\section{#1}\setcounter{lemma}{0}}

\title{Non-commutative groups as prescribed polytopal symmetries}
\author{Alexandru Chirvasitu, Frieder Ladisch, and Pablo Sober\'on}

\begin{document}

\date{}

\newcommand{\Addresses}{
  \bigskip
  \footnotesize

  \textsc{Department of Mathematics, University at Buffalo, Buffalo,
    NY 14260-2900, USA}\par\nopagebreak \textit{E-mail address}:
  \texttt{<EMAIL>}

  \medskip

  \textsc{Institut für Mathematik, Universität Rostock,
          18051 Rostock,
          Germany}\par\nopagebreak \textit{E-mail address}:
     \texttt{<EMAIL>}

  \medskip

  \textsc{Department of Mathematics, Baruch College, City University of New York,
    New York, NY 10010, USA}\par\nopagebreak \textit{E-mail address}:
  \texttt{<EMAIL>}

}}

\maketitle

\begin{abstract}

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope.  We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with  $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$.  We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.

\end{abstract}

\noindent {\em Key words: polytope, symmetric polytope, combinatorial automorphism, geometric automorphism}

\vspace{.5cm}

\noindent{MSC 2010: 52B15; 52B11}

\tableofcontents

\subsection*{Acknowledgements}

A.C. is grateful for partial support through NSF grant DMS-1801011.
P.S. is grateful for partial support through NSF grant DMS-1851420.
F.L. is grateful for support by DFG, Project SCHU1503/7-1.

\section*{Introduction}\label{se.intro}

Polyhedra and their symmetries have been a rich subject of study.  In this paper we are interested in the symmetries of convex polytopes, the convex hulls of finite sets of points in $\mathbb{R}^d$.  For a given polytope~$\cP$, we consider the following two symmetry groups: The \emph{geometric symmetry group} $G(\cP)$ consists of all (Euclidean) isometries of the ambient space that map $\cP$ onto itself.  The \emph{combinatorial symmetry group} $\Gamma(\cP)$ of $\cP$ consists of all the automorphisms of the face lattice of $\cP$. The group $\Gamma(\cP)$ can be identified with the group of all permutations of the vertices of $\cP$ which map faces of $\cP$ to faces of $\cP$.  It has recently been established that every finite group is isomorphic to the combinatorial automorphism group of a convex polytope $\cP$.  This result was first proved by Schulte and Williams \cite{sw} and later by Doignon \cite{Doignon:2018ch} with a simplified proof.  In Doignon's paper, it was shown that every finite group is the combinatorial automorphism group of a $0/1$-polytope.

Given a group $G$, there are many different convex polytopes whose symmetry group is exactly $G$.  This leads to a rich family of extremal problems: \textit{given a group $G$ and a parameter $\lambda( \cdot )$, determine the maximum/minimum value that $\lambda(P)$ can take, where $\cP$ ranges over the polytopes whose (geometric or combinatorial) symmetry group is exactly $G$.} In this note we are interested in finding polytopes $\cP$ of minimal dimension with prescribed symmetry group.  We give a lower bound for the dimension of a polytope having an elementary abelian group of given order as combinatorial automorphism group (\Cref{pr.simpl-sph}).  A consequence is that for every $d$, there are finite groups which can not be realized as combinatorial automorphism groups of convex polytopes of dimension less than or equal to $ d$.
It turns out that for elementary abelian groups, our lower bound is sharp (\Cref{th:cpd-sharp}). While our results also yield lower bounds for the dimension of a polytope with some other prescribed finite group as combinatorial automorphism group, these lower bounds are probably not optimal when the group is not elementary abelian.

We also show that finding a polytope of smallest dimension with a prescribed group as \emph{geometric} symmetry group reduces to a problem in the representation theory of finite groups.  Using this, we can compute \emph{exactly} the smallest dimension of a polytope having a given abelian group as geometric symmetry group (\Cref{th.bd}), and similarly for other classes of groups. For example, the symmetric group $S_n$ on $n$ letters is the geometric (and combinatorial) symmetry group of a regular simplex of dimension $n-1$, and this is in fact the smallest dimension of a polytope with geometric symmetry group $S_n$ (see \Cref{th.bd_sn_etal}).

