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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 13
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 31898)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 13
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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string | meta
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\section{Introduction}
Grand Unified Theories (GUTs) are based on large symmetry groups, the smallest of which is an ${\rm SU(5)}$ model
with an additional, possibly approximate, ${\rm Z}_2$ symmetry.
When such large symmetries are broken in a cosmological setting, several kinds of topological defects
can be produced. The ensuing cosmology will depend critically on the interactions of the different
defects. In particular, the ${\rm SU(5)} \times {\rm Z}_2$ symmetry breaking leads to the existence of magnetic
monopoles and domain walls in the aftermath of the phase transition.
We expect the magnetic monopoles to interact with domain walls, potentially resolving the magnetic
monopole over-abundance problem \cite{Dvali:1997sa}. To investigate this idea further,
we study the interactions of ${\rm SU(5)}$ monopoles and ${\rm Z}_2$ domain walls in this paper.
The interaction of monopoles and domain walls was also studied in \cite{Pogosian:1999zi} with the domain wall
structure given by
\begin{equation}
\Phi = \tanh \left ( \frac{z}{w} \right ) \Phi_0
\label{q=0wall}
\end{equation}
where the order parameter $\Phi$ is in the adjoint representation of ${\rm SU(5)}$, $\Phi_0$ is its constant
vacuum expectation value (VEV), and
$w$ is the width of the domain wall. By numerical evaluation it was found that monopoles hitting
this domain wall will unwind and spread on the wall. Subsequently \cite{Pogosian:2000xv,Vachaspati:2001pw,Pogosian:2001fm,Vachaspati:2003zp}, however, it was found
that the model actually has several domain wall solutions, including the one in Eq.~(\ref{q=0wall}),
and that the lightest (stable) wall has a different structure (see Sec.~\ref{sec:wall}). Hence the
interaction of the stable wall and the monopole needs to be revisited.
In Sec.~\ref{model} we provide details of the ${\rm SU(5)} \times {\rm Z}_2$ model, the monopole solution,
the wall solutions, and finally our scheme for setting up a configuration with a monopole and
a domain wall together. This provides us with initial conditions that we numerically evolve in
Sec.~\ref{evolution}. The complexity of the field equations and the problem requires some special
numerical techniques that we briefly
describe in Sec.~\ref{evolution}.
Our results are summarized in Sec.~\ref{conclusions}. Essentially we find that there are two
internal space polarizations for the monopole with respect to the wall. One of the polarizations
is able to pass through the wall with only some kinematic changes in its motion.
The monopole with the other polarization is unable to pass through the domain
wall and unwinds on the wall, radiating away its gauge fields. The disappearance of this
monopole is further explained in Sec.~\ref{conclusions}.
\section{The Model}
\label{model}
The ${\rm SU(5)}$ model we consider is given by the Lagrangian:
\begin{equation}
L = -{1\over 4} X^a_{\mu\nu}X^{a\mu\nu} + {1\over 2} D_\mu\phi^a D^\mu\phi^a - V(\Phi )
\label{lagrangian}
\end{equation}
where $\Phi = \phi^a T^a$ ($a = 1,...,24$), $X^a_{\mu \nu}$ are the gauge field strengths defined as
\begin{equation}
X_{\mu \nu} = \partial_\mu X_\nu - \partial_\nu X_\mu - i g [X_\mu, X_\nu] \,,
\end{equation}
$X_\mu = X^a_\mu T^a$ are the gauge fields and $g$ is the coupling constant. $T^a$ are the generators
of ${\rm SU(5)}$ normalized by Tr($T^aT^b$) = $\delta_{a b}/2$. The covariant derivative is given by
\begin{equation}
D_\mu \phi^a = \partial_\mu \phi^a - i g [X_\mu ,\Phi]^a \,.
\label{covariantderivative}
\end{equation}
The most general renormalizable ${\rm SU(5)}$ potential is
\begin{equation}
V(\Phi ) = -m^2 {\rm Tr}\Phi^2 + \gamma {\rm Tr}\Phi^3
+ h ({\rm Tr} \Phi^2 )^2 +
\lambda {\rm Tr} \Phi^4 -V_0 \,,
\label{potential}
\end{equation}
and we will assume that $\gamma$ vanishes giving the model an additional ${\rm Z}_2$ symmetry\footnote{For the purpose of our analysis it is sufficient to assume that $\gamma$ is small enough to give an approximate ${\rm Z}_2$ symmetry that allows for sufficiently long lived domain walls.}. For $\lambda \ge 0$ and $h+7\lambda /30 \ge 0$, the potential has its global minimum at \cite{Ruegg:1980gf}
\begin{equation}
\label{eq.vev}
\Phi_0 = \frac{\eta}{2\sqrt{15}} \text{diag}(2,2,2,-3,-3),
\end{equation}
with $\eta=m/\sqrt{h+7\lambda /30}$. The VEV, $\Phi_0$,
spontaneously breaks the ${\rm SU(5)}$ symmetry to ${\rm SU(3)}\times {\rm SU(2)} \times {\rm U(1)}/ \Z6$.
In what follows, the four diagonal generators of ${\rm SU(5)}$ are chosen to be
\begin{equation}
\begin{split}
\lambda_3 &= \frac{1}{2} \text{diag}(1,-1,0,0,0), \\
\lambda_8 &= \frac{1}{2 \sqrt{3}} \text{diag}(1,1,-2,0,0), \\
\tau_3 &= \frac{1}{2} \text{diag}(0,0,0,1,-1), \\
Y &= \frac{1}{2\sqrt{15}} \text{diag}(2,2,2,-3,-3).
\end{split}
\end{equation}
We use $a=1,2,3$ to denote generators $T^a = \tau_a = \text{diag}(0,0,0,\sigma_a/2)$ where $\sigma_a$ are the Pauli spin matrices.
\subsection{The monopole}
\label{sec:monopole}
Let us consider a magnetic monopole whose winding lies in the 4-5 block of $\Phi$. This is possible \cite{Wilkinson:1977yq} if we take the VEV along one of the radial directions far away from the monopole to be
\begin{align}
\label{eqMinf}
\Phi_\infty &=& \frac{\eta}{2\sqrt{15}} \text{diag}(2,-3,2,2,-3) \nonumber \\
&=& \eta \sqrt{\frac{5}{12}}(\lambda_3 + \tau_3) + \frac{\eta}{6}(Y-\sqrt{5}\lambda_8).
\end{align}
The monopole ansatz for the scalar field can be written as \cite{Pogosian:2000xv}
\begin{equation}
\label{eq.phimonopole}
\Phi_M(r) = P(r) \sum_{a=1}^3 x^a \tau_a + M(r) \left( \frac{\sqrt{3}}{2}\lambda_3 - \frac{1}{2}\lambda_8 \right) + N(r) Y,
\end{equation}
while the non-zero gauge fields can be written as
\begin{equation}
X_i^a = \epsilon_{ij}^a \frac{x^j}{g r^2} (1-K(r)) , \ (a=1,2,3)
\label{eq.gaugemonopole}
\end{equation}
and $P(r), M(r), N(r),$ and $K(r)$ are profile functions that depend only on the spherical radial coordinate $r=\sqrt{x^2+y^2+z^2}$ and satisfy the boundary conditions:
\begin{align}
\label{eq.profilebc}
\lim_{r \rightarrow \infty} r P(r) &=& \eta \sqrt{\frac{5}{12}},
\ M(\infty) = \eta \frac{\sqrt{5}}{3}, \nonumber
\\ N(\infty) &=& \frac{\eta}{6},~~K(\infty) = 0.
\end{align}
The profile functions for the monopole alone were evaluated numerically and are shown in Fig.~\ref{fig.monparam}.
\begin{figure}[tbh]
\hspace*{-0.025\textwidth}\includegraphics[width=0.55\textwidth]{monparam.png}
\caption{The profile functions for the monopole alone, evaluated numerically.}
\label{fig.monparam}
\end{figure}
The non-Abelian magnetic field can be defined as \cite{Dokos:1979vu}
\[
B^k = -\frac{1}{2} \epsilon^{i j k} X_{ij}
\]
with the associated energy density given by ${\rm Tr} (B_k B^k)$. Far away from the centre, the monopole field becomes $B^k \rightarrow Q x^k /(g r^3)$, with $Q = \tau^j {x}^j /r$.
The monopole charge $Q$ includes a component along the generator of the unbroken ${\rm U(1)}$ symmetry
($\Phi_\infty$ of Eq.~(\ref{eqMinf})), as well as ${\rm SU(2)}$ and ${\rm SU(3)}$ magnetic charges. The ${\rm U(1)}$ part of
the magnetic field, which is a defining feature of a topological ${\rm SU(5)}$ monopole, is given by
\begin{equation}
B^k_Y = -\frac{1}{2} \epsilon^{i j k} X^a_{ij} {\hat \phi}^a
\label{eq.bfield}
\end{equation}
where ${\hat \phi}^a \equiv \phi^a / \sqrt{\phi^b \phi^b}$.
\subsection{The wall}
\label{sec:wall}
Without loss of generality \cite{Vachaspati:2001pw}, the domain wall solution can be taken to be diagonal at all $z$ and written in terms of the diagonal generators of ${\rm SU(5)}$ as
\begin{equation}
\label{eq.dw}
\Phi_{DW} (z) = a(z) \lambda_3 + b(z) \lambda_8 + c(z) \tau_3 + d(z) Y \,.
\end{equation}
In each of the two disconnected parts of the vacuum manifold ${\cal M}$ there are a total of 10 different diagonal VEVs corresponding to all possible permutations of 2's and 3's in Eq.~(\ref{eq.vev}). Topology dictates that there must be a domain wall separating any pair of VEVs from the two disconnected parts of ${\cal M}$. However, not every such pair of VEVs corresponds to a stable domain wall solution. For instance, as shown in \cite{Pogosian:2000xv}, the wall across which $\Phi_0$ goes to $-\Phi_0$ is unstable and will decay into a lower energy stable wall. The stable domain walls are obtained when both 3's in Eq.~(\ref{eq.vev}) change into 2's across the wall.
\begin{figure}[tbh]
\hspace*{-0.025\textwidth}\includegraphics[width=0.55\textwidth]{wallparam.png}
\caption{The wall profile functions for Cases 1 and 2. Note that the profile function $c(z)$ goes to zero in Case 2, which gives an unbroken ${\rm SU(2)} \subset {\rm SU(3)}$ symmetry in the 4-5 block of $\Phi^{(2)}_+$.}
\label{fig.wallparam}
\end{figure}
Let us choose the boundary condition at $z = -\infty$ as
\begin{align}
\label{eq.-inf}
\Phi_- &=& \Phi(z = -\infty) = \frac{\eta}{2\sqrt{15}} \text{diag}(2,-3,2,2,-3) \nonumber \\
&=& \eta \sqrt{\frac{5}{12}}(\lambda_3 + \tau_3) + \frac{\eta}{6}(Y-\sqrt{5}\lambda_8) \,.
\end{align}
For this choice of $\Phi_-$, there are three different choices of $\Phi(z=+\infty)$, proportional to
\begin{align}
\nonumber
&&\text{diag}(3,-2,-2,3,-2) \\
\nonumber
&&\text{diag}(-2,-2,3,3,-2) \\
&&\text{diag}(3,-2,3,-2,-2) \,,
\label{3VEVs}
\end{align}
that lead to stable domain walls. For the purpose of understanding the monopole-wall interactions, it is sufficient to consider only two of the above, corresponding to the two
distinct entries in the 4-5 block of $\Phi$. We take
the first to be the same as in \cite{Pogosian:2000xv}, subsequently referred to as Case 1:
\begin{align}
\label{eq.inf1}
\Phi^{(1)}_+ &=& \frac{\eta}{2\sqrt{15}} \text{diag}(3,-2,-2,3,-2) \nonumber \\
&=& \eta \sqrt{\frac{5}{12}}(\lambda_3 + \tau_3) - \frac{\eta}{6}(Y-\sqrt{5}\lambda_8) \,.
\end{align}
The value of the field in the core of this wall is proportional to $\text{diag}(1,-1,0,1,-1)$. The other case, subsequently referred to as Case 2, has
\begin{align}
\label{eq.inf2}
\Phi^{(2)}_+ &=& \frac{\eta}{2\sqrt{15}} \text{diag}(3,-2,3,-2,-2) \nonumber \\
&=& \eta \frac{\sqrt{15}}{6}\lambda_3 + \frac{\eta}{6}(4Y-\sqrt{5}\lambda_8) \,,
\end{align}
with the field in the wall being proportional to $\text{diag}(1,-1,1,0,-1)$.
A novel feature
of these walls is that the unbroken symmetry groups on either side of the wall are isomorphic
to each other but they are realized along different directions of the initial ${\rm SU(5)}$ symmetry
group. Hence the wall is the location of a clash of symmetries \cite{Davidson:2002eu}.
Note that the symmetry within the wall is [${\rm SU(2)} \times {\rm U(1)}$]$^2$.
The ${\rm SU(2)}$'s correspond to rotations in the 1-3 and 2-5 blocks and the ${\rm U(1)}$'s
to rotations along $\sigma_3$ in the 1-2 and 3-5 blocks. Therefore the symmetry group within
the wall is 8-dimensional, and is smaller than the 12-dimensional symmetry outside the
wall\footnote{For simplest domain walls, such as kinks in $\lambda \Phi^4$, the full
symmetry of the Lagrangian is restored inside the core. However, the symmetry inside stable domain walls in
$\SU{N} \times {\rm Z}_2$ is always lower than that of the vacuum \cite{Vachaspati:2001pw}.}.
Also note that the symmetry in the 4-5 block is different for the $\Phi^{(1)}_+$ and $\Phi^{(2)}_+$ vacua.
This is going to be of direct relevance for the fate of the monopoles.
The profile functions $a(z)$, $b(z)$, $c(z)$ and $d(z)$ for both cases are shown in Fig.~\ref{fig.wallparam}. In each case, they are linear combinations of two functions $F_+(z)$ and $F_-(z)$ defined by the alternative way of writing the domain wall solution \cite{Pogosian:2000xv}
\begin{equation}
\Phi_{DW} = \frac{\Phi_+(z) - \Phi_-(z)}{2} F_-(z) + \frac{\Phi_+(z) + \Phi_-(z)}{2} F_+(z) \, ,
\end{equation}
where $F_+(\pm \infty)=1$, $F_-(\pm \infty) = \pm 1$. For a general choice of parameters, functions $F_\pm(z)$ must be found numerically. For $h/\lambda = -3/20$, they are known in closed form \cite{Pogosian:2000xv}: $F_+ (x) = 1$, $F_-(x) = \tanh (mx/\sqrt{2})$. Correspondingly, for this value of $h/\lambda$, the four functions $a(z)$, $b(z)$, $c(z)$, and $d(z)$ are either constant or describe a transition from one constant value to another. For $h/\lambda \ne -3/20$ the ``constant'' functions develop a small bump around $z=0$ as can be seen in Fig.~\ref{fig.wallparam}.
