at most 8 vertices) implies Theorem \[theorem:conway\], which states the only possible configurations for graphs satisfying (P2-3) are attained when $n=3$ or $n=5$, hence solving the “not 99-graph problem" for all possible values of $n$.
$G$ has no red triangles - proof of Lemma 1
===========================================
We will prove by case analysis that $G$ satisfying (P2-3) has no red triangles. In the following, there are several graph illustrations. Red and blue edges are denoted by colored lines. Every edge in our graphs is either red or blue. In case an edge does not appear in our illustrations it is either undecided or irrelevant to the argument. Let us assume by contradiction the existence of a red triangle, then there is some red triangle $\Delta(ABC)$:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A);
As every red edge is a part of a unique red quadrilateral, there is some red quadrilateral containing $AB$. As a quadrilateral cannot contain all edges of a triangle, there are two possible cases:
- Case 1 - Each edge of $\Delta(ABC)$ lies in a unique quadrilateral.
- Case 2 - There are two edges of $\Delta(ABC)$ sharing a quadrilateral.
Case 1 is impossible.
*Proof*: Let us assume the first case holds, then there are vertices $D$ and $E$, different than $A$, $B$ and $C$, such that $AB\in\Box(ABED)$:
\* Note that quadrilateral $\Box(ABED)$, refers to the cycle of length 4 comprised of the sides $AB,BE,ED,DA$.
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E);
The edge $AC$ also lies in a unique quadrilateral, which by our assumption, is edge wise disjoint to $\Box(ABED)$. Therefore, there are vertices $F,G\in V$, such that $AC$ is in the red quadrilateral $\Box(ACGF)$. Is it possible that $\{D,E\}\cap\{F,G\}\neq \emptyset$? Assume that $D$ is shared by the two quadrilaterals. This implies that $\Box(ACGF)$ is either $\Box(ACDF)$ or $\Box(ACGD)$. Clearly, as $\Box(ACGD)$ and $\Box(ABED)$ share $AD$, in contradiction to (P3), it has to be that $D\neq F$. Thus, we collapse to $\Box(ACGF) = \Box(ACDF)$, in which case $CD$ is red, and we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (C) edge \[above\] node (D);
This is impossible since now $BE$ is shared by both $\Box(ABED)$ and $\Box(CBED)$, a contradiction to (P3). Similarly, we reach a contradiction by assuming $E\in\{F,G\}$, as this implies either $EA$ or $EC$ are red. Thus, assuming we are in case one implies that $\{F,G\}\cap\{D,E\}=\emptyset$. So we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{}; (F) at (-2,2) [F]{}; (G) at (0,3.6) [G]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (A) edge \[above\] node (F); (F) edge \[above\] node (G); (G) edge \[above\] node (C);
Similar analysis for $BC$, implies the existence of $H,I\notin\{A,B,C,D,E,F,G\}$, such that:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{}; (F) at (-2,2) [F]{}; (G) at (0,3.6) [G]{}; (H) at (4,3.6) [H]{}; (I) at (6,2) [I]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (A) edge \[above\] node (F); (F) edge \[above\] node (G); (G) edge \[above\] node (C); (C) edge \[above\] node (H); (H) edge \[above\] node (I); (I) edge \[above\] node (B);
Note that $GD$ is blue, for if otherwise, we have edge $AD$ in both quadrilaterals $\Box(DABE)$ and $\Box(DAFG)$. Similarly, $GI$ and $ID$ are blue. We have:
\(A) at (-1,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{}; (F) at (-2,2) [F]{}; (G) at (0,3.6) [G]{}; (H) at (4,3.6) [H]{}; (I) at (6,2) [I]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (A) edge \[above\] node (F); (F) edge \[above\] node (G); (G) edge \[above\] node (C); (C) edge \[above\] node (H); (H) edge \[above\] node (I); (I) edge \[above\] node (B);
\(G) edge \[above\] node (I); (D) edge \[above\] node (G); (I) edge \[above\] node (D);
Now, if $AG$ is red, then $\Box(AGCB)$ is a red quadrilateral sharing $AB$ with $\Box(ABED)$, contradicting (P3). Also, if $AI$ is red, then we have a red $\Box(ACHI)$ sharing $CH$ with $\Box(BCHI)$. Therefore, $AG$ and $AI$ must be blue, and we have:
\(A) at (-1,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{}; (F) at (-2,2) [F]{}; (G) at (0,3.6) [G]{}; (H) at (4,3.6) [H]{}; (I) at (6,2) [I]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (A) edge \[above\] node (F); (F) edge \[above\] node (G); (G) edge \[above\] node (C); (C) edge \[above\] node (H); (H) edge \[above\] node (I); (I) edge \[above\] node (B);
\(G) edge \[above\] node (I); (D) edge \[above\] node (G); (I) edge \[above\] node (D); (A) edge \[above\] node (G); (I) edge \[above\] node (A);
Implying that $IG$ is a blue edge shared by $\Delta(AGI)$ and $\Delta(DGI)$ contradicting (P2). Hence, case 1 is impossible. Q.E.D.
Case 2 is impossible.
*Proof*: We assume now in contradiction that case 2 holds, and w.l.o.g., that $AC$ and $BC$ lie in the same quadrilateral. Then we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (F) at (2,4) [F]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (F) edge \[above\] node (A); (F) edge \[above\] node (B);
As $AB$ is part of a unique quadrilateral, there are $D,E$, such that $AB\in\Box(ABED)$. It is easy to check that $D,E\notin\{A,B,C,F\}$. We have established the fundamental structure of case 2:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (F) at (2,4) [F]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (F) edge \[above\] node (A); (F) edge \[above\] node (B);
It is simple to see that $DC$ and $EC$ must be blue, for if $DC$ was red, then we would have quadrilaterals $\Box(DCBE)$ and $\Box(ABED)$ share $BE$. Similarly, $EC$ must be blue, then:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (F) at (2,4) [F]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (F) edge \[above\] node (A); (F) edge \[above\] node (B);
\(D) edge \[above\] node (C); (E) edge \[above\] node (C);
Also, we have $FD$ blue, for if otherwise, $FD$ was red, and then $\Box(FBED)$ and $\Box(ABED)$ would be red quadrilaterals sharing $BE$. Similarly $EF$ must be blue, then we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (F) at (2,4) [F]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (F) edge \[above\] node (A); (F) edge \[above\] node (B);
\(D) edge \[above\] node (C); (E) edge \[above\] node (C); (F) edge \[above\] node (D); (F) edge \[above\] node (E);
Thus, $FC$ must be red, for if otherwise, it would be shared by $\Delta(FDC),\Delta(FEC)$. Omitting the blue edges from the illustration for simplicity, we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (C) at (2,2) [C]{}; (F) at (2,4) [F]{}; (D) at (0,-2) [D]{}; (E) at (4,-2) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (A); (A) edge \[above\] node (D); (B) edge \[above\] node (E); (D) edge \[above\] node (E); (F) edge \[above\] node (A); (F) edge \[above\] node (B); (F) edge \[above\] node (C);
Thus, $\Box(FCBA)$ and $\Box(ABED)$ are red quadrilaterals, sharing $AB$, a contradiction to (P3). Q.E.D.
We have thus established the impossibility of all different configurations that support a red triangle, hence there are no red triangles, proving Lemma 1.
G has no blue quadrilaterals - proof of Lemma 2
===============================================
We will follow similar case analysis arguments to prove the impossibility of $G$ satisfying (P2-3) having a blue quadrilateral. Assume in contradiction that $G$ satisfies (P2-3) and contains a blue quadrilateral, then we have the following substructure of $G$:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (D) at (0,2) [D]{}; (C) at (4,2) [C]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (D); (D) edge \[above\] node (A);
Can two edges, w.l.o.g., $AD$ and $AB$ reside in the same triangle? The answer is no, for if otherwise we would have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (D) at (0,2) [D]{}; (C) at (4,2) [C]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (D); (D) edge \[above\] node (A); (D) edge \[above\] node (B);
Contradicting (P2), as $DB$ is shared by the triangles $\Delta(ADB)$ and $\Delta(DCB)$. Thus, we have each edge of $\Box(ABCD)$ on a different triangle. Say $DC$ is a part of $\Delta(DCE)$, for $E\notin\{A,B,C,D\}$, then we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (D) at (0,2) [D]{}; (C) at (4,2) [C]{}; (E) at (2,4) [E]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (D); (D) edge \[above\] node (A); (E) edge \[above\] node (D); (E) edge \[above\] node (C);
Note that $E$ cannot be connected by a blue edge to any other vertex of $ABCD$, for if otherwise, assume w.l.o.g. that it is connected to $A$ by a blue edge, then we have $DE$ in both triangles $\Delta(ADE)$ and $\Delta(DEC)$. The previous statement, together with the fact that each blue edge belongs to a unique triangle, implies the existence of 3 other vertices, $F, G$ and $H$, forming the following structure:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (D) at (0,2) [D]{}; (C) at (4,2) [C]{}; (E) at (2,4) [E]{}; (F) at (-2,1) [F]{}; (G) at (6,1) [G]{}; (H) at (2,-2) [H]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (D); (D) edge \[above\] node (A); (E) edge \[above\] node (D); (E) edge \[above\] node (C); (F) edge \[above\] node (D); (F) edge \[above\] node (A); (H) edge \[above\] node (A); (H) edge \[above\] node (B); (G) edge \[above\] node (C); (G) edge \[above\] node (B);
To end our argument we note that $DG$ must be red, for if otherwise $CG$ would lie in both triangles $\Delta(DCG)$ and $\Delta(CGB)$. Similarly, $GH$ and $HD$ must be red, but then we have:
\(A) at (0,0) [A]{}; (B) at (4,0) [B]{}; (D) at (0,2) [D]{}; (C) at (4,2) [C]{}; (E) at (2,4) [E]{}; (F) at (-2,1) [F]{}; (G) at (6,1) [G]{}; (H) at (2,-2) [H]{};
\(A) edge \[above\] node (B); (B) edge \[above\] node (C); (C) edge \[above\] node (D); (D) edge \[above\] node (A); (E) edge \[above\] node (D); (E) edge \[above\] node (C); (F) edge \[above\] node (D); (F) edge \[above\] node (A); (H) edge \[above\] node (A); (H) edge \[above\] node (B); (G) edge \[above\] node (C); (G) edge \[above\] node (B);
\(D) edge \[above\] node (G); (G) edge \[above\] node (H); (H) edge \[above\] node (D);
Then $\Delta(DGH)$ is a red triangle, and by Lemma 1, we reach a contradiction. Hence, there are no blue quadrilaterals.
Proof of Theorem 2
==================
Let us denote the number of blue edges by $b$ and the number of red edges by $r$. As $b$ and $r$ are respectively the edges and none edges of a simple graph $G$, we must have $r + b = {n\choose 2}$. Assuming $G$ satisfies (P2-3), by Lemma 1, the graph induced by the red edges is triangle free. It is a well known fact that a triangle free graph has at most $n^2/4$ edges. Hence, $r\le n^2/4$. Lemma 2 implies that the graph induced by the blue edges is quadrilateral free. It is known that a quadrilateral free graph has at most $(1+\sqrt{4n-3})n/4$ edges [@C4], hence $b\le\dfrac{n}{4}(1+\sqrt{4n-3})$. We have then established that:
${n\choose 2} = r + b \le \dfrac{n^2}{4} + \dfrac{n}{4}(1+\sqrt{4n-3})$. Which implies that $n<9$.
Q.E.D.
Acknowledgments
===============
We would like to thank Brendan Rooney for notifying us that we, in fact, solved the alternative interpretation of Conwel’s 99-graph problem.
[^1]: Technion, Department of Computer Science, saarzehavi@gmail.com
[^2]: Technion, Department of Computer Science, ivodavid@gmail.com
|
---
abstract: 'This paper addresses the sharpness of a weighted $L^{2}$-estimate for the Fourier extension operator associated to the circle, obtained by J. Bennett, A. Carbery, F. Soria and A. Vargas in 2006. A point left open in their paper was the necessity of a certain $\log R$-factor in the bound. Here, I show that the factor is necessary for all $1/2$-Ahlfors-David regular weights on the circle, but it can be removed for $s$-Ahlfors-David regular weights with $s \neq 1/2$.'
address: 'School of Mathematics, University of Edinburgh'
author:
- Tuomas Orponen
title: 'On Ahlfors-David regular weighted bounds for the extension operator associated to the circle'
---
Introduction
============
This paper is concerned with the Fourier extension operator $g \mapsto \widehat{g d\sigma}$ associated to the circle $S^{1} \subset {\mathbb{R}}^{2}$, defined for all $g \in L^{1}(S^{1})$ by $$\widehat{gd\sigma}(x) = \int_{S^{1}} e^{-2\pi i x \cdot \xi} g(\xi) \, d\sigma(\xi).$$ Here $\sigma$ is the length measure on $S^{1}$. The following weighted inequality for $\widehat{\cdot d\sigma}$ was established by J. Bennett, A. Carbery, F. Soria and A. Vargas [@BCSV] in 2006: $$\label{bcsv} \int_{S^{1}} |\widehat{gd\sigma}(Rx)|^{2} \, d\mu x \lesssim \frac{\log R}{R}\|g\|_{2}^{2}\sup_{R^{-1} \leq \alpha \leq R^{-2/3}} \frac{\mu(T(\alpha,\alpha^{2}R))}{\alpha}.$$ By assumption, the measure $\mu$ is supported on $S^{1}$, and the notation $T(\alpha,\beta)$ stands for an arbitrary rectangle with dimensions $\alpha \times \beta$; so, the $\sup$ is taken over all the scales $R^{-1} \leq \alpha \leq R^{-2/3}$ and over all $(\alpha \times \alpha^{2}R)$-rectangles in ${\mathbb{R}}^{2}$, with any orientation. The quantity $\|g\|_{2}^{2}$ appearing on the right hand side of is the square of the **unweighted** $L^{2}$-norm of $g$ on $S^{1}$. As usual, the notation $A \lesssim B$ means that $A \leq CB$ for some absolute constant $C \geq 1$.
One motivation to study inequalities of the form stems from fact (see [@BCSV Proposition 1] or [@CSV]) that the best constant $A$ in the inequality $$\int_{\{|x| \leq 1\}} |\widehat{gd\sigma}(Rx)|^{2} \, d\mu x \lesssim A \cdot \frac{\|g\|_{2}^{2}}{R}$$ is comparable with the best constant $B$ in the inequality $$\label{multiplier} \int_{\{|x| \leq 1\}} |S^{1/R}f(x)|^{2} \, d\mu x \leq B \int_{{\mathbb{R}}^{2}} |f(x)|^{2} \, dx,$$ where $S^{1/R}$ is the Fourier multiplier in ${\mathbb{R}}^{2}$ with symbol in $\Phi(|\xi| - R)$, and $\Phi$ is a non-negative normalised smooth bump function in one variable.
The necessity of the $\log R$-factor in was left open in [@BCSV], and the primary purpose of the present paper is to address this issue. By definition, a measure $\mu$ on ${\mathbb{R}}^{2}$ is *$s$-Ahlfors-David regular*, if $$\label{ahlforsDavid} \mu(B(x,r)) \sim r^{s}, \qquad x \in {\operatorname{spt}}\mu, \: 0 < r \leq {\operatorname{diam}}({\operatorname{spt}}\mu),$$ where $A \sim B$ is shorthand for $A \lesssim B \lesssim A$. The constant $$M_{R}(\mu) := \sup_{R^{-1} \leq \alpha \leq R^{-2/3}} \frac{\mu(T(\alpha,\alpha^{2}R))}{\alpha}$$ is readily computed for $s$-Ahlfors-David regular measures supported on $S^{1}$. Here are the numbers:
\[constants\] Let $\mu$ be an $s$-Ahlfors-David regular measure supported on $S^{1}$. Then, $$M_{R}(\mu) \sim \begin{cases} R^{(2 - s)/3}, & \text{if } 1/2 \leq s \leq 1,\\ R^{1 - s}, & \text{if } 0 \leq s \leq 1/2. \end{cases}$$
The point to observe is that the number $s = 1/2$ has a special role. Now, consider the quantity $${\mathcal{E}}(\mu,R) := \sup_{\|g\|_{2} = 1} R \cdot \int_{S^{1}} |\widehat{g d\sigma}(Rx)|^{2} \, d\mu x.$$ Inequality can be restated as ${\mathcal{E}}(\mu,R) \lesssim M_{R}(\mu) \cdot \log R$. The main result of the paper, below, shows that the $(\log R)$-factor can be dispensed with for all $s$-Ahlfors-David regular measures with $s \neq 1/2$, but, on the other hand, the factor is necessary for **all** $1/2$-Ahlfors-David regular measures:
\[main\] Let $0 \leq s \leq 1$, and let $\mu$ be an $s$-Ahlfors-David regular measure on $S^{1}$. Then, there exist arbitrarily large $R \geq 1$ such that $${\mathcal{E}}(\mu,R) \sim \begin{cases} M_{R}(\mu), & \text{if } s \neq \tfrac{1}{2},\\ M_{R}(\mu) \cdot \log R, & \text{if } s = \tfrac{1}{2}. \end{cases}$$
Even if there is a slight improvement over for the upper bound in the case $s \neq \tfrac{1}{2}$, the proof is still very much the same as in [@BCSV]; so, this part of the result is included mainly to demonstrate the special role of $s = 1/2$ (nevertheless, a proof is included in Section \[sNeqHalf\]). The upper bound in the case $s = \tfrac{1}{2}$ is precisely , so the main point of the whole paper is to establish the lower bound $${\mathcal{E}}(\mu,R) \gtrsim M_{R}(\mu) \cdot \log R \sim R^{1/2} \cdot \log R$$ for arbitrarily large $R \geq 1$. Note that this also gives a lower bound for the best constant $B$ in the inequality , for all $1/2$-Ahlfors-David regular measures $\mu$ supported on the unit circle.
The cases $s \neq \tfrac{1}{2}$ {#sNeqHalf}
===============================
The constants $M_{R}(\mu)$ in Proposition \[constants\] are computed as follows:
Fix $R^{-1} \leq \alpha \leq R^{-2/3}$. Then, it is possible to choose a rectangle $T(\alpha,\alpha^{2}R)$ so that $T(\alpha,\alpha^{2}R)$ contains an arc $J \subset S^{1}$ of length $\ell(J) \sim \alpha^{2}R$ (to see this, consider first the “hardest” case $\alpha = R^{-2/3}$). It follows that $$\frac{\mu(T(\alpha,\alpha^{2}R))}{\alpha} \gtrsim \frac{\alpha^{2s}R^{s}}{\alpha} = \alpha^{2s - 1}R^{s}.$$ Now, depending on whether $s \leq 1/2$ or $s \geq 1/2$, the expression above is maximised by choosing either $\alpha = R^{-1}$ or $\alpha = R^{-2/3}$ – and, of course, the expression is independent of $\alpha$ when $s = 1/2$. These choices give the lower bounds in Proposition \[constants\]. The upper bounds are obtained by observing that $T(\alpha,\alpha^{2}R)$ is always contained in a ball of radius $\sim \alpha^{2}R$.
Next, I sketch the proof of the upper bounds in Theorem \[main\]:
The proof of in [@BCSV] is based on the following representation of the extension operator. Suppose that $g \in L^{1}(S^{1})$ is so smooth that $$g(e^{2\pi i \theta}) = \sum_{k \in {\mathbb{Z}}} a_{k}e^{2\pi i k \theta}, \qquad \theta \in [0,1].$$ Then $$\label{representation} \widehat{g d\sigma}(Re^{2\pi i\theta}) = \sum_{k \in {\mathbb{Z}}} a_{k}J_{k}(R)e^{2\pi i k \theta}, \qquad \theta \in [0,1],$$ where $J_{k}$ is the $k^{th}$ Bessel function of the first kind. Next, assuming (momentarily) that the non-zero Fourier coefficients of $g$ are supported on the interval $\{R/2,\ldots,R\}$, one decomposes $g$ into $\sim \log R$ pieces $g_{p}$ such that the non-zero Fourier coefficients of $\widehat{g_{p}}(k)$ are supported on those indices $k$ with $R - k \sim 4^{p}R^{1/3}$, $1 \leq 4^{p} \lesssim R^{2/3}$; for a fixed $p$, denote the set of such indices by $A_{p}$ (as in [@BCSV]). At this point, the proof divides into the cases $s > 1/2$ and $s < 1/2$ (the case $s = 1/2$ being already covered in [@BCSV]).
The case $s > 1/2$
------------------
Choosing a small constant $\beta > 0$ and using the linearity of the extension operator, one finds that $$\begin{aligned}
R \cdot \int_{S^{1}} |\widehat{gd\sigma}(Rx)|^{2} \, d\mu x & = R \cdot \int_{S^{1}} \left| \sum_{1 \leq 4^{p} \lesssim R^{2/3}} 2^{-\beta p}2^{\beta p} \widehat{g_{p} d\sigma}(Rx) \right|^{2} \, d\mu x \notag\\
&\label{form7} \lesssim_{\beta} \sum_{1 \leq 4^{p} \lesssim R^{2/3}} 2^{2\beta p} R \cdot \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\mu x. \end{aligned}$$ Next, using the representation , the integrals can be written as $$\label{form8} \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\mu x = \sum_{j,k \in A_{p}} a_{j}\overline{a_{k}}J_{j}(R)J_{k}(R)\hat{\mu}(j - k).$$ Since $|j - k| \lesssim 4^{p}R^{1/3}$ for $j,k \in A_{p}$, one then finds a smooth function $P_{p} \colon {\mathbb{R}}\to {\mathbb{R}}$ satisfying $\widehat{P_{p}}(j - k) = 1$ for all $j,k \in A_{p}$, and $$|P_{p}(t)| \lesssim_{N} \frac{4^{p}R^{1/3}}{(1 + 4^{p}R^{1/3}|t|)^{N}}, \qquad N \in {\mathbb{N}}.$$ Then $$\begin{aligned}
\int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\mu x & = \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \mu \ast P_{p}(x) \, d\sigma x\\
& \leq \|\mu \ast P_{p}\|_{L^{\infty}} \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\sigma x \end{aligned}$$ The first factor is bounded by $$\|\mu \ast P_{p}\|_{L^{\infty}} \lesssim 4^{p(1 - s)}R^{(1 - s)/3}$$ using the growth bound $\mu(B(x,r)) \lesssim r^{s}$ (or see Lemma \[convolutionEstimates\] below for details), while for the second factor, one has (using and $\hat{\sigma}(j - k) = \delta_{j,k}$) $$\int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\sigma x = \sum_{k \in A_{p}} |a_{k}|^{2}|J_{k}(R)|^{2} \lesssim \frac{2^{-p}}{R^{2/3}} \sum_{k \in A_{p}} |a_{k}|^{2} \leq \frac{2^{-p}}{R^{2/3}}\|g\|_{2}^{2}.$$ Here the uniform bound $$|J_{k}(R)| \lesssim R^{-1/2} \cdot \min\left\{k^{1/6},\left|\frac{|R| + |k|}{|R| - |k|}\right|^{1/4} \right\}$$ was used, see [@BCSV Lemma 5] (or use the techniques in the proof of Lemma \[average\] below to deduce the result). So, all in all, $$R \cdot \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\mu x \lesssim R^{(2 - s)/3} \cdot 2^{p(1-2s)}\|g\|_{2}^{2} \sim M_{R}(\mu) \cdot 2^{p(1 - 2s)}\|g\|_{2}^{2}.$$ Thus, if $2\beta < 2s - 1$, one may infer the sum on line adds up to a constant times $M_{R}(\mu) \cdot \|g\|_{2}^{2}$, as desired. A similar argument takes care of functions $g$, whose Fourier support lies in $\{R,\ldots,3R/2\}$, $\{-R,\ldots,-R/2\}$ and $\{-3R/2,\ldots,-R\}$. The remaining cases, where ${\operatorname{spt}}\hat{g} \subset \{|k| > 3R/2\}$ or ${\operatorname{spt}}\hat{g} \subset \{|k| < R/2\}$, have already been dealt with in [@BCSV (2), Proposition 6]. This completes the proof in the case $s > 1/2$, because now any function $g$ can be split up into at most six pieces, each one of which has been handled separately above.
The case $s < 1/2$
------------------
One proceeds almost as above, with the single difference that instead of the factors $2^{\beta p}$ and $2^{-\beta p}$, one introduces $2^{\beta p}/R^{\beta/3}$ and $(2^{\beta p}/R^{\beta/3})^{-1}$. Since $$\sum_{1 \leq 4^{p} \lesssim R^{2/3}} \left(\frac{2^{\beta p}}{R^{\beta/3}}\right)^{2} \lesssim_{\beta} 1, \qquad \beta > 0,$$ one obtains the following analogue of : $$R \cdot \int_{S^{1}} |\widehat{g d\sigma}(Rx)|^{2} \, d\mu x \lesssim_{\beta} \sum_{1 \leq 4^{p} \lesssim R^{2/3}} \left(\frac{2^{p}}{R^{1/3}} \right)^{-2\beta} R \cdot \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\mu x, \quad \beta > 0.$$ Next, the proof continues as in the case $s > 1/2$ until one has reached the estimate $$R \cdot \int_{S^{1}} |\widehat{g_{p}d\sigma}(Rx)|^{2} \, d\mu x \lesssim R^{(2 - s)/3} \cdot 2^{p(1-2s)}\|g\|_{2}^{2}.$$ The numbers here need to be interpreted as $$R^{(2 - s)/3} \cdot 2^{p(1-2s)} = R^{1 - s} \cdot \left(\frac{2^{p}}{R^{1/3}}\right)^{1 - 2s} \sim M_{R}(\mu) \cdot \left(\frac{2^{p}}{R^{1/3}}\right)^{1 - 2s},$$ so that finally $$R \cdot \int_{S^{1}} |\widehat{g d\sigma}(Rx)|^{2} \, d\mu x \lesssim M_{R}(\mu) \cdot \|g\|_{2}^{2} \sum_{1 \leq 4^{p} \lesssim R^{2/3}} \left(\frac{2^{p}}{R^{1/3}}\right)^{1 - 2s -2\beta} \lesssim_{\beta} M_{R}(\mu) \cdot \|g\|_{2}^{2},$$ as long as $2\beta < 1 - 2s$. The rest of the proof is similar to the case $s > 1/2$.
I postpone the discussion of the sharpness of the bounds until the end of the next section.
The case $s = \tfrac{1}{2}$
===========================
In this section, $\mu$ is an $1/2$-Ahlfors-David regular probability measure on $[0,1] \cong S^{1}$, unless otherwise stated, and $R \geq 1$ is large. What follows is a construction of a function $g \in L^{2}(S^{1})$ with $\|g\|_{2} = 1$ such that for appropriately chosen radii $r \sim R$, one has $$\int_{S^{1}} |\widehat{g d\sigma}(r x)|^{2} \, d\mu x \gtrsim \frac{\log R}{R^{1/2}}\|g\|_{2}^{2}.$$ I recall from the previous section that if $g$ has the representation $$g(e^{2\pi i \theta}) = \sum_{k \in {\mathbb{Z}}} a_{k}e^{2\pi i k \theta}, \qquad \theta \in [0,1],$$ then $$\widehat{g d\sigma}(re^{2\pi i\theta}) = \sum_{k \in {\mathbb{Z}}} a_{k}J_{k}(r)e^{2\pi i k \theta}, \qquad \theta \in [0,1],$$ where $J_{k}$ is the $k^{th}$ Bessel function of the first kind. The construction of $g$ is based on these formulae, so one needs some understanding about the asymptotic behaviour of $J_{k}(r)$, for large $k$ and $r$. This is given by the next lemma:
\[average\] Assume that $R - k \sim 4^{p}R^{1/3}$, where $C \leq 4^{p} \leq R^{2/3}/C$ for some large enough absolute constant $C \geq 1$. Then $$A_{k}(R) := \frac{1}{R^{1/3}} \int_{R - R^{1/3}}^{R + R^{1/3}} |J_{k}(r)| \, dr \gtrsim \frac{1}{2^{p/2}R^{1/3}}.$$
In brief, the point here is that when $R - k \sim 4^{p}R^{1/3}$ and $r \sim R$, the function $r \mapsto J_{k}(r)$ oscillates roughly between $-2^{-p/2}R^{-1/3}$ and $2^{-p/2}R^{-1/3}$. Moreover, for $p$ large enough, the frequency of the oscillation is so high that an interval of length $\sim R^{1/3}$ contains a “peak” of $r \mapsto J_{k}(r)$.
To make the argument precise, one needs a fair understanding of the asymptotic behaviour of $J_{k}(r)$, which is fortunately contained in Erdélyi’s treatise [@E]. Namely, (10) on [@E p. 107] gives the asymptotic expansion $$\label{form4} J_{k}(k \lambda) = \left(\frac{\lambda}{2}k^{2/3} \phi'(\lambda) \right)^{-1/2}{\operatorname{Ai}}(-k^{2/3}\phi(\lambda))[1 + O(1/k)],$$ which holds (quoting Erdelyi) *uniformly in $\lambda$, $0 < \lambda < \infty$, as $k \to \infty$, ${\operatorname{Re}}k \geq 0$, except that the error term needs some modification near zeros of ${\operatorname{Ai}}(-k^{2/3}\phi(\lambda))$.* The function $\phi$ is defined as the unique solution to the differential equation $$\phi(\lambda)(\phi'(\lambda))^{2} = 1 - \frac{1}{\lambda^{2}},$$ so that $\phi' > 0$, and $\phi'$ is bounded uniformly away from zero and infinity on $[1/2,3/2]$, see [@E p. 98]. The function ${\operatorname{Ai}}$ is the *Airy function*, which for non-positive arguments has the relatively simple expression $$\label{airy} {\operatorname{Ai}}(-t) = \frac{\sqrt{t}}{3}\left(J_{\tfrac{1}{3}}\left(\tfrac{2}{3}x^{\tfrac{3}{2}}\right) + J_{-\tfrac{1}{3}}\left(\tfrac{2}{3}x^{\tfrac{3}{2}}\right) \right), \quad t \geq 0.$$
Some inconvenience is caused by the fact that the function ${\operatorname{Ai}}(-k^{2/3}\phi(\lambda))$ has, indeed, zeroes in the region relevant to the proof, so one has to get acquainted with the meaning of the “some modification” of the error term. This modification is shown in (15) of [@E p. 102] (for the convenience of the reader interested in tracking the reference, I mention that the functions $Y_{0}$ and $Y_{2}$ in this equation are defined in (3) of [@E p. 98], whereas $y_{0}(\lambda)$ is related to $J_{k}(k\lambda)$ on the last line of [@E p. 106]). Decoding Erdélyi’s notation, the end result looks like $$\begin{aligned}
J_{k}(k \lambda) = & \left(\frac{\lambda}{2}k^{2/3}\phi'(\lambda) \right)^{-1/2} {\operatorname{Ai}}(-k^{2/3}\phi(\lambda))[1 + O(1/k)] \notag\\
&\label{form5} \qquad + O(1/k)\left(\frac{\lambda}{2}k^{2/3}\phi'(\lambda) \right)^{-1/2}{\operatorname{Bi}}(-k^{2/3}\phi(\lambda)). \end{aligned}$$ So, in comparison with , there is the added term on line , where, for non-positive arguments, $${\operatorname{Bi}}(-t) = \sqrt{\frac{t}{3}}\left(J_{-\tfrac{1}{3}}\left(\tfrac{2}{3}x^{\tfrac{3}{2}}\right) - J_{\tfrac{1}{3}}\left(\tfrac{2}{3}x^{\tfrac{3}{2}}\right) \right), \quad t \geq 0.$$ Fortunately, for $t \geq 0$, one has, $|Ai(-t)|, |{\operatorname{Bi}}(-t)| \leq C$ for some absolute constant $C$, so one can now deduce the weaker expansion $$\label{form6} J_{k}(k\lambda) = \left(\frac{\lambda}{2}k^{2/3}\phi'(\lambda) \right)^{-1/2} {\operatorname{Ai}}(-k^{2/3}\phi(\lambda)) + O(k^{-4/3}),$$ valid for $\lambda \in [1/2,3/2]$ and for large enough $k \geq 0$. In particular, with $\lambda = r/k$, $r \in (R - R^{1/3},R + R^{1/3})$ and $3R/4 \leq k \leq 4R/3$, say, one has $$\begin{aligned}
\int_{R - R^{1/3}}^{R + R^{1/3}} |J_{k}(r)| \, dr & = \int_{R - R^{1/3}}^{R + R^{1/3}} \left| J_{k} \left(k \cdot \tfrac{r}{k} \right) \right| \, dr\\
& \geq \int_{R - R^{1/3}}^{R + R^{1/3}} \left| \left(\frac{r}{4k} k^{2/3}\phi'\left(\frac{r}{k}\right)\right)^{-1/2}{\operatorname{Ai}}\left(-k^{2/3}\phi\left(\frac{r}{k}\right)\right)\right| \, dr - O(1/R)\\
& \sim \frac{1}{R^{1/3}} \int_{R - R^{1/3}}^{R + R^{1/3}} \left|{\operatorname{Ai}}\left(-k^{2/3}\phi\left(\frac{r}{k}\right) \right) \right| \, dr - O(1/R), \end{aligned}$$ for large enough $R \geq 1$. It remains to show that $$\label{core} \int_{R - R^{1/3}}^{R + R^{1/3}} \left|{\operatorname{Ai}}\left(-k^{2/3}\phi\left(\frac{r}{k}\right) \right) \right| \, dr \gtrsim \frac{R^{1/3}}{2^{p/2}},$$ assuming that $R - k \sim 4^{p}R^{1/3}$, and $3R/4 \leq k \leq R - R^{1/3}$. The latter condition ensures that $r/k \geq 1$ for all $r$ in the domain of integration, which means that equation (4) on [@E p. 105] is available: it gives that $$k \cdot \frac{2}{3}\left(\phi\left(\frac{r}{k}\right) \right)^{3/2} = \left(r^{2} - k^{2} \right)^{1/2} - k \cdot \cos^{-1} \left(\frac{k}{r}\right) =: f_{k}(r).$$ The derivative of $f_{k}$ is simply $$\label{diff1} f_{k}'(r) = \frac{\sqrt{r^{2} - k^{2}}}{r} \sim \frac{\sqrt{r - k}}{R^{1/2}}$$ for $R/2 \leq k \leq r \leq 2R$, so that in particular $$\label{diff2} f_{k}'(r) \sim \frac{\sqrt{4^{p}R^{1/3}}}{R^{1/2}} = \frac{2^{p}}{R^{1/3}}$$ for $r \in (R - R^{1/3},R + R^{1/3})$ and $R - k \sim 4^{p}R^{1/3}$. Hence, by a change of variable, $$\begin{aligned}
\int_{R - R^{1/3}}^{R + R^{1/3}} \left|{\operatorname{Ai}}\left(-k^{2/3}\phi\left(\frac{r}{k}\right) \right) \right| \, dr & = \int_{R - R^{1/3}}^{R + R^{1/3}} \left|{\operatorname{Ai}}\left(-\left(\frac{3}{2}\right)^{2/3}f_{k}(r)^{2/3} \right) \right| \, dr\\
& \gtrsim \frac{R^{1/3}}{2^{p}} \int_{R - R^{1/3}}^{R + R^{1/3}} \left|{\operatorname{Ai}}\left(-\left(\frac{3}{2}\right)^{2/3}f_{k}(r)^{2/3} \right) \right|f_{k}'(r) \, dr\\
& = \frac{R^{1/3}}{2^{p}} \int_{f_{k}(R - R^{1/3})}^{f_{k}(R + R^{1/3})} |{\operatorname{Ai}}(-cr^{2/3})| \, dr, \end{aligned}$$ where $c = (3/2)^{2/3}$. Write $a_{k} := f_{k}(R - R^{1/3})$ and $b_{k} := f_{k}(R + R^{1/3})$. Because $f_{k}(k) = 0$, one has (using ) $$\begin{aligned}
a_{k} & = \int_{k}^{R - R^{1/3}} f_{k}'(r) \, dr \sim \frac{1}{R^{1/2}} \int_{k}^{R - R^{1/3}} \sqrt{r - k} \, dr\\
& \sim \frac{(R - R^{1/3} - k)^{3/2}}{R^{1/2}} \sim \frac{(4^{p}R^{1/3})^{3/2}}{R^{1/2}} = 2^{3p}, \end{aligned}$$ and $b_{k} - a_{k} \sim 2^{p}$, using . Consequently, by definition of ${\operatorname{Ai}}$, $$\int_{a_{k}}^{b_{k}} |{\operatorname{Ai}}(-cr^{2/3})| \, dr = \int_{a_{k}}^{b_{k}} \frac{\sqrt{r^{2/3}}}{3} \left|J_{\tfrac{1}{3}}(r) + J_{-\tfrac{1}{3}}(r)\right| \, dr \sim 2^{p} \int_{a_{k}}^{b_{k}} \left|J_{\tfrac{1}{3}}(r) + J_{-\tfrac{1}{3}}(r)\right| \, dr.$$ Finally, one can has the following well-known asymptotic expansion for low-order Bessel functions: $$J_{\alpha}(r) = \sqrt{\frac{2}{\pi r}}\left(\cos\left(r - \frac{\alpha \pi}{2} - \frac{\pi}{4}\right) + O(1/r) \right).$$ In particular, $$\left|J_{\tfrac{1}{3}}(r) + J_{-\tfrac{1}{3}}(r)\right| \gtrsim \frac{1}{\sqrt{r}}\left|\cos\left(r - \frac{5\pi}{12}\right) + \cos\left(r - \frac{\pi}{12}\right)\right| - O(r^{-3/2}),$$ which, combined with the previous estimates, gives $$\begin{aligned}
\int_{R - R^{1/3}}^{R + R^{1/3}} & \left|{\operatorname{Ai}}\left(-k^{2/3}\phi\left(\frac{r}{k}\right) \right) \right| \, dr\\
& \gtrsim \frac{R^{1/3}}{2^{3p/2}}\int_{a_{k}}^{b_{k}}\left|\cos\left(r - \frac{5\pi}{12}\right) + \cos\left(r - \frac{\pi}{12}\right)\right| - O(2^{-9p/4}). \end{aligned}$$ Finally, it is clear that the last integral is $\sim b_{k} - a_{k} \sim 2^{p}$. This proves and the lemma.
Now, the construction of the function $g$ can begin. Since the theorem claims that a suitable $g$ can be constructed for **any** $1/2$-dimensional measure $\mu$, it is natural that $g$ should somehow be derived from the measure itself. For the time being, it is convenient to think that $\mu$ is supported on $[0,1] \subset {\mathbb{R}}$ instead of $S^{1}$. Consider the following Littlewood-Paley decomposition of $\mu$. Let $\phi \colon {\mathbb{R}}\to [0,1]$ be a smooth radially decreasing function with $\phi(\xi) = 1$ for $|\xi| \leq R^{1/3}/2$ and $\phi(\xi) = 0$ for $|\xi| \geq R^{1/3}$. Moreover, choose $\phi$ so that $\phi = \hat{\eta}$ for some function $\eta \colon {\mathbb{R}}\to {\mathbb{R}}$ with $\eta(t) \sim R^{1/3}$ for $|t| \leq cR^{-1/3}$, and $$\label{form1} \eta(t) \lesssim_{N} R^{1/3} \sum_{k = 1}^{\infty} \frac{\chi_{\{| \cdot | \leq 2^{k}R^{-1/3}\}}}{2^{kN}}, \qquad N \in {\mathbb{N}}.$$ As usual, define $$\eta_{4^{-p}}(t) := 4^{p}\eta(4^{p}t).$$ Here are some standard estimates for $\mu \ast \eta_{4^{-p}}$:
\[convolutionEstimates\] Suppose that $\mu$ is an $s$-Ahlfors-David regular measure on $[0,1]$. Then $$\mu \ast \eta_{4^{-p}}(t) \gtrsim R^{(1-s)/3}4^{p(1-s)}, \qquad t \in {\operatorname{spt}}\mu,$$ and $$\mu \ast \eta_{4^{-p}}(t) \lesssim R^{(1 - s)/3}4^{p(1 - s)}, \qquad t \in {\mathbb{R}}.$$
To obtain the lower bound, use $$\mu(B(t,cR^{-1/3}4^{-p})) \gtrsim R^{-s/3}4^{-ps},$$ combined with the fact that $\eta_{4^{-p}}(t) \sim R^{1/3}4^{p}$ for $|t| \leq cR^{-1/3}4^{-p}$. The upper bound follows from (with $N = 2$) and $$\mu(B(t,2^{k}R^{-1/3}4^{-p})) \lesssim 2^{ks}4^{-ps}R^{-s/3},$$ which is automatically valid for all $t \in {\mathbb{R}}$ (and not just $t \in {\operatorname{spt}}\mu$).
The lemma has the useful corollary that for some large enough constant $K \geq 1$, $$\label{form2} \Delta_{p}(\mu)(t) := \mu \ast (\eta_{4^{-p - K}} - \eta_{4^{-p}})(t) \gtrsim R^{(1-s)/3}4^{p(1-s)}, \qquad t \in {\operatorname{spt}}\mu.$$ Observe that the Fourier coefficients of $\Delta_{p}(\mu)$ are supported on $$D_{p} := D_{p}^{l} \cup D_{p}^{r} := -[cR^{1/3}4^{p},CR^{1/3}4^{p}] \cup [cR^{1/3}4^{p},CR^{1/3}4^{p}]$$ for some constants $c,C > 0$, depending only on $K$. The next step is to choose a collection of $\sim \log R$ disjoint sets $D_{p}$, with $CR^{1/3}4^{p} \leq R$, with $p$ so large that Lemma \[average\] is applicable. Denote the family of these sets by ${\mathcal{D}}$. Given $D_{p} \in {\mathcal{D}}$, define the Fourier coefficients of $g_{r}$ on $R + D_{p}^{l}$ as follows: $$a_{k}(r) := \widehat{g_{r}}(k) := \frac{\widehat{\Delta_p(\mu)}(k - R){\operatorname{sgn}}J_{k}(r)}{2^{p} A_{k}(R)}, \qquad k \in R + D_{p}^{l}.$$ Observe that the sets $R + D_{p}^{l}$ are disjoint for various $D_{p} \in {\mathcal{D}}$, and the Fourier coefficients of $g_{r}$ are only defined “to the left from $R$” (the reason for this is that the lower bound in Lemma \[average\] is only valid in the region $k < R$). This completes the definition of $g_{r}$, so all other Fourier coefficients are simply zero. Next, assume that $R \in {\mathbb{N}}$, and observe that $$\begin{aligned}
\frac{1}{R^{1/3}} & \int_{R - R^{1/3}}^{R + R^{1/3}} \int_{S^{1}} |\widehat{g_{r}d\sigma}(rx)|^{2} \, d\mu x \, dr\\
& = \int_{0}^{1} \frac{1}{R^{1/3}} \int_{R - R^{1/3}}^{R + R^{1/3}} \left|\sum_{D_{p} \in {\mathcal{D}}} \sum_{k \in R + D_{p}^{l}} a_{k}J_{k}(r)e^{2\pi i (k - R) \theta} \right|^{2} \, dr \, d\mu \theta\\
& \geq \int_{0}^{1} \left| \sum_{D_{p} \in {\mathcal{D}}} \sum_{k \in R + D_{p}^{l}} \frac{1}{R^{1/3}} \int_{R - R^{1/3}}^{R + R^{1/3}} \frac{\widehat{\Delta_p(\mu)}(k - R)|J_{k}(r)|}{2^{p} A_{k}(R)}e^{2\pi i(k - R)\theta}\right|^{2} \, d\mu \theta\\
& = \int_{0}^{1} \left| \sum_{D_{p} \in {\mathcal{D}}} \frac{1}{2^{p}} \sum_{k \in {\mathcal{D}}_{p}^{l}} \widehat{\Delta_{p}(\mu)}(k)e^{2\pi i k\theta}\right|^{2} \, d\mu \theta\\
& \geq \int_{0}^{1} \left(\sum_{D_{p} \in {\mathcal{D}}} \frac{1}{2^{p}} {\operatorname{Re}}\sum_{k \in {\mathcal{D}}_{p}^{l}} \widehat{\Delta_{p}(\mu)}(k)e^{2\pi i k\theta} \right)^{2} \, d\mu \theta. \end{aligned}$$ Because the function $\Delta_{p}(\mu)$ is real-valued, one has $$\begin{aligned}
{\operatorname{Re}}\widehat{\Delta_{p}(\mu)}(k)2^{2\pi ik\theta} = \frac{\widehat{\Delta_{p}(\mu)}(k)e^{2\pi k\theta} + \widehat{\Delta_{p}(\mu)}(-k)e^{2\pi i(-k)\theta}}{2}, \end{aligned}$$ which implies that $${\operatorname{Re}}\sum_{k \in D_{p}^{l}} \widehat{\Delta_{p}(\mu)}(k)e^{2\pi i k\theta} = \frac{1}{2} \sum_{k \in D_{p}} \widehat{\Delta_{p}(\mu)}(k)e^{2\pi i k\theta} = \frac{\Delta_{p}(\mu)(\theta)}{2},$$ recalling that the Fourier coefficients of $\Delta_{p}(\mu)$ are supported on $D_{p}$. For every $D_{p} \in {\mathcal{D}}$, the lower bound with $s = 1/2$ now gives $$\begin{aligned}
\frac{1}{R^{1/3}} & \int_{R - R^{1/3}}^{R + R^{1/3}} \int_{S^{1}} |\widehat{g_{r}d\sigma}(rx)|^{2} \, d\mu x \, dr\\
& \gtrsim \int_{0}^{1} \left(\sum_{D_{p} \in {\mathcal{D}}} \frac{1}{2^{p}} \cdot R^{1/6}2^{p} \right)^{2} \, d\mu \theta \gtrsim R^{1/3} \cdot (\log R)^{2}. \end{aligned}$$ In particular, there exists a radius $r \in (R - R^{1/3},R + R^{1/3})$ such that $$\label{form3} \int_{S^{1}} |\widehat{g_{r}d\sigma}(rx)|^{2} \, d\mu x \gtrsim R^{1/3} \cdot (\log R)^{2}.$$ It remains to give a uniform upper bound for the $L^{2}$-norms of the functions $g_{r}$. By Plancherel and Lemma \[average\], $$\begin{aligned}
\|g_{r}\|_{2}^{2} = \sum_{D_{p} \in {\mathcal{D}}} \sum_{k \in R + D_{p}^{l}} \frac{|\widehat{\Delta_{p}(\mu)}(k - R)|^{2}}{4^{p}A_{k}(R)^{2}} \lesssim R^{2/3} \sum_{D_{p} \in {\mathcal{D}}} \frac{1}{2^{p}} \sum_{k \in {\mathcal{D}}_{p}} |\widehat{\Delta_{p}(\mu)}(k)|^{2}, \end{aligned}$$ because $R - k \sim 4^{p}R^{1/3}$ for $k \in R + D_{p}^{l}$ and $p$ was chosen large enough to begin with. The inner sum on the right hand side is the squared $L^{2}$-norm of $\Delta_{p}(\mu) = \mu \ast (\eta_{4^{-p - K}} - \eta_{4^{-p}})$, which can be bounded by estimating separately the $L^{2}$-norms of $\mu \ast \eta_{4^{-p - K}}$ and $\mu \ast \eta_{4^{-p}}$. For instance, applying the upper bound from Lemma \[convolutionEstimates\], one finds that $$\|\mu \ast \eta_{4^{-p}}\|_{2}^{2} \lesssim R^{1/6} \cdot 2^{p} \int (\mu \ast \eta_{4^{-p}})(t) \, dt = R^{1/6} \cdot 2^{p},$$ since $\int \eta = 1$ and $\mu$ is a probability measure. Finally, $$\|g_{r}\|_{2}^{2} \lesssim R^{2/3} \sum_{D_{p} \in {\mathcal{D}}} R^{1/6} \sim R^{5/6} \cdot \log R,$$ which in combination with shows that $$\int_{S^{1}} |\widehat{g_{r} d\sigma}(rx)|^{2} \, d\mu x \gtrsim \frac{\log R}{R^{1/2}} \cdot R^{5/6} \cdot \log R \gtrsim \frac{\log R}{R^{1/2}} \cdot \|g_{r}\|_{2}^{2}.$$
Sharpness of the bounds for $s \neq \tfrac{1}{2}$
-------------------------------------------------
The sharpness of the bound in the case $s > 1/2$ is easily seen using a Knapp type example; more precisely, take $g(x) = e^{2\pi a \cdot x}\chi_{J}$, where $J$ is an arc of length $\ell(J) \sim R^{-1/3}$ and $a \in {\mathbb{R}}^{2}$ will be chosen momentarily. Then $\|g\|_{2}^{2} \sim \ell(J) = R^{-1/3}$, and $|\widehat{gd\sigma}(x)| \gtrsim R^{-1/3}$ for $x \in T$, where $T$ is a rectangle with dimensions $R^{1/3} \times R^{2/3}$. Choosing $a$ appropriately, this rectangle can be placed so that the intersection $(R \cdot S^{1}) \cap T$ is an arc of length $\sim R^{2/3}$, where $R \cdot S^{1} = \{|x| = R\}$. In particular, $$R \cdot \int_{S^{1}} |\widehat{g d\sigma}(Rx)|^{2} \, d\mu x \gtrsim R^{1/3} \cdot \mu(\{x : Rx \in T\}) \gtrsim R^{(1 - s)/3} \sim R^{(2 - s)/3} \cdot \|g\|_{2}^{2},$$ where the right hand side is $\sim M_{R}(\mu) \cdot \|g\|_{2}^{2}$ under the assumption $s > 1/2$.
In the case $s < 1/2$, the easiest way (at this point, at least) is probably to review the proof of the case $s = 1/2$. The functions $g_{r}$ can be defined exactly as before, and using yields $$\int_{S^{1}} |\widehat{g_{r}d\sigma}(rx)|^{2} \, d\mu x \gtrsim R^{4/3 - 2s}$$ instead of , for some $r \in (R - R^{1/3},R + R^{1/3})$. On the other hand, applying Lemma \[convolutionEstimates\] as above yields the uniform bound $\|g_{r}\|_{2}^{2} \lesssim R^{4/3 - s}$. Then, if $s < 1/2$, Proposition \[constants\] implies that $$R \cdot \int_{S^{1}} |\widehat{g_{r}d\sigma}(rx)|^{2} \, d\mu x \gtrsim R^{1 - s}\|g_{r}\|_{2}^{2} \sim M_{R}(\mu)\|g\|_{2}^{2},$$ completing the proof of Theorem \[main\].
[1234]{} <span style="font-variant:small-caps;">J. Bennett, A. Carbery, F. Soria and A. Vargas</span>: *A Stein conjecture for the circle*, Math. Ann. **336** (2006), pp. 671–695 <span style="font-variant:small-caps;">A. Carbery, F. Soria, A. Vargas</span>: *Localisation and weighted inequalities for spherical Fourier means*, J. Anal. Math. **103** (2007), pp. 133–156 <span style="font-variant:small-caps;">A. Erdélyi</span>: *Asymptotic expansions*, Dover (1956)
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abstract: 'We analyze the spin flip loss for ultracold neutrons in magnetic bottles of the type used in experiments aiming at a precise measurement of the neutron lifetime, extending the one-dimensional field model used previously by Steyerl $\textit{et al.}$ \[Phys. Rev. C $\mathbf{86}$, 065501 (2012)\] to two dimensions for cylindrical multipole fields. We also develop a general analysis applicable to three dimensions. Here we apply it to multipole fields and to the bowl-type field configuration used for the Los Alamos UCN$\tau$ experiment. In all cases considered the spin flip loss calculated exceeds the Majorana estimate by many orders of magnitude but can be suppressed sufficiently by applying a holding field of appropriate magnitude to allow high-precision neutron lifetime measurements, provided other possible sources of systematic error are under control.'
author:
- 'A. Steyerl'
- 'K. K. H. Leung'
- 'C. Kaufman'
- 'G. Müller'
- 'S. S. Malik'
title: Spin flip loss in magnetic confinement of ultracold neutrons for neutron lifetime experiments
---
Introduction {#sec:I}
============
The neutron lifetime $\tau_{n}$ is an important parameter in nuclear physics, particle physics, and cosmology. $\tau_{n}$ can be combined with the neutron $\beta$-decay ($n\rightarrow p+e^{-}+\bar{\nu}_{e}$) correlation coefficients to determine the universal weak interaction vector and axial-vector coupling constants whose values allow searches for semi-leptonic scalar and tensor currents beyond the Standard Model [@BHA01; @CIR01; @GAR01]. A $\tau_{n}$ of reliable precision is also needed for calculations of the neutrino flux expected from solar and reactor sources, including detection efficiencies [@MEN01; @ZHA01], as well as in Big Bang nucleosynthesis calculations. At present, we are confronted by an apparent discrepancy of about three standard deviations between the average $\tau_{n}$ from ultracold neutron (UCN) storage experiments and the $\tau_{n}$ from cold neutron beam experiments. It is the leading source of uncertainty in predictions of the primordial abundance of $^{4}$He [@COC01; @IOC01; @MAT01]. For reviews of $\tau_{n}$ experiments see [@ABE01; @DUB01; @NIC01; @PAU01; @WIE01; @SEE01].
One of the most promising methods to achieve higher precision for the storage-type experiments employs UCN storage in magnetic bottles. In suitable non-uniform magnetic field configurations the neutrons in one spin state should, in principle, experience no losses other than $\beta$-decay, provided that depolarization, i.e. the non-adiabatic transition to the non-storable spin state, is sufficiently suppressed. In most existing and proposed magnetic trap systems the low-field seeking state with spin parallel to the magnetic field is being stored and a spin-flip transition to the anti-parallel state results in a loss by escape from the bottle or by interaction with the bottle walls.
Until recently, UCN depolarization estimates [@VLA01; @POK01] have been based on Majorana’s quasi-classical result [@MAJ01] for a particle with spin moving with constant velocity vector through an infinitely extended non-uniform magnetic field of specific form. For field parameters as currently used for magnetic UCN storage the probability $D$ of a spin flip away from the field direction would be of order $D\sim e^{-10^{6}}$, thus immeasurably small.
Walstrom $\textit{et al.}$ [@WAL01], in 2009, pointed out that the values of $D$ for confined, rather than freely moving, neutrons are much larger. For a particular vertical path in the field of the Los Alamos gravito-magnetic UCN trap they calculated $D\sim 10^{-20}-10^{-23}$, which is much larger than the Majorana estimate but still negligible in any actual or projected neutron lifetime experiment.
In [@STE01; @STE02] we extended this theory to general orbits with both vertical and horizontal velocity components, using the model of an ideal Halbach magnetic field $\mathbf{B}$ where the magnitude $B$ only depends on the vertical position. We found that the lateral motion in the plane where the Halbach field rotates is of critical importance. Taking it into account increases the spin-flip loss thus calculated by some 10 orders of magnitude to $D\sim 10^{-12}$ for a field minimum (holding field) of $B_{h}\approx 5$ mT. This translates into a spin-flip loss rate that is a fraction $\sim\! 10^{-4}$ of the $\beta$-decay rate and decreasing rapidly with larger holding field.
The analysis in [@WAL01; @STE01] is based on the following concepts: For the one-dimensional (1D) field model of [@STE01], the potential $V(z)=gz-\mu B(z)/m=gz+|\mu|B(z)/m$ for the high-field repelled $|+\rangle$ spin state of a neutron with mass $m$ and negative magnetic moment $\mu=-60.3$ neV/T depends only on the vertical $z$ coordinate. In this model the neutrons are exposed to a uniform gravitational field $-g\hat{\mathbf{z}}$ and a non-uniform magnetic field of magnitude $B(z)$. They perform an oscillatory motion with turning points (TP) at the lower and upper horizontal equipotential surfaces (ES) where $v_{z}=0$ and the potential is $V=(E/m)-v^{2}_{\perp}/2$. Here $E$ is the neutron energy; $v_{z}$ and $v_{\perp}=\sqrt{v^{2}_x+v^{2}_{y}}$ are the vertical and horizontal velocity components, respectively. $v_{\perp}$ is constant for the 1D field model.
As the particle moves from one TP to the next, starting out in a pure $|+\rangle$ spin state, its wave component for the $|-\rangle$ spin state increases. It may change over many orders of magnitude [@WAL01; @STE01], peaking at critical points where the field magnitude $B$ is small and the vector $\mathbf{B}$ rotates rapidly in the reference frame of the moving particle. The spin flip probability is “measured” only at the next TP where, in the Copenhagen interpretation, the wave function collapses and UCNs in the $|+\rangle$ state return to the trapping region while the $|-\rangle$ projection separates in space and quickly becomes lost. Conceptually, the “measurement” could be made by an ideal neutron detector placed just next to the TP, which would intersect the UCNs in the “wrong” spin state as they exit the storage space. This “measurement’ resets the UCN wave function to a pure initial $|+\rangle$ state for the next lap where the sequence of wave evolution and collapse at the following TP is repeated.
Outline {#sec:II}
=======
In the present article we extend this approach to the analysis of depolarization in cylindrical multipole fields such as those described in Refs. [@HUF01; @EZH01; @LEU01; @LEU02; @BEC01; @MAT02], where the trapping fields are generated by Halbach arrays of permanent magnets [@EZH01; @LEU01; @LEU02; @BEC01] or, for [@HUF01; @MAT02] and, earlier [@PAU02], by superconducting currents. For these cylindrical configurations we use a 2D field model which enables us to obtain a semi-analytic expression for the ensemble-averaged spin-flip loss and which can be analyzed with no need to involve simulations. The results are consistent with the only experimental spin-flip probabilities with varying holding field available so far [@LEU02].
For the cylindrical $2 N$-pole we approximate the field in cylindrical coordinates $r$, $\phi$, $\zeta$ as follows: $$\begin{aligned}
\label{1}
B_{r}&=B_{max} (r/R)^{N-1}\sin{\left(N\phi\right)},\nonumber\\
B_{\phi}&=B_{max} (r/R)^{N-1}\cos{\left(N\phi\right)},\nonumber\\
|\mathbf{B}|&=B(r)=\sqrt{B^{2}_{\zeta}+B^{2}_{max}(r/R)^{2 N-2}}\,,\end{aligned}$$ where $N\ge 2$. $\zeta$ points along the cylinder axis and the holding field $B_{\zeta}$ is considered constant. $B_{max}$ is the trapping field magnitude at the wall and the radius $R$ (typically $\sim\! 5$ cm) is much smaller than the length, which is of order $1$ m. This justifies the neglect of gravity for horizontal configurations of this type [@HUF01] since the gravitational energy varies little over the trap radius. To assess the merits of model (\[1\]) in general we have performed 3D simulations including gravity both for the vertical and the horizontal cylindrical multipole configurations.
As a second application we extend the previous analysis [@STE01] of depolarization for a 1D field model of the Los Alamos UCN$\tau$ trap [@WAL01; @SAL01; @SAU01] to the actual field in this magneto-gravitational trap with its asymmetrically double-curved wall in the shape of a bowl. As in [@WAL01] we approximate the field for the curved arrays of permanent magnets by that of the corresponding infinite planar array tangent to the bowl surface at the closest point on the bowl surface. We also use the same expressions for the flat-wall field, dubbed “smooth” \[Eq. (5) of [@WAL01]\], “one-way ripple” \[Eq. (7)\] and “two-way ripple” \[Eq. (8)\]. The “one-way ripple” takes into account the finite magnet size and the “two-way ripple” also includes the effect of iron shims between the magnets, a design feature not implemented for the current UCN$\tau$ system (status of 2016).
The theoretical approach is outlined in Sec. \[sec:III\], where we derive a first-order approximation to the spin-flip probability from the spin-dependent Schrödinger equation (SE), and in Sec. \[sec:IV\] where we average these results over the ensemble of orbits in the field configurations of UCN$\tau$ and of multipole bottles. In Sec. \[sec:V\] we derive a higher-order solution of the spin-dependent SE and show that it deviates very little from the first-order approximation. These various approaches are semi-classical since the field $\mathbf{B}(t)$ acting on the neutron spin is determined by the classical motion of the particle through the field. However, the results have been shown [@WAL01; @STE01] to be consistent also with a fully quantum mechanical analysis starting from the spin and space dependent SE. We show in the Appendix that this equivalence also holds for our extension to arbitrary field configurations.
As in [@WAL01; @STE01], we use the Wentzel-Kramers-Brillouin (WKB) approximation [@MOR01] to solve the SE. This is justified since the spatial variation of field variables is much slower than the variation of the UCN wave function. The scales are of order cm for gravity and $\mathbf{B}$, and of order $\mu$m or less for the wavelength. Thus the wave function for spin state $|+\rangle$ can be expressed in the WKB form except at a TP $z^{\prime}=0$, where its amplitude $1/\sqrt{k^{\prime}_{+}(z^{\prime})}$ diverges since the wave number $k^{\prime}_{+}$ vanishes. (See Eq. (\[A12\]) for details.)
Semi-classical approach {#sec:III}
=======================
Neutron lifetime experiments based on magnetic storage require that the spin follows the changes of field direction along the neutron path for a time much longer than the neutron lifetime, implying that the probability $|\alpha(t)|^{2}$ for spin $|+\rangle$, parallel to the local field, is always much larger than the small spin-flipped part $|\beta(t)|^{2}$. Therefore, in order to separate large terms in the SE from the small ones it is advantageous to use a reference system which rotates with the field experienced by the moving particle. Thus we use the SE for spin 1/2 with quantization axis in the local field direction, as in [@WAL01; @STE01]: $$\begin{aligned}
\label{2}
i\hbar \frac{d}{dt}&\left(\alpha(t) \chi^{+}(t)+\beta(t) \chi^{-}(t)\right)\nonumber\\
&=|\mu|B\left(\alpha(t) \chi^{+}(t)-\beta(t) \chi^{-}(t)\right),\end{aligned}$$ where $$\begin{aligned}
\label{3}
&\chi^{+}=\left( \begin{array}{c}
c \\
e_{+}s \\
\end{array} \right)\quad\textrm{and}\quad\chi^{-}=\left( \begin{array}{c}
e_{-}s \\
-c \\
\end{array} \right)\end{aligned}$$ are the spinors aligned in the direction of or opposite to the local magnetic field $\mathbf{B}$, respectively. For the Los Alamos “bowl” we choose the $x$ direction (defined as the direction of the toroidal holding field $\mathbf{B}_{h}(z)$ measured in the vertical symmetry plane) as the fixed quantization axis relative to which the local field direction is given by the polar angle $\theta=\arccos(B_{x}/B)$ and the azimuthal angle $\phi=\arctan(B_{z}/B_{y})$. In (\[3\]) we have used $c=\cos{(\theta/2)}$, $s=\sin{(\theta/2)}$ and $e_{\pm}=e^{\pm i\phi}$.
Differentiating $\chi^{+}(t)$ and $\chi^{-}(t)$ we get [@STE01] $$\label{4}
\dot{\chi}^{+}=A_{pp}\chi^{+}+A_{pm}\chi^{-},\,\,\,\dot{\chi}^{-}=A_{mp}\chi^{+}+A_{mm}\chi^{-},$$ with time dependent coefficients $$\begin{aligned}
\label{5}
A_{pp}&=\frac{i}{2}\dot{\phi}(1-\cos\theta),\,\,\,A_{pm}=-\frac{1}{2}e_{+}(\dot{\theta}+i\dot{\phi}\sin\theta)\nonumber\\
A_{mp}&=-A^{\ast}_{pm},\,\,\,A_{mm}=A^{\ast}_{pp}=-A_{pp}.\end{aligned}$$ $A_{mp}$ and $A_{pp}$ are small quantities to be treated as small perturbations.
Using Eqs. (\[3\])-(\[5\]) in (\[2\]) we obtain for the terms with $\chi^{+}$ in first order $$\label{6}
\dot{\alpha}+\frac{i\omega_{L}}{2}\alpha=0$$ and for those with $\chi^{-}$ $$\label{7}
\dot{\beta}-\frac{i\omega_{L}}{2}\beta=-\alpha A_{pm}=\frac{\alpha}{2}e_{+}(\dot{\theta}+i\dot{\phi}\sin\theta),$$ where $\omega_{L}=2|\mu|B/\hbar$ is the local Larmor frequency. Equations (\[6\]) and (\[7\]) correspond to Eqs. (57) and (58) of [@STE01]. The solutions of (\[6\]) and (\[7\]) are [@WAL01; @STE01], in WKB approximation, $$\begin{aligned}
\alpha(t)&=e^{-i\Theta/2}\label{8},\\
\beta(t)&=-\frac{i A_{pm}}{\omega_{L}}e^{-i\Theta/2}=\frac{i}{2\omega_{L}}e_{+}(\dot{\theta}+i\dot{\phi}\sin\theta)e^{-i\Theta/2},\label{9}\end{aligned}$$ where $\Theta=\int_{0}^{t}\omega_{L}(t^{\prime})\,dt^{\prime}$ is twice the phase angle accumulated since the previous TP. Since $\alpha$ and $\beta$ have the same phase, $-\Theta/2$, the wave components $\alpha$ and $\beta$ propagate between TPs as a unit (they do not run apart). This feature had also been noted for the 1D field model [@STE01].
From (\[9\]), the probability of finding the neutron in the spin-flipped state along the way to the next TP is $$\label{10}
p(t)=|\beta(t)|^{2} =\frac{\dot{\theta}^{2}(t)+\dot{\phi}^{2}(t)\sin^{2}\theta(t)}{4\omega^{2}_{L}(t)}=\frac{\Omega^{2}(t)}{4\omega^2_{L}(t)}.$$ $\Omega(t)$ is the frequency of field rotation about an axis normal to the plane defined by $\mathbf{B}$ and $\dot{\mathbf{B}}$ and can be expressed as $\Omega=|\mathbf{B}\mathbf{\times}\dot{\mathbf{B}}|/B^{2}$. This form holds since $\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}\theta$ is the squared projection of vector $\dot{\mathbf{B}}$, as drawn from the tip of vector $\mathbf{B}$, onto the unit sphere. For any trajectory we determine the arrival time at, and the position of TPs from the condition that the velocity component along the gradient of potential $V$ vanishes, $\mathbf{v\cdot \nabla}V=0$, since, at a TP, the orbit is tangential to the ES.
Equations (\[9\]) and (\[10\]) make use of an approximation which we will discuss in Sec. \[sec:V\] in connection with the higher-order solution given in (\[25\]) and (\[27\]). In short, this analysis shows the following features of $p(t)$: Starting from zero at a TP, $p(t)$ increases to the value given in (\[10\]) within a short time of order $\mu$s. Eq. (\[10\]) holds over the entire remainder, typically $0.01-0.1$ s, of the motion to the next TP where $p(t)$ is “measured”. Equation (\[10\]) shows that the result depends only on the local field variables $\Omega(t)$ and $\omega_{L}(t)$; it is independent of the path history.
For the sequence of TPs encountered along a neutron path of total duration $T_{tot}$ we determine the spin-flip rate between consecutive TPs at $t_{i-1}$ and $t_{i}$ by dividing $p(t_{i})$, from (\[10\]), by the time interval $\Delta T_{i}=t_{i}-t_{i-1}$. Taking into account the probability $\Delta T_{i}/T_{tot}$ of finding the particle on this path element the spin-flip rate for the entire path becomes $$\label{11}
1/\tau_{dep}=\frac{1}{T_{tot}}\sum_{i=1}^{n}\,p(t_{i}),$$ where $n$ is the number of TPs encountered. Finally, the depolarization rate measured in the experiments is the ensemble average over all paths, which is determined by the source characteristics and by spectral cleaning. We assume an isotropic Maxwell spectrum, thus an energy independent phase space density (PSD). The Boltzmann factor $e^{-E/k_{B}T}$ is close to unity since UCN energies $E$, which are of order $\lesssim 10^{-7}$ eV, are much lower than $k_{B}T$ even for a low trap temperature $T$. The corresponding velocity dependence of the spectrum is $f(v)\sim v^{2}$.
Relation (\[10\]) is also obtained in a fully quantum mechanical approach using the space and spin dependent SE in WKB approximation [@WAL01; @STE01]. This equivalence holds for any field geometry as shown in Appendix $A$.
Ensemble average of spin-flip loss {#sec:IV}
==================================
In actual magnetic UCN storage systems it is difficult to make sure that the spectrum is isotropic and fills phase space uniformly up to the trapping limit. In practice this would require the complete removal of UCNs with energies slightly exceeding the limit. These tend to be in quasi-stable orbits lasting for times of the order of $\tau_{n}$, thus affecting the precision of a measurement of $\tau_{n}$. Deviations from the Maxwell spectrum may also be due to the characteristics of the UCN source and of UCN transport to the trap but we will disregard these differences since they are expected to be of minor importance for the specific loss due to spin flip.
1D field model {#sec:IV.A}
--------------
For the 1D field model of Ref. [@STE01] the condition of constant PSD was taken into account as follows: The spin-flip probability was averaged over the vertical velocity component $v_{z0}$ in the plane $z=z_{0}$ where the gravitational downward force, $-mg$, is balanced by the magnetic upward force, $-\mu d|\mathbf{B}|/dz$. $z_{0}$ is the 1D equivalent of an elliptic fixed point $O$ and $z=z_{0}$ is the only plane where UCNs of any energy $E$ can reside, down to $E=0$ and up to the maximum value for trapping. (Here and henceforth we set the potential $V=0$ at $O$ and assume that there are no other potential minima in the trapping region; this is the case for the field configurations presently used or proposed.) For any other height, the lower energy limit is non-zero. Therefore, to include all possible orbits in the averaging process we have to choose $z_{0}$ as the reference height, and since the statistical distribution of $v_{z0}$ values is uniform for uniform PSD, the mean depolarization rate is given as the average of $1/\tau_{dep}$ as a function of $v_{z0}$.
To analyze the depolarization between consecutive TPs we start trajectories from a TP, not from the fixed point $O$; so we have to connect the statistics at $z_{0}$, which is given by a uniform distribution of $v_{z0}$, with the distribution $P(z)$ of launching height $z$ for given $v_{z0}$. $P(z)$ follows from energy conservation: The potential at the launching point is $V(z)=v^{2}_{z0}/2$, thus $(dV/dz)dz=v_{z0}dv_{z0}$. Therefore, to represent constant spacing in velocity space ($\Delta v_{z0}=$ const.) for orbits launched at height $z$ we have to choose the density $P(z)=1/\Delta z$ as $$\label{12}
P(z)\propto \frac{\Delta v_{z0}}{\Delta z}=\frac{|dV(z)/dz|}{v_{z0}}\sim \frac{|dV(z)/dz|}{\sqrt{V(z)}}.$$
General 3D field models {#sec:IV.B}
-----------------------
Now we adapt this result to arbitrary 3D field models as are relevant for the Los Alamos bowl [@WAL01]. The asymmetry introduced by the choice of two different radii of curvature along the rows of Halbach field magnets helps to randomize the orbits although fully mixing phase flow cannot be achieved. To select a representative sample of orbits for depolarization calculations we assume that most particles with energy $E$ below the trapping limit will, at some time, be found at rest. (In practice it suffices to require $v\ll v_{max}\approx 3$ m/s.) In this case the particle has just reached the ES $V=E/m=$ constant. It will then be accelerated back into the region of lower potential in the direction perpendicular to the ES.
Taking these points as the initial position for simulated trajectories starting from rest we make sure that the particle remains within the volume bounded by the ES with potential $V=E/m$ (as long as no spin flip to the $|-\rangle$ state takes place). The statistical distribution of launch points in space is determined by the following extension of (\[12\]) to 3D geometry: We relate the launching point to the fixed point $O$ in the same way as for the 1D field model. Equating the initial energy $E=m V(x,y,z)$ with the kinetic energy at $O$, $E=m v^{2}_{0}/2$, we differentiate $V(x,y,z)=v^{2}_{0}/2$. This gives $\nabla V\,ds=v_{0}\,dv_{0}$ where $s$ is the coordinate perpendicular to the ES $V(x,y,z)=$ const. at the launching point. Furthermore, the potential in the immediate vicinity of $O$ is constant, thus the spatial density is stationary. Hence, to satisfy uniformity of PSD the density in velocity space must be uniform: $\Delta v_{0}=$ const., and we have to choose the spatial density of launching points according to $$\label{13}
\frac{1}{\Delta s}\propto\frac{\Delta v_{0}}{\Delta s}\sim \frac{|\nabla V(x,y,z)|}{\sqrt{V(x,y,z)}}.$$ We take the last form of (\[13\]) as the (not normalized) probability distribution $P(x,y,z$) representing the number of launch points per unit volume at position ($x,y,z$). In the simulations we use von Neumann’s acceptance or rejection method to implement this probability distribution. For this purpose we need the fraction $P/P_{max}$ where $P_{max}$ is the maximum value of $P$ for the ensemble of trajectories.
We use this method to select random launching points for simulated orbits in UCN$\tau$ and use the three field models for the Halbach array, “smooth”, “one-way ripple” and “two-way ripple” and the toroidal-shaped holding field coil geometry described in [@WAL01]. Averaging the spin-flip rate (\[11\]) over a sample of some $10^{3}$ orbits, each of duration $T_{tot}=10$ s with $n\approx 100$ TPs, for four values of holding field $B_{x0}$ at the bowl bottom, we obtain the depolarization rate $\langle 1/\tau_{dep} \rangle$ shown in Table \[table:one\]. The results are consistent, within a factor of two, with those given in Fig. 3 of [@STE01] for the 1D field model: $\langle 1/\tau_{dep}\rangle\approx 4\times 10^{-9}$ s$^{-1}$ for $B_{x0}=5$ mT and $\langle 1/\tau_{dep}\rangle\approx 5\times 10^{-11}$ s$^{-1}$ for $B_{x0}=50$ mT. Fig. \[fig:one\] shows as squares the 1D calculation of [@STE01] and as circles the present 3D calculations for the smooth-field model. In the range $B_{x0}\gtrsim 5$ mT the data are represented reasonably well by the proportionality $\langle 1/\tau_{dep}\rangle\sim B^{-2}_{x0}$ as indicated by the dashed line.
--------------- ------------ ---------------- ----------------
$B_{x0} [T]$ smooth one-way ripple two-way ripple
\[1ex\] $0.1$ $0.022(1)$ $0.023(1)$ $0.024(1)$
$0.03$ $0.32(1)$ $0.33(1)$ $0.31(1)$
$0.01$ $1.6(1)$ $1.7(1)$ $1.9(1)$
$0.001$ $9.4(1)$ - -
\[0.5ex\]
--------------- ------------ ---------------- ----------------
: [Mean depolarization rate $\langle 1/\tau_{dep}\rangle$ $[10^{-9}/s]$ for the Los Alamos UCN$\tau$ field]{}
\[table:one\]
Table \[table:one\] shows that the field ripple has a minor effect on the net depolarization loss. This is plausible since the ripple only affects the immediate vicinity, of order mm, of the wall which contributes little to the spin flip since the adiabatic condition is well satisfied in the region of high field strength near the wall.
Cylindrical multipole fields {#sec:IV.C}
----------------------------
### Results for the field model of equation (\[1\]) {#sec:IV.C.1}
We use the field of an ideal cylindrical multipole given in Eq. (\[1\]), which does not take into account the deviations induced in the actual designs by the discrete geometry of electric currents [@HUF01; @MAT02], or by the permanent magnet blocks with constant magnetization within each block in the schemes of Refs. [@EZH01; @LEU01; @LEU02; @WAL01; @BEC01].
Equation (\[1\]) neglects gravity and assumes a uniform holding field $B_{\zeta}$ in the axial direction. With these simplifications, the field is determined only by the polar coordinates $r$ and $\phi$ in the plane perpendicular to the axis. Moreover, the field magnitude $B$ and the force $-|\mu|\,dB/dr$ acting on a $|+\rangle$ spin UCN depend only on $r$. In this central force field the equipotential lines are concentric cylindrical shells. Energy $E$ and angular momentum $L_{\zeta}$ about the symmetry axis are conserved. (For vertical systems, $L_{z}$ is conserved also in the presence of gravity as well as for variable $B_{\zeta}$, as for end fields, as long as the potential $V$ remains cylindrically symmetric.)
The orbits of the 2D hexapole ($2 N=6$) are ellipses. Analytic expressions for the orbits, in terms of elliptic integrals, exist also for the quadrupole ($2 N=4$), decapole ($2 N=10$) and for $2 N=14$ [@GOL01]. Alternatively, the radial equation of motion, $\dot{r}=\sqrt{2\left(E-V_{eff}(r)\right)/m}$, in the effective potential $V_{eff}(r)=V(r)+L^{2}_{\zeta}/(2 m^{2}r^{2})$ is readily solved numerically for any $N(\ge 2)$. There are two apsidal radii, $r_{min}$ and $r_{max}$, and the orbits are symmetric about the angular positions of these TPs. Therefore, it suffices to analyze only the path section between consecutive TPs.
![(Color online) Comparison of spin flip loss rate calculated in [@STE01] for the Los Alamos UCN$\tau$ trap [@WAL01] using a 1D field model (blue squares) with the present 3D calculation for the smooth field given in Eq. (5) of [@WAL01] (red circles). In the range $B_{x0}\gtrsim 5$ mT the data are represented reasonably well by the power law $\langle 1/\tau_{dep} \rangle\sim B^{-2}_{x0}$ as shown by the dashed line.[]{data-label="fig:one"}](Fig1){width="75mm"}
We average over all possible orbits confined within the trap radius $R$ and subject to the requirement of uniform PSD as follows. Choosing the radius $r_{1}<R$ of an ES we consider all orbits which turn around at $r_{1}$. Subset $a$ of these orbits comes from the inside and has $0\le r_{min}\le r_{1}$ and $r_{max}=r_{1}$. The other subset $b$ of orbits comes from the outside and has $r_{min}=r_{1}$ and $r_{1}\le r_{max}\le R$. In case $a$ ($b$) the region exterior to the storage space for spin $|+\rangle$ is the range $r>r_{1}$ ($r<r_{1}$). In either case, a spin-flipped UCN entering this “forbidden zone” is attracted toward the high field at the wall and considered as lost.
The classification $a$ or $b$ is determined by the peripheral velocity $v_{1}$ at $r_{1}$: For group $a$ the range of $v_{1}$ is between $0$ (for the radial path from or toward the center $r=0$, for which the angular momentum is zero) and $v_{c}=\sqrt{r_{1} F(r_{1})/m}$ with centripetal force $F(r_{1})=\,m dV/dr_{1}$. In the latter limit the path is circular with radius $r_{1}$. For group $b$, $v_{1}$ ranges from $v_{c}$ (circular) to $v_{2}$ for the limiting path skirting the wall ($r_{max}(v_{2})=R$). In each case, the second turning radius and the time $\Delta t(r_{1},v_{1})$ it takes from one TP to the next are found numerically from the radial equation of motion.
To determine the statistical weight of a given orbit with one TP at radius $r_{1}$ we have to modify the strategy used for the UCN$\tau$ field. In that case we considered only the sample of orbits where the particle starts from rest at the ES with the highest potential, $V=E/m$, reached for given energy $E$.
For the cylindrical multipole field (\[1\]) only regular orbits exist and releasing a particle from rest would cover only the subset (of measure zero) of trajectories with angular momentum zero, which oscillate radially through $O$ (the axis $r=0$). However, field (\[1\]) is an idealization and in the physical situations field irregularities such as “ripples” and stray fields in the axial $\zeta$ direction are unavoidable. As far as the statistics of orbits perturbed in this way goes, the following strategy appears justified: For a path turning around at $r_{1}$ with peripheral velocity $v_{1}$ we consider the ES of radius $\rho$ such that $V(\rho)=E/m=V(r_{1})+v_{1}^{2}/2$ and relate the statistical weight for radius $\rho$ to the uniform phase-space density at $O$. As in Secs. \[sec:IV.A\] and \[sec:IV.B\] we differentiate the energy balance between $r=\rho$ and $r=0$, $E/m= V(\rho)=v^{2}_{0}/2$, and obtain the proportionality $$\label{14}
\frac{\Delta v_{0}}{\Delta \rho}\sim \frac{dV(\rho)/d\rho}{\sqrt{V(\rho)}}.$$ Multiplying by $\rho$ to take into account the number of allowed points along the circle with radius $\rho$ we derive for the weighting factor for radius $\rho$, and therefore also for the probability $P(r_{1},v_{1})$ for an orbit with apsidal radius $r_{1}$ and apsidal velocity $v_{1}$: $$\label{15}
P(r_{1},v_{1})=\rho\,\frac{dV(\rho)/d\rho}{\sqrt{V(\rho)}}$$ where, by definition of $V(\rho)$ and using (\[1\]), $$\begin{aligned}
\label{16}
V(\rho)&=V(r_{1})+v_{1}^{2}/2\nonumber\\
&=(|\mu|/m)\left(\sqrt{B^{2}_{\zeta}+B^{2}_{max}(\rho/R)^{2 N-2}}-B_{\zeta}\right).\end{aligned}$$ To evaluate $dV(\rho)/d\rho$ in (\[15\]) we have to take into account the dependence of $V(\rho)$ and of $$\label{17}
\rho=R\left\{\frac{[m V(r_{1})+|\mu|B_{\zeta}\,+m v_{1}^{2}/2]^{2}-|\mu|^{2}B^{2}_{\zeta}}{|\mu|^{2}B^{2}_{max}} \right\}^{1/(2 N - 2)}$$ (from (\[16\])) on $r_{1}$ and $v_{1}$: $$\label{18}
\frac{dV(\rho)}{d\rho}=\frac{\partial V(\rho)/\partial r_{1}}{\partial \rho/\partial r_{1}}+\frac{\partial V(\rho)/\partial v_{1}}{\partial \rho/\partial v_{1}}=2 \frac{dV(r_{1})/dr_{1}}{\partial \rho/\partial r_{1}},$$ where we have used $\partial \rho/\partial v_{1}=[v_{1}/(dV(r_{1})/dr_{1})]$ $\times (\partial \rho/\partial r_{1})$ which follows from (\[17\]) with the help of $\partial V(\rho)/\partial r_{1}=dV(r_{1})/dr_{1}$ and $\partial V(\rho)/\partial v_{1}=v_{1}$. The result is $$\begin{aligned}
\label{19}
&P(r_{1},v_{1})\sim\nonumber\\
&\frac{\mu^{2}B^{2}_{t}(r_{1})+|\mu| B(r_{1}) m v_{1}^{2}+m^{2}v_{1}^{4}/4}{\left(|\mu|B(r_{1})+m v_{1}^{2}/2\right)\sqrt{|\mu | [B(r_{1})-B_{\zeta}]+m v_{1}^{2}/2}},\end{aligned}$$ where $B_{t}(r)=B_{max}(r/R)^{2 N - 2}$ and $B(r)=\sqrt{B_{\zeta}^{2}+B^{2}_{t}}$ are the multipole fields without and with holding field $B_{\zeta}$, respectively.
Weight factor (\[19\]) determines how the depolarization rate from (\[10\]) is averaged over all paths. At TPs we have $\dot{\theta}=0$ since $\theta$ depends only on $r$ and $\dot{r}=0$. (Here we measure $\theta$ from the $\zeta$-axis.) The angular velocity of trapping field rotation experienced by a neutron moving through a TP is $\dot{\phi}=(N-1)v_{1}/r_{1}$. Thus, averaging $p/\Delta T$ from (\[10\]) over the ensemble of paths the overall depolarization rate becomes $$\begin{aligned}
\label{20}
\left\langle 1/\tau_{dep}\right\rangle=&\frac{(N-1)^{2}}{\nu}\int_{r_{1}=0}^{R}dr_{1}\frac{\sin^{2}\theta(r_{1})}{4 r^{2}_{1}\omega^{2}_{L}(r_{1})}\nonumber\\
&\times\int_{v_{1}=0}^{v_{2}(r_{1})}\,dv_{1}\,P(r_{1},v_{1})\frac{v_{1}^{2}}{\Delta t(r_{1},v_{1})},\end{aligned}$$ with Larmor frequency $\omega_{L}(r_{1})$ at radius $r_{1}$ and normalization constant $\nu=\int_{0}^{R}dr_{1}\int_{0}^{v_{2}(r_{1})}dv_{1}\,P(r_{1},v_{1})$.
![(Color online) Mean spin-flip rates calculated from Eq. (\[20\]) for a cylindrical multipole trap vs. order $2N$. The purple, red and black points (down triangles, closed circles and diamond symbols) represent integer $N$; the intervening orange, blue and green points (open circles, squares and up triangles) are for half-integral $N$.[]{data-label="fig:two"}](Fig2){width="85mm"}
Numerical results for a wide range of multipole orders $2N$ are shown in Fig. \[fig:two\] for $R=4.7$ cm and $B_{max}=1.3$ T. These are typical values for multipole traps; we keep these parameters the same for hypothetical systems where only the multipole order $2 N$ is varied and use $B_{\zeta}=0.001$ T, $0.01$ T and $0.1$ T for the holding field. To show the behavior of $\langle 1/\tau_{dep}\rangle$ vs. $N$ more clearly we have added the half-integral values $N=5/2$ and $N=7/2$ which cannot be realized with magnetic fields.
Our calculation for the octupole ($2N=8$) at $B_{\zeta}=1$ mT gives $\langle\tau^{-1}_{dep}\rangle=1.5\times 10^{-5}$ s$^{-1}$. This result is consistent with the order of magnitude $\tau_{dep}=(4\pm 16)\times 10^{4}$ s$^{-1}$ given in Table IV of [@LEU02] for a solenoid current $3$ A which corresponds to $B_{\zeta}\gtrsim 1$ mT. (In experiment [@LEU02] some depolarization may have been caused by reflection on the Fomblin-coated wall at the bottom of the trap.)
The depolarization rates calculated from (\[20\]) for the multipole traps are about $10^{2}$ times those for the 3D UCN$\tau$ field for the same holding field. The difference may be attributed to the small radius $R=4.7$ cm used. The dimensions of the UCN$\tau$ field are larger, $\sim\!0.5$ m, and therefore the average field gradient is smaller. Our calculations for a cylindrical multipole with large radius $R=1$ m, a value similar to the multipole design of Ref. [@MAT02], gives $\approx 10^{2}$ times lower spin-flip losses for the same values of $B_{\zeta}$.
We have evaluated expression (\[20\]) for the mean depolarization rate, taking into account all possible flight paths subject to the condition of constant PSD and confined to a cylinder of radius $R$. This was possible since all orbits are regular for model field (\[1\]).
By contrast, the orbits in the actual magnetic traps are perturbed and may show instability. In this case we rely on sampling. For instance, for the Los Alamos UCN$\tau$ system with its field asymmetry we have considered, in Sec. \[sec:IV.B\], only orbits for which the particle velocity and angular momentum vanish at some time.
Applying the same method to 3D simulations for vertical multipole configurations including gravity would not provide a proper sample of orbits since these systems conserve angular momentum about the vertical axis, $L_{z}=0$. All paths launched from rest would be confined to vertical planes passing through the central axis, as stated earlier.
### 3D simulations for multipole fields {#sec:IV.C.2}
We include orbits with $L_{z}\neq 0$ as follows. Choosing a random initial position $Q$ within the trap volume a particle is launched with initial velocity vector $\mathbf{v}_{1}$ tangential to the ES at $Q$ and pointing in a random direction within the launch plane. To conform to a uniform distribution in velocity space the endpoint of $\mathbf{v}_{1}$ is uniformly distributed within the area of a disk whose radius is determined by the maximum velocity for particle trajectories confined to the trap volume. This and the following operations correspond to those described for the 2D field model (\[1\]) in Sec. \[sec:IV.C.1\] but averages over the circular ESs of the latter model are now replaced by averages over ESs of general shape in 3D space. Based on (\[13\]), this leads to an approximation for the weight factor $P(\mathbf{r}_{1},\mathbf{v}_{1})$ for launch at position $\mathbf{r}_{1}$ and initial velocity $\mathbf{v}_{1}$ and, finally, to the mean depolarization rate $\langle 1/\tau_{dep} \rangle$ by averaging (\[11\]) over some $10^{3}$ orbits, each of duration $T_{tot}=10$ s with $n\gtrsim 500$ TPs.
------------------------ ------------- ----------- -------------
$2N$ 4 8 20
\[1ex\] HOPE, vertical $[4.1(1)]$ $0.34(1)$ $[0.78(1)]$
HOPE, horizontal $[0.61(1)]$ $0.15(1)$ $[0.89(1)]$
NIST, mark 2 $0.43(1)$
NIST, mark 3 $0.021(1)$
\[0.5ex\]
------------------------ ------------- ----------- -------------
: Mean depolarization rate $\langle 1/\tau_{dep}\rangle$ $[10^{-9}/s]$ from 3D simulations for cylindrical multipoles with gravity and end coils included
\[table:two\]
We approximate the effect of spectral cleaning in UCN storage experiments by specifying the largest energy $E_{max}$ of stored particles. The calculations of $\langle 1/\tau_{dep} \rangle$ shown in Table \[table:two\] use $E_{max}\approx 0.8$ times the value $|\mu|B_{high}$ for the highest field $B_{high}$ in the trap and are based on the field parameters of the following two designs:
\(a) The HOPE octupole magnet [@LEU02] has bore radius $4.7$ cm and its axis oriented vertically or horizontally. For the vertical configuration we assume activation of only the bottom solenoid with a maximum axial field of $1.4$ T while gravity provides the cap. With both end field solenoids activated in the horizontal configuration we assume fields of $1.4$ T on both ends, separated by a distance of $1.13$ m, without activating the long holding field solenoid. The radial confinement field is $B_{max}=1.3$ T at the trap wall.
\(b) Two horizontal Ioffe type quadrupole magnets have been used at NIST [@BRO01; @YAN01] (versions mark 2 and mark 3). For mark 2 (mark 3) we use maximal fields of, axially: $1.4$ T ($4.0$ T), and radially: $1.3$ T ($3.9$ T), a bore radius of $5$ cm ($5$ cm) and a separation of $0.4$ m ($0.76$ m) between the centers of the end field solenoids; for mark 2, the latter include “bucking” coils [@BRO01; @YAN01] causing the axial field to drop off more quickly to a minimum of $0.1$ T at the trap center. For mark 3, the minimum field is $0.6$ T.
Since the depolarization rates depend only weakly on the details of the field distribution, such as field ripples, we use the smoothed fields and approximate the solenoid fields by their values on the solenoid axis, neglecting the variations of field magnitude and direction over the bore cross section. The results are shown in Table \[table:two\].
For the HOPE-type system we include, in square brackets, also systems with the same geometries and maximum fields but different multipole order $2 N$. We observe the same tendency as for the 2D calculations of Fig. \[fig:two\]: The depolarization rates for $2 N=4$ and $2 N=20$ are higher than in the intermediate range ($2 N\sim 8$). This can be explained by higher field gradients, near the axis for low $N$ and near the wall for high $N$.
As a further general feature, the magnitudes and $N$ dependence of $\langle 1/\tau_{dep}\rangle$ from the 3D simulations in Table \[table:two\] are well approximated by the 2D results from Fig. \[fig:two\] if we use values of holding field close to their minima: $0.08$ T/$0.1$ T for HOPE (vertical configuration with $B_{min}=0.013$ T/horizontal with $B_{min}=0.09$ T) [@LEU02] and $0.1$ T ($0.6$ T) for NIST mark 2 (mark 3) with $B_{min}\approx 0.1$ T ($0.6$ T) [@BRO01; @YAN01].
Finally, $\langle 1/\tau_{dep}\rangle$ approximately scales like $B^{-2}_{\zeta}$. A similar increase of $\langle 1/\tau_{dep} \rangle$ with decreasing holding field is also seen for UCN$\tau$ as shown by the dashed line in Fig. \[fig:one\]. As an application of scaling in an experiment, we could deliberately lower the holding field to enhance the depolarization loss to a measurable level to verify that the loss for the actual field is negligible.
A higher-order solution {#sec:V}
=======================
In this section we compare the first-order approximation for the spin-dependent SE, Eqs. (\[6\]), (\[7\]), with a higher-order approach where we retain all the terms with $A_{pm}$ and $A_{pp}$ \[given in (\[5\])\], which arose from the transformation to the reference system rotating with the field: $$\begin{aligned}
&\dot{\alpha}+\frac{i\omega_{L}}{2}\alpha=-A_{pp} \alpha+A^{\ast}_{pm}\beta,\label{21}\\
&\dot{\beta}-\frac{i\omega_{L}}{2}\beta=-A_{pm} \alpha+A_{pp}\beta.\label{22}\end{aligned}$$ The coupled first-order ODEs (\[21\]) and (\[22\]) can be solved by direct numerical integration with initial conditions $\alpha(0)=1$, $\beta(0)=0$ for a particle starting in the $|+\rangle$ state at $t=0$.
Alternatively, we can use the perturbation approach developed in [@STE03] for searches for a permanent electric dipole (EDM) of the neutron, to solve the SE for spin $1/2$ up to second order of small perturbations. In the EDM case the UCN spin state is perturbed by magnetic field inhomogeneities and a strong static electric field. In Eqs. (\[21\]) and (\[22\]) the perturbations are the terms on the RHS, which are much smaller than those on the left.
To facilitate comparison with [@STE03] we define $\Sigma_{pp}(t)=-2 i A_{pp}(t)$ (real-valued), $\Sigma(t)=-2 i A_{pm}(t)$ (complex), $\omega_{1}(t)=\omega_{L}(t)+\Sigma_{pp}(t)$ and $\Theta_{1}(t)=\int_{0}^{t}\omega_{1}(t^{\prime})\,dt^{\prime}$. In practical cases, $\Sigma_{pp}$ is at least $10^{4}$ times smaller than $\omega_{L}$; thus $\omega_{1}$ is very close to $\omega_{L}$.
The transformations $\alpha(t)=u(t)\,e^{-i\Theta_{1}(t)/2}$, $\beta(t)=w(t)\,e^{i\Theta_{1}(t)/2}$ turn Eqs. (\[21\]), (\[22\]) into $$\begin{aligned}
\label{23}
&i\dot{u}(t)=\frac{1}{2}\Sigma^{\ast}(t)\,w(t)\,e^{i\Theta_{1}(t)}\textrm{,}\nonumber\\
&i\dot{w}(t)=\frac{1}{2}\Sigma(t)\,u(t)\,e^{-i\Theta_{1}(t)}.\end{aligned}$$ These coupled ODEs for $u$ and $w$ have the same form as Eqs. (7) for $\alpha_r$ and $\beta_{r}$ in [@STE03]. The only difference is the arguments of the exponential functions. In [@STE03], the SE was transformed into the reference frame rotating at constant frequency $\omega_{0}$ for constant applied Larmor field. In the present case, $\omega_{1}(t)$ can be an arbitrary function of $t$; thus the phase factor $e^{\pm i\omega_{0}t}$ is replaced by $e^{\pm i\Theta_{1}(t)}$.
We combine the two first-order ODEs (\[23\]) into the single second-order ODE for $u(t)$: $$\label{24}
\ddot{u}(t)-\left(i\omega_{1}(t)+\frac{\dot{\Sigma}^{\ast}(t)}{\Sigma^{\ast}(t)}\right)\dot{u}(t)=-\frac{1}{4}|\Sigma(t)|^{2}u(t)$$ which has the same form as Eq. (8) of [@STE03] with $\omega_{0}$ in the first term of the expression in brackets replaced by $\omega_{1}(t)$. This additional time dependence does not affect the method of solving (\[24\]) since the second term is time-dependent in either case. The initial conditions, $u(0)=1$ and $w(0)=0$, are the same as for the EDM case with initial spin up ($\alpha_{r}(0)=1$, $\beta_{r}(0)=0$).
Following the steps (11) to (18) of [@STE03] we derive $$\begin{aligned}
\label{25}
w(t)&=\frac{2 i \dot{u}(t)}{\Sigma^{\ast}(t)}\,e^{-i \Theta_{1}(t)}=-\frac{i}{2}\left(\Sigma_{i}(t)-\Sigma_{i}(0) \right)\nonumber\\
&=-\frac{i}{2}\int_{0}^{t}dt^{\prime}\,e^{-i\Theta_{1}(t^{\prime})}\,\Sigma(t^{\prime})\end{aligned}$$ where $\Sigma_{i}(t)=\int dt\,e^{-i\Theta_{1}(t)}\,\Sigma(t)$.
The initial value, $w(0)=0$ at $t=0$, satisfies the required initial condition $\beta(0)=0$. As $t$ increases, the integral in (\[25\]) rapidly increases on a time scale of order $t_{min}=1/\omega_{1}(0)\approx 1/\omega_{L}(0)$. In practical application this is a very short time since the Larmor frequency $\omega_{L}$ is large everywhere inside the trap volume, even at the field minimum where $B$ is the holding field. For $B(0)=0.001$ T, $t_{min}=\pi\hbar/(|\mu|B(0))\approx 30$ $\mu$s.
For times $t\gg t_{min}$ it is advantageous to change the integration variable in (\[25\]) from $t^{\prime}$ to $\Theta_{1}$, with $dt^{\prime}=d\Theta_{1}/\omega_{1}$, and to integrate by parts: $$\begin{aligned}
\label{26}
w(t)=&\left[\frac{\Sigma(t)\,e^{-i\Theta_{1}(t)}}{2\omega_{1}(t)} \right]^t_{0}-\frac{1}{2}\int_{0}^{t}\,dt^{\prime}\,e^{-i\Theta_{1}(t^{\prime})}\frac{d}{dt^{\prime}}\left[\frac{\Sigma(t^{\prime})}{\omega_{1}(t^{\prime})} \right].\end{aligned}$$ We did not employ the WKB approximation to derive Eq. (\[26\]) but its use enables us to evaluate it analytically: We can neglect the last term in (\[26\]) since the field variable $\Sigma/\omega_{1}$ varies slowly on the wavelength scale and get, with (\[5\]), $$\begin{aligned}
\label{27}
\beta(t)&=w(t)\,e^{i\Theta_{1}(t)/2}=\frac{\Sigma(t)\,e^{-i\Theta_{1}(t)/2}}{2 \omega_{1}(t)}\nonumber\\
&=\frac{i}{2\omega_{1}(t)}e_{+}(\dot{\theta}+i\dot{\phi}\sin\theta)e^{-i\Theta_{1}(t)/2}.\end{aligned}$$
In (\[27\]) we have set the integration constant from the lower limit $t=0$ of the first term in (\[26\]) equal to zero, and an equivalent approximation had also been made in deriving the first-order solution (\[9\]) which differs from (\[27\]) only by the replacement of $\omega_{1}$ by $\omega_{L}$. The detailed justification in [@STE01], below Eq. (28), can be summarized as follows. Equations (\[9\]) and (\[27\]) are semi-classical since the SE is solved for the time dependent field $\mathbf{B}(t)$ determined from the classical equations of motion. A fully quantum mechanical treatment requires the solution of the spin and space dependent SE as in [@WAL01; @STE01] and in Appendix A. In this quantum analysis the exact solution for the wave function near a TP involves the Airy functions and the TP is blurred into a non-zero region (typically of order $\mu$m). So is the starting time at a TP. Thus, in (\[26\]) the initial value of $w(t)$ at $t_{0}\approx 0$, which is $\sim\! e^{-i \Theta_{1}(t_{0})}\approx e^{-i\omega_{1}(0)t_{0}}$, should be averaged over the rapidly varying phase $\omega_{1}(0)t_{0}$ with the result $w(0)=\left\langle e^{-i\Theta_{1}(t_{0})}\right\rangle_{t_{0}} =0$. This holds except for the initial few micrometers of the path subsequent to a TP.
Outside of this region we have, from (\[27\]) and (\[5\]), $$\label{28}
|\beta(t)|^{2}=|w(t)|^{2}=\frac{|\Sigma(t)|^{2}}{4\omega^{2}_{1}(t)}=\frac{|A_{pm}(t)|^{2}}{\omega^{2}_{1}(t)}=\frac{\Omega^{2}(t)}{4 \omega^{2}_{1}(t)}.$$ Eq. (\[28\]) agrees with the first-order solution (\[10\]) if we replace $\omega_{L}$ by $\omega_{1}$. As we have seen, in practical cases the difference between $\omega_{L}$ and $\omega_{1}$ is negligible.
![(Color online) Magnitude squared of spin-flip amplitude $\beta$ for UCNs released from rest at position $(x,y,z)=(-0.1,0.2,0.3)$ m in the Los Alamos UCN$\tau$ “smooth field” [@WAL01]. The analytic results, Eqs. (\[10\]) and (\[28\]) (blue solid), and the numerical solutions of ODE (\[9\]) (red dashed) and of ODEs (\[21\]), (\[22\]) (green dot-dashed) closely agree. The first two turning points, shown by arrows, are at $0.2108$ s (when the particles pass the field minimum at a close distance) and $0.2291$ s (when they are reflected at the high field near the wall). The reset of $\beta$ to zero at these turning points and its subsequent fast increase, within microseconds or less, to the values given by the curves are not shown.[]{data-label="fig:three"}](Fig3){width="85mm"}
![(Color online) Spin-flip probability $|\beta|^{2}$ for UCNs launched in UCN$\tau$ at time $t=0$ from a TP at $(x,y,z)=(-0.1,0.2,0.2)$ m at velocity $(v_{x},v_{y},v_{z})=(0,-2,0)$ m/s (tangential to the local equipotential surface). The numerical integrations of ODE (\[9\]) (red dashed) and of ODEs (\[21\]), (\[22\]) (green dotted) show the rapid increase, within a few Larmor periods, from $\beta=0$ at $t=0$ to the asymptotic behavior given by Eqs. (\[10\]) and (\[28\]) (blue solid). The ensuing evolution of $|\beta|^{2}$ up to the next TP (shown by the arrow) is practically unaffected by the transient behavior at $t\approx 0$.[]{data-label="fig:four"}](Fig4){width="85mm"}
Fig. \[fig:three\] shows the time-dependence of spin-flip probability $|\beta(t)|^{2}$ for a particle released from rest at an arbitrary position in UCN$\tau$, here $(x,y,z)=(-0.1,0.2,0.3)$ m, calculated in three ways: (a) The analytic results from Eqs. (\[10\]) and (\[28\]) (solid curve), and the numerical solutions (b) of differential Eq. (\[9\]) (dashed) and (c) of ODEs (\[21\]) and (\[22\]) (dot-dashed). The three curves closely agree except within short time intervals $\delta t \lesssim 10$ $\mu$s subsequent to passage though TPs at non-zero velocity, such as the TPs marked by the arrows. The deviations (not shown in Fig. \[fig:three\]) are due to the reset to $\beta=0$ at a TP. They are shown in detail in Fig. \[fig:four\] for a UCN moving away from a TP at $(x,y,z)=(-0.1,0.2,0.2)$ m at initial velocity $2$ m/s tangential to the local ES. As expected from Eq. (\[26\]), $|\beta|^{2}$ jumps, within a few Larmor periods ($\lesssim 10$ $\mu$s), from $0$ to the asymptotic curve given by (\[10\]) (or (\[28\])). The remainder of the wave evolution up to the next TP, shown by the arrow, is practically unaffected by the transient at $t\approx 0$.
As a crucial test of the validity of numerical integration of (\[21\]), (\[22\]) we verified the norm, $|\alpha(t)|^{2}+|\beta(t)|^{2}$, to be $1$ within $1$ ppm. The demands on the precision of numerical integration of (\[9\]), (\[21\]) and (\[22\]) become more stringent for paths through regions of higher magnetic field since large Larmor frequencies require short time steps.
Summary and conclusions {#sec:VI}
=======================
The spin-flip loss in magnetic storage of UCNs in the Los Alamos UCN$\tau$ permanent magnet trap had been analyzed theoretically in [@WAL01] for neutrons on a specific vertical path, and in [@STE01] for arbitrary motion. In the latter work we used a 1D model for the trapping field. In the present article we have extended this analysis to arbitrary orbits in arbitrary fields in 3D space and report calculations of mean spin-flip rates for the UCN$\tau$ system and for multipole fields such as the cylindrical octupole of the HOPE project [@LEU01; @LEU02] and the Ioffe-type quadrupole trapping fields of [@HUF01; @BRO01; @YAN01]. We have also investigated a simplified 2D field model for the cylindrical multipole fields and shown that it yields analytic results for depolarization probabilities which are consistent with the more elaborate 3D simulations. In all cases relevant to magnetic UCN storage we have established agreement between the semi-classical approach, solving the spin-dependent SE for the time-dependent field seen by the particle in a classical orbit, and a fully quantum mechanical analysis based on the space and time dependent SE solved in WKB approximation. The relative difference between a first-order treatment (in Sec. \[sec:III\]) and a higher-order analysis (in Sec. \[sec:V\]) of depolarization in the semi-classical framework is at most on the order of $10^{-4}$ in practical applications.
We confirm and generalize the earlier conclusions of [@WAL01; @STE01] relating to “Majorana spin flip at zeros of the magnetic field”. Magnetic UCN traps avoid locations of vanishing field by applying a holding field $B_{h}$ perpendicular to the trapping field. For typical values of $B_{h}$ we calculate spin-flip probabilities which are greater, by many orders of magnitude, than the Majorana prediction [@MAJ01] which had been derived for an infinitely extended field rather than trapping fields of finite extent. For the magnetic traps investigated we have found an approximate power law behavior of spin-flip loss rate as a function of $B_{h}$: $1/\tau_{dep}\sim B^{-2}_{h}$ (to be compared with the exponential behavior, $\sim e^{-\pi\xi/2}$ with adiabaticity parameter $\xi=\omega_{L}/\Omega$, for the Majorana field model [@MAJ01]). This implies that $B_{h}$ can be made large enough to render spin-flip loss negligible as compared to other possible sources of systematic error in neutron lifetime experiments, foremost that due to marginal trapping (leaving aside the more fundamental question raised in [@GRE01] whether or not neutron lifetime values derived from storage experiments should indeed be identical to those from beam-type experiments). We have shown that this conclusion is not restricted to the simplified 1D field model previously used in [@STE01] but holds also for the actual 3D fields in suitable magnetic neutron bottles.
We are grateful to P. Huffman for very helpful discussions.
Quantum analysis {#sec:A}
================
The space and spin dependent wave function $\Psi$ for a neutron with energy $E$ moving in a gravito-magnetic trapping field satisfies the SE $$\label{A1}
E\Psi=\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+m g z+|\mu|\boldsymbol{\sigma \cdot B}(x,y,z) \right] \Psi$$ where $\Psi=\alpha^{(3)}(x,y,z)\chi^{+}+\beta^{(3)}(x,y,z)\chi^{-}$ and $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are Pauli matrices. Superscript $(3)$ indicates that $\alpha^{(3)}(x,y,z)$, for the spin-up wave (relative to the local magnetic field direction), and $\beta^{(3)}(x,y,z)$, for the spin-down wave, are functions of the three spatial coordinates.
The derivatives of $\chi^{+}$ and $\chi^{-}$ with respect to $j=x,y,z$ are of the same form as the temporal derivatives (\[4\]). In terms of the spin angles $\theta$ and $\phi$ and of $e_{\pm}=e^{\pm i\phi}$ as defined below Eq. (\[3\]) we have $$\begin{aligned}
&\chi^{+}_{j}=\frac{i}{2}\phi_{j}(1-\cos\theta)\chi^{+}-\frac{1}{2}e_{+}(\theta_{j}+i\phi_{j}\sin\theta)\chi^{-},\label{A2}\\
&\chi^{-}_{j}=\frac{1}{2}e_{-}(\theta_{j}-i\phi_{j}\sin\theta)\chi^{+}-\frac{i}{2}\phi_{j} (1-\cos\theta)\chi^{-}\label{A3},\end{aligned}$$ where the subscript $j$ denotes partial differentiation.
Keeping only the dominant contributions, as in Eqs. (\[6\]) and (\[7\]), the Laplacian in (\[A1\]) reads $$\begin{aligned}
\label{A4}
&\nabla^{2}\Psi=(\alpha^{(3)}_{xx}+\alpha^{(3)}_{yy}+\alpha^{(3)}_{zz})\chi^{+}+\\
&\left\{(\beta^{(3)}_{xx}+\beta^{(3)}_{yy}+\beta^{(3)}_{zz})-e_{+}\sum_{j=1}^{3}\alpha^{(3)}_{j}(\theta_{j}+i\phi_{j}\sin\theta)\right\}\chi^{-}\nonumber.\end{aligned}$$ Thus, in WKB approximation the spatial wave functions satisfy $$\begin{aligned}
\nabla^{2}\alpha^{(3)}+k^{2}_{+}\alpha^{(3)}&=0,\label{A5}\\
\nabla^{2}\beta^{(3)}+k^{2}_{-}\beta^{(3)}&=e_{+}\sum_{j=x,y,x}\alpha^{(3)}_{j}(\theta_{j}+i\phi_{j}\sin\theta),\label{A6}\end{aligned}$$ where $$\label{A7}
k^{2}_{\pm}(x,y,z)=\frac{2m}{\hbar^{2}}\left[E-mgz\mp |\mu|B(x,y,z) \right]$$ are the squared local wave numbers for the ($+$) and ($-$) spin state, respectively.
Now we consider a UCN with spin ($+$) starting at time $t=0$ at a TP and arriving at $t=\Delta T$ at the next TP which we label $U$. At $U$ the UCN momentarily moves along the local ES and we introduce a local Cartesian system of coordinates centered at $U$ with $x^{\prime}$ and $y^{\prime}$ in the plane of this ES. $z^{\prime}$ points away from the direction into which the UCN is reflected.[^1] Coordinate system $x^{\prime}$, $y^{\prime}$, $z^{\prime}$ is defined for the narrow space where the particle motion can be considered linear and uniform.
Since the $\alpha^{(3)}$ and $\beta^{(3)}$ constituents of the wave function move as a unit the wave numbers $k_{x^{\prime}}$ and $k_{y^{\prime}}$ are the same for both. Thus we put $$\begin{aligned}
\label{A8}
&\alpha^{(3)}(x^{\prime},y^{\prime},z^{\prime})=\alpha(z^{\prime})\,e^{ik_{x^{\prime}}x^{\prime}}\,e^{ik_{y^{\prime}}y^{\prime}},\nonumber\\
&\beta^{(3)}(x^{\prime},y^{\prime},z^{\prime})=\beta(z^{\prime})\,e_{+}\,e^{ik_{x^{\prime}}x^{\prime}}\,e^{ik_{y^{\prime}}y^{\prime}}.\end{aligned}$$ As in [@STE01], $e_{+}=e^{i\phi}$ can be interpreted as a Bloch-wave modulation due to the field rotation.
Substituting (\[A8\]) in (\[A5\]) and (\[A6\]) we obtain $$\label{A9}
\frac{d^{2}\alpha(z^{\prime})}{d z^{\prime 2}}+k^{\prime 2}_{+}(z^{\prime})\alpha(z^{\prime})=0$$ and $$\begin{aligned}
\label{A10}
&\left[\frac{d^{2}\beta(z^{\prime})}{d z^{\prime 2}}+k^{\prime 2}_{-}\beta(z^{\prime})\right]e^{ik_{x^{\prime}}x^{\prime}}\,e^{ik_{y^{\prime}}y^{\prime}}\nonumber\\
&=\sum_{j=x^{\prime}\textrm{,}y^{\prime}\textrm{,}x^{\prime}}\alpha^{(3)}_{j}(x^{\prime},y^{\prime},z^{\prime})(\theta_{j}+i\phi_{j}\sin\theta),\end{aligned}$$ where the wave numbers for the $z^{\prime}$ direction are given by $$\label{A11}
k^{\prime 2}_{\pm}=k^{2}_{\pm}-k^{2}_{x^{\prime}}-k^{2}_{y^{\prime}}$$ with $k^{2}_{\pm}$ defined in (\[A7\]). Among the components of $k_{+}$ and $k_{-}$, the $z^{\prime}$ component $k^{\prime}_{+}(z^{\prime})$ plays a special role. Even within the narrow space where the primed system of coordinates has been defined, $k^{\prime}_{+}$ is not constant. It becomes zero at the TP $z^{\prime}=0$ and, in this semi-classical picture, is only defined for $z^{\prime}\le 0$.
In WKB approximation the solution of (\[A9\]) is $$\label{A12}
\alpha(z^{\prime})=\frac{1}{\sqrt{k^{\prime}_{+}(z^{\prime})}}\,e^{iX^{\prime}_{+}(z^{\prime})}$$ with $X^{\prime}_{+}(z^{\prime})=\int^{z^{\prime}}k^{\prime}_{+}(u)du$. (The exact quantum solution in form of an Airy function has no singularity at $z^{\prime}=0$ and decays exponentially in the classically forbidden zone $z^{\prime}>0$.)
To solve (\[A10\]) we substitute the WKB approximation for the partial derivatives on the RHS, $\alpha^{(3)}_{x^{\prime}}=ik_{x^{\prime}}\alpha^{(3)}$, $\alpha^{(3)}_{y^{\prime}}=ik_{y^{\prime}}\alpha^{(3)}$, $\alpha^{(3)}_{z^{\prime}}=ik^{\prime}_{+}(z^{\prime})\alpha^{(3)}$, and implement the total time derivative in the form $d/dt=v_{x^{\prime}}(\partial/\partial x^{\prime})+v_{y^{\prime}}(\partial/\partial y^{\prime})+v_{z^{\prime}}(\partial/\partial z^{\prime})$ with velocity $\mathbf{v}=(\hbar/m)\mathbf{k}_{+}$. Employing also (\[A12\]), (\[A10\]) becomes $$\label{A13}
\frac{d^{2}\beta(z^{\prime})}{dz^{\prime 2}}+k^{\prime 2}_{-}\beta(z^{\prime})=\frac{m}{\hbar}\frac{i}{\sqrt{k^{\prime}_{+}(z^{\prime})}}(\dot{\theta}+i\dot{\phi}\sin\theta)\,e^{iX^{\prime}_{+}(z^{\prime})}.$$
The solution of (\[A13\]) has been outlined in [@WAL01; @STE01]. The phase factor for the wave $\beta(z^{\prime})$ is the same as for $\alpha(z^{\prime})$: $e^{i X^{\prime}_{+}}(z^{\prime})$. Therefore, in WKB approximation we have $d^{2}\beta(z^{\prime})/dz^{\prime 2}=-k^{\prime 2}_{+}(z^{\prime})\beta(z^{\prime})$ and the solution of (\[A13\]) becomes $$\label{A14}
\beta(z^{\prime})=\frac{m}{\hbar}\frac{i}{\sqrt{k^{\prime}_{+}(z^{\prime})}}\frac{\dot{\theta}+i\dot{\phi}\sin\theta}{k^{\prime 2}_{-}-k^{\prime 2}_{+}(z^{\prime})}e^{iX^{\prime}_{+}(z^{\prime})}.$$
The depolarization loss measured at TP $U$ is given by the probability current for spin-flipped UCNs, $$\label{A15}
j_{-}(z^{\prime})=-\frac{\hbar}{m}\real\left[i\beta^{\ast}(z^{\prime})\left(\frac{d\beta}{dz^{\prime}} \right) \right],$$ leaving the storage space at $z^{\prime}=0$ in the positive $z^{\prime}$ direction. With (\[A14\]) this current is $$\label{A16}
j_{-}(z^{\prime})=\frac{m}{\hbar}\frac{\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}\theta}{[k^{\prime 2}_{-}-k^{\prime 2}_{+}(z^{\prime})]^{2}}=\frac{\hbar}{m}\frac{\Omega^{2}}{4\omega^{2}_{L}},$$ evaluated at $z^{\prime}=0$ (i.e., for the field variables $\Omega$ and $\omega_{L}$ at the particle position at time $t=\Delta T$). In the last step of (\[A16\]) we have used the Larmor frequency $\omega_{L}=\hbar (k^{\prime 2}_{-}-k^{\prime 2}_{+})/(2 m)$.
To evaluate the spin-flip loss rate $1/\tau_{dep}$ between the consecutive TPs we divide the current (\[A16\]) by the number $\mathcal{N}$ of ($+$) spin UCNs moving between the TPs in a channel with unit cross section centered at the trajectory. The cross section of this channel is measured parallel to the ES at every point along the path. The $\mathcal{N}$ particles within this volume contribute to loss current (\[A16\]), their decay rate $-\dot{\mathcal{N}}$ equaling $j_{-}(0)$.
Denoting the wave number perpendicular to the ESs traversed along the way by $k^{\prime}_{+}(t)$ and using the WKB form $|\alpha(t)|^{2}=1/k^{\prime}_{+}(t)$ as the particle density we have $$\begin{aligned}
\label{A17}
\mathcal{N}&=\int_{\textrm{channel}}|\alpha(t)|^{2}\,d(volume)\nonumber\\
&=\int_{0}^{\Delta T}\frac{1}{k^{\prime}_{+}(t)}\frac{\hbar k^{\prime}_{+}(t)\,dt}{m}=\frac{\hbar}{m}\Delta T.\end{aligned}$$ As for the 1D field model of [@STE01], $\mathcal{N}$ is given directly by the travel time $\Delta T$. Using (\[A16\]), the depolarization rate becomes $$\label{A18}
1/\tau_{dep}=-\dot{\mathcal{N}}/\mathcal{N}=\frac{m}{\hbar}\frac{j_{-}(0)}{\Delta T}=\frac{\Omega^{2}}{4\omega^{2}_{L}\,\Delta T},$$ evaluated for the field at the endpoint $U$. This agrees with the semi-classical result $1/\tau_{dep}=p(\Delta T)/\Delta T$ with $p(t)$ given by Eq. (\[10\]). Generalizing this result to arbitrary time $t$, we choose the UCN position at $t$ as the center of reference system $x^{\prime},y^{\prime},z^{\prime}$, with $z^{\prime}$ normal to the local ES, and use (\[A14\]), (\[A15\]) to obtain the identity $$\label{A19}
\frac{m}{\hbar} j_{-}(t)=\frac{\Omega^{2}(t)}{4\omega^{2}_{L}(t)}=p(t).$$ This shows that the semi-classical and the quantum approaches to depolarization are equivalent, with $p(t)$ directly corresponding to $m/\hbar$ times the probability current $j_{-}(t)$.
There is an open question of interpretation: In the derivation of (\[A13\]) we used a total time derivative in the form $\dot{f}=v_{x^{\prime}}(\partial f/\partial x^{\prime})+v_{y^{\prime}}(\partial f/\partial y^{\prime})+v_{x^{\prime}}(\partial f/\partial z^{\prime})$ with velocity $\mathbf{v}$ referring to the particle’s motion along its classical path. In this sense, the quantum approach of this Appendix does involve classical concepts. Use of the WKB method is not the only approximation made.
A similar caveat applies to the possibility of going to higher-order approximations in this quantum approach. In the semi-classical analysis we were allowed to add, in (\[21\]) and (\[22\]), terms such as $A_{pp}\beta$ which are of second order and had been neglected in the first-order approach of Eqs. (\[6\]) and (\[7\]). However, adding the corresponding second-order contributions in the quantum treatment would require that we also add the second-order quantities neglected in the WKB approximation used to derive the ODEs (\[A5\]) and (\[A6\]). These would be replaced by coupled non-linear PDEs of high complexity. At this stage the semi-classical and quantum approaches clearly diverge.
[100]{}
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[^1]: There are special cases where the path curvature at a TP equals the curvature of the ES. These are locations where two TPs coincide and the trajectory may proceed on either side of the ES, depending on the exact initial conditions. In this limit, the direction “away” is ill-defined, but for a continuous spectral distribution in phase space these paths represent a negligible fraction of the ensemble.
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---
abstract: 'The quark matter created in relativistic nuclear collisions is interpreted as a nearly-perfect fluid. The recent efforts to explore its finite-density properties in the beam energy scan programs motivate one to revisit the issue of the local rest frame fixing in off-equilibrium hydrodynamics. I first investigate full second-order relativistic hydrodynamics in the Landau and the Eckart frames. Then numerical hydrodynamic simulations are performed to elucidate the effect of frame choice on flow observables in relativistic nuclear collisions. The results indicate that the flow can differ in the Landau and the Eckart frames but charged particle and net baryon rapidity distributions are mostly frame independent when off-equilibrium kinetic freeze-out is considered.'
author:
- Akihiko Monnai
bibliography:
- 'frame.bib'
title: Landau and Eckart frames for relativistic fluids in nuclear collisions
---
Introduction {#sec1}
============
The existence of the strongly-coupled quark-gluon plasma [@Yagi:2005yb] as a high-temperature phase of QCD has been partly motivated by a number of relativistic hydrodynamic analyses of high-energy nuclear collisions at BNL Relativistic Heavy Ion Collider [@Arsene:2004fa; @Back:2004je; @Adams:2005dq; @Adcox:2004mh] and CERN Large Hadron Collider [@Aamodt:2010pa; @ATLAS:2011ah; @Chatrchyan:2012wg]. Modern versions of such analyses incorporate the effects of viscosity to take account of off-equilibrium processes in the system, which play important roles in quantitative understanding of the experimental data [@Wang:2016opj].
The theoretical framework of relativistic dissipative hydrodynamics, however, is still not completely understood, partially because one has to introduce relaxation effects to the off-equilibrium processes to avoid violating stability and causality [@Hiscock:1983zz; @Hiscock:1985zz; @Hiscock:1987zz]. Such extended frameworks are called the second-order theory [@Israel:1976tn; @Israel:1979wp; @Muronga:2001zk; @Muronga:2003ta; @Muronga:2006zw; @Muronga:2006zx; @Koide:2006ef; @Tsumura:2006hn; @Tsumura:2007wu; @Tsumura:2009vm; @Tsumura:2011cj; @Tsumura:2012ss; @Baier:2007ix; @Romatschke:2009kr; @Bhattacharyya:2008jc; @Natsuume:2007ty; @Lublinsky:2009kv; @Betz:2009zz; @Monnai:2010qp; @Monnai:2018rgs; @Molnar:2013lta; @Denicol:2012cn; @Denicol:2012vq; @Denicol:2019iyh; @PeraltaRamos:2009kg; @Calzetta:2010au; @Aguilar:2017ios; @Florkowski:2015lra; @Jaiswal:2015mxa; @Tinti:2016bav; @Harutyunyan:2018cmm; @Mitra:2019jld] as opposed to the traditional linear response theory [@Landau; @Eckart:1940te], which is also known as the first-order theory, because the off-equilibrium correction of the respective order in terms of dissipative currents is taken into account in the entropy current of those theories.
Non-relativistic hydrodynamic flow can be defined as a local flux of particles. In relativistic systems, however, the definition of the flow is not trivial because the energy and the conserved number can flow separately in the presence of dissipative processes. There are two distinctive ways of defining the local rest frame for the flow: the Landau (or energy) frame [@Landau] and the Eckart (or conserved charge/particle) frame [@Eckart:1940te]. There have been decades of debate over the eligibility of the two definitions of the local rest frame [@Tsumura:2006hn; @Tsumura:2007wu; @Tsumura:2009vm; @Tsumura:2011cj; @Tsumura:2012ss; @Van:2007pw; @Van:2011yn; @Osada:2011gx; @Minami:2012hs; @Oliinychenko:2015lva; @Sagaceta-Mejia:2018cao]. Most of the numerical analyses of hydrodynamic models for relativistic nuclear collisions so far do not give explicit consideration to the frame because the diffusion or the dissipation current is neglected, but the Landau frame is often considered to be a preferred choice when there is a theoretical need. This could be owing to the fact that the primary conserved charge in nuclear collisions is the net baryon number, which is often small at high energies; the Eckart frame cannot be defined when conserved charges are approximated to be negligible. There are several calculations [@Monnai:2012jc; @Kapusta:2014dja; @Denicol:2018wdp; @Li:2018fow] that include the effects of baryon diffusion, which intrinsically implies that the Landau frame is chosen.
The beam energy scan (BES) programs are being performed at RHIC. The exploration of mid-to-low beam energy regime is also planned at facilities including GSI Facility for Antiprotons and Ion Research (FAIR), JINR Nuclotron-based Ion Collider fAcility (NICA), CERN Super Proton Synchrotron (SPS), and JAEA/KEK Japan Proton Accelerator Research Complex (J-PARC) in order to elucidate the QCD phase structure at finite densities. It would be insightful to come back to the question of the flow frame in hydrodynamic models and investigate whether the choice of the frame can affect observables in those experiments.
In this paper, full second-order hydrodynamic equations are investigated in the Landau and the Eckart frames. Stability and causality conditions in the two frames are shown to be related to the correspondences between the first- and the second-order transport coefficients in those frames. Then the implications of a frame choice on the hydrodynamic evolution in heavy-ion systems are discussed focusing on the baryon diffusion and the energy dissipation. Numerical analyses are performed for rapidity distribution because the effects of the net baryon number would appear most prominently in the direction of the collision. Fragments of the shattered nuclei are the source of the conserved charge.
The paper is organized as follows. Full second-order relativistic dissipative hydrodynamics is investigated in the Landau and the Eckart frames in Sec. \[sec2\]. Causality and stability conditions are discussed in Sec. \[sec3\]. Sec. \[sec4\] presents numerical demonstration of the effects of a frame choice in nuclear collisions. Discussion and conclusions are presented in Sec. \[sec5\]. The natural unit $c = \hbar = k_B = 1$ and the mostly-negative Minkowski metric $g^{\mu \nu} = \mathrm{diag}(+,-,-,-)$ is used in this paper.
Relativistic hydrodynamics in Landau and Eckart Frames {#sec2}
======================================================
The ideal hydrodynamic flow is uniquely determined since the local fluxes of the energy and the charge densities are in the same direction, *i.e.*, the directions of the eigenvector of the energy-momentum tensor and the conserved current match as $T^{\mu \nu} u_\nu = e u^\mu$ and $N^\mu = n u^\mu$. Here $e$ is the energy density and $n$ is the conserved charge density. On the other hand, the presence of the vector dissipative currents lead to the separation of the two local fluxes in relativistic systems. The Landau frame is chosen in the direction of the total energy flux so the dissipation of energy does not appear explicitly, $T^{\mu \nu} u^L_\nu = e_L u_L^\mu$. The Eckart frame is the choice of flow where the total conserved charge flux is diffusion-less $N^\mu = n_E u_E^\mu$. Here the subscripts $L$ and $E$ represent the Landau and the Eckart frames, respectively. The energy-momentum tensor, the conserved charge current, and the entropy current $s^\mu$ are assumed to be invariant under frame transformations [@Israel:1979wp].
The tensor decompositions read $$\begin{aligned}
T^{\mu \nu} &=& e_L u_L^\mu u_L^\nu - (P_L + \Pi_L) \Delta_L^{\mu \nu} + \pi_L^{\mu \nu}, \\
N^\mu &=& n_L u_L^\mu + V_L^\mu ,\end{aligned}$$ in the Landau frame and $$\begin{aligned}
T^{\mu \nu} &=& e_E u_E^\mu u_E^\nu - (P_E + \Pi_E) \Delta_E^{\mu \nu} \nonumber \\
&+& W_E^\mu u_E^\nu + W_E^\nu u_E^\mu + \pi_E^{\mu \nu}, \\
N^\mu &=& n_E u_E^\mu ,\end{aligned}$$ in the Eckart frame. Here $\Pi$ is the bulk pressure, $\pi^{\mu \nu}$ is the shear stress tensor, $W^\mu$ is the energy dissipation, $V^\mu$ is the baryon diffusion, and $\Delta^{\mu \nu} = g^{\mu \nu} - u^\mu u^\nu$ is the space-like projection. It can be immediately seen that the two frames become identical in the ideal hydrodynamic limit. The dissipative corrections to the energy and the conserved charge densities are neglected for simplicity [@Monnai:2018rgs]. Also I consider a system with a single charge conservation though the extension to general systems should be a straightforward task.
In the following arguments, the vector dissipative currents $W_E^\mu$ and $V_L^\mu$ are considered and the shear and bulk viscous corrections are set aside for simplicity. When the dissipative corrections are much smaller than the equilibrium variables, the difference in the thermodynamic variables of the two frames $\Delta n_{E-L} = n_E - n_L$ and $\Delta e_{E-L} = e_E - e_L$ are, at a given space-time point, $$\begin{aligned}
\Delta n_{E-L} &=& \frac{1}{2} \frac{V_L^\mu V^L_\mu}{n_L} + \mathcal{O}(\delta^3), \label{eq:dn}\\
\Delta e_{E-L} &=& - \frac{W_E^\mu W^E_\mu}{e_E + P_E} + \mathcal{O}(\delta^3), \label{eq:de}\end{aligned}$$ where the correction is of the second order in dissipative quantities. They indicate that the corrections to other thermodynamic variables, *i.e.*, the pressure $P$, the entropy density $s$, the temperature $T$, and the chemical potential $\mu$ are of the second order. The corrections to the transport coefficients should also be of the second order because they are functions of the energy and the conserved charge densities. Hereafter the subscripts $L$ and $E$ are dropped for those variables for simplicity unless otherwise required. The flow difference $\Delta u^\mu_{E-L} = u_E^\mu - u_L^\mu$ is $$\begin{aligned}
\Delta u^\mu_{E-L} = \frac{V_L^\mu }{n} + \mathcal{O}(\delta^2) = - \frac{W_E^\mu}{e+P} + \mathcal{O}(\delta^2), \end{aligned}$$ where the leading order correction is of the first order.
The macroscopic variables are estimated using the conservation laws $\partial_\mu T^{\mu \nu} = 0$ and $\partial_\mu N^\mu = 0$, the equation of state $P=P(e,n_B)$, and the constitutive relations for the dissipative currents. In the Landau frame, the second-order causal expression of the baryon diffusion, based on an extended Israel-Stewart framework [@Israel:1976tn; @Israel:1979wp; @Monnai:2010qp], reads $$\begin{aligned}
V_L^\mu &=& \kappa_V \nabla_L^\mu \frac{\mu}{T} - \tau_V (\Delta_{L})^{\mu}_{\ \nu} D_L V_L^\nu \nonumber \\
&+& \chi_V^a V_L^\mu D_L \frac{\mu}{T} + \chi_V^b V_L^\mu D_L \frac{1}{T} + \chi_V^c V_L^\mu \nabla^L_\nu u_L^\nu \nonumber \\
&+& \chi_V^d V_L^\nu \nabla^L_\nu u_L^\mu + \chi_V^e V_L^\nu \nabla_L^\mu u^L_\mu,
\label{eq:diffusion}\end{aligned}$$ where $\kappa_V \geq 0$ is the baryon conductivity, $\tau_V \geq 0$ is the relaxation time for the baryon diffusion, and $\chi_V^{a,b,c,d,e}$ are second-order transport coefficients. $D = u^\mu \partial_\mu$ and $\nabla^\mu = \Delta^{\mu \nu} \partial_\nu$ are the time- and the space-like derivatives, respectively. Similarly, in the Eckart frame, the energy dissipation reads $$\begin{aligned}
W_E^\mu &=& - \kappa_W \bigg( \nabla_E^\mu \frac{1}{T} + \frac{1}{T} D_E u_E^\mu \bigg) - \tau_W (\Delta_{E})^{\mu}_{\ \nu} D_E W_E^\nu \nonumber \\
&+& \chi_W^a W_E^\mu D_E \frac{\mu}{T} + \chi_W^b W_E^\mu D_E \frac{1}{T} + \chi_W^c W_E^\mu \nabla^E_\nu u_E^\nu \nonumber \\
&+& \chi_W^d W_E^\nu \nabla^E_\nu u_E^\mu + \chi_W^e W_E^\nu \nabla_E^\mu u^E_\nu,
\label{eq:dissipation}\end{aligned}$$ where $\kappa_W \geq 0$ is the energy conductivity and $\tau_W \geq 0$ is the relaxation time for the energy dissipation, and $\chi_W^{a,b,c,d,e}$ are second-order transport coefficients. For the full expression of the second-order hydrodynamic equations including the scalar and the tensor dissipative currents, see for example Ref. [@Monnai:2010qp].
The second law of thermodynamics implies that the entropy production is expressed in a quadratic form. It can be written in the Landau frame as $$\begin{aligned}
\partial_\mu s^\mu = - \frac{V_L^\mu V^L_\mu }{\kappa_V} \geq 0,\end{aligned}$$ and in the Eckart frame as $$\begin{aligned}
\partial_\mu s^\mu = - \frac{W_E^\mu W^E_\mu}{\kappa_W} \geq 0,\end{aligned}$$ with the mostly-minus metric. The first and the second order transport coefficients of the two frames are related by the identification of the entropy production: $$\begin{aligned}
\kappa_V &=& \kappa_W \bigg( \frac{n}{e+P} \bigg)^2, \label{kappaLE}\\
\tau_V &=& \tau_W - \frac{\kappa_W}{(e+P)T}, \label{tauLE} \\
\chi_V^a &=& \chi_W^a - \frac{\tau_W nT}{e+P}, \\
\chi_V^b &=& \chi_W^b + \tau_W T - \frac{\kappa_W}{e+P}, \\
\chi_V^c &=& \chi_W^c + \frac{\kappa_W}{(e+P)T}, \\
\chi_V^d &=& \chi_W^d + \frac{\kappa_W}{(e+P)T}, \\
\chi_V^e &=& \chi_W^e . \label{chieLE}\end{aligned}$$ See Appendix \[sec:A\] for the derivation. Those relations indicate that the full second-order terms are necessary in addition to the conventional Israel-Stewart terms for understanding the frame dependence of relativistic dissipative hydrodynamics, because the vanishing second-order transport coefficients in one frame lead to non-vanishing ones in the other frame, except for $\chi_V^e$ and $\chi_W^e$.
CAUSALITY AND STABILITY OF SECOND-ORDER HYDRODYNAMICS {#sec3}
=====================================================
In this section, causality and stability conditions of the relativistic full second-order hydrodynamic equations are investigated in the Landau and the Eckart frames. A plane wave perturbation $\delta Q = \delta \bar{Q} e^{i(\omega t - kx)}$ is considered for a macroscopic variable $Q$ around global equilibrium where $u^\mu = (1,0,0,0)$. The perturbed equations of motion are used to analyze the hydrodynamic modes [@Hiscock:1985zz; @Hiscock:1987zz].
Landau Frame
------------
In the Landau frame, the perturbed energy-momentum tensor and the conserved charge current are $$\begin{aligned}
\delta T^{\mu \nu} &=& (e+P) (\delta u^\mu u^\nu + u^\mu \delta u^\nu) \nonumber \\
&+& \delta e u^\mu u^\nu - \delta P g^{\mu \nu}, \\
\delta N^\mu &=& n \delta u^\mu + \delta n u^\mu + \delta V^\mu ,\end{aligned}$$ which follow the conservation law and the constitutive relation $$\begin{aligned}
\delta V^\mu &=& \kappa_V \nabla^\mu \delta \alpha - \tau_V \Delta ^{\mu \nu} D \delta V_\nu ,\end{aligned}$$ where $\alpha = \mu/T$. The longitudinal and the transverse modes relevant to the diffusion are given by $$\begin{aligned}
\mathcal{M}^L_{xx}
\begin{pmatrix}
\delta e\\
\delta n\\
\delta u^x \\
\delta V^x \\
\end{pmatrix}
&=&0 , \end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^L_{xy}
\begin{pmatrix}
\delta u^y \\
\delta V^y \\
\end{pmatrix}
=0 , \
\mathcal{M}^L_{xz}
\begin{pmatrix}
\delta u^z \\
\delta V^z \\
\end{pmatrix}
=0 , \end{aligned}$$ where $$\begin{aligned}
\mathcal{M}^L_{xx} = \begin{pmatrix}
i\omega&0&-ikh&0\\
-ik \frac{\partial P}{\partial e}|_n&-ik\frac{\partial P}{\partial n}|_e&i\omega h&0\\
0&i\omega&-ikn&-ik\\
-ik\kappa_V \frac{\partial \alpha}{\partial e}|_n&-ik\kappa_V \frac{\partial \alpha}{\partial n}|_e&0&1+i \omega \tau_V
\end{pmatrix} , \nonumber \\\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^L_{xy} = \mathcal{M}^L_{xz} =
\begin{pmatrix}
i\omega h &0 \\
0&1+i\omega \tau_V
\end{pmatrix} ,\end{aligned}$$ using the enthalpy density $h = e+P$. They have non-trivial solutions when the matrices have vanishing determinants. The longitudinal equations $\det(\mathcal{M}^L_{xx}) = 0$ lead to $$\begin{aligned}
\omega^2 - c_s^2 k^2 = \frac{i \kappa_V (c_2 \omega^2 - c_4 k^2) k^2 }{\omega (1+i \tau_V \omega)} , \label{eq:MLxx}\end{aligned}$$ where the sound velocity is defined as $$\begin{aligned}
c_s^2 = \frac{\partial P}{\partial e}\bigg|_n + \frac{n}{h} \frac{\partial P}{\partial n}\bigg|_e , \label{eq:cs2}\end{aligned}$$ and the thermodynamic coefficients as $$\begin{aligned}
c_2 &=& \frac{\partial \alpha}{\partial n} \bigg|_e ,\\
c_4 &=& \frac{\partial \alpha}{\partial n} \bigg|_e \frac{\partial P}{\partial e}\bigg|_n - \frac{\partial \alpha}{\partial e}\bigg|_n \frac{\partial P}{\partial n}\bigg|_e .\end{aligned}$$ The Routh-Hurwitz stability criteria [@Routh; @Hurwitz] indicate that the $\mathrm{Im} (\omega)$ stays semi-positive when $c_2 \geq 0$ and $c_s^2 c_2 - c_4 \geq 0$. Those conditions are satisfied in thermodynamic systems since the former follows from the thermodynamic requirement that the fugacity should increase as the number density increases at a fixed energy density and the latter from $$\begin{aligned}
c_s^2 c_2 - c_4 &=& \frac{\beta}{h}
\bigg( \frac{\partial P}{\partial \alpha}\bigg|_\beta \frac{\partial \alpha}{\partial n} \bigg|_e - \frac{\partial P}{\partial \beta}\bigg|_\beta \frac{\partial \alpha}{\partial e} \bigg|_n \bigg)^2 \geq 0 ,\end{aligned}$$ where $\beta = 1/T$, using the definition of the sound velocity (\[eq:cs2\]) and the thermodynamic properties $$\begin{aligned}
\frac{\partial P}{\partial \alpha}\bigg|_\beta &=& \frac{n}{\beta} , \label{eq:dpda}\ \
\frac{\partial P}{\partial \beta}\bigg|_\alpha = - \frac{h}{\beta} ,\end{aligned}$$ and $$\begin{aligned}
\frac{\partial \beta}{\partial n} \bigg|_e &=& - \frac{\partial \alpha}{\partial e} \bigg|_n \label{eq:bnae}.\end{aligned}$$
Although it is possible to analytically solve the quartic equation, the general solutions are complicated. Here asymptotic forms at small $k$ are considered for more physical arguments. The propagating modes are, up to the leading order in real and imaginary parts, $$\begin{aligned}
\omega = \pm c_s k + i \frac{\kappa_V (c_s^2 c_2 - c_4 )}{2 c_s^2} k^2 ,\end{aligned}$$ and the non-propagating mode is $$\begin{aligned}
\omega = \frac{i}{\tau_V} ,\end{aligned}$$ aside from the trivial $\omega = 0$. They satisfy the causality condition $$\begin{aligned}
\bigg | \frac{\partial \mathrm{Re} (\omega)}{\partial k} \bigg | \leq 1.\end{aligned}$$ The stability condition $$\begin{aligned}
\mathrm{Im} (\omega) \geq 0,\end{aligned}$$ is satisfied for $c_2 c_s^2 - c_4\geq 0$, which is consistent with the Routh-Hurwitz stability conditions.
The solutions to the transverse equations $$\begin{aligned}
\det(\mathcal{M}^L_{xy}) &=& \det(\mathcal{M}^L_{xz}) \nonumber \\
&=& i \omega h (1+i\omega \tau_V) = 0, \label{eq:MLxy}\end{aligned}$$ are the non-propagating modes $\omega = 0$ and $\omega = i/\tau_V$. The causality and stability conditions are trivially satisfied. Those results of the longitudinal and the transverse modes indicate that the second-order diffusive hydrodynamics is causal and stable in the Landau frame.
Eckart Frame
------------
In the Eckart frame, the energy-momentum tensor and the conserved charge current are expressed as $$\begin{aligned}
\delta T^{\mu \nu} &=& (e+P) (\delta u^\mu u^\nu + u^\mu \delta u^\nu) + \delta e u^\mu u^\nu \nonumber \\
&-& \delta P g^{\mu \nu} + \delta W^\mu u^\nu + \delta W^\nu u^\mu, \\
\delta N^\mu &=& n \delta u^\mu + \delta n u^\mu ,\end{aligned}$$ and the energy dissipation current as $$\begin{aligned}
\delta W^\mu &=& - \kappa_W \beta_\mathrm{eq} D \delta u^\mu - \kappa_W \nabla^\mu \delta \beta - \tau_W \Delta^{\mu \nu} D \delta W_\nu . \nonumber \\\end{aligned}$$ The perturbed equations of motion are $$\begin{aligned}
\mathcal{M}^E_{xx}
\begin{pmatrix}
\delta e\\
\delta n\\
\delta u^x \\
\delta W^x \\
\end{pmatrix}
&=&0 , \end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^E_{xy}
\begin{pmatrix}
\delta u^y \\
\delta W^y \\
\end{pmatrix}
=0 , \
\mathcal{M}^E_{xz}
\begin{pmatrix}
\delta u^z \\
\delta W^z \\
\end{pmatrix}
=0 , \end{aligned}$$ where $$\begin{aligned}
\mathcal{M}^E_{xx} = \begin{pmatrix}
i\omega&0&-ik h&-ik\\
-ik \frac{\partial P}{\partial e}|_n&-ik \frac{\partial P}{\partial n}|_e&i\omega h&i\omega \\
0&i\omega&-ikn&0\\
ik\kappa_W \frac{\partial \beta}{\partial e}|_n&ik\kappa_W \frac{\partial \beta}{\partial n}|_e&i \omega \kappa_W \beta &1+i \omega \tau_W
\end{pmatrix} , \nonumber \\\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^E_{xy} = \mathcal{M}^E_{xz} =
\begin{pmatrix}
i\omega h&i \omega \\
i \omega \kappa_W \beta &1+i\omega \tau_V
\end{pmatrix} .\end{aligned}$$ The longitudinal equations $\det(\mathcal{M}^E_{xx}) = 0$ lead to $$\begin{aligned}
\omega^2 - c_s^2 k^2 = \frac{i \kappa_W (d_2 \omega^2 - d_4 k^2) k^2 }{\omega [1+i (\tau_W - \kappa_W \beta/h ) \omega]} , \label{eq:MExx}\end{aligned}$$ where $$\begin{aligned}
d_2 &=& \frac{n}{h} \bigg( \frac{\partial \beta}{\partial n} \bigg|_e + \frac{\beta}{h} \frac{\partial P}{\partial n}\bigg|_e \bigg),\\
d_4 &=& \frac{n}{h} \bigg( \frac{\partial \beta}{\partial n} \bigg|_e \frac{\partial P}{\partial e}\bigg|_n - \frac{\partial \beta}{\partial e}\bigg|_n \frac{\partial P}{\partial n}\bigg|_e \bigg) .\end{aligned}$$ Here, $\mathrm{Im} (\omega)$ stays semi-positive when $d_2 \geq 0$, $c_s^2 d_2 - d_4 \geq 0$, and $$\begin{aligned}
\tau_W - \frac{\beta}{h} \kappa_W \geq 0, \label{eq:twkw}\end{aligned}$$ according to the Routh-Hurwitz stability criteria. The first two conditions are again satisfied in thermodynamic systems as $$\begin{aligned}
d_2 &=& \frac{n^2}{h^2} \frac{\partial \alpha}{\partial n}\bigg|_e \geq 0,\\
c_s^2 d_2 - d_4 &=& \frac{n^2 \beta}{h^3}
\bigg( \frac{\partial P}{\partial \alpha}\bigg|_\beta \frac{\partial \alpha}{\partial n} \bigg|_e - \frac{\partial P}{\partial \beta}\bigg|_\beta \frac{\partial \alpha}{\partial e} \bigg|_n \bigg)^2 \geq 0,\nonumber \\\end{aligned}$$ using the relations (\[eq:dpda\]) and (\[eq:bnae\]). Note that $d_2 = c_2 n^2/h^2$ and $d_4 = c_4 n^2/h^2$. The third condition is also consistent with the ones reported in Ref. [@Hiscock:1987zz; @Osada:2011gx].
The results indicate that second-order dissipative hydrodynamics is stable in the Eckart frame if the transport coefficients satisfy the condition (\[eq:twkw\]). It can be immediately seen that the first-order theory is unstable in the Eckart frame by taking the limit of vanishing relaxation time $\tau_W\to 0$.
It is important to note that the space-like projection of the energy-momentum conservation law leads to $$\begin{aligned}
(e+P)Du^\mu &=& \nabla^\mu P
- W^\mu \nabla_\nu u^\nu \nonumber \\
&-& W^\nu \nabla_\nu u^\mu - \Delta^{\mu \nu} DW_\nu , \label{eq:emcperp}\end{aligned}$$ which is also used to convert the thermodynamic forces (\[eq:thermoforce\]). The higher order terms in the identity is important even in the stability analyses of the first-order theory because if one neglects the correction by truncation and use it to remove the acceleration term in the energy dissipation current, the equation can become seemingly “stable" at the first order. This is because the relaxation term-like correction originating from the last term in Eq. (\[eq:emcperp\]) is effectively introduced by the procedure at the second order even though it is not apparent. The prefactor before this effective relaxation term is $\kappa_W/(e+P)T$, which is the minimum value of the relaxation time required for hydrodynamic stability. The constitutive relation is qualitatively modified and thus cannot be regarded as a first-order theory.
The asymptotic forms of the propagating and the non-propagating modes at small $k$ are $$\begin{aligned}
\omega = \pm c_s k + i \frac{\kappa_W (c_s^2 d_2 - d_4 )}{2 c_s^2} k^2 ,\end{aligned}$$ and $$\begin{aligned}
\omega = \frac{i}{\tau_W - \kappa_W \beta/h} ,\end{aligned}$$ aside from $\omega = 0$. Those modes are causal and stable if the Routh-Hurwitz criteria are satisfied.
The transverse equations $$\begin{aligned}
\det(\mathcal{M}^E_{xy}) &=& \det(\mathcal{M}^E_{xz}) \nonumber \\
&=& i\omega [h+i\omega (\tau_V - \kappa_W \beta)] = 0, \label{eq:MExy}\end{aligned}$$ have the non-propagating solutions $\omega = i/(\tau_W - \kappa_W \beta/h)$ and $\omega = 0$. One can see that all the modes satisfy the stability and causality conditions if the relaxation time is sufficiently larger than the conductivity (\[eq:twkw\]).
Comparing the two frames, the characteristic equations in the Landau frame (\[eq:MLxx\]) and (\[eq:MLxy\]) and their solutions are equivalent to those in the Eckart frame (\[eq:MExx\]) and (\[eq:MExy\]) under the identification of the conductivities (\[kappaLE\]) and the relaxation times (\[tauLE\]) that follow from the matching of the entropy production. The relation of the relaxation times in the two frames implies that the Eckart stability condition on $\tau_W$ is closely related with the fact that $\tau_V$ is semi-positive in the other frame.
Numerical Application to Heavy-Ion Collisions {#sec4}
=============================================
The effects of a frame choice on relativistic nuclear collisions are demonstrated by solving the energy dissipative and the baryon diffusive hydrodynamic equations. For this purpose, a non-boost invariant (1+1)-dimensional hydrodynamic system is considered [@Monnai:2012jc]. Full (3+1)-dimensional calculations for quantitative analyses of the data sets from the beam energy scan experiments is beyond the scope of the current study and will be presented elsewhere.
The hydrodynamic model
----------------------
Hydrodynamics system are characterized with the equation of state and the transport coefficients. The equation of state at finite baryon density [@Monnai:2019hkn] is based on lattice QCD [@Bazavov:2014pvz; @Bazavov:2012jq; @Ding:2015fca; @Bazavov:2017dus] and the hadron resonance gas model. The strangeness and the electric charge are not considered here for simplicity and left for future studies.
The transport coefficients are chosen as $\kappa_W = c_W (e+P)$, $\tau_W = \tilde{c}_W \kappa_W / (e+P)T$, and $\chi_W^{a,b,c,d,e} = 0$ in the Eckart frame. The model conductivity is motivated by the non-equilibrium statistical operator method for the $\phi^4$-theory [@Hosoya:1983id] coupled with the lower bound of shear viscosity conjectured in the gauge-string correspondence [@Kovtun:2004de]. $c_W = 10$ and $\tilde{c}_W = 2$ are used for demonstration. Those in the Landau frame is obtained using the relations (\[kappaLE\])-(\[chieLE\]).
The initial conditions are parametrically constructed as $$\begin{aligned}
e(\tau_\mathrm{th},\eta_s) &=& a_1 \exp(-a_2 \eta_s^2 - a_3 \eta_s^4), \\
n_B(\tau_\mathrm{th},\eta_s) &=& n_B^+(\eta_s) + n_B^-(\eta_s),\end{aligned}$$ where $$\begin{aligned}
&&n_B^\pm(\eta_s) = \nonumber \\
&&\begin{cases}
b_1 \exp[-b_2 (\eta_s \mp \eta_0)^2 - b_3 (\eta_s \mp \eta_0)^4] & \mathrm{for} \ \pm \eta_s > \eta_0, \\
b_1 \exp[-\tilde{b}_2 (\eta_s \mp \eta_0)^2 - \tilde{b}_3 (\eta_s \mp \eta_0)^4] & \mathrm{for} \ \pm \eta_s \leq \eta_0,
\end{cases}\nonumber \\\end{aligned}$$ at $\tau_\mathrm{th} = 3$ fm/$s$. The parameters are tuned to roughly reproduce the SPS data for 17.3 GeV Pb+Pb collisions [@Appelshauser:1998yb] without dissipative corrections. Here $a_1= 7.19$ (GeV/fm$^{3}$), $a_2 = 0.8$, and $a_3 = 0.05$ for the energy density and $b_1 = 0.45$ (1/fm$^{3}$), $b_2 = 0.4$, $b_3 = 4.0$, $\tilde{b}_2 = 0.55$, $\tilde{b}_3 = 2.3$, and $\eta_0 = 0.69$ for the net baryon density. It should be noted again that they are for demonstration and not for full quantitative analyses of the data because even though the results exhibit fair agreement with the data, the transverse expansion and the hadronic transport are not taken into account here. The prolonged space-time evolution may partially mimic the transport effects. The initial values of the energy dissipation and the baryon diffusion currents are set to zero to allow comparison of the effects of those processes coming from hydrodynamic evolution.
The kinetic freeze-out is estimated using the Cooper-Frye formula [@Cooper:1974mv] with off-equilibrium corrections to the phase-space distribution functions [@Teaney:2003kp; @Monnai:2009ad]. It reads $$\begin{aligned}
E_i\frac{dN^i}{d^3p} = \frac{g_i}{(2\pi)^3} \int_\Sigma p_i^\mu d\sigma_\mu (f^0_i + \delta f_i), \label{eq:CF}\end{aligned}$$ where $g_i$ is the degeneracy, $\Sigma$ is the freeze-out hypersurface, and $d\sigma_\mu$ is the freeze-out hypersurface element. $f_i^0$ is the equilibrium (Bose-Einstein or Fermi-Dirac) phase-space distribution function for the $i$-th particle species and $\delta f_i$ is the off-equilibrium distortion of the distribution function. The expression of $\delta f$ in the Landau and the Eckart frames are shown in Appendix \[sec:B\]. The hypersurface is determined with the freeze-out energy density $e_\mathrm{f} = 0.4$ GeV/fm$^3$.
Space-time evolution {#sec4B}
--------------------
![The space-time rapidity dependences of (a) the entropy density and (b) the net baryon density at the initial time (thin solid line) and those after ideal (thick solid line), baryon diffusive (dashed line), and energy dissipative (dotted line) hydrodynamic evolutions at $\tau = 20$ fm/$c$.[]{data-label="fig:1"}](entropy.pdf "fig:"){width="3.3in"} ![The space-time rapidity dependences of (a) the entropy density and (b) the net baryon density at the initial time (thin solid line) and those after ideal (thick solid line), baryon diffusive (dashed line), and energy dissipative (dotted line) hydrodynamic evolutions at $\tau = 20$ fm/$c$.[]{data-label="fig:1"}](netbaryon.pdf "fig:"){width="3.3in"}
First, I investigate the off-equilibrium hydrodynamic evolution in the Landau and the Eckart frames and compare them with the ideal hydrodynamic evolution. The entropy and the net baryon distributions at the initial time and $\tau = 20$ fm/$c$ are shown in Fig. \[fig:1\]. It should be noted that the lifetime of the fireball is longer in the current geometry owing to the lack of transverse expansion. The effect of baryon diffusion or energy dissipation is small on the entropy density for the current choice of transport coefficients.
The effect on the net baryon density, on the other hand, is visible. The baryon diffusion causes stronger stopping because the fugacity gradients induce net baryon diffusion from forward to mid-rapidity regions. At the edges near $|\eta_s| \sim 2$, the baryon diffusion is in the outward direction. The energy dissipation, on the other hand, is less trivial because of the interplay of the temperature gradient and the acceleration terms. The temperature gradients carry the energy density towards forward rapidity regions while the acceleration correction prevents flow convection and keep the density in the mid-rapidity region. The effects cancel at the first-order in the limit of vanishing chemical potential as seen in (\[eq:thermoforce\]). The off-equilibrium deformation of the net baryon distribution in the Eckart frame can be mainly caused by the deceleration of flow as seen in Fig. \[fig:2\] near mid-rapidity. The off-equilibrium evolutions of the net baryon distribution in the Landau frame and in the Eckart frame are quantitatively similar to each other. This can be a consequence of the fact that the frame-dependence of the thermodynamic quantities are of second order.
The difference between the flow rapidity $Y_f$ and the space-time rapidity $\eta_s$ (Fig. \[fig:2\]) implies that the Landau flow is closer to the ideal flow than the Eckart flow. Here the flow rapidity is defined as $$\begin{aligned}
u^\mu = (\cosh Y_f,0,0,\sinh Y_f),\end{aligned}$$ which reduces to the boost-invariant flow when $Y_f - \eta_s = 0$. The flow is affected more in the Eckart frame possibly because the energy dissipation is directly coupled to the equation of motion for flow acceleration (\[eq:emcperp\]). At forward space-time rapidity $|\eta_s| > 1.5$, the Eckart flow is faster then the Landau flow because of the peak position in the net baryon distribution.
![The space-time rapidity dependence of the difference between the flow and the space-time rapidities at the initial time (thin solid line) and those after ideal (thick solid line), baryon diffusive (dashed line), and energy dissipative (dotted line) hydrodynamic evolutions at $\tau = 20$ fm/$c$.[]{data-label="fig:2"}](flow.pdf){width="3.3in"}
Charged particle and net baryon rapidity distributions
------------------------------------------------------
The charged hadron rapidity distributions are shown in Fig. \[fig:3\]. The effect of energy dissipation in the Eckart frame is visible while that of baryon diffusion is negligible when the off-equilibrium correction at freeze-out (\[eq:CF\]) is not taken into account. The difference comes from the difference in the Landau and the Eckart flow and the lack of the $\delta f$ corrections. When the correction is incorporated, the effect of energy dissipation becomes small and similar to that of baryon diffusion as found in Fig. \[fig:3\] (b).
![The rapidity distributions of charged particles (a) without and (b) with $\delta f$ correction at freeze-out for the ideal hydrodynamic system (solid line) compared to those for the systems with baryon diffusion in the Landau frame (dashed line) and with energy dissipation in the Eckart frame (dotted line).[]{data-label="fig:3"}](dnchdy.pdf "fig:"){width="3.3in"} ![The rapidity distributions of charged particles (a) without and (b) with $\delta f$ correction at freeze-out for the ideal hydrodynamic system (solid line) compared to those for the systems with baryon diffusion in the Landau frame (dashed line) and with energy dissipation in the Eckart frame (dotted line).[]{data-label="fig:3"}](dnchdy_df.pdf "fig:"){width="3.3in"}
The net baryon rapidity distribution with baryon diffusion in the Landau frame and with energy dissipation in the Eckart frame are shown in Fig. \[fig:4\]. The off-equilibrium effects are visible in both frames without the $\delta f$ correction. This is consistent with the observation of hydrodynamic evolution of the net baryon density in Sec. \[sec4B\]. The baryon stopping is larger in the Eckart frame because of the flow deceleration. The effect of $\delta f$ correction at freeze-out is found to enhance the baryon stopping caused by the baryon diffusion. Again the net baryon distributions in the two frames become close to each other once the off-equilibrium correction at freeze-out is properly taken into account.
![The rapidity distributions of net baryon number (a) without and (b) with $\delta f$ correction at freeze-out for the ideal hydrodynamic system (solid line) compared to those for the systems with baryon diffusion in the Landau frame (dashed line) and with energy dissipation in the Eckart frame (dotted line).[]{data-label="fig:4"}](dnbdy.pdf "fig:"){width="3.3in"} ![The rapidity distributions of net baryon number (a) without and (b) with $\delta f$ correction at freeze-out for the ideal hydrodynamic system (solid line) compared to those for the systems with baryon diffusion in the Landau frame (dashed line) and with energy dissipation in the Eckart frame (dotted line).[]{data-label="fig:4"}](dnbdy_df.pdf "fig:"){width="3.3in"}
It is worth noting that the effect of the $\delta f$ correction is larger in the Eckart frame for the charged particle distribution while it is larger in the Landau frame for the net baryon distribution (Fig. \[fig:5\]). The results suggest that an adequate treatment of $\delta f$ corrections are important for qualitative understanding of the flow observables.
![The ratios of the rapidity distributions with and without $\delta f$ correction for (a) charged particles and (b) for net baryon number in the Landau frame (dashed line) and in the Eckart frame (dotted line).[]{data-label="fig:5"}](dnchdy_ratio.pdf "fig:"){width="3.2in"} ![The ratios of the rapidity distributions with and without $\delta f$ correction for (a) charged particles and (b) for net baryon number in the Landau frame (dashed line) and in the Eckart frame (dotted line).[]{data-label="fig:5"}](dnbdy_ratio.pdf "fig:"){width="3.2in"}
Discussion and Conclusions {#sec5}
==========================
The baryon diffusive and the energy dissipative hydrodynamics at the second-order in the Landau and the Eckart frames have been discussed. The system is stable at the second order when the relaxation time is semi-positive in the Landau frame and it is larger than the minimum value in the Eckart frame. The mode analyses implies that causality is also satisfied in the long wave length limit. The transport coefficients of the two frames at the linear and the second order are shown to be related. The full second-order terms are found to be necessary for a consistent matching. The results are generic and independent of the individual derivation method of the hydrodynamic equations of motion.
The frame dependence is tested in a numerical hydrodynamic model of relativistic heavy-ion collisions. The net baryon number is chosen as the conserved charge of the system and the space-time evolutions of a QCD medium in the Landau and the Eckart frames are compared to that of the inviscid system. The space-time rapidity distribution of the entropy density is not much affected by the dissipative currents while that of net baryon density is visibly modified. The effects of the baryon diffusion and the energy dissipation is found to be quantitatively similar for those thermodynamic variables. The flow, on the other hand, is implied to be different in the Landau and the Eckart frames.
The charged particle distribution is estimated in both frames. The result is found to be mostly unaffected by the baryon diffusion for the chosen set of transport coefficients. The distribution for the energy dissipation is also not modified much owing to the cancellation of the effects of the flow deceleration and the off-equilibrium correction at freeze-out. A larger baryon stopping is observed in the net baryon distribution owing to the fugacity gradient for the Landau frame and also to the flow deceleration in the Eckart frame. The $\delta f$ correction is found to increase the baryon stopping effect of baryon diffusion so that the difference between the net baryon distributions of the two frames becomes small.
The results indicate that the hydrodynamic estimation of the observables may not depend much on the choice of the local rest frame in relativistic nuclear collisions. It would be important to investigate other observables that are directly dependent on the flow, such as thermal photons with blue shifting, to elucidate the issue of the Landau and the Eckart frames in hydrodynamic models.
It is worth noting that studied in the present numerical analyses are the finite temperature and chemical potential regions near the QCD transition explored by relativistic nuclear collisions. One should be careful when determining a frame in the zero temperature or chemical potential limit. A careful treatment of the equation of state and the transport coefficients may also become important in such cases.
Future prospects include the application to the full (3+1) dimensional analyses of the beam energy scan data of flow-related observables to extract relations between the initial conditions and the transport coefficients in each frame to investigate the validity of the choice of the local rest frame more quantitatively.
The author is grateful for the valuable comments by T. Kunihiro. The work of A.M. was supported by JSPS KAKENHI Grant Number JP19K14722.
ENTROPY PRODUCTION IN LANDAU AND ECKART FRAMES {#sec:A}
==============================================
The relation between the transport coefficients can be determined by the identification of the entropy production of the Landau and the Eckart frames: $$\begin{aligned}
\partial_\mu s^\mu = - \frac{V_L^\mu V^L_\mu}{\kappa_V} = - \frac{W_E^\mu W^E_\mu}{\kappa_W} .\end{aligned}$$ The entropy production in the Landau frame up to the next-to-leading order is $$\begin{aligned}
\partial_\mu s^\mu &=& - \kappa_V \nabla_\mu^L \frac{\mu}{T} \nabla^\mu_L \frac{\mu}{T} \nonumber \\
&+& 2 \tau_V \nabla_\mu^L \frac{\mu}{T} D_L V_L^\mu - 2 \chi_W^a \nabla_\mu^L \frac{\mu}{T} V_L^\mu D_L \frac{\mu}{T} \nonumber \\
&-& 2 \chi_V^b \nabla_\mu^L \frac{\mu}{T} V_L^\mu D_L \frac{1}{T} - 2 \chi_V^c \nabla_\mu^L \frac{\mu}{T} V_L^\mu \nabla^L_\nu u_L^\nu \nonumber \\
&-& 2 \chi_W^d \nabla_\mu^L \frac{\mu}{T} V_L^\nu \nabla^L_\nu u_L^\mu - 2 \chi_W^e \nabla_\mu^L \frac{\mu}{T} V_L^\nu \nabla_L^\mu u^L_\nu + \mathcal{O}(\delta^4). \nonumber \\\end{aligned}$$
The entropy production in the Eckart frame can be expressed using the variables in the Landau frame as, up to the same order, $$\begin{aligned}
\partial_\mu s^\mu &=& - \frac{W_E^\mu W^E_\mu}{\kappa_W} \nonumber \\
&=& - \kappa_W \bigg( \frac{n}{e+P} \bigg)^2 \nabla_\mu^L \frac{\mu}{T} \nabla^\mu_L \frac{\mu}{T} \nonumber \\
&+& 2 \bigg[ \tau_W - \frac{\kappa_W}{(e+P)T} \bigg] \nabla_\mu^L \frac{\mu}{T} D_L V_L^\mu \nonumber \\
&-& 2 \bigg[ \chi_W^a - \frac{\tau_W nT}{e+P} \bigg] \nabla_\mu^L \frac{\mu}{T} V_L^\mu D_L \frac{\mu}{T} \nonumber \\
&-& 2 \bigg[ \chi_W^b + \tau_W T - \frac{\kappa_W}{(e+P)} \bigg] \nabla_\mu^L \frac{\mu}{T} V_L^\mu D_L \frac{1}{T} \nonumber \\
&-& 2 \bigg[ \chi_W^c + \frac{\kappa_W}{(e+P)T} \bigg] \nabla_\mu^L \frac{\mu}{T} V_L^\mu \nabla^L_\nu u_L^\nu \nonumber \\
&-& 2 \bigg[ \chi_W^d + \frac{\kappa_W}{(e+P)T} \bigg] \nabla_\mu^L \frac{\mu}{T} V_L^\nu \nabla^L_\nu u_L^\mu \nonumber \\
&-& 2 \chi_W^e \nabla_\mu^L \frac{\mu}{T} V_L^\nu \nabla_L^\mu u^L_\nu + \mathcal{O}(\delta^4).\end{aligned}$$ It should be noted that the thermodynamic forces of the energy dissipation and the baryon diffusion are mutually convertible using the hydrodynamic identity derived from the Gibbs-Duhem relation and energy-momentum conservation as $$\begin{aligned}
&\bigg( \nabla_E^\mu \frac{1}{T} + \frac{1}{T} D_E u^\mu \bigg) = \frac{n}{e+P} \nabla_E^\mu \frac{\mu}{T} \nonumber \\
&- \frac{1}{(e+P)T} [W_E^\mu \nabla^E_\nu u_E^\nu + W_E^\nu \nabla^E_\nu u_E^\mu + (\Delta_{E})^{\mu}_{\ \nu} D_E W_E^\nu] \nonumber \\
&= \frac{n}{e+P} \nabla_L^\mu \frac{\mu}{T} - \frac{n}{e+P} \bigg( u_L^\mu \frac{V_L^\nu}{n} \nabla^L_\nu \frac{\mu}{T} + \frac{V_L^\mu}{n} D^L \frac{\mu}{T} \bigg) \nonumber \\
&+ \frac{1}{nT} \bigg[V_L^\mu \nabla^L_\nu u_L^\nu + V_L^\nu \nabla^L_\nu u_L^\mu + (\Delta_{L})^{\mu}_{\ \nu} D_L V_L^\nu \nonumber \\
&+ \frac{n}{e+P} V_L^\mu D_L \frac{e+P}{n} \bigg] + \mathcal{O}(\delta^3) , \label{eq:thermoforce}\end{aligned}$$ where $$\begin{aligned}
D_L \frac{e+P}{n} &=& - \frac{e+P}{n} T D_L \frac{1}{T} +T D_L \frac{\mu}{T} .\end{aligned}$$ The correspondences between the transport coefficients in the two frames can be obtained as Eqs. (\[kappaLE\])-(\[chieLE\]).
FREEZE-OUT WITH OFF-EQUILIBRIUM DISTRIBUTION {#sec:B}
============================================
The distribution function in relativistic systems with the energy dissipation and the baryon diffusion is estimated using the Grad’s moment method [@Grad] based on Ref. [@Israel:1979wp; @Monnai:2010qp]. The distribution can be decomposed into the equilibrium and the off-equilibrium parts as $$\begin{aligned}
f_0^i &=& \{\exp[(p^\mu u_\mu - b_i \mu_B)/T]\mp 1\}^{-1}, \\
\delta f^i &=& -f_0^i (1\pm f_0^i) (b_i p_i^\mu \varepsilon_\mu^B + p_i^\mu p_i^\nu \varepsilon_{\mu\nu}),\end{aligned}$$ where $b_i$ is the quantum number for baryons. The upper sign is for bosons and the lower one for fermions. If the auxiliary vector and tensor $\varepsilon_\mu^B$ and $\varepsilon_{\mu \nu}$ are expressed in terms of macroscopic dissipative currents, $$\begin{aligned}
\varepsilon_\mu^{L;B} = D_V V^L_\mu, \ \ \varepsilon^L_{\mu \nu} = B_V (V^L_\mu u^L_\nu + V^L_\nu u^L_\mu),\end{aligned}$$ in the Landau frame and $$\begin{aligned}
\varepsilon_\mu^{E;B} = D_W W^E_\mu, \ \ \varepsilon^E_{\mu \nu} = B_W (W^E_\mu u^E_\nu + W^E_\nu u^E_\mu),\end{aligned}$$ in the Eckart frame. The coefficients can be determined by the self-consistency condition that the off-equilibrium distribution reproduces the respective dissipative current within the framework of kinetic theory. They are, $$\begin{aligned}
D_W = - 2J_{31}^B \mathcal{J}_2^{-1}, \ \ B_W = J_{21}^{BB} \mathcal{J}_2^{-1},\end{aligned}$$ and $$\begin{aligned}
D_V = 2J_{41} \mathcal{J}_2^{-1}, \ \ B_V = -J_{31}^B \mathcal{J}_2^{-1},\end{aligned}$$ where $$\begin{aligned}
\mathcal{J}_2 = 2(J_{31}^B J_{31}^B-J_{41}J_{21}^{BB}).\end{aligned}$$ Here the moments are defined as $$\begin{aligned}
J^{B...B}_{k l} &=& \frac{1}{(2l+1)!!} \sum_i \int \frac{(b_i ... b_i) d^3p}{(2\pi)^3 E_i} \nonumber \\
&\times& [m_i^2 - (p\cdot u)^2]^{l} (p\cdot u)^{k-2l} f_0^i(1\pm f_0^i) .\end{aligned}$$
The off-equilibrium corrections are essential for conserving the energy-momentum and the net baryon number during the conversion from fluid to particles at freeze-out. The underlying equations of state for the hydrodynamic model and relativistic kinetic theory should be the same for successful conversion. The hadron gas with all resonances below 2 GeV in mass [@Tanabashi:2018oca] is used for the numerical estimation of the distortion coefficients to match the constructions of $\delta f$ and the equation of state.
|
---
author:
- 'Colin M. Hardy'
- 'Philip W. Livermore'
- Jitse Niesen
bibliography:
- 'allrefs.bib'
title: Constraints on the magnetic field within a stratified outer core
---
Mounting evidence from both seismology and experiments on core composition suggests the existence of a layer of stably stratified fluid at the top of Earth’s outer core. In this work we examine the structure of the geomagnetic field within such a layer, building on the important but little known work of [@malkus1979dynamo]. We assume (i) an idealised magnetostrophic spherical model of the geodynamo neglecting inertia, viscosity and the solid inner core, and (ii) a strongly stratified layer of constant depth immediately below the outer boundary within which there is no spherically radial flow. Due to the restricted dynamics, Malkus showed that the geomagnetic field must obey certain a condition which is a refined and more restrictive version of the well known condition of @Taylor_63 which holds on an infinite set of azimuthal rings within the stratified layer. By adopting a spectral representation with truncation $N$ in each direction, we show that this infinite class collapses to a discrete set of $O(N^2)$ Malkus constraints. Although fewer than the $N^3$ degrees of freedom of the magnetic field, their nonlinear nature makes finding a magnetic field that obeys such constraints, here termed a [*Malkus state*]{}, a challenging task. Nevertheless, such Malkus states when constrained further by geomagnetic observations have the potential to probe the interior of the core.
By focusing on a particular class of magnetic fields for which the Malkus constraints are linear, we describe a constructive method that turns any purely-poloidal field into an exact Malkus state by adding a suitable toroidal field. We consider poloidal fields following a prescribed smooth profile within the core that match a degree-13 observation-derived model of the magnetic field in epoch 2015 or a degree-10 model of the 10000-yr time averaged magnetic field. Despite the restrictions of the Malkus constraints, a significant number of degrees of freedom remain for the unknown toroidal field and we seek extremal examples. The Malkus state with the least toroidal energy has in both cases a strong azimuthal toroidal field, about double the magnitude of that observed from the poloidal field at the core-mantle boundary. For the 2015 field for a layer of depth 300 km, we estimate a root mean squared azimuthal toroidal field of $3$ mT with a pointwise maximum of 8 mT occurring at a depth of about 70 km.
Introduction
============
The question of whether or not Earth’s liquid outer core contains a stratified layer just below its outer boundary has long been debated [@whaler1980does; @Braginsky_67; @braginsky1987waves; @hardy2019stably; @gubbins2007geomagnetic]. A stratified layer may result from the pooling of buoyant elements released from the freezing of the solid inner core [@braginsky2006formation; @bouffard2019chemical], diffusion from the mantle above [@jeanloz1990nature; @buffett_seagle_2010] or sub-adiabatic thermal effects [@Pozzo_etal_2012]. Within a strongly stratified layer, the dynamics would be very different to the remainder of the convecting core because spherical radial motion would be suppressed [@braginsky1999dynamics; @davies2015constraints; @cox2019penetration]. In terms of using observations of the changing internal geomagnetic field as a window on the dynamics within the core, the existence of a stratified layer is crucial because motion confined to the stratified layer such as waves may have a pronounced geomagnetic signature, which may be falsely interpreted as emanating from the large-scale dynamo process ongoing beneath.
Observational constraints on the stratified layer are largely from seismology, where analysis of a specific ‘SmKS’ class of waves has revealed a localised decrease in wave velocities in the outermost $100-300 \text{km}$ of the core [@helffrich2013causes; @lay1990stably; @helffrich2010outer], suggesting that the outermost part of the core has a different density and/or elasticity than the rest of the core. However, this evidence is far from conclusive because not all studies agree that a stratified layer is necessary to explain seismic measurements [@irving2018seismically], and there are inherent uncertainties due to the remoteness of the core [@alexandrakis_eaton_2010]. So far, observational geomagnetism has offered equivocal evidence for stratified layers. Time dependent observational models can be explained by simple core flow structures on the core-mantle boundary (CMB) which have either no layer [@Holme_2015; @Amit_2014] (upwelling at the CMB is permitted), or a strongly stratified layer (in which all radial motion is suppressed), [@Lesur_etal_2015].
A complementary approach to understanding the observational signature of a stratified layer is by numerical simulation of a stratified geodynamo model [@nakagawa2011effect]. Models of outer core dynamics have demonstrated that dynamo action can be sensitive to variations in the assumed background state of a fully convective outer core, and that the presence of stably stratified layers can significantly alter the dynamics and morphology of the resultant magnetic field [@glane2018enhanced; @christensen2018geodynamo; @olson2018outer]. Hence comparisons between the magnetic fields from stratified models with the geomagnetic field can be used to infer compatibility with the presence of a stratified layer. This has been used to constrain the possible thickness of a stratified layer such that it is consistent with geomagnetic observations. [@yan2018sensitivity] find that unstratified dynamo simulations significantly underpredict the octupolar component of the geomagnetic field. Their model endorses the presence of a thin stably stratified layer, as the resultant magnetic field can be rendered Earth-like by the inclusion of 60-130 km layer. However, the results are rather sensitive to both the strength of stratification and layer depth, with a thicker layer of 350 km resulting in an incompatible octupole field. Similarly [@olson2017dynamo] find that stratified model results compare favorably with the time-averaged geomagnetic field for partial stratification in a thin layer of less than 400 km, but unfavorable for stratification in a thick 1000 km layer beneath the CMB. Additionally, in terms of dynamics, [@Braginsky_93; @Buffett_2014] show that MAC (Magnetic, buoyancy (Archimedean) and Coriolis forces) waves in the [*hidden ocean*]{} at the top of the core provide a mechanism for the 60 year period oscillations detected in the geomagnetic field [@roberts200760]. The model of [@buffett2016evidence] suggests that MAC waves underneath the CMB are also able to account for a significant part of the fluctuations in length of day (LOD) [@gross2001combined; @holme2005geomagnetic] through explaining the dipole variation, but are contingent on the existence of a stratified layer at the top of the core with a thickness of at least 100 km. However, not all stratified dynamo model results champion this scenario for the Earth. It has been found that the inclusion of a thin stable layer in numerical models can act to destablise the dynamo, through generating a thermal wind which creates a different differential rotation pattern in the core [@stanley2008effects]. Additionally many distinctive features of the geomagnetic field are not reproduced, as strong stratification leads to the disappearance of reverse flux patches and suppression of all non-axisymmetric magnetic field components [@mound2019regional; @christensen2008models].
One reason why there is no clear message from existing geodynamo models is perhaps that they all have been run in parameter regimes very far from Earth’s core [@Roberts_Aurnou_2011]. Two important parameters, the Ekman and Rossby numbers, quantify the ratio of rotational to viscous forces $E \sim 10^{-15}$ and the ratio of inertial to viscous forces $R_o \sim 10^{-7}$ respectively [@Christensen_2015]. These parameters being so small causes difficulties when attempting to numerically simulate the geodynamo because they lead to small spatial and temporal scales that need to be resolved in any direct numerical simulation, but are extremely computationally expensive to do so. Despite this challenge, numerical models have been used with great success to simulate aspects of the geodynamo, reproducing features such as torsional oscillations [@Wicht_Christensen_2010] that are consistent with observational models [@gillet2010fast], geomagnetic jerks [@aubert2019geomagnetic] and allowing predictions of the Earth’s magnetic field strength [@christensen2009energy]. Recent simulations have been able to probe more Earth-like parameter regimes than previously possible, achieving very low Ekman numbers of $E = 10^{-7} - 10^{-8}$ [@schaeffer2017turbulent; @aubert2019approaching]. However despite this progress, these simulations remain in parameter regimes vastly different to that of the Earth [@Christensen_2015], posing the inescapable question of how representative of the Earth they really are, as force balances can still vary significantly between the simulation regime and the correct regime of the Earth [@wicht2019advances], with the ability to simultaneously reproduce Earth-like field morphology and reversal frequency still beyond current capabilities [@christensen2010conditions]. The assessments conducted by [@sprain2019assessment] highlight that present geodynamo models able unable to satisfactorily reproduce all aspects of Earth’s long term field behaviour.
In this paper we consider the approach proposed by [@Taylor_63], based on the assumption that the inertia-free and viscosity-free asymptotic limit is more faithful to Earth’s dynamo than adopting numerically-expedient but nevertheless inflated parameter values. This amounts to setting the values of $R_o$ and $E$ to zero, which simplifies the governing equations significantly, enabling numerical solutions at less computational expense and importantly for us, analytic progress to be made. The resulting dimensionless magnetostrophic regime then involves an exact balance between the Coriolis force, pressure, buoyancy and the Lorentz force associated with the magnetic field ${{\bf B}}$ itself: $${\bm{\hat{z}}} \times {\bm{u}} = -{{\bm{\nabla}}}p + F_B{\bm{\hat r}} + ({{{\boldsymbol \nabla}}\times}{\bm{B}}) \times {\bm{B}}, \label{eqn:magneto}$$ where $F_B$ is a buoyancy term that acts in the unit radial direction ${\bm{\hat r}}$ [@Fearn_98].
Throughout this paper we consider the magnetostrophic balance of \[eqn:magneto\]. @Taylor_63 showed that, as a consequence of this magnetostrophic balance, the magnetic field must obey at all times $t$ the well-known condition $$T(s,t) \equiv \int_{C(s)} (({{{\boldsymbol \nabla}}\times}{{\bf B}}) \times {{\bf B}})_\phi ~ s \text{d}\phi \text{d}z =0,\label{eqn:Taylor}$$ for any geostrophic cylinder $C(s)$ of radius $s$, aligned with the rotation axis, where $(s,\phi,z)$ are cylindrical coordinates. This constraint applies in the general case for fluids independent of stratification. It was first shown by [@malkus1979dynamo] how can be refined within a stratified layer of constant depth, which in the limit of zero radial flow leads to a more strict constraint. This constraint now applies on every axisymmetric ring coaxial with the rotation axis that lies within the layer and is known as the [*[Malkus constraint]{}*]{} $$M(s,z,t) \equiv \int_0^{2\pi} (({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi} ~ \text{d}\phi =0,$$ for any $s$ and $z$ within the layer. Magnetic fields that satisfy the Taylor or Malkus constraints respectively are termed Taylor or Malkus states.
The associated timescale over which the dominant force balance described by the magnetostrophic equations evolves is $\sim 10^4$ years. However observations show changes in the geomagnetic field on much shorter timescale of years to decades [@jackson2015geomagnetic]. This vast discrepancy in timescales motivates distinguishing between the slowly evolving background state and perturbations from it and considering these two features separately. The theoretically predicted magnetostrophic timescale, represented by Taylor or Malkus states, describes the slow evolution of the magnetic field, and may explain dynamics such as geomagnetic reversals and also the longstanding dominance of the axially symmetric dipolar component of the field. Although rapid dynamics such as MHD waves occur on a much shorter timescale, they cannot be considered in isolation as their structure depends critically upon the background state that they perturb. Thus although insightful models of perturbations can be based upon simple states [e.g. @Malkus_67], ultimately a close fit to the observed geomagnetic field requires accurate knowledge of the background state. It is the search for such a state that is explored in this paper. Dynamical models of a non-stratified background state, produced by evolving the magnetic field subject to Taylor’s constraint, have appeared very recently [@Wu_Roberts_2015; @roberts2018magnetostrophic; @li2018taylor] and are currently restricted to axisymmetry, although the model of [@li2018taylor] can be simply extended to a three dimensional system. These models can additionally be used to probe the effect of incorporating inertia driven torsional waves within this framework [@roberts2014modified].
In this paper we adopt a different strategy and explore the use of both the Taylor and Malkus constraints as a tool for analytically constraining instantaneous structures of the magnetic field throughout Earth’s core. This method ignores any dynamics and asks simply whether we can find a set of magnetic fields which satisfy the necessary constraints: Taylor’s constraint in the interior and Malkus’s constraint in the stratified layer, which will provide plausible background geomagnetic states. However, constructing Malkus states is a non-trivial task. Firstly we need to establish whether such fields can even exist, and if so how numerous they are, before we are able to construct examples of Malkus states. Since we are geophysically motivated, we also wish to determine whether such fields can be compatible with geomagnetic observations.
Our task is a challenging one: even finding magnetic fields that exactly satisfy the comparatively simple case of Taylor’s constraint has proven to be difficult in the 55 years since the seminal paper of @Taylor_63, although notable progress has been made in axisymmetry [@Hollerbach_Ierley_91; @Soward_Jones_83] and in 3D [@Jault_Cardin_99] subject to imposing a specific symmetry. Recently, significant progress has been made in this regard by presenting a more general understanding of the mathematical structure of Taylor’s constraint in three dimensions [@livermore2008structure]. This method was implemented by [@livermore2009construction] to construct simple, large scale magnetic fields compatible with geomagnetic observations. It is this which provides the foundation for the work presented here.
The remainder of this paper is structured as follows. In section 2 we present a new, more general derivation of the condition required to be satisfied with a stratified layer of fluid, which under an idealised limit reduces to what is known as Malkus’ constraint. In section 3 we summarise the method for discretising and constructing a Taylor state before extending this to Malkus states in section 4. In section 5 we prove that an arbitrary poloidal field can be transformed into a Malkus state through the addition of an appropriate toroidal field and show how this is a useful approach due to the resultant equations being linear. In section 6 we present our results for an Earth like magnetic field satisfying all relevant constraints, within the linear framework. In section 7 we discuss these results with regard to Earth’s internal field, specifically our estimate of toroidal field strength, before concluding in section 8.
Derivation of Malkus’ constraint {#sec:Malk_derivation}
================================
Within stably stratified fluids radial flows are suppressed, hence in the limit of strong stratification radial fluid velocities are negligibly small [@braginsky1999dynamics; @davies2015constraints]. We proceed within this idealistic limit and require that $u_r=0$ within a region of stratified fluid that is a volume of revolution: we represent the proposed stratified layer within Earth’s core as a spherically symmetric layer of constant depth. We assume further that the system is in magnetostrophic balance; that is, rapidly rotating with negligible inertia and viscosity. The resulting constraint was first derived by [@malkus1979dynamo], however, here we present an alternative and more straightforward derivation courtesy of Dominique Jault (personal communication).
We use the condition for incompressible flow that ${{\bm{\nabla}}}\cdot {\bm{u}} = 0$ and the standard toroidal poloidal decomposition within spherical coordinates $(r,\theta,\phi)$. From the condition that there is no spherically-radial component of velocity then ${\bm{u}}$ must be purely toroidal and hence can be written as $${\bm{u}} = {{\bm{\nabla}}}\times (\mathcal{T}(r,\theta,\phi){\bm{\hat{r}}})= \frac{1}{r\sin\theta} {\dfrac{\partial \mathcal{T}}{\partial \phi}} {\bm{\hat{\theta}}} - \frac{1}{r} {\dfrac{\partial \mathcal{T}}{\partial \theta}} {\bm{\hat{\phi}}}.$$ Therefore the cylindrically-radial velocity, written in spherical coordinates, is$$u_s=\sin\theta u_r+\cos\theta u_\theta = \frac{\cos\theta}{r\sin\theta} {\dfrac{\partial \mathcal{T}}{\partial \phi}}$$ and so $$\int_0^{2\pi} u_s ~ \text{d}\phi = \frac{\cos\theta}{r\sin\theta} \int_0^{2\pi} {\dfrac{\partial \mathcal{T}}{\partial \phi}} ~ \text{d}\phi = 0.$$ Now, since ${\bm{\hat{\phi}}} \cdot ({\hat{\bf z}}\times {\bm{u}}) = u_s $ then, from the azimuthal component of the magnetostrophic \[eqn:magneto\] we have $$u_s = -{\dfrac{\partial p}{\partial \phi}} +(({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi}.$$ Integrating this around any circle in a plane orthogonal to ${\bm{\hat{r}}}$ centred on the rotation axis, (as illustrated by the red rings in \[fig:constraint\_surfaces\]), and using the single-valued nature of pressure, gives Malkus’ constraint,
$$\underbrace{\int_0^{2\pi}u_s ~ \text{d}\phi}_{=0} = -\underbrace{\int_0^{2\pi}{\dfrac{\partial p}{\partial \phi}} ~ \text{d}\phi}_{=0} +\int_0^{2\pi} (({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi} ~d\phi = 0, \nonumber$$
or equivalently requiring that the Malkus integral $M$ is zero: $$M(s,z,t) \equiv \int_0^{2\pi} (({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi} ~ \text{d}\phi =0. \label{eqn:Malkcon}$$
We are also able to generalise this constraint from considering the idealistic limit of requiring $u_r=0$ within the stratified fluid to the more general situation of permitting $u_r \neq 0$, where we express the Malkus integral in terms of the radial flow. Now, the flow ${{\bf u}}$ is no longer purely toroidal and hence $$\begin{aligned}
M(s,z,t) =\int_0^{2\pi} u_s ~ \text{d}\phi = \int_0^{2\pi} u_\theta \cos\theta \text{d}\phi + \int_0^{2\pi} u_r \sin\theta ~ \text{d}\phi. \end{aligned}$$
We now use the condition for incompressible flow that ${{\bm{\nabla}}}\cdot {\bm{u}} = 0,$ $$0 = {{\bm{\nabla}}}\cdot {{\bf u}}= \frac{1}{r^2}{\dfrac{\partial (r^2u_r)}{\partial r}} +\frac{1}{r\sin\theta}{\dfrac{\partial (u_\theta \sin\theta)}{\partial \theta}} + \frac{1}{r\sin\theta} {\dfrac{\partial u_\phi}{\partial \phi}},$$
$$\Rightarrow \int_0^{2\pi} \left(\frac{\sin\theta}{r}{\dfrac{\partial (r^2u_r)}{\partial r}} + {\dfrac{\partial (u_\theta \sin\theta)}{\partial \theta}} \right) \text{d}\phi = - \int_0^{2\pi} {\dfrac{\partial u_\phi}{\partial \phi}} \text{d}\phi = 0.$$ Now integrating over $[0, \theta]$ we find $$\int_0^{2\pi}u_\theta \text{d}\phi = \frac{1}{\sin\theta}\int_0^\theta \frac{\sin\theta'}{r} \int_0^{2\pi}{\dfrac{\partial (r^2u_r)}{\partial r}} \text{d}\phi \text{d}\theta' = - \frac{1}{r\sin\theta}\int_0^\theta \sin \theta' {\dfrac{\partial }{\partial r}}\left( r^2 \int_0^{2\pi} u_r d\phi\right) ~ \text{d}\theta'$$
$$\Rightarrow M = -\frac{1}{r\tan\theta} \int_0^\theta {\dfrac{\partial }{\partial r}} \left(r^2\int_0^{2\pi} u_r \sin\theta' \text{d}\phi \right) \text{d}\theta' + \int_0^{2\pi} u_r \sin\theta ~ \text{d}\phi.$$
In the above derivation, no assumption has been made about stratification and this equation holds as an identity in the magnetostrophic regime independent of stratification. In the case considered by Malkus, $M=0$ is recovered in the limit of $u_r \rightarrow 0$.
It is clear that Malkus’ constraint is similar to Taylor’s constraint except now not only does the azimuthal component of the Lorentz force need to have zero average over fluid cylinders, it needs to be zero for the infinite set of constant-$z$ slices of these cylinders (here termed [*Malkus rings*]{}, see figure \[fig:constraint\_surfaces\]) that lie within the stratified region. In terms of the flow, the increased restriction of the Malkus constraint arises because it requires zero azimuthally-averaged $u_s$ at any given value of $z$, whereas Taylor’s constraint requires only that the cylindrically averaged $u_s$ vanishes and allows outward flow at a given height to be compensated by inward flow at another. We note that all Malkus states are Taylor states, but the converse is not true.
[.48]{} ![ Geometry of constraint surfaces \[fig:constraint\_surfaces\]](Tay_Malk_domain_dash.png "fig:"){width="55.00000%"}
[.48]{} ![ Geometry of constraint surfaces \[fig:constraint\_surfaces\]](Tay_Malk_cylinders2_rings_neartop_label.png "fig:"){width="98.00000%"}
Geometry and representation of a stratified magnetostrophic model {#sec:fulldom}
=================================================================
The physical motivation for applying Malkus’ constraint arises from seeking to represent a realistic model for the magnetic field in the proposed stratified layer within Earth’s outer core. Hence we compute solutions for the magnetic field in the Earth-like configuration illustrated in \[fig:fulldomain\], consisting of a spherical region in which Taylor’s constraint applies, representing the convective region of Earth’s core, surrounded by a spherical shell in which Malkus’ constraint applies, representing the stratified layer immediately beneath the CMB. Our method allows a free choice of inner radius $r_{SL}$, so in order to agree with the bulk of seismic evidence [@helffrich2010outer; @helffrich2013causes; @lay1990stably], the value $r_{SL}=0.9R$ is chosen for the majority of our solutions, where $R$ is the full radius of the core (3845 km). However due to the uncertainty which exists for the thickness of Earth’s stratified layer [@Kaneshima_2017], we also probe how sensitive our results are to layer thickness, considering $r_{SL}=0.85R$ and $r_{SL}=0.95R$ as well. The Earth’s inner core is neglected throughout, since incorporating it would lead to additional intricacies due to the cylindrical nature of Taylor’s constraint which leads to a distinction between regions inside and outside the tangent cylinder [@Livermore_Hollerbach_2012; @livermore2008structure]. Since the focus here is on the outermost reaches of the core, we avoid such complications.
The method used to construct the total solution for the magnetic field throughout Earth’s core that is consistent with the Taylor and Malkus constraints is sequential. Firstly, we use a regular representation of the form shown in \[eqn:torpolexpan\] to construct a Malkus state in the stratified layer. Secondly, we construct a Taylor state which matches to the Malkus state at $r=r_{SL}$; overall the magnetic field is continuous but may have discontinuous derivatives on $r=r_{SL}$. We note that any flow driven by this magnetic field through the magnetostrophic balance may also be discontinuous at $r=r_{SL}$ because in general $u_r \neq 0$ in the inner region but $u_r = 0$ is assumed in the stratified region. Considerations of such dynamics lie outside the scope of the present study focussed only on the magnetic constraints, but imposing continuity of $u_r$ for example would clearly require additional constraints.
As a pedogogical exercise we also construct some Malkus states within a fully stratified sphere ($r_{SL} = 0$), as detailed in \[sec:Apa\_both\]. Without the complications of matching to a Taylor state, the equations take a simpler form and we present some first examples in \[sec:Ap\_sol\_simp\]. Dynamically, sustenance of a magnetic field within a fully stratified sphere is of course ruled out by the theory of [@Busse_75a], which provides a strictly positive lower bound for the radial flow as a condition on the existence of a dynamo. Nonetheless it can be insightful to first consider the full sphere case, as it facilitates the consideration of fundamental principles of the magnetic field and Malkus constraint structure, and allows direct comparisons to be made with similar full sphere Taylor states. In what follows we represent a magnetic field by a sum of toroidal and poloidal modes with specific coefficients $$\label{eqn:Brep} {\bm{B}} = \sum_{l=1}^{L_{max}} \sum_{m=-l}^{l} \sum_{n=1}^{N_{max}} a_{l,n}^m {{\bm{\mathcal{T}}} }_{l,n}^m + b_{l,n}^m {{\bm{\mathcal{S}}}}_{l,n}^m$$ where ${\bm{\mathcal{T}}}_{l,n}^m={{{\boldsymbol \nabla}}\times}(T_{l,n}(r) Y_l^m {\hat{\bf r}})$, ${\bm{\mathcal{S}}}_{l,n}^m={{{\boldsymbol \nabla}}\times}{{{\boldsymbol \nabla}}\times}(S_{l,n}(r) Y_l^m {\hat{\bf r}})$, $N_{max}$ is the radial truncation of the poloidal and toroidal field. In the above, $Y_l^m$ is a spherical harmonic of degree $l$ and order $m$, normalised to unity by its squared integral over solid angle. Positive or negative values of $m$ indicate respectively a $\cos m\phi$ or $\sin m\phi$ dependence in azimuth. The scalar functions ${{T} }_{l,n}^m$ and ${{S} }_{l,n}^m$, $n\ge 1$, are respectively chosen to be the functions $\chi_{l,n}$ and $\psi_{l,n}$ composed of Jacobi polynomials [@li2010optimal; @Li_etal_2011]. They are orthogonal, and obey regularity conditions at the origin and the electrically insulating boundary condition at $r=R$ $$\frac{d \mathcal{S}_l^m}{dr} + l \mathcal{S}_l^m/R = \mathcal{T}_l^m = 0. \label{eqn:bc}$$ We note that this description is convenient but incomplete when used within the spherical shell, for which the magnetic field does not need to obey regularity at the origin. For simplicity, we nevertheless use this representation in both layers, although restricting the domain of the radial representation to $[0,r_{SL}]$ for the inner region.
Discretisation of the Taylor constraint {#sec:Tay_disc}
=======================================
Since the Malkus constraint forms a more restrictive constraint which encompasses the Taylor constraint it is useful for us to first summarise the structure of the Taylor constraint in a full sphere. The integral given in \[eqn:Taylor\], which Taylor’s constraint requires to be zero, is known as the Taylor integral. Although applied on an infinite set of surfaces, [@livermore2008structure] showed that Taylor’s constraint reduces to a finite number of constraint equations for a suitably truncated magnetic field expansion $$\mathcal{S}_l^m(r)=r^{l+1}\sum_{j=0}^{N_{max}}c_j r^{2j} ~~~~ \text{and} ~~~~ \mathcal{T}_l^m(r)=r^{l+1}\sum_{j=0}^{N_{max}}d_j r^{2j}, \label{eqn:torpolexpan}$$ which is an expanded version of for some $c_j$ and $d_j$. The Taylor integral itself then collapses to a polynomial of finite degree which depends upon $s^2$ [@lewis1990physical] and the coefficients $a_{l,n}^m, b_{l,n}^m$, and takes the form
$$\label{eqn:Taypoly} T(s) = s^2\sqrt{R-s^2}Q_{D_{T}}(s^2)=0,$$
for some polynomial $Q_{D_{T}}$ of maximum degree $D_T$.
Taylor’s constraint is now equivalent to enforcing that the coefficients of all powers of $s$ in the polynomial $Q_{D_{T}}$ equal zero, as this ensures $T(s)$ vanishes identically on every geostrophic contour. This reduces the infinite number of constraints to a finite number of simultaneous, coupled, quadratic, homogeneous equations. This reduction is vital as it gives a procedure for enforcing Taylor’s constraint in general, and allows the implementation of a method to construct magnetic fields which exactly satisfy this constraint, known as Taylor states, as demonstrated by [@livermore2009construction]. In the next section we see how, with some relatively simple alterations this procedure can be extended to the construction of exact Malkus states.
Malkus states {#sec:Malk_state}
=============
This section outlines some general properties of the mathematical structure of Malkus’ constraints and provides the methodology for constructing the first known Malkus states; we also address the questions of existence and uniqueness of solutions and the dimension of the resultant solution space.
Along similar lines as we showed for Taylor’s constraints in \[sec:Tay\_disc\], on adopting the representation the Malkus integral reduces to a multinomial in $s^2$ and $z$ [@lewis1990physical] and we require $$M(s,z) = Q_{D_{M}}(s^2,z) = 0$$ for some finite degree multinomial $Q_{D_{M}}$ in $s$ and $z$. Note that the Taylor integral is simply a z-integrated form of $Q_{D_{M}}$. Equating every multinomial term in $Q_{D_{M}}(s^2,z)$ to zero results in a finite set of constraints that are nonlinear in the coefficients $a_{l,n}^m$ and $b_{l,n}^m$. The number of constraints can be quantified for a given truncation following a similar approach as that employed by [@livermore2008structure] for Taylor’s constraint, by tracking the greatest exponent of the dimension of length. This analysis is conducted in \[sec:enum\_con\] and results in the number of Malkus constraints given by $$\label{eq:Malk_numcon}
C_M= {C_T}^2+3C_T+2,$$ where the number of Taylor constraints for an equivalent magnetic field is $C_T = L_{max} + 2N_{max} - 2$ (after the single degeneracy due to the electrically insulating boundary condition is removed) [@livermore2008structure]. Therefore we find that as expected the Malkus’ constraints are more numerous than Taylor’s constraints. It is significant to notice that $C_M \gg C_T$ and in particular for high degree/resolution systems $C_M \approx {C_T}^2$.
In order to satisfy these constraints, the magnetic field has $2L_{max}N_{max}(L_{max}+2)$ degrees of freedom (this being the number of unknown spectral coefficients within the truncation of $(L_{max}, N_{max})$. In axisymmetry the number of degrees of freedom reduces to $2N_{max}L_{max}$.
If we truncate the magnetic field quasi uniformly as $N= \mathcal{O}(L_{max}) \approx \mathcal{O}(N_{max})$, then we observe that at high $N$ the number of constraints ($O(N^2)$ Malkus constraints; $O(N)$ Taylor constraints) is exceeded by the number of degrees of freedom of $N^3$. A simple argument based on linear algebra suggests that many solutions exist at high $N$, however this may be misleading because the constraints are nonlinear and it is not obvious *a priori* whether any solutions exist, or if they do, how numerous they might be. We consider a simple example in \[sec:Apb\], which shows the structure of constraint equations that arise. The example highlights that degeneracy of the constraint equations plays a far more significant role for the Malkus constraints compared with the Taylor constraints, which only have a single weak degeneracy due to the electrically insulating boundary condition [@livermore2008structure]. However, due to the complex nature of these nonlinear equations, at present we have no theory to predict which constraints will be degenerate and hence the total number of independent constraints.
Because of the apparent uncertainty of the existence of Malkus states, it is instructive to identify whether imposing strong symmetry is useful to identify very simple examples. Owing to symmetries inherent in the spherical harmonics, many classes of simple Taylor states exist, as outlined by [@livermore2009construction]: for example any field that is either symmetric or anti-symmetric with respect to a rotation of $\pi$ radians about the $x$-axis is a Taylor state. Due to the absence of averaging in $z$, such symmetric magnetic fields do not automatically satisfy the Malkus constraints. However some simple classes of field are guaranteed to be Malkus states, such as single spherical harmonic modes, axisymmetric purely toroidal or poloidal fields since the integrand itself $(({{{\boldsymbol \nabla}}\times}{{\bf B}}) \times {{\bf B}})_\phi$ is zero. Also equatorially symmetric purely toroidal or poloidal fields comprising either only cosine or only sine dependence in azimuth are Malkus states as the resultant integrand is anti-symmetric with respect to a rotation of $\pi$ radians and hence the azimuthal average over $[0,2\pi]$ causes the Malkus integral to vanish.
Finding a Malkus state
======================
Owing to the nonlinear albeit finite nature of the Malkus constraints, it is far from obvious whether any solutions exist beyond those of the simple structure explored above. In the next section, we demonstrate the existence of a class of solutions with arbitrarily complex lateral structure.
A special class of Malkus states {#sec:theo}
--------------------------------
Here we demonstrate that within the class of magnetic fields that all contain a known poloidal component (but whose toroidal component is unknown) then there exists systems where all the Malkus constraints are linear in the unknown spectral parameters. A formal statement of this fact is given in the theorem given below.
Any arbitrary, prescribed, polynomial poloidal field can be transformed into a Malkus state through the addition of an appropriate polynomial toroidal field.
\[proof:linear\] We prove below that by considering an arbitrary, prescribed, truncated polynomial poloidal field, the addition of a specific choice of toroidal modes renders the Malkus constraints linear in the unknown toroidal coefficients. By taking a sufficient number of such modes such that the degrees of freedom exceed the number of constraints, it follows that for the general case (barring specific degenerate cases) by solving the linear system the resultant magnetic field is a Malkus state.
To show this, because the Malkus constraint is quadratic in the magnetic field, we introduce the concept of a magnetic field interaction. In general there are three possible field interactions within the Malkus integral, toroidal-toroidal, poloidal-poloidal and toroidal-poloidal, respectively $$M = \sum_{l_1,l_2}^{L_{max}} \sum_{m}^{L_{max}} \left( [{\bm{T}}_{l_l}^m, {\bm{T}}_{l_2}^m] + [{\bm{S}}_{l_l}^m, {\bm{S}}_{l_2}^m] + [{\bm{T}}_{l_l}^m, {\bm{S}}_{l_2}^m] \right)$$ where
$$\begin{aligned}
\label{eq:tortor}
[{\bm{T}}_{l_l}^m, {\bm{T}}_{l_2}^m] &= \int_0^{2\pi} \frac{l_1(l_1+1)\mathcal{T}_{l_l}^m \mathcal{T}_{l_2}^m}{r^3 \sin\theta}\left({Y}_{l_l}^m{\dfrac{\partial {Y}_{l_2}^m}{\partial \phi}}\right) s ~ \text{d}\phi +sc, \\
[{\bm{S}}_{l_l}^m, {\bm{S}}_{l_2}^m] &= \int_0^{2\pi} \frac{l_1(l_1+1)\mathcal{S}_{l_l}^m (\frac{\text{d}^2}{\text{d}r^2}-l_2(l_2+1)/r^2)\mathcal{S}_{l_2}^m}{r^3 \sin\theta}\left({Y}_{l_l}^m{\dfrac{\partial {Y}_{l_2}^m}{\partial \phi}}\right) s ~ \text{d}\phi +sc,\nonumber \\
[{\bm{T}}_{l_l}^m, {\bm{S}}_{l_2}^m] &= \int_0^{2\pi} \frac{1}{r^3}\left( l_1(l_1+1){T}_{l_l}^m \frac{\text{d}\mathcal{S}_{l_2}^m}{\text{d}r} Y_{l_1}^m {\dfrac{\partial {Y}_{l_2}^m}{\partial \theta}}\right.
\left.- l_2(l_2+1)\mathcal{S}_{l_2}^m \frac{\text{d}T_{l_1}^m}{\text{d}r} Y_{l_2}^m {\dfrac{\partial {Y}_{l_1}^m}{\partial \theta}} \right) s ~ \text{d}\phi, \nonumber
\end{aligned}$$
where $sc$ is the symmetric counterpart given by interchanging the vector harmonics and hence the positions of $l_1$ and $l_2$ [@livermore2008structure]. Note that there is no poloidal-toroidal interaction since the curl of a poloidal vector is toroidal and (${\bm{\mathcal{T}_1}} \times {\bm{\mathcal{T}_2}})_\phi = 0,$ for any two toroidal vectors ${\bm{\mathcal{T}_1}}$ and ${\bm{\mathcal{T}_2}}$.
For the situation we consider of a given poloidal field, then the only non-linearity within the unspecified coefficients arises from the toroidal-toroidal interactions, which results in quadratic dependence, just as for the general case with unprescribed poloidal field. However, by restricting attention to toroidal fields that result in no toroidal-toroidal interaction, the unknown toroidal coefficients appear only in a linear form through the toroidal-poloidal interactions. Axisymmetric modes are the simplest set of toroidal modes which are non-self-interacting, however there are too few of them (within the truncation) to solve the resulting linear system which is over-constrained (see \[fig:new\_con\_dof\]).
Therefore we require additional non-axisymmetric toroidal modes, which we choose such that the total set of toroidal modes remains non-self-interacting. This is achieved by exploiting the previously noted observations that any single harmonic is a Malkus state and that the set of equatorially symmetric toroidal modes $T_l^l$ is a Malkus state (and therefore has no self-interaction). Owing additionally to azimuthal symmetry, the modes $${T_1^0}, T_2^0, \cdots, T_1^{-1}, T_{1}^1,T_2^{-2}, T_{2}^2, \dots,$$ that is, the modes $T^m_l$ with $m = 0$ or $m = \pm l$, have no self-interactions. Each harmonic mode may be expanded in radial modes up to the truncation $N_{max}$. The non-interacting nature of the modes may be confirmed from \[eq:tortor\].
The addition of these nonaxisymmetric modes increases the number of degrees of freedom from the axisymmetric case by a factor of three such that it is now larger than the number of constraints (which are now all linear). This can be shown in general since for a toroidal field truncated at $L_1, N_1$ and a poloidal field truncated at $L_2, N_2$ the number of Taylor constraints is equal to half of the maximum degree of the polynomial in $s$, (i.e. $C_T = \frac{1}{2}(L_1+L_2+2N_1+2N_2) - 2$) [@livermore2008structure] and the maximum number of Malkus constraints we have shown is given in terms of $C_T$ by \[eq:Malk\_numcon\]. This results in a situation where if the poloidal field is fixed at a chosen resolution then for a toroidal field truncated quasi uniformly as $N= \mathcal{O}(L_{max}) \approx \mathcal{O}(N_{max})$ then we can see that the number of Malkus constraints scales as $\frac{9}{4} N^2$, which importantly, grows slower than the number of degrees of freedom for the non-axisymmetric linear system which scales as $3 N^2$. Hence it is guaranteed that at a sufficiently large resolution toroidal field representation then there will be more degrees of freedom than constraints.
Therefore, barring degenerate cases, Malkus states exist. Compared with the case of a purely axisymmetric toroidal field, the number (but not the specific form) of linear constraints remains unaltered by the addition of these extra non-axisymmetric modes. It is worth noting that the depth of the stratified layer does not enter into above derivation. The magnetic field solution in fact satisfies the Malkus constraints everywhere within its region of definition: in our case, this is the full sphere $0 \le r \le R$.
provides a specific example of the number of constraints given a poloidal field of degree $13$. It demonstrates two important things. Firstly, that due to degeneracy (for which we have no explanation) the independent linear constraints (red triangles) are much fewer than the full set of linear constraints (red squares). Secondly, that the number of degrees of freedom exceed the number of independent constraints at $L_{max}=N_{max} \geq 10$ if we consider the non-axisymmetric toroidal basis (blue circles) but is not exceeded at any truncation if we adopt the axisymmetric toroidal basis (blue stars). In particular, taking a non-axisymmetric toroidal field with truncation $L_{max}=N_{max}=13$ gives an infinite set of Malkus states. We note that the above deviation is based upon a polynomial representation, which is sufficient for our purposes here. However, we know that any continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function, and hence it can be extended to include an arbitrary magnetic field structure by expressing the relevant scalars in a polynomial basis of suitably large truncation.
We need to match the Malkus state (physically defined within the stratified layer) to a Taylor state in the region beneath. One way of proceeding is to simply evaluate the Malkus state beneath the stratified layer (where it also satisfies Taylor constraint); however this effectively imposes additional constraints on the inner region and is overly restrictive. Instead, we impose the same profile of poloidal field and expand the toroidal component of the Taylor state in the same set of spherical harmonic modes as used for the Malkus state. Such a choice also renders the Taylor constraints linear in the unknown toroidal coefficients.
![This graph compares the number of constraints to degrees of freedom (DOF) as a function of toroidal field spherical harmonic resolution with $L_{max}=M_{max}=N_{max}$, given a fixed poloidal field of $L_{max}=M_{max}=13$. This illustrates that for the non-axisymmetric linear system we construct then the number of degrees of freedom (red) exceeds the number of independent constraints (red triangles) for a toroidal field of resolution $L_{max}=N_{max} \geq 10$. \[fig:new\_con\_dof\]](new_dof_vs_constraints5.pdf){width="70.00000%"}
Further geophysical constraints
-------------------------------
In order to construct a Malkus state according to the above procedure, we need to completely specify the poloidal field. Following [@livermore2009construction], we downwards continue observation-derived models inside the core $r \le R$ by assuming a profile for each poloidal harmonic of degree $l$ that minimises the Ohmic dissipation within the modelled core $$\label{eqn:polprofile}
(2l+3)r^{l+1} - (2l+1)r^{l+3}.$$ We adopt two choices of observation-derived model. First, we use the CHAOS-6 model [@Finlay_etal_2016] at epoch 2015 evaluated to degree 13, the maximum obtainable from geomagnetic observations without significant interference due to crust magnetism [@Kono2015Geointro]. Second, we use the time-averaged field over the last 10000 years from the CALS10k.2 model [@constable2016persistent], which although is defined to degree 10 it has power concentrated mostly at degrees 1–4 because of strong regularisation of sparsely-observed ancient magnetic field structures. Recalling that the magnetostrophic state that we seek is defined over millenial timescales, this longer average provides on the one hand a better approximation to the background state, but on the other a much lower resolution. Even within these geomagnetically consistent Malkus states, there are nevertheless multiple degrees of freedom remaining. This raises the question of which of the multiple possible solutions are most realistic for the Earth, and motivates us to incorporate additional conditions to distinguish ‘Earth-like’ solutions.
We determine specific solutions through optimising the toroidal field $\bf T$ through either its Ohmic dissipation or its energy, respectively $$Q = \frac{\eta}{\mu_0}\int_V ({{{\boldsymbol \nabla}}\times}{\bf T})^2 dV, \qquad \mathcal{E} = \frac{1}{2\mu_0} \int_V {\bf T}^2 dV,$$ where $\eta \approx 1$ m$^2$s$^{-1}$ is magnetic diffusivity and $\mu_0=4\pi \times 10^{-7}~\text{NA}^{-2}$ is the permeability of free space. Both of these target functions are quadratic in the magnetic field, and so seeking a minimal value subject to the now linear constraints is straightfoward. In our sequential method to find a matched Malkus-Taylor state, we first optimise the Malkus state, and then subsequently find an optimal matching Taylor state.
Of the dissipation mechanisms in the core: Ohmic, thermal and viscous, the Ohmic losses are believed to dominate. On these grounds, the most efficient arrangement of the geomagnetic field would be such that Ohmic dissipation $Q$ is minimised. It is worth noting that in general our procedure is not guaranteed to provide the Malkus state field with least dissipation, but only an approximation to it, since we effectively separately optimise for the poloidal and toroidal component with least dissipation. In terms of finding a Malkus state with minimum toroidal field energy, this is useful in allowing us to determine the weakest toroidal field which is required in order to transform the imposed poloidal field into a Malkus state.
In \[sec:Ap\_sol\_simp\] we compare the method of finding the weakest toroidal field required to make a Malkus state, between using only selected toroidal modes, and all toroidal modes (resulting in a nonlinear system). For low truncation, minimisation of the toroidal energy subject to these nonlinear constraints is computationally solvable, and the two approaches produce comparable results. This suggests that estimates for the lower bound of Earth’s toroidal field strength obtained using our linearised approach will not differ greatly from related full non-linear optimisation (that is computationally infeasible).
An Earth-like example {#sec:highres}
=====================
We now present some visualisations of the specific class of Malkus states discussed above with minimal toroidal field energy for which the system of equations which enforce the constraints is linear. The geometry assumed here is as illustrated in \[fig:fulldomain\], with a Malkus state in the stratified layer in the region $0.9R < r \leq R$, matching to an inner Taylor state. We shall show the adjustment of the imposed poloidal field structure to a Malkus state by the required additive toroidal field. The strength of this toroidal field will be shown by contour plots of its azimuthal component. We note that the radial component of the magnetic field is defined everywhere by the imposed poloidal field, with the smooth degree 2 radial profile defined in \[eqn:polprofile\].
Magnetic field at 2015 {#sec:Tay_Malk_sol}
----------------------
We begin by showing in \[fig:CMB\_field\_basemap\_cor\_both\] both the radial and azimuthal structure, $B_r$ and $B_\phi$, of the CHAOS-6 model at epoch 2015 on the CMB, $r=R$. Of note is that at the truncation to degree 13, the azimuthal component is about half as strong as the radial component.
[.47]{} ![Magnetic field at the CMB based on the poloidal field from CHAOS-6 at epoch 2015. Visualised using the Mollweide projection and centred on the Greenwich meridian. \[fig:CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_rot__phi_new2_a_r.pdf "fig:"){width="90.00000%"}
[.47]{} ![Magnetic field at the CMB based on the poloidal field from CHAOS-6 at epoch 2015. Visualised using the Mollweide projection and centred on the Greenwich meridian. \[fig:CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_rot__phi_new2_a_phi.pdf "fig:"){width="90.00000%"}
summarises the strength of toroidal field (in terms of its azimuthal root mean squared value over solid angle) as a function of radius, for different toroidal truncations $L_{max} = N_{max}$ (shown in different colours). The toroidal field is required to be four orders of magnitude stronger in the stratified layer in order to satisfy the more restrictive Malkus constraints, compared with the inner region in which the weaker Taylor constraint applies, and adopts a profile that is converged by degree 13. The strong toroidal field throughout the stratified layer occurs despite the electrically insulating boundary condition at the outer boundary that requires the toroidal field to vanish. Within the stratified layer, the azimuthal toroidal field strength attains a maximum rms value of 2.5 mT at a radius of about $0.98R$ or a depth of about 70 km, about double the observed value at the CMB, and locally exceeds the imposed azimuthal poloidal magnetic field (of rms $0.28$ mT at this radius).
![\[fig:Tor\_Pol\_field\_radial\_profile\] The root mean squared azimuthal field strength (defined over solid angle) as a function of radius, comparing the strengths of the poloidal field (red) and toroidal field (blue, green, magneta and cyan) for toroidal fields with maximum spherical harmonic degree, order and radial resolution, 13 – 16 respectively. The poloidal field is the degree 13 field of minimum Ohmic dissipation compatible with the CHAOS-6 model at epoch 2015 [@Finlay_etal_2016]. ](Toroidal_strength_radial_profile.pdf){width="70.00000%"}
shows $B_\phi$ for both the total field and the toroidal component in isolation, using a toroidal truncation of 13 (corresponding to the blue line in \[fig:Tor\_Pol\_field\_radial\_profile\].) The top row shows the structure at the radius of maximum rms toroidal field ($r=0.98R$), demonstrating that the additive toroidal field component (of maximum 8 mT) dominates the total azimuthal field. The bottom row shows a comparable figure at $r=0.7R$, in the inner region where only Taylor’s constraint applies. Plotted on the same scale, the required additive toroidal component is tiny compared with the imposed poloidal field. This highlights again that the Malkus constraint is much more restrictive than the Taylor constraint.
[.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](Malkus_Toroidal_2_a2_r098_2.pdf "fig:"){width=".85\linewidth"}
[.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](Malkus_Total_2_a2_r098_2.pdf "fig:"){width=".85\linewidth"}
[.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](tor07_2.pdf "fig:"){width=".85\linewidth"}
[.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](tot07_2.pdf "fig:"){width=".85\linewidth"}
For comparison, \[Tay\_fullsphere\] shows an equivalent solution to \[fig:fullsolution\](a,b) but in the absence of stratification (where the magnetic field satisfies only Taylor’s constraint). The toroidal contribution to the azimuthal field is very weak (note the colourbar range is reduced from that of \[fig:fullsolution\](a,b) from 8 to 0.04 mT) and is of very large scale. This further highlights the weakness of the Taylor constraints compared with the Malkus constraints.
[.47]{} ![Azimuthal field for an unstratified comparative case, for which the magnetic field satisfies only Taylor’s constraint. \[Tay\_fullsphere\]](Tay_tor098.pdf "fig:"){width="85.00000%"}
[.47]{} ![Azimuthal field for an unstratified comparative case, for which the magnetic field satisfies only Taylor’s constraint. \[Tay\_fullsphere\]](Tay_tot098.pdf "fig:"){width=".85\linewidth"}
Time averaged field over the past ten millenia
----------------------------------------------
Here we show results for a poloidal field that is derived from the 10000-year time averaged field from the CALS10k.2 model [@constable2016persistent]. The model is only available up to spherical harmonic degree 10, hence we adopt a truncation of $L_{max} = N_{max} = 10$ for the toroidal field. Due to the absence of small-scale features in the field (caused by regularisation) the maximum value of $B_r$ is reduced to about $1/2$ of the comparable value from the degree-13 CHAOS-6 model from epoch 2015, and similarly the azimuthal field to about $1/6$ of its value. We note that over a long enough time span, the magnetic field is generally assumed to average to an axial dipole: a field configuration that is both a Malkus state and one in which the azimuthal component vanishes. Thus a small azimuthal component is consistent with such an assumption.
[.47]{} ![Magnetic field at the CMB based on the 10000-year time average field from CALS10k.2 \[fig:CALS\_CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_r_2_a.pdf "fig:"){width="90.00000%"}
[.47]{} ![Magnetic field at the CMB based on the 10000-year time average field from CALS10k.2 \[fig:CALS\_CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_r_2_a_phi.pdf "fig:"){width="90.00000%"}
Contours of the azimuthal field within the stratified layer (at $r=0.97R$) are shown in \[fig:CALS\_Min\_total\_r095\_phi\_rot\_both\], which is approximately the radius at which the maximum rms azimuthal toroidal field occurs. As before, the toroidal field dominates the azimuthal component, and its rms (1.66 mT) is about double that on the CMB (0.085 mT). Although its maximum absolute value is about 3 mT, less than the 8 mT found in the 2015 example above, this is consistent with the overall reduction in structure of the imposed poloidal field.
[.47]{} ![The azimuthal component of the Malkus state magnetic field within the stratified layer at a radius of $r=0.97R$, approximately the radius of maximum rms azimuthal toroidal field. \[fig:CALS\_Min\_total\_r095\_phi\_rot\_both\]](CALSMalkus_Toroidal_2_a2_r097_2.pdf "fig:"){width="90.00000%"}
[.47]{} ![The azimuthal component of the Malkus state magnetic field within the stratified layer at a radius of $r=0.97R$, approximately the radius of maximum rms azimuthal toroidal field. \[fig:CALS\_Min\_total\_r095\_phi\_rot\_both\]](CALSMalkus_Total_2_a2_r097_2.pdf "fig:"){width="90.00000%"}
Discussion
==========
Estimates of the internal magnetic field strength {#sec:tor_min}
-------------------------------------------------
Estimating the magnetic field strength inside the core is challenging, because observations made on Earth’s surface, using a potential-field extrapolation, only constrain the poloidal magnetic structure down to the CMB and not beneath, but also even this structure is visible only to about spherical harmonic degree 13. Furthermore, within the framework of such an extrapolation, the toroidal field is zero on the CMB. Estimating the field within the core beyond these surface values requires insight from numerical models or observations of physical mechanisms that are sensitive to the interior field.
Based on numerical models, [@Zhang_Fearn_93] suggest that a criterion for stability of the geomagnetic field is a toroidal field no more than 10 times that of the poloidal field, resulting an approximate upper bound of 5 mT. In a more recent and conflicting study, [@sreenivasan2017damping] suggest that the mean toroidal field is approximately 10 mT or higher, since this intensity is required for the slow magnetostrophic waves present to be able to originate from small-scale motions in the core. Observation-based studies based on electric field measurements [@shimizu1998observational] suggest a toroidal field strength at the CMB of anywhere in the range of 1-100 times that of the poloidal magnetic field there (i.e. up to about 100 mT). [@Buffett_2010] calculated a core averaged field strength of 2.5 mT from measurements of tidal dissipation; @gubbins2007geomagnetic estimated the toroidal field strength of 1 mT as compatible with patches of reversed magnetic flux. Lastly, the magnetic signature of both torsional and Rossby waves have led to respective estimates of at least 2 mT for $B_s$ within the core and therefore an RMS strength of $4$ mT assuming isotropy [@gillet2010fast], and an RMS estimate of $B_\phi$ of 12 mT [@Hori_etal_2015]. The strong toroidal field within our 2015 models of up to 8 mT (and rms $B_\phi$ of $2.5$ mT) within the stratified layer (at radius $r=0.98R$ or a depth of about 70 km) is in agreement with the majority of these estimates. This maximum value is notably about 8 times stronger than the observed radial field on the CMB. In both the 2015 and the 10000-yr averaged model, the rms toroidal field within the stratified layer was about double the radial field on the CMB. Interestingly, the azimuthal component of our solution within the inner unstratified region is about 100 times weaker, demonstrating the extent to which Malkus’ constraint is far more restrictive than Taylor’s constraint.
Limitations of our model
------------------------
Our model does not produce a formal lower bound on the azimuthal component of a magnetic field that (a) satisfies both the Malkus and Taylor constraints in their relevant regions along with (b) constraints on the radial field at the CMB. Instead, our results give only an upper bound on the lower bound [e.g. @jackson2011ohmic] because we have made a variety of simplifying assumptions, the most notable of which are (i) we have restricted ourselves to a subspace of Malkus states for which the constraints are linear (ii) we have imposed the entire poloidal profile and (iii) we have used a regular basis set for all magnetic fields even within the stratified layer when this is not strictly necessary. However, we show for the example considered in \[sec:Ap\_sol\_simp\] that in this case assumption (i) does not have a significant impact and our estimate is close to the full nonlinear lower bound. It may be that the other assumptions also do not cause our azimuthal field estimates to deviate significantly from the actual lower bound. Leaving aside the minimum toroidal field suggested by our model, our analysis allows two statements to be made on the weakness of the Malkus constraints, and the ability of magnetic structures assumed on the CMB to probe the magnetic structure within the stratified layer. Firstly, our method can find a toroidal field that converts any poloidal field into a Malkus state within a stratified layer of any depth. This means that we cannot use consistency of observation-derived models of the radial field with the Malkus constraints as a discriminant to test the probe the existence (and depth) of a stratified layer: all such models are consistent. Second, even if a stratified layer is assumed, the lateral radial magnetic field structure at the bottom of the layer is unconstrained by its structure at the top because we can find a Malkus state assuming any poloidal profile within the layer. Thus using considerations of the Malkus constraints, models of the surface magnetic field, such as CHAOS-6, cannot be downwards-continued further than the CMB into a stratified layer beneath.
Model robustness
----------------
There remains much uncertainty over the depth of any stably stratified layer at the top of the Earth’s core [@hardy2019stably]. Hence it is natural to consider how our results may change if the layer were to be of a different thickness to the $10\%$ of core radius used, as such we also calculated minimum toroidal-energy solutions matched to CHAOS-6 in epoch 2015 for layer thicknesses of $5\%$ and $15\%$. We find very little dependence of the field strengths internal to the layer on the depth of the layer itself, with our root mean square azimuthal field taking peak values of 2.7, 2.5 and 2.4 mT for thicknesses of $5\%$, $10\%$ and $15\%$ respectively.
The resolution of poloidal field also impacts significantly our optimal solutions. This has already been identified in the comparison between the degree-13 2015 model, and the degree-10 10,000-yr time-averaged model, that respectively resulted in rms azimuthal field estimates of 2.5 and 1.2 mT. Interestingly, for very long time-averaging windows the magnetic field is widely supposed to converge to an axial dipole, and assuming a simple poloidal profile is itself an exact Malkus state, with zero azimuthal field strength. We can further test
the effect of resolution by considering maximum poloidal degrees of 6 and 10 for the 2015 model to compare with our solution at degree 13. We find that our estimates for the root mean square azimuthal field (taken over their peak spherical surface) were 1.6 and 2.2 mT respectively. In all these calculations, the spherical harmonic degree representing the toroidal field was taken high enough to ensure convergence. Thus stronger toroidal fields are apparently needed to convert more complex poloidal fields into a Malkus state. This has important implications for the Earth, for which we only know the degree of the poloidal field to about $13$ due to crustal magnetism. Our estimates of the azimuthal field strength would likely increase if a full representation of the poloidal field were known.
Ohmic dissipation
-----------------
Our method can be readily amended to minimise the toroidal Ohmic dissipation, rather than the toroidal energy. In so doing, we provide a new estimate of the lower bound of Ohmic dissipation within the core. Such lower bounds are useful geophysically as they are linked to the rate of entropy increase within the core, which has direct implications for: core evolution, the sustainability of the geodynamo, the age of the inner core and the heat flow into the mantle [@jackson2011ohmic]. The poloidal field with maximum spherical harmonic degree 13 that we use, based on CHAOS-6 [@Finlay_etal_2016] and the minimum Ohmic dissipation radial profile [@Book_Backus_etal_96] has by itself an Ohmic dissipation of 0.2 GW. [@Jackson_Livermore_2009] showed that by adding additional constraints for the magnetic field, a formal lower bound on the dissipation could be raised to 10 GW, and even higher to 100 GW with the addition of further assumptions about the geomagnetic spectrum. This latter bound is close to typical estimates of 1 - 15 TW [@Jackson_Livermore_2009; @jackson2011ohmic].
The addition of extra conditions derived from the assumed dynamical balance, namely Taylor constraints, were considered by @jackson2011ohmic by adopting a very specific magnetic field representation. These constraints alone raised their estimate of the lower bound from 0.2 to 10 GW, that is, by a factor of 20. In view of the much stronger Malkus constraints (compared to the Taylor constraints), we briefly investigate their impact here.
We follow our methodology and find an additive toroidal field of minimal dissipation (rather than energy) that renders the total field a Malkus state. The dissipation is altered from $0.2$ to $0.7$ GW. That this increase is rather small (only a small factor of about 3) is rather disappointing, but is not in contradiction to our other results. It is generally true that the Malkus constraints are more restrictive than the Taylor constraint, but this comparison can only be made when the same representation is used for both. The method of @jackson2011ohmic assumed a highly restrictive form, so that in fact their Taylor states were apparently actually more tightly constrained than our Malkus states and thus produced a higher estimate of the lower bound. Despite our low estimate here, additional considerations of Malkus constraint may increase the highest estimates of @Jackson_Livermore_2009 well into the geophysically interesting regime.
Further extensions
------------------
The Malkus states we have computed, which match to field observations, provide a plausible background state at the top of the core. It may be interesting for future work to investigate how waves thought to exist within such a stratified layer [@Buffett_2014] may behave when considered as perturbations from such a background state, and whether they remain valid suggestions for explaining secular variation in the geomagnetic field. Similarly, combining our analysis with constraints on $B_s$ from torsional wave models [@gillet2010fast] may be insightful, and would combine aspects of both long and short-term dynamics. It is worth noting though, that we have investigated only static Malkus states without consideration of dynamics: we do not require the magnetic field to be either steady or stable, both of which would apply additional important conditions. An obvious extension to this work then is to investigate the fluid flows which are generated by the Lorentz force associated with these fields. This would then allow a consideration of how such flows would modify the field through the induction equation. These dynamics are however, are still relatively poorly known even for the much simpler problem of Taylor states. Recent progress by [@hardy2018determination] now allows a full calculation of the flow driven by a Taylor state. A general way to discover stationary and stable Taylor states comparable with geomagnetic observations is still out of reach, and currently the only way to find a stable Taylor state is by time-stepping [e.g. @li2018taylor]. The well established test used to determine whether the appropriate magnetostrophic force balance is achieved within numerical dynamo simulations is ‘Taylorisation’, which represents a normalised measure of the magnitude of the Taylor integral \[eqn:Taylor\] and hence the departure from the geophysically relevant, magnetostrophic regime [e.g. @Takahashi_etal_2005].
We propose an analogous quantity termed ‘Malkusisation’ defined in the same way, in terms of the Malkus integral:
$$\text{Malkusization} = \frac{|\int_0^{2\pi} ([{{{\boldsymbol \nabla}}\times}{{\bf B}}] \times {{\bf B}})_\phi d\phi|}{ \int_0^{2\pi} | ([{{{\boldsymbol \nabla}}\times}{{\bf B}}] \times {{\bf B}})_\phi | d\phi }$$
This quantity is expected to be very small within a stratified layer adjacent to a magnetostrophic dynamo, provided stratification is sufficiently strong. The recently developed dynamo simulations of [@olson2018outer; @stanley2008effects] which incorporate the presence of a stratified layer can utilise the computation of this quantity to access the simulation regime.
Finally, we note that the appropriate description of a stratified layer may in fact need to be more complex than a single uniform layer that we assume. Numerical simulations of core flow with heterogeneous CMB heat flux by [@mound2019regional] find that localised subadiabatic regions that are stratified are possible amid the remaining actively convecting liquid. If indeed local rather than global stratification is the more appropriate model for the Earth’s outermost core then the condition of requiring an exact Malkus state would not apply, and the constraints on the magnetic field would be weakened by the existence of regions of non-zero radial flow.
Conclusion
==========
In this paper we have shown how to construct magnetic fields that are consistent with geomagnetic observations, a strongly stratified layer and the exact magnetostrophic balance thought to exist within Earth’s core. To do this, we derived the Malkus constraints that must be satisfied by such a magnetic field, whose structure gives insight into the nature of the Earth’s magnetic field immediately beneath the CMB, where a layer of stratified fluid may be present. For a fixed magnetic field resolution, although the Malkus constraints are more numerous than the Taylor constraints, many solutions compatible with geomagnetic observations still exist. By making further assumptions about the field structure, we estimate that the toroidal field within the stratified layer is about 8 mT, significantly stronger than the 1 mT of the radial field inferred from degree-13 observations.
Acknowledgements
================
This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Fluid Dynamics at the University of Leeds under Grant No. EP/L01615X/1. P.W.L. acknowledges partial support from NERC grant NE/G014043/1. The authors would also like to thank Dominique Jault and Emmanuel Dormy for helpful discussions, as well as the Leeds Deep Earth group for useful comments. Figures were produced using matplotlib [@Hunter_2007].
Full sphere Malkus states {#sec:Apa_both}
=========================
Simple Example {#sec:Apb}
--------------
Here we consider a simple example of an axisymmetric magnetic field in a full sphere of radius $R$, consisting of four modes: a toroidal $l=1$, $n=1$ mode, a toroidal $l=1$, $n=2$ mode, a poloidal $l=1$, $n=1$ mode and a poloidal $l=1$, $n=2$ mode, each of which has an unspecified corresponding coefficient $\alpha_{l,n}$ and $\beta_{l,n}$ for toroidal and poloidal modes respectively. Through this we demonstrate the form of the linear constraints which arise from Malkus’ constraint in this case. It is significant to note the vital role of degeneracy within these constraints in permitting a solution.
Through computing the Malkus integral and enforcing that this is zero for all values of $s$ and $z$ by requiring that the coefficients of all powers of $s$ and $z$ vanish we obtain a series of simultaneous equations:
$$\left(-\frac{11}{8}\beta_{1,2}+2\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{140}\left(\frac{77}{69}\beta_{1,2}+\frac{56}{759}\beta_{1,1}\right)\alpha_{1,1}=0,$$
$$\left(\frac{319}{84}\beta_{1,2}-\frac{10}{3}\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{70}\beta_{1,2}\alpha_{1,1}=0,$$
$$\left(-\frac{165}{56}\beta_{1,2}+\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{140}\beta_{1,2}\alpha_{1,1}=0,$$
$$\left(\frac{319}{84}\beta_{1,2}-\frac{10}{3}\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{70}\beta_{1,2}\alpha_{1,1}=0,$$
$$\left(-\frac{165}{28}\beta_{1,2}+2\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{70}\beta_{1,2}\alpha_{1,1}=0,$$
$$\left(-\frac{165}{56}\beta_{1,2}+\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{140}\beta_{1,2}\alpha_{1,1}=0.$$
Although there are 6 equations here, there are only two independent conditions:
$$\alpha_{1,1}\beta_{1,2}+\frac{5}{2}\alpha_{1,2}\beta_{1,2}=0, ~~~~ \text{and} ~~~~ \alpha_{1,2}\beta_{1,1}+\frac{11}{7}\alpha_{1,2}\beta_{1,2}=0.$$ If both $\beta_{1,2}$ and $\alpha_{1,2}$ are nonzero, then these become linear constraints.
Hence, in this case we can see that there are 4 degrees of freedom, 6 constraint equations but only 2 independent constraints. This means that while on first inspection the system appears to be overconstrained with no solution, there are in fact multiple Malkus state solutions, with the solution space being spanned by two degrees of freedom ($\beta_{1,2}, \alpha_{1,2}$) with the other coefficients determined in terms of these by the relationships:
$$\alpha_{1,1}=-\frac{5}{2}\alpha_{1,2} ~~~~ \text{and} ~~~~ \beta_{1,1}=-\frac{11}{7}\beta_{1,2}.$$
Despite the significant degenercy in the Malkus constraints, they are notably more restrictive than the Taylor constraints for this truncation of $L_{max}=1$, $N_{max}=2$, for which the Taylor integral is identically zero and so provides no restriction.
Solution of a low-resolution system {#sec:Ap_sol_simp}
-----------------------------------
We now present the first known non-trivial solution of a Malkus state. Here we consider a full sphere magnetic field truncated at $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$, and impose a minimum Ohmic dissipation poloidal profile that matches the CHAOS-6 model (to degree 3) at $r=R$.We seek a toroidal field using all spherical harmonic modes within the truncation $L_{max}=3,~N_{max}=3$ (described by 45 degrees of freedom) that when added to this poloidal field satisfies the Malkus constraints. Of the 72 nonlinear constraint equations, only 42 are independent. Thus the number of degrees of freedom exceed the number of independent constraints, although since the constraints are nonlinear it is not immediate that a solution exists. However, using the computer algebra software Maple, we find the solution that minimises toroidal field strength as well as satisfying all the constraints, which is visualised in \[fig:fixpol\_nonlin\_all\]. We cannot generalise this procedure to higher resolutions because of the numerical difficulty in finding optimal solutions in such a nonlinear problem.
[.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](r09.pdf "fig:"){width="100.00000%"}
[0.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](phi_r09.pdf "fig:"){width="100.00000%"}
[.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](r08.pdf "fig:"){width="100.00000%"}
[0.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](phi_r08.pdf "fig:"){width="100.00000%"}
For comparison we also compute the solution using the method described in \[sec:theo\], which owing to the specific choice of toroidal spherical harmonic modes results in a linear system. The qualitative similarity between these solutions is important in giving an insight into how important it is that we make the (necessary) choice for our higher resolution Earth-like solutions, of only including modes that result in a linear system. Quantitatively this holds too, with rms values of $B_\phi$ of 0.21 and 0.23 mT for the non-linear and linear solutions respectively. Hence we suggest that the estimates for the lower bound of Earth’s toroidal field strength we have calculated would not be significantly different were it possible to solve the full non-linear system.
![Linear solution for $B_\phi$ at $r=0.9 R$, using the method outlined in \[sec:theo\] and used for the Earth-like solutions](phi_r09_linear_2.pdf){width="40.00000%"}
Enumeration of constraints {#sec:enum_con}
==========================
In order to determine the number of Malkus constraints, we calculate the maximum possible exponent in dimension of length within the Malkus integral. Since each constraint equation arises from ensuring a coefficient of a different exponent vanishes, enumerating all possibilities gives the maximum number of constraints.
There are three possible non-zero interactions whose sum comprise the Malkus integral, toroidal-toroidal, toroidal-poloidal and poloidal-poloidal as defined in \[eq:tortor\]. Since the poloidal field definition contains two curls whereas the toroidal field only one, then this extra derivative reduces the maximum exponent by one for interations involving a poloidal field as opposed to a toroidal one. This means that the maximal case is determined by the toroidal-toroidal interaction, $[{\bm{\mathcal{T}_1}},{\bm{\mathcal{T}}}_2]$. Since the Malkus integrand is identical to the Taylor integrand, we observe that the maximum radial exponent in the Malkus integrand $(({{{\boldsymbol \nabla}}\times}{\bm{\mathcal{T}_1}})\times {\bm{\mathcal{T}_2}})_\phi$ is $2L_{max}+4N_{max}-1$, as derived by [@livermore2008structure]. This is then reduced by two due to the property that the interaction of two toroidal harmonics that have identical spherical harmonic degrees and orders is zero [@livermore2008structure]. This requires that one of the two modes has an $L_{max}$ of at least one smaller than the other, hence resulting in a maximum possible degree in $r$ of $2L_{max}+4N_{max}-3$.
Now under a transform in coordinate systems we note that $r^n$ in spherical coordinates can be expressed as $s^jz^k$ in cylindrical coordinates, where $n=j+k$. Since only even values of $j$ are present this results in $L_{max}+2N_{max}-2 = C_T$ non-trivial constraint equations in this dimension. There is no such restriction on $k$, which can take all values up to the maximum of $2L_{max}+4N_{max}-3 = 2C_T+1$.
Each one of the constraints arises from a coefficient of a term with a different combination of exponents in $s$ and $z$, explicitly, these terms have the following form: $$\begin{aligned}
&(A_{C_T,0}z^0 + A_{C_T,1}z) s^{2C_T} +(A_{C_T-1,0}z^0+A_{C_T-1,1}z+A_{C_T-1,2}z^2+A_{C_T-1,3}z^3)s^{2(C_T-1)}\nonumber \\&+(A_{C_T-2,0}z^0+\dots+A_{C_T-2,5}z^5)s^{2(C_T-2)} +\dots \nonumber\\& +(A_{1,0}+\dots+A_{1,2C_T-1}z^{2C_T-1})s^2+(A_{0,0}+\dots+A_{0,2C_T+1}z^{2C_T+1}).\end{aligned}$$ Hence from the summation of the total number of these terms for every combination of $j$ and $k$, with $j$ even, such that $j+k \leq 2C_T+1$ we have the following expression for the maximum number of Malkus constraints, $$C_M = 2 \sum_{n=0}^{C_T} (n+1) = (C_T+1)(C_T+2) = C_T^2 + 3C_T + 2.$$
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abstract: 'We calculate the structure of the finitely generated groups $\hoz{2}{\spl{2}{\Z[1/m]}}$ when $m$ is a multiple of $6$. Furthermore, we show how to construct homology classes, represented by cycles in the bar resolution, which generate these groups and have prescribed orders. When $n\geq 2$ and $m$ is the product of the first $n$ primes, we combine our results with those of Jun Morita to show that the projection $\st{2}{\Z[1/m]}\to \spl{2}{\Z[1/m]}$ is the universal central extension. Our methods have wider applicability: The main result on the structure of the second homology of certain rings is valid for rings of $S$-integers with sufficiently many units. For a wide class of rings $A$, we construct explicit homology classes in $\hoz{2}{\spl{2}{A}}$, functorially dependent on a pair of units, which correspond to symbols in $K_2(2,A)$.'
address: 'School of Mathematical Sciences, University College Dublin'
author:
- Kevin Hutchinson
bibliography:
- 'h2sl2.bib'
title: 'The second homology of $\mathrm{SL}_2$ of $S$-integers'
---
Introduction {#sec:intro}
============
We calculate the structure of the finitely generated groups $\hoz{2}{\spl{2}{\Z[1/m]}}$ – the Schur multiplier of $\spl{2}{\Z[1/m]}$ – when $m$ is a multiple of $6$ (Theorem \[thm:z\] below). Furthermore, we show how to construct explicit homology classes, in the bar resolution, which generate these groups and have prescribed orders (sections \[sec:classes\] and \[sec:h2\]). Our methods have wider applicability, however: The main result on the structure of the second homology of certain rings is valid for rings of $S$-integers with sufficiently many units. The homology classes which we construct make sense over any ring in which $6$ is a unit.
For a ring $A$ satisfying some finiteness conditions the homology groups $\hoz{2}{\spl{n}{A}}$ are naturally isomorphic to the $K$-theory group $K_2(A)$ when $n$ is sufficiently large. However, $n=2$ is rarely sufficiently large, even when $A$ is a field.
We review some background results (see Milnor [@mil:intro] for details). For a commutative ring $A$, the *unstable* $K_2$-groups of the ring $A$, $K_2(n,A)$, are defined to be the kernel of a surjective homomorphism $\st{n}{A}\to E_n(A)$ where $\st{n}{A}$ is the rank $n-1$ Steinberg group of $A$ and where $E_n(A)$ is the subgroup of $\spl{n}{A}$ generated by elementary matrices. There are compatible homomorphisms $\st{n}{A}\to \st{n+1}{A}$, $E_n(A)\to E_{n+1}(A)$, and taking direct limits as $n\to\infty$, we obtain a surjective map $\sti{A}\to E(A)$ whose kernel is $K_2(A):=\lim K_2(n,A)$. In fact, $K_2(A)$ is central in $\sti{A}$ and the extension $$1\to K_2(A)\to \sti{A}\to E(A)\to 1$$ is the universal central extension of $E(A)$ and hence $\hoz{2}{E(A)}\cong K_2(A)$. Furthermore, for a commutative ring $A$, $E(A)=\mathrm{SL}(A)=\lim \spl{n}{A}$.
When $A$ satisfies some reasonable finiteness conditions these statements remain true when $K_2(A)$, $\sti{A}$ and $E(A)$ are replaced with $K_2(n,A)$, $\st{n}{A}$ and $E_n(A)$ for all sufficiently large $n$. In particular, when $F$ is a field with at least $10$ elements $\hoz{2}{\spl{2}{F}}\cong K_2(2,F)$.
When $F$ is a global field and when $S$ is a nonempty set of primes of $F$ containing the infinite primes, we let $\ntr{S}$ denote the corresponding ring of $S$-integers. (For example if $F=\Q$ and $1<m\in \Z$, we have $\Z[1/m]=\ntr{S}$ where $S$ consists of the primes dividing $m$ and the infinite prime.) Now the groups $\hoz{2}{\spl{2}{\ntr{S}}}$ and $K_2(2,\ntr{S})$ are finitely-generated abelian groups which satisfy $$\lim_S \hoz{2}{\spl{2}{\ntr{S}}}=\hoz{2}{\spl{2}{F}}\mbox{ and }
\lim_S K_2(2,\ntr{S})=K_2(2,F).$$ It is natural to guess that we might have $\hoz{2}{\spl{2}{\ntr{S}}}\cong K_2(2,\ntr{S})$ when $S$ is sufficiently large in some appropriate sense. The example of $\ntr{S}=\Z$, when $\hoz{2}{\spl{2}{\Z}}=0$ while $K_2(2,\Z)\cong\Z$ shows that some condition on $S$ will be required.
In the current paper, rather than comparing $\hoz{2}{\spl{2}{\ntr{S}}}$ to $K_2(2,\ntr{S})$ directly, we introduce a convenient proxy for $K_2(2,\ntr{S})$ which we denote $\tilde{K}_2(2,\ntr{S})$ (see section \[sec:main\] below for definitions). There are natural maps $$\hoz{2}{\spl{2}{\ntr{S}}}\to \tilde{K}_2(2,\ntr{S}),\quad
K_2(2,\ntr{S})\to \tilde{K}_2(2,\ntr{S})$$ and the structure of the group $\tilde{K}_2(2,\ntr{S})$ is easy to describe (see Lemma \[lem:tilde\]): $$\tilde{K}_2(2,\ntr{S})\cong K_2(\ntr{S})_+\oplus \Z^r$$ where $K_2(\ntr{S})_+$ is the subgroup of totally positive elements of $K_2(\ntr{S})$ and $r$ is the number of real embeddings of $F$.
Our main theorem (\[thm:s\]) states that when $S$ is sufficiently large (see the statement for more details) that the natural map $ \hoz{2}{\spl{2}{\ntr{S}}}\to \tilde{K}_2(2,\ntr{S})$ is an isomorphism. In the case $F=\Q$, the condition that $S$ be sufficiently large reduces to the requirement that $2,3\in S$. In particular, when $6|m$, we obtain isomorphisms $$\hoz{2}{\spl{2}{\Z[1/m]}}\cong \tilde{K}_2(2,\Z[1/m])\cong \Z\oplus\left( \oplus_{p|m}\F{p}^\times
\right).$$
Jun Morita ([@morita:zs]) proved isomorphisms of the form $$K_2(2,\Z[1/m])\cong \tilde{K}_2(2,\Z[1/m])$$ for certain integers $m$ (eg. if $m$ is the product of the first $n$ prime numbers). Combining Morita’s results with those above we deduce that $$\hoz{2}{\spl{2}{\Z[1/m]}}\cong K_2(2,\Z[1/m])$$ for such $m$, and that, consequently, the extension $$1\to K_2(2,\Z[1/m])\to \st{2}{\Z[1/m]}\to \spl{2}{\Z[1/m]}\to 1$$ is a universal central extension.
The main tool we use to prove Theorem \[thm:s\] is the expression of $\spl{2}{\ntr{S\cup \{\mathfrak{p}\}}}$ as an amalgamated product $$\spl{2}{\ntr{S}}\ast_{\tcong{\ntr{S}}{\mathfrak{p}}} H(\mathfrak{p})$$ associated to the action of $\spl{2}{\ntr{S\cup \{\mathfrak{p}\}}}$ on the Serre tree corresponding to the discrete valuation of the prime ideal $\mathfrak{p}$. This decomposition gives a Mayer-Vietoris sequence in homology. Analysis of the terms and the maps in low dimension yields, for $S$ sufficiently large, an exact sequence $$\xymatrix{
\hoz{2}{\spl{2}{\ntr{S}}}\ar[r]
&\hoz{2}{\spl{2}{\ntr{S\cup \{\mathfrak{p}\}}}}\ar^-{\delta}[r]
& \hoz{1}{k(\mathfrak{p})} \to 0.
}$$ where the map $\delta$ is essentially the tame symbol of $K$-theory (see Theorem \[thm:mv\]). This analysis requires, in particular, the deep and beautiful theorem of Vaserstein and Liehl ([@vaserstein:sl2] and [@liehl]) and the solution of the congruence subgroup problem for $\mathrm{SL}_2$ (Serre, [@serre:sl2]).
In the later part of the paper, we tackle an old question in $K_2$-theory; namely, how to write down natural homology classes in $\hoz{2}{\spl{2}{A}}$, depending functorially on a pair of units $u,v\in A^\times$, which correspond, under the map $\hoz{2}{\spl{2}{A}}\to K_2(2,A)$ when it exists, to the symbols $c(u,v)\in K_2(2,A)$. The answer to the corresponding question for $\hoz{2}{\spl{3}{A}}$ and $K_2(3,A)$ is well-known, namely the homology class (in the bar resolution) $$\left([\diag{u}{u^{-1}}{1}|\diag{v}{1}{v^{-1}}]-[\diag{v}{1}{v^{-1}}|\diag{u}{u^{-1}}{1}]
\right)\otimes 1$$ corresponds to the symbol $\mil{u}{v}\in K_2(3,A)$, at least up to sign. There is no such simple expression in the case of $K_2(2,A)$. The symbols $c(u,v)$ are easily and naturally described in terms of the generators of the Steinberg group, but the corresponding natural homology classes, even in the case of a field, have no known simple construction. Since $K_2(2,\Z)$ is infinite cyclic with generator $c(-1,-1)$ while $\hoz{2}{\spl{2}{\Z}}=0$ it follows that there can be no simple universal expression defined over the ring $\Z$. The homology classes, $C(u,v)$, that we construct in section \[sec:classes\] below are not very elegant (though it seems unlikely that they can be greatly improved on). To begin with, the construction of the representing cycles requires the presence of a unit $\lambda$ such that $\lambda^2-1$ is also a unit, although the resulting homology classes can be shown quite generally to be independent of the choice of $\lambda$. Furthermore, the representing cycles consist usually of $32$ terms and hence are far from simple.
However, the cycles we construct are explicit and functorial for homomorphisms of rings. We prove (Theorem \[thm:symb\]) that they map to the symbols $c(u,v)\in K_2(2,A)$ when $A$ is a field. We can thus use them to write down provably non-trivial homology classes in $\hoz{2}{\spl{2}{A}}$ for more general rings $A$. In particular, in section \[sec:h2\], we use them to write down explicit elements of the groups $\hoz{2}{\spl{2}{\ntr{S}}}$ with given order and to construct generators of the groups $\hoz{2}{\spl{2}{\Z[1/m]}}$ when $m$ is divisible by $6$.
Preliminaries and notation
==========================
Notation
--------
For a Dedekind Domain $A$ with field of fractions $F$, $\cl{A}$ denotes the ideal classgroup of $A$. If $\mathfrak{p}$ is a nonzero prime ideal of $A$, $v_{\mathfrak{p}}:F^\times\to\Z$ denotes the corresponding discrete value. For a global field $F$ and a nonempty set of primes $S$ of $F$ we let $\ntr{S}$ denote the ring of $S$-integers: $$\ntr{S}:=\{ a\in F^\times\ |\ v_{\mathfrak{p}}(a)\geq 0\mbox{ for all }\mathfrak{p}\not\in S\}.$$
For a finite abelian group $M$, $\psyl{M}{p}$ denotes the Sylow $p$-subgroup of $M$.
For a commutative ring $A$, we let $\sgr{A}:= \Z[A^\times/(A^\times)^2]$ be the group ring of the group of square classes of units. For $a\in A^\times$, the square class of $a$ will be denoted $\an{a}\in \sgr{A}$. Furthermore, the element $\an{a}-1$ in the augmentation ideal, $\aug{A}\subset \sgr{A}$, will be denoted $\pf{a}$.
Elementary matrices
-------------------
We will have occasion to refer to the following facts:
For a commutative ring $A$, and any $x\in A$ we define the elementary matrices $$E_{12}(x):=\matr{1}{x}{0}{1}, E_{21}(x):=\matr{1}{0}{x}{1}\in \spl{2}{A}.$$ Let $E_2(A)$ be the subgroup of $\spl{2}{A}$ generated by $E_{12}(x)$, $E_{21}(y)$, $x,y\in A$.
The following theorem of Vaserstein and Liehl will be essential below. Its proof relies on the resolution of the congruence subgroup problem for $\mathrm{SL}_2$ (see Serre [@serre:sl2]).
\[thm:vl\] Let $K$ be a global field and let $S$ be a set of places of $K$ of cardinality at least $2$ and containing all archimedean places. Let $$\mathcal{O}_S:= \{ x\in K\ |\ v(x)\geq 0 \mbox{ for all }v\not\in S\}$$ be the ring of $S$-integers of $K$. Let $I_1$ and $I_2$ be nonzero ideals of $\mathcal{O}_S$. Let $$\tilde{\Gamma}(I_1,I_2):= \left\{ \matr{a}{b}{c}{d}\in \spl{2}{\mathcal{O}_S}\ | \
b\in I_1,c\in I_2, a-1,b-1\in I_1I_2\right\}$$ Then $\tilde{\Gamma}(I_1,I_2)$ is generated by the elementary matrices $$E_{12}(x), x\in I_1\mbox{ and } E_{21}(y), y\in I_2.$$
\[prop:e2\] Let $A$ be a commutative ring.
1. $E_2(A)=\spl{2}{A}$ if $A$ is a field or a Euclidean domain or if $A=\mathcal{O}_S$ is the ring of $S$-integers in a global field and $|S|\geq 2$.
2. $E_2(A)$ is perfect if there exists $\lambda_1,\ldots,\lambda_n\in A^\times$ and $b_1,\ldots,
b_n\in A$ such that\
$\sum_{i=1}^nb_i(\lambda_i^2-1)=1$ in $A$.
In particular, $E_2(A)$ is perfect if there exists $\lambda\in A^\times$ such that $\lambda^2-1\in
A^\times$ also.
<!-- -->
1. This is standard linear algebra in the case of a Euclidean Domain or a field, and the theorem of Vaserstein-Liehl in the case of $S$-integers.
2. For $\lambda\in A^\times$, let $$D(\lambda):=\matr{\lambda}{0}{0}{\lambda^{-1}}\in\spl{2}{A}.$$
Note that $D(\lambda)\in E_2(A)$ since $$D(\lambda)=w(\lambda)w(-1)\mbox{ where } w(\lambda):=
\matr{0}{\lambda}{-\lambda^{-1}}{0}=E_{12}(\lambda)E_{21}(-\lambda^{-1})E_{12}(\lambda).$$ Then $$D(\lambda)E_{12}(x)D(\lambda)^{-1}= E_{12}(\lambda^2x)$$ and hence, for any $b\in A$ we have $$[D(\lambda),E_{12}(bx)]=D(\lambda)E_{12}(bx)D(\lambda)^{-1}E_{12}(-bx)=E_{12}((\lambda^2-1)bx).$$
Thus $$E_{12}(x)=E_{12}(\sum_i(\lambda_i^2-1)b_ix)=\prod_iE_{12}((\lambda_i^2-1)b_ix)=
\prod_i[D(\lambda_i),E_{12}(b_ix)].$$
On the other hand, the groups $E_2(\F{2})=\spl{2}{\F{2}}$ and $E_2(\F{3})=\spl{2}{\F{3}}$ are not perfect. It follows that if the ring $A$ admits a homomorphism to $\F{2}$ or $\F{3}$ then $E_2(A)$ is not perfect. In particular, the group $E_2(\Z)$ is not perfect.
In [@swan:special], R. Swan showed that $E_2(A)\not=\spl{2}{A}$ for $A=\Z[\sqrt{-5}]$.
Indeed, when $A$ is the ring of integers in a quadratic imaginary number field then $E_2(A)\not=\spl{2}{A}$ except in the five cases that $A$ is a Euclidean Domain (see [@vaserstein:sl2]).
Homology of Groups
------------------
For any group $G$, $F_\bullet(G)$ will denote the (right) bar resolution of $\Z$ over $\Z[G]$: i.e. for $n\geq 1$, $F_n(G)$ is the free right $\Z[G]$-module with generators $[g_n|\cdots |g_1]$, $\g_i\in G$, and $F_0(G)=\Z[G]$ (regarded as a right $\Z[G]$-module). The boundary homomorphism $d_n:F_n(G)\to F_{n-1}(G)$ is given by $$d_n([g_n|\cdots |g_1])=[g_n|\cdots |g_2]g_1+\sum_{i=1}^{n-1}(-1)^{n-i}[g_{n-1}|\cdots |g_{i+1}g_{i}|\cdots |g_1]
+(-1)^n[g_{n-1}|\cdots|g_1]$$ for $n\geq 2$ and $d_1([g]):= g-1$.
We let $\bar{F}_\bullet(G)$ denote the complex $\{\bar{F}_n(G)\}_{n\geq 0}$ where $$\bar{F}_n(G):= F_n(G)\otimes_{\Z[G]}\Z.$$ Thus $\hoz{n}{G}\cong H_n(\bar{F}_\bullet(G))$.
We will require the following standard “centre kills” argument from group homology:
\[lem:ck\] Let $G$ be a group and let $M$ be a $\Z[G]$-module. Suppose that $g\in Z(G)$ has the property that $g-1$ acts as an automorphism on $M$. Then $\ho{i}{G}{M}=0$ for all $i\geq 0$.
The functor $K_2(2,A)$ {#sec:k2}
======================
In this section, we review some of the theory of the functor $K_2(2,A)$ for commutative rings $A$.
Definitions
-----------
Let $A$ be a commutative ring.
We let $A^\times$ act by automorphisms on $\spl{2}{A}$ as follows: Let $$M(a):=\matr{a}{0}{0}{1}\in \gl{2}{A}.$$ and define $$a\ast X:= X^{M(a)}=M(a)^{-1}XM(a)$$ for $a\in A^\times$, $X\in \spl{2}{A}$.
In particular, we have $$a\ast E_{12}(x)= E_{12}(a^{-1}x)\mbox{ and } a\ast E_{21}(x)=E_{21}(ax)$$ for all $a\in A^\times$, $x\in A$.
The rank one Steinberg group $\st{2}{A}$ is defined by generators and relations as follows: The generators are the terms $$x_{12}(t)\mbox{ and } x_{21}(t), \quad t\in A$$ and the defining relations are
1. $$x_{ij}(s)x_{ij}(t)=x_{ij}(s+t)$$ for $i\not=j\in \{ 1,2\}$ and all $s,t\in A$, and
2. For $u\in A^\times$, let $$w_{ij}(u):= x_{ij}(u)x_{ji}(-u^{-1})x_{ij}(u)$$ for $i\not=j\in \{ 1,2\}$. Then $$w_{ij}(u)x_{ij}(t)w_{ij}(-u)=x_{ji}(-u^{-2}t)$$ for all $u\in A^\times$, $t\in A$.
There is a natural surjective homomorphism $\phi:\st{2}{A}\to E_2(A)$ defined by $\phi(x_{ij}(t))=E_{ij}(t)$ for all $t$. It is easily verified that the formulae $$a\ast x_{12}(t)=x_{12}(a^{-1}t)\mbox{ and }a\ast x_{21}(t)=x_{21}(at)$$ define an action of $A^\times$ on $\st{2}{A}$ by automorphisms. Clearly the homomorphism $\phi$ is equivariant with respect to this action.
By definition $K_2(2,A)$ is the kernel of $\phi$. It inherits an action of $A^\times$.
For $u\in A^\times$ and for $i\not=j\in \{ 1,2\}$, we let $$h_{ij}(u):= w_{ij}(u)w_{ij}(-1).$$ Note that $$\phi(w_{12}(u))=\matr{0}{u}{-u^{-1}}{0} \mbox{ and } \phi(h_{12}(u))=\matr{u}{0}{0}{u^{-1}}.$$
Note that, from the definitions and defining relation (1), for any $a\in A$ and for any unit $u$ we have $$x_{ij}(a)^{-1}=x_{ij}(-a)\mbox{ and } w_{ij}(u)^{-1}=w_{ij}(-u).$$
The defining relation (2) above thus immediately gives the following conjugation formula.
\[lem:conjw\] Let $A$ be a commutative ring. Let $a\in A$ and $u\in A^\times$. For $i\not=j\in
\{ 1,2\}$ $$x_{ij}(a)^{w_{ij}(-u)}=x_{ji}(-u^{-2}a).$$
Since the right-hand-side is unchanged by $u\rightarrow -u$, we deduce:
\[cor:conjw\] Let $A$ be a commutative ring. Let $a\in A$ and $u\in A^\times$. For $i\not=j\in
\{ 1,2\}$ $$x_{ij}(a)^{w_{ij}(u)^{-1}}=x_{ji}(-u^{-2}a)=x_{ij}(a)^{w_{ij}(u)}.$$ and $$x_{ji}(a)^{w_{ij}(u)}=x_{ij}(-u^2a).$$
Form the definition of $h_{ij}(u)$, we then obtain:
\[cor:conjh\] Let $A$ be a commutative ring. Let $a\in A$ and $u\in A^\times$. For $i\not=j\in
\{ 1,2\}$ $$x_{ij}(a)^{h_{ij}(u)}=x_{ij}(u^{-2}a)\mbox{ and }
x_{ij}(a)^{h_{ij}(u)^{-1}}=x_{ij}(u^2a).$$
Symbols
-------
In particular, for $u,v\in A^\times$ the *symbols* $$c(u,v):= h_{12}(u)h_{12}(v)h_{12}(uv)^{-1}$$ lie in $K_2(2,A)$.
The elements $c(u,v)$ are central in $\st{2}{A}$. We let $C(2,A)$ denote the subgroup of $K_2(2,A)$ generated by these symbols.
Note that for $a,u\in A^\times$ we have $$a\ast w_{12}(u)=w_{12}(a^{-1}u)\mbox{ and } a\ast w_{21}(u)=w_{21}(au)$$ and hence $$a\ast h_{12}(u)=h_{12}(a^{-1}u)h_{12}(a^{-1})^{-1}\mbox{ and }
a\ast h_{21}(u)=h_{21}(au)h_{21}(a)^{-1}.$$
It follows easily that $$a\ast c(u,v) =c(u,a^{-1})^{-1}c(u,a^{-1}v).$$
Thus the abelian group $C(2,A)$ is a module over the group ring $\Z[A^\times]$ with this action.
\[lem:symb\] Let $A$ be a commutative ring. Then $$a^2\ast c(u,v)=c(u,v)$$ for all $a,u,v\in A^\times$.
In particular, $C(2,A)$ is naturally an $\sgr{A}$-module.
We have $h_{ij}(u)=h_{ji}(u)^{-1}$ in $\st{2}{A}$. Thus $$c(u,v)= h_{12}(u)h_{12}(v)h_{12}(uv)^{-1}=h_{21}(u)^{-1}h_{21}(v)^{-1}h_{21}(uv).$$
Thus $$\begin{aligned}
a*c(u,v)&=& h_{21}(a)h_{21}(au)^{-1}h_{21}(a)h_{21}(av)^{-1}h_{21}(auv)h_{21}(a)^{-1}\\
&=& h_{21}(a)h_{21}(au)^{-1}h_{21}(a)\cdot\left( h_{21}(u)h_{21}(u)^{-1}\right)\cdot
h_{21}(av)^{-1}h_{21}(auv)h_{21}(a)^{-1}\\
&=& h_{21}(a)c(u,a)^{-1}c(u,av)h_{21}(a)^{-1}\\
&=& c(u,a)^{-1}c(u,av)\\
&=& a^{-1}\ast c(u,v).\\\end{aligned}$$
The symbols $c(u,v)$ satisfy the following properties (see [@mat:pres], or also [@steinberg:chev])
\[prop:mat\] Let $A$ be a commutative ring. Then
1. $c(u,v)=1$ if $u=1$ or $v=1$.
2. $c(u,v)=c(v^{-1},u)$ for all $u,v\in A^\times$.
3. $c(u,vw)c(v,w)=c(uv,w)c(u,v)$ for all $u,v,w\in A^\times$.
4. $c(u,v)=c(u,-uv)$ for all $u,v\in A^\times$
5. $c(u,v)=c(u,(1-u)v)$ whenever $u,1-u,v\in A^\times$.
Combining the result of Lemma \[lem:symb\] with Proposition \[prop:mat\] (3), we see that the square class $\an{a}\in \sgr{A}$ acts on $C(2,A)$ via $$\an{a}c(u,v)= c(u,a)^{-1}c(u,av)= c(au, v)c(a,v)^{-1}.$$
Futhermore Proposition \[prop:mat\] (4) is equivalent to $$\an{v}c(u,-u)=1\mbox{ for all }u,v\in A^\times$$ and Proposition \[prop:mat\] (5) is equivalent to $$\an{v}c(u,1-u)=1 \mbox{ for all }u,v\in A^\times.$$
We will use the following property of symbols ([@mat:pres]):
\[lem:square\] If $u,v,w$ are units in $A$, then $$c(u,v^2w)=c(u,v^2)c(u,w)$$ and $$c(u,v^2)=c(u,v)c(v,u)^{-1}=c(u^2,v).$$
Furthermore, we have the following theorem of Matsumoto and Moore ([@mat:pres],[@moore:pres]):
\[thm:mm\] Let $F$ be an infinite field. Then
1. The sequence $$1\to K_2(2,F)\to \st{2}{F}\to \spl{2}{F}\to 1$$ is the universal central extension of the perfect group $\spl{2}{F}$.
In particular, $K_2(2,F)\cong\hoz{2}{\spl{2}{F}}$ naturally.
2. $K_2(2,F)$ has the following presentation: It is generated by the symbols $c(u,v)$, $u,v\in F^\times$, subject to the five relations of Proposition \[prop:mat\].
The stabilization homomorphism $K_2(2,F)\to K_2(F)$
---------------------------------------------------
For a field $F$, the Theorem of Matsumoto also gives a presentation of $K_2(n,F)$ for all $n\geq 3$. In particular, it follows that $K_2(F)=\milk{2}{F}$, the second Milnor $K$-group of the field $F$. The stabilization map $K_2(2,F)\to K_2(F)$ is surjective and sends the symbols $c(u,v)$ to the symbols $\{ u,v\}$ of algebraic $K$-theory.
Let $\gw{F}$ be the Grothendieck-Witt ring of isometry classes of nondegerate quadratic forms over $F$. It is generated by the classes $\an{a}$ of $1$-dimensional forms and the map $\sgr{F}\to\gw{F}$ sending $\an{a}\to \an{a}$ is a surjection of rings. The fundamental ideal $I(F)$ of $\gw{F}$ is the ideal generated by the elements $\pf{a}:=\an{a}-1$.
There is a natural surjective homomorphism of $\sgr{F}$-modules $$K_2(2,F)\to I^2(F),\quad c(u,v)\mapsto \pf{u}\pf{v}.$$
Furthermore, by a theorem of Milnor ([@milnor:quad]) there is also a surjective map $\milk{2}{F}\to I^2(F)/I^3(F)$ sending the symbol $\{ u,v\}$ to the class of $\pf{u}\pf{v}$. The kernel of this map is precisely $2\milk{2}{F}$.
By a result essentially due to Suslin ([@sus:tors], but see also [@mazz:sus]) for an infinite field $F$, we also have the following description of $K_2(2,F)$:
\[thm:susmazz\] Let $F$ be an infinite field. The maps $K_2(2,F)\to K_2(F)$, $K_2(2,F)\to I^2(F)$ induce an isomorphism of $\sgr{F}$-modules $$K_2(2,F)\to \milk{2}{F}\times_{I^2(F)/I^3(F)}I^2(F),
\quad c(u,v)\mapsto [u,v]:= (\mil{u}{v},\pf{u}\pf{v}).$$
\[cor:mwk\] Let $F$ be an infinite field. There is a natural short exact sequence of $GW(F)$-modules $$0\to I^3(F)\to K_2(2,F)\to \milk{2}{F}\to 0.$$
Milnor-Witt $K$-theory {#sec:mwk}
----------------------
The homology of the special linear group of a field is related to the Milnor-Witt $K$-theory of the field (see, for example, [@hutchinson:tao3]).
Milnor-Witt $K$-theory of a field $F$ is a $\Z$-graded algebra $\mwk{\bullet}{F}$ generated by symbols $[ u]$, $u\in F^\times$ in degree $1$ and a symbol $\eta$ in degree $-1$, satisfying certain relations (see [@morel:trieste] for details). It arises naturally as a ring of operations in stable $\mathbb{A}^1$-homotopy theory.
A deep theorem of Morel asserts:
\[thm:morel\]\[[@morel:puiss]\] There is a natural isomorphism of graded rings $$\mwk{\bullet}{F}\cong \milk{\bullet}{F}\times_{I^{\bullet}(F)/I^{\bullet +1}(F)}I^\bullet(F).$$
(Here, when $n<0$, $\mwk{n}{F}:=0$ and $I^n(F):=W(F)$, the Witt ring of the field.)
The theorem of Suslin on the structure of $K_2(2,F)$ quoted above, implies
There is a natural isomorphism $K_2(2,F)\cong \mwk{2}{F}$, sending $c(u,v)$ to $[u][v]$.
The map from $\hoz{2}{\spl{2}{F}}$ to $K_2(2,F)$ {#sec:map}
================================================
Let $A$ be a commutative ring for which $E_2(A)=\spl{2}{A}$ is a perfect group. Suppose further that the group extension $$\xymatrix{
1\ar[r]
& K_2(2,A)\ar[r]
& \st{2}{A}\ar^-{\phi}[r]
& \spl{2}{A}\ar[r]
& 1
}$$ is a central extension.
Let $s:\spl{2}{A}\to \st{2}{A}$ be a section of $\phi$. Then there is a corresponding $2$-cocycle $f_s:\spl{2}{A}\times \spl{2}{A}\to K_2(2,A)$ defined by $$f_s(X,Y):= s(X)s(Y)s(XY)^{-1}.$$ This yields a cohomology class $f\in \coh{2}{\spl{2}{A}}{K_2(2,A)}$ which is independent of the choice of section $s$.
However, since $\hoz{1}{\spl{2}{A}}=0$, the universal coefficient theorem tells us that there is a natural isomorphism $$\coh{2}{\spl{2}{A}}{K_2(2,A)}\cong \mathrm{Hom}(\hoz{2}{\spl{2}{A}},K_2(2,A))$$ described as follows: Let $z\in \coh{2}{\spl{2}{A}}{K_2(2,A)}$ be represented by the $2$-cocycle $h$. Then $h$ induces a homomorphism $$\bar{F}_2\to K_2(2,F),\quad \sum_in_i[X_i|Y_i]\mapsto \prod_ih(X_i,Y_i)^{n_i}$$ which vanishes on boundaries, and thus in turn induces a homomorphism $$\bar{h}: \hoz{2}{\spl{2}{A}}\to K_2(2,F).$$
In particular, the cocycle $f_s$ above induces the homomorphism $$\hoz{2}{\spl{2}{A}}\to K_2(2,F), \quad \sum_in_i[X_i|Y_i]\mapsto \prod_if_s(X_i,Y_i)^{n_i}$$
This homomorphism is an isomorphism precisely when the central extension is universal. In particular, it is an isomorphism when $A$ is an infinite field, by the theorem of Matsumoto-Moore.
We now specialise to the case of a field $F$.
For our calculations, we will use the following section $s:\spl{2}{F}\to \st{2}{F}$: $$s\left(\matr{a}{b}{c}{d}\right):=
\left\{
\begin{array}{ll}
x_{12}(ab)h_{12}(a), & \mbox{ if } c=0,\\
x_{12}(ac^{-1})w_{12}(-c^{-1})x_{12}(dc^{-1}), & \mbox{ if }c\not=0.\\
\end{array}
\right.$$
Note that, in particular, we have $$s(E_{ij}(a))=x_{ij}(a)\mbox{ and } s(D(u))=h_{12}(u)$$ when $i\not= j\in \{ 1,2\}$, $a\in A$ and $u\in F^\times$.
Furthermore, functoriality of the constructions above guarantee that the induced homomorphism $$\bar{f}:\hoz{2}{\spl{2}{F}}\to K_2(2,F)$$ is a map of $\Z[F^\times]$-modules. Recall that this homomorphism is induced by the homomorphism $$\bar{F}_2(\spl{2}{F})\to K_2(2,F), [X|Y]\mapsto f_s(X,Y)=s(X)s(Y)s(XY)^{-1}.$$
\[lem:f\] Let $F$ be a field. Let $u,v\in F^\times$ and $a,b\in F$. Let $$X=\matr{u}{a}{0}{u^{-1}}, \quad Y= \matr{v}{b}{0}{v^{-1}}$$ Then $
f_s(X,Y)=c(u,v).
$
We have, $$s(X)=x_{12}(au)h_{12}(u),\quad s(Y)=x_{12}(bv)h_{12}(v)\mbox{ and } s(XY)= x_{12}(bu^2v+au)h_{12}(uv).$$
Thus $$\begin{aligned}
f(X,Y)&=& x_{12}(au)h_{12}(u)x_{12}(bv)h_{12}(v)h_{12}(uv)^{-1}x_{12}(-bu^2v-au)\\
&=& x_{12}(au)x_{12}(bv)^{h_{12}(u)^{-1}}h_{12}(u)h_{12}(v)h_{12}(uv)^{-1}x_{12}(-bu^2v-au)\\
&=& x_{12}(au)x_{12}(bu^2v)c(u,v)x_{12}(-bu^2v)x_{12}(-au)\mbox{\quad by Corollary
\ref{cor:conjh}}\\
&=& c(u,v)\mbox{\quad since $c(u,v)$ is central.} \\\end{aligned}$$
\[cor:square\] Let $F$ be a field. Let $a,b\in F^\times$. Then $$\left([D(a)|D(b)]-[D(b)|D(a)]\right)\otimes 1\in F_2(\spl{2}{F})\otimes \Z$$ is a cycle and the corresponding homology class maps to $c(a^2,b)$ under the natural isomorphism $\hoz{2}{\spl{2}{k}}\cong K_2(2,F)$ induced by $f_s$.
The first statement in immediate since $D(a)D(b)=D(ab)=D(b)D(a)$.
The image of this cycle is $$f_s(D(a),D(b))\cdot f_s(D(b),D(a))^{-1}=c(a,b)c(b,a)^{-1}=c(a^2,b)$$ by Lemma \[lem:square\].
The Mayer-Vietoris sequence
===========================
Throughout this section $A$ will denote a Dedekind Domain with field of fractions $K$.
The groups $H(I)$
-----------------
We collect together some basic and well-known facts about certain subgroups of $\spl{2}{K}$ (see for example [@serre:sl2 p. 520]).
Let $I$ be a fractional ideal of $A$.
We consider the lattice $\Lambda=\Lambda_I:=A\oplus I \subset K\oplus K=K^2$.
Let $H(I)$ denote the the subgroup $$\{ M\in \spl{2}{K}\ | M\cdot \Lambda= \Lambda\}=
\left\{ \matr{a}{b}{c}{d}\in \spl{2}{K}\ |\ a,d\in A, c\in I,d\in I^{-1}
\right\}=
\tilde{\Gamma}(I,I^{-1}).$$
Note that, in particular, $H(A)=\spl{2}{A}$.
We also note that if $J$ is any nonzero fractional ideal of $A$, then $$H(I)= \{ M\in \spl{2}{K}\ | M\cdot (J\Lambda)= J\Lambda\}$$ where $$J\Lambda=J\cdot(A\oplus I)=J\oplus IJ.$$
\[lem:hi\] Let $I$ be a fractional ideal of the $A$.
1. Suppose that $I'=aI$ where $0\not=a\in K$. Then $H(I')=H(I)^{M(a)}$ where $$M(a)=\matr{a}{0}{0}{1}\in\gl{2}{K}.$$
2. Suppose $I$ is an integral ideal. Let $$A'= \{ r\in K\ | v_{\mathfrak{q}}(r)\geq 0\mbox{ for all }\mathfrak{q}\not|I\}.$$ Then there exists $M\in\spl{2}{A'}$ such that $H(I^2)=\spl{2}{A}^M$. In particular, $H(I^2)\cong \spl{2}{A}$.
1. This follows from the observation that multiplication by $M(a)$ induces an isomorphism of lattices $A\oplus I'\cong a\cdot(A\oplus I)$, and hence conjugation by $M(a)$ induces an isomorphism of the stabilizers.
2. We first observe that, since $I^{-1}\cdot \Lambda_I=I^{-1}\oplus I$, $H(I^2)$ is the stabilizer of $I^{-1}\oplus I$.
There exists an integral ideal $J$ of $A$ satisfying: $I+J=A$ and $IJ=xA$ for some nonzero $x\in A$. So $J=xI^{-1}$. Thus multiplication by $M(x)$ induces an isomorphism $I^{-1}\oplus I\cong J\oplus I$.
Choose $a\in I, b\in J$ with $a+b=1$. Consider the short exact sequence of $A$-modules $$\xymatrix{
0\ar[r]
&xA\ar^-{f}[r]
&J\oplus I\ar^{g}[r]
&A\ar[r]
&0\\
}$$ where $g(y)=(y,-y)$ and $f(y,z)=y+z$. There is a splitting $A\to J\oplus I$ given by $y\mapsto (by,ay)$. This gives an isomorphism of $A$-modules $$J\oplus I\cong xA\oplus A,\quad (y,z)\mapsto (ay-bz,y+z);$$ i.e.multiplication by $$N:= \matr{a}{-b}{1}{1}\in\spl{2}{A}$$ induces an isomorphism of lattices $J\oplus I\cong xA\oplus A$.
Now, multiplication by $M(x)^{-1}$ induces an isomorphism $xA\oplus A\cong A\oplus A$.
Putting all of this together, multiplication by $$M:=M(x)^{-1}NM(x)=\matr{a}{-b/x}{x}{1}\in\spl{2}{K}$$ induces an isomorphism of lattices $I^{-1}\oplus I\cong A\oplus A$, and thus conjugation by $M$ induces an isomorphism of stabilizers as required.
Finally, we note that since $xA=IJ$ and $bA=JK$ for some integral ideal $K$, $(b/x)A=KI^{-1}$ and hence $b/x\in A'$. Thus $M\in\spl{2}{A'}$ as claimed.
Let $I$ be a fractional ideal of $A$. Suppose that the class of $I$ in $\cl{A}$ is a square. Then $H(I)\cong \spl{2}{A}$.
In particular, the ideal $\tilde{\mathfrak{p}}:=\mathfrak{p}A_{\mathfrak{p}}$ in $A_{\mathfrak{p}}$ is a principal ideal with generator $\pi$, say. It follows from Lemma \[lem:hi\] that $$H(\tilde{\mathfrak{p}})= \spl{2}{A_{\mathfrak{p}}}^{M(\pi)}.$$
Let $\mathfrak{p}$ be a nonzero prime ideal of $A$. Let $n\geq 1$ and let $\pi\in A$ satisfy $v_{\mathfrak{p}}(\pi)=1$. We let $\gamma_{\pi,n}:H(\mathfrak{p})\to \spl{2}{A/\mathfrak{p}^n}$ be the composite $$\xymatrix{
H(\mathfrak{p})\ar[r]
&H(\tilde{\mathfrak{p}})\ar^-{\cong}_-{\mathrm{conj}_{M(1/\pi)}}[r]
&\spl{2}{A_{\mathfrak{p}}}\ar[r]
&\spl{2}{A_{\mathfrak{p}}/\tilde{\mathfrak{p}}^n}\ar[r]^-{\cong}
&\spl{2}{A/\mathfrak{p}^n}
}$$
The map $\gamma_{\pi,n}$ is surjective for all $n$ and the kernel of this map is independent of the choice of $\pi$.
By definition, we have $$\gamma_{\pi,n}\left(\matr{a}{b}{c}{d}\right)=\matr{\bar{a}}{\bar{\pi b}}{\bar{c/\pi}}{\bar{d}}$$ where $$\bar{x}:= x+\tilde{\mathfrak{p}}^n\in A_{\mathfrak{p}}/\tilde{\mathfrak{p}}^n\cong A/\mathfrak{p}^n.$$
Since $\spl{2}{A/\mathfrak{p}^n}$ is generated by elementary matrices, we need only show how to lift these. We begin by observing that $\pi A=\mathfrak{p}J$ where $J$ is an ideal not contained in $\mathfrak{p}$. It follows that $A=\mathfrak{p}^n+J$ for any $n\geq 1$; i.e. the map $J\to A/\mathfrak{p}^n$ is surjective.
Thus, given any $x\in A$ there exists $x'\in J$ with $\bar{x'}=\bar{x}$. Since $x'\in
J$ it follows that $x'/\pi\in J\cdot (\mathfrak{p}J)^{-1}=\mathfrak{p}^{-1}$. Hence $E_{12}(x'/\pi)\in H(\mathfrak{p})$ and $$\gamma_{\pi,n}(E_{12}(x'/\pi))= E_{12}(\bar{x'})= E_{12}(\bar{x}).$$ Of course, we also have $E_{21}(\pi x)\in H(\mathfrak{p})$ and $\gamma_{\pi,n}(E_{21}(\pi x))=E_{21}(\bar{x})$. This proves the surjectivity statement.
For the second part, suppose that $\pi'\in A$ also satisfies $v_{\mathfrak{p}}(\pi')=1$. Then $\pi'=\pi\cdot u$ for some $u\in A_{\mathfrak{p}}^\times$. From the definition, we have $$\gamma_{\pi',n}= f\circ \gamma_{\pi,n}$$ where $f$ is conjugation by $M(\bar{u}^{-1})$ on $\spl{2}{A/\mathfrak{p}^n}$. It follows at once that $\ker{\gamma_{\pi',n}}=\ker{\gamma_{\pi,n}}$ as claimed.
We let $\tpcong{A}{\mathfrak{p}^n}$ denote the kernel of the $\gamma_{\pi,n}$ (for any choice of $\pi$). Thus, for all $n\geq 1$, there is a short exact sequence $$1\to \tpcong{A}{\mathfrak{p}^n}\to H(\mathfrak{p})\to \spl{2}{A/\mathfrak{p}^n}\to 1.$$
Note that $$\tpcong{A}{\mathfrak{p}^n}=
\left\{ \matr{a}{b}{c}{d}\in H(\mathfrak{p})\ |\ a-1,d-1\in \mathfrak{p}^n, c\in \mathfrak{p}^{n+1},
b\in \mathfrak{p}^{n-1}\right\}.$$ In particular, for all $n\geq 1$ we have $$\pcong{A}{\mathfrak{p}^{n+1}}\subset \tpcong{A}{\mathfrak{p}^n}\subset
\tcong{A}{\mathfrak{p}^n}\subset \spl{2}{A}.$$
For a field $F$, we will use the notation $$\bor(F):= \left\{ \matr{a}{b}{0}{a^{-1}}\in \spl{2}{F}\right\}\mbox{ and }
\bor'(F):= \left\{ \matr{a}{0}{c}{a^{-1}}\in \spl{2}{F}\right\}.$$ Of course, these two subgroups of $\spl{2}{F}$ are naturally isomorphic.
We will need the following result below.
\[lem:bor’\] There is a natural short exact sequence $$1\to \tpcong{A}{\mathfrak{p}}\to \tcong{A}{\mathfrak{p}}\to \bor'(k(\mathfrak{p}))\to 1.$$
This is immediate from the fact that the image of $\tcong{A}{\mathfrak{p}}$ in $\spl{2}{A/\mathfrak{p}}=\spl{2}{k(\mathfrak{p})}$ under the map $\gamma_{\pi,1}$ is precisely $\bor'(k(\mathfrak{p}))$.
The Mayer-Vietoris sequence
---------------------------
Let $\mathfrak{p}$ be a nonzero prime ideal of $A$ and let $v=v_{\mathfrak{p}}$ be the associated discrete valuation. We let $k(\mathfrak{p})$ or $k(v)$ denote the residue field $A/\mathfrak{p}$. We will further suppose that the class of $\mathfrak{p}$ has finite order in $\cl{A}$. Thus $\mathfrak{p}^n=xA$ for some $n\geq 1$ and $x\in A$. (This condition is automatically satisfied when $K$ is a global field.)
Let $$\pcong{A}{\mathfrak{p}}:=\ker{\spl{2}{A}\to \spl{2}{k(\pi)}}=
\left\{ \matr{a}{b}{c}{d}\in\spl{2}{A}\ :\ 1-a,1-d,b,c\in\mathfrak{p}\right\}$$ and let $$\tcong{A}{\mathfrak{p}}:= \left\{ \matr{a}{b}{c}{d}\in\spl{2}{A}\ :\ c\in\mathfrak{p}\right\}.$$
We let $\tilde{\mathfrak{p}}$ denote the extension of $\mathfrak{p}$ to the localization $A_{\mathfrak{p}}$, which is thus a discrete valuation ring with unique (principal) nonzero prime ideal $\tilde{\mathfrak{p}}$.
The action of $\spl{2}{K}$ on the Serre tree associated to the valuation $v$ ([@serre:trees Chapter II]) yields a decomposition $$\begin{aligned}
\label{amal1}
\spl{2}{K}=\spl{2}{A_{\mathfrak{p}}}\star_{\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}}
H(\tilde{\mathfrak{p}})\end{aligned}$$ of $\spl{2}{K}$ as the sum of $\spl{2}{A_{\mathfrak{p}}}$ and $H(\tilde{\mathfrak{p}})$ amalgamated along their intersection\
$\spl{2}{A_{\mathfrak{p}}}\cap H(\tilde{\mathfrak{p}})
=
\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}$.
Let $$A':=\{ a\in K\ |\ v_{\mathfrak{q}}(a)\geq 0 \mbox{ for all prime ideals }\mathfrak{q}\not=\mathfrak{p}
\}.$$ Note that since $\mathfrak{p}^n=xA$ by assumption, $A'=A[1/x]$.
Since $A[1/x]$ is dense in $K$ in the $\mathfrak{p}$-adic topology, and since $$\spl{2}{A[1/x]}\cap\spl{2}{A_{\mathfrak{p}}}=\spl{2}{A},\quad
\spl{2}{A[1/x]}\cap H(\tilde{\mathfrak{p}})= H(\mathfrak{p})$$ there is also an induced decomposition $$\spl{2}{A[1/x]}=\spl{2}{A}\star_{\tcong{A}{\mathfrak{p}}}H(\mathfrak{p}).$$
For convenience, in the remainder of this section we will set $$G:=\spl{2}{A[1/x]},\
G_1:= \spl{2}{A},\ G_2:= H(\mathfrak{p})\mbox{ and } \Gamma_0:= \tcong{A}{\mathfrak{p}}.$$ Thus $G=G_1\star_{\Gamma_0}G_2$ and this decomposition gives rise a short exact sequence of $\Z[G]$-modules: $$\xymatrix{
0\ar[r]
&\Z[G/\Gamma_0]\ar^-{\alpha}[r]
&\Z[G/G_1]\oplus \Z[G/G_2]\ar^-{\beta}[r]
&\Z\ar[r]
&0.}$$ where $\alpha$ is the map $$\alpha: \Z[G/\Gamma_0]\to \Z[G/G_1]\oplus \Z[G/G_2],\ g\Gamma_0\mapsto (gG_1,gG_2)$$ and $\beta$ is the unique $\Z[G]$-homomorphism $$\beta: \Z[G/G_1]\oplus \Z[G/G_2]\to \Z,\ (G_1,0)\mapsto -1, (0,G_2)\mapsto 1.$$
This short exact sequence of $\Z[G]$-modules gives rise to a long exact sequence in homology. Combining this with the isomorphisms of Shapiro’s lemma, $\ho{r}{G}{\Z[G/H]}\cong\hoz{r}{H}$, gives us the *Mayer-Vietoris exact sequence* of the amalgamated product: $$\xymatrix{
\cdots \ar^-{\delta}[r]
&\hoz{r}{\Gamma_0}\ar^-{\alpha}[r]
&\hoz{r}{G_1}\oplus\hoz{r}{G_2}\ar^-{\beta}[r]
&\hoz{r}{G}\ar^-{\delta}[r]
&\cdots
}$$
The maps $\alpha$ and $\beta$ in this sequence can be described as follows: Let $\iota_1:\Gamma_0\to G_1$ and $\iota_2:\Gamma_0\to G_2$ be the natural inclusions. Then $$\alpha(z)=(\iota_1(z),\iota_2(z))\mbox{ for all } z\in \hoz{r}{\Gamma_0}.$$ Likewise, let $j_1:G_1\to G$ and $j_2:G_2\to G$ be the natural inclusions. Then $$\beta(z_1,z_2)=j_2(z_2)-j_1(z_1)\mbox{ for all } z_1\in\hoz{r}{G_1}, \ z_1\in\hoz{r}{G_1}.$$
The amalgamated product decomposition (\[amal1\]) – i.e. taking the case $A=A_{\mathfrak{p}}$ – also gives rise to a Mayer-Vietoris sequence $$\xymatrix{
\cdots \ar^-{\delta}[r]
&\hoz{r}{\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}}\ar^-{\alpha}[r]
&\hoz{r}{\spl{2}{A_{\mathfrak{p}}}}\oplus\hoz{r}{H(\tilde{\mathfrak{p}})}\ar^-{\beta}[r]
&\hoz{r}{\spl{2}{K}}\ar^-{\delta}[r]
&\cdots
}$$
The connecting homomorphism
---------------------------
As above, let $\mathfrak{p}$ be a prime ideal of the Dedekind Domain $A$ and let $$\delta:\hoz{2}{\spl{2}{K}}\to \hoz{1}{\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}}$$ be the connecting homomorphism in the Mayer-Vietoris sequence associated to the decomposition $$\spl{2}{K}=\spl{2}{A_{\mathfrak{p}}}\star_{\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}}
H(\tilde{\mathfrak{p}})
=\spl{2}{A_{\mathfrak{p}}}\star_{\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}}\spl{2}{A_{\mathfrak{p}}}^{M(\pi)}.$$
\[prop:delta\] Let $\rho:\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}\to
k(\tilde{\mathfrak{p}})^\times$ be the (surjective) map $$\matr{a}{b}{c}{d}\mapsto a\pmod{\tilde{\mathfrak{p}}}.$$ Then the composite homomorphism, $\Delta$ say, $$\xymatrix{
K_2(2,K)\cong \hoz{2}{\spl{2}{K}}\ar^-{\delta}[r]
&\hoz{1}{\tcong{A_{\mathfrak{p}}}{\tilde{\mathfrak{p}}}}\ar^-{\rho}[r]
&\hoz{1}{k(\tilde{\mathfrak{p}})^\times}\cong k(\tilde{\mathfrak{p}})^\times
}$$ is the map $$c(a,b)\mapsto (-1)^{v(a)v(b)}\frac{b^{v(a)}}{a^{v(b)}}\pmod{\tilde{\mathfrak{p}}}.$$
In fact, the isomorphisms in the statement of Proposition \[prop:delta\] are canonical only up to sign. We have made our choices so that the sign is $+1$; but the choice of sign does not materially affect our main results.
Before proving Proposition \[prop:delta\], we require
\[lem:k2gen\] Let $K$ be a field with discrete valuation $v$. Then $K_2(2,K)$ is generated by the set $C_v:=\{ c(x,u)\ |\ v(u)=0, v(x)=1\}$.
Let $D$ be the subgroup of $K_2(2,K)$ generated by $C_v$. Let $a,b\in K^\times$. We must prove that $c(a,b)\in D$.
Since $$c(a,b)=c(b^{-1},a)=c(a^{-1},b^{-1})=c(b,a^{-1})$$ we can assume that $v(a),v(b)\geq 0$.
We will prove the result by induction on $n=v(a)+v(b)\geq 0$.
If $n=0$, then $v(a)=v(b)=0$ and choosing $\pi\in K^\times$ with $v(\pi)=1$ we have $$c(a,b)= c(\pi a,b)^{-1}c(\pi a,b)c(\pi,a)\in D.$$
On the other hand, suppose that $v(a),v(b)>0$. If $0<v(b)\leq v(a)$ then $a=bc$ with $0\leq v(c)<v(a)$ and hence $$c(a,b)=c(bc,b)=c(-c,b)\in D$$ by the inductive hypothesis. An analogous argument applies to the case $0<v(a)<v(b)$.
Since $c(a,b)=c(b^{-1},a)$, we can reduce to the case where $v(b)=0$ and $v(a)\geq 2$. Then let $a=a'\pi$ where $v(\pi)=1$ and $1\leq v(a')<v(a)$. We have $$c(a,b)=c(a'\pi,b)= c(a',\pi b)c(\pi,b)c(a',\pi)^{-1}$$ which lies in $D$ by the induction hypothesis (using the argument for the case $v(a),v(b)>0$ for the first term).
By Lemma \[lem:k2gen\], we must prove that $$\Delta(c(x,u))= u\pmod{\tilde{\mathfrak{p}}}$$ whenever $v(u)=0$, $v(x)=1$.
We note that it is enough to prove that $\Delta(c(x,u^2))=u^2\pmod{\tilde{\mathfrak{p}}}$ whenever $v(u)=0$, $v(x)=1$. For if $u\in K$ is not a square, choose an extension $v'$ of $v$ to $K':=K(\root {} \of {u})$. Then there is a natural map of Mayer Vietoris exact sequences inducing a commutative square $$\xymatrix{
\hoz{2}{\spl{2}{K}}\ar^-{\delta}[r]\ar[d]
&k(v)^\times\ar@{(->}^-{i}[d]\\
\hoz{2}{\spl{2}{K'}}\ar^-{\delta'}[r]
&k(v')^\times
}$$ so that $i(\Delta(c(x,u))=\Delta'(c(x,u))=\bar{u}\in k(v')^\times$ since $u$ is a square in $K'$, and thus $\Delta(c(x,u))=\bar{u}\in k(v)^\times$.
Now, by Corollary \[cor:square\], the symbol $c(x,u^2)\in K(2,K)$ corresponds to the homology class represented by the cycle $$Z:=\left([D(x)|D(u)]-[D(u)|D(x)]\right)\otimes 1 \in {F}_2(G)\otimes_{\Z[G]} \Z$$ where $G=\spl{2}{K}$.
Recall that the Mayer-Vietoris sequence is the long exact homology sequence derived from the short exact sequence of complexes $$0\to F_\bullet(G)\otimes_{\Z[G]}\Z[G/\Gamma_0]\to
F_\bullet(G)\otimes_{\Z[G]}\left(\Z[G/G_1]\oplus\Z[G/G_2]\right)\to
F_\bullet(G)\otimes_{\Z[G]}\Z\to 0.$$
Now the cycle $Z$ lifts to $$\left([D(x)|D(u)]-[D(u)|D(x)]\right)\otimes (1\cdot G_1,0)\in F_2(G)\otimes
\left(\Z[G/G_1]\oplus\Z[G/G_2]\right).$$
Under the boundary map $d_2$, this is sent to $$[D(u)]\otimes (D(x)\cdot G_1-1\cdot G_1,0)\in
F_1(G)\otimes
\left(\Z[G/G_1]\oplus\Z[G/G_2]\right)$$ since $D(u)\in \Gamma_0\subset G_1$.
Now let $$w:=\matr{0}{1}{-1}{0}\in G_1.$$ Then $$w\cdot D(x)=w^{M(x)}\in G_2.$$ Thus $(D(x)\cdot G_1-1\cdot G_1,0)$ is the image of $$w^{-1}\cdot (w^{M(x)}\Gamma_0-\Gamma_0)=D(x)\Gamma_0-w^{-1}\Gamma_0$$ under the map $\alpha:\Z[G/\Gamma_0]\to \Z[G/G_1]\oplus\Z[G/G_2]$.
Thus the homology class $\delta(Z)\in \hoz{1}{\Gamma_0}$ is represented by the cycle $$[D(u)]\otimes (D(x)\Gamma_0-w^{-1}\Gamma_0)
=\left([D(u)]D(x)-[D(u)]w^{-1}\right)\otimes \Gamma_0 \in
F_1(G)\otimes_{\Z[G]}\Z[G/\Gamma_0].$$
This, in turn, is the image of $$\left([D(u)]D(x)-[D(u)]w^{-1}\right)\otimes 1 \in
F_1(G)\otimes_{\Z[\Gamma_0]}\Z$$ under the natural isomorphism $$F_\bullet(G)\otimes_{\Z[\Gamma_0]}\Z\cong F_\bullet(G)\otimes_{\Z[G]}\Z[G/\Gamma_0].$$ For a group $H$ we let $C_\bullet(H)$ denote the right homogeneous resolution of $H$. The isomorphism $F_\bullet(H)\to C_\bullet(H)$ of complexes of right $\Z[H]$-modules is given by $$[h_n|\cdots |h_1]\mapsto (h_n\cdot h_{n-1}\cdots h_1,\ldots, h_1,1).$$
Thus the cycle $\left([D(u)]D(x)-[D(u)]w^{-1}\right)\otimes 1 \in
F_1(G)\otimes_{\Z[\Gamma_0]}\Z$ corresponds to the cycle $$\left((D(ux),D(u))-(D(u)w^{-1},w^{-1})\right)\otimes 1\in C_1(G)\otimes_{\Z[\Gamma_0]}\Z.$$
To construct an augmentation-preserving map of $\Z[\Gamma_0]$-resolutions from $C_\bullet(G)$ to $C_\bullet(\Gamma_0)$, we choose any set-theoretic section $s:G/\Gamma_0\to G$ of the natural surjection $G\to G/\Gamma_0, g\mapsto g\Gamma_0$. For $g\in G$ we let $\bar{g}:=s(g\Gamma_0)^{-1}g\in \Gamma_0$. Then the map $$\tau:C_\bullet(G)\to C_\bullet(\Gamma_0), (g_n,\ldots,g_0)\mapsto (\bar{g}_n,\ldots,\bar{g}_0)$$ is an augmentation preserving map of $\Z[\Gamma_0]$-complexes.
We further specify that we the section $s$ satisfies $$s(D(u)w^{-1}\Gamma_0)=w^{-1}\mbox{ and } s(D(x)\Gamma_0)=D(x)$$ for all $u$ with $v(u)=0$. Then $$\tau\left((D(ux),D(u))-(D(u)w^{-1},w^{-1})\right)=(D(u),1)-(D(u^{-1}),1)\in C_1(\Gamma_0)$$ since $wD(u)w^{-1}=D(u^{-1})$ in $G$.
Finally, the homology class $$\left((D(u),1)-(D(u^{-1}),1)\right)\otimes 1 \in C_1(\Gamma_0)\otimes_{\Z[\Gamma_0]}\Z$$ corresponds to the element $$D(u)\cdot D(u^{-1})^{-1}=D(u^2)\in \Gamma_0/[\Gamma_0,\Gamma_0]$$ under the isomorphism $\hoz{1}{\Gamma_0}\cong \Gamma_0/[\Gamma_0,\Gamma_0]$, and hence maps to $u^2\pmod{\tilde{\mathfrak{p}}}\in k(\tilde{\mathfrak{p}})^\times$ under the map $\rho$.
The abelianization of some congruence subgroups
-----------------------------------------------
\[prop:tcong\] Let $A$ be a ring of $S$-integers in a global field $K$. Suppose that $|S|\geq 2$ and that there exists $\lambda\in A^\times$ such that $\lambda^2-1\in A^\times$ also. Let $\mathfrak{p}$ be a nonzero prime ideal of $A$.
Then the map $\rho:\tcong{A}{\mathfrak{p}}\to k(\mathfrak{p})^\times$ induces an isomorphism $$\hoz{1}{\tcong{A}{\mathfrak{p}}}\cong k(\mathfrak{p})^\times.$$
The map $\rho$ induces a short exact sequence $$\xymatrix{
1\ar[r]
&\ucong{A}{\mathfrak{p}}\ar[r]
& \tcong{A}{\mathfrak{p}}\ar^-{\rho}[r]
&k(\mathfrak{p})^\times\ar[r]
&1
}$$ where $$\ucong{A}{\mathfrak{p}}:=\left\{ \matr{a}{b}{c}{d}\in \spl{2}{A}\ |\
\c,a-1,d-1 \in \mathfrak{p}\right\}=\tilde{\Gamma}(A,\mathfrak{p})$$ in the notation of Theorem \[thm:vl\].
Since $k(\mathfrak{p})^\times$ is an abelian group, it follows that $$[\tcong{A}{\mathfrak{p}},\tcong{A}{\mathfrak{p}}]\subset \ucong{A}{\mathfrak{p}}$$
On the other hand, by Theorem \[thm:vl\], $\ucong{A}{\mathfrak{p}}=
\tilde{\Gamma}(A,\mathfrak{p})$ is generated by elementary matrices $E_{12}(x), x\in A$, $E_{21}(y), y\in \mathfrak{p}$. However, $$E_{12}(x)=[D(\lambda), E_{12}(x/(\lambda^2-1))], E_{21}(y)=[D(\lambda), E_{21}(y/(\lambda^2-1))\in
[\tcong{A}{\mathfrak{p}},\tcong{A}{\mathfrak{p}}].$$
So $[\tcong{A}{\mathfrak{p}},\tcong{A}{\mathfrak{p}}]=\ucong{A}{\mathfrak{p}}$ as required.
For a field $k$ we let $\lsl{2}{k}$ denote the $3$-dimensional vector space of $2\times 2$ trace zero matrices.
\[lem:congindex\] Let $A$ be a Dedekind domain and let $\mathfrak{p}$ be a maximal ideal. Then for any $m\geq 1$ there are natural isomorphisms of groups $$\frac{\tpcong{A}{\mathfrak{p}^n}}{\tpcong{A}{\mathfrak{p}^{n+1}}}\cong
\lsl{2}{k(\mathfrak{p})}\cong \frac{\pcong{A}{\mathfrak{p}^n}}{\pcong{A}{\mathfrak{p}^{n+1}}}$$
From the definitions of $\pcong{A}{\mathfrak{p}^n}$ and $\tpcong{A}{\mathfrak{p}^n}$ we have $$\frac{\tpcong{A}{\mathfrak{p}^n}}{\tpcong{A}{\mathfrak{p}^{n+1}}}\cong
\ker{\spl{2}{A/\mathfrak{p}^{n+1}}\to \spl{2}{A/\mathfrak{p}^{n}}}\cong
\frac{\pcong{A}{\mathfrak{p}^n}}{\pcong{A}{\mathfrak{p}^{n+1}}}.$$
Let $\pi\in \mathfrak{p}\setminus\mathfrak{p}^2$. For any $n\geq 1$, the group $\mathfrak{p}^n/\mathfrak{p}^{n+1}$ is a $1$-dimensional $k(\mathfrak{p})$-vector spaces with basis $\{ \pi^n+\mathfrak{p}^{n+1}\}$.