The results described so far affirmatively answer two questions of Schulte, Sober\'on and Williams \cite[Open Questions~1 and 2]{ssw}.
We also consider the following problem suggested by Schulte, Sober\'on, and Williams: Instead of solving instances of the general extremal problems described above, one can look for polytopes with fixed symmetry groups that satisfy additional geometric properties.  This was explored in the cited paper~\cite{ssw}, where it was established that every finite abelian group of even order is the automorphism group of a \textit{centrally symmetric} polytope.  Moreover, the involution that corresponds to the central symmetry can be prescribed in advance.  We generalize this result to non-abelian groups with a central involution, thereby answering another question from \cite{ssw} (see \Cref{th.sym}).  The methods we use come from representation theory and geometric group theory.  Representation theory has been used previously to study convex polytopes, as can be seen in \cite{Guralnick:2006gk, Friese:2016ka, Baumeister:2009dfa} and the references therein.

Our main results answer all the open questions \cite[Open Questions 1, 2 and 3]{ssw} affirmatively.  In \Cref{sec.openqu}, we propose some new questions, which are suggested by our results.

\section{Preliminaries}

Given a polytope $\cP$, we denote by $\Gamma(\cP)$ its group of combinatorial automorphisms and by $G(\cP)$ its group of geometric symmetries.  One of our main tools we use is the following theorem.

\begin{theorem}[\cite{ssw}]\label{theorem-symmetrybreaks}
  Let $d\geq 3$, let $Q$ be a convex $d$-polytope with (combinatorial) automorphism group $\Gamma(Q)$, and let $\Gamma$ be a subgroup of $\Gamma(Q)$. Then there exists a convex $d$-polytope~$\mathcal{P}$ with the following properties:
\begin{enumerate}
\item  $\Gamma(\mathcal{P})=\Gamma$.
\item  \label{it:tess} $\mathcal{P}$ is isomorphic (as an abstract polytope) to a face-to-face tessellation $\mathcal{T}$ of the $(d-1)$-sphere $\mathbb{S}^{d-1}$ by spherical convex $(d-1)$-polytopes.
\item $\skel_{d-2}(\mathcal{C}(Q))$ is a subcomplex of $\skel_{d-2}(\mathcal{P})$.
\item  \label{it:geom} If $\Gamma$ is a subgroup of the (geometric) symmetry group $G(Q)$ of $Q$, then the tessellation $\mathcal{T}$ on $\mathbb{S}^{d-1}$ in \labelcref{it:tess}  can be chosen in such a way that $G(\mathcal{T})=\Gamma=\Gamma(\mathcal{T})$.
\end{enumerate}
\end{theorem}

In the statement above, $\skel_{d-2}(\cdot )$ stands for the $(d-2)$-dimensional skeleton of a polytope, and $\mathcal{C}(Q)$ for the barycentric subdivision of the boundary complex of $Q$. The proof of \Cref{theorem-symmetrybreaks} is done by taking a barycentric subdivision of $Q$ and then adding faces in order to break the undesired symmetries in $\cP$.  The method preserves any geometric symmetries of $Q$ that were present in $\Gamma$.  This method actually gives a stronger result.  Not only is the group of combinatorial automorphisms of $\cP$ equal to $\Gamma$, but the group of automorphisms of $\skel_{1}(\cP)$ (as a graph) is equal to $\Gamma$.  The reason for this is that the only tool that $\cite{ssw}$ uses to determine that extra symmetries have been broken is an analysis of the degrees of the vertices in $\skel_{1}(\cP)$.  It follows that in \Cref{th.sym} below, the combinatorial automorphism group~$\Gamma(\cP)$ can be replaced by the automorphism group of the $1$-skeleton (as a graph) of our polytope.

The following corollary will be useful a few times:
\begin{corollary}\label{cor.geosym-givengroup}
  Let $d\geq 3$ and let $\Gamma$ be a finite subgroup of the orthogonal group $\orth_d(\bR)$.  Then there exists a convex $d$-polytope~$\cP$ such that $G(\cP) = \Gamma(\cP) = \Gamma$.
\end{corollary}
\begin{proof}
  The group~$\Gamma$ acts on the unit sphere $\bS^{d-1}\subset \bR^d$.  If we take a sufficiently large union $F$ of finitely many $\Gamma$-orbits, then the convex hull $\cQ = \conv(F)$ will be a convex $d$-polytope.  By construction, $\Gamma \leq G(\cQ)$, so the result follows from~\Cref{theorem-symmetrybreaks}, \labelcref{it:geom}.
\end{proof}