\subsection{Monopole and Wall}
As our initial configuration, we take the monopole to be on the $z=-\infty$ side, far away from the wall. In this case, the ansatz for the initial combined field configuration of the wall and the monopole can be written as \cite{Pogosian:2000xv}
\begin{align}
\label{eq.phimdm}
\Phi_{M+DW} &=& P(r) \frac{c(z')}{c(-\infty)} \sum_{a=1}^3 x^a \tau_a + N(r) \frac{d(z')}{d(-\infty)} Y \nonumber \\
&+& M(r) \left( \frac{\sqrt{3}}{2} \frac{a(z')}{a(-\infty)} \lambda_3 - \frac{1}{2} \frac{b(z')}{b(-\infty)} \lambda_8 \right)
\end{align}
where $z' = \gamma (z- z_0)$, $\gamma=1/\sqrt{1-v^2}$ is the boost factor, $v$ is the wall velocity and $z_0$ is the initial position of the wall. The monopole is at $x=0=y=z$. It is easy to check that, far away from the monopole, the profile functions take on the values in Eq.~(\ref{eq.profilebc}) and $\Phi_{M+DW} \rightarrow \Phi_{DW}$. Close to the monopole, $z' \rightarrow -\infty$, since the monopole is initially very far from the wall, and $\Phi_{M+DW} \rightarrow \Phi_M$ as desired. We work in the temporal gauge, $X_0^a=0$, and with the initial ansatz for the gauge fields given by Eq.~(\ref{eq.gaugemonopole}) for both cases.
It is instructive to examine the difference in the nature of the magnetic field in Cases 1 and 2. As mentioned in Sec.~\ref{sec:monopole}, the charge of our monopole along the $z$-direction, $Q = (1/2)\text{diag}(0,0,0,1,-1)$, is a combination of the ${\rm U(1)}$, the ${\rm SU(2)}$, and the ${\rm SU(3)}$ magnetic charges. Since the broken symmetry in our model is along the generator of (hypercharge) ${\rm U(1)}$, the magnetic field, as defined in Eq.~(\ref{eq.bfield}), corresponds solely to the ${\rm U(1)}$ component of the charge.
In Case 1, ${\rm Tr}(Q \Phi)$ is the same on both sides of the wall and the ${\rm U(1)}$ magnetic field is unaffected by the presence of the DW. In Case 2, however, ${\rm Tr}(Q \Phi^{(2)}_+)=0$ and there is no magnetic field corresponding to the unbroken ${\rm U(1)}$ on the $z=+\infty$ side of the wall. Instead, the gauge field on that side is associated with an ${\rm SU(2)}$ that is part of the unbroken ${\rm SU(3)}$. We note that, while the magnetic \emph{energy density} associated with the gauge field is unaffected by the presence of the wall, it is specifically the ${\rm U(1)}$ magnetic field that is a defining feature of a topologically stable monopole.
\begin{figure}[tbh]
\hspace*{-0.025\textwidth}\includegraphics[width=0.55\textwidth]{magnetic.png}
\caption{The magnetic field ${\bf B}_Y$ (defined in Eq.~(\ref{eq.bfield})) multiplied by $r^2$ for Cases 1 and 2, where at each point $r^2B_Y^z$ and $r^2B_Y^x$ are plotted as a vector. In Case 1, there is a magnetic field associated with the unbroken ${\rm U(1)}$ symmetry on both sides of the wall. In Case 2, the magnetic field becomes associated with the ${\rm SU(2)} \subset {\rm SU(3)}$ on the $z=+\infty$ side on the wall, while its ${\rm U(1)}$ component vanishes. Note that it is the ${\rm U(1)}$ magnetic field that characterizes a topologically stable monopole.
}
\label{fig.magnetic}
\end{figure}
The magnetic field, as defined in Eq.~(\ref{eq.bfield}), is plotted for both cases in Fig.~\ref{fig.magnetic}, where the vectors have components $r^2 B_Y^z$ and $r^2 B_Y^x$. This plot shows that, in Case 1, there is a ${\rm U(1)}$ magnetic field on both sides of the wall falling off as $r^2$ as expected, while in Case 2 the ${\rm U(1)}$ magnetic field is zero on the $z=+\infty$ side of the wall.
\section{Evolution}
\label{evolution}
\begin{figure*}[htb]
\includegraphics[width=0.7\textwidth]{case1mp.png}
\caption{The potential and magnetic energy densities in the $xz$ plane for the colliding monopole and wall in Case 1. We see that the monopole passes through the wall and the energy densities remain localized. Additionally, we see the magnetic energy density is unchanged before and after the collision.}
\label{fig.case1mp}
\end{figure*}
\begin{figure}[tbh]
\hspace*{-0.025\textwidth}\includegraphics[width=0.55\textwidth]{case1phi.png}
\caption{The scalar field $\phi^a$ in the $xz$ plane for the colliding monopole and wall in Case 1, where at each point $\phi^3$ and $\phi^1$ are plotted as a vector. In this case the scalar field arrangement in direction and magnitude remains virtually unchanged.}
\label{fig.case1phi}
\end{figure}
Let us consider an initial monopole-wall configuration given by Eq.~(\ref{eq.phimdm}) in which VEV at $z=-\infty$ is given by $\Phi_-$ in Eq.~(\ref{eq.-inf}). As mentioned in the previous Section, there are 2 types of boundary conditions at $z=+\infty$, given by Eqs.~(\ref{eq.inf1}) and (\ref{eq.inf2}), dubbed Case 1 and Case 2, leading to 2 different outcomes of the monopole-wall collision.
Before considering the two cases in detail, let us note that initially, when the monopole and the wall are
very far away from each other, the field configuration has just three non-zero gauge fields and six scalar
fields corresponding to the generators that appear in Eq.~(\ref{eq.phimdm}). Because these six generators
form a closed algebra, it follows from the equations of motion that the subsequent evolution does
not involve fields corresponding to the other 18 generators. Namely, the scalar and the gauge field equations are
\begin{align}
D_\mu D^\mu \phi^a &=& - {\partial V / \partial \phi^a}
\label{eom:scalar}
\\
D_\mu X^{\mu \nu a} &=& g f_{a b c} (D^\nu \Phi)^b \phi^c
\label{eom:gauge}
\end{align}
where $f_{a b c}$ are the ${\rm SU(5)}$ structure constants defined by $[T^a,T^b] = i f_{a b c} T^c$. Let $\cal{C}$ be the set of indices of the 6 generators that appear in the initial field configuration given by Eq.~(\ref{eq.phimdm}). Since the 6 generators form a closed algebra,
$f_{abc} = 0$ for $a \notin \cal{C}$ and $b,c \in \cal{C}$. Now let $\phi^a$ and $X_\mu^a$ be fields corresponding to any $a \notin \cal{C}$. If $\phi^a$ and $X_\mu^a$ are zero at the initial time, they will remain zero if $f_{abc} = 0$ for $b,c \in \cal{C}$ and ${\partial V / \partial \phi^a} \ne 0$. The former condition is satisfied as mentioned above, while the latter holds since ${\rm Tr}[T^aT^b] \propto \delta_{ab}$ and ${\rm Tr}[T^aT^bT^cT^d] = 0$ for $b,c,d \in \cal{C}$, as we have checked by explicit evaluation. Thus, for our purposes, it is sufficient\footnote{Although the field components for $a \not\in \cal{C}$ continue to vanish during evolution if they vanish initially, we cannot exclude the possibility that the fields in these other directions play a role if they did not vanish initially.} to consider only $a \in \cal{C}$.
Our numerical implementation is based on techniques developed in \cite{Pogosian:1999zi}. First, the DW and the monopole profile functions are found via numerical relaxation. The monopole is initialized at the centre of the lattice, and then the DW functions are boosted to give the DW a velocity towards the monopole and put into the initial configuration given by Eqs.~(\ref{eq.phimdm}) and (\ref{eq.gaugemonopole}). With the initial time derivatives simply determined from the Lorentz boost factor, this initial configuration is evolved forward in time using a staggered leapfrog code together with absorbing boundary conditions. We work in Cartesian coordinates, which offer superior stability, but take advantage of the axial symmetry of our configuration to restrict the lattice to just three spacings along one of the Cartesian dimensions \cite{Alcubierre:1999ab}. We choose this to be the $y$ direction. We then use a $256 \times 256$ lattice grid for the $x$ and $z$ coordinates. Additionally, the axial symmetry allows us to solve only for positive $x$ and use reflection to find the fields at negative $x$. The units of length are set by $\eta=1$ and we take each lattice spacing to correspond to half of a length unit. In these units, the range of $x$ and $z$ axis for a $256 \times 256$ grid is $[-64,64]$. Note that, in some figures, we do not plot the entire range. The radius of the monopole core is about $10$ length units and is about the same as a half of the domain wall width. At the initial time, the wall is 30 length units away from the center of the monopole.
\subsection{Case 1: the monopole passes through}
It is not difficult to predict that the monopole in Case 1 will pass through the wall. The monopole winding is due to the fields in the ${\rm SU(2)}$ subgroup corresponding to generators $\tau_a$, $a=1,..,3$. In Eq.~(\ref{eq.phimdm}), these fields are multiplied by the function $c(z)$ which has the same value at $z=\pm \infty$ and, as known from \cite{Pogosian:2000xv}, is approximately constant across the domain wall. Only $b(z)$ and $d(z)$ change signs across the wall, but these are irrelevant for the winding of the monopole. Thus, the presence of the wall is of no qualitative consequence to the winding of the monopole or its profile functions. The only effect is the small change in $c(z)$ around $z=0$ (note that, as mentioned earlier, $c(z)$ is strictly a constant when $h/\lambda = -3/20$).
We numerically collide the monopole and the wall by giving the wall an initial velocity of $0.8$ (in speed of light units) and choosing parameters $\eta = 1$, $h = -\lambda/5$, and $\lambda=0.5$ for $V(\Phi)$.
Fig.~\ref{fig.case1mp} shows the potential and magnetic energy densities as the wall hits the monopole in Case 1. In addition, we plot the scalar field configuration in Fig.~\ref{fig.case1phi}, where each point is a vector with components $\phi^3$ and $\phi^1$. These figures show that the magnetic energy density and the scalar field configuration remain unchanged after the collision, and that the potential energy densities corresponding to the monopole and the domain wall remain localized.
We performed the simulations for a range of model parameters and found that, depending on the value of $h/\lambda$, the monopole may be weakly attracted or repelled by the wall leading to a small time delay or advance as the monopole goes through. The monopole may also cause ripples along the wall, losing some of its kinetic energy, but no
structural change occurs. For $h/\lambda=-3/20$, the scattering is completely trivial, with no time delay or advance.
\subsection{Case 2: the monopole unwinds}
\begin{figure*}[tbh]
\includegraphics[width=\textwidth]{case2mp.png}
\caption{The potential and magnetic energy densities in the $xz$ plane for the colliding monopole and wall in Case 2. We can see that as the domain wall and monopole collide, the potential energy contained by the monopole disappears and the monopole begins to radiate away its magnetic energy in a hemispherical wave. Note that the middle and final plots for the magnetic energy density have a much smaller scale as the ripples are not visible at the original scale.}
\label{fig.case2mp}
\end{figure*}
\begin{figure*}[tbh]
\includegraphics[width=\textwidth]{case2phi.png}
\caption{The scalar field $\phi^a$ in the $xz$ plane for the colliding monopole and wall in Case 2. At each point in the first row, $\phi^3$ and $\phi^1$ are plotted as a vector. In the second row, the direction of the arrow is given by $\tan^{-1}(\phi^3/\phi^1)$ and the color represents the magnitude of the field $|\phi| = \sqrt{\phi^a\phi^a}$ for $a=1,2,3$. The first row shows the monopole unwinding as the wall sweeps past it, and the second shows how the fields arrange themselves to unwind the monopole.
}
\label{fig.case2phi}
\end{figure*}
As in Case 1, it is possible to guess the outcome of the monopole-wall collision without doing numerical simulations. For this, we note that $\Phi^{(2)}_+$ has an ${\rm SU(2)}$ symmetry in the 4-5 block, which means that there is no topology that can support the winding. Thus, the monopole cannot exist in that corner of the matrix. An equivalent way to see this is to note that the function $c(z)$, which multiplies the three relevant monopole scalar fields, goes to zero at $z=+\infty$ (see Fig.~\ref{fig.wallparam}), effectively erasing the monopole.
Additional insight can be gained by noting that the long range
magnetic field of the monopole transforms into an ${\rm SU(3)}$ magnetic field on the far side of
the wall. More explicitly, the ${\rm U(1)}$ magnetic field is given by Eq.~(\ref{eq.bfield}) with $X_{ij}^a$
determined using the solution in Eq.~(\ref{eq.gaugemonopole}). Since $X_{ij}$ only has components
in the $\tau^a$ directions, it lies in the 4-5 block. However, the 4-5 block is entirely within the unbroken ${\rm SU(3)}$
on the right-hand side of the wall. Thus the long range magnetic field of the monopole is
purely ${\rm SU(3)}$ on the right-hand side of the wall and, from the
vantage point of someone there, there is no ${\rm U(1)}$ magnetic field emerging
from the left-hand side of the wall. However, a ${\rm U(1)}$ magnetic field is an essential feature of a topological
monopole. Thus, from the right-hand side of the wall, there is no magnetic monopole in the system,
only some source of ${\rm SU(3)}$ magnetic flux.
Doing the numerical simulation with the parameters chosen as before, we plot the potential and magnetic energy densities as the wall hits the monopole in Fig.~\ref{fig.case2mp}. This figure shows that the potential energy for the monopole disappears as the wall and monopole collide, and the magnetic energy that was stored in the monopole radiates away in a hemispherical wave. The collision was simulated with initial wall velocities ranging from $0.1$ to $0.99$ for $h/\lambda = -1/5$, and initial wall velocities of $0.6$, $0.8$ and $0.99$ for $h/\lambda = -3/20$ and $1/5$. In all of these cases, the result of the collision was unchanged.
In Fig.~\ref{fig.case2phi}, we show the $a=1,2,3$ components of the scalar field using two different representations. In the first row, the fields $\phi^3$ and $\phi^1$ are plotted as a vector. The plot shows that the components of the field that are responsible for the winding vanish on the $z=+\infty$ side of the wall. In the second row of Fig.~\ref{fig.case2phi}, the color represents the magnitude $|\phi| \equiv \sqrt{\phi^a\phi^a}$, $a=1,2,3$, and the direction of the arrow is given by $\tan^{-1}(\phi^3/\phi^1)$. Even though $|\phi|$ becomes very small, it is not strictly zero at a finite distance from the wall, and so one can still define the direction of the arrow in this way. One can see that initially the field has a hedgehog configuration across the wall. However, as the wall sweeps along, the fields on the $z=+\infty$ side of the wall rotate around in such a way as to unwind the monopole. In the final step, all fields that are non--zero are pointing in one direction, and therefore the monopole winding is gone.
\section{Conclusions}
\label{conclusions}
In a Grand Unified model there can be several types of defects, including magnetic monopoles and
domain walls. In the aftermath of the cosmological phase transition in which the Grand
Unified symmetry is spontaneously broken to the standard model symmetry, the monopoles and
walls will interact\footnote{Scattering of fermions and GUT domain walls was studied in \cite{Steer:2006ik}}. We have studied these interactions explicitly in an ${\rm SU(5)} \times {\rm Z}_2$
GUT, taking into account that the model has several different types of domain walls, and that only the lowest
energy wall is expected to be cosmologically relevant. Even this stable
wall has several different orientations in internal space, two of which are distinct for the purposes of
monopole-wall interaction.
The first wall (Case 1 above) is found to be transparent to the monopole. This is simply because
the domain wall mainly resides in a certain block of field space, while the winding of the monopole
resides in a different non-overlapping block. The interactions between the monopole and the wall are very weak, and only affect the dynamics of the monopole as it passes through the wall. Depending on the parameters, the monopole might be
attracted or repelled by the wall leading to a time delay or advance as the monopole goes through.
The second wall (Case 2 above) is opaque to the monopole. When the monopole hits the
wall its energy is transformed into radiation on the other side of the wall, as seen in
Fig.~\ref{fig.case2mp}. A useful way to picture this system is to consider
a magnetic monopole that is located inside a spherical domain wall.