The required isomorphism $$\lsl{2}{k(\mathfrak{p})}\to
\ker{\spl{2}{A/\mathfrak{p}^{n+1}}\to \spl{2}{A/\mathfrak{p}^{n}}}$$ is then the map $$\matr{\bar{a}}{\bar{b}}{\bar{c}}{\bar{d}}\mapsto \matr{1+a\pi^n}{b\pi^n}{c\pi^n}{1+d\pi^n}$$ where $a,b,c,d\in A$ map to $\bar{a},\bar{b},\bar{c},\bar{d}\in k(\mathfrak{p})$.
\[cor:congindex\] Let $A$ be a Dedekind domain and let $\mathfrak{p}$ be a maximal ideal. Suppose that $k(\mathfrak{p})$ is a finite field with $q$ elements. Then $$[\tpcong{A}{\mathfrak{p}}:\tpcong{A}{\mathfrak{p}^n}]=q^{3(n-1)}=
[\pcong{A}{\mathfrak{p}}:\pcong{A}{\mathfrak{p}^n}]$$ for all $n\geq 1$.
\[lem:crt\] Suppose that $I$ and $J$ are comaximal ideals in $A$; i.e. $I+J=A$. Then the composite map $\pcong{A}{I}\to \spl{2}{A}\to \spl{2}{A/J}$ is surjective.
By the Chinese Remainder Theorem the map $A/IJ\to (A/I)\times (A/J)$, $a\mapsto (a+I,a+J)$ is an isomorphism of rings. It follows that the map $$\spl{2}{A/IJ}\to \spl{2}{A/I}\times \spl{2}{A/J}, \quad X\mod{IJ}\mapsto (X\mod{I},X\mod{J})$$ is an isomorphism of groups and hence that $$\spl{2}{A}\to \spl{2}{A/I}\times \spl{2}{A/J}, \quad X\mapsto (X\mod{I},X\mod{J})$$ is a surjective group homomorphism. This implies the statement of the Lemma.
\[lem:index\] Suppose that $k(\mathfrak{p})$ is a finite field with $q$ elements. We have $$[\spl{2}{A}:\pcong{A}{\mathfrak{p}}]=q(q^2-1)=
[\spl{2}{A}:\tpcong{A}{\mathfrak{p}}]$$
The first equality follows from the isomorphism $$\frac{\spl{2}{A}}{\pcong{A}{\mathfrak{p}}}\cong \spl{2}{k(\mathfrak{p})}.$$
For the second inequality, denote by $C$ the image of the map $$\tpcong{A}{\mathfrak{p}}\to \spl{2}{A}\to \spl{2}{A/\mathfrak{p}^2}.$$
Then $C$ fits into a short exact sequence $$1\to W\to C\to T\to 1$$ where $$T=\left\{ \matr{a}{0}{0}{d}\in \spl{2}{A/\mathfrak{p}^2}\ | a-1,d-1\in \mathfrak{p}\right\}
\cong k(\mathfrak{p})$$ and $$W= \left\{ \matr{1}{b}{0}{1}\in \spl{2}{A/\mathfrak{p}^2}\ \right \}\cong A/\mathfrak{p}^2.$$
It follows that $\card{C}=[\tpcong{A}{\mathfrak{p}}:\pcong{A}{\mathfrak{p}^2}]=q^3$. Since $[\spl{2}{A}:\pcong{A}{\mathfrak{p}^2}]=\card{\spl{2}{A/\mathfrak{p}^2}}=q^4(q^2-1)$, the second equality follows.
\[prop:pcong\] Let $A$ be a ring of $S$-integers in a global field $K$ where $|S|\geq 2$. Let $\mathfrak{p}$ be a nonzero prime ideal and let $p>0$ be the characteristic of the residue field $k(\mathfrak{p})$.
Suppose that $\mathfrak{p}^n=xA$ for some $n\geq 1$ and $x\in A$. Suppose further that there exist $\lambda\in A[1/x]^\times$ such that $\lambda^2-1\in A[1/x]^\times$ also. Then $\hoz{1}{\pcong{A}{\mathfrak{p}}}$ and $\hoz{1}{\tpcong{A}{\mathfrak{p}}}$ are finite abelian $p$-groups.
Let $\Gamma$ denote either $\pcong{A}{\mathfrak{p}}$ or $\pcong{A}{\mathfrak{p}}$. By Proposition 2 of [@serre:sl2] the commutator subgroup $[\Gamma,\Gamma]$ contains a principal congruence subgroup $\pcong{A}{I}$ for some ideal $I$ of $A$. There exists a nonzero ideal $J$ of $A$ such that $I$ factors as $\mathfrak{p}^mJ$ where $J\not\subset\mathfrak{p}$ and $m\geq 1$. Since $\pcong{A}{\mathfrak{p}^{m+1}J}\subset \pcong{A}{\mathfrak{p}^mJ}$, we can suppose without loss that $m\geq 2$.
By definition, $\pcong{A}{I}$ is the kernel of the natural map $\spl{2}{A}\to \spl{2}{A/I}$. This map is surjective since $\spl{2}{A/I}=E_2(A/I)$.
Since $J\not\subset \mathfrak{p}$, it follows that $A=\mathfrak{p}^m+ J$ and hence, by Lemma \[lem:crt\], the map $\pcong{A}{\mathfrak{p}^m}\to \spl{2}{A/J}$ is surjective. Since $\pcong{A}{\mathfrak{p}^m}\subset \Gamma$, it follows that the map $\Gamma\to\spl{2}{A/J}$ is surjective.
However, since $\spl{2}{A/J}=E_2(A/J)$ and since $x+J$ is a unit in $A/J$ the hypotheses of the proposition ensure that all elementary matrices are commutators and hence that $\spl{2}{A/J}$ is a perfect group. It then follows that the natural map $[\Gamma,\Gamma]\to \spl{2}{A/J}$ is surjective.
Thus $[\Gamma,\Gamma]/\pcong{A}{I}$ surjects onto $\spl{2}{A/J}$ and hence $$\card{\spl{2}{A/J}}\mbox{ divides }\left[[\Gamma,\Gamma]:\pcong{A}{I}]\right].$$
It follows that $$[\spl{2}{A}:[\Gamma,\Gamma]]|\card{\spl{2}{A/\mathfrak{p}^m}}=(q^2-1)q^{3m-2}.$$
Since $[\spl{2}{A}:\Gamma]=q(q^2-1)$ by Lemma \[lem:index\] it follows that $\card{\Gamma/[\Gamma,\Gamma]}$ divides $q^{3m-3}$, and so is a power of $p$ as claimed.
The second homology of congruence subgroups
-------------------------------------------
\[lem:borel\] Let $k$ be a finite field of characteristic $p$ and let $M$ be an $\spl{2}{k}$-module. Then, for all $i\geq 0$, the natural map $$\psyl{\ho{i}{\bor(k)}{M}}{p}\to \psyl{\ho{i}{\spl{2}{k}}{M}}{p}$$ is an isomorphism.
As in the proof of [@hut:cplx13 Cor 3.10.2].
\[prop:h2\] Let $A$ be a ring of $S$-integers in a global field $K$ where $|S|\geq 2$. Let $\mathfrak{p}$ be a nonzero prime ideal and let $p>0$ be the characteristic of the residue field $k(\mathfrak{p})$. Suppose that $\mathfrak{p}^m=xA$ for some $m\geq 1$, $x\in A$. Suppose further that there exist $\lambda\in A[1/x]^\times$ such that $\lambda^2-1\in A[1/x]^\times$ also.
Then the natural maps $$\iota_1:\hoz{2}{\tcong{A}{\mathfrak{p}}}\to\hoz{2}{\spl{2}{A}}$$ and $$\iota_2:\hoz{2}{\tcong{A}{\mathfrak{p}}}\to\hoz{2}{H(\mathfrak{p})}$$ are surjective.
Let $k=k(\mathfrak{p})$. There is a commutative diagram of group extensions $$\xymatrix{
1\ar[r]
&\pcong{A}{\mathfrak{p}}\ar[r]\ar^-{\mathrm{id}}[d]
&\tcong{A}{\mathfrak{p}}\ar[r]\ar^-{\iota_1}[d]
&\bor(k)\ar[r]\ar^-{\iota}[d]
&1\\
1\ar[r]
&\pcong{A}{\mathfrak{p}}\ar[r]
&\spl{2}{A}\ar[r]
&\spl{2}{k}\ar[r]
&1\\
}$$ and (using Lemma \[lem:bor’\]) $$\xymatrix{
1\ar[r]
&\tpcong{A}{\mathfrak{p}}\ar[r]\ar^-{\mathrm{id}}[d]
&\tcong{A}{\mathfrak{p}}\ar[r]\ar^-{\iota_2}[d]
&\bor'(k)\ar[r]\ar^-{\iota}[d]
&1\\
1\ar[r]
&\tpcong{A}{\mathfrak{p}}\ar[r]
&H(\mathfrak{p})\ar^-{\gamma_{\pi,1}}[r]
&\spl{2}{k}\ar[r]
&1.\\
}$$
We give the argument for $\iota_1$. The analogous argument for $\iota_2$ is achieved by replacing $\bor(k)$ with $\bor'(k)$.
The top group extension gives rise to a spectral sequence $$E^2_{i,j}(\tcong{A}{\mathfrak{p}})=\ho{i}{\bor(k)}{\hoz{j}{\pcong{A}{\mathfrak{p}}}}\Rightarrow
\hoz{i+j}{\tcong{A}{\mathfrak{p}}}$$ and the lower one gives rise to the spectral sequence $$E^2_{i,j}(\spl{2}{A})=\ho{i}{\spl{2}{k}}{\hoz{j}{\pcong{A}{\mathfrak{p}}}}\Rightarrow
\hoz{i+j}{\spl{2}{A}}.$$ The map of extensions induces a natural map of spectral sequences compatible with the map $\iota_1$ on abutments.
For $H=\tcong{A}{\mathfrak{p}}$ or $\spl{2}{A}$, the image of the edge homomorphism $E^\infty_{0,j}(H)\to \hoz{j}{H}$ is equal to the image of $\hoz{j}{\pcong{A}{\mathfrak{p}}}\to \hoz{j}{H}$. Thus, comparing the $E^\infty$-terms of total degree $2$, we obtain a commutative diagram of the form $$\xymatrix{
\hoz{2}{\pcong{A}{\mathfrak{p}}}\ar[r]\ar^-{\mathrm{id}}[d]
&\hoz{2}{\tcong{A}{\mathfrak{p}}}\ar[r]\ar^-{\iota_1}[d]
&C(\tcong{A}{\mathfrak{p}})\ar[r]\ar[d]
&0\\
\hoz{2}{\pcong{A}{\mathfrak{p}}}\ar[r]
&\hoz{2}{\spl{2}{A}}\ar[r]
&C(\spl{2}{A})\ar[r]
&0\\
}$$ where, for $H=\tcong{A}{\mathfrak{p}}$ or $\spl{2}{A}$, $C(H)$ is a group fitting into an exact sequence $$0\to E^{\infty}_{1,1}(H)\to C(H)\to E^\infty_{2,0}(H)\to 0.$$
Since $\hoz{1}{\pcong{A}{\mathfrak{p}}}$ is a finite abelian $p$-group, it follows from Lemma \[lem:borel\] that the natural maps $$E^2_{i,1}(\tcong{A}{\mathfrak{p}})=\ho{i}{\bor(k)}{\hoz{1}{\pcong{A}{\mathfrak{p}}}}\to
\ho{i}{\spl{2}{k}}{\hoz{1}{\pcong{A}{\mathfrak{p}}}}=E^2_{i,1}(\spl{2}{A})$$ are all isomorphisms.
Since, furthermore, all these groups $E^2_{i,1}$ are finite abelian $p$-groups, it follows that the differentials $$d^2_{i,0}:E^2_{i,0}(H)\to E^2_{i-1,1}(H)$$ factor through $\psyl{E^2_{i,0}(H)}{p}$, for $H=\tcong{A}{\pi}$ or $\spl{2}{A}$.
By Lemma \[lem:borel\] again, we have natural isomorphisms $$\psyl{E^2_{i,0}(\tcong{A}{\mathfrak{p}})}{p}=\psyl{\hoz{i}{\bor(k)}}{p}\cong
\psyl{\hoz{i}{\spl{2}{k}}}{p}=\psyl{E^2_{i,0}(\spl{2}{A})}{p}.$$
Thus, we have $$E^\infty_{1,1}(\tcong{A}{\mathfrak{p}})\cong E^\infty_{1,1}(\spl{2}{A})$$ since $$E^\infty_{1,1}(H)=\coker{d^2:\psyl{E^2_{3,0}(H)}{p}\to E^2_{1,1}(H)}$$ when $H=\tcong{A}{\mathfrak{p}}$ or $\spl{2}{A}$.
Finally, for $H=\tcong{A}{\mathfrak{p}}$ or $\spl{2}{A}$, we have $$E^\infty_{2,0}(H)=\ker{d^2:E^2_{2,0}(H)\to E^2_{0,1}(H)}.$$ But straightforward calculations (see, for example, [@hut:cplx13 Section 3]) show that $$E^2_{2,0}(\tcong{A}{\mathfrak{p}})=\hoz{2}{\bor(k)}= \psyl{\hoz{2}{\bor(k)}}{p}
= \psyl{\hoz{2}{\spl{2}{k}}}{p}= \hoz{2}{\spl{2}{k}}=E^2_{2,0}(\spl{2}{A}).$$
Thus $$E^\infty_{2,0}(\tcong{A}{\mathfrak{p}})\cong E^\infty_{2,0}(\spl{2}{A}).$$
Hence the map $C(\tcong{A}{\mathfrak{p}})
\to C(\spl{2}{A})$ is an isomorphism, and the result follows.
An exact sequence for the second homology of $\mathrm{SL}_2$ of $S$-integers
----------------------------------------------------------------------------
Let $K$ be a global field and let $S\subset T$ be nonempty sets of primes of $K$ containing the infinite primes. Then there is a natural short exact sequence $$\xymatrix{
0\ar[r]
& K_2(\ntr{S})\ar[r]
& K_2(\ntr{T})\ar^-{\sum\tau_{\mathfrak{p}}}[r]
& \oplus_{\mathfrak{p}\in T\setminus S}k(\mathfrak{p})^\times\ar[r]
&0.
}$$
In this section, we demonstrate an analogous exact sequence for $\hoz{2}{\spl{2}{\ntr{S}}}$, at least when $S$ is sufficiently large.
\[thm:mv\] Let $A$ be a ring of $S$-integers in a global field $K$ where $|S|\geq 2$. Let $\mathfrak{p}$ be a nonzero prime ideal and let $p>0$ be the characteristic of the residue field $k(\mathfrak{p})$.
Suppose also that there exists $\lambda\in A^\times$ such that $\lambda^2-1\in A^\times$ also.
Let $x\in A$ and $m\geq 1$ such that$\mathfrak{p}^m=xA$.
Then there is a natural exact sequence $$\xymatrix{
\hoz{2}{\spl{2}{A}}\ar[r]
& \hoz{2}{\spl{2}{A[1/x]}}\ar^-{\delta}[r]
& \hoz{1}{k(\mathfrak{p})^\times}\ar[r]
& 0
}$$
Here the map $\delta$ fits into a commutative diagram $$\xymatrix{
\hoz{2}{\spl{2}{A[1/x]}}\ar^-{\delta}[r]\ar[d]
& \hoz{1}{k(\mathfrak{p})^\times}\ar^-{\cong}[d]\\
\hoz{2}{\mathrm{SL}(K)}=\milk{2}{K}\ar^-{\tau_{\mathfrak{p}}}[r]
&k(\mathfrak{p})^\times\\
}$$ where $\tau_{\mathfrak{p}}:\milk{2}{K}\to k(\mathfrak{p})^\times$ is the tame symbol $$\tau_{\mathfrak{p}}(\{ x,y\})=(-1)^{v(x)v(y)}x^{-v(y)}y^{v(x)}\pmod{\mathfrak{p}}\in k(\mathfrak{p})^\times.$$
By Proposition \[prop:h2\] the map $\iota_2:\hoz{2}{\tcong{A}{\mathfrak{p}}}\to \hoz{2}{H(\mathfrak{p})}$ is surjective.
Thus the Mayer-Vietoris sequence yields the exact sequence $$\xymatrix{
\hoz{2}{\spl{2}{A}}\ar[r]
& \hoz{2}{\spl{2}{A[1/x]}}\ar^-{\delta}[r]
& \hoz{1}{\tcong{A}{\mathfrak{p}}}\ar[r]
& 0.
}$$
The remaining statements of the theorem follow from Proposition \[prop:delta\] and Proposition \[prop:tcong\].
In this proof, the hypothesis that $\lambda,\lambda^2-1\in A^\times$ is needed so that Proposition \[prop:tcong\] is validly applied.
The first part of the proof only requires the weaker condition that $\lambda,\lambda^2-1\in A[1/x]^\times$. For example, taking $A=\Z[1/3]$, $\mathfrak{p}= 2A$, $x=2$ and $\lambda=3$, we obtain an exact sequence $$\hoz{2}{\spl{2}{\Z[1/3]}}\to\hoz{2}{\spl{2}{\Z[1/6]}}\to \hoz{1}{\tcong{\Z[1/3]}{2}}\to 0.$$
In this sequence, $$\hoz{2}{\spl{2}{\Z[1/3]}}\cong \Z\mbox{ and } \hoz{2}{\spl{2}{\Z[1/6]}}\cong \Z\oplus \Z/2$$ by the calculations of Adem-Naffah, [@adem:naffah], and Tuan-Ellis, [@tuanellis].
Thus $$\hoz{1}{\tcong{\Z[1/3]}{2}}\not= 0$$ while $\hoz{1}{k(2)^\times}=\hoz{1}{\F{2}^\times}=0$, so that the conclusion of Proposition \[prop:tcong\] is false in the case $A=\Z[1/3]$ and $\mathfrak{p}=2A$.
Theorem \[thm:mv\] is not valid for more general Dedekind Domains $A$, even when there is a unit $\lambda$ such that $\lambda^2-1$ is also a unit.
For example, let $k$ be an infinite field and let $K=k(t)$, $A=k[t]$, $\mathfrak{p}=tA$, $x=t$. It is shown in [@hut:laurent Theorem 4.1] that the cokernel of the natural map $$\hoz{2}{\spl{2}{A}}\to\hoz{2}{\spl{2}{A[1/t]}}$$ is isomorphic to $\mwk{1}{k}$, the first Milnor-Witt $K$-group of the residue field $k$. It seems reasonable to suppose that this statement should be true for a larger class of Dedekind Domains.
Note that there is a natural surjective map $\mwk{1}{k}\to \milk{1}{k}\cong k^\times$ which is an isomorphism when the field $k$ is finite. However, in general, the kernel of this homomorphism is $I^2(k)$ (see section \[sec:mwk\] above).
\[cor:mv\] Let $K$ be a global field. Let $S$ be a set of primes of $K$ containing the infinite primes. Suppose that $\card{S}\geq 2$ and that $\ntr{S}$ contains a unit $\lambda$ such that $\lambda^2-1$ is also a unit.
Let $T$ be any set of primes containing $S$. Then there is a natural exact sequence $$\hoz{2}{\spl{2}{\ntr{S}}}\to \hoz{2}{\spl{2}{\ntr{T}}}\to \oplus_{\mathfrak{p}\in T\setminus S}
k(\mathfrak{p})^\times\to 0.$$
We proceed by induction on $\card{T\setminus S}$.
The case $\card{T\setminus S}=1$ is just Theorem \[thm:mv\].
The inductive step follows immediately by applying the snake lemma to the commutative diagram with exact rows $$\xymatrix{
&\hoz{2}{\spl{2}{\ntr{T'}}}\ar[r]\ar@{>>}[d]
&\hoz{2}{\spl{2}{\ntr{T}}}\ar[r]\ar[d]
&k(\mathfrak{q})^\times\ar[r]\ar^-{\mathrm{id}}[d]
&0\\
0\ar[r]
&\oplus_{\mathfrak{p}\in T'}k(\mathfrak{p})^\times\ar[r]
&\oplus_{\mathfrak{p}\in T}k(\mathfrak{p})^\times\ar[r]
&k(\mathfrak{q})^\times\ar[r]
&0\\
}$$ where $\mathfrak{q}$ is any element in $T\setminus S$ and $T'=T\setminus\{ \mathfrak{q}\}$.
\[cor:mv2\] Let $K$ be a global field. Let $S$ be a set of primes of $K$ containing the infinite primes. Suppose that $\card{S}\geq 2$ and that $\ntr{S}$ contains a unit $\lambda$ such that $\lambda^2-1$ is also a unit.
Then there is a natural exact sequence $$\hoz{2}{\spl{2}{\ntr{S}}}\to \hoz{2}{\spl{2}{K}}\to
\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\to 0.$$
Since $$K=\lim_{S\subset T}\ntr{T}$$ this follows from Corollary \[cor:mv\] by taking (co)limits.
The main theorem {#sec:main}
================
Let $K$ be a global field. In this section, we use the results above to prove our main theorem which identifies $\hoz{2}{\spl{2}{\ntr{S}}}$ with a certain subgroup of $K_2(2,K)$, which we now describe.
For a prime $\mathfrak{p}$ of $K$, we denote by $T_{\mathfrak{p}}$ the composite $$\xymatrix{
K_2(2,K)\ar@{>>}[r]
&\milk{2}{K}\ar^-{\tau_{\mathfrak{p}}}[r]
&k(\mathfrak{p})^\times
}$$ where $\tau_{\mathfrak{p}}$ is the tame symbol, as above.
When $S$ is a nonempty set of primes of $K$ containing the infinite primes, we set $$\tilde{K}_2(2,\ntr{S}):=\ker{ K_2(2,K)\to \oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times}.$$
We begin by noting that this group is closely related to $K_2(\ntr{S})$:
For any global field $K$ and for any nonempty set $S$ of primes which contains the infinite primes there is a natural exact sequence $$0\to I^3(K)\to \tilde{K}_2(2,\ntr{S})\to K_2(\ntr{S})\to 0.$$
In particular, $\tilde{K}_2(2,\ntr{S})\cong K_2(\ntr{S})$ if $K$ is of positive characteristic or is a totally imaginary number field.
Apply the snake lemma and Corollary \[cor:mwk\] (1) to the map of short exact sequences $$\xymatrix{
0\ar[r]
&\tilde{K}_2(2,\ntr{S})\ar[r]\ar[d]
&K_2(2,K)\ar[r]\ar[d]
&\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\ar^-{\mathrm{id}}[d]\ar[r]
&0\\
0\ar[r]
&K_2(\ntr{S})\ar[r]
&\milk{2}{K}\ar[r]
&\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\ar[r]
&0.\\}$$
The second statement follows from the fact that, for a global field $K$, $I^3(K)\cong \Z^{r(K)}$ where $r(K)$ is the number of distinct real embeddings of $K$.
\[exa:z\] Consider the global field $K=\Q$.
For any set $S$ of prime numbers, we will set $$\Z_S:=\Z[\{ 1/p\}_{p\in S}]=\ntr{S\cup \{\infty\}}.$$
The kernel of the surjective map $$K_2(2,\Q)\to \oplus_{p}\F{p}^\times$$ is an infinite cyclic direct summand with generator $c(-1,-1)$.
It follows that for any set $S$ of prime numbers $$\tilde{K}_2(2,\Z_S)\cong \Z\oplus \left(\oplus_{p\in S}\F{p}^\times\right).$$
More generally, we have the following description of the groups $\tilde{K}_2(2,\ntr{S})$:
For a global field $K$, let $\Omega$ be the set of real embeddings of $K$. For $\sigma\in \Omega$, there is a corresponding homomorphism $$T_\sigma:K_2(K)\to \mu_2,\quad \{ a,b\}\mapsto
\left\{
\begin{array}{ll}
-1, &\mbox{ if } \sgn{\sigma(a)},\sgn{\sigma(b)}<0\\
1,&\mbox{ otherwise }
\end{array}
\right.$$
Let $$K_2(K)_+:=\ker{\oplus_{\sigma\in\Omega}T_\sigma:K_2(K)\to\mu_2^\Omega}$$ and let $K_2(\ntr{S})_+=K_2(\ntr{S})\cap K_2(K)_+$.
\[lem:tilde\] Let $K$ be a global field. Let $S$ be a nonempty set of primes of $K$ including the infinite primes. Then $$\tilde{K}_2(2,\ntr{S})\cong K_2(\ntr{S})_+\oplus \Z^{\Omega}.$$
By classical quadratic form theory, the group $I^n(\R)$ is infinite cyclic with generator $\pf{-1}^n=(-2)^{n-1}\pf{-1}$.
It is shown in [@hut:laurent] that for a global field $K$ the natural surjective map $$K_2(2,K)\to I^2(\R)^{\Omega}\cong \Z^{\Omega},\quad c(u,v)\mapsto
(\pf{\sgn{\sigma(u)}}\pf{\sgn{\sigma(u)}})_{\sigma\in \Omega}$$ has kernel isomorphic to $K_2(K)_+$, where this isomorphism is realised by restricting the natural map $K_2(2,K)\to K_2(K)$. Furthermore, the composite map $I^3(K)\to K_2(2,K)\to I^2(\R)^{\Omega}$ induces an isomorphism $$I^3(K)\cong I^3(\R)^{\Omega}=2\cdot(I^2(\R)^{\Omega}).$$
Since $I^3(K)\subset \tilde{K}_2(2,\ntr{S})$, the image of the map $$\tilde{K}_2(2,\ntr{S})\to I^2(\R)^{\Omega}\cong \Z^{\Omega}$$ contains a full sublattice.
On the other hand, the kernel of this map is isomorphic – via the map $K_2(2,K)\to K_2(K)$ – to $K_2(\ntr{S})\cap K_2(K)_+$.
It is natural to ask, of course, about the relation between $\tilde{K}_2(2,\ntr{S})$ and $K_2(2,\ntr{S})$.
It is a theorem of van der Kallen ([@vanderkallen:stab]) that when $K$ is a global field and when $S$ contains all infinite places and $\card{S}\geq 2$ then the stabilization map $$K_2(2,\ntr{S})\to K_2(\ntr{S})$$ is always surjective.
We deduce:
Let $K$ be a global field and let $S$ be a nonempty set of primes of $K$ containing the infinite primes. Then the image of the natural map $K_2(2,\ntr{S})\to K_2(2,K)\cong K_2(2,K)$ lies in $\tilde{K}_2(2,\ntr{S})$.
Furthermore, when $\card{S}\geq 2$, and when there exist units $u_\sigma\in \ntr{S}^\times$, $\sigma\in \Omega$ satisfying $$\sgn{\tau(u_\sigma)}=(-1)^{\delta_{\sigma,\tau}},$$ the resulting natural map $K_2(2,\ntr{S})\to
\tilde{K}_2(2,\ntr{S})$ is surjective; i.e. the image of the map $K_2(2,\ntr{S})\to K_2(2,K)$ is precisely $\tilde{K}_2(2,\ntr{S})$.
The diagram $$\xymatrix{
&{K}_2(2,\ntr{S})\ar[r]\ar[d]
&K_2(2,K)\ar[r]\ar[d]
&\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\ar^-{\mathrm{id}}[d]
&\\
0\ar[r]
&K_2(\ntr{S})\ar[r]
&\milk{2}{K}\ar[r]
&\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\ar[r]
&0\\}$$ commutes.
Our hypothesis on units ensures that the map $$\tilde{K}_2(2,\ntr{S})\to K_2(2,K)\to I^2(\R)^{\Omega}$$ is surjective.
Since we also have $$K_2(\ntr{S})_+\subset K_2(\ntr{S})\subset \tilde{K}_2(2,\ntr{S})$$ by the result of van der Kallen, the second statement follows.
One would expect that the resulting map $K_2(2,\ntr{S})\to \tilde{K}_2(2,\ntr{S})$ is very often an isomorphism. It seems to be difficult, however, to prove this in any given instance. In the case $K=\Q$, Jun Morita, [@morita:zs Theorems 2,3] has proved:
\[thm:morita\] Let $S$ be any of the following sets of primes numbers:
$S=\{p_1,\ldots, p_n\}$, the set of the first $n$ successive prime numbers, or $S$ is one of $\{2,5\}$, $\{ 2,3,7\}$, $\{ 2,3,11\}$, $\{ 2,3,5,11\}$, $\{ 2,3,13\}$, $\{ 2,3,7,13\}$, $\{2,3,17\}$, $\{ 2,3,5,19\}$.
Then $K_2(2,\Z_S)$ is central in $\st{2}{\Z_S}$ and the natural map $$K_2(2,\Z_S)\to \tilde{K}_2(2,\Z_S)\cong \Z\oplus \left(\oplus_{p\in S}\F{p}^\times\right)$$ is an isomorphism.
\[lem:k2o\] Let $K$ be a global field and let $S$ be a nonempty set of primes of $K$ containing the infinite primes. Then the image of the natural map $$\xymatrix{
\hoz{2}{\spl{2}{\ntr{S}}}\to \hoz{2}{\spl{2}{K}}\ar^-{\cong}[r]
&K_2(2,K)
}$$ lies in $\tilde{K}_2(2,\ntr{S})$.
The diagram $$\xymatrix{
\hoz{2}{\spl{2}{\ntr{S}}}\ar[r]\ar[d]
&\hoz{2}{\mathrm{SL}(\ntr{S})}\ar[d]\\
\hoz{2}{\spl{2}{K}}\ar[r]\ar^-{\cong}[d]
&\hoz{2}{\mathrm{SL}(K)}\ar^-{\cong}[d]\\
K_2(2,K)\ar[r]
&\milk{2}{K}\\
}$$ commutes. But $\hoz{2}{\mathrm{SL}(\ntr{S})}\cong K_2(\ntr{S})$ and the natural map $K_2(\ntr{S})\to K_2(K)=\milk{2}{K}$ induces an isomorphism $$K_2(\ntr{S})\cong\ker{\milk{2}{k}\to \oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times}.$$
If $K$ is a global field and and if $S$ is a nonempty set of primes containing the infinite primes we will let $$\mathcal{K}_S:=\ker{\hoz{2}{\spl{2}{\ntr{S}}}\to \hoz{2}{\spl{2}{K}}}.$$
Note that $$\mathcal{K}_S:=\ker{\hoz{2}{\spl{2}{\ntr{S}}}\to \tilde{K}_2(2,\ntr{S})}$$ since $\tilde{K}_2(2,\ntr{S})\subset K_2(2,K)\cong \hoz{2}{\spl{2}{K}}$.
In general, the kernels $\mathcal{K}_S$ can be arbitrarily large, even in the case $K=\Q$:
The calculations of Adem-Naffah, [@adem:naffah], show that the ranks of the groups $\hoz{2}{\spl{2}{\Z[1/p]}}$ grow with linearly $p$ when $p$ is a prime number. On the other hand, the rank of $\hoz{2}{\spl{2}{\Q}}$ is $1$.
\[lem:h2o\] Let $K$ be a global field. Let $S$ be a set of primes of $K$ containing the infinite primes. Suppose that $\card{S}\geq 2$ and that $\ntr{S}$ contains a unit $\lambda$ such that $\lambda^2-1$ is also a unit.
Then
1. The natural map $$\hoz{2}{\spl{2}{\ntr{S}}}\to\tilde{K}_2(2,\ntr{S})$$ is surjective.
2. If $T\supset S$, then the natural map $\mathcal{K}_S\to\mathcal{K}_T$ is surjective.
1. By Corollary \[cor:mv2\], we have a commutative diagram with exact rows $$\xymatrix{
&\hoz{2}{\spl{2}{\ntr{S}}}\ar[r]\ar[d]
&\hoz{2}{\spl{2}{K}}\ar[r]\ar^-{\cong}[d]
&\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\ar[r]\ar^-{\mathrm{id}}[d]
&0\\
0\ar[r]
&\tilde{K}_2(2,\ntr{S})\ar[r]
&K_2(2,K)\ar[r]
&\oplus_{\mathfrak{p}\not\in S}k(\mathfrak{p})^\times\ar[r]
&0.\\
}$$
2. Apply the snake lemma to the diagram $$\xymatrix{
&\hoz{2}{\spl{2}{\ntr{S}}}\ar[r]\ar[d]
&\hoz{2}{\spl{2}{\ntr{T}}}\ar[r]\ar[d]
&\oplus_{\mathfrak{p}\in T\setminus S}k(\mathfrak{p})^\times\ar[r]\ar^-{\mathrm{id}}[d]
&0\\
0\ar[r]
&\tilde{K}_2(2,\ntr{S})\ar[r]
&\tilde{K}_2(2,\ntr{T})\ar[r]
&\oplus_{\mathfrak{p}\in T\setminus S}k(\mathfrak{p})^\times\ar[r]
&0.\\
}$$
Note, on the other hand, that the map $$0=\hoz{2}{\spl{2}{\Z}} \to \tilde{K}_2(2,\Z)\cong K_2(2,\Z)\cong \Z$$ cannot be surjective.
\[thm:s\] Let $K$ be a global field.
1. There exists a finite set $S$ of primes of $K$ satisfying
1. $S$ contains all infinite primes and $\card{S}\geq 2$.
2. There exists a unit $\lambda$ of $\ntr{S}$ such that $\lambda^2-1$ is also a unit.
3. The natural map $\hoz{2}{\spl{2}{\ntr{S}}}\to \tilde{K}_2(2,\ntr{S})$ is an isomorphism.
2. If $T$ is any set of primes containing $S$ then $\hoz{2}{\spl{2}{\ntr{T}}}\cong \tilde{K}_2(2,\ntr{T})$; i.e. there is a natural short exact sequence $$\xymatrix{
0\ar[r]
& \hoz{2}{\spl{2}{\ntr{T}}}\ar[r]
& K_2(2,K)\ar^-{\sum{T_{\mathfrak{p}}}}[r]
&\oplus_{\mathfrak{p}\not\in T}k(\mathfrak{p})^\times\ar[r]
& 0.\\
}$$
1. Let $S_0$ be any set of primes satisfying (a) and (b). Since $\hoz{2}{\spl{2}{\ntr{S}}}$ is a finitely-generated abelian group, so also is $\mathcal{K}_{S_0}$. Since $\hoz{2}{\spl{2}{K}}=\lim_T\hoz{2}{\spl{2}{\ntr{T}}}$, the limit being taken over finite sets $T$ of primes, it follows that there is a finite set of primes $S$ containing $S_0$ for which $$\mathcal{K}_{S_0}=\ker{\hoz{2}{\spl{2}{\ntr{S_0}}}\to\hoz{2}{\spl{2}{\ntr{S}}}}.$$
By Lemma \[lem:h2o\] it follows that $\mathcal{K}_S=0$ and hence that $\hoz{2}{\spl{2}{\ntr{S}}}\cong \tilde{K}_2(2,\ntr{S})$ as required.
2. This is immediate from Lemma \[lem:h2o\].
\[lem:z\] Let $K=\Q$ and let $S=\{ 2,3,\infty\}$. Then $S$ satisfies conditions (a)-(c) of Theorem \[thm:s\] (1).
The set $S=\{ 2,3,\infty\}$ clearly satisfies conditions (a) and (b).
Observe that $$\ntr{S}=\Z_{\{ 2,3\}}=\Z\left[ \frac{1}{2},\frac{1}{3}\right]=\Z\left[\frac{1}{6}\right].$$
By Lemma \[lem:h2o\], the natural map $$\hoz{2}{\spl{2}{\Z[1/6]}}\to \tilde{K}_2(2,\Z[1/6])
\cong \Z\oplus \F{3}^\times$$ (see Example \[exa:z\]) is surjective.
On the other hand, the calculations of Tuan and Ellis, [@tuanellis], show that $$\hoz{2}{\spl{2}{\Z[1/6]}}\cong \Z\oplus \Z/2\Z.$$
It follows that the natural map above is an isomorphism.
In view of Theorem\[thm:s\] (2) and Example \[exa:z\], we immediately deduce:
\[thm:z\] Let $T$ be any set of prime numbers containing $2, 3$. Then there is an isomorphism $$\hoz{2}{\spl{2}{\Z_T}}\cong \Z\oplus\left(\oplus_{p\in T}\F{p}^\times\right).$$
In particular, if $m\in \Z$ and if $6|m$ then $$\hoz{2}{\spl{2}{\Z[1/m]}}\cong \Z\oplus\left(\oplus_{p|m}\F{p}^\times\right).$$
Combining this with Morita’s Theorem (\[thm:morita\]) we deduce:
\[prop:morita\] Let $S$ be any of the following sets of primes numbers:
$S=\{p_1,\ldots, p_n\}$, the set of the first $n$ successive prime numbers, or $S$ is one of $\{ 2,3,7\}$, $\{ 2,3,11\}$, $\{ 2,3,5,11\}$, $\{ 2,3,13\}$, $\{ 2,3,7,13\}$, $\{2,3,17\}$, $\{ 2,3,5,19\}$.
Then the natural map $$\hoz{2}{\spl{2}{\Z_{S}}}\to K_2(2,\Z_{S})$$ is an isomorphism and $$1\to K_2(2,\Z_{S})\to \st{2}{\Z_{S}}\to \spl{2}{\Z_{S}}\to 1$$ is the universal central extension of $\spl{2}{\Z_{S}}$.
Since $K_2(2,\Z_S)$ is central in $\st{2}{\Z_S}$, there is a natural homomorphism $\hoz{2}{\spl{2}{\Z_{S}}}\to K_2(2,\Z_{S})$ through which the map $\hoz{2}{\spl{2}{\Z_{S}}}\to K_2(2,\Q)$ factors.
Since $\hoz{2}{\spl{2}{\Z_{S}}}$ and $K_2(2,\Z_{S})$ are both isomorphic to $\tilde{K}_2(2,\Z_S)\subset K_2(2,\Q)$, the result follows immediately.
Some $2$-dimensional homology classes {#sec:classes}
=====================================
In this section we construct explicit cycles in the bar resolution of $\spl{2}{A}$ which represent homology classes in $\hoz{2}{\spl{2}{A}}$. We show that these classes map to the symbols $c(u,v)\in K_2(2,A)$, when $A$ is a field.
The homology classes $C(a,b)$
-----------------------------
Let $A$ be a commutative ring and let $a\in A^\times$. We define the following elements of $\spl{2}{A}$: $$w:=\matr{0}{1}{-1}{0},\quad G_a:=\matr{0}{-1}{1}{a+a^{-1}},\quad
H_a:= E_{21}(a)=\matr{1}{0}{a}{1}.$$
Note that $$wG_a= \matr{1}{a+a^{-1}}{0}{1}=E_{12}(a+a^{-1}).$$ We also define $$R_a:=H_aG_aH^{-1}_a=H_{a}G_aH_{-a}=\matr{a}{-1}{0}{a^{-1}}.$$ Thus, by definition, $$H_aG_a=R_aH_a=\matr{0}{-1}{1}{a^{-1}}.$$
Let $$\Theta_a:= [H_a|G_a]-[R_a|H_a]+[w^{-1}|wG_a]\in \bar{F}_2(\spl{2}{A})=\bar{F}_2.$$
Then $$d_2(\Theta_a)= [w^{-1}]+[wG_a]-[R_a]
\in \bar{F}_1.$$
Now let $a,b\in A^\times$. Then $$\begin{aligned}
d_2(\Theta_{ab}-\Theta_a-\Theta_b+\Theta_1)=
\left( [R_a]+[R_b]-[R_{ab}]\right)+\left( [wG_{ab}]-[wG_a]-[wG_b]+[wG_1]-[R_1]\right).\end{aligned}$$
Now $$[R_a]+[R_b]=[R_aR_b]+d_2\left([R_a|R_b]\right)$$ and $$[R_{ab}]= [R_aR_b]+[(R_aR_b)^{-1}]-d_2\left([R_aR_b|(R_aR_b)^{-1}(R_{ab})]\right).$$
Hence $$[R_a]+[R_b]-[R_{ab}]=-[(R_aR_b)^{-1}R_{ab}]+d_2\left([R_a|R_b]+[R_aR_b|(R_aR_b)^{-1}(R_{ab})]\right)$$ where $$(R_aR_b)^{-1}R_{ab}=\matr{1}{(ab)^{-1}(a+b^{-1}-1)}{0}{1}=E_{12}((ab)^{-1}(a+b^{-1}-1)).$$
Putting this together, we deduce $$\begin{aligned}
&&d_2(\Theta_{ab}-\Theta_a-\Theta_b+\Theta_1 - [R_a|R_b]-[R_aR_b|(R_aR_b)^{-1}(R_{ab})])\\
&=& [wG_{ab}]-[wG_a]-[wG_b]+[wG_1]-[R_1]+[(R_aR_b)^{-1}(R_{ab})]\\
&=& [E_{12}(ab+(ab)^{-1})]-[E_{12}(a+a^{-1})]-[E_{12}(b+b^{-1})]\\
&&+[E_{12}(2)]-[E_{12}(-1)]+
[E_{12}((ab)^{-1}(a+b^{-1}-1))].\end{aligned}$$
Now suppose that there exists $\lambda\in A^\times$ such that $\lambda^2-1\in A^\times$. Let $$D(\lambda):=\matr{\lambda}{0}{0}{\lambda^{-1}}\in\spl{2}{A}.$$
Recall that for any $x\in A$ $$D(\lambda)E_{12}(x)D(\lambda)^{-1}= E_{12}(\lambda^2x)$$ and hence for any $x\in A$ we have $$\begin{aligned}
E_{12}(x)&=& D(\lambda)E_{12}(x')D(\lambda)^{-1}E_{12}(x')^{-1}\\
&=&D(\lambda)E_{12}(x')(E_{12}(x')D(\lambda))^{-1}= [D(\lambda),E_{12}(x')].\end{aligned}$$ where $x':= x/(\lambda^2-1)$.
Now if $G$ is any group and if $g,h\in G$ then $$D_2([(gh)(hg)^{-1}|hg]-[g|h]+[h|g])= [(gh)(hg)^{-1}]=\left[ [g,h]\right].$$
Thus, we define $$\Psi_x=\Psi_{x,\lambda}:=\left[E_{12}(x)|E_{12}(x')D(\lambda)\right]-\left[ D(\lambda)| E_{12}(x')\right]
+\left[ E_{12}(x')|D(\lambda)\right]\in \bar{F}_2.$$ By the preceeding remarks, we have $d_2(\Psi_{x,\lambda})=[E_{12}(x)]\in \bar{F}_1$ for any $x\in A$.
From all of these calculations we deduce:
\[prop:fab\] Let $A$ be a commutative ring. Suppose that there exists $\lambda\in A^\times$ such that $\lambda^2-1\in A^\times$. Let $a,b\in A^\times$. Then $$\begin{aligned}
F(a,b)_\lambda:&=& [R_a|R_b]+[R_aR_b|(R_aR_b)^{-1}(R_{ab})]+\Theta_a+\Theta_b-\Theta_{ab}-\Theta_1 \\
&+&\Psi_{ab+(ab)^{-1}}-\Psi_{a+a^{-1}}-\Psi_{b+b^{-1}}+\Psi_{2}-\Psi_{-1}+\Psi_{(ab)^{-1}(a+b^{-1}-1)}\in \bar{F}_2 \end{aligned}$$ is a cycle, representing a homology class $C(a,b)_{\lambda}\in \hoz{2}{\spl{2}{A}}$.
The cycles $F(a,b)_{\lambda}$ are clearly functorial in the sense that if $\psi:A\to B$ is a homomorphism of commutative rings and if $a,b,\lambda,\lambda^2-1\in A^\times$ then $$\psi_*(F(a,b)_{\lambda})=F(\psi(a),\psi(b))_{\psi(\lambda)}\in \bar{F}_2(\spl{2}{B}).$$
More generally, suppose that $\Lambda=(\lambda_1,\ldots, \lambda_n,b_1,\ldots, b_n)\in (A^\times)^n\times (A^n)$ satisfies $$\sum_{i=1}^n(\lambda_i^2-1)b_i=1$$
Then for any $x\in A$ $$E_{12}(x)=
\prod_i[D(\lambda_i),E_{12}(b_ix)].$$ by the proof of Proposition \[prop:e2\].
Since $$[\prod_{i=1}^nc_i]=\sum_{i=1}^n[c_i]-d_2\left(\sum_{k=1}^{n-1}[c_1\cdots c_k|c_{k+1}]\right)$$ in $\bar{F}_1(A)$, we can easily write down an element $\Psi_{x,\Lambda}\in \bar{F}_2(A)$ satisfying $d_2(\Psi_{x,\Lambda})=[E_{12}(x)]$ and thus construct cycles $F(a,b)_{\Lambda}$.
Specialising to the case $a=b=-1$ we obtain: $$F(-1,-1)_{\lambda}=[R_{-1}|R_{-1}]+[E_{12}(2)|E_{12}(-3)]+\Psi_{-3}-\Psi_{-1}\\
+2( \Theta_{-1}-\Theta_1+\Psi_{2}-\Psi_{-2}).$$
As we will see, when $A$ is a field with at least four elements, the homology class $C(a,b)_{\lambda}$ does not depend on the choice of $\lambda$. In fact, this is the case for many commutative rings. For example, we have:
\[lem:indep\] Let $A$ be a commutative ring. Suppose there exists $n\in \Z$ such that $n,n^4-1\in A^\times$.
Then, for any $a,b\in A^\times$, the homology class $C(a,b)_{\lambda}\in \hoz{2}{\spl{2}{A}}$ is independent of the choice of $\lambda$.
Suppose that $\lambda,\mu\in A^\times$ satisfy the condition $\lambda^2-1,\mu^2-1
\in A^\times$.
Let $a,b\in A^\times$. Note that $F(a,b)_{\lambda}-F(a,b)_\mu$ is a sum or difference of terms of the form $\Psi_{x,\lambda}-\Psi_{x,\mu}$, $x\in A$. We will show that each such term is a boundary.
We begin by observing that, for any $x\in A$, the elements $\Psi_{x,\lambda}$ and $\Psi_{x,\mu}$ lie in $\bar{F}_2(\bor)$ where $$\bor:= \left\{\matr{u}{y}{0}{u^{-1}}\in\spl{2}{A}\ |\ u\in A^\times\right\}$$ is the subgroup of upper-triangular matrices in $\spl{2}{A}$.
Note that $$d_2(\Psi_{x,\lambda}-\Psi_{x,\mu})=[E_{12}(x)]-[E_{12}(x)]=0$$ so that $\Psi_{x,\lambda}-\Psi_{x,\mu}$ represents a homology class in $\hoz{2}{\bor}$. We will show that it represents the trivial class.
Let $T:= \{ D(u)\ |\ u\in A^\times \}$ be the group of diagonal matrices and let $\pi:\bor\to T$ be the natural surjective homomorphism sending $D(u)E_{12}(z)$ to $D(u)$. Then $$U:=\ker{\pi}=\{ E_{12}(y)\ |\ y\in A\}$$ is the group of unipotent matrices.
We have $T\cong A^\times$ via $D(u)\leftrightarrow u$ and $U\cong A$, via $E_{12}(x)\leftrightarrow x$.
Note that $$\begin{aligned}
\pi(\Psi_{x,\lambda})&=& \pi\left( \left[E_{12}(x)|E_{12}(x')D(\lambda)\right]-
\left[ D(\lambda)| E_{12}(x')\right]\right)\\
&=& [I|D(\lambda)]-[D(\lambda)|I]+[I|D(\lambda)]\in \bar{F}_2(T).\end{aligned}$$ Since $$d_3([X|I|I]=[I|I]-[I|X]\mbox{ and } d_3([I|I|X])=[X|I]-[I|I]$$ it follows easily that $\pi(\Psi_{x,\lambda}-\Psi_{x,\mu})\in d_3(\bar{F}_3(T))$. Thus $\pi(\Psi_{x,\lambda}-\Psi_{x,\mu})$ represents the trivial homology class in $\hoz{2}{T}$.
To conclude, we will show that our hypotheses are enough to ensure that $\pi$ induces an isomorphism $\hoz{2}{\bor}\cong \hoz{2}{T}$.
We consider the Hochschild-Serre spectral sequence associated to the short exact sequence $$1\to U\to \bor \to T\to 1.$$
This has the form $$E^2_{i,j}=\ho{i}{T}{\hoz{j}{U}}\Rightarrow \hoz{i+j}{\bor}$$ $D(u)\in T$ acts by conjugation on $U\cong A$ as multiplication by $u^2$. Thus the induced action of $D(u)$ on $\hoz{2}{U}\cong U\bigwedge_{\Z}U\cong A\bigwedge_{\Z}A$ is multiplication by $u^4$.
In particular, $D(n)$ acts as multiplication by $n^2$ on $\hoz{1}{U}$, and as multiplication by $n^4$ on $\hoz{2}{U}$.
Since $T$ is abelian, and since $n^2-1$, $n^4-1$ are units in $A$, it follows from the “centre kills” argument that $\ho{i}{T}{\hoz{j}{U}}=0$ for $1\leq j\leq 2$.
Thus, from the spectral sequence, the map $\pi$ induces an isomorphism $\hoz{n}{\bor}\cong\hoz{n}{T}$ for $n\leq 2$.
Since $2^4-1=3\cdot 5$, the condition of the Lemma \[lem:indep\] is satisfied by any ring in which $2,3$ and $5$ are units.
A variation
-----------
We describe a slightly more compact $2$-cycle $\tilde{F}(a,b)_{\lambda}$ in the case where $a^2-1, b^2-1$ and $(ab)^2-1$ are all units in $A$.
Suppose that $a\in A$ is a unit such that $a^2-1\in A^\times$ also. Let $$\tilde{H}_a=\matr{\frac{1}{1-a}}{\frac{a}{1-a}}{\frac{a}{1+a}}{\frac{1}{1+a}}\in \spl{2}{A}.$$
Then $$\tilde{H}_aG_a\tilde{H}_a^{-1}=\matr{a}{0}{0}{a^{-1}}=D(a).$$
Thus if we let $$\tilde{\Theta}_a:=[\tilde{H}_a|G_a]-[D(a)|\tilde{H}_a]+[w^{-1}|wG_a]\in \bar{F}_2$$ then $$d_2(\tilde{\Theta}_a)=[w^{-1}]+[wG_a]-[D(a)].$$
If $a^2-1, b^2-1, (ab)^2-1\in A^\times$ then $$\begin{aligned}
d_2(\tilde{\Theta}_a+\tilde{\Theta}_b-\tilde{\Theta}_{ab}-\Theta_1)&=& [D(ab)]-[D(a)]-[D(b)]\\
&&+[E_{12}(a+a^{-1})]+[E_{12}(b+b^{-1})]-[E_{12}(ab+(ab)^{-1})]+[E_{12}(-1)]-[E_{12}(2)]\\
&=& d_2(-[D(a)|D(b)]+\Psi_{a+a^{-1}}+\Psi_{b+b^{-1}}-\Psi_{ab+(ab)^{-1}}+\Psi_{-1}-\Psi_2).\\\end{aligned}$$
We deduce:
\[prop:var\] If $a,b,\lambda,a^2-1,b^2-1,(ab)^2-1,\lambda^2-1\in A^\times$ then $$\begin{aligned}
\tilde{F}(a,b)_\lambda:= [D(a)|D(b)]+\tilde{\Theta}_a+\tilde{\Theta}_b-\tilde{\Theta}_{ab}-\Theta_1
+\Psi_{ab+(ab)^{-1}}-\Psi_{a+a^{-1}}-\Psi_{b+b^{-1}}+\Psi_{2}-\Psi_{-1}\end{aligned}$$ is a cycle representing a homology class $\tilde{C}(a,b)_\lambda\in \hoz{2}{\spl{2}{A}}$.
Symbols as homology classes
---------------------------
In this section, the map of sets $s:\spl{2}{F}\to\st{2}{F}$ and the homomorphism $\bar{f}:\hoz{2}{\spl{2}{F}}\to K_2(2,F)$ are those described in section \[sec:map\] above.
\[thm:symb\] Let $F$ be a field with at least four elements. Let $\lambda\in F^\times\setminus \{ \pm 1\}$.
1. Let $a,b\in F^\times$. Then $$\bar{f}(C(a,b)_{\lambda})=c(a,b).$$
2. Suppose further that $a,b,ab\not\in \{ \pm 1\}$. Then $$\bar{f}(\tilde{C}(a,b)_{\lambda})=c(a,b).$$
We begin by noting that, by Lemma \[lem:f\], we have $$\bar{f}(\Psi_x)=1 \forall x\in F\mbox{ and } \bar{f}([R_aR_b|(R_aR_b)^{-1}(R_{ab})])=1$$ since $c(1,v)=c(u,1)=1$ in $K_2(2,F)$.
Also, by Lemma \[lem:f\], $$\bar{f}([R_a|R_b])=\bar{f}([D(a)|D(b)]=c(a,b).$$
1. For any $u\in F^\times$ $$\begin{aligned}
\bar{f}([H_u|G_u])&=& s(H_u)s(G_u)s(H_uG_u)^{-1}\\
&=& x_{21}(u)\cdot w_{12}(-1)x_{12}(u+u^{-1})\cdot x_{12}(-u^{-1})w_{12}(1)\\
&=&x_{21}(u)\cdot (w_{12}(-1)x_{12}(u)w_{12}(1))\\
&=&x_{21}(u)x_{12}(u)^{w_{12}(1)}\\
&=&x_{21}(u)x_{21}(-u)=1\mbox{\quad by Lemma \ref{lem:conjw}}.\\ \end{aligned}$$ and $$\begin{aligned}
\bar{f}([w^{-1}|wG_u])&=& s(w^{-1})s(wG_u)s(G_u)^{-1}\\
&=& w_{12}(-1)x_{12}(u+u^{-1})\cdot\left(w_{12}(-1)x_{12}(u+u^{-1})\right)^{-1}=1.\end{aligned}$$
Furthermore $$\begin{aligned}
\bar{f}([R_u|H_u])&=& s(R_u)s(H_u)s(R_uH_u)^{-1}\\
&=&x_{12}(-u)h_{12}(u)x_{21}(u)x_{12}(-u^{-1})w_{12}(1)\\
&=&h_{12}(u)x_{12}(-u^{-1}) x_{21}(u)x_{12}(-u^{-1})w_{12}(1)\mbox{ since $x_{12}(-u)^{h_{12}(u)}
=x_{12}(-u^{-1})$}\\
&=& h_{12}(u)w_{12}(-u^{-1})w_{12}(1).\\
\end{aligned}$$
Now $$w_{12}(-u^{-1})w_{12}(1)=w_{12}(-u^{-1})w_{12}(-1)w_{12}(-1)^{-1}w_{12}(-1)^{-1}
=h_{12}(-u^{-1})h_{12}(-1)^{-1}$$ and hence $$\bar{f}([R_u|H_u])=c(u,-u^{-1})=c(-u,u)=1.$$
Thus $$\bar{f}(\Theta_u)=1$$ for all units $u$.
Putting all of this together gives $\bar{f}(F(a,b)_\lambda)=c(a,b)$ as required.
2. We must show that $\bar{f}(\tilde{\Theta}_a)=1$ whenever $a,a^2-1\in F^\times$.
As above, we have $\bar{f}([w^{-1}|wG_a])=1$.
Now, $$\begin{aligned}
s(D(a))=h_{12}(a),\quad s(\tilde{H}_a)=x_{12}\left(\frac{1+a}{a(1-a)}\right)
w_{12}\left(\frac{1+a}{-a}\right)x_{12}(a^{-1}),\\
\mbox{ and }\quad
s(D(a)\tilde{H}_a)=x_{12}\left(\frac{a(1+a)}{1-a}\right)w_{12}(-(1+a))x_{12}(a^{-1}).\end{aligned}$$ Thus $$\begin{aligned}
\bar{f}([D(a)|\tilde{H}_a])&=& s(D(a))s(\tilde{H}_a)s(D(a)\tilde{H}_a)^{-1}\\
&=&h_{12}(a)x_{12}\left(\frac{1+a}{a(1-a)}\right)
w_{12}\left(\frac{1+a}{-a}\right)x_{12}(a^{-1})x_{12}(-a^{-1})w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=&h_{12}(a)x_{12}\left(\frac{1+a}{a(1-a)}\right)
w_{12}\left(\frac{1+a}{-a}\right)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=&h_{12}(a)w_{12}\left(\frac{1+a}{-a}\right)x_{21}\left(\frac{-a}{1-a^2}\right)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\\end{aligned}$$ using $$x_{12}\left(\frac{1+a}{a(1-a)}\right)^{
w_{12}\left(\frac{1+a}{-a}\right)}= x_{21}\left(\frac{-a}{1-a^2}\right).$$
Since, by Lemma \[lem:conjw\], $$x_{21}\left(\frac{-a}{1-a^2}\right)^{w_{12}(1+a)}= x_{12}\left(\frac{a(1+a)}{1-a}\right),$$ this gives $$\begin{aligned}
\bar{f}([D(a)|\tilde{H}_a])&=&h_{12}(a)
w_{12}\left(\frac{1+a}{-a}\right)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{1-a}\right)x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=&h_{12}(a)
w_{12}\left(\frac{1+a}{-a}\right)w_{12}(1+a)\\
&=&h_{12}(a)h_{12}\left(\frac{1+a}{-a}\right)h_{12}(-(1+a))^{-1}\\
&=&c(a,-(1+a)a^{-1})=c(a,1+a).\end{aligned}$$
Now $$s(\tilde{H}_aG_a)=s\left(\matr{\frac{a}{1-a}}{\frac{a^2}{1-a}}{\frac{1}{1+a}}{\frac{a^{-1}}
{1+a}}\right)= x_{12}\left(\frac{a(1+a)}{1-a}\right)w_{12}(-(1+a))x_{12}(a^{-1}).$$
So $$\begin{aligned}
\bar{f}([\tilde{H}_a|G_a])&=& s(\tilde{H}_a)s(G_a)s(\tilde{H}_aG_a)^{-1}\\
&=&x_{12}\left(\frac{1+a}{a(1-a)}\right)
w_{12}\left(\frac{1+a}{-a}\right)x_{12}(a^{-1})w_{12}(-1)x_{12}(a+a^{-1})
x_{12}(-a^{-1})w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=&x_{12}\left(\frac{1+a}{a(1-a)}\right)
w_{12}\left(\frac{1+a}{-a}\right)x_{12}(a^{-1})w_{12}(-1)x_{12}(a)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=& w_{12}\left(\frac{1+a}{-a}\right)x_{21}\left(\frac{-a}{1-a^2}\right)w_{12}(-1)
x_{21}(-a^{-1})x_{12}(a)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\ \end{aligned}$$ using $$x_{12}\left(\frac{1+a}{a(1-a)}\right)^{w_{12}\left(\frac{1+a}{-a}\right)}= x_{21}\left(\frac{-a}{1-a^2}\right)
\mbox{ and }
x_{12}(a^{-1})^{w_{12}(-1)}=x_{21}(-a^{-1}).$$
Since $x_{12}(-a)w_{12}(a)=x_{21}(-a^{-1})x_{12}(a)$, we thus have
$$\begin{aligned}
\bar{f}([\tilde{H}_a|G_a])
&=& w_{12}\left(\frac{1+a}{-a}\right)x_{21}\left(\frac{-a}{1-a^2}\right)w_{12}(-1)
x_{12}(-a)w_{12}(a)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=& w_{12}\left(\frac{1+a}{-a}\right)w_{12}(-1)x_{12}\left(\frac{a}{1-a^2}\right)
x_{12}(-a)w_{12}(a)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right).\\\end{aligned}$$
Since $$x_{12}\left(\frac{a}{1-a^2}\right)
x_{12}(-a)=x_{12}\left(\frac{a}{1-a^2}-a\right)= x_{12}\left(\frac{a^3}{1-a^2}\right),$$ we obtain $$\begin{aligned}
\bar{f}([\tilde{H}_a|G_a])
&=& w_{12}\left(\frac{1+a}{-a}\right)w_{12}(-1)
x_{12}\left(\frac{a^3}{1-a^2}\right)w_{12}(a)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{a-1}\right).\\\end{aligned}$$
The conjugation rules of Corollary \[cor:conjw\] give $$x_{12}\left(\frac{a^3}{1-a^2}\right)w_{12}(a)w_{12}(1+a)=
w_{12}(a)x_{12}\left(\frac{-a}{1-a^2}\right)w_{12}(1+a)=
w_{12}(a)w_{12}(1+a)x_{12}\left(\frac{a(1+a)}{1-a}\right).$$
Therefore $$\begin{aligned}
\bar{f}([\tilde{H}_a|G_a])
&=& w_{12}\left(\frac{1+a}{-a}\right)w_{12}(-1)
w_{12}(a)w_{12}(1+a)
x_{12}\left(\frac{a(1+a)}{1-a}\right)x_{12}\left(\frac{a(1+a)}{a-1}\right)\\
&=& w_{12}\left(\frac{1+a}{-a}\right)w_{12}(-1)
w_{12}(a)w_{12}(1+a)\\
&=&h_{12}\left(\frac{1+a}{-a}\right)h_{12}(a)h_{12}(-(1+a))^{-1}\\
&=& c(-(1+a)a^{-1},a)=c(1+a,a).\end{aligned}$$
Putting this together, we get $$\begin{aligned}
\bar{f}(\tilde{\Theta}_a)= \bar{f}([\tilde{H}_a|G_a])\cdot \bar{f}([D(a)|\tilde{H}_a])^{-1}
= c(1+a,a)c(a,1+a)^{-1}\\
= c(a^2,1+a)=c((-a)^2,1+a)
= c(-a,1+a)c(1+a,-a)=1.\\\end{aligned}$$
Applications: generators for $\hoz{2}{\spl{2}{\Z[1/m]}}$ {#sec:h2}
========================================================
The general principle is the following:
\[lem:ui\] Let $m=q_1q_2\cdots q_n$ where $q_1,\ldots,q_n$ are distinct primes. Suppose that the positive integers $u_{2},\ldots,u_{n}\in \Z[1/m]^\times$ satisfy the conditions
1. $u_i$ is a primitive root modulo $q_i$ for $i\geq 2$,
2. When $i\not= j\in \{ 2,\ldots,n\}$, $$q_i^{v_{q_j}(u_i)}\cong 1\pmod{q_j} .$$
Then there is a direct sum decomposition $$\tilde{K}_2(2,\Z[1/m])\cong \Z\oplus \Z/(q_2-1)\oplus \cdots \oplus\Z/(q_n-1)$$ with the property that infinite cyclic factor is generated by $c(-1,-1)$ and the factor $\Z/(q_i-1)$ is generated by $c(u_{i},q_i)$.
The isomorphism $$\tilde{K}_2(2,\Z[1/m])\cong \Z\oplus\left(\oplus_{i=2}^n\F{q_i}^\times\right)$$ is induced by the map $$\sigma: \tilde{K}_2(2,\Z[1/m])\to \Z,\quad
c(a,b)\mapsto
\left\{
\begin{array}{ll}
1,& a<0\mbox{ and }b<0\\
0,& \mbox{ otherwise}
\end{array}
\right.$$ and the tame symbols $T_{p_i}:K(2,\Q)\to\F{p_i}^\times$.
Now $$\begin{aligned}
T_{p_i}(c(u_{i},q_i))=\tau_{q_i}(\{ u_i,q_i\})=u_i\pmod{q_i}=w_i\end{aligned}$$ while for $j\not=i$ $$\begin{aligned}
T_{q_j}(c(u_{i},q_i))=q_i^{v_{q_j}(u_i)}{\pmod{q_j}}\equiv 1\pmod{q_j}.\end{aligned}$$
It is not known whether there must exist units satisfying condition (1) in general, but exceptions, if they exist, are rare.
If units $u_i$ are found satisfying condition (1), then it can always be arranged for condition (2) to hold; namely, multiply $u_i$ by a high power of $m_i$ where $m_i= (\prod_{k=1}^nq_k)/q_i$.
Combining Lemma \[lem:ui\] with Theorems \[thm:z\] and \[thm:symb\], we deduce:
Let $m=q_1\cdots q_n$ be distinct primes satisfying $q_1<q_2<\cdots <q_n$ and $q_1=2,q_2=3$. Let $u_2,\ldots,u_n$ be as in Lemma \[lem:ui\]. There is a direct sum decomposition $$\hoz{2}{\spl{2}{\Z[1/m]}}\cong \Z\oplus \left(\oplus_{i=2}^n \Z/(q_i-1)\Z\right)$$ where the first summand corresponds to the subgroup of $\hoz{2}{\spl{2}{\Z[1/m]}}$ generated by the homology class $C(-1,-1)$, and the summand $\Z/(q_i-1)\Z$ corresponds to the subgroup generated by the homology class $C(u_{i},q_i)$.
In the case $m=6$, we can take $u_2=2$. We deduce that the cyclic factors of $$\hoz{2}{\spl{2}{\Z[1/6]}}\cong \Z\oplus\Z/2$$ are generated by the homology classes $C(-1,-1)$ and $C(2,3)$.
In the case $m=30$, then the units $u_2=2$, $u_3=2$ satisfy the necessary congruences. Thus the cyclic factors of $$\hoz{2}{\spl{2}{\Z[1/30]}}\cong \Z\oplus\Z/2\oplus\Z/4$$ are generated by the homology classes $C(-1,-1)$, $C(2,3)$ and $C(2,5)$.
By Theorem \[thm:z\], we have $$\hoz{2}{\Z[1/42]}\cong \Z\oplus \F{3}^\times\oplus \F{7}^\times\cong\Z\oplus\Z/2\oplus\Z/6.$$ The first factor is generated by the homology class $C(-1,-1)$. Furthermore, $u_2=2=u_3$ satisfy the congruences of Lemma \[lem:ui\]. It follows that the homology classes $C(2,3)$ and $C(2,7)$ generate the second and third cyclic factors.
Let $\omega$ be a primitive cube root of unity and let $p$ be a rational prime which is congruent to $1$ modulo $3$. Let $\mathcal{O}=\Z[\omega,\frac{1}{3p}]$.
Observe that $\omega\in
\mathcal{O}^\times$ and $\omega^2-1= \sqrt{-3}\omega\in \mathcal{O}^\times$ also. Then $p\mathcal{O}=\mathfrak{p}_1\mathfrak{p}_2$ where $k(\mathfrak{p}_i)\cong \F{p}$ for $i=1,2$. Since $K_2(\Z[\omega])=0$, we have $$\begin{aligned}
K_2(\mathcal{O})\cong \tilde{K}_2(2,\mathcal{O})&\cong &k(\mathfrak{p}_1)^\times\oplus
k(\mathfrak{p}_2)^\times\oplus k(\mathfrak{q})^\times\\
& \cong& \F{p}^\times\oplus\F{p}^\times\oplus\F{3}^\times \\\end{aligned}$$ where $\mathfrak{q}=\sqrt{-3}\mathcal{O}$.
By Lemma \[lem:h2o\] and Theorem \[thm:symb\] the natural map $$\hoz{2}{\spl{2}{\mathcal{O}}}\to K_2(\mathcal{O})$$ is surjective and the homology class $C(-\omega,p)$ maps, via the tame symbol, to the element $-\bar{\omega} \in k(\mathfrak{p}_i)^\times$ of order $6$, while the class $C(3,p)$ maps to $\bar{3}\in k(\mathfrak{p}_i)^\times\cong \F{p}^\times$.
|
---
abstract: 'The local structure of (Al$_{0.06}$V$_{0.94}$)$_2$O$_3$ in the paramagnetic insulating (PI) and antiferromagnetically ordered insulating (AFI) phase has been investigated using hard and soft x-ray absorption techniques. It is shown that: 1) on a local scale, the symmetry of the vanadium sites in both the PI and the AFI phase is the same; and 2) the vanadium $3d$ - oxygen $2p$ hybridization, as gauged by the oxygen $1s$ absorption edge, is the same for both phases, but distinctly different from the paramagnetic metallic phase of pure V$_2$O$_3$. These findings can be understood in the context of a recently proposed model which relates the long range monoclinic distortion of the antiferromagnetically ordered state to orbital ordering, if orbital short range order in the PI phase is assumed. The measured anisotropy of the x-ray absorption spectra is discussed in relation to spin-polarized density functional calculations.'
author:
- 'P. Pfalzer'
- 'J. Will'
- 'A. Nateprov'
- 'M. Klemm'
- 'V. Eyert'
- 'S. Horn'
- 'A. I. Frenkel'
- 'S. Calvin'
- 'M. L. denBoer'
bibliography:
- 'AlV2O3.bib'
title: 'Evidence for short range orbital order in paramagnetic insulating (Al,V)$_2$O$_3$'
---
Introduction {#introduction .unnumbered}
============
The metal-insulator transition (MIT) in V$_2$O$_3$ has been intensively investigated and discussed for many years as an example of a classical Mott transition. However, such a picture is blurred by the complexity of the V$_2$O$_3$ phase diagram [@McWhan73] which involves magnetic and structural transitions coinciding with the MIT, as a function of temperature, doping (Cr, Al, Ti), pressure, and oxygen stoichiometry. At room temperature pure V$_2$O$_3$ is in a paramagnetic metallic (PM) phase which x-ray diffraction (XRD) shows to be trigonal. At about 180 K there occurs a transition to an antiferromagnetically insulating (AFI) monoclinic phase. Doping with Cr or Al results in the formation of a paramagnetic insulating (PI) phase. The lattice parameters change but long-range trigonal symmetry is preserved, as indicated by XRD.[@McWhan69_1; @Spalek90] The role of electronic correlations in the interplay between changes in the physical structure, the magnetic properties, and the electronic structure at the various transitions needs further investigation. It is still not certain whether the electronic transition is driven by structural changes or vice-versa. Although recent LDA + DMFT calculations[@Held01] show the importance of electronic correlations in a description of the electronic structure, a description of the MIT in V$_2$O$_3$ must take into account the relationship between physical structure and the electronic and magnetic properties of the system.
The Mott-Hubbard picture of V$_2$O$_3$, in particular a description in terms of a one band Hubbard model,[@Rozenberg95] is based on a level scheme of Castellani *et al.* for the electronic structure of V$_2$O$_3$.[@Cast78] The crystal field generated by the O octahedron splits the V $3d$ states into upper e$_g$ and lower $t_{2g}$ states, and the latter are further split into $a_{1g}$ and $e_g^\pi$ states due to the trigonal symmetry of the lattice. Interactions between nearest vanadium neighbors along the $c$-direction (vertical pairs) splits the $a_{1g}$ states into bonding and antibonding molecular orbitals. According to Castellani *et al.* the bonding $a_{1g}$ orbital is fully occupied while the antibonding $a_{1g}$ orbital shifts energetically above the $e_g^\pi$ states. This leaves only one electron per vanadium atom to occupy the $e_g^\pi$ states, leading to orbital degeneracy, which is susceptible to a degeneracy-lifting process such as the Jahn-Teller effect or orbital ordering. Indeed, Bao *et al.*[@Bao98; @Bao97] suggested that such orbital ordering occurs in V$_2$O$_3$. This conclusion was based on neutron scattering experiments which show disagreement between the propagation vector characterizing the AFI phase and the propagation vector expected from magnetic short range order in the PM and PI phases of pure V$_2$O$_3$ and its alloys. The latter propagation vector is identical to that of a spin density wave in vanadium-deficient V$_2$O$_3$. The suggested orbital ordering would distinguish the AFI phase of V$_2$O$_3$ from all the other phases and prevent a unified description of the MIT in this compound. The validity of Castellani’s model has been called into question by soft x-ray absorption[@Mueller97; @Park00]and band structure calculations.[@Ezhov99]Near-edge x-ray absorption fine structure (NEXAFS) measurements of the V $2p$ and O $1s$ edges provides information on the unoccupied states near the Fermi level. Müller *et al.* concluded from such a NEXAFS study of the different phases of V$_2$O$_3$ and (V,Cr)$_2$O$_3$ that all the insulating phases have, within experimental error, identical local electronic structures.[@Mueller97] In addition, angular resolved NEXAFS measurements in the metallic and insulating phase of V$_2$O$_3$ are inconsistent with the assumption that the first excited states are purely $e_g^\pi$. Rather, the isotropy of the absorption spectra observed in the metallic phase suggests the first excited states are a mixture of $e_g^\pi$ and $a_{1g}$, while the anisotropy observed in the insulating phases suggests these states have increased $a_{1g}$ character. These conclusions were confirmed by LDA+U band structure calculations by Ezhov *et al.*[@Ezhov99] and NEXAFS studies by Park *et al.*[@Park00] Using a model fit to the V $2p$, Ezhov *et al.* showed that orbital occupancy of the $e_g^\pi$ states is larger than one and changes at both MIT’s. This renders the one band Hubbard model[@Rozenberg95] often applied to the MIT in V$_2$O$_3$ inadequate.
Important for a unified view of the MIT in V$_2$O$_3$ is the understanding of the relationship between electronic, magnetic, and structural changes at the MIT. Recently, a model was proposed[@Shiina01; @Mila00] that takes into account degrees of freedom of molecular orbitals formed by vertical V-V pairs and their interaction within the $ab$-plane, offering a consistent picture of the magnetic and structural properties of the AFI phase.
In this paper we show using EXAFS and NEXAFS techniques that, on a local scale, the structural and electronic properties of the AFI and PI phase are the same. This fact is attributed to short range orbital order in the PI phase, consistent with recent model calculations[@Shiina01; @Mila00] and the characteristics of magnetic short range order observed by neutron scattering.[@Bao98] Spin polarized density functional theory calculations presented here reflect the anisotropy of the O $2p$ density of states observed in the insulating phases of the V$_2$O$_3$ phase diagram.
Experimental and Theoretical Methodology {#experimental-and-theoretical-methodology .unnumbered}
========================================
Measurements were performed on an Al-doped V$_2$O$_3$ single crystal about 2.5 mm by 1.5 mm by 1 mm grown by chemical transport using TeCl$_4$ as transport agent and a mixture of V$_2$O$_3$ and Al$_2$O$_3$ powder corresponding to a nominal concentration x = 0.1 in (V$_{1-x}$Al$_x$)$_2$O$_3$. Energy dispersive x-ray scattering showed the actual Al concentration in the single crystal was 6 atomic percent, presumably due to lower transport rate of Al compared to vanadium during crystal growth. XRD showed the expected trigonal structure with lattice parameters $c_{hex} = 13.81(1)$ Å and $a_{hex} = 4.985(5)$ Å, both smaller than those found by Joshi *et al.* ($c_{hex} \approx
13.89$ Å and $a_{hex} \approx 5.00$ Å at 6.2 at%).[@Joshi77] The resulting $c/a$ ratio of 2.7703 is much smaller than in the metallic phase (2.828). Resistivity measurements, the small value of the $c/a$ ratio, and the Al concentration show the sample was in the PI phase at room temperature. The transition from PI to AFI occurred at 165 K as determined by magnetic susceptibiblity measurements using a Quantum Devices SQUID magnetometer. Laue diffraction measurements were performed on the single crystal at room temperature and at 77 K. The room temperature trigonal symmetry was, as expected, broken at low temperature.
![\[structure\] Local structure of V$_2$O$_3$. Black (grey) spheres represent vanadium (oxygen) ions. Only those oxygen ions which form an octahedron around the central vanadium atom V$_0$ are shown. The hexagonal $c$ axis is along the line V$_{1'}$-V$_0$-V$_1$.](V2O3_NN.eps){width="8.5cm"}
The EXAFS measurements were performed at beam line X23B at the National Synchrotron Light Source at Brookhaven National Laboratory. Spectra at different temperature points above and below the transition temperatures of the pure and doped sample were taken using a closed cycle helium cryostat. The sample was held under vacuum to reduce thermal leakage and air-absorption and prevent water condensation. The x-ray absorption spectra were measured in fluorescence yield mode using a Lytle detector;[@Lytle84] background radiation was filtered by a Ti foil. In fluorescence mode self-absorption effects may strongly damp the EXAFS amplitude even at normal incidence. As the primary effect of self-absorption is to reduce the EXAFS amplitude,[@Troeger92; @Pfalzer99] but leave the phase unchanged, the distances of neighboring atoms obtained by EXAFS, which are determined largely by the phase, are unaffected. We corrected for self-absorption using a generalization of the method of Tröger *et al.*[@Troeger92; @Goulon82] to large detector surfaces, as described elsewhere.[@Pfalzer99] This correction is quite reliable, as was shown in Ref. by demonstrating that for Al-doped V$_2$O$_3$ the absorption was isotropic in the $ab$ plane after the correction, as expected for trigonal symmetry.
For the EXAFS measurements the sample was oriented so that the $(11\bar{2}0)$ plane (in hexagonal notation) was the surface perpendicular to the incident beam. In this geometry the orientation of the polarization vector $\vec{E}$ of the incoming x-rays with respect to the hexagonal $c$ axis can be changed by rotating the sample around the surface normal. Measurements were made with $\vec{E}$ parallel and perpendicular to $\vec{c}_{hex}$. As described by Frenkel *et al.*,[@Frenkel97] this facilitates measuring the two different components of the anisotropic absorption coefficient: $$\mu = \mu_\perp sin^2\theta + \mu_\parallel cos^2\theta ,$$ where $\mu_\perp$ ($\mu_\parallel$) is the absorption coefficient for $\vec{E}$ perpendicular (parallel) to $\vec{c}_{hex}$ and $\theta$ is the angle between $\vec{E}$ and $\vec{c}_{hex}$. Since different scattering paths contribute to $\mu_\perp$ and $\mu_\parallel$, measurement of these two independent quantities achieves better separation of paths. After correcting for self-absorption as described above standard EXAFS analysis was performed. For the Fourier transform a window in $k$ space from 3 Å$^{-1}$ to 12 Å$^{-1}$ was used ($k$ being the wavevector of the photoelectron).
Measured spectra were compared to model spectra calculated with FEFF8.[@Ankudinov98] The calculation requires estimated atomic positions, provided by models of the structure. To construct these structural models we used the lattice parameters provided by our XRD measurements. Then, for the trigonal model, we used the atomic positions tabulated by Wyckoff[@Wyckoff63] for the pure compound. For the monoclinic model, we started with the measured hexagonal lattice parameters and tilted the $c_{hex}$-axis by 1.995$^\circ$, the same amount as in the AFI phase of undoped V$_2$O$_3$,[@Urbach95] to reproduce the monoclinic distortion. The resulting pseudo-hexagonal lattice vectors were converted to monoclinic ones using the conversion matrix of Ref. . Relative atomic positions for the monoclinic phase were also taken from Ref. . The structure refinement was performed by varying the interatomic distances to all vanadium atoms up to the fourth nearest neighbor, as shown in Fig. \[structure\], as well as to the nearest two shells of oxygen atoms. A few double scattering paths with high scattering amplitudes were included in the fits. The nearest oxygen shell forms a (distorted) octahedron around the absorbing vanadium atom V$_0$ and is responsible for the well-separated peak of the Fourier transformed spectra between 1 and 2 Å. Fitting was carried out in $r$ space using $k^3$ weighting.
![\[EXAFS-ksp\] EXAFS spectra of (Al$_{0.06}$V$_{0.94}$)$_2$O$_3$ in $k$-space. The spectra above (dashed line) and below (solid) the PI to AFI transition are virtually the same for Al-doped V$_2$O$_3$ for both $\vec{E} \perp \vec{c}_{hex}$ and $\vec{E} \parallel
\vec{c}_{hex}$ (a and b), while pronounced changes are observed in pure V$_2$O$_3$ which is in the PM state at room temperature (c), showing the two insulating phases have virtually identical local physical structure which differs from that of the metal.](q-space-changes_bw.eps){width="8.5cm"}
NEXAFS measurements on the O $1s$ edge were performed under ultra high vacuum at the U41-1/PGM beamline at the BESSY 2 storage ring using the same crystal in the same geometry as for the EXAFS measurements. The signal was monitored by measuring the total electron yield; this is somewhat surface-sensitive, so care was taken to prepare and maintain clean surfaces characteristic of the bulk.
The experiments were complemented by electronic structure calculations based on density functional theory (DFT) in the local density approximation (LDA). The calculations used the augmented spherical wave (ASW) method in its scalar-relativistic implementation,[@wkg; @revasw] which was already applied in Ref. . The present study included the paramagnetic metallic V$_2$O$_3$, paramagnetic insulating (V$_{0.962}$Cr$_{0.038}$)$_2$O$_3$, and antiferromagnetic insulating V$_2$O$_3$ phase using the crystal structure data of Dernier [@Dernier70; @remark] as well as Dernier and Marezio.[@DM70] Note that, since DFT is a ground state theory, differences between the phases were taken into account only via the different crystal structure. In order to study the effect of spin-polarization separately from the monoclinic distortion in the AFI state, a complementary set of calculations with enforced spin-degeneracy was performed for this structure.
In order to account for the openness of the crystal structures, empty spheres, i.e. pseudo atoms without a nucleus, were included to model the correct shape of the crystal potential in large voids. Optimal empty sphere positions and radii of all spheres were automatically determined by the recently developed sphere geometry (SGO) algorithm.[@vpop] As a result, 8 and 16 empty spheres with radii ranging from 1.78 to 2.42 $a_B$ were included in the trigonal and monoclinic cell, respectively, keeping the linear overlap of vanadium and oxygen spheres below 16.5%. The basis set comprised V $4s$, $4p$, $3d$ and O $2s$, $2p$ as well as empty sphere states. Fast self-consistency was achieved by an efficient algorithm for convergence acceleration.[@mixpap] Brillouin zone sampling was done using an increasing number of $\vec{k}$ points ranging from 28 to 2480 and 108 to 2048 points within the respective irreducible wedges, ensuring convergence of the results with respect to the fineness of the $k$-space grid.
Results {#results .unnumbered}
=======
In Fig. \[EXAFS-ksp\] we compare the EXAFS (in $k$ space) of a pure V$_2$O$_3$ sample and the Al-doped sample above (dashed lines) and below (solid lines) their respective transition temperatures. For the Al-doped sample EXAFS spectra were measured with the polarization vector $\vec{E}$ oriented along and perpendicular to the hexagonal $c$-axis. The EXAFS is very similar for both orientations for both the PI and the AFI phase of this sample \[see Fig. \[EXAFS-ksp\](a) and 2(b)\], indicating that this transition, involving a long range monoclinic distortion and magnetic order, is accompanied by only minor changes in local structure. In contrast, for pure V$_2$O$_3$, the large differences in the EXAFS oscillations apparent in Fig. \[EXAFS-ksp\](c) show that the local environment of the absorbing atoms is very different above and below the MIT. For the PI and AFI phases of the Al-doped sample, small differences are apparent at high $k$; these presumably are due to the distinctly different Debye-Waller factors at the measuring temperatures of 30 K and 180 K, respectively.
![\[EXAFS-rsp\] FT Magnitudes of the EXAFS spectra of (Al$_{0.06}$V$_{0.94}$)$_2$O$_3$ (solid line) compared to calculated EXAFS using a monoclinic model (dotted lines) and a trigonal model (long dashes), for two geometries: (a) $\vec{E}$ parallel to $\vec{c}_{hex}$, in which significant differences between the models are expected. The monoclinic model fits the data much better than the trigonal one in both the PI (T = 180 K) and AFI (T = 30 K) phases. (b) $\vec{E}$ perpendicular to $\vec{c}_{hex}$. In this geometry many paths contribute and little difference is expected between the models, as observed. Dotted vertical lines indicate the R range used in the fit. The positions of the peaks are not corrected by the scattering phase shifts ($0.3-0.5$ Å).](r-space-fits_bw.eps){width="8.5cm"}
Fitting procedures, further discussed below, provide local interatomic distances which confirm that the local structure is virtually the same in the PI and AFI phase of the Al-doped sample, but also in the AFI phase of pure V$_2$O$_3$. The fit also shows that both the PI and AFI phases of the Al-doped V$_2$O$_3$ have a local symmetry which corresponds to a monoclinic lattice. Fig. \[EXAFS-rsp\] compares the measured EXAFS (in $r$ space, where $r$ is the interatomic distance) for both orientations of $\vec{E}$ to $\vec{c}_{hex}$ to spectra derived from model calculations for monoclinic and trigonal structures. It is evident that the monoclinic model fits the data better. The trigonal model (long dashes) fits the data very poorly in the PI phase at 180 K as well as in the AFI phase at 30 K, while the monoclinic model (dotted lines) fits the data very well in both cases, especially doing a much better job in describing the distorted oxygen octahedron (peak at 1.7 Å) and the next vanadium neighbors (V$_1$ and V$_{1'}$ contributions are around 2.2 Å and below 4 Å in Fig. \[EXAFS-rsp\]). This visual observation is confirmed by the fact that the reduced chi-squared of the trigonal model is 38.1, while that of the monoclinic model is 10.7. The better fit of the monoclinic model than the trigonal model is particularly apparent in the $\vec{E}$ parallel to $\vec{c}_{hex}$ orientation. This is expected because the EXAFS measured in this orientation is more sensitive to the structural differences between the phases; the major effect of the reduction in symmetry during the transition from PI or PM to AFI is a tilt of the $c_{hex}$ axis and a “rotation” of V$_0$-V$_1$ next neighbor pairs all located along this axis. This orientation also achieves good separation of scattering paths, since there is only a single V next neighbor atom (V$_1$ in Fig. \[structure\]) along the $c_{hex}$ axis. In fact, the main contribution to the scattered intensity at 2.2 Å in Fig. \[EXAFS-rsp\](a) is due to the V$_0$-V$_1$ single scattering path. The broad peaks in the spectra taken with the polarization vector $\vec{E}$ of the x-rays perpendicular to the $c_{hex}$ axis \[Fig. \[EXAFS-rsp\](b)\] contain a large number of paths and therefore fit equally well to both models.
--------------------- ------- -------------------- ---------------- ------ ------ -------------------- ------
[V$_2$O$_3$+Cr]{} [V$_2$O$_3$]{}
trig. mon. PI AFI PI AFI
V$_1$ 2.72 2.72 2.76 2.79 2.75 2.74
i 2.86 2.96 2.91 2.86
V$_2$ ii 2.91 2.87 2.97 2.93 2.92 2.88
iii 3.00 3.10 3.05 2.99
i 3.43 3.44 3.42 3.44
\[-1.5ex\][V$_3$]{} ii \[-1.5ex\][3.44]{} 3.45 3.46 3.44 \[-1.5ex\][3.45]{} 3.46
i 3.62 3.69 3.72 3.63
V$_4$ ii 3.69 3.71 3.78 3.81 3.70 3.73
iii 3.72 3.79 3.82 3.74
--------------------- ------- -------------------- ---------------- ------ ------ -------------------- ------
: \[distances\]Distances in Å from the central ion V$_0$ to nearby V ions (as labeled in Fig. \[structure\]) for various V compounds. The calculated values for (Al$_{0.06}$V$_{0.94}$)$_2$O$_3$ are obtained using hexagonal lattice vectors measured by XRD and relative atomic positions, assuming a trigonal (trig)[@Dernier70] and monoclinic (mon)[@DM70] lattice. (For the monoclinic lattice the ions V$_2$, V$_3$ and V$_4$ become non-degenerate and hence are identified separately in the table.) The values measured by EXAFS for the PI and AFI phases are those obtained using the fits shown in Fig. \[EXAFS-rsp\]. Distances for V$_2$O$_3$+Cr, specifically (Cr$_{0.038}$V$_{0.962}$)$_2$O$_3$, in the PI phase were measured by Dernier.[@Dernier70] Values for the AFI phase of pure V$_2$O$_3$ are calculated from the monoclinic lattice vectors and atomic positions published by Dernier and Marezio.[@DM70]
Table \[distances\] lists the distances from a central reference ion, labeled V$_0$ in Fig. \[structure\], to nearby V ions for the various phases of the V$_2$O$_3$ compounds. The values measured by EXAFS for the PI and AFI phases are those obtained using the fits shown in Fig. \[EXAFS-rsp\]. For comparison, we include interatomic distances calculated for (Al$_{0.06}$,V$_{0.94}$)$_2$O$_3$ obtained using hexagonal lattice vectors measured by XRD and relative atomic positions, assuming a trigonal[@Dernier70] and monoclinic lattice.[@DM70] Distances for (Cr$_{0.038}$V$_{0.962}$)$_2$O$_3$ in the PI phase were measured by Dernier[@Dernier70] using XRD and those for the AFI phase of pure V$_2$O$_3$ are calculated from the monoclinic lattice vectors and atomic positions published by Dernier and Marezio.[@DM70] Uncertainties are typically $\pm$0.02 Å. It is evident that the interatomic distances measured by EXAFS for the PI and AFI phases (Al$_{0.06}$V$_{0.94}$)$_2$O$_3$ are essentially the same. In addition, these distances are in correspondence with, though consistently slightly larger than, those obtained for the AFI phase of pure V$_2$O$_3$, confirming that all insulating phases have essentially the same local structure.
![\[NEXAFS-O1s\] NEXAFS spectra of the O $1s$ edge of (Al$_{0.06}$V$_{0.94}$)$_2$O$_3$ as a function of angle between $\vec{E}$ and the $c_{hex}$ axis, showing large but identical angular anisotropy for (a) the PI and (b) the AFI phase. For comparison, this angular dependence almost vanishes for pure V$_2$O$_3$ in its metallic state (c) (from Ref. , energy-shifted to facilitate comparison). All spectra were normalized to the large maximum at about 532 eV.](NEXAFS_O1s.eps){width="8.5cm"}
NEXAFS spectra of the O $1s$ edge in the same Al-doped Al$_2$O$_3$ crystal are presented in Fig. \[NEXAFS-O1s\] (a) and (b). Large changes in the spectra are apparent as the sample is rotated from a geometry with $\vec{E}$ parallel to $\vec{c}_{hex}$ ($\vartheta=0^\circ$) to one with $\vec{E}$ perpendicular to $\vec{c}_{hex}$ ($\vartheta=90^\circ$). It is evident that cooling from the PI to the AFI phase has no significant effect on the spectra or their dependence on angle. These spectra and their angular dependence are similar to those of pure V$_2$O$_3$ in its low temperature AFI phase,[@Mueller97] but different from those of pure V$_2$O$_3$ in its high temperature PM phase, which has a much weaker angular dependence \[Fig. \[NEXAFS-O1s\](c)\].
Discussion {#discussion .unnumbered}
==========
The EXAFS and O edge soft x-ray absorption measurements presented here consistently show that the structure, at least on a local scale, and the V-O hybridization in the PI and AFI phase of (Al,V)$_2$O$_3$ are similar, but distinctly different from the metallic phase. The most obvious difference between the metallic and insulating phases is the anisotropy of the O $1s$ x-ray absorption spectra, reflecting the V $3d$ - O $2p$ hybridization. These findings suggest that the structural change at the MI transition (long or short range order) is directly connected with the emergence of the insulating state, e.g. via changes of hybridization.
![\[LDA\] Partial oxygen $2p$ densities of states predicted by the density functional calculations described in the text for different crystal structures of (V$_{1-x}$Cr$_x$)$_2$O$_3$ with various c/a ratios and symmetries: (a) the trigonal paramagnetic phase ($x=0$); (b) paramagnetic insulating phase with a larger trigonal distortion corresponding to $x=0.038$; (c) the monoclinic paramagnetic phase of V$_2$O$_3$; and (d) the monoclinic antiferromagnetic phase. Solid (dashed) lines indicate O $2p_z$ ($2p_x$ or $2p_y$) states. For the magnetically ordered phase (d) spin up and spin down states have been added to facilitate comparison and the vertical scale has been expanded.](v2o3_O_2p_bandstruktur_v2.eps){width="8.5cm"}
To estimate the effect of purely structural changes on the hybridization, we show in Fig. \[LDA\] partial (projected) O $2p_z$ and $2p_{x,y}$ densities of states (DOS) predicted by the electronic structure calculations. The V $3d$ partial DOS predictions for the spin-degenerate cases (see Ref. ) are in good agreement with those of Mattheiss[@mattheiss94] and Ezhov [*et al.*]{}[@Ezhov99] The V $3d$ and O $2p$ partial DOS presented in Fig. \[LDA\] are very similar for all phases, except for a slight decrease of the O $2p$ band width in the PI phase as compared to the PM phase. In particular, the experimentally observed optical band gap is not reproduced by the calculations and all phases are predicted to be metallic. These discrepancies between experiment and calculations are usually attributed to the fact that on-site electron-electron correlations are not fully taken into account by LDA. The calculations also predict that the O $2p$ DOS of all phases is isotropic, i.e. the projected $2p_z$ and $2p_{x,y}$ DOS are similar, implying that NEXAFS spectra of the O $1s$ edge with the polarization vector $\vec{E}$ parallel or perpendicular to $\vec{c}_{hex}$ should also be alike. Experimentally, this is only observed for the metallic phase of V$_2$O$_3$, as shown in Fig. \[NEXAFS-O1s\](c), but not for the insulating phases. However, the calculations for (V$_{0.962}$Cr$_{0.038}$)$_2$O$_3$ \[Fig. \[LDA\](b)\] described above did not include the local distortions observed by EXAFS in the PI phase and the calculations for monoclinic V$_2$O$_3$ \[Fig. \[LDA\](c)\] artificially enforced a spin-degenerate state. If the antiferromagnetic ground state of the monoclinic phase is taken into account by using a spin-polarized calculation \[Fig. \[LDA\](d)\], the partial DOS displays various band shifts, although the experimentally observed optical band gap is still not reproduced. In particular, the O $2p_z$ partial DOS now differs substantially from the $2p_{x,y}$ near 0.3 eV, 1.7 eV, and 4 eV and is, therefore, more compatible with the observed anisotropy of the O $1s$ spectra. In this context it should be noted that according to neutron scattering measurements antiferromagnetic short range order persists in both the PI and the PM phase, although the anisotropy of the O $1s$ spectra is only observed in the PI and AFI phases.
To address the effect of electronic correlations on the electronic structure of V$_2$O$_3$, recently LDA was combined with dynamical mean field theory (DMFT) and calculated and measured photoemission and x-ray absorption spectra were compared on the basis of the calculated V $3d$ spectral weight.[@Held01] The electron-electron interaction shifts the $a_{1g}$ and $e_g^\pi$ states with respect to each other. This would cause an anisotropy in the oxygen $1s$ spectra, since the $a_{1g}$ and $e_g^\pi$ states hybridize differently with the O $2p_z$ and $2p_{x,y}$ states. According to LDA the V $a_{1g}$ hybridize primarily with the O $2p_z$ orbital along the $c_{hex}$ axis, while the $e_g^\pi$ hybridize primarily with the O $2p_{x,y}$ in–plane orbitals. Therefore polarization-dependent excitations of O $1s$ core electrons into unoccupied O $2p_z$ and O $2p_{x,y}$ states provide information on their hybridization with V $a_{1g}$ and $e_g^\pi$ states respectively. Based on this we concluded earlier[@Mueller97]that the observed anisotropy in the AFI phase of pure V$_2$O$_3$ results from an increase in $a_{1g}$ character in the unoccupied DOS, causing a corresponding increase of weight of O $p_z$ hybridized states above $\epsilon_F$. This interpretation is consistent with the conclusion of Park,[@Park00] based on measurements of the V $L_{2,3}$ edges, that there is an increase in $e_g^\pi$ occupancy during the transition from the metallic to the insulating state in V$_2$O$_3$. On the other hand, the relative shifts of the $a_{1g}$ and $e_g^\pi$ states predicted from LDA + DMFT calculations are not sufficient to account for the observed anisotropy of the O $1s$ spectra in the insulating state.
Recently Shiina[@Shiina01] attributed the monoclinic lattice distortion in the AFI phase to orbital ordering which would make the three originally equivalent magnetic bonds in the $ab$-plane inequivalent and cause a monoclinic lattice distortion. Evidence for orbital fluctuations in the PI phase was provided by neutron scattering measurements,[@Bao98] which showed that magnetic short range order was limited to nearest neighbor distances, resulting in a first order transition from the PI to the AFI phase. Given the results of Ref. , orbital fluctuations in the PI phase could cause a dynamic monoclinic distortion. Assuming the time scale for such fluctuations is long compared to the time scale of the x-ray absorption process, EXAFS and soft x-ray absorption would measure an instantaneous structure and the PI phase would appear monoclinic, while XRD, which measures on a much longer time scale, would see a trigonal lattice. In the AFI phase the monoclinic distortion might become static although, on a local scale, still the same as in the PI phase. This model would account for the fact that neither EXAFS nor soft x-ray absorption observe differences between the PI and the AFI phase, in contrast to XRD. The spectroscopic results are, however, also consistent with static short range orbital order in the PI phase, which would, accordingly, be an orbital glass, with an disorder-order transition to the ordered AFI phase. The local monoclinic distortion we have found in (V$_{0.94}$Al$_0.06$)$_2$O$_3$ is reminiscent of the much smaller, but significant monoclinic distortion found in the metallic phase of pure V$_2$O$_3$.[@Frenkel97] From the fact that this monoclinic distortion is not detected in XRD measurements, Frenkel *et al.*[@Frenkel97] set an upper limit of 40 Å on the size of possible monoclinic domains and concluded that the MIT contains both an order-disorder and a displacive component. The monoclinic distortion in the metallic phase was determined to be about 30% of that in the antiferromagnetically ordered insulating phase. From the above discussion it can be concluded that their data suggest orbital fluctuations are also present in the metallic phase of V$_2$O$_3$, although local distortions are less prominent than in the PI phase, possibly due to better screening in the metallic phase.
Conclusion {#conclusion .unnumbered}
==========
The x-ray absorption measurements presented here show that both the PI and AFI insulating phases of V$_2$O$_3$ are distinguished from the PM phase by: (i) the presence of local or long range distortion of the lattice (probably connected to short or long range orbital order, respectively) and (ii) differences in the V $3d$ - O $2p$ hybridization, accompanied by corresponding band shifts. Both the distortion and the hybridization appear to be independent of the presence of antiferromagnetic correlations, which are present in all phases.[@Bao98] The similarity of the PI and the AFI phases, at least on a local scale, suggests a common route from their insulating behavior to the metallic behavior of the PM phase. Interactions between orbital degrees of freedom, which lead to an orbitally ordered state in the AFI phase and orbital short range order in the PI phase, appear to be an an important fingerprint of the MIT. The characteristic differences between V $3d$ - O $2p$ hybridization in the metallic and the insulating phases suggest that those changes in hybridization play a role in the MIT. Such changes might be due to strong anharmonic contributions to the temperature-dependent phonon spectrum.
We appreciate valuable assistance in the measurements and analysis from J.Kirkland at NSLS and Ch. Jung and M. Mast at BESSY and enlightening discussions with P. Riseborough and K.-H. Höck. This work was supported in part by the BMBF under contract number 0560GWAA and the DFG under contract number HO-955/2 and SFB484 and by the US DOE Contract DEFG02-91-ER45439. The NSLS is supported by the DOE.
|
---
abstract: 'We study inhomogeneous semilinear parabolic equations with source term $f$ independent of time $u_{t}=\Delta u+u^{p}+f(x)$ on a metric measure space, subject to the conditions that $f(x)\geq 0$ and $u(0,x)=\varphi (x)\geq 0$. By establishing Harnack-type inequalities in time $t$ and some powerful estimates, we give sufficient conditions for non-existence, local existence, and global existence of weak solutions. This paper generalizes previous results on Euclidean spaces to general metric measure spaces.'
address:
- 'Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK.'
- 'Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.'
- 'Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.'
author:
- 'Kenneth J. Falconer'
- Jiaxin Hu
- Yuhua Sun
title: Inhomogeneous parabolic equations on unbounded metric measure spaces
---
[^1]
Introduction {#SecIntr}
============
In recent years, the study of PDEs on self-similar fractals has attracted increasing interest, see for example [@Dalry; @Fal99; @FalHu01; @Kig01; @STR06]. In this paper we investigate a class of nonlinear diffusions with source terms on general metric measure spaces. Diffusion is of fundamental importance in many areas of physics, chemistry, and biology. Applications of diffusion include: (1) Sintering, i.e. making solid materials from powder (powder metallurgy, production of ceramics); (2) Catalyst design in the chemical industry; (3) Steel can be diffused (e.g. with carbon or nitrogen) to modify its properties; (4) Doping during production of semiconductors; (5) The well-known Black-Scholes Model in Financial Mathematics that is closely related to option pricing can be transformed to a parabolic equation.
Let $\left( M,d,\mu \right) $ be a *metric measure* space, that is, $\left( M,d\right) $ is a locally compact separable metric space and $\mu $ is a Radon measure on $M$ with full support. We consider the following nonlinear diffusion equation with a source term $f$ on $(M,d,\mu )$: $$u_{t}=\Delta u+u^{p}+f(x),\;t>0\text{ and }x\in M, \label{eq}$$with initial value $$u(0,x)=\varphi (x), \label{eq:init}$$where $p>1$ and $f,\varphi :M\rightarrow \mathbb{R}$ are non-negative measurable functions. With an appropriate interpretation of weak solutions of (\[eq\]) on $(M,d,\mu )$, we shall investigate the non-existence (or blow-up) of solutions, the local and global existence of weak solutions to (\[eq\])-(\[eq:init\]), as well as the regularity of these solutions. Although we were partially motivated by a series of earlier papers [Bandle00,FalHu01,Fujita66, GHL03, Wei80, Wei81,zhang98]{}, there are new ideas in this paper. In particular, we have used the theory of heat kernels on metric measure spaces.
Recall the definition of the heat kernel which will be central to our approach. A function $k(\cdot ,\cdot ,\cdot ):\mathbb{R}_{+}\times M\times
M\rightarrow \mathbb{R}$ is called a *heat kernel* if the following conditions $(k1)-(k4)$ are fulfilled: for $\mu $-almost all [$(x,y)\in
M\times M$ and for all ]{} $t,s>0,$
1. *Markov property*[: $k(t,x,y)>0$, and $\int_{M}k(t,x,y)d\mu (y)\leq 1$;]{}
2. *symmetry*[: $k(t,x,y)=k(t,y,x)$;]{}
3. *semigroup property*[: $k(s+t,x,z)=\int_{M}k(s,x,y)k(t,y,z)d\mu (y)$;]{}
4. *normalization*[: for all $f\in L^{2}(M,\mu )$]{}$${\lim\limits_{t\rightarrow 0^{+}}\int_{M}k(t,x,y)f(y)d\mu (y)=f(x)}\text{ in
the }{L^{2}(M,\mu )}\text{{-norm.}}$$
We assume that the heat kernel $k(t,x,y)$ considered in this paper is jointly continuous in $x,y,$ and hence the above formulae in $(k1)-(k4)$ hold for *every* [$(x,y)\in M\times M$. ]{}
Two typical examples of heat kernels in $\mathbb{R}^{n}$ are the Gauss-Weierstrass and the Cauchy-Poisson kernels: $$\begin{aligned}
k(t,x,y) &=&\frac{1}{(4\pi t)^{n/2}}\exp {\left( -\frac{|x-y|^{2}}{4t}\right) }, \\
k(t,x,y) &=&\frac{C_{n}}{t^{n}}\left( 1+\frac{|x-y|^{2}}{t^{2}}\right)
^{-(n+1)/2}\text{ \ }\left( C_{n}=\frac{\Gamma \big({\textstyle\frac{1}{2}}(n+1)\big)}{\pi ^{(n+1)/2}}\right) .\end{aligned}$$Jointly continuous sub-Gaussian heat kernels exist on many basic fractals, for example, on the Sierpínski gasket, see Barlow and Perkins [BP88]{}, and on Sierpínski carpets, see Barlow and Bass [@BB99; @Bar98]. For other fractals see [@HK99; @Kig01]. For non-sub-Gaussian heat kernels, see [@BBCK09; @ChenK08].
A heat kernel $k$ is called *conservative* if it satisfies
1. *conservative property:*[ $\int_{M}k(t,x,y)d\mu (y)=1$, for all $t>0$ and all $x\in M$.]{}
We will also assume that the heat kernel satisfies the following estimates
1. *two-sided bounds*[: there exist constants ]{}$\alpha
,\beta >0$ such that for all $t>0$ and all $x,y\in M,$[ $$\frac{1}{t^{\alpha /\beta }}\Phi _{1}\left( \frac{d(x,y)}{t^{1/\beta }}\right) \leq k(t,x,y)\leq \frac{1}{t^{\alpha /\beta }}\Phi _{2}\left( \frac{d(x,y)}{t^{1/\beta }}\right) \label{hk-est}$$w]{}here $\Phi _{1}$ and $\Phi _{2}$ are strictly positive and non-increasing functions on $[0,\infty )$.
It turns out that the parameter $\alpha $ in (\[hk-est\]) is the *fractal dimension*, and $\beta $ is the *walk dimenison* of $M$, see [@GHL03].
Two-sided estimates (\[hk-est\]) hold on various fractals where $$\Phi _{i}(s)=C_{i}\exp (-c_{i}s^{\beta /(\beta -1)})\text{ \ }(\text{for all
}s\geq 0)$$for constants $C_{i},c_{i}>0$ $(i=1,2)$ and $\beta >2$ is the walk dimension$.$
To prove the regularity of solutions, we need to assume that the heat kernel $k$ is Hölder continuous in the space variables:
1. *Hölder continuity*[: there exist constants $L>0$, $\nu \geq 1$ and $0<\sigma \leq 1$ such that]{}$${\ }\left\vert {k(t,x_{1},y)-k(t,x_{2},y)}\right\vert {\leq Lt^{-\nu
}d(x_{1},x_{2})^{\sigma }}$$f[or all $t>0$ and all $x_{1},x_{2},y\in M$.]{}
Given a heat kernel $k$, the operator $\Delta $ in (\[eq\]) is interpreted as the *infinitesimal generator* of the *heat semigroup* $\left\{
K_{t}\right\} _{t\geq 0}$ in $L^{2}:=L^{2}(M,\mu )$. Thus we let $$K_{t}g(x)=\int_{M}k(t,x,y)g(y)d\mu (y)\ (t>0,g\in L^{2}), \label{semi:1}$$and define $\Delta $ by$$\Delta g=\lim_{t\downarrow 0}\frac{K_{t}g-g}{t}\text{ (in }L^{2}\text{-norm).} \label{generaror}$$Observe that $\{K_{t}\}_{t>0}$ is a strongly continuous and contractive semigroup in $L^{2}$, that is, for all $s,t\geq 0$ and all $g\in L^{2},$ $$\begin{aligned}
K_{s+t} &=&K_{s}K_{t}, \label{semi:2} \\
\lim_{t\rightarrow 0^{+}}||K_{t}g-g||_{2} &=&0, \notag \\
||K_{t}\phi ||_{q} &\leq &||\phi ||_{q}\text{ \ (for all }1\leq q\leq \infty
\text{).} \notag\end{aligned}$$The domain of the operator $\Delta $ is dense in $L^{2}.$
A function $u(t,x)$ is termed a *weak solution* to (\[eq\])-([eq:init]{}) if it satisfies the following integral equation $$u(t,x)=K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau
+\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau , \label{weak solution}$$where $K_{t}$ is the heat semigroup defined in (\[semi:1\]).
The structure of this paper is as follows. In Section \[non\], we show the non-existence of weak solution to (\[eq\])-(\[eq:init\]). In Section [exis]{}, we obtain sufficient conditions for the local and global existence of solutions for a range of parameters $p$, source terms $f$ and intial values $\varphi $. The critical exponents $p$ depend only on the fractal dimension $\alpha $ and the walk dimension $\beta $. Finally, in Section [regular]{}, we investigate the Hölder continuity of weak solutions.
**Notation***.* The letters $C,C_{i}(i=1,2,\ldots )$ denote positive constants whose values are unimportant and may differ at different occurrences.
Non-existence of solutions {#non}
==========================
In this section we give sufficient conditions for the non-existence of essentially bounded solutions. The exponents $p=1+\beta
/\alpha $ (where $\alpha ,\beta >0)$, and $p=\alpha /(\alpha -\beta )$, where $\alpha >\beta >0$ occur in the heat kernel bounds (\[hk-est\]), play a crucial rôle in our analysis, see Theorem \[Non-existence\]. First, we establish Lemma \[Harnack\], where condition $(k6)$ is our only assumption on the heat kernel $k$ (we do not need the conservative property of $k$ at this stage.)
The following properties the functions $\Phi _{1}$ and $\Phi _{2}$ in condition $(k6)$ may or may not hold: there exist positive constants $a_{i},b_{i}$ and $c_{i}$ such that, for all $s,t\geq 0,$ $$\begin{aligned}
\Phi _{1}(s) &\geq &a_{1}\Phi _{2}(a_{2}s), \label{general1} \\
\Phi _{2}(s+t) &\geq &b_{1}\Phi _{2}(b_{2}s)\Phi _{2}(b_{3}t),
\label{general4} \\
\Phi _{1}^{p}(s) &\geq &c_{1}\Phi _{2}(c_{2}s). \label{general2}\end{aligned}$$Note that if (\[general1\]) holds, then $0<a_{1}\leq 1$ by letting $s=0$ and using the fact that $\Phi _{2}(0)\geq \Phi _{1}(0).$ Without loss of generality, we may assume that $a_{2}>1$ in (\[general1\]), since if ([general1]{}) holds for some $a_{2}\leq 1,$ it also holds for any constant $a_{2}>1$ by the monotonicity of $\Phi _{2}.$
The *Gauss-type* functions $\Phi _{1}$ and $\Phi _{2}$ $$\begin{aligned}
\Phi _{1}(s) &=&C_{1}\exp (-C_{2}s^{\gamma }), \notag \\
\Phi _{2}(s) &=&C_{3}\exp (-C_{4}s^{\gamma }),\;s\geq 0, \label{G-func}\end{aligned}$$for constants $\gamma >0$ and $C_{i}>0 \,(1\leq i\leq 4)$ satisfy properties (\[general1\])-(\[general2\]). The *Cauchy-type* functions $$\begin{aligned}
\Phi _{1}(s) &=&C_{1}\left( 1+s\right) ^{-\gamma }, \notag \\
\Phi _{2}(s) &=&C_{2}\left( 1+s\right) ^{-\gamma },\;s\geq 0 \label{C-func}\end{aligned}$$for constants $\gamma >0$ and $C_{i}>0\,(i=1,2)$, satisfy properties ([general1]{}) and (\[general4\]), but not (\[general2\]) if $p>1.$
Condition $(k6)$ and inequality (\[general1\]) lead to the following key lemma.
\[Harnack\] Assume that the heat kernel $k$ satisfies condition $(k6)$ and (\[general1\]). Then, for all non-negative measurable functions $g$ on $M$ and for all $t>0,x\in M,$ $$\begin{aligned}
K_{t}g(x) &\geq &A_{1}K_{Bt}g(x), \label{harnack1} \\
\int_{0}^{t}K_{\tau }g(x)d\tau &\geq &A_{2}tK_{B^{2}t}g(x), \label{harnack2}\end{aligned}$$where $A_{1}=a_{1}a_{2}^{-\alpha }<1,A_{2}=a_{1}a_{2}^{-2\alpha
}(1-a_{2}^{-\beta })<1$ and $B=a_{2}^{-\beta }<1.$ Consequently, for all non-negative measurable functions $\varphi ,$ $$K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }g(x)d\tau \geq A\left[
K_{B_{1}t}\varphi (x)+tK_{B_{^{1}}t}g(x)\right] , \label{eqn-3}$$where $A=\min \left\{ A_{1}^{2},A_{2}\right\} <1$ and $B_{1}=B^{2}=a_{2}^{-2\beta }.$
It follows from condition $(k6)$ and (\[general1\]) that $$\begin{aligned}
K_{t}g(x) &=&\int_{M}k(t,x,y)g(y)d\mu (y) \notag \\
&\geq &\int_{M}\frac{1}{t^{\alpha /\beta }}\Phi _{1}\left( \frac{d(x,y)}{t^{1/\beta }}\right) g(y)d\mu (y) \notag \\
&\geq &a_{1}\int_{M}\frac{1}{t^{\alpha /\beta }}\Phi _{2}\left( a_{2}\frac{d(x,y)}{t^{1/\beta }}\right) g(y)d\mu (y). \label{eqn-1}\end{aligned}$$which gives that, using $(k6)$ again, $$\begin{aligned}
K_{t}g(x) &\geq &a_{1}a_{2}^{-\alpha }\int_{M}k\left( a_{2}^{-\beta
}t,x,y\right) g(y)d\mu (y) \\
&=&a_{1}a_{2}^{-\alpha }K_{a_{2}^{-\beta }t}g(x)=A_{1}K_{Bt}g(x),\end{aligned}$$proving (\[harnack1\]).
To show (\[harnack2\]), we see from (\[eqn-1\]) that for all $\tau \in
\lbrack a_{2}^{-\beta }t,t],$ using the monotonicity of $\Phi _{2}$ and condition $(k6),$ $$\begin{aligned}
K_{\tau }g(x) &\geq &a_{1}\int_{M}\frac{1}{\tau ^{\alpha /\beta }}\Phi
_{2}\left( a_{2}\frac{d(x,y)}{\tau ^{1/\beta }}\right) g(y)d\mu (y) \\
&\geq &a_{1}\int_{M}\frac{1}{t^{\alpha /\beta }}\Phi _{2}\left( a_{2}\frac{d(x,y)}{(a_{2}^{-\beta }t)^{1/\beta }}\right) g(y)d\mu (y) \\
&\geq &a_{1}a_{2}^{-2\alpha }\int_{M}k\left( a_{2}^{-2\beta }t,x,y\right)
g(y)d\mu (y) \\
&=&a_{1}a_{2}^{-2\alpha }K_{B^{2}t}g(x).\end{aligned}$$Therefore,$$\begin{aligned}
\int_{0}^{t}K_{\tau }g(x)d\tau &\geq &\int_{a_{2}^{-\beta }t}^{t}K_{\tau
}g(x)d\tau \\
&\geq &\int_{a_{2}^{-\beta }t}^{t}a_{1}a_{2}^{-2\alpha }K_{B^{2}t}g(x)d\tau
\\
&=&a_{1}a_{2}^{-2\alpha }\left( 1-a_{2}^{-\beta }\right) tK_{B^{2}t}g(x),\end{aligned}$$proving (\[harnack2\]).
Finally, replacing $t$ by $Bt,$ we see from (\[harnack1\]) that $K_{Bt}\varphi (x)\geq A_{1}K_{B^{2}t}\varphi (x)=A_{1}K_{B_{1}t}\varphi (x),$ and thus $$K_{t}\varphi (x)\geq A_{1}K_{Bt}\varphi (x)\geq AK_{B_{1}t}\varphi (x).
\label{eqn-2}$$Adding (\[harnack2\]) and (\[eqn-2\]), we obtain (\[eqn-3\]).
Lemma \[Harnack\] gives the following estimate (\[nonexistence-bound\]) that plays an important rôle in proving the non-existence of global bounded solutions.
\[Non-existence\] Assume that the heat kernel $k$ satisfies conditions $(k6)$ and (\[general1\]). Let $u(t,x)$ be a non-negative essentially bounded solution of (\[weak solution\]) in $(0,T)\times M$. Then, for all $(t,x)\in (0,T)\times M,$ $$t^{1/(p-1)}K_{B_{1}t}\varphi (x)+t^{p/(p-1)}K_{B_{1}t}f(x)\leq C_{1},
\label{nonexistence-bound}$$where $B_{1}=a_{2}^{-2\beta }$ as before, and $C_{1}$ depends only on $p$ (and in particular is independent of $T,\varphi $ and $f$).
Observe that by condition $(k1)$ and using a weighted Hölder inequality$, $ for all $t>0,x\in M$ and for all non-negative functions $g,$ $$\begin{aligned}
K_{t}\left( g^{p}\right) (x) &=&\int_{M}k(t,x,y)g^{p}(y)d\mu (y) \\
&\geq &\left[ \int_{M}k(t,x,y)g(y)d\mu (y)\right] ^{p}=\left[ K_{t}g(x)\right] ^{p}.\end{aligned}$$It follows from (\[weak solution\]) and (\[eqn-2\]) that $$\begin{aligned}
u(t,x) &\geq &\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau \notag \\
&\geq &A\int_{0}^{t}K_{B_{1}(t-\tau )}u^{p}(\tau ,x)d\tau \notag \\
&\geq &A\int_{0}^{t}\left[ K_{B_{1}(t-\tau )}u(\tau ,x)\right] ^{p}d\tau .
\label{iter}\end{aligned}$$From (\[weak solution\]) and (\[eqn-3\]), we see that $$\begin{aligned}
u(t,x) &\geq &K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau \notag \\
&\geq &A\left( K_{B_{1}t}\varphi (x)+tK_{B_{1}t}f(x)\right) .
\label{estimate-1}\end{aligned}$$Starting from (\[estimate-1\]), we shall apply (\[iter\]) repeatedly to deduce the desired inequality (\[nonexistence-bound\]). Indeed, we obtain from (\[iter\]) and (\[estimate-1\]) that, using the semigroup property (\[semi:2\]) of $\left\{ K_{t}\right\} _{t\geq 0}$ and the elementary inequality $(a+b)^{p}\geq a^{p}+b^{p}$ for all $p\geq 1$ and $a,b\geq 0$, $$\begin{aligned}
u(t,x) &\geq &A\int_{0}^{t}\left[ K_{B_{1}(t-\tau )}u(\tau ,x)\right]
^{p}d\tau \\
&\geq &A\int_{0}^{t}\left[ K_{B_{1}(t-\tau )}\left\{ A\left( K_{B_{1}\tau
}\varphi +\tau K_{B_{1}\tau }f\right) \right\} (x)\right] ^{p}d\tau \\
&=&A^{p+1}\int_{0}^{t}[K_{B_{1}t}\varphi (x)+\tau K_{B_{1}t}f(x)]^{p}d\tau \\
&\geq &A^{p+1}\left\{ t\left( K_{B_{1}t}\varphi (x)\right)
^{p}+\int_{0}^{t}\tau ^{p}(K_{B_{1}t}f(x))^{p}d\tau \right\} \\
&=&A^{p+1}\left\{ t\left( K_{B_{1}t}\varphi (x)\right) ^{p}+\frac{1}{1+p}t^{1+p}(K_{B_{1}t}f(x))^{p}\right\} .\end{aligned}$$Repeating the above procedure, we obtain that for all $n\geq 1,$ $$\begin{aligned}
u(t,x) &\geq &A^{1+p+\cdots +p^{n}}\left\{ \frac{t^{1+p+\cdots
+p^{n-1}}[K_{Bt}\varphi (x)]^{p^{n}}}{(1+p)^{p^{n-2}}(1+p+p^{2})^{p^{n-3}}\cdots (1+p+\cdots +p^{n-1})}\right. \\
&&+\left. \frac{t^{1+p+\cdots +p^{n}}[K_{Bt}f(x)]^{p^{n}}}{(1+p)^{p^{n-1}}(1+p+p^{2})^{p^{n-2}}\cdots (1+p+\cdots +p^{n})}\right\} .\end{aligned}$$It follows that $$\begin{aligned}
A^{(p^{n+1}-1)/(p-1)p^{n}}& t^{(p^{n}-1)/(p-1)p^{n}}K_{Bt}\varphi (x) \notag
\\
\leq & u(t,x)^{p^{-n}}\prod_{i=2}^{n}(1+p+\cdots +p^{i-1})^{p^{-i}},
\label{bound1} \\
A^{(p^{n+1}-1)/(p-1)p^{n}}& t^{(p^{n+1}-1)/(p-1)p^{n}}K_{Bt}f(x) \notag \\
\leq & u(t,x)^{p^{-n}}\prod_{i=1}^{n}(1+p+\cdots +p^{i})^{p^{-i}}.
\label{bound2}\end{aligned}$$Since $$\begin{aligned}
\log \prod_{i=2}^{n}(1+p+\cdots +p^{i-1})^{p^{-i}} &\leq &\sum_{i=2}^{\infty
}\frac{1}{p^{i}}\log (ip^{i})<+\infty , \\
\log \prod_{i=1}^{n}(1+p+\cdots +p^{i})^{p^{-i}} &\leq &\sum_{i=1}^{\infty }\frac{1}{p^{i}}\log ((i+1)p^{i})<+\infty ,\end{aligned}$$and that $u(t,x)$ is essentially bounded on $(0,T)\times M$, we pass to the limit as $n\rightarrow \infty $ in (\[bound1\]) and (\[bound2\]), and conclude that $$\begin{aligned}
t^{1/(p-1)}K_{Bt}\varphi (x) &\leq &C_{1}/2, \label{varphi-bound} \\
t^{p/(p-1)}K_{Bt}f(x) &\leq &C_{1}/2, \label{f-bound}\end{aligned}$$for some $C_{1}>0.$ Adding (\[varphi-bound\]) and (\[f-bound\]), we obtain (\[nonexistence-bound\]).
We are now in a position to obtain the main results of this section.
\[Non-existence1\]Assume that the heat kernel $k$ satisfies conditions $(k6)$ and (\[general1\]). Then the problem (\[eq\])-(\[eq:init\]) does not have any essentially bounded global solution in each of the following cases:
1. [if $p<1+\frac{\beta }{\alpha }$ and if either $\varphi
(x)\gvertneqq 0$ or $f(x)\gvertneqq 0$.]{}
2. [if $\alpha \leq \beta $ and if $f(x)\gvertneqq 0$;]{}
3. [if $\alpha >\beta $ and $p<\frac{\alpha }{\alpha -\beta }(>1+\frac{\beta }{\alpha })$ and if $f(x)\gvertneqq 0$. ]{}
We prove the results by contradiction. Assume that $u(t,x)$ is a non-negative essentially bounded global solution. Replacing $B_{1}t$ by $t$, we see from (\[nonexistence-bound\]) that for all $x\in M$ and $t>0,$ $$t^{1/(p-1)}K_{t}\varphi (x)+t^{p/(p-1)}K_{t}f(x)\leq C_{1},
\label{simple-form}$$where $0<C_{1}< \infty$ is independent of $\varphi $ and $f$.
*Proof of Case (i):* If $\varphi (x)\gvertneqq 0$, we see from $(k6)$, using Fatou’s lemma, that $$\begin{aligned}
\liminf_{t\rightarrow \infty }\,t^{\alpha /\beta }K_{t}\varphi (x) &\geq
&\liminf_{t\rightarrow \infty }\int_{M}\Phi _{1}\left( \frac{d(x,y)}{t^{1/\beta }}\right) \varphi (y)d\mu (y) \\
&\geq &C_{2}.\end{aligned}$$where $C_{2}=1$ if $||\varphi ||_{1}=\infty $, and $C_{2}=\Phi
_{1}(0)||\varphi ||_{1}$ if $||\varphi ||_{1}<\infty $. However, as $1/(p-1)>\alpha /\beta $, this is impossible by using (\[simple-form\]). Hence, (\[eq\])-(\[eq:init\]) does not have any global essentially bounded solution.
If $f(x)\gvertneqq 0$, observe that $u(t+t_{0},x)$ is a weak solution of (\[weak solution\]) with initial data $\varphi (x)=u(t_{0},x)$. We may find $t_{0}>0$ such that $u(t_{0},x)\gvertneqq 0$. Repeating the above argument, we again see that (\[eq\])-(\[eq:init\]) does not have any global essentially bounded solution.
*Proof of Case (ii):* Observe that by (\[weak solution\]) and ([harnack2]{}),$$u(t,x)\geq \int_{0}^{t}K_{\tau }f(x)d\tau \geq A_{2}tK_{B_{1}t}f(x).
\label{eqn-4}$$
We distinguish two cases: $\alpha <\beta $ and $\alpha =\beta .$
$\bullet $ The case $\alpha <\beta $ . It follows from (\[eqn-4\]) and $(k6)$ that$$\begin{aligned}
\liminf_{t\rightarrow \infty }t^{(\alpha/\beta)-1}u(t,x) &\geq
&A_{2}\liminf_{t\rightarrow \infty }t^{\alpha/\beta}K_{B_{1}t}f(x) \notag \\
&\geq &A_{2}\liminf_{t\rightarrow \infty }t^{\alpha/\beta}\int_{M}\frac{1}{(B_{1}t)^{\alpha /\beta }}\Phi _{1}\left( \frac{d(x,y)}{(B_{1}t)^{1/\beta }}\right) f(y)d\mu (y) \notag \\
&\geq &C_{3}, \label{case2-1}\end{aligned}$$where $C_{3}=1$ if $||f||_{1}=\infty $ and $C_{3}=A_{2}B_{1}^{-\alpha /\beta
}\Phi _{1}(0)>0$ if $||f||_{1}<\infty $. However, since $u$ is globally essentially bounded and $\alpha/\beta<1,$ we see$$\liminf_{t\rightarrow \infty }t^{(\alpha/\beta)-1}u(t,x)=0,$$a contradiction.
$\bullet $ The case $\alpha =\beta $. For $t>1,$ it follows from ($k6$), ( \[general1\]) and the monotonicity of $\Phi _{2}$ that$$\begin{aligned}
u(t,x) &\geq &\int_{0}^{t}K_{\tau }f(x)d\tau \notag \\
&\geq &\int_{0}^{t}d\tau \int_{M}\tau ^{-1}\Phi _{1}\left( \frac{d(x,y)}{\tau ^{1/\beta }}\right) f(y)d\mu (y) \notag \\
&\geq &a_{1}\int_{1}^{t}d\tau \int_{M}\tau ^{-1}\Phi _{2}\left( a_{2}\frac{d(x,y)}{\tau ^{1/\beta }}\right) f(y)d\mu (y) \notag \\
&\geq &a_{1}\int_{1}^{t}\tau ^{-1}d\tau \int_{M}\Phi
_{2}(a_{2}d(x,y))f(y)d\mu (y). \label{case2-2}\end{aligned}$$Since [$f(x)\gvertneqq 0$]{}, we can find a point $x\in M$ such that $$\int_{M}\Phi _{2}(a_{2}d(x,y))f(y)d\mu (y)>0.$$Passing to the limit as $t\rightarrow \infty $ in (\[case2-2\]) this contradicts that $u$ is globally essentially bounded.
*Proof of Case (iii):* It follows from (\[simple-form\]) and ($k6$) that $$\begin{aligned}
\liminf_{t\rightarrow \infty }C_{1}t^{\alpha /\beta \,-\,p/(p-1)} &\geq
&\liminf_{t\rightarrow \infty }t^{\alpha /\beta }K_{t}f(x) \\
&\geq &\liminf_{t\rightarrow \infty }\int_{M}\Phi _{1}\left( \frac{d(x,y)}{t^{1/\beta }}\right) f(y)d\mu (y) \\
&\geq &C_{4},\end{aligned}$$where $C_{4}=1$ if $||f||_{1}=\infty $ and $C_{4}=\Phi _{1}(0)||f||_{1}$ if $||f||_{1}<\infty $. However, this is impossible since $\frac{\alpha }{\beta }-\frac{p}{p-1}<0$. The proof is complete.
In Theorem \[Non-existence1\], we do not know in general if there exists any essentially bounded global solution for two the critical cases $p=1+\frac{\beta }{\alpha }$ $(\alpha ,\beta >0)$ and $p=\frac{\alpha }{\alpha -\beta }$ $(\alpha >\beta >0)$.
However, Theorem \[Non-existence1\] $(i)$ may be improved to include the critical exponent $p=1+\frac{\beta }{\alpha }$ under further assumptions [([general4]{}) and ]{}(\[general2\]) on the heat kernel $k.$ We first need the following property.
If $\Phi _{2}$ satisfies (\[general4\]), then for all $t>0$ and all $x,y\in M$, $$\frac{\Phi _{2}\left( d(x,y)t^{-1/\beta }\right) }{\Phi _{2}\left(
b_{2}d(x,0)t^{-1/\beta }\right) }\geq b_{1}\Phi _{2}\left(
b_{3}d(y,0)t^{-1/\beta }\right) , \label{general3}$$where the constants $b_{i}\, (i=1,2,3)$ are as in (\[general4\]).
Since $\Phi _{2}$ is strictly positive and decreasing on $[0,\infty )$ and $d(x,y)\leq d(x,0)+d(y,0),$ we have $$\Phi _{2}\left( d(x,y)t^{-1/\beta }\right) \geq \Phi _{2}\left(
d(x,0)t^{-1/\beta }+d(y,0)t^{-1/\beta }\right) . \label{monotone}$$It follows from (\[general4\]) that $$\Phi _{2}\left( d(x,0)t^{-1/\beta }+d(y,0)t^{-1/\beta }\right) \geq
b_{1}\Phi _{2}\left( b_{2}d(x,0)t^{-1/\beta }\right) \Phi _{2}\left(
b_{3}d(y,0)t^{-1/\beta }\right) ,$$which combines with (\[monotone\]) to give (\[general3\]).
\[T-nonexis2\] Assume that the heat kernel $k$ satisfies conditions $(k5),(k6)$ and (\[general1\]), [(\[general4\]) and ]{}(\[general2\]). Then (\[eq\])-(\[eq:init\]) does not have any essentially bounded global solutions if $p\leq 1+\frac{\beta }{\alpha }$ and if either $\varphi
(x)\gvertneqq 0$ or $f(x)\gvertneqq 0$.
In view of Theorem \[Non-existence1\] $(i)$ it is enough to consider the critical exponent $p=1+\beta /\alpha .$ We only consider the case $\varphi
(x)\gvertneqq 0$ (the case $f(x)\gvertneqq 0$ may be treated in a similar way). Then (\[simple-form\]) becomes $$t^{\alpha /\beta }K_{t}\varphi (x)+t^{1+\alpha /\beta }K_{t}f(x)\leq C_{1}.$$From condition $(k6)$ $$\int_{M}\varphi (y)d\mu (y)\leq C_{2}, \label{case1-0}$$where $C_{2}=C_{1}/\Phi _{1}(0)$. For any $t_{0}>0$, the function $v(t,x)\equiv u(t+t_{0},x)$ is a weak solution to (\[weak solution\]) with initial data $\varphi (x)=u(t_{0},x)$. Repeating the procedure of ([case1-0]{}), we have that for all $t_{0}>0,$ $$\int_{M}u(t_{0},y)d\mu (y)\leq C_{2}. \label{case1-1}$$
We claim that there exist positive constants $\gamma ,\rho $ possibly depending on $t_{0}$ and $\varphi $ such that, for all $x\in M,$ $$u(t_{0},x)\geq \rho k(\gamma ,x,0). \label{case1-2}$$To see this, observe that $$\Phi _{2}\left( d(x,0)\gamma ^{-1/\beta }\right) \geq k(\gamma ,x,0)\gamma
^{\alpha /\beta },$$and thus, using (\[general3\]) and setting $\gamma =(a_{1}b_{2})^{-\beta
}t_{0},$ $$\begin{aligned}
\Phi _{2}\left( a_{2}d(x,y)t_{0}^{-1/\beta }\right) &\geq &b_{1}\Phi
_{2}\left( a_{1}b_{3}d(y,0)t_{0}^{-1/\beta }\right) \Phi _{2}\left(
a_{1}b_{2}d(x,0)t_{0}^{-1/\beta }\right) \\
&\geq &b_{1}\Phi _{2}\left( a_{1}b_{3}d(y,0)t_{0}^{-1/\beta }\right)
k(\gamma ,x,0)\gamma ^{\alpha /\beta }.\end{aligned}$$Using (\[weak solution\]) and (\[general1\]), $$\begin{aligned}
u(t_{0},x) &\geq &\int_{M}k(t_{0},x,y)\varphi (y)d\mu (y) \\
&\geq &t_{0}^{-\alpha /\beta}\int_{M}\Phi _{1}\left( d(x,y)t_{0}^{-1/\beta
}\right) \varphi (y)d\mu (y) \\
&\geq &a_{1}t_{0}^{-\alpha /\beta}\int_{M}\Phi _{2}\left(
a_{2}d(x,y)t_{0}^{-1/\beta }\right) \varphi (y)d\mu (y) \\
&\geq &a_{1}b_{1}\left( \frac{\gamma }{t_{0}}\right) ^{\alpha
/\beta}k(\gamma ,x,0)\int_{M}\Phi _{2}\left( a_{1}b_{3}d(y,0)t_{0}^{-1/\beta
}\right) \varphi (y)d\mu (y),\end{aligned}$$and hence, inequality (\[case1-2\]) holds by setting$$\rho :=a_{1}b_{1}\left( \frac{\gamma }{t_{0}}\right) ^{\alpha /\beta
}\int_{M}\Phi _{2}\left( a_{1}b_{3}d(y,0)t_{0}^{-1/\beta }\right) \varphi
(y)d\mu (y),$$proving our claim.
Consider $v(t,x)\equiv u(t+t_{0},x)$ such that $u(t_{0},x)\gvertneqq 0$. Applying (\[case1-2\]), we obtain $$\begin{aligned}
v(t,x) &\geq &\int_{M}k(t,x,y)u(t_{0},y)d\mu (y) \\
&\geq &\rho \int_{M}k(t,x,y)k(\gamma ,y,0)d\mu (y) \\
&=&\rho k(t+\gamma ,x,0),\end{aligned}$$which yields that, using (\[weak solution\]), $(k5)$ and Fubini’s theorem, $$\begin{aligned}
\int_{M}v(t,x)d\mu (x) &\geq &\int_{M}d\mu (x)\int_{0}^{t}d\tau
\int_{M}k(t-\tau ,x,y)v^{p}(\tau ,y)d\mu (y) \notag \\
&=&\int_{0}^{t}d\tau \int_{M}v^{p}(\tau ,y)d\mu (y) \notag \\
&\geq &\rho ^{p}\int_{0}^{t}d\tau \int_{M}k^{p}(\tau +\gamma ,y,0)d\mu (y).
\label{case1-3}\end{aligned}$$As $p=1+\beta /\alpha $, we see from (\[general2\]) and $(k6)$ that $$\begin{aligned}
k^{p}(\tau +\gamma ,y,0) &\geq &(\tau +\gamma )^{-(1+\alpha /\beta )}\Phi
_{1}^{p}\left( d(y,0)(\tau +\gamma )^{-1/\beta }\right) \\
&\geq &c_{1}(\tau +\gamma )^{-(1+\alpha /\beta )}\Phi _{2}\left(
c_{2}d(y,0)(\tau +\gamma )^{-1/\beta }\right) \\
&=&c_{1}c_{2}^{-\alpha }(\tau +\gamma )^{-1}[c_{2}^{-\beta }(\tau +\gamma
)]^{-\alpha /\beta }\Phi _{2}\left( c_{2}d(y,0)(\tau +\gamma )^{-1/\beta
}\right) \\
&\geq &c_{1}c_{2}^{-\alpha }(\tau +\gamma )^{-1}k(c_{2}^{-\beta }(\tau
+\gamma ),y,0),\end{aligned}$$which combines with (\[case1-3\]) to give that $$\int_{M}v(t,x)d\mu (x)\geq c_{1}c_{2}^{-\alpha }\rho ^{p}\int_{0}^{t}(\tau
+\gamma )^{-1}d\tau . \label{case1-4}$$Passing to the limit as $t\rightarrow \infty $, we conclude that $$\int_{M}v(t,x)d\mu (x)\rightarrow \infty ,$$which contradicts (\[case1-1\]).
We note that our results agree with the earlier ones where $M=\mathbb{R}^{n}$ and $\mu $ is Lebesgue measure, and where the heat kernel $k$ is the Gauss-Weierstrass function (so that $\Delta $ is the usual Laplacian), see [@Fujita66; @Wei80; @Wei81; @Bandle00]. See also [@FalHu01] where $M$ is a fractal and $\mu $ is $\alpha $-dimenisonal Hausdorff measure, and where $k$ is the Gauss-type heat kernel on $M.$
Existence of solutions {#exis}
======================
In this section we give sufficient conditions for local existence and global existence of weak solutions.
\[existence\] Suppose that the heat kernel $k$ satisfies $(k6)$. Let $b(t)$ be a continuously differentiable function on $[0,T_{0})$ satisfying $$b^{\prime }(t)=b^{p}(t)\left[ \int_{0}^{t}\frac{||K_{\tau }f||_{\infty }}{b(\tau )}d\tau +||K_{t}\varphi ||_{\infty }\right] ^{p-1} \label{b(t)-ode}$$with initial value $b(0)=1$. If $$\int_{0}^{T_{0}}\left[ \int_{0}^{s}\frac{||K_{\tau }f||_{\infty }}{b(\tau )}d\tau +||K_{s}\varphi ||_{\infty }\right] ^{p-1}ds\leq \frac{1}{p-1},
\label{existence-estimate}$$then (\[eq\])-(\[eq:init\]) has a non-negative local solution $u\in
L^{\infty }((0,T),M)$ for all $0< T<T_{0}$, provided that $
||\varphi ||_{\infty }<\infty $.
\[R-exis\]By Peano’s theorem, there exists some $T_{0}>0$ and some continuous differentiable function $b(t)$ such that (\[b(t)-ode\]) holds in $[0,T_{0})$. Clearly, such a $b(t)$ is non-decreasing in $[0,T_{0})$. On the other hand, condition (\[existence-estimate\]) may be verified for some specific cases. For example, if $\left( k5\right) $ holds and if $f=0,\varphi =C>0,$ then $$b(t)=[1-(p-1)C^{p-1}t]^{-1/(p-1)}$$satisfies (\[b(t)-ode\]) in $[0,T_{0})$ where $T_{0}=(p-1)^{-1}C^{-(p-1)}$, and (\[existence-estimate\]) also holds. As an another example, let $f=1,\varphi =0$ and assume $\left( k5\right) $ holds. Then, for $p=2,$ we see that $b(t)=1/\cos t$ satisfies (\[b(t)-ode\]) for $t\in \lbrack 0,\pi
/2)$, and that (\[existence-estimate\]) holds.
Define $$a(t)=b(t)\int_{0}^{t}\frac{||K_{\tau }f||_{\infty }}{b(\tau )}d\tau .$$ Note that $a(0)=0$ and $a(t)\geq 0$ for $t\in \lbrack 0,T_{0})$. Incorporating this into (\[b(t)-ode\]), we get $$b^{\prime }(t) =b(t)\left[ a(t)+b(t)||K_{t}\varphi ||_{\infty }\right]
^{p-1} \label{b(t)}.$$ Moreover, $$\begin{aligned}
a^{\prime }(t) &=& ||K_{t}f||_{\infty } + \frac{b'(t)a(t)}{b(t)}\nonumber\\
&=& ||K_{t}f||_{\infty }+a(t)\left[ a(t)+b(t)||K_{t}\varphi
||_{\infty }\right] ^{p-1}.\end{aligned}$$ Together with the initial conditions, these differential equations are equivalent to $$\begin{aligned}
a(t) &=&\int_{0}^{t}||K_{\tau }f||_{\infty }d\tau +\int_{0}^{t}a(\tau
)(a(\tau )+b(\tau )||K_{\tau }\varphi ||_{\infty })^{p-1}d\tau ,
\label{a(t)-b(t)1} \\
b(t) &=&1+\int_{0}^{t}b(\tau )(a(\tau )+b(\tau )||K_{\tau }\varphi
||_{\infty })^{p-1}d\tau . \label{a(t)-b(t)2}\end{aligned}$$
Let $
\mathcal{H}$ be the family of continuous functions $u$ satisfying $$K_{t}\varphi (x)\leq u(t,x)\leq a(t)+b(t)K_{t}\varphi (x)\ \text{ for all }(t,x)\in \lbrack 0,T_{0})\times M. \label{sub-sup}$$ Define $$\mathcal{F}u(t,x)=K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau
+\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau . \label{def-1}$$We claim that if $u\in \mathcal{H}$, then $\mathcal{F}u\in
\mathcal{H}$, that is, $$K_{t}\varphi (x)\leq \mathcal{F}u(t,x)\leq a(t)+b(t)K_{t}\varphi (x)\quad
(0\leq t <T_{0}, x \in M). \label{def-2}$$Observe that, using $(k1)$,$$\begin{aligned}
&&\hspace{-1.5cm} \int_{0}^{t}K_{t-\tau }[a(\tau )+b(\tau )K_{\tau }\varphi ]^{p}(x)d\tau \\
&=&\int_{0}^{t}d\tau \int_{M}k(t-\tau ,x,y)\left[ a(\tau )+b(\tau )K_{\tau
}\varphi (y)\right] ^{p}d\mu (y) \\
&\leq &\int_{0}^{t}[a(\tau )+b(\tau )||K_{\tau }\varphi ||_{\infty }]^{p-1}
\left[ a(\tau )+b(\tau )K_{t}\varphi (x)\right] d\tau .\end{aligned}$$It follows from (\[def-1\]) and (\[sub-sup\]) that $$\begin{aligned}
\mathcal{F}u(t,x) &\leq &K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau
+\int_{0}^{t}K_{t-\tau }[a(\tau )+b(\tau )K_{\tau }\varphi ]^{p}(x)d\tau
\notag \\
&\leq &\left[ \int_{0}^{t}||K_{\tau }f||_{\infty }d\tau +\int_{0}^{t}a(\tau
)[a(\tau )+b(\tau )||K_{\tau }\varphi ||_{\infty }]^{p-1}d\tau \right]
\notag \\
&&+\left[ 1+\int_{0}^{t}b(\tau )[a(\tau )+b(\tau )||K_{\tau }\varphi
||_{\infty }]^{p-1}d\tau \right] K_{t}\varphi (x)\\
&=& a(t)+b(t)K_{t}\varphi (x)\end{aligned}$$ using (\[a(t)-b(t)1\]) and (\[a(t)-b(t)2\]), so (\[def-2\]) holds, proving our claim.
For $n=0,1,2,\cdots ,$ define $$\begin{aligned}
u_{0}(t,x) &=&K_{t}\varphi (x), \\
u_{n+1}(t,x) &=&\mathcal{F}u_{n}(t,x).\end{aligned}$$Using (\[def-1\]) inductively, it follows that the sequence $\{u_{n}(t,x)\} $ is non-decreasing in $n$, and, for all $n\geq 0$ and all $x\in M,t\in \lbrack 0,T_{0}),$ satisfies $$K_{t}\varphi (x)\leq u_{n}(t,x)\leq a(t)+b(t)K_{t}\varphi (x).$$Let $u(t,x):=\lim\limits_{n\rightarrow \infty }u_{n}(t,x)$. Note that $K_{t}\varphi (x)\leq u(t,x)\leq a(t)+b(t)K_{t}\varphi (x)$. Using the monotone convergence theorem, we have $$\lim_{n\rightarrow \infty }\int_{0}^{t}d\tau \int_{M}k(t-\tau
,x,y)u_{n}^{p}(\tau ,y)d\mu (y)=\int_{0}^{t}d\tau \int_{M}k(t-\tau
,x,y)u^{p}(\tau ,y)d\mu (y).$$Since $u_{n}(t,x)$ satisfies $$u_{n+1}(t,x)=K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau
+\int_{0}^{t}K_{t-\tau }u_{n}^{p}(\tau ,x)d\tau , \label{limits}$$we pass to the limit as $n\rightarrow \infty $ to obtain $$u(t,x)=K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau
+\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau ,$$which shows that $u(t,x)$ is a non-negative local solution of (\[eq\])-(\[eq:init\]) for $t\in \lbrack 0,T_{0})$.
Since $a(t)$, $b(t)$ are differentiable functions on $[0,T_{0})$, we see from (\[sub-sup\]) that for all $t\in \lbrack 0,T_{0})$, $$||u(t,\cdot )||_{\infty }\leq ||a(t)+b(t)K_{t}\varphi ||_{\infty }<\infty .$$The proof is complete.
Recall that, by Theorem \[Non-existence1\], (\[eq\])-(\[eq:init\]) does not have any essentially bounded global weak solution [if $\alpha >\beta $ and $p<\frac{\alpha }{\alpha -\beta }$ and if $f(x)\gvertneqq 0$]{}. However, we can show that (\[eq\])-(\[eq:init\]) possesses a essentially bounded global solution if [$p>\frac{\alpha }{\alpha -\beta },$ for small functions ]{}$f$ and $\varphi $ (cf. [@zhang98] for Euclidean spaces). To do this, we need some integral estimates which are consequences of measure bounds for small and large balls.
Recall that a measure $\mu $ on a metric measure space is [*upper $\alpha $-regular*]{} if there exist some $C,\alpha >0$ such that $$\mu (B(x,r))\leq Cr^{\alpha }\text{ \ (for all }x\in M,r>0\text{),} \label{upper reg}$$ and is $\alpha $-*regular* if there exists a constant $C>0$ such that for all $x\in M$ and all $r>0,$ $$C^{-1}r^{\alpha }\leq \mu (B(x,r))\leq Cr^{\alpha }\text{ \ (for all }x\in M,r>0\text{).}
\label{volume}$$It was shown in [@GHL03 Theorem 3.2] that if the heat kernel $k$ satisfies $(k5),(k6)$ with $\Phi _{2}(s)$ satisfying$$\int_{0}^{\infty }s^{\alpha -1}\Phi _{2}(s)ds<\infty , \label{phi}$$then the measure $\mu $ is $\alpha $-*regular*. Note that, by the monotonicity of $\Phi _{2},$ condition (\[phi\]) implies that $s^{\alpha }\Phi _{2}(s)\leq C<\infty $ for all $s\in \lbrack 0,\infty
).$
\[sup-radial\] Assume that $\mu $ is upper $\alpha $-regular and $x_{0}$ is a reference point in $M$. If $0 < \lambda _{1}< \alpha $ and $\lambda
_{1}+\lambda _{2}>\alpha ,$ then there exists a constant $C_{0}>0$ such that$$\int_{M}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x_{0})^{\lambda
_{2}}]}d\mu (y)\leq C_{0}\quad \text{{\rm (for all }}x\in M). \label{sup-dis}$$
For each $x\in M,$ let $\Omega _{1}=\left\{ y\in M:d(y,x)\geq
d(y,x_{0})\right\} $ and $\Omega _{2}=M\setminus \Omega _{1}.$ Then $$\int_{\Omega _{1}}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x_{0})^{\lambda _{2}}]
}d\mu (y)
\leq \int_{M}\frac{1}{d(y,x_{0})^{\lambda _{1}}[1+d(y,x_{0})^{\lambda
_{2}}]}d\mu (y)$$ and $$\int_{\Omega _{2}}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x_{0})^{\lambda _{2}}]
}d\mu (y) \leq \int_{M}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x)^{\lambda _{2}}]}d\mu
(y).$$ Routine estimates using upper regularity (\[upper reg\]) now give uniform bounds on these integrals near $x_0$ and $x$ (since $ \lambda _{1}< \alpha $) and for large $d(y,x_{0})$ and $d(y,x)$ (since $\lambda
_{1}+\lambda _{2}>\alpha$), to give (\[sup-dis\]).
Assume that $\mu $ is upper $\alpha $-regular and $x_{0}$ is a reference point in $M$. If $0< \lambda _{1}<\alpha $ and $\lambda _{2}>\alpha ,$ then there exists a constant $C_{1}>0$ such that$$\int_{M}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x_{0})^{\lambda _{2}}]}d\mu
(y)\leq \frac{C_{1}}{1+d(x,x_{0})^{\lambda _{1}}}. \label{eqn-10}$$
Fix $x\in M.$ If $d(x,x_{0})\leq 1,$ then (\[eqn-10\]) directly follows from (\[sup-dis\]), since$$\int_{M}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x_{0})^{\lambda _{2}}]}d\mu
(y)\leq C_{0}\leq \frac{2C_{0}}{1+d(x,x_{0})^{\lambda _{1}}}.$$
Assume that $d(x,x_{0})\geq 1$. If $d(y,x_{0})\geq d(x,x_{0})/2$, we have that $$\frac{1}{1+d(y,x_{0})^{\lambda _{2}}}\leq \frac{C}{[1+d(y,x_{0})^{\lambda
_{2}-\lambda _{1}}]\left[ 1+d(x,x_{0})^{\lambda _{1}}\right] }$$where $C$ is independent of $x_{0},y$. Using Proposition \[sup-radial\], it follows that $$\begin{aligned}
&&\int_{d(y,x_{0})\geq d(x,x_{0})/2}\frac{1}{d(y,x)^{\lambda
_{1}}[1+d(y,x_{0})^{\lambda _{2}}]}d\mu (y) \nonumber \\
&\leq &\frac{C}{1+d(x,x_{0})^{\lambda _{1}}}\int_{d(y,x_{0})\geq
d(x,x_{0})/2}\frac{1}{d(y,x)^{\lambda _{1}}[1+d(y,x_{0})^{\lambda
_{2}-\lambda _{1}}]}d\mu (y) \nonumber \\
&\leq &\frac{C_{1}}{1+d(x,x_{0})^{\lambda _{1}}}. \label{eqn-11}\end{aligned}$$If $d(y,x_{0})<d(x,x_{0})/2,$ then $$d(y,x)^{-\lambda _{1}} \leq \left[ d(x,x_{0})-d(y,x_{0})\right] ^{-\lambda
_{1}}
\leq \left[ d(x,x_{0})/2\right] ^{-\lambda _{1}}\leq \frac{2^{\lambda
_{1}+1}}{1+d(x,x_{0})^{\lambda _{1}}},$$ and hence, $$\int_{d(y,x_{0})<d(x,x_{0})/2}\frac{1}{d(y,x)^{\lambda
_{1}}[1+d(y,x_{0})^{\lambda _{2}}]}d\mu (y)\leq \frac{C_{2}}{1+d(x,x_{0})^{\lambda _{1}}}. \label{eqn-12}$$where we have used that $\int_{M}\frac{1}{1+d(y,x_{0})^{\lambda _{2}}}d\mu (y)<\infty $ as $\lambda _{2}>\alpha $.
Adding (\[eqn-11\]) and (\[eqn-12\]) we see that (\[eqn-10\]) also holds if $d(x,x_{0})\geq 1.$
We now show the global existence of weak solutions for small $\varphi $ and $f.$
\[global\] Let $\alpha >\beta >0$ and suppose that the heat kernel $k$ satisfies $(k5),(k6)$ and that $\Phi _{2}$ satisfies (\[phi\]). Let $\lambda >\alpha $ and let $x_{0}$ be a reference point in $M$. The for each $p>\alpha/(\alpha -\beta)$ there exists $\delta >0$ such that if $$0<\varphi (x),f(x)\leq \frac{\delta }{1+d(x,x_{0})^{\lambda }}$$ then (\[eq\])-(\[eq:init\]) has an essentially bounded global solution.
Recall that conditions $(k5),(k6)$ and (\[phi\]) imply that $\mu $ is $
\alpha $-regular. Let the map $\mathcal{F}$ be defined as in (\[def-1\]): $$\mathcal{F}u(t,x)=K_{t}\varphi (x)+\int_{0}^{t}K_{\tau }f(x)d\tau
+\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau .$$For $\epsilon >0$, let $S_{\varepsilon }$ be the complete subset of the Banach space $L^{\infty }\left( \lbrack 0,\infty
)\times M \right)$ given by $$S_{\varepsilon }=\left\{ u\in L^{\infty }\left( \lbrack 0,\infty
)\times M \right) :0\leq u(t,x)\leq \frac{\varepsilon }{1+d(x,x_{0})^{\alpha -\beta }}
\right\}$$ We will use the contraction principle to show that, for appropriately small $\epsilon$ and $\delta$, there exists a global solution in $S_{\varepsilon }$.
For $\lambda>\alpha$, we claim that there exists $C_{2}>0$ such that, for all $0\leq g(x)\leq \delta/(1+d(x,x_{0})^{\lambda })$, we have $$K_{t}g(x)\leq \frac{C_{2}\delta }{1+d(x,x_{0})^{^{\alpha }}}\text{ for all }
x\in M\text{ and all }t>0. \label{Kg}$$ To see this, let $x\in M.$ If $d(x,x_{0})\leq 1$, then (\[Kg\]) is clear since $$\begin{aligned}
K_{t}g(x) &=&\int_{M}k(t,x,y)g(y)d\mu (y) \\
&\leq &\int_{M}\frac{\delta }{1+d(y,x_{0})^{\lambda }}k(t,x,y)d\mu (y) \\
&\leq &\delta \int_{M}k(t,x,y)d\mu (y)\leq \delta \\
&\leq &\frac{2\delta }{1+d(x,x_{0})^{^{\alpha }}}.\end{aligned}$$So assume $d(x,x_{0})>1.$ We have, using condition $\left( k6\right)
, $ $$\begin{aligned}
K_{t}g(x) &\leq &\int_{M}\frac{\delta }{1+d(y,x_{0})^{\lambda }}k(t,x,y)d\mu
(y) \notag \\
&\leq &\delta \left\{ \int_{\Omega _{1}}\frac{1}{1+d(y,x_{0})^{\lambda }}
\frac{1}{t^{\alpha /\beta }}\Phi _{2}\left( \frac{d(y,x)}{t^{1/\beta }}
\right) d\mu (y)\right. \notag \\
&&\quad +\left. \int_{\Omega _{2}}\frac{1}{1+d(y,x_{0})^{\lambda }}k(t,x,y)d\mu
(y)\right\}, \label{eqn-30}\end{aligned}$$ where $\Omega _{1}=\left\{ y\in M:d(y,x_{0})\leq d(x,x_{0})/2\right\} $ and $
\Omega _{2}=M\setminus \Omega _{1}.$ For $y\in \Omega _{1},$ we have, noting from (\[phi\]) that $s^{\alpha} \Phi_2(s)$ is bounded, $$\begin{aligned}
\frac{1}{t^{\alpha /\beta }}\Phi _{2}\left( \frac{d(y,x)}{t^{1/\beta }}
\right) &=&\frac{1}{d(y,x)^{\alpha }}\left( \frac{d(y,x)}{t^{1/\beta }}
\right) ^{\alpha }\Phi _{2}\left( \frac{d(y,x)}{t^{1/\beta }}\right) \\
&\leq &\frac{C}{d(y,x)^{\alpha }}\leq \frac{2^{\alpha }C}{d(x,x_{0})^{\alpha
}} \\
&\leq &\frac{2^{\alpha +1}C}{1+d(x,x_{0})^{\alpha }},\end{aligned}$$and hence, using that $\int_{M}\frac{d\mu (y)}{1+d(y,x_{0})^{
\lambda }}<+\infty $ for $\lambda >\alpha ,$ $$\begin{aligned}
\int_{\Omega _{1}}\frac{1}{1+d(y,x_{0})^{\lambda }}\frac{1}{t^{\alpha /\beta
}}\Phi _{2}\left( \frac{d(y,x)}{t^{1/\beta }}\right) d\mu (y) &\leq &\frac{2^{\alpha +1}C}{1+d(x,x_{0})^{\alpha }}\int_{\Omega _{1}}\frac{d\mu (y)}{1+d(y,x_{0})^{\lambda }} \notag \\
&\leq &\frac{C}{1+d(x,x_{0})^{\alpha }}. \label{eqn-31}\end{aligned}$$ For $y\in \Omega _{2}$, $$\begin{aligned}
\int_{\Omega _{2}}\frac{1 }{1+d(y,x_{0})^{\lambda }}k(t,x,y)d\mu (y) &\leq &\frac{2^{\lambda }}{1+d(x,x_{0})^{\lambda }}\int_{\Omega _{2}}k(t,x,y)d\mu
(y) \notag \\
&\leq &\frac{C}{1+d(x,x_{0})^{\alpha }}. \label{eqn-32}\end{aligned}$$using that $\lambda >\alpha$. Adding (\[eqn-31\]) and (\[eqn-32\]), we see that (\[Kg\]) follows from (\[eqn-30\]), proving our claim.
Observe that by $\left( k6\right) $ and (\[phi\]),$$\begin{aligned}
\int_{0}^{t}k(\tau ,x,y)d\tau &\leq &\int_{0}^{t}\frac{1}{\tau ^{\alpha
/\beta }}\Phi _{2}\left( \frac{d(y,x)}{\tau ^{1/\beta }}\right) d\tau \notag
\\
&=&\frac{\beta }{d(y,x)^{\alpha -\beta }}\int_{d(x,y)/t^{1/\beta }}^{\infty
}s^{\alpha -\beta -1}\Phi _{2}(s)ds \notag \\
&\leq &\frac{\beta }{d(y,x)^{\alpha -\beta }}\int_{0}^{\infty }s^{\alpha
-\beta -1}\Phi _{2}(s)ds \notag \\
&\leq &\frac{C}{d(y,x)^{\alpha -\beta }}, \label{eqn-33}\end{aligned}$$since $$\int_{0}^{\infty }s^{\alpha -\beta -1}\Phi _{2}(s)ds
\leq \Phi _{2}(0)\int_{0}^{1}s^{\alpha -\beta -1}ds+\int_{1}^{\infty
}s^{\alpha -1}\Phi _{2}(s)ds
<+\infty,$$using the monotonicity of $\Phi_2$ and (\[phi\]).
Therefore, using (\[eqn-33\]) and (\[eqn-10\]) with $
\lambda _{1}=\alpha -\beta >0$ and $\lambda _{2}=\lambda >\alpha ,$ $$\begin{aligned}
\int_{0}^{t}K_{\tau }f(x)d\tau &=&\int_{M}\left[ \int_{0}^{t}k(\tau,x,y)d\tau \right] f(y)d\mu (y) \notag \\
&\leq &\int_{M}\frac{C}{d(y,x)^{\alpha -\beta }}\frac{\delta }{
\big(1+d(y,x_{0})^{\lambda }\big)}d\mu (y) \notag \\
&\leq &\frac{C\delta }{1+d(x,x_{0})^{\alpha -\beta }} \label{eqn-35}\end{aligned}$$for all $x\in M$ and $t>0.$ Similarly, for $u\in S_{\varepsilon },$ we have that, using (\[eqn-10\]) with $\lambda _{1}=\alpha -\beta ,\lambda
_{2}=p(\alpha -\beta )>\alpha ,$ $$\begin{aligned}
\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau &\leq
&\int_{0}^{t}\int_{M}k(t-\tau ,x,y)\frac{\varepsilon ^{p}}{(1+d(y,x_{0})^{\alpha -\beta })^{p}}d\mu (y)d\tau \notag \\
&\leq &\int_{M}\frac{C}{d(y,x)^{\alpha -\beta }}\frac{\varepsilon ^{p}}{(1+d(y,x_{0})^{\alpha -\beta })^{p}}d\mu (y) \notag \\
&\leq &C\varepsilon ^{p}\int_{M}\frac{1}{d(y,x)^{\alpha -\beta }}\frac{1}{1+d(y,x_{0})^{(\alpha -\beta )p}}d\mu (y) \notag \\
&\leq &\frac{C\varepsilon ^{p}}{1+d(x,x_{0})^{\alpha -\beta }}
\label{eqn-36}\end{aligned}$$for all $x\in M$ and $t>0.$ It follows from (\[Kg\]), ([eqn-35]{}), (\[eqn-36\]) that if $u\in S_{\varepsilon },$ then $$\begin{aligned}
\mathcal{F}u(t,x) &\leq &\frac{C_{2}\delta }{1+d(x,x_{0})^{^{\alpha }}}+\frac{C\delta+C\varepsilon ^{p}}{1+d(x,x_{0})^{\alpha -\beta }} \\
&\leq &\frac{C_{1}(\delta +\varepsilon ^{p})}{1+d(x,x_{0})^{\alpha -\beta }}
\\
&\leq &\frac{\varepsilon }{1+d(x,x_{0})^{\alpha -\beta }}\end{aligned}$$provided that $C_{1}\left( \delta +\varepsilon ^{p}\right) \leq \varepsilon $, in which case $\mathcal{F}S_{\varepsilon }\subset S_{\varepsilon }$.
Next we show that $\mathcal{F}$ is contractive on $S_{\varepsilon }.$ Indeed, for $u_{1},u_{2}\in S_{\varepsilon }$, we have $$\left\vert \mathcal{F}u_{1}(t,x)-\mathcal{F}u_{2}(t,x)\right\vert \leq
\int_{0}^{t}\int_{M}k(t-\tau ,x,y)\left\vert u_{1}^{p}(\tau
,y)-u_{2}^{p}(\tau ,y)\right\vert d\mu (y)d\tau .$$Using the elementary inequality $$|a^{p}-b^{p}|\leq p\max \{a^{p-1},b^{p-1}\}|a-b|\text{ for }a,b\geq 0,p>1,$$and the definition of $S_{\varepsilon }$, we obtain, using ([eqn-33]{}) and (\[sup-dis\]), that $$\begin{aligned}
&&\hspace{-1.5cm}\left\vert \mathcal{F}u_{1}(t,x)-\mathcal{F}u_{2}(t,x)\right\vert \\
&\leq &||u_{1}-u_{2}||_{\infty}\int_{0}^{t}\int_{M}k(t-\tau ,x,y)\frac{p\varepsilon
^{p-1}}{[1+d(y,x_{0})^{\alpha -\beta }]^{p-1}}d\mu (y)d\tau \\
&\leq &||u_{1}-u_{2}||_{\infty}\int_{M}\frac{C}{d(y,x)^{\alpha -\beta }}\frac{p\varepsilon ^{p-1}}{1+d(y,x_{0})^{\left( \alpha -\beta \right) (p-1)}}d\mu
(y) \\
&\leq &C_{3}p\varepsilon ^{p-1}||u_{1}-u_{2}||_{\infty}.\end{aligned}$$Thus if $\epsilon$ is small enough to ensure that both $C_{3}p\varepsilon ^{p-1}<1$ and $C_{1} \varepsilon ^{p} < \varepsilon $, and then $\delta$ is chosen small enough so that $C_{1}\left( \delta +\varepsilon ^{p}\right) \leq \varepsilon $, applying Banach’s contraction principle to $\mathcal{F}$ on the complete set $S_{\varepsilon }$ implies that (\[weak solution\]) and thus (\[eq\])-(\[eq:init\]) has a global positive solution in $S_{\varepsilon }$.
Regularity {#regular}
==========
In this section, we discuss the regularity of weak solutions. We show that weak solutions are Hölder continuous in the spatial variable $x$ if the source term $f$ and initial value $\varphi $ are both Hölder continuous**.** We adapt the method used in [FalHu01]{}.
In order to obtain the regularity of weak solutions, we need to assume that the function $\Phi _{2}$ in condition $(k6)$ satisfies the following assumption: $$\int_{0}^{\infty }s^{\alpha }\Phi _{2}(s)ds<\infty ,\text{ }
\label{general5}$$where $\alpha $ is as in condition $(k6).$ Since $\Phi _{2}$ is non-increasing on $[0,\infty ),$ condition (\[general5\]) implies that $
s^{1+\alpha }\Phi _{2}(s)=o(1)$ as $s\rightarrow \infty .$
Clearly, the Gauss-type function $\Phi _{2}$ defined as in (\[G-func\]) satisfies condition (\[general5\]) for all $\gamma >0$ whilst the Cauchy-type function $\Phi _{2}$ defined as in (\[C-func\]) satisfies condition (\[general5\]) for all $\gamma >1+\alpha .$
Note that condition (\[general5\]) is stronger than (\[phi\]), and hence it implies that $\mu $ is $\alpha $-regular$.$
\[int-estimate\] Assume that $\mu $ is upper $\alpha $-regular. If $\Phi
_{2}$ satisfies (\[general5\]) then, for all $\lambda \in (0,1],$ $$\int_{M}d(x,y)^{\lambda }\Phi _{2}\left( \frac{d(x,y)}{t^{1/\beta }}\right)
d\mu (y)\leq C_{2}t^{(\alpha +\lambda )/\beta } \quad \text{ \ (for all }x\in M,t>0\text{)}
\label{int}$$for some constant $C_{2}$.
Let $g(r)=r^{\lambda }\Phi _{2}\left( \frac{r}{t^{1/\beta }}\right) $ for $
r>0.$ From (\[int\]) $g(r) = o(r^{-\alpha})$ so, by a standard argument using $\alpha$-regularity and integration by parts (see [@FalHu01 Proposition 4.1]), it follows that $$\begin{aligned}
\int_{M}d(x,y)^{\lambda }& \Phi _{2}\left( \frac{d(x,y)}{t^{1/\beta }}
\right) d\mu (y)
= \int_{M} g\big(d(x,y)\big) d\mu (y)\\
& \leq C_{1}\int_{0}^{\infty }r^{\alpha }|g^{\prime }(r)|dr \\
& =C_{1}\int_{0}^{\infty }r^{\alpha }\left\vert \lambda r^{\lambda -1}\Phi
_{2}\left( \frac{r}{t^{1/\beta }}\right) +r^{\lambda }\Phi _{2}^{\prime
}\left( \frac{r}{t^{1/\beta }}\right) t^{-1/\beta }\right\vert dr \\
& \leq C_{2}t^{(\alpha +\lambda )/\beta }\left[ \int_{0}^{\infty }\lambda
s^{\alpha +\lambda -1}\Phi _{2}(s)ds+\int_{0}^{\infty }s^{\alpha +\lambda
}\left( -\Phi _{2}^{\prime }(s)\right) ds\right] .\end{aligned}$$ By an easy calculation, the last integral $$\begin{aligned}
\int_{0}^{\infty }s^{\alpha +\lambda }\left( -\Phi _{2}^{\prime }(s)\right)
ds &=&-\left. s^{\alpha +\lambda }\Phi _{2}(s)\right\vert _{0}^{\infty
}+(\alpha +\lambda )\int_{0}^{\infty }s^{\alpha +\lambda -1}\Phi _{2}(s)ds \\
&=&(\alpha +\lambda )\int_{0}^{\infty }s^{\alpha +\lambda -1}\Phi
_{2}(s)ds\leq C_{3}\end{aligned}$$using (\[general5\]). Therefore, $$\int_{M}d(x,y)^{\lambda }\Phi \left( \frac{d(x,y)}{t^{1/\beta }}\right) d\mu
(y)\leq C_{2}t^{(\alpha +\lambda )/\beta },$$as desired.
We now show the Hölder continuity of weak solutions of (\[weak solution\]).
Assume that $\varphi $, $f\in L^{1}(M)$ are Hölder continuous with exponents $\theta _{1},\theta _{2}\in (0,1]$ respectively: for all $x_{1},x_{2}\in M,$ $$\begin{aligned}
\left\vert \varphi (x_{1})-\varphi (x_{2})\right\vert &\leq
&C_{5}d(x_{1},x_{2})^{\theta _{1}}, \label{varphi-holder} \\
\left\vert f(x_{1})-f(x_{2})\right\vert &\leq &C_{6}d(x_{1},x_{2})^{\theta
_{2}}, \label{f-holder}\end{aligned}$$where $C_{5},C_{6}>0$. Assume that the heat kernel $k$ satisfies $(k5)-(k7)$ and that $\Phi _{2}$ satisfies (\[general5\]) with $\lambda =\max \left\{
\theta _{1},\theta _{2}\right\} $. Let $u(t,x)$ be a non-negative weak solution to (\[eq\])-(\[eq:init\]) that is bounded in $(0,T)\times M$ for some $T>0$. Then $u(t,x)$ is Hölder continuous: for all $x_{1},x_{2}\in M$ and all $t\in (0,T),$ $$|u(t,x_{1})-u(t,x_{2})|\leq Cd(x_{1},x_{2})^{\theta }, \label{u-cts}$$where $\theta =\theta _{1}\sigma /(\theta _{1}+\nu \beta )$ and $C>0$ may depend on $T$ but is independent of $t,x$.
From $(k6)$, (\[varphi-holder\]), and (\[int\]), there exists $C>0$ such that for all $t>0$ and $x\in M,$ $$\begin{aligned}
\int_{M}k(t,x,y)\left\vert \varphi (y)-\varphi (x)\right\vert d\mu (y) &\leq
&C_{5}t^{-\alpha /\beta }\int_{M}d(x,y)^{\theta _{1}}\Phi _{2}\left( \frac{d(x,y)}{t^{1/\beta }}\right) d\mu (y) \notag \\
&\leq &Ct^{\theta _{1}/\beta }. \label{eqn-7}\end{aligned}$$By (\[weak solution\]) it is enough to show that each of the functions $u_{0},u_{1},u_{2}$ is Hölder continuous in $(0,T)\times M$, where $$\begin{aligned}
u_{0}(t,x) &=&K_{t}\varphi (x), \\
u_{1}(t,x) &=&\int_{0}^{t}K_{\tau }f(x)d\tau , \\
u_{2}(t,x) &=&\int_{0}^{t}K_{t-\tau }u^{p}(\tau ,x)d\tau .\end{aligned}$$We first show the Hölder continuity of $u_{0}$. Indeed, for $t>0$ and $x_{1},x_{2}\in M,$ we see from $(k7)$ that $$\begin{aligned}
|u_{0}(t,x_{1})-u_{0}(t,x_{2})| &=&\left\vert
\int_{M}(k(t,x_{1},y)-k(t,x_{2},y))\varphi (y)d\mu (y)\right\vert \notag \\
&\leq &Lt^{-\nu }d(x_{1},x_{2})^{\sigma }||\varphi ||_{1} \notag \\
&\leq &L||\varphi ||_{1}d(x_{1},x_{2})^{\sigma -\nu s_{0}}
\label{u0-estimate1}\end{aligned}$$if $t\geq d(x_{1},x_{2})^{s_{0}}$, where $s_{0}>0$ will be specified later on. On the other hand, if $t\leq d(x_{1},x_{2})^{s_{0}}$, we have, using $(k5)$, (\[eqn-7\]) and (\[varphi-holder\]), that $$\begin{aligned}
|u_{0}(t,x_{1})-u_{0}(t,x_{2})| &\leq &\left\vert
\int_{M}k(t,x_{1},y)(\varphi (y)-\varphi (x_{1}))d\mu (y)\right. \\
&&\left. +\left[ \varphi (x_{1})-\varphi (x_{2})\right] -\int_{M}k(t,x_{2},y)(\varphi (y)-\varphi (x_{2}))d\mu (y)\right\vert \\
&\leq &2Ct^{\theta _{1}/\beta }+C_{5}d(x_{1},x_{2})^{\theta _{1}} \\
&\leq &C\left[ d(x_{1},x_{2})^{s_{0}\theta _{1}/\beta
}+d(x_{1},x_{2})^{\theta _{1}}\right] .\end{aligned}$$Combining this with (\[u0-estimate1\]), it follows that $$\begin{aligned}
|u_{0}(t,x_{1})-u_{0}(t,x_{2})| &\leq &C\left[ d(x_{1},x_{2})^{\sigma -\nu
s_{0}}+d(x_{1},x_{2})^{s_{0}\theta _{1}/\beta }+d(x_{1},x_{2})^{\theta _{1}}\right] \notag \\
&\leq &Cd(x_{1},x_{2})^{\theta _{1}\sigma /(\theta _{1}+\nu \beta )},
\label{u0-holder1}\end{aligned}$$for all $t>0$ and $x_{1},x_{2}\in M$ with $d(x_{1},x_{2})\leq 1$, where $s_{0}=\sigma /\left( \nu +\frac{\theta _{1}}{\beta }\right) $ so that $\sigma -\nu s_{0}=s_{0}\theta _{1}/\beta $, and where we have used the fact that $\theta _{1}\geq s_{0}\theta _{1}/\beta $ for $\sigma \leq 1\leq \nu $ and $\beta \geq 1$.
Next we show the Hölder continuity of $u_{1}$. As with (\[eqn-7\]), we have from $(k6)$, (\[f-holder\]) and (\[int\]) that$$\int_{M}k(\tau ,x_{1},y)\left\vert f(y)-f(x_{1})\right\vert d\mu (y)\leq
C\tau ^{\theta _{2}/\beta },$$which yields that, using $(k5)$ and (\[f-holder\]), $$\begin{aligned}
\left\vert u_{1}(t,x_{1})-u_{1}(t,x_{2})\right\vert &=&\left\vert
\int_{0}^{t}\left[ K_{\tau }f(x_{1})-K_{\tau }f(x_{2})\right] d\tau
\right\vert \notag \\
&=&\left\vert \int_{0}^{t}d\tau \int_{M}k(\tau ,x_{1},y)(f(y)-f(x_{1}))d\mu
(y)\right. \notag \\
&&\left. +t\left[ f(x_{1})-f(x_{2})\right] -\int_{0}^{t}d\tau \int_{M}k(\tau
,x_{2},y)(f(y)-f(x_{2}))d\mu (y)\right\vert \notag \\
&\leq &2C\int_{0}^{t}\tau ^{\theta _{2}/\beta }d\tau
+C_{6}td(x_{1},x_{2})^{\theta _{2}} \notag \\
&=&Ct^{\theta _{2}/\beta +1}+C_{6}td(x_{1},x_{2})^{\theta _{2}} \notag \\
&\leq &C\left[ d(x_{1},x_{2})^{s_{1}+s_{1}\theta _{2}/\beta
}+d(x_{1},x_{2})^{s_{1}+\theta _{2}}\right] \label{u1-estimate1}\end{aligned}$$if $t\leq d(x_{1},x_{2})^{s_{1}}$, where $s_{1}>0$ will be chosen later.
On the other hand, if $t>d(x_{1},x_{2})^{s_{1}},$ and setting $t_{1}=d(x_{1},x_{2})^{s_{1}}$, we obtain, using $(k7)$, that $$\begin{aligned}
\left\vert \int_{t_{1}}^{t}\left[ K_{\tau }f(x_{1})-K_{\tau }f(x_{2})\right]
d\tau \right\vert &\leq &\int_{t_{1}}^{t}d\tau \int_{M}\left\vert k(\tau
,x_{1},y)-k(\tau ,x_{2},y)\right\vert \left\vert f(y)\right\vert d\mu (y)
\notag \\
&\leq &\int_{t_{1}}^{t}L\tau ^{-\nu }d(x_{1},x_{2})^{\sigma }||f||_{1}d\tau
\notag \\
&\leq &L\frac{t_{1}^{1-\nu }-t^{1-\nu }}{\nu -1}d(x_{1},x_{2})^{\sigma
}||f||_{1} \notag \\
&\leq &\frac{L}{\nu -1}d(x_{1},x_{2})^{s_{1}(1-\nu )+\sigma }||f||_{1}.
\label{Pf-estimate1}\end{aligned}$$It follows from (\[Pf-estimate1\]) and (\[u1-estimate1\]) that $$\begin{aligned}
\left\vert u_{1}(t,x_{1})-u_{1}(t,x_{2})\right\vert &\leq &\left\vert
\int_{0}^{t_{1}}K_{\tau }f(x_{1})-K_{\tau }f(x_{2})d\tau \right\vert
+\left\vert \int_{t_{1}}^{t}K_{\tau }f(x_{1})-K_{\tau }f(x_{2})d\tau
\right\vert \notag \\
&\leq &C\left[ d(x_{1},x_{2})^{s_{1}+s_{1}\theta _{2}/\beta
}+d(x_{1},x_{2})^{s_{1}+\theta _{2}}+d(x_{1},x_{2})^{s_{1}(1-\nu )+\sigma }\right] \notag \\
&\leq &Cd(x_{1},x_{2})^{\sigma (\theta _{2}+\beta )/(\theta _{2}+\nu \beta )}
\label{u1-holder1}\end{aligned}$$if $d(x_{1},x_{2})\leq 1$, where $s_{1}=\sigma \beta /(\theta _{2}+\nu \beta
)$ so that $s_{1}+s_{1}\theta _{2}/\beta =s_{1}(1-\nu )+\sigma $, and where we have used the fact that $$s_{1}+\theta _{2}\geq s_{1}+s_{1}\theta _{2}/\beta$$for $\sigma \leq 1\leq \nu $ and $\beta \geq 1$.
Finally, we show the Hölder continuity of $u_{2}.$ Since $u(t,x)$ is bounded on $(0,T)\times M$, we see that $$\int_{t-\eta }^{t}d\tau \int_{M}k(t-\tau ,x,y)u^{p}(\tau ,y)d\mu (y)\leq
C\eta .$$Hence, using $(k7)$, we obtain $$\begin{aligned}
\left\vert u_{2}(t,x_{1})-u_{2}(t,x_{2})\right\vert &=&\left\vert
\int_{t-\eta }^{t}d\tau \int_{M}k(t-\tau ,x_{1},y)u^{p}(\tau ,y)d\mu
(y)\right. \\
&&-\int_{t-\eta }^{t}d\tau \int_{M}k(t-\tau ,x_{2},y)u^{p}(\tau ,y)d\mu (y)
\\
&&+\left. \int_{0}^{t-\eta }d\tau \int_{M}(k(t-\tau ,x_{1},y)-k(t-\tau
,x_{2},y))u^{p}(\tau ,y)d\mu (y)\right\vert \\
&\leq &2C\eta +L\int_{0}^{t-\eta }d\tau \int_{M}|t-\tau |^{-\nu
}d(x_{1},x_{2})^{\sigma }u^{p}(\tau ,y)d\mu (y) \\
&\leq &C(\eta +\eta ^{1-\nu }d(x_{1},x_{2})^{\sigma }).\end{aligned}$$Taking $\eta =d(x_{1},x_{2})^{\sigma /\nu }$, we thus have $$\left\vert u_{2}(t,x_{1})-u_{2}(t,x_{2})\right\vert \leq
Cd(x_{1},x_{2})^{\sigma /\nu }. \label{u2-holder1}$$Combining (\[u0-holder1\]), (\[u1-holder1\]) and (\[u2-holder1\]), we conclude that $$|u(t,x_{1})-u(t,x_{2})|\leq Cd(x_{1},x_{2})^{\theta _{1}\sigma /(\theta
_{1}+\nu \beta )}.$$for all $t\in (0,T)$ and $x_{1},x_{2}\in M$ with $d(x_{1},x_{2})\leq 1,$ for some $C>0$, where we have used that $$\theta _{1}\sigma /(\theta _{1}+\nu \beta )\leq \sigma /\nu \leq \sigma
(\theta _{2}+\beta )/(\theta _{2}+\nu \beta ).$$The proof is complete.
Finally, one may show that if the heat kernel $k$ satisfies $(k5)$, if $\left\Vert f\right\Vert _{\infty }<\infty $ and if $\varphi (x)$ satisfies$$|K_{t+\delta }\varphi (x)-K_{t}\varphi (x)|\leq C\delta \quad \text{ (for all } t>0,x\in M),$$then the essentially bounded weak solution $u$ of ([weak solution]{}) is Lipschitz continuous in time $t$ on $(0,T)\times M,$ that is,$$|u(t+\delta ,x)-u(t,x)|\leq C_{1}\delta \quad (t\in (0,T), \delta >0, x\in M).$$ We omit the details, which are similar to the special case considered in [@FalHu01].
We note that, unlike the blow-up and the existence, the regularity of solutions is not related to the Hausdorff dimension $\alpha $ and the walk dimension $\beta .$
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M.T. Barlow, R.F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. **361** (2009), 1963-1999.
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K. Dalrymple, R.S. Strichartz and J.P. Vinson, Fractal differential equations on the Sierpínski gasket, J. Fourier Anal. Appl. **5** (1999), 203-284.
K.J. Falconer, Semilinear PDEs on self-similar fractals, Comm. Math. Phys. **206** (1999), 235-245.
K.J. Falconer and J. Hu, Nonlinear diffusion equations on unbounded fractal domains, J. Math. Anal. Appl. **256** (2001), 606-624.
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[^1]: JH was supported by NSFC (Grant No. 11071138).\
KJF thanks the Department of Mathematical Sciences, Tsinghua University, for their hospitality
|
---
abstract: |
The complex Wishart ensemble is the statistical ensemble of $M \times N$ complex random matrices with $M \geq N$ such that the real and imaginary parts of each element are given by independent standard normal variables. The Marcenko–Pastur (MP) density $\rho(x; r), x \geq 0$ describes the distribution for squares of the singular values of the random matrices in this ensemble in the scaling limit $N \to \infty$, $M \to \infty$ with a fixed rectangularity $r=N/M \in (0, 1]$. The dynamical extension of the squared-singular-value distribution is realized by the noncolliding squared Bessel process, and its hydrodynamic limit provides the two-parametric MP density $\rho(x; r, t)$ with time $t \geq 0$, whose initial distribution is $\delta(x)$. Recently, Blaizot, Nowak, and Warcho[ł]{} studied the time-dependent complex Wishart ensemble with an external source and introduced the three-parametric MP density $\rho(x; r, t, a)$ by analyzing the hydrodynamic limit of the process starting from $\delta(x-a), a > 0$. In the present paper, we give useful expressions for $\rho(x; r, t, a)$ and perform a systematic study of dynamic critical phenomena observed at the critical time $t_{\rm c}(a)=a$ when $r=1$. The universal behavior in the long-term limit $t \to \infty$ is also reported. It is expected that the present system having the three-parametric MP density provides a mean-field model for QCD showing spontaneous chiral symmetry breaking.
0.2cm
[**Keywords**]{} Marcenko–Pastur law $\cdot$ Wishart random-matrix ensemble $\cdot$ Wishart process $\cdot$ Random-matrix ensemble with an external source $\cdot$ Hydrodynamic limit $\cdot$ Dynamic critical phenomena $\cdot$ Spontaneous chiral symmetry breaking
author:
- 'Taiki Endo [^1], Makoto Katori [^2]'
date: 8 January 2020
title: '**Three-Parametric Marcenko–Pastur Density**'
---
Introduction and Main Results
=============================
\[sec:introduction\]
Marcenko–Pastur law {#thm:MP_law}
-------------------
Assume that $M, N \in \N :=\{1,2, \dots\}$, $M \geq N$. Consider $M \times N$ complex random matrices $K=(K_{jk})$ such that the real and the imaginary parts of elements are i.i.d. and normally distributed with mean $\mu=0$ and variance $\sigma^2=1/2$. The normal distribution with mean $\mu$ and variance $\sigma^2$ is denoted by $N(\mu, \sigma^2)$ and when a random variable $X$ obeys $N(\mu, \sigma^2)$, we write it as $X \sim N(\mu, \sigma^2)$. Then the present setting is described as $$\Re K_{jk} \sim N(0, 1/2), \quad \Im K_{jk} \sim N(0, 1/2),
\quad j=1, \dots, M, \quad k=1, \dots, N.$$ We consider a statistical ensemble of $N \times N$ Hermitian random matrices $L$ defined by $$L= K^{\dagger} K,
\label{eqn:L1}$$ where $K^{\dagger}$ denotes the Hermitian conjugate of $K$. This ensemble of random matrices is called the [*complex Wishart random-matrix ensemble*]{} or the [*chiral Gaussian unitary ensemble*]{} (chGUE) (see, for instance, [@For10]). We denote the eigenvalues of $L$ as $X_j^N, j=1, \dots, N$, which are nonnegative, since $L$ is nonnegative definite by definition; $X_j^N \in \Rp := \{x \in \R: x \geq 0\}$. The positive square roots of them, $\sqrt{X_j^N}, j=1, \dots, N$ are called [*singular values*]{} of random rectangular matrices $K$. In other words, the eigenvalue distribution of the Hermitian random matrices $L$ can be regarded as the distribution of squares of singular values of the rectangular complex random matrices $K$ in the complex Wishart random-matrix ensemble.
Let $\cC_{\rm c}(\R)$ be the set of all continuous real-valued function with compact support on $\R$. We consider the empirical measure defined by $$\Xi^N(dx)= \frac{1}{N} \sum_{j=1}^N
\delta_{X_j^N /M}(dx),
\quad x \in \Rp,
\label{eqn:Xi1}$$ where $\delta_y(dx)$ denotes a Dirac measure concentrated on $y$ such that $\int_{\R} f(x) \delta_y(dx)=f(y)$ for all $f \in \cC_{\rm c}(\R)$. Then we take the double limit $N \to \infty$, $M \to \infty$ for each fixed value of the rectangularity $$r=\lim_{\substack{N \to \infty, \cr M \to \infty}} \frac{N}{M} \in (0, 1].
\label{eqn:scaling1}$$ We can prove that in this scaling limit (\[eqn:scaling1\]), the empirical measure (\[eqn:Xi1\]) converges weakly to a deterministic measure $\rho(x) dx, x \in \Rp$ in the sense that $\int_{\Rp} f(x) \Xi^N(dx) \to \int_{\Rp} f(x) \rho(x) dx$ as $N \to \infty$ for any $f \in \cC_{\rm c}(\R)$. Moreover, the probability density $\rho$ in the limit measure has a finite support in $\R$ and it is explicitly given as a function of the parameter $ r \in (0, 1]$ as [@MP67] $$\rho(x; r)
= \frac{\sqrt{(x-x_{\rm L}(r))(x_{\rm R}(r)-x)}}{2 \pi r x}
\b1_{(x_{\rm L}(r), x_{\rm R}(r))}(x)
\label{eqn:MP1}$$ with $$x_{\rm L}(r) :=(1-\sqrt{r})^2, \quad
x_{\rm R}(r) :=(1+\sqrt{r})^2.
\label{eqn:xL_xR_r}$$ Here $\b1_{\Lambda}(x)$, $\Lambda \subset \R$ is an indicator function such that $\b1_{\Lambda}(x)=1$ if $x \in \Lambda$, and $\b1_{\Lambda}(x)=0$ otherwise. This convergence theorem is known as the [*Marcenko–Pastur law*]{} for the Wishart random-matrix ensemble [@MP67; @For10; @AGZ10] and we call (\[eqn:MP1\]) the [*Marcenko–Pastur (MP) density*]{} in this paper.
Dynamical extension of MP density {#thm:dynamical_MP_density}
---------------------------------
A dynamical extension of the eigenvalue distribution of the Wishart random-matrix ensemble is realized by the solution $\{ {X}^N_j(t) \in \Rp : t \geq 0, j=1, 2, \dots, N\}$ of the following system of stochastic differential equations (SDEs), $$\begin{aligned}
d {X}^N_j(t) &=2 \sqrt{ {X}^N_j(t)} dB_j(t) + 2 (\nu+1) dt
\nonumber\\
& \quad
+ 4 {X}^N_j(t) \sum_{\substack{1 \leq k \leq N, \cr k \not=j}}
\frac{1}{ {X}^N_j(t)- {X}^N_k(t)} dt,
\quad j=1, 2, \dots, N, \quad t \geq 0,
\label{eqn:W_process}\end{aligned}$$ where $\nu=M-N$ and $B_j(t), t \geq 0$ are independent one-dimensional standard Brownian motions starting from $x^N_j \in \Rp$, $j=1, \dots, N$. We assume that $0 \leq x^N_1 \leq x^N_2 \leq \cdots \leq x^N_N < \infty$. This one-parameter family ($\nu > 0$) of $N$-particle stochastic processes was called (the eigenvalue process of) the [*Wishart process*]{} by Bru [@Bru91]. It is also called the [*Laguerre process*]{} or the [*noncolliding squared Bessel process*]{} [@KO01; @KT04; @KT11].
We set $\nu=(1-r) M =(1-r) N/r$, $r \in (0, 1]$, and consider the empirical measure of the solution of SDEs (\[eqn:W\_process\]), $$\Xi^N_t(dx) :=\frac{1}{N} \sum_{j=1}^N \delta_{ {X}^N_j(t)/M}(dx),
\quad x \in \Rp, \quad t \geq 0.
\label{eqn:empirical2}$$ If the initial empirical measure satisfies some moment conditions and converges weakly to a measure, $\Xi^N_0(dx)=(1/N) \sum_{j=1}^N \delta_{x^N_j/M}(dx)
\to \xi(dx)$ in the limit $N \to \infty$, $M \to \infty$ with $r=N/M$ fixed in $(0, 1]$, it is proved that $\Xi^N_t(dx)$ converges weakly to a time-dependent deterministic measure, which we denote here as $\rho_{\xi}(x; r, t), t \geq 0$ [@CDG01; @BNW13]. We define the [*Green’s function*]{} (the resolvent) $G_{\xi}(z; r, t)$ by the Stieltjes transform of $\rho_{\xi}$, $$G_{\xi}(z; r, t) := \int_{\R} \frac{\rho_{\xi}(x; r, t)}{z-x} dx,
\quad z \in \C \setminus \R.$$ Then we can prove that this solves the following nonlinear partial differential equation (PDE) [@CDG01; @BNW13], $$\frac{\partial G_{\xi}}{\partial t}
=-\frac{\partial G_{\xi}}{\partial z}
+ r \left\{
\frac{\partial G_{\xi}}{\partial z}-2 z G_{\xi} \frac{\partial G_{\xi}}{\partial z}
- G_{\xi}^2 \right\},
\quad t \in [0, \infty),
\label{eqn:PDE1}$$ under the initial condition, $G_{\xi}(z; r, 0)=\int_{\R} \xi(dx)/(z-x)$, $z \in \C \setminus \R$. Once the Green’s function $G_{\xi}(z; r, 0), z \in \C \setminus \R$ is determined, we can obtained the density function following the Sokhotski-Plemelj theorem, $$\rho_{\xi}(x; r, t)
=- \Im \left[ \lim_{\varepsilon \to 0}
\frac{1}{\pi} G_{\xi}(x+i \varepsilon; r, t) \right],
\quad i :=\sqrt{-1}.
\label{eqn:SPtheorem}$$ We regard (\[eqn:PDE1\]) as the analogy of the complex Burgers equations in the inviscid limit ([*i.e.*]{}, the (complex) one-dimensional Euler equation), and we call the limit process the [*hydrodynamic limit*]{} of the Wishart process [@BNW13; @BNW14].
The simplest case of the dynamical extension of the MP density (\[eqn:MP1\]) with (\[eqn:xL\_xR\_r\]) is obtained by setting the initial distribution, $$\xi(dx)= \delta_0(dx) := \delta(x) dx,
\label{eqn:xi_0}$$ that is, all particles are concentrated on the origin; $x^N_j=0, j \in \N$. By the method of complex characteristics, Blaizot, Nowak, and Warcho[ł]{} [@BNW13] showed that the Green’s function for (\[eqn:xi\_0\]), $G_{\delta_0}(z)=G_{\delta_0}(z;r,t)$, is given by the solution of the equation $$z=\frac{1}{G_{\delta_0}(z)} + \frac{t}{1-rt G_{\delta_0}(z)},
\quad z \in \C \setminus \R, \quad r \in (0, 1], \quad t \geq 0.
\label{eqn:G_eq_0}$$ They solved (\[eqn:G\_eq\_0\]) and using the Sokhotski-Plemelj theorem (\[eqn:SPtheorem\]) derived the time-dependent extension of (\[eqn:MP1\]), $$\rho(x; r, t) := \rho_{\delta_0}(x; r, t)
= \frac{\sqrt{(x-x_{\rm L}(r, t))(x_{\rm R}(r, t)-x)}}{2 \pi r t x}
\b1_{(x_{\rm L}(r, t), x_{\rm R}(r, t))}(x)
\label{eqn:MP_t}$$ with $$x_{\rm L}(r, t) :=(1-\sqrt{r})^2t, \quad
x_{\rm R}(r, t) :=(1+\sqrt{r})^2t,
\quad t \in (0, \infty).
\label{eqn:xL_xR_r_t}$$ Since $\rho(x; r, 1)$ is equal to the original MP density (\[eqn:MP1\]), the above provides a dynamical derivation of the Marcenko–Pastur law. The dependence on $t$ of $\rho(x; r, t)$ given by (\[eqn:MP\_t\]) with (\[eqn:xL\_xR\_r\_t\]) is very simple, but we regard this as the [*two-parametric*]{} MP density in the present paper.
Main results {#sec:main_results}
------------
Blaizot, Nowak, and Warcho[ł]{} [@BNW14] have studied the hydrodynamic limit of the Wishart eigenvalue-process starting from one-parameter family ($a \geq 0$) of the initial distribution, $$\xi(dx)= \delta_a(dx) := \delta(x-a) dx.$$ They showed that the Green’s function, $G(z)= G_{\delta_a}(z; r, t)$, $a \geq 0$, is obtained by the solution of the equation, $$z =\frac{1}{G_{\delta_a}(z)}
+\frac{t}{1-r t G_{\delta_a}(z)}
+\frac{a}{(1- r t G_{\delta_a}(z))^2},
\quad z \in \C \setminus \R, \quad r \in (0,1], \quad t \geq 0, \quad a \geq 0.
\label{eqn:z_t_1}$$ They claimed in [@BNW14] that a proper solution of this equation yields $\rho_{\delta_a}(x; r, t)$ via the Sokhotski-Plemelj theorem and showed an illustration (Fig.1 in [@BNW14]) of the time dependence of this density function for a special case with $r=1$ and $a=1$. The explicit formula of $\rho_{\delta_a}(x; r, t)$ was, however, not given there. See also Section 8 in [@LWZ16] and Section 3 in [@FG16] for implicit expressions of $G_{\delta_a}$.
We write the solution discussed in [@BNW14] as $\rho(x; r, t, a)$ and call it the [*three-parametric Marcenko-Pastur (MP) density*]{}. The purpose of the present paper is to report useful expressions and detailed analysis of this density function $\rho(x; r, t, a)$ on $\Rp$ with three parameters $r \in (0, 1], t \geq 0$ and $a \geq 0$.
The main theorem of the present paper is the following.
\[thm:main1\] Let $$\begin{aligned}
S(x; r, t, a) &:=
4 a x^3 - \{8 a^2+4a(3r+2)t-t^2\} x^2
\nonumber\\
& \quad
+2 [2 a^3-2a^2(5r-2) t + a\{r(6r-1)+1\} t^2 -(r+1)t^3 ] x
\nonumber\\
& \quad
+(r-1)^2 t^2 \{a^2-a(4r-2)t+t^2\}.
\label{eqn:S1}\end{aligned}$$ For $r \in (0, 1], t >0, a \geq 0$, consider the case such that the cubic equation with respect to $x$, $$S(x; r, t, a)=0,
\label{eqn:cubiceq1}$$ has three real solutions, $x_1 \leq x_2 \leq x_3$, where $x_j=x_j(r, t, a), j=1,2,3$. Define $$x_{\rm L}(r, t, a) := x_2(r, t, a),
\quad
x_{\rm R}(r, t, a) :=x_3(r, t, a).
\label{eqn:x_LR_3MP}$$ Put $$\begin{aligned}
g &= g(x; r, t, a)
\nonumber\\
& :=
-2 x^3+ 3 \{(2r+1)t+6a \} x^2
-3 \Big[ (r-1)\{(2r+1)t-3a\} t - \sqrt{- 3 S} \Big] x +2(r-1)^3 t^3,
\label{eqn:g1}\end{aligned}$$ with $S=S(x; r, t, a)$ given by (\[eqn:S1\]), and define $$\begin{aligned}
\varphi &= \varphi(x; r, t, a)
\nonumber\\
& := - \frac{2}{3} \{x-(r-1)t\}
-\frac{2^{1/3}}{3}
\frac{x^2+\{3a-(2r+1)t\}x+t^2(r-1)^2}{g^{1/3}}
-\frac{g^{1/3}}{3 \times 2^{1/3}}.
\label{eqn:varphi}\end{aligned}$$ Then the three-parametric MP density is given by $$\rho(x; r, t, a)
= \frac{\sqrt{(x-f_{\rm L}(x; r, t, a))(f_{\rm R}(x; r, t, a)-x)}}{2 \pi r x t}
\b1_{(x_{\rm L}(r, t, a), x_{\rm R}(r, t, a))}(x)
\label{eqn:3_MP1}$$ with $$f_{\rm L}(x; r, t, a) := \left( \frac{\sqrt{d_-}+\sqrt{d_+}}{2} -\sqrt{d_0} \right)^2,
\quad
f_{\rm R}(x; r, t, a) := \left( \frac{\sqrt{d_-}+\sqrt{d_+}}{2} + \sqrt{d_0} \right)^2,
\label{eqn:fLR}$$ where $$\begin{aligned}
& d_- = t-a-2\sqrt{a \varphi}, \quad
d_+ = t-a+2\sqrt{a \varphi},
\nonumber\\
& d_0
= \varphi+x+\frac{t-a}{2} + \frac{1}{2} \sqrt{d_- d_+}.
\label{eqn:d1}\end{aligned}$$
0.5cm [**Remark 1**]{} The formula (\[eqn:3\_MP1\]) for the present three-parametric MP density seems to be similar to the original MP density (\[eqn:MP1\]) and the two-parametric MP density (\[eqn:MP\_t\]). We should note, however, that $f_{\rm L}$ and $f_{\rm R}$ appearing in (\[eqn:3\_MP1\]) are not equal to the endpoints $x_{\rm L}$ and $x_{\rm R}$ of the support of density and they depend on $x$ as shown by (\[eqn:fLR\]) with (\[eqn:g1\]), (\[eqn:varphi\]) and (\[eqn:d1\]). We can see that $$\varphi(x; r, t, 0) := \lim_{a \to 0} \varphi(x; r, t, a)=-x+(r-1)t,$$ and hence $d_{\pm} \to t, d_0 \to rt$ as $a \to 0$. Then as $a \to 0$, $f_{\rm L} \to x_{\rm L}(r; t)$, $f_{\rm R} \to x_{\rm R}(r; t)$ with (\[eqn:xL\_xR\_r\_t\]); that is, the dependence of $f_{\rm L}$ and $f_{\rm R}$ on $x$ vanishes only in this limit. Theorem \[thm:main1\] states that for general $a> 0$, the endpoints $x_{\rm L}$ and $x_{\rm R}$ of the support for the three-parametric MP density are given by the suitably chosen solutions (\[eqn:x\_LR\_3MP\]) of the cubic equation (\[eqn:cubiceq1\]) with (\[eqn:S1\]) as proved in Section \[sec:support\] below. That is, the formula (\[eqn:3\_MP1\]) is universal, but the choice of solutions (\[eqn:x\_LR\_3MP\]) depends on the parameters $r, t, a$. The equation (\[eqn:z\_t\_1\]) seems to be a simple perturbation of (\[eqn:G\_eq\_0\]), but the solution turns out to have rich structures, by which we can describe dynamic critical phenomena at time $t=t_{\rm c} :=a$ for $a >0$, when $r=1$, as shown below. 0.3cm From the view point of the original random matrix theory, Theorem \[thm:main1\] gives the limit theorem for the eigenvalue distribution of random matrix $L$ given by (\[eqn:L1\]) in the scaling limit (\[eqn:scaling1\]), in which $M \times N$ rectangular complex random matrices $K=(K_{jk})$ are distributed as $$\begin{aligned}
\Re K_{kk} &\sim N(\sqrt{Ma}, t/2), \quad k=1, \dots, N,
\nonumber\\
\Re K_{jk} &\sim N(0, t/2), \quad j=1, \dots, M, \quad k=1, \dots, N, \quad j \not=k,
\nonumber\\
\Im K_{jk} &\sim N(0, t/2), \quad j=1, \dots, M, \quad k=1, \dots, N,
\label{eqn:K2}\end{aligned}$$ $t >0, r \in (0, 1]$. By (\[eqn:L1\]), (\[eqn:K2\]) gives $$\bE[L_{jk}]=M(a+t) \delta_{jk}, \quad
j, k=1, \dots, N.$$ When $a > 0$, such an ensemble of random matrices will be called the [*Wishart ensemble with an external source*]{} or the [*non-centered Wishart ensemble*]{}, since even at $t=0$, the diagonal elements of $L$ have positive means, $\bE[L_{jj}]=Ma >0, j=1, \dots, N$, [@BH17; @KMW11; @DKRZ12; @HK13; @For13].
In Figure \[fig:histogram\], we compare two histograms for the empirical measures (\[eqn:empirical2\]) of the eigenvalues of matrices $L=K^{\dagger} K$ given by $K$ of size $1000 \times 300$ (with the rectangularity $r=300/1000=0.3$), whose elements are following the probability law (\[eqn:K2\]) with different parameters $(t, a)$. When $(t, a)=(1, 0)$, the distribution of eigenvalues has a maximum at $x \simeq 0.4$. When $(t, a)=(1, 1)$, due to an external source at $x=a=1$, the distribution is shifted to the positive direction having a maximum at $x \simeq 1$ and becomes broader. The former is well fitted by the original MP density $\rho(x; r=0.3)$ and the latter is by the three-parametric MP density $\rho(x; r=0.3, t=1, a=1)$ given by (\[eqn:3\_MP1\]).
![ Two histograms for the empirical measures (\[eqn:empirical2\]) with two different sets of parameters $(t, a)=(1,0)$ and $(1,1)$ are superposed in order to compare each other. They show the distributions of the eigenvalues of $L=K^{\dagger} K$ given by $K$ of size $1000 \times 300$, whose elements are randomly generalized following the probability law (\[eqn:K2\]) with $(t, a)=(1, 0)$ and $(1, 1)$. The original MP density $\rho(x; r=0.3)$ and the three-parametric MP density $\rho(x; r=0.3, t=1, a=1)$ are shown by a thin curve and a thick curve, respectively. Due to an external source at $x=a=1$, the eigenvalue distribution with $(t, a)=(1,1)$, which is well fitted by the three-parametric MP density $\rho(x; r=0.3, t=1, a=1)$, is shifted to the positive direction and becomes broader compare with the original MP density $\rho(x; r=0.3)$. []{data-label="fig:histogram"}](Fig1_Endo_Katori.eps){width="60.00000%"}
For each values of rectangularity $r \in (0, 1]$ and strength of an external source $a \geq 0$, we can show time evolution of the support $(x_{\rm L}(r, t, a), x_{\rm R}(r, t, a) )$ of $\rho(x; r, t, a)$ on the $(x, t)$-plane, $(\Rp)^2$. Figure \[fig:support\_r0.3\] shows the domains $$\cD(r, a) := \{ (x_{\rm L}(r, t, a), x_{\rm R}(r, t, a)) : t \geq 0\}
\subset (\Rp)^2$$ for $(r, a)=(0.3, 0)$ and $(r, a)=(0.3, 1)$. On the other hand, Figure \[fig:support\_r1\] shows the domains for $(r, a)=(1, 0)$ and $(r, a)=(1, 1)$. As demonstrated by these figures, we can prove the following qualitative change of the domain when $a>0$ and $r=1$.
[c]{}
![ For $r=0.3$, time evolution of the support $(x_{\rm L}, x_{\rm R})$ is shown on the $(x, t)$-plane for the two-parametric MP density $\rho(x; r=0.3, t) := \rho(x; r=0.3, t, a=0)$ in the left, and for the three-parametric MP density with $a=1$, $\rho(x; r=0.3, t, a=1)$ in the right. The supports are extended in time, but the left edges of supports are kept to be positive, $x_{\rm L} > 0$, for all $t > 0$. []{data-label="fig:support_r0.3"}](Fig2left_Endo_Katori.eps){width="80.00000%"}
![ For $r=0.3$, time evolution of the support $(x_{\rm L}, x_{\rm R})$ is shown on the $(x, t)$-plane for the two-parametric MP density $\rho(x; r=0.3, t) := \rho(x; r=0.3, t, a=0)$ in the left, and for the three-parametric MP density with $a=1$, $\rho(x; r=0.3, t, a=1)$ in the right. The supports are extended in time, but the left edges of supports are kept to be positive, $x_{\rm L} > 0$, for all $t > 0$. []{data-label="fig:support_r0.3"}](Fig2right_Endo_Katori.eps){width="80.00000%"}
[c]{}
![ For $r=1$, time evolution of the support $(x_{\rm L}, x_{\rm R})$ is shown on the $(x, t)$-plane for the two-parametric MP density $\rho(x; r=1, t) := \rho(x; r=1, t, a=0)$ in the left, and for the three-parametric MP density with $a=1$, $\rho(x; r=1, t, a=1)$ in the right. In the two-parametric MP density, the support starts from the singleton $\{ 0 \}$ at $t=0$ and the left edge of support $x_{\rm L}$ is identically zero; $x_{\rm L} \equiv 1$ for $t \geq 0$. On the other hand, in the three-parametric MP density with $a=1$, the support starts from the singleton $\{ 1 \}$ at $t=0$, and $x_{\rm L} >0$ when $t < t_{\rm c}=1$. As $t \nearrow t_{\rm c}=1$, however, $x_{\rm L} \searrow 0$ continuously, and then $x_{\rm L} \equiv 1$ for $t \geq t_{\rm c}=1$. We regard $t_{\rm c}=1$ as a critical time. []{data-label="fig:support_r1"}](Fig3left_Endo_Katori.eps){width="80.00000%"}
![ For $r=1$, time evolution of the support $(x_{\rm L}, x_{\rm R})$ is shown on the $(x, t)$-plane for the two-parametric MP density $\rho(x; r=1, t) := \rho(x; r=1, t, a=0)$ in the left, and for the three-parametric MP density with $a=1$, $\rho(x; r=1, t, a=1)$ in the right. In the two-parametric MP density, the support starts from the singleton $\{ 0 \}$ at $t=0$ and the left edge of support $x_{\rm L}$ is identically zero; $x_{\rm L} \equiv 1$ for $t \geq 0$. On the other hand, in the three-parametric MP density with $a=1$, the support starts from the singleton $\{ 1 \}$ at $t=0$, and $x_{\rm L} >0$ when $t < t_{\rm c}=1$. As $t \nearrow t_{\rm c}=1$, however, $x_{\rm L} \searrow 0$ continuously, and then $x_{\rm L} \equiv 1$ for $t \geq t_{\rm c}=1$. We regard $t_{\rm c}=1$ as a critical time. []{data-label="fig:support_r1"}](Fig3right_Endo_Katori.eps){width="80.00000%"}
\[thm:support\] Assume that $a > 0$.
[[(i)]{}]{} If and only if $r=1$, $\cD(r,a)$ touches the origin $x=0$. Otherwise, the left edge of ${\rm supp} \, \rho(x; r, t, a)$ is strictly positive; $x_{\rm L}(r, t, a) > 0$, $r \in (0, 1)$.
[[(ii)]{}]{} When $r=1$, there is a critical time $$t_{\rm c}(a)=a$$ such that $x_{\rm L}(1, t, a) >0$ while $0 \leq t < t_{\rm c}(a)$, and $x_{\rm L}(1, t, a) \equiv 0$ for $t \geq t_{\rm c}(a)$. In particular, just before the critical time $t_{\rm c}(a)$, the left edge of ${\rm supp} \, \rho(x; r, t, a)$ behaves as $$x_{\rm L}(1, t, a) \simeq \frac{4}{27 a^2}(t_{\rm c}(a)-t)^{\nu}
\quad
\mbox{with $\nu=3$
\quad as $t \nearrow t_{\rm c}(a)$}.$$
In the case with $r=1$, the dynamic critical phenomena at the critical time $t=t_{\rm c}(a)$ are observed in the vicinity of the origin as follows.
\[thm:critical\] When $r=1$, the three-parametric MP density shows the following dynamic critical phenomena at $t=t_{\rm c}(a)$.
[[(i)]{}]{} For $0 < t < t_{\rm c}(a)$, $$\rho(x; 1, t, a) \simeq C_1(t, a) (x-x_{\rm L}(1, t, a))^{\beta_1}
\quad
\mbox{with $\displaystyle{\beta_1 = \frac{1}{2}}$
\quad as $x \searrow x_{\rm L}(1, t, a)$},$$ where $$C_1(t, a) \simeq \frac{9 a}{4 \pi}(t_{\rm c}(a)-t)^{-\gamma_1}
\quad
\mbox{with $\displaystyle{\gamma_1=\frac{5}{2}}$
\quad as $t \nearrow t_{\rm c}(a)$}.$$
[[(ii)]{}]{} At $t=t_{\rm c}(a)$, $$\rho(x; 1, t_{\rm c}(a), a) \simeq \frac{\sqrt{3}}{2 \pi} a^{-2/3} x^{-\gamma_2}
\quad
\mbox{with $\displaystyle{\gamma_2 = \frac{1}{3}}$
\quad as $x \searrow 0$}.$$
[[(iii)]{}]{} For $t > t_{\rm c}(a)$, $$\rho(x; 1, t, a) \simeq C_2(t, a) x^{-\gamma_3}
\quad
\mbox{with $\displaystyle{\gamma_3 = \frac{1}{2}}$
\quad as $x \searrow 0$},$$ where $$C_2(t, a) \simeq \frac{1}{\pi t_{\rm c}(a)} (t-t_{\rm c}(a))^{\beta_2}
\quad
\mbox{with $\displaystyle{\beta_2 = \frac{1}{2}}$
\quad as $t \searrow t_{\rm c}(a)$}.$$
0.5cm [**Remark 2**]{} The [*critical exponents*]{} $\nu=3$ and $\gamma_2=1/3$ can be read in the argument given by [@BNW14]. Using the expressions given in Theorem \[thm:main1\] here we prove them as well as determining other critical exponents and critical amplitudes. The amplitude $C_1$ of $\rho$ in the subcritical time-region ($0<t < t_{\rm c}(a)$) diverges with the critical exponent $\gamma_1=5/2$ as $t \nearrow t_{\rm c}(a)$. Then both at the critical time ($t=t_{\rm c}(a)$) and in the supercritical time-region ($t > t_{\rm c}(a)$) $\rho$ diverges as $x \searrow 0$, but the critical exponents are different. In the supercritical time-region, the amplitude $C_2$ of the diverging $\rho$ with the critical exponent $\gamma_3=1/2$ vanishes as $t \searrow t_{\rm c}(a)$ (with the exponent $\beta_2=1/2$). Consequently the divergence of $\rho$ as $x \searrow 0$ is weakened at $t=t_{\rm c}(a)$ having a smaller value of exponent as $\gamma_2=1/3 < \gamma_3=1/2$. See Fig. \[fig:rho\_critical\]. The proof of Proposition \[thm:critical\] given in Subsection \[sec:proof3\] implies the scaling relation $$\nu=\beta_2+\gamma_1.$$ 0.3cm
![ Critical behavior of the three-parametric MP density $\rho$ is shown for $r=1$ and $a=1$ with the critical time $t_{\rm c}(1)=1$. The dashed curve denotes the emergence of $\rho$ at $x = x_{\rm L} \simeq 0.028$ with the critical exponent $\beta_1=1/2$ at a subcritical time ($t=0.5 t_{\rm c}(1)$). The divergence of $\rho$ as $x \searrow 0$ at the critical time $t=t_{\rm c}(1)$ is shown by a solid curve and that at a supercritical time ($t=1.5 t_{\rm c}(1)$) by a dotted curve. The former with the critical exponent $\gamma_2=1/3$ is weaker than the latter with $\gamma_3=1/2$. []{data-label="fig:rho_critical"}](Fig4_Endo_Katori.eps){width="65.00000%"}
The functions $S(x; r, t, a)$, $g(x; r, t, a)$, $\varphi(x; r, t, a)$, $f_{\rm L}(x; r, t, a)$, and $f_{\rm R}(x; r, t, a)$, which appear in Theorem \[thm:main1\], are all [*homogeneous as multivariate functions of $x, t, a$*]{} for each fixed value of $r \in (0, 1]$. This fact implies the following scaling property of the three-parametric MP density, $$\rho(\kappa x; r, \kappa t, \kappa a)
=\frac{1}{\kappa} \rho(x; r, t, a),
\quad r \in (0, 1],
\label{eqn:rho_scaling}$$ for an arbitrary parameter $\kappa >0$. By this property, the following long-term behavior of the three-parametric MP density is readily concluded.
\[thm:long\_term\] For $a \geq 0$, $$\lim_{t \to \infty} \rho(y; r, t, a) dy \Big|_{y=tx}
= \rho(x; r) dx,$$ where $\rho(x; r)$ is given by (\[eqn:MP1\]) with (\[eqn:xL\_xR\_r\]).
The long-term behavior of the present three-parametric MP density is given by a dilatation of the original MP density by factor $t$. In this sense, the original Marcenko–Pastur law is [*universal*]{} and it describes the large-scale and long-term behavior of the Wishart ensemble and process. 0.3cm
![ Time evolution of the hydrodynamical density $\rho^{\rm chiral}(x; r, t, a)$ of the QCD Dirac operator in the critical case, which is obtained as (\[eqn:rho\_chiral\]) from the present three-parametric MP density with $r=1$. For $a=1$, $\rho^{\rm chiral}$ is plotted for $t=0.5$ (the thinnest curve), 1.0, 1.5 and 3 (the thickest curve). When $0 \leq t \leq t_{\rm c}=1$, the density at the origin $\rho^{\rm chiral}(0; r=1, t, a=1) =0$, while it becomes positive for $t > t_{\rm c}$. For $t > 3 t_{\rm c}$, $\rho^{\rm chiral}$ shows a relaxation in the sense of (\[eqn:converge\]) to the universal density following Wigner’s semicircle law (\[eqn:Wigner\]). []{data-label="fig:chiral"}](Fig5_Endo_Katori.eps){width="70.00000%"}
[**Remark 3**]{} So far we have studied the density $\rho$ of eigenvalues of random matrices $L$ given by (\[eqn:L1\]) in the hydrodynamic limit. On the other hand, when the present random matrix ensemble, chGUE, is applied as a model to the [*quantum chromodynamics (QCD)*]{} in high energy physics, the density $\rho^{\rm chiral}$ of the positive-signed and negative-signed singular values of random rectangular matrix $K$ have been discussed [@JNPZ99; @LWZ16; @FG16]. For the transformation from $\rho$ to $\rho^{\rm chiral}$, see Eq.(3.34) in [@FG16], for instance. The present three-parametric MP density $\rho(x; r, t, a)$ given by Theorem \[thm:main1\] provides the following hydrodynamical description of the time-depending spectrum for the QCD Dirac operator with parameters $r \in (0, 1]$ and $a \geq 0$, $$\rho^{\rm chiral}(x; r, t, a)
=2 |x| \rho(x^2; r, t, a^2), \quad x \in \R, \quad t > 0,
\label{eqn:rho_chiral}$$ under the initial state $$\rho^{\rm chiral}(x; r, 0, a)
=\delta(x+a)+\delta(x-a), \quad x \in \R.$$ Figure \[fig:chiral\] shows the time evolution of (\[eqn:rho\_chiral\]) in the critical case $r=1$ with $a=1$. By (\[eqn:rho\_chiral\]), Proposition \[thm:critical\] (ii) and (iii) give the following for $r=1$, $$\begin{aligned}
\rho^{\rm chiral}(x; 1, t_{\rm c}(a), a)
&\simeq \frac{\sqrt{3}}{\pi} a^{-4/3} |x|^{1/\delta}
\quad \mbox{with $\delta=3$ as $|x| \to 0$},
\nonumber\\
\rho^{\rm chiral}(0; 1, t, a)
&\simeq \frac{2}{\pi t_{\rm c}(a^2)}(t-t_{\rm c}(a^2))^{\beta_2}
\quad \mbox{with $\displaystyle{\beta_2=\frac{1}{2}}$
as $t \searrow t_{\rm c}(a)$}.
\label{eqn:MF1}\end{aligned}$$ Moreover, Proposition \[thm:long\_term\] implies through (\[eqn:rho\_chiral\]) that, when $r=1$, $$\begin{aligned}
\lim_{t \to \infty} \rho^{\rm chiral}(y; 1, t, a) dy
\Big|_{y=\sqrt{t} x}
&= \lim_{t \to \infty} \rho(t x^2; 1, t, a^2) d(t x^2)
\nonumber\\
&= \rho(x^2; 1) d x^2 = 2 x \rho(x^2; 1) dx
\nonumber\\
&= \rho^{\rm Wigner}(x) dx, \quad x \in \R,
\label{eqn:converge}\end{aligned}$$ where $$\rho^{\rm Wigner}(x)=\frac{1}{\pi} \sqrt{4-x^2}
\b1_{(-2, 2)}(x)
\label{eqn:Wigner}$$ is the density function describing [*Wigner’s semicircle law*]{} (see, for instance, [@For10]). As mentioned in [@LWZ16], the time evolution of $\rho^{\rm chiral}$ from the two-peak shape with zero density at the origin $(0 \leq t \leq t_{\rm c}(a))$ to the universal shape $\rho^{\rm Wigner}$ after $t \sim 3 t_{\rm c}(a)$ via a critical shape at $t=t_{\rm c}(a)$ can be interpreted as a transition from an initial state with restored chiral symmetry to a final state with [*spontaneous chiral symmetry breaking*]{}. In [@JNPZ99], we find the argument that the present system with the density $\rho^{\rm chiral}(x; r, t, a)$ give a [*mean-field model for QCD*]{} and $\delta=3$ and $\beta_2=1/2$ in (\[eqn:MF1\]) are the mean-field values for the scaling exponents describing a condensation of light quarks to create massive constituents. 0.3cm The paper is organized as follows. In Section \[sec:proofs\] we give proofs of theorems and propositions given above. More precisely, Subsections \[sec:solving\] and \[sec:support\] are devoted to the proof of Theorem \[thm:main1\]. The proofs of Propositions \[thm:support\], \[thm:critical\], and \[thm:long\_term\] are given in Subsections \[sec:proof2\], \[sec:proof3\], and \[sec:proof4\], respectively. Concluding remarks are given in Section \[sec:concluding\].
Proofs of Theorem and Propositions
==================================
\[sec:proofs\]
Solving the algebraic equations for the density and its Hilbert transform {#sec:solving}
-------------------------------------------------------------------------
For the Green’s function $G_{\delta_a}(z)=G_{\delta_a}(z; r, t)$, $a \geq 0$, we put $$\begin{aligned}
R &=R(x) := \lim_{\varepsilon \to 0} \Re G_{\delta_a}(x+i \varepsilon),
\nonumber\\
I &=I(x) := - \lim_{\varepsilon \to 0} \Im G_{\delta_a}(x+i \varepsilon),\end{aligned}$$ that is, $\lim_{\varepsilon \to 0} G(x+i \varepsilon)=R(x) - i I(x)$. For the three-parametric MP density $\rho(x) :=\rho(x; r, t,a)$, its [*Hilbert transform*]{} is defined by $$\begin{aligned}
\cH[\rho](x)
&= \frac{1}{\pi} \dashint_{\R}
\frac{\rho(y)}{x-y} dy
\nonumber\\
&:= \frac{1}{\pi} \lim_{\varepsilon \to 0}
\left\{
\int_{-\infty}^{x-\varepsilon} \frac{\rho(y)}{x-y} dy +
\int_{x+\varepsilon}^{\infty} \frac{\rho(y)}{x-y} dy
\right\}.\end{aligned}$$ The Sokhotski-Plemelj theorem states that $$\rho(x)=\frac{I(x)}{\pi}, \quad
\cH [\rho(\cdot)](x) = \frac{R(x)}{\pi}.
\label{eqn:SP}$$
Let $$A =\frac{1}{R^2+I^2}, \qquad
B =\frac{1}{(1-rt R)^2+(rt I)^2}.
\label{eqn:AB1}$$ By definition, we obtain the equation, $$A=\frac{(rt)^2}{2 r t R-1 +1/B}.
\label{eqn:AB2}$$ For the equation (\[eqn:z\_t\_1\]), we obtain the following.
\[thm:equations\] The equation (\[eqn:z\_t\_1\]) for the complex-valued function $G_{\delta_a}(z), a \geq 0$ is equivalent with the following system of equations for the real-valued functions $A$ and $B$, $$\begin{aligned}
x & =R A+ (1-rt R)t B + a \left[ (1-rt R)^2-(rt I)^2 \right] B^2,
\nonumber\\
0 & =[A-rt^2B]-2a(1-rt R)rt B^2.
\label{eqn:equations1}\end{aligned}$$
[*Proof*]{} We put $z=x+i \varepsilon, x, \varepsilon \in \R$ in (\[eqn:z\_t\_1\]) and take the limit $\varepsilon \to 0$. Then the real part and the imaginary part of the obtained equation give (\[eqn:equations1\]). 0.3cm
Before solving the system of equations (\[eqn:equations1\]) for general $a \geq 0$, first we solve it for the special case $a=0$. In this case (\[eqn:equations1\]) with (\[eqn:SP\]) and (\[eqn:AB1\]) are simplified as $$\begin{aligned}
x &= \frac{R_0}{R_0^2+(\pi \rho_0)^2}
+\frac{(1-r t R_0)t}{(1-rtR_0)^2+(rt)^2(\pi \rho_0)^2}
\nonumber\\
&= \left\{ \frac{1}{R_0^2+(\pi \rho_0)^2}- \frac{rt^2}{(1-rtR_0)^2+(rt)^2(\pi \rho_0)^2} \right\} R_0
+\frac{t}{(1-rtR_0)^2+(rt)^2(\pi \rho_0)^2},
\nonumber\\
0 &=\frac{1}{R_0^2+(\pi \rho_0)^2}- \frac{rt^2}{(1-rtR_0)^2+(rt)^2(\pi \rho_0)^2},\end{aligned}$$ for $R_0 :=R(x; r,t,0)$ and $\rho_0 :=\rho(x; r, t, 0)$. They give two different expressions for $x$, $$\begin{aligned}
x &= \frac{1}{rt \{R_0^2+(\pi \rho_0)^2\}},
\label{eqn:a0_x1}
\\
x &= \frac{t}{(1-rtR_0)^2+(rt)^2(\pi \rho_0)^2}.
\label{eqn:a0_x2}\end{aligned}$$ From (\[eqn:a0\_x1\]), we have the relation $$(\pi \rho_0)^2=\frac{1}{r t x}- R_0^2.
\label{eqn:a0_eq1}$$ Combining this with (\[eqn:a0\_x2\]), we have $x=t/(1-2 r t R_0 + rt/x)$, which is solved as $$R_0=\frac{x+(r-1)t}{2 rt x}.
\label{eqn:R_0}$$ Put (\[eqn:R\_0\]) into (\[eqn:a0\_eq1\]), we obtain $$(\pi \rho_0)^2
= \frac{1}{(2 rt x)^2}
\{-x^2+2(r+1)tx-(r-1)^2 t^2 \}
= \frac{(x-x_{\rm L})(x_{\rm R}-x)}{(2 r t x)^2},
\label{eqn:sq_2_rho}$$ where $x_{\rm L}=x_{\rm L}(r, t)$ and $x_{\rm R}=x_{\rm R}(r, t)$ are given by (\[eqn:xL\_xR\_r\_t\]) and the two-parametric MP density (\[eqn:MP\_t\]) is obtained as the positive square root of (\[eqn:sq\_2\_rho\]) for $x_{\rm L} \leq x \leq x_{\rm R}$.
The above calculation suggests that it will be easier to obtain $R$ than $I$. By (\[eqn:SP\]) and the first equation of (\[eqn:AB1\]), if $$\frac{1}{A} - R^2 \geq 0,$$ then $$\rho(x)=\frac{1}{\pi} \sqrt{ \frac{1}{A}-R^2}.
\label{eqn:rho_Z1}$$ Hence if we can express $A$, not using $I$, but using only $R$ and parameters $r, t, a$, then the obtained $R$ determines the density function $\rho$. Actually we will show that this strategy is successful in the following.
By eliminating $A$ in (\[eqn:equations1\]), we obtain a quadratic equation for $B$ as $$2a (1-rtR) B^2+(t-a)B-x=0.
\label{eqn:quadratic1}$$ We choose the following solution of (\[eqn:quadratic1\]), $$B=\frac{a-t+\sqrt{D}}{4a(1-rtR)},
\label{eqn:B1}$$ with $$D=D(x; r, t, a) =8a(1-rtR) x +(t-a)^2,
\label{eqn:Det1}$$ by the following reason. If we put $a=0$, (\[eqn:quadratic1\]) gives $B|_{a=0}=x/t$. On the other hand, (\[eqn:Det1\]) gives $$\sqrt{D} = -(a-t)+4a(1-rtR_0) \frac{x}{t}+\rO(a^2),$$ and hence (\[eqn:B1\]) has the correct limit in $a \to 0$; $\lim_{a \to 0} B=x/t$. If we put (\[eqn:B1\]) into (\[eqn:AB2\]), then we have $$A=\frac{(rt)^2 \{2(2rt R-1)x-a+t-\sqrt{D}\} }{2\{(2rt R-1)^2x+t(2rt R-1)-a\}}.
\label{eqn:A1}$$ The function $A$ is indeed expressed by $R$ and parameters $r, t, a$ apart from $I$.
We find that the second equation of (\[eqn:equations1\]) gives $$tB=\frac{A}{rt}-2a(1-rtR)B^2,$$ and if we use this equation, the quadratic equation (\[eqn:quadratic1\]) for $B$ is written as $$x=\frac{A}{rt}-aB.
\label{eqn:aAB1}$$ Now we put the expression (\[eqn:B1\]) for $B$ and the expression (\[eqn:A1\]) for $A$ into (\[eqn:aAB1\]). Then we obtain the following equation for $R$, $$\begin{aligned}
& 8 x (rt R-1)
\Big\{ 4r^2t^2 xR^2 +2rt^2R -4rt x R + (x -a-t) \Big\}
\nonumber\\
& \quad \times
\Big[ 8r^3 t^3 x^2 R^3
-8r^2t^2 x \{2x+(r-1)t \}R^2 \nonumber \\
& \quad \qquad +2rt [ 5x^2+ \{(6r-5)t -a\} x+(r-1)^2t^2 ] R \nonumber \\
& \quad \qquad -[ 2x^2+ \{ (4r -3)t -2a \} x +(r-1) t \{(2r-1)t-a\} ] \Big] =0.
\label{eqn:Eq1}\end{aligned}$$
We want to obtain $R=R(x; r, t, a)$ which solves (\[eqn:Eq1\]) and satisfy the following [*continuity condition with respect to $a$*]{}, $$\lim_{a \to 0} R(x; r, t, a)=R_0
\label{eqn:R_02}$$ with (\[eqn:R\_0\]). This is given as a real solution of the cubic equation obtained from the last factor in (\[eqn:Eq1\]), $$\begin{aligned}
& 8r^3 t^3 x^2 R^3
-8r^2t^2 x \{2x+(r-1)t \}R^2 \nonumber \\
& \quad \qquad +2rt [ 5x^2+ \{(6r -5)t -a\} x+(r-1)^2t^2 ] R \nonumber \\
& \quad \qquad -[ 2x^2+ \{(4r -3)t -2a\} x +(r-1) t \{(2r-1)t-a\} ] =0.
\label{eqn:Eq2}\end{aligned}$$ Applying the Cardano formula, we obtain the solution as $$R(x; r, t, a)
=\frac{2x+(r-1)t}{3rt x}
- \frac{x^2 +\{ 3a -(2r+1)t \} x + (r-1)^2 t^2}
{3 \times 2^{2/3}r t g^{1/3} x}
-\frac{g^{1/3}}{6 \times 2^{1/3}rt x},
\label{eqn:R_solution}$$ where $g=g(x; r, t, a)$ is given by (\[eqn:g1\]) with (\[eqn:S1\]).
\[thm:R\_to\_R0\] The solution (\[eqn:R\_solution\]) of (\[eqn:Eq2\]) satisfies the continuity condition (\[eqn:R\_02\]) with (\[eqn:R\_0\]).
[*Proof*]{} If we set $a=0$, (\[eqn:g1\]) and (\[eqn:S1\]) become $$g_0 :=g(x; r, t, 0)
= -2 x^3+3(2r+1) t x^2-3 \Big[(r-1)(2r+1)t -\sqrt{-3 S_0} \Big] x + 2(r-1)^3 t^3,$$ with $$S_0 := S(x; r, t, 0)=
t^2 \{x^2-2(r+1) t x+(r-1)^2 t^2\}.
\label{eqn:h0}$$ In this case, the equality $$g_0=\frac{1}{4} \left\{ x -(r-1) t + \frac{1}{t} \sqrt{- 3 S_0} \right\}^3
\label{eqn:g0}$$ is established. By putting (\[eqn:g0\]) with (\[eqn:h0\]) into (\[eqn:R\_solution\]) with $a=0$, we can verify (\[eqn:R\_02\]) with (\[eqn:R\_0\]). 0.3cm
We set $$\varphi=\varphi(x; r, t, a) :=2 x \{ r t R(x; r, t, a)-1\}.
\label{eqn:varphiA}$$ Then it is easy to verify that the expression (\[eqn:R\_solution\]) for $R$ is written as (\[eqn:varphi\]) for $\varphi$ and that (\[eqn:A1\]) gives $$\frac{1}{A}-R^2=\frac{1}{(2 r t x)^2}
\Big[ 2(t-a+\sqrt{D})x -\varphi^2 \Big],$$ with $$D=-4 a \varphi + (t-a)^2.
\label{eqn:Det2}$$ Therefore, if $$2(t-a+\sqrt{D})x -\varphi^2 \geq 0,
\label{eqn:conditionA}$$ then (\[eqn:rho\_Z1\]) gives $$\rho(x)=\frac{\sqrt{2(t-a+\sqrt{D})x -\varphi^2}}{2 \pi r t x}.
\label{eqn:rhoA1}$$
For the expression (\[eqn:3\_MP1\]) for $\rho(x)$ given in Theorem \[thm:main1\], we perform the further calculation as follows. It is easy to verify the equality, $$2(t-a+\sqrt{D})=\left(\sqrt{d_-}+\sqrt{d_+}\right)^2,$$ where $d_{\pm}$ are defined by (\[eqn:d1\]). Therefore, we can see that $$\begin{aligned}
& 2(t-a+\sqrt{D}) x-\varphi^2
\nonumber\\
& \quad
=\left\{ \sqrt{x} \left(\sqrt{d_-}+\sqrt{d_+}\right) -\varphi \right\}
\left\{ \sqrt{x} \left(\sqrt{d_-}+\sqrt{d_+}\right) + \varphi \right\}
\nonumber\\
& \quad
=\left\{\left( \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right)^2-d_0 \right\}
\left\{d_0 - \left( \sqrt{x}-\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right)^2 \right\}
\nonumber\\
& \quad
= \left( \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} + \sqrt{d_0} \right)
\left( \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} - \sqrt{d_0} \right)
\nonumber\\
& \quad
\times
\left( \sqrt{d_0} + \sqrt{x}-\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right)
\left( \sqrt{d_0} - \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right)
\nonumber\\
& \quad
=(x-f_{\rm L})(f_{\rm R}-x),\end{aligned}$$ where $d_0$ is given by (\[eqn:d1\]), and $f_{\rm L}=f_{\rm L}(x; r, t, a)$ and $f_{\rm R}=f_{\rm R}(x; r, t, a)$ are given by (\[eqn:fLR\]). Hence (\[eqn:rhoA1\]) is written as (\[eqn:3\_MP1\]), provided that the condition (\[eqn:conditionA\]) is equivalent with the condition $$x_{\rm L}(r, t, a) \leq x \leq x_{\rm R}(r, t, a),
\label{eqn:conditionB}$$ where $x_{\rm L}(r, t, a)$ and $x_{\rm R}(r, t, a)$ are defined by (\[eqn:x\_LR\_3MP\]).
Determining the support of density function {#sec:support}
-------------------------------------------
Assume that $t >0, x >0$. Then if and only if the condition (\[eqn:conditionA\]) is satisfied, $\rho$ given by (\[eqn:rhoA1\]) is positive or zero. And if and only if $\rho \geq 0$, its Hilbert transform $R/\pi$ given by the second equation of (\[eqn:SP\]) and $\varphi$ defined by (\[eqn:varphiA\]) are real valued. By the explicit expression (\[eqn:varphi\]) with (\[eqn:g1\]) for $\varphi$, the following is obvious.
\[thm:rho\_real\] If and only if $S(x; r, t, a) \leq 0$, $\rho(x; r, t, a) \geq 0$.
0.3cm
Now we prove the following.
\[thm:rho\_zero\] If $S(x; r, t, a)=0$, then $\rho(x; r, t, a)=0$.
[*Proof*]{} The formula (\[eqn:rhoA1\]) with (\[eqn:Det2\]) is written as $$\rho(x)=\frac{\sqrt{\varphi F(\varphi)}}{2 \pi r t x \sqrt{2(a-t+\sqrt{D})x + \varphi^2}},
\label{eqn:rho_Z}$$ with $$\begin{aligned}
F(\varphi) &:= \frac{1}{\varphi}
\{2(a-t+\sqrt{D})x+\varphi^2 \} \{2(t-a+\sqrt{D})x-\varphi^2\}
\nonumber\\
&= -\varphi^3+4(t-a)x \varphi-16a x^2.
\label{eqn:F1}\end{aligned}$$ Hence we consider the condition of $F(\varphi)=0$. For $\varphi$ defined by (\[eqn:varphiA\]), the cubic equation (\[eqn:Eq2\]) is written as $$\begin{aligned}
H(\varphi) &:= \varphi^3 + 2 \{x-(r-1)t \} \varphi^2
+ [ x^2+\{(-2r+3)t-a\} x +(r-1)^2 t^2 ] \varphi
\nonumber\\
& \quad
+t x \{x + (r-1)(a-t) \} =0.\end{aligned}$$ Therefore, $$F(\varphi)=0 \quad \Longleftrightarrow \quad
\widetilde{F}(\varphi) := F(\varphi)+H(\varphi) = 0.$$ Note that the cubic terms of $\varphi$ are canceled and $\widetilde{F}(\varphi)$ is reduced to be quadratic in $\varphi$. We obtain $$\begin{aligned}
\widetilde{F}(\varphi)
&= 2 \{x-(r-1)t \} \varphi^2
+[ x^2 +\{(-2r+7) t - 5a\} x +(r-1)^2 t^2 ] \varphi
\nonumber\\
& \quad
+ x [(t-16a)x+(r-1)t(a-t)]
\nonumber\\
&=2 \{x-(r-1)t \} (\varphi-\varphi_-)(\varphi-\varphi_+),\end{aligned}$$ where $$\begin{aligned}
\varphi_- &= \varphi_-(x; r, t, a)=
-\frac{x^2+\{(7-2r)t-5a\} x+(r-1)^2 t^2 + \sqrt{\Delta}}{4\{x-(r-1)t\}},
\nonumber\\
\varphi_+ &= \varphi_+(x; r, t, a)=
-\frac{x^2+\{(7-2r)t-5a\} x+(r-1)^2 t^2 - \sqrt{\Delta}}{4\{x-(r-1)t\}},
\label{eqn:varphi_pm}\end{aligned}$$ with $$\begin{aligned}
\Delta &=x^4 + 2 \{-(2r-3)t + 59 a \} x^3
+[ \{2r(3r-8)+35\} t^2-2a(58r-33) t + 25 a^2 ] x^2
\nonumber\\
& \quad -2 (r-1)^2 \{(2r-3) t + a \} t^2 x + (r-1)^4 t^4.\end{aligned}$$ The condition $F(\varphi)=0$ is thus written as $F(\varphi_+) F(\varphi_-)=0$. On the other hand, by (\[eqn:F1\]) and (\[eqn:varphi\_pm\]), we can show that $$\begin{aligned}
& F(\varphi_+) F(\varphi_-) = H(\varphi_+) H(\varphi_-)
\nonumber\\
& \quad = -\frac{
\{4 x^3 - \{100 a+(12r+13)t\} x^2 + (r-1) t \{(12r+13) t-25 a\} x
- 4 (r-1)^3 t^3\} x^2 }{8\{x-(r-1)t\}^3}
\nonumber\\
& \qquad \times S(x; r, t, a).\end{aligned}$$ Note that $x-(r-1)t >0$, if $x>0$, since $r \in (0, 1], t \geq 0$. Hence the statement of Lemma is concluded.
0.3cm [*Proof of Theorem \[thm:main1\]*]{} Assume that $t > 0$. By definition (\[eqn:x\_LR\_3MP\]) of $x_{\rm L}(r, t, a)$ and $x_{\rm R}(r, t, a)$, and by Lemmas \[thm:rho\_real\] and \[thm:rho\_zero\], we can conclude that the condition (\[eqn:conditionA\]) is equivalent with the condition (\[eqn:conditionB\]). Under the condition $2(t-a+\sqrt{D})x-\varphi^2>0$ (equivalently, $x_{\rm L}(r, t, a) < x < x_{\rm R}(r, t, a)$), if $\varphi F(\varphi) \not=0$, then $2(a-t+\sqrt{D})x+\varphi^2 \not=0$ by the equality given by the first line of (\[eqn:F1\]). Hence $\rho(x)$ given by (\[eqn:rho\_Z\]) is finite for $x>0$. The proof of Theorem \[thm:main1\] is thus complete.
Proof of Proposition \[thm:support\] {#sec:proof2}
------------------------------------
The constant term in the cubic function $S(x; r, t, a)$ of $x$ given by (\[eqn:S1\]) becomes 0 for arbitrary $t > 0$ and $a \geq 0$, if and only if $r=1$. This implies Proposition \[thm:support\] (i).
When $r=1$, the cubic equation (\[eqn:cubiceq1\]) with (\[eqn:S1\]) becomes $$x \{ 4 a x^2- (8 a^2+20 a t - t^2) x + 4(a-t)^3 \} =0.$$ For $t >0, a>0$, this equation has three real solutions; $x=0$ and $$\begin{aligned}
x =x_{\pm}
&:= \frac{1}{8a} \{
8a^2+20at-t^2
\pm \sqrt{(8a^2+20at-t^2)^2-64a(a-t)^3} \}
\nonumber\\
&= \frac{1}{8a} \{
8a^2+20at-t^2
\pm \sqrt{t} (8a+t)^{3/2} \}.\end{aligned}$$ When $0 < t < a$, $8a^2+20at-t^2 > 7a^2+20at>0$, $$\sqrt{t} (8a+t)^{3/2}
=\sqrt{(8a^2+20at-t^2)^2-64a(a-t)^3}
< 8a^2+20at-t^2,$$ and hence $0 < x_- < x_+$. On the other hand, when $t \geq a$, $$\sqrt{t} (8a+t)^{3/2}
=\sqrt{(8a^2+20at-t^2)^2-64a(a-t)^3}
\geq |8a^2+20at-t^2|,$$ and hence $x_- < 0 < x_+$. Then we can conclude the following by the definitions of $x_{\rm L}(r, t, a)$ and $x_{\rm R}(r, t, a)$ given by (\[eqn:x\_LR\_3MP\]).
\[eqn:transition1\] Assume that $a > 0$.
[(i)]{} When $0 < t < a$, $$\begin{aligned}
x_{\rm L}(1, t, a) &= \frac{1}{8 a} \{
8 a^2 + 20 a t - t^2 -\sqrt{t} (8a+t)^{3/2} \},
\nonumber\\
x_{\rm R}(1, t, a) &= \frac{1}{8 a} \{
8 a^2 + 20 a t - t^2 +\sqrt{t} (8a+t)^{3/2} \},
\label{eqn:edges_t<a}\end{aligned}$$ and $$0 < x_{\rm L}(1, t, a) < x_{\rm R}(1, t, a).$$
[(ii)]{} When $t \geq a$, $$\begin{aligned}
x_{\rm L}(1, t, a) &= 0,
\nonumber\\
x_{\rm R}(1, t, a) &= \frac{1}{8 a} \{
8 a^2 + 20 a t - t^2 +\sqrt{t} (8a+t)^{3/2} \} > 0.\end{aligned}$$
Put $t=a-\varepsilon, 0 < \varepsilon \ll 1$ in $x_{\rm L} $ given by (\[eqn:edges\_t<a\]). Then it is easy to verify that $$x_{\rm L}(1, t, a)=\frac{4}{27} \frac{\varepsilon^3}{a^2}
+ \frac{8}{81} \frac{\varepsilon^4}{a^3}
+\frac{52}{729} \frac{\varepsilon^5}{a^4}
+\rO(\varepsilon^6).
\label{eqn:xLexp}$$ Hence Proposition \[thm:support\] (ii) is proved.
Proof of Proposition \[thm:critical\] {#sec:proof3}
-------------------------------------
In the case with $r=1$, we have the following expressions from (\[eqn:S1\]), (\[eqn:g1\]), (\[eqn:varphi\]), and (\[eqn:Det2\]), $$\begin{aligned}
S_1(x) &:= S(x; 1, t, a)
=x [ 4ax^2-(8a^2+20 a t-t^2) x +4(a-t)^3],
\nonumber\\
g_1(x) &:= g(x; 1, t, a)
= x \left[ -2 x^2+9(t+2a)x + 3 \sqrt{-3 S_1(x)} \right],
\nonumber\\
\varphi_1(x) &:= \varphi(x; 1, t, a)
= -\frac{2}{3} x - \frac{2^{1/3}}{3} \frac{\{x+3(a-t)\}x}{g_1(x)^{1/3}}
-\frac{g_1(x)^{1/3}}{3 \times 2^{1/3}},
\nonumber\\
D_1(x) &:= - 4 a \varphi_1(x)+(a-t)^2.
\label{eqn:r1_functions}\end{aligned}$$ Here we write $\rho_1(x):= \rho(x; 1, t, a)$.
First assume $0 < t < t_{\rm c}(a)=a$. Put $x_{\rm L} := x_{\rm L}(1, t, a)$ and let $0 < \delta \ll 1$. Since $S_1(x_L)=0$, we have the expansions in the form, $$\begin{aligned}
S_1(x_{\rm L}+\delta)
&= c_1 \delta + c_2 \delta^2 + \rO(\delta^3),
\nonumber\\
g_1(x_{\rm L}+\delta)
&= g_1(x_{\rm L})+c_3 \delta^{1/2}+c_4 \delta + \rO(\delta^{3/2}),
\nonumber\\
\varphi_1(x_{\rm L}+\delta)
&= \varphi_1(x_{\rm L})+ c_5 \delta + \rO(\delta^{3/2}),\end{aligned}$$ where $c_j, j=1, \dots, 5$ are functions of $t, a, x_{\rm L}$, but independent of $\delta$. It should be noted that, in the expansion of $\varphi_1(x_{\rm L}+\delta)$, the coefficient of term $\delta^{1/2}$ is proportional to $$d:= \{x_{\rm L}+3(a-t)\}x_{\rm L}
\left( \frac{2}{g_1(x_{\rm L})} \right)^{1/3}
-\left( \frac{g_1(x_{\rm L})}{2} \right)^{1/3},$$ where $g_1(x_{\rm L})=\{-2 x_{\rm L}+9(t+2a)\}x_{\rm L}^{2}$, and we can show that $d \propto S_1(x_{\rm L})=0$. Then, if we note $\rho_1(x_{\rm L})=0$, (\[eqn:rhoA1\]) gives $$\rho_1(x_{\rm L}+\delta)
=\frac{\delta^{1/2}}{\sqrt{2} \pi t x_{\rm L}}
\sqrt{ t-a+\sqrt{D_1(x_{\rm L})}
-\left( \frac{2a x_{\rm L}}{\sqrt{D_1(x_{\rm L})}}
+\varphi_1(x_{\rm L}) \right) c_5} +\rO(\delta).
\label{eqn:rho_expansion}$$ By (\[eqn:xLexp\]) with $\varepsilon := a-t$, we see that $$\begin{aligned}
\varphi_1(x_{\rm L}) &= - \frac{4}{9 a} \varepsilon^2 + \rO(\varepsilon^3),
\nonumber\\
D_1(x_{\rm L}) &= \frac{25}{9} \varepsilon^2 + \rO(\varepsilon^3),
\nonumber\\
c_5 &= - \frac{5a}{3} \varepsilon^{-1} + \rO(1).\end{aligned}$$ Hence by (\[eqn:rho\_expansion\]), Proposition \[thm:critical\] (i) is proved.
Next assume $t=t_{\rm c}(a)=a$. Then (\[eqn:r1\_functions\]) gives $$\begin{aligned}
S_1(x) &= a(4x-27a) x^2
=-27 a^2 x^2+\rO(x^3),
\nonumber\\
g_1(x) &= 2 \times 3^3 a x^2 + \rO(x^3),
\nonumber\\
\varphi_1(x) &= - a^{1/3} x^{2/3} + \rO(x),
\nonumber\\
D_1(x) &= - 4 a \varphi_1(x) = 4 a^{4/3} x^{2/3} +\rO(x).\end{aligned}$$ Then (\[eqn:rho\_expansion\]) proves Proposition \[thm:critical\] (ii).
Finally assume $t > t_{\rm c}(a)=a$. Then (\[eqn:r1\_functions\]) gives $$\begin{aligned}
S_1(x) &= - 4 |\varepsilon|^3 x+\rO(x^2),
\nonumber\\
g_1(x) &= 6 \sqrt{3} |\varepsilon|^{3/2} x^{3/2} + \rO(x^2),
\nonumber\\
\varphi_1(x) &= -\frac{t}{|\varepsilon|} x + \rO(x^{3/2}),
\nonumber\\
D_1(x) &= |\varepsilon|^2 + \rO(x),\end{aligned}$$ where $|\varepsilon|=-\varepsilon=t-a$. Then (\[eqn:rho\_expansion\]) proves Proposition \[thm:critical\] (iii). The proof of Proposition \[thm:critical\] is hence complete.
Long-Term Scaling and Proof of Proposition \[thm:long\_term\] {#sec:proof4}
-------------------------------------------------------------
It is obvious that the functions $S(x; r, t, a)$, $g(x; r, t, a)$, $\varphi(x; r, t, a)$, $f_{\rm L}(x; r, t, a)$, and $f_{\rm R}(x; r, t, a)$, which appeared in Theorem \[thm:main1\], are all homogeneous as multivariate functions of $x, t, a$ for each fixed value of $r \in (0, 1]$. Actually, we see that, for an arbitrary parameter $\kappa>0$, $$\begin{aligned}
& S(\kappa x; r, \kappa t, \kappa a) = \kappa^4 S(x; r, t, a), \quad
\nonumber\\
&
g(\kappa x; r, \kappa t, \kappa a) = \kappa^3 g(x; r, t, a), \quad
\varphi(\kappa x; r, \kappa t, \kappa a) = \kappa \varphi (x; r, t, a),
\nonumber\\
& f_{\rm L}(\kappa x; r, \kappa t, \kappa a) = \kappa f_{\rm L}(x; r, t, a),
\quad
f_{\rm R}(\kappa x; r, \kappa t, \kappa a) = \kappa f_{\rm R}(x; r, t, a).\end{aligned}$$ Then the scaling property of the three-parametric MP density (\[eqn:rho\_scaling\]) is concluded. If we set $\kappa=1/t$, replace $x$ by $t x =: y$, and take the limit $t \to \infty$ for a fixed $a >0$, Proposition \[thm:long\_term\] is proved.
Concluding Remarks
==================
\[sec:concluding\] In the present paper, we have studied the time-dependent complex Wishart ensemble of random matrices with an external source. Following Blaizot, Nowak, and Warcho[ł]{} [@BNW14], we have considered the hydrodynamic limit of the process of the squared-singular-values of random complex rectangular matrices with a rectangularity $r \in (0, 1]$. We solved the algebraic equation (\[eqn:z\_t\_1\]) for the Green’s function $G_{\xi}$, which is equivalent with the nonlinear PDE for $G_{\xi}$ (\[eqn:PDE1\]) under the initial distribution $\xi(dx)=\delta_a(dx), a > 0$; a delta measure concentrated at $x=a >0$. This algebraic equation (\[eqn:z\_t\_1\]) was given and its solution was studied by [@BNW14], but explicit expressions for the density function has not been available. In this paper we called the density function of this system the [*three-parametric Marcenko–Pastur (MP) density*]{}, $\rho(x; r, t, a), r \in (0, 1], t >0, a \geq 0$, and gave useful expressions to $\rho(x; r, t, a)$ (Theorem \[thm:main1\]). As an application of the result, the dynamic critical phenomena were clarified (Propositions \[thm:support\] and \[thm:critical\]), which are observed at the critical time $t_{\rm c}(a)=a$, if and only if $r=1$ and $a>0$. There we have introduced six kinds of [*critical exponents*]{}, $$\nu=3, \quad
\beta_1=\beta_2=\frac{1}{2}, \quad
\gamma_1=\frac{5}{2}, \quad
\gamma_2 =\frac{1}{\delta}=\frac{1}{3}, \quad
\gamma_3=\frac{1}{2},$$ which represent the singularities of the dynamic critical phenomena.
The present results can be regarded as [*macroscopic descriptions*]{} of the system and the critical phenomena. [*Microscopic descriptions*]{} have been also studied in several papers [@KMW11; @DKRZ12; @HK13; @For13] for the similar systems and the associated dynamic critical phenomena. Connection between these two kinds of descriptions [@BNW14] and the universality of such dynamic critical phenomena will be studied in more detail in the future. As mentioned in Remark 3 given at the end of Section \[sec:main\_results\], in the context of high energy physics, the present macroscopic description can be regarded as a mean-field approximation for more precise theory of QCD which exhibits spontaneous chiral symmetry breaking.
As emphasized in [@BNW13; @BNW14], the MP density of the Wishart random-matrix ensemble has been used in a broad range of mathematical sciences, physics, and information theory (see the references in [@BNW13; @BNW14]). It is expected that the non-centered Wishart ensembles/processes and the present three-parametric MP density will be also useful in many applications, where the mean zero condition cannot be assumed.
[**Acknowledgements**]{} The present authors thank Hiroya Baba for useful discussion when the present study was started. They also thank anonymous referees very much, who suggested them to discuss the present results in the context of the application of chGUE to QCD. This work was supported by the Grant-in-Aid for Scientific Research (C) (No.19K03674), (B) (No.18H01124), and (S) (No.16H06338) of Japan Society for the Promotion of Science.
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[^1]: Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: taiki@phys.chuo-u.ac.jp
[^2]: 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
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abstract: 'We consider a directed polymer interacting with a diluted pinning potential restricted to a line. We characterize explicitely the set of disorder configurations that give rise to localization of the polymer. We study both relevant cases of dimension $1+1$ and $1+2$. We also discuss the case of massless effective interface models in dimension $2+1$.'
author:
- |
É. Janvresse, T. de la Rue, Y. Velenik\
Laboratoire de Mathématiques Raphaël Salem\
UMR-CNRS 6085, Université de Rouen\
`Elise.Janvresse@univ-rouen.fr,`\
`Thierry.de-la-Rue@univ-rouen.fr,`\
`Yvan.Velenik@univ-rouen.fr`
bibliography:
- 'JRV04c.bib'
title: Pinning by a sparse potential
---
Key words: Polymer, interface, random environment, pinning, localization.
AMS subject classification: 60K35, 60K37, 82B41.
It is customary, when modelling a disordered physical system, to assume that the disorder is sampled from some suitable random distribution. Of course there is a high degree of arbitrariness in the choice of this distribution, and one hopes that only qualitative features are relevant. Then, in the best possible cases, one can prove results that hold for almost every disorder configuration. However, there are several drawbacks with such an approach : First, it would be desirable to avoid these additional assumptions on the distribution of the disorder, and second, even with an almost sure result, we are left clueless about the validity of the desired property when an explicit disorder configuration is given. Therefore, it would be very valuable if one could instead characterize the set of realizations of the environment for which a specific property holds, or at least give some sufficient conditions. This is a much more ambitious program, and it is probably doomed to fail in general. In this paper, we give a simple example of a problem where such an approach can actually be pursued.
An important physical problem, which has received much attention recently from the mathematical community, is that of a directed polymer in a random environment, see, e.g., [@CoShYo04] and references therein. The latter is modelled by an exponential perturbation of the path measure of a $d$-dimensional random walk (or Brownian motion), depending on the realization of a random environment. The perturbation is such that visits of the random walk to regions where the environment takes positive values are rewarded, while visits to regions of opposite sign are penalized. Among the many questions of interest, there is the problem of studying the superdiffusive behaviour of the path in dimension $1+1$[^1]. Heuristically, one expects that there will be an “optimal tube” in the environment, inside which the landscape looks particularly good from the random walk point of view, and along which the random walk will localize. It is natural to split this problem into two : 1) Establish the existence of such an “optimal tube”, whatever that really means, and 2) Prove that given such an “optimal tube”, there is pinning of the polymer along the tube. Once we accept that such a splitting is natural, one can start to build simpler models for both points separately, in order to gain a better understanding of these issues. For the second part, a natural simplification is as follows : Consider a path $Y : \mathbb{N}\to\mathbb{Z}^d$, and perturb the path measure of a random walk $X$ by rewarding each intersection between the paths $X$ and $Y$. The question is then to understand under which conditions on the path $Y$ there will be pinning of the polymer $X$, i.e, there will be a positive density of such intersections. This might then shed some light on the properties of this “optimal tube” one should look for when analyzing the random environment. The only result we are aware of in this direction is due to Ioffe and Louidor [@IoLo04], who consider the situation where $Y$ is itself a random walk (its increments having possibly a different law from those of $X$), and proved that pinning occurs in dimension $1+1$ for almost all realizations of $Y$. This is however “only” an almost sure result, and as such it does not tell what is the set of measure $0$ (w.r.t. the law of the random walk $Y$) of paths which do not lead to pinning, which is most unfortunate since the “optimal tube” is expected to behave quite differently from a random walk trajectory. It would thus be very interesting to get an explicit characterization of the paths $Y$ for which pinning occurs, or at least sufficient conditions.
We hope to come back to this issue in the future. In the present work, we analyze a much simpler situation. Namely, here the pinning potential is restricted to a single line, and the disorder comes from the fact that this potential is diluted. More precisely, let $\omega\in\{0,1\}^\mathbb{N}$, $\eta>0$, and let ${\mathbb{P}}_0$ denote the law of an aperiodic, symmetric random-walk on $\mathbb{Z}^d$ starting from $0$, with increments of finite variance. Our main interest is in the following perturbation of ${\mathbb{P}}_0$, $${\mathbb{P}}_{N,\eta}^\omega (X) {\stackrel{{\scriptscriptstyle\text{def}}}{=}}(Z_{N,\eta}^\omega)^{-1} \;\exp\left(\eta\sum_{i\in\Lambda_N} {\mathbf{1}}_{(X_i=0)} \omega_i \right) \, {\mathbb{P}}_0 (X),$$ where $\Lambda_N{\stackrel{{\scriptscriptstyle\text{def}}}{=}}\{1,\ldots, N\}$ and $Z_{N,\eta}^\omega$ is the partition function used to normalize ${\mathbb{P}}_{N,\eta}^\omega$ to a probability measure. This measure models the interaction of a directed polymer in $1+d_2$ dimensions, interacting with an attractive diluted potential restricted to the line $x_2=\dots=x_{d_2+1} = 0$. The central question is under which conditions does such a potential localize the polymer, i.e., when is it true that $$\liminf_{N\to\infty} \frac1N {\mathbb{E}}_{N,\eta}^\omega\left( \sum_{i=1}^N {\mathbf{1}}_{X_i = 0} \right) > 0 \text{ ?}$$ When this happens, we say that there is pinning of the polymer by the potential. It has been known for a long-time that in the special case $\omega\equiv 1$, the polymer is pinned for any value of $\eta>0$ when $d_2=1$ or $d_2=2$, but is not pinned for small enough values of $\eta$ in higher dimensions (see *e.g.* [@Bu81] for a special case, and [@BoVe01] for a more general treatment). The general case of a discrete-time Markov chain interacting with a (possibly random) potential restricted to a line was recently investigated by Alexander and Sidoravicius [@AlSi05]. In particular, they compared the effect of an i.i.d. random potential with the constant potential given by its average. One of their main results is that pinning of the polymer is strictly enhanced by the presence of such disorder. The situation we consider is a simple particular case of their setting, but our result is stronger since we work with a fixed (arbitrary) environment.
Of course, diluting the potential only makes it less likely for the polymer to be pinned, so there is still delocalization at small values of $\eta$ in dimensions $3$ and higher, for arbitrary $\omega$. In this work, we therefore restrict our attention to dimensions $1$ and $2$. Rather remarkably, in this case it is possible to obtain a very simple characterization of the set of environments for which pinning occurs, see Theorem \[thm\_Main\] below and its corollary. Before stating the result, we also introduce another case where the same question can be investigated: $2+1$-dimensional massless effective interface models. In this case, let $\Lambda\Subset\mathbb{Z}^2$, let $V:\mathbb{R}\to\mathbb{R}$ such that $0 < c_- \leq V'' \leq c_+ < \infty$, and let $\eta\geq 0$. We are interested in the measure (on $\mathbb{R}^{\Lambda}$) defined by[^2] $$\begin{gathered}
\label{EffInt}
{\mathbb{P}}_{\Lambda,\eta}^\omega (dX) {\stackrel{{\scriptscriptstyle\text{def}}}{=}}(Z_{\Lambda,\eta}^\omega)^{-1} \;\exp \Bigl( -\tfrac12\sum_{\substack{i \sim j\\ i,j\in\Lambda}} V(X_i-X_j) + \sum_{\substack{i \sim j\\ i\in\Lambda,j\not\in\Lambda}} V(X_i) \Bigr) \\
\times \prod_{i\in\Lambda} \left( dX_i + \eta \omega_i \delta_0(dX_i) \right) ,\end{gathered}$$ where $\delta_0$ is the Dirac mass at $0$ and $\omega\in\{0,1\}^{\mathbb{N}^2}$. In the special case $\Lambda = \Lambda_N {\stackrel{{\scriptscriptstyle\text{def}}}{=}}\{1,\ldots,N\}^2$, we simply write ${\mathbb{P}}_{N,\eta}^\omega$ and $Z_{N,\eta}^\omega$. This models a two-dimensional interface in a three-dimensional medium, interacting with an attractive (diluted) potential located in the plane $x_3=0$. The basic question is the same as above: Determine under which conditions the interface is localized by the potential. The case $\omega\equiv1$ has been studied in details recently, see [@BoVe01] and reference therein. It turns out that in this case too, an explicit description of the set of disorder configurations leading to pinning can be obtained.
![A simulation of the 1+1-dimensional process. The environment has density $.8$ on the first and last third of the interval, and $0$ inbetween.[]{data-label="fig:path"}](path.eps){width="12cm" height="2cm"}
The following theorem and its corollary are valid for all three cases of dimensions $1+1$, $1+2$ and $2+1$.
\[thm\_Main\] Let $0<\delta<1$ and $\eta>0$. For all $N>N_0(\delta,\eta)$, and for all $\omega$ such that $$\label{density}
\sum_{i\in\Lambda_N}\omega_i > \delta |\Lambda_N|,$$ we have for some $C=C(\delta,\eta)>0$ $${\mathbb{E}}_{\eta,N}^\omega\left[\sum_{i\in\Lambda_N} {\mathbf{1}}_{(X_i=0)}\,\omega_i\right] > C |\Lambda_N|.$$
\[cor\_Main\] For any $\eta>0$, $$\label{eq_densityOfVists}
\liminf_{N\to\infty} |\Lambda_N|^{-1} {\mathbb{E}}_{\eta,N}^\omega\left[\sum_{i\in\Lambda_N}^N{\mathbf{1}}_{(X_i=0)}\,\omega_i\right] > 0 ,$$ if and only if $$\liminf_{N\to\infty} |\Lambda_N|^{-1} \sum_{i\in\Lambda_N} \omega_i > 0.$$
Note that this is a sensible definition of pinning, since if pinning does not hold in this sense, then there exists a sequence of increasing boxes, such that along this sequence the density of pinned sites goes to zero. Of course, along other sequences there can be a strictly positive density of pinned sites (examples are easily constructed).
<span style="font-variant:small-caps;">Step 1.</span> It is enough to prove that $Z_{N,\eta}^\omega/Z_{N,0}^\omega>D^{|\Lambda_N|}$, for some $D=D(\delta,\eta)>1$. Indeed, $$\label{eq_identity}
\log \frac{Z_{N,\eta}^\omega}{Z_{N,0}^\omega} = \int_0^\eta {\mathbb{E}}_{\tilde\eta,N}^\omega\left[\sum_{i\in\Lambda_N} {\mathbf{1}}_{(X_i=0)}\,\omega_i\right]\,d\tilde\eta,$$ and, since the expectation is increasing in $\tilde\eta$, we obtain $${\mathbb{E}}_{\eta,N}^\omega\left[\sum_{i\in\Lambda_N} {\mathbf{1}}_{(X_i=0)}\,\omega_i\right] > \dfrac{|\Lambda_N|\log D}{\eta}.$$ Of course, in the cases of dimensions $1+1$ and $1+2$, $Z_{N,0}^\omega=1$.
<span style="font-variant:small-caps;">Step 2.</span> We first treat the cases of polymers, that is dimensions $1+d_2$, $d_2=1$ or $2$. Let $$\Omega_N{\stackrel{{\scriptscriptstyle\text{def}}}{=}}\{i\in\Lambda_N,\ \omega_i=1\}.$$ Writing $$\exp\left(\eta\sum_{i\in\Lambda_N} {\mathbf{1}}_{(X_i=0)}\, \omega_i\right)=\prod_{i\in\Lambda_N} \left((e^\eta-1){\mathbf{1}}_{(X_i=0)}{\mathbf{1}}_{i\in\Omega_N}+1\right)$$ and expanding the product, we get $$\begin{aligned}
\nonumber
Z_{N,\eta}^\omega & = {\mathbb{E}}_0\left[\exp\left(\eta\sum_{i\in\Lambda_N} {\mathbf{1}}_{(X_i=0)}\, \omega_i\right)\right]\\
& = \sum_{A\subset\Omega_N}(e^\eta-1)^{|A|}\,{\mathbb{P}}_0(X_i\equiv 0\text{ on }A).
\label{decomposition}\end{aligned}$$
<span style="font-variant:small-caps;">Step 2.1.</span> It is convenient to number the sites of $\Omega_N$, $\Omega_N=\{t_1<\cdots<t_{|\Omega_N|}\}$. Restricting the sum to the subsets $A$ of fixed cardinality $r$ (to be chosen later), and using the Markov property for the random walk, we obtain the following lower bound for the partition function (where $t_0=\ell_0{\stackrel{{\scriptscriptstyle\text{def}}}{=}}0$). $$\label{lower-bound}
Z_{N,\eta}^\omega \ge (e^\eta-1)^r\sum_{0<l_1<\cdots<l_r\le|\Omega_N|}\,\prod_{i=1}^r{\mathbb{P}}_0\left(X_{t_{\ell_i}-t_{\ell_{i-1}}}=0\right),$$ which yields, by the local CLT, for some $c>0$, $$\label{lower-bound2}
Z_{N,\eta}^\omega \ge (e^\eta-1)^r\sum_{0<l_1<\cdots<l_r\le|\Omega_N|}\,\prod_{i=1}^r\dfrac{c}{(t_{\ell_i}-t_{\ell_{i-1}})^{d_2/2}}.$$
<span style="font-variant:small-caps;">Step 2.2.</span> We begin by considering the simpler case of dimension $1+1$. Observe that, by Jensen inequality, we have $$\label{jensen}
\prod_{i=1}^r\dfrac{1}{\sqrt{t_{\ell_i}-t_{\ell_{i-1}}}} = \exp\left(-\dfrac{r}{2}\sum_{i=1}^r\dfrac{1}{r}\log(t_{\ell_i}-t_{\ell_{i-1}})\right)\ge \exp\left(-\dfrac{r}{2}\log\dfrac{N}{r}\right).$$ Setting $r=|\Omega_N|/K$ for some integer $K$ to be chosen later, and observing that the number of terms in the RHS of is at least $K^r$, we obtain $$Z_{N,\eta}^\omega \ge \left(Kc(e^\eta-1)\sqrt{\dfrac{r}{N}}\right)^r.$$ Using $|\Omega_N|\ge \delta N$, and choosing $K$ large enough, the conclusion follows.
<span style="font-variant:small-caps;">Step 2.3.</span> We now turn to the more delicate case of dimension $1+2$. Although the above argument involving Jensen inequality is too rough to conclude now, it suggests that the worst possible environment $\omega$ with a fixed density $\delta$ occurs when $t_i-t_{i-1}\equiv \delta^{-1}$.
We introduce $\Delta_i {\stackrel{{\scriptscriptstyle\text{def}}}{=}}t_i-t_{i-1}$, and $$\begin{aligned}
\Psi(\Delta_1,\ldots,\Delta_{|\Omega_N|}) &{\stackrel{{\scriptscriptstyle\text{def}}}{=}}\sum_{0<\ell_1<\cdots<\ell_r\le|\Omega_N|} \,\prod_{i=1}^r\dfrac{1}{(t_{\ell_i}-t_{\ell_{i-1}})}\\
& =\sum_{0<\ell_1<\cdots<\ell_r\le|\Omega_N|} \frac1{(\Delta_1+\cdots+\Delta_{\ell_1})\cdots (\Delta_{\ell_{r-1}+1}+\cdots+\Delta_{\ell_r})} .\end{aligned}$$ Instead of working directly with the function $\Psi$, it is convenient to consider a periodized version defined by (see Figure \[figure\]) $$\begin{gathered}
{\Psi_{\textrm{per}}}({\tilde\Delta}_1,\Delta_2,\ldots,\Delta_{|\Omega_N|}) {\stackrel{{\scriptscriptstyle\text{def}}}{=}}\\ \sum_{0<\ell_1<\cdots<\ell_r\le|\Omega_N|} \frac1{(\Delta_{\ell_r+1}+\cdots+\Delta_{|\Omega_N|}+{\tilde\Delta}_1+\Delta_2+\cdots+\Delta_{\ell_1})}\times\\
\prod_{i=2}^r \frac1{(\Delta_{\ell_{i-1}+1}+\cdots+\Delta_{\ell_i})} ,\end{gathered}$$ where ${\tilde\Delta}_1 {\stackrel{{\scriptscriptstyle\text{def}}}{=}}N+1 - \sum_{i=2}^{|\Omega_N|}\Delta_i$.
We are going to determine the (unique) minimum of the function ${\Psi_{\textrm{per}}}$, seen as a function on $\mathbb{R}_+^{|\Omega_N|}$, restricted to the manifold ${\tilde\Delta}_1+\sum_{i=2}^{|\Omega_N|} \Delta_i = N+1$.
Notice first that ${\Psi_{\textrm{per}}}$ is a convex function. Indeed, the function $(x_1,\ldots,x_k) \mapsto (x_1\cdots x_k)^{-1}$ is convex on $\mathbb{R}_+^k$, the composition of a convex function with an affine function is convex, and the sum of convex functions is also convex, as well as its restriction to an affine subspace.
We claim that the point $\Delta_i \equiv \Delta {\stackrel{{\scriptscriptstyle\text{def}}}{=}}(N+1)/|\Omega_N|$ is a (strict) local minimum of ${\Psi_{\textrm{per}}}$, and therefore its unique minimum. To prove this, it is enough, by symmetry, to show that $$\label{minloc}
{\Psi_{\textrm{per}}}(\Delta,\Delta,\Delta,\ldots,\Delta) - {\Psi_{\textrm{per}}}(\Delta+h,\Delta-h,\Delta,\ldots,\Delta) \leq 0 ,$$ for all $h$ small enough. Indeed, each allowed configuration $({\tilde\Delta}_1,\Delta_2,\ldots,\Delta_{|\Omega_{N}|})$ can be written as $(\Delta+h_1,\Delta+h_2-h_1,\ldots,\Delta+h_{|\Omega_{N}|-1}-h_{|\Omega_{N}|-2},\Delta-h_{|\Omega_{N}|-1})$, and equation together with invariance of ${\Psi_{\textrm{per}}}$ under cyclic permutation of the variables, ensure that all partial derivatives with respect to the $h_i$’s are nonnegative.
Observing that ${\Psi_{\textrm{per}}}$ is also invariant under the transformation $$({\tilde\Delta}_1,\Delta_2,\ldots,\Delta_{|\Omega_N|}) \mapsto (\Delta_{|\Omega_N|},\ldots,\Delta_2,{\tilde\Delta}_1),$$ we have that $${\Psi_{\textrm{per}}}(\Delta+h,\Delta-h,\Delta,\ldots,\Delta) = {\Psi_{\textrm{per}}}(\Delta-h,\Delta+h,\Delta,\ldots,\Delta) .$$ Therefore, the claim follows by convexity.
We need to compare $\Psi$ and ${\Psi_{\textrm{per}}}$. Noticing that $\Delta_{\ell_r+1}+\cdots+\Delta_{|\Omega_N|}+{\tilde\Delta}_1 \geq \Delta_1$, we immediately get that $$\Psi(\Delta_1,\ldots,\Delta_{|\Omega_N|}) \geq {\Psi_{\textrm{per}}}({\tilde\Delta}_1,\Delta_2,\ldots,\Delta_{|\Omega_N|}) \geq {\Psi_{\textrm{per}}}(\Delta,\ldots,\Delta) .$$ Therefore $Z_{N,\eta}^\omega \geq \left( c(e^\eta -1) \right)^r \, {\Psi_{\textrm{per}}}(\Delta,\ldots,\Delta)$. It only remains to find a bound on ${\Psi_{\textrm{per}}}(\Delta,\ldots,\Delta)$. Let $K>0$; this number will be chosen later. We have that, for all $N$ large enough, $$\begin{aligned}
{\Psi_{\textrm{per}}}(\Delta,\ldots,\Delta)
&\geq \frac1N \sum_{\substack{0<\ell_1<\ell_2<\ldots<\ell_r\\|\ell_i-\ell_{i-1}| \leq K}}\, \prod_{i=2}^r \frac1{(\ell_i-\ell_{i-1})\Delta}\\
&= \frac1{\Delta^{r-1}N} \sum_{\substack{0<\ell_1<\ell_2<\ldots<\ell_{r-1}\\|\ell_i-\ell_{i-1}| \leq K}}\, \prod_{i=2}^{r-1} \frac1{\ell_i-\ell_{i-1}} \, \sum_{k=1}^K \frac1k\\
&\geq \frac1{\Delta^{r-1}N} \log K \, \sum_{\substack{0<\ell_1<\ell_2<\ldots<\ell_{r-1}\\|\ell_i-\ell_{i-1}| \leq K}}\, \prod_{i=2}^{r-1} \frac1{\ell_i-\ell_{i-1}}\\
&\geq \frac{(\log K)^{r-1}}{\Delta^{r-1} N} .\end{aligned}$$ Choosing now $K$ large enough (which is possible as soon as $N>N_0(\delta,\eta)$), we conclude the proof of the theorem.
\[figure2\]
<span style="font-variant:small-caps;">Step 3.</span> We finally consider the case of dimension $2+1$. Let $\Omega_N {\stackrel{{\scriptscriptstyle\text{def}}}{=}}\{i\in\Lambda_N, \omega_i=1\}$. Expanding the product in , we obtain a representation similar to , $$\label{expansion}
\frac{Z_{N,\eta}^\omega}{Z_{N,0}^\omega} = \sum_{A\subset\Omega_N} \eta^{|A|}\, \frac {Z_{\Lambda_N \setminus A,0}^\omega}{Z_{N,0}^\omega} .$$ Let us partition $\Lambda_N$ into cells of sidelength $K$ (to be chosen later). We suppose, to ease notations, that this partitionning can be done exactly; the general case is treated in a straightforward way.
Let $0<\rho<\delta/(2-\delta)$, and let us say that a cell is *good* if it contains at least $\rho K^2$ sites of $\Omega_N$. Clearly, there is at least a density $\rho/(1+\rho)$ of good cells, since otherwise $$\sum_{i\in\Lambda_N} \omega_i \leq \left( 1-\frac{\rho}{1+\rho} \right) \frac{|\Lambda_N|}{K^2}\, \rho K^2 + \frac{\rho}{1+\rho}\frac{|\Lambda_N|}{K^2}\, K^2 = \frac{2\rho}{1+\rho} |\Lambda_N| <\delta |\Lambda_N| .$$ Now, let us say that a row of cells is *good* if the number of good cells in this row is at least $\zeta N/K$, where $\zeta$ is some small enough constant. A similar computation shows that for the class of environments we consider, there must be at least a fraction $\zeta/(1+\zeta)$ of good rows.
Returning to , we see that we must find a reasonable lower bound on the ratio of partition functions in the RHS, for a large enough class of sets $A$. Let us denote by $\mathbf{A}$ the class of sets $A$ containing exactly one site in each good cell located in a good row. The good rows can be numbered $\mathcal{R}_1,\ldots,\mathcal{R}_g$, with $g\geq \zeta/(1+\zeta) N/K$. $A_k$, the set of sites of $A\in\mathbf{A}$ belonging to the row $\mathcal{R}_k$, can then also be ordered according to their first coordinate, $A_k = \{t_{k,1},\ldots,t_{k,n_k}\}$, where $n_k\geq \zeta N/K$. For each $k$, let also $t_{k,0}$ be a site of $\mathbb{Z}^2\setminus\Lambda_N$ neighbour of the leftmost cell of $\mathcal{R}_k$. We need the following result from [@DeVe00] : For any $B\Subset\mathbb{Z}^2$ and $t\in B$, $$\frac {Z_{B\setminus \{t\},0}^\omega}{Z_{B,0}^\omega} \geq \frac{c}{\sqrt{\log(1+d(t,B^c))}}.$$ From this, we obtain, setting $A_{k,i} = A \setminus \{ t_{\ell,j} : \ell<k, \text{ or } \ell=k \text{ and } j\leq i \}$, that $$\begin{aligned}
\frac {Z_{\Lambda_N \setminus A,0}^\omega}{Z_{N,0}^\omega}
&= \prod_{k=1}^g \prod_{i=1}^{n_k} \frac {Z_{\Lambda_N \setminus A_{k,i-1},0}^\omega}{Z_{\Lambda_N \setminus A_{k,i},0}^\omega}\\
&\geq \prod_{k=1}^g \prod_{i=1}^{n_k} \frac{c}{\sqrt{\log |t_{k,i}-t_{k,i-1}|}}\,.\end{aligned}$$ We can now conclude exactly as in Step 2.2. Indeed, the innermost product is of the same type as in , except that we have an additional log which only helps us.
The *if* part follows immediately from Theorem \[thm\_Main\]. To prove the converse, it is enough to bound the indicator function in by 1.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to M. Lagouge and G. Giacomin for pointing out a mistake in an earlier version of this paper.
[^1]: We use the terminology “dimension $d_1+d_2$" when considering a $d_1$-dimensional (directed) object in a $(d_1+d_2)$-dimensional space.
[^2]: In this paper, we only consider the so-called $\delta$-pinning. However, given the very rough nature of the bounds we are after, there is no difficulty in treating more general potentials.
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abstract: 'We calculate the Chern-Simons invariants of the twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of the twist knot cone-manifold structures. Following the general instruction of Hilden, Lozano, and Montesinos-Amilibia, we here present the concrete formulae and calculations. We use the Pythagorean Theorem, which was used by Ham, Mednykh, and Petrov, to relate the complex length of the longitude and the complex distance between the two axes fixed by two generators. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic twist knot orbifolds. We also derive some interesting results. The explicit formula of $A$-polynomials of twist knots are obtained from the complex distance polynomials. Hence the edge polynomials corresponding to the edges of the Newton polygons of A-polynomials of twist knots can be obtained. In particular, the number of boundary components of every incompressible surface corresponding to slope $-4n+2$ appears to be $2$.'
address:
- |
Department of Science, Hongik University, 94 Wausan-ro, Mapo-gu, Seoul, 121-791\
Department of Mathematical Sciences, Seoul National University, San 56-1 Shinrim-dong Kwanak-gu Seoul 151-747\
Korea
- |
Department of Mathematics Education, Hongik University, 94 Wausan-ro, Mapo-gu, Seoul, 121-791\
Korea
author:
- 'Ji-Young Ham, Joongul Lee'
title: 'Explicit formulae for Chern-Simons invariants of the twist knot orbifolds and Edge polynomials of twist knots'
---
Introduction
============
In the 1970s, Chern and Simons [@CS] defined an invariant of a compact $3\ (\text{mod}\ 4)$ - dimensional Riemannian manifold, M, which is now called the Chern-Simons invariant, $\text{\textnormal{cs}}(M)$. In the 1980s, Meyerhoff [@Mey1] extended the definition of $\text{\textnormal{cs}}(M)$ to cusped manifolds. It is the integral of a certain 3-form and an invariant of the Riemannian connection on a principal tangent bundle of M.
Various methods of finding the Chern-Simons invariant using ideal triangulations have been introduced [@N1; @N2; @Z1; @CMY1; @CM1; @CKK1] and implemented [@SnapPy; @Snap]. But, for orbifolds, to our knowledge, there does not exist a single convenient program which computes Chern-Simons invariant. The Chern-Simons invariant can also be obtained by eta-invariant, $\eta(M)$. $\text{\textnormal{cs}}(M)=\frac{3}{2} \eta(M)$ (mod $\frac{1}{2}$) [@CGHN1; @Y1]. But it is easier to compute Chern-Simons invariants than $\eta$-invariants.
Instead of working on complicated combinatorics of 3-dimensional ideal tetrahedra to find the Chern-Simons invariants of the twist knot orbifolds, we deal with simple one dimensional singular loci. To make the computation simpler, we express the complex length of the singular locus in terms of the complex distance between the two axes fixed by two generators. To find out the complex length of the singular locus, we start from working on $\text{\textnormal{SL}}(2,C)$. The singular locus appears in $\text{\textnormal{SL}}(2,C)$ as a series of matrix multiplications. If we first calculate the complex distance and recover the complex length of the singular locus from the complex distance using Lemma \[lem:pytha\], the multiplication needed in $\text{\textnormal{SL}}(2,C)$ can be cut down approximately by half. Similar methods for volumes can be found in [@HMP]. We use the Schläfli formula for the generalized Chern-Simons function on the family of a twist knot cone-manifold structures [@HLM3]. In [@HLM2] a method of calculating the Chern-Simons invariants of two-bridge knot orbifolds were introduced but without explicit formulae. Similar approaches for $SU(2)$-connections can be found in [@KK1] and for $\text{\textnormal{SL}}(2,C)$-connections in [@KK2]. Explicit integral formulae for Chern-Simons invariants of the Whitehead link (the two component twist link) orbifolds and their cyclic coverings are presented in [@A; @A1]. A brief explanation for twist knot cone-manifolds are in [@HMP]. You can also refer to [@CHK; @T1; @K1; @P2; @HLM1; @PW].
The main purpose of the paper is to find the explicit and efficient formulae for Chern-Simons invariants of the twist knot orbifolds. For two-bridge hyperbolic link, there exists an angle $\alpha_0 \in [\frac{2\pi}{3},\pi)$ for each link $K$ such that the cone-manifold $K(\alpha)$ is hyperbolic for $\alpha \in (0, \alpha_0)$, Euclidean for $\alpha=\alpha_0$, and spherical for $\alpha \in (\alpha_0, \pi]$ [@P2; @HLM1; @K1; @PW]. We will use the Chern-Simons invariant of the lens space $L(4n+1,2n+1)$ calculated in [@HLM2]. The following theorem gives the formulae for $T_m$ for even integers $m$. For odd integers $m$, we can replace $T_m$ by $T_{-m-1}$ as explained in Section \[sec:twist\]. So, the following theorem actually covers all possible hyperbolic twist knots. We exclude the non-hyperbolic case, $n=0,\ -1$.
\[thm:main\] Let $T_{2n}$ be a hyperbolic twist knot. Let $T_{2n}(\alpha)$, $0 \leq \alpha < \alpha_0$ be a hyperbolic cone-manifold with underlying space $S^3$ and with singular set $T_{2n}$ of cone-angle $\alpha$. Let $k$ be a positive integer such that $k$-fold cyclic covering of $T_{2n}(\frac{2 \pi}{k})$ is hyperbolic. Then the Chern-Simons invariant of $T_{2n}(\frac{2 \pi}{k})$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd) is given by the following formula:
$$\begin{split}
\text{\textnormal{cs}} \left(T_{2n} \left(\frac{2 \pi}{k} \right)\right) & \equiv \frac{1}{2} \text{\textnormal{cs}}\left(L(4n+1,2n+1) \right) \\
&+\frac{1}{4 \pi^2}\int_{\frac{2 \pi}{k}}^{\alpha_0} Im \left(2*\log \left(M^{-2}\frac{A+iV}{A-iV}\right)\right) \: d\alpha \\
& +\frac{1}{4 \pi^2}\int_{\alpha_0}^{\pi}
Im \left(\log \left(M^{-2}\frac{A+iV_1}{A-iV_1}\right)+\log \left(M^{-2}\frac{A+iV_2}{A-iV_2}\right)\right) \: d\alpha,
\end{split}$$
where for $A=\cot{\frac{\alpha}{2}}$, $V$ $(Im(V) \leq 0)$, $V_1$, and $V_2$ are zeroes of the complex distance polynomial $P_{2n}=P_{2n}(V,B)$ which is either given recursively by
$$P_{2n} = \begin{cases}
\left(\left(4 B^4-8 B^2+4\right) V^2-4 B^4+8 B^2-2\right) P_{2(n-1)} -P_{2(n-2)}, \
\text{if $n>1,$} \\
\left(\left(4 B^4-8 B^2+4\right) V^2-4 B^4+8 B^2-2\right) P_{2(n+1)}-P_{2(n+2)}, \
\text{if $n<-1,$}
\end{cases}$$
with initial conditions $$\begin{split}
P_{-2} (V,B) & =\left(2 B^2-2\right) V+2 B^2-1,\\
P_{0} (V,B) & = 1, \\
P_{2} (V,B) & =\left(4 B^4-8 B^2+4\right) V^2+\left(2-2 B^2\right) V-4 B^4+6 B^2-1 \\
\end{split}$$
or given explicitly by $$P_{2n} = \begin{cases}
\sum_{i=0}^{2n}
\left(\begin{smallmatrix}
n+ \lfloor \frac{i}{2} \rfloor \\
i
\end{smallmatrix}\right)
\left(2 (B^2-1) (1-V)\right)^{i}
\left(1+2 (V-1)^{-1}\right)^{\lfloor \frac{1+i}{2} \rfloor}
\qquad
\text{if $n \geq 0$} \\
\sum_{i=0}^{-2n-1}
\left(\begin{smallmatrix}
-n+ \lfloor \frac{(i-1)}{2} \rfloor \\
i
\end{smallmatrix}\right)
\left(2 (B^2-1) (V-1) \right)^i
\left(1+ 2 (V-1)^{-1}\right)^{\lfloor \frac{1+i}{2} \rfloor}
\text{if $n<0$}
\end{cases}$$
where $B=\cos{\frac{\alpha}{2}}$ and $V_1$ and $V_2$ approach common $V$ as $\alpha$ decreases to $\alpha_0$ and they come from the components of $V$ and $\overline{V}$.
We here present some derived results. Theorem \[thm:A-polynomial-re\] gives the recursive formulae of A-polynomial of twist knots. In [@HS Theorem 1], Hoste and Shanahan presented the recursive formulae of A-polynomials of the twist knots with the opposite orientation. Theorem \[thm:A-polynomial\] gives the explicit formulae of A-polynomials of twist knots. In [@Mat Theorem1.1], Mathews presented the explicit formulae of A-polynomials of the twist knots with the opposite orientation. In case $n \leq 0$ of [@Mat], there is a typo; $\left(\frac{M^2-1}{L+M^2}\right)^i$ has to be changed into $\left(\frac{1-M^2}{L+M^2}\right)^i$ [@Mat1]. For each side of slop $a/b$ of the Newton polygon of $A_{2n}$, we can obtain the edge polynomial. By substituting t for $L^b M^a$ of each term appearing along the side edge, we have the edge polynomial, $f_{a/b}(t)$. Theorem \[thm:edge-polynomial\] gives the edge polynomials of twist knots. In [@CL Corollary 11.5], Cooper and Long showed that the edge polynomials of a two-bridge knot are all $\pm (t-1)^k (t+1)^l$. We pin them down in case of twist knots. Corollary \[cor:boundary\] tells the number of boundary components in case of slope $-4n+2$ of twist knots. From [@HT], we know that the number of boundary components of two-bridge knots are one or two. We pin them down in case of slope $-4n+2$ of twist knots. Proofs of derived results are in Section \[sec:poly\].
\[thm:A-polynomial-re\] A-polynomial $A_{2n}=A_{2n}(L,M)$ is given recursively by
$$A_{2n} = \begin{cases}
A_u A_{2(n-1)}
-M^4 \left(1+L M^2\right)^4 A_{2(n-2)} \ \ \
\text{if $n>1$} \\
A_u A_{2(n+1)}
-M^4 \left(1+L M^2\right)^4 A_{2(n+2)} \ \ \
\text{if $n<-2$}
\end{cases}$$
with initial conditions $$\begin{split}
A_{-4} (L,M) & =1-L+2 L M^2+2 L M^4+L^2 M^4-L^2 M^6-L M^8+L M^{10}+2 L^2 M^{10}\\
&+2 L^2 M^{12}-L^2 M^{14}+L^3 M^{14},\\
A_{-2} (L,M) & =1+L M^6,\\
A_{0} (L,M) & = -1, \\
A_{2} (L,M) & =L-L M^2-M^4-2 L M^4-L^2 M^4-L M^6+L M^8, \\
\end{split}$$ where $$A_u=1-L+2 L M^2+M^4+2 L M^4+L^2 M^4+2 L M^6-L M^8+L^2 M^8.$$
\[thm:A-polynomial\] A-polynomial $A_{2n}=A_{2n}(L,M)$ is given explicitly by
$$A_{2n} = \begin{cases}
- M^{2 n} \left(1+L M^2\right)^{2 n} \sum_{i=0}^{2n}
\left(\begin{smallmatrix}
n+ \lfloor \frac{i}{2} \rfloor \\
i
\end{smallmatrix}\right)
\left(1-M^2\right)^i \left(1+L M^2\right)^{-i} \\
\times (L-1)^{\lfloor \frac{i}{2} \rfloor}\left(L M^2-M^{-2}\right)^{\lfloor \frac{1+i}{2} \rfloor}
\qquad
\text{if $n \geq 0$} \\
M^{-2 n} \left(1+L M^2\right)^{-2 n-1} \sum_{i=0}^{-2n-1}
\left(\begin{smallmatrix}
-n+ \lfloor \frac{(i-1)}{2} \rfloor \\
i
\end{smallmatrix}\right)
\left(M^2-1\right)^i \left(1+L M^2\right)^{-i} \\
\times (L-1)^{\lfloor \frac{i}{2} \rfloor}\left(L M^2-M^{-2}\right)^{\lfloor \frac{1+i}{2} \rfloor} \qquad
\text{if $n<0$}
\end{cases}$$
\[thm:edge-polynomial\] When $n > 0$, the edge polynomials of twist knots are $$\begin{aligned}
\begin{cases}
&\pm (t-1) \ \ \ \text{if the slope is $-4n$} \\
&-(t-1)^n \ \ \ \text{if the slope is $4$ and n is even} \\
& \pm (t-1)^n \ \ \ \text{if the slope is $4$ and n is odd}.
\end{cases}\end{aligned}$$
When $n < 0$, the edge polynomials of twist knots are $$\begin{aligned}
\begin{cases}
&t+1 \ \ \ \ \ \ \ \ \ \ \text{if the slope is $-4n+2$} \\
&\pm (t-1)^{-n-1} \ \ \ \text{if the slope is $4$ and n is even} \\
& (t-1)^{-n-1} \ \ \ \text{if the slope is $4$ and n is odd}.
\end{cases}\end{aligned}$$
\[cor:boundary\] The number of boundary components of every incompressible surface corresponding to slope $-4n+2$ of twist knots are $2$.
Twist knots {#sec:twist}
===========
reflection
A knot $K$ is the twist knot if $K$ has a regular two-dimensional projection of the form in Figure \[fig:T2n\]. $K$ has 2 left-handed vertical crossings and $m$ right-handed horizontal crossings. We will denote it by $T_m$. One can easily check that the slope of $T_m$ is $2/(2m+1)$ which is equivalent to the knot with slope $(m+1)/(2m+1)$ [@S1]. For example, Figure \[fig:knot\] shows two different regular projections of knot $6_1$; one with slope $2/9$ (left) and the other with slope $5/9$ (right). Note that $T_m$ and its mirror image have the same fundamental group up to orientation and hence have the same fundamental domain up to isometry in $\mathbb{H}^3$. It follows that $T_m(\alpha)$ and its mirror image have the same fundamental set upto isometry in $\mathbb{H}^3$ and have the same Chern-Simons invariant up to sign. Since the mirror image of $T_m$ is equivalent to $T_{-m-1}$, when $m$ is odd we will use $T_{-m-1}$ for $T_m$. Hence a twist knot can be represented by $T_{2n}$ for some integer $n$ with slope $2/(4n+1)$ or $(2n+1)/(4n+1)$.
Let us denote by $X_{2n}$ the exterior of $T_{2n}$. In [@HS], the fundamental group of $X_{2n}$ is calculated with 2 right-handed vertical crossings as positive crossings instead of two left-handed vertical crossings. The following theorem is tailored to our purpose. The following theorem can also be obtained by reading off the fundamental group from the Schubert normal form of $T_{2n}$ with slope $(2n+1)/(4n+1)$ [@R1].
\[prop:fundamentalGroup\] $$\pi_1(X_{2n})=\left\langle s,t \ |\ swt^{-1}w^{-1}=1\right\rangle,$$ where $w=(ts^{-1}t^{-1}s)^n$.
The complex distance polynomial and A-polynomial {#sec:poly}
================================================
Given a twist knot $T_{2n}$ and a set of generators, $\{s,t\}$, we identifiy the set of representations, $R$, of $\pi_1 (X_{2n})$ into $SL(2, {\mathbb{C}})$ with $R\left(\pi_1 (X_{2n})\right)=\{(\eta(s),\eta(t)) \ |\ \eta \in R\}$. Since the defining relation of $\pi_1 (X_{2n})$ gives the defining equation of $R\left(\pi_1 (X_{2n})\right)$ [@R3], $R\left(\pi_1 (X_{2n})\right)$ can be thought of an affine algebraic set in ${\mathbb{C}}^{2}$. $R\left(\pi_1 (X_{2n})\right)$ is well-defined upto isomorphisms which arise from changing the set of generators. We say elements in $R$ which differ by conjugations in $SL(2, {\mathbb{C}})$ are equivalent.
We use two coordinates to give the structure of the affine algebraic set to $R\left(\pi_1 (X_{2n})\right)$. Equivalently, for some $O \in SL(2, {\mathbb{C}})$, we consider both $\eta$ and $\eta^{\prime}=O^{-1} \eta O$:
For the complex distance polynomial, we use for the coordinates
$$\eta(s)=\left[\begin{array}{cc}
({M+1/M})/2 & e^{\frac{\rho}{2}} ({M-1/M})/2 \\
e^{-\frac{\rho}{2}} ({M-1/M})/2 & ({M+1/M})/2
\end{array} \right],$$
$$\eta(t)=\left[\begin{array}{cc}
({M+1/M})/2 & e^{-\frac{\rho}{2}} ({M-1/M})/2 \\
e^{\frac{\rho}{2}} ({M-1/M})/2 & ({M+1/M})/2
\end{array} \right],$$
and for the A-polynomial,
$$\begin{array}{ccccc}
\eta^{\prime}(s)=\left[\begin{array}{cc}
M & 1 \\
0 & M^{-1}
\end{array} \right]
\text{,} \ \ \
\eta^{\prime}(t)=\left[\begin{array}{cc}
M & 0 \\
t & M^{-1}
\end{array} \right].
\end{array}$$
In [@HMP], the complex distance polynomial of $T_{2n}$ is presented recursively. Theorem \[thm:cpolynomial-ex\] gives it explicitly. It is the defining polynomial of algebraic set $R\left(\pi_1 (X_{2n})\right)$ with the set of generators given in Proposition \[prop:fundamentalGroup\] and with the coordinates $\eta(s)$ and $\eta(t)$ in $SL(2, {\mathbb{C}})$. The actual computation in [@HMP] is done after setting $M=e^{\frac{i \alpha}{2}}$ and then the variables are changed to $B=\cos \frac{\alpha}{2}$ and $V=\cosh \rho$.
[@HMP Theorem 3.1] \[thm:cpolynomial\] For $B=\cos \frac{\alpha}{2}$, $V=\cosh \rho$ is a root of the following complex distance polynomial $P_{2n}=P_{2n}(V,B)$ which is given recursively by
$$P_{2n} = \begin{cases}
\left(\left(4 B^4-8 B^2+4\right) V^2-4 B^4+8 B^2-2\right) P_{2(n-1)} -P_{2(n-2)} \
\text{if $n>1$} \\
\left(\left(4 B^4-8 B^2+4\right) V^2-4 B^4+8 B^2-2\right) P_{2(n+1)}-P_{2(n+2)} \
\text{if $n<-1$}
\end{cases}$$
with initial conditions $$\begin{split}
P_{-2} (V,B) & =\left(2 B^2-2\right) V+2 B^2-1,\\
P_{0} (V,B) & = 1, \\
P_{2} (V,B) & =\left(4 B^4-8 B^2+4\right) V^2+\left(2-2 B^2\right) V-4 B^4+6 B^2-1.\\
\end{split}$$
\[thm:cpolynomial-ex\] For $B=\cos \frac{\alpha}{2}$, $V=\cosh \rho$ is a root of the following complex distance polynomial $P_{2n}=P_{2n}(V,B)$ which is given explicitly by
$$P_{2n} = \begin{cases}
\sum_{i=0}^{2n}
\left(\begin{smallmatrix}
n+ \lfloor \frac{i}{2} \rfloor \\
i
\end{smallmatrix}\right)
\left(2 (B^2-1) (1-V)\right)^{i}
\left(1+2 (V-1)^{-1}\right)^{\lfloor \frac{1+i}{2} \rfloor}
\qquad \ \ \
\text{if $n \geq 0$} \\
\sum_{i=0}^{-2n-1}
\left(\begin{smallmatrix}
-n+ \lfloor \frac{(i-1)}{2} \rfloor \\
i
\end{smallmatrix}\right)
\left(2 (B^2-1) (V-1) \right)^i
\left(1+ 2 (V-1)^{-1}\right)^{\lfloor \frac{1+i}{2} \rfloor}
\text{if $n<0$}.
\end{cases}$$
We write $f_{2n}$ for the claimed formula and show that $f_{2n}=P_{2n}$. $f_{2n}$ can be rewritten as $$\begin{aligned}
f_{2n}=\begin{cases} \sum_{j=0}^{n}
\left(\begin{smallmatrix}
n+j \\
2j
\end{smallmatrix}\right)
\left(2(B^2-1) (V-1)\right)^{2j} \left(1+2(V-1)^{-1} \right)^{j}\\
- \sum_{j=0}^{n}
\left(\begin{smallmatrix}
n+j \\
2j+1
\end{smallmatrix}\right)
\left(2(B^2-1) (V-1)\right)^{2j+1} \left(1+2(V-1)^{-1} \right)^{j+1} \ \text{if $n \geq 0$} \\
\sum_{j=0}^{-n-1}
\left(\begin{smallmatrix}
-n-1+j \\
2j
\end{smallmatrix}\right)
\left(2(B^2-1) (V-1)\right)^{2j} \left(1+2(V-1)^{-1} \right)^{j} \\
+ \sum_{j=0}^{-n-1}
\left(\begin{smallmatrix}
-n+j \\
2j+1
\end{smallmatrix}\right)
\left(2(B^2-1) (V-1)\right)^{2j+1} \left(1+2(V-1)^{-1} \right)^{j+1} \ \text{if $n < 0$}.
\end{cases}\end{aligned}$$
Now, the theorem follows by solving the recurrence formula with the initial conditions given in Theorem \[thm:cpolynomial\]:
$$P_{2n} = \begin{cases}
\left[\left(\left(2 (B^2-1) (V-1)\right)^2 -2 (B^2-1) (V-1) \right) \left(1+2(V-1)^{-1} \right) +1\right]\\
\times h_{n-1}-h_{n-2} \qquad
\text{if $n>1$} \\
\left(2 \left(B^2-1\right) (V-1) \left(1+2(V-1)^{-1} \right)+1\right) h_{-n-1}-h_{-n-2} \qquad
\text{if $n<-1$}
\end{cases}$$
where $$\begin{aligned}
h_{n}& =
\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}
\left(\begin{smallmatrix}
n+ 1 \\
2k+1
\end{smallmatrix}\right)
\left(2(B^2-1)^2 (V-1)^2 \left(1+2(V-1)^{-1} \right)+1\right)^{n-2k} \\
&\times \left(\left(2(B^2-1)^2 (V-1)^2\left(1+2(V-1)^{-1} \right)+1\right)^2-1\right)^{k}\\
= & \sum_{i=0}^{\lfloor \frac{n}{2} \rfloor}
\left(\begin{smallmatrix}
n- i \\
i
\end{smallmatrix}\right) (-1)^i
\left[ \sum_{j=0}^{n-2i}
\left(\begin{smallmatrix}
n- 2i \\
j
\end{smallmatrix}\right) 2^{n-2i-j}
\left(\left(2(B^2-1) (V-1)\right)^2 \left(1+2(V-1)^{-1} \right)\right)^{j} \right] \\
= & \sum_{j=0}^{n} \left[\sum_{i=0}^{\lfloor \frac{n-j}{2} \rfloor}
\left(\begin{smallmatrix}
n- i \\
i
\end{smallmatrix}\right)
\left(\begin{smallmatrix}
n- 2i \\
j
\end{smallmatrix}\right) (-1)^i 2^{n-2i-j}\right]
\left(\left(2(B^2-1) (V-1)\right)^2 \left(1+2(V-1)^{-1} \right)\right)^{j} \\
= & \sum_{j=0}^{n}
\left(\begin{smallmatrix}
n+1+j \\
2j+1
\end{smallmatrix}\right)
\left(\left(2(B^2-1) (V-1)\right)^2 \left(1+2(V-1)^{-1} \right)\right)^{j}. \end{aligned}$$ $f_{2n}$ can be obtained by simplifying the above formula.
Using Theorem 4.4 in [@HMP], we get the following Lemma \[lem:pytha\] which relates the zeroes of the complex distance polynomial, $P_{2n}=P_{2n}(V,B)$, and the zeroes of the A-polynomial, $A_{2n}=A_{2n}(L,M)$. Theorem \[thm:A-polynomial-re\] (resp. Theorem \[thm:A-polynomial\]) can be obtained from $P_{2n}$ of Theorem \[thm:cpolynomial\] (resp. Theorem \[thm:cpolynomial-ex\]) by replacing $V$ with $\left((M^2+1) (LM^2-1)\right)/\left((M^2-1) (LM^2+1)\right)$ using the equality in Lemma \[lem:pytha\] and $B$ with $\left(M+M^{-1}\right)/2$ and by clearing denominators.
\[lem:pytha\] $$\begin{aligned}
iV&=A \frac{LM^2-1}{LM^2+1} \ \ \ and \ \ \ L=M^{-2} \frac{A+iV}{A-iV},\\ V&=\frac{(M^2+1) (LM^2-1)}{(M^2-1) (LM^2+1)} \ \ \ and \ \ \ L=M^{-2} \frac{VM^2-V+M^2+1}{-VM^2+V+M^2+1}.\end{aligned}$$
$(1)$ is Theorem 4.4 in [@HMP].
Since $$\begin{aligned}
\frac{A}{i} &=\frac{\cos{\frac{\alpha}{2}}}{i\sin{\frac{\alpha}{2}}}
=\frac{\cosh{\frac{i\alpha}{2}}}{\sinh{\frac{i\alpha}{2}}}
=\frac{\frac{M+M^{-1}}{2}}{\frac{M-M^{-1}}{2}}
=\frac{M^2+1}{M^2-1},\end{aligned}$$ we get the first equality of $(2)$. By solving the first equality for $L$, we get the second equality of $(2)$.
By Lemma \[lem:edge-polynomial\], we have Newton polygons, $NP$, associated to $A_{2n}$ in Figure \[fig:NewtonPolygon\]. Let us only consider nonzero slopes if $n \neq -1$.
When $n >0$, $NP$ has two slopes $-4n$ and $4$. When the slope is $-4n$ and the edge has the term $-M^{4n}$ on it, there are only two terms of $A_{2n}$ appearing along the edge of the slope. From the explicit formula of $A_{2n}$ of Theorem \[thm:A-polynomial\], they are $-M^{4n}$ and $L$. The term $-M^{4n}$ occurs when $i=2n$. We get the term $L$ by adding two terms $(n-1) L$ which occurs when $i=2n-1$ and $-nL$ which occurs when $i=2n$. From $L-M^{4n}$ by substituting $Lt$ for $M^{4n}$ and dividing by $L$, we get the edge polynomial $1-t$. Hence, on the right above edge with the same slope $-4n$, we have $t-1$ for the edge polynomial. When the slope is $4$ and the term $-M^{4n}$ is on the edge, from the explicit formula of $A_{2n}$ of Theorem \[thm:A-polynomial\], $ n LM^{4n+4}$ is on the edge and the term occurs when $i=2n$. We can get the edge polynomial from the sum of the terms which appear on this edge by substituting $t$ for $LM^{4}$ and dividing by $M^{4n}$. Hence the coefficient of $LM^{4n+4}$ is the coefficient of $t$ of the edge polynomial. Since we know the constant term is $-1$ and the coefficient of $t$ is $n$, we get the edge polynomial $(t-1)^n$ when $n$ is odd and $-(t-1)^n$ when $n$ is even because the fact that the edge polynomials of two-bridge knots are up to sign the product of some powers of $t-1$ and some powers of $t+1$ [@CL] and the coefficient conditions forces the power of $t+1$ to be zero. Hence, on the right below edge with the same slope $4$, we have $-(t-1)^n$ for the edge polynomial.
When $n <0$, $NP$ has two slopes $-4n+2$ and $4$. When the slope is $-4n+2$ and the edge has the term $1$ on it, there are only two terms of $A_{2n}$ appearing along the edge of the slope. From the explicit formula of $A_{2n}$ of Theorem \[thm:A-polynomial\], they are $1$ and $LM^{-4n+2}$. The term $1$ occurs when $i=2n$. We get the term $LM^{-4n+2}$ by adding two terms $(n+1) LM^{-4n+2}$ which occurs when $i=-2n-2$ and $-nLM^{-4n+2}$ which occurs when $i=-2n-1$. From $1+LM^{-4n+2}$ by substituting $t$ for $LM^{-4n+2}$, we get the edge polynomial $t+1$. Hence, on the right above edge with the same slope $-4n+2$, we have $t+1$ for the edge polynomial. When the slope is $4$ and the term $LM^{-4n+2}$ is on the edge, from the explicit formula of $A_{2n}$ of Theorem \[thm:A-polynomial\], $(n+1) L^2M^{-4n+6}$ is on the edge. We get the term $(n+1) L^2M^{-4n+6}$ by adding two terms $\frac{(n+1)(n+2)}{2} L^2M^{-4n+6}$ which occurs when $i=-2n-2$ and $\frac{-n (n+1)}{2} L^2M^{-4n+6}$ which occurs when $i=-2n-1$. We can get the edge polynomial from the sum of the terms which appear on this edge by substituting $t$ for $LM^{4}$ and dividing by $LM^{-4n+2}$. Hence the coefficient of $L^2M^{-4n+6}$ is the coefficient of $t$ of the edge polynomial. Since we know the constant term is $1$ and the coefficient of $t$ is $n+1$, we get the edge polynomial $(t-1)^{-n-1}$ when $n$ is odd and $-(t-1)^{-n-1}$ when $n$ is even because the fact that the edge polynomials of two-bridge knots are up to sign the product of some powers of $t-1$ and some powers $t+1$ [@CL] and the coefficient conditions forces the power of $t+1$ to be zero. Hence, on the right below edge with the same slope $4$, we have $(t-1)^{-n-1}$ for the edge polynomial.
\[lem:edge-polynomial\] The Newton polygons associated to $A_{2n}$ are polygons in Figure \[fig:NewtonPolygon\].
The lemma is true when $n=-2$, $-1$, or $1$.
Since the Newton polygon of $A_{2n}$ has ones on the corners up to sign [@CL2], to determine the shape of the Newton polygon, we only need to consider $A_{2n}$ modulo $2$. We will use the recursive formula of $A_{2n}$ of Theorem \[thm:A-polynomial-re\]. In modulo $2$, $A_u$ has $6$ terms and $M^4 \left(1+L M^2\right)^4$ has two terms $M^4+L^4M^{12}$. The lemma can be proved by induction. You just have to combine six copies of Newton polygons of $A_{2n-1}$ (if $n>1$) or $A_{2n+1}$ (if $n<-2$) shifted by $A_u$ and two copies of Newton polygons of $A_{2n-2}$ (if $n>1$) or $A_{2n+1}$ (if $n<-2$) shifted by $M^4+L^4M^{12}$ removing double points.
![Newton polygons of $A_{2n}$.[]{data-label="fig:NewtonPolygon"}](NewtonPolygon-1.pdf)
\[Proof of Corollary \[cor:boundary\]\] The number of boundary components of two-bridge knots are one or two [@HT]. Hence the number of boundary components of every incompressible surface corresponding to slope $-4n+2$ are bounded above by $2$.
The orders of roots, roots of unity, of $f_{a/b}$ divide the number of boundary components of every incompressible surface corresponding to slope $a/b$ [@CCGLS1]. Hence, when the slope is $-4n+2$, since there is a single root of unity of degree $2$ by Theorem \[thm:edge-polynomial\], the number of boundary components of every incompressible surface corresponding to slope $-4n+2$ is bounded below by $2$.
Generalized Chern-Simons function {#sec:CSfunction}
=================================
The general references for this section are [@HLM3; @HLM2; @Y1] and [@MeyRub1]. We introduce the generalized Chern-Simons function on the family of a twist knot cone-manifold structures. For the oriented knot $T_{2n}$, we orient a chosen meridian $s$ such that the orientation of $s$ followed by orientation of $T_{2n}$ coincides with orientation of $S^3$. Hence, we use the definition of Lens space in [@HLM2] so that we can have the right orientation when the definition of Lens space is combined with the following frame field. On the Riemannian manifold $S^3-T_{2n}-s$ we choose a special frame field $\Gamma$. A *special* frame field $\Gamma=(e_1,e_2,e_3)$ is an orthonomal frame field such that for each point $x$ near $T_{2n}$, $e_1(x)$ has the knot direction, $e_2(x)$ has the tangent direction of a meridian curve, and $e_3(x)$ has the knot to point direction. A special frame field always exists by the proposition $3.1$ of [@HLM3]. From $\Gamma$ we obtain an orthonomal frame field $\Gamma_{\alpha}$ on $T_{2n}(\alpha)-s$ by the Schmidt orthonormalization process with respect to the geometric structure of the cone manifold $T_{2n}(\alpha)$. Moreover it can be made special by deforming it in a neighborhood of the singular set and $s$ if necessary. $\Gamma^{\prime}$ is an extention of $\Gamma$ to $S^3-T_{2n}$. For each cone-manifold $T_{2n}(\alpha)$, we assign the real number:
$$I\left(T_{2n}(\alpha)\right)=\frac{1}{2} \int_{\Gamma(S^3-T_{2n}-s)}Q-\frac{1}{4 \pi} \tau(s,\Gamma^{\prime})-\frac{1}{4 \pi} \left(\frac{\beta \alpha}{2 \pi}\right),$$
where $-2 \pi \leq \beta \leq 2 \pi$, $Q$ is the Chern-Simons form:
$$Q=\frac{1}{4 \pi^2} \left(\theta_{12} \wedge \theta_{13} \wedge \theta_{23} + \theta_{12} \wedge \Omega_{12} + \theta_{13} \wedge \Omega_{13} + \theta_{23} \wedge \Omega_{23} \right),$$
and
$$\tau(s,\Gamma^{\prime})=-\int_{\Gamma^{\prime}(s)} \theta_{23},$$
where ($\theta_{ij}$) is the connection $1$-form, ($\Omega_{ij}$) is the curvature $2$-form of the Riemannian connection on $T_{2n}(\alpha)$ and the integral is over the orthonomalizations of the same frame field. When $\alpha = \frac{2 \pi}{k}$ for some positive integer, $I \left(T_{2n}\left(\frac{2 \pi}{k}\right)\right)$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd) is independent of the frame field $\Gamma$ and of the representative in the equivalence class $\overline{\beta}$ and hence an invariant of the orbifold $T_{2n}\left(\frac{2 \pi}{k}\right)$. $I \left(T_{2n}\left(\frac{2 \pi}{k}\right)\right)$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd) is called *the Chern-Simons invariant of the orbifold* and is denoted by $\text{\textnormal{cs}} \left(T_{2n}\left(\frac{2 \pi}{k}\right) \right)$.
On the generalized Chern-Simons function on the family of the twist knot cone-manifold structures we have the following Schläfli formula.
(Theorem 1.2 of [@HLM2]) \[thm:schlafli\] For a family of geometric cone-manifold structures, $T_{2n}(\alpha)$, and differentiable functions $\alpha(t)$ and $\beta(t)$ of $t$ we have $$dI \left(T_{2n}(\alpha)\right)=-\frac{1}{4 \pi^2} \beta d \alpha.$$
Proof of the theorem \[thm:main\] {#sec:proof}
=================================
For $n \geq 1$ and $M=e^{i \frac{\alpha}{2}}$ ($B=\cos \frac{\alpha}{2}$), $A_{2n}(M,L)$ and $P_{2n}(V,A)$ have $2n$ component zeros, and for $n < -1$, $-(2n+1)$ component zeros. The component which gives the maximal volume is the geometric component [@HMP; @D1; @FK1] and in [@HMP] it is identified. For each $T_{2n}$, there exists an angle $\alpha_0 \in [\frac{2\pi}{3},\pi)$ such that $T_{2n}$ is hyperbolic for $\alpha \in (0, \alpha_0)$, Euclidean for $\alpha=\alpha_0$, and spherical for $\alpha \in (\alpha_0, \pi]$ [@P2; @HLM1; @K1; @PW]. Denote by $D(T_{2n}(\alpha))$ be the set of common zeros of the discriminant of $A_{2n}(L, e^{ \frac{i \alpha}{2}})$ over $L$ and the discriminant of $P_{2n}(V,\cos \frac{\alpha}{2})$ over $V$. Then $\alpha_0$ will be one of $D(T_{2n}(\alpha))$.
On the geometric component we can calculate the Chern-Simons invariant of an orbifold $T_{2n}(\frac{2 \pi}{k})$ (mod $\frac{1}{k}$ if $k$ is even or mod $\frac{1}{2k}$ if $k$ is odd), where $k$ is a positive integer such that $k$-fold cyclic covering of $T_{2n}(\frac{2 \pi}{k})$ is hyperbolic: $$\begin{aligned}
\text{\textnormal{cs}}\left(T_{2n} \left(\frac{2 \pi}{k} \right)\right)
& \equiv I \left(T_{2n} \left(\frac{2 \pi}{k} \right)\right)
\ \ \ \ \ \ \ \ \ \ \ \ \left(\text{mod} \ \frac{1}{k}\right) \\
& \equiv I \left(T_{2n}( \pi) \right)
+\frac{1}{4 \pi^2}\int_{\frac{2 \pi}{k}}^{\pi} \beta \: d\alpha
\ \ \ \ \left(\text{mod} \ \frac{1}{k}\right) \\
& \equiv \frac{1}{2} \text{\textnormal{cs}}\left(L(4n+1,2n+1) \right) \\
&+\frac{1}{4 \pi^2}\int_{\frac{2 \pi}{k}}^{\alpha_0} Im \left(2*\log \left(M^{-2}\frac{A+iV}{A-iV}\right)\right) \: d\alpha \\
& +\frac{1}{4 \pi^2}\int_{\alpha_0}^{\pi}
Im \left(\log \left(M^{-2}\frac{A+iV_1}{A-iV_1}\right)+\log \left(M^{-2}\frac{A+iV_2}{A-iV_2}\right)\right) \: d\alpha \\
& \left( \text{mod} \ \frac{1}{k}\ \text{if $k$ is even or } \text{mod} \ \frac{1}{2k}\ \text{if $k$ is odd} \right)\end{aligned}$$ where the second equivalence comes from Theorem \[thm:schlafli\] and the third equivalence comes from the fact that $I \left(T_{2n}(\pi)\right) \equiv \frac{1}{2} \text{\textnormal{cs}}\left(L(4n+1,2n+1) \right)$ $\left(\text{mod }\frac{1}{2}\right)$, Lemma \[lem:pytha\], and geometric interpretations of hyperbolic and spherical holonomy representations.
The fundamental set of the two-bridge link orbifolds are constructed in [@MR2]. The following theorem gives the Chern-Simons invariant of the Lens space $L(4n+1,2n+1)$.
(Theorem 1.3 of [@HLM2]) \[thm:Lens\] $$\begin{aligned}
\text{\textnormal{cs}} \left(L \left(4n+1,2n+1\right)\right) \equiv \frac{6n+4}{8n+2} && (\text{mod}\ 1).\end{aligned}$$
Chern-Simons invariants of the hyperbolic twist knot orbifolds and of their cyclic coverings
============================================================================================
The table \[table1-1\] (resp. the table \[table1-2\]) gives the approximate Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ (resp. for $n$ between $-9$ and $-2$) and for $k$ between $3$ and $10$, and of its cyclic covering, $cs \left(M_k (T_{2n})\right)$. We used Simpson’s rule for the approximation with $10^4$ ($5 \times 10^3$ in Simpson’s rule) intervals from $\frac{2 \pi}{k}$ to $\alpha_0$ and $10^4$ ($5 \times 10^3$ in Simpson’s rule) intervals from $\alpha_0$ to $\pi$. The table \[tab2\] gives the approximate Chern-Simons invariant of $T_{2n}$ for each n between $-9$ and $9$ except the unknot, the torus knot, and the amphicheiral knot. We again used Simpson’s rule for the approximation with $10^4$ ($5 \times 10^3$ in Simpson’s rule) intervals from $0$ to $\alpha_0$ and $10^4$ ($5 \times 10^3$ in Simpson’s rule) intervals from $\alpha_0$ to $\pi$. We used Mathematica for the calculations. We record here that our data in table \[tab2\] and those obtained from SnapPy match up up to six decimal points.
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_4(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k( T_4)\right)$
----- ------------------------------------------------------------ -------------------------------------------------
3 0.0875301 0.262590
4 0.144925 0.579699
5 0.0784576 0.392288
6 0.0351571 0.210943
7 0.00506505 0.0354553
8 0.108039 0.864313
9 0.0218112 0.196301
10 0.0530574 0.530574
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
&
$k$ $\text{\textnormal{cs}} \left(T_6(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_6)\right)$
----- ------------------------------------------------------------ -------------------------------------------------
3 0.0449535 0.134860
4 0.0876043 0.350417
5 0.0165337 0.0826684
6 0.138167 0.829004
7 0.0120078 0.0840545
8 0.0430876 0.344700
9 0.0121250 0.109125
10 0.0876213 0.876213
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_8(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_8)\right)$
----- ------------------------------------------------------------ -------------------------------------------------
3 0.0161266 0.0483799
4 0.0536832 0.214733
5 0.0817026 0.408513
6 0.103012 0.618074
7 0.0481239 0.336867
8 0.00768503 0.0614802
9 0.0322210 0.289989
10 0.0521232 0.521232
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
&
$k$ $\text{\textnormal{cs}} \left(T_{10} (\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left( M_k (T_{10})\right)$
----- ---------------------------------------------------------------- -----------------------------------------------------
3 0.162697 0.488091
4 0.0320099 0.128040
5 0.0597580 0.298790
6 0.0809665 0.485799
7 0.0260276 0.182193
8 0.110559 0.884475
9 0.0100766 0.0906893
10 0.0299660 0.299660
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_{12} (\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{12})\right)$
----- ---------------------------------------------------------------- ----------------------------------------------------
3 0.148360 0.445081
4 0.0170833 0.0683334
5 0.0447221 0.223610
6 0.0658884 0.395330
7 0.0109281 0.0764969
8 0.0954474 0.763579
9 0.0505121 0.454609
10 0.0148406 0.148406
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
&
$k$ $\text{\textnormal{cs}} \left(T_{14} (\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{14})\right)$
----- ---------------------------------------------------------------- ----------------------------------------------------
3 0.137750 0.413249
4 0.00620422 0.0248169
5 0.0337900 0.168950
6 0.0549355 0.329613
7 0.0713932 0.499752
8 0.0844779 0.675823
9 0.0395376 0.355839
10 0.00386414 0.0386414
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_{16} (\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{16})\right)$
----- ---------------------------------------------------------------- ----------------------------------------------------
3 0.129617 0.388850
4 0.247931 0.991725
5 0.0254880 0.127440
6 0.0466221 0.279733
7 0.0630741 0.441518
8 0.0761526 0.609221
9 0.0312096 0.280887
10 0.0955375 0.955375
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
&
$k$ $\text{\textnormal{cs}} \left(T_{18} (\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{18})\right)$
----- ---------------------------------------------------------------- ----------------------------------------------------
3 0.123198 0.369593
4 0.241431 0.965725
5 0.0189709 0.0948543
6 0.0400981 0.240588
7 0.0565432 0.395803
8 0.0696194 0.556955
9 0.0246862 0.222176
10 0.0890057 0.890057
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $2$ and $9$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-1"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_{-4} (\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-4})\right)$
----- ---------------------------------------------------------------- ----------------------------------------------------
3 0.0200144 0.0600431
4 0.186811 0.747246
5 0.00166667 0.00833333
6 0.0504622 0.302773
7 0.0163442 0.114410
8 0.116990 0.935921
9 0.0292902 0.263612
10 0.0595432 0.595432
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
&
$k$ $\text{\textnormal{cs}} \left(T_{-6}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-6})\right)$
----- --------------------------------------------------------------- ----------------------------------------------------
3 0.0749433 0.224830
4 0.0126376 0.0505506
5 0.0873477 0.436738
6 0.140792 0.844753
7 0.0376998 0.263898
8 0.0862114 0.689691
9 0.0133130 0.119817
10 0.0552937 0.552937
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_{-8}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-8})\right)$
----- --------------------------------------------------------------- ----------------------------------------------------
3 0.109659 0.328978
4 0.0564153 0.225661
5 0.0330919 0.165460
6 0.0205610 0.123366
7 0.0130371 0.0912595
8 0.00816423 0.0653138
9 0.00482762 0.0434486
10 0.00244289 0.0244289
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
&
$k$ $\text{\textnormal{cs}} \left(T_{-10}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-10})\right)$
----- ---------------------------------------------------------------- -----------------------------------------------------
3 0.133696 0.401087
4 0.0832469 0.332988
5 0.0603968 0.301984
6 0.0480386 0.288232
7 0.0405999 0.284200
8 0.0357763 0.286210
9 0.0324710 0.292239
10 0.0301075 0.301075
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_{-12}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-12})\right)$
----- ---------------------------------------------------------------- -----------------------------------------------------
3 0.150597 0.451792
4 0.101087 0.404347
5 0.0784041 0.392020
6 0.0661095 0.396657
7 0.0587029 0.410920
8 0.0538979 0.431183
9 0.0506045 0.455441
10 0.0482492 0.482492
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
&
$k$ $\text{\textnormal{cs}} \left(T_{-14}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-14})\right)$
----- ---------------------------------------------------------------- -----------------------------------------------------
3 0.162874 0.488621
4 0.113753 0.455011
5 0.0911448 0.455724
6 0.0788794 0.473276
7 0.0000589596 0.000412717
8 0.0666914 0.533532
9 0.00784780 0.0706302
10 0.0610519 0.610519
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
[ll]{}
$k$ $\text{\textnormal{cs}} \left(T_{-16}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-16})\right)$
----- ---------------------------------------------------------------- -----------------------------------------------------
3 0.00545933 0.0163780
4 0.123196 0.492785
5 0.000626850 0.00313425
6 0.0883767 0.530260
7 0.00956397 0.0669478
8 0.0762009 0.609607
9 0.0173633 0.156270
10 0.0705667 0.705667
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
&
$k$ $\text{\textnormal{cs}} \left(T_{-18}(\frac{2 \pi}{k})\right)$ $\text{\textnormal{cs}} \left(M_k (T_{-18})\right)$
----- ---------------------------------------------------------------- -----------------------------------------------------
3 0.0126607 0.0379822
4 0.130503 0.522012
5 0.00795573 0.0397786
6 0.0957144 0.574286
7 0.0169085 0.118359
8 0.0835496 0.668397
9 0.0247059 0.222353
10 0.0779144 0.779144
: Chern-Simons invariant of the hyperbolic twist knot orbifold, $\text{\textnormal{cs}} \left(T_{2n} (\frac{2 \pi}{k})\right)$ for $n$ between $-9$ and $-2$ and for $k$ between $3$ and $10$, and of its cyclic covering, $\text{\textnormal{cs}} \left(M_k (T_{2n})\right)$.[]{data-label="table1-2"}
[cc]{}
2n $\alpha_0$ $\text{\textnormal{cs}}\left(T_{2n}\right)$
---- ------------ ---------------------------------------------
4 2.57414 0.344023
6 2.75069 0.277867
8 2.84321 0.242222
10 2.90026 0.220016
12 2.93897 0.204869
14 2.96697 0.193882
16 2.98817 0.185550
18 3.00477 0.179014
: Chern-Simons invariant of $T_{2n}$ for $n$ between $2$ and $9$ and for $n$ between $-9$ and $-2$).[]{data-label="tab2"}
&
2n $\alpha_0$ $\text{\textnormal{cs}}\left(T_{2n}\right)$
----- ------------ ---------------------------------------------
-4 2.40717 0.346796
-6 2.67879 0.444846
-8 2.80318 0.492293
-10 2.87475 0.0200385
-12 2.92130 0.0382117
-14 2.95401 0.0510293
-16 2.97825 0.0605519
-18 2.99694 0.0679043
: Chern-Simons invariant of $T_{2n}$ for $n$ between $2$ and $9$ and for $n$ between $-9$ and $-2$).[]{data-label="tab2"}
*Acknowledgements.* The authors would like to thank Alexander Mednykh and Hyuk Kim for their various helps, Nathan Dunfield and Daniel Mathews for their prompt helps and anonymous referees.
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abstract: 'We point out that a primordial magnetic field can be generated in the electroweak phase transition by a non-Abelian bootstrap, where the field is generated by currents of $W''$s, which in turn are extracted from the vacuum by the magnetic field. This magnetic field is produced as a vortex condensate at the electroweak phase transition. It becomes stringy as a consequence of the dynamical evolution due to magnetohydrodynamics.'
author:
- |
Poul Olesen[^1]\
[*The Niels Bohr Institute*]{}\
[*Blegdamsvej 17, Copenhagen Ø, Denmark* ]{}
title: 'Non-Abelian bootstrap of primordial magnetism'
---
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There is much evidence for the existence of a primordial magnetic field. There are several proposals for how these fields are generated, as reviewed for example in the paper [@hector]. One possibility is genesis at the electroweak phase transition, as first discussed by Vachaspati [@tand]. In his case the magnetic field is generated from properties of the Higgs field. In this note we shall discuss another possible mechanism for generation at the electroweak phase transition, namely a non-Abelian bootstrap mechanism whereby a magnetic field is generated from currents coming from charged $W'$s which in turn are generated from the magnetic field. This self organized mechanism relies heavily on properties that are generic for non-Abelian vector fields. The proposed mechanism is therefore also of interest in principle, since it may give direct observational information on non-Abelian field theory of vectors.
We start by considering a simple model with an SU(2) massive vector field, $${\cal L}=-\frac{1}{4}~F_{\mu\nu}^2-m_W(T)^2~W^\dagger_\mu W_\mu.~~W_\mu=
\frac{1}{\sqrt{2}}~(A_\mu^1+iA_\mu^2),
\label{model}$$ where $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu +ig[A_\mu,A_\nu],~~A_\mu=
A_\mu^a ~
\sigma_a/2.$$ This is the same model as considered some time ago by Ambjørn and the author in [@amb], except that the mass depends on the temperature $T$. It is assumed that there is a phase transition, $$m_W(T)=0~~{\rm for}~~T>T_c,~~m_W(T)\neq 0~~{\rm otherwise.}$$ The magnetic field $$f_{\mu\nu}=\partial_\mu A_\nu^3 -\partial_\nu A_\mu^3
\label{f}$$ is given by [@amb] $$ef_{12}=m_W(T)^2+2e^2~|W|^2,
\label{field}$$ where we used the ansatz [@amb] $W=W_1,~W_2=iW_1\equiv iW, W_3=W_0=0,
W=W(x_1,x_2)$. The result (\[field\]) arizes from minimizing the energy written in the form $${\rm Energy~density}=
|(D_1+iD_2)W|^2+\frac{1}{2}\left(f_{12}-\frac{m_W(T)^2}{e}-2e|W|^2\right)^2+
\frac{m_W(T)^2}{e}f_{12}-\frac{m_W(T)^4}{2e^2 }.$$ In this equation $$D_a=\partial_a-ieA_a^3.$$
We carry out the minimization by requiring that the two positive quadratic terms vanish like in the Bogomol’nyi limit. We immediately obtain Eq. (\[field\]). The equation of motion for $W=|W|\exp(i\chi)$ can be obtained from the vanishing of the first term, $$(D_1+iD_2)W=0,$$ from which we get by use also of Eq. (\[field\]) $$-(\partial_1^2+\partial_2^2)\ln |W|=m_W(T)^2+2e^2|W|^2-\epsilon_{ij}\partial_i
\partial_j\chi,
\label{eqofmotion}$$ where $\chi$ is the phase of $W$. The relative plus sign between the two terms on the right hand side of Eq. (\[field\]) reflects the antiscreening[^2] of this solution. Because of this sign there is no single vortex solution. Instead Eq. (\[eqofmotion\]) has periodic solutions, corresponding to a lattice of vortices. As usual in each periodicity domain the term $\epsilon_{ij}\partial_i\partial_j\chi$ gives a delta function corresponding to the delta function coming from the zero of $|W|$ on the left hand side of (\[eqofmotion\]).
The result (\[field\]) is now considered as a self organized solution of the equations of motion which shows the possibility of creating a magnetic field from the non-Abelian dynamics. The energy is taken from the expansion energy of the universe.
The order of magnitude of the field below the temperature $T_c$ is $$f_{12}\sim m_W^2/e\sim 10^{24}~G,$$ which is a large field, of the same order of magnitude as the one found by Vachaspati [@tand]. Each flux tube has a dimension of order $1/m_W$. For $T>T_c$ the solution of the equation of motion (\[eqofmotion\]) can be found explicitly in terms Weierstrass’ p-function, and it can be shown that it corresponds to a zero energy solution [@po]. By a nonperturbative gauge transformation one can transform this solution to the perturbative ground state $A_\mu^a=0$.
The solution of Eq. (\[eqofmotion\]) is a bootstrap type of solution, because the magnetic field is inherent in the vacuum and is extracted from “emptiness” by the appearance of the mass in the phase transition and is kept alive by currents from the $W'$s. In other words, the magnetic field is generated by $W-$currents, $$\partial_1f_{12}=2e\partial_1|W|^2=-j_2~~{\rm and}~~\partial_2f_{12}=
2e\partial_2|W|^2=j_1,$$ and these $W'$s in turn are generated by the magnetic field, because of the non-Abelian instability discussed a long time ago [@nkn], according to which the magnetic field exceeding the magnitude $m_W^2/e$ is unstable unless stabilized by $W'$s from the vacuum. Thus the magnetic field and the vector bosons are interwoven in the structure of the solution of Eq. (\[eqofmotion\]) and only exist because of one another.
The energy density is given by $${\cal E}=\frac{m_W(T)^2}{e}~f_{12}-\frac{1}{2}~\frac{m_W(T)^4}{e^2}$$ We see that this energy is smaller than the no condensate energy $f_{12}^2/2$ due to the negative contribution from the $W$ condensate.
Considering the vortices as strings we can compute the string tension by integrating the energy density over a single quadratic[^3] domain with area $c/m_w^2$, where $c$ is a numerical constant. The result is $${\rm string~tension}\equiv\sigma (T)=(2\pi-c/2)~\frac{m_W(T)^2}{e^2},
\label{tension}$$ where we used the quantization of the flux $${\rm Flux}=\int_{\rm domain}f_{12}d^2 x=2\pi/e.$$ Thus we see that the string tension[^4] vanishes above the critical temperature, where the field contents of the solution becomes non-perturbative pure gauge fields, as discussed in [@po].
As the universe expands the strings develop according to the magnetohydrodynamic (MHD) field equations. A long time ago we showed [@mhdpo] that in the limit of infinite conductivity (“ideal” MHD) these equations are satisfied by Nambu-Goto strings. Later this was discussed including dissipative effects by Schubring [@schubing]. Also, numerically the turbulent plasma governed by the MHD equations has been found to be extremely intermittent with the vorticity concentrated in thin vortex types with the magnetic field concentrated also in thin vortex types [@aake]. Therefore the stringy initial behavior exhibited by the vortex condensate discussed above fits well with the subsequent MHD governed develpment of the universe. In the string picture the magnetic field is given by [@mhdpo],[@schubing] $$B_i({\bf x},t)=\sum_{\rm strings}b\int d\sigma \frac{\partial z_i(\sigma,y)}
{\partial\sigma}~
\delta^3 (x-z(\sigma,t)),$$ where $f_{12}=B_3$ etc. and $b$ is the magnetic flux. In this equation there should be a sum over all the strings in the vortex lattice. The string coordinates satisfy $$\frac{\partial z_i}{\partial t}\frac{\partial z_i}{\partial \sigma}=0,~~
{\rm and}~~\frac{\partial^2z_i}{\partial \sigma^2}=\frac{1}{v_0^2}
\frac{\partial^2z_i}{\partial t^2}.$$ Here $v_0$ is a maximum transverse velocity.
The string tension from the gauge theory is temperature dependent and vanishes above the critical temperature. This phenomenon was found in string theory for the Nambu-Goto string long time ago by Pisarski and Alvarez [@pisarski], where the critical temperature is the deconfinement (Hagedorn) temperature. Their result would be obtained from Eq. (\[tension\]) if $$m_W(T)^2\propto \sqrt{1-(T/T_c)^2}$$ In our case, the exitence of the critical temperature indicates that the vortex/string picture breaks down above this temperature. Of course, even in the field theory case this temperature would also correspond to deconfinement if monopoles exist. They would be confined below the critical temperature, and released above the critical temperature because of zero string tension.
So far we have consideed the simple model (\[model\]) with a temperature dependent mass. We shall now consider the standard electroweak theory with a Higgs field $\phi$, where the magnetic field turns out to be given by [@ew] $$f_{12}=\frac{g~\phi_0(T)^2}{2\sin \theta}+2g\sin\theta |W|^2$$ in the Bogomol’nyi limit where the Higgs mass equals the $Z$ mass. For the realistic mass case a much more complicated perturbative treatment is necessary. For simplicity we shall therefore stick to the Bogomol’nyi limit. The equations of motion are [@ew] $$-(\partial_1^2+\partial_2^2)\ln |W|=\frac{g^2}{2}\phi^2+2g^2|W|^2-
\epsilon_{ij}\partial_i\partial_j\chi,$$ which is analogous to Eq. (\[eqofmotion\]), and $$(\partial_1^2+\partial_2^2)\ln\phi=\frac{g^2}{4\cos^2\theta}~(\phi^2-\phi_0(T)^2)
+g^2|W|^2.$$ It has been proven mathematically that these coupled equations have periodic solutions [@math]-[@math5]. Here $\phi_0(T)$ vanishes above the critical temperature and the solution then becomes degenerate with the perturbative vacuum [@po].
The string tension can again be obtained by integrating the energy density over one domain in the plane. The result is the same as in Eq. (\[tension\]). Thus the previous discussion can be repeated for the electroweak theory, at least in the Bogomol’nyi limit. Again the string tension vanishes above the critical temperature.
We end this discussion with some remarks on the chiral anomaly effect on the evolution of the primordial magnetic field [@chiral1]-[@chiral6]. The inclusion of this effect will modify the MHD equations by adding an effective electric current. Also, hyperfields are relevant above the electroweak phase transition, and there may be magnetic helicity above and below this transition. In ideal MHD helicity is conserved, but this is not valid when the Ohmic resistance is included, and the helicity will ultimately decay. It is clear that our solution is not born with helicity, since for this solution ${\bf AB}=0$, but due to fluctuations from the full MHD equations there will always be some helicity [@it].
It is always a problem for primordial magnetic fields generated from particle physics that the initial scale is small. Even though the expansion of the universe increases this scale in general this is not enough for the generation of realistic scales. Therefore the phenomenon of inverse cascading, i.e. the drift of energy towards larger scales, is important [^5]. Often this phenomenon is linked with (conserved) helicity [@h]. However, with vanishing helicity there is still an inverse cascade in freely decaying MHD, moving energy from smaller to larger scales, as discussed recently [@ic1]-[@ic3]. Thus, helicity is [*not*]{} a necessary condition for an inverse cascade to occur. More explicitly it was found numerically by Zrake [@ic1] that the energy scales in a self-similar manner, which was shown by the author [@ic3] to be an [*exact*]{} consequence of the standard MHD equations for freely decaying turbulence when dissipation is included. The energy density should satisfy $${\cal E}(k,t)=\sqrt{\frac{t_0}{t}}~{\cal E}
\left(k\sqrt{\frac{t}{t_0}},t_0\right)
\label{scale}$$ According to this formula (which is one of the few known exact results in HD and MHD) energy is moved from smaller to larger scales as time passes. This is obviously important in order to increase the scale of the magnetic field on the top of the expansion of the universe. These results have to be modified if chiral MHD turbulence is taken into account, as discussed recently in [@chiral6].
For completeness we display the magnetic energy for an expanding flat universe with the metric $$d\tau^2=dt^2-a(t)^2d{\bf x}^2=a(\tilde{t})^2\left(d\tilde{t}^2-
d{\bf x}^2\right).$$ Here $t$ is the Hubble time and $$\tilde{t}=\int dt/a(t)$$ is the conformal time corresponding to the expansion parameter $a(t)$. Eq. (\[scale\]) is then replaced by $${\cal E }_B(k,\tilde{t})=\left(\frac{a(\tilde{t_0})}{a(\tilde{t})}\right)^4~
\sqrt{\frac{\tilde{t_0}}{\tilde{t}}}~{\cal E}_B
\left(k\sqrt{\frac{\tilde{t}}{\tilde{t_0}}},\tilde{t_0}\right).$$ Again we see a drift towards large distances as the univese expands.
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In conclusion we have shown that a primordial magnetic field can be generated in the electroweak phase transition by a non-Abelian vector bootstrap. The resulting field consists of a set of antiscreening vortices which, because they have to follow the MHD equations, develop in a stringy manner. Due to the inverse cascade the field may survive at large distances so as to have a realistic scale at the present time. If it turns out that the relevant field is generated as a hyperfield above the electroweak phase transition it may still receive additional contributions from the bootstrap mechanism discussed here when the universe passes the electroweak phase transition and the hyperfield turns into a magnetic field.
[X]{} D. Grasso and H. R. Rubinstein, Phys. Rep. 348 (2001) 163 T. Vachaspati, Phys. Lett. B 265 (1991) 258 J. Ambjørn and P. Olesen, Phys. Lett. B 214 (1988) 565 P. Olesen, Phys. Lett. B 268 (1991) 389; arXiv:1605.00603 N. K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376 P. Olesen, Phys. Lett. B 366 (1996) 117 D. Schubring, Phys. Rev. D 91 (2015) 043518 A. Nordlund, private communication (1996) R. D. Pisarski and O. Alvarez, Phys. Rev. D 26 (1982) 3735 J. Ambjørn and P. Olesen, Nucl. Phys. B 315 (1989) 606;\
Phys. Lett. B 218 (1989) 67; Nucl. Phys. B 330 (1990) 193 J. Spruck and Y. Yang, Comm. Math. Phys. 144 (1992) 1 J. Spruck and Y. Yang, Comm. Math. Phys. 144 (1992) 215 Y. Yang, [*Solitons in Field Theory and Nonlinear Analysis*]{}, Springer, New York, 2001 D. Bartolucci and G. Tarantello, Comm. Math. Phys. 229 (2002) 3 D. Bartolucci and F. De Marchis, J. Math. Phys. 53 (2012) 073704 M. E. Shaposhnikov, Nucl. Phys. B 299 (1988) 797 M. Giovannini and M. E. Shaposhnikov, Phys. Rev. D 57 (1998) 2186 M. Dvornikov and V. B. Semikoz, JCAP 1202 040 (2012) V. B. Semikoz, A. Yu. Smirnov, and D. D. Sokoloff, Phys. Rev. D 93 (2016) 103003 K. Kamada and A. J. Long, Phys. Rev. D 93 (2016) 083520; arXiv:1610.03074 P. Pavlovic, N. Leite, and G. Sigl, arXiv:1612.07382 L. Campanelli, Phys. Rev. D 70 (2004) 083009 A. Pouquet, U. Frisch, and J. Lorat, J. Fluid Mech. 77 (1976) 321 J. Zrake, Astrophys. J. 794 (2014) L26 A. Brandenburg, T. Kahniashvili, and A. G. Tevzadze, Phys. Rev. Lett. 114 (2015) 075001 P. Olesen, arXiv:1509.08962
[^1]: email: 137olesen@gmail.com
[^2]: This is the anti-Lenz’ law according to which the magnetic field will be enhanced by the current. The necessary energy is produced by the $W-$condensate.
[^3]: For simplicity we consider a quadratic domain. Energy may be minimized by other types of domains. The constant $c$ is close to $2\pi$ if $m_W^2>>2e²|W|^2$.
[^4]: In a certain non-perturbatively defined gauge the strings still exist [@po] with zero string tension above the critical temperature. This vacuum string configuration is, however, degenerate with the perturbative vacuum.
[^5]: A more appropriate term is perhaps drift towards the infrared, i.e. small $k$.
|
---
abstract: 'An encryption scheme of JPEG images in the bitstream domain is proposed. The proposed scheme preserves the JPEG format even after encrypting the images, and the file size of encrypted images is the exact same as that of the original JPEG images. Several methods for encrypting JPEG images in the bitstream domain have been proposed. However, since some marker codes are generated or lost in the encryption process, the file size of JPEG bitstreams is generally changed due to the encryption operations. The proposed method inputs JPEG bitstreams and selectively encrypts the additional bit components of the Huffman code in the bitstreams. This feature allows us to have encrypted images with the same data size as that recoded in the image transmission process, when JPEG images are replaced with the encrypted ones by the hooking, so that the image transmission are successfully carried out after the hooking.'
author:
-
-
title: |
Bitstream-Based JPEG Image Encryption\
with File-Size Preserving
---
JPEG, Encryption, File-size preserving, Bitstream-based
Introduction
============
Due to the spread of digital cameras and smart phones, opportunities to use digital images are increasing. Generally, captured images are immediately JPEG encoded and stored. These images are not only stored on personal devices, but also are often uploaded to cloud providers, such as social networks, cloud photo storage services, and so on. Most the cloud providers accept only limited file formats like JPEG. In addition, such cloud environments are based on the reliability of the providers, but they are not a reliable situation in terms of privacy preserving for users. Therefore, various image encrypting methods have been studied for compressed image data.
Encryption then Compression(EtC) systems [@AEtCSfJMJS; @APSIPA3; @APSIPA15; @APSIPA16; @APSIPA17; @APSIPA18; @APSIPA19; @APSIPA24; @APSIPA25; @APSIPA2017Chuuman; @2018-1] are methods to encrypt for images before encoding. Some of these methods have the compatibility with international compression standards, and enable privacy preserving decompression and compression.The compression performances of EtC systems are almost as same as one of the original images, but the file sizes are slightly different from those of compressed images without encryption.
Several bitstream based encryption methods have also been proposed[@ESwGMCfJ2I; @Imaizumi1; @PSofJ2IwGMC; @LpBsbJE; @JEwFSP; @EJIRUBwFC]. For JPEG 2000 images, some bitstream-based encryption methods have been proposed in consideration of the generation of special marker codes and without changing the file size [@ESwGMCfJ2I; @Imaizumi1; @PSofJ2IwGMC]. Even for JPEG images, some bitstream-based encryption methods have been proposed, but the file size of the encrypted bitstream has changed by the occurrences or disappearances of the JPEG marker code [@LpBsbJE; @JEwFSP; @EJIRUBwFC]. For example, in [@LpBsbJE], encryption is performed while keeping the file size by rearranging the run lengths of AC coefficients. However, occurrences or disappearances of the pseudo marker code have not been studied. In this case, the encrypted bitstream can not be correctly decoded or the file size of the encrypted bitstream is changed. On the other hand, in the methods of [@JEwFSP; @EJIRUBwFC], bitstream-based block scrambling and coefficient scrambling are implemented. In these methods, in order to accurately hold the JPEG format, encryption processing is executed the byte stuffing operation which prevent accidental generation of markers by the arithmetic encoding procedures. As a result, It has been shown that the file size changes by several bytes before and after encryption.
In this paper, we propose a bitstream-based JPEG encryption scheme that makes the file size exactly equal to the original. The proposed method guarantees a constant file size by providing a mechanism to avoid occurrences / disappearances of the JPEG marker code. This feature allows us to have encrypted images with the same data size as that stored in the image transmission process, when JPEG images are replaced with the encrypted ones by the hooking, so that the image transmission are successfully carried out after the hooking.
JPEG Bitstream and Its Byte Stuffing {#sec:fileSizeChange}
====================================
JPEG Bitstream
--------------
[Figure \[fig:bitstream\]]{}(a) shows the structure of a JPEG bitstream. SOI and EOI are the marker codes which correspond to “Start of Image” and “End of Image”, respectively. JPEG bitstreams have some marker segments (“Segment” in [Fig.\[fig:bitstream\]]{}(a)) which store information to be used for data decoding such as quantization tables, Huffman tables, and so on. Each marker segment starts with a marker code. The marker codes are special two-byte codes where the first byte is “FF” and the second byte is a value between “01” and “FE”.
[Figure \[fig:bitstream\]]{}(b) shows the structure of the “image data” in [Fig.\[fig:bitstream\]]{}(a). “Image data” consists of multiple MCUs (Minimum Coded Unit). [Figure \[fig:bitstream\]]{}(b) is an example in the case of 4:2:0 color subsampling. Therefore, each MCU has four Y(luminance) blocks, one subsampled Cb block, and one subsampled Cr block. Moreover, each block has $DC_k$ which is the difference value from the DC coefficient of previous block, and 63 AC coefficients $AC_{k,n}$ ($n=1\cdots63$). Each coefficient has a Huffman code part corresponding to the group number ($g$) that determines the range of values, and an additional bits part for uniquely identifying values within the range. In the case of AC coefficients, the Huffman code also includes the run length ($r$) of the zero value until a significant value exists.
[Figure \[fig:bitstream\]]{} (c) is an example of Huffman code and additional bits. Since each data is composed of variable length bits, they are stored in byte units.
![Structure of a JPEG bitstream[]{data-label="fig:bitstream"}](JPEGall.pdf "fig:")\
(a) JPEG bitstream\
![Structure of a JPEG bitstream[]{data-label="fig:bitstream"}](MCU.pdf "fig:")\
(b) MCU\
![Structure of a JPEG bitstream[]{data-label="fig:bitstream"}](no_padding.pdf "fig:")\
(c) Representation with byte-aligned code
Difficulty of file size preserving
----------------------------------
[Figure \[fig:ff00\]]{} illustrates an example of entropy-coded data segment, where $DC_k$ in [Fig.\[fig:ff00\]]{}(a) is entropy-coded DC coefficient data and $AC_{k, 1}$ is the entropy-coded data of the first AC component in the $k$-th block, respectively. Each data have Huffman code part and additional bits part. To generate the JPEG bitstream, byte-based packing is first applied to the entropy-coded data in [Fig.\[fig:ff00\]]{}(a), as shown in [Fig.\[fig:ff00\]]{}(b). The byte “FF”, i.e. “11111111”, which corresponds to a marker code, may be produced due to the byte-based packing. Therefore, finally, in order to ensure that the marker does not occur within an entropy-coded segment, any “FF” byte in either a Huffman or additional bits, is followed by a “stuffed” zero byte, whose operation is called as ‘byte stuffing’, as shown in [Fig.\[fig:ff00\]]{}(c)[@JPEG]. Note that the file size of the stream (b), is not the same as that of the stream (c). We have to consider the byte stuffing operation to preserve the same file size when JPEG images are encrypted.
![Byte stuffing in entropy-coded data segment[]{data-label="fig:ff00"}](dctCoeffs.pdf "fig:")\
(a) Entropy coded DCT coeffcients\
![Byte stuffing in entropy-coded data segment[]{data-label="fig:ff00"}](byteArray.pdf "fig:")\
(b) Byte-based packing\
![Byte stuffing in entropy-coded data segment[]{data-label="fig:ff00"}](byteStuffing.pdf "fig:")\
(c) Byte stuffing
Proposed Structure
==================
We propose a new bitstream-based JPEG image encryption method which allows us to exactly preserve the same file size as the original JPEG bitstream.
Outline of the proposed structure
---------------------------------
Bitstreams encrypted by the proposed method have not only the same file sizes, but also the compatibility with JPEG decoders. Some of only additional bits fields that satisfy conditions are encrypted to keep the compatibility with JPEG decoders. [Figure \[fig:proposed\]]{} illustrates the outline of the proposed method. The procedure of the proposed method is summarized as follows.
1. Analysis, byte-by-byte, the entropy-coded data segment and extract additional bits from a byte that satisfies two conditions: the byte includes both Huffman code and additional bits, and the Huffman code includes at least one “0” bit.
2. Generate a random binary sequence with a secret key.
3. Carry out exclusive-or operation between only extracted additional bits and the random sequence generated in 2), and replace the additional bits with the result.
4. Produce an encrypted bitstream by combining the encrypted additional bits with other data without any encryption.
![Outline of the proposed method[]{data-label="fig:proposed"}](proposed.pdf "fig:"){width="3.3in"}\
Encryption considering occurrences or disappearances of ‘FF00’ {#sec:condition}
--------------------------------------------------------------
If all additional bits are simply encrypted, there is a possibility to generate or lose “FF”. Therefore, in the proposed method, only limited additional bits are encrypted based on exclusive-or operation with a random binary sequence.
In [Fig.\[fig:check\]]{}, the additional bits in [Fig.\[fig:ff00\]]{}(c) were replaced with ’x’. Using [Fig.\[fig:check\]]{}, we describe an outline of determination whether encryption is possible or not.
### The first byte
The data consist of a 5-bit Huffman code and 3-bit additional bits. Even if all the additional bits are 1, the entire byte never becomes “FF”. Therefore, the additional bits part is able to be encrypted.
### The second byte
The data consist of 3-bit additional bits and a 5-bit Huffman code. In the original data, since the additional bit was “111” and the remaining Huffman code was ‘111111’’, “FF” was composed as the whole byte. If any bit of the additional bit is changed to 0 by encryption, the entire byte is not ‘FF’ and ‘00’ of the third byte is not inserted. Since this causes a file size change, the additional bits of the second byte are not encrypted.
### The third byte
The data are padding data because the second byte is “FF”. Since this is not an additional bit, it is not encrypted.
### The last byte
The data consist of a 1-bit Huffman code and 7-bit additional bits. Even if all the additional bits are 1, the entire byte never becomes “FF”. Therefore, the additional bits part is able to be encrypted.
![Example for determination whether encrypt or not[]{data-label="fig:check"}](check.pdf "fig:"){width="8cm"}\
By analyzing the Huffman code in the byte as described above, it is possible to determine whether encryption is possible or not. In summary, the following bytes are not encrypted.
1. The whole 8 bits are Huffman codes.
2. The whole 8 bits are additional bits.
3. “00” byte immediately after “FF”.
4. Huffman code and additional bits are included and the all bits of Huffman code are ‘1’.
Experimental Results and Discussion
===================================
Some simulations were carried out to demonstrate the effectiveness of the proposed method. For the simulations, we used the reference software distributed by JPEG[@reference] with 4:2:0 chroma subsampling.
Image quality evaluation of encrypted image
-------------------------------------------
First, the image quality of encrypted images was evaluated. [Figure \[fig:proposedResult\]]{}(a) and (b) were standard images, “lena” and “mandrill” encoded by JPEG with Q-factor 80, respectively. [Figure \[fig:proposedResult\]]{}(c) to (f) were decoded images with a standard JPEG decoder from the encrypted images by the proposed scheme. In [Fig.\[fig:proposedResult\]]{}(c) and (d), the additional bits in only DC components were encrypted. On the other hand, in [Fig.\[fig:proposedResult\]]{}(e) and (f), the additional bits in both DC and AC components were encrypted. The encrypted image in [Fig.\[fig:proposedResult\]]{}(b) has slightly visible information on the original one, because the AC components were the same as the original ones. On the other hand, the encrypted image in [Fig.\[fig:proposedResult\]]{}(c) had less visible information than [Fig.\[fig:proposedResult\]]{}(b).
------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------
![Original and encrypted results (lena, mandrill)[]{data-label="fig:proposedResult"}](lena.jpg "fig:"){width="4cm"} ![Original and encrypted results (lena, mandrill)[]{data-label="fig:proposedResult"}](mandrill.jpg "fig:"){width="4cm"}
\(a) Original ($Q=80$) (b)Original ($Q=80$)
(lena) (mandrill)
![Original and encrypted results (lena, mandrill)[]{data-label="fig:proposedResult"}](lena_dc_conv.jpg "fig:"){width="4cm"} ![Original and encrypted results (lena, mandrill)[]{data-label="fig:proposedResult"}](mandrill_dc_conv.jpg "fig:"){width="4cm"}
\(c) DC component only \(d) DC component only
(lena) (mandrill)
![Original and encrypted results (lena, mandrill)[]{data-label="fig:proposedResult"}](lena_both_conv.jpg "fig:"){width="4cm"} ![Original and encrypted results (lena, mandrill)[]{data-label="fig:proposedResult"}](mandrill_both_conv.jpg "fig:"){width="4cm"}
\(e) Both Components \(f) Both Components
(lena) (mandrill)
------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------
The number of bytes to be encrypted
-----------------------------------
[Table \[tab:count\]]{} indicates the ratio of encryption applied to additional bits in “image data”. Data excluded from encryption are data which satisfy conditions 2) and 4) of [**\[sec:condition\]**]{}. Due to the increase of Q-factors, the proportion of encryption targets decreased. This is because the number of bytes corresponding to the condition 2) necessity increases due to the increase of Q-factors.
\(a) $Q=50$\
target
--------- ----- -------- ------
DC only 70 4,729 98.5
AC only 172 9,591 98.2
Both 197 13,467 98.6
: The number of bytes to encrypt / bytes to be excluded from encryption (Lena image)[]{data-label="tab:count"}
\
(b) $Q=80$\
target
--------- ----- -------- ------
DC only 420 6,306 93.8
AC only 460 20,931 97.8
Both 741 26,104 97.2
: The number of bytes to encrypt / bytes to be excluded from encryption (Lena image)[]{data-label="tab:count"}
\
(c) $Q=95$\
target
--------- ------- -------- ------
DC only 1,758 7,152 80.2
AC only 3,225 59,063 94.8
Both 4,336 65,274 93.8
: The number of bytes to encrypt / bytes to be excluded from encryption (Lena image)[]{data-label="tab:count"}
File-size preserving
--------------------
Next, we compare our method with the previous works[@EJIRUBwFC; @AEtCSfJMJS], in terms of the file sizes. [Table \[tab:changeSize\]]{} shows the file sizes of encrypted JPEG images under various conditions. From this table, JPEG images encrypted by the proposed method had exactly the same file sizes as those of the original ones. However, other encryption methods could not preserve the same file sizes, because they do not consider the effect of byte stuffing.
Q-factor 50 80 95
------------------- --------------- --------------- -----------------
Original 24,279 43,879 106,548
**Proposed** **24,279(0)** **43,879(0)** **106,548(0)**
Cheng[@EJIRUBwFC] 24,281(+2) 43,865(-14) 106,553(+5)
EtC[@AEtCSfJMJS] 24,767(+488) 44,487(+608) 108,262(+1,714)
: Length of original and encrypted images (difference)\[byte\], for lena image[]{data-label="tab:changeSize"}
Conclusion
==========
In this paper, we have proposed an encryption method that allows us to preserve the same file size before and after the encryption. In the encryption process, the proposed method considers the effect of byte stuffing and guarantees a constant file size by providing a mechanism to avoid the occurrence or disappearance of the JPEG marker code. By preserving the file size, it can be expected that the image transmission are successfully carried out after the hooking encryption process.
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J. Zhou, X. Liu, O. C. Au, and Y. Y. Tang, “Designing an efficient image encryption-then-compression system via prediction error clustering and random permutation,” IEEE Trans. on information forensics and security, vol. 9, no. 1, pp. 39-50, 2014.
O. Watanabe, A. Uchida, T. Fukuhara, and H. Kiya, “An encryption-then-compression system for JPEG 2000 standard,” IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015, pp. 1226-1230.
K. Kurihara, S. Shiota, and H. Kiya, “An encryption-then-compression system for jpeg standard,” Picture Coding Symposium (PCS), 2015, pp. 119-123.
K. Kurihara, M. Kikuchi, S. Imaizumi, S. Shiota, and H. Kiya, “An Encryption-then-Compression System for JPEG / Motion JPEG Standard,” IEICE Trans. Fundamentals, vol.E98-A, no.11, pp.2238-2245, November 2015.
K. Kurihara, O. Watanabe, and H. Kiya, “An encryption-then- compression system for jpeg XR standard,” in IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB), 2016, pp. 1-5.
K. Kurihara, S. Imaizumi, S. Shiota, and H. Kiya, “An encryption-then-compression system for lossless image compression standards,” IEICE Trans. Inf. & Syst., vol. E100-D, no. 1, pp. 52-56, 2017.
T. Chuman, K. Kurihara, and H. Kiya, “On the security of block scrambling-based etc systems against jigsaw puzzle solver attacks,” IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017, pp. 2157-2161.
T. Chuman, K. Kurihara, and H. Kiya, “Security evaluation for block scrambling-based etc systems against extended jigsaw puzzle solver attacks,” IEEE International Conference on Multimedia and Expo (ICME), 2017, pp. 229-234.
Tatsuya Chuman, Kenta Iida, and Hitoshi Kiya, “Image Manipulation on Social Media for Encryption-then-Compression Systems,” Proc. APSIPA Annual Summit and Conference, Kuala Lumpur, Malaysia, 14th December, 2017.
Tatsuya Chuman, Kenta Kurihara, and Hitoshi Kiya “On the Security of Block Scrambling-based EtC Systems against Extended Jigsaw Puzzle Solver Attacks,” IEICE Trans. Inf. & Sys., vol.E101-D, no.1, pp.37-44, January 2018.
Ikeda, H., Iwamura, K.: Selective encryption scheme and mode to avoid generating marker codes in JPEG2000 code streams with block cipher. In: Proceedings of IEEE WAINA, pp. 593–600 (2011)
H. Kiya, S. Imaizumi, and O. Watanabe, “Partial-Scrambling of Image Encoded Using JPEG2000 without Generating Marker Codes,” Proc. IEEE International Conference on Image Processing(ICIP), no.WA-P1.3, 17th September, 2003.
Hitoshi Kiya, Shoko Imaizumi, and Osamu Watanabe, “Partial-Scrambling of Image Encoded Using JPEG2000 without Generating Marker Codes,” Proc. IEEE International Conference on Image Processing, no.WA-P1.3, Barcelona, Spain, 17th September, 2003.
Unterweger, Andreas and Uhl, Andreas: “Length-preserving Bit-stream-based JPEG Encryption, ” Proceedings of the on Multimedia and Security, MM&\#38;Sec ’12, pp. 85–90, 2012.
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abstract: |
Coherent bremsstrahlung (CBS) is a specified type of radiation at colliders with short bunches. In the present paper we calculate the main characteristics of CBS for the BEPC collider. At this collider $dN_\gamma \sim 3\cdot 10^{8} \; dE_\gamma /
E_\gamma$ photons of CBS will be emitted for a single collision of the beams in the energy range $E_\gamma \stackrel {<} {\sim}\;
240$ eV.
It seems that CBS can be a potential tool for optimizing collisions and for measuring beam parameters. Indeed, the bunch length $\sigma_z$ can be found from the CBS spectrum because critical energy $E_c \propto 1/\sigma_z$; the horizontal transverse bunch size $\sigma_x$ is related to $dN_\gamma \propto
1/\sigma_x^2$. Besides, CBS may be very useful for a fast control over an impact parameter $R$ between the colliding bunch axes because a dependence of $dN_\gamma$ on $R$ has a very specific behavior.
It seems quite interesting to investigate this type of radiation at the BEPC collider (for example, in the range of a visible light with the rate about $3\cdot 10^{14}$ photons per second) and to apply it for the fast beam control.
author:
- |
Y.B.Ding\
[*Graduate School, USTC at Beijing, Academia Sinica,*]{}\
[*Beijing 100039, China*]{}\
and\
[*Department of Physics, University of Milan, INFN, 20133 Milan, Italy*]{}\
E-mail:ding@mi.infn.it\
V.G. Serbo\
[*Novosibirsk State University, 630090 Novosibirsk, Russia*]{}\
E-mail: serbo@math.nsc.ru\
and\
[*Department of Physics, University of Milan, INFN, 20133 Milan, Italy*]{}
date: 'June 12, 1998'
title: '**Coherent bremsstrahlung at the BEPC collider[^1]**'
---
-1.3cm
[**Keywords**]{}: colliding beams, bremsstrahlung, coherent bremsstrahlung, beam parameter measurements, BEPC
Three types of radiation at colliders
=====================================
Let us speak, for definiteness, about emission by electrons moving through a positron bunch. If the photon energy is large enough, one deals with the ordinary (incoherent) [**bremsstrahlung**]{}.
If the photon energy becomes sufficiently small, the radiation is determined by the interaction of the electron with the collective electromagnetic field of the positron bunch. It is known (see, e.g. §77 in Ref. [@Landau2]) that the properties of this coherent radiation are quite different depending on whether the electron deflection angle $\theta_d$ is large enough or rather small as compared with the typical emission angle[^2] $\theta_r \sim 1/\gamma_e$.
It is easy to estimate the ratio of these angles. The electric [**E**]{} and magnetic [**B**]{} fields of the positron bunch are approximately equal in magnitude, $\mid {\bf E}\mid \approx \mid
{\bf B}\mid \sim eN_p /(\sigma_z (\sigma_x +\sigma_y)) \sim 200$ G. These fields are transverse and they deflect the electron into the same direction. In such fields the electron moves around a circumference of radius $\rho \sim \gamma_e m_e c^2/(eB)$ and bents on the angle $\theta_d \sim \sigma_z /\rho$. On the other hand, the radiation angle $\theta_r$ corresponds to a length $l_\rho =\rho /\gamma_e \sim m_e c^2 /(eB)$. Therefore, the ratio of these angles is determined by the dimensionless parameter $\eta$ $$\eta={r_e N_p\over \sigma_x+\sigma_y} \sim {\theta_d \over
\theta_r } \sim {\sigma_z \over l_\rho}\,.
\label{1}$$
We call a positron bunch [*long*]{} if $\eta \gg 1$. The radiation in this case is usually called [**beamstrahlung**]{}. Its properties are similar to those for the ordinary synchrotron radiation in an uniform magnetic field (see, e.g. review [@Chen]).
We call a positron bunch [*short*]{} if $\eta \ll 1$. In this case the motion of the electron can be assumed to remain rectilinear over the course of the collision. The radiation in the field of a short bunch differs substantially from the synchrotron one. In some respect it is similar to the ordinary bremsstrahlung, which is why we called it [**coherent bremsstrahlung (CBS)**]{}.
In most colliders the parameter $\eta$ is either much smaller then 1 (all the $pp$, $\bar p p$ and relativistic heavy-ion colliders, some $e^+e^-$ colliders and B-factories) or $\eta \sim 1$ (e.g., LEP). Only for linear $e^+e^-$ colliders $\eta \gg 1$. Therefore, the CBS has a very wide region of applicability.
Below we use the following parameters for the BEPC collider $$N_e=N_p=2\cdot 10^{11}, \;
E_e=E_p=2 \; \mbox{GeV},\;
\sigma_x= 890 \;\mu\mbox{m},\; \sigma_y=37 \; \mu\mbox{m},\;
\sigma_z=5\; \mbox{cm} \,.
\label{2}$$ Therefore, for BEPC the parameter $\eta$ is equal to $$\eta =0.608,
\label{3}$$ so for the BEPC collider our calculation gives, strictly speaking, an estimate only (though we have some reason to believe that the real parameter is determined not by the relation $\eta
\ll 1$, but by the relation $\eta \ll 10$ - see Ref. [@PL92]).
A classical approach to CBS was given in Ref. [@Bassetti]. A quantum treatment of CBS based on the rigorous concept of colliding wave packets were considered in [@YaF], some applications of CBS to modern colliders in [@Ginz; @PL92; @PS; @ESS1; @ESS2]. A new method to calculate CBS based on the equivalent photon approximation for the collective electromagnetic field of the oncoming bunch is presented in [@ESS1]. This method is much more simple and transparent as that previously discussed. It allows to calculate not only the classical radiation but to take into account quantum effects in CBS as well [@ESS3].
Distinctions of the CBS from the usual bremsstrahlung and from the beamstrahlung
=================================================================================
In the usual bremsstrahlung the number of photons emitted by electrons is proportional to the number of electrons and positrons: $$dN_{\gamma}\; \propto \;N_e\; N_p\; {dE_{\gamma} \over E_{\gamma}} \ .
\label{4}$$ With decreasing photon energies the coherence length $\sim
4\gamma ^2_e \hbar c /E_\gamma $ becomes comparable to the length of the positron bunch $\sigma_z$. At photon energies $$E_{\gamma} \stackrel{<}{\sim} E_c= 4 {\gamma^2_e \hbar c \over
\sigma_z}
\label{5}$$ the radiation arises from the interaction of the electron with the positron bunch as a whole, but not with each positron separately. The quantity $E_c$ is called [*the critical photon energy*]{}. Therefore, the positron bunch is similar to a “particle” with the huge charge $e\, N_p$ and with an internal structure described by the form factor of the bunch. The radiation probability is proportional to the squared number of positrons $N_p^2$ and the number of the emitted photons is given by $$dN_{\gamma}\; \propto \;N_e\; N^2_p\; {dE_{\gamma} \over E_ {\gamma}} \ .
\label{6}$$
The CBS differs strongly from the beamstrahlung in the soft part of its spectrum. As one can see from (\[4\]) the total number of CBS photons diverges in contrast to the beamstrahlung for which (as well as for the synchrotron radiation) the total number of photons is finite.
Experimental status
=====================
The ordinary bremsstrahlung was used for luminosity measuring (for example, at the VEPP-4, HERA and LEP colliders).
The beamstrahlung has been observed in a single experiment at SLC [@Bon] in which it has been demonstrated that it can be used for measuring a transverse bunch size of the order of 5 $\mu$m.
The main characteristics of the CBS have been calculated only recently and an experiment for its observation is now under preparation at VEPP-2M (Novosibirsk).
Qualitative description of CBS
===============================
We start with the standard calculation of bremsstrahlung (see [@BLP], §93 and §97) at $ep$ collisions. This process is defined by the block diagram of Fig. \[Fig1\], which gives the radiation of the electron (we do not consider the similar block diagram which gives the radiation of positron, the interference of these two block diagrams is neglegible).
fig1.tex
We denote the 4-momentum of the virtual photon by $\hbar q = (\hbar \omega/c,\hbar {\bf q})$. The main contribution to the cross section is given by the region of small values $(- q^2)$. In this region the given reaction can be represented as a Compton scattering of the equivalent photon (radiated by the positron) on the electron. Therefore, one obtains $$d\sigma_{e^-e^+\to e^-e^+\gamma}=
dN_{EP} (\omega, E_p)\, d\sigma_{e
\gamma} (\omega,E_e, E_\gamma).
\label{7}$$ Here $$dN_{EP} (\omega, E_p) \approx {\alpha \over \pi} {d\omega\over
\omega} \int^{(-q^2)_{\max}} _{(-q^2)_{\min}} {d(-q^2)\over
(-q^2)}= {\alpha \over \pi}{d \omega\over \omega}
\ln{m^2_e\over(m_p\hbar\omega /E_p)^2}\ .
\label{8}$$ is the number of equivalent photons (EP) with the frequency $\omega$ generated by the positron.
For the cross section (\[7\]) we obtain (the case of the $e^-$ radiation only) $$d\sigma_{e^-e^+\to e^-e^+\gamma}
\approx {16\over 3}\alpha r^2_e\, \left(1-y+{3\over
4}\,y^2\right)\, \ln{{4E_e E_p (1-y)\over m_e m_p
c^4\,y}} \; {dE_\gamma\over E_\gamma} \,,\;\;
y={E_\gamma \over E_e}\, .
\label{9}$$
Just as in the standard calculations we can estimate the number of CBS photons using equivalent photon approximation. Taking into account that the number of EP increases by a factor $\sim
N_p$ compared to the ordinary bremsstrahlung we get (using $
d(-q^2) \to d^2 q_\bot / \pi$) $$dN_{EP} \sim N_p \; {\alpha \over \pi^2}\; {d\omega \over
\omega} \; {d^2 q_\bot \over q_\bot^2}.
\label{10}$$
fig2.tex
Since the impact parameter $\varrho\, \sim \, 1/ q_\bot$, we can rewrite this expression in another form $$dN_{EP} \sim N_p \; {\alpha \over \pi^2}\; {d\omega \over
\omega} \; {d^2 \varrho \over \varrho^2} \ .
\label{11}$$ It is not difficult to estimate the region which gives the main contribution $
|\varrho_x | \sim \sigma_x, \;
|\varrho_y| \sim \sigma_y \ .
$ Integrating over this region we obtain estimates for $dN_{EP}$ and for the “effective cross section” $$dN_{EP} \sim N_p \; {\alpha \over \pi}\; {d\omega \over
\omega} \;{\sigma_x \sigma _y \over \sigma_x^2 +\sigma_y^2 } \,,
\;
d \sigma _{{\rm eff}} \sim N_p\, \alpha\, r_e^2\; {\sigma_x
\sigma _y \over \sigma_x^2 +\sigma_y^2 } {dE_\gamma \over
E_\gamma}.
\label{12}$$
To illustrate the transition from the ordinary bremsstrahlung to CBS we present in Fig. \[Fig2\] the photon spectrum for the BEPC collider. In the region of $E_\gamma \sim 100$ eV the number of photons dramatically increases by about 8 orders of magnitude.
Possible applications
=====================
Coherent bremsstrahlung was not observed yet. Therefore, one can speak about applications of CBS on the preliminary level only. Nevertheless, even now we can see such features of CBS which can be useful for applications. They are the following.
A huge number of the soft photons whose spectrum is determined by the length of the positron bunch are emitted. The number of CBS photons for [**a single collision**]{} of the beams is (see Refs. [@PL92; @ESS1] for details) $$dN_{\gamma }=N_{0}\Phi (E_{\gamma}/E_{c}){dE_{\gamma}\over
E_{\gamma}}.
\label{13}$$ Here for the flat Gaussian bunches (e.g. at $a_y^2= \sigma
_{ey}^2 + \sigma _{py}^2 \ll a_x^2= \sigma _{ex} ^2 + \sigma
_{px} ^2\;$) constant $N_0$ is equal to $$N_0={8\over 3\pi}\;\alpha N_e \left({r_e N_p
\over a_x}\right) ^2\;{\arcsin (\sigma _{ex}/a_x)^2 + \arcsin
(\sigma _{ey}/a_y)^2 \over [1-(\sigma _{ex}/a_x)^4] ^{1/2}},
\label{14}$$ and for the flat and identical Gaussian bunches it is $$N_0={8\over 9\sqrt{3}}\;\alpha N_e \left({r_e N_p \over
\sigma _x}\right) ^2 \approx 0.5 \alpha N_e \eta^2\,.
\label{15}$$ The function $$\Phi (x)={3\over 2}\;\int _0^\infty {1+z^2\over (1+z)^4}\; \mbox
{exp} [-x^2(1+z)^2]\;dz;$$ $$\Phi (x)=1 \;\;\mbox{at}\;\; x \ll 1;\;\;\; \Phi
(x)=(0.75/x^2)\cdot \mbox{e}^{-x^2} \;\;\mbox{at}\;\; x\gg 1;
\label{16}$$ some values of this function are: $\Phi (x)=$ 0.80, 0.65, 0.36, 0.10, 0.0023 for $x=$ 0.1, 0.2, 0.5, 1, 2 (see Ref. [@YaF]).
In Table 1 we give the parameters $E_c$ and $N_0$ for the BEPC collider as well as for some colliders now under development for comparison
Table 1
BEPC KEKB [@PS] LHC, $pp$ [@Ginz] LHC, $Pb\,Pb$ [@ESS2]
------------ ----------------- --------------------------------- ------------------- -----------------------
$E$ (GeV) 2 8/3.5 7000 574000
$E_c$ (eV) 240 40000/7400 590 90
$N_0 $ 2.7$\cdot 10^8$ 20$\cdot 10^6\;$/ 8$\cdot 10^6$ 80 50
Specific features of CBS — a sharp dependence of spectrum (\[13\]) on the positron bunch length, an unusual behavior of the CBS photon rate in dependence on the impact parameter between axes of the colliding bunches, an azimuthal asymmetry and polarization of photons — can be very useful for an operative control over collisions and for measuring bunch parameters.
It may be convenient for BEPC to use the CBS photons in the range of [*visible light*]{} $E_\gamma \sim 2-3$ eV $\ll E_c=240$ eV. In this region the rate of photons will be $${dN_\gamma \over \tau} \approx 3\cdot 10^{14}\; {dE_\gamma \over
E_\gamma} \;\; \;\mbox{photons$\;$ per $\;$ second}
\label{17}$$ (here $\tau=0.8 \; \mu$s is time between collisions of bunches at a given interaction region), and it is possible to use a polarization measurement without difficulties.
Collisions with the nonzero impact parameter of bunches
=======================================================
If the electron bunch axis is shifted in the vertical direction by a distance $R_{y}$ from the positron bunch axis, the luminosity $L(R_{y})$ (as well as the number of events for the usual reactions) decreases very quickly: $$L(R_{y})=L(0)\exp \left(-{R^{2}_{y}\over 4\sigma ^{2}_{y}}\right).
\label{18}$$ In contrast, for the BEPC collider the number of CBS photons increases almost two times. The increase reaches 75 % at $R_{y}\approx 3\,\sigma _{y}$. After that, the rate of photons decreases, but even at $R_{y}= 15\;\sigma_y$ the ratio $dN_\gamma
(R_y) /dN_\gamma(0) = 1.01$. The corresponding results are presented in Fig 3 (for the calculation we used formulae from Ref. [@YaF]).
fig3.tex
The effect does not depend on the photon energy. It can be explained in the following way.
At $R_{y}=0$ a considerable portion of the electrons moves in the region of small impact parameters where electric and magnetic fields of the positron bunch are small. For $R_{y}$ such as $\sigma^2_{y}\ll R^2_{y}\ll \sigma^2_{x}$, these electrons are shifted into the region where the electromagnetic field of the positron bunch are larger, and, therefore, the number of emitted photons increases. For large $R_y$ (at $\sigma^2_x \ll
R^2_y \ll \sigma^2_z$), fields of the positron bunch are $|{\bf E}|
\approx |{\bf B}| \propto\; 1/R_y$ and, therefore, $dN_\gamma
\propto \; 1/R_y^2$, i.e. the number of emitted photons decreases but very slowly.
This feature of CBS can be used for a fast control over impact parameters between beams (especially at the beginning of every run) and over transverse beam sizes. For the case of long bunches, such an experiment has already been performed at the SLC (see Ref. [@Bon]).
Azimuthal asymmetry and polarization
====================================
If the impact parameter between beams is nonzero, an azimuthal asymmetry of the CBS photons appears, which can also be used for operative control over beams. For definiteness, let the electron bunch axis be shifted in the vertical direction by the distance $R_{y}$ from the positron bunch axis. When $R_{y}$ increases, the electron bunch is shifted into the region where the electric field of the positron bunch is directed almost in a vertical line. As a result, the equivalent photons (produced by the positron bunch) obtain a linear polarization in the vertical direction. The mean degree of such a polarization $l$ for the BEPC collider is shown in Fig 4:
fig4.tex
Let us define the azimuthal asymmetry of the emitted photons by the relation $$A= {dN_{\gamma}(\varphi=0)-dN_{\gamma}(\varphi=\pi/2)\over
dN_{\gamma}(\varphi=0)+dN_{\gamma}(\varphi=\pi/2)},
\label{19}$$ where the azimuthal angle $\varphi $ is measured with respect to the horizontal plane. It is not difficult to obtain that this quantity does not depend on photon energy and is equal to: $$A={2(\gamma_e \theta)^2 \over 1+(\gamma_e \theta)^4} \; l,
\label{20}$$ where $\theta$ is the polar angle of the emitted photon. From Fig 3 one can see that when $R_{y}$ increases, the fraction of photons emitted in the horizontal direction becomes greater than the fraction of photons emitted in the vertical direction.
If the equivalent photons have the linear polarization (and $l$ is its mean degree), then the CBS photons get also the linear polarization in the same direction. Let $l^{(f)}$ be the mean degree of CBS photon polarization. The ratio $l^{(f)}/l$ varies in the interval from 0.5 to 1 when $E_\gamma$ increases (see Table 2).
Table 2
$E_\gamma /E_c$ 0 0.2 0.4 0.6 0.8 1 1.5 2
----------------- ----- ----- ------ ------ ------ ------ ------ ------
$l^{(f)}/l$ 0.5 0.7 0.81 0.86 0.89 0.94 0.96 0.97
Discussion
==========
Coherent bremsstrahlung is a new type of radiation at storage rings which was not observed yet. Therefore, if it will be observed at BEPC, it will be a pioneer work in this field. Besides, there is another interesting coincidence. The Lorentz factor $\gamma_e=E_e/(m_e c^2)=4\cdot 10^3$ at the BEPC accelerator is of the same order as the Loretz factor at the LHC collider ( $\gamma_p=7\cdot 10^3$ for pp collisions and $\gamma_{Pb}=3\cdot 10^3$ for PbPb collisions ). It means that CBS spectrum at BEPC will be similar to that at LHC. As a result, an experience in observation and application of CBS at BEPC may be important for Large Hadron Colliders as well.
A serious problem for the observation of CBS may be [**the background due to synchrotron radiation on the external magnetic field**]{} of the accelerator. This background depends strongly on the details of the magnetic layout of the collider and can not be calculated in a general form. To distinguish CBS from syncrotron radiation (SR) one can use such tricks as:
[*(i)*]{} According to Eqs. (\[6\], \[13\], \[15\]), the number of the emitted CBS photons is proportional to the number of electrons $N_e$ and to the squared number of positrons $N_p^2$, i.e. $dN_{\gamma}^{CBS} \propto N_e N_p^2$. As for SR, the number of SR photons is proportional to the number of electrons only $dN_{\gamma}^{SR} \propto N_e$. Therefore, if one will observe that $dN_{\gamma}$ changes when one changes the positron current, it will be the sign that the observed radiation is caused by CBS but not SR.
[*(ii)*]{} If one has a possibility to shift position of positron buch in the vertical direction, on can clearly distinguish CBS from SR. Indeed, SR of electrons does not change in this case, but the number of CBS photons changes considerable as it can be seen from Fig. 3.
Acknowledgements {#acknowledgements .unnumbered}
================
We are very grateful to Jin Li for providing us the BEPC parameters and to Chuang Zhang for useful discussions. Y.B.D. acknowledges the fellowship by INFN and V.G.S. acknowledges the fellowship by the Italian Ministry of Forein Affairs.
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[^1]: This work was partly supported by the National Natural Science Foundation of China (NSFC) and by Russian Foundation for Basic Research (code 96-02-19114)
[^2]: We use the following notation: $N_e$ and $N_p$ are the numbers of electrons and positrons in the bunches; $\sigma_z$ is the longitudinal, $\sigma_x$ and $\sigma_y$ are the horizontal and vertical transverse sizes of the positron bunch; $\gamma_e=E_e/(m_ec^2)$ is the electron Lorentz factor; $E_c=4\gamma_e^2 \hbar c/\sigma_z $ is the characteristic (critical) energy for the coherent bremsstrahlung photons; $r_e=e^2/(m_e c^2)$.
|
---
author:
- |
CHEN Yong,CHEN Jianmin\
School of Mathematics and Computing Science, Hunan University of Science and Technology,\
Xiangtan, Hunan, 411201, P.R.China. chenyong77@gmail.com
title: 'On the Imbedding Problem for Three-state Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues'
---
[**Abstract** ]{}\
For an indecomposable $3\times 3$ stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the $3\times 3$ stochastic matrix in [@jhson] is pointed out.
[ **keywords:** ]{} imbedding problem; three-state time homogeneous Markov chains; negative eigenvalues\
[ **MSC numbers:** ]{} 60J10; 60J27; 60J20
Introduction
============
The imbedding problem for finite Markov chains has a long history and was first posed by Elfving [@Elfv], which has applications to population movements in social science [@Sil], credit ratings in mathematical finance [@irw], and statistical inference for Markov processes [@BS; @Met]. For a review of the imbedding problem, the reader can refer to [@cart].
According to Kingman [@irw; @King], the imbedding problem is completely solved for the case of $2\times 2$ matrices by D. G. Kendall, who proved that a $2\times 2$ transition probability matrix is compatible with a continuous Markov process if and only if the sum of the two diagonal entries is larger than $1$.
The explicit description of the imbedding problem for the $3\times 3$ stochastic matrix with distinct eigenvalues or with coinciding positive eigenvalues is shown by Johansen in [@jhson]. When the common eigenvalue is negative, P. Carette provides several necessary and sufficient conditions to characterize the imbedded stochastic matrix [@cart Theorem 3.3, Theorem 3.6]. But in contrast to the above conclusions of Kingman or Johansen, it is not clear-cut.
Let ${\mathsf{I}}$ be the identity matrix. Let ${\mathsf{P}},\,{\mathsf{P}}^{\infty}$ be the $3\times 3$ stochastic matrix and its limiting probability matrix respectively. Theorem 3.3 ( resp. Theorem 3.6,) of [@cart] needs to calculate square root of the matrix ${\mathsf{P}}^{\infty}-{\mathsf{I}}$ (resp. ${\mathsf{P}}$), and then to test whether each of the off-diagonal elements satisfies an inequality. But the matrix square root is many-valued, just like the matrix logarithm [@Sil p22].
In the present paper, a new necessary and sufficient condition is shown, which overcomes the difficulty of uncountably many versions of logarithm or square root (Theorem \[main\]). We chiefly rely on an alternate parameterization of the transition rate matrix (Eq.(\[parameter\])) for the proof. At the same time, for a fixed ${\mathsf{P}}^{\infty}$, the exact lower bound of the eigenvalue of ${\mathsf{P}}$ that makes ${\mathsf{P}}$ embeddable is given (Remark \[rem\])[^1].
In the more general context of time-inhomogeneous Markov chains, the imbedding problem is dealt with by some authors [@fryS; @fry1; @fry2; @fug; @jhr]. But we only focus on the time-homogeneous Markov chains here.
The imbedding problem for 3-order transition Matrix with coinciding negative eigenvalues
========================================================================================
The transition matrix ${\mathsf{P}}$ is called embeddable if there is a transition rate matrix ${\mathsf{Q}}$ for which ${\mathsf{P}}=e^{{\mathsf{Q}}}$.
Let ${\mathsf{P}}=(p_{ij})$ be a $3\times 3$ transition probability matrix. Suppose ${\mathsf{P}}$ is indecomposable, i.e., its state space does not contain two disjoint closed sets[@chung P17]. Let the unique stationary probability distribution be $\mu'=(\mu_1,\mu_2,\mu_3)$ with $\mu_1+\mu_2+\mu_3=1$. Let $\vec{e}=(1,1,1)'$
\[lem0\] Suppose ${\mathsf{P}}$ is indecomposable. If ${\mathsf{Q}}$ is a transition rate matrix such that ${\mathsf{P}}=e^{h{\mathsf{Q}}}$, $h>0$, then ${\mathsf{Q}}$ has $\mu'{\mathsf{Q}}=0$.
If a distribution $\nu$ has $\nu'{\mathsf{Q}}=0$ (i.e., the left eigenvector with eigenvalue $0$), then ${\mathsf{P}}=e^{h{\mathsf{Q}}}=\sum^{\infty}_{n=0}\frac{1}{n!}h^n{\mathsf{Q}}^n$ implies that $\nu'{\mathsf{P}}=\nu'$. Since ${\mathsf{P}}$ is indecomposable, one has that $\nu=\mu$.
\[lem1\] Suppose the transition matrix ${\mathsf{P}}$ is embeddable with eigenvalues ${\left\{1,\,\lambda,\,\lambda\right\}}$, $\lambda<0$. Then ${\mathsf{P}}$ is diagonalizable and satisfies $$\label{PP}
{\mathsf{P}}={\mathsf{P}}^{\infty}+\lambda({\mathsf{I}}-{\mathsf{P}}^{\infty}),$$ where ${\mathsf{P}}^{\infty}=\vec{e}\mu'$ is the limiting probability matrix, ${\mathsf{I}}$ is the identity matrix.[^2]
If ${\mathsf{Q}}$ is a transition rate matrix such that ${\mathsf{P}}=e^{{\mathsf{Q}}}$, then ${\mathsf{Q}}$ has a pair of conjugate complex eigenvalues and is diagonalizable. Since ${\mathsf{P}}=e^{{\mathsf{Q}}}$, ${\mathsf{P}}$ is diagonalizable too.
The eigenvalues of ${\mathsf{P}}$ are ${\left\{1,\,\lambda,\,\lambda\right\}}$, thus the rank of the matrix $\lambda{\mathsf{I}}-{\mathsf{P}}$ is $1$. Since $(\lambda{\mathsf{I}}-{\mathsf{P}})\vec{e}=(\lambda-1)\vec{e}$, the three rows of $\lambda{\mathsf{I}}-{\mathsf{P}}$ are equal. Then $$\label{mat}
{\mathsf{P}} =
\left[ \begin{array}{lll}
1-(x+y)&x&y\\
z&1-(y+z)&y\\
z&x&1-(z+x)
\end{array}
\right],$$ and $1-\lambda=x+y+z$. Note that $\mu'=(z,\,x,\,y)/(x+y+z)$, this ends the proof.
\[cor2\] Suppose ${\mathsf{P}}$ satisfies the condition of Lemma \[lem1\], then the stationary probability distribution is positive and all elements of $\,{\mathsf{P}}$ are positive.
Since ${\mathsf{P}}=\lambda{\mathsf{I}}+(1-\lambda){\mathsf{P}}^{\infty}$, one has that $\lambda+(1-\lambda)\mu_i\geqslant 0,\,i=1,2,3$. Then $\mu_i\geqslant \frac{-\lambda}{1-\lambda}>0$, and the off-diagonal elements satisfy $p_{ij}=(1-\lambda)\mu_j>0,\,i\neq j$. In addition, it was shown by Goodman that each of the diagonal elements of an embeddable matrix dominates the determinant, and that this determinant is positive: $p_{ii}\geqslant \det {\mathsf{P}}> 0$ [@jhr; @gdm].
The conclusions of Lemma \[lem1\] and Corollary \[cor2\] also appear in [@cart]. By Lemma \[lem1\], ${\mathsf{P}}$ is completely determined by its stationary distribution and the coinciding eigenvalues.
\[prop3\] Suppose that ${\mathsf{P}}$ satisfies Eq.(\[PP\]). ${\mathsf{P}}$ can be imbedded if and only if there exists a transition rate matrix ${\mathsf{Q}}$ such that it has $\mu'{\mathsf{Q}}=0$ and eigenvalues $\theta$ and $\bar{\theta}$ and it holds that $e^{\theta h}=\lambda$ for some $h\in {\mathbb{R}}^+$.
The proof of Proposition \[prop3\] is presented in Section \[prf\]. Here $\theta$ is a complex eigenvalue $-p+{\mathrm{i}}q$ with $\frac{q}{p}=\frac{(2k+1)\pi}{-\log{\left\vert\lambda\right\vert}}$, $k\in {\mathbb{Z}}^+$.
By the Runnenberg condition in [@Sil; @Run], the complex eigenvalue $-p+{\mathrm{i}}q$ of the transition rate matrix ${\mathsf{Q}}$ satisfies that $$\frac{{\left\vertq\right\vert}}{p} \leqslant\frac{1}{\sqrt{3}},$$ which the reader can also refer to [@cy] for detail. Then $${\left\vert\lambda\right\vert}\leqslant e^{-\sqrt{3} \pi}\doteq 0.0043,$$ and ${\mathsf{P}}$ is almost equal to its limiting probability matrix by Eq.(\[PP\]).
For a transition rate matrix $${\mathsf{Q}}= \left[
\begin{array}{lll}
-a_2-a_3&a_2 &a_3\\
b_1&-b_1-b_3&b_3\\
c_1&c_2 &-c_1-c_2
\end{array}
\right ],$$ suppose that it has $\mu'{\mathsf{Q}}=0$, that is to say, $$\label{flux}
\mu_1a_2-\mu_2b_1=\mu_2 b_3 - \mu_3 c_2 = \mu_3 c_1 - \mu_1 a_3.$$ Let $$\label{gamma}
\hspace{-9mm} \nu=\frac{\mu_1a_2-\mu_2b_1}{2},\,
\gamma=\frac{\mu_1a_2+\mu_2b_1}{2},\,
\delta=\frac{\mu_2b_3+\mu_3c_2}{2},\,
\kappa=\frac{\mu_3c_1+\mu_1a_3}{2} .$$ We propose an alternate parameterization of the transition rate matrix as $$\label{parameter}
{\mathsf{Q}}=\left[
\begin{array}{lll}
-\frac{\kappa+\gamma}{\mu_1}&\frac{\gamma+\nu}{\mu_1}&
\frac{\kappa-\nu}{\mu_1}\\
\frac{\gamma-\nu}{\mu_2}&-\frac{\gamma+\delta}{\mu_2}&
\frac{\delta+\nu}{\mu_2}\\
\frac{\kappa+\nu}{\mu_3}&\frac{\delta-\nu}{\mu_3}&
-\frac{\delta+\kappa}{\mu_3}
\end{array}
\right ],$$ where $$\label{vkg}
\kappa,\,\gamma,\,\delta\geqslant 0,\quad \kappa+\gamma,\,\gamma+\delta,\,\delta+\kappa>0, \quad\text{ and} \quad {\left\vert\nu\right\vert}\leqslant\kappa,\,\gamma,\,\delta.$$ The re-parameterization is one-to-one. Then the transition rate matrix ${\mathsf{Q}}$ with $\mu'{\mathsf{Q}}=0$ must satisfy Eq.(\[parameter\]). [^3] Let the eigen-equation of ${\mathsf{Q}}$ be $\lambda(\lambda^2 + \alpha \lambda +\beta)=0$. Then we have that $$\begin{aligned}
\alpha &=&\frac{\kappa+\gamma}{\mu_1}+
\frac{\gamma+\delta}{\mu_2}+\frac{\delta+\kappa}{\mu_3},\label{alpha}\\
\beta &=& \frac{ \kappa\gamma + \gamma\delta + \delta\kappa+\nu^2}{\mu_1\mu_2\mu_3 }.\label{beta}\end{aligned}$$ and the eigenvalue is $$\theta =-\frac{\alpha}{2}+{\mathrm{i}}\sqrt{\beta-\alpha^2/4}\,.$$ Let the ratio between the imaginary ($q$) and real ($-p$) parts of the nonzero eigenvalues be $$\begin{aligned}
\label{H-func}
H(\kappa,\gamma,\delta,\nu)&\triangleq & \frac{{\left\vertq\right\vert}}{p} \nonumber \\
&=&\sqrt{\frac{4\beta-\alpha^2}{\alpha^2}}
=\sqrt{4\frac{\beta}{\alpha^2}-1}\nonumber \\
&=&\sqrt{\frac{4}{\mu_1\mu_2\mu_3}\frac{\kappa\gamma + \gamma\delta + \delta\kappa+\nu^2}{(\frac{\kappa+\gamma}{\mu_1}+
\frac{\gamma+\delta}{\mu_2}+\frac{\delta+\kappa}{\mu_3})^2}-1}.\end{aligned}$$
Since there may be many different transition rate matrices such that $H(\kappa,\gamma,\delta,\nu)=\frac{(2k+1)\pi}{-\log{\left\vert\lambda\right\vert}},\,k\in{\mathbb{Z}}^+$, the solution of ${\mathsf{P}}=e^{{\mathsf{Q}}}$ is not unique.
\[prop4\] For the given $\mu'=(\mu_1,\,\mu_2,\,\mu_3)$, if $\nu=0$ and $\kappa:\gamma:\delta=\frac{1}{\mu_2}:\frac{1}{\mu_3}:\frac{1}{\mu_1}$, then $H(\kappa,\gamma,\delta,\nu)=0$. Note that when $$\kappa,\,\gamma,\,\delta\geqslant 0,\quad \kappa+\gamma,\,\gamma+\delta,\,\delta+\kappa>0, \quad\text{ and} \quad {\left\vert\nu\right\vert}\leqslant\kappa,\,\gamma,\,\delta,$$ $H(\kappa,\gamma,\delta,\nu)$ is a continuous real function of $(\kappa,\gamma,\delta,\nu)$.[^4] Therefore, $H(\kappa,\gamma,\delta,\nu)$ takes the value $\frac{\pi}{-\log{\left\vert\lambda\right\vert}}$ if the maximum of $H(\kappa,\gamma,\delta,\nu)$ is greater than or equal to $\frac{\pi}{-\log{\left\vert\lambda\right\vert}}$.
Therefore, an optimization problem is formulated, and the next Proposition solves it. We say that three positive numbers $a,\,b,\,c$ satisfy the triangle inequality if $a + b > c,\,b + c > a,\,a + c >b$. Let $$m=\min{\left\{\mu_1,\,\mu_2,\,\mu_3\right\}}.$$
\[lem5\] Suppose that $$F(x_1, x_2, x_3, \mu_1, \mu_2, \mu_3)=\frac{x_1x_2+x_2x_3+x_3x_1+(\min{\left\{x_1,x_2, x_3\right\}})^2}{(\frac{x_1+x_2}{\mu_1}+
\frac{x_2+x_3}{\mu_2}+\frac{x_3+x_1}{\mu_3})^2},$$ with $x_1, x_2, x_3\geqslant0$ and $x_1+x_2,\,x_2+x_3,\,x_3+x_1 >0$. Then the maximum of $F(x_1, x_2, x_3, \mu_1, \mu_2, \mu_3)$ is $$\label{maximum}
\left\{
\begin{array}{ll}
\frac{1}{(\frac{1}{\mu_1}+\frac{1}{\mu_2}+\frac{1}{\mu_3})^2}, &\mbox{\, when\, } \frac{1}{\mu_1},\frac{1}{\mu_2},\frac{1}{\mu_3} \mbox{\, satisfy the triangle inequality, }\\
\frac{\mu_1\mu_2\mu_3}{4(1-m )}, &\mbox{otherwise}.
\end{array}
\right.$$ The maximum is attained at the points $$\label{maximumpoint}
\left\{
\begin{array}{ll}
x_1= x_2= x_3, &\mbox{\, when\, } \frac{1}{\mu_1},\frac{1}{\mu_2},\frac{1}{\mu_3} \mbox{\,satisfy the triangle inequality, }\\
x_1= x_2=\frac{\frac{1}{\mu_2}+\frac{1}{\mu_3}}{\frac{2}{\mu_1}-\frac{1}{\mu_2}-\frac{1}{\mu_3}} x_3, &\mbox{\, when\, } \frac{1}{\mu_1}\geqslant\frac{1}{\mu_2}+\frac{1}{\mu_3} ,\\
x_2= x_3=\frac{\frac{1}{\mu_3}+\frac{1}{\mu_1}}{\frac{2}{\mu_2}-\frac{1}{\mu_3}-\frac{1}{\mu_1}} x_1, &\mbox{\, when\, } \frac{1}{\mu_2}\geqslant\frac{1}{\mu_3}+\frac{1}{\mu_1},\\
x_3= x_1=\frac{\frac{1}{\mu_1}+\frac{1}{\mu_2}}{\frac{2}{\mu_3}-\frac{1}{\mu_1}-\frac{1}{\mu_2}} x_2, &\mbox{\, when\, } \frac{1}{\mu_3}\geqslant\frac{1}{\mu_1}+\frac{1}{\mu_2}.
\end{array}
\right.$$
Proof of Proposition \[lem5\] is presented in Section \[prf\].
\[main\] Suppose that ${\mathsf{P}}$ is a $3\times 3$ stochastic matrix with eigenvalues ${\left\{1,\lambda,\lambda\right\}},\,\lambda<0,$ and that ${\mathsf{P}}$ satisfies Eq.(\[PP\]). Then ${\mathsf{P}}$ can be imbedded if and only if $$\label{eq2}
\sqrt{\frac{4\mu_1\mu_2\mu_3}{(\mu_1\mu_2+\mu_1\mu_3+\mu_2\mu_3)^2}-1}\geqslant \frac{\pi}{-\log{\left\vert\lambda\right\vert}},\mbox{\, when\, } \frac{1}{\mu_1},\frac{1}{\mu_2},\frac{1}{\mu_3} \mbox{\,satisfy the triangle inequality, }$$ or $$\hspace{-9mm} \sqrt{ \frac{m}{1-m}}\geqslant \frac{\pi}{-\log{\left\vert\lambda\right\vert}},\mbox{\, otherwise. }$$
[*Proof of Theorem \[main\].*]{} Clearly, if the function $H(\kappa,\gamma,\delta,\nu)$ reaches its maximum then $\nu=\min{\left\{\kappa,\gamma,\delta\right\}}$. From Proposition \[prop3\] we have to find a transition rate matrix $Q(\kappa, \gamma, \delta, \nu)$ of the form just above (\[parameter\]), with eigenvalues $\theta(\kappa, \gamma, \delta, \nu)$ and $\bar{\theta}(\kappa, \gamma, \delta, \nu)$ for which $e^{\theta h} = \lambda$ for some $h > 0$. Using Proposition \[lem5\] we can find $(\kappa_0, \gamma_0, \delta_0, \nu_0)$, so that for the given $\mu',\,\lambda$ we have $$H(\kappa_0, \gamma_0, \delta_0, \nu_0)= \frac{\pi}{-\log{\left\vert\lambda\right\vert}}=H_0.$$ The corresponding transition rate matrix $Q(\kappa_0, \gamma_0, \delta_0, \nu_0)$ has eigenvalue $\theta(\kappa_0, \gamma_0, \delta_0, \nu_0)=\theta_0$ and $\bar{\theta}_0$, where $$\theta_0=-\frac{\alpha_0}{2}+{\mathrm{i}}\sqrt{\beta_0-\alpha_0^2/4}=\frac{\alpha_0}{2}[-1+{\mathrm{i}}H_0]=\frac{\alpha_0}{2}[-1+{\mathrm{i}}\frac{\pi}{-\log{\left\vert\lambda\right\vert}}].$$ We finally choose $h =-2 \log{\left\vert\lambda\right\vert} /\alpha_0$ and find $$\begin{aligned}
\theta_0 h &=-\frac{2\log{\left\vert\lambda\right\vert}}{\alpha_0} \frac{\alpha_0}{2}[-1+{\mathrm{i}}\frac{\pi}{-\log{\left\vert\lambda\right\vert}}]\\
&= \log{\left\vert\lambda\right\vert}+{\mathrm{i}}\pi\end{aligned}$$ which satisfies $e^{\theta_0 h} = \lambda $.
If $\mu_1+\mu_2+\mu_3=1$, and $\frac{1}{\mu_1},\,\frac{1}{\mu_2},\,\frac{1}{\mu_3}$ satisfy the triangle inequality, then it can be shown easily that $$4\mu_1\mu_2\mu_3\geqslant(\mu_1\mu_2+\mu_1\mu_3+\mu_2\mu_3)^2\geqslant 3\mu_1\mu_2\mu_3,$$ i.e., the term inside the $\sqrt{\cdot}$ of Eq.(\[eq2\]) is positive.
\[rem\] For a fixed ${\mathsf{P}}^{\infty}=\vec{e}\mu' $, what are the possible values of $\lambda<0$ that make the stochastic matrix ${\mathsf{P}}$ embeddable? In [@cart Theorem 3.7], P. Carette puts the above question and shows that there exists $\Lambda<0$ such that ${\mathsf{P}}$ is imbeddable if and only if $\Lambda\leqslant \lambda <0$. As a consequence of Theorem \[main\], the exact value of $\Lambda$ is that $$\hspace{-4mm} \Lambda=\left\{
\begin{array}{ll}
-\exp{\left\{-\frac{\pi}{\sqrt{b}}\right\}}, &\mbox{when\,} \frac{1}{\mu_1},\frac{1}{\mu_2},\frac{1}{\mu_3} \mbox{\,satisfy the triangle inequality,\, }\\
-\exp{\left\{-\sqrt{\frac{1-m}{m}}\,\pi\right\}}, &\mbox{otherwise},
\end{array}
\right.$$ where $b=\frac{4\mu_1\mu_2\mu_3}{(\mu_1\mu_2+\mu_1\mu_3+\mu_2\mu_3)^2}-1,\, m=\min{\left\{\mu_1,\,\mu_2,\,\mu_3\right\}}$.
Proof of the propositions {#prf}
-------------------------
\[cor0\] If ${\mathsf{P}}$ satisfies Eq.(1), then its right eigenvectors with eigenvalue $\lambda$ span the orthogonal complement of $\mu$.
If $f$ is a right eigenvector of ${\mathsf{P}}$ with eigenvalue $\lambda$ then $$\lambda f= {\mathsf{P}} f = \vec{e} \mu' f +\lambda ({\mathsf{I}}-\vec{e} \mu') f =(1-\lambda)\vec{e} \mu' f + \lambda f,$$ so that $\mu' f=0$. Hence the eigenvectors span the orthogonal complement of $\mu$.
[*Proof of Proposition \[prop3\].*]{} The necessity. Since $\lambda<0$ and ${\mathsf{P}}=e^{h{\mathsf{Q}}}$, ${\mathsf{Q}}$ has complex eigenvalues $\theta$ and $\bar{\theta}$ such that $e^{\theta h}=\lambda$. It follows from Lemma \[lem0\] that ${\mathsf{Q}}$ has $\mu' {\mathsf{Q}}=0$.
The sufficiency. Since ${\mathsf{P}}$ satisfies Eq.(\[PP\]), one obtains that $$\label{f-mat}
{\mathsf{P}}={\mathsf{F}}\,{\mathrm{diag}}{\left\{1,\,\lambda,\,\lambda\right\}}{\mathsf{F}}^{-1},$$ where ${\mathsf{F}}=[\vec{e},\,\varphi_1,\,\varphi_2]$, $\varphi_1,\,\varphi_2$ are any two linear independent vectors in the orthogonal complement of $\mu$ by Lemma \[cor0\].
Denote by $f+{\mathrm{i}}g$ the eigenvector of ${\mathsf{Q}}$ with eigenvalue $\theta=p+{\mathrm{i}}q,\,q\neq 0$. Clearly $f,\,g$ are linear independent. $\mu'{\mathsf{Q}}=0$ implies that $f,\,g$ span the orthogonal complement of $\mu$.
Let ${\mathsf{P}}_h=e^{h{\mathsf{Q}}}$. Hence $e^{h{\mathsf{Q}}}=\sum_{n=0}^{\infty}\frac{h^n}{n!}{\mathsf{Q}}^n$ implies that if $\mu'{\mathsf{Q}}=0 $ then $\mu'{\mathsf{P}}_h=\mu'$ and that $${\mathsf{P}}_h(f+{\mathrm{i}}g)=e^{\theta h}(f+{\mathrm{i}}g)={\lambda}(f+{\mathrm{i}}g).$$ Hence ${\mathsf{P}}_h f={\lambda}f,\, {\mathsf{P}}_h g={\lambda}g$. Any transition rate matrix ${\mathsf{Q}}$ has ${\mathsf{Q}}\vec{e}=0$, so that ${\mathsf{P}}_h \vec{e}=\vec{e}$. Thus one obtains that $$\label{f2-mat}
{\mathsf{P}}_h=[\vec{e},\,f,\,g]\,{\mathrm{diag}}{\left\{1,\,\lambda,\,\lambda\right\}}[\vec{e},\,f,\,g]^{-1}.$$ Note that in Eq.(\[f-mat\]), one can choose that the matrix ${\mathsf{F}}=[\vec{e},\,f,\,g]$, which implies that ${\mathsf{P}}_h={\mathsf{P}}$.
\[lemchen\] Suppose that $f(x)=x+\frac{a}{x}$ with $0<x\leqslant c$, where $a,\,c>0$ are two constants. Then the minimum of $f(x)$ is $$\label{minm}
\left\{
\begin{array}{ll}
2\sqrt{a}, &\mbox{\,when \,} \sqrt{a}\leqslant c ,\\
c+\frac{a}{c}, &\mbox{\, when\, } \sqrt{a}> c.
\end{array}
\right.$$ The minimum is attained at the point $$\label{min1}
\left\{
\begin{array}{ll}
x=\sqrt{a}, &\mbox{\, when\, } \sqrt{a}\leqslant c ,\\
x=c, &\mbox{\, when\, } \sqrt{a}> c.
\end{array}
\right.$$
It is trivial.
Denote that $$\begin{aligned}
\mathcal{D}={\left\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:\, x_1,\,x_2,\,x_3\geqslant0, x_1+x_2,\,x_2+x_3,\,x_3+x_1 >0\right\}},\\
\mathcal{E}_1=\mathcal{D} \cap {\left\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:\, x_1\leqslant x_2\leqslant x_3\right\}}.\end{aligned}$$ Since $F(x_1,x_2,x_3)=F(rx_1,rx_2,rx_3),\,\forall r>0$, the existence of the maximum of $F(x_1,x_2,x_3)$ on $\mathcal{D}$ is equal to the existence on the unit sphere which holds true since the unit sphere is compact.[^5]
\[lem11\] Restricted on $\mathcal{E}_1$, the maximum of $F(x_1,x_2,x_3)$ is $$\label{maximum1}
\left\{
\begin{array}{ll}
\frac{\mu_1\mu_2\mu_3}{4(1-\mu_1 )}, &\mbox{\,when \,} \frac{1}{\mu_1}\geqslant \frac{1}{\mu_2}+\frac{1}{\mu_3} ,\\
\frac{1}{(\frac{1}{\mu_1}+\frac{1}{\mu_2}+\frac{1}{\mu_3})^2}, &\mbox{\, when\, } \frac{1}{\mu_1}<\frac{1}{\mu_2}+\frac{1}{\mu_3}.
\end{array}
\right.$$ The maximum is attained at the points $$\label{maximumpoint1}
\left\{
\begin{array}{ll}
x_1=x_2=\frac{\frac{1}{\mu_2}+\frac{1}{\mu_3}}{\frac{2}{\mu_1}-\frac{1}{\mu_2}-\frac{1}{\mu_3}} x_3, &\mbox{\, when\, } \frac{1}{\mu_1}\geqslant\frac{1}{\mu_2}+\frac{1}{\mu_3} ,\\
x_1=x_2=x_3, &\mbox{\, when\, } \frac{1}{\mu_1}<\frac{1}{\mu_2}+\frac{1}{\mu_3}.
\end{array}
\right.$$
Restricted on $\mathcal{E}_1$, i.e., $x_1\leqslant x_2\leqslant x_3$, we obtain that $$F(x_1,x_2,x_3)=\frac{x_1x_2+x_2x_3+x_3x_1+x_1^2}{(\frac{x_1+x_2}{\mu_1}+
\frac{x_2+x_3}{\mu_2}+\frac{x_3+x_1}{\mu_3})^2}=\frac{(x_1+x_2)(x_3+x_1)}{(\frac{x_1+x_2}{\mu_1}+
\frac{x_2+x_3}{\mu_2}+\frac{x_3+x_1}{\mu_3})^2}.$$ Let $r=x_1+x_2,\,s=x_2+x_3,\,t=x_3+x_1$. Then $0<r\leqslant t\leqslant s,$ and $$F(x_1,x_2,x_3)=\frac{rt}{(\frac{r}{\mu_1}+
\frac{s}{\mu_2}+\frac{t}{\mu_3})^2}\triangleq G(r,s,t)\leqslant G(r,t,t).$$ That is to say, $G(r,s,t)$ attains its maximum when $s=t$ since it is a decreasing function of $s$ . Let $w=\sqrt{\frac{r}{t}}$. Then $0<w\leqslant 1$ and we have that $$\begin{aligned}
G(r,t,t)&=&\frac{rt}{[\frac{1}{\mu_1}r+
(\frac{1}{\mu_2}+\frac{1}{\mu_3})t]^2}\nonumber \\
&=&\big[\,\frac{\mu_1}{w+\mu_1(\frac{1}{\mu_2}+\frac{1}{\mu_3})\frac{1}{w}}\,\big]^2 \triangleq L(w).\end{aligned}$$ It follows from Lemma \[lemchen\] that when $\mu_1(\frac{1}{\mu_2}+\frac{1}{\mu_3})>1$, the maximum of $L(w)$ is $\frac{1}{(\frac{1}{\mu_1}+\frac{1}{\mu_2}+\frac{1}{\mu_3})^2}$, and when $\mu_1(\frac{1}{\mu_2}+\frac{1}{\mu_3})\leqslant1$, the maximum of $L(w)$ is $\frac{\mu_1}{4(\frac{1}{\mu_2}+\frac{1}{\mu_3})}$. In addition, the direct computation yields the maximum points (\[maximumpoint1\]) from Lemma \[lemchen\].
[*Proof of Proposition \[lem5\].*]{} In order to prove Proposition 2.5 we first assume that $\mu_i$ satisfies the triangular inequality. There is no loss of generality in assuming that $x_1 \leqslant x_2 \leqslant x_3$ because if this is not the case, the arguments can be permuted to satisfy the restriction. This will leave the value of $F$ invariant if also $\mu_i$ are permuted correspondingly, but the triangular inequality condition is invariant to permutations of $\mu_i$. Thus from Lemma \[lem11\] we get that $F$ is bounded by the restricted maximum: $$F(x_1, x_2, x_3)\leqslant \frac{1}{(\frac{1}{\mu_1}+\frac{1}{\mu_2}+\frac{1}{\mu_3})^2}.$$ If, however, $\mu_i$ does not satisfy the triangular inequality, then, without loss of generality we can assume that $$\frac{1}{\mu_1}\geqslant \frac{1}{\mu_2}+\frac{1}{\mu_3}.$$ Now on the set $\mathcal{E}_1$ we can apply Lemma \[lem11\] and find that the function is bounded by the unrestricted maximum $$F(x_1, x_2, x_3)\leqslant \frac{\mu_1\mu_2\mu_3}{4(1-\mu_1 )}.$$ Now when $\frac{1}{\mu_1}\geqslant \frac{1}{\mu_2}+\frac{1}{\mu_3}$, we have $\mu_1\leqslant \frac{\mu_2\mu_3}{\mu_2+\mu_3}\leqslant\min(\mu_2,\,\mu_3)$, so that $\mu_1=m=\min(\mu_1,\,\mu_2,\,\mu_3)$ and $$F(x_1, x_2, x_3)\leqslant \frac{\mu_1\mu_2\mu_3}{4(1-\mu_1 )}\leqslant\frac{\mu_1\mu_2\mu_3}{4(1-m )}.$$
The imbedding problem for 3-order transition matrix with positive eigenvalues or complex eigenvalues
====================================================================================================
Suppose ${\mathsf{P}}$ be an indecomposable $3\times 3$ transition probability matrix with eigenvalues ${\left\{1,\,\lambda_1,\,\lambda_2\right\}}$. Then it satisfies that $$\label{eigen2}
{\mathsf{P}}^2-(\lambda_1+\lambda_2){\mathsf{P}}+\lambda_1\lambda_2{\mathsf{I}}=(\lambda_1-1)(\lambda_2-1){\mathsf{P}}^{\infty}.$$
It is exactly the Eq.(1.15) of [@jhson]. Johansen S. points out that the condition of the imbedding problem can be given in terms of ${\mathsf{P}}$ and ${\mathsf{P}}^{\infty}$ (or $\mu$) in [@jhson]. The following two propositions are the direct corollary of Proposition 1.2, Proposition 1.4 of [@jhson] and Eq.(\[eigen2\]).
Let ${\mathsf{P}}$ be an indecomposable $3\times 3$ transition probability matrix with positive eigenvalues ${\left\{1,\,\lambda_1,\,\lambda_2\right\}}$. ${\mathsf{P}}$ can be imbedded if and only if $$\label{positive}
{p_{ij}} \geqslant {\mu_j} \frac{(\lambda_2-1)\log \lambda_1-(\lambda_1-1)\log \lambda_2}{\log \lambda_2-\log \lambda_1},\quad i\neq j.$$ If $\lambda_1=\lambda_2=\lambda$, then the right hand side of (\[positive\]) is in the sense of limit, i.e., fix the value of $\lambda_1$, and let $\lambda_2\rightarrow \lambda_1$, one has that $${p_{ij}} \geqslant {\mu_j}( \lambda\log \lambda-\lambda +1 ),\quad i\neq j.$$
Let ${\mathsf{P}}$ be an indecomposable $3\times 3$ transition probability matrix with complex eigenvalues ${\left\{1,\,\lambda_1,\,\lambda_2\right\}}$, $\lambda_1=r e^{{\mathrm{i}}\theta}\,,\lambda_2=r e^{-{\mathrm{i}}\theta},\,\theta\in (0,\,\pi)$. ${\mathsf{P}}$ can be imbedded if and only if $${p_{ij}} \geqslant{\mu_j}( 1-r\cos \theta +\frac{\sin\theta}{\theta}\,r\log r),\quad i\neq j.$$ or $${p_{ij}} \leqslant{\mu_j}( 1-r\cos \theta +\frac{\sin\theta}{2\pi-\theta}\,r\log r),\quad i\neq j.$$
The two propositions appear implicitly in [@jhson] and seem neater than Eq.(1.11-1.13,1.17) of Reference [@jhson].
If ${\mathsf{P}}$ is reversible, then $\frac{p_{ij}}{\mu_j} =\frac{p_{ji}}{\mu_i} $. One should only test half number of the inequalities.
Conclusion
==========
In the present paper, we solve the imbedding problem for 3-order transition matrix with coinciding negative eigenvalues by means of an alternate parameterization of the transition rate matrix, which is different from the traditional way to calculate the matrix logarithm or the matrix square root.
Acknowledgements {#acknowledgements .unnumbered}
================
Many thanks to the anonymous referee for the helpful comments and suggestions leading to the improvement of the paper. Thanks to Jianmin Chen, one of my undergraduate students, for showing me the proof of the key Lemma \[lem11\]. This work is supported by Hunan Provincial Natural Science Foundation of China (No 10JJ6014).
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Chen Y., On the monotonicity of fluctuation spectra for three-state Markov processes, Fluctuation and Noise Letters, 7 (3) L181-192, (2007)
Chung K. L., Markov chains with stationary Transition Probabiities, Second Edition, Springer-Verlag, Berlin, 1976
Elfving G., Zur Theorie der Markoffschen Ketten, Acta Soc. Sci. Finn. A, 2, 1-17. (1937)
Frydman H., Singer B., Total positivity and the embedding problem for Markov chains, Math. Proc. Camb. Phil. Soc. 86, 339-344. (1979)
Frydman H., The embedding problem for Markov chains with 3 states, The Mathematical Proceedings of the Cambridge Philosophical Society, 87, 285-294. (1980)
Frydman H., A structure of the bang-bang representation for $3\times3$ embeddable matrices, Z. Wahrscheinlichkeitstheorie Verw. Geb., 53, 305-316. (1980)
Fuglede B., On the Imbedding Problem for Stochastic and Doubly Stochastic Matrices, Probab. Th. Rel. Fields 80, 241-260. (1988) Goodman G., An intrinsic time for non-stationary finite Markov chains. Z. Wahrscheinlichkeitstheorie Verw. Geb., 16, 165-180. (1970)
Israel R., Rosenthal J., Wei J., Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings, Math. Finance 11, 245-265. (2001) Jiang D.-Q., Qian M., Qian M.-P., Mathematical Theory of Nonequilibrium Steady States, Springer, Berlin, 2004
Johansen S., Some Results on the Imbedding Problem for Finite Markov Chains, J. Lond. Math. Soc., 8, 345-351. (1974)
Johansen S., Ramsey F. L., A bang-bang representation for 3x3 embeddable stochastic matrices. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 107-118. (1979)
Kalpazidou S., Cycle representations of Markov processes. Second edition. Springer, New York, 2006
Kingman J. F. C., The imbedding problem for finite Markov chains, Z. Wahrscheinlichkeitstheorie Verw. Geb. 1, 14-24. (1962)
McCausland W. J., Time reversibility of stationary regular finite-state Markov chains, Journal of Econometrics 136, 303-318. (2007)
Metzner P., Dittmer E., Jahnke T., Schütte C, Generator estimation of Markov jump processes, Journal of Computational Physics, Volume 227, Issue 1, 10 November 2007, 353-375. (2007)
Qian M.-P., Qian M., The decomplsition into a detailed blanced part and a circulation part of an irreversible stationary Markov chain, Scientia Sinica, Special Issue(II), 69-79.(1979)
Runnenberg J. T., “On Elfving’s Problem of Imbedding a Time-discrete Markov Chain in a Continuous Time One for Finitely Many States." Proceedings, Koninklijke Nederlandse Akademie van Wetenschappen, ser. A, Mathematical Sciences, 65(5), 536-541. (1962)
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[^1]: The existence of the lower bound is shown in [@cart Theorem 3.7]
[^2]: This conclusion is asserted in [@jhson].
[^3]: This re-parameterization of the transition rate matrix appears in [@cy], and appears in [@qianq; @jdq; @Kalpa; @McM] implicitly, which is named the cycle decomposition by some authors.
[^4]: It is needed that the term inside the $\sqrt{\cdot}$ of Eq.(\[H-func\]) is positive.
[^5]: The points $(1,0,0),\,(0,1,0),\,(0,0,1)$ are not in $\mathcal{D}$, but the values of $F(x_1,x_2,x_3)$ are zero at these points, which do not alter the maximum of $F(x_1,x_2,x_3)$.
|
---
abstract: 'We construct a natural smooth compactification of the space of smooth genus-one curves with $k$ distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, i.e. the space of stable genus-zero maps $\ov\M_{0,k}(\Pn,d)$. In fact, our compactification is obtained from the singular space of stable genus-one maps $\ov\M_{1,k}(\Pn,d)$ through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component $\ov\M_{1,k}^0(\Pn,d)$ of $\ov\M_{1,k}(\Pn,d)$. A number of applications of these desingularizations in enumerative geometry and Gromov-Witten theory are described in the introduction.'
author:
- 'Ravi Vakil[^1] and Aleksey Zinger[^2]'
title: 'A Desingularization of the Main Component ofthe Moduli Space of Genus-One Stable Maps into $\Pn$'
---
Introduction {#intro_sec}
============
Background and Applications {#background_subs}
---------------------------
The space of degree-$d$ genus-$g$ curves with $k$ distinct marked points in $\Pn$ is generally not compact, but admits a number of natural compactifications[^3]. Among the most prominent compactifications is the moduli space of stable genus-$g$ maps, $\ov\M_{g,k}(\Pn,d)$, constructed in [@Gr] and [@FuP]. It has found numerous applications in classical enumerative geometry and is a central object in Gromov-Witten theory. However, most applications in enumerative geometry and some results in GW-theory have been restricted to the genus-zero case. The reason for this is essentially that the genus-zero moduli space has a particularly simple structure: it is smooth and contains the space of smooth genus-zero curves as a dense open subset. On the other hand, the moduli spaces of positive-genus stable maps fail to satisfy either of these two properties. In fact, $\ov\M_{g,k}(\Pn,d)$ can be arbitrarily singular according to [@murphy]. It is thus natural to ask whether these failings can be remedied by modifying $\ov\M_{g,k}(\Pn,d)$, preferably in a way that leads to a range of applications. As announced in [@summ] and shown in this paper, the answer is yes if $g\!=\!1$.\
We denote by $\M_{1,k}(\Pn,d)$ the subset of $\ov\M_{1,k}(\Pn,d)$ consisting of the stable maps that have smooth domains. This space is smooth and contains the space of genus-one curves with $k$ distinct marked points in $\Pn$ as a dense open subset, provided $d\!\ge\!3$. However, $\M_{1,k}(\Pn,d)$ is not compact. Let $\ov\M_{1,k}^0(\Pn,d)$ be the closure of $\M_{1,k}(\Pn,d)$ in the compact space $\ov\M_{1,k}(\Pn,d)$. While $\ov\M_{1,k}^0(\Pn,d)$ is not smooth, it turns out that a natural sequence of blowups along loci disjoint from $\M_{1,k}(\Pn,d)$ leads to a desingularization of $\ov\M_{1,k}^0(\Pn,d)$, which will be denoted by $\wt\M_{1,k}^0(\Pn,d)$.\
The situation is as good as one could possibly hope. A general strategy when attempting to desingularize some space is to blow up the “most degenerate” locus, then the proper transform of the “next most degenerate locus”, and so on. This strategy works here, but with a novel twist: we apply it to the entire space of stable maps $\ov\M_{1,k}(\Pn,d)$. The most degenerate locus is in fact an entire irreducible component, and blowing it up removes it[^4]. Hence one by one we erase the “bad” components of $\ov\M_{1,k}(\Pn,d)$. Each blowup of course changes the “good” component $\ov\M_{1,k}^0(\Pn,d)$, and miraculously at the end of the process the resulting space $\wt\M_{1,k}^0(\Pn,d)$ is nonsingular. We note that this cannot possibly be true for an arbitrary $g$, as $\M_{g,k}(\Pn,d)$ behaves quite badly according to [@murphy]. The sequential blowup construction itself is beautifully simple. It is completely described in the part of Subsection \[descr\_subs\] ending with the main theorem of the paper, Theorem \[main\_thm\]. However, showing that $\wt\M_{1,k}^0(\Pn,d)$ is in fact smooth requires a considerable amount of preparation (which takes up Subsections \[curveprelim\_subs\]-\[map1prelim\_subs\]) and is finally completed in Subsection \[map1blconstr\_subs\].\
The desingularization $\wt\M_{1,k}^0(\Pn,d)$ of $\ov\M_{1,k}^0(\Pn,d)$ possesses a number of “good” properties and has a variety of applications to enumerative algebraic geometry and Gromov-Witten theory. It has already been observed in [@Fo] that the cohomology of $\wt\M_{1,k}^0(\Pn,d)$ behaves in a certain respect like the cohomology of the moduli space of genus-one curves, $\ov\cM_{1,k}$. The space $\wt\M_{1,k}^0(\Pn,d)$ can be used to count genus-one curves in $\Pn$, mimicking the genus-zero results of [@KM] and [@RT] (though perhaps not their simple recursive formulas). Proceeding analogously to the genus-zero case (e.g. as in [@P], [@enumtang], and [@genuss0pr]), Theorem \[main\_thm\] can then be used to count genus-one curves with tangency conditions and singularities. In all cases, such counts can be expressed as integrals of natural cohomology classes on $\ov\M_{1,k}^0(\Pn,d)$ or $\wt\M_{1,k}^0(\Pn,d)$. Integrals on the latter space can be computed using the localization theorem of [@ABo], as $\wt\M_{1,k}^0(\Pn,d)$ is smooth and inherits a torus action from $\Pn$ and $\ov\M_{1,k}(\Pn,d)$.\
We next discuss two types of applications of Theorem \[main\_thm\] in Gromov-Witten theory, as well as a bonus result of this paper, Theorem \[cone\_thm\]. It is shown in [@g1comp] and [@g1comp2] that the space $\ov\M_{1,k}^0(\Pn,d)$ has a natural generalization to arbitrary almost Kahler manifolds and gives rise to new symplectic [*reduced genus-one GW*]{}-invariants. These reduced invariants are yet to be constructed in algebraic geometry. However, the spaces $\wt\M_{1,k}^0(\Pn,d)$ do possess a number of “good” properties and give rise to algebraic invariants of algebraic manifolds; see the first and last sections of [@summ]. It is not clear whether these are the same as the reduced genus-one invariants, but it may be possible to verify this by using Theorem \[cone\_thm\].\
Theorem \[main\_thm\] also has applications to computing Gromov-Witten invariants of complete intersections, once it is combined with Theorem \[cone\_thm\]. Let $a$ be a nonnegative integer. For a general $s\!\in\!H^0(\Pn,\O_{\Pn}(a))$, $$Y\equiv s^{-1}(0) \subset \Pn$$ is a smooth hypersurface. We denote its degree-$d$ Gromov-Witten invariant by $\GW_{g,k}^Y(d;\cdot)$, i.e. $$\GW_{g,k}^Y(d;\psi)\equiv
\blr{\psi,\big[\ov\M_{g,k}(Y,d)\big]^{vir}}
\qquad \text{for all} \qquad \, \psi\!\in\!H^*\big(\ov\M_{g,k}(Y,d);\Q\big).$$ Suppose $\U$ is the universal curve over $\ov\M_{g,k}(\Pn,d)$, with structure map $\pi$ and evaluation map $\ev$: $$\xymatrix{\U \ar[d]^{\pi} \ar[r]^{\ev} & \Pn \\
\ov\M_{g,k}(\Pn,d).}$$ It can be shown that $$\label{genus0_e}
\GW_{0,k}^Y(d;\psi)
=\blr{\psi\cdot e\big(\pi_*\ev^* \O_{\Pn}(a)\big), \big[\ov\M_{0,k}(\Pn,d)\big]}$$ for all $\psi\!\in\!H^*(\ov\M_{0,k}(\Pn,d);\Q)$; see [@Bea] for example. The moduli space $\ov\M_{0,k}(\Pn,d)$ is a smooth orbifold and $$\pi_*\ev^* \O_{\Pn}(a) \lra\ov\M_{0,k}(\Pn,d)$$ is a locally free sheaf, i.e. a vector bundle. The right-hand side of \_ref[genus0\_e]{} can be computed via the classical localization theorem of [@ABo]. The complexity of this computation increases quickly with the degree $d$, but it has been completed in full generality in a number of different of ways; see [@Ber], [@Ga], [@Gi], [@Le], and [@LLY].\
If $n\!=\!4$, so $Y$ is a threefold, then $$\label{genus1_e}
\GW_{1,k}^Y(d;\psi)=
\frac{d(a\!-\!5)\!+\!2}{24}\GW_{0,k}^Y(d;\psi)
+\blr{\psi\cdot e(\pi_*\ev^*\O_{\Pn}(a)),\big[\ov\M_{1,k}^0(\Pf,d)\big]}$$ for all $\psi\!\in\!H^*(\ov\M_{1,k}(\Pf,d);\Q)$; see [@LZ Crl. \[g1gw-cy\_crl\]]. This decomposition generalizes to arbitrary complete intersections $Y$ and perhaps even to higher-genus invariants. The sheaf $$\label{sheaf_e}
\pi_*\ev^* \O_{\Pn}(a) \lra\ov\M_{1,k}^0(\Pf,d)$$ is not locally free. Nevertheless, its euler class is well-defined: the euler class of every desingularization of this sheaf is the same, in the sense of [@g1cone Subsect. \[g1cone-appr\_subs\]]. This euler class can be geometrically interpreted as the zero set of a sufficiently good section of the cone $$\V_{1,k}^d\lra\ov\M_{1,k}^0(\Pf,d),$$ naturally associated to the sheaf \_ref[sheaf\_e]{}[^5]; see the second part of the next subsection and Lemma \[conesheaf\_lmm\].\
One would hope to compute the last expression in \_ref[genus1\_e]{} by localization. However, since the variety $\ov\M_{1,k}^0(\Pf,d)$ and the cone $\V_{1,k}^d$ are singular, the localization theorem of [@ABo] is not immediately applicable in the given situation. Let $$\ti\pi\!: \wt\M_{1,k}^0(\Pf,d)\lra\ov\M_{1,k}^0(\Pf,d)$$ be the projection map. As a straightforward extension of the main desingularization construction of this paper, we show that the cone $$\ti\pi^*\V_{1,k}^d\lra\wt\M_{1,k}^0(\Pf,d)$$ contains a vector bundle $$\wt\V_{1,k}^d\lra\wt\M_{1,k}^0(\Pf,d)$$ of rank $da=\rk\,\V_{1,k}^d|_{\M_{1,k}^0(\Pf,d)}$; see Theorem \[cone\_thm\]. It then follows that $$\label{euler_e}\begin{split}
\blr{\psi\cdot e\big(\pi_*\ev^*\O_{\Pn}(a)\big),
\big[\ov\M_{1,k}^0(\Pf,d)\big]}
&\equiv \blr{\psi\cdot e\big(\V_{1,k}^d),\big[\ov\M_{1,k}^0(\Pf,d)\big]}\\
&=\blr{\ti\pi^*\psi\cdot e(\wt\V_{1,k}^d),\big[\wt\M_{1,k}^0(\Pf,d)\big]}.
\end{split}$$ The last expression above is computable by localization.\
[*Remark:*]{} Another approach to computing positive-genus Gromov-Witten invariants has been proposed in [@MaP]. In contrast to the approach of [@LZ], it applies to arbitrary-genus invariants, but can at present be used to compute invariants of only low-dimensional and/or low-degree complete intersections.\
The main desingularization construction of this paper is the subject of Section \[map1bl\_sec\], but its key aspects are presented in the next subsection. The construction itself and its connections with Sections \[curvebl\_sec\] and \[map0bl\_sec\] are outlined in Subsection \[outline\_subs\]. We suggest that the reader return to Subsections \[descr\_subs\] and \[outline\_subs\] before going through the technical details of the blowup constructions in Sections \[curvebl\_sec\]-\[map1bl\_sec\]. In the next subsection, we also describe a natural sheaf over $\wt\M_{1,k}^0(\Pn,d)$ which is closely related to the sheaf $\pi_*\ev^* \O_{\Pn}(a)$ over $\ov\M_{1,k}^0(\Pn,d)$. It is shown to be locally free in Section \[cone\_sec\]. Finally, all the data necessary for applying the localization theorem of [@ABo] to $\wt\M_{1,k}^0(\Pn,d)$ and $e(\wt\V_{1,k}^d)$ is given in Subsection \[local\_subs\].\
Through this article we work with Deligne-Mumford stacks. They can also be thought of as analytic orbivarieties. As we work with reduced scheme structures throughout the paper, we will call such objects simply varieties. Also, all immersions will be assumed to be from smooth varieties. (The notion of “immersion” is often called “unramified” in algebraic geometry.)\
The authors would like to thank Jun Li for many enlightening discussions.
Description of the Desingularization {#descr_subs}
------------------------------------
The moduli space $\ov\M_{1,k}(\Pn,d)$ has irreducible components of various dimensions. One of these components is $\ov\M_{1,k}^0(\Pn,d)$, the closure of the stratum $\M_{1,k}^0(\Pn,d)$ of stable maps with smooth domains. We now describe natural subvarieties of $\ov\M_{1,k}(\Pn,d)$[^6] which contain the remaining components of $\ov\M_{1,k}(\Pn,d)$. They will be indexed by the set $$\begin{gathered}
\A_1(d,k) \equiv\big\{\si\!=\!(m;J_P,J_B)\!: m\!\in\!\Z^+,\, m\!\le\!d;~
[k]\!=\!J_P\!\sqcup\!J_B\big\},\\
\hbox{where}\qquad
[k]=\{1,\ldots,k\}.\end{gathered}$$ For each $\si\!\in\!\A_1(d,k)$, let $\M_{1,\si}(\Pn,d)$ be the subset of $\ov\M_{1,k}(\Pn,d)$ consisting of the stable maps $[\cC,u]$ such that $\cC$ is a smooth genus-one curve $E$ with $m$ smooth rational components attached directly to $E$, $u|_E$ is constant, the restriction of $u$ to each rational component is non-constant, and the marked points on $E$ are indexed by the set $J_P$. Here $P$ stands for “principal component”, $B$ stands for “bubble component”, and $\A$ stands for “admissible set”. Figure \[m3\_fig1\] shows the domain of an element of $\M_{1,\si}(\Pn,d)$, where $\si\!=\!(3;\{2\},\{1\})$, from the points of view of symplectic topology and of algebraic geometry. In the first diagram, each shaded disc represents a sphere; the integer next to each rational component $\cC_i$ indicates the degree of $u|_{\cC_i}$. In the second diagram, the components of $\cC$ are represented by curves, and the pair of integers next to each component $\cC_i$ shows the genus of $\cC_i$ and the degree of $u|_{\cC_i}$. In both diagrams, the marked points are labeled in bold face. Let $\ov\M_{1,\si}(\Pn,d)$ be the closure of $\M_{1,\si}(\Pn,d)$ in $\ov\M_{1,k}(\Pn,d)$. The space $\ov\M_{1,\si}(\Pn,d)$ has a number of irreducible components. These components are indexed by the splittings of the degree $d$ into $m$ positive integers and of the set $J_B$ into $m$ subsets. However, we do not need to distinguish these components.\
It is straightforward to check that $$\ov\M_{1,k}(\Pn,d) =\ov\M^0_{1,k}(\Pn,d)\cup
\bigcup_{\si\in\A_1(d,k)}\!\!\!\!\!\ov\M_{1,\si}(\Pn,d).$$ Dimensional considerations imply that if $\si\!=\!(m;J_P,J_B)\!\in\!A_1(d,k)$ and $m\!\le\!n$, then $\ov\M_{1,\si}(\Pn,d)$ is a union of components of $\ov\M_{1,k}(\Pn,d)$. The converse holds as well: $\ov\M_{1,\si}(\Pn,d)$ is contained in $\ov\M_{1,k}^0(\Pn,d)$ if $m\!>\!n$ by [@g1comp Theorem \[g1comp-str\_thm\]]. However, we will use the entire collection $\A_1(d,k)$ of subvarieties of $\ov\M_{1,k}(\Pn,d)$ to construct $\wt\M_{1,k}^0(\Pn,d)$. The independence of the indexing set $\A_1(d,k)$ of $n$ leads to a number of good properties being satisfied by our blowup construction; see (2) of Theorem \[main\_thm\] and the second part of this subsection. It may also be possible to use this construction to define reduced genus-one GW-invariants in algebraic geometry; this is achieved in symplectic topology in [@g1comp2].\
(-1.1,-1.8)(10,1.25) (0,-4)[(5,-1.5)(2.5,1.5)(7.5,-1.5)[.2]{} (5,-3.3)[2]{}[60]{}[120]{}(5,0.3)[2]{}[240]{}[300]{} (5,-4)[1]{}(5,-3)[.2]{}(4,-4)[.2]{} (6.83,.65)[1]{}(6.44,-.28)[.2]{} (3.17,.65)[1]{}(3.56,-.28)[.2]{}]{} (7,.4)(5.2,-4.2) (.2,-.9)[$d_1$]{}(3.1,2.3)[$d_2$]{}(7.8,-2.5)[$d_3$]{} (15,-1)[3]{}[-60]{}[60]{}(16.93,1.3)[.2]{} (17.6,1.4) (17,-1)(22,-1)(16.8,-2)(21,-3)(16.8,0)(21,1) (15.2,-3.5)(22.4,1) (23.4,-1)(22.4,-3) (18.9,-2.5)[.2]{}(18.6,-3) (33,-1)
We define a partial ordering $\prec$ on the set $\A_1(d,k)$ by $$\label{map1ord_e}
\si'\!\equiv\!\big(m';J_P',J_B'\big)\prec \si\!\equiv\!\big(m;J_P,J_B\big)
\qquad\hbox{if}\quad
\si'\!\neq\!\si,~~~ m'\!\le\!m, ~~\hbox{and}~~ J_P'\!\subseteq\!J_P.$$ This relation is illustrated in Figure \[map1ord\_fig\], where an element $\si$ of $\A_1(d,k)$ is represented by an element of the corresponding space $\M_{1,\si}(\Pn,d)$. We indicate that the degree of the stable map on every bubble component is positive by shading the disks in the figure. We show only the marked points lying on the principal component. The exact distribution of the remaining marked points between the components is irrelevant.\
(-1.1,-1.8)(10,1.25) (0,-4)[(5,-1.5)(2.5,1.5) (5,-3.3)[2]{}[60]{}[120]{}(5,0.3)[2]{}[240]{}[300]{} (5,-4)[1]{}(5,-3)[.2]{} (6.83,.65)[1]{}(6.44,-.28)[.2]{}]{} (11.5,-1.5)
$\prec$
(15,-4)[(5,-1.5)(2.5,1.5)(7.5,-1.5)[.2]{} (5,-3.3)[2]{}[60]{}[120]{}(5,0.3)[2]{}[240]{}[300]{} (5,-4)[1]{}(5,-3)[.2]{} (3.17,.65)[1]{}(3.56,-.28)[.2]{}]{} (22,.4) (25,-4)[(5,-1.5)(2.5,1.5) (5,-3.3)[2]{}[60]{}[120]{}(5,0.3)[2]{}[240]{}[300]{} (5,-4)[1]{}(5,-3)[.2]{} (6.83,.65)[1]{}(6.44,-.28)[.2]{} (3.17,.65)[1]{}(3.56,-.28)[.2]{}]{}
Choose an ordering $<$ on $\A_1(d,k)$ extending the partial ordering $\prec$. The desingularization $$\label{desing_e2}
\ti\pi\!: \wt\M_{1,k}^0(\Pn,d)\lra\ov\M_{1,k}^0(\Pn,d)$$ is constructed by blowing up $\ov\M_{1,k}(\Pn,d)$ along the subvarieties $\ov\M_{1,\si}(\Pn,d)$ and their proper transforms in the order specified by $<$. In other words, we first blow up $\ov\M_{1,k}(\Pn,d)$ along $\ov\M_{1,\si_{\min}}(\Pn,d)$, where $$\si_{\min}\equiv(1;\eset,[k])$$ is the smallest element of $\A_1(d,k)$. We then blow up the resulting space along the proper transform of $\ov\M_{1,\si_2}(\Pn,d)$, where $\si_2$ is the smallest element of $\A_1(d,k)\!-\!\{\si_{\min}\}$. We continue this procedure until we blow up along the proper transform of $\ov\M_{1,\si_{\max}}(\Pn,d)$, where $$\si_{\max}=(d;[k],\eset)$$ is the largest element of $\A_1(d,k)$. The variety resulting from this last blowup is the proper transform $\wt\M_{1,k}^0(\Pn,d)$ of $\ov\M_{1,k}^0(\Pn,d)$, as all other irreducible components of $\ov\M_{1,k}(\Pn,d)$ have been “blown out of existence”.\
The first interesting case of this construction, i.e. for $\ov\M_{1,0}^0(\P^2,3)$, is described in detail in [@summ]. The space $\wt\M_{1,0}^0(\P^2;3)$ is a smooth compactification of the space of smooth plane cubics. It has a richer structure than the naive compactification, $\P^9$, does.
\[main\_thm\] Suppose $n,d\!\in\!\Z^+$, $k\!\in\!\bar\Z^+$, $<$ is an ordering on the set $\A_1(d,k)$ extending the partial ordering $\prec$, and $$\ti\pi\!: \wt\M_{1,k}^0(\Pn,d)\lra\ov\M_{1,k}^0(\Pn,d)$$ is the blowup of $\ov\M_{1,k}^0(\Pn,d)$ obtained by blowing up $\ov\M_{1,k}(\Pn,d)$ along the subvarieties $\ov\M_{1,\si}(\Pn,d)$ and their proper transforms in the order specified by $<$.\
(1) The variety $\wt\M_{1,k}^0(\Pn,d)$ is smooth and is independent of the choice of ordering $<$ extending $\prec$.\
(2) For all $m\!\le\!n$, the embedding $\ov\M_{1,k}^0(\P^m,d)\!\lra\!\ov\M_{1,k}^0(\Pn,d)$ lifts to an embedding $$\wt\M_{1,k}^0(\P^m,d)\lra\wt\M_{1,k}^0(\Pn,d)$$ and the image of the latter embedding is the preimage of $\ov\M_{1,k}^0(\P^m,d)$ under $\ti\pi$.\
(3) The blowup locus at every step of the blowup construction is a smooth subvariety in the corresponding blowup of $\ov\M_{1,k}(\Pn,d)$.\
(4) All fibers of $\ti\pi$ are connected.
[*Remark:*]{} While in Section \[map1bl\_sec\] we analyze the blowup construction starting with the reduced scheme structure on $\ov\M_{1,k}(\Pn,d)$, Theorem \[main\_thm\] applies to the standard scheme structure on $\ov\M_{1,k}(\Pn,d)$ as well. It is known that $\M_{1,k}^0(\Pn,d)$ is a smooth stack (as such maps are unobstructed, see for example [@enumtang Prop. 5.5(c)]). Thus, its scheme-theoretic closure, $\ov\M_{1,k}^0(\Pn,d)$, is reduced. During the blowup process all other components of $\ov\M_{1,k}(\Pn,d)$ are “blown out of existence”, as is any non-reduced scheme structure.\
In Theorem \[main\_thm\] and throughout the rest of the paper we denote by $\bar\Z^+$ the set of nonnegative integers. We analyze the sequential blowup construction of Theorem \[main\_thm\] in Section \[map1bl\_sec\] using the inductive assumptions ($I1$)-($I15$) of Subsection \[map1blconstr\_subs\]. One of these assumptions, ($I3$), implies the second part of the first statement of Theorem \[main\_thm\], as different choices of an ordering $<$ extending the partial ordering $\prec$ correspond to different orders of blowups along subvarieties that are disjoint. For example, suppose $$\si_{\eset}=\big(2;\eset,\{1,2\}\big), \qquad
\si_1=\big(2;\{1\},\{2\}\big), \quad\hbox{and}\quad
\si_2=\big(2;\{2\},\{1\}\big).$$ While $\ov\M_{1,\si_1}(\Pn,d)$ and $\ov\M_{1,\si_2}(\Pn,d)$ do intersect in $\ov\M_{1,2}(\Pn,d)$, their proper transforms are disjoint after the blowup along $\ov\M_{1,\si_{\eset}}(\Pn,d)$. The second statement of Theorem \[main\_thm\] follows immediately from the description of the blowup construction in this and the next subsections, as each step of the construction commutes with the embeddings of the moduli spaces induced by the embedding $\P^m\!\lra\!\Pn$.\
The main claim of this paper is that $\wt\M_{1,k}^0(\Pn,d)$ is a smooth variety. The structure of the space of maps from curves with no contracted genus-one components $$\M_{1,k}^{\eff}(\Pn,d) \equiv \ov\M_{1,k}(\Pn,d)
- \bigcup_{\si\in\A_1(d,k)}\!\!\!\!\! \ov\M_{1,\si}(\Pn,d)$$ is well understood; see [@H Chapter 27] for example. In particular, $\M_{1,k}^{\eff}(\Pn,d)$ is smooth. Below we describe the structure of the complement $\part\wt\M_{1,k}^0(\Pn,d)$ of $\M_{1,k}^{\eff}(\Pn,d)$ in $\wt\M_{1,k}^0(\Pn,d)$.\
If $J$ is a finite set and $g$ is a nonnegative integer, we denote by $\ov\cM_{g,J}$ the moduli space of stable genus-$g$ curves with $|J|$ marked points, which are indexed by the set $J$. Similarly, we denote by $\ov\M_{g,J}(\Pn,d)$ the moduli space of stable maps from genus-$g$ curves with marked points indexed by $J$ to $\Pn$. If $j\!\in\!J$, let $$\ev_j\!: \ov\M_{g,J}(\Pn,d)\lra\Pn$$ be the evaluation map at the marked point labeled by $j$.\
If $\si\!=\!(m;J_P,J_B)$ is an element of $\A_1(d,k)$, we define $$\begin{aligned}
{1}
\ov\cM_{\si;P} &= \ov\cM_{1,[m]\sqcup J_P} \qquad\hbox{and}\\
\ov\M_{\si;B}(\Pn,d)&=
\Big\{(b_1,\ldots,b_m)\in\prod_{i=1}^m\ov\M_{0,\{ 0 \}\sqcup J_i}(\Pn,d_i)\!:
d_i\!>\!0,~\sum_{i=1}^md_i\!=\!d;~ \bigsqcup_{i=1}^m J_i\!=\!J_B;\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~~
\ev_0(b_{i_1})\!=\!\ev_0(b_{i_2})~\forall\, i_1,i_2\!\in\![m]\Big\}.\end{aligned}$$ There is a natural node-identifying surjective immersion $$\io_{\si}\!: \ov\cM_{\si;P}\!\times\!\ov\M_{\si;B}(\Pn,d)
\lra \ov\M_{1,\si}(\Pn,d) \subset \ov\M_{1,k}(\Pn,d).$$ As before, $P$ denotes “principal component”, and $B$ denotes “bubble components”. This immersion descends to the quotient: $$\bar\io_{\si}\!: \big(\ov\cM_{\si;P}\!\times\!\ov\M_{\si;B}(\Pn,d)\big) \big/G_{\si}
\lra \ov\M_{1,\si}(\Pn,d),$$ where $G_{\si}\!\equiv\!S_m$ is the symmetric group on $m$ elements. If $m\!\ge\!3$, $\bar\io_{\si}$ is not an isomorphism as some subvarieties of the left side are identified. An example of a point on the right which is the image of two points on the left is given in Figure \[doublept\_fig\]. In addition to the conventions used in Figure \[m3\_fig1\], in the first, symplectic-topology, diagram of Figure \[doublept\_fig\] we leave the components of the domain on which the map is constant unshaded. The subvarieties identified by the map $\bar\io_{\si}$ get “unidentified” after taking the proper transform of $\ov\M_{1,\si}(\Pn,d)$ in the blowup of $\ov\M_{1,k}(\Pn,d)$ at the step corresponding to $$\ti\si=\max\big\{\si'\!\in\!\A_1(d,k)\!: \si'\!\prec\!\si\big\}.$$ This is insured by the inductive assumption ($I13$) in Subsection \[map1blconstr\_subs\] and implies the third statement of Theorem \[main\_thm\]. For example, if $m\!=\!3$ and $k\!=\!0$ as in Figure \[doublept\_fig\], the “identified” subvarieties are “unidentified” after the blowup of the proper transform of $\ov\M_{1,(2;\eset,\eset)}(\Pn,d)$.\
(-1.1,-1.8)(10,1.25) (5,-1.5)(1.5,2.5) (6.8,-1.5)[2]{}[150]{}[210]{}(3.2,-1.5)[2]{}[330]{}[30]{} (2.5,-1.5)[1]{}(3.5,-1.5)[.2]{} (7.5,-1.5)[1]{}(6.5,-1.5)[.2]{} (1.09,-.09)[1]{}(1.79,-.79)[.2]{} (1.09,-2.91)[1]{}(1.79,-2.21)[.2]{} (8.91,-.09)[1]{}(8.21,-.79)[.2]{} (8.91,-2.91)[1]{}(8.21,-2.21)[.2]{} (-.4,-2.5)[$d_1$]{}(-.4,.4)[$d_2$]{} (10.4,-2.5)[$d_3$]{}(10.4,.4)[$d_4$]{} (15,-1)[3]{}[-60]{}[60]{}(15.4,-4) (16.8,.4)(21,.4)(15.6,.4) (16.8,-2.4)(21,-2.4)(15.6,-2.4) (18.5,-.4)(20,2.5)(21.3,2.4) (19.5,.6)(22.5,-.4)(23.8,-.2) (19.5,-2.6)(22.5,-1.6)(23.8,-1.8) (18.5,-1.6)(20,-4.5)(21.3,-4.4) (32.5,-1)
[*Remark:*]{} Throughout the paper, we use $\M$ (fraktur font) to denote moduli spaces of stable [*maps*]{}, of genus-zero or one, into $\Pn$. We use $\cM$ (calligraphic font) to denote moduli spaces of stable [*curves*]{}.\
For each $i\!\in\![m]$, let $$\pi_i\!: \ov\M_{\si;B}(\Pn,d)\lra
\!\!\bigsqcup_{d_i>0,J_i\subset J_B}\!\!\!\!\!\!\ov\M_{0,\{0\}\sqcup J_i}(\Pn,d_i)$$ be the natural projection onto the $i$th component. We put $$F_{\si;B}=\bigoplus_{i=1}^m \pi_i^*L_0,$$ where $L_0\!\lra\!\ov\M_{0,\{0\}\sqcup J_i}(\Pn,d_i)$ is the universal tangent line bundle for the marked point $0$. In Subsections \[curve1bl\_subs\] and \[map0blconstr\_subs\], we construct blowups \[ravi1\] $$\begin{aligned}
{1}
\ti\pi_{\si;P}\!\equiv\!\pi_{1,([m],J_P)}
&:\wt\cM_{\si;P}\!\equiv\!\wt\cM_{1,([m],J_P)}
\lra \ov\cM_{\si;P}\!\equiv\!\ov\cM_{1,[m]\sqcup J_P}
\qquad\hbox{and}\\
\ti\pi_{\si;B}\!\equiv\!\pi_{0,([m],J_B)}
&:\wt\M_{\si;B}(\Pn,d)\!\equiv\!\wt\M_{0,([m],J_B)}(\Pn,d)
\lra \P F_{\si;B}\!\equiv\!\P F_{([m],J_B)}.\end{aligned}$$ We also construct a section $$\label{spacesec_e1}
\wt\cD_{\si;B}\!\equiv\!\wt\cD_{([m],J_B)}
\in \Ga\big(\wt\M_{\si;B}(\Pn,d);
\E_{\si;B}^*\!\otimes\!\ti\pi_{\si;B}^*\pi_{\P F_{\si;B}}^*\ev_0^*T\Pn\big),$$ where $$\ev_0\!: \ov\M_{\si;B}(\Pn,d)\lra\Pn \qquad\hbox{and}\qquad
\pi_{\P F_{\si;B}}\!: \P F_{\si;B}\lra\ov\M_{\si;B}(\Pn,d)$$ are the natural evaluation map and the bundle projection map, respectively, and $$\E_{\si;B}\!\equiv\!\ti\E \lra
\wt\M_{\si;B}(\Pn,d)\!\equiv\!\wt\M_{0,([m],J_B)}(\Pn,d)$$ is a line bundle. This line bundle is the sum of the tautological line bundle $$\ga_{\si;B} \lra \P F_{\si;B}$$ and all exceptional divisors. The section $\wt\cD_{\si;B}$ is transverse to the zero section. Thus, its zero set, $$\label{spacedfn_e1}
\wt\cZ_{\si;B}(\Pn,d) \equiv \wt\cD_{\si;B}^{-1}(0)
\subset \wt\M_{\si;B}(\Pn,d),$$ is a smooth subvariety. The boundary $\part\wt\M_{1,k}^0(\Pn,d)$ of $\wt\M_{1,k}^0(\Pn,d)$ is a union of smooth divisors: $$\part\wt\M_{1,k}^0(\Pn,d)=\bigcup_{\si\in\A_1(d,k)}\!\!\!\!\!
\wt\cZ_{\si}(\Pn,d)\big/G_{\si},
\qquad\hbox{where}\quad
\wt\cZ_{\si}(\Pn,d)=\wt\cM_{\si;P}\!\times\!\wt\cZ_{\si;B}(\Pn,d);$$ see the inductive assumptions ($I7$) and ($I8$) in Subsection \[map1blconstr\_subs\] and Figure \[bdlift\_fig\]. By the inductive assumption ($I6$) and ($I7$), the normal bundle of $\wt\cZ_{\si}(\Pn,d)$ in $\wt\M_{1,k}(\Pn,d)$ is the quotient of the line bundle $$\L_{\si;P}\otimes \ti\pi_{\si;B}^*\ga_{\si;B}\lra
\wt\cM_{\si;P}\!\times\!\wt\cZ_{\si;B}(\Pn,d)$$ by the $G_{\si}$-action, where $$\L_{\si;P}\!\equiv\!\L \lra \wt\cM_{\si;P}\!\equiv\!\wt\cM_{1,([m],J_P)}$$ is the universal tangent line bundle constructed in Subsection \[curve1bl\_subs\]. Thus we conclude that $\wt\M_{1,k}^0(\Pn,d)$ is smooth, as the open subset $\M_{1,k}^{\eff}(\Pn,d)$ is smooth, and its complement is a union of smooth divisors whose normal sheaves are line bundles (i.e. with their reduced induced scheme structure, they are Cartier divisors).\
(-1.1,-1.8)(10,1.25) (10,0)[$\wt\cM_{\si;P}~\times~\wt\cZ_{\si;B}(\Pn,d)$]{} (25,0)[$\wt\M_{1,k}^0(\Pn,d)$]{} (10,-4)[$\ov\cM_{\si;P}~\times~\ov\M_{\si;B}(\Pn,d)$]{} (25,-4)[$\ov\M_{1,k}(\Pn,d)$]{} (16,0)(21,0)(16,-4)(21,-4) (6.5,-1)(6.5,-3)(12.5,-1)(12.5,-3)(25,-1)(25,-3) (18.5,.7)(18.5,-3.4) (7.6,-2)(13.6,-2) (25.5,-2)
[*Remark 1:*]{} In the Gromov-Witten theory, the symbol $\E$ is commonly used to denote the Hodge vector bundle of holomorphic differentials. It is the zero vector bundle in the genus-zero case. The line bundles over moduli spaces of genus-zero curves and maps we denote by $\E$, with various decorations, play roles analogous to that of the Hodge line bundle over moduli spaces of genus-one curves. The most overt parallel is described at the end of Subsection \[curvebldata\_subs\]. There are deeper, more subtle, connections as well; compare the structural descriptions of Lemmas \[deriv0str\_lmm\] and \[map1bl\_lmm2\], for example.\
[*Remark 2:*]{} Throughout this paper, the symbols $\cD$ and $\D$, with various decorations, denote vector bundle sections related to derivatives of holomorphic maps into $\Pn$ and of holomorphic bundle sections. In most cases, such bundle sections are viewed as vector bundle homomorphisms.\
The final claim of Theorem \[main\_thm\] follows from the fact that $\ov\M_{1,k}^0(\Pn,d)$ is unibranch (locally irreducible). If $\pi\!:Y\!\lra\!X$ is a surjective birational map of irreducible varieties, and $\pi^{-1}(x)$ is not connected for some $x\!\in\!X$, then $X$ is not unibranch at $x$.\
We next describe a desingularization of the sheaf $\pi_*\ev^*\O_{\Pn}(a)$ and of the corresponding cone $\V_{1,k}^d$ over $\ov\M_{1,k}^0(\Pn,d)$. Let $\wt\U\!=\!\ti\pi^*\U$ be the pullback of $\U$ by $\ti\pi$: $$\xymatrix{\wt\U \ar[d]^{\pi} \ar[r]^{\ti\pi}& \U \ar[d]^{\pi} \ar[r]^{\ev} & \Pn \\
\wt\M_{1,k}^0(\Pn,d) \ar[r]^{\ti\pi}& \ov\M_{1,k}^0(\Pn,d).}$$ For each $\si\!\in\!\A_1(d,k)$, let $$\V_{\si;B}\lra \ov\M_{\si;B}(\Pn,d)$$ be the cone induced by the sheaf $\O_{\Pn}(a)$, similarly to $\V_{g,k}^d$; see Subsection \[conehomomor\_subs\] for details. It is a vector bundle of rank $da\!+\!1$. We note that $$\ti\pi^*\V_{1,k}^d\big|_{\wt\cZ_{\si}(\Pn,d)}
=\pi_B^*\big\{\ti\pi_{\si;B}^*
\pi_{\P F_{\si;B}}^*\V_{\si;B}|_{\wt\cZ_{\si;B}(\Pn,d)}\big\}\big/G_{\si},$$ where $$\pi_B\!:\wt\cM_{\si;P}\!\times\!\wt\cZ_{\si;B}(\Pn,d)\lra
\wt\cZ_{\si;B}(\Pn,d)$$ is the projection map. Let $\cL\!=\!\ga^{*\otimes a}$, where $\ga\!\lra\!\Pn$ is the tautological line bundle.
\[cone\_thm\] Suppose $d,n,a\!\in\!\Z^+$ and $k\!\in\!\bar\Z^+$.\
(1) The sheaf $\pi_*\ti\pi^*\ev^*\O_{\Pn}(a)$ over $\wt\M_{1,k}^0(\Pn,d)$ is locally free and of the expected rank, i.e. $da$.\
(2) If $\wt\V_{1,k}^d\!\subset\!\ti\pi^*\V_{1,k}^d$ is the corresponding vector bundle and $\si\!\in\!\A_1(d,k)$, then there exists a surjective bundle homomorphism $$\wt\D_{\si;B}\!: \ti\pi_{\si;B}^*\pi_{\P F_{\si;B}}^*\V_{\si;B}|_{\wt\cZ_{\si;B}(\Pn,d)}
\lra \E_{\si;B}^*\!\otimes\!\ti\pi_{\si;B}^*\pi_{\P F_{\si;B}}^*\ev_0^*\cL$$ over $\wt\cZ_{\si;B}(\Pn,d)$ such that $$\wt\V_{1,k}^d\big|_{\wt\cZ_{\si}(\Pn,d)}
= \big(\pi_B^*\, \ker \wt\D_{\si;B}\big) \big/G_{\si}.$$ (3) $\ti\pi_*\pi_*\ti\pi^*\ev^*\O_{\Pn}(a)=\pi_*\ev^*\O_{\Pn}(a)$ over $\ov\M_{1,k}^0(\Pn,d)$.
We prove the first two statements of this theorem by working with the cone $$p\!:\ov\M_{1,k}(\cL,d)\lra \ov\M_{1,k}(\Pn,d).$$ The sheaves $\pi_*\ev^*\O_{\Pn}(a)$ and $\pi_*\ti\pi^*\ev^*\O_{\Pn}(a)$ are the sheaves of (holomorphic) sections of $$\V_{1,k}^d\equiv \ov\M_{1,k}(\cL,d)\big|_{\ov\M_{1,k}^0(\Pn,d)}
\lra \ov\M_{1,k}^0(\Pn,d)$$ and $\ti\pi^*\V_{1,k}^d$, respectively; see Lemma \[conesheaf\_lmm\]. In Subsection \[conebl\_subs\], we lift the blowup construction of Subsection \[map0blconstr\_subs\] to $\ov\M_{1,k}(\cL,d)$. In particular, we blow up $\ov\M_{1,k}(\cL,d)$ along the subvarieties $$\ov\M_{1,\si}(\cL,d) = p^{-1}\big(\ov\M_{1,\si}(\Pn,d)\big),
\qquad\si\in\A_1(d,k),$$ and their proper transforms. The end result of this construction, which we denote by $\wt\M_{1,k}^0(\cL,d)$, is smooth for essentially the same reasons that $\ov\M_{1,k}^0(\Pn,d)$ is. The only additional input we need is Lemma \[cone1bl\_lmm2\], which is a restatement of the key result concerning the structure of the cone $\V_{1,k}^d$ obtained in [@g1cone]. The bundle $$\ti{p}\!:\wt\M_{1,k}^0(\cL,d)\lra\wt\M_{1,k}^0(\Pn,d)$$ of vector spaces of the same rank contains $\wt\M_{1,k}^0(\Pn,d)$ as the zero section. Thus, $\ti{p}$ is a vector bundle. There is a natural inclusion $$\wt\M_{1,k}^0(\cL,d) \lra \ti\pi^*\ov\M_{1,k}(\cL,d).$$ All sections of $\ti\pi^*\ov\M_{1,k}(\cL,d)$ must in fact be sections of $\wt\M_{1,k}^0(\cL,d)$ and thus the sheaf $\pi_*\ti\pi^*\ev^*\O_{\Pn}(a)$ is indeed locally free. The bundle map $$\wt\D_{\si;B}\equiv\wt\D_{([m],J_B)}$$ of the second statement of Theorem \[cone\_thm\] is described in Subsection \[conehomomor\_subs\]. It is the “vertical” part of the natural extension of the bundle map $\wt\cD_{\si;B}$ from stable maps into $\Pn$ to stable maps into $\cL$. Finally, the last statement of Theorem \[cone\_thm\] is a consequence of the last statement of Theorem \[main\_thm\]; see Lemma \[pushfor\_lmm3\]. At this point, this observation does not appear to have any applications though.\
[*Remark:*]{} By applying the methods of Section \[cone\_sec\] and of [@g1cone], it should be possible to show that the standard scheme structure on $\ov\M_{1,k}(\Pn,d)$ is in fact reduced.
Outline of the Main Desingularization Construction {#outline_subs}
--------------------------------------------------
The main blowup construction of this paper is contained in Subsections \[map1prelim\_subs\] and \[map1blconstr\_subs\]. It is a sequence of [*idealized*]{} blowups along smooth subvarieties. In other words, the blowup locus $\ov\M_{1,\si}^{\si-1}$ at each step comes with an [*idealized*]{} normal bundle $\N^{\ide}$. It is a vector [*bundle*]{} (of the smallest possible rank) containing the normal cone $\N$ for $\ov\M_{1,\si}^{\si-1}$. After taking the usual blowup of the ambient space along $\ov\M_{1,\si}^{\si-1}$, we attach the [*idealized exceptional divisor*]{} $$\cE^{\ide}\equiv\P\N^{\ide}$$ along the usual exceptional divisor $$\cE\equiv\P\N\subset \cE^{\ide}.$$ The blowup construction summarized in Theorem \[main\_thm\] is contained in the idealized blowup construction of Section \[map1bl\_sec\]. The latter turns out to be more convenient for describing the proper transforms of $\ov\M_{1,k}^0(\Pn,d)$, including at the final stage, i.e. $\wt\M_{1,k}^0(\Pn,d)$.\
The ambient space $\ov\M_{1,k}^{\si}$ at each step $\si\!\in\!\{0\}\!\sqcup\!\A_1(d,k)$ of the blowup construction contains a subvariety $\ov\M_{1,\si^*}^{\si}$ for each $\si^*\!\in\!\A_1(d,k)$. We take $\ov\M_{1,\si}^{\si}$ to be the idealized exceptional divisor for the idealized blowup just constructed, i.e. along $\ov\M_{1,\si}^{\si-1}$. If $\si^*\!<\!\si$ or $\si^*\!>\!\si$, $\ov\M_{1,\si^*}^{\si}$ is the proper transform of $\ov\M_{1,\si^*}^{\si^*}$ or $\ov\M_{1,\si^*}(\Pn,d)$, respectively.\
Every immersion $\io_{\si^*}$ of Subsection \[descr\_subs\] comes with an [*idealized*]{} normal bundle $\N_{\io_{\si^*}}^{\ide}$. It is a vector bundle of the smallest possible rank containing the normal cone to the immersion $\io_{\si}$ (see Definition \[virtvar\_e\]). It is given by $$\N_{\io_{\si^*}}^{\ide} = \bigoplus_{i\in[m^*]}\!\!
\pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0
\qquad\hbox{if}\quad \si^*=(m^*;J_P^*,J_B^*),$$ where $$\pi_P,\pi_B\!:\ov\cM_{\si^*;P}\!\times\!\ov\M_{\si^*;B}(\Pn,d)
\lra \ov\cM_{\si^*;P},\ov\M_{\si^*;B}(\Pn,d)$$ are the component projection maps. In the case of Figure \[m3\_fig1\], $\N_{\io_{\si^*}}^{\ide}$ is a rank-three vector bundle encoding the potential smoothings of the three nodes. At each step $\si$ of the blowup construction, $\io_{\si^*}$ induces an immersion $\io_{\si,\si^*}$ onto $\ov\M_{1,\si^*}^{\si}$. Like the domain of $\io_{\si^*}$, the domain of $\io_{\si,\si^*}$ splits as a Cartesian product. If $\si^*\!>\!\si$, the second component of the domain does not change from the previous step, while the first is modified by blowing up along a collection of disjoint subvarieties, as specified by the inductive assumption ($I9$) in Subsection \[map1blconstr\_subs\]. The idealized normal bundle $\N_{\io_{\si,\si^*}}^{\ide}$ is obtained from $\N_{\io_{\si-1,\si^*}}^{\ide}$ by twisting the first factor in each summand by a subset of the exceptional divisors, as specified by the inductive assumption ($I11$). These blowup and twisting procedures correspond to several interchangeable steps in the blowup construction of Subsection \[curve1bl\_subs\]. For $\si^*\!=\!\si$, the first component in the domain of $\io_{\si-1,\si}$ has already been blown up all the way to $\wt\cM_{\si;P}$ and the first component of every summand of $\N_{\io_{\si-1,\si}}^{\ide}$ has already twisted to the universal tangent line bundle $\L$, i.e. $$\N_{\io_{\si-1,\si}}^{\ide}=
\bigoplus_{i\in[m]} \pi_P^*\L \!\otimes\!
\pi_B^*\pi_i^*L_0
=\pi_P^*\L\otimes \pi_B^*F_{\si;B}
\lra \wt\cM_{\si;P}\!\times\!\ov\M_{\si;B}(\Pn,d),$$ if $\si\!=\!(m;J_P,J_B)$. In particular, the domain for $\io_{\si,\si}$, $$\P\N_{\io_{\si-1,\si}}^{\ide}=\wt\cM_{\si;P}\!\times\!\P F_{\si;B},$$ still splits as a Cartesian product! The idealized normal bundle for $\io_{\si,\si}$ is the tautological line for $\P\N_{\io_{\si-1,\si}}^{\ide}$: $$\N_{\io_{\si,\si}}^{\ide}=\ga_{\N_{\io_{\si-1,\si}}^{\ide}}
=\pi_P^*\L\!\otimes\!\pi_B^*\ga_{F_{\si;B}}
\equiv \pi_P^*\L\!\otimes\!\pi_B^*\ga_{\si;B}.$$ On the other hand, if $\si^*\!<\!\si$, the domain of $\io_{\si,\si^*}$ is obtained from the domain of $\io_{\si-1,\si^*}$ by blowing up the second component along a collection of disjoint subvarieties, as specified by the inductive assumption ($I4$) in Subsection \[map1blconstr\_subs\]. This corresponds to several interchangeable steps of the blowup construction in Subsection \[map0blconstr\_subs\]. By the time we are done with the last step of the blowup construction in Subsection \[map1blconstr\_subs\], $\P F_{\si;B}$ has been blown up all the way to $\wt\M_{\si;B}(\Pn,d)$. In the $\si^*\!<\!\si$ case, $$\N_{\io_{\si,\si^*}}^{\ide} = \N_{\io_{\si-1,\si^*}}^{\ide},$$ since $\ov\M_{1,\si^*}^{\si-1}$ is transverse to $\ov\M_{1,\si}^{\si-1}$.\
We study the proper transform $\ov\M_{1,(0)}^{\si}$ of $\ov\M_{1,k}^0(\Pn,d)$ in $\ov\M_{1,k}^{\si}$ by looking at the structure of $$\bar\cZ_{\si^*}^{\si}=\io_{\si,\si^*}^{\,-1}\big(\ov\M_{1,(0)}^{\si}\big).$$ Given a finite set $J$, there are natural bundle sections $$s_j\in\Ga(\ov\cM_{1,J};L_j^*\!\otimes\!\E^*),~~~j\!\in\!J,
\qquad\hbox{and}\qquad
\cD_0\in\Ga\big(\ov\M_{0,\{0\}\sqcup J}(\Pn,d);L_0^*\!\otimes\!\ev_0^*T\Pn\big);$$ see Subsection \[curvebldata\_subs\] and \[map0str\_subs\], respectively. By Lemma \[map1bl\_lmm2\], the intersection of $$\bar\cZ_{\si^*}^0\equiv\io_{\si^*}^{\,-1}\big(\ov\M_{1,k}^0(\Pn,d)\big)$$ with the main stratum $\cM_{\si^*;P}\!\times\!\M_{\si^*;B}(\Pn,d)$ of $\ov\cM_{\si^*;P}\!\times\!\ov\M_{\si^*;B}(\Pn,d)$ is $$\cZ_{\si^*}^0=\big\{b\!\in\!\cM_{\si^*;P}\!\times\!\M_{\si^*;B}(\Pn,d)\!:
\ker\cD_{\si^*}|_b\!\neq\!\{0\}\big\}$$ where $$\begin{gathered}
\cD_{\si^*} \in \Ga\big(\ov\cM_{\si^*;P} \!\times\!\ov\M_{\si^*;B}(\Pn,d);
\Hom(\N_{\io_{\si^*}}^{\ide},\pi_P^*\E^*\!\otimes\!\pi_B^*\ev_0^*T\Pn)\big),\\
\cD_{\si^*}\big|_{\pi_P^*L_i\otimes\pi_B^*\pi_i^*L_0}
=\pi_P^*s_i\!\otimes\!\pi_B^*\pi_i^*\cD_0,
\qquad\forall\,i\!\in\![m^*].\end{gathered}$$ In addition, if $\N\bar\cZ_{\si^*}^{\si}\!\subset\!\N_{\io_{\si,\si^*}}^{\ide}$ is the normal cone for the immersion $\io_{\si,\si^*}|_{\bar\cZ_{\si^*}^{\si}}$ into $\ov\M_{1,(0)}^{\si}$, then $$\N\bar\cZ_{\si^*}^0\big|_{\cZ_{\si^*}^0}=\ker \cD_{\si^*}\big|_{\cZ_{\si^*}^0}$$ and $\N\bar\cZ_{\si^*}^0$ is the closure of $\N\bar\cZ_{\si^*}^0\big|_{\cZ_{\si^*}^0}$ in $\N_{\io_{\si^*}}^{\ide}$. By Lemma \[virimmer\_lmm3\], $\N\bar\cZ_{\si^*}^{\si}$ is still the closure of $\N\bar\cZ_{\si^*}^0\big|_{\cZ_{\si^*}^0}$, but now in $\N_{\io_{\si,\si^*}}^{\ide}$, for all $\si\!<\!\si^*$. In Subsection \[curve1bl\_subs\], we construct a non-vanishing section $$\ti{s}_i\in \Ga(\ov\cM_{1,J};\L^*\!\otimes\!\E^*) \approx \Ga(\ov\cM_{1,J};\C)$$ obtained by twisting $s_i$ by some exceptional divisors. Since $\ti{s}_i$ agrees with $s_i$ on $\cM_{\si^*;P}$, we can replace $s_i$ with $\ti{s}_i$ in the descriptions of $\cD_{\si^*}$, $\cZ_{\si^*}^0$, and $\N\bar\cZ_{\si^*}^0\big|_{\cZ_{\si^*}^0}$ above. In particular, $\N\bar\cZ_{\si^*}^{\si^*-1}$ is the closure of $$\begin{gathered}
\N\bar\cZ_{\si^*}^0\big|_{\cZ_{\si^*}^0}=
\pi_P^*\L\!\otimes\!\pi_B^*\ker\cD_{\si^*;B}\big|_{\cZ_{\si^*}^0}
\subset \pi_P^*\L\!\otimes\!\pi_B^*F_{\si^*;B}, \qquad\hbox{where}\\
\cD_{\si^*;B} \in \Ga\big(\ov\M_{\si^*;B}(\Pn,d);
\Hom(F_{\si^*;B},\ev_0^*T\Pn)\big), \qquad
\cD_{\si^*;B}\big|_{\pi_i^*L_0}=\pi_i^*\cD_0, \quad\forall\,i\!\in\![m^*].\end{gathered}$$\
The bundle homomorphism $\cD_{\si^*;B}$ induces a section $$\wt\cD_0\in\Ga\big(\P F_{\si^*;B};
\ga_{\si;B}^*\!\otimes\!\pi_{\P F_{\si^*;B}}^*\ev_0^*T\Pn\big).$$ By the previous paragraph and Lemma \[virimmer\_lmm3\], $\bar\cZ_{\si^*}^{\si^*}$ is the closure of $$\begin{gathered}
\wt\cM_{\si^*;P} \times \wt\cD_0^{-1}(0)\!\cap\!\P F_{\si^*;B}\big|_{\M_{\si^*;B}(\Pn,d)}
\subset \wt\cM_{\si^*;P} \times\P F_{\si^*;B} \qquad\hbox{and}\\
\N\bar\cZ_{\si^*}^{\si^*}=
\pi_P^*\L\!\otimes\!\pi_B^*\ga_{\si;B}\big|_{\bar\cZ_{\si^*}^{\si^*}}.\end{gathered}$$ Since $\wt\M_{1,k}^0(\Pn,d)\!\equiv\!\wt\M_{1,k}^{\si_{\max}}(\Pn,d)$ is the proper transform of $\ov\M_{1,(0)}^{\si^*}$ in $\ov\M_{1,k}^{\si_{\max}}$, $$\wt\cZ_{\si^*}\equiv \io_{\si_{\max},\si^*}^{\,-1}\big(\wt\M_{1,k}^0(\Pn,d)\big)$$ is still the closure of $$\wt\cM_{\si^*;P} \times \wt\cD_0^{-1}(0)\!\cap\!\P F_{\si^*;B}\big|_{\M_{\si^*;B}(\Pn,d)}
\subset \wt\cM_{\si^*;P} \times\wt\M_{\si^*;B}(\Pn,d).$$ On the other hand, in the process of constructing the blowup $\wt\M_{\si;B}(\Pn,d)$ of $\P F_{\si^*;B}$ in Subsection \[map0blconstr\_subs\], we also define a bundle section $$\wt\cD_{\si^*;B}\in
\Ga\big(\wt\M_{\si^*;B}(\Pn,d);
\E_{\si;B}^*\!\otimes\!\ti\pi_{\si;B}^*\pi_{\P F_{\si^*;B}}^*\ev_0^*T\Pn\big)$$ by twisting $\wt\cD_0$ by the exceptional divisors. In particular, $$\wt\cD_{\si^*;B}^{-1}(0)\!\cap\!\P F_{\si^*;B}\big|_{\M_{\si^*;B}(\Pn,d)}
=\wt\cD_0^{-1}(0)\!\cap\!\P F_{\si^*;B}\big|_{\M_{\si^*;B}(\Pn,d)}.$$ Since $\wt\cD_{\si^*;B}$ is transverse to the zero set, we conclude that $$\wt\cZ_{\si^*} = \wt\cM_{\si^*;P} \times \wt\cD_{\si^*;B}^{-1}(0),$$ as stated in Subsection \[descr\_subs\].\
Finally, the role played by the blowup construction of Subsection \[curve0bl\_subs\] in the blowup construction of Section \[map0bl\_sec\] is similar to the role played by the construction of Subsection \[curve1bl\_subs\] in the construction of Section \[map1bl\_sec\]. In the case of Section \[map0bl\_sec\], we blow up a moduli space of genus-zero stable maps, $\P F_{(\aleph,J)}$, along certain subvarieties $\wt\M_{0,\vr}^0$ and their proper transforms. These subvarieties are images of natural node-identifying immersions $\io_{0,\vr}$. The domain of $\io_{0,\vr}$ splits as the Cartesian product of a moduli space of genus-zero curves and a moduli space of genus-zero maps, defined in Subsections \[curve0bl\_subs\] and \[map0prelim\_subs\], respectively. As we modify $\wt\M_{0,\vr}^0$ by taking its proper transforms in the blowups of $\P F_{(\aleph,J)}$ constructed in Subsection \[map0blconstr\_subs\], the first factor in the domain of the corresponding immersion changes by blowups along collections of smooth disjoint subvarieties, as specified by the inductive assumption ($I6)$. This change corresponds to several interchangeable steps in the blowup construction of Subsection \[curve0bl\_subs\]. By the time we are ready to blow up the proper transform of $\wt\M_{0,\vr}^0$, the first component of the domain of the corresponding immersion has been blown up all the way to $\wt\cM_{0,\rho_P}$, the end result in the blowup construction of Subsection \[curve0bl\_subs\].\
In the blowup construction of Subsection \[map0blconstr\_subs\], we twist a natural bundle section $$\wt\cD_0\in\Ga\big(\P F_{(\ale,J)};
\ga_{(\aleph,J)}^*\!\otimes\!\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big)$$ by the exceptional divisors to a bundle section $$\wt\cD_{(\aleph,J)}\in
\Ga\big(\wt\M_{(\ale,J)}(\Pn,d);
\wt\E^*\!\otimes\!\pi_{0,(\ale,J)}^*\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big).$$ The two sections enter in an essential way in the main blowup construction of this paper. It is also essential that $\wt\cD_{(\aleph,J)}$ is transverse to the zero set. The section $\wt\cD_0$ is transverse to the zero set outside of the subvarieties $\wt\M_{0,\vr}^0$ and vanishes identically along $\wt\M_{0,\vr}^0$. Its derivative in the normal direction to $\io_{0,\vr}$ is described by Lemma \[map0bl\_lmm2\], using Lemma \[deriv0str\_lmm\]. The bundle sections $s_i$ over a moduli space of genus-zero curves defined in Subsection \[curvebldata\_subs\] and modified in Subsection \[curve0bl\_subs\] enter into the expression of Lemma \[deriv0str\_lmm\]. In fact, this expression is identical to the expression for $\cD_{\si^*}$ above, i.e. in the genus-one case. We use Lemma \[map0bl\_lmm2\] to show that with each newly twisted version of $\wt\cD_0$ is transverse to the zero set outside of the proper transforms of the remaining subvarieties $\ov\M_{0,\vr}^0$, i.e. the ones that have not been blown up yet; see the inductive assumption ($I4$) in Subsection \[map0blconstr\_subs\]. In particular, at the end of the blowup of Subsection \[map0blconstr\_subs\], we end up with a twisted version of $\wt\cD_0$, which we call $\wt\cD_{(\aleph,J)}$, which is transverse to the zero set.
Localization Data {#local_subs}
-----------------
Suppose the group $G\!=\!(S^1)^{n+1}$ or $G\!=\!(\C^*)^{n+1}$ acts in a natural way on the projective space $\Pn$. In particular, the fixed locus consists of $n\!+\!1$ points, which we denote by $p_0,\ldots,p_n$, and the only curves preserved by $G$ are the lines passing through pairs of fixed points. The $G$-action on $\Pn$ lifts to an action on $\ov\M_{1,k}(\Pn,d)$ and on $\wt\M_{1,k}^0(\Pn,d)$. The fixed loci of these two actions that are contained in $\M_{1,k}^{\eff}(\Pn,d)$ and their normal bundles are the same and are described in [@H Sects. 27.3 and 27.4]. We note that the four-term exact sequence [@H (27.6)] applies to such loci.\
In this subsection, we describe the fixed loci of the $G$-action on $\wt\M_{1,k}^0(\Pn,d)$ that are contained in $\part\wt\M_{1,k}^0(\Pn,d)$ and their normal bundles. To simplify the discussion, we ignore all automorphism groups until the very end of this subsection.\
The boundary fixed loci $\wt\cZ_{\ti{\Ga}}$ will be indexed by [*refined decorated rooted trees*]{} $\ti\Ga$. Figure \[graph\_fig\] shows such a tree $\ti\Ga$ and the corresponding decorated graph $\Ga\!=\!\pi(\ti\Ga)$. In [@H Section 27.3] the fixed loci $\cZ_{\Ga}$ of the $G$-action on $\ov\M_{g,k}(\Pn,d)$ are indexed by decorated graphs $\Ga$. If $\Ga$ is a decorated graph such that $\cZ_{\Ga}$ is a $G$-fixed locus contained in $\part\ov\M_{1,k}(\Pn,d)$, we will have $$\cZ_{\Ga}\cap\ov\M_{1,k}^0(\Pn,d)
=\ti\pi\Big(\bigsqcup_{\pi(\ti\Ga)=\Ga}\!\! \wt\cZ_{\ti{\Ga}}\Big),$$ where $\ti\Ga$ denotes a refined decorated rooted tree.\
(0,-2)(10,2) (10,0)[.3]{}(10,.7) (10,0)(7,2)(7,2)[.2]{} (8.6,1.5)(7.3,2.5) (10,0)(6,0)(6,0)[.2]{} (7.7,.4)(6.3,.6) (6,0)(3,2)(3,2)[.2]{} (4.6,1.4)(3.4,2.5) (6,0)(3,-2)(3,-2)[.2]{} (4.4,-.6)(3.2,-1.3) (10,0)(7,-1.5)(7,-1.5)[.2]{} (7.7,-.7)(6.8,-.9) (7,-1.5)(5,-3)(4.7,-3.1) (10,0)(8,-3)(8,-3)[.2]{} (8.5,-1.6)(7.5,-3.3) (10,0)(10,-3)(10,-3.5) (10,0)(12,-2)(12,-2)[.2]{} (11,-1.5)(12.5,-2.2) (10,0)(14,0)(14,0)[.2]{} (14,0)(17,-1)(17,-1)[.2]{} (16.2,-.3)(17.5,-1) (14,0)(17,1)(17,1)[.2]{} (15.6,.9)(17.2,1.6) (17,1)(20,2)(20,2)[.2]{} (18.6,1.9)(20.2,2.6) (10,0)(13,2)(13,2)[.2]{} (13,2)(16,3)(16,3)[.2]{} (14.6,3)(16.2,3.6) (13,2)(14,4)(14.3,4.4) (30,0)[.2]{} (30,0)(27,2)(27,2)[.2]{} (28.6,1.5)(27.3,2.5) (30,0)(26,0)(26,0)[.2]{} (27.7,.4)(26.3,.6) (26,0)(23,2)(23,2)[.2]{} (24.6,1.4)(23.4,2.5) (26,0)(23,-2)(23,-2)[.2]{} (24.4,-.6)(23.2,-1.3) (30,0)(27,-1.5)(27,-1.5)[.2]{} (27.7,-.7)(26.8,-.9) (27,-1.5)(25,-3)(24.7,-3.1) (30,0)(28,-3)(28,-3)[.2]{} (28.5,-1.6)(27.5,-3.3) (30,0)(30,-3)(30,-3.5) (30,0)(32,-2)(32,-2)[.2]{} (31,-1.5)(32.5,-2.2) (30,0)(33,-1)(33,-1)[.2]{} (32.2,-.3)(33.5,-1) (30,0)(33,1)(33,1)[.2]{} (32,1.1)(33.2,1.6) (33,1)(36,2)(36,2)[.2]{} (34.6,1.9)(36.2,2.6) (30,0)(32,2)(32,2)[.2]{} (31,1.5)(32.2,2.6) (30,0)(30.5,2.5)(30.5,3) (27.5,4) (28.3,4.2)[A]{}(30,0)[B]{}
We now formally describe what we mean by a refined decorated rooted tree and its corresponding decorated graph. A [*graph*]{} consists of a set $\Ver$ of [*vertices*]{} and a collection $\Edg$ of [*edges*]{}, i.e. of two-element subsets of $\Ver$. In Figure \[graph\_fig\], the vertices are represented by dots, while each edge $\{v_1,v_2\}$ is shown as the line segment between $v_1$ and $v_2$. A graph is a [*tree*]{} if it contains no [*loops*]{}, i.e. the set $\Edg$ contains no subset of the form $$\big\{\{v_1,v_2\},\{v_2,v_3\},\ldots,\{v_N,v_1\}\big\},
\qquad v_1,\ldots,v_N\!\in\!\Ver,~ N\!\ge\!1.$$ A tree is [*rooted*]{} if $\Ver$ contains a distinguished element $v_0$. It is represented by the large dot in the first diagram of Figure \[graph\_fig\]. A rooted tree is [*refined*]{} if $\Ver\!-\!\{v_0\}$ contains two, possibly empty, distinguished subsets $\Ver_+$ and $\Ver_0$ such that $$\Ver_+\!\cap\!\Ver_0=\eset \quad\hbox{and}\quad
\{v_0,v\}\!\in\!\Edg ~~\forall\, v\!\in\!\Ver_+\!\cup\!\Ver_0.$$ We put $$\Edg_+=\big\{\{v_0,v\}\!: v\!\in\!\Ver_+\big\} \quad\hbox{and}\quad
\Edg_0=\big\{\{v_0,v\}\!: v\!\in\!\Ver_0\big\}.$$ The elements of $\Edg_+$ and $\Edg_0$ are shown in the first diagram of Figure \[graph\_fig\] as the thick solid lines and the thin dashed lines, respectively. Finally, a [*refined decorated rooted tree*]{} is a tuple $$\label{treedfn_e}
\ti\Ga = \big(\Ver,\Edg;v_0;\Ver_+,\Ver_0;\mu,\d,\eta \big),$$ where $(\Ver,\Edg;v_0;\Ver_+,\Ver_0)$ is refined rooted tree and $$\mu\!:\Ver\!-\!\Ver_0\lra\big\{0,\ldots,n\}, \qquad
\d\!: \Edg\!-\!\Edg_0\lra\Z^+, \quad\hbox{and}\quad
\eta\!: \{1,\ldots,k\}\lra\Ver$$ are maps such that\
${}\quad$ (i) $\mu(v_1)\!=\!\mu(v_2)$ and $\d(\{v_0,v_1\})\!=\!\d(\{v_0,v_2\})$ for all $v_1,v_2\!\in\!\Ver_+$;\
${}\quad$ (ii) if $v_1\!\in\!\Ver_+$, $v_2\!\in\!\Ver\!-\!\Ver_0\!-\!\Ver_+$, and $\{v_0,v_2\}\!\in\!\Edg$, then $$\mu(v_1)\neq\mu(v_2) \qquad\hbox{or}\qquad
\d(\{v_0,v_1\})\!\neq\!\d(\{v_0,v_2\});$$ ${}\quad$ (iii) if $\{v_1,v_2\}\!\in\!\Edg$ and $v_2\!\not\in\!\Ver_0\!\cup\!\{v_0\}$, then $$\mu(v_2)\!\neq\!\mu(v_1) \quad\hbox{if}~~~ v_1\!\not\in\!\Ver_0
\qquad\hbox{and}\qquad
\mu(v_2)\!\neq\!\mu(v_0) \quad\hbox{if}~~~ v_1\!\in\!\Ver_0;$$ ${}\quad$ (iv) if $v_1\!\in\!\Ver_0$, then $\{v_1,v_2\}\!\in\!\Edg$ for some $v_2\!\in\!\Ver\!-\!\{v_0\}$ and $$\val(v_1)\equiv \big|\{v_2\!\in\!\Ver\!: \{v_1,v_2\}\!\in\!\Edg\}\big|
+ \big|\{l\!\in\![k]\!: \eta(l)\!=\!v_1\big|\ge 3;$$ ${}\quad$ (v) $\sum_{e\in\Edg_+}\!\!\d(e)\ge2$.\
In Figure \[graph\_fig\], the value of the map $\mu$ on each vertex, not in $\Ver_0$, is indicated by the number next to the vertex. Similarly, the value of the map $\d$ on each edge, not in $\Edg_0$, is indicated by the number next to the edge. The elements of the set $[k]\!=\![3]$ are shown in bold face. Each of them is linked by a line segment to its image under $\eta$. The first condition above implies that all of the thick edges have the same labels, and so do their vertices, other than the root $v_0$. By the second condition, the set of thick edges is a maximal set of edges leaving $v_0$ which satisfies the first condition. By the third condition, no two consecutive vertex labels are the same. By the fourth condition, there are at least two solid lines, at least one of which is an edge, leaving from every vertex which is connected to the root by a dashed line. The final condition implies that either the set $\Edg_+$ contains at least two elements or its only element is marked by at least 2.\
A [*decorated graph*]{} is a tuple $$\Ga=\big(\Ver,\Edg;g,\mu,\d,\eta\big),$$ where $(\Ver,\Edg)$ is a graph and $$g\!:\Ver\lra\bar\Z^+, \quad \mu\!:\Ver\lra\big\{0,\ldots,n\}, \quad
\d\!: \Edg\lra\Z^+, \quad\hbox{and}\quad \eta\!: \{1,\ldots,k\}\lra\Ver$$ are maps such that $$\mu(v_1)\neq\mu(v_2) \qquad\hbox{if}\quad \{v_1,v_2\}\!\in\!\Edg.$$ The domain $[k]$ of the map $\eta$ can be replaced by any finite set. A decorated graph can be represented graphically as in the second diagram of Figure \[graph\_fig\]. In this case, every vertex $v$ should be labeled by the pair $(g(v),\mu(v))$. However, we drop the first entry if it is zero. If $\ti\Ga$ is a refined decorated rooted tree as in \_ref[treedfn\_e]{}, the corresponding decorated graph $\Ga$ is obtained by identifying all elements of $\Ver_0$ with $v_0$, dropping $\Edg_0$ from $\Edg$, and setting $$g(v) =\begin{cases}
1,& \hbox{if}~v\!=\!v_0;\\
0,& \hbox{otherwise}.
\end{cases}$$ In terms of the first diagram in Figure \[graph\_fig\], this procedure corresponds to contracting the dashed edges and adding $1$ to the label for $v_0$.\
The fixed locus $\cZ_{\Ga}$ of $\ov\M_{1,k}(\Pn,d)$ consists of the stable maps $u$ from a genus-one nodal curve $\Si_u$ with $k$ marked points into $\Pn$ that satisfy the following conditions. The components of $\Si_u$ on which the map $u$ is not constant are rational and correspond to the edges of $\Ga$. Furthermore, if $e\!=\!\{v_1,v_2\}$ is an edge, the restriction of $u$ to the component $\Si_{u,e}$ corresponding to $e$ is a degree-$\d(e)$ cover of the line $$\P^1_{p_{\mu(v_1)},p_{\mu(v_2)}}\subset\Pn$$ passing through the fixed points $p_{\mu(v_1)}$ and $p_{\mu(v_2)}$. The map $u|_{\Si_{u,e}}$ is ramified only over $p_{\mu(v_1)}$ and $p_{\mu(v_2)}$. In particular, $u|_{\Si_{u,e}}$ is unique up to isomorphism. The remaining, contracted, components of $\Si_u$ correspond to the vertices $v\!\in\!\Ver$ such that $$\val(v)+g(v)\ge 3.$$ For such a vertex $v$, $g(v)$ specifies the genus of the component corresponding to $v$. The map $u$ takes this component to the fixed point $\mu(v)$. Thus, $$\cZ_{\Ga}\approx \ov\cM_{\Ga}\!\equiv\!\prod_{v\in\Ver}\!\!\ov\cM_{g(v),\val(v)};$$ see [@H Section 27.3]. For the purposes of this definition, $\ov\cM_{0,1}$ and $\ov\cM_{0,2}$ denote one-point spaces. For example, in the case of the second diagram in Figure \[graph\_fig\], $$\cZ_{\Ga}\approx
\ov\cM_{\Ga} \!\equiv \ov\cM_{1,10} \!\times\! \ov\cM_{0,3}
\!\times\! \ov\cM_{0,2}^2 \!\times\! \ov\cM_{0,1}^8 \approx \ov\cM_{1,10}.$$ In this case, $\cZ_{\Ga}$ is a locus in $\ov\M_{1,3}(\Pn,22)$, with $n\!\ge\!3$.\
If $\ti\Ga$ is a refined decorated rooted tree as in \_ref[treedfn\_e]{}, we put $$\Edg(v_0)=\big\{\{v_0,v_1\}\!\in\!\Edg\!: v_1\!\in\!\Ver\big\}
\qquad\hbox{and}\qquad
J_{v_0}=\big\{l\!\in\![k]\!: \mu(l)\!=\!v_0\big\}.$$ Similarly, for each $v\!\in\!\Ver_0$, we set $$\Edg(v)=\big\{\{v,v_1\}\!\in\!\Edg\!: v_1\!\in\!\Ver\!-\!\{v_0\}\big\}
\qquad\hbox{and}\qquad
J_v=\big\{l\!\in\![k]\!: \mu(l)\!=\!v\big\}.$$ If $e\!=\!\{v,v_1\}$ is an element of $\Edg(v)$ for some $v\!\in\!\Ver_0$ or of $\Edg(v_0)\!-\!\Edg_0$ with $v\!=\!v_0$, let $(\Ver_e,\Edg_e)$ be the branch of the tree $(\Ver,\Edg)$ beginning at $v$ with the edge $e$. We put $$J_e=\big\{l\!\in\![k]\!: \mu(l)\!\in\!\Ver_e\!-\!\{v\}\big\}
\qquad\hbox{and}\qquad
d_e=\sum_{e'\in\Edg_e}\!\!\!\d(e').$$ Let $\ti\Ga_e$ be the decorated graph defined by $$\begin{gathered}
\ti\Ga_e=\big(\Ver_e,\Edg_e;g_e\!\equiv\!0,\mu_e,
\d_e\!\equiv\!\d|_{\Edg_e},\eta_e),
\qquad\hbox{where}\\
\mu_e(v')=\begin{cases}
\mu(v'),&\hbox{if}~v'\!\neq\!v;\\
\mu(v_0),&\hbox{if}~v'\!=\!v;
\end{cases} \qquad
\eta_e\!:\{0\}\!\sqcup\!J_e\lra\Ver_e, \quad
\eta_e(l)=\begin{cases}
\eta(l),& \hbox{if}~l\!\in\!J_e;\\
v,& \hbox{if}~l\!=\!0;
\end{cases}\end{gathered}$$ see Figure \[graph\_fig2\] for two examples.\
(0,-2)(10,2) (10,0)[.3]{}(10,.7) (10,0)(7,2)(7,2)[.2]{} (8.6,1.5)(7.3,2.5) (10,0)(6,0)(6,0)[.2]{} (7.7,.4)(6.3,.6) (6,0)(3,2)(3,2)[.2]{} (4.6,1.4)(3.4,2.5) (6,0)(3,-2)(3,-2)[.2]{} (4.4,-.6)(3.2,-1.3) (10,0)(7,-1.5)(7,-1.5)[.2]{} (7.7,-.7)(6.8,-.9) (7,-1.5)(5,-3)(4.7,-3.1) (10,0)(8,-3)(8,-3)[.2]{} (8.5,-1.6)(7.5,-3.3) (10,0)(10,-3)(10,-3.5) (10,0)(12,-2)(12,-2)[.2]{} (11,-1.5)(12.5,-2.2)(11.7,-1.1) (10,0)(14,0)(14,0)[.2]{} (14,0)(17,-1)(17,-1)[.2]{} (15.5,-.9)(17.5,-1) (14,0)(17,1)(17,1)[.2]{} (15.6,.9)(17.2,1.6)(16,.2) (17,1)(20,2)(20,2)[.2]{} (18.6,1.9)(20.2,2.6) (10,0)(13,2)(13,2)[.2]{} (13,2)(16,3)(16,3)[.2]{} (14.6,3)(16.2,3.6) (13,2)(14,4)(14.3,4.4) (30,1.5)[.2]{}(29.8,2.1) (30,1.5)(34,1.5)(34,1.5)[.2]{} (32.5,1.9)(34,2.1) (30,1.5)(32,3.5)(32.4,3.6) (27.2,1.7)[$\ti\Ga_{e_1}\!=$]{} (30,-2)[.2]{}(29.8,-1.4) (30,-2)(34,-2)(34,-2)[.2]{} (32.5,-1.6)(34,-1.4) (34,-2)(38,-2)(38,-2)[.2]{} (36.5,-1.6)(38,-1.4) (30,-2)(32,-3.5)(32.4,-3.4) (27.2,-1.8)[$\ti\Ga_{e_2}\!=$]{}
If $e$ is an element of $\Edg(v_0)\!-\!\Edg_0$ or of $\Edg(v)$ for some $v\!\in\!\Ver_0$, let $$\cZ_{\ti\Ga_e} \subset \ov\M_{0,\{0\}\sqcup J_e}(\Pn,d_e)$$ be the fixed locus corresponding to the decorated graph $\ti\Ga_e$. We put $$\begin{gathered}
\si(\ti\Ga)=\big(|\Edg(v_0)|;J_{v_0},[k]\!-\!J_{v_0}\big)\in\A_1(d,k),
\qquad
\wt\cM_{\ti\Ga;P}=\wt\cM_{\si(\ti\Ga);P};\\
\bar\cZ_{\ti\Ga;B}=\!\!
\prod_{e\in\Edg(v_0)-\Edg_0}\!\!\!\!\!\!\!\!\!\!\!\! \cZ_{\ti\Ga_e}
~\times \prod_{v\in\Ver_0}\!\! \Big(\ov\cM_{0,\{0\}\sqcup\Edg(v)\sqcup J_v}
\times\! \prod_{e\in\Edg(v)}\!\!\!\!\!\! \cZ_{\ti\Ga_e}\Big)
\subset \ov\M_{\si(\ti\Ga);B}(\Pn,d);\\
F_{\ti\Ga;B}=\bigoplus_{e\in\Edg_+}\!\!\!L_{e;0}\subset F_{\si(\ti\Ga);B}
\lra \bar\cZ_{\ti\Ga;B},\end{gathered}$$ where $L_{e;0}\!\lra\!\cZ_{\ti\Ga_e}$ is the tangent line bundle for the marked point $0$. If $e\!=\!\{v_0,v_1\}$ is an element of $\Edg_+$, let $$\mu_+(\ti\Ga)=\mu(v_1), \qquad \d_+(\ti\Ga)=\d(e),
\quad\hbox{and}\quad
\dim_+(\ti\Ga)=\begin{cases}
|\Edg_+|\!-\!2,& \hbox{if}~\d_+(\ti\Ga)\!=\!1;\\
|\Edg_+|\!-\!1,& \hbox{if}~\d_+(\ti\Ga)\!\ge\!2.
\end{cases}$$ By the assumption (i) above, the numbers $\mu_+(\ti\Ga)$ and $\d_+(\ti\Ga)$ are independent of the choice of $e\!\in\!\Edg_+$. Furthermore, if $e,e'\!\in\!\Edg_+$, then the line bundles $L_{e;0}$ and $L_{e';0}$ are $G$-equivariantly isomorphic. Thus, $$F_{\ti\Ga;B} \approx {\mathbb}{C}^{|\Edg_+|}\otimes L_{e;0}
\qquad\hbox{if}\quad e\!\in\!\Edg_+.$$ The group $G$ acts trivially on ${\mathbb}{C}^{|\Edg_+|}$. Let $$\begin{gathered}
F_{\ti\Ga;B}'=
\begin{cases}
\big\{(w_e)_{e\in\Edg_+}\!\in\!\C^{\Edg_+}:\sum_{e\in\Edg_+}\!w_e\!=\!0\big\},
&\hbox{if}~ \d_+(\ti\Ga)\!=\!1;\\
\C^{\Edg_+}, &\hbox{if}~ \d_+(\ti\Ga)\!\ge\!2;
\end{cases}\\
\wt\cZ_{\ti\Ga;B}=\P\big(F_{\ti\Ga;B}'\!\otimes\!L_{e;0}\big)
\approx \bar\cZ_{\ti\Ga;B}\times{\mathbb}{P}^{\dim_+(\ti\Ga)}.\end{gathered}$$ While the moduli space $\wt\M_{\si(\ti\Ga);B}(\Pn,d)$ is a blowup of $\P F_{\si(\ti\Ga);B}$, none of the blowup loci intersects $\wt\cZ_{\ti\Ga;B}$. Thus, $$\wt\cZ_{\ti\Ga;B} \subset \wt\M_{\si(\ti\Ga);B}(\Pn,d).$$ In fact, $$\wt\cZ_{\ti\Ga;B} \subset \wt\cZ_{\si(\ti\Ga);B}(\Pn,d).$$ We put $$\wt\cZ_{\ti\Ga}=\wt\cM_{\ti\Ga;P}\times \wt\cZ_{\ti\Ga;B}.$$ By the above, $\wt\cZ_{\ti\Ga}$ is a fixed point locus in $\wt\M_{1,k}^0(\Pn,d)$. For example, in the case of the first diagram in Figure \[graph\_fig2\], $$\begin{gathered}
\si(\ti\Ga)=\big(7;\{2\},\{1,3\}\big), \qquad
\wt\cM_{\ti\Ga;P}=\wt\cM_{1,([7],\{2\})}, \\
\bar\cZ_{\ti\Ga;B}=\Big(\ov\cM_{0,2}^6\!\times\!\ov\cM_{0,1}^5\!\times\!\ov\cM_{0,3}\Big)
\times\Big(\ov\cM_{0,3}^2\!\times\!\ov\cM_{0,2}^4\!\times\!\ov\cM_{0,1}^3\Big)
\approx\{pt\};\\
\rk\, F_{\ti\Ga;B}=\rk\, F_{\ti\Ga;B}'=3, \qquad
\wt\cZ_{\ti\Ga;B}\approx\P^2, \qquad
\wt\cZ_{\ti\Ga}\approx \wt\cM_{1,([7],\{2\})}\!\times\!\P^2.\end{gathered}$$ The weight of the $G$-action on the line $L_{e;0}$ is $1/2$ of the weight of the $G$-action on $T_{p_0}\P_{p_0,p_1}^1$; see [@H Sects 27.1 and 27.2].\
We next describe the equivariant normal bundle ${{\mathcal}N}\wt\cZ_{\ti\Ga}$ of $\wt\cZ_{\ti\Ga}$ in $\wt\M_{1,k}^0(\Pn,d)$. Let $$\N_{\ov\M_{\si(\ti\Ga);B}(\Pn,d)}\bar\cZ_{\ti\Ga;B} \lra \bar\cZ_{\ti\Ga;B}$$ be the normal bundle of $\bar\cZ_{\ti\Ga;B}$ in $\ov\M_{\si(\ti\Ga);B}(\Pn,d)$. This normal bundle can easily be described using [@H Section 27.4]. Let $$F_{\ti\Ga;B}^- = F_{\si(\ti\Ga);B} \big/ (F_{\ti\Ga;B}'\!\otimes\!L_{e;0})
\approx \!\!
\bigoplus_{e'\in\Edg(v_0)-\Edg^+}\!\!\!\!\!\!\!\!\!\!\!\!\!\! L_{e';0}~~
\oplus
\begin{cases}
L_{e;0},& \hbox{if}~\d_+(\ti\Ga)\!=\!1;\\
\{0\},& \hbox{if}~\d_+(\ti\Ga)\!\ge\!2,
\end{cases}$$ where $e$ is an element of $\Edg_+$. The normal bundle of $\wt\cZ_{\ti\Ga;B}$ in $\wt\M_{\si(\ti\Ga);B}(\Pn,d)$ is given by $$\N_{\wt\M_{\si(\ti\Ga);B}(\Pn,d)}\wt\cZ_{\ti\Ga;B}=
\N_{\ov\M_{\si(\ti\Ga);B}(\Pn,d)}\bar\cZ_{\ti\Ga;B}
\oplus \ga_{\dim_+\ti\Ga}^*\!\otimes\!L_{e;0}^*\!\otimes\!F_{\ti\Ga;B}^-,$$ where $\ga_{\dim_+\ti\Ga}\!\lra\!\P^{\dim_+\ti\Ga}$ is the tautological line bundle. Since none of the exceptional divisors intersects $\wt\cZ_{\ti\Ga;B}$, $$\label{bunrestr_e1}
\E_{\si;B}\big|_{\wt\cZ_{\ti\Ga;B}} = \ga_{\dim_+\ti\Ga}\!\otimes\!L_{e;0}.$$ Since the section $\wt\cD_{\si;B}$ is transverse to the zero set, the normal bundle of $\wt\cZ_{\ti\Ga;B}$ in $\wt\cZ_{\si(\ti\Ga);B}(\Pn,d)$ is $$\N_{\wt\cZ_{\si(\ti\Ga);B}(\Pn,d)}\wt\cZ_{\ti\Ga;B} =
\N_{\wt\M_{\si(\ti\Ga);B}(\Pn,d)}\wt\cZ_{\ti\Ga;B}
\big/ \big(\ga_{\dim_+\ti\Ga}^*\!\otimes\!L_{e;0}^*\!\otimes\!T_{\mu(v_0)}\Pn\big)$$ by \_ref[spacesec\_e1]{} and \_ref[spacedfn\_e1]{}. Finally, $${{\mathcal}N}\wt\cZ_{\ti\Ga}=
\N_{\wt\cZ_{\si(\ti\Ga);B}(\Pn,d)}\wt\cZ_{\ti\Ga;B}
\oplus \L_{\si(\ti\Ga);P}\!\otimes\!\ga_{\dim_+\ti\Ga}\!\otimes\!L_{e;0},$$ since the normal bundle of $\cZ_{\si(\ti\Ga)}(\Pn,d)$ in $\ov\M_{1,k}^0(\Pn,d)$ is $\L_{\si(\ti\Ga);P}\!\otimes\!\ga_{\si(\ti\Ga);B}$.\
In order to compute the last number in \_ref[euler\_e]{}, we also need to determine the restriction of the vector bundle $\wt\V_{1,k}^d$ to $\wt\cZ_{\ti\Ga}$. By Theorem \[cone\_thm\] and \_ref[bunrestr\_e1]{}, there is a short exact sequence of vector bundles: $$0\lra \wt\V_{1,k}^d\big|_{\wt\cZ_{\ti\Ga}} \lra
\V_{\si(\ti\Ga);B}^d\big|_{\wt\cZ_{\ti\Ga;B}} \lra
\ga_{\dim_+\ti\Ga}^*\!\otimes\!L_{e;0}^*\!\otimes\!\cL_{\mu(v_0)} \lra 0$$ over $\wt\cZ_{\ti\Ga}$. This exact sequence describes the euler class of the restriction of $\wt\V_{1,k}^d$ to $\wt\cZ_{\ti\Ga}$.\
If $\si\!=\!(m;J_P,J_B)\!\in\!\A_1(d,k)$, $$\blr{c_1^{|m|+|J_P|}(\L_{\si;P}^*),\wt\cM_{\si;P}}
=\frac{m^{|J_P|}\cdot(m\!-\!1)!}{24},$$ by [@g1desing2]. This is the only intersection number on $\wt\cM_{\si;P}$ needed for computing the last number in \_ref[euler\_e]{} and the integrals of the cohomology classes on $\ov\M_{1,k}^0(\Pn,d)$ that count elliptic curves in $\Pn$ passing through specified constraints. For more general enumerative problems, such as counting curves with tangency conditions, as in [@enumtang], and with singularities, as in [@genuss0pr], one would need to compute the intersection numbers of the form $$\Blr{c_1^{\be_0}(\L_{\si;P}^*)\cdot\prod_{l\in J_P}\!\psi_l^{\be_l},\wt\cM_{\si;P}},
\qquad\hbox{where}\qquad \be_0+\sum_{l\in J_P}\be_l=|m|+|J_P|.$$ The argument in [@g1desing2] gives a recursive formula for a generalization of such numbers. The recursion is on $|m|\!+\!|J_P|$, i.e. the total number of marked points. The starting data for the recursion are the numbers $$\Blr{\prod_{l=1}^{l=k}\!\psi_l^{\be_l},\ov\cM_{1,k}},
\qquad\hbox{where}\qquad \sum_{l=1}^{l=k}\be_l=k,$$ which in turn are computable from the genus-one string and dilaton equations; see [@H Section 25.2].\
In the above discussion we ignored all automorphism groups. As in [@H Chapter 27], the rational function for each refined decorated rooted tree $\ti\Ga$ obtained following the above algorithm and applying the localization theorem of [@ABo] should be divided by the order of the appropriate automorphism group $\bA_{\ti\Ga}:$ $$\big|\bA_{\ti\Ga}\big|=\big|\Aut(\ti\Ga)\big|\cdot
\prod_{e\in\Edg-\Edg_0}\!\!\!\!\!\!\! \d(e).$$ For example, in the case of the first diagram in Figure \[graph\_fig2\], $$\big|\bA_{\ti\Ga}\big|= 1\cdot \big(1^3\cdot2^5\cdot3^3)=864.$$
Blowups of Moduli Spaces of Curves {#curvebl_sec}
==================================
Blowups and Subvarieties {#curveprelim_subs}
------------------------
In this section we construct blowups of certain moduli spaces of genus-one and genus-zero curves; see Subsections \[curve1bl\_subs\] and \[curve0bl\_subs\]. The former appear in Subsection \[map1blconstr\_subs\] as the first factor in the domain of the proper transforms of the immersion $\io_{\si}$ of Subsection \[descr\_subs\]. The latter play the analogous role in Subsection \[map0blconstr\_subs\], where we blow up certain moduli spaces of genus-zero maps. In turn, these last blowups describe the second factor of the domain of maps induced by $\io_{\si}$ in Subsection \[map1blconstr\_subs\]; see Subsection \[outline\_subs\] for more details.\
We begin by introducing convenient terminology and reviewing standard facts from algebraic geometry. If $\ov\cM$ is a smooth variety and $Z$ is a smooth subvariety of $\ov\cM$, let $$\N_{\ov\cM}Z\equiv T\ov\cM|_Z\big/TZ$$ be the normal bundle of $Z$ in $\ov\cM$. We denote by $$\pi_Z^{\perp}\!: T\ov\cM|_Z \lra \N_{\ov\cM}Z$$ the quotient projection map.
\[ag\_dfn1\] Let $\ov\cM$ be a smooth variety.\
(1) Smooth subvarieties $X$ and $Y$ of $\ov\cM$ [intersect properly]{} if $X\!\cap\!Y$ is a smooth subvariety of $\ov\cM$ and $$T(X\!\cap\!Y)=TX|_{X\cap Y}\cap TY|_{X\cap Y}.
\footnote{In other words, the scheme-theoretic intersection of $X$ and $Y$ is smooth.
If the set-theoretic intersection $X\!\cap\!Y$ is smooth,
the second part of this condition is also equivalent to the injectivity of
the natural homomorphism $$TX|_{X\cap Y}/T(X\!\cap\!Y)\lra T\ov\cM/TY.$$ }$$ (2) If $Z$ is a smooth subvariety of $\ov\cM$, properly intersecting subvarieties $X$ and $Y$ of $\ov\cM$ [intersect properly relative to $Z$]{} if $$\pi_Z^{\perp}\big( T(X\!\cap\!Y)|_{X\cap Y\cap Z}\big)
=\pi_Z^{\perp}\big( TX|_{X\cap Y\cap Z}\big)\cap \pi_Z^{\perp}\big( TY|_{X\cap Y\cap Z}\big)
\subset \N_{\ov\cM}Z.$$\
For example, if $X$ and $Y$ are two smooth curves in a projective space that intersect without being tangent to each other, then $X$ and $Y$ intersect properly (but not transversally, unless the dimension of the projective space is $2$). If $X$, $Y$, and $Z$ are three distinct lines that lie a plane, then they intersect properly pairwise, but $X$ and $Y$ do not intersect properly relative to $Z$.
\[subvarcoll\_dfn\] If $\ov\cM$ is a smooth variety, a collection $\{\ov\cM_{\rho}\}_{\rho\in\A}$ of smooth subvarieties is [properly intersecting]{} if $\ov\cM_{\rho_1}$ and $\ov\cM_{\rho_2}$ intersect properly relative to $\ov\cM_{\rho_3}$ for all $\rho_1,\rho_2,\rho_3\!\in\!\A$.
If $Z$ is a smooth subvariety of $\ov\cM$, let $$\pi\!:\Bl_Z\ov\cM\lra\ov\cM$$ be the blowup of $\ov\cM$ along $Z$. If $X$ is a subvariety of $\ov\cM$, we denote by $\Pr_ZX$ the proper transform of $X$ in $\Bl_Z\ov\cM$, i.e. the closure of $\pi^{-1}(X\!-\!Z)$ in $\Bl_Z\ov\cM$.
\[ag\_lmm1\] Let $\ov\cM$ be a smooth variety.\
(1) If $X$ and $Z$ are properly intersecting subvarieties of $\ov\cM$, then $\Pr_ZX$ is a smooth subvariety of $\Bl_Z\ov\cM$ and $$\Pr_ZX=\Bl_{X\cap Z}X.$$ (2) If $X$, $Y$, and $Z$ are pairwise properly intersecting subvarieties of $\ov\cM$ and $X$ and $Y$ intersect properly relative to $Z$, then $\Pr_ZX$ and $\Pr_ZY$ are properly intersecting subvarieties of $\Pr_Z\ov\cM$ and $$\Pr_ZX \cap \Pr_ZY = \Pr_Z(X\!\cap\!Y).$$ (3) If $X$, $Y$, $Z$, and $Z'$ are pairwise properly intersecting subvarieties of $\ov\cM$ and $X$ and $Y$ intersect properly relative to $Z$ and $Z'$, then $\Pr_ZX$ and $\Pr_ZY$ intersect properly relative to $\Pr_ZZ'$.
\[subvarcoll\_crl\] If $\ov\cM$ is a smooth variety, $\{\ov\cM_{\rho}\}_{\rho\in\A}$ is a properly intersecting collection of subvarieties of $\ov\cM$, and $\rho\!\in\!\A$, then $\{\Pr_{\ov\cM_{\rho}}\ov\cM_{\rho'}\}_{\rho'\in\A-\{\rho\}}$ is a properly intersecting collection of subvarieties of $\Bl_{\ov\cM_{\rho}}\ov\cM$.
Moduli Spaces of Genus-One and Zero Curves {#curvebldata_subs}
------------------------------------------
In this subsection, we describe natural subvarieties of moduli spaces of genus-one and -zero curves and natural bundle sections over these moduli spaces. These bundle sections and their twisted versions introduced in the next two subsections are used in Subsections \[map0blconstr\_subs\] and \[map1blconstr\_subs\] to describe the structure of the proper transforms of $\ov\M_{1,k}^0(\Pn,d)$. Below we also state the now-standard facts about these objects that are used in the next two subsections.\
If $I$ is a finite set, let $$\label{g0and1curvsubv_e}\begin{split}
\A_1(I) &=\big\{\big(I_P,\{I_k\!:k\!\in\!K\}\big)\!:
K\!\neq\!\eset;~I\!=\!\bigsqcup_{k\in\{P\}\sqcup K}\!\!\!\!\!I_k;~
|I_k|\!\ge\!2 ~\forall\, k\!\in\!K\big\};\\
\A_0(I) &=\big\{\big(I_P,\{I_k\!:k\!\in\!K\}\big)\!:
K\!\neq\!\eset;~I\!=\!\bigsqcup_{k\in\{P\}\sqcup K}\!\!\!\!\!I_k;~
|I_k|\!\ge\!2 ~\forall\, k\!\in\!K;~|K|\!+\!|I_P|\!\ge\!2\big\}.
\end{split}$$ If $\rho\!=\!(I_P,\{I_k\!:k\!\in\!K\})$ is an element of $\{(I,\eset)\}\!\sqcup\!\A_1(I)$, we denote by $\cM_{1,\rho}$ the subset of $\ov\cM_{1,I}$ consisting of the stable curves $\cC$ such that\
${}\quad$ (i) $\cC$ is a union of a smooth torus and $|K|$ projective lines, indexed by $K$;\
${}\quad$ (ii) each line is attached directly to the torus;\
${}\quad$ (iii) for each $k\!\in\!K$, the marked points on the line corresponding to $k$ are indexed by $I_k$.\
Let $\ov\cM_{1,\rho}$ be the closure of $\cM_{1,\rho}$ in $\ov\cM_{1,I}$. Figure \[g1curv\_fig\] illustrates this definition, from the points of view of symplectic topology and of algebraic geometry. In the first diagram, each circle represents a sphere, or $\P^1$. In the second diagram, the irreducible components of $\cC$ are represented by curves, and the integer next to each component shows its genus. Similarly, if $$\rho\!=\!(I_P,\{I_k\!:k\!\in\!K\})\in
\big\{(I,\eset)\big\}\!\sqcup\!\A_0(I),$$ let $\cM_{0,\rho}$ be the subset of $\ov\cM_{0,\{0\}\sqcup I}$ consisting of the stable curves $\cC$ such that\
${}\quad$ (i) the components of $\cC$ are indexed by the set $\{P\}\!\sqcup\!K$;\
${}\quad$ (ii) for each $k\!\in\!K$, the component $\cC_k$ of $\cC$ is attached directly to $\cC_P$;\
${}\quad$ (iii) for each $k\!\in\!K$, the marked points on $\cC_k$ are indexed by $I_k$.\
We denote by $\ov\cM_{0,\rho}$ the closure of $\cM_{0,\rho}$ in $\ov\cM_{0,\{0\}\sqcup I}$. This definition is illustrated in Figure \[g0curv\_fig\]. In this case, we do not indicate the genus of the irreducible components in the second diagram, as all of the curves are rational.
(-1.1,-1.8)(10,1.3) (0,-4)[(5,-1.5)(2.5,1.5)(7.5,-1.5)[.2]{}(2.5,-1.5)[.2]{} (5,-3.3)[2]{}[60]{}[120]{}(5,0.3)[2]{}[240]{}[300]{} (5,-4)[1]{}(5,-3)[.15]{} (5,-5)[.2]{}(4,-4)[.2]{}(6,-4)[.2]{} (6.83,.65)[1]{}(6.44,-.28)[.15]{} (5.9,1.04)[.2]{}(7.76,.26)[.2]{} (3.17,.65)[1]{}(3.56,-.28)[.15]{} (4.1,1.04)[.2]{}(2.24,.26)[.2]{}]{} (7,.4)(2.4,-3.7) (1.1,-2.7)(2,.3) (3.1,.5)(6,1.8) (7.7,-2.5)(7.7,-4.2) (5.3,-4.5) (15,-1)[3]{}[-60]{}[60]{}(16.93,1.3)[.2]{}(16.93,-3.3)[.2]{} (17.6,1.4) (17.6,-3.4) (16.8,0)(22.05,1.25)(18.9,.5)[.2]{}(21,1)[.2]{} (18.9,1.2)(21,1.7) (17,-1)(22,-1)(19.5,-1)[.2]{}(21,-1)[.2]{} (19.5,-.3)(21,-.3) (16.8,-2)(22.05,-3.25)(18.9,-2.5)[.2]{} (19.95,-2.75)[.2]{}(21,-3)[.2]{} (18.9,-1.8)(19.95,-2.05) (21,-2.3) (16.1,-3.5)[$1$]{}(22.5,1.2)[$0$]{}(22.4,-1)[$0$]{}(22.5,-3.2)[$0$]{} (32,-1)
---------------------------
$I_P\!=\!\{i_1,i_2\}$
$K\!=\!\{1,2,3\}$
$I_1\!=\!\{i_3,i_4\}$
$I_2\!=\!\{i_5,i_6\}$
$I_3\!=\!\{i_7,i_8,i_9\}$
---------------------------
(-1.1,-1.8)(10,1.3) (5,-1.5)[1.5]{} (5,-3)[.2]{}(4.9,-3.6) (6.06,-2.56)[.2]{}(6.6,-2.9) (2.5,-1.5)[1]{}(3.5,-1.5)[.15]{} (2.5,-.5)[.2]{}(2.9,0) (2.5,-2.5)[.2]{}(2.3,-3) (5,1)[1]{}(5,0)[.15]{} (4,1)[.2]{}(3.6,.7) (6,1)[.2]{}(6.6,1) (7.5,-1.5)[1]{}(6.5,-1.5)[.15]{} (7.5,-.5)[.2]{}(7.9,0) (8.5,-1.5)[.2]{}(9.1,-1.5) (7.5,-2.5)[.2]{}(8,-2.9) (18,2)(18,-4)(18,1.5)[.2]{}(17.5,1.3) (18,-3.5)[.2]{}(17.5,-3.6) (16.8,0)(22.05,1.25)(18.9,.5)[.2]{}(21,1)[.2]{} (18.9,1.2)(21,1.7) (17,-1)(22,-1)(19.5,-1)[.2]{}(21,-1)[.2]{} (19.5,-.3)(21,-.3) (16.8,-2)(22.05,-3.25)(18.9,-2.5)[.2]{} (19.95,-2.75)[.2]{}(21,-3)[.2]{} (18.9,-1.8)(19.95,-2.05) (21,-2.3) (32,-1)
\[curvstr\_lmm\] If $g\!=\!0,1$ and $I$ is a finite set, the collection $\{\ov\cM_{g,\rho}\}_{\rho\in\A_g(I)}$ is properly intersecting.
We define a partial ordering on the sets $\A_g(I)$ for $g\!=\!0,1$ by setting $$\label{partorder_e}
\rho'\!\equiv\!\big(I_P',\{I_k'\!: k\!\in\!K'\}\big)
\prec \rho\!\equiv\!\big(I_P,\{I_k\!: k\!\in\!K\}\big)$$ if $\rho'\!\neq\!\rho$ and there exists a map $\vph\!:K\!\lra\!K'$ such that $I_k\!\subset\!I_{\vph(k)}'$ for all $k\!\in\!K$. This condition means that the elements of $\cM_{\rho'}$ can be obtained from the elements of $\cM_{\rho}$ by moving more points onto the bubble components or combining the bubble components; see Figure \[partorder\_fig\]. In the $g\!=\!0$ case, we define the bubble components to be the components not containing the marked point $0$.\
(-1.1,-1.8)(10,1.3) (-2,-1)[3]{}[-60]{}[60]{}(-.07,1.3)[.2]{}(-.07,-3.3)[.2]{} (.6,1.4)
$i_1$
(.6,-3.4)
$i_2$
(-.2,0)(6.05,1.25)(1.8,.4)[.2]{}(3.05,.65)[.2]{} (4.3,.9)[.2]{}(5.55,1.15)[.2]{} (1.8,1.1)
$i_3$
(2.9,0)
$i_4$
(4.2,1.6)
$i_5$
(5.5,.5)
$i_6$
(-.2,-2)(5.05,-3.25)(1.9,-2.5)[.2]{} (2.95,-2.75)[.2]{}(4,-3)[.2]{} (1.9,-1.8)
$i_7$
(2.95,-2.05)
$i_8$
(4,-2.3)
$i_9$
(-.9,-3.5)[$1$]{}(6.45,1.3)[$0$]{}(5.5,-3.2)[$0$]{} (8,-1)
$\prec$
(9,-1)[3]{}[-60]{}[60]{}(10.93,1.3)[.2]{}(10.93,-3.3)[.2]{} (11.6,1.4)
$i_1$
(11.6,-3.4)
$i_2$
(10.8,0)(17.05,1.25)(14.05,.65)[.2]{} (15.3,.9)[.2]{}(16.55,1.15)[.2]{} (12,-1)[.2]{}(12.7,-.8)
$i_3$
(13.9,0)
$i_4$
(15.2,1.6)
$i_5$
(16.5,.5)
$i_6$
(10.8,-2)(16.05,-3.25)(12.9,-2.5)[.2]{} (13.95,-2.75)[.2]{}(15,-3)[.2]{} (12.9,-1.8)
$i_7$
(13.95,-2.05)
$i_8$
(15,-2.3)
$i_9$
(10.1,-3.5)[$1$]{}(17.45,1.3)[$0$]{}(16.5,-3.2)[$0$]{} (19,-1)[3]{}[-60]{}[60]{}(20.93,1.3)[.2]{}(20.93,-3.3)[.2]{} (21.6,1.4)
$i_1$
(21.6,-3.4)
$i_2$
(20.8,0)(27.05,1.25)(22.8,.4)[.2]{}(24.05,.65)[.2]{} (25.3,.9)[.2]{}(26.55,1.15)[.2]{} (22.8,1.1)
$i_3$
(23.9,0)
$i_4$
(25.2,1.6)
$i_5$
(26.5,.5)
$i_6$
(21.95,-.48)[.2]{}(21.4,-.7)
$i_7$
(21.95,-1.52)[.2]{}(22.6,-1.6)
$i_8$
(21.6,-2.5)[.2]{}(21,-2.4)
$i_9$
(20.1,-3.5)[$1$]{}(27.45,1.3)[$0$]{} (29,-1)[3]{}[-60]{}[60]{}(30.93,1.3)[.2]{}(30.93,-3.3)[.2]{} (31.6,1.4)
$i_1$
(31.6,-3.4)
$i_2$
(30.8,0)(36.05,1.25)(32.9,.5)[.2]{}(35,1)[.2]{} (32.9,1.2)
$i_3$
(35,1.7)
$i_4$
(31,-1)(36,-1)(33.5,-1)[.2]{}(35,-1)[.2]{} (33.5,-.3)
$i_5$
(35,-.3)
$i_6$
(30.8,-2)(36.05,-3.25)(32.9,-2.5)[.2]{} (33.95,-2.75)[.2]{}(35,-3)[.2]{} (32.9,-1.8)
$i_7$
(33.95,-2.05)
$i_8$
(35,-2.3)
$i_9$
(30.1,-3.5)[$1$]{}(36.5,1.2)[$0$]{}(36.4,-1)[$0$]{}(36.5,-3.2)[$0$]{}
In the blowup constructions of the next two subsections we will twist certain line bundles over moduli spaces of curves and homomorphisms between them. In the rest of this subsection we describe the relevant starting data.\
For each $i\!\in\!I$, let $L_i\!\lra\!\ov\cM_{1,I}$ be the universal tangent line bundle at the marked point labeled $i$. Let $\E\!\lra\!\ov\cM_{1,I}$ be the Hodge line bundle of holomorphic differentials. The natural pairing of tangent vectors with cotangent vectors induces a section $$s_i\in\Ga\big(\ov\cM_{1,I};\Hom(L_i,\E^*)\big).$$ Explicitly, $$\begin{gathered}
\big\{s_i([\cC;w])\big\}([\cC,\psi])=\psi_{x_i(\cC)}w \qquad\hbox{if}\\
[\cC]\!\in\!\ov\cM_{1,I}, \quad [\cC,w]\!\in\!L_i|_{\cC}\!=\!T_{x_i(\cC)}\cC,
\quad [\cC,\psi]\!\in\!\E|_{\cC}\!=\!H^0(\cC;T^*\cC),\end{gathered}$$ and $x_i(\cC)\!\in\!\cC$ is the marked point on $\cC$ labeled by $i$.\
In the genus-zero case, the line bundle $L_0\!\lra\!\ov\cM_{0,\{0\}\sqcup I}$ will be one of the substitutes for $\E$. We note that for every $p\!\in\!\P^1$, there is a natural isomorphism between the tangent space $T_p\P^1$ of $\P^1$ at $p$ and the space of holomorphic differentials $H^0(\P^1;T^*\P^1\!\otimes\!\O(2p))$ on $\P^1$ that have a pole of order two at $p$. More precisely, let $w$ be a meromorphic function on $\P^1$ such that $p$ is the only zero of $w$ and this zero is a simple one. We can then view $w$ as a coordinate around $p$ in $\P^1$. Every tangent vector $v\!\in\!T_p\P^1$ can be written as $$v=c_w(v)\frac{\part}{\part w}, \qquad c_w(v)\in\C.$$ We define the isomorphism $$\psi\!: T_p\P^1 \lra H^0(\P^1;T^*\P^1\!\otimes\!\O(2p)) \qquad\hbox{by}\qquad
v \lra \psi_v=\frac{c_w(v)\, dw}{w^2}.$$ If $w'$ is another meromorphic function on $\P^1$ such that $p$ is the only zero of $w'$ and this zero is a simple one, then $$w'=\frac{w}{\al w\!+\!\be} \quad\Lra\quad
dw'=\frac{\be\, dw}{(\al w\!+\!\be)^2} , ~~
c_{w'}(v)= \frac{c_w(v)}{\be}
\quad\Lra\quad
\frac{c_{w'}(v)\, dw'}{{w'}^2} = \frac{c_w(v)\, dw}{w^2}.$$ Thus, the isomorphism $\psi$ is well-defined. If $i\!\in\!I$, we define the section $$\begin{gathered}
s_i \in \Ga\big(\ov\cM_{0,\{0\}\sqcup I};\Hom(L_i,L_0^*)\big)
\quad\hbox{by}\quad
\big\{s_i([\cC;w])\big\}([\cC,v])=\psi_v\big|_{x_i(\cC)}w\\
\hbox{if}\qquad
[\cC]\!\in\!\ov\cM_{0,\{0\}\sqcup I}, \quad [\cC,w]\!\in\!L_i|_{\cC}\!=\!T_{x_i(\cC)}\cC,
\quad [\cC,v]\!\in\!L_0|_{\cC}\!=\!T_{x_0(\cC)}\cC.\end{gathered}$$\
We note that in both cases the section $s_i$ vanishes precisely on the curves for which the point $i$ lies on a bubble component. In fact, $$\label{curvezero_e}
s_i^{-1}(0)=\!\sum_{\rho\in\B_g(I;i)}\!\!\!\!\!\ov\cM_{1,\rho},
\qquad\hbox{where}\qquad
\B_g(I;i)=\big\{ \big(I_P,\{I_B\}\big)\!\in\!\A_g(I) \!: i\!\in\!I_B\big\}.$$
A Blowup of a Moduli Space of Genus-One Curves {#curve1bl_subs}
----------------------------------------------
Let $I$ and $J$ be finite sets such that $I$ is nonempty. In this subsection, we construct a blowup $$\pi_{1,(I,J)}\!:\wt\cM_{1,(I,J)} \lra \ov\cM_{1,I\sqcup J}$$ of the moduli space $\ov\cM_{1,I\sqcup J}$, $|I|\!+\!1$ line bundles $$\ti\E,\, \ti{L}_i \lra \wt\cM_{1,(I,J)}, \qquad i\!\in\!I,$$ and $|I|$ nowhere vanishing sections $$\ti{s}_i\in\Ga\big(\wt\cM_{1,(I,J)};\Hom(\ti{L}_i,\ti\E^*)\big),
\qquad i\!\in\!I.$$ Since the sections $\ti{s}_i$ do not vanish, all $|I|\!+\!1$ bundles $\ti{L}_i$ and $\ti\E^*$ are explicitly isomorphic. They will be denoted by $\L$ and called the universal tangent line bundle.\
The smooth variety $\wt\cM_{1,(I,J)}$ is obtained by blowing up some of the subvarieties $\ov\cM_{1,\rho}$, defined in the previous subsection, and their proper transforms in an order consistent with the partial ordering $\prec$. The line bundle $\ti\E$ is the sum of the Hodge line bundle $\E$ and all exceptional divisors. For each given $i\!\in\!I$, $\ti{L}_i$ is the tangent line bundle $L_i$ for the marked point $i$ minus some of these divisors. The section $\ti{s}_i$ is induced from the pairing $s_i$ of the previous subsection.\
With $I$ and $J$ as above and $\A_g(I\!\sqcup\!J)$ as in \_ref[g0and1curvsubv\_e]{}, let $$\label{smallcolldfn_e}
\A_g(I,J) =\big\{\big((I_P\!\sqcup\!J_P),\{I_k\!\sqcup\!J_k\!: k\!\in\!K\}\big)\!\in\!\A_g(I\!\sqcup\!J)\!:
~I_k\!\neq\!\eset~ \forall\, k\!\in\!K \big\}.$$ We note that if $\rho\!\in\!\A_g(I\!\sqcup\!J)$, then $\rho\!\in\!\A_g(I,J)$ if and only if every bubble component of an element of $\cM_{\rho}$ carries at least one element of $I$. Furthermore, $$\label{curve1zero_e}
\B_g(I\!\sqcup\!J;i)\subset \A_g(I,J) \qquad\forall\,i\!\in\!I.$$ If $|I|\!+\!|J|\!\ge\!2$, with respect to the partial ordering $\prec$ the set $\A_1(I,J)$ has a unique minimal element: $$\rho_{\min}\equiv\big(\eset,\{I\!\sqcup\!J\}\big).$$ Let $<$ be an ordering on $\A_1(I,J)$ extending the partial ordering $\prec$. We denote the corresponding maximal element by $\rho_{\max}$. If $\rho\!\in\!\A_1(I,J)$, we put $$\label{minusdfn_e}
\rho\!-\!1=
\begin{cases}
\max\{\rho'\!\in\!\A_1(I,J)\!: \rho'\!<\!\rho\},&
\hbox{if}~ \rho\!\neq\!\rho_{\min};\\
0,& \hbox{if}~\rho\!=\!\rho_{\min},
\end{cases}$$ where the maximum is taken with respect to the ordering $<$.\
We now describe the starting data for the inductive blowup procedure involved in constructing the space $\wt\cM_{1,(I,J)}$ and the line bundle $\L$ over $\wt\cM_{1,(I,J)}$. Let $$\ov\cM_{1,(I,J)}^0=\ov\cM_{1,I\sqcup J}, \qquad
\E_0=\E\lra\ov\cM_{1,(I,J)}^0, \quad\hbox{and}\quad
\ov\cM_{1,\rho}^0=\ov\cM_{1,\rho} ~~~\forall \,\rho\!\in\!\A_1(I,J).$$ For each $i\!\in\!I$, let $$L_{0,i}=L_i\lra\ov\cM_{1,(I,J)}^0 \qquad\hbox{and}\qquad
s_{0,i}=s_i\in\Ga\big(\ov\cM_{1,(I,J)}^0;\Hom(L_{0,i},\E_0^*)\big).$$ By \_ref[curvezero\_e]{}, $$s_{0,i}^{\,-1}(0)=\!\sum_{\rho^*\in\B_1(I\sqcup J;i)}\!\!\!\!\!\!\!\! \ov\cM_{1,\rho^*}^0.$$\
Suppose $\rho\!\in\!\A_1(I,J)$ and we have constructed\
${}\quad$ ($I1$) a blowup $\pi_{\rho-1}\!:\ov\cM_{1,(I,J)}^{\rho-1}\!\lra\!\ov\cM_{1,(I,J)}^0$ of $\ov\cM_{1,(I,J)}^0$ such that $\pi_{\rho-1}$ is an isomorphism outside of the preimages of the spaces $\ov\cM_{1,\rho'}^0$ with $\rho'\!\le\!\rho\!-\!1$;\
${}\quad$ ($I2$) line bundles $L_{\rho-1,i}\!\lra\!\ov\cM_{1,(I,J)}^{\rho-1}$ for $i\!\in\!I$ and $\E_{\rho-1}\!\lra\!\ov\cM_{1,(I,J)}^{\rho-1}$;\
${}\quad$ ($I3$) sections $s_{\rho-1,i}\!\in\!
\Ga(\ov\cM_{1,(I,J)}^{\rho-1};\Hom(L_{\rho-1,i},\E_{\rho-1}^*))$ for $i\!\in\!I$.\
For each $\rho^*\!>\!\rho\!-\!1$, let $\ov\cM_{1,\rho^*}^{\rho-1}$ be the proper transform of $\ov\cM_{1,\rho^*}^0$ in $\ov\cM_{1,(I,J)}^{\rho-1}$. We assume that\
${}\quad$ ($I4$) the collection $\{\ov\cM_{1,\rho^*}^{\rho-1}\}_{\rho^*\in\A_1(I,J),\rho^*>\rho-1}$ is properly intersecting;\
${}\quad$ ($I5$) for all $i\!\in\!I$, $$s_{\rho-1,i}^{\,-1}(0)=\sum_{\rho^*\in\B_1(I\sqcup J;i),\rho^*>\rho-1}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \ov\cM_{1,\rho^*}^{\rho-1}.$$ The assumption ($I5$) means that we will gradually be killing the zero locus of the section $s_i$. We note that all five assumptions are satisfied if $\rho\!-\!1$ is replaced by $0$.\
If $\rho$ is as above, let $$\ti\pi_{\rho}\!:\ov\cM_{1,(I,J)}^{\rho}\lra\ov\cM_{1,(I,J)}^{\rho-1}$$ be the blowup of $\ov\cM_{1,(I,J)}^{\rho-1}$ along $\ov\cM_{1,\rho}^{\rho-1}$. We denote by $\ov\cM_{1,\rho}^{\rho}$ the corresponding exceptional divisor. If $\rho^*\!>\!\rho$, let $\ov\cM_{1,\rho^*}^{\rho}\!\subset\!\ov\cM_{1,(I,J)}^{\rho}$ be the proper transform of $\ov\cM_{1,\rho^*}^{\rho-1}$. If $$\label{rhodfn_e}
\rho=\big(I_P\!\sqcup\!J_P,\{I_k\!\sqcup\!J_k\!: k\!\in\!K\}\big)$$ and $i\!\in\!I$, we put $$\label{bundtwist_e}
L_{\rho,i}=\begin{cases}
\ti\pi_{\rho}^{\,*}L_{\rho-1,i},& \hbox{if}~ i\!\not\in\!I_P;\\
\ti\pi_{\rho}^{\,*}L_{\rho-1,i}\otimes\O(-\ov\cM_{1,\rho}^{\rho}),& \hbox{if}~ i\!\in\!I_P;
\end{cases} \qquad
\E_{\rho}=\ti\pi_{\rho}^{\,*}\,\E_{\rho-1}\otimes\O(\ov\cM_{1,\rho}^{\rho}).$$ The section $\ti\pi_{\rho}^{\,*}s_{\rho-1,i}$ induces a section $$\ti{s}_{\rho,i}\in
\Ga\big(\ov\cM_{1,(I,J)}^{\rho};\Hom(L_{\rho,i},\ti\pi_{\rho}^{\,*}\E_{\rho-1}^*)\big).$$ This section vanishes along $\ov\cM_{1,\rho}^{\rho}$, by the inductive assumption ($I5$) if $i\!\not\in\!I_P$. Thus, $\ti{s}_{\rho,i}$ induces a section $$s_{\rho,i}\in \Ga\big(\ov\cM_{1,(I,J)}^{\rho};\Hom(L_{\rho,i},\E_{\rho}^*)\big).$$ We have now described the inductive step of the procedure. It is immediate that the requirements ($I1$)-($I3$) and ($I5$) are satisfied, with $\rho\!-\!1$ replaced by $\rho$, are satisfied. Corollary \[subvarcoll\_crl\] and the assumption ($I4$) imply that the assumption ($I4$) with $\rho\!-\!1$ replaced by $\rho$ is also satisfied.\
We conclude the blowup construction after the $\rho_{\max}$ step. Let $$\wt\cM_{1,(I,J)}=\ov\cM_{1,(I,J)}^{\rho_{\max}}; \qquad
\ti\E=\E_{\rho_{\max}}; \qquad
\ti{L}_i=L_{\rho_{\max},i},\quad
\ti{s}_i=s_{\rho_{\max},i} \qquad\forall\, i\!\in\!I.$$ By ($I5$), with $\rho\!-\!1$ replaced by $\rho_{\max}$, and \_ref[curve1zero\_e]{}, the section $\ti{s}_i$ does not vanish. We note that by ($I1$), the stratum $$\cM_{1,(I,J)}\subset\ov\cM_{1,(I,J)}$$ consisting of the smooth curves is a Zariski open subset of $\ov\cM_{1,(I,J)}^{\rho}$ for all $\rho\!\in\!\{0\}\!\sqcup\!\A_1(I,J)$.\
By the next lemma, different extensions of the partial order $\prec$ to an order $<$ on $\A_1(I,J)$ correspond to blowing up along disjoint subvarieties in different orders. Thus, the end result of the above blowup construction is well-defined, i.e. independent of the choice of the ordering $<$ extending the partial ordering $\prec$.
\[curve1bl\_lmm\] Suppose $\rho,\rho'\!\in\!\A_1(I,J)$ are such that $\rho\!\not\prec\!\rho'$ and $\rho'\!\not\prec\!\rho$. If $\rho\!\neq\!\rho'$, then the spaces $\ov\cM_{1,\rho}^{\ti\rho}$ and $\ov\cM_{1,\rho'}^{\ti\rho}$ are disjoint for some $\ti\rho\!\prec\!\rho,\rho'$.
[*Proof:*]{} (1) Suppose $$\rho=\big(I_P\!\sqcup\!J_P,\{I_k\!\sqcup\!J_k\!: k\!\in\!K\}\big)
\qquad\hbox{and}\qquad
\rho'=\big(I_P'\!\sqcup\!J_P',\{I_k'\!\sqcup\!J_k'\!: k\!\in\!K'\}\big).$$ For each $k\!\in\!K$ and $k'\!\in\!K'$, let $$\begin{gathered}
\rho_k\!=\!\big((I\!-\!I_k)\!\sqcup\!(J\!-\!J_k),\{I_k\!\sqcup\!J_k\}\big)\in \A_1(I,J)
\qquad\hbox{and}\\
\rho_{k'}'\!=\!\big((I\!-\!I_{k'}')\!\sqcup\!(J\!-\!J_{k'}'),\{I_{k'}'\!\sqcup\!J_{k'}'\}\big)
\in \A_1(I,J).\end{gathered}$$ By definition, $\ov\cM_{1,\rho_k}^0$ and $\ov\cM_{1,\rho_{k'}'}^0$ are divisors in $\ov\cM_{1,(I,J)}^0\!=\!\ov\cM_{1,I\sqcup J}$, $$\ov\cM_{1,\rho}^0=\bigcap_{k\in K}\ov\cM_{1,\rho_k}^0,
\qquad\hbox{and}\qquad
\ov\cM_{1,\rho'}^0=\bigcap_{k'\in K'}\ov\cM_{1,\rho_{k'}'}^0.$$ Furthermore, if $\ov\cM_{1,\rho_k}^0\!\cap\!\ov\cM_{1,\rho_{k'}'}^0\!\neq\!\eset$, then either $$I_k\!\sqcup\!J_k \subset I_{k'}'\!\sqcup\!J_{k'}',
\qquad\hbox{or}\qquad
I_k\!\sqcup\!J_k \supset I_{k'}'\!\sqcup\!J_{k'}',
\qquad\hbox{or}\qquad
(I_k\!\sqcup\!J_k) \cap (I_{k'}'\!\sqcup\!J_{k'}')=\eset.$$ (2) Suppose $\ov\cM_{1,\rho}^0\!\cap\!\ov\cM_{1,\rho'}^0\!\neq\!\eset$. By the above, there exist decompositions $$K=K_+\sqcup K_0\sqcup \bigsqcup_{l'\in K_+'}\!\!\!K_{l'}
\qquad\hbox{and}\qquad
K'=K_+'\sqcup K_0'\sqcup \bigsqcup_{l\in K_+}\!\!\!K_l'$$ and a bijection $\vph\!:K_0\!\lra\!K_0'$ such that $$\begin{gathered}
I_k\!\sqcup\!J_k \subsetneq I_{l'}'\!\sqcup\!J_{l'}'
\quad\forall\, k\!\in\!K_{l'},\, l'\!\in\!K_+', \qquad
I_l\!\sqcup\!J_l\supsetneq I_{k'}'\!\sqcup\!J_{k'}'
\quad\forall\, k'\!\in\!K_l',\, l\!\in\!K_+,\\
\hbox{and}\qquad
I_k\!\sqcup\!J_k = I_{\vph(k)}'\!\sqcup\!J_{\vph(k)}' \quad\forall\, k\!\in\!K_0.\end{gathered}$$ We note that the subsets $K_+$ and $K_+'$ of $K$ and $K'$ are nonempty. For example, if $K_+$ were empty, then we would have $\rho'\!\prec\!\rho$, contrary to our assumptions. Let $$\ti\rho= \big( \ti{I}_P\!\sqcup\!\ti{J}_P,
\{\ti{I}_k\!\sqcup\!\ti{J}_k\!: k\!\in\!K_0\!\sqcup\!K_+\!\sqcup\!K_+'\}\big)
\in \A_1(I,J)$$ be given by $$\ti{I}_P \sqcup \ti{J}_P = (I_P\!\cap\!I_P') \sqcup (J_P\!\cap\!J_P'),
\qquad
\ti{I}_k\!\sqcup\!\ti{J}_k =\begin{cases}
I_k\!\sqcup\!J_k,& \hbox{if}~ k\!\in\!K_0\!\sqcup\!K_+;\\
I_k'\!\sqcup\!J_k',& \hbox{if}~ k\!\in\!K_+'.\\
\end{cases}$$ For example, if $\rho$ corresponds to the second diagram on the right side of Figure \[partorder\_fig\] and $\rho'$ corresponds to either the first or the third diagram on the right side, then $\ti\rho$ corresponds to the diagram on the left side of Figure \[partorder\_fig\]. By definition, $\ti\rho\!\prec\!\rho,\rho'$. Furthermore, $$\ov\cM_{1,\rho}^0\!\cap\!\ov\cM_{1,\rho'}^0 \subset \ov\cM_{1,\ti\rho}^0.$$ Thus, by Lemma \[curvstr\_lmm\], Corollary \[subvarcoll\_crl\], and (2) of Lemma \[ag\_lmm1\], $$\ov\cM_{1,\rho}^{\ti\rho}\cap\ov\cM_{1,\rho'}^{\ti\rho}
\subset \ov\cM_{1,(I,J)}^{\ti\rho}$$ is the closure of the empty set.
A Blowup of a Moduli Space of Genus-Zero Curves {#curve0bl_subs}
-----------------------------------------------
Suppose $\ale$ is a nonempty finite set and $\vr\!=\!(I_l,J_l)_{l\in\ale}$ is a tuple of finite sets such that $I_l\!\neq\!\eset$ and $|I_l|\!+\!|J_l|\!\ge\!2$ for all $l\!\in\!\ale$. Let $$\ov\cM_{0,\vr}= \prod_{l\in\ale}\! \ov\cM_{0,\{0\}\sqcup I_l\sqcup J_l}
\qquad\hbox{and}\qquad
F_{\vr}=\bigoplus_{l\in\ale} \pi_l^*L_0 \lra \ov\cM_{0,\vr},$$ where $L_0\!\lra\!\ov\cM_{0,\{0\}\sqcup I_l\sqcup J_l}$ is the universal tangent line bundle for the marked point $0$ and $$\pi_l\!: \ov\cM_{0,\vr} \lra \ov\cM_{0,\{0\}\sqcup I_l\sqcup J_l}$$ is the projection map. In this subsection, we construct a blowup $$\pi_{0,\vr}\!:\wt\cM_{0,\vr}\lra \P F_{\vr}$$ of the projective bundle $\P F_{\vr}$ over $\ov\cM_{0,\vr}$. We also construct line bundles $$\ti\E,\, \ti{L}_{(l,i)}\lra \wt\cM_{0,\vr},
\qquad i\!\in\!I_l,\, l\!\in\!\ale,$$ and nowhere vanishing sections $$\ti{s}_{(l,i)}\in\Ga\big(\wt\cM_{0,\vr};\Hom(\ti{L}_{(l,i)},\ti\E^*)\big),
\qquad i\!\in\!I_l,\, l\!\in\!\ale.$$ In particular, all line bundles $\ti{L}_{(l,i)}$ and $\ti\E^*$ are explicitly isomorphic. They will be denoted by $\L$ and called the universal tangent line bundle.\
Similarly to the previous subsection, the smooth variety $\wt\cM_{0,\vr}$ is obtained by blowing up the subvarieties $\ov\cM_{0,\rho}$ defined below and their proper transforms in an order consistent with a natural partial ordering $\prec$. The line bundle $\ti\E$ is the sum of the tautological line bundle $$\ga_{\vr}\lra\P F_{\vr}$$ and all exceptional divisors. For every $l\!\in\!\aleph$ and $i\!\in\!I_l$, $\ti{L}_{(l,i)}$ is $\pi_l^*L_i$ minus some of these divisors. The section $\ti{s}_{(l,i)}$ is induced from the pairings $s_i$ of Subsection \[curvebldata\_subs\].\
With $\vr$ as above and $\A_0(I_l,J_l)$ as in \_ref[smallcolldfn\_e]{}, let $$\label{vr0setdfn_e}\begin{split}
\A_0(\vr) =\big\{ \big(\ale_+,(\rho_l)_{l\in\ale}\big)\!:~
&\ale_+\!\subset\!\ale,~\ale_+\!\neq\!\eset;~
\rho_l\!\in\!\{(I_l\!\sqcup\!J_l,\eset)\}\!\sqcup\!\A_0(I_l,J_l)~\forall\, l\!\in\!\ale;\\
&\rho_l\!=\!(I_l\!\sqcup\!J_l,\eset)~\forall\, l\!\in\!\ale\!-\!\ale_+;
\big(\ale_+,(\rho_l)_{l\in\ale}\big)\!\neq\!
\big(\ale,(I_l\!\sqcup\!J_l,\eset)_{l\in\ale}\big)\big\}.
\end{split}$$ We define a partial ordering on $\A_0(\vr)$ by setting $$\label{rho0dfn_e}
\rho'\!\equiv\!\big(\ale_+',(\rho_l')_{l\in\ale}\big)
\prec \rho\!\equiv\!\big(\ale_+,(\rho_l)_{l\in\ale}\big)$$ if $\rho'\!\neq\!\rho$, $\ale_+'\!\subset\!\ale_+$, and for every $l\!\in\!\ale$ either $\rho_l'\!=\!\rho_l$, $\rho_l'\!\prec\!\rho_l$, or $\rho_l'\!=\!(I_l\!\sqcup\!J_l,\eset)$. Let $<$ be an ordering on $\A_0(\vr)$ extending the partial ordering $\prec$. We denote the corresponding minimal and maximal elements of $\A_0(\vr)$ by $\rho_{\min}$ and $\rho_{\max}$, respectively. If $\rho\!\in\!\A_0(\vr)$, we define $$\rho\!-\!1 \in \{0\} \!\sqcup\! \A_0(\vr)$$ as in \_ref[minusdfn\_e]{}.\
If $\rho\!\in\!\A_0(\vr)$ is as in \_ref[rho0dfn\_e]{}, let $$\begin{gathered}
\ov\cM_{0,\rho}=\prod_{l\in\ale}\ov\cM_{0,\rho_l},
\qquad
F_{\rho}=\bigoplus_{l\in\ale^+} \pi_l^*L_0\big|_{\ov\cM_{0,\rho}}
\subset F_{\vr},\\
\hbox{and}\qquad
\wt\cM_{0,\rho}^0\!=\!\P F_{\rho} \subset \wt\cM_{0,\vr}^0\!\equiv\!\P F_{\vr}.\end{gathered}$$ The spaces $\wt\cM_{0,\vr}^0$ and $\wt\cM_{0,\rho}^0$ can be represented by diagrams as in Figure \[g0curv\_fig2\]. The trees of circles attached to the vertical lines correspond to the tuples $\rho_l$, with conventions as in the first, symplectic-topology, diagram in Figure \[g0curv\_fig\]. For each such tree, the marked point $0$ is the point on the line. We indicate the elements of $\ale_+\!\subset\!\ale$ with plus signs next to these points. Note that by \_ref[vr0setdfn\_e]{}, every dot on a vertical line for which the corresponding tree of circles contains more than one circle must be labeled with a plus sign. From Lemma \[curvstr\_lmm\], we immediately obtain
(-2,-2.4)(10,1.3) (0,2.5)(0,-5.5) (1.2,1.5)[1.2]{}(0,1.5)[.25]{}(-.8,1.5) (2.05,2.35)[.2]{}(2.05,.65)[.2]{} (1.2,-1.5)[1.2]{}(0,-1.5)[.25]{}(-.8,-1.5) (1.2,-.3)[.2]{}(2.4,-1.5)[.2]{}(1.2,-2.7)[.2]{} (1.2,-4.5)[1.2]{}(0,-4.5)[.25]{}(-.8,-4.5) (1.2,-3.3)[.2]{}(2.4,-4.5)[.2]{}(1.2,-5.7)[.2]{} (7.5,-1.5)
---------------------------------------------------
$\aleph\!=\!\{1,2,3\}$
$|I_1\!\sqcup\!J_1|\!=\!2$
$|I_2\!\sqcup\!J_2|\!=\!|I_3\!\sqcup\!J_3|\!=\!3$
$\aleph_+\!=\!\{1,2,3\}$
$\rho_1\!=\!(I_1\!\sqcup\!J_1,\eset)$
$\rho_2\!=\!(I_2\!\sqcup\!J_2,\eset)$
$\rho_3\!=\!(I_3\!\sqcup\!J_3,\eset)$
---------------------------------------------------
(20,2.5)(20,-5.5) (21.2,1.5)[1.2]{}(20,1.5)[.25]{}(19.2,1.5) (22.05,2.35)[.2]{}(22.05,.65)[.2]{} (21.2,-1.5)[1.2]{}(20,-1.5)[.25]{}(19.2,-1.5) (21.2,-.3)[.2]{} (23.4,-1.5)[1]{}(22.4,-1.5)[.15]{} (24.11,-.79)[.2]{}(24.11,-2.21)[.2]{} (21.2,-4.5)[1.2]{}(20,-4.5)[.25]{}(19.2,-4.5) (21.2,-3.3)[.2]{}(22.4,-4.5)[.2]{}(21.2,-5.7)[.2]{} (29,-1.5)
---------------------------------------------------
$\aleph\!=\!\{1,2,3\}$
$|I_1\!\sqcup\!J_1|\!=\!2$
$|I_2\!\sqcup\!J_2|\!=\!|I_3\!\sqcup\!J_3|\!=\!3$
$\aleph_+\!=\!\{2,3\}$
$\rho_1\!=\!(I_1\!\sqcup\!J_1,\eset)$
$\rho_2\!\neq\!(I_2\!\sqcup\!J_2,\eset)$
$\rho_3\!=\!(I_3\!\sqcup\!J_3,\eset)$
---------------------------------------------------
\[curv0str\_lmm\] Suppose $\ale$ is a nonempty finite set and $\vr\!=\!(I_l,J_l)_{l\in\ale}$ is a tuple of finite sets such that $I_l\!\neq\!\eset$ and $|I_l|\!+\!|J_l|\!\ge\!2$ for all $l\!\in\!\ale$. If $\A_0(\vr)$ is as above, the collection $\{\wt\cM_{0,\rho}\}_{\rho\in\A_0(\vr)}$ is properly intersecting.
We now describe the starting data for the sequential blowup construction of this subsection. Let $$\E_0\!=\!\ga_{\vr} \lra \wt\cM_{0,\vr}^0\!=\!\P F_{\vr}
\qquad\hbox{and}\qquad
L_{0,(l,i)}\!=\!\pi_{0,\vr}^*\pi_l^*L_i \lra \wt\cM_{0,\vr}^0
~~~\forall\,i\!\in\!I_l,\,l\!\in\!\ale.$$ We take $$s_{0,(l,i)}\in\Ga\big(\wt\cM_{0,\vr}^0;\Hom(L_{0,(l,i)},\E_0^*)\big)$$ to be the section induced by $\pi_{0,\vr}^{\,*}\pi_l^*s_i$, where $s_i$ is the natural homomorphism described in Subsection \[curvebldata\_subs\]. It follows immediately from \_ref[curvezero\_e]{} that $$s_{0,(l,i)}^{~-1}(0)=\!\sum_{\rho^*\in\B_0(\vr;l,i)}\!\!\!\!\! \wt\cM_{0,\rho^*}^0,
\qquad\hbox{where}$$ $$\begin{split}
\B_0(\vr;l,i)=\Big\{\big(\ale_+,(\rho_{l'})_{l'\in\ale}\big)\!\in\!\A_0(\vr)\!:
\ale_+\!=\!\ale\!-\!\{l\} ~\hbox{and}~
\rho_{l'}\!=\!(I_{l'}\!\sqcup\!J_{l'},\eset) ~\forall l'\!\in\!\ale, ~~~\hbox{or}&\\
\ale_+\!=\!\ale,~\rho_l\!\in\!\B_0(I_l\!\sqcup\!J_l;i),~
\rho_{l'}\!=\!(I_{l'}\!\sqcup\!J_{l'},\eset) ~\forall l'\!\in\!\ale\!-\!\{l\}&\Big\}.
\end{split}$$\
The rest of the construction proceeds as in Subsection \[curve1bl\_subs\]. The analogue of \_ref[bundtwist\_e]{} now is $$\begin{gathered}
\label{bundletwist_e2a}
L_{\rho;(l,i)}=\begin{cases}
\ti\pi_{\rho}^*L_{\rho-1,(l,i)},& \hbox{if}~
l\!\not\in\!\ale_+ ~\hbox{or}~
\rho_l\!\neq\!(I_l\!\sqcup\!J_l,\eset),\,i\!\not\in\!I_{l,P};\\
\ti\pi_{\rho}^*L_{\rho-1,(l,i)}\otimes
\O(-\wt\cM_{0,\rho}^{\rho}),& \hbox{otherwise};
\end{cases} \\
\label{bundletwist_e2b}
\E_{\rho}=\ti\pi_{\rho}^*\,\E_{\rho-1}\!\otimes\!\O(\wt\cM_{0,\rho}^{\rho}).\end{gathered}$$ As before, we take $$\begin{gathered}
\wt\cM_{0,\vr}=\wt\cM_{0,\vr}^{\rho_{\max}}; \qquad
\ti\E=\E_{\rho_{\max}};\\
\ti{L}_{(l,i)}=L_{\rho_{\max},(l,i)} \quad\hbox{and}\quad
\ti{s}_{(l,i)}=s_{\rho_{\max},(l,i)} \qquad\forall~ i\!\in\!I_l,~l\!\in\!\ale.\end{gathered}$$ The analogue of the inductive assumption ($I5$) insures that each section $\ti{s}_{(l,i)}$ does not vanish. The statement and the proof of Lemma \[curve1bl\_lmm\] remain valid in the present setting, with only minor changes. Thus, the end result of the above blowup construction is again well-defined, i.e. independent of the choice of the ordering $<$ extending the partial ordering $\prec$.
A Blowup of a Moduli Space of Genus-Zero Maps {#map0bl_sec}
=============================================
Blowups and Immersions {#map0prelim_subs1}
----------------------
In this section we construct blowups of certain moduli spaces of genus-zero maps; see Subsections \[map0prelim\_subs\] and \[map0blconstr\_subs\]. As outlined in Subsection \[outline\_subs\], these blowups appear in Subsection \[map1blconstr\_subs\] as the second factor in the domain of the immersions induced by the immersions $\io_{\si}$ of Subsection \[descr\_subs\].\
As in Section \[curvebl\_sec\], we begin by introducing convenient terminology and reviewing standard facts from algebraic geometry. If $\ov\M$ is a variety, we denote its Zariski tangent space and its tangent cone by $T\ov\M$ and $TC\ov\M$, respectively. If $X$ is a smooth variety (but not necessarily equidimensional), we recall that a morphism $\io_X\!:X\!\lra\!\ov\M$ is an [immersion]{} if the differential of $\io_X$, $$d\io_X\!: TX\lra \io_X^*TC\ov\M,$$ is injective at every point of $X$. Let $$\Im^s\,\io_X\equiv \big\{p\!\in\!\ov\M\!: |\io_X^{-1}(p)|\!\ge\!2\big\}
\qquad\hbox{and}\qquad \N_{\io_X}\equiv \io_X^*TC\ov\M\big/ \Im\, d\io_X$$ be [the singular locus of $\io_X$]{} and [the normal cone of $\io_X$ in $\ov\M$]{}, respectively. We denote by $$\pi_{\io_X}^{\perp}\!: \io_X^*TC\ov\M \lra \N_{\io_X}$$ the projection map. If $Z$ is a subvariety of $\ov\M$, let $$\io_Z\!: Z\lra\ov\M$$ the inclusion map.
\[ag\_dfn2\] Let $\ov\M$ be a variety.\
(1) An immersion $\io_X\!:X\!\lra\!\ov\M$ is [properly self-intersecting]{} if for all $x_1,x_2\!\in\!X$ such that $\io_X(x_1)\!=\!\io_X(x_2)$ and sufficiently small neighborhoods $U_1$ of $x_1$ and $U_2$ of $x_2$ in $X$ $$TC_{\io_X(x_1)}\big(\io_X(U_1)\!\cap\!\io_X(U_2)\big) =
\Im\,d\io_X|_{x_1} \cap \Im\,d\io_X|_{x_2} \subset TC_{\io_X(x_1)}\ov\M.
~\footnote{We emphasize that intersections are taken to be set-theoretic
intersections unless otherwise noted.}$$ (2) If $\io_X\!:X\!\lra\!\ov\M$ and $\io_Y\!:Y\!\lra\!\ov\M$ are immersions such that $\io_X$ is properly self-intersecting, $\io_X$ is [properly self-intersecting relative to $\io_Y$]{} if for all $x_1,x_2\!\in\!X$ and $y\!\in\!Y$ such that $$\io_X(x_1) =\io_X(x_2)=\io_Y(y)$$ and for all sufficiently small neighborhoods $U_1$ of $x_1$ and $U_2$ of $x_2$ in $X$, $$\pi_{\io_Y}^{\perp}\big|_y\big(TC_{\io_Y(y)} (\io_X(U_1)\!\cap\!\io_X(U_2))\big)
= \pi_{\io_Y}^{\perp}\big|_y\Im\,d\io_X|_{x_1} \cap
\pi_{\io_Y}^{\perp}\big|_y\Im\,d\io_X|_{x_2}
\subset \N_{\io_Y}\big|_y.$$
This definition generalizes Definition \[ag\_dfn1\]; see the paragraph following the latter for some examples.
\[immercoll\_dfn\] If $\ov\M$ is a variety, a collection $\{\io_{\vr}\!:X_{\vr}\!\lra\!\ov\M\}_{\vr\in\A}$ of immersions is [properly self-intersecting]{} if for all $\rho_1,\rho_2,\rho_3\!\in\!\A$ the immersion $\io_{\rho_1}\!\sqcup\!\io_{\rho_2}$ is properly self-intersecting relative to $\io_{\rho_3}$.
\[ag\_lmm2a\] Suppose $\ov\M$ is a variety and $Z$ is a smooth subvariety of $\ov\M$.\
(1) If $\io_X\!:X\!\lra\!\ov\M$ is an immersion such that the immersion $\io_X\!\sqcup\!\io_Z\!:X\!\sqcup\!Z\!\lra\!\ov\M$ is properly self-intersecting, then $\io_X$ lifts to an immersion $$\Pr_Z\io_X\!: \Bl_{\io_X^{-1}(Z)}X \lra \ov\M
\qquad\st\qquad \Im\,\Pr_Z\io_X=\Pr_Z\,\Im\,\io_X.$$ (2) If in addition $\io_X$ is properly self-intersecting relative to $\io_Z$, then $\Pr_Z\io_X$ is properly self-intersecting and $$\Im^s\,\Pr_Z\io_X=\Pr_Z\,\Im^s\,\io_X.$$ (3) If in addition $\io_Y\!:Y\!\lra\!\ov\M$ is an immersion such that $\io_X\!\sqcup\io_Y\!\sqcup\!\io_Z$ is properly self-intersecting and $\io_X$ is properly self-intersecting relative to $\io_Y$, then $\Pr_Z\io_X$ is properly self-intersecting relative to $\Pr_Z\io_Y$. Furthermore, $$\big\{\Pr_Z\io_X\big\}^{-1}\big(\Pr_Z\Im\,\io_Y\big)=
\Pr_{\io_X^{-1}(Z)}\io_X^{-1}(\Im\,\io_Y).$$\
[*Remark:*]{} Since we always require that the blowup locus be smooth, an implicit conclusion of (1) of Lemma \[ag\_lmm2a\] is that $\io_X^{-1}(Z)$ is a smooth subvariety of $X$; this is immediate from the local situation. Note that $X$ itself is smooth, as it is the domain of the immersion $\io_X$.
\[immercoll\_crl\] If $\ov\M$ is a variety, $\{\io_{\vr}\!:X_{\vr}\!\lra\!\ov\M\}_{\vr\in\A}$ is a properly self-intersecting collection of immersions, and $\vr\!\in\!\A$ is such that $\io_{\vr}$ is an embedding, then $\{\Pr_{\Im\,\io_{\vr}}\io_{\vr'}\}_{\vr'\in\A-\{\vr\}}$ is a properly self-intersecting collection of immersions into $\Bl_{\Im\,\io_{\vr}}\ov\M$.
\[ag\_lmm2b\] Suppose $\ov\M$ is a smooth variety, $Z$ is a smooth subvariety of $\ov\M$, $\io_X\!:X\!\lra\!\ov\M$ is an immersion such that the immersion $\io_X\!\sqcup\!\io_Z$ is properly self-intersecting. Let $$\io_X^{-1}(Z) = \bigsqcup_{\vr\in\A} Z_{\vr}$$ be the decomposition of $\io_X^{-1}(Z)$ into path components. If there exist a splitting $$\N_{\io_X}=\bigoplus_{i\in I}L_i \lra X$$ and a subset $I_{\vr}$ of $I$ for each $\vr\!\in\!\A$ such that $$\label{ag_lmm2b_e}
\io_X|_{Z_{\vr}}^{\,*}TZ\big/TZ_{\vr}
=\bigoplus_{i\in I-I_{\vr}}\!\!\!L_i|_{Z_{\vr}} \qquad\forall\,\vr\!\in\!A,$$ then $$\N_{\Pr_Z\io_X}=\bigoplus_{i\in I}
\Big(\pi^*L_i\otimes \bigotimes_{i\in I_{\vr}}\O(-E_{\vr})\Big),$$ where $E_{\vr}$ is the component of the exceptional divisor for the blowup $\pi\!: \Bl_{\io_X^{-1}(Z)}X\!\lra\!X$ that projects onto $Z_{\vr}$.
We note that by (1) of Definition \[ag\_dfn2\], the homomorphism $$\io_X|_{Z_{\vr}}^{\,*}TZ\big/TZ_{\vr} \lra
\N_{\io_X}\!\equiv\!\io_X^*T\ov\M\big/ \Im\, d\io_X$$ induced by the inclusions is injective. Thus, we can identify $\io_X|_{Z_{\vr}}^{\,*}TZ\big/TZ_{\vr}$ with a subbundle of $\N_{\io_X}$, as we have done in Lemma \[ag\_lmm2b\].
Moduli Spaces of Genus-Zero Maps {#map0str_subs}
--------------------------------
In this subsection, we describe natural subvarieties of the moduli space of genus-zero maps and a natural bundle section over them. This bundle section induces other bundle sections, introduced in the next two subsections, that are used in the blowup construction of Subsection \[map1blconstr\_subs\] to describe the structure of the proper transforms of $\ov\M_{1,k}^0(\Pn,d)$; see Subsection \[outline\_subs\] for more details. Below we also state two well-known facts in the Gromov-Witten theory, Lemmas \[g0mapstr\_lmm1\] and \[g0mapstr\_lmm2\], and a more recent result, Lemma \[deriv0str\_lmm\].\
If $d\!\in\!\Z^+$ and $J$ is a finite set, let $$\begin{gathered}
\label{g0map_e}
\A_0(d,J)=\big\{(m;J_P,J_B)\!: m\!\in\!\Z^+,\,m\!\le\!d;~
J_P\!\subset\!J; m\!+\!|J_P|\!\ge\!2\big\};\\
\ov\M_{0,(0,J)}(\Pn,d)=\ov\M_{0,\{0\}\sqcup J}(\Pn,d).\notag\end{gathered}$$ If $\si\!=\!(m;J_P,J_B)$ is an element of $\A_0(d,J)$, let $\M_{0,\si}(\Pn,d)$ be the subset of $\ov\M_{0,\{0\}\sqcup J}(\Pn,d)$ consisting of the stable maps $[\Si,u]$ such that\
${}\quad$ (i) the components of $\Si$ are $\Si_i\!=\!{\mathbb}{P}^1$ with $i\!\in\!\{P\}\!\sqcup\![k]$;\
${}\quad$ (ii) $u|_{\Si_P}$ is constant and the marked points on $\Si_P$ are indexed by the set $\{0\}\!\sqcup\!J_P$;\
${}\quad$ (iii) for each $i\!\in\![m]$, $\Si_i$ is attached to $\Si_P$ and $u|_{\Si_i}$ is not constant.\
We denote by $\ov\M_{0,\si}(\Pn,d)$ the closure of $\M_{0,\si}(\Pn,d)$ in $\ov\M_{0,\{0\}\sqcup J}(\Pn,d)$. Figure \[g0map\_fig\] illustrates this definition, from the points of view of symplectic topology and of algebraic geometry. In the first diagram, each disk represents a sphere, and we shade the components on which the map $u$ is non-constant. In the second diagram, the irreducible components of $\Si$ are represented by lines, and the integer next to each component shows the degree of $u$ on that component. In both cases, we indicate the marked points lying on the component $\Si_P$ only.\
(-1.1,-2.2)(10,1.5) (5,-1.5)[1.5]{} (2.5,-1.5)[1]{}(3.5,-1.5)[.15]{} (5,1)[1]{}(5,0)[.15]{} (7.5,-1.5)[1]{}(6.5,-1.5)[.15]{} (5,-3)[.2]{}(4.9,-3.6) (6.06,-2.56)[.2]{}(6.6,-2.9) (18,2)(18,-4)(18,1.5)[.2]{}(18.6,1.6) (18,-3.2)[.2]{}(18.5,-3.3) (16.8,0)(22.05,1.25)(22.6,1.2) (17,-1)(22,-1)(22.5,-1.1) (16.8,-2)(22.05,-3.25)(22.6,-3.3) (32,-1)
We define a partial ordering on the set $\A_0(d,J)$ by setting $$\label{partorder_e2}
\si'\!\equiv\!(m';J_P',J_B') \prec \si\!\equiv\!(m;J_P,J_B)
\qquad\hbox{if}\quad \si'\!\neq\!\si,~m'\!\le\!m,~J_P'\!\subset\!J_P.$$ Similarly to Subsection \[curvebldata\_subs\], this condition means that the elements of $\M_{0,\si'}(\Pn,d)$ can be obtained from the elements of $\M_{0,\si}(\Pn,d)$ by moving more points onto the bubble components or combining the bubble components; see Figure \[partorder\_fig2\]. As in the $g\!=\!0$ case of Subsection \[curvebldata\_subs\], the bubble components are the components not containing the marked point $0$.
(-0.3,-2.2)(10,1.5) (10,-1.5)[1.5]{} (8.23,.27)[1]{}(8.94,-.44)[.15]{} (11.77,.27)[1]{}(11.06,-.44)[.15]{} (10,-3)[.2]{}(9.9,-3.6) (15,-1)
$\prec$
(22,-1.5)[1.5]{} (19.5,-1.5)[1]{}(20.5,-1.5)[.15]{} (22,1)[1]{}(22,0)[.15]{} (23.5,-1.5)[.2]{}(24.2,-1.5) (22,-3)[.2]{}(21.9,-3.6) (30,-1.5)[1.5]{} (27.5,-1.5)[1]{}(28.5,-1.5)[.15]{} (30,1)[1]{}(30,0)[.15]{} (32.5,-1.5)[1]{}(31.5,-1.5)[.15]{} (30,-3)[.2]{}(29.9,-3.6)
\[g0mapstr\_lmm1\] If $\si_1,\si_2\!\in\!\A_0(d,J)$, $\si_1\!\neq\!\si_2$ $\si_1\!\not\prec\!\si_2$, and $\si_2\!\not\prec\!\si_1$, then $$\begin{gathered}
\ov\M_{0,\si_1}(\Pn,d) \cap \ov\M_{0,\si_2}(\Pn,d)
\subset \ov\M_{0,\ti\si(\si_1,\si_2)}(\Pn,d),\\
\hbox{where}\qquad
\ti\si(\si_1,\si_2)=\max\big\{\si'\!\in\!\A_0(d,J)\!: \si'\!\prec\!\si_1,\si_2\big\}.\end{gathered}$$ If $\ti\si(\si_1,\si_2)$ is not defined, $\ov\M_{0,\si_1}(\Pn,d)$ and $\ov\M_{0,\si_2}(\Pn,d)$ are disjoint.
For example, if $\si_1$ and $\si_2$ correspond to the two diagrams on the right of Figure \[partorder\_fig2\], then $\ti\si(\si_1,\si_2)$ corresponds to the diagram on the left of Figure \[partorder\_fig2\].\
If $\si\!=\!(m;J_P,J_B)$ is an element of $\A_0(d,J)$, let $$\begin{gathered}
\ov\M_{\si;B}(\Pn,d) \subset \prod_{i\in[m]}~\bigsqcup_{d_i>0,J_i\subset J_B}
\!\!\!\!\!\!\!\!\ov\M_{0,\{0\}\sqcup J_i}(\Pn,d_i) \qquad\hbox{and}\\
\pi_i\!: \ov\M_{\si;B}(\Pn,d)\lra
\!\!\bigsqcup_{d_i>0,J_i\subset J_B}\!\!\!\!\!\!\!\!\ov\M_{0,\{0\}\sqcup J_i}(\Pn,d_i),
\quad i\!\in\![m],\end{gathered}$$ be as in Subsection \[descr\_subs\]. Since each of the spaces $\ov\M_{0,\{0\}\sqcup J_i}(\Pn,d_i)$ is smooth and each of the evaluation maps $$\ev_0\!: \ov\M_{0,\{0\}\sqcup J_i}(\Pn,d_i)\lra \Pn$$ is a submersion, the space $\ov\M_{\si;B}(\Pn,d)$ is smooth. We denote by $$\label{g0nodeiden_e}
\io_{\si}\!: \ov\cM_{0,\{0\}\sqcup[m]\sqcup J_P}\!\times\!\ov\M_{\si;B}(\Pn,d)
\lra \ov\M_{0,\si}(\Pn,d) \subset \ov\M_{0,\{0\}\sqcup J}(\Pn,d)$$ the natural node-identifying map. It descends to an immersion $$\bar\io_{\si}\!:
\big(\ov\cM_{0,\{0\}\sqcup[m]\sqcup J_P}\!\times\!\ov\M_{\si;B}(\Pn,d)\big)
\big/S_m \lra \ov\M_{0,\{0\}\sqcup J}(\Pn,d).$$ Let $$\pi_P,\pi_B\!: \ov\cM_{0,\{0\}\sqcup [m]\sqcup J_P}\!\times\!\ov\M_{\si;B}(\Pn,d)
\lra \ov\cM_{0,\{0\}\sqcup [m]\sqcup J_P},\ov\M_{\si;B}(\Pn,d)$$ be the natural projection maps. The following lemma can be easily deduced from [@P].
\[g0mapstr\_lmm2\] If $d\!\in\!\Z^+$ and $J$ is a finite set, the collections $\{\io_{\si}\}_{\si\in\A_0(d,J)}$ and $\{\bar\io_{\si}\}_{\si\in\A_0(d,J)}$ of immersions are properly self-intersecting. If $\si\!\in\!\A_0(d,J)$ is as in \_ref[partorder\_e2]{}, $$\Im^s\,\bar\io_{\si} \subset \bigcup_{\si'\prec\si}\! \ov\M_{0,\si'}(\Pn,d)
\qquad\hbox{and}\qquad
\N_{\io_{\si}}=\bigoplus_{i\in[m]} \pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0.$$ If in addition $\si'\!\in\!\A_0(d,J)$, $\si'\!\prec\!\si$, and $\si'$ is as in \_ref[partorder\_e2]{}, then $$\begin{gathered}
\io_{\si}^{-1} \big( \ov\M_{0,\si'}(\Pn,d) \big)
=\Big(\bigcup_{\rho\in\A_0(\si;\si')} \!\!\!\!\!\!\!\ov\cM_{0,\rho}\Big)
\times \ov\M_{\si;B}(\Pn,d),
\qquad\hbox{where}\\
\A_0(\si;\si')=\big\{ \rho\!=\!\big(I_P\!\sqcup\!J_P',\{I_k\!\!\sqcup\!J_k\!: k\!\in\!K\})
\!\in\!\A_0([m],J_P)\!: |K|\!+\!|I_P|\!=\!m'\big\}\end{gathered}$$ and $\A_0([m],J_P)$ and $\ov\cM_{0,\rho}$ are as in Subsection \[curvebldata\_subs\]. Finally, if $\rho\!\in\!\A_0(\si;\si')$ is as above, $$\io_{\si} \big|_{\ov\cM_{0,\rho}\times\ov\M_{\si;B}(\Pn,d)}^*
T\ov\M_{0,\si'}(\Pn,d)
\big/T\big(\ov\cM_{0,\rho}\!\times\!\ov\M_{\si;B}(\Pn,d)\big)
=\bigoplus_{i\in[m]-I_P}\!\!\!\!\!\!\pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0.$$\
We finish this subsection by describing a natural bundle section $$\cD_0\in\Ga\big(\ov\M_{0,\{0\}\sqcup J}(\Pn,d),\Hom(L_0;\ev_0^*T\Pn)\big)$$ which plays a central role in the rest of the paper. An element $[b]\!\in\!\ov\M_{0,\{0\}\sqcup J}(\Pn,d)$ consists of a prestable nodal curve $\Si$ with marked points and a map $u\!:\Si\!\lra\!\Pn$. One of the marked points is labeled by $0$. We denote it by $x_0(b)$. We define $\cD_0$ by $$\cD_0\big|_b = du|_{x_0(b)}\!: T_{x_0(b)}\Si\lra T_{\ev_0(b)}\Pn.$$ If $\U\!\lra\!\ov\M_{0,\{0\}\sqcup J}(\Pn,d)$ is the universal curve and $\ev\!:\U\!\lra\!\Pn$ is the natural evaluation map, then $\cD_0|_b$ is simply the restriction of $d\ev|_{x_0(b)}$ to the vertical tangent bundle of $\U$. The bundle section $\cD_0$ vanishes identically along the subvarieties $\ov\M_{0,\si}(\Pn,d)$ with $\si\!\in\!\A_0(d,J)$.
\[deriv0str\_lmm\] If $d\!\in\!\Z^+$ and $J$ is a finite set, the section $\cD_0$ is transverse to the zero set on the complement of the subvarieties $\ov\M_{0,\si}(\Pn,d)$ with $\si\!\in\!\A_0(d,J)$. Furthermore, for every $$\si\!\equiv\!(m;J_P,J_B) \in \A_0(d,J),$$ the differential of $\cD_0$, $$\na\cD_0\!: \N_{\io_{\si}} \lra \io_{\si}^*\,\Hom(L_0,\ev_0^*T\Pn)
=\pi_P^*L_0^*\!\otimes\!\pi_B^*\ev_0^*T\Pn,$$ in the normal direction to the immersion $\io_{\si}$ is given by $$\na\cD_0\big|_{\pi_P^*L_i\otimes\pi_B^*\pi_i^*L_0}
= \pi_P^*s_i\!\otimes\!\pi_B^*\pi_i^*\cD_0
\qquad\forall\, i\!\in\![m],$$ where $s_i$ is the homomorphism defined in Subsection \[curvebldata\_subs\].
The first claim of the lemma is an immediate consequence of the fact that $$H^1\big(\Si;u^*T\Pn\!\otimes\!\O(-2z)\big)=\{0\}$$ for every genus-zero stable map $(\Si,u)$ and a smooth point $z\!\in\!\Si$ such that the restriction of $u$ to the irreducible component of $\Si$ containing $z$ is not constant. The second statement of the lemma follows from Theorem 2.8 in [@g2n2and3].
Initial Data {#map0prelim_subs}
------------
If $\ale$ and $J$ are finite sets and $d$ is positive integer, let $$\begin{split}
\ov\M_{0,(\aleph,J)}(\Pn,d)&=
\Big\{(b_l)_{l\in\ale}\in\prod_{l\in\ale}\ov\M_{0,\{0\}\sqcup J_l}(\Pn,d_l)\!:
d_l\!\in\!\Z^+,~\sum_{l\in\ale}d_l\!=\!d;~ \bigsqcup_{l\in\ale} J_l\!=\!J;\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~~
\ev_0(b_l)\!=\!\ev_0(b_{l'})~\forall\, l,l'\!\in\!\ale\Big\};
\end{split}$$ $$\begin{split}
\M_{0,(\aleph,J)}(\Pn,d)&=
\Big\{(b_l)_{l\in\ale}\in\prod_{l\in\ale}\M_{0,\{0\}\sqcup J_l}(\Pn,d_l)\!:
d_l\!\in\!\Z^+,~\sum_{l\in\ale}d_l\!=\!d;~ \bigsqcup_{l\in\ale} J_l\!=\!J;\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~~
\ev_0(b_l)\!=\!\ev_0(b_{l'})~\forall\, l,l'\!\in\!\ale\Big\},
\end{split}$$ where $\M_{0,\{0\}\sqcup J_l}(\Pn,d_l)$ is the subset of $\ov\M_{0,\{0\}\sqcup J_l}(\Pn,d_l)$ consisting of stable maps with smooth domains. For each $l\!\in\!\ale$, let $$\pi_l\!: \ov\M_{0,(\ale,J)}(\Pn,d)\lra
\!\!\bigsqcup_{d_l>0,J_l\subset J}\!\!\!\!\ov\M_{0,\{0\}\sqcup J_l}(\Pn,d_l)$$ be the projection map. We put $$F_{(\ale,J)}=\bigoplus_{l\in\ale} \pi_l^*L_0,$$ where $L_0\!\lra\!\ov\M_{0,\{0\}\sqcup J_l}(\Pn,d_l)$ is the universal tangent line bundle for the marked point $0$. In the next subsection, we construct a blowup $$\pi_{0,(\ale,J)}\!: \wt\M_{0,(\ale,J)}(\Pn,d) \lra \P F_{(\ale,J)}$$ of the projective bundle $\P F_{(\ale,J)}$ over $\ov\M_{0,(\ale,J)}(\Pn,d)$ and a line bundle $$\ti\E \lra \wt\M_{0,(\ale,J)}(\Pn,d).$$ We also describe a natural bundle section $$\wt\cD_{(\ale,J)} \in \Ga\big( \wt\M_{0,(\ale,J)}(\Pn,d);
\ti\E^*\!\otimes\!\pi_{0,(\ale,J)}^*\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big),$$ where $$\pi_{\P F_{(\ale,J)}}\!: \P F_{(\ale,J)} \lra \ov\M_{0,(\ale,J)}(\Pn,d)$$ is the bundle projection map. This section is transverse to the zero set.\
Similarly to Subsection \[curve0bl\_subs\], the smooth variety $\wt\M_{0,(\ale,J)}(\Pn,d)$ is obtained by blowing up the subvarieties $\wt\M_{0,\vr}^0(\Pn,d)$ defined below and their proper transforms in an order consistent with a natural partial ordering $\prec$. The line bundle $\ti\E$ is the sum of the tautological line bundle $$\ga_{(\ale,J)}\lra\P F_{(\ale,J)}$$ and all exceptional divisors. The section $\wt\cD_{(\ale,J)}$ is induced from the sections $\pi_l^*\cD_0$, with $l\!\in\!\ale$, where $\cD_0$ is as in Subsection \[map0str\_subs\].\
If $\ale$, $J$, and $d$ are as above, let $$\begin{split}
\A_0(\ale;d,J) = \big\{ \big((\si_l)_{l\in\ale},J_B\big)\!:
\,&(\si_l,\eset)\!\in\!\{(0,\eset)\}\!\sqcup\!\A_0(d_l,J_{l,P}),~
(\si_l)_{l\in\ale}\!\neq\!(0)_{l\in\ale};\\
&\sum_{l\in\ale}d_l\!=\!d,~\bigsqcup_{l\in\ale}J_{l,P}\!=\!J\!-\!J_B\big\}.
\end{split}$$ We define a partial ordering $\prec$ on $\A_0(\ale;d,J)$ by setting $$\label{vrdfn_e2}
\vr'\!\equiv\!\big((\si_l')_{l\in\ale},J_B'\big)
\prec \vr\!\equiv\!\big((\si_l)_{l\in\ale},J_B)$$ if $\vr'\!\neq\!\vr$ and for every $l\!\in\!\aleph$ either $\si_l'\!=\!\si_l$, $(\si_l',\eset)\!\prec\!(\si_l,\eset)$, or $\si_l'\!=\!0$. If $\vr\!\in\!\A_0(\ale;d,J)$ is as in \_ref[vrdfn\_e2]{}, we put $$\ale_P(\vr)=\big\{l\!\in\!\aleph\!: \si_l\!\neq\!0\big\}
\qquad\hbox{and}\qquad
\ale_S(\vr)=\big\{l\!\in\!\aleph\!: \si_l\!=\!0\big\}.$$ Here $P$ and $S$ stand for the subsets of principal and secondary elements of $\aleph$, respectively; see the next paragraph. Note that $$\begin{gathered}
\vr'\prec \vr ~~~\Lra~~~ \ale_P(\vr')\subset\ale_P(\vr), \qquad
\ale_P(\vr)\neq\eset~~~\forall\,\vr\!\in\!\A_0(\ale;d,J),\qquad\hbox{and}\notag\\
\label{vrdfn_e3}
\vr=\big((m_l;J_{l,P})_{l\in\ale_P(\vr)},(0)_{l\in\ale_S(\vr)},J_B\big)\end{gathered}$$ for some $m_l$ and $J_{l,P}$. Choose an ordering $<$ on $\A_0(\ale;d,J)$ extending the partial ordering $\prec$. We denote the corresponding minimal and maximal element by $\vr_{\min}$ and $\vr_{\max}$, respectively. For every $\vr\!\in\!\A_0(\ale;d,J)$, define $$\vr\!-\!1 \in \{0\}\!\sqcup\!\A_0(\ale;d,J)$$ as in \_ref[minusdfn\_e]{}.\
If $\vr\!\in\!\A_0(\ale;d,J)$ is as in \_ref[vrdfn\_e2]{}, let $$\begin{split}
\ov\M_{0,\vr}(\Pn,d) =
\Big\{(b_l)_{l\in\ale}\in\prod_{l\in\ale}\ov\M_{0,(\si_l,J_{l,B})}(\Pn,d_l)\!:
\sum_{l\in\ale}d_l\!=\!d;~\bigsqcup_{l\in\ale}J_{l,B}\!=\!J_B;\quad&\\
\ev_0(b_{l_1})\!=\!\ev_0(b_{l_2})~\forall\, l_1,l_2\!\in\!\ale&\Big\}
\subset\ov\M_{0,(\ale,J)}(\Pn,d).
\end{split}$$ With $*\!=\!P,S$, we define $$F_{\vr;*}=\bigoplus_{l\in\ale_*(\vr)}
\!\!\! \pi_l^*L_0\Big|_{\ov\M_{0,\vr}(\Pn,d)}
\subset F_{(\ale,J)}\big|_{\ov\M_{0,\vr}(\Pn,d)}.$$ Let $$\wt\M_{0,\vr}^0(\Pn,d)=\P F_{\vr;P}
\subset \wt\M_{0,(\ale,J)}^0(\Pn,d)\!\equiv\!\P F_{(\ale,J)}.$$ From Lemma \[g0mapstr\_lmm1\], we immediately obtain
\[map0bl\_lmm1a\] If $\vr_1,\vr_2\!\in\!\A_0(\ale;d,J)$, $\vr_1\!\neq\!\vr_2$ $\vr_1\!\not\prec\!\vr_2$, and $\vr_2\!\not\prec\!\vr_1$, then $$\begin{gathered}
\wt\M_{0,\vr_1}^0(\Pn,d) \cap \wt\M_{0,\vr_2}^0(\Pn,d)
\subset \wt\M_{0,\ti\vr(\vr_1,\vr_2)}^0(\Pn,d),\\
\hbox{where}\qquad
\ti\vr(\vr_1,\vr_2)=\max\big\{\vr'\!\in\!\A_0(\ale;d,J)\!: \vr'\!\prec\!\vr_1,\vr_2\big\}.\end{gathered}$$ If $\ti\vr(\vr_1,\vr_2)$ is not defined, $\wt\M_{0,\vr_1}^0(\Pn,d)$ and $\wt\M_{0,\vr_2}^0(\Pn,d)$ are disjoint.
With $\vr$ as \_ref[vrdfn\_e3]{}, let $$\vr_P=\big([m_l],J_{l,P}\big)_{l\in\ale_P(\vr)}, \quad
\ale_B(\vr)=\ale_S(\vr)\sqcup\!\bigsqcup_{l\in\ale_P(\vr)}\!\!\!\![m_l], \quad
J_B(\vr)=J_B, \quad\hbox{and}\quad G_{\vr}=\prod_{l\in\ale_P(\vr)}\!\!\!\!\!S_{m_l}.$$ With $\wt\cM_{0,\vr_P}^0$ as in Subsection \[curve0bl\_subs\], we denote by $$\io_{0,\vr}\!: \wt\cM_{0,\vr_P}^0 \times
\ov\M_{0,(\ale_B(\vr),J_B(\vr))}(\Pn,d) \lra \wt\M_{0,\vr}^0(\Pn,d)
\subset \wt\M_{0,(\ale,J)}^0(\Pn,d)$$ the natural node-identifying map induced by the immersions $\io_{(\si_l,J_{l,B})}$ in \_ref[g0nodeiden\_e]{}. It descends to an immersion $$\bar\io_{0,\vr}\!:
\big(\wt\cM_{0,\vr_P}^0 \!\times\!
\ov\M_{0,(\ale_B(\vr),J_B(\vr))}(\Pn,d)\big)\big/G_{\vr}
\lra \wt\M_{0,(\ale,J)}^0(\Pn,d).$$ Let $$\pi_P,\pi_B\!: \wt\cM_{0,\vr_P}^0 \times \ov\M_{0,(\ale_B(\vr),J_B(\vr))}(\Pn,d)
\lra \wt\cM_{0,\vr_P}^0,\ov\M_{0,(\ale_B(\vr),J_B(\vr))}(\Pn,d)$$ be the projection maps.\
For the rest of this section, as well as for Section \[map1bl\_sec\], we take $$\begin{gathered}
\ov\M_{0,(\ale,J)}=\ov\M_{0,(\ale,J)}(\Pn,d), ~~~
\M_{0,(\ale,J)}=\M_{0,(\ale,J)}(\Pn,d), ~~~
\wt\M_{0,(\ale,J)}^0=\wt\M_{0,(\ale,J)}^0(\Pn,d) \quad\forall~(\ale,J);\\
\wt\M_{0,\vr}^0=\wt\M_{0,\vr}^0(\Pn,d)
\qquad\forall~\vr\!\in\!\A_0(\ale;d,J).\end{gathered}$$
\[map0bl\_lmm1\] If $\ale$ and $J$ are finite sets and $d\!\in\!\Z^+$, the collections $$\{\io_{0,\vr}\}_{\vr\in\A_0(\ale;d,J)} \qquad\hbox{and}\qquad
\{\bar\io_{0,\vr}\}_{\vr\in\A_0(\aleph;d,J)}$$ of immersions are properly self-intersecting. If $$\vr^*\!\equiv\!\big((m_l^*;J_{l,P}^*)_{l\in\ale_P(\vr^*)},
(0)_{l\in\ale_S(\vr^*)},J_B^*\big) \in \A_0(\aleph;d,J),$$ then $$\begin{gathered}
\Im^s\,\bar\io_{0,\vr^*} \subset \bigcup_{\vr'\prec\vr^*}\! \wt\M_{0,\vr'}^0
\qquad\hbox{and}\\
\N_{\io_{0,\vr^*}}= \!\bigoplus_{l\in\ale_P(\vr^*)} \bigoplus_{i\in[m_l^*]}
\!\!\pi_P^*L_{0,(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0
~\oplus~
\pi_P^*\E_0^*\!\otimes\pi_B^*\!\bigoplus_{l\in\ale_S(\vr^*)}\!\!\!\!\!\pi_l^*L_0,\end{gathered}$$ where $\E_0$ is as in Subsections \[curve0bl\_subs\]. If $\vr,\vr^*\!\in\!\A_0(\ale;d,J)$, $\vr$ is as \_ref[vrdfn\_e3]{}, and $\vr\!\prec\!\vr^*$, then $$\io_{0,\vr^*}^{~-1} \big( \wt\M_{0,\vr}^0\big)
=\Big(\bigcup_{\rho\in\A_0(\vr^*;\vr)} \!\!\!\!\!\!\! \wt\cM_{0,\rho}^0 \Big)
\times \ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))},$$ where $$\begin{split}
\A_0(\vr^*;\vr)=\Big\{ \rho\!=\!\big(\ale_P(\vr),
\big(I_{l,P}^*\!\sqcup\!J_{l,P},\{I_{l,k}^*\!\sqcup\!J_{l,k}^*\!: k\!\in\!K_l^*\}
\big)_{l\in\ale_P(\vr^*)}\big)\!\in\!\A_0(\vr_P^*)\!: \qquad\qquad&\\
|K_l^*|\!+\!|I_{l,P}^*|\!=\!m_l~\forall\, l\!\in\!\ale_P(\vr^*)&\Big\}
\end{split}$$ and $\A_0(\vr_P^*)$ and $\wt\cM_{0,\rho}^0$ are as in Subsection \[curve0bl\_subs\]. Finally, if $\rho\!\in\!\A_0(\vr^*;\vr)$ is as above, $$\begin{split}
&\io_{0,\vr^*} \big|_{\wt\cM_{0,\rho}^0\times\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}}^*
T\ov\M_{0,\vr}
\big/T\big(\wt\cM_{0,\rho}^0\!\times\!\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}\big)\\
&\qquad
=\!\bigoplus_{l\in\ale_P(\vr^*)-\ale_P(\vr)} \bigoplus_{i\in[m_l^*]}
\!\!\pi_P^*L_{0,(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0
~\oplus~\!\bigoplus_{l\in\ale_P(\vr)} \bigoplus_{i\in[m_l^*]-I_{l,P}}
\!\!\!\!\!\!\!\!\!\pi_P^*L_{0,(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0.
\end{split}$$\
The normal bundle $\N_{\io_{0,\vr^*}}$ for the immersion $\io_{0,\vr^*}$ splits into horizontal and vertical bundles: $$\N_{\io_{0,\vr^*}} = \N_{\io_{0,\vr^*}}^{\bot} \oplus \N_{\io_{0,\vr^*}}^{\top}.$$ It is immediate from the definitions that $$\N_{\io_{0,\vr^*}}^{\top}=
\io_{0,\vr^*}^{\,*}\big(\ga_{(\aleph,J)}^{\,*}\!\otimes\!F_{\vr;S}\big)
=\pi_P^*\E_0^*\!\otimes\pi_B^*\!\!\bigoplus_{l\in\ale_S(\vr^*)}\!\!\!\!\!\pi_l^*L_0.$$ The horizontal normal bundle $\N_{\io_{0,\vr^*}}^{\bot}$ is the pullback of the normal bundle for the node identifying immersion $$\ov\cM_{0,\vr_P^*} \times
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))} \lra \ov\M_{0,\vr^*}(\Pn,d)
\subset \ov\M_{0,(\ale,J)}$$ induced by the immersions $\io_{(\si_l^*,J_{l,B}^*)}$ in \_ref[g0nodeiden\_e]{} by the bundle projection map $\pi_{\P F_{\vr;P}}$. The normal bundle for this immersion is the sum of component-wise normal bundles given by Lemma \[g0mapstr\_lmm2\]. The remaining claims of Lemma \[map0bl\_lmm1\] follow easily from the corresponding statements of Lemma \[g0mapstr\_lmm2\] as well.\
We note that for every $\vr^*\!\in\!\A_0(\ale;d,J)$, $$\A_0(\vr_P^*)= \bigsqcup_{\vr\prec\vr^*}\!\A_0(\vr^*;\vr).$$ Furthermore, if $\vr_1,\vr_2\!\in\!\A_0(\ale;d,J)$ are such that $\vr_1,\vr_2\!\prec\!\vr^*$, then $$\rho_1^*\!\in\!\A_0(\vr^*;\vr_1), \quad \rho_2^*\!\in\!\A_0(\vr^*;\vr_2), \quad
\rho_1^*\!\prec\!\rho_2^* \qquad\Lra\qquad \vr_1\!\prec\!\vr_2.$$ Thus, we can choose an ordering $<$ on $\A_0(\vr_P^*)$ extending the partial ordering $\prec$ of Subsection \[curve0bl\_subs\] such that $$\vr_1\!<\!\vr_2, \quad \rho_1^*\!\in\!\A_0(\vr^*;\vr_1), \quad
\rho_2^*\!\in\!\A_0(\vr^*;\vr_2) \qquad\Lra\qquad \rho_1^*\!<\!\rho_2^*,$$ whenever $\vr_1,\vr_2\!\in\!\A_0(\ale;d,J)$ are such that $\vr_1,\vr_2\!\prec\!\vr^*$. In the next subsection, we will refer to the blowup construction of Subsection \[curve0bl\_subs\] corresponding to such an ordering.\
Via the projection maps $\pi_l$, the bundle sections $\cD_0$ of Subsection \[map0str\_subs\] induce a linear bundle map $$\cD_{(\ale,J)}\!: F_{(\ale,J)}\lra \ev_0^*T\Pn$$ over $\ov\M_{0,(\ale,J)}$. In turn, this homomorphism induces a bundle section $$\wt\cD_0\in\Ga\big(\wt\M_{0,(\ale,J)}^0;
\E_0^*\!\otimes\!\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big),
\quad\hbox{where}\quad
\E_0=\ga_{(\aleph,J)}\lra \wt\M_{0,(\ale,J)}^0.$$ This section vanishes identically on the subvarieties $\wt\M_{0,\vr}^0$ of $\wt\M_{0,(\ale,J)}^0$ with $\vr\!\in\!\A_0^*(\ale;d,J)$.
\[map0bl\_lmm2\] The section $\wt\cD_0$ is transverse to the zero set on the complement of the subvarieties $\wt\M_{0,\vr^*}^0$ with $\vr^*\!\in\!\A_0(\ale;d,J)$. Furthermore, for every $\vr^*\!\in\!\A_0(\ale;d,J)$ as in Lemma \[map0bl\_lmm1\], the differential of $\wt\cD_0$, $$\na\wt\cD_0\!: \N_{\io_{0,\vr^*}} \lra
\io_{0,\vr^*}^{\,*}
\big( \E_0^*\!\otimes\!\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big)
=\pi_P^*\E_0^*\!\otimes\!\pi_B^*\ev_0^*T\Pn,$$ in the normal direction to the immersion $\io_{0,\vr^*}$ is given by $$\begin{gathered}
\na\wt\cD_0\big|_{\pi_P^*L_{0,(l,i)}\otimes\pi_B^*\pi_{(l,i)}^*L_0}
=\pi_P^*s_{0,(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*\cD_0
~~~\forall~ i\!\in\![m_l^*],\,l\!\in\!\ale_P(\vr^*),\\
\hbox{and}\qquad
\na\wt\cD_0\big|_{\N_{\io_{0,\vr^*}}^{\top}}
=\pi_P^*\id\!\otimes\!\pi_B^*\cD_{(\ale_B(\vr^*),J_B(\vr^*))},\end{gathered}$$ where $s_{0,(l,i)}$ is the homomorphism defined in Subsection \[curve0bl\_subs\].
This lemma follows immediately from Lemma \[deriv0str\_lmm\].
Inductive Construction {#map0blconstr_subs}
----------------------
We are now ready to describe the inductive assumptions for our construction of the blowup $$\pi_{0,(\ale,J)}\!:
\wt\M_{(\ale,J)}^{\vr_{\max}} \!\equiv\! \wt\M_{0,(\ale,J)}(\Pn,d)
\lra \wt\M_{(\ale,J)}^0 \!\equiv\! \wt\M_{0,(\ale,J)}^0(\Pn,d).$$ Suppose $\vr\!\in\!\A_0(\ale;d,J)$ and we have constructed\
${}\quad$ ($I1$) a blowup $\pi_{\vr-1}\!:
\wt\M_{(\ale,J)}^{\vr-1}\!\lra\!\wt\M_{(\ale,J)}^0$ such that $\pi_{\vr-1}$ is an isomorphism outside of the preimage of the spaces $\wt\M_{0,\vr'}^0$ with $\vr'\!\le\!\vr\!-\!1$;\
${}\quad$ ($I2$) a line bundle $\E_{\vr-1}\!\lra\!\wt\M_{(\ale,J)}^{\vr-1}$;\
${}\quad$ ($I3$) a section $\wt\cD_{\vr-1} \in
\Ga\big( \wt\M_{(\ale,J)}^{\vr-1};
\E_{\vr-1}^*\!\otimes\!\pi_{\vr-1}^{~*}\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big)$.\
For each $\vr^*\!>\!\vr\!-\!1$, let $$\wt\M_{0,\vr^*}^{\vr-1}\equiv\wt\M_{0,\vr^*}^{\vr-1}(\Pn,d)
\subset \wt\M_{(\ale,J)}^{\vr-1}$$ be the proper transform of $\wt\M_{0,\vr^*}^0$ in $\wt\M_{(\ale,J)}^{\vr-1}$. We assume that\
${}\quad$ ($I4$) the section $\wt\cD_{\vr-1}$ is transverse to the zero set on the complement of the subvarieties $\wt\M_{0,\vr^*}^{\vr-1}$ with $\vr^*\!>\!\vr\!-\!1$ and vanishes identically along these subvarieties;\
${}\quad$ ($I5$) if $\vr_1,\vr_2\!\in\!\A_0(\ale;d,J)$ are such that $\vr_1\!\neq\!\vr_2$, $\vr_1\!\not\prec\!\vr_2$, $\vr_2\!\not\prec\!\vr_1$, and $\vr\!-\!1\!<\!\vr_1,\vr_2$, then $$\wt\M_{0,\vr_1}^{\vr-1} \cap \wt\M_{0,\vr_2}^{\vr-1} ~
\begin{cases}
\subset \wt\M_{0,\ti\vr(\vr_1,\vr_2)}^{\vr-1}, &\hbox{if}~ \ti\vr(\vr_1,\vr_2)\!>\!\vr\!-\!1;\\
=\eset, &\hbox{otherwise},
\end{cases}$$ where $\ti\vr(\vr_1,\vr_2)$ is as in Lemma \[map0bl\_lmm1\].\
We also assume that for all $\vr^*\!\in\!\A_0(\ale;d,J)$ such that $\vr^*\!>\!\vr\!-\!1$:\
${}\quad$ ($I6$) the domain of the $G_{\vr^*}$-invariant immersion $\io_{\vr-1,\vr^*}$ induced by $\io_{0,\vr^*}$ is $$\begin{gathered}
\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)} \times
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))},
\qquad\hbox{where}\\
\rho_{\vr^*}(\vr\!-\!1)=
\begin{cases}
\max\big\{\rho\!\in\!\A_0(\vr^*;\vr')\!:
\vr'\!\le\!\vr\!-\!1,\vr'\!\prec\!\vr^*\big\},
&\begin{aligned}
&\hbox{if}~\exists \vr'\!\in\!\A_0(\ale;d,J)\\
&~\hbox{s.t.}~\vr'\!\le\!\vr\!-\!1,\vr'\!\prec\!\vr^*;
\end{aligned}\\
0,& \hbox{otherwise};
\end{cases}\end{gathered}$$ ${}\quad$ ($I7$) if $\vr'\!\in\!\A_0(\ale;d,J)$ is such that $\vr\!-\!1\!<\!\vr'\!\prec\!\vr^*$, then $$\io_{\vr-1,\vr^*}^{~-1} \big( \wt\M_{0,\vr'}^{\vr-1}\big)
=\Big(\bigcup_{\rho\in\A_0(\vr^*;\vr')} \!\!\!\!\!\!\!
\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)} \Big)
\times \ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))};$$ ${}\quad$ ($I8$) $\Im^s\,\bar\io_{\vr-1,\vr^*}\subset\bigcup_{\vr-1<\vr'\prec\vr^*}
\wt\M_{0,\vr'}^{\vr-1}$, where $$\bar\io_{\vr-1,\vr^*}\!:
\Big(\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)}\!\times\!