Now there is a topological magnetic monopole inside the wall, but only an ${\rm SU(3)}$ magnetic
flux from the outside. In particular, there is no topological magnetic monopole as seen from
the outside. Therefore the spherical wall itself must carry the topological charge of an
antimonopole\footnote{The correspondence between spherical domain walls and global monopoles in $\SU{N}$ has previously been noted in \cite{Pogosian:2001fm}.}.
If the spherical wall shrinks, either it can annihilate the magnetic monopole within it and radiate away
the energy, or the monopole can escape the wall, in which case the wall would then
collapse into an antimonopole so that the total topological charge of the system continues
to vanish. Our explicit numerical evolution shows that annihilation occurs for the parameter ranges
we have considered.
Our results have bearing on cosmology as they explicitly show the possible destruction
of magnetic monopoles.
In the case where the ${\rm Z}_2$ symmetry is approximate, the walls will eventually
decay away, and it is possible that these interactions could lead to a universe that is free of magnetic monopoles.
Estimates in \cite{Dvali:1997sa} indicate that this possibility is worth investigating in more detail.
With several types of domain walls and monopoles simultaneously forming in a phase
transition \cite{Pogosian:2002ua,Antunes:2003be,Antunes:2004ir}, and with the complex nature of both the
inter-wall \cite{Pogosian:2001pq} and monopole-wall interaction, the fate of the monopoles will remain uncertain
until a comprehensive simulation of the GUT phase transition is performed. We leave this for a future study.
\acknowledgments
LP and MB are supported by the National Sciences and Engineering Research Council of Canada.
TV gratefully acknowledges the Clark Way Harrison Professorship at Washington University during the
course of this work, and was supported by the DOE at ASU.
|
{
"timestamp": "2015-06-01T02:14:19",
"yymm": "1505",
"arxiv_id": "1505.08170",
"language": "en",
"url": "https://arxiv.org/abs/1505.08170"
}
|
\section{INTRODUCTION}
In 1977, Blandford and Znajek argued for force-free electrodynamics as a reasonable mathematical framework describing a black hole magnetosphere (\cite{BZ77}). The force-free magnetosphere in a Kerr background has received considerable attention off late due to its expected properties with regards to its ability to extract energy and angular momentum from a rotating black hole in an astrophysical setting (for example see \cite{TT14}, \cite{GT14}, \cite{ZYL14} and \cite{LRS14}).
The basic mechanism we expect is that of a plasma interacting with its own electromagnetic field near the vicinity of the event horizon of the black hole. The plasma relaxes, by possibly ejecting helicity, thereby producing a force-free magnetosphere. Jet formation in radio-loud, active galactic nuclei could be explained if indeed force-free electrodynamics permitted solutions with the desired features.
In \cite{MD11} we were successful in creating an exact solution that allowed the extraction of energy via a electromagnetic poynting flux. Although, mathematically consistent, it was not clear how this solution could be physically realized in nature. The possible decomposition of the net electromagnetic current into leptonic flows and a baryonic fluid remains an unanswered question. If the actual ejection of current and energy happens at the event horizon, it is necessary to match an outgoing exterior solution with a relevant infalling interior solution. In this case, the interior fields must satisfy the Znajek regularity condition at the event horizon \cite{ZNA77}.
In this paper however, we have a different purpose. In \cite{MD07} we obtained the first exact solution to the force-free magnetosphere in a Kerr background. The resulting current vector was proportional to the in-falling principle null geodesic vector field
\begin{equation}
n= \frac{r^2+a^2}{\Delta}\;\partial_t- \partial_r +\frac{a}{\Delta}\;\partial_\varphi.
\label{ndef}
\end{equation}
The vector field above is written in the standard Boyer-Lindquist coordinates. This corresponding solution did satisfy the Znajek regularity condition and hence was well defined at the event horizon of the black hole. Using the symmetries of the equations of electrodynamics in the case of axis-symmetric and stationary solutions, we were successful in finding a similar solution wherein the current vector was proportional to the out-going null vector field of the Newman-Penrose tetrad
\begin{equation}
l= \frac{r^2+a^2}{\Delta}\;\partial_t+ \partial_r +\frac{a}{\Delta}\;\partial_\varphi.
\label{ldef}
\end{equation}
However, unlike the previous case, this new solution was not valid at the event horizon \cite{MD11}.
In the sections below, we will now construct two local, linearly independent, exact solutions to the force-free magnetosphere, wherein the current vectors are proportional to the linear combinations of $m$ and $m^\star$ (the complex conjugate of $m$) where
$$m = \frac{1}{\sqrt{2} \rho} \left(i a \sin \theta, \;0, \;1,\; \frac{i}{\sin \theta}\right).$$
These new solutions are local in the sense that they are not strictly valid at the event horizon and the symmetry axis of the Kerr black hole. The point to also note is that the Newman-Penrose tetrad in a Kerr background are precisely the vectors $n, l, m$ and $m^\star$. We will have then succeeded in establishing a direct connection between fore-free electrodynamics and the Newman-Penrose tetrad.
After presenting a brief overview of force-free electrodynamics in a Kerr background in the next section, we move onto presenting our new solution in section \ref{omchoices}. We then conclude by discussing the immediate properties of this new solution.
\section{Basic Equations of Force-free Dynamics}
In the Boyer-Lindquist coordinates, the Kerr metric takes the form:
$$ds^2=( \beta^2 - \alpha^2 )\;dt^2 \;+ $$
\begin{equation}
\;2 \;\beta_\varphi \;d\varphi
\;dt
+\gamma_{rr}\; dr^2 + \;\gamma_{\theta \theta}\; d\theta^2 +
\;\gamma_{\varphi\varphi}\;d\varphi^2 \;,
\end{equation}
where the metric coefficients are given by
$$\beta_\varphi \; \equiv g_{t \varphi}\; = \;\frac{-2Mr a
\sin^2\theta}{\rho^2}\;,\;\;\;\gamma_{rr} =
\frac{\rho^2}{\Delta}\;,$$
$$\beta^2-\alpha^2 \;= \;g_{tt} \;=\; -1 + \frac{2Mr}{\rho^2}\;,$$
$$
\gamma_{\theta \theta} = \rho^2, \;\; {\rm and}\;\; \gamma_{\varphi \varphi} = \frac{\Sigma^2 \sin^2\theta}{\rho^2}\;.
$$
Here,
$$\rho^2 = r^2 + a^2
\cos^2\theta\;,\;\;\;\Delta = r^2 -2 M r + a^2$$
and
$$
\Sigma^2 = (r^2 + a^2)^2 -\Delta \; a^2 \sin^2\theta\;.
$$
Also
$$
\alpha^2 = \frac{\rho^2 \Delta}{\Sigma^2}, \;\;\; \beta^2 =
\frac{\beta_\varphi^2}{\gamma_{\varphi \varphi}}\;,\;\sqrt{\gamma} = \sqrt{\frac{\rho^2\;\Sigma^2}{\Delta}}\;\sin \theta\;,$$
and
$$\sqrt{-g}=\alpha\; \sqrt{\gamma} = \rho^2 \sin\theta\;.$$
Maxwell's equations in any spacetime can be written as
\begin{equation}
\nabla_\beta \star \;F^{\alpha \beta} = 0 \;, \;{\rm and} \;
\nabla_\beta F^{\alpha \beta} = I^\alpha\;.
\label{maxeq}
\end{equation}
Here $F^{\alpha \beta}$ is the Maxwell stress tensor, $I^\alpha$ is the electric current density vector and $\nabla$ is the covariant derivative of the geometry. $\star \;F$ is
the two form defined by
\begin{equation}
\star \;F^{\alpha \beta} \equiv \frac{1}{2}\epsilon^{\alpha \beta \mu \nu} F_{\mu \nu}\;.
\end{equation}
Here, $\epsilon_{\alpha \beta \mu \nu}$ is the completely antisymmetric Levi-Civita tensor density of spacetime such that $
\epsilon_{0123}= \sqrt{-g}= \alpha \sqrt{\gamma}
$. In this paper, we choose to work with electric and magnetic fields rather than the covariant formalism of electrodynamics. Thus we define
$ E$ and $ B$ vector fields by
\begin{equation}
F_{ \mu \nu} =
\left[\begin{array}{cccc}
0& -E_1& -E_2 & -E_3\\
E_1 & 0 & \sqrt{\gamma}\; B^3 & - \sqrt{\gamma}\; B^2\\
E_2& - \sqrt{\gamma} \;B^3 & 0 & \sqrt{\gamma} \;B^1\\
E_3 & \sqrt{\gamma}\;B^2 & - \sqrt{\gamma} \;B^1 & 0\\
\end{array}\right] \;.
\label{fdown}
\end{equation}
In the remainder of this section, we simply summarize the equations of force-free dynamics in a 3+1 formalism. The details behind the calculations can be found in \cite{MD05} and \cite{K04}. When, as in the case of the Kerr metric in Boyer-Lindquist coordinates, $\partial_t$ corresponds to the Killing vector that generates time translations, eq.(\ref{maxeq}) become the familiar set of Maxwell equations with the following modifications:
\begin{equation}
\tilde \nabla \cdot B = 0\;,
\label{divb}
\end{equation}
\begin{equation}
\partial_t B + \tilde \nabla \times E = 0\;,
\label{faraday}
\end{equation}
\begin{equation}
\tilde \nabla \cdot D = \rho\;,
\label{maxcharge}
\end{equation}
and
\begin{equation}
-\partial_t D + \tilde \nabla \times H = J\;.
\label{maxcurrent}
\end{equation}
Here, $\rho = \alpha I^t$ and $J^k = \alpha I^k$, also,
\begin{equation}
\alpha D = E- \beta \times B
\label{contitutive1}
\end{equation}
and
\begin{equation}
H = \alpha B - \beta \times D\;.
\label{contitutive2}
\end{equation}
The generalized cross product of vector fields are defined by
\begin{equation}
(A \times B)^i \equiv \; \epsilon^{ijk} \; A_j\; B_k\;,
\end{equation}
and
$\tilde \nabla$ corresponds to the induced covariant derivative of the 3-D absolute space defined by surfaces of constant $t$. Now we impose the requirements of force-free electrodynamics. By definition, here
\begin{equation}
F_{\nu \alpha}\; I^\alpha =0\; .
\label{divtemforce}
\end{equation}
In the 3+1 formalism, this condition reduces to
\begin{equation}
E \cdot J = 0
\label{fofree1}
\end{equation}
and
\begin{equation}
\rho E + J \times B = 0.
\label{fofree2}
\end{equation}
Eqs. (\ref{divb}) through (\ref{fofree2}) define the entire content of what we mean by force-free electrodynamics in a Kerr background.
Finally, for the electromagnetic field to be well defined at the event horizon of the Kerr black hole, the following Znajek regularity condition should be satisfied (\cite{ZNA77}):
\begin{equation}
H_\varphi \left|_{r_+} = \frac{\sin^2\theta}{\alpha}\; B^r\; (2Mr\; \Omega -a) \right |_{r_+}.
\label{znaregcond}
\end{equation}
Now we impose the symmetries of the background metric onto the electromagnetic fields and currents. Specifically, we require that all electrodynamics quantities are time-independent and axis-symmetric. It is then not difficult to show that there exists a vector $\omega =\Omega \partial_\varphi$ such that
\begin{equation}E = - \omega \times B\;.
\label{omdefeqn}
\end{equation}
The vanishing of the curl of $E$ for time-independent solutions gives that
$$\tilde \nabla_B \Omega =0\;.$$
Therefore, the magnetic field can be written as
\begin{equation}
B_P = \frac{\Lambda}{\sqrt\gamma}\left(-\Omega_{,\theta} \;\partial_r + \Omega_{,r}\;
\partial_\theta \right)\;.
\label{bpexplicit}
\end{equation}
The subscript $P$ on $B$ indicates the poloidal ($r$ and $\theta$) components of $B$. In standard notation
$$\Lambda = -\frac{d A_\varphi}{d\Omega}\;,$$
where $A_\varphi \equiv A_T$ is the toroidal ($\varphi$ component) of the vector potential $A$. Clearly, this is possible only because $A_\varphi$ depends only on the value of $\Omega$. Eq.(\ref{omdefeqn}) implies that $E_T=0$, and $E_P \cdot B_P= 0$. On the other hand, for force-free dynamics $ E_P \cdot J_P =0$. Thus, we can conclude that $ B_P \propto J_P$. This along with eq.(\ref{maxcurrent}) implies that the toroidal component of $H$ must satisfy
\begin{equation}
\tilde \nabla_B H_\varphi =0\;.
\label{hphistream}
\end{equation}
I.e.,
$H_\varphi$ is also dependent only on the value of $\Omega$.
Eq.(\ref{fofree2}) is trivial when projected to plane spanned by $\partial_\varphi$ and $B_P$. However, when projected along $E_P$, eq.(\ref{fofree2}) gives the only remaining master constraint equation for time-independent, axis-symmetric, force-free electrodynamics.
The resulting constraint equation reduce to a manageable form when written in terms of the streaming function $\Omega$. To facilitate this define
\begin{equation}
\chi(\Omega)=\partial_t +\omega =\partial_t + \Omega\; \partial_\varphi.
\label{kerrcanvf}
\end{equation}
Let
\begin{equation}
\chi_\varphi (\Omega) \equiv g (\partial_t + \Omega\; \partial_\varphi,\partial_\varphi) = g_{t\varphi} + \Omega \; g_{\varphi \varphi}
\label{kerrgeospc1}
\end{equation}
and
\begin{equation}
\chi_t (\Omega)\equiv g(\partial_t + \Omega\; \partial_\varphi,\partial_t)= g_{t t} + \Omega \; g_{t \varphi}\;.
\label{kerrgeospc2}
\end{equation}
In terms of $\Omega$ and $\Lambda$
$$(\rho E + J \times B ) \cdot E_p= 0$$
is satisfied provided
\begin{widetext}
\begin{equation}
\frac{1}{2 \Lambda } \frac{d H_\varphi^2}{d \Omega}=\alpha \gamma_{\varphi \varphi} \Omega \;\tilde \nabla \cdot \left(\frac{\Lambda}{\alpha \gamma_{\varphi \varphi}} \chi_\varphi (\Omega) \tilde \nabla\Omega\right)+
\alpha \gamma_{\varphi \varphi} \tilde \nabla \cdot \left(\frac{\Lambda}{\alpha \gamma_{\varphi \varphi}} \chi_t (\Omega) \tilde \nabla\Omega\right)\;,
\label{consteqlong1}
\end{equation}
or equivalently
\begin{equation}
\frac{1}{2 \Lambda } \frac{d H_\varphi^2}{d \Omega}=\alpha \gamma_{\varphi \varphi} \tilde \nabla \cdot \left(\frac{\Lambda}{\alpha \gamma_{\varphi \varphi}} \chi^2(\Omega) \tilde \nabla\Omega\right)-\Lambda \chi_\varphi (\Omega) (\tilde \nabla \Omega)^2\;.
\label{consteqlong2}
\end{equation}
Here $\chi^2(\Omega)= g(\chi(\Omega), \chi(\Omega))$ and $(\tilde \nabla \Omega)^2= \gamma^{ij}\Omega_{,i}\Omega_{,j}$.
\end{widetext}
The main point and difficulty here is that, just like the left hand side of the equation above (see eq.(\ref{hphistream})), the right hand side of the equation above must depend only on the streaming function $\Omega$. The functions $\Lambda$ and $\Omega$ determine every other electromagnetic quantity uniquely.