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}\Big)\big/G_{\vr^*}
\lra \wt\M_{(\ale,J)}^{\vr-1},$$ is the immersion map induced by $\io_{\vr-1,\vr^*}$.\
Furthermore, we assume that\
${}\quad$ ($I9$) the collections $\{\io_{\vr-1,\vr^*}\}_{\vr^*\in\A_0(\ale;d,J),\vr^*>\vr-1}$ and $\{\bar\io_{\vr-1,\vr^*}\}_{\vr-1\in\A_0(\ale;d,J),\vr^*>\vr-1}$ of immersions are properly self-intersecting.\
Finally, for all $\vr^*\!\in\!\A_0(\ale;d,J)$ such that $\vr^*\!>\!\vr\!-\!1$:\
${}\quad$ ($I10$) $\io_{\vr-1,\vr^*}^{\,*}\E_{\vr-1}=\pi_P^*\E_{\rho_{\vr^*}(\vr-1)}$, where $$\pi_P, \, \pi_B\!: \wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)} \times
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}
\lra \wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)}, \,
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}$$ are the two projection maps;\
${}\quad$ ($I11$) if $\vr^*$ is as in Lemma \[map0bl\_lmm1\], then the normal bundle for the immersion $\io_{\vr-1,\vr^*}$ is given by $$\begin{split}
\N_{\io_{\vr-1,\vr^*}}
&= \N_{\io_{\vr-1,\vr^*}}^{\bot}\!\oplus\!\N_{\io_{\vr-1,\vr^*}}^{\top} \\
&\equiv\bigoplus_{l\in\ale_P(\vr^*)}\bigoplus_{i\in[m_l^*]}
\!\!\pi_P^*L_{\rho_{\vr^*}(\vr-1),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0~\oplus~
\pi_P^*\E_{\rho_{\vr^*}(\vr-1)}^*\!\otimes\!
\bigoplus_{l\in\aleph_S(\vr^*)}\!\!\!\!\!\pi_l^*L_0,
\end{split}$$ where $L_{\rho_{\vr^*}(\vr-1),(l,i)},\E_{\rho_{\vr^*}(\vr-1)}\!\lra\!
\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)}$ are the line bundles constructed in Subsection \[curve0bl\_subs\];\
${}\quad$ ($I12$) the differential of $\wt\cD_{\vr-1}$, $$\na\wt\cD_{\vr-1}\!: \N_{\io_{\vr-1,\vr^*}}
\lra \io_{\vr-1,\vr^*}^{\,*}\big(\E_{\vr-1}^{\,*}\!\otimes\!
\pi_{\vr-1}^{~*}\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big)
=\pi_P^*\E_{\rho_{\vr^*}(\vr-1)}^*\!\otimes\!\pi_B^*\ev_0^*T\Pn,$$ in the normal direction to the immersion $\io_{\vr-1,\vr^*}$ is given by $$\begin{gathered}
\na\wt\cD_{\vr-1}
\big|_{\pi_P^*L_{\rho_{\vr^*}(\vr-1),(l,i)}\otimes\pi_B^*\pi_{(l,i)}^*L_0}
=\pi_P^*s_{\rho_{\vr^*}(\vr-1),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*\cD_0
\quad\forall~i\!\in\![m_l^*],\,l\!\in\!\ale_P(\vr^*)\\
\hbox{and}\qquad
\na\wt\cD_{\vr-1}\big|_{\N_{\io_{\vr-1,\vr^*}}^{\top}}
=\pi_P^*\id\!\otimes\!\pi_B^*\cD_{(\ale_B(\vr^*),J_B(\vr^*))},\end{gathered}$$ where $s_{\rho_{\vr^*}(\vr-1),(l,i)}$ is the homomorphism defined in Subsection \[curve0bl\_subs\].\
By the inductive assumption ($I4$), the loci on which the sections $\wt\cD_{\vr}$ fail to be transverse to the zero set shrink and eventually disappear. For each $\vr$, the behavior of $\wt\cD_{\vr}$ in the directions normal to the “bad” locus is described by ($I12$). By the inductive assumption ($I5$), if $\vr_1$ and $\vr_2$ are non-comparable elements of $(\A_0(\ale;d,J),\prec)$, the proper transforms of $\wt\M_{0,\vr_1}^0$ and $\wt\M_{0,\vr_2}^0$ become disjoint by the time either is ready to be blown up for any ordering $<$ extending the partial ordering $\prec$. Similarly to Subsections \[curve1bl\_subs\] and \[curve0bl\_subs\], ($I5$) will imply that the end result of the present blowup construction is independent of the choice of an extension $<$. By ($I6$), our blowup construction modifies each immersion $\io_{0,\si^*}$ by changing the first factor of the domain according to the blowup construction of Subsection \[curve0bl\_subs\], until a proper transform of the image of $\io_{0,\si^*}$ is to be blown up; see below. By ($I8$), by the time this happens the immersion $\bar\io_{0,\si^*}$ induced by $\io_{0,\si^*}$ transforms into an embedding. Thus, all blowup loci are smooth.\
We note that all of the assumptions ($I1$)-($I12$) are satisfied if $\vr\!-\!1$ is replaced by $0$. In particular, ($I5$) is a restatement of Lemma \[map0bl\_lmm1a\], while ($I4$) and ($I12$) are the two parts of Lemma \[map0bl\_lmm2\]. The statements ($I7$)-($I11$), with $\vr\!-\!1$ replaced by $0$, are contained in Lemma \[map0bl\_lmm1\].\
If $\vr\!\in\!\A_0(\ale;d,J)$ is as above, let $$\ti\pi_{\vr}\!: \wt\M_{(\ale,J)}^{\vr} \lra \wt\M_{(\ale,J)}^{\vr-1}$$ be the blowup of $\wt\M_{(\ale,J)}^{\vr-1}$ along $\wt\M_{0,\vr}^{\vr-1}$, which is a smooth subvariety by the inductive assumption ($I8$). We denote the exceptional divisor for this blowup by $\wt\M_{0,\vr}^{\vr}$. If $\vr^*\!>\!\vr$, let $\wt\M_{0,\vr^*}^{\vr}\!\subset\!\wt\M_{(\ale,J)}^{\vr}$ be the proper transform of $\wt\M_{0,\vr^*}^{\vr-1}$. We put $$\label{bundletwist_e3}
\E_{\vr}= \ti\pi_{\vr}^*\E_{\vr-1}\otimes\O\big(\wt\M_{0,\vr}^{\vr}\big).$$ The section $\ti\pi_{\vr}^*\wt\cD_{\vr-1}$ vanishes identically along the divisor $\wt\M_{0,\vr}^{\vr}$. Thus, it induces a section $$\wt\cD_{\vr}\in \Ga
\big( \wt\M_{(\ale,J)}^{\vr}; \E_{\vr}^*\!\otimes\!
\pi_{\vr}^*\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big),$$ where $\pi_{\vr}=\pi_{\vr-1}\!\circ\ti\pi_{\vr}$.\
The inductive assumptions ($I1$)-($I3$), with $\vr\!-\!1$ replaced by $\vr$, are clearly satisfied, while ($I5$), ($I8$), and ($I9$) follow from (2) of Lemma \[ag\_lmm2a\] and Corollary \[immercoll\_crl\]. On the other hand, by ($I6$), the domain of the immersion $\io_{\vr-1,\vr}$ is $$\wt\cM_{0,\vr_P}^{\rho_{\vr}(\vr-1)} \times
\ov\M_{0,(\ale_B(\vr),J_B(\vr))}
=\wt\cM_{0,\vr_P} \times \ov\M_{0,(\ale_B(\vr),J_B(\vr))},$$ where $\wt\cM_{0,\vr_P}\!\lra\!\wt\cM_{0,\vr_P}^0$ is the blowup constructed in Subsection \[curve0bl\_subs\]. By ($I11$), the normal bundle for the immersion $\io_{\vr-1,\vr}$ is given by $$\begin{split}
\N_{\io_{\vr-1,\vr}} &= \bigoplus_{l\in\ale_P(\vr)}\bigoplus_{i\in[m_l]} \!\pi_P^*L_{\rho_{\vr}(\vr-1),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0~\oplus~
\pi_P^*\E_{\rho_{\vr}(\vr-1)}^*\!\otimes\!
\bigoplus_{l\in\aleph_S(\vr)}\!\!\!\!\pi_l^*L_0\\
&= \bigoplus_{l\in\ale_P(\vr)}\bigoplus_{i\in[m_l]}
\!\pi_P^*\L\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0~\oplus~
\pi_P^*\L\!\otimes\!
\bigoplus_{l\in\aleph_S(\vr)}\!\!\!\!\pi_l^*L_0\\
&=\pi_P^*\L \otimes \pi_B^*F_{(\ale_B(\vr),J_B(\vr))},
\end{split}$$ where $\L\!\lra\!\wt\cM_{0,\vr_P}$ is the universal tangent line bundle constructed in Subsection \[curve0bl\_subs\]. We also note that by ($I10$), $$\io_{\vr-1,\vr}^{\,*}
\big(\E_{\vr-1}^{\,*}\!\otimes\!\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn\big)
=\pi_P^*\L \otimes \pi_B^*\pi_{\P F_{(\ale_B(\vr),J_B(\vr))}}^*\ev_0^*T\Pn.$$ By ($I12$), the differential of $\wt\cD_{\vr-1}$ in the normal direction to the immersion $\io_{\vr-1,\vr}$ is given by $$\na\wt\cD_{\vr-1} =\pi_P^*\id\otimes \pi_B^*\cD_{(\ale_B(\vr),J_B(\vr))}.$$ Thus, if $$\io_{\vr,\vr}\!: \P\N_{\io_{\vr-1,\vr}}
\approx \wt\cM_{0,\vr_P} \times \wt\M_{0,(\ale_P(\vr),J_B(\vr))}^0
\lra \wt\M_{0,\vr}^{\vr}\subset \wt\M_{(\ale,J)}^{\vr}$$ is the immersion induced by $\io_{\vr-1,\vr}$, then $$\begin{split}
\io_{\vr,\vr}^{\,*}\wt\cD_{\vr} \!=\! \pi_B^*\wt\cD_0
&\in \Ga\big(\wt\cM_{0,\vr_P}\!\times\!\wt\M_{0,(\ale_P(\vr),J_B(\vr))}^0;
\io_{\vr,\vr}^{~*}\big(\E_{\vr}^* \!\otimes\!\pi_{\vr}^*\pi_{\P F_{(\ale,J)}}^*\ev_0^*T\Pn
\big)\big)\\
&=\Ga\big(\wt\cM_{0,\vr_P}\!\times\!\wt\M_{0,(\ale_B(\vr),J_B(\vr))}^0;
\pi_B^*(\ga_{(\ale_B(\vr),J_B(\vr))}^*\!\otimes\!
\pi_{\P F_{(\ale_B(\vr),J_B(\vr))}}^*\ev_0^*T\Pn)\big).
\end{split}$$ Lemmas \[map0bl\_lmm1\] and \[map0bl\_lmm2\] thus imply that the restriction of the section $\wt\cD_{\vr}$ to the exceptional divisor $\wt\M_{0,\vr}^{\vr}$ is transverse to the zero set away from the subvarieties $\wt\M_{0,\vr^*}^{\vr}$ with $\vr^*\!>\!\vr$. Thus, by the inductive assumption ($I4$) as stated above, ($I4$) is satisfied with $\vr\!-\!1$ replaced by $\vr$.\
We now verify that the remaining inductive assumptions are satisfied. If $\vr\!<\!\vr^*$, but $\vr\!\not\prec\!\vr^*$, $$\rho_{\vr^*}(\vr) = \rho_{\vr^*}(\vr\!-\!1)
\qquad\hbox{and}\qquad
\ov\M_{0,\vr^*}^{\vr-1}\cap\ov\M_{0,\vr}^{\vr-1} =\eset,$$ by definition and by ($I5$), respectively. It then follows that $$\begin{gathered}
\io_{\vr,\vr^*}=\io_{\vr-1,\vr^*}, \qquad
\wt\M_{0,\vr^*}^{\vr}\!\cap\!\wt\M_{0,\vr'}^{\vr}
=\wt\M_{0,\vr^*}^{\vr-1}\!\cap\!\wt\M_{0,\vr'}^{\vr-1} \quad\forall \vr'\!>\!\vr, \\
\io_{\vr,\vr^*}^{\,*}\E_{\vr}=\io_{\vr-1,\vr^*}^{\,*}\E_{\vr-1},
\qquad \N_{\io_{\vr,\vr^*}}=\N_{\io_{\vr-1,\vr^*}},
\quad\hbox{and}\quad \na\wt\cD_{\vr}=\na\wt\cD_{\vr-1}.\end{gathered}$$ Thus, the inductive assumptions ($I6$), ($I7$), and ($I10$)-($I12$), as stated above, imply the corresponding statements with $\vr\!-\!1$ replaced by $\vr$.\
Suppose that $\vr\!\prec\!\vr^*$. By ($I6$) and (1) of Lemma \[ag\_lmm2a\], the domain of the immersion $\io_{\vr,\vr^*}$ induced by the immersion $\io_{\vr-1,\vr^*}$ is the blowup of $$\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)} \times
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}$$ along the preimage of $\ov\M_{0,\vr}^{\vr-1}$ under $\io_{\vr-1,\vr^*}$ in $$\pi_{\rho_{\vr^*}(\vr-1)}\!\times\!\id\!:
\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)} \!\times\!
\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))} \lra
\wt\cM_{0,\vr_P^*}^0 \!\times\! \ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}.$$ By ($I7$), this preimage is $$\Big(\bigcup_{\rho\in\A_0(\vr^*;\vr)} \!\!\!\!\!\!\!
\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)}\Big)
\times \ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}.$$ By the last paragraph of Subsection \[curve0bl\_subs\] and the second paragraph after Lemma \[map0bl\_lmm1\], $$\wt\cM_{0,\rho_1}^{\rho_{\vr^*}(\vr-1)}
\cap \wt\cM_{0,\rho_2}^{\rho_{\vr^*}(\vr-1)} = \eset
\qquad \forall\, \rho_1,\rho_2\!\in\!\A_0(\vr^*;\vr), \, \rho_1\!\neq\!\rho_2.$$ Thus, by the construction of Subsection \[curve0bl\_subs\], the blowup of $\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)}$ along $$\bigcup_{\rho\in\A_0(\vr^*;\vr)}\!\!\!\!\!\!\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)}$$ is $\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr)}$, as needed for the inductive statement ($I6$), with $\vr\!-\!1$ replaced by $\vr$. For the same reasons, ($I10$), \_ref[bundletwist\_e2b]{}, and \_ref[bundletwist\_e3]{} imply that $$\begin{split}
\io_{\vr,\vr^*}^{\,*}\E_{\vr}
&=\io_{\vr-1,\vr^*}^{\,*}\E_{\vr-1}
\otimes \io_{\vr,\vr^*}^{\,*}\O\big(\wt\M_{0,\vr}^{\vr}\big)\\
&=\pi_P^* \E_{\rho_{\vr^*}(\vr-1)}
\otimes \bigotimes_{\rho\in\A_0(\vr^*;\vr)} \!\!\!\!\!\!\!
\pi_P^* \O\big(\wt\cM_{0,\rho}^{\rho}\big)
=\pi_P^* \E_{\rho_{\vr^*}(\vr)}.
\end{split}$$ Thus, the inductive statement ($I10$), with $\vr\!-\!1$ replaced by $\vr$, is satisfied. The assumption ($I7$) is checked similarly, using (3) of Lemma \[ag\_lmm2a\].\
We next determine the normal bundle for the immersion $\io_{\vr,\vr^*}$. The restrictions of the line bundles $L_{\rho_{\vr^*}(\vr-1),(l,i)}$ and $\E_{\rho_{\vr^*}(\vr-1)}$ to the complement of the exceptional divisors in $\wt\cM_{0,\vr_P^*}^{\rho_{\vr^*}(\vr-1)}$ are $\pi_{\rho_{\vr^*}(\vr-1)}^*L_{0,(l,i)}$ and $\pi_{\rho_{\vr^*}(\vr-1)}^*\E_0$, by the construction of Subsection \[curve0bl\_subs\]. Thus, by the last statement of Lemma \[map0bl\_lmm1\], ($I11$), and the inductive assumptions ($I1$) above and in Subsection \[curve0bl\_subs\], $$\begin{split}
&\io_{\vr-1,\vr^*}\big|_{\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)}
\times\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}}^*
T\ov\M_{0,\vr}^{\vr-1}
\big/T\big(\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)}
\!\times\!\ov\M_{0,(\ale_B(\vr^*),J_B(\vr^*))}\big)\\
&~=\!\bigoplus_{l\in\ale_P(\vr^*)-\ale_P(\vr)} \bigoplus_{i\in[m_l^*]}
\!\!\pi_P^*L_{\rho_{\vr^*}(\vr-1),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0
~\oplus~\!\bigoplus_{l\in\ale_P(\vr)} \bigoplus_{i\in[m_l^*]-I_{l,P}}
\!\!\!\!\!\!\!\!\!\pi_P^*L_{\rho_{\vr^*}(\vr-1),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0
\end{split}$$\
for all $\rho\!\in\!\A_0(\vr^*;\vr)$ as in the statement of Lemma \[map0bl\_lmm1\]. Let $$I_P(\rho)=\big\{(l,i)\!:l\!\in\!\ale_P(\vr),\, i\!\in\!I_{l,P}\big\}.$$ From Lemma \[ag\_lmm2b\], we then obtain $$\begin{split}
\N_{\io_{\vr,\vr^*}}
&=\bigoplus_{l\in\ale_P(\vr^*)}\bigoplus_{i\in[m_l^*]} \!\!
\bigg(\big(\pi_P^*L_{\rho_{\vr^*}(\vr-1),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0\big)
\!\otimes\!\pi_P^*
\O\Big(~~~~~-\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{\rho\in\A_0(\vr^*;\vr),(l,i)\in I_P(\rho)}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)}\Big)\bigg)\\
&\qquad\qquad\qquad\qquad\qquad\oplus~
\pi_P^*\E_{\rho_{\vr^*}(\vr-1)}^*\!\otimes\!
\bigoplus_{l\in\aleph_S(\vr^*)}\!\!\!\!\!\pi_l^*L_0\!\otimes\!\pi_P^*
\O\Big(-\!\!\!\!\!\!\!\!
\sum_{\rho\in\A_0(\vr^*;\vr)}\!\!\!\!\!\!\!\wt\cM_{0,\rho}^{\rho_{\vr^*}(\vr-1)}\Big)\bigg)\\
&=\bigoplus_{l\in\ale_P(\vr^*)}\bigoplus_{i\in[m_l^*]} \!\!
\big(\pi_P^*L_{\rho_{\vr^*}(\vr),(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*L_0\big)
~\oplus~\pi_P^*\E_{\rho_{\vr^*}(\vr)}^*\!\otimes\!
\bigoplus_{l\in\aleph_S(\vr^*)}\!\!\!\!\!\pi_l^*L_0.
\end{split}$$ The last equality above follows from \_ref[bundletwist\_e2a]{} by the same argument as in the previous paragraph. We have thus verified that the inductive assumption ($I11$), with $\vr\!-\!1$ replaced by $\vr$, is satisfied. Finally, the inductive assumption ($I12$) and the continuity of the two bundle sections involved in the identity in ($I12$), with $\vr\!-\!1$ replaced by $\vr$, imply ($I12$) with $\vr\!-\!1$ replaced by $\vr$.\
We conclude this construction after the blowup at the $\vr_{\max}$ step. Let $$\wt\M_{0,(\ale,J)}(\Pn,d)=\wt\M_{(\ale,J)}^{\vr_{\max}}, \qquad
\ti\E=\E_{\vr_{\max}}, \qquad \wt\cD_{(\ale,J)}=\wt\cD_{\vr_{\max}}.$$ By the inductive assumption ($I4$), applied with $\vr\!-\!1$ replaced by $\vr_{\max}$, the section $\wt\cD_{(\ale,J)}$ is transverse to the zero set. As in the previous two subsections, the final result of this blowup construction is independent of the order $<$ chosen to extend the partial ordering $\prec$ on $\A_0^*(\ale;d,J)$, as can be seen from ($I5$).
A Blowup of a Moduli Space of Genus-One Maps {#map1bl_sec}
============================================
Idealized Blowups and Immersions {#map1prelim_subs1}
--------------------------------
In this section we describe the main blowup construction of this paper. This is the sequential idealized blowup construction for $\ov\M_{1,k}(\Pn,d)$ with the initial data and the inductive step specified in Subsections \[map1prelim\_subs\] and \[map1blconstr\_subs\], respectively. This construction is outlined in Subsections \[descr\_subs\] and \[outline\_subs\].\
In contrast to the situations in Sections \[curvebl\_sec\] and \[map0bl\_sec\], the variety $\ov\M_{1,k}(\Pn,d)$ is singular. In order to describe the structure of $\ov\M_{1,k}(\Pn,d)$, we introduce the notion of [*idealized normal bundle*]{} for an immersion. Recall that the domain of an immersion is assumed to be a smooth variety.
\[virtvar\_e\] Suppose $\ov\M$ is a variety and $\io_X\!:X\!\lra\!M$ is an immersion. An [idealized normal bundle]{} for the immersion $\io_X$ is a vector bundle $\N_{\io_X}^{\ide}$ over $X$ such that $\N_{\io_X}\!\subset\!\N_{\io_X}^{\ide}$.
[*Remark:*]{} An idealized normal bundle is of course not unique; an idealized normal bundle plus any other vector bundle is still an idealized normal bundle. If the image of $\io_X$ is an irreducible component of $\ov\M$, an idealized normal bundle of the smallest possible rank still need not be unique; it can be twisted by any divisor in $X$ disjoint from the preimage under $\io_X$ of the other components of $\ov\M$. For each of the immersions we encounter in the next subsection, there is a natural choice for $\N_{\io_X}^{\ide}$. These idealized normal bundles also transform in a natural way under blowups and proper immersions, as described in Lemma \[virimmer\_lmm2\] below.\
Suppose $\ov\M$ is a variety, $Z$ is a smooth subvariety of $\ov\M$, and $\N_{\io_Z}^{\ide}$ is an idealized normal bundle for the embedding $\io_Z$ of $Z$ into $\ov\M$. Let $$\cE_Z\!\equiv\!\P\N_{\io_Z} \subset \Bl_Z\ov\M$$ be the exceptional divisor for the blowup of $\ov\M$ along $Z$. We denote by $\Bl_Z^{\ide}\ov\M$ the variety obtained by identifying $\Bl_Z\ov\M$ with $$\cE_Z^{\ide}\equiv\P\N_{\io_Z}^{\ide}$$ along $\cE_Z$. We will call $$\pi^{\ide}\!: \Bl_Z^{\ide}\ov\M\lra \ov\M \qquad\hbox{and}\qquad
\cE_Z^{\ide}\subset\Bl_Z^{\ide}\ov\M$$ [the idealized blowup of $\ov\M$ along $Z$]{} and [the idealized exceptional divisor for $\pi^{\ide}$]{}, respectively. More generally, we will call $$\pi\!: \wt\M \lra \ov\M$$ an [idealized blowup of $\ov\M$]{} if $\pi$ is a composition of idealized blowups along smooth subvarieties. In practice, [*idealized blowup*]{} is simply a convenient term. In the inductive assumption ($I1$) in Subsection \[map1blconstr\_subs\] below, it can be replaced by [*morphism of varieties*]{}, as the remaining inductive assumptions describe all the relevant properties of this morphism. Let $$\ga_Z\lra \cE_Z^{\ide}$$ be the tautological line bundle. Note that the normal bundle of $\cE_Z\!\subset\!\cE_Z^{\ide}$ in $$\Pr_Z\ov\M=\Bl_Z\ov\M$$ is $\ga_Z|_{\cE_Z}$. (This observation implies the first statement of Lemma \[virimmer\_lmm2\].)\
Our strategy is as follows. We begin with a space with a properly self-intersecting collection of immersions, each with an idealized normal bundle. These are the immersions $\io_{\si}$ with $\si\!\in\!\A_1(d,k)$ defined in Subsection \[descr\_subs\]; their images are the subvarieties $\ov\M_{1,\si}(\Pn,d)$ of $\ov\M_{1,k}(\Pn,d)$. The idealized normal bundle for the immersion $\io_{\si}$ is the direct sum of the deformation spaces of the nodes between the contracted genus-one curve and the non-contracted genus-zero curves that are identified by $\io_{\si}$. At each stage, one of our immersions is an embedding, and we blow it up, replacing it with its idealized exceptional divisor. The exceptional divisor of the blowup of the main component is the intersection of the new main component with the idealized exceptional divisor. Then after each step, we have a new properly self-intersecting collection of immersions. Moreover, there is a natural idealized normal bundle to each of the proper transforms of the immersions we have “yet to blow up”.\
We now say this more explicitly. The following two lemmas are direct extensions of Corollary \[immercoll\_crl\] and Lemma \[ag\_lmm2b\]. The first lemma states that if we have a properly self-intersecting collection of immersions, one of which is an embedding, then upon blowing up the embedding, we still have a properly self-intersecting collection of immersions. It is immediate from the definition of “properly self-intersecting”, by checking in local coordinates.\
The second part of the second lemma follows from Lemma \[ag\_lmm2b\] with only one change. Instead of writing $$\N_{\io_X} =\bigoplus_{i\in I}L_i \qquad\hbox{and}\qquad
\N_{\Pr_Z\io_X}=\bigoplus_{i\in I}
\Big(\pi^*L_i\otimes \bigotimes_{i\in I_{\vr}}\O(-E_{\vr})\Big)$$ as in the statement of Lemma \[ag\_lmm2b\], we are saying that if we have a natural inclusion $\N_{\io_X}\!\subset\!\bigoplus_{i \in I}L_i$, then we get a natural conclusion $$\N_{\Pr_Z\io_X} \subset \bigoplus_{i\in I}
\Big(\pi^*L_i\otimes \bigotimes_{i\in I_{\vr}}\O(-E_{\vr})\Big).$$ The vector bundles on the right are the original idealized normal bundle and the new idealized normal bundle, respectively.
\[virimmer\_lmm\] Suppose $\ov\M$ is a variety, $\{\io_{\si}\!:X_{\si}\!\lra\!\ov\M\}_{\si\in\A}$ is a properly self-intersecting collection of immersions, and $\si\!\in\!\A$ is such that $\io_{\si}$ is an embedding. If $\N_{\io_{\si}}^{\ide}$ is an idealized normal bundle for $\io_{\si}$, then $$\big\{\Pr_{\Im\,\io_{\si}}\io_{\si'}\big\}_{\si'\in\A-\{\si\}}
\cup\big\{\io_{\cE_{\Im\io_{\si}}^{\ide}}\big\}$$ is a properly self-intersecting collection of immersions into $\Bl_{\Im\,\io_{\si}}^{\ide}\ov\M$.
\[virimmer\_lmm2\] If $\ov\M$ is a variety, $Z$ is a smooth subvariety of $\ov\M$, and $\N_{\io_Z}^{\ide}$ is an idealized normal bundle for $\io_Z$, then $$\N_{\io_{\cE_Z^{\ide}}}^{\ide}=\ga_Z$$ is an idealized normal bundle for the immersion $\io_{\cE_Z^{\ide}}$. Suppose in addition that $\io_X$, $\A$, $Z_{\vr}$, and $E_{\vr}$ are as in Lemma \[ag\_lmm2b\] and $\N_{\io_X}^{\ide}$ is an idealized normal bundle for $\io_X$. If there exist a splitting $$\N_{\io_X}^{\ide}=\bigoplus_{i\in I}L_i \lra X$$ and a subset $I_{\vr}$ of $I$ for each $\vr\!\in\!\A$ such that \_ref[ag\_lmm2b\_e]{} holds, then $$\N_{\Pr_Z\io_X}^{\ide}=\bigoplus_{i\in I}
\Big(\pi^*L_i\otimes \bigotimes_{i\in I_{\vr}}\O(-E_{\vr})\Big)$$ is an idealized normal bundle for the immersion $\Pr_Z\io_X$.
\[prsub\_dfn\] Suppose $\ov\M$ is a variety, $\io_X\!:X\!\lra\!\ov\M$ is an immersion, $\ov\M^0$ is a subvariety in $\ov\M$, and $TC\ov\M^0\!\subset\!T\ov\M$ is the tangent cone of $\ov\M^0$ in $\ov\M$ ($TC\ov\M^0$ not necessarily reduced). The subvariety $\ov\M^0$ is [proper relative to $\io_X$]{} if $$d\io_X\,TC\io_X^{-1}(\ov\M^0)
= \io_X^*TC\ov\M^0\cap \Im\, d\io_X\subset \io_X^*T\ov\M$$ and the map $$\label{prsub_dfn_e}
\io_X^*TC\ov\M^0|_{\io_X^{-1}(\ov\M^0)}\big/\Im\,d\io_X|_{TC\io_X^{-1}(\ov\M^0)}
\lra \io_X^*T\ov\M/\Im\,d\io_X \subset \N_{\io_X}^{\ide}$$ induced by inclusions is injective, with its image being reduced.
The left-hand side of \_ref[prsub\_dfn\_e]{} denotes the family of cones over $\io_X^{-1}(\ov\M^0)$ such that for each $x\!\in\!\io_X^{-1}(\ov\M^0)$ $$\io_X^*TC\ov\M^0|_{\io_X^{-1}(\ov\M^0)}\big/\Im\,d\io_X|_{TC\io_X^{-1}(\ov\M^0)}
\Big|_x$$ is the quotient by the minimal vector subspace of $\Im\,d\io_X|_x\!=\!d\io_X(T_xX)$ containing the cone $\Im\,d\io_X|_{T_xC\io_X^{-1}(\ov\M^0)}$. If $TC\io_X^{-1}(\ov\M^0)$ is a vector bundle, the two conditions in (2) of Definition \[prsub\_dfn\] are equivalent.\
If $\ov\M^0$ is a subvariety of $\ov\M$ which is proper relative to an immersion $\io_X\!:X\!\lra\!\ov\M$, we denote by $$\N_{\io_X|\ov\M^0}\subset \io_X^*T\ov\M/\Im\,d\io_X
\subset \N_{\io_X}^{\ide}$$ the image of the homomorphism \_ref[prsub\_dfn\_e]{}. We will call $\N_{\io_X|\ov\M^0}$ [the normal cone of $\io_X|_{\io_X^{-1}(\ov\M^0)}$ in $\ov\M^0$]{}.
\[virimmer\_lmm3\] Suppose $\ov\M$ is a variety, $\io_X\!:X\!\lra\!\ov\M$ is an immersion with an idealized normal bundle $\N_{\io_X}^{\ide}$, $\ov\M^0$ is a subvariety of $\ov\M$ which is proper relative to $\io_X$, and $$\cZ\subset\ov\cZ\!\equiv\!\io_X^{-1}(\ov\M^0)$$ is such that $\N_{\io_X|\ov\M^0}$ is the closure of $\N_{\io_X|\ov\M^0}|_{\cZ}$ in $\N_{\io_X}^{\ide}$.\
(1) If $X$ is a smooth subvariety of $\ov\M$, then $\Pr_X\ov\M^0$ is proper relative to the immersion $\io_{\cE_X^{\ide}}$, $$\cE_X^{\ide}\cap\Pr_X\ov\M^0 \subset \cE_X$$ is the closure of $\P\N_{\io_X|\ov\M^0}|_{\cZ}$ in $\cE_X^{\ide}$, and $$\N_{\io_{\cE_X^{\ide}}|\Pr_X\ov\M^0}=\ga_X|_{\cE_X^{\ide}\cap\Pr_X\ov\M^0}.$$ (2) If $Z$ is a smooth subvariety of $\ov\M$ disjoint from $\io_X(\cZ)$ and $\N_{\io_Z}^{\ide}$ is an idealized normal bundle for $\io_Z$, then $\Pr_Z\ov\M^0$ is a proper subvariety of $\Bl_Z^{\ide}\ov\M$ relative to the immersion $\Pr_Z\io_X$ and $\N_{\Pr_Z\io_X|\Pr_Z\ov\M^0}$ is the closure of $\N_{\io_X|\ov\M^0}|_{\cZ}$ in $\N_{\Pr_Z\io_X}^{\ide}$.
The first part of (1) essentially follows from the universal property of blowing up: if $\ov \M$ is blown up along $Z$, then the proper transform of $\ov \M^0$ in $\ov \M$ (the scheme-theoretic closure of $\ov\M^0\!-\!Z$ in the blowup) is the blowup of $\ov \M^0$ along $\ov\M^0\!\cap\!Z$, and the normal bundle to the exceptional divisor in $Bl_{\ov \M^0 \cap Z} \ov \M^0$ is the restriction of the normal bundle of the exceptional divisor in $Bl_Z \ov \M$. The statement (1) is the etale-local version of this. Part (2) is clear by working in local coordinates.
Preliminaries {#map1prelim_subs}
-------------
In this subsection, we state a number of known facts concerning the moduli space $\ov\M_{1,k}(\Pn,d)$ that insure that the inductive requirements of the next subsection are satisfied at the initial stage of the inductive construction. Lemmas \[map1bl\_lmm1a\]-\[map1bl\_lmm1b3\], with the exception of one statement, are straightforward to check from the definitions (and [@P] in some cases). We show that the last statement of Lemma \[map1bl\_lmm1b1\] is a reinterpretation of a standard fact concerning moduli spaces of stable maps.\
Let $(\A_1(d,k),\prec)$ be the partially ordered set of triples described in Subsection \[descr\_subs\]. It has a unique minimal element and a unique maximal element: $$\si_{\min}=(1;\eset,[k]) \qquad\hbox{and}\qquad \si_{\max}=(d;[k],\eset).$$ Let $<$ be an order on $\A_1(d,k)$ extending the partial ordering $\prec$. For every $\si\!\in\!A_1(d,k)$, we define $$\si\!-\!1 \in \{0\}\!\sqcup\!\A_1(d,k)$$ as in \_ref[minusdfn\_e]{}. For each element $\si\!=\!(m;J_P,J_B)$ of $\A_1(d,k)$, let $$\ov\M_{1,\si}^0 \!\equiv\! \ov\M_{1,\si}(\Pn,d)
\subset \ov\M_{1,k}^0 \!\equiv\! \ov\M_{1,k}(\Pn,d)$$ be the subvarieties defined in Subsection \[descr\_subs\].\
[*Warning:*]{} Note that $\ov\M_{1,k}^0$ denotes the entire moduli space $\ov\M_{1,k}(\Pn,d)$ and not the main component $\ov\M_{1,k}^0(\Pn,d)$. Similarly to Sections \[curvebl\_sec\] and \[map0bl\_sec\], the superscript $0$ indicates the $0$th stage in the blowup process.
\[map1bl\_lmm1a\] If $\si_1\!=\!(m_1;J_{1;P},J_{1;B})$ and $\si_2\!=\!(m_2;J_{2;P},J_{2;B})$ are elements of $\A_1(d,k)$, $\si_1\!\neq\!\si_2$, $\si_1\!\not\prec\!\si_2$, and $\si_2\!\not\prec\!\si_1$, then $$\begin{gathered}
\ov\M_{1,\si_1}^0 \cap \ov\M_{1,\si_2}^0\subset \ov\M_{1,\ti\si(\si_1,\si_2)}^0,
\qquad\hbox{where}\\
\ti\si(\si_1,\si_2)=\big(\!\min(m_1,m_2);J_{1;P}\!\cap\!J_{2;P},J_{1;B}\!\cup\!J_{2;B}\big).\end{gathered}$$\
With $\si$ as above, we define $$I_P(\si)=\ale_B(\si)=[m], \qquad J_P(\si)=J_P, \qquad J_B(\si)=J_B,
\qquad G_{\si}=S_m.$$ As in Subsection \[descr\_subs\], we denote by $$\begin{gathered}
\io_{0,\si}\!: \ov\cM_{1,(I_P(\si),J_P(\si))}^0 \times
\ov\M_{0,(\ale_B(\si),J_B(\si))} \lra
\ov\M_{1,\si}^0\subset\ov\M_{1,k}^0,\\
\hbox{where}\qquad \ov\M_{0,(\ale_B(\si),J_B(\si))}=\ov\M_{0,(\ale_B(\si),J_B(\si))}(\Pn,d),\end{gathered}$$ the natural node-identifying map and by $$\bar\io_{0,\si}\!:
\big(\ov\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!
\ov\M_{0,(\ale_B(\si),J_B(\si))}\big)\big/G_{\si}\lra \ov\M_{1,k}^0$$ the induced immersion. Let $$\pi_P, \, \pi_B\!:
\ov\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!
\ov\M_{0,(\ale_B(\si),J_B(\si))}\lra
\ov\cM_{1,(I_P(\si),J_P(\si))}^0, \ov\M_{0,(\ale_P(\si),J_B(\si))}$$ be the two projection maps.
\[map1bl\_lmm1b1\] If $d,n\!\in\!\Z^+$ and $k\!\in\!\bar\Z^+$, the collections $\{\io_{0,\si}\}_{\si\in\A_1(d,k)}$ and $\{\bar\io_{0,\si}\}_{\si\in\A_1(d,k)}$ of immersions are properly self-intersecting. If $\si^*\!=\!(m^*;J_P^*,J_B^*)\!\in\!\A_1(d,k)$, $$\Im^s\,\bar\io_{0,\si^*} \subset \bigcup_{\si'\prec\si^*}\! \ov\M_{1,\si'}
\qquad\hbox{and}\qquad
\N_{\io_{0,\si^*}}^{\ide}=\bigoplus_{i\in[m^*]}\!\! \pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0$$ is an idealized normal bundle for $\io_{0,\si^*}$.
We deduce the last claim of this lemma from the deformation-obstruction exact sequence (24.2) in [@H] as follows. Suppose $$\begin{gathered}
[\Si,u]=\io_{0,\si^*}\big([\Si_P]\!\times\![\Si_B,u_B]\big)
\in \ov\M_{1,\si^*}^0, \qquad\hbox{where}\\
[\Si_B,u_B]=\big([\Si_i,u_i]\big)_{i\in[m^*]}\in\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}.\end{gathered}$$ By [@H (24.2)], there exists a natural homomorphism $$j_{\Si,u}\!: T\ov\M_{1,k}(\Pn,d)\big|_{[\Si,u]}=\Def(\Si,u) \lra \Def(\Si),$$ where $\Def(\Si,u)$ and $\Def(\Si)$ denote the deformations of the stable-map pair $(\Si,u)$ and the deformations of the curve $\Si$ (with its marked points), respectively. As $[\Si,u]$ is considered as the image of $[\Si_P]\!\times\![\Si_B,u_B]$ under $\io_{0,\si^*}$, there are $m^*$ distinguished nodes of $\Si$. These are the nodes of $\Si$ that do not correspond to either the nodes of $\Si_P$ or the nodes of any of the curves $\Si_i$ with $i\!\in\![m^*]$; see Figure \[idebund\_fig\]. Let $$\Def(\Si_P,\Si_B)\subset \Def(\Si)$$ be the deformations of $\Si$ that do not smooth out the distinguished nodes of $\Si$. Since the smoothing of a given node of $\Si$ is parametrized by the tensor product of the tangent lines to the two branches of $\Si$ at the node, we have an exact sequence $$0 \lra \Def(\Si_P,\Si_B)\lra \Def(\Si) \stackrel{j_{\Si}}{\lra}
\N_{\io_{0,\si^*}}^{\ide}\big|_{[\Si,u]} \lra 0.$$ We denote by $$\Def\big(\Si_P,(\Si_B,u_B)\big)\subset
T\ov\M_{1,k}(\Pn,d)\big|_{[\Si,u]}=\Def(\Si,u)$$ the kernel of the map $$j_{\Si}\!\circ j_{\Si,u}\!: \Def(\Si,u) \lra \N_{\io_{0,\si^*}}^{\ide}\big|_{[\Si_P]\times[\Si_B,u_B]}.$$ The space $\Def\big(\Si_P,(\Si_B,u_B)\big)$ consists of deformations of $(\Si,u)$ that do not smooth out the $m^*$ distinguished nodes of $\Si$. Thus, $$\begin{split}
\Def\big(\Si_P,(\Si_B,u_B)\big) &\approx\Def(\Si_P)\oplus\Def(\Si_B,u_B)\\
&=T\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^0|_{[\Si_P]} \oplus
T\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}|_{[\Si_B,u_B]}.
\end{split}$$ The isomorphism from the right-hand side to the left-hand side is given by $d\io_{0,\si^*}$. Thus, the homomorphism $j_{\Si}\!\circ\!j_{\Si,u}$ induces an injection $$\N_{\io_{0,\si^*}}|_{[\Si,u]} \equiv
TC\ov\M_{1,k}(\Pn,d)\big|_{[\Si,u]}\big/\Im\,d\io_{0,\si^*}
\lra \N_{\io_{0,\si^*}}^{\ide}\big|_{[\Si_P]\times[\Si_B,u_B]},$$ as needed.
(-1.1,-3.1)(10,1.4) (5,-1.5)(1.5,2.5) (6.8,-1.5)[2]{}[150]{}[210]{}(3.2,-1.5)[2]{}[330]{}[30]{} (2.5,-1.5)[1]{}(3.5,-1.5)[.2]{} (1.79,-.79)[.25]{}(1.25,-.8) (1.79,-2.21)[.25]{}(1.23,-2.3) (6.5,-1.5)[.25]{}(7,-1.5) (10,2)(10,-5) (11,2)[1]{}(10,2)[.25]{} (11,-1.5)[1]{}(10,-1.5)[.25]{} (11,-5)[1]{}(10,-5)[.25]{} (12.42,-3.58)[1]{}(11.71,-4.29)[.15]{} (12.42,-6.42)[1]{}(11.71,-5.71)[.15]{} (13,2)(13,-1.5)(14,-5) (8.4,-1.5) (15.5,-1.5)(20.5,-1.5)(18,-1) (27,-1.5)(1.5,2.5) (28.8,-1.5)[2]{}[150]{}[210]{}(25.2,-1.5)[2]{}[330]{}[30]{} (24.5,-1.5)[1]{}(25.5,-1.5)[.15]{} (29.5,-1.5)[1]{}(28.5,-1.5)[.25]{}(28.3,-1.7)[A3]{} (23.09,-.09)[1]{} (23.09,-2.91)[1]{} (30.91,-.09)[1]{}(30.21,-.79)[.15]{} (30.91,-2.91)[1]{}(30.21,-2.21)[.15]{} (23.79,-.79)[.25]{}(23.95,-.95)[A1]{} (23.79,-2.21)[.25]{}(24,-2.4)[A2]{} (27,-7)
\[map1bl\_lmm1b2\] If $d$, $n$, $k$, and $\si^*$ are as in Lemma \[map1bl\_lmm1b1\], $\si\!\in\!\A_1(d,k)$ is as above, and $\si\!\prec\!\si^*$, then $$\begin{gathered}
\io_{0,\si^*}^{~-1} \big( \ov\M_{1,\si}^0\big)
=\Big(\bigcup_{\rho\in\A_P(\si^*;\si)} \!\!\!\!\!\!\!\! \ov\cM_{1,\rho}^0 \Big)
\times \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))},
\qquad\hbox{where}\\
\A_P(\si^*;\si)=\Big\{
\rho\!=\!\big(I_P\!\sqcup\!J_P,\{I_k\!\sqcup\!J_k\!: k\!\in\!K\}\big)
\!\in\!\A_1\big(I_P(\si^*),J_P(\si^*)\big)\!: |K|\!+\!|I_P|\!=\!m\Big\}\end{gathered}$$ and $\A_1(I_P(\si^*),J_P(\si^*))$ and $\ov\cM_{1,\rho}^0\!\equiv\!\ov\cM_{1,\rho}$ are as in Subsection \[curvebldata\_subs\]. Furthermore, if $\rho\!\in\!\A_P(\si^*;\si)$ is as above, $$\io_{0,\si^*} \big|_{\ov\cM_{1,\rho}^0\times
\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}}^* T\ov\M_{1,\si}^0
\big/T\big(\ov\cM_{1,\rho}^0\!\times\!\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}\big)
=\bigoplus_{i\in[m]-I_P}\!\!\!\!\!\!\pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0.$$
\[map1bl\_lmm1b3\] If $d$, $n$, $k$, $\si$, and $\si^*$ are as above, then $$\begin{gathered}
\io_{0,\si}^{~-1} \big( \ov\M_{1,\si^*}^0\big)
=\ov\cM_{1,(I_P(\si),J_P(\si))}^0
\times \Big(\bigcup_{\vr\in\A_B(\si;\si^*)} \!\!\!\!\!\!\!\! \ov\M_{0,\vr} \Big),
\qquad\hbox{where}\\
\A_B(\si;\si^*)=\big\{ \vr\!=\!\big((\si_l)_{l\in\ale_B(\si)},J_B\big)
\!\in\!\A_0\big(\ale_B(\si);d,J_B(\si)\big)\!:
\big|\ale_B(\vr)\big|\!=\!m^*\big\},\end{gathered}$$ and $\A_0(\ale_B(\si);d,J_B(\si))$, $\ale_B(\vr)$, and $\ov\M_{0,\vr}\!\equiv\!\ov\M_{0,\vr}(\Pn,d)$ are as in Subsection \[map0prelim\_subs\]. Furthermore, if $\vr\!\in\!\A_B(\si;\si^*)$ is as above, $$\io_{0,\si} \big|_{\ov\cM_{1,(I_P(\si),J_P(\si))}^0\times
\ov\M_{0,\vr}}^* T\ov\M_{1,\si^*}^0
\big/T\big(\ov\cM_{1,(I_P(\si),J_P(\si))}^0\!\times\!\ov\M_{0,\vr}\big)
=\bigoplus_{i\in\aleph_P(\vr)}\!\!\!\!\pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0,$$ where $\aleph_P(\vr)\!\subset\!\aleph_B(\si)$ is as in Subsection \[map0prelim\_subs\].
We note that for every $\si^*\!\in\!\A_1(d,k)$, $$\A_1\big(I_P(\si^*),J_P(\si^*)\big)= \bigsqcup_{\si\prec\si^*}\!\A_P(\si^*;\si).$$ Furthermore, if $\si_1,\si_2\!\in\!\A_1(d,k)$ are such that $\si_1,\si_2\!\prec\!\si^*$, then $$\rho_1\!\in\!\A_P(\si^*;\si_1), \quad \rho_2\!\in\!\A_P(\si^*;\si_2), \quad
\rho_1\!\prec\!\rho_2 \qquad\Lra\qquad \si_1\!\prec\!\si_2.$$ Thus, we can choose an ordering $<$ on $\A_1(I_P(\si^*),J_P(\si^*))$ extending the partial ordering $\prec$ of Subsection \[curve1bl\_subs\] such that $$\si_1\!<\!\si_2, \quad \rho_1\!\in\!\A_P(\si^*;\si_1),
\quad \rho_2\!\in\!\A_P(\si^*;\si_2) \qquad\Lra\qquad \rho_1\!<\!\rho_2,$$ whenever $\si_1,\si_2\!\in\!\A_1(d,k)$ are such that $\si_1,\si_2\!\prec\!\si^*$. In the next subsection, we will refer to the blowup construction of Subsection \[curve1bl\_subs\] corresponding to such an ordering.\
Similarly, if $\si'\!\in\!\A_1(d,k)$, $$\A_0\big(\ale_B(\si');d,J_B(\si')\big)= \bigsqcup_{\si'\prec\si}\!\A_B(\si';\si).$$ Furthermore, if $\si_1,\si_2\!\in\!\A_1(d,k)$ are such that $\si'\!\prec\!\si_1,\si_2$, then $$\vr_1\!\in\!\A_B(\si';\si_1), \quad \vr_2\!\in\!\A_B(\si';\si_2), \quad
\vr_1\!\prec\!\vr_2 \qquad\Lra\qquad \si_1\!\prec\!\si_2.$$ Thus, we can choose an ordering $<$ on $\A_0(\ale_B(\si');d,J_B(\si'))$ extending the partial ordering $\prec$ of Subsection \[map0prelim\_subs\] such that $$\si_1\!<\!\si_2, \quad \vr_1\!\in\!\A_B(\si';\si_1),
\quad \vr_2\!\in\!\A_B(\si';\si_2) \qquad\Lra\qquad \vr_1\!<\!\vr_2,$$ whenever $\si_1,\si_2\!\in\!\A_1(d,k)$ are such that $\si'\!\prec\!\si_1,\si_2$. In the next subsection, we will refer to the blowup construction of Subsection \[map0blconstr\_subs\] corresponding to such an ordering.\
We denote by $\ov\M_{1,(0)}^0$ the main component $\ov\M_{1,k}^0(\Pn,d)$ of the moduli space $\ov\M_{1,k}(\Pn,d)$. If $\si\!\in\!\A_1(d,k)$, we put $$\begin{aligned}
{1}
&\ov\cZ_{\si}^0= \io_{0,\si}^{~-1} \big( \ov\M_{1,(0)}^0\big)
\equiv \io_{0,\si}^{~-1} \big( \ov\M_{1,(0)}^0\!\cap\ov\M_{1,\si}^0\big);\\
&\cZ_{\si}^0= \io_{0,\si}^{~-1} \big( \ov\M_{1,(0)}^0\!\cap\M_{1,\si}^0\big)
\subset\ov\cZ_{\si}^0, \quad\hbox{where}\quad
\M_{1,\si}^0=\M_{1,\si}^0(\Pn,d).\end{aligned}$$ We denote by $\N\ov\cZ_{\si}^0\!\subset\!\N_{\io_{0,\si}}^{\ide}$ the normal cone $\N_{\io_{0,\si}|\ov\M_{1,(0)}^0}$ for $\io_{0,\si}|_{\ov\cZ_{\si}^0}$ in $\ov\M_{1,(0)}^0$. Its structure is described in Lemma \[map1bl\_lmm2\] below. Let $$\cD_{0,\si} \in \Ga\big(\ov\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!
\ov\M_{0,(\ale_B(\si),J_B(\si))};
\Hom(\N_{\io_{0,\si}}^{\ide},\pi_P^*\E_0^*\!\otimes\!\pi_B^*\ev_0^*T\Pn)\big)$$ be the section defined by $$\cD_{0,\si}\big|_{\pi_P^*L_i\otimes\pi_B^*\pi_i^*L_0}
=\pi_P^*s_{0,i}\!\otimes\!\pi_B^*\pi_i^*\cD_0,
\qquad\forall\,i\!\in\![m],$$ where $s_{0,i}$ and $\cD_0$ are as in Subsections \[curve1bl\_subs\] and \[map0str\_subs\], respectively.
\[map1bl\_lmm2\] For all $\si\!\in\!\A_1(d,k)$, $\ov\M_{1,(0)}^0$ is a proper subvariety of $\ov\M_{1,k}^0$ relative to the immersions $\io_{0,\si}$ and $\bar\io_{0,\si}$. Furthermore, $$\begin{gathered}
\cZ_{\si}^0=\big\{b\!\in\!
\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!\M_{0,(\ale_B(\si),J_B(\si))}\!:
\ker\cD_{0,\si}|_b\!\neq\!\{0\}\big\}\\
\hbox{and}\qquad
\N\ov\cZ_{\si}^0\big|_{\cZ_{\si}^0}=\ker\cD_{0,\si}\big|_{\cZ_{\si}^0}.\end{gathered}$$ Finally, $\ov\cZ_{\si}^0$ is the closure of $\cZ_{\si}^0$ in $\ov\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!\ov\M_{0,(\ale_B(\si),J_B(\si))}$ and $\N\ov\cZ_{\si}^0$ is the closure of $\N\ov\cZ_{\si}^0\big|_{\cZ_{\si}^0}$ in $\N_{\io_{0,\si}}^{\ide}$.
This lemma is a consequence of [@g1comp Theorem \[g1comp-str\_thm\]] and related results. In particular, the first claim in the second sentence of Lemma \[map1bl\_lmm2\] is a special case of the first statement of [@g1comp Theorem \[g1comp-str\_thm\]]. The second claim is nearly a special case of the last statement of [@g1comp Theorem \[g1comp-str\_thm\]], but some additional comments are required. [@g1comp Theorem \[g1comp-str\_thm\]] by itself is a purely topological statement, as it describes the topological structure of a neighborhood of each stratum of $\io_{0,\si}(\ov\cZ_{\si}^0)$ in $\ov\M_{1,(0)}^0$. On the other hand, by Subsection 4.1 in [@g1], $\N\ov\cZ_{\si}^0\big|_{\cZ_{\si}^0}$ is contained in $\ker\cD_{0,\si}$. The second claim in the second sentence of Lemma \[map1bl\_lmm2\] can then be obtained from a dimension count and a comparison of the gluing construction used in the proof of [@g1comp Theorem \[g1comp-str\_thm\]] with the analysis of limiting behavior in [@g1 Subsect. 4.1]. This comparison implies that the gluing parameter in the analytic construction of [@g1comp] agrees to the first two orders in the zero limit with the smoothing parameter in algebraic geometry. Thus, $\N\ov\cZ_{\si}^0\big|_{\cZ_{\si}^0}$ must be equal to $\ker\cD_{\si,0}$. Alternatively, suppose that $d\!\le\!n$. If the moduli space $\ov\M_{1,\si}^0$ is nonempty, then $m\!\le\!n$ and thus for a Zariski open subset $\cZ_{\si;1}$ of $\cZ_{\si}^0$ $$\label{map1bl_lmm2e3}
1\le\dim\N\ov\cZ_{\si}^0\big|_{\cZ_{\si;1}}=1
=\dim\ker\cD_{0,\si}\big|_{\cZ_{\si;1}}
\qquad\Lra\qquad
\N\ov\cZ_{\si}^0\big|_{\cZ_{\si;1}}=\ker\cD_{0,\si}\big|_{\cZ_{\si;1}}.$$ Since $\cD_{0,\si}$ is transverse to the zero set over $\cZ_{\si}^0$, the second claim in the second sentence of the lemma follows from \_ref[map1bl\_lmm2e3]{}, if $d\!\le\!n$. The general case follows from the observation that $$\ov\M_{1,\si}(\Pn,d)=\big\{[\Si,u]\!\in\!\ov\M_{1,\si}(\P^{n+d},d)\!:
u(\Si)\!\subset\!\Pn\big\}$$ and the $d\!\le\!n$ case.\
The first claim in the last sentence of Lemma \[map1bl\_lmm2\] can be obtained by combining the first statement of [@g1comp Theorem \[g1comp-str\_thm\]], the $m\!=\!1$ case of [@g2n2and3 Theorem 2.8], and the Implicit Function Theorem. It also follows immediately from the last claim of Lemma \[map1bl\_lmm2\]. The latter can be deduced from [@g1comp Theorem \[g1comp-str\_thm\]] as follows. Suppose first that $m\!\le\!n$. In this case, [@g2n2and3 Theorem 2.8] implies that $\ov\cZ_{\si}^0$ admits a stratification $$\ov\cZ_{\si}^0=\cZ_{\si;1}\sqcup \bigsqcup_{\al\in\A}\cZ_{\si;\al}$$ such that $\cZ_{\si;1}$ is a Zariski open subset of $\ov\cZ_{\si}^0$, $$\begin{gathered}
\cZ_{\si;1}\subset\cZ_{\si}^0, \qquad
\dim\,\N\ov\cZ_{\si}^0|_b\!=\!1 ~~~\forall\, b\!\in\!\cZ_{\si;1},\notag\\
\label{nofuzz_e1}
\hbox{and}\qquad
\max\big\{\!\dim\,\N\ov\cZ_{\si}^0|_b\!: b\!\in\!\cZ_{\si;\al}\big\}
\le \codim_{\ov\cZ_{\si}^0}\cZ_{\si;\al} ~~~\forall\,\al\!\in\!\A;\end{gathered}$$ see the next paragraph. Let $$\wt\cZ_{\si}^0= \P \N\ov\cZ_{\si}^0 \subset
\P\N_{\io_{0,\si}}^{\ide}\big|_{\ov\cZ_{\si}^0}$$ be the exceptional divisor for the blowup of $\ov\M_{1,(0)}^0$ along $\ov\M_{1,\si}^0$. Since all irreducible components of $\wt\cZ_{\si}^0$ must be of the same dimension, $\wt\cZ_{\si}^0$ must be the closure of $\wt\cZ_{\si}^0|_{\cZ_{\si}^0}$ by \_ref[nofuzz\_e1]{}. This closure property remains valid even if we do not assume that $m\!\le\!n$ for the following reason. Let $pt\!\in\!\P^{n+d}$ be any point not contained in $\Pn$. Let $$\pi\!: \P^{n+d}-\{pt\}\lra\Pn$$ be the corresponding linear projection. It induces projection maps $$\begin{aligned}
{1}
&\vph\!: \big\{[\Si,u]\!\in\!\ov\M_{1,k}^0(\P^{n+d},d)\!: pt\!\not\in\!u(\Si)\big\}
\lra \ov\M_{1,k}^0(\Pn,d) \qquad\hbox{and}\\
&\ti\vph\!: \big\{[\Si,u;v]\!\in\!\wt\cZ_{\si}^0(\P^{n+d},d)\!: pt\!\not\in\!u(\Si)\big\}
\lra \wt\cZ_{\si}^0(\Pn,d).\end{aligned}$$ The latter map takes $\wt\cZ_{\si}^0(\P^{n+d},d)|_{\cZ_{\si}^0(\P^{n+d},d)}$ to $\wt\cZ_{\si}^0(\Pn,d)|_{\cZ_{\si}^0(\Pn,d)}$. Since the closure of $$\wt\cZ_{\si}^0(\P^{n+d},d)|_{\cZ_{\si}^0(\P^{n+d},d)}$$ contains $\wt\cZ_{\si}^0(\Pn,d)$, it follows that so does the closure of $\wt\cZ_{\si}^0(\Pn,d)|_{\cZ_{\si}^0(\Pn,d)}$. This observation implies the last claim of Lemma \[map1bl\_lmm2\].\
We conclude this subsection by briefly describing the stratification mentioned above. A stratum $\M_{\Ga_B}$ of $\ov\M_{0,(\ale_B(\si),J_B(\si))}$ corresponds to a tuple $\Ga_B\!\equiv\!(\Ga_{B;l})_{l\in\aleph_B(\si)}$ of dual graphs, all of which are trees. The vertices of $\Ga_{B;l}$ correspond to the irreducible components of the domain of the stable map $b_l$ in the definition of $\ov\M_{0,(\ale_B(\si),J_B(\si))}$ at the beginning of Subsection \[map0prelim\_subs\]. Each vertex $v$ of $\Ga_{B;l}$ is labeled by a nonnegative integer, which specifies the degree of the stable map $b_l$ on the corresponding component $\Si_v$. There is an edge in $\Ga_{B;l}$ between two vertices if and only if the two corresponding components of the domain share a node. In addition, there are tails attached at some vertices of $\Ga_{B;l}$, which are labeled by the indexing set for marked points of the map $b_l$, i.e. $J_{l,P}$ in the notation of Subsection \[map0prelim\_subs\]. Let $v_l^*$ be the vertex of $\Ga_{B;l}$ to which the tail corresponding to the marked point $0$ is attached. If the degree of $v_l^*$ is positive, let $$\chi_l(\Ga_B)\equiv\chi_l(\Ga_{B;l})=\{v_l^*\}.$$ Otherwise, denote by $\chi_l(\Ga_B)$ the set of positive-degree vertices of $\Ga_{B;l}$ that are not separated from $v_l^*$ by a positive-degree vertex. Suppose $$b\!\equiv\!(b_l)_{l\in\ale_B(\si)} \in \M_{\Ga_B}\equiv
\ov\M_{0,(\ale_B(\si),J_B(\si))} \cap\prod_{l\in\ale_B(\si)}\!\!\!\!\!\M_{\Ga_{B;l}},
\qquad\hbox{with}\qquad b_l\!=\![\Si_l,u_l]$$ as in the paragraph preceding Lemma \[deriv0str\_lmm\]. If $l\!\in\!\ale_B(\si)$ and $v\!=\!v_l^*$, let $$\Im\,\cD_v|_b=\Im\,\cD_0|_{b_l}\equiv\Im\, du_l|_{x_0(b_l)}\subset T_{\ev_0(b)}\Pn.$$ If $v$ is a vertex of $\Ga_{B;l}$ different from $v_l^*$, we denote by $\Im\,\cD_v|_b$ the image of $d\{u_l|_{\Si_v}\}$ at the node of $\Si_v$ corresponding to the edge of $\Ga_{B;l}$ that leaves $v$ on the unique path from $v$ on $v_l^*$ in $\Ga_{B;l}$. Note that if $v\!\in\!\chi_l(\Ga_B)$, the image of this node under $u_l$ is $\ev_0(b)$. We set $$\chi(\Ga)=\bigsqcup_{l\in\ale_B(\si)}\!\!\!\!\chi_l(\Ga_B).$$ With $b$ as above, let $$\codim\,\cD|_b=\big|\chi(\Ga_B)\big|-
\dim\hbox{Span}\big\{\Im\,\cD_v|_b\!:v\!\in\!\chi_l(\Ga_B),\,l\!\in\!\ale_B(\si)\big\}.$$ For each pair $\al\!=\!(\Ga_B,\mu)$, where $\mu\!\in\!\Z^+$ is such that $$\label{alcond_e}
\max\big(1,|\chi(\Ga_B)|\!-\!n\big)\le \mu\le|\chi(\Ga_B)|,$$ we put $$\cZ_{\Ga_B;\al}=\big\{b\!\in\!\M_{\Ga_B}\!: \codim\,\cD|_b\!=\!\mu\big\}.$$ By the first statement of [@g1comp Theorem \[g1comp-str\_thm\]], $$\ov\cZ_{\si}^0=\bigsqcup_{\al}\cZ_{\si;\al},
\qquad\hbox{where}\qquad
\cZ_{\si;\al}=\ov\cM_{1,(I_P(\si),J_P(\si))}^0\!\times\!\cZ_{\Ga_B;\al}.$$ The disjoint union is taken over all pairs $\al\!=\!(\Ga,\mu)$ as described above. From transversality as in the first claim of Lemma \[deriv0str\_lmm\], it is easy to see that $$\label{codimest_e}\begin{split}
\codim_{\M_{\Ga_B}} \cZ_{\Ga_B;\al}
&=\big(n-(|\chi(\Ga_B)|\!-\!\mu)\big)\mu\\
&\ge n-(|\chi(\Ga_B)|\!-\!\mu);
\end{split}$$ see the end of [@g1cone Subsect. \[g1cone-g1str\_subs\]], for example. The above inequality follows from the first inequality in \_ref[alcond\_e]{}. By \_ref[codimest\_e]{}, if $m\!=\!|\ale_B(\si)|\!\le\!n$, $$\begin{split}
\codim_{\ov\cZ_{\si}^0} \cZ_{\si;\al}
&= \codim_{\M_{\Ga_B}} \cZ_{\Ga_B;\al} +
\codim_{\ov\M_{0,(\ale_B(\si),J_B(\si))}}\M_{\Ga_B}\\
&\qquad\qquad\qquad\qquad\qquad
- \codim_{\ov\cM_{1,(I_P(\si),J_P(\si))}^0\times\ov\M_{0,(\ale_B(\si),J_B(\si))}}
\ov\cZ_{\si}^0 \\
&\ge \big(n\!-\!|\chi(\Ga_B)|\!+\!\mu\big)+\big(|\chi(\Ga_B)|\!-\!|\ale_B(\si)|\big)
-\big(n\!-\!|\ale_B(\si)|\!+\!1\big)
=\mu\!-\!1.
\end{split}$$ On the other hand, by the last statement of [@g1comp Theorem \[g1comp-str\_thm\]], $$\max\big\{\!\dim\,\N\ov\cZ_{\si}^0|_b\!: b\!\in\!\cZ_{\si;\al}\big\}= \mu.$$ We conclude that $$\max\big\{\!\dim\,\N\ov\cZ_{\si}^0|_b\!: b\!\in\!\cZ_{\si;\al}\big\}
\le \codim_{\ov\cZ_{\si}^0} \cZ_{\si;\al} +1.$$ The equality holds if and only if $\mu\!=\!1$ and $\Ga_B$ is a tuple of one-vertex graphs, i.e. the image of $\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\M_{\Ga_B}$ under $\io_{0,\si}$ is contained in $\M_{1,\si}$. This observation concludes the proof of the stratification claim made in the previous paragraph.
Inductive Construction {#map1blconstr_subs}
----------------------
This subsection is the analogue of Subsection \[map0blconstr\_subs\] in the present situation. Suppose $\si\!\in\!\A_1(d,k)$ and we have constructed\
${}\quad$ ($I1$) an idealized blowup $\pi_{\si-1}\!:\ov\M_{1,k}^{\si-1}\!\lra\!\ov\M_{1,k}^0$ such that $\pi_{\si-1}$ is an isomorphism outside of the preimages of the subvarieties $\ov\M_{1,\si'}^0$ with $\si'\!\le\!\si\!-\!1$;\
${}\quad$ ($I2$) for each $\si'\!\in\!\{(0)\}\!\sqcup\!\A_1(d,k)$, a subvariety $\ov\M_{1,\si'}^{\si-1}$ of $\ov\M_{1,k}^{\si-1}$ such that $$\ov\M_{1,k}^{\si-1}=\ov\M_{1,(0)}^{\si-1}
\cup\bigcup_{\si'\in\A_1(d,k)} \!\!\!\!\!\!\! \ov\M_{1,\si'}^{\si-1}, \qquad
\pi_{\si-1}\big(\ov\M_{1,\si'}^{\si-1}\big)=\ov\M_{1,\si'}^0
~~~\forall\, \si'\!\in\!\{(0)\}\!\sqcup\!\A_1(d,k),$$ and $\ov\M_{1,\si^*}^{\si-1}$ is the proper transform of $\ov\M_{1,\si^*}^0$ for $\si^*\!=\!(0)$ and for all $\si^*\!\in\!\A_1(d,k)$ such that $\si^*\!>\!\si\!-\!1$.\
We assume that\
${}\quad$ ($I3$) for all $\si_1,\si_2\!\in\!\A_1(d,k)$ such that $\si_1\!\neq\!\si_2$, $\si_1\!\not\prec\si_2$, $\si_2\!\not\prec\si_1$, and $\si\!-\!1\!<\!\si_1,\si_2$, $$\ov\M_{1,\si_1}^{\si-1}\cap \ov\M_{1,\si_2}^{\si-1} ~
\begin{cases}
\subset\ov\M_{1,\ti\si(\si_1,\si_2)}^{\si-1}, &
\hbox{if}~ \ti\si(\si_1,\si_2)\!>\!\si\!-\!1;\\
=\eset,& \hbox{otherwise},\\
\end{cases}$$ where $\ti\si(\si_1,\si_2)$ is as in Lemma \[map1bl\_lmm1a\].\
We also assume that for every $\si'\!\in\!A_1(d,k)$ such that $\si'\!\le\!\si\!-\!1$:\
${}\quad$ ($I4$) $\ov\M_{1,\si'}^{\si-1}$ is the image of a $G_{\si'}$-invariant immersion $$\begin{gathered}
\io_{\si-1,\si'}\!: \wt\cM_{1,(I_P(\si'),J_P(\si'))}
\times \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}
\lra \ov\M_{1,k}^{\si-1},
\qquad\hbox{where}\\
\vr_{\si'}(\si\!-\!1)=
\begin{cases}
\max\big\{\vr\!\in\!\A_B(\si';\si^*)\!: \si'\!\prec\!\si^*\!\le\!\si\!-\!1\big\},
&\hbox{if}~\exists\si^*\!\in\!\A_1(d,k)
~\hbox{s.t.}~\si'\!\prec\!\si^*\!\le\!\si\!-\!1;\\
0,& \hbox{otherwise},
\end{cases}\end{gathered}$$ and $\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}
\!\equiv\!\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}(\Pn,d)$ is the blowup $\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^0(\Pn,d)$ constructed in Subsection \[map0blconstr\_subs\];\
${}\quad$ ($I5$) if $\si^*\!\in\!\A_1(d,k)$ is such that $\si\!-\!1\!<\!\si^*$ and $\si'\!\prec\!\si^*$, then $$\io_{\si-1,\si'}^{\,-1}\big(\ov\M_{1,\si^*}^{\si-1}\big)
= \wt\cM_{1,(I_P(\si'),J_P(\si'))} \times \Big(\bigcup_{\vr\in\A_B(\si';\si^*)}
\!\!\!\!\!\!\!\!\!\wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}\Big),$$ where $\wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}\!\equiv\!\wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}(\Pn,d)$ is the subvariety of $\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}$ described in Subsection \[map0blconstr\_subs\];\
${}\quad$ ($I6$) an idealized normal bundle for the immersion $\io_{\si-1,\si'}$ is given by $$\N_{\io_{\si-1,\si'}}^{\ide} = \pi_P^*\L \otimes
\pi_B^*\pi_{\vr_{\si'}(\si-1)}^{~*}\ga_{(\ale_B(\si'),J_B(\si'))},$$ where $$\pi_B,\, \pi_P\!:
\wt\cM_{1,(I_P(\si'),J_P(\si'))}
\times \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}
\lra \wt\cM_{1,(I_P(\si'),J_P(\si'))}, \,
\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}$$ are the two projection maps and $\L\!\lra\!\wt\cM_{1,(I_P(\si'),J_P(\si'))}$ is the universal tangent line bundle of Subsection \[curve1bl\_subs\];\
${}\quad$ ($I7$) $\ov\cZ_{\si'}^{\si-1}\!\equiv\!
\io_{\si-1,\si'}^{~-1}(\ov\M_{1,(0)}^{\si-1})$ is the closure of $$\cZ_{\si'}^{\si-1}\equiv \wt\cM_{1,(I_P(\si'),J_P(\si'))} \times
\Big(\wt\cD_{\vr_{\si'}(\si-1)}^{\,-1}(0)
-\bigcup_{\stackrel{\vr\in\A_B(\ale_B(\si');d,J_B(\si'))}{\vr_{\si'}(\si-1)<\vr}}
\!\!\!\!\!\! \wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}\Big)$$ in $\wt\cM_{1,(I_P(\si'),J_P(\si'))}
\!\times\! \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}$ and $$\N\ov\cZ_{\si'}^{\si-1}\equiv\N_{\io_{\si-1,\si'}|\ov\M_{1,(0)}^{\si-1}}
=\N_{\io_{\si-1,\si'}}^{\ide}\big|_{\ov\cZ_{\si'}^{\si-1}}$$ is the normal cone for $\io_{\si-1,\si'}|\ov\M_{1,(0)}^{\si-1}$ in $\ov\M_{1,k}^{\si-1}$;\
${}\quad$ ($I8$) the immersion map $$\bar\io_{\si-1,\si'}\!:\big( \wt\cM_{1,(I_P(\si');J_P(\si'))}
\!\times\! \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}\big)\big/G_{\si'}
\lra \ov\M_{1,k}^{\si-1}$$ induced by $\io_{\si-1,\si'}$ is an embedding.\
Furthermore, we assume that for every $\si^*\!\in\!A_1(d,k)$ such that $\si^*\!>\!\si\!-\!1$:\
${}\quad$ ($I9$) the domain of the $G_{\si^*}$-invariant immersion $\io_{\si-1,\si^*}$ induced by $\io_{0,\si^*}$ is $$\begin{gathered}
\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\times \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))},
\qquad\hbox{where}\\
\rho_{\si^*}(\si\!-\!1)=
\begin{cases}
\max\big\{\rho\!\in\!\A_P(\si^*;\si')\!:
\si'\!\le\!\si\!-\!1,\si'\!\prec\!\si^*\big\},
& \begin{split}
&\hbox{if}~\exists\si'\!\in\!\A_1(d,k)\\
&~\hbox{s.t.}~\si'\!\le\!\si\!-\!1,\si'\!\prec\!\si^*;
\end{split}\\
0,& \hbox{otherwise},
\end{cases}\end{gathered}$$ and $\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\!\lra\!\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}$ is the blowup constructed in Subsection \[curve1bl\_subs\];\
${}\quad$ ($I10$) if $\si'\!\in\!\A_1(d,k)$ is such that $\si\!-\!1\!<\!\si'\!\prec\!\si^*$, then $$\io_{\si-1,\si^*}^{~-1} \big( \wt\M_{1,\si'}^{\si-1} \big)
=\Big(\bigcup_{\rho\in\A_P(\si^*;\si')} \!\!\!\!\!\!\!
\wt\cM_{1,\rho}^{\rho_{\si^*}(\si-1)} \Big)
\times \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))};$$ ${}\quad$ ($I11$) if $\si^*$ is as in Lemma \[map1bl\_lmm1b1\], an idealized normal bundle for the immersion $\io_{\si-1,\si^*}$ is given by $$\N_{\io_{\si-1,\si^*}}^{\ide}=
\bigoplus_{i\in[m^*]} \pi_P^*L_{\rho_{\si^*}(\si-1),i} \!\otimes\!
\pi_B^*\pi_i^*L_0,$$ where $$\pi_P,\, \pi_B\!:
\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\!\times\! \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))} \lra
\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}, \,
\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}$$ are the two projection maps and $L_{\rho_{\si^*}(\si-1),i}\!\lra\!\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}$ is the line bundle constructed in Subsection \[curve1bl\_subs\];\
${}\quad$ ($I12$) $\ov\cZ_{\si^*}^{\si-1}\!\equiv\!
\io_{\si-1,\si^*}^{~-1}(\ov\M_{1,(0)}^{\si-1})$ is the closure of $\cZ_{\si^*}^0$ in $\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\!\times\!\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}$ and the normal cone $$\N\ov\cZ_{\si^*}^{\si-1}\equiv\N_{\io_{\si-1,\si^*}|\ov\M_{1,(0)}^{\si-1}}$$ for $\io_{\si-1,\si^*}|\ov\cZ_{\si^*}^{\si-1}$ is the closure of $\N\ov\cZ_{\si^*}^0\big|_{\cZ_{\si^*}^0}$ in $\N_{\io_{\si-1,\si^*}}^{\ide}$;\
${}\quad$ ($I13$) $\Im^s\,\bar\io_{\si-1,\si^*}\subset\bigcup_{\si-1<\si'\prec\si^*}
\wt\M_{0,\si'}^{\si-1}$, where $$\bar\io_{\si-1,\si^*}\!:\big(
\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\!\times\! \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}\big)\big/G_{\si^*}
\lra \ov\M_{1,k}^{\si-1},$$ is the immersion map induced by $\io_{\si-1,\si^*}$.\
Finally, we assume that\
${}\quad$ ($I14$) the collections $\{\io_{\si-1,\si'}\}_{\si'\in\A_1(d,k)}$ and $\{\bar\io_{\si-1,\si'}\}_{\si'\in\A_1(d,k)}$ of immersions are properly self-intersecting;\
${}\quad$ ($I15$) for all $\si'\!\in\!\A_1(d,k)$, the subvariety $\ov\M_{1,(0)}^{\si-1}$ of $\ov\M_{1,k}^{\si-1}$ is proper relative to the immersions $\io_{\si-1,\si'}$ and $\bar\io_{\si-1,\si'}$.\
By the inductive assumption ($I3$), if $\si_1$ and $\si_2$ are non-comparable elements of $(\A_1(d,k),\prec)$, the proper transforms of $\ov\M_{1,\si_1}^0$ and $\ov\M_{1,\si_2}^0$ become disjoint by the time either is ready to be blown up for any ordering $<$ extending the partial ordering $\prec$. Similarly to the three blowup constructions encountered previously, ($I3$) will imply that the end result of the present blowup construction is independent of the choice of an extension $<$. By ($I9$), our blowup construction modifies each immersion $\io_{0,\si^*}$ by changing the first factor of the domain according to the blowup construction of Subsection \[curve1bl\_subs\], until a proper transform of the image of $\io_{0,\si^*}$ is to be blown up; see below. By ($I11$) and ($I13$), in the process, the singular locus of $\io_{0,\si^*}$ disappears and the first component in every summand of $\N_{\io_{0,\si^*}}^{\ide}$ gets twisted to $\L$. In particular, all blowup loci are smooth. On the other hand, by the inductive assumptions ($I7$) and ($I8$), for $\si'\!\le\!\si\!-\!1$ the intersection of the proper transform of $\ov\M_{1,(0)}^0$ with the proper transform of the exceptional divisor $\ov\M_{1,\si'}^{\si'}$ is an embedding of a subvariety of a smooth variety. The singular locus of this subvariety is annihilated by the time the entire blowup construction is complete, according to the inductive assumptions ($I7$) above and the inductive assumption ($I4$) in Subsection \[map0blconstr\_subs\]. These assumptions imply that the proper transform of $\ov\M_{1,(0)}^0$ after the final blowup step is smooth.\
We note that all of the assumptions ($I1$)-($I15$) are satisfied if $\si\!-\!1$ is replaced by $0$. In particular, ($I4$) and ($I12$) are restatements of Lemmas \[map1bl\_lmm1a\] and \[map1bl\_lmm2\], respectively. The statements ($I10$), ($I11$), and ($I13$)-($I15$), with $\si\!-\!1$ replaced by $0$, are contained in Lemmas \[map1bl\_lmm1b1\] and \[map1bl\_lmm1b2\].\
If $\si\!\in\!\A_1(d,k)$ is as above, let $$\ti\pi_{\si}\!: \ov\M_{1,k}^{\si}\lra\ov\M_{1,k}^{\si-1}$$ be the idealized blowup of $\ov\M_{1,k}^{\si-1}$ along $\ov\M_{1,\si}^{\si-1}$, which is a smooth subvariety by the inductive assumption ($I13$). We denote the idealized exceptional divisor, $$\cE_{\ov\M_{1,\si}^{\si-1}}^{\ide}=\P\N_{\bar\io_{\si-1,\si}}^{\ide},$$ by $\ov\M_{1,\si}^{\si}$. For each $\si'\!\in\!\{(0)\}\!\sqcup\!(\A_1(d,k)\!-\!\{\si\})$, we denote by $$\ov\M_{1,\si'}^{\si} \subset \Bl_{\ov\M_{1,\si}^{\si-1}}\ov\M_{1,k}^{\si-1}
\subset \ov\M_{1,k}^{\si}$$ the proper transform of $\ov\M_{1,\si'}^{\si-1}$. Let $\pi_{\si}\!=\!\pi_{\si-1}\!\circ\!\ti\pi_{\si}$.\
The inductive assumptions ($I1$) and ($I2$), with $\si\!-\!1$ replaced by $\si$, are clearly satisfied, while ($I3$), ($I8$) for $\si'\!\neq\!\si$, and ($I13$)-($I15$) follow from (2) of Lemma \[ag\_lmm2a\], Corollary \[immercoll\_crl\], and Lemma \[virimmer\_lmm3\]. On the other hand, by ($I9$), the domain of the immersion $\io_{\si-1,\si}$ is $$\ov\cM_{1,(I_P(\si),J_P(\si))}^{\rho_{\si}(\si-1)}
\!\times\! \ov\M_{0,(\ale_B(\si),J_B(\si))}
=\wt\cM_{1,(I_P(\si),J_P(\si))}
\!\times\! \ov\M_{0,(\ale_B(\si),J_B(\si))}.$$ By ($I11$), the chosen idealized normal bundle for the immersion $\io_{\si-1,\si}$ is given by $$\label{sinormal_e}
\N_{\io_{\si-1,\si}}^{\ide}=
\bigoplus_{i\in[m]} \pi_P^*L_{\rho_{\si}(\si-1),i} \!\otimes\!
\pi_B^*\pi_i^*L_0
=\pi_P^*\L\otimes \pi_B^*F_{(\ale_B(\si),J_B(\si))}.$$ Thus, the domain of the immersion $\io_{\si,\si}$ induced by $\io_{\si-1,\si}$ is $$\P\N_{\io_{\si-1,\si}}^{\ide} = \wt\cM_{1,(I_P(\si),J_P(\si))}
\!\times\! \wt\M_{0,(\ale_B(\si),J_B(\si))}^0
=\wt\cM_{1,(I_P(\si),J_P(\si))}
\!\times\! \wt\M_{0,(\ale_B(\si),J_B(\si))}^{\vr_{\si}(\si)}.$$ By the first statement of Lemma \[virimmer\_lmm2\], an idealized normal bundle for the embedding $\io_{\si,\si}$ is the tautological line bundle over $\P\N_{\io_{\si-1,\si}}^{\ide}$, i.e. $$\N_{\io_{\si,\si}}^{\ide}
= \pi_P^*\L \!\otimes\! \pi_B^*\ga_{(\ale_B(\si),J_B(\si))}
= \pi_P^*\L \!\otimes\! \pi_B^*\pi_{\vr_{\si}(\si)}^*\ga_{(\ale_B(\si),J_B(\si))}.$$ Thus, the inductive assumptions ($I4$) and ($I6$), with $\si'\!=\!\si$ and $\si\!-\!1$ replaced by $\si$, are satisfied. The same is the case with ($I8$), since the map $\bar\io_{\si-1,\si}$ is an embedding by ($I13$).\
We also note that by the first statement of Lemma \[map1bl\_lmm1b3\], the inductive assumptions ($I1$) and ($I2$), and the last statement of Lemma \[ag\_lmm2a\], $$\io_{\si-1,\si}^{~-1} \big( \ov\M_{1,\si^*}^{\si-1}\big)
=\wt\cM_{1,(I_P(\si),J_P(\si))}
\times \Big(\bigcup_{\vr\in\A_B(\si;\si^*)} \!\!\!\!\!\!\!\! \ov\M_{0,\vr} \Big)$$ for all $\si^*\!\in\!\A_1(d,k)$ such that $\si\!\prec\!\si^*$. In addition, by the last statement of Lemma \[map1bl\_lmm1b3\] $$\io_{\si-1,\si} \big|_{\wt\cM_{1,(I_P(\si),J_P(\si))}\times
\ov\M_{0,\vr}}^* T\ov\M_{1,\si^*}^{\si-1}
\big/T\big(\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\ov\M_{0,\vr}\big)
\subset\N_{\io_{\si-1,\si}}^{\ide}$$ is a vector bundle for all $\vr\!\in\!\A_B(\si;\si^*)$ and $$\io_{\si-1,\si} \big|_{\cM_{1,(I_P(\si),J_P(\si))}\times
\M_{0,\vr}}^* T\ov\M_{1,\si^*}^{\si-1}
\big/T\big(\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\M_{0,\vr}\big)
=\bigoplus_{i\in\aleph_P(\vr)}\!\!\!\!\pi_P^*L_i\!\otimes\!\pi_B^*\pi_i^*L_0.$$ Thus, by the first equality in \_ref[sinormal\_e]{}, $$\begin{split}
\io_{\si-1,\si} \big|_{\wt\cM_{1,(I_P(\si),J_P(\si))}\times
\ov\M_{0,\vr}}^* T\ov\M_{1,\si^*}^{\si-1}
\big/T\big(\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\ov\M_{0,\vr}\big)
&=\pi_P^*\L\!\otimes\bigoplus_{i\in\aleph_P(\vr)}\!\!\!\!\pi_B^*\pi_i^*L_0\\
&=\pi_P^*\L \otimes\pi_B^*F_{\vr;P}.
\end{split}$$ It follows that $$\begin{split}
\io_{\si,\si}^{~-1} \big( \ov\M_{1,\si^*}^{\si}\big)
&=\bigcup_{\vr\in\A_B(\si;\si^*)} \!\!\!\!\!\!\!\!
\P\big(\pi_P^*\L \otimes\pi_B^*F_{\vr;P}\big)\big|_{
\wt\cM_{1,(I_P(\si),J_P(\si))}\times\ov\M_{0,\vr}} \\
&=\wt\cM_{1,(I_P(\si),J_P(\si))}\times
\Big(\bigcup_{\vr\in\A_B(\si;\si^*)} \!\!\!\!\!\!\!\!\P F_{\vr;P}\Big)\\
&\equiv\wt\cM_{1,(I_P(\si),J_P(\si))}\times
\Big(\bigcup_{\vr\in\A_B(\si;\si^*)}\!\!\!\!\!\!\!\!\wt\M_{0,\vr}^0\Big)
=\wt\cM_{1,(I_P(\si),J_P(\si))}\times
\Big(\bigcup_{\vr\in\A_B(\si;\si^*)}\!\!\!\!\!\!\!\!\wt\M_{0,\vr}^{\vr_{\si}(\si)}\Big),
\end{split}$$ as needed for the inductive assumption ($I5$) with $\si\!-\!1$ replaced by $\si$ and $\si'\!=\!\si$.\
Furthermore, by ($I12$), $\N\ov\cZ_{\si}^{\si-1}$ is the closure of $$\begin{split}
\N\ov\cZ_{\si}^0\big|_{\cZ_{\si}^0}
&\equiv \ker\cD_{0,\si}\big|_{
\cM_{1,(I_P(\si),J_P(\si))}^0 \times\M_{0,(\ale_B(\si),J_B(\si))}}\\
&= \pi_P^*\L \otimes
\pi_B^*\ker\cD_{(\ale_B(\si),J_B(\si))}\big|_{\M_{0,(\ale_B(\si),J_B(\si))}}
\end{split}$$ in $\pi_P^*\L\!\otimes\!\pi_B^*F_{(\ale_B(\si),J_B(\si))}$, where $\cD_{(\ale_B(\si),J_B(\si))}$ is the bundle homomorphism described in Subsection \[map0prelim\_subs\]. Thus, by the first statement of Lemma \[virimmer\_lmm3\], $$\ov\cZ_{\si}^{\si}\equiv\io_{\si,\si}^{~-1}(\ov\M_{1,(0)}^{\si})$$ is the closure of $$\cM_{1,(I_P(\si),J_P(\si))}\times
\big\{b\!\in\!\P F_{(\ale_B(\si),J_B(\si))}|_{\M_{0,(\ale_B(\si),J_B(\si))}}\!:
\wt\cD_0b\!=\!0\big\}$$ in $\wt\cM_{1,(I_P(\si),J_P(\si))}
\!\times\! \wt\M_{0,(\ale_B(\si),J_B(\si))}^{\vr_{\si}(\si)}$. The inductive assumption ($I7$), with $\si'\!=\!\si$ and $\si\!-\!1$ replaced by $\si$, now follows from the first statement of Lemma \[map0bl\_lmm2\].\
We next verify that the inductive assumptions ($I4$)-($I7$) hold for $\si'\!<\!\si$, with $\si\!-\!1$ replaced by $\si$. If $\si'\!\not\prec\!\si$, then $$\vr_{\si'}(\si)=\vr_{\si'}(\si\!-\!1)
\qquad\hbox{and}\qquad
\ov\M_{1,\si'}^{\si-1}\cap\ov\M_{1,\si}^{\si-1}=\eset,$$ by definition and ($I3$), respectively. It then follows that $$\begin{gathered}
\io_{\si,\si'}=\io_{\si-1,\si'}, \qquad
\N_{\io_{\si,\si'}}^{\ide}=\N_{\io_{\si-1,\si'}}^{\ide},\\
\hbox{and}\qquad \ov\M_{1,\si'}^{\si}\cap\ov\M_{1,\si^*}^{\si}
=\ov\M_{1,\si'}^{\si-1}\cap\ov\M_{1,\si^*}^{\si-1}
\quad\forall\, \si^*\!\in\!\{(0)\}\!\sqcup\!\A_1(d,k).\end{gathered}$$ Thus, the inductive assumptions ($I4$)-($I7$), as stated above, imply the corresponding statements with $\si\!-\!1$ replaced by $\si$.\
Suppose that $\si'\!\prec\!\si$. By ($I4$) and (1) of Lemma \[ag\_lmm2a\], the domain of the immersion $\io_{\si,\si'}$ induced by $\io_{\si-1,\si'}$ is the blowup of $$\wt\cM_{1,(I_P(\si'),J_P(\si'))}
\times \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}$$ along the preimage of $\ov\M_{1,\si}^{\si-1}$ under $\io_{\si-1,\si'}$ in $$\id\!\times\!\pi_{\vr_{\si'}(\si-1)}\!:
\wt\cM_{1,(I_P(\si'),J_P(\si'))}
\!\times\! \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}
\lra
\wt\cM_{1,(I_P(\si'),J_P(\si'))}
\!\times\! \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^0.$$ By ($I5$), this preimage is $$\wt\cM_{1,(I_P(\si'),J_P(\si'))} \times
\Big(\bigcup_{\vr\in\A_B(\si';\si)}\!\!\!\!\!\!\!\!\wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}\Big).$$ By the inductive assumption ($I5$) in Subsection \[map0blconstr\_subs\] and the second paragraph after Lemma \[map1bl\_lmm1b3\], $$\wt\M_{0,\vr_1}^{\vr_{\si'}(\si-1)} \cap \wt\M_{0,\vr_2}^{\vr_{\si'}(\si-1)}
=\eset
\qquad\forall\, \vr_1,\vr_2\!\in\!\A_B(\si';\si),\, \vr_1\!\neq\!\vr_2.$$ Thus, by the construction of Subsection \[map0blconstr\_subs\], the blowup of $\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si-1)}$ along $$\bigcup_{\vr\in\A_B(\si';\si)}\!\!\!\!\!\!\!\!\wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}$$ is $\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si)}$, as needed for the inductive statement ($I4$), with $\si\!-\!1$ replaced by $\si$. The inductive requirement ($I5$) is obtained by the same reasoning, using the last statement of Lemma \[ag\_lmm2a\].\
Since $\ov\M_{1,\si}^{\si-1}$ is not contained in $\ov\M_{1,\si'}^{\si-1}$, the bundle homomorphism $$\io_{\si-1,\si'}^{\,*}T\ov\M_{1,\si}^{\si-1}\lra \N_{\io_{\si-1,\si'}}^{\ide}$$ must be surjective on every fiber over $\io_{\si-1,\si'}^{-1}(\ov\M_{1,\si}^{\si-1})$ by ($I14$). Thus, the inductive assumption ($I6$), for $\si'\!<\!\si$, continues to hold. Furthermore, by ($I7$) and the last statement of Lemma \[ag\_lmm2a\], $\ov\cZ_{\si'}^{\si}$ is the closure of $$\begin{split}
&\wt\cM_{1,(I_P(\si'),J_P(\si'))} \times
\Big(\wt\cD_{\vr_{\si'}(\si-1)}^{~-1}(0)
-\bigcup_{\stackrel{\vr\in\A_B(\ale_B(\si');d,J_B(\si'))}{\vr_{\si'}(\si-1)<\vr}}
\!\!\!\!\!\! \wt\M_{0,\vr}^{\vr_{\si'}(\si-1)}\Big)\\
&\qquad\qquad\qquad
= \wt\cM_{1,(I_P(\si'),J_P(\si'))} \times
\Big(\wt\cD_{\vr_{\si'}(\si-1)}^{~-1}(0)
-\bigcup_{\vr\in\A_B(\si';\si)}\!\!\!\!\!\! \wt\M_{0,\vr}^{\vr}
-\bigcup_{\stackrel{\vr\in\A_B(\ale_B(\si');d,J_B(\si'))}{\vr_{\si'}(\si)<\vr}}
\!\!\!\!\!\! \wt\M_{0,\vr}^{\vr_{\si'}(\si)}\Big)
\end{split}$$ in $\wt\cM_{1,(I_P(\si'),J_P(\si'))}
\!\times\! \wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si)}$. By the construction of Subsection \[map0blconstr\_subs\], $$\wt\cD_{\vr_{\si'}(\si-1)}
\big|_{\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si)}
-\bigcup_{\vr\in\A_B(\si';\si)}\! \wt\M_{0,\vr}^{\vr}}
=\wt\cD_{\vr_{\si'}(\si)}
\big|_{\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si)}
-\bigcup_{\vr\in\A_B(\si';\si)}\! \wt\M_{0,\vr}^{\vr}}.$$ Since $\wt\cD_{\vr_{\si'}(\si)}$ is transverse to the zero set outside of the subvarieties $\wt\M_{0,\vr}^{\vr_{\si'}(\si)}$ with $\vr\!>\!\vr_{\si'}(\si)$ by the inductive requirement ($I4$) in Subsection \[map0blconstr\_subs\], we conclude that the first part of the inductive assumption ($I7$), with $\si\!-\!1$ replaced by $\si$, is satisfied. The second part follows from the last statement of Lemma \[virimmer\_lmm3\].\
It remains to verify the inductive assumption ($I9$)-($I12$), with $\si\!-\!1$ replaced by $\si$. Suppose $\si^*\!\in\!\A_1(d,k)$ is such that $\si\!<\!\si^*$. If $\si\!\not\prec\!\si^*$, then $$\rho_{\si^*}(\si)=\rho_{\si^*}(\si\!-\!1)
\qquad\hbox{and}\qquad
\ov\M_{1,\si^*}^{\si-1}\cap\ov\M_{1,\si}^{\si-1}=\eset,$$ by definition and ($I3$), respectively. It then follows that $$\begin{gathered}
\io_{\si,\si^*}=\io_{\si-1,\si^*}, \qquad
\N_{\io_{\si,\si^*}}^{\ide}=\N_{\io_{\si-1,\si^*}}^{\ide},\\
\hbox{and}\qquad \ov\M_{1,\si^*}^{\si}\cap\ov\M_{1,\si'}^{\si}
=\ov\M_{1,\si^*}^{\si-1}\cap\ov\M_{1,\si'}^{\si-1}~~~\forall
\si'\!\in\!\{(0)\}\!\sqcup\!\A_1(d,k).\end{gathered}$$ Thus, the inductive assumptions ($I9$)-($I12$), as stated above, imply the corresponding statements with $\si\!-\!1$ replaced by $\si$.\
Suppose that $\si\!\prec\!\si^*$. By ($I9$) and (1) of Lemma \[ag\_lmm2a\], the domain of the immersion $\io_{\si,\si^*}$ induced by $\io_{\si-1,\si^*}$ is the blowup of $$\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\times \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}$$ along the preimage of $\ov\M_{1,\si}^{\si-1}$ under $\io_{\si-1,\si^*}$ in $$\pi_{\rho_{\si^*}(\si-1)}\!\times\!\id\!:
\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}
\!\times\! \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}
\lra \ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^0
\!\times\! \ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}$$ By ($I10$), this preimage is $$\Big(\bigcup_{\rho\in\A_P(\si^*;\si)}\!\!\!\!\!\!
\ov\cM_{1,\rho}^{\rho_{\si^*}(\si-1)}\Big) \times\ov\M_{0,(\ale_B(\si^*),J_B(\si^*))}.$$ By Lemma \[curve1bl\_lmm\] and the paragraph after Lemma \[map1bl\_lmm1a\], $$\ov\cM_{1,\rho_1}^{\rho_{\si^*}(\si-1)}\cap\ov\cM_{1,\rho_2}^{\rho_{\si^*}(\si-1)}
=\eset
\qquad\forall\, \rho_1,\rho_2\!\in\!\A_P(\si^*;\si),\,
\rho_1\!\neq\!\rho_2.$$ Thus, by the construction of Subsection \[curve1bl\_subs\], the blowup of $\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}$ along $$\bigcup_{\rho\in\A_P(\si^*;\si)}\!\!\!\!\!\!
\ov\cM_{1,\rho}^{\rho_{\si^*}(\si-1)}$$ is $\ov\cM_{1,(I_P(\si^*),J_P(\si^*))}^{\rho_{\si^*}(\si-1)}$, as needed for the inductive statement ($I9$), with $\si\!-\!1$ replaced by $\si$. The inductive assumptions ($I10$) and ($I11$) are verified similarly, using the last statement of Lemma \[ag\_lmm2a\] and Lemma \[virimmer\_lmm2\]. The argument for ($I11$) is nearly identical to the verification of the inductive assumption ($I11$) in Subsection \[map0blconstr\_subs\]. Finally, the inductive requirement ($I12$), with $\si\!-\!1$ replaced by $\si$, follows from the last statement of Lemma \[virimmer\_lmm3\], along with the assumptions ($I1$) and ($I2$).\
We conclude this blowup construction after the $\si_{\max}$ step and put $$\wt\M_{1,k}^0(\Pn,d)=\ov\M_{1,(0)}^{\si_{\max}}, \qquad
\ti\pi=\pi_{\si_{\max}}\big|_{\ov\M_{1,(0)}^{\si_{\max}}}, \quad\hbox{and}\quad
\wt\cZ_{\si}(\Pn,d)=\ov\cZ_{\si}^{\si_{\max}}.$$ The inductive assumptions ($I1$)-($I8$) imply that $$\ti\pi\!: \wt\M_{1,k}^0(\Pn,d) \lra \ov\M_{1,k}^0(\Pn,d)$$ is a desingularization as described in Subsection \[descr\_subs\]. By ($I3$), the final result of this blowup construction is independent of the choice of full ordering $<$ extending the natural partial ordering $\prec$ on $\A_1(d,k)$.
Proof of Theorem \[cone\_thm\] {#cone_sec}
==============================
Pushforwards of Vector Bundles {#pushfor_subs}
------------------------------
In this section we prove Theorem \[cone\_thm\] by lifting the construction of Section \[map1bl\_sec\] from stable maps into $\Pn$ to stable maps into (the total space of) the line bundle $\cL$.\
Let $\tau\!:\cL\!\lra\!\Pn$ be the bundle projection map. We denote by $\ov\M_{1,k}(\cL,d)$ the moduli space of degree-$d$ stable maps from genus-one curves with $k$ marked points into $\cL$. The projection map $\tau$ induces a morphism, $$p\!: \ov\M_{1,k}(\cL,d)\lra\ov\M_{1,k}(\Pn,d),
\qquad [\Si,u]\lra[\Si,\tau\circ u].$$ Since no fiber of $\cL$ contains the image of a non-constant holomorphic map, the ghost components of $(\Si,\tau\circ u)$ are precisely the same as the ghost components of $(\Si,u)$. We note that $$p^{-1}([\Si,u])=H^0(\Si;u^*\cL)\big/\Aut(\Si,u).$$ In particular, $p$ is a bundle of vector spaces, but of two possible ranks: $da$ and $da\!+\!1$. Let $\S_{\cL}$ denote the sheaf of (holomorphic) sections of $$p|_{\ov\M_{1,k}^0(\Pn,d)}\!:
\ov\M_{1,k}(\cL,d)\big|_{\ov\M_{1,k}^0(\Pn,d)}\lra\ov\M_{1,k}^0(\Pn,d).$$ Similarly, denote by $\ti{S}_{\cL}$ the sheaf of sections of $$p\!: \ti\pi^*\ov\M_{1,k}(\cL,d)\lra\wt\M_{1,k}^0(\Pn,d),$$ where $\ti\pi\!:\wt\M_{1,k}^0(\Pn,d)\!\lra\!\ov\M_{1,k}^0(\Pn,d)$ is the desingularization map of Theorem \[main\_thm\]:\
(-3.5,-2.2)(10,.5) (15,0)[$\ov\M_{1,k}(\cL,d)$]{}(15,-4)[$\ov\M_{1,k}(\Pn,d)$]{} (15,-1)(15,-3)(15.6,-2) (6,-1)(6,-3)(6.6,-2) (6,0)[$\wt\pi^*\ov\M_{1,k}(\cL,d)$]{} (6,-4)[$\wt\M_{1,k}^0(\Pn,d)$]{} (8.5,-4.1)(12.2,-4.1)(10.2,-3.5) (8.9,-.1)(12.4,-.1)(10.4,.5)
\[conesheaf\_lmm\] With notation as in Theorem \[cone\_thm\] and above,\
${}\quad$ (1) the sheaves $\S_{\cL}$ and $\pi_*\ev_*\O_{\Pn}(a)$ over $\ov\M_{1,k}^0(\Pn,d)$ are isomorphic;\
${}\quad$ (2) the sheaves $\ti{\S}_{\cL}$ and $\pi_*\ti\pi^*\ev_*\O_{\Pn}(a)$ over $\wt\M_{1,k}^0(\Pn,d)$ are isomorphic.
Let $\U_{\cL}$ be the universal curve over $\ov\M_{1,k}(\cL,d)|_{\ov\M_{1,k}^0(\Pn,d)}$, with structure map $\pi_{\cL}$ and evaluation map $\ev_{\cL}$. The projection map $\tau$ induces a morphism $\ti{p}_{\cL}$ on $\U_{\cL}$ so that the diagram\
(-1,-2.2)(10,1.2) (10.2,0)[$\U_{\cL}$]{}(20,0)[$\U$]{} (11,0)(19.2,0)(15,.6) (14,2)[$\cL$]{}(10.5,0.5)(13.5,1.8)(11.5,1.5) (24,2)[$\Pn$]{}(20.3,0.5)(23.3,1.8)(21,1.2) (14.8,2)(23.2,2)(19,2.5) (10,-4)[$\ov\M_{1,k}(\cL,d)|_{\ov\M_{1,k}^0(\Pn,d)}$]{}(20,-4)[$\ov\M_{1,k}^0(\Pn,d)$]{} (14.5,-4)(17.3,-4)(16,-3.5) (10,-1)(10,-3)(9.3,-2) (20,-1)(20,-3)(20.5,-2)
commutes. Suppose $W\!\subset\!\ov\M_{1,k}^0(\Pn,d)$ is an open subset.\
(i) An element $$s\in \big\{\pi_*\ev_*\O_{\Pn}(a)\big\}(W)\equiv H^0\big(\pi^{-1}(W);\ev^*\cL\big)$$ induces a morphism $\ti{s}\!:\pi^{-1}(W)\!\lra\!\cL$ so that $\ev\!=\!\tau\!\circ\ti{s}$. In turn, $\ti{s}$ induces morphisms $f_s$ and $\ti{f}_s$ to $\ov\M_{1,k}(\cL,d)|_{\ov\M_{1,k}^0(\Pn,d)}$ and $\U_{\cL}$,\
(-1,-2.2)(10,.3) (10.2,0)[$\pi^{-1}(W)$]{}(20,0)[$\U_{\cL}$]{} (12.2,0)(19.2,0)(14.8,.6) (10,-4)[$W$]{}(20,-4.2)[$\ov\M_{1,k}(\cL,d)|_{\ov\M_{1,k}^0(\Pn,d)}$]{} (11,-4)(15.3,-4)(14,-3.5) (10,-1)(10,-3)(9.4,-2) (20,-1)(20,-3)(20.7,-2) (25,0)[$\cL$]{} (20.8,0)(24.5,0)(22.6,.6)
so that $\ti{s}\!=\!\ev_{\cL}\!\circ\ti{f}_s$. Then, $$\ev\circ\ti{p}\circ\ti{f}_s=\tau\circ\ev_{\cL}\!\circ\ti{f}_s
=\tau\circ\ti{s}=\ev\!: \pi^{-1}(W)\lra\Pn
\qquad\Lra\qquad p\circ f_{s}=\id_W,$$ since $\pi\circ\ti{p}\circ\ti{f}_s=p\circ f_{s}\!\circ\pi$. Thus, $f_s\!\in\!\S_{\cL}(W)$. It is immediate that the map $$\big\{\pi_*\ev_*\O_{\Pn}(a)\big\}(W)\lra\S_{\cL}(W),
\qquad s\lra f_s,$$ induces a sheaf homomorphism.\
(ii) Conversely, let $\si\!\in\!\S_{\cL}(W)$, i.e. $\si\!:W\!\lra\!\ov\M_{1,k}(\cL,d)$ is a morphism such that $p\circ\si=\id_W$. Since $\U_{\cL}\!=\!p^*\U$, $$\pi^{-1}(W)\equiv\U|_W=\si^*\U_{\cL}.$$ Thus, $\si$ lifts to a morphism $$\ti\si\!: \pi^{-1}(W)=\si^*\U_{\cL}\lra \U_{\cL}.$$ Let $g_{\si}\!=\!\ev_{\cL}\!\circ\ti\si$. Then, $$\tau\circ g_{\si}=\tau\circ\ev_{\cL}\!\circ\ti\si
=\ev\circ\ti{p}\circ\ti\si=\ev,$$ i.e. $g_{\si}\!\in\!H^0(\pi^{-1}(W);\ev^*\cL)$. It is immediate that the map $$\S_{\cL}(W)\lra\big\{\pi_*\ev_*\O_{\Pn}(a)\big\}(W),
\qquad \si\lra g_{\si},$$ induces a sheaf homomorphism. Furthermore, $$g_{f_s}=s \quad\forall\,s\in\big\{\pi_*\ev_*\O_{\Pn}(a)\big\}(W)
\qquad\hbox{and}\qquad
f_{g_{\si}}=\si \quad\forall\,\si\in\S_{\cL}(W).$$ These observations imply the first statement of Lemma \[conesheaf\_lmm\]. The second claim is proved similarly.\
Let $$\ov\M_{1,k}^0(\cL,d)\subset \ov\M_{1,k}(\cL,d)$$ be the closure of the locus of maps from smooth domains. We show in Subsection \[conebl\_subs\] that the proper transform $\wt\M_{1,k}^0(\cL,d)$ of $\ov\M_{1,k}^0(\cL,d)$ in $$\ti\pi^*\ov\M_{1,k}(\cL,d)\lra \wt\M_{1,k}^0(\Pn,d)$$ is smooth. Similarly to the case of $\wt\M_{1,k}^0(\Pn,d)$, the main stratum of $\wt\M_{1,k}^0(\cL,d)$, $$\M_{1,k}^{\eff}(\cL,d)\equiv
\ov\M_{1,k}(\cL,d)\big|_{\M_{1,k}^{\eff}(\Pn,d)}
=\wt\M_{1,k}^0(\cL,d)-
\bigcup_{\si\in\A_1(d,k)}\!\!\!\!\!\!p^{-1}\big(\Im\,\io_{\si_{\max},\si}\big),$$ is smooth. On the other hand, by the inductive assumption ($I1$) and the last paragraph of Subsection \[conebl\_subs\], for each $\si\!\in\!\A_1(d,k)$ $$\wt\M_{1,k}^0(\cL,d)\cap p^{-1}\big(\Im\,\io_{\si_{\max},\si}\big)$$ is the image of a smooth variety under the bundle homomorphism $\bar{j}_{\si_{\max},\si}$ lifting the embedding $\bar\io_{\si_{\max},\si}$ of Subsection \[map1blconstr\_subs\]. Thus, $$\wt\M_{1,k}^0(\cL,d)\cap p^{-1}\big(\Im\,\io_{\si_{\max},\si}\big)$$ is a smooth subvariety of $\wt\M_{1,k}^0(\cL,d)$. As its normal cone in $\wt\M_{1,k}^0(\cL,d)$ is a line bundle by the inductive assumption ($I1$) of Subsection \[conebl\_subs\] for every $\si\!\in\!\A_1(d,k)$, we conclude that the entire space $\wt\M_{1,k}^0(\cL,d)$ is smooth. Furthermore, the fibers of $$\ti{p}\!: \wt\M_{1,k}^0(\cL,d)\lra \wt\M_{1,k}^0(\Pn,d)$$ are vector spaces of the same rank and $\wt\M_{1,k}^0(\cL,d)$ contains $\wt\M_{1,k}^0(\Pn,d)$ as the zero section. Thus, $\ti{p}$ is a vector bundle.\
Lemma \[conesheaf\_lmm\] and the previous paragraph imply (1) of Theorem \[cone\_thm\]. The second claim of this theorem is obtained in the last paragraph of Subsection \[conebl\_subs\]. Finally, (3) of Theorem \[cone\_thm\] follows from (4) of Theorem \[main\_thm\] and the following lemma.
\[pushfor\_lmm3\] Suppose $\ti\pi\!:\wt\M\!\lra\!\ov\M$ is a morphism between varieties, $\U\!\lra\!\ov\M$ is a flat family of curves, $\cL\!\lra\!\U$ is a line bundle, and $\pi\!:\ti\U\!\lra\!\wt\M$ and $\wt\cL\!\lra\!\wt\U$ are the pullbacks of $\U$ and $\cL$ via $\ti\pi$:\
(-3,-2.2)(10,1) (12.3,0)[$\U$]{} (15.7,2)[$\cL$]{}(12.8,0.5)(15.2,1.8)(12.3,-4)[$\ov\M$]{} (12.3,-1)(12.3,-3)(12.9,-2) (6.8,-1)(6.8,-3)(7.4,-2) (7.8,0)[$\wt\U\!=\!\wt\pi^*\U$]{} (11.2,2)[$\wt\cL\!=\!\ti\pi^*\cL$]{}(7,0.5)(9.4,1.8) (6.8,-4)[$\wt\M$]{} (7.8,-4.1)(11.3,-4.1)(9.6,-3.5) (9.8,-.1)(11.6,-.1)(10.7,.5)
If the morphism $\ti\pi$ is surjective and its fibers are compact and connected, then $$\ti\pi_*\pi_*\wt\cL=\pi_*\cL.$$\
Since $\cL$ is locally trivial, Lemma \[pushfor\_lmm3\] follows from $$\ti\pi_*\O_{\wt\M}=\O_{\ov\M}.$$ In turn, this identity follows from the fact that every holomorphic function on a compact connected variety is constant. Thus, if $W\!\subset\!\ov\M$ is any open subset and $\ti{f}$ is a holomorphic function on $\ti\pi^{-1}(W)\!\subset\!\wt\M$, then $\ti{f}$ is constant on the fibers of $\ti\pi$, i.e. $\ti{f}\!=\!\ti\pi^*f$ for some holomorphic function $f$ on $W$.
Construction of Bundle Homomorphism {#conehomomor_subs}
-----------------------------------
In this subsection we describe the surjective bundle homomorphism that appears in the second statement of Theorem \[cone\_thm\]; see Proposition \[conehomomor\_prp\]. The construction of this homomorphism is similar to the construction of the homomorphism $\wt\cD_{(\ale,J)}$ in Subsections \[map0prelim\_subs\] and \[map0blconstr\_subs\].\
Let $\cL\!\lra\!\Pn$ be a line bundle as in Subsection \[descr\_subs\]. If $J$ is a finite set, let $$\V_0=\ov\M_{0,\{0\}\sqcup J}(\cL,d) \lra \ov\M_{0,\{0\}\sqcup J}(\Pn,d)$$ be the corresponding cone. In particular, if $[\Si,u]\!\in\!\ov\M_{0,\{0\}\sqcup J}(\Pn,d)$, then $$\V_0\big|_{[\Si,u]}=H^0(\Si;u^*\cL)\big/\Aut(\Si,u).$$ In this, genus-[*zero*]{}, case, this is a vector bundle of the expected rank. Let $$\na^u\!: \Ga(\Si;u^*\cL)\lra\Ga(\Si;T^*\Si\!\otimes\!u^*\cL)$$ be the pullback of the standard Hermitian connection in $\cL$ by $u$. We define $$\begin{gathered}
\begin{split}
\D_0&\in\Ga\big(\ov\M_{0,\{0\}\sqcup J}(\Pn,d);
\Hom(L_0\!\otimes\!\V_0,\ev_0^*\cL)\big)\\
&\quad=\Ga\big(\ov\M_{0,\{0\}\sqcup J}(\Pn,d);
\Hom(L_0,\Hom(\V_0,\ev_0^*\cL))\big)\\
&\quad=\Ga\big(\ov\M_{0,\{0\}\sqcup J}(\Pn,d);
\Hom(\V_0,\Hom(L_0,\ev_0^*\cL))\big)
\end{split}\\
\hbox{by}\qquad \D_0\xi = \na^u\xi|_{x_0(\Si,u)} \quad\forall\,\xi\!\in\!H^0(\Si;u^*\cL),\end{gathered}$$ where $x_0(\Si,u)\!\in\!\Si$ is the marked point labeled by $0$ as before. We note that $\D_0$ vanishes identically on the subvarieties $\ov\M_{0,\si}(\Pn,d)$ with $\si\!\in\!\A_0(d,J)$ defined in Subsection \[map0str\_subs\].\
If $\aleph$ and $J$ are finite sets, let $$p_{(\aleph,J)}\!:\V_{(\aleph,J)} \lra \ov\M_{0,(\aleph,J)}(\Pn,d)$$ be the vector bundle induced by $\cL$, where $\ov\M_{0,(\aleph,J)}(\Pn,d)$ is as in Subsection \[map0prelim\_subs\]. It is immediate that $$\V_{(\ale,J)} = \big\{
(\xi_i)_{i\in\ale}\!\in\!\bigoplus_{i\in\ale}\pi_i^*\V_0\!:
\ev_0(\xi_i)\!=\!\ev_0(\xi_{i'})~\forall\,i,i'\!\in\!\ale\big\}
=\ov\M_{0,(\ale,J)}(\cL,d).$$ Note that for every $\si\!=\!(m;J_P,J_B)\!\in\!\A_0(d,J)$, $$\io_{\si}^*\V_0\!=\!\pi_B^*\V_{([m],J_B)}
\lra\ov\cM_{0,\{0\}\sqcup[m]\sqcup J_B} \!\times\!\ov\M_{0,([m],J)}(\Pn,d),$$ where $\io_{\si}$ is as in Subsection \[map0str\_subs\].
\[conederiv\_lmm1\] If $d\!\in\!\Z^+$, $J$, $\cL$, and $\V_0$ are as above, the bundle homomorphism $$\D_0\in\Ga\big(\ov\M_{0,\{0\}\sqcup J}(\Pn,d);
\Hom(\V_0,L_0^*\!\otimes\!\ev_0^*\cL)\big)$$ is surjective on the complement of the subvarieties $\ov\M_{0,\si}(\Pn,d)$ with $\si\!\in\!\A_0(d,J)$. Furthermore, for every $$\si\!\equiv\!(m;J_P,J_B) \in \A_0(d,J),$$ the differential of $\D_0$, $$\na\D_0\!: \N_{\io_{\si}} \lra \io_{\si}^*\,\Hom(\V_0,L_0^*\!\otimes\!\ev_0^*\cL)
=\pi_P^*L_0^*\!\otimes\!\pi_B^*\Hom(\V_{([m],J_B)},\ev_0^*\cL),$$ in the normal direction to the immersion $\io_{\si}$ is given by $$\na\D_0\big|_{\pi_P^*L_i\otimes\pi_B^*\pi_i^*L_0}
= \pi_P^*s_i\!\otimes\!\pi_B^*\pi_i^*\D_0 \qquad\forall\,i\in[m],$$ where $s_i$ is the homomorphism defined in Subsection \[curvebldata\_subs\].
Lemma \[conederiv\_lmm1\] can viewed as the analogue of Lemma \[deriv0str\_lmm\] for vector bundle sections. The first claim of Lemma \[conederiv\_lmm1\] is an immediate consequence of the fact that $$H^1\big(\Si;u^*\cL\!\otimes\!\O(-2z)\big)=\{0\}$$ for every genus-zero stable map $(\Si,u)$ and a smooth point $z\!\in\!\Si$ such that the restriction of $u$ to the irreducible component of $\Si$ containing $z$ is not constant. The second statement follows from [@g1cone Lemma \[g1cone-derivest\_lmm\]].\
With notation as in Subsection \[map0prelim\_subs\], let $$\begin{split}
\D_{(\ale,J)} &\in \Ga\big(\ov\M_{0,(\ale,J)}(\Pn,d);
\Hom(\V_{(\ale,J)},\Hom(F_{(\ale,J)},\ev_0^*\cL))\big)\\
&\quad = \Ga\big(\ov\M_{0,(\ale,J)}(\Pn,d);
\Hom(F_{(\ale,J)},\Hom(\V_{(\ale,J)},\ev_0^*\cL))\big)\\
&\quad =\Ga\big(\ov\M_{0,(\ale,J)}(\Pn,d);
\Hom(F_{(\ale,J)}\!\otimes\!\V_{(\ale,J)},\ev_0^*\cL)\big)
\end{split}$$ be the homomorphism defined by $$\D_{(\ale,J)}\big|_{\pi_i^*L_0\otimes\pi_j^*\V_0}=
\begin{cases}
\pi_i^*\D_0,&
\hbox{if}~j\!=\!i;\\
0,&\hbox{otherwise};
\end{cases}
\qquad\forall\, i,j\!\in\!\ale.$$\
It induces a section $$\begin{split}
\wt\D_0 \in& \Ga\big(\wt\M_{0,(\ale,J)}^0(\Pn,d);
\Hom(\ga_{(\ale,J)},\pi_{\P F_{(\ale,J)}}^*\Hom(\V_{(\ale,J)},\ev_0^*\cL))\big)\\
& =\Ga\big(\wt\M_{0,(\ale,J)}^0(\Pn,d); \Hom(\pi_{\P F_{(\ale,J)}}^*\V_{(\ale,J)},
\E_0^*\!\otimes\!\pi_{F_{(\ale,J)}}^*\ev_0^*\cL)\big).
\end{split}$$ This section vanishes identically on the subvarieties $\wt\M_{0,\vr}^0(\Pn,d)$ of $\wt\M_{0,(\ale,J)}^0(\Pn,d)$ with $\vr\!\in\!\A_0(\ale;d,J)$, defined in Subsection \[map0prelim\_subs\].
\[conederiv\_lmm2\] The bundle homomorphism $$\wt\D_0\in\Ga\big(\wt\M_{0,(\ale,J)}^0(\Pn,d); \Hom(\pi_{\P F_{(\ale,J)}}^*\V_{(\ale,J)},
\E_0^*\!\otimes\!\pi_{F_{(\ale,J)}}^*\ev_0^*\cL)\big)$$ is surjective on the complement of the subvarieties $\wt\M_{0,\vr^*}^0(\Pn,d)$ with $\vr^*\!\in\!\A_0(\ale;d,J)$. Furthermore, for every $\vr^*\!\in\!\A_0(\ale;d,J)$ as in Lemma \[map0bl\_lmm1\], the differential of $\wt\D_0$, $$\begin{split}
\na\wt\D_0\!: \N_{\io_{0,\vr^*}} \lra& \io_{0,\vr^*}^{\,*}
\Hom\big(\pi_{\P F_{(\ale,J)}}^*\V_{(\ale,J)},
\E_0^*\!\otimes\!\pi_{F_{(\ale,J)}}^*\ev_0^*\cL\big)\\
& =\pi_P^*\E_0^*\!\otimes\!
\pi_B^*\Hom(\V_{(\ale_B(\vr^*),J_B(\vr^*))},\ev_0^*\cL),
\end{split}$$ in the normal direction to the immersion $\io_{0,\vr^*}$ is given by $$\begin{gathered}
\na\wt\D_0\big|_{\pi_P^*L_{0,(l,i)}\otimes\pi_B^*\pi_{(l,i)}^*L_0}
= \pi_P^*s_{0,(l,i)}\!\otimes\!\pi_B^*\pi_{(l,i)}^*\D_0
\qquad\forall\,i\!\in\![m_l^*],\, l\!\in\!\ale_P(\vr^*),\\
\hbox{and}\qquad
\na\wt\D_0\big|_{\N_{\io_{0,\vr^*}}^{\top}}
=\pi_P^*\id\!\otimes\!\pi_B^*\D_{(\ale_B(\vr^*),J_B(\vr^*))},\end{gathered}$$ where $s_{0,(l,i)}$ is the homomorphism defined in Subsection \[curve0bl\_subs\].
This lemma follows immediately from Lemma \[conederiv\_lmm1\].
\[conehomomor\_prp\] With notation as above, there exists a surjective bundle homomorphism $$\wt\D_{(\ale,J)} \in \Ga\big(\wt\M_{0,(\ale,J)}(\Pn,d);
\Hom(\pi_{0,(\ale,J)}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale,J)},\wt\E^*\!\otimes\!
\pi_{0,(\ale,J)}^*\pi_{\P F_{(\ale,J)}}^*\ev_0^*\cL)\big)$$ such that $$\begin{gathered}
\wt\D_{(\ale,J)}\big|_{\P F_{(\ale,J)}^0}
=\wt\D_0\big|_{\P F_{(\ale,J)}^0}, \qquad\hbox{where}\\
\P F_{(\ale,J)}^0 =\P F_{(\ale,J)}-
\bigcup_{\vr\in\A_0(\ale;d,J)}\!\!\!\!\!\!\!\!\wt\M_{0,\vr}^0(\Pn,d)
\subset \wt\M_{0,(\ale,J)}^0(\Pn,d),\wt\M_{0,(\ale,J)}(\Pn,d).\end{gathered}$$\
In fact, in the notation of Subsection \[map0blconstr\_subs\], for every $\vr\!\in\!\{0\}\!\sqcup\!\A_0(\ale;d,J)$ there exists a bundle homomorphism $$\wt\D_{\vr} \in \Ga\big(\wt\M_{0,(\ale,J)}^{\vr};
\Hom(\pi_{\vr}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale,J)},\E_{\vr}^*\!\otimes\!
\pi_{\vr}^*\pi_{\P F_{(\ale,J)}}^*\ev_0^*\cL)\big)$$ such that\
${}\quad$ (i) the restrictions of $\wt\D_{\vr}$ and $\wt\D_0$ to $\P F_{(\ale,J)}^0$ agree;\
${}\quad$ (ii) $\wt\D_{\vr}$ is surjective outside of the subvarieties $\wt\M_{0,\vr^*}^{\vr}$ with $\vr^*\!>\!\vr$;\
${}\quad$ (iii) $\wt\D_{\vr}$ vanishes identically on the subvarieties $\wt\M_{0,\vr^*}^{\vr}$ with $\vr^*\!>\!\vr$;\
${}\quad$ (iv) for each $\vr^*\!>\!\vr$, the differential of $\wt\D_{\vr}$ in the normal direction to the immersion $\io_{\vr,\vr^*}$\
${}\qquad\quad$ is given as in the statement of Lemma \[conederiv\_lmm2\], but with $s_{0,(l,i)}$ replaced by $s_{\rho_{\vr^*}(\vr),(l,i)}$.\
Similarly to the construction of the bundle sections $\wt\cD_{\vr}$ in Subsection \[map0blconstr\_subs\], we construct the bundle homomorphisms $\wt\D_{\vr}$ inductively starting with $\wt\D_0$ and twisting by the exceptional divisor at each step. The inductive assumptions (i)-(iv) are analogous to ($I3$), ($I4$), and ($I12$) in Subsection \[map0blconstr\_subs\] and are verified similarly. Of course, we take $$\wt\D_{(\ale,J)}=\wt\D_{\vr_{\max}}.$$
Structure of the Cone $\V_{1,k}^d$ {#conestr_subs}
----------------------------------
In this subsection we describe the structure of the cone $$p_0\!:\ov\M_{1,k}(\cL,d)\lra\ov\M_{1,k}(\Pn,d),$$ restating the primary structural result of [@g1cone].\
For each element $\si\!=\!(m;J_P,J_B)$ of $\A_1(d,k)$, let $$\V_{1,\si}^0 \!\equiv\! \ov\M_{1,\si}(\cL,d)
=p_0^{-1}\big(\ov\M_{1,\si}^0\big)\equiv p_0^{-1}\big(\ov\M_{1,\si}(\Pn,d)\big)
\subset \V_{1,k}^0 \!\equiv\! \ov\M_{1,k}(\cL,d).$$ The subvarieties[^7] $\ov\M_{1,\si}(\cL,d)$ of $\ov\M_{1,k}(\cL,d)$ can also be defined analogously to the subvarieties $\ov\M_{1,\si}(\Pn,d)$ of $\ov\M_{1,k}(\Pn,d)$; see the beginning of Subsection \[descr\_subs\]. Similarly to Subsection \[map1prelim\_subs\], let $$j_{0,\si}\!: \ov\cM_{1,(I_P(\si),J_P(\si))}^0 \times \V_{(\ale_B(\si),J_B(\si))} \lra
\V_{1,\si}^0\subset\V_{1,k}^0$$ be the natural node-identifying immersion so that the diagram\
(-1.1,-1.8)(10,.5) (10,0)[$\ov\cM_{1,(I_P(\si),J_P(\si))}^0~\times~\V_{(\ale_B(\si),J_B(\si))}$]{} (25,0)[$\V_{1,\si}^0~\subset~\V_{1,k}^0$]{} (10.4,-4)[$\ov\cM_{1,(I_P(\si),J_P(\si))}^0~~\times~~\ov\M_{0,(\ale_B(\si),J_B(\si))}$]{} (25,-4)[$\ov\M_{1,\si}^0~\subset~\ov\M_{1,k}^0$]{} (17.2,0)(22,0)(18.5,-4)(21.5,-4) (6,-1)(6,-3.2)(14,-1)(14,-3.2)(22.7,-1)(22.7,-3) (19.5,.7)(20,-3.4) (6.6,-2)(15,-2) (23.4,-2)
commutes.
\[cone1bl\_lmm1\] If $d,n\!\in\!\Z^+$ and $k\!\in\!\bar\Z^+$, the collection $\{j_{0,\si}\}_{\si\in\A_1(d,k)}$ of immersions is properly self-intersecting. For every $\si\!\in\!\A_1(d,k)$, $$\N_{j_{0,\si}}^{\ide}=\big\{\id\!\times\!p_{0,\si}\big\}^*
\N_{\io_{0,\si}}^{\ide}$$ is an idealized normal bundle for $j_{0,\si}$.
The differential $dp_0$ of $p_0$ induces a surjective linear map $$\Im\, dj_{0,\si}\lra \Im\,d\io_{0,\si}.$$ Since the fibers of $p_0$ are vector spaces, it follows that $dp_0$ induces an injection $$j_{0,\si}^*TC\V_{1,k}^0\big/\Im\, dj_{0,\si}\lra
\io_{0,\si}^*TC\ov\M_{1,k}^0\big/\Im\, d\io_{0,\si}.$$ Thus, Lemma \[cone1bl\_lmm1\] follows from Lemma \[map1bl\_lmm1b1\].\
We denote by $\V_{1,(0)}^0$ the main component $\ov\M_{1,k}^0(\cL,d)$ of the moduli space $\ov\M_{1,k}(\cL,d)$. If $\si\!\in\!\A_1(d,k)$, we put $$\W_{\si}^0= j_{0,\si}^{~-1} \big(\V_{1,(0)}^0\big)
\equiv j_{0,\si}^{~-1} \big( \V_{1,(0)}^0\!\cap\V_{1,\si}^0\big).$$ Note that $$\big\{\id\!\times\!p_{0,\si}\big\}\big(\W_{\si}^0\big)
=\bar\cZ_{\si}^0\equiv\io_{0,\si}^{-1}\big(\ov\M_{1,(0)}^0\big).$$ Let $\N\W_{\si}^0\!\subset\!\N_{j_{0,\si}}^{\ide}$ be the normal cone $\N_{j_{0,\si}|\V_{1,(0)}^0}$ for $j_{0,\si}|_{\W_{\si}^0}$ in $\V_{1,(0)}^0$. Its structure is described in Lemma \[cone1bl\_lmm2\] below. Let $$\begin{split}
\D_{0,\si} &\in \Ga\big(\ov\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!
\ov\M_{0,(\ale_B(\si),J_B(\si))};
\Hom(\pi_B^*\V_{(\ale_B(\si),J_B(\si))},\Hom(\N_{\io_{0,\si}}^{\ide},
\pi_P^*\E_0^*\!\otimes\!\pi_B^*\ev_0^*\cL))\big)\\
&\quad=\Ga\big(\ov\cM_{1,(I_P(\si),J_P(\si))}^0 \!\times\!
\ov\M_{0,(\ale_B(\si),J_B(\si))};
\Hom(\N_{\io_{0,\si}}^{\ide},\pi_P^*\E_0^*\!\otimes\!
\pi_B^*\Hom(\V_{(\ale_B(\si),J_B(\si))},\ev_0^*\cL))\big)
\end{split}$$ be the section defined by $$\D_{0,\si}\big|_{\pi_P^*L_i\otimes\pi_B^*\pi_i^*L_0}
=\pi_P^*s_i\!\otimes\!\pi_B^*\pi_i^*\D_0,
\qquad\forall\,i\!\in\![m],$$ where $s_i$ and $\D_0$ are as in Subsections \[curvebldata\_subs\] and \[conehomomor\_subs\], respectively. If $\xi\!\in\!\pi_B^*\V_{(\ale_B(\si),J_B(\si))}$, we will view $\D_{0,\si}\xi$ as a homomorphism $$\D_{0,\si}\xi\!:
\N_{j_{0,\si}}^{\ide}\big|_{\xi}\!=\!
\N_{\io_{0,\si}}^{\ide}\big|_{\{\id\times p_{0,\si}\}(\xi)}
\lra \pi_P^*\E_0^*\!\otimes\!\pi_B^*\ev_0^*\cL
\big|_{\{\id\times p_{0,\si}\}(\xi)}.$$
\[cone1bl\_lmm2\] For all $\si\!\in\!\A_1(d,k)$, $\V_{1,(0)}^0$ is a proper subvariety of $\V_{1,k}^0$ relative to the immersion $j_{0,\si}$. The homomorphism $$\N\W_{\si}^0 \lra \{\id\!\times\!p_{0,\si}\}^*\N\bar\cZ_{\si}^0$$ induced by $dp_0$ is injective. Furthermore, $$\begin{gathered}
\W_{\si}^0\big|_{\cZ_{\si}^0}=\big\{
\xi\!\in\!\pi_B^*\V_{(\ale_B(\si),J_B(\si))}|_{\cZ_{\si}^0}\!:
\ker\,\{\D_{0,\si}\xi\}
\big|_{\N\bar\cZ_{\si}^0|_{\{\id\times p_{0,\si}\}(\xi)}}\!\neq\!\{0\}
\big\}\\
\hbox{and}\qquad
\N\W_{\si}^0\big|_{\xi}=
\ker\big\{\D_{0,\si}\xi\big\}
\big|_{\N\bar\cZ_{\si}^0|_{\{\id\times p_{0,\si}\}(\xi)}}
\subset\N_{j_{0,\si}}^{\ide} \qquad\forall\,\xi\!\in\!\W_{\si}^0\big|_{\cZ_{\si}^0}.\end{gathered}$$ Finally, $\W_{\si}^0$ is the closure of $\W_{\si}^0|_{\cZ_{\si}^0}$ in $\ov\cM_{1,(I_P(\si),J_P(\si))}^0\!\times\!\V_{(\ale_B(\si),J_B(\si))}$ and $\N\W_{\si}^0$ is the closure of $\N\W_{\si}^0\big|_{\W_{\si}^0|_{\cZ_{\si}^0}}$ in $\N_{j_{0,\si}}^{\ide}$.
Since the fibers of $p_0$ are vector spaces, the first two sentences of this lemma follow from Lemma \[map1bl\_lmm2\]. The middle claim of Lemma \[cone1bl\_lmm2\] is a restatement of [@g1cone Lemma \[g1cone-g1conebdstr\_lmm\]]. The remaining claims of the lemma follow from [@g1cone Lemma \[g1cone-g1conebdstr\_lmm\]] by dimension counting, similarly to the argument following Lemma \[map1bl\_lmm2\].\
[*Remark:*]{} It may appear that the statement of Lemma \[cone1bl\_lmm2\] depends on the choice of a hermitian connection (or metric) in the line bundle $\cL\!\lra\!\Pn$. As explained in detail in [@g1cone Subsect. \[g1cone-g1conelocalstr\_subs2\]], the dependence is only on the holomorphic structure of $\cL$, as the case should be.
Desingularization Construction {#conebl_subs}
------------------------------
In this subsection we lift the inductive blowup construction of Subsection \[map1blconstr\_subs\] to the cone $$p_0\!:\V_{1,k}^0\lra\ov\M_{1,k}^0.$$ For each $\si\!\in\!\A_1(d,k)$, let $\V_{1,k}^{\si}\!\equiv\!\pi_{\si}^*\V_{1,k}^0$ be the pullback of $\V_{1,k}^0$ to $\ov\M_{1,k}^{\si}$:\
(-1.5,-2)(10,.3) (10,0)[$\V_{1,k}^{\si}\!\equiv\!\pi_{\si}^*\V_{1,k}^0$]{}(18,0)[$\V_{1,k}^0$]{} (10,-4)[$\ov\M_{1,k}^{\si}$]{}(18,-4)[$\ov\M_{1,k}^0$]{} (13,0)(17,0)(11.5,-4)(16.5,-4) (10,-.8)(10,-3)(18,-.8)(18,-3) (15,.6)(14,-3.5) (10.7,-2)(18.7,-2)
For each $\si'\!\in\!\A_1(d,k)$, let $$\V_{1,\si'}^{\si}=\V_{1,k}^{\si}\big|_{\ov\M_{1,\si'}^{\si}}
=\pi_{\si}^{-1}\big(\V_{1,k}^0\big|_{\ov\M_{1,\si'}^0}\big).$$ The bundle homomorphisms $j_{0,\si'}$ lift to bundle homomorphisms onto $\V_{1,\si'}^{\si}$ covering the immersion $\io_{\si,\si'}$ of Subsection \[map1blconstr\_subs\]:\
(-3,-2.4)(10,.5) (8,0)[$\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times
\pi_{\vr_{\si'}(\si)}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))}$]{} (25,0)[$\V_{1,\si'}^{\si}~\subset~\V_{1,k}^{\si}$]{} (6.5,-4)[$\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times
~~~~~\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si)}$]{} (25,-4)[$\ov\M_{1,\si'}^{\si}~\subset~\ov\M_{1,k}^{\si}$]{} (18,0)(22,0)(15.3,-4)(21.2,-4) (2,-1)(2,-3.2)(12,-1)(12,-3.2)(22.7,-1)(22.7,-3) (20,.7)(18,-3.4) (2.6,-2)(13.1,-2) (23.4,-2) (-6,-1.5)[$\si'\!\le\!\si\!:$]{}
(-3,-2.5)(10,.5) (8,0)[$\ov\cM_{1,(I_P(\si'),J_P(\si'))}^{\rho_{\si'}(\si)}
\times\V_{(\ale_B(\si'),J_B(\si'))}$]{} (25,0)[$\V_{1,\si'}^{\si}~\subset~\V_{1,k}^{\si}$]{} (8.6,-4)[$\ov\cM_{1,(I_P(\si'),J_P(\si'))}^{\rho_{\si'}(\si)}\times
\ov\M_{0,(\ale_B(\si'),J_B(\si'))}$]{} (25,-4)[$\ov\M_{1,\si'}^{\si}~\subset~\ov\M_{1,k}^{\si}$]{} (15.3,0)(22,0)(16.3,-4)(21.2,-4) (5,-1)(5,-3.2)(12,-1)(12,-3.2)(22.7,-1)(22.7,-3) (19,.7)(19,-3.4) (5.6,-2)(13.1,-2) (23.4,-2) (-6,-1.5)[$\si'\!>\!\si\!:$]{}
The collection $\{\io_{\si,\si'}\}_{\si'\in\A_1(d,k)}$ of immersions is properly self-intersecting by the inductive assumption ($I14$) of Subsection \[map1blconstr\_subs\]. Thus, by the same argument as in the paragraph following Lemma \[cone1bl\_lmm1\], so is the collection $\{j_{\si,\si'}\}_{\si'\in\A_1(d,k)}$. Furthermore, $$\label{idebundle_e}
\N_{j_{\si,\si'}}^{\ide}=\big\{\id\!\times\!p_{\si,\si'}\big\}^*
\N_{\io_{\si,\si'}}^{\ide}$$ is an idealized normal bundle for $j_{\si,\si'}$. These two observations also follow from Lemma \[cone1bl\_lmm1\] by induction using Lemmas \[virimmer\_lmm\] and \[virimmer\_lmm2\].
\[coneblstr\_lmm\] If $\si\!\in\!\A_1(d,k)$, $\V_{1,\si}^{\si-1}$ is a smooth subvariety of $\V_{1,k}^{\si-1}$ and $$p_{\si}\!: \V_{1,k}^{\si}\lra\ov\M_{1,k}^{\si}$$ is the idealized blowup of $\V_{1,k}^{\si-1}$ along $\V_{1,\si}^{\si-1}$.
Recall from Subsection \[map1blconstr\_subs\] that the immersion $$\bar\io_{\si-1,\si}\!:
\big(\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\ov\M_{0,(\ale_B(\si),J_B(\si))}\big)\big/G_{\si}
\lra \ov\M_{1,\si}^{\si-1}\subset\ov\M_{1,k}^{\si-1}$$ induced by $\io_{\si-1,\si}$ is an embedding and $$\ti\pi_{\si}\!: \ov\M_{1,k}^{\si}\lra\ov\M_{1,k}^{\si-1}$$ is the idealized blowup along $\ov\M_{1,\si}^{\si-1}$. Thus, the immersion $$\bar{j}_{\si-1,\si}\!:
\big(\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\V_{(\ale_B(\si),J_B(\si))}\big)\big/G_{\si}
\lra \V_{1,\si}^{\si-1}\subset \V_{1,k}^{\si-1}$$ induced by $j_{\si-1,\si}$ is also an embedding and $\V_{1,\si}^{\si-1}$ is a smooth subvariety of $\V_{1,k}^{\si-1}$. Let $$\ti\pi_{\si}\!: \V\lra\V_{1,k}^{\si-1}$$ be the idealized blowup along $\V_{1,k}^{\si-1}$. Since $$\N_{j_{\si-1,\si}}^{\ide}=\big\{\id\!\times\!p_{\si-1,\si}\big\}^*
\N_{\io_{\si-1,\si}}^{\ide}$$ and the linear map $$j_{\si-1,\si}^*TC\V_{1,k}^{\si-1}\big/\Im\, dj_{\si-1,\si}\lra
\io_{\si-1,\si}^*TC\ov\M_{1,k}^{\si-1}\big/\Im\, d\io_{\si-1,\si}$$ induced by $dp_{\si-1}$ is injective, $p_{\si-1}$ lifts to a map $p$ over the blowdown maps $\ti\pi_{\si}$:\
(-1.5,-2)(10,.3) (10,0)[$\V$]{}(18,0)[$\V_{1,k}^{\si-1}$]{} (10,-4)[$\ov\M_{1,k}^{\si}$]{}(18,-4)[$\ov\M_{1,k}^{\si-1}$]{} (10.8,0)(16.6,0)(11.5,-4)(16.5,-4) (10,-.8)(10,-3)(18,-.8)(18,-3) (13.5,.6)(14,-3.5) (10.5,-2)(19.1,-2)
Then $p$ and the top arrow[^8] $\ti\pi_{\si}$ factor through a morphism $f$ to $\ti\pi_{\si}^*\V_{1,k}^{\si-1}$:\
(-2.5,-2)(10,1.2) (5,3)[$\V$]{} (5.5,2.9)(16.7,.4)(5,2.3)(9,-3.2)(5.2,2.6)(9,.7) (6.8,-1)(12,2) (8,1.8) (10,0)[$\ti\pi_{\si}^*\V_{1,k}^{\si-1}$]{}(18,0)[$\V_{1,k}^{\si-1}$]{} (10,-4)[$\ov\M_{1,k}^{\si}$]{}(18,-4)[$\ov\M_{1,k}^{\si-1}$]{} (11.8,0)(16.6,0)(11.5,-4)(16.5,-4) (10,-.8)(10,-3)(18,-.8)(18,-3) (13.5,.5)(14,-3.5) (10.7,-2)(19.1,-2)
We show in the next paragraph that $f$ is an isomorphism. Since $\ti\pi_{\si}^*\V_{1,k}^{\si-1}\!=\!\V_{1,k}^{\si}$, this implies the second statement of Lemma \[coneblstr\_lmm\].\
By construction, the maps $$\ti\pi_{\si}\!: \ov\M_{1,k}^{\si}\lra\ov\M_{1,k}^{\si-1} \qquad\hbox{and}\qquad
\ti\pi_{\si}\!: \V\lra\V_{1,k}^{\si-1}$$ are isomorphisms on the complements of the idealized exceptional divisors $$\ov\M_{1,\si}^{\si}\equiv\cE_{\ov\M_{1,\si}^{\si-1}}^{\ide}\subset\ov\M_{1,k}^{\si}
\qquad\hbox{and}\qquad
\V_{1,\si}^{\si}\equiv\cE_{\V_{1,\si}^{\si-1}}^{\ide}\subset\V.$$ Thus, $f\!: \V\!\lra\!\ti\pi_{\si}^*\V_{1,k}$ is an isomorphism over the complement of $\ov\M_{1,\si}^{\si}$ in $\ov\M_{1,k}^{\si}$. In particular, $f$ is linear on all fibers of $p$. Furthermore, $$\ti\pi_{\si}^*\V_{1,k}^{\si-1}\big|_{\ov\M_{1,\si}^{\si}}
=\big\{(\ell,v)\!\in\!\P\N_{\ov\M_{1,\si}^{\si-1}}^{\ide}\!\times\!\V_{1,k}^{\si-1}\!:
\ti\pi_{\si}(\ell)\!=\!p_{\si-1}(v)\big\}.$$ On the other hand, since $$\N_{\V_{1,\si}^{\si-1}}^{\ide}=p_{\si-1}^*\N_{\ov\M_{1,\si}^{\si-1}}^{\ide}$$ by \_ref[idebundle\_e]{}, we have $$\V_{1,\si}^{\si}=p_{\si-1}^*\P\N_{\ov\M_{1,\si}^{\si-1}}^{\ide}
=\big\{(v,\ell)\!\in\!\V_{1,k}^{\si-1}\!\times\!\P\N_{\ov\M_{1,\si}^{\si-1}}^{\ide}\!:
p_{\si-1}(v)\!=\!\ti\pi_{\si}(\ell)\big\}.$$ Thus, the restriction of $f$ to $\V_{1,\si}^{\si}$ must interchange $v$ and $\ell$, i.e. it is a vector bundle isomorphism over $\ov\M_{1,k}^{\si}$. Finally, $\ov\M_{1,\si}^{\si-1}$ is a smooth subvariety of $\V_{1,k}^{\si-1}$ and $$T\big(\ov\M_{1,k}^{\si-1}\!\cap\!\V_{1,\si}^{\si-1}\big)=
T\ov\M_{1,\si}^{\si-1}
=TC\ov\M_{1,k}^{\si-1}\cap T\V_{1,\si}^{\si-1}\subset TC\V_{1,k}^{\si-1}.$$ Thus, similarly to (1) of Lemma \[ag\_lmm2a\], the proper transform of $\ov\M_{1,k}^{\si-1}$ in $\V$ is the blowup of $\ov\M_{1,k}^{\si-1}$ along $$\ov\M_{1,k}^{\si-1}\!\cap\!\V_{1,\si}^{\si-1}=\ov\M_{1,\si}^{\si-1},$$ i.e. $\V$ contains $\ov\M_{1,k}^{\si}$ as the zero section. The map $f$ must be the identity on $\ov\M_{1,k}^{\si}$. Since $f$ is a linear isomorphism on all fibers of $p$ by the above, it then follows that $f$ is an isomorphism everywhere.\
[*Remark:*]{} If $\V_{1,k}^{\si-1}$ is a vector bundle over $\ov\M_{1,k}^{\si-1}$, the second statement of Lemma \[coneblstr\_lmm\] applies to standard blowups of $\ov\M_{1,k}^{\si-1}$ and $\V_{1,k}^{\si-1}$ as well. However, the second statement does not generally apply to standard blowups in the setting of Lemma \[coneblstr\_lmm\], as the analogue of the morphism $f$ may not be surjective.\
By the inductive assumption ($I1$) of Subsection \[map1blconstr\_subs\], the projection map $\pi_{\si}$ is an isomorphism outside of the subvarieties $\V_{1,\si'}^{\si}$ with $\si'\!\le\!\si$. We denote by $$\V_{1,(0)}^{\si}\subset \V_{1,k}^{\si}$$ the proper transform of $\V_{1,(0)}^0$. For each $\si'\!\in\!\A_1(d,k)$, let $$\W_{\si'}^{\si}=j_{\si,\si'}^{-1}\big(\V_{1,(0)}^{\si}\big)
=j_{\si,\si'}^{-1}\big(\V_{1,(0)}^{\si}\!\cap\!\V_{1,\si'}^{\si}\big).$$ By the inductive assumption ($I15$) of Subsection \[map1blconstr\_subs\], $\ov\M_{1,(0)}^{\si}$ is a proper subvariety of $\ov\M_{1,k}^{\si}$ with respect to the immersion $\io_{\si,\si'}$. Thus, by the same argument as in the paragraph following Lemma \[cone1bl\_lmm2\], the subvariety $\V_{1,(0)}^{\si}$ of $\V_{1,k}^{\si}$ is proper with respect to the immersion $j_{\si,\si'}$. Furthermore, if $$\N\W_{\si'}^{\si} \equiv \N\W_{j_{\si,\si'}|\V_{1,(0)}^{\si}}
\subset \N_{j_{\si',\si}}^{\ide}$$ denotes the normal cone for $j_{\si',\si}|_{\W_{\si'}^{\si}}$ in $\V_{1,(0)}^{\si}$, then the homomorphism $$\N\W_{\si'}^{\si} \lra \{\id\!\times\!p_{\si,\si'}\}^*\N\bar\cZ_{\si'}^{\si}$$ induced by $dp_{\si}$ is injective. These two observations also follow from Lemma \[cone1bl\_lmm2\] by induction using Lemma \[virimmer\_lmm3\].\
If $\si'\!\in\!\A_1(d,k)$, let $$\wt\cZ_{\si';B}^0=\wt\cD_0^{-1}(0)\cap
\P F_{(I_P(\si'),J_P(\si'))}\big|_{\M_{0,(\ale_B(\si),J_B(\si))}}.$$ By the inductive assumptions ($I7$) in Subsection \[map1blconstr\_subs\] and ($I4$) in Subsection \[map0blconstr\_subs\], $$\ov\cZ_{\si'}^{\si}\equiv\io_{\si,\si'}^{\,-1}\big(\ov\M_{1,(0)}^{\si}\big)$$ is the closure of $\wt\cM_{1,(I_P(\si'),J_P(\si'))}\!\times\!\wt\cZ_{\si';B}^0$ in $$\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times\wt\M_{0,(\ale_B(\si'),J_B(\si'))}^{\vr_{\si'}(\si)}$$ for all $\si\!\in\!\A_1(k,d)$ such that $\si'\!\le\!\si$.\
Suppose $\si\!\in\!\{0\}\!\cup\!\A_1(d,k)$ and $\si'\!\in\!\A_1(d,k)$. We claim that\
${}\quad$ ($I1$) if $\si'\!\le\!\si$, then $\W_{\si'}^{\si}$ is the closure of $$\begin{split}
\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times \ker\wt\D_0|_{\wt\cZ_{\si';B}^0}
&\subset \wt\cM_{1,(I_P(\si'),J_P(\si'))}\times
\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))},\\
&\qquad \wt\cM_{1,(I_P(\si'),J_P(\si'))}\times
\pi_{\vr_{\si'}(\si)}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))}
\end{split}$$ in $\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times
\pi_{\vr_{\si'}(\si)}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))}$ and $$\N\W_{\si'}^{\si}=\N_{j_{\si,\si'}}^{\ide}\big|_{\W_{\si'}^{\si}};$$ ${}\quad$ ($I2$) if $\si'\!>\!\si$, then $\W_{\si'}^{\si}$ and $\N\W_{\si'}^{\si}$ are the closures of $$\begin{gathered}
\W_{\si'}^0|_{\cZ_{\si'}^0}\subset
\ov\cM_{1,(I_P(\si'),J_P(\si'))}\!\times\!\V_{(\ale_B(\si'),J_B(\si'))},
\ov\cM_{1,(I_P(\si'),J_P(\si'))}^{\rho_{\si'}(\si)}\!\times\!\V_{(\ale_B(\si'),J_B(\si'))}\\
\hbox{and}\qquad\
\N\W_{\si'}^0\big|_{\W_{\si'}^0|_{\cZ_{\si'}^0}}
\subset \N_{j_{0,\si'}}^{\ide}|_{\W_{\si'}^0|_{\cZ_{\si'}^0}}
\subset \N_{j_{0,\si'}}^{\ide},\N_{j_{\si,\si'}}^{\ide}\end{gathered}$$ in $\ov\cM_{1,(I_P(\si'),J_P(\si'))}^{\rho_{\si'}(\si)}\!\times\!\V_{(\ale_B(\si'),J_B(\si'))}$ and in $\N_{j_{\si,\si'}}^{\ide}$, respectively.\
If $\si\!=\!0$, the assumption ($I1$) is trivially satisfied, while ($I2$) constitutes part of Lemma \[cone1bl\_lmm2\]. Suppose $\si\!\in\!\A_1(d,k)$ and the two assumptions hold with $\si$ replaced by $\si\!-\!1$. By Lemma \[coneblstr\_lmm\], $\V_{1,k}^{\si}$ is the idealized blowup of $\V_{1,k}^{\si-1}$ along $\V_{1,\si}^{\si-1}$. Thus, by the last statement of Lemma \[virimmer\_lmm3\] both of the inductive assumptions continue to hold for $\si'\!\neq\!\si$.\
On the other hand, let $$\begin{split}
\cZ_{\si;B}&= \big\{b\!\in\!\M_{0,(\ale_B(\si),J_B(\si))}\!:
\ker\cD_{(\ale_B(\si),J_B(\si))}\!\neq\!0\big\}, \\
\W_{\si;B}^0&=\big\{
\xi\!\in\!\V_{(\ale_B(\si),J_B(\si))}|_{\cZ_{\si;B}}\!:
\ker\,\{\D_{(\ale_B(\si),J_B(\si))}\xi\}
\big|_{\ker\cD_{(\ale_B(\si),J_B(\si))}|_{p_{\si-1,\si}(\xi)}}\!\neq\!\{0\}\big\},
\qquad\hbox{and}\\
\N\W_{\si;B}^0&=
\big\{(\xi,\ups)\!:\xi\!\in\!\W_{\si;B}^0, \,
\ups\!\in\!\ker\{\D_{0,\si}\xi\}\big|_{\ker\cD_{(\ale_B(\si),J_B(\si))}|_{p_{\si-1,\si}(\xi)}}\big\}
\subset p_{\si-1,\si}^{\,*}F_{(\ale_B(\si),J_B(\si))}.
\end{split}$$ By the inductive assumption ($I12$) in Subsection \[map1blconstr\_subs\], $$\begin{gathered}
\ov\cZ_{\si}^{\si-1}\cap\big(
\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\M_{0,(\ale_B(\si),J_B(\si))}\big)
=\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\cZ_{\si;B} \qquad\hbox{and}\\
\N\bar\cZ_{\si}^0|_{\wt\cM_{1,(I_P(\si),J_P(\si))}\times\cZ_{\si;B}}
=\pi_P^*\L\otimes\pi_B^*\ker\cD_{0,(\ale_B(\si),J_B(\si))}.\end{gathered}$$ By the inductive assumption ($I2$) above, Lemma \[cone1bl\_lmm2\], and the inductive assumption ($I11$) in Subsection \[map1blconstr\_subs\], $\W_{\si}^{\si-1}$ and $\N\W_{\si}^{\si-1}$ are the closures of $$\begin{gathered}
\W_{\si}^0\big|_{\cZ_{\si}^0}=\big\{
\xi\!\in\!\pi_B^*\V_{(\ale_B(\si),J_B(\si))}|_{\cZ_{\si}^0}\!:
\ker\,\{\D_{0,\si}\xi\}
\big|_{\N\bar\cZ_{\si}^0|_{\{\id\times p_{0,\si}\}(\xi)}}\!\neq\!\{0\}
\big\}\\
\hbox{and}\qquad
\N\W_{\si}^0\big|_{\W_{\si}^0|_{\cZ_{\si}^0}}=
\big\{(\xi,\ups)\!:\xi\!\in\!\W_{\si}^0\big|_{\cZ_{\si}^0},~
\ups\!\in\!\ker\{\D_{0,\si}\xi\}
\big|_{\N\bar\cZ_{\si}^0|_{\{\id\times p_{0,\si}\}(\xi)}}\big\}
\subset\N_{j_{0,\si}}^{\ide}, \N_{j_{\si-1,\si}}^{\ide}\end{gathered}$$ in $\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\V_{(\ale_B(\si),J_B(\si))}$ and in $$\N_{j_{\si-1,\si}}^{\ide}= \pi_P^*\L\otimes
\pi_B^*p_{\si-1,\si}^*F_{(\ale_B(\si),J_B(\si))}.$$ As before, $$\pi_P,\pi_B\!: \wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\V_{(\ale_B(\si),J_B(\si))}
\lra \wt\cM_{1,(I_P(\si),J_P(\si))},\V_{(\ale_B(\si),J_B(\si))}$$ are the projections onto the principle and bubble components. The bundle homomorphisms $s_i$ and $\ti{s}_i$ of Subsection \[curve1bl\_subs\] agree on $$\cM_{1,(I_P(\si),J_P(\si))}\subset \ov\cM_{1,(I_P(\si),J_P(\si))},
\wt\cM_{1,(I_P(\si),J_P(\si))}.$$ The homomorphism $\ti{s}_i$ is an isomorphism from $\ti{L}_i$ to $\ti{E}^*$ over $\wt\cM_{1,(I_P(\si),J_P(\si))}$, and both line bundles are isomorphic to $\L$. It follows that $\W_{\si}^{\si-1}$ and $\N\W_{\si}^{\si-1}$ are the closures of $$\wt\cM_{1,(I_P(\si),J_P(\si))}\times\W_{\si;B}^0 \qquad\hbox{and}\qquad
\pi_P^*\L\otimes\pi_B^*\N\W_{\si;B}^0$$ in $\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!\V_{(\ale_B(\si),J_B(\si))}$ and in $$\N_{j_{\si-1,\si}}^{\ide}=\pi_P^*\L\otimes
\pi_B^*p_{\si-1,\si}^{\,*}F_{(\ale_B(\si),J_B(\si))}.$$ Thus, by the first statement of Lemma \[virimmer\_lmm3\], $$\W_{\si}^{\si}\equiv j_{\si,\si}^{\,-1}\big(\V_{1,(0)}^{\si}\big)$$ is the closure of $$\begin{split}
\P\big(\pi_P^*\L\!\otimes\!\pi_B^*\N\W_{\si;B}^0\big)
&=\wt\cM_{1,(I_P(\si),J_P(\si))}\times \P\N\W_{\si;B}^0
=\wt\cM_{1,(I_P(\si),J_P(\si))}\times
\ker\wt\D_0|_{\wt\cZ_{\si;B}^0}\\
&\quad\subset\P \N_{j_{\si-1,\si}}^{\ide}
=\wt\cM_{1,(I_P(\si),J_P(\si))}\!\times\!
\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si),J_B(\si))},
\end{split}$$ i.e. the first part of the inductive assumption ($I1$) for $\si'\!=\!\si$ is satisfied. Furthermore, by the second part of (1) of Lemma \[virimmer\_lmm3\], $$\N\W_{\si}^{\si}=\ga_{\V_{1,\si}^{\si-1}}\big|_{\W_{\si}^{\si}}
=\big\{\id\!\times\!p_{\si,\si}\big\}^*\ga_{\ov\M_{1,\si}^{\si-1}}\big|_{\W_{\si}^{\si}}
=\big\{\id\!\times\!p_{\si,\si}\big\}^*\N_{\io_{\si,\si}}^{\ide}\big|_{\W_{\si}^{\si}}
=\N_{j_{\si,\si}}^{\ide}\big|_{\W_{\si}^{\si}}.$$ We have thus verified the second part of the inductive assumption ($I1$) for $\si'\!=\!\si$.\
Since the immersions $\bar{i}_{\si,\si'}$ with $\si'\!\le\!\si$ are embeddings by the inductive assumption ($I8$) in Subsection \[map1blconstr\_subs\], so are the immersions $$\bar{j}_{\si,\si'}\!: \big(\wt\cM_{1,(I_P(\si'),J_P(\si'))}\!\times\!
\pi_{\vr_{\si'}(\si)}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))}\big)
\big/G_{\si'}\lra \V_{1,\si'}^{\si}\subset\V_{1,k}^{\si}$$ induced by $j_{\si,\si'}$. In particular, all of the morphisms $$\begin{split}
\bar{j}_{\si_{\max},\si'}\!: \big(\wt\cM_{1,(I_P(\si'),J_P(\si'))}\!\times\!
\pi_{0,(\ale_B(\si'),J_B(\si'))}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))}\big)
\big/G_{\si'} \qquad\qquad\qquad&\\
\lra \V_{1,\si'}^{\si_{\max}}\subset\V_{1,k}^{\si_{\max}}=
\ti\pi^*\ov\M_{1,k}(\cL,d)&
\end{split}$$ are embeddings. On the other hand, by the inductive assumption ($I1$), $$\wt\W_{\si'}\equiv\W_{\si'}^{\si_{\max}}\equiv
j_{\si_{\max},\si'}^{\,-1}\big(\wt\M_{1,k}^0(\cL,d)\big)
\equiv j_{\si_{\max},\si'}^{\,-1}\big(\V_{1,(0)}^{\si_{\max}}\big)$$ is the closure of $$\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times
\ker\wt\D_0|_{\wt\cZ_{\si';B}^0}
\subset \wt\cM_{1,(I_P(\si'),J_P(\si'))}\!\times\!
\pi_{0,(\ale_B(\si'),J_B(\si'))}^*\pi_{\P F_{(\ale,J)}}^*\V_{(\ale_B(\si'),J_B(\si'))}.$$ By Proposition \[conehomomor\_prp\] and the inductive assumption ($I8$) in Subsection \[map1blconstr\_subs\], this closure is $$\begin{gathered}
\wt\cM_{1,(I_P(\si'),J_P(\si'))}\times\ker\wt\D_{(I_P(\si'),J_P(\si'))}|_{\wt\cZ_{\si';B}},
\qquad\hbox{where}\\
\wt\cZ_{\si';B}=\wt\cD_{(I_P(\si'),J_P(\si'))}^{-1}(0).\end{gathered}$$ Since the bundle section $\wt\cD_{(I_P(\si'),J_P(\si'))}$ is transverse to the zero set, $\wt\cZ_{\si';B}$ is a smooth subvariety of $\wt\M_{0,(I_P(\si'),J_P(\si'))}(\Pn,d)$ and $$\wt\W_{\si'}\lra \wt\cM_{1,(I_P(\si'),J_P(\si'))}\times\wt\cZ_{\si';B}$$ is a smooth vector bundle by Proposition \[conehomomor\_prp\]. We conclude that $$\wt\M_{1,k}^0(\cL,d)\cap \V_{1,\si'}^{\si_{\max}}$$ is a smooth subvariety of $\wt\M_{1,k}^0(\cL,d)$ for all $\si'\!\in\!\A_{1,k}(k,d)$. Its normal cone is a line bundle by the inductive assumption ($I1$).\
[*Department of Mathematics, Stanford University, Stanford, CA 94305-2125*]{}\
vakil@math.stanford.edu\
[*Department of Mathematics, SUNY, Stony Brook, NY 11794-3651*]{}\
azinger@math.sunysb.edu\
[\[MirSym\]]{}
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[^1]: Partially supported by an NSF grant DMS–0228011
[^2]: Partially supported by an NSF Postdoctoral Fellowship
[^3]: We call a space $\ov\M$ a [*compactification*]{} of $\M$ if $\ov\M$ is compact and contains $\M$. In particular, $\M$ need not be dense in $\ov\M$.
[^4]: Blowing up an irreducible component of a stack will result in the component being removed (or “blown out of existence”), and the remainder of the stack is blown up along its intersection with the component in question.
[^5]: $\V_{1,k}^d$ is a variety such that the fibers of the projection map to $\ov\M_{1,k}^0(\Pf,d)$ are vector spaces, but not necessarily of the same dimension.
[^6]: In fact, these will be substacks of the stack $\ov\M_{1,k}(\Pn,d)$. They can also be thought of as analytic sub-orbivarieties of the analytic orbivariety $\ov\M_{1,k}(\Pn,d)$. As we work with reduced scheme structures throughout the paper, we will call such objects simply varieties.
[^7]: typo fixed
[^8]: added “the top arrow” to make clear which $\ti \pi_{\si}$ you mean. — R
|
---
abstract: 'We show that the moduli space of semi-stable sheaves on a smooth quadric surface, having dimension $1$, multiplicity $4$, Euler characteristic $2$, and first Chern class $(2, 2)$, is the blow-up at two points of a certain hypersurface in a weighted projective space.'
address: 'Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea'
author:
- Mario Maican
title: On two moduli spaces of sheaves supported on quadric surfaces
---
Let ${\mathbf M}$ be the moduli space of Gieseker semi-stable sheaves ${\mathcal F}$ on ${\mathbb P}^1 \times {\mathbb P}^1$ having Hilbert polynomial $P_{{\mathcal F}}(m) = 4m + 2$, relative to the polarization $\O(1,1)$, and first Chern class $c_1({\mathcal F}) = (2, 2)$. Let ${\operatorname{M}}_{{\mathbb P}^3}(m^2+3m+2)$ be the moduli space of Gieseker semi-stable sheaves ${\mathcal F}$ on ${\mathbb P}^3$ having Hilbert polynomial $P_{{\mathcal F}}(m) = m^2 + 3m + 2$. Such sheaves are supported on quadric surfaces. The purpose of this note is to show that ${\operatorname{M}}_{{\mathbb P}^3}(m^2+3m+2)$ is isomorphic to a certain hypersurface in a weighted projective space (see Proposition \[hypersurface\]) and to give an elementary proof of a result of Chung and Moon [@chung_moon] stating that ${\mathbf M}$ is the blow-up of ${\operatorname{M}}_{{\mathbb P}^3}(m^2+3m+2)$ at two regular points.
Let $l$, $m$, $n$ be positive integers. Let $V$ be a vector space over ${\mathbb C}$ of dimension $l$. The reductive group $G = \big( {\operatorname{GL}}(n, {\mathbb C}) \times {\operatorname{GL}}(m, {\mathbb C}) \big)/{\mathbb C}^*$ acts by conjugation on the vector space ${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}V)$ of $m \times n$-matrices with entries in $V$. The resulting good quotient $${\operatorname{N}}(V; m, n) = {\operatorname{N}}(l; m, n) = {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}V)^{{\scriptstyle \operatorname{ss}}}/\!\!/ G$$ is called a *Kronecker moduli space*. Kronecker moduli spaces arise from the study of moduli spaces of torsion-free sheaves, as in [@drezet_reine]. According to [@asterisque Corollary 3.7] and [@chung_moon Lemma 5.2], the map $${\operatorname{Hom}}(2\O_{{\mathbb P}^3}(-1), 2\O_{{\mathbb P}^3})^{{\scriptstyle \operatorname{ss}}}{\longrightarrow}{\operatorname{M}}_{{\mathbb P}^3}(m^2 + 3m + 2), \qquad \langle {\varphi}\rangle \longmapsto \langle {\mathcal Coker}({\varphi}) \rangle,$$ is a good quotient modulo $\big( {\operatorname{GL}}(2, {\mathbb C}) \times {\operatorname{GL}}(2, {\mathbb C}) \big)/{\mathbb C}^*$. Thus, the above moduli space is isomorphic to ${\operatorname{N}}(4; 2, 2)$. According to [@asterisque Remark 3.9], ${\operatorname{M}}_{{\mathbb P}^3}(m^2 + 3m + 2)$ is rational; this result was reproved in [@chung_moon] using the wall-crossing method.
\[projection\] Assume that ${\operatorname{N}}(l; m, n)$ contains stable points. Then the same is true of ${\operatorname{N}}(k; m, n)$ for all integers $k > l$, and, moreover, ${\operatorname{N}}(k; m, n)$ is birational to $\AA^{(k-l)mn} \times {\operatorname{N}}(l; m, n)$.
Let $U$, $V$ be vector spaces over ${\mathbb C}$ of dimension $k - l$, respectively, $l$, and put $W = U \oplus V$. The projection of $W$ onto the second factor induces a $G$-equivariant projection $$\pi \colon {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}W) {\longrightarrow}{\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}V).$$ From King’s criterion of semi-stability [@king] we see that $$\pi^{-1} \big({\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}V)^{{\scriptstyle \operatorname{s}}}\big) \subset {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}W)^{{\scriptstyle \operatorname{s}}}.$$ The left-hand-side, denoted by $E$, is a trivial $G$-linearized vector bundle over ${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}V)^{{\scriptstyle \operatorname{s}}}$ with fiber ${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}U)$. The geometric quotient map $${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^m {\otimes}V)^{{\scriptstyle \operatorname{s}}}{\longrightarrow}{\operatorname{N}}(V; m, n)^{{\scriptstyle \operatorname{s}}}$$ is a principal $G$-bundle, so we can apply [@huybrechts_lehn Theorem 4.2.14] to deduce that $E$ descends to a vector bundle $F$ over ${\operatorname{N}}(V; m, n)^{{\scriptstyle \operatorname{s}}}$. Clearly, $F$ is the geometric quotient of $E$ by $G$, hence $F$ is isomorphic to an open subset of ${\operatorname{N}}(W; m, n)^{{\scriptstyle \operatorname{s}}}$. We conclude that ${\operatorname{N}}(W; m, n)$ is birational to $\AA^{(k-l)mn} \times {\operatorname{N}}(V; m, n)$.
\[main\_result\]
1. For $l \ge 3$, ${\operatorname{N}}(l; 2, 2)$ is rational.
2. For $l \ge 3$ and $n \ge 1$, ${\operatorname{N}}(l; n, n+1)$ is rational.
According to [@drezet_reine Lemma 25], ${\operatorname{N}}(3; 2, 2)$ is isomorphic to ${\mathbb P}^5$. Identifying ${\mathbb P}^5$ with the space of conic curves in ${\mathbb P}^2$, the stable points correspond to irreducible conics. Applying Lemma \[projection\], yields (i).
According to [@modules_alternatives Propositions 4.5 and 4.6], the subset of ${\operatorname{N}}(3; n, n+1)$ of matrices whose maximal minors have no common factor is isomorphic to the subset of ${\operatorname{Hilb}}_{{\mathbb P}^2}(n(n+1)/2)$ of schemes that are not contained in any curve of degree $n-1$. Thus, ${\operatorname{N}}(3; n, n+1)$ is birational to ${\operatorname{Hilb}}_{{\mathbb P}^2}(n(n+1)/2)$, so it is rational. Moreover, ${\operatorname{N}}(3; n, n+1)$ consists only of stable points. Applying Lemma \[projection\], yields (ii).
\[square\_rational\] For $l \ge 3$ and $n \ge 1$, ${\operatorname{N}}(l; n, n)$ is a rational variety.
The argument is inspired by [@asterisque Remark 3.9]. In view of [@drezet_reine Section 3], ${\operatorname{N}}(3; n, n)$ contains stable points. This is due to the fact that we have the inequality $x < n/n < 1/x$, where $x$ is the smaller solution to the equation $x^2 - 3x +1 = 0$. Thus, we are in the context of Lemma \[projection\], which asserts that ${\operatorname{N}}(l; n, n)$ is rational for $l \ge 3$ if ${\operatorname{N}}(3; n, n)$ is rational. We may, therefore, restrict to the case when $l = 3$. Let $V$ be a vector space over ${\mathbb C}$ with basis $\{ x, y, z \}$. An element ${\varphi}\in {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)$ can be written uniquely in the form ${\varphi}= {\varphi}_1 x + {\varphi}_2 y + {\varphi}_3 z$, where ${\varphi}_1, {\varphi}_2, {\varphi}_3 \in {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n)$. Let $${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)_0 \subset {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)^{{\scriptstyle \operatorname{s}}}$$ be the open invariant subset given by the condition that ${\varphi}_1$ be invertible. Let $X \subset {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)_0$ be the closed subset given by the condition ${\varphi}_1 = I$. The group ${\operatorname{PGL}}(n, {\mathbb C})$ acts on $X$ by conjugation. The composite map $$X \hookrightarrow {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)_0 {\longrightarrow}{\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)_0/G$$ is surjective and its fibers are precisely the ${\operatorname{PGL}}(n, {\mathbb C})$-orbits. Thus, it factors through a bijective morphism $$X/{\operatorname{PGL}}(n, {\mathbb C}) {\longrightarrow}{\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}V)_0/G.$$ In characteristic zero, bijective morphisms of irreducible varieties are birational. We have reduced to the following problem. Let $U$ be a complex vector space of dimension $2$ and let ${\operatorname{PGL}}(n, {\mathbb C})$ act on ${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)$ by conjugation. Then the resulting good quotient is rational.
Choose a basis $\{ y, z \}$ of $U$. An element $\psi \in {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)$ can be uniquely written in the form $\psi = y \psi_1 + z \psi_2$, where $\psi_1, \psi_2 \in {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n)$. Let $${\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)_0 \subset {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)$$ be the open invariant subset given by the conditions that $\psi$ have trivial stabilizer and that $\psi_1$ be invertible and have distinct eigenvalues. Let $Y \subset {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)_0$ be the closed subset given by the condition that $\psi_1$ be a diagonal matrix. Let $S, T \subset {\operatorname{PGL}}(n, {\mathbb C})$ be the image of the canonical embedding of the group of permutations of $n$ elements, respectively, the subgroup of diagonal matrices. Then $H = S T$ is a closed subgroup of ${\operatorname{PGL}}(n, {\mathbb C})$ leaving $Y$ invariant. The composite map $$Y \hookrightarrow {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)_0 {\longrightarrow}{\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)_0/{\operatorname{PGL}}(n, {\mathbb C})$$ is surjective and its fibers are precisely the $H$-orbits. Thus, it factors through a bijective morphism $$Y/H \longrightarrow {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n {\otimes}U)_0/{\operatorname{PGL}}(n, {\mathbb C})$$ that must be birational. We have reduced the problem to showing that $Y/H$ is rational.
Let $Y_0 \subset Y$ be the open $H$-invariant subset given by the condition that all entries of $\psi_2$ be non-zero. Concretely, $Y_0 = D \times E$, where $D, E \subset {\operatorname{Hom}}({\mathbb C}^n, {\mathbb C}^n)$ are the subset of invertible diagonal matrices with distinct entries on the diagonal, respectively, the subset of matrices without zero entries. The normal subgroup $T \le H$ acts trivially on $D$, hence $(D \times E)/T$ is a trivial bundle over $D$ with fiber $E/T$. The induced action of $S = H/T$ is compatible with the bundle structure. The stabilizer in $S$ of any $\psi_1 \in D$ acts trivially on the fiber over $\psi_1$, because it is trivial. It follows that $(D \times E)/T$ descents to a fiber bundle $F$ over $D/S$. Clearly, $F$ is isomorphic to $(D \times E)/H$, hence $(D \times E)/H$ is birational to $D/S \times E/T$. Both $D/S$ and $E/T$ are rational, namely $D/S$ is isomorphic to an open subset of ${\operatorname{S}}^n (\AA^1) {\simeq}\AA^n$, while $E/T {\simeq}(\AA^1 \setminus \{ 0 \})^{n^2-n+1}$. In conclusion, $Y/H$ is rational.
Let $r > 0$ and $\chi$ be integers. Let ${\operatorname{M}}_{{\mathbb P}^2}(r, \chi)$ denote the moduli space of Gieseker semi-stable sheaves on ${\mathbb P}^2$ having Hilbert polynomial $P(m) = rm+\chi$. It is well known that ${\operatorname{M}}_{{\mathbb P}^2}(r, 0)$ is birational to ${\operatorname{N}}(3; r, r)$ and, if $r$ is even, ${\operatorname{M}}_{{\mathbb P}^2}(r, r/2)$ is birational to ${\operatorname{N}}(6; r/2, r/2)$. We obtain the following.
The moduli spaces ${\operatorname{M}}_{{\mathbb P}^2}(r, 0)$ and, if $r$ is even, ${\operatorname{M}}_{{\mathbb P}^2}(r, r/2)$, are rational.
The rationality of ${\operatorname{M}}_{{\mathbb P}^2}(3, 0)$ and ${\operatorname{M}}_{{\mathbb P}^2}(4, 2)$ is already known from [@lepotier_revue].
The maps $$\det \colon {\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}V) {\longrightarrow}{\operatorname{S}}^2 V, \qquad \det({\varphi}) = {\varphi}_{11} {\varphi}_{22} - {\varphi}_{12} {\varphi}_{21},$$ and $${\operatorname{e}}\colon {\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}V) {\longrightarrow}\Lambda^4 V, \qquad {\operatorname{e}}({\varphi}) = {\varphi}_{11} \wedge {\varphi}_{22} \wedge {\varphi}_{12} \wedge {\varphi}_{21}$$ are semi-invariant in the sense that for any $(g, h) \in G$ and ${\varphi}\in {\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}V)$, $$\det((g, h) {\varphi}) = \det(g)^{-1} \det(h) \det({\varphi}), \qquad {\operatorname{e}}((g, h) {\varphi}) = \det(g)^{-2} \det(h)^2 {\operatorname{e}}({\varphi}).$$ Using King’s criterion of semi-stability [@king], it is easy to see that ${\varphi}$ is semi-stable if and only if $\det({\varphi}) \neq 0$ and is stable if and only if $\det({\varphi})$ is irreducible in ${\operatorname{S}}^* V$. In the case when $\dim(V) = 3$, the isomorphism ${\operatorname{N}}(V; 2, 2) \to {\mathbb P}({\operatorname{S}}^2 V)$ of [@drezet_reine] is given by $\langle {\varphi}\rangle \mapsto \langle \det({\varphi}) \rangle$.
In the sequel we will assume that $\dim(V) = 4$ and that $m = 2$, $n = 2$. Choose bases $\{ x, y, z, w \}$ of $V$ and $\{ v_1, v_2, v_3, v_4 \}$ of $V^*$. Consider the semi-invariant functions $$\epsilon, \rho \colon {\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}V) {\longrightarrow}{\mathbb C}, \qquad \epsilon({\varphi}) = {\operatorname{i}}_{v_1 \wedge v_2 \wedge v_3 \wedge v_4} {\operatorname{e}}({\varphi}),$$ $$\rho({\varphi}) = {\operatorname{i}}_{v_1 \wedge v_2 \wedge v_3 \wedge v_4}
({\operatorname{i}}_{v_1} \det({\varphi}) \wedge {\operatorname{i}}_{v_2} \det({\varphi}) \wedge {\operatorname{i}}_{v_3} \det({\varphi}) \wedge {\operatorname{i}}_{v_4} \det({\varphi})).$$ Here ${\operatorname{i}}_v$ denotes the internal product with a vector $v \in V^*$.
\[functional\_relation\] We have the relation $\epsilon^2 = \rho$.
Let $\{ v_1', v_2', v_3', v_4' \}$ be another basis of $V^*$ and let $\upsilon \in {\operatorname{GL}}(4, {\mathbb C})$ be the change-of-basis matrix. With respect to this basis we define the functions $\rho'$ and $\epsilon'$ as above. Then $\epsilon'({\varphi}) = \det(\upsilon) \epsilon({\varphi})$ and $\rho'({\varphi}) = \det(\upsilon)^2 \rho({\varphi})$, hence $\epsilon({\varphi})^2 = \rho({\varphi})$ if and only if $\epsilon'({\varphi})^2 = \rho'({\varphi})$. Put $U = {\operatorname{span}}\{ x, y, z \}$ and let $$\pi \colon {\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}V) {\longrightarrow}{\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}U)$$ be the morphism induced by the projection of $V = U \oplus {\mathbb C}w$ onto the first factor. It is enough to verify the relation on the Zariski open subset given by the condition that $\det(\pi({\varphi}))$ be irreducible. Changing, possibly, the basis of $U$, we may assume that $\det(\pi({\varphi})) = x^2 - yz$. Since $\pi({\varphi})$ is stable, and since ${\operatorname{N}}(U; 2, 2)$ is isomorphic to ${\mathbb P}({\operatorname{S}}^2 U)$, we have $$\pi({\varphi}) \sim \left[
\begin{array}{cc}
x & y \\
z & x
\end{array}
\right], \quad \text{so we may write} \quad
{\varphi}= \left[
\begin{array}{cc}
x + aw & y + bw \\
z + cw & x + dw
\end{array}
\right].$$ We have $$\det({\varphi}) = x^2 - yz + (a+d)xw - cyw - bzw + (ad-bc)w^2,$$ $${\operatorname{e}}({\varphi}) = (d-a) x \wedge y \wedge z \wedge w.$$ Since we are free to choose the basis of $V^*$, we choose $\{ v_1, v_2, v_3, v_4 \}$ to be the dual of $\{x, y, z, w\}$. We have $$\begin{aligned}
{\operatorname{i}}_{v_1} \det({\varphi}) & = \frac{\partial}{\partial x} \det({\varphi}) = 2x + (a+d)w, \\
{\operatorname{i}}_{v_2} \det({\varphi}) & = \frac{\partial}{\partial y} \det({\varphi}) = -z - cw, \\
{\operatorname{i}}_{v_3} \det({\varphi}) & = \frac{\partial}{\partial z} \det({\varphi}) = -y - bw, \\
{\operatorname{i}}_{v_4} \det({\varphi}) & = \frac{\partial}{\partial w} \det({\varphi}) = (a+d)x - cy - bz + 2(ad - bc)w,\end{aligned}$$ $$\epsilon({\varphi}) = d - a, \qquad \rho({\varphi}) = \left|
\begin{array}{cccc}
2 & 0 & 0 & a + d \\
0 & 0 & -1 & -c \\
0 & -1 & 0 & -b \\
a + d & -c & -b & 2(ad - bc)
\end{array}
\right| = (a-d)^2.$$ In conclusion, $\epsilon({\varphi})^2 = (d - a)^2 = \rho({\varphi})$.
Consider the action of ${\mathbb C}^*$ on ${\operatorname{S}}^2 V \oplus \Lambda^4 V$ given by $t (q, p) = (tq, t^2 p)$ and let ${\mathbb P}$ denote the weighted projective space $\big( ({\operatorname{S}}^2 V \oplus \Lambda^4 V) \setminus \{ 0 \} \big)/{\mathbb C}^*$. Consider the map $$\eta \colon {\operatorname{N}}(V; 2, 2) {\longrightarrow}{\mathbb P}, \qquad \eta(\langle {\varphi}\rangle) = \langle \det({\varphi}), {\operatorname{e}}({\varphi}) \rangle.$$ Choose coordinates on ${\mathbb P}$ given by the choice of basis $\{ x, y, z, w \}$ of $V$. In view of Proposition \[functional\_relation\], the image of $\eta$ is contained in the hypersurface $H \subset {\mathbb P}$ given by the equation ${\operatorname{res}}(q) = p^2$, where ${\operatorname{res}}(q)$ denotes the resultant of the quadratic form $q$.
\[hypersurface\] Assume that $\dim(V) = 4$. Then the map $\eta \colon {\operatorname{N}}(V; 2, 2) \to H$ is an isomorphism.
The singular points of the cone $\hat{H} \subset {\operatorname{S}}^2 V \oplus \Lambda^4 V$ over $H$ are of the form $(q, 0)$, where $q \in {\operatorname{S}}^2 V$ is a singular point of the vanishing locus of the resultant. It follows that $\hat{H}$ is regular in codimension $1$. From Serre’s criterion of normality we deduce that $H$ is normal (condition S2 is satisfied because $\hat{H}$ is a hypersurface in a smooth variety). Normality is inherited by a good quotient, hence $H = (\hat{H} \setminus \{ 0 \})/{\mathbb C}^*$ is normal, too. In view of the Main Theorem of Zariski, it is enough to show that $\eta$ is bijective. Since ${\operatorname{N}}(V; 2, 2)$ is complete, and since ${\operatorname{N}}(V; 2, 2)$ and $H$ are irreducible of the same dimension, it is enough to show that $\eta$ is injective.
Assume that $\eta(\langle {\varphi}_1 \rangle) = \eta(\langle {\varphi}_2 \rangle)$. Varying ${\varphi}_1$ and ${\varphi}_2$ in their respective orbits, we may assume that $\det({\varphi}_1) = \det({\varphi}_2)$ and ${\operatorname{e}}({\varphi}_1) = {\operatorname{e}}({\varphi}_2)$. If $\det({\varphi}_1)$ is reducible, say $\det({\varphi}_1) = u u'$ for some $u, u' \in V$, then it is easy to see that $${\varphi}_1 \sim \left[
\begin{array}{cc}
u & u_1 \\
0 & u'
\end{array}
\right], \qquad {\varphi}_2 \sim \left[
\begin{array}{cc}
u & u_2 \\
0 & u'
\end{array}
\right]$$ for some $u_1, u_2 \in V$. But then $\langle {\varphi}_1 \rangle = \langle {\varphi}_2 \rangle = \langle {\operatorname{diag}}(u, u') \rangle$. Assume now that $\det({\varphi}_1)$ is irreducible. There exists a vector $w \in V$ and a subspace $U \subset V$ such that $V = U \oplus {\mathbb C}w$ and $\det(\pi({\varphi}_1))$ is irreducible (notations as at Proposition \[functional\_relation\]). As mentioned at Proposition \[functional\_relation\], we may choose a basis $\{ x, y, z \}$ of $U$ such that $\det(\pi({\varphi}_1))= x^2 - yz$, forcing $$\pi({\varphi}_1) \sim \pi({\varphi}_2) \sim \left[
\begin{array}{cc}
x & y \\
z & x
\end{array}
\right].$$ Thus, we may write $${\varphi}_1 = \left[
\begin{array}{cc}
x + a_1 w & y + b_1 w \\
z + c_1 w & x + d_1 w
\end{array}
\right], \qquad {\varphi}_2 = \left[
\begin{array}{cc}
x + a_2 w & y + b_2 w \\
z + c_2 w & x + d_2 w
\end{array}
\right].$$ The relation $\det({\varphi}_1) = \det({\varphi}_2)$ yields the relations $b_1 = b_2$, $c_1 = c_2$, $a_1 + d_1 = a_2 + d_2$. The relation ${\operatorname{e}}({\varphi}_1) = {\operatorname{e}}({\varphi}_2)$ yields the relation $a_1 - d_1 = a_2 - d_2$. We conclude that ${\varphi}_1 = {\varphi}_2$, hence $\langle {\varphi}_1 \rangle = \langle {\varphi}_2 \rangle$.
\[double\_cover\] It was already known to Le Potier [@asterisque Remark 3.8] that the map $$\det \colon {\operatorname{N}}(V; 2, 2) {\longrightarrow}{\mathbb P}({\operatorname{S}}^2 V)$$ is a double cover branched over the locus of singular quadratic surfaces.
In the sequel, we will use the abbreviations $\O(r, s) = \O_{{\mathbb P}^1 \times {\mathbb P}^1}(r, s)$, $\omega = \omega_{{\mathbb P}^1 \times {\mathbb P}^1}$, and ${\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}= {\mathcal Ext}^1_{\O}({\mathcal F}, \omega)$ for a sheaf ${\mathcal F}$ on ${\mathbb P}^1 \times {\mathbb P}^1$ of dimension $1$. We quote below [@chung_moon Proposition 3.8].
\[resolutions\] The sheaves ${\mathcal F}$ giving points in ${\mathbf M}$ are precisely the sheaves having one of the following three types of resolution: $$\label{type_0}
0 {\longrightarrow}2\O(-1, -1) \overset{{\varphi}}{{\longrightarrow}} 2\O {\longrightarrow}{\mathcal F}{\longrightarrow}0,$$ $$\label{type_1}
0 {\longrightarrow}\O(-2, -1) {\longrightarrow}\O(0, 1) {\longrightarrow}{\mathcal F}{\longrightarrow}0,$$ $$\label{type_2}
0 {\longrightarrow}\O(-1, -2) {\longrightarrow}\O(1, 0) {\longrightarrow}{\mathcal F}{\longrightarrow}0.$$
This proposition was proved in [@chung_moon] by the wall-crossing method, however, it was also nearly obtained in [@ballico_huh]. At [@ballico_huh Lemma 20] it is mistakenly claimed that all sheaves in ${\mathbf M}$ have resolution (\[type\_0\]). At a closer inspection, the argument of [@ballico_huh Lemma 20] shows that the sheaves in ${\mathbf M}$ satisfying the conditions $\H^0({\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(1, 0)) = 0$ and $\H^0({\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(0, 1)) = 0$ are precisely the sheaves given by resolution (\[type\_0\]). Indeed, the exact sequence (50) in [@ballico_huh] reads $$\label{ballico_huh_sequence}
0 {\longrightarrow}{\mathcal H} {\longrightarrow}2\O {\longrightarrow}{\mathcal F}{\longrightarrow}0,$$ where ${\mathcal H}$ is a locally free sheaf of rank $2$ and determinant $\omega$. Dualizing this sequence, we get the exact sequence $$0 {\longrightarrow}2\O(-2, -2) {\longrightarrow}{\mathcal H}^{{\scriptscriptstyle \operatorname{D}}}{\simeq}{\mathcal H}^* {\otimes}\omega {\simeq}{\mathcal H} {\otimes}\det({\mathcal H})^* {\otimes}\omega {\simeq}{\mathcal H} {\longrightarrow}{\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}{\longrightarrow}0.$$ From this we get the relations $${\operatorname{h}}^1({\mathcal H}(1, 0)) = {\operatorname{h}}^0({\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(1, 0)) \quad \text{and} \quad {\operatorname{h}}^1({\mathcal H}(0, 1)) = {\operatorname{h}}^0({\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(0, 1)).$$ The vanishing of $\H^1({\mathcal H}(1, 0))$ and $\H^1({\mathcal H}(0, 1))$ implies that ${\mathcal H} {\simeq}2\O(-1, -1)$, in which case (\[ballico\_huh\_sequence\]) yields resolution (\[type\_0\]).
According to [@rendiconti Theorem 13], if ${\mathcal F}$ gives a point in ${\mathbf M}$, then ${\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(0, 1)$ and ${\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(1, 0)$ give points in the moduli space ${\mathbf M}'$ of semi-stable sheaves on ${\mathbb P}^1 \times {\mathbb P}^1$ having Hilbert polynomial $P(m) = 4m$ and first Chern class $c_1 = (2, 2)$. We claim that the sheaves ${\mathcal E}$ giving points in ${\mathbf M}'$ and satisfying the condition $\H^0({\mathcal E}) \neq 0$ are precisely the structure sheaves of curves $E \subset {\mathbb P}^1 \times {\mathbb P}^1$ of type $(2, 2)$. By the argument of [@ballico_huh Lemma 9], $\O_E$ gives a stable point in ${\mathbf M}'$. Conversely, if ${\mathcal E}$ gives a point in ${\mathbf M}'$ and $\H^0({\mathcal E}) \neq 0$, then, by the argument of [@dedicata Proposition 2.1.3], there is an injective morphism $\O_C \to {\mathcal E}$ for a curve $C \subset {\mathbb P}^1 \times {\mathbb P}^1$. If $C$ did not have type $(2, 2)$, then the semi-stability of ${\mathcal E}$ would get contradicted. Thus, $C$ has type $(2, 2)$ and, comparing Hilbert polynomials, we see that $\O_C {\simeq}{\mathcal E}$. In conclusion, if $\H^0({\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(0, 1)) \neq 0$, then ${\mathcal F}{\simeq}\O_E(0, -1)^{{\scriptscriptstyle \operatorname{D}}}{\simeq}\O_E(0, 1)$, hence ${\mathcal F}$ has resolution (\[type\_1\]). If $\H^0({\mathcal F}^{{\scriptscriptstyle \operatorname{D}}}(1, 0)) \neq 0$, then ${\mathcal F}{\simeq}\O_E(-1, 0)^{{\scriptscriptstyle \operatorname{D}}}{\simeq}\O_E(1, 0)$, hence ${\mathcal F}$ has resolution (\[type\_2\]).
We denote by ${\mathbf M}_0, {\mathbf M}_1, {\mathbf M}_2 \subset {\mathbf M}$ the subsets of sheaves given by resolution (\[type\_0\]), (\[type\_1\]), respectively, (\[type\_2\]). Clearly, ${\mathbf M}_0$ is open and ${\mathbf M}_1$, ${\mathbf M}_2$ are divisors isomorphic to ${\mathbb P}^8$. Let ${\operatorname{Hom}}(2\O(-1,-1), 2\O)_0$ denote the subset of injective morphisms.
\[generic\_quotient\] The canonical map from below is a good quotient modulo $G$: $$\gamma \colon {\operatorname{Hom}}(2\O(-1,-1), 2\O)_0 {\longrightarrow}{\mathbf M}_0, \qquad \gamma({\varphi}) = \langle {\mathcal Coker}({\varphi}) \rangle.$$
According to [@buchdahl Lemma 1], for any coherent sheaf ${\mathcal F}$ on ${\mathbb P}^1 \times {\mathbb P}^1$ there is a spectral sequence converging to ${\mathcal F}$ in degree zero and to $0$ in degrees different from zero, similar to the Beilinson spectral sequence. Its first level $\operatorname{E}_1^{ij}$ is given by $$\operatorname{E}_1^{ij} = 0 \quad \text{if $i > 0$ or $i < -2$},$$ $$\operatorname{E}_1^{0j} = \H^j({\mathcal F}) {\otimes}\O, \quad \operatorname{E}_1^{-2,j} = \H^j({\mathcal F}(-1,-1)) {\otimes}\O(-1, -1),$$ and by the exact sequences $$\H^j({\mathcal F}(0, -1)) {\otimes}\O(0, -1) {\longrightarrow}\operatorname{E}_1^{-1,j} {\longrightarrow}\H^j({\mathcal F}(-1, 0)) {\otimes}\O(-1, 0).$$ If ${\mathcal F}$ gives a point in ${\mathbf M}_0$, then $$\H^0({\mathcal F}) {\simeq}{\mathbb C}^2, \quad \H^1({\mathcal F}) = 0, \quad \H^0({\mathcal F}(-1,-1)) = 0, \quad \H^1({\mathcal F}(-1, -1)) {\simeq}{\mathbb C}^2,$$ $$\H^0({\mathcal F}(0, -1)) = 0, \quad \H^1({\mathcal F}(0, -1)) = 0, \quad \H^0({\mathcal F}(-1, 0)) = 0, \quad \H^1({\mathcal F}(-1, 0)) = 0.$$ Thus, $\operatorname{E}_1$ has only two non-zero terms: $\operatorname{E}_1^{-2, 1} = 2 \O(-1, -1)$ and $\operatorname{E}_1^{0,0} = 2\O$. The relevant part of $\operatorname{E}_2$ is represented in the following table: $$\xymatrix
{
2 \O(-1, -1) \ar[drr]^-{{\varphi}} & 0 & 0 \\
0 & 0 & 2\O
}$$ The sequence degenerates at $\operatorname{E}_3$, hence ${\varphi}$ is injective and ${\mathcal Coker}({\varphi}) {\simeq}{\mathcal F}$. This shows that resolution (\[type\_0\]) can be obtained from the Beilinson spectral sequence of ${\mathcal F}$. Arguing as at [@dedicata Theorem 3.1.6], we can see that resolution (\[type\_0\]) can be obtained for local flat families of sheaves in ${\mathbf M}_0$, hence $\gamma$ is a categorical quotient. By the uniqueness of the categorical quotient, we deduce that $\gamma$ is a good quotient map.
We fix vector spaces $V_1$ and $V_2$ over ${\mathbb C}$ of dimension $2$ and we make the identifications $${\mathbb P}^1 \times {\mathbb P}^1 = {\mathbb P}(V_1) \times {\mathbb P}(V_2), \quad \H^0(\O(r, s)) = {\operatorname{S}}^r V_1^* {\otimes}{\operatorname{S}}^s V_2^*, \quad V = V_1^* {\otimes}V_2^*.$$ Let $${\mathbf W}\subset {\operatorname{Hom}}\big(2\O(-1, -1) \oplus \O(-1, 0) \oplus \O(0, -1), \ \O(-1, 0) \oplus \O(0, -1) \oplus 2\O \big)$$ be the open subset of injective morphisms $\psi$ for which ${\mathcal Coker}(\psi)$ is Gieseker semi-stable. We represent $\psi$ by a matrix $$\psi = \left[
\begin{array}{c:c}
\psi_{11} & \psi_{12} \\
\hdashline
\psi_{21} & \psi_{22}
\end{array}
\right] = \left[
\begin{array}{cc:cc}
1 {\otimes}u_{12} & 1 {\otimes}v_{12} & a_1 & 0 \\
u_{11} {\otimes}1 & v_{11} {\otimes}1 & 0 & a_2 \\
\hdashline
f_{11} & f_{12} & u_{21} {\otimes}1 & 1 {\otimes}u_{22} \\
f_{21} & f_{22} & v_{21} {\otimes}1 & 1 {\otimes}v_{22}
\end{array}
\right],$$ where $a_1, a_2 \in {\mathbb C}$, $u_{ij}, v_{ij} \in V_j^*$, $f_{ij} \in V$. The algebraic group $${\mathbf G}= \big( {\operatorname{Aut}}(2\O(-1, -1) \oplus \O(-1, 0) \oplus \O(0, -1)) \times {\operatorname{Aut}}(\O(-1, 0) \oplus \O(0, -1) \oplus 2\O) \big)/{\mathbb C}^*$$ acts on ${\mathbf W}$ by conjugation. We represent elements of ${\mathbf G}$ by pairs $(g, h)$, where $$g = \left[
\begin{array}{cc}
g_{11} & 0 \\
g_{21} & g_{22}
\end{array}
\right], \qquad h = \left[
\begin{array}{cc}
h_{11} & 0 \\
h_{21} & h_{22}
\end{array}
\right],$$ $g_{11} \in {\operatorname{Aut}}(2\O(-1, -1))$, $h_{22} \in {\operatorname{Aut}}(2\O)$, etc.
\[global\_quotient\] The canonical map $\theta \colon {\mathbf W}\to {\mathbf M}$, $\theta(\psi) = \langle {\mathcal Coker}(\psi) \rangle$ is a good quotient modulo ${\mathbf G}$.
Let ${\mathbf W}_0 \subset {\mathbf W}$ be the open subset given by the condition that $\psi_{12}$ be invertible. Concretely, ${\mathbf W}_0$ is the set of morphisms $\psi$ such that $\psi_{12}$ is invertible and $\alpha(\psi) = \psi_{21} - \psi_{22} \psi_{12}^{-1} \psi_{11}$ is injective. In view of Proposition \[resolutions\], ${\mathbf M}_0 = \theta({\mathbf W}_0)$. The restricted map $\theta_0 \colon {\mathbf W}_0 \to {\mathbf M}_0$ is the composition $${\mathbf W}_0 \overset{\alpha}{{\longrightarrow}} {\operatorname{Hom}}(2\O(-1, -1), 2\O)_0 \overset{\gamma}{{\longrightarrow}} {\mathbf M}_0,$$ where $\gamma$ is the good quotient map from Corollary \[generic\_quotient\]. Let ${\mathbf G}_0 \unlhd {\mathbf G}$ be the closed normal subgroup given by the conditions $g_{11} = c I$, $h_{22} = c I$, $c \in {\mathbb C}^*$. We have the relation $\alpha(h \psi g^{-1}) = h_{22}^{} \alpha(\psi) g_{11}^{-1}$, hence $\alpha$ is constant on the orbits of ${\mathbf G}_0$. Since any $\psi \in {\mathbf W}_0$ is equivalent to $$\left[
\begin{array}{cc}
0 & I \\
\alpha(\psi) & 0
\end{array}
\right],$$ it follows that the fibers of $\alpha$ are precisely the ${\mathbf G}_0$-orbits, and that $\alpha$ has a section. We deduce that $\alpha$ is a geometric quotient modulo ${\mathbf G}_0$. Since $\gamma$ is a good quotient modulo ${\mathbf G}/{\mathbf G}_0$, we conclude that $\theta_0$ is a good quotient modulo ${\mathbf G}$. Let ${\mathbf M}^{{\scriptstyle \operatorname{s}}}_0 \subset {\mathbf M}_0$ be the subset of stable points. Since $\gamma^{-1}({\mathbf M}_0^{{\scriptstyle \operatorname{s}}}) \to {\mathbf M}_0^{{\scriptstyle \operatorname{s}}}$ is a geometric quotient modulo ${\mathbf G}/{\mathbf G}_0$, we deduce that $\theta^{-1}({\mathbf M}_0^{{\scriptstyle \operatorname{s}}}) \to {\mathbf M}_0^{{\scriptstyle \operatorname{s}}}$ is a geometric quotient modulo ${\mathbf G}$.
Assume now that $\psi \in {\mathbf W}\setminus {\mathbf W}_0$. Denote ${\mathcal F}= {\mathcal Coker}(\psi)$. Then $\psi_{12} \neq 0$, otherwise ${\mathcal Coker}(\psi_{22})$ would be a destabilizing subsheaf of ${\mathcal F}$. Thus, ${\mathbf W}\setminus {\mathbf W}_0$ is the disjoint union of two subsets ${\mathbf W}_1$ and ${\mathbf W}_2$. The former is given by the relations $a_1 \neq 0$, $a_2 = 0$; the latter is given by the relations $a_1 = 0$, $a_2 \neq 0$. Assume that $\psi \in {\mathbf W}_1$. Then $u_{11}$, $v_{11}$ are linearly independent, otherwise ${\mathcal F}$ would have a destabilizing quotient sheaf of slope zero. Likewise, $u_{22}$, $v_{22}$ are linearly independent, otherwise ${\mathcal F}$ would have a destabilizing subsheaf of slope $1$. Consider the morphism $$\xi \in {\operatorname{Hom}}(2\O(-1,-1) \oplus \O(0, -1), \ \O(0, -1) \oplus 2\O),$$ $$\xi = \left[
\begin{array}{ccc}
u_{11} {\otimes}1 & v_{11} {\otimes}1 & 0 \\
f_{11} - a_1^{-1} u_{21} {\otimes}u_{12} & f_{12} - a_1^{-1} u_{21} {\otimes}v_{12} & 1 {\otimes}u_{22} \\
f_{21} - a_1^{-1} v_{21} {\otimes}u_{12} & f_{22} - a_1^{-1} v_{21} {\otimes}v_{12} & 1 {\otimes}v_{22}
\end{array}
\right].$$ Clearly, ${\mathcal F}{\simeq}{\mathcal Coker}(\xi)$. Applying the snake lemma to the exact diagram $${\small
\xymatrix@C+=28pt
{
& 0 \ar[d] & 0 \ar[d]
\\
0 \ar[r] & \O(0, -1) \ar[d] \ar[r]^-{\tiny \left[ \!\!\! \begin{array}{c} 1 {\otimes}u_{22} \\ 1 {\otimes}v_{22} \end{array} \!\!\! \right]} &
2\O \ar[d] \ar@{->>}[rr]^-{\tiny \left[ \!\!\! \begin{array}{cc} -1 {\otimes}v_{22} & \!\!\! 1 {\otimes}u_{22} \end{array} \!\!\! \right]} & & \O(0, 1)
\\
0 \ar[r] & 2\O(-1, -1) \oplus \O(0, -1) \ar[d] \ar[r]^-{\xi} & \O(0, -1) \oplus 2\O \ar@{->>}[rr] \ar[d] & & {\mathcal F}\\
\O(-2, -1) \ar@{^{(}->}[r]^-{\tiny \left[ \!\!\! \begin{array}{r} -v_{11} {\otimes}1 \\ u_{11} {\otimes}1 \end{array} \!\!\! \right]} &
2\O(-1, -1) \ar[d] \ar[r]^-{\tiny \left[ \!\!\! \begin{array}{cc} u_{11} {\otimes}1 & \!\! v_{11} {\otimes}1 \end{array} \!\!\! \right]} & \O(0, -1) \ar[rr] \ar[d] & & 0
\\
& 0 & 0
}}$$ we obtain resolution (\[type\_1\]). This shows that $\theta({\mathbf W}_1) \subset {\mathbf M}_1$. It is now easy to see that the restricted map ${\mathbf W}_1 \to {\mathbf M}_1$ is surjective and that its fibers are precisely the ${\mathbf G}$-orbits. By symmetry, the same is true of the restricted map ${\mathbf W}_2 \to {\mathbf M}_2$.
Let ${\mathbf M}^{{\scriptstyle \operatorname{s}}}\subset {\mathbf M}$ be the open subset of stable points and ${\mathbf W}^{{\scriptstyle \operatorname{s}}}= \theta^{-1}({\mathbf M}^{{\scriptstyle \operatorname{s}}})$. We have proved above that the fibers of the restricted map $\theta^{{\scriptstyle \operatorname{s}}}\colon {\mathbf W}^{{\scriptstyle \operatorname{s}}}\to {\mathbf M}^{{\scriptstyle \operatorname{s}}}$ are precisely the ${\mathbf G}$-orbits. Since ${\mathbf M}^{{\scriptstyle \operatorname{s}}}$ is normal (being smooth), we can apply [@popov_vinberg Theorem 4.2] to deduce that $\theta^{{\scriptstyle \operatorname{s}}}$ is a geometric quotient modulo ${\mathbf G}$. Finally, since ${\mathbf M}= {\mathbf M}_0 \cup {\mathbf M}^{{\scriptstyle \operatorname{s}}}$, we deduce that $\theta$ is a good quotient map.
Choose bases $\{ u_1, v_1 \}$ of $V_1^*$ and $\{ u_2, v_2 \}$ of $V_2^*$. Then $x = u_1 {\otimes}u_2$, $y = v_1 {\otimes}u_2$, $z = u_1 {\otimes}v_2$, $w = v_1 {\otimes}v_2$ form a basis of $V$. An easy calculation shows that the set of injective morphisms $${\operatorname{Hom}}(2\O(-1, -1), 2\O)_0 \subset {\operatorname{Hom}}({\mathbb C}^2, {\mathbb C}^2 {\otimes}V)$$ is the subset of matrices whose determinant is not a multiple of $xw - yz$. Thus, $${\operatorname{Hom}}(2\O(-1, -1), 2\O)_0/\!\!/G {\simeq}{\operatorname{N}}(V; 2, 2) \setminus {\det}^{-1} \{ \langle xw - yz \rangle \}.$$ According to Remark \[double\_cover\], $\det^{-1} \{ \langle xw - yz \rangle \}$ consists of two points $\nu_1$ and $\nu_2$, where $\epsilon(\nu_1) = 1$, $\epsilon(\nu_2) = -1$. We saw at Corollary \[generic\_quotient\] that $\gamma$ induces an isomorphism $${\operatorname{Hom}}(2\O(-1, -1), 2\O)_0/\!\!/G {\longrightarrow}{\mathbf M}_0.$$ The inverse of this isomorphism is denoted by $$\beta_0 \colon {\mathbf M}_0 {\longrightarrow}{\operatorname{N}}(V; 2, 2) \setminus \{ \nu_1, \nu_2 \}.$$ It is natural to ask whether ${\mathbf M}$ is the blow-up of ${\operatorname{N}}(V; 2, 2)$ at $\nu_1$ and $\nu_2$. This is, indeed, one of the main results in [@chung_moon], where a blowing-down map $\beta \colon {\mathbf M}\to {\operatorname{N}}(V; 2, 2)$ is constructed via Fourier-Mukai transforms of sheaves, in view of the identification of ${\operatorname{N}}(V; 2, 2)$ with ${\operatorname{M}}_{{\mathbb P}^3}(m^2 + 3m + 2)$. We give below an alternate construction.
\[blowing\_down\] The map $\beta_0$ extends to a blowing-down map $\beta \colon {\mathbf M}\to {\operatorname{N}}(V; 2, 2)$ with exceptional divisor ${\mathbf M}_1 \cup {\mathbf M}_2$ and blowing-up locus $\{ \nu_1, \nu_2 \}$.
Recall that on ${\mathbf M}_0 = {\mathbf W}_0/\!\!/{\mathbf G}$, $\beta_0$ is induced by the map sending $\psi$ to $$\psi_{21} - a_1^{-1} \left[
\begin{array}{c}
u_{21} {\otimes}1 \\
v_{21} {\otimes}1
\end{array}
\right] \left[
\begin{array}{cc}
1 {\otimes}u_{12} & 1 {\otimes}v_{12}
\end{array}
\right] - a_2^{-1} \left[
\begin{array}{c}
1 {\otimes}u_{22} \\
1 {\otimes}v_{22}
\end{array}
\right] \left[
\begin{array}{cc}
u_{11} {\otimes}1 & v_{11} {\otimes}1
\end{array}
\right].$$ Equivalently, $\beta_0$ is induced by the map sending $\psi$ to $$a_2 \psi_{21} - a_1^{-1} a_2 \left[
\begin{array}{c}
u_{21} {\otimes}1 \\
v_{21} {\otimes}1
\end{array}
\right] \left[
\begin{array}{cc}
1 {\otimes}u_{12} & 1 {\otimes}v_{12}
\end{array}
\right] - \left[
\begin{array}{c}
1 {\otimes}u_{22} \\
1 {\otimes}v_{22}
\end{array}
\right] \left[
\begin{array}{cc}
u_{11} {\otimes}1 & v_{11} {\otimes}1
\end{array}
\right]$$ which is defined on ${\mathbf W}_0 \cup {\mathbf W}_1$. Clearly, this map factors through a morphism ${\mathbf M}_0 \cup {\mathbf M}_1 \to {\operatorname{N}}(V; 2, 2)$, which maps ${\mathbf M}_1$ to the class of the matrix $$\left[
\begin{array}{c}
1 {\otimes}u_2 \\
1 {\otimes}v_2
\end{array}
\right] \left[
\begin{array}{cc}
u_1 {\otimes}1 & v_1 {\otimes}1
\end{array}
\right] = \left[
\begin{array}{cc}
x & y \\
z & w
\end{array}
\right],$$ that is, to $\nu_1$. Analogously, $\beta_0$ extends to a morphism defined on ${\mathbf M}_0 \cup {\mathbf M}_2$, which maps ${\mathbf M}_2$ to the class of the matrix $$\left[
\begin{array}{c}
u_1 {\otimes}1 \\
v_1 {\otimes}1
\end{array}
\right] \left[
\begin{array}{cc}
1 {\otimes}u_2 & 1 {\otimes}v_2
\end{array}
\right] = \left[
\begin{array}{cc}
x & z \\
y & w
\end{array}
\right],$$ that is, to $\nu_2$. The two morphisms constructed thus far glue to a morphism $\beta \colon {\mathbf M}\to {\operatorname{N}}(V; 2, 2)$. Since $\nu_1$ and $\nu_2$ are smooth points, $\beta$ is a blow-down.
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---
abstract: 'Recently it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and propose a universal measure of complexity based on the Gell-Mann’s view of complexity. Our universal measure of complexity bases on a non-linear transformation of time-dependent entropy, where the system state with the highest complexity is the most distant from all the states of the system of lesser or no complexity. We have shown that the most complex is optimally mixed states consisting of pure states i.e., of the most regular and most disordered which the space of states of a given system allows. A parsimonious paradigmatic example of the simplest system with a small and a large number of degrees of freedom, is shown to support this methodology. Several important features of this universal measure are pointed out, especially its flexibility (i.e., its openness to extensions), ability to the analysis of a system critical behavior, and ability to study the dynamic complexity.'
author:
- |
Jaros[ł]{}aw Klamut\
Faculty of Physics\
University of Warsaw,\
Pasteur Str. 5\
PL-02-093 Warsaw, Poland\
Ryszard Kutner$^*$\
Faculty of Physics\
University of Warsaw,\
Pasteur Str. 5\
PL-02-093 Warsaw, Poland\
`ryszard.kutner@fuw.edu.pl` Zbigniew R. Struzik\
Faculty of Physics, University of Warsaw,\
Pasteur Str. 5, PL-02-093 Warsaw, Poland\
\
University of Tokyo, Bunkyo-ku, Tokyo 113-8655, Japan\
\
Advanced Center for Computing and Communication\
RIKEN, 2-1 Hirosawa, Wako 351-0198, Saitama, Japan
title: Towards a Universal Measure of Complexity
---
Introduction {#section:Introduction}
============
Analysis of the concept of complexity is a non-trivial task due to its diversity, arbitrariness, uncertainty, and contextual nature [@Nicolis; @JKSD; @Pincus; @Dorogov1; @AlBer; @Dorogov; @GrasPro; @RichMo; @PBEH]. There are many different levels/scales, faces, and types of complexity researched with very different technologies/techniques and tools [@XYZ; @Wiki1] (and refs. therein). In the context of dynamical systems, Grassberger has suggested [@PeGra] that a slow convergence of the entropy to its extensive asymptotic limit is a signature of complexity. This idea was materialized [@BNT; @PBR] further by information and statistical mechanics techniques. It generalizes many previous approaches to complexity, unifying physical ideas with ideas from learning and coding theory [@Borda]. There is even a connection of this approach to algorithmic or Kolmogorov complexity. The hidden pattern can be an essence of complexity [@JPC; @GreGol; @HGSchus; @HHL]. Technics adapted from the theories of information and computation have brought physical science (in particular, the region extended between classical determinism and deterministic chaos) to discovering hidden patterns and quantifying their dynamic structural complexity [@Wolfram].
We must remember that the complexity also depends on the imposed conditions (e.g., boundary or initial) as well as the restrictions adopted. It creates a challenge for every complexity study. It concerns the complexity that can appear in the movement of a single entity and collection of entities braided together. These entities can be as irreducible or straightforward systems as well as complex systems.
When we talk about complexity, we mean irreducible complexity which we can no longer be divided into smaller sub-complexities. We refer to this as a primary complexity. Considering the primary complexity here, we mean one that can be expressed at least in an algorithmic way – it is an effective complexity if it also contains a logic depth [@GMan; @GeMan; @GeManLl; @AMSz; @Jaguar]. We should take into account that our models (analytical and numerical) and theories describing reality are not fully deterministic. The evolution of a complex system is potentially multi-branched and the selection of an alternative trajectory (or branch selection) bases on randomly taken decisions.
One of the essential questions concerning a complex system is the problem of its stability/robustness and the question of stationarity of its evolution [@RKJM]. What is more, the relationship between complexity and disorder on the one hand, and complexity and pattern on the other are important questions – especially in the context of irreversible processes, where non-linear ones running away from equilibrium play a central role. Financial markets can be a spectacular example of these processes [@Sornet; @SorQui; @WSGKS; @WSzGKS; @KDWLGWKS; @AJ; @AJ2; @Ross1; @Ross2; @Zamb; @VSI].
To the central question: whether entropy is a direct measure of complexity, we answer negatively. In our opinion, the measure of complexity is appropriately, non-linearly transformed entropy. This work is devoted to finding this transformation.
Definition of a universal measure of complexity and its properties
==================================================================
In this Section, we translate the Gell-Mann general qualitative concept of complexity into the language of mathematics, and we present its consequences.
Gell-Mann concept of complexity {#section:complexity}
-------------------------------
The problem of defining a universal measure of complexity is urgent. For this work, the Gell-Mann concept [@GMan; @GelMan] of complexity is the inspirational starting point. We apply this concept to irreversible processes, by assuming that both [*a fully ordered as well as fully disordered systems cannot be the complex*]{}. The fully ordered system essentially has no complexity because of maximal possible symmetry of the system, but the fully disordered system contains no information as it entirely dissipates. Hence, the maximum of complexity should be sought somewhere in between these pure extreme states. This point of view allows introducing a formal quantitative phenomenological complexity measure based on entropy as a parameter of order [@Bert; @Sornet]. This measure reflects the dynamics of the system through the dependence of entropy on time. The vast majority of works analyzing the general aspects of complexity, including its basis, is based on information theory and computational analysis. Such an approach needs the supplementing with a provision allowing to return from a bit to physical representation – only then will one allow physical interpretations, including understanding of the causes of complexity.
We define the macroscopic phenomenological partial measure of complexity as a non-linear function of entropy $S$ of the order of $(m,n)$, $$\begin{aligned}
CX(S;m,n)&\stackrel{\rm def.}{=}&(S^{max} - S)^m(S-S^{min})^n \nonumber \\
&=&CX(S;m-1,n-1)\left[(S-S^{arit})^2-\left(\frac{Z}{2}\right)^2\right],~m,n \ge 1,
\label{rown:CXS}\end{aligned}$$ where $S^{min}$ and $S^{max}$ are minimal and maximal values of entropy $S$, respectively, $S^{arit}=\frac{S^{min}+S^{max}}{2}$, and the entropic span $Z\stackrel{\rm def.}{=}S^{max}-S^{min}$, whereas $m$ and $n$ are natural numbers[^1]. They define the order $(m,n)$ of the partial measure of complexity $CX$. Let us add that this formula is also met for mesoscopic scale. In other words, complexity appears in all systems for which entropy we can build. Notably, $S^{max}$ does not have to concern the state of thermodynamic equilibrium of the system. It may refer to the state for which entropy reaches its maximum value in the observed time interval. However, in this work, we are limited only to systems having a state of thermodynamic equilibrium. Below we discuss the definition (\[rown:CXS\]), indicating that it satisfies all properties of the measure of complexity. Of course, when $m=0$ and $n=1$ then $CX$ simply becomes $S-S^{min}$, i.e. the entropy of the system (the constant is not important here). However, when m = 1, n = 0, we get the information contained in the system (constant does not play a role here). Definition (1) gives us a lot more – showing this is the purpose of this work.
The partial measure of complexity given by Eq. (\[rown:CXS\]) is determined with the accuracy of the additive constant of $S$ i.e., this constant does not contribute to the measure of the complexity of the system.
The partial measure of complexity can also be expressed using specific entropy as follows, $$\begin{aligned}
CX(s;m,n)=\frac{1}{N^{m+n}}CX(Ns;m,n)
\label{rown:specific}\end{aligned}$$ where $N$ is the number of objects that make up the system and specific entropy $s=S/N$. As one can see, the partial measure of complexity scaling with $N$ is taking place. Thus, the partial measure of complexity we can base both on entropy and specific entropy. The latter approach we can use if we compare the complexities of systems consisting of a different number of objects.
However, often extraction of an additional multiplicative constant (e.g., particle number) to have $s$ independent of $N$, is a technical difficulty or even impossible especially for non-extensive systems. Then, it is more convenient to use the entropy of the system instead of the specific entropy. It is also important to realize that determining extreme entropy values (or extreme specific entropy values) of actual systems can be complicated and requires additional dedicated tools/technologies, algorithms, and models.
The entropy can be here both additive (the Boltzmann-Gibbs thermodynamic one [@ETJ] and Shanon information [@Borda]) and non-additive entropy (Rényi [@BeckSchl], Tsallis [@Tsallis]). Apparently, the measure $CX(S)$ is a concave (or convex up) function of entropy $S$, which disappears on the edges at points $S = S^{min}$ and $S = S^{max}$.
It has a maximum $$\begin{aligned}
CX^{max}=CX(S=S_{CX}^{max};m,n)=m^m~n^n\left(\frac{Z}{m+n}\right)^{m+n}
\label{rown:CXSmax2}\end{aligned}$$ at point $$\begin{aligned}
S = S_{CX}^{max}=\overline{S}=\frac{mS^{min}+nS^{max}}{m+n}=\frac{\frac{1}{m}S^{max}+\frac{1}{n}S^{min}}
{\frac{1}{m}+\frac{1}{n}}
\label{rown:CXSmax}\end{aligned}$$ as at this point $\frac{dCX(S)}{dS}\mid _{\overline{S}}~=0$ and $\frac{d^2CX(S)}{dS^2}\mid _{\overline{S}}~<0$. The quantity $CX^{max}$ is well suited for global measurement of complexity because (at a given order $(m,n)$), it depends only on the entropy span $Z$. Perhaps $CX^{max}$ would also be a good candidate for measure of the logic depth of complexity.
The most complex structure
--------------------------
The question now arises about the structure of the system corresponding to entropy $S_{CX}^{max}$ given by Eq. (\[rown:CXSmax\]). The answer is given by the following constitutive equation, $$\begin{aligned}
S\left(Y=Y^{CX^{max}}\right)=S_{CX}^{max},
\label{rown:constitS}\end{aligned}$$ where $Y$ is the set of variables and parameters (e.g., thermodynamic) on which the state of the system depends. However, $Y=Y^{CX^{max}}$ is a set of such values of these variables and parameters which are the solution of Eq. (\[rown:constitS\]). This solution gives the entropy value $S=S_{CX}^{max}$ that maximizes the measure of complexity, that is $CX=CX^{max}$. Hence, with the value of $Y^{CX^{max}}$ we can finally answer the key question: what structure/pattern is behind it, or how the structure of maximum complexity looks.
There are a few comments to be made regarding constitutive Equation (\[rown:constitS\]) itself. It is a (non-linear) transcendental equation in the untangled form relative to the $Y$. This equation should be solved numerically because we do not expect it to have an analytical solution for maximally complex systems. An instructive example of a specific form of this equation and its solution for a specific physical problem is presented in Sec. \[section:idgas\].
Eq. (\[rown:CXSmax\]) legitimizes the measure of complexity we have introduced. Namely, its maximum value falls on the weighted average entropy value, which describes the optimal mixture of completely ordered and completely disordered phases. To the left of $\overline{S}$, we have a phase with dominance of order and to the right a phase with dominance of disorder. The transition between both phases at $\overline {S}$ is continuous. Thus, we can say that the partial measure of complexity we have introduced also defines a certain type of phase diagram in $S$ and $CX$ variables (phase diagram plain). The more detailed information is given in Section \[section:entropic\].
Evolution of the partial measure of complexity
----------------------------------------------
Differentiating Eq. (\[rown:CXS\]) over time $t$ we get the following non-linear dynamics equation, $$\begin{aligned}
\frac{dCX(S(t);m,n)}{dt}=\chi _{CX} (S;m,n)\frac{dS(t)}{dt}=(m+n)\left(S_{CX}^{max}-S(t)\right) CX(S(t);m-1,n-1)\frac{dS(t)}{dt},
\label{rown:CXSt}\end{aligned}$$ where the entropic $S$-dependent (non-linear) susceptibility is defined by $$\begin{aligned}
\chi _{CX}(S;m,n)\stackrel{\rm def.}{=}\frac{\partial CX(S;m,n)}{\partial S}=(m+n)\left(S_{CX}^{max}-S(t)\right)CX(S(t);m-1,n-1)
\label{rown:CXStd}\end{aligned}$$ and $\frac{dS(t)}{dt}$ can be expressed, for example, using the right-hand side of the master Markov equation (see ref. [@Kamp] for details). However, we must realize that the dependence of entropy on time can be, in general, a non-monotonic, because real systems are not isolated (cf. the schematic plot in Fig. \[figure:CXSt\]). One can see how the dynamics of complexity is controlled in a non-linear way by the evolution of entropy of the system.
In concluding this Section, we state that Eqs. (\[rown:CXS\]) – (\[rown:CXSt\]) provide together technology for studying the multiscale aspects of complexity, including the dynamic complexity. However, it is still a simplified approach as we show in Section \[section:conclus\].
Properties of the partial measure of complexity {#section:arbitZ}
-----------------------------------------------
It is worth paying attention to Eqs. (\[rown:CXS\]) - (\[rown:CXSmax\]). As one can see, for a fixed span of $Z$ there may be systems of different complexity. In other words, the complexity description only using $CX^{max}$ is insufficient, because there can be many systems with the same entropic span. From this point of view, we assume systems as equivalent, i.e., belonging to the same complexity class $(Z,m,n)$, if they have the same span. However, we can distinguish them as, in general, they differ in the location of $S_{CX}^{max}$. We can say that a given entropic class has greater potential complexity if it has a larger $CX^{max}$. In a given class, the complexity has a larger the system that lies closer to $CX^{max}$, i.e., its current entropy $S$ is closer to $S_{CX}^{max}$. For a given $CX$ with Eq. (1) we get (for each order $(m, n)$) two $S$ solutions: one on the left and the other on the right of $CX^{max}$ (except when $S = S_{CX}^{max}$). That is, all values of the complexity partial measure have doubly degenerated except for $CX^{max}$.
A distinction should be made between two cases of measuring complexity: (i) $Z<m+n$ and (ii) $Z>m+n$. It is particularly evident when we consider the ratio of both types of complexity measures for $m+n>1$, $$\begin{aligned}
\frac{CX^{max}(Z_i)}{CX^{max}(Z_{ii})}=\left(\frac{Z_i}{Z_{ii}}\right)^{m+n}<1,
\label{rown:ratiomax}\end{aligned}$$ where $Z_i$ belongs to case (i) while $Z_{ii}$ to case (ii). As one can see, the greater the exponent $m+n$, the greater the difference between $CX^{max}(Z_{ii})$ and $CX^{max}(Z_i)$.
The alternate form of Eq. (\[rown:CXS\]), $$\begin{aligned}
CX(\Delta ) = \left(\frac{n}{n+m} Z +\Delta \right)^n~\left(\frac{m}{n+m} Z - \Delta \right)^m,
\label{rown:CXSDelta}\end{aligned}$$ where deviation $\Delta =\Delta (t)=S(t)-S_{CX}^{max}$, makes the operating of the $CX$ coefficient easier in the vicinity of $S_{CX}^{max}$, where the parabolic expansion is valid. We have then, $$\begin{aligned}
CX(\Delta ) \approx CX^{max}\left[1-\frac{1}{2mn}\left((n+m)\frac{\Delta }{Z}\right)^2\right]\approx CX^{max}\exp \left(-\frac{1}{2mn}\left((n+m)\frac{\Delta }{Z}\right)^2\right),
\label{rown:CXSDelta2}\end{aligned}$$ that is a Gaussian form, which has variance $\sigma ^2=\frac{nm}{(n+m)^2}Z^2$. As can be seen, only in the narrow range of $S$ around $S_{CX}^{max}$ the measure of complexity $CX$ is symmetrical regardless of the order $(m, n)$.
In fact, only the location of the maximum of $CX(S)$ is determined (for a given range of $S$) by the ratio of $m$ to $n$. However, to have dependence of coefficient $CX$ on entropy in the entire entropy range $S^{min}\leq S\leq S^{max}$, it is necessary to determine two extreme values of entropy ($S^{min}$ and $S^{max}$) and two exponents ($n$ and $m$). In general case, to find these parameters and exponents it is still far from trivial because they have a contextual (and not a universal) character.
However, in a particular situation, when the maximum complexity is symmetrical, i.e., when $n=m$, we get $$\begin{aligned}
S_{CX}^{max}=\overline{S}=\frac{S^{min}+S^{max}}{2}
\label{rown:SCX2}\end{aligned}$$ and $$\begin{aligned}
CX^{max}=\left(\frac{Z}{2}\right)^{2n}.
\label{rown:CX2}\end{aligned}$$
Definition (\[rown:CXS\]) of the partial measure of complexity applies both to single- and multi-particle issues because entropy can also be built even for a very long single-particle trajectory. Moreover, Definition (\[rown:CXS\]) emphasizes our point of view that any evolving system for which one can introduce the concept of entropy and which have a state of thermodynamic equilibrium (for which entropy reaches a global maximum), contains at least a signature of complexity. For systems of negligible complexity, i.e., for which $S\approx S^{min}$ or $S\approx S^{max}$, the coefficient $CX(S)$ is close to zero. It does not mean, however, that we cannot locate $S_{CX}^{max}$ near $S^{min}$ or $S^{max}$. It is sufficient then to have strongly asymmetric situations when $n \ll m$ or $n \gg m$, respectively.
### Significant partial measure of complexity
We consider the partial measure of complexity to be significant when the entropy of the system locates between two inflection points of the $CX(S;m,n)$ curve, i.e., in the range $S_{ip}^-\leq S \leq S_{ip}^+$. This case occurs for $n,m\geq 2$. We then obtain, $$\begin{aligned}
S^{min}<S_{ip}^{\mp }= S^{min}+\frac{n(n-1)}{\sqrt{n(n+m-1)}}\frac{S^{max}-S^{min}}{\sqrt{n(n+m-1)}\pm \sqrt{m }}<S^{max},
\label{rown;S-+ip}\end{aligned}$$ see Fig. \[figure:CXS\](d) for details.
There are two different cases where a single inflection point is present. Namely, $$\begin{aligned}
S^{min}<S_{ip}^- =\frac{2S^{max}+m(m-1)S^{min}}{2+m(m-1)}<\overline{S}, & \mbox{for $m\geq 2$,~$n=1$},
\label{rown:S-ip}\end{aligned}$$ and $$\begin{aligned}
\overline{S}<S_{ip}^+ =\frac{2S^{min}+n(n-1)S^{max}}{2+n(n-1)}<S^{max}, & \mbox{for $m=1,~n\geq 2,$}.
\label{rown:S+ip}\end{aligned}$$ The case defined by Eq. (\[rown:S-ip\]) we present in Fig. \[figure:CXS\](b), while defined by Eq. (\[rown:S+ip\]) we present in Fig. \[figure:CXS\](c).
For $n=m=1$ the curve $CX(S;m,n)$ vs. $S$ has no inflection points and it looks like a horseshoe (cf. Fig. \[figure:CXS\](a)).
Notably, we can equivalently write, $$\begin{aligned}
S^{min}<S_{ip}^{\mp }= S^{max}-\frac{m(m-1)}{\sqrt{m(n+m-1)}}\frac{S^{max}-S^{min}}{\sqrt{m(n+m-1)}\mp \sqrt{n}}<S^{max}, & \mbox{for $n,m\geq 2$}.
\label{rown:AltS}\end{aligned}$$
Let’s consider the span $Z_{ip}=S_{ip}^+-S_{ip}^-$ of the two-phase area. From Eq. (\[rown;S-+ip\]) or equivalently from Eq. (\[rown:AltS\]) we obtain that, $$\begin{aligned}
Z_{ip}= \frac{2\sqrt{nm}}{(n+m)\sqrt{n+m-1}}Z.
\label{rown:Spanip}\end{aligned}$$ As one can see, span $Z_{ip}$ depends linearly on span $Z$ and in a non-trivial way on exponents $n$ and $m$. Thus, with the $Z$ set, only $Z_{ip}$’s non-trivial dependence on the order $(m,n)$ of measure of complexity $CX$ occurs, which is different from $CX^{max}$ dependence. In other words, $Z_{ip}$ is less sensitive to complexity than $CX^{max}$.
The significant partial measure of complexity ranges between two inflection points only for case $n,m \geq 2$ (cf. Fig. \[figure:CXS\](d)). Indeed, a mixture of phases is observed in this area. For areas where $S^{min}\leq S <S_{ip}^-$ and $S_{ip}^+ < S \leq S^{max}$ we have (practically speaking) only single phases, ordered and disordered, respectively (see Section \[section:entropic\] for details). The case defined by Eq. (\[rown;S-+ip\]) and equivalently by Eq. (\[rown:AltS\]) are the most general that takes into account the fullness of complexity behavior as a function of entropy. Other cases impoverish the description of complexity. Therefore, we will continue to consider the situation when $n,m\geq 2 $.
![Schematic plots of measure of complexity $CX(S;m,n)$ vs. $S$ for four characteristic cases: (**a**) Case $n=m=1$ where no inflection points, $S_{ip}^{\mp}$ are present. (**b**) Case $m=2$ and $n=1$ where a single inflection point $S_{ip}^+$ is present. (**c**) Case $m=1$ and $n=2$ where a single inflection point $S_{ip}^-$ is present. (**d**) Case $m=2$ and $n=2$ where both inflection points are present. The shape of the curve, containing two inflection points, is typical for partial measures of complexity, characterized by exponents $m, n\geq 2$.[]{data-label="figure:CXS"}](FIG_01.pdf){width="10cm"}
The choice of any of the $CX(S;m,n)$ forms (i.e. exponents $n$ and $m$) is somewhat arbitrary function of state of the system as it depends on the function of state, that is on the entropy. In our opinion, the shape of the $CX(S;m,n)$ coefficient vs. $S$ we present in Fig. \[figure:CXS\](b) is the most appropriate because only then the significant complexity (ranging between $S_{ip}^-$ and $S_{ip}^+$) is well defined.
In generic case we should use, however, the definition (\[rown:CXS\]). Then, we can define the order of the partial complexity using the pair of exponents $(n,m)$ . The introduction of the order of the partial complexity is in line with our perception of the existence of multiple levels of (full) complexity.
We only discover the nature of the $CX$ factor, i.e. its dynamics and in particular its dynamical structures, when we analyze the entropy dynamics $S(t)$ (see Fig. \[figure:CXSt\] for details).
![Schematic plot of the partial measure of complexity $CX(S;m,n)$ vs. $S$ and $t$. The red curve shows the dependence of entropy $S$ on time $t$. The black curve represents $CX(S(t);m,n)$ in three dimensions. The blue curve represents projection of the black curve on the $(S,CX)$ plane. Indeed, different variants of this blue curve we show in Fig. \[figure:CXS\]. The non-monotonic dependence of the entropy on time visible here indicates the open nature of the system. However, this non-monotonicity is not visible through the blue curve. For instance, the three local maxima of the black curve, colapses to one of the blue curve.[]{data-label="figure:CXSt"}](FIG_02_ver6.pdf){width="10cm"}
The measurability of the partial measure of complexity is necessary to characterize it quantitatively and to be able to compare different complexities. Following Gell-Mann [@GelMan], we must identify the scales at which we perform the analysis and thus determine coarse-graining to define the entropy. Its dependence on complexity cannot be ruled out.
However, the question of direct measurement of the partial measure of complexity in an experiment (real or numerical) remains open.
Remarks on the entropic susceptibility {#section:entropic}
--------------------------------------
An essential tool for studying phase transitions is the system susceptibility – in our case, the entropic susceptibility of the partial measure of complexity. It plays the role of the order parameter here.
The plot of susceptibility $\chi _{CX}$ vs. $S$ is presented in Figure \[figure:Suscepti\]. Four phases are visible (numbered from 1 to 4).
![Schematic plot of the entropic (non-equilibrium) susceptibility $\chi _{CX}(S;m,n)$ of the partial measure of complexity vs. $S$ at arbitrary fixed order $(m=2,n=2)$. The finite susceptibility value at the $S_-^{ip}$ and $S_+$ phase transition points may be considered to correspond to finite susceptibility value in absorbing non-equilibrium phase transition in the model of direct percolation at a critical point in the presence of an external field [@HHL]. However, the situation presented here is richer, because susceptibility changes its sign, smoothly passing through zero at $S=S_{CX}^{max}$. At this point, the system is extremely robust.[]{data-label="figure:Suscepti"}](FIG_03.pdf){width="10cm"}
Phase number 1 is almost entirely ordered – the disordered phase input is residual. At point $S_-^{ip}$, there is a phase transition to the mixed-phase marked with number 2 still with the predominance of the ordered phase. At the $S_-^{ip}$ transformation point, the entropic susceptibility reaches a local maximum. By further increasing the entropy of the system, it enters phase 3 as a result of phase transformation at the very specific $S_{CX}^{max}$ transformation point. At this point, the entropic susceptibility of the partial measure of complexity disappears. This mixed phase (number 3) is already characterized by the advantage of the disordered phase over the ordered one. Finally, the last transformation which occurs at $S_+^{ip}$, leads the system to the dominating phase by the disordered phase – the input of the ordered phase is residual here. At this transformation point, the susceptibility reaches a local minimum. Intriguingly, entropic susceptibility can have both positive and negative value passing smoothly through zero at $S=S_{CX}^{max}$ where system is exceptionally robust. The presence of phases with positive and negative entropic susceptibility is an extremely intriguing phenomenon.
It should be emphasized that the values of local extremes of the entropic susceptibility of the partial measure of complexity are finite and not divergent, as in the case of (equilibrium and non-equilibrium) phase transitions in the absence of an external field. From this point of view, Definition (1) should be treated as a simplified definition of complexity measure. Extending this definition to describe the critical behavior of a system is possible – however, this is a topic for future work.
Universal full measure of complexity
------------------------------------
Universal the full measure of complexity $X$ is a weighted sum of the partial measures of complexity $CX(S;m,n)$ for individual scales. That is, $$\begin{aligned}
X(S;m_0,n_0)=\sum_{m\geq m_0,n\geq n_0}w(m,n)CX(S;m,n),~m_0,n_0\geq 0,
\label{rown:umsX}\end{aligned}$$ where $w(m,n)$ is a normalized weight, which must be given in an explicit form. This form is to some extent imposed by the power-law form of partial complexity. Namely, we can assume $$\begin{aligned}
w(m,n)=\left(1-\frac{1}{M}\right)^2\frac{1}{M^{m-m_0+n-n_0}},~M>1,
\label{rown:mn00}\end{aligned}$$ which seems to be a particularly simple because $$\begin{aligned}
\frac{w(m+1,n)}{w(m,n)}=\frac{w(m,n+1)}{w(m,n)}=\frac{1}{M},
\label{rown:WNmn}\end{aligned}$$ independently of $m$ and $n$.
From Eqs. (\[rown:umsX\]) and (\[rown:mn00\]) we directly obtain $$\begin{aligned}
X(S;m_0,n_0)=\left(1-\frac{1}{M}\right)^2\frac{(S^{max}-S)^{m_0}}{1-\frac{S^{max}-S}{M}}\frac{(S-S^{min})^{n_0}}{1-\frac{S-S^{min}}{M}}.
\label{rown:CXmnwN}\end{aligned}$$ The $M$ parameter is chosen here so as to obtain a finite value of $X(S;m_0,n_0)$ for any value of $S$. This occurs when $M>S^{max}-S^{min}=Z$. Apparently, $m_0,n_0\geq 1$ is the natural lower limit of $m_0,n_0$, because only then $X(S;m_0,n_0)$ disappears for $S=S^{min},S^{max}$, like this should be. We still assume more strongly that $m_0,n_0\geq 2$, which has already been mentioned above. We emphasize that $X$ does not scale with $N$ as opposed to partial measures of complexity.
Note that for $M \gg Z$, both measures of complexity have approximate values $X(S; m0, n0)\approx CX(S; m0, n0)$. Important differences between these two measures only appear for $Z/M$ close to 1, because only then the denominator in Eq. (\[rown:CXmnwN\]) plays an important role. Of course, $M$ is a free parameter, and possibly its specific value could be obtained from some additional (e.g., external) conditions.
In Fig. \[figure:FIG\_4\_extra\] we compare the behavior of the partial (black curve) and full measure (orange curve) of complexity, where we used the entropy instead of the specific entropy. Whether $CX$ lies below or above $X$ depends both on $M$ parameter (determining the weight at which individual measures of partial complexity enter the full measure of complexity), as well as on the $Z/M$ ratio.
![Comparison of the partial measure of complexity $CX(S;m=2,n=2)$ and full measure of complexity $X(S;m_0=2,n_0=2)$ for the symmetric case of $m=n=m_0=n_0$. In addition, we assumed that $S^{min}=0, S^{max}=8$ and $M=10$ for both measures. For both cases Eqs. (\[rown:SCX2\]), (\[rown:CX2\]) and (\[rown:CXmnwN\]) are valid. Vertical dashed lines indicate inflection points: black for the $CX$ curve, orange for the $X$ curve, while $S_{CX}^{max}=S_{X}^{max}=4$.[]{data-label="figure:FIG_4_extra"}](FIG_04.pdf){width="10cm"}
We continue to determine the entropic susceptibility of the full measure of complexity, $$\begin{aligned}
\chi _X&=&\frac{dX(S;m_0,n_0)}{dS}=(m_0+n_0)(S_{CX}^{max}-S)\chi_x (S;m_0-1,n_0-1)\nonumber \\
&+&\frac{2}{M^2}X(S;m_0,n_0)\frac{S-S^{arit}}{(1-\frac{S^{max}-S}{M})(1-\frac{S-S^{min}}{M})}.
\label{rown:dXS}\end{aligned}$$
![Schematic plot of the entropic (non-equilibrium) susceptibility $\chi_{X}(S;m,n)$ of the full measure of complexity vs. $S$ at arbitrary fixed order $(m_0=2,n_0=2)$. As expected from Fig. \[figure:FIG\_4\_extra\], turning points of $CX$ (cf. Fig. \[figure:Suscepti\]) lie within the $S$ interval bounded by inflection points of $X$.[]{data-label="figure:FIG_5"}](FIG_chi_X.pdf){width="10cm"}
Thus, the evolution of $X$ is governed by an equation analogous to Eq. (\[rown:CXSt\]) except that $\chi _{CX} $ present in this equation should be replaced by $\chi _X $ given by Eq. (\[rown:dXS\]).
Distribution non-equilibrium entropies {#section:entropy}
--------------------------------------
Distribution entropy is understood as the entropy based on coarse-grained probability distributions – this type of entropy is most often used [@Kamp; @BeckSchl; @LLZh]. A very characteristic example is the entropy class built on time-dependent probability distributions, $\{p_j(t)\}$, satisfying master (Markovian), or M-equation, in presence of detailed balance conditions. We give here two very characteristic (nonequivalent) examples of this type of entropies[^2] $$\begin{aligned}
S(t)=S_0\left[1-\sum_{j}p_{j}^{eq}f\left(\frac{p_j(t)}{p_{j}^{eq}}\right)\right]
\label{rown:entrop1}\end{aligned}$$ and $$\begin{aligned}
S(t)=-S_0\ln{\sum_{j}p_{j}^{eq}f\left(\frac{p_j(t)}{p_{j}^{eq}}\right)},
\label{rown:entrop2}\end{aligned}$$ where $p_j(t)$ is a probability of finding a system in state $j$ at time $t$, while $p_j^{eq}$ is a corresponding equilibrium probability. We are considering only discrete states here. The function $S_0f(x)\geq 0$, where domain $0\leq x\leq \infty $, is a non-negative convex function obeying $S_0\frac{d^2f}{dx^2}\geq 0$. It can be shown [@Kamp] that entropies defined in this way meet the law of entropy increase, i.e. its derivative $$\begin{aligned}
\frac{dS(t)}{dt}\geq 0;
\label{rown:st0}\end{aligned}$$ therefore $S(t)\rightarrow S^{max}$ from below when $p_j(t)\rightarrow p_j^{eq}$. Eq. (\[rown:st0\]) is the key property of entropy. Let us add that for $p=p^{eq}$ entropy defined by equations (\[rown:entrop1\]) and (\[rown:entrop2\]) disappear. In other words, these entropies are negative and grow to zero as the system tends to equilibrium. It is worth paying attention to the possibility of defining generalized information gain, whereby this information gain is calculated here relative to the equilibrium distribution. We can write, $$\begin{aligned}
\Delta I(p(t),p^{eq})=-S(t),
\label{rown:Kull}\end{aligned}$$ where $p(t)=\{p_j(t)\}$ and $p^{eq}(t)=\{p_j^{eq}(t)\}$. Furthermore, entropy $S(t)$ is closely related to partition function. Therefore, in this approach, the entropy is a base function.
Most often the function $f(x)$ is selected in the form, $$\begin{aligned}
f(x)=x^{\alpha },
\label{rown:fx}
\end{aligned}$$ coupled with a constant $S_0=\frac{1}{\alpha -1}$. With these choices the entropy given by Eq. (\[rown:entrop1\]) is called Tsallis entropy and the entropy given by Eq. (\[rown:entrop2\]) Rényi entropy. Usually, the entropic index $\alpha $ is denoted by $q$ in the case of Tsallis entropy.
Both Tsallis and Rényi entropies converge to Boltzmann-Gibbs-Shannon (BGS) entropy[^3] when the entropy index tends to 1. We remind that the BGS entropy has two basic features:
- obeys the Boltzmann $H$-theorem (for dilute binary interacting gases),
- it is an additive quantity that becomes extensive for a gas or a solid.
The Tsallis and Rényi entropies are particularly useful that can be used in both extensive/linear, non-extensive/nonlinear, as well as equilibrium/non-equilibrium physics and also beyond the physics domain.
Finger print of complexity in simplicity {#section:idgas}
========================================
Let’s consider the perfect gas at a fixed temperature which is initially closed in the left half of an isolated container. Then the partition next is removed, and the gas undergoes a spontaneous expansion. We are dealing here (practically speaking) with an irreversible process even for a small number of particles (at least the order of $10^2$).
Let’s recall the definition of ’perfect gas.’ It is a gas of particles that cannot ‘see’ each other, i.e., there are no interactions between them. Thus, from a physical point of view, it is a dilute gas at high temperature. We further assume that all particles have the same kinetic energy. The legitimate question is whether such a gas will expand after the partition is removes? We notice that the thermodynamic force is at work here, being roughly proportional to the difference in the number of particles in the right and left parts of the container. This force causes the expansion process. Thus, we are dealing with the simplest paradigmatic irreversible process [@KTH]. The particles remain stuck in the final state and will not leave it (with accuracy to slight fluctuations in the number of particles in the right half of the container). Such a final state of the whole system is referred to as the equilibrium state. The simple coarse-grain description of the system allows us to introduce here the concept of configuration entropy.
Note that the macroscopic state of the system (generally, the non-equilibrium one) can be described by the instantaneous number of particles in the left ($N_L$) and right ($N_R$) parts of the container, with $N=N_L+N_R$, where $N$ is the total fixed number of particles. It allows one to define the weight of the macroscopic state $\Gamma (N_L)$, also called thermodynamic probability. This is the number of ways to arrange the $N_L$ particles in the left part of the container and $N_R=N-N_L$ in the right. Hence, $$\begin{aligned}
\Gamma (N_L)=\frac{N!}{N_L!(N-N_L)!}.
\label{rown:Gama}\end{aligned}$$ We do not distinguish here permutations of particles inside each part of the container separately. We take into account only permutations of particles located in different halves of the container. This is because our resolution is too small here to observe the location of particles inside each container separately. Such a coarse-graining creates an information barrier: more information can mask the complexity of the system. We will not be able to see the complexity because we will not be able to construct entropy. This creates a paradoxical situation: the surplus of information makes the task difficult and does not facilitate obtaining the insight into the system. Here we have an analogy with chaotic dynamics, where chaos is visible only in the Poincare surface cross-section of the phase space and not in the entire phase space.
The configuration entropy at a given time $t$ we define as follows, $$\begin{aligned}
S(N_L(t))=\ln \Gamma (N_L(t)),
\label{rown:St}\end{aligned}$$ where $\Gamma (N_L)$ is given by Eq. (\[rown:Gama\]). The above expression can be used both for the equilibrium and non-equilibrium states.
It can be demonstrate using the Stirling formula that for large $N$, entropy is reduced to the BGS form, $$\begin{aligned}
\ln \Gamma (N_L)=-N\left[p_L(t)\ln p_L(t) + p_R(t)\ln p_R(t) \right]=Ns(t),
\label{rown:SNt}\end{aligned}$$ where $p_J(t)\stackrel{\rm def.}{=}\frac{N_J(t)}{N},~J=L,R$, and $s(t)$ is a specific entropy. The law of entropy increase (\[rown:st0\]) is also fulfilled here, as expected.
We now obtain the equation for determining $N_L^{CX^ {max}}$, i.e. the number of particles in the left part of the container that maximizes the partial complexity measure $CX$. To this end, we substitute $N_L=N_L^{CX^{max}}$ into the right side of Eq. (\[rown:St\]) and in Eq. (\[rown:SCX2\]) we put $S^{max}$ equal to the right-hand side of Eq. (\[rown:St\]) for $N_L=N/2=N^{eq}$. Hence, we obtain a constitutive equation, $$\begin{aligned}
S(N_L=N_L^{{CX}^{max}})=S_{CX}^{max},
\label{rown:constit}\end{aligned}$$ where $N_L^{{CX}^{max}}$ is our sought quantity. We assumed to the right of the above equation (as it is commonly used), that $S^{min}$ = 0. Thus, only the relation $S^{max}=S(N^{eq})$ is taken into account. Equation (\[rown:constit\]) is an example of Eq. (\[rown:constitS\]), where $N_L^{CX^{max}}$ plays the role of $Y^{CX^{max}}$. This equation has the following explicit form, $$\begin{aligned}
\left[\Pi_{j=1}^{N-N_L^{CX^{max}}}\left(1+\frac{N_L^{CX^{max}}}{j}\right)\right]^2=\Pi_{j=1}^{N/2}\left(1+\frac{N}{2j}\right), ~\mbox{for $n=m=2$}.
\label{rown:Pi}\end{aligned}$$ Just deriving Eq. (\[rown:Pi\]) was the primary purpose of this example. This is a transcendental equation which exact analytical solution is unknown. When deriving Eq. (\[rown:Pi\]) we used the initial condition for the entropy that is, $S(t=0)=S^{min}=\ln \Gamma (N_L=N)=0$, which follows from Eqs. (\[rown:Gama\]) and (\[rown:St\]). Even for such a simple toy model, determining the partial measure of complexity is a non-trivial task, also because $N_L$ is different from $N/2$ (as we show below).
The numerical solutions of Eq. (\[rown:Pi\]), i.e. the relationship of $N_L^{CX^{max}}$ to $N$, are shown in Fig. \[figure:LN\] (for simplicity, $L$ defining the vertical axis on the plot means $N_L^{CX^{max}}$). Both solutions (small circles above and below the solid straight line) show that $N_L^{CX^{max}}$ is significantly different from $N/2$. Thus, the most complex state is significantly different from the equilibrium state.
![Dependence of $L(=N_L^{CX^{max}})$ vs. $N$. There are two solutions of Eq. (\[rown:Pi\]): one marked with blue circles and the other with orange ones. Above $N\approx 10^2$, both dependencies are linear, which is particularly clearly confirmed in Fig. ( \[figure:LNdir\]). That is, in a log-log scale, their slopes equal 1. However, in linear scale, the directional coefficients of these straight lines equal $0.11$ and $0.89$, respectively. It is clearly shown in Fig. \[figure:LNdir\]. Only the solution with orange circles is realistic, because the chance that $89\% $ of particles will pass in the finite time to the second part of the container (as indicated by the solution marked with blue circles) is negligibly small The black solid tangent straight line indicates a reference case $N_L^{CX^{max}}=N/2$.[]{data-label="figure:LN"}](Klamut_FIG_1_NEW.pdf){width="10cm"}
![Directional coefficient of linear dependencies $L$ vs.$N$ as a function o $N$. For $N$ greater than $10^2$, no $N$-dependence of this coefficient is observed. Both solutions (having $L/N=0.11$ and $L/N=0.89$) are mutually symmetric about the straight horizontal line $L/N=1/2$ but only the solution $L/N=0.89$ we consider as realistic. The black horizontal straight solid line indicates a reference case $N_L^{CX^{max}}=N/2$.[]{data-label="figure:LNdir"}](Klamut_FIG_2_NEW.pdf){width="10cm"}
Having the $N_L^{{CX}^{max}}$ dependence on $N$, we obtain the dependence of complexity $CX^{Xmax}$ on $N$ order $(m=2,n=2)$. We can write, $$\begin{aligned}
CX^{max}=\left(\frac{S(N/2)}{2}\right)^4=\left[\frac{1}{2N}\ln\left( \Pi_{j=1}^{N/2}\left(1+\frac{N}{2j}\right)\right)\right]^4,
\label{rown:CXmaxgas}\end{aligned}$$ as in our case $S^{max}=S(N/2)$ equals the logarithm of the right-hand side of Eq. (\[rown:Pi\]) divided by $N$. Notably, Eq. (\[rown:CXmaxgas\]) is based on Eq. (\[rown:CX2\]).
In Fig. \[figure:CXmaxN\] we present the dependence of $CX^{max}$ on $N$. $CX^{max}$ is a non-extensive function – it reaches the plateau for $N\gg 1$. For $ N\approx 10^4 $ the plateau is achieved with a good approximation. It can be said that $CX^{max}$ behaves like specific complexity measure. This is important for researching complexity. Namely, systems can attain complexity already on a mesoscopic scale. Although the absolute value of the complexity measure is relatively small, it is evident and possesses a structure related to the current inflection point there (near $N=10$).
This example shows that even such a simple arrangement of non-interacting objects may have non-disappearing non-equilibrium complexity. A necessary (but not sufficient) condition is the possibility of constructing entropy and the presence of a time arrow.
![Dependence of $CX^{max}$ on $N$. As one can see, $CX^{max}$ is a non-extensive function – it reaches the plateau for $N\gg 1$. For $ N\approx 10^4 $ the plateau is achieved with a good approximation. This is an important issue for researching complexity. Namely, systems can attain complexity already on a mesoscopic scale.[]{data-label="figure:CXmaxN"}](Klamut_FIG_3.pdf){width="10cm"}
Concluding remarks {#section:conclus}
==================
In many recent publications [@Pincus; @PBEH; @RichMo; @LLZh] it is argued that entropy can be a direct measure of complexity. Namely, the smaller value of entropy indicates more regularity or lower system complexity, while its larger value indicates more disorder, randomness and higher system complexity. However, according to Gell-Mann, a more disorder means less and not more system complexity. These two viewpoints are contradictory – this is a serious problem, which we addressed.
Our motivation in solving the above problem was based on the Gell-Mann’s view of complexity. This is because fail to agree that the loss of information by the system as it approaches equilibrium, increases its complexity; notably, $\Delta I(p^{eq},p^{eq})$ takes its minimum value then and complexity must decrease. In addition, the differences in definition (\[rown:CXS\]) eliminate the useless dependence of complexity on the additive constant that may appear in the definition of entropy. It can be said that the system state with the highest complexity is the most state most distant from all the states of the system of lesser or no complexity.
Thus, in the sense of Gell-Mann, the measure of complexity should supply a complementary information to the entropy or its monotonic mapping.
Therefore, in this work, we presented a methodology which allows building a universal measure of complexity as a function of system state based on the non-linearly transformed entropy. This is a non-extensive measure. This measure should meet a number of conditions/axioms that we indicated at work. A parsimonious example, of the simplest system with a small and a large number of degrees of freedom,is presented to support our methodology. As a result of this approach, we have shown that (generally speaking) the most complex are optimally mixed states consisting of pure states, i.e., of the most regular and most disordered, which the space of states of a given system allows.
We should pay attention to an essential issue regarding the definition of the phenomenological partial measure of complexity given by definition (\[rown:CXS\]). This definition is open in the sense that if the description of complexity requires, for example, one additional quantity, then the definition (\[rown:CXS\]) takes on an extended form, $$\begin{aligned}
CX(S,E;m_1,n_1,m_2,n_2)\stackrel{\rm def.}{=}(S^{max} - S)^{m_1}(S-S^{min})^{n_1}(E^{max}-E)^{m_2}(E-E^{min})^{n_2} \geq 0,
\label{rown:def2D}\end{aligned}$$ whereby $E^{min}\leq E\leq E^{max}$ this new quantity was marked. This definition has still an open character. Specifically, this definition also allows (if the situation requires it) to replace one quantity with another, e.g., entropy with free energy or considering some derivatives (e.g., of the type $\frac{\partial S}{\partial E}$). Openness and substitutability should be the key features of the measure of complexity. What’s more, exponents $ m_j,~n_j,~j = 1,2,$ determine the order of complexity, i.e. its level or scale. We emphasize that the introduced measure of complexity can describe isolated and closed systems (although in contact with the reservoir), as well as open systems that can change their elements.
From Eqs. (\[rown:def2D\]) and (\[rown:umsX\]) we get the phenomenological universal full measure of complexity in the form which extends Eq. (\[rown:CXmnwN\]), $$\begin{aligned}
X(S;m_1^0,n_1^0,m_2^0,n_2^0)&=&\left(1-\frac{1}{M_1}\right)^2\frac{(S^{max}-S)^{m_1^0}}{1-\frac{S^{max}-S}{M_1}}\frac{(S^{min}-S)^{n_1^0}}{1-\frac{S^{min}-S}{M_1}} \nonumber \\
&\times &\left(1-\frac{1}{M_2}\right)^2\frac{(E^{max}-E)^{m_2^0}}{1-\frac{E^{max}-E}{M_2}}\frac{(E^{min}-E)^{n_2^0}}{1-\frac{E^{min}-E}{M_2}}\ge 0.
\label{rown:extend_def21}\end{aligned}$$
Definitions of measures of complexity (\[rown:CXS\]) and (\[rown:CXmnwN\]) and their possible extensions are universal and useful. It is due to entropy associated not only with thermodynamics (Carnot, Clausius, Kelvin) and statistical physics (Boltzmann, Gibbs, Planck, Rényi, Tsallis) but also with the information approach (Shannon, Kolmogorov, Lapunov, Takens, Grassberger, Hantschel, Procaccia) and with the approach from the side of cellular automata (von Neumann, Ulam, Turing, Conway, Wolfram, et al.), i.e., with any representation of the real world using a binary string. Today, we have already several very effective methods for counting entropy of such strings as well as other macroscopic characteristics sensitive to the organization and self-organization of systems, as well as to their synchronization (synergy, coherence), competition, reproduction, adaptation – all of them sometimes having local and sometimes global characters.
Our definitions of complexity also goes out to meet the research of the complexity of the biologically active matter. In this, especially research on the consciousness of the human brain can get a fresh impulse. The point is that most researchers believe that the main feature of conscious action is a maximum complexity, i.e., critical complexity [@Sornett] – in our approach it would be $CX^{max}$. It separates the phase with the predominance of ordered states from the one containing the predominance of disordered states, as it should be. However, to achieve this strictly, one would have to go from positive integers $m$ and $n$ to positive continuous exponents, $\alpha $ and $\beta $, respectively. The criticality would be recoverable for $\alpha $ or $\beta $ smaller than 1. Then it would be possible to show the singular behavior of the entropic susceptibility. As such, our approach might serve to study the evolution/dynamics of consciousness.
Therefore, we hope that our approach will enable (i) the universal classification of complexity, (ii) analysis of a system critical behavior and its applications, and (iii) study the dynamic complexity. All these constitute create the background of science of complexity.\
**Author contributions**: RK and ZRS conceptualised the work,; RK wrote the draft and conducted the formal analysis and prepared the draft of figures; JK provided numerical calculations and provided the final figures; ZRS finalised the manuscript. All authors read and approved the final manuscript.\
**Funding**: One of the authors of the work (ZRS) benefited from the financial support of the ZIP Program. This program of integrated development activities of the University of Warsaw is implemented under the operational program Knowledge Education Development, priority axis III. Higher education for economy and development, action: 3.5 Comprehensive university programs, from April 2, 2018 to March 31, 2022, based on the contract signed between the University of Warsaw and the National Center for Research and Development. The program is co-financed by the European Union from the European Social Fund; <http://zip.uw.edu.pl/node/192>. Apart from that, there was no other financial support.\
**Conflicts of interest**: The authors declare no conflict of interest.
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[^1]: An extension to real positive numbers is possible but this is not the case in this paper.
[^2]: More specifically: one should talk about specific entropy. However, we continue to leave the word ’specific,’ as this does not lead to confusion in this work. In addition, entropies given by Equations (22) and (23) belong to the category of relative entropies. We can call them generalized Kullback-Leibler entropies [@PBEH].
[^3]: More precisely, entropies given by formulas (\[rown:entrop1\]) and (\[rown:entrop2\]) converge to Kullback-Leibler relative entropy. In our case, this entropy, we calculate relative to the state of thermodynamic equilibrium of the system.
|
---
abstract: 'In this paper, we investigate the block that has an abelian defect group of rank $2$ and its Brauer correspondent has only one simple module. We will get an isotypy between the block and its Brauer correspondent. It will generalize the result of Kessar and Linckelmann ([@KL]).'
author:
- Xueqin Hu
date: 'School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China'
title: '**Blocks with abelian defect groups of rank $2$ and one simple module**'
---
Introduction
============
Let $p$ be a prime and $\mathcal{O}$ a complete discrete valuation ring having an algebraically closed residule field $k$ of characteristic $p$ and a quotient field $\mathcal{K}$ of characteristic $0$. We will always assume that $\mathcal{K}$ is big enough for the finite groups below.
Let $G$ be a finite group and $b$ a block of $\mathcal{O}G$ with a defect group $P$. Denote by $\mathrm{Irr}_\mathcal{K}(G,b)$ and $\mathrm{IBr}(G,b)$ the set of irreducible ordinary characters in $b$ and the set of irreducible Brauer characters in $b$ respectively. Set $l_G(b)=|\mathrm{IBr}(G,b)|$. Let $c$ be the Brauer correspondent of $b$ in $N_G(P)$. In [@KL], Kessar and Linckelmann investigated the block $b$ under the assumptions that $l_{N_G(P)}(c)=1$ and $P$ is elementary abelian of rank $2$. They showed that the inertial quotient of $b$ is abelian and there is an isotypy between $b$ and $c$ all of whose signs are positive.
In this note, we will generalize these results to the blocks with defect groups of rank $2$.
\[MT\] Keep the notation as above. Assume that $P$ is abelian of rank $2$ and $l_{N_G(P)}(c)=1$. Then the inertial quotient of $b$ is abelian and there is an isotypy between $b$ and $c$.
These results are well-known when either $p$ is $2$ or the inertial quotient of $b$ is trivial. Therefore, we may assume that $p$ is odd and the inertial quotient of $b$ is non-trivial throughout this paper.
The structure of the block $c$
==============================
Keep the notation as above. In this section, we will investigate the structure of the inertial quotient of $b$ and irreducible ordinary characters of the block $c$.
Given a positive integer $a$, denote by $C_a$ the cyclic group of order $a$. We will use $[-\,\ ,\,\ -]$ to represent the commutator. Assume that $P=C_{p^n}\times C_{p^m}$ for some positive integers $n,m$. We will fix a maximal $b$-Brauer pair $(P,b_P)$. For any $Q\leq P$, denote by $(Q,b_Q)$ the unique $b$-Brauer pair contained in $(P,b_P)$. Let $E$ be the inertial quotient of $b$ associated with $(P,b_P)$, namely, $E=N_G(P,b_P)/C_G(P)$.
\[E is abelian\] The inertial quotient $E$ is abelian if $l_{N_G(P)}(c)=1$.
Let $\Phi(P)$ be the Frattini subgroup of $P$. So $P/\Phi(P)$ is $C_p\times C_p$. Set $H$ to be $N_G(P,b_P)$. Then $\Phi(P)\unlhd H$ and denote $H/\Phi(P)$ by $\bar{H}$. For any subset $X$ of $\mathcal{O}H$, $\bar{X}$ denotes the image of $X$ under the canonical map $\mathcal{O}H\longrightarrow\mathcal{O}\bar{H}$.
Since $l_H(b_P)=1$, $l_{\bar{H}}(\bar{b}_P)=1$ and $\bar{b}_P$ is a block of $\bar{H}$ with defect group $\bar{P}=C_p\times C_p$. Let $\hat{C}$ be the subgroup of $H$ such that $\hat{C}/\Phi(P)=C_{\bar{H}}(\bar{P})$. Hence, $\hat{C}=\{x\in H~|~[P,x]\subseteq\Phi(P)\}$. It is clear that $P=[P,\hat{C}]\times C_P(\hat{C})$. So $P=C_P(\hat{C})$ since $[P,\hat{C}]\leq\Phi(P)$. This means $C_{\bar{H}}(\bar{P})=\bar{C}_G(P)$. Hence, $(\bar{P},\bar{b}_P)$ is a maximal $\bar{b}_P$-Brauer pair of $\mathcal{O}\bar{H}\bar{b}_P$. By [@KL Proposition 5.2], $N_{\bar{H}}(\bar{P},\bar{b}_P)/C_{\bar{H}}(\bar{P})$ is abelian. It is evident that $E$ is isomorphic to $N_{\bar{H}}(\bar{P},\bar{b}_P)/C_{\bar{H}}(\bar{P})$. We are done.
By [@DJ Lemma 2] and the structure of blocks with normal defect groups, $E$ is a direct product of two isomorphic groups. Next, we will show that $E$ acts diagonally on $P$. This can be deduced from the following general fact.
\[diagonal action\] Let $D$ be an abelian $p$-group of rank $2$ and $F\leq\mathrm{Aut}(P)$ an abelian $p^\prime$-group which is a direct product of two isomorphic subgroups. Then we have the decompositions $F=F_1\times F_2$ and $D=D_1\times D_2$ such that $F_1$ acts faithfully on $D_1$ and centralises $D_2$ and $F_2$ acts faithfully on $D_2$ and centralises $D_1$ and $F_1\cong F_2$. In particular, $F_1$ and $F_2$ are cyclic groups of order dividing $(p-1)$.
We will exhibit it by induction on $|D|$. When $D$ is elementary abelian, it is actually done in [@KL Proposition 5.3]. We may assume that $n\geq 2$ or $m\geq 2$. Let $\Phi(D)$ be the Frattini subgroup of $D$. So $D/\Phi(D)$ is $C_p\times C_p$. Let $\pi$ be the canonical map from $F$ to $\mathrm{Aut}(D/\Phi(D))$. For any subset $X$ of $F$, $\bar{X}$ denotes the image of $X$ under $\pi$. It is clear that $\pi$ is injective. So there exist two subgroups $F_1$ and $F_2$ of $F$ and two subgroups $D_1$ and $D_2$ of $D$ containing $\Phi(D)$ satisfying the properties $\bar{F}=\bar{F}_1\times \bar{F}_2$ and $D/\Phi(D)=D_1/\Phi(D)\times D_2/\Phi(D)$ and $\bar{F}_1$ acts faithfully on $D_1/\Phi(D)$ and centralises $D_2/\Phi(D)$ and $\bar{F}_2$ acts faithfully on $D_2/\Phi(D)$ and centralises $D_1/\Phi(D)$ and $\bar{F}_1\cong \bar{F}_2$. Hence, $D_1$ and $D_2$ are $F$-stable and they fulfill
\(i) $D_1=[D_1,F_1]\cdot\Phi(D)$ and $[D_1,F_2]\subseteq\Phi(D)$ and $F_1$ acts faithfully on $D_1$;
\(ii) $D_2=[D_2,F_2]\cdot\Phi(D)$ and $[D_2,F_1]\subseteq\Phi(D)$ and $F_2$ acts faithfully on $D_2$;
\(iii) $D_1\cap D_2=\Phi(D)$ and $D_1/\Phi(D)\cong C_p\cong D_2/\Phi(D)$ and $D=D_1\cdot D_2$.
Suppose that $\Phi(D)$ is cyclic. Then $D=C_p\times C_{p^m}$ with $m\geq 2$ and $\Phi(D)=C_{p^m-1}$. Since $D_2=[D_2,F_1]\times C_{D_2}(F_1)$ and $[D_2,F_1]\subseteq\Phi(D)$, $\Phi(D)=[D_2,F_1]\times C_{\Phi(D)}(F_1)$. Then either $[D_2,F_1]=1$ or $C_{\Phi(D)}(F_1)=1$ by the assumption that $\Phi(D)$ is cyclic. If $[D_2,F_1]=1$, then $\Phi(D)\leq D_2\leq C_D(F_1)$. Clearly, $D=[D,F_1]\times C_D(F_1)$ and $D_2$ is a maximal subgroup of $D$. Thus, $D_2=C_D(F_1)$ and $[D,F_1]=[D_1,F_1]$. Since $F_1$ and $F_2$ commute with each other and $[D_1,F_2]\subseteq\Phi(D)\subseteq C_D(F_1)$, $[[D_1,F_1],F_2]=1$. So $[D_1,F_1]\leq C_P(F_2)$. Since $F_1$ acts faithfully on $D_1$ and $D_1=[D_1,F_1]\times C_{D_1}(F_1)$, $F_1$ acts faithfully on $[D_1,F_1]$. Thus, the decompositions $F=F_1\times F_2$ and $D=[D_1,F_1]\times D_2$ are what we want. We may assume that $C_{\Phi(D)}(F_1)=1$. Then $\Phi(D)=[D_2,F_1]$ and $D_2=\Phi(D)\times C_{D_2}(F_1)$. If $C_{\Phi(D)}(F_2)=1$, we can get $\Phi(D)=[D_1,F_2]$ and $D_1=\Phi(D)\times C_{D_1}(F_2)$ similarly. Then $D=C_{D_1}(F_2)\times C_{D_2}(F_1)$ which is impossible. So $C_{\Phi(D)}(F_2)\neq1$. Then replacing $D_2$ by $D_1$ in the previous argument, we can obtain the decompositions that we need.
Suppose $\Phi(D)$ is of rank $2$. Then both $D_1$ and $D_2$ are of rank $2$. Let $K$ be subgroup of $F$ consisting of automorphisms acting trivially on $D_1$. Then $D=[D,K]\times C_D(K)$ and $D_1\leq C_D(K)$. Hence, $K$ has to be trivial since $D_1$ has rank $2$. This means $F$ acts faithfully on $D_1$. By induction, we have $D_1=D_{11}\times D_{12}$ and $F=F_{11}\times F_{12}$ such that $F_{11}$ acts faithfully on $D_{11}$ and centralises $D_{12}$ and $F_{12}$ acts faithfully on $D_{12}$ and centralises $D_{11}$ and $F_{11}\cong F_{12}$. Then $D=[D,F_{11}]\times C_{D}(F_{11})$ and $D_{11}=[D_{11},F_{11}]\leq[D,F_{11}]$ and $D_{11}\leq C_{D}(F_{12})$. In particular, $C_{[D,F_{11}]}(F_{12})\neq 1$. But $[D,F_{11}]$ is cyclic. Then $[D,F_{11}]\leq C_D(F_{12})$ and moreover $C_D(F_{12})=[D,F_{11}]\times(C_D(F_{11})\cap C_D(F_{12}))$. But $C_D(F_{12})$ is also cyclic. We have $[D,F_{11}]=C_D(F_{12})$. Similarly, we can prove that $[D,F_{12}]=C_D(F_{11})$. Then the decompositions $D=[D,F_{11}]\times [D,F_{12}]$ and $F=F_{11}\times F_{12}$ are what we want. We are done.
Hence, by Lemma \[diagonal action\], we have $E=E_1\times E_2$ and $P=P_1\times P_2$ such that
\(i) $E_1$ acts faithfully on $P_1$ and centralises $P_2$;
\(ii) $E_2$ acts faithfully on $P_1$ and centralises $P_1$;
\(iii) $E_1\cong E_2$ are cyclic groups of order $l$, which $l$ is a positive integer dividing $(p-1)$.
We can easily describe the source algebra of the block $c$ by the structure theory of blocks with normal defect groups and the structure of inertial quotient $E$. It is well-known that there exists a central extension
$$\xymatrix@C=0.5cm{
1 \ar[r] & Z \ar[r] & \tilde{E} \ar[r] & E \ar[r] & 1 }$$ with $Z$ cyclic $p^\prime$-group such that there is an irreducible ordinary character $\theta$ of $Z$ which is covered by a unique irreducible character of $\tilde{E}$. Let $e_\theta\in\mathcal{O}Z$ be the central idempotent corresponding to $\theta$. Set $N=P\rtimes\tilde{E}$. Then $\mathcal{O}Ne_\theta$ is the source algebra of the block $c$. Note that $e_\theta$ is still a block of $C_N(R)$ for any $R\leq P$. The following lemma gives some information about the degrees and number of irreducible ordinary characters of $\mathcal{O}Ne_\theta$, which is similar with [@KL Proposition 5.3]. We will skip the proof.
\[irreducible character of c\] Set $A$ to be $\mathcal{O}Ne_\theta$. Then the degree of an element of $\mathrm{Irr}_\mathcal{K}(A)$ is either $l$ or $l^2$ and $\mathrm{Irr}_\mathcal{K}(A)$ has $p^n+p^m-1$ elements of degree $l$ and $\frac{p^n-1}{l}\cdot\frac{p^m-1}{l}$ elements of degree $l^2$.
The extension of local system
=============================
Keep the notation as above. In this section, we will use the so-called $(G,b)$-local system introduced by Puig and Usami in [@PU] to prove the main theorem.
First, let us recall some notation and state the definition of $(G,b)$-local system under our setting (see [@PU]).
Let $\mathcal{CF}_\mathcal{K}(G)$ be the vector space of $\mathcal{K}$-valued class functions of $G$ and $\mathcal{BCF}_\mathcal{K}(G)$ be the vector space of $\mathcal{K}$-valued class functions on the set $G_{p^\prime}$ of $p^\prime$-elements of $G$. It is clear that the set of irreducible ordinary characters of $G$ is a $\mathcal{K}$-basis of $\mathcal{CF}_\mathcal{K}(G)$ and the set of irreducible Brauer characters of $G$ is a $\mathcal{K}$-basis of $\mathcal{BCF}_\mathcal{K}(G)$. For $\chi,\chi^\prime\in\mathcal{CF}_\mathcal{K}(G)$, we denote by $\langle\chi,\chi^\prime\rangle$ the inner product of $\chi$ and $\chi^\prime$.
Let $u$ be a $p$-element of $G$. we have the well-known surjective $\mathcal{K}$-linear map $d_G^u:\mathcal{CF}_\mathcal{K}(G)\longrightarrow
\mathcal{BCF}_\mathcal{K}(C_G(u))$ defined by $d_G^u(\chi)(s)=\chi(us)$ for any $\chi\in\mathcal{CF}_{\mathcal{K}}(G)$ and $s\in C_G(u)_{p^\prime}$. It has a section $e_G^u:\mathcal{BCF}_\mathcal{K}(C_G(u))\longrightarrow
\mathcal{CF}_\mathcal{K}(G)$ such that for $\varphi\in\mathcal{BCF}_\mathcal{K}(C_G(u))$, $e_G^u(\varphi)(g)=0$ if the $p$-part of $g$ is not conjugate to $u$ in $G$.
For the block $b$, let $\mathcal{CF}_\mathcal{K}(G,b)$ be the subspace of $\mathcal{CF}_\mathcal{K}(G)$ generated by the elements in $\mathrm{Irr}_\mathcal{K}(G,b)$ and $\mathcal{L}_\mathcal{K}(G,b)$ the group of generalized characters in $b$. Also, let $\mathcal{CF}_\mathcal{K}^\circ(G,b)=
\mathcal{CF}_\mathcal{K}(G,b)\cap\mathrm{Ker}(d_G^1)$ and $\mathcal{L}_\mathcal{K}^\circ(G,b)=
\mathcal{L}_\mathcal{K}(G,b)\cap\mathrm{Ker}(d_G^1)$.
(Puig-Usami [@PU 3.2]) With the above notation and assumption. Let $X$ be an $E$-stable non-empty set of subgroups of $P$ and assume that $X$ contains any subgroup of $P$ containing an element of $X$. Let $\Gamma$ be a map over $X$ sending $Q\in X$ to a bijective isometry $$\Gamma_Q:\mathcal{BCF}_{\mathcal{K}}(C_N(Q),e_\theta)\longrightarrow
\mathcal{BCF}_\mathcal{K}(C_G(Q),b_Q).$$ If $\Gamma$ satisfies the following conditions, then $\Gamma$ is called a $(G,b)$-[[local system]{}]{} over $X$.
\(i) For any $Q\in X$, any $\eta\in\mathcal{BCF}_\mathcal{K}(C_N(Q),e_\theta)$ and any $s\in E$, we have $\Gamma_Q(\eta)^s=\Gamma_{Q^s}(\eta^s)$.
\(ii) For any $Q\in X$ and any $\eta\in\mathcal{L}_\mathcal{K}(C_N(Q),e_\theta)$, the sum $$\sum\limits_{u}e_{C_G(Q)}^u(\Gamma_{Q\cdot\langle u\rangle}
(d_{C_N(Q)}^u(\eta)))$$ where $u$ runs over a set of representatives $U_Q$ for the orbits of $C_E(Q)$ in $P$, is a generalized character of $C_G(Q)$.
Let $\Gamma$ be a $(G,b)$-local system over $X$. Such $\Gamma$ always exists by [@PU 3.4.2]. For any $Q\in X$, we have a map $\Delta_Q:
\mathcal{CF}_\mathcal{K}(C_N(Q),e_\theta)\longrightarrow
\mathcal{CF}_\mathcal{K}(C_G(Q),b_Q)$ defined by $$\Delta_Q(\eta)=
\sum\limits_{u\in U_Q}e_{C_G(Q)}^u(\Gamma_{Q\cdot\langle u\rangle}
(d_{C_N(Q)}^u(\eta))).$$ Then by [@PU 3.3 and 3.4] $\Delta_Q$ gives a perfect isometry between the block $e_\theta$ of $C_N(Q)$ and the block $b_Q$ of $C_G(Q)$ and $\Delta_Q(\lambda\ast\eta)=\lambda\ast\Delta_Q(\eta)$ for any $\lambda\in\mathcal{CF}_\mathcal{K}(P)^{C_E(Q)}$ and $\eta\in\mathcal{CF}_\mathcal{K}(C_N(Q))$. Here, $\mathcal{CF}_\mathcal{K}(P)^{C_E(Q)}$ denotes the set of $C_E(Q)$-stable elements of $\mathcal{CF}_\mathcal{K}(P)$ and $\ast$ denotes the $\ast$-construction of charaters due to Brou$\acute{\mathrm{e}}$ and Puig (see [@BP]). Hence, if $X$ contains the trivial subgroup $1$ of $P$, then $\Delta_1$ induces a perfect isometry between the block $e_\theta$ of $N$ and the block $b$ of $G$. Moreover, this is an isotypy in the sense of [@B] by [@WZZ Proposition 2.7].
In [@PU], Puig and Usami developed a criterion for the extendibility of the $(G,b)$-local system. With the notation above. Suppose that $1\not\in X$ and let $Q$ be a maximal subgroup of $P$ such that $Q\not\in X$. Denote by $X^\prime$ the union of $X$ and the $E$-orbit of $Q$. For any subset $Y$ of $\mathcal{O}C_N(Q)$, denote by $\bar{Y}$ the image of $Y$ under the canonical map from $\mathcal{O}C_N(Q)$ to $\mathcal{O}C_N(Q)/Q$. We have the similar notation for $\mathcal{O}C_G(Q)$. So $\bar{e}_\theta$ and $\bar{b}_Q$ are the blocks of $\bar{C}_N(Q)$ and $\bar{C}_G(Q)$ respectively. Set $\Delta_Q^\circ=
\sum\limits_{u\in U_Q-Q}e_{C_G(Q)}^u\circ\Gamma_{Q\cdot\langle u\rangle}
\circ d_{C_N(Q)}^u$ (see [@PU 3.6.2]). By [@PU Proposition 3.7 and Remark 3.8], $\Delta_Q^\circ$ induces a bijective isometry $$\bar{\Delta}_Q^\circ:
\mathcal{CF}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)\cong
\mathcal{CF}_\mathcal{K}^\circ(\bar{C}_G(Q),\bar{b}_Q)$$ such that $\bar{\Delta}_Q^\circ(\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta))=
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_G(Q),\bar{b}_Q)$. Clearly, $\bar{\Delta}_Q^\circ(\lambda\ast\eta)=
\lambda\ast\bar{\Delta}_Q^\circ(\eta)$ for $\lambda\in\mathrm{Irr}_\mathcal{K}(\bar{P})^{C_E(Q)}$ and $\eta\in\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ (see [@W05 Case 2.2]) and $\bar{\Delta}_Q^\circ$ is $N_E(Q)$-stable. The following is the key criterion of extendibility.
([@PU Proposition 3.11])\[extendibility\] With the notation above, the $(G,b)$-local system $\Gamma$ over $X$ can be extended to a $(G,b)$-local system $\Gamma^\prime$ over $X^\prime$ if and only if $\bar{\Delta}_Q^\circ$ can be extended to an $N_E(Q)$-stable bijective isometry $$\bar{\Delta}_Q:\mathcal{CF}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_{\theta})\cong
\mathcal{CF}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ such that $\bar{\Delta}_Q(\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta))=
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$.
In order to prove Theorem \[MT\], it suffices to show that there is a $(G,b)$-local system over the set of all the subgroups $P$. Hence, by Proposition \[extendibility\], we can assume that there is a $(G,b)$-local system $\Gamma$ over $X$ such that $1\not\in X$ and $Q$ is a maximal subgroup of $P$ such that $Q\not\in X$.
\[MT’\] With the notation above and assumptions of Section $2$. Then $\bar{\Delta}_Q^\circ$ can be extended to an $N_E(Q)$-stable bijective isometry $$\bar{\Delta}_Q:\mathcal{CF}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_{\theta})\cong
\mathcal{CF}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ such that $\bar{\Delta}_Q(\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta))=
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$.
By the structure of $E$ and $P$, $C_E(Q)$ has only three possibilities: $1$, $E$ and $E_1$ or $E_2$. So we will divided the proof into $3$ cases.
Assume that $C_E(Q)=1$.
Then the blocks $e_\theta$ of $C_N(Q)$ and $b_Q$ of $C_G(Q)$ are nilpotent. By the same argument as in [@PU 4.4], $\bar{\Delta}_Q^\circ$ can be extended to an $N_E(Q)$-stable bijective isometry $\bar{\Delta}_Q$.
Assume that $C_E(Q)=E$.
Then $Q$ has to be trivial subgroup of $P$ and $N_E(Q)=E$. So $\bar{C}_N(Q)=N$ and $\bar{C}_G(Q)=G$ and we have a bijective isometry $$\bar{\Delta}^\circ:\mathcal{CF}_\mathcal{K}^\circ(N,e_\theta)\longrightarrow
\mathcal{CF}_\mathcal{K}^\circ(G,b)$$ such that $\bar{\Delta}^\circ(\mathcal{L}_\mathcal{K}^\circ(N,e_\theta))=
\mathcal{L}_\mathcal{K}^\circ(G,b)$.
The following technique we adopt to extend $\bar{\Delta}^\circ$ is essentially due to Kessar and Linckelmann (see [@KL Theorem 4.1]).
By Lemma \[irreducible character of c\], we have the following disjoint union $$\mathrm{Irr}_\mathcal{K}(N,e_\theta)=\Lambda_1\cup\Lambda_2,$$ where $\Lambda_1$ consists of irreducible ordinary characters of dimension $l$ and $\Lambda_2$ consists of irreducible ordinary characters of dimension $l^2$. Hence, $|\Lambda_1|=p^n+p^m-1$ and $|\Lambda_2|=\frac{p^n-1}{l}\cdot\frac{p^m-1}{l}$. We can assume that $n\geq 2$. Then $|\Lambda_1|>2$ and $|\Lambda_2|>2$. Choose an element $\psi_i\in\Lambda_i$ and set $\Lambda_i^\prime=\Lambda_i-\{\psi_i\}$ for $i=1,2$. Since $l_{N}(e_\theta)=1$, it is easy to see $$\mathcal{B}=\{\psi_1-\psi_1^\prime\,|\,\psi_1^\prime\in\Lambda_1^\prime\}
\cup\{\psi_2-\psi_2^\prime\,|\,\psi_2^\prime\in\Lambda_2^\prime\}
\cup\{\psi_2-l\psi_1\}$$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ(N,e_\theta)$. Since $p$ is odd, $|\Lambda_i^\prime|\geq 3$ for $i=1,2$. So by the same argument in [@PU 4.4], for any $i=1,2$, there exists a subset $\Omega_i=\{\chi_{\psi_i},\chi_{\psi_i^\prime}\,|\,
\psi_i^\prime\in\Lambda_i^\prime\}$ of $\mathrm{Irr}_\mathcal{K}(G,b)$ and $\delta_i\in\{\pm 1\}$ such that $\bar{\Delta}^\circ(\psi_i-\psi_i^\prime)=
\delta_i(\chi_{\psi_i}-\chi_{\psi_i^\prime})$. Since $\langle\psi_1-\psi_1^\prime,
\psi_2-\psi_2^\prime\rangle=0$ for any $\psi_1^\prime\in\Lambda_1^\prime$ and $\psi_2^\prime\in\Lambda_2^\prime$, $\{\chi_{\psi_1},\chi_{\psi_1^\prime}\,|\,
\psi_1^\prime\in\Lambda_1^\prime\}$ and $\{\chi_{\psi_2},\chi_{\psi_2^\prime}\,|\,
\psi_2^\prime\in\Lambda_2^\prime\}$ have trivial intersection. Denote $\psi_2-l\psi_1$ by $\mu$. Then $\langle\mu,
\psi_1-\psi_1^\prime\rangle=-l$ for all $\psi_1^\prime\in\Lambda_1^\prime$. Thus $$\label{equation E}
\begin{array}{ll}
\bar{\Delta}^\circ(\mu)=
\delta_1(a-l)\chi_{\psi_1}+
\delta_1a\sum\limits_{\psi_1^\prime\in\Lambda_1^\prime}\chi_{\psi_1^\prime}+
\Xi
\end{array}$$ for some integer $a$ and some element $\Xi\in\mathcal{L}_\mathcal{K}(N,e_\theta)$ not involving any of elements in $\Omega_1$. Since $\langle\mu,
\psi_2-\psi_2^\prime\rangle=1$ and $\bar{\Delta}^\circ(\psi_2-\psi_2^\prime)=
\delta_2(\chi_{\psi_2}-\chi_{\psi_2^\prime})$, $\Xi$ must involve one of the two characters occuring in $\bar{\Delta}^\circ(\psi_2-\psi_2^\prime)$ for any $\psi_2^\prime\in\Lambda_2^\prime$. Taking norms on both sides in equation (\[equation E\]), we have $$\label{inequation norm}
\begin{array}{ll}
&1+l^2\geq(a-l)^2+(p^n+p^m-2)a^2=
(p^n+p^m-1)a^2-2la+l^2\\
\Longleftrightarrow
&1\geq(p^n+p^m-1)a^2-2la
\end{array}$$
Suppose that $a\leq0$. Since $a$ is integer and $p^n+p^m-1,l$ are positive integers, $a$ has to be $0$.
Suppose that $a>0$. Since $p^n+p^m-1>2l$, $(p^n+p^m-1)a^2-2la>(p^n+p^m-1)(a^2-a)$. This forces $a=1$. Hence, $a=0$ or $1$. Notice that $\Xi\neq 0$. This implies (\[inequation norm\]) is a proper inequality. So $a$ must be $0$. Then equation (\[equation E\]) becomes $$\bar{\Delta}^\circ(\mu)=-\delta_1l\chi_{\psi_1}+\Xi.$$ Comparing norms, we have $\langle\Xi,\Xi\rangle=1$.
For any $\psi_2^\prime,\psi_2^{\prime\prime}\in\Lambda_2^\prime$, $$\langle\bar{\Delta}^\circ(\mu),
\delta_2(\chi_{\psi_2}-\chi_{\psi_2^\prime})\rangle=
\langle\mu,
\psi_2-\psi_2^\prime\rangle=1$$ and $$\langle\bar{\Delta}^\circ(\mu),
\delta_2(\chi_{\psi_2^{\prime}}-\chi_{\psi_2^{\prime\prime}})\rangle=
\langle\mu,
\psi_2^\prime-\psi_2^{\prime\prime}\rangle=0.$$ Then $\Xi=\delta_2\chi_{\psi_2}$. But $\bar{\Delta}^\circ(\mu)(1)=0$. This forces $\delta_1=\delta_2$. Since $\mathcal{B}$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ(N,e_\theta)$, $\mathrm{Irr}_\mathcal{K}(G,b)=\Omega_1\cup\Omega_2$. Hence, we get a bijective isometry $\bar{\Delta}$ from $\mathcal{L}_\mathcal{K}(N,e_\theta)$ to $\mathcal{L}_\mathcal{K}(G,b)$ mapping $\psi_i$ and $\psi_i^\prime$ to $\chi_{\psi_i}$ and $\chi_{\psi_i^\prime}$ respectively, where $i=1,2$. In particular, $l_G(b)=l_N(e_\theta)=1$. Clearly, it is an extension of $\bar{\Delta}^\circ$. Since $\bar{\Delta}^\circ$ is $E$-stable and $l_G(b)=l_N(e_\theta)=1$, $\bar{\Delta}$ is also $E$-stable.
Assume that $C_E(Q)=E_i$ for some $i=1,2$.
We can assume that $C_E(Q)=E_1$ and then $1\neq Q\leq P_2$ and $N_E(Q)=E$. It suffices to prove that $\bar{\Delta}_Q^\circ$ can extend to an $E_2$-stable bijective isometry $\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$.
By [@W14 Theorem 1], $|\mathrm{Irr}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)|=
|\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)|$ and $l_{\bar{C}_N(Q)}(\bar{e}_\theta)=l_{\bar{C}_G(Q)}(\bar{b}_Q)$ since the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ has a cyclic hyperfocal subgroup. It is clear $C_N(Q)=(P_1\rtimes\tilde{E}_1)\times P_2$ and $C_N(Q)\unlhd N$, where $\tilde{E}_1$ is the preimage of $E_1$ in $\tilde{E}$. Hence, $E_1$ is the inertial quotient of the block $e_\theta$ of $C_N(Q)$ and $P_1$ is a hyperfocal subgroup with respect to $E_1$. By [@W14 Theorem 1], $l_{C_N(Q)}(e_\theta)=l$. We will claim that $N$ acts transitively on $\mathrm{IBr}(C_N(Q),e_\theta)$. Indeed, this holds because $l_N(e_\theta)=1$ by the assumption and $N/C_N(Q)\cong E_2$ is a cyclic group of order $l$.
Denote by $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ the subset of $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)$ consisting of characters covering $\theta$. Then $|\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta|=l$ and we set $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta=\{\tau_i\,|\, i=1,2,\cdots,l\}$, which is transitively acted by $N$. Hence, we can write $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ as $\{\tau^a\,|\,a\in E_2\}$ for any $\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$. By Clifford theorem, we have $\mathrm{Res}_Z^{\tilde{E}_1}(\tau_i)=\theta$ for any $i$ and $\mathrm{Ind}_Z^{\tilde{E}_1}(\theta)
=\sum\limits_{i=1}^l\tau_i$. Let $M$ be a representative of $\tilde{E}_1$-orbit of $\mathrm{Irr}_\mathcal{K}(P_1)-\{1_{P_1}\}$, where $1_{P_1}$ is the trivial character of $P_1$. Then $$\mathrm{Irr}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)=
\{\tau_i\bar{\zeta}_j\,|\, \bar{\zeta}_j\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2),
i=1,2,\cdots,l\}\cup
\{\mathrm{Ind}_{P_1\times Z}^{P_1\rtimes\tilde{E}_1}(\xi\theta)\bar{\zeta}_j\,|\,
\xi\in M,\bar{\zeta}_j\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}.$$ We will write $\mathrm{Ind}_{P_1\times Z}^{P_1\rtimes\tilde{E}_1}(\xi\theta)$ and $\bar{\chi}\cdot 1_{\bar{P}_2}$ as $\mathrm{Ind}(\xi)$ and $\bar{\chi}$ respectively for simplicity. Here, $\bar{\chi}$ is an element of $\mathcal{CF}_\mathcal{K}(\overline{P_1\rtimes\tilde{E}_1})$. Clearly, $\mathrm{Ind}(\xi)$ is $N$ and $E_2$-stable for any $\xi\in M$. Similar to the argument of [@W05 Case 2], $$\{(\sum\limits_{i=1}^l\tau_i-\mathrm{Ind}(\xi))\bar{\zeta}\,|\,\xi\in M,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cup
\{\tau_i-\tau_i\bar{\zeta}\,|\,i=1,2,\cdots,l,
1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}$$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$.
Assume that $\bar{P}_2=1$, i.e., $Q=P_2$.
Set $H=N_G(Q,b_Q)$. Then $H=C_G(Q)N_G(P,b_P)$ and $b_Q$ is still a block of $H$. Let $d$ be the Brauer correspondent of the block $b_Q$ of $H$ in $N_H(P)$. Then $l_{N_H(P)}(d)=1$ by the assumption. We claim that $l_H(b_Q)=1$.
Indeed, considering the canonical map from $\mathcal{O}H$ to $\mathcal{O}(H/Q)$, denote by $\bar{X}$ the image of $X$ under this canonical map for any subset $X$ of $\mathcal{O}H$. Then $\bar{b}_Q$ is still a block of $\bar{C}_G(Q)$ and $\bar{H}/\bar{C}_G(Q)$ is a cyclic group of order $l$. By [@KR Lemma 3.5], $\mathrm{Br}_{\bar{P}}(\bar{b}_Q)=\overline{\mathrm{Br}_P(b_Q)}$. Since $l_{N_H(P)}(d)=1$, $\bar{d}$ is still a block of $\bar{N}_H(P)$. Therefore, $\mathrm{Br}_{\bar{P}}(\bar{b}_Q)=\bar{d}$ is a block of $\bar{N}_H(P)$. Suppose that the blocks of $\bar{H}$ covering the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ have the same defect group $\bar{P}$. Then $\bar{b}_Q$ is a block of $\bar{H}$ since $\mathrm{Br}_{\bar{P}}(\bar{b}_Q)$ is a block of $\bar{N}_H(P)$ and $N_{\bar{H}}(\bar{P})=\bar{N}_H(P)$. Hence, it has a defect group $\bar{P}$ which is cyclic by our assumption. In particular, we have $l_H(b_Q)=l_{\bar{H}}(\bar{b}_Q)=l_{N_H(P)}(d)=1$ since $N_{\bar{H}}(\bar{P})=\bar{N}_H(P)$. Consequently, the argument follows from the lemma below.
[*[**Lemma 3.4**]{} Let $L$ be a normal subgroup of $K$ such that $K/L$ is a cyclic $p^\prime$-group. Let $i$ be a $K$-stable block of $L$ with defect group $D$. For any block $e$ of $K$ covering $i$, $e$ has defect group $D$.*]{}
[*Proof.*]{} We will prove it by induction on $K/L$. Let $M\leq K$ such that $M$ contains $L$ and $|M/L|$ is a prime. Then $M\trianglelefteq K$ and $K/M$ is still a cyclic $p^\prime$-group. Denote by $M[i]$ the subgroup of $M$ consisting of elements acting on $\mathcal{O}Li$ as inner automorphisms. Therefore, $M[i]=M$ or $L$. Let $f$ be a block of $M$ covered by $e$. So $f$ covers the block $i$ of $L$. If $M[i]=M$, then $\mathcal{O}Mf$ and $\mathcal{O}Li$ are source algebra equivalent by [@K90 Theorem 7]. In particular, the block $f$ has defect group $D$. If $M[b]=L$, then $f=i$ by [@D Theorem 3.5] and certainly they have the same defect group. In conclusion, $D$ is a defect group of the block $f$. Let $K_f$ be the stabilizer of $f$ in $K$. Then blocks of $K_f$ covering $f$ have defect group $D$ by induction. So is $e$.
Moreover, we claim that there is a regular $E_2$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$, namely, $H$ acts transitively on it.
Indeed, since the block $\bar{b}_Q$ of $\bar{H}$ has a cyclic defect group, it must be nilpotent. By [@P11 Theorem 3.13], the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ is basic Morita equivalent to its Brauer correspondent. Note that the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ is not nilpotent since $l>1$. This implies that every irreducible Brauer character of the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ can be uniquely lifted to an irreducible ordinary character by the theory of cyclic blocks.
On the other hand, since $l_{C_G(Q)}(b_Q)=l$ and $l_H(b_Q)=1$ and $H/C_G(Q)\cong E_2$ has order $l$, $H$ acts transitively on $\mathrm{IBr}(C_G(Q),b_Q)$. Combining this with the argument above, there exits a regular $H$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$. We are done.
Assume that $|M|=1$.
Then $\mathrm{rank}_\mathcal{O}
(\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta))=1$ and $\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)=
\mathbb{Z}(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. Since there is a regular $E_2$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$, $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)=\{\chi_0\}\cap
\{\chi_1,\chi_2,\cdots,\chi_l\}$ such that $\chi_0$ is $E_2$-stable and $E_2$ acts regularly on $\{\chi_1,\chi_2,\cdots,\chi_l\}$. Then we have $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta_0\chi_0-\sum\limits_{i=1}^l\delta_i\chi_i$$ for some $\delta_0,\delta_i\in\{\pm1\},i=1,2,\cdots,l$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, we have $\delta_1=\delta_2=\cdots=\delta_l=\delta_0$. If we write $\{\tau_1,\tau_2,\cdots,\tau_l\}$ and $\{\chi_1,\chi_2,\cdots,\chi_l\}$ as $\{\tau^a\,|\,a\in E_2\}$ and $\{\chi^a\,|\,a\in E_2\}$ respectively, then we can define a bijective isometry as below $$\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ $$\mathrm{Ind}(\xi)\mapsto\delta_0\chi_0$$ $$\tau^a\mapsto\delta_0\chi^a.$$ It is evident that it is an extension of $\bar{\Delta}_Q^\circ$ and $E_2$-stable. We are done for this case.
Assume that $|M|\geq2$.
Then there are at least two different $\xi_1,\xi_2\in M$. So $\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2)\in
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ and $\langle\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2),
\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2)\rangle=2$. Then there exist $\chi_1\neq\chi_2\in
\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2))=
\delta(\chi_1-\chi_2)$$ for some $\delta\in\{\pm1\}$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, we have ${^a}(\delta\chi_1-\delta\chi_2)=\delta(\chi_1-\chi_2)$ for any $a\in E_2$. This means that $\chi_1$ and $\chi_2$ are both $E_2$-stable.
If there is a $\xi_3\in M$ different from $\xi_1$ and $\xi_2$, then there is a $\chi_3\in\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ different from $\chi_1$ and $\chi_2$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_3))=
\delta\chi_1-\delta\chi_3~\mathrm{or}~
-\delta\chi_2+\delta\chi_3$$ and $\chi_3$ is $E_2$-stable; then we may choose the notation in such a way that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2))=
\delta(\chi_1-\chi_2)
~\mathrm{and}~
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_3))=
\delta(\chi_1-\chi_3)$$ for some $E_2$-stable elements $\chi_1,\chi_2,\chi_3$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q).$
If $|M|\geq 4$, then for any $\xi\in M-\{\xi_1,\xi_2,\xi_3\}$, there is a unique $\chi\in\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)-
\{\chi_1,\chi_2,\chi_3\}$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi))=
\delta(\chi_1-\chi)$$ and $\chi$ is $E_2$-stable.
In conclusion, we have an injective isometry $$\Phi:\mathbb{Z}\{\mathrm{Ind}(\xi)\,|\,\xi\in M\}\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ mapping $\mathrm{Ind}(\xi)$ to $\delta\chi_\xi$ such that $$\Phi(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi^\prime))=
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi^\prime))$$ and $\chi_\xi$ is $E_2$-stable for any $\xi,\xi^\prime\in M$.
Denote $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)-
\{\chi_\xi\,|\,\xi\in M\}$ by $\Omega$. Then $|\Omega|=l$ and $E_2$ acts on $\Omega$. Since there is a regular $E_2$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$, $E_2$ acts regularly on $\Omega$. This means that $\Omega$ can be represented as $\{\chi^a\,|\,a\in E_2\}$ for some $\chi\in\Omega$.
Now we fix an element $\xi$ of $M$. Suppose that $\chi$ does not get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. Then there is $\xi^\prime\in M$ such that $\langle\chi,\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi^\prime)-
\sum\limits_{i=1}^l\tau_i)\rangle\neq 0$ since $\{\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i\,|\,
\xi\in M\}$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ
(\bar{C}_N(Q),\bar{e}_\theta)$. Hence, $\chi$ has to get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)-
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi^\prime)-\sum\limits_{i=1}^l\tau_i)$ which is $\delta(\chi_\xi-\chi_{\xi^\prime})$. This is impossible. So $\chi$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $\xi\in M$. Since $\bar{\Delta}_Q^\circ$ and $\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i$ are $E_2$-stable, $\chi^a$ has to get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $a\in E_2$ and $\xi\in M$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i\rangle=1+l$ and $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi^\prime)\rangle=1$, $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta\chi_\xi-\sum\limits_{a\in E_2}\delta_a\chi^a~\mathrm{or}~
-\delta\chi_{\xi^\prime}-\sum\limits_{a\in E_2}\delta_a\chi^a$, where $\delta_a\in\{\pm1\}$ for any $a\in E_2$. Note that the last situation can happen if and only if $|M|=2$. By switching $\chi_\xi$ and $\chi_{\xi^\prime}$ if necessary, we can assume that $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta\chi_\xi-\sum\limits_{a\in E_2}\delta_a\chi^a$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable and $E_2$ acts regularly on $\Omega$, $\delta_a$ is equal to $\delta$ for any $a\in E_2$. Then we can define an $E_2$-stable bijective isometry as follows $$\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ $$\mathrm{Ind}(\xi)\mapsto\delta\chi_\xi$$ $$\tau^a\mapsto\delta\chi^a.$$ It is clear that $\bar{\Delta}_Q$ is an extension of $\bar{\Delta}_Q^\circ$.
$\bar{P}_2>1$, namely, $Q$ is a non-trivial proper subgroup of $P_2$.
Then $\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}\in
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ for any $\xi\in M$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in
\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$.
Now we fix an element $\xi\in M$. Since $p$ is odd, $|\bar{P}_2|\geq 3$. Then there are at least two elements $\bar{\zeta}$ and $\bar{\zeta}^\prime$ of $\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ different from $1_{\bar{P}_2}$. With the same argument in the first three paragraphs in Case 3.1.2, we can get a subset $\{\chi_\xi,\chi_{\bar{\zeta}}\,|\,
1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})=
\delta(\chi_\xi-\chi_{\bar{\zeta}})$$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, where $\delta\in\{\pm1\}$.
Given any $1\neq a\in E_2$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, ${^a}(\mathrm{Ind}(\xi)\bar{\zeta})=\mathrm{Ind}(\xi)({^a}\bar{\zeta})$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, this means ${^a}\chi_\xi-{^a}\chi_{\bar{\zeta}}=\chi_\xi-\chi_{{^a}\bar{\zeta}}$. Hence, we have $\chi_\xi$ is $E_2$-stable and ${^a}\chi_{\bar{\zeta}}=\chi_{{^a}\bar{\zeta}}$. On the other hand, $$(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})\bar{\zeta}=
(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}^2)-
(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}).$$ Since $\bar{\Delta}_Q^\circ$ is compatible with $\ast$-structure, using $\bar{\Delta}_Q^\circ$ on both sides in the above equality, we can get $$\delta(\chi_\xi-\chi_{\bar{\zeta}})\ast\bar{\zeta}=
\delta(\chi_{\bar{\zeta}}-\chi_{\bar{\zeta}^2}).$$ Therefore, $\chi_{\bar{\zeta}}=\chi_\xi\ast\bar{\zeta}$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$.
Suppose that there is another element $\xi^\prime$ of $M$ different from $\xi$. Similarly, we can get a subset $\{\chi_{\xi^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ such that $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi^\prime)-
\mathrm{Ind}(\xi^\prime)\bar{\zeta})=
\delta^\prime(\chi_{\xi^\prime}-\chi_{\xi^\prime}\ast\bar{\zeta})$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ and $\chi_{\xi^\prime}$ is $E_2$-stable, where $\delta^\prime\in\{\pm 1\}$. Assume that $\{\chi_\xi\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_{\xi^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\neq\emptyset$. Then there is $\bar{\zeta}_0\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ such that $\chi_\xi=\chi_{\xi^\prime}\ast\bar{\zeta}$. If $\bar{\zeta}_0=1_{\bar{P}_2}$, then $\chi_\xi=\chi_{\xi^\prime}$. This implies that $\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}=
\pm(\mathrm{Ind}(\xi^\prime)-\mathrm{Ind}(\xi^\prime)\bar{\zeta})$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$. This is impossible. Then $\bar{\zeta}_0$ is non-trivial. But it implies that $\chi_{\xi^\prime}\ast\bar{\zeta}_0^2=\chi_{\xi^\prime}$ since $\langle\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}_0,
\mathrm{Ind}(\xi^\prime)-\mathrm{Ind}(\xi^\prime)\bar{\zeta}_0\rangle=0$. It is well-known that $\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\backslash\{1_{\bar{P}_2}\}$ acts freely on irreducible ordinary characters of height zero in the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ (see [@R §1]). Hence, $\bar{\zeta}_0^2=1_{\bar{P}_2}$ since the defect group of the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ is cyclic. But it is impossible because $p$ is odd. Then $$\{\chi_\xi\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_{\xi^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}=\emptyset$$ for any different $\xi,\xi^\prime\in M$. It is clear that $\chi_\xi\ast\bar{\zeta}$ is an irreducible ordinary character in the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ by [@BP Corollary]. Then we get an injective isometry $$\Psi:\mathbb{Z}\{\mathrm{Ind}(\xi)\bar{\zeta}\,|\,
\xi\in M,\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}
\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ mapping $\mathrm{Ind}(\xi)\bar{\zeta}$ to $\delta_\xi(\chi_\xi\ast\bar{\zeta})$ such that $\Psi(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})=
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})$ and $\chi_\xi$ is $E_2$-stable for any $\xi\in M$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, where $\delta_\xi\in\{\pm 1\}$.