\section{Canonical choices of $\Omega$}
\label{omchoices}
Due to stationarity and axis-symmetry, the Kerr metric mixes the standard Boyer-Lindquist $t$ and $\varphi$ coordinates in a particular simple way:
$$g_{tt} + \frac{1}{a \sin^2 \theta } \;g_{t \varphi} = -1 ,$$
$$g_{t \varphi} + \frac{1}{a \sin^2 \theta } \;g_{\varphi \varphi} = \frac{r^2+a^2}{a} ,$$
and
$$ g_{t \varphi} + \frac{a}{r^2 + a^2}\; g_{\varphi \varphi} = \frac{a}{r^2+a^2}\; \sin^2 \theta \Delta ,$$
$$g_{t t} + \frac{a}{r^2 + a^2}\; g_{t \varphi} = - \frac{a}{r^2 + a^2} \; \Delta .$$
Thus, there are two special choices for $\Omega$ in Kerr geometry in the Boyer-Lindquist coordinate system that makes $\chi_t$ and $\chi_\varphi$ exceptionally simple (i.e., written in terms of well known functions of the spacetime geometry). However, {\it a priori}, there is no guaranty that these choices are consistent with eq.(\ref{consteqlong1}).
\subsection{Recovering Our Previous Solution}
When
$$
\Omega \equiv \Omega_1 = \frac{1}{a \sin^2 \theta},
$$
we have that
$$\chi_t (\Omega_1) = -1$$
and
$$\chi_\varphi (\Omega_1) = \frac{r^2 + a^2}{a}\;.$$
In \cite{MD07}, we have shown that this is indeed a valid choice for $\Omega$ since in this case
$$\frac{1}{2 \Lambda } \frac{d H_\varphi^2}{d \Omega}=\frac{-2}{a^3 \sin\theta} \;\frac{d}{d\theta}\left[\frac{\Lambda \cos\theta}{\sin^4 \theta}\right],$$
and the right hand side above,
like $\Omega$, is only a function of $\theta$ for any arbitrary but sufficiently regular $\Lambda(\theta)$ (so as to make the solutions well defined at the poles). This can be easily integrated to give
\begin{equation}
H_\varphi=\alpha B_{\varphi} =\frac{2}{a^2} \Lambda \frac{\cos\theta}{\sin^4\theta}.
\label{hphi_1}
\end{equation}
It also easily checked that the above equation satisfies the Znajek regularity condition. The other non-trivial components of the electric and magnetic fields are
\begin{equation}
B^r=\frac{2}{a} \Lambda \frac{\cos{\theta}}{\sqrt{\gamma}\sin^3{\theta}}.
\label{radb_1}
\end{equation}
\begin{equation}
E_\theta =-\frac{2}{a^2} \Lambda \frac{\cos{\theta}}{\sin^5{\theta}}.
\end{equation}
The electromagnetic current in this configuration is given by
\begin{equation}
I =-\frac{2}{a^2\; \alpha\; \sqrt\gamma} \;\;\frac{d}{d \theta}\left[\Lambda \frac{\cos\theta}{\sin^4\theta}\right]n,
\end{equation}
where $n$ is the infalling principle null geodesic of the Kerr geometry explicitly given in eq.(\ref{ndef}).
In \cite{MD11}, we showed that the inherent assumed symmetries of the fields (axis-symmetry and time- independence) implied that there exists a dual solution with the same $\Omega = \Omega_1$ such that the current vector was proportional to the outgoing principle null geodesic of the kerr geometry $l$ (see eq.(\ref{ldef})).
Here the explicit forms of the electromagnetic field components are not very different except for a few sign changes (see \cite{MD11} for details). This dual solution however does not satisfy the Znajek regularity condition. Here, the surfaces of constant $\Omega$ are cones (surfaces of constant Boyer-Lindquist coordinate $\theta$). Recently, the solution presented here was extended to the $\varphi$ dependent case by Brennan et. al. (see \cite{BGJ13}).
\subsection{A New Class Of Solutions}
Now consider a second canonical choice
$$
\Omega \equiv \Omega_2 (r) = \frac{a}{r^2 +a^2}.
$$
In this case, we have that
$$\chi_\varphi (\Omega_2) = \frac{a}{r^2 + a^2} \sin^2\theta \Delta$$
and
$$\chi_t (\Omega_2)=-\frac{\Delta}{r^2 + a^2}\;.$$
In this case, once again we find that the right hand side of eq.(\ref{consteqlong1}) becomes
$$
\frac{1}{2 \Lambda } \frac{d H_\varphi^2}{d \Omega} = -\frac{\Delta \Omega }{a^2} \frac{d}{dr}\left[\Lambda \Delta \Omega \Omega_{,r}\right].$$
Clearly, the right hand side above,
like $\Omega$, is only a function of $r$ for any arbitrary but sufficiently regular $\Lambda(r)$. This is also easily integrated to give
\begin{equation}
a H_\varphi = a \alpha B_\varphi =\pm \sqrt{C^2 - (\Lambda \Delta \Omega \Omega_{,r})^2}\;.
\label{rdephphi}
\end{equation}
Here $C$ is an integration constant large enough to ensure that $H_\varphi$ is real.
Eq.(\ref{bpexplicit}) gives the following explicit solution for the poloidal components of the magnetic field:
$$B_P = \frac{-2 a r \Lambda}{\sqrt{\gamma}\; (r^2+a^2)^2} \;\partial_\theta.$$
Here the only non-zero component of the electric field given by eq.(\ref{omdefeqn}) takes the form
$$E_P= \frac{-2 a^2 r \Lambda}{(r^2+a^2)^3}\; dr.$$
The components of the dual fields ($D$ and $H$) are given by
$$D= D_P = \frac{\Lambda}{\alpha \gamma_{\varphi \varphi}} \;\chi_\varphi (\Omega) \;d\Omega$$
which in our case reduces to
$$D= -\frac{\Lambda}{\alpha \gamma_{\varphi \varphi}} \;\frac{2 r a^2 }{(r^2+a^2)^3} \;\sin^2\theta \Delta\; dr\;.$$
Also, in general
$$H_P = -\chi_t (\Omega) \;\frac{B_P}{\alpha}$$
which for our particular choice of $\Omega_2$ reduces to
$$H_P = -\frac{2 a r \Lambda}{ (r^2+a^2)^3}\;\frac{\Delta}{\rho^2 \sin \theta} \;\partial_\theta.$$
It is important to note that there is still a choice in the explicit form for $\Lambda$. Here, the surfaces of constant $\Omega$ are spheres (surfaces of constant Boyer-Lindquist coordinate $r$).
\section{Discussion And Conclusion}
For the remainder of the section, we restrict our discussion the solution generated by
$\Omega= \Omega_2 (r)$.
Since $B^r =0$ everywhere, the Znajek regularity condition will require that $H_\varphi$ vanish at the event horizon.
I.e., either $C=0$ or $\Lambda$ diverges at the event horizon since $\Lambda \approx {\rm const}/\Delta$ near the event horizon. If $C=0$, we do not have a ``real" valued $H_\varphi$. On the other hand if $\Lambda$ diverges at the event horizon, our solution becomes invalid at the event horizon (see expression for $E_r$ above). I.e., either way, our solution is not valid at the event horizon.
It turns out that the symmetry axis of the Kerr geometry is problematic for our new solution as well. For the Maxwell tensor to be well defined along the poles ($\theta = 0, \pi$) we must have that $\sqrt{\gamma} B^\theta /\sin\theta$ must be finite along the axis. This is not true in our case. Therefore, the solution presented here is only valid away from the poles and the event horizon of the black hole.
\begin{widetext}
Incidently, in the exterior geometry, the solution here is magnetically dominated since
$$B^2 - E^2 = g_{\theta \theta} (B^\theta)^2 + g^{\varphi \varphi} (B_\varphi)^2 - g^{rr} (E_r)^2 > g_{\theta \theta} (B^\theta)^2 - g^{rr} (E_r)^2$$
$$=\frac{\Lambda^2 (\Omega_{,r})^2}{(r^2+a^2)^2\gamma}\left [\rho^2(r^2+a^2)^2 - \Sigma^2 a^2 \sin^2 \theta \right] >\frac{\Lambda^2 (\Omega_{,r})^2}{\gamma}(r^2-a^2)>0.$$
\end{widetext}
It would appear that the force-free solution we have obtained here by mere observation has limited utility. However, there are other interesting features contained in this solution. To see this, we first compute the current density vector. When $\Omega = \Omega_2$, the current density vector is given by
$$I = \frac{F^\prime}{\rho^2 \sin \theta} \left (\sin \theta,\; 0,\; \frac{F }{a^2 H_\varphi},\; \frac{1}{a \sin \theta}\right),$$
where $F= \Lambda \Delta \Omega \Omega_{,r}$ and $F^\prime = dF/dr$.
Then
$$I^2 = g(I,I) = \frac{(F^\prime)^2}{a^4\rho^2 \sin^2 \theta} \left(1+ \frac{F^2}{C^2-F^2}\right)$$
which means that the current vector is spacelike since
$$C^2-F^2 = (a H_\varphi)^2.$$
As was mentioned above, we had previously obtained solutions where the current density vector was proportional to the principal geodesics $n,l$ of the Kerr geometry. Here, it turns out that the current vector is related to the remaining two vectors in the canonical null tetrad of the Kerr geometry, specifically, $m$ and $m^\star$, where
$$m = \frac{1}{\sqrt{2} \rho} \left(i a \sin \theta, \;0, \;1,\; \frac{i}{\sin \theta}\right)\;.$$
Here $\rho = r + i a \cos\theta$ (this is consistent with the notation that $\rho^2 = \rho \bar \rho \equiv (r + i a \cos\theta)(r -i a \cos\theta)$).
Clearly then, the current density vector above can be written in terms of the real and imaginary parts of $m$ as follows:
$$I = \frac{F^\prime}{a\rho^2 \sin \theta} \left[ {\rm Im} \;(\sqrt{2} \rho m) + \frac{F}{a H_\varphi} {\rm Re} \;(\sqrt{2} \rho m) \right].$$
The associated dual solution described in general in \cite{MD11} is generated by the same $\Omega_2$ and has a current vector of the form:
$$\tilde I = \frac{F^\prime}{a\rho^2 \sin \theta} \left[ {\rm Im} \;(\sqrt{2} \rho m) -\frac{F}{a H_\varphi} {\rm Re} \;(\sqrt{2} \rho m) \right].$$
In this case, the $\pm$ sign in eq.(\ref{rdephphi}) is exactly what distinguishes a solution from its dual.
It is indeed remarkable that there exists, four linearly independent solutions to the time-independent, stationary, axis-symmetric, force-free magnetosphere in a Kerr geometry where the current density vectors are proportional to $n$ and $l$, and two other albeit local solutions proportional the linear combinations of the remaining bases vectors $m$ and $m^\star$ in the null tetrad of the Newman-Penrose formalism. Since $B^r = 0$ in the new solutions presented here (solutions generated by $\Omega_2$), they do not allow an exchange of energy and or angular momentum with the Black hole. This feature is an expected one from Christodoulou's general calculations since at the event horizon $\Omega_2 |_{r_+} = a/(r_+ ^2 + a^2) \equiv \Omega_H$, the angular velocity of the event horizon.
|
{
"timestamp": "2015-06-01T02:14:20",
"yymm": "1505",
"arxiv_id": "1505.08172",
"language": "en",
"url": "https://arxiv.org/abs/1505.08172"
}
|
\section*{ABSTRACT}
Since the arrival of genetic typing methods in the late 1960's, researchers have puzzled at the clinical consequence of observed strain mixtures within clinical isolates of \emph{Plasmodium falciparum}. We present a new statistical model that infers the number of strains present and the amount of admixture with the local population (panmixia) using whole-genome sequence data. The model provides a rigorous statistical approach to inferring these quantities as well as the proportions of the strains within each sample. Applied to 168 samples of whole-genome sequence data from northern Ghana, the model provides significantly improvement fit over models implementing simpler approaches to mixture for a large majority (129/168) of samples. We discuss the possible uses of this model as a window into within-host selection for clinical and epidemiological studies and outline possible means for experimental validation.
\section*{INTRODUCTION}
The protozoan parasite \emph{Plasmodium falciparum} (Pf) is the cause of the vast majority of fatal malaria cases, killing at least half a million people a year \cite{Hay2009,Snow2005,World2008}. The parasite's ability to develop resistance to drugs and the rapid spread of that resistance across geographically-separated populations presents a constant threat to international control efforts \cite{Wootton2002,Mita2009,Payne1987}. While research has elucidated many genetic factors in resistance, much of genetic epidemiology of the parasite - including the effective recombination rate and the rate of gene flow across populations - is still unclear \cite{Sidhu2002,Mita2009,Roper2004}.
Since the late 1960's, researchers focused on the structure of Pf clinical infections have struggled to understand the implications of multiplicity of infection (MOI), where multiple strains appear to be present within a single patient's bloodstream \cite{Wilson1969,Mcgregor1972,Jamjoom1983,Conway1991,Auburn2012}. While MOI-focused studies implicate increased or decreased levels of MOI with a range of conditions, including clinical severity \cite{Muller2001}, age-specific severity \cite{Henning2004,Smith1999a,Farnert1999,Stirnadel1999}, parasitemia levels during pregnancy \cite{Beck2001}, and other effects \cite{Beck1999,Smith1999b,Paganotti2004,Mayengue2009}, there is no broad consensus about MOI's role, if any, in controling the course of an infection. Still, a wide variety of studies and genetic assays -- most typically by typing of the \emph{mdr} gene -- show MOI as a reliable feature of clinical Pf isolates \cite{Auburn2012}.
The appearance of whole-genome sequencing (WGS) technologies applied to Pf extracted directly from infected patients' bloodstreams provides an unprecedented window into the structure of genetic mixture within samples \cite{Manske2012,Auburn2011}. Initial work on understanding the structure of this mixture data shifted focus from estimating MOI to analysis based on inbreeding coefficients \cite{Auburn2012,O'Brien2014,Weir1984}. These metrics, a special type of $F$-statistic, provide an estimate of the departure of within-sample allele frequencies from those expected under a Hardy-Weinberg-type equilibrium with the nearby population. In this perspective, each patient's bloodstream is taken to be a subpopulation exhibiting a degree of admixture of all of the strains from the local environment, ranging from a perfectly random sampling of all nearby strains to the repeated sampling of just single strain.
The initial study applying WGS to clinical Pf isolates collected from eight countries on three continents shows that the parasite exhibits significant population structure at continental scales, with the amount of subpopulation structure varying significantly among regions \cite{Manske2012}. Employing novel F-statistics to measure the inbreeding coefficient, this work also argued that the degree of mixture varies significantly across populations, with highly mixed samples occurring relatively frequently in west Africa but only occasionally in Papua New Guinea. Importantly, the authors suggested an association between increased levels of observed mixture with increases in transmission intensity in the local environment. Transmission intensity, the rate at which individuals are infected with Pf, likely determines some part the frequency of out-crossing within parasite populations and so may be critical to understanding gene flow and strategies for resistance control \cite{Guerra2008}.
In this paper, we present a new statistically rigorous model that synthesizes these two distinct and previously disparate approaches to analyzing \emph{P. falciparum} clinical mixtures: assessing the number of distinct genetic types within a sample (the MOI approach) and measuring the degree of panmixia with respect to the local population (the inbreeding coefficient approach). The model centers around how these two sub-models contribute to generate the observed within-sample non-reference allele frequency as it relates to the population-level non-reference allele frequency for single nucleotide polymorphisms (SNPs). For clarity, we will deprecate the use of \emph{non-reference} in front of the term allele frequency, since they are all calibrated in this fashion. We will use the acronyms WSAF to denote the within-sample allele frequency and PLAF to denote population-level allele frequency to avoid confusion about the particularly frequency being indicated.
The essential structure of the model is to explain observed `bands' that emerge when examining the WSAF for SNPs as a function of their PLAF (Figure \ref{example}). The model posits that the number of these bands results as a direct consequence of the number of distinct strains present within a sample and that the degree of admixture with the local population determines bands' slopes. To distinguish from the inbreeding coefficient, we refer to the degree of admixture as the panmixia coefficient. The collection of bands are then modeled jointly as a finite mixture.
Figure \ref{diagram} lays out the important components of the model. In the simplest case a sample is composed of a single, unmixed strain, and all SNPs exhibit a WSAF of zero or one (see Figure \ref{diagram}(a)), depending on whether they agree with the reference. Consequently the WSAF is independent of PLAF, leading to two flat bands at these values. We call these samples unmixed. In the case where finite number of strains mixed within a sample, then for each variant position some number of those strains will possess a reference allele and some will not. Which strains carry non-reference alleles and those strains' proportion in the sample mixture then determine the WSAF for each SNP. Observed across many SNPs, this leads to the apparent bands of constant WSAF across the PLAF. It follows that for $K$ component strains there are $2^K$ possible combinations of biallelic states, leading to that number of apparent WSAF bands.
The complementary banding structure of panmixia arises when a fraction of the Pf organisms present within the blood are randomly sampled from the local population. In its simplest formulation, the panmixture model represents the admixture of two distinct Pf populations: a single strain, representing $1 - \alpha$ of the within-sample genomes, and a random sample of strains from the local population, representing $\alpha$ of the remaining genomes. In the case of perfect panmixia ($\alpha=1$), a sample would be comprised of organisms evenly sampled from the ambient population and the plot of WSAF against PLAF would become a single line at $y=x$. In this data set, we do not observe any sample close perfect panmixia but observe several instances of apparent partial panmixia with a single dominant strain (Figure \ref{example}(c) and \ref{diagram}(c)). The $\alpha$ tilt in the WSAF arises from the fact that for this proportion of organisms the probability of sampling non-reference allelele is proportional to the PLAF, absent any other population structure. These samples, with a single strain and a degree of panmixia, we call panmixed. In more complex cases, where there is more than one dominant strain, the total number of bands is still determined by the number of these strains. However, now panmixia tilts each of these bands equally leading to complex mixtures (Figures \ref{example}(d) and \ref{diagram}(d)).
The paper proceeds as follows. We first detail the structure of the WGS data, introduce some notation, and the essential mathematical structure of the model. We then present an extensive simulation study on the performance of the model and then an examination of its application to field isolates collected from northern Ghana. We conclude by discussing the strengths and weakness of the model, some possible improvements, and what consequences this analysis may have for understanding the etiology of clinical malaria.
\section*{DATA, NOTATION, AND MODEL}
\subsection*{Data}
The WGS data come from Illumina HiSeq sequencing applied to \emph{P. falciparum} extracted from $235$ clinical blood samples collected from infected patients from the Kassena-Nankana district (KND) region of Upper East region of northern Ghana. Collection occurred over approximately $2$ years, from June 2009 to June 2011. The full sequencing protocol and collection regime are described in \cite{Manske2012}. After quality control measures, sequencing was performed on $235$ samples, and, following a documented protocol using comparison against world-wide variation, $198,181$ single-nucleotide polymorphisms (SNPs) were called within each sample \cite{Manske2012}. Each call provides the number of reference and non-reference read counts observed at each variant position within the genome, ascertained against the the $3\mbox{D}7$ reference \cite{Gardner2002}. For this project, we additionally filtered these data. First, multiallelic positions were reclassed as biallelic. We then excluded positions that exhibited no variation within the KND samples, any level of missingness (no read counts observed), or minor allele frequency less than $0.01$. To remove low quality samples, we removed thoses possesed more than $4,000$ SNPs called with fewer than $20$ read counts, following an inflection point observed in Supplementary Figure S1(a). These cleaning measures left $2,429$ SNPs in $168$ samples. More than $95\%$ of remaining samples' sequencing was completed without PCR amplification. We observe little apparent population structure among the samples, evidenced either by principal components analysis or a neighbor-joining tree of the pairwise difference among samples (Supplementary Figures S2). The data preparation scripts are available with the source code for the model, \href{https://github.com/jacobian1980/pfmix/}{https://github.com/jacobian1980/pfmix/}.
\subsection*{Notation}
Following the data preparation and cleaning, our analysis begins with a set of $N$ clinical samples, each composed of $M$ SNPs. At each SNP $j$ within each clinical sample $i$, we observe $r_{ij}$ reads that agree with the reference genome and $n_{ij}$ reads that do not agree with the reference. For a sample $j$. we write the complete data across all SNPs as $\mathcal{D}_i = [(r_{i1},n_{i1}),\cdots,(r_{iM},n_{iM})]$. For each SNP $j$, we associate a PLAF $p_j$. The collection of all $p_j$ we refer to as $\mathcal{P}$.
Conditional upon the number of strains $K$, there are $2^K$ bands, indexed by $r=1,\cdots,2^K$. The full collection of bands we call $\mathcal{Q}$, with $q_{ijr}$ indicating the WSAF for band $r$ at SNP $j$ in sample $i$. The probability of a SNP lying within the distinct bands across the PLAF is specified by a mixture component $\lambda_r$, which is a function of the PLAF, and so is often written $\lambda_r(p_j)$. The degree of panmixia in a sample is given by $\alpha$, a value between zero and one. A complete list of the model parameters is given in Table \ref{notation}.
\subsection*{Model}
Statistically, the model takes the form of a finite mixture model, with the mixture components associated with individual bands \cite{Redner1984,Mclachlan2004}. We take a Bayesian approach to inference and so lay out the model by giving an overall rationale for the decomposition of the posterior distribution and then justifying the appropriate choice of probability distributions for each of the terms \cite{Gelman2013}.
\subsubsection*{Decomposition}
We assume that samples are independent of each other and that the SNP data for each sample depends solely on $K$, the WSAF $\mathcal{Q}$, the PLAF $\mathcal{P}$, and a shape parameter $\nu$. As samples are independent, we will deprecate sample-specific subscripts for the model parameters. Considering the data for a single sample, $\mathcal{D}_i$, the posterior distribution can then be written as:
\begin{eqnarray}
\mathbb{P}(\mathcal{Q},\mathcal{P}, \mathcal{W}, \alpha, \nu, K | \mathcal{D}_i) & \propto & \mathbb{P}(\mathcal{D}_i | \mathcal{Q}, \mathcal{P}, \mathcal{W}, \alpha, \nu, K ) \cdot \mathbb{P}(\mathcal{Q},\mathcal{P}, \mathcal{W}, \alpha, \nu, K) \\
& = & \mathbb{P}(\mathcal{D}_i | \mathcal{Q}, \mathcal{P}, \nu, K) \cdot \mathbb{P}(\mathcal{Q}, \mathcal{P}, \nu, K, \mathcal{W}, \alpha) \mbox{.}
\label{init_decomp}
\end{eqnarray}
We also assume that the WSAF, $\mathcal{Q}$, depends only on the PLAF, $\mathcal{P}$, the panmixia coefficient $\alpha$, the number of strains $K$, and their proportions within the sample, $\mathcal{W}$, allowing the right-hand side of Equation \ref{init_decomp} to be further decomposed, by noting that
\begin{eqnarray}
\mathbb{P}(\mathcal{Q}, \mathcal{P}, \nu, K, \mathcal{W}, \alpha) & = & \mathbb{P}(\mathcal{Q}|\mathcal{P}, \nu, K, \mathcal{W}, \alpha) \cdot \mathbb{P}(\mathcal{P}, \nu, K, \mathcal{W}, \alpha)
\label{second_decomp} \mbox{.}
\end{eqnarray}
While $\mathcal{W}$ clearly depends on the number of strains, $K$, the remaining parameters we take to be independent of this value and of each other. This means that the last right-hand side term in Equation \ref{second_decomp} becomes:
\begin{eqnarray}
\mathbb{P}(\mathcal{P}, \nu, K, \mathcal{W}, \alpha) & = & \mathbb{P}(\mathcal{P}) \cdot \mathbb{P}(\nu) \cdot \mathbb{P}(\mathcal{W}|K) \cdot \mathbb{P}(K) \cdot \mathbb{P}(\alpha) \mbox{.}
\label{third_decomp}
\end{eqnarray}
Substituting Equations \ref{second_decomp} and \ref{third_decomp} into Equation \ref{init_decomp}, yields the final decomposition:
\begin{eqnarray}
\mathbb{P}(\mathcal{Q},\mathcal{P}, \mathcal{W}, \alpha, \nu, K | \mathcal{D}_i) & \propto & \mathbb{P}(\mathcal{D}_i | \mathcal{Q}, \mathcal{P}, \nu, K) \cdot \mathbb{P}(\mathcal{Q}|\mathcal{P}, \nu, K, \mathcal{W}, \alpha) \cdot \nonumber \\
& & \hspace{1.5cm}\mathbb{P}(\mathcal{P}) \cdot \mathbb{P}(\nu) \cdot \mathbb{P}(\mathcal{W}|K) \cdot \mathbb{P}(K) \cdot \mathbb{P}(\alpha) \mbox{.}
\end{eqnarray}
We now specify each of the terms on the right-hand side above as probability distributions.
\subsubsection*{Likelihood : $\mathbb{P}(\mathcal{D}_i | \mathcal{Q}, \mathcal{P}, \nu, K)$}
Within band $r$, the WSAF at SNP $j$ in sample $i$ is $q_{ijr}$. Supposing that read counts at $j$ are identically and independently distributed with probability $q_{ijr}$, we model the probability of the data $(r_{ij},n_{ij})$ as a Beta-binomial distribution, allowing us to model greater dispersion than expected under a pure binomial. We parameterize this distribution in terms of $q_{ijr}$ and $\nu$ rather than the more commonly used shape and scale parameters, $\alpha$ and $\beta$. The relationship between the two parameterization is $q_{ijr} \cdot \nu = \alpha$ and $(1-q_{ijr}) \cdot \nu = \beta$. We use this parameterization as it allows us to write the model in terms of the mean allele frequency that defines each band. The additional $\nu$ is a shape parameter that serves as a proxy for the variance. These parameters give a likelihood expression:
\begin{eqnarray}
\mathbb{P}(n_{ij}, r_{ij}|r, q_{ijr}, \nu) &=& {n_{ij} + r_{ij} \choose n_{ij} } \cdot \frac{\mbox{B}(n_{ij} + q_{ijr} \cdot \nu,r_{ij} + (1-q_{ijr})\cdot \nu)}{\mbox{B}( q_{ijr}\cdot \nu, (1-q_{ijr})\cdot \nu)} \mbox{,}
\label{likelihood}
\end{eqnarray}
where $\mbox{B}$ is the beta function.
As any SNP could lie within any band, we employ a novel version of the finite mixture model to capture this segregation. Fixing the number of strains to $K$, there are then $2^K$ ways that the strains can be assorted into non-reference and reference allele states at any given position $j$. A given band $r$ arises from $C_r$ strains exhibiting the non-reference allele and $2^K - C_r$ strains exhibiting the reference allele. Supposing no population structure among the strains, the probability that a given SNP will be in that band is simply the probability of drawing $C_r$ non-reference alleles and $2^K-C_r$ reference alleles, conditional upon $p_j$:
\begin{eqnarray*}
\mathbb{P}(\mbox{SNP }j \in \mbox{band } r|p_j) &=& p_j^{C_r} \cdot (1- p_j)^{2^K-C_r} \\
& = & \lambda_r(p_j) \mbox{.}
\label{lambda}
\end{eqnarray*}
Consequently, the density of the mixture coefficients for each band varies across the PLAF but such that they sum to $1$ across all bands at any position $j$:
\begin{eqnarray*}
\mathbb{P}(\mathcal{D}_{ij} | \mathcal{Q}, \mathcal{P}, \nu, K) & = & \sum_{r=1}^{2^K} \mathbb{P}(\mbox{SNP }j \in \mbox{band } r|p_j) \cdot \mathbb{P}(n_{ij}, r_{ij}|r, q_{ijr}, \nu) \\
& = & \sum_{r=1}^{2^K} \lambda_r(p_j) \cdot \mathbb{P}(n_{ij}, r_{ij}|r, q_{ijr}, \nu) \mbox{.}
\end{eqnarray*}
Following from the assumption of no population structure, SNPs will assort into bands independently. This leads to a product form for the likelihood of sample's data, $\mathcal{D}_i$:
\begin{eqnarray}
\mathbb{P}(\mathcal{D}_i | \mathcal{Q}, \mathcal{P}, \nu, K) & = & \prod_{j=1}^M \bigg[ \sum_{r=1}^{2^K} \lambda_r(p_j) \cdot \mathbb{P}(n_{ij}, r_{ij}|r, q_{ijr}, \nu) \bigg] \mbox{.}
\label{likelihood_2}
\end{eqnarray}
\subsubsection*{Band structure: $\mathbb{P}(\mathcal{Q}|\mathcal{P}, \nu, K, \mathcal{W}, \alpha) $}
The full mixture model contains two distinct subcomponents that we call the simple mixture model and the panmixture model, respectively. Both models generalize the unmixed case, though naturally in different ways. We first describe the unmixed model and then layout the two extensions before showing how these can be combined to create the full model. In practice, we only fit data using the full model and allow it to indicate the number of strains, their proportions, and the degree of panmixia. We do not know the number of strains \emph{a priori} so we employ metrics applied to the posterior distribution inferred with different values of $K$ to determine it. However, for the purpose of detailing the model, we assume that $K$ is known. \\
\noindent \textbf{Unmixed model} - In an unmixed sample only one strain is present and there is no panmixia, and so $K=1$ and $\alpha=0$. Consequently, we expect all SNPs to exhibit WSAF either zero or one (Figure \ref{diagram}(a)) depending on whether they agree with the reference or not. There are then two bands, $r=1,2$ and $q_{ij1} = 0$ and $q_{ij2} = 1$. \\
\noindent \textbf{Simple mixture model} - The simple mixture model assumes that a finite number $K$ of distinct strains, $s_1, \cdots, s_K$, are combined together in the sample with proportions, $\mathcal{W} = [w_1,\cdots, w_K]$ but that $\alpha=0$. Naturally, $\sum_{k} w_k = 1$ and for each SNP $j$, the probability of being within band $r$ is given by $\lambda_r(p_j)$, as above. Band $r$ is defined by a vector $v_r = [\mathbf{1}_{\{s_1 \in r\}}, \cdots, \mathbf{1}_{\{s_K \in r\}}]$, where $\mathbf{1}_{\{s_k \in r\}}$ is an indicator function of whether strain $k$ exhibits a non-reference allele within the sample. The WSAF of $q_{ijr}$ is then given by the sum of all of proportions of strains that exhibit a non-reference allele:
\begin{eqnarray}
q_{ijr} &=& \sum_{k=1}^K w_k \cdot \mathbf{1}_{\{s_k \in r\}} \mbox{.}
\label{simple_model}
\end{eqnarray}
Taken across all $r$ bands, this leads to $2^K$ bands with zero slope and corresponding proportions $(0,w_1,\cdots,w_K,w_1+w_2,w_1+w_3,\cdots,1)$.