At the same time, $\tau-\tau\bar{\zeta}\in
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ for any $\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$. Take an element $\tau$ of $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$. With the same arguments as above, we can get an element $\chi_\tau$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ and $\delta_\tau\in\{\pm1\}$ such that $\bar{\Delta}_Q^\circ(\tau-\tau\bar{\zeta})=
\delta_\tau(\chi_\tau-\chi_\tau\ast\bar{\zeta})$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$. Choosing any $1\neq a\in E_2$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, we have $$\delta_{{^a}\tau}(\chi_{{^a}\tau}-\chi_{{^a}\tau}\ast{^a}\bar{\zeta})=
\bar{\Delta}_Q^\circ({^a}\tau-{^a}\tau({^a}\bar{\zeta}))=
{^a}(\bar{\Delta}_Q^\circ(\tau-\tau\bar{\zeta}))=
\delta_\tau({^a}\chi_\tau-{^a}\chi_\tau\ast{^a}\bar{\zeta}).$$ Then $\chi_{{^a}\tau}={^a}\tau$ or $\chi_{{^a}\tau}={^a}\chi_\tau\ast{^a}\bar{\zeta}$. If $\chi_{{^a}\tau}={^a}\chi_\tau\ast{^a}\bar{\zeta}$, then ${^a}\chi_\tau=\chi_{{^a}\tau}\ast{^a}\bar{\zeta}$. Therefore, $\chi_{{^a}\tau}=\chi_{{^a}\tau}\ast{^a}(\bar{\zeta}^2)$, which is impossible. Hence, $\chi_{{^a}\tau}={^a}\chi_\tau$ and $\delta_{{^a}\tau}=\delta_\tau$ for any $a\in E_2$ since $E_2$ acts transitively on $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$. And we denote $\delta_\tau$ by $\delta$. By the facts that $\langle\tau-\tau\bar{\zeta},
\tau^\prime-\tau^\prime\bar{\zeta}^\prime\rangle=0$ and $\langle\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta},
\tau-\tau\bar{\zeta}^\prime\rangle=0$ for any $\tau\neq\tau^\prime\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ and $\xi\in M$ and $\bar{\zeta},\bar{\zeta}^\prime\in
\mathrm{Irr}_\mathcal{K}(\bar{P}_2)-\{1_{\bar{P}_2}\}$, we can get $$\{\chi_\tau\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_{\tau^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}=\emptyset$$ and $$\{\chi_\xi\ast\bar{\zeta}\,|\,
\xi\in M, \bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_\tau\ast\bar{\zeta}\,|\,
\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}=\emptyset.$$ Hence, we have a well-defined $E_2$-stable bijective isometry as below $$\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ $$\mathrm{Ind}(\xi)\bar{\zeta}\mapsto\delta_\xi\chi_\xi\ast\bar{\zeta}$$ $${^a}\tau\bar{\zeta}\mapsto\delta{^a}\chi_\tau\ast\bar{\zeta}.$$ It suffices to show that $\bar{\Delta}_Q$ is an extension of $\bar{\Delta}_Q^\circ$, namely, $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta_\xi\chi_\xi-\delta\sum\limits_{i=1}^l\chi_{\tau_i}$$ for any $\xi\in M$.
Choose an element $\xi$ of $M$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\tau-\tau\bar{\zeta}\rangle=-1$, then at least $\chi_\tau$ and $\chi_\tau\ast\bar{\zeta}$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ and $\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)-\{1_{\bar{P}_2}\}$.
Keep the notation as above. Suppose that there are $\tau$ and $\bar{\zeta}$ such that $\chi_\tau\ast\bar{\zeta}$ gets involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\tau\bar{\zeta}-\tau\bar{\zeta}^\prime\rangle=0$ for any $\bar{\zeta}^\prime\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ different from $\bar{\zeta}$ and $1_{\bar{P}_2}$, $\chi_\tau\ast\bar{\zeta}$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $\bar{\zeta}$. At the same time, since $\bar{\Delta}_Q^\circ$ is $E_2$-stable and $\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i$ is $E_2$-stable, we have ${^a}(\chi_\tau\ast\bar{\zeta})$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $a\in E_2$ and $\bar{\zeta}$. Then there are at least $l\cdot(|\bar{P}_2|-1)$ different irreducible characters involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. This is impossible since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i\rangle=1+l$ and $|\bar{P}_2|-1\geq 2$ and $l>1$.
So for any $\xi\in M$, $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
a_\chi\chi-\delta\sum\limits_\tau\chi_\tau$. Here, $a_\chi\in\{\pm 1\}$ and $\chi$ is an element of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)-
\{\chi_\tau\,|\,
\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta\}$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}\rangle=1$ for any $\bar{\zeta}\neq 1_{\bar{P}_2}$, we have $a_\chi=\delta_\xi$ and $\chi=\chi_\xi$. We are done.
Then the proof of Theorem \[MT\] will follow by Theorem \[MT’\] and [@PU 3.4.2].
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|
---
abstract: |
We study the volume of the nodal set of eigenfunctions of the Laplacian on the ${m}$-dimensional sphere. It is well known that the eigenspaces corresponding to ${E_{n}}=n(n+{m}-1)$ are the spaces ${\mathcal{E}_n}$ of spherical harmonics of degree $n$, of dimension ${\mathcal{N}}$. We use the multiplicity of the eigenvalues to endow ${\mathcal{E}_n}$ with the Gaussian probability measure and study the distribution of the ${m}$-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to $\sqrt{{E_{n}}}$. One of our main results is bounding the variance of the volume to be $O(\frac{{E_{n}}}{\sqrt{{\mathcal{N}}}})$.
In addition to the volume of the nodal set, we study its Leray measure. We find that its expected value is $n$ independent. We are able to determine that the asymptotic form of the variance is $\frac{const}{{\mathcal{N}}}$.
address: 'Centre de recherches mathématiques (CRM), Université de Montréal C.P. 6128, succ. centre-ville Montréal, Québec H3C 3J7, Canada'
author:
- Igor Wigman
title: On the distribution of the nodal sets of random spherical harmonics
---
[^1]
Introduction
============
Let $M$ be a smooth compact manifold and $f$ a real valued function on $M$. We define its nodal set to be the subset of $M$, where $f$ vanishes and we are interested mainly in the nodal sets of eigenfunctions of the Laplacian on $M$. It is known [@Cheng], that generically, the nodal sets are smooth submanifolds of $M$ with codimension $1$. For example, if $M$ is a surface, the nodal sets are [*lines*]{}. One is interested in studying their volume (i.e. the length of the nodal line for the $2$-dimensional case), and other nodal properties for highly excited eigenstates. It was conjectured by Yau that the volume of the nodal set is bounded from above and below by a multiple of the square root of the Laplace eigenvalue. The lower bound was proven by Bruning and Gromes [@Bruning-Gromes] and Bruning [@Bruning] for the planar case. Donnelly-Fefferman’s celebrated result [@Donnelly-Fefferman] resolved Yau’s conjecture for real analytic metrics. However the general case of a smooth manifold is still open.
Spherical Harmonics
-------------------
In this paper, we study the nodal sets for the eigenfunctions of the Laplacian $\Delta$ on the ${m}$-dimensional unit sphere ${{\mathcal{S}}^{m}}$. It is well known that the eigenvalues $E$ of the Laplace equation $$\Delta f +E f = 0$$ on ${{\mathcal{S}}^{m}}$ are all the numbers of the form $$\label{eq:eigval def} E = {E_{n}}= n(n+{m}-1),$$ where $n$ is an integer. Given a number ${E_{n}}$, the corresponding eigenspace is the space ${\mathcal{E}_n}$ of the spherical harmonics of degree $n$. Its dimension is given by $$\label{eq:eigspcdim asymp} {\mathcal{N}}= {\mathcal{N}}_{n}=
\frac{2n+{m}-1}{n+{m}-1} {n+{m}-1 \choose
{m}-1} \sim \frac{2}{({m}-1)!}n^{{m}-1}.$$ Given an integral number $n$, we fix an $L^2({{\mathcal{S}}^{m}})$ orthonormal basis of ${\mathcal{E}_n}$ $$\eta_{1}(x) = \eta_{1}^{n} (x), \, \eta_{2}(x) = \eta_{2}^{n}
(x),\ldots ,\eta_{{\mathcal{N}}}(x) = \eta_{{\mathcal{N}}}^{n} (x),$$ giving an identification ${\mathcal{E}_n}\cong{\mathbb{R}}^{{\mathcal{N}}}$. For further reading on the spherical harmonics we refer the reader to [@AAR], chapter 9.
Random model
------------
We consider a [*random eigenfunction*]{} $$\label{eq:rand eigfnc def} f(x)= \sqrt{{\frac{|{{\mathcal{S}}^{m}}|}{{\mathcal{N}}}}}
\sum\limits_{k=1}^{{\mathcal{N}}} a_{k}\eta_{k}(x),$$ where $a_k$ are Gaussian $N(0,1)$ i.i.d. which we assume to be defined on the same sample space $\Omega$. That is, we use the identification ${\mathcal{E}_n}\cong{\mathbb{R}}^{{\mathcal{N}}}$ to induce the Gaussian probability measure $\upsilon$ on ${\mathcal{E}_n}$ as $$d\upsilon(f) = e^{-\frac{1}{2}\|\vec{a}\|^2}\frac{da_{1} \cdot\ldots\cdot
da_{{\mathcal{N}}}}{(2\pi)^{{\mathcal{N}}/2}},$$ where $\vec{a} = (a_{i})\in{\mathbb{R}}^{{\mathcal{N}}}$ are as in .
Note that $\upsilon$ is invariant with respect to the orthonormal basis for ${\mathcal{E}_n}$. As usual, for any random variable $X$ on $\Omega$, we denote its expectation ${\mathbb{E}}X$. For example, with the normalization factor in , for every [*fixed*]{} point $x\in{{\mathcal{S}}^{m}}$, one has $$\label{eq:E(f(x)^2)=1} {\mathbb{E}}[f(x)^2] =
{\frac{|{{\mathcal{S}}^{m}}|}{{\mathcal{N}}}}\sum\limits_{i=1}^{{\mathcal{N}}} \eta_i(x) ^2 = 1,$$ a simple corollary from the addition theorem (see ).
Any characteristic $X(L)$ of the nodal set $$L=L_{f}=\{x\in{{\mathcal{S}}^{m}}:\: f(x)=0 \}$$ is a random variable defined on the same sample space $\Omega$. We are interested in the distribution of two different characteristics. The most natural characteristic of the nodal set $L_{f}$ of $f$ is, of course, its $({m}-1)$-dimensional volume ${\mathcal{Z}}={\mathcal{Z}}(f)$. The study of the distribution of the random variable ${\mathcal{Z}}$ for a random $f\in{\mathcal{E}_n}$ is one of the goals of the present paper.
Berard [@Berard] showed that the expected volume ${\mathbb{E}}{\mathcal{Z}}$ is $${\mathbb{E}}{\mathcal{Z}}(f) = const\cdot \sqrt{{E_{n}}}$$ (see proposition \[prop:exp len\]) and Neuheisel [@Neuheisel] proved that as $n\rightarrow\infty$, $$\label{eq:var bnd Neuheisel}
{\operatorname{Var}}({\mathcal{Z}}) = O\bigg(\frac{{E_{n}}}{n^{\frac{({m}-1)^2}{3{m}+1}}}\bigg) =
O\bigg(\frac{{E_{n}}}{{\mathcal{N}}^{\frac{{m}-1}{3{m}+1}}}\bigg).$$
\[rem:Neuh sphr mod\] Rather than taking $a_{k}$ standard Gaussian i.i.d., Neuheisel assumes that the vector $\vec{a}=(a_{k})\in{\mathbb{R}}^{{\mathcal{N}}}$ is chosen uniformly on the unit sphere ${\mathcal{S}}^{{\mathcal{N}}-1}$. However, it is easy to see, that, since ${\mathcal{Z}}(f)={\mathcal{Z}}(b\cdot f)$ for any constant $b\in{\mathbb{R}}$, both of those models are equivalent.
The volume of the nodal line of a random eigenfunction on the torus $$\mathcal{T}^{{m}} = {\mathbb{R}}^{{m}}/{\mathbb{Z}}^{{m}}$$ was studied by Rudnick and Wigman [@RW]. In this case, it is not difficult to see that the expectation is given by ${\mathbb{E}}{\mathcal{Z}}(f^{\mathcal{T}^{m}}) = const\cdot \sqrt{E}$. Moreover, they prove that as the eigenspace dimension ${\mathcal{N}}$ grows to infinity, the variance is bounded by $${\operatorname{Var}}{\mathcal{Z}}(f^{\mathcal{T}^{m}}) =
O\bigg(\frac{E}{\sqrt{{\mathcal{N}}}}\bigg),$$ which, in particular, implies that the tails of the distribution of the normalized random variable $\frac{{\mathcal{Z}}}{{\mathbb{E}}{\mathcal{Z}}}$ die.
More generally, one may also consider a random model of eigenfunctions for a generic compact manifold $M$. Of course, for generic manifolds, one does not expect the Laplacian to have any multiplicities, so that we cannot introduce a Gaussian ensemble on the eigenspace. Let $E_{j}$ be the eigenvalues and $\phi_{j}$ the corresponding eigenfunctions. It is well known that the $E_{j}$ are discrete, $E_{j}\rightarrow\infty$ and $L^{2}(M) = span\{\phi_{j} \}$.
In this case, rather than considering random eigenfunctions, one considers random [*combinations*]{} of eigenfunctions with growing energy window of either type $$f^{L}(x) = \sum\limits_{E_{j}\in [0,E]} a_{j}\phi_{j}(x)$$ (called the long range), or $$f^{S}(x) = \sum\limits_{\sqrt{E_{j}}\in [\sqrt{E},\sqrt{E}+1]} a_{j}\phi_{j}(x),$$ (called the short range), as $E\rightarrow\infty$. Berard [@Berard] found that $${\mathbb{E}}{\mathcal{Z}}(f^{L}) \sim c_{M} \cdot \sqrt{E}$$ and recently Zelditch [@Z1] proved that $${\mathbb{E}}{\mathcal{Z}}(f^{S}) \sim c_{M} \cdot \sqrt{E},$$ notably with the same constant $c_{M}$ for both the long and the short ranges.
Berry [@Berry; @2002] computed the expected length of nodal lines for isotropic, monochromatic random waves in the plane, which are eigenfunctions of the Laplacian with eigenvalue ${E_{n}}$. He found that the expected length (per unit area) is again of size approximately $\sqrt{{E_{n}}}$ and argued that the variance should be of order $\log {E_{n}}$.
Leray nodal measure {#sec:ler meas def}
-------------------
Another property of the nodal line we consider is its [*Leray measure*]{} (also called [*the microcanonical measure*]{}). Given a function $f$ on ${{\mathcal{S}}^{m}}$, we define the Leray nodal measure to be $$\label{eq:leray meas def}
{\mathcal {L}}(f):=\lim\limits_{\epsilon\to 0} \frac 1{2\epsilon} {\operatorname{meas}}\{x\in {{\mathcal{S}}^{m}}: |f(x)|<\epsilon \},$$ provided that the last limit exists. One may write the definition of the Leray nodal measure formally as $$\label{eq:leray meas form}
{\mathcal {L}}(f):=\int\limits_{{{\mathcal{S}}^{m}}} \delta(f(x)) dx,$$ where $\delta$ is the Dirac delta function.
As is well known, the limit exists when $\nabla f\neq 0$ on the nodal set in which case $${\mathcal {L}}(f) = \int_{\{x:f(x)=0\}} \frac{d\nu'(x)}{|\nabla f(x)|},$$ where $\nu'$ is the Riemannian hypersurface measure on the nodal set. This holds almost always on ${\mathcal{E}_n}$ (see section \[sec:sing func\]).
The distribution of the Leray nodal measure on the sphere was also considered by Neuheisel. As in case of the volume, one may compute the expected value $${\mathbb{E}}{\mathcal {L}}= \frac{|{{\mathcal{S}}^{m}}|}{\sqrt{2\pi}}$$ using a rather standard computation (see proposition \[prop:exp ler meas\]) and Neuheisel proved that the variance is bounded by $$\label{eq:var Ler bnd Neuheisel}
{\operatorname{Var}}{\mathcal {L}}= O\bigg(\frac{1}{n^{\frac{{m}-1}{2}}}\bigg) = O(\frac{1}{\sqrt{{\mathcal{N}}}})$$
Here, as in the case for the volume, Neuheisel considered a slightly different variation of the random model (see remark \[rem:Neuh sphr mod\]). Even though the Leray nodal measure is not invariant under dilations, i.e. $${\mathcal {L}}(b\cdot f) = \frac{1}{b}{\mathcal {L}}(f),$$ those models are still equivalent [*asymptotically*]{}, as ${\mathcal{N}}\rightarrow\infty$.
The Leray measure ${\mathcal {L}}(f^{\mathcal{T}^{m}})$ for the random eigenfunctions on the torus $\mathcal{T}^{n}$ was considered by Oravecz, Rudnick and Wigman [@ORW]. The expectation is given by $${\mathbb{E}}{\mathcal {L}}(f^{\mathcal{T}^{m}}) = \frac{1}{\sqrt{2\pi}}.$$ These authors were able to establish the variance to be asymptotic to $${\operatorname{Var}}{\mathcal {L}}(f^{\mathcal{T}^{m}})\sim c\cdot \frac{1}{{\mathcal{N}}}$$ for some $c>0$, for $m=2$ and $m \ge 5$.
The expectation
---------------
\[prop:exp ler meas\] For $n$ sufficiently large, the expectation of the Leray nodal measure of the random eigenfunction is given by $${\mathbb{E}}{\mathcal {L}}(f) = \frac{|{{\mathcal{S}}^{m}}|}{\sqrt{2\pi}}.$$
One has \[prop:exp len\] $$\label{eq:Elen=c sqrt(E)} {\mathbb{E}}{\mathcal{Z}}(f) = c_{{m}}
\cdot\sqrt{{E_{n}}},$$ with the constant $c_{{m}}$ defined by $$\label{eq:c def} c_{{m}} =
\frac{2\pi^{{m}/2}}{\sqrt{{m}}\Gamma(\frac{m}{2})}.$$
Statement of the main results
-----------------------------
Our main results concern the variance of the Leray nodal measure ${\mathcal {L}}$ and the volume ${\mathcal{Z}}$ of the nodal set. We improve on Neuheisel’s results and , and need to use some of the steps in his work; however because some of the arguments in Neuheisel contain gaps, we need to redo them, partially accounting for the length of this paper.
For ${\mathcal {L}}$ we were able to determine its asymptotics precisely.
\[thm:var ler meas\] As $n\rightarrow\infty$, the variance of the Leray nodal measure is asymptotic to $$\label{eq:var ler meas} {\operatorname{Var}}{\mathcal {L}}(f) \sim
\frac{2^{{m}-2}\pi^{\frac{{m}-2}{2}}\Gamma(\frac{{m}}{2})
|{{\mathcal{S}}^{m}}|}{({m}-1)!} \cdot \frac{1}{{\mathcal{N}}}.$$
One should compare the asymptotic result to Neuheisel’s bound .
Note that unlike the torus, our proof here works for any dimension ${m}\ge 2$, including ${m}=3,\, 4$. The reason is that for the sphere, the so-called two point function $u$ (to be defined, see ) is related to the ultraspherical polynomials, a standard family of orthogonal polynomials [@SZ]. In particular, using Hilb’s asymptotics for the ultraspherical polynomials, it is easy to show that the 4th moment of $u$ is dominated by its second moment (see lemmas \[lem:2nd mom Qn\] and \[lem:4th mom Qn\]).
Unlike the spherical case, the two point function for the $d$-dimensional torus is related to the distribution of points $$\vec{n} = (n_{1},\ldots n_{d})\in {\mathbb{Z}}^d$$ so that $$\| \vec{n} \|^2
= n_{1}^2+\ldots +n_{d}^2 = \frac{E}{4\pi^2}.$$ For $d \ge 5$ a strong equidistribution result for $\vec{n}$ implies in particular that the 4th moment of $u$ is dominated by its second moment. For the two-dimensional case we used a special result due to Zygmund. The remaining cases $d=3,\, 4$ are, to our best knowledge, open.
Concerning the volume, we have the following result:
\[thm:var length\] One has $$\label{eq:var length} {\operatorname{Var}}{\mathcal{Z}}(f) =
O\bigg(\frac{{E_{n}}}{\sqrt{{\mathcal{N}}}}\bigg),$$ asymptotically as $n\rightarrow\infty$.
Note that theorem \[thm:var length\] implies that the variance of the [*normalized*]{} random variable $\tilde{{\mathcal{Z}}}:=\frac{{\mathcal{Z}}}{{\mathbb{E}}{\mathcal{Z}}}$ with expectation $1$, vanishes as $n\rightarrow\infty$. Thus, in particular, the tails of the distribution of $\tilde{{\mathcal{Z}}}$ “die", that is, for every $\epsilon>0$, most of the mass of $\tilde{{\mathcal{Z}}}$ is concentrated in $[1-\epsilon, 1+\epsilon]$. In addition, theorem \[thm:var length\] bounds the “typical" size of the tail of the distribution of $\tilde{{\mathcal{Z}}}$ (and thus of ${\mathcal{Z}}$). One should compare to , obtained by Neuheisel. Partially motivated by the recent result [@GW] for the analogous ensemble of random one dimensional trigonometric polynomials, it may be possible to improve the bound to ${E_{n}}/{\mathcal{N}}$ .
On the proofs of the main results
---------------------------------
The spherical case offers some marked differences from that of the torus [@ORW] and [@RW]. Unlike the torus, which is identified with the unit square with its sides pairwise glued, the sphere possesses a nontrivial geometry. In the course of the proof of the main results, one has to study the joint distribution of the gradients $\nabla f(x)$ and $\nabla f(y)$ as random vectors, where $x,y\in {{\mathcal{S}}^{m}}$ are [*fixed*]{}, and $f\in{\mathcal{E}_n}$ is randomly chosen. The main obstacle here is that for different points $x\in{{\mathcal{S}}^{m}}$, the gradients live in different spaces, namely, the tangent spaces $T_{x}({{\mathcal{S}}^{m}})$ which are, in general, different.
One then has to canonically identify the spaces $T_{x}({{\mathcal{S}}^{m}})$ via a family of isometries $\phi_{x}$ smooth w.r.t. $x$. In reality, such a choice is not possible for every $x$, and we treat this complication in section \[sec:orthonorm bas corr mat exp\].
Once the geometric problems are resolved, the treatment of the so-called [*two-point*]{} (or, alternatively, [*covariance*]{}) function and its derivatives is more standard, related to the well-known ultraspherical polynomials (see [@SZ] or appendix \[sec:ultrasph pol\]). In particular, we find that the geometrical structure of the so-called [*singular sets*]{} on ${{\mathcal{S}}^{m}}$ is less complicated than the singular set on the torus (see section \[sec:sing set\]).
Some Conventions
----------------
Throughout the paper, the letters $x,y$ and $z$ will denote either points on the sphere ${{\mathcal{S}}^{m}}$ or spherical variables and $t$ will denote a real variable. For $x,\, y\in{{\mathcal{S}}^{m}}$, $d(x,y)$ will stand for the spherical distance between $x$ and $y$. The letters $\mu$, $\nu$, $\upsilon$ will be reserved for measures, where the measure $\nu$ will stand for the uniform measure on ${{\mathcal{S}}^{m}}$ so that $d\nu(x) = dx$.
Finally, given a set $S$, we denote its volume by $|S|$. For example, $$\label{eq:sphere vol}
|{{\mathcal{S}}^{m}}| = \frac{2\pi^{\frac{{m}+1}{2}}}{\Gamma(\frac{{m}+1}{2})}.$$ In this manuscript we will use the notations $A\ll B$ and $A=O(B)$ interchangeably.
Plan of the paper
-----------------
This paper is organized as follows. Section \[sec:expectation\] is devoted to the computation of the expected value of the Leray nodal measure and the volume, that is, proving propositions \[prop:exp ler meas\] and \[prop:exp len\], where the rest of the paper focuses on the variance of those characteristics, i.e. proving theorems \[thm:var ler meas\] and \[thm:var length\]. The treatment of the variance in both cases will be divided into two steps. First, we express it in an integral form in section \[sec:int form sec mom\]. We treat the integrals obtained in section \[sec:int form sec mom\] throughout section \[sec:asymp var\]. In case of the Leray measure, we will be able to give a precise asymptotic expression. In the case of volume, we give an upper bound.
Appendix \[sec:ultrasph pol\] will introduce the reader to the ultraspherical polynomials and will also provide all the necessary background we will need in this paper. The goal of appendix \[sec:sing func rare\] is to prove that the set of “bad" (singular) eigenfunctions in the space of all the eigenfunctions, is “rare" in some strong sense. Finally, appendix \[sec:f (x)(y)gr f(x)(y) sp\] will prove a particular nondegeneracy result for the distribution of the eigenfunctions and its gradients, needed to give meaning to the integral formula obtained for the variance of the volume given in section \[sec:int form sec mom\].
Acknowledgements
----------------
The author wishes to thank Zeév Rudnick for initiating this research and for his help and support while conducting it. Many stimulating discussions with Mikhail Sodin, Dmitry Jakobson and Stéphane Nonnenmacher are appreciated. The author is grateful to Sherwin Maslowe for proofreading this paper. I would like to thank CRM Analysis laboratory and its members for their support. Some part of this research was done during the author’s visit to the Bielefeld University, supported by SFB 701: Spectral Structures and Topological Methods in Mathematics. Finally, I wish to thank the anonymous referee for his comments and suggestions.
Expectation {#sec:expectation}
===========
In this section we prove propositions \[prop:exp ler meas\] and \[prop:exp len\]. As a start, we wish to stay away from the set of the singular functions discussed in section \[sec:sing func\].
The singular functions {#sec:sing func}
----------------------
In this section we define the notion of the singular functions and formulate the intuitive statement that they are “rare". The proofs are given in appendix \[sec:sing func rare\].
\[def:sing func\] An eigenfunction $f\in {\mathcal{E}_n}$ is [*singular*]{} if $\exists x\in {{\mathcal{S}}^{m}}$ with $f(x)=0$ and $\nabla f(x)
= \vec{0}$. An eigenfunction $f\in {\mathcal{E}_n}$ is [*nonsingular*]{} if $\nabla f\neq \vec 0$ on the nodal set.
A nonsingular eigenfunction has no self-intersections. We denote $Sing\subseteq {\mathcal{E}_n}$ to be the set of all the singular eigenfunctions. First, we claim that as a set, $Sing$ is [*“small"*]{}.
\[lem:Sing codim 1\] The set $Sing$ has codimension $1$ in ${\mathcal{E}_n}$.
Now, given $x\in{{\mathcal{S}}^{m}}$ and $b\in{\mathbb{R}}$, we denote ${\mathcal P}_{b}^{x}$ to be the set of all the eigenfunctions which attain the value $b$ at the point $x$. That is, $$\label{eq:PPxa def} {\mathcal P}^{x}_{b} = \{f\in{\mathcal{E}_n}:\: f(x) = b \}.$$ The set ${\mathcal P}^{x}_{b}$ is a hyperplane in ${\mathcal{E}_n}$.
Moreover, given $(x,y)\in{{{\mathcal{S}}^{m}}}\times{{\mathcal{S}}^{m}}$ and $b=(b_1,b_2)\in{\mathbb{R}}^2$ we denote $$\label{eq:PPxya def} {\mathcal P}^{x,y}_{b} = \{f\in{\mathcal{E}_n}:\: f(x) =
b_{1},\, f(y) = b_{2} \}.$$ For $x\ne \pm y$, ${\mathcal P}^{x,y}_{b}$ is an affine subspace of ${\mathcal{E}_n}$ of codimension $2$, as it is easy to see from the addition theorem (see section \[eq:two-pnt func\]).
The next couple of lemmas establish the fact that the intersections of $Sing$ with ${\mathcal P}^{x}_{b}$ and ${\mathcal P}^{x,y}_{b}$ for $x\ne \pm y$, are of codimension $1$. Lemma \[lem:Sing codim 1 Paxy\] is essential while treating the variance of the Leray nodal measure (section \[sec:ler var int form\]).
\[lem:Sing codim 1 Pax\] For every $x\in{{\mathcal{S}}^{m}}$ and $b\in{\mathbb{R}}$, the set $$Sing_{b}^x := Sing\cap {\mathcal P}_{b}^{x}$$ has codimension $1$ in ${\mathcal P}_{b}^{x}$.
\[lem:Sing codim 1 Paxy\] If $x,y\in{{\mathcal{S}}^{m}}$ and $x\ne \pm y$, then for every $b=(b_1,b_2)\in{\mathbb{R}}^2$, the set $Sing_{b}^{x,y} :=
Sing\cap {\mathcal P}_{b}^{x,y}$ has codimension $1$ in ${\mathcal P}_{b}^{x,y}$.
The proofs of all the lemmas of this section are given in appendix \[sec:sing func rare\].
Two-point function {#eq:two-pnt func}
------------------
We define the so called [*two-point*]{} function, also referred in the literature as the [*covariance*]{} function $$\label{eq:u(x,y) def} u(x,y) =u^{m}_{n}(x,y)={\mathbb{E}}\big[ f(x)f(y)\big]
= \frac{|{{\mathcal{S}}^{m}}|}{{\mathcal{N}}}\sum\limits_{k=1}^{{\mathcal{N}}}\eta_{k}
(x) \eta_{k} (y).$$
The addition theorem [@AAR], page 456, theorem 9.6.3 implies that $$\label{eq:u(x,y) def ult} u(x,y) = {Q_{n}^{{m}}}(\cos{d(x,y)}),$$ where $${Q_{n}^{{m}}}:[-1,1]\rightarrow{\mathbb{R}}$$ are the [*normalized*]{} ultraspherical polynomials defined and studied in appendix \[sec:ultrasph pol\]. Recall that $d(x,y)$ is the spherical distance so that $$\cos{d(x,y)} = \langle x,y\rangle,$$ thinking of ${{\mathcal{S}}^{m}}$ as being embedded into ${\mathbb{R}}^{{m}+1}$.
It is immediate that $u$ is rotationally invariant, i.e. $$\label{eq:rot inv u} u(Rx,Ry) = u(x,y),$$ where $R$ is any rotation on ${{\mathcal{S}}^{m}}$. In case $y$ is not specified, it is taken to be the northern pole $N\in{{\mathcal{S}}^{m}}$, that is $$\label{eq:u(x) def} u(x) := u(x,N).$$
For every $t\in [-1,1]$, $|{Q_{n}^{{m}}}(t) | \le 1$ and $|{Q_{n}^{{m}}}(t)| =1$, if and only if $t=\pm 1$. Therefore $$\label{eq:u(x,y)=1 iff x=pm y} (u(x,y)=\pm 1) \Leftrightarrow (x=\pm
y),$$ and $$\label{eq:u(x)=1 iff x=N,S} (u(x)= \pm 1 ) \Leftrightarrow (x\in
\{N, S\} ),$$ where $N$ and $S$ are the northern and the southern poles respectively.
Leray nodal measure {#leray-nodal-measure}
-------------------
We will need the following definitions from [@ORW], section 3.
For $\epsilon>0$, set $${\mathcal {L}}_\epsilon(f):=\frac 1{2\epsilon}{\operatorname{meas}}\{x: |f(x)|<\epsilon\}\;.$$ so that ${\mathcal {L}}(f) = \lim_{\epsilon\to 0} {\mathcal {L}}_\epsilon(f)$.
For $\alpha>0$, $\beta>0$ let $${\mathcal{E}_n}(\alpha,\, \beta) = \{f\in {\mathcal{E}_n}:\: |f(x)| \leq \alpha
\Rightarrow\ |\nabla f (x)| > \beta \} \;.$$
The sets ${\mathcal{E}_n}(\alpha,\, \beta)$ are open, and have the monotonicity property $$\alpha_1 >\alpha_2 \Rightarrow {\mathcal{E}_n}(\alpha_1,\beta) \subseteq
{\mathcal{E}_n}(\alpha_2,\beta)$$ and $$\beta_1>\beta_2 \Rightarrow {\mathcal{E}_n}(\alpha,\beta_1) \subseteq
{\mathcal{E}_n}(\alpha,\beta_2) \;.$$ Moreover, for any sequence $\alpha_n,\beta_n \to 0$ we have $${\mathcal{E}_n}\setminus Sing =
\bigcup\limits_{n} {\mathcal{E}_n}(\alpha_n,\, \beta_n) \;.$$
We have (cf. [@ORW], lemma 3.1)
\[lem:ler meas bnd\] For $f\in {\mathcal{E}_n}(\alpha,\, \beta)$ and $0<\epsilon < \alpha$, we have $${\mathcal {L}}_\epsilon(f) \ll \sqrt{{E_{n}}}$$ where the constant involved in the $'\ll'$-notation depends only on $\alpha$ and $\beta$.
To prove lemma \[lem:ler meas bnd\], we will need lemma 3.2 from [@ORW].
\[lem:Kac\] Let $g(t)$ be a trigonometric polynomial on $[0,2\pi]$ of degree at most $M$ so that there are $\alpha>0$, $\beta>0$ such that $|g'(t)|>\beta$ whenever $|g(t)|<\alpha$. Then for all $0<\epsilon<\alpha$ we have $$\frac 1 {2\epsilon}{\operatorname{meas}}\{t\in [0,2\pi]: |g(t)|<\epsilon \} \ll
\frac{M}{\beta},$$ where the constant in the $'\ll'$-notation may depend on ${m}$ only.
Let $(\phi_{1},\ldots \phi_{{m}})$ be the standard multi-dimensional spherical coordinates so that $x\in{{\mathcal{S}}^{m}}$ is parameterized by $$x = (\cos{\phi_{1}},\sin{\phi_{1}} \cos{\phi_{2}},\,\ldots,\,
\sin{\phi_{1}}\ldots \sin{\psi_{{m}}})$$ for $(\phi_{1},\ldots \phi_{{m}}) \in R:=[0,\pi]\times \ldots
[0,\pi]\times [0,2\pi]$. It is well-known that for $\phi_{i} \ne
0,\pi,2\pi$, $\big\{ \frac{\partial}{\partial \phi_{k}} \big\}$ is an orthogonal basis of $T_{x}({{\mathcal{S}}^{m}})$ and we have $$\bigg\| \frac{\partial}{\partial \phi_{k}}\bigg\| = \sin{\phi_{1}}
\cdot \ldots\cdot \sin{\phi_{k-1}},$$ so that the Jacobian $$J=J(\phi_{1},\,\ldots,\,\phi_{{m}}
)=\frac{D x}{D(\phi_{1},\ldots,\, \phi_{{m}})}$$ satisfies $$J = \sin\phi_{1} ^{{m}-1}\cdot \sin{\phi_{2}}
^{{m}-2}\cdot\ldots\cdot \sin\phi_{{m}-1}.$$
Let $0<\epsilon <\alpha$. We write $${\operatorname{meas}}\{x\in{{\mathcal{S}}^{m}}:\: |f(x)|
<\epsilon\}$$ as an integral $$\label{eq:meas f<eps sph int}
\begin{split}
{\operatorname{meas}}\{x\in{{\mathcal{S}}^{m}}:\: |f(x)| <\epsilon\} = \int\limits_{{{\mathcal{S}}^{m}}}
\chi\bigg(\frac{f(x)}{\epsilon} \bigg) dx =
\int\limits_{A_{\epsilon} }
|J(\phi_{1},\,\ldots,\,\phi_{{m}})| d\phi_{1}\cdot\ldots\cdot
d\phi_{{m}}
\end{split}$$ in the spherical coordinates, where we denoted $$A_{\epsilon} := \{
P \in R :\: |f(P)| < \epsilon\}.$$ For $P\in R$, $1\le k \le
{m}$ we define $p_k(P) = \frac{1}{\big\|
\frac{\partial}{\partial \phi_{k}}\big\|} \frac{\partial f}{\partial
\phi_{k}} (P)$, so that $$\label{eq:grad=p1^2+p2^2} \|\nabla f(x) \| ^2 = p_{1}^2 + p_{2}
^2+\ldots + p_{{m}}^2.$$
We decompose $$A_{\epsilon} = W_{1} \cup W_{2}\cup\ldots \cup
W_{{m}}$$ with $$W_{k} := \big\{P\in A_{\epsilon}:\: |p_{k} (P)| = \max\limits_{j} {|p_{j} (P)|}
\big\}.$$ Note that on $W_{k}$, $$|p_{i} (P)| >
\frac{\beta}{\sqrt{{m}}},$$ by and $\|\nabla f(x)\| > \beta$ on $A_{\epsilon}$.
Note that for $\phi_{k}$, $k\ne k_{0}$ fixed, $g(\phi_{k_{0}}):=f(\phi_{1},\,\ldots ,\, \phi_{{m}})$ is a trigonometric polynomial in $\phi_{k_{0}}$ on either $[0,\pi]$ or $[0,2\pi]$ of degree $\le n\le\sqrt{{E_{n}}}$ with derivative $$g'(\phi) = \bigg\| \frac{\partial}{\partial
\phi_{k_{0}}}\bigg\|\cdot p_1 (P)$$ so that on $W_{k_{0}}$, $|g'(\phi)| > \big\|
\frac{\partial}{\partial
\phi_{k_{0}}}\big\|\frac{\beta}{\sqrt{{m}}}$. Thus lemma \[lem:Kac\] implies $${\operatorname{meas}}\{\theta :\: |g(\phi_{k_{0}})| < \epsilon \} \ll
\frac{\sqrt{{E_{n}}}}{\big\| \frac{\partial}{\partial
\phi_{k_{0}}}\big\| }\cdot \epsilon.$$
Therefore the contribution of $W_{k_{0}}$ to the integral is $$\begin{split}
&\int\limits_{W_{1} } |J| d{\phi_{1}}\cdot\ldots\cdot
d{\phi_{{m}}} \le \int\limits {\operatorname{meas}}\{\phi :\:
|g(\phi_{k_{0}})| < \epsilon \} d\phi_{1} \cdot \ldots
\hat{d\phi_{k_{0}}} \ldots d\phi_{{m}} \\&\ll \int\limits
\frac{|J|}{\big\| \frac{\partial}{\partial \phi_{k_{0}}}\big\|}
\cdot \sqrt{{E_{n}}}\epsilon d\phi_{1} \cdot \ldots
\hat{d\phi_{k_{0}}} \ldots d\phi_{{m}} \ll \epsilon
\sqrt{{E_{n}}}.
\end{split}$$ which concludes the proof of the lemma.
We conclude the section with a formal derivation of proposition \[prop:exp ler meas\]. A rigorous proof proceeds along the same lines as the proof of theorem 4.1 in [@ORW] (see section 4.2), using lemmas \[lem:ler meas bnd\], \[lem:Sing codim 1\] and \[lem:Sing codim 1 Pax\]. We omit it here.
Given a function $f\in{\mathcal{E}_n}$, we write its Leray nodal measure formally as $${\mathcal {L}}(f) = \int\limits_{{{\mathcal{S}}^{m}}} \delta(f(x)) dx,$$ see .
Then, taking the expected value of both sides and changing the order of the expectation and the limit, we obtain $$\label{eq:ELf=intE} {\mathbb{E}}{\mathcal {L}}(f) = \int\limits_{{{\mathcal{S}}^{m}}}
{\mathbb{E}}\delta(f(x)) dx.$$
Now, for each [*fixed*]{} $x\in{{\mathcal{S}}^{m}}$, the random variable $v=f(x)$ is a linear combination of Gaussian random variables, and therefore, Gaussian itself. Its mean is zero and its variance is $1$ by . Writing the Gaussian probability density function explicitly, we have $${\mathbb{E}}\delta(f(x)) = {\mathbb{E}}\delta(v) = \int\limits_{-\infty}^{\infty}
\delta(a) \frac{e^{-\frac{1}{2}a^2}}{\sqrt{2\pi}}da =
\frac{1}{\sqrt{2\pi}}.$$
To finish the proof of this proposition we integrate the last equality on the sphere and substitute it into .
Choice of orthonormal bases for $T_{x}({{\mathcal{S}}^{m}})$ {#sec:orthonorm bas corr mat exp}
------------------------------------------------------------
For every $x\in{{\mathcal{S}}^{m}}$ we will need to identify $$\label{eq:Tx(S)=R^m} \phi_{x}: T_{x}({{\mathcal{S}}^{m}}) \cong {\mathbb{R}}^{{m}},$$ so that given a smooth function $f$ on ${{\mathcal{S}}^{m}}$, the function $$\nabla f(x) \in{\mathbb{R}}^{{m}},$$ is, under the identification , [*almost everywhere smooth*]{} (i.e, $C^k$, if $f$ is $C^{k+1}$) of argument $x$.
Since we will be typically interested in the length of the gradient, we will require the identifications to be length preserving, namely, isometries. This is naturally accomplished, given a choice $$B_{x}=\{e_{1}^{x},\ldots,\, e_{{m}}^{x}\}$$ of an [*orthonormal*]{} basis of $T_{x}({{\mathcal{S}}^{m}})$ for every $x\in{{\mathcal{S}}^{m}}$, so that for every vector $e_i$, all of its coordinates satisfy the appropriate smoothness condition. To do so, we consider the sphere without its southern pole $S$ $$R:={{\mathcal{S}}^{m}}\setminus \{ S \}.$$ Choosing an [*arbitrary*]{} orthonormal basis $B=B_{N}$ corresponding to the northern pole provides such a basis $B_{x}$ for every $x\in R$ by means of the parallel transport of $B$ along the unique geodesic linking $N$ and $x$ on $R$. We choose an arbitrary orthonormal basis $B_{S}$ of the tangent plane $T_{S}({{\mathcal{S}}^{m}})$ of ${{\mathcal{S}}^{m}}$ at the southern pole. It doesn’t affect any of the computations below, and we will neglect it from this point on.
Let $g(x):{{\mathcal{S}}^{m}}\rightarrow{\mathbb{R}}$ be any smooth function. We will use the notation $$\frac{\partial}{\partial e_{i}} g (x) = \frac{\partial}{\partial
e_{i}^{x}} g\vert_{x}$$ for the directional derivative of $g(x)$ at $x$ along $e_{i}^{x}$, i.e. $$\frac{\partial}{\partial e_{i}} g (x) = \langle \nabla g (x),
e_{i}^{x} \rangle.$$
In case of ambiguity, i.e. if we deal with a two variable function $$h(x,y):{{\mathcal{S}}^{m}}\times{{\mathcal{S}}^{m}}\rightarrow{\mathbb{R}},$$ we write $\frac{\partial}{\partial e_{i}^{x}} h$ or $\frac{\partial}{\partial e_{i}^{y}} h$ for the derivative of $g$ as a function of $x$ with $y$ constant, or vice versa respectively. Similarly, we will use the notation $\nabla_{x} g(x,y)\in
T_{x}({{\mathcal{S}}^{m}})$ and $\nabla_{y} g(x,y) \in T_{y}({{\mathcal{S}}^{m}})$ to denote the gradient of $g(x,y)$ as a function of $x$ or $y$ respectively.
Note that with the choice of the identifications as above, we have $$\label{eq:gradx d = -grady d} \nabla_{x} d(x,y)|_{(x,N)} = -\nabla_{y}
d(x,y)|_{(x,N)},$$ which is going to be useful in simplifying the covariance matrix $\Sigma$ (see section \[sec:corr mat, var\]).
In fact, for all our purposes, it is sufficient to make the choice of the orthonormal bases [*locally*]{}. Such a choice is possible for any manifold.
The covariance matrix, expectation {#sec:corr mat, exp}
----------------------------------
Given a point $x\in{{\mathcal{S}}^{m}}$ we consider the random vector $(v,w)\in{\mathbb{R}}\times{\mathbb{R}}^{{m}}$ $$(v,w) = (f(x),\nabla f(x)),$$ where we use the identification . It is easy to see that being a linear transformation of a mean zero Gaussians, its distribution is a mean zero Gaussian as well.
We claim that the covariance matrix of $(v,w)$ is $$\label{eq:exp covar mat} \tilde{\Sigma}_{(m+1)\times (m+1)} :=\left(
\begin{matrix}{\mathbb{E}}f(x)^2 &{\mathbb{E}}\big[ f(x) \nabla f(x) \big] \\
{\mathbb{E}}\big[ f(x) \nabla f(x) \big]^{t} &{\mathbb{E}}\big[ \nabla f(x) ^ t \nabla f(x) \big] \end{matrix} \right)= \left(\begin{matrix} 1 &0 \\
0 &\frac{{E_{n}}}{{m}} I_{{m}} \end{matrix}\right).$$
First, $$\label{eq:E(f(x)^2)=1 pr} {\mathbb{E}}f(x)^2 = u(x,x) = 1,$$ by the definition , of $u(x,y)$ and .
Next, we have $${\mathbb{E}}(f(x)\nabla f(x)) = \frac{1}{2}\nabla {\mathbb{E}}(f(x)^2) = \nabla 1/2 =
\vec{0}$$ by .
Finally, we compute ${\mathbb{E}}\big[\nabla f(x) ^ t \nabla f(x)\big]$. For $i\ne j$, we have $${\mathbb{E}}\bigg[\frac{\partial}{\partial e_i^x} f(x)\frac{\partial}{
\partial e_j^x} f(x) \bigg] = \bigg[\frac{\partial}{\partial e_i^x \partial e_j^y}
u(x,\,y)\bigg]\bigg|_{x=y} = 0,$$ computing the second partial derivative explicitly in local coordinates (see section \[sec:orthonorm bas corr mat exp\] for an explanation of the partial derivatives notations).
For $i=j$, we have by the rotational symmetry on ${{\mathcal{S}}^{m}}$, $$\begin{split}
&{\mathbb{E}}\bigg(\big(\frac{\partial}{\partial e_i^x} f(x)\big)^2\bigg) =
\frac{1}{{m}|{{\mathcal{S}}^{m}}|}\int\limits_{{{\mathcal{S}}^{m}}}{\mathbb{E}}\bigg(\nabla
f(x) \cdot \nabla f(x)\bigg) dx \\&= \frac{1}{{m}|{{\mathcal{S}}^{m}}|}
{\mathbb{E}}\bigg[\int\limits_{{{\mathcal{S}}^{m}}} \nabla f(x) \cdot \nabla f(x)
dx\bigg] = -\frac{1}{{m}|{{\mathcal{S}}^{m}}|} {\mathbb{E}}\bigg[\int\limits_{{{\mathcal{S}}^{m}}} f(x) \cdot \triangle f(x) dx\bigg] \\&=
\frac{{E_{n}}}{{m}|{{\mathcal{S}}^{m}}|} {\mathbb{E}}\bigg[\int\limits_{{{\mathcal{S}}^{m}}}
f(x)^2 dx\bigg]= {\frac{{E_{n}}}{{m}}}\cdot
\frac{1}{|{{\mathcal{S}}^{m}}|}\int\limits_{{{\mathcal{S}}^{m}}} {\mathbb{E}}[f(x)^2] dx =
{\frac{{E_{n}}}{{m}}},
\end{split}$$ by the divergence theorem and . Thus $$\label{eq:corr grads x} {\mathbb{E}}(\nabla f(x)^{t} \nabla f(x)) = {\mathbb{E}}(\nabla
f(y)^{t} \nabla f(y)) = {\frac{{E_{n}}}{{m}}}I_m.$$
Riemannian volume
-----------------
Let $\chi$ be the indicator function of the interval $[-1,1]$. For $\epsilon>0$, we define the random variable $${\mathcal{Z}}_\epsilon(f):=\frac{1}{2\epsilon} \int_{{{\mathcal{S}}^{m}}}
\chi\bigg(\frac{f(x)}\epsilon\bigg) |\nabla f(x)|dx \;.$$
\[lem: formula for Z\] Suppose that $f\in {\mathcal{E}_n}$ is nonsingular. Then $${\operatorname{vol}}(f^{-1}(0))=\lim_{\epsilon \to 0} {\mathcal{Z}}_\epsilon(f) \;.$$
Lemma \[lem: formula for Z\] implies that the expectation of the volume and its second moments are given by the following.
\[cor:formulas for moments\] The first and second moments of the volume ${\mathcal{Z}}(f)$ of the nodal set of $f$ are given by $${\mathbb{E}}({\mathcal{Z}}) = {\mathbb{E}}(\lim_{\epsilon\to 0} {\mathcal{Z}}_\epsilon),\qquad
{\mathbb{E}}({\mathcal{Z}}^2) = {\mathbb{E}}(\lim_{\epsilon_1,\epsilon_2\to 0} {\mathcal{Z}}_{\epsilon_1}
{\mathcal{Z}}_{\epsilon_2}) \;.$$
\[lem:Zeps=O(sqrt(E))\] For every $f\in{\mathcal{E}_n}$ and $\epsilon>0$, one has $${\mathcal{Z}}_{\epsilon}(f) = O(\sqrt{{E_{n}}}),$$ where the constant involved in the $'O'$ notation depends only on ${m}$.
To prove lemma \[lem:Zeps=O(sqrt(E))\], we use lemma 3.3 from [@RW].
\[lem:Kaclength\] Let $g(t)$ be a trigonometric polynomial on $[0,2\pi]$ of degree at most $M$. Then for all $\epsilon>0$ we have $$\frac 1 {2\epsilon}\int\limits_{\{t: |g(t)|\leq \epsilon
\}}|g'(t)|dt \leq 6M \;.$$
We write ${\mathcal{Z}}_{\epsilon}$ in the multi-dimensional spherical coordinates (see the proof of lemma \[lem:Zeps=O(sqrt(E))\]) as $${\mathcal{Z}}_\epsilon(f):=\frac{1}{2\epsilon} \int\limits_{R}
\chi\bigg(\frac{f(\phi_{1},\,\ldots,\,\phi_{{m}})}\epsilon\bigg)
\big\|\nabla f(\phi_{1},\,\ldots,\,\phi_{{m}})\big\| \cdot
|J| d\phi_{1}\cdot\ldots\cdot d\phi_{{m}}.$$ Note that in the spherical coordinates, for $\phi_{k}\ne 0,\pi,2\pi$ $$\nabla f(\phi_{1},\,\ldots,\, \phi_{{m}}) =
\bigg(\frac{1}{\big\|\frac{\partial}{\partial
\phi_{k}}\big\|}\frac{\partial f}{\partial\phi_{k}} \bigg)_{1\le k
\le {m}},$$ in the orthonormal basis associated to $\big\{\frac{\partial}{\partial\phi_{k}}\big\}$. Thus $$\|\nabla f \| \cdot |J|\ll \sum\limits_{k=1}^{{m}} \bigg|
\frac{\partial f}{\partial \phi_{k}} \bigg|.$$
Note that for $\phi_{k}$, $k\ne k_{0}$ fixed, $f(\phi_{1},\,\ldots,\, \phi_{{m}})$ is a $1$-variable trigonometric polynomial in $\phi_{k_{0}}$ of degree $\le
n\ll\sqrt{{E_{n}}}$. Therefore, $$\begin{split}
{\mathcal{Z}}_{\epsilon} (f) &\ll \sum\limits_{k_{0}=1}^{{m}}
\int\limits_{} \frac{1}{2\epsilon}\bigg[\int\limits_{\{\theta: \:
|f(\phi_{1},\,\ldots,\, \phi_{{m}})| < \epsilon \}}
\bigg|\frac{\partial f(\phi_{1},\,\ldots,\,
\phi_{{m}})}{\partial\phi_{k_{0}}}\bigg| d\phi_{k_{0}} \bigg]
d\phi_{1} \cdot\ldots \hat{d\phi_{k_{0}}}\ldots \cdot
d\phi_{{m}}
\\&\ll \sqrt{{E_{n}}},
\end{split}$$ by lemma \[lem:Kaclength\].
Now we are in a position to prove the main result of this section, namely proposition \[prop:exp len\].
We saw that $${\mathbb{E}}{\mathcal{Z}}(f) = {\mathbb{E}}\lim\limits_{\epsilon\rightarrow 0}
{\mathcal{Z}}_{\epsilon} (f)$$ by corollary \[cor:formulas for moments\]. Lemma \[lem:Zeps=O(sqrt(E))\] and the dominated convergence theorem allow us to exchange the order of taking expectation and the limit to obtain $$\label{eq:Elen=lim Eleneps} {\mathbb{E}}{\mathcal{Z}}(f) =
\lim\limits_{\epsilon\rightarrow 0} {\mathbb{E}}{\mathcal{Z}}_{\epsilon}(f).$$
By Fubini’s theorem, $$\label{eq:Eleneps=int K1eps} {\mathbb{E}}{\mathcal{Z}}_{\epsilon}(f) =
{\mathbb{E}}\bigg[\frac{1}{2\epsilon} \int_{{{\mathcal{S}}^{m}}}
\chi\bigg(\frac{f(x)}\epsilon\bigg) |\nabla f(x)|dx \bigg] =
\int\limits_{{{\mathcal{S}}^{m}}} K_{\epsilon}^{1} (x) dx,$$ where $K^{1}_{\epsilon}(x)$ is defined by $$\label{eq:K1eps def} K^{1}_{\epsilon} (x) := {\mathbb{E}}\bigg[
\frac{1}{2\epsilon} \chi\bigg(\frac{f(x)}{\epsilon}\bigg) |\nabla
f(x)| \bigg] = \frac{1}{2\epsilon} \int\limits_{{\mathcal{E}_n}}
\chi\bigg(\frac{f(x)}\epsilon\bigg) |\nabla f(x)| d\upsilon(f).$$
We write $K^{1}_{\epsilon}$ in terms of the random vector $(v,w)$, introduced in section \[sec:corr mat, exp\] as $$K^{1}_{\epsilon} (x) =
\frac{1}{2\epsilon}\int\limits_{{\mathbb{R}}\times{\mathbb{R}}^{{m}}} \chi\bigg (
\frac{v}{a}\bigg)\|w \| d\mu(v,w),$$ where $d\mu(v,w)$ is the joint probability density function of $(v,w)$, namely mean zero Gaussian with covariance $\tilde{\Sigma}$ given by . Writing the Gaussian probability explicitly, we have $$\begin{split}
K^{1}_{\epsilon} (x) &=
\frac{1}{2\epsilon}\int\limits_{{\mathbb{R}}\times{\mathbb{R}}^{{m}}} \chi\bigg (
\frac{v}{\epsilon}\bigg)\|w \| \exp\bigg(-\frac{1}{2}
(v,w)\tilde{\Sigma}^{-1}(v,w)^{t} \bigg)
\frac{dvdw}{(2\pi)^{({m}+1)/2}\sqrt{\det{\tilde{\Sigma}}}}
\\&= \frac{1}{2\epsilon}\int\limits_{-\epsilon}^{\epsilon} \exp\big(-\frac{1}{2}v^2 \big) dv
\int\limits_{{\mathbb{R}}^{{m}}} \|w\| \exp\bigg( -\frac{1}{2}
\frac{\|w\|^2{m}}{{E_{n}}}\bigg)\frac{{m}^{{m}/2}
dw}{(2\pi)^{({m}+1)/2}{E_{n}}^{{m}/2}} \\&=
\frac{1}{2\epsilon}\int\limits_{-\epsilon}^{\epsilon}
\exp\big(-\frac{1}{2}v^2 \big) dv \cdot
\frac{\sqrt{{E_{n}}}}{\sqrt{{m}}}\int\limits_{{\mathbb{R}}^{{m}}}
\|w'\| \exp\bigg( -\frac{1}{2} \| w'\|^2\bigg)\frac{
dw'}{(2\pi)^{({m}+1)/2}},
\end{split}$$ changing the variables $$w=\sqrt{\frac{{E_{n}}}{{m}}}w' .$$
Following and , we integrate the last expression and take the limit $\epsilon\rightarrow 0$ to obtain $${\mathbb{E}}{\mathcal{Z}}(f) = c_{{m}} \sqrt{{E_{n}}},$$ where $$\label{eq:c exp int} c_{{m}} = \frac{|{{\mathcal{S}}^{m}}|}{\sqrt{{m}}(2\pi)^{({m}+1)/2}}\int\limits_{{\mathbb{R}}^{{m}}}
\|w'\| \exp\bigg( -\frac{1}{2} \| w'\|^2\bigg) dw'.$$ Finally, substituting $$\int\limits_{{\mathbb{R}}^{{m}}} \|w'\|
\exp\bigg( -\frac{1}{2} \| w'\|^2\bigg) dw' =
\sqrt{2}(2\pi)^{{m}/2}\frac{\Gamma\bigg(\frac{{m}+1}{2}
\bigg)}{\Gamma\bigg(\frac{{m}}{2} \bigg)}$$ (see e.g. [@RW], page 7) and into the last expression yields .
An integral formula for the second moment {#sec:int form sec mom}
=========================================
Covariance matrices, second moment {#sec:corr mat, var}
----------------------------------
Similarly to the computation of the expected volume, we will naturally encounter a random vector on $${\mathbb{R}}\times{\mathbb{R}}\times{\mathbb{R}}^{{m}}\times{\mathbb{R}}^{{m}},$$ defined as $$(f(x), f(y), \nabla f(x),\nabla f(y)),$$ for some [*fixed*]{} $x,y\in{{\mathcal{S}}^{m}}$, where we again use the identification . We will use the rotational symmetry of the sphere to reduce the discussion to the case $y=N$ is the northern pole. Thus we consider $$\label{eq:Z rand vec def} Z:=(v_1,v_2,w_1,w_2) = (f(x), f(N), \nabla
f(x),\nabla f(N))$$ for some $y\in {{\mathcal{S}}^{m}}$.
It is obvious that the joint distribution of this vector, is mean zero Gaussian. It remains, therefore, to compute the covariance matrix. We need the following notations.
Let $D=D(x)$ be the vector in ${\mathbb{R}}^{{m}}$ defined by $$D(x) = \nabla_x u(x,y)|_{(x,N)} \in T_x({{\mathcal{S}}^{m}})\cong {\mathbb{R}}^{{m}}.$$ Note that for $x\ne\pm N$, we may use to obtain $$D(x)= {Q_{n}^{{m}}}{'}(d(x,N)) \sin(d(x,N))\nabla_x d(x,\, y)|_{(x,N)}.$$ It is then clear from that we then have $$\label{eq:D2=-D1} \nabla_x u(x,y)|_{(x,N)} = -\nabla_{y} u(x,y)|_{(x,N)}.$$
Finally, let $$H=H(x)=(h_{ij})$$ be the ${m}\times
{m}$ matrix defined as $$\label{eq:pseudo-Hessian of u} H = \nabla_x\nabla_y u(x,y)|_{(x,N)},$$ i.e. $H=(h_{jk})$ with entries given by $$h_{jk} = \frac{\partial^2}{\partial e_{j}^{x}\partial e_{k}^{y}}
u(x,y)|_{(x,N)}.$$
We will be in particular interested in the conditional distribution of $$Z_1=(w_1,w_2) = (\nabla f(x),\nabla f(N)),$$ conditioned upon $f(x)=f(N)=0$.
For the variance computation of the Leray nodal measure, we will need the distribution of the random vector $$\tilde{Z}:=(v_{1},\, v_{2}) = (f(x), f(N)).$$ It is distributed mean zero Gaussian as well.
The covariance matrices of the random vectors above are given in the following lemma.
\[eq:Z covar mat\] Let $x\in{\mathcal{S}}$. Then
1. \[it:covar mat ler\] The distribution of the random vector $\tilde{Z}=(v_1,v_2)$ is mean zero Gaussian with covariance matrix given by $$\label{eq:A blk def} A = \left(\begin{matrix} 1 &u(x,N) \\ u(x,N)
&1 \end{matrix}\right).$$
2. \[it:covar mat len\] The covariance matrix of the random vector $Z$ is the $(2{m}+2)\times (2{m}+2)$ matrix $$\Sigma = \left(\begin{matrix}A & B \\ B^{t} & C
\end{matrix}\right),$$ where $A\in M_{2\times 2}$ is given by , $B\in
M_{2\times 2{m}}$ is given by $$\label{eq:B blk def} B = \left(\begin{matrix} \vec{0} &-D(x) \\
D(x) &\vec{0} \end{matrix}\right),$$ and $C\in M_{2{m}\times 2{m}}$ is given by $$\label{eq:C blk def} C = \left(\begin{matrix} {\frac{{E_{n}}}{{m}}}I_{{m}} &H \\
H^{t} &{\frac{{E_{n}}}{{m}}}I_{{m}} \end{matrix}\right)$$ with the “pseudo-Hessian" matrix $H=(h_{jk})$ of $u$ given by . The distribution of $Z$ is nondegenerate for $x\ne\pm N$ (this is equivalent to $\Sigma$ being invertible).
3. \[it:red covar mat\] The covariance matrix of the conditional distribution of $Z_1$, conditioned upon $v_1=v_2=0$ is given by $$\label{eq:Omega def} \Omega = \bigg[\left(\begin{matrix}
{\frac{{E_{n}}}{{m}}}I &H \\ H^{t} &{\frac{{E_{n}}}{{m}}}I
\end{matrix}\right) - \frac{1}{1-u^2} \left(\begin{matrix}D^t D &-uD^t D \\ -uD^t D &D^t D \end{matrix}
\right)\bigg].$$ We call the matrix $\Omega$ the “reduced covariance matrix" of $Z_1$, and one has $$\label{eq:Jac iden det(Sigma)} \det{\Sigma} = \det{A}\det{\Omega} =
(1-u^2)\det{\Omega}.$$
Part of the lemma is evident from the definition of the two-point function. It is also clear that part of the lemma implies part , since one computes the covariance matrix $\Omega$ of the conditional distribution from $\Sigma$ employing $$\Sigma^{-1} =
\left(\begin{matrix}* &* \\ * &\Omega^{-1} \end{matrix} \right).$$
The nondegeneracy of the distribution of the random vector $Z$ for $x\ne\pm y$ follows directly from appendix \[sec:f (x)(y)gr f(x)(y) sp\]. The matrix $\Sigma$ is then invertible, being the covariance of a nonsingular joint Gaussian distribution.
It remains, therefore, to prove part of the lemma. It is clear that the block $A$ is the same as the covariance matrix in part , i.e. given by .
Now by the definition, $$B = \left(\begin{matrix} {\mathbb{E}}(f(x) \nabla f(x)) &{\mathbb{E}}(f(x)\nabla f(N)) \\
{\mathbb{E}}(f(N)\nabla f(x)) &{\mathbb{E}}(f(N)\nabla f(N))\end{matrix}\right),$$ and we have already seen that $${\mathbb{E}}(f(x)\nabla f(x)) = \vec{0}$$ in section \[sec:corr mat, exp\] as well as $${\mathbb{E}}(f(N)\nabla f(N)) = \vec{0}.$$ Also $${\mathbb{E}}(f(N)\nabla f(x)) = \nabla_x {\mathbb{E}}(f(x)f(N)) = \nabla_x u(x,y)|_{(x,N)} =
D(x),$$ and similarly $${\mathbb{E}}(f(x)\nabla f(N)) = -D(x),$$ which finishes the proof of .
Finally, we compute $C$. By the definition, $$C = \left(\begin{matrix} {\mathbb{E}}(\nabla f(x)^{t} \nabla f(x)) &{\mathbb{E}}(\nabla f(x)^{t} \nabla f(N)) \\
{\mathbb{E}}(\nabla f(N)^{t}\nabla f(x)) &{\mathbb{E}}(\nabla f(N)^{t}\nabla
f(N))\end{matrix}\right).$$
We have already computed that ${\mathbb{E}}(\nabla f(x)^{t} \nabla f(x))$ and ${\mathbb{E}}(\nabla f(N)^{t} \nabla f(N))$ are given by . Finally, $${\mathbb{E}}(\nabla f(x)^{t} \nabla f(N)) = \nabla_x\nabla_y {\mathbb{E}}[f(x)f(N)] =
\nabla_x\nabla_y u(x,\, y)|_{(x,N)} = H,$$ and similarly $${\mathbb{E}}(\nabla f(N)^{t} \nabla f(x)) = H^t.$$ This implies and finishes the proof of the lemma.
Leray nodal measure {#sec:ler var int form}
-------------------
\[prop:ler var int form\] The second moment of the Leray nodal measure is given by $$\label{eq:ler var int form} {\mathbb{E}}{\mathcal {L}}(f)^2 = \frac{|{{\mathcal{S}}^{m}}|}{2\pi}\int\limits_{{{\mathcal{S}}^{m}}} \frac{d x}{\sqrt{1-u(x)^2}} ,$$ where $u(x)$ is the two-point function given by and .
As in the case of expectation, we give a formal derivation of proposition \[prop:ler var int form\], omitting a rigorous treatment. A rigorous proof is obtained following the lines of the proof of theorem 5.1 in [@ORW] (see section 5.3), using lemma \[lem:Sing codim 1 Paxy\] in our case. The convergence of the integral on the RHS of , necessary to the proof, follows from and lemma \[lem:1/sqrt(1-u\^2) int asymp\].
We write the Leray measure as again, so that $$\label{eq:ler var dbl atom frm}
\begin{split}
{\mathbb{E}}{\mathcal {L}}(f)^2 &= {\mathbb{E}}\bigg
[\int\limits_{{{\mathcal{S}}^{m}}}\int\limits_{{{\mathcal{S}}^{m}}} \delta(f(x))\delta(f(y))
dx dy \ \bigg] = \int\limits_{{{\mathcal{S}}^{m}}\times{{\mathcal{S}}^{m}}} {\mathbb{E}}\big[
\delta(f(x)) \delta(f(y)) \big] dx dy \\&= |{{\mathcal{S}}^{m}}|\int\limits_{{{\mathcal{S}}^{m}}} {\mathbb{E}}\big[ \delta(f(x)) \delta(f(N)) \big] dx,
\end{split}$$ by the rotational symmetry of the sphere. Now, for a [*fixed*]{} $x\in{{\mathcal{S}}^{m}}$ with $x\ne \pm N$, the random variables $v_{1}:=f(x)$ and $v_{2} := f(N)$ are multivariate mean zero Gaussian with covariance matrix $A$ given by .
Thus, writing the Gaussian measure explicitly, we obtain $$\begin{split}
{\mathbb{E}}\big( \delta(f(x)) \delta(f(y)) &= E \big[\delta(v_1)\delta(v_2)
\big] \\&= \int \limits_{{\mathbb{R}}^{2}} \delta(a_1) \delta(a_2)
\exp(-\frac{1}{2}a A^{-1}a^{t})\frac{da}{2\pi\sqrt{\det{A}}} \\&=
\frac{1}{2\pi\sqrt{\det{A}}} = \frac{1}{2\pi \sqrt{1-u(x,y)^2}}.
\end{split}$$ Plugging this into yields .
Riemannian volume
-----------------
\[prop:justif ord chng\] The second moment of ${\mathcal{Z}}(f)$ is given by $$\label{eq:int form sec mom} {\mathbb{E}}({\mathcal{Z}}^2) = |{{\mathcal{S}}^{m}}|\int_{{{\mathcal{S}}^{m}}}
K(x) dx$$ where $$\label{eq:K(x) def}
K(x) = \frac {1} {\sqrt{1-u^2}}
\int_{{\mathbb{R}}^{{m}}\times{\mathbb{R}}^{m}} \| w_1\| \| w_2\|
\frac{\exp(-\frac {1}{2} (w_{1},w_{2})\Omega^{-1}
(w_{1},w_{2})^{t})}{\sqrt{\det\Omega}} \frac{dw_1
dw_2}{(2\pi)^{{m}+1}},$$ where $\Omega=\Omega(x)$ is defined by .
Denote $$\label{eq:def K eps(x,y)} K_{\epsilon_1, \epsilon_2} (x,y) :=
\frac{1}{4\epsilon_1\epsilon_2} \int_{{\mathcal{E}_n}} \|\nabla f(x) \|
\|\nabla f(y) \| \chi \bigg( \frac{f(x)}{\epsilon_1} \bigg) \chi
\bigg( \frac{f(y)}{\epsilon_2} \bigg) d\upsilon(f) \;.$$
To prove the proposition we will need the following lemma (cf. lemma 5.3 in [@RW]).
\[lem:bound on K eps(x,y)\] For $(x,\, y)\in{{\mathcal{S}}^{m}}\times{{\mathcal{S}}^{m}}$ with $x\ne y$, one has the inequality $$\label{eq:bnd krnl eps} K_{\epsilon_1,\, \epsilon_2} (x,\,
y)\ll_{{m}} \frac{{E_{n}}}{\sqrt{1-u(x,y)^2}},$$ where the implied constant depends only on the dimension ${m}$.
The proof is almost identical to the proof of lemma 5.3 of [@RW] .
Write $f(x) = \langle f,\, U(x) \rangle$, where $U(x)$ is the unit vector $$U(x) = \sqrt{{\frac{|{{\mathcal{S}}^{m}}|}{{\mathcal{N}}}}} \big(\eta_i (x)\big)_{i} \in
S^{{\mathcal{N}}-1},$$ where $\big\{ \eta_i (x)\}_{i=1}^{{\mathcal{N}}}$ is the $L_2$ orthonormal basis of ${\mathcal{E}_n}$ chosen, and where we identify the function $f$ with a vector in ${\mathbb{R}}^{{\mathcal{N}}}$ via . Note that $$\langle U(x),U(y) \rangle = u(x,y)$$ is the cosine of the angle between $U(x)$ and $U(y)$.
We have $$\nabla f(x) = DU \cdot f$$ where the derivative $DU$ is a ${m}\times {\mathcal{N}}$ matrix. Equivalently, $$\big(\nabla f(x)\big)_i =
\bigg\langle f, \bigg(\frac{\partial}{\partial e_i} U(x) \bigg)
\bigg\rangle ,\quad 1\le i\le {m}.$$
By the triangle and Cauchy-Schwartz inequalities, $$\|\nabla f (x)
\| \le \sum_{i=1}^{{m}} \| f\|\cdot \bigg\|
\bigg(\frac{\partial}{\partial e_i} U(x) \bigg) \bigg\| \ll
\sqrt{{E_{n}}} \| f \|,$$ due to $$\bigg\| \bigg(\frac{\partial}{\partial e_i} U(x) \bigg) \bigg\|^2 =
{\mathbb{E}}\bigg[ \bigg(\frac{\partial f}{\partial e_{i}}\bigg)^2 \bigg] =
{\frac{{E_{n}}}{{m}}},$$ by .
Therefore $$\label{eq:mult int Keps} K_{\epsilon_1,\epsilon_2} (x,y) \ll
\frac{{E_{n}}}{4\epsilon_1
\epsilon_2}\int\limits_{\substack{|f(x)| < \epsilon_1 \\
|f(y)|<\epsilon_2}} \|f\|^2 e^{-\| f \| ^2/2} df \;.$$
Consider the plane $\pi\subset {\mathbb{R}}^{{\mathcal{N}}}$ spanned by $U(x)$ and $U(y)$. The domain of the integration is all the vectors $f\in{\mathbb{R}}^{\mathcal{N}}$ so that the projection of $f$ on $\pi$ falls into the parallelogram $P$ defined by the perpendiculars $l_{x}^{\pm}$ and $l_{y}^{\pm}$ to the endpoints of $\pm U(x)$ and $\pm U(y)$. Denote the angle $\alpha$ between the sides of $P$, computed as $$\cos{\alpha} = \langle U(x), \,U(y) \rangle = u(x,y).$$ We claim that the area of $P$ is $$\mbox{area}(P) = 4\epsilon_1 \epsilon_2
\frac{1}{\sqrt{1-u(x,y)^2}} \;.$$
To see that, we assume, with no loss of generality that $\epsilon_{2} \cos{\alpha} \le \epsilon_{1}$ (otherwise exchange between $x$ and $y$) and $\alpha\in (0,\frac{\pi}{2})$. Now if furthermore, $$\epsilon_{2} \le \epsilon_{1} \cos{\alpha},$$ then the line $l_{y}^{+}$ does not intersect the interval $[0, \epsilon_{1} U(y)]$, and the sides of $P$ are easily seen to have lengths $\frac{2\epsilon_{1}}{\sin{\alpha}}$ and $\frac{2\epsilon_{2}}{\sin{\alpha}}$, and the angle between the sides of $P$ is $\alpha$, so that our claim follows. Otherwise (namely if $\epsilon_{2} > \epsilon_{1} \cos{\alpha}$), a little trigonometric computation shows that the lengths of the sides of $P$ are again $\frac{2\epsilon_{1}}{\sin{\alpha}}$ and $\frac{2\epsilon_{2}}{\sin{\alpha}}$ and the angle between the sides of $P$ is $\alpha$.
Write the multiple integral in as the iterated integral $$\label{eq:mult int K rep} \int_{P} \left(\int_{p+\pi^{\perp}} \|
f\|^2 e^{-\| f \|^2/2} df\right) dp \;,$$ where the variable $p$ runs over all the points of the parallelepiped $P$. The inner integral in is $O(1)$. Indeed, note that for every $f_1\in\pi^{\perp}$, $$\begin{split}
\| p+f_1\|^2 e^{-\|p+f_1\| ^2/2}
&= (\|p\|^2+\|f_1\|^2)e^{-(\|p\|^2+\|f_1\| ^2)/2} \\
&\ll (1+\|f_1\|^2) \cdot e^{-\|f_1\| ^2/2}\;,
\end{split}$$ since $\|p\|^2 e^{-\| p \| ^2/2}$ is bounded. Our claim follows from convergence of the integral $\int_{{\mathbb{R}}^{{\mathcal{N}}-2}}
(1+\|w\|^2)e^{-\| w\|^2/2}dw $. Therefore $$\int_{\substack{|f(x)| < \epsilon_1 \\
|f(y)|<\epsilon_2}} \|f\|^2 e^{-\| f \| ^2/2} df \ll area(P) \ll
\epsilon_1\epsilon_2 \frac{1}{\sqrt{1-u(x,y)^2}} \;.$$ Substituting the last estimate into proves .
We give a formal derivation of proposition \[prop:justif ord chng\]. Having lemma \[lem:bound on K eps(x,y)\] in our hands, a rigorous proof of proposition \[prop:justif ord chng\] is identical to the proof of proposition 5.2 of [@RW] and we omit it here. In the course of the proof one shows that $$K(x)=\lim_{\epsilon_1,\epsilon_2\to
0}K_{\epsilon_1,\epsilon_2}(x,N)\;.$$ Therefore, taking the limit $\epsilon_1,\epsilon_2\to 0$ in , we obtain
\[cor:ker bnd ler ker\] If $u(x)^2 \ne 1$ then $$K(x) \ll \frac{{E_{n}}}{\sqrt{1-u(x)^2}}\;.$$
Corollary \[cor:formulas for moments\] allows us to write an expression for the second moment formally as $${\mathbb{E}}{\mathcal{Z}}(f)^2 = {\mathbb{E}}\bigg[ \int\limits_{{{\mathcal{S}}^{m}}\times{{\mathcal{S}}^{m}}}
\delta(f(x)) \| \nabla f(x)\| \delta(f(y)) \|\nabla f(y) \| dx dy
\bigg],$$ and changing the order of taking the integration, we obtain $$\begin{split}
\label{eq:sec mom len intexp} {\mathbb{E}}{\mathcal{Z}}(f)^2 &=
\int\limits_{{{\mathcal{S}}^{m}}\times{{\mathcal{S}}^{m}}} {\mathbb{E}}\bigg[ \delta(f(x))\cdot \|
\nabla f(x)\| \cdot \delta(f(y))\cdot \|\nabla f(y) \|\bigg] dx dy
\\&= |{{\mathcal{S}}^{m}}|\int\limits_{{{\mathcal{S}}^{m}}} {\mathbb{E}}\bigg[ \delta(f(N))\cdot \| \nabla
f(N)\| \cdot \delta(f(x))\cdot \|\nabla f(x) \|\bigg] dx,
\end{split}$$ by the rotational symmetry of the sphere. In fact, the integrand $${\mathbb{E}}\bigg[ \delta(f(x))\cdot \|
\nabla f(x)\| \cdot \delta(f(y))\cdot \|\nabla f(y) \|\bigg]$$ depends on $d(x,y)$ only (this is the isotropic property of the random ensemble ${\mathcal{E}_n}$).
Now for a fixed $x\in{{\mathcal{S}}^{m}}$ with $x\ne\pm
N$, the joint distribution of the random vector $Z$ defined as in is Gaussian with mean zero and covariance $\Sigma = \Sigma(x)$ as in lemma \[eq:Z covar mat\]. Thus we may write $$\begin{split} &{\mathbb{E}}\bigg[ \delta(f(x)) \cdot \| \nabla f(x)\|\cdot \delta(f(N)) \cdot\|\nabla
f(N) \|\bigg] \\&= \int\limits_{{\mathbb{R}}^{2}\times{\mathbb{R}}^{2{m}}}
\delta(v_{1}) \cdot \|w_{1}\| \cdot \delta(v_{2})\cdot \| w_{2} \|
\exp(-\frac{1}{2}(v,w)\Sigma^{-1}(v,w)^{t})\frac{dv
dw}{(2\pi)^{{m}+1} \sqrt{1-u^2}\sqrt{\det{\Omega}}},
\end{split}$$ substituting the explicit expression for the Gaussian measure and using (recall that $\Omega=\Omega(x)$ is defined by ).
Substituting $v_1=v_2=0$, we have $$\begin{split}
&{\mathbb{E}}\bigg[ \delta(f(x)) \cdot\| \nabla f(x)\|\cdot \delta(f(N))
\cdot\|\nabla f(N) \|\bigg] \\&= \int\limits_{{\mathbb{R}}^{2{m}}}
\|w_{1}\| \| w_{2} \| \exp(-\frac{1}{2}w\Sigma^{-1}w^{t})\frac{
dw}{(2\pi)^{{m}+1} \sqrt{1-u^2}\sqrt{\det{\Omega}}}.
\end{split}$$ To obtain the statement of the proposition, we integrate the last expression over ${{\mathcal{S}}^{m}}$ and plug it into .
Asymototics of the variance {#sec:asymp var}
===========================
In this section we prove theorems \[thm:var ler meas\] and \[thm:var length\].
Leray nodal measure {#leray-nodal-measure-1}
-------------------
Here we use the ultraspherical or Gegenbauer polynomials (see appendix \[sec:ultrasph pol\] for details).
Using proposition \[prop:ler var int form\], and proposition \[prop:exp ler meas\], we obtain $$\label{eq:var int [-1,1]}
\begin{split}
{\operatorname{Var}}({\mathcal{Z}}) = \frac{|{{\mathcal{S}}^{m}}|}{2\pi}\int\limits_{{{\mathcal{S}}^{m}}}
\frac{dx}{\sqrt{1-u(x)^2}} - \frac{|{{\mathcal{S}}^{m}}|^2}{2\pi} =
\frac{|{{\mathcal{S}}^{m}}|}{2\pi}\int\limits_{-1}^{1} \bigg(
\frac{1}{\sqrt{1-{Q_{n}^{{m}}}(t)^2}} - 1\bigg)d\mu(t),
\end{split}$$ where $\mu=\mu_{{m}}$ is the measure on $I:=[-1,1]$ defined by $$\label{eq:mu def} d\mu(t) =
\frac{2\pi^{{m}/2}}{\Gamma(\frac{{m}}{2})} \cdot
(1-t^2)^{\frac{{m}-2}{2}} dt.$$
It is easy to check that $\mu = g_{*}\nu$, where $g:{{\mathcal{S}}^{m}}\rightarrow I$ is the function $$\label{eq:g sphr [-1,1] def} g(x):= \cos{d(x,N)},$$ and $d$ is the spherical distance (recall that $\nu$ is the uniform measure on ${{\mathcal{S}}^{m}}$).
Lemma \[lem:1/sqrt(1-u\^2) int asymp\] together with conclude the proof of the theorem once noting .
\[lem:1/sqrt(1-u\^2) int asymp\] One has the following asymptotics $$\int\limits_{-1}^{1}\bigg[\frac{1}{\sqrt{1-{Q_{n}^{{m}}}(t) ^2}} -
1\bigg]d\mu(t) = 2^{m-2} \pi^{m/2} \Gamma(\frac{m}{2})
\frac{1}{n^{{m}-1}} + O(\epsilon({m};n)),$$ where $\epsilon({m};n)$ is given by $$\label{eq:eps def} \epsilon({m};n) := \begin{cases}
\frac{\log{n}}{n^2},\; &{m}=2 \\n^{-{m}},
&{m}\ge 3\end{cases},$$ and $\mu$ is the measure defined by .
To prove lemma \[lem:1/sqrt(1-u\^2) int asymp\], we will divide the domain of the integral (i.e. the interval $I:=[-1,1]$) into two subintervals: $B := [-1+\frac{c_{0}}{n^2}, 1-\frac{c_{0}}{n^2}]$ with $c_{0}$ constant, and $B^{c}:= I \setminus B$. We will show that the main contribution to the integral in comes from $B$, [*bounding*]{} the contribution of $B^{c}$ to that integral.
We will reuse this partition while proving theorem \[thm:var length\] (see section \[sec:var len bnd proof\]). This justifies devoting a separate section (namely, section \[sec:sing set\]) to the treatment of $B^{c}$. In analogy to the situation of [@ORW] (cf. section 6.1) and [@RW] (cf. section 6.2), we will call $B$ and $B^{c}$ the [*nonsingular*]{} and the [*singular*]{} intervals respectively. The proof of lemma \[lem:1/sqrt(1-u\^2) int asymp\] will be finally given in section \[sec:proof of 1/sqrt(1-u\^2) asymp\].
The singular and nonsingular intervals, as well as some of their properties will be given in section \[sec:sing set\]. The proof of lemma \[lem:1/sqrt(1-u\^2) int asymp\] will be finally given in section \[sec:proof of 1/sqrt(1-u\^2) asymp\].
The singular interval {#sec:sing set}
---------------------
In the course of the proofs of theorems \[thm:var ler meas\] and \[thm:var length\], we are going to deal with the function $$h(t)=\frac{1}{\sqrt{1-({Q_{n}^{{m}}}(t))^2}}$$ defined on $[-1,1]$. We wish to expand it into the Taylor polynomial of $f(s)=\frac{1}{\sqrt{1-s^2}}$ around $s=0$ as $$\label{eq:1/sqrt(1-u^2) exp nonsing}
\frac{1}{\sqrt{1-({Q_{n}^{{m}}}(t))^2}} = 1+\frac{({Q_{n}^{{m}}}(t))^2}{2}
+ O(({Q_{n}^{{m}}}(t))^4).$$
To be able to justify the expansion above, we will have to bound ${Q_{n}^{{m}}}(t)$ away from $\pm 1$, as in corollary \[cor:|Q|<eps0\]. This corollary provides us with a subinterval $B\subseteq [-1,1]$ (which will be referred as the [*nonsingular*]{} interval) of large measure $\mu$, such that ${Q_{n}^{{m}}}(t)$ is bounded away from $\pm 1$ for all $t\in B$. Giving a special treatment to its complement (referred as the [*singular*]{} interval, even though it is in fact a union of two disjoint intervals), we will show that its contribution is negligible (see sections \[sec:sing set contr ler\] and \[sec:sing set contr len\]). We give a rigorous treatment below.
Let $I$ be the interval $I=[-1,1]$. Choose any $0<\epsilon_0 < 1$ and the constant $c_0>0$ guaranteed by corollary \[cor:|Q|<eps0\], corresponding to $\epsilon_0$, assuming that $n$ is large enough in the sense of corollary \[cor:|Q|<eps0\]. We fix $\epsilon_0$ and $c_0$ throughout the rest of the paper and define the nonsingular interval $$B=B_{n} := \big[-1+\frac{c_0}{n^2},1-\frac{c_0}{n^2}\big].$$ Corollary \[cor:|Q|<eps0\] implies that the expansion holds on $B$ with the constant involved in the $'O'$-notation dependent only on $\epsilon_0$.
By an explicit computation, it is clear that $$\label{eq:bnd nonsing set} \mu(B^{c}) \ll n^{- {m}},$$ where $\mu$ is the measure on $I$ defined by .
Recall that $\mu$ is the measure on $[-1,1]$ induced from the uniform measure $\nu$ on ${{\mathcal{S}}^{m}}$ by $g:{{\mathcal{S}}^{m}}\rightarrow [-1,1]$ defined by . We also define the [*spherical*]{} nonsingular set $$SB := g^{-1} (B),$$ and the [*spherical singular set*]{} $$SB^c := {{\mathcal{S}}^{m}}\setminus
SB.$$ Since, as it was mentioned earlier, $\mu=g_{*}\nu$, it is evident that $$\label{eq:meas SBc bnd} \nu(SB^c) = \mu(B^c) = O(n^{-{m}}).$$
The set $SB$ is analogous to the [*nonsingular*]{} set in the sense of [@ORW] (cf. section 6.1) and [@RW] (cf. section 6.2). The structure of $SB$ on the sphere (i.e., its projection $B$ into $[-1,1]$ by $g$) is by far simpler than that of the singular set on the torus, due to the lack of problems of arithmetic nature.
Proof of lemma \[lem:1/sqrt(1-u\^2) int asymp\] {#sec:proof of 1/sqrt(1-u^2) asymp}
-----------------------------------------------
We write $$\int\limits_{-1}^{1}\frac{d\mu(t)}{\sqrt{1-({Q_{n}^{{m}}}(t)) ^2}} dt =
\int\limits_{B} + \int\limits_{B^c}.$$ This, together with lemmas \[lem:bnd int 1/sqrt(1-u\^2) sing\] and \[lem:contr nonsing int 1/sqrt(1-u\^2)\] imply the result.
### The contribution of the singular interval $B^{c}$ {#sec:sing set contr ler}
\[lem:bnd int 1/sqrt(1-u\^2) sing\] One has $$\label{eq:bnd int 1/sqrt(1-u^2) sing}
\int\limits_{B^c}\frac{d\mu(t)}{\sqrt{1-({{Q_{n}^{{m}}}}
(t))^2}} \ll n^{-{m}}.$$
We will bound the contribution of the integral on $$B^{c}\cap [0,1]
= [1-\frac{c_0}{n^2},1],$$ the rest being similar. Furthermore, we may assume by symmetry, that ${Q_{n}^{{m}}}(t) \ge 0$ so that $$\frac{1}{\sqrt{1-{Q_{n}^{{m}}}(t)^2}} \ll \frac{1}{\sqrt{1-{Q_{n}^{{m}}}(t)}} .$$ In what follows we will, consistently with appendix \[sec:ultrasph pol\], adapt the notation $$\alpha:=\frac{{m}-2}{2}.$$
Writing $t=\cos{\psi}$, we have $\phi\in [0,\frac{c_1}{n}] $ for some constant $c_1>0$. Substituting into Hilb’s generalized asymptotic formula (see lemma \[lem:Hilb asymp gen\]), we have $$Q_{n}^{{m}} (\cos{\psi}) =
C\cdot\sqrt{\frac{\psi}{\sin{\psi}}}
\frac{J_{\alpha}(n\psi)}{(\sin{\psi})^{\alpha}}+O(\psi^2),$$ for some constant $C=C_{n}^{{m}}$, using the normalization defined by . Taking the limit $\phi\rightarrow 0$, the value of the constant $C$ is easily seen to be $$\label{eq:C def} C = \bigg[\lim\limits_{\phi\rightarrow
0}\frac{J_{\alpha}(n\phi)}{\phi^{\alpha}}\bigg]^{-1}=n^{-\alpha}
\tilde{C},$$ where $$\label{eq:tildeC def} \tilde{C} = \tilde{C}^{{m}} :=
\bigg[\lim\limits_{\phi\rightarrow
0}\frac{J_{\alpha}(\phi)}{\phi^{\alpha}}\bigg]^{-1}\ne 0,$$ since ${Q_{n}^{{m}}}(1)\ne 0$ (one can obtain an explicit expression for this constant using the expansion of the Bessel function into power series, see e.g. [@OL], page 57).
Thus, the contribution of the singular interval to the integral, is, for $n$ large enough $$\begin{split}
\label{eq:1/sqrt(1-Q^2) [0,1/n] hilb subs}
&\int\limits_{1-\frac{c_{0}}{n^2}}^{1}\ll\int\limits_{0}^{c_1/n}\frac{(\sin{\phi})^{{m}-1}}{\sqrt{1-{Q_{n}^{{m}}}(\cos{\phi})}}
d\phi \ll \int\limits_{0}^{c_1/n}
\frac{\phi^{{m}-1}}{\sqrt{1-C\cdot\sqrt{\frac{\phi}{\sin{\phi}}}
\frac{J_{\alpha}(n\phi)}{(\sin{\phi})^{\alpha}}+O(\phi^2)}} d\phi
\\&= n^{-{m}}\int\limits_{0}^{c_1}
\frac{\psi^{{m}-1}}{\sqrt{1-C\cdot (1+O(\frac{\psi}{n})^2)
\frac{J_{\alpha}(\psi)}{(\frac{\psi}{n})^{\alpha}}+O((\frac{\psi}{n})^2)}}
d\psi \\&= n^{-{m}}\int\limits_{0}^{c_1}
\frac{\psi^{{m}-1}}{\sqrt{1-\tilde{C}
\frac{J_{\alpha}(\psi)}{\psi^{\alpha}}+O((\frac{\psi}{n})^2)}}
d\psi,
\end{split}$$ by .
We claim that $$\label{eq:1-CJ>>phi^2} 1-\tilde{C}
\frac{J_{\alpha}(\psi)}{\psi^{\alpha}} \gg_{c_{1}} \psi^{2}.$$
Having proved would imply that $$\int_{1-\frac{c_{0}}{n^2}}^{1} \ll n^{-{m}} \int_{0}^{c_{1}}
\psi^{{m}-2}d\psi \ll n^{-{m}},$$ which is the statement of the lemma.
To see , it is sufficient to show that $$\lim\limits_{\psi\rightarrow 0} \frac{1-\tilde{C}
\frac{J_{\alpha}(\psi)}{\psi^{\alpha}}}{\psi^2} > 0$$ and $$\label{eq:|C*J/sin|<1} \bigg|\tilde{C}
\frac{J_{\alpha}(\psi)}{\psi^{\alpha}} \bigg| < 1$$ for every $\psi\in (0,c_1]$. However the former inequality follows from the Bessel function expansion into power series around $\psi=0$ (see [@OL], page 57, (9.09)) $$1-\tilde{C} \frac{J_{\alpha}(\psi)}{\psi^{\alpha}} =
a_0\psi^2+O(\psi^4) ,$$ for some constant $a_0 > 0$, so that the limit is positive.
To see , we note that in the course of establishing , we showed $${Q_{n}^{{m}}}(\cos{\frac{\psi}{n}}) = \tilde{C}
\frac{J_{\alpha}(\psi)}{\psi^{\alpha}}
+O\bigg(\big(\frac{\psi}{n}\big)^2\bigg).$$ Therefore, if is not satisfied, taking $n$ large enough would contradict $|{Q_{n}^{{m}}}(t)| \le 1$.
### The contribution of the nonsingular interval $B$
\[lem:contr nonsing int 1/sqrt(1-u\^2)\] $$\label{eq:contr nonsing int 1/sqrt(1-u^2)}
\int\limits_{B}\bigg[\frac{1}{\sqrt{1-({Q_{n}^{{m}}}(t))^2}} - 1\bigg]
d\mu(t) = 2^{m-2} \pi^{m/2}
\Gamma(\frac{m}{2})\cdot\frac{1}{n^{{m}-1}}+O(\epsilon({m};
n)),$$ where $\epsilon({m};n)$ is given by .
On $B$ we may write $$\frac{1}{\sqrt{1-({Q_{n}^{{m}}}(t))^2}} = 1 +
\frac{({Q_{n}^{{m}}}(t))^2}{2} + O\big({Q_{n}^{{m}}}(t)^4\big)$$ (see section \[sec:sing set\]). Integrating, we obtain $$\begin{split}
&\int\limits_{B}\bigg[\frac{1}{\sqrt{1-({Q_{n}^{{m}}}(t))^2}} - 1
\bigg] d\mu(t) = \frac{1}{2}\int\limits_{B} ({Q_{n}^{{m}}}(t))^2
d\mu(t) + O\bigg(\int\limits_{B} ({Q_{n}^{{m}}}(t))^4 d\mu(t)\bigg)
\\&= O(\mu(B^c)) + (\frac{1}{2}\int\limits_{-1}^{1}
({Q_{n}^{{m}}}(t))^2 d\mu(t) +O(\mu(B^{c}))) +
O\bigg(\int\limits_{-1}^{1} ({Q_{n}^{{m}}}(t))^4 d\mu(t)\bigg) \\&=
\frac{1}{2}(2^{m-1} \pi^{m/2}
\Gamma(\frac{m}{2})\frac{1}{n^{{m}-1}}+O(n^{-{m}}))+O(\epsilon({m};n))
\\&=
2^{m-2} \pi^{m/2} \Gamma(\frac{m}{2}) \cdot
\frac{1}{n^{{m}-1}}+O(\epsilon({m};n)),
\end{split}$$ as stated, by and lemmas \[lem:2nd mom Qn\] and \[lem:4th mom Qn\].
Riemannian volume {#sec:var len bnd proof}
-----------------
The goal of this section is to prove theorem \[thm:var length\].
### Plan of the proof of theorem \[thm:var length\]
We have by proposition \[prop:justif ord chng\], $$\label{eq:val int form} {\operatorname{Var}}({\mathcal{Z}}(f)) = |{{\mathcal{S}}^{m}}|\int_{{{\mathcal{S}}^{m}}}
K(x) dx - c_{{m}}{E_{n}},$$ where $$K(x) = \frac {1} {\sqrt{1-u^2}}
\int_{{\mathbb{R}}^{{m}}\times{\mathbb{R}}^{m}} \| w_1\| \| w_2\|
\frac{\exp(-\frac {1}{2} (w_{1},w_{2})\Omega^{-1}
(w_{1},w_{2})^{t})}{\sqrt{\det\Omega}} \frac{dw_1
dw_2}{(2\pi)^{{m}+1}},$$ and $c_{{m}}$ is a constant given by .
As in case of the Leray nodal measure, we divide the integration range into the nonsingular set $SB$ and its complement $SB^{c}$ (see section \[sec:sing set\]). We bound the corresponding contributions to the integral separately (see lemmas \[lem:int kern Bsing\] and \[lem:int kern Bcnonsing\]). Using corollary \[cor:ker bnd ler ker\], it is easy to relate the contribution of $SB^{c}$ to the last integral in , which we already bounded while treating the variance of the Leray nodal measure (lemma \[lem:bnd int 1/sqrt(1-u\^2) sing\]).
It then remains to bound the contribution of the integral on $SB$. Here we may write $\frac{1}{\sqrt{1-u^2}} = 1+ O(u^2)$ and one may show that, up to an admissible error, we may replace it by $1$. We will define a new matrix $S$ by $$\Omega = {\frac{{E_{n}}}{{m}}}(I-S),$$ and notice that substituting $S=0$ into the integral, the identity matrix $I$ recovers the square of the expected volume $({\mathbb{E}}{\mathcal{Z}})^2$. Bounding the variance is then equivalent to “bounding" the matrix $S$ in some [*average*]{} sense.
To quantify the last statement we set $\sigma(x)$ to be the spectral norm of the matrix $S(x)$. We will show that the variance is bounded by $${E_{n}}\cdot\bigg(\int\limits_{{{\mathcal{S}}^{m}}}\sigma(x) dx +
O(\frac{1}{{\mathcal{N}}}) \bigg).$$ To bound $\int\sigma(x)$, we use the trivial inequality $\sigma(x) \le \sqrt{{\operatorname{tr}}{S^2}}$. We will prove that $\int {\operatorname{tr}}(S(x)^2) \ll \frac{1}{{\mathcal{N}}} $, and together with the Cauchy-Schwartz inequality this implies the statement of the theorem.
### A bound for the contribution on the singular interval $SB^c$ {#sec:sing set contr len}
\[lem:int kern Bsing\] One has $$\int\limits_{SB^{c}} K(x) dx \ll {E_{n}}\epsilon(m;n),$$ where $\epsilon(m;n)$ is defined by .
We use corollary \[cor:ker bnd ler ker\] to write $$\int\limits_{SB^{c}} K(x) dx \ll_{{m}} {E_{n}}\int\limits_{SB^{c}} \frac{dx}{\sqrt{1-u(x)^2}} = {E_{n}}\int\limits_{B^{c}} \frac{d\mu(t)}{\sqrt{1-{Q_{n}^{{m}}}(t)}} \ll
{E_{n}}\epsilon({m};n),$$ obtaining the last inequality by lemma \[lem:bnd int 1/sqrt(1-u\^2) sing\].
### A bound for the contribution on the nonsingular interval $SB$
\[lem:int kern Bcnonsing\] $$\int\limits_{SB} K(x) dx = \frac{1}{|{{\mathcal{S}}^{m}}|}({\mathbb{E}}({\mathcal{Z}}))^2 +
O(\frac{{E_{n}}}{\sqrt{{\mathcal{N}}}}).$$
Define $\Omega_1=\Omega_1 (x)$ by $\Omega = {\frac{{E_{n}}}{{m}}}\cdot
\Omega_{1}$. The matrix $\Omega_{1}$ is symmetric, and positive for a set of $x\ne\pm N$, since $\Omega$ is such. Therefore it has a positive definite square root $P_{1}^2 =
\Omega_{1}$.
Intuitively, $\Omega_{1}$ approximates the identity matrix $I$. To quantify this intuitive statement, we introduce the matrix $$\label{eq:S def} S=I-\Omega_{1} = {{\frac{{m}}{{E_{n}}}}} \frac{1}{1-u^2}
\left(\begin{matrix} D^{t} D &-uD ^{t} D-(1-u^2)H \\
-uD^{t} D - (1-u^2)H^{t} & D^{t} D \end{matrix} \right),$$ and its spectral norm $\sigma=\sigma(x)$, i.e $$\sigma = \max\limits_{1 \le i \le 2{m}} |\alpha_{i}|,$$ where $\alpha_i$ are the eigenvalues of $S$. Note that, since $\Omega_{1}$ is positive definite, $S \ll I$ in the sense that all its eigenvalues are in $(-\infty, 1)$.
Changing the coordinates $$w = \sqrt{{\frac{{E_{n}}}{{m}}}}zP_1,$$ we write the definition of $K(x)$ as $$\label{eq:K chng var P1} K(x) = \frac{{E_{n}}}{{m}\sqrt{1-u^2}} \int\limits_{{\mathbb{R}}^{2{m}}} \| (zP_{1})_{1}
\| \cdot \| (zP_{1})_{2} \| e^{-\frac{1}{2}\|z \|^{2}}
\frac{dz}{(2\pi)^{{m}+1}},$$ where for $a\in{\mathbb{R}}^{2{m}}$ we write $(a)_{1}\in{\mathbb{R}}^{{m}}$ and $(a)_{2}\in{\mathbb{R}}^{{m}}$ to denote either the first or the last ${m}$ coordinates.
We claim that $$\label{eq:P1=I+O(sig)} P_{1} = I(1+O(\sigma)).$$ This follows from that fact that if $S \sim diag(\alpha_{i})$ then $P_1 \sim diag(\sqrt{1-\alpha_{i}})$ so that $$P_1-I\sim diag(\sqrt{1-\alpha_{i}}-1) \ll diag(|\alpha_{i}|) \ll
\sigma I,$$ since $\sqrt{1-y} -1 < |y|$ on $(-\infty, 1)$.
Moreover, by the definition of the spherical nonsingular set, on $SB$, $u(x)$ is bounded away from $1$, so that one may expand $$\label{eq:1/sqrt(1-u^2) exp nonsing const} \frac{1}{\sqrt{1-u^2}} =
1+O(u^2),$$ where the constant involved in the $'O'$ notation is absolute.
Substituting and into , we obtain $$K(x) = \frac{{E_{n}}}{{m}(2\pi)^{{m}+1}
}\int\limits_{{\mathbb{R}}^{2{m}}} \| (z)_{1} \| \cdot \|
(z)_{2} \| e^{-\frac{1}{2}\|z \|^{2}}
(1+O(u^2))(1+O(\sigma))^2 dz.$$
Continuing, we have $$\begin{split}
K(x) &= \frac{{E_{n}}}{{m}(2\pi)^{{m}+1} } \int
\limits_{{\mathbb{R}}^{{m}}\times {\mathbb{R}}^{{m}}} \|(z)_{1} \| \cdot \|
(z)_{2} \| e^{-\frac{1}{2}(\|z_{1} \|^{2}+\|z_{2} \|^{2})}
dz_{1}dz_{2} (1+O(u^2)+O(\sigma)+O(\sigma^2))
\\&= \bigg( \frac{\sqrt{{E_{n}}}}{\sqrt{{m}}
(2\pi)^{\frac{{m}+1}{2}}} \int\limits_{{\mathbb{R}}^{{m}}}
\|z'\| e^{-\frac{1}{2}\|z' \|^{2}} dz' \bigg)^2
(1+O(u^2)+O(\sigma)+O(\sigma^2)) \\&=
\frac{1}{|{{\mathcal{S}}^{m}}|^2}({\mathbb{E}}{\mathcal{Z}})^2 (1+O(u^2)+O(\sigma)+O(\sigma^2)),
\end{split}$$ by and .
Integrating on $SB$, we obtain $$\int\limits_{SB} K(x)dx - \frac{1}{|{{\mathcal{S}}^{m}}|}({\mathbb{E}}{\mathcal{Z}})^2 \ll
{E_{n}}\bigg(\frac{1}{\sqrt{{\mathcal{N}}}}+\int\limits_{SB} u^2dx +
O(\nu(SB^c))\bigg),$$ by and lemma \[lem:1st sec mom sigma\].
To bound the last expression, we use , as well as, by the definition of the two-point function, we have $$\int\limits_{SB} u^2 dx \le \int\limits_{{{\mathcal{S}}^{m}}} ({Q_{n}^{{m}}}(\cos{d(x,N)}))^2 dx = \int\limits_{-1}^{1} ({Q_{n}^{{m}}}(t))^2
d\mu(t) \ll \frac{1}{{\mathcal{N}}},$$ by lemma \[lem:2nd mom Qn\] and . This concludes the proof of the lemma.
\[lem:1st sec mom sigma\] For a fixed ${m}$, as $n\rightarrow\infty$, one has
1. \[it:int sigma\^2 < 1/N\] $$\int\limits_{SB} \sigma (x) ^2dx \ll \frac{1}{{\mathcal{N}}}.$$
2. \[it:int sigma < 1/sqrt(N)\] $$\int\limits_{SB} \sigma (x)dx \ll \frac{1}{\sqrt{{\mathcal{N}}}}.$$
Part \[it:int sigma < 1/sqrt(N)\] of the lemma clearly follows from part \[it:int sigma\^2 < 1/N\] by the Cauchy-Schwartz inequality. Thus we are only to prove part \[it:int sigma\^2 < 1/N\].
To prove the statement, we recall that $\sigma$ is by the definition the spectral norm of $S$, defined by . To bound $\int\sigma^2$, we use the trivial inequality $\sigma\le {\operatorname{tr}}(S^2)$.
Since, by the definition of the nonsingular set $SB$, the two-point function $u(x)$ is bounded away from $1$, we may disregard the $1-u^2$ altogether. We define the matrix $$S_{1} := \frac{{E_{n}}}{{m}} (1-u^2)S = \left(\begin{matrix} D^{t} D &-uD ^{t} D-(1-u^2)H \\
-uD^{t} D - (1-u^2)H^{t} & D^{t} D \end{matrix} \right).$$ We claim that $$\label{eq:tr(S1^2)<<n^3} \int\limits_{{{\mathcal{S}}^{m}}} {\operatorname{tr}}{S_{1}^2} dx =
O(n^{5-{m}}).$$ This is sufficient for the statement of the present lemma, since then $$\begin{split} \int\limits_{SB} \sigma^2 dx \ll
\frac{1}{{E_{n}}^2}\int\limits_{SB} {\operatorname{tr}}{S_1^2} dx \le
\frac{1}{{E_{n}}^2}\int\limits_{{{\mathcal{S}}^{m}}} {\operatorname{tr}}{S_1^2} dx \ll
\frac{1}{n^4}\cdot n^{5-{m}} \ll \frac{1}{{\mathcal{N}}}.
\end{split}$$
Now the elements of the matrix $S^2$ are bounded by elements either of the form $$\frac{\partial u}{\partial e_{i_{1}}^{z}} \vert_{(x,N)}\cdot
\frac{\partial u}{\partial e_{i_{2}}^{z}} \vert_{(x,N)} \cdot
\frac{\partial u}{\partial e_{i_{3}}^{z}} \vert_{(x,N)} \cdot
\frac{\partial u}{\partial e_{i_{4}}^{z}} \vert_{(x,N)},$$ the form $$\frac{\partial u}{\partial e_{i_{1}}^{z}} \vert_{(x,N)} \cdot
\frac{\partial u}{\partial e_{i_{2}}^{z}} \vert_{(x,N)} \cdot
\frac{\partial ^{2} u}{\partial e_{i_{3}}^{z} \partial
e_{i_{4}}^{z}} \vert_{(x,N)},$$ or the form $$\frac{\partial ^{2} u}{\partial e_{i_{1}}^{z} \partial
e_{i_{2}}^{z}} \vert_{(x,N)} \cdot \frac{\partial ^{2} u}{\partial
e_{i_{3}}^{z}
\partial e_{i_{4}}^{z}} \vert_{(x,N)},$$ where in all the expressions above $z$ may be either $x$ or $y$ (see section \[sec:orthonorm bas corr mat exp\] for an explanation of the partial derivatives notations).
Using the Cauchy-Schwartz inequality again and the symmetry with respect to the variables, it suffices to prove the inequalities $$\label{eq:1st der int sph bnd}
\int\limits_{{{\mathcal{S}}^{m}}}\bigg(\frac{\partial u}{\partial e^{x}_1}
(x)\bigg)^4 dx \ll n^{5-{m}}$$ and $$\label{eq:2nd der int sph bnd}
\int\limits_{{{\mathcal{S}}^{m}}}\bigg(\frac{\partial^2 u}{\partial
e^{x}_{1}\partial e^{x}_{2}} (x)\bigg)^2 dx \ll n^{5-{m}}.$$
We may compute the partial derivative in (assuming $x\ne\pm N$) as $$\frac{\partial}{\partial e_{1}^{x}} {Q_{n}^{{m}}}(\cos{d(x,N)}) =
-{Q_{n}^{{m}}}{'}(\cos{d(x,N)}) \sin{d(x,N)} \frac{\partial}{\partial
e_{1}^{x}} d(x,N),$$ so that, since $\frac{\partial}{\partial e_{1}^{x}} d(x,y) $ is obviously bounded on ${{\mathcal{S}}^{m}}$, it is sufficient to bound $$\int\limits_{{{\mathcal{S}}^{m}}} \big({Q_{n}^{{m}}}{'}(\cos{d(x,N)})
\sin{d(x,y)}\big)^4 dx = \int\limits_{-1}^{1}
\big({Q_{n}^{{m}}}{'}(t)\big)^{4} (1-t^2)^2 d\mu(t),$$ and thus follows from lemma \[lem:4th mom der Qn\], recalling the definition of the measure $\mu$.
As for , we write the second partial derivative in the integrand as $$\begin{split}
\frac{\partial^2 u}{\partial e^{x}_{1}\partial e^{x}_{2}} (x) &=
{Q_{n}^{{m}}}{''}(\cos{d(x,N)}) \sin^2{d(x,N)}
\frac{\partial}{\partial e_{1}^{x}} d(x,N) \frac{\partial}{\partial
e_{2}^{x}}d(x,N) \\&- {Q_{n}^{{m}}}{'}(\cos{d(x,N)}) \cdot
\frac{\partial}{\partial e_{2}^{x}} \bigg[ \sin{d(x,N)}
\frac{\partial}{\partial e_{1}^{x}} d(x,N) \bigg],
\end{split}$$ so that, using a similar argumentation, we conclude that the integral in is bounded by $$\begin{split}
&\ll \int\limits_{{{\mathcal{S}}^{m}}} \big( {Q_{n}^{{m}}}{''}(\cos{d(x,N)})
\big) ^2 (\sin{d(x,N)})^4 dx+\int\limits_{{{\mathcal{S}}^{m}}} \big( {Q_{n}^{{m}}}{'}(\cos{d(x,N)}) \big)
^2 dx \\&= \int\limits_{-1}^{1}\big(
{Q_{n}^{{m}}}{''}(t) \big) ^2 (1-t^2)^2 d\mu(t)+\int\limits_{-1}^{1} \big(
{Q_{n}^{{m}}}{'}(t)\big)^2 d\mu(t).
\end{split}$$ Therefore, follows from lemmas \[lem:2nd mom der Qn\] and \[lem:2th mom sec der Qn\].
### Concluding the proof of theorem \[thm:var length\]
We write as, $${\operatorname{Var}}({\mathcal{Z}}) =
\int\limits_{SB^c}K(x)dx+\bigg(\int\limits_{SB}K(x)dx-{\mathbb{E}}({\mathcal{Z}})^2\bigg)
\ll {E_{n}}\epsilon({m}; n) +
\frac{{E_{n}}}{\sqrt{{\mathcal{N}}}} \ll
\frac{{E_{n}}}{\sqrt{{\mathcal{N}}}},$$ by lemmas \[lem:int kern Bsing\] and \[lem:int kern Bcnonsing\], where we use $$\epsilon({m};n) \ll
\frac{{E_{n}}}{\sqrt{{\mathcal{N}}}},$$ due to and .
Legendre and ultraspherical polynomials {#sec:ultrasph pol}
=======================================
The ultraspherical (or Gegenbauer) polynomials $P^{\alpha}_{n}(t):[-1,1]\rightarrow{\mathbb{R}}$ of degree $n$ generalize the Legendre polynomials $P_{n}(t) = P_{n}^{0} (t)$. We use the corresponding [*normalized*]{} polynomials $Q^{m}_{n} (t)$ for an integral $m\ge 0$, which differ from $P^{\alpha}_{n}$ (for a suitably chosen $\alpha$), by a constant, defined by $$Q^{m}_{n}(1)=1.$$
Definition and basic facts
--------------------------
The Legendre polynomials $P_{n}$ are the unique polynomials of degree $n$, orthogonal on $[-1,1]$ (w.r.t. the trivial weight function), normalized by $P_{n}(t)=1$. More generally, for a real $$\alpha>-1,$$ we define the ultraspherical polynomials $P_{n}^{\alpha}(t)$, being, up to a constants, the unique sequence polynomials of degree $n$, pairwise orthogonal w.r.t. the weight function on $[-1,1]$ defined by $$\label{eq:omega def}
\omega(t) = (1-t^2)^{\alpha}.$$ It is defined uniquely by the normalizing condition $$\label{eq:P(1) def} P_{n}^{\alpha}(1) =
\frac{\Gamma(n+\alpha+1)}{\Gamma(n+1) \cdot \Gamma(\alpha+1)},$$ once we know that $t=1$ is not a zero of $P_{n}^{\alpha}$, see [@SZ], chapter 3.3. The ultraspherical polynomials is a particular case $\alpha=\beta$ of a more general class of polynomials, usually referred to as the Jacobi polynomials $P_{n}^{\alpha,\beta}$ (see e.g. [@SZ] for more information).
While studying the spherical harmonics on the $m$-dimensional sphere, we are interested in the ultraspherical polynomials with $\alpha = \frac{m-2}{2}$, and moreover, we would like to normalize it by setting its value at $1$ to be $1$. That is, we define $$\label{eq:Q def} Q_{n}^{m}(t) := \frac{P_{n}^{\alpha}
(t)}{P_{n}^{\alpha} (1)},$$ where $$\label{eq:alpha def} \alpha := \frac{m-2}{2}.$$ For example, $$Q_{n}^{2}(t) = P_{n}(t) = P_{n}^{0}(t)$$ are the usual Legendre polynomials.
Throughout the section, we fix an integral number $m\ge 2$, and use the associated value of $\alpha$, defined by . It is well known that $Q_{n}^{m}$ is either even or odd, for the even and odd values of $n$ respectively, and $|Q_{n}^{m}(t)|$ has a maximum at $t=\pm 1$.
The function $v=P_{n}^{\alpha}(t)$ satisfies the differential equation ( [@SZ], page 60, (4.2.1)) $$\label{eq:diff eq ultrsph} (1-t^2)v'' - mtv'+n(n+m-1)v=0.$$ Due to its linear nature, it is also satisfied by $v=Q_{n}^{m}(t)$. The following recurrence relation ( [@SZ], page 83, (4.7.27)) will prove itself as very useful $$\label{eq:recc ultrsph}
(1-t^2)P_{n}^{\alpha} {'} (t)
+ntP_n^{\alpha}(t)-(n+\alpha)P_{n-1}^{\alpha}(t) = 0.$$ Note that this recurrence relation is not satisfied by $Q^{m}(t)$ due to the different normalization constants for $P_{n}$ and $P_{n-1}$.
Some basic results
------------------
Recall the definition and of the measure $\mu=\mu_{m}$ and the weight function $\omega_{m}$ respectively. We note that $$d\mu(t) =
\frac{2\pi^{m/2}}{\Gamma(\frac{m}{2})} \cdot \omega(t) dt,$$ so for purposes of giving an upper bound only, we may disregard the difference between $d\mu$ and $\omega dt$.
Concerning the $2$nd and the $4$th moments of the ultraspherical polynomials, we have the following:
\[lem:2nd mom Qn\] For $m$ [*fixed*]{}, the second moment of the normalized ultraspherical polynomials is $$\int\limits_{-1}^{1} Q_{n}^{m} (t) ^2 d \mu(t) = 2^{m-1} \pi^{m/2}
\Gamma(\frac{m}{2}) \cdot \frac{1}{n^{m-1}} +O(n^{-m}),$$ as $n\rightarrow\infty$.
One has ( [@SZ], (4.3.3)) $$\label{eq:2nd mom P} \int\limits_{-1}^{1} P^{\alpha}_{n} (t)^2
\omega(t) dt = \frac{2^{m-1}}{2n+m-1}
\frac{\Gamma(n+\frac{m}{2})^2}{\Gamma(n+1)\Gamma(n+m-1)} ,$$ and implies that $$\label{eq:P(1) asymp} P^{m}_{\alpha} (1) =
\frac{\Gamma(n+\frac{m}{2})}{\Gamma(n+1)\Gamma(\frac{m}{2})} \sim c
\cdot n^{\alpha}.$$ Thus, using the definition of the normalized ultraspherical polynomials, we obtain $$\begin{split}
\int\limits_{-1}^{1} Q^{m}_{n} (t)^2 d\mu(t) &=
\frac{2^{m-1}}{2n+m-1}
\frac{\Gamma(n+1)\Gamma(\frac{m}{2})^2}{\Gamma(n+m-1)} \cdot
\frac{2\pi^{m/2}}{\Gamma(\frac{m}{2})}
\\&= \frac{2^m \pi^{m/2} \Gamma(\frac{m}{2})}{2n+m-1}
\frac{n!}{(n+m-2)!} \\&= 2^{m-1} \pi^{m/2} \Gamma(\frac{m}{2}) \cdot
\frac{1}{n} \cdot\frac{1}{n^{m-2}}(1+O(\frac{1}{n})) \\&= 2^{m-1}
\pi^{m/2} \Gamma(\frac{m}{2}) \frac{1}{n^{m-1}} + O(n^{-m}),
\end{split}$$ as stated.
\[lem:Hilb asymp gen\] $$\label{eq:Hilb asymp gen} (\frac{1}{2} \sin{\theta})^{\alpha}
P^{\alpha}_{n}(\cos{\theta}) = N^{-\alpha}
\frac{\Gamma(n+\alpha+1)}{n!}
\bigg(\frac{\theta}{\sin{\theta}}\bigg)^{1/2}J_{\alpha}(N\theta)+\delta(\theta),$$ uniformly for $0\le\theta\le\pi/2$, where $N=n+\frac{m-1}{2}$, $J_{\alpha}$ is the Bessel $J$ function of order $\alpha$ and the error term is $$\begin{split} \delta(\theta) \ll \begin{cases}
\theta^{1/2} O(n^{-3/2}), \: &cn^{-1} < \theta < \pi/2 \\
\theta^{\alpha+2}O(n^{\alpha}), \: &0<\theta < cn^{-1}.
\end{cases}
\end{split}$$
#### Remark:
It is clear, that $$\label{eq:hilb const sim 1} n^{-\alpha}
\frac{\Gamma(n+\alpha+1)}{n!} = 1+O(\frac{1}{n}),$$ so we will usually omit this factor.
\[cor:|Q|<eps0\] For every $\epsilon_{0} > 0$, there exists a constant $c_{0}>0$ depending on $m$ only, such that if $c\ge
c_0$ and $t\in [0,1-\frac{c}{n^2}]$, where $n$ is large enough so that the interval above is not empty, one has $$|Q_{n}^{m}(t)| < \epsilon_{0}.$$
Let $t=\cos{\theta}$. Then if $0 \le t < 1-\frac{c_0}{n^2}$, $\theta
> C_{0}\cdot\frac{\sqrt{c_0}}{n}$ for some absolute constant $C_0>0$. Lemma \[lem:Hilb asymp gen\] implies that one has $$|P^{\alpha}_{n} (\cos{\theta})| \le C
\frac{1}{\sin^{\alpha}{\theta}} |J_{\alpha}(N \theta)|$$ for some absolute constant $C>0$. We bound it by $$|P^{\alpha}_{n} (\cos{\theta})| \le C_1
\frac{n^{\alpha}}{c_0^{\alpha/2}} |J_{\alpha} (N\theta)|,$$ so that implies that $$|Q^{m}_{n} (\cos{\theta})| \le \frac{C_2}{c_0^{\alpha/2}}
|J_{\alpha} (N\theta)| < \epsilon_{0},$$ provided that we choose $c_{0}$ large enough, since $J_{\alpha}$ is bounded.
\[lem:4th mom Qn\] The $4$th moment of the ultraspherical polynomials satisfies $$\int\limits_{-1}^{1} Q^{m}_{n}(t)^4 d\mu(t) \ll \epsilon(m;n),$$ where $\epsilon(m;n)$ is defined by .
We will limit ourselves to the interval $[0,1]$. To prove the statement there, we invoke the generalized Hilb’s asymptotics (lemma \[lem:Hilb asymp gen\]).
We have, using , that $$(\sin{\theta})^{m-2} (P^{\alpha}_{n}(\cos{\theta}))^4 \ll
J_{\alpha}^4 (N\theta)\frac{\theta^2}{(\sin{\theta})^{m}} +
\frac{\delta^4(\theta)}{\sin^{m-2}{\theta}}$$ and claim that $$\label{eq:4th mom P} \int\limits_{-1}^{1} P^{\alpha}_{n}(t)^4
d\mu(t) \ll \begin{cases} \frac{\log{n}}{n^2},\; m=2\\ n^{m-4},\;
m\ge 3\end{cases}.$$
We have $$\label{eq:int Hilb asymp subs} \int\limits_{-1}^{1}
P^{\alpha}_{n}(t)^4 d\mu(t) \ll \int\limits_{0}^{\pi/2}
J_{\alpha}^4(N\theta) \frac{\theta^2}{(\sin{\theta})^{m-1}}d\theta +
\int\limits_{0}^{\pi/2} \frac{\delta^4(\theta)}{\sin^{m-3}{\theta}}
d\theta$$
The contribution of the main term in to the integral in is $$\begin{split} &\ll \int\limits_{0}^{\pi/2} J_{\alpha}^4(N\theta)
\frac{1}{\theta^{m-3}}d\theta = N^{m-4}\int\limits_{0}^{\frac{\pi
N}{2}} \frac{J_{\alpha}^4(\phi)}{\phi ^{m-3}} d\phi \\&=
N^{m-4}\bigg[\int\limits_{0}^{1} \frac{J_{\alpha}^4(\phi)}{\phi
^{m-3}} d\phi+\int\limits_{1}^{\frac{\pi
N}{2}}\frac{d\phi}{\phi^{m-1}}\bigg],
\end{split}$$ using the well known decay $$J_{\alpha}(y)\ll\frac{1}{\sqrt{y}}$$ of the Bessel J functions at infinity.
The first integral involved in the expression above is $O(1)$, since $J_{\alpha}$ vanishes with multiplicity (at least) $\alpha$ at zero (it follows, for example, from Hilb’s formula). The second one is bounded by $$\ll \begin{cases} \log{n},\; m=2 \\ 1,\; m\ge 3\end{cases}.$$ Therefore the contribution of the main term in to the LHS of is dominated by the RHS of .
The contribution of the error term in is at most $$n^{2m-4}\int\limits_{0}^{1/n} \theta^{m+7} d\theta +
n^{-6}\int\limits_{1/n}^{\pi/2} \theta^{5-m}d\theta = O(n^{m-12}).$$
We obtain the statement of the lemma by using and with .
Moments of the derivatives of the ultraspherical polynomials {#sec:mom der Qn}
------------------------------------------------------------
\[lem:2nd mom der Qn\] $$\int\limits_{-1}^{1} Q_{n}^{m}{'}(t)^2 d\mu(t) \ll
\frac{\log{n}}{n^{m-4}}.$$
We will bound the integral on $[0,1]$ only, having a similar bound on $[-1,0]$. By , the statement of the lemma is equivalent to $$\int\limits_{0}^{1} P_{n}^{\alpha}{'}(t)^2 d\mu(t) \ll n^2\log{n}.$$ We rewrite the last integral using as $$\int\limits_{0}^{1} \frac{\bigg((n+m/2-1)P_{n-1}^{\alpha} (t) -
ntP_{n}^{\alpha}(t)\bigg)^2 }{(1-t^2)^2} d\mu(t).$$
To give a bound, we partition the range of the integration into $2$ subranges: $$\label{eq:2 subrang 2nd mom der} \int\limits_{0}^{1} =
\int\limits_{0}^{1-1/n^2} + \int\limits_{1-1/n^2}^{1}.$$
To bound the second integral in , we define $$\label{eq:f numer def} f(t) := (n+m/2-1)P_{n-1}^{\alpha} (t) -
ntP_{n}^{\alpha}(t) = (1-t^2) P_{n}^{\alpha}{'}(t).$$ Computing the derivative $f'(t)$ and using again we obtain $$\label{eq:f' comp} f'(t) = (m-2)tP_{n}^{\alpha} {'} (t) - n(n+m-1)
P_{n}^{\alpha} (t).$$
We claim that this implies $$\label{eq:f' = O(...)} f'(t) \ll n^2 P_{n}^{\alpha}(1) \ll
n^{\frac{m+2}{2}},$$ the second inequality being a consequence of . To see the first inequality of , we note that it is sufficient to show that $$\label{eq:tP O est} tP_{n}^{\alpha} {'} (t) \ll n^{\frac{m+2}{2}},$$ by . With no loss of generality we may assume that $t=1$ or $P_{n}^{\alpha} {'}$ has a local extremum, i.e. $P_{n}^{\alpha}
{''}(t)=0$. In both cases the equation implies $$t P_{n}^{\alpha}{'} (t) = P_{n}^{\alpha}(t) O(n^2),$$ which implies .
Now using the linear Taylor approximation of $f(t)$ around $t=1$ with , the second integral in is, since $f(1)=0$, $$\ll n ^{m+2}\int\limits_{1-1/n^2}^{1} \frac{(t-1)^2}{(1-t^2)^2}
\cdot (1-t^2)^{\frac{m-2}{2}} dt\ll n ^{m+2} \int\limits_{1-1/n^2}^{1}
(1-t^2)^{\frac{m-2}{2}} dt \ll n ^2.$$
In order to bound the first integral in , we employ the generalized Hilb’s asymptotics . The integrand is (taking the change of variables $t=\cos{\theta}$ and into the account), $$\label{eq:hilb appl 2nd mom der}
\begin{split}
&n^2 \frac{\bigg( (1+O(\frac{1}{n})) P_{n-1}^{\alpha} (\cos{\theta})
- \cos{\theta} P_{n}^{\alpha}(\cos{\theta})\bigg)^2
}{(\sin{\theta})^{3}} \cdot (\sin{\theta})^{2\alpha} \\ &\ll
n^2\cdot\frac{\frac{\sin{\theta}}{\theta}\cdot\bigg(\big(1+O(\frac{1}{n})\big)J_{\alpha}((N-1)\theta)-
\big(1+O(\theta^2)+O(\frac{1}{n})\big)J_{\alpha}(N\theta)\bigg)^2}{(\sin{\theta})^3}
+ n^2\frac{\delta^2(\theta)}{(\sin{\theta})^3} \\ &\ll n^2
\frac{\bigg(J_{\alpha}(N\theta) -
J_{\alpha}((N-1)\theta)\bigg)^2}{\theta^3}+
\frac{1}{\theta^3}+O(n^2\theta)
+n^2\frac{\delta^2(\theta)}{\theta^3},
\end{split}$$ and the integration range is essentially $[\frac{1}{n},\frac{\pi}{2}]$.
The contribution of the last error term in is $$\ll n^2 \cdot n^{-3}\int\limits_{1/n}^{\pi/2}
\frac{\theta}{\theta^{3}}d\theta \ll 1,$$ the other ones being trivially bounded by $O(n^2)$.
The contribution of the main term in is $$\begin{split}
&n^2 \int\limits_{1/n}^{\pi/2}
\frac{\big(J_{\alpha}(N\theta)-J_{\alpha}((N-1)\theta)\big)^2}{\theta^3}
d\theta \ll n^4 \int\limits_{1}^{\frac{n\pi}{2}}
\frac{\bigg(J_{\alpha}(\phi)-J_{\alpha}(\phi(1-\frac{1}{N}))\bigg)^2}{\phi^3}d\phi
\\&\ll n^4\cdot
\frac{1}{n^2}\int\limits_{1}^{\frac{n\pi}{2}}\frac{\phi^2}{\phi^3}d\phi\ll
n^2\log{n},
\end{split}$$ due to the boundness of the derivative $ J_{\alpha}'(t)$. As it was stated, this is equivalent to the statement of the lemma.
\[lem:4th mom der Qn\] One has $$\int\limits_{-1}^{1} Q_{n}^{m}{'}(t)^4 (1-t^2)^{2} d\mu(t) \ll
\begin{cases} n^{2} \log{n},\; &m=2 \\ \frac{1}{n^{m-4}},\; &m\ge 3 \end{cases}.$$
The proof of the lemma is similar to the one of lemma \[lem:2nd mom der Qn\].
We will bound the integral only on $[0,1]$, having a similar bound on $[-1,0]$. The statement of the lemma is equivalent to $$\int\limits_{0}^{1} \frac{\bigg((n+m/2-1)P_{n-1}^{\alpha} (t) -
nxP_{n}^{\alpha}(t)\bigg)^4 }{(1-t^2)^2} d\mu(t) \ll
\begin{cases} n^{2}\log{n}, &m=2 \\ n^{m},\; &m\ge 3 \end{cases},$$ using and .
We partition the range of the integration into $2$ subranges: $$\label{eq:2 subrang 4th mom der} \int\limits_{0}^{1} =
\int\limits_{0}^{1-\frac{1}{n^2}} +
\int\limits_{1-\frac{1}{n^2}}^{1}.$$
To bound the second integral in we use the definition of the function $f(t)$, as well as the inequality , as in the course of proof of lemma \[lem:2nd mom der Qn\]. Thus the integral is $$\ll n^{2(m+2)}\int\limits_{1-1/n^2}^{1} \frac{(1-t)^4}{(1-t)^2}
(1-t)^{\frac{m-2}{2}}dt \ll n^{2(m+2)}n^{-(m+4)} = n^m.$$
To bound the first integral in , we employ the generalized Hilb’s asymptotics . The integrand is (taking into consideration the change of variables $t=\cos{\theta}$), $$\label{eq:hilb appl 4th mom der}
\begin{split}
&n^4 \frac{\bigg( (1+O(\frac{1}{n})) P_{n-1}^{\alpha} (\cos{\theta})
- \cos{\theta} P_{n}^{\alpha}(\cos{\theta})\bigg)^4
}{\sin{\theta}^{3}\cdot \sin{\theta}^{2\alpha}} \cdot (\sin{\theta})^{4\alpha} \\
&\ll
n^4\cdot\frac{\frac{\sin{\theta}}{\theta}\cdot\bigg(\big(1+O(\frac{1}{n})\big)J_{\alpha}((N-1)\theta)-
\big(1+O(\theta^2)+O(\frac{1}{n})\big)J_{\alpha}(N\theta)\bigg)^4}{(\sin{\theta})^{m+1}}
+ n^4\frac{\delta^4(\theta)}{(\sin{\theta})^{m+1}} \\ &\ll n^4
\frac{\bigg(J_{\alpha}(N\theta) -
J_{\alpha}((N-1)\theta)\bigg)^4}{\theta^{m+1}}+
\frac{1}{\theta^{m+1}}+O(n^4\frac{1}{\theta^{m-7}})
+n^4\frac{\delta^4(\theta)}{\theta^{m+1}},
\end{split}$$ and the integration range is (up to a constant) $[\frac{1}{n},\frac{\pi}{2}]$.
The contribution of the last error term in is $$\ll
\frac{n^4}{n^6}\int\limits_{1/n}^{\pi/2}\frac{\theta^2}{\theta^{m+1}}
d\theta = n^{-2}\int\limits_{1/n}^{\pi/2}
\frac{d\theta}{\theta^{m-1}} \ll \max{(n^{m-4}\log{n},1)},$$ the other ones being trivially bounded by $O(n^m)$.
The contribution of the main term in is $$\label{eq:4th mom der hilb p=nt}
\begin{split}
&n^4 \int\limits_{1/n}^{\frac{\pi}{2}}
\frac{\big(J_{\alpha}(N\theta)-J_{\alpha}((N-1)\theta)\big)^4}{\theta^{m+1}}
d\theta \ll n^{m+4} \int\limits_{1}^{\frac{\pi N}{2}}
\frac{\bigg(J_{\alpha}(\phi)-J_{\alpha}(\phi(1-\frac{1}{N}))\bigg)^4}{\phi^{m+1}}d\phi.
\end{split}$$
Let $g(\phi)$ be the function $$g(t) := J_{\alpha}(\phi)-J_{\alpha}(\phi(1-\frac{1}{N})).$$ Then by the mean value theorem, $$g(\phi) = \frac{\phi}{N} J_{\alpha}'(s),$$ where $\phi(1-\frac{1}{N}))<s < \phi$, and using the decay $$|J_{\alpha}'(s)| \ll \frac{1}{\sqrt{s}},$$ we obtain $$\label{eq:g decay} |g(\phi)| \ll \frac{\sqrt{\phi}}{N}.$$
Substituting into , we have that the contribution is $$\begin{split}
&\ll n^{m+4}\cdot \frac{1}{n^4}\int\limits_{1}^{\frac{\pi n}{2}}
\frac{\phi^2}{\phi^{m+1}}d\phi \ll n^{m}\int\limits_{1}^{\frac{\pi
n}{2}}\frac{d\phi}{\phi^{m-1}} \ll \begin{cases} n^{2}\log{n} ,\;
&m=2 \\ n^{m},\; &m\ge 3\end{cases},
\end{split}$$ which concludes the proof of the lemma.
\[lem:2th mom sec der Qn\] $$\int\limits_{-1}^{1} Q_{n}^{m} {''}(t)^2 (1-t^2)^2 d\mu(t) \ll
\frac{1}{n^{m-5}}$$
We use the differential equation to write the integral as $$\begin{split}
&\int\limits_{-1}^{1} \big(mt Q_{n}^{m}{'}(t)-n(n+m-1)Q_{n}^{m}
(t)\big)^2 d\mu(t) \\ &\ll \int\limits_{-1}^{1}
\big((Q_{n}^{m}{'}(t))^2 d\mu(t) + n^4\int\limits_{-1}^{1} Q_{n}^{m}
(t)^2 d\mu(t) \ll \frac{1}{n^{m-5}},
\end{split}$$ by lemmas \[lem:2nd mom Qn\] and \[lem:2nd mom der Qn\].
The singular functions are “rare" {#sec:sing func rare}
=================================
In this section we give the proofs of lemmas \[lem:Sing codim 1\], \[lem:Sing codim 1 Pax\] and \[lem:Sing codim 1 Paxy\] (see section \[sec:sing func\]).
\[not:bigcirc,arc,S2\] Here and in appendix \[sec:f (x)(y)gr f(x)(y) sp\] we adapt the following notations. Let $x$ and $y$ on the sphere ${{\mathcal{S}}^{m}}$ such that $x\ne\pm y$.
1. Denote $\bar{xy}$ the (unique) big circle through $x$ and $y$.
2. The smaller arc of $\bar{xy}$ connecting $x$ to $y$ will be denoted by $\breve{xy}$.
3. Let $z\in{{\mathcal{S}}^{m}}$ be a point not lying on the plane $\Pi=\Pi(x,y)$ defined by $O$, $x$ and $y$. We denote $\mathcal{S}^2 =
\mathcal{S}^2 (x,y,z)$ the (unique) $2$-dimensional big sphere containing $O$, $x$, $y$ and $z$, i.e. $$\mathcal{S}^2 :=
{{\mathcal{S}}^{m}}\cap\Pi(x,y,z) .$$ (Note that there is no ambiguity in notations for ${m}=2$).
We also recall the fact that if $\mathcal{S}^2\subseteq{{\mathcal{S}}^{m}}$ is any big sphere, then for any two points $x,y\in\mathcal{S}^2$, $$\bar{xy}_{\mathcal{S}^2} = \bar{xy}_{{{\mathcal{S}}^{m}}}.$$ In particular, the shortest path between $x$ and $y$ on ${{\mathcal{S}}^{m}}$ passes inside $\mathcal{S}^2$ and $$\nabla_{x} d_{{{\mathcal{S}}^{m}}} (x,y)
=\nabla_{x} d_{\mathcal{S}^2} (x,y) \in T_{x} (\mathcal{S}^2)$$ under the natural embedding $$T_{x} (\mathcal{S}^2) \subseteq T_{x}
({{\mathcal{S}}^{m}}).$$
The following simple geometric lemma will prove itself as quite useful.
\[lem:d(x,xi)=d(x,xi’),d(y,xi)=d(y,xi’)\] Let $x,y\in {{\mathcal{S}}^{m}}$ such that $x\ne y$ and $\xi\ne \xi'\in{{\mathcal{S}}^{m}}$ such that $d(x,\xi)=d(x,\xi')$ and $d(y,\xi)=d(y,\xi')$. Denote $v:=\nabla_{x}d(x,y)$ and $$v_{1}=v_{1}(\xi,\xi') = \nabla_{x}
d(x,\xi)-\nabla_{x} d(x,\xi').$$ Then for all $\xi$ and $\xi'$, $v\perp v_{1}$, and moreover the vectors $v_{1}$ span $v^{\perp}$ in $T_{x}({{\mathcal{S}}^{m}})$.
To see the claim of the lemma, we first note that it is obvious for ${m}=2$. For higher dimensions, it follows from the fact that any $2$-dimensional big sphere is given by $\mathcal{S}^2 =
{{\mathcal{S}}^{m}}\cap\Pi$, where $\Pi$ is a $3$-dimensional linear subspace of ${\mathbb{R}}^{m+1}$, i.e. one direction vector orthogonal to the plane containing $\bar{xz}$.
Recall that the set $Sing\subseteq {\mathcal{E}_n}$ is the set of singular functions (see definition \[def:sing func\]).
We define the map $$\Psi:{\mathcal{E}_n}\times {{\mathcal{S}}^{m}}\rightarrow {\mathbb{R}}\times{\mathbb{R}}^{{m}}$$ by $$(f,x)\mapsto (f(x), \nabla f(x)),$$ using the isometry $T_{x}({{\mathcal{S}}^{m}}) \cong {\mathbb{R}}^{{m}}$ again, so that $$Sing = \pi_{{\mathcal{E}_n}} (\Psi^{-1}(0,\vec{0})).$$ We claim that $\Psi$ is submersion. Having this claim in our hands would imply $$\Psi^{-1} (0,\vec{0}) \le {\mathcal{N}}-1$$ by the submersion theorem. Therefore $$\dim(Sing) \le {\mathcal{N}}-1$$ as well.
To see that $\Psi$ is indeed a submersion, we compute its differential to be $$d\Psi = \left( \begin{matrix} \eta_{1}(x) &\eta_{2} (x) &\ldots
&\eta_{{\mathcal{N}}} (x) &* \\ \nabla\eta_1(x) &\nabla\eta_2(x)
&\ldots &\nabla\eta_{{\mathcal{N}}} (x) &
* \end{matrix}\right),$$ where $\{\eta_{k}\}$ is the orthonormal basis of ${\mathcal{E}_n}$, which appears in the definition of $f$. Denote the matrix $A_{({m}+ 1)\times {\mathcal{N}}}$ with the first ${\mathcal{N}}$ columns of $d\Psi$. We claim that $A$ is of full rank, i.e. $rk(A)={m}+1$. To see that we compute the Gram matrix of its rows to be $$A\cdot A^{t} = \left(\begin{matrix} 1 &0 \\
0 &\frac{{E_{n}}}{{m}} I_{{m}} \end{matrix}\right),$$ see section \[sec:corr mat, exp\]. Since it is clearly invertible, we conclude that $rk(A)={m}+1$.
Recall that we defined ${\mathcal P}^{x}_{b}$ and ${\mathcal P}^{x,y}_{b}$ in section \[sec:sing func\] (see and ).
Define $B^{x}_{b}\subseteq Sing\cap{\mathcal P}^{x}_{b} $ to be the set of function having $\pm x$ as their singular point, that is $$B^{x}_{b} = \big\{ f\in{\mathcal P}_{b}^{x}:\: f(x)=0 ,\, \nabla f(x) = 0
\big\} \cup \big\{ f\in{\mathcal{E}_n}:\: f(-x)=0 ,\, \nabla f(-x) = 0
\big\}.$$ It is obvious that $B^{x}_{b}$ is nonempty only if $b=0$. Since $f(-y)=\pm f(y)$ for every $y\in{{\mathcal{S}}^{m}}$, $f\in B^{x}_{b}$ implies that $f$ is singular at $x$. The set $B^{x}_{b}$ is of codimension ${m}$, since the covariance matrix is invertible, so that the Gaussian distribution of $\nabla f(x)$ conditioned upon $f(x)=0$, is nonsingular.
Next, we define $$\bar{B}^{x}_{b}:=
Sing\cap{\mathcal P}^{x}_{b}\setminus B^{x}_{b}.$$ To prove the statement of the lemma, we need to prove that $\bar{B}^{x}_{b}$ is of codimension $1$ in $Sing\cap{\mathcal P}_{b}^{x}$. To do so, we follow closely the proof of lemma \[lem:Sing codim 1\]. This time we define $$\Psi_{x}:{\mathcal{E}_n}\times {{\mathcal{S}}^{m}}\setminus \{\pm x \} \rightarrow
{\mathbb{R}}^2\times{\mathbb{R}}^{{m}}$$ by $$(f,y)\mapsto (f(x),f(y), \nabla f(y)),$$ satisfying $$\bar{B}^{x}_{b} = \pi_{{\mathcal{E}_n}} (\Psi^{-1}(b,0,\vec{0})).$$
Using a similar dimensional approach, it is sufficient to prove that $\Psi_{x}$ is a submersion. The differential of $\Psi_{x}$ is $$d\Psi_{x} = \left( \begin{matrix} \eta_{1}(x) &\eta_{2} (x) &\ldots
&\eta_{{\mathcal{N}}} (x) &* \\ \eta_{1}(y) &\eta_{2} (y) &\ldots
&\eta_{{\mathcal{N}}} (y) &*\\ \nabla\eta_1(y) &\nabla\eta_2(y)
&\ldots &\nabla\eta_{{\mathcal{N}}} (y) &
* \end{matrix}\right).$$ Assume by contradiction that the vectors $(\eta_{k}(x),\eta_{k}
(y),\, \nabla\eta_{k}(y))$ satisfy a nontrivial linear functional. Since $\eta_{k}$ span the whole space ${\mathcal{E}_n}$, that functional is satisfied by $$\big({Q_{n}^{{m}}}(\cos{d(x,\xi)}),{Q_{n}^{{m}}}(\cos{d(y,\xi)}),\nabla
{Q_{n}^{{m}}}(\cos{d(y,\xi)})\big),$$ for every $\xi\in{{\mathcal{S}}^{m}}$. The surjectivity of $d\Psi_{x}$ then follows from lemma \[lem:dPsix surj eq\].
Finally, we note that $$Sing\cap{\mathcal P}^{x}_{b} = B^{x}_{b}\cup \bar{B}^{x}_{b},$$ which concludes the proof of this lemma.
\[lem:dPsix surj eq\] For every $x\in{{\mathcal{S}}^{m}}$, $y\ne \pm x$, the only solutions in $\alpha,\beta\in{\mathbb{R}}$, $C\in{\mathbb{R}}^{{m}}$ for $$\label{eq:bas rel singpxa} \alpha Q_{n}^{m}(\cos{d(x,\xi)})+\beta
Q_{n}^{m}(\cos{d(y,\xi)})- Q_{n}^{m}{'}(\cos{d(y,\xi)})
\sin{d(y,\xi)} \langle C,\, \nabla_{y} d(y,\xi)\rangle = 0$$ are $\alpha=\beta=0$, $C=\vec{0}$.
It is obvious that either $\alpha=0$ or $\beta=0$, imply that $\alpha=\beta=0$, $C=\vec{0}$. Therefore we may assume that $\alpha=-1$, $\beta\ne 0$.
Substituting $\xi$ in and, in addition, any $\xi'\ne
\xi$ not lying on $\bar{xy}$ with $d(\xi',x)=d(\xi,x)$ and $d(\xi',y)=d(\xi,y)$, we obtain that $C$ is collinear to any $v_1=v_{1}(\xi,\xi')\in T_{y}({{\mathcal{S}}^{m}})$ of the form $$v_{1} =
\nabla_{y}d(y,\xi)-\nabla_{y}d(y,\xi').$$ Lemma \[lem:d(x,xi)=d(x,xi’),d(y,xi)=d(y,xi’)\] implies that $C$ is collinear to $v:=\nabla _{y} d(x,y)$.
We restrict ourselves to any two-dimensional big sphere $\mathcal{S}^2 \subseteq {{\mathcal{S}}^{m}}$ containing $x$ and $y$. Knowing that $C\parallel v$, for $\xi\in\mathcal{S}^2$ on the big circle perpendicular to $\nabla_{y} d(x,y)$, is $$Q_{n}^{m} (bt) = \frac{1}{\beta} Q_{n}^{m} (t),$$ by the spherical cosine theorem, where we denote $t:=\cos{d(y,\xi)}$ and $b:=\cos{d(x,y)}$. It is clear that it is only possible if $b=1$, that is $x=\pm y$, which is a contradiction.
To prove the statement of the lemma, we partition the set $Sing\cap{\mathcal P}^{x,y}_{b}$ into $$Sing\cap{\mathcal P}^{x,y}_{b} = B^{x}_{b} \cup B^{x,y}_{b}\cup
\bar{B}^{x,y}_{b},$$ defining appropriately each of the sets above and proving the statements regarding each of them separately.
First, similarly to the proof of lemma \[lem:Sing codim 1 Pax\], we treat the set $$B^{x}_{b}\subseteq Sing\cap{\mathcal P}^{x,y}_{b}$$ of function having $\pm x$ as their singular point, that is $$B^{x}_{b} = \big\{ f\in{\mathcal P}_{b}^{x,y}:\: f(x)=0 ,\, \nabla f(x) = 0
\big\} \cup \big\{ f\in{\mathcal{E}_n}:\: f(-x)=0 ,\, \nabla f(-x) = 0
\big\}.$$ It is easy to see (exactly as in case of lemma \[lem:Sing codim 1 Pax\]) that $B^{x}_{b}$ has codimension $\ge 1$ in ${\mathcal P}^{x,y}_{b}$.
Next, we treat the case when the function $f$ has a singular point on $D\subseteq {{\mathcal{S}}^{m}}$, a distinguished codimension $1$ set of points on the sphere we are about to define. Let $A_1 := \bar{xy}$ be the big circle linking $x$ to $y$ and $A_2\subseteq {{\mathcal{S}}^{m}}$ be the set of all the points $z$ such that the spherical angle $\angle xzy =
\frac{\pi}{2}$ is right angle. Define $$D=D_{x,y}=A_1\cup A_2
\setminus \{ \pm x \}.$$ It is clear that $D$ is a codimension one set on the sphere satisfying $\pm x\notin
D$, $\pm y \in D$.
Define $B^{x,y}_{b}\subseteq Sing\cap{\mathcal P}^{x,y}_{b} $ to be the set of singular functions having a $D$-point as their singular point, that is $$B=B^{x,y}_{b} = \big\{ f\in{\mathcal P}_{b}^{x,y}:\: \exists z\in D:f(z)=0
,\, \nabla f(z) = 0 \big\}.$$
We claim that $B$ has codimension at least $1$ in ${\mathcal P}_{b}^{x,y}$. To see that we define the map $$\tilde{\Psi}_{b_1}^{x,y}:{\mathcal{E}_n}\times
D\rightarrow{\mathbb{R}}^{2}\times{\mathbb{R}}^{{m}}$$ by $$(f,z)\mapsto (f(x),f(z),\nabla f (z)).$$ It is clear that $$B \subseteq \pi_{{\mathcal{E}_n}}
(({\tilde{\Psi}^{x,y}_{b_1}}) ^{-1} (b_{1},0,\vec{0})).$$ Moreover, $\tilde{\Psi}^{x,y}_{b_1}$ is a submersion (see the proof of lemma \[lem:Sing codim 1 Pax\]).
Therefore, $({\tilde{\Psi}^{x,y}_{b_1} }){^{-1}} (b_{1},0,\vec{0})$ is of codimension ${m}+2$ in ${\mathcal{E}_n}\times D$, i.e. of dimension ${\mathcal{N}}-3$, so that $B$ is of codimension $\ge 1$ in ${\mathcal P}_{a}^{x,y}$.
Finally, we treat the “generic" case. We define the set $$\bar{B}=\bar{B}^{x,y}_{b} := Sing\cap{\mathcal P}^{x,y}_{b}\setminus
(B^{x}_{b} \cup B^{x,y}_{b})$$ of functions in $Sing\cap{\mathcal P}^{x,y}_{b}$ having the set of their singular points outside of $\{\pm x\}\cup D$ (i.e. having at least one singular point there).
We define $${\Psi}_{b}^{x,y}:{\mathcal{E}_n}\times {{\mathcal{S}}^{m}}\setminus (D\cup \{\pm
x\})\rightarrow{\mathbb{R}}^{3}\times{\mathbb{R}}^{{m}}$$ by $$(f,z)\mapsto (f(x),f(y),f(z),\,\nabla f (z)).$$ It is obvious that $$\bar{B} = \pi_{{\mathcal{E}_n}} (({\Psi}^{x,y}_{b_1}) ^{-1}
(b_{1},b_{2},0,\vec{0})).$$
As before, to prove that $\bar{B}$ is of codimension $1$, it is sufficient to prove that ${\Psi}_{b}^{x,y}$ is a submersion. To see that ${\Psi}_{b}^{x,y}$ is a submersion, we compute its differential to be $$d\Psi = \left( \begin{matrix} \eta_{1}(x) &\eta_{2} (x) &\ldots
&\eta_{{\mathcal{N}}} (x) &* \\ \eta_{1}(y) &\eta_{2} (y) &\ldots
&\eta_{{\mathcal{N}}} (y) &* \\ \eta_{1}(z) &\eta_{2} (z) &\ldots
&\eta_{{\mathcal{N}}} (z) &* \\ \nabla\eta_1(z) &\nabla\eta_2(z)
&\ldots &\nabla\eta_{{\mathcal{N}}} (z) &
* \end{matrix}\right).$$ Its surjectivity follows from lemma \[lem:dPsixy surj eq\].
This concludes the proof of this lemma.
\[lem:dPsixy surj eq\] Let $x$, $y$ and $z$ be points on the sphere ${{\mathcal{S}}^{m}}$. Suppose that $x\ne y$, $z\notin \bar{xy}$, and $\angle xzy \ne \frac{\pi}{2}$. Then the only solution to $$\label{eq:bas lin dep 3 pnt} \alpha f(x)+\beta f(y)+\gamma
f(z)+\langle C,\, \nabla_{z} f(z) \rangle = 0$$ for every $f\in{\mathcal{E}_n}$ is $\alpha=\beta=\gamma = 0 $, $C=\vec{0}$.
Substituting $$f(z) = {Q_{n}^{{m}}}(\cos{d(z,\, \xi)})$$ with $\xi \in{{\mathcal{S}}^{m}}$, is $$\label{eq:bas lin dep 3 pnt subs Pn}
\begin{split}
&\alpha {Q_{n}^{{m}}}(\cos{d(x,\, \xi)})+\beta {Q_{n}^{{m}}}(\cos{d(y,\,
\xi)})+{Q_{n}^{{m}}}(\cos{d(z,\, \xi)})\\&-{Q_{n}^{{m}}}{'} (\cos{d(z,\,
\xi)})\sin{d(z,\, \xi)}\langle C,\, \nabla_{z} d(z,\xi) \rangle = 0.
\end{split}$$ for $\xi \ne \pm z$.
Comparing the equality for $\xi$ not lying on $\bar{yz}$ and any $\xi'$ with $d(x,\xi)=d(x,\eta')$ and $d(z,\xi)=d(z,\xi')$, we obtain $$\label{eq:bas relat subs harm symm pair}
\begin{split}
&{Q_{n}^{{m}}}{'}(\cos{d(z,\xi)})\sin(d(z,\xi))\langle C,\nabla_{z}
d(z,\xi)- \nabla_{z} d(z,\xi')\rangle \\&= \beta \bigg[{Q_{n}^{{m}}}(\cos{d(y,\xi')}) - {Q_{n}^{{m}}}(\cos{d(y,\xi)}) \bigg].
\end{split}$$
Let $$\xi''\ne\xi'''\in{{\mathcal{S}}^{m}}$$ with $\xi''\ne \xi$ be the unique pair of points with $$d(z,\xi'')=d(z,\xi''')=d(z,\xi)$$ and $$\nabla_{z}d(z,\xi'') - \nabla_{z}d(z,\xi''')=\nabla_{z}d(z,\xi) -
\nabla_{z}d(z,\xi').$$ In particular, we have $$d(x,\xi'')=d(x,\xi''').$$
Substituting $\xi''$ and $\xi'''$ into , as we may do, yields $$\label{eq:bas relat subs harm dual symm pair}
\begin{split}
&{Q_{n}^{{m}}}{'}(\cos{d(z,\xi)})\sin(d(z,\xi))\langle C,\nabla_{z}
d(z,\xi'')- \nabla_{z} d(z,\xi''')\rangle \\&= \beta
\bigg[{Q_{n}^{{m}}}(\cos{d(y,\xi''')}) - {Q_{n}^{{m}}}(\cos{d(y,\xi'')})
\bigg],
\end{split}$$ and comparing to we see that either $\beta = 0$ or $$\label{eq:4 symm pnts sat} {Q_{n}^{{m}}}(\cos{d(y,\xi)}) - {Q_{n}^{{m}}}(\cos{d(y,\xi')}) - {Q_{n}^{{m}}}(\cos{d(y,\xi'')}) + {Q_{n}^{{m}}}(\cos{d(y,\xi''')})=0.$$
Suppose by contradiction that the latter holds. We restrict ourselves to any big two-dimensional sphere $\mathcal{S}^2(x,y,z)\subseteq{{\mathcal{S}}^{m}}$ (recall notation \[not:bigcirc,arc,S2\]). Let $d>0$ be a small number and $\phi\ne\phi'\in \mathcal{S}^2$ be the (unique) points which satisfy $$d(z,\phi) = d(z,\phi') = d$$ and $\bar{\phi\phi'}\perp \bar{xz}$. We may approach to $\phi$ by $\mathcal{S}^2$-points $\xi$ and $\xi''$ and to $\phi'$ by $\xi'$ and $\xi'''$ of the form as above with the additional requirement $$d(\xi,\bar{xz})=d(\xi',\bar{xz})=d(\xi'',\bar{xz})=d(\xi''',\bar{xz})=d.$$
Dividing by $d(\xi,\xi'') =
d(\xi',\xi''')$, and taking the limit as $\xi\rightarrow\phi$, we obtain $$\label{eq:4 symm pnts sat lim}
{Q_{n}^{{m}}}(\cos{d(y,\phi)}) \sin{d(y,\phi)}
\frac{\partial}{\partial e^{\phi}} d(y,\phi) = {Q_{n}^{{m}}}(\cos{d(y,\phi')}) \sin{d(y,\phi')} \frac{\partial}{\partial
e^{\phi'}} d(y,\phi '),$$ where $e^{\phi}$ and $e^{\phi'}$ are the unit tangent vectors in the directions $\breve{\phi\xi''}$ and $\breve{\phi'\xi'''}$ respectively.
Denote $d_0:=d(y,z) $, $d_1 := d(y,\phi) $ and $d_2:=d(y,\phi')$. Let $\delta$ be the angle $\delta = \angle xzy$. We have $\delta\ne
0,\frac{\pi}{2}$ by the assumptions of the lemma. We compute $$c_1=c_1(d):=\cos{d_1} = \cos d_0\cos d +
\cos{\delta}\sin{d_0}\sin{d}$$ and $$c_2=c_2(d):=\cos{d_2} = \cos d_0\cos d
-\cos{\delta}\sin{d_0}\sin{d},$$ by the spherical cosine theorem. It is obvious that the LHS of is an analytic function of $c_1$, and the RHS is the same function evaluated at $c_2=g(c_{1})$ for some analytic function $g$. The function $g$ is defined on an neighbourhood of $\cos{d_{0}}$ satisfying $g(\cos{d_{0}})=\cos{d_{0}}$. Therefore, lemma \[lem:P(g(x))=P(x)=>g’=+-1\] implies that $$g'(\cos{d_0}) = \pm 1.$$
On the other hand, computing the derivative explicitly, we have $$g'(\cos{d_0}) =
\frac{-\cos{d_0}\sin{d_0}-\cos{\delta}\cos{d_0}\sin{d_0}}{-\cos{d_0}\sin{d_0}+\cos{\delta}\cos{d_0}\sin{d_0}}
= \frac{1+\cos{\delta}}{1-\cos{\delta}},$$ which, clearly, under the assumptions of the lemma, cannot be equal to $\pm 1$, and therefore we obtain the necessary contradiction. This proves that $\beta=0$. By the symmetry, we have $\alpha=0$ as well.
Thus implies that $$C\perp v_{1} (\xi):=\nabla_{z} d(z,\xi)-\nabla_{z} d(z,\xi'),$$ for every $\xi,\xi'$ of the form above. However, for every $\xi$, the vectors $$v_1(\xi,\xi') \in T_{z}({{\mathcal{S}}^{m}})$$ are all orthogonal to $v:=\nabla_{z} d(x,z)$, the vector in the direction of $\bar{xz}$, and moreover, they span the orthogonal complement $v^{\perp}$, by lemma \[lem:d(x,xi)=d(x,xi’),d(y,xi)=d(y,xi’)\].
Therefore $C$ must be collinear to $v$. Similarly we may argue that $C$ is collinear to $v':=\nabla_{z} d(z,y)$. However, $v$ and $v'$ are not collinear by the assumptions of the present lemma, so that $C=0$. Knowing that, $\gamma=0$ is easy to obtain.
\[lem:P(g(x))=P(x)=>g’=+-1\]
Let $f(t)$ an analytic, not identically vanishing function, and $g(t)$ a differentiable function defined on an neighbourhood $I$ of $t_0\in I$ such that $g(t_0)=t_0$. Suppose that we have on $I$ $$f(g(t))=f(t).$$ Then $g'(t_0)=\pm 1$.
We have by the chain rule, $$f'(g(t))g'(t) = f'(t).$$ Therefore, if $f'(t_0)\ne 0$ then $g'(t_0)=1$ and we are done. Otherwise, we continue differentiating to obtain $$f''(g(t))g'^2(t) +f'(g(t))g''(t)= f''(t)$$ so that if $f''(t_0)\ne 0$, we have $g'^2(t_0)=1$ and we are done again. Otherwise we continue differentiating until we encounter the first derivative $f^{(k)}(t_0) \ne 0$ implying $g'^{k}(t) = 1$. Such a number $k$ exists, since $f$ is [*analytic*]{}.
Non degeneracy of point value and gradient distribution {#sec:f (x)(y)gr f(x)(y) sp}
=======================================================
In this section we prove that for $\pm N \ne x \in{{\mathcal{S}}^{m}}$, the distribution of the random vector $Z$ defined in section \[sec:corr mat, var\], is [*nonsingular*]{} Gaussian.
Let $x \ne \pm N\in{{\mathcal{S}}^{m}}$ and $V=V_{x}$ be vector space $$V = {\mathbb{R}}^2\times T_{x} ({{\mathcal{S}}^{m}})\times T_{N}({{\mathcal{S}}^{m}}).$$ Define the subspace $$U = U_{x,\, n} \subseteq V$$ by $$U = \{\big(f(x),\,f(N),\, \nabla f(x),\, \nabla f(y) \big)
:\:f\in{\mathcal{E}_n}\}.$$ Then one has $$U = V,$$ provided that $n$ large enough. That is, the distribution of the random vector $$V=\big(f(x),\,f(N),\, \nabla f(x),\, \nabla f(N) \big)$$ is Gaussian nondegenerate and one may identify $$U \cong {\mathbb{R}}^{2{m}+2},$$ as in section \[sec:orthonorm bas corr mat exp\].
Let $x\ne \pm N$. We assume by contradiction, that $U$ is a proper subspace of $V$, i.e. there is a nontrivial functional $h:V\rightarrow{\mathbb{R}}$ vanishing on $U$.
We wish to work with coordinates and employ the orthonormal bases for $T_{x}({{\mathcal{S}}^{m}})$ and $T_{N} ({{\mathcal{S}}^{m}})$ chosen in section \[sec:orthonorm bas corr mat exp\], so that under the corresponding identification, one has .
By our assumption, there exist numbers $\alpha,\, \beta\in{\mathbb{R}}$ and vectors $C,\, D\in{\mathbb{R}}^{2}$ so that $$\label{eq:bas relat} \alpha f(x)+\beta f(N) + \langle C,\,
\nabla f(x) \rangle + \langle D,\, \nabla f(N) \rangle = 0.$$ We know that for every $\eta\in{{\mathcal{S}}^{m}}$, the function $$f(x):= {Q_{n}^{{m}}}(\cos{d(x,\eta)}),$$ is a spherical harmonic lying in ${\mathcal{E}_n}$. For this particular function is for $\eta\ne \pm x,\pm N$, $$\label{eq:bas relat subs harm}
\begin{split} &\alpha {Q_{n}^{{m}}}(\cos{d(x,\,
\eta)})+\beta {Q_{n}^{{m}}}(\cos{d(N,\, \eta)}) \\&- {Q_{n}^{{m}}}{'}(\cos{d(x,\,\eta)})\sin{(d(x,\,\eta))}\cdot\langle C,\,
\nabla_{x} d(x,\eta) \rangle \\&- {Q_{n}^{{m}}}{'}(\cos{d(N,\,\eta)})\sin{(d(N,\,\eta))}\cdot\langle D,\,
\nabla_{N} d(N,\eta) \rangle = 0.
\end{split}$$
First choose $\eta\in{{\mathcal{S}}^{m}}$ not lying on $\bar{xN}$ and compare for $\eta$ and any $\eta'\ne\eta$ satisfying $d(x,\,\eta') = d(x,\,\eta)$ and $d(y,\,\eta') =
d(y,\,\eta)$. We obtain $$\begin{split}
&{Q_{n}^{{m}}}{'} (\cos{d(x,\,\eta)})\sin{(d(x,\,\eta))}\cdot\langle
C,\, \nabla_{x} d(x,\eta) \rangle \\&+ {Q_{n}^{{m}}}{'}
(\cos{d(N,\,\eta)})\sin{(d(N,\,\eta))}\cdot\langle D,\, \nabla_{N}
d(N,\eta) \rangle \\&= {Q_{n}^{{m}}}{'}
(\cos{d(x,\,\eta)})\sin{(d(x,\,\eta))}\cdot\langle C,\, \nabla_{x}
d(x,\eta') \rangle \\&+ {Q_{n}^{{m}}}{'}
(\cos{d(N,\,\eta)})\sin{(d(N,\,\eta))}\cdot\langle D,\, \nabla_{N}
d(N,\eta') \rangle.
\end{split}$$
Equivalently, $$\label{eq:relat subs sides chng}
\begin{split}
&{Q_{n}^{{m}}}{'} (\cos{d(x,\,\eta)})\sin{(d(x,\,\eta))}\cdot\langle
C,\, \nabla_{x} d(x,\eta) -\nabla_{x} d(x,\eta')\rangle \\&=
-{Q_{n}^{{m}}}{'}(\cos{d(N,\,\eta)})\sin{(d(N,\,\eta))}\cdot\langle
D,\, \nabla_{N} d(N,\eta) -\nabla_{N} d(N,\eta')\rangle.
\end{split}$$
For every $\eta$, the vectors $$v_1(\eta) = \nabla_{x} d(x,\eta) -
d(x,\eta') \in T_{x}({{\mathcal{S}}^{m}})$$ are all orthogonal to $v:=\nabla_{x}
d(x,N)$, the vector in the direction of $\bar{xN}$, and moreover, they span the orthogonal complement $v^{\perp}$ by lemma \[lem:d(x,xi)=d(x,xi’),d(y,xi)=d(y,xi’)\]. We claim that the equality implies that $$C \perp
sp\{v_1 (\eta)\}$$ and thus $C$ and $v$ are collinear. Similarly, $D$ and $v':=\nabla_{N} d(x,N)$ are collinear, and since we identify $v$ with $-v'$, that implies $C=\lambda D$ are collinear.
Suppose otherwise. Let $v_{0} = v_{1} (\eta_{0})$ such that $\langle
C,\, v_{0} \rangle \ne 0$, and consider the two-dimensional sphere $\mathcal{S}^2\subseteq{{\mathcal{S}}^{m}}$ defined by $\bar{xN}$ and $v_0$. For $\eta\in \mathcal{S}^2$, one has $$v_{1}(\eta)\parallel v_{0}.$$ We fix $d=d(N,\eta)$ so that $\cos{d}$ is a zero of ${Q_{n}^{{m}}}{'}$. Then the RHS of vanishes and our assumptions imply that $\cos{d(N,\eta)}$ is a zero of ${Q_{n}^{{m}}}{'}$.
However the function $\cos{d(x,\,\eta)}$ is a continuous nonconstant function of $\eta$ on the arc $$A:=\{\eta :\: d(N,\eta)=d \}\subseteq\mathcal{S}^2,$$ and therefore its image contains an interval, contradicting the finiteness of number of zeros of ${Q_{n}^{{m}}}{'}$. Therefore $$C \parallel v,$$ which proves our claim, i.e. $C=\lambda D$ for some $\lambda\in {\mathbb{R}}$.
Substituting the last equality into with $\eta\in{{\mathcal{S}}^{m}}$ such that $$d(x,\eta)=d(N,\eta),$$ implies $\lambda = -1$, i.e. $$\label{eq:C=-D} C=-D .$$
Now substitute $\eta\rightarrow x$ in to obtain $$\label{eq:bas relat subs harm eta=x} \alpha +\beta {Q_{n}^{{m}}}(\cos{d}) + {Q_{n}^{{m}}}{'}(\cos{d})\sin{(d)}\cdot\langle C,\,
\nabla_{N} d(N,x) \rangle = 0,$$ where $d=d(x,N)$. We obtain similarly $$\label{eq:bas relat subs harm eta=y} \alpha {Q_{n}^{{m}}}(\cos{d})
+\beta - {Q_{n}^{{m}}}{'} (\cos{d})\sin{(d)}\cdot\langle C,\,
\nabla_{x} d(x,N) \rangle = 0,$$ upon substitution $\eta\rightarrow N$. Since in our identification, we have $\nabla_{x} d(x,N) = - \nabla_{N} d(x,N)$, together with imply $$\label{eq:alpha=beta} \alpha=\beta,$$ since $$({Q_{n}^{{m}}}(\cos{d}) \ne 1) \Leftarrow (d\ne 0,\pi)
\Leftrightarrow (x\ne\pm N).$$
We claim that $\alpha=0$ and $C=0$. Assume otherwise. Consider any two-dimensional sphere $\mathcal{S}^2$ containing $\bar{xN}$, and the big circle $E\subseteq \mathcal{S}^2$ defined by $$E = \{\eta\in\mathcal{S}^2:\: d(x,\,\eta) = d(N,\,\eta) \}.$$ On $E$, is, substituting and $$\label{eq:bas relat subs harm on E} 2\alpha {Q_{n}^{{m}}}(\cos{d(x,\eta)}) + {Q_{n}^{{m}}}{'} (\cos{d(x,\eta)}) \sin{d(x,\eta)}
\langle C,\, \nabla_{N} d(N,\, \eta) -\nabla_{x} d(x,\, \eta)
\rangle = 0.$$ It is clear that the vector $$v=\nabla_{N} d(N,\, \eta) -\nabla_{x}
d(x,\, \eta)$$ is collinear to $\nabla_x d(x,N)$, which, as we have seen, collinear to $C$. In particular, $\alpha =0$ if and only if $C=0$ and thus we may assume by contradiction $\alpha\ne 0$ and $C\ne 0$.
Since $x\ne \pm N$, the point $\eta\in E$ lying on $\breve{xN}$, satisfies $$d(x,\eta) < \frac{\pi}{2}$$ and the point $\eta' = -\eta
\in E$ satisfies $$d(x,\eta) > \frac{\pi}{2}.$$ Therefore there exists a point $\eta_0 \in E$ with $$d(x,\eta_{0}) = \frac{\pi}{2}.$$ Then either ${Q_{n}^{{m}}}(\cos{d(x,\eta_{0})})=0$ or ${Q_{n}^{{m}}}{'}
(\cos{d(x,\eta_{0})})=0$, depending on whether $n$ is even or odd. However, the equality implies then $${Q_{n}^{{m}}}(\cos{d(x,\eta_{0})})={Q_{n}^{{m}}}{'}
(\cos{d(x,\eta_{0})})=0.$$ This contradicts the fact that ${Q_{n}^{{m}}}$ does not have any double zeros, since then the differential equation satisfied by ${Q_{n}^{{m}}}$ would imply ${Q_{n}^{{m}}}\equiv
0$.
[99]{}
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[^1]: The author is supported by a CRM ISM fellowship
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---
abstract: 'We present the results of an experimental study of vortex dynamics in non-twinned $YBa_2Cu_3O_{6,87}$ crystal. It is found that critical currents $J_c$ and $J_{c,dyn}$, which correspond to the pinning force in the thermal creep and flux flow mode, respectively, non-monotonically vary with the magnetic field. However, the minimum in the $J_{c,dyn}(H)$ dependence is observed in higher fields, compared with the minimum position $H_{OD}$ in the $J_c(H)$ dependence. Considering that the field $H_{OD}$ corresponds to the static order-disorder transition, this difference is explained by partial dynamic ordering of the vortex solid. It is concluded that finite transverse barriers guarantee finite density of transverse displacements of vortex lines $u_t\simeq c_La_0$ suitable for preservation of the disordered state of the moving vortex solid.'
author:
- 'A. V. Bondarenko'
- 'A. A. Zavgorodniy'
- 'D. A. Lotnik'
- 'M. A. Obolenskii'
- 'R. V. Vovk'
- 'Y. Biletskiy'
bibliography:
- '/bondarenko/tex.sample/paper.bib'
title: 'Quasi-static and dynamic order-disorder transition in presence of strong pinning'
---
The interaction of static and dynamic elastic media with chaotic pinning potential is one the chapters of solid state physics, which includes dislocations in solids, charge density waves, Vigner crystals, and vortex lattices (VL’s) in Type-II superconductors. The VL’s are the most appropriate objects for the experimental study of elastic media, because it is easy to change the strength of pinning potential in superconductors, as well as the elasticity and motion velocity of VL’s. An important feature of the VL’s is the non-monotonous field variation of the pinning force $F_p$, which is observed in low-T$_c$ (NbSe$_2$ [@Bhattacharya93; @Higgins96], V$_3$Si [@Gapud03]) middle-$T_c$ (MgB$_2$ [@Pissas02; @Kim04]), and high-T$_c$ (BiSrCaCuO [@Khaikovich96], YBaCuO [@Kupfer98; @Pissas00]) superconductors. The increase of the pinning force can be explained by softening of the elastic moduli of VL’s in vicinity of the upper critical field $H_{c2}(T)$ [@Higgins96] or the melting line $H_m(T)$ [@Kwok94] that causes better adaptation of the vortex lines to the pinning landscape. Some alternative models [@Ertas97; @Rosenstein07] suggest formation of an ordered vortex solid (VS) in low fields, which transforms into a disordered one in some magnetic field $H_{OD}$, though the nature of the order-disorder (OD) transition and the mechanism of increasing the force $F_p$ may be different. These models are supported by correlation between the field $H_{OD}$ corresponded to the structural OD transition [@Cubbit93] and the onset of the $F_p$ increase [@Khaikovich96] in BiCaSrCuO crystals. An actual problem of the VS phase is the nature of its ordering under an increased vortex velocity $v$. The “shaking temperature” model [@Koshelev94] suggests that transverse vortex displacements $u_t$ induced by the disorder reduce with increased velocity, $u_t\propto 1/v$; and the increase of the velocity above some critical value $v_c$ results in a dynamic transition from the disordered to ordered state. It was later justified [@Giamarchi96] that the increase in $v$ leads to a suppression of the pinning in the longitudinal (with respect to $\textbf{v}$) direction only, while pinning barriers remain finite in the transverse direction. The effect of motion on the transverse barriers, phase state and pinning force of vortex solid is still controversial issue, and this subject first of all requires additional reliable experimental studies. The goal of this work is experimental study of vortex dynamics in the presence of strong pinning.
The measurements were performed on detwinned [YBa$_2$Cu$_3$O$_{7-\delta}$ ]{}crystal, annealed in an oxygen atmosphere at 500$^{\circ}$C for one week. Such anneal corresponded to an oxygen deficiency $\delta\simeq$ 0.13 [@Otterlo00] and $T_c\simeq$ 91.8 K. The crystal then was held at room temperature for 7 days to form clusters of oxygen vacancies, which reduced the tension of the field $H_{OD}$ [@Liang98]. The field variation of the pinning force was studied through measurement of the current-voltage characteristics, $E(J)$, using the standard four-probe method with dc current. The investigated sample had rectangular shape with smooth surfaces; its dimensions were 3.5$\times$0.4$\times$0.02 mm with the smallest dimension along the $c$ axis; the current was applied along the largest dimension; and the distance between the current and potential contacts, and between the potential contacts was about 0.5 mm. The measurements were performed at a temperature of 86.7 K in the field $\textbf{H}\parallel \textbf{c}$.
![\[fig:1\] $E(J)$ curves presented in the linear (a) and semi-logarithmical scale (b), and $\rho_d(J)$ curves presented in the semi-logarithmical scale (c). The inset in panel (b) shows the $E(J)$ dependencies measured upon increase (light symbols) and decrease (dark symbols) of the current.](Fig1){width="3.2in"}
Fig. \[fig:1\] shows the $v(J)=cE(J)/B$ dependencies and current variation of the normalized dynamic resistance $\rho_d(J)\equiv[dE(j)/dJ]/\rho_{BS}$, where $\rho_{BS}=\rho_N
B/B_{c2}$[@Kupfer98]. At low currents, the electric field increases exponentially with an increase in current and the resistance $\rho_d$ is much lower than one. This increase in $v$ and low dynamic resistance indicate the presence of thermally activated vortex creep. At high currents, the $v(J)$ dependence is linear and the value of $\rho_d$ is close to 1, indicating the presence of the flux flow mode. The critical current in the thermal creep mode $J_E$ can be characterized by the voltage criteria of $E = 1~\mu$V/cm and $E =100~\mu$V/cm, and the dynamic critical current $J_{c,dyn}$ can be determined by extrapolating the linear parts of the $v(J)$ dependence, corresponded to the flux flow mode, to zero voltage [@Kokubo07]. Field variation of the currents $J_E$ and $J_{c,dyn}$ normalized by their values in a field of 0.5 kOe are shown in Fig. \[fig:2\]a. It is seen that the currents $J_c$ and $J_{c,dyn}$ start to increase in the fields above 1.25 kOe and 2.5 kOe, respectively, which are substantially smaller in comparison with the fields $H_{c2}$ and $H_m$. Therefore this increase can not be caused by better adaptation of the vortices to the pinning landscape induced by softening of the elastic moduli. Obtained field variation of the currents $J_E$ and $J_{c,dyn}$, and the peculiarities of vortex dynamics can be explained in frames of the model proposed by Ertas and Nelson [@Ertas97]. It is assumed that the OD transition occurs when transverse displacements of vortex lines exceed the value of $c_La_0$, where $a_0\simeq\sqrt{\Phi_0/B}$ is intervortex distance, $\Phi_0$ is the flux quantum, and $c_L$ is the Lindemann number. The field is defined by equality of energies $E_{el}(H_{OD}) = E_p(H_{OD})$, where $E_p$ is the pinning energy, $E_{el}\simeq c_L^2\varepsilon\varepsilon_0a_0$ is increase of the elastic energy caused by displacements $u_{t}=c_{L}a_0$, $\varepsilon$ is the anisotropy parameter, $\varepsilon_0 =
(\Phi_0/4\pi\lambda)^2$ is the line tension of vortex line and $\lambda$ is the penetration depth. As evident from Fig. \[fig:2\]a, the minimum position does not depend on the driving force within the creep regime in agreement with magnetization measurements [@Kupfer98; @Pissas00]. This means that the value of ratio $E_{el}/E_p$, and, therefore the energy $E_p$, is not changed, indicating that minimum in the $J_E(H)$ curve corresponds to static OD transition, $H_{OD}\simeq$ 1.25 kOe.
![\[fig:2\] (a) Field variation of the current $J_{c,dyn}$ and $J_E$ normalized by their values in a field of 0.5 kOe. (b) Field variation of the velocities $v_p$ and $v_{min}$ correspondent to the peak and minimum position in the $\rho_d(J)$ dependencies, respectively. The inset in panel (b) shows sketch of the transverse vortex displacement $u_{t,L}$ correspondent to the Lindemann criteria. Dash and solid circles correspond to the lower and upper boundaries of the displacements $u_{t,L}$ (see the text), respectively, in the static VS in magnetic field $H >
H_{OD}$. Dot ellipses show evolution of the maximal displacements $u_{t,L}$ upon increase of the velocity $v$. Dashed region corresponds to the displacements $u_{t,L}$ in the dynamic VS. ](fig_22){width="3.2in"}
Estimations presented below show that static OD transition in our sample is caused by vortex interaction with the clusters of oxygen vacancies rather than with the isolated oxygen vacancies. Indeed, for the point disorder the pinning energy is [@Blatter94; @Ertas97] $E_p\simeq
(\gamma\varepsilon^2\varepsilon_0\xi^4)^{1/3}(L_0/L_c)^{1/5}$, where $L_0\simeq 2\varepsilon a_0$ is the length of longitudinal fluctuations, $L_c\simeq\varepsilon\xi (J_0/J_d)^{1/2}$ is the correlation length, $J_0=4c\varepsilon_0/3\sqrt{3}\xi\Phi_0$ is the depairing current, $\xi$ is the coherence length, and $\gamma\simeq(J_c\Phi_0/c)^2L_c$ is the disorder parameter. Using realistic for the [YBa$_2$Cu$_3$O$_{7-\delta}$ ]{}superconductor parameters ($\lambda$ = 500 nm, $\xi$ = 4 nm, and $\varepsilon$ = 1/7) and experimental value of the depinning current $J_{c,dyn}<$ 5 kA/cm$^2$ we obtain the energy $E_p <$ 2$\cdot$10$^{-16}$ erg, which is about 25 times smaller compared to the elastic energy $E_{el}\simeq
c_L^2\varepsilon\varepsilon_0a_0\simeq$ 5$\cdot$10$^{-15}$ erg estimated for the $c_L$ = 0.2 and $H_{OD}$ = 1.25 kOe. The pinning energy induced by vortex interaction with the clusters of oxygen vacancies equals the condensation energy $U_c\approx(H_c^2/8\pi)V_{cl}$, where $H_c=\Phi_0/2\sqrt{2}\pi\lambda\xi$ is the thermodynamic critical field and $V_{cl}$ is the volume of clusters. For spherical clusters with radius $r\simeq\xi$ we obtain the energy $E_p\simeq
U_c\approx$ 10$^{-14}$ erg, which is suitable for occurrence of the OD transition in the field of 1.25 kOe.
As it is shown in Fig. \[fig:2\]b, minimum in the $J_{c,dyn}(H)$ curve occurs in a field of 2.5 kOe, which is about two times exceeds the value of $H_{OD}$. Also, above the minimum position, the current $J_{c,dyn}(H)$ increases with the field more gradually in comparison with increase of the current $J_E$ above the OD transition. This difference can be explained by suppression of the longitudinal and conservation of the transverse pinning barriers, as it was theoretically predicted in [@Giamarchi96]. In frames of the “shaking temperature” model [@Koshelev94], this means conservation of the transverse (with respect to vector $\textbf{v}$) $u_{\perp}$ and reduction of the parallel $u_{\parallel}\propto 1/v$ component of the displacements $u_t =
\sqrt{(u_{\parallel})^2+(u_{\perp})^2}$ with increased velocity $v$. In magnetic field $\textbf{H}\parallel \textbf{c}$ and in presence of the chaotic pinning potential, spatial distribution of the displacements is isotropic; and in the field $H > H_{OD}$, the displacements $u_{t,L}$, which correspond to the Lindemann criteria, fall in the interval $c_La_0(H_{OD}) > u_{t,L} >
c_La_0(H)$, as it is shown schematically in the inset of Fig. 2b. Density of the displacements (the number of vortex displacements $u_{t,L}$ per unit length of vortex line) $n_{t,L}$ is proportional to the area of ring confined by the upper (solid circle) and lower (dashed circle) boundary of the displacements $u_{t,L}$. Reduction of the component $u_{\parallel}$ with increased velocity $v$ leads to reduction of the upper boundary (dotted lines for velocities $v_2 > v_1\neq 0$), and thus to reduction of the density $n_{t,L}$. It is important, that for any finite velocity $v$ the component $u_{\parallel}$ is finite, and thus the cross-hatched area at the diagram, which corresponds to the displacements $u_{t,L}$, and the density $n_{t,L}$ is also finite. Increase of the field reduces the lower boundary of the displacements $u_{t,L}$, and therefore the density $n_{t,L}$ increases.
Considering that the displacements $u_{t,L}$ produce the dislocations in the VS phase, and increase of the density $n_{t,L}$ results in an increase of the current $J_{c,dyn}$ [@Bondarenko08a], the field variation of the currents $J_E$ and $J_{c,dyn}$ can be explained in the following way. In low fields, the ordered VS phase, which is characterized by the absence of dislocation and realization of the 1D pinning, is formed, and the currents $J_E$ and $J_{c,dyn}$ decrease with increased field due to enhancement of the vortex-vortex interaction, making difficult to fit the vortices in the pinning landscape. Above the OD transition, the VS phase contains dislocations that results in occurance of the 3D pinning [@Ertas97], and thus the current $J_E$ increases at the transition point $H_{OD}$ due to dimensional crossover in the pinning [@Kes86; @Brandt86]. Further increase of the current $J_E$ with magnetic field is caused by increase of the density $n_{t,L}$, as it was found in [@Bondarenko08a]. The density $n_{t,L}$ is smaller in the moving VS phase than in the static VS phase, but it is finite and increases with the field. Therefore, the $J_{c,dyn}(H)$ dependence is determined by competition between decrease of the pinning force caused by enhancement of the vortex-vortex interaction and increase of the pinning force associated with increase of the density $n_{t,L}$. In our measurements, the former mechanism dominates in magnetic fields $H\leq$ 2 kOe, while the last one dominates in the fields $H\geq$ 3 kOe.
Proposed interpretation agrees with numerical simulations of 2D [@Faleski96; @Moon96; @Olson00; @Kolton99] and 3D [@Otterlo00] VL’s in the presence of strong pinning. First, it was shown that in the flux flow mode the disordered state of the VL’s is preserved [@Faleski96; @Kolton99; @Otterlo00; @Olson00], and the transverse barriers remain finite [@Moon96; @Olson00]. Second, the $v(J)$ curves cross one another near the OD transition [@Otterlo00]. Third, our interpretation implies that cross-hatched area in the diagram collapses to a segment at $v\rightarrow\infty$, indicating that moving VS can be ordered in agreement with conclusion in [@Otterlo00]. Finally, the onset of ordering of the moving VS phase is manifested as a peak in the $\rho_d(J)$ curves, and the end of ordering corresponds to value of the resistance $\rho_d(J)$=1 [@Faleski96; @Kolton99], and in our measurements peak in the $\rho_d(J)$ curves appears in the fields $H > H_{OD}$. Following computer simulations, we determined the field variation of the velocities $v_p$ and $v_{min}$, which correspond to the peak and minimum positions in the $\rho_d(J)$ curves respectively. As it is shown in Fig. 3b, the velocity $v_p$ and the difference $\Delta v = v_{min} - v_p$ increase with the field indicating that the critical velocity of the ordering as well as the interval of velocities $\Delta v$, in which the ordering realizes, increase with the field. This behavior is plausible considering that the lower boundary of the displacements $u_{t,L}$ decreases with the increased field that requires higher $v$’s to decrease the amplitude below this boundary. Also, the difference between the upper and lower boundary of the displacements $u_{t,L}$, $\Delta u = c_L[a_0(H_{OD})-a_0(H)]$, increases with the field that results in increase of the difference $\Delta v$.
Our interpretation allows explaining occurrence of the hysteresis effect in the curve $v(J)$ measured with the increased and decreased current in a field of 1.5 kOe, and absence of the hysteresis effect below and quite above the OD transition. Indeed, in close vicinity to the OD transition, $(H/H_{OD} - 1) << 1$, the density $n_{t,L}$ in the dynamic VS is much smaller than in static VS, and small increase of the velocity $v$ leads to dynamic transition into the ordered state. In this case the “shaking temperature” model predicts the hysteresis effect, which reflects the “overheated state” of the ordered dynamic VS. The decrease in density $n_{t,L}$ quite above the OD transition is not dramatic, and transition from strongly disordered static VS to less disordered dynamic VS occurs in a wide interval of velocities $\Delta v$ without hysteresis. It is important to notice that the $E(J)$ curves measured after zero field cooling coincide with the $E(J)$ curves measured after non zero field cooling, that indicates the absence of metastable states in the VS. This agrees with experimental studies of the YBaCuO crystals: the metastable states exist in vicinity of the vortex sold - vortex liquid transition, but they disappear below this transition [@Fendrich96].
Recent quantitative theory of the dynamic VS by Rosenstein and Zhuravlev [@Rosenstein07] predicts jump-like increase of the pinning force at the OD transition. It is evident that this theory does not describe our results because increase of the currents $J_E$ and $J_{c,dyn}(H)$ occurs in different fields, and field variation of the currents does not show the jump-like increase.
The obtained field variation of the currents $J_E$ and $J_{c,dyn}$, occurrence of the hysteresis effect in close vicinity to the OD transition, and absence of the metastable states in the VS are different from that in superconductors with weak bulk pinning. For example, in crystals NbSe$_2$ [@Henderson96] and MgB$_2$ [@Kim04], the current $J_{c,dyn}$ increases in a jump-like manner at the OD transition [@Henderson96; @Henderson98], and the hysteresis effect occurs in rather wide interval of magnetic fields and it is caused by presence of the metastable states in the VS [@Henderson96; @Kim04], which are induced the effect of surface barriers [@Paltiel00]. The surface barriers in the NbSe$_2$ [@Banerjee00] and MgB$_2$ [@Pissas02] cause asymmetry of the magnetization loops, and this asymmetry reflects a difference in the barriers for vortex entrance and exit of samples [@Burlachkov93]. The magnetization loops of the YBaCuO crystals are symmetric indicating negligible effect of the surface barriers. Therefore obtained field variation of the current $J_E$ corresponds to equilibrium quasistatic VS.
In conclusion, we determined field variation of the critical currents in the quasistatic and dynamic vortex solid. The currents non-monotonously vary with the field, but minimum position in the $J_{c,dyn}(H)$ dependence is shifted to higher fields in comparison with the minimum in the $J_E(H)$ dependence. The difference is interpreted by partial ordering of the vortex solid with increased vortex velocity. The disordered state of the dynamic vortex solid is attributed to preservation of finite transverse pinning barriers that guarantees presence of the transverse vortex displacements suitable for formation of dislocations. This interpretation allows explaining observed increase of the critical current of the dynamic ordering.
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---
abstract: 'In this work we study neighborhoods of curves in surfaces with positive self-intersection that can be embeeded as a germ of neighborhood of a curve on the projective plane.'
author:
- 'M. Falla Luza'
- 'P. Sad'
title: Positive Neighborhoods of Curves
---
Introduction
============
We study in this paper neighborhoods of compact, smooth, holomorphic curves of complex surfaces which have positive self intersection number. Our main purpuse is to give a condition that guarantees the existence of an embedding of a neighborhood of the curve into the projective plane. The first example of a result on this problem comes from [@FA]; in that paper the authors showed that if the curve has genus 0 and self intersection number equal to 1 then the existence of three different fibrations over it implies that some neighborhood is diffeomorphic to a neighborhood of the line in the projective plane. In this paper we consider curves of self-intersection $d^2$ with $d \geq 2$.
Since a fibration over a curve of genus 0 is defined by a local submersion over $\Pp^1$ (that is, defined in a neighborhood of the curve), we may wonder if in the case of higher genus the existence of a number of local submersions is enough to guarantee an embedding into the projective plane $\Pp^2$. It is in fact a necessary condition.
In order to discuss this, let us suppose that a curve $C$ contained in some surface $S$ can be embedded in $\mathbb P^2$ as a curve $C_0$ of degree $d \geq 2$ (we have of course to start with $C\cdot C= d^2$ in $S$). It is easy to find infinitely many submersions in a neighborhood of $C_0$. For example, we take two curves $\{A=0\}$ and $\{B=0\}$ of the same degree $l\in {\mathbb N}$ which cross each other in $l^2$ distinct points not in $C_0$. It can be seen that the map $A/B$, which is well defined outside $\{A=0\}\cap \{B=0\}$, has no multiple fibers so that it has only a finite number of critical points; if $C_0$ avoids all these points then $A/B$ is a submersion in some neighborhood of $C_0$ and the restriction of $A/B$ to $C_0$ is a ramified map from $C_0$ to $\Pp^1$ of degree $l.d$. We will be particularly interested in the case $l=1$, that is, $A=0$ and $B=0$ are lines whose common point is not in $C_0$; the submersion $A/B$ will be called a [*pencil submersion*]{} and the restriction of $A/B$ to $C_0$ is a ramified map of degree $d$ (any local submersion that leaves such a trace in $C_0$ is in fact a pencil submersion). We see that to be equivalent to a neighborhood of $C_0$, a neighborhood of $C$ has to carry also many submersions to $\Pp^1$, the surprising feature in [@FA] is that only three submersions are needed. The converse is not true as we can see in the following example.
\[fake-example\] Consider the rational curve in $\mathbb P^2$ defined in affine coordinates by the equation $y^2=x^2(x+1)$; it is a smooth rational curve except for the node at the point $(0,0)$. We blow up first at a point in the curve different from $(0,0)$, and then we blow up at $(0,0)$. The strict transform is a smooth rational curve $C$ of self intersection number equal to 4 with many local submersions (which come from submersions constructed in the plane as above), but its neighborhood can not be embedded in the plane: given a submersion constructed using $l=1$ as above (before blow up’s), we notice that it induces a ramified map from $C$ to $\Pp^1$ of degree 3; but for a conic $C_0$ in the plane (which has of course self intersection number equal to 4), the ramified map induced by any local submersion is of even degree.
A more refined question would be: can we obtain an embedding once it is assumed the existence of three local submersions in a neighborhood of $C$ whose restrictions to $C$ are meromorphic maps of degree (a multiple of) $d$? We give a partial negative answer in Section \[sec-examples\].
We introduce then an extra condition (also a necessary one). A curve $C$ that has an embedding $\phi:C \rightarrow C_0\subset \mathbb P^2$ carries naturally a special set of meromorphic maps ${\bf G}_{\phi}$ =$\{G_{|_{C_0}}\circ {\phi},\,\, G \,\,\,pencil\, submersion\}$. A set $\{F_i\}$ of submersions defined in a neighborhood of $C$ whose restrictions to $C$ have no common critical points is [**projective at C**]{} if ${F_i}_{|_C}\in {\bf G}_{\phi}$. The submersions are called [**independet**]{} if the singularities of the correspondent pencils on $\Pp^2$ are not aligned. We may state then our main result:
\[main-thm\] The existence of a projective triple of independent submersions at $C$ implies the existence of an embedding of a neighborhood of $C$ into the projective plane.
The submersions in the statement of the Theorem are supposed to produce different fibrations; we remark that if $F$ is a submersion over $\Pp^1$ and $T$ is a Moebius transformation, then $F$ and $T\circ F$ induce the same fibration.
The fibers of a submersion define a regular foliation in a neighborhood of $C$, which is generically transverse to $C$ with tangency points at the critical points of the restriction to the curve; the submersion is a meromorphic first integral for the foliation. The converse does not hold, that is, this type of foliation may not have a first integral, see [@MEZ].
We mention that the study of neighborhoods of curves has already been pursued when the self-intersection is not positive as we can see in [@GRA], [@SAV] and [@UE].
This paper is organized as follows: Section \[sec-examples\] presents some examples and it is followed by Section \[sec-const-mer-maps\] where we discuss how to built meromophic maps starting from two different pencil submersions. This allows (Section \[sec-new-foliation\]) to show the existence of foliations defined in a neighborhood of the curve which have this curve as an invariant set, and finally in Section \[sec-proof-thm\] we prove our theorem.
Examples {#sec-examples}
========
This Section has two parts. In the first part we give examples of surfaces containing smooth curves of self-intersection number $d^2$ which are not embeddable in the plane, although they are fibered by submersions whose restrictions to the curves are meromorphic functions of a degree multiple of $d$. Once this is done, we give examples which satisfy the extra condition of our Theorem but have only one or two fibrations and do not embed them in the plane.
Separating branches and examples with 3 fibrations
--------------------------------------------------
We will use the following construction. Let us consider a curve H with an ordinary singularity $P$ with $m$ branches $L_1,\dots,L_m$. For each branch $L_j$ we take a neighborhood $V_j$ which is biholomorphic to a bidisc $D_j$ by means of a biholomorphism $\phi_j:D_j\rightarrow V_j$ ; we assume that $\delta L_j \cap V_i =\emptyset$ for all $i\ne j$. We fix a neighborhood $V$ of $H\setminus \cup_1^m L_j$. Finally we take the disjoint union of $V$ with all th $D_j$, and glue $D_j$ to $V$ using the restriction of the map $\phi_j$ to $\phi_j^{-1}(V\cap V_j)$. In this way the union of the sets $V_j$, which contains $P$, is replaced by $m$ copies of the bidisc, and there is a new curve $H^{\prime}$ replacing $H$ inside a new surface without the ordinary singularity. As for the self-intersection number $H^{\prime}\cdot H^{\prime}$, we have that $H^{\prime}\cdot H^{\prime}= H\cdot H -m(m-1)= (H\cdot H -m^2) +m$. Also any holomorphic foliation $\mathcal F$ defined in $V\cup_1^m V_j$ induces naturally a holomorphic foliation in the new surface which is $\mathcal F$ in $V$ and $\phi_j^*(\mathcal F)$ in each $D_j$. We refer to this construction as [*separating branches of $H$ at P*]{}.
![Separating branches[]{data-label="fig:viaduto"}](viaduto.png)
Let us consider then a smooth plane curve $C^{\prime}$ of degree $d^{\prime}$ and genus $g(C^{\prime})= \dfrac{(d^{\prime}-1)(d^{\prime}-2)}{2}$; it can be also immersed in the plane as a curve $C$ of degree $d$ for any $d>2g(C^{\prime})$ with a number $s$ of nodal points such that $d^2-3d-2s={d^{\prime}}^2 -3d^{\prime}$. We choose $d-d^{\prime}=k^2$ for some $k\in \mathbb N$ such that
(i) $d^{\prime}$ divides $k^3-k^2$,
(ii) $d^{\prime}$ does not divide $2k^2$;
we choose also three pencils $d\left(\dfrac {u_j}{v_j}\right)=0$ of curves of degree $k$ whose sets of $k^2$ base points lie in the regular part of $C$ and are two by two disjoint. After blowing up at these $3k^2$ points and separating branches at the nodal points of $C$ we get a curve $\tilde C$ containing in some surface with self-intersection number equal to $d^2-3k^2-2s = {d^{\prime}}^2$; the maps $\dfrac {u_j}{v_j}$ become submersions whose restrictions to $\tilde C$ are meromorphic maps of degree $k.d- k^2$, which is a multiple of $d^{\prime}$ because of (i).
A neighborhood of $\tilde C$ is not equivalent to a neighborhood of $C^{\prime}$ in the plane. In fact, let us take a linear pencil $\mathcal L$ in the plane with base point outside $C$ and transverse to the branches at each nodal point (this is before blow-up‘s and separation of branches). We have $2d=Tang(\mathcal L,C)+ \chi(C)$; since $tang(\mathcal L,C,P)=2$ for each nodal point $P$, we get $2d = 2s+\chi(C)+ tang(\mathcal L,C)$, where the last term counts the tangencies with the regular part of $C$. These tangencies persist when we blow up and separate branches; therefore, if a neighborhood of $\tilde C$ is equivalent to a neighborhood of $C^{\prime}$, we get in this neighborhood a foliation $\mathcal L^{\prime}$ with $Tang(\mathcal L^{\prime}, C^{\prime})=tang(\mathcal L,C)=2d-2s-\chi(C)$. It follows that $(deg(\mathcal L^{\prime})+2)d^{\prime}=2d-2s-\chi(C)+\chi(C^{\prime})=2d-2s+2g(C)-2g(C^{\prime})$ and since $g(C)-s=g(C^{\prime})$, we conclude that $(deg(\mathcal L^{\prime})+2)d^{\prime}=2d= 2d^{\prime}+2k^2$, a contradiction because of (ii).
We remark that when $d=4, d^{\prime}=3$ or $d=3,d^{\prime}=2$ the construction can be done with $k=1$ because $d>2g(C^{\prime})$ (and obviously (i) is satisfied in both cases). When $d=4, d^{\prime}=3$ we have also that (ii) holds true. In the general case $d^{\prime}>3$ we may choose $k=d^{\prime}+1$ for example in order to get both (i) and (ii) satisfied.
The special case $d=3,d^{\prime}=2$ (and $k=1$) can be treated with a small difference in what concerns the proof that the neighborhood of $\tilde C$ is not equivalent to a neighborhood of $C$: we select $\mathcal L$ as the pencil whose base point is the node point $P$ of $C$. Since $Tang(\mathcal L,C)= 6=tang(\mathcal L,C,P)$, we see that there is no other point of tangency between $\mathcal L$ and $C$. We get then $(deg(\mathcal L^{\prime})+2).2= 4+2$ (after separating the branches at $P$ we obtain 2 radial singularities belonging to $\tilde C$ and a fortiori to $C^{\prime}$) and therefore $deg(\mathcal L^{\prime})=1$. But is impossible for a foliation of degree 1 in the plane to have 2 radial singularities.
It would be nice to have examples where the degree induced by the submersions on the curve is exactly $d^{\prime}$.
Special Examples
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A construction already presented in [@SA] of a pair (curve, surface) with a submersion whose restriction to the curve is a ramification map given a priori (we will say that the submersion is a [**lifting**]{} of a ramification map). We start with a line bundle of Chern class $n\in \mathbb N$ over a curve $C$; the lines of the bundle define a foliation $\mathcal L$ in the total space of the bundle. Let $f: C \rightarrow \Pp^1$ be a ramified map with simple critical points and let $p$ be one of these points. There exists an involution $i$ defined in a neighborhood of $p$ in $C$ by $f(q)= f(i(q))$ for $q$ close to $p$.
We fix a neighborhood $U$ of $p$ and a holomorphic diffeomorphism $H:U \rightarrow {\mathbb D}\times {\mathbb D}$ such that: 1) $H(p)= (0,0)$; 2) $H(C \cap U)=\{(z_1,0)\in {\mathbb D}\times {\mathbb D}\}$; 3)$H$ takes $\mathcal L$ to the foliation $dz_1=0$ and 4) $h:= H|_{C \cap U}$ conjugates $i$ to the involution $z \longmapsto -z$, that is: $h(i(q))=-h(q)$. We take also a biholomorphism $\psi$ from ${\mathbb D}\times {\mathbb D}$ to a neighborhood of $(0,0)$ with the properties: 1) $\psi (0,0)=(0,0)$; 2) $\psi(z_1,0)= (z_1,0)$; 3) $\psi(\{1/2<|z_1|<1\}\times {\mathbb D})$ is saturated by leaves of the foliation $dZ_2-Z_1dZ_1=0$ and 4) $\psi$ is a holomorphic diffeomorphism when restricted to $\{1/2 < |z_1| <1\}\times {\mathbb D}$ that sends the foliation $dz_1=0$ to the foliation $dZ_2-Z_1dZ_1=0$. Put $H_1= \psi \circ H$.
We remove from the total space of the line bundle the fibers over the points of $h^{-1}(\{|z_1|\le 1/2\})$ and glue $\psi({\mathbb D}\times {\mathbb D})$ to the remaining set using $H_1$. In this way we get a new holomorphic surface which contains $C$ (the same curve we started with) and a holomorphic foliation transverse to $C$ except at $p$, where the tangency is simple. Furthermore, the “local holonomy” of the new foliation at $p$ is exactly $i$. We repeat the same procedure for all critical points of $f$. At the end, we have a holomorphic surface that contains $C$ and a holomorphic foliation which is transverse to $C$ except at the critical points of $f$; we may even assume that the self-intersection number of $C$ is $n\in \mathbb N$. The map $f$ can be extended along the leaves (because of its compatibility with the involutions involved), producing the desired lifting. A similar construction can be made if the critical points are not simple.
Let us give two examples of pairs (curve, surface) which are not embeddable in the projective plane.
We have already noticed that, in order to be embeddable in the projective plane, all the submersions defined in the neighborhood of the curve must have as restrictions maps whose degrees are multiple of $d$ (here $d^2$ is the self-intersection number of the curve in the surface). This does not happen in the example given in the Introdution. We give now another example of a different nature. Take $C\subset \mathbb P^2$. We start by claiming that there exists a ramification $f:C\rightarrow \Pp^1$ of degree $(d-1)d$ such that the set of poles is not contained in any curve of degree $d-1$. In order to see this, let us start with a ramification map $f_0:C\rightarrow \Pp^1$ defined as the restriction of $\dfrac {1}{Q_0}$ to $C$, where $Q_0$ is a polynomial of degree $d-1$ which intersects $C$ transversely at $l=(d-1)d$ different points $P_1,\dots,P_l$. Let us consider nearby points $P_1^{\prime},\dots,P_l^{\prime}$ and apply Riemann-Roch’s theorem to $D=P_1^{\prime} +\dots +P_l^{\prime}$: $l(D)\ge (d-1)d-g+1$; if we want to have $l(D)>1$, we ask for $(d-1)d-g+1>1$, or $(d-1)d >\dfrac{(d-1)(d-2)}{2}$, which is always true when $d>1$. In fact, from the proof of Riemann-Roch’s theorem , since $(P_1^{\prime},\dots,P_l^{\prime})$ is close to $(P_1,\dots,P_l)$, we may choose a meromorphic function close to $f_0$, so its polar divisor is $D$. On the other hand, the points $(P_1^{\prime},\dots,P_l^{\prime})$ which belong to a curve of degree $d-1$ are contained in a subvariety of dimension $\dfrac{d(d+1)}{2}-1$, and all we have to do is check if $(d-1)d > \dfrac{d(d+1)}{2}-1$, which is obvious if $d \ge 3$ (we remark that there are not two different curves of degree $d-1$ passing through the $(d-1)d$ points $P_1^{\prime},\dots,P_l^{\prime}$). We select then $P_1^{\prime},\dots,P_l^{\prime}$ outside this subvariety in order to get the ramification map $f$ and take a lifting $F$ defined in a surface $S$. We prove then the statement: there is no embedding $\Phi: S\rightarrow {\mathbb P^2}$. In fact, the submersion $F\circ{\Phi^{-1}}$ defined in a neighborhood of $C_0 \subset {\mathbb P^2}$ extends to $\mathbb P^2$ as a meromorphic function (holomorphic in a neighborhood of $C_0$). We observe that, for $d\ge 3$, given two embeddings $\phi_i:C\rightarrow {\mathbb P^2}$, i=1,2 there exists an automorphism $T\in Aut(\mathbb P^2)$ such that $T(\phi_1(C))= \phi_2(C)$, see Appendix. Then the map $\Phi|_{C}:C \rightarrow C_0$ comes from a linear map on $\Pp^2$ and poles of $f$ are the intersection of $C$ with a curve of degree $d-1$, which is impossible.
We present now an example of a non embeddable pair (curve, surface) with a set of two fibrations which is projective at the curve. We start with a projective, smooth curve $C$ and select two pencil submersions; let $f_1$ and $f_2$ be the associated ramification maps of $C$. The tangencies between the pencils are obviously pieces of the common line. We will replace one of these pieces by a non-invariant curve of tangencies between two new foliations. The idea is the same used above to realize ramification maps; the homeomorphism $\psi$ is going to be changed. The point $p$ this time is a point of tangency, and the coordinate chart $H$ sends the foliations associated to the submersions to two foliations $(dz_1=0, \mathcal H)$. We consider in $\mathbb C^2$ a couple of foliations ($dZ_1=0, \mathcal H^{\prime}: d(Z_1-Z_2(Z_2-Z_1))=0)$, which have $Z_1=2Z_2$ as non-invariant line of tangencies. The homeomorphism $\psi$ is choosed in order to satisfy: 1) $\psi(0,0)=(0,0)$ and $\psi(z_1,0)=(z_1,0)$; 2) $\psi|_{\{1/2<|z_1|<1\}\times \mathbb D}$ is a holomorphic diffeomorphism over its image that sends $(dz_1=0,\mathcal H)$ to $(dZ_1=0, \mathcal H^{\prime})$. We put again $H_1=\psi \circ H$,which is the new glueing map. We can see that $C^2$ does not change and so the germ of surface is not isomorphic to $(C, \Pp^2)$.
We could also use in the construction the pair of foliations ($dZ_1=0, \mathcal H^{\prime}: d(Z_1-Z_2(Z_2-Z_1^{k+1}))=0)$ for $k\in \mathbb N$, but the curve $C$ will have self-intersection number equal to $d^2-k$.
Constructing meromorphic maps {#sec-const-mer-maps}
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Let us once more describe the setting we are going to analyse. We have a curve $C$ contained in some surface $S$ with $C\cdot C= d^2$ and $d\in \mathbb N$. There exist three submersions $F$, $G$ and $H$ defined in $S$ and taking values in $\Pp^1$ which define foliations $\mathcal F$, $\mathcal G$ and $\mathcal H$ generically transversal to $C$ whose leaves are the levels curves. In order to simplify the exposition, we assume that all tangencies with $C$ are simple and distinct (when we look to the tangencies for any pair of foliations). We denote $f=F|_{C}$, $g=G|_{C}$ and $h=H|_{C}$, all of them ramification maps from $C$ to $\Pp^1$ whose ramification points correspond to the tangency points of the foliations (because $F$, $G$ and $H$ are submersions). Furthermore, we assume that $C$ embedds into $\mathbb P^2$ by a map $\phi:C\rightarrow C_0$; $C_0$ is a smooth algebraic curve of degree $d$. In order to complete the picture, we select pencil submersions $F_0$, $G_0$ and $H_0$ (with associated foliations ${\mathcal F}_0$, ${\mathcal G}_0$ and ${\mathcal H}_0$), with singular points not aligned, which restric to $C_0$ as $d$ to $1$ maps $f_0$, $g_0$ and $h_0$ to $\Pp^1$ and ask $\{f,g,h\}$ to be conjugated by $\phi$ to $\{f_0,g_0,h_0\}$: $f_0 \circ \phi = f$, $g_0 \circ \phi = g$ and $ h_0 \circ \phi = h$. We remark that $\phi(tang(\mathcal F,C))= tang (\mathcal F_0,C_0)$ once more because these tangency points are exactly the ramification points of $f$ and $f_0$ (we have also that $\phi(tang(\mathcal G,C))= tang (\mathcal G_0,C_0)$ and $\phi(tang(\mathcal H,C))= tang (\mathcal H_0,C_0)$). For simplicity, we will assume that $F_{|_C}, G_{|_C}$ and $H_{|_C}$ have only simple critical points.
Any pair of foliations defined by projective submersions at $C$ are generically transverse to each other along $C$.
Let $\mathcal F$ and $\mathcal G$ be two projective submersions at $C$. From [@BRU] we have $$tang(\mathcal F,\mathcal G)\cdot C= N_{\mathcal F} \cdot C + N_{\mathcal G}\cdot C + K_S \cdot C$$ where $tang(\mathcal F,\mathcal G)$ is the curve of tangencies between the foliations, $N_{\mathcal F}$ (resp. $N_{\mathcal G}$) is the normal bundle associated to $\mathcal F$ (resp. $\mathcal G$) and $K_S$ is the canonical bundle of $S$. Since $$\begin{aligned}
N_{\mathcal F}\cdot C &=& \chi(C)+ tang ({\mathcal F},C)= 3d-d^2 + d^2-d\\
N_{\mathcal G}\cdot C &=& \chi(C)+ tang ({\mathcal G},C) =3d-d^2 + d^2-d\\
-K_S \cdot C_0 &=& \chi(C) + C\cdot C = 3d-d^2 + d^2\\\end{aligned}$$ we conclude that $
tang(\mathcal F,\mathcal G)\cdot C = d
$ so that $\mathcal F$ and $\mathcal G$ are not tangent to each other along $C$.
We observe that the Lemma is not true for $d=1$ (see [@FA]).
In this Section we will see how to associate to a pair of submersions, say $F, G$, a meromorphic map $\Phi_{F,G}$. It is defined initially as a biholormorphism from a neghborhood of the set $C\setminus (A \cup \phi^{-1}(A_0))$ to a neighborhood of $C_0\setminus (A_0 \cup \phi(A))$, where $A= tang(\mathcal F, C)\cup tang (\mathcal G,C) \cup (tang(\mathcal F, \mathcal G)\cap C)$ and $A_0= tang(\mathcal F_0, C_0)\cup tang (\mathcal G_0,C_0) \cup (tang(\mathcal F_0, \mathcal G_0)\cap C_0)$. Given a point $p\in C\setminus (A \cup {\phi}^{-1}(A_0))$, the foliations $\mathcal F$ and $\mathcal G$ are transverse to each other and to $C$ in a neighborhood of this point and the foliations $\mathcal F_0$ and $\mathcal G_0$ are transverse to each other and to $C_0$ in a neighborhood of $\phi(p)$; therefore, for $q\in S$ close to $p$ we may associate the points $q_{\mathcal F}$ and $q_{\mathcal G}$ where the leaves of $\mathcal F$ and $\mathcal G$ intersect $C$. The leaves of $\mathcal F_0$ and $\mathcal G_0$ through $\phi(q_{\mathcal F})$ and $\phi(q_{\mathcal G})$ will intersect (by definition) at the point $\Phi_{F,G}(q)$. It can be seen that this maps extends biholomorphically to the points of $tang(\mathcal F,C)$ and $tang(\mathcal G,C)$, essentially because the foliations $\mathcal F$ and $\mathcal G$ are transverse to each other at those points. From now on we change $A$ and $A_0$ to $A= tang(\mathcal F, \mathcal G)\cap C$ and $A_0=tang(\mathcal F_0, \mathcal G_0)\cap C_0$ and analyse the behavior of $\Phi_{F,G}$ at points of $A \cup {\phi^{-1}}(A_0)$. We distinguish two cases
(A) $\phi(p)\in tang(\mathcal F_0,\mathcal G_0,C_0)$.
(B) $\phi(p)\notin tang(\mathcal F_0,\mathcal G_0,C_0)$.
\[extension\] $\Phi_{F,G}$ extends meromorphically to a neighborhood of $C$.
[**Case A:**]{} We may assume, choosing conveniently the coordinates $(x,y)$ around $p$ and affine coordinates $(X,Y)$, that
- $p=(0,0)$, $C$ is $y=0$ and $\mathcal F$ is defined by $dx=0$;
- $\phi(p)=(0,0)$, $\mathcal F_0$ is defined by $dX=0$, $\mathcal G_0$ is the radial pencil with $(0,1)$ as base point ($X=0$ is a common fiber of $\mathcal F_0$ and $\mathcal G_0$);
- $C_0$ is defined by $Y=h(X)$ with $h(0)=0$, $h^{\prime}(0)=0$ and $\phi(x)= (x,h(x))$.
The leaf of $\mathcal F$ (respec. $\mathcal G$) through a point $(x,y)$ crosses the $x$-axis at $x$ (respectively $\xi(x,y)$ for a holomorphic function $\xi$ such that $\xi(x,0)=x$. It follows that $$\Phi_{F,G}(x,y)= \left(x,1-\dfrac{x(1-h(\xi(x,y))}{\xi (x,y))}\right)= \left(x,\dfrac{u(x,y)}{\xi(x,y)}\right)$$
The expression defines a meromorphic map in a neighborhood of $(0,0)$. There are two possible cases:
- $\bf A_1$: the germs $x$ and $\xi$ are relatively prime; the line of poles of $\Phi_{F,G}(x,y)$ is $\xi (x,y)=0$ and has multiplicity 1. We write $\xi(x,y)-x=y\,A_1(x,y)$ for some holomorphic function $A_1(x,y)$; the $\mathcal G$-fiber may be transversal to the $\mathcal F$-fiber (when $A_1(0,0)\neq 0$) or tangent to it (in which case $A_1(0,0)\neq 0$).
- $\bf A_2$: the germs $x$ and $\xi$ have a common factor; write $\xi(x,y)=x(1+y\,A_2(x,y))$, thus $\Phi_{F,G}(x,y)$ is a holomorphic map ($\mathcal F$ and $\mathcal G$ have $x=0$ as a common fiber), but it may be non-injective (unless $A_2(0,0)\neq 0$).
[**Case B:**]{} We assume:
- $p=(0,0)$, $C$ is $y=0$ and $\mathcal F$ is defined by $dx=0$;
- $\phi(p)=(0,0)$, $\mathcal F_0$ is defined by $dX=0$ and $\mathcal G_0$ is defined by $dY-dX=0$ (in affine coordinates);
- $C_0$ is defined by $Y=h(X)$ with $h(0)=0$, $h^{\prime}(0)=0$ and $\phi(x)= (x,h(x))$.
We have then $$\Phi_{F,G}(x,y)= (x,\xi(x,y)-x +h(\xi(x,y))$$ It follows that $\Phi_{F,G}$ is a holomorphic map in a neighborhood of $p$; writing $\xi(x,y)-x=y\,B(x,y)$, we see that $\Phi_{F,G}$ is a local biholomorphism when $B(0,0)\neq 0$, that is, the fibers of $\mathcal F$ and $\mathcal G$ are transversal at $p$.
An important consequence for us is that the pull-back by $\Phi_{F,G}$ of a holomorphic foliation $\mathcal L$ on $\Pp^2$ is also a holomorphic foliation in $S$. In the next Section we describe the singularities of ${\Phi^*_{F,G}}(\mathcal L)$.
New foliations on S {#sec-new-foliation}
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Let us take a foliation $\mathcal L$ on $\mathbb P^2$ defined by $\omega= LdP-d.PdL=0$, where $P(X,Y)=\sum_{i+j \leq d} a_{ij}X^iY^j$ is a polynomial of degree $d$ such that $C_0=\{P=0\}$ (we may assume $a_{0d}\neq 0$) and $L$ is a linear polynomial such that $L=0$ is transverse to $C_0$ . The singularities of $\mathcal L$ contained in $C_0$ are supposed to be disjoint of $A_0\cup \phi(A)$.
We proceed to compute the multiplicity $Z(\mathcal L^{*},C,p)$ along $C$ of $p$ as a singularity of $\mathcal L^{*}=\Phi_{F,G}^{*}(\mathcal L)$ at the points where $\Phi_{F,G}$ maybe fails to be a biholomorphism. In order to make the computation easier, we take $L(X,Y)=X+b$.
\[Indices-Z\] With notation of the proof of Proposition \[extension\], we have
- Case A1: $Z(\mathcal L^{*},C,p)= d+mult_0(A_1(x,0))$.
- Case A2: $Z(\mathcal L^{*},C,p)= mult_0(A_2(x,0))$.
- Case B: $Z(\mathcal L^{*},C,p)=mult_0(B(x,0))$.
[**Case A1**]{}: $x$ and $\xi$ are relatively prime. It follows that $$P(\Phi_{F,G}(x,y))=\dfrac{yv(x,y)}{\xi(x,y)^d}$$ In fact, $P(\Phi_{F,G}(x,0))=0$ and $
P(X,Y)=a_{0d}Y^d + \sum_{j\leq d-1}a_{ij}X^iY^j
$ and therefore $$P(\Phi_{F,G}(x,y))= a_{0d}\dfrac{u^d}{\xi^d}+ \dfrac{\sum_{d-j\geq 1}a_{ij}x^iu^j{\xi}^{d-j}}{\xi^d}$$ In particular, $v(x,0)=x^{d-1}A_1(x,0)+\dots$. We have also $L(\Phi_{F,G}(x,y))=x+b$, $b\neq 0$, so that $$\Phi_{F,G}^{*}\,\omega = \dfrac{1}{\xi^{d+1}}[(x+b){\xi}(yd\,v + vd\,y) - d.yv((x+b)d\xi + {\xi}d\,x)]$$ Therefore $\mathcal L^{*}$ is defined by $ (x+b){\xi}(yd\,v + vd\,y) - d.yv((x+b)d\xi + {\xi}d\,x)=0$ near the point $p$ and $$Z(\mathcal L^{*},C,p)= 1+mult_0(v(x,0))= d+mult_0(A_1(x,0))$$ We observe that $Z(\mathcal L^{*},C,p)>0$ when the case ${\bf A1}$ is present.
[**Case A2**]{}: $\xi$ divides $x$ ($\mathcal F$ and $\mathcal G$ share the leaf passing through $p$). Let us write as before $\xi(x,y)= x(1+yA_2(x,y))$; it follows that $$\Phi_{F,G}(x,y)=(x, \dfrac {yA_2(x,y) +h(\xi(x,y))}{1+yA_2(x,y)})$$ Writing $P(\Phi_{F,G}(x,y))= yv(x,y)$, we see that $v(x,0)=A_2(x,0)+\dots$ and $$\Phi_{F,G}^{*}\,\omega = (x+b)(vdy + ydv) - d.yvdx$$ We conclude that $$Z(\mathcal L^{*},C,p)=mult_0(v(x,0))=mult_0(A_2(x,0))$$ Let us notice that $Z(\mathcal L^{*},C,p)=0$ implies that $A_2(0,0)\neq 0$, that is, $\Phi_{F,G}(x,y)$ is a local biholomorphism at $p$.
[**Case B**]{}: $\phi(p)\notin tang(\mathcal F_0,\mathcal G_0) \cap C_0$. We have
$$\Phi_{F,G}(x,y)=(x,\xi-x+h(\xi(x,y))$$ Writing $P(\Phi_{F,G}(x,y))= yv(x,y)$, we see that $v(x,0)=B(x,0)+\dots$ and $$\Phi_{F,G}^{*}\,\omega = (x+b)(vdy + ydv) - d.yvdx$$ We conclude that $$Z(\mathcal L^{*},C,p)=mult_0(v(x,0))_0=mult_0(B(x,0))$$ Again, $Z(\mathcal L^{*},C,p)=0$ implies that $B(0,0)\neq0$, that is, $\Phi_{F,G}(x,y)$ is a local biholomorphism at the point p.
We intend now to see the implications of having two maps $\Phi_{F,G}$ and $\Phi_{F,H}$ simultaneously; the fibrations $\mathcal F$, $\mathcal G$ and $\mathcal H$ are associated to pencil submersions $\mathcal F_0$, $\mathcal G_0$ and $\mathcal H_0$. Let us call $B= tang(\mathcal F,\mathcal H)\cap C$ and $B_0= tang(\mathcal F_0,\mathcal H_0)\cap C_0$. We consider two foliations $\mathcal I$ and $\mathcal L$ on $\mathbb P^2$ like before. Remark that $Z(\mathcal{I}, C_0)= Z(\mathcal{L}, C_0)=d$. We will assume: 1) all singularities of $\mathcal I$ and $\mathcal L$ lie outside the set $K= A_0 \cup \phi(A)\cup B_0 \cup \phi(B)$; 2) all curves of tangencies between $\mathcal I$ and $\mathcal L$ cross $C_0$ outside the set $K$. We denote $\mathcal I^*= \Phi_{F,G}^*(\mathcal I)$ and $\mathcal L^*= \Phi_{F,H}^*(\mathcal L)$. We will use again the formulae from [@BRU] to compute numerical invariants associated to tangent lines between two foliations. We have: $$tang(\mathcal I,\mathcal L)\cdot C_0= N_{\mathcal I} \cdot C_0 + N_{\mathcal L}\cdot C_0 + K_{\mathbb P^2}\cdot C_0=2d^2 -d,$$ since $\mathcal I$ and $\mathcal H$ have degree $d-1$ and $K_{\mathbb P^2}\cdot C_0 =-3d$.
Let us call $\mathcal Z_1(\mathcal I^*,C)$ ($\mathcal Z_1(\mathcal L^*,C))$ the set of points where $\Phi_{F,G}$ is not a local biholomorphism (respectively $\Phi_{F,H}$ is not a local biholomorphism). We define $Z_1(\mathcal I^*,C)$ as the sum of all indexes $Z(\mathcal I^*,C,p)$ at points of $\mathcal Z_1(\mathcal I^*,C)$ (we put $Z_1(\mathcal L^*,C)$ for the correspondent sum at points of $\mathcal Z_1(\mathcal L^*,C)$).
As for the foliations $\mathcal I^*$ and $\mathcal L^*$, we have that $$\begin{aligned}
tang(\mathcal I^*,\mathcal L^*)\cdot C&=& N_{\mathcal I^*} \cdot C + N_{\mathcal L^*}\cdot C + K_S \cdot C \\
&=& Z(\mathcal I^*,C) + Z(\mathcal L^*,C) + 2d^2-3d\\
&=& Z_1(\mathcal I^*,C) + Z_1(\mathcal L^*,C) + 2d^2-d\end{aligned}$$ We conclude therefore that $$tang(\mathcal I^*,\mathcal L^*)\cdot C = Z_1(\mathcal I^*,C) + Z_1(\mathcal L^*,C)+ tang(\mathcal I,\mathcal L)\cdot C_0$$ Observe that curves $C$ and $C_0$ appear as components of the tangency locus in both sides of the last equation, thus we cancel $d$ from the equation and consider, from now, tangency loci besides $C$ and $C_0$. This formula suggests that $tang(\mathcal I^*,\mathcal L^*)\cap C$ may be also computed looking at the points of $\phi^{-1}(tang(\mathcal I, \mathcal L)\cap C) \cup \mathcal Z_1(\mathcal I^*,C) \cup \mathcal Z_1(\mathcal L^*,C)$.
Our aim is to prove that $\Phi_{F,G}$ and $\Phi_{F,H}$ are everywhere local biholomorphisms. First of all we have to associate the tangencies between $\mathcal I$ and $\mathcal L$ to tangencies between $\mathcal I^*$ and $\mathcal L^*$. There is a little difficulty here because $\mathcal I^*$ and $\mathcal L^*$ are obtained from $\mathcal I$ and $\mathcal L$ using different pull-back’s; the pre-image by $\phi$ of a point of tangence between $\mathcal I$ and $\mathcal L$ might not be a point of tangency between $\mathcal I^*$ and $\mathcal L^*$. We take the foliations $\mathcal I$ and $\mathcal L$ defined by the equations $LdP-d.PdL=0$ and $(L+ a)dP + d.PdL=0$; their curve of tangencies is defined by $dL\wedge dP=0$ (besides the curve $C_0$). When intersecting with $C_0$, these are the points of tangency of $\mathcal F_0$ with $C_0$.
\[intersection-at-tangencies\] Let $\phi(p)$ be a point of tangency between $\mathcal F_0$ and $C_0$. Then $(tang(\mathcal I^*,\mathcal L^*), C)_p=1$.
We may take local coordinates $(x,y)$ around $p$ and $(r,s)$ around $\phi(p)$ such that
- $C=\{y=0\}$ and $C_0= \{s=0\}$.
- $\mathcal F$ and $\mathcal F_0$ are defined by $d(y-x^2)=0$ and $d(s-r^2)=0$ respectively.
The foliations $\mathcal I$ and $\mathcal L$ are defined as $ su\,d(s-r^2)-(s-r^2 + \delta)\,d(su)=0$ and $su\,d(s-r^2)-(s-r^2 + a+\delta)\,d(su)=0$, where $u$ is a holomorphic function such that $u(0,0)\ne 0$ and $\delta \neq 0$. Let us write $\Phi_{F,G}(x,y)= (f(x,y),yA(x,y))$ and $\Phi_{F,H}(x,y)= (g(x,y),yB(x,y))$; we have $A(0,0)\neq 0$, $B(0,0)\ne 0$ and $(f(x,0),0)=(g(x,0),0)= \phi(x)$.
The foliations $\mathcal I^*=\Phi_{F,G}^*{\mathcal I}$ and $\mathcal L^*=\Phi_{F,H}^*{\mathcal L}$ are defined as $$\begin{aligned}
yAu\,d(yA-f^2) - (yA-f^2 +\delta)\,d(yAu)&=&0,\\
yBu\,d(yB-g^2) - (yB-g^2 + a+ \delta)\,d(yBu)&=&0\end{aligned}$$ We see easily that the curve of tangencies is given by $ABau^2{\phi}{\phi^{\prime}}y + y^2(...)=0$, so that the component different from $C=\{y=0\}$ crosses $C$ at $p$ transversaly.
We proceed now to examine the points of tangency between $\mathcal I^*$ and $\mathcal L^*$ that possibly appear at ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$. If we denote their number as $tang_1 (\mathcal I^*,\mathcal L^*)$, we have seen that $$tang_1 (\mathcal I^*,\mathcal L^*) = Z_1(\mathcal I^*,C)+ Z_1(\mathcal L^*,C).$$ In fact, we have seen that out of ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$ tangency curves correspond to each other when restricted to $C$ and $C_0$.
[**We claim that this equality holds at each point of**]{} ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$. Let us consider some point $p\in {\mathcal Z}_1(\mathcal I^*,C)$; since $\Phi_{F,G}$ is not a local biholomorphism, we have as explained before the possibilities $\bf A1$, $\bf A2$ and $\bf B$, the first two occuring when $\phi(p)\in tang({\mathcal F}_0,{\mathcal G}_0 \cap C_0$. If $p$ is $\bf A1$ or $\bf A2$ for $\Phi_{F,G}$, then $p$ is $\bf B$ for $\Phi_{F,H}$ (in the same way, when $q\in {\mathcal Z}_1(\mathcal L^*,C)$ is $\bf A1$ or $\bf A2$ for $\Phi_{F,H}$, then $q$ is $\bf B$ for $\Phi_{F,G}$. It may happen also that $p$ is $\bf B$ for $\Phi_{F,G}$ and $\Phi_{F,H}$. The reason is that we are supposing the submersions $F$, $G$ and $H$ to be independent so that we are in case $\bf A$ for maps $\Phi_{F,G}$ and $\Phi_{F,H}$ simultaneously.
[**Case 1: $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is ${\bf A1}$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$.**]{} The local equations for $\mathcal I^*$ and $\mathcal L^*$ at $p$ are $$\begin{aligned}
(x+b)\xi(ydv+vdy)-d.yv \{(x+b)d\xi+{\xi}dx\}&=&0\\
(x+b^{'})(v^{'}dy+ydv^{'})-d.yv^{'}dx&=&0.\end{aligned}$$ The line of tangencies has equation $$(b-b^{'}){\xi}vv^{'}-(x+b)(x+b^{'})[vv^{'}{\xi}_x+{\xi}(v^{'}v_x-vv^{'}_x)]=0$$ We observe that $mult_0({\xi}vv^{'})=mult_0(v)+1+mult_0(v^{'})=Z(\mathcal I^*,C,p)+Z(\mathcal L^*,C,p)$ (it may happen $Z(\mathcal L^*,C,p)=0$).
[**Case 2: $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf A2$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$.**]{} The local equations are $$\begin{aligned}
(x+b)(ydv+vdy)-d.yvdx&=&0\\
(x+b^{'})(v^{'}dy+ydv^{'})-d.yv^{'}dx&=&0\end{aligned}$$ The line of tangencies has equation $$(b-b^{'})vv^{'}-(x+b)(x+b^{'})[v^{'}v_x-vv^{'}_x]=0$$ We remark that $mult_0(vv^{'})=mult_0(v)+mult_0(v^{'})=Z(\mathcal I^*,C,p)+Z(\mathcal L^*,C,p)$ (it may happen that $Z(\mathcal L^*,C,p)=0$).
[**Case 3: $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf B$ for $\Phi_{F,G}$ and $\Phi_{F,H}$.**]{} The conclusion is the same as above: $mult_0(vv^{'})=mult_0(v)+mult_0(v^{'})=Z(\mathcal I^*,C,p)+Z(\mathcal L^*,C,p)$. (it may happen $Z(\mathcal L^*,C,p)=0$).
The remaining cases (when $p\in {\mathcal Z}_1(\mathcal L^*,C,p)$): $p$ is $\bf A1$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$; $p$ is $\bf A2$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$; $p$ is $\bf B$ for both $\Phi_{F,H}$ and $\Phi_{F,G}$ are entirely similar.
We conclude from $tang_1 (\mathcal I^*,\mathcal L^*) = Z_1(\mathcal I^*,C)+ Z_1(\mathcal L^*,C)$ (and the fact that $b$, $b'$ are generic) that the terms $[vv^{'}{\xi}_x+{\xi}(v^{'}v_x-vv^{'}_x)]$ (first case) and $[v^{'}v_x-vv^{'}_x]$ (second and third cases) have the same multiplicities at $0$ as ${\xi}vv^{'}$ and $vv^{'}$ respectively; [**the claim is proved**]{}.
Let us make explicit the relations between the several multiplicities involved before.
- Case 1: we write $v(x,0)=ax^l+\dots$ and $v^{'}(x,0)=cx^m+\dots$. It follows that $[vv^{'}{\xi}_x+{\xi}(v^{'}v_x-vv^{'}_x)]=ac(1-m+l)x^{m+l}+\dots$. Since $mult_0({\xi}vv^{'})=m+l+1$ necessarily $m=l+1$. Using $v(x,0)=x^{d-1}A_1(x,0)$ (and $v^{'}(x,0)=B^{'}(x,0)$) we get: $$mult_0(A_1(x,0))+d = mult_0(B^{'}(x,0))$$
- Case 2: we write again $v(x,0)=ax^l+\dots$ and $v^{'}(x,0)=cx^m+\dots$. Then $[v^{'}v_x-vv^{'}_x)]=ac(l-m)x^{m+l-1}+\dots $. Since $mult_0(vv^{'})=m+l$, we see that $l=m$. Using $v(x,0)=A_2(x,0)$ and $v^{'}(x,0)=B^{'}(x,0)$ $$mult_0(A_2(x,0))=mult_0(B^{'}(x,0))$$
- Case 3: it is analogous to Case 2 and we find $$mult_0(B(x,0))=mult_0(B^{'}(x,0))$$
There are correspondent equalities when $p$ is $\bf A_1$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$($mult_0(A^{'}_1(x,0))+d=mult_0(B(x,0))$) or $p$ is $\bf A_2$ for $\Phi_{F,H}$ and $\bf B$ for $\Phi_{F,G}$($mult_0(A^{'}_2(x,0))=mult_0(B(x,0)$).
Proof of Theorem \[main-thm\] {#sec-proof-thm}
=============================
Let us take some $C^{\infty}$ pertubation $\tilde C$ of $C$ and look to the curves $\Phi_{F,G}(\tilde C)$ and $\Phi_{F,H}(\tilde C)$, which are $C^{\infty}$ pertubations of $C_0$; we ask $\tilde C$ to be a holomorphic smooth curve with $({\tilde C}.{C})_p=1$ when passing through each $p\in {\mathcal Z_1}(\mathcal I^*,C) \cup {\mathcal Z_1}(\mathcal L^*,C)$ and ask also that $\Phi_{F,G}$ and $\Phi_{F,H}$ be holomorphic along these (local) holomorphic curves. Let us observe again that $\Phi_{F,G}$ is not a local biholomorphism at a point $p\in {\mathcal Z_1}(\mathcal I^*,C)$ (and $\Phi_{F,H}$ is not a local biholomorphism at a point $p\in {\mathcal Z_1}(\mathcal L^*,C)$ as well).
We proceed now to prove that for any $p\in {\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$ one has $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q \geq 2$$ for $q$ close to $\phi(p)$. Observe that in principle this number should be equal to $({\tilde C}.C)_{p}+ ({\tilde C}.C)_{P}=2$. Let us go back to the cases we discussed in the last Section.
- $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf A1$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$. The pertubation $\tilde C$ near $p$ has to be contained in some small sector around $C$, where $\Phi_{F,G}$ is holomorphic. Since $$\Phi_{F,G}=\left(x,\dfrac{yA_1(x,y)+xh(\xi(x,y))}{x+yA_1(x,y)}\right)$$ when we put $y={\epsilon}x$ (for $\tilde C$) we see that $\sum_{q} (\Phi_{F,G}({\tilde C}).C_0)_q$ is the number of solutions (near $\phi(p)$) to the equation $$\dfrac{\epsilon\,x\,A_1(x,\epsilon\,x)+xh(\xi(x,\epsilon\,x))}{x+\epsilon\,x\,A_1(x,\epsilon\,x)}=h(x)$$ which is $= mult_0(A_1(x,0))$. In order to estimate $\sum_{q} (\Phi_{F,H}({\tilde C}).C_0)_q$ we use $$\Phi_{F,H}(x,y)=(x,yB^{'}(x,y)+h(\xi(x,y)))$$ and we have to find the number of solutions of $$\epsilon\,x\,B^{'}(x,\epsilon\,x)+h(\xi(x,\epsilon\,x))=h(x)$$ (remember that now $p$ is of $B$ type for $\Phi_{F,H}$), which is readily seen to be $1+mult_0(B^{'}(x,0))$. We have seen before that $mult_0(B^{'}(x,0))= mult_0(A_1(x,0))+d$, so for $q$ close to $\phi(p)$: $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q= 2mult_0(A_1(x,0))+d+1$$ which is strictly bigger than 2 when $d>1$.
- $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf A2$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$. We have $$\Phi_{F,G}(x,y)=\left(x, \dfrac {yA_2(x,y) +h(\xi(x,y))}{1+yA_2(x,y)}\right)$$ and $$\Phi_{F,H}(x,y)=(x,yB^{'}(x,y)+h(\xi(x,y)))$$ Using again $y=\epsilon x$, we get $\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q=1+mult_0A_2(x,0)$ and $\sum_q(\Phi_{F,H}({\tilde C}).C_0)_{\phi(q)}=1+mult_0(B^{'}(x,0))$ Therefore for $q$ close to $\phi(p)$: $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q= 2+2mult_0(A_2(x,0))$$
- $p\in {\mathcal Z}_1(\mathcal I^*,C)$ is $\bf B$ for $\Phi_{F,G}$ and $\bf B$ for $\Phi_{F,H}$. Similarly we find for $q$ close to $\phi(p)$: $$\sum_q(\Phi_{F,G}({\tilde C}).C_0)_q + (\Phi_{F,H}({\tilde C}).C_0)_q= 2+2mult_0(B(x,0))$$
The remaining cases are analogous. We conclude that Case ${\bf A1}$ never appears and that Cases $\bf A2$ and $\bf B$ are present only at points $p$ where $\Phi_{F,G}$ is a local biholomorphisms.
Appendix: Automorphisms of a plane curve {#appendix}
========================================
We present a proof of the following theorem given by Jose Felipe Voloch.
\[Theorem1\] Let $C$ be a smooth plane curve of degree $d\geq 3$, then every automorphism of $C$ is linear, i.e. it comes from an element of $Aut(\mathbb{P}^2)$.
The case $d=3$ is a consequence of Legendre’s normal form, see [@HOU], so we focus on the case $d \geq 4$. Before proving the theorem we give some useful remarks and lemmas based on exercises $17$ and $18$ of [@AR]. We say that the set $S=\{p_1, \ldots, p_k\} \subseteq \mathbb{P}^2$ of distinct points **impose independent conditions on curves of degree $n$** if $h^0(\mathbb{P}^2, \mathcal{I}_S (n)) = h^0(\mathbb{P}^2, \mathcal{O}(n)) - k$.
Any set of $n+1$ points impose independent conditions on curves of degree $n$. On the other hand ${n+2}$ points impose independent conditions if and only if they are not aligned.
Take first $S=\{p_1, \ldots, p_{n+1}\}$ and denote $S_k =\{p_1, \ldots, p_k\}$. Taking the product of $n$ lines through another point we see that $H^0(\mathbb{P}^2, \mathcal{I}_{S_{i+1}} (n))$ is strictly contained in $H^0(\mathbb{P}^2, \mathcal{I}_{S_{i}} (n))$, therefore $h^0(\mathbb{P}^2, \mathcal{I}_{S} (n))= h^0(\mathbb{P}^2, \mathcal{O}(n)) - (n+1)$.
Consider now a set $S=\{p_1, \ldots, p_{n+1}, p_{n+2} \}$. If they are over a line $L$ and $E$ is a curve of degree $n$ passing through $n+1$ of them Bezout’s theorem implies $L \subseteq E$. This shows that $S$ fails to impose independent conditions on curves of degree $n$. Suppose now that every curve of degree $n$ passing by $n+1$ points contains also the other point of $S$. If they are not aligned, we can take for example the curve $E$ formed by lines joining $p_{n+1}$ with points $p_1, \ldots, p_{n}$, thus $p_{n+2}$ must be on this curve and we can assume that $p_n$, $p_{n+1}$ and $p_{n+2}$ are aligned. If some $p_j$, $j=1, \ldots, n-1$ is not on this line we consider $E'$ obtained from $E$ replacing $\overline{p_{n+1}, p_j}$ by a generic line passing by $p_{n+1}$, thus $E'$ contains $(n+1)$ points but not $S$, contradiction.
Let $D$ be an effective divisor on $C$ of degree $m$. We use previous lemma in order to study meromorphic functions on $C$ having $D$ as polar divisor. Changing the fiber if necessary we will assume from now that $D$ has not multiple points. We recall that $l(D)$ is the dimension of the space of meromorphic functions f such that $(f) +D \geq 0$ and $i(D)$ is the dimension of the space of holomorphic forms $\omega$ such that $(\omega) \geq D$.
If $m \leq d-2$ then $l(D)=1$.
Recall (see [@R]) that holomorphic $1-$forms on $C=\{P=0\}$ are generated by elements $\frac{x^i y^j}{P_y}dx $ with $i+j \leq d - 3$. By the previous lemma the dimension of the space of polynomials vanishing at $D$ is $g(C) - m$, thus $i(D) = g(C) -m$ and Riemann-Roch gives $l(D) = 1$.
If $m=d-1$ and $l(D) \geq2$ then $D= E- p$ where $p \in C$ and $E \in |\mathcal{O}_C(1)|$.
Once again Riemann-Roch theorem gives $i(D) = g-d + l(D) \geq g-(d-1)+1$ then points of $D$ do not impose independent conditions and they must be aligned. We conclude by noting that intersection of a line with $C$ is a divisor of degree $d$.
Finally we have
$|\mathcal{O}_C(1)|$ is the only linear system of degree $d$ and dimension $3$.
Let $D \in |\mathcal{O}_C(1)|$ be an aligned divisor of degree $d$ on $C$. Then points of $D$ fails to impose independent conditions on curves of degree $d-3$ and $i(D) = g(C) - (d-2)$ or equivalently $l(D) = 3$. If $A$ is another effective divisor of degree $d$ and $l(A)=3$ then any subset of $d-1$ points are aligned. We conclude that $A \in |\mathcal{O}_C(1)|$ and is linearly equivalent to $D$.
**Proof of Theorem \[Theorem1\]:** Let $\phi : C \rightarrow C$ be an automorphism of $C$. Last proposition implies that for any line $L$ on $\mathbb{P}^2$, points of $\phi(L \cap C)$ determine a line $L' \subseteq \mathbb{P}^2$, thus $\phi$ comes from an automorphism of $\check{\mathbb{P}}^2$ which corresponds to an element of $Aut(\mathbb{P}^2)$.
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---
abstract: |
The quadratic numerical range $W^2(A)$ is a subset of the standard numerical range of a linear operator which still contains its spectrum. It arises naturally in operators which have a $2 \times 2$ block structure, and it consists of at most two connected components, none of which necessarily convex. The quadratic numerical range can thus reveal spectral gaps, and it can in particular indicate that the spectrum of an operator is bounded away from $0$.
We exploit this property in the finite-dimensional setting to derive Krylov subspace type methods to solve the system $Ax = b$, in which the iterates arise as solutions of low-dimensional models of the operator whose quadratic numerical ranges is contained in $W^2(A)$. This implies that the iterates are always well-defined and that, as opposed to standard FOM, large variations in the approximation quality of consecutive iterates are avoided, although $0$ lies within the convex hull of the spectrum. We also consider GMRES variants which are obtained in a similar spirit. We derive theoretical results on basic properties of these methods, review methods on how to compute the required bases in a stable manner and present results of several numerical experiments illustrating improvements over standard FOM and GMRES.
author:
- 'Andreas Frommer, Birgit Jacob, Karsten Kahl, Christian Wyss, Ian Zwaan[^1]'
bibliography:
- 'qfom.bib'
title: 'Krylov type methods exploiting properties of the quadratic numerical range[^2]'
---
Introduction
============
It is well known that Krylov subspace methods for a linear system $Ax=b$ with a nonsingular matrix $A \in {\mathbb{C}}^{n \times n}$ tend to converge slowly or even diverge or fail in situations where $0$ lies in the “interior” of the spectrum $\sigma(A)$ of $A$. Specifically, if $0$ is contained in the numerical range (or field of values) of $A$, a convex set which contains $\sigma(A)$, we know that methods based on a Galerkin variational characterization like FOM, the full orthogonalization method, can fail due to the non-existence of certain iterates which manifests itself numerically by huge variations in magnitude and associated stability problems. In methods which are based on residual minimization like GMRES, the generalized minimal residual method, stagnation can occur in such cases. Related to this, classical convergence theory for Krylov subspace methods, in particular for the non-Hermitian case, typically assumes that $0$ is not contained in the numerical range and then gets quantitative results on convergence speed in which the distance of the numerical range to 0 enters as a parameter, see, e.g., [@EisenstatElmanSchultz1983; @SaadSchultz1986; @Starke97] and the discussion and references in the books [@LiesenStrakos2013; @Saad2003].
In this paper we study modifications of the FOM method, and also of GMRES, which converge stably and smoothly when the [*quadratic numerical range*]{}, a subset of the standard numerical range, splits into two parts which do not contain 0. The quadratic numerical range arises naturally for matrices which have a canonical $2 \times 2$ block structure. Analgously to standard Krylov subspace methods, these modifications are also based on projections. By projecting onto a larger space than the Krylov subspace we manage to preserve the gap in the quadratic numerical range and thus shield the projected matrices away from singularity. At the same time we do not require more matrix vector multiplications as in standard Krylov subspace methods, i.e. one per iteration.
This paper is organized as follows: Section \[numerical\_range:sec\] reviews those properties of the numerical range and the FOM and GMRES method which are important for the sequel. Section \[quadratic\_fom:sec\] first introduces the quadratic numerical range and then develops the new modified projection methods termed quadratic FOM and quadratic GMRES. This section also contains first elements of an analysis. In Section \[algorithm:sec\] we then discuss how the new methods can be realized as efficient algorithms before we give some numerical examples in Section \[numerics:sec\].
Numerical range and FOM {#numerical_range:sec}
=======================
Regardless of the dimension, $n$, we will always denote by $\langle \cdot,\cdot \rangle$ the standard sesquilinear inner product on ${\mathbb{C}}^n$ and $\| \cdot \|$ the associated norm. For a linear operator $A \in {\mathbb{C}}^{n \times n}$ the numerical range (or field of values) $W(A)$ is the set of all its Rayleigh quotients $$W(A) = \{ \tfrac{{\langle Ax, x\rangle}}{{\langle x, x\rangle}}: x \in {\mathbb{C}}^n, x \neq 0 \} = \left\{ {\langle Ax, x\rangle}: x \in {\mathbb{C}}^n, \| x \| = 1 \right\} .$$
$W(A)$ is a compact convex set (see [@HornJohnson2013], e.g.) which contains the spectrum ${\mathrm{spec}}(A)$. If $A$ is normal, $A^*A = AA^*$, then $W(A)$ is actually the convex hull of ${\mathrm{spec}}(A)$. For non-normal $A$, the numerical range $W(A)$ can be much larger than the convex hull of the spectrum. If for some $m \leq n$ the matrix $V = [v_1 \mid \cdots \mid v_m] \in {\mathbb{C}}^{n \times m}$ is an orthonormal matrix, i.e. $V^*V = I_m$, the identity on ${\mathbb{C}}^m$, then the numerical range of the “projected” matrix $V^*AV \in {\mathbb{C}}^{m \times m}$ is contained in that of $A$, since for all $y \in {\mathbb{C}}^m, y\neq 0$ we have ${\langle y, y\rangle} = {\langle Vy, Vy\rangle}$ and thus $$\tfrac{{\langle V^*AVy, y\rangle}}{{\langle y, y\rangle}} = \tfrac{{\langle AVy, Vy\rangle}}{{\langle y, y\rangle}} = \tfrac{{\langle AVy, Vy\rangle}}{{\langle Vy, Vy\rangle}} \in W(A).$$
For future use we state this observation as a lemma.
\[W\_enclosure:lem\] Let $A \in {\mathbb{C}}^{n \times n}$ be arbitrary and let $V \in {\mathbb{C}}^{n \times m}$ be orthonormal. Then $$W(V^*AV) \subseteq W(A).$$
We continue by summarizing the properties of two Krylov subspace methods, namely FOM [@Saad1981] GMRES [@SaadSchultz1986], which are relevant for this work. Proofs and further details can be found in [@Saad2003], e.g.
A Krylov subspace method for solving the linear system $$Ax=b, \enspace A \in {\mathbb{C}}^{n \times n}, b \in {\mathbb{C}}^n,$$ takes its $k$th iterate from the affine subspace $x^{(0)} + {\mathcal{K}}^{(k)}(A,r^{(0)})$, where $r^{(0)} = b-Ax^{(0)}$ and $${\mathcal{K}}^{(k)}(A,r^{(0)}) = {\mathrm{span}}\{r^{(0)},Ar^{(0)},\ldots,A^{k-1}r^{(0)}\}.$$ Krylov subspaces are nested and the Arnoldi process (see [@Saad2003], e.g.), iteratively computes an orthonormal basis $v^{(1)},v^{(2)},\ldots $ for these subspaces. Collecting the vectors into an orthonormal matrix $V^{(k)} = [v^{(1)} \mid \cdots \mid v^{(k)}]$, the Arnoldi process can be summarized by the Arnoldi relation $$\label{Arnoldi_relation:eq}
AV^{(k)} = V^{(k+1)} \underline{H}^{(k)}, k=1,2,\ldots.$$ where $\underline{H}^{(k)} \in {\mathbb{C}}^{(k+1) \times k}$ collects the coefficients resulting from the orthonormalization process. It has upper Hessenberg structure. Denoting by $H^{(k)}$ the $k \times k$ matrix obtained from $\underline{H}^{(k)}$ by removing the last row, we see that $$H^{(k)} = (V^{(k)})^*AV^{(k)}.$$ The full orthogonalization method (FOM) is the Krylov subspace method with iterate $x^{(k)}_{{\mathtt{fom}}}$ characterized variationally via $$x^{(k)}_{{\mathtt{fom}}} \in x^{(0)} + {\mathcal{K}}^{(k)}(A,r^{(0)}), \enspace r^{(k)}_{\mathtt{fom}}= b-Ax^{(k)}_{{\mathtt{fom}}} \perp {\mathcal{K}}^{(k)}(A,r^{(0)}),$$ which gives $$x^{(k)}_{{\mathtt{fom}}} = x^{(0)} + V^{(k)} (H^{(k)})^{-1} (V^{(k)})^* r^{(0)},$$ provided $H^{(k)}$ is nonsingular. Note that since $v_1$ is a multiple of $r^{(0)}$ we have $$\label{e1:eq}
(V^{(k)})^*r^{(0)} = \|r^{(0)}\|e_1^k,$$ where $e_1^k$ denotes the first canonical unit vector in $\mathbb C^k$.
For an arbitrary (nonsingular) matrix $A$, the matrix $H^{(k)}$ can become singular in which case the $k$-th FOM iterate does not exist. An important consequence of Lemma \[W\_enclosure:lem\] is therefore that such a breakdown of FOM cannot occur if $0 \not \in W(A)$, and, moreover, that $H^{(k)}$ will have no eigenvalues with modulus smaller than the distance of $W(A)$ to $0$. On the other hand, if $0 \in W(A)$, even when $H^{(k)}$ is nonsingular, it can become arbitrarily ill-conditioned, which then typically yields large residuals for the corresponding iterates and which is observed in practice as irregular convergence behavior. We can interprete FOM as the method which for each $k$ builds a reduced model $H^{(k)}$ of dimension $k$ of the original matrix and then obtains its iterate $x^{(k)}_{{\mathtt{fom}}}$ by lifting the solution of the corresponding reduced system $H^{(k)} \xi_k = (V^{(k)})^*r^{(0)}$ back to the full space as a correction to the initial guess $x^{(0)}$, $x^{(k)}_{{\mathtt{fom}}} = x^{(0)} + V^{(k)}\xi_k$. This interpretation will serve as a guideline for our development of the “quadratic” FOM method in section \[quadratic\_fom:sec\].
The generalized minimal residual method (GMRES) is the Krylov subspace method with iterate $x^{(k)}_{{\mathtt{gmres}}}$ characterized variationally via $$x^{(k)}_{{\mathtt{gmres}}} \in x^{(0)} + {\mathcal{K}}^{(k)}(A,r^{(0)}), \enspace r^{(k)}_{\mathtt{gmres}}= b-Ax^{(k)}_{{\mathtt{gmres}}} \perp A \cdot {\mathcal{K}}^{(k)}(A,r^{(0)}),$$ This implies that the residual $b-Ax^{(k)}_{{\mathtt{gmres}}}$ is smallest in norm among all possible residuals $b-Ax$ with $x \in x^{(0)} + {\mathcal{K}}^{(k)}(A,r^{(0)})$, i.e. $x^{(k)}_{\mathtt{gmres}}$ solves the least squares problem $$x^{(k)}_{\mathtt{gmres}}= {\mathrm{argmin}}_{x \in x^{(0)}+\mathcal{K}^{(k)}(A,r^{(0)})} \| b - Ax\| = x^{(0)} + {\mathrm{argmin}}_{y \in \mathcal{K}^{(k)}(A,r^{(0)})} \| r^{(0)}-Ay \|.$$ To obtain an efficient algorithm it is important to see that this $n \times k$ least squares problem can be reduced to a $(k+1)\times k$ system due to the Arnoldi relation : We have that $x^{(k)}_{{\mathtt{gmres}}} = x{_{}^{(0)}}+V^{(k)} \xi^{(k)}$ where $\xi^{(k)}$ solves $$\label{reduced_ls:eq}
\xi^{(k)}= {\mathrm{argmin}}_{\xi \in {\mathbb{C}}^k} \| (V^{(k+1)})^* r^{(0)} - \underline{H}^{(k)} \xi \|,$$ where $(V^{(k+1))^*}r^{(0)} = \|r^{(0)}\|e_1^{k+1}$. In case that $H^{(k)}$ is nonsingular, one can use the normal equation for to characterize $\xi_k = (\hat{H}^{(k)})^{-1} e_1^k$, where $$\label{reduced_model_gmres:eq}
\hat{H}^{(k)} = H^{(k)} + |h_{k+1,k}|^2((H^{(k)})^{-*}e_k)e_k^*, \mbox { where } h_{k+1,k} \mbox{ is the } (k+1,k) \mbox{ entry of } \underline{H}^{(k)}.$$ This means that the GMRES approach constructs a reduced model $\hat{H}^{(k)}$ which differs by the FOM model by a matrix of rank 1. The eigenvalues of $\hat{H}^{(k)}$ are called the [*harmonic Ritz values*]{} of $A$ w.r.t. ${\mathcal{K}}^{(k)}(A,r^{(0)})$, i.e. the values $\mu$ for which $$A^{-1}x - \tfrac{1}{\mu}x \perp A{\mathcal{K}}^{(k)}(A,r^{(0)}) \enspace \mbox{ for some } x \in A{\mathcal{K}}^{(k)}(A,r^{(0)}), x \neq 0.$$ They are the inverses of the Ritz values of $A^{-1}$ w.r.t the subspace $A{\mathcal{K}}(A,r^{(0)})$ which implies $$\mu^{-1} \in W(A^{-1}).$$ With $\rho$ denoting the numerical radius of $A^{-1}$, i.e. $\rho = \max \{ | \omega |: \omega \in W(A^{-1}) \}$ we see that $|\mu | \geq \rho^{-1}$. In this sense, as opposed to FOM, the GMRES approach shields the eigenvalues of the reduced model $\hat{H}^{(k)}$ away from $0$. Note that if $H^{(k)}$ is singular, GMRES stagnates, i.e. $x^{(k)}_{\mathtt{gmres}}= x^{(k-1)}_{\mathtt{gmres}}$.
Quadratic numerical range, QFOM and QGMRES {#quadratic_fom:sec}
==========================================
We now assume that $A \in {\mathbb{C}}^{n \times n}$ has a “natural” block decomposition of the form $$\label{A_block_structure:eq}
A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \enspace \mbox{ with } A_{ij} \in {\mathbb{C}}^{n_i \times n_j}, i,j=1,2, \, n_1+n_2 = n, \, n_1,n_2\ge 1.$$ All vectors $x$ from ${\mathbb{C}}^{n}$ are endowed with the same block structure $$x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}, \enspace x_i \in {\mathbb{C}}^{n_i}, i=1,2.$$ The definition of the quadratic numerical range goes back to [@LangerTretter1998], where it was introduced as a tool to localize spectra of block operators in Hilbert space.
\[ranges:def\] The quadratic numerical range $W^2$ of $A$ is given as $$W^2(A) = \bigcup_{\|x_1\| = \|x_2\| = 1} {\mathrm{spec}}\left( \begin{bmatrix} x_1^*A_{11}x_1 & x_1^*A_{12}x_2 \\ x_2^*A_{21}x_1 & x_2^*A_{22}x_2 \end{bmatrix} \right).$$
The following basic properties are, e.g., proved in [@TretterBook2008]
\[W2\_properties:lem\] We have
- $W^2(A)$ is compact,
- $W^2(A)$ has at most two connected components,
- ${\mathrm{spec}}(A) \subseteq W^2(A) \subseteq W (A),$
- If $n_1, n_2\ge 2$, then $W(A_{11}), W(A_{22}) \subseteq W^2(A)$.
The following counterpart of Lemma \[W\_enclosure:lem\] holds.
\[W2\_enclosure:lem\] Let $A \in {\mathbb{C}}^{n \times n}$ have block structure and assume that $V_1 \in {\mathbb{C}}^{n_1 \times m_1}, V_2 \in {\mathbb{C}}^{n_2 \times m_2}$ with $m_i \leq n_i, i=1,2$ have orthonormal columns. Put $V = [\begin{smallmatrix} V_1 & 0 \\ 0 & V_2 \end {smallmatrix} ] \in {\mathbb{C}}^{n \times m}$ with $m=m_1+m_2$. Then $$W^2(V^*AV) \subseteq W^2(A), \mbox{ where } V^*AV = \begin{bmatrix} V_1^*A_{11}V_1 & V_1^*A_{12}V_2 \\ V_2^*A_{21}V_1 & V_2^*A_{22}V_2 \end{bmatrix} \in {\mathbb{C}}^{m \times m}.$$
Let $y_i \in {\mathbb{C}}^{m_i}$ for $i=1,2$ with $\|y_i\| = 1$. Then $ x_i := V_{i}y_i$ satisfies $\|x_i\| = 1, i=1,2$, and since $$\begin{bmatrix} (y_1)^*V_1^*A_{11}V_1 y_1 & (y_1)^*V_1^*A_{12}V_2 y_2 \\ (y_2)^*V_2^*A_{21}V_1 y_1 & (y_2)^*V_2^*A_{22}V_2 y_2\end{bmatrix}
\ = \
\begin{bmatrix} x_1^*A_{11}x_1 & x_1^*A_{12}x_2 \\ x_2^*A_{21}x_1 & x_2^*A_{22}x_2 \end{bmatrix}$$ we obtain $W^2(V^*AV) \subseteq W^2(A)$.
Our approach is now to build a Krylov subspace type method where, as opposed to FOM, the iterates are obtained by inverting a reduced model of $A$ whose quadratic numerical range is contained in that of $A$. In this manner, if $0 \not \in W^2(A)$ with $\delta =
\min \{|\mu|: \mu \in W^2(A) \}$ denoting the distance of $0$ to $W^2(A)$, no eigenvalue of the reduced model will have modulus smaller than $\delta$. In cases where $0 \in W (A)$ and $0 \not \in W^2(A)$ this bears the potential of obtaining smoother and faster convergence than with FOM and, as it will turn out experimentally, also faster than with GMRES.
We project the Krylov subspace ${\mathcal{K}}^{(k)}(A,r^{(0)})$ onto its first $n_1$ and last $n_2$ components, respectivley, denoted ${\mathcal{K}}^{(k)}_1(A,r^{(0)}) \subseteq {\mathbb{C}}^{n_1}$ and ${\mathcal{K}}^{(k)}_2(A,r^{(0)}) \subseteq {\mathbb{C}}^{n_2}$. Clearly, $${\mathcal{K}}^{(k)}(A,r^{(0)}) \subseteq {\mathcal{K}}^{(k)}_1(A,r^{(0)}) \times {\mathcal{K}}^{(k)}_2(A,r^{(0)}) =: {\mathcal{K}}^{(k)}_\times(A,r^{(0)}),$$ and $\dim {\mathcal{K}}^{(k)}(A,r{_{}^{(0)}}) \leq \dim {\mathcal{K}}_{\times}^{(k)}(A,r^{(0)}) =: d_\times^{(k)} \leq 2k$. Note that the dimension $d_i^{(k)}$ of either ${\mathcal{K}}^{(k)}_i(A,r^{(0)})$ may be less than $k$ and that $d{_{\times}^{(k)}} = d{_{1}^{(k)}} + d{_{2}^{(k)}}$.
We can obtain an orthonormal basis for each of the ${\mathcal{K}}{_{i}^{(k)}}(A,r{_{}^{(0)}})$ as the columns of the matrix $V{_{i}^{(k)}}$ which arises from the QR-decomposition of the respective block of the matrix $V{_{}^{(k)}}$ from the Arnoldi process, i.e. $$\label{V_relations:eq}
V{_{}^{(k)}}\!=\! \begin{bmatrix} V{_{1}^{(k)}}R{_{1}^{(k)}} \\ V{_{2}^{(k)}}R{_{2}^{(k)}} \end{bmatrix},
V{_{i}^{(k)}} \in {\mathbb{C}}^{n_i \times d{_{i}^{(k)}}} \mbox{ orthonorm.}, \, R{_{i}^{(k)}} \in {\mathbb{C}}^{d{_{i}^{(k)}} \times k} \mbox{ upper triang.} $$
Note that with this definition of $V{_{i}^{(k)}}$ we have the useful property that $V{_{i}^{(k+1)}}$ arises from $V{_{i}^{(k)}}$ by the addition of a new last column, just in the way $V{_{}^{(k+1)}}$ arises from $V{_{}^{(k)}}$, with the exception that the new last column could be empty, i.e. there is no new last column, when the last column of the $i$th block in $V^{(k)}$ is linearly dependent of the other columns. Similarly $R{_{i}^{(k+1)}}$ arises from $R{_{i}^{(k)}}$ by adding a new last column and a new last row (if it is not empty).
We now introduce variational characterizations based on the space ${\mathcal{K}}_{\times}^{(k)}(A,r^{(0)}).$
QFOM
----
[*Quadratic FOM*]{} imposes a Galerkin condition using ${\mathcal{K}}_{\times}^{(k)}(A,r^{(0)})$.
\[qfom:def\] The $k$-th [*quadratic FOM (“QFOM”) iterate*]{} $x_{{\mathtt{qfom}}}^{(k)}$ is defined variationally through $$\label{qfom_variational:eq}
x_{{\mathtt{qfom}}}^{(k)} \in x^{(0)} + {\mathcal{K}}^{(k)}_\times(A,r^{(0)}),\enspace b-A x_{{\mathtt{qfom}}}^{(k)} \perp {\mathcal{K}}_{(\times)}^{(k)}(A,r^{(0)}).$$
The columns of the matrix $$V_\times^{(k)} = \begin{bmatrix} V^{(k)}_1 & 0 \\ 0 & V^{(k)}_2 \end{bmatrix}$$ form an orthonormal basis of ${\mathcal{K}}^{(k)}_\times(A,r^{(0)})$. Defining the reduced model $H_\times^{(k)}$ of $A$ as $$\label {W2_model:eq}
H_\times^{(k)} = (V{_{\times}^{(k)}})^* A V{_{\times}^{(k)}} = \begin{bmatrix} (V^{(k)}_1)^*A_{11}V^{(k)}_1 & (V^{(k)}_1)^*A_{12}V^{(k)}_2 \\ (V^{(k)}_2)^*A_{21}V^{(k)}_1 & (V^{(k)}_2)^*A_{22}V^{(k)}_2 \end{bmatrix}$$ we see that if $H_\times^{(k)}$ is nonsingular, the QFOM iterate $x^{(k)}_{{\mathtt{qfom}}}$ according to Definition \[qfom:def\] exists and can be represented as $$\label{qfom_representation:eq}
x_{{\mathtt{qfom}}}^{(k)} = x^{(0)} + V_\times^{(k)} (H_\times^{(k)})^{-1} (V_\times^{(k)})^* r^{(0)}.$$
Instead of we now have $$\label{e2:eq}
(V_\times^{(k)})^* r^{(0)} = \begin{bmatrix} \|r^{(0)}_1\| e^{d^{(k)}_1}_1 \\ \|r^{(0)}_2\|e^{d^{(k)}_2}_1 \end{bmatrix}, \mbox{ where } r^{(0)} = \begin{bmatrix} r^{(0)}_1 \\ r^{(0)}_2 \end{bmatrix}.$$ If $H_\times^{(k)}$ is singular, the $k$-th QFOM iterate does not exist. We will show in section \[algorithm:sec\] that computing $x^{(k)}_{{\mathtt{qfom}}}$ costs $k$ matrix-vector multiplications with $A$ plus additional arithmetic operations of order $\mathcal{O}(k^3)$. The cost is therefore the same as for standard FOM in terms of matrix-vector multiplications, and the additional cost is also of the same order (though with a larger constant).
Analysis of QFOM
----------------
The following theorem summarizes some basic properties of QFOM. Recall that the [*grade*]{} of a vector $v$ with respect to a square matrix $A$ is the first index $g(v)$ for which ${\mathcal{K}}^{(g(v))}(A,v) = {\mathcal{K}}^{(g(v)+1)}(A,v)$. We know (see [@Saad2003], e.g.) that then ${\mathcal{K}}^{(g(v))}(A,v) = {\mathcal{K}}^{(g(v)+i)}(A,v)$ for all $i \geq 0$ and that $A^{-1}v \in {\mathcal{K}}^{(g(v))}(A,v)$, provided $A$ is nonsingular.
\[QFOM\_properties:thm\] Let $A$ be nonsingular. Then
- There exists an index $k_{\max} \leq g(r{_{}^{(0)}})$ such that $A^{-1}r{_{}^{(0)}} \in {\mathcal{K}}{_{\times}^{(k_{\max})}}(A,r{_{}^{(0)}})$, and if $H^{(k_{\max})}_\times$ is nonsingular, $x_{{\mathtt{qfom}}}^{(k_{\max})}$ exists and $x_{\mathtt{qfom}}^{(k_{\max})} = A^{-1}b$.
- The inclusion $W^2(H_\times^{(k)} ) \subseteq W^2(A)$ holds for $k=1,\ldots,k_{\max}$, where the $2 \times 2$ block structure of $H_\times^{(k)}$ is given in .
- If $0 \not \in W^2(A)$, then $x_{\mathtt{qfom}}^{(k)}$ exists for $k=1,\ldots,k_{\max}$, i.e. $H_{k}^\times$ is nonsingular for all $k=1,\ldots,k_{\max}$.
To show (i), let $g$ be the grade of $r{_{}^{(0)}}$ w.r.t. $A$ and let $k_{\max}\leq g$ be the smallest index $k$ for which ${\mathcal{K}}{_{}^{(g)}}(A,r{_{}^{(0)}}) \subseteq {\mathcal{K}}{_{\times}^{(k)}}(A,r{_{}^{(0)}})$. Since $A$ is nonsingular, there exists $y^* \in {\mathcal{K}}{_{\times}^{(k_{\max})}}(A,r^{(0)})$ with $Ay^* = r{_{}^{(0)}}$, i.e. $y^* = A^{-1}r{_{}^{(0)}}$. As a consequence, $x^* = A^{-1}b = x{_{}^{(0)}}+ y^* \in x{_{}^{(0)}} + {\mathcal{K}}{_{\times}^{(k_{\max})}}(A,r{_{}^{(0)}}{})$ satisfies the variational characterization from Definition \[qfom:def\] just as $x{_{{\mathtt{qfom}}}^{(k_{\max})}}$ does. If $H^{(k_{\max})}_\times$ is nonsingular there is exactly one vector from $x^{(0)}+{\mathcal{K}}^{(k_{\max})}_{\times}(A,r^{(0)})$ which satisfies which gives $x_{{\mathtt{qfom}}}^{(k_{\max})} = x^*$.
Part (ii) follows directly from Lemma \[W2\_enclosure:lem\]. Finally, part (iii) is an immediate consequence of part (ii) and the spectral enclosure property stated as Lemma \[W2\_properties:lem\](iii).
More far-reaching results seem to be difficult to obtain. In particular, the absence of a polynomial interpolation property—which we discuss in the sequel—makes it impossible to follow established concepts from standard Krylov subspace theory.
The FOM iterates satisfy a polynomial interpolation property: We know that $(H^{(k)})^{-1} = q(H^{(k)})$ where $q$ is the polynomial of degree at most $k-1$ which interpolates the function $z \to z^{-1}$ on the eigenvalues in the Hermite sense, i.e. up to the $j-1$st deriviative if the multiplicity of the eigenvalue in the minimal polynomial is $j$; see [@Higham2008]. We have that $$V^{(k)}(H^{(k)})^{-1}(V^{(k)})^*r^{(0)} = V^{(k)}q(H^{(k)})(V^{(k)})^*r^{(0)} = q(A) r^{(0)},$$ where the last, important equality holds because $V^{(k)}(V^{(k)})^*$ represents the orthogonal projector on ${\mathcal{K}}_m(A,r^{(0)})$, thus implying that for all powers $j=0,\ldots,k-1$ we have $V^{(k)} (H^{(k)})^j (V^{(k)})^* r^{(0)} = V^{(k)}((V^{(k)})^*AV^{(k)})^j(V^{(k)})^*r^{(0)} = A^j r^{(0)}$. As a consequence $$\label{interpolation_property:eq}
x^{(k)}_{\mathtt{fom}}= x^{(0)} + q(A)r^{(0)}.$$ Since $\hat{H}^{(k)}$ differs from $H^{(k)}$ only in its last column, the same argument as above shows that an analogue of holds for the GMRES iterates, where now $q$ interpolates on the spectrum of $\hat{H}^{(k)}$. This interpolation property is very helpful in the analysis of the FOM and GMRES method, but there is no analog for QFOM. Indeed, while we can express $(H_\times^{(k)})^{-1}$ as a polynomial $q$ of degree at most $d_1^{(k)} + d_2^{(k)} -1 \leq 2k-1$ in $H_\times^{(k)}$, the matrix $V_\times^{(k)} (V_\times^{(k)})^*$ is an orthogonal projector on $K_\times^{(k)}(A,r^{(0)})$ which contains $K^{(k)}(A,r^{(0)})$ but not necessarily the higher powers $A^i r^{(0)}$ for $i \geq k$. Therefore, we cannot conclude that $(V_\times^{(k)}) (H_\times^{(k)})^i (V_\times^{(k)})^*r^{(0)} = (V_\times^{(k)}) ((V_\times^{(k)})^*AV_\times^{(k)})^i (V_\times^{(k)})^*r^{(0)}$ would be equal to $A^ir^{(0)}$ for $i \geq k$, and therefore, since the degree of the polyonomial $q$ is likely to be larger than $k-1$ don’t get $V_\times^{(k)} q(H_\times^{(k)})(V_\times^{(k)})^*r^{(0)} =
q(A)r^{(0)}$.
To finish this section, we look at the very extreme case in which $W^2(A)$ consists of just one or two points, and we show that in this case QFOM obtains the solution after just one iteration in a larger number of cases than standard FOM or GMRES does. So assume $W^2(A) = \{ \lambda_1,\lambda_2\}$, where $\lambda_1 = \lambda_2$ is allowed.
\[W2\_extreme:lem\] Let $n_1, n_2\ge 2$. $W^2(A) = \{\lambda_1, \lambda_2\}$ iff $$\label{form_A:eq}
A = \begin{bmatrix} \lambda_1 I & A_{12} \\A_{21} & \lambda_2 I \end{bmatrix}, \mbox{ where } A_{12} = 0 \mbox{ or } A_{21} = 0,$$ (up to a permutation of $\lambda_1$, $\lambda_2$ on the diagonal).
For $x_i \in {\mathbb{C}}^{n_i}, \|x_i\| = 1, i=1,2$ denote $$\alpha = x_1^*A_{11}x_1, \beta = x_1^*A_{12}x_2, \gamma = x_2^*A_{21}x_1, \delta = x_2^*A_{22}x_2.$$ Then $\lambda \in W^2(A)$ iff $$\label{quadratic:eq}
(\lambda-\alpha)(\lambda-\delta)-\beta\gamma = 0
$$ for $\alpha,\beta,\gamma,\delta$ associated with such $x_1,x_2$. Now, if $A$ is of the form , then $\beta\gamma = 0$, $\alpha=\lambda_1$ and $\delta=\lambda_2$, which immediately gives that is sufficient to get $W^2(A) = \{ \lambda_1,\lambda_2\}$.
To prove necessity, assume $W^2(A)= \{\lambda_1, \lambda_2\}$. Since $W(A_{ii}) \subseteq W^2(A)$ for $i=1,2$ by Lemma \[W2\_properties:lem\](iv) and since the numerical range is convex, this implies $W(A_{11})=\{\mu_1\}$, $W(A_{22})=\{\mu_2\}$ with $\mu_1,\mu_2\in \{\lambda_1,\lambda_2\}$. Consequently $A_{11} = \mu_1 I, A_{22} = \mu_2 I$. For a proof by contradiction assume now that both $A_{12}$ and $A_{21}$ are nonzero. Then there exist normalized vectors $x_1,x_2, y_1,y_2$ such that $x_1^*A_{12}x_2 \neq 0$ and $y_2^*A_{21} y_1 \neq 0$. For $\epsilon \in {\mathbb{R}}$, consider $z_1 = x_1 + \epsilon y_1, z_2 = x_2+ \epsilon y_2$. Then $z_1^*A_{12}z_2 \neq 0$ for $\epsilon \neq 0$ small enough and $$z_2^*A_{21}z_1 = x_2^*A_{21}x_1 + \epsilon (x_2^*A_{21}y_1 + y_2^*A_{21}x_1) + \epsilon^2 y_2^*A_{21}y_1.$$ This quadratic function in $\epsilon$ is nonzero for sufficiently small $\epsilon \neq 0$. Thus, for $\epsilon \neq 0$ sufficiently small, taking the normalized versions of $z_1,z_2$ we get that the corresponding $\beta$ and $\gamma$ are both nonzero. Consequently the expression $$(\lambda-\mu_1)(\lambda-\mu_2)-\beta\gamma$$ is nonzero for $\lambda=\mu_1\in W^2(A)$, but zero at the same time by . Thus at least one of the matrices $A_{12}, A_{21}$ is zero. It follows that $W^2(A)=\{\mu_1,\mu_2\}$ and consequently $\mu_1=\lambda_1$ and $\mu_2=\lambda_2$ up to a permutation of $\lambda_1$, $\lambda_2$.
With these preparations we obtain the following result.
Assume that $n_1, n_2\ge 2$ and $0\notin W^2(A) = \{\lambda_1,\lambda_2\}$ and consider the linear system $$A x = b.$$ Without loss of generality we assume that iterations start with the initial guess $x^{(0)} = 0$. We also denote by $x^{*} = A^{-1}b$ the solution of the system. Then
- $x_{\mathtt{fom}}^{(1)} = x^{*}$ if $b$ is an eigenvector of $A$. In all other cases, $x_{\mathtt{fom}}^{(2)} = x^{*}$.
- $x_{\mathtt{qfom}}^{(1)} = x^{*}$ if $A_ {12}b_2$ is collinear to $b_1$ (or $0$). In all other cases, $x_{\mathtt{qfom}}^{(2)} = x^{*}$.
By Lemma \[W2\_extreme:lem\] we know that $A$ has the form $$A = \begin{bmatrix} \lambda_1 I & A_{12} \\ 0 & \lambda_2 I \end{bmatrix} \mbox{ or }
A = \begin{bmatrix} \lambda_1 I & 0 \\ A_{21} & \lambda_2 I \end{bmatrix},$$ and we focus on the first case. The second case can be treated in a completely analogous manner. We first note that if $\lambda_1 \neq \lambda_2$, the eigenvectors to the eigenvalue $\lambda_1$ are of the form $\left[ \begin{smallmatrix} x_1 \\ 0 \end{smallmatrix} \right]$ and the eigenvectors to the eigenvalue $\lambda_2$ are given by $\left[ \begin{smallmatrix} (\lambda_2-\lambda_1)^{-1}A_{12}x_2 \\ x_2 \end{smallmatrix} \right]$. If $\lambda_1 = \lambda_2$, all vectors of the form $\left[ \begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \right] $ with $A_{12}x_2 = 0$ are eigenvectors. The theorem thus asserts that the situations where FOM gets the solution in the first iteration is a true subset of the situations in which QFOM obtains the solution in its first iteration.
To proceed, we observe that the minimal polynomial of $A$ is $p(z) = (z-\lambda_1)(z-\lambda_2)$ in all cases except for the case where $\lambda_1 = \lambda_2$ and $A_{12} = 0$, i.e. when $A = \lambda_1 I$ with minimal polynomial $p(z) = (z-\lambda_1)$. Since $x_{\mathtt{fom}}^{(1)} \in {\mathcal{K}}^{(1)}(A,b)$, which is spanned by $b$, FOM obtains the solution $x^{*}$ in the first iteration exactly in the case where $b$ is an eigenvector of $A$. If $b$ is not an eigenvector of $A$, then the minimal polynomial is $p(z) = (z-\lambda_1)(z-\lambda_2)$ so that the grade of $b$ is 2, and FOM obtains the solution §$x^{*}$ in its second iteration.
If $b_1 \neq 0$ and $b_2 \neq 0$, the first iteration of QFOM obtains $x^{(1)}_{\mathtt{qfom}}$ as $$\begin{aligned}
x^{(1)}_{\mathtt{qfom}}&=& \begin{bmatrix} \tfrac{1}{\|b_1\|}b_1 & 0 \\ 0 & \tfrac{1}{\|b_2\|}b_2 \end{bmatrix}
\begin{bmatrix} \lambda_1 & \tfrac{1}{\|b_1\|\, \|b_2\|}b_1^*A_{12}b_2 \\ 0 & \lambda_2 \end{bmatrix}^{-1} \begin{bmatrix} {\|b_1\|} \\ {\|b_2\|} \end{bmatrix} \\
&=&
\begin{bmatrix} \tfrac{1}{\|b_1\|}b_1 & 0 \\ 0 & \tfrac{1}{\|b_2\|}b_2 \end{bmatrix}
\begin{bmatrix} \tfrac{1}{\lambda_1} & -\tfrac{1}{\lambda_1\lambda_2}\tfrac{1}{\|b_1\|\, \|b_2\|}b_1^*A_{12}b_2 \\ 0 & \tfrac{1}{\lambda_2} \end{bmatrix} \begin{bmatrix} {\|b_1\|} \\ {\|b_2\|} \end{bmatrix} \\
&=&
\begin{bmatrix} \tfrac{1}{\lambda_1}(b_1-\tfrac{1}{\lambda_2}\tfrac{1}{\|b_1\|^2}b_1b_1^*A_{12}b_2) \\ \tfrac{1}{\lambda_2} b_2 \end{bmatrix},\end{aligned}$$ which is equal to the solution $$x^{*} = \left[ \begin{matrix} \tfrac{1}{\lambda_1}(b_1-\tfrac{1}{\lambda_2}A_{12}b_2) \\ \tfrac{1}{\lambda_2} b_2 \end{matrix} \right]$$ exactly when the projector $\tfrac{1}{\|b_1\|^2}b_1b_1^*$ acts as the identity on $A_{12}b_2$, i.e. when $A_{12}b_2$ is zero or collinear to $b_1$. A similar observation holds if $b_1 = 0$ or $b_2 = 0$. In all other cases, by Theorem \[QFOM\_properties:thm\] we have $x^{(2)}_{\mathtt{qfom}}= x^{*}$ since the grade of $b$ then equals 2.
QGMRES and QQGMRES
------------------
In principle, we can proceed in a manner similar to QFOM to derive a “quadratic” GMRES method. Variationally, its iterates $x{_{{\mathtt{qgmr}}}^{(k)}}$ would be characterized by $$\label{full_qgmres:eq}
x{_{{\mathtt{qgmr}}}^{(k)}} \in x^{(0)} + {\mathcal{K}}^{(k)}_\times(A,r^{(0)}), \enspace b-Ax{_{{\mathtt{qgmr}}}^{(k)}} \perp A {\mathcal{K}}^{(k)}_\times(A, r^{(0)}),$$ which is equivalent to minimizing the norm of the residual $\|b-Ax\|$ for $x \in x^{(0)} + K_\times^{(k)}(A,r^{(0)})$. Thus, as for standard GMRES, we can get $x^{(k)}_{\mathtt{qgmr}}$ as $x^{(0)} + V_\times^{(k)} \eta_k$ where $\eta_k$ solves the least squares problem $$\label{ls_qgmres:eq}
\eta_k = {\mathrm{argmin}}_{\eta \in {\mathbb{C}}^{d{_{\times}^{(k)}}}} \| r^{(0)}- AV^{(k)}_\times\eta \|.$$ However, as opposed to standard GMRES, it is not possible to recast this $n \times d{_{\times}^{(k)}}$ least squares problem into one with a reduced first dimension, since an analogon to the Arnoldi relation does not hold for the product spaces $\mathcal{K}_\times^{(k)}(A,r^{(0)})$. In particular, for $x^{(k)} \in x^{(0)} + {\mathcal{K}}^{(k)}_\times(A,r^{(0)})$, the residual $r^{(k)} = r^{(0)} - Ax^{(k)}$ need not be contained in $\mathcal{K}{_{\times}^{(k+1)}}(A,r^{(0)})$. This fact prevents approaches based on the variational characterization to be realized with cost depending exclusively on $k$ and not on $n$.
As an alternative, we thus suggest an approach similar to truncated GMRES (see [@Saad2003], e.g.). We project the $n \times d{_{\times}^{(k)}}$ least squares problem onto a $d{_{\times}^{(k+1)}} \times d{_{\times}^{(k)}}$ least squares problem by minimizing, instead of the whole residual $\|b-Ax^{(k)}\|$, only its orthogonal projection on $\mathcal{K}^{(k+1)}_\times(A,r^{(0)})$.
The $k$-th [*quadratic quasi GMRES (“QQGMRES”)*]{} iterate $x^{(k)}_{{\mathtt{qqgmr}}}$ is the solution of the least squares problem $$\label{ls1:eq}
x^{(k)}_{{\mathtt{qqgmr}}} = {\mathrm{argmin}}_{x \in x^{(0)} + {\mathcal{K}}{_{\times}^{(k)}}(A,b)} \| (V{_{\times}^{(k+1)}})^*(b - Ax) \|.$$
Computationally, we have that $x^{(k)}_{{\mathtt{qqgmr}}} = x^{(0)} + V_\times^{(k)} \zeta_k$, where $\zeta_k$ solves the $d{_{\times}^{(k+1)}} \times d{_{\times}^{(k)}}$ least squares problem $$\label{ls2:eq}
\zeta_k = {\mathrm{argmin}}_{\zeta \in {\mathbb{C}}^{d{_{\times}^{(k)}}}} \| (V^{(k+1)}_\times)^*r^{(0)} - (V^{(k+1)}_\times)^* A V_\times^{(k)} \zeta \|,$$ where $$\label{ls_reduced_model:eq}
\underline{H}{_{\times}^{(k)}} = (V{_{\times}^{(k+1)}})^* A V{_{\times}^{(k)}} = \begin{bmatrix} (V{_{1}^{(k+1)}})^*A_{11}V{_{1}^{(k)}} & (V{_{1}^{(k+1)}})^*A_{12}V{_{2}^{(k)}} \\ (V{_{2}^{(k+1)}})^*A_{21}V{_{1}^{(k)}} & (V{_{2}^{(k+1)}})^*A_{22}V{_{2}^{(k)}} \end{bmatrix}$$ and where the structure of $(V_{k+1}^\times)^*r^{(0)}$ is given in .
Analysis of QGMRES and QQGMRES
------------------------------
As for QFOM, there is no polynomial interpolation property for QGMRES nor for QQGMRES. We can again present only simple first elements of an analysis.
As solutions to least squares problems, the iterates $x{_{{\mathtt{qgmr}}}^{(k)}}$ and $x{_{{\mathtt{qqgmr}}}^{(k)}}$ are always defined. They are uniquely defined in case of QGMRES, since $AV{_{\times}^{(k)}}$ has full rank since $V{_{\times}^{(k)}}$ has full rank. For QQMRES we have
The matrix $\underline{H}{_{\times}^{(k)}}$ from has full rank if $0 \not \in W^2(A)$.
The matrix $\underline{H}{_{\times}^{(k)}}$ is obtained from $H{_{\times}^{(k)}}$ by complementing it with two rows, one after each block, and $H{_{\times}^{(k)}}$ is nonsingular by Theorem \[QFOM\_properties:thm\](iii). Thus, $\underline{H}{_{\times}^{(k)}}$ has full rank $d{_{\times}^{(k)}} = d{_{1}^{(k)}} + d{_{2}^{(k)}}$.
QGMRES and QQGMRES also both have a finite termination property.
Let $k_{\max} \leq g(r{_{}^{(0)}})$ be as in the proof of Theorem \[QFOM\_properties:thm\]. Then $x{_{{\mathtt{qgmr}}}^{(k_{\max})}} = A^{-1}b$. Provided $\underline{H}{_{\times}^{(k_{\max})}}$ has full rank, we also have $x{_{{\mathtt{qqgmr}}}^{(k_{\max})}} = A^{-1}b$.
As in the proof of Theorem \[QFOM\_properties:thm\], we have that $x^* = A^{-1}b = x^{(0)} + y^*$ with $y^* = A^{-1}r^{(0)}$ being contained in ${\mathcal{K}}_\times^{(k_{\max})}(A,r^{(0)})$. So $x^*$ satisfies the variational characterization with residual norm 0, and as such it is unique. This implies that $x^*$ is identical to the QGMRES iterate $x{_{{\mathtt{qgmr}}}^{(k_{\max})}}$. For QQGMRES, we write $y^* \in {\mathcal{K}}_\times^{(k_{\max})}(A,r^{(0)})$ as $y^* = V{_{\times}^{(k)}}\zeta$. This $\zeta$ is a solution of the least squares problem , yielding the minimal value 0 for the resiudal norm. If $\underline{H}{_{\times}^{(k_{\max})}}$ has full rank, the solution of the least squares problem is unique. And since the QQGMRES iterate $x{_{{\mathtt{qqgmr}}}^{((k_{\max})}}$ is obtained by solving this least squares problem, it is equal to $x^*$.
Trivially, QGMRES gets iterates $x{_{{\mathtt{qgmr}}}^{(k)}}$ whose residuals $r{_{{\mathtt{qgmr}}}^{(k)}}$ are smaller in norm than $r{_{{\mathtt{gmres}}}^{(k)}}$, i.e. the residual of the iterate $x{_{{\mathtt{gmres}}}^{(k)}}$ of standard GMRES, since QGMRES minimizes the residual norm over a larger subspace. Moreover, since QQGMRES minimizes over the same subspace as QGMRES, but minimizes the norm of the projection of the residual rather than the norm of the residual itself, we also have that $\|r{_{{\mathtt{qgmr}}}^{(k)}}\| \leq \|r{_{{\mathtt{qqgmr}}}^{(k)}}\|$. Finally, note that we cannot expect the relation $\|r{_{{\mathtt{qqgmr}}}^{(k)}}\| \leq \|r{_{{\mathtt{gmres}}}^{(k)}}\|$ to hold in general.
Algorithmic aspects {#algorithm:sec}
===================
An important practical question is how one can compute $V^{(k)}_\times$ and $H^{(k)}_\times$ efficiently and in a stable manner. Interestingly, for the special case where $A_{21} = I$ and $A_{22} = 0$, which arises in the linearization of quadratic eigenvalue problems, this question has been treated in many papers, and recently the [*two-level orthogonal Arnoldi method*]{} has emerged as a cost-efficient and at the same time stable algorithm; see [@KressnerRoman2014; @LuSuBai2016; @MeerbergenPerez2018]. In the following, we describe how the two-level orthogonal Arnoldi method generalizes to general $2\times 2$ block matrices with minor changes. Generalizing the stability analysis is not as straightforward, and a detailed analysis is beyond the scope of this paper. The main idea is that we refrain from directly computing the orthogonal Arnoldi basis $V{_{}^{(k)}}$ from , but rather compute/update the orthonormal bases $V{_{1}^{(k)}},V{_{2}^{(k)}}$ of its block components while at the same time updating $H{_{\times}^{(k)}}$.
Assume that no breakdown occurs and no deflation is necessary. Then we have (see ) $$V{_{}^{(k)}} = \begin{bmatrix} V{_{1}^{(k)}}R{_{1}^{(k)}} \\ V{_{2}^{(k)}} R{_{2}^{(k)}} \end{bmatrix},$$ where the $V{_{i}^{(k)}}$ have $k$ orthonormal columns, and the $R{_{i}^{(k)}} \in {\mathbb{C}}^{k \times k}$ are upper triangular. Since the columns of $V{_{}^{(k)}}$ are orthonormal, too, this implies $$\label{sum_R:eq}
(R{_{1}^{(k)}})^*R{_{1}^{(k)}} + (R{_{2}^{(k)}})^*R{_{2}^{(k)}} = (V{_{}^{(k)}})^*V{_{}^{(k)}} = I,$$ showing that the matrix $\left[ \begin{smallmatrix} R{_{1}^{(k)}} \\ R{_{2}^{(k)}} \end{smallmatrix} \right] \in {\mathbb{C}}^{2k \times k}$ also has orthonormal columns. Writing the Arnoldi relation in terms of the block components gives $$\label{recursion:eq}
\begin{split}
A_{11} V{_{1}^{(k)}} R{_{1}^{(k)}} + A_{12} V{_{2}^{(k)}} R{_{2}^{(k)}}
&= V{_{1}^{(k+1)}} R{_{1}^{(k+1)}} \underline H{_{}^{(k)}} =: V{_{1}^{(k+1)}} \underline{H}{_{1}^{(k)}}, \\
A_{21} V{_{1}^{(k)}} R{_{1}^{(k)}} + A_{22} V{_{2}^{(k)}} R{_{2}^{(k)}}
&= V{_{2}^{(k+1)}} R{_{2}^{(k+1)}} \underline H{_{}^{(k)}} =: V{_{2}^{(k+1)}} \underline{H}{_{2}^{(k)}},
\end{split}$$ where the matrices $$\underline{H}{_{i}^{(k)}} := R{_{i}^{(k+1)}} \underline H{_{}^{(k)}} \in {\mathbb{C}}^{(k+1)\times k},\qquad i=1,2,$$ are upper Hessenberg.
The relation reveals that $V{_{i}^{(k+1)}}$ can be obtained as an update of $V{_{i}^{(k)}}$ by adding a new last column, and $\underline H{_{i}^{(k)}}$ as an update of $\underline H{_{i}^{(k-1)}}$ by adding a new last column and a new last row. Thus, the new column of $V{_{i}^{(k+1)}}$ arises from the orthonormalization of the last column of $A_{i1} V{_{1}^{(k)}} R{_{1}^{(k)}} + A_{i2} V{_{2}^{(k)}} R{_{2}^{(k)}}$ against all columns of $V{_{i}^{(k)}}$ and it is nonzero. The upper-Hessenberg matrix $\underline H{_{1}^{(k)}}$ is obtained from $\underline H{_{1}^{(k-1)}}$ by first adding a new last row of zeros and then adding a new last column holding the coefficients from the orthonormalization. To obtain a viable computational scheme, it remains to show that $R{_{i}^{(k+1)}}$ as well as $\underline H{_{}^{(k)}}$ (which we need to get the QFOM or QGMRES iterates) can also be obtained from these quantities. We do so by establishing how to get them as updates from $H{_{}^{(k-1)}}$ and $R{_{i}^{(k)}}$, noting that in the very first step we have $$R{_{i}^{(1)}} = \|b_i\|,\qquad V{_{i}^{(1)}} = b_i/\|b_i\|,\qquad i=1,2,$$ unless $b_i = 0$ in which case we let the corresponding $R{_{i}^{(1)}}$ be zero and let $V{_{i}^{(1)}}$ be a random unitary vector. For $k > 1$ we write $$R{_{i}^{(k+1)}} =
\begin{bmatrix}
R{_{i}^{(k)}} & r{_{i}^{(k+1)}} \\
0 & \rho{_{i}^{(k+1)}}
\end{bmatrix}
\quad\text{and}\quad
\underline H{_{}^{(k)}} =
\begin{bmatrix}
\underline H{_{}^{(k-1)}} & h{_{}^{(k)}} \\
0 & \eta{_{}^{(k)}}
\end{bmatrix},$$ where $R{_{i}^{(k)}}$ and $\underline H{_{}^{(k-1)}}$ are known, and the remaining quantities are to be determined. Since $\underline H{_{i}^{(k)}}$ equals $$\label{HK:eq}
R{_{i}^{(k+1)}} \underline H{_{}^{(k)}}
= \begin{bmatrix}
R{_{i}^{(k)}} \underline H{_{}^{(k-1)}} & R{_{i}^{(k)}} h{_{}^{(k)}} + \eta{_{i}^{(k)}} r{_{i}^{(k+1)}} \\
0 & \eta{_{}^{(k)}} \rho{_{i}^{(k+1)}}
\end{bmatrix}
= \begin{bmatrix}
\underline H{_{i}^{(k-1)}} & h{_{i}^{(k)}} \\
0 & \eta{_{i}^{(k)}}
\end{bmatrix},$$ it follows, using , that $$\begin{aligned}
[(R{_{1}^{(k)}})^* \; 0] \underline H {_{1}^{(k)}} + [(R{_{2}^{(k)}})^* \;0] \underline H{_{2}^{(k)}}
&=& \big( (R{_{1}^{(k)}})^* [R{_{1}^{(k)}}\; r{_{1}^{(k+1)}}] + (R{_{2}^{(k)}})^* [R{_{2}^{(k)}}\; r{_{2}^{(k+1)}}] \big) \underline H{_{}^{(k)}} \\
&=& [I\; 0] \underline H{_{}^{(k)}}
\, = \, H{_{}^{(k)}}.\end{aligned}$$ Hence, we see that $$\label{h_update:eq}
h{_{}^{(k)}} = (R{_{1}^{(k)}})^* h{_{1}^{(k)}} + (R{_{2}^{(k)}})^* h{_{2}^{(k)}},$$ which allows for the computation of $h{_{}^{(k)}}$ from known quantities. Once $h{_{}^{(k)}}$ is known, can be used to compute $$\widetilde r{_{i}^{(k+1)}} = \eta{_{}^{(k)}} r{_{i}^{(k+1)}} = h{_{i}^{(k)}} - R{_{i}^{(k)}} h{_{}^{(k)}},$$ at which point $\eta{_{}^{(k)}}$ and the $\rho{_{i}^{(k)}}$ are the only remaining quantities to be determined. Letting $\eta{_{}^{(k)}}$ be real valued (and nonnegative) allows its computation in at least two different ways. The first is to consider the bottom right entry of which gives $$\begin{aligned}
(\eta{_{}^{(k)}})^2
&=& \|\eta{_{}^{(k)}} r{_{1}^{(k+1)}}\|^2 + |\eta{_{}^{(k)}} \rho{_{1}^{(k+1)}}|^2 + \|\eta{_{}^{(k)}} r{_{2}^{(k+1)}}\|^2 + |\eta{_{}^{(k)}} \rho{_{2}^{(k+1)}}|^2 \\
&=& \|\widetilde r{_{1}^{(k+1)}}\|^2 + |\eta{_{1}^{(k)}}|^2 + \|\widetilde r{_{2}^{(k+1)}}\|^2 + |\eta{_{2}^{(k)}}|^2.\end{aligned}$$ The second possibility is to determine $\eta{_{}^{(k)}}$ from the $(k+1,k+1)$ entry of the equality $(\underline H{_{}^{(k)}})^* \underline H {_{}^{(k)}} = (\underline H{_{1}^{(k)}})^* \underline H{_{1}^{(k)}} + (\underline H{_{2}^{(k)}})^* \underline H{_{2}^{(k)}}$, which results in $$(\eta{_{}^{(k)}})^2 + \| h{_{}^{(k)}}\|^2
= \|h{_{1}^{(k)}}\|^2 + |\eta{_{1}^{(k)}}|^2 + \| h{_{2}^{(k)}}\|^2 + |\eta{_{2}^{(k)}}|^2,$$ using . The first method may be preferred, since it guarantees that the computed $(\eta{_{}^{(k)}})$ is nonnegative, even with roundoff errors. Once $\eta{_{}^{(k)}}$ has been determined, we get $\rho{_{i}^{(k)}}$ as $\rho{_{i}^{(k)}} = \eta{_{i}^{(k)}}/\eta{_{}^{(k)}}$ from . Putting everything together yields the following proposition.
In iteration $k$, the quantities $V{_{i}^{(k+1)}}, R{_{i}^{(k+1)}}$ and $\underline H{_{i}^{(k)}}$ as well as $\underline H{_{}^{(k)}}$ can be obtained from those of iteration $k-1$ at cost comparable to one matrix-vector multiplication with $A$, $2k$ vector scalings and additions with vectors of length $n$ and additional $\mathcal O(k^2)$ arithmetic operations.
Computing the last column of $V{_{i}^{(k)}}R{_{i}^{(k)}}$ costs $k$ vector scalings and additions with vectors of length $n_i$ for $i=1,2$, which is comparable to $k$ scalings and additions with vectors of length $n$. Multiplication of these last columns with the $A_{ij}$ in amounts to one matrix vector multiplication with $A$. Orthogonalizing the two resulting blocks against all columns of $V{_{k}^{(i)}}$ costs again $k$ scalings and additions of vectors of size $n_1$ and $n_2$ which corresponds to additional $k$ such operations on vectors of length $n$. All other necessary updates as described before require $\mathcal{O}(k^2)$ operations.
In the standard Arnoldi process, when $\eta{_{}^{(k)}} = 0$, we know that we have reached the maximum size of the Krylov subspace, i.e. $k$ is equal to the grade of the initial residual $r{_{}^{(0)}}$, and that $A^{-1}b$ is contained in ${\mathcal{K}}{_{}^{(k)}}(A,r{_{}^{(0)}})$. Since by we have $\eta{_{i}^{(k)}} = \rho{_{i}^{(k)}} \eta{_{}^{(k)}}$, $i=1,2$, we see that the two-level orthogonal Arnoldi method also stops when $\eta{_{}^{(k)}} = 0$. However, the reverse statement need not necessarily be true, i.e. we can have $\eta{_{i}^{(k)}} = 0$ for $i=1,2$ without having $\eta{_{}^{(k)}} = 0$. This would represent a serious breakdown of the two-level orthogonal Arnoldi process. Of course, exact zeros rarely appear in a numerical computation, but near breakdowns should be dealt with appropriately. In our implementation, we simply chose to replace a block vector corresponding to some $\eta{_{i}^{(k)}} \approx 0$ by a vector with just random entries. This makes the book-keeping much easier, since then $d{_{i}^{(k)}} = k$ for all $k$ and $i=1,2$, while keeping $V{_{\times}^{(k)}}$ as a subspace of our approximation space.
The full algorithm is summarized in Algorithm \[alg:qkrylov\]. We assume no deflation is necessary and no breakdown occurs for simplicity, but we can deal with this in practice in two ways. When $\widetilde v{_{i}^{(k+1)}}$ is (numerically) linear dependent, we can either set $v{_{i}^{(k+1)}}$ to some random vector and set $\eta{_{i}^{(k)}}$ to zero, or we can set $V{_{i}^{(k+1)}} = V{_{i}^{(k)}}$ and $\underline H{_{i}^{(k)}} = [H{_{i}^{(k-1)}}\; h{_{i}^{(k)}}]$. The former approach requires less bookkeeping, but the latter approach can safe space and time. Another simplification compared to a practical implementation is the use of classical Gramm–Schmidt for the orthogonalization, instead of repeated Gram–Schmidt or modified Gram–Schmidt. However, the algorithm does show how to avoid unnecessary recomputation of quantities. In particular, we avoid recomputing matrix-vector products by updating the products $W{_{ij}^{(k)}} = A_{ij} V{_{j}^{(k)}}$, $Z{_{ij}^{(k,k)}} = (V{_{i}^{(k)}})^* A_{ij} V{_{j}^{(k)}}$, and $Z{_{ij}^{(k+1,k)}} = (V{_{i}^{(k+1)}})^* A_{ij} V{_{j}^{(k)}}$. Since this updating approach requires more memory, it should only be used if that extra memory is available, and if matrix-vector products with $A$ are sufficiently expensive.
[ ]{}
$\underline H{_{}^{(0)}} = []$ and $\beta = (\|b_1\|^2 + \|b_2\|^2)^{-1/2}$ $x{_{\texttt{qfom}}^{(k_{\max})}}$
From the pseudocode of the algorithm we can determine the computational cost per iteration as follows. We count one matrix-vector multiplication with each of the blocks $A_{11}$, $A_{12}$, $A_{21}$, and $A_{22}$, which equals one matrix-vector multiplication with $A$. Then we have an orthogonalization cost of $\mathcal O((n_1 + n_2)k) = \mathcal O(nk)$, which equals the orthogonalization cost in the standard Arnoldi process. Updating the $Z_{ij}$ costs $\mathcal O(nk)$ floating-point operations per iteration, but does not have an equivalent cost in Arnoldi. The same is true for updating the matrices $\underline H{_{}^{(k)}}$ and $R{_{i}^{(k+1)}}$ for $i = 1,2$, although the cost is limited to $\mathcal O(k)$ flops in this case. Computing $c{_{\texttt{qfom}}^{(k)}}$ and $d{_{\texttt{qfom}}^{(k)}}$ takes $\mathcal O(k^3)$ floating-point operations, while computing the approximation $x{_{\texttt{qfom}}^{(k)}}$ and its residual $r{_{\texttt{qfom}}^{(k)}}$ require $\mathcal O(nk)$. Clearly, computing the approximation and its residual is expensive, but there is no need to do it in every iteration. For example, in a restarted version of the QFOM algorithm, we may decide to compute them only once per restart, after the inner loop reaches $k_{\max}$. When we add everything together, we see that QFOM has the same asymptotic cost as FOM, although QFOM does require more memory.
With minor changes, we can change the code of Algorithm \[alg:qkrylov\] to compute the QQGMRES approximation instead of the QFOM approximation. One downside of QQGMRES is that we cannot guarantee that its approximation, or even the residual norm of its approximation, is better than that of GMRES. We can remedy this problem by interpolating between the GMRES and the QQGMRES solution. Let $r{_{\texttt{gmres}}^{(k)}} = b - A x{_{\texttt{gmres}}^{(k)}}$ and $r{_{\texttt{qqgmr}}^{(k)}} = b - A x{_{\texttt{qqgmr}}^{(k)}}$, then $$\begin{gathered}
\|b - A (\alpha x{_{\texttt{gmres}}^{(k)}} + (1-\alpha) x{_{\texttt{qqgmr}}^{(k)}})\|^2
= \|\alpha r{_{\texttt{gmres}}^{(k)}} + (1-\alpha) r{_{\texttt{qqgmr}}^{(k)}}\|^2 \\
= \alpha^2 \|r{_{\texttt{gmres}}^{(k)}} - r{_{\texttt{qqgmr}}^{(k)}}\|^2 + 2\alpha (\Re\{(r{_{\texttt{gmres}}^{(k)}})^* r{_{\texttt{qqgmr}}^{(k)}}\} - \|r{_{\texttt{qqgmr}}^{(k)}}\|^2) + \|r{_{\texttt{qqgmr}}^{(k)}}\|^2.\end{gathered}$$ Hence, the residual norm of the interpolated approximation is minimized for $$\alpha_{\texttt{opt}} = \frac{\|r{_{\texttt{qqgmr}}^{(k)}}\|^2 - \Re\{(r{_{\texttt{gmres}}^{(k)}})^* r{_{\texttt{qqgmr}}^{(k)}}\}}{\|r{_{\texttt{gmres}}^{(k)}} - r{_{\texttt{qqgmr}}^{(k)}}\|^2}$$ if $r{_{\texttt{gmres}}^{(k)}} \neq r{_{\texttt{qqgmr}}^{(k)}}$. The residual norm of the approximation $x{_{\texttt{opt}}^{(k)}}$ corresponding $\alpha_\texttt{opt}$ is $$\|r_{\texttt{opt}}\|^2
= \frac{\|r{_{\texttt{gmres}}^{(k)}}\|^2 \|r{_{\texttt{qqgmr}}^{(k)}}\|^2 - \Re\{(r{_{\texttt{gmres}}^{(k)}})^* r{_{\texttt{qqgmr}}^{(k)}}\}^2}{\|r{_{\texttt{gmres}}^{(k)}} - r{_{\texttt{qqgmr}}^{(k)}}\|^2},$$ and satisfies $\|r_{\texttt{opt}}\| \le \min \{ \|r_{\texttt{gmres}}\|, \|r_{\texttt{qqgmr}}\| \}$.