\noindent \textbf{Panmixture model} - As mentioned above, in its simplest case, the panmixture model represents the admixture of two distinct Pf populations: a single strain, representing $1-\alpha$ of the within-sample orgnaisms, and a random sample of strains from the local population, for the remaining $\alpha$ organisms. When $\alpha = 0$ the model reduces to the unmixed case. We will refer the single strain as the dominant strain, although, conditional upon $\alpha$, it may represent only a small proportion of the sample's population. For each position $j$, there are still only two bands: the higher one corresponding to the non-reference allele being present in the dominant strain, and the lower one corresponding to its absence. However, the WSAF for these bands varies according to $p_j$ with slope $\alpha$. To resolve $q_{ijr}$, first consider the upper band, $r=2$. At any position $j$, $1-\alpha$ of the reads come from the dominant strain. The remaining reads, each sampled randomly from the local population, each have probability $p_j$ of being non-reference. This leads to $q_{ij2} = (1-\alpha) + \alpha \cdot p_j$. For the lower band, the dominant strain contributes no non-reference reads so $q_{ij1} = \alpha \cdot p_j$.\\
\noindent \textbf{Complex mixture model} - The complex model synthesizes the simple mixture and panmixture models. In this case, at position $j$, $\alpha$ of the reads are sampled randomly from the across the local population, contributing a fraction of $\alpha \cdot p_j$ non-reference alleles. The state of the remaining reads are determined by $\mathcal{W}$ as in Equation $\ref{simple_model}$. For band $r$ at position $j$, the WSAF is then given by
\begin{eqnarray}
q_{ijr} & = & (1-\alpha) \cdot \bigg( \sum_{k=1}^K w_k \cdot \mathbf{1}_{\{s_k \in r\}} \bigg)+ \alpha \cdot p_j \mbox{.}
\label{full_model}
\end{eqnarray}
There are then $2^K$ bands with proportions $(0,w_1,\cdots,w_K,w_1+w_2,w_1+w_3,\cdots,1)$ and slope $\alpha$.
\subsubsection*{Priors}
For the remaining four probability distributions we place the following vague prior distributions:
\begin{eqnarray*}
\mathcal{W}|K & \sim & \mbox{\small{DIRICHLET}}(\mathbf{1}_K) \\
\alpha & \sim & \mbox{\small{UNIFORM}}(0,1) \\
\nu & \sim & \mbox{\small{EXPONENTIAL}}(5)\\
K & \sim & \mbox{zero-truncated } \mbox{\small{POISSON}}(2) \mbox{,}
\end{eqnarray*}
where $\mathbf{1}_K$ is a vector of $K$ ones.
\subsection*{Inference}
We infer the model parameters using a standard Bayesian Markov chain Monte Carlo (MCMC) approach \cite{Gilks2005,Geyer1992} with one exception: we first calculate maximum-likelihood estimates (MLE) for $\mathcal{P}$ across all samples and then treat these as fixed when inferring the remaining parameters \cite{Scholz1985}. This choice is motivated by statistical expedience and computational speed. Except for $\mathcal{P}$, the parameters of the model are independent across samples and so this approximation enables the algorithm to infer parameters in parallel rather than jointly. This avoids the difficulties of performing inference on the number of strains within samples simultaneously, which would require an involved trans-dimensional MCMC scheme (such as reversible jump MCMC, \cite{Green1995}) acting jointly across all samples. Running in parallel also increases the computational speed of the implementation by at least an order of magnitude. Since the sample collection is large enough that $\mathcal{P}$ is nearly independent of any given sample, we do not expect this approximation to significantly bias inference.
For each SNP $j$, the MLE derives from treating the non- and reference reads within a sample as coming from a binomial distribution with parameter $p_j$. This leads to:
\begin{eqnarray*}
\hat{p}_j & = & \displaystyle \sum_{i}^N n_{ij} \bigg/ \displaystyle \sum_{i}^N (n_{ij} + r_{ij})\mbox{.}
\end{eqnarray*}
To infer $K$ for each sample, we employ a Bayesian Information Criterion (BIC) \cite{Posada2004,Chen1998} and harmonic mean estimator to the Bayes Factor (hmeBF) \cite{Lavine1999,Kass1995} as metrics for model selection. To find the maximum likelihood value for use with the BIC, we initially implemented a separate estimation algorithm but found no significant difference with using the highest value observed from the posterior samples. In simulations, we observe that the BIC and hmeBF provide similar guidance, with the BIC frequently indicating a smaller $K$. For the simulation study and empirical data example, we provide only the BIC result.
Conditional on $\mathcal{P}$ and $K$, we implement a Metropolis-Hastings algorithm to draw samples from the posterior distribution \cite{Gilks2005}. For each of the three parameters, $\alpha$, $\mathcal{W}$, and $\nu$, we propose new values directly from the prior distribution, leading to Metropolis-Hastings ratios almost solely dependent on the ratio between the likelihood and priors for the proposed state to those for the current. The inference scheme is implemented in set of scripts for the R computing language, and can be found under the Academic Free License at \href{https://github.com/jacobian1980/pfmix/}{https://github.com/jacobian1980/pfmix/s}. For a single sample, a sufficiently long MCMC run takes approximately 20 minutes on a single high-performance computing core.
\section*{RESULTS}
\subsection*{Simulations under the model}
To demonstrate the efficacy of our implementation, we present a simulation study examining the algorithm's performance. We consider two distinct aspects of the inference separately: how well the model infers the number of strains, and, conditional upon that number, how well it infers the model's other parameters. We simulate data from the model in the following way. Conditional upon $M$,$\alpha$, $K$ and the sum of the read counts, $C$, we draw a vector of probabilities, $\mathcal{W}$, from a uniform Dirichlet distribution. We combine the values of $\mathcal{W}$ in all possible permutations to create the $2^K$ bands and assign the PLAF for the SNPs evenly from $1/M$ to $1$, so that the $j^{\mbox{\tiny{th}}}$ SNP has PLAF $\frac{j}{M}$. For each SNP, we first probabilistically select the band it occupies according to according to Equation \ref{lambda}. Conditional upon selecting $r$, we then simulate read counts according to the likelihood (Equation \ref{likelihood}) with $q_{ijr}$ according to Equation \ref{full_model}. For all simulations, we set $\nu=10$. We run the simulation across the range of values for $M$,$\alpha$, $K$ and $C$. For each parameter set, we create $10$ independent realizations.
\subsubsection*{Number of components}
Figure \ref{fig:comp_sim} shows performance of the algorithm for inferring the number of components increases in precision with the number of SNPs and the number of reads. Conditional upon $\alpha$, the simulations indicate that the number of SNPs, $M$, to be the largest determinant of performance, with the sum of the read counts, $C$, playing an important supporting role. Inference of the number of underlying strains, $K$, is generally strong for low panmixture levels (small $\alpha$ values), but is noticeably more conservative for $\alpha=0.5$, likely due to the bands becoming increasingly tightly packed as panmixia increases. In general, inference is slightly conservative, likely owing to the BIC estimator's bias toward parsimony \cite{Findley1991}.
\subsubsection*{Parameters}
Figure \ref{fig:comp_sim_para} shows similar performance for inference of the strain proportions $\mathcal{W}$ and $\alpha$. For $\mathcal{W}$, we report the mean squared deviation. For $\alpha$, we report the absolute normalized deviation to account for relative difference from the true value. For both parameters, we observe that the number of SNPs is the strongest determinant of accuracy, with $M = 150$ ensuring moderately strong performance. High $\alpha$ moderately decreases the quality of inference for the strain proportions.
\subsection*{Clinical samples from northern Ghana}
Applying the algorithm to the $168$ high-quality samples from KND, we observe $K$ range $1$ to $7$, with $\alpha$ falling between $0$ and $0.14$, and a moderate correlation between $K$ and $\alpha$ (Figure \ref{fig:portrait}). The largest subset of samples were unmixed, with $K=1$ and $\alpha < 0.01$, though the majority of samples exhibit moderate levels of mixture, with $K=2,3,4$ and $0.01\le \alpha \le 0.03$. A small number of samples exhibit complex mixtures, with $K>4$ and $\alpha$ typically greater than $0.02$. These results confirm the presence of mixtures within Pf clinical isolates, but also indicate, possibly more complex patterns involving interactions between the number of dominant strains and the degree of panmixia.
We observe that for most samples the $95\%$ credible interval for $\alpha$ is within a small percentage of the median value. For $\mathcal{W}_i$, we observe a similarly tight posterior distribution, particularly for samples with $K\le3$. The posterior uncertainty increases together with increasing $K$ and increasing $\alpha$. For a small number of samples, the model initially produced unusually low values for $\nu$, indicating a bimodal Beta-binomial distribution inconsistent with a mixture of strains and consequently suspect values for $K$. For these, we bounded $\nu$ such that it ensured a unimodal distribution and then recovered results consistent the remaining samples.
To visually inspect the quality of the results, we generate figures for each of the samples showing the observed WSAF and PLAF data, the inferred model structure, and data simulated under the inferred model following the observed PLAF. We show examples of these plots for three typical samples in Figures \ref{fig:some}. Nearly all samples (158/168), across all different mixture patterns, show strong visual correspondence between the observed and model-simulated data. We also observe a strong correlation between the inferred number of components and a quasi-maximum likelihood estimate for the inbreeding coefficient for each sample (Figure \ref{fig:f_stat}) \cite{O'Brien2014}.
For each sample, we compare the full model to two reduced versions under the restrictions $\alpha=0$ and $K=1$, respectively. These restrictions correspond to the cases where the model becomes the simple mixture model, with $2^K$ bands but no admixture with the local population, and the panmixture model, where there is a single strain with some local population admixture, respectively. For numerical stability reasons, we set the former restriction as $\alpha=0.001$. For $60\%$ of samples (108/168), the BIC selects the full model over either of the restricted models. For 58 samples the BIC criterion selected the $K=1$ restriction, while for 23 samples it selected $\alpha=0$. Taken in aggregate across all samples, the criterion overwhelmingly selects the full model over either of the restricted models.
\section*{DISCUSSION}
The model captures two dimensions of within-sample mixture for \emph{P. falciparum} that had previously modeled separately: the number of strains and the degree of admixture with the local population. Evidenced by the comparison of the full model with restricted sub-models, this approaches provides a marked improvement over both more restricted approaches in capturing the structure of mixture in clinical samples. While the model provides a more involved qualitative understanding of the samples, the strong correlation between the inbreeding coefficient and the inferred number of strains shows that the model produces results consistent with previous methods.
In order to perform inference, the model makes a number of simplifying assumptions that may be violated in practice. The model presumes that SNPs are unlinked and consequently independent for the purpose of calculating the likelihood. Given the high recombination rate of \emph{P. falciparum} this assumption may hold for the majority of pairs of SNPs, but neglects correlations that appear locally ($\sim$ 10 kB). However, we expect that this independence assumption serves to moderately weaken the inferential power of the model rather than cause any type of bias since it fails to include possibly informative data, rather than posit a possibly misspecified model. More problematic is the model's implicit assumption of limited population structure. In the case of the KND samples, and perhaps in much of west Africa, this assumption appears supported \cite{Anderson2000,Manske2012}. In other contexts, specifically south-east Asia, recent population bottlenecks and selection suggest that this assumption will be violated \cite{Miotto2013}. The consequences on this model inference are unknown but can likely be partially resolved with appropriate simulation studies.
The model presents an important new tool for interrogating the biology of clinical Pf infections. In particular, how the number of component strains and the panmixia coefficient relate to the infection parameters, such as seasonality, transmission intensity, and outcrossing, and evolutionary parameters such as the rate of change within sections of the Pf genome, could have implications for understanding the genetic epidemiology of Pf. The model also presents a means for clarifying the poorly detailed structure of intra-host infection dyanmics, such as strain selection or density-dependent selection \cite{Kwiatkowski1991}, by resolving how the number of strains, the mixture proportions, and the panmixia coefficient change within the course of an infection or in response to drug intervention.
An unexpected structural consequence of the model is that power to infer additional strains diminishes as the panmixia coefficient ($\alpha$) increases. This results from the simplifying assumption that $1-\alpha$ of the reads come from the dominant strains while the remaining $\alpha$ fraction are sampled randomly from the local population. Geometrically, we see that as $\alpha$ increases that the bands will get progressively closer together as they approach panmixia, making them harder for the model to distinguish. Also, as $\alpha$ increases to one, the fraction of reads representing the dominant strains diminishes, reducing power to infer these components. We observe this pattern strongly in simulations (Figure \ref{fig:comp_sim_para}): for $\alpha = 0.5$ or greater, the model consistently infers too few components. While this deficiency may be overcome in some fashion with additional SNPs or read counts, the geometry indicates there may be fundamental limits on any model's ability to discriminate the true number of components in the high panmixia regime.
In principle, the model can be explicitly tested against experiment. Laboratory facilities with the capacity to store many field strains ($> 100$) could generate artificial samples in an experimental analog of our simulation procedure, as follows. Starting with $N$ unmixed strains, identified using inbreeding coefficients, they could create mixtures, they would need to first fix the required sequencing volume as $\eta$, and the parameters for panmixia ($\alpha$), number of component strains ($K$), and their mixture parameters, $\mathcal{S}$ and $\mathcal{W}$. For the finite mixture component, they would then combine volumes of $\eta \cdot \mathcal{W}$ from the $K$ strains. For the panmixture component, they would then fix some large number but experimentally feasible number of strains (say $100$) and randomly sample from all of them a volume of $\eta/100$. Finally, combining these into final sample and applying WGS sequencing, will yield data that we hypothesize will closely follow the integrated model outlined above, with $\nu$ capturing the experimental variation. Naturally, consistent results would indicate the sufficiency of the model, but not it's necessity, holding out the possibility of a more minimal description. These results could be further compared against other next-generation technologies, such as single-cell sequencing, that have been deployed to understand Pf clinical mixtures \cite{Nair2014}.
The method works efficiently in practice (the properties of a single sample can be inferred in minutes on a standard laptop) but a number of possible improvements could strengthen its statistical performance. Most immediately, creating a full Bayesian approach rather than the parallelizing implementation here - while likely not improving the parametric inference for individual samples - would provide the full posterior distribution across all samples for more considered model comparison. In that same line, more refined approaches to inferring the number of strains within samples, either via a reversible jump MCMC approach or methods for rigorously estimating Bayes factors \cite{Green1995}, would provide researchers more power to resolve the structure of mixture, though likely at the cost of significantly more computation.
The model does not perform haplotype phasing to resolve the sequence of the underlying strains \cite{Stephens2001,Howie2012,O'Brien2014b}. The analysis here suggests that a method for estimating haplotypes would be straight-forward for some samples (unmixed ones, for instance) but difficult if not impossible for others (when $\alpha$ is greater than $0.5$). Researchers may be particularly interested in whether, in these phased samples, particular stretches of the genome appear more or less frequently in the dominant strains than others, indicating immunological or environmental selection. This is also an avenue for statistical development.
The two phenomenologies of mixture that the model captures - a finite mixture of distinct strains and an inbred population admixture - cannot be immediately associated with any specific aspect of the infection process. A number of variables appear plausible in determining these relationships, including transmission intensity, the length of infection, the immunological status of the infected individual, and within-host density dependent selection. Together with WGS data, this new approach can serve as a means for biological researchers to directly resolve these hypotheses and resolve the consequence of mixture in \emph{P. falciparum} infections.
\subsection*{Authorship}
JO designed and implemented the study and wrote the manuscript. ZI assisted in the design of the study and commented on the manuscript. LA-E collected the data and commented on the manuscript and the study.
\section*{Tables}
\begin{table}[!ht]
\begin{center}
\begin{tabular}{l|l}
Parameter & Definition \\
\hline $N$ & Number of samples \\
$M$ & Number of SNPs \\
$K$ & Number of strains \\
$i = 1,\cdots,N$ & Index for samples \\
$j = 1,\cdots,M$ & Index for SNPs \\
$r = 1,\cdots 2^K$ & Index for bands / strain mixtures \\
$p_j$ & (Non-reference) allele frequency for SNP $j$ \\
$\mathcal{P} = [p_j]$ & The PLAF for all SNPs \\
$\mathcal{Q} = [q_{ij}] $ & Within-sample allele frequency for SNP $j$ in sample $i$ \\
$\alpha$ & Degree of panmixia within a sample, panmixia coefficient\\
$\mathcal{S} = [s_1,\cdots,s_K]$ & Strains in a sample \\
$\mathcal{W} = [w_1,\cdots,w_K]$ & Strain proportions in a sample \\
$\lambda_r$ & Band proportions within sample \\
$\nu$ & Variation parameter for Beta-binomial \\
WSAF & Within-sample allele frequency \\
PLAF & Population-level allele frequency
\end{tabular}
\end{center}
\caption{Parameters and definitions for the model and its description.}
\label{notation}
\end{table}
\begin{table}[!ht]
\begin{center}
\begin{tabular}{c|rrrr}
Parameter & Values: & & & \\
\hline M & 50 & 150 & 500 & 2500\\
C & 10 & 25 & 100 & 250 \\
$\alpha$ & 0.01 & 0.1 & 0.5 & \\
$K$ & 1 & 3 & & \\
\end{tabular}
\caption{Table of simulated parameter values: $C$ the number of read counts while $M$, $K$ and $\alpha$ are as in Table \ref{notation}.}
\label{table:parameters}
\end{center}
\end{table}
\newpage
\section*{Figures}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.45]{diagram_2.pdf}
\end{center}
\caption{Four representative samples with WSAF for each SNP plotted against the PLAF, showing an absence of mixture (a), a partially panmixed sample (b), a simple mixture (c), and a complex mixture (d).}
\label{example}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.45]{diagram.pdf}
\end{center}
\caption{The essential structure of the model comprises four distinct states, relating the WSAF to the PLAF: no mixture (upper left); simple mixture (lower left); panmixture (upper right); and complex mixture (lower right).}
\label{diagram}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.3]{sim_read_counts.pdf} &
\includegraphics[scale=0.3]{sim_snps.pdf} \\
\end{tabular}
\end{center}
\caption{Performance for inference of number of components}
\label{fig:comp_sim}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[scale=0.23]{sim_reads_p.pdf}
\caption{}
\label{fig:gull}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[scale=0.23]{sim_snps_p.pdf}
\caption{}
\label{fig:gull}
\end{subfigure} \\
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[scale=0.23]{sim_reads_alpha.pdf}
\caption{}
\label{fig:gull}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[scale=0.23]{sim_snps_alpha.pdf}
\caption{}
\label{fig:gull}
\end{subfigure}%
\caption{Performance for parameter inference: (a) mean squared deviation for $\mathcal{W}$ by number of read counts; (b) mean squared deviation for $\mathcal{W}$ by number of SNPs; (c) absolute normalized deviation for $\alpha$ by number of read counts; and (d) absolute normalized deviation for $\alpha$ by number of SNPs.}
\label{fig:comp_sim_para}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.37]{sample_portrait.pdf} &
\end{tabular}
\end{center}
\caption{The frequency of number of inferred strains per sample (lef) and the posterior median value of $\alpha$ by the number of inferred strains (right).}
\label{fig:portrait}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.37]{some_samples.pdf} &
\end{tabular}
\end{center}
\caption{Examples of real data. For three samples (rows), we present the observed data WSAF plotted against the PLAF (first column), a diagram of the inferred model indicating the bands, proportions, and $\alpha$ (second column), and data simulated under the inferred model. $\alpha$ and $\mathcal{W}$ are the maximum \emph{a posteriori} values. In the second column, the model's PLAF-varying mixture densities are shown in grey scale, with black equal to one. }
\label{fig:some}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.37]{num_fstat.pdf} &
\end{tabular}
\end{center}
\caption{Boxplot of the F-statistic inbreeding coefficient ($1-F_{is}$) for each sample grouped by the number of inferred strains. }
\label{fig:f_stat}
\end{figure}
\clearpage
|
{
"timestamp": "2015-06-01T02:14:19",
"yymm": "1505",
"arxiv_id": "1505.08171",
"language": "en",
"url": "https://arxiv.org/abs/1505.08171"
}
|
\section{Introduction}
The description of the space of orbits of multidimensional toric action $(\mathbb{C}^*)^d \lefttorightarrow \mathbb{P}^n$ was a part of a hypergeometric project in [GKZ] and led to a development of a notion of a \textbf{secondary polytope}. The space of orbits (together with the limit degenerations) is a toric variety which is dual to the \textbf{secondary polytope}. In [GKZ] this space is called the \textbf{Chow quotient}.
In [A], this space was broadly generalised to the moduli space consisting of multiple irreducible components, called the \textbf{modular Chow quotient}. Virtual fundamental classes for this space are defined in the work [M].
In our work we construct from the combinatorics of secondary polytope the \textbf{Secondary operad}: cyclic operad with operations corresponding to the strata of the real version of modular Chow quotient for $d=2$. It corresponds to the Morse theory in the $1$-dimensional setting, and the complex version (which is still unclear) would correspond to the Losev-Manin commutativity equations [LM],[L].
The relations between the various problems of enumerative geometry discussed in this paper are presented in the following tables:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Theory: & Morse & $\mathbb{C}^*$ action & enumeration\\
& & & of curves\\
\hline
Enumerative problem: & integral & invariant & Gromov-Witten\\
& curves & rational curves & \\
\hline
Moduli space: & A-$\infty$ operad & Losev-Manin & Deligne-Mumford\\
\hline
Algebraic structure: & $\partial^2 = 0$ & commutativity & WDVV\\
& & equations & \\
\hline
\end{tabular}
\end{center}
The relation between the commutativity equations and the Morse theory is explained in [L], and relation between commutativity equations and WDVV is explained in [LM]. We'd like (ideally) to construct the analogue of the Gromov-Witten theory for the mappings of the surfaces to an algebraic variety $X$. There are various difficulties, but one of the ways of doing it might be looking for its equivariant analogies.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Theory: & 2-Morse* & $(\mathbb{C}^*)^2$ action & enumeration\\
& & &of surfaces\\
\hline
Enumerative problem: & integral leaves & ? & ?\\
\hline
Moduli space: & ? & Chow quotient & Alexeev's\\
& & & stable pairs\\
\hline
Algebraic structure: & Secondary operad & ? & ?\\
\hline
\end{tabular}
\end{center}
* - By $2$-Morse we mean the description of integral leaves of the pair of commuting gradient-like vector fields. This theory seems to lack argument of general position (which is present in $1$-Morse), so it seems one should think only of the complexifiable actions. Nevertheless, we can still develop the corresponding algebraic structures.
The "?" signs in the table above correspond to the fact that we don't really understand, what kind of enumerative problems should be considered: should we enumerate surfaces passing through the number of points, or the number of curves. The construction of Alexeev suggests the latter, but precise problem is unclear.
However, "Morse-like" real version of the problem can be considered without any deformation, and this advantage allows us to describe the structure for the real case.
The paper is organised as follows. In Section $2$ we will construct the formula for the expected codimension of a toric degeneration, show that it sometimes gives non-positive result and suggest a program of getting rid of such "bad" degenerations. Section $3$ is a brief exposition of the theory of secondary polytopes and Alexeev's modular Chow quotient. In the Section $4$ we will define perturbed regularity condition and use it to fulfill our program and define the secondary operad. In the last section we will discuss questions our work opens.
The algebro-geometric part of the paper is mostly sketchy, because we don't prove anything new in the algebro-geometric realm. The reader interested in the rigorous constructions of the moduli spaces of surfaces should definitely take a look in the works [A],[M]. However, we hope that the algebraic structures from this enumerative geometry after development would incorporate these formalisms and show further directions for the appropriate generalisations.
\textbf{Acknowledgements.} I want to thank my scientific advisor, Andrei Losev, for presenting me this question and constantly keeping my interest to the subject, even when it seemed to be completely obscure for both of us. I want to thank Petr Pushkar and Dmitry Korb for the discussions and counterexamples strongly influencing my intuition in this area and Valery Alexeev, who's minicourse on the moduli spaces of surfaces at HSE gave me the more rigorous basis for thinking about the geometric counterpart of the problem. Also I want to thank Anastasiya Goryatchkina, who helped me with vector graphics for this paper.
The article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. The author was also supported in part by the Moebius Contest Foundation for Young Scientists and by the Simons Foundation.
\section{Expected codimension}
Let us consider a real smooth compact manifold $M$ with a gradient-like vector field $v$ in a general position. Let us consider two critical points $A$, $B$. $n_A^B$ is defined as a number of integral curves of the field $v$, starting at $A$ and ending in $B$, taken with appropriate signs (if their number is infinite, $n$ is taken to be $0$). The main theorem of the Morse theory states that $n^2 = \sum \limits_{B} n_A^B n_B^C = 0$.
Let us present the sketch of proof (or, maybe, explanation) of this theorem. For the detailed exposition of the Morse theory the reader can take a look in [Mil].
Let us consider the space of integral curves between $A$ and $C$. If it is $k$-dimensional, then its boundary is $k-1$-dimensional and consists of degenerated curves. The degeneration in the general position (i.e. degenerations which occur in the codimension one) are the degenerations with one intermediate critical point ($B$). Taking $k=1$ we can conclude that the space of curves between $A$ and $C$ consists of the union of circles and line segments, and boundary components of the line segments are precisely the elements of the sum $\sum \limits_{B} n_A^B n_B^C$. For each line segment, the contributions of its ends have an opposite signs, hence the sum vanishes.
If we consider the fundamental cycle of the space of integral curves $\Phi_A^C$, then by the same argument, the following Maurer-Cartan relation holds: $\partial (\Phi_A^C) = \sum \limits_{B} \Phi_A^B \Phi_B^C$ for any $A$, $C$.
\begin{center}
\includegraphics[width=6.0cm, viewport=100 280 550 730, clip]{1.pdf}
\textit{Degeneration of an integral curve in a Morse theory.}
\end{center}
The same program can (and should) be applied to the leaves of an action of a pair of commuting vector fields (or, in the complex case, to the orbits of $(\mathbb{C}^*)^2$-action).
The closure of an orbit of $(\mathbb{C}^*)^d$ is a toric variety (and in the real case, closure of the leaf is a real toric variety). So, at first we will describe an applicable combinatorial formalism for the degenerations.
\definition \textbf{Affine fan} $F$ is the finite decomposition of the space $\mathbb{R}^d$ into the union of convex (maybe not bounded) polytopes. The fan is always assumed to be non-empty.
Affine fan encodes the following variety: let us take a vertex $v$ of a fan $F$. Link of this vertex is a toric fan $T(v)$. Let us take the union of $T(v)$'s and glue* them along the components of the boundary, corresponding to the edges of the fan $F$.
* - gluing in algebraic geometry can be done in multiple ways, but there is the most "economic" one, which is seminormal.
\definition \textbf{Infinity of the affine fan} $Inf(F)$ is a proper toric fan, constructed from the affine fan as follows: take any point in the space $\mathbb{R}^d$ and start rescaling it with constant tending to zero. Limit of this procedure is called infinity of the affine fan.
Affine fan encodes the flat equivariant degeneration of a toric variety, encoded by its infinity.
\begin{center}
\includegraphics[width=6.0cm, viewport=150 250 500 600, clip]{2.pdf}
\textit{Affine fan and its infinity.}
\end{center}
Let us consider the $2$-dimensional affine fan $F$. Let us denote by $v(F)$ the number of its vertices, and $e(F)$ the number of its internal edges.
Let $X$ be an algebraic variety with $(\mathbb{C}^*)^2$-action (corr. its real part).
\definition \textbf{Expected codimension} of $F$ is a jump of an invariant euler characteristic of a normal sheaf to the general leaf and the degeneration of type $F$.
\lemma It depends only on $F$ and calculated as
$Codim(F) = 2 v(F) - e(F) - 2$.
\proof
The restriction of a tangent bundle of $X$ is flat, so the jump of an invariant euler characteristic of a normal sheaf can be calculated as a minus jump of an invariant euler characteristic of a tangent bundle. It proves the first statement.
An invariant euler characteristic of the tangent bundle is an expected dimension of equivariant automorphisms - which is $2 \times$ number of toric components $-$ number of relations, i.e. internal edges. The non-degenerated leaf has a $2$-dimensional space of automorphisms (the torus itself), hence the jump equals $2 v(F) - e(F) - 2$. $\blacksquare$
\begin{center}
\includegraphics[width=12.0cm, viewport=-200 400 800 800, clip]{3.pdf}
\textit{Various degenerations of codimension 1. Diagrams in the upper part of the picture show the combinatorics of the leaves.}
\end{center}
So, by the analogy with the Morse theory, one could possibly assume the following \textbf{naive conjecture}:
Let us consider $M$ - real smooth compact manifold with a \textit{good} action of two commuting gradient-like vector fields. Let us consider, additionally, that $0$-dimensional and $1$-dimensional orbits of the group $\mathbb{R}^2$ of flows of these vector fields are isolated. Then, for any cycle $C$ of $1$-dimensional orbits we can consider the number of integral leaves $n_C$.
\textbf{Naive conjecture}: Then, the following equality holds (the sum is taken with appropriate signs) for any cycle $C$ of $1$-dimensional orbits.
\begin{equation}
\sum \limits_{F|codim(F)=1} \#F(C)
\end{equation}
where $\#F(C)$ is a number of degenerated toric varieties of type $F$ with the boundary $C$.
This naive conjecture is not true and cannot be true in any reasonable way and the simplest way to see it is the following:
\textit{Expected codimension sometimes happen to be zero or even negative}.
The example is given below.
\begin{center}
\includegraphics[width=10.0cm, viewport=0 300 550 730, clip]{4.pdf}
\textit{Zero codimension degeneration.}
\end{center}
In the next section we will see that this situation is typical and the degeneration \textit{actually has} more deformations than the non-degenerated leaf. These deformations belong to the other irreducible component of the moduli space, constructed by Alexeev.
Let us denote by $\Phi_C$ the fundamental class of the space of leaves with the boundary $C$ and by the $F(\Phi_{C_1}, ..., \Phi_{C_n})$ the fundamental class of the space of degenerated leaves of type $F$ with components with boundaries $C_1, ..., C_n$.
\textbf{Idea of the treatment:} An equation $\partial(\Phi_C) = \sum \limits_{F|codim(F)=1} F(\Phi_{C_1}, ..., \Phi_{C_n})$ cannot hold, because there are also boundary components, corresponding to the $codim(F) < 1$. But if we consider, for example, $F(\Phi_{C_1}, ..., \Phi_{C_n})$ for $codim(F) < 1$, we will get the class of wrong dimension. We need only the class of the locus of such degenerated toric varieties, which occur as the degenerations of the regular leaves. The idea is to deform this locus to the homotopy-equivalent component of the boundary.
\section{Secondary polytopes and Alexeev's modular Chow quotient}
This part of exposition mainly follows [GKZ].
Let us consider a set $A$ of points of a lattice $\mathbb{Z}^2$. Denote by $X$ the projective space $\mathbb{P}(\mathbb{C}^A)$ together with the action of $(\mathbb{C}^*)^2$, defined as follows: for a basis vector of $\mathbb{C}^A$, corresponding to the point $a \in A$ with coordinates $(x, y)$ it acts with the character $z^x w^y$.
The closure of the general orbit $Z \in X$ is a $2$-dimensional toric variety, which moment polytope coincides with the convex hull of $A$.
\definition Irreducible component of $Z$ in the $(\mathbb{C}^*)^2$-invariant locus of the Chow scheme is called \textbf{Chow quotient} of X.