Numerical experiments {#numerics:sec}
=====================
The Hain-Lüst operator
----------------------
Hain-Lüst operators appear in magnetohydrodynamics [@HainLuest58], and their spectral properties, in particular their quadratic numerical range, were investigated in a series of papers, e.g., in [@LangerTretter1998; @MuhammadMarletta12; @MuhammadMarletta13]. We consider the Hain-Lüst operator $${\mathcal{A}}= \begin{bmatrix} -{\mathcal{L}}& I \\ I & q \end{bmatrix}$$ acting on $L^2([0,1])\times L^2([0,1])$ where ${\mathcal{L}}=d^2/dx^2$ is the Laplace operator on $[0,1]$ with Dirichlet boundary conditions, $I$ is the identity operator, and $q$ denotes multiplication by the function $q(x)=-3+2e^{2\pi ix}$. The domain of ${\mathcal{A}}$ is $D({\mathcal{A}})=(H^2([0,1])\cap H^1_0([0,1])) \times L^2([0,1])$.
We consider a discretization of ${\mathcal{A}}$, approximating function values at an equispaced grid for both blocks, i.e. we take $x_j=jh$, $j=0,\dots,N+1$, $h=1/(N+1)$ and obtain, using finite differences, the discretized Hain-Lüst operator $$A = \begin{bmatrix} \frac{1}{h^2}L & I \\ I & Q \end{bmatrix} \in {\mathbb{C}}^{2N\times 2N},$$ with $L = \mbox{tridiag}(-1,2,-1) \in {\mathbb{C}}^{N \times N}$ and $ Q = -3I + 2{\mathrm{diag}}(e^{2h\pi i}, \dots, e^{2hN\pi i}) \in {\mathbb{C}}^{N \times N}$, see [@MuhammadMarletta13] for more details.
Note that $\tfrac{1}{h^2}L$ is Hermitian and that $Q$ is normal, so the numerical ranges of these diagonal blocks of $A$ satisfy $$\begin{aligned}
W_1 &:=& W(\tfrac{1}{h^2} L ) \, =\, \tfrac{1}{h^2}[2-2\cos(\pi h), 2 + 2\cos(\pi h)] \, =: \, [\alpha_{\min}(h), \alpha_{\max}(h)], \\
W_2 &=& W(Q) \, =\, {\mathrm{conv}}\{-3+2e^{2\pi h j}, j=1,\ldots,N\} \subseteq C(-3,2),\end{aligned}$$ where $C(-3,2)$ is the circle with center $-3$ and radius $2$. Since both numerical ranges $W_1$ and $W_2$ are contained in the convex set $W(A)$ we see that $0 \in W(A)$. The following argumentation shows that, with the possible exception of very large values for $h$, we have $0 \not \in W^2(A)$: Any $\lambda \in W^2(A)$ satisfies $$\label{2x2:eq}
(\lambda-x_1^*\tfrac{1}{h^2}Lx_1)(\lambda-x_2^*Qx_2) = (x_1^*x_2)(x_2^*x_1),$$ for some $x_1,x_2$ with $\|x_1\| = \|x_2\| = 1$. Assume that $\lambda$ lies within the strip $a < \Re(\lambda) < b$ with $-1 < a <0$ and $0<b< \alpha_{\min}(h)$. Then we have ${\textcolor{SunsetRedL}{d(\lambda,W_1) > \alpha_{\min}(h)-b}}$ as well as ${\textcolor{SunsetRedL}{d(\lambda,W_2) > a+1}}$ for the distances of $\lambda$ to the sets $W_1,W_2$. Taking absolute values in and using the bound $|x^*_1x_2| \leq 1$ we thus see that $\lambda$ from this strip cannot be in $W^2(A)$ if $(a+1)(\alpha_{\min}(h)-b) > 1$. This is the case, for example, if $b < \alpha_{\min}(h) -2$ and $a > -\tfrac{1}{2}$. Note that $\lim_{h \to 0}\alpha_{\min}(h) = \pi^2$.
+ table \[x=RST, y=FOM\] ;
+ table \[x=RST, y=GMRES\] ;
+ table \[x=RST, y=QFOM\] ;
+ table \[x=RST, y=QQGMRES\] ;
+ table \[x=RST, y=InterpQQGMRES\] ;
+ table \[x=RST, y=FOM\] ;
+ table \[x=RST, y=GMRES\] ;
+ table \[x=RST, y=QFOM\] ;
+ table \[x=RST, y=QQGMRES\] ;
+ table \[x=RST, y=InterpQQGMRES\] ;
In all our examples we chose the right hand side $b$ as $b = Ae$ where $e$ is the vector of all ones, and our initial guess is always $x^0 = 0$. Figure \[fig:hl1\] shows convergence plots for FOM, GMRES, QFOM, QQGMRES and the interpolated QQGMRES method as described at the end of Section \[algorithm:sec\]. The figure displays the relative norm of the residual as a function of the invested matrix-vector multiplications. In the left part, we took $N=1\,023$, the right part is for $N = 16\,383$. We restarted every method after $m=50$ iterations to avoid that the arithmetic work and the storage related with the (two-level) Arnoldi process becomes too expensive. Note that the figure displays the residual norms at the end of each cycle only, which makes the convergence of some of the methods, in particular FOM, to appear smoother than it actually is. Two major observations can be made: On the one side, the FOM type methods yield significantly larger residals than the GMRES type methods. For $N=1\,023$, the “quadratic methods” still make progress in the later cycles while their “non-quadratic” counter parts then basically stagnate. There is no such difference visible for dimension $N=16\,383$; convergence for all methods is very slow. In a second numerical experiment we therefore report results of a geometric multigrid method as an attempt to cope with large condition numbers. For a given discretization with step size $h=1/(N+1)$ with $N+1 =
2^k$ we construct the system at the next coarser level to be the discretizaton with $h_c = 2h = 1/(N_c+1)$ with $N_c+1 = 2^{k-1}$. We stop descending the grid hierarchy when we reach $N=7$, where we solve the corresponding $14 \times 14$ system by explicit inversion of $A$. Interpolation between two levels of the grid hierarchy is done using standard linear interpolation from the neighboring grid points; restriction is the standard adjoint of interpolation. For the smoothing iteration we test one or five steps of standard GMRES versus one or two steps of QFOM. We always performed V-cycles with pre-smoothing. The left part of Figure \[hl2:fig\] gives the resulting convergence plots for the multigrid methods for $N=1\,023$, the right part for $N=16\,383$.
+ table \[x=it, y=gmresfive\] ;
+ table \[x=it, y=gmresone\] ;
+ table \[x=it, y=qfomtwo\] ;
+ table \[x=it, y=qfomone\] ;
+ table \[x=it, y=gmresfive\] ;
+ table \[x=it, y=gmresone\] ;
+ table \[x=it, y=qfomtwo\] ;
+ table \[x=it, y=qfomone\] ;
From these plots it is apparent that QFOM is a well-working smoothing iteration for the multigrid method, whereas GMRES is not, even not for larger numbers of smoothing steps per iteration. As a complement to these results, Figure \[hl3:fig\] illustrates the mesh size independence of the convergence behavior of the multigrid method with QFOM smoothing. It shows that the number of iterations required to reduce the initial residual by a factor of $10^{-12}$ is basically independent of $h$.
+ table \[x=N, y=it\] ;
The Schwinger model
-------------------
Our second example is the Schwinger model in two-dimensions that arises in computations of quantum electrodynamics (QED). QED models the interactions of electrons and photons and is oftentimes used as a simpler model problem for the $4$-dimensional problems of quantum chromodynamics (QCD). It is a quantum field theory, meaning that physical quantities arise as expected values of solutions of partial differential equations whose coefficients are coming from the quantum background field, i.e., they are stochastic quantities obeying a given distribution. The Schwinger model is a discretization of the Dirac equation $$\mathcal{D}\psi = \left(\sigma_1 \otimes \left(\partial_x + A_x\right) + \sigma_2 \otimes \left(\partial_y + A_y \right)\right)\psi = \varphi,$$ on a regular, 2-dimensional $N \times N$ cartesian lattice, where the spin structure[^3] is encoded by the Pauli matrices $$\sigma_1 = \begin{pmatrix} & 1 \\ 1 & \end{pmatrix},\ \ \sigma_2 = \begin{pmatrix} & i \\ -i & \end{pmatrix} \text{\ \ and\ \ } \sigma_3 = \begin{pmatrix} 1 & \\ & -1 \end{pmatrix}$$ and $A_{\mu}$ encodes the background gauge field. In the Schwinger model we have $A_{\mu} \in \mathbb{R}$. Using a central covariant finite difference discretization for the first order derivatives, and introducing a scaled second-order stabilization term one writes the action of the discretized operator $D \in \mathbb{C}^{2N^2 \times 2N^2}$ of the Schwinger model at any lattice site $x$ on a spinor $\psi(x) \in \mathbb{C}^{2}$ as $$\label{schwinger1:eq}
\left.
\begin{array}{rcl} \displaystyle
\left(D\psi\right)(x) & = & \left(m_0 + 2\right)\psi(x) \\
\displaystyle & & \displaystyle \mbox{} + \frac{1}{2}\sum_{\mu \in \{x,y\}} \left(\left(I-\sigma_{\mu}\right)\otimes U_{\mu}(x)\right)\psi(x+e_{\mu}) \\
\displaystyle & & \displaystyle\mbox{} + \frac{1}{2}\sum_{\mu \in \{x,y\}} \left(\left(I+\sigma_{\mu}\right)\otimes \overline{U_{\mu}(x-e_{\mu})}\right)\psi(x-e_{\mu}).
\end{array}
\right\}$$ In here [$U_\mu$ correspond to a discrete version of the stochastically varying gauge field with $ U_\mu(x) \in \mathbb{C}, |U_\mu(x)| = 1$ for all $x$]{}, and $m_0$ sets the mass of the simulated theory. The naming convention of this formula is depicted in \[fig:namingconvention\], and [we refer to the textbook [@GattLang2010], e.g., for further details.]{}
[The canonical $2\times 2$ block structure of the Schwinger model matrix arises from the spin structure: We reorder the unknowns in $\psi$ according to spin, i.e., we take $$\psi = \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix},$$ where $\psi_1 \in \mathbb{C}^{N^2}$ collects all the spin 1 components $\psi_1(x)$ of $\psi(x) = \left[ \begin{smallmatrix} \psi_1(x) \\ \psi_2(x) \end{smallmatrix} \right] \in \mathbb{C}^2$ at all lattice sites, and similarly for $\psi_2$. Then the reordered discretized Schwinger model matrix, acting on the reordered vector $\left( \begin{smallmatrix} \psi_1(x) \\ \psi_2(x) \end{smallmatrix} \right) $, is given as $$ D = \begin{pmatrix} A & B \\ -B^{*} & A \end{pmatrix}.$$ ]{}
Here, the diagonal blocks $A$ correspond to the discretized second order stabilization term and are thus called gauge Laplace operators, while the off-diagonal blocks $B$ correspond to the central finite covariant difference discretization of the Dirac equation. Using we see that the action of [the blocks $A$ and $B$ on a vector $\psi_1, \psi_2$ is given as]{} $$\begin{aligned}
(A \psi_{1})(x) &= (m_0 + 2) \psi_{1}(x)
\begin{aligned}[t]
&- \frac{1}{2} \sum_{\mu \in \{x,y\} } U_{\mu}(x) \psi_1(x+e_{\mu}) \\
&- \frac{1}{2} \sum_{\mu \in \{x,y\} } \overline{U_{\mu}(x-e_{\mu})}\psi_1(x-e_{\mu}),
\end{aligned}
\\
(B \psi_{2})(x) &=
\begin{aligned}[t]
&- \frac{1}{2} \left( U_{x}(x) \psi_1(x+e_{x}) + i \cdot U_{y}(x) \psi_1(x+e_{y})\right)\\
&+ \frac{1}{2} \left(\overline{U_{x}(x-e_{x})}\psi_1(x-e_{x}) - i \cdot \overline{U_{y}(x-e_{y})} \psi_1(x-e_{y})\right).
\end{aligned}\end{aligned}$$ From this we see that the mass parameter $m_0$ induces a shift by a multiple of the identity in $A$, which we make explicit in writing $A = A_0 + m_0I$.
In our tests we consider the “symmetrized” operator $Q := \Sigma_3 D$ with $\Sigma_3 = \sigma_3 \otimes I_{N \cdot N}$. Due to $A^{*} = A, B^{*} = -B$ this operator $$Q = \begin{pmatrix} A & B \\ B^* & -A \end{pmatrix} = \begin{pmatrix} A_0 + m_0I & B \\ B^* & -A_0-m_0I \end{pmatrix}$$ is hermitian, but indefinite.
The quadratic range $W_2(Q)$ has two connected components to the left and right of $0$ on the real axis, provided $m_0 > -\alpha_{\min}$, the smallest eigenvalue of $A_0$. This can be seen as follows: Let $x_1, x_2 \in {\mathbb{C}}^{N \times N }$ be two normalized vectors and let $$\begin{pmatrix} x_1^*Ax_1 & x_1^*Bx_2 \\ x_2^*B^*x_1 & -x_2^*Ax_2 \end{pmatrix} =: \begin{pmatrix} \alpha_1 & \beta \\ \overline{\beta} & -\alpha_2 \end{pmatrix}.$$ Then any eigenvalue $\lambda$ of this matrix satisfies $$\begin{aligned}
&& (\lambda-\alpha_1)(\lambda+\alpha_2) \, = \, |\beta|^2 \\
&\Longrightarrow & (\Re(\lambda)-\alpha_1)(\Re(\lambda)+\alpha_2) = |\beta|^2 + \Im(\lambda)^2.\end{aligned}$$ The last equality cannot be satisfied if $-\alpha_2 < \Re(\lambda) < \alpha_1$. In particular, if $m_0 > -\alpha_{\min}$, the equality cannot be satisfied if $|\Re(\lambda)| < m_0 +\alpha_{\min}$, since $\alpha_1, \alpha_2 \geq m_0 +\alpha_{\min}$.
For our tests we use a gauge configuration obtained by a heatbath algorithm excluding the fermionic action, which results in the smallest eigenvalue $\alpha_{\min}$ of $A_0$ being approximately $0.11$. \[fig:schwinger\] reports results for two different choices of $m_0$. As in the first example we perform a restart after every $50$ iterations. The first choice for $m_0$ is $m_0 = -0.1 > -\alpha_{\min}$, so that the quadratic range indeed has two connected components with a gap around $0$. The second is $m_0 = -0.22 < -\alpha_{\min}$, so that $W^2(Q)$ consists of only one component containing $0$. The figure shows that a marked improvement can be observed for the “quadratic” methods if the quadratic range consists indeed of two different connected components (left plot), whereas this advantage is lost to a large extent for the second choice for $m_0$, where $W^2(Q)$ does not indicate a spectral gap (right plot). In this case, the system is also severely ill-conditioned, so that the convergence of all methods considered is much slower. We also note that for this example and for both choices for $m_0$, interpolated QQGMRES does not differ substantially from standard GMRES. Without showing the corresponding convergence plots, let us at least mention that when decreasing $m_0$ from $-0.1$ to $-0.22$ we observe for a long time a convergence behavior very similar to that for the largest value $-0.1$, even when $m_0$ is already smaller than $-\alpha_{\min}$.
+ table \[x=RST, y=FOM\] ;
+ table \[x=RST, y=GMRES\] ;
+ table \[x=RST, y=QFOM\] ;
+ table \[x=RST, y=QQGMRES\] ;
+ table \[x=RST, y=InterpQQGMRES\] ;
+ table \[x=RST, y=FOM\] ;
+ table \[x=RST, y=GMRES\] ;
+ table \[x=RST, y=QFOM\] ;
+ table \[x=RST, y=QQGMRES\] ;
+ table \[x=RST, y=InterpQQGMRES\] ;
[^1]: Bergische Universität Wuppertal, Wuppertal, Germany .
[^2]: This version dated .
[^3]: The $\sigma$-matrices are generators of a Clifford algebra and arise in the derivation of the Dirac equation from the Klein-Gordon equation. They give rise to the internal spin (i.e., angular momentum) degrees of freedom of the fields $\psi$ [@GattLang2010]. Note that although our discussion is limited to this particular choice of generators, all the results that follow extend to any other of the admissible choices of the $\sigma$-matrices.
|
---
abstract: 'In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$, fully described by Dajczer and Florit [@DF2] in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal $n$-dimensional submanifolds in space forms of any codimension are locally cylinders provided that they carry a totally geodesic distribution of rank $n-2\geq2,$ which is contained in the relative nullity distribution, such that the length of the mean curvature vector field is constant along each leaf. The case of dimension $n=3$ turns out to be special. We show that there exist elliptic three-dimensional submanifolds in spheres satisfying the above properties. In fact, we provide a parametrization of three-dimensional submanifolds as unit tangent bundles of minimal surfaces in the Euclidean space whose first curvature ellipse is nowhere a circle and its second one is everywhere a circle. Moreover, we provide several applications to submanifolds whose mean curvature vector field has constant length, a much weaker condition than being parallel.'
author:
- 'A.E. Kanellopoulou and Th. Vlachos'
title: On the mean curvature of submanifolds with nullity
---
Introduction
============
A fundamental concept in the theory of submanifolds is the index of relative nullity introduced by Chern and Kuiper [@chrn]. At a point $x\in M^n$ the *index of relative nullity* $\nu(x)$ of an isometric immersion $f\colon M^n\to{\mathbb{Q}}_c^m$ is the dimension of the *relative nullity* tangent subspace $\D_f(x)$ of $f$ at $x$, that is the kernel of the second fundamental form $\alpha^f$ at that point. Here, $\mathbb{Q}_c^m$ is the simply connected space form with curvature $c$, that is, the Euclidean space ${\mathbb{R}}^m,$ the sphere $\mathbb{S}^m$ or the hyperbolic space $\mathbb{H}^m$, according to whether $c=0,c=1$ or $c=-1,$ respectively. The kernels form an integrable distribution along any open subset where the index is constant and the images under $f$ of the leaves of the foliation are totally geodesic submanifolds in the ambient space.
Cylinders are the simplest examples of submanifolds with positive index of relative nullity. An isometric immersion $f\colon M^n\to {\mathbb{R}}^m$ is said to be a $k$-*cylinder* if the manifold $M^n$ splits as a Riemannian product $M^n=M^{n-k}\times{\mathbb{R}}^k$ and there is an isometric immersion $g\colon M^{n-k}\to{\mathbb{R}}^{m-k}$ such that $f=g\times{\rm{id}}_{{\mathbb{R}}^k}$. A frequent theme in submanifold theory is to find geometric conditions for an isometric immersion with index of relative nullity $\nu\geq k>0$ at any point to be a $k$-cylinder.
A fundamental result asserting that an isometric immersion $f\colon M^n\to{\mathbb{R}}^m$ of a Riemannian manifold with positive index of relative nullity must be a cylinder is Hartman’s theorem [@har] that requires the Ricci curvature of $M^n$ to be nonnegative. Even for hypersurfaces, the same conclusion does not hold if instead we assume that the Ricci curvature is nonpositive. Notice that the latter is always the case if $f$ is a minimal immersion. Counterexamples easy to construct are the complete irreducible ruled hypersurfaces of any dimension discussed in [@dg p. 409].
The cylindricity of minimal submanifolds was studied in [@hsv; @D] under global assumptions. These results are truly global in nature since there are plenty of (noncomplete) examples of minimal submanifolds of any dimension $n$ with constant index $\nu=n-2$ that are not part of a cylinder on any open subset. They can be all locally parametrically described in terms of a certain class of elliptic surfaces (see [@DF2 Th. 22]). Some of the many papers containing characterizations of submanifolds as cylinders without the requirement of minimality are [@dg0; @gf; @har; @ma].
In this paper, we deal with nonminimal $n$-dimensional submanifolds of arbitrary codimension and index of relative nullity $\nu\geq n-2$ at any point. Our aim is to provide geometric conditions, in terms of the mean curvature, for an isometric immersion to be a cylinder. The choice of the geometric condition is inspired by the observation that cylinders are endowed with a totally geodesic distribution contained in the relative nullity distribution, such that the mean curvature is constant along each leaf. Throughout the paper, the *mean curvature* of an isometric immersion $f$ is defined as the length $H=\| {\mathcal{H}}\|$ of the *mean curvature vector field* given by ${\mathcal{H}}=\mathrm{trace}(\alpha^f)/n.$
The following result provides a characterization of cylinders of dimension $n\geq4.$
[[**.** ]{}]{}\[main\] Let $f\colon M^n \to \mathbb{Q}^{n+p}_c, n\geq4,$ be an isometric immersion such that $M^n$ carries a totally geodesic distribution $D$ of rank $n-2$ satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^n$. If the mean curvature of $f$ is constant along each leaf of $D$, then $f$ is minimal, or $c=0$ and $f$ is locally a $(n-2)$-cylinder over a surface on the open subset where the mean curvature is positive. Moreover, the submanifold is globally a cylinder if the leaves of $D$ are complete.
It is interesting that the above theorem fails for substantial three-dimensional submanifolds of codimension $p\geq2$. Being substantial means that the codimension cannot be reduced. We show that besides cylinders, there exist elliptic three-dimensional submanifolds in spheres satisfying the properties assumed in Theorem \[main\]. Thus the submanifolds being three-dimensional is special. The notion of elliptic submanifolds was introduced in [@DF2]. In fact, the following result allows a parametrization of them in terms of minimal surfaces in the Euclidean space, the so-called *bipolar parametrization*, using the following construction.
Let $g\colon L^2 \to {\mathbb{R}}^{n+1}, n\geq5,$ be a minimal surface. The map $\Phi_g\colon T^1L\to \mathbb{S}^n$ defined on the unit tangent bundle of $L^2$ and given by \[start\] \_g(x,w)=g\_[\*\_x]{}w parametrizes (outside singular points) an immersion with index of relative nullity at least one at any point.
[[**.** ]{}]{}\[main1\] Let $f\colon M^3 \to \mathbb{Q}^{3+p}_c$ be an isometric immersion such that $M^3$ carries a totally geodesic distribution $D$ of rank one satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^3$. If the mean curvature of $f$ is constant along each integral curve of $D$, then one of the following holds:
\(i) The immersion $f$ is minimal.
\(ii) $c=0$ and $f$ is locally a cylinder over a surface.
\(iii) $c=1$ and the immersion $f$ is elliptic and locally parametrized by , where $g\colon L^2 \to {\mathbb{R}}^{n+1}, n\geq5,$ is a minimal surface whose first curvature ellipse is nowhere a circle and the second curvature ellipse is everywhere a circle.
Minimal surfaces satisfying the conditions in part (iii) of the above theorem can be constructed using the Weierstrass representation by choosing appropriately the holomorphic data. It is worth noticing that minimal surfaces in the Euclidean space that satisfy the Ricci condition, or equivalently are locally isometric to a minimal surface in ${\mathbb{R}}^3,$ fulfill these conditions (see Section 6 for details). These surfaces were classified by Lawson [@Law].
The above results allow us to provide applications to submanifolds with constant mean curvature and not necessarily constant but positive index of relative nullity.
Having constant mean curvature is a much weaker restriction on the mean curvature vector field than being parallel in the normal bundle. One can check that three-dimensional elliptic submanifolds described in Theorem \[main1\] do not have parallel mean curvature vector field along the totally geodesic distribution. Combining this with Theorem \[main\], it follows that a submanifold is locally a cylinder provided that it carries a totally geodesic distribution of rank $n-2\geq1$ that is contained in the relative nullity distribution, along which the mean curvature vector field is parallel in the normal connection.
Opposed to the fact that there is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$ (see [@DF2]), we prove the following result for submanifolds with constant and positive mean curvature.
[[**.** ]{}]{}\[main2\] Let $f\colon M^n \to \mathbb{Q}^{n+p}_c,n\geq3,$ be a nonminimal isometric immersion with index of relative nullity $\nu\geq n-2$ at any point. If the mean curvature of $f$ is constant and either $n\geq4$ or $n=3$ and $p=1$, then $c=0$. Moreover, there exists an open dense subset $V\subseteq M^n$ such that every point of which has a neighborhood $U\subseteq V$ so that $f(U)$ is an open subset of the image of a cylinder over a surface in ${\mathbb{R}}^{p+2}$, or over a curve in ${\mathbb{R}}^{p+1}$ with constant first Frenet curvature.
The following is an immediate consequence of the above result due to real analyticity of hypersurfaces with constant mean curvature.
[[**.** ]{}]{}\[main2’\] Let $f\colon M^n \to \mathbb{Q}^{n+1}_c, n\geq3,$ be a nonminimal isometric immersion with index of relative nullity $\nu\geq n-2.$ If the mean curvature of $f$ is constant, then $c=0$ and $f(M)$ is an open subset of the image of a cylinder over a surface in ${\mathbb{R}}^3$ of constant mean curvature.
The next result extends Corollary 1 in [@cfgms] for hypersurfaces in every space form without any global assumption.
[[**.** ]{}]{}\[T5\] Let $f\colon M^n \to \mathbb{Q}^{n+1}_c, n\geq3,$ be an isometric immersion with constant mean curvature. If $M^n$ has sectional curvature $K\leq c$, then either $f$ is minimal, or $c=0$ and $f(M)$ is an open subset of the image of a cylinder over a surface in ${\mathbb{R}}^3$ of constant mean curvature. In the latter case, $f$ is a cylinder over a circle provided that $M^n$ is complete.
The following rigidity result that was proved in [@dg0] for $c=0$ is another consequence of our main results.
[[**.** ]{}]{}\[C6\] Any nonminimal isometric immersion $f\colon M^n \to \mathbb{Q}^{n+1}_c,n\geq3,$ with constant mean curvature is rigid, unless $c=0$ and $f(M)$ is an open subset of the image of a cylinder over a surface in ${\mathbb{R}}^3$ of constant mean curvature.
Our next result extends to any dimension a well-known theorem for constant mean curvature surfaces due to Klotz and Osserman [@ko] (see [@Luis] for another extension).
[[**.** ]{}]{}\[T7\] Let $f\colon M^n \to \mathbb{Q}_c^{n+1},n\geq3,$ be an isometric immersion with constant mean curvature, where $c=0$ or $c=1$. If $M^n$ is complete and its extrinsic curvature does not change sign, then $f$ is either minimal, totally umbilical or a cylinder over a sphere of dimension $1\leq k < n.$
For submanifolds with constant mean curvature of codimension two, we prove the following.
[[**.** ]{}]{}\[T8\] Let $f\colon M^n \to {\mathbb{R}}^{n+2}, n\geq3,$ be a nonminimal isometric immersion with constant mean curvature. If the sectional curvature of $M^n$ is nonpositive, then there exists an open dense subset $V\subseteq M^n$ such that every point of which has a neighborhood $U\subseteq V$ where one of the following holds:
\(i) The neighborhood $U$ splits as a Riemannian product $U=M^2\times W^{n-2}$ such that $f|_U=g\times j$ is a product, where $g\colon M^2 \to{\mathbb{R}}^4$ is a surface with constant mean curvature and $j\colon W^{n-2}\to {\mathbb{R}}^{n-2}$ is the inclusion.
\(ii) The immersion on $U$ is a composition $f|_U=h\circ F,$ where $h=\gamma\times id_{{\mathbb{R}}^{n-1}} \colon {\mathbb{R}}\times{\mathbb{R}}^n\to{\mathbb{R}}^{n+2}$ is cylinder over a unit speed plane curve $\gamma(s)$ with curvature $k(s)$ and $F\colon M^n \to{\mathbb{R}}^{n+1}$ is a hypersurface. Moreover, the mean curvature $H_F$ of $F$ is given by $$H_F^2=H_f^2-\frac{1}{n^2}k^2\circ F_a\left(1-\<\xi, a\>^2\right)^2,$$ where $F_a$ and $\<\xi, a\>$ are the height functions, with respect to $a=\partial/\partial s,$ of $F$ and its Gauss map $\xi$, respectively.
\(iii) The neighborhood $U$ splits as a Riemannian product $U=M^2_1\times M_2^2\times W^{n-4}$ such that $f|_U=g_1\times g_2\times j$ is a product, where $g_i\colon M^2_i \to{\mathbb{R}}^3, i=1,2,$ are surfaces with constant mean curvature and $j\colon W^{n-4}\to {\mathbb{R}}^{n-4}$ is the inclusion.
For constant sectional curvature submanifolds with constant mean curvature of codimension two, we prove the following theorem that extends results in [@enomoto1; @dtt].
[[**.** ]{}]{}\[T11\] Let $f\colon M_{\tilde{c}}^n\to\mathbb{Q}_c^{n+2},n\geq3,$ be an isometric immersion of a Riemannian manifold of constant sectional curvature $\tilde c.$ If the mean curvature of $f$ is constant and $n\geq4$, or $n=3$ and $c=\tilde{c},$ then one of the following holds:
\(i) $f$ is totally geodesic or totally umbilical.
\(ii) $\tilde{c}=c=0$ and $f=g\times j,$ where $g\colon M^2\to{\mathbb{R}}^4$ is a flat surface with constant mean curvature and $j\colon W\to{\mathbb{R}}^{n-2}$ is an inclusion.
\(iii) $\tilde{c}=0, c=-1$ and $f$ is a composition $f=i\circ F$, where $i\colon {\mathbb{R}}^{n+1}\to \mathbb H^{n+2}$ is the inclusion as a horosphere and $F\colon M^n_{\tilde{c}} \to {\mathbb{R}}^{n+1}$ is cylinder over a circle.
Cylinder theorems for complete minimal Kähler submanifolds were proved in [@dr; @fur]. For Kähler submanifolds with constant mean curvature, we prove the following results.
[[**.** ]{}]{}\[T9\] Let $f\colon M^n \to {\mathbb{R}}^{n+1},n\geq4,$ be an isometric immersion with constant mean curvature. If $M^n$ is Kähler, then $f$ is either minimal or $f(M)$ is an open subset of the image of a cylinder over a surface in ${\mathbb{R}}^3$ with constant mean curvature.
[[**.** ]{}]{}\[T10\] Let $f\colon M^n \to {\mathbb{R}}^{n+2}, n\geq4,$ be a nonminimal isometric immersion of a Kähler manifold $M^n$ with constant mean curvature. If the Ricci curvature or the holomorphic curvature of $M^n$ is nonpositive, then there exists an open dense subset $V\subseteq M^n$ such that every point of which has a neighborhood $U\subseteq V$ where $f|_U$ is as in Theorem \[T8\].
The paper is organized as follows: In Section 2, we recall well-known results about the relative nullity distribution, totally geodesic distributions that are contained in the relative nullity distribution, as well as results about their splitting tensor. In Section 3, we fix the notation, give some preliminaries and prove auxiliary results that will be used in the proofs of our main theorems. Section 4 is devoted to the proof of Theorem \[main\]. In Section 5, we recall the notion of elliptic submanifolds, as well as the associated notion of higher curvature ellipses. We also discuss the polar and bipolar surfaces of elliptic submanifolds. In Section 6, we study the case of three-dimensional submanifolds. We provide a parametrization for these submanifolds in terms of certain elliptic surfaces, the so-called *polar parametrization* (see Theorem \[main0\]). Based on this, we give the proof of Theorem \[main1\]. We conclude this section by showing that minimal surfaces in the Euclidean space that are locally isometric to a minimal surface in ${\mathbb{R}}^3$ satisfy the conditions in part (iii) of Theorem \[main1\]. In Section 7, we prove Theorem \[main2\] and the applications of our main results on submanifolds with constant mean curvature. In addition, we provide examples of submanifolds as in part (ii) of Theorems \[T8\] and \[T11\].
The relative nullity distribution
=================================
In this section, we recall some basic facts from the theory of [isometric immersions ]{}that will be used throughout the paper.
Let $M^n,n\geq3,$ be a Riemannian manifold and let $f\colon M^n\to{\mathbb{Q}}^m_c$ be an isometric immersion into a space form ${\mathbb{Q}}^m_c$. The *relative nullity* subspace $\Delta_f(x)$ of $f$ at $x\in M^n$ is the kernel of its second fundamental form $\alpha^f\colon TM\times TM\to N_fM$ with values in the normal bundle, that is, $$\Delta_f(x)=\left\{X\in T_xM:\alpha^f(X,Y)=0\;\;\text{for all}\;\;Y\in T_xM\right\}.$$ The dimension $\nu(x)$ of $\Delta_f(x)$ is called the *index of relative nullity* of $f$ at $x\in M^n$.
A smooth distribution $D\subset TM$ on $M^n$ is *totally geodesic* if $\n_TS\in \Gamma(D)$ whenever $T,S\in \Gamma(D)$. Let $D$ be a smooth distribution on $M^n$ and $D^{\perp}$ denote the distribution on $M^n$ that assigns to each $x\in M^n$ the orthogonal complement of $D(x)$ in $T_xM$. We write $X=X^v+X^h$ according to the orthogonal splitting $TM=D\oplus D^{\perp}$ and denote ${\nabla}^{h}_XY = (\nabla_X Y)^h$ for all $X,Y\in TM$, where $\n$ is the Levi-Civitá connection on $M^n.$ The *splitting tensor* $C\colon D\times D^{\perp}\to D^{\perp}$ is given by $$C(T,X)=-{\nabla}^{h}_XT$$ for any $T\in D$ and $X\in D^{\perp}$.
When $D$ is a totally geodesic distribution such that $D(x)\subseteq \D_f(x)$ for all $x\in M^n,$ the following differential equation for the tensor ${\mathcal{C}}_T=C(T,\cdot)$ is well-known to hold (cf. [@da] or [@dg]): \[C1\] \^[h]{}\_S \_T=\_T\_S+\_[\_ST]{} +cS,TI, where $I$ is the identity endomorphism of $D^{\perp}$. Here $\nabla^{h}_S {\mathcal{C}}_T \in \Gamma ({\rm{End}} (D^\perp))$ is defined by $$(\nabla^{h}_S {\mathcal{C}}_T)X=\nabla^{h}_S {\mathcal{C}}_TX-{\mathcal{C}}_T\nabla^{h}_SX$$ for all $T,S\in D$ and $X\in D^{\perp}$. The Codazzi equation gives \[cod\] \_TA\_=A\_\_T+ A\_[\^\_T]{} for any $T\in D,$ where the shape operator $A_{\xi}$ with respect to the normal direction $\xi$ is restricted to $D^{\perp}$ and $\n^{\perp}$ stands for the normal connection of $f$. In particular, the endomorphism $A_\xi\circ{\mathcal{C}}_T$ of $D^\perp$ is symmetric, that is, \[C3\] A\_\_T=\_T\^tA\_.
For later use, we recall the following well-known results (cf. [@da]).
[[**.** ]{}]{}\[cylinder\] Let $f\colon M^n\to\mathbb{Q}^m_c$ be an isometric immersion such that $M^n$ carries a smooth totally geodesic distribution $D$ of rank $0<k< n$ satisfying $D(x)\subseteq \D_f(x)$ for all $x\in M^n.$ If the splitting tensor $C$ vanishes, then $c=0$ and $f$ is locally a $k$-cylinder.
[[**.** ]{}]{}\[nu0\] For an isometric immersion $f\colon M^n\to{\mathbb{Q}}_c^m$, the following assertions hold:
\(i) The index of relative nullity $\nu$ is upper semicontinuous. In particular, the subset $$M_0=\{x\in M^n:\nu(x)=\nu_0\},$$ where $\nu$ attains its minimum value $\nu_0$ is open.
\(ii) The relative nullity distribution $x\mapsto\Delta_f(x)$ is smooth on any subset of $M^n$ where $\nu$ is constant.
\(iii) If $U\subseteq M^n$ is an open subset where $\nu$ is constant, then $\D_f$ is a totally geodesic (hence integrable) distribution on $U$ and the restriction of $f$ to each leaf is totally geodesic.
Auxiliary results
=================
The aim of this section is to prove several lemmas that will be used in the proofs of our main results.
Throughout this section, we assume that $f\colon M^n \to \mathbb{Q}^{n+p}_c,n\geq3,$ is a nonminimal isometric immersion such that $M^n$ carries a smooth totally geodesic distribution $D$ of rank $n-2$ satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^n$. We also suppose that the mean curvature of $f$ is constant along each leaf of $D$.
Hereafter we work on the open subset where the mean curvature is positive and we choose a local orthonormal frame $\xi_{n+1}, \dots, \xi_{n+p}$ in the normal bundle $N_fM$, such that $\xi_{n+1}$ is collinear to the mean curvature vector field. We also choose a local orthonormal frame $e_1, \dots, e_n$ in the tangent bundle $TM$ such that $e_1, e_2$ span $D^\perp$ and diagonalize $A_{\xi_{n+1}}|_{D^\perp}$, where $A_{\xi_{n+1}}$ denotes the shape operator of $f$ with respect to $\xi_{n+1}$. Then, we have $A_{\xi_{n+1}}e_i=k_ie_i, \,\ i=1,2,
$ and consequently the mean curvature is given by $nH=k_1 +k_2,$ where $k_1,k_2$ are the principal curvatures.
Since the mean curvature is positive, at least one of the principal curvatures $k_1$ and $k_2$ has to be different from zero. In the sequel, we assume without loss of generality, that $k_1\neq 0$ and we define the function $$\rho=-\frac{k_2}{k_1}.$$ On the open subset where the mean curvature is positive we have $$\label{k1k2}
k_1=-\frac{nH}{\rho-1}\,\ \,\ \text{and}\,\ \,\ k_2=\frac{n\rho H}{\rho-1}.$$ We use the above mentioned notation throughout the paper.
The following lemma gives the form of the splitting tensor.
[[**.** ]{}]{}\[char\] On the open subset where the mean curvature is positive, the splitting tensor is given by $${\mathcal{C}}_T=\psi_1(T)L_1 + \psi_2(T)L_2$$ for any $T\in\Gamma(D),$ where $\psi_1, \psi_2$ are 1-forms dual to the vector fields $\n_{e_2}e_2, \n_{e_1}e_2,$ respectively, and $L_1$, $L_2$ $\in\Gamma({\rm{End}}(D^\perp))$ are defined by $L_1e_1=\rho e_1=-L_2e_2$ and $L_1e_2=e_2 =L_2e_1.$ Moreover, the following hold: $$\begin{aligned}
& T(k_1)=\rho k_1\psi_1(T)+\sum_{\alpha=n+2}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_1\>, \label{1}\\
& T(k_2)=k_2\psi_1(T)-\sum_{\alpha=n+2}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_1\>, \label{2}\\
& (k_1-k_2)\omega(T)=k_2\psi_2(T) + \sum_{\alpha={n+2}}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_2\>, \label{3} \\
& (k_1-k_2)\omega(T)=-\rho k_1\psi_2(T)+ \sum_{\alpha={n+2}}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_2\> \label{4}
\iffalse{& \<\n_{e_r}e_1,e_s\>=0, \,\ {\text{for all \,\ }}\,\ s\geq3 \label{5}\\
& \<\n_{e_r}e_2,e_s\>=0, \,\ {\text{for all \,\ }}\,\ s\geq3 \label{6}}\fi\end{aligned}$$ for any $T\in\Gamma(D)$, where $\omega$ denotes the connection form given by $\omega=\<\n e_1,e_2\>$.
From the Codazzi equation we have $$\big(\n_{T}A_{\xi_{n+1}}\big)e_i - \big(\n_{e_i}A_{\xi_{n+1}}\big)T=A_{\nap_T{\xi_{n+1}}}e_i - A_{\nap_{e_i}{\xi_{n+1}}}T$$ for any $T\in\Gamma(D)$ and $i=1,2.$ The above is equivalent to the following $$\begin{aligned}
& T(k_1)=k_1\<\n_{e_1}e_1,T\>+\sum_{\alpha=n+2}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_1\>, \\
& T(k_2)=k_2\<\n_{e_2}e_2,T\>-\sum_{\alpha=n+2}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_1\>, \\
& (k_1-k_2)\omega(T)=k_2\<\n_{e_1}e_2,T\> + \sum_{\alpha={n+2}}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_2\>, \\
& (k_1-k_2)\omega(T)=k_1\<\n_{e_2}e_1,T\>+ \sum_{\alpha={n+2}}^{n+p}\<\nap_{T}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_2\>.
\iffalse{& \<\n_{e_r}e_1,e_s\>=0, \,\ {\text{for all \,\ }}\,\ s\geq3 \label{5}\\
& \<\n_{e_r}e_2,e_s\>=0, \,\ {\text{for all \,\ }}\,\ s\geq3 \label{6}}\fi\end{aligned}$$
Using the assumption that the mean curvature is constant along each leaf of the distribution $D$, the first two equations imply $$\<\n_{e_1}e_1,T\>=\rho\<\n_{e_2}e_2,T\>$$ for any $T\in\Gamma(D).$ Additionally, the last two equations yield $$\<\n_{e_2}e_1,T\>=-\rho\<\n_{e_1}e_2,T\>.$$ The above conclude the proof of the lemma.
[[**.** ]{}]{}\[eq\] Let $e_r, r\geq3,$ be an orthonormal frame of the distribution $D.$ Then the functions $u_r:=\psi_1(e_r)$ and $v_r:=\psi_2(e_r)$ satisfy $$2\rho(u_ru_s + v_rv_s)-c\delta_{rs} = \frac{\rho-1}{nH}\sum_{\alpha=n+2}^{n+p}\<\nap_{e_r}\xi_{n+1},\xi_{\alpha}\>\big(u_s\<A_{\xi_\alpha}e_1,e_1\> -v_s\<A_{\xi_\alpha}e_1,e_2\> \big)\label{second}$$ for all $ r, s\geq3$, where $\delta_{rs}$ is the Kronecker delta.
Using Lemma \[char\], we have $$\begin{aligned}
(\n^h_{e_r}{\mathcal{C}}_{e_s})=e_r(u_s)L_1+e_r(v_s)L_2+u_s\n^h_{e_r}L_1+v_s\n^h_{e_r}L_2\label{dc}\end{aligned}$$ for any $r,s\geq3$. A direct computation yields $$\begin{aligned}
&(\n^h_{e_r}L_1)e_1=-(\n^h_{e_r}L_2)e_2=e_r(\rho)e_1+(\rho-1){\omega(e_r)}e_2,\label{dl11}\\
&(\n^h_{e_r}L_1)e_2=(\n^h_{e_r}L_2)e_1=(\rho-1){\omega(e_r)}e_1.\label{dl12}\end{aligned}$$ Equations and imply that \[ero\] e\_r()=-(-1)u\_r + \_[=n+2]{}\^[n+p]{}\_[e\_r]{}\_[n+1]{},\_A\_[\_]{}e\_1,e\_1. From equation (\[C1\]) we know that the splitting tensor satisfies $$\begin{aligned}
&(\n_{e_r}^h{\mathcal{C}}_{e_s})e_i={\mathcal{C}}_{e_s}\circ{\mathcal{C}}_{e_r}e_i + {\mathcal{C}}_{\n_{e_r}e_s}e_i +c\delta_{rs}e_i \label{crs1}\end{aligned}$$ for any $r,s\geq3$ and $i=1,2$.
Let $\omega_{rs}$ be the connection form given by $\omega_{rs}=\<\n e_r,e_s\> $ for all $r,s\geq3$. Using equations -, we find that for $i=1$ is equivalent to $$\begin{aligned}
\label{rv1}
\rho e_r(u_s)=&\rho(2\rho-1)u_ru_s-\rho v_rv_s -(\rho-1)v_s\omega(e_r)\qquad \nonumber\\
&-u_s\frac{(\rho-1)^2}{nH} \sum_{\alpha=n+2}^{n+p}\<\nap_{e_r}\xi_{n+1},\xi_{\alpha}\>\<A_{\xi_\alpha}e_1,e_1\>
+ \rho\sum_{t\geq3}^n\omega_{st}(e_r)u_t+c\delta_{rs}\end{aligned}$$ and $$\begin{aligned}
e_r(v_s)=\rho u_r v_s +u_s v_r-(\rho-1)u_s{\omega(e_r)}+\sum_{t\geq3}^n\omega_{st}(e_r)v_t \qquad \qquad \qquad \qquad\label{ru}\end{aligned}$$ for all $r,s\geq3$. Moreover, equation for $i=2$ implies that $$\begin{aligned}
e_r(u_s)=&u_ru_s-\rho v_rv_s+(\rho-1)v_s{\omega(e_r)}+\sum_{t\geq3}^n\omega_{st}(e_r)u_t+c\delta_{rs} \qquad \qquad \qquad \label{rv2}\end{aligned}$$ for all $r,s\geq3$.
Combining and , we obtain $$2\rho u_ru_s + \rho v_rv_s - c\delta_{rs}- v_s(\rho+1){\omega(e_r)}=u_s\frac{\rho-1}{nH} \sum_{\alpha=n+2}^{n+p}\<\nap_{e_r}\xi_{n+1},\xi_{\alpha}\> \<A_{\xi_\alpha}e_1,e_1\>.$$ Using , equation is written as \[omega\] (+1)[(e\_r)]{}=-v\_r- \_[=n+2]{}\^[n+p]{}\_[e\_r]{}\_[n+1]{},\_A\_[\_]{}e\_1,e\_2 and now the desired equation follows directly from the above two equations.
We recall that the *first normal space* $N_1^f(x)$ of the immersion $f$ at a point $x\in M^n$ is the subspace of its normal space $N_fM(x)$ spanned by the image of its second fundamental form $\alpha^f$ at $x$, that is, $$N_1^f(x)={\rm{span}}\left\{\alpha^f(X,Y): X,Y \in T_xM\right\}.$$ The rank condition and the symmetry of the second fundamental form imply that $\dim N_1^f(x)\leq3$ for all $x\in M^n$.
We consider the open subset $$M_3=\left\{x\in M^n:\dim N_1^f(x)=3\right\}.$$
[[**.** ]{}]{}\[M3\] If the open subset $M_3^*:=M_3\smallsetminus\{x\in M^n: H(x)=0\}$ is nonempty, then the splitting tensor vanishes on it.
On the subset $M_3^*,$ we consider the orthogonal splitting $
N_1^f=\hat{N}^f_1 \oplus {\rm{span}}\{{\mathcal{H}}\}.$ We choose the local frame such that $\xi_{n+1}$ is collinear to the mean curvature vector field ${\mathcal{H}}$, and $ \xi_{n+2},\xi_{n+3}$ span the plane bundle $\hat{N}^f_1$. Then, we have $$\text{trace}A_{\xi_{n+2}}|_{D^\perp}=0=\text{trace}A_{\xi_{n+3}}|_{D^\perp}.$$ Hence, we obtain $$A_{\xi_{n+2}}|_{D^\perp}\circ J=J^t\circ A_{\xi_{n+2}}|_{D^\perp} \,\ \text{and} \,\ A_{\xi_{n+3}}|_{D^\perp}\circ J=J^t\circ A_{\xi_{n+3}}|_{D^\perp},$$ where $J$ denotes the unique, up to a sign, almost complex structure acting on the plane bundle $D^\perp$.
Equation implies that for any $T\in\Gamma(D)$ we have $$A_{\xi_{n+2}}|_{D^\perp}\circ{\mathcal{C}}_T={\mathcal{C}}_T^t\circ A_{\xi_{n+2}}|_{D^\perp} \,\ \text{and} \,\ A_{\xi_{n+3}}|_{D^\perp}\circ{\mathcal{C}}_T={\mathcal{C}}_T^t\circ A_{\xi_{n+3}}|_{D^\perp}.$$ Since $\hat{N}^f_1$ is a plane bundle, the above imply that ${\mathcal{C}}_T\in {\rm{span}}\{I,J\}\subseteq {\rm{End}}(D^\perp)$. This, combined with Lemma \[char\], yields $$(\rho-1)\psi_1(T)=0 \,\ \text{and} \,\ (\rho-1)\psi_2(T)=0$$ for any $T\in \Gamma(D)$. Thus, the splitting tensor vanishes identically on $M_3^*$ and this concludes the proof of the lemma.
Hereafter, we assume that $M_3$ is not dense on $M^n$ and we consider the open subset $$M_2=\left\{x\in M^n\smallsetminus\overline{M}_3:\dim N_1^f(x)=2\right\}.$$ In the sequel, we assume that the open subset $M_2^*:=M_2\smallsetminus\{x\in M^n: H(x)=0\}$ is nonempty. We choose a local orthonormal frame such that $\xi_{n+1}$ and $\xi_{n+2}$ span the plane bundle $N_1^f$ on this subset and $\xi_{n+1}$ is collinear to the mean curvature vector field. Thus, there exist smooth functions $\lambda,\mu$ such that $$\begin{aligned}
A_{\xi_{n+2}}e_1=\lambda e_1 + \mu e_2, \,\ A_{\xi_{n+2}}e_2=\mu e_1 - \lambda e_2 \,\,\,\, \text{and}\,\,\,\, \lambda^2+\mu^2>0.\end{aligned}$$
We proceed with some auxiliary lemmas.
[[**.** ]{}]{}\[n+2\] The plane bundle $N_1^f$ is parallel in the normal connection along the distribution $D$ on the subset $M_2^*$. Moreover, the following hold: $$\begin{aligned}
&\mu \psi_1(T)=-\lambda \psi_2(T), \label{Nostr1}\\
&\mu \phi(T)=-(\lambda^2+\mu^2)\frac{\rho-1}{nH}\psi_2(T),\label{Nostr2}\\
&T(\mu)+2\lambda\omega(T)+(\rho+1)\lambda \psi_2(T)=0, \label{rm}\\
&T(\lambda)-2\mu\omega(T)-\mu \rho \psi_2(T)-\lambda \psi_1(T)=\frac{n\rho H}{\rho-1}\phi(T), \label{rl1}\\
&T(\lambda)-2\mu\omega(T)-\mu \psi_2(T)-\lambda\rho \psi_1(T)=\frac{nH}{\rho-1}\phi(T) \label{rl2}\end{aligned}$$ for any $T\in \Gamma(D)$, where $\phi$ is the normal connection form given by $\phi=\<\nabla^\perp\xi_{n+1},\xi_{n+2}\>.$
Equation (\[cod\]) implies that $$\<\nabla^\perp_T\xi_{\alpha}, \xi\>=0 \;\;\text{if}\;\; \alpha=n+1,n+2$$ for any $T\in \Gamma(D)$ and any $\xi \in \Gamma({N^f_1}^{\perp})$. Thus, the subbundle $N_1^f$ is parallel in the normal connection along the distribution $D.$
Moreover, from we have $$(\n_TA_{\xi_{n+2}})e_i=A_{\xi_{n+2}}\circ{\mathcal{C}}_Te_i+ A_{\n^{\perp}_T\xi_{n+2}}e_i, \;\;i=1,2,$$ for any $T\in \Gamma(D)$. Bearing in mind the form of the splitting tensor given in Lemma \[char\], the above equations yield directly , and the following $$\begin{aligned}
&T(\mu)+2\lambda\omega(T)+\lambda\rho \psi_2(T)-\mu \psi_1(T)=0, \\
&T(\mu)+2\lambda\omega(T)-\mu\rho \psi_1(T)+\lambda \psi_2(T)=0\end{aligned}$$ for any $T\in \Gamma(D)$. Subtracting the above equations, we obtain (\[Nostr1\]). Equation follows by substrating , and using (\[Nostr1\]). Finally, plugging (\[Nostr1\]) into the first of the above equations, we have (\[rm\]) and this completes the proof.
We now suppose that the subset ${M_3\cup M_2}$ is not dense on $M^n$ and we consider the open subset $$M_1=\left\{x\in M^n\smallsetminus \overline{M_3\cup M_2}: \dim N_1^f(x)=1\right\}.$$
[[**.** ]{}]{}\[M1\] If the subset $M_1^*:=M_1\smallsetminus\{x\in M^n: H(x)=0\}$ is nonempty, then $c=0$ and $f|_{M_1^*}$ is locally a cylinder over a surface in ${\mathbb{R}}^{p+2}$ or over a curve in ${\mathbb{R}}^{p+1}$.
On the subset $M_1^*$ we choose a local orthonormal frame $\xi_{n+1},\dots,\xi_{n+p}$ in the normal bundle such that $\xi_{n+1}$ is collinear to the mean curvature vector field. Then we have $A_{\xi_\alpha}=0$ for all $\alpha\geq n+2.$ The Codazzi equation yields $$A_{\nap_{e_i}\xi_\alpha}e_r=A_{\nap_{e_r}\xi_\alpha}e_i$$ for all $\alpha\geq n+2,$ $i=1,2,$ and $r\geq3$. Thus, we obtain $\nap_{e_r}\xi_{n+1}=0$ and Lemma \[eq\] gives $$2\rho(u_ru_s + v_rv_s)=c\delta_{rs}
\label{first2}$$ for all $r\geq 3$. Moreover, equation becomes $$e_r(\rho)=-\rho(\rho-1)u_r.$$ Differentiating (\[first2\]) with respect to $e_r$ and using the above along with equations and , we obtain $$\rho u_r(\rho-3)(u_r^2+v_r^2)-2c\rho u_r + 2\rho\sum_{s\geq3}^n\omega_{rs}(e_r)\big(u_su_r +v_sv_r\big)=0$$ for all $r\geq3$. In view of (\[first2\]), the above equation simplifies to the following $$c(\rho+1)u_r=0.$$
Now we prove that $c=0$. Arguing indirectly, we suppose that $c\neq0.$ Assume that the open set of points where $\rho\neq-1$ is nonempty. On this subset, we have $u_r=0$ for all $r\geq3.$ Thus, equation becomes $2\rho v_r^2=c$ for all $r\geq3$. Using , equation yields $2\rho^2 v_r^2=c(\rho+1),$ which is a contradiction. Assume now that the set of points where $\rho=-1$ has nonempty interior. On this subset, yields $u_r=0$ and equation implies that $v_r=0,$ which contradicts the assumption that $c\neq0.$
Hence, $c=0$ and equation becomes $$\rho(u_r^2+v_r^2)=0$$ for all $r\geq3.$ If $\rho\neq0,$ then the splitting tensor vanishes and Proposition \[cylinder\] implies that $f$ is locally a cylinder over a surface. If the subset of points where $\rho=0$ has nonempty interior, then the Codazzi equation implies that the tangent bundle splits as an orthogonal sum of two parallel distributions one of which has rank $n-1$. Thus, the manifold splits locally as a Riemannian product by the De Rham decomposition theorem. Since the second fundamental form is adapted to this splitting, the result follows from [@da Th. 8.4] and the proof is completed.
Submanifolds of dimension $n\geq4$
==================================
We are now ready to give the proof of our first main result.
*Proof of Theorem \[main\]:* If the open subset $M_3^*$ is nonempty, then Lemma \[M3\] implies that the splitting tensor vanishes identically on it. Then, by Proposition \[cylinder\] the immersion $f$ is locally a cylinder over a surface on $M_3^*$.
Now we assume that the subset $M_3$ is not dense on $M^n$ and suppose that $M_2^*$ is nonempty. Hereafter, we work on $M_2^*$. Due to the choice of the local orthonormal frame $\xi_{n+1}, \xi_{n+2}$ in the normal subbundle $N_1^f$, using (\[Nostr1\]) and (\[Nostr2\]), equation of Lemma \[eq\] takes the following form $$v_rv_s\left(\lambda^2+\mu^2\right)\Big(2\rho-\left(\lambda^2+\mu^2\right)\frac{(\rho-1)^2}{n^2H^2}\Big)=c\mu^2 \delta_{rs} \label{compare}$$ for any $r,s\geq3$.
We claim that $v_r=0$ for any $r\geq3$. In fact, at points where $$2\rho-\left(\lambda^2+\mu^2\right)\frac{(\rho-1)^2}{n^2H^2}\neq0,$$ it follows from that $$\begin{aligned}
v_r^2=\frac{c\mu^2}{\left(\lambda^2+\mu^2\right)\left(2\rho-\left(\lambda^2+\mu^2\right)\frac{(\rho-1)^2}{n^2H^2}\right)}\end{aligned}$$ for any $r\geq3$ and $v_rv_s=0$ for $r\neq s\geq3$. Thus, $v_r=0$ for any $r\geq3$ at those points.
It remains to prove that the same holds on the subset $U\subseteq M_2^*$ of points where $$2\rho-\left(\lambda^2+\mu^2\right)\frac{(\rho-1)^2}{n^2H^2}=0.$$ Notice that because of , the subset $U$ is the set of points where \[hypothesis\] \^2+\^2=-2k\_1k\_2. In order to prove that $v_r=0$ for any $r\geq3$ on $U$, we assume that the interior of $U$ is nonempty. We suppose to the contrary that there exists $r_0\geq3$ such that $v_{r_0}\neq0$ on an open subset of $U$. Differentiating with respect to $e_{r_0}$ and using , , , and , we obtain $$\lambda^2u_{r_0}-\lambda \mu v_{r_0} + (\rho+1)k_1k_2 u_{r_0}=\lambda(k_1-2k_2)\phi(e_{r_0}).$$ Multiplying by $\mu$ the above and using , we find that $$\mu u_{r_0}\left(\lambda^2+ (\rho+1)k_1k_2\right)=
\lambda v_{r_0}\Big(\mu^2- (\lambda^2+\mu^2)(k_1-2k_2)\frac{\rho-1}{nH}\Big).$$ Taking into account , and , the above yields $$\lambda v_{r_0}(\rho+1)(\lambda^2+\mu^2)=0.$$ Due to , we conclude that $\lambda=0$ and consequently $\mu\neq0$. Then, it follows from that $u_s=0$ for any $s\geq3$. Equations and for $s= r_0$ imply that $$\rho v^2_{r_0}+(\rho-1)v_{r_0}\omega(e_{r_0})-c=0
\;\;{\text {and}}\;\;
\rho v^2_{r_0}-(\rho-1)v_{r_0}\omega(e_{r_0})-c=0,$$ respectively. Hence $\omega(e_{r_0})=0$, and consequently yields $$\rho v_{r_0}+\frac{\rho-1}{nH}\mu\phi(e_{r_0})=0.$$ Using , and we find that $\rho=0$, which contradicts . Thus we have proved the claim that $v_r=0$ for any $r\geq3$.
Now, we claim that $u_r=0$ for any $r\geq3$. Equation implies that $
\mu u_r=0
$ for any $r\geq3$. Obviously, the function $u_r$ vanishes at points where $\mu\neq0$.
We assume that the set of points where $\mu=0$ has nonempty interior and we argue on this subset. Since $\lambda\neq0$ on this subset, it follows from that ${\omega(e_r)}=0$ for any $r\geq3,$ and consequently and yield [(e\_r)]{}=u\_r e\_r()=(+1)u\_r\[28\] for all $r\geq3$. Using the first of the above equations, (\[second\]) is written equivalently as $$\begin{aligned}
\label{c=0}
u_r^2\Big(2\rho-\frac{\lambda^2(\rho-1)^2}{n^2H^2}\Big)=c\end{aligned}$$ for all $r\geq3$.
Since we already proved that $v_r=0$ for all $r\geq3$, Lemma \[char\] implies that the image of the splitting tensor $C\colon D\to {\rm{End}}(D^\perp)$ satisfies $\dim {\rm{Im}}\, C\leq1.$ Thus, $\dim\ker\, C\geq n-3.$
We now suppose that $\dim\ker\, C=n-3$. Then, there exists a unique $r_0\geq3$ such that $u_{r_0}\neq0$ and $u_s=0$ for any $s\neq r_0$. Thus, equation implies that $c=0$ and $$2\rho=\frac{\lambda^2(\rho-1)^2}{n^2H^2}.$$ On account of , the above equation becomes $\lambda^2=-2k_1k_2>0.$ Differentiating this equation with respect to $e_{r_0}$ and using the second of , , and , we obtain $2\lambda^2 + k_1k_2=0,$ which contradicts the previous equation. Thus, the splitting tensor vanishes identically on the subset $M_2^*$ and consequently, by Proposition \[cylinder\], the immersion $f$ is locally a cylinder over a surface.
If the open subset $M_1^*$ is nonempty, then Lemma \[M1\] implies that $f$ is locally a cylinder over a surface or a curve and this completes the proof.
Elliptic submanifolds
=====================
In this section, we recall from [@DF2] the notion of elliptic submanifolds of a space form as well as several of their basic properties.
Let $f\colon M^n\to{\mathbb{Q}}_c^m$ be an isometric immersion. The $\ell^{th}$*-normal space* $N^f_\ell(x)$ of $f$ at $x\in M^n$ for $\ell\ge 1$ is defined as $$N^f_\ell(x)={\mbox{span}}\left\{\alpha^f_{\ell+1}\left(X_1,\ldots,X_{\ell+1}\right):
X_1,\ldots,X_{\ell+1}\in T_xM\right\}.$$ Here $\alpha^f_2=\alpha^f$ and for $s\geq 3$ the so-called $s^{th}$*-fundamental form* is the symmetric tensor $\alpha^f_s\colon TM\times\cdots\times TM\to N_fM$ defined inductively by $$\alpha^f_s(X_1,\ldots,X_s)=\pi^{s-1}\left(\nabla^\perp_{X_s}\cdots
\nabla^\perp_{X_3}\alpha^f(X_2,X_1)\right),$$ where $\pi^k$ stands for the projection onto $(N_1^f\oplus\cdots\oplus N_{k-1}^f)^{\perp}$.
An isometric immersion $f\colon M^n\to{\mathbb{Q}}_c^m$ is called *elliptic* if $M^n$ carries a totally geodesic distribution $D$ of rank $n-2$ satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^n$ and there exists an (necessary unique up to a sign) almost complex structure $J\colon D^\perp\to D^\perp$ such that the second fundamental form satisfies $$\alpha^f(X,X)+\alpha^f(JX,JX)=0$$ for all $X\in D^\perp$. Notice that $J$ is orthogonal if and only $f$ is minimal.
Assume that $f\colon M^n\to{\mathbb{Q}}_c^m$ is substantial and elliptic. Assume also that $f$ is *nicely curved* which means that for any $\ell\geq 1$ all subspaces $N^f_\ell(x)$ have constant dimension and thus form subbundles of the normal bundle. Notice that any $f$ is nicely curved along connected components of an open dense subset of $M^n$. Then, along that subset the normal bundle splits orthogonally and smoothly as \[splits\] N\_fM=N\^f\_1N\^f\_[\_f]{}, where all $N^f_\ell$’s have rank two, except possibly the last one that has rank one in case the codimension is odd. Thus, the induced bundle $f^*T{\mathbb{Q}}_c^m$ splits as $$f^*T{\mathbb{Q}}_c^m=f_*D\oplus N^f_0\oplus N^f_1\oplus \cdots \oplus N^f_{\tau_f},$$ where $N^f_0=f_*D^\perp$. Setting $$\tau^o_f = \left\{\begin{array}{l}
\tau_f\;\;\;\;\;\;\;\;\;\;\;\mbox{if}\;\;m-n\;\;\; \mbox{is even}\\
\tau_f-1\;\;\;\;\; \mbox{if}\;\;m-n\;\;\; \mbox{is odd}
\end{array} \right.$$ it turns out that the almost complex structure $J$ on $D^\perp$ induces an almost complex structure $J_\ell$ on each $N_\ell^f$, $0\leq \ell\leq\tau^o_f$, defined by $$J_\ell\alpha_{\ell+1}^f\left(X_1,\ldots,X_\ell,X_{\ell+1}\right)
=\alpha_{\ell+1}^f\left(X_1,\ldots,X_\ell,J X_{\ell+1}\right),$$ where $\alpha_1^f=f_*$.
The *$\ell^{th}$-order curvature ellipse* $\mathcal{E}_\ell^f(x)\subset N^f_\ell(x)$ of $f$ at $x\in M^n$ for $0\leq\ell\leq\tau^o_f$ is $$\mathcal{E}_\ell^f(x)=\big\{\alpha^f_{\ell+1}(Z_{\theta},\dots,Z_{\theta}):
Z_{\theta}=\cos\theta Z+\sin\theta J Z\;\;\mbox{and}\;\;\theta\in [0,\pi)
\big\},$$ where $Z\in D^\perp(x)$ has unit length and satisfies $\<Z,JZ\>=0$. From ellipticity such a $Z$ always exists and $\mathcal{E}_\ell^f(x)$ is indeed an ellipse.
We say that the curvature ellipse $\mathcal{E}_\ell^f$ of an elliptic submanifold $f$ is a *circle* for some $0\leq\ell\leq\tau^o_f$ if all ellipses $\mathcal{E}_\ell^f(x)$ are circles. That the curvature ellipse $\mathcal{E}_\ell^f$ is a circle is equivalent to the almost complex structure $J_\ell$ being orthogonal. Notice that $\mathcal{E}_0^f$ is a circle if and only if $f$ is minimal.
Let $f\colon M^n\to{\mathbb{Q}}_c^{m-c}, c\in \{0,1\},$ be a substantial nicely curved elliptic submanifold. Assume that $M^n$ is the saturation of a fixed cross section $L^2\subset M^n$ to the foliation of the distribution $D.$ The subbundles in the orthogonal splitting (\[splits\]) are parallel in the normal connection (and thus in ${\mathbb{Q}}_c^{m-c}$) along $D$. Hence each $N^f_\ell$ can be seen as a vector bundle along the surface $L^2$.
A *polar surface* to $f$ is an immersion $h$ of $L^2$ defined as follows:
- If $m-n-c$ is odd, then the polar surface $h\colon L^2\to{\mathbb{S}}^{m-1}$ is the spherical image of the unit normal field spanning $N^f_{\tau_f}$.
- If $m-n-c$ is even, then the polar surface $h\colon L^2\to{\mathbb{R}}^m$ is any surface such that $h_*T_xL=N^f_{\tau_f}(x)$ up to parallel identification in ${\mathbb{R}}^m$.
Polar surfaces always exist since in case ${\rm(b)}$ any elliptic submanifold admits locally many polar surfaces.
The almost complex structure $J$ on $D^\perp$ induces an almost complex structure $\tilde J$ on $TL$ defined by $P\circ\tilde J=J\circ P$, where $P\colon TL \to D^\perp$ is the orthogonal projection. It turns out that a polar surface to an elliptic submanifold is necessarily elliptic. Moreover, if the elliptic submanifold has a circular curvature ellipse then its polar surface has the same property at the “corresponding" normal bundle. As a matter of fact, up to parallel identification it holds that \[eqp\] N\_s\^h=N\_[\^o\_f-s]{}\^f J\^h\_s=(J\^f\_[\^o\_f-s]{})\^t, 0s\^o\_f. In particular, the polar surface is nicely curved.
A *bipolar surface* to $f$ is any polar surface to a polar surface to $f$. In particular, if we are in case $f\colon M^3\to{\mathbb{S}}^{m-1}$, then a bipolar surface to $f$ is a nicely curved elliptic surface $g\colon
L^2\to{\mathbb{R}}^m$.
Three-dimensional submanifolds
==============================
In this section, we study the case of three-dimensional submanifolds and we provide the proof of Theorem \[main1\]. To this purpose, we need the following results.
[[**.** ]{}]{}\[flat\] Let $f\colon M^3 \to \mathbb{Q}^{3+p}_c$ be an isometric immersion such that $M^3$ carries a totally geodesic distribution $D$ of rank one satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^3$. If the mean curvature of $f$ is constant along each integral curve of $D$ and the normal bundle of $f$ is flat, then $f$ is minimal, or $c=0 $ and $f$ is locally a cylinder.
We assume that $f$ is nonminimal. If the open subset $M_3^*$ is nonempty, then Lemma \[M3\] and Proposition \[cylinder\] imply that the immersion $f$ is locally a cylinder over a surface.
We now assume that the open subset $M_2^*$ is nonempty and we argue on it. Having flat normal bundle implies that $\mu=0$ and according to we obtain $v_3=0$. Consequently, is written as \[a3\] e\_3(u\_3)=u\_3\^2+c. Comparing equations and , we obtain $$\phi(e_3)=\frac{\rho-1}{nH}\lambda u_3.$$ Thus, \[e3l\] e\_3()=(+1)u\_3 and consequently equation becomes \[ero’\] e\_3()=u\_3(-1)(-), where $\tau $ is the function given by $$\tau=\frac{\lambda^2(\rho-1)^2}{n^2H^2}.$$ Moreover, equation is written as $u_3^2(2\rho-\tau)=c.$ Differentiating with respect to $e_3$ and using equations , and , we derive that $$u_3^2(\rho+1)(\rho-\tau)=0.$$
Now we claim that $u_3=0.$ Arguing indirectly, we suppose that $u_3\neq0$ on an open subset. We observe that $\rho\neq-1$ due to our assumption and equation . Hence $\rho=\tau,$ or equivalently $\rho n^2H^2=\lambda^2(\rho-1)^2$ and $e_3(\rho)=0$ by . Thus $e_3(\lambda)=0$, which contradicts since $\lambda\neq0.$ This proves the claim that $u_3=0$ and consequently the splitting tensor vanishes. That the immersion $f$ is locally a cylinder on $M_2^*$ follows from Proposition \[cylinder\].
If the open subset $M_1^*$ is nonempty, then Lemma \[M1\] implies that the immersion $f$ is locally a cylinder over a surface or over a curve. Thus the proof is completed.
[[**.** ]{}]{}\[sphelliptic\] Let $f\colon M^3 \to \mathbb{Q}^{3+p}_c$ be a nonminimal isometric immersion such that $M^3$ carries a totally geodesic distribution $D$ of rank one satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^3$. If the mean curvature of $f$ is constant along each integral curve of $D$ and $f$ is not locally a cylinder, then the splitting tensor of $f$ is an almost complex structure on $D^\perp.$ Moreover, $f$ is a spherical elliptic submanifold with respect to this almost complex structure and its first curvature ellipse is a circle.
Since by assumption the immersion $f$ is not a cylinder on any open subset, it follows from Lemma \[M3\], Proposition \[cylinder\] and Lemma \[M1\] that the open subsets $M_3^*$ and $M_1^*$ are both empty.
Proposition \[flat\], implies that the immersion $f$ has nonflat normal bundle on $M_2^*.$ Thus, we have $\mu\neq0$ and $\rho\neq-1.$ Using and , equations , , , , and are written as $$\begin{aligned}
&\omega(e_3)=-\frac{\rho-\tau}{\rho+1}v_3,\nonumber\\
&e_3(\rho)=\frac{\lambda}{\mu}(\rho-1)(\rho-\tau)v_3,\label{e3ro}\\
&e_3(\mu)=-\frac{\lambda}{\rho+1}\left(2\tau+\rho^2+1\right)v_3,\nonumber\\
&e_3(\lambda)=\Big(\frac{2\mu}{\rho+1}\tau-\frac{2\mu\rho}{\rho+1}-\frac{\lambda^2}{\mu}(\rho+1)\Big)v_3,\label{3la}\\
&e_3(v_3)=\frac{\lambda}{\mu(\rho+1)}\left((\rho-1)\tau -(2\rho^2+\rho+1)\right)v_3^2,\nonumber \\
&(\lambda^2+\mu^2)(2\rho-\tau)v_3^2=c\mu^2, \label{comp1}\end{aligned}$$ respectively, where $\tau$ is the function given by $$\tau=(\lambda^2+\mu^2)\frac{(\rho-1)^2}{n^2H^2}.$$ By differentiating equation and using all the above equations, we obtain $$\lambda(\lambda^2+\mu^2)\Big(\rho(5\rho^2+6\rho+5)-(4\rho^2+2\rho+4)\tau - 2\tau^2\Big)v_3^3=c\lambda\mu^2v_3.$$
We claim that $\lambda v_3=0.$ Arguing indirectly, we assume that the open subset where $\lambda v_3\neq0$ is nonempty. Thus, comparing the above equation with equation , we derive that $\tau=\rho.$ This along with imply that $e_3(\tau)=e_3(\rho)=0.$ By the definition of $\tau,$ it follows that $e_3(\lambda^2+\mu^2)=0.$ Using the above equations it is easy to see that $$e_3(\lambda^2+\mu^2)=-2\frac{\lambda}{\mu}(\lambda^2+\mu^2)(\rho+1)v_3,$$ which is a contradiction and this proves our claim.
Now we claim that $v_3$ cannot vanish on any open subset. Arguing indirectly, we suppose that $v_3\neq 0$ on an open subset. Then equation implies that $u_3=0. $ By Lemma \[char\], the splitting tensor vanishes and consequently the immersion $f$ would be a cylinder by Proposition \[cylinder\]. This contradicts our assumption.
Since we already proved that $\lambda v_3=0$, we obtain $\lambda=0$ and equation implies that $u_3=0.$ It follows from equation that \[romi\] \^2=. In particular, we have $\rho>0.$ This, along with equation yield \[roci\] v\_3\^2=c. Hence, $c=1.$ We now observe that the splitting tensor satisfies ${\mathcal{C}}_3^2=-I,$ where $I$ is the identity endomorphism of $D^{\perp},$ that is, ${\mathcal{C}}_3$ is an almost complex structure $J\colon D^\perp\to D^\perp.$ Using equation and the fact that the shape operator $A_{\xi_5}$ satisfies $A_{\xi_5}e_i=\mu e_j$ for $i\neq j=1,2,$ we easily verify that the second fundamental form of $f$ satisfies $\alpha^f(Je_1,e_2)=\alpha^f(e_1,Je_2).$ This is equivalent to the ellipticity of the immersion $f.$
In order to prove that the first curvature ellipse of $f$ is a circle, it is sufficient to prove that the vector fields $\alpha^f(e_1,e_1)$ and $\alpha^f(e_1,Je_1)$ are of the same length and perpendicular. Obviously, they are perpendicular since $$\alpha^f(e_1,e_1)=k_1\xi_4 \;\; \text{and} \;\;
\alpha^f(e_1,Je_1)=\mu v_3\xi_5.$$ Using equations and , we obtain $$\frac{\|\alpha^f(e_1,Je_1)\|^2}{\|\alpha^f(e_1,e_1)\|^2}=\rho v_3^2.$$ Bearing in mind equation , we conclude that the first curvature ellipse is a circle and this completes the proof.
The following result parametrizes all three-dimensional submanifolds in spheres that carry a totally geodesic distribution of rank one, contained in the relative nullity distribution, such that the mean curvature is constant along each integral curve. This parametrization, given in terms of their polar surfaces, was introduced in [@DF2] as the *polar parametrization*.
[[**.** ]{}]{}\[main0\] Let $h\colon L^2\to \mathbb{Q}_c^{N+1}, c\in\{0,1\},N\geq 5,$ be a nicely curved elliptic surface of substantial even codimension, such that the curvature ellipses $\mathcal{E}_{\tau_h-2}^h, \mathcal{E}_{\tau_h}^h$ are circles and $\mathcal{E}_{\tau_h-1}^h$ is nowhere a circle. Then, the map $\Psi_h\colon M^3\to \mathbb{S}^{N+c}$ defined on the circle bundle $M^3=UN^h_{\tau_h}$ by $\Psi_h(x,w)=w$ is a nonminimal elliptic isometric immersion with polar surface $h$. Moreover, $M^3$ carries a totally geodesic distribution $D$ of rank one satisfying $D(p)\subseteq\D_{\Psi_h}(p)$ for any $p\in M^3$ such that the mean curvature of $\Psi_h$ is constant along each integral curve of $D.$
Conversely, let $f\colon M^3 \to \mathbb{S}^{3+p}, p\geq2,$ be a substantial nonminimal isometric immersion such that $M^3$ carries a totally geodesic distribution $D$ of rank one satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^3$. If the mean curvature of $f$ is constant along each integral curve of $D$, then f is elliptic and there exists an open dense subset of $M^3$ such that for each point there exist a neighborhood $U,$ and a local isometry $F\colon U\to UN^h_{\tau_h}$ such that $f=\Psi_h\circ F,$ where $h$ is a polar surface to $f$ with curvature ellipses as above.
Let $h\colon L^2\to \mathbb{Q}_c^{N+1}, c\in\{0,1\},$ be a substantial elliptic surface, where $N=2m+3, m\geq1$. We choose a local orthonormal frame $e_1, e_2$ in the tangent bundle of $L^2$ such that the almost complex structure $J$ of the elliptic surface is given by $$Je_1=be_2 \,\ \, \text{and} \,\ \, Je_2=-\frac{1}{b}e_1,$$ where $b$ is a positive smooth function.
We argue for the case where $m\geq2.$ The case where $m=1$ can be handled in a similar manner. We know from that the normal bundle splits orthogonally as $$N_hL=N^h_1\oplus\cdots\oplus N^h_{m-1}\oplus N^h_m\oplus N^h_{m+1}.$$ Let $\zeta_3,\dots,\zeta_{2m+4}$ be an orthonormal frame in the normal bundle, defined on an open subset $V\subseteq L^2,$ such that $\zeta_{2s+1}, \zeta_{2s+2}$ span the plane subbundle $N^h_s$ for any $1\leq s\leq m+1.$ The corresponding normal connection forms $\omega_{\alpha\beta}$ are given by $\omega_{\alpha\beta}=\<\nap \zeta_\alpha,\zeta_\beta\>, \alpha,\beta=3,\dots,2m+4.$
Due to our hypothesis, we may choose the frame such that $$\alpha^h_m(e_1,\dots,e_1)=\kappa_{m-1}\zeta_{2m-1}, \,\ \,\ \alpha^h_m(e_1,\dots,e_1,e_2)=\frac{\kappa_{m-1}}{b}\zeta_{2m}$$ and $$\alpha^h_{m+2}(e_1,\dots,e_1)=\kappa_{m+1}\zeta_{2m+3}, \,\ \,\ \alpha^h_{m+2}(e_1,\dots,e_1,e_2)=\frac{\kappa_{m+1}}{b}\zeta_{2m+4},$$ where $\kappa_{m-1}, \kappa_{m+1}$ denote the radii of the circular curvature ellipses $\mathcal{E}^h_{m-1}, \mathcal{E}^h_{m+1},$ respectively. Since the curvature ellipse $\mathcal{E}^h_{m}$ is nowhere a circle, we may choose $\zeta_{2m+1}, \zeta_{2m+2}$ to be collinear to the major and minor axes of this ellipse, respectively. Thus, we may write $$\alpha^h_{m+1}(e_1,\dots,e_1)=v_{11}\zeta_{2m+1}+v_{12}\zeta_{2m+2} \,\ \,\ \text{and} \,\ \,\ \alpha^h_{m+1}(e_1,\dots,e_1,e_2)=v_{21}\zeta_{2m+1}+v_{22}\zeta_{2m+2},$$ where $v_{ij}$ are smooth functions such that \[axes\] b\^2v\_[21]{}v\_[22]{}+v\_[11]{}v\_[12]{}=0, \_m=(v\_[11]{}\^2+b\^2v\_[21]{}\^2)\^[1/2]{}, \_[m]{}=(v\_[12]{}\^2+b\^2v\_[22]{}\^2)\^[1/2]{} and $\kappa_m,\mu_m$ denote the lengths of the semi-axes of the curvature ellipse $\mathcal{E}^h_{m}$.
Bearing in mind the definition of the higher fundamental forms, their symmetry and the ellipticity of the surface $h,$ we have $$\alpha_{s+1}^h\left(e_1,\dots,e_1,e_2\right)=\left(\nap_{e_2}\alpha_s^h\left(e_1,\dots,e_1\right)\right)^{N^h_s}=\left(\nap_{e_1}\alpha_s^h\left(e_1,\dots,,e_1,e_2\right)\right)^{N^h_s},$$ $$\alpha_{s+1}^h\left(e_1,\dots,e_1\right)=-b^2\left(\nap_{e_2}\alpha_s^h\left(e_1,\dots,e_1, e_2\right)\right)^{N^h_s}=\left(\nap_{e_1}\alpha_s^h\left(e_1,\dots,,e_1\right)\right)^{N^h_s}$$ for $s=m,m+1.$ From these we obtain $$\begin{aligned}
&\omega_{2m-1,2m+1}(e_1)=\frac{v_{11}}{\kappa_{m-1}},\label{-11} \,\ \,\ \omega_{2m-1,2m+2}(e_1)=\frac{v_{12}}{\kappa_{m-1}},\\
&\omega_{2m-1,2m+1}(e_2)=\frac{v_{21}}{\kappa_{m-1}},\label{-12} \,\ \,\ \omega_{2m-1,2m+2}(e_2)=\frac{v_{22}}{\kappa_{m-1}},\\
&\omega_{2m,2m+1}(e_1)=\frac{b v_{21}}{\kappa_{m-1}},\label{01} \,\ \,\ \omega_{2m,2m+2}(e_1)=\frac{b v_{22}}{\kappa_{m-1}},\\
&\omega_{2m,2m+1}(e_2)=-\frac{v_{11}}{b \kappa_{m-1}},\label{02} \,\ \,\
\omega_{2m,2m+2}(e_2)=-\frac{v_{12}}{b \kappa_{m-1}}, \\
&\omega_{2m+1,2m+3}(e_1)=\frac{b \kappa_{m+1}}{\kappa_m \mu_m}v_{22},\label{131} \,\ \,\ \omega_{2m+1,2m+3}(e_2)=\frac{\kappa_{m+1}}{b \kappa_m \mu_m}v_{12}, \\
&\omega_{2m+1,2m+4}(e_1)=-\frac{\kappa_{m+1}}{\kappa_m \mu_m}v_{12},\label{141} \,\ \,\ \omega_{2m+1,2m+4}(e_2)=\frac{\kappa_{m+1}}{\kappa_m \mu_m}v_{22},\\
&\omega_{2m+2,2m+3}(e_1)=-\frac{b \kappa_{m+1}}{\kappa_m \mu_m}v_{21}, \,\ \,\ \omega_{2m+2,2m+3}(e_2)=-\frac{\kappa_{m+1}}{b \kappa_m \mu_m}v_{11},\label{.}\\
&\omega_{2m+2,2m+4}(e_1)=\frac{\kappa_{m+1}}{\kappa_m \mu_m}v_{11}, \,\ \,\
\omega_{2m+2,2m+4}(e_2)=-\frac{\kappa_{m+1}}{\kappa_m \mu_m}v_{21}. \label{..} \end{aligned}$$
Let $\Pi\colon M^3\to L^2$ the natural projection of the circle bundle $$M^3=UN^h_{\tau_h}=\left\{(x,\delta)\in N^h_{m+1}\colon \|\delta\|=1, x\in L^2\right\}.$$ We parametrize $\Pi^{-1}(V)$ by $V\times{\mathbb{R}}$ via the map $$(x,\theta)\mapsto\left(x,\cos\theta \zeta_{2m+3}(x)+\sin\theta \zeta_{2m+4}(x)\right)$$ and consequently, we have $$\Psi_h(x,\theta)=\cos \theta \zeta_{2m+3}+\sin \theta \zeta_{2m+4}.$$
We notice that $\nap N^h_{m+1}\subseteq N_m^h\oplus N_{m+1}^h.$ Then we easily find that $$\begin{aligned}
\Psi_{h_*}E_i&=\big(\cos\theta\omega_{2m+3,2m+1}(e_i)+\sin\theta\omega_{2m+4,2m+1}(e_i)\big)\zeta_{2m+1}\\
&+\big(\cos\theta\omega_{2m+3,2m+2}(e_i)+\sin\theta\omega_{2m+4,2m+2}(e_i)\big)\zeta_{2m+2},\end{aligned}$$ where the vector fields $E_i\in TM,i=1,2,$ are given by $$E_i=e_i-\omega_{2m+3,2m+4}(e_i)\frac{\partial}{\partial\theta}.$$ Using equations , , and , we obtain \[dPsi\] \_[h\_\*]{}E\_1=((- b v\_[22]{}+v\_[12]{})\_[2m+1]{} +(b v\_[21]{}- v\_[11]{})\_[2m+2]{}) $$\begin{aligned}
\label{dPsi}
\Psi_{h_*}E_1&=\frac{\kappa_{m+1}}{\kappa_m\mu_m}\left(\left(- b v_{22}\cos\theta+v_{12}\sin\theta \right)\zeta_{2m+1}\right.\\
&\left.+\left(b v_{21}\cos\theta - v_{11}\sin\theta \right)\zeta_{2m+2}\right)\nonumber\end{aligned}$$ and \[dPsi’\] \_[h\_\*]{}E\_2=(-(+ v\_[22]{})\_[2m+1]{}+(+ v\_[21]{})\_[2m+2]{}) $$\begin{aligned}
\label{dPsi'}
\Psi_{h_*}E_2&=\frac{\kappa_{m+1}}{\kappa_m\mu_m}\left(-\left(\frac{v_{12}}{b}\cos\theta+ v_{22}\sin\theta\right)\zeta_{2m+1}\right.\\
&\left.+\left(\frac{v_{11}}{b}\cos\theta + v_{21}\sin\theta\right)\zeta_{2m+2}\right).\nonumber\end{aligned}$$ Additionally, we have \[theta\] \_[h\_\*]{}(/)=-\_[2m+3]{}+\_[2m+4]{}.
It follows that the normal bundle of the isometric immersion $\Psi_h$ is given by $$N_{\Psi_h}M=c\ {\rm{span}} \{h\}\oplus N^h_1\oplus\cdots\oplus N^h_{m-2}\oplus N^h_{m-1}.$$ It is easy to see that the first normal bundle of $\Psi_h$ is $
N_1^{\Psi_h}=N^h_{m-1}.
$ Moreover, it follows easily that the distribution $D={\rm{span}}\{\partial/\partial\theta\}$ is contained in the nullity distribution $\D_{\Psi_h}$ of $\Psi_h.$ In particular, from equation and the Gauss formula we derive that $
\n_{\partial/\partial\theta}\partial/\partial\theta=0.
$ This implies that the distribution $D$ is totally geodesic.
It remains to show that the mean curvature of the immersion $\Psi_h$ is constant along each integral curve of $D$. The shape operator $A_{\zeta_{2m-j}}$ of $\Psi_h$ with respect to the normal direction $\zeta_{2m-j}, j=0,1,$ is given by the Weingarten formula as \[-dPsi\] -\_[h\_\*]{}(A\_[\_[2m-j]{}]{}E\_i)=\_[e\_i]{}\_[2m-j]{}-(\_[e\_i]{}\_[2m-j]{})\^[N\^h\_[m-2]{}N\^h\_[m-1]{}]{} =(\_[e\_i]{}\_[2m-j]{})\^[N\^h\_m]{},i=1,2, since $\zeta_{2m-1},\zeta_{2m}\in N_{m-1}^h$. Here, $\tilde{\n}$ stands for the induced connection of the induced bundle $h^*T\mathbb{Q}_c^{N+1}.$ Using -, equations yield $$\begin{aligned}
&\Psi_{h_*}\left(A_{\zeta_{2m-1}}E_1\right)=-\frac{1}{\kappa_{m-1}}\left(v_{11}\zeta_{2m+1}+v_{12}\zeta_{2m+2}\right),\label{po}\\
&\Psi_{h_*}\left(A_{\zeta_{2m-1}}E_2\right)=-\frac{1}{\kappa_{m-1}}\left(v_{21}\zeta_{2m+1}+v_{22}\zeta_{2m+2}\right),\label{popo}\\
&\Psi_{h_*}\left(A_{\zeta_{2m}}E_1\right)=-\frac{b}{\kappa_{m-1}}\left(v_{21}\zeta_{2m+1}+v_{22}\zeta_{2m+2}\right),\label{popopo}\\
&\Psi_{h_*}\left(A_{\zeta_{2m}}E_2\right)=\frac{1}{b \kappa_{m-1}}\left(v_{11}\zeta_{2m+1}+v_{12}\zeta_{2m+2}\right).
\label{popopopo}\\ \nonumber\end{aligned}$$
We may set \[2m-1\] A\_[\_[2m-1]{}]{}E\_i=\_[i1]{}E\_1+\_[i2]{}E\_2 A\_[\_[2m]{}]{}E\_i=\_[i1]{}E\_1+\_[i2]{}E\_2, i=1,2, where $\lambda_{ij}$ and $\gamma_{ij}$ are smooth functions on the manifold $M^3$. From equations , , and the first one of , we obtain $$\lambda_{11}=\frac{1}{\kappa_{m-1}\kappa_{m+1}}\left(\left(v_{11}^2+v_{12}^2\right)\cos\theta+ b\left(v_{11}v_{21}+v_{12}v_{22}\right)\sin\theta\right)$$ and $$\lambda_{22}=\frac{1}{\kappa_{m-1}\kappa_{m+1}}\left(- b^2\left(v_{21}^2+v_{22}^2\right)\cos\theta+ b\left(v_{11}v_{21}+v_{12}v_{22}\right)\sin\theta\right).$$ Hence $${\rm trace} A_{\zeta_{2m-1}}=\frac{1}{\kappa_{m-1}\kappa_{m+1}}\left(\left(v^2_{11}+v^2_{12}- b^2v_{21}^2-b^2v_{22}^2\right)\cos\theta+ 2b\left(v_{11}v_{21}+v_{12}v_{22}\right)\sin\theta\right).$$
Similarly, from equations , , and the second of , we find that $$\gamma_{11}=\frac{1}{\kappa_{m-1}\kappa_{m+1}}\left(b\left(v_{11}v_{21}+v_{12}v_{22}\right)\cos\theta+ b^2\left(v_{21}^2+v_{22}^2\right)\sin\theta\right)$$ and $$\gamma_{22}=\frac{1}{\kappa_{m-1}\kappa_{m+1}}\left(b\left(v_{11}v_{21}+v_{12}v_{22}\right)\cos\theta-\left(v_{11}^2+v_{12}^2\right)\sin\theta\right).$$ Then, it follows that $${\rm trace} A_{\zeta_{2m}}=\frac{1}{\kappa_{m-1}\kappa_{m+1}}\left(2b\left(v_{11}v_{21}+v_{12}v_{22}\right)\cos\theta-\left(v^2_{11}+v^2_{12}- b^2v_{21}^2-b^2v_{22}^2\right) \sin\theta\right).$$ Thus, the mean curvature of the isometric immersion $\Psi_h$ is given by $$\|{\mathcal{H}}_{\Psi_h}\|^2=\frac{1}{(3\kappa_{m-1}\kappa_{m+1})^2}\Big(\big(v_{11}^2+v_{12}^2+b^2v_{21}^2+b^2v_{22}^2\big)^2-4\big(v_{11}^2+b^2v_{21}^2\big)^2\big(v_{12}^2+b^2v_{22}^2\big)^2\Big).$$ Using equations , the above equation becomes $$\|{\mathcal{H}}_{\Psi_h}\|=\frac{|\kappa^2_m-\mu^2_m|}{3\kappa_{m-1}\kappa_{m+1}}.$$ It is obvious that the mean curvature of the isometric immersion $\Psi_h$ is constant along each integral curve of the distribution $D.$ This completes the proof of the direct statement of the theorem for $m\geq2.$ The case $m=1$ can be treated in a similar manner. In this case, the mean curvature of $\Psi_h$ is given by $$\|{\mathcal{H}}_{\Psi_h}\|=\frac{|\kappa^2_1-\mu^2_1|}{3\kappa_{2}^2}.$$
Conversely, let $f\colon M^3\to\mathbb{S}^{3+p}$ be a nonminimal isometric immersion. Suppose that $M^3$ carries a totally geodesic distribution $D$ of rank one satisfying $D(x)\subseteq \D_f(x)$ for any $x\in M^3$ such that the mean curvature is constant along each integral curve of $D.$ From Proposition \[sphelliptic\], we know that $f$ is an elliptic submanifold and its first curvature ellipse is a circle. Hereafter, we work on a connected component of an open dense subset where $f$ is nicely curved.
We consider a polar surface $h\colon L^2\to \mathbb{Q}^{p-c+4}_c$ to the immersion $f,$ where $c=0$ if $p$ is even and $c=1$ if $p$ is odd. We notice that $\tau^0_f=\tau_h-1.$ Using equations , we conclude that the curvature ellipse $\mathcal{E}^h_{\tau_h-2}$ of the surface $h$ is a circle and the curvature ellipse $\mathcal{E}^h_{\tau_h-1}$ is nowhere a circle.
We claim that the last curvature ellipse $\mathcal{E}^h_{\tau_h}$ is a circle. We notice that $N^h_{\tau_h}={\rm{span}}\{\xi,\eta \},$ where the sections $\xi,\eta$ of the normal bundle $N_hL$ are given by $\xi=f\circ \pi\;\;\text{and}\;\; \eta=f_*e_3\circ\pi.$ Here $\pi$ denotes the natural projection $\pi\colon M^3\to L^2$ onto the fixed cross section $L^2\subset M^3$ to the foliation generated by the distribution $D.$
Let $X_1,\dots,X_{\tau_h}\in TL$ be arbitrary vector fields. By we have $N^h_{\tau_h-1}=N^f_0=f_*D^\perp.$ Thus, there exists $X\in \Gamma(D^\perp)$ such that $$\alpha^h_{\tau_h}\left(X_1,\dots,X_{\tau_h}\right)=f_*X.$$ For every vector field $Y\in TL$ there exists a vector field $Z\in \Gamma(D^\perp)$ such that $Y=\pi_*Z.$ Then we have $$\begin{aligned}
\alpha^h_{\tau_h+1}(X_1,\dots,X_{\tau_h},Y)&=\left(\nap_Y\alpha^h_{\tau_h}(X_1,\dots,X_{\tau_h})\right)^{N^h_{\tau_h}}\nonumber\\
&= -\<f_*X,f_*Z\>\xi-\<f_*X, \tilde{\n}_Zf_*e_3\>\eta.\end{aligned}$$ Using the Gauss formula and the definition of the splitting tensor, the above equation becomes $$\alpha^h_{\tau_h+1}(X_1,\dots,X_{\tau_h},Y)=-\<X,Z\>\xi+\<X,{\mathcal{C}}_3Z\>\eta.$$ From Proposition \[sphelliptic\], we know that the splitting tensor in the direction of $e_3$ is the almost complex structure $J_0^f\colon D^\perp\to D^\perp$ of $f.$ Hence, we obtain $$\alpha^h_{\tau_h+1}(X_1,\dots,X_{\tau_h},Y)=-\<X,Z\>\xi+\<X,J^f_0Z\>\eta.$$ On account of $\pi_*\circ J^f_0=J^h_0\circ \pi_*,$ we have $J^h_0Y=\pi_*J^f_0Z.$ Thus, it follows that $$\alpha^h_{\tau_h+1}(X_1,\dots,X_{\tau_h},J^h_0Y)=-\<X,J_0^hZ\>\xi-\<X,Z\>\eta.$$ Since $\xi,\eta $ is an orthonormal frame of the subbundle $N^h_{\tau_h},$ it is now obvious that the normal vector fields $\alpha^h_{\tau_h+1}(X_1,\dots,X_{\tau_h+1},Y)$ and $\alpha^h_{\tau_h+1}(X_1,\dots,X_{\tau_h},J_0^hY)$ are of the same length and perpendicular. Hence, the last curvature ellipse of the polar surface $h$ is a circle.
We observe that the isometric immersion $f$ is written as the composition $f=\Psi_h\circ F,$ where $F\colon U\to UN_{\tau_h}^h$ is the local isometry given by $F(x)=(\pi(x),f(x)), \, x\in U,
$ and $U$ is the saturation of the cross section $L^2\subset M^3$. This completes the proof.
[[**.** ]{}]{}[*It follows from the computation of the mean curvature of the submanifold $\Psi_h$ in the proof of Theorem \[main0\], that the mean curvature is constant by properly choosing the elliptic surface $h.$ Ejiri [@Ejiri] proved that tubes in the direction of the second normal bundle of a pseudoholomorphic curve in the nearly Kähler sphere ${\mathbb{S}}^6$ have constant mean curvature. Opposed to our case, the index of relative nullity of these tubes is zero.*]{}
*Proof of Theorem \[main1\]:* Assume that the isometric immersion $f$ is neither minimal nor locally a cylinder. Proposition \[sphelliptic\] implies that $f$ is spherical. Thus, from Theorem \[main0\] we know that for each point on an open dense subset there exist an elliptic surface $h\colon L^2\to \mathbb{Q}^{p-c+4}_c,$ where $c=0$ if $p$ is even and $c=1$ if $p$ is odd, a neighborhood $U$ and a local isometry $F\colon U\to UN^h_{\tau_h}$ such that $f=\Psi_h\circ F.$ In fact, the elliptic surface $h$ is a polar to $f.$ Moreover, we know that the curvature ellipses $\mathcal{E}^h_{\tau_h-2}$ and $\mathcal{E}^h_{\tau_h}$ are circles, while the curvature ellipse $\mathcal{E}^h_{\tau_h-1}$ is nowhere a circle.
Now, we consider a bipolar surface $g$ to $f,$ that is, a polar surface to the elliptic surface $h.$ Then it follows from equations that the curvature ellipse $\mathcal{E}^g_0$ of $g$ is a circle. This means that the bipolar surface is minimal. Furthermore, its first curvature ellipse is nowhere a circle and the second one is a circle. That the isometric immersion $f$ is locally parametrized by follows from the fact that $f=\Psi_h\circ F$ and $N_0^g=N_{\tau_h}^h.$ This completes the proof.
Minimal surfaces
----------------
The following proposition provides a way of constructing minimal surfaces in ${\mathbb{R}}^6$ that satisfy the properties that are required in Theorem \[main1\](iii).
[[**.** ]{}]{}\[Ricci\] Let $\hat{g}\colon M^2\to {\mathbb{R}}^6$ be the minimal surface defined by $$\hat{g}=\cos\varphi g_\theta\oplus\sin\varphi g_{\theta+\pi/2},$$ where $g_\theta, \theta\in [0,\pi), $ is the associated family of a simply connected minimal surface $g\colon M^2\to {\mathbb{R}}^3$ with negative Gaussian curvature, and $\oplus$ denotes the orthogonal sum with respect to an orthogonal decomposition of ${\mathbb{R}}^6$. If $\varphi\neq \pi/4,$ then its first curvature ellipse is nowhere a circle and its second curvature ellipse is a circle.
Let $g\colon M\to{\mathbb{R}}^n$ be an oriented minimal surface. The complexified tangent bundle $TM\otimes \mathbb{C}$ is decomposed into the eigenspaces $T^{\prime}M$ and $T^{\prime \prime}M$ of the complex structure $J$, corresponding to the eigenvalues $i$ and $-i.$ The $r$-th fundamental form $\alpha^g_r$, which takes values in the normal subbundle $N_{r-1}^g$, can be complex linearly extended to $TM\otimes\mathbb{C}$ with values in the complexified vector bundle $N_{r-1}^g\otimes \mathbb{C}$ and then decomposed into its $(p,q)$-components, $p+q=r,$ which are tensor products of $p$ differential 1-forms vanishing on $T^{\prime \prime}M$ and $q$ differential 1-forms vanishing on $T^{\prime}M.$ The minimality of $g$ is equivalent to the vanishing of the $(1,1)$-component of the second fundamental form. Hence, the $(p,q)$-components of $\alpha^g_r$ vanish unless $p=r$ or $p=0$.
It is known (see [@v Lem. 3.1]) that the curvature ellipse of order $r-1$ is a circle if and only if the $(r,0)$-component of $\alpha^g_r$ is isotropic, that is $$\<\alpha^g_r(X,\dots,X),\alpha^g_r(X,\dots,X)\>=0$$ for any $X\in T^{\prime}M,$ where $\<\cdot,\cdot\>$ denotes the bilinear extension over the complex numbers of the Euclidean metric.
*Proof of Proposition \[Ricci\]:* We choose a local tangent orthonormal frame $e_1,e_2$ such that the shape operator $A$ of $g$ satisfies $AE=k\bar{E},$ where $E=e_1+ie_2$ and $k$ is a positive smooth function. The associated family satisfies $g_{\theta_*}=dg\circ J_\theta,$ where $J_\theta=\cos\theta I+\sin\theta J$ and $I$ is the identity endomorphism of the tangent bundle. Then we have \[dghat\] \_\*E=e\^[-i]{}(ǧ\_\*E,-iǧ\_\*E).
Using the Gauss formula and the fact that the shape operator $A_\theta$ of $g_{\theta}$ is given by $A_\theta=A\circ J_\theta$, we find that the second fundamental form $\hat{\a}$ of $\gh$ satisfies \[ff2\] (E,E)=2ke\^[-i]{}(Ň, -iŇ), where $N$ is the Gauss map of $g$. It is obvious that $\hat{\a}(E,E)$ is not isotropic if $\v\neq \pi/4$, which implies that the first curvature ellipse of $\gh$ is nowhere a circle.
Differentiating with respect to $E$ and using the Weingarten formula, we obtain $$\tilde{\n}_E\hat{\a}(E,E)=2e^{-i\t}E(k)\left(\cos\v N,-i\sin\v N\right)-2k^2e^{-i\t}\left(\cos\v g_*\bar{E},-i\sin\v g_*\bar{E}\right),$$ where $\tilde{\n}$ is the connection of the induced bundle of $\gh$. Since $\gh_*E$ and $\gh_*\bar{E}$ span $N_0^\gh\otimes{\mathbb{C}},$ the above equation along with yield $$\big(\tilde{\n}_E\hat{\a}(E,E)\big)^{N_0^\gh\otimes{\mathbb{C}}}=-2k^2e^{-2i\t}\cos2\v \gh_*\bar{E}.$$ Equation implies that $
N_1^{\gh}\otimes{\mathbb{C}}={\rm{span}}_{{\mathbb{C}}}\{\xi,\eta\},
$ where $\xi=(N,0)$ and $\eta=(0,iN)$. It follows directly that $$\big(\tilde{\n}_E\hat{\a}(E,E)\big)^{N_1^\gh\otimes{\mathbb{C}}}=2e^{-i\t}E(k)\left(\cos\v N,-i\sin\v N\right).$$ Using the above and since the $(3,0)$-component of the third fundamental form of $\gh$ is given by $$\hat{\a}_3(E,E,E)=\big(\tilde{\n}_E\hat{\a}(E,E)\big)^{\left(N^{\gh}_0\otimes{\mathbb{C}}\oplus N^{\gh}_1\otimes{\mathbb{C}}\right)^{\perp}},$$ we obtain $$\hat{\a}_3(E,E,E)=k^2e^{-i\t}\sin2\v\left(-\sin\v g_*\bar{E},i\cos\v g_*\bar{E}\right).$$ Thus the (3,0)-component of the third fundamental form of $\gh$ is isotropic, and the proof is completed.
Submanifolds with constant mean curvature
=========================================
In this section, we provide the proofs of the applications of our main results to submanifolds with constant mean curvature.
*Proof of Theorem \[main2\]:* The manifold $M^n$ is written as the disjoint union of the subsets $$M_{n-i}=\left\{x\in M^n:\nu(x)=n-i\right\}, \,\ \,\ i=1,2.$$
Assume that the subset $M_{n-2}$ is nonempty. Then, using Proposition \[nu0\] it follows from Theorem \[main\] for $n\geq4$, or Theorem \[main1\] for $n=3$ and $p=1,$ that the isometric immersion $f$ is locally a cylinder over a surface on $M_{n-2}$.
Suppose that the interior ${\rm{int}}(M_{n-1})$ of the subset $M_{n-1}$ is nonempty. It follows from the Codazzi equation that the relative nullity distribution is parallel in the tangent bundle along ${\rm{int}}(M_{n-1})$. Thus, the tangent bundle splits as an orthogonal sum of two parallel orthogonal distributions of rank one and $n-1$ on ${\rm{int}}(M_{n-1})$. By the De Rham decomposition theorem, ${\rm{int}}(M_{n-1})$ splits locally as a Riemannian product of two manifolds of dimension one and $n-1.$ Then, the Gauss equation yields $c=0$. Since the second fundamental form is adapted to the orthogonal decomposition of the tangent bundle, it follows that $f$ is a cylinder over a curve in ${\mathbb{R}}^{p+1}$ with constant first Frenet curvature (see [@da Th. 8.4]).
We observe that the open subset $V={\rm{int}}(M_{n-1})\cup M_{n-2}$ is dense on $M^n$, and this completes the proof.
In order to proceed to the proofs of the applications of our main results, we need to recall Florit’s estimate of the index of relative nullity for isometric immersions with nonpositive extrinsic curvature. The *extrinsic curvature* of an [isometric immersion ]{}$f\colon M^n\to\tilde{M}^{n+p}$ for any point $x\in M^n$ and any plane $\sigma\in T_xM$ is given by $$K_f(\sigma)=K_M(\sigma)-K_{\tilde{M}}(f_*\sigma),$$ where $K_M$ and $K_{\tilde{M}}$ are the sectional curvatures of $M^n$ and $\tilde{M}^{n+p},$ respectively. Florit [@f] proved that the index of relative nullity satisfies $\nu\geq n-2p$ at points where the extrinsic curvature of $f$ is nonpositive.
*Proof of Corollary \[T5\]:* We have that the index of relative nullity of $f$ satisfies $\nu\geq n-2.$ Theorem \[main2\] implies that $c=0$ and, on an open dense subset, $f$ splits locally as a cylinder over a surface in ${\mathbb{R}}^3$ of constant mean curvature. By real analyticity, the splitting is global. If $M^n$ is complete, then the surface is also complete with nonnegative Gaussian curvature. That the surface is a cylinder over a circle follows from [@ko].
*Proof of Corollary \[C6\]:* Assume that the hypersurface is nonrigid. Then, the well-known Beez-Killing Theorem (see [@da]) implies that the index of relative nullity satisfies $\nu\geq n-2.$ The result follows from Corollary \[main2’\].
*Proof of Theorem \[T7\]:* We suppose that the hypersurface is nonminimal.
We first assume that the extrinsic curvature is nonnegative. If $c=0$, a result of Hartman [@hart] asserts that $f(M^n)=\mathbb{S}_R^{k}\times{\mathbb{R}}^{n-k},$ where $1\leq k\leq n.$ If $c=1,$ then $M^n$ is compact by the Bonnet-Myers theorem. According to [@nomichu Th. 2], $f$ is totally umbilical.
In the case of nonpositive extrinsic curvature, the result follows from Corollary \[T5\].
*Proof of Theorem \[T8\]:* According to the aforementioned result due to Florit [@f], we have $\nu\geq n-4.$ Clearly the manifold $M^n$ is written as the disjoint union of the subsets $$M_{n-i}=\left\{x\in M^n:\nu(x)=n-i\right\}, \,\ \,\ i=1,2,3.$$
We distinguish the following cases.
*Case I*: We suppose that the subset $M_{n-4}$ is nonempty. According to Proposition \[nu0\], this subset is open. Using [@fz Th. 1], we have that on an open dense subset of $M_{n-4}$ the immersion $f$ is locally a product $f=f_1\times f_2$ of two hypersurfaces $f_i\colon M^{n_i}\to {\mathbb{R}}^{n_i+1},i=1,2,$ of nonpositive sectional curvature. The assumption that $f$ has constant mean curvature implies that both hypersurfaces have constant mean curvature as well. Each hypersurface $f_i,i=1,2,$ has index of relative nullity $n_i-2.$ Then it follows from Corollary \[main2’\] that the submanifold is locally as in part (iii) of the theorem.
*Case II*: Assume that the interior of the subset $M_{n-3}$ is nonempty. Due to [@fz1 Th. 1], on an open dense subset of ${\rm{int}}(M_{n-3})$, $f$ is written locally as a composition $f=h\circ F$, where $h=\gamma\times id_{{\mathbb{R}}^{n-1}} \colon {\mathbb{R}}\times{\mathbb{R}}^n\to{\mathbb{R}}^{n+2}$ is cylinder over a unit speed plane curve $\gamma(s)$ with nonvanishing curvature $k(s)$ and $F\colon M^n \to{\mathbb{R}}^{n+1}$ is a hypersurface. The second fundamental form of $f$ is given by $$\a^f(X,Y)=h_*\a^F(X,Y)+\a^h\left(F_*X,F_*Y\right),\;\;X,Y\in TM.$$ From this we obtain $
k\< F_*T,\partial/\partial s\>^2=0
$ for any $T \in \Delta_f.$ This implies that the height function $F_a=\<F, \partial/\partial s\>$ is constant along the leaves of $\Delta_f.$ Then, the mean curvature vector field of $f$ is given by $$n{\mathcal{H}}_f=nH_Fh_*\xi+k\circ F_a\|{\mbox{grad}}F_a\|^2\eta,$$ where $\xi,\eta $ stand for the Gauss maps of $F$ and $h$, respectively. Using that $$\|{\mbox{grad}}F_a\|^2=1-\<\xi, a\>^2,$$ it follows that the mean curvature of $F$ is given as in part (ii) of the theorem.
*Case III*: Suppose that the subset $M_{n-2}\cup M_{n-1}$ has nonempty interior. Then Theorem \[main2\] implies that the submanifold is locally as in part (i) of the theorem, and this completes the proof.
*Proof of Theorem \[T11\]:* It follows from [@da Th. 5.1] that $ \tilde{c}\geq c$ if $n\geq4$. We distinguish the following cases.
*Case I*: We assume that $\tilde{c}>c.$ From [@O' Lem. 8] or [@dt Prop. 9], we have that the second fundamental form splits orthogonally and smoothly as $$\alpha^f(\cdot,\cdot)=\beta (\cdot,\cdot)+\sqrt{\tilde{c}-c}\ \<\cdot,\cdot\>\eta,$$ where $\eta $ is a unit normal vector field and $\beta$ is a flat bilinear form. Thus, the shape operator $A_\xi,$ associated to a unit normal vector field $\xi$ perpendicular to $\eta,$ has $\mathrm{rank}A_\xi\leq1.$ The mean curvature $H$ of $f$ is given by $$H^2=\frac{k^2}{n^2} + \frac{\tilde{c}-c}{n},$$ where $k=\mathrm{trace}A_\xi.$ Obviously, the function $k$ is constant. If $k=0,$ then $f$ is totally umbilical.
Assume now that $k\neq0.$ Let $X$ be a unit vector field such that $A_\xi X=kX$. The Codazzi equation $$(\n_XA_{\eta})T-(\n_TA_{\eta})X=A_{\n_X^\perp\xi}T-A_{\n_T^\perp\xi}X$$ implies that $$\n_T^\perp\xi=\n_T^\perp\eta=0$$ for any $T\in \ker A_\xi$. Moreover, from the Codazzi equation $$(\n_XA_{\xi})T-(\n_TA_{\xi})X=A_{\n_X^\perp\eta}T-A_{\n_T^\perp\eta}X$$ it follows that $$\n_T X=0 \;\;\text{and}\;\;\<\n_X T, X\>=\<\n^\perp_T\xi, \eta\>$$ for any $T\in \ker A_\xi$. Hence the orthogonal distributions $D^1=\mathrm{span}\{X\}$ and $D^{n-1}=\ker A_\xi$ are parallel. By the De Rham decomposition theorem, the manifold splits locally as a Riemannian product $M_{\tilde{c}}^n=M^1\times M^{n-1}$. Consequently, we have $\tilde{c}=0$ and $c=-1$. Clearly $M^{n-1}$ is flat and the second fundamental form is adapted to this decomposition. Then it follows that $f$ is a composition $f=i\circ F$, where $i\colon {\mathbb{R}}^{n+1}\to \mathbb H^{n+2}$ is the inclusion as a horosphere and $F\colon M^n_{\tilde{c}} \to {\mathbb{R}}^{n+1}$ is the cylinder over a circle (see [@da Th. 8.4]).
*Case II*: We suppose that $c=\tilde{c}.$ It is known that $\nu\geq n-2$ (see Example 1 and Corollary 1 in [@Moo]). Then, the result follows from Theorem \[main2\].
If $n=3,$ then Theorem \[main1\] implies that either $c=0$ and $f(M)$ is an open subset of a cylinder over a flat surface $g\colon M^2\to {\mathbb{R}}^4$ of constant mean curvature, or $c=1$ and $f$ is parametrized by . In the latter case, by Proposition \[sphelliptic\] we have that $f$ is either totally geodesic or elliptic. However, the ellipticity of $f$ implies that the sectional curvature cannot be equal to one. This completes the proof.
*Proof of Theorem \[T9\]:* Assume that $f$ is nonminimal. According to Abe [@abe], the index of relative nullity satisfies $\nu\geq n-2.$ Thus, Corollary \[main2’\] and Proposition \[cylinder\] conclude the proof.
*Proof of Theorem \[T10\]:* Using [@fz2], it follows that $\nu\geq n-4.$ The rest of the proof is omitted since it is similar to the proof of Theorem \[T8\].
The following example produces submanifolds satisfying the conditions in part (ii) of Theorem \[T8\] or \[T10\].
[[**.** ]{}]{}[*Let $F=g\times id_{{\mathbb{R}}^{n-2}}\colon U\times {\mathbb{R}}^{n-2}\to {\mathbb{R}}^{n+1}$ be a cylinder over a rotational surface $g(x,\theta)=(x\cos \theta, x\cos \theta, \varphi(x)), (x,\theta)\in U,$ where $\varphi(x)$ is a smooth function. We consider a cylinder $h=\gamma \times id_{{\mathbb{R}}^n}$ in ${\mathbb{R}}^{n+2}$ over a unit speed plane curve $\gamma$ with curvature $k$. Then the isometric immersion $f=h\circ F$ satisfies the conditions in part (ii) of Theorems \[T8\] and \[T10\], with constant constant curvature $H$ and $a=(1,0,\dots,0)$, if the function $\varphi(x)$ solves the ordinary differential equation $$\varphi \varphi^{\prime \prime}-1-\varphi^{\prime \, 2}=\pm\varphi\sqrt{(1+\varphi^{\prime \, 2})\left(n^2H^2(1+\varphi^{\prime \, 2})^2-k^2\right)}.$$ In particular, $g$ can be chosen as a Delaunay surface and $\gamma$ as the curve with curvature $k=c_0(1+\varphi^{\prime \, 2})$ for a constant $c_0$ such that $0<|c_0|<n|H|$.* ]{}
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Athina Eleni Kanellopoulou\
University of Ioannina\
Department of Mathematics\
Ioannina–Greece\
e-mail: alinakanellopoulou@gmail.com
Theodoros Vlachos\
University of Ioannina\
Department of Mathematics\
Ioannina–Greece\
e-mail: tvlachos@uoi.gr
|
---
abstract: 'The classical Gibbs-Donnan equilibrium describes excess osmotic pressure associated with confined colloidal charges embedded in an electrolyte solution. In this work, we extend this approach to describe the influence of multivalent ion binding on the equilibrium force acting on a charged rod translocating between two compartments, thereby mimicking ionic effects on force balance during *in vitro* DNA ejection from bacteriophage. The subtle interplay between Gibbs-Donnan equilibrium and adsorption equilibrium leads to a non-monotonic variation of the ejection force as multivalent salt concentration is increased, in qualitative agreement with experimental observations.'
author:
- 'M. Castelnovo'
- 'A. Evilevitch'
title: 'Binding Effects in Multivalent Gibbs-Donnan Equilibrium'
---
Introduction
============
The classical Gibbs-Donnan equilibrium is usually invoked to explain various behaviors in ionic systems: swelling of polyelectrolyte gels, membrane potentials and osmotic shock experiments of cells or viruses [@refgen]. The physics behind these phenomena is associated with some colloidal charge (usually synthetic or bio-polyelectrolytes) confined within a subpart of the system, while small co- *and* counter-ions can diffuse freely in and out of this region [@donnan]. Due to the electroneutrality inside the compartment, this confinement leads to some excess of counterions relative to the outer compartment, and therefore an additional osmotic pressure is set up inside. This effect is nowadays well-known and characterized experimentally on model systems [@raspaud].
From a theoretical point of view, the Gibbs-Donnan approach is a simple and versatile way of incorporating electrostatic effects up to leading order in *inhomogeneous* systems, through the use of ionic chemical potential balance between two *homogeneous* phases and the electroneutrality condition. In systems either where colloidal charge is large or where the buffer solution contains multivalent ions, the Gibbs-Donnan theory has to be extended to take into account the electrostatic binding of counterions onto the colloidal charge [@reiss]. Motivated by recent *in vitro* experiments measuring multivalent salt influence on the force ejecting DNA from bacteriophage Lambda [@alexnew], we propose in this Letter such an extension in order to include multivalent ion binding effects in the classical Gibbs-Donnan approach. Indeed, we expect the interplay between Z-valent ion partitioning between inside and outside the viral capsid (pure Donnan effect), and the binding statistics onto DNA inside and outside (pure Langmuir-like adsorption statistics), to give raise to non-trivial translocation behaviors. In a slightly different context for example, the balance of these two effects has been shown by Klein Wolterink *et al.* to produce non-monotonic variation of the radius of gyration of a strongly charged polyelectrolyte star with respect to salt concentration [@borisov]. As will be shown in this work, this simple generalization of Gibbs-Donnan approach is able to explain at least qualitatively the non-monotonicity of ejecting force as function of multivalent salt concentration. In particular, we find that the repulsive force decreases and then increases, upon increase in the added salt concentration. The minimum in the force is associated to the charge neutralization point of DNA, which correlates well with experimental results [@alexnew].
Model
=====
![Sketch of the model. The two-sided arrows indicate different equilibria taking place in the system[]{data-label="cartoon"}](figure1.eps)
The model system considered in this work is depicted in figure \[cartoon\]. It consists of a cavity of volume $V_0$ embedded into a larger volume $V>>V_0$. The whole volume $V$ is filled with a (Z:1) electrolyte at an adjustable concentration. The cavity is permeable to both species (Z-valent cations and monovalent anions) of the electrolyte. A translocation gate on the cavity allows a rigid rod bearing $L$ uniformly distributed negative charges to move between inside and outside the cavity. For a given configuration, there are $L_{in}$ and $L-L_{in}$ negative charges respectively inside and outside the cavity. Multivalent cations are expected to bind onto the charged rod, due to strong electrostatic interactions. The free energy of this system is written as $F_{tot}=F_{layers}+F_{free}+F_{neutral}$ where the first term is associated with the Z-cation adsorbed layers on the rod, the second term is associated to remaining free ions in the solution, and finally the last term takes into account any non-ionic effects. It can include for example the effect of a neutral osmotic pressure difference between inside and outside the cavity due to neutral polymers that cannot enter the cavity, as in the experimental setup used by Evilevitch *et al.* [@alexold; @alexnew], or bending effects in the case where the rod has a finite bending modulus [@rob1; @rob2; @tzlil].
Following the spirit of the original Gibbs-Donnan approach, the specific features of electrostatic binding are neglected, and the binding of Z-valent cations onto the rod is approximated by Langmuir-like adsorption isotherms [@donnan]. Each negative charge of the rod is therefore assumed to be a potential discrete binding site for a single Z-valent cation. The energy gain associated with a single binding event is given by $-kT\epsilon_A$. This energy is of order $\epsilon_A\simeq Zl_B/b$, where $b$ is the typical size of rod unit and $l_B$ the Bjerrum length. This neglects any correlation effects among adsorbed Z-valent ions. It can be checked *a posteriori* that including a certain level of correlations does not change the qualitative picture drawn by this simple model. Denoting by $N_{Ain}$ and $N_{Aout}$ respectively the number of adsorbed ions on the rod inside and outside the cavity, the free energy of the adsorbed layers is $F_{layers}=F_A(N_{Ain},L_{in})+F_A(N_{Aout},L-L_{in})$, where the adsorption free energy of ions on a one dimensional lattice reads $$\frac{F_A(N_A,L)}{kT}=N_A \ln \frac{N_A}{L}+ (L-N_A) \ln \left(1-\frac{N_A}{L}\right)-N_A\epsilon_A$$ The first two terms describe the mixing entropy of adsorbed ions, and the last one is the adsorption energy. Assuming a unique value $b^3$ for the molar volume of each species (Z-cations, anions, and negative charges on the rod) for the sake of simplicity, the free energy of remaining free ions in the solution is given by the sum of translational free energies $F_T(N,V)=kTN(\ln (Nb^3/V)-1)$ of Z-cations and anions, respectively, inside and outside the cavity: $$\begin{aligned}
F_{free} & = & \bigg\{ F_T(N_{Zin}-N_{Ain},V_0-b^3L_{in})+F_T(ZN_{Zin}-L_{in},V_0-b^3L_{in})\bigg\}\nonumber\\
& & +\bigg\{F_T(N_Z-N_{Zin}-N_{Aout},V-V_0-b^3(L-L_{in}))\nonumber \\
& & +F_T(Z(N_Z-N_{Zin})-(L-L_{in}),V-V_0-b^3(L-L_{in}))\bigg\}\end{aligned}$$ Note that the number of Z-cations inside the cavity is $N_{Zin}$, including both free and adsorbed ions, while the total number of Z-cations in the solution (both inside and outside the cavity, free and adsorbed) is $N_Z$. Following Donnan, electroneutrality conditions have been taken into account inside and outside the cavity, so that the numbers of inner and outer anions are $N_{-in}=ZN_{Zin}-L_{in}$ and $N_{-out}=Z(N_Z-N_{Zin})-(L-L_{in})$.
At equilibrium, the free energy of the system is minimum with respect to $N_{Ain},N_{Aout},N_{Zin}$ and $L_{in}$. The first three minimization conditions mean simply that the adsorbed layers are in equilibrium with the free Z-ions of the solution, both inside and outside the cavity (Langmuir balance), and that free Z-ions inside the cavity are in equilibrium with free Z-ions outside the cavity (Gibbs-Donnan balance). These equations are written in the limit $V>>V_0$, introducing species concentration $n_{Zin}=N_{Zin}/V_0,n_Z=N_Z/V,\phi_{in}=N_{Ain}/L_{in},\phi_{out}=N_{Aout}/(L-L_{in}),l_{in}=L_{in}/V_0,l=L/V$ $$\begin{aligned}
\label{phiin}\frac{\phi_{in}}{1-\phi_{in}} & = & (n_{Zin}-\phi_{in}l_{in})b^3e^{\epsilon_A}\\
\label{phiout}\frac{\phi_{out}}{1-\phi_{out}} & = & (n_{Z}-\phi_{out}l)b^3e^{\epsilon_A}\\
\label{donnan1}\left(n_{Zin}-\phi_{in}l_{in}\right)\left(Zn_{Zin}-l_{in}\right)^Z & = & \left(n_Z-\phi_{out}l\right)\left(Zn_{Z}-l\right)^Z\end{aligned}$$ For a given value of $l_{in}$, these equations are solved self-consistently and their solutions allow calculation of properties of interest for the translocation problem, like the ionic contribution to the force acting on the rod $f_{ionic}=\frac{\partial (F_{layers}+F_{free})}{\partial L_{in}b}$. Under equilibrium conditions, the value of $l_{in}$ is set by minimization of the free energy with respect to $L_{in}$. Introducing the force contribution associated with non-ionic features $f_{neutral}=\frac{\partial F_{neutral}}{\partial L_{in}b}$, this equation is simply the force balance on the rod $f_{neutral}+f_{ionic}=0$. In this Letter, we focus on the behavior of the ionic force for different solution conditions at fixed $l_{in}$, rather than determining the equilibrium partition of the rod inside and outside the capsid. Indeed, the value of $l_{in}$ is dependent on the choice of neutral force contribution $f_{neutral}$, which is beyond the scope of this work [@alexnew].
Results and discussion
======================
In the absence of any analytical solutions for the equilibrium equations \[phiin\]-\[donnan1\], we solved them numerically for a given representative set of parameters. As it is seen in figure \[plotforce\], the ionic force is always positive and increasing with respect to the rod length inside the cavity. This has to be contrasted with the net attractive force found in the work of Zandi *et al.* [@roya], where the binding particles are only present inside the cavity and are not allowed to bind on the outer part of the rod. Now, increasing multivalent salt concentration leads to a non-monotonic variation of the force at *fixed inner length* in the present model, which is the main result of this work: the force is first decreasing at low salt concentration, and then increasing at higher salt concentration. The multivalent salt threshold corresponds to the neutralization point of the rod $\phi_{in,out}\sim 1/Z$, as it is seen on figure \[plotconc\]. This behavior is interpreted below by the interplay between Gibbs-Donnan and Langmuir contribution to the ionic force.
![Ionic force acting on the rod *vs* rod length inside $L_{in}$, for different multivalent salt concentration. Parameters: $L=100,V_0=10^3b^3,V=10^9b^3,b=2nm,Z=4,\epsilon_A=2.5$. *Left panel*: Total force; the large arrow and numbers highlight the variation of the force-length plots as multivalent salt concentration is increased. Note that the force decreases at first ($1\rightarrow 2 \rightarrow 3$) upon increase in $n_Z$, and then increases ($4\rightarrow 5 \rightarrow 6$). *Upper right panel*: Gibbs-Donnan contribution to the ionic force. *Lower right panel*: Langmuir contribution to the ionic force.[]{data-label="plotforce"}](figure2.eps)
The ionic force is rewritten as the sum of three main contributions $f_{ionic}=f_{GD}+f_{L}+f_{\Delta \Pi}$. The three terms are respectively associated with the Gibbs-Donnan force contribution, the Langmuir force contribution and the osmotic force [@castelosmotic]. These forces read $$\begin{aligned}
\label{fgd}\frac{f_{GD}b}{kT} & = & -\ln \frac{Zn_{Zin}-l_{in}}{Zn_Z-l} \\
\label{fl}\frac{f_{L}b}{kT} & = & \ln\frac{1-\phi_{in}}{1-\phi_{out}}\\
\label{fo}\frac{f_{\Delta \Pi}b}{kT} & = & b^3(\left(n_{Zin}-\phi_{in}l_{in}\right)+\left(Zn_{Zin}-l_{in}\right)- \left(n_Z-\phi_{out}l\right)-\left(Zn_{Z}-l\right))\end{aligned}$$ The first contribution Eq. \[fgd\] is associated with the free ions partitioning between inside and outside the cavity. In the absence of ion binding on the rod, there is a depletion of anions and an excess of Z-cations inside the cavity, due to the negative charge of the rod. Within classical Gibbs-Donnan approach, the net force is repulsive. The effect of ion binding on the rod at low multivalent ion concentration is mainly to reduce the net negative charge of the rod, without affecting the sign of the force. Above the neutralization threshold concentration $n_Z^*$, for which $\phi_{out}\sim 1/Z$, the net charge of the rod is positive, due to the binding of Z-cations. In this case, there is a depletion of *free* Z-cations and an excess of anions inside the cavity. However, in contrast to the previous case, there is still an overall excess of Z-cations inside (fig. \[plotconc\]), a growing number of them being involved in the binding on the rod leading to its overcharging. The important proportion of adsorbed Z cations, not contributing to translational entropy inside the cavity, is now favorable to the rod being inside the cavity. As a consequence, the Gibbs-Donnan force is attractive as shown in figure \[plotforce\]. This interpretation does not take into account the free energy of adsorbed layers, however, which compensates this negative Gibbs-Donnan force to give a net repulsive force.
Indeed the second contribution, Eq. \[fl\], comes from the mixing entropy of adsorbed ions along the rod. The sign of the force is mainly given by the relative value of inner and outer adsorption degrees $\phi_{in}$ and $\phi_{out}$. At low multivalent concentration, Gibbs-Donnan equilibrium favors free Z-cations excess inside, thereby increasing inner adsorption relative to the outside environment. Therefore the Langmuir force tends to pull the rod inside. This is the quasistatic effect seen in simulations of reference [@roya] for particular conditions, though with a larger amplitude, due to the asymetrical binding on the rod. Above the neutralization threshold, the situation is reversed, and outer adsorption is favored $\phi_{out}>\phi_{in}$ (fig. \[plotconc\]). As a consequence, Langmuir force is now repulsive, favoring rod ejection from the cavity.
The last contribution Eq. \[fo\] in the ionic force is the osmotic force, first introduced in [@castelosmotic], and it is proportional to the osmotic pressure difference between inside and outside the cavity. It arises from the fact that, in order to insert the rod which has a finite volume inside the cavity, one has to perform a pressure work. Within the conditions chosen in this work, this osmotic force is one order of magnitude smaller than Gibbs-Donnan and Langmuir forces, and is therefore disregarded in the present analysis.
![Adsorbed and free Z-cation concentrations. The value of parameters are the same as in figure \[plotforce\]. *Left panel*: Inner *vs* outer adsorbed degree of Z-cations for concentrations $n_Z b^3=0.01,0.02,0.03,0.04,0.05,0.06$ ($\phi_{in,out}$ increase with $n_Z$).*Upper right panel*: Ratio of free Z-cation concentration inside/outside *vs* rod length inside. Symbols are defined in figure \[plotforce\].*Lower right panel*: Ratio of total Z-cation concentration inside/outside *vs* rod length inside. Symbols are defined in figure \[plotforce\].[]{data-label="plotconc"}](figure3.eps)
For the chosen values of parameters (rod length, adsorption energy, etc...), the magnitude of the ionic force is of order of a few tenths of a piconewton, which is quite small at the molecular level. This comes mainly from the fact that the net entropic forces discussed previously arise from ionic concentration gradients between two compartments and two parts of the rod, unlike asymetrical situations where binding is limited to the inner part of the rod for example. Increasing the length and adsorption energy at fixed $V_0$ produces larger ionic force amplitudes. Added to the neutral force contribution, the ionic force can still lead to substantial changing of behaviors, as is discussed below.
Applications and limitations
============================
As already mentioned in the introduction of this Letter, the present model was originally designed in order to evaluate the leading order of multivalent ionic effects on *in vitro* DNA ejection from bacteriophage [@alexnew]. Within these experiments, DNA ejection is triggered using solubilized specific receptor in a solution containing neutral polymers producing an osmotic force that is able to balance the DNA ejection force. Changing ionic conditions of the solution at constant polymer concentration allows us to address the influence of ionic environment on the ejection force. The present model is of course a crude simplification of the real system. However it is able to predict qualitatively the non-monotonicity of ejection force, as it is observed in the relevant experiments. Within this model, the minimum in the force is associated to the neutralization of charged rod $\phi \sim 1/Z$. Using Eq. \[phiout\], the Z-valent threshold concentration scales like $n_{Z}\sim e^{-\epsilon_A}$. This implies that this threshold concentration decreases with the valency of counterions, a fact that is observed experimentally for three different valencies: $Na$ ($Z=1$), $Mg$ ($Z=2$), spermine ($Z=4$). Quantitative comparison of predicted and measured threshold is however difficult since the experimental buffer contains more than one multivalent salt. Our model can easily be extended to include more salt types and their binding competition with different energies $\epsilon_{A\, i}$ ($i=1,2...$). This modified model leads qualitatively to the same results as the one derived for one multivalent salt (data not shown). The quantitative difference between the two models is the precise location of ionic force minimum or neutralization threshold. Since the main effect is already observed within the present simple one-salt model, we prefer to stick to this model, for the sake of clarity. Note that Gibbs-Donnan balance has already been used by Odijk and co-workers in order to describe DNA packing in a model-bacteriophage [@odijk3]. But the presence of multivalent ions as well as their influence on DNA ejection has been disregarded.
The main effect that is not included explicitly within this model is the presence of correlations between bound ions, changing the effective adsorption energy [@shklovskii]. However, we don’t expect this effect to change drastically the results derived within our model: indeed, the non-monotonicity of the ionic force is related within our model to the presence of neutralization of the rod and possible overcharging, and not to its precise description. We checked with non-linear adsorption energy that prohibits the rod from being unrealistically too overcharged, that the qualitative behavior is unchanged. In particular, we observed that including only correlations and neglecting Gibbs-Donnan balance does not lead to non-monotonicity of ionic force, so that the experimentally observed minimum of the force can not be explained solely by invoking correlation effects. Therefore both basic ingredients of our model, Gibbs-Donnan balance and adsorption statistics, are necessary to observe the non-monotonicity of the force. Notice that for similar reasons we don’t expect correlations of free ions, like the leading order Debye-Huckel correlations [@refgen], to change qualitatively the results obtained in this Letter.
In order to interpret the results of DNA ejection experiments within this model, an additional implicit assumption has been made: the ionic effects can be decoupled from other contributions, termed “neutral” within this work. This assumption is valid within the simple geometry shown in figure \[cartoon\]. In a more realistic model of viral bacteriophage, where DNA is arranged in a spool-like fashion within the viral cavity, the neutral contribution would include bending energy as well as direct electrostatic repulsion between neighbouring DNA turns in the spool [@rob1; @rob2; @tzlil]. Up to now except in aforementioned works of Odijk and co-workers [@odijk3], the inclusion of electrostatic effects in such models has been based on phenomenological expressions describing osmotic pressure of hexagonal phases of DNA as measured in the experiments of [@raspaud2] for example. Since the simple extended Gibbs-Donnan model presented here is qualitatively consistent with these expressions[^1], we expect our model to describe correctly the leading order of multivalent ionic effects.
The range of application of the simple model proposed in this Letter is larger than the ejection experiment just mentioned. Indeed it is relevant to the description of translocation properties of polyelectrolytes[@mutu], whenever the separating interface is permeable to mobile free ions. We showed in this paper that the force balance acting on the rod can be displaced by changing ionic conditions. Within a biological context, the applicability of the present model for the translocation of bio-polyelectrolytes between different cell compartments, for example, is limited by the presence of ionic channels on the interface (cell compartment membrane) that actively regulate the ionic gradients accross the interface. Therefore the pure Gibbs-Donnan balance is no longer obtained, and new models have to be developed.
Fruitful discussions at early stages of this work with W.M. Gelbart are greatfully acknowledged.
[0]{}
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Raspaud E., Da Conceicao M. and Livolant F. , 84:2533, 2000.
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Evilevitch A. , Castelnovo M. , Knobler C. M. and Gelbart W. M. , 2005.
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Evilevitch A. , Castelnovo M. , Knobler C. M. and Gelbart W. M. , 108:6838, 2004.
Purohit P.K., Inamdar M.M., Grayson P.D., Squires T.D., Kondev J. and Phillips R. .
Purohit P.K., Kondev J. and Phillips R. , 100:3173, 2003.
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[^1]: M. Castelnovo, unpublished results
|
---
abstract: 'Formation of fermion bag solitons is an important paradigm in the theory of hadron structure. We report here on our non-perturbative analysis of this phenomenon in the $1+1$ dimensional massive Gross-Neveu model, in the large $N$ limit. Our main result is that the extremal static bag configurations are reflectionless, as in the massless Gross-Neveu model. Explicit formulas for the profiles and masses of these solitons are presented. We also present a particular type of self-consistent reflectionless solitons which arise in the massive Nambu-Jona-Lasinio models, in the large-$N$ limit.'
address: |
$^a$Physics Department, University of Haifa at Oranim, Tivon 36006, Israel \
$^b$Physics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel \
$^c$[*Talk delivered by JF.*]{}
author:
- 'Joshua Feinberg$^{a,b,c}$ and Shlomi Hillel$^b$'
title: 'Fermion Bag Solitons in the Massive Gross-Neveu and Massive Nambu-Jona-Lasinio Models in $1+1$ Dimensions: Inverse Scattering Analysis'
---
Introduction
============
An important dynamical mechanism, by which fundamental particles acquire masses, is through interactions with vacuum condensates. Thus, a massive particle may carve out around itself a spherical region [@sphericalbag] or a shell [@shellbag] in which the condensate is suppressed, thus reducing the effective mass of the particle at the expense of volume and gradient energy associated with the condensate. This picture has interesting phenomenological consequences [@sphericalbag; @mackenzie].
This dynamical distortion of the homogeneous vacuum condensate configuration, namely, formation of fermion bag solitons, was demonstrated explicitly by Dashen, Hasslacher and Neveu (DHN) [@dhn] many years ago, who studied semiclassical bound states in the $1+1$ dimensional Gross-Neveu (GN) model [@gn], using the inverse scattering method [@inverse]. Following DHN, Shei [@shei] has applied the inverse scattering method to study solitons in the $1+1$ dimensional Nambu-Jona-Lasinio (NJL) model [@njl] in the large-$N$ limit.
Fermion bags in the GN model were discussed in the literature several other times since the work of DHN, using alternative methods [@others; @papa; @josh1]. For a review on these and related matters (with an emphasis on the relativistic Hartree-Fock approximation) see [@thiesreview]. For a more recent review of static fermion bags in the GN model (with an emphasis on reflectionless backgrounds and supersymmetric quantum mechanics) see [@bagreview]. The large-$N$ semiclassical DHN spectrum of these fermion bags turns out to be essentially correct also for finite $N$, as analysis of the exact factorizable S-matrix of the GN model reveals [@Smatrix].
A variational calculation of these effects in the $1+1$ dimensional massive generalization of the Gross-Neveu model, which we will refer to as MGN, was carried in [@FZMGN] a few years ago, and more recently in [@Thies]. Very recently, we studied static fermion bags in the MGN model [@FH], which we obtained using the inverse-scattering formalism, thus avoiding the need to choose a trial variational field configuration. Our main result in [@FH] is that the extremal static bag configurations are reflectionless, as in the massless Gross-Neveu model. In the next section we briefly review the results of [@FH], leaving technical details out. Then, in Section 3, we show that a subclass of the reflectionless solitons of [@FH] arise self-consistently in the $1+1$ dimensional massive NJL (MNJL) model. The latter extends the results of [@shei; @FZ-NJL] for the massless NJL model. Solitons in the MNJL model were also recently studied in [@Thies], where a derivative expansion was carried out around a particular soliton background of the corresponding massless NJL model.
Solitons in the Massive Gross-Neveu Model
=========================================
One way of writing the action for the MGN model is S =d\^2x{\_[a=1]{}\^N|\_a\_a -[12g\^2]{}(\^2-2M)}, \[lagrangian\] where the $\psi_a$ ($a=1,\ldots,N$,) are massive Dirac fermions and $\si$ is an auxiliary field. Integrating the $\si$ out results in an equivalent form of (\[lagrangian\]), with quartic fermion self-interactions.
An obvious symmetry of (\[lagrangian\]) with its $N$ Dirac spinors is $U(N)$. Actually, (\[lagrangian\]) is symmetric under the larger group $O (2N)$ [@dhn] (see also Section 1 of [@bagreview]). The fact that the symmetry group of (\[lagrangian\]) is $O (2N)$ rather than $ U(N)$ is related to the fact that it is invariant against charge-conjugation, like the massless GN model. Consequently, the energy eigenvalues of the Dirac equation associated with (\[lagrangian\]), $[i\notpa-\si (x)]\,\psi = 0$, come in $\pm\om$ pairs.
As usual, the theory (\[lagrangian\]) can be rewritten with the help of the scalar flavor singlet auxiliary field $\si(x)$. Also as usual, we take the large $N$ limit holding $\lambda\equiv Ng^2$ fixed. Integrating out the fermions, we obtain the bare effective action S\[\] =-[12g\^2]{} d\^2x (\^2-2M) -iN [log]{}(i-). \[fermout\] Noting that $\gam_5 ( i\notpa -\si )= -(i\notpa +\si)\gam_5
$, we can rewrite the $\tr~{\rm log}(i\notpa -\si )$ as ${1\over 2}
\tr~{\rm log}\left[-(i\notpa -\si)(i\notpa +\si)\right]$. In this paper we focus on static soliton configurations. If $\si$ is time independent, the latter expression may be further simplified to $ {T\over 2}\int {d\om \over 2 \pi} [\tr~{\rm log} (h_+-
\om^2)+\tr~{\rm log} (h_- -\om^2)]$ where $h_{\pm} \equiv -\pa_x^2 + \si^2 \pm \si'\,,$ and where $T$ is an infra-red temporal regulator. As it turns out, the two Schrödinger operators $h_{\pm}$ are isospectral (see Appendix A of [@bagreview] and Section 2 of [@josh1]) and thus we obtain S\[\] = -[12g\^2]{} d\^2x (\^2-2M) - iNT \_[-]{}\^ [d2 ]{} [log]{}(h\_- -\^2). \[effective\]
In contrast to the standard massless GN model, the MGN model studied here is not invariant under the $Z_2$ symmetry $\psi\rightarrow \gamma_5 \psi$, $\sigma \rightarrow -\sigma$, and the physics is correspondingly quite different. As a result of the $Z_2$ degeneracy of its vacuum, the GN model contains a soliton (the so called CCGZ kink [@ccgz; @dhn; @others; @josh1; @bagreview]) in which the $\sigma$ field takes on equal and opposite values at $x=\pm\infty$. The stability of this soliton is obviously guaranteed by topological considerations. With any non-zero $M$ the vacuum value of $\sigma$ is unique and the CCGZ kink becomes infinitely massive and disappears. If any soliton exists at all, its stability has to depend on the energetics of trapping fermions.
Let us briefly recall the computation of the unique vacuum of the MGN model. We shall follow [@FZMGN]. For an earlier analysis of the MGN ground state (as well as its thermodynamics), see [@klimenko]. Setting $\si$ to a constant we obtain from (\[effective\]) the renormalized effective potential (per flavor) $V(\si,\mu) = {\si^2\over 4\pi}~ {\rm log}~ {\si^2\over
e\mu^2} +
{1\over \lambda(\mu)}~\left[{\si^2\over 2} -
M(\mu)\si\right]\,,$ where $\mu$ is a sliding renormalization scale with $\lambda(\mu)=Ng^2(\mu)$ and $M(\mu)$ the running couplings. By equating the coefficient of $\si^2$ in two versions of $V$, one defined with $\mu_1$ and the other with $\mu_2$, we find immediately that ${1\over\lambda(\mu_1)} - {1\over\lambda(\mu_2)} =
{1\over \pi}~{\rm
log}
~{\mu_1\over\mu_2}$ and thus the coupling $\lambda$ is asymptotically free, just as in the GN model. Furthermore, by equating the coefficient of $\sigma$ in $V$ we see that the ratio ${M(\mu)\over\lambda(\mu)}$ is a renormalization group invariant. Thus, $M$ and $\lambda$ have the same scale dependence.
Without loss of generality we assume that $M(\mu)>0$ and thus the absolute minimum of $V(\si,\mu)$, namely, the vacuum condensate $m=\langle\si\rangle$, is the unique (and positive) solution of the gap equation ${dV\over d\si}~{\Big|_{\si=m}}= m\left[ {1\over \pi}~{\rm log}
~{m\over\mu} + {1\over \lambda(\mu)}\right] -
{M(\mu)\over\lambda(\mu)} = 0\,.$ Referring to (\[lagrangian\]), we see that $m$ is the mass of the fermion. Using the explicit scale dependence of $\lambda(\mu)$, we can re-write the gap equation as ${m\over\lambda(m)}= {M(\mu)\over\lambda(\mu)}$, which shows manifestly that $m$, an observable physical quantity, is a renormalization group invariant. This equation also implies that $M(m)=m$, which makes sense physically.
Fermion bags correspond to inhomogeneous solutions of the saddle-point equation ${\delta S \over \delta \sigma(x,t)}=0$. In particular, static bags $\sigx$ are the extremal configurations of the energy functional (per flavor) $\ce [\sigx] = - {S[\sigx]\over NT}\,,$ subjected to the boundary condition that $\sigx$ relaxes to its unique vacuum expectation value $m$ at $x=\pm\infty$. More specifically, we have to evaluate the energy functional of a static configuration $\sigx$, obeying the appropriate boundary conditions at spatial infinity, which supports $K$ pairs of bound states of the Dirac equation at energies $\pm\om_n$, $n=1,\ldots ,K$ (where, of course, $\om_n^2<m^2$). The bound states at $\pm\om_n$ are to be considered together, due to the charge conjugation invariance of the GN model. Due to Pauli’s exclusion principle, we can populate each of the bound states $\pm\om_n$ with up to $N$ fermions. In such a typical multiparticle state, the negative frequency state is populated by $N-h_n$ fermions (i.e., by $h_n$ holes or antiparticles) and the positive frequency state contains $p_n$ fermions (or particles). We shall refer to the total number of particles and antiparticles trapped in the n-th pair of bound states $\nu_n = p_n+h_n $ as the valence, or occupation number of that pair.
The energy functional $\ce [\sigx]$ is, in principle, a complicated and generally unknown functional of $\sigx$ and of its derivatives (which furthermore, requires regularization). Thus, the extremum condition ${\del\ce [\si]\over \del\sigx} =0$, as a functional equation for $\sigx$, seems intractable. The considerable complexity of the functional equations that determine the extremal $\sigx$ configurations is the source of all difficulties that arise in any attempt to solve the model under consideration. DHN found a way around this difficulty in the case of the GN model [@dhn]. They have used inverse scattering techniques [@inverse] to express the (regulated) energy functional $\ce [\si]$ in terms of the so-called “scattering data” associated with, e.g., the hamiltonian $h_-$ mentioned above (and thus with $\sigx$), and then solved the extremum condition ${\del\ce [\si]\over \del\sigx} =0$ with respect to those data.
The scattering data associated with $h_-$ are [@inverse] the reflection amplitude $r(k)$ of the Schrödinger operator $h_-$ at momentum $k$, the number $K$ of bound states in $h_-$ and their corresponding energies $0<\om_n^2\leq m^2\,,(n=1,\ldots, K)$, and also additional $K$ parameters $\{c_n\}$, where $c_n$ has to do with the normalization of the $n$th bound state wave function $\psi_n$ of $h_-$. More precisely, the $n$th bound state wave function, with energy $\om_n^2$, must decay as $\psi_n(x)\sim
{\rm const.}\exp -\kappa_n x$ as $x\rightarrow\infty$, where $0<\kappa_n = \sqrt{m^2 -\om_n^2}\,.$ If we impose that $\psi_n (x)$ be normalized, this will determine the constant coefficient as $c_n$. (With no loss of generality, we may take $c_n>0$.) Recall that $r(-k) = r^*(k)$, since the Schrödinger potential $V(x) = \si^2(x) - \si'(x)$ is real. Thus, only the values of $r(k)$ for $k>0$ enter the scattering data. The scattering data are independent variables, which determine $V(x)$ uniquely, assuming $V(x)$ belongs to a certain class of potentials which fall-off fast enough toward infinity. (Since the MGN does not bear topological solitons, neither $h_-$ nor $h_+$ can have a normalizable zero energy eigenstate. Thus, all the $\om_n$ are strictly positive.)
We can apply directly the results of DHN in order to write down that part of $\ce [\sigx]$ which is common to the MGN and GN models, i.e., $\ce [\sigx]$ with its term proportional to $M$ removed, in terms of the scattering data. For lack of space we shall not write DHN’s expression for the energy functional explicitly. Suffice it is to mention at this point that the “DHN-part” of $\ce [\sigx]$ depends on the reflection amplitude only via certain regular dispersion integrals of the quantity $\log [1-|r(k)|^2]$. The well-known reflectionless nature of solitons in the GN model is a direct consequence of this simple fact.
In order to complete the task of expressing the effective action of the MGN model in terms of the scattering data, we have to find such a representation for the remaining piece of $\ce [\sigx]$ proportional to $M$, namely, for $-{M\over\lambda}\int_{-\infty}^\infty \left(\sigx -m\right)
\, dx$. The latter integral cannot be expressed in terms of the scattering data based on the trace identities of the Schrödinger operator $h_-$ discussed in Appendix B of [@dhn]. Evidently, new analysis is required to obtain its representation in terms of the scattering data. Happily enough, we were able to obtain such a representation in [@FH], which reads \[sigmatext\] \_[-]{}\^(-m) dx = [1]{}\_[-]{}\^[im-k]{}dk + \_[n=1]{}\^K ([m - \_n m + \_n]{}). Thus, the $M$-dependent part of $\ce [\sigx]$, like its “DHN-part”, depends on the reflection amplitude only via the combination $\log [1-|r(k)|^2]$. Combining these two terms together, it follows that $\delta \ce^{reg} [\sigx]/\delta r(k) =
F(k)\, r^*(k)/(1-|r(k)|^2)$, where $F(k)$ is a calculable function, which does not vanish identically. Thus, $r(k) \equiv 0$ is the unique solution of the variational equation $\delta \ce^{reg} [\sigx]/\delta r(k) = 0$. Static extremal bags in the MGN model are [*reflectionless*]{}, as their counterparts in the GN model.
Explicit formulas for reflectionless $\sigx$ configurations with an arbitrary number $K$ of pairs of bound states are displayed in Appendix B of [@bagreview]. In particular, the one which supports a single pair of bound states at energies $\pm\om_b$ ($\kappa = \sqrt{m^2-\om_b^2}$), the one originally discovered by DHN, is \[dhnsoliton\] = m + .
We see that the formidable problem of finding the extremal $\sigx$ configurations of the energy functional $\ce [\si]$ is reduced to the simpler problem of extremizing an ordinary function $\ce(\om_n, c_n) =
\ce\left[\si (x; \om_n, c_n)\right]$ with respect to the 2$K$ parameters $\{c_n, \om_n\}$ that determine the reflectionless background $\sigx$. If we solve this ordinary extremum problem, we will be able to calculate the mass of the fermion bag. This we did in detail in [@FH]. Let us sketch the procedure and state the final result:
The bare regulated energy function $\ce(\om_n)$ which depends on the bare couplings $\lambda$ and $M$ and on the UV-cutoff $\Lambda$ explicitly can be renormalized, in a manner essentially similar to the effective potential, as was described above. $\ce $ is independent of the $c_n$’s, which appear in the scattering data. (The latter are thus flat directions for the energy function and determine the collective coordinates of the soliton.) The renormalized energy function thus obtained is a sum of the form $\sum_{n=1}^K
f(\om_n,\nu_n)$ where $f(\om,\nu)$ is a known function, which depends also on the physical mass $m$ explicitly, and also through the RG-invariant ratio $\gamma = {1\over\lambda (m)} = {M(\mu)\over m\lambda(\mu)}$. Thus, the extremum condition fixes each $\om_n$ in terms of the number of the total number $\nu_n$ of particles and holes trapped in the bound states of the Dirac equation at $\pm\om_n$, and not by the numbers of trapped particles and holes separately (see (\[thetaext\])). This fact is a manifestation of the underlying $O(2N)$ symmetry, which treats particles and holes symmetrically. Moreover, it indicates [@FH] that this pair of bound states gives rise to an $O(2N)$ antisymmetric tensor multiplet of rank $\nu_n$ of soliton states. (As it turns out, only tensors of ranks $0<\nu_n<N$ correspond to viable solitons [@FH].) The soliton as a whole is therefore the tensor product of all these antisymmetric tensor multiplets. Finally, we showed in [@FH] that only the irreducible ($K=1$) soliton was protected by energy conservation and $O(2N)$ symmetry against decaying into lighter solitons (or free massive fermions). Its profile is given by (\[dhnsoliton\]), where $\kappa = m\sin \theta$ (or, equivalently, $\om = m\cos \theta$), with $0<\theta <\pi/2$, and where $\theta$ is determined by the extremum condition \[thetaext\] + = [2 N]{}. The left-hand side of (\[thetaext\]) is a monotonically increasing function. Therefore, (\[thetaext\]) has a unique solution in the interval $[0,\pi/2]$. This solution is evidently smaller than $\theta^{\rm GN} =
{\pi\nu\over 2N}$, the corresponding value of $\theta$ in the GN model for the same occupation number. Thus, the corresponding bound state energy $\om = m\cos\theta$ in the MGN model is higher than its GN counterpart, and thus less bound. The soliton mass (i.e., the renormalized energy function, evaluated at the solution of (\[thetaext\])) is () = Nm ( [2]{}+ [1 + 1 - ]{}). \[solitonmass\] This coincide with the corresponding results of variational calculations presented in [@FZMGN; @Thies], which were based on (\[dhnsoliton\]) as a trial configuration. In fact, it was realized in [@Thies] that the trial configuration (\[dhnsoliton\]) is an exact solution of the extremum condition ${\del\ce [\si]\over \del\sigx} =0$, provided (\[thetaext\]) is used to fix $\kappa = m\sin\theta $.
Reflectionless solitons in the Massive Nambu-Jona-Lasinio Model
===============================================================
It is natural to inquire whether the results of the previous section carry over to the phenomenologically interesting MNJL model. The action for the MNJL model may be written as a generalization of (\[lagrangian\]), $S =\int d^2x\,\left\{\sum_{a=1}^N\,\bar\psi_a\left[i\notpa-(\si + i\pi
\gamma_5)\right]\psi_a
-{1\over 2g^2}\left(\si^2 +\pi^2-2M\si\right)\right\}\,,$ \[lagrangian-njl\] where $\pix$ is a pseudo-scalar auxiliary field. (Here we assumed that the $2\times 2$ chiral mass matrix does not have a pseudo-scalar component, but this does not restrict the generality of our discussion in any way. This particular orientation of the mass matrix can be always reached at by performing a global - and therefore, anomaly free- chiral rotation in the $\sigma-\pi$ plane.)
As in our discussion of the MGN model, we can integrate out the fermions, and obtain the bare effective action S\[\] =-[12g\^2]{} d\^2x (\^2+\^2-2M) -iN [log]{}(i-- i\_5). \[fermout-njl\] As before, we take the large $N$ limit, holding $\lambda\equiv Ng^2$ fixed. Unlike the NJL model, with its continuum of degenerate vacua, the ground state of the MNJL model (\[fermout-njl\]) is unique, as in the MGN model. It corresponds to a constant field configuration, where $\pi=0$ and where $\si = m$ is determined by an equation identical to the one which arises in the MGN model.
Shei [@shei] has studied static solitons in the NJL model (i.e., $M=0$ in (\[fermout-njl\])) using inverse scattering techniques. Similarly to DHN’s results for the GN model, he concluded that extremal soliton profiles are reflectionless. Some of his results were rederived in [@FZ-NJL], using a certain method based on properties of the diagonal resolvent of the Dirac operator, (which was applied first to the GN model in [@josh1]).
Could Shei’s analysis be extended to study solitons in the MNJL model, similarly to the extension of DHN’s inverse scattering analysis to the MGN model? Could it be that the self-consistent static soliton backgrounds in the MNJL model are reflectionless? It seems that all we need in order to answer these questions is a generalization of (\[sigmatext\]) to the case in which the Dirac operator involves a pseudo-scalar background $\pix$. Unfortunately, we were not able (so-far) to find such a generalization, and therefore we cannot answer these questions in general at the moment. However, we were able to find a particular family of self-consistent reflectionless static solitons in the MNJL model by making an educated guess, as we shall now explain.
The spectrum of the Dirac equation associated with (\[fermout-njl\]) is [*not*]{} invariant against charge-conjugation, unless $\pix\equiv 0$. Thus, the bound states corresponding to a static soliton background are not paired, in general. In particular, as has been shown by Shei, there exist solitons in the NJL model which bind fermions into a single bound state. However, he has also found solitons with charge-conjugation-invariant spectrum (see Eqs.(3.25)-(3.28) in [@shei]), with a pair of bound states $\pm\om_b$, in which $\pix=0$ identically, and $\sigx$ is given by (\[dhnsoliton\]), which thus coincide with the DHN solitons in the GN model, for which $\om_b=
m \cos\left({\pi\nu\over 2N}\right)$ and $\cm_{\rm DHN} (\nu) =
{2 Nm\over \pi}\sin\left({\pi\nu\over 2N}\right)$. However, unlike in the GN model, in the NJL model we must choose $p=h={\nu\over 2}$ (i.e., a soliton of this type must trap an equal number of fermions and anti-fermions). The reason for this restriction is not hard to understand physically: the total chiral rotation $\Delta\theta$, namely, the difference in $\arctan {\pix\over\sigx}$ between the two ends of the one dimensional space, must be related to the fermion number charge $n_f$ deposited in the soliton according to $\Delta\theta = -{2\pi n_f\over N}$ [@goldwil] (see also Eqs. (5.10) and (5.22) in [@FZ-NJL]). The soliton profile $(\sigx,\pix)$ under consideration, starts at the vacuum point $(m,0)$ at $x=-\infty$ and returns to it at $x=+\infty$. Thus, $\Delta\theta = n_f = p-h = 0$ for this soliton.
Now, any static soliton profile in the MNJL model must start at the [*unique*]{} vacuum $(m,0)$ at $x=-\infty$ and return to it at $x=+\infty$. Thus, it should bring about null total chiral rotation, precisely as Shei’s charge-conjugation-invariant configuration does. Therefore, if the MNJL bears [*reflectionless*]{} static solitons, they must be of this form (or charge-conjugate invariant generalizations thereof, with more pairs of paired bound states). The only thing that should change compared to the NJL model is the quantization condition, relating $\om_b$ and $\nu$.
We have verified that this is indeed the case, simply by substituting this configuration into the static inhomogeneous saddle-point equations associated with (\[fermout-njl\]). Varying (\[fermout-njl\]) with respect to $\pix$ we [*obviously*]{} obtain an equation identical in form to that of the NJL model. (For the latter, see the second equation in (5.1) in [@FZ-NJL]). Using the explicit expressions for the entries of the diagonal resolvent of the Dirac operator with a reflectionless $(\sigx, \pix)$ background with two bound states (Eqs. (4.13) and (4.14) in [@FZ-NJL] with paired bound states $\om_2 = -\om_1 $), we see that $\pix\equiv 0$ is indeed a solution of that equation. (Here, having $p=h={\nu\over 2}$ is essential.). This $\pi$-equation leaves $\om_1$ an undetermined function of $\nu$. We still have to vary with respect to $\sigx$. Substituting the explicit expressions for the appropriate entries of the diagonal resolvent of the Dirac operator (Eqs.(4.13), (4.14) and (2.10) of [@FZ-NJL]) in the saddle-point equation arising from variation with respect to $\sigx$, and using the simplifying identity Eq. (2.24) of [@jmp], we arrive simply at the static saddle-point equation of the MGN model, which is solved by $\sigx$ given by (\[dhnsoliton\]) and the quantization condition (\[thetaext\]), leading to soliton mass (\[solitonmass\]). Thus, a restricted subset of the extremal reflectionless solitons of the MGN model appear, not surprisingly, also in the MNJL model. For these solitons $\pix\equiv 0$. The question whether these solitons exhaust all possibilities in the two-dimensional MNJL model remains open.
References {#references .unnumbered}
==========
[99]{}
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|
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address: |
$^{1}$ Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany\
$^{2}$ Leibniz Universität Hannover, D-30167 Hannover, Germany\
$^{3}$ Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada\
$^{4}$ Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1, Canada\
$^{5}$ Perimeter Institute For Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada
bibliography:
- 'template.bib'
---
Introduction
============
Black holes (BHs) are very interesting “stars” in the Universe where both strong gravity and macroscopic quantum behavior are expected to coexist. Classical BHs in General Relativity (GR) have been thought to have only three hairs, i.e., mass, angular momentum, and charge, making observational predictions for BHs relatively easy [@1968CMaPh...8..245I; @1971PhRvL..26..331C] (compared to other astrophysical compact objects). For astrophysical BHs, due to the effect of ambient plasma, this charge is vanishingly small, leaving us with effectively two hairs for isolated black holes, with small accretion rates. In other words, finding conclusive deviations from [*s*tandard]{} predictions of these 2-parameter models, may be interpreted as fingerprints of a quantum theory of gravity or other possible deviations from GR. For example, the quasinormal modes (QNMs) of spinning BHs, which have been widely-studied over the past few decades (a subject often referred to as BH spectroscopy), only depend on the mass and spin of the Kerr BH (e.g., [@Kokkotas:1999bd]). The ringdown of the perturbations of the BH is regarded as a superposition of these QNMs, and thus can be used to test the accuracy of GR predictions and no-hair theorem (e.g., see [@Isi:2019aib]). As a result, precise detection of QNMs from the ringdown phase (from BH mergers or formation) in gravitational wave (GW) observations may enable us to test the classical and quantum modifications to GR (e.g., [@Bhagwat:2019dtm]).
A concrete path towards this goal is paved through the study of “GW echoes”, a smoking gun for near-horizon modifications of GR which are motivated from the resolutions of the proposed resolutions to the BH information paradox and dark energy problems [@Almheiri:2013hfa; @PrescodWeinstein:2009mp]. The list of these models include wormholes [@Cardoso:2016rao], gravastars [@Mazur:2004fk], fuzzballs [@Lunin:2001jy], 2-2 holes [@Holdom:2016nek], Aether Holes [@PrescodWeinstein:2009mp], Firewalls [@Almheiri:2013hfa] and the Planckian correction in the dispersion relation of gravitational field [@Oshita:2018fqu; @Oshita:2019sat].
The possibility of observing GW echoes was first proposed shortly after the first detection of GWs by LIGO [@Cardoso:2016rao; @Cardoso:2016oxy; @Cardoso:2019rvt], which has led to several observational searches [@Abedi:2016hgu; @Uchikata:2019frs; @Conklin:2017lwb; @Westerweck:2017hus; @Nielsen:2018lkf; @Abedi:2018npz; @Salemi:2019uea; @Holdom:2019bdv; @Ashton:2016xff; @Abedi:2017isz; @Abedi:2018pst]. Tentative evidence for and/or detection of these echoes can be seen in the results reported by different groups [@Abedi:2016hgu; @Conklin:2017lwb; @Westerweck:2017hus; @Nielsen:2018lkf; @Abedi:2018npz; @Salemi:2019uea; @Uchikata:2019frs; @Holdom:2019bdv] from O1 and O2 LIGO observations of binary BH and neutron star mergers, but the origin and the statistical significance of these signals remain controversial [@Westerweck:2017hus; @Ashton:2016xff; @Abedi:2017isz; @Abedi:2018pst; @Salemi:2019uea], motivating further investigation.
Given their uncertain theoretical and observational status, GW echoes are gathering much attention from those who are interested in the observational signatures of quantum gravity, and the field remains full of excitement, controversy and confusion. In this review article, we aim to bring some clarity to this situation, from its background, to its current status, and into its future outlook.
The review article is organized as follows: In the next section, we provide builds the motivation to investigate the quantum signatures from BHs. In Sec. \[sec:QBHs\], we discuss theoretical models of quantum BHs, starting from the BH information loss paradox, and then its proposed physical resolutions that lead to observable signatures. In Sec. \[sec:echo\_predictions\], we review how to predict the GW echoes from spinning BHs based on the Chandrasekhar-Detweiler (CD) equation, and also review the Boltzmann reflectivity model [@Oshita:2019sat; @Wang:2019rcf] for quantum black holes. Sec. \[sec:echo\_searches\] is devoted to the echo searches, where we summarize positive, negative, and mixed reported outcomes, and attempt to provide a balanced and unified census. In Sec. \[sec:future\_prospects\], we discuss the future prospects for advancement in theoretical and observational studies of quantum black holes, while Sec. \[sec:final\_words\] concludes the review article.
Throughout the article, we use the following notations:
Symbol Description
----------------- -----------------------------------------------
$a$ spin parameter
$\bar{a}$ non-dimensional spin parameter ($a/(GM)$)
$c$ speed of light
$\hbar$ Planck constant
$k_{\rm B}$ Boltzmann constant
$G$ gravitational constant
$M_{\text{Pl}}$ Planck mass
$E_{\text{Pl}}$ Planck energy
$l_{\text{Pl}}$ Planck length
$M$ mass of a balck hole or exotic compact object
$M_{\odot}$ solar mass ($1.988 \times 10^{30}$ kg)
$r_g$ Schwarzschild radius
$T_{\rm H}$ Hawking temperature
Furthermore, unless noted otherwise, we use the natural Planck units with $\hbar = c = 1= G=1$.
Invitation {#sec:invitation}
==========
Classical BHs
-------------
Schwarzschild and Kerr spacetimes are solutions of general relativity giving the spacetime configurations of BHs, which are the most dense objects in our universe. In some regions, matter accumulates and attracts more matter with its gravity which is classically always attractive. At the end, the force is so strong that even the light cannot escape from those regions, where then BHs form. The first and most important feature (the definition of BHs) is the formation of horizon. Inside the (event) horizon, all the light cones are directed into the singularity, and nothing can escape, unless it could travel faster than the speed of light. Therefore, horizons stand as the causal boundaries of BHs in Einstein’s theory of Relativity.
Realistic BHs in the sky have different hairs (mass, spin and charge), and their dynamics share more complicated structure, thus, have different kinds of horizons. To list some of them, event horizons are defined as the boundaries where no light can escape to the infinite future. However, for a dynamically evolving BH, event horizons are teleological, i.e. we cannot predict them until we have the entire history of the spacetime. Apparent horizons, however, are predictable at a specific time without knowing the future. Any surface has two null normal vectors and if expansion of both of them are negative, the surface is called “trapped”. Apparent horizons are the outermost of all the trapped surfaces, which is why they are also known as the “marginally outer trapped surface”.
Here is a simple example to distinguish these two horizons — we start with a Schwarzschild BH at time $t_1$, now the event and apparent horizons coincide at the Schwarzschild radius. We throw a spherical null shell into the BH and let it cool down at $t_2$. This process is perfectly described by Vaidya metric [@Poisson:2009pwt]. The apparent horizon changes immediately when the shell falls into the BH but the event horizon starts to expand earlier, even before the shell reaches it. It is because that after throwing the shell, the gravity of BH increases. Thus it is harder for light to escape from the BH to infinity. In other words, particles might be doomed to fall into a singularity, even before they had a chance to meet the infalling gravitating matter that is responsible for their fate. Therefore, the event horizon is modified earlier than the apparent horizon. While this result is counter-intuitive, it is a result of the formal definition of the event horizons, which requires the information about the entire history of spacetime, in particular, the future!
Beyond the horizons, another intriguing trait of BHs is the curvature singularity, which sits at the centre of the BHs. Horizons can also be singular, but usually only coordinate singularities and (in classical General Relativity) removable by changing to a proper coordinate system. However, the singularities inside the BHs are where the general relativity breaks down and so far we do not have any good physics to describe them. We cannot chase the information lost into these singularities (using standard physics), which leads to the information paradox (more on this later).
Back in November 1784, John Michell, an English clergyman, advanced the idea that light might not be able to escape from a very massive object (at a fixed density). For example, light cannot escape from the surface of a star with the density of the sun, if it was 500 times bigger than the sun. Albert Einstein, later in 1915, developed general relativity. Soon after this, Karl Schwarzschild solved the Einstein vacuum field equation under spherical symmetry with a singular mass at the center, which was the first solution for BHs, the Schwarzschild metric.
While 20th century saw a golden age of general relativity with blooming of dozens of different BH solutions, the existence of BHs was not directly confirmed until one century later in 2015. LIGO-Virgo collaboration reported unprecedented detection of GWs from the binary BH merger events [@TheLIGOScientific:2016agk; @TheLIGOScientific:2016pea; @Abbott:2016blz; @Abbott:2016nmj; @Abbott:2017vtc; @Abbott:2017oio; @TheLIGOScientific:2017qsa; @Abbott:2017gyy]. Numerical relativity is consistent with LIGO data at least up to quite near the horizon range. But the detection has not confirmed the existence of the horizons. We will discuss in this article how the detection opens a window for searching for quantum nature of the BHs beyond the general relativity.
### Schwarzschild spacetime
The Schwarzschild spacetime was the first exact solution in the Einstein theory of general relativity. It models a static gravitational field outside a mass which has spherical symmetry, zero charge and rotation. Karl Schwarzschild found this solution in 1915, and four months later, Johannes Droste published a more concrete study on this independently. The metric in the Schwarzschild coordinate is: $$ds^2= -\left(1-\frac{2M}{r} \right)dt^2+\left(1-\frac{2M}{r} \right)^{-1} dr^2 + r^2 d \Omega^2,$$ where $M$ is the mass of the centre object, $2M$ is Schwarzschild radius and $d \Omega ^2 =d \theta^2 + \sin^2\theta d\phi^2$ is the metric on a 2-sphere. The metric describes gravitational field outside any spherical object without charges. If the radius of the central object is smaller than the Schwarzschild radius, the object is then too dense to be stable, and will go through a gravitational collapse and form the Schwarzschild BH.
Later in 1923, G.D.Birkhoff proved that any spherically symmetric solution of the vacuum Einstein field equation must be static and asymptotically flat. Hence, Schwarzschild metric is the only solution in that case. For any static solution, the event horizon always coincides with the apparent horizon at $r=2M$. In general relativity, Schwarzschild metric is singular at the horizon but , as stated above, this is only a coordinate artifact. That is to say, a free falling observer feels no drama going through the horizon. It takes the observer a finite amount of proper time but infinite coordinate time. Particularly, we can remove the singularity by a proper coordinate transformation. In contrast, the origin $r=0$ is the intrinsic curvature singularity. The scalar curvature is infinite and the general relativity is no longer valid at this point.
### Kerr spacetime
The Kerr spacetime [@PhysRevLett.11.237], discovered by Roy Kerr, is a realistic generalization of the Schwarzschild spacetime. It describes the gravitational field of an empty spacetime outside a rotating object. The spacetime is stationary and has axial symmetry. The metric in the Boyer-Lindquist coordinate is:
$$\begin{aligned}
ds^2 & =-\left(1-\frac{2Mr}{\rho ^2}\right) dt^2 + \frac{\rho^2}{\Delta} dr^2 + \rho^2 d\theta^2 +\left(r^2+a^2+\frac{2Mra^2}{\rho^2} \sin^2{\theta} \right)\sin^2{\theta} d \phi^2 - \frac{4Mra\sin^2{\theta}}{\rho^2}dt d\phi, \\
& = -\frac{\rho^2\Delta}{\Sigma} dt^2 + \frac{\Sigma}{\rho^2}\sin^2{\theta}(d\phi - \omega dt)^2 +\frac{\rho^2}{\Delta}dr^2 + \rho^2 d\theta^2,\end{aligned}$$
where $a=J/M$, $\rho^2=r^2+a^2\cos^2{\theta}$, $\Delta=r^2-2Mr+a^2$, $\Sigma=(r^2+a^2)^2-a^2 \Delta \sin^2{\theta}$ and $\omega=-\frac{g_{t\phi}}{g_{\phi\phi}}=\frac{2Mar}{\Sigma}$. The Cartesian coordinates can be defined as $$x=\sqrt{r^2+a^2}\sin{\theta}\cos{\phi}, \quad
y=\sqrt{r^2+a^2}\sin{\theta}\sin{\phi}, \quad
z=r\cos{\theta}.$$ There are two singularities easily reading from the coordinate where the $g^{rr}$ and $g_{tt}$ vanish. The first one gives $r_{\pm} = M \pm \sqrt{M^2-a^2}$ corresponding to the horizon analog to the Schwarzschild metric. The larger root $r_+ = M + \sqrt{M^2-a^2}$ is the event horizon, while the other root is inner apparent horizon.
The second singularity is related to an interesting effect in the Kerr spacetime called frame-dragging effect: When reaching close to the Kerr BHs, the observers even with zero angular momentum (ZAMOs) will co-rotate with the BHs because of the swirling of spacetime from the rotating body. We assume that $u^{\alpha}$ is the four-velocity of ZAMOs, and from the conservation of angular momentum $g_{\phi t} \dot{t} +g_{\phi \phi}\dot{\phi} =0 $, where an overdot is differentiation with respect to the proper time of the observers $\tau$. Thus, $\frac{d\phi}{dt}=-\frac{g_{t\phi}}{g_{\phi\phi}}$. Because of this frame-dragging effect, there is a region of spacetime where static observers cannot exist, no matter how much external force is applied. This region is known as the “ergosphere” $r \leq M+\sqrt{M^2-a^2\cos^2{\theta}}$. The rotation also leads to another interesting feature, called “superradiance”. That is, we can extract energy from scattering waves off the Kerr BHs. The exact formalism of superradiance is defined and discussed in Sec. \[super\].
Finally, the Kerr spacetime also possesses a curvature singularity at the origin $\rho^2=r^2+a^2 \cos^2{\theta}$. However, in contrast to Schwarzschild case, this singularity can be avoided since it is a ring at r=0 and $\theta = \pi/2$, where z=0 and $x^2+y^2=a^2$. In principle, observers can go through the ring without hitting the singularity. However, it is widely believed that the inner horizon, $r_-$ in Kerr spacetime is subject to an instability which would dim the analytic extension of Kerr metric beyond $r_-$ unphysical [@Poisson:1989zz].
### Blue-shift near horizon
As shown in the metric, different observers have different proper time. Hence, in the general relativity, the clocks at a gravitational field tick in a different speed in a different spacetime point. This is the blue(red)-shift effect, and it is extremely strong close to the dense object, especially near horizon.
Assuming static clocks in the Schwarzschild spacetime $ds^2=-d\tau^2=-(1-2M/r_o)dt^2$, where $\tau$ is the proper (clock) time of an observer at distance $r_o$. Hence, $t$ is the proper time of an observer at infinity. The shifted wavelength $\lambda_{o}$ measured by observers at $r_o$ compared to observers at infinite is $$\frac{ \lambda_{o}}{ \lambda_{\infty}} =\frac{d\tau}{dt}=\left(1-\frac{2M}{r_o} \right)^{1/2}.$$
### Thermodynamics of Semi-classical BH
Jacob Bekenstein and Stephen Hawking first proposed that the entropy of BHs is related to the area of their event horizons divided by the Planck area [@Bekenstein:1972tm; @Bekenstein:1973ur; @Bekenstein:1974ax; @Gibbons:1976ue; @Hawking:1978jz]. Furthermore, in 1974, Stephen Hawking showed that rather than being totally black, BHs emit thermal radiation at the Hawking temperature, $T_{\rm H} = \frac{\kappa}{2\pi}$, where $\kappa$ is the surface gravity at the horizon [@Hawking:1974rv; @Hawking:1974sw; @PhysRevD.13.2188]. This then lead to the celebrated Bekenstein-Hawking entropy formula $S_{\rm BH} = \frac{A}{ 4 }$ [@Gibbons:1976ue; @Hawking:1978jz], where $A$ is the area of the event horizon. However, the nature of microstates of BHs that are enumerated by this entropy remains so far unknown. String theory associates it with higher dimensional fuzzball solutions, as discussed later in Sec. \[fuzzball\]. Loop quantum gravity relates the quantum geometries of the horizon to the microstates [@Rovelli:1996dv]. Both these approaches can give the right Bekenstein-Hawking entropy, given specific assumptions and idealizations.
Interestingly, not only the entropy exists for the BHs, but also Brandon Carter, Stephen Hawking and James Bardeen [@Bardeen:1973gs] discovered the four laws of BH thermality analogous to the four laws of thermodynamics. The latter is presented in the parentheses.
- **The zeroth law**: A stationary BH has constant surface gravity $\kappa$. (A thermal equilibrium system has a constant temperature $T_{\rm H}$.)
- **The first law**: A small change of mass for a stationary BH is related to the changes in the horizon area A, the angular momentum J, and the electric charge Q: $dM = \frac{\kappa}{8\pi} dA + \Omega d J + \Phi d Q$, where $\Omega$ is the angular velocity and $\Phi$ is the electrostatic potential (Energy conservation: $dE=TdS-PdV-\mu dN$).
- **The second law**: The area of event horizon $A$ never decreases in general relativity (The entropy of isolated systems never decreases).
- **The third law**: BHs with a zero surface gravity cannot be achieved (Matter in a zero temperature cannot be reached).
Membrane Paradigm {#sec:membrane}
-----------------
As mentioned above, in classical general relativity, freely falling observers experience no drama as they cross the event BH horizons, at least not until they reach the singularity inside the BH. However, to a distant and static observer outside a BH, any infalling objects are frozen at the horizons due to the blue-shift effect. Hence, the BH interior can be regarded as an irrelevant region for the static observers. Based on this complementary picture near horizon, in 1986, Kip S. Thorne, Richard H. Price and Douglas A. Macdonald published the idea of *membrane paradigm* [@Thorne:1986iy]. They use a classically radiating membrane to model the BHs, which is motivated as a useful tool to study physics outside BHs without involving any obscure behavior within BH interior.
Let us introduce a spherical membrane located infinitesimally outside the Schwarzschild radius, a.k.a. the [*stretched horizon*]{}. When the membrane is sufficiently thin, one can use the Israel junction condition to nicely embed the membrane in the Schwarzschild spacetime. The condition is $$(K^{(+)} f_{ab} -K^{(+)}_{ab}) - (K^{(-)} f_{ab} -K^{(-)}_{ab}) = 8 \pi T_{ab},$$ where $f_{ab}$ is the induced metric of the membrane, $K^{(\pm)}_{ab}$ is the extrinsic curvatures on its two sides, and $T_{ab}$ is its stress tensor. The infalling observer will cross the horizon and enter the BH interior without possibility of seeing the membrane. However the static observer outside the BH can remove irrelevant interior region from the remaining spacetime with a membrane. Assuming reflection symmetry $K^+_{ab} = -K^-_{ab},$ the Israel junction condition on the membrane becomes $$K^{(+)} f_{ab} -K^{(+)}_{ab} = 4 \pi T_{ab},
\label{membrane_junc_out}$$ where the stress tensor $T_{ab}$ is no longer zero but has contribution from the extrinsic curvature on the membrane. Rewriting the left hand side of (\[membrane\_junc\_out\]), one can obtain the following relation $$T^a_b = \frac{1}{4 \pi} \left( -\sigma^a_b + \delta^a_b \left( \frac{\theta}{2} + \kappa \right) \right),$$ where $\sigma^a_b$ is the shear, $\theta$ is the expansion, and $\kappa$ is the surface gravity at the horizon. From the analogy with the energy momentum tensor of the 2-dimensional compressible fluid, one can read that the shear viscosity $\eta$ and bulk viscosity $\zeta$ are given by $\eta = 1/(16 \pi)$ and $\zeta = -1/(16 \pi)$, respectively. The negativity of the bulk viscosity implies gravitational instability for the expansion or compression at the horizon. The viscosity at the membrane lead to the thermal dissipation of infalling gravitational waves into the horizon and in this sense, the causality of a classical BH is dual to the viscosity of a 2-dimensional fluid at the stretched horizon. This was the earliest example of fluid-gravity correspondence, which is now active area of research.
Modifying Einstein gravity which revises the structure of BHs can provide a modified structure of the thin-shell membrane. For example, by adapting the transport properties of the membrane fluid, we can investigate various models of quantum BHs. As an example, we provide a simple idea [@Oshita:2019sat] which relates the reflectivity of the “horizon” of quantum BHs to viscosity in the context of membrane paradigm. We start by perturbing the Schwarzschild spacetime, whose metric is $g_{\mu \nu}^{\rm Sch}$. Within Regge-Wheeler formalism [@Regge:1957td], the axial axisymmetric perturbation $g_{\mu\nu} = g^{\rm Sch}_{\mu\nu}(r) +\delta g_{\mu\nu}(r,\theta,t)$ take teh form: $$\begin{aligned}
\delta g_{t \phi}&= \epsilon e^{- i \omega t} h_0(r) y(\theta),\label{htphi} \\
\delta g_{r \phi}&= \epsilon e^{- i \omega t} h_1(r) y(\theta),\end{aligned}$$ where other $\delta g_{\mu \nu}$ components vanish, and $\epsilon \ll 1$ controls the order of perturbation. The membrane stands at $r=r_0+ \epsilon R(t,\theta)$, where $r_0$ is its unperturbed position. We apply the Israel junction conditions $K_{ab}-K f_{ab} = - 4 \pi T_{ab}$ to Brown-York stress tensor as defined in [@Jacobson:2011dz]. The indexes $\mu, \nu$ run over $(t,r,\theta,\phi)$ in the 4d spacetime, while $a, b$ run over $(t,\theta,\phi)$ on the 3d membrane. We further assume that $T_{ab}$ is the energy stress tensor of a viscous fluid: $$\begin{aligned}
T_{ab}= [\rho_0+ \epsilon \rho_1(t,\theta) ] u_{a} u_{b}+ \nonumber\\ [ p_0+ \epsilon p_1(t,\theta)-\zeta \Theta] \gamma_{ab} -2 \eta \sigma_{ab},\\
\sigma_{ab}=\frac{1}{2}(u_{a;c} \gamma^c_b+u_{b;c}\gamma^c_a- \Theta \gamma_{ab}), \\
\gamma_{ab} \equiv h_{ab}+u_{a}u_{b}, \quad \Theta \equiv u^a_{;a},\label{theta}\end{aligned}$$ where $\rho_0$ and $p_0$ ($\rho_1$ and $p_1$) are background (perturbation on) membrane density and pressure, and $u_a$, $\eta$ and $\zeta$ are fluid velocity, shear viscosity, and bulk viscosity, respectively. Plugging Eqs. (\[htphi\]-\[theta\]) into the the Israel junction condition, we find in the zeroth order in $\epsilon$: $$\begin{aligned}
\rho_0(r_0)&= -\frac{\sqrt{f(r_0)}}{4 \pi r_0 },\\
p_0(r_0)&= \frac{\sqrt{f(r_0)} (g(r_0)+r_0 g'(r_0))}{8 \pi r_0 g(r_0)},\end{aligned}$$ where $g(r_0) = (1-2M/r_0)^{1/2}$ and $f(r_0) = 1-2M/r_0$. Assuming $u_{\phi}=0$, equation of ${\theta \phi}$ component gives in next order of $\epsilon$: $$\omega h_1(r) =-8 i \pi \eta [h_1(r) +(r-r_g)h'_1(r)].
\label{boundary}$$ We can further use $\psi_{\omega} = \frac{1}{r} \left(1-\frac{2M}{r}\right) h_1(r)$ and the tortoise coordinate $x=r+2M\log[r/(2M) -1]$ to rewrite Eq. (\[boundary\]) as $$\omega\psi_{\omega} = 16 i \pi \eta \frac{\partial \psi_{\omega}}{ \partial x}.
\label{boundaryrw}$$ For the classical BHs with a purely ingoing boundary condition $\psi_{\omega} \propto e^{-i\omega x}$ at the horizon, Eq. (\[boundaryrw\]) gives $\eta =\frac{1}{16 \pi}$, which is consistent with the standard membrane paradigm. If instead we assume there is no longer horizon but a reflective surface with $\psi_{\omega} = A_{out} e^{i\omega x} + A_{in} e^{-i\omega x} $, Eq. (\[boundaryrw\]) gives: $$\frac{A_{\rm out}}{A_{\rm in}} =\frac{1-16 \pi \eta}{1+ 16 \pi \eta} e^{-2i\omega x}.$$ Which relates the reflectivity of the membrane to the viscosity of the surface fluid.
Dawn of Gravitational Wave Astronomy
------------------------------------
From 2015 onwards, the LIGO/Virgo collaboration reported unprecedented GW observations from binary BH merger events [@TheLIGOScientific:2016agk; @TheLIGOScientific:2016pea; @Abbott:2016blz; @Abbott:2016nmj; @Abbott:2017vtc; @Abbott:2017oio; @TheLIGOScientific:2017qsa; @Abbott:2017gyy]. It is the first time that humankind can detect GWs after one century of the Einstein’s general theory of gravity. In 2017, Rainer Weiss, Kip Thorne and Barry C. Barish won the Nobel Prize in Physics “for decisive contributions to the LIGO detector and the observation of Gravitational Waves”.
The first and most prominent binary BH merger signal seen by LIGO, GW150914, matches well with predictions of numerical relativity simulations that settle into Kerr metric, but contrary to original claims, it could not confirm the existence of the event horizons [@Cardoso:2016rao]. However, it opened a new front to test general relativity in strong gravity regime and Kerr-like spacetimes (e.g., Quantum BHs) from modified gravity, which is the main topic of this review article.
This is the dawn of GW astronomy, and we stand at the threshold of a new age. We are detecting even more compact binary merger events with a better sensitivity from the O3 run of LIGO/Virgo. Future experiments such as Einstein Telescope, Cosmic Explorer, and LISA are expected to improve this by orders of magnitude. More studies on the echo-emission mechanism as well as observational strategies will be crucial for taking advantage of these new observations, to shed light on the nature of quantum BHs. It is our point of view that the best bet is on a sustained synergy between theory and observation, relying on well-motivated theoretical models (such as the Boltzmann reflectivity, aether holes, 2-2 holes, or fuzzballs, discussed in this review) to provide concrete templates for data analysis, which in turn could be used to pin down the correct theory underlying quantum BHs. With some luck, this has the potential to revolutionize our understanding of fundamental physics and quantum gravity.
Quantum Gravity and Equivalence Principle
-----------------------------------------
The Einstein’s general theory of relativity is classical. However, in the Einstein field equation $G_{\mu \nu} = 8 \pi G T_{\mu \nu}$, the classical spacetime geometry is related to stress energy tensor of quantum matter. For decades, scientist have tried to reconcile this inconsistency by embedding general relativity (or its generalizations) within some quantum mechanical framework, i.e. quantum gravity.
Conventional approach to quantizing Einstein gravity fails because it is not renormalizable. This implies that making predictions for observables, such as scattering cross-sections, requires knowledge of infinitely many parameters at high energies, leading to loss of [*predictivity*]{}. In the modern language, general relativity could at best be an effective field theory, and requires UV-completion beyond a cutoff near (or below) Planck energy (e.g., [@Donoghue:1994dn]).
Most proposals for this UV-completion involve replacing spacetime geometry with a more fundamental degree of freedom, such as strings (string theory) [@Polchinski:1998rq], discrete spins (loop quantum gravity) [@Ashtekar:2004eh], spacetime atoms (causal sets) [@Bombelli:1987aa], or tetra-hydra (causal dynamical triangulation) [@Ambjorn:2004qm]. More exotic possibilities include Asymptotic Safety [@Niedermaier:2006wt], Quadratic Gravity [@Holdom:2015kbf], and Fakeon approach [@Anselmi:2017ygm] that introduce a non-perturbative or non-traditional quantization schemes for 4d geometry. Yet another possibility is to modify the symmetry structure of General Relativity in the UV, as is proposed in Lorentz-violating (or Horava-Lifshitz) quantum gravity [@Horava:2009uw].
While proponents of these various proposals (with varying degrees of popularity) have claimed limited success in empirical explanations of some natural phenomena, it should be fair to say that none can objectively pass muster of [*predicitivity*]{}. As such, for now, the greatest successes of these proposals remain in the realm of Mathematics.
Due to this lack of concrete predictivity, the EFT estimates (discussed above) are instead commonly used to argue that the quantum gravitational effects should only show up at Planck scale $ \sim 10^{-35} meter$ or $10^{28}$ eV, which is far from anything accessible by current experiments. However, such arguments miss the possibilities of non-perturbative effects (such as phase transitions) which depend on a more comprehensive understanding of the full phase space of the specific quantum gravity proposal.
For example, it has been shown that the non-perturbative quantum gravitational effects may lead to Planck-scale modifications of the classical BH horizons [@Mathur:1997wb]. Proposed models like gravastars [@Mazur:2004fk], fuzzballs [@Lunin:2001jy; @Lunin:2002qf; @Mathur:2005zp; @Mathur:2008nj; @Mathur:2012jk], aether BHs [@PrescodWeinstein:2009mp], and firewalls [@Braunstein:2009my; @Almheiri:2012rt] amongst others [@Barcelo:2015noa; @Kawai:2017txu; @Giddings:2016tla] all drastically alter the standard structure of the BH stretched horizons with a non-classical surface. Soon after the first reported detection of gravitational waves, [@Cardoso:2016rao] discerned that Planck-length structure modification around horizons leads to a similar waveform as in classical GR, but followed by later repeating signal — *echoes* — in the ringdown from the reflective surface that replaces the classical horizon. This discovery equals a new road leading to Rome — quantum nature of gravity — and has sparked off a novel area of modeling and searching for signatures from Quantum BHs. The next section will discuss the quantum theories of BH models and possible road maps to probe them, inspired by the detection of binary BH merger events in gravitational waves.
Quantum BHs {#sec:QBHs}
===========
Evaporation of BHs and the Information Paradox
----------------------------------------------
It was already recognized by Stephen Hawking in the 1970s that the evaporation of a BH leads to an apparent breakdown of the unitarity of quantum mechanics. Here, we will briefly review this problem, which is known as the BH information loss paradox [@Hawking:1976ra]. In the context of quantum field theory in curved spacetime, the energy flux out of a BH horizon is obtained by specifying a proper vacuum state and fixing the (classical) background spacetime. However, a radiating BH must lose its mass in time, and so fixing the background is valid only for a much shorter timescale than the evaporation timescale. One can roughly estimate the lifetime of a BH as follows: The energy expectation value of a Hawking particle is of the order of the Hawking temperature $T_H \equiv (8 \pi M)^{-1}$, which would be emitted over the timescale of $t \sim M$. Then we can estimate the luminosity of the BH as $$\frac{d M}{dt} \sim \frac{- T_H}{M} \sim - ( M^2)^{-1},$$ and this gives its lifetime $t_{\text{life}}$ $$t_{\text{life}} \sim M^3.
\label{lifetimeBH}$$ To be consistent with the result of a more rigorous calculation (see e.g. [@Fabbri:2005mw]), we need a factor of about $10^{5}$ in (\[lifetimeBH\]) $$t_{\text{life}} \simeq 10^{5} M^3 \sim 10^{75} \left( \frac{M}{M_{\odot}} \right)^3 \text{[sec]},$$ which is much much longer than the cosmic age of $\sim 4 \times 10^{17} \ [\text{sec}]$ for astrophysical BHs whose mass are $ \gtrsim M_{\odot}$. It may be true that BHs evaporate due to the Hawking radiation, at least, until reaching the Planck mass. However, the gravitational curvature near the horizon eventually reaches the Planckian scale and the classical picture of background gravitational field would break down. As such, the possibility of leaving a “remnant" after the evaporation has been discussed (see e.g. [@Aharonov:1987tp; @Banks:1992is; @Giddings:1994pj; @Adler:2001vs]), but the most natural possibility would be that only Hawking radiation is left after the completion of the BH evaporation.
If the Hawking evaporation just leaves the “thermal" radiation afterwards, one can immediately understand why the evaporation process is paradoxical. Let us suppose that a pure quantum state collapses into a BH and it radiates Hawking quanta until the BH evaporates. If the final state is a thermal mixed state, the evaporation is a process which transforms a pure to mixed state. Therefore, if the final state of any BH is a completely thermal state, one can say that the evaporation process is a non-unitary process. The information loss paradox can be also explained from the geometric aspect using the Penrose diagram. In quantum mechanics, the time-evolution of a quantum state is described by a unitary operator, $\hat{U}$, that maps an initial quantum state $\ket{\text{in}}$ on a past Cauchy surface $\Sigma_{\text{i}}$ into a final quantum state $\ket{\text{f}}$ on a future Cauchy surface $\Sigma_{\text{f}}$. Since the unitary operator gives a reversible process, one can also obtain the initial state from the final state as $$\ket{\text{in}} = \hat{U}^{\dag} \ket{\text{f}}.$$ Although this is true in a flat space, the argument is very controversial in the existence of an evaporating BH. Assuming a gravitational collapse forms a horizon and singularity, then it eventually evaporates, leaving behind a thermal radiation, the Penrose diagram describing the whole process is given by Fig. \[haw1\]. Let us consider three quantum states: an initial quantum state $\ket{\text{in}}$ on $\Sigma_{\text{i}}$, an intermediate quantum state $\ket{\text{mid}}$ on $\Sigma_{\text{m}}$, and a final state $\ket{\text{f}}$ on $\Sigma_{\text{f}}$, where $\Sigma_{\text{i}}$, $\Sigma_{\text{m}}$, and $\Sigma_{\text{f}}$ are the Cauchy surfaces and $\Sigma_{\text{m}}$ intersects the future horizon $H^+$ and so one can split it into the exterior and interior regions as $\Sigma_{\text{m}} \equiv \Sigma_{\text{ext}} \cup \Sigma_{\text{int}}$ (see Fig. \[haw1\]).
The final quantum state $\ket{\text{f}}$ is determined by information on the exterior part of the intermediate Cauchy surface $\Sigma_{\text{ext}}$ rather than that on the whole intermediate Cauchy surface $\Sigma_{\text{m}}$, which leads to the information loss paradox. To see this in more detail, let us consider an initial pure quantum state $$\displaystyle \ket{\text{in}} = \sum_{i} c^{\rm in}_i \ket{\psi_i},$$ where $\left\{ c^{\rm in}_i \right\}$ is an initial vector in the Hilbert space. The intermediate state is still a pure state due to the unitary evolution of $\ket{\text{in}}$ $$\ket{\text{mid}} = \displaystyle \hat{U} \ket{\text{in}} = \sum_{i,j} c_{i,j} \ket{\psi_i}_{\text{int}} \otimes \ket{\psi_j}_{\text{ext}},$$ the time-evolution from $\Sigma_{\text{m}}$ to $\Sigma_{\text{f}}$ is non-unitary, provided that the final state on $\Sigma_{\text{f}}$ is obtained by the unitary evolution of the exterior intermediate state. The density matrix of the exterior intermediate state, denoted by $\hat{\rho}_{\text{ext}}$, is obtained by tracing over all the internal basis states: $$\hat{\rho}_{\text{ext}} = \displaystyle \sum_{k} \bra{\psi_k}_{\text{int}} \ket{\text{mid}} \braket{\text{mid} | \psi_k}_{\text{int}} = \displaystyle \sum_{k,j,j'} c_{k,j} c^{\ast}_{k,j'} \ket{\psi_j}_{\text{ext}} \bra{\psi_{j'}}_{\text{ext}}.
\label{densitymatext}$$ The resulting density matrix, (\[densitymatext\]), is independent of the interior orthogonal basis $\left\{ \ket{\psi_j}_{\text{int}} \right\}$ due to the tracing operation. Therefore, the loss of the interior information results in a non-unitary evolution and an initial quantum state evolves to a mixed state after the BH evaporation.
![The Penrose diagram describing an evaporating BH. []{data-label="haw1"}](haw1.png){height="80mm"}
BH complementarity
------------------
The BH complementarity has been one of the leading proposals for the retrieval of BH information, which was first put forth by by Susskind, Thorlacius, and Uglum [@Susskind:1993if]. According to a distant observer, due to the infinite redshift at a BH horizon, the Hawking radiation involves modes of transplanckian frequency whose energy can be arbitrarily large in the vicinity of the horizon. In the BH complementarity proposal, the energetic modes form the membrane, which can absorb, thermalize, and reemit information, on the BH horizon. They argue that such a picture regarding the retrieval of BH information by the stretched horizon is consistent with the following three plausible postulates:\
\
[*P*ostulate 1 (unitarity)]{}— According to a distant observer, the formation of a BH and the evaporation process can be described by the standard quantum theory. There exists a unitary S-matrix which describes a process from infalling matter to outgoing non-thermal radiation.\
\
[*P*ostulate 2 (semi-classical equations)]{}— Outside the stretched horizon of a massive BH, physics can be approximately described by a set of semi-classical field equations.\
\
[*P*ostulate 3 (degrees of freedom)]{}— For a distant observer, the number of microscopic states of a BH can be estimated by $\exp{S (M)}$, where the exponent $S(M)$ is the Beksntein-Hawking entropy.\
\
On the other hand, it has been presumed that a freely infalling observer would not observe anything special when passing through the horizon due to the equivalence principle. In this sense, there are two totally different and seemingly inconsistent scenarios that co-exist in the BH complementarity. However, the contradiction arises only when attempting to compare the experiments performed inside and outside horizon, which might be impossible due to a backreaction of the high-energy modes near the stretched horizon [@Susskind:1993mu].
Firewalls
---------
In 2012, Almheiri, Marolf, Polchinski and Sully (AMPS) argued [@Almheiri:2012rt] that the Postulates 1-3 in the BH complementarity and the Equivalence principle of GR are mutually inconsistent for an *old BH* [@Page:1993df; @Page:1993wv; @Page:2013dx], provided that the monogamy of entanglement is satisfied. Then they argued that the “most conservative” resolution is a violation of the equivalence principle near the BH and its horizon should be replaced by high-energetic quanta, so called “firewall”, to avoid the inconsistency. Before introducing the original firewall argument in more detail, let us review a theorem in quantum information theory, the monogamy of entanglement. Let us consider three independent quantum systems, A, B, and C. The strong subadditivity relation of entropy is given by $$S_{AB} + S_{BC} \geq S_{B} + S_{ABC}.$$ If A and B is fully entangled, we have $$S_{AB} = 0 \ \ \text{and} \ \ S_{ABC} = S_C.$$ Then the strong subadditivity relation reduces to $$S_{B} + S_{C} - S_{BC} \leq 0.
\label{mutual1}$$ Since the left hand side in (\[mutual1\]) is the mutual information of $B$ and $C$, denoted by $I_{BC}$, and it is a non-negative quantity, (\[mutual1\]) reduces to $$I_{BC} = S_{B} + S_{C} - S_{BC} = 0,$$ which means that the quantum system B cannot fully correlate with C when B and A are fully entangled mutually. Therefore, any quantum system cannot fully entangle with other two quantum systems simultaneously. This is the monogamy of entanglement that is an essential theorem in the firewall argument.
Let us consider an old BH, whose origin is a gravitational collapse of a pure state, with early Hawking particles A, late Hawking particle B, and infalling particle inside the horizon C. In order for the final state of the BH to be pure state, A and B should be fully entangled mutually, that is a necessary condition for the Postulate 1. On the other hand, created pair particles , B and C, are also fully entangled according to the quantum field theory in classical background (Postulate 2). That is, imposing the Postulate 1 and 2 inevitably results in that B is fully and simultaneously entangled with both A and C, which obviously contradicts with the monogamy of entanglement. In order to avoid this contradiction, AMPS argued that there is no interior of BHs and the horizons should be replaced by energetic boundaries that the entanglement of Hawking pairs are broken. They called these boundaries “firewalls”. According to this proposal, any object falling into a BH would burn up at the firewall, which contradicts the equivalence principle (in vacuum) and replaces the BH complementarity proposal. Although there are some updates of this proposal, based on ER=EPR conjecture [@Almheiri:2013hfa; @Papadodimas:2012aq; @Maldacena:2013xja; @Susskind:2013lpa; @Bousso:2012as], backreaction due to gravitational schockwaves [@Yoshida:2019qqw], and quantum decohenrence of Hawking pair due to the interior tidal force [@Oshita:2016pbh]), they do remain speculative, and at the level of toy models. However, on general grounds, if quantum effects lead to such an energetic wall at the stretched horizon, it could contribute to the reflectivity of BH which may be observable by merger events leading to the formation of BHs.
Gravastars
----------
The gravitational vacuum condensate star (gravastar) was proposed as a final state of gravitational collapse by Mazur and Mottola [@Mazur:2004fk]. According to the proposal, the resulting state of gravitational collapse is a cold compact object whose interior is a de Sitter condensate, which is separated from the outside black hole spacetime by a null surface. In this state, there is no singularity (with the exception of the null boundary) and no event horizon, which avoids the BH information loss paradox. Such gravitational condensation could be caused by quantum backreaction at the Schwarzschild horizon $r=r_g$ even for an arbitrarily large-mass collapsing object. One might wonder why the backreaction can lead to such a drastic effect for any mass since the tidal force which acts on an infalling test body can be arbitrarily weak for an arbitrarily large mass at the Schwarzschild radius. The argument is that considering a photon with asymptotic frequency $\omega$ near the Schwarzschild radius, the (infinite) blue-shift effect by which the local energy is enhanced as $\hbar \omega / \sqrt{1-r_g/r}$, could lead to a drastic effect at the Schwarzschild radius. This is unavoidable since any object is immersed in quantum vacuum fluctuations and virtual particles always exist around them. From this argument, the gravitational condensation has been expected to take place at the final stage of gravitational collapse. The authors in [@Mazur:2004fk] also estimate the entropy on the surface of gravastar by starting with a simplified vacuum condenstate model which consists of three different equations of state $$\begin{aligned}
0 \leq r < r_1, ~~~~~~~~~~~ &\rho = -p,\\
r_1 < r < r_1+\delta r, ~~~~~~~~~~~ &\rho=p,\\
r_1+\delta r < r, ~~~~~~~~~~ &\rho=p=0,\end{aligned}$$ where $r_1$ is the radius of interior region and $\delta r$ is the thickness of the thin-shell of the gravastar. Then the obtained entropy of the shell was found out to be $S \sim 10^{57} g k_{\text{B}} \left( M/M_{\odot} \right)^{3/2}$, where $g$ is a dimensionless constant. Recently, the derivation of gravastar-like configuration was performed by Carballo-Rubio [@Carballo-Rubio:2017tlh]. He derived the semi-classical Tolman-Oppenheimer-Volkoff (TOV) equation by taking into account the polarization of quantum vacuum and solved it to obtain the exact solution of an equilibrium stellar configuration. It also has its de Sitter interior and thin-shell near the Schwarzschild radius, which is consistent with the original gravastar proposal [@Mazur:2004fk].
From the observational point of view, the shadows of a gravastar was investigated in [@Sakai:2014pga] where they argue the shadows of a BH and gravastar could be distinguishable. In addition, tests of gravastar with GW observations have been discussed in e.g. [@Pani:2009ss; @Cardoso:2016oxy; @Conklin:2017lwb].
\[fuzzball\]Fuzzballs
---------------------
Samir D. Mathur has proposed fuzzballs [@mathur2005fuzzball] as description of true microstates of the quantum BHs from string theory. A fuzzball state has the BH mass inside a horizon-sized region and a smooth (but higher-dimensional) geometry. Here are some crucial features of the conjecture:
1. Different fuzzball geometries represent different microstates of the quantum BH — fuzzball. Application the AdS/CFT duality [@Maldacena:1997re] suggests that the counting of the microstates is consistent with the Bekenstein-Hawking entropy.
2. Fuzzballs do not possess horizons. Instead, they end with smooth “caps” near where the horizons would have been. Every microstate has almost the same geometry outside the would-be horizon matching the classical BH picture for the outside observers. But the microstates differ from each other near the would-be horizons.
3. Fuzzball solves the information paradox by removing the horizon and singularity. The horizon is replaced by fuzzy matter and no longer vacuum. The particles created near the would-be horizon now have access to the information of fuzzball interior. Moreover, the higher-dimensional spacetime ends smoothly around the would-be horizon and is singularity-free. The infalling particles at the low frequencies interact with the “fuzz” for a relatively long time scale, while high frequency ones excite the microstates and lose their energy the same as in the classical BHs case. Hence, the traditional horizons only show up effectively from the point of view of an outside observers, over relatively short time scale $\lesssim M \log(M)$.
How do these higher dimensional “microstates” with the smooth and horizonless geometries looks like? We, for the first time, show a specific reduced 4D fuzzball solution has an associated 4D effective fluid near the would-be horizon. The anisotropic pressure of the fluid is crucial to the horizonless geometry.
Applying Kaluza-Klein reduction of non-supersymmetric microstates of the D1-D5-KK system [@giusto2007non]. the metric in 4D is $$\begin{aligned}
\label{equ1}
ds_4^2=-\frac{f^2}{\sqrt{AD}}(dt+c_1 c_5 \omega)^2+\sqrt{AD}[\frac{dr^2}{\Delta}+d\theta^2+\frac{\Delta}{f^2} \sin^2\theta d\varphi^2]\\
\Delta=r^2-r_0^2, \ f^2=\Delta+r_0^2n^2\sin^2\theta,\\
A=f^2+2p[(r-r_0)+n^2 r_0 (1+\cos\theta) ],\\
B=f^2+2\frac{r_0 (r-r_0) (n^2-1)}{p-r_0 (1+n^2)}[(r-r_0)+n^2 r_0 (1-\cos\theta)], \\
C=2\frac{r_0 \sqrt{r_0(r+r_0)} n (n^2-1)}{p-r_0 (1+n^2)}[(r-r_0)+(p+ r_0) (1-\cos\theta)],\\
G=\frac{Af^2-C^2}{B^2},\ D=B c_1^2 c_5^2- f^2(c_1^2 s_5^2+s_1^2 c_5^2)+ \frac{G f^2}{A} s_1^2 s_5^2,\\
J^2=\frac{r_0^3 p (r+r_0) n^2 (n^2-1)^2}{p-r_0 (1+n^2)}, \ \omega ^2=\frac{2J \sin^2\theta(r-r_0)}{f^2} d\varphi,\end{aligned}$$ where parameters $c_1$, $c_5$, $s_1$, $s_5$, $r_0$, n and p are related to the mass, angular momentum and charges of the solution.
### \[sec2.1\]Asymptotic behavior
Here, we study the asymptotic behavior of metric. As shown in Table \[table1\], it behaves exactly like Schwarzschild metric when setting K=1, M=$r_{\rm g}$. They have different $g_{t\varphi}$ compared with Kerr metric. As stated, it resembles the Schwarzschild BH far away, but present different geometry close to the horizon.
Metric Fuzzball Kerr BH Sch. BH
---------------------- -------------------------------------------------------- ---------------------------------------- -----------------------------------------
$g_{tt}$ $-K+\frac{M}{K^3} \frac{1}{r} +O( \frac{1}{r^2})$ $-1+r_g\frac{1}{r} +O( \frac{1}{r^2})$ $-1+r_g \frac{1}{r} +O( \frac{1}{r^2})$
$g_{rr}$ $K+\frac{M}{K} \frac{1}{r} +O( \frac{1}{r^2})$ $1+r_g\frac{1}{r} +O( \frac{1}{r^2})$ $1+r_g \frac{1}{r} +O( \frac{1}{r^2})$
$g_{\theta\theta}$ $Kr^2+\frac{M}{K}r +O( 1)$ $r^2+O(1)$ $r^2+O(1)$
$g_{\varphi\varphi}$ $Kr^2\sin^2\theta +\frac{M}{K}r+O(1)$ $r^2\sin^2\theta +O(1)$ $r^2\sin^2\theta +O(1)$
$g_{t\varphi}$ $ 0+\frac{4 c_1 c_5 J}{K}\frac{1}{r}+O(\frac{1}{r^2})$ $ N r+ L \frac{1}{r}+O(\frac{1}{r^2})$ 0
: \[table1\]Asymptotical behavior of metric
$$\begin{aligned}
K=\sqrt{c_1^2 c_5^2-c_1^2 s_5^2-s_1^2 c_5^2}\\
M=\frac{p}{K}+\frac{c_1^2 c_5^2 J^2}{r_0^2 p n^2 (n^2-1)}\\
N=-2r_g \alpha r \sin^2\theta\\
L=2r_g \alpha^3 \cos^2\theta \sin^3\theta\end{aligned}$$
### \[sec2.2\]Matter field
We can now study the effective 4d matter stress tensor from the Einstein tensor of the 4d fuzzball geometry (\[equ1\]). For a sample choice of parameters, outlined in Table \[table2\], the (diagonalized) Energy-stress tensor $T^{\mu}_{\ \nu}$ is $$\left( \begin{array}{cccc}
-\rho=-\frac{5981}{1461}f(r,\theta)-g(r,\theta) & 0 & 0 & 0 \\
0 & P_1=f(r,\theta) & 0 & 0 \\
0 & 0 & P_2=-f(r,\theta) & 0 \\
0 & 0 & 0 & P_3=g(r,\theta)\\ \end{array} \right)$$
$r_0$ $p$ m $c_1$ $c_5$ $s_1$ $s_5$
------- ----- --- ------- ------- ------- ----------------------
$1$ $4$ 2 $2$ $1$ $1$ $\frac{1}{\sqrt{2}}$
: \[table2\]Parameters Setting
where $f(r,\theta)$ and $g(r,\theta)$ are functions of coordinate r and $\theta$. The concrete expressions depend on parameter setting. $\rho$, $P_1$, $P_2$ and $P_3$ are the energy density and anisotropic pressure of the matter field. Their behavior near horizon is shown in Fig. \[fuzzmatter\]. Pressure $P_1=-P_2$ is an analytic result and true for any parameter setting, while relationship between the energy and pressure: $-\rho=-\frac{5981}{1461}f(r,\theta)-g(r,\theta)=-\frac{5981}{1461}P_1-P_3$ is a numerical approximation. The approximation is exact far away from the fuzzball with parameters in Table \[table2\] and for any given $\theta$ except 0 and $\pi$. The relationship changes near the horizon shown as in Fig \[fuzzhorizon\] after averaging over $\theta$. At around $r \sim 1000$, we have radial pressure equals tangential pressure $P_1=P_3$. Similar to the metric, matter fields are singular at $r=r_0$, $\theta=0$ and $\pi$.
![\[fuzzmatter\]Near-horizon matter field of fuzzball solution from paper [@giusto2007non]. Shown in the figure, $P_1=-P_2$. Asymptotically $P_1=-P_3$.](fuzzmatter.jpg){width="0.8\columnwidth"}
![\[fuzzhorizon\]Near-horizon relationship between density and pressure of the fuzzball solution from paper [@giusto2007non]. The dashed line is the asymptotic behavior, and the solid line is the real behavior of $\frac{P_1}{P_3 - \rho}$](fuzzhorizon.jpg){width="0.8\columnwidth"}
The matter field has an anisotropic pressure. It is not traceless so it cannot be a simple electromagnetic field. We also checked that it cannot be a single scalar field. However, most fuzzball microstates still remain intractable, with no clear dimensional reduction or 4d geometry. To circumvent this obstacle, we have proposed a “mock fuzzball” spacetime [@wang:2016mock] which captures the horizonless feature of the model with an anisotropic fluid. This conjecture leads to an interesting application to dynamic binary quantum BH merger simulations, as discussed later in Sec. \[numerical\].
\[sec3\]Mock Fuzzballs
----------------------
![\[figmockf\] The geometry of Schwarzschild mock fuzzball. Red curve shows that the metric is modified around the horizon, and the shaded area is removed from the spacetime.](mockfuzz.pdf){width="\columnwidth"}
Here, we introduce a “mock” fuzzball geometry, based on the motivation to build a generic and macroscopic metric which captures important, coarse-grained properties of the fuzzball, i.e, the metric has neither horizon nor singularity and spacetime ends around the stretched horizon. Fig. \[figmockf\] shows how the causal diagram of a Schwarzschild BH is changed to remove the horizon. We study different mock fuzzballs, and check the corresponding matter fields, a swell as potential observable effect. The simplest case of the mock fuzzball [^1] is given by: $$\begin{aligned}
\label{simmock}
ds^2=-(1-\frac{2M}{r}+b) dt^2+\frac{1}{1-\frac{2M}{r}} dr^2+r^2 d\Omega^2, \ r>2M\end{aligned}$$ where parameter $ 0 < b \ll 1$ is only important around $2M$. It ensures that $g_{tt}$ doesn’t vanish and thus remove the horizon. The geometry resembles a traditional BH far away. Besides, this metric is only valid where $r>2M$ to imitate a fuzzball metric which ends around the stretched horizon. The corresponding matter field is $$\begin{aligned}
\label{fakematter}
P_r&=T^{r}_{\ r}= -\frac{bM}{4 \pi r^2(-2M+r+br)}, \\
P_t&=T^{\theta}_{\ \theta} = T^{\varphi}_{\ \varphi}=\frac{bM(-M+r+br)}{8 \pi r^2(-2M+r+br)^2}, \\ &\textnormal{other components vanish}\end{aligned}$$ The anisotropic behavior of the pressure near the stretched horizon is shown as in Fig. \[schmfuzz\] with $b=0.01$ and $M=\frac{1}{2}$. The absolute value of two pressures are equal at $r= \frac{3M}{1+b}$. Tangential pressure is much larger than radial pressure near horizon, and opposite far away. It doesn’t have any singularity at r=2M since all fields reach an extremum there.
![\[schmfuzz\] Anisotropic behavior of pressure of Schwarzschild mock fuzzball versus proper distance from horizon. The absolute value of two pressures are equal at $r= \frac{3M}{1+a}$. Radial pressure (dashed line) is much larger than tangential(solid line) pressure near horizon (proper length=0), and opposite far away. The figure shows that when approaching horizon, both pressure reach the extremum, hence have no singularity at r=2M.](schmfuzz.jpg){width="0.45\columnwidth"}
### Other mock fuzzballs
Besides the simplest case introduced in the last section, we can modify other terms in Schwarzschild metric to recover energy density which fuzzballs in Sec. \[fuzzball\] actually have. We also study charged and rotating BHs.
- Schwarzschild metric
The simplest mock fuzzball studied above has no energy density. However, we can recover energy density by changing $g_{\theta\theta}$ and $g_{\varphi\varphi}$ within spherical symmetry: $$\begin{aligned}
ds^2=-(1+b-\frac{2M}{r}) dt^2+\frac{1}{1-\frac{2M}{r}} dr^2+(2 d M + (1 - d) r)^2 d\Omega^2, \ r>2M\\
\rho=-T^{t}_{\ t}=\frac{d}{4M^2}+O(r-2M)\\
P_r= T^{r}_{\ r}=-\frac{1}{4M^2}+O(r-2M)\\
P_t=T^{\theta}_{\ \theta}=T^{\varphi}_{\ \varphi}=\frac{1 + 2 b - 2 b d}{16 b M^2}+O(r-2M)\\
\textnormal{other components vanish}\end{aligned}$$ Small b and d ensures that $\rho \ll P_r \ll P_t$. In addition, $b>0,\ d<1$ ensure finite Ricci scalar without curvature singularity at $r>2M$.
Another possible modification to recover energy density is to assume that mass has a time dependence: $$\begin{aligned}
ds^2=-(1+b-\frac{2M(t)}{r}) dt^2+\frac{1}{1-\frac{2M(t)}{r}} dr^2+r^2 d\Omega^2, \ r>2M\\
\rho=-G^{t}_{\ t}=-\frac{\sqrt{-b M(t)^3 M'(t)^2}}{\sqrt{2 b M(t)^3 \sqrt{r-2M(t)}} }+\frac{1}{8M(t)^2}+)(r-2M(t))^\frac{1}{2}\\
P_r= G^{r}_{\ r}=-\frac{\sqrt{-b M(t)^3 M'(t)^2}}{\sqrt{2 b M(t)^3 \sqrt{r-2M(t)}} }-\frac{1}{8M(t)^2}+O(r-2M(t))^\frac{1}{2}\\
P_t=G^{\theta}_{\ \theta}=G^{\varphi}_{\ \varphi}=-\frac{3M'(t)^2}{b (r-2M(t))^2}+\frac{M'(t)^2-b M(t)m''(t)}{a^2M(t)(r-M(t))}\\
+\frac{b^2+2b^3-4M'(t)^2-8bM'(t)^2+8bM(t)m''(t)}{16b^3M(t)^2}+O(r-2M(t))\\
\textnormal{other components vanish}\end{aligned}$$
- Extremal BH metric
For an extremal BH, the fuzzball has another interesting property: Proper length from somewhere near stretched horizon to “horizon” is finite, in contrast to the infinite throat in the traditional picture. Parameter $c$ here captures the finite throat. $$\begin{aligned}
ds^2=-(b+(1-\frac{r_q}{r})^2) dt^2+\frac{1}{c+(1-\frac{r_q}{r})^2} dr^2+r^2 d\Omega^2, \ r>2M\\
\rho=-T^{t}_{\ t}=\frac{1-c}{{r_q}^2}+O(r-r_q)\\
P_r= T^{r}_{\ r}=-\frac{-1+c}{{r_q}^2}+O(r-r_q)\\
P_t=T^{\theta}_{\ \theta}=T^{\varphi}_{\ \varphi}=\frac{c}{b {r_q}^2}+O(r-r_q)\\
\textnormal{other components vanish}\end{aligned}$$
- Non-Extremal BH metric $$\begin{aligned}
ds^2=-(a+1-\frac{r_s}{r}+(\frac{r_q}{r})^2) dt^2+\frac{1}{c+1-\frac{r_s}{r}+(\frac{r_q}{r})^2} dr^2+r^2 d\Omega^2, \ r>2M\\
\rho=-T^{t}_{\ t}=\frac{c r^2 - rq^2}{{r}^4}\\
P_r= T^{r}_{\ r}=-\frac{(1 + a) c r^4 + (-1 + a) r^2 {r_q}^2 - c r^2 {r_q}^2 -{r_q}^4 - a r^3 rs +
r {r_q}^2 rs}{r^4 ({r_q}^2 + r (r + a r - rs))}\\
Pt=T^{\theta}_{\ \theta}=T^{\varphi}_{\ \varphi} \textnormal{ dropped for simplicity}\\
\textnormal{other components vanish}\end{aligned}$$
### What does an infalling observer see?
Assuming Einstein field equations, mock fuzzball geometries can only be sourced by matter fields with exotic (and anisotropic) equations of state. Considering simplest Schwarzschild mock fuzzball: $$ds^2=-(1-\frac{2M}{r}+b) dt^2+\frac{1}{1-\frac{2M}{r}} dr^2+r^2 d\Omega^2,$$ There is no longer vacuum outside the horizon, which may potentially lead to observable effects, depending on how strongly the “fuzz” matter can interact with the detectors. To visualize the signal, we assume a geodesic observer radially falling towards the stretched horizon with zero velocity at infinity. We calculate (see Appendix \[fuzzballflux\] for details) two observable scalars: energy density $\mathcal{U}$ and energy flux $\mathcal{F}$, as seen by the observer: $$\begin{aligned}
\mathcal{U}&= T_{\mu \nu} u^{\mu} u^{\nu}=-\frac{4 b M^2}{r^2 (b r-2 M+r)^2},\\
\mathcal{F}&= -T_{\mu \nu} u^{\mu} a^{\nu}=-\frac{ b \left(\frac{2M}{r}\right)^{3/2}}{(b r-2 M+r)^2},\end{aligned}$$ where $T_{\mu \nu}$ is from Einstein field equation of the mock fuzzball, $a^{\nu}$ is the unit detector area vector and $u^{\mu}$ is the four-velocity of the observer with $a_\mu u^\mu=0$. Both energy density and flux are finite and vanish when parameter $b$ vanishes.
To visualize observable energy density and flux, we choose parameters $M=1$ and $b=0.1$ and compare it with the signal of Hawking radiation as Fig. \[energy\] and Fig. \[flux\]. Both energy density and flux of the mock fuzzball are larger than those of Hawking radiation near the horizon, but drops faster away from the horizon. Energy density of the mock fuzzball is negative while Hawking radiation is positive. Flux of the mock fuzzball flows into BH while Hawking radiation flows into BH near horizon and changes direction away from the horizon.
Specially, energy density and flux of fuzzball are finite near horizon while Hawking radiation diverges. This is because we use Page’s approximation of the Hawking radiation [@page1982thermal], which assumes the Hartle-Hawking state for expectation value of stress-energy tensor: $$\begin{aligned}
T_{tr}&=\frac{-7.44 10^5}{4 \pi r(r-2M)\frac{1}{M^2}},\\
T_{rr}&=\frac{1}{1-2M/r} \alpha^{-1} (\beta+960\frac{M^6}{r^6}),\\
T_{tt}&=(1-2M/r) 3 \alpha^{-1} (\beta-2112\frac{M^6}{r^6}),\\
\alpha&=368640 {\pi}^2 M^2,\\
\beta&=192 \left(\frac{M}{r}\right)^5+80 \left(\frac{M}{r}\right)^4+32 \left(\frac{M}{r}\right)^3+12 \left(\frac{M}{r}\right)^2+\frac{4M}{r}+1,\end{aligned}$$
The Hartle-Hawking state is a thermal equilibrium states of particles while the Hawking radiation is not in equilibrium. The inconsistency leads to infinite flux and energy density of Hawking radiation. More precise correction can be found in [@parker2009quantum].
![\[energy\] The energy density of mock fuzzball, compared with that of Hawking radiation as seen by a radially infalling observer along a geodesic. Here the orange curve is positive and the black curve is negative. The energy density of mock fuzzball is larger than the Hawking radiation near horizon and drops faster than the Hawking radiation away from the horizon.](energy_density.png){width="\columnwidth"}
![\[flux\] The energy flux seen by an observer falling through fuzzball geometry, compared to that of Hawking radiation. Here the orange curve is positive, while the black curves are negative. The flux of mock fuzzball is larger than the Hawking radiation near horizon and drops faster than the Hawking radiation away from the horizon.](flux.jpg){width="\columnwidth"}
Aether Holes and Dark Energy
----------------------------
In 2009, Prescod-Weinstein, Afshordi, and Balogh [@prescod2009stellar] studied the spherically symmetric solutions of the Gravitational Aether proposal for solving the old cosmological constant problem [@Afshordi:2008xu; @Aslanbeigi:2011si]. Surprisingly, they showed that if one sets Planck-scale boundary conditions for aether near the horizons of stellar mass BHs, its pressure will match the observed pressure of dark energy at infinity.
In the Gravitational Aether proposal [@Afshordi:2008xu; @Aslanbeigi:2011si], the modified Einstein field equation is given by $$\begin{aligned}
\frac{1}{8 \pi G'}G_{\mu \nu}=T_{\mu \nu}-\frac{1}{4}T ^{\alpha}_{\ \alpha}g_{\mu \nu}+T'_{\mu \nu},\\
T'_{\mu \nu}=p'(u'_{\mu}u'_{\nu}+g_{\mu \nu}),\end{aligned}$$ where $G'=\frac{4}{3}G_N$, and then energy-momentum tensor of aether is assumed to be a perfect fluid with stress-energy tensor $T'_{\mu \nu}$ without energy density. Here, quantum vacuum energy decouples from the gravity, as only the traceless part of the matter energy-momentum tensor appears on the right-hand side of the field equations. It can be shown that the Bianchi identity and energy-momentum conservation completely fix the dynamics, and thus the theory has no additional free parameters, or dynamical degrees of freedom, compared to General Relativity.
The modified Schwarzschild metric is the vacuum solution with spherical symmetry in modified equations, and identical to a traditional equations sourced by the aether perfect fluid. Far away from the would-be horizon but close enough to the origin ($2M \ll r \ll |p_0|^{-1/2}$), the solution has the form $$ds^2=-(1+4\pi p_0 r^2)dt^2 +dr^2 +r^2 d\Omega^2$$ which can be compared to the de Sitter metric $$ds^2=-(1-\frac{8}{3}\pi \rho_\Lambda r^2)dt^2 + (1-\frac{8}{3}\pi \rho_\Lambda r^2)^{-1} dr^2 +r^2 d\Omega^2$$
We see that assuming $p_0=-\frac{2}{3}\rho_\Lambda$, the $g_{\rm tt}$’s agree with each other. Therefore, the Newtonian observers (for $2M \ll r \ll |p_0|^{-1/2}$) will experience the same acceleration as in the de-Sitter metric with the cosmological constant. However, on larger scales, one has to take into account the effects of multiple black holes and other matter in the Universe. The Planckian boundary conditions at the (would-be) horizon relates the pressure of the aether to the mass of the astrophysical BHs, $-p_0 \sim M^{-3}$ [@prescod2009stellar]. In particular, the BH masses within the range $10~ M_{\odot}-100~ M_{\odot}$, which correspond to the most astrophysical BHs in galaxies, yield aether pressures comparable to the pressure of Dark Energy, inferred from cosmic acceleration. Moreover, Ricci scalar is inversely proportional to $g_{tt}$, so the event horizon where $g_{tt}=0$ has a curvature singularity, which is reminiscent of the firewall and fuzzball proposals discussed above.
In particular, the fuzzball paradigm is a good approach to remove the singularity. On the one hand, fuzzball gives an extra anisotropic matter field similar to the aether theory, which stands as a good evidence that quantum effects can modify the Einstein field equation with extra sources of 4d energy-momentum like aether. Furthermore, fuzzball is a regular and horizonless geometry, which might indicate the singularity is removable in the full quantum picture of BHs.
2-2 holes
---------
In general relativity, gravitational collapse of ordinary matter will always leads to singularities behind trapping horizons [@1965PhRvL..14...57P]. In [@Holdom:2016nek], Holdom and Ren revisited this problem with the asymptotically free quadratic gravity, which could be regarded as a UV completion of general relativity [@Holdom:2016nek]. The quantum quadratic gravity (QQG), whose action is given by $$S_{\text{QQG}} = \int d^4 x \sqrt{-g} \left( \frac{1}{2} {\cal M}^2 R - \frac{1}{2 f_2^2} C_{\mu \nu \alpha \beta} C^{\mu \nu \alpha \beta} + \frac{1}{3 f_0^2} R^2 \right),
\label{QQG_action}$$ is famously known to be not only asymptotically free, but also perturbatively renormalizable [@Stelle:1976gc; @Voronov:1984; @Fradkin:1981iu; @Avramidi:1985ki]. However, it suffers from a spin-2 ghost due to the higher derivative terms, which is commonly regarded as a pathology of the theory. In [@Holdom:2016nek], it is proposed that the ghost may not be problematic when ${\cal M}$ is sufficiently small, so that the poles in the perturbative propagators fall into the non-perturbative regime, and the perturbative analysis of ghosts is not reliable. Then it is conjectured [@Holdom:2015kbf] that the full graviton propagator in the IR, when ${\cal M} \lesssim \Lambda_{QQG}$, the spin-2 ghost pole is absent in an analogy with the quantum chromodynamics (QCD) where the gluon propagator, describing off-shell gluons, also does not have a pole. Here $\Lambda_{QQG}$ is a certain critical value in QQG, analogous to confinment scale $\Lambda_{QCD}$ in QCD. Based on this conjecture, the asymptotically free quadratic action in (\[QQG\_action\]) may involve small quadratic corrections at super-Planckian scale, and so the super-Planckian gravity might be governed by the classical action $$S_{CQG} = \frac{1}{16 \pi} \int d^4x \sqrt{-g} \left( M_{\text{Pl}}^2 R -\alpha C_{\mu \nu \alpha \beta} C^{\mu \nu \alpha \beta} + \beta R^2 \right).
\label{CQG_action}$$ Since gravitational collapse would involve the super-Planckian energy scale, applying the classical action (\[CQG\_action\]) to such a situation is interesting from a point of view of the quantum gravitational phenomenology. Then the authors in [@Holdom:2016nek] found a solution of horizonless compact object, so-called 2-2 hole, in the classical quadratic gravity. 2-2 holes have an interior with a shrinking volume and a timelike curvature singularity at the origin. It also has a thin-shell configuration, leading to non-zero reflectivity at the would-be horizon, which may cause the emission of GW echoes [@Conklin:2017lwb]. Recently, 2-2 holes sourced by thermal gases were also investigated in [@Holdom:2019ouz; @Ren:2019afg].
Non-violent Unitarization {#sec:non-violent}
-------------------------
A separate class of possible approaches to the BH information paradox involves a violation Postulate 2 in BH complementarity, i.e. non-locality of field equations well outside the stretched horizon, which is dubbed as “nonviolent unitarization” by Steve Giddings [@Giddings:2017mym]. Such a possibility would allow for transfer of information outside horizon around the Page time (e.g., [@Bardeen:2018omt; @Bardeen:2018frm]), but could also lead to large scale observable deviations from general relativistic predictions in GW and electromagnetic signals [@Giddings:2019vvj]. However, it is not clear whether this non-locality is only limited to BH neighborhoods, and if not, how it could affect precision experimental/observational tests in other contexts. Moreover, in contrast to GW echoes that we shall discuss next, it is hard to provide concrete predictions for astrophysical observations in the nonviolent unitarization scenarios.
Gravitational Wave Echoes: Predictions {#sec:echo_predictions}
======================================
GW echoes may be one of the observable astrophysical signals, a smoking gun, so to speak, for the quantum gravitational processes near BH horizons. A number of models of Exotic Compact Objects (ECOs) that we discussed above are expected to emit GW echoes. Some examples are wormholes [@Cardoso:2016rao], gravastars [@Cardoso:2016oxy], and 2-2 holes [@Holdom:2016nek]. Moreover, even Planckian correction in the dispersion relation of gravitational field [@Oshita:2018fqu; @Oshita:2019sat; @Wang:2019rcf] and the BH area quantization [@Cardoso:2019apo] may also lead to echo signals. Not only the specific models to reproduce GW echoes but also comprehensive modeling of echo spectra in non-spinning case [@Mark:2017dnq], in spinning case [@Wang:2019rcf; @Conklin:2019fcs], and in a semi-analytical way [@Testa:2018bzd; @Maggio:2019zyv] have been investigated, which enable us to easily obtain echo spectra. In this section, we review the details of GW echoes by starting with the Chandrasekhar-Detweiler (CD) equation [@Chandrasekhar:1976zz; @Detweiler:1977gy] that is a wave equation with a purely real angular momentum barrier in the Kerr spacetime. We also provide a short review of the GW ringdown signal, that is followed by the GW echo, and the superradiance of spinning BHs. The superradiance with a high reflectivity at the would-be horizon may cause the ergoregion instability, which we shall also discuss separately.
On the equations governing the gravitational perturbation of spinning BHs
-------------------------------------------------------------------------
The GW ringdown is one of the most important signals to probe the structure of BH since it mainly consists of discrete QNMs of BH characterized by mass and spin. In this subsection, we review that the QNMs can be obtained by looking for specific complex frequencies such that the mode functions of GWs satisfy the outgoing boundary condition. Let us start with the CD equation [@Chandrasekhar:1976zz; @Detweiler:1977gy] that is the wave equation for a spin-$s$ field and has a purely real angular momentum barrier: $$\left[ \frac{d^2}{dr^{\ast} {}^2} - {\cal V}_{ij} \right] {}_s X_{lm} (r^{\ast}, \omega) = - T,
\label{SN_equation}$$ where $T$ is the source term and the potential ${\cal V}_{ij}$ with $i,j = \pm 1$ is given by $$\begin{aligned}
\begin{split}
{\cal V}_{ij} &= \frac{-K^2}{(r^2+a^2)^2} + \frac{\rho^4 \Delta}{(r^2+a^2)^2} \left[ \frac{\lambda (\lambda+2)}{g+ b_i \Delta} -b_i \frac{\Delta}{\rho^8} + \frac{(\kappa_{ij} \rho^2 \Delta - h) (\kappa_{ij} \rho^2 g -b_i h)}{\rho^4 (g+b_i \Delta) (g-b_i \Delta)^2} \right]\\
&+ \left[ \frac{r \Delta a m/\omega}{(r^2+a^2)^2 \rho^2} \right]^2 - \frac{\Delta}{(r^2+a^2)} \frac{d}{dr} \left[ \frac{r \Delta a m/\omega}{(r^2+a^2)^2 \rho^2} \right].
\end{split}
\label{CD_potential_1}\end{aligned}$$ The functions in (\[CD\_potential\_1\]) are defined by $$\begin{aligned}
b_{\pm1} &\equiv \pm 3 (a^2 - am/\omega),\\
\kappa_{ij} &\equiv j \left\{ 36 M^2 -2\lambda \left[ (a^2-am/\omega) (5\lambda+6) -12 a^2 \right] + 2 b_i \lambda (\lambda +2) \right\}^{1/2},\\
\rho^2 &\equiv r^2+a^2-am/\omega,\\
\Xi_{i} & \equiv \frac{\Delta^2}{\rho^8} (F+b_i),\\
\Theta_{ij} &\equiv i \omega +\frac{1}{F-b_i} \left( \frac{\Delta}{\rho^2} \frac{dF}{dr} -\kappa_{ij} \right),\\
\begin{split}
\kappa &\equiv (\lambda^2 (\lambda+2)^2 +144 a^2\omega^2 (m-a\omega)^2 -a^2 \omega^2 (40 \lambda^2-48 \lambda)+a\omega m (40 \lambda^2 +48 \lambda))^{1/2}\\
&~~~~+12 i \omega M,
\end{split}\\
F & \equiv \frac{\lambda \rho^4 +3 \rho^2 (r^2-a^2) -3r^2 \Delta}{\Delta},\\
g & \equiv \lambda \rho^4 +3 \rho^2 (r^2-a^2) -3r^2 \Delta,\\
h & \equiv g' \Delta - g \Delta'.\end{aligned}$$ Equation (\[CD\_potential\_1\]) gives four potentials, $(i,j) = (-1,+1), \ (+1, -1), \ (+1,+1)$, and $(-1,-1)$. One has to use the different potentials in order to cover the whole frequency space with the CD potentials because $1/(g+ b_{+1} \Delta)$ and $1/(g+ b_{-1} \Delta)$ in (\[CD\_potential\_1\]) are singular in different frequency regions (see FIG. \[CD\_potential\]).
![The CD potentials with $(i,j) = (-1,+1)$ and $(+1,-1)$ for $\bar{a} = 0.8$ and $\ell = m = 2$. The potential ${\cal V}_{+1,-1}$ (${\cal V}_{-1,+1}$) is singular for $r_g \omega = 0.5$ $(1.7)$ in this case. []{data-label="CD_potential"}](CD_potential.png){height="45mm"}
The CD equation is obtained as the generalized Darboux transformation of the Teukolsky equation [@Chandrasekhar:1975zza][^2]. In the asymptotic regions, $r^{\ast} \to \pm \infty$, the CD equation reduces to the following wave equation $$\begin{aligned}
\begin{cases}\displaystyle
\left( \frac{d^2}{dr^{\ast} {}^2} + \tilde{\omega}^2 \right) {}_s X_{lm} = - T, \ &\text{for} \ r^{\ast} \to - \infty,\\
\displaystyle
\left( \frac{d^2}{dr^{\ast} {}^2} + \omega^2 \right) {}_s X_{lm} = -T, \ &\text{for} \ r^{\ast} \to + \infty,
\end{cases}
\label{SN_equation_asmp}\end{aligned}$$ where $\tilde{\omega} \equiv \omega - m \Omega_{\rm H}$, in terms horizon angular frequency $\Omega_{\rm H} \equiv a / (2 M r_+)$, and horizon outre radius $r_+ \equiv M + \sqrt{M^2 - a^2}$ of the Kerr BH. In the following, we will omit the subscripts of $l$, $m$, and $s$ for brevity. One can read that the homogeneous solutions of (\[SN\_equation\_asmp\]) are given by the superposition of ingoing and outgoing modes $$X=
\begin{cases}
A e^{i \tilde{\omega} r^{\ast}} + B e^{-i \tilde{\omega} r^{\ast}} & \text{for} \ r^{\ast} \to - \infty,\\
C e^{i \omega r^{\ast}} + D e^{-i \omega r^{\ast}} & \text{for} \ r^{\ast} \to +\infty,
\end{cases}$$ where $A$, $B$, $C$, and $D$ are arbitrary constants. The QNMs can be found by looking for the complex frequencies at which the homogeneous solution satisfies the outgoing boundary condition of $A=D = 0$. This is equivalent to looking for the zero-points of the Wronskian between the two homogeneous solutions $X_{+}$ and $X_{-}$ $$W_{\text{BH}} \equiv X_{-} \frac{d X_{+}}{dr^{\ast}} - X_+ \frac{d X_-}{d r^{\ast}} = 2i \omega A_{\text{in}} (\omega) = 2i\tilde{\omega} B_{\text{out}} (\omega),$$ where the two homogeneous solutions satisfy the following boundary conditions $$\begin{aligned}
&X_- \sim
\begin{cases}
e^{-i \tilde{\omega} r^{\ast}} \ &\text{for} \ r^{\ast} \to - \infty,\\
A_{\text{out}} e^{i \omega r^{\ast}} + A_{\text{in}} e^{-i \omega r^{\ast}} \ &\text{for} \ r^{\ast} \to + \infty,
\label{modein}
\end{cases}\\
&X_+ \sim
\begin{cases}
B_{\text{in}} e^{-i \tilde{\omega} r^{\ast}} + B_{\text{out}} e^{i \tilde{\omega} r^{\ast}} \ &\text{for} \ r^{\ast} \to - \infty,\\
e^{i \omega r^{\ast}} \ &\text{for} \ r^{\ast} \to + \infty.
\label{modeout}
\end{cases}\end{aligned}$$ Recently, the QNMs of Kerr spacetime were precicely investigated in [@Casals:2019vdb] by using the method developed by Mano, Suzuki, and Takasugi [@10.1143/PTP.95.1079; @Mano:1996mf; @Sasaki:2003xr; @Casals:2018eev] that enables us to obtain the solution of the Teukolsky equation in an analytic way.
\[super\]Transmission and reflection coefficients of the angular momentum barrier
---------------------------------------------------------------------------------
The GW echoes are results of multiple reflections in the [*c*avity]{} between the would-be horizon (e.g., fuzzball/firewall) and angular momentum barrier (see Fig. \[echo\_pic\_1\]) and so the amplitude of echoes is mainly determined by the reflectivities of the would-be horizon and angular momentum barrier. In this subsection, we review the calculation of the reflectivity of angular momentum barrier.
![GW echoes following a BBH merger from a cavity of membrane/firewall-angular momentum barrier [@Abedi:2016hgu]. []{data-label="echo_pic_1"}](echo_pic.pdf){width="70.00000%"}
From the mode functions (\[modein\], \[modeout\]), one can obtain the energy conservation law for the incident, reflected, and transmitted waves by using another Wronskian relation $$\tilde{W}_{\text{BH}} \equiv X \frac{d X^{\ast}}{dr^{\ast}} - X^{\ast} \frac{d X}{dr^{\ast}},$$ which is constant for real frequency due to the reality of the angular momentum barrier in the CD equation. Then we obtain the following relations by using $\tilde{W}_{\text{BH}} (- \infty) = \tilde{W}_{\text{BH}} (+ \infty)$ for $X_-$ and $X_+$ $$\begin{aligned}
&1-\left| \frac{A_{\text{out}}}{A_{\text{in}}} \right|^2 = \frac{\tilde{\omega}}{\omega} \frac{1}{|A_{\text{in}}|^2},\\
&1-\left| \frac{B_{\text{in}}}{B_{\text{out}}} \right|^2 = \frac{\omega}{\tilde{\omega}} \frac{1}{|B_{\text{out}}|^2}.\end{aligned}$$ From the above relations, one can read that the energy reflectivity and transmissivity for inward incident waves $$I_{\text{ref}}^{\leftarrow} \equiv |A_{\text{out}}/A_{\text{in}}|^2, \ \ I_{\text{trans}}^{\leftarrow} \equiv (\tilde{\omega}/\omega) |{1}/A_{\text{in}}|^2,
\label{ref/tra_in}$$ respectively, and those for outward incident waves are given by $$I_{\text{ref}}^{\rightarrow} \equiv |B_{\text{in}}/B_{\text{out}}|^2, \ \ I_{\text{trans}}^{\rightarrow} \equiv (\omega/\tilde{\omega}) |{1}/B_{\text{out}}|^2.
\label{ref/tra_out}$$ From (\[ref/tra\_in\], \[ref/tra\_out\]), one can calculate the reflectivity/transmissivity of the angular momentum barrier by numerically solving the homogeneous CD equation (\[SN\_equation\]). In the spinning case $\bar{a} > 0$, the energy reflectivity is greater than $1$ for $-m \Omega_H< \tilde{\omega} < 0$, a phenomenon that is often referred to as BH [*superradiance*]{}. The superradiance can be characterized by the amplification factor $Z \equiv (I_{\rm ref}^{\leftarrow/\rightarrow})^2-1$, and when only interested in the low frequency region, one can use the analytic expression [@1973JETP...37...28S] $$\displaystyle
Z \simeq 4 Q \beta_{sl} \prod_{k=1}^{l} \left( 1+ \frac{4 Q^2}{k^2} \right) [\omega (r_+ -r_-)]^{2l +1},
\label{sta-san}$$ where $r_-$ is the radius of inner horizon, $\sqrt{\beta_{sl}} \equiv \frac{(l-s)! (l+s)!}{(2l)! (2l+1)!!}$ and $Q \equiv- \frac{r_+^2 +a^2}{r_+-r_-} \tilde{\omega}$. To give a few examples of $I_{\text{ref}}^{\leftarrow}/I_{\text{ref}}^{\rightarrow}$, we numerically calculate it in the frequency range $0.001 \leq 2M \omega \leq 2$, which is shown in the FIG. \[ampl1\]. As can be seen from FIG. \[ampl1\], the energy flux reflectivity exceeds $1$, which means that the energy of a spinning BH is extracted by reflected radiation, within the superradiance regime. We will discuss the ergoregion instability caused by the superradiance and the reflectivity of would-be horizon in Sec. \[subsec\_ergoinst\].
![The amplification factor for $\bar{a} = 0.8$ and $\ell = m = 2$. In the right panel, low frequency region (solid) is calculated with the potential ${\cal V}_{-1,+1}$ and the higher frequency region (dashed) is calculated from ${\cal V}_{+1,-1}$ in the CD equation.[]{data-label="ampl1"}](amplification1.png){height="45mm"}
Transfer function of echo spectra and geometric optics approximation
--------------------------------------------------------------------
When the GW echo is caused by an incident wave packet repeatedly reflected between the cavity, one can use the geometric optics approximation to predict the GW echo signal, which was first pioneered in [@Mark:2017dnq]. Let us start with the calculation of the Green’s function of GW ringdwon signal $G_{\text{BH}}$ by using the CD equation. It satisfies $$\left( \frac{d^2}{dr^{\ast} {}^2} - {\cal V} \right) G_{\text{BH}}(r^{\ast}, r^{\ast} {}') = \delta (r^{\ast}-r^{\ast} {}'),$$ where we omit the subscripts of ${\cal V}_{ij}$. Once imposing the outgoing boundary condition, the Green’s function $G_{\text{BH}}$ is uniquely determined as $$G_{\text{BH}} (r^{\ast}, r^{\ast} {}') = \frac{X_- (r^{\ast}_{<}) X_+ (r^{\ast}_>)}{W_{\text{BH}}},$$ where $r^{\ast}_{<} \equiv \text{min} (r^{\ast}, r^{\ast} {}')$ and $r^{\ast}_{>} \equiv \text{max} (r^{\ast}, r^{\ast} {}')$. Therefore, when there is no reflectivity at the horizon, the Fourier mode of GWs at infinity and at the horizon can be obtained as $$\begin{aligned}
\lim_{r^{\ast} \to \infty} X(r^{\ast}, \omega) &= -X_+ (r^{\ast}) \int^{+ \infty}_{-\infty} d r^{\ast} {}' \frac{X_- (r^{\ast} {}') T (r^{\ast} {}')}{W_{\text{BH}}} \equiv X_+ Z_{\infty} (\omega),
\label{Z_inf}\\
\lim_{r^{\ast} \to -\infty} X(r^{\ast}, \omega) &= -X_- (r^{\ast}) \int^{+ \infty}_{-\infty} d r^{\ast} {}' \frac{X_+ (r^{\ast} {}') T (r^{\ast} {}')}{W_{\text{BH}}} \equiv X_-
Z_{\text{BH}} (\omega),
\label{Z_BH}\end{aligned}$$ where $T(r^*)$ is the source for the inhomogeneous CD equation. If there is no reflection near the horizon, the relevant observable spectrum is only $Z_{\infty}$, and $Z_{\text{BH}}$ is irrelevant for observation. On the other hand, if reflection at the would-be horizon is caused by a certain mechanism, $Z_{BH}$ is also observable in addition to $Z_{\infty}$.
One can obtain echo spectra by using the geometric optics approximation, which should be reliable as long as the would-be horizon and angular momentum barrier are well separated in tortoise coordinates, $r^*$. The amplitude of the first echo, $Z_{\text{echo}}^{(1)}$, can be estimated by $$Z_{\text{echo}}^{(1)} \simeq {\cal T}_{\text{BH}}^{\rightarrow} {\cal R} e^{-2i \tilde{\omega} r^{\ast}_0} Z_{\text{BH}} (\omega),$$ and the second echo may have the amplitude of $$Z_{\text{echo}}^{(2)} \simeq {\cal T}_{\text{BH}}^{\rightarrow} {\cal R}^2 {\cal R}_{\text{BH}}^{\rightarrow} e^{-2\times 2i \tilde{\omega} r^{\ast}_0} Z_{\text{BH}} (\omega).$$ As such, one can obtain the amplitude of $n$-th echo as $$Z_{\text{echo}}^{(n)} = {\cal T}_{\text{BH}}^{\rightarrow} {\cal R}^n ({\cal R}_{\text{BH}}^{\rightarrow})^{n-1} e^{-2 n i \tilde{\omega} r^{\ast}_0} Z_{\text{BH}} (\omega),$$ where ${\cal R}_{\rm BH}^{\rightarrow} \equiv B_{\rm in}/B_{\rm out}$ and ${\cal T}_{\rm BH}^{\rightarrow} \equiv \sqrt{\omega/ |\tilde{\omega}|} B_{\rm out}^{-1}$. Since only the reflectivity and transmissivity of outgoing waves are involved in the echoes, we will not use ${\cal R}_{\rm BH}^{\leftarrow}$ and ${\cal T}_{\rm BH}^{\leftarrow}$ in the following, and so omit the symbol $\rightarrow$. Summing up all contributions from $n=1$ to $n=\infty$, one obtains $$\displaystyle
\sum_{n=1}^{\infty} Z_{\text{BH}}^{(n)} = \frac{{\cal T}_{\text{BH}} {\cal R} e^{-2i \tilde{\omega} r^{\ast}_0}}{1-{\cal R} {\cal R}_{\text{BH}} e^{-2i\tilde{\omega} r^{\ast}_0}} Z_{\text{BH}} \equiv {\cal K} (\omega) Z_{\text{BH}}.$$ Note that here we assume that $|{\cal R} {\cal R}_{\text{BH}}| < 1$, as otherwise the infinite sum of the geometric series does not converge. Finally, the spectrum taking into account the reflection at the would-be horizon is obtained $$X (r^{\ast}, \omega) = (Z_{\infty} + {\cal K} Z_{\text{BH}}) e^{i \omega r^{\ast}} = (1+ {\cal K} Z_{\text{BH}} / Z_{\infty}) Z_{\infty} e^{i \omega r^{\ast}}.$$ When ${\cal R} = 0$ we have ${\cal K} = 0$ and so it reduces to (\[Z\_inf\]). Once we specify a specific form of the source term $T$, one can obtain $Z_{\text{BH}}/ Z_{\infty}$. For example, let us assume the source term located at $r^{\ast} = r^{\ast}_s$ $$T(r^{\ast}) = S (\omega) \delta (r^{\ast}-r^{\ast}_s),$$ where $S (\omega)$ is a non-singular function in terms of frequency. Substituting this source term in (\[Z\_inf\]) and (\[Z\_BH\]), one obtains $$\frac{Z_{\text{BH}}}{Z_{\infty}} = \frac{{\cal R}_{\text{BH}} + e^{-2i\tilde{\omega} r^{\ast}_s}}{{\cal T}_{\text{BH}}}.$$ Note that this is independent of the function $S(\omega)$. Therefore, we finally obtain the following transfer function $$\begin{aligned}
X &= e^{i \omega r^{\ast}} Z_{\infty} \left( 1+ {\mathcal K} \frac{Z_{\text{BH}}}{Z_{\infty}} \right) =e^{i \omega r^{\ast}}
Z_{\infty} \left( 1+ {\mathcal K}^{+}_{\text{echo}} + {\mathcal K}^{-}_{\text{echo}} \right),
\label{Xtilde}\\
{\mathcal K}^{+}_{\text{echo}} &\equiv \frac{{\mathcal R}_{\text{BH}} {\mathcal R} e^{-2i \tilde{\omega} r_0^{\ast}}}{1-{\mathcal R} {\mathcal R}_{\text{BH}} e^{-2i \tilde{\omega} r_0^{\ast}}},
\label{K+}\\
{\mathcal K}^{-}_{\text{echo}} &\equiv \frac{e^{-2i\tilde{\omega} r^{\ast}_s} {\mathcal R} e^{-2i \tilde{\omega} r_0^{\ast}}}{1-{\mathcal R} {\mathcal R}_{\text{BH}} e^{-2i \tilde{\omega} r_0^{\ast}}}.
\label{K-}\end{aligned}$$ As discussed in [@Oshita:2019seis1], actually ${\cal K}_{\text{echo}}^+$ and ${\cal K}_{\text{echo}}^-$ represent two different trajectories of GWs in the cavity. Here we are interested in outgoing incident waves that is related to ${\cal K}_{\text{echo}}^+$ and so in the following we discard ${\cal K}_{\text{echo}}^-$ from the transfer function, which does not change the qualitative feature of resulting echo signals.
Once we determine the spectrum of injected GWs, $Z_{\infty}$, one can obtain a template of GW echoes. Using the spectrum of GW ringdown may be a good approximation to obtain a realistic template. In this case, $Z_{\infty}$ is given by [@Berti:2005ys] $$\begin{aligned}
Z_{\infty} &= \frac{2 G M}{D_o} \tilde{A}_{lm0} \left[ e^{i \phi_{lm0}} S_{lm0} (\theta) \alpha_+ + e^{-i \phi_{lm0}} S^{\ast}_{lm0} (\theta) \alpha_- \right],
\label{Z_inf_QNM}\\
\alpha_{\pm} &\equiv \frac{-\text{Im}[\omega_{lm0}]}{\text{Im}[\omega_{lm0}]^2 + (\omega \pm \omega_{lm0})^2},\end{aligned}$$ where $D_o$ is the distance between the GW source and observer, $\omega_{lm0}$ is the most long-lived QNM, $\tilde{A}_{lm0}$ is the initial ringdown amplitude, $\phi_{lm0}$ is the phase of ringdown GWs, and $\theta$ is the observation angle. The amplitude $\tilde{A}_{lm0}$ is proportional to $\sqrt{\epsilon_{\text{rd}}}$, where $\epsilon_{\text{rd}} \equiv E_{\text{GW}}/M$ and $E_{\text{GW}}$ is the total energy of GW ringdown [@Berti:2005ys]. To give a few examples, the ringdown $+$ echo spectra are shown in FIG. \[spe\_echo\_ring\].
Ergoregion Instability and the QNMs of quantum BH {#subsec_ergoinst}
-------------------------------------------------
As pointed out in the previous subsection, one should check if $|{\cal R} {\cal R}_{\text{BH}}| < 1$ is satisfied when calculating the transfer function in the geometric optics picture. This is physically important to understand the ergoregion instability caused by the reflection at the would-be horizon. Since the common ratio of the geometric series is ${\cal R} {\cal R}_{\text{BH}}$, the echo amplitude may be amplified and diverges when $|{\cal R} {\cal R}_{\text{BH}}| > 1$. This is nothing but the ergoregion instability that prevents BHs from having high spins. One can also derive the criterion from the QNMs of ECOs. The echo QNMs $\omega_n$ can be obtain by looking for the poles of the Green’s function of echo GWs. That is, one can look for the poles from the zero points of the denominator of the transfer function $$\begin{aligned}
1- {\cal R} {\cal R}_{\text{BH}} e^{-2i (\omega_n - m \Omega_H) r^{\ast}_0}=0,\end{aligned}$$ and we obtain $$\omega_n = \frac{2 \pi n +(\delta+ \delta')}{\Delta t_{\text{echo}}} + m \Omega_H + i \frac{\ln{|{\cal R} {\cal R}_{\text{BH}}|}}{\Delta t_{\text{echo}}},$$ where $\Delta t_{\text{echo}} \equiv 2 |r^{\ast}_0|$, $\delta \equiv \text{arg} [{\cal R}]$ and $\delta' \equiv \text{arg} [{\cal R}_{\text{BH}}]$. Then we obtain the real and imaginary parts of the echo QNMs $$\begin{aligned}
\text{Re} [\omega_n] &\simeq \frac{2 \pi n +(\delta+ \delta')}{\Delta t_{\text{echo}}} + m \Omega_H,\\
\text{Im} [\omega_n] &\simeq \left. \frac{\ln{|{\cal R} {\cal R}_{\text{BH}}|}}{\Delta t_{\text{echo}}} \right|_{\omega = \text{Re}[\omega_n]}.\end{aligned}$$ The positivity of the imaginary part of QNMs, which leads to the instability, is equivalent to having $|{\cal R} {\cal R}_{\text{BH}}| > 1$. Furthermore, we can see that the real parts of the QNM frequencies depend on the phases of ${\cal R}$ and ${\cal R}_{\rm BH}$, while their imaginary part depends on their absolute values. We can also rewrite the imaginary part in terms of the amplification factor $$\text{Im} [\omega_n] \simeq \frac{\ln|{\cal R}|}{\Delta t_{\text{echo}}} + \frac{\ln{(1+Z)}}{2 \Delta t_{\text{echo}}} \simeq \frac{\ln|{\cal R}|}{\Delta t_{\text{echo}}} + \frac{Z}{2 \Delta t_{\text{echo}}}.$$ Then we obtain the analytic form of the imaginary part of QNMs in the low-frequency regime $(M \omega \ll 1)$ $$\text{Im} [\omega_n] \simeq \frac{\ln|{\cal R}|}{\Delta t_{\text{echo}}} + \frac{2Q}{\Delta t_{\text{echo}}} \beta_{sl} \prod_{k=1}^{l} \left( 1+ \frac{4 Q^2}{k^2} \right) \left[\left( \frac{2 \pi n +(\delta+ \delta')}{\Delta t_{\text{echo}}} + \Omega_H \right) (r_+ -r_-) \right]^{2l +1},$$ where we used (\[sta-san\]). This is the generalization of the analytic form of QNMs [@Oshita:2019seis1]. This analytic form is well consistent with numerically obtained QNMs in the low frequency region as is shown in FIG. \[qnm1\].
![QNMs with $|R| = 0.7$ (red filled circles) and $0.9$ (blue filled circles), $r^{\ast}_0 = -40 M$, $\bar{a}=0.9$, and $\ell = m= 2$.[]{data-label="qnm1"}](QNM_num_ana.png){height="55mm"}
Echoes from Planckian correction to dispersion relation and Boltzmann Reflectivity {#subsec:boltzmann}
----------------------------------------------------------------------------------
The Boltzmann reflection of a BH horizon has been discussed in the context of (stimulated) Hawking radiation from the path integral approach [@PhysRevD.13.2188], quantum tunneling approach [@Srinivasan:1998ty; @Vanzo:2011wq], and Feynman propagator approach [@Padmanabhan:2019yyg]. Recently, two of us studied the reflectivity of a BH for incident GWs from a Lorentz violating dispersion relation and argued that it can be approximated by a Boltzmann-like reflectivity [@Oshita:2018fqu]. More recently, three of us used general arguments from thermodynamic detailed balance, fluctuation-dissipation theorem, and CP-symmetry to show that the reflectivity of quantum BH horizons should be universally given by a Boltzmann factor [@Oshita:2019sat; @Wang:2019rcf]: $$\left.
\frac{{\cal F}_{\rm out}}{{\cal F}_{\rm in}}\right|_{\rm horizon} = \exp\left(-\frac{\hbar \tilde{\omega}}{k_B T_{\rm H}}\right) \label{Bolt_flux}$$ The reflection of quantum BH might be understood as Hawking radiation stimulated by enormous number of incoming gravitons, and if that is so, having the dependence of the reflectivity on the Hawking temperature $T_{\rm H}$ is natural. Furthermore, one can also avoid the ergoregion instability in this model [@Oshita:2019sat; @Wang:2019rcf]. In this subsection, we briefly review the Boltzmann reflectivity model from both theoretical and phenomenological aspects.
### Boltzmann reflectivity from dissipation
The dissipative effects at the apparent horizon have been discussed from the point of view of the membrane paradigm [@Thorne:1986iy; @Jacobson:2011dz], the fluctuating geometry around a BH [@Parentani:2000ts; @Barrabes:2000fr], and the minimal length uncertainty principle [@Brout:1998ei]. Our approach to derive the Boltzmann reflectivity starts with a heuristic assumption to model the dissipative effects, which are expected in any thermodynamic system from fluctuation-dissipation theorem. Let us assume that the wave equation governing the perturbation of BH is given by [@Oshita:2019sat]: $$\left[ -i \frac{\tilde{\gamma} \Omega (r^{\ast})}{E_{\text{Pl}}} \frac{d^2}{dr^{\ast} {}^2} + \frac{d^2}{dr^{\ast} {}^2} + \tilde{\omega}^2 - V(r^{\ast}) \right]
\psi_{\tilde{\omega}} (r^{\ast}) = 0,
\label{we1}$$ where $\tilde{\gamma}$ is a dimensionless dissipation parameter, $\Omega (x) \equiv |\tilde{\omega}|/ \sqrt{|g_{00} (x)|}$ is the blueshifted (or proper) frequency, and $V$ is the angular momentum barrier. The form of the dissipation term is expected from the fluctuation-dissipation theorem near the horizon, where the Hawking radiation (quantum fluctuation/dissipation) and the incoming GWs (stimulation) are blue shifted. This dissipative modification to the dispersion relation becomes dominant only when the blueshift effect is so intense that the proper frequency is comparable to the Planck energy, $\Omega \sim E_{\text{Pl}}$. Furthermore, from a phenomenological point of view, the dissipative term in (\[we1\]) is similar to the viscous correction to sound wave propagation in terms of shear viscosity, $\nu$, in Navier-Stokes equation, $-i (4/3) \nu \Omega \nabla^2$ (e.g., [@Liberati:2013usa]).
Let us solve the modified wave equation by imposing a physically reasonable boundary condition (see FIG. \[mode\_func\]): $$\begin{aligned}
\psi_{\tilde{\omega}} \sim \text{constant.} \ \ \text{for} \ \ r^{\ast} \to - \infty,\end{aligned}$$
![Mode function obtained by solving (\[we1\]) with the boundary condition of $\psi_{\tilde{\omega}} = \text{constant.}$ for $r^{\ast} \to - \infty$.[]{data-label="mode_func"}](mode_function.png){height="75mm"}
The constant boundary condition in the limit of $r^{\ast} \to - \infty$ means that the energy flux carried by the ingoing GWs cannot go through the horizon, and is either absorbed or reflected. That is consistent with the BH complementarity [@Susskind:1993if] or the membrane paradigm [@Thorne:1986iy; @PhysRevD.18.3598]. Although there is no unique choice of wave equation around a Kerr BH, we here choose the CD equation that has a purely real angular momentum barrier. The modified CD equation is assumed to have the form of $$\left( \frac{- i \tilde{\gamma} |\tilde{\omega}|}{\sqrt{\delta (r)} E_{\text{Pl}}} \frac{d^2}{dr^{\ast} {}^2} + \frac{d^2}{dr^{\ast} {}^2} - {\cal V} \right) \psi_{\tilde{\omega}} =0,\label{SNeq}$$ where $\sqrt{\delta(r)} \equiv \sqrt{1-r_g/r +(a/r)^2}$ is the blue shift factor in terms of the co-rotating frame [@Frolov:2014dta; @Poisson:2009pwt]. In the near horizon limit ($r^{\ast} \to - \infty$, see below for details), the CD equation reduces to the following form in the limit of $r^{\ast} \to -\infty$: $$\left(-i \frac{\tilde{\gamma} |\tilde{\omega}|}{Q E_{\text{Pl}}} e^{-\kappa_+ r^{\ast}} \frac{d^2}{dr^{\ast} {}^2} + \frac{d^2}{dr^{\ast} {}^2} - \tilde{\omega}^2 \right) \psi_{\tilde{\omega}} = 0,
\label{SN_near_horizon}$$ where $\kappa_+$ is the surface acceleration at the outer horizon, $Q$ is defined as $$Q \equiv \exp \left[ \frac{1}{2} \frac{\sqrt{1-\bar{a}^2}}{r_+^2/r_g^2 +\bar{a}^2/4} \left( - \frac{r_+}{r_g} + \frac{r_-^2/r_g^2 +\bar{a}^2/4}{2\sqrt{1-\bar{a}^2}} \log{(1-\bar{a}^2)} \right) + \frac{1}{2} \log \sqrt{1-\bar{a}^2} -\log \left(\frac{r_+}{r_g}\right) \right],$$ and $r_{\pm} \equiv M(1 \pm \sqrt{1 - \bar{a}^2})$. The solution of (\[SN\_near\_horizon\]) which satisfies the aforementioned boundary condition is $$\displaystyle
\lim_{r^{\ast} \to -\infty} \psi_{\tilde{\omega}} = {}_2 F_1 \left[ -i \frac{\tilde{\omega}}{\kappa_+}, i \frac{\tilde{\omega}}{\kappa_+}, 1, -i \frac{Q E_{\text{Pl}} e^{\kappa_+ r^{\ast}}}{\tilde{\gamma} |\tilde{\omega}|} \right],$$ and one can read that in the intermediate region, $- \kappa_+^{-1} \log \left[Q E_{\text{Pl}} / (\tilde{\gamma} |\tilde{\omega}|) \right] \ll r^{\ast} \ll \pm \kappa_{+}^{-1}$, $\psi_{\tilde{\omega}}$ can be expressed as the superposition of outgoing and ingoing modes $$\begin{aligned}
\psi_{\tilde{\omega}}=
\begin{cases}
e^{\pi \tilde{\omega}/(2 \kappa_+)}A_+ e^{-i \tilde{\omega} r^{\ast}} + e^{-\pi \tilde{\omega}/(2 \kappa_+)} A_+^{\ast} e^{i \tilde{\omega} r^{\ast}} \ &\text{for} \ \tilde{\omega} > 0,\\
e^{-\pi \tilde{\omega}/(2 \kappa_+)}A_- e^{-i \tilde{\omega} r^{\ast}} + e^{\pi \tilde{\omega}/(2 \kappa_+)} A_-^{\ast} e^{i \tilde{\omega} r^{\ast}} \ &\text{for} \ \tilde{\omega} < 0,
\end{cases}\end{aligned}$$ where $A_{\pm}$ has the form of $$A_{\pm} \equiv \left(\frac{\tilde{\gamma} |\tilde{\omega}|}{Q E_{\text{Pl}}} \right)^{i \tilde{\omega}/ \kappa_+} \times \frac{\Gamma (-2i \tilde{\omega}/ \kappa_+)}{\Gamma (-i \tilde{\omega}/\kappa_+) \Gamma (1-i \tilde{\omega}/\kappa_+)} = e^{i \tilde{\omega} r^{\ast}_0} \times \frac{\Gamma (-2i \tilde{\omega}/ \kappa_+)}{\Gamma (-i \tilde{\omega}/\kappa_+) \Gamma (1-i \tilde{\omega}/\kappa_+)},$$ and $$\begin{aligned}
r^{\ast}_0 \equiv \frac{1}{\kappa_+} \ln{\left( \tilde{\gamma} | \tilde{\omega}|/(Q E_{\text{Pl}}) \right)}.
\label{xf}\end{aligned}$$ Therefore, the energy reflectivity is given by $$\begin{aligned}
|{\cal R}|^2 =
\begin{cases}
e^{-2\pi \tilde{\omega}/\kappa_+} \ &\text{for} \ \tilde{\omega} > 0,\\
e^{2\pi \tilde{\omega}/\kappa_+} \ &\text{for} \ \tilde{\omega} < 0,
\end{cases}\end{aligned}$$ and finally we obtain $${\cal R} = \exp{\left[- \frac{|\tilde{\omega}|}{2 T_H} +i\delta_{\text{wall}} \right]},\label{Bolt_amp}$$ where $\delta_{\text{wall}}$ is the phase shift at the would-be horizon and it is determined by $A_+$ or $A_-$. Equation (\[Bolt\_amp\]) then reproduces the Boltzmann energy flux reflectivity in (\[Bolt\_flux\]). As we noted earlier, the same result can be independently derived using thermodynamic detailed balance or CP symmetry near BH horizons.
When we further modify the dispersion relation by adding a quartic correction term $$\tilde{\Omega}^2 = \tilde{K}^2 +i \tilde{\gamma} \tilde{\Omega} \tilde{K}^2 -C_d^2 \tilde{K}^4,
\label{dispersion}$$ where $C_d$ is a constant parameter and $\tilde{\Omega}$ and $\tilde{K}$ are the proper frequency and proper wavenumber, respectively, the exponent of the Boltzmann factor is modified, and the analytic form can be obtain for $C_d \gg \tilde{\gamma}$ by using the WKB approximation [@Oshita:2018fqu] $$\label{Ref_dis}
|{\cal R}| \simeq \exp{\left[ - \frac{\sqrt{2 + 4 C_d^2 /\tilde{\gamma}^2}}{\pi (1+4 C_d^2/\tilde{\gamma}^2)} \left( \frac{|\tilde{\omega}|}{2 T_H} \right) \right]}.$$ One of the essential differences between the modified dispersion relation model, that could give the Boltzmann reflectivity at the would-be horizon, and the Exotic Compact Object (ECO) model is the reflection radius $r^{\ast}_0$. In the former case, $r^{\ast}_0$ depends on the frequency of incoming GWs, $\tilde{\omega}$, and so the reflection surface is not uniquely determined. This is because the reflection takes place when the frequency $|\tilde{\omega}|$ reaches the Planckian frequency at which the modification in the dispersion relation becomes dominant. Therefore, the reflection radius depends on the initial (asymptotic) frequency of incoming GWs (see Equation \[xf\]). On the other hand, in the ECO scenario, the reflection radius would be fixed and it would stand at $\sim$ a Planck proper length outside the horizon. In this case, the reflection radius is given by $$r^{\ast}_0 \simeq \frac{1}{\kappa_+} \ln{\left( \frac{M}{E_{\text{Pl}}} \right)},$$ which depends only on the mass of BH. For a detailed discussion of how one can observationally distinguish these scenarios, we refer the reader to [@Oshita:2019seis1].
### Phenomenology of Boltzmann reflectivity
Here we summarize some interesting phenomenological aspects of the Boltzmann reflectivity model, and refer the reader to [@Wang:2019rcf] for more details.
Nearly all the previous studies of GW echoes assume a constant reflectivity model, in which the ratio of outgoing to ingoing flux at the horizon is assumed to be independent of frequency. In contrast, the echo spectrum in the Boltzmann reflectivity model can be significantly different. This difference can be seen in the sample echo spectra for both models, shown in FIG. \[spe\_echo\_ring\]. As can be seen from the spectra, the echo amplitude is highly excited near $m \times$ the horizon frequency, since the Boltzmann reflectivity is sharply peaked around $\omega \simeq m\Omega_{\rm H} \pm T_{\rm H}$ and is exponentially suppressed outside this range. In the extremal limit $\bar{a}\to 1$, the Hawking temperature becomes zero and so the frequency range in which $|{\cal R}| \sim 1$ vanishes. Therefore, the peaks in echo spectrum is highly suppressed for a highly spinning BH (see FIG. \[ref\_bol\]) [^3].
![Spectra of echo $+$ ringdown with $\bar{a} = 0.4$, $\ell=m=2$, and $M= 2.7 M_{\odot}$, $\epsilon_{\text{rd}} = 0.01$, $D_o = 40$ Mpc, and $\theta = 90^\circ$. The left panel shows the spectrum in the constant reflectivity model with $|{\cal R}|=0.5$ and the right panel shows the spectrum in the Boltzmann reflectivity model with $\tilde{\gamma} =1$.[]{data-label="spe_echo_ring"}](spectrum_echo_ringdown.png){height="55mm"}
![The energy reflection rate $|{\cal R}|^2$ in the Boltzmann reflectivity model.[]{data-label="ref_bol"}](reflectivity_bol.png){height="65mm"}
This nature of the Boltzmann reflectivity suppresses the ergoregion instability at least up to the Thorne limit $\bar{a} \leq 0.998$. In [@Oshita:2019seis1], a more general case is investigated, where the Hawking temperature in the Boltzmann factor is replaced by the quantum horizon temperature, $T_{\rm H} \to T_{\rm QH}$ (e.g., as in Equation \[Ref\_dis\] above) $${\cal R} = \exp{\left(- \frac{|\tilde{\omega}|}{2T_{\rm QH}} \right)},$$ and the ratio $T_{\rm H} / T_{\rm QH}$ is constrained from the ergoregion instability by using $|{\cal R} {\cal R}_{\text{BH}}| < 1$. The constraint is $T_{\rm H} / T_{\rm QH} \gtrsim 0.5$ up to the Thorne limit [@Oshita:2019seis1] and so the Boltzmann reflectivity $(T_{\rm H} / T_{\rm QH} =1)$ is safe up to $\bar{a} \lesssim 0.998$. As an example, we show a time domain function of ringdown and echo phases with $T_{\rm H} / T_{\rm QH} = 0.6$ in FIG. \[time\_domain\] by implementing the inverse Fourier transform of $X = Z_{\infty} (1+ {\cal K}_{\text{echo}}^{+})$, where we choose $Z_{\infty}$ so that it reproduces the ringdown phase [@Oshita:2019seis1].
Other notable phenomenological properties of quantum BHs with Boltzmann echoes are [@Wang:2019rcf]:
- The QNMs of the quantum BH are approximately those of a cavity with a [complex]{} length $|r_0^*|+i(4 T_{\rm QH})^{-1}$.
- For $\tilde{\gamma} \sim 1$ (i.e. Planck-scale modifications), the first $\sim 20$ echo amplitudes decay as inverse time $1/t$, and then exponentially.
- Each QNM of the classical BH can be written as a superposition QNMs of the quantum BH for $t < \Delta t_{\rm echo}$. The superposition can be approximated as a geometric series, leading to a closed-form expression for echo waveforms. In particular, the first 20 echoes have approximate temporal Lorentzian envelopes around their peaks, whose width grows linearly width echo number.
The parameter dependence of echo spectrum in the Boltzmann reflectivity model and its consistency with the tentative detection of echo in GW170817 are also investigated in [@Oshita:2019seis1] in more detail.
![The time domain function with $M=2.7 M_{\odot}$, $\bar{a} = 0.7$, $D_o=40$ Mpc, $\epsilon_{\text{rd}} =0.01$, $\theta=90^\circ$, and $\ell =m = 2$ in the Boltzmann reflectivity model with $\tilde{\gamma}=1$ and $T_{\rm H} / T_{\rm QH} = 0.6$.[]{data-label="time_domain"}](time_domain.png){height="85mm"}
Gravitational Wave Echoes: Observations {#sec:echo_searches}
=======================================
*For the first time in modern science history we are able to probe the smallest possible theoretical scales or highest possible theoretical energies through GW echoes.*
The direct observation of GWs [@Abbott:2016blz] was a scientific breakthrough that has opened a vast new frontier in astronomy, providing us with possible tests of General relativity in the extreme physical conditions near the BH horizons. Motivated by the resolutions of BH information paradox that propose alternatives to BH horizons (see Section \[sec:QBHs\] above), several groups have searched the LIGO/Virgo public data for GW echoes [@Cardoso:2016rao; @Cardoso:2016oxy] (see Section \[sec:echo\_predictions\] and [@Cardoso:2019rvt] for a review), which has led to claims (and counter-claims) of tentative evidence and/or detection [@Abedi:2016hgu; @Conklin:2017lwb; @Westerweck:2017hus; @Nielsen:2018lkf; @Abedi:2018npz; @Salemi:2019uea; @Uchikata:2019frs; @Holdom:2019bdv]. While the origins of these tentative signals remain controversial [@Westerweck:2017hus; @Nielsen:2018lkf; @Ashton:2016xff; @Abedi:2017isz; @Abedi:2018pst; @Salemi:2019uea] they motivate further investigation using improved statistical and theoretical tools, and well as new observations.
In astrophysics, GW echoes from quantum BHs can be seen as a transient signal, coming from the post-coalescence phase of the binary BH merger (Fig. \[echo\_pic\_1\]) or formation of a BH (e.g., via collapse of a hypermassive neutron star; Fig. \[echo\_pic\_2\]). This section will summarize the current status of observational searches for echoes [@Abedi:2020sgg].
![GW echoes following a collapse of binary neutron star merger event from a cavity of membrane-angular momentum barrier [@Abedi:2018npz]. []{data-label="echo_pic_2"}](echo_pic_2.pdf){width="70.00000%"}
In order to properly model echoes, we need a full knowledge of quantum BH nonlinear dynamics, which is so far nonexistent. Therefore, any strategy to search for echoes requires parametrizing one’s ignorance, which has so far taken many shapes and form. Indeed, we need to keep a balance between having a simple tractable model (which may simply miss the real signal), or an exhaustive complex model (which may dilute a weak signal with look-elsewhere effects). Current search methods can be generally split into two: Parametrized template-based methods [@Abedi:2016hgu; @Uchikata:2019frs; @Westerweck:2017hus; @Nielsen:2018lkf; @Lo:2018sep; @Tsang:2018uie], and “model-agnostic” coherent methods [@Conklin:2017lwb; @Abedi:2018npz; @Salemi:2019uea; @Holdom:2019bdv].
Out of these, 8 studies find some observational evidence for echoes [@Abedi:2016hgu; @Conklin:2017lwb; @Westerweck:2017hus; @Abedi:2018npz; @Nielsen:2018lkf; @Salemi:2019uea; @Uchikata:2019frs; @Holdom:2019bdv], 3 are comment notes [@Ashton:2016xff; @Abedi:2017isz; @Abedi:2018pst], and 3 more [@Uchikata:2019frs; @Tsang:2019zra; @Lo:2018sep] found no significant echo signals in the binary BH merger events. We can sort them into eight independent groups with 1. positive [@abedi2016echoes; @Abedi:2018npz; @Uchikata:2019frs; @Conklin:2017lwb; @Holdom:2019bdv], 2. mixed [@Westerweck:2017hus; @Nielsen:2018lkf; @Salemi:2019uea], and 3. negative [@Uchikata:2019frs; @Lo:2018sep; @Tsang:2019zra] results.
\[Positive\]Positive Results
----------------------------
### \[Echoes from Abyss:O1\]Echoes from the Abyss: Echoes from binary BH mergers O1 by Abedi, Dykaar, and Afshordi (ADA) [@Abedi:2016hgu]
The first search for echoes from Planck-scale modifications of general relativity near BH event horizons using the public data release by the Advanced LIGO GW observatory was developed by Abedi, Dykaar, and Afshordi (ADA) [@Abedi:2016hgu]. In this search, a naive phenomenological template for echoes was introduced, leading to tentative evidence at false detection probability of 1% (or $\simeq 2.5\sigma$ significance level[^4] shown in Figs. \[SNR\], \[SNR\_fig\] and \[Histogramloglog\]) for the presence of echoes [@Abedi:2016hgu]. This work was followed by comments, discussion, and controversy about the origin of this signal [@Ashton:2016xff; @Westerweck:2017hus; @Abedi:2017isz; @Abedi:2018pst; @Nielsen:2018lkf; @Salemi:2019uea]. The ADA model was also later tested for LIGO/Virgo O2 independent events[@Uchikata:2019frs], which interestingly, yielded a similar percent-level p-value as O1 (see Section \[Uchikata\] below). The ADA search was the first phenomenological time-domain echo template search applied to real GW observations [@Abedi:2016hgu]. Using a standard GR inspiral-merger-ringdown template $M(t)$, a naive model including five free parameters was proposed:
![Maximized SNR$^2$ around the expected time of merger echoes Eq. (\[t\_echo\_meas\]), for the combined (top) and GW150914 (bottom) events. The significance and p-values of the peaks within the gray rectangle are specified in this plot [@Abedi:2016hgu].[]{data-label="SNR"}](SNR_TOTAL_1.pdf){width="70.00000%"}
{width="\textwidth"}
![Average number of background peaks higher than a particular SNR-value within a time-interval $2\% \times \overline{\Delta t}_{\rm echo}$ (gray rectangle in Figs. \[SNR\] and \[SNR\_fig\]) for combined (left) and GW150914 (right) events [@Abedi:2016hgu]. The red dots show the observed SNR peak at $t_{\rm echo} = 1.0054 \Delta t_{\rm echo}$ (Figs. \[SNR\] and \[SNR\_fig\]). The correspondence between SNR values and their significance is indicated in horizontal bar.[]{data-label="Histogramloglog"}](Histogramloglog.pdf){width="70.00000%"}
For a Kerr BH with final mass $M_{\rm BH}$ and dimensionless spin parameter $a$, the time delay of echoes from Planck-scale modifications of general relativity is: [@Abedi:2016hgu; @Abedi:2018npz]: $$\begin{aligned}
\Delta t_{\rm echo} \simeq \frac{4 G M_{\rm BH}}{c^3} \left(1+\frac{1}{\sqrt{1-\bar{a}^2}}\right) \times \ln\left(M_{\rm BH} \over M_{\rm planck}\right) \nonumber \\
\simeq 0.126~ {\rm sec} \left(\frac{M_{\rm BH}}{67~ M_{\odot}}\right) \left(1+\frac{1}{\sqrt{1-\bar{a}^2}}\right). \label{eq.0.1}\end{aligned}$$ For the final BH (redshifted) masses and spins reported by the LIGO collaboration for each merger event, echo time delays $\Delta t_{\rm echo}$ and their errors constrained inside $1\sigma$ error are as follows [@Abedi:2016hgu]: $$\begin{aligned}
\Delta t_{{\rm echo}, I }({\rm sec})
=\left\{
\begin{matrix}
0.2925 \pm 0.00916 & I= {\rm GW150914} \\
0.1778 \pm 0.02789 & I={\rm GW151012} \\
0.1013 \pm 0.01152 & I={\rm GW151226}
\end{matrix}
\right. \ \ \ \ \ \label{t_echo_meas}
\end{aligned}$$
### Search {#sec_template}
In this analysis, ADA devised an echo waveform using theoretical best-fit waveform of Hanford $M_{H,I}(t)$ and Livingston $M_{L,I}(t)$ detectors (in real time series) for the BBH events, provided by the LIGO and Virgo collaborations. The search used the observed data release for the two detectors, $h_{H,I}(t)$ and $h_{L,I}(t)$ respectively, at 4096 Hz and for 32 sec duration. The devised phenomenological echo waveform which was then constructed using five free parameters:
1. $\Delta t_{\rm echo}$: Time-interval between successive echoes, within their $1\sigma$ range (Eq. \[t\_echo\_meas\]).
2. $t_{\rm echo}$: Time of arrival of the first echo, which is related to $\Delta t_{\rm echo}$ with corrections $\sim\pm \mathcal{O}(1\%) \times \Delta t_{\rm echo}$ due to the non-linear dynamics of the merger.
3. $t_0$: Truncation time for GR template with a smooth cut-off function, $$\Theta_I(t, t_{0})\equiv\frac{1}{2}\left\{1+ \tanh\left[\frac{1}{2} \omega_I(t)(t-t_{\rm merger}-t_{0})\right] \right\},$$ where $\omega_I(t)$ is frequency of GR template as a function of time [@TheLIGOScientific:2016src] and $t_{\rm{merger}}$ is the time of maximum amplitude of the template. It is assumed that $t_{0}$ vary within the range $t_{0} \in (-0.1,0) \overline{\Delta t}_{\rm echo}$. Having this definition, a truncated template is introduced: $$\begin{aligned}
{\cal M}_{T,I}^{H/L} (t, t_{0}) \equiv\Theta_I(t, t_{0}) {\cal M}_{I}^{H/L} (t).\end{aligned}$$
4. $\gamma$: Damping factor of successive echoes, varying between $0.1$ and $0.9$.
5. $A$: Over-all amplitude of the echo waveform with respect to the main event. This free parameter is fitted assuming a flat prior.
The search model of echoes having all the free parameters, assuming a $(-1)^{n+1}$ factor due to the phase flip at each reflection, is given by: $$M_{TE,I}^{H/L}(t) \equiv
~ A\displaystyle\sum_{n=0}^{\infty}(-1)^{n+1}\gamma^{n} {\cal M}_{T,I}^{H/L}(t+t_{\rm merger}-t_{\rm echo}-n\Delta t_{\rm echo},t_{0}). \label{template}$$ Fig. (\[template\_echoes\]) shows this template using the best fit parameters within the range given above along with the main merger event GW150914.
![Original GW template of GW150914 [@Abedi:2016hgu], along with best fit echoes template \[template\].\[template\_echoes\]](GW150914_template_echoes.pdf){width="100.00000%"}
Once this analysis has been completed for GW150914 (loudest event of O1), it has been repeated for rest of the events combined via SNR maximization: $$\begin{aligned}
{\rm SNR}^2_{total} \equiv \sum_I {\rm SNR}^2_I. \label{snr_total}
\end{aligned}$$ The proposed combination takes same $\gamma$ and $t_0/\overline{\Delta t}_{\rm echo}$ for all events, keeping $\Delta t_{\rm echo}$ and $A$’s as free. The results are shown in Fig’s (\[SNR\]-\[Histogramloglog\]) and Tables \[table\_1\]-\[table\_2\].
Range GW150914 Combined
------------------------------------------------------- ------------- ------------- ------------- --
$(t_{\rm echo} -t_{\rm merger})/ \Delta t_{\rm echo}$ (0.99,1.01) 1.0054 1.0054
$\gamma$ (0.1,0.9) 0.89 0.9
$t_{0}/\overline{\Delta t}_{\rm echo}$ (-0.1,0) -0.084 -0.1
Amplitude 0.0992 0.124
SNR$_{\rm max}$ 4.21 6.96
p-value $0.11$ $0.011$
significance 1.6$\sigma$ 2.5$\sigma$
: Comparing the expected theoretical values of echo time delays $\Delta t_{\rm echo}$’s of each merger event (Eq. \[t\_echo\_meas\]), to their best combined fit within the 1$\sigma$ credible region, and the contribution of each event to the combined SNR for the echoes (Eq. \[snr\_total\]) [@Abedi:2016hgu]. []{data-label="table_2"}
GW150914 GW151012 GW151226
---------------------------------- --------------- --------------- ---------------
$\Delta t_{\rm echo, pred}$(sec) 0.2925 0.1778 0.1013
$\pm$ 0.00916 $\pm$ 0.02789 $\pm$ 0.01152
$\Delta t_{\rm echo, best}$(sec) 0.30068 0.19043 0.09758
$|A_{\rm best, I}|$ 0.091 0.34 0.33
SNR$_{\rm best, I}$ 4.13 4.52 3.83
: Comparing the expected theoretical values of echo time delays $\Delta t_{\rm echo}$’s of each merger event (Eq. \[t\_echo\_meas\]), to their best combined fit within the 1$\sigma$ credible region, and the contribution of each event to the combined SNR for the echoes (Eq. \[snr\_total\]) [@Abedi:2016hgu]. []{data-label="table_2"}
- *Energy estimation*[@Abedi:2016hgu]: Given a best-fit template for the echoes, one can provide an estimate for their total GW energy:
$$\begin{aligned}
E^{I}_{\rm{echoes}}/(M_{\odot}c^{2})=\left\{
\begin{matrix}
0.029 & I=\rm{GW150914}, \\
0.16 & I=\rm{GW151012}, \\
0.047 & I=\rm{GW151226}, \\
\end{matrix}
\right.\nonumber \\ \label{angle}\end{aligned}$$
### \[Uchikata\]Uchikata et al. [@Uchikata:2019frs] analysis based on the template of Abedi, Dykaar, and Afshordi (ADA) [@Abedi:2016hgu] for O1 and O2
Uchikata et al. [@Uchikata:2019frs] have examined GW echo signals for nine binary BH merger events observed by Advanced LIGO and Virgo during the first and second observation runs (O1 and O2 respectively). They have used several models for a number of searches leading to positive and negative results. In this part we bring their positive results and discuss the rest in part \[Uchikatanegative\]. In this search the critical p-value as 0.05 corrsponding to $2\sigma$ significance, (p-value below/above this value) indicates echo signals (are likely/unlikely) to be present in the data.
SNR is evaluated using a matched filter analysis defined using $$\begin{aligned}
\rho \equiv (x|h)=4\rm{Re} \left( \int_{f_{\rm{min}}}^{f_{\rm{max}}} df \frac{\tilde{x}(f)\tilde{h}^{*}(f)}{S_{n}(f)} \right)\end{aligned}$$ where $\tilde{\rm{x}}(\rm{f})$ and $\tilde{\rm{h}}(\rm{f})$ are observed data and template in frequency domain respectively, and $\rm{S}_{\rm{n}}(\rm{f})$ is the noise power spectrum of detector. In this analysis they assume frequency band of $\rm f_{max}=2048$ Hz, $\rm f_{min}=40$ Hz and normalization condition of $\rm (h|h)=1$ [@Uchikata:2019frs]. They then perform an echo search by maximizing SNR, following [@Abedi:2016hgu]. Fig. \[Uchikata1\] presents $\rm SNR^{2}$ for the best fit parameters of the event GW150914 (C01)[^5] with additional consideration of the best fit initial phase[^6] to the ADA [@Abedi:2016hgu] template.
![$\rm SNR^{2}$ with respect to $T=(t_{\rm{echo}}-t_{\rm{merger}})/\Delta t_{\rm{echo}}$. Solid, dashed, and dotted lines correspond to $\rho^{2}$ for combined (Hanford and Livingston), Hanford, and Livingston, respectively, for the best fit parameters of the event GW150914 [@Uchikata:2019frs].\[Uchikata1\]](Uchikata1.pdf){width="100.00000%"}
In this part, we show the results using the same template given by ADA [@Abedi:2016hgu] except the cut-off parameter $t_{0}$ as described in \[sec\_template\] has been fixed to its best fit value, and set the search region of $\Delta t_{techo}$ to its 90% (rather than 68%) credible regions in $(a, M)$ space. Similar to ADA, the initial phase of the template is also fixed to zero. Since variation of $t_{0}$ weakly affects SNR and has an advantage in saving computational costs, they fixed $t_{0} = -0.1\Delta t_{echo}$.
Here are the results:
1. *O1 events (reanalysis of Westerweck et al.* [@Westerweck:2017hus]): Since Uchikata et al. [@Uchikata:2019frs] have followed the same background estimation as Westerweck et al. [@Westerweck:2017hus] the results are compared to their O1 results for p-values in Table \[table\_3\]. It is seen that they are almost consistent within the Poisson errors for all events, confirming a marginal p-value of 3%-5% for ADA echoes [@Abedi:2016hgu].
Event Westerweck et al. [@Westerweck:2017hus] Uchikata et al. [@Uchikata:2019frs]
---------- ----------------------------------------- -------------------------------------
GW150914 $0.238 \pm 0.043$ $0.157 \pm 0.035$
GW151012 $0.063 \pm 0.022$ $0.047 \pm 0.019$
GW151226 $0.476 \pm 0.061$ $0.598 \pm 0.069$
Total $0.032 \pm 0.016$ $0.055 \pm 0.021$
: P-values along with Poisson errors for O1 events [@Uchikata:2019frs].[]{data-label="table_3"}
2. *O2 events*:
Analysis of Uchikata et al. [@Uchikata:2019frs] show that the six independent BBH O2 events in Table \[table\_4\] have similarly small p-values for ADA echoes as O1. As shown in this table, the total p-value for the six O2 events is 0.039. Combining O2 with O1 events shown in Table \[table\_3\], leads to the total p-value of 0.047.
Event Uchikata et al. [@Uchikata:2019frs]
---------- ------------------------------------- --
GW170104 0.071
GW170608 0.079
GW170729 0.567
GW170814 0.024
GW170818 0.929
GW170823 0.055
Total 0.039
: P-values for O2 events [@Uchikata:2019frs].[]{data-label="table_4"}
### Echoes from the Abyss: Binary neutron star merger GW170817 [@Abedi:2018npz]
A binary neutron star merger event collapsing into a black hole (Fig. \[echo\_pic\_2\]) can also enable us to test general relativity through GW echoes. Although, the current LIGO/Virgo/KAGRA detector sensitivity is blind to post-merger ringdown frequency of GWs, they can be sensitive to low frequency echoes harmonics where the ringdown frequency is suppressed by $ln(M_{\rm BH}/M_{\rm planck})$ in Eq. \[eq.0.1\] [@Cardoso:2016oxy]. In other words, since the final mass of BNS merger (2-3 $M_{\odot}$) is much smaller than that of the binary BH mergers [@Abedi:2016hgu], the lowest harmonics $ n/\Delta t_{\rm echo}$ ($\simeq n \times 80$ Hz) of echo chamber are shifted to the regime of LIGO sensitivity, for small $n$. Therefore, as first suggested by [@Conklin:2017lwb], an optimal model-agnostic search strategy could consist of looking for periodically spaced-harmonics in the frequency space.
Using this model-agnostic search applied to the cross-power spectrum of the two LIGO detectors, Abedi and Afshordi [@Abedi:2018npz] found a tentative detection of echoes around 1.0 sec after the BNS merger, at $f_{echo}\simeq 72$ Hz (see Figs. \[NS-NS\_11\], \[NS-NS\_3d\], \[NS-NS\_4\], and \[NS-NS\_9\]) using GW event data GW170817 provided by LIGO/Virgo collaboration [@TheLIGOScientific:2017qsa; @Abbott:2017dke]. As it is shown in Figs. \[NS-NS\_11\] and \[NS-NS\_3d\], the main signal is also accompanied by secondary lower significance resonances at 73 Hz and $t-t_{\rm{merger}}$ = 32.9 sec. It is worth noting that after this detection, Gill et al. [@Gill:2019bvq] used independent Astrophysical considerations, based GW170817 electromagnetic follow-ups, to determine that the remnant of GW170817 must have collapsed into a BH after $t_{\rm coll}=0.98^{+0.31}_{-0.26}$ sec, which coincides with the detected GW echo signal at 1.0 second (see Fig. \[NS-NS\_9\]). This fining of Abedi and Afshordi is consistent with a $2.6-2.7$ $M_{\odot}$ “BH” remnant with dimensionless spin $0.84-0.87$. For this signal, considering all the “look-elsewhere” effects, a significance of $4.2\sigma$[^7] (see Fig. \[NS-NS\_10\]), or a false alarm probability of $1.6\times10^{-5}$ has been reported, i.e. a similar cross-correlation within the expected frequency/time window after the merger cannot be found more than 4 times in 3 days of GW data. Total energy of detected GW echoes signal using simple assumptions is around $\sim 10^{-2}\ M_{\odot} c^2$.
{width="100.00000%"}
{width="\textwidth"}
{width="100.00000%"}
![Amplitude-time plot of first echo peak at 1.0 sec after the merger at frequency of 72 Hz [@Abedi:2018npz]. After this detection Gill et al. [@Gill:2019bvq] with independent Astrophysical considerations have also determined that the remnant of GW170817 must have collapsed to a BH after $t_{\rm coll}=0.98^{+0.31}_{-0.26}$ sec. Error-bar (in blue) is the time of collapse considering this independent observation in [@Gill:2019bvq] compared to the detected signal of echoes which is also as a consequence of BH collapse. The shaded region is 0-1 sec prior range after the merger.[]{data-label="NS-NS_9"}](peak_in_short_time.pdf){width="50.00000%"}
![Average number of background peaks higher than a particular -X(t,f) within a frequency-intervals of 63-92 Hz and time-intervals of 1 sec. the observed $-X(t_{\rm{peak}},f_{\rm{peak}})$ peak at 1.0 sec after the merger is marked by red square. The horizontal bar shows the relation between $X(t,f)$ values and their significance [@Abedi:2018npz].[]{data-label="NS-NS_10"}](Histogram.pdf){width="50.00000%"}
### MODEL-AGNOSTIC SEARCH FOR ECHOES
In this part, we describe the method that leads to the detection of [@Abedi:2018npz], in some detail. The final mass of GW170817 is within $\sim 2-3 M_{\odot}$ which could form either a BH or a neutron star (NS). A BNS merger can end up in four possible ways: [@Abbott:2017dke]:
1. A prompt collapse to a BH.
2. A formation of a BH within $\lesssim$ 1 sec from a hypermassive NS.
3. Collapse to a BH on timescales of $10$ - $10^{4}$ sec from a supramassive NS.
4. A stable neutron star. [@TheLIGOScientific:2017qsa].
Abedi and Afshordi [@Abedi:2018npz] have considered the possibility of first and second scenario.
As it was pointed out in part \[Echoes from Abyss:O1\] the time delay for Planck scale echoes is given by, [@Abedi:2018npz], $$\begin{aligned}
&\Delta t_{\rm echo} \simeq \frac{4 G M_{\rm BH}}{c^3}\left(1+\frac{1}{\sqrt{1-\bar{a}^2}}\right) \times \ln\left(M_{\rm BH} \over M_{\rm planck}\right)& \nonumber\\ &\simeq 4.7~ {\rm msec} \left(M_{\rm BH} \over 2.7~ M_\odot \right) \left(1+\frac{1}{\sqrt{1-\bar{a}^2}}\right).& \label{delay}\end{aligned}$$
There are two natural frequencies for the waveform of the echoes: The resonance frequencies (natural harmonics) of the echo chamber (formed by the angular momentum barrier and the near-horizon quantum structure), and the BH ringdown (or classical QNM) frequencies. The high frequency harmonics that are initially excited by the merger event decay quickly, while the low frequency harmonics live for longer time [@Maggio:2017ivp; @Bueno:2017hyj; @Wang:2018gin]. While the former captures the repeat period of the echoes, the latter describes the echo internal structure. Given that the ringdown frequencies are not resolved by LIGO detector, we can roughly approximate the observable signal as a sum of Dirac delta functions, repeating with the period $\Delta t_{\rm echo}$: $$\begin{aligned}
h(t) \propto \sum_n \delta_D(t-n\Delta t_{\rm echo} -t_0) \Rightarrow
h_{f} \propto \sum_n \delta_D(f- n f_{\rm echo}). \label{h_f_res}\end{aligned}$$ Therefore, the method searches for coherent periodic peaks of equal amplitude in cross-power spectrum of the two detectors at integer multiples of $f_{\rm echo} \equiv \Delta t_{\rm echo}^{-1} $, in following steps:
1. Fundamental frequency of echoes $f_{\rm echo}= \Delta t_{\rm echo}^{-1}$ within the 90% credible region range for final BH mass and spin is given by: $$63 \leq f_{{\rm echo} }({\rm Hz}) \leq 92.$$
2. The prior range for echoes search is $0 <t-t_{\rm{merger}} \leq 1~ \rm{sec}$.
3. Using amplitude spectral density (ASD) Wiener filter (rather than whiten) the data by dividing by noise variance PSD=ASD$^2$ (rather than ASD): $$\begin{aligned}
H(t,f)={\rm Spectrogram}\left[ {\rm IFFT}\left( \frac{{\rm FFT}(h_{H}(t- \delta t))}{PSD_{H}} \right) \right], \nonumber\\
L(t,f)={\rm Spectrogram}\left[ {\rm IFFT}\left( \frac{{\rm FFT}(h_{L}(t))}{PSD_{L}} \right) \right].\end{aligned}$$ where $\delta t$ is the time shift between detectors.
4. Cross-correlating the obtained spectrograms and sum over all the resonance frequencies of $n\times f$, $$\begin{aligned}
X(t,f)=\sum_{n=1}^{10} \Re\left[H(t,nf) \times L^*(t,nf)\right]. \label{x_def}\end{aligned}$$ Since the polarizations of the LIGO detectors are opposite for GW170817, the real GW signals appears as peaks in $-X(t,f)$ (see Figs. \[NS-NS\_11\] and \[NS-NS\_4\]).
The simplicity of the method (not having any arbitrary or [*ad-hoc*]{} cuts or parameters) along with its high significance are reasons for making this finding more interesting and reliable.
### “GW echoes through new windows” by Conklin et al. [@Conklin:2017lwb; @Holdom:2019bdv]
In this search, three methods (named as I, II, III) that are based on general properties of echoes has been suggested. Conklin et al. [@Conklin:2017lwb] have mostly focused on Method II, which is based on frequency windows, while the other two methods use time windows. Window functions in these methods allow us to find quasiperiodic structures in time and/or frequency domains. Method II turns out to be the most successful one. Accordingly, in this paper we just review this method. The search methods become more optimal using correlations of data in multiple detectors. Using the suggested method and search [@Conklin:2017lwb] find significant evidence for GW echoes, which is shown in Table \[table\_5\] (and see Fig. \[Holdom2\] for GW170104) for both O1 and O2 LIGO/Virgo observations.
![Correlation vs. echo time delay $\Delta t_{\rm echo}$ vs. $\rm N_{E}$ for GW170104 using method II [@Conklin:2017lwb]. Here $N_{E}$ is the number of frequency steps between spikes.[]{data-label="Holdom2"}](Holdom2.pdf){width="90.00000%"}
Event (method) Best-fit $\Delta t_{\rm echo}$ (sec) p-value Bandpass $(f_{min},f_{max})\Delta t_{\rm echo}$ Window parameters for average
---------------- -------------------------------------- --------------- ------------------------------------------------- -------------------------------------
GW151226 (I) 0.0786 $<0.0013$[^8] (34,62)[^9] $N_{E}$=(1-29), (5-29), (9-29)[^10]
GW151226 (II) 0.0791 0.0076 (12,58) $N_{E}$=(260,270)
GW170104 (II) 0.201 $<0.0018$ (16,62) $N_{E}$=(100,125,150,175,200)
GW170608 (II) 0.0756 $<0.004$ (14,60) $N_{E}$=(140,200,260)
GW170814 (II) 0.231 0.04 (12,58) $N_{E}$=(170,190)[^11]
GW170814 (III) 0.228 0.0077 (30,80) $N_{E}=10-17$, $t_{w}=40,80$[^12]
The time delay of echoes, $\Delta t_{\rm echo}$ which is similar to the Planckian echoes in Eq. \[eq.0.1\], is paramaterized in [@Conklin:2017lwb] using $$\begin{aligned}
\left. \Delta t_{\rm echo}/M \right|_{CHR}=-\eta \left(1+\frac{1}{\sqrt{1-\bar{a}^{2}}}\right)\ln\left( r_{0}-r_{+} \over M\right)\end{aligned}$$ where $r_{0}$ is the location of the quantum structure outside $r_{+}$. Here $\eta=2$ corresponds to proper Planck length (Eq. \[eq.0.1\]).
Taking into account of errors in final mass, spin, and redshift it is realised that (see Fig \[Holdom1\]) the echoes found are consistent with $\eta=1.7$ [@Conklin:2017lwb; @Holdom:2019bdv]. Best fit properties of the peaks for O1 and O2 events also shown in Table \[table\_6\] [@Holdom:2019bdv]. Alternatively, this result can be interpreted as the energy scale for reflection from quantum horizons to be $6 \pm 2$ orders of magnitude below Planck energy [@Oshita:2016pbh].
![Determination of $\eta$ for the events in O1 and O2 [@Holdom:2019bdv].[]{data-label="Holdom1"}](Holdom1.pdf){width="90.00000%"}
Event (method) Best-fit $\Delta t_{\rm echo}$ (sec) $N_{E}$ $\Delta t_{\rm echo}/M$
---------------- -------------------------------------- --------- -------------------------
GW150914 (II) 0.251 200 806
GW151012 (II) 0.145 160 826
GW151226 (II) 0.0791 783 270
GW170104 (II) 0.201 150 831
GW170608 (II) 0.0756 200 862
GW170729 (II) 0.489 180/170 1240
GW170809 (II) 0.235 170 845
GW170814 (II) 0.231 200 878
GW170817 (II) 0.00719 250 663
GW170818 (II) 0.275 140 933
: Echoes best fit time delays and corresponding $N_{E}$ using method II for O1 and O2 events [@Holdom:2019bdv].[]{data-label="table_6"}
### Comment on: “Gravitational wave echoes through new windows by Conklin et al. [@Conklin:2017lwb; @Holdom:2019bdv]”
Conklin et al. [@Conklin:2017lwb] remark that “We have not found signals for the two earlier events, GW150914 and GW151012, which play a significant role in ADA results in [@Abedi:2016hgu]”. While in their updated search [@Holdom:2019bdv] they indicate existence of signals for both GW150914 and GW151012, although with no p-value estimation which is crucial in this search. Therefore, it is hard to truly evaluate the significance of echoes reported in [@Holdom:2019bdv]. Moreover, it is not clear how much the choices made in Method II in [@Conklin:2017lwb] might have been affected by [*a posteriori*]{} statistics.
\[Mixed\]Mixed Results
----------------------
### \[Results of Westerweck et al.\]Results of Westerweck et al. and Nielsen et al. [@Westerweck:2017hus; @Nielsen:2018lkf]
Westerweck et al. [@Westerweck:2017hus] re-analysed the same model proposed by ADA [@Abedi:2016hgu], using more background data and a modified procedure. They focused on the data analysis methods of ADA [@Abedi:2016hgu] and their significance estimation, namely the concerns presented in [@Ashton:2016xff] suggesting a different significance estimate using 4096 seconds of LOSC data.
The results of p-value estimation and comparison with ADA results are shown in Table \[table\_7\]. In addition, Nielsen et al. [@Nielsen:2018lkf] have searched for echoes signals in GW data via Bayesian model selection probabilities, comparing signal and no-signal hypotheses using ADA model [@Abedi:2016hgu]. Accordingly, calculation of Bayes factors for the ADA model in O1 events presented in Table \[table\_8\].