\note It is a toric variety w.r.t to the torus $(\mathbb{C}^*)^A/(C^*\times(\mathbb{C}^*)^2)$ which naturally parametrises general orbits and their possible degenerations.
\definition Connected component of $Z$ in the $(\mathbb{C}^*)^2$-invariant locus of the Chow scheme is called \textbf{modular Chow quotient} of X.
\note This definition is different from the original definition in [A], as it doesn't take into account the stack structure of the modular Chow quotient (which we won't use).
Now we shall construct a toric fan of the Chow quotient.
Let us consider a space $Fun(A) = \mathbb{R}^A$ of real-valued functions on $A$.
The space of affine functions $Aff(\mathbb{R}^2)$ is naturally mapped to $Fun(A)$ via restriction. We denote by $V = Fun(A)/Aff(\mathbb{R}^2)$.
For any $v \in Fun(A)$ let us consider a minimal concave function $f$ on the convex hull of $A$ such that $f(x,y) \geq v(a)$ for $\forall a \in A$ with coordinates $(x, y)$. It is obviously piecewise-linear, with the domains of linearity being convex polytopes $\{P_1, ..., P_k\}$, $\bigcup P_i = Conv(A)$.
It depends only on a class of $v$ in $V$.
Denote the corresponding decomposition of $Conv(A)$ into the union of polytopes by $D(v) = \{P_1, ..., P_k\}$
This data is combinatorial and induces the stratification of a space $V$ into a union of open convex polyhedral cones.
\definition This collection of cones is called the \textbf{secondary fan} of $A$.
\note Elements of $V$ naturally encode $1$-parametric orbits in the torus $(\mathbb{C}^*)^A/(\mathbb{C}^*\times(\mathbb{C}^*)^2)$, and $D(v)$ encodes the corresponding degeneration of a (moment polytope of a) leaf while moving along this $1$-parametric orbit.
\definition The decomposition $D = \{P_1, ..., P_k\}$ is called \textbf{regular} if it comes from the construction as $D(v)$ for some $v$, i.e. if it corresponds to some degeneration of a general orbit.
\note In the modular Chow quotient, the leafs of the degeneration deform independently, hence no condition of regularity is imposed.
\definition An affine fan $F$ in the space $(\mathbb{R}^2)^*$ is called \textbf{normal} to the decomposition $D$ in the space $\mathbb{R}^2$ if its vertices correspond to the polygons of $D$ and the edges are normal to the boundaries between corresponding polygons.
\begin{center}
\includegraphics[width=6.0cm, viewport=0 200 550 500, clip]{8.pdf}
\includegraphics[width=6.0cm, viewport=100 230 550 730, clip]{9.pdf}
\textit{Normal fan to the decomposition.}
\end{center}
\theorem Normal fan to the decomposition $D$ exists if and only if this decomposition is regular.
\proof
Let us assume that the decomposition is regular w.r.t. to the concave function $f$. Consider affine functions $f_i = f|_P$ on each $P_i$. Turn them into linear functions $\gamma_i \in (\mathbb{R}^2)^*$. It will be the set of vertices of the affine fan. For two neighboring polygons $P_i$ and $P_j$ consider an edge between $\gamma_i$ and $\gamma_j$. It will be orthogonal to their common boundary, because $f_i = f_j$ on it. It is not hard to see that concavity of function $f$ guarantees that the resulting object will be an affine fan.
Conversely, having an affine fan, we get the set of $\gamma_i$'s for free. Let $g_i$'s be any affine functions with the linear parts $\gamma_i$'s. Then, $g_i - g_j|_{P_i\cap P_j}$ are constants $a_{ij}$, satisfying the cocycle condition $a_{i_1 i_2} + ... + a_{i_s i_1} = 0$ for any $P_{i_1}, ..., P_{i_s}$ located along a vertex in a decomposition. $H^1(Conv(A), \mathbb{R}) = 0$, hence this cocycle is a coboundary, so we can produce another set $f_i = g_i + const_i$ such that $f_i = f_j$ on $P_i \cap P_j$. Concavity of the constructed piecewise-linear function follows (by the local checks on a boundaries between $P_i$ and $P_j$'s) from the fact that $\gamma_i$'s are the vertices of an affine fan. $\blacksquare$
So, we are in a need for the new object, which would be dual to the non-regular decompositions. From the fact that two affine fans which are combinatorially equivalent and have parallel edges correspond to the same degenerated toric variety we come to the following
\definition \textbf{Graph with directions (GwD)} is an oriented graph with outgoing edges (one can think that they go to the vertex "infinity"). Each edge is endowed with an additional data: the direction, i.e. unit vector in $\mathbb{R}^2$. It is allowed to change orientation of an edge together with changing the direction of an edge to the opposite.
We will use the wording "outgoing direction along an edge $e$ from a vertex $v$" for the direction of $e$ if it is oriented from $v$ and for the minus direction of $e$ otherwise.
We will consider only graphs such that outgoing directions from any vertex positively generate $\mathbb{R}^2$.
\note It is the first (but not the last) place in the paper when giving a definition in the dimension $d > 2$ would be hard (and we don't even know how to do it right).
\definition \textbf{Positive representation of a GwD} $\Gamma$ is an affine fan $F$ with vertices corresponding to the vertices of $\Gamma$, edges corresponding to the edges of $\Gamma$ and parallel to the directions of those edges (orientation-preserving).
They are assumed to be equivalent if they differ only by a parallel transport.
\begin{center}
\includegraphics[width=10.0cm, viewport=-100 300 700 600, clip]{7.pdf}
\textit{Different positive representations of the same GwD.}
\end{center}
\definition \textbf{Representation of a GwD} $\Gamma$ is a set of points $p_i$'s corresponding to the vertices of $\Gamma$, such that vectors $e_{ij} = p_i - p_j$ are parallel to the directions of edges between vertices $i$ and $j$. No conditions about orientation are imposed, $p_i$'s can even coincide (and edge between such points would be parallel to anything).
\note Representations of $\Gamma$ obviously form a vector space $Rep(\Gamma)$, and set of positive representations $PRep(\Gamma)$ is a convex polyhedral cone.
The reader could notice that we actually defined some invariant of the GwD - the dimension of its space of representations. The calculation of dimensions shows that it is expected to be $2v - e - 2$, where $v$ denotes the number of vertices and $e$ denotes the number of internal edges - it coincides with the formula of the expected codimension.
\definition GwD $\Gamma$ is called \textbf{too rigid} if $dim(Rep(\Gamma))$ is greater than expected.
\note Degenerations corresponding to the too rigid GwD's are the boundary components, connected to the other irreducible components of modular Chow quotient. It is (maybe in not so explicit way) can be found in [A].
\note Those GwD's wouldn't exist for a general set of directions. However, we cannot simply deform the directions - because they are rational edges of toric fans.
\begin{center}
\includegraphics[width=8.0cm, viewport=0 350 550 700, clip]{5.pdf}
\textit{Too rigid graphs doesn't exist for a general set of directions.}
\end{center}
We proceed by construction of the secondary operad.
\section{Secondary operad}
Let $A$ be a (finite) set in $\mathbb{R}^2$. Assume that there are no three points in $A$ on the same line (this condition is technical and will be discussed later).
\definition The basis operations of a \textbf{secondary operad} are any polygons with vertices in $A$, decomposed into a union of convex polygons with vertices in $A$ with non-intersecting interiors. The composition is the gluing of two polygons by a set of edges.
\textbf{Sign convention:} the set of internal edges of the polygonal decompositions is ordered. Changing an order shifts the sign by the sign of the permutation. The parity of the basis element is counted as the parity of the number of edges plus one, hence the composition is of degree $0$. There might be different conventions about $\mathbb{Z}$-grading.
\note For a regular decomposition $D$ the sign data is equivalent to the orientation on the polyhedral cone $PRep(F)$, where $F$ is a normal fan to $D$ (coming from the exact sequence $0 \rightarrow Rep(F) \rightarrow (\mathbb{R}^2)^v \rightarrow (\mathbb{R})^e \rightarrow 0$). Hence, any regular decomposition of codimension $1$ has a canonical sign data, as it has the $1$-dimensional $PRep$.
\textit{Assume that all GwDs dual to the decompositions of} $A$ \textit{are not too rigid (i.e., all decompositions are regular).}
\note It is rare situation and it doesn't follow from the conditions we imposed on $A$.
Then, the following works:
\definition The differential $\partial$ is defined as follows: for any convex polytope $P$ it returns $\partial(P) = ( \sum \limits_{D|_{codim(D) = 1}}D)$ the sum over decompositions of $P$ (with the canonical sign data). For a decomposition $\partial(D)$ is defined by Leibniz rule.
\lemma $\partial^2 = 0$
\proof The sign data from the Leibniz rule is equivalent to the orientation by outward normal vector: for any decomposition $D$ of codimension 2 there is $2$-dimensional cone of representations, which has boundary, consisting of two edges - corresponding to the two summands of type $D$ in $\partial^2 (P)$ with the opposite signs.
\note Reformulating geometrically: consider the space of representations of GwD's dual to the decompositions of $P$, and glue them along the boundary components - when a part of the representation of a GwD shrink into a point, glue this boundary component to another space of representations with GwD with this part shrinked. Then the operations of secondary operad will correspond to the (oriented) facets of this space, $\partial$ is just the dual cellular differential and $\partial^2 = 0$ is obvious.
\note The space constructed above is actually a toric fan for a real version of Chow quotient.
\begin{center}
\includegraphics[width=12.0cm, viewport=0 400 650 750, clip]{10.pdf}
\textit{Shrinks of the codim=2 GwD and $\partial^2 = 0$.}
\end{center}
Now we finally move to the case where too rigid graphs occur.
Let us consider the set of all possible directions $S$ occuring in all decompositions of $A$. Let us consider the (germ of) deformation $S_t$ general enough (each direction is being deformed smoothly). Then, no too rigid graphs can be constructed using this set of directions for $t \neq 0$.
\note This is the main reason why we ask $A$ to have no triples of points on a same line - we don't want the same directions to occur in a different parts of GwD.
\definition The decomposition $D$ is called \textbf{perturbedly regular} if the dual GwD with deformed directions admits a positive realisation for $t \neq = 0$.
\note No too rigid graphs are dual to the perturbedly regular decompositions, but some irregular decompositions will become perturbedly regular.
\begin{center}
\includegraphics[width=16.0cm, viewport=0 600 700 800, clip]{6.pdf}
\textit{Some irregular decompositions will become perturbedly regular.}
\end{center}
\definition $\partial(P) = ( \sum \limits_{D|_{codim(D) = 1}}D)$, where the sum is taken over all perturbedly regular decompositions of codimension 1.
\theorem $\partial^2 = 0$
\proof The same argument as in the previous theorem should be applied to the same space with $0 < t \ll 1$.
Now, as we defined an operad, we will discuss why do we think that this definition is right.
Consider the following situation: the leaf deforms to the degeneration corresponding to the too rigid graph. Then these degenerations form a locus in an irreducible component of the modular Chow Quotient. The fundamental class of this locus should be the summand in the $\partial(\Phi)$, but we want it to be the composition in an operad, so we deform it to the union of boundary components of this component.
\definition \textbf{Perturbation} of a too rigid graph is a set of all perturbed graphs who's realisations tend to this too rigid graph as $t \rightarrow 0$.
\note So, the definition of $\partial$ can be reformulated as follows: we take the sum over all $F$'s with $codim \leq 1$, but then perturb too rigid summands.
\claim {Let us take the real part of the modular Chow quotient. Consider too rigid decomposition $D$ and the corresponding irreducible component $Z$ (which is just the product of modular quotients for each $P_i \in D$). Then the intersection $L$ with the main component can be deformed (without moving a boundary) to the union of the boundary components from the perturbation of $D$.}
\textit{Sketch of the proof}
$Z$ is a toric variety w.r.t. to the torus $\prod \limits_i(\mathbb{C}^*)^{P_i}/((\mathbb{C}^*)^2\times(\mathbb{C}^*))$, and $L$ is a closure of an orbit of the subtorus which is given by the image of $(\mathbb{C}^*)^{A}/((\mathbb{C}^*)^2\times(\mathbb{C}^*))$. It can be easily seen (and contained in [A]) that the quotient torus (which controls the deformations) is the cokernel of the map $((\mathbb{C}^*)^2)^v \rightarrow (\mathbb{C}^*)^e$.
It agrees with the fact that the space of first-order deformations of the directions modulo the subspace of deformations not destroying too rigid graph is the cokernel of the complex $(\mathbb{R}^2)^v \rightarrow (\mathbb{R})^e$ (and these too complexes can be organised into a commutative diagram via exponentiation). So, from the first-order deformation of the directions one obtains the $1$-parametric subgroup in the normal quotient torus.
Being deformed along this (general enough) $1$-parametric subgroup, $L$ tends to the union of boundary components. The boundary component is being tended to by this $1$-parametric subgroup if and only if it is contained in the perturbation.
\note In the complex version these components might have some integer coefficients which are to be calculated.
\section{Questions}
\question What is the complex analogue of the presented structure? From the dimension one we know that it should be a formal series in some variables. Will these variables correspond to the cohomology of $X$ or the space of curves on $X$? How to do it?
\question What should one do if the set $A$ has triples of points on a same line? Even if the perturbation can be done in such a way that avoids too rigid graphs (and it seems it can), it is still unclear what should be used instead of Leibniz rule. The structure becomes definitely something richer than just a cyclic operad.
\question There is a problem with a formalism of virtual fundamental classes for this theory - the normal sheaf to the surface can have second cohomology, hence the deformation theory is not perfectly obstructed. The work [M] solves this problem (in a way we don't fully understand). Can one use these virtual fundamental classes to prove that structures of the type considered in this paper actually act on any projective variety $X$ with an action of $(\mathbb{C}^*)^2$?
\question It is unclear how to do the same formalism of perturbations in the dimension greater than $2$.
|
{
"timestamp": "2015-06-01T02:14:04",
"yymm": "1505",
"arxiv_id": "1505.08157",
"language": "en",
"url": "https://arxiv.org/abs/1505.08157"
}
|
"\\section{Introduction}\n\nLevel-set percolation for the (massive and massless) Gaussian free field(...TRUNCATED)
| {"timestamp":"2015-06-01T02:14:12","yymm":"1505","arxiv_id":"1505.08169","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nStudies of CPT and Lorentz violation in the standard model \\cite{Collad(...TRUNCATED)
| {"timestamp":"2015-06-01T02:14:03","yymm":"1505","arxiv_id":"1505.08156","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\r\n\r\nDespite of the fact that some chemically peculiar (CP) stars have \"(...TRUNCATED)
| {"timestamp":"2015-06-01T02:14:05","yymm":"1505","arxiv_id":"1505.08158","language":"en","url":"http(...TRUNCATED)
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"\\section*{Abstract}\nA mass of traces of human activities show rich dynamic patterns.\nIn this art(...TRUNCATED)
| {"timestamp":"2015-06-01T02:14:06","yymm":"1505","arxiv_id":"1505.08159","language":"en","url":"http(...TRUNCATED)
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"\\section{The arithmetic applications}\n\\label{sec:arith-applic}\n\n\n\\subsection{Setup and runni(...TRUNCATED)
| {"timestamp":"2015-06-01T02:14:10","yymm":"1505","arxiv_id":"1505.08165","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\r\n\r\nWe assume that the reader is familiar with basic notation and termin(...TRUNCATED)
| {"timestamp":"2015-06-01T02:14:07","yymm":"1505","arxiv_id":"1505.08162","language":"en","url":"http(...TRUNCATED)
|
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