Event [@Abedi:2016hgu] original 16s (32s) widened priors 16s (32s)
----------- ------------------ -------------------- --------------------------
GW150914 0.11 0.199 (0.238) 0.705 (0.365)
GW151012 - 0.056 (0.063) 0.124
GW151226 - 0.414 (0.476) 0.837
GW170104 - 0.725 0.757
(1,2) - 0.004 0.36
(1,3) - 0.159 0.801
(1,2,3) 0.011 0.020 (0.032) 0.18 (0.144)
(1,3,4) - 0.199 (0.072) 0.9 (0.32)
(1,2,3,4) - 0.044 (0.032) 0.368 (0.112)
: Comparison of p-values obtained in [@Abedi:2016hgu] and using larger portion of data (4096 seconds of LOSC data) [@Westerweck:2017hus]. This data is divided into segments of 16 or 32 seconds length. Here different combinations of the events are considered, denoted as (GW150914, GW151012, GW151226, GW170104) $\rightarrow (1, 2, 3, 4)$. Having the original priors, the Poisson errors (as suggested in [@Abedi:2018pst]): for GW150914 our p-values are $0.199 \pm 0.028$ ($0.238 \pm 0.043$), and for (1,2,3) our p-values are $0.02 \pm 0.009$ ($0.032 \pm 0.016$). The Poisson errors for the full combination (1,2,3,4) with original priors, are $0.044 \pm 0.013$ ($0.032 \pm 0.016$). The comparison of p-values using widened priors are also shown in this table.[]{data-label="table_7"}
Event Log Bayes factor Max SNR
---------- ------------------ ---------
GW150914 -1.8056 2.86
GW151012 1.2499 5.5741
GW151226 0.4186 4.07
: Results of Bayes factor [@Nielsen:2018lkf] using ADA model. Gaussian noise hypothesis is preferred for negative values of Log Bayes factor. Echoes hypothesis is preferred for positive values of Log Bayes factor. Log Bayes values of $<$ 1 are “not worth more than a bare mention” [@Nielsen:2018lkf].[]{data-label="table_8"}
In the following we explain the results given in different plots:
1. In Fig. \[Julian1\] it is shown that depending on the overall amplitude of the injection, the signal either can be recovered or it would be difficult to recover.
![This plot shows whether it is possible to recover the potential signals with a variety of amplitudes in [@Westerweck:2017hus]. Here it can be seen that amplitudes less than A=0.1 in ADA [@Abedi:2016hgu] are difficult to be identified in data, while amplitude twice this value would be clearly identifiable.[]{data-label="Julian1"}](Julian1.pdf){width="50.00000%"}
2. Fig. \[Julian2\] shows injected and recovered values for $\gamma$ having different overall amplitude $A$ used in [@Abedi:2016hgu]. Although this plot shows a preference for $\gamma = 1$, having high value of $\gamma$ can be recovered easily.
![This plot shows injected and recovered values for $\gamma$. The diagonal line is accurate recovery [@Westerweck:2017hus]. The preference for $\gamma = 1$ (dashed line) at lower injection amplitudes can be clearly seen.[]{data-label="Julian2"}](Julian2.pdf){width="50.00000%"}
3. Fig. \[Julian3\] shows injection of echoes signals into the Gaussian noise having different overall amplitudes A.
![In this plot echoes signals are injected into Gaussian noise and are analysed for different overall amplitudes A [@Westerweck:2017hus]. Injection amplitudes above $10^{-22}$ of the peak strain, can be accurately recovered. For the injections with lower amplitudes no matter what value they take, recovered amplitudes are likely to be around $10^{-22}$. Horizontal lines show the amplitudes reported in [@Abedi:2016hgu]. The injections are made with $\gamma = 0.8$.[]{data-label="Julian3"}](Julian3.pdf){width="50.00000%"}
### Comment on: “Low significance of evidence for BH echoes in gravitational wave data” [@Westerweck:2017hus]
1. *Comment on search strategy*:
As described in former part in Fig. \[Julian2\], Westerweck et al. [@Westerweck:2017hus] demonstrated the preference of $\gamma=1$ at lower injection amplitudes for ADA model [@Abedi:2016hgu]. This is expected result as $\gamma\rightarrow 1$ extends the template to infinity. In other words, it extends the template range to infinite time, which is clearly dominated by noise. However, $\gamma = 1$ is still far from the best-fit $\gamma = 0.9$ on edge of the prior where at least 90% of energy goes to first 11 echoes. Besides, initial waveform must change significantly in subsequent echoes. Indeed, repeated template that does not damp ($\gamma=1$) is not physical. One solution to this problem might be to use a finite range of data (which is used by Westerweck et al. [@Westerweck:2017hus]) making the result strongly dependent on what portion of data has been taken.
It should be also noted that using 1-sigma range for errors in $\Delta \rm t_{echo}$, implies a 32% chance for the signal to be missed that causes reduced significance by diluting $\rm SNR^{2}$ from some events.
Westerweck et al. [@Westerweck:2017hus] have found that the least significant event LVT151012 which is now called GW151012 has the most contribution to tentative evidence for echoes. This peculiar finding does not disfavor echoes as there is no simply reasonable justification that significance of echoes should be directly related to the significance of main event. Additionally, Wang et al. [@Wang:2018gin] have shown that by changing only $\pm 20\%$ of frequency of initial condition of echoes the $\rm SNR^{2}$ for echoes can change by 3 orders of magnitude. As BBH events have different component spins and mass ratios we might expect a significant diversity in relative echo signal amplitude for each of them. Interestingly, as will be discussed in next part \[Results of Salemi et al.\] mass ratio of BBH events appears to show correlation with the echo amplitude [@Abedi:2020sgg].
2. *Comment on abstract and conclusion*:
The most crucial comments for Westerweck et al. [@Westerweck:2017hus] goes to their abstract and conclusion (also provided in [@Abedi:2018pst]). Although ADA [@Abedi:2018pst] strongly acknowledge the analysis by Westerweck et al. [@Westerweck:2017hus], which is a careful re-evaluation of ADA analysis, the Abstract/Conclusion of Westerweck et al. [@Westerweck:2017hus] misrepresents their finding. The most critical point of this misrepresentation is in Abstract claiming “a reduced statistical significance ... entirely consistent with noise”. Contrasted to this claim in their Table I (Table \[table\_7\] in this paper) they found p-value=0.020 for the noise hypothesis, with the same model and data as in ADA (as opposed to 0.011 in ADA [@Abedi:2016hgu]). However, if one follows standard nomenclature (e.g., <https://en.wikipedia.org/wiki/P-value#Usage> or [@Goodman]), p-values $< 0.05$ disfavour noise hypothesis, providing “moderate to strong” Bayesian evidence for echoes, which is contrary to what they state in their Abstract.
To conclude, considering all the critiques of Westerweck et al. [@Westerweck:2017hus], we see NO evidence that their improved analysis with p-value$=0.020\pm 0.009$ has reduced the significance of echoes, entirely consistent with p-value $=0.011$ of ADA [@Abedi:2016hgu]. The fact that completely independent events of O2 also show a low p-value$=0.039$ for ADA echoes (see Table \[table\_4\] [@Uchikata:2019frs]) further boosts the statistical evidence for ADA model in LIGO/Virgo data.
### \[Results of Salemi et al.\]Results of Salemi et al. [@Salemi:2019uea]
Another independent group [@Salemi:2019uea] has found similar post-merger GW signals that can be attributed to GW echoes. However, the setup of this methodology, which is based on coherent WaveBurst (cWB) [@cWB] method, was not originally developed to search for echoes. This search, which is independent of the waveform models, has been developed based on coherent excess power in events from the GWTC-1 (catalog of compact binary coalescence). Here, loose bounds on the duration and bandwidth of the signal leads to evaluation of coherent response of independent detectors.
This search has focused on detected features as deviations from GR and has presented the method to obtain their significance. It appears that from eleven events reported in the GWTC-1, two of them (GW151012 and GW151226) in Figs. \[Salami1\] and \[Salami2\] respectively, show an excess of coherent energy after the merger ($\Delta t \simeq 0.2$ s and $\simeq 0.1$ s, respectively) with p-values (0.004 and 0.03, respectively). However, [@Salemi:2019uea] have shown that (Fig. \[Salami3\]) the post-merger signal from GW151012 favours different sky location than that of the main event.
![Reconstruction of cWB for the event GW151012 via color-coded time-frequency maps [@Salemi:2019uea]. The upper plot shows the squared coherent network SNR and the plot bellow shows the normalized residual noise energy. The residual plot is given after the reconstructed signal was subtracted from the data. In this plots a secondary cluster occurring 200 ms after the merger (consistent with echo times predicted and seen by ADA, Equation \[t\_echo\_meas\]). The dashed vertical lines denote coalescence time for GW151012 (the network has used the Livingston detector time as a reference).\[Salami1\]](Salami1.pdf){width="70.00000%"}
![Reconstruction of cWB for the event GW151226, as color-coded time-frequency maps [@Salemi:2019uea]. The upper plot shows the squared coherent network SNR and the plot bellow shows the normalized residual noise energy. The residual plot is given after the reconstructed signal was subtracted from the data. In this plots a secondary cluster occurring 100 ms after the merger (consistent with echo times predicted and seen by ADA, Equation \[t\_echo\_meas\]). The dashed vertical lines denote coalescence time for GW151226 (the network has used the Livingston detector time as a reference).\[Salami2\]](Salami2.pdf){width="70.00000%"}
![Maximum posteriori probability for time delay between Hanford (H) and Livingston (L) in line-of-sight frame for the main event GW151012 (blue contour) and the secondary signal (green contour) [@Salemi:2019uea].\[Salami3\]](Salami3.pdf){width="70.00000%"}
In TableI \[table\_9\], $\rm SNR^{minR}_{pc}$, and its upper and lower bounds are presented. In the last column of this table, estimated p-values for postmerger features are reported. Along with, $2\sigma$ upper and lower bounds for SNR, upper and lower bounds for the p-values are reported.
Event source SNR SNR$_{\rm{pc}}^{\rm{minR}}\left\{^{\rm{SNR}^{\rm{sup}}_{\rm{pc}}}_{\rm{SNR}^{\rm{inf}}_{\rm{pc}}}\right\}$ p-value$_{\rm{pc}}\left\{^{\rm{P}_{\rm{sup}}}_{\rm{P}_{\rm{inf}}}\right\}$
---------- -------- ------ ------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------- -- -- --
GW150914 BBH 25.2 $5.72^{6.92}_{5.64}$ $0.94\pm 0.02^{0.95}_{0.71}$
GW151012 BBH 10.5 $6.60^{6.54}_{6.26}$ $0.0037\pm 0.0014^{0.0068}_{0.0042}$
GW151226 BBH 11.9 $4.40^{4.41}_{4.36}$ $0.025\pm 0.005^{0.03}_{0.02}$
GW170104 BBH 13.0 $5.29^{5.30}_{3.95}$ $0.07\pm 0.01^{0.31}_{0.07}$
GW170608 BBH 14.1 $1.69^{1.75}_{1.64}$ $0.51\pm 0.02^{0.54}_{0.49}$
GW170729 BBH 10.2 $4.81^{4.86}_{3.43}$ $0.09\pm0.01^{0.35}_{0.08}$
GW170809 BBH 11.9 $3.89^{4.71}_{3.88}$ $0.28\pm 0.01^{0.28}_{0.11}$
GW170814 BBH 17.2 $5.98^{6.02}_{5.94}$ $0.10\pm 0.01^{0.11}_{0.09}$
GW170817 BNS 29.3 $0.21^{0.21}_{0.21}$ $0.55\pm 0.01^{0.56}_{0.55}$
GW170818 BBH 8.6 $1.97^{2.04}_{1.76}$ $0.87\pm 0.02^{0.91}_{0.86}$
GW170823 BBH 10.8 $3.11^{3.54}_{2.69}$ $0.60\pm 0.02^{0.74}_{0.44}$
### Hint of dependence of significance of echoes on binary BH mass ratio,\
*Comment on: Results of Salemi et al.* [@Salemi:2019uea]
In this part, we first re-examine the interpretation of Salemi et al. [@Salemi:2019uea] about the signals they found and then give a more conclusive support for echoes hypothesis.
- *Comment on: Results of Salemi et al.* [@Salemi:2019uea]
Salemi et al. [@Salemi:2019uea] disfavoured echoes hypothesis pointing that post-merger signal of GW151012 has arrived from a different sky location than that of the main event. However, the p-value$\sim 0.004$ of this secondary signal, disfavours two signals being unrelated. We see that all the secondary (post-merger) clusters they claim as signals in Figs. \[Salami1\] and \[Salami2\] are nearly monochromatic. That means the waveform of these signals are quasi-periodic, leading to degeneracies in inferred time-delays. So looking again to the secondary signal of GW151012, the null (residual) plot in Fig. \[Salami1\] shows the peak of the cluster (which is mostly responsible to make a different sky localization) is at $\sim 130$ Hz which corresponds to 7.7 m sec time delay. This 7.7 m sec is the same time delay of first peak and second peak in Fig. \[Salami3\] for post-coalescence signal (green), confirming that the monochromatic degeneracy in Fig. \[Salami3\] might have caused the different sky localization. Fig. \[Salami4\] shows better interpretation of cause of this error.
- *Dependence of significance of echoes on binary BH mass ratio*
Consider a system of binary BHs (BBH), with progenitor masses $m_{1}$ and $m_{2}$. It is known that these systems consist of two almost equal mass BHs $m_{1}\sim m_{2}$. However, the inevitable diversity in the initial conditions, specifically binary mass ratio, can lead to different echo properties. Here, we review the evidence for correlation between the significance of echoes and the progenitor BBH mass ratio (first presented in [@Abedi:2020sgg]):
1. We have used LIGO parameter estimation samples for BBH events provided in [@GWTC-1]. Then obtained mass ratios by weighting all events as equal. We used full m1 vs m2 distribution samples for “Overall\_posterior”. We fit these posterior points to a straight line shown in Fig. \[mass ratio error\].\
2. We take p-values reported in [@Salemi:2019uea] for each event post-coalescence signal and plot the best fit line of mass ratio vs $(-\log(p-value))^{0.5}$. We used least square method [@least-square-method] in fitting a straight line consisting all the posterior points provided by LIGO. Finally, we use the slope of best fit line as our primary parameter in determining significance.\
3. Finally, we obtain the significance (Fig. \[histogram-slope\]) assuming that there is no relation between p-value and mass ratio by taking random p-values for BBH Catalog events within the uniform range $0<(-\log(\rm{p-value}))^{0.5}<2.5$ ($0.0019<$ p-values$<1$). Indeed, having no relation between p-value and mass ratio of events shall end up with zero slope in large number of random selections. Therefore, in order to find a false detection rate for correlation, number of slopes higher than the actual measured slope is calculated (see Fig. \[histogram-slope\]). Accounting for the “look elsewhere” effect, we find tentative hint of mass-ratio dependence of echo significance reported in [@Salemi:2019uea] at false detection probability of 1%.
![Plot of mass ratio dependence of p-values in [@Salemi:2019uea]. Vertical lines are error bars for 50% credible region and central points are best value of mass ratio obtained from posteriors distribution. Because of relation of p-value to error function erf(SNR) we took roughly SNR $\sim \sqrt{-\log(\rm p-value)}$ as horizontal axis [@Abedi:2020sgg].[]{data-label="mass ratio error"}](mass-ratio-50-percent.pdf){width="70.00000%"}
![Histogram of slopes for uniform random choices of $0<(-\log(\rm{p-value}))^{0.5}<2.5$. We see that only 1.3% of these random realizations, the slope can exceed the observed value [@Abedi:2020sgg]. []{data-label="histogram-slope"}](Slope--p-value.pdf){width="60.00000%"}
Table \[table\_20\] shows events and p-value of their post-coalescence signal reported in [@Salemi:2019uea] versus expected Planckian echo time delays and average mass ratios. At a glance it is seen that smallest mass ratios go to smallest p-values. This can happen with 1/10 chance out of 10 BBH events. For two most significant events, being also the most extreme BBHs, the random chance becomes p-value$=\frac{1}{9}\times \frac{1}{10} = 0.011$ which is consistent with the statistics of regression analysis in Fig. \[histogram-slope\].
Event p-value $\pm 2\sigma$ average mass ratio
---------- ----------------------- -------------------- -- --
GW150914 $0.94 \pm 0.02$ 0.86
GW151012 $0.0037 \pm 0.0014$ 0.58
GW151226 $0.025 \pm 0.005$ 0.56
GW170104 $0.07 \pm 0.01$ 0.65
GW170608 $0.51 \pm 0.02$ 0.68
GW170729 $0.09 \pm 0.01$ 0.68
GW170809 $0.28 \pm 0.01$ 0.68
GW170814 $0.10 \pm 0.01$ 0.82
GW170818 $0.87 \pm 0.02$ 0.75
GW170823 $0.60 \pm 0.02$ 0.74
\[Negative\]Negative Results
----------------------------
### Template-based gravitational-wave echoes search using Bayesian model selection by Lo et al. [@Lo:2018sep]
Lo et al. [@Lo:2018sep] found that using a wider range of priors (listed in Table \[table\_16\]) compared to that of ADA model [@Abedi:2016hgu], including the main event template in the echo template, and keeping just three echoes, leads to lower significance on echo signals evaluated by the Bayes factor using Bayesian analysis.
Parameter Prior range
----------------------------- ---------------
A \[0.0, 1.0\]
$\gamma$ \[0.0, 1.0\]
$t_{0}$ (sec) \[-0.1,0.01\]
$t_{\rm echo}$ (sec) \[0.05, 0.5\]
$\Delta t_{\rm echo}$ (sec) \[0.05, 0.5\]
: Prior range proposed by Lo et al. [@Lo:2018sep] of the echo parameters of ADA model [@Abedi:2016hgu].[]{data-label="table_16"}
They have considered two hypotheses as the null hypothesis $\mathcal{H}_{0}$ and alternative hypothesis $\mathcal{H}_{1}$, $$\begin{aligned}
\mathcal{H}_{0} \equiv \rm{No\ echoes\ in\ the\ data } \Rightarrow d=n+h_{\rm IMR} \nonumber \ \ \ \ \ \ \ \ \ \\
\mathcal{H}_{1} \equiv \rm{There\ are\ echoes\ in\ the\ data } \Rightarrow d=n+h_{\rm IMRE} \nonumber \\\end{aligned}$$ where $d$ and $n$ indicate the GW data, and the instrumental noise respectively and $h_{\rm IMR}$, and $h_{\rm IMRE}$ are the inspiral-merger-ringdown (IMR) gravitational-wave signal and inspiral-merger-ringdown-echo (IMRE) gravitational-wave signal respectively.
The log Bayes factor $\ln \mathcal{B}$ in their search method gives the detection statistics which determines whether there is an IMRE signal or an IMR signal in data. From the Bayesian perspective if the log Bayes factor, is greater than 0, we can conclude that the data favor the alternative hypothesis.
The relation between p-value and null distribution of detection statistic $\ln \mathcal{B}$ is given by $$\begin{aligned}
\rm p-value=Pr(\ln \mathcal{B} \geq \ln \mathcal{B}_{\rm detected}|\mathcal{H}_{0}) \nonumber \\
=1-\int_{-\infty}^{\ln \mathcal{B}_{\rm detected}} p(\ln \mathcal{B}|\mathcal{H}_{0}) d\ln \mathcal{B},\end{aligned}$$ where $\ln \mathcal{B}_{\rm detected}$ is the detection statistic which is obtained using a segment of data in analysis, and $p(\ln \mathcal{B}|\mathcal{H}_{0})$ is called the null distribution of $\ln \mathcal{B}$, i.e., the distribution of $\ln \mathcal{B}$ assuming that $\mathcal{H}_{0}$ is true.
Lo et al. have injected an IMRE injection of template with echo parameters discussed earlier for the event GW150914 into simulated Gaussian noise. The detection statistic for this injection is given as follows, $$\begin{aligned}
\ln \mathcal{B}_{\rm detected,Gaussian}=-0.2576<0.\end{aligned}$$ Therefore, this finding indicates that the data slightly favor the null hypothesis from Bayesian analysis point of view. While the p-value and the corresponding statistical significance, for Gaussian noise, show that the data favors echo hypothesis, $$\begin{aligned}
{\rm p-value} = 0.01275,\nonumber \\
{\rm statistical~ significance} = 2.234~ \sigma .\end{aligned}$$
Table \[table\_17\] shows the values of the detection statistic $\ln \mathcal{B}$ vs corresponding statistical significance in Gaussian and O1 backgrounds. Therefore, for a detection of gravitational-wave echoes having statistical significance $\geq 5\sigma$, the detection threshold would be, $$\begin{aligned}
\ln \mathcal{B}_{\rm threshold,Gaussian}=1.9,\nonumber \\
\ln \mathcal{B}_{\rm threshold,O1}=5.7,\end{aligned}$$ for Gaussian noise and O1 noise respectively.
Statistical significance Detection statistic (Gaussian noise) Detection statistic (O1 noise)
-------------------------- -------------------------------------- --------------------------------
$1\sigma$ -0.9 0.1
$2\sigma$ -0.4 1.5
$3\sigma$ 1.1 4.0
$4\sigma$ 1.5 5.4
$5\sigma$ 1.9 5.7
: Detection statistic $\ln \mathcal{B}$ vs its corresponding statistical significances shown for both Gaussian and O1 backgrounds [@Lo:2018sep].[]{data-label="table_17"}
Table \[table\_18\] shows the detection statistics and the corresponding statistical significance and p-value for the O1 events. This table also shows that the ordering of the events by their statistical significance is consistent with what has been reported by Nielsen et al. [@Nielsen:2018lkf].
$$\begin{aligned}
\ln \mathcal{B}_{\rm O1}^{\rm (cat)}=-1.1,\end{aligned}$$
Event Detection statistic p-value Statistical significance $(\sigma)$
---------- --------------------- --------- -------------------------------------
GW150914 -1.3 0.806 $<1$
GW151012 0.4 0.0873 1.4
GW151226 -0.2 0.254 $<1$
: The detection statistic and its corresponding statistical significance and p-value for O1 events [@Lo:2018sep]. The ordering of events by their statistical significance is consistent with what reported by Nielsen et al. [@Nielsen:2018lkf][]{data-label="table_18"}
### Comments on Lo et al. [@Lo:2018sep]
In their analysis, Lo et al. [@Lo:2018sep] have included both the main event, as well as the ADA echo waveform in their template, but they used expanded priors in Table \[table\_16\]. Although expanding the ADA priors covers a larger space of possibilities, it tends to dilute marginal signals and bury them in the noise. For example, there is no good physical interpretation for repeating echoes that do not damp (with $\gamma=1$), as they violate energy conservation.
For these reasons, it is not surprising that Lo et al. [@Lo:2018sep] find a smaller Bayes factors, due to their expanded priors. However, it is well-known that expanding prior into nonphysical regimes will artificially lower Bayesian evidence for any model, especially since $\gamma=1$ is a (formal) singularity of the likelihood function.
### \[Uchikatanegative\]Results of Uchikata et al. [@Uchikata:2019frs]
As discussed in Section \[Uchikata\] above, Uchikata et al. [@Uchikata:2019frs] can approximately reproduce the evidence for ADA echoes in both O1 and O2 events. Here, we present their search using an alternative template that failed to find any evidence for echoes. In order to build the latter, they considered Kerr spacetime, replacing the event horizon with a reflective membrane. They then used the transmissivity of the Kerr angular momentum barrier ${\cal T}_{\rm BH}(\omega)$ to filter the ADA template, which acts as a hi-pass filter (truncating the low frequency part of the ADA phenomenological waveform \[template\]). Moreover, the overall phase shift of the waveform as a free parameter is taken into account contrary to ADA search. Using this template, they have found no significant echo signals in the binary BH merger events. The background estimation, has used the same method provided by Westerweck et al. [@Westerweck:2017hus].
1. The results in Table \[table\_10\] gives p-values for all events. The combined p-value is well above the critical p-value 0.05. In other words, echo signals using this model do not exist in the data, or their amplitudes are too small to be detected within the current detector sensitivity.
-----------------------------
Data version
-----------------------------
: Obtained P-values for each event along with total p-value [@Uchikata:2019frs]. A hyphen means that 4096-second of data are not available. []{data-label="table_10"}
\
Event C01 C02
---------- ------- -------
GW150914 0.992 0.984
GW151012 0.646 0.882
GW151226 0.276 -
GW170104 0.717 0.677
GW170608 - 0.488
GW170729 - 0.575
GW170814 - 0.472
GW170818 - 0.976
GW170823 - 0.315
Total 0.976 0.921
: Obtained P-values for each event along with total p-value [@Uchikata:2019frs]. A hyphen means that 4096-second of data are not available. []{data-label="table_10"}
2. Since the phase shift at the membrane (due to the reflection and boundary condition) is model dependent, it is physically reasonable to assume a total phase shift as a parameter (see Fig. \[Uchikata1\] that has used extra phase parameter for GW150914). In contrast, former studies [@Abedi:2016hgu; @Westerweck:2017hus] only considered phase inversion (or Dirichlet boundary conditions) at the reflective membrane. Therefore, the results of two cases, when the phase shift is fixed to $\pi$ (result 1) and when it is a free parameter (result 2), respectively, in Table \[table\_11\] are compared. In this table we see that, p-values become slightly larger (taking GW151226 as an exception) when the phase shift due to the reflection has taken as a free parameter.
Event Result 1 Result 2
---------- ---------- ----------
GW150914 0.638 0.992
GW151012 0.417 0.646
GW151226 0.953 0.276
GW170104 0.213 0.717
Total 0.528 0.976
: P-value for each event and total p-value [@Uchikata:2019frs]. Result 1 is the case when the phase shift is fixed to $\pi$, and result 2 is the case when the total phase shift is also a parameter.[]{data-label="table_11"}
### Comment on Negative results of Uchikata et al. [@Uchikata:2019frs]
Model provided by Uchikata et al. [@Uchikata:2019frs] substantially truncates the low frequency part of the GR waveform (which is the basis of ADA template \[template\]). However, one may argue that the GR waveform has already been filtered once by the transmissivity of the angular momentum barrier, ${\cal T}_{\rm BH}(\omega)$ as it is what is seen by observers at infinity. In fact, [@Wang:2019rcf] have shown that both GR signal (main event) and echoes can be constructed from a superposition of the QNMs of the quantum BH, which are essentially the modes trapped between the angular momentum barrier and the quantum membrane (see Section \[subsec\_ergoinst\] above). Therefore, both the GR signal and the echoes pass through same barrier and are thus truncated by the same ${\cal T}_{\rm BH}(\omega)$. This implies that ${\cal T}_{\rm BH}(\omega)$ cancels in the ratio of echo to main signal waveform, and in contrast to Uchikata et al. [@Uchikata:2019frs], no truncation is needed.
Indeed, as evidenced by their own analysis (Section \[Uchikata\] above), the low-frequency part of the ADA template is necessary to obtain a significant signal. This is physically justified since the high frequencies leak out of the angular momentum barrier quickly, leading to a rapid decay, while echoes can last much longer at lower frequencies [@Wang:2018gin].
### \[Results of Tsang et al.\]Results of Tsang et al. [@Tsang:2019zra]
Tsang et al. [@Tsang:2018uie; @Tsang:2019zra] proposed a morphology-independent search method which consists of a large number of free parameters for echoes compared to ADA model [@Abedi:2016hgu] (49 versus 5). They search for echoes in all the significant events in (GWTC-1), and found that for all the events, the ratios of evidences for signal versus noise and signal versus glitch do not rise above their respective background. Only the smallest p-value=3% goes to the event GW170823. Hence they found no significant evidence for echoes in GWTC-1. The results of search are given in Table \[table\_12\] and \[table\_13\].
Event Log $B_{S/N}$ $p_{S/N}$ Log $B_{S/G}$ $p_{S/N}$
---------- --------------- ----------- --------------- -----------
GW150914 2.32 0.26 2.95 0.43
GW151012 -0.59 0.70 0.35 0.88
GW151226 -0.67 0.72 2.48 0.53
GW170104 1.09 0.44 3.80 0.28
GW170608 -0.90 0.75 0.90 0.82
GW170823 6.11 0.03 5.29 0.11
Combined - 0.34 - 0.57
: Log Bayes factors for signal versus noise and signal versus glitch, and the corresponding p-values, for events seen in two detectors of GWTC-1 [@Tsang:2019zra]. The bottom row shows the combined p-values for all these events together.[]{data-label="table_12"}
Event Log $B_{S/N}$ $p_{S/N}$ Log $B_{S/G}$ $p_{S/N}$
--------------- --------------- ----------- --------------- -----------
GW170729 4.24 0.67 5.64 0.62
GW170809 9.05 0.31 12.69 0.09
GW170814 8.75 0.33 8.54 0.34
GW170817 11.05 0.19 10.30 0.20
GW170817$+1s$ 6.19 0.52 9.39 0.27
GW170818 10.39 0.23 9.36 0.27
Combined - 0.47 - 0.22
: Same as Table \[table\_12\], while for the events that three detectors has been involved [@Tsang:2019zra]. In the case of GW170817, in order to cover 1.0 sec after the merger for echoes found by Abedi and Afshordi [@Abedi:2018npz], additional search as first echo being from this time has been set. For this particular event, latter prior choice has been taken for combined p-values.[]{data-label="table_13"}
### Comment on: Results of Tsang et al. [@Tsang:2019zra]
Tsang et al. [@Tsang:2019zra] have developed a model which consists of a large number of free parameters (49 by our count). Indeed, a larger space of possiblities leads to lower significance. So it is not surprising to get large p-values out of large free-parameter space. Indeed, all the SNRs for echoes reported in ADA would be below the detection threshold reported by [@Tsang:2018uie], given their large number of parameters (see [@Abedi:2020sgg] for more detail).
A concordant picture of Echoes
------------------------------
In order to have an optimal search for echoes, one may want to take the following guidelines into consideration:
1. Have a good physical model (or you will not find them!)
2. Use a simple template (avoid too many arbitrary choices)
3. Avoid [*a posteriori*]{} statistics (don’t look at data to make your model)
Based on the positive (Section \[Positive\]), mixed (Section \[Mixed\]), and negative (Section \[Negative\]) results, one may offer the following general observations:
1. Coherent searches appear to give more significant evidence for echoes
2. Template searches can find evidence for echoes, if they include lower frequencies
3. Models with a large number of free parameters and/or wider priors can weaken the echoes below the detection threshold
An executive summary of these observations is shown in Tables \[table\_positive\] and \[table\_fail\] as positive evidence (p-value$\leq 0.05$) and failed results, respectively.
Authors Method Data p-value
--- ----------------------- -------- ------ ---------
1 [@Abedi:2016hgu]
2 [@Conklin:2017lwb]
3 [@Westerweck:2017hus]
4 [@Nielsen:2018lkf]
5 [@Uchikata:2019frs]
6 [@Uchikata:2019frs]
7 [@Salemi:2019uea]
8 [@Abedi:2018npz]
9 [@Gill:2019bvq]
Authors Method Data possible caveat
--- ----------------------- -------- ------ -----------------
1 [@Westerweck:2017hus]
2 [@Nielsen:2018lkf]
3 [@Uchikata:2019frs]
4 [@Salemi:2019uea]
5 [@Lo:2018sep]
6 [@Tsang:2018uie]
The study of post-merger GW observations with the above-mentioned motivations have lead to tentative signals, at varying levels of significance, by different groups. We shall outline several similarities amongst these findings below. However, it is also important to note that these similarities do not mean that the signals found are the same, but it does provide a [*preponderance of corroborating evidence*]{} for GW echoes in current observations.
### \[Five independent groups, Five independent methods, identical results\]Five independent groups, Five independent methods, identical results!
1. In [@Abedi:2016hgu] (Table II), using the reported masses and spins of LIGO O1 events, the time delays of 0.1 sec and 0.2 sec for GW151226 and GW151012 were predicted respectively for Planckian echoes. These happen to be exactly the same as the times for post-merger signals found in [@Salemi:2019uea].
2. Results of [@Abedi:2016hgu; @Uchikata:2019frs; @Westerweck:2017hus; @Nielsen:2018lkf; @Salemi:2019uea; @Abedi:2018npz] all are consistent with Planckian echoes at p-values of ${\cal O}(\%)$.
3. Furthermore, the reconstructed detector responses for GW151226 and GW151012 [@SalemiGW151012; @SalemiGW151226] in [@Salemi:2019uea] give consistent amplitudes (0.33, 0.34)$\times$(maximum amplitude of main event) comparing with [@Abedi:2016hgu] (Table II). Energy reported in [@Abedi:2016hgu] (Appendix A) is also consistent with strength of signals found in [@Salemi:2019uea]. Finally, SNR reported for GW151012 in [@Abedi:2016hgu] (Table II and Fig. 6) has highest value which is also consistent with highest significance event in [@Salemi:2019uea].
4. Log Base factor values in Table II of [@Nielsen:2018lkf], where they found positive evidence for ADA echoes in GW151012 and GW151226 (where GW151012 is more significant) is consistent with significance of signals found in [@Salemi:2019uea].
5. Also note that the echo signal of GW150914 [@Abedi:2016hgu] at time delay 0.3 sec had narrowest time window ($\pm 3\%$ in Table II) and smallest energy (Table II) compared to GW151226 and GW151012, which could explain its absence in [@Salemi:2019uea], and no evidence (negative Log Base factor value) in [@Nielsen:2018lkf] Table II.
6. Nevertheless, the residual signal in [@SalemiGW150914] which is a supporting results for [@Salemi:2019uea] is consistent with 300 m sec echo signal time delay in [@Abedi:2016hgu] (table II).
7. The percent-level evidence of Uchikata et al. [@Uchikata:2019frs] for ADA echoes in O1 events (shown in Table \[table\_3\]) are consistent with the results of other groups [@Abedi:2016hgu; @Westerweck:2017hus; @Nielsen:2018lkf; @Salemi:2019uea].
8. The results of Uchikata et al. [@Uchikata:2019frs] for O2 events shown in Table \[table\_4\] are given $\sim 4\%$ overall p-value which are small as O1 events.
9. Lo et al. [@Lo:2018sep] Table \[table\_18\] by adding main event to the ADA waveform and keeping only three echoes with larger prior ranges also found similar ordering of events by their statistical significance with what reported by Nielsen et al. [@Nielsen:2018lkf].
### Other findings
Here we present similarities found for echoes in binary neutron star merger GW170817:
1. We note that the time-scale of 1.0 sec after merger for collapse into BH (first reported by Abedi and Afshordi [@Abedi:2018npz]) is now also independently found from purely Astrophysical considerations by [@Gill:2019bvq], who found $t_{\rm coll}=0.98^{+0.31}_{-0.26}$ second.
2. Along with echo signal found by Abedi and Afshordi [@Abedi:2018npz] another group [@Conklin:2017lwb] claimed evidence for an echo frequency of $f'_{\rm echo} \simeq (0.00719 ~{\rm sec}) ^{-1}= 139$ Hz for GW170817, with a p-value of $1/300$. Noting the proximity of this value to the second harmonic of Abedi and Afshordi finding with echo frequency $2\times f_{\rm echo} = 144$ Hz, it is feasible that the two different methods are seeing (different harmonics of) the same echo signal. However, the method applied in [@Conklin:2017lwb] is sub-optimal, as they whiten (rather than Wiener filter in [@Abedi:2018npz]) the data, and thus could underestimate the significance of the correlation peak they found (see [@Abedi:2018pst] for further discussion).
To contrast, let us point out two apparent inconsistencies:
1. The results of Conklin et al. [@Conklin:2017lwb; @Holdom:2019bdv] are finding non-Planckian echoes signal which might not be consistent with the results of other groups [@Abedi:2016hgu; @Westerweck:2017hus; @Nielsen:2018lkf; @Salemi:2019uea; @Abedi:2018npz]. However, it may not be appropriate to do a one-to-one comparison of [@Conklin:2017lwb; @Holdom:2019bdv] to other studies, as the employed method is significantly different.
2. Results of Salemi et al. [@Salemi:2019uea] show that using their reconstruction of cWB, the post merger signal of GW151012 seen in Fig. \[Salami1\] appear to come from a different sky location, compared to the main event signal. However, as we discussed above (Section \[Results of Salemi et al.\]), this might be due to time-delay degeneracy in the cWB monochromatic signals.
### Independent confirmations of model predictions
Here, we outline model predictions that have been confirmed using independent data by independent groups
1. [*Binary BH mergers:*]{}
Uchikata et al. [@Uchikata:2019frs] have used ADA model [@Abedi:2016hgu] to search for echoes for both O1 (first observing run) and O2 (second observing run) while the original search of ADA only covers O1. The results for O2 (with p-value=0.039) in Tables \[table\_3\] and \[table\_4\] show similar evidence as O1 (with p-value=0.055).
2. [*Binary neutron star merger:*]{}
After detection of echoes signal with $4.2\sigma$ significance around 1 sec after BNS merger GW170817 [@Abedi:2018npz] where Abedi and Afshordi claim that it has collapsed to BH at this time, Gill et al. [@Gill:2019bvq] with independent Astrophysical consideration have also determined that the remnant of GW170817 must have collapsed to a BH after $t_{\rm coll}=0.98^{+0.31}_{-0.26}$ sec. Error-bar for this observation compared to the detected signal of echoes by Abedi and Afshordi as a consequence of BH collapse is shown in Fig. \[NS-NS\_9\].
### Concerns about ADA searches
1. *Concerns about errors in $\Delta t_{echo}$*: The original ADA search [@Abedi:2016hgu] had used an ad-hoc method for finding symmetric 1-sigma errors for $\Delta t_{echo}$, which would miss $\sim 1/3$ of Planckian echoes. Furthermore, the actual LIGO posteriors for these parameters (which are now publicly available, even though they were not at the time), are not Gaussian. A fully Bayesian Bayesian search, using actual mass and spin posteriors would avoid these short-comings.
2. *Concerns about keeping $t_{0}=-0.1$ and $\gamma=0.9$ fixed*: Uchikata et al. [@Uchikata:2019frs] have fixed $t_{0}$ at its best fit value of O1 in order to search in O2. It might be a good idea to keep these parameters at their best-fit values (from O1 or O2), to make perform more efficient searches in e.g., O3 events.
3. *Concerns about mass ratio dependence of echoes overall amplitude*: As pointed out in Section \[Results of Salemi et al.\] (Fig. \[mass ratio error\]), it appears that the amplitudes of echo signals may depend on the BBH mass ratio, which should be taken into account if one wants to optimally combine echo signals in different events.
### On negative GW Echo searches
For attempts that fail to yield any evidence for echoes [@Tsang:2018uie; @Uchikata:2019frs], we again point out that an optimal echo search should us a simple model with minimum number of free parameters. The current positive results turn out to be weak signals with SNR$\sim 4$, which is below the threshold for those searches that consist of many free parameters ($SNR>8$ for 49 free parameters). In addition, negative results of Uchikata et al. [@Uchikata:2019frs] indicate that the echo signals found in [@Abedi:2016hgu] mostly consist of low frequency modes, which is independently confirmed in [@Abedi:2018npz], and Uchikata et al’s own search in both O1 and O2, using the original ADA template.
### Non-Gaussianity of backgrounds
In order to search for signals of a given echo model, we need a proper understanding of background behaviour. Only then we might be able to determine the best statistical methodology. Since ADA [@Abedi:2016hgu] used different parts of LIGO data to get a combined significance, one may think about what would be the best method of combination of separate sets of data with different background behaviour. Then, we can also ask how the search changes by including three or higher number of detectors.
It was already observed that LIGO noise vary significantly and is very non-gaussian over long time-scales (see Fig’s 14-15 in [@TheLIGOScientific:2016src]). This either non-stationary or non-Gaussian background makes the interpretation of p-value ambiguous, particularly in finding marginal echo signals which are often near the detection threshold. Therefore, one must examine how much this varying background affects the inferred significance of a detection. This is studied by looking at other different minute-long stretches of data within a minute of the main events [@Abedi:2016hgu]. As can be seen in Fig. \[Histogram-non-gaussian\], the variation of p-values at the tail of the distribution is much higher than what is expected from Poisson statistics of the SNR peaks. This becomes more interesting when we see that the smallest p-value is coming from the range which is closest to the main event. This might be because the marginal LVT151012 (now called GW151012) detection is over a minimum of the LIGO (combined) detector noise.
![p-value distribution for combined events of different stretches of data within 1 minute of the main events. Surprisingly, the blue line which is closest to the main event, and has used to define p-value in [@Abedi:2016hgu] (Fig. \[Histogramloglog\]), happens to give the smallest p-value. The shaded region represents the Poisson error range for blue histogram. This shows that the variation in p-values is clearly much larger. This behaviour is interpreted as non-gaussianity and/or non-stationarity of the LIGO noise. In this plot the y-axis on the left (right) shows p-value (number of higher peaks) within the mentioned range of data. In each histogram the total number of “peaks” is $(38-9)/0.02= 1450$.[]{data-label="Histogram-non-gaussian"}](Histogram-non-gaussian.pdf){width="50.00000%"}
Future Prospects {#sec:future_prospects}
================
Towards Synergistic Statistical Methodologies
---------------------------------------------
As we summarized in the previous sections, the past fours years have witnessed hundreds of theoretical studies focusing on model-building for echoes, as well as dozens of observational searches and statistical methodologies. However, in spite of remarkable progress on both fronts, the theoretical and observational tracks have largely developed independently. However, it appears that both tracks have become mature enough, so much so that the time is ripe for a synergistic convergence. For example, Bayesian methods developed in [@Nielsen:2018lkf; @Lo:2018sep] applied to a superposition of QNMs of quantum BHs (as outlined in [@Wang:2019rcf]) would put coherent methods developed by [@Conklin:2017lwb; @Abedi:2018pst] on more sound statistical [*and*]{} physical footings. The analogy will be with helio- or astro-seismology, where modeling a dense spectrum of QNM frequencies can be used to infer the intrenal structure of the compact objects [@Oshita:2019seis1].
The real challenge will be in allowing enough freedom in our best physical models, in order to capture all the remaining theoretical uncertainties, [*but not any more!*]{}
\[numerical\] Echoes in Numerical Relativity
--------------------------------------------
Most studies of echoes have so far focused on the linear perturbation theory around the final BH for simplicity, but in reality the mergers start with the highly nonlinear binary BH inspiral. Hence, we need a covariant numerical implementation of binary quantum BHs within a highly-nonlinear dynamical spacetime to fully address the entire dynamics, especially the initial conditions. There are several possible approaches borrowed from numerical relativity which can be modified to either include the quantum boundary condition or the full dynamics of binary quantum BHs.
For instance, the effective one body (EOB) formalism [@Buonanno:1998gg; @Damour:2008yg] is a concrete strategy which only needs to solve ordinary differential equations rather than to perform the costly 3d numerical relativity simulations. It uses higher-order post-Newtonian expansion in a resummed form (different from the usual the Taylor-expansion), to include the non-perturbative result using a conservative description of binary BHs dynamics, radiation-reaction and emitted GW waveform. One possible approach, that is currently underway, is to capture the nonlinear effects in echoes by modifying the boundary condition in the EOB codes to implement the quantum BH dynamics.
Another route is to directly modify numerical relativity codes that have successfully produced waveforms for BBH merger events. A concrete strategy could be incorporating the mock fuzzball energy-momentum tensor (Section \[sec3\]) as a source for Einstein equations, directly into the numerical relativity codes. If the fuzzball “fluid” manages to stay just outside the apparent BH horizons in a dynamical setting, then it can potentially generate echoes in a fully nonlinear numerical simulation of quantum BBH merger.
Recently, [@Okounkova:2019dfo; @Okounkova:2019zjf] presented the first numerical simulation of BBH mergers in Chern-Simon gravity. They start with the modified action and predict the dynamics order by order. It is possible that a similar iterative approach can be applied to model boundary conditions at apparent horizons, or evolution of mock fuzzballs.
Quantum Gravity, Holography, and Echoes
---------------------------------------
As we discussed in Section \[sec:echo\_predictions\] above, any modification of event horizons that could lead to echoes should be a non-perturbative modification of general relativity, and can only be fully captured by a non-perturbative description of quantum gravity. A possible example of this is the fuzzball program in string theory (Section \[sec:QBHs\] above). But more generally, what can non-perturbative approaches to quantum gravity tell us about BH echoes?
One of our greatest insights into the dynamics of quantum gravity has come from the Holographic Principle, that extending Bekenstein-Hawking area law for entropy of BHs [@Bekenstein:1973ur], suggests the entire dynamics of a quantum gravitational system should be captured on its boundary. The most concrete realization of this principle was proposed by Juan Maldacena [@Maldacena:1997re], in the form a conjectured duality between quantum gravity in Anti-de Sitter (AdS) spacetime and a Conformal field theory (CFT), commonly known as AdS/CFT correspondence or conjecture. It proposes that CFT in spacetime of d-1 dimension, at the asymptotic boundary of an AdS spacetime is mathematically equivalent to string theory (or quantum gravity) within the bulk AdS in d dimension. This topic has been extremely fruitful over the past two decades, offering many synergies between seemingly disparate notions in geometry and quantum information. For example, the Ryu-Takayanagi conjecture [@Ryu:2006ef] relates the entanglement entropy of boundary CFTs with the areas of extremal surfaces in the bulk AdS, generalizing the notion of Bekenstein-Hawking BH enetropy to arbitrary geometries.
An intriguing connection between AdS/CFT and echoes is the appearance of echo times: $$\Delta t_{\rm echo} = t_{\rm scrambling} = \frac{\ln(S_{\rm BH})}{2\pi T_{\rm H}},$$ as “scrambling time”, in the AdS/CFT literature [@Sekino:2008he]. Here, $S_{\rm BH}$ and $T_{\rm H}$ are the entropy and temperature of the BH respectively. The scrambling time refers to the time it takes to destroy quantum entanglements in a chaotic system, while BHs (and their CFT duals) are conjectured to be fast scramblers, i.e. the most efficient in destroying entanglement (e.g., [@Maldacena:2015waa]). Interestingly, Saraswat and Afshordi [@Saraswat:2019npa] have recently shown that the scrambling time (computed using Ryu-Takayanagi conjecture in a dynamical setting) is identical to the Planckian echo times, for generic charged AdS BHs. Could this imply that echoes could be a generic property of (possibly a certain class of) quantum chaotic systems?
Another possible connection could come in the form of the fluid-gravity correspondence, e.g., in the context of membrane paradigm discussed in Section \[sec:membrane\]. For example, in [@Oshita:2019sat], we have argued that Boltzmann reflectivity of GW echoes, implies that viscosity of the boundary fluid should vanish at small frequencies $\hbar \omega \ll k T$. One may also speculate that other holographic manifestations of BH echoes may appear in the Kerr/CFT conjecture [@Castro:2010fd], Braneworld BHs [@Dey:2020lhq], or as Regge poles of the boundary plasma in AdS/CFT.
Einstein Telescope, Cosmic Explorer
-----------------------------------
The Einstein Telescope (ET) [@Punturo:2010zz] and Cosmic Explore (CE) [@PhysRevD.91.082001] are the third-generation ground based GW detectors. The ET consists of three underground detectors with three arms $10$ kilometers long and CE will be realized with two arms 40 kilometers long, which are 10 times longer than Advanced LIGO’s. These next-generation GW detectors might allow us to observe some Planckian signatures from quantum BHs such as GW echoes from merger events leading to a remnant BH. We plot the spectra of GW echoes and ringdown with the sensitivity curves of Advanced LIGO, ET, and CE in FIG. \[third\_gene\]. The detection of GW echoes with the third generation GW observatories are discussed in [@Testa:2018bzd; @Maggio:2019zyv], and it may be possible to distinguish ECOs with $|{\cal R}| \lesssim 0.3$ from BHs with at $2\sigma$ level when SNR $\sim 100$ in ringdown, which would be possible for the third-generation GW detectors. The relative error on the reflectivity of would-be horizon is also investigated in [@Testa:2018bzd; @Maggio:2019zyv], and the relative error for measurement of relectivity in ground-based detectors is approximately given by $$\left| \frac{\Delta {\cal R}}{1-\cal R} \right| \simeq 0.5 \times \left(\frac{8}{\rho_{\text{ringdown}}} \right),$$ where $M = 30 M_{\odot}$, $\rho_{\text{ringdown}}$ is the SNR in the ringdown phase, while the distance between the top of the angular momentum barrier and the would-be horizon is assumed to be longer than $50 M$ in the tortoise coordinate. For comparison, we note that the loudest detected BBH event, GW150914, has $\rho_{\text{ringdown}} \simeq 8$.
![Spectra of ringdown and echo phases with the reflectivity of $|{\cal R}| = 0.99$, $0.9$, $0.6$, and $0.3$. We set $D_o = 40$ Mpc, $\bar{a} =0.1$, $\ell = m= 2$, $M = 4 M_{\odot}$, $\theta = 20^{\circ}$, and $\epsilon_{\rm rd} = 0.1 \%$.[]{data-label="third_gene"}](third_generation_echo.png){width="100.00000%"}
The detectability of GW echoes from failed supernovae, leading to the formation of BHs, with the third-generation GW observatories is also discussed in [@Oshita:2019seis1]. Calculating the SNR of GW spectrum consisting of echo and ringdown, $\rho_{\rm ringdown + echo}$, in the Boltzmann reflectivity model, the horizon distance $D_{\rm h}$, defined as the distance where $\rho_{\rm ringdown + echo} = 8$, is estimated. Given the optimistic case in the Boltzmann reflectivity model, $T_{\rm H}/ T_{\rm QH} = e^{15 (\bar{a}-1)}$, the horizon distance can be estimated as $D_{\rm h} \sim 10$ Mpc for the Advanced LIGO at design sensitybity and $D_{\rm h} \sim 100$ Mpc for the third-generation detectors such as ET and CE. Therefore, the authors in [@Oshita:2019seis1] argue that the searching for GW echoes, sourced by failed supernovae within our Galaxy and nearby galaxies, may be possible. However, in the case of $T_{\rm QH} = T_{\rm H}$, the horizon distance is less than or comparable with $10$ Mpc and so the echo search with failed supernovae would be restricted to within the Local Group. For the comparison, the strain amplitude of GW echoes in $T_{\rm QH}/ T_{\rm H} = 1$ and $T_{\rm QH}/ T_{\rm H} = e^{15 (\bar{a} -1)}$ are shown[^13] in FIG. \[third\_gene\_boltz\].
![Spectra of ringdown and echo phases in the Boltzmann reflectivity model with $\bar{a} = 0.1$, $\epsilon_{\rm rd} = 6 \times 10^{-7}$, $M = 2.4 M_{\odot}$, $\theta = 90^\circ$, and $D_o = 1$ Mpc. Here we also assume $\gamma = 10^{-10}$, $T_{\rm H} / T_{\rm QH} = 1$ (left) and $T_{\rm H} / T_{\rm QH} = 1.37 \times 10^{-6}$ (right).[]{data-label="third_gene_boltz"}](third_detect_boltzmann.png){width="100.00000%"}
LISA
----
The Laser Interferometer Space Antena (LISA) is planned to be the first GW observatory in space. It will have three satellites separated by millions of kilometers and their orbits maintain near-equilateral triangular formation. LISA might enable us to reach high-precision detection of ringdown in SNR $\sim {\cal O} (10^3)$, which puts stronger constraints on the reflectivity of BHs [@Testa:2018bzd; @Maggio:2019zyv].
Recently, a novel proposal to discriminate BH horizons based on the tidal heating was proposed in [@Datta:2019epe]. One of the main targets of the LISA mission is precision measurements of extreme-mass-ratio inspirals (EMRIs), in which the tidal heating could be important. The (partial) absorption of ECOs or BHs plays the role of dissipation at the surface, by which tides back-react on the orbital trajectory. It is argued that this tidal heating is responsible for a large dephasing between the orbits of a BH and ECO. This dephasing accumulates over the timescale of months and the accumulation speed is faster for a higher spin. The authors in [@Datta:2019epe] also found a proportionality relation between the dephasing $\delta \phi$ and energy reflectivity $|{\cal R}|^2$.
In order to make use of this scheme to put strong constraints on the reflectivity of ECOs, one has to obtain accurate EMRI waveforms by properly taking into account the tidal heating for orbiting objects, which may decrease systematic errors in data analysis.
Not only the tidal heating, but also the tidal deformability contributes to the GW Fourier phase and it can be characterized by the tidal Love number $k$. The Love number of ECO of mass $M$ may scale as $1/|\log{\delta}|$, where $\delta \equiv r_0 - r_{\rm h}$, with $r_{\rm h}$ is the BH horizon radius of mass $M$ and $r_0$ is the radius of the ECO. So the $k-\delta$ relation is $$\delta = r_{\rm h} e^{-1/k},$$ and assuming this relation, one can infer the near-horizon structure characterized by $\delta$ from the measurement of the Love number $k$. For instance, if the Love number of the order of $k \sim 10^{-2}$ is measured by LISA from a supermassive BH binary signal, leading to the formation of a BH of $M \sim 10^6 M_{\odot}$, it yields the resolution of $\delta \sim l_{\rm Pl}$.
However, the authors in [@Addazi:2018uhd] point out that the statistical and quantum mechanical uncertainties in measurements of near-horizon lead to some difficulty to measure $\delta$ precisely. The former one comes from the fact that the statistical uncertainty in $\delta$ is proportional to $1/k$, and the inferred value of $k$, where the inferred value of $\delta$ is comparable with its statistical uncertainty, is around $k \sim 0.2$. Therefore, any inferred value of $\delta$, derived from $k$ that is smaller than $\sim 0.2$, would be dominated by the statistical uncertainty. The latter one comes from the uncertainty principle in quantum mechanics. Once precisely measuring $\delta \sim l_{\rm Pl}$, it may lead to the uncertainty in the mass of the ECO, which then leads to the uncertainty in the binding energy. This results in the uncertainty in the orbital and GW frequencies, which means that one cannot measure $\delta$ precisely if it is much shorter than $l_{\rm Pl}$.
Pulsar Timing Arrays
--------------------
Following their first discovery in 1968 [@Hewish:1968bj], over 2000 pulsars have now been detected by radio telescopes across the world. The pulsars’ intrinsic properties, as well as propagation effects in the interstellar medium, can influence the arrival times of pulsar pulses. Therefore, pulsar timing arrays (PTA) can be used as a detection tool for BH binaries [@Hobbs:2009yy], and thus, might be used to detect singatures of echoes from quantum BHs. In particular, millisecond pulsars stand out for their unparalleled stability (comparable to atomic clocks!) without being subject to starquakes and accretion. To give an explicit example, we show the spectrum of GW echoes predicted by the Boltzmann reflectivity model [@Oshita:2019sat; @Wang:2019rcf] with the sensitivity curve of International Pulsar Timing Array (IPTA) and Square Kilometre Array (SKA) (FIG. \[IPTA\_SKA\]). The lower curve in Fig. [@Hewish:1968bj] is for a $3 \time 10^9 M_{\odot}$ BH merger at $D_o = 1$ Gpc. Given that this mass is comparable to that of M87 supermassive BH, located at 16 Mpc, we expect $\sim 2 \times 10^5$ of such BHs at < Gpc. Assuming that each BH merges once every Hubble time $\sim 10^{10}$ years, and that echoes last for 20 years (from simple mass scaling), the chances of detecting such a loud event with PTAs at any time is 0.1%. However, fainter events will be more prevalent as their number increases as SNR$^{-3/2}$ from volume scaling. Furthermore, increase in supermassive BH merger activity observed at high redshifts shall boost this statistics.
![Spectra of GW echoes in the Boltzmann reflectivity model with $\bar{a}=0.6$, $\ell=m=2$, $D_o= 1$ Gpc, and $\gamma =1$. The gray line shows the case of $M=3 \times 10^{9} M_{\odot}$, $\epsilon_{\rm rd} = 0.005$, $T_{\rm H}/T_{\rm QH} = 0.1$ and the black line shows one for $M= 8 \times 10^{9} M_{\odot}$, $\epsilon_{\rm rd} = 0.01$, $T_{\rm H}/T_{\rm QH} = 0.05$. We also plot the PSD for the IPTA (blue) and SKA (red).[]{data-label="IPTA_SKA"}](IPTA_SKA.png){width="80.00000%"}
PTAs are anticipated to detect the low frequency GW signal from supermassive BBH within the next few years [@Hobbs:2009yy]. We expect that the first GW detection will be a stochastic background of supermassive BH binaries. With any luck, this shall lead to new insights into the nature of quantum BHs and gravity.
Final Word {#sec:final_words}
==========
In this review article, we provided a comprehensive overview of the theoretical motivations for why quantum black holes in our universe may have different observable properties, in contrast to their classical counterparts in Einstein’s theory of general relativity. The most prominent and potentially observable smoking gun for these quantum black holes comes in the form of gravitational wave echoes, which have been the subject of intense theoretical and observational scrutiny over the past few years. We provided a concise account of theoretical predictions, as well as the exciting and confusing state of observational searches for echoes in LIGO/Virgo observations. We closed by article by our vision of the future of “Quantum Black Holes in the Sky”, via a synergy of statistical methodology, quantum gravity, and numerical relativity, and in light of the next generation of gravitational wave observatories.
While this review article focuses on the gravitational wave echoes, as arguably the most concrete and promising signature of quantum black holes, other possible observable signatures can be (and should be) explored. For example, interactions of photons or neutrinos with near-horizon quantum structure could lead to signatures in radio images in Event Horizon Telescope observations [@Rummel:2019ads], or ultra high energy neutrinos in Ice Cube observatory [@Afshordi:2015foa], respectively. However, these signals will be suppressed if Boltzmann reflectivity is assumed, as they have $\hbar \omega \gg k T_{\rm H}$. Another alternative to echoes may come through non-localities in non-violent unitarization, which would be observable far from the horizon (see Section \[sec:non-violent\]). However, it is arguably difficult to pin down concrete predictions in this scenario.
To conclude, the world of Quantum Black Holes remains a wide open and largely uncharted territory, spanning from the dark corners of obscure mathematical structures to the nitty-gritty details of gravitational wave detector noise. It also holds the promise to crack the century-old puzzle of quantum gravity, and yet be imminently testable in the next few years. Therefore, the study of “Quantum Black Holes in the Sky” remains extremely exciting, active, and confusing, and is bound to provide us with new surprises in the new decade, and beyond.
Acknowledgments {#sec::acknowledgments .unnumbered}
===============
We would like to thank Michael Balogh, Ofek Birnholtz, Avery Broderick, Vitor Cardoso, Ramit Dey, Hannah Dykaar, Will East, Steve Giddings, Bob Holdom, Badri Krishnan, Lam Hui, Luis Lehner, Luis Longo, Samir Mathur, Emil Mottola, Shinji Mukohyama, Rob Myers, Ramesh Narayan, Alex Nielsen, Paolo Pani, Joe Polchinski (RIP), Chanda Prescod-Weinstein, Jing Ren, Krishan Saraswat, Rafael Sorkin, Daichi Tsuna, Yasaman Yazdi, Huan Yang, Aaron Zimmerman, and many others for discussions and/or collaborations about Quantum Black Holes over the past decade. N. O. is supported by JSPS Overseas Research Fellowships and the Perimeter Institute for Theoretical Physics.
Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Innovation. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw- openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Instituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.
Energy density and flux seen by a radially free falling observer in mock fuzzball\[fuzzballflux\]
=================================================================================================
We start with the geodesic of a radially falling observer in the mock fuzzball. $$\begin{aligned}
\label{line}u^{\mu}u_{\mu}=-1&=-(1-\frac{2M}{r}+b)(\frac{dt}{d\tau})^2-\frac{1}{1-\frac{2M}{r}}(\frac{dr}{d\tau})^2,\\
\label{lineinitial}-1&=-(1-\frac{2M}{r}+b)(\frac{dt}{d\tau})^2,\\
\label{conservation}(1-\frac{2M}{r}+b)\frac{dt}{d\tau}&=\text{constant}.\end{aligned}$$ Where $\tau$ is the proper time of the observer. $\theta$ and $\varphi$ vanish with a radially observer. Eq. \[line\] is from the line element, Eq. \[lineinitial\] is initial condition of Eq. \[line\] : the observer rest at infinite. Eq. \[conservation\] is energy conservation. Eq. \[line\]-\[conservation\] give four-velocity: $$\begin{aligned}
u^{\mu}=(\frac{r}{b r-2M+r},- \sqrt{\frac{2M (r-2 M)}{r (b r-2 M+r)}},0,0).\end{aligned}$$ Then we consider area vector $a_{\mu}=A(-u^1,u^0,0,0)$ which is normal to four-velocity $a^{\mu}u_{\mu}=0$. We determine $A=\sqrt{\frac{1-2M/r+b}{1-2M/r}}$ by normalization of the area vector $a^{\mu}a_{\mu}=1$. In the end, the energy density and flux are defined as: $$\begin{aligned}
\mathcal{U}&= T_{\mu \nu} u^{\mu} u^{\nu}=-\frac{4 b M^2}{r^2 (b r-2M+r)^2},\\
\mathcal{F}&= -T_{\mu \nu} u^{\mu} a^{\nu}=-\frac{ b \left(\frac{2M}{r}\right)^{3/2}}{(b r-2 M+r)^2}.\end{aligned}$$
[^1]: It turns out that this toy model spacetime coincides with the one proposed earlier by [@Damour:2007ap].
[^2]: Recently, it was found out that the CD equation with $(i,j) = (+1, \pm 1)$ is related to $(-1,\mp1)$ by the Darboux transformation.
[^3]: Note that in order to set the initial conditions of the QNMs of the quantum BH cavity, we choose a superposition that reproduces the time-evolution of the dominant QNM of the classical BH for $t \lesssim \Delta t_{\rm echo}$.
[^4]: In [@Abedi:2016hgu], 2-tailed gaussian probability assigned to significance, e.g., p-value $= 68\%$ and $95\%$ correspond to $1\sigma$ and $2\sigma$ respectively.
[^5]: There are two versions of noise subtraction in LIGO open data, called C01 and C02 [@gw-openscience]
[^6]: Additional consideration of the best fit initial phase has given negative results and is discussed in negative result part below \[Uchikatanegative\]. Therefore, this plot only justifies Uchikata et al. [@Uchikata:2019frs] reevaluation of ADA search [@Abedi:2016hgu] in Fig. \[SNR\], while belongs to the discussion in part \[Uchikatanegative\].
[^7]: In this paper, a 1-tailed gaussian probability to assign a significance to a p-value is used, e.g., p-value= 84% and 98% correspond to $1\sigma$ and $2\sigma$ respectively.
[^8]: upper bounds are just limited by the number of trials
[^9]: the bandpass ranges in units of Hz are: (433, 789), (152, 733), (80, 308), (185, 794), (52, 251), (132, 351)
[^10]: $(i-j)$ means echoes $i$ through $j$ were used
[^11]: the whole time range used was shifted 10 seconds later
[^12]: the explicit sets used: ($N_{E}$, $t_{w}/M$) = (15,40),(10,80),(15,80),(3-15,40),(5-17,40),(3-15,80)
[^13]: We here assume that the energy fraction of ringdown phase is $\epsilon_{\rm rd} = 6 \times 10^{-7}$ although it highly depends on the detail of nonlinear gravitational collapse.
|
---
address: |
University of Warsaw, Faculty of Physics,\
Hoża 69, 00-681 Warsaw, Poland
author:
- 'D. SOKO[Ł]{}OWSKA'
title: DARK MATTER AND 125 GEV HIGGS FOR IDM
---
Introduction
============
According to the standard cosmological model around 25% of the Universe is made of cold, non-baryonic, neutral and very weakly interacting particles. The Inert Doublet Model (IDM) is one of the simplest extensions of the Standard Model (SM) that can provide such Dark Matter (DM) candidate. The scalar sector in the IDM is extended with respect to the SM-like Higgs doublet $\Phi_S$ by a second $SU(2)$ doublet, $\Phi_D$, which is odd under a $D\;(Z_2)$ symmetry: $\Phi_S\to \Phi_S, \Phi_D \to - \Phi_D, \textrm{SM fields} \to \textrm{SM fields}$.[@Cao:2007rm; @Barbieri:2006dq]
The IDM can provide a viable DM candidate in agreement with collider constraints and relic density measurements in three regions of DM mass: $M_{DM} \lesssim 10 {\,\mbox{GeV}}$, $40 {\,\mbox{GeV}}\lesssim M_{DM} \lesssim 160 {\,\mbox{GeV}}$ and $M_{DM} \gtrsim 500 {\,\mbox{GeV}}$.[@Dolle:2009fn] Further constraints for the DM candidate can come from direct and indirect detection experiments. However, as for now there is no agreement how to consistently interpret various reported signals and the exclusion limits.[@Bergstrom:2012fi]
In this work we set constraints on scalar DM from the IDM by using the LHC Higgs data and WMAP relic density measurements. Combining the $h \to \gamma \gamma$ data for the SM-like Higgs with the WMAP results excludes a large part of the IDM parameter space, setting limits on DM that are stronger or comparable to these obtained by the DM detection experiments.
The Inert Doublet Model
=======================
The IDM is defined as a 2HDM with a $D$-symmetric potential and vacuum state: $$\begin{array}{c}
V=-{\frac}{1}{2}\left[m_{11}^2(\Phi_S^\dagger\Phi_S)\!+\! m_{22}^2(\Phi_D^\dagger\Phi_D)\right]+
{\frac}{\lambda_1}{2}(\Phi_S^\dagger\Phi_S)^2\!
+\!{\frac}{\lambda_2}{2}(\Phi_D^\dagger\Phi_D)^2\\[2mm]+\!\lambda_3(\Phi_S^\dagger\Phi_S)(\Phi_D^\dagger\Phi_D)\!
\!+\!\lambda_4(\Phi_S^\dagger\Phi_D)(\Phi_D^\dagger\Phi_S) +{\frac}{\lambda_5}{2}\left[(\Phi_S^\dagger\Phi_D)^2\!
+\!(\Phi_D^\dagger\Phi_S)^2\right],\\[2mm]
\langle\Phi_S\rangle =\frac{1}{\sqrt{2}} \begin{pmatrix}0\\ v\end{pmatrix}\,,\qquad \langle\Phi_D\rangle = \frac{1}{\sqrt{2}}
\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \quad v = 246 \textrm{ GeV},
\end{array}\label{dekomp_pol}$$ and Yukawa interaction set to Model I.[@Cao:2007rm; @Barbieri:2006dq]
In the IDM only one doublet, $\Phi_S$, is involved in the EW symmetry breaking. It provides a SM-like Higgs boson $h$, which has tree-level couplings to fermions and gauge bosons like in the SM with possible deviation from the SM in loop couplings. The second doublet, $\Phi_D$, is inert and contains four dark scalars $H, A, H^\pm$, that have no couplings to fermions. The lightest particle coming from this doublet is stable, being a good DM candidate.
The IDM can be described by the masses of scalar particles and their physical couplings: $\lambda_{345}=\lambda_3 + \lambda_4 + \lambda_5$ is related to $hHH$ and $hhHH$ vertices, while $\lambda_3$ gives $h H^+ H^-$ and $\lambda_2$: $HHHH$. Parameters of the IDM are constrained by various theoretical and experimental conditions. In our analysis we use vacuum stability constraints, that ensure the potential is bounded from below. We also demand, that the state (\[dekomp\_pol\]) is the global, and not just a local minimum.[@Krawczyk:2010; @Sokolowska:2011aa] Parameters of the potential should also fulfill perturbative unitarity bounds.[@Kanemura:1993]
The value of the Higgs boson mass, $M_h = 125 {\,\mbox{GeV}}$, and above conditions provide the following constraints for the parameters of the potential: $\lambda_1 = 0.258, \; m_{22}^2\lesssim 9\cdot10^4{\,\mbox{GeV}}^2, \; \lambda_3, \lambda_{345} > -\sqrt{\lambda_1\lambda_2} \geqslant -1.47, \; \lambda_{2}^{\textrm{max}} = 8.38$.[@Swiezewska:2012]
Masses of dark particles are constrained by the LEP measurements and EWPT to be: ${M_{H^{\pm}}}+M_H>M_{W},\ {M_{H^{\pm}}}+ M_A>M_W,\ M_H+M_A >M_Z,\ 2{M_{H^{\pm}}}> M_Z,\ {M_{H^{\pm}}}>70{\,\mbox{GeV}}$ with an excluded region where simultaneously $M_H< 80{\,\mbox{GeV}},\ M_A< 100{\,\mbox{GeV}}\ \textrm{and}\ M_A - M_H> 8{\,\mbox{GeV}}$.[@Gustafsson:2009]
The diphoton decay rate ${R_{\gamma\gamma}}$ in the IDM
=======================================================
${R_{\gamma\gamma}}$ is the ratio of the diphoton decay rate of the Higgs particle $h$ observed at the LHC to the SM prediction. The current measured values of ${R_{\gamma\gamma}}$ are ${R_{\gamma\gamma}}= 1.65\pm0.24\mathrm{(stat)}^{+0.25}_{-0.18}\mathrm{(syst)}$ for ATLAS and ${R_{\gamma\gamma}}= 0.79^{+0.28}_{-0.26}$ for CMS.[@ATLAS:2013oma; @CMStalk] They are in 2$\sigma$ agreement with the SM value ${R_{\gamma\gamma}}= 1$, however a deviation from that value is still possible and would be an indication of physics beyond the SM.
The ratio ${R_{\gamma\gamma}}$ in the IDM is given by: $$\label{rgg}
R_{\gamma \gamma}:=\frac{\sigma(pp\to h\to \gamma\gamma)^{\textrm{IDM}}}{\sigma(pp\to h\to \gamma\gamma)^{\textrm {SM}}}
\approx \frac{\Gamma(h\to \gamma\gamma)^{\mathrm{IDM}}}{\Gamma(h\to \gamma\gamma)^{\mathrm{SM}}}\frac{\Gamma(h)^{\mathrm{SM}}}{\Gamma(h)^{\mathrm{IDM}}} \, ,$$ where $\Gamma(h)^{\mathrm{SM}/\mathrm{IDM}}$ are the total decay widths of $h$ in the SM and the IDM, while $\Gamma(h\to \gamma\gamma)^{\mathrm{SM}/\mathrm{IDM}}$ are the respective partial decay widths for $h\to\gamma\gamma$. In the IDM two sources of deviation from ${R_{\gamma\gamma}}=1$ are possible. First is a $H^\pm$ contribution to the partial decay width:[@Djouadi:2005; @Arhrib:2012] $$\Gamma(h \rightarrow \gamma\gamma)^{\mathrm{IDM}}=\frac{G_F\alpha^2M_h^3}{128\sqrt{2}\pi^3}\bigg|\mathcal{M}^{\mathrm{SM}}+\delta\mathcal{M}^{\mathrm{IDM}}({M_{H^{\pm}}},\lambda_3)\bigg|^2\,, \label{Hloop}$$ where $\mathcal{M}^{\textrm{SM}}$ is the SM amplitude and $\delta\mathcal{M}^{\textrm{IDM}}$ is the $H^\pm$ contribution. The interference between $\mathcal{M}^{\textrm{SM}}$ and $\delta\mathcal{M}^{\textrm{IDM}}$ can be either constructive or destructive. The second source of deviations are possible invisible decays $h\to HH, AA$, which can strongly augment the total decay width $\Gamma^{\textrm{IDM}}(h)$ with respect to the SM case. If $h$ can decay invisibly then ${R_{\gamma\gamma}}$ is always below 1.[@Swiezewska:2012eh; @Arhrib:2012] For $M_H>M_h/2$ (and $M_A>M_h/2$) the invisible channels are closed, and ${R_{\gamma\gamma}}>1$ is possible. ${R_{\gamma\gamma}}$ depends only on the masses of the dark scalars and $\lambda_{345}$ (or $\lambda_3$), so setting a lower bound on ${R_{\gamma\gamma}}$ leads to upper and lower bounds on ${\lambda_{345}}$ as functions of $M_{H,A,H^\pm}$.[@Krawczyk:2013jta]
#### $HH, AA$ decay channels open
In this region, the invisible decay channels have stronger influence on the value of ${R_{\gamma\gamma}}$ than the contribution from $H\pm$ loop.[@Swiezewska:2012eh] If we demand that ${R_{\gamma\gamma}}> 0.7$, we get allowed values of $\lambda_{345}$ that are small, typically in range $(-0.04,0.04)$. For ${R_{\gamma\gamma}}> 0.8$ the allowed values of $\lambda_{345}$ are smaller than for ${R_{\gamma\gamma}}>0.7$. The condition ${R_{\gamma\gamma}}> 0.9$ strongly limits the allowed parameter space of the IDM. The allowed $A,H$ mass difference is $\delta_A \lesssim 2$ GeV, and values of $\lambda_{345}$ are smaller than in the previous cases. Requesting larger ${R_{\gamma\gamma}}$ leads to the exclusion of the whole region of masses, apart from $M_H \approx M_A \approx M_h/2$.[@Krawczyk:2013jta]
#### $AA$ decay channel closed
When the $AA$ decay channel is closed, the values of ${R_{\gamma\gamma}}$ do not depend on the value of $M_A$, while the charged scalar contribution becomes more relevant. If ${R_{\gamma\gamma}}>0.7$ then an exact value of ${M_{H^{\pm}}}$ is not crucial for the obtained limits on $\lambda_{345}$, and allowed values of $|\lambda_{345}|$ are of order $ 0.02$. For ${R_{\gamma\gamma}}>0.8$ the obtained bounds are different for ${M_{H^{\pm}}}= 70$ GeV and 120 GeV. Smaller ${M_{H^{\pm}}}$ leads to stronger limits, requiring $|\lambda_{345}|\sim 0.005$, while larger ${M_{H^{\pm}}}$ allow $|\lambda_{345}| \sim 0.015$. Larger value of ${R_{\gamma\gamma}}$ leads to smaller allowed values of $\lambda_{345}$. In the case of ${R_{\gamma\gamma}}>0.9$ a large region of DM masses is excluded, as it is not possible to obtain the requested value of ${R_{\gamma\gamma}}$ for any value of $\lambda_{345}$ if $M_H \lesssim 45 {\,\mbox{GeV}}$.[@Krawczyk:2013jta]
#### Invisible decay channels closed
If $M_A,M_H>M_h/2$, the invisible channels are closed and the only modification to ${R_{\gamma\gamma}}$ comes from the charged scalar loop (\[Hloop\]). Enhancement in ${R_{\gamma\gamma}}$ is possible when $\lambda_3<0$.[@Arhrib:2012; @Swiezewska:2012eh] Unitarity and positivity limits on ${\lambda_3}$ and $\lambda_{345}$ constrain the allowed values of ${M_{H^{\pm}}}$ and $M_H$ for a given value of ${R_{\gamma\gamma}}$. For ${R_{\gamma\gamma}}^\textrm{max}=1.01$ masses of ${M_{H^{\pm}}}\gtrsim 700$ GeV are excluded, and if ${R_{\gamma\gamma}}^\textrm{max} =1.02$ this bound is stronger, forbidding ${M_{H^{\pm}}}\gtrsim 480$ GeV. Also, even a small deviation from ${R_{\gamma\gamma}}=1$ requires a relatively large $\lambda_{345}$, if the mass difference $\delta_{H^\pm}$ is of the order $(50-100)$ GeV. Small values of $|\lambda_{345}|$ are preferred if $\delta_{H^\pm}$ is small.[@Krawczyk:2013jta]
Combining ${R_{\gamma\gamma}}$ and relic density constraints on DM \[consequences\]
===================================================================================
Here we compare the limits on the $\lambda_{345}$ parameter obtained from ${R_{\gamma\gamma}}$ with those coming from the requirement that the DM relic density is in agreement with the WMAP measurements: $0.1018<\Omega_{DM} h^2<0.1234$. If this condition is fulfilled, then $H$ constitutes 100% of DM in the Universe. Values of $\Omega_H h^2 <0.1018$ are allowed if $H$ is a subdominant DM candidate.
#### Low DM mass
In the IDM the low DM mass region corresponds to masses of $H$ below 10 GeV, while the other dark scalars are heavier, $M_A \approx M_{H^+} \approx 100$ GeV. To obtain the proper relic density, the $HHh$ coupling $(\lambda_{345})$ has to be large, for example $|\lambda_{345}| = (0.35-0.41)$ for CDMS-II favoured mass $M = 8.6$ GeV. The coupling allowed by ${R_{\gamma\gamma}}\sim 0.7$, i.e. $|\lambda_{345}| \sim 0.02$, is an order of magnitude smaller than needed for ${\Omega_{DM}h^2}$ and thus we can conclude that the low DM mass region cannot be accommodated in the IDM with recent LHC results.
![Comparison of the values of ${R_{\gamma\gamma}}$ and region allowed by the relic density measurements for the medium DM mass with (left) $HH$ invisible channel open and $M_A = M_{H^\pm} = 120$ GeV, (central) with $HH$ invisible channel closed and $\delta_A = \delta_{H^\pm} = 50$ GeV and heavy DM mass (right) with $\delta_A = \delta_H^\pm = 1 {\,\mbox{GeV}}$. Red bound: region in agreement with WMAP. Grey area: excluded by WMAP. $\delta_{A,\pm} = M_{A,H^\pm}-M_H$.[]{data-label="midOmega"}](RggWMAPopen-eps-converted-to.pdf){width="0.9\linewidth"}
![Comparison of the values of ${R_{\gamma\gamma}}$ and region allowed by the relic density measurements for the medium DM mass with (left) $HH$ invisible channel open and $M_A = M_{H^\pm} = 120$ GeV, (central) with $HH$ invisible channel closed and $\delta_A = \delta_{H^\pm} = 50$ GeV and heavy DM mass (right) with $\delta_A = \delta_H^\pm = 1 {\,\mbox{GeV}}$. Red bound: region in agreement with WMAP. Grey area: excluded by WMAP. $\delta_{A,\pm} = M_{A,H^\pm}-M_H$.[]{data-label="midOmega"}](RggWMAPclosed-eps-converted-to.pdf){width="0.9\linewidth"}
![Comparison of the values of ${R_{\gamma\gamma}}$ and region allowed by the relic density measurements for the medium DM mass with (left) $HH$ invisible channel open and $M_A = M_{H^\pm} = 120$ GeV, (central) with $HH$ invisible channel closed and $\delta_A = \delta_{H^\pm} = 50$ GeV and heavy DM mass (right) with $\delta_A = \delta_H^\pm = 1 {\,\mbox{GeV}}$. Red bound: region in agreement with WMAP. Grey area: excluded by WMAP. $\delta_{A,\pm} = M_{A,H^\pm}-M_H$.[]{data-label="midOmega"}](RggWMAPclosed){width="0.9\linewidth"}
#### Medium DM mass: invisible decay channels open
We first consider a case with $M_A = M_{H^\pm} = 120$ GeV and $M_h/2>M_H>50 {\,\mbox{GeV}}$. Red bound in the left panel of figure \[midOmega\] denotes the WMAP-allowed range of ${\Omega_{DM}h^2}$. If we consider $H$ as a subdominant DM candidate with $\Omega_H h^2 < {\Omega_{DM}h^2}$ then also the regions below and above red bounds in figure \[midOmega\] are allowed. This usually corresponds to larger values of $\lambda_{345}$. For a large portion of the parameter space limits for $\lambda_{345}$ from ${R_{\gamma\gamma}}$, even for the least stringent case ${R_{\gamma\gamma}}>0.7$, cannot be reconciled with the WMAP-allowed region, where $|\lambda_{345}| \sim 0.1$, excluding $M_H\lesssim 53 {\,\mbox{GeV}}$.
#### Medium DM mass: invisible decay channels closed
Here we choose $\delta_{H^\pm} = \delta_A = 50$ GeV and $M_H$ varying between $M_h/2$ and $83$ GeV. Figure \[midOmega\] (central panel) gives the WMAP-allowed range with the corresponding values of ${R_{\gamma\gamma}}$. The absolute values of $\lambda_{345}$ that lead to the proper relic density are in general larger than in the case of $M_H<M_h/2$. From figure \[midOmega\] it can be seen that this region of $M_H$ is consistent with ${R_{\gamma\gamma}}<1$, and that ${R_{\gamma\gamma}}>1$ and $\Omega_{DM} h^2$ constraints cannot be fulfilled for the middle DM mass region. If the IDM is the source of all DM in the Universe and $M_H \approx(63-83)$ GeV then the maximal value of ${R_{\gamma\gamma}}$ is around $0.98$. A subdominant DM candidate, which corresponds to larger $\lambda_{345}$, is consistent with ${R_{\gamma\gamma}}>1$.
#### Heavy DM mass: almost degenerated particle spectra
In this case it is possible to get ${R_{\gamma\gamma}}>1$ and be with agreement with WMAP, as shown in right panel in figure \[midOmega\] for $M_H \gtrsim 500$ GeV and $\delta_A = \delta_\pm = 1 {\,\mbox{GeV}}$, although deviation from ${R_{\gamma\gamma}}=1$ is very small.
Summary
=======
The DM candidate from the IDM is consistent with the WMAP results on the DM relic density and in three regions of masses it can explain 100 % of the DM in the Universe. In a large part of the parameter space it can also be considered as a subdominant DM candidate. Measurements of the diphoton ratio ${R_{\gamma\gamma}}$ done at the LHC set strong limits on masses of the DM and other dark scalars, and their self-couplings.
We can exclude *the low DM mass region* in the IDM, i.e. $M_H \lesssim 10$ GeV, as values of $|\lambda_{345}|$ needed for the proper $\Omega_{DM} h^2$ are an order of magnitude larger than those allowed by assuming that ${R_{\gamma\gamma}}>0.7$. In *the medium mass region* ${R_{\gamma\gamma}}>1$ favours degenerated $H$ and $H^\pm$. When the mass difference is large, $\delta_{H^\pm} \approx 50$ GeV, then values of $|\lambda_{345}|$ that provide ${R_{\gamma\gamma}}>1$ are bigger than those allowed by WMAP. We conclude it is not possible to have ${R_{\gamma\gamma}}>1$ and all DM in the Universe explained by the IDM in the medium DM mass region. If ${R_{\gamma\gamma}}>1$ then $H$ may be a subdominant DM candidate. If ${R_{\gamma\gamma}}<1$ then $M_H\approx (63-80)$ GeV can explain 100% of DM in the Universe. For *heavy DM particles* it is possible to obtain ${R_{\gamma\gamma}}>1$ and fulfill WMAP bounds, although deviation from ${R_{\gamma\gamma}}=1$ is small.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the grant NCN OPUS 2012/05/B/ST2/03306 (2012-2016).
References {#references .unnumbered}
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|
---
abstract: 'Reasoned by its dynamical behavior, the memristor enables a lot of new applications in analog circuit design. Since some realizations are shown (e.g. 2007 by Hewlett Packard), the development of applications with memristors becomes more and more interesting. Whereas most of the research was done in the direction of memristor applications in neural networks and storage devices, less publications deal with practical applications of analog memristive circuits. But this topic is also promising further applications. Therefore, this article proposes a frequency dependent rectifier memristor bridge for different purposes (e.g. using as a programmable synaptic membrane voltage generator for Spike-Time-Dependent-Plasticity) and describes the circuit theory. In this context it is shown that the Picard Iteration is one possibility to solve the system of nonlinear state equations of memristor circuits analytically. An intuitive picture of how a memristor works in a network in general is given as well. In this context some research on the dynamical behavior of a HP memristor should be done.'
author:
- 'Oliver Pabst, Torsten Schmidt'
bibliography:
- 'Quellen.bib'
title: 'Sinusoidal analysis of memristor bridge circuit: rectifier for low frequencies'
---
Introduction {#Memsysteme}
============
usage of memristors in analog circuit design enables new applications. In [@pershin2011analog], an ADC consisting of memristors has been proposed. Another applications are an automatic gain control circuit [@wey2010automatic] ,programmable analog circuits [@pershin2009practical], an electrical potentiometer [@pershin2009practical] or oscillators [@bahgat2012memristor; @talukdar2011memristor]. In [@merrikh2011memristor] memristors are used for basic arithmetic operations. In this paper a frequency dependent rectifier memristor bridge is presented. Therefore, a general description of memristive systems will be given first. Then the HP memristor is presented and some analysis about its dynamical behavior will be shown in \[Time\_response\] and in \[Properties\] properties of memristors are given. In \[Graetz Schaltung mit Memristoren\], the memristor bridge is presented, which is solved analytically by Picard Iteration in\[PicardGraetz\]. Finally, the results of the present investigations are summarized.
In General, a memristive system is described by the two equations [@di2009circuit] $$V(t) = M(\underline{x},I,t)\cdot I(t)$$ $$\frac{\mathrm d\underline{x}}{\mathrm dt}=\underline{f}(\underline{x},I,t)
\label{Zustandsgleichung}$$ where $V(t)$ is the applied voltage and $I(t)$ is the current through the device. $M(\underline{x},I,t)$ is called memristance and relates voltage and current. Thus, similar to a linear resistor its dimension is Ohm. $\underline{x}$ is a vector of the dimension n which consists of the internal state variables. The dynamics of this vector is specified by the state equation \[Zustandsgleichung\].
HP memristor
============
The HP memristor was realized in 2007 by a team of researchers of Hewlett Packard around Stanley Williams [@drakakis2010memristors; @strukov2008missing]. While this memristor is passive, latest realization of a memristor is an active one on the base of niobium oxide [@pickett2012sub]. The HP memristor is made of a titanium dioxide layer which is located between two platinum electrodes. This layer is in the dimension of several nanometers and if there is an oxygen dis-bonding, its conductance will rise instantaneously. However, without doping, the layer behaves as an isolator. The area of oxygen dis-bonding will be referred to space-charge region and changes its extension if an electrical field is applied. This is done by drifting the charge carriers. The smaller the insulating layer the higher the conductance of the memristor. Also, the tunnel effect plays a crucial role. Without an external influence there is no change in the extension of the space-charge region. The internal state $x$ is the normed extent of the space-charge region and can be described by the equation $$x:=\frac{w}{D}, \ 0\le x\le 1, \ x\in \mathbb{R}$$ where $\omega$ is the absolute extent of the space-charge region and $D$ is the absolute extent of the titanium dioxide layer. The memristance can be described by the following equation [@strukov2008missing]: $$M(x)={\mathrm{R_{on}}}\cdot x+{\mathrm{R_{off}}}\cdot (1-x) \ .
\label{HPMemristanz}$$ ${\mathrm{R_{on}}}$ is the resistance of the maximum conducting state and ${\mathrm{R_{off}}}$ represents the opposite case. The vector of internal state of the HP-memristor is one dimensional. That is why the scalar notation is used. The state equation is $$\frac{\mathrm dx}{\mathrm dt}=f(x,I,t)=\frac{\mu_v\cdot {\mathrm{R_{on}}}}{D^2}\cdot {\mathrm{I_M}}(t)
\label{Zustandsgleichung1}$$ where $\mu_v$ is the oxygen vacancy mobility and $I_M(t)$ is the current through the device. The following quantizations are used for memristors in this paper. $$\begin{aligned}
{\mathrm{R_{on}}}&=100\, \Omega, & \mu_v&=10^{-14}\, \frac{m^2}{s\cdot V}\\
{\mathrm{R_{off}}}&=16\, k\Omega, & D&=10^{-8} \, m \ .\end{aligned}$$ These values are given as an example by the Hewlett Packard Team. The ratio of maximum and minimum value of the memristance can be as large as e.g. $160$.
Window function
---------------
The extension of space-charge region is physically limited. Therefore, a window function shall be established to take account of saturation which is needed for a more realistic model. Several shapes of window functions are possible e.g. [@biolek2009spice]. In the following, the window function $${\mathrm{f_W}}(x)=\left\{\begin{array}{cl} \left( 4\cdot x\cdot (1-x) \right.)^{\frac{1}{4}}, & \mbox{if}\ 0< x < 1 \\ 0, & \mbox{else} \end{array} \right.$$ is used [@joglekar2009elusive]. The current mathematical description of this window function shows a weakness in numerical calculations. Excluding external influences, in the case of $\{ x\le 0$ or $x\ge 1\}$ the window function will be 0. Therefore, a state change of the system will not be possible. But considering ${\mathrm{V_M}}$, the voltage over the memristor, for the two cases $x \le 0$ and ${\mathrm{V_M}}>0$ or $x \ge 1$ and ${\mathrm{V_M}}<0$, a state change should be possible. The window function will result in: $${\mathrm{f_W}}(x,{\mathrm{V_M}})=\left\{\begin{array}{cl} (4\cdot x\cdot (1-x))^{\frac{1}{4}}, & \mbox{if}\ \epsilon< x < 1-\epsilon \\ & \mbox{or}\ x\le \epsilon \ and \ {\mathrm{V_M}}>0 \\ & \mbox{or}\ x\ge 1-\epsilon \ and \ {\mathrm{V_M}}<0 \\ 0, & \mbox{else} \end{array} \right. \ .$$ A better stability for numerical simulations is achieved by introducing $\epsilon$ and therefore an extended restriction of definition area. For all simulations in this paper $\epsilon$ equals $0.02$.
Polarity of the HP memristor {#Symboldefinition}
----------------------------
An agreement concerning the memristor polarity will be introduced. The extension of the space-charge region is going together with a decrease of the memristance and vice versa. The polarity determines the behavior of the memristor. According to the direction of the applied field, if the memristance decreases the memristor will be “forward biased". The other direction should be called “reverse biased". Because of the minimum resistance and the time lag, both definitions are main intuitive. For a memristor the algebraic sign of ${\mathrm{V_M}}$ equals that of ${\mathrm{I_M}}$ and the polarity just depends on it. Including the window function and the algebraic signs, Eq. \[Zustandsgleichung1\] extends to $$\frac{\mathrm dx}{\mathrm dt}=\frac{\mu_v\cdot{\mathrm{R_{on}}}}{D^2} \cdot {\mathrm{f_W}}\big(x,+\left(-\right){\mathrm{V_M}}\big)\cdot \big(+\left(-\right){\mathrm{I_M}}(t)\big)$$ if the memristor is forward (reverse) biased.
Time response of a HP memristor {#Time_response}
===============================
Change of the internal state
----------------------------
The calculations of this part will be done by neglecting the window function. The aim is to investigate the time behavior of an AC sine current source ${\mathrm{I_S}}$ which is connected in series with a HP memristor. Regarding the state equation (\[Zustandsgleichung1\]) $$\frac{\mathrm dx}{\mathrm dt}=\underbrace{\frac{\mu_v\cdot {\mathrm{R_{on}}}}{D^2}}_{=const}\cdot \underbrace{I_0\cdot \sin(\omega\cdot t)}_{{\mathrm{I_S}}={\mathrm{I_M}}} \ ,$$ dividing of the variables on both sides $$\int_{{\mathrm{x_{0}}}}^{{\mathrm{x_{0}}}+\Delta x}{dx'}=const\cdot I_0\cdot \int_{{\mathrm{t_{0}}}}^{{\mathrm{t_{0}}}+\Delta t}{\sin(\omega\cdot t')dt'}, \ \Delta t\ge 0$$ and performing integration leads to $$\Delta x=-\frac{I_0\cdot const}{\omega}\cdot \Big(\ cos\big(\omega\cdot ({\mathrm{t_{0}}}+\Delta t)\big)-cos(\omega\cdot {\mathrm{t_{0}}})\Big) \ .
\label{Sinusstromquelle_Aenderungx}$$ whereas ${\mathrm{x_{0}}}:=x({\mathrm{t_{0}}})$. At $({\mathrm{t_{0}}}+\Delta t)$ the extension of the space-charge region is $({\mathrm{x_{0}}}+\Delta x)$. In the case of a memristor which is connected in series with a sine current source, the change of the space charge region $\Delta x$ does not depend on the initial state ${\mathrm{x_{0}}}$ (see Eq. \[Sinusstromquelle\_Aenderungx\]). There will be an other result if there is a sine voltage source instead. In this case the state equation is $$\frac{\mathrm dx}{\mathrm dt}=\underbrace{\frac{\mu_v\cdot{\mathrm{R_{on}}}}{D^2}}_{=const}\cdot \underbrace{\frac{V_0\cdot \sin (\omega\cdot t)}{M(x)}}_{{\mathrm{I_M}}} \ .$$ and because of $M(x)$ it leads by integration to a quadratic equation. The relevant solution of this quadratic equation for $\Delta x$ with ${\mathrm{t_{0}}}=0$ is $$\Delta x=\left(r-{\mathrm{x_{0}}}\right)-\sqrt{\left(r-{\mathrm{x_{0}}}\right)^2+b\cdot \big(\cos(\omega\cdot\Delta t)-1\big)}
\label {Deltax}$$ whereas $r:=\frac{{\mathrm{R_{off}}}}{{\mathrm{R_{off}}}-{\mathrm{R_{on}}}}$ and $b:=\frac{2\cdot V_0\cdot const}{\omega\cdot ({\mathrm{R_{off}}}-{\mathrm{R_{on}}})}$. The second solution is irrelevant because for $\Delta t=0$ the change of the space charge region $\Delta x$ has to be zero. Reasoned by the disregard of the window function, the usage of equation (\[Deltax\]) is bounded because the solution has to be within the physical limits. But the bottom line is that $\Delta x$ depends on the initial state ${\mathrm{x_{0}}}$, if there is a voltage source in series with the memristor. As shown before, for a supplied current source it does not. Fig. \[MemStrom\](a) illustrates these insight. Since the memristance is determined by the internal state, the change of the memristance behaves similarly. As you can see in figure \[MemStrom\](b) for a sine voltage source the change of the memristance depends strongly on the internal memristance, however for a sine current source it does not. Note, in the case of a sinusoidal signal, the maximum change of the space charge region $\Delta x_{max}$ will be reached at the end of one half period. This is because the direction of the change of the space charge region depends on the algebraic sign of the external signal.
Highest frequency for fully state change {#maximale Frequenz}
----------------------------------------
To estimate the dynamical behavior of the HP-memristor in circuits the frequency ${\mathrm{f_{cut}}}$ for sinusoidal signals should be introduced. This is the highest frequency for which a memristor will be able to change from lowest to highest memristance or vice versa. It means that this state change completes exactly at the end of one half period ($\Delta{\mathrm{t_{cut}}}= 0.5\cdot T_{cut}$). Note, for the used memristor model and same conditions, the required time for changing from a state A to a state B is the same as for the reverse process. ${\mathrm{f_{cut}}}$ depends on the amplitude of the supplied source. Neglecting the window function the calculations should be performed for a single memristor in series with a supplied sine voltage source. This example allows rough estimates for circuits which are more complex. Taking into account that $\omega$ equals $2\cdot \pi\cdot f$ and $T$ equals $\frac{1}{f}$. Integration of Eq. \[Deltax\] and solving for $f$ with $t=0$ leads to $${\mathrm{f_{cut}}}=\frac{1}{\pi}\cdot\frac{V_0\cdot const}{\left(\frac {\Delta x^2}{2}+{\mathrm{x_{0}}}\cdot \Delta x\right)\cdot\left({\mathrm{R_{on}}}-{\mathrm{R_{off}}}\right)+\Delta x\cdot{\mathrm{R_{off}}}}$$ whereat ${\mathrm{f_{cut}}}$ is direct proportional to the amplitude $V_0$. : Considering the restricted domain of $x$ the maximal frequency for switching from highest memristance $x(t=0)=0.02$ to lowest memristance $x(t=\frac{T}{2})=0.98$ should be calculated. Therefore $\Delta x$ is equal $0.96$. Calculating by using $V_0=30\,V$ leads to $ {\mathrm{f_{cut}}}\approx 12.35675\, Hz$. Testing by solving the state equation numerically leads to following results:\
$x(t=\frac{1}{2\cdot f})=0.98$ for $f=12.35675\,Hz$\
$x(t=\frac{1}{2\cdot f})=0.9424$ for $f=12.4\, Hz$\
If the frequency is higher than ${\mathrm{f_{cut}}}$, saturation wouldn’t be reached. Considering the window function, the result of solving the state equation by simulation is $x(t=\frac{1}{2\cdot f})=0.98$ for $f=10.75\, Hz$. Therefore the calculations without window function are reasonable for rough estimates. Note, for $f> {\mathrm{f_{cut}}}$ if the frequency is increasing the ratio of maximum and minimum value of the memristance will decrease. Note, for a supplied sine current source the calculation for ${\mathrm{f_{cut}}}$ is also possible (Conversion of Eq. \[Sinusstromquelle\_Aenderungx\]). As an example, for an amplitude $I_0=\frac{30 \, V}{{\mathrm{R_{off}}}}=1.9 \, mA$, ${\mathrm{f_{cut}}}$ is about $6.2 \, Hz$.
Properties {#Properties}
==========
To summarize, the value of memristance depends on the current load of the past [@pershin2011memory]. If there is no current flow, the internal state will be retained. From this it follows that a memristor acts like a nonvolatile memory, whereas the range of values is continuous [@chua2011resistance; @sinha2011evolving]. That is an interesting fact comparable to transistor memory technology. The difference between highest and lowest possible memristance is relative large [@strukov2008missing]. Indeed in the true sense, the memristor is no switch but it could be used for switching operations. Therefore, depending on the direction and the benchmark the memristor passes higher potentials and locks for lower ones. Subject to the time shift the behavior of the memristor is frequency dependent. For $\lim_{f \rightarrow \infty}$ it behaves like a linear resistor [@di2009circuit] because the change of the internal state couldn’t follow the rapid voltage change. For sufficiently small frequencies the nonlinearities are dominating whereas the time shift is direct proportional to the amplitude of the signal.
Memristor bridge circuit {#Graetz Schaltung mit Memristoren}
========================
This circuit (Fig. \[Greatz\]) contains four HP memristors, one AC voltage source and one load resistor. Per definition for a positive voltage, memristors $M_{1,4}$ are forward biased, while $M_{2,3}$ are reverse biased. In [@kimmemristor] and [@Cohen2012harmonics] this circuit is also been presented, but there are differences in application. While this paper deals with periodic signals and their specifics at different frequencies, in [@kimmemristor] pulses are used for synaptic weight programming. In [@Cohen2012harmonics] the focus lies on generation of nth-order harmonics and the effect of frequency doubling by using this circuit.
Mathematical describtion {#Knotenspannungen}
------------------------
![[*Schematic of the memristor bridge circuit.*]{}[]{data-label="Greatz"}](memristorbridge.pdf){width="\textwidth"}
$$\begin{aligned}
{\mathrm{V_M}}_1&={\mathrm{V_S}}-V_{10}\\
{\mathrm{V_M}}_2&=V_{10}\\
{\mathrm{V_M}}_3&={\mathrm{V_S}}-V_{20}\\
{\mathrm{V_M}}_4&=V_{20}\\
{\mathrm{V_{RL}}}&=V_{10}-V_{20}$$
At this point the new notation for the memristance $$M(x_n):=M_n, \ n\in \mathbb{N}$$ should be established. The voltages $$V_{10}:=V_{10}(x_1,x_2,x_3,x_4)={\mathrm{V_S}}\cdot \frac{{\mathrm{Num}}+M_2\cdot M_3 \cdot {\mathrm{R_L}}}{{\mathrm{Den}}}$$ $$V_{20}:=V_{20}(x_1,x_2,x_3,x_4)={\mathrm{V_S}}\cdot \frac{{\mathrm{Num}}+M_1\cdot M_4 \cdot {\mathrm{R_L}}}{{\mathrm{Den}}}$$ are defined from the nodes to ground, whereas the notations $$\begin{aligned}
{\mathrm{Num}}\ :=& \ (M_1+M_3+{\mathrm{R_L}}) \cdot M_2\cdot M_4 \\
{\mathrm{Den}}\ :=& \ \big(M_1+M_2\big)\cdot \big(M_3\cdot M_4+M_3\cdot {\mathrm{R_L}}+M_4\cdot {\mathrm{R_L}}\big) \notag \\ \ &+ M_1\cdot M_2 \cdot \big(M_3+M_4\big) \ .\end{aligned}$$ are used for a better overview. The structure of the circuit implies a nonlinear system of differential equations of the fourth order. The four state equations are $$\begin{aligned}
\frac{\mathrm dx_n}{\mathrm dt}=\frac{\mu_v\cdot{\mathrm{R_{on}}}}{D^2}\cdot {\mathrm{f_W}}(x_n,\pm {\mathrm{V_M}}_n)\cdot \underbrace{\pm \frac{{\mathrm{V_M}}_n}{M(x_n)}}_{{\mathrm{I_M}}_n}, \ & n \le 4 \ .\end{aligned}$$ For $x_{1,4}$ the sign is “$+$" and for $x_{2,3}$ the sign is negative.
Simulation results and functionality
------------------------------------
Using these state equations, numerical simulations are possible. From this point ${\mathrm{V_S}}$ should be a sine AC voltage source and $R_L=1 \, k\Omega$.
Regarding Fig. \[URL-Graetz-f-0,5\], dependent on the frequency a qualitative difference for the voltage over the load ${\mathrm{V_{RL}}}$ is detectable. For low frequencies (represented by $f=0.5\, Hz$ in the case of $V_0=30\, V$) the output signal is almost exclusive positive, comparable with a rectifier circuit. Therefore, the frequency of the output signal is twice as high as the one of the input signal. Reasoned by the time shift of the change of state, which is subject to a HP memristor, there are temporary short negative peaks. Using a higher excitation frequency (represented by $f=30\, Hz$ for $V_0=30\, V$) there is no rectifier function detectable.
By means of figure \[MemristanceM2-Graetz-f-0,5\], this frequency selective behavior can be better understood. For low frequencies ($f\le {\mathrm{f_{cut}}}$), a complete state change will happen, the memristors are going into saturation. Beginning with an initial value $M_{1,4}$ is decreasing, while $M_{2,3}$ is increasing until saturation is reached. When the sine voltage source reaches its negative half period the process will be reversed until saturation is reached again and so on. During one period there is a change for the relational operator between both memristances. So every half period, the potential in node one is going to be higher than the potential in node two, which implies that ${\mathrm{V_{RL}}}$ is always going to be positive. For $f>{\mathrm{f_{cut}}}$, the memristors do not completely switch from lowest to highest memristance. It follows that they are not going into saturation and the memristors will just reach the initial state after one period. So there is no change for the relational operator between $M_{1,4}$ and $M_{2,3}$ during one period and the relation of the memristances just depend on the initial conditions (See figure \[MemristanceM2-Graetz-f-0,5\](b)). For example, if the initial states are equal for all four memristors, then $M_{1,4}$ would be less than $M_{2,3}$. So ${\mathrm{V_{RL}}}$ would be positive for the first half period too. But at the second half period, the Memristances $M_2$ and $M_3$ keep higher than $M_4$ and $M_1$. So the potential at node “2" is higher than at node “1", which implies that ${\mathrm{V_{RL}}}$ is negative. There are two reasons why the absolute value for the amplitude of $V_{RL}$ decreases for increasing frequency. On the one hand for equal initial conditions the maximum potential difference between node “1" and “2" decreases by increasing the frequency. On the other hand, dependent on the initial conditions, there is may a higher voltage drops across the memristors. Note, the maximum of the amplitude of $V_{RL}$ is not at $t=\frac{T}{4}$. At this time the amplitude of the sine source begins to decrease, but the change of the memristor states is continuing, which has firstly more impact.
The lower the frequencies the more the circuit behaves like a Graetz circuit. For very high frequencies the circuit is going to behave like a Wheatstone bridge. This can be proven by the voltage divider. For $\lim_{{\mathrm{R_L}}\rightarrow \infty}$ it simply is $${\mathrm{V_{RL}}}=V_{10}-V_{20}={\mathrm{V_S}}\cdot \left(\frac{M_2}{M_1+M_2}-\frac{M_4}{M_3+M_4}\right) \ ,$$ and had to be zero for equal resistances, if the conclusion is true.
Simulations illustrate that for e.g. $f=50\, kHz$ and $x_n(0)=0.5$ the amplitude of $V_{RL}$ equals $0.5 \, mV$, while the ratio between highest and lowest memristance of e.g. $M(x_1)$ equals $1.0002$. Using a supply periodic square wave voltage source with $${\mathrm{V_S}}= sgn( \sin(\omega\cdot t)).$$ is another interesting example. The results for this are shown in Fig. \[rechteck\]. A frequency dependent behavior is also detectable. For higher frequencies the voltage over the load resistor is serrated. For low frequencies this voltage is almost constant with negative peaks.
The condition that the initial states of all memristors are equal leads to the in Fig. \[rechteck\] presented behavior. But what happens if the circuit is supplied by a periodic square wave voltage and the initial states are not equal? As it is described in chapter \[Time\_response\] and shown in Fig. \[Spiketime\](a), for a supplied periodic voltage and higher frequencies the variation of memristance depends on the initial conditions. So the voltage curve changes from the saw tooth form. As represented in Fig. \[Spiketime\](b) it is possible to create a curve which is similar to synaptic membrane voltages which are used for Spike-Time-Dependent-Plasticity [@linares2009memristance]. The amplitude of the voltage varies for different initial conditions, which could be set by a low frequency or DC voltage.
Possible applications {#Moegliche Anwendungen}
---------------------
Depending on the frequency, there are two significant species. Fusing a common Graetz circuit and a Wheatstone circuit in one device is one application possibility. Replacing the independent source with a controlled one leads to a further application. The source is controlled by the load voltage and its frequency is variable. Keeping in mind, the amplitude of the output signal decreases by increasing the frequency. Because of that, a frequency controlled regulator is conceivable. If the amplitude exceeds a predefined value then the frequency should be increased to prevent a further amplification. Using the circuit as a saw tooth generator is also possible. Like mentioned before, the presented circuit also could be used as a programmable synaptic membrane voltage generator for Spike-Time-Dependent-Plasticity.
Picard Iteration {#PicardGraetz}
================
At this point, the Picard Iteration should be introduced as a possibility to solve common memristive systems analytically. One advantage related to e.g. the Volterra-series expansion is that the Picard Iteration converges more rapidly. In this chapter, the application possibility of this Iteration should be shown on the memristor bridge circuit. In favor, the general description of the Picard Iteration should be given first: A nonlinear, dynamical system with $$\frac{\mathrm dx}{\mathrm dt}=f\big( t,x(t)\big), \ x(t_0)=x_0$$ is given, whereas $x(t)$ is Lipschitz continuous and an element of a Banach space. Then, the iteration is given by $$x^{[k+1]}:=x_0+\int_{t_0}^{t}{f\big(t^{'},x^{[k]}(t^{'})\big)\cdot \mathrm dt^{'}}, \ t\in \{t_0,t_0+\epsilon\} \ .$$ whereas $k$ denotes the order of the iterative steps.
In the case of the memristor bridge circuit, $x(t)$ is the normed extent of the space-charge region. The iteration should be performed without considering the window function. The window function suppresses changes of the state equation within the marginal area of the memristor. Therefore for $x_n^{[l+1]}$ the argument of the window function would be of the same iteration step. This would lead to a recursive mathematical expression and shows one weakness of the Picard Iteration. The iteration should be only performed for higher frequencies, thus, the window function can be neglected. By setting the initial conditions it should be noted that the internal states stay within the domain. Attention should be paid to the case $x>1$ because the memristance would become negative. For memristive systems using the HP memristor model, the Picard iteration could be performed by $$x_n^{[k+1]}:=x_0+\int_{t_0}^{t}{\frac{\mu_v\cdot {\mathrm{R_{on}}}}{D^2}\cdot \frac{{\mathrm{V_M}}_n^{[k]}}{M(x_n^{[k]})} \cdot \mathrm dt^{'}}, \ t\in \{t_0,t_0+\epsilon\}
\label{Picard2}$$ with ${\mathrm{V_M}}_n^{[k]}={\mathrm{V_M}}_n(t,x_1^{[k]},x_2^{[k]},...,x_n^{[k]})$. Similar to chapter \[Graetz Schaltung mit Memristoren\] the used notation for the memristance $M(x_n^{[k]})$ is $M_n^{[k]}$ with $n,k\in \mathbb{N}$. ${\mathrm{Num}}^{[k]}$ and ${\mathrm{Den}}^{[k]}$ should established as well. The only difference to their analogons in chapter \[Graetz Schaltung mit Memristoren\] is the usage of $M(x_n^{[k]})$ instead of $M(x_n)$.\
: Using the initial states $x^{[1]}_n=x_n(0)$, $\frac{\mathrm dx^{[1]}_n}{\mathrm dt}=0$ as arguments for the first iteration step, then the equations for the currents are to be $$\begin{aligned}
{\mathrm{I_M}}_1^{[1]} \ & ={\mathrm{V_S}}\cdot \underbrace{\frac{1}{M_1^{[1]}}\cdot \Bigg(1-\frac{{\mathrm{Num}}^{[1]}+M_2^{[1]}\cdot M_3^{[1]} \cdot {\mathrm{R_L}}}{{\mathrm{Den}}^{[1]}}\Bigg)}_{{\mathrm{M_{R}}}_1^{[1]}} \\
{\mathrm{I_M}}_2^{[1]} \ & ={\mathrm{V_S}}\cdot \underbrace{\frac{1}{M_2^{[1]}}\cdot \frac{{\mathrm{Num}}^{[1]}+M_2^{[1]}\cdot M_3^{[1]} \cdot {\mathrm{R_L}}}{{\mathrm{Den}}^{[1]}}}_{{\mathrm{M_{R}}}_2^{[1]}} \\
{\mathrm{I_M}}_3^{[1]} \ & ={\mathrm{V_S}}\cdot \underbrace{\frac{1}{M_3^{[1]}}\cdot \Bigg(1-\frac{{\mathrm{Num}}^{[1]}+M_1^{[1]}\cdot M_4^{[1]} \cdot {\mathrm{R_L}}}{{\mathrm{Den}}^{[1]}}\Bigg)}_{{\mathrm{M_{R}}}_3^{[1]}} \\
{\mathrm{I_M}}_4^{[1]} \ & ={\mathrm{V_S}}\cdot \underbrace{\frac{1}{M_4^{[1]}}\cdot \frac{{\mathrm{Num}}^{[1]}+M_1^{[1]}\cdot M_4^{[1]} \cdot {\mathrm{R_L}}}{{\mathrm{Den}}^{[1]}}}_{{\mathrm{M_{R}}}_4^{[1]}} \ .\end{aligned}$$ : $$\frac{\mathrm dx^{[2]}_n}{\mathrm dt}=\pm const\cdot V_0 \cdot \sin(\omega \cdot t) \cdot {{\mathrm{M_{R}}}_n^{[1]}}$$ By dividing of the variables this equation is $$\int_{x^{[1]}_n}^{x^{[2]}_n}{\mathrm dx^{'}}=\pm const\cdot V_0 \cdot {\mathrm{M_{R}}}_n^{[1]}\cdot \int_{0}^{t}{\sin(\omega \cdot t^{'}) \mathrm dt^{'}}$$ and leads by integration to $$x^{[2]}_n=x^{[1]}_n\pm \frac{const\cdot V_0}{\omega} \cdot {\mathrm{M_{R}}}_n^{[1]}\cdot (1-\cos(\omega \cdot t))$$ whereat for $x^{[2]}_{1,4}$ the sign is “$+$" and for $x^{[2]}_{2,3}$ the sign is negative. For this circuit for the second step the Picard Iteration gives a good quality solution (regarding Fig. \[Graetzpicard\]). Therefore no further iteration steps should be performed. The voltage over the load resistor can be solved by $$\begin{aligned}
{\mathrm{V_{RL}}}^{[2]} \ & ={\mathrm{V_S}}\cdot \frac{{\mathrm{R_L}}}{{\mathrm{Den}}^{[2]}}\cdot (M_2^{[2]}\cdot M_3^{[2]}- M_1^{[2]}\cdot M_4^{[2]}) \ .\end{aligned}$$
![[*${\mathrm{V_{RL}}}$ over time t, analytical and numerical solution.*]{}](pic.png "fig:"){width="\textwidth"} \[Graetzpicard\]
conclusions
===========
At this paper two similar circuits consisting of HP memristors are presented. For both, a frequency selective behavior is presented. In contrast to higher frequencies for low frequencies there is a rectifier like behavior. This functional change with frequency is reasoned by the delay of changing of the internal state caused by an external source. To estimate the dynamical behavior in circuits, the time behavior of a single memristor in series with a periodic source was investigated. The configuration of the circuits leads to a nonlinear system of differential equations which describes the internal states. Using the Picard Iteration is one possibility to solve this system analytically. The frequency selective behavior can be used to realize two modes in one circuit (Graetz and Wheatstone circuit). Another applications are a frequency controlled regulator or a programmable synaptic membrane voltage generator for Spike-Time-Dependent-Plasticity.
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abstract: 'Black hole entropy calculations which are based on counting of microstates and based on modified dispersion relations in the framework of loop quantum gravity are considered. We suggest that the inconsistency of two approaches can be explained by different ways. This inconsistency can affect the definition and constancy of the Immirzi parameter or order of the modification constants of dispersion relations. Possible results of these effects are discussed.'
address: 'Department of Physics, Ankara University, Faculty of Sciences, 06100, Tandoğan-Ankara, Turkey\'
author:
- 'Ö. Aç[i]{}k'
- 'Ü. Ertem'
title: Effect of Modified Dispersion Relations on Immirzi Parameter
---
Introduction
============
In Loop Quantum Gravity (LQG) [@Rovelli; @Thiemann], there is a free parameter $\gamma$, called Immirzi parameter that has no effect in the classical theory but has an effect in the quantum theory. So, it reflects the quantization ambiguity of the theory. LQG is based on a connection formulation of general relativity whose phase space variables are an $SU(2)$ connection and a densitized triad. However, there is a one parameter family of canonical transformations which lead to the same Hamiltonian formulation of general relativity. The parameter that labels the family of canonical transformations is the Immirzi parameter. Different values of $\gamma$ are reflected in the different forms of the Hamiltonian constraint. If $\gamma$ is selected as a complex number (more specifically the complex number $i$), then the Hamiltonian constraint is simpler than in the ADM formulation and quantization can be easier, but in this case, one has some extra reality constraints. The pure complexity of $\gamma$ is not a necessity and if it is selected as a real number, then the Hamiltonian constraint is more complicated, but it is still manageable for quantization and there is no reality constraints in this case [@Rovelli; @Thiemann; @RovThi]. However, in quantum theory $\gamma$ does not vanish and still present in the spectrum calculations of operators. In LQG, geometric operators such as area and volume have discrete eigenvalues and $\gamma$ appears in the spectrum of these operators. So, finding the value of $\gamma$ means determining the area and volume quantums. Hence, there is a quantization ambiguity for quantum gravity and choosing a value of the Immirzi parameter needs an explanation, namely it must be fixed by theoretical or experimental ways.
Immirzi parameter appears also in the computation of black hole entropy in the framework of LQG, because of the relation between the area and entropy of a black hole. So, one can determine the value of $\gamma$ by comparing black hole entropy-area relation found in LQG to the semiclassical Bekenstein-Hawking (BH) entropy-area relation [@ABCK]. Black hole entropy-area relation have been found by counting of microscopic states for a fixed area value, and besides the linear area term, there is also an $ln$ correction term in the found relation [@DomLew; @Meissner].
On the other hand, in LQG, some theoretical calculations reveal the presence of corrections to energy-momentum relations, so it implies some modifications of dispersion relations [@GamPul; @MorUrr; @MorUrr2; @Smolin]. This kind of modification effects may be observed at gamma ray and ultra high energy cosmic ray threshold anomalies [@MagSmo]. Modified dispersion relations can be understood in the framework of Deformed Special Relativity (DSR) which refers that there is an observer independent invariant energy scale, Planck energy, besides the invariant speed of light [@MagSmo; @AmSmSt; @KowNow; @KowGlik; @KowNow2; @FrKGSm; @KGSmo]. But modifications of dispersion relations implies some modifications to particle localization limit and this results some corrections to black hole entropy-area relation [@ACArzLM]. In this case also, there is an $ln$ correction term, but this time a term proportional to square root of area also exists. So, for consistency one must consider the entropy-area relation with correction terms in determining the Immirzi parameter $\gamma$. This indicates some possibilities like restrictions on modifications to dispersion relations, or different values of $\gamma$ for different scales.
In this paper, by comparing the two different approaches of finding black hole entropy in the framework of LQG, we discuss the explanations for the inconsistencies between the two approaches and find some possibilities about non-constancy of Immirzi parameter and orders of modification constants of dispersion relations. Organization of the paper is as follows. In section 2, we summarized the procedure of finding Immirzi parameter with counting microscopic states of a black hole in the framework of LQG. Section 3 discusses how the entropy-area relation can be modified with modification of dispersion relations. In section 4, we find an equation for $\gamma$ with considering modified entropy-area relation. This section also includes some limits to coefficients of modification terms of dispersion relations for the consistency with fixed $\gamma$. Some all order modifications of dispersion relations are also discussed for comparison with coefficient limits. In section 5, possible effects of scale dependence of $\gamma$ is argued and section 6 concludes the paper.
Immirzi Parameter from Black Hole Entropy
=========================================
Black holes in general relativity obey some laws that resemble the thermodynamical principles. In this sense, the area of the event horizon is related to entropy. The Bekenstein-Hawking formula gives the entropy of a black hole which is proportional to horizon area of the black hole $A$; $$\begin{aligned}
S=\frac{A}{4L_{p}^2}\end{aligned}$$ where $L_{p}$ is the Planck length. A quantum theory of gravity must provide a mechanism for microscopic states of a black hole which explains this entropy relation. In LQG framework, the fixed horizon area of a black hole $A$ can be obtained from different intersections of edges of a spin network with the horizon. Spin networks are the basis for kinematical Hilbert space of LQG and they are eigenstates of the geometric operators. That different possibilities of intersections constitute the microscopic states of a black hole. Edges of a spin network are labeled by $SU(2)$ representations $j=1/2,1,3/2,...$. So, different microscopic states represents the different intersections with different spin labels which result the same area value. Area operator has discrete eigenvalues and the area spectrum includes the Immirzi parameter $\gamma$; $$\begin{aligned}
A=8\pi\gamma L_{p}^2\sum_{i}\sqrt{j_{i}(j_{i}+1)}\end{aligned}$$ where the sum is over intersections. Thus $\gamma$ will appear in entropy-area relation and can be fixed by comparing with BH entropy for large area values.
Calculations about counting of microscopic states of a black hole has been achieved by several people [@DomLew; @Meissner]. The result is that entropy-area relation includes an $ln$ correction term; $$\begin{aligned}
S=\frac{\gamma_{0}}{\gamma}
\frac{A}{4L_{p}^2}-\frac{1}{2}\ln(\frac{A}{L_{p}^2})+O(\frac{L_{p}^2}{A})\end{aligned}$$ where $\gamma_{0}$ satisfies the equation; $$\begin{aligned}
\sum_{i}(2j_{i}+1) \exp(-2\pi\gamma_{0}\sqrt{j_{i}(j_{i}+1)})=1\end{aligned}$$ The solution of this equation can be found approximately as $\gamma_{0}=0.27398...$. By comparing with BH entropy for large $A/L_{p}^2$ values, it can be seen that $\gamma$ must be equal to $\gamma_{0}$. Fixing $\gamma$ means fixing the quantum of area, and one can find from (2) that minimum possible area value of a surface. But this determination of $\gamma$ is valid only for large area values. If it is also valid for small area values then there can not be correction terms rather than the $ln$ term to entropy in calculations for finding entropy-area relations by using different methods. But in the next section we will see that if modified dispersion relations are considered there is a correction term to entropy which is proportional to square root of area and this will effect the fixing of $\gamma$.
Black Hole Entropy from Modified Dispersion Relations
=====================================================
Several theoretical calculations about light propagation and neutrino propagation in LQG [@GamPul; @MorUrr; @MorUrr2; @Smolin] predict that the usual relation between energy and momentum which comes from special relativity, may be modified at Planck scales in the form of $$\begin{aligned}
E^2=p^2+m^2+\alpha_1 L_{p} E^3+\alpha_2 L_{p}^2 E^4+O(L_{p}^3 E^5)\end{aligned}$$ where $\alpha_1$ and $\alpha_2$ are constants of order one. This kind of modification of dispersion relations can be explained by alternative possibilities [@KGSmo]. Some of them are; (i) No effect of Planck scale phenomena can be observed in low energies and hence modification of dispersion relations has no results for observable phenomena, (ii) Lorentz invariance breaks down and there is a preferred frame at the Planck scale, (iii) Relativity of inertial frames maintained but Planck length or Planck energy becomes an observer independent quantity. This possibility is called Deformed Special Relativity (DSR). Experimentally, modification effects of dispersion relations may be observed by gamma ray and ultra high energy cosmic ray thresholds [@MagSmo].
Such a modification causes an effect to the Plank scale particle localization limit [@ACArzLM]. An absolute limit on the localization of a particle of energy is given by $E\geq\frac{1}{\delta x}$. But, if one considers (5), then particle localization limit can be found as follows; $$\begin{aligned}
E\geq \frac{1}{\delta x}-\alpha_1 \frac{L_p}{(\delta x)^2}+
(\frac{11}{8}\alpha_{1}^2-\frac{3}{2}\alpha_2)\frac{L_{p}^2}{(\delta
x)^3}+O(\frac{L_{p}^3}{(\delta x)^4}).\end{aligned}$$
The particle localization limit must be considered to derive the BH entropy-area relation. So if (6) is valid then black hole entropy relation will change because of modification terms. This has been calculated in [@ACArzLM] and found that modified entropy is $$\begin{aligned}
S\simeq\frac{A}{4L_{p}^2}+\alpha_1 \sqrt{\pi} \frac{\sqrt{A}}{L_p}
+(\frac{3}{2}\alpha_2 -\frac{11}{8}\alpha_{1}^2) \pi
\ln\frac{A}{L_{p}^2}.\end{aligned}$$ If both $\alpha_1$ and $\alpha_2$ are vanish then entropy is equal to BH entropy. If only $\alpha_1$ vanish then there is only the $ln$ correction term and that is consistent with the entropy corrections which are found from counting of microscopic states (which is mentioned in [@ACArzPro; @ACArzLM]), but this correspondence fixes the value of $\alpha_2$. Generally if $\alpha_1$ and $\alpha_2$ are different from zero then there is a correction term which is proportional to square root of area. From these discussions one can conclude that the $\alpha_1$ coefficient of modified dispersion relations must be zero, but we will see in the next section that this is not the only possibility. On the other hand, the existence of the square root area term will restrict the order of $\alpha_1$ because of the constancy of the Immirzi parameter.
Immirzi Parameter from Modified Black Hole Entropy
==================================================
We have seen that there are two manifestations of black hole entropy in the framework of LQG. One from counting of microstates and one from modification of dispersion relations. Both includes a leading term corresponding to BH entropy and an $ln$ correction term. But, while there is a square root of area term in the modification of dispersion relations approach, there is no such a term in the counting of microstates approach. For the consistency of the two approaches, these two entropy relations must be equal.
The second term of equation (3) and the third term of equation (7) are $ln$ correction terms, so we must equalize the coefficients of these terms and we find a relation between $\alpha_1$ and $\alpha_2$ modification constants of dispersion relations $$\begin{aligned}
12\alpha_2 - 11\alpha_{1}^2=-\frac{4}{\pi}.\end{aligned}$$ If there is no Planck order modification to dispersion relations namely $\alpha_1=0$, so two entropy relations are consistent, then $\alpha_2$ must be equal to $-\frac{1}{3\pi}$. On the other hand, if $\alpha_1 \neq 0$ then we must equalize the first term of (3) and the first two terms of (7), $$\begin{aligned}
\frac{\gamma_{0}}{\gamma}
\frac{A}{4L_{p}^2}=\frac{A}{4L_{p}^2}+\alpha_1 \sqrt{\pi}
\frac{\sqrt{A}}{L_p}.\end{aligned}$$ In this case we can not fix the value of Immirzi parameter to $\gamma_0$. By using (2) and the definition $\sum
\sqrt{j_{i}(j_{i}+1)}=J$, one can find that $\gamma$ is equal to $$\begin{aligned}
\gamma=[\frac{1}{\sqrt{2J}\gamma_0}(\alpha_{1}\pm\sqrt{\alpha_{1}^2+2J
\gamma_0})]^{-2}.\end{aligned}$$ So, $\gamma$ is dependent to $\alpha_1$, and it is also dependent to $J$, but this means that the value of the Immirzi parameter changes with the number of intersections of edges, hence with the scale determined by the area of the corresponding surface. However, if $J^{1/2}\gg\alpha_1$ then (10) transforms to $\gamma\simeq\gamma_0$, namely for the large area values $\gamma$ goes to $\gamma_0$. This is expected from the counting of microstates approach. But for the small values of $J$, $\gamma$ is changing, this is a contradiction with the constancy of $\gamma$. If it is constant, then it must be equal to same quantity for all area values. The smallest value of $J$ is comes from an edge with $j=1/2$ and it is $J_{min}=\sqrt{3}/2$. So, if $\gamma$ is a constant and is equal to $\gamma_0$, then $\alpha_1 \ll J_{min}^{1/2}$, and this means that $$\begin{aligned}
\alpha_1 \ll 1.\end{aligned}$$ So, for the constancy of $\gamma$ there is no need to $\alpha_1 =0$, but it must be much smaller than 1.
On the other hand, if $\alpha_1$ is order one, then $\gamma$ can not be a constant for all area values, and it changes with $J$ and $\alpha_1$. This means that for small area values, the area spectrum must have additional dependence on $j_i$’s and also depends on $\alpha_1$; $$\begin{aligned}
A=16\pi L_{p}^2 \gamma_{0}^2 (\frac{J}{\alpha_{1}\pm
\sqrt{\alpha_{1}^2+2J\gamma_0}})^2.\end{aligned}$$ For large area values that is $J^{1/2}\gg\alpha_1$ , (12) converges to (2) where $\gamma$ equals to $\gamma_0$. Then, in the small area regime, the area spectrum must depends on the different $\gamma$-sectors of the theory. Hence, if $\alpha_1$ is order one, then $\gamma$ will be a scale-dependent parameter, and its values are exactly determined by $\alpha_1$ and the scale of $J$.
Comparisons with Some All-Order Dispersion Relations
----------------------------------------------------
Some all-order modified dispersion relations have been considered in the frameworks of $\kappa$-Minkowski space-time and Deformed Special Relativity [@ACArzLM]. Various models predict different $\alpha_1$ and $\alpha_2$ coefficients. For consistency, this coefficients in the models must satisfy some requirements mentioned above.
In the framework of $\kappa$-Minkowski space-time, dispersion relations are given by $$\begin{aligned}
\cosh(E/E_p)-\cosh(m/E_p)-\frac{p^2}{2E_{p}^2}\exp(E/E_p)=0\nonumber.\end{aligned}$$ In this case $\alpha_1=-1/2$. So if this theory is true, then in the lack of $\sqrt{A}$ term in black hole entropy from counting of microstates, area spectrum must change with (12), namely depends on the different $\gamma$-sectors of the theory.
Another possibility for dispersion relations is given by $$\begin{aligned}
\cosh(\sqrt{2}E/E_p)-\cosh(\sqrt{2}m/E_p)-\frac{p^2}{E_{p}^2}\cosh(\sqrt{2}E/E_p)=0\nonumber.\end{aligned}$$ This case has $\alpha_1=0$ and $\alpha_2=-5/18$. By vanishing of $\alpha_1$, this is consistent with two different entropy calculations, but the value of $\alpha_2$ is inconsistent with predictions from consistency of entropy relations discussed above.
In the case of Deformed Special Relativity, dispersion relations are given by $$\begin{aligned}
\frac{E^2}{(1-E/E_p)^2}-\frac{p^2}{(1-E/E_p)^2}-m^2=0\nonumber.\end{aligned}$$ This is the case of both $\alpha_1$ and $\alpha_2$ vanish, but still there are some modifications to dispersion relations. Vanishing of $\alpha_2$ is inconsistent with $ln$ correction terms in entropy relation found from counting of microstates.
The true modification of dispersion relations can only be decided from experiments and observations which are mentioned in [@MagSmo]. Then, one can know the exact modification coefficients and compare the results with the consistency conditions mentioned above. On the other hand, if $\gamma$ is scale-dependent, then this must be observed by future measurements of different scale area values which then must have values of different $\gamma$-sectors of the quantum theory.
Possible Effects of Scale-Dependent Immirzi Parameter
=====================================================
In the presence of the first order Planck scale modifications to the dispersion relations, Immirzi parameter $\gamma$ must satisfy (10) for the consistency of entropy relations that are calculated by two different ways. This means that $\gamma$ depends on $J$ and has a scale dependence. Scale dependence of $\gamma$ affects the area spectrum and area eigenvalues must have an extra $J$ dependence. So, for small scales area spectrum changes with different $\gamma$ values, but for $J^{1/2}\gg\alpha_1$ area eigenvalues are only affected by multiplication with $\gamma_0$. These are also relevant for the spectrum of volume and length operators, since they also depend on $\gamma$, and are affected similarly by changing of $\gamma$.
On the other hand, $\gamma$ enters the classical theory by Holst’s modification of Hilbert-Palatini action; $$\begin{aligned}
S_H=-\frac{1}{32\pi G}\int(R_{ab}\wedge \ast
e^{ab}-\frac{\Lambda}{6}\ast 1-\frac{2}{\gamma}R_{ab}\wedge e^{ab})\end{aligned}$$ where $\ast$ is the Hodge star operator, $e^{a}$ is 1-form basis, $R_{ab}$ is curvature 2-form and $\Lambda$ is the cosmological constant. The last term can be written as a sum of a torsion square term and an exact term which is topological Nieh-Yan class $R_{ab}\wedge e^{ab}=T^{a}\wedge T_a-d(e^{a}\wedge T_a)$ [@CandZan]. Second term is a boundary term, so $\gamma$ controls the width of the fluctuations of the torsion [@FreidStar]. The mean value of torsion is zero (in the non-existence of matter), but it can fluctuate about the mean value. So, scale dependence of $\gamma$ means scale dependence of the width of fluctuations of torsion at the quantum level.
Coupling of spinors with (13) gives non-zero torsion and Immirzi parameter has an effect on non-minimal fermion interaction term. The coupling constant is dependent on $\gamma$ [@RovPer; @FrMiTa], but it is shown in [@Mercuri] that if the inverse of the coupling constant is equal to $\gamma$ then the last term of (13) and the non-minimal coupling term together turn to a boundary term. So in this case $\gamma$ has no effect on classical theory. In [@CTY], it is argued that coupling (16) with quadratic spinor Lagrangian indicates that $\gamma$ is the ratio between scalar and pseudo-scalar contributions in the theory. With scale-dependence of $\gamma$, this ratio also has scale dependence.
Another appearance of $\gamma$ is in Loop Quantum Cosmology (LQC) [@Bojowald]. Because of the volume operator has a dependence on $\gamma$, operator of the inverse scale factor $a^{-1}$ has also a dependence on $\gamma$. The density operator $d=a^{-3}$ can be constructed from the inverse scale factor [@Bojowald2]; $$\begin{aligned}
d_j(a)=a^{-3}p(\frac{3a^2}{\gamma L_{p}^2 j})^6\end{aligned}$$ where $p(q)$ is a function derived in [@Bojowald3]. If $a^2\ll
\frac{1}{3}\gamma L_{p}^2 j$ then $d_j(a)\sim a^{12}$ and if $a^2\gg
\frac{1}{3}\gamma L_{p}^2 j$ then $d_j(a)\sim a^{-3}$. This is a possible explanation for the inflationary phase in the early universe without using scalar fields. But if $\gamma$ changes with $j$ like (12) then (17) has an extra $j$ dependence and this may effect the early evolution of the universe in the framework of LQC.
Summary and Conclusion
======================
Immirzi parameter can be calculated from counting of microstates of a black hole by comparing the found entropy relation with the BH formula. In counting of microstates approach, the entropy has an $ln$ correction term. In large scales, this correction term is negligible and $\gamma$ is strictly equal to a number shown as $\gamma_0$. On the other hand, black hole entropy is also calculated from dispersion relations and modification of dispersion relations induces some modifications to BH entropy. But, in this case one has an additional correction term which is proportional to square root of the area besides the $ln$ correction term. For consistency, these two entropy relations found from different approaches must coincide. Comparing the two entropies indicates some possibilities about the Immirzi parameter and order of the modification constants of the dispersion relations. These possibilities are as follows:
- $\alpha_1$ must be zero, and hence no Planck order modifications to the dispersion relations, so two entropy calculations are consistent, but this time $\alpha_2$ must be equal to $-\frac{1}{3\pi}$.
- $\alpha_1$ can be different from zero, but must be $\ll 1$, then two approaches are consistent, and $\gamma\sim\gamma_0$.
- If $\alpha_1\sim1$, then the calculations for counting of microstates of a black hole must be modified with a square root of area term.
- If $\alpha_1\sim1$ and counting of microstates approach is right, then $\gamma$ must be scale-dependent and hence it has different values for small scales and converges to $\gamma_0$ for large area values.
Each of these possibilities give rise to the consistency of the entropy relations. The last possibility has some effects. If $\gamma$ changes with scale, then spectrums of area and volume operators have an extra $j$ dependence. It effects also width of torsional fluctuations. Varying of $\gamma$ changes the spectrum of $d_j$ operator in LQC, and effects the early evolution of the universe. The correct case about the consistency must be decided by the near future experiments.
This work was supported in part by the Scientific and Technical Research Council of Turkey (TÜBİTAK).
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---
abstract: 'There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mZ^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mZ^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq \mZ^n$ such that $\mZ^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $\(\zeta(6) \prod_{k=4}^n \zeta(k)\)^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.'
address:
- ' INRIA, Domaine de Voluceau, 78153 Rocquencourt, France and Tsinghua University, Institute for Advanced Study, Beijing 100084, China'
- 'Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia'
author:
- 'Phong Q. Nguyen'
- 'Igor E. Shparlinski'
title: 'Counting Co-Cyclic Lattices'
---
Introduction
============
Let $\cI_{n,V}$ be the set of subgroups $L$ of $\mZ^n$ such that $[\mZ^n:L] = V$. In other words, $\cI_{n,V}$ is the (finite) set of full-rank integer lattices $\subseteq \mZ^n$ of determinant (or co-volume) equal to $V$. Let $$\cI_{n,\le V}= \cup_{1 \le v \le V} \cI_{n,v}.$$ A classical result (see [@GiGr06; @Sc68]) states that when $n$ is fixed and $V$ grows to $\infty$, $$\begin{aligned}
\label{eq:nblattices}
\#\cI_{n,\le V} \sim \Xi_{2,n} \frac{V^n}{n} \end{aligned}$$ where we define $$\Xi_{m,n} = \prod_{k=m}^n \zeta(k) \mand
\Xi_{m} = \prod_{k=m}^\infty \zeta(k)$$ and $\zeta$ is the [*Riemann zeta-function*]{}. This is a special case of subgroup growth for the group $\mZ^n$.
We are interested in counting special subsets of $\cI_{n,\le V}$. More precisely, $\cI_{n,\le V}$ can naturally be partitioned with respect to the (finite Abelian) factor group $\mZ^n/L$ for $L \in \cI_{n,\le V}$. For any finite Abelian group $G$, we denote by $\cL_{n,G}$ the finite set of full-rank integer lattices $L \subseteq \mZ^n$ such that $\mZ^n/L \simeq G$. Then $$\cI_{n,V} = \cup_{\#G = V} \cL_{n,G}\mand \cI_{n,\le V} = \cup_{\#G \le V} \cL_{n,G}.$$ The sets $\cL_{n,G}$ have attracted significant interest in complexity theory. In a seminal work [@Aj96], Ajtai has discovered the first worst-case to average-case reduction for lattice problems: he proves that for the special case $G = (\mZ/q\mZ)^m$, when $\#G$ is sufficiently large, finding very short non-zero vectors (with non-negligible probability) in a random lattice $L \in \cL_{n,G}$ chosen with uniform distribution is as hard as finding short non-zero vectors in any lattice of dimension $m$. Ajtai’s reduction has recently been generalised [@GIN] to any finite Abelian group $G$ of sufficiently large order, which motivates to study the cardinals of $\cL_{n,G}$, depending on $G$.
We settle this question for cyclic groups $G$. More precisely, we give an asymptotic formula for the cardinality $N_n(V)$ of the the subset of $\cI_{n,\le V}$ formed by all [*co-cyclic lattices*]{}, that is, full-rank integer lattices $L$ such that $\mZ^n/L$ is cyclic: $$N_n(V) = \# \cup_{\substack{G~\mathrm{cyclic}\\ \#G \le V}} \cL_{n,G},$$ which is the subset of $\cI_{n,\le V}$ formed by all [*co-cyclic lattices*]{}, that is, full-rank integer lattices $L$ such that $\mZ^n/L$ is cyclic. Such lattices have been previously studied from a complexity point of view in [@PaSch; @Trol].
Throughout the paper we use the notion of [*natural density*]{}. We recall that the natural density of a property $\cP$ in a family of objects ordered according to their “size” (such as lattices of determinant up to $V$, groups of order up to $V$ and so on) is defined as the limit as $V\to \infty$ of the ratio of the number of such objects of size at most $V$ satisfying the property $\cP$ to the total number of such objects of size at most $V$.
For example, our results, coupled with , show that the natural density of co-cyclic lattices of a fixed dimension $n$ tends asymptotically (as $n$ grows) to $$\begin{aligned}
\label{eq:density}
\frac{1}{\zeta(6) \Xi_{4} } \approx 85\%. \end{aligned}$$ Hence, “most” integer lattices of sufficiently large dimension are co-cyclic.
We also determine the density of integer lattices with squarefree index in $\mZ^n$, which are special cases of co-cyclic lattices, see also the discussion in Section \[sec:gen\]. More precisely, we obtain an asymptotic formula on the cardinality $$N^\sharp_n(V) = \# \cup_{\substack{\#G \le V\\\#G~\mathrm{squarefree}}} \cL_{n,G}.$$ Coupled with , we obtain that the natural density of full-rank integer lattices a fixed dimension $n$ with squarefree determinant of a fixed dimension $n$ tends asymptotically (as $n$ grows) to $$\begin{aligned}
\label{eq:density sf}
\frac{1}{\Xi_{3}} \approx 71.7\%.\end{aligned}$$
Main Results
============
For $n \ge 2$ we define $\vartheta_n$ by the absolutely converging product $$\vartheta_n
= \prod_{p} \(1 + \frac{p^{n-1} -1}{p^{n+1} - p^{n}}\),$$ where hereafter $p$ always runs through prime numbers.
\[thm:NnV\] For any fixed $n \ge 2$, we have $$N_n(V) = \frac{\vartheta_n }{n} V^n +
O\( V^{n-1+o(1)}\),$$ as $V\to \infty$.
Similarly, we define $$\rho_n
= \frac{6}{\pi^2} \prod_{p} \(1 + \frac{p^{n-1} -1}{p^{n+1} - p^{n-1}}\) .$$
\[thm:NnV-SF\] For any fixed $n \ge 2$, we have $$N^\sharp_n(V) = \frac{\rho_n +o(1)}{n} V^n$$ as $V\to \infty$.
We note that it is easy to get an explicit bound on error term in the asymptotic formula of Theorem \[thm:NnV-SF\].
It is also interesting to study the asymptotic behaviour of the constant $\vartheta_n$ as $n \to \infty$. We define $$\vartheta = \prod_p \left( 1+ \frac{1}{p^2-p} \right) = \frac{\zeta(2)\zeta(3)}{\zeta(6)}= \frac{ 315 \zeta(3)}{2 \pi^4} =1.94359\ldots$$ We note that $\vartheta$ appears in the asymptotic formula of Landau [@Lan]: $$\label{eq:Landau}
\sum_{d\le t} \frac{1}{\varphi(d)} =
\vartheta \left( \log t + \gamma -
\sum_{p}\frac{\log p}{p^2-p+1} \right ) + O\( \frac{\log t}{t} \),$$ where $\varphi(d)$ is the Euler function and $\gamma$ is the [*Euler-Mascheroni constant*]{} (a more recent reference is [@Mont]).
\[thm:theta\] For any $n \ge 2$, we have $$\vartheta\(1 - \frac{7\cdot 2^{-n}}{6} +O(3^{-n})\) \ge \vartheta_n \ge \vartheta\(1 -
\frac{5\cdot 2^{-n}}{3} +O(3^{-n})\) .$$
By combining Theorems \[thm:NnV\] and \[thm:theta\] with , we obtain .
We now define $$\rho= \prod_p \left( 1+ \frac{1}{p^2-1} \right) = \zeta(2).$$ Finally, we also have
\[thm:rho\] For any $n \ge 2$, we have $$\rho_n = \rho\(1 - 2^{-n-1} +O(3^{-n})\) .$$
By combining Theorems \[thm:NnV\] and \[thm:rho\] with , we obtain .
One can easily get tighter bounds in Theorems \[thm:theta\] and \[thm:rho\].
Proofs of Main Results
======================
Proof of Theorem \[thm:NnV\] {#sec:proof1}
----------------------------
Given an integer $q$, we say that two $n$-dimensional vectors ${\mathbf{a}}, {\mathbf{b}} \in \Z^n$ are [*equivalent*]{} modulo $q$, if for some integer $\lambda$ with $\gcd(\lambda,q) =1$ we have $${\mathbf{a}} \equiv \lambda {\mathbf{b}} \pmod q.$$ We also say that a vector ${\mathbf{a}}=(a_1,\ldots, a_n) \in \Z^n$ is [*primitive*]{} modulo $q$, if $\gcd(a_1,\ldots, a_n, q) = 1$.
Let $A_n(q)$ be the number of distinct non-equivalent primitive modulo $q$ vectors ${\mathbf{a}}=(a_1,\ldots, a_n) \in \Z^n$.
Let $L \in \cI_{n,q}$. Paz and Schnorr [@PaSch] have proved that $L$ is co-cyclic if and only if there exist a vector ${\mathbf{a}}=(a_1,\ldots, a_n) \in \Z^n$ primitive modulo $q$, such that $L = \cL_n(q,{\mathbf{a}})$ where $$\begin{split}
\label{eq:Lnq}
\cL_n(q, {\mathbf{a}}) = \{{\mathbf{x}} &=(x_1,\ldots, x_n) \in \Z^n~:\\
&~a_1 x_1 + \ldots + a_nx_n \equiv 0 \pmod q\}.
\end{split}$$ It follows that $N_n(V)$ satisfies: $$N_n(V) = \sum_{q \le V} A_n(q)$$ Using the M[ö]{}bius function $\mu(d)$, see [@HW Section 16.3], we write $$\label{eq:An1}
A_n(q) =\frac{1}{\varphi(q)} \sum_{d\mid q}\mu(d)
\sum_{\substack{a_1, \ldots, a_n =1 \\
d \mid \gcd(a_1,\ldots, a_n)}}^q 1
=\frac{q^n}{\varphi(q)} \sum_{d\mid q}\frac{\mu(d)}{d^n}.$$
Using the well-known identity (see [@HW Theorem 62]) $$\varphi(q) = q \prod_{p\mid q}\(1 - \frac{1}{p}\),$$ and since $$\sum_{d\mid q}\frac{\mu(d)}{d^n} = \prod_{p\mid q}\(1 - \frac{1}{p^n}\)$$ we derive from $$\label{eq:An2}
A_n(q)= q^{n-1} \prod_{p\mid q}\(1 + \frac{p^{n-1} -1}{p^n - p^{n-1}}\).$$
Let $f_n(d)$ be the multiplicative function defined as $$f_n(d) = \prod_{p \mid d} \frac{p^{n-1} -1}{p^n - p^{n-1}}$$ if $d$ is squarefree and $f(d) = 0$ otherwise. From , we see that $$A_n(q)= q^{n-1} \sum_{d \mid q} f_n(d).$$ Therefore, changing the order of summation and writing $q = kd$, we derive $$\label{eq:Nn}
N_n(V) = \sum_{d \le V} f_n(d) d^{n-1} \sum_{k \le V/d} k^{n-1}.$$ We now observe that $$\frac{p^{n-1} -1}{p^n - p^{n-1}} \le \frac{2}{p}.$$ Hence, for any integer $d$ we have $$f(d) \le \frac{2^{\omega(d)}}{d},$$ where $\omega(d)$ is the number of prime divisors of $d$. Recalling the well-known bound $$\omega(d) = O\(\frac{\log d}{\log \log d}\)$$ (which follows immediately from the trivial inequality $\omega(d)! \le d$ and Stirling’s formula) we obtain $$\label{eq:fd}
f(d) \le d^{-1+o(1)}, \qquad \text{as}\ d\to \infty.$$
Since, for a fixed $n$, $$\sum_{k \le V/d} k^{n-1} = \frac{1}{n} (V/d)^n + O\((V/d)^{n-1}\)$$ we now derive from and that $$\label{eq:Nn prelim}
\begin{split}
N_n(V) &= \frac{1}{n} V^n \sum_{d \le V} \frac{f_n(d)}{d} +
O\( V^{n-1} \sum_{d \le V} f_n(d)\)\\
& = \frac{1}{n} V^n \sum_{d \le V} \frac{f_n(d)}{d} +
O\( V^{n-1+o(1)}\) ,
\end{split}$$ as $V\to \infty$. Finally, using again, we obtain $$\sum_{d \le V} \frac{f_n(d)}{d} = \vartheta_n + O\( V^{n-1+o(1)}\),$$ where $\vartheta_n$ is given by the absolutely converging series $$\vartheta_n = \sum_{d =1}^\infty \frac{f_n(d)}{d} =
\prod_{p} \(1 + \frac{f_n(p)}{p}\)
= \prod_{p} \(1 + \frac{p^{n-1} -1}{p^{n+1} - p^{n}}\).$$ which concludes the proof.
Proof of Theorem \[thm:NnV-SF\]
-------------------------------
For two functions $F(t)$ and $G(t)$ depending on a parameter $t$ we write $F(t) \sim G(t)$ as an equivalent of $$\lim_{t\to \infty} F(t)/G(t) = 1.$$
We need the following classical result of Prachar [@Prach] which asserts that for any integers $d> a \ge 1$ with $\gcd(d,a)=1$ and arbitrary $\varepsilon >0$ we have $$\# \{k\le x~:~k\equiv a \pmod d, \ k~\text{squarefree}\}
\sim \frac{6x}{\pi^2d}\prod_{p\mid d}\(1-\frac{1}{p^2}\)^{-1},$$ as $x \to \infty$, provided $d \le x^{2/3-\varepsilon}$, where the product is taken over all prime divisors $p$ of $d$; see also [@Hool Theorem 3]. In particular, under the same condition on $d$ and $x$, we have $$\label{eq:SF coprime}
\begin{split}
\# \{k\le x~ :~&\gcd(k,d)=1, \ k~\text{squarefree}\}\\
&\sim \frac{6\varphi(d)x}{\pi^2 d}\prod_{p\mid d}\(1-\frac{1}{p^2}\)^{-1}\sim \frac{6x}{\pi^2}\prod_{p\mid d}\(1+\frac{1}{p}\)^{-1},
\end{split}$$ as $x \to \infty$ (one can certainly prove directly as well).
As in the proof of Theorem \[thm:NnV\] we write $$N^\sharp_n(V) = \sum_{\substack{q \le V\\ q~\text{squarefree}}} A_n(q) .$$ Furthermore, instead of , we derive $$N^\sharp_n(V) = \sum_{d \le V} f_n(d) d^{n-1}
\sum_{\substack{k \le V/d\\\gcd(k,d)=1\\ k~\text{squarefree}}} k^{n-1}$$ (recall that the function $f(d)$ is supported only on squarefree integers).
For $d > V^{1/2}$ we estimate the sum over $k$ trivially as $O((V/d)^{n})$. Thus, recalling , we see that the total contribution from such terms is $$\label{eq:large d}
\begin{split}
\sum_{V^{1/2} < d \le V} f_n(d) &d^{n-1}
\sum_{\substack{k \le V/d\\\gcd(k,d)=1\\ k~\text{squarefree}}} k^{n-1}\\
&=
V^n \sum_{V^{1/2} < d \le V} d^{-2+o(1)}
= V^{n-1/2+o(1)}.
\end{split}$$
For $d \le V^{1/2}$, the asymptotic formula applies to the sums over $k$, so similarly to , via partial summation, we derive that the total contribution from such terms is $$\label{eq:small d}
\begin{split}
\sum_{d \le V^{1/2}} f_n(d) &d^{n-1}
\sum_{\substack{k \le V/d\\\gcd(k,d)=1\\ k~\text{squarefree}}} k^{n-1}\\
&\sim \frac{6 V^n }{\pi^2n }\sum_{d\le V^{1/2}} \frac{f_n(d)}{d}
\prod_{p\mid d}\(1+\frac{1}{p}\)^{-1},
\end{split}$$ as $V\to \infty$.
Since $$\begin{split}
\sum_{d\le V^{1/2}} \frac{f_n(d)}{d}
&\prod_{p\mid d}\(1+\frac{1}{p}\)^{-1}
\sim \sum_{d=1}^\infty \frac{f_n(d)}{d}
\prod_{p\mid d}\(1+\frac{1}{p}\)^{-1} \\
& = \prod_{p} \(1 + \frac{f_n(p)}{p} \(1+\frac{1}{p}\)^{-1} \) \\
& = \prod_{p} \(1 + \frac{f_n(p)}{p+1} \) = \prod_{p} \(1 + \frac{p^{n-1} -1}{p^{n+1} - p^{n-1}}\) ,
\end{split}$$ the result now follows from and .
Proof of Theorem \[thm:theta\]
------------------------------
First we note that for any $k \ge 2$ $$\label{eq:zeta}
\zeta(k) = 1 + \frac{1}{2^{k}} + \frac{1}{3^{k}} + O\(\int_{3}^\infty\frac{1}{t^{k}}dt\)
= 1 + 2^{-k} + O(3^{-k})$$ We have, $$\frac{ \vartheta_n }{\vartheta} = \prod_p \frac{ p^2-p+1-1/p^{n-1}}{p^2-p+1} = \prod_p \( 1- \frac{1}{p^{n-1}(p^2-p+1)}\) .$$ Since $p^2-p+1 \ge p$, we have $$1- \frac{1}{p^{n-1}(p^2-p+1)} \ge 1- \frac{1}{p^{n}} .$$ Hence, we now see that $$\begin{split}
\frac{ \vartheta_n }{\vartheta} &\ge
\( 1- \frac{1}{3\cdot 2^{n-1}}\) \prod_{p\ge 3} \( 1- \frac{1}{p^{n}} \) \\
& = \( 1- \frac{1}{3\cdot 2^{n-1}}\) \( 1- \frac{1}{3^{n}}\)^{-1} \zeta(n)^{-1}.
\end{split}$$ Thus, using we obtain $$\frac{ \vartheta_n }{\vartheta} \ge
1- \frac{1}{3\cdot 2^{n-1}} - \frac{1}{2^n} + O(3^{-n})$$ On the other hand, since $p^2-p+1 \le p^2$, we also have $$\begin{split}
\frac{ \vartheta_n }{\vartheta} & \le \( 1- \frac{1}{3\cdot 2^{n-1}}\) \prod_{p\ge 3}
\( 1- \frac{1}{p^{n+1}}\) \\
& = \( 1- \frac{1}{3\cdot 2^{n-1}}\) \( 1- \frac{1}{3^{n+1}}\)^{-1}\zeta(n+1)^{-1}.
\end{split}$$
Proof of Theorem \[thm:rho\]
----------------------------
We have, $$\frac{\rho_n }{\rho} = \prod_p \( 1- \frac{1}{p^{n+1}}\) = \frac{1}{\zeta(n+1)}.$$ Using , we conclude the proof.
Comparison with random finite Abelian groups
============================================
Motivation
----------
In any context involving finite Abelian groups, it is interesting to study if these finite Abelian groups behave like random finite Abelian groups in various families in terms of natural density.
There are at least two distributions worth considering, which we discuss here.
The uniform distribution
------------------------
Let $a(n)$ denote the number of non-isomorphic Abelian groups of order $n$. It is well-known that if $n=p_1^{k_1} \ldots p_s^{k_s}$ is a prime number factorisation of $n$ then $$a(n) = \prod_{i=1}^s \cP(k_i)$$ where $\cP(k)$ is the number integer partitions of $k$ (as it is obvious that $a(n)$ is a multiplicative function and also $a(p^k) = \cP(k)$ for any integer power of a prime $p$), and also $$\sum_{\#G \le x} 1= \sum_{n \le x} a(n) = A_1 x + A_2 x^{1/2} + A_3 x^{1/3} + R(x),$$ where $$A_j = \prod_{k=1, k \ne j}^{\infty} \zeta(k/j), \quad j=1,2,3,$$ and the best result for the error term is $R(x) \ll x^{1/4+o(1)}$ (see [@RoSa06]).
Since clearly there is a unique (up to isomorphism) cyclic subgroup of order $n$ and furthermore, all subgroups of square-free order are cyclic, we see that the natural density of cyclic groups is $$\frac{1}{A_1} = \frac{1}{\Xi_2} \approx 44\% .$$ Furthermore, recalling that there are $\(\zeta(2)^{-1}+o(1)\)V$ squarefree number up to $V$ (see [@HW Section 18.6]), we conclude that the natural density of groups of square-free order is $$\frac{1}{\zeta(2) A_1} =\frac{1}{\zeta(2) \Xi_2} \approx 26\% .$$
The Cohen-Lenstra distribution
------------------------------
According to the Cohen-Lenstra heuristics [@CoLe84], a given finite Abelian group $G$ occurs with mass inversely proportional to the order $\#\mathrm{Aut}(G)$ of its automorphism group $\mathrm{Aut}(G)$, similarly to many other mass formulas. Let $$T(V) = \sum_{\#G \le V} 1/\#\mathrm{Aut}(G),$$ with $G$ running over all finite Abelian groups of order at most $V$, up to isomorphism; Furthermore, let $T(V; \fP)$ is the same sum as $T(V)$ restricted to the groups $G$ satisfying $\fP$. Then the natural density of a property $\fP$ is defined here as the limit $$\label{eq:Nat Dens}
\Delta(\fP) = \lim_{V\to \infty} T(V; \fP)/T(V),$$ provided it exists.
It is shown in [@CoLe84] that the denominator is asymptotically equivalent to: $$T(V) \sim \Xi_2 \log V.$$ If $\fP$ is the property $\fP_{cycl}=\text{``{\em $G$ is cyclic}''}$, then the numerator is: $$\begin{aligned}
T(V; \fP_{cycl}) & = \sum_{n \le V} \frac{1}{\varphi(n)}\\
& =
\vartheta \left( \log V + \gamma -
\sum_{p}\frac{\log p}{p^2-p+1} \right ) + O\( \frac{\log x}{x} \),\end{aligned}$$ see . Hence, with respect to the Cohen-Lenstra distribution, the natural density of cyclic groups is $$\label{eq:cycl dens}
\Delta(\fP_{cycl}) = \vartheta \Xi_2^{-1} = \zeta(6)^{-1} \Xi_4^{-1} \approx 85\%.$$
Hence, the cyclicity of the factor group $\mZ^n/L$ (when $L$ is full-rank integer lattices $L$ in $\mZ^n$) behaves as predicted by the Cohen-Lenstra heuristics.
If $\fP_{sf}$ is the property $\fP_{sf}=\text{``{\em $\#G$ is square-free}''}$, then: $$T(V; \fP_{sf}) = \sum_{n \le V} \frac{\mu^2(n)}{\varphi(n)}.$$ Ward [@Wa27 Equation (2.2)] has shown that, as $V$ grows to $\infty$: $$\sum_{n \le V} \frac{\mu(n)^2}{\varphi(n)} = \log V + c + o(V^{-1/2})$$ for some absolute constant $c$. Hence, with respect to the Cohen-Lenstra distribution, the natural density of groups of square-free order is $$\Delta(\fP_{sf}) = \Xi_2^{-1} \approx 44\%,$$ which differs from the natural density of co-cyclic lattices.
More generally, it is also possible to obtain the natural density when $\fP$ is the property $\fP_{\le r}=\text{``{\em $G$ has rank at most $r$}''}$. To do so, we rely on results from [@CoLe84] for $p$-groups, for which the Cohen-Lenstra distribution is a probability distribution: the probability that a random $p$-group has rank $r$ is exactly: $$P(p,r) = p^{-r^2} \frac{\prod_{i=1}^{\infty} (1-p^{-i})}{\prod_{i=1}^{r} (1-p^{-i})^2}.$$ It follows that the density of $\fP_{\le r}$ is: $$\begin{aligned}
\Delta(\fP_{\le r}) & = \prod_{p} \sum_{k=0}^r P(p,k)\\ & = \prod_{p}
\( \sum_{k=0}^r p^{-k^2} \frac{(1-p^{-1})}
{\prod_{i=1}^{k} (1-p^{-i})^2} \prod_{i=2}^{\infty} (1-p^{-i}) \) \\
& = \Xi_2^{-1} \prod_{p} \( \sum_{k=0}^r p^{-k^2} \frac{(1-p^{-1})}
{\prod_{i=1}^{k} (1-p^{-i})^2} \).\end{aligned}$$ This proves again that $\Delta(\fP_{\le 1}) = \vartheta \Xi_2^{-1}$, which is the previous natural density $\Delta(\fP_{cycl})$ of cyclic groups . We also have: $$\begin{aligned}
\Delta(\fP_{\le 2})& = \Xi_2^{-1} \prod_{p} \( 1+\frac{1}{p(p-1)} + \frac{p^{-4}}{(1-p^{-1})(1-p^{-2})^2} \) \\
& = \Xi_2^{-1}
\prod_{p} \left( 1+ \frac{p^4-p^2+1}{p(p-1)^3(p+1)^2} \right) \approx 99.5\%.
\end{aligned}$$ It is shown in [@FuGo15] that for all $p \ge 2$: $$1-p^{-1}-p^{-2} \le \prod_{i\ge 1} (1-p^{-i}) \le 1.$$ It follows that for all $p \ge 2$: $$P(p,k) \le \frac{p^{-k^2}}{1- p^{-1}-p^{-2}} \le 4 p^{-k^2}.$$ Therefore, $$\sum_{k\ge r} P(p,k) \le \sum_{k \ge r} 4p^{-k^2} \le 8p^{-r^2}.$$ (provided the above sums and products converge). Hence the natural density of the property $\fP_{\ge r}=\text{``{\em $G$ has rank at least $r$}''}$ satisfies: $$\begin{aligned}
\Delta(\fP_{\ge r}) & = 1- \prod_{p} \sum_{j=0}^{r-1} P(p,j)
= 1- \prod_{p} \(1 - \sum_{j\ge r} P(p,j)\) \\
& \le 1- \prod_{p} \(1 - 8p^{-r^2}\) = 1- \exp\(\sum_{p} 8p^{-r^2}\)\\
& \le 1- \exp\(8 (\zeta(r^2)-1)\).\end{aligned}$$ Since $\zeta(r^2)-1) = 2^{-r^2} + O(3^{-r^2}))$, see we obtain $$\Delta(\fP_{\ge r}) \le 8\cdot 2^{-r^2} + O(3^{-r^2}) .$$
Comparison with groups of points of elliptic curves over prime fields
---------------------------------------------------------------------
We note that Gekeler [@Ge08] settled analogous questions for elliptic curves over prime finite fields: if $q$ is a large random prime and $E$ runs over all elliptic curves over $\mF_q$, then
- The natural density of cyclic $E$ is $$\prod_{p} \left( 1 - \frac{1}{(p^2-1)p(p-1)} \right) \approx 0.81 .$$
- The natural density of $E$ such that $\#E$ is squarefree is $$\prod_{p} \left( 1 - \frac{p^3-p-1}{(p^2-1)p^2(p-1)} \right) \approx 0.44.$$
Comments and Open Questions {#sec:gen}
===========================
We have studied the cardinality $N_n(V)$ of the subset of $\cI_{n,\le V}$ formed by all lattices $L$ such that $\mZ^n/L$ is cyclic. More generally, we define $\fG_m(V)$ as the set of groups $G$ with $m$ invariant factors (that is, of rank $m$, where the rank is defined as the minimal size of a generating set) and of order $\# G \le V$. It is natural to study the cardinality $N_{n,m}(V)$ of the subset $\cup_{G\in \fG_m(V)} \cL_{n,G}$ of $\cI_{n,V}$ formed by all lattices $L$ such that $\mZ^n/L$ has exactly $m$ invariant factors. We are now interested in $m \ge 2$, since we already know $N_{n,1}(V) = N_n(V)$.
Let $G = \mZ/{q_1}\mZ \times \cdots \times \mZ/{q_m}\mZ$ be a finite Abelian group with $m$ invariant factors: $q_{m} > 1$ and each $q_{i+1}$ divides $q_i$. By analogy with Section \[sec:proof1\], we say that two $n$-dimensional vectors ${\mathbf{a}}, {\mathbf{b}} \in G^n$ are [*equivalent*]{} modulo $G$, if there is an automorphism $\tau$ of $G$ such that $$b_i = \tau(a_i), \qquad 1 \le i \le n.$$ We also say that a vector ${\mathbf{a}}=(a_1,\ldots, a_n) \in G^n$ is [*primitive*]{} modulo $G$ if the components $a_1,\ldots, a_n$ generate $G$.
Let $A_{n}(G)$ be the number of distinct non-equivalent primitive modulo $G$ vectors ${\mathbf{a}}=(a_1,\ldots, a_n) \in G^n$. Then: $$N_{n,m}(V) = \sum_{G\in \fG_m(V)} A_{n}(G).$$ Indeed, $L \in \cI_{n,\#G}$ satisfies $\mZ^n/L \simeq G$ if and only there exists a vector ${\mathbf{a}}=(a_1,\ldots, a_n) \in G^n$ primitive modulo $G$, such that: $$L = \{{\mathbf{x}}=(x_1,\ldots, x_n) \in \Z^n~:~
a_1 x_1 + \ldots + a_nx_n = 0 \ \mathrm{in}\ G \}.$$ Note that when $n$ is sufficiently large with respect to $\#G$, most elements of $G^n$ are primitive modulo $G$: more precisely, Pak [@Pa99] shows that for any $k > 0$, if $(g_1,\dots,g_n) \in G^n$ is picked uniformly at random where $n > (k+1) \log \#G +2$, then $g_1,\dots,g_n$ generate the whole group $G$ with probability at least $1-1/\#G^k$, which implies that: $$A_{n}(G) = \frac{\#G^n ( 1 + O(1/\#G^k))}{\#\mathrm{Aut}(G)}.$$ In particular, $$A_{n}(G) \sim \frac{\#G^n}{\#\mathrm{Aut}(G)}$$ as $\# G \to \infty$. Now, there are classical formulas for $\#\mathrm{Aut}(G)$ when $G$ is a finite Abelian group (see [@HiRh07; @Le09; @Le10]):
\[frac:G p-group\] If $$G = \prod_{i=1}^k (\mZ/{p^{e_i}}\mZ)^{r_i}$$ is a finite Abelian $p$-group in standard form, that is, $k \ge 0$, $e_1 > \cdots > e_k > 0$, $r_i > 0$, then: $$\#\mathrm{Aut}(G) = \left( \prod_{i=1}^k \left( \prod_{s=1}^{r_i} (1-p^{-s}) \right) \right) \left( \prod_{1 \le i,j \le k} p^{\min(e_i,e_j) r_i r_j} \right).$$
\[frac:G1 G2\] If $G_1$ (resp. $G_2$) is a finite Abelian $p_1$-group (resp. $p_2$-group), with distinct primes $p_1 \ne p_2$, then $$\#\mathrm{Aut}(G_1 \times G_2) = \#\mathrm{Aut}(G_1) \times \#\mathrm{Aut}(G_2).$$
Thus, given a decomposition of $G$ as a product of $p$-groups, we have a closed formula for $ \#\mathrm{Aut}(G)$, but if one is given an invariant factor decomposition instead, then one must first convert it into a $p$-groups decomposition. We conclude with posing several open problems which might be addressed using these formulas:
\[prob:NnmV\] Obtain an asymptotical formula for $N_{n,m}(V)$ for $m \ge 2$.
Show that $N_{n,m}(V) = o( \#I_{n,\le V}) $ when $m$ is sufficiently large.
\[prob:Ajt\] Show that the Ajtai [@Aj96] lattices, that are the classes $\cL_{n,G}$ for $G = (\mZ/q\mZ)^m$, form a negligible fraction of $I_{n,\le V}$.
Here we make a few comments regarding Problem \[prob:Ajt\]. For $G = (\mZ/q\mZ)^m$, where $$q =\prod_{i=1}^\nu p_i^{e_i}$$ is the prime number factorisation of $q$, each $p$-group of $G$ is of the form $(\mZ/{p_i^{e_i}}\mZ)^{m}$ whose automorphism group, by Fact \[frac:G p-group\] has order $$\#\mathrm{Aut}((\mZ/p_i^{e_i}\mZ)^{m})
= p_i^{e_i m^2} \prod_{s=1}^{m} (1-p_i^{-s}), \qquad i =1, \ldots, \nu.$$ Hence, then by Fact \[frac:G1 G2\] $$\#\mathrm{Aut}((\mZ/q\mZ)^m) =q^{m^2} \prod_{i=1}^\nu \prod_{s=1}^{m} (1-p_i^{-s}).$$ Note that $\mathrm{Aut}((\mZ/q\mZ)^m) \simeq \mathrm{GL}_m (\mZ/q\mZ)$. We also note that due to the application in worst-case to average-case reductions (see [@Aj96]) we are especially interested in the regime where $n$ is of order $m \log m$ and $q$ is of order $m$.
Note that [@GIN] shows how to efficiently sample a random lattice from the uniform distribution over $\cL_{n,G}$, given any $G$ for which the factorization of $\#G$ is known. The knowledge of approximate values of $N_{n,m}(V)$, see Problem \[prob:NnmV\], may lead to an effective sampling of random lattice from the uniform distribution over $\cI_{n,\le V}$, which is an open problem.
Acknowledgements {#acknowledgements .unnumbered}
================
P. Q. Nguyen would like to thank Christophe Delaunay for very helpful discussions on [@CoLe84].
During the preparation of this paper, P. Q. Nguyen was supported in part by China’s 973 Program, Grant 2013CB834205, and NSFC’s Key Project, Grant 61133013, and I. E. Shparlinski was supported in part by ARC grants DP130100237 and DP140100118.
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---
abstract: 'We consider the leading non-analytic temperature dependence of the specific heat and temperature and momentum dependence of the spin susceptibility for two dimensional fermionic systems with non-circular Fermi surfaces. We demonstrate the crucial role played by Fermi surface curvature. For a Fermi surface with inflection points, we demonstrate that thermal corrections to the uniform susceptibility in $D=2$ change from $\chi_{s} \propto T$ to $\chi_{s} \propto T^{2/3}$ for generic inflection points, and to $\chi_{s} \propto T^{1/2}$ for special inflection points along symmetry directions. Errors in previous work are corrected. Application of the results to $Sr_2RuO_4$ is given.'
author:
- 'Andrey V. Chubukov$^{1}$ and Andrew J. Millis$^{2}$'
title: 'Nonanalytic corrections to the specific heat and susceptibility of a non-Galilean-Invariant Two-Dimensional Fermi Liquid'
---
Introduction
============
L. D. Landau’s “Fermi liquid theory” provides a robust and accurate description of the leading low temperature, long wavelength behavior of a wide range of systems of interacting fermions in two and three spatial dimensions. In Landau’s original work [@Landau57] it was assumed that the temperature ($T$) and momentum ($q$) corrections to the leading Fermi liquid behavior were analytic functions of $(T/T_{F})^{2}$ and $(q/k_{F})^{2}$, with the Fermi momentum $k_{F}$ set by the inter-particle spacing and the Fermi temperature $T_{F}\sim v_{F}k_{F}$ with $v_{F}$ a typical measured electron velocity. However, subsequent work revealed that in dimensions $d=2$ and $d=3$ the leading temperature and momentum dependences of measurable quantities including the specific heat and spin susceptibility are in fact non-analytic functions of $T^{2}$ and $q^{2}$. The nonanalyticities have been studied in detail for Galilean-invariant systems (spherical Fermi surface or circular Fermi line) (see Ref for a list of references). In this paper we extend the analysis to a more general class of systems, still described by Fermi liquid theory but with a Fermi surface of arbitrary shape.
The extension is of interest in order to allow comparisons to systems such as $Sr_2RuO_4$ [@Bergemann03] and quasi-one-dimensional organic conductors [@organic], in which lattice effects are important. The extension also provides further insight into a fundamental theoretical issue: a crucial finding of the previous analysis [@Chubukov05; @Chubukov06] was that for two dimensional systems the non-analytical term in the specific heat coefficient $\delta C(T)/T \propto T$ arise solely from “backscattering” processes at any strength of fermion-fermion interaction. For the spin susceptibility, the situation is more complex: in addition to the backscattering contributions to $\delta \chi_s (T) \propto T$ and $\delta \chi_s (q) \propto q$ there are contributions of third and higher order in the interaction which involve non-backscattering processes [@ch_masl_latest]. If the interaction is not too strong the backscattering terms dominate.
Unlike scattering at a general angle, the kinematics of backscattering is effectively one-dimensional, and depends sensitively on the shape of the Fermi surface. We shall show that the crucial parameter is the Fermi surface curvature, and that for two dimensional systems in which the Fermi surface possesses an inflection point, the power laws change from $\left|(T,q)\right|$ to $\left|(T,q)^{1/2}\right|$ or $\left|(T,q)^{2/3}\right|$ according to whether the inflection point is or is not along a reflection symmetry axis of the material. Similar effects occur in three dimensional systems, but the effects will be weaker as the singularities there are only logarithmic.
The importance of the Fermi surface geometry was previously noted by Fratini and Guinea, who showed that the presence of inflection points changes the power law for the spin susceptibility $\chi_s (T)$ [@Fratini02]. However, we believe that their calculation treated the kinematics of the backscattering incorrectly; as a result, they found that, in $2D$, the presence of the inflection point changes the temperature dependence only by a logarithm, from $\chi_s (T) \propto T$ to $\chi_s (T) \propto T \log T$, instead of the $T^{\left(\frac{1}{2},\frac{1}{3}\right)}$ found here.
The paper is organized as follows. Section II introduces the model and defines notation. In Section III, we demonstrate the physical origin of the results, via a calculation of the long range dynamical correlations of a Fermi liquid. Section IV presents results for the specific heat of a multiband system. In Section V we calculate the nonanalyticities in the momentum and temperature dependence of $\chi$. Section VI shows how new power laws emerge for Fermi surfaces with inflection points, and why one-dimensionality affects the powers. Section VII presents estimates of the size of the nonanalytic terms in $Sr_2RuO_4$. Section VIII is a conclusion.
Model
=====
We study fermions moving in two dimensions in a periodic potential. As discussed at length in Ref. \[\], because we are concerned with the low $T$ properties of a Fermi liquid we may adopt a quasiparticle picture. Lifetime effects are not important, and the quasiparticle weight ($z$) factors may be absorbed in interaction constants. We may therefore consider several bands, labeled by band index $a$, of quasiparticles moving with (renormalized) dispersion $\varepsilon_{p}^a$. The Fermi surfaces are defined by the condition $\varepsilon_{\mathbf{k}}^a=\mu$; an example is shown in Fig \[fsfig\]. We parameterize the position at the Fermi surface by a coordinate $s^a$. For vectors $\mathbf{k}$ near a particular Fermi surface point $\mathbf{k}_{F}$ we will write$$\varepsilon_{\mathbf{k}}^a-\mu=v_{F}^a(s^a)\left( k_{\Vert}+\frac{k_{\bot}^{2}}{2k_{0}(s^a)}\right) \label{ep}$$ with $v_{F}(s^a)=\left\vert \partial\varepsilon_{\mathbf{k}}^a/\partial
\mathbf{k}\right\vert $ the Fermi velocity at the point $(s^a)$ and the components of $\mathbf{k}$ parallel and perpendicular to the Fermi velocity (Fermi surface normal) given by
$$\begin{aligned}
k_{\Vert} & =\left( \mathbf{k}-\mathbf{k}_{F}\right) \cdot\widehat
{\mathbf{v}}_{F}(s^a)\label{ppar}\\
k_{\bot} & =\left( \mathbf{k}-\mathbf{k}_{F}\right) \times\widehat
{\mathbf{v}}_{F}(s^a)\label{pperp}$$
$k_0^{-1}(s^a)$ is the curvature of the Fermi surface at the point $s^a$. For a circular Fermi surface, $k_0=k_F$ independent of $s$; but in general $k_0\neq k_F$ and both depend on $s$.
It is sometimes convenient to use the variables $\varepsilon_k$ and $\theta_k$, where $\theta_k$ is the angle determining the direction of the Fermi velocity ${\bf v}_F(k)=\partial \varepsilon_k/\partial {\bf k}$ relative to some fixed axis. Eq \[ep\] shows that the Jacobean of the transformation is $$d^2k=\frac{k_0(k)}{v_F(k)}d\varepsilon_k d\theta_k.
\label{jacobean}$$
In a non-Galilean-invariant system the Fermi surface may contain [*inflection points*]{} at which the curvature vanishes, i.e. $k_0\rightarrow \infty$. If the inflection point does not lie on an axis of reflection symmetry of the Brillouin zone, the dispersion (measured in terms of the difference of the momentum from an inflection point) is $$\varepsilon_{\mathbf{k}}-\mu=v_{F}\left( k_{\Vert}+\frac{k_{\bot}^{3}}{k^2_{1}}\right) \label{ep_1}$$ However, if the inflection point lies on a symmetry axis, then only even powers in $k_\bot$ may occur and $$\varepsilon_{\mathbf{k}}-\mu=v_{F}\left( k_{\Vert}+\frac{k_{\bot}^{4}}{k^3_{2}}\right) \label{ep_11}$$ Here $k_1$ and $k_2$ are coefficients expected in general to be $\sim k_F$.
The fermions interact. We assume that the $T\rightarrow 0$, long wavelength properties are described by the Fermi liquid theory, so that at low energies the interactions may be parameterized by the fully reducible Fermi surface to Fermi surface scattering amplitude $\Gamma_{\alpha,\beta,\gamma\delta}(k,p;k,p)$. For particles near the Fermi surface, $|{\bf k}|, |{\bf p}| \approx k_F$, and $\Gamma$ depends on the angle $\theta$ between ${\bf k}$ and ${\bf p}$ and on the band indices $a$ (for $k$) and $b$ (for $p$). Backscattering corresponds to $\theta = \pi$. It is often useful to decompose $\Gamma_{\alpha,\beta;\gamma,\delta}(\pi)$ into charge and spin components $$\Gamma^{ab}_{\alpha,\beta;\gamma,\delta}(\pi) =
\Gamma^{ab,c} \delta_{\alpha\gamma}\delta_{\beta \delta}
+ \Gamma^{ab,s} {\bf \sigma}_{\alpha\gamma}{\bf \sigma}_{\beta \delta}
\label{m13_1}$$
It is also instructive to make contact with second order perturbation theory for a model in which the particles are subject to a spin-independent interaction $$H_{int}=\sum_{\mathbf{q}}U(\mathbf{q})\rho_{\mathbf{q}}\rho_{-\mathbf{q}}\label{Hint}$$ with charge density operator $$\rho_{\mathbf{q}}=\sum_{\mathbf{p}\alpha}c_{\mathbf{p+q},\alpha}^{\dag
}c_{p,\alpha}\label{rho}$$ The leading perturbative result is then $$\Gamma^c = U(0) -\frac{U(2k_{F})}{2}, ~~~\Gamma^s = - \frac{U(2k_{F})}{2}
\label{m13_2}$$
If the interaction is local and only one band is relevant, i.e., $U(q) = U$, $\Gamma^c = -\Gamma^s = U/2$, and $$\Gamma_{\alpha,\beta;\gamma,\delta}(\pi) =
U \left(\delta_{\alpha\gamma}\delta_{\beta \delta} -
\delta_{\alpha\delta}\delta_{\beta \gamma}\right)
\label{new_1}$$
Physical Origin of Nonanalyticities; role of curvature
======================================================
Previous studies of the isotropic case demonstrated [@Chubukov05; @Chubukov06] that the nonanalyticities in the specific heat and spin susceptibility arise from the long spatial range dynamical correlations characteristic of Fermi liquids. These are of two types. One involves slow ($|\Omega|<v_Fq$) long wavelength fluctuations and is expressed mathematically in terms of the long wavelength limit of the polarizibility $\delta\Pi_{LW}\equiv lim_{q\rightarrow 0}\Pi(q,\Omega)-\Pi(q,0)\sim \left|\Omega\right|/q$. The $1/q$ behavior of polarizibility gives rise to a long-range correlation between fermions which decays as $|\Omega|/r$ at distances $1 \ll rk_F < E_F/|\omega|$. The other involves processes with momentum transfer $q \approx 2p_F$, and is expressed mathematically in terms of the polarizibility $\delta \Pi \equiv \Pi(q,\Omega)-\Pi(2k_F,\Omega)\sim \Omega/
\sqrt{2k_F-q}$ (for $|\Omega| <2k_F-q$. The $1/\sqrt{2k_F-q}$ behavior of $\delta \Pi$ gives rise to an oscillation with a slowly decaying envelope, $\cos(2k_{F} r-\pi/4) |\Omega|/\sqrt{r}$ , again at distances $1 \ll rp_F < E_F/|\omega|$ and again leading to singular behavior. We now compute these processes in the multiband model defined above, and then show how they affect thermodynamic variables.
Consider the long wavelength process first. The non-analyticity comes from a particle-hole pair excitation in which both particle and hole are in the same band, and have momenta in the vicinity of Fermi surface points $s_a^*$ satisfying $\vec{v}_F(s_a^*)\cdot {\vec q}=0$. Choosing one of these points as the origin of coordinates we have $$\begin{aligned}
\delta \Pi^{aa}_{s_a^*}&\equiv&\Pi_{s_a^*}(q,\Omega)-\Pi_{s_a^*}(q,0)=T\sum_n\int \frac{dk_\Vert dk_\bot}{(2\pi)^2}\frac{1}{i\omega_n-v_F(s_a^*)\left(k_\Vert+\frac{(k_\bot -q/2)^2}{2k_0(s_a^*)}\right)}
\nonumber\\
&\times&\left(\frac{1}{i\omega_n+i\Omega_n-v_F(s_a^*)\left(k_\Vert+\frac{(k_\bot+q/2)^2}{2k_0(s_a^*)}\right)}-\frac{1}{i\omega_n-v_F(s_a^*)\left(k_\Vert+\frac{(k_\bot + q/2)^2}{2k_0(s_a^*)}\right)}\right)
\label{Pi0}\end{aligned}$$ Performing the integral over $k_\Vert$ and the Matsubara sum as usual yields $$\delta \Pi^{aa}_{s_a^*}=\int \frac{dk_\bot}{2\pi}\frac{1}{2\pi v_F(s_a^*)}\frac{i\Omega}{i\Omega-\frac{v_F(s_a^*) k_\bot q}{k_0(s_a^*)}}$$ We see that for the part of $\delta \Pi$ which is even in $\Omega$ the integral is indeed dominated by $k_\bot \sim \Omega k_0(s_a^*)/v_F(s_a^*)$, so the approximation of expanding near this point is justified, and we obtain the nonanalytic long wavelength contribution as a sum over all Fermi points $s_a^*$ satisfying $\vec{v}_F(s_a^*)\cdot\vec{q}=0$ with coefficients determined by the local Fermi velocity and local curvature: $$\delta \Pi^{aa}_{LW}(q,\Omega)=\frac{\left|\Omega\right|}{q}\left(\sum_{s_a^*}\frac{k_0(s_a^*)}{4\pi v_F^2(s_a^*)}\right)
\label{pilong}$$ For a circular Fermi surface, two points satisfy $\vec{v}_F (s_a^*)\cdot\vec{q}=0$, $k_0=k_F$ and Eq. (\[pilong\]) reduces to the familiar result $k_F |\Omega|/(2 \pi v^2_F q)$. If the curvature vanishes, then use of Eqs (\[ep\_1\]) or (\[ep\_11\]) in Eq (\[Pi0\]) yields a $\delta \Pi_{LW}\sim\left( \Omega/q\right)^{1/2,1/3}$ respectively.
We next consider the “$2k_F$” process. Here the situation is a little different. The singularities in general come from processes connecting two Fermi points with parallel tangents (the importance of parallel tangents points has been noted in other contexts [@Altshuler95])–for example the points $A,B$ or $A,C$ shown in Fig \[fsfig\]. For a given vector ${\bf q}$ we denote as ${\bf Q}$ the closest vector which is parallel to ${\bf q}$ and connects two “parallel tangents” points. Symmetry ensures that the leading dependence of $\delta \Pi^{ab}_Q=\Pi({\bf q})-\Pi({\bf Q})$ involves only the $q_{\Vert} =({\bf q}-{\bf Q})\cdot {\bf Q}/|{\bf Q}|$. Labeling the initial and final of the two Fermi points connected by $Q$ as $s_{1,2}$ and noting that for systems with inversion symmetry the points come in pairs symmetric under interchange to the band indices we have $$\begin{aligned}
&& \delta \Pi^{ab}_Q \equiv \Pi({\bf q},\Omega)-\Pi({\bf Q},0)=T\sum_n\int \frac{dk_\Vert dk_\bot}{(2\pi)^2}
\frac{1}{i\omega_n-v_F(s_1)\left(k_\Vert+\frac{k_\bot^2}{2k_0(s_1)}\right)} \times
\nonumber\\
&&\left(\frac{1}{i\omega_n+i\Omega_n-v_F(s_2)\left(k_\Vert+q_\Vert+\frac{k_\bot^2}{2k_0(s_2)}\right)}-\frac{1}{i\omega_n-v_F(s_2)\left(k_\Vert+\frac{k_\bot^2}{2k_0(s_2)}\right)}\right)+(1 \leftrightarrow 2)
\label{PiQ}\end{aligned}$$ Here the [*sign*]{} of $v_F$ and $k_0$ becomes important. For two points “on the same side” of the Fermi surface (e.g. points $A$ and $B$ in Fig \[fsfig\]) the two velocities and the two curvatures have the same sign (a change of $k_\Vert$ either increases or decreases both energies), the integrations proceed as in the analysis of Eq (\[Pi0\]), and the different position of the $q$ means that to the order of interest there is no singular non-analytical term. On the other hand, if the two velocities have opposite sign (e.g. points $A$, $C$ in Fig \[fsfig\]) then after integrating over $k_\Vert$ and $k_\bot$ we obtain (for $\Omega>0$) $$\delta \Pi_Q=\frac{\sqrt{k_{avg}}}{\left|4 v_{F1}v_{F2}\right|}T\sum_{\omega_n>0 \hspace{.02in} or \hspace{0.02in} \omega_n<-\Omega}
\frac{sgn(\omega)}{\sqrt{\frac{2i\omega}{v_{avg}}+\frac{i\Omega}{v_{F2}}-q_\Vert}} +\left(1\leftrightarrow2\right)$$ Here $v_{F1} = v_F (s_1), ~v_{F2} = v_F (s_2)$, and $$\begin{aligned}
\frac{1}{k_{avg}}&=&\frac{1}{2}\left(\frac{1}{k_0(s_1)}+\frac{1}{k_0(s_2)}\right)\\
\frac{1}{v_{avg}}&=&\frac{1}{2}\left(\frac{1}{\left|v_{F1}\right|}+\frac{1}{\left|v_{F2}\right|}\right)
\label{vavg}\end{aligned}$$ Completing the evaluation yields $$\delta \Pi_Q= \frac{\sqrt{k_{avg}}}{4\pi \left|( v_{F1}\right| +
\left| v_{F2}\right|)}
~\left(\sqrt{q_\Vert + \frac{i\Omega}{|v_{F1}|}} +
\sqrt{q_\Vert - \frac{i\Omega}{|v_{F1}|}}+1\leftrightarrow2\right)
\label{new_2}$$ The singular $|\Omega|/\sqrt{q}$ behavior of the dynamic $\delta \Pi_Q$ only holds when $q_{\Vert} <0$ and is obtained by expanding (\[new\_2\]) in $\Omega/q_{\Vert}$. On the contrary, the singular behavior of the static $\Pi_Q \propto \sqrt{q_{\Vert}}$ holds at $q_{\Vert} >0. $ The dynamic part of $\Delta \Pi_Q$ behaves as $\Omega^2/(q_{\Vert})^{3/2}$ at $q_{\Vert} >0$.
Specific Heat
=============
This section treats the non-analyticity in the specific heat. The Galilean-invariant case studied previously is simple enough that the corrections can be evaluated, with no approximations beyond the usual ones of Fermi Liquid theory [@Chubukov06]. The evaluation confirms that in two dimensions the non-analytical contributions to the specific heat involve only the backscattering amplitude $\Gamma(\pi)$. In work prior to that reported in Ref \[\] this conclusion was reached by approximate calculations in which it was assumed that the non-analytical contributions were governed by the backscattering amplitude only, and then the assumption was shown [*a fortiori*]{} to be consistent. In the non-Galilean-invariant case of interest here, a complete analysis along the lines of Ref \[\] is not possible. We will follow earlier work and assume that the effects arise only from backscattering processes, and then show that the assumption is self consistent. We specialize for ease of writing to a momentum-independent vertex $\Gamma$ (see (\[new\_1\])), but keep band indices. For a two-dimensional system the diagram which gives the nonanalytic term in the thermodynamic potential $\Xi$ is [@Chubukov06] shown in Fig \[Omega\]. We may write the resulting diagram schematically as $$\Xi=
-\frac{1}{2}
\sum_{abcd}\int (dqd\Omega)\Gamma_{abcd}^2\Pi_{ab}(q,\Omega)\Pi_{cd}(q,\Omega)
\label{xi}$$ Now, singularities leading to non-analytical terms may arise for $q\rightarrow0$, in which case we must consider only the intraband contribution to $\Pi$, and $q\rightarrow Q$, where $Q$ is one of the “parallel tangents” vectors mentioned in the previous section, in which case the band indices of polarizabilities may be different. Let us consider first the small $q$ singularities. We have $$\Xi_{LW}=
-\frac{1}{2}
\sum_{ab}\int (dqd\Omega)\Gamma_{aabb}^2\Pi_{aa}(q,\Omega)\Pi_{bb}(q,\Omega)
\label{xi1}$$ Substituting from Eq (\[pilong\]) gives $$\Xi_{LW}=
-\frac{1}{2}
T\sum_\Omega\int \frac{qdqd\theta}{(2\pi)^2}\sum_{ab}\Gamma_{aabb}^2\frac{\Omega^2}{q^2}\frac{k_0^a(\theta)k_0^b(\theta)}{(4\pi)^2 v_{F,a}^2v_{F,b}^2}
\label{chilong}$$ The integral over $q$ is logarithmic and is cut by $\Omega$; the analytical continuation and integral of frequencies may then be performed and we obtain $$\left.\frac{\delta C}{T}\right|_{LW}=- \frac{3\zeta(3)}{\pi^3}
\sum_{ab}\int\frac{d\theta}{2\pi}
\frac{\Gamma_{aabb}^2k_0^a(\theta)k_0^b(\theta)}{v_{F,a}^2v_{F,b}^2}$$
We now turn to the parallel tangents part of the calculation, finding $$\Xi_Q=
-\frac{1}{2}
\sum_{ab}\int (dqd\Omega)\Gamma_{abba}^2\Pi_{ab}(Q+q,\Omega)\Pi_{ba}(Q+q,\Omega)
\label{xiq}$$ Note that the symmetry of the vertex means that $\Gamma_{abba}^2=\Gamma_{aabb}^2$. Again substituting $\Delta \Pi_Q$ instead of $\Pi_{ab}(Q+q,\Omega)$ and evaluating the integrals explicitly we find [@comm_2] (note that to obtain the non-analytical behavior it is sufficient to expand $\delta \chi_Q$ for $|\Omega| <<q$) $$\Xi_Q=
-\frac{1}{2}
\sum_{ab}\Gamma_{abba}^2T\sum_\Omega\int \frac{dq_\Vert dq_\bot}{(2\pi)^2}\frac{\Omega^2}{(4\pi)^2 v_{Fa}^2v_{Fb}^2}\frac{k_{avg}}{|q_\Vert|}
\label{xiq2}$$ Now noting that $k_{avg}=k_0^ak_0^b/(k_0^a+k_0^b)$ and that $dq_\bot=(k_0^a+k_0^b)d\theta$ we see that $\Xi_Q$ and $\Xi_{LW}$ give identical contributions despite the apparently different kinematics.
Adding the two contributions, we obtain $$\frac{\delta C}{T} =- \frac{3\zeta(3)}{2 \pi^3}
\sum_{ab}\int\frac{d\theta}{2\pi}
\frac{\Gamma_{aabb}^2k_0^a(\theta)k_0^b(\theta)}{v_{F,a}^2v_{F,b}^2}
\label{apr4_1}$$ Note that the integrals over the Fermi surface contain $k^2_0 = k^2_0
(\theta_k)$ rather than the product of two $k_0$ factors at different points along the Fermi surface. This is a direct consequence of the fact that only backscattering contributes to (\[eq32\]) and (\[eq33\]). For a one band model with an isotropic Fermi surface, Eq. (\[apr4\_1\]) reduces to the result in [@Chubukov05]. For a generic interaction, the calculation goes through as before with $\Gamma^2_{aabb}$ replaced the components of the fully renormalized, symmetrized Fermi surface to Fermi surface backscattering amplitude so that $$\frac{\delta C}{T} =- \frac{3\zeta(3)}{2 \pi^3}
\sum_{ab}\int\frac{d\theta}{2\pi}
\frac{\left( \Gamma^{ab,c}(\pi)^2 + 3\Gamma^{ab,s}(\pi)\right)^2k_0^a(\theta)k_0^b(\theta)}{v_{F,a}^2v_{F,b}^2}
\label{apr4_1_1}$$
Susceptibility
==============
=0.8
Overview
--------
This section presents calculations of the non-analytical momentum and temperature dependence of the spin susceptibility of a two dimensional non-Galilean-invariant Fermi liquid system. with a Fermi surface without inflection points. In contrast to the specific heat, there are two classes of contributions to the nonalnayticities in the susceptibility [@ch_masl_latest]. One is of second order in the fullly renormalized interaction amplitudes, involves only the backscattering, and is treated here. The other, which we do not study here, is of third and higher orders, and involve averages of the interaction function over a wide range of angles (analogously to similar contributions to the specific heat of a [*three*]{} dimensional Fermi liquid [@Chubukov06]). The former process is dominant at weak coupling, and involves an integral of the square of the curvature over the Fermi arc. The latter process has a less singular dependence on the curvature.
Even to the order at which we work, many diagrams contribute (see Fig \[diagramsforchi\]); we evaluate one in detail to illustrate the basic ideas behind the calculation and then simply present the result for the sum of all diagrams. Consider for definiteness the “vertex correction” diagram – diagram 3 in Figure \[diagramsforchi\]. The analytical expression corresponding to this diagram is (we use a condensed notation in which (dk) stands for an integral over momentum, normalized by $(2\pi)^2$, and a sum over the corresponding Matsubara frequency) $$\delta\chi(q,0)=-4\sum_{ab} \int(dk)(dl) G^a(k+q)G^a(k)\Lambda^{ab}_k(l) G^a(k+l+q)G^a(k+l)
\label{chibasic}$$ and $\Lambda$ is the product of the interaction vertices and internal polarization bubble: $$\Lambda^{ab}_k(q)=\int(dp)\left(\Gamma^{aabb} (\theta) \right)^2 G^b(p+q/2)G^b(p-q/2)]
\label{lambdadef}$$ where $\theta$ is the angle between ${\bf k}$ and ${\bf p}$, and we have used the fact that ${\bf k}$ and ${\bf p}$ are near the Fermi surface. A complete calculation in the Galilean-invariant case shows that, just as for the specific heat, the non-analytical momentum dependence of the susceptibility arises from the regions of small $l$ and $l\sim 2k_F$. We consider these in turn.
Small l
-------
Here we choose a particular point $s_k$ on the Fermi surface and integrate over $\varepsilon_k$ and the corresponding frequency. We parametrize the position on the Fermi surface by the angle $\phi$ between ${\mathbf v}_k$ and ${\bf q}$ and use Eq \[jacobean\]. We adopt coordinates $l_\Vert$ and $l_\bot$ denoting directions parallel and perpendicular to ${\bf v}(s_k)$ and obtain $$\begin{aligned}
\delta \chi_{LW}(q)&=& - \frac{4}{ 2\pi }\sum_{ab} \int \frac{d\theta_k k^a_0 (\theta_k)}{2\pi (v^a_k)^2}
\int \frac{dl_\Vert dl_\bot}{(2\pi)^2}T\sum_\Omega
\frac{i\Omega}{\left({\bf v}^a_k\cdot {\bf q}\right)^2} \Lambda^{ab}_k(l_\Vert,l_\bot,\Omega)
\nonumber \\
&&\left(\frac{1}{\frac{i\Omega}{v^a_k}-{\hat {\bf v}}_k\cdot {\bf q}-l_\Vert}+\frac{1}{\frac{i\Omega}{v^a_k}
+{\hat {\bf v}}_k\cdot {\bf q}-l_\Vert}-\frac{2}{\frac{i\Omega}{v^a_k}-{\hat{\bf v}}_k\cdot {\bf q}-l_\Vert}\right)
\label{chismallq1}\end{aligned}$$ We may similarly evaluate $\Lambda$, proceeding from Eq (\[lambdadef\]). Choosing as origin the point ${\bf p}=-{\bf k}$, defining coordinates $p_\Vert$ and $p_\bot$ antiparallel and perpendicular to ${\bf v}_k$, integrating over $p_\Vert$ and the corresponding loop frequency gives $$\Lambda^{ab}_k(l_\Vert,l_\bot,\Omega)=\frac{\Gamma^{aabb}(\pi)^2}{2\pi (v^b_k)^2}\int\frac{dp_\bot}{2\pi}\frac{i\Omega}{\frac{i\Omega}{v^b_k} +l_\Vert-\frac{p_\bot l_\bot}{k^b_0}}
\label{Lambda2}$$
Viewed as a function of $l_\Vert$ the second line in Eq (\[chismallq1\]) decays rapidly ($\sim l_\Vert^3$) at large $l_\Vert$ and has poles only in the half plane $sgn Im l_\Vert=sgn\Omega$. Evaluation of the $l_\Vert$ integral by contour methods, closing the contour in the half plane $sgn l_\Vert=-sgn\Omega$ shows that nonanalyticities can only arise from singularities of $\Lambda$. Reference to Eq (\[Lambda2\]) shows that these can only arise from momenta ${\bf p}$ satisfying ${\bf v}_p\cdot{\bf v}_k<0$. In the Galilean-invariant case the integral could be evaluated exactly; the resulting non-analytical terms were found to be determined by a very small region around ${\bf p}=-{\bf k}$, i.e., around $\theta = \pi$ in (\[lambdadef\]). Here we assume that this is the case, and show [*a fortiori*]{} that the assumption is consistent.
Performing the integral over $l_\Vert$ yields $$\begin{aligned}
\delta \chi_{LW} (q)&=&-4i \sum_{ab}\frac{(\Gamma^{aabb} (\pi))^2}{(2\pi)^2}
\int\frac{d\theta_{k} k^a_0 (k)}{2\pi (v^a_kv^b_k)^2} T\sum_\Omega
\int\frac{dl_\bot dp_\bot}{(2\pi)^2}\left(\frac{\Omega}{{\bf v}^a_{k}\cdot {\bf q}}\right)^2 \text{sgn} \Omega
\nonumber \\
&&\left(\frac{1}{\frac{2i\Omega}{v_{avg}}-{\hat {\bf v}}_{k}\cdot {\bf q}-\frac{l_\bot p_\bot}{k^b_0}}
+\frac{1}{\frac{2i\Omega}{v_{avg}}+{\hat {\bf v}}_{k}\cdot {\bf q}-\frac{l_\bot p_\bot}{k^b_0}}
-\frac{2}{\frac{2i\Omega}{v_{avg}}-\frac{l_\bot p_\bot}{k^b_0}}
\right)
\label{chitriad3}\end{aligned}$$ with $v_{avg}$ defined in Eq \[vavg\]. Performing the sum over frequency and rescaling each of $l_\bot ,p_\bot$ by $\sqrt{k^b_0}$ yields $$\begin{aligned}
\delta \chi_{LW}(q)&=&-\sum_{ab} \frac{\left(\Gamma^{aabb} (\pi)\right)^2}{8\pi^3}
\int \frac{d\theta_k k^a_0k^b_0}{2\pi (v^a_k v^b_k)^2}\int_{-\Lambda}^\Lambda
\frac{dl_\bot dp_\bot}{(2\pi)^2}~\frac{v_{avg}^3}{(v^a)^2 }
\nonumber \\
&&\frac{\left(\Upsilon(l_\bot,p_\bot;{\hat {\bf v}}\cdot {\bf q})
+\Upsilon(l_\bot,p_\bot;-{\hat {\bf v}}\cdot {\bf q})-2\Upsilon(l_\bot,p_\bot;0)\right)}{({\hat{\mathbf v}}_k\cdot {\bf q})^2}
+...
\label{chismallq2}\end{aligned}$$ with $$\Upsilon(x,y;z)=(xy-z)^2 \log |xy-z|
\label{12_1}$$
Eq (\[chismallq2\]) is based on an expansion for small $l_\bot ,p_\bot$. The integrals over these quantities are cut off by other physics above a cutoff scale $\Lambda$ which we have written as a hard cutoff. The ellipsis denotes other terms arising from physics at and beyond the cutoff scale, which lead to additional, regular contributions to $\delta \chi$ involving positive, even powers of $q$. Evaluation of Eq \[chismallq2\] yields (details are given in Appendix \[app:details\]) $$\delta \chi_{LW}(q)=-\sum_{ab} \frac{\left(\Gamma^{aabb} (\pi)\right)^2\left|{\bf q}\right|}{
48\pi^3}
\int \frac{d\theta_k k^a_0 k^b_0}{2\pi (v^a_k)^2(v^b_k)^2}\frac{v_{avg}^3}{(v^a)^2 }
\left|{\hat {\mathbf v}}_k\cdot {\hat q}\right|+...
\label{chi3}$$ where $v_k =v_F (\theta_k)$, $k_0 = k_0 (\theta_k)$, ${\hat {\mathbf v}}_k\cdot {\hat q} = \cos (\theta_k - \theta_q)$, and $\theta_q$ is the angle between the direction of ${\bf q}$ and the direction of $\theta =0$; the ellipsis again denotes analytical terms. We see that the non-analytical term is explicitly independent of the cutoff, confirming the consistency of our analysis. An alternative evaluation of $\delta \chi_{LW}(q)$ is presented in the Appendix \[app:compl\].
For a circular Fermi surface, $k_0 = k_F$, $v_F$ is a constant, and Eq. (\[chi3\]) reduces to the previously known result [@Chubukov05]. However, the previously published computations are arranged in a way which apparently does not invoke the curvature at all. In Appendix C we show that the previous method does in fact involve the curvature, and leads to results equivalent to those presented here.
$2p_F$ processes
----------------
To evaluate the contribution of $2p_F$ processes we could proceed from Eq (\[chibasic\]) but expanding $\Lambda$ in ${\tilde q} = q-Q$, where, we remind, $Q =2k_{F}{\hat q}$. As before the products of $G$ produce an expression with all poles in the same half plane. Exploiting the non-analyticity of $\Pi({\bf q}+{\bf {\tilde q}},\Omega)$ we obtain an expression which has a nonanalytic part which evaluates to the same expression as Eq (\[chi3\]). Instead of presenting the details of this calculation, we present an alternative approach due to Belitz, Kirkpatrick, and Vojta [@bkv], in which one partitions the diagram into two “triads”, using the explicit form of $\Lambda_k$ $$\delta \chi (q) =-4\sum_{ab}\int(dl)(dk_1)(dk_2)\Gamma^{aabb}(\theta)^2\left[G^a(k_1+q)G^a(k_1)G^a(k_1+l)\right]
\left[G^b(k_2+q)G^b(k_2)G^b(k_2+l)\right]
\label{chitriad}$$ where $\theta$ is the angle between ${\bf k}_1$ and ${\bf k}_2$. Choosing a particular point on the Fermi surface and evaluating the integral over $\varepsilon_{k_1}$ and the associated frequency yields ($q_\Vert$ is the component of ${\bf q}$ parallel to the direction chosen for $k_1$) $$\begin{aligned}
\delta \chi& &=-4\int\frac{k_0(\theta_1)d\theta_{1}}{2\pi (v^a)^2}\frac{\Omega}{({\bf v^a}\cdot {\bf q})}
\left(\frac{1}{(\frac{i\Omega}{v^a}-l_\Vert)}-\frac{1}{\frac{i\Omega}{v^a}
-{\hat {\mathbf v}}\cdot {\mathbf q }-l_\Vert)}\right)
\nonumber \\
&&\Gamma^{aabb}(\theta_1)^2\left[G^b(k_2+q)G^b(k_2)G^b(k_2+l)\right]
\label{chitriad2}\end{aligned}$$ As in the previous calculation, singular contributions can only come from regions where $k_2$ is directed oppositely to $k_1$. Choosing as origin of $k_2$ the point diametrically opposite $k_1$, introducing parallel and perpendicular components as before and integrating over $k_{2\Vert}$, the associated frequency, and $l_\Vert$ we get $$\begin{aligned}
\delta \chi&=&\sum_{ab}4i \int\frac{k^a_0(\theta_1)d\theta_{1}}{2\pi (v^a)^2(v^b)^2}\frac{\Gamma^2 (\pi)}{(2\pi)^2} T\sum_\Omega
\int\frac{dk_\bot dl_\bot}{(2\pi)^2}\left(\frac{\Omega^2 \text{sgn} \Omega}{\left|({\bf v^a}\cdot {\bf q})({\bf v}^b \cdot {\bf q})\right|}\right)
\nonumber \\
&&\left(\frac{1}{\frac{2i\Omega}{v_{avg}}-{\hat {\mathbf v}}\cdot {\bf q}-\frac{l_\bot k_\bot}{k^b_0}}
+\frac{1}{\frac{2i\Omega}{v_{avg}}+{\hat {\mathbf v}}\cdot {\bf q}
-\frac{l_\bot k_\bot}{k^b_0}}-\frac{2}{\frac{2i\Omega}{v_{avg}}-\frac{l_\bot k_\bot}{k^b_0}}\right)
\label{chitriad3_1}\end{aligned}$$ Eq (\[chitriad3\_1\]) is seen to be of precisely the same form as Eq (\[chitriad3\]) and gives the same result; the only difference is the dependence on orbital index. Integrating over frequency and combining the results from small $q$ and $2k_F$ contributions gives an answer whose orbital dependent part depends on the velocities via the combination $$\frac{v^a(v^b)^3 +(v^a)^3v^b+2(v^a)^2(v^b)^2}{(v^a+v^b)^3}\left( {\hat {\mathbf v}}\cdot {\mathbf q}\right)
=v_{avg} {\hat {\mathbf v}}\cdot {\bf q}$$
Final Result
------------
Collecting the small $q$ and $2k_F$ contributions from all diagrams in Fig. \[diagramsforchi\] we find for the spin susceptibility $$\delta\chi_s(q) =\sum_{ab} \frac{v_{avg}|q|}{6\pi^{3}}~
\int_{0}^{2\pi} \frac{k^a_0(\theta_k) k^b_0 (\theta_k)d \theta_k}{2\pi (v^a)^2(v^b)^2} (\Gamma^{ab,s} (\pi))^2
\left|{\hat{\mathbf v}}\cdot {\hat {\mathbf q}}\right|
\label{eq32}$$ At $q=0$ and $T>0$, we have $$\delta\chi_s(T) = \sum_{ab} \frac{T}{\pi^{3}}
\int_{0}^{2\pi} \frac{d \theta_kk^a_0(\theta_k) k^b_0 (\theta_k)}{2\pi (v^a)^2(v^b)^2} (\Gamma^{ab,s} (\pi))^2). \label{eq33}$$ For an isotropic Fermi surface and one band, this again reduces to the result in [@Chubukov05]. To make contact with previous work, which considered a simplified interaction with indentical spin and charge components, $\Gamma^{ab,s} (\pi)$ has to be replaced by $\Gamma^{ab} (\pi)/2$. Higher order powers of $\Gamma$ do contribute to $|q|$ and $T$ terms in $\chi_s$, in distinction to $C(T)/T$, and at strong coupling, the non-analytic terms in the spin susceptibility are not expressed entirely via $\Gamma^2_s (\pi)$ [@ch_masl_latest]. the terms of order $\Gamma^3$ have a less singular dependence on the curvature. At weak and moderate coupling, though, Eqs. (\[eq32\]) and (\[eq33\]) should be sufficient. Finally, for the charge susceptibility $\chi_c (q,T)$ non-analytic contributions from individual diagrams are all cancelled out: the full $\chi_c (q,T)$ is an analytic function of both arguments.
Fermi surfaces with Inflection Points {#results}
=====================================
We see from Eq. (\[eq32\]) and (\[eq33\]) that as long as $k_0$ is finite all along the Fermi surface, the anisotropy of the Fermi surface affects the prefactors for $|q|$ and $T$ terms, but do not change the functional forms of the non-analytic terms in the specific heat and spin susceptibility. New physics, however, emerges when the Fermi surface develops inflection points at which $k_0(\theta_k)$ diverges. Inflection points are a generic feature of realistic Fermi surfaces of two dimensional materials. In this section we show how inflection points emerge and then indicate the modifications they make to the results presented above.
Inflection points in commonly occurring models
----------------------------------------------
We first note that many quasi-one dimensional organic materials have a band dispersion described by $$H_{organic}=-2ta\left(|k_x|-k_{F}\right)-2t'cos(k_yb)
\label{Horganic}$$ with $|t'|<<t$, lattice constants $a,b$ not too different, and a third dimensional coupling weaker than $t'$ by an order of magnitude. In this case $k_0=2t'cos(k_yb) +{\cal O}(t'^2/t)$ obviously vanishes at $k_yb\approx \pm \pi/2$. Thus inflection points are generic to quasi one dimensional materials.
We now consider the fully two dimensional $t-t'$ model, with quasiparticle dispersion $$\varepsilon_{k} = -2t (\cos k_{x} + \cos k_{y}) + 4 t^{\prime}\cos k_{x} \cos
k_{y} - \mu\label{d1}$$ We assume that $t$ and $t^{\prime}$ are positive and $\mu$ is negative; $t$ should be larger than $2 t^{\prime}$ for stability.
Consider now the Fermi surface crossing along the $(0,0)\rightarrow (\pi,0)$ direction (if it exists). The crossing occurs at $$-(2t-4t')cos(k_x)=\mu+2t$$ the velocity is along $x$ and the curvature may be read off by expanding to second order in $k_y$, giving $$k_0({\hat x})=\frac{1}{(2t-4t'cos(k_x))}
\label{k0x}$$ which is manifestly positive.
On the other hand, at the Fermi surface crossing along the diagonal $k_x=k_y$ we find $$k_0({\hat x}+{\hat y})=\frac{1}{2tcos(k_x)-4t'}
\label{k0diag}$$ which is positive at small $k_x$ but changes sign as the Fermi surface approaches the point $(\pi/2,\pi/2)$. We therefore conclude that for chemical potentials in the appropriate range inflection points must exist because the curvature has opposite sign at two points on the Fermi surface.
spin susceptibility and specific heat
-------------------------------------
We now analyze how the inflection points affect the nonanalytic terms in the spin susceptibility and specific heat. For simplicity, we restrict to one-band syatems. Quite generally, near each of the inflection points $k_0(\theta)$ behaves as $$k_0(\theta) \propto(\theta- \theta_{0})^{-1}$$ At $\theta= \theta_{0}$, the curvature diverges, i.e., there is no quadratic term in the expansion of the quasiparticle energy in deviations from the Fermi surface. In the generic case, the dispersion is then $$\varepsilon_{k} = v_{F} (\theta_{0}) k_{\Vert} + A k^{3}_{\bot} \label{eq34}$$ where, as before, the directions $k_\Vert$ and $k_\bot$ are along and transverse to the direction of the Fermi velocity at $\theta= \theta_{0}$. For a special situation when $\theta_{0}$ coincides with a reflection symmetry axis for $\varepsilon
_{k}$, i.e., when $\mathbf{v}_{F} (\theta_{0})$ is directed along the Brillouin zone diagonal in $t-t^{\prime}$ dispersion, the expansion of $\varepsilon_{k}$ in the direction transverse to the zone diagonal holds in even powers of $k_{\bot}$, i.e., at $\theta= \theta_{0}$, $$\varepsilon_{k} = v_{F} (\theta_{0}) k_{\Vert} + B k^{4}_{\bot} \label{eq35}$$
In both cases, a formal integration over $\theta$ in Eqs. (\[eq32\]) and (\[eq33\]) yields divergences. The divergences are indeed artificial and are cut by either $A$ or $B$ terms in the dispersion. The effect on $\delta
\chi(q,T)$ and $\delta C(T)/T$ can be easily estimated if we note that the angle integrals diverge as $\int d \theta k^2_0 (\theta) \sim \int d \theta/(\theta-\theta_{0})^{2}$. In a generic case, described by (\[eq34\]), $1/|\theta- \theta_{0}|$ has to be replaced by $1/[|\theta-
\theta_{0}| + A k_{\bot,typ}]$. The angle integral then yields $1/ k_{\bot,typ}$. It follows from Eq. (\[eq34\]) that $k_{\bot, typ} \sim(k_{\Vert,typ})^{1/3}$. It also follows from our consideration above that typical values of $k_{\Vert}$ are of order $|q|$. Combining the pieces, we find that the integral diverges as $|q|^{1/3}$, or $$\delta\chi(q) \propto \Gamma^{2} (\pi) |q|^{2/3} \label{eq36}$$ Similarly, at finite $T$ we obtain $$\delta\chi(T) \propto \Gamma^2 (\pi) |T|^{2/3} \label{eq37}$$ And the specific heat $$\delta C(T)/T \propto \Gamma^2 (\pi) |T|^{2/3} \label{eq37_a}$$
For special inflection points along symmetry direction, the analogous consideration shows the angle integral yields $1/(k_{\bot,typ})^{2}$. At the same time, it follows from Eq. (\[eq35\]) that $k_{\bot,typ} \sim
(k_{\Vert,typ})^{1/4} \sim|q|^{1/4}$. Then the angle integral diverges as $|q|^{1/2}$, and $$\delta\chi(q) \propto \Gamma^2 (\pi) |q|^{1/2} \label{eq36'}$$ while $$\delta\chi(T) \propto \Gamma^2 (\pi) |T|^{1/2} \label{eq37'}$$ and $$\delta C(T)/T \propto \Gamma^2 (\pi) |T|^{1/2} \label{eq37'_a}$$ The results, Eqs. (\[eq36\] -(\[eq37’\_a\])), differ from the results by Fratini and Guinea [@Fratini02]. They obtained $\chi(T) \propto T \log T$ for a generic inflection point, and $\chi(T) \propto T^{3/4} \log T$ for a special inflection point. In our calculations, a similar $T \log^{2} T$ behavior for a generic case would hold if the non-analytic terms were coming from vertices $\Gamma (\theta)$ with arbitrary $\theta$ rather than $\theta = \pi$. Then, e.g., the coefficient for the $|q|$ term in the spin susceptibility would be given by a double integral $\int d \theta_{1} |k_0(\theta_{1})|~\int d \theta_{2} |k_0
(\theta_{2})|$. Each integral diverges logarithmically, and the correction would then scale as $T \log^{2} T$. The emergence of the anomalous power of temperature or momenta for a generic inflection point in our calculations is the direct consequence of the one-dimensionality of the relevant interaction. We see therefore that the anisotropy of the Fermi surface is an ideal tool to probe the fundamental 1D nature of the non-analyticities in a Fermi liquid.
Application to $Sr_2RuO_4$
==========================
A crucial and so far unresolved question related to the results reported here and in previous papers is the observability of the effects. Evidence for the $T^3lnT$ nonanalyticities expected in three dimensional materials have been observed in the specific heat of $^3He$ [@Abel66] (indeed this observation played a crucial role in stimulating the theoretical literature) and similar effects have been noted in the specific heat of the heavy fermion material $UPt_3$ [@Stewart84]. More recently, a linear in $T$ behavior has been observed in the specific heat of fluid monolayers He$^{3}$ adsorbed on graphite [@casey], but to our knowledge no evidence for nonanalytic terms in the susceptibility has been reported.
We consider here $Sr_2RuO_4$, a highly anisotropic layered compound for which detailed information about the shape of the Fermi surface, the quasiparticle mass enhancements, the susceptibility and optical conductivity is available [@Bergemann03; @Ingle05; @Lee06]. These data imply [@Bergemann03] that the material is “strongly correlated”, in the sense that Fermi velocities and susceptibilities are substantially renormalized from the predictions of band theory. The data also suggest that the dynamical self energy is at most weakly momentum-dependent, because the shapes of the Fermi surface deviate only slightly from those found in band structure calculations, implying that the self energy has a much stronger frequency dependence than momentum dependence. We extract Fermi surface shape and Fermi velocities from quantum oscillation data [@Bergemann03] and cross-check with photoemission data where available. We use available susceptibility [@Bergemann03] and optical data [@Lee06] to estimate the interaction functions.
It is also useful to consider the leading low-T behavior of the specific heat coefficient, which in a Fermi liquid is given in terms of fundamental constants and a sum over bands of the average of the inverse of the Fermi velocity as $$\frac{C}{T}=\frac{2\pi^2}{3} \sum_\lambda I_\lambda
\label{Cfl}$$ with $$I_\lambda=\oint \frac{d \theta}{4\pi^2} \frac{k_{F,\lambda}(\theta)\sqrt{1+\left(\frac{dk_F(\theta)}{k_F(\theta)d\theta}\right)^2}}{\left|v_{F,\lambda}(\theta)\right|}
\label{Idef}$$
For Galilean-invariant fermions, it is customary to use the relation (we use units where $\hbar=0$) $v_F=k_F/m$ to define a band mass $$m_\lambda=2\pi I_\lambda
\label{mlambdadef}$$ so $C/T=(\pi/3)\sum_\lambda m_\lambda$. To obtain the specific heat in conventional units ($mJ/mol/K^2$) one must multiply by $k_B^2$ and by the Avogardo number.
We begin with the results for the Fermi surface shape and Fermi velocities. In $Sr_2RuO_4$, the relevant electrons are the three $t_{2g}$ symmetry $Ru$ d-orbitals and there are accordingly three bands at the Fermi surface, conventionally labeled as $\alpha,\beta,\gamma$. The Fermi surface [*shape*]{}, shown in Fig \[fsfig\] is well described by the two dimensional tight binding model $$\begin{aligned}
\varepsilon_{\gamma}(k_x,k_y)&=&-2t_{1\gamma}\left(cos(k_x)+cos(k_y)\right)-4t_{2\gamma}\left(cos(k_x)cos(k_y)\right) - \varepsilon_{0\gamma} \label{egamma} \\
\varepsilon_{\alpha,\beta}&=&-(t_{1\alpha} + t_{2\alpha})\left(cos(k_x) + cos(ky)\right) - \varepsilon_{0xz}
\nonumber \\
&&\pm \sqrt{\left((t_{1\alpha} - t_{2\alpha})(cos(k_x)- cos(k_y)\right)^2 + 16t_{3\alpha}^2sin(k_x)^2sin(k_y)^2}
\label{eab}\end{aligned}$$ (couplings in the third dimension are an order of magnitude smaller).
A detailed quantum oscillation study has been performed by Bergemann and collaborators [@Bergemann03]. These authors present in Table 4 of their work a tight-binding parametrization which reproduces the [*shape*]{} of the Fermi surface. They also present results for the mass enhancements in each Fermi surface sheet, which may be converted into experimental estimates for $I_\lambda$. The shape, of course, does not depend on the magnitudes of the tight binding parameters. We accordingly rescale these in order to obtain velocities (more precisely, integrals $I_\lambda$) corresponding to the data reported by Bergemann et al[@Bergemann03].
band $\varepsilon_0$ $t_1$ $t_2$ $t_3$ $\frac{m_\lambda}{m_e}$
---------- ----------------- ------- ------- ------- ------------------------- -- -- -- -- -- --
$\alpha$ 0.13 0.13 0.013 0.02 2.5
$\beta$ 0.16 0.15 0.013 0.02 5.8
$\gamma$ 0.012 0.079 0.032 0 16
: Tight binding band parameters (in \[eV\]) which reproduce the shape and, approximately, the Fermi velocities of the three bands at the Fermi surface of $Sr_2RuO_4$. Parameters are taken from Table $4$ of Ref [@Bergemann03] and then renormalized to produce sheet-dependent quasiparticle mass enhancements approximately consistent with experiment. Last column: mass parameter computed using Eqs (\[Idef\]), (\[eab\]).
\[default\]
A few remarks about the velocities and masses are in order. First, the calculated $\gamma$ band properties depend very sensitively on how close the $\gamma$ ($xy$)-derived band approaches the van Hove points $(\pi,0)$, $(0,\pi)$. Published band calculations [@Oguchi95; @Mazin97; @Liebsch00] show wide variations in the position of the the singularity relative to the Fermi level. Second, the $'mass'$ derived from the specific heat involves both the velocity and the geometrical properties of the Fermi surface. The mass for the $\alpha$ band is small because of its small size, even though its velocity is relatively small. Third, and most important, the curvature of the $\alpha,\beta$ bands depends very sensitively on the parameters $t_{2\alpha},t_{3\alpha}$; the velocities also depend somewhat on these parameters. A recent angle-resolved photoemission experiment [@Ingle05] reports that the $\alpha$ band Fermi velocity at the zone face crossing point is $v_\alpha=1.02eV-\AA$; the parameterization used here gives an essentially identical value.
We now turn to the Landau interaction function. A complete experimental determination is not available, but considerable partial information exists. Bergemann and co-workers [@Bergemann03] have determined, for each band, the spin polarization induced by a uniform external magnetic field, so the $"L=0"$ spin channel Landau parameters may be estimated. Optical conductivity data [@Lee06] provide some information on the $"L=1"$ spin-symmetric channel current response. General arguments suggest that the charge compressibility is only weakly renormalized in correlated oxide materials, allowing a rough estimate of the $"L=0"$ charge channel interaction. We will use this information to estimate the scattering amplitudes and hence the nonanalytic terms in the susceptibilities. These estimations are certainly subject to large uncertainties, but we hope they will give a reasonable idea of the magnitude of the effects.
In the three-band material of present interest the interaction function is a symmetric $3\times3$ matrix with components $\Gamma^{a,b}$ labelled by orbital or band indices, which should then be decomposed into charge (symmetric) and spin (antisymmmetric) components and into the angular harmonics appropriate to the tetragonal symmmetry of the material. We begin with the isotropic $"L=0"$ spin channel. We assume (consistent with the usual practice in transition metal oxides) that the deviations from $O(3)$ symmetry, while crucial for electronic properties such as the band structure and conductivity, are not crucial for the local interactions, which arise from the physics of the spatially well localized $d$ electrons. This implies that the interactions are invariant under permutations of orbitals, so that it is reasonable to assume that the two-particle irreducible spin channel interaction takes the simple Slater-Kanamori form with two parameters, which we write as $\Gamma^{a,a}\equiv U_{eff}$ and $\Gamma^{a\neq b}\equiv J_{eff}$. Thus the physical static susceptibilities are given by $${\mathbf \chi}=\left({\mathbf \chi}^{-1}_0+{\mathbf \Gamma}(U_{eff},J_{eff})\right)^{-1}
\label{chidef}$$ and fix the parameters $U_eff$ and $J_{eff}$ by comparing measured susceptibilities to the values predicted by the renormalized tight binding parameters.
Ref [@Bergemann03] presents (as mass enhancements) data for the spin susceptibility of each band (obtained from the spin splitting of the Fermi surfaces); finding $\chi^{\alpha,\alpha}/\chi_0^{\alpha,\alpha}\approx 1.2$, $\chi^{\beta,\beta}/\chi_0^{\beta,\beta}\approx 1.3$ and $\chi^{\gamma,\gamma}/\chi_0^{\gamma,\gamma}\approx 1.6$. We estimate $U_{eff}\approx 0.033$ and $J_{eff}=-0.008$, where $\chi_0$ is the susceptibility implied by the quantum oscillations Fermi surface and mass. This implies that the dimensionless Landau interaction parameters (in the limit $\omega/k\rightarrow 0$ limit $A^{ab}\equiv \Gamma^{ab}\sqrt{ \chi^{a}\chi^b}$ $${\mathbf A}^{spin}=\left(\begin{array}{ccc}-0.053& 0.040 & 0.10 \\0.04 &- 0.13& 0.17 \\0.10 & 0.17 & -0.40\end{array}\right)
\label{gamanti}$$
The uncertainties in the off diagonal components are large, perhaps $50\%$, but because the interactions enter squared, the contribution of the off diagonal components is not very significant. The dominant term is the $\gamma-\gamma$ band interaction, as expected because it has the largest mass and the largest susceptibility enhancement, but that all of the other contributions taken together make a non-negligible contribution to the interaction. Finally, we note that the spin channel renormalizations are not large, so use of the second order result is not unreasonable.
We now turn to the charge channel, beginning with the compressibility. There is no experimental information available. However, it is generally believed that for systems, such as transition metal oxides, with strong local interactions the total charge susceptibility is not strongly renormalized, so that the Landau parameter acts to undo the effects of the mass enhancement. Further, if the $J$ (orbital non-diagonal) component of the interaction is not too small relative to the $U$ (orbital diagonal component) then a residual interaction acts to shift the levels such that the ratio of occupancies of each of the three $t_{2g}$ orbitals remains constant under chemical potential shifts. Taking as unrenormalized value the susceptibilities folllowing from the tight binding parameters given in Ref [@Bergemann03] we then obtain $${\mathbf A}^{S,0}=-\left(\begin{array}{ccc}0.40 & 0.034 & 0.026 \\0.034 & 0.8 & 0.026 \\0.026 & 0.026 & 0.80\end{array}\right)
\label{Fc}$$ with again considerable uncertainty in the off diagonal components.
Finally, we turn to the current renormalization. The optical conductivity is commonly presented in the extended Drude form $$\sigma(\Omega)=\frac{\frac{e^2}{\hbar c}D_{band}}{-i\Omega\frac{m^*(\Omega)}{m}+\Gamma(\Omega)}
\label{drudedef}$$ where $c$ is the mean interplane spacing and $m^*/m$ has the meaning of an optical mass enhancement defined with respect to a reference value determined by $D_{band}$. In a Fermi liquid at low temperatures, $\Gamma(\Omega \rightarrow 0)$ is very small and (assuming tetragonal symmetry) $$D_{band}\frac{m}{m^*(0)}\equiv D=\sum_\lambda
\oint \frac{d \theta}{4\pi^2} k_{F,\lambda}(\theta)
\sqrt{1+\left(\frac{dk_F(\theta)}{k_F(\theta)d\theta}\right)^2}
\left|v_{F,\lambda}(\theta)\right|\left(1+\frac{F^{1S}_\lambda}{2}\right)
\label{ddef}$$ Note that the numerical value of the mass enhancement $m^*/m^0$ depends on the choice of reference value $D_{band}$ but that $D$ is a physically meaningful quantity determined directly from the data.
The room temperature conductivity of $Sr_2RuO_4$ has been measured [@Lee06]. These authors chose the value $e^2D_{band}/\hbar c$ (which they denote as $\omega_p^2/4\pi$) to correspond to $\omega_p^2\approx 8\times 10^{8} cm^{-2}$ and find $m^*/m(\Omega \rightarrow 0)$ (which they denote as $\lambda$) to be $\approx 3.5$. This implies that $D\approx 0.13eV$, somewhat smaller than the value $0.18$ obtained from Eq \[ddef\], implying that the average over all bands is $F^{1S} \approx -0.55$. The temperature dependence of $D$ in $Sr_2RuO_4$ has not been measured, but it seems reasonable that $D$ should decrease as $T$ decreases, implying a further increase in the magnitude of $F^{1S}$. Determining the temperature dependence of the optical mass is therefore an important issue.
Ref [@Bergemann03] presents, as masses, data for the cyclotron resonance frequencies for the different Fermi surface sheets. These masses should be essentially equivalent to the $D$ values quoted above. Bergemann et. al. emphasize that the frequencies are subject to large errors, and that the results should be regarded as tentative. The quoted cyclotron masses correspond to $D$ values about a factor of two larger than those implied by the measured fermi velocities, and about a factor of $3$ larger than the values inferred from the optical data. In view of the stated large uncertainties in the measurement and the qualitative incosistency with the optical data, we disregard the cyclotron resonance measurements here.
Now, the crucial object for the specific heat is the backscattering amplitude. A negative $F^{1S}$ implies a positive backscattering amplitude, so as a rough approximation to the effects of the current channel Landau renormalization we add the interaction corresponding to $F^{1S}=-0.6$ to the diagonal components of Eq \[Fc\].
The crucial points emerging from this analysis are that the reducible interactions in the charge channel are of order unity, whereas those in the spin channel are somewhat smaller, implying a larger nonanalyticity in the specific heat than in the susceptibility. Substituting the interaction amplitudes into Eqs. \[apr4\_1\_1\], \[eq33\] and performing the fermi surface averages then yields the following estimates $$\begin{aligned}
\gamma(T)&=&36 ~mJ/mol-K^2\left(1-0.0015T[K]\right) \\
\chi(T)[Si/Volume]&=&1.5\times 10^{-4} \left(1-.00001 T[K]\right)\end{aligned}$$ The small magnitude of the corrections (especially to the spin susceptibiltiy) follows from the small prefactors in Eq (\[eq33\]) and the not too large Landau renormalizations. The size of the effect is increased by the relatively small curvatures of the $\alpha$ and, especially, $\beta$ bands, and we note that substantial increases in the coefficients occur if the mixing coefficients $t_{2\alpha},t_{3\alpha}$ in Eq \[eab\] are reduced. We expect the results to be valid above a (still not well determined) scale probably $\sim 1-2K$ at which the Fermi surface warping becomes important enough to make the material three dimensional, and below the scale at which Fermi liquid theory breaks down, and we see that temperatures of order $10K$ lead to $20\%$ deviations in the value of the specific heat coefficient and to $1\%$ changes in $\chi$.
Replacing $Sr$ by $Ca$ leads to a dramatic (factor $\sim 100$ in $Sr_{0.5}Ca_{1.5}RuO_4$) enhancement of the susceptibility. It seems likely that this increase is not due to a decrease in the fermi velocities, but must be interpeted as a dramatic increase in the spin Landau parameter, suggesting perhaps that nonanalytic $T$-dependence of $\chi$ might be more easily observed in $Ca$ doped materials, although in this case disorder effects would need to be considered.
Conclusions
===========
In this paper, we studied non-analytic terms in the spin susceptibility and specific heat in 2D systems with anisotropic, non-circular Fermi surfaces. For systems with circular Fermi surfaces, the non-analytic terms in $\chi_{s} (q,T)$ and $C(T)/T$ are linear in $max (q,T)$. We argued that the anisotropy of the Fermi surface serves as a testing ground to verify the theoretical prediction that the non-analytic terms originate from a single 1D scattering amplitude which combines two 1D interaction processes for particles at the Fermi surface in which the transferred momenta are either $0$ or $2k_{F}$, and, simultaneously, the total moment is zero. We obtained explicit expressions for the non-analytic momentum and temperature dependences of the spin susceptibility and the specific heat in systems with non-circular Fermi surfaces and demonstrated that for the Fermi surfaces with inflection points, the the non-analytic temperature and momentum dependences are $\chi_{s} \propto max (q^{2/3}, T^{2/3})$, $C(T)/T \propto T^{2/3}$ in a generic case, and as $\chi_{s} \propto max (q^{1/2}, T^{1/2})$, $C(T)/T \propto T^{1/2}$ for the special cases when the inflection points are located along symmetry axis for the quasiparticle dispersion. We estimated the order of magnitude of the effects in the quasi two dimensional material $Sr_2RuO_4$.
It is our pleasure to thank C. Bergemann, D.L. Maslov, A. Mackenzie and N. Ingles for useful conversations. The research is supported by NSF Grant No. DMR 0240238 (AVC) and DMR 0431350 (AJM), and AJM thanks the DPMC at the University of Geneva for hospitality while this work was completed.
The details of the evaluation of Eq.(\[chi3\]) {#app:details}
==============================================
For simplicity, we neglect band index, i.e., set $v^a_k = v^b_k = v_k$, and $k^a_0 (k) = k^b_0 (k) = k_0 (k)$. Using (\[12\_1\]), Eq. (\[chismallq2\]) is re-expressed as $$\begin{aligned}
&&\delta \chi_{LW}(q) =-\frac{\Gamma^2 (\pi)}{64\pi^5}
\int \frac{d\theta_k k_0 (k)}{2\pi v^3_k}\int_{-\Lambda}^\Lambda \int_{-\Lambda}^\Lambda \frac{dx dy}{
\epsilon^2}~\times
\nonumber \\
&& \left[(xy-\epsilon)^2 \log{(xy-\epsilon)^2} + (xy+\epsilon)^2 \log{(xy+\epsilon)^2} - 2 x^2 y^2 \log{x^2y^2}\right]
\label{12_2}\end{aligned}$$ where $\epsilon = ({\bf v}_k \cdot {\bf q}) k_0 (k)/v_k$. Rescaling $x = \sqrt{|\epsilon|} {\bar x}, ~y = \sqrt{|\epsilon|} {\bar y}$, substituting into (\[12\_2\]) and dropping irrelevant terms confined to high energies, we obtain $$\delta \chi_{LW}(q) = -\frac{\Gamma^2 (\pi)}{64\pi^5}
\int \frac{d\theta_k k_0 (k) |\epsilon|}{2\pi v^3_k} Z
\label{12_3}$$ where $$Z = \int_{-\Lambda}^\Lambda \int_{-\Lambda}^\Lambda
d{\bar x} d{\bar y} \left[({\bar x} {\bar y}-\epsilon)^2 \log{({\bar x} {\bar y}-\epsilon)^2} + ({\bar x} {\bar y}+\epsilon)^2 \log{({\bar x} {\bar y}+\epsilon)^2} - 2 ({\bar x} {\bar y})^2 \log{({\bar x} {\bar y})^2}\right]
\label{12_21}$$ Introducing further ${\bar x} = \sqrt{2r} \cos \phi/2$ and ${\bar y} = \sqrt{2r} \sin \phi/2$, we rewrite $Z$ as $$\begin{aligned}
Z = 2 \int_{0}^\pi d \psi \int_{0}^\Lambda
&dr& \left[(r \sin{\phi} -1)^2 \log{(r \sin{\phi} -1)^2} +
(r \sin{\phi} +1)^2 \log{(r \sin{\phi} -1)^2} \right. \nonumber \\
&&\left. - 2 r^2 \sin^2{\phi} \log{ r^2 \sin^2{\phi}}\right]
\label{12_31}\end{aligned}$$ Subtracting the irrelevant large $r$ contribution $6 + \log{ r^2 \sin^2{\phi}}$ from the integrand in (\[12\_31\]), we the universal part of $Z$ in the form $$\begin{aligned}
Z = 2 \int_{0}^\pi d \psi \int_{0}^\Lambda
&dr& \left[ r^2 \sin^2{\phi} \log \left(1 - \frac{1}{r^2 \sin^2{\phi}}\right)^2
+ 2 r \sin{\phi} \log\left(\frac{1 + \frac{1}{r \sin{\phi}}}{1 -\frac{1}{r \sin{\phi}}}\right)^2 \right. \nonumber \\
&&\left. -6 + \log \frac{ (r^2 \sin^2{\phi} -1)2}{r^4 \sin^4{\phi}}\right]
\label{12_4}\end{aligned}$$ One can make sure that the integral over $r$ vanishes if we set the upper limit at $\Lambda = \infty$. as the integrant depends on $r$ only via $r \sin \phi$, the finite contribution to the integral comes from $\lambda \sin \phi = O(1)$, i.e. from a narrow range of $\phi$ either near zero or near $\pi$. The contributions from these two regions are equal. Resticting with the contribution from small $\phi$, expanding $\sin \phi \approx \phi$ and introducing $z = r \phi$ and $t = \Lambda \phi$, we obtain from (\[12\_4\]) $$Z = 4 \int_{0}^\infty \frac{dt}{t} \int_{0}^t
dz \left[ z^2 \log \left(1 - \frac{1}{z^2}\right)^2
+ 2 z\log\left(\frac{1 + \frac{1}{z}}{1 -\frac{1}{z}}\right)^2 -6 + \log \left(\frac{z^2 -1}{z^2}\right)^2\right]
\label{12_5}$$ Changing the order of the integration, we obtain for the universal part of $Z$ $$\begin{aligned}
Z &=& -4 \int_{0}^\infty dz \log z \left[ z^2 \log \left(1 - \frac{1}{z^2}\right)^2
+ 2 z\log\left(\frac{1 + \frac{1}{z}}{1 -\frac{1}{z}}\right)^2 -6 + \log \left(\frac{z^2 -1}{z^2}\right)^2\right] \nonumber \\
&&= \frac{4\pi^2}{3}
\label{12_6}\end{aligned}$$ Substituting this into (\[12\_3\]) and using the definition of $\epsilon$, we reproduce (\[chi3\]).
An alternative evaluation of $\delta\chi_{LW} (q)$ {#app:compl}
==================================================
In this Appendix we present a complementary evaluation of $\delta\chi_{LW} (q)$ using a somewhat different computational procedure. We again restrict to one band. The point of departure are Eqs. (\[chibasic\]) and (\[lambdadef\]), which we re-write at $T=0$ as $$\begin{aligned}
\delta\chi_{LW} (q) & = -4 \int\int\int\int\frac{d^{2}k~d^{2}q~d\omega
d\Omega }{(2\pi)^{6}}~ \Gamma^{2} \Pi(l,\Omega ) G_{0}(\mathbf{k},\omega ) \times\nonumber\\
& G_{0}(\mathbf{k}+\mathbf{l},\omega +\Omega )~ G_{0}(\mathbf{k}+\mathbf{q}+\mathbf{l},\omega +\Omega )~G_{0}(\mathbf{k}+\mathbf{q},\omega )
\label{23_1}\end{aligned}$$
We first integrate over internal momenta $\mathbf{k}$ and frequency $\omega$ in the fermionic propagator. Expanding the result in $q^{2}$, we obtain $$\delta\chi_{LW} (q) \propto \Gamma^2 q^{2} \int_{0}^{\infty}d \Omega
\Omega \int d^{2} l \int d \theta_{1} \frac{k_0 (\theta_{1})}{v_{F}
(\theta_{1})}~ \frac{1}{ (i \Omega - v_{F} l \cos\theta_{1})^{5}} \Pi(l,
\Omega) \label{eq27}$$ where $\theta_{1}$ is the angle between $\mathbf{l}$ and $\mathbf{k}$. Directing $l_{x}$ and $l_{y}$ along and transverse to $\mathbf{k}$ and substituting the polarization operator we obtain $$\begin{aligned}
\delta\chi_{LW} (q) & \propto \Gamma^2 q^{2} \int_{0}^{\infty}d
\Omega \Omega^{2}_{m} \int d l_{x} \int d \theta_{1} \frac{k_0(\theta_{1})}{v_{F} (\theta_{1})}~ \frac{1}{ (i \Omega - v_{F} (\theta_{1})
l_{x})^{5}}~\times\nonumber\\
& \int d l_{y} \int d \theta~\frac{k_0 (\theta_{1} + \theta)}{ v_{F}
(\theta_{1} + \theta)}\frac{1}{i\Omega - v_{F} (\theta_{1} + \theta)
(l_{x} \cos\theta+ l_{y} \sin\theta)} \label{eq28}$$ \[$\theta$ is the angle between two internal momenta $\mathbf{p}$ and $\mathbf{k}$\]. We now integrate over $l_{y}$ and then over $\theta$. The full result for this 2D integral depends on particular forms of $k_0(\theta)$ and $v_{F} (\theta)$. However, we only need from the integral over $d l_y d \theta$ the term which is non-analytic in the lower half-plane of $l_{x}$ (this will allow us to avoid a degenerate pole at $v_{F} l_{x} = i \Omega$). One can easily verify that the non-analyticity comes from the integration near $\theta= \pi$ which yields, instead of the second line in (\[eq28\]) $$i ~\frac{k_0(\theta_{1}-\pi)}{2 v^{2}_{F} (\theta_{1}-\pi)} \log\left[
i\Omega + v_{F} (\theta_{1} -\pi) l_{x}\right]$$ For the Fermi surfaces with inversion symmetry (which we will only consider) $k_0 (\theta_{1}-\pi) = k_0 (\theta_{1})$ and $v_{F} (\theta_{1}-\pi) = v_{F}
(\theta_{1})$ (we recall that $v_{F} (\theta)$ is the modulus of the Fermi velocity at a particular $\theta$). Substituting this result into (\[eq28\]) and extending the integral over $l_{x}$ onto the lower half-plane, we obtain Eqn (\[chi3\]).
We also verified that the same result could be obtained by evaluating the singular part of $\Pi(l, \Omega)$ by explicitly expanding near $\mathbf{p} = -
\mathbf{k}$ and expanding the dispersion $\varepsilon_{p} = \varepsilon_{-k + l}$ to second order in $l$. In this computation, one power of $k_0(\theta)$ comes from expanding the dispersion, while the other comes from the Jacobean of the transformation from $d^{2}k$ to $d\varepsilon_{k} d\theta$.
Reevaluation of $\delta\chi_{LW} (q)$ for an isotropic Fermi surface {#app:B}
====================================================================
In this Appendix, we reconsider a previously published [@Chubukov05] evaluation of $\delta\chi_{LW} (q)$. Although this evaluation leads to results identical to those we presented in the body of the paper for a circular Fermi surface, it apparently does not invoke the curvature explicitly. Here we deconstruct this analysis, showing how the curvature actually enters even when Fermi surface is circular.
We begin from Eq (\[23\_1\]). The analysis presented in the main body of the paper involves choosing a direction for ${\bf l}$, and then performing the integral over ${\bf k}$, which picked out points with a definite relationship to ${\bf l}$ and involved the curvature in a direct way, and finally integrating over $l$. On the other hand, the “conventional” analysis involves first fixing the direction of ${\bf k}$, integrating over the magnitude of $k$ and over $q$, and then averaging over the direction of ${\bf k}$. In this method one expands $\epsilon_{k}$, $\epsilon_{k+l}$, $\epsilon
_{k+q}$ and $\epsilon_{k+l +q}$ in (\[23\_1\]) to linear order in the deviations from the Fermi surface as $\epsilon_{k} = v_{F} (k-k_{F})$, $\epsilon_{k+l} =
\epsilon_{k} + v_{F} l_{x}$, etc. Because the Green functions have been linearized the curvature apparently does not enter, in contrast to the previous derivation, where the dependence of the Green function lines on curvature was essential.
Integrating over $k$ and over the corresponding Matsubara frequency, and expanding the result in powers of $q$, we obtain at $T=0$, neglecting regular terms $$\delta\chi_{LW} (q) \propto \Gamma^{2} q^{2}
\int_{0}^{\infty}d \Omega
\Omega \int dl_{x} \frac{1}{ (i \Omega - v_{F} l_{x})^{5}} ~\int d
l_{y} \Pi({\bf l}, \Omega) \label{eq1}$$ The key point is that the curvature dependence is hidden in the polarizibility $\Pi$, but in the circular Fermi surface limit this dependence is hidden. To make the curvature dependence manifest we use the fact that only backscattering contributes and evaluate the polarization bubble $\Pi(l, \Omega) = \int d^{2} t d \omega^{\prime}G_{0} (t, \omega^{\prime})
G_{0} (l+t, \Omega + \omega^{\prime})$ by expanding near $\mathbf{t} = -\mathbf{k}$. Introducing $\mathbf{t} + \mathbf{k} = \mathbf{p}$ and assuming that $p$ is small, we expand the dispersions $\epsilon_{t} = \epsilon_{-k_x + \mathbf{p}}$ and $\epsilon_{t+l} = \epsilon_{-k_{x} + \mathbf{p} + \mathbf{l}}$ to second order in $p$: $$\epsilon_{t} = \epsilon_{-k_{x} + \mathbf{p}} = - v_{F} \left(p_{x} + \frac
{p^{2}_{y}}{2k_0}\right); ~~ \epsilon_{t+l} = \epsilon_{-k_{x} + \mathbf{p} +
\mathbf{l}} = - v_{F} \left(p_{x} + l_{x} + \frac{(p_{y} + l_{y})^{2}}{2k_0}
\right)
\label{eq6}$$ Substituting this expansion into the bubble and integrating over $p_{y}$ we obtain $$\Pi(l, \Omega) = i \frac{\Omega k_0}{2\pi^{2} v^2_{F} l_{y}}~\log{\frac{A
l_{y} - (i\Omega + v_{F} l_{x})}{-A l_{y} - (i\Omega + v_{F} l_{x})}}
\label{eq7}$$ where $k_0 A/v_F \sim k_F$ is the upper limit of the integral over $p_y$. Integrating next $\Pi (l, \Omega)$ over $l_{y}$, we find the same branch cut singularity as in “conventional” approach $$\int dl_{y} \Pi(l, \Omega) = \frac{k_0 \Omega}{\pi v^2_{F}}~\log[i\Omega+
v_{F} l_{x}] \label{eq8}$$ Substituting this result into (\[23\_1\]) and using the fact that $d^2 k$ in (\[23\_1\]) can be re- expressed as $(k_0/v_F) d \epsilon_k d \theta$, we reproduce Eq. (\[chi3\]) for a circular Fermi surface, and also reproduce Eq. (4.18) in \[a\], but with $k_0 /v_F$ instead of $m$.
For completeness, we also show that $\delta \chi_{2k_F} (q)$ in systems with a circular Fermi surface can also be obtained with and without the curvature. A “conventional” computation \[a\] expresses $\delta \chi_{2k_F} (q)$ in terms of the curvature. An alternative computational scheme involves the same “triad” method that we used in the main text. In this scheme, the original expansion near $2k_F$ momentum transfer contains the curvature, but it disappears from the answer at the latest stage. Performing the same integrations over $\epsilon_k$, the corresponding frequency and $l_y$ as in the main text, we find (keeping $\Gamma = \Gamma (\theta)$) $$\begin{aligned}
&&\delta\chi_{2k_{F}} (q) \propto\sqrt{k_0} \int d \theta \Gamma^{2} (\theta)
\int_{v_F |q|}^{\infty}d\Omega
\Omega^{2}\nonumber \\
&& \times \int\frac{dl_{x}}{(l_{x} - i
\Omega)^{2} (i\Omega - v_F(l_{x} \cos\theta+ \frac{k_0}{2} \sin^{2}\theta))^{3/2}} \label{eq22}$$ For $\cos\theta<0$, the two double poles are in different half-planes of $l_{x}$. Integrating over $l_{x}$, we then obtain $$\delta\chi_{2k_{F}} (q) \propto\sqrt{k_0} \int_{\pi/2}^{\pi}
d \theta \Gamma^{2} (\theta)~\int_{v_F |q|}^{\infty} \frac{d\Omega\Omega^{2}}
{(\frac{v_F k_0}{2}
\sin^{2}\theta+ i\Omega (1-\cos\theta))^{5/2}} \label{eq23}$$ Since relevant $\Omega \sim v_{F} |q|$, the $\theta$ integral is confined to $\theta=\pi$. Expanding near $\pi$ we obtain $$\int_{\pi/2}^{\pi} \frac{d \theta \Gamma^{2} (\theta)}{(\frac{v_F k_0}{2} \sin^{2}\theta+
i\Omega (1-\cos\theta))^{5/2}} \approx \Gamma^{2} (\pi) \int_{0}^{\infty}
\frac{dx}{(\frac{v_F k_0}{2} x^{2} - 2 i \Omega)^{5/2}} = -\frac{U^{2}(\pi)}{3 \Omega^{2} \sqrt{v_F k_0}} \label{eq24}$$ Substituting this into (\[eq23\]), we find that $k_0$ is canceled out, and $$\delta\chi^{2k_{F}} (q) \propto \Gamma^{2} (2k_{F}) \int_{v_{F} |q|}^{E_{F}} d
\Omega\rightarrow \Gamma^{2} (\pi) |q|$$ Restoring the prefactor, we reproduce the same result as in the main text, but with $m v_F$ instead of $k_0$.
[99]{} L. D. Landau, Sov. Phys. JETP [**3**]{} 920 (1957) and [**5**]{} 101 (1957).
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---
abstract: 'The full off-shell one loop renormalization for all divergent amplitudes up to dimension 6 in the Abelian Higgs-Kibble model, supplemented with a maximally power counting violating higher-dimensional gauge-invariant derivative interaction $\sim g ~ \phi^\dagger \phi (D^\mu \phi)^\dagger D_\mu \phi$, is presented. This allows one to perform the complete renormalization of radiatively generated dimension 6 operators in the model at hand. We describe in details the technical tools required in order to disentangle the contribution to UV divergences parameterized by (generalized) non-polynomial field redefinitions. We also discuss how to extract the dependence of the $\beta$-function coefficients on the non-renormalizable coupling $g$ in one loop approximation, as well as the cohomological techniques (contractible pairs) required to efficiently separate the mixing of contributions associated to different higher-dimensional operators in a spontaneously broken effective field theory.'
author:
- 'D. Binosi'
- 'A. Quadri'
date: 'August 23, 2019'
title: |
Off-shell renormalization\
in the presence of dimension 6 derivative operators.\
II. UV coefficients
---
Introduction
============
In this paper we continue the study of the off-shell renormalization of the Abelian Higgs-Kibble model supplemented by the maximally power counting violating dimension 6 operator $\phi^\dagger \phi (D^\mu \phi)^\dagger D_\mu \phi$. In particular, we will show here how to evaluate the one-loop divergent coefficients associated to [*all*]{} dimension 6 operators which are radiatively generated.
The general aspects of the formalism needed to achieve this result have been explained in details in [@BQ:2019a], to which we refer the reader for a thorough exposition of the technical tools required within the Algebraic Renormalization approach to the problem [@Piguet:1995er; @Ferrari:1999nj; @Grassi:1999tp; @Grassi:2001zz; @Quadri:2003ui; @Quadri:2003pq; @Quadri:2005pv; @Bettinelli:2007tq; @Bettinelli:2007cy; @Bettinelli:2008ey; @Bettinelli:2008qn; @Anselmi:2012qy; @Anselmi:2012jt; @Anselmi:2012aq] we use. The present paper describes in a self-contained way the procedure developed in [@BQ:2019a] from an operational point of view. In particular we show how to disentangle the contributions to UV divergences parameterzied by unphysical (generalized) non-polynomial field redefinitions from those associated to the renormalization of physical gauge-invariant operators in the evaluation of one-loop $\beta$-functions.
To systematically compute the (one-loop) UV coefficients in spontaneously broken effective field theories possessing (dimension 6) derivative operators, it is convenient to first renormalize an associated auxiliary model, the so-called $X$-theory, which is obtained by describing the scalar physical degree of freedom in terms of the gauge-invariant field coordinate $$\begin{aligned}
v X_2 \sim \phi^\dagger \phi - \frac{v^2}{2},
\label{constr.intr}\end{aligned}$$ $v$ being the vacuum expectation value of the Higgs scalar $\phi$.
Then, in the $X$-theory all higher dimensional operators in the classical action are required to vanish at $X_2=0$. Thus, the operator $\frac{g}{v\Lambda} \phi^\dagger \phi (D^\mu \phi) D_\mu \phi$ (with the energy scale $\Lambda$ much higher than the electroweak scale $v$) will be expressed as $\frac{g}{\Lambda}X_2 (D^\mu \phi) D_\mu \phi$; going on-shell with the field $X_2$ and an additional Lagrange multiplier $X_1$ enforcing algebraically the constraint in eq[constr.intr]{}, we get back the original operator. Two external sources are then required in order to formulate in a mathematically consistent way the $X$-theory [@BQ:2019a]: one is coupled to the constraint $v X_2 - \phi^\dagger \phi - \frac{v^2}{2}$ and is denoted by $\bar c^*$; the second, called $T_1$, is required to close the algebra of operators, implementing the $X_2$-equation of motion at the quantum level.
The important point is that, unlike in the ordinary formalism, in the $X$-theory all 1-PI amplitudes, with the exception of those involving insertions of the $T_1$ source, exhibit a manifest weak power-counting [@Ferrari:2005va]: only a finite number of divergent amplitudes exist at each loop order (although increasing with the loop number, as expected in a general effective field theory setting). As for $T_1$-dependent amplitudes, they can be recovered by resumming the $T_1$-insertions on the Green’s functions at $T_1=0$, which, sometimes, can be even done in a closed form.
Once the renormalization of the $X$-theory is achieved, one goes on-shell with the $X_{1}$ and $X_2$ fields, which amounts to a suitable mapping of the sources $\bar c^*$ and $T_1$ onto operators depending on $\phi$ and its covariant derivatives. Then, one can immediately read off the UV coefficients of the higher dimensional gauge-invariant operators in the target theory, as now everything is expressed in terms of the original $\phi$ field. We hasten to emphasize that since we are working off-shell the effects of generalized field redefinitions, that are present already at one-loop order, and are not even polynomial for the model at hand [@BQ:2019a], need to be correctly accounted for. This is automatically done through the cohomologically trivial invariants of the $X$-theory. In fact, as we will show, the associated coefficients are gauge-dependent (as we will explicitly check by evaluating all the coefficients both in Feynman and Landau gauge), being instrumental in ensuring crucial cancellations leading to the gauge-independence of the coefficients associated to gauge-invariant operators. Notice in fact that since the ensuing analysis is based on cohomological results valid for anomaly-free gauge theories, the computational approach presented here can be readily extended to the electroweak gauge group $\rm SU(2) \times U(1)$ and, more generally, to any non-anomalous non-Abelian gauge group.
The paper is organized as follows. Our notations and conventions are described in Sect. \[sec.not\]. After providing in Sect. \[sec.map\] a brief reminder on the structure of the mapping to the target theory, we proceed to evaluate the coefficients of the cohomologically trivial invariants relevant for dimension 6 operators in Sect. \[sec:ct\]. Sect. \[sec:ps\], \[sec:ms\] and \[sec:gi\] are then devoted to the evaluation of the coefficients of the three classes of gauge invariant operators appearing in the theory: those only depending on the external sources, those mixing external sources and fields and those that only depend on the fields. We finally apply the mapping to the target theory in Sect. \[sec:map\] thereby computing the coefficients of all the UV divergent operators up to dimension $6$ in the original (target) theory. This allows us to construct (Sect. \[sec.beta\]) the full $\beta$ functions of the theory. Our conclusions and outlook are presented in Sect. \[sec:concl\]. The paper ends with two appendices: Appendix \[app:list\] contains the list of all the independent invariants needed for renormalizing the theory, while the relevant $X$-theory divergent one-loop amplitudes up to dimension $6$ are given in Appendix \[app:UVdivamp\].
Notations and setup {#sec.not}
===================
The tree-level vertex functional in the $X$-formalism can be written as [@BQ:2019a] $$\begin{aligned}
\G^{(0)} & = \int \!\mathrm{d}^4x \, \Big [ -\frac{1}{4} F^{\mu\nu} F_{\mu\nu} + (D^\mu \phi)^\dagger (D_\mu \phi) - \frac{M^2-m^2}{2} X_2^2 - \frac{m^2}{2v^2} \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2 \nonumber \\
& - \bar c (\square + m^2) c + \frac{1}{v} (X_1 + X_2) (\square + m^2) \Big ( \phi^\dagger \phi - \frac{v^2}{2} - v X_2 \Big ) \nonumber \\
& + \frac{g}{\Lambda} X_2 (D^\mu \phi)^\dagger (D_\mu \phi) + \T(D^\mu \phi)^\dagger (D_\mu \phi) \nonumber \\
& + \frac{b^2}{2\xi} - b \Big ( \partial A + \frac{e v}{\xi} \chi \Big ) + \bar{\omega}\Big ( \square \omega + \frac{e^2 v}{\xi} (\sigma + v) \omega\Big ) \nonumber \\
& + \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} - v X_2 \Big ) + \sigma^* (-e \omega \chi) + \chi^* e \omega (\sigma + v) \Big ].
\label{tree.level}\end{aligned}$$ The first line is the action of the Abelian Higgs-Kibble model in the $X$-formalism. Besides the usual scalar field $\phi \equiv \frac{1}{\sqrt{2}} (\sigma + v + i \chi)$, with $v$ its vacuum expectation value (v.e.v.), one also adds a singlet field $X_2$ that provides a gauge-invariant parameterization of the physical scalar mode. When going on-shell with the field $X_1$, that plays the role of a Lagrange multiplier, one recovers the constraint[^1] $X_2 \sim \frac{1}{v} ( \phi^\dagger \phi - v^2/2)$. Inserting the latter back into the first line of Eq.(\[tree.level\]), the $m^2$-term cancels out and one is left with the usual Higgs quartic potential with coefficient $\sim M^2/2v^2$.
Hence, Green’s functions in the target theory have to be $m^2$-independent, a fact that provides a very strong check of the computations, due to the ubiquitous presence of $m^2$ both in Feynman amplitudes and invariants.
The $X_{1,2}$-system comes together with a [*constraint*]{} BRST symmetry, ensuring that the number of physical degrees of freedom in the scalar sector remains unchanged in the $X$-formalism w.r.t. the standard formulation relying only on the field $\phi$ [@Quadri:2006hr; @Quadri:2016wwl]. More precisely, the vertex functional (\[tree.level\]) is invariant under the following BRST symmetry: $$\begin{aligned}
\s X_1 = v c; \, \quad \s \phi = \s X_2 = \s c = 0; \quad \s \bar c = \phi^\dagger\phi - \frac{v^2}{2} - v X_2 \, .
\label{u1.brst}\end{aligned}$$ The associated ghost and antighost fields $c, \bar c$ are free. The constraint BRST differential $\s$ anticommutes with the [*gauge group*]{} BRST symmetry of the classical action after the gauge-fixing introduced in the fourth line of Eq.(\[tree.level\]): $$\begin{aligned}
s A_\mu = \partial_\mu \omega \, ; \qquad s \omega = 0 \, ; \qquad s \bar{\omega} = b \, ; \qquad s b =0 \, ; \quad s \phi = i e \omega \phi. \end{aligned}$$ Here $\omega$ ($\bar \omega$) is the U(1) ghost (antighost); the latter field is paired into a BRST doublet with the Lagrange multiplier field $b$, enforcing the $R_\xi$ gauge-fixing condition $$\begin{aligned}
{\cal F}_\xi = \partial A + \frac{ev}{\xi} \chi.
\label{g.f.cond}\end{aligned}$$ The two BRST symmetries can both be lifted to the corresponding Slavnov-Taylor identities at the quantum level, provided one introduces the antifields, [*i.e.*]{}, the external sources coupled to the relevant BRST transformation that are non-linear in the quantized fields. The antifield couplings are displayed in the last line of Eq.(\[tree.level\]). Then the ST identity for the constraint BRST symmetry is $$\begin{aligned}
{\cal S}_{\scriptscriptstyle{C}}(\G) \equiv \int \!\mathrm{d}^4 x \, \Big [ v c \frac{\delta \G}{\delta X_1}
+ \frac{\delta \G}{\delta \bar c^*}\frac{\delta \G}{\delta \bar c} \Big ] =
\int \!\mathrm{d}^4 x \, \Big [ v c \frac{\delta \G}{\delta X_1}
-(\square + m^2) c \frac{\delta \G}{\delta \bar c^*} \Big ] = 0,
\label{sti.c} \end{aligned}$$ where in the latter equality we have used the fact that both the ghost $c$ and the antighost $\bar c$ are free: $$\begin{aligned}
\frac{\delta \G}{\delta \bar c} = -(\square + m^2) c \, \, ,
\qquad
\frac{\delta \G}{\delta c} = (\square + m^2) \bar c \, . \end{aligned}$$ Hence Eq.(\[sti.c\]) reduces to the $X_1$-equation of motion $$\begin{aligned}
\frac{\delta \G}{\delta X_1}=
\frac{1}{v} (\square + m^2)
\frac{\delta \G}{\delta \bar c^*}.
\label{X1.eq}\end{aligned}$$ Finally, the ST identity (equivalently the BV master equation) associated to the gauge group BRST symmetry reads $$\begin{aligned}
& {\cal S}(\G) = \int \mathrm{d}^4x \, \Big [
\partial_\mu \omega \frac{\delta \G}{\delta A_\mu} + \frac{\delta \G}{\delta \sigma^*} \frac{\delta \G}{\delta \sigma} + \frac{\delta \G}{\delta \chi^*} \frac{\delta \G}{\delta \chi}
+ b \frac{\delta \G}{\delta \bar \omega} \Big ] = 0.
\label{sti} \end{aligned}$$
The third line of Eq.(\[tree.level\]) contains the derivative dim.6 operator $$X_2 (D^\mu \phi)^\dagger D_\mu \phi \sim \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) (D^\mu \phi)^\dagger D_\mu \phi$$ together with the source $T_1$ required to define the $X_2$-equation at the quantum level in the presence of such an additional non power-counting renormalizable interaction: $$\begin{aligned}
\frac{\delta \G}{\delta X_2} = \frac{1}{v} (\square + m^2) \frac{\delta \G}{\delta \bar c^*} + \frac{g}{\Lambda} \frac{\delta \G}{\delta T_1} - (\square + m^2)X_1 - (\square + M^2) X_2 - v \bar c^* \, .
\label{X2.eq}\end{aligned}$$ Notice that the terms in the third line of Eq.(\[tree.level\]) respect both BRST symmetries and thus they do not violate either the $X_1$-equation (\[X1.eq\]) or the ST identity (\[sti\]).
The set of the functional identities holding in this theory is completed by:
- The $b$-equation: $$\begin{aligned}
\frac{\delta \G^{(0)}}{\delta b} = \frac{b}{\xi} - \partial A - \frac{e v}{\xi} \chi ;
\label{b.eq}\end{aligned}$$
- The antighost equation: $$\begin{aligned}
\frac{\delta \G^{(0)}}{\delta \bar \omega} = \square \omega + \frac{e v}{\xi} \frac{\delta \G^{(0)}}{\delta \chi^*} .
\label{antigh.eq}\end{aligned}$$
In what follows subscripts denote functional differentiation w.r.t. fields and external sources. Moreover, if not otherwise stated, amplitudes will be denoted as, [*e.g.*]{}, $\G^{(1)}_{\chi\chi}$, meaning $$\begin{aligned}
\G^{(1)}_{\chi\chi} \equiv \left .
\frac{\delta^2 \G^{(1)}}{\delta \chi(-p) \delta \chi(p)} \right |_{p=0}.\end{aligned}$$ A bar denotes the UV divergent part of the corresponding amplitude in the Laurent expansion around $\epsilon=4-D$, with $D$ the space-time dimension. Dimensional regularization is always implied, with amplitudes evaluated by means of the packages [FeynArts]{} and [FormCalc]{} [@Hahn:2000kx; @Hahn:2000jm]. As already remarked, all amplitudes will be evaluated in the Feynman ($\xi=1$, with $\xi$ the gauge fixing parameter) and Landau ($\xi=0$) gauge; this will allow to explicitly check the gauge cancellations in gauge invariant operators.
The UV divergent contributions to one-loop amplitudes form a local functional (in the sense of formal power series) aptly denoted by $\overline{\G}^{(1)}$. In particular, $\overline{\G}^{(1)}$ belongs to the kernel of ${\cal S}_0$ i.e. $$\begin{aligned}
{\cal S}_0(\overline{\G}^{(1)}) =0,
\label{uv.div.1loop.st}\end{aligned}$$ where ${\cal S}_0$ is the linearized ST operator $$\begin{aligned}
{\cal S}_0 (\overline{\G}^{(1)}) & = \int \!\mathrm{d}^4 x \, \Big [ \partial_\mu \omega \frac{\delta \overline{\G}^{(1)}}{\delta A_\mu} + e\omega(\sigma+v)\frac{\delta \overline{\G}^{(1)}}{\delta \chi} -e\omega\chi\frac{\delta \overline{\G}^{(1)}}{\delta \sigma}
+ b \frac{\delta \overline{\G}^{(1)}}{\delta \bar \omega} \nonumber \\
& + \frac{\delta \G^{(0)}}{\delta \sigma} \frac{\delta \overline{\G}^{(1)}}{\delta \sigma^*} + \frac{\delta \G^{(0)}}{\delta \chi} \frac{\delta \overline{\G}^{(1)}}{\delta \chi^*} \Big ] \nonumber \\
& = s \overline{\G}^{(1)} + \int \!\mathrm{d}^4 x \, \Big [ \frac{\delta \G^{(0)}}{\delta \sigma} \frac{\delta \overline{\G}^{(1)}}{\delta \sigma^*} + \frac{\delta \G^{(0)}}{\delta \chi} \frac{\delta \overline{\G}^{(1)}}{\delta \chi^*} \Big ],
\label{S0}\end{aligned}$$ which acts as the BRST differential $s$ on the fields of the theory while mapping the antifields into the classical equations of motion of their corresponding fields. Then, the nilpotency of ${\cal S}_0$ ensures that $\overline{\G}^{(1)}$ is the sum of a gauge-invariant functional ${\overline{{\cal I}}}^{(1)}$ and a cohomologically trivial contribution ${\cal S}_0(\overline{Y}^{(1)})$: $$\begin{aligned}
\overline{\G}^{(1)} =
{\overline{{\cal I}}}^{(1)}_\mathrm{gi} + {\cal S}_0(\overline{Y}^{(1)}) .
\label{uv.div.1loop.vf}\end{aligned}$$
\[sec.map\]Mapping on the external sources
==========================================
As a result of the previous Section, we only need to determine the invariants contributing to ${\overline{{\cal I}}}^{(1)}_\mathrm{gi}$ and $\overline{Y}^{(1)}$ that will induce in the target theory operators of dimension less or equal to $6$.
To that end we first need to consider how the mapping affects the external sources $\bar c^*,T_1$.
The $X_1$- and $X_2$-equations and at loop order $n \geq 1$ for $\G^{(n)}$ read $$\begin{aligned}
\frac{\delta \G^{(n)}}{\delta X_1} &= \frac{1}{v} (\square + m^2) \frac{\delta \G^{(n)}}{\delta \bar c^*};&
\frac{\delta \G^{(n)}}{\delta X_2} &= \frac{1}{v} (\square + m^2) \frac{\delta \G^{(n)}}{\delta \bar c^*} +
\frac{g}{\Lambda} \frac{\delta \G^{(n)}}{\delta T_1},
\label{X.eqs.loops}\end{aligned}$$ thus implying that the whole dependence on $X_1$ and $X_2$ can only arise through the combinations $$\begin{aligned}
&\tbarc = \bar c^* + \frac{1}{v} (\square + m^2) (X_1+ X_2);& &{\cal T}_1 = T_1 + \frac{g}{\Lambda} X_2.&
\label{X2.subst}\end{aligned}$$ In particular, eq[X.eqs.loops]{} states that the 1-PI amplitudes involving at least one $X_1$ or $X_2$ external legs are uniquely fixed in terms of amplitudes involving neither $X_1$ or $X_2$.
We now turn to the analysis of how the right-hand side of Eqs.(\[X2.subst\]) is transformed under the mapping. For that purpose we need to impose the equations of motion for $X_{1,2}$. At the one-loop level, we can restrict to tree-level equations of motion for these fields. As already discussed, the $X_1$-equation of motion enforces the constraint $X_2 = \frac{1}{v} \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )$. Once one takes into account this constraint, the $X_2$-equation of motion in turn yields $$\begin{aligned}
(\square + m^2) (X_1 + X_2) = - (M^2 - m^2) X_2 + \frac{g}{\Lambda} (D^\mu \phi)^\dagger D_\mu \phi - v \bar c^*.
\label{X2.EOM}\end{aligned}$$ By substituting the above expressions for $X_{1,2}$ into the replacement rules we arrive at the sought-for mapping transformation (at zero external sources): $$\begin{aligned}
& \tbarc \rightarrow - \frac{(M^2 - m^2)}{v^2} \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) + \frac{g}{v \Lambda} (D^\mu \phi)^\dagger D_\mu \phi;&
& {\cal T}_1 \rightarrow \frac{g}{v \Lambda} \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ).
\label{repl.fin.1}\end{aligned}$$
Since the right-hand side of the above equation contains operators of dimension at least $2$, in order to obtain target operators of up to dimension $6$ it is clear that we need to consider amplitudes with up to $3$ external sources $\bar c^*$ and $T_1$. Equivalently, we can assign dimension $2$ to both $\bar c^*$ and $T_1$ and use it in order to identify the mixed fields-external sources invariants that will contribute to target operators of up to dimension $6$. For instance $\int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )$ would project onto $$\begin{aligned}
\int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) \rightarrow &
- \frac{(M^2 - m^2)}{v^2} \int \d \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2 \nonumber \\
&+ \frac{g}{v \Lambda} \int \d (D^\mu \phi)^\dagger D_\mu \phi\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ),\end{aligned}$$ whereas $\int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2$ would give rise to $$\begin{aligned}
\int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2 \rightarrow
- \frac{(M^2 - m^2)}{v^2} \int \d \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^3,\end{aligned}$$ where we have neglected the covariant kinetic term in the first term of eq[repl.fin.1]{} since it would generate a dimension $8$ operator.
Finally, the coefficients of the three possible types of invariants contributing to the $X$-theory functional ${\overline{{\cal I}}}^{(1)}_\mathrm{gi}$ will be indicated with $\cfgi{i}$ (combinations of the field strength, its derivatives and $\phi$ and its covariant derivatives of up to dimension $6$), $\cfxt{i}$ (combinations of external sources and fields) or $\cfps{i}$ (combinations of external sources only). The complete list of invariants is reported in Appendix \[app:list\].\
\[sec:ct\]Cohomologically trivial invariants
============================================
Before addressing the evaluation of the coefficients of the gauge invariants, it is necessary to fix the coefficients $\cfct{i}$ of the cohomologically trivial invariants contributing to ${\cal S}_0(\overline{Y}^{(1)})$. Taking into account the bounds on the dimensions, this requires to consider two invariants at $T_1=0$, namely $$\begin{aligned}
\cfct{0} {\cal S}_0 \! \int \d [\sigma^*(\sigma + v) + \chi^* \chi];&
&\cfct{1}\, {\cal S}_0\! \int \d \, (\sigma^* \sigma + \chi^* \chi).\end{aligned}$$
Generalized field redefinitions
-------------------------------
To begin with let us observe that eq[S0]{} implies $$\begin{aligned}
\cfct{1}\, {\cal S}_0\! \int \d \, (\sigma^* \sigma + \chi^* \chi) \supset - e v \cfct{1} \int \d
\chi^* \omega.
\label{rho1}\end{aligned}$$ Therefore, the coefficient $\cfct{1}$ associated to this invariant is controlled by the single amplitude $\overline{\Gamma}^{(1)}_{\chi^* \omega}$. Indeed, eq[rho1]{} demands that $$\begin{aligned}
ev \cfct{1} = - \overline{\Gamma}^{(1)}_{\chi^* \omega},\end{aligned}$$ or, using the result , $$\begin{aligned}
\cfct{1} = \frac{M_A^2}{8 \pi^2 v^2} \div (1-\dl),
\label{cfct.1}\end{aligned}$$ with $\delta_{\xi0}=\delta_{00}=1$ in the Landau gauge and $\delta_{\xi0}=\delta_{10}=0$ in the Feynman gauge. Notice that this result implies that there are no [*pure*]{} field redefinitions in Landau gauge, [*i.e.*]{}, the v.e.v. renormalizes in the same way as the fields, as we will soon show.
Finally, repeated insertions of the source $T_1$ resum to $$\begin{aligned}
\cfct{1} {\cal S}_0 \! \int \d \, \frac{1}{1+T_1} (\sigma^* \sigma + \chi^* \chi).
\label{cfct.1.resum}\end{aligned}$$ A comment is in order here. In the standard formalism one should consider the effect of the generalized field redefinitions in the target theory, which, as explained in Ref.[@BQ:2019a], is the one induced by eq[cfct.1.resum]{}. This implies that the fields $\sigma$ and $\chi$ undergo the transformation $$\begin{aligned}
& \sigma \rightarrow \sigma + \frac{\cfct{1}}{1+ \frac{g}{\Lambda v} \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )} \sigma;&
& \chi \rightarrow \chi + \frac{\cfct{1}}{1+ \frac{g}{\Lambda v} \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )} \chi.\end{aligned}$$ This would be a rather involved task, which is however simplified in the approach developed here, since all the combinatorics is automatically taken into account via the renormalization of the $X$-theory, through the cohomologically trivial invariant eq[cfct.1.resum]{}.
Tadpoles {#sec:tadpoles}
--------
The tadpoles $\overline{\G}^{(1)}_{\sigma}, \overline{\G}^{(1)}_{\bar c^*}$ allow to fix the coefficients of three invariants: $$\begin{aligned}
& \cfct{0}\, {\cal S}_0 \! \int \d [\sigma^*(\sigma + v) + \chi^* \chi] + \cfgi{1} \int \d \, \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )
+ \cfps{1} \int \d \bar c^*
\nonumber \\
& \supset \int \d \left[ ( - m^2 v \cfct{0} + v \cfgi{1} ) \sigma + ( \cfct{0} v^2 + \cfps{1}
) \bar c^* \right ].
\label{l1.bc}\end{aligned}$$ Indeed, eq[l1.bc]{} gives rise to the equations
$$\begin{aligned}
- m^2 v \cfct{0} + v \cfgi{1} &=\overline{\Gamma}^{(1)}_{\sigma};\label{tadsa}\\
\cfct{0} v^2 +
\cfps{1}
&=\overline{\Gamma}^{(1)}_{\bar c^*}.
\label{tadsb}\end{aligned}$$
Direct inspection of the one-loop results and shows that, in the Feynman gauge, it is consistent to set $\left .\cfct{0}\right |_{\xi=1}=0$, thus yielding the results
$$\begin{aligned}
\left.\cfgi{1}\right|_{\xi=1} & = \frac{1}{v} \left . \overline{\Gamma}^{(1)}_{\sigma} \right |_{\xi=1} =
\frac{1}{16 \pi^2 v^2}\div \left[ m^2 ( M^2 + M_A^2) + 2 (M^4 + 3 M_A^4) \right], \\
\left. \cfps{1}
\right|_{\xi=1} & = \left . \overline{\Gamma}^{(1)}_{\bar c^*} \right |_{\xi=1} = - \frac{M^2+ M_A^2 }{16 \pi^2}\div .\end{aligned}$$
On the other hand, since $\cfgi{1}$ must be gauge invariant, eq[tadsa]{} implies $$\begin{aligned}
\cfct{0}= \frac{1}{m^2v} \Big ( v\cfgi{1}-
\overline{\Gamma}^{(1)}_{\sigma}
\Big ) = \frac{M_A^2}{16 v^2\pi^2} \div \dl,
\label{cfct.0}\end{aligned}$$ whereas eq[tadsb]{} furnishes a consistency condition that can be easily checked. Notice in particular that eq[tadsb]{} shows that $\cfps{1}$ is gauge independent (as it should) since the gauge dependence in $\overline{\Gamma}^{(1)}_{\bar c^*}$ is cancelled by the one in $\cfct{0}$. Finally, using eq[cfct.0]{} and the gauge independence of $\cfps{1}$, eq[tadsb]{} can be recast in the form $$\begin{aligned}
-\frac{m^2}{v} \Big ( \left . \overline{\Gamma}^{(1)}_{\bar c^*} \right |_{\xi=0} -
\left . \overline{\Gamma}^{(1)}_{\bar c^*} \right |_{\xi=1} \Big ) =
\left . \overline{\Gamma}^{(1)}_{\sigma} \right |_{\xi=0} -
\left . \overline{\Gamma}^{(1)}_{\sigma} \right |_{\xi=1}
\label{cc.bc}.\end{aligned}$$
Next, we need to consider the insertion of one and two sources $T_1$ on tadpole amplitudes. Starting from a single insertion, the relevant projection equation becomes $$\begin{aligned}
& \cfct{0T_1} {\cal S}_0 \int \d
T_1 [\sigma^*(\sigma + v) + \chi^* \chi] +
\cfxt{2} \int \d T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) +
\cfps{7}
\int \d \bar c^* T_1
\nonumber \\
& \supset\int \d \Big [ ( - m^2 v \cfct{0T_1} + v \cfxt{2} ) T_1 \sigma + (v^2 \cfct{0T_1} +
\cfps{7}
) \bar c^* T_1 \Big ].
\label{proj.1}\end{aligned}$$ As before, one obtains two equations
$$\begin{aligned}
- m^2 v \cfct{0T_1} + v \cfxt{2}&=\overline{\Gamma}^{(1)}_{T_1\sigma},\label{Ta}\\
v^2 \cfct{0T_1} + \cfps{7}
&=\overline{\Gamma}^{(1)}_{\bar c^* T_1},\label{Tb}
\end{aligned}$$
which is most easily solved in the Feynman gauge in which $\left.\cfct{0T_1}\right|_{\xi=1}=0$, and therefore, using the results and ,
$$\begin{aligned}
\left.\cfxt{2}\right|_{\xi=1} &= \frac{1}{v} \left . \overline{\Gamma}^{(1)}_{T_1\sigma} \right |_{\xi=1}
= - \frac{1}{8 \pi^2 v^2} \Big [ m^2( M^2 + M_A^2) + 2 (M^4 - 3 M_A^4) \Big ] \div , \\
\left. \cfps{7}
\right|_{\xi=1} &= \left . \overline{\Gamma}^{(1)}_{\bar c^* T_1} \right |_{\xi=1} = \frac{(M^2+ M_A^2)}{8 \pi^2} \div .
\label{theta.2bcT1}
\end{aligned}$$
Then, using the fact that $\theta_2$ is gauge invariant, eq[Ta]{} can be used to fix the coefficient $\cfct{0T_1}$, obtaining $$\begin{aligned}
\cfct{0T_1}&=\frac{1}{m^2v} \Big ( v\cfxt{2}-
\overline{\Gamma}^{(1)}_{T_1\sigma}
\Big ) = -\frac{M_A^2}{8\pi^2v^2} \div \dl,\end{aligned}$$ which, once inserted in eq[Tb]{} shows that $\cfps{7}$ is gauge invariant, thus allowing to recast the condition in the form $$\begin{aligned}
-\frac{m^2}{v} \Big ( \left . \overline{\Gamma}^{(1)}_{\bar c^*T_1} \right |_{\xi=0} -
\left . \overline{\Gamma}^{(1)}_{\bar c^*T_1} \right |_{\xi=1} \Big ) =
\left . \overline{\Gamma}^{(1)}_{T_1\sigma } \right |_{\xi=0} -
\left . \overline{\Gamma}^{(1)}_{T_1\sigma } \right |_{\xi=1},
\label{cc.bcT}\end{aligned}$$ in complete analogy with eq[cc.bc]{}.
Finally, for the case of two $T_1$-insertions, the relevant projection equation reads $$\begin{aligned}
& \cfct{0T_1^2} \int \d
T_1^2 {\cal S}_0 \int \d [\sigma^*(\sigma + v) + \chi^* \chi] +
\cfxt{12} \int \d T_1^2 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) +
\frac{\cfps{11}}{2}
\int \d \bar c^* T_1^2
\supset
\nonumber \\
& \int \d \Big [ ( - m^2 v \cfct{0T_1^2} + v \cfxt{12} ) \sigma T_1^2 + (v^2 \cfct{0T_1^2} + \frac{\cfps{11}}{2}
) \bar c^* T_1^2 \Big ],
\label{proj.2}\end{aligned}$$ giving rise to the conditions
$$\begin{aligned}
2 (- m^2 v \cfct{0T_1^2} + v \cfxt{12} ) &=\overline{\Gamma}^{(1)}_{\sigma T_1T_1},\label{T2a}\\
2 v^2 \cfct{0T_1^2} + \cfps{11} &=\overline{\Gamma}^{(1)}_{\bar c^* T_1T_1}. \label{T2b}
\end{aligned}$$
In the Feynman gauge $\left.\cfct{0T_1^2}\right|_{\xi=1}=0$, so that, using eqs[GsT1T1]{}[Gc\*T1T1]{}
$$\begin{aligned}
& \left.\cfxt{12}\right|_{\xi=1} = \frac{1}{2v} \left . \overline{\Gamma}^{(1)}_{ \sigma T_1T_1} \right |_{\xi=1}
= \frac{1}{16 \pi^2 v^2} \Big [ m^2 (3 M^2+2 M_A^2)+6 (M^4+M_A^4) \Big ] \div , \label{theta12}\\
& \left. \cfps{11}
\right|_{\xi=1} = \left . \overline{\Gamma}^{(1)}_{\bar c^* T_1T_1} \right |_{\xi=1} = -\frac{3 M^2+ 2 M_A^2}{8\pi^2}\div.
\label{theta.bcT12}\end{aligned}$$
Using then the gauge independence of $\cfxt{12}$ we obtain, from eq[T2a]{} $$\begin{aligned}
\cfct{0T_1^2}&=\frac{1}{2 m^2v} \Big ( 2 v\cfxt{12}-\overline{\Gamma}^{(1)}_{\sigma T_1T_1} \Big ) = \frac{M_A^2}{8\pi^2v^2} \div \dl,\end{aligned}$$ which, once inserted in eq[T2b]{} shows that $\cfps{11}$ is also gauge invariant, so that the condition reads $$\begin{aligned}
-\frac{m^2}{v} \Big ( \left . \overline{\Gamma}^{(1)}_{\bar c^*T_1T_1} \right |_{\xi=0} -
\left . \overline{\Gamma}^{(1)}_{\bar c^*T_1T_1} \right |_{\xi=1} \Big ) =
\left . \overline{\Gamma}^{(1)}_{\sigma T_1T_1} \right |_{\xi=0} -
\left . \overline{\Gamma}^{(1)}_{\sigma T_1T_1} \right |_{\xi=1}.
\label{cc.bcTT}\end{aligned}$$ We remark that resummation of the $T_1$-insertions is not at work for the tadpoles in the Landau gauge since the loop with a massless Goldstone field in $\overline{\G}^{(1)}_{\bar c^*}$ and $\overline{\G}^{(1)}_{\sigma}$ happens to be zero in dimensional regularization.
In the Landau gauge there is no pure field redefinition since $\left . \cfct{1} \right |_{\xi=0}=0$. On the other hand the invariant $$\begin{aligned}
\cfct{0}\, {\cal S}_0 \! \int \d [\sigma^*(\sigma + v) + \chi^* \chi],\end{aligned}$$ shows that in Landau gauge also the v.e.v. $v$ renormalizes in the same way as the field $\phi$. This is a well-known fact in spontaneously broken gauge theories [@Sperling:2013eva].
The pure external sources sector {#sec:ps}
================================
We now move to the pure external sources sector. These invariants, which are reported in eq[ESinv]{}, cannot depend on the gauge, as we will explicitly show.
Linear terms
------------
$\cfps{1}$ has been already fixed in Eq.(\[tadsb\]). $\cfps{2}$ can be fixed by looking at the $T_1$-tadpole : $$\begin{aligned}
\cfps{2} = \overline{\G}^{(1)}_{T_1}
= - \frac{(M^4-3 M_A^4)}{16 \pi^2}\div.
\label{theta.T1}\end{aligned}$$ Notice that there are no contributions from cohomologically trivial invariants since there are no linear couplings for $T_1$ at tree-level. Consequently $\overline{\G}^{(1)}_{T_1}$ is the same both in Landau and in Feynman gauge.
Bilinears
---------
$\cfps{3}$ is fixed by the 2-point $\bar c^*$-amplitude eq[Gc\*c\*]{}: $$\begin{aligned}
\cfps{3} = \overline{\G}^{(1)}_{\bar c^* \bar c^*} = \frac{1}{8 \pi^2}\div \, .
\label{theta.bc2}\end{aligned}$$ Notice that $\overline{\G}^{(1)}_{\bar c^* \bar c^*}$ does not develop momentum-dependent divergences and that it does not depend on the gauge.
This is clearly not the case for $\overline{\G}^{(1)}_{T_1T_1}$ as eq[GT1T1]{} shows; we can then read off the coefficients of the different bilinear invariants, obtaining $$\begin{aligned}
& \cfps{4} = \overline{\G}^{(1)}_{T_1T_1}
= \frac{3}{16 \pi^2} (M^4 + M_A^4)\div , \nonumber \\
& \cfps{5} = - \left . \frac{\partial \overline{\G}^{(1)}_{T_1T_1}}{\partial p^2} \right |_{p=0}
= \frac{3}{32\pi^2} (M^2 + M_A^2)\div , \qquad
& \cfps{6} = \left . \frac{\partial \overline{\G}^{(1)}_{T_1T_1}}{\partial p^4} \right |_{p=0}
= \frac{1}{32\pi^2}\div .\end{aligned}$$
We notice that $\cfps{6}$ has been included for completeness but does not contribute to operators of dim. $\leq 6$ in the target theory, rather to dim.$8$ operators.
Finally, $\cfps{7}$ has been fixed in eq[theta.2bcT1]{}, while the $p^2$-coefficient of the amplitude $\overline{\Gamma}^{(1)}_{\bar c^* T_1}$, see , is gauge independent and implies $$\begin{aligned}
\cfps{8} = - \left . \frac{\partial \overline{\G}^{(1)}_{T_1\bar c^*}}{\partial p^2} \right |_{p=0}=
\frac{1}{16 \pi^2} \div .\end{aligned}$$
Trilinears
----------
While $\cfps{11}$ has been fixed in eq[theta.bcT12]{}, it turns out that the remaining trilinears do not receive contributions from cohomologically trivial invariants. In particular we find $$\begin{aligned}
\cfps{9} = 0\end{aligned}$$ since ${\Gamma}^{(1)}_{\bar c^*\bar c^*\bar c^*}$ is UV finite, and, using the results and $$\begin{aligned}
\cfps{10} &= \overline{\Gamma}^{(1)}_{\bar c^* \bar c^* T_1} = -\frac{1}{4\pi^2}\div;
&
\cfps{12} &= \overline{\G}^{(1)}_{T_1T_1T_1} =
-\frac{3 M^4}{4\pi^2}\div.\end{aligned}$$
The mixed external sources-field sector {#sec:ms}
=======================================
The $\cfxt{1}$ and $\cfxt{2}$ coefficients
------------------------------------------
The coefficients $\cfxt{1}$ and $\cfxt{2}$ can be fixed by evaluating the three-point functions $\overline{\G}^{(1)}_{\bar c^* \chi\chi}$ and $\overline{\G}^{(1)}_{T_1 \chi\chi}$ at zero momentum. Since $$\begin{aligned}
& \cfct{0}\, {\cal S}_0 \int \d \left[\sigma^* (\sigma + v) + \chi^* \chi\right] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi) +
\cfxt{1} \int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )
\nonumber \\
& \supset
\int \d \Big ( \cfct{0}
+ \cfct{1} +
\frac{\cfxt{1}}{2}
\Big )\bar c^* \chi^2,\end{aligned}$$ one arrives at the relation $$\begin{aligned}
2 \cfct{0}
+ 2 \cfct{1} +
\cfxt{1} =
\overline{\G}^{(1)}_{\bar c^* \chi\chi }.\end{aligned}$$ Then, using eqs[cfct.1]{}[cfct.0]{}[Gc\*chch]{}, we immediately obtain the result $$\begin{aligned}
\cfxt{1} = \overline{\G}^{(1)}_{\bar c^* \chi\chi } - 2 ( \cfct{0}+\cfct{1}) = - \frac{m^2 + M^2+ M_A^2}{8 \pi^2 v^2} \div,
\label{theta1}\end{aligned}$$ which, due to the compensation of the gauge parameter dependence between the amplitude and the coefficients $\cfct{0}$ and $\cfct{1}$ turns out to be gauge independent, as it should. In a similar fashion, considering the combination $$\begin{aligned}
& \cfct{0T_1} {\cal S}_0 \int \d T_1 \left[\sigma^* (\sigma + v) + \chi^* \chi\right]
+ \cfxt{2} \int \d T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) \supset
\int \d
\Big (
- \cfct{0T_1} \frac{m^2}{2} +
\frac{\cfxt{2}}{2}
\Big )
T_1 \chi^2,\end{aligned}$$ we get $$\begin{aligned}
- \cfct{0T_1} m^2 +\cfxt{2}=\overline{\G}^{(1)}_{T_1 \chi\chi},\end{aligned}$$ or, using the result $$\begin{aligned}
\cfxt{2} =
-\frac{m^2 (M^2 + M_A^2) + 2 (M^4 - 3 M_A^4)}{8 \pi^2 v^2}\div,\end{aligned}$$ and again one obtains the gauge independence of this parameter as a result of the cancellation of the gauge-dependence between the 1-PI amplitude and the coefficient $\cfct{0T_1}$.
The validity of these results can be checked against the relations provided by 1-PI amplitudes involving one source and one external $\sigma$-field. For example considering the $\bar c^*\sigma$ case, we find $$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \Big ( \int \d (\sigma^* \sigma + \chi^* \chi) \Big ) +
\cfxt{1} \int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )
\nonumber \\
& \supset
\int \d v \Big ( 2 \cfct{0} + \cfct{1}
+ \cfxt{1}
\Big )\bar c^* \sigma,\end{aligned}$$ yielding the relation $$\begin{aligned}
v (2 \cfct{0} + \cfct{1} + \cfxt{1}) = \overline{\G}^{(1)}_{\bar c^* \sigma},\end{aligned}$$ which can be checked directly using eqs[cfct.1]{}[cfct.0]{}[theta1]{}. Notice that $\overline{\G}^{(1)}_{\bar c^* \sigma}$ is the same in Feynman and Landau gauge, see eq[Gc\*s]{}; therfore, since $\cfxt{1}$ is gauge independent, so must be the combination $2 \cfct{0} + \cfct{1}$, as can be easily verified.
Considering the $T_1\sigma$ amplitudes, we find instead $$\begin{aligned}
& \cfct{0T_1} {\cal S}_0 \Big ( \int \d T_1 (\sigma^* (\sigma + v) + \chi^* \chi) \Big )
+
\cfxt{2} \int \d T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )\nonumber \\
&\supset
\int \d \Big ( - v m^2 \cfct{0T_1}
+ v \cfxt{2}
\Big ) T_1 \sigma.
$$ Thus we get $$\begin{aligned}
- v m^2 \cfct{0T_1} + v \cfxt{2} = \overline{\G}^{(1)}_{T_1 \sigma},\end{aligned}$$ which can be immediately verified using the one-loop result .
The $\cfxt{3}$ and $\cfxt{5}$ coefficients
------------------------------------------
In order to fix $\cfxt{3}$ and $\cfxt{5}$, we need the amplitude $\overline{\G}^{(1)}_{\bar c^* \chi \chi}$, which can be decomposed in form factors according to $$\begin{aligned}
& \overline{\G}^{(1)}_{\bar c^* \chi \chi}(p_1,p_2) =
\ff{0}{\bar c^* \chi\chi}+
\ff{1}{\bar c^* \chi\chi}(p_1^2+p_2^2)+
\ff{2}{\bar c^* \chi\chi}(p_1{\cdot} p_2).\end{aligned}$$ We find
$$\begin{aligned}
&\cfxt{3} \int \d \bar c^* (D^\mu \phi)^\dagger D_\mu \phi \supset
\cfxt{3} \int \d
\frac{\bar c^*}{2} \partial^\mu \chi \partial_\mu \chi, \\
& \cfxt{5} \int \d \bar c^* \Big [
(D^2\phi)^\dagger \phi + \mathrm{h.c.}\Big ] \supset
\cfxt{5} \int \d
\bar c^* \chi \square \chi,\end{aligned}$$
which, using the result eq[Gc\*chch]{}, implies the following identifications $$\begin{aligned}
\cfxt{3} &= -\ff{2}{\bar c^* \chi\chi} =
- \frac{1}{16 \pi^2}
\frac{g}{v \Lambda}
\Big ( 2 + \frac{g v}{\Lambda} \Big )
\div;&
\cfxt{5} &= -\ff{1}{\bar c^* \chi\chi} =
- \frac{1}{16 \pi^2}
\frac{g}{\Lambda v}\div
\, . \end{aligned}$$ Notice that both coefficients are the same in Landau and Feynman gauge, as expected.
In this case a consistency check is provided by the three-point function $\overline{\Gamma}^{(1)}_{\bar c^* A_\mu A_\nu}$, since one has $$\begin{aligned}
& \cfxt{3} \int \d \bar c^* (D^\mu \phi)^\dagger D_\mu \phi + \cfxt{5} \int \d \bar c^* \Big [
(D^2\phi)^\dagger \phi + h.c.\Big ] \supset
\int \d
\frac{M_A^2}{2} \Big ( \cfxt{3} - 2 \cfxt{5} \Big ) \bar c^* A^2,\end{aligned}$$ so that $$\begin{aligned}
M_A^2 ( \cfxt{3} - 2 \cfxt{5} ) g_{\mu\nu} =
\left .
\overline{\G}^{(1)}_{\bar c^* A_\mu A_\nu}(p_1,p_2)
\right |_{p_1=p_2=0},\end{aligned}$$ as can be easily verified with the help of eq[Gc\*AA]{}.
The $\cfxt{4}$ and $\cfxt{6}$ coefficients
------------------------------------------
In order to fix $\cfxt{4}$ and $\cfxt{6}$ we need the amplitude $\overline{\G}^{(1)}_{T_1 \chi \chi}$, which we decompose as before according to $$\begin{aligned}
\overline{\G}^{(1)}_{T_1 \chi \chi} (p_1,p_2)=
\ff{0}{T_1 \chi\chi}+
\ff{1}{T_1 \chi\chi}(p_1^2+p_2^2)+
\ff{2}{T_1 \chi\chi}({p_1{\cdot}p_2})
+ {\cal O}(p_i^4),\end{aligned}$$ and the dots denote terms of order $p^4$, which are not needed.
There are two projections to be considered, namely $T_1\partial^\mu\chi\partial_\mu \chi$ and $T_1 \chi \square \chi$, to which the cohomologically trivial invariants can also contribute. To begin with, observe that $$\begin{aligned}
\cfct{1} {\cal S}_0 \int \d \frac{1}{1+T_1}(\sigma^* \sigma + \chi^* \chi)
& = \cfct{1} {\cal S}_0\int \d
(1 - T_1 + \cdots) (\sigma^* \sigma + \chi^* \chi) \nonumber \\
& \supset\cfct{1} \int \d \Big (T_1 \partial^\mu\chi \partial_\mu\chi
+ T_1 \chi \square \chi
\Big ).
$$ On the other hand we have $$\begin{aligned}
& \cfct{0} {\cal S}_0
\int \d
( \sigma^* (\sigma + v) + \chi^* \chi )
+
\cfct{0T_1} {\cal S}_0
\int \d T_1
[ \sigma^* (\sigma + v) + \chi^* \chi ] \nonumber \\
& \supset
\int \d \Big [
\cfct{0} T_1
\partial^\mu \chi \partial_\mu \chi
- \cfct{0T_1} T_1 \chi \square \chi
\Big ].\end{aligned}$$
Therefore we obtain $$\begin{aligned}
\cfct{1}-\cfct{0T_1}+\cfxt{4} &= -\ff{2}{T_1 \chi\chi};&
2 (\cfct{1} + \cfct{0} ) +\cfxt{6} = - \ff{1}{T_1 \chi\chi} \, ,\end{aligned}$$ from which, using eq[GT1chch]{}, we finally get the values
$$\begin{aligned}
& \cfxt{4} =
-\frac{1}{32 \pi^2 v^2}
\Big [
4 m^2 + M_A^2 \Big ( 4 - 3 \frac{g^2 v^2}{\Lambda^2} \Big ) +
M^2 \Big ( 4 + 3 \frac{g^2 v^2}{\Lambda^2} \Big )
\Big ]\div, \nonumber \\
& \cfxt{6} = -
\frac{1}{16 \pi^2 v^2}
\Big [ m^2 - M_A^2+ M^2 \Big (1 + 2 \frac{gv}{\Lambda} \Big ) \Big ]
\div.\end{aligned}$$
Similarly to what we have done in the previous case, we can check the results above using the three-point function $\overline{\Gamma}^{(1)}_{T_1 A_\mu A_\nu}$. Indeed we have $$\begin{aligned}
& \cfxt{4} \int \d T_1 (D^\mu \phi)^\dagger D_\mu \phi + \cfxt{6} \int \d T_1 \Big [
(D^2\phi)^\dagger \phi + \mathrm{h.c.}\Big ] + \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] \nonumber \\
&+\cfct{0T_1} {\cal S}_0 \int \d T_1 [\sigma^* (\sigma + v) + \chi^* \chi]
\supset \int \d
\frac{M_A^2}{2} [\cfxt{4} - 2 \cfxt{6} +
2 ( \cfct{0} + \cfct{0T_1} )] T_1 A^2,\end{aligned}$$ implying the consistency condition $$\begin{aligned}
M_A^2 \Big [ \cfxt{4} - 2 \cfxt{6} +
2 ( \cfct{0} + \cfct{0T_1} )
\Big ] g_{\mu\nu} =
\left .
\overline{\G}^{(1)}_{T_1 A_\mu A_\nu}(p_1,p_2)
\right |_{p_1=p_2=0}.\end{aligned}$$ the validity of which can be easily verified with the help of eq[GT1AA]{}.
The $\cfxt{7}$ and $\cfxt{8}$ coefficients
------------------------------------------
In this sector the relevant projections are
$$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi)+
\cfxt{1} \int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) \nonumber \\
&+ \cfxt{7}\int \d \bar c^*
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2\supset
\int \d
\Big ( \cfct{0} + \cfct{1} + \frac{\cfxt{1}}{2}
+ v^2 \cfxt{7}
\Big ) \bar c^* \sigma^2,\\
& \cfct{0T_1}{\cal S}_0\int \d T_1 [\sigma^* (\sigma+v) + \chi^* \chi]
- \cfct{1} {\cal S}_0 \int \d
T_1 (\sigma^* \sigma + \chi^* \chi)
+\cfxt{2} \int \d T_1\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) \nonumber \\
& + \cfxt{8}\int \d T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2\supset
\int \d \Big ( m^2 \cfct{1}- \frac{5}{2} \cfct{0T_1} m^2+ \frac{\cfxt{2}}{2} +v^2 \cfxt{8}\Big ) T_1 \sigma^2,\end{aligned}$$
yielding the relations $$\begin{aligned}
& 2(\cfct{0} + \cfct{1}) + \cfxt{1}
+ 2 v^2 \cfxt{7} =
\overline{\G}^{(1)}_{\bar c^* \sigma\sigma};&
& 2 m^2 \cfct{1}- 5 \cfct{0T_1} m^2+\cfxt{2} +2 v^2 \cfxt{8} = \overline{\G}^{(1)}_{T_1 \sigma \sigma} \, ,\end{aligned}$$ and, finally, the values $$\begin{aligned}
&\cfxt{7} = 0;&
&\cfxt{8} = -\frac{1}{8\pi^2 v^4}
\Big [ m^4+2m^2(M^2+M_A^2) + 2 (M^4- 3 M_A^4) \Big ] \div,\end{aligned}$$ see eqs[Gc\*ss]{}[GT1ss]{}.
The $\cfxt{9}$ and $\cfxt{10}$ coefficients
-------------------------------------------
The fact that the function $\overline{\G}^{(1)}_{\bar c^* A_\mu A_\nu}$ turns out to be momentum independent, see eq[Gc\*AA]{}, implies immediately that $$\begin{aligned}
\cfxt{9}=0.\end{aligned}$$ Next, in order to extract the coefficient $\cfxt{10}$ one needs first to change the variables to the contractible pairs basis, as explained in [@BQ:2019a]. To this end, one replaces the derivatives of the gauge field with a linear combination of the complete symmetrization over the Lorentz indices and a contribution depending on the field strength: $$\begin{aligned}
\partial_{\nu_1 \dots \nu_\ell} A_\mu = \partial_{(\nu_1 \dots \nu_\ell} A_{\mu)}+\frac{\ell}{\ell+1}\partial_{(\nu_1 \dots \nu_{\ell-1}} F_{\nu_\ell) \mu},
\label{cp.gauge}\end{aligned}$$ where $(\dots)$ denote complete symmetrization. In the present case it is therefore sufficient to consider the monomial $T_1 \partial^\mu A^\nu \partial_\mu A_\nu$ since, due to eq[cp.gauge]{} $$\begin{aligned}
T_1 \partial^\mu A^\nu \partial_\mu A_\nu =
T_1 \partial^{(\mu} A^{\nu)}\partial_{(\mu} A_{\nu)} +
\frac{T_1}{4} F^{\mu\nu} F_{\mu\nu}. \end{aligned}$$
Then, after we decompose the amplitude $\overline{\G}^{(1)}_{T_1 A_\mu A_\nu}$ according to $$\begin{aligned}
\overline{\G}^{(1)}_{T_1 A_\mu A_\nu}(p_1,p_2) & = [ \gamma^0_{T_1 AA} -2 \gamma^1_{T_1 AA} (p_1{\cdot} p_2) +
\gamma^2_{T_1 AA} (p_1^2+p_2^2)] g^{\mu\nu} \nonumber \\
& + \gamma^3_{T_1 AA}
p_1^\mu p_2^\nu +
\gamma^4_{T_1 AA}
p_1^\nu p_2^\mu,\end{aligned}$$ eq[GT1AA]{} gives $$\begin{aligned}
\cfxt{10}= \frac{\gamma^1_{TAA}}{4} =
-\frac{M_A^2}{128 \pi^2}\frac{g^2}{v^2 \Lambda^2} \div.
\label{cfxt.10}\end{aligned}$$
The $\cfxt{11}$, $\cfxt{12}$ and $\cfxt{13}$ coefficients
---------------------------------------------------------
The coefficient $\cfxt{12}$ has been fixed already, see eq[theta12]{}; on the other hand, $\cfxt{11}$ is determined by the projection of $$\begin{aligned}
& \cfxt{11} \int \d
\bar c^* T_1
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )
- \cfct{1} {\cal S}_0\int \d T_1 (\sigma^* \sigma + \chi^* \chi)+
\cfct{0T_1}
{\cal S}_0 \int \d T_1 [\sigma^* (\sigma+v) + \chi^* \chi]
\nonumber \\
& \supset \int \d
\Big ( v \cfxt{11} - v \cfct{1} + 2 v \cfct{0T_1} \Big ) \bar c^* T_1 \sigma \, .\end{aligned}$$ yielding $$\begin{aligned}
\cfxt{11} = \frac{1}{v}
\Big ( \overline{\G}^{(1)}_{\bar c^* T_1 \sigma} + v \cfct{1} - 2 v \cfct{0T_1} \Big ) =
\frac{1}{4 \pi^2 v^2}(m^2+ M^2 + M_A^2)\div,\end{aligned}$$ where the one-loop result has been used. Finally, $$\begin{aligned}
\cfxt{13} \int \d
(\bar c^*)^2 \Big (
\phi^\dagger \phi - \frac{v^2}{2}
\Big ) \supset \int \d
\cfxt{13} v \sigma (\bar c^*)^2,\end{aligned}$$ which implies$$\begin{aligned}
\cfxt{13}= \frac{1}{2v}
\overline{\G}^{(1)}_{\bar c^*\bar c^* \sigma } = 0,\end{aligned}$$ as this amplitude turns out to be UV finite.
The gauge-invariant field sector {#sec:gi}
================================
The last sector we need to consider is finally the one of gauge invariants containing only the fields.
The $\cfgi{2}$ and $\cfgi{3}$ coefficients
------------------------------------------
While the coefficient $\cfgi{1}$ has been already fixed, see eq[l1.bc]{}, $\cfgi{2}$ and $\cfgi{3}$ can be determined by considering the two- and three-point $\sigma$ amplitudes. The relevant projection equation are
$$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi)+
\cfgi{1} \int \d
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) \nonumber \\
&+\cfgi{2} \int \d
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2\supset \int \d \Big( v^2\cfgi{2}+\frac12\cfgi{1}- m^2 \cfct{1} - \frac52 m^2\cfct{0}\Big)\sigma^2, \\
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi)+
\cfgi{2} \int \d
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2 \nonumber \\
&+\cfgi{3} \int \d
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^3\supset \int \d \Big(2 v^3 \cfgi{3} + 2 v \cfgi{2}
- \frac{3 m^2}{v} \cfct{1} - \frac{4 m^2}{v} \cfct{0}\Big )\sigma^3,\end{aligned}$$
yielding
$$\begin{aligned}
& 2 v^2\cfgi{2}+\cfgi{1}-2 m^2 \cfct{1} - 5 m^2\cfct{0} =
\overline{\G}^{(1)}_{\sigma\sigma},\\
& 6 v^3 \cfgi{3} + 6 v \cfgi{2}
- \frac{9 m^2}{v} \cfct{1} - \frac{12 m^2}{v} \cfct{0} = \overline{\G}^{(1)}_{\sigma\sigma\sigma}.
$$
eqs[Gss]{}[Gsss]{} implies then the following results
$$\begin{aligned}
\cfgi{2} & = \frac{1}{16 \pi^2 v^4}
\Big [ m^4 + 2 m^2 (M^2 + M_A^2) + 2 (M^4+3 M_A^4) \Big ] \div, \nonumber \\
\cfgi{3} & = 0.\end{aligned}$$
The values of these coefficients can be checked by looking at the $\overline{\G}^{(1)}_{\sigma \chi\chi}$ and $\overline{\G}^{(1)}_{\sigma\sigma \chi\chi}$ amplitudes, for which the projection equation $$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi)+\cfgi{2} \int \d
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2 \nonumber \\
&\supset \int\d \Big( v \cfgi{2} - \frac{3 m^2}{2v} \cfct{1} - \frac{2 m^2}{v} \cfct{0} \Big)\sigma\chi^2+\int\d \Big(\frac12 \cfgi{2} - \frac{ m^2}{v^2} \cfct{1}- \frac{ m^2}{v^2} \cfct{0}\Big)\sigma^2\chi^2,\end{aligned}$$ gives rise to the consistency conditions $$\begin{aligned}
& 2 v \cfgi{2} - \frac{3 m^2}{v} \cfct{1} - \frac{4 m^2}{v} \cfct{0} = \overline{\G}^{(1)}_{\sigma \chi\chi},
\nonumber \\
& 2 \cfgi{2} - \frac{4 m^2}{v^2} \cfct{1}
- \frac{4 m^2}{v^2} \cfct{0}
=
\overline{\G}^{(1)}_{\sigma\sigma \chi\chi},\end{aligned}$$ which, using eqs[Gschch]{}[Gsschch]{}, can be easily proven to be fulfilled.
The $\cfgi{4}$ and $\cfgi{5}$ coefficients
------------------------------------------
These coefficients are fixed by the 2-point Goldstone amplitude, which is controlled by the invariants $$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi) +
\cfgi{1} \int \d \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )
\nonumber \\
&+ \cfgi{4} \int \d (D^\mu \phi)^\dagger D_\mu \phi
+\cfgi{5} \int \d \phi^\dagger
[ (D^2)^2 +
D^\mu D^\nu D_\mu D_\nu +
D^\mu D^2 D_\mu ] \phi
\nonumber \\
& \supset
\int \d
\Big [
\frac{1}{2} \Big ( \cfgi{1} - m^2 \cfct{0} \Big ) \chi^2 +
\Big ( \cfct{0}+\cfct{1}+\frac{\cfgi{4}}{2} \Big )\partial^\mu \chi \partial_\mu \chi
+ \frac{3}{2} \cfgi{5} \chi \square^2 \chi
\Big ],\end{aligned}$$ which gives rise to the following projections $$\begin{aligned}
& \cfgi{1} - m^2 \cfct{0} = \left . \overline{\G}^{(1)}_{\chi\chi} \right |_{p^2=0};&
& 2(\cfct{0}+\cfct{1})+\cfgi{4}
= \left . \frac{\partial\overline{\G}^{(1)}_{\chi\chi}}{\partial p^2} \right |_{p^2=0};&
&3 \cfgi{5} = \left . \frac{\partial\overline{\G}^{(1)}_{\chi\chi}}{\partial (p^2)^2} \right |_{p^2=0}.\end{aligned}$$ From the one-loop expression reported in , we then obtain the gauge-independent coefficients
$$\begin{aligned}
& \cfgi{4} = -
\frac{1}{32 \pi^2 v^2}
\Big [ \frac{gv}{\Lambda} \Big (4 - \frac{gv}{\Lambda} \Big )M^2
+ M_A^2 \Big (16 +
12 \frac{gv}{\Lambda} + 3 \frac{g^2 v^2}{\Lambda^2}
\Big ) \Big ] \div,
\label{l4} \\
& \cfgi{5} = \frac{g^2}{96 \pi^2 \Lambda^2} \div .\label{l5}\end{aligned}$$
The $\cfgi{6}$ and $\cfgi{7}$ coefficients
------------------------------------------
The relevant Green’s function for fixing these coefficients is the four-point Goldstone amplitude since $$\begin{aligned}
&\cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi)+\cfgi{2} \int \d
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2
\nonumber \\
&
+\cfgi{6} \int \d \Big ( \phi^\dagger \phi- \frac{v^2}{2} \Big ) \Big ( \phi^\dagger D^2 \phi + (D^2\phi)^\dagger \phi \Big ) +\cfgi{7} \int \d \Big ( \phi^\dagger \phi- \frac{v^2}{2} \Big ) (D^\mu \phi)^\dagger D_\mu \phi
\nonumber \\
& \supset \int \d
\Big \{ \Big [ \frac{\cfgi{2}}{4} - (\cfct{0}+\cfct{1}) \frac{m^2}{2v^2}
\Big ] \chi^4 + \frac{\cfgi{6}}{2} \chi^3 \square \chi
+ \frac{\cfgi{7}}{4} \chi^2 \partial^\mu \chi \partial_\mu \chi
\Big \},\end{aligned}$$ yielding $$\begin{aligned}
6 \cfgi{2} - \frac{12 m^2}{v^2} (\cfct{0}+ \cfct{1})
- 3 \cfgi{6} \sum_{i=1}^4
p_i^2
- \cfgi{7}
\sum_{i<j} p_i p_j
=
\overline{\G}^{(1)}_{\chi\chi\chi\chi}(p_i).
\label{4chi.proj}\end{aligned}$$ Notice that we keep the momentum dependence of the four point $\chi$ amplitude on the right-hand side. A remark is in order here. Before attempting to extract the coefficients of the momenta polynomia on the left-hand side of eq[4chi.proj]{}, we need to take into account the fact that [FeynArts]{} and [FormCalc]{} internally implement momentum conservation, so the amplitude is only known on the hyperplane $\sum_i p_i = 0$. Hence we eliminate $p_4$ in favor of the remaining momenta, $p_4 = -\sum_{i=1}^3 p_i$, so that eq[4chi.proj]{} becomes $$\begin{aligned}
6 \cfgi{2} - \frac{12 m^2}{v^2} (\cfct{0}+ \cfct{1})
-
( 6 \cfgi{6} - \cfgi{7} )
\Big ( \sum_{i=1}^3 p_i^2 + \sum_{i<j} p_i p_j \Big )
=
\overline{\G}^{(1)}_{\chi\chi\chi\chi}(p_1,p_2,p_3).
\label{4chi.mom.cons}\end{aligned}$$ Then the condition $$\begin{aligned}
6 \cfgi{2} - \frac{12 m^2}{v^2} (\cfct{0}+ \cfct{1}) =
\left . \overline{\G}^{(1)}_{\chi\chi\chi\chi} \right |_{p_i=0},\end{aligned}$$ is easily verified, see eq[Gchchchch]{}. On the other hand, we notice that the restriction of $\overline{\G}^{(1)}_{\chi^4}$ on the momentum conservation hyperplane only fixes the combination $6 \cfgi{6} - \cfgi{7} $, and an additional amplitude needs to be considered to fix the two coefficients separately.
To this end, let us consider the two point $\sigma$-amplitude, with the following projection on the derivative-dependent sector $$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi) +
\cfgi{4} \int \d (D^\mu \phi)^\dagger D_\mu \phi\nonumber \\
&+\cfgi{6} \int \d \Big ( \phi^\dagger \phi- \frac{v^2}{2} \Big ) \Big ( \phi^\dagger D^2 \phi + (D^2\phi)^\dagger \phi \Big )\supset\int \d \Big [ \Big ( \frac{\cfgi{4}}{2} + \cfct{0} + \cfct{1} \Big )\partial^\mu \sigma \partial_\mu \sigma
+ v^2 \cfgi{6} \sigma \square \sigma \Big ],\end{aligned}$$ leading to the condition $$\begin{aligned}
2 v^2 \cfgi{6} - \cfgi{4}
-2 ( \cfct{0} + \cfct{1} ) = -
\left . \frac{\partial \overline{\G}_{\sigma\sigma}^{(1)}}{\partial p^2} \right |_{p^2=0}.\end{aligned}$$ This gives then the result, see eq[Gss]{} $$\begin{aligned}
\cfgi{6} = \frac{1}{64 \pi^2 v^3}
\frac{g}{\Lambda}
\Big [ 4 m^2 + (M^2 - 3 M_A^2)
\Big ( 4 + \frac{gv}{\Lambda} \Big )
\Big ] \div,\end{aligned}$$ which in combination with eqs[4chi.mom.cons]{}[Gchchchch]{} yields finally $$\begin{aligned}
\cfgi{7}=
\frac{1}{32 \pi^2 v^3}
\frac{g}{\Lambda}
\Big [ 2 m^2 \Big ( 2 + \frac{gv}{\Lambda} \Big )
- M^2 \Big ( 4 - 5 \frac{gv}{\Lambda} \Big ) - 3 M_A^2 \Big (
12 + 5 \frac{gv}{\Lambda}
\Big )
\Big ] \div.\end{aligned}$$ We can check this result against the projections on the monomials $\sigma \chi \square \chi, \sigma \partial^\mu \chi \partial_\mu \chi$, namely (we use integration by parts in the last line) $$\begin{aligned}
& \cfgi{6} \int \d \Big ( \phi^\dagger \phi -
\frac{v^2}{2} \Big )
\Big ( \phi^\dagger D^2 \phi +
(D^2\phi)^\dagger \phi
\Big ) +
\cfgi{7} \int \d \, \Big ( \phi^\dagger \phi -
\frac{v^2}{2} \Big ) (D^\mu \phi)^\dagger D_\mu \phi
\nonumber \\
& \supset\int \d \, \Big ( v \cfgi{6}
\sigma \chi \square \chi +
\frac{v \cfgi{7}}{2} \sigma \partial^\mu \chi \partial_\mu \chi
+ \frac{\cfgi{6} v}{2} \chi^2 \square \sigma
\Big ) \nonumber \\
& =
\int \d \, \Big [
2 v \cfgi{6} \sigma \chi \square \chi
+
\Big (
v \cfgi{6} + \frac{v \cfgi{7}}{2}
\Big )
\sigma \partial^\mu \chi \partial_\mu \chi
\Big ].
\label{proj.sigma2chi}\end{aligned}$$ After eliminating the $\sigma$-momentum in favour of the remaining tow by using momentum conservation, the resulting amplitude can be expanded as $$\begin{aligned}
\overline{\G}^{(1)}_{\sigma \chi \chi}(p_1,p_2) =
\gamma_{\sigma\chi\chi}+
\gamma^1_{\sigma\chi\chi} (p_1^2 + p_2^2)
+ \gamma^2_{\sigma\chi\chi} p_1{\cdot} p_2 + {\cal O}(p^4)\end{aligned}$$ eq[proj.sigma2chi]{} then implies the consistency conditions $$\begin{aligned}
& 2 v\cfgi{6}+ \gamma^1_{\sigma\chi\chi} = 0;&
2 v \cfgi{6} + v \cfgi{7}
+ \gamma^2_{\sigma\chi\chi} = 0,\end{aligned}$$ which can be easily verified using the result eq[Gschch]{}.
The $\cfgi{8}$ and $\cfgi{9}$ coefficients
------------------------------------------
These coefficients are controlled by the $AA$ amplitude which also provides a non-trivial example of the contractible pairs technique. Indeed, the two-point function of the Goldstone field fixes the coefficient $\cfgi{5}$ via the projection on the monomial $\int \d \chi \square^2 \chi$; on the other hand, the $\cfgi{5}$ invariant admits also a non-trivial expansion in power of the gauge field, precisely accounting for the non-transverse form factors of $\overline{\G}^{(1)}_{A^\mu A^\nu}$.
To see this in detail, observe that the relevant invariants are $$\begin{aligned}
& \cfct{0} {\cal S}_0 \int \d [\sigma^* (\sigma + v) + \chi^* \chi] +
\cfct{1} {\cal S}_0 \int \d (\sigma^* \sigma + \chi^* \chi) \nonumber \\
&+\cfgi{5} \int \d \phi^\dagger[ (D^2)^2 + D^\mu D^\nu D_\mu D_\nu + D^\mu D^2 D_\mu ] \phi + \frac{\cfgi{8}}{2} \int \d F_{\mu\nu}^2 +
\cfgi{9} \int \d \partial^\mu F_{\mu\nu} \partial_\rho F^{\rho\nu}\nonumber \\
& \supset
\int \d \Big [
\Big ( \cfct{0} + \frac{\cfgi{4}}{2} \Big ) e^2 v^2 A^2
- \frac{\cfgi{5}}{2} e^2 v^2
(2 A^\mu \partial_\mu \partial A +
A^\mu \square A_\mu ) +
\frac{\cfgi{8}}{2} (\partial^\mu A^\nu - \partial^\nu A^\mu)^2\nonumber \\
&
+ \cfgi{9} (\square A^\mu - \partial^\mu (\partial A))^2
\Big ]\end{aligned}$$ There are no contribution of order $p^4$ in $\overline{\G}^{(1)}_{A^\mu A^\mu}$, see eq[GAA]{}, so $$\begin{aligned}
\cfgi{9}=0.\end{aligned}$$ The remaining terms give the projection equation $$\begin{aligned}
\Big [ e^2 v^2 ( 2 \cfct{0} + \cfgi{4}) + ( 2 \cfgi{8} + e^2 v^2
\cfgi{5}) p^2 \Big ] g^{\mu\nu}
+ 2 \Big ( e^2 v^2 \cfgi{5} - \cfgi{8} \Big ) p^\mu p^\nu
= \overline{\G}^{(1)}_{A^\mu A^\nu}(p).\end{aligned}$$ Notice that in the right-hand side of the above equation we keep the momentum dependence of the two point gauge amplitude. From eqs[cfct.0]{}[l4]{}[l5]{}[GAA]{}, we see that the above equation is verified with $$\begin{aligned}
\cfgi{8} = - \frac{M_A^2}{96 \pi^2 v^2}
\Big ( 2 + 2 \frac{gv}{\Lambda} +
\frac{g^2 v^2}{\Lambda^2} \Big ) \div,\end{aligned}$$ which implies that $\cfgi{8}$ is gauge-independent, as it should.
The $\cfgi{10}$ coefficient
---------------------------
This coefficient can be obtained in much the same way as $\cfxt{10}$, [*i.e.*]{}, by the contractible pair method. Parameterize the amplitude $\overline{\G}^{(1)}_{\sigma A_\mu A_\nu}$ according to $$\begin{aligned}
\overline{\G}^{(1)}_{\sigma A_\mu A_\nu}(p_1,p_2)= & [ \gamma^0_{\sigma AA} -2 \gamma^1_{\sigma AA} p_1\cdot p_2 +
\gamma^2_{\sigma AA} (p_1^2+p_2^2)] g^{\mu\nu}
+ \gamma^3_{\sigma AA}
p_1^\mu p_2^\nu +
\gamma^4_{\sigma AA}
p_1^\nu p_2^\mu, \end{aligned}$$ and extract $\cfgi{10}$ through the form factor $\gamma^1_{\sigma AA}$: $$\begin{aligned}
\cfgi{10} =
\int \d F^2_{\mu\nu}
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) \supset\cfgi{10}
\int \d 2 \sigma \partial^\mu A^\nu \partial_\mu A_\nu. \end{aligned}$$ We obtain, see eq[GsAA]{}, $$\begin{aligned}
\cfgi{10}= \frac{\gamma^1_{\sigma AA}}{4v} =
\frac{M_A^2}{128 \pi^2}\frac{g^2}{v^2\Lambda^2}
\Big ( -4 + \frac{gv}{\Lambda} \Big )
\div.\end{aligned}$$ Notice in particular that the combination $$\begin{aligned}
\cfgi{10}+\frac{g}{v\Lambda} \cfxt{10} = - \frac{M_A^2}{32 \pi^2} \frac{g^2}{\Lambda^2 v^2}\div,\end{aligned}$$ correctly reproduces the coefficient $c_{\cal O}^{(1)}$ of [@BQ:2019a].
Mapping
=======
\[sec:map\]Renormalization coefficients
---------------------------------------
We are now in a position to evaluate the renormalization coefficients of the operators of dimension less or equal to $6$ in the target theory. For that purpose one simply needs to map the invariants depending on the external sources by applying the substitution rules and collecting the contributions to the operator one is interested in.
Notice that all the coefficients obtained must be gauge-invariant (as a consequence of the gauge-invariance of the $\cfxt{i},\cfps{i}$ and $\cfgi{i}$ coefficients); in addition they must not depend on $m^2$. The latter is a highly non-trivial check of the computations, due to the ubiquitous presence of $m^2$ in the projections as well as in the amplitudes.
In what follows, we list here the results for all possible operators, reinstating the correct $D$-dimensional dependence on the ’t Hooft mass $\mu$.
- $\phi^\dagger \phi - \frac{v^2}{2}$
$$\begin{aligned}
\widetilde{\lambda}_1&=\frac{1}{v^2} \Big [(M^2-m^2) \cfps{1} + \frac{gv}{\Lambda} \cfps{2} + v^2 \cfgi{1}\Big ] & \nonumber \\
&=\frac{\mu^{-\epsilon}}{16 \pi^2 v^2} \Big \{ M^4 \Big ( 3 - \frac{gv}{\Lambda} \Big ) + M_A^2 \Big [ M^2 + 3 M_A^2 \Big (2 + \frac{gv}{\Lambda} \Big ) \Big ]\Big \} \div.\end{aligned}$$
- $ \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2$
$$\begin{aligned}
\widetilde{\lambda}_2&= \frac{(m^2-M^2)^2}{2 v^4}
\cfps{3} +
\frac{g^2}{2\Lambda^2 v^2}
\cfps{4} +
\frac{g}{\Lambda v^3}(m^2- M^2)\cfps{7} +
\frac{m^2-M^2}{v^2}
\cfxt{1}+ \frac{g}{\Lambda v} \cfxt{2} + \cfgi{2}
\nonumber \\
&= \frac{\mu^{-\epsilon}}{32 \pi^2 v^4}
\Big \{
4 M_A^2 M^2 \Big ( 1 - \frac{gv}{\Lambda} \Big )
+ 3 M_A^4 \Big (
4 + 8 \frac{gv}{\Lambda} +
\frac{g^2 v^2}{\Lambda^2}
\Big )\nonumber \\
&+
M^4 \Big ( 10 -12 \frac{gv}{\Lambda} + 3 \frac{g^2 v^2}{\Lambda^2} \Big )
\Big \} \div.\end{aligned}$$
- $\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^3$ $$\begin{aligned}
\widetilde{\lambda}_3&= \frac{(m^2-M^2)^3}{6 v^6} \cfps{9} +
\frac{g (m^2-M^2)^2}{2 \Lambda v^5}\cfps{10}
+ \frac{g^2(m^2- M^2)}{2 \Lambda^2 v^4}\cfps{11}+
\frac{g^3}{6 \Lambda^3 v^3}
\cfps{12} \nonumber \\
& +
\frac{m^2-M^2}{v^2}\cfxt{7} +
\frac{g}{v\Lambda}\cfxt{8}+
\frac{g(m^2-M^2)}{\Lambda v^3}\cfxt{11}+
\frac{g^2}{\Lambda^2 v^2}\cfxt{12} +
\frac{(m^2-M^2)^2}{v^4}
\cfxt{13} + \cfgi{3} \nonumber \\
& = - \frac{\mu^{-\epsilon}}{16 \pi^2 v^5}
\frac{g}{\Lambda}
\Big [
2 M^2 M_A^2 \Big( 2 - \frac{gv}{\Lambda} \Big )
- 6 M_A^4\Big( 2 + \frac{gv}{\Lambda} \Big )
+ M^4
\Big( 10 -9 \frac{gv}{\Lambda}
+ 2 \frac{g^2v^2}{\Lambda^2}\Big )
\Big ] \div.\end{aligned}$$
- $(D^\mu \phi)^\dagger D_\mu \phi$ $$\begin{aligned}
\widetilde\lambda_4&=\frac{g}{\Lambda v}\cfps{1} + \cfgi{4} \nonumber \\
&=-\frac{\mu^{-\epsilon}}{32 \pi^2 v^2}
\Big [ M^2 \frac{gv}{\Lambda}
\Big ( 6 - \frac{gv}{\Lambda} \Big ) +
M_A^2
\Big (
16 + 14 \frac{gv}{\Lambda}
+
3\frac{g^2v^2}{\Lambda^2}
\Big )
\Big ]\div.\end{aligned}$$
- $\phi^\dagger [ (D^2)^2 + D^\mu D^2 D_\mu + D^\mu D^\nu D_\mu D_\nu ] \phi$ $$\begin{aligned}
\widetilde{\lambda}_5&\equiv \cfgi{5} = \frac{\mu^{-\epsilon}}{96 \pi^2}\frac{g^2}{\Lambda^2} \div .\end{aligned}$$
- $\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) ( \phi^\dagger D^2\phi + \mathrm{h.c.})$ $$\begin{aligned}
\widetilde{\lambda}_6&= \frac{g^2}{2\Lambda^2v^2}
\cfps{5} +
\frac{g}{\Lambda v^3}(m^2- M^2)
\cfps{8}+
\frac{m^2-M^2}{v^2} \cfxt{5}
+\frac{g}{\Lambda v}\cfxt{6}
+\cfgi{6} \nonumber \\
& = -\frac{\mu^{-\epsilon}}{16\pi^2v^2}\frac{g^2 M^2}{\Lambda^2}\div.\end{aligned}$$
- $\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) (D^\mu \phi)^\dagger D_\mu \phi$ $$\begin{aligned}
\widetilde{\lambda}_7&= \frac{g (m^2- M^2)}{\Lambda v^3}\cfps{3}+
\frac{g^2}{\Lambda^2v^2}( \cfps{5}+\cfps{7})+
\frac{2 g}{\Lambda v^3}(m^2-M^2)\cfps{8}
+\frac{g}{\Lambda v}( \cfxt{1}+
\cfxt{4} ) \nonumber \\
&+
\frac{m^2-M^2}{v^2} \cfxt{3}
+\cfgi{7} \nonumber \\
& = - \frac{\mu^{-\epsilon}}{32 \pi^2 v^3}
\frac{g}{\Lambda}
\Big [
M^2\Big ( 16 -14 \frac{gv}{\Lambda} + 3 \frac{g^2 v^2}{\Lambda^2} \Big ) + M_A^2
\Big ( 36 + 8\frac{gv}{\Lambda} - 3 \frac{g^2 v^2}{\Lambda^2} \Big )
\Big ] \div.\end{aligned}$$
- $F^{\mu\nu} F_{\mu\nu}$ $$\begin{aligned}
\widetilde{\lambda}_8&\equiv\cfgi{8} = - \frac{\mu^{-\epsilon}}{96 \pi^2 v^2}M_A^2
\Big ( 2 + 2 \frac{gv}{\Lambda} +
\frac{g^2 v^2}{\Lambda^2} \Big ) \div .\end{aligned}$$
- $\partial^\mu F_{\mu\nu} \partial^\rho F_{\rho\nu}$ $$\begin{aligned}
\widetilde{\lambda}_9\equiv \cfgi{9}=0.\end{aligned}$$
- $\Big (
\phi^\dagger \phi - \frac{v^2}{2}
\Big ) F_{\mu\nu}^2$ $$\begin{aligned}
\widetilde{\lambda}_{10}&=-\frac{M^2-m^2}{v^2}\cfxt{9} + \frac{g}{v\Lambda}\cfxt{10}+\cfgi{10} \nonumber \\
&= -\frac{\mu^{-\epsilon}}{32 \pi^2} \frac{g^2 M_A^2}{\Lambda^2 v^2 }\div.\end{aligned}$$
\[sec.beta\]$\beta$-functions
-----------------------------
It is now immediate to construct the $\beta$-functions of the theory. Renormalization implies that the running of the coupling $\widetilde{\lambda}_i$ in the target theory is determined by the corresponding $\beta$-function $\beta_i$ $$\begin{aligned}
\beta_i=(4\pi)^2\frac{\mathrm{d}}{d\log\mu}\widetilde{\lambda}_i.\end{aligned}$$ Then, taking into accounts only terms proportional to the beyond the SM coupling $g$ we can write $$\begin{aligned}
\beta_i\supseteq-(4\pi)^2C_i,\end{aligned}$$ where the coefficients $C_i$ are obtained from the corresponding $\widetilde{\lambda}_i$ dropping terms proportional to the power counting renormalizable couplings and replacing $g/\Lambda$ with $\widetilde{\lambda}_7$ as dictated by eqs[tree.level]{}[g.invs]{}.
In the linear approximation we finally obtain $$\begin{aligned}
\beta_i\supseteq c_i\widetilde{\lambda}_7,\end{aligned}$$ with $$\begin{aligned}
c_1&=\frac1{v}(M^4-3M^4_A);&
c_2&=\frac2{v^3}(3M^4+M^2M^2_A-6M^4_A),\nonumber \\
c_3&=\frac2{v^5}(5M^4+2M^2M^2_A-6M^4_A);&
c_4&=\frac1{v}(3M^2+7M^2_A),\nonumber \\
c_5&=0;&
c_6&=0,\nonumber \\
c_7&=\frac2{v^3}(4M^2+9M^2_A);&
c_8&=\frac{1}{3v}M^2_A,\nonumber \\
c_9&=0;&
c_{10}&=0.\end{aligned}$$
Conclusions {#sec:concl}
===========
We have presented the explicit evaluation of all the UV coefficients of dimension less or equal to 6 operators in an Abelian spontaneously broken gauge theory supplemented with a maximally power counting violating derivative interaction of dimension 6. This has been possible by following the methodology put forward in a companion paper [@BQ:2019a], in which one constructs an auxiliary theory based on the $X$-formalism in which a power-counting can be established (thus limiting the number of divergent diagrams one has to consider at each loop order) together with a mapping onto the original theory.
In particular, a separation of the gauge-dependent contributions, associated to the cohomologically trivial invariants, from the genuine physical renormalizations of gauge invariant operators has been achieved, and we have explicitly checked in two different gauges (Feynman and Landau) our results in order to explicitly verify the gauge independence of the coefficients of gauge invariant operators. In this respect it should be clear the pivotal role played by the field redefinitions for the correct identification of the gauge dependent coefficients of the cohomologically trivial invariants and, consequently, of the coefficients of the gauge invariant operators. Purely gauge fixed on-shell calculations will completely miss their contributions, running the risk of obtaining gauge dependent results even in the case of ostensibly gauge invariant quantities. As an example we have derived the complete set of one-loop $\beta$-functions of the model which, after the field renormalization is carried out, can be read immediately from the renormalization coefficient of the corresponding operator.
The techniques presented here and in [@BQ:2019a] are suitable to be generalized to: the inclusion of the complete set of dimension 6 operators; the extension of higher orders in the loop expansion; the extension of non-Abelian case, and, in particular, to the Standard Model effective field theory in which dimension 6 operators are added to the usual SU(2)$\times$U(1) action. This latter generalization would be especially interesting, as it would allow to better understand the remarkable cancellations and regularities discovered when evaluating the one-loop anomalous dimensions for this model, and which have been linked to holomorphicity [@Cheung:2015aba], and/or remnants of embedding supersymmetry [@Elias-Miro:2014eia]. Work in this direction is currently underway and we hope to report soon on this and related issues.
List of invariants {#app:list}
==================
Pure external sources invariants
--------------------------------
The invariants in this sector are $$\begin{aligned}
&\cfps{1} \int \d \bar c^* ;& &
\cfps{2} \int \d T_1 , \nonumber \\
& \cfps{3} \int \d \frac{1}{2} (\bar c^*)^2 ;
& & \cfps{4} \int \d \frac{1}{2} T_1^2 , \nonumber \\
& \cfps{5} \int \d \frac{1}{2} T_1\square T_1 ;
& & \cfps{6} \int \d \frac{1}{2} T_1\square^2 T_1 \nonumber \\
& \cfps{7} \int \d \bar c^* T_1 ;
& & \cfps{8} \int \d \bar c^* \square T_1 , \nonumber \\
& \cfps{9} \int \d \frac{1}{3!}(\bar c^*)^3 ;
& & \cfps{10} \int \d \frac{1}{2} (\bar c^*)^2 T_1 , \nonumber \\
& \cfps{11} \int \d \frac{1}{2} (\bar c^*) T_1^2 ;
& & \cfps{12} \int \d \frac{1}{3!}T_1^3 .
\label{ESinv}\end{aligned}$$
Notice that $\cfps{6}$ has been inserted for completeness but does not contribute to dim. 6 operators in the target theory.
Mixed field-external sources invariants
---------------------------------------
The invariants in this sector are $$\begin{aligned}
& \cfxt{1} \int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big );&
& \cfxt{2} \int \d T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ),
\nonumber \\
& \cfxt{3} \int \d \bar c^* (D^\mu \phi)^\dagger D_\mu \phi;&
& \cfxt{4} \int \d T_1 (D^\mu \phi)^\dagger D_\mu \phi, \nonumber \\
& \cfxt{5} \int \d \bar c^* \Big [ (D^2 \phi)^\dagger \phi + \mathrm{h.c.} \Big ];&
& \cfxt{6} \int \d T_1 \Big [ (D^2 \phi)^\dagger \phi + \mathrm{h.c.} \Big ], \nonumber \\
& \cfxt{7} \int \d \bar c^* \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2 ;&
& \cfxt{8} \int \d T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big )^2,
\nonumber \\
& \cfxt{9} \int \d \bar c^*
F_{\mu\nu}^2;&
& \cfxt{10} \int \d T_1
F_{\mu\nu}^2,
\nonumber \\
& \cfxt{11} \int \d
\bar c^* T_1 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big );&
& \cfxt{12} \int \d
T_1^2 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ),\nonumber \\
& \cfxt{13} \int \d
(\bar c^*)^2 \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ).&
\label{mix.invs}
\end{aligned}$$ Notice that the use of the contractible pair basis allows us to re-express the (otherwise present) invariants $$\begin{aligned}
&\cfxt{14} \int \d \bar c^* \square
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big );&
&\cfxt{15} \int \d T_1\square
\Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ),\end{aligned}$$ in terms of the above, since one has $$\begin{aligned}
\square \Big ( \phi^\dagger \phi - \frac{v^2}{2} \Big ) =
(D^2\phi)^\dagger \phi + \phi^\dagger (D^2\phi) + 2 (D^\mu \phi)^\dagger D_\mu \phi,\end{aligned}$$ and therefore $$\begin{aligned}
\cfxt{14}&=2\cfxt{3}+\cfxt{5};& \cfxt{15}&=2\cfxt{4}+\cfxt{6}.\end{aligned}$$
Gauge invariants depending only on the fields
---------------------------------------------
The invariants in this sector are $$\begin{aligned}
& \cfgi{1} \int \d \Big (
\phi^\dagger \phi - \frac{v^2}{2}
\Big );&
& \cfgi{2} \int \d \Big (
\phi^\dagger \phi - \frac{v^2}{2}
\Big )^2, \nonumber \\
& \cfgi{3} \int \d \Big (
\phi^\dagger \phi - \frac{v^2}{2}
\Big )^3; &
& \cfgi{4} \int \d
(D^\mu \phi)^\dagger D_\mu \phi, \nonumber \\
& \cfgi{5} \int \d \phi^\dagger
[ (D^2)^2 +
D^\mu D^\nu D_\mu D_\nu +
D^\mu D^2 D_\mu ] \phi;&
& \cfgi{6} \int \d \Big ( \phi^\dagger \phi
- \frac{v^2}{2} \Big )
\Big ( \phi^\dagger D^2 \phi +
(D^2\phi)^\dagger \phi \Big ),
\nonumber \\
& \cfgi{7} \int \d
\Big ( \phi^\dagger \phi
- \frac{v^2}{2} \Big ) (D^\mu \phi)^\dagger D_\mu \phi; &
& \frac{\cfgi{8}}2 \int \d F_{\mu\nu}^2 \, , \nonumber \\
& \cfgi{9} \int \d \partial^\mu F_{\mu\nu} \partial^\rho F_{\rho\nu}; &
& \cfgi{10}
\int \d \Big ( \phi^\dagger \phi
- \frac{v^2}{2} \Big )
F_{\mu\nu}^2.
\ \label{g.invs}\end{aligned}$$ These invariants are the only ones appearing also in the target theory; in that case the associated coefficient will be indicated as $\widetilde{\lambda}_i$ (with $i=1,\dots,10$).
UV divergent ancestor amplitudes {#app:UVdivamp}
================================
Tadpoles {#tadpoles}
--------
$$\begin{aligned}
\overline{\G}^{(1)}_{\bar c^*} &=
- \frac{M^2+(1 - \delta_{\xi0} )M_A^2 }{16 \pi^2}\div, \label{Gc*}\\
\overline{\G}^{(1)}_{T_1} &=
- \frac{(M^4-3 M_A^4)}{16 \pi^2}\div, \label{GT1}\\
\overline{\G}^{(1)}_{\sigma} &=\frac{1}{16 \pi^2 v}
\Big [ m^2 M^2 + (1 - \delta_{\xi0} ) m^2 M_A^2 + 2 (M^4 + 3 M_A^4) \Big ]\div .\label{Gs} \end{aligned}$$
Two-point functions
-------------------
$$\begin{aligned}
\overline{\Gamma}^{(1)}_{\chi^* \omega}&=\frac{eM_A^2}{8 \pi^2 v} \div (\dl-1),\label{Gch*om}\\
\overline{\G}^{(1)}_{\chi\chi} &= \frac{1}{32 \pi^2 v^2} \Big \{ 2 m^2 (M^2 + M_A^2) + 4 (M^4+ 3 M_A^4)
- \frac{1}{16 \pi^2 v^2} M_A^2( m^2 + 2 p^2) \frac{\delta_{\xi0}}{\epsilon}
\nonumber \\
&
- \Big [ \frac{gv }{\Lambda} M^2 \Big ( 4 - \frac{gv }{\Lambda} \Big )
+ M_A^2 \Big (
8 + 12 \frac{gv}{\Lambda} +
3 \frac{g^2v^2}{\Lambda^2} \Big ) \Big ] p^2+
\frac{g^2 v^2}{\Lambda^2} p^4 \Big \} \div, \label{Gchch}\\
\overline{\G}^{(1)}_{\sigma\sigma} &= \frac{1}{16 \pi^2 v^2}
\Big \{
2 m^4 + m^2 (5 M^2+ M_A^2) + 6 (M^4+ 3 M_A^4) \nonumber \\
&
- \Big [ 4 M_A^2 + 2 \frac{gv}{\Lambda} (m^2+ 2 M^2)
\Big ]
p^2 +
\frac{g^2 v^2}{\Lambda^2} p^4
\Big \} \div
- \frac{ M_A^2 (m^2 + 2 p^2)}{16 \pi^2 v^2}
\frac{\delta_{\xi0}}{\epsilon},\label{Gss}\\\
\overline{\G}^{(1)}_{A_\mu A_\nu} &=
-\frac{M_A^2}{32 \pi^2 v^2}
\Big \{ M^2 \frac{gv}{\Lambda} \Big (
4 - \frac{gv}{\Lambda} \Big ) +
M_A^2 \Big [
4( 4 - \delta_{\xi0} ) + 12 \frac{gv}{\Lambda}
+ 3 \frac{g^2v^2}{\Lambda^2}\nonumber \\
&+\frac{1}{3} \Big ( 2 + \frac{gv}{\Lambda} \Big )^2 p^2
\Big ]
\Big \} \frac{g^{\mu\nu}}{\epsilon}
+ \frac{M_A^2}{24 \pi^2 v^2}
\Big ( 1 + \frac{gv}{\Lambda} +
\frac{g^2v^2}{\Lambda^2 }\Big )
\frac{p^\mu p^\nu}{\epsilon},
\label{GAA}\\
\overline{\G}^{(1)}_{\bar c^* \bar c^*} &= \frac{1}{8 \pi^2}\div; \label{Gc*c*}\\
\overline{\Gamma}^{(1)}_{\bar c^* T_1} &= \frac{1}{16 \pi^2} \Big [ 2 M^2+ 2 M_A^2 (1 - \delta_{\xi0}) - p^2 \Big ]\div,
\label{Gc*T1}\\
\overline{\G}^{(1)}_{T_1T_1}(p^2) &=
\frac{1}{32 \pi^2} \Big [ 6 (M^4 + M_A^4) - 3 (M^2+M_A^2)p^2
+ p^4 \Big ] \div,
\label{GT1T1}\\
\overline{\G}^{(1)}_{T_1\sigma}(p^2) &= -\frac{1}{32 \pi^2 v}
\Big \{ 4 m^2 (M^2 + M_A^2) + 8 (M^4 - 3 M_A^4) \nonumber \\
& - 2 \Big (
m^2 + M^2 - M_A^2 + 2 M^2 \frac{gv}{\Lambda} \Big ) p^2 +
\frac{gv}{\Lambda} p^4
\Big \} \div +
\frac{\delta_{\xi0}}{8 \pi^2 v} M_A^2 (m^2-p^2) \div, \label{GT1s}\ \\
\overline{\G}^{(1)}_{\bar c^*\sigma}(p^2) = &
\frac{1}{16 \pi^2 v} \Big [ -2 (m^2+M^2) + \frac{gv}{\Lambda} p^2 \Big ] \div.\label{Gc*s}\end{aligned}$$
Three-point functions
---------------------
$$\begin{aligned}
\overline{\Gamma}^{(1)}_{\bar c^* \bar c^* T_1} &= -\frac{1}{4\pi^2}\div, \label{Gc*c*T1}\\
\left . \overline{\Gamma}^{(1)}_{\bar c^* T_1 T_1} \right |_{p_1=p_2=0} &=
-\frac{3 M^2 + 2 M_A^2}{8 \pi^2}\div + \frac{M_A^2}{4 \pi^2} \frac{\delta_{\xi0}}{\epsilon}, \label{Gc*T1T1}\\
\overline{\Gamma}^{(1)}_{T_1T_1T_1} &= -\frac{3 M^4}{4\pi^2}\div, \label{GT1T1T1}\\
\overline{\G}^{(1)}_{\bar c^* T_1 \sigma} &= \frac{m^2+M^2+ \frac{M_A^2}{2} }{4 \pi^2 v}\div
-\frac{M_A^2}{8 \pi^2 v}\frac{\delta_{\xi0}}{\epsilon},\label{Gc*T1s}\\
\overline{\G}^{(1)}_{\bar c^* A_\mu A_\nu}(p_1,p_2)
&= -
\frac{M_A^2}{16 \pi^2}\frac{g^2}{\Lambda^2} g_{\mu\nu}\div, \label{Gc*AA}\\
\overline{\G}^{(1)}_{T_1 A_\mu A_\nu}(p_1,p_2)
&=
\frac{M_A^2}{32 \pi^2 v^2}
\Big [
\frac{gv}{\Lambda} \Big ( 8 - 3 \frac{gv}{\Lambda} \Big ) M^2 -
\Big ( 8 + 4 \delta_{\xi0} - 3 \frac{g^2v^2}{\Lambda^2} \Big
) M_A^2 \nonumber \\
& + \frac{2}{3} \frac{gv}{\Lambda} \Big ( 1 + 2 \frac{g v}{\Lambda} \Big ) (p_1^2+p_2^2) + 2 \frac{g^2 v^2}{\Lambda^2} p_1{\cdot}p_2 \Big ] g_{\mu\nu}\div, \nonumber \\
& - \frac{1}{96 \pi^2 v} \frac{g}{\Lambda} M_A^2 \Big ( 2 + \frac{gv}{\Lambda} \Big ) (p_{1\mu}p_{1\nu} + p_{2\mu}p_{2\nu})\div
+ \frac{M_A^2}{16 \pi^2}\frac{g^2}{\Lambda^2}p_{1\mu} p_{2\nu}, \label{GT1AA} \\
\overline{\G}^{(1)}_{\sigma A_\mu A_\nu}(p_1,p_2) &=
-\frac{M_A^2}{16 \pi^2 v^3}
\Big [
3 \Big ( 4 + 8 \frac{gv}{\Lambda} + 3 \frac{g^2v^2}{\Lambda^2} \Big ) M_A^2
- \frac{g^2v^2}{\Lambda^2} m^2 + \frac{gv}{\Lambda} \Big ( 8 - 3 \frac{gv}{\Lambda} \Big ) M^2 \nonumber \\
& + \frac{1}{6} \frac{gv}{\Lambda} \Big ( 4 - 10 \frac{gv}{\Lambda} +3 \frac{g^2v^2}{\Lambda^2} \Big ) (p_1^2+ p_2^2)
- \frac{g^2v^2}{\Lambda^2} \Big ( 4 - \frac{gv}{\Lambda} \Big ) p_1 {\cdot} p_2
\Big ]
g_{\mu\nu} \div \nonumber \\
&
+ \frac{1}{48 \pi^2 v^2} \frac{g}{\Lambda} M_A^2 \Big ( 2 + 7 \frac{gv}{\Lambda} \Big ) (p_{1\mu}p_{1\nu} + p_{2\mu}p_{2\nu})\div
+ \frac{M_A^2}{8 \pi^2 v} \frac{g^2}{\Lambda^2} p_{1\mu} p_{2\nu}\div, \label{GsAA}\\
\overline{\G}^{(1)}_{\bar c^* \chi \chi} (p_1,p_2)
&= \Big [
- \frac{m^2 + M^2 - M_A^2}{8 \pi^2 v^2} -
\frac{M_A^2}{8 \pi^2 v^2} \delta_{\xi0}
+ \frac{1}{16 \pi^2}\frac{g}{\Lambda v} (p_1^2+ p_2^2)\nonumber \\
&+\frac{1}{16 \pi^2} \frac{g}{\Lambda v}
\Big ( 2 + \frac{g v}{\Lambda} \Big ) p_1 {\cdot} p_2
\Big ] \div, \label{Gc*chch}\\
\overline{\G}^{(1)}_{T_1 \chi \chi}(p_1,p_2) &= \Big \{
-\frac{m^2 (M^2 + M_A^2) + 2 (M^4 - 3 M_A^4)}{8 \pi^2 v^2} +
\frac{m^2 M_A^2}{8 \pi^2 v^2} \delta_{\xi0}
\nonumber \\
&
+\frac{1}{32 \pi^2 v^2}
\Big [ 4 m^2 +
(M^2- M_A^2)
\Big (4 + 3
\frac{g^2v^2}{\Lambda^2}\Big)
+ 4 M_A^2\delta_{\xi0}
\Big ] p_1 {\cdot} p_2 \nonumber \\
&
+ \frac{1}{16 \pi^2 v^2}
\Big [m^2-3M_A^2+ M^2
\Big(1+2 \frac{gv}{\Lambda} \Big )\Big ]
(p_1^2+p_2^2)
\Big \}\div + {\cal O}(p^4), \label{GT1chch}\\
\left . \overline{\G}^{(1)}_{\bar c^* \sigma \sigma} \right |_{p_1=p_2=0} &= -\frac{1}{8 \pi^2 v^2} (m^2 + M^2 - M_A^2) \div
- \frac{M_A^2}{8 \pi^2 v^2} \frac{\delta_{\xi0}}{\epsilon}, \label{Gc*ss}\end{aligned}$$
$$\begin{aligned}
\left . \overline{\G}^{(1)}_{T_1 \sigma \sigma} \right |_{p_1=p_2=0} &= -\frac{1}{8 \pi^2 v^2} (2 m^4 + 5 m^2 M^2+6 M^4+3m^2 M_A^2-18 M_A^4)\div
+
\frac{3m^2 M_A^2}{8 \pi^2 v^2}
\frac{\delta_{\xi0}}{\epsilon},\\
\left . \overline{\G}^{(1)}_{\sigma T_1 T_1} \right |_{p_1=p_2=0} &= \frac{1}{8\pi^2 v}\Big[ m^2 (3 M^2 + 2 (1-\delta_{\xi 0} )M_A^2) + 6 (M^4+ M_A^4) \Big ] \div \,
,\label{GsT1T1}\\
\overline{\G}^{(1)}_{\bar c^* \bar c^* \sigma} &= 0,\label{GT1ss}\\
\left .
\overline{\G}^{(1)}_{\sigma\sigma\sigma}
\right |_{p_1=p_2=0}& =
\frac{3}{8\pi^2 v^3}
\Big ( m^4 + 2 m^2 M^2+ 2 M^4 - m^2M_A^2
(1 - \delta_{\xi0}) + 6 M_A^4\Big ) \div, \label{Gsss}\\\
\overline{\G}^{(1)}_{\sigma\chi\chi} (p_1,p_2)
& =
\frac{1}{8\pi^2 v^3}
\Big ( m^4 + 2 m^2 M^2+ 2 M^4 - m^2M_A^2
(1 - \delta_{\xi0}) + 6 M_A^4\Big ) \div
\nonumber \\
& - \frac{1}{32 \pi^2 v^2}
\frac{g}{\Lambda} \Big [
4 m^2 + (M^2 - 3 M_A^2) \Big ( 4 + \frac{gv}{\Lambda} \Big )
\Big ] (p_1^2+p_2^2)
\nonumber \\
&
-\frac{1}{16 \pi^2 v^2}
\frac{g}{\Lambda}
\Big [
3 \frac{gv}{\Lambda} M^2 +
m^2 \Big ( 4 + \frac{gv}{\Lambda} \Big )
- 3 M_A^2 \Big ( 8 + \frac{3 gv}{\Lambda}
\Big )
\Big ] p_1 {\cdot} p_2 \div \,.\label{Gschch}\end{aligned}$$
Four-point functions
--------------------
$$\begin{aligned}
\left . \overline{\G}^{(1)}_{\sigma\sigma\chi\chi} \right |_{p_i=0} & =
\frac{1}{8 \pi^2 v^4}
\Big ( m^4+2 m^2 M^2 + 2 M^4- 2m^2 M_A^2 (1 - \delta_{\xi0} ) + 6 M_A^4 \Big ) \div, \label{Gsschch}\\
\overline{\G}^{(1)}_{\chi\chi\chi\chi}(p_1,p_2,p_3) & =
\frac{3}{8 \pi^2 v^4}
\Big ( m^4 + 2 m^2 M^2 + 2 M^4 - 2 m^2 M_A^2 + 6 M_A^4 \Big ) \div \nonumber \\
&+\frac{3}{4 \pi^2 v^4}(m^2 - M_A^2) M_A^2 \frac{\delta_{\xi0}}{\epsilon}
-
\frac{1}{16 \pi^2 v^3}
\frac{g}{\Lambda}
\Big [ 3 M_A^2 \frac{gv}{\Lambda} +
\Big ( 8 - \frac{gv}{\Lambda} \Big )
M^2 \nonumber \\
&+ \Big ( 4 -\frac{gv}{\Lambda} \Big ) m^2\Big ]
\Big ( \sum_{i=1}^3 p_i^2 + \sum_{i<j} p_i p_j \Big ) + {\cal O}(p_i^4).\label{Gchchchch}\end{aligned}$$
[24]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
J. High Energ. Phys. (2019) 2019: 32, 1904.06692.
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[^1]: Going on-shell with $X_1$ yields the condition $$\begin{aligned}
(\square + m^2) \Big (
\phi^\dagger \phi - \frac{v^2}{2} - v X_2 \Big ) = 0 \, ,\end{aligned}$$ so that the most general solution is $X_2 = \frac{1}{v} \Big (
\phi^\dagger \phi - \frac{v^2}{2} \Big ) + \eta,$ $\eta$ being a scalar field of mass $m$. However in perturbation theory the correlators of the mode $\eta$ with any gauge-invariant operators vanish [@Binosi:2019olm], so that one can safely set $\eta =0$.
|
---
author:
- 'Ulrich Bunke[^1]'
title: 'On the functoriality of Lott’s secondary analytic index'
---
0.2cm
0.2cm
-1cm
\[section\] \[prop\][Lemma]{} \[prop\][Definition]{} \[prop\][Theorem]{} \[prop\][Corollary]{} \[prop\][Assumption]{} \[prop\][Conjecture]{} \[prop\][Problem]{} \[prop\][Fact]{}
Introduction
============
In [@lott99] for a smooth manifold $M$ J. Lott has defined a secondary $K$-group $\bar{K}_R^0(M)$ of local systems of $R$-modules ($R$ is some ring) such that their complexifications (we must fix a representation $\rho:R\rightarrow {{\mbox{\rm End}}}({{\bf C}}^n)$ in order to define this notion) have explicitly trivialized characteristic classes. If $p:E\rightarrow B$ is a smooth fibre bundle with closed fibres, then he defines a push-forward $p_!:\bar{K}_R^0(E)\rightarrow \bar{K}_R^0(B)$. This construction involves the analytic torsion form of [@bismutlott95]. We will review Lott’s construction in Section \[lott\]. In [@ma99] X. Ma has studied the behaviour of the analytic torsion form under iterated fibre bundles. We review his result in Section \[ma\].
The goal of the present note is to verify that Ma’s result indeed implies that the push-forward of Lott is functorial. We now describe this assertion in detail. Let $p:E\rightarrow B$ be a fibration with fibre $Z$ which is in fact an iterated fibration. So we assume that there are fibrations $p_1:E=:E_1\rightarrow E_2$ with fibre $Z_1$ and $p_2:E_2\rightarrow B$ with fibre $Z_2$ such that $Z\rightarrow Z_2$ is a fibration with fibre $Z_1$. We will verify the following:
\[main\] We have $p_!=(p_2)_!\circ (p_1)_!$ as maps from $\bar{K}_R^0(E)$ to $\bar{K}_R^0(B)$.
The details of the proof are contained in Section \[pf\]. [*We do now claim any kind of originality here because the argument just amounts to combine the construction of Lott and the result of Ma mentioned above.*]{}
Secondary $K$-Groups and analytic push-forward {#lott}
==============================================
This section reviews Sections 2.1 and 2.2 of [@lott99]. Suppose that $R$ is a right-Noetherian ring which is right-regular, i.e. every finitely generated right-$R$-module has a finite resolution by finitely generated projective right-$R$-modules. We fix a representation $\rho:R\rightarrow {{\mbox{\rm End}}}({{\bf C}}^n)$ such that ${{\bf C}}^n$ becomes a flat left $R$-module. If $V$ is a right-$R$-module, then $V_{{\bf C}}:=V\otimes_{R,\rho} {{\bf C}}^n$ is called its complexification.
Let $M$ be a connected smooth manifold. If ${{\cal F}}$ is a local system (a locally constant sheaf) of finitely generated right-$R$-modules, then ${{\cal F}}_{{\bf C}}$ (the fibrewise complexification) is the sheaf of parallel sections of a flat complex vector bundle which we denote by $(F_{{\bf C}},\nabla^{F_{{\bf C}}})$.
Let $(E,\nabla^E)$ be a flat complex vector bundle. If we choose a hermitean metric $h^E$ (we consider $h^E$ is a section of the flat bundle ${{\mbox{\rm Hom}}}_{{\bf C}}(E,E^*)$) on $E$, then we can define the characteristic forms $$\begin{aligned}
\omega(\nabla^{E},h^{E})&:=&(h^E)^{-1} \nabla^{{{\mbox{\rm Hom}}}(E,E^*)}h^{E}\\
c_k(\nabla^{E},h^{E})&=&(2\pi\imath)^{-\frac{k-1}{2}} 2^{-k} {{\rm Tr}}\:
\omega(\nabla^{E},h^{E})\\
c(\nabla^{E},h^{E})&=&\sum_{j=0}^\infty
\frac{1}{j! }c_{2j+1}(\nabla^{E},h^{E})\ . \end{aligned}$$ The form $c(\nabla^{E},h^{E})$ is closed and represents the characteristic class $c(\nabla^E)\in H_{dR}^{odd}(M)$ of the flat vector bundle $
(E,\nabla^E)$ which is independent of the choice of $h^E$.
The abelian group $\hat{K}^0_R(M)$ is generated by triples $f=({{\cal F}},h^{F_{{\bf C}}},\eta)$, where
1. ${{\cal F}}$ is a local system of finitely generated right-$R$-modules,
2. $h^{F_{{\bf C}}}$ is a hermitean metric of the corresponding flat complex vector bundle $(F_{{\bf C}},\nabla^{F_{{\bf C}}})$, and
3. $\eta\in\Omega^{ev}(M)/{{\rm image}}(d)$,
subject to the following relations : If $$0\rightarrow {{\cal F}}_1\rightarrow {{\cal F}}_2\rightarrow {{\cal F}}_3\rightarrow 0$$ is an exact sequence of local systems of finitely generated right-$R$-modules, $h^{(F_i)_{{\bf C}}}$ are hermitean metrics, $\eta_i\in
\Omega^{ev}(M)/{{\rm image}}(d)$, and we form $f_i:=({{\cal F}}_i,h^{(F_i)_{{\bf C}}},\eta_i)$, then $f_2\sim f_1+f_3$ if $$\eta_2=\eta_1+\eta_3+{{\cal T}}({{\cal C}},h^{{{\cal C}}})\ ,$$ where ${{\cal T}}({{\cal C}},h^{{{\cal C}}})$ is the analytic torsion form associated to the exact complex of flat complex vector bundles $({{\cal C}},h^{{{\cal C}}})$ $$0\rightarrow (F_1)_{{\bf C}}\rightarrow (F_2)_{{\bf C}}\rightarrow (F_3)_{{\bf C}}\rightarrow 0$$ equipped with the metric $h^{{{\cal C}}}$ induced by $h^{(F_i)_{{\bf C}}}$. For the details of the definition of the torsion form ${{\cal T}}({{\cal C}},h^{{\cal C}})$ we refer to [@bismutlott95], Sec. 2, or [@lott99], A.3. The appearence of the torsion form in the equivalence relation is explained by the relation $$d{{\cal T}}({{\cal C}},h^{{{\cal C}}})=\sum_{i=1}^3 (-1)^i c((F_i)_{{\bf C}},h^{(F_i)_{{\bf C}}})\ .$$
Lott shows that $$f \mapsto c(\nabla^{F_{{\bf C}}},h^{F_{{\bf C}}})-d\eta$$ extends to a map $c^\prime:\hat{K}^0_R(M)\rightarrow \Omega^{odd}(M)$, and he defines $$\bar{K}^0_R(M):=\ker(c^\prime)\ .$$ The assignment $M\mapsto \bar{K}^0_R(M)$ yields a homotopy invariant contravariant functor from the category of manifolds to abelian groups.
We now consider a smooth fibre bundle $p:E\rightarrow B$ with compact fibre $Z$. If ${{\cal F}}$ is a locally constant sheaf of finitely generated right-$R$-modules, then we can form the sheaves ${{\cal R}}^ip_*{{\cal F}}$ on $B$ which are again locally constant sheaves of finitely generated right-$R$-modules.
If we choose a fibrewise Riemannian metric $g^Z$ (i.e. a metric on the vertical bundle $TZ$), then we can compute $({{\cal R}}^ip_*{{\cal F}})_{{\bf C}}={{\cal R}}^ip_*({{\cal F}}_{{\bf C}})$ (this equality holds because ${{\bf C}}^n$ is a flat $R$-module) using the fibrewise de Rham complex twisted with $F_{{\bf C}}$. The metric $g^Z$ and a hermitean meric $h^{F_{{\bf C}}}$ induce $L^2$-scalar products. Identifying the stalk $({{\cal R}}^ip_*({{\cal F}})_{{\bf C}})_b$ (which is the fibre of the flat complex vector bundle $(R^ip_*{{\cal F}})_{{\bf C}}$) with harmonic forms we obtain metrics $h^{(R^ip_*{{\cal F}})_{{\bf C}}}$.
We further choose a horizontal distribution $T^HE$. It induces a connection $\nabla^{TZ}$ on the vertical bundle. Let $e(TZ,\nabla^{TZ})$ be the associated Euler form. We refer to [@bismutlott95] for the definition of the analytic torsion form ${{\cal T}}(T^HE,g^Z,h^{F_{{\bf C}}})\in\Omega^{ev}(B)$.
J. Lott defines the push-forward $p_!:\bar{K}^0_R(M)\rightarrow
\bar{K}^0_R(B)$ by the assignment: $$({{\cal F}},h^F,\eta)\mapsto
\sum_p ({{\cal R}}^ip_*({{\cal F}}),h^{(R^ip_*{{\cal F}})_{{\bf C}}},0) +
(0,0,\int_Ze(TZ,\nabla^{TZ})\wedge \eta-{{\cal T}}(T^HE,g^Z,h^{F_{{\bf C}}}))\ .$$ Lott proves well-definedness and independence of $T^HE$ and $g^Z$.
Analytic torsion form and iterated fibrations {#ma}
=============================================
This section reviews the result of [@ma99]. Let $p:E\rightarrow B$ be a fibration with fibre $Z$ which is in fact an iterated fibration. We assume that there are fibrations $p_1:E:=E_1\rightarrow E_2$ with fibre $Z_1$ and $p_2:E_2\rightarrow B$ with fibre $Z_2$ such that $Z\rightarrow Z_2$ is a fibration with fibre $Z_1$. By $TZ$, $TZ_1$, and $TZ_2$, we denote the corresponding vertical bundles. We choose vertical Riemannian metrics $g^Z$, $g^{Z_1}$, $g^{Z_2}$.
Furthermore, we choose horizontal bundles $T^HE$, $T^HE_1$, and $T^HE_2$, for $p$, $p_1$, $p_2$. We obtain connections $\nabla^{TZ}$, $\nabla^{TZ_1}$, and $\nabla^{TZ_2}$. We identify $TZ$ with $TZ_1\oplus p_1^* TZ_2$ (using $T^HE_1$) and obtain another connection ${}^0\nabla^{TZ}:=\nabla^{TZ_1}\oplus p_1^*\nabla^{TZ_2}$ on $TZ$. By $\tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ})$ we denote the corresponding transgression of the Euler form such that $$d\tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ})=e(TZ,{}^0\nabla^{TZ})-
e(TZ,\nabla^{TZ})\ .$$
The main result of X. Ma is the formula $$\begin{aligned}
&&{{\cal T}}(T^HE,g^Z,h^{F_{{\bf C}}})-\int_{Z_2}
e(TZ_2,\nabla^{TZ_2})\wedge {{\cal T}}(T^HE_1,g^{Z_1},h^F)\\
&&-\sum_{i}(-1)^i{{\cal T}}(T^HE_2,g^{Z_2},h^{(R^ip_*{{\cal F}})_{{\bf C}}})\\
&&+\int_{Z}\tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ})\wedge
c(\nabla^{F_{{\bf C}}},h^{F_{{\bf C}}})\\ &=&{{S}}\:\: (mod\: {{\rm image}}(d))\ ,\end{aligned}$$ where ${{S}}$ is a higher analytic torsion invariant associated to the Leray spectral sequence of the family of fibrations $Z_b\rightarrow (Z_1)_b$, $b\in
B$.
We now describe ${{S}}$ in detail. The spectral sequence $({{\cal E}}_r,d_r)$ is associated to the composition $(p_2)_*\circ (p_1)_*$. Its second term is ${{\cal E}}_2^{p,q}={{\cal R}}^p(p_2)_* {{\cal R}}^q(p_1)_*({{\cal F}})$, and it converges to ${{\cal R}}^{p+q}p_*({{\cal F}})$. Since ${{\bf C}}^n$ is a flat $R$-module complexification commutes with taking the spectral sequence. In particular, we obtain flat complex vector bundles $(E_r^{p,q})_{{\bf C}}$. The differentials $d_r$ of the spectral sequence induce corresponding bundle homomorphisms such that we obtain complexes of flat complex vector bundles $$\dots\stackrel{d_r}{\rightarrow}
(E_r^{p,q})_{{\bf C}}\stackrel{d_r}{\rightarrow}
(E_r^{p+r,q+1-r})_{{\bf C}}\stackrel{d_r}{\rightarrow}\dots$$ with cohomology $(E^{p,q}_{r+1})_{{\bf C}}$. $h^{(R^q(p_*)_1{{\cal F}})_{{\bf C}}}$ induces metrics $h^{(E_2)_{{\bf C}}^{p,q}}$. Now we obtain inductively metrics on the cohomology groups $h^{(E^{p,q}_{r+1})_{{\bf C}}}$.
In order to save notation we denote by ${{\cal D}}_r$ the direct sum of complexes above at the level $r$ and by $h^{{{\cal D}}_r}$ the induced metric.
Let $${{\cal V}}:\dots\stackrel {d_{i-1}}{\rightarrow} V_i\stackrel{d_i}{\rightarrow}
V_{i+1}\stackrel{d_{i+1}}{\rightarrow}\dots$$ be a finite complex of flat complex vector bundles equipped with hermitean metrics $h^{V_i}$ (we write $h^{{{\cal V}}}$ for the whole collection). We further choose hermitean metrics on the flat cohomology bundles $h^{H^i}$ (and we again write $h^{{{\cal H}}}$ for this collection). We form the short exact sequences $$\begin{aligned}
{{\cal C}}_i&:& 0\rightarrow \ker(d_i)\rightarrow
V_i\rightarrow {{\rm image}}(d_i)\rightarrow 0\\
{{\cal D}}_i&:&0\rightarrow {{\rm image}}(d_{i-1}) \rightarrow
\ker(d_i) \rightarrow H^i \rightarrow 0\end{aligned}$$ where all spaces have induced hermitean metrics $h^{{{\cal C}}_i}$, $h^{{{\cal D}}_i}$. We define $${{\cal T}}({{\cal V}},h^{{{\cal V}}},h^{{{\cal H}}}):=\sum_{i} (-1)^i \left(
{{\cal T}}({{\cal C}}_i,h^{{{\cal C}}_i})+{{\cal T}}({{\cal D}}_i,h^{{{\cal D}}_i})\right)\ .$$
If $V$ is a flat complex vector bundle with a filtration $0\subset V_0\subset V_1\subset\dots\subset V_n=V$ by flat subbundles, then we consider the short exact sequences $${{\cal E}}_i:0\rightarrow V_i \rightarrow V_{i+1}\rightarrow
Gr_{i+1}(V):=V_{i+1}/V_i\rightarrow 0\ .$$ If we further choose hermitean metrics $h^{V}$ (inducing $h^{V_i}$ by restriction) and $h^{Gr_i(V)}$, then we define metrics $h^{{{\cal E}}_i}$ and $${{\cal T}}(V,Gr(V),h^V,h^{Gr(V)}):=\sum_i {{\cal T}}({{\cal E}}_i,h^{{{\cal E}}_i})\ .$$
We can now define $${{S}}:=\sum_{r=2}^\infty
{{\cal T}}({{\cal D}}_r,h^{{{\cal D}}_r},h^{{{\cal D}}_{r+1}})-\sum_{k=0}^\infty(-1)^k
{{\cal T}}((R^kp_*{{\cal F}})_{{\bf C}},\oplus_{p+q=k}({{\cal E}}_\infty^{p,q})_{{\bf C}},
h^{(R^kp_*{{\cal F}})_{{\bf C}}},h^{({{\cal E}}_\infty^{p,q})_{{\bf C}}})\ .$$ In the last term we use the natural identification $Gr_{k-p}(R^kp_*{{\cal F}})_{{\bf C}}\stackrel{\cong}{\rightarrow} ({{\cal E}}_\infty^{p,q})_{{\bf C}}$.
Verification of Theorem \[main\] {#pf}
================================
The group $\bar{K}^0_R(E)$ is generated by elements $f:=({{\cal F}},h^{F_{{\bf C}}},\eta)$ with $d\eta=c(\nabla^{F_{{\bf C}}},h^{F_{{\bf C}}})$. Let us write out a representative of $(p_1)_!([f])$. We obtain $$\sum_{q}(-1)^q({{\cal R}}^q(p_1)_*{{\cal F}},h^{(R^q(p_1)_*{{\cal F}})_{{\bf C}}},0)+
(0,0,\int_{Z_1}e(TZ_1,\nabla^{TZ_1})\wedge \eta-{{\cal T}}(T^HE_1,g^{Z_1},h^{F_{{\bf C}}}))\
.$$ A representative of $(p_2)_!\circ (p_1)_!([f])$ is given by $$\begin{aligned}
&&\sum_{p,q} (-1)^{p+q}
({{\cal R}}^p(p_2)_*{{\cal R}}^q(p_1)_*{{\cal F}},h^{(E_2)_{{\bf C}}^{p,q}},0)\\ &&+(0,0,\int_{Z_2}
e(TZ_2,\nabla^{TZ_2})\wedge \left(\int_{Z_1}e(TZ_1,\nabla^{TZ_1})\wedge
\eta-{{\cal T}}(T^HE_1,g^{Z_1},h^{F_{{\bf C}}})\right))\\
&&-\sum_{q}(-1)^q(0,0,{{\cal T}}(T^HE_2,g^{Z_2},h^{(R^q(p_1)_*{{\cal F}})_{{\bf C}}}))\ .\end{aligned}$$ We must show that this expression represents the same element as $$\sum_i(-1)^i ({{\cal R}}^ip_*{{\cal F}},h^{(R^ip_*{{\cal F}})_{{\bf C}}},0) +
(0,0,\int_Ze(TZ,\nabla^{TZ})\wedge \eta-{{\cal T}}(T^HE,g^Z,h^{F_{{\bf C}}}))\ .$$ We first compare the terms involving the form $\eta$. Indeed we have $$\begin{aligned}
\lefteqn{\int_{Z_2} e(TZ_2,\nabla^{TZ_2})\wedge \left(
\int_{Z_1}e(TZ_1,\nabla^{TZ_1})\wedge
\eta \right)- \int_Ze(TZ,\nabla^{TZ})\wedge
\eta}&&\\
&=& \int_Z d\tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ}) \wedge \eta\\
&=& \int_Z \tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ}) \wedge d\eta\\
&=& \int_Z \tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ}) \wedge
c(\nabla^{F_{{\bf C}}},h^{F_{{\bf C}}})\:(mod\:{{\rm image}}(d))\ .\end{aligned}$$ Thus it remains to show that $$\begin{aligned}
&&\sum_{p,q} (-1)^{p+q}
({{\cal R}}^p(p_2)_*{{\cal R}}^q(p_1)_*{{\cal F}},h^{(E_2)_{{\bf C}}^{p,q}},0)\\ &&-
(0,0,\int_{Z_2} e(TZ_2,\nabla^{TZ_2})\wedge {{\cal T}}(T^HE_1,g^{Z_1},h^{F_{{\bf C}}}))\\
&&-\sum_{q}(-1)^q(0,0,{{\cal T}}(T^HE_2,g^{Z_2},h^{(R^q(p_1)_*{{\cal F}})_{{\bf C}}}))\\
&&-\sum_p (-1)^p({{\cal R}}^pp_*({{\cal F}}),h^{(R^pp_*{{\cal F}})_{{\bf C}}},0) \\
&&+(0,0,{{\cal T}}(T^HE,g^Z,h^{F_{{\bf C}}})\\
&&+ \int_Z \tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ}) \wedge
c(\nabla^{F_{{\bf C}}},h^{F_{{\bf C}}})\end{aligned}$$ represents the trivial element in $\hat{K}^0_R(B)$.
Let $${{\cal V}}:\dots\stackrel {d_{i-1}}{\rightarrow} {{\cal V}}_i\stackrel{d_i}{\rightarrow}
{{\cal V}}_{i+1}\stackrel{d_{i+1}}{\rightarrow} \dots$$ be a finite complex of local systems of finitely generated right-$R$-modules over $B$. We fix hermitean metrics $h^{(V_i)_{{\bf C}}}$ (we write $h^{{{\cal V}}_{{\bf C}}}$ for the whole collection). Since ${{\bf C}}^n$ is a flat $R$-module we can interchange the operation of complexification and of taking fibrewise cohomology. We let $H^i$ denote the flat complex vector bundle obtained from the complexification of the cohomology sheaves ${{\cal H}}^i({{\cal V}})$. We further choose hermitean metrics $h^{H^i}$ (and we write $h^{{{\cal H}}}$ for this collection). We consider the short exact sequences $$\begin{aligned}
{{\cal C}}_i&:& 0\rightarrow \ker(d_i)\rightarrow {{\cal V}}_i \rightarrow
{{\rm image}}(d_i)\rightarrow 0\\
{{\cal D}}_i&:&0\rightarrow {{\rm image}}(d_{i-1}) \rightarrow
\ker(d_i) \rightarrow {{\cal H}}^i({{\cal V}}) \rightarrow 0\ ,\end{aligned}$$ where the corresponding complexes of flat complex vector bundles $({{\cal C}}_i)_{{\bf C}}$, $({{\cal D}}_i)_{{\bf C}}$ have induced hermitean metrics $h^{({{\cal C}}_i)_{{\bf C}}}$, $h^{({{\cal D}}_i)_{{\bf C}}}$. In $\hat{K}^0_R(B)$ we have $$\begin{aligned}
\sum_{i}(-1)^i ({{\cal V}}_i,h^{(V_i)_{{\bf C}}},0)&=&
\sum_{i}(-1)^i
({{\cal H}}^i({{\cal V}}),h^{H^i},-{{\cal T}}(C_i,h^{({{\cal C}}_i)_{{\bf C}}})-{{\cal T}}(D_i,h^{({{\cal D}}_i)_{{\bf C}}}))\\
&=& \sum_{i}(-1)^i
({{\cal H}}^i({{\cal V}}),h^{H^i},0)-(0,0,{{\cal T}}(({{\cal V}})_{{\bf C}},h^{({{\cal V}})_{{\bf C}}},h^{{{\cal H}}_{{\bf C}}}))\ .\end{aligned}$$ Using this we compute $$\begin{aligned}
&&\sum_{p,q} (-1)^{p+q}
({{\cal R}}^p(p_2)_*{{\cal R}}^q(p_1)_*{{\cal F}},h^{(E_2)_{{\bf C}}^{p,q}},0)\\ &=&\sum_{p,q}
(-1)^{p+q}
({{\cal E}}_3^{p,q},h^{(E_3)_{{\bf C}}^{p,q}},0)-(0,0,{{\cal T}}({{\cal D}}_2,h^{{{\cal D}}_2},h^{{{\cal D}}_{3}}))\\
&=&\dots\\ &=&\sum_{p,q} (-1)^{p+q}
({{\cal E}}_\infty^{p,q},h^{(E_\infty)_{{\bf C}}^{p,q}},0)-\sum_{r=2}^\infty(0,0,{{\cal T}}({{\cal D}}_r,h^{{{\cal D}}_r},h^{{{\cal D}}_{r+1}}))\ .\end{aligned}$$
Let now ${{\cal V}}$ be a local system of finitely generated right-$R$-modules which is filtered by local systems of submodules $0\subset {{\cal V}}_0\subset
{{\cal V}}_1\subset\dots\subset {{\cal V}}_n={{\cal V}}$. We fix a hermitean metric $h^{V_{{\bf C}}}$ which induces metrics $h^{(V_i)_{{\bf C}}}$. Furthermore we fix metrics $h^{Gr_i({{\cal V}})_{{\bf C}}}$. In $\hat{K}^0_R(B)$ we have $$({{\cal V}},h^{V_{{\bf C}}},0)=\sum_{i}(Gr_i({{\cal V}}),g^{Gr_i({{\cal V}})_{{\bf C}}},0)-(0,0,{{\cal T}}(V_{{\bf C}},Gr({{\cal V}})_{{\bf C}},h^{V_{{\bf C}}},h^{Gr({{\cal V}})_{{\bf C}}}))\ .$$
Using this observation we further compute $$\begin{aligned}
\lefteqn{
\sum_{p+q=k} (-1)^{k}
({{\cal E}}_\infty^{p,q},h^{(E_\infty)_{{\bf C}}^{p,q}},0)}&&\\
&=&
\sum_{i}(Gr_i({{\cal R}}^kp_*{{\cal F}}),h^{Gr_i(R^kp_*{{\cal F}})_{{\bf C}}},0)\\
&=&({{\cal R}}^kp_*{{\cal F}},h^{(R^kp_*{{\cal F}})_{{\bf C}}},0)+(0,0,{{\cal T}}((R^kp_*{{\cal F}})_{{\bf C}},\oplus_{p+q=k}({{\cal E}}_\infty^{p,q})_{{\bf C}},
h^{(R^kp_*{{\cal F}})_{{\bf C}}},h^{Gr(R^kp_*{{\cal F}})_{{\bf C}}}))\ .\end{aligned}$$
Thus it remains to show that $$\begin{aligned}
&&-\sum_{r=2}^\infty {{\cal T}}({{\cal D}}_r,h^{{{\cal D}}_r},h^{{{\cal D}}_{r+1}})\\
&&+
\sum_{k=0}^\infty (-1)^k{{\cal T}}((R^kp_*{{\cal F}})_{{\bf C}},\oplus_{p+q=k}({{\cal E}}_\infty^{p,q})_{{\bf C}},
h^{(R^kp_*{{\cal F}})_{{\bf C}}},h^{Gr(R^kp_*{{\cal F}})_{{\bf C}}})\\
&&- \int_{Z_2} e(TZ_2,\nabla^{TZ_2})\wedge
{{\cal T}}(T^HE_1,g^{Z_1},h^{F_{{\bf C}}})\\
&&-\sum_{p}(-1)^p{{\cal T}}(T^HE_2,g^{Z_2},h^{(R^p(p_1)_*{{\cal F}})_{{\bf C}}})\\
&&+{{\cal T}}(T^HE,g^Z,h^{F_{{\bf C}}})\\
&& + \int_Z \tilde e(TZ,\nabla^{TZ},{}^0\nabla^{TZ}) \wedge
c(\nabla^{F_{{\bf C}}},h^{F_{{\bf C}}})\end{aligned}$$ is an exact form. But this is exactly the assertion of X. Ma. This finishes the verification.
[1]{}
J. M. Bismut and J. Lott. Flat vector bundles, direct images, and higher real analytic torsion. , 8(1995), 291–363.
J. Lott. Secondary analytic indices. Preprint, 1999.
X. Ma. Functoriality of real analytic torsion form. IHES-Preprint M/99/03, 1999.
[^1]: Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, E-mail:bunke@uni-math.gwdg.de
|
---
author:
- 'Hironori <span style="font-variant:small-caps;">Sakai</span>$^{1}$[^1], Yo <span style="font-variant:small-caps;">Tokunaga</span>$^{1}$, Tatsuya <span style="font-variant:small-caps;">Fujimoto</span>$^{1}$, Shinsaku <span style="font-variant:small-caps;">Kambe</span>$^{1}$, Russell E. <span style="font-variant:small-caps;">Walstedt</span>$^{1}$, Hiroshi <span style="font-variant:small-caps;">Yasuoka</span>$^{1}$, Dai <span style="font-variant:small-caps;">Aoki</span>$^{2}$, Yoshiya <span style="font-variant:small-caps;">Homma</span>$^{2}$, Etsuji <span style="font-variant:small-caps;">Yamamoto</span>$^{1}$, Akio <span style="font-variant:small-caps;">Nakamura</span>$^{1}$, Yoshinobu <span style="font-variant:small-caps;">Shiokawa</span>$^{1,2}$, Kunihisa <span style="font-variant:small-caps;">Nakajima</span>$^{3}$, Yasuo <span style="font-variant:small-caps;">Arai</span>$^{3}$, Tatsuma D. <span style="font-variant:small-caps;">Matsuda</span>$^{1}$, Yoshinori <span style="font-variant:small-caps;">Haga</span>$^{1}$, Yoshichika <span style="font-variant:small-caps;">Ōnuki</span>$^{1,4}$'
date: 'Submission date: 18 Mar. 2005: Acceptance: 30 Mar. 2005'
title: 'Anisotropic superconducting gap in transuranium superconductor PuRhGa$_{5}$: Ga NQR study on a single crystal'
---
The recent discovery of Pu-based superconductors PuTGa$_{5}$ (T=Co, Rh) has stimulated interest in further experiments on transuranium compounds. A lot of attention is now focused on these Pu heavy-fermion compounds, which have relatively high superconducting (SC) transition temperatures, [*i.e.,*]{} [$T_{\rm c}$]{}$\simeq$ 18 K [@sarrao] (PuCoGa$_{5}$) and [$T_{\rm c}$]{}$\simeq$ 9 K [@wastin] (PuRhGa$_{5}$). The PuTGa$_{5}$ systems belong to a large family of “115 compounds", which all crystallize in the same quasi-two-dimensional structure. In this family, the Ce115 isomorphs CeT’In$_{5}$ (T’=Co, Ir, Rh) [@cecoin5; @ceirin5; @cerhin5], which are well-known 4[*f*]{} heavy fermion systems with antiferromagnetic tendencies, show $d$-wave superconductivity with relatively low [$T_{\rm c}$]{} values: [$T_{\rm c}$]{}=0.4 K and 2.3 K for CeIrIn$_{5}$ and CeCoIn$_{5}$, respectively.[@cecoin5kohori; @ceirin5zheng; @cerh115mito; @ce115curro] On the other hand, the U115 [@tokiwa; @ikeda; @tokiwa2] and Np115 [@colineau; @aoki; @aoki2; @aoki3] compounds usually show itinerant magnetic order or Pauli paramagnetism, but up to now have not shown any superconductivity. Band calculations [@maehira; @maehira2] and de Haas-van Alphen experiments [@tokiwa; @tokiwa3; @ikeda2; @aoki; @aoki2] have revealed that An115 (An=U, Np, Pu,$\cdots$) compounds often have cylindrical Fermi surfaces, reflecting a quasi-two-dimensional character.
To elucidate the relatively high [$T_{\rm c}$]{} values exhibited by PuTGa$_5$, it is important to determine the pairing symmetry, which reflects the SC pairing mechanism. Nuclear Magnetic Resonance (NMR and NQR) is a microscopic probe well-suited to this task. In particular, the $T$ dependence of NQR (zero-field) spin-lattice relaxation rates ([$1/T_{1}$]{}) can provide crucial information regarding the SC pairing symmetry. In this letter, we report $^{69,71}$Ga NMR/NQR experimental results above and below the SC transition in PuRhGa$_{5}$.
A single crystal of PuRhGa$_{5}$ has been prepared using the Ga flux method. [@haga] The dimensions of the single crystal are $1\times 2\times 3$ mm$^3$. This crystal has been attached with varnish to a silver cap for thermal contact, coated with epoxy resin, and sealed tightly in a polyimid tube in order to avoid radiation contamination. The sealed sample was then mounted into an rf coil. NQR/NMR measurements have been carried out in the temperature range 1.5-300 K using a phase-coherent, pulsed spectrometer, which has been installed in the radiation controlled area. $T_{1}$ was measured for the Ga NQR lines using the saturation-recovery method. The recovery of nuclear magnetization $M(t)$ from a saturation pulse comb was single-exponential type, $\{M(t)-M(\infty)\}/M(\infty) \propto \exp(-3t/T_{1})$, in the whole temperature range. The superconductivity of PuRhGa$_{5}$ has been reported to be influenced by self-radiation damage due to spontaneous $\alpha$ decay, [*i.e.*]{}, [$T_{\rm c}$]{} decreases gradually with $dT_{\rm c}/dt\simeq -0.2 \sim -0.5$ K/month.[@wastin] Accordingly, in order to study the superconductivity of PuRhGa$_{5}$, brief and timely experimental measurements are required. In our experiments, the bulk [$T_{\rm c}$]{} was determined by both [*in situ*]{} ac susceptibility ($\chi_{\rm ac}$) measurements using the rf coil, and the sudden decrease of the NQR signal intensity, as mentioned below. [$T_{\rm c}$]{} was found to be 8.5 K in zero field at the starting point of the present experiments, which was one month after the synthesis of the single crystal. All the NMR/NQR experiments reported here were conducted within a single month.
![\[crystalstructure\]Crystal structure of PuRhGa$_{5}$.](Fig1.eps){width="5cm"}
PuRhGa$_{5}$ crystalizes in the tetragonal HoCoGa$_{5}$ structure with lattice parameters of $a =$ 4.2354 and $c =$ 6.7939 Å[@wastin], as shown in Fig. \[crystalstructure\]. This structure can be viewed as alternating PuGa$_{3}$ and RhGa$_{2}$ layers stacked along the $c$ axis. There are two crystallographically inequivalent Ga sites, which are denoted Ga(1) (the $1c$ site) and Ga(2) (the $4i$ site), respectively. The Ga(1) site is surrounded by four Pu atoms in the $c$ plane. On the other hand, the Ga(2) site is surrounded by two Pu and two Rh atoms in the $a$ plane. Therefore, in an applied field ($H_{0}$), the Ga(2) sites split into two magnetically inequivalent sites, except for the case of $H_{0}\parallel$ $c$ axis.
 Field-swept NMR spectrum in PuRhGa$_{5}$ at 10 K at a frequency of 84.1 MHz with the applied field ($H_{0}$) parallel to the $c$ axis. Spectral assignments are denoted by arrows, and the zero-shift positions $^{69,71}K=0$ are also indicated. The two NMR lines near $^{69,71}K=0$ originate from residual Ga flux. (b) Enlarged spectra around 90 kOe at a same frequency of 84.1MHz at different angles ($\theta$) between $H_{0}$ and the $c$ axis. ](Fig2.eps){width="8cm"}
Figure \[NMRspectrum\](a) shows a field-swept NMR spectrum in the normal state at $T = 10$ K and a frequency of 84.1 MHz, with $H_{0}$ parallel to the $c$-axis. This spectrum is well explained if we associate two sets of lines to $^{69}$Ga(2) and $^{71}$Ga(2) sites, where each set has a center line and two quadrupolar satellite lines denoted by arrows in Fig.\[NMRspectrum\]. Since the Ga(2) sites have a lower local symmetry than the Ga(1), with four-fold axial symmetry, each center line lies at an asymmetric position between the satellites. This is due to the second-order quadrupolar effect. The Knight shift ($K$), electric field gradient parameter ($\nu_{\rm EFG}\equiv e^2qQ/2h$), and asymmetry parameter ($\eta$) of the Ga(2) site have been estimated from second-order perturbation theory, where $eQ$ is nuclear quadrupole moment and $eq$ is the largest electric field gradient. [@abragam] The calculated $K$, $\nu_{\rm EFG}$ and $\eta$ at 10 K are $K \simeq 0.19$ %, $^{69}\nu_{\rm EFG} = 28.2$ MHz, $^{71}\nu_{\rm EFG} = 17.4$ MHz, and $\eta \simeq 0.42$, respectively. Here, the ratio $^{69}\nu_{\rm EFG}/^{71}\nu_{\rm EFG} \simeq 1.6$ is consistent with that of the nuclear quadrupole moments ($^{69}Q/^{71}Q = 1.59$). These values of $\nu_{\rm EFG}$ and $\eta$ for Ga(2) sites are similar to those of the U115 [@haru] and Np115[@kambe; @sakai] families, [*e.g.*]{}, $^{69}\nu_{\rm EFG}$ and $\eta$ at the same site are 27.5 MHz and 0.2 for UPtGa$_{5}$, and $\sim$27.3 MHz and $\sim$0.29 for NpCoGa$_{5}$, respectively.
It should be noted here that the above assignment of Ga(2) NMR lines has been confirmed further by the field orientation dependence ($\theta$) of the spectra in Fig. \[NMRspectrum\](b), where $\theta$ is an angle between $H_{0}$ and the $c$ axis. As seen in Fig.\[NMRspectrum\](b), a small increase of $\theta$ causes a splitting of each NMR line. If these lines were ascribed to Ga(1) sites, such a splitting could not be expected, because the Ga(1) sites remain equivalent under any applied field. The $^{69,71}$Ga(1) NMR signals are very weak due to a short $T_{2}$, and not seen in Fig.\[NMRspectrum\](a) under this experimental condition.
![\[NQRspectrum\]$^{69}$Ga(2) NQR spectrum with rf field parallel to $c$ axis at zero field in PuRhGa$_{2}$. The inset shows the temperature dependence of the $^{69}$Ga(2) NQR frequency. ](Fig3.eps){width="6cm"}
Next, we turn to the zero-field NQR experiments. The $^{69}$Ga(2) NQR spectrum of the single crystal has been observed at $\nu_{\rm Q}\simeq 29.15$ MHz as shown in Fig.\[NQRspectrum\]. Here, the value of $\nu_{\rm Q}$ is consistent with the expected frequency, $\nu_{\rm Q} = \nu_{\rm EFG} \sqrt{1+\eta^{2}/3}$, extracted from $\nu_{\rm EFG}$ and $\eta$ in the NMR results. The narrow NQR line width ($\sim$20 kHz) assures us that this sample is still of good quality, [*i.e.*]{}, broadening effects due to defects and/or impurities are not serious at this point. The temperature dependence of $\nu_{\rm Q}$ is shown in the inset of Fig.\[NQRspectrum\]. The value of $\nu_{\rm Q}$ increases gradually as temperature decreases from 300 K, and saturates below $\sim$50 K. The slight temperature dependence of $\nu_{\rm Q}$ is caused by that of $\nu_{\rm EFG}$, because the value of $\eta$ is found to be temperature-independent from NMR results. Note that the $^{71}$Ga(2) NQR line has been observed concurrently at the frequency of 18.38 MHz at $T = 10$ K.
![\[invT1\]Temperature dependence of [$1/T_{1}$]{} for the $^{69}$Ga(2) NQR line in PuRhGa$_{5}$. The inset shows the temperature dependence of [*in situ*]{} ac susceptibility using the rf coil. The solid curve has been calculated assuming a line-node gap with $2\Delta_{0}(0)\sim 5k_{\rm B}T_{\rm c}$ and a residual DOS $N_{\rm res}/N_{0}\sim0.25$. The broken curve shows the same calculation without any residual DOS. The dotted line shows Korringa-like behavior for comparison.](Fig4.eps){width="8cm"}
$T_{1}$ has been measured for both $^{69}$Ga(2) and $^{71}$Ga(2) NQR lines in order to clarify the $T_{1}$ process. The ratio $^{69}T^{-1}/^{71}T^{-1}$ of data so obtained is found to be nearly equal to $^{69}\gamma_{\rm n}^{2}/^{71}\gamma_{\rm n}^{2}$ below $\sim$150 K, where $\gamma_{\rm n}$ denotes the nuclear gyromagnetic ratio. This indicates that the magnetic dipole relaxation mechanism is dominant over quadrupolar relaxation, at least in the low temperature range. The inset to Fig.\[invT1\] shows the result of [*in situ*]{} $\chi_{\rm ac}$ measurements. Here, this measurement was performed before and after NQR $T_{1}$ measurements. The onset [$T_{\rm c}$]{} was determined to be 8.5 K from $\chi_{\rm ac}$ data, where the distribution of [$T_{\rm c}$]{} was estimated to be less than $\sim$0.5 K. In the duration of a single week to obtain NQR $T_{1}$ data, the onset [$T_{\rm c}$]{} was unchanged within less than $\sim$0.1 K. It is noted that the NQR signal intensity was weakened suddenly below 8.5 K, because the rf pulse was attenuated due to the Meissner effect.
Figure \[invT1\] shows the temperature dependence of [$1/T_{1}$]{}obtained for the $^{69}$Ga(2) NQR line. In the normal state from [$T_{\rm c}$]{} to $\sim$ 30 K, $1/T_{1}$ is approximately proportional to $T$, [*i.e.*]{}, Korringa-like behavior associated with typical coherent Fermi liquid state. The values of $1/T_{1}$ above $\sim$30 K fall below Korringa-like behavior, which is thought to be related to a reduction in the coherence of the Fermi liquid state. This Korringa-like behavior is also confirmed by preliminary high-field NMR $T_{1}$ measurements at $H_{0}\sim$110 kOe ($\parallel c$), where [$T_{\rm c}$]{} is suppressed down to $\sim$4 K. The present $T_{1}$ results suggest that the SC state in PuRhGa$_{5}$ sets in after a Fermi liquid state is established below $\sim$30 K. Further discussion of $T_{1}$ in the normal state will be presented elsewhere.
In the SC state, we have found that [$1/T_{1}$]{} shows no coherence peak just below [$T_{\rm c}$]{}, but decreases $\propto$ $T^{3}$ as $T$ decreases, as seen in Fig.\[invT1\]. Besides, a deviation from $T^{3}$ behavior is also observed for $T$ well below [$T_{\rm c}$]{}. The [$1/T_{1}$]{} behavior we found cannot be explained in terms of a fully open SC gap with $s$-wave symmetry. These results strongly suggest that PuRhGa$_{5}$ has an anisotropic gap in the SC state. We can reproduce the behavior of [$1/T_{1}$]{} under the assumption of $d$-wave symmetry with line nodes, which is widely accepted to be realized in the Ce115 family. This calculation has been done in the following way: An anisotropic gap function is assumed as a polar function $\Delta(\theta,\phi)=\Delta_{0}\cos\theta$, where $\theta$ and $\phi$ mean angular parameters on the Fermi surface, and the temperature dependence of $\Delta_{0}$ is assumed to be BCS-like. Then, the temperature dependence of [$1/T_{1}$]{} below [$T_{\rm c}$]{}has been calculated from the following integral, $$\Bigl(\frac{1}{T_{1}}\Bigr) / \Bigl (\frac{1}{T_{1}}\Bigr)_{T=T_{\rm c}}= \frac{2}{k_{\rm B}T}\int \Bigl\langle \frac{N_{\rm s}(E)^{2}}{N_{0}^{2}} \Bigr\rangle _{\theta, \phi}f(E)[1-f(E)] dE,$$ where $N_{\rm s}(E)=N_{0} E/\sqrt{E^{2}-\Delta(\theta,\phi)^{2}}$ with $N_{0}$ being the density of states (DOS) in the normal state , $f(E)$ is the Fermi distribution function, and $\langle\cdots\rangle_{\theta,\phi}$ means the anglar average over the Fermi surface. From the best fit of the experimental data to this calculation, we evaluate the SC gap $2\Delta_{0}(T\rightarrow0) \simeq 5 k_{\rm B}$[$T_{\rm c}$]{}with a residual DOS ($N_{\rm res}/N_{\rm 0}\simeq 0.25$) in the SC state. This gap estimate is similar to that of CeIrIn$_{5}$ with [$T_{\rm c}$]{}$\simeq 0.4$ K, [@ceirin5zheng] while it is smaller than that of CeCoIn$_{5}$, [*i.e.*]{}, $2\Delta_{0}(0)\simeq 8 k_{\rm B}T_{\rm c}$ with [$T_{\rm c}$]{}$\simeq 2.3$ K. [@cecoin5kohori] In the high-[$T_{\rm c}$]{} cuprates, the value of $2\Delta_{0}(0)$ often reaches $\sim10 k_{\rm B}T_{\rm c}$. PuRhGa$_{5}$ would be classified as an intermediate-coupling superconductor.
The finite residual DOS in the $d$-wave SC state is mostly caused by potential scattering associated with nonmagnetic impurities, distortion, and contamination with a secondary phase etc. [@ishida; @kitaoka] In Pu compounds, the influence of unavoidable potential scattering coming from self-radiation damage is also expected. In the literature, the relation between the size of the residual DOS and the reduction rate of [$T_{\rm c}$]{} has been calculated based on an anisotropic SC model, where the scattering is treated in the unitarity (strong) limit.[@hotta; @miyake] Using the relation with $N_{\rm res}/N_0 \simeq 0.25$ observed in our sample, we can obtain $T_{\rm c}/T_{\rm c0} \simeq 0.94$, where $T_{\rm c0}$ is the intrinsic value of [$T_{\rm c}$]{}without potential scattering effects. We can estimate the maximum (intrinsic) $T_{\rm c0} = 9.0$ K using the $T_{\rm c}=8.5$ K in our sample observed a month after synthesis. This $T_{\rm c0}$ is consistent with the observed [$T_{\rm c}$]{} just after synthesis. The aging effect on the residual DOS in the SC gap will be an interesting effect to study in Pu-based superconductors.
In summary, we have succeeded in $^{69,71}$Ga NMR/NQR measurements for the Ga(2) site of transuranium superconductor PuRhGa$_{5}$. The $^{69,71}$Ga(2) NQR lines have been found at frequencies consistent with an analysis based on NMR results. The NQR [$1/T_{1}$]{} in the SC state shows no coherence peak just below [$T_{\rm c}$]{}, but obeys a $T^{3}$ behavior below [$T_{\rm c}$]{}. Such a result is strong evidence that PuRhGa$_{5}$ is an unconventional superconductor with an anisotropic SC gap. Assuming a $d$-wave symmetry, the SC gap $\Delta_{0}(0) \simeq 5k_{\rm B}T_{\rm c}$ with $N_{\rm res}/N_0 \simeq 0.25$ has been evaluated. To determine the parity of SC pairing in PuRhGa$_{5}$, Knight shift measurements are in progress.
We thank Dr. T. Hotta and Dr. T. Maehira for helpful discussions. This work was supported in part by the Grant-in-Aid for Scientific Research of MEXT (Grants. No. 14340113).
-Added note. After completion of this paper, an NMR/NQR study on the closely related compound PuCoGa$_{5}$ by Curro [*et al.*]{}, has appeared [@curro]. It is useful to note some contrasting points in the behavior of these two Pu superconductors. First, the $T_1$ results of Curro [*et al.*]{}, are also analyzed in terms of $d$-wave superconductivity, finding a SC gap value 2$\Delta_0 \simeq 8k_{\rm B}T_{\rm c}$ ([*c.f.*]{}, $5k_{\rm B}T_{\rm c}$ for PuRhGa$_{5}$). Secondly, the $T$-variation of $(T_1T)^{-1}$ for PuCoGa$_{5}$ shows a monotonic increase right down to $T_{\rm c}$, whereas the Rh isomorph shows a Korringa-like behavior below $\sim$30 K.
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[^1]: E-mail address: piros@popsvr.tokai.jaeri.go.jp
|
---
abstract: 'Complex interactions between entities are often represented as edges in a network. In practice, the network is often constructed from noisy measurements and inevitably contains some errors. In this paper we consider the problem of estimating a network from multiple noisy observations where edges of the original network are recorded with both false positives and false negatives. This problem is motivated by neuroimaging applications where brain networks of a group of patients with a particular brain condition could be viewed as noisy versions of an unobserved true network corresponding to the disease. The key to optimally leveraging these multiple observations is to take advantage of network structure, and here we focus on the case where the true network contains communities. Communities are common in real networks in general and in particular are believed to be presented in brain networks. Under a community structure assumption on the truth, we derive an efficient method to estimate the noise levels and the original network, with theoretical guarantees on the convergence of our estimates. We show on synthetic networks that the performance of our method is close to an oracle method using the true parameter values, and apply our method to fMRI brain data, demonstrating that it constructs stable and plausible estimates of the population network.'
address:
- 'Department of Statistics, University of California Davis, Davis, CA 95616, USA\'
- 'Department of Statistics, University of Michigan, Ann Arbor, MI 48109, USA\'
- 'Department of Statistics, University of Michigan, Ann Arbor, MI 48109, USA\'
author:
-
-
-
bibliography:
- 'allref.bib'
title: Estimating a network from multiple noisy realizations
---
Introduction
============
Optimal estimates and the role of noise {#sec:known parameters}
=======================================
The estimation algorithm {#sec: algorithm}
========================
Numerical results {#sec: numerical results}
=================
Discussion {#sec:discussion}
==========
We have proposed a novel way to estimate an underlying “population” network from its multiple noisy realizations, leveraging the underlying community structure. In contrast to most previous work (with the notable exceptions of ), our algorithm does not vectorize the network or reduce it to global summaries; the procedure is designed specifically for network data, and thus tends to outperform methods that do not respect the underlying network structure. While we focused on the stochastic block model as the underlying network structure, because of its simple form and its role as an approximation to any exchangeable network model, this assumption is not essential. An extension to the degree-corrected stochastic block model is left as future work, and we believe in practice the algorithm will work well for any network with community structure. On the other hand, the assumption of independent noise is important and unlikely to be relaxed. The assumption of false positive and false negative probability matrices being piecewise constant is also important, as it allows us to significantly reduce the number of parameters and estimate them using the shared information within each block, but clearly many other ways to impose sharing information are possible, perhaps through a general low rank formulation. We leave exploring such a formulation for future work.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by NSF grant DMS-1521551 and ONR grant N000141612910 to E. Levina. We thank our collaborators in Stephan Taylor’s lab in Psychiatry at the University of Michigan for providing a processed version of the data.
Estimation error {#ap: estimation error}
================
Convergence of the EM algorithm {#ap: convergence}
===============================
Extension to Weighted Graphs {#ap: weighted}
============================
|
---
author:
- |
Jakob Ablinger\
Research Institute for Symbolic Computation (RISC),\
Johannes Kepler University, Altenbergerstra[ß]{}e 69, A–4040, Linz, Austria\
E-mail:
- |
Johannes Blümlein\
Deutsches Elektronen–Synchrotron (DESY),\
Platanenallee 6, D-15738 Zeuthen, Germany\
E-mail:
- |
Peter Marquard\
Deutsches Elektronen–Synchrotron (DESY),\
Platanenallee 6, D-15738 Zeuthen, Germany\
E-mail:
- |
\
Deutsches Elektronen–Synchrotron (DESY),\
Platanenallee 6, D-15738 Zeuthen, Germany\
E-mail:
- |
Carsten Schneider\
Research Institute for Symbolic Computation (RISC),\
Johannes Kepler University, Altenbergerstra[ß]{}e 69, A–4040, Linz, Austria\
E-mail:
title: 'Massive three loop form factors in the planar limit[^1] '
---
Introduction
============
The top quark, being the heaviest particle of the Standard Model (SM), plays a significant role in understanding the electro-weak symmetry breaking (EWSB). Besides, its heftiness generates a strong potential for hidden beyond the SM (BSM) physics scenarios. Hence, a detailed study on top quark observables is always a crucial topic. On the other hand, the abundance of top quark pair production at the high energy colliders allows us to obtain accurate measurements. Especially at the future linear or circular electron-positron colliders, the experimental accuracy for this channel will reach ultimate precision. In order to match the experimental accuracy, precise predictions are required on the theoretical side as well. Perturbative quantum chromodynamics (QCD) effects constitute the major contributions in precision physics and one of the main ingredients of QCD corrections is the form factor. Form factors are the matrix elements of local composite operators between physical states. In scattering cross-sections, they provide important contributions to the virtual corrections. The vector and axial-vector massive form factors are of importance for the forward-backward asymmetry of bottom or top quark pair production at electron-positron colliders while, the scalar and pseudo-scalar ones may shed light on the decay of a Higgs boson to a pair of heavy quarks. They are also important to inspect the properties of the top quark [@Abe:1995hr; @D0:1995jca] during the high luminosity phase of the LHC [@HLHC] and the experimental precision studies at future high energy $e^+ e^-$ colliders [@Accomando:1997wt].
In this note, we present both the color–planar and complete light quark non-singlet three-loop contributions to the massive form factors for vector, axial-vector, scalar and pseudo-scalar currents. Our results except for the vector current, are presented in [@Ablinger:2018yae] and for the vector current, including the technical details, will be presented elsewhere [@FORMF2]. In [@Bernreuther:2004ih; @Bernreuther:2004th; @Bernreuther:2005rw; @Bernreuther:2005gw], the two-loop QCD corrections to the massive vector, axial-vector form factors, the anomaly contributions, and the scalar and pseudo-scalar form factors were first presented. In [@Gluza:2009yy], an independent computation led to a cross-check of the vector form factor, including the additional ${\mathcal O}({\varepsilon})$ terms in the dimensional parameter ${\varepsilon}= (4-D)/2$. The contributions up to ${\mathcal O}({\varepsilon}^2)$ for all the massive two-loop form factors were obtained recently in Ref. [@Ablinger:2017hst]. The color–planar contributions to the massive three-loop vector form factor have been computed in [@Henn:2016tyf; @Henn:2016kjz] and the complete light quark contributions in [@Lee:2018nxa]. In a parallel and independent computation in [@Lee:2018rgs], the authors also have obtained both the color–planar and complete light quark non-singlet three-loop massive form factors for the aforementioned currents. In [@Grozin:2017aty], the large $\beta_0$ limit has been considered.
Notation
========
The notations follow those used in Ref. [@Ablinger:2018yae; @Ablinger:2017hst]. To summarize, we consider the decay of a virtual massive boson of momentum $q$ into a pair of heavy quarks of mass $m$, momenta $q_1$ and $q_2$ and color $c$ and $d$, through a vertex indicated by $I=V,A,S,P$ for a vector, an axial-vector, a scalar and a pseudo-scalar boson, respectively. Here $q^2 = (q_1+q_2)^2$ is the center of mass energy squared and the dimensionless variable $s$ is defined by $$s = \frac{q^2}{m^2}\,.$$ By studying the Lorentz structure, the following general form of the amplitudes for the vector and axial-vector currents can be obtained $$\begin{aligned}
-i \delta_{cd} ~ \bar{u}_c (q_1) ~ \Big[
v_Q \Big( \gamma^{\mu} ~F_{V,1} + \frac{i}{2 m} \sigma^{\mu \nu} q_{\nu} ~ F_{V,2} \Big)
+
a_Q \Big( \gamma^{\mu} \gamma_5~F_{A,1} + \frac{1}{2 m} q^{\mu} \gamma_5 ~ F_{A,2} \Big)
\Big] ~ v_d (q_2) , \end{aligned}$$ and for the scalar and pseudo-scalar currents $$\begin{aligned}
- \frac{m}{v} \delta_{cd} ~ \bar{u}_c (q_1) ~ \Big[ s_Q \, F_{S} + i p_Q \gamma_5 \, F_{P} \Big] ~ v_d (q_2) \,.\end{aligned}$$ $\bar{u}_c (q_1)$ and $v_d (q_2)$ are the bi-spinors of the quark and the anti-quark, respectively. The scalar objects, $F_{I}$, with $I =V, A, S, P$, are the corresponding form factors, expanded in the strong coupling constant $\alpha_s = g_s^2/(4\pi)$ as follows $$F_{I} = \sum_{n=0}^{\infty} \asr^n F_{I}^{(n)} \,.$$ $\sigma^{\mu\nu} = \frac{i}{2} [\gamma^{\mu},\gamma^{\nu}]$ and $v_Q, a_Q, s_Q, p_Q$ are the vector, axial-vector, scalar and pseudo-scalar coupling constant, respectively. $v = (\sqrt{2} G_F)^{-1/2}$ is the SM vacuum expectation value of the Higgs field, with $G_F$ being the Fermi constant. Finally, we multiply appropriate projectors as provided in [@Ablinger:2017hst], to obtain the unrenormalized form factors. Next, the trace over the color and spinor indices is performed. For later purposes we denote the number of colors by $N_c$. $n_l$ and $n_h$ are the number of light and heavy quarks, respectively.
Since we use dimensional regularization [@tHooft:1972tcz], the important factor for axial-vector and pseudo-scalar currents, is a proper definition of $\gamma_5$ in $D$ space-time dimensions. As both the color-planar and complete light quark contributions belong to the so-called non-singlet case, where the axial-vector or pseudo-scalar vertex is connected to open heavy quark lines, both $\gamma_5$-matrices appear in the same chain of Dirac matrices. Hence we can conveniently use an anti-commuting $\gamma_5$ in $D$ space-time dimensions, with $\gamma_5^2 = 1$. This also implies the well-known Ward identity, $$\label{eq:cwi}
q^{\mu} \Gamma_{A,cd}^{\mu, \sf ns} = 2 m \Gamma_{P,cd}^{\sf ns} \,,$$ which in terms of the form factors, takes the following form $$\label{eq:cwiFF}
2 F_{A,1}^{\sf ns} + \frac{s}{2} F_{A,2}^{\sf ns} = 2 F_{P}^{\sf ns} \,.$$ Here, the non-singlet contributions are denoted by ${\sf ns}$. For convenience, we introduce the Landau variable [@Barbieri:1972as] $$\label{eq:varxp}
x=\frac{\sqrt{q^2-4m^2}-\sqrt{q^2}}{\sqrt{q^2-4m^2}+\sqrt{q^2}}\quad \leftrightarrow
\quad s = \frac{q^2}{m^2}=-\frac{(1-x)^2}{x}.$$
Computational details
=====================
We follow the generic procedure to compute the form factors. The Feynman diagrams are generated using [QGRAF]{} [@Nogueira:1991ex]. The [QGRAF]{} output is then processed using [Q2e/Exp]{} [@Harlander:1997zb; @Seidensticker:1999bb] and [FORM]{} [@Vermaseren:2000nd; @Tentyukov:2007mu]. The color algebra has been performed using [Color]{} [@vanRitbergen:1998pn]. By decomposing the dot products among the loop and external momenta, the diagrams can be expressed in terms of a linear combination of a large set of scalar integrals.
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam1a.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam3a.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam4a.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam6b.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam8a.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam24a.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam18b.eps){width="100.00000%"}
![The color-planar topologies[]{data-label="fig:cptopologies"}](fig/fam16a.eps){width="100.00000%"}
These integrals are then reduced using integration by parts identities (IBPs) [@Chetyrkin:1981qh; @Laporta:2001dd] with the help of the program [Crusher]{} [@CRUSHER] to obtain 109 master integrals (MIs), out of which 96 appear in the color-planar case. In the color-planar limit, the families of integrals can be represented by eight topologies, shown in Figure \[fig:cptopologies\], whereas for the complete light quark contributions, three more topologies, cf. Figure \[fig:nltopo\], are required [^2].
![The $n_l$ topologies[]{data-label="fig:nltopo"}](fig/fam1c.eps){width="100.00000%"}
![The $n_l$ topologies[]{data-label="fig:nltopo"}](fig/fam3b.eps){width="100.00000%"}
![The $n_l$ topologies[]{data-label="fig:nltopo"}](fig/fam20b.eps){width="100.00000%"}
Finally, to compute the MIs, we use the method of differential equations [@Kotikov:1990kg; @Remiddi:1997ny; @Henn:2013pwa; @Ablinger:2015tua]. For a recent review on the computational methods of loop integrals in quantum field theory, see [@Blumlein:2018cms]. The basic idea is to obtain a set of differential equations of the MIs by performing differentiation w.r.t $x$ and then to use the IBP relations. The first step to solve the corresponding linear system of differential equations is to find out whether the system is first order factorizable or not. Using the package [Oresys]{} [@ORESYS], based on Zürcher’s algorithm [@Zuercher:94; @NewUncouplingMethod], we have found that the present system is indeed first order factorizable in $x$-space. Without any need to choose a special basis, we can now simply solve the system in terms of iterated integrals of whatsoever alphabet, cf. Ref. [@FORMF2] for details. The differential equations are solved order by order in $\varepsilon$ successively, starting at the leading pole terms $\propto 1/\varepsilon^3$. The successive solutions in $\varepsilon$ contribute to the inhomogeneities in the next order. We compute the master integrals block-by-block, where for an $m \times m$ system, $l$ single inhomogeneous ordinary differential equations are obtained, where $1\leq l \leq m$. The orders of these differential equations are $m_1, \ldots, m_l$ such that $m_1 + \cdots + m_l = m$. We have solved these differential equations using the variation of constant. The other $m-l$ solutions result from the former solution immediately. The constants of integration are determined using boundary conditions at $x = 1$. The calculation is performed by intense use of [HarmonicSums]{} [@Ablinger:2011te; @HSUM; @Ablinger:2014rba; @Ablinger:2010kw; @Ablinger:2013hcp; @Ablinger:2013cf; @Ablinger:2014bra], which uses the package [Sigma]{} [@Schneider:sigma1; @Schneider:sigma2]. Finally, all the MIs have been checked numerically using [FIESTA]{} [@Smirnov:2008py; @Smirnov:2009pb; @Smirnov:2015mct].
The non-homogeneous contributions contain only rational functions of $x$ and hence the results can be written in terms of iterative integrals. While integration over a letter is a straightforward algebraic manipulation, often $k$-th powers of a letter, $k \in \mathbb{N}$, appear which needs to be transformed to the letter by partial integration. This method is partially related to the method of hyperlogarithms [@Brown:2008um; @Ablinger:2014yaa]. We obtain up to weight [w=6]{} real-valued iterated integrals over the alphabet $$\begin{aligned}
\frac{1}{x},~~
\frac{1}{1-x},~~
\frac{1}{1+x},~~
\frac{1}{1-x+x^2},~~
\frac{x}{1-x+x^2},\end{aligned}$$ i.e. the usual harmonic polylogarithms (HPLs) [@Remiddi:1999ew] and their cyclotomic extension [@Ablinger:2011te], including the respective constants in the limit $x \rightarrow 1$, i.e. the multiple zeta values (MZVs) [@Blumlein:2009cf] and the cyclotomic constants [@Ablinger:2011te; @Ablinger:2017tqs; @Ablinger:2018xyz]. The use of shuffle algebra [@Blumlein:2003gb], implemented in [HarmonicSums]{}, reduces the number of functions accordingly, which facilitates numerical evaluation. In the MZVs and cyclotomic cases, there are proven reduction relations to weight [w = 12]{} [@Blumlein:2009cf] and [w = 6]{} [@Ablinger:2017tqs; @Ablinger:2018xyz], respectively, which have been used. The 188 cyclotomic constants which appear up to [w = 6]{}, reduce to 23 constants. Note that there are more conjectured relations, cf. [@Henn:2015sem], based on PSLQ [@PSLQ]. If these conjectured relations are used, only MZVs remain as constants in all form factors. The analytic result for the different form factors in terms of HPLs and cyclotomic HPLs [@Remiddi:1999ew; @Ablinger:2011te] can be analytically continued outside $x~\in~[0,1[$ by using the mappings $x \rightarrow -x, x \rightarrow (1-x)/(1+x)$ on the expense of extending the cyclotomy class in cases needed.
Ultraviolet renormalization and universal infrared structure
============================================================
To perform the ultraviolet (UV) renormalization of the form factors, we choose a mixed scheme. The heavy quark mass and wave function have been renormalized in the on-shell (OS) renormalization scheme. We renormalize the strong coupling constant in the $\overline{\rm MS}$ scheme, by setting the universal factor $S_\varepsilon =
\exp(-\varepsilon (\gamma_E - \ln(4\pi))$ for each loop order to one at the end of the calculation. The required renormalization constants are already well known and denoted by $Z_{m, {\rm OS}}$ [@Broadhurst:1991fy; @Melnikov:2000zc; @Marquard:2007uj; @Marquard:2015qpa; @Marquard:2016dcn], $Z_{2,{\rm OS}}$ [@Broadhurst:1991fy; @Melnikov:2000zc; @Marquard:2007uj; @Marquard:2018rwx] and $Z_{a_s}$ [@Tarasov:1980au; @Larin:1993tp] for the heavy quark mass, wave function and strong coupling constant, respectively. For all the currents, the renormalization of the heavy-quark wave function and the strong coupling constant are multiplicative, while the renormalization of massive fermion lines has been taken care of by properly considering the counter terms. For the scalar and pseudo-scalar currents, presence of the heavy quark mass in the Yukawa coupling employs another overall mass renormalization constant, which also has been performed in OS renormalization scheme.
The universal behavior of infrared (IR) singularities of the massive form factors was first investigated in [@Mitov:2006xs] considering the high energy limit. Later in [@Becher:2009kw], a general argument was provided to factorize the IR singularities as a multiplicative renormalization constant. Its structure is constrained by the renormalization group equation (RGE), as follows, $$F_{I} = Z (\mu) F_{I}^{\mathrm{fin}} (\mu)\, ,$$ where $F_{I}^{\mathrm{fin}}$ is finite as ${\varepsilon}\rightarrow 0$. The RGE for $Z(\mu)$ reads $$\label{eq:rgeZ}
\frac{d}{d \ln \mu} \ln Z({\varepsilon}, x, m, \mu) = - \Gamma (x,m,\mu) \,,$$ where $\Gamma$ is the corresponding cusp anomalous dimension, which is by now available up to three-loop order [@Grozin:2014hna; @Grozin:2015kna]. Notice that $Z$ does not carry any information regarding the vertex. Both $Z$ and $\Gamma$ can be expanded in a perturbative series in $\alpha_s$ as follows $$Z = \sum_{n=0}^{\infty} \asr^n Z^{(n)} \,, \qquad
\Gamma = \sum_{n=0}^{\infty} \asr^{n+1} \Gamma_{n}$$ and one finds the following solution for Eq. (\[eq:rgeZ\]) $$\begin{aligned}
\label{eq:solnZ}
Z &= 1 + \asr \Bigg[ \frac{\Gamma_0}{2 {\varepsilon}} \Bigg]
+ \asr^2 \Bigg[ \frac{1}{{\varepsilon}^2} \Big( \frac{\Gamma_0^2}{8} - \frac{\beta_0 \Gamma_0}{4} \Big) + \frac{\Gamma_1}{4 {\varepsilon}} \Bigg]
\nonumber\\
&+ \asr^3 \bigg[ \frac{1}{{\varepsilon}^3} \left( \frac{\Gamma_0^3}{48} - \frac{\beta_0 \Gamma_0^2}{8} + \frac{\beta_0^2 \Gamma_0}{6} \right)
+ \frac{1}{{\varepsilon}^2} \left( \frac{\Gamma_0 \Gamma_1}{8} - \frac{\beta_1 \Gamma_0}{6} \right)
+ \frac{1}{{\varepsilon}} \left( \frac{\Gamma_2}{6} \right) \bigg]
+ {\cal O} (\alpha_s^4) \,.\end{aligned}$$ Eq. (\[eq:solnZ\]) correctly predicts the IR singularities for all massive form factors at the three-loop level.
Results and checks
==================
We finally obtain the color–planar and the complete light quark non–singlet ($n_l$) contributions for the three-loop massive form factors for vector, axial-vector, scalar and pseudo-scalar currents. The expressions, except for the vector current, are attached as supplemental material along with the publication [@Ablinger:2018yae]. The corresponding results for vector current will be available in [@FORMF2].
![The $O(\varepsilon^0)$ contribution to the vector three-loop form factors $F_{V,1}^{(3)}$ (left) and $F_{V,2}^{(3)}$ (right) as a function of $x$. Dash-dotted line: leading color contribution of the non-singlet form factor; Full line: sum of the complete non-singlet $n_l$-contributions for $n_l =5$ and the color-planar non-singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.[]{data-label="Fig:VF12"}](plots/vectorF1.pdf "fig:"){width="49.00000%"} ![The $O(\varepsilon^0)$ contribution to the vector three-loop form factors $F_{V,1}^{(3)}$ (left) and $F_{V,2}^{(3)}$ (right) as a function of $x$. Dash-dotted line: leading color contribution of the non-singlet form factor; Full line: sum of the complete non-singlet $n_l$-contributions for $n_l =5$ and the color-planar non-singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.[]{data-label="Fig:VF12"}](plots/vectorF2.pdf "fig:"){width="49.00000%"}
![The $O(\varepsilon^0)$ contribution to the axial-vector three-loop form factors $F_{A,1}^{(3)}$ (left) and $F_{A,2}^{(3)}$ (right) as a function of $x$. Dash-dotted line: leading color contribution of the non-singlet form factor; Full line: sum of the complete non-singlet $n_l$-contributions for $n_l =5$ and the color-planar non-singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.[]{data-label="Fig:AV12"}](plots/avectorG1.pdf "fig:"){width="49.00000%"} ![The $O(\varepsilon^0)$ contribution to the axial-vector three-loop form factors $F_{A,1}^{(3)}$ (left) and $F_{A,2}^{(3)}$ (right) as a function of $x$. Dash-dotted line: leading color contribution of the non-singlet form factor; Full line: sum of the complete non-singlet $n_l$-contributions for $n_l =5$ and the color-planar non-singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.[]{data-label="Fig:AV12"}](plots/avectorG2.pdf "fig:"){width="49.00000%"}
In Figures \[Fig:VF12\]–\[Fig:FSP\] we illustrate the behaviour of the $O(\varepsilon^0)$ parts of the different form factors as a function of $x \in [0,1]$. We also show their small- and large-$x$ expansions. The latter representations are obtained using [HarmonicSums]{}. The different limits are characterized as follows :
*Low energy region* ($x \rightarrow 1$): In the space-like case ($q^2 < 0$) we have expanded the form factors, redefining $x=e^{i\phi}$, $\phi=0$.
*High energy region* ($x \rightarrow 0$): Here we expand the form factors up to ${\cal O} (x^4)$. The chirality flipping form factors $F_{V,2}$ and $F_{A,2}$ vanish and the effect of $\gamma_5$ gets nullified in this limit implying $F_{V,1}=F_{A,1}$ and $F_S=F_P$. In the small quark mass limit, the form factors satisfy the Sudakov evolution equation. A detailed study has been performed in [@Mitov:2006xs; @Ahmed:2017gyt] to predict part of the vector form factors in this limit from the then available components up to three and four loop level, respectively.
*Threshold region* ($x \rightarrow -1$): We define $\beta = \sqrt{1 -
\frac{4m^2}{q^2}}$ and expand the form factors around $\beta = 0$.
For the numerical evaluation of the HPLs and cyclotomic HPLs in the Kummer representation, we use the [GiNaC]{}-package [@Vollinga:2004sn; @Bauer:2000cp].
![The $O(\varepsilon^0)$ contribution to the scalar and pseudo-scalar three-loop form factors $F_S^{(3)}$ (left) and $F_P^{(3)}$ (right) as a function of $x$. Dash-dotted line: leading color contribution of the non-singlet form factor; Full line: sum of the complete non-singlet $n_l$-contributions for $n_l =5$ and the color-planar non-singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.[]{data-label="Fig:FSP"}](plots/scalar.pdf "fig:"){width="49.00000%"} ![The $O(\varepsilon^0)$ contribution to the scalar and pseudo-scalar three-loop form factors $F_S^{(3)}$ (left) and $F_P^{(3)}$ (right) as a function of $x$. Dash-dotted line: leading color contribution of the non-singlet form factor; Full line: sum of the complete non-singlet $n_l$-contributions for $n_l =5$ and the color-planar non-singlet form factor; Dashed line: large $x$ expansion; Dotted line: small $x$ expansion.[]{data-label="Fig:FSP"}](plots/pscalar.pdf "fig:"){width="49.00000%"}
We have performed a series of further checks. Through an explicit computation, the Ward identity Eq. (\[eq:cwiFF\]) has been checked. By maintaining the gauge parameter $\xi$ to first order throughout the calculation, a partial check on gauge invariance has been achieved. After $\alpha_s$-decoupling, the UV renormalized results satisfy the universal IR structure, confirming again the correctness of all pole terms. Finally, we have compared our results with those of Ref. [@Lee:2018rgs], which has been obtained using different methods, and agree by adjusting the respective conventions.
[**Acknowledgment.**]{} This work was supported in part by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15). We would like to thank M. Steinhauser for providing their yet unpublished results in electronic form and A. De Freitas and V. Ravindran for discussions. The Feynman diagrams have been drawn using [Axodraw]{} [@Vermaseren:1994je].
\[2\][\#2]{}
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[^1]: DESY 18–118, DO–TH 18/14
[^2]: Only sub-topologies with a maximum of eight propagators contribute.
|
---
abstract: 'We study the constructions of Dicke states of identical particles of spin-$1$, $3/2$ and $2$ in the number representation with given particle number $N$ and magnetic quantum number $M$. The complete bases and corresponding coefficients in the Dicke states are given, in terms of which the Dicke states are explicitly expressed in the number representation. As a byproduct, a rule of how to construct all the anti-symmetric states in these high spin systems is given. Finally, by employing the negativity as the entanglement measure, we explore the entanglement properties for spin-$1$ cases including certain pure states of two particles and many-particle Dicke states.'
author:
- 'Wan-Fang Liu'
- 'Zheng-Da Hu'
title: 'Constructions of Dicke states in high spin multi-particle systems'
---
Introduction {#sec:introduction}
============
The Dicke state, put forward by Dicke in $1954$, is a multi-particle state of spin-$1/2$ with the maximal total angular momentum [@Dicke1]. During the past decades, it is under extensive researches and some new features have been found. Especially, it has become a basic state as the development of quantum information science. Based on the Dicke states, one can construct several new quantum states such as GHZ states, W states, squeezed spin states and spin coherence states, which are very important in quantum information theory [@G2; @Du3; @Ck; @Wang5; @Ki6]. The original Dicke state focuses on the spin-$1/2$ case. However, the situations of spin $s$ or angular momentum $j$ more than $1/2$ have emerged their importance and attracted much attention as the development of low temperature physics. The system of many $^{23}Na$ atoms trapped in a optical lattice is spin-$1$ [Yi]{}, and the system of many $^{132}C_{s}$ or $^{135}B_{a}$ atoms is spin-$%
3/2$ [@St; @Ho; @Zhou]. For these high-spin systems, there may exist some hidden symmetries, strong quantum fluctuations and novel phases [@Wu]. For instance, Haldane predicted that the one dimensional Heisenberg chain has a spin gap for integer value of spin [@Hal1; @Hal2]. Wang *et al*. studied the entanglement properties in a spin-$1$ Heisenberg chain [Wang7,Wang8]{}. The eigenstates and magnetic response in spin-$1$ and $2$ Bose-Einstein condensates were discussed by Koashi [@Koa]. These systems with high spin have more spin orientations and more quantum eigenstates such that richer physical phenomena can emerge. Therefore, it is desirable to construct the basic quantum states based on Dicke states for the high-spin cases.
It is well known that the single particle is a qubit with only two magnetic components for a spin-$1/2$ many-body system. Thus, the configuration of the Dicke state $|J,M\rangle$ for spin-$1/2$ is simplest. For certain given total spin $J$ and total magnetic component $M$, one can find the explicit form for the its Dicke state by means of a binary linear equation group accompanied with normalization and symmetry constraints. However, for high-spin many-body systems, the construction of the Dicke state is not a easy task due to much more spin components. The conventional approach of $3j$ symbol in quantum mechanics is appropriate only for the case of small particle number $N$ and one should seek a different route for the case of large particle number $N$. In a word, the investigation of Dicke states for high-spin cases is a nontrivial but troublesome task which may be considered as a supplement to the modern quantum mechanics. Motivated by the construction of spin-$1/2$ Dicke state, in this work, we explore the Dicke states of identical spin-$s$ particles with $s=1$, $3/2$ and $2$.
This paper is ornanized as follows. In Sec. \[sec:concl\], we find those complete bases and corresponding coefficients in the Dicke state $%
|J,M\rangle $. In Sec. III, we study the anti-symmetric states in high spin systems. In Sec. IV, in terms of the negativity, the entanglement of two spin-$1$ particle is discussed. We conclude in Sec. V.
Construction of Dicke states in high spin multi-particle system {#sec: C of Dicke}
===============================================================
Dicke states in spin-1/2 multi-particle system
----------------------------------------------
First, let us recall the derivation process of Dicke states in spin-$1/2$ multi-particle system [@Dicke1]. For a multi-particle system consisting of identical spin-$1/2$ particles, the states $|J,M\rangle $, which possess the maximal total spin angular momentum, are termed as the Dicke states. Obviously, the states $|J,J\rangle $ and $|J,-J\rangle $ possess the simple form $$\begin{aligned}
|J,J\rangle =& \underset{N}{\underbrace{|\frac{1}{2}\rangle \cdots \otimes
\cdots |\frac{1}{2}\rangle }}, \notag \\
|J,-J\rangle =& \underset{N}{\underbrace{|-\frac{1}{2}\rangle \cdots \otimes
\cdots |-\frac{1}{2}\rangle }}.\end{aligned}$$By introducing the collective raising and lowering operators $J_{\pm }=\sum
s_{i\pm }$ and using the relation $$\begin{aligned}
J_{\pm }|J,M\rangle &=&\sqrt{(J\mp M)(J\pm M+1)}|J,M\pm 1\rangle , \notag \\
s_{i\pm }|s,m_{s}\rangle &=&\sqrt{(s\mp m_{s})(s\pm m_{s}+1)}|s,m_{s}\pm
1\rangle ,\end{aligned}$$one can obtain all the other Dicke states $|J,M\rangle $ from the states $%
|J,\pm J\rangle $ although the repeating process may be tedious. Here, $s $ is the spin of single particle, $s_{i\pm }$ are raising and lowering operators for the $i$th particle, and $m_{s}$ is the eigenvalue of $s_{z}$. However, this method does not have any advantage for multi-qubit system, especially, for the high-spin systems. Thereby, it is desirable to find other ways to express Dicke states of multi-particle systems. In what follows, we will demonstrate an alternative approach via the number representation.
Supposing the eigenstates of the operator $s_{z}$ of spin-$1/2$ angular momentum are $|\frac{1}{2}\rangle $ and $|-\frac{1}{2}\rangle $ and the numbers of particles occupying the two states are $n_{1}$ and $n_{2}$, respectively, we employ the number representation $\{|n_{1},n_{2}\rangle \}$ with $$|n_{1},n_{2}\rangle =\sqrt{\frac{n_{1}!n_{2}!}{N!}}\sum P(\underset{n_{1}}{|%
\underbrace{\frac{1}{2}\cdots \frac{1}{2}}}\underset{n_{2}}{\underbrace{-%
\frac{1}{2}\cdots -\frac{1}{2}}}\rangle ), \label{eq2}$$where $N=n_{1}+n_{2}$ is the total particle number and $P$ denotes permutation operations between two particles with different states. Then, the Dicke state $|J,M\rangle $ can be expressed as $$|J,M\rangle =|n_{1},n_{2}\rangle , \label{eq1}$$where $J=\frac{N}{2}$ is the maximal azimuthal quantum number of $S=\sum
s_{i}$, $M$ is the spin magnetic quantum number of $S_{z}$ whose value can be $M=J$, $J-1$, $\cdots $, $1-J$, $-J$. It is straightforward to obtain $$|J,\frac{N}{2}\rangle =|N,0\rangle ,\text{ }|J,-\frac{N}{2}\rangle
=|0,N\rangle . \label{eq3}$$In terms of Eq. (\[eq2\]) and the conservation of quantum numbers in different single particle states, one can easily obtain the constraint equation set $$n_{1}+n_{2}=N, \\
\frac{n_{1}}{2}-\frac{n_{2}}{2}=M. \label{eq4}$$Therefore, once $N$ and $M$ are given, the explicit form of the Dicke state can be easily derived according to Eq. (\[eq4\]) and Eq. (\[eq2\]). Finally, the Dicke state has a form
$$|J,M\rangle =|n_{1},n_{2}\rangle =\sqrt{\frac{(\frac{N}{2}+M)!(\frac{N}{2}%
-M)!}{N!}}\sum P(\underset{\frac{N}{2}+M}{|\underbrace{\frac{1}{2}\cdots
\frac{1}{2}}}\underset{\frac{N}{2}-M}{\underbrace{-\frac{1}{2}\cdots -\frac{1%
}{2}}}\rangle ).$$
\[sec: C of Dicke copy(1)\]
Dicke states of identical spin-1 particles
------------------------------------------
For simplicity, the states with maximal total angular momentum for $%
N$ identical particles of high spin ( $s_{i}>1/2$) are called generalized Dicke states $|J,M\rangle $ here. Following the approach for the above spin-$%
1/2$ case, we can express $|J,M\rangle $ in the number representation as
$$|J,M\rangle =\sum_{k=0}^{\max
}C_{k,n_{1},n_{0},n_{-1}}|n_{1},n_{0},n_{-1}\rangle , \label{eq5}$$
where $k$ is a parameter directly related with $n_{0}$, $$\begin{aligned}
\max =& \frac{1}{2}(J-|M|-\min ), \notag \\
\min =& \frac{1}{2}[(-1)^{J-|M|+1}+1], \label{eq6}\end{aligned}$$and $C_{k,n_{1},n_{0},n_{-1}}$ are the superposition coefficients with $%
n_{1},$ $n_{0}$ and $n_{-1}$ denoting the numbers of particles in states $%
|\uparrow \rangle $, $|0\rangle $ and $|\downarrow \rangle $, respectively, and $$|n_{1},n_{0},n_{-1}\rangle =\sqrt{\frac{n_{1}!n_{0}!n_{-1}!}{N!}}\underset{p}%
{\sum }P(\underset{n_{1}}{|\underbrace{1\cdots 1}}\underset{n_{0}}{%
\underbrace{0\cdots 0}}\underset{n_{-1}}{\underbrace{-1\cdots -1}}\rangle ).
\label{eq7}$$
It should be noted that the form of Eq. (\[eq5\]) is different from that of Eq. (\[eq1\]) such that the values of $n_{1},$ $n_{0}$ and $n_{-1}$ are not unique for specific $N$ and $M$, and they shall be determined with the help of Eq. (\[eq6\]). This is the very difference for Dicke states of high-spin systems from those of the spin-$1/2$ system. For the special cases $M=\pm J$ and $M=\pm (J-1)$, it is easy to check that $\max =\min =0$ and $%
\max =0,$ $\min =1$, respectively, in which case $n_{1},$ $n_{0}$ and $%
n_{-1} $ are uniquely determined. For other cases, we should find all the values of $n_{1},$ $n_{0}$ and $n_{-1}$ as well as $C_{k,n_{1},n_{0},n_{-1}}$.
Generally, three equations are needed to determine $n_{1},$ $n_{0}$ and $%
n_{-1}$, and it is not difficult to find the first and the second equations as follows $$\begin{aligned}
n_{1}+n_{0}+n_{-1}& =N, \notag \\
n_{1}-n_{-1}& =M. \label{eq8}\end{aligned}$$Here, we find the third equation given by $$n_{0}=\min +2k, \label{eq9}$$with $k=0,1,\cdots \max $. According to Eq. (\[eq6\]) and Eq. (\[eq9\]), the number of elementary states contained in the basis $%
\{|n_{1},n_{0},n_{-1}\rangle \}$ for state $|J,M\rangle $ is $\max +1$. Using the normalization condition, we also obtain the coefficients $$C_{k,n_{1},n_{0},n_{-1}}=\frac{(J-|M|)!}{2^{-n_{0}}}\sqrt{\frac{N!}{%
n_{1}!n_{0}!n_{-1}!}}\overset{J-|M|}{\underset{l=1}{\prod }}\frac{1}{\sqrt{%
(2N-l+1)l}}. \label{eq10}$$It is worth nothing that an arbitrary combination of max$+1$ elementary states does not change the value of $M$. As a result, the arbitrary combination also applies to the construction of the Dicke state $|J,M\rangle
$. However, since $|J,M-1\rangle $ and $|J,M\rangle $ must satisfy the relation $$|J,M-1\rangle =\frac{\hat{J}_{-}|J,M\rangle }{\sqrt{(J+M)(J-M+1)}},
\label{eq11}$$the basis of $|J,M\rangle $ and that of $|J,M-1\rangle $ also satisfy certain relations. Thereby, once the basis of the Dicke state $|J,M\rangle $ is determined, that of $|J,M-1\rangle $ is specified as well.
For an illustration, we consider the case of particle number $N=10$ and obtain all $\{|n_{1},n_{0},n_{-1}\rangle \}$ and the corresponding $%
C_{k,n_{1},n_{0},n_{-1}}$ respect to different values of $M$, which is listed in the following Table \[T1\] and \[T22\]. For convenience, we have omitted the subscripts $k,n_{1},n_{0},n_{-1}$ in the coefficients $%
C_{k,n_{1},n_{0},n_{-1}}.$
$%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{$\ M=9$} & \multicolumn{2}{|l|}{$\ \ \ M=8$} &
\multicolumn{2}{|l|}{$\ \ \ M=7$} & \multicolumn{2}{|l|}{$\ \ \ M=6$} &
\multicolumn{2}{|l|}{$\ \ \ M=5$} \\ \hline
$C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$
\\ \hline
1 & 9, 1, 0 & 0.2294 & 9, 0, 1 & 0.3794 & 8, 1, 1 & 0.0964 & 8, 0, 2 & 0.2155
& 7, 1, 2 \\ \hline
& & 0.9733 & 8, 2, 0 & 0.9177 & 7, 3, 0 & 0.5452 & 7, 2, 1 & 0.6584 & 6, 3,
1 \\ \hline
& & & & & & 0.8328 & 6, 4, 0 & 0.7212 & 5, 5, 0 \\ \hline
\end{tabular}%
$
$%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{$\ \ \ M=4$} & \multicolumn{2}{|l|}{$\ \ \ M=3$} &
\multicolumn{2}{|l|}{$\ \ \ M=2$} & \multicolumn{2}{|l|}{$\ \ \ M=1$} &
\multicolumn{2}{|l|}{$\ \ \ M=0$} \\ \hline
$C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$ & $C$ & $n\_[1]{},n\_[0]{},n\_[-1]{}$
\\ \hline
0.0556 & 7, 0, 3 & 0.1472 & 6, 1, 3 & 0.0408 & 6, 0, 4 & 0.1225 & 5, 1, 4 &
0.0369 & 5, 0, 5 \\ \hline
0.3606 & 6, 2, 2 & 0.5100 & 5, 3, 2 & 0.2829 & 5, 2, 3 & 0.4473 & 4, 3, 3 &
0.2611 & 4, 2, 4 \\ \hline
0.7212 & 5, 4, 1 & 0.7212 & 4, 5, 1 & 0.6325 & 4, 4, 2 & 0.6929 & 3, 5, 2 &
0.6031 & 3, 4, 3 \\ \hline
0.5889 & 4, 6, 0 & 0.4451 & 3, 7, 0 & 0.6533 & 3, 6, 1 & 0.5238 & 2, 7, 1 &
0.6607 & 2, 6, 2 \\ \hline
& & & & 0.3024 & 2, 8, 0 & 0.1746 & 1, 9, 0 & 0.3531 & 1, 8, 1 \\ \hline
& & & & & & & & 0.0744 & 0, 10, 0 \\ \hline
\end{tabular}%
$
For the case of negative $M$, we need only to exchange the values between $%
n_{1}$ and $n_{-1}$. For instance, when $J=10$, $M=-1$, in terms of the above table, we can conveniently construct the Dicke state $|10,-1\rangle $ as $$|10,-1\rangle =0.1225|4,1,5\rangle +0.4473|3,3,4\rangle +0.6929|2,5,3\rangle
+0.5238|1,7,2\rangle +0.1746|0,9,1\rangle . \label{eq11'}$$
\[sec: C of Dicke copy(2)\]
Dicke states of identical spin-$3/2$ particles
----------------------------------------------
For the case of spin-$3/2$, the states have the form similar to the case of spin-$1$, which reads $$|J,M\rangle =\sum_{k=k_{0}}^{\max
}C_{k,n_{1},n_{2},n_{3},n_{4}}|n_{1},n_{2},n_{3},n_{4}\rangle , \label{eq12}$$where $C_{k,n_{1},n_{2},n_{3},n_{4}}$ are the coefficients with $n_{1}$, $%
n_{2}$, $n_{3}$ and $n_{4}$ denoting the occupation numbers of particles in states $|\frac{3}{2}\rangle $, $|\frac{1}{2}\rangle $, $|-\frac{1}{2}\rangle
$and $|-\frac{3}{2}\rangle $ respectively, and $k$ related to $n_{2}$ and $%
n_{3}$. The basis $\{|n_{1},n_{2},n_{3},n_{4}\rangle \}$ has the form as
$$|n_{1},n_{2},n_{3},n_{4}\rangle =\sqrt{\frac{n_{1}!n_{2}!n_{3}!n_{4}!}{N!}}%
\underset{P}{\sum }P(\underset{n_{1}}{|\underbrace{\frac{3}{2}\cdots \frac{3%
}{2}\rangle }}\underset{n2}{|\underbrace{\frac{1}{2}\cdots \frac{1}{2}%
\rangle }}\underset{n3}{|\underbrace{-\frac{1}{2}\cdots -\frac{1}{2}\rangle }%
}\underset{n_{4}}{|\underbrace{-\frac{3}{2}\cdots -\frac{3}{2}}}\rangle ).
\label{eq15}$$
With some calculations, we also derive $$k_{0}=\frac{1}{2}\{\frac{1}{2}(\alpha _{1}+|\alpha _{1}|)+\frac{1}{2}%
[1-(-1)^{\frac{1}{2}(\alpha _{1}+|\alpha _{1}|)}]\}, \label{eq17}$$with $\alpha _{1}=\frac{N}{2}-|M|$, and $$\begin{aligned}
\max =& \frac{1}{2}(J-|M|-\min ), \notag \\
\min =& \frac{1}{2}[(-1)^{J-|M|+1}+1]. \label{eq16}\end{aligned}$$First, the two basic constraint equations is as follows $$\begin{aligned}
n_{1}+n_{2}+n_{3}+n_{4}=& N, \notag \\
3n_{1}+n_{2}-n_{3}-3n_{4}=& 2M. \label{eq18}\end{aligned}$$The third constraint equation is found to be $$n_{2}-n_{3}=(-1)^{\frac{|M|-M}{2|M|}}\gamma _{_{1}}, \label{eq19}$$with $\gamma _{_{1}}=J-|M|-3k$. For specific value of $k$, we also find the fourth constraint equation as $$n_{2}+n_{3}=|\gamma _{_{1}}|-2(k_{1}-1), \label{eq20}$$with $$m_{k}=\frac{1}{2}[\frac{1}{2}(\beta _{1}-|\beta _{1}|)+k+1+|\frac{1}{2}%
(\beta _{1}-|\beta _{1}|)+k+1|]+\frac{1}{2}(\gamma _{1}+|\gamma _{1}|)
\label{eq21}$$and $\beta _{1}=k-\frac{1}{2}(\alpha _{1}+|\alpha _{1}|)$. Then, we obtain a set of constraint equations $$\underset{k=k_{0}}{\overset{\max }{\sum }}\overset{m_{k}}{\underset{k_{1}=1}{%
\sum }}\left\{
\begin{array}{c}
n_{1}+n_{2}+n_{3}+n_{4}=N \\
3n_{1}+n_{2}-n_{3}-3n_{4}=2M \\
n_{2}-n_{3}=(-1)^{\frac{|M|-M}{2|M|}}\gamma _{_{1}} \\
n_{2}+n_{3}=|\gamma _{_{1}}|-2(k_{1}-1)%
\end{array}%
\right. \label{eq22}$$with $k_{0}\leq k\leq \max $ and $1\leq k_{1}\leq m_{k}$. The number of states to form a complete basis is $\underset{k=k_{0}}{\overset{\max }{\sum }%
}m_{k}$ and the normalized coefficients are $$C_{k,n_{1},n_{2},n_{3},n_{4}}=\frac{(J-|M|)!}{3^{-(n_{2}+n_{3})/2}}\sqrt{%
\frac{N!}{n_{1}!n_{2}!n_{3}!n_{4!}}}\overset{J-|M|}{\underset{l=1}{\prod }}%
\frac{1}{\sqrt{(3N-l+1)l}}. \label{eq23}$$Again, for an illustration, we consider the case of particle number $N=6$, and list all these parameters in the Dicke states for different values of $M$ in Table \[T33\] and \[T44\]
$%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{$\ \ M=9$} & \multicolumn{2}{|l|}{$\ \ M=8$} &
\multicolumn{2}{|l|}{$\ \ \ M=7$} & \multicolumn{2}{|l|}{$\ \ \ \ M=6$} &
\multicolumn{2}{|l|}{$\ \ \ \ M=5$} \\ \hline
$C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ \\ \hline
1 & 6, 0, 0, 0 & 1 & 5, 1, 0, 0 & 0.9393 & 4, 2, 0, 0 & 0.8135 & 3, 3, 0, 0
& 0.6301 & 2, 4, 0, 0 \\ \hline
& & & & 0.3430 & 5, 0, 1, 0 & 0.0857 & 5, 0, 0, 1 & 0.1715 & 4, 1, 0, 1
\\ \hline
& & & & & & 0.5752 & 4,1,1,0 & 0.2100 & 4,0,2,0 \\ \hline
& & & & & & & & 0.7276 & 3, 2, 1, 0 \\ \hline
\end{tabular}%
$
$%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{$\ \ \ \ M=4$} & \multicolumn{2}{|l|}{$\ \ \ \ M=3$} &
\multicolumn{2}{|l|}{$\ \ \ \ M=2$} & \multicolumn{2}{|l|}{$\ \ \ \ M=1$} &
\multicolumn{2}{|l|}{$\ \ \ \ M=0$} \\ \hline
$C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{}$ \\ \hline
0.4125 & 1, 5, 0, 0 & 0.1982 & 0, 6, 0, 0 & 0.2763 & 1, 4, 0, 1 & 0.1825 &
0, 5, 0, 1 & 0.1825 & 1, 3, 0, 2 \\ \hline
0.2510 & 3, 2, 0, 1 & 0.2954 & 2, 3, 0, 1 & 0.0752 & 3, 1, 0, 2 & 0.1361 &
2, 2, 0, 2 & 0.0203 & 3, 0, 0, 3 \\ \hline
0.1025 & 4, 0, 1, 1 & 0.0284 & 4, 0, 0, 2 & 0.1303 & 3, 0, 2, 1 & 0.0641 &
3, 0, 1, 2 & 0.3872 & 0, 4, 1, 1 \\ \hline
0.7531 & 2, 3, 1, 0 & 0.1706 & 3, 0, 3, 0 & 0.3707 & 0, 5, 1, 0 & 0.1666 &
2, 0, 4, 0 & 0.1825 & 2, 0, 3, 1 \\ \hline
0.4348 & 3, 1, 2, 0 & 0.6267 & 1, 4, 1, 0 & 0.3908 & 2, 2, 1, 1 & 0.4713 &
1, 3, 1, 1 & 0.1825 & 2, 1, 1, 2 \\ \hline
& & 0.2412 & 3, 1, 1, 1 & 0.3908 & 2, 1, 3, 0 & 0.3333 & 2, 1, 2, 1 & 0.3872
& 1, 1, 4, 0 \\ \hline
& & 0.6267 & 2, 2, 2, 0 & 0.6769 & 1, 3, 2, 0 & 0.4999 & 0, 4, 2, 0 & 0.5476
& 1, 2, 2, 1 \\ \hline
& & & & & & 0.5772 & 1, 2, 3, 0 & 0.5476 & 0, 3, 3, 0 \\ \hline
\end{tabular}%
$
For negative values of $M$, one may perform the exchanges $%
n_{1}\rightleftarrows n_{4}$, $n_{2}\rightleftarrows n_{3}$ in the case of positive $M$. For instance, when $J=6$, $M=-1$, using the results of above table, we obtain
$$\begin{aligned}
|6,-1\rangle =& 0.1825|1,0,5,0\rangle +0.1361|2,0,2,2\rangle
+0.0641|2,1,0,3\rangle +0.1666|0,4,0,2\rangle +0.4713|1,1,3,1\rangle \notag
\\
\quad&+0.3333|1,2,1,2\rangle +0.4999|0,2,4,0\rangle +0.5772|0,3,2,1\rangle.
\label{eq24}\end{aligned}$$
\[sec: C of Dicke copy(3)\]
Dicke states of identical spin-2 particles
------------------------------------------
For the case of spin-$2$, the Dicke states in the number representation are given by
$$|J,M\rangle =\sum_{k=k_{0}}^{\max
}C_{k,n_{1},n_{2},n_{3},n_{4},n_{5}}|n_{1},n_{2},n_{3},n_{4},n_{5}\rangle ,
\label{eq25}$$
with $J=2N$ and the number states $$|n_{1},n_{2},n_{3},n_{4},n_{5}\rangle =\sqrt{\frac{%
n_{1}!n_{2}!n_{3}!n_{4}!n_{5}!}{N!}}\underset{P}{\sum }P(\underset{n_{1}}{|%
\underbrace{2\cdots 2\rangle }}\underset{n2}{\underbrace{|1\cdots 1\rangle }}%
\underset{n_{3}}{\underbrace{|0\cdots 0}}\underset{n4}{\underbrace{|-1\cdots
-1\rangle }}\underset{n5}{\underbrace{|-2\cdots -2}}\rangle ), \label{eq26}$$where $n_{1}$, $n_{2}$, $n_{3}$, $n_{4}$ and $n_{5}$ denote the numbers of particles in states $|2\rangle $, $|1\rangle $, $|0\rangle $, $|-1\rangle $ and $|-2\rangle $ respectively, $k$ is related to $n_{2}$ and $n_{4}$, and $%
k_{0}$ and $\max $ are given by $$k_{0}=\left\{
\begin{array}{c}
\frac{1}{3}(\alpha _{2}+|\alpha _{2}|),\text{ if }\frac{1}{2}(\alpha
_{2}+|\alpha _{2}|)=3n^{\prime }, \\
\frac{1}{3}(\alpha _{2}+|\alpha _{2}|+1),\text{ if }\frac{1}{2}(\alpha
_{2}+|\alpha _{2}|)=3n^{\prime }+1, \\
\frac{1}{3}(\alpha _{2}+|\alpha _{2}|+2),\text{ if }\frac{1}{2}(\alpha
_{2}+|\alpha _{2}|)=3n^{\prime }+2,%
\end{array}%
\right. \label{eq27}$$$$\max =\left\{
\begin{array}{c}
\frac{2}{3}(2N-|M|),\text{ if }2N-|M|=3n^{\prime \prime }, \\
\frac{2}{3}(2N-|M|-\frac{1}{2}),\text{ if }2N-|M|=3n^{\prime \prime }-1, \\
\frac{2}{3}(2N-|M|-1),\text{ if }2N-|M|=3n^{\prime \prime }-2.%
\end{array}%
\right. \label{eq28}$$Here, $n^{\prime }$ and $n^{\prime \prime }$ are integers and $\alpha
_{2}=N-|M|$. It is easy to verify that $$|J,J\rangle =|N,0,0,0,0\rangle , \notag \\
|J,-J\rangle =|0,0,0,0,N\rangle , \label{eq29}$$To specify $n_{1}$, $n_{2}$, $n_{3}$, $n_{4}$ and $n_{5}$ as well as $%
C_{k,n_{1},n_{2},n_{3},n_{4},n_{5}}$, five equations are needed. First, the two basic equations are $$\begin{aligned}
n_{1}+n_{2}+n_{3}+n_{4}+n5=& N, \notag \\
2n_{1}+n_{2}-n_{4}-2n_{5}=& 2M. \label{eq30}\end{aligned}$$We derive the other three equations as $$\begin{aligned}
&n_{2}-n_{4}= (-1)^{\frac{|M|-M}{2|M|}}\gamma _{_{2}},\text{ }k_{0}\leq
k\leq \max ,\quad \gamma _{_{2}}=J-|M|-2k, \notag \label{eq31} \\
&n_{2}+n_{4}= \gamma _{_{2}}+2(k_{1}-1),\quad 1\leq k_{1}\leq m_{k}, \notag
\\
&n_{3}= 2(k_{2}+1)+\frac{1}{2}[1-(-1)^{k}]-2,\quad 0\leq k_{2}\leq
m_{k}-k_{1},\end{aligned}$$where $$m_{k}=\frac{1}{2}\{\frac{1}{2}(\beta _{2}+|\beta _{2}|)+\frac{2k+3+(-1)^{k}}{%
4}+|\frac{1}{2}(\beta _{2}+|\beta _{2}|)+\frac{2k+3+(-1)^{k}}{4}|\}+\frac{1}{%
2}(\gamma _{2}-|\gamma _{2}|), \label{eq33}$$with $\beta _{2}=k-\frac{1}{2}(\alpha _{2}+|\alpha _{2}|)$. Note that $k$, $%
k_{1}$ and $k_{2}$ are all nonnegative integers. Besides, we find that the number of states to form a complete basis is $\overset{\max }{\underset{%
k=k_{0}}{\sum }}\frac{1}{2}m_{k}(m_{k}+1)$ and the normalized coefficients read as $$C_{k,n_{1},n_{2},n_{3},n_{4},n_{5}}=(\frac{3}{2})^{n_{3}/2}\frac{(J-|M|)!}{%
3^{-(n_{2}+n_{3}+n_{4})/2}}\sqrt{\frac{N!}{n_{1}!n_{2}!n_{3}!n_{4!}!n_{5}}}%
\overset{J-|M|}{\underset{l=1}{\prod }}\frac{1}{\sqrt{(4N-l+1)l}}.
\label{eq34}$$
Here, we focus on the case $N=5$ for example and derive all the ($n_{1},$ $%
n_{2},$ $n_{3},$ $n_{4},$ $n_{5})$ and $C_{k,n_{1},n_{2},n_{3},n_{4},n_{5}}$ for different values of $M$ as follows. In terms of the table above, all the five-particle Dicke states can be obtained.
$%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{$\ \ \ \ M=9$} & \multicolumn{2}{|l|}{$\ \ \ \ \ \ M=8$}
& \multicolumn{2}{|l|}{$\ \ \ \ \ \ \ \ M=7$} & \multicolumn{2}{|l|}{$\ \ \
\ \ \ \ M=6$} & \multicolumn{2}{|l|}{$\ \ \ \ \ \ \ \ M=5$} \\ \hline
$C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$
& $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ \\
\hline
1 & 4, 1, 0, 0, 0 & 0.9177 & 3, 2, 0, 0, 0 & 0.7493 & 2, 3, 0, 0, 0 & 0.5140
& 1, 4, 0, 0, 0 & 0.2570 & 0, 5, 0, 0, 0 \\ \hline
& & 0.3974 & 4, 0, 1, 0, 0 & 0.6489 & 3, 1, 1, 0, 0 & 0.7710 & 2, 2, 1, 0, 0
& 0.7038 & 1, 3, 1, 0, 0 \\ \hline
& & & & 0.1325 & 4, 0, 0, 1, 0 & 0.0321 & 4, 0, 0, 0, 1 & 0.0718 & 3, 1,
0, 0, 1 \\ \hline
& & & & & & 0.2726 & 3, 0, 2, 0, 0 & 0.5279 & 2, 1, 2, 0, 0 \\ \hline
& & & & & & 0.2570 & 3, 1, 0, 1, 0 & 0.3519 & 2, 2, 0, 1, 0 \\ \hline
& & & & & & & & 0.1760 & 3, 1, 0, 1, 0 \\ \hline
\end{tabular}%
$
$%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|l|}{$\ \ \ \ M=4$} & \multicolumn{2}{|l|}{$\ \ \ \ \ \ M=3$}
& \multicolumn{2}{|l|}{$\ \ \ \ \ \ \ \ M=2$} & \multicolumn{2}{|l|}{$\ \ \
\ \ \ \ M=1$} & \multicolumn{2}{|l|}{$\ \ \ \ \ \ \ \ M=0$} \\ \hline
$C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$
& $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ & $C$ & $n\_[1]{},n\_[2]{},n\_[3]{},n\_[4]{},n\_[5]{}$ \\
\hline
0.1113 & 0, 4, 1, 0, 0 & 0.1285 & 1, 3, 0, 0, 1 & 0.1008 & 0, 4, 0, 0, 1 &
0.2138 & 0, 3, 1, 0, 1 & 0.0510 & 1, 2, 0, 0, 2 \\ \hline
0.6677 & 2, 2, 0, 0, 1 & 0.5452 & 0, 3, 2, 0, 0 & 0.2138 & 1, 2, 1, 0, 1 &
0.0267 & 2, 1, 0, 0, 2 & 0.3058 & 0, 2, 2, 0, 1 \\ \hline
0.3634 & 1, 2, 2, 0, 0 & 0.2570 & 0, 4, 0, 1, 0 & 0.5238 & 0, 2, 3, 0, 0 &
0.2268 & 1, 1, 2, 0, 1 & 0.1665 & 0, 3, 0, 1, 1 \\ \hline
0.0556 & 1, 3, 0, 1, 0 & 0.1363 & 2, 1, 1, 0, 1 & 0.4938 & 0, 3, 1, 1, 0 &
0.3928 & 0, 1, 4, 0, 0 & 0.0312 & 2, 0, 1, 0, 2 \\ \hline
0.2361 & 3, 0, 1, 0, 1 & 0.4721 & 1, 1, 3, 0, 0 & 0.0089 & 3, 0, 0, 0, 2 &
0.1512 & 1, 2, 0, 1, 1 & 0.1529 & 1, 0, 3, 0, 1 \\ \hline
0.3855 & 2, 0, 3, 0, 0 & 0.5452 & 1, 2, 1, 1, 0 & 0.0926 & 2, 0, 2, 0, 1 &
0.6415 & 0, 2, 2, 1, 0 & 0.2052 & 0, 0, 5, 0, 0 \\ \hline
0.4451 & 2, 1, 1, 1, 0 & 0.0321 & 3, 0, 0, 1, 1 & 0.2268 & 1, 0, 4, 0, 0 &
0.2469 & 0, 3, 0, 2, 0 & 0.2497 & 1, 1, 1, 1, 1 \\ \hline
0.06423 & 3, 0, 0, 2, 0 & 0.2361 & 2, 0, 2, 1, 0 & 0.1008 & 0, 4, 0, 0, 1 &
0.0926 & 2, 0, 1, 1, 1 & 0.6116 & 0, 1, 3, 1, 0 \\ \hline
& & 0.1574 & 2, 1, 0, 2, 0 & 0.0873 & 2, 1, 0, 1, 1 & 0.3208 & 1, 0, 3, 1, 0
& 0.4994 & 0, 2, 1, 2, 0 \\ \hline
& & & & 0.5238 & 1, 1, 2, 1, 0 & 0.3704 & 1, 1, 1, 2, 0 & 0.0510 & 2, 0,
0, 2, 1 \\ \hline
& & & & 0.2469 & 1, 2, 0, 2, 0 & 0.0617 & 2, 0, 0, 3, 0 & 0.3058 & 1, 0,
2, 2, 0 \\ \hline
& & & & 0.1512 & 2, 0, 1, 2, 0 & & & 0.1665 & 1, 1, 0, 3, 0 \\ \hline
\end{tabular}%
$
According to the expression of the superposition coefficients $C$ of spin-$1$, $3/2$ and $2$, it is not difficult to conclude that the coefficients for spin-$1/2$ case can be expressed as $$C_{k,n_{1},n_{2}}=\sqrt{\frac{N!}{n_{1}!n_{2}!}}(J-|M|)!\overset{J-|M|}{%
\underset{l=1}{\prod }}\frac{1}{\sqrt{(N-l+1)l}}, \label{eq35}$$By solving Eq. (\[eq4\]) to obtain $$J-|M|=\left\{
\begin{array}{c}
n_{1}\text{ if }M>0, \\
n_{2}\text{ if }M<0,%
\end{array}%
\right. \label{eq36}$$and substituting Eq. (\[eq36\]) into Eq. (\[eq35\]), the coefficients are simplified as$$C_{k,n_{1},n_{2}}\equiv 1, \label{eq37}$$which agrees with the setup in Eq. (\[eq1\]).
Anti-symmetric states in high spin systems
==========================================
As a natural byproduct, we proceed to discuss the anti-symmetric states in these high spin systems. It is well known that any two particles should not be in the same state in an anti-symmetric state. Therefore, we can conclude that anti-symmetric states exist only for particle number less than $2s+1$ for spin-$s$ systems. For the case of electrons of spin-$1/2$ , the particle number of collective anti-symmetric spin states is only $2$. Similarly, one can check that the upper limits for particle numbers of spin $1$, $3/2$ and $%
2$ are $3$, $4$ and $5$, respectively. Based on this fact, the number of anti-symmetric states for high spin-$s$ systems is $$C(2s+1,2s+1)+C(2s+1,2s)+\cdots +C(2s+1,2)=2^{2s+1}-(2s+2), \label{tas}$$where $C(n,k)$ means the $k$-combinations of $n$.
For convenience, we mark the states of single particle as $|\alpha \rangle ,$ $|\beta \rangle \cdots \in \{|-s\rangle ,\cdots ,|s\rangle \}$. Then, for the case of two particles, the $C_{2s+1}^{2}$ elementary anti-symmetric states can be write as $$|\psi (i,j)\rangle _{AS}=\sqrt{\frac{1}{2}}\underset{P}{\sum }\delta
_{P}P(|\alpha \rangle |\beta \rangle ),\text{ }|\alpha \rangle \neq |\beta
\rangle , \label{an}$$where the symbol $P$ denotes all possible permutations and $\delta _{P}$ (with the initial value +1) changes its sign between $+1$ and $-1$ after every permutation. Due to the fact that the state $|J=2s-1,2s-1\rangle $ is anti-symmetric, so all states belonging to the subspace with total spin $%
J=2s-1$ are anti-symmetric. It is obvious that in the two-qubit system there is only a elementary anti-symmetric state
$$|J=0,M=0\rangle =\sqrt{\frac{1}{2}}(|\uparrow \rangle |\downarrow \rangle
-|\downarrow \rangle |\uparrow \rangle ),$$
and in two-qudit system with $J=1$ there is three elementary anti-symmetric states as
$$\begin{aligned}
|J &=&1,1\rangle =\sqrt{\frac{1}{2}}(|\uparrow \rangle |0\rangle -|0\rangle
|\uparrow \rangle ), \notag \\
|J &=&1,0\rangle =\sqrt{\frac{1}{2}}(|\uparrow \rangle |\downarrow \rangle
-|\downarrow \rangle |\uparrow \rangle ), \notag \\
|J &=&1,-1\rangle =\sqrt{\frac{1}{2}}(|0\rangle |\downarrow \rangle
-|\downarrow \rangle |0\rangle ),\end{aligned}$$
For the case with $s>1,$ in the subspace with total spin $J=2s-1,$ the four states $$\begin{aligned}
|J &=&2s-1,\pm (2s-1)\rangle =\sqrt{\frac{1}{2}}(|\pm s\rangle |\pm s\mp
1\rangle -|\pm s\mp 1\rangle |\pm s\rangle ), \notag \\
|J &=&2s-1,\pm (2s-2)\rangle =\sqrt{\frac{1}{2}}(|\pm s\rangle |\pm s\mp
2\rangle -|\pm s\mp 2\rangle |\pm s\rangle ),\end{aligned}$$ are elementary anti-symmetric states given by Eq. (\[an\]), and the rest $%
4s-5$ states are linear superposition of those elementary anti-symmetric states. For the case including three particles, there is $C_{2s+1}^{3}$ elementary anti-symmetric states$$|\psi (i,j,k)\rangle _{AS}=\sqrt{\frac{1}{3!}}\underset{P}{\sum }\delta
_{P}P(|\alpha \rangle |\beta \rangle |\gamma \rangle ),\text{ }|\alpha
\rangle \neq |\beta \rangle \neq |\gamma \rangle .$$
To conclude, for the $2s+1$ particles, there is only one elementary anti-symmetric state as
$$|\psi \lbrack 1,2\cdots (2s+1)\text{th}]\rangle _{AS}=\sqrt{\frac{1}{(2s+1)!}%
}\underset{P}{\sum }\delta _{P}P(|-s\rangle \otimes |-s+1\rangle \cdots
\otimes \cdots |s-1\rangle \otimes |s\rangle ),$$
which is just the state $|J=0,M=0\rangle $. In particular, for the case of spin-1, the anti-symmetric state can be written as$$|\psi (1,2,3)\rangle _{AS}=\sqrt{\frac{1}{3!}}(|\uparrow \rangle |0\rangle
|\downarrow \rangle -|\uparrow \rangle |\downarrow \rangle |0\rangle
+|\downarrow \rangle |0\rangle |\uparrow \rangle -|\downarrow \rangle
|\uparrow \rangle |0\rangle -|0\rangle |\uparrow \rangle |\downarrow \rangle
+|0\rangle |\downarrow \rangle |\uparrow \rangle ).$$
For instance, the particle number may be $2$, $3$, $4$ and $5$ for $s=2$, and one can construct all the anti-symmetric states as $$\begin{aligned}
|\psi (i,j)\rangle _{AS}=& \sqrt{\frac{1}{2!}}\underset{P}{\sum }\delta
_{P}P(|\alpha \rangle |\beta \rangle ),\text{ }|\alpha \rangle \neq |\beta
\rangle , \notag \\
|\psi (i,j,k)\rangle _{AS}=& \sqrt{\frac{1}{3!}}\underset{P}{\sum }\delta
_{P}P(|\alpha \rangle |\beta \rangle |\gamma \rangle ),\text{ }|\alpha
\rangle \neq |\beta \rangle \neq |\gamma \rangle , \notag \\
|\psi (i,j,k,l)\rangle _{AS}=& \sqrt{\frac{1}{4!}}\underset{P}{\sum }\delta
_{P}P(|\alpha \rangle |\beta \rangle |\gamma \rangle |\eta \rangle ),|\alpha
\rangle \neq |\beta \rangle \neq |\gamma \rangle \neq |\eta \rangle , \notag
\\
|\psi (i,j,k,l,m)\rangle _{AS}=& \sqrt{\frac{1}{5!}}\underset{P}{\sum }%
\delta _{P}P(|2\rangle |1\rangle |0\rangle |-1\rangle |-2\rangle ),
\label{eq38}\end{aligned}$$where $|\alpha \rangle $, $|\beta \rangle $, $|\gamma \rangle $, $|\eta
\rangle \in $ $\{|2\rangle ,$ $|1\rangle ,$ $|0\rangle ,$ $|-1\rangle $, $%
|-2\rangle \},$ and $|2\rangle ,$ $|1\rangle ,$ $|0\rangle ,$ $|-1\rangle $ and $|-2\rangle $ are the eigenstates of single spin magnetic quantum number $m_{s}$ with eigenvalues $2$, $1$, $0$, $-1$ and $-2$, respectively. According Eq. (\[tas\]), there totally exist $26$ different anti-symmetric states for spin-$2$ systems. Generally speaking, due to the Pauli exclusion principle, the number of anti-symmetric states compared to that of all the Dicke states is very limited.
Entanglement of two qudits in system with many particles
========================================================
In this section, we study the entanglement for the case of two qudits. The entanglement criteria proposed by Peres-Horodecki [@P1; @P2] is adopted. For states with certain symmetries in the high spin systems, this criterion is good enough to measure the entanglement. However, the states discussed by us are beyond this requirement. In order to quantify the entanglement, Vidal and Werner proposed a entanglement measure termed as negativity [@Wi]. The first thing need to do is that one obtains the density matrix $\rho
_{ij} $ of two qudits in the basis {$|\uparrow \downarrow \rangle ,$ $%
|00\rangle ,$ $|\downarrow \uparrow \rangle ,$ $|\uparrow \rangle |0\rangle ,
$ $|0\rangle |\uparrow \rangle ,$ $|0\rangle |\downarrow \rangle ,$ $%
|\downarrow \rangle |0\rangle |,$ $|\uparrow \rangle |\uparrow \rangle ,$ $%
|\downarrow \rangle |\downarrow \rangle $}. Next, one can perform partial transpose (PT) to $\rho _{ij}$, and obtain the matrix $\rho _{ij}^{T}$ in the basis spanned by {$|\uparrow \rangle |\uparrow \rangle ,$ $|0\rangle
|0\rangle ,$ $|\downarrow \rangle |\downarrow \rangle ,$ $|\uparrow \rangle
|0\rangle ,$ $|0\rangle |\downarrow \rangle ,$ $|0\rangle |\uparrow \rangle
, $ $|\downarrow \rangle |0\rangle ,$ $|\uparrow \rangle |\downarrow \rangle
,$ $|\downarrow \rangle |\uparrow \rangle $}. The negativity is then defined as $$\mathcal{N}(\rho _{ij})=\underset{i}{\sum }|\lambda _{i}|, \label{eq39}$$where $\lambda _{i}$ are the negative eigenvalues of $\rho _{ij}^{T}$. If $%
\mathcal{N}(\rho _{ij})>0$, then the two particles stay in the entangled state. However, in the $9$ basis, the density of two particles generally has $81$ elements. Different from the case of many spin-$1/2$ particles, these elements can not be represented by the expectation value of the collective operators of system. This leads to certain difficulties in the calculation of the entanglement. However, in the following, we will show that the number of effective elements will be greatly reduced for some special states.
Entanglement of specific states in the system with two spin-$1$ particles
-------------------------------------------------------------------------
Let us begin with the system with two spin-1 particles. We discuss the entanglement of two particle with spin-$1$. The first state considered is the generalized symmetric Bell state of two qudits, which is an important state for the qudit teleportation scheme [@Sych; @Be]. Its form is given by $$|B_{G}\rangle =\sqrt{\frac{1}{3}}(|\uparrow \rangle |\uparrow \rangle
+|0\rangle |0\rangle +|\downarrow \rangle |\downarrow \rangle ),
\label{eq40}$$where $s_{z}|\uparrow \rangle =|\uparrow \rangle $, $s_{z}|0\rangle =0$, and $s_{z}|\downarrow \rangle =|\downarrow \rangle $. It is easy to check that this state is a maximally entangled state and the negativity equal 1 in this state. In order to compare with the state presented above, we study the negativity of another state with a generalized form $$|\psi _{1}\rangle =\sqrt{\frac{1}{3}}[|\uparrow \rangle |\uparrow \rangle
+c_{1}\frac{1}{\sqrt{2}}(|\uparrow \rangle |\downarrow \rangle +|\downarrow
\rangle |\uparrow \rangle )+c_{2}|0\rangle |0\rangle +|\downarrow \rangle
|\downarrow \rangle ], \label{eq41}$$where coefficients $c_{1}$ and $c_{2}$ satisfy the relation $%
|c_{1}|^{2}+|c_{2}|^{2}=1$. After the PT, we can give the matrix density of two particles in a block diagonal form as $$\rho _{ij}^{T}=\mathrm{diag}(C_{5\times 5},D_{4\times 4}), \label{eq411}$$with $C$ and $D$ given by $$C=\left(
\begin{array}{ccccc}
\frac{1}{3} & 0 & \frac{|c_{1}|^{2}}{6} & \frac{c_{1}^{\ast }}{3\sqrt{2}} &
\frac{c_{1}}{3\sqrt{2}} \\
0 & \frac{|c_{2}|^{2}}{3} & 0 & 0 & 0 \\
\frac{|c_{1}|^{2}}{6} & 0 & \frac{1}{3} & \frac{c_{1}}{3\sqrt{2}} & \frac{%
c_{1}^{\ast }}{3\sqrt{2}} \\
\frac{c_{1}}{3\sqrt{2}} & 0 & \frac{c_{1}^{\ast }}{3\sqrt{2}} & \frac{%
|c_{1}|^{2}}{6} & \frac{1}{3} \\
\frac{c_{1}^{\ast }}{3\sqrt{2}} & 0 & \frac{c_{1}}{3\sqrt{2}} & \frac{1}{3}
& \frac{|c_{1}|^{2}}{6}%
\end{array}%
\right) ,\quad D=\left(
\begin{array}{cccc}
0 & \frac{c_{2}c_{1}^{\ast }}{3\sqrt{2}} & \frac{c_{2}}{3} & 0 \\
\frac{c_{1}c_{2}^{\ast }}{3\sqrt{2}} & 0 & 0 & \frac{c_{2}^{\ast }}{3} \\
\frac{c_{2}^{\ast }}{3} & 0 & 0 & \frac{c_{1}c_{2}^{\ast }}{3\sqrt{2}} \\
0 & \frac{c_{2}}{3} & \frac{c_{2}c_{1}^{\ast }}{3\sqrt{2}} & 0%
\end{array}%
\right) . \label{eq42}$$Specially, for combination ($c_{1}=\sqrt{1/3}$ and $c_{2}$=$\sqrt{2/3}$), the state can be reduced to even spin coherent state of two qudits $$|\psi _{e}\rangle =\sqrt{\frac{1}{3}}(|2,2\rangle +|2,0\rangle +|2,-2\rangle
). \label{eq43}$$The negativity in this state is $0.8221$, where $|2,0\rangle $ is the Dicke state $$|2,0\rangle =\sqrt{\frac{1}{6}}(|\uparrow \rangle |\downarrow \rangle
+|\downarrow \rangle |\uparrow \rangle )+\sqrt{\frac{2}{3}}|0\rangle
|0\rangle , \label{eq44}$$in which state the negativity is $0.833$. It is worth mentioning that this even spin coherent state can be generate by one-axis twisting model or the two-axis counter model with the initial state $|2,-2\rangle $. For another state $$|\psi _{2}\rangle =\frac{1}{2}(|\uparrow \rangle |\downarrow \rangle
+|\downarrow \rangle |\uparrow \rangle )+\frac{1}{\sqrt{2}}|0\rangle
|0\rangle , \label{eq45}$$with $M=0$, we obtain the negativity with value $0.9571.$
There are two interested generalized singlet Bell states which are the elementary states of dimmer states [@Sych; @Chu; @Ar] $$|B_{s\pm }\rangle =\sqrt{\frac{1}{3}}(|\uparrow \rangle |\downarrow \rangle
+|\downarrow \rangle |\uparrow \rangle \pm |0\rangle |0\rangle ).
\label{eq46}$$Also, we obtain that the negativity is $1$. \[sec: C of Dicke copy(5)\]
Entanglement of two qudits in the Dicke states of many spin-1 particles
-----------------------------------------------------------------------
In the nine bases, the reduced density matrix of the two spins in the Dicke states can be written as $$\rho _{ij}=\mathrm{diag}(T_{1},T_{2,}\text{ }T_{3},\text{ }a_{8},\text{ }%
a_{9}) \label{eq47}$$with $T_{1},T_{2}$ and $T_{3}$ $$T_{1}=\left(
\begin{array}{ccc}
a_{1} & c_{1} & b_{3} \\
c_{1}^{\ast } & a_{2} & c_{2} \\
b_{3} & c_{2}^{\ast } & a_{3}%
\end{array}%
\right) ,\text{ }T_{2}=\left(
\begin{array}{cc}
a_{4} & b_{1} \\
b_{1}^{\ast } & a_{5}%
\end{array}%
\right) ,\text{ }T_{3}=\left(
\begin{array}{cc}
a_{6} & b_{2} \\
b_{2}^{\ast } & a_{7}%
\end{array}%
\right) .\text{ } \label{eq48}$$
After the PT, the density matrix $\rho _{ij}$ transfers to $$\rho _{ij}^{T}=\text{diag}(T_{1}^{\prime },T_{2,}^{\prime }\text{ }%
T_{3}^{\prime },\text{ }a_{1},\text{ }a_{3}), \label{eq49}$$with $$T_{1}^{\prime }=\left(
\begin{array}{ccc}
a_{8} & b_{1} & b_{3} \\
b_{1}^{\ast } & a_{2} & b_{2} \\
b_{3} & b_{2}^{\ast } & a_{9}%
\end{array}%
\right) ,\text{ }T_{2}^{\prime }=\left(
\begin{array}{cc}
a_{4} & c_{1} \\
c_{1}^{\ast } & a_{6}%
\end{array}%
\right) ,\text{ }T_{3}^{\prime }=\left(
\begin{array}{cc}
a_{5} & c_{2} \\
c_{2}^{\ast } & a_{7}%
\end{array}%
\right) . \label{eq50}$$Here, we have considered that the elements $a_{1}$ and $a_{3}$ are always positive since they characterize the probability of finding two particles in the states $|\uparrow \rangle |\downarrow \rangle $ and $|\downarrow \rangle
|0\rangle $, which have no relationship with entanglement. In order to obtain the entanglement, we calculated the $17$ useful elements as follows $$\begin{aligned}
a_{2} =&1-2\underset{k}{\sum }|C_{k,n_{1},n_{0},n_{-1}}|^{2}\frac{N-n_{0}}{N%
}+\frac{2}{N(N-1)}\underset{k}{\sum }|C_{k,n_{1},n_{0},n_{-1}}|^{2}%
\{n_{1}n_{-1}+\frac{1}{2}[n_{1}(n_{1}-1)+n_{-1}(n_{-1}-1)]\}, \notag
\label{eq51} \\
a_{3} =&\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}n_{1}n_{-1}, \notag \\
a_{4} =&a_{5}=\frac{1}{2}\underset{k}{\sum }|C_{k,n_{1},n_{0},n_{-1}}|^{2}%
\frac{N-n_{0}}{N}+\frac{M}{2N}-\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}[n_{1}n_{-1}+n_{1}(n_{1}-1)], \notag \\
a_{6} =&a_{7}=\frac{1}{2}\underset{k}{\sum }|C_{k,n_{1},n_{0},n_{-1}}|^{2}%
\frac{N-n_{0}}{N}-\frac{M}{2N}-\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}[n_{1}n_{-1}+n_{-1}(n_{-1}-1)], \notag \\
a_{8} =&\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}n_{1}(n_{1}-1), \notag \\
a_{9} =&\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}n_{-1}(n_{-1}-1), \notag \\
c_{1} =&c_{2}=\frac{1}{N(N-1)}\underset{k}{\sum }C_{k,n_{1},n_{0},n_{-1}}^{%
\ast }C_{k,n_{1}+1,n_{0}-2,n_{-1}+1}\sqrt{n_{0}(n_{0}-1)(n_{1}+1)(n_{-1}+1)},
\notag \\
b_{1} =&\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}n_{0}n_{1}, \notag \\
b_{2} =&\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}n_{0}n_{-1}, \notag \\
b_{3} =&\frac{1}{N(N-1)}\underset{k}{\sum }%
|C_{k,n_{1},n_{0},n_{-1}}|^{2}n_{1}n_{-1}.\end{aligned}$$
![The negativity of the Dicke states $|J,M\rangle $ with positive $M$. The number of particles from top to bottom is $20, 30,...,80$.[]{data-label="f1"}](fig1.eps)
We can obtain the negative eigenvalues of three matrixes by solving the eigenvalue equation. Substituting those eigenvalues into the formula of the negativity, the entanglement can be calculated. Specially, for the case $M=0$, the negativity can be reduced to simpler form given by [@Wang6]. The properties of entanglement in the Dicke states with $20-80$ particles are shown in Fig. \[f1\]. We observe that as $|M|$ decreases, the negativity is a monotone increasing function, and comparing with others states, the state $|J,0\rangle $ possesses the maximal entanglement. This property is different from the concurrence of the Dicke states of multi-particle for the spin-$1/2$ case. In addition, as $|M|$ increases, the maximal value of the negativity decreases. Considering that those Dicke states $|J,M\rangle $ with $|M|<N-1$ are linear combination of different states $%
\{|n_{1},n_{0},n_{-1}\rangle \}$, the states $|J,M\rangle $ with $|M|<N-1$ actually forms a subspace. We can construct the states which are equal probability combination of different states $|n_{1},n_{0},n_{-1}\rangle $ in the subspace mentioned above. The negativities in these states are compared with those of the Dicke states. The consequences show that, with the equal probability combination, there are some advantages in the generation of negativity, specially for the cases $M=0$. Here, we consider $N=30, 80$, and present the negativities of two cases, as shown Fig. \[f2\].
![Comparison of negativity between the Dicke states and the states with equal probability combination of all states $|n_{1},n_{2},n_{3}\rangle $ in the subspace $M$. We take $N=30$ (on the left side) and $N=80$ (on the right side). The solid line and dashed line correspond to the Dicke states.[]{data-label="f2"}](fig2.eps)
Conclusions {#sec:concl}
===========
In summary, we have investigated the construction of Dicke states for high-spin particles based on that of Dicke states for the spin-$1/2$ case. For three high-spin cases (spin-$1$, $3/2$ and $2$) with given particle numbers and spin magnetic quantum numbers, the sets of constraint equations are found to determine all the basis states in the number representation as well as the corresponding normalized superposition coefficients, in terms of which the Dicke states are explicitly expressed in the number representation. As a byproduct, we give a rule to construct all the anti-symmetric states in these high-spin systems and show that the number of anti-symmetric states is rather limited. Finally, in terms of the negativity, the entanglement properties for spin-$1$ cases including specific pure states of two particles and the Dicke states of many particles are discussed. Our results may contribute to the applications of high-spin systems in quantum information science due to the crucial importance of Dicke states.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge fruitful discussions with Professor Yang Ming. This work is supported by the Natural Science Foundation of Anhui Province of China (Grant No. 1408085QA15), the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174), the National Natural Science Foundation of China (Grant No. 11504140), the natural science foundation of Jiangsu province of China (Grant No. BK20140128) and the Fundamental Research Funds for the Central Universities(Grant No. JUSRP51517).
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|
---
abstract: |
A warped extra dimension model predicts an extra scalar particle beyond the Standard Model which is called a radion. Although interactions of the radion are similar to those of the Higgs boson in the Standard Model, a relatively light radion ($\lesssim 100~{\rm
GeV}$) is not severely constrained from the Higgs search experiments at the LHC. In this paper we study discovery potential of the radion at a photon collider as an option of ILC. Owing to the trace anomaly of the energy-momentum tensor, both a production of radion in $\gamma \gamma$ collision and its decay to gluon pair are enhanced sizably. We find that the photon collider has a sensitivity for discovering the radion in low-mass region up to $\Lambda_\phi\sim 3~{\rm TeV}$, where $\Lambda_\phi$ is a scale parameter which suppresses the interactions of radion to the Standard Model particles.
---
OCHA-PP-320\
April 2014\
[ **Production and decay of radion in Randall-Sundrum model at a photon collider** ]{}
Gi-Chol Cho$^a$ and Yoshiko Ohno$^b$
$^a$[*Department of Physics, Ochanomizu University, Tokyo 112-8610, Japan*]{}\
$^b$[*Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan*]{}\
Introduction
============
A warped extra dimension model proposed by Randall and Sundrum (RS) is one of the attractive candidates to solve the gauge hierarchy problem in the Standard Model (SM) naturally [@Randall:1999ee]. The model is given in the five-dimensional space-time where one warped extra dimension is compactified on the orbifold $S^1/Z_2$. The space-time metric is given by $$\begin{aligned}
ds^2 = e^{- 2 k y} \eta_{\mu\nu} dx^\mu dx^\nu - dy^2~, \end{aligned}$$ where $x^\mu (\mu = 0, 1,2, 3$), $y$ and $k$ denote the coordinate of four-dimensional space-time, that of a fifth dimension, and the ${\rm
AdS_5}$ curvature, respectively. The Minkowski metric is $\eta_{\mu\nu}={\rm diag}(+1,-1,-1,-1)$ and $e^{-2ky}$ is called a warp factor. Two 3-branes are located at $y=0$ and $\pi r_c$. They are called the UV brane ($y=0$) and the IR brane ($y=\pi r_c$). In the original RS model all the SM fields are confined on the IR brane and only graviton is allowed to propagate into the bulk. There are some variants of the RS model where some of SM fields propagate in the extra dimension [@Davoudiasl:1999tf; @Davoudiasl:2000wi]. With this setup, the four dimensional Planck mass $M_{\rm pl}$ is expressed by a fundamental parameter $M$ in the five dimensional Einstein-Hilbert action as $$\begin{aligned}
M_{\rm pl}^2=\frac{M^3}{k}(1-e^{-2ky})\label{eq:hr},\end{aligned}$$ As a result, an effective mass parameter in the IR brane is given as $M_{\rm pl} e^{-k \pi r_c}$, and the gauge hierarchy problem could be solved naturally when the distance between UV and IR branes are stabilized by $kr_c \sim 12$. Goldberger and Wise have proposed an attractive mechanism to stabilize the distance between two branes introducing a bulk scalar field which has scalar potentials on both branes [@Goldberger:1999uk]. Minimizing the scalar potentials on the branes, the distance between two branes takes appropriate value ($kr_c\sim 12$) without fine-tuning of the parameters in the scalar potentials. In four dimensional effective theory of the original RS model, there are two new particles beyond the Standard Model. One is a spin-2 graviton (and its Kaluza-Klein excitations) and the other is a scalar-field radion $\phi$ which is a metric fluctuation along the extra dimension. The radion acquires the mass of the order of the electroweak scale due to the Goldberger-Wise mechanism and it could be a lightest extra particle in the RS model [@Goldberger:1999uk; @Kribs:2006mq]. The radion, therefore, is expected to be the first signature of warped extra dimension models in direct search experiments such as the LHC. Phenomenology of the radion can be characterized by two parameters, a radion mass $m_\phi$ and a scale parameter $\Lambda_\phi$. The search experiments of the Higgs boson at the LHC give stringent constraints on the parameters of the radion. As will be shown later, the radion couples to the trace part of the energy-momentum tensor of the SM. Thus it is known that the interactions of the radion to the SM particles such as electroweak gauge bosons ($W^\pm,Z$) and fermions are similar to those of the Higgs boson, except for the scale parameters in the couplings. On the other hand, the interactions of radion to photons and gluons have additional source from the trace anomaly of the energy-momentum tensor in addition to the 1-loop contributions from $W$ boson and/or fermions as in the SM. Furthermore, there could be a mixing between the radion and the SM Higgs boson through the scalar-curvature mixing term in the four-dimensional effective action [@Giudice:2000av; @Csaki:2000zn]. According to these characteristic features of the radion interactions, there have been studied the phenomenological aspects of the radion in various colliders [@Dominici:2002jv; @Desai:2013pga; @Battaglia:2003gb; @Cheung:2003ze; @Abbiendi:2004vx; @Bae:2001id; @Mahanta:2000ci]. It has been reported that the Higgs boson mass $m_h$ is $126.0\pm 0.4{\rm (stat.)}\pm 0.4{\rm
(syst.)}~{\rm GeV}$ at ATLAS [@Aad:2012tfa] and $125.3\pm 0.4{\rm (stat.)} \pm 0.5{\rm
(syst.)}~{\rm GeV}$ at CMS [@Chatrchyan:2012ufa], respectively, and there is no signature of any other scalar particles up to $600~{\rm GeV}$ from the $ZZ$ mode. Thus one can apply the results of the Higgs search experiments at the LHC to constrain the mass and couplings of the radion. Recently, the bounds on the parameters $m_\phi$ and $\Lambda_\phi$ were studied in light of the Higgs boson discovery at the LHC using $pp\to h\to \gamma \gamma, ZZ, W^+W^-$ [@Desai:2013pga; @Cho:2013mva]. It was found that constraint on the parameter $\Lambda_\phi$ for the high-mass region of the radion ($m_\phi \lesssim 1~{\rm TeV}$) from the $ZZ$ mode is much severe than results in the previous studies [@Gunion:2003px; @Barger:2011qn]. It is, however, pointed out that the radion in low-mass region ($m_\phi \sim 100~{\rm GeV}$) is not constrained at the LHC, i.e., the Higgs search in the $\gamma \gamma$ channel at the LHC is less sensitive to a relatively light radion, since the $\phi \to
gg$ mode dominates over the other decay modes in this region which suppresses the branching ratio of $\phi \to \gamma \gamma$. Then it is worth examining possibilities to search for the radion in the low-mass region in collider experiments. In this paper, we study production and decay of the radion at a photon collider which has been proposed as an option of an $e^+ e^-$ linear collider such as the ILC [@Adolphsen:2013kya]. It has been studied that the photon collider has an advantage to distinguish the radion produced in the $\gamma \gamma$ collision from the SM Higgs boson production [@Gunion:2004nx; @Chaichian:2001gr; @Cheung:2000rw] supposing the SM Higgs boson is relatively heavy. The production of radion in the light-by-light scattering at the LHC has been discussed in ref. [@Fichet:2013gsa]. We show that the decay of radion in low-mass region into gluon pair is a promissing channel for its discovery at photon collider. This paper is organized as follows. In Sec. 2, we briefly review the interactions of radion to SM fields with emphasis on the difference from those of the SM Higgs boson. Production and decay of the radion at the photon collider are discussed in Sec. 3. We show our numerical results in Sec. 4 where a discovery potential of the radion at the photon collider is discussed quantitatively. Sec. 5 is devoted to summary and discussion.
Interactions
============
The radion field represents a fluctuation of the distance between the UV and IR branes. Taking account of a fluctuation along the fifth dimension, the metric is written as [@Csaki:2000zn] $$\begin{aligned}
ds^2 = e^{-2(ky + F(x,y))}\eta_{\mu\nu}dx^\mu dx^\nu -(1+2F(x,y))^2
dy^2,
\label{metric}\end{aligned}$$ where $F(x,y)$ is a scalar perturbation. A canonically normalized radion field $\phi$ is given by [@Csaki:2000zn] $$\begin{aligned}
F(x,y)=\frac{\phi}{\Lambda_\phi} e^{2k(y-\pi r_c)}, \end{aligned}$$ where the scale parameter $\Lambda_\phi$ is ${\cal O}({\rm
TeV})$ [@Csaki:2000zn; @Csaki:1999mp]. The radion couples to the trace part of the energy-momentum tensor of the SM. Then, the interaction Lagrangian of the radion is given by $$\begin{aligned}
{\cal L}_{\rm int} =
\frac{\phi}{\Lambda_\phi} \tracemumu,
\label{eq:interaction_Lagrangian}\end{aligned}$$ where $\tracemumu$ is the trace of energy-momentum tensor of the SM which is given as $$\begin{aligned}
\tracemumu &=& -2 m_W^2 W_\mu^+ W^{- \mu} - m_Z^2 Z_\mu Z^\mu
+ \sum_f m_f \bar{f} f
+ (2m_h^2 h^2 - \partial_\mu h \partial^\mu h)
\nonumber \\
&+& \frac{\beta_{\rm QED}}{2e} F_{\mu\nu}F^{\mu\nu}
+ \frac{\beta_{\rm QCD}}{2g_s} G^a_{\mu\nu}G^{a\mu\nu}
+ \cdots,
\label{eq:trace_SM_EMtensor}
\\
\beta_{\rm QED}&=&
\left(\frac{1}{16\pi}\right)^2 \left(\frac{19}{6}g_2^3
-\frac{41}{6}g_Y^3 \right),
\\
\beta_{\rm QCD}&=&
\left(\frac{1}{16\pi}\right)^2\left(11-\frac{2}{3}n_f \right)
g_s^3, \end{aligned}$$ where the first line in r.h.s. of (\[eq:trace\_SM\_EMtensor\]) is obtained from the energy-momentum tensor of the SM. Two terms in the second line of (\[eq:trace\_SM\_EMtensor\]) come from the trace anomaly for photons ($F_{\mu\nu}$) and gluons ($G^a_{\mu\nu},a=1,\cdots, 8$), respectively, and the ellipsis denote the higher-order terms. The number of active quark-flavors is denoted by $n_f$[^1]. We can see from (\[eq:trace\_SM\_EMtensor\]) that, except for the trace anomaly terms, the interactions of the radion to the SM fields are very similar to those of the SM Higgs boson. The interactions of the radion are, however, suppressed by the scale parameter $\Lambda_\phi \sim O({\rm TeV})$ which corresponds to the Higgs vacuum expectation value, $v =246 {\rm GeV}$, in the interactions of the SM Higgs boson to other SM fields. It is worth mentioning that, in general, the radion and the Higgs boson can mix after the electroweak symmetry breaking through the scalar-curvature term in the four-dimensional effective action [@Giudice:2000av; @Csaki:2000zn]. In our following study, however, we do not consider the mixing between the radion and the Higgs boson, since the current experimental results of the Higgs searches at the LHC tell us that the measured branching ratios of the Higgs boson are consistent with those in the SM [@Aad:2012tfa; @Chatrchyan:2012ufa], so that we can neglect the radion-Higgs mixing in a good approximation.
Production and decay of radion at a photon collider
===================================================
The production cross section of the radion in the $\gamma\gamma$ collision, and the branching fractions are obtained from the interaction Lagrangian in eq. (\[eq:interaction\_Lagrangian\]). In the photon collider, the high energy photons are obtained from electron beams through the inverse Compton scattering. The convoluted production cross section with energy distribution of photon beams is given by [@Chaichian:2001gr; @Cheung:2000rw; @Cheung:1992jn; @Telnov:1998qj] $$\begin{aligned}
\sigma(s)=\int_{m_\phi/\sqrt{s}}^{x_{max}}dz\Bigg[
2z\int^{x_{max}}_{z^2/x_{max}}\frac{dx}{x}\ f_\gamma(x)
f_\gamma\Big(\frac{z^2}{x}\Big)\Bigg]
\times \hat{\sigma}_{\gamma\gamma\rightarrow \phi}(\hat{s}),
\label{sigma}\end{aligned}$$ where $\sqrt{s}$ and $\sqrt{\hat{s}}$ represent the center-of-mass energy of electron pair and $\gamma\gamma$ systems, respectively. A momentum fraction of a photon against the electron momentum is denoted by $x$, and $z$ is defined by $z^2=\hat{s}/s$. The production cross section of the radion in the $\gamma\gamma$ annihilation $\hat{\sigma}_{\gamma\gamma\rightarrow \phi}(\hat{s})$ can be expressed using the decay rate of $\phi \to \gamma\gamma$ as $$\begin{aligned}
\hat{\sigma}_{\gamma\gamma\rightarrow \phi}(\hat{s})
&=&
\cfrac{4\pi^2}{\hat{s}}\ \delta(\sqrt{\hat{s}}-m_\phi)\ \Gamma(\phi\rightarrow
\gamma\gamma)\notag \\[5pt]
&=&
\cfrac{4\pi^2}{zs\sqrt{s}}\ \delta\bigg(z-\frac{m_\phi}{\sqrt{s}}\bigg)\
\Gamma(\phi\rightarrow \gamma\gamma)\ .
\label{cs_hat}\end{aligned}$$ A function $f_\gamma(x)$ in (\[sigma\]) is the unpolarized photon flux from the laser back-scattering $$\begin{aligned}
f_\gamma(x) = \frac{1}{D(\xi)}\bigg(
1-x+\frac{1}{1-x}-\frac{4x}{\xi(1-x)}+\frac{4x^2}{\xi^2(1-x)^2}
\bigg) ,\end{aligned}$$ where $$\begin{aligned}
D(\xi)=
\bigg(
1 - \frac{4}{\xi} - \frac{8}{\xi^2}
\bigg)
\ln(1+\xi) + \frac{1}{2} + \frac{8}{\xi} - \cfrac{1}{2(1+\xi)^2}\ . \end{aligned}$$ A parameter $\xi$ is chosen to be $\xi=4.8$, then $D(\xi)=1.8$ and $x_{max}=0.83$ [@Cheung:1992jn]. In Fig. \[fig:cs\], the production cross section (\[sigma\]) is given as a function of $m_\phi$ for $\Lambda_\phi=1~{\rm TeV}$ (a) and $3~{\rm TeV}$ (b), respectively. Three curves in black, red and blue in each figure correspond to $\sqrt{s}=250~{\rm GeV}$, $500~{\rm GeV}$ and $1~{\rm TeV}$, respectively. For each $\sqrt{s}$, the cross section is enhanced around $m_\phi=150-200~{\rm GeV}$. Note that the threshold energy of electron collision for the radion production is somewhat smaller than $m_\phi$ since the radion is produced via collision of back-scattered photons. For example, in Fig. \[fig:cs\] (a), the production cross section is larger than $1~{\rm fb}$ for $m_\phi \lesssim 400~{\rm GeV}$ ($\sqrt{s}=500~{\rm GeV}$) and $m_\phi \lesssim 820~{\rm GeV}$ ($\sqrt{s}=1~{\rm TeV}$).
The decay widths of the radion to the SM particles are easily calculated from eq. (\[eq:interaction\_Lagrangian\]): $$\begin{aligned}
\Gamma(\phi\rightarrow gg) &=&
\frac{\alpha_s^2 m_\phi^3}{32\pi^3\Lambda_\phi^2}
\left| b_{\rm QCD}+x_t\left\{1+(1-x_t)f(x_t)\right\}\right|^2,
\label{radiongg}
\\
\Gamma(\phi\rightarrow \gamma\gamma) &=&
\frac{\alpha_{\rm em}^2 m_\phi^3}{256\pi^3\Lambda_\phi^2}
\Bigg| b_2 + b_Y - \left\{2+3x_W+3x_W(2-x_W)f(x_W)\right\}
\nonumber \\
& & \qquad \qquad \qquad \qquad \qquad
\qquad +\frac{8}{3}x_t\left\{1+(1-x_t)f(x_t)\right\}\Bigg|^2,
\label{radiongamgam}
\\
\Gamma(\phi\rightarrow Z\gamma) &=&
\frac{\alpha_{\rm em}^2 m_\phi^3}{128\pi^3 s_{\rm w}^2
\Lambda_\phi^2}\Bigg(1-\frac{m_Z^2}{m_\phi^2}\Bigg)^3\nonumber\\
& & \qquad \qquad \qquad
\times \Bigg| \sum_f N_f \frac{Q_f}{c_{\rm W}} \hat{v}_f \
A_{1/2}^\phi(x_f,\lambda_f) +A_1^\phi(x_W,\lambda_W)\Bigg|^2,
\label{radionZgamma}
\\
\Gamma(\phi\rightarrow f\bar{f}) &=&
\frac{N_c m_f^2 m_\phi}{8\pi\Lambda_\phi^2}
(1-x_f)^{3/2},\\
\Gamma(\phi\rightarrow W^+ W^-) &=&
\frac{m_\phi^3}{16\pi\Lambda_\phi^2}\sqrt{1-x_W}
\Big(1-x_W+\frac{3}{4}x_W^2\Big),\\
\Gamma(\phi\rightarrow ZZ) &=&
\frac{m_\phi^3}{32\pi\Lambda_\phi^2}\sqrt{1-x_Z}
\Big(1-x_Z+\frac{3}{4}x_Z^2\Big),\\
\Gamma(\phi\rightarrow hh) &=&
\frac{m_\phi^3}{32\pi\Lambda_\phi^2}\sqrt{1-x_h}
\Big(1+\frac{1}{2}x_h\Big)^2,\end{aligned}$$ where $(b_{\rm QCD},b_2,b_Y)=(7, 19/6,-41/6)$. A symbol $f$ denotes all quarks and leptons. Two variables $x_i$ and $\lambda_i$ are defined as $x_i = 4m_i^2/m_\phi^2\ (i=t,f,W,Z,h)$ and $\lambda_i = 4m_i^2/m_Z^2\ (i=f,W)$. The gauge couplings for QCD and QED are given by $\alpha_s$ and $\alpha_{\rm em}$, respectively. The factor $N_f$ is the number of active quark-flavors in the 1-loop diagrams and $N_c$ is 3 for quarks and 1 for leptons. $Q_f$ and $\hat{v}_f$ denote the electric charge of the fermion and the reduced vector coupling in the $Zf\bar{f}$ interactions $\hat{v}_f=2I_f^3-4Q_f s_W^2$, where $I^3_f$ denotes the weak isospin and $s_W^2\equiv \sin^2{\theta_W},\ c_W^2=1-s_W^2$. The form factors $A^\phi_{1/2}(x,\lambda)$ and $A^\phi_1
(x,\lambda)$ are given by $$\begin{aligned}
A^\phi_{1/2}(x,\lambda)&=
I_1(x,\lambda)-I_2(x,\lambda)\ ,\\
A^\phi_1(x,\lambda) &=
c_W\Bigg\{4\Bigg(3-\frac{s_W^2}{c_W^2}\Bigg)I_2(x,\lambda)
+\Bigg[\Bigg(1+\frac{2}{x}\Bigg)\frac{s_W^2}{c_W^2}-\Bigg(5+\frac{2}{x}
\Bigg)\Bigg]I_1(x,\lambda)
\Bigg\}\ \notag.\end{aligned}$$ The functions $I_1(x,\lambda)$ and $I_2(x,\lambda)$ are $$\begin{aligned}
I_1(x,\lambda) &= \frac{x\lambda}{2(x-\lambda)}
+\frac{x^2\lambda^2}{2(x-\lambda)^2}[f(x^{-1})-f(\lambda^{-1})]
+\frac{x^2\lambda}{(x-\lambda)^2}[g(x^{-1})-g(\lambda^{-1})]\ ,
\notag\\
I_2(x,\lambda) &=
-\frac{x\lambda}{2(x-\lambda)}[f(x^{-1})-f(\lambda^{-1})]\ ,
\label{loop}\end{aligned}$$ where the loop functions $f(x)$ and $g(x)$ in (\[radiongg\]), (\[radiongamgam\]) and (\[loop\]) are given by [@Cheung:2000rw; @Djouadi:2005gi] $$\begin{aligned}
f(x) &=
\begin{cases}
\left\{\sin^{-1}\Bigg(\cfrac{1}{\sqrt{x}}\Bigg)\right\}^2 & ,\quad x\geq 1 \\[2mm]
-\cfrac{1}{4}\left(\log\cfrac{1+\sqrt{1-x}}{1-\sqrt{1-x}}-i\pi\right)^2&,
\quad x < 1
\end{cases}
\qquad,\\
\notag\\
g(x) &=
\begin{cases}
\sqrt{x^{-1}-1}\sin^{-1}\sqrt{x} &, \quad x\leq 1
\\[2mm]
\cfrac{\sqrt{1-x^{-1}}}{2}\left(\log\cfrac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}}-i\pi\right)&,
\quad x > 1
\end{cases}
\qquad.\end{aligned}$$ The branching ratio of the radion for all possible decay modes as a function of $m_\phi$ is shown in Fig. \[fig\_br\]. The mass of Higgs boson is fixed at $m_h=125~{\rm GeV}$ in the figure. We can see from Fig. \[fig\_br\] that the dominant decay mode is $\phi
\to gg$ for $m_\phi \lesssim 150~{\rm GeV}$ while it is altered by $\phi \to W^+W^-$ for $2 m_W \lesssim m_\phi$. The decay into $ZZ$ or $hh$ are subdominant for $2m_i \lesssim m_\phi$ ($i=Z,h$). We note here that the dominant decay mode of the radion at low-mass region is $\phi \to gg$ while that of the SM Higgs boson is $h\to b\bar{b}$. This is because that the interaction in the former is enhanced by the trace anomaly of the energy-momentum tensor as mentioned in a previous section.
![ The production cross sections of the radion at a photon collider given in Eq. [(\[sigma\])]{} as a function of radion mass $m_\phi$ with $\Lambda_\phi=1{\rm TeV\ (a)}$ and $\Lambda_\phi=3{\rm TeV\ (b)}$. The curves correspond to the electron beam energy $\sqrt{s} =
250{\rm \ GeV}\ ({\rm black}),\ 500{\rm \ GeV}\ ({\rm red})$ and $1{\rm \ TeV}\ ({\rm blue})$. []{data-label="fig:cs"}](cs_gam_r_1.eps "fig:") (a)
\[fig:cs\_gam\_r\_1\]
![ The production cross sections of the radion at a photon collider given in Eq. [(\[sigma\])]{} as a function of radion mass $m_\phi$ with $\Lambda_\phi=1{\rm TeV\ (a)}$ and $\Lambda_\phi=3{\rm TeV\ (b)}$. The curves correspond to the electron beam energy $\sqrt{s} =
250{\rm \ GeV}\ ({\rm black}),\ 500{\rm \ GeV}\ ({\rm red})$ and $1{\rm \ TeV}\ ({\rm blue})$. []{data-label="fig:cs"}](cs_gam_r_3.eps "fig:") (b)
\[fig:cs\_gam\_r\_2\]
![\[fig:br\] The branching ratio of radion for each decay modes as a function of $m_\phi$. The mass of the SM Higgs boson is fixed at $m_h=125$ GeV. []{data-label="fig_br"}](br.eps)
Numerical analysis
==================
In this section, we discuss a discovery potential of the radion in low-mass region ($m_\phi \lesssim 150~{\rm GeV}$) at the photon collider focusing on the signal process $\gamma\gamma \to \phi\to gg$, which is expected to be enhanced by the trace anomaly in both production and decay processes. At the photon collider, the leading background is $\gamma\gamma \to h
\to gg$. Then we survey the model parameter space $(m_\phi, \Lambda_\phi)$ by requiring a significance $S/\sqrt{B}>5$ by defining both $S$ and $B$ as $$\begin{aligned}
S &=& \int \mathcal{L}_{\it eff}\ dt\ \times \sigma(e e \rightarrow
\phi)\times {\rm Br}(\phi\rightarrow gg),
\\
B &=& \int \mathcal{L}_{\it eff}\ dt\ \times \sigma(e e \rightarrow
h)\times {\rm Br}(h\rightarrow gg). \end{aligned}$$ The production cross section $\sigma(e^+e^-\rightarrow \phi)$ is identical to that in (\[sigma\]), and $\sigma(e^+e^-\rightarrow h)$ is obtained replacing $\hat{\sigma}_{\gamma\gamma\rightarrow \phi}(\hat{s})$ in (\[cs\_hat\]) by $\hat{\sigma}_{\gamma\gamma\rightarrow h}(\hat{s})$. The effective photon luminosity $\int \mathcal{L}_{\it eff}\ dt$ is assumed to be $1/3$ of the electron luminosity following the TESLA technical design report [@Badelek:2001xb]. In our numerical analysis, we fix the Higgs boson mass $m_h$ at $125~{\rm GeV}$ for simplicity. Although the dominant decay channel is $\phi \to gg$ for $m_\phi
\lesssim 150~{\rm GeV}$, it is altered by $\phi \to VV$ ($V=W^\pm,Z$) for $2 m_W \lesssim m_\phi$. It is very hard to study the radion production and decay using the $W^+W^-$ channel since $\gamma\gamma\to W^+W^-$ occurs at the tree level in the SM and it overwhelms the signal process. Thus we use the $ZZ$ channel instead of $W^+W^-$, and estimate $S/\sqrt{B}$ assuming that $B$ is dominated by the $h\to ZZ$ channel as a reference in the high-mass region.
In Fig. \[fig:sig\], we show the signal region on $m_\phi$-$\Lambda_\phi$ plane where $S/\sqrt{B}>5$ is expected for various $\sqrt{s}$ and the effective integrated luminosity of back-scattered photon $\int {\cal L}_{\it eff} dt$. The expected beam luminosity of ILC is $(0.75,\, 1.8,\, 3.6) \times 10^{34} {\rm cm^{-2}s^{-1}}$, for $\sqrt{s}=250~{\rm GeV}$, $500~{\rm GeV}$, $1~{\rm
TeV}$ [@Adolphsen:2013kya; @Behnke:2013xla]. Therefore we used $\int {\cal L}_{\it eff} dt=80~{\rm fb}^{-1}$, $160~{\rm fb}^{-1}$ and $330~{\rm fb}^{-1}$ in Fig. \[fig:sig\] (a), (b) and (c). In the figure, black and red regions correspond to $S/ \sqrt{B} > 5$ for the $\phi \to gg$ and $\phi \to ZZ$ channels, respectively. It is also shown the excluded region on $(m_\phi, \Lambda_\phi)$ plane [@Cho:2013mva] taking account of the Higgs search experiments at the LHC [@CMS-PAS-HIG-13-002; @CMS-PAS-HIG-13-003; @CMS-PAS-HIG-13-001]. Turquoise, purple and magenta regions are excluded from the $pp\to h\to
ZZ$, $pp\to h\to W^+W^-$ and $pp\to h\to \gamma\gamma$ in the Higgs searches at the LHC.
From the figure, we find that the significance $S/\sqrt{B}$ at least $5$ is achieved in both the $gg$ and $ZZ$ channels. When $m_\phi \lesssim 150~{\rm GeV}$, $S/\sqrt{B} > 5$ is possible for $\Lambda_\phi\lesssim 3~{\rm TeV}$ in (a), (b) and (c). On the other hand, the $ZZ$ mode is available only for $180~{\rm
GeV}\lesssim m_\phi$. In the $ZZ$ mode, although there are sizable regions with $S/\sqrt{B}>5$ in (b) and (c), these regions are entirely disfavored from the Higgs search experiments at the LHC. We, therefore, expect that the photon collider has a good chance for discovery of the radion with $m_\phi \lesssim 150~{\rm GeV}$ but it has no sensitivity to search for the radion if the mass is larger than $180~{\rm GeV}$.
We have so far focused on the gluon final state in the radion decay. In the experiment, the two gluons in the final state are observed as two jets which also contain quarks. Since $\gamma\gamma\to q\bar{q}$ occurs at the tree level, it is very crucial to separate the gluon final states from $\gamma\gamma \to jj$. The detectors at ILC are aiming to achieve a high efficiency of tagging the $b$ and $c$ quark flavors [@Behnke:2013xla; @Behnke:2013lya]. Subtracting $b$ and $c$ jets from the data, and requiring appropriate kinematical cuts, it is expected to obtain two gluons in dijet data with a certain efficiency. A more quantitative estimation on the background (including $\gamma\gamma\to q\bar{q}$) processes is necessary based on the Monte Carlo simulation, which will be done elsewhere [@Ohno:sim].
![ Signal regions on $(m_\phi,\ \Lambda_\phi)$ plane for various beam energy of electron $\sqrt{s}=250$ GeV (a), $500$ GeV (b) and 1 TeV (c). The excluded regions from the recent results from the LHC are also shown [@Cho:2013mva]. Black and red regions denote parameter regions where signal significance $S/\sqrt{B}>5$ for $\gamma \gamma \rightarrow \phi \rightarrow gg$ and $\gamma \gamma \rightarrow \phi \rightarrow ZZ$ processes, respectively. Region in turquoise, purple and magenta show the $95\%$ CL excluded region from $pp\rightarrow h\rightarrow ZZ,\
pp\rightarrow h\rightarrow W^+W^-$ and $pp\rightarrow h\rightarrow
\gamma\gamma$ processes at the LHC.[]{data-label="fig:sig"}](sig250.eps "fig:") (a)
\[fig:sig250\]
![ Signal regions on $(m_\phi,\ \Lambda_\phi)$ plane for various beam energy of electron $\sqrt{s}=250$ GeV (a), $500$ GeV (b) and 1 TeV (c). The excluded regions from the recent results from the LHC are also shown [@Cho:2013mva]. Black and red regions denote parameter regions where signal significance $S/\sqrt{B}>5$ for $\gamma \gamma \rightarrow \phi \rightarrow gg$ and $\gamma \gamma \rightarrow \phi \rightarrow ZZ$ processes, respectively. Region in turquoise, purple and magenta show the $95\%$ CL excluded region from $pp\rightarrow h\rightarrow ZZ,\
pp\rightarrow h\rightarrow W^+W^-$ and $pp\rightarrow h\rightarrow
\gamma\gamma$ processes at the LHC.[]{data-label="fig:sig"}](sig500.eps "fig:") (b)
\[fig:sig500\]
![ Signal regions on $(m_\phi,\ \Lambda_\phi)$ plane for various beam energy of electron $\sqrt{s}=250$ GeV (a), $500$ GeV (b) and 1 TeV (c). The excluded regions from the recent results from the LHC are also shown [@Cho:2013mva]. Black and red regions denote parameter regions where signal significance $S/\sqrt{B}>5$ for $\gamma \gamma \rightarrow \phi \rightarrow gg$ and $\gamma \gamma \rightarrow \phi \rightarrow ZZ$ processes, respectively. Region in turquoise, purple and magenta show the $95\%$ CL excluded region from $pp\rightarrow h\rightarrow ZZ,\
pp\rightarrow h\rightarrow W^+W^-$ and $pp\rightarrow h\rightarrow
\gamma\gamma$ processes at the LHC.[]{data-label="fig:sig"}](sig1000.eps "fig:")\
(c)
\[fig:sig1000\]
Summary
=======
We have studied production and decay of the radion in the RS model at a photon collider as an option of $e^+e^-$ linear collider (ILC). Owing to the trace anomaly of the energy-momentum tensor, the interactions of the radion to photons and gluons are much enhanced. Focusing on the gluon final states in the radion decay, which is a dominant decay mode in the low-mass region of the radion, we investigated the model parameter space where the significance $S/\sqrt{B}>5$, and found that it could be achieved for $\Lambda_\phi
\lesssim 3~{\rm TeV}$ and $m_\phi \lesssim 150~{\rm GeV}$, without conflicting the constraints from the LHC experiments. To be more realistic, it is necessary to estimate both signal and background (including $\gamma\gamma\to q\bar{q}$) processes using the Monte Carlo simulation. The photon collider could be a good stage to look for the radion in the low-mass region which LHC experiment does not cover.
[**Acknowledgements**]{}
The work of G.C.C is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (No.24104502) and from the Japan Society for the Promotion of Science (No.21244036).
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[^1]: We take $n_f=6$ in our analysis.
|
---
abstract: 'We describe the concept of *logical scaffolds*, which can be used to improve the quality of software that relies on AI components. We explain how some of the existing ideas on runtime monitors for perception systems can be seen as a specific instance of logical scaffolds. Furthermore, we describe how logical scaffolds may be useful for improving AI programs beyond perception systems, to include general prediction systems and agent behavior models.'
author:
- Nikos Aréchiga
- Jonathan DeCastro
- Soonho Kong
- Karen Leung
bibliography:
- 'FoMLaS19.bib'
title: Better AI through Logical Scaffolding
---
Introduction
============
Recent progress in AI has led to possible deployment in a wide variety of important domains. This includes safety-critical cyberphysical systems such as automobiles [@bojarski_end_2016] and airplanes [@julian_policy_2016], but also decision making systems in diverse domains including legal [@perry_predictive_2013] and military applications [@cummings_artificial_2017].
Current AI programs differ from traditional programs in their reliance on data. The specification, input-output semantics, and executable generation procedure are all data driven [@karpathy_software_2017].
Unlike traditional software development, in AI programs a specification is not formally articulated. Indeed, in many of the most promising recent applications of AI, such as vision and human intent prediction, it is not feasible to write a formal specification. Instead, an implicit specification is provided via a test set, and the goal is to achieve a certain performance over the test set.
Traditional software development specifies the input-output semantics of the program in a programming language. In AI programs, the engineer provides a training dataset, and the program must match the input-output statistics of the dataset.
Instead of using a compiler to translate the programming language constructs to machine code, an engineer provides a “skeleton” in the form of a powerful function approximator (such as a neural network). The engineer then uses an optimization procedure to search for the parameterization that provides the best approximation to the input-output statistics of the training data. A cross-validation set is used to check generalizability of the learned function to unseen data.
This paradigm has proven to be powerful, especially in domains in which it is difficult to formally articulate the specification for the program, much less write a declarative program. However, this approach suffers from the following drawbacks.
1. [**Implicit specifications**]{} Since the specification is given implicitly as a desired performance over a test set, it is difficult for the tests to ascertain whether the program is providing the right answers for the right reasons [@koopman_toward_2018]. For this reason, the test set may fail to test the right things. This type of implicit specification is inadequate for use in a safety case, and it will be difficult to ascertain that programs tested in this way will be safe to deploy.
2. [**Non-representative training set**]{} Since the program seeks to match the statistical input-output properties of the training dataset, deficiencies of this training set will extend to deficiencies of the program. For example, scenarios that occur rarely in the training set may be fairly common in the real world, leading to degraded performance in deployment [@cui_class-balanced_2019]. In this sense, the training set may fail to train for realistic scenarios.
3. [**Robustness and sensitivity to adversarial attacks**]{} Since the model is templated by a functional template with many degrees of freedom, it is common for the process to result in programs that are susceptible to extreme sensitivity to irrelevant features of the input. The literature on *adversarial examples* demonstrates how slight perturbations to an input can lead to incorrect results with high confidence [@goodfellow_explaining_2015].
We propose to attack these issues by the use of *logical scaffolds*, which are lightweight formal properties that provide some information about the relationship of the program inputs and outputs. These logical scaffolds can be written in languages for which monitoring algorithms exist, such as Signal Temporal Logic [@maler_monitoring_2004], Signal Convolutional Logic [@silvetti_signal_2018], Timed Quality Temporal Logic [@dokhanchi_evaluating_2018] and many others. Logical scaffolds may arise from a number of different sources, including a formalization of physical laws, domain knowledge, and common sense.
Logical scaffolds are more general than related work such as reasonableness monitors [@gilpin_reasonableness_2018] and model assertions [@kang_model_2018] because scaffolds can be used for different types of AI programs beyond merely perception. Furthermore, recent work in smoothly differentiable formulations of STL and MTL [@leung_backpropagation_2019; @pant_smooth_2017; @mehdipour_arithmetic-geometric_2019] enable the scaffolds to become part of the training process directly.
Logical scaffolds
=================
Informally, a logical scaffold is a predicate that encodes something that is believed to be true about the input-output relation of an AI program. It is *not* a complete specification. If a specification existed, the scaffold would be a logical consequence of the complete specification. In other words, the scaffold is a consequence of correct functionality. As such, it constitutes a necessary, but not sufficient, condition for correct behavior.
Most of the existing literature on monitoring runtime properties is centered around perception systems. These runtime monitors can be formalized as scaffolds, but the key idea can be generalized beyond perception to include applications in explainable intent prediction and expressible behavior modeling.
Sources of logical scaffolds are as diverse as the sources of human intuition about the application domain. The following is not an exhaustive list.
- **Perception**
- Commonsense notions of label consistency, like the properties monitored in [@dokhanchi_evaluating_2018] and [@kang_model_2018], in which class labels are not expected to mutate or drop out between frames.
- Class-specific information, such as the intuition that a mailbox should not be seen crossing the road [@gilpin_reasonableness_2018].
- **Intent prediction and behavior modeling**
- Physics-derived knowledge, such as knowledge of maximum speeds or actuator capabilities, for example, that on an icy road, other vehicles may be out of control or less able to brake and swerve.
- Natural expectations that pedestrians and vehicles are unlikely to seek damage to themselves, unless they are out of control.
The challenges related to implicit specifications are ameliorated by using logical scaffolds at training and testing time. For generative models, such as intent predictors and agent models, we can impose an understandable structure on the latent space, as described below. At testing time, we are able to use parametric logical scaffolds to learn explanations of the input-output behavior of the system, which can be used to check that the system is passing its tests for the right reasons.
The challenges related to non-representative training sets as well as robustness and sensitivity to adversarial attacks can be ameliorated by using the scaffolds at deployment.
The work of Kang et al. [@kang_model_2018] has shown how hand crafted runtime monitors can be used to flag scenarios in which the program fails. These scenarios can then be added to the training set, yielding greatly improved performance. This approach has a flavor of active learning, in which the scenarios that are most difficult for the program are fed back for further study. Here, we propose that the monitors need not be hand crafted, but automatically generated from logical scaffolds that express a variety of properties. Conversely, in [@dokhanchi_evaluating_2018], Dokhanchi et al. automatically generate monitors from Timed Quality Temporal Logic that check for label stability, i.e., ensuring that labels do not mutate across frames.
Training
--------
There may be many ways that logical scaffolds can be used at training time. In this work we consider training generative models that make use of a latent space.
Generative models are models that can generate data that is similar to the data they are trained on. Important examples of generative models for contemporary applications include the following.
1. Intent predictors, which are used by autonomous vehicles or other robots to predict future trajectories of other agents, such as automobiles, pedestrians and bicycles.
2. Reactive agents, which seek to generate appropriate behaviors for a specific environment. Examples may include simulations agents that subject an autonomous vehicle, drone, or robot to challenging but realistic behaviors, as well as decision-making agents for different applications.
3. Scenario generators, which seek to synthesize testing and simulation environments that may be challenging but still realistic.
We can use logical scaffolds expressed in a differentiable logic (e.g. [@leung_backpropagation_2019], [@mehdipour_arithmetic-geometric_2019], [@pant_smooth_2017]) and use them to add structure to the latent space. A diagram of this idea is shown in Figure \[x2latent\].
Testing
-------
At test time, we are interested in finding out whether the model is producing the correct outputs for the correct reasons. To accomplish this, we make use of *parametric* logical scaffolds, i.e. scaffolds with free parameters.
We sample from the latent space, and prompt the model to generate an output. Then, we take a bank of pSTL formulas, and fit values for each of the parameters. Then, we check to see which of the parameters have clusters that correspond to the clusters of the original latent space. The corresponding STL formulas are “explanations” of the latent space clusters. A diagram of this idea is shown in Figure \[latent2x\].
Deployment
----------
At deployment, the logical scaffold can be used to detect anomalies. These anomalies can later be fed back into the training procedure for the next iteration of the system. The use of logical scaffolds for runtime improvement is predicated on the fact that these scaffolds are written in formal languages that support runtime monitoring.
The work of [@kang_model_2018] has already demonstrated how hand-crafted runtime monitors can be used to greatly improve the performance of single shot detectors, and the work of [@dokhanchi_evaluating_2018] has developed a special-purpose runtime logic to monitor the stability of class labels.
The idea of augmenting AI programs with knowledge of physics (“Newton + Hinton”[^1]) is not new. We believe that even greater impact can be obtained from the broader principle of systematically developing logical scaffolds that encode physics domain-specific knowledge, and common sense.
Conclusions and Future Work
===========================
We have outlined a technique of “logical scaffolding”, which involves conditions that are necessary for correctness, but not sufficient. We have outlined how these logical scaffolds can be used to improve the performance and reliability of AI systems at training, testing, and deployment.
In future work, we will explore case studies in perception, behavior modeling, and scenario generation.
[^1]: Thanks to Adrien Gaidon for this term
|
---
abstract: 'We theoretically and experimentally demonstrate coherence phenomena in optical parametric amplification inside a cavity. The mode splitting in transmission spectra of phase-sensitive optical parametric amplifier is observed. Especially, we show a very narrow dip and peak, which are the shape of $\delta $ function, appear in the transmission profile. The origin of the coherence phenomenon in this system is the interference between the harmonic pump field and the subharmonic seed field in cooperation with dissipation of the cavity.'
author:
- 'Hongliang Ma, Chenguang Ye, Dong Wei, Jing Zhang$^{\dagger }$'
title: 'Coherence Phenomena in the Phase-sensitive Optical Parametric Amplification inside a Cavity'
---
$Introduction$*.* — Coherence and interference effects play very important roles in determining the optical properties of quantum systems. Electromagnetically induced transparency (EIT) [@one] in quantum-mechanical atomic systems is a well understood and thoroughly studied subject. EIT has been utilized in a variety of applications, such as lasing without inversion [@two], slow and stored light [@two1; @two2], enhanced nonlinear optics [@three], and quantum computation and communication [@four]. Relying on destructive quantum interference, EIT is a phenomenon where the absorption of a probe laser field resonant with an atomic transition is reduced or even eliminated by the application of a strongly driving laser to an adjacent transition. Since EIT results from destructive quantum interference, it has been recently recognized that similar coherence and interference effects also occur in classical systems, such as plasma [@five], coupled optical rsonators[@six], mechanical or electric oscillators [@seven]. In particular, the phenomenology of the EIT and dynamic Stark effect is studied theoretically in a dissipative system composed by two coupled oscillators under linear and parametric amplification using quantum optics model in Ref. [@eight]. The classical analog of EIT is not only helpful to understand deeply the physical meaning of EIT phenomenon, but also offers a number of itself important applications, such as slow and stored light by coupled optical resonators[@nine].
In this Letter, we extend the model in Ref. [@eight] and present a new system - phase-sensitive optical parametric amplifier (OPA) to demonstrate coherence effects theoretically and experimentally. We observe mode splitting in transmission spectra of OPA. Especially, we show a very narrow dip and peak, which are the shape of $\delta $ function, appear in the transmission profile. This phenomenon results from the interference between the harmonic pump field and the subharmonic seed field in OPA. The destructive and constructive interference correspond to optical parametric deamplifier and amplifier respectively, which are in cooperation with dissipation of the cavity. The absorptive and dispersive response of an optical cavity for the probe field is changed by optical parametric interaction in the cavity. Phase-sensitive optical parametric amplifier presents a number of new characteristics of coherence effects.
$Theoretical$ $model.$ — Consider the interaction of two optical fields of frequencies $\omega $ and $2\omega $, denoted by subharmonic and harmonic wave (the pump), which are coupled by a second-order, type-I nonlinear crystal in a optical cavity as shown in Fig.1. The cavity is assumed to be a standing wave cavity, and only resonant for the subharmonic field with dual-port of transmission $T_{HR}$ and $T_c$, internal losses $A$ and length $L$ (roundtrip time $\tau =2L/c$). We consider both the subharmonic seed beam $a^{in}$ and harmonic pump beam $%
\beta ^{in}$ are injected into the back port ($T_{HR}$ mirror) of the cavity, where the relative phase between the injected field is adjusted by a movable mirror outside the cavity. $T_{HR}$ mirror is a high reflectivity mirror at the subharmonic wavelength, yet has a high transmission coefficient at the harmonic wavelength and $T_c$ mirror has a high reflectivity coefficient for the harmonic wave. The harmonic wave makes a double pass through the nonlinear medium. The equation of motion for the mean value of the subharmonic intra-cavity field can then be derived by the semiclassical method [@ten] as $$\tau \frac{da}{dt}=-i\tau \Delta a-\gamma a+g\beta ^{in}a^{\ast
}+\sqrt{2\gamma _{in}}a^{in}. \label{degen}$$ The decay rate for internal losses is $\gamma _l=A/2 $ and the damping associated with coupling mirror and back mirror is $\gamma
_c=T_c/2 $ and $\gamma _{in}=T_{HR}/2 $, respectively. The total damping is denoted by $\gamma =\gamma _{in}+\gamma _c+\gamma _l$. $\Delta $ is the detuning between the cavity-resonance frequency $\omega _c$ and the subharmonic field frequency $\omega $. The strength of the interaction is characterized by the nonlinear coupling parameter $g$. Eq.\[degen\] is complemented with the boundary conditions $a^{out}=\sqrt{2\gamma _c}a$ and $%
a^{ref}=-a^{in}+\sqrt{2\gamma _{in}}a$ to create propagating beams, where $a^{out}$ is the transmitted field from the coupling mirror $T_c$ and $%
a^{ref}$ is the reflected field from the back mirror $T_{HR}$. The phase-sensitive optical parametric amplifier always operates below the threshold of optical parametric oscillation (OPO) $\beta
_{th}^{in}=\gamma /g$. Eq.\[degen\] ignores the third-order term[@eleven] describing the conversion losses due to harmonic generation. For simplicity, we assume that the phase of the pump field is zero in any case, i.e, $\beta ^{in}$ is real and positive value. The intra-cavity field $a$ and the injected field $a^{in}$ are expressed as $a =$ $%
\alpha \exp \left( -i\phi \right) $ and $a^{in} =$ $%
A_{in}\exp \left( -i\varphi \right) $ respectively. Here, $\alpha $ and $%
A_{in}$ are real values, $\phi $ and $\varphi $ are the relative phase between the intra-cavity field and the pump field as well as between the seed field and the pump field, respectively. If the harmonic pump is turned off, the throughput for the non-impedance matched subharmonic seed beam is given by $ a_{no~pump}^{out}
=2\sqrt{\gamma _c\gamma _{in}}A_{in}/(\gamma +i\tau \Delta )$. The subharmonic seed beam is subjected to either amplification or de-amplification, depending on the chosen relative phase between the subharmonic field and the pump field.
$Case1:$ Consider the transmitted intensity of the subharmonic seed beam as a function of the detuning $\Delta $ between the subharmonic field frequency and the cavity-resonance frequency, and keep the pump field of frequency $%
\omega _p=2\omega $ constant. Setting the derivative to zero ($d\alpha /dt=0$) and separating the real and image part of Eq.\[degen\], the steady state solutions of the amplitude and relative phase of the intra-cavity field are given by $$\begin{aligned}
-\gamma \alpha +g\beta ^{in}\alpha \cos 2\phi +\sqrt{2\gamma _{in}}%
A_{in}\cos \left( \phi -\varphi \right) &=&0, \label{steady} \\
-\tau\Delta \alpha +g\beta ^{in}\alpha \sin 2\phi +\sqrt{2\gamma _{in}}%
A_{in}\sin \left( \phi -\varphi \right) &=&0. \nonumber\end{aligned}$$ When the amplitude and relative phase of the subharmonic seed beam are given, the transmitted intensity of the subharmonic beam is obtained from Eq.\[steady\] and the boundary condition. Fig.2(a) shows a Lorentzian profile of the subharmonic transmission when the pump field is absent. This corresponds to the typical transmitted spectrum of the optic al empty cavity. When the injected subharmonic field is out of phase ($\varphi =\pi /2)$ with the pump field, the subharmonic transmission profile is shown in Fig.2(b,c,d) for different pump powers, in which there is a symmetric mode splitting. The transmitted power of the subharmonic beam is normalized to the power in the absence of the pump and zero detuning. The transmission spectra show that the dip becomes deeper and two peaks higher as the pump intensity increases. The origin of mode splitting in transmission spectra of OPA is destructive interference in cooperation with dissipation of the cavity. If the subharmonic field resonates in the cavity perfectly, i.e. $\Delta =0$, the subharmonic intra-cavity field and the pump filed are exactly out of phase and will interfere destructively to produce the deamplification for the subharmonic field in the nonlinear crystal. Thus a dip appears at the zero detuning of the transmission profile. If the subharmonic field is not quite resonant in the cavity perfectly, that is, the subharmonic field’s frequency is not exactly an integer multiple of the free spectral range (but close enough to build up a standing wave), the phase difference between the subharmonic intra-cavity field and the pump field will not be exactly out of phase and will increase as the detuning increases. The subharmonic intra-cavity field will change from deamplification to amplification as the phase difference increases. Thus we see that the transmission profile has two symmetric peaks at two detuning frequencies. When the phase of the injected subharmonic field is deviated from out of phase with the pump field, i.e. $\varphi =\pi /2\pm \theta $, an asymmetric mode splitting in the subharmonic transmission profile is illustrated in Fig.2(e,f), in which the dip is deviated from the zero detuning of the transmission profile and two peaks have different amplitude.
$Case2:$ Consider the subharmonic transmission profiles when the frequency of the pump field is fixed at $\omega _p=2(\omega
_c+\Omega )$. When scanning the frequency of the the subharmonic seed beam, an idler field in the OPA cavity will be generated with the frequency $\omega _i=\omega _p-\omega $ due to energy conservation. The equation of motion of OPA become frequency-nondegenerate and is given by
$$\begin{aligned}
\tau \frac{da}{dt} &=&-i\tau \Delta a-\gamma a+g\beta ^{in}a_i^{\ast }+\sqrt{%
2\gamma _{in}}a^{in}, \label{nondegen} \\
\tau \frac{da_i}{dt} &=&-i\tau \Delta _ia_i-\gamma a_i+g\beta
^{in}a^{\ast } \nonumber\end{aligned}$$
where $a_i$ is the idler field in the OPA cavity. $\Delta _i$ is the detuning between the cavity-resonance frequency $\omega _c$ and the idler field frequency $\omega _i$. Thus the subharmonic transmission profile in this case is obtained from Eq.\[nondegen\] for $\omega
\neq \omega _i$ and Eq.\[degen\] for $\omega =\omega _i$. When $\Omega =0$, so $\Delta =-\Delta _i$, the stationary solution of the subharmonic and idle field is given by solving the mean-field equations of Eq.\[nondegen\] and using the input-output formalisms. We obtain $$\begin{aligned}
A^{out} &=&\frac{2\sqrt{\gamma _c\gamma _{in}}}{i\tau\Delta +\gamma -\frac{%
(g\beta ^{in})^2}{i\tau\Delta +\gamma }}A^{in}, \\
A_i^{out} &=&\frac{2\sqrt{\gamma _c\gamma _{in}}g\beta ^{in}}{\left(
-i\tau\Delta +\gamma \right) ^2-(g\beta ^{in})^2}A^{in*}. \nonumber\end{aligned}$$ We will record the total output power including the subharmonic and idle field. The transmitted power of the subharmonic beam is given by
$$P_{out}^{nor}=\left\{
\begin{array}{c}
\left| \frac \gamma {i\tau\Delta +\gamma -\frac{(g\beta
^{in})^2}{i\tau\Delta +\gamma }}\right| ^2+\left| \frac{\gamma
g\beta ^{in}}{\left( -i\tau\Delta
+\gamma \right) ^2-(g\beta ^{in})^2}\right| ^2\qquad \\
\qquad \qquad \qquad \qquad \mathrm{if}\quad \omega \neq \omega _i\quad \\
\frac{\gamma ^2}{(\gamma \pm g\beta ^{in})^2}\quad \qquad
\mathrm{if}\quad \omega =\omega _i.
\end{array}
\right.$$
Here, $\pm $ corresponds to the deamplifier and amplifier in frequency-degenerate OPA. Fig.3(a) and (b) show that the very narrow dip and peak, which is the shape of $\delta $ function, appear in the transmission profile. This novel coherence phenomena results in that the destructive and constructive interference are established only in the point of $\omega =\omega _i$, and completely destroyed in the other frequencies.
$Experiment$. — The experimental setup is shown schematically in Fig.4. A diode-pumped intracavity frequency-doubled continuous-wave(cw) ring Nd:YVO$_{%
\text{4}}$/KTP single-frequency green laser severs as the light sources of the pump wave (the second-harmonic wave at $532$ $nm$) and the seed wave (the fundamental wave at $1064$ $nm$) for OPA. The green beam doubly passes the acousto-optic modulator (AOM) to shift the frequency 440 MHz. The infrared beam doubly passes AOM to shift the frequency around 220 MHz. We actively control the relative phase between the subharmonic and the pump field by adjusting the phase of the subharmonic beam with a mirror mounted upon a piezoelectric transducer (PZT). Both beams are combined together by a dichroic mirror and injected into the OPA cavity. OPA consists of periodically poled KTiOPO$_4$ (PPKTP) crystal (12 $mm$ long) and two external mirrors separated by $63$ $mm$. Both end faces of crystal are polished and coated with an antireflector for both wavelengths. The crystal is mounted in a copper block, whose temperature was actively controlled at millidegrees kelvin level around the temperature for optical parametric process (31.3${%
{}^{\circ }}$C). The input coupler M1 is a $30$ $mm$ radius-of-curvature mirror with a power reflectivity $99.8\%$ for $1064$ $nm$ in the concave and a total transmissivity $70\%$ for $532$ $nm$, which is mounted upon a PZT to adjust the cavity length. The output wave is extracted from M2, which is a $30-mm$ radius-of-curvature mirror with a total transmissivity $3.3\%$ for $1064$ $nm$ and a reflectivity $99\%$ for $532$ $nm$ in the concave. Due to the large transmission of input coupler at $532$ $nm$, the pump field can be thought as only passes the cavity twice without resonation. The measured cavity finesse was $148$ with the PPKTP crystal, which indicates the total cavity loss of $%
4.24$%. Due to the high nonlinear coefficient of PPKTP, the measured threshold power is only $35$ $mW$.
First, we fix the frequency of the subharmonic and the pump field with $%
\omega _p=2\omega $ and scan cavity length, which corresponds to the condition of case 1. Figure 5 shows the experimental results: (a) without the pump field, (b) $\varphi =\pi /2$ and $\beta ^{in}/\beta _{th}^{in}=0.33$, (c) $\varphi =\pi /2$ and $\beta ^{in}/\beta _{th}^{in}=0.71$, (d) $%
\varphi =\pi /2$ and $\beta ^{in}/\beta _{th}^{in}=0.9$, (e) $\varphi =\pi /2-0.07$ and $\beta ^{in}/\beta _{th}^{in}=0.9$, (f) $\varphi =\pi /2+0.07$ and $\beta ^{in}/\beta _{th}^{in}=0.9$. It can be seen that the experimental curves are in good agreement with the theoretical results shown in Fig.2, which are obtained with the experimental parameters.
Then, we fix the cavity length and frequency of the pump field and scan the frequency of the subharmonic field by the AOM, which corresponds to the condition of case 2. The output including the subharmonic and idle field is detected by a photodiode. There is a beat-note signal in the photocurrent with frequency proportional to the detuning. The very narrow dip and peak appeared in a broad Lorentzian profile are observed experimentally as shown in Fig.6. The insets in Fig.6 show the enlarged narrow dip and peak by reducing the scanned range of frequency, which present the square shape. Because the measurement of transmission profile is dynamic processes, the shape of $%
\delta $ function for the narrow dip and peak in the theoretical model becomes square shape in experiment. The width of the square shape is $\sim$2KHz which is estimated from the voltage on VCO (Voltage-Controlled Oscillator) of AOM.
$Conclusion.$ — We reported the theoretical and experimental results of coherence phenomena in the phase-sensitive optical parametric amplification inside a cavity. The splitting in transmission spectra of OPA was observed. Mode splitting, as well known, occurs not only in coupled quantum system, but also in coupled optical resonators and in coupled mechanical and electronic oscillators. To the best of our knowledge, we first observed mode splitting experimentally in the optical parametric process. This system will be important for practically optical and photonic applications such as optical filters, delay lines, and closely relate to the coherent phenomenon of EIT predicted for quantum systems. OPA also has a important application as squeezed light source. Our results may help us to investigate quantum noise spectrum.
$^{\dagger} $Corresponding author’s email address: jzhang74@yahoo.com, jzhang74@sxu.edu.cn
J. Zhang thanks Prof. Kunchi Peng and Changde Xie for the helpful discussions.This research was supported in part by National Natural Science Foundation of China (Approval No.60178012), Program for New Century Excellent Talents in University, Natural Science Foundation of Shanxi Province, and the Research Fund for the Returned Abroad Scholars of Shanxi Province.
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Fig.1 Schematic of optical parametric amplifier in standing-wave cavity.
Fig.2 The theoretical results for case 1, (a) without the pump field injection; (b) $\varphi =\pi /2$ and $\beta ^{in}/\beta _{th}^{in}=0.33$; (c) $%
\varphi =\pi /2$ and $\beta ^{in}/\beta _{th}^{in} =0.71$; (d) $\varphi =\pi /2$ and $\beta ^{in}/\beta _{th}^{in} =0.9$; (e) $\varphi =\pi /2-0.07$ and $\beta ^{in}/\beta _{th}^{in} =0.9$; (f) $\varphi =\pi /2+0.07$ and $\beta ^{in}/\beta _{th}^{in} =0.9$.
Fig.3 The theoretical results for case 2, (a) $\varphi =\pi /2$ for deamplification; (b) $\varphi =0$ for amplification.
Fig.4 Schematic of the experimental setup. DC: dichroic mirror; $\lambda /2$, half-wave plate; T-C, temperature controller, HV-AMP, high voltage amplifier.
Fig.5 The experimental results for case 1 corresponding to Fig.2.
Fig.6 The experimental results for case 2 corresponding to Fig.3.
|
---
abstract: 'A perturbative analysis is used to investigate the effect of rotation on the instability of a steady accretion shock (SASI) in a simple toy-model, in view of better understanding supernova explosions in which the collapsing core contains angular momentum. A cylindrical geometry is chosen for the sake of simplicity. Even when the centrifugal force is very small, rotation can have a strong effect on the non-axisymmetric modes of SASI by increasing the growth rate of the spiral modes rotating in the same direction as the steady flow. Counter-rotating spiral modes are significantly damped, while axisymmetric modes are hardly affected by rotation. The growth rates of spiral modes have a nearly linear dependence on the specific angular momentum of the flow. The fundamental one-armed spiral mode ($m=1$) is favoured for small rotation rates, whereas stronger rotation rates favour the mode $m=2$. A WKB analysis of higher harmonics indicates that the efficiency of the advective-acoustic cycles associated to spiral modes is strongly affected by rotation in the same manner as low frequency modes, whereas the purely acoustic cycles are stable. These results suggest that the linear phase of SASI in rotating core-collapse supernovae naturally selects a spiral mode rotating in the same direction of the flow, as observed in the 3D numerical simulations of Blondin & Mezzacappa (2007). This emphasizes the need for a 3D approach of rotating core-collapse, before conclusions on the explosion mechanisms and pulsar kicks can be drawn.'
author:
- Tatsuya Yamasaki and Thierry Foglizzo
title: |
Effect of Rotation on the Stability of a Stalled Cylindrical Shock\
and its Consequences for Core-Collapse Supernovae
---
Introduction\[intro\]
=====================
Despite extensive studies, the explosion mechanism of core-collapse supernovae is still elusive. According to the delayed explosion scenario, the shock is first stalled at a distance of a few hundred kilometers, and then revived after neutrinos diffuse out of the proto neutron star. Unfortunately, numerical simulations suggest that neutrino heating may not be efficient enough, at least in spherical symmetry [@lieb05].
Recent studies have shown that the spherical stalled shock is unstable against non radial perturbations with a low degree $l=1,2$ even if the flow is convectively stable. This result was demonstrated using axisymmetric numerical simulations [@blo03; @blo06; @sch06; @sch08; @ohn06] and linear stability analyses [@gal05; @fog07; @yam07]. Some numerical simulations have shown that this hydrodynamical instability, often called SASI, may assist the revival of the shock and trigger a successful explosion, powered either by neutrino heating [@mar07] or by acoustic waves [@bur06]. Some observed properties of young neutron stars may also be the consequences of SASI, such as their distribution of velocities [@sch04; @sch06] or their spin [@blo07a; @blo07b].
Until now, most studies of SASI have assumed that the unperturbed flow is purely radial and not rotating. Since the angular momentum of massive stars is likely to be large [@heg05], it is desirable to understand how the properties of SASI are affected by rotation. In this Letter, the effect of rotation on the linear stage of SASI is investigated using a perturbative analysis in order to shed light on one of the surprising results observed by [@blo07a] in their 3D numerical simulations: the development of SASI seems to systematically favour a spiral mode rotating in the same direction as the accretion flow. As a consequence of momentum conservation, this mode diminishes and may even reverse the angular momentum acquired by the proto-neutron star from the stationary flow. Incidentally, the present linear study does not address another surprising result of [@blo07a], that a spiral mode of SASI always dominate the axisymmetric mode even without rotation. Following an approach similar to @fog07 (hereafter FGSJ07), we first compute the eigenfrequencies by solving accurately a boundary value problem between the shock surface and the accretor surface; in a second step, we use the same WKB method as in FGSJ07 to measure the stability of purely acoustic and advective-acoustic cycles in this region. This approach is different from [@lam07], which is based on the approximate derivation of a dispersion relation. Rather than the complexity of describing the non-spherical shape of a shock deformed by rotation [@yam05], we have chosen, as a first step, to solve the much simpler problem of a cylindrical accretion shock. This flow is simple enough to allow for a complete coverage of the parameter space and a physical insight of the main effects of rotation on SASI. Once characterized, these effects can be transposed into the more complex geometry of a rotating stellar core.
Formulation\[calculations\]
===========================
Following a similar desire of simplification, [@blo07b] have limited their 2D simulation domain $(r,\varphi)$ to the vicinity of the equatorial region $(\theta=\pi/2)$ of a non-rotating spherical flow, where the poloidal velocity of a symmetric mode is $v_\theta=0$. Their calculation neglected the perturbation of density induced by the term $(\rho/ r)(\partial v_\theta/\partial\theta)$ in the equation of continuity, implicitly decoupling the structure of perturbations in the vertical direction from their structure in the equatorial plane. In order to study the effects of rotation in a simple setup and maintain a self consistent set of equations, we have chosen to adopt a cylindrical geometry invariant along the $z$-axis of rotation. The accretor and the shock are thus cylindrical in our toy-model.
Except for the geometry of the flow and the presence of rotation, the assumptions are the same as in FGSJ07. In the equations describing the flow (Appendix A), we have assumed that i) the free-falling supersonic flow is cold, ii) gravity is Newtonian and self-gravity is neglected, iii) the shock is adiabatic, iv) neutrino heating is neglected and the cooling rate per volume is approximated as ${\cal L} \propto \rho^{\beta-\alpha}P^{\alpha}$ with $\alpha=3/2$, and $\beta=5/2$, where $\rho$ is the density and $P$ the pressure, v) the accreting material is described by a perfect gas with a uniform adiabatic index $\gamma=4/3$, vi) the condition ${\cal M}=0$ is imposed at the inner boundary, where ${\cal M}\equiv -v_r/c$ is the Mach number associated to the radial velocity $v_r$ and adiabatic sound speed $c$, vii) the flow is inviscid and its specific angular momentum $L$ is conserved.
We have adopted values of the parameters typical for the core-collapse problem: the radius of the inner boundary is $r_*=50$ \[km\] and the gravitational potential is $\Phi=-1.3GM_{\odot}/r$, where $G$ is the gravitational constant and $M_{\odot}$ the solar mass. By adopting the same gravitational potential as for a spherical accretor, the Bernoulli equation is unchanged.
The stability of the accretion flow is investigated for various values of the specific angular momentum $L$, measured by the rotation frequency $f_{\rm p}\equiv L/(2\pi r_{\rm p}^2)$ extrapolated at a radius $r_{\rm p}\sim 10$\[km\] by reference to young pulsars. The range of rotation rates considered corresponds to $0\le f_{\rm p}\le 10^{3}$\[Hz\]. We have chosen to compare instability timescales in cylindrical models with different rotation rates but identical shock radii and mass accretion rates, in order to keep geometrical factors constant. This is made possible by adapting the intensity of cooling accordingly, by a modest amount ($<15\%$) over the range of rotation rates considered. We observe that the advection time from the shock to the accretor is hardly affected by rotation (a few percent). This is because the centrifugal force $L^2/r^3$ is much smaller than gravity $-d\Phi/dr$: $$\begin{aligned}
{L^2\over r^3 |d\Phi/dr|}&=&4.6\times 10^{-2} \left({r_*\over r}\right)\left({f_{\rm p}\over 10^3{\rm [Hz]}}\right)^2.\end{aligned}$$ The perturbations superimposed upon the steady solution are proportional to $\exp\{-i(\omega t -m\theta -k_z z)\}$. We adopt the variables $\delta S,\delta q,\delta f,\delta h$ defined by $$\begin{aligned}
\delta S&\equiv&{2\over\gamma-1}{\delta c\over c}-{\delta\rho\over\rho},\label{eq5}
\\
\delta q&\equiv&\delta\left(\int ^r\frac{\cal L}{\rho v_r}dr'\right),
\label{eq2}\\
\delta f&\equiv& v_r \delta v_r +\frac{L}{r}\delta v_{\theta}+\frac{2}{\gamma -1}c\delta c -\delta q,
\label{eq3}\\
\delta h&\equiv& \frac{\delta v_r}{v_r}+\frac{\delta \rho}{\rho},
\label{eq4}\end{aligned}$$ where $\delta$ denotes the Eulerian perturbation and $v_\theta$ is the azimuthal velocity. Then the differential system describing the perturbations becomes particularly compact: $$\begin{aligned}
\frac{{\rm d} \delta f}{{\rm d} r}&=&\frac{i\omega c^2}{v_r(1-{\cal M}^2)}\left\{{\cal M}^2 \delta h
-{\cal M}^2 \frac{\omega'}{\omega}\frac{\delta f}{c^2}\right.\nonumber\\
&&\left.+[1+(\gamma -1){\cal M}^2 ]\frac{\delta S}{\gamma}-\frac{\delta q}{c^2}\right\},
\label{eq6}\\
\frac{{\rm d} \delta h}{{\rm d} r}&=&\frac{i\omega'}{v_r(1-{\cal M}^2)}\left\{\frac{\mu^2}{c^2}\frac{\omega'}{\omega}\delta f-{\cal M}^2 \delta h
-\delta S +\frac{\delta q}{c^2}\right\},
\label{eq7}\\
\frac{{\rm d}\delta S}{{\rm d} r}&=&\frac{i\omega'}{v_r}\delta S +\delta\left(\frac{\cal L}{Pv_r}\right),
\label{eq8}\\
\frac{{\rm d}\delta q}{{\rm d} r}&=&\frac{i\omega'}{v_r}\delta q +\delta\left(\frac{\cal L}{\rho v_r}\right),
\label{eq9}\end{aligned}$$ where $\mu$ and $\omega'$ are defined by $$\begin{aligned}
\mu^2&\equiv& 1-\frac{c^2}{\omega'^2}(1-{\cal M}^2)\left(\frac{m^2}{r^2}+k_z^2 \right),
\label{eq10}\\
\omega'&\equiv&\omega-\frac{mL}{r^2}.
\label{eq11}\end{aligned}$$ These equations are solved by imposing the Rankine-Hugoniot relations for the perturbed quantities, which are written as, $$\begin{aligned}
\frac{\delta f_{\rm sh}}{\omega}&=&iv_{r,1} \Delta\zeta\left(1-\frac{v_{r,\rm sh}}{v_{r,1}}\right),
\label{eq12}\\
\delta h_{\rm sh}&=&-i\frac{\omega'}{v_{r,\rm sh}}\Delta\zeta\left(1-\frac{v_{r,\rm sh}}{v_{r,1}}\right),
\label{eq13}\\
\frac{\delta S_{\rm sh}}{\gamma}&=&i\frac{\omega' v_{r,1}}{c_{\rm sh}^2}\Delta\zeta\left(1-\frac{v_{r,\rm sh}}{v_{r,1}}\right)^2
-\frac{{\cal L}_{\rm sh}-{\cal L}_1}{\rho_{\rm sh} v_{r,\rm sh}}\frac{\Delta \zeta}{c_{\rm sh}^2}\nonumber\\
&&+\left(1-\frac{v_{r,\rm sh}}{v_{r,1}}\right)\frac{\Delta\zeta}{c_{\rm sh}^2}\left(\frac{v_{r,1} v_{r,\rm sh}}{r_{\rm sh}}+\frac{L^2}{r_{\rm sh}^3}-\frac{{\rm d} \Phi}{{\rm d} r}\right),
\label{eq14}\\
\delta q_{\rm sh}&=&-\frac{{\cal L}_{\rm sh}-{\cal L}_1}{\rho_{\rm sh} v_{r,\rm sh}}\Delta\zeta,
\label{eq15}\end{aligned}$$ where the subscripts ’${\rm sh}$’ and ’1’ refer to the values just below and above the shock, respectively; $\Delta \zeta$ is the radial displacement of the shock surface. Since we have assumed that the flow above the shock is cold, the cooling rate ${\cal L}_1=0$. In addition to these equations, we impose the condition that the radial velocity perturbation vanishes at the inner boundary ($\delta v_r =0$). The derivations of the basic equations and the boundary conditions are shown in the Appendix A.
When rotation is neglected ($L=0$, $\omega'=\omega$), we remark the formal resemblance between the above formulation and the formulation of FGSJ07 describing a spherical flow. The only difference is the expression for the parameter $\mu^2$ in Eq. (\[eq10\]), where $(m^2+r^2k_z^2)$ replaces $l(l+1)$, and a geometrical factor $2$ in the boundary condition for the entropy perturbation ($v_{r,\rm sh}v_{r,1}$ replaces $2v_{r,\rm sh}v_{r,1}$ in Eq. \[eq14\]).
For a small value of the angular momentum $L$, we also remark that the effect of the centrifugal force $L^2 /r^3$ on the stationary flow is quadratic, and so is its effect on the entropy perturbation in Eq. (\[eq14\]). The only first order effect of rotation on the differential system satisfied by $\delta f,\delta h,\delta S,\delta q$, is the Doppler shift described by $\omega'$ in Eq. (\[eq11\]).
Results\[results\]
==================
Given the resemblance of formulations, let us first check that the stability properties of the cylindrical flow without rotation resemble those of the spherical flow studied by FGSJ07 despite the different geometry. In the spherical problem, the axisymmetric mode $m=0$ and the spiral modes $\pm m$ of a given degree $l$ have exactly the same growth rate (FGSJ07). In the cylindrical flow, the spiral modes $\pm m$ are also degenerate without rotation. A numerical resolution of the eigenfrequencies shows that the instability is dominated by the mode $m=\pm1$ if $r_{\rm sh}/r_*\ge2$, and by a larger $|m|$ for smaller shock radius, exactly as observed in the spherical flow (Fig. 6 of FGSJ07). The growth rate of the axisymmetric mode ($m=0$, $k_z>0$) however, is expected to differ from the spiral modes ($m>0$, $k_z=0$) in a cylindrical flow. For example if $r_{\rm sh}/r_*=5$, the most unstable axisymmetric mode is twice as slow as the most unstable spiral mode.
The dependence of the growth rate on the specific angular momentum is shown in Fig. \[fig1\] for the spiral modes $m=\pm1,\pm2$ in a flow where $r_{\rm sh}/r_*=5$. The growth rate of the modes rotating in the same direction as the flow ($m>0$) is increased by rotation, whereas the counter-rotating modes ($m<0$) are stabilized. The increase of the growth rate is almost proportional to the specific angular momentum. In a rotating flow with $L\sim 200\times 2\pi\cdot 10^{12}[{\rm cm}^2/{\rm s}]$ ($f_{\rm p}=200[{\rm Hz}]$), the growth rate of the fundamental mode $m=1$ is twice its value in a non rotating flow of same size. The strong effect of rotation on the growth rate of SASI does not seem to depend on the presence of a corotation radius $r_{\rm co}$, defined by ${\rm Re}(\omega) -mL/r_{\rm co}^2=0$, also displayed in Fig. \[fig1\]. We also investigated the effects of rotation on the axisymmetric modes ($m=0$) for various values of the vertical wave number $k_z$, and found that their growth rates are hardly affected, by less than one percent of $|v_{r,\rm sh}|/r_{\rm sh}$ in our study (Fig. \[fig2\]). A global overview of the parameter space of the cylindrical SASI is displayed in Fig. \[fig3\], which indicates the azimuthal wavenumber $m$ of the most unstable mode for a wide range of shock radii and specific angular momentum. The asymmetric one-armed spiral mode is unstable in most of the parameter space, and always more unstable than without rotation. The growth rate of the mode $m=2$ exceeds that of the mode $m=1$ (by less than $10\%$ in the example of Fig. 1), as the specific angular momentum is increased.
Discussion\[discussion\]
========================
Instability mechanism {#mechanism}
---------------------
As underlined in Sect. 3, the dynamical effect of the centrifugal force on the stationary flow is modest. We anticipated in Sect. 2 that the only linear effect of angular momentum is a Doppler shift of the eigenfrequency $\omega'=\omega-m\Omega(r)$, where $\Omega(r)$ is the local rotation frequency. This leaves the axisymmetric mode $m=0$ unaffected and explains the relative insensitivity of its growth rate with respect to the rotation rate, at least for moderate angular momentum. The strong effect of rotation on the growth rate of the spiral modes can thus be traced back to this Doppler shifted frequency. What is the mechanism of the instability ? As seen in the previous section, the destabilizing role of rotation does not seem related to the presence or absence of a corotation radius, thus discarding a Papaloizou-Pringle mechanism [@gol85]. Two possibilities have been proposed for the mechanism of SASI without rotation; one is the advective-acoustic mechanism [@fog00; @fog01; @fog02] and the other is the purely acoustic mechanism [@blo06]. Up to now, there is no satisfactory direct argument for the mechanism of the modes with a long wavelength. FGSJ07 used a WKB approximation to prove that the instability of the modes with a short wavelength is due to an advective-acoustic mechanism and extrapolated this conclusion to the modes with a long wavelength, which are the most unstable. This method, recalled in Appendix B, is based on the identification of acoustic waves and advected waves at a radius immediately below the shock surface, and the measurement of their coupling coefficients, above this radius due to the shock, and below this radius due to the flow gradients. These coupling processes are responsible for the existence of two cycles, namely a purely acoustic cycle characterized by an efficiency ${\cal R}$, and an advective-acoustic cycle characterized by an efficiency ${\cal Q}$.
By using the same method, the present study does not address directly the instability mechanism of long wavelength modes. However, the WKB approximation enables us to describe, in a conclusive manner, the instability mechanism of short wavelength modes affected by rotation. First we checked that when the shock distance is increased ($r_{\rm sh}=20r_*$), the overtones are also unstable and their growth rate is an oscillatory function of the frequency similar to Fig. 7 of FGSJ07. The effect of rotation on the advective-acoustic cycle is illustrated by Fig 3, for the spiral modes $m=\pm1$ corresponding to the $10$-th overtone, as a function of the rotation rate. The cycle efficiency ${\cal Q}$ is strongly amplified by rotation if $m>0$, while strongly damped if $m<0$. The stabilization of the counter-rotating spiral coincides with a marginally stable cycle ${\cal Q}\sim1$. The calculation of the amplification factor ${\cal R}$ of perturbations during each purely acoustic cycle indicates its stability (${\cal R}<1$). Contrary to the expectation of @lam07 (see next subsection), rotation clearly favours the spiral mode of the advective-acoustic cycle.
This consequence of rotation established unambiguously for short wavelength perturbations is identical to the influence of rotation on the fundamental mode of SASI: we consider this a new hint that the advective-acoustic mechanism can be extrapolated to low frequencies. The detailed analysis of the consequences of the Doppler shifted frequency on the increase of the advective-acoustic efficiency ${\cal Q}$ will be presented elsewhere ([@yam08], in preparation).
Comments on the Results of Laming (2007) {#comment}
----------------------------------------
The effect of rotation on the growth rate of SASI, established in Sect. 3 in a cylindrical geometry, is qualitatively similar to the effect conjectured by [@lam07] (hereafter L07). Nevertheless, their investigation about the instability mechanism led them to a different interpretation of the roles of the acoustic and advective-acoustic cycles.
We must point out a fundamental difference between the method of L07 and ours: by using a WKB approximation, we have carefuly defined the range of validity of our method, namely short wavelength modes. This guarantees that the advective-acoustic interpretation of the instability mechanism is physical and robust, at least in some parameter range. In contrast, the existence of a purely acoustic instability is still a conjecture because the domain of validity of the method used by L07 is ambiguous: their analytical derivation of a dispersion relation when advection is included requires to neglect terms of order $(v_r /\omega r)$ while terms of order ${\cal M}$ are retained. This approximation is not supported by the results of their Fig. 2, which indicates that $(v_{r,{\rm sh}}/\omega r_{\rm sh})$ is comparable to or larger than ${\cal M}_{\rm sh}$ for the modes $l=0$ and $l=1$. An accurate description of this acoustic mode, even in a simplified set up, would be useful to gain confidence in its possible existence.
In addition to the question of the validity of the approximations used by L07, we find that our results invalidate their reasoning concerning the instability mechanism. They proposed that the advective-acoustic mechanism would be essential if $r_{\rm sh}/r_*\ge 10$, whereas a purely acoustic unstable process would be dominant for small shock radii, and they argued that rotation is a key ingredient to discriminate between the two mechanisms. When rotation is included, its effect on SASI has been attributed by L07 to a purely acoustic mechanism, despite the results of their Table 3. However, their view that rotation cannot possibly enhance the growth of the advective-acoustic cycle is clearly incorrect, at least for the short wavelength modes (our Fig. \[fig4\]).
Consequences of rotation on supernova explosions {#consequence}
------------------------------------------------
The perturbative study of a simple cylindrical configuration has enabled us to cover a large parameter space of shock radii and rotation rates, in order to (i) demonstrate the linear selection of non-axisymmetric modes, (ii) establish a correlation between the preferred direction of the spiral SASI and the rotation of the collapsing core, (iii) identify the advective-acoustic mechanism at work for short wavelength spiral perturbations.
The fact that rotation favours a spiral mode $m=1,2$ in a cylindrical flow seems directly connected to the property observed by [@blo07a] in their 3D simulations including rotation. Tracing back the main influence of rotation to the local Doppler shifted frequency $\omega-m\Omega$, we may indeed expect a similar destabilization of the spiral modes with a positive value of $m$, a stabilization of the counter-rotating ones, and a comparatively weak influence on the axisymmetric modes.
Even a moderate amount of angular momentum results in a shortening of the growth time of SASI through the destabilization of a non-axisymmetric mode. The promising consequences of SASI on both the explosion mechanisms and the pulsar kick could thus be considerably modified, since they were established on the basis of axisymmetric numerical simulations [@bur06; @bur07; @mar07; @sch04; @sch06]. Our study suggests that the effect of rotation on the linear phase of SASI can be safely neglected only for slowly rotating progenitors with a specific angular momentum $L\ll 2\pi \cdot 10^{14}$ cm$^2/$s. Although a fast growth of SASI might be helpful to an early shock revival, the dynamical effects of a spiral mode $m=1$, and even $m=2$, on the possible explosion mechanisms are not known yet.
If the direction of the kick were determined by the geometry of the most unstable $l=1$ SASI mode, our perturbative approach would suggest a kick-spin misalignment. The strength of the equatorial kick may be diminished by the domination of a symmetric mode $m=2$. It is worth noting however that the relationship between the timescale of the most unstable SASI mode and the onset of explosion is not straightforward, and should be evaluated by future 3D numerical simulations. Our linear approach modestly aims at guiding our intuition for the interpretation of these simulations.
Derivation of the Basic Equations
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Basic Equations
---------------
The basic equations describing the flow are $$\begin{aligned}
\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mbox{\boldmath $v$})=0,\label{cont}\\
\frac{\partial \mbox{\boldmath $v$}}{\partial t}+\mbox{\boldmath $w$}\times \mbox{\boldmath $v$}
+\nabla \left(\frac{|\mbox{\boldmath $v$}|^2}{2}+\frac{c^2}{\gamma-1}+\Phi \right)=\frac{c^2}{\gamma} \nabla S,\label{mom}\\
\frac{\partial S}{\partial t}+\mbox{\boldmath $v$}\cdot \nabla S=\frac{\cal L}{P}.\label{energy}\end{aligned}$$ Small amplitude perturbations are superimposed onto the above equations. From the transverse components $(\theta,z)$ of the Euler equation, the vorticity can be expressed as follows: $$\begin{aligned}
\delta w_\theta&=&{ik_z\over v_r}\left(\delta f-{L\over r}\delta v_\theta+\delta q-{c^2\over\gamma}\delta S\right)-{i\omega'\over v_r}\delta v_z,\label{wthet}\\
\delta w_z&=&i\omega{\delta v_\theta\over v_r}+{im\over rv_r}\left({c^2\over\gamma}\delta S-\delta f-\delta q\right).\label{wzed}\end{aligned}$$ From the definition of the vorticity vector, $$\begin{aligned}
\delta w_r&\equiv&{im\over r}\delta v_z-ik_z\delta v_\theta,\\
{{\rm d} r\delta v_\theta\over{\rm d} r}&=&im\delta v_r+r\delta w_z,\label{drvt}\\
{{\rm d} \delta v_z\over{\rm d} r}&=&ik_z\delta v_r-\delta w_\theta.\label{dvz}\end{aligned}$$ Using the two equations (\[wthet\]) and (\[wzed\]) and the definition of vorticity leads to: $$\begin{aligned}
\left({{\rm d}\over{\rm d} r}-{i\omega'\over v_r}\right)(r\delta w_r)&=&0.\label{conswr}\end{aligned}$$ The radial Euler equation combined with Eq. (\[eq8\]) and (\[eq9\]) is $$\begin{aligned}
{{\rm d} \delta f\over {\rm d} r}&=&i\omega\delta v_r + {L\over r}\delta w_z+{i\omega'\over v_r}
\left({c^2\over\gamma}\delta S-\delta q\right).\label{dfdr}\end{aligned}$$ Guided by the conservation of $\delta K$ in a radial flow [@fog01], let us define the quantities $\delta K_1,\delta K_2$ as follows: $$\begin{aligned}
\delta K_1&\equiv&v_r r\delta w_z-im\left({c^2\over\gamma}\delta S-\delta q\right),\label{defk1}\\
&=&i\omega r\delta v_\theta-im\delta f,\label{dk1}\\
\delta K_2&\equiv&v_r\delta w_\theta+ik_z\left({c^2\over\gamma}\delta S-\delta q\right),\\
&=&-i\omega' \delta v_z+ik_z\left(\delta f-{L\over r}\delta v_\theta\right),\label{dk2}\end{aligned}$$ where Eq. (\[dk1\]) and Eq. (\[dk2\]) are deduced from Eqs. (\[wthet\]) and (\[wzed\]). The flow quantities $\delta v_\theta$, $\delta v_z$, $\delta w_\theta$, $\delta w_z$ can be expressed with $\delta K_1$ and $\delta K_2$ using Eqs. (\[defk1\]-\[dk2\]): $$\begin{aligned}
\delta v_\theta &=& {m\over r}{\delta f\over\omega}-{i\delta K_1\over \omega r},\label{dvtheta}\\
\delta v_z &=& k_z{\delta f\over\omega}+{i\over\omega'}\left(\delta K_2+{Lk_z\over\omega r^2}\delta K_1\right),\\
\delta w_\theta&=&{ik_z\over v_r}\left(\delta q -{c^2\over \gamma}\delta S\right)+{\delta K_2\over v_r},\\
\delta w_z&=&-{im\over v_r r}\left(\delta q -{c^2\over \gamma}\delta S\right)+{\delta K_1\over rv_r}.\label{dwz}\end{aligned}$$ Using Eq. (\[drvt\]), (\[dvz\]) and (\[dfdr\]), we can prove that $$\begin{aligned}
\left({{\rm d}\over{\rm d} r}-{i\omega'\over v_r}\right)\delta K_1&=&0,\label{consk1}\\
\left({{\rm d}\over{\rm d} r}-{i\omega'\over v_r}\right)\delta K_2&=&-{2L\over r^2}\delta w_r.\label{consk2}\end{aligned}$$ The perturbations of radial velocity, sound speed and density are related to $f,h,\delta S,\delta q$ as follows: $$\begin{aligned}
{\delta v_r\over v_r}&=&{1\over 1-{\cal M}^2}\left(\delta h+\delta S-{\delta f\over c^2}+{L\over rc^2}\delta v_\theta
-{\delta q\over c^2}\right),\\
{\delta c^2\over c^2}&=&{\gamma-1\over 1-{\cal M}^2}\left({\delta f\over c^2}-{L\over rc^2}\delta v_\theta
+{\delta q\over c^2}-{\cal M}^2 \delta h-{\cal M}^2\delta S\right),\\
{\delta \rho\over \rho}&=&{1\over 1-{\cal M}^2}\left({\delta f\over c^2}-{L\over rc^2}\delta v_\theta+{\delta q\over c^2}
-{\cal M}^2 \delta h-\delta S\right).\end{aligned}$$ The continuity of mass flux is $$\begin{aligned}
{{\rm d} \delta h\over{\rm d} r}={i\omega'\over v_r}{\delta\rho\over \rho}-{im\over r v_r}\delta v_\theta.\end{aligned}$$ The differential system satisfied by $\delta f, \delta h,\delta S, \delta q$ is $$\begin{aligned}
{{\rm d} \delta f\over{\rm d} r}&=&{i\omega c^2\over v_r(1-{\cal M}^2)}\left\lbrace
{\cal M}^2 \delta h -{\cal M}^2{\omega'\over\omega}{\delta f\over c^2}
+\left\lbrack1+(\gamma-1){\cal M}^2\right\rbrack{\delta S\over\gamma}-{\delta q\over c^2}
\right\rbrace\nonumber\\
&&+{L\over v_r r^2}{\delta K_1\over 1-{\cal M}^2},\label{appdf}\\
{{\rm d} \delta h\over{\rm d} r}&=&{i\omega'\over v_r(1-{\cal M}^2)}\left(
{\mu^{2} \over c^{2}}{\omega'\over \omega} \delta f
-{\cal M}^2 \delta h- \delta S+{\delta q\over c^2}\right)\nonumber\\
&&-{\delta K_1\over\omega r^2 v_r}
\left( m+{L\omega'\over c^2(1-{\cal M}^2)}\right)
\label{dhp},\\
{{\rm d} \delta S\over{\rm d} r}&=&{i\omega'\over v_r}\delta S
+\delta\left({{\cal L}\over Pv_r}\right), \label{dsp}
\\
{{\rm d} \delta q\over {\rm d} r}&=&{i\omega'\over v_r}\delta q
+\delta\left({{\cal L}\over \rho v_r}\right). \label{difq}\end{aligned}$$ The transverse velocity perturbations $\delta v_\theta,\delta v_z$ at the shock are expressed in Appendix A.2. by: $$\begin{aligned}
\delta v_{\theta,{\rm sh}}&=&{im\over r_{\rm sh}}\Delta\zeta(v_{r,1}-v_{r,{\rm sh}}),\label{theta}\\
\delta v_{z,{\rm sh}}&=&ik_z\Delta\zeta(v_{r,1}-v_{r,{\rm sh}}). \label{vertical}\end{aligned}$$ Together with the boundary conditions (Eqs. (\[eq12\]-\[eq15\])) established in Appendix A.2., we deduce from the definition of $\delta w_r,\delta K_1,\delta K_2$ that these three quantities vanish at the shock. From the conservation Eqs. (\[conswr\]), (\[consk1\]), (\[consk2\]), we conclude that $\delta w_r$, $\delta K_1$ and $\delta K_2$ are uniformly zero throughout the flow. The flow quantities $\delta v_\theta$, $\delta v_z$, $\delta w_\theta$ expressed in Eqs. (\[dvtheta\]-\[dwz\]) are thus simplified accordingly, and the differential system (\[appdf\]-\[difq\]) is transformed into the simpler Eqs. (\[eq6\]-\[eq9\]).
Boundary Conditions
-------------------
The Rankine-Hugoniot relation is written as
$$\begin{aligned}
\rho_1({\bf v}_1 -{\bf v}_{\rm s})\cdot \frac{\bf n}{|{\bf n}|}&=&\rho_{\rm sh}({\bf v}_{\rm sh} -{\bf v}_{\rm s})\cdot \frac{\bf n}{|{\bf n}|},
\label{b1}\\
\rho_1\left\{({\bf v}_1 -{\bf v}_{\rm s})\cdot \frac{\bf n}{|{\bf n}|}\right\}^2+\frac{\rho_1 c_1^2}{\gamma}&=&\rho_{\rm sh}\left\{({\bf v}_{\rm sh} -{\bf v}_{\rm s})\cdot \frac{\bf n}{|{\bf n}|}\right\}^2+\frac{\rho_{\rm sh} c_{\rm sh}^2}{\gamma},
\label{b2}\\
({\bf v}_1 -{\bf v}_{\rm s})\cdot {\bf t_1}&=&({\bf v}_{\rm sh} -{\bf v}_{\rm s})\cdot {\bf t_1},
\label{b3}\\
({\bf v}_1 -{\bf v}_{\rm s})\cdot {\bf t_2}&=&({\bf v}_{\rm sh} -{\bf v}_{\rm s})\cdot {\bf t_2},
\label{b4}\\
\frac 12 \left\{({\bf v}_1 -{\bf v}_{\rm s})\cdot \frac{\bf n}{|{\bf n}|}\right\}^2 +\frac{c_1^2}{\gamma-1}&=&\frac 12
\left\{({\bf v}_{\rm sh} -{\bf v}_{\rm s})\cdot \frac{\bf n}{|{\bf n}|}\right\}^2
+\frac{c_{\rm sh}^2}{\gamma-1},
\label{b5}\end{aligned}$$
where ${\bf v}_{\rm s}$ is the velocity vector of the shock surface and ${\bf n}$, ${\bf t_1}$ and ${\bf t_2}$ are the vector normal and tangent to the shock surface which is written at first order as follows,
$$\begin{aligned}
{\bf n}&=&\left(1,-\frac{1}{r_{\rm s}}\frac{\partial r_{\rm s}}{\partial \theta},
-\frac{\partial r_{\rm s}}{\partial z}\right),\\
{\bf t_1}&=&\left(\frac{\partial r_{\rm s}}{\partial \theta},
r_{\rm s},0\right),\\
{\bf t_2}&=&\left(\frac{\partial r_{\rm s}}{\partial z},0,1\right).\end{aligned}$$
Considering small perturbations of the above Eqs, (\[b1\])-(\[b5\]), we obtain $$\begin{aligned}
\rho_{\rm sh}v_{r,{\rm sh}}\delta h_{\rm sh}+i\omega' \Delta \zeta(\rho_{\rm sh}-\rho_1)=\Delta \zeta \left[\frac{d}{dr} (\rho v_r)_1 -\frac{d}{dr}(\rho v_r)_{\rm sh}\right] ,\\
v_{r,{\rm sh}}^2 \delta \rho_{\rm sh}+2\rho_{\rm sh}v_{r,{\rm sh}}\delta v_{r,{\rm sh}}+\frac{2}{\gamma}\rho_{\rm sh}c_{\rm sh}\delta c_{\rm sh}+\delta \rho_{\rm sh}\frac{c_{\rm sh}^2}{\gamma} \nonumber\\
=\Delta \zeta\left[\frac{d}{dr}(\rho v_r^2 +P)_1 -\frac{d}{dr}(\rho v_r^2 +P)_{\rm sh}\right],\\
\delta f_{\rm sh}-v_{{\theta},{\rm sh}}\delta v_{\theta,{\rm sh}}+\delta q_{\rm sh}+i\omega' \Delta \zeta(v_{\rm sh}-v_1) \nonumber\\
=\Delta \zeta \left[\frac{d}{dr}\left(\frac{v_r^2}{2}+\frac{c^2}{\gamma-1}\right)_1 -\frac{d}{dr}\left(\frac{v_r^2}{2}+\frac{c^2}{\gamma-1}\right)_{\rm sh}\right],\end{aligned}$$ and Eqs. (\[theta\])-(\[vertical\]). Using the relations in the steady flow, $$\begin{aligned}
\frac{d}{dr}(\rho v_r)&=&-\frac{\rho v_r}{r},\\
\frac{d}{dr}(\rho v_r^2 +P)&=&\rho\frac{d\Phi}{dr}-\frac{\rho v_r^2}{r},\\
\frac{d}{dr}\left(\frac{v_r^2 +v_{\theta}^2}{2}+\frac{c^2}{\gamma-1}\right)&=&\frac{\cal L}{\rho v_r}+\frac{d\Phi}{dr},\end{aligned}$$ we obtain the boundary conditions (\[eq12\])-(\[eq15\]).
WKB Method for the Calculation of the Amplification Coefficients
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In order to interpret the complex eigenfrequency $\omega$ of a given eigenmode, we compute the efficiency of the advective-acoustic and purely acoustic cycles associated to the real frequency $\omega_r\equiv$ Re$(\omega)$ of this eigenmode. The coefficient ${\cal R}(\omega_r)$ is defined by the amplification of perturbations after one purely acoustic cycle, initiated at the shock by an acoustic wave propagating downward. ${\cal Q}(\omega_r)$ and ${\cal Q}^q(\omega_r)$ measure the amplification of pertubations through an advective-acoustic cycle, initiated at the shock by the advection of an entropy perturbation $\delta S$ with $\delta q=0$, or a heat perturbations $\delta q$ with $\delta S=0$ respectively. Each of these coefficients ${\cal R}$, ${\cal Q}$ and ${\cal Q}^q$ is the product of the coupling coefficient at the shock (${\cal R}_{\rm sh}$, ${\cal Q}_{\rm sh}$, or ${\cal Q}_{\rm sh}^q$), multiplied by the coupling coefficient through the flow (${\cal R}_{\nabla}$, ${\cal Q}_{\nabla}$ or ${\cal Q}^q_{\nabla}$). The technique of calculation of each factor is the same as that described in the Appendix D of FGSJ07. The calculations are based on the decomposition of the variables onto the basis of advected and acoustic perturbations, which is exact when the flow is uniform. Even when the flow is moderately inhomogeneous, a similar decomposition is obtained using a WKB approximation. Since the definitions of the variables $\delta f,\delta h,\delta S,\delta q$ employed in this paper are slightly different from those in FGSJ07, the decomposition is modified as follows: $$\begin{aligned}
\delta f=\delta f^{+}+\delta f^{-}+\delta f^{S}+\delta f^{q},\\
\delta h=\delta h^{+}+\delta h^{-}+\delta h^{S}+\delta h^{q}.\end{aligned}$$ The superscripts $+$, $-$, $S$, $q$ refer to the contributions of the ingoing and outgoing acoustic wave, the advected quantities $\delta S$ and $\delta q$, respectively. Adopting the WKB approximation, the quantities $\delta f^{\pm},\delta h^{\pm}$ associated with the acoustic waves satisfy the differential system (\[eq6\]-\[eq9\]) where $\delta S=0$ and $\delta q=0$, and the radial derivatives are replaced by a multiplication by $ik_{\pm}$: $$\begin{aligned}
ik_{\pm}\delta f^{\pm}=\frac{i\omega c^2}{v_r (1-{\cal M}^2)}\left({\cal M}^2 \delta h^{\pm}
-{\cal M}^2 \frac{\omega'}{\omega}\frac{\delta f^{\pm}}{c^2}\right),\\
ik_{\pm}\delta h^{\pm}=\frac{i\omega'}{v_r (1-{\cal M}^2)}\left(\frac{\mu^2}{c^2}\frac{\omega'}{\omega}\delta f^{\pm}-{\cal M}^2 \delta h^{\pm}\right).\end{aligned}$$ The dispersion relation of acoustic waves corresponds to: $$\begin{aligned}
k_{\pm}=\frac{\omega'}{c}\frac{{\cal M}\mp \mu}{1-{\cal M}^2}.\end{aligned}$$ The advected quantities satisfy the differential system (\[eq6\]-\[eq9\]) where the radial derivatives are replaced by a multiplication by $ik_0$, where $k_0\equiv \omega'/v$. $$\begin{aligned}
ik_0 \delta f^S=\frac{i\omega c^2}{v_r(1-{\cal M}^2)}\left\{{\cal M}^2 \delta h^S
-{\cal M}^2 \frac{\omega'}{\omega}\frac{\delta f^S}{c^2}
+[1+(\gamma -1){\cal M}^2 ]\frac{\delta S}{\gamma}\right\},\\
ik_0\delta h^S=\frac{i\omega'}{v_r(1-{\cal M}^2)}\left(\frac{\mu^2}{c^2}\frac{\omega'}{\omega}\delta f^S-{\cal M}^2 \delta h^S-\delta S\right),\\
ik_0 \delta f^q=\frac{i\omega c^2}{v_r(1-{\cal M}^2)}\left({\cal M}^2 \delta h^q
-{\cal M}^2 \frac{\omega'}{\omega}\frac{\delta f^q}{c^2}
-\frac{\delta q}{c^2}\right),\\
ik_0\delta h^q=\frac{i\omega'}{v_r(1-{\cal M}^2)}\left(\frac{\mu^2}{c^2}\frac{\omega'}{\omega}\delta f^q-{\cal M}^2 \delta h^q+\frac{\delta q}{c^2}\right).\end{aligned}$$ Solving these two sets of equations leads to: $$\begin{aligned}
\delta h^{\pm}={\pm}\frac{\omega'}{\omega}\frac{\mu}{{\cal M}c^2}\delta f^{\pm},\\
\delta f^{S}=\frac{\omega}{\omega'}\frac{1-{\cal M}^2}{1-\mu^2 {\cal M}^2}\frac{c^2}{\gamma}\delta S,\\
\delta h^{S}=\frac{\omega'}{\omega}\frac{\mu^2}{c^2}\delta f^S -\delta S,\\
\delta f^{q}=-\frac{\omega}{\omega'}\frac{1-{\cal M}^2}{1-\mu^2 {\cal M}^2}\delta q,\\
\delta h^{q}=\frac{1-\mu^2}{1-\mu^2 {\cal M}^2}\frac{\delta q}{c^2}.\end{aligned}$$ The coupling coefficients ${\cal R}_{\rm sh}$, ${\cal Q}_{\rm sh}$ and ${\cal Q}^q_{\rm sh}$ are obtained by decomposing the variables at the boundary described by Eqs. (\[eq12\]-\[eq15\]) onto the basis of acoustic and advected perturbations, immediately below the shock: $$\begin{aligned}
\delta f_{\rm sh}&=& \delta f_{\rm sh}^++\delta f_{\rm sh}^-+\delta f_{\rm sh}^S+\delta f_{\rm sh}^q,\\
{\cal R}_{\rm sh}&\equiv& {\delta f_{\rm sh}^+\over \delta f_{\rm sh}^-},\\
{\cal Q}_{\rm sh}&\equiv& {\delta f_{\rm sh}^S\over \delta f_{\rm sh}^-},\\
{\cal Q}^q_{\rm sh}&\equiv& {\delta f_{\rm sh}^q\over \delta f_{\rm sh}^-}.\end{aligned}$$ The three coupling coefficients ${\cal R}_{\nabla}$, ${\cal Q}_{\nabla}$ and ${\cal Q}^q_{\nabla}$ are calculated by measuring numerically, at a radius $R$ immediately below the shock ($R= r_{\rm sh}$), the acoustic feedback $\delta f^-(R)$ that would be produced, either by an ingoing purely acoustic perturbation $\delta f^+$, or a purely advective perturbation $\delta f^S$, or $\delta f^q$. Each of these three coefficients is calculated by integrating the differential system (\[eq6\]-\[eq9\]) from the radius $R$ down to the accretor surface. For example, the boundary condition used at $r=R$ for the calculation of ${\cal Q}_{\nabla}$ involves a perturbation of entropy and vorticity $\delta f^S(R)$, and the right amount of acoustic feedback $\delta f^-(R)$, $$\begin{aligned}
\delta f(R)&=&\delta f^S(R)+\delta f^-(R),\end{aligned}$$ such that the inner boundary condition at the accretor surface is satisfied. The coupling coefficient ${\cal Q}_\nabla$ measures the efficiency of this acoustic feedback: $$\begin{aligned}
{\cal Q}_\nabla&\equiv&{\delta f^-(R)\over \delta f^S(R)}.\end{aligned}$$ Since ${\cal Q}^q$ is negligible compared to both ${\cal Q}$ and ${\cal R}$, we discuss only ${\cal R}$ and ${\cal Q}$ in the text. The WKB decomposition is a good approximation when the inhomogeneity caused by the convergence of the flow, gravity and cooling is moderate within a wavelength of the perturbation. Since we use this decomposition immediately below the shock front, the approximation is valid when the inhomogeneity of the steady flow just below the shock is sufficiently small. The amplification coefficients ${\cal Q}$ and ${\cal R}$ illustrated in our Fig. \[fig4\] were computed in a flow with a large shock radius ($r_{\rm sh}=20r_{\ast}$), for a short wavelength mode (tenth overtone), in order to obtain reliable results.
Blondin, J. M., Mezzacappa, A., & DeMarino, C. 2003, , 584, 971 Blondin, J. M., & Mezzacappa, A. 2006, , 642, 401 Blondin, J. M., &Mezzacappa, A. 2007, Nature, 445, 58 Blondin, J. M., & Shaw, S. 2007, , 656, 366 Burrows, A., Livne, E., Dessart, L., Ott, C.D., & Murphy, J. 2006, , 640, 878 Burrows, A., Livne, E., Dessart, L., Ott, C.D., & Murphy, J. 2007, , 655, 416 Foglizzo, T., & Tagger, M. 2000, , 363, 174 Foglizzo, T. 2001, , 368, 311 Foglizzo, T. 2002, , 392, 353 Foglizzo, T., Galletti, P., Scheck, L., & Janka, H.-Th. 2007, , 654, 1006 (FGSJ07) Galletti, P., Foglizzo, T. 2005, in Proc. SF2A-2005 Meeting, ed. F. Casoli et al. (Les Ulis: EDP Sciences), 487 Goldreich, P., & Narayan, R. 1985, , 213, 7 Heger, A., Woosley, S. E., & Spruit, H. C. 2005, , 626, 350 Laming, J.M. 2007, , 659, 1449 (L07) Liebendörfer, M., Rampp, M., Janka, H.Th., & Mezzacappa, A. 2005, , 620, 840 Marek, A., & Janka, H.-Th. 2007, submitted to , (astro-ph/0708.3372) Ohnishi, N., Kotake K., & Yamada, S. 2006, , 641, 1018 Scheck, L., Plewa, T., Janka, H.-Th., Kifonidis, K., & Müller, E. 2004, , 92, 011103 Scheck, L., Kifonidis, K., Janka, H.-Th., & Müller, E. 2006, , 457, 963 Scheck, L., Janka, H.-Th., Foglizzo, T., & Kifonidis, K. 2008, , 477, 931 Yamasaki, T., & Foglizzo, T. 2008, in preparation Yamasaki, T., & Yamada, S. 2005, , 623,1000 Yamasaki, T., & Yamada, S. 2007, , 656, 1019
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abstract: 'The most direct way to express arbitrary dependencies in datasets is to estimate the joint distribution and to apply afterwards the argmax-function to obtain the mode of the corresponding conditional distribution. This method is in practice difficult, because it requires a global optimization of a complicated function, the joint distribution by fixed input variables. This article proposes a method for finding global maxima if the joint distribution is modeled by a kernel density estimation. Some experiments show advantages and shortcomings of the resulting regression method in comparison to the standard Nadaraya-Watson regression technique, which approximates the optimum by the expectation value.'
author:
- |
Steffen Kühn\
Technische Universität Berlin\
Chair of Electronic Measurement and Diagnostic Technology\
Einsteinufer 17, 10585 Berlin, Germany\
steffen.kuehn@tu-berlin.de
title: Kernel Regression by Mode Calculation of the Conditional Probability Distribution
---
Introduction
============
Regression is a very important method in engineering and science for the estimation of the dependencies between two or more variables on the basis of some given sample points. The best known regression method is certainly the parametric regression technique after Legendre and Gauss, which minimizes the squared error between a model – often a polynom – and the samples.
The least squares method is fast and well suitable for strongly linearly correlated data, but seldom appropriate for high-dimensional problems with difficult, unknown, and non-linear dependencies. For these problems, non-parametric regression techniques – like kernel or Nadaraya-Watson regression methods – are more suitable ([@Nadaraya64; @Watson64]). The first step of Nadaraya-Watson regression is to estimate the unknown joint density distribution $p_{\boldsymbol{X},\boldsymbol{Y}}(\boldsymbol{x},\boldsymbol{y})$ of the given sample data $D = \{(\boldsymbol{x}_1,\boldsymbol{y}_1),\ldots,(\boldsymbol{x}_n,\boldsymbol{y}_n)\}$ by a kernel-density estimator [@Scott92]. The resulting model distribution has in the most general case the form $$\tilde{p}_{\boldsymbol{X},\boldsymbol{Y}}(\boldsymbol{x},\boldsymbol{y}) = \sum\limits_{i=1}^{m} a_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)\psi_i(\boldsymbol{y} - \boldsymbol{y}_i)
\label{f0}$$ with $m \leq n$ and $\sum_{i=1}^{m} a_i = 1$. Furthermore, the kernel functions $\phi_i$ and $\psi_i$ have to be normalized so that the integrals over all values for $\boldsymbol{x}$ and $\boldsymbol{y}$ are one. With this model, the conditional distribution $\tilde{p}_{\boldsymbol{Y},\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$ can be easily derived: $$\begin{split}
\tilde{p}_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x}) &= \frac{\tilde{p}_{\boldsymbol{X},\boldsymbol{Y}}(\boldsymbol{x},\boldsymbol{y})}{\tilde{p}_{\boldsymbol{X}}(\boldsymbol{x})} = \frac{\tilde{p}_{\boldsymbol{X},\boldsymbol{Y}}(\boldsymbol{x},\boldsymbol{y})}{\int\limits_{\boldsymbol{y}} \tilde{p}_{\boldsymbol{X},\boldsymbol{Y}}(\boldsymbol{x},\boldsymbol{y}) \mathrm{d}\boldsymbol{y}} \\
& = \frac{\sum\limits_{i=1}^{m} a_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)\psi_i(\boldsymbol{y} - \boldsymbol{y}_i)}{\sum\limits_{i=1}^{m} a_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)}.
\label{f1}
\end{split}$$ This distribution represents the relative probabilities for realizations of $\boldsymbol{y}$ given a vector $\boldsymbol{x}$. But for a regression, we do not need a probability distribution, but a single vector. The most intuitive choice is, of course, the mode of the conditional distribution, that means the value $\boldsymbol{y}$ for which $\tilde{p}_{\boldsymbol{Y}|\boldsymbol{X}}$ becomes maximal. For this case, the regression function $\tilde{f}(\boldsymbol{x})$ takes the specific form $$\tilde{\boldsymbol{y}} = \tilde{f}(\boldsymbol{x}) = \argmax\limits_{ \boldsymbol{y}}\{\tilde{p}_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})\}.\label{f2}$$ The difficulty is, however, that the maximization is not easy to calculate, because the expression (\[f1\]) is highly non-linear.
On the other hand, the expected value of $\tilde{p}_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})$ regarding $\boldsymbol{y}$ is easy to calculate: $$\int\limits_{\boldsymbol{y}} \boldsymbol{y}\,\tilde{p}_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{y}|\boldsymbol{x})\, \mathrm{d}\boldsymbol{y}= \frac{\sum\limits_{i=1}^{m} a_i\,\boldsymbol{y}_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)}{\sum\limits_{i=1}^{m} a_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)}.$$ The idea of Nadaraya and Watson was to approximate expression (\[f2\]) by $$\tilde{\boldsymbol{y}} \approx \frac{\sum\limits_{i=1}^{m} a_i\,\boldsymbol{y}_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)}{\sum\limits_{i=1}^{m} a_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)},$$ which is often sufficiently good. But there are also some potential problems.
![Joint distributions with unambiguous and ambiguous relations between $x$ and $y$.[]{data-label="fig1"}](surfaceschnitt){width="\columnwidth"}
Figure \[fig1\] demonstrates this for two different joint distributions. Case (A) is uncritical and describes essentially a hyperbolic tangent function. Case (B), however, causes problems. Here, the variables $x$ and $y$ are non-linearly correlated too, but the underlying dependency cannot be described by a function. The difficulty becomes obvious when considering the conditional distributions $p_{X|Y}(x|-0.6)$ and $p_{Y|X}(y|3)$, which are no longer unimodal. A computation of the expected value would yield in both cases zero, which is far away from probable values for $y = -0.6$ or $x = 3$.
In contrast to the Nadaraya-Watson method, the calculation of expression (\[f2\]) should not lead to problems. The $\argmax$-calculation cannot resolve the ambiguousness, of course, due to the fact that two global maxima exist, but it is better to return only *one* maximum than a completely incorrect value. Especially for high-dimensional tasks, this shortcoming of the Nadaraya-Watson method can be annoying, because the occurrence of ambiguousness is difficult to recognize. To overcome this “curse of compromise”, the next section proposes a method for solving expression (\[f2\]) directly if the density estimation is given in the form (\[f0\]).
Finding Global Maxima for Kernel Density Estimations
====================================================
The fundamental idea of the here proposed method to find the global maximum of a probability density function is to utilize its special properties. In general, the *global* maximum of an arbitrary function, which is, for example, given as a piece of code, can be found only by trial and error. In principle, this can be also applied to find the global maximum of a probability density function. But this “blind” search would be very inefficient and the remaining likelihood to find still better values does not become zero, regardless of how long the algorithm runs.
But for probability densities $h(\boldsymbol{y})$, this remaining likelihood can be reduced very fast by using $h$-distributed sample points, instead of evenly distributed samples. Why is it so? To answer this question, we assume that $q$ accordingly distributed sample points $\boldsymbol{y}_i$ with $i=1,\ldots,q$ have been generated. For each $\boldsymbol{y}_i$, there is a percentage $\alpha_i$ for more improbable realizations $\boldsymbol{y}$. Let $\alpha_i$ be $99\%$. The probability that all other $q-1$ generated samples have lower $\alpha$-values is $(99\%/100\%)^{q-1}$. For $q = 10000$, this probability is only $2.27\,10^{-44}$! In practice, this means that it is impossible not to come close to the global maximum with $10000$ $h$-distributed sample points. That is all.
Fortunately, the generation of accordingly distributed samples is not very difficult for kernel density estimations like expression (\[f0\]). In the first step, we insert for the calculation of expression (\[f2\]) the given value $\boldsymbol{x}$ and get $$\tilde{\boldsymbol{y}} = \argmax\limits_{\boldsymbol{y}}\left(h(\boldsymbol{y})\right)
\label{f4}$$ with $$h(\boldsymbol{y}) := \sum\limits_{i=1}^{m} b_i\,\psi_i(\boldsymbol{y} - \boldsymbol{y}_i)
\label{f6}$$ and $$b_i := \frac{a_i\,\phi_i(\boldsymbol{x} - \boldsymbol{x}_i)}{\sum_{j=1}^{m} \phi_j(\boldsymbol{x} - \boldsymbol{x}_j)}.$$ Note that $h$ fulfills the requirements for a probability density function. Furthermore, the $b_i$ can be interpreted as probabilities[^1] because of $\sum_{j=1}^{m} b_j = 1$.
In the next step, we generate a dataset $$D' = \{\boldsymbol{y}_1',\ldots,\boldsymbol{y}_q'\} \label{f5}$$ of $h$-distributed random samples. For this purpose, we can utilize the distribution function $H$ of the density function $h$: $$\begin{split}
H(\boldsymbol{y}) = \int\limits_{-\boldsymbol{\infty}}^{\boldsymbol{y}} \sum\limits_{i=1}^{m} b_i\,\psi_i(\boldsymbol{z} - \boldsymbol{y}_i) \mathrm{d}\boldsymbol{z} & = \sum\limits_{i=1}^{m} b_i\,\int\limits_{-\boldsymbol{\infty}}^{\boldsymbol{y}}\psi_i(\boldsymbol{z} - \boldsymbol{y}_i) \mathrm{d}\boldsymbol{z} \\
& = \sum\limits_{i=1}^{m} b_i\,\Psi_i(\boldsymbol{y} - \boldsymbol{y}_i).
\end{split}$$ The distribution functions $\Psi_i$ for the kernels $\psi_i$ are usually known or at least easy to calculate. The generation of the $k=1,\ldots,q$ random samples (\[f5\]) can be performed in three stages:
1. Choose randomly one of the $m$ kernels $\psi_i$ corresponding to the probabilities $b_i$.
2. Let $j$ be the choice of the first stage. Generate now a $\psi_j$-distributed random value using the distribution function $\Psi_j$.
3. Add the kernel center $\boldsymbol{y}_j$ to the random value from stage two to get a random sample $\boldsymbol{y}_k'$
After that, we calculate the function values $h(\boldsymbol{y}_k')$ for all $k=1,\ldots,q$ of dataset (\[f5\]). The argument $\boldsymbol{y}_k'$ for which $h(\boldsymbol{y}_k')$ becomes maximal is then a good starting point for a local optimization method like gradient ascent [@Duda00], for example.
Implementation Example
======================
The subsequent Matlab code[^2] snippet implements the described method for multidimensional Gaussian kernels with diagonal covariance matrix.
function [xm,pxm] = findMax(para,q)
m = length(para.a);
d = length(para.x(1,:));
cdf = zeros(1,m);
for (i=2:m) cdf(i) = cdf(i-1) + para.a(i-1); end
xr = zeros(q,d); yr = zeros(q,1);
for (i=1:q)
rv = rand(1); lvi = find(cdf < rv); ri = lvi(length(lvi));
xr(i,:) = randn(1,d).*sqrt(para.s(ri)) + para.x(ri);
yr(i) = KDE(xr(i,:),para);
end
[pxm,xi] = max(yr); xm = xr(xi,:);
The parameters of the expression (\[f6\]) are combined into the structure `para` with three elements: `para.a` are the weights $b_i$, `para.s` the standard deviations, and `para.x` the centers of the Gaussian kernels. The function `KDE` calculates the estimated density value for a given `x`:
function y = KDE(x,para);
m = length(para.a); y = 0;
for (i=1:m)
y = y + para.a(i)*gauss(x,para.x(i,:),para.s(i,:));
end
function y = gauss(x,m,s)
y = prod(1./(sqrt(2*pi)*s)).*exp(sum(-(x-m).^2./(2*s.^2)));
The gradient ascent is not performed in this example.
![Contour plots for a two-dimensional kernel density estimation with two global maxima at about $(1\,1)^T$ and $-(1\,1)^T$. Both were found by the algorithm (black marks). For the left hand plot, $q$ was $100$ and for the right hand plot $1000$.[]{data-label="fig2"}](q100-100 "fig:"){width="0.5\columnwidth"} ![Contour plots for a two-dimensional kernel density estimation with two global maxima at about $(1\,1)^T$ and $-(1\,1)^T$. Both were found by the algorithm (black marks). For the left hand plot, $q$ was $100$ and for the right hand plot $1000$.[]{data-label="fig2"}](q1000-100 "fig:"){width="0.5\columnwidth"}
Figure \[fig2\] shows the results of an experiment with function `findMax` for different values of `q`. The parameters of the probability density function $h(\boldsymbol{y})$ were
para.a = [0.45,0.45,0.1];
para.x = [[1,1];[-1,-1];[-1.5,1.5]];
para.s = [[1,1];[1,1];[0.5,0.5]];
That means that there are two global maxima – at approximately $(1\,1)^T$ and $-(1\,1)^T$. For this reason, both could be the result returned by the algorithm. But only one of these possibilities is returned per step. The plots also show that the distribution of the computed points becomes more compact with increasing size of $q$. Figure \[fig3\] shows the result for a more complex density with $80$ kernels and several local maxima.
![The result for a more complex density with $80$ kernels and several local maxima. $q$ was $100$ (left) and $1000$ (right).[]{data-label="fig3"}](q100-100-complex.pdf "fig:"){width="0.5\columnwidth"} ![The result for a more complex density with $80$ kernels and several local maxima. $q$ was $100$ (left) and $1000$ (right).[]{data-label="fig3"}](q1000-100-complex.pdf "fig:"){width="0.5\columnwidth"}
Regression Experiments
======================
This section investigates the properties of the described method. The first experiment compares the standard Nadaraya-Watson method and the proposed method with computation of the mode in view of its ability to estimate a clear functional dependency between $\boldsymbol{x}$ und $\boldsymbol{y}$. For this purpose, a dataset of $n = 1000$ random sample points of the function $y = \sin(x^{\frac{8}{5}})$ in the interval $[0,2\,\pi]$ was generated. Furthermore, a slight, Gaussian distributed noise with a standard deviation of $\sigma_N = 0.2$ was added to the $y$-values. The dataset is shown on the left of Figure \[fig4\].
![Two datasets.[]{data-label="fig4"}](D1.pdf "fig:"){width="0.5\columnwidth"} ![Two datasets.[]{data-label="fig4"}](D2.pdf "fig:"){width="0.5\columnwidth"}
Before applying the two regression methods, the distribution of the data has to be modeled by a kernel density. Different types of kernels can be applied. One of the simplest is the $d$-dimensional Gaussian kernel $$g(\boldsymbol{x},\boldsymbol{s}) = \prod\limits_{k=1}^{d}\frac{1}{\sqrt{2\,\pi} s_k} \exp\left(-\frac{x_k^2}{2\,s_k^2}\right)$$ with $\boldsymbol{s} = (s_1\,\ldots\,s_d)^T$ as only free parameter. Its application to the two dimensional problem of Figure \[fig4\] yields $$\tilde{p}(\boldsymbol{x}) = \frac{1}{n} \sum\limits_{i=1}^{n} g(\boldsymbol{x}-\boldsymbol{x}_i,\boldsymbol{s})$$ with $\boldsymbol{x} = (x\,y)^T$. For high-dimensional problems, the smoothness $\boldsymbol{s}$ has to be automatically optimized with respect to a certain quality measurement, such as the self-contribution [@Duin76] for example. Another method is plug-in estimation. A good overview about this topic is given by @Turlach93 or @Scott92.
For the two-dimensional dataset in Figure \[fig4\], it is still possible to estimate the smoothness parameter visually. For $\boldsymbol{s} = (0.1\,0.1)^T$, the resulting density is drawn as contour plot in Figure \[fig5\] at the top. Furthermore, the picture provides the result of the Nadaraya-Watson regression (left) and of the proposed method (right) as white dotted lines. Every point represents the outcome for a single given value $x$.
![The results of the Nadaraya-Watson method (left) and of the optimization method (right). The density estimation above shows a clear dependency between $x$ and $y$ – contrary to the other below.[]{data-label="fig5"}](NW1 "fig:"){width="0.5\columnwidth"} ![The results of the Nadaraya-Watson method (left) and of the optimization method (right). The density estimation above shows a clear dependency between $x$ and $y$ – contrary to the other below.[]{data-label="fig5"}](GO1 "fig:"){width="0.5\columnwidth"}
![The results of the Nadaraya-Watson method (left) and of the optimization method (right). The density estimation above shows a clear dependency between $x$ and $y$ – contrary to the other below.[]{data-label="fig5"}](NW2 "fig:"){width="0.5\columnwidth"} ![The results of the Nadaraya-Watson method (left) and of the optimization method (right). The density estimation above shows a clear dependency between $x$ and $y$ – contrary to the other below.[]{data-label="fig5"}](GO2 "fig:"){width="0.5\columnwidth"}
The picture demonstrates too that the Nadaraya-Watson regression performs clearly better. This is only at the first glance surprising. Formally, both regression methods should give the same results, because expected value and maximum are identical for the true conditional probability distribution $$p_{Y|X}(y|x) = g(y - \sin(x^{\frac{8}{5}}),\sigma_N).$$ But both regression techniques have different susceptibilities to estimation errors and the property of the effective value to average out noise leads to a much smoother curve progression.
This advantage becomes a shortcoming if the dependencies within the data are ambiguous. To demonstrate this, a second dataset was generated (Figure \[fig4\], right). Now, for most values of $x$ two values of $y$ with different emphasis are reasonable. The Nadaraya-Watson regression calculates for every $x$ a “compromise”. This can lead to very improbable values for $y$. The optimization method however chooses always the most probable value and is because of this immune to this effect.
Conclusion
==========
If a dependency between some data is clear and unambiguous, the standard Nadaraya-Watson method or – still better – the local linearizing Nadaraya-Watson approach [@Cleveland79] should be preferred for the modeling. But for numerous real life applications, it cannot be guaranteed that this condition is fulfilled because, for example, the data may be collected online. Another difficulty is a high dimensionality. The dependency may be simple and unambiguous between some of the vector components, but between others it may not. Every input-output combination has to be checked, what is mostly impracticable.
For such general and complex cases, the proposed method is more suitable, because the assumption of unambiguousness is not necessary. The approach returns always a prediction that is probable in respect to the knowledge given by the sample data. For some applications, this property is more important than continuity of the curve and its smoothness.
An example for such a scenario is machine control. The data are in this case measurements from actuators and sensors. The controller continuously has to solve the problem which actuator values leads to the desired sensor values. For this purpose, already one good setting is sufficient, regardless of whether several possibilities exist or not. An average value or a “compromise”, however, is mostly a bad decision.
[7]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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Richard O. Duda, Peter E. Hart, and David G. Stork. *Pattern Classification*. John Wiley & Sons, Inc., 2000.
R.P.W. Duin. On the choice of the smoothing parameters for parzen estimators of probability density functions. *IEEE Transactions on Computers*, Vol. C-25, No. 11:0 1175–1179, 1976.
E. A. Nadaraya. On estimating regression. *Theory of Probability and its Applications*, Vol. 9:0 141–142, 1964.
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[^1]: Many $b_i$ are very low for a given value $\boldsymbol{x}$. The corresponding kernels should be omitted to improve the computation speed.
[^2]: The code was tested with Matlab 7.1.
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