A few times, we will use the following standard result from representation theory~\cite[Theorem~2.13]{huppCT}.
\begin{proposition}
\label{pr.inv_innprod}
Let $\rho\colon \Gamma\to \GL(d,\bR)$ be a representation of the finite group $\Gamma$.  Then there is a $\Gamma$-invariant inner product on $\bR^d$.  Equivalently, the representation $\rho$ is similar to a representation $\Gamma\to \orth_d(\bR)$.
\end{proposition}
\begin{proof}
  Let $(\cdot,\cdot)$ be an arbitrary inner product on $\bR^d$.  Then $[x,y]:= \sum_{g\in \Gamma} (\rho(g)x,\rho(g)y)$ yields a $\Gamma$-invariant inner product on $\bR^d$.  When $S=(s_1,\dotsc,s_d)$ is an orthonormal basis of $\bR^d$ with respect to $[\cdot,\cdot]$, then the corresponding matrices $S^{-1}\rho(g)S$ are orthogonal.
\end{proof}
A general consequence of this result is that it does not make much of a difference for the questions considered in this paper, whether we deal with the geometric symmetry group of polytopes, or the group of affine symmetries of polytopes (the affine maps sending a polytope to itself).

\section{Symmetric polytopes}\label{subse.sym}

Our first main result answers \cite[Open Question 3]{ssw} affirmatively.

\begin{theorem}\label{th.sym}
  Let $\Gamma$ be a finite group and $\sigma\in \Gamma$ a central order-two element. Then, there is a centrally symmetric polytope $\cP$ with
  \begin{equation*}
    G(\cP)=\Gamma(\cP)\cong \Gamma
  \end{equation*}
  and $\sigma$ acting as the central symmetry.
\end{theorem}
\begin{proof}
  Let $G=\langle\sigma\rangle$ and $\rho:G\to \{\pm 1\}\subset \bR^{\times}$ the non-trivial representation, whose one-dimensional real carrier space we denote by $V$. Upon equipping the induced representation
  \begin{equation*}
    W:=\mathrm{Ind}_G^{\Gamma}V
  \end{equation*}
  with a $\Gamma$-invariant inner product
  (\Cref{pr.inv_innprod}),
  we get an embedding of $\Gamma$ into the orthogonal group of $W$.
  Moreover, $\sigma$ acts as $-1$ on $W$ by construction.
  By \Cref{cor.geosym-givengroup},
  it follows that there is a polytope~$\cP$ such that
  $G(\cP)=\Gamma(\cP)\cong \Gamma$.
\end{proof}

\section{The convex polytope dimension}\label{subse.1-2}

Our next result shows that the ``convex polytope dimension of a group'' is a meaningful parameter.

\begin{definition}
  Let $G$ be a group.  We define $\cpd(G)$ (the \textit{convex polytope dimension of $G$}) to be the minimum integer $d$ such that there is a convex polytope $\mathcal{P}$ in $\mathbb{R}^d$ such that $\Gamma (\mathcal{P}) = G$.
\end{definition}

The results in \cite{sw} show that this parameter is well defined.  In this section we will prove that it can be arbitrarily large, and is therefore interesting to compute.  Note that when $A$ is a subgroup of $B$ and $\cpd(B)\geq 3$, then $\cpd(A) \le \cpd(B)$.  This is a direct consequence of \Cref{theorem-symmetrybreaks}.  On the other hand,
the only groups with $\cpd(B)=2$ are the dihedral groups of order $2n\geq 6$, and the only groups with $\cpd(B)\leq 1$ are the groups of order $1$ and $2$. So when $B$ is a dihedral group and $A\leq B$ is cyclic of order $n\geq 3$, or the Klein four-group, then $\cpd(B)=2$, but $\cpd(A)=3$.  Aside from these exceptions, $\cpd(\cdot)$ is monotone.

\begin{remark}\label{re:act-on-sphr}
  As recalled in \cite[Section 2]{ssw}, to every convex polytope $P$ one can associate its {\it boundary complex}, consisting of its proper faces. This complex
  \begin{itemize}
  \item gives a tesselation of the boundary $\partial P$, which, topologically, is a sphere;
  \item has a barycentric subdivision giving a triangulation of said sphere, i.e., a {\it simplicial sphere};
  \item depends only on the combinatorial data of how the proper faces of $P$ are glued along facets.
  \end{itemize}
  It follows that the combinatorial symmetry group $\Gamma(\cP)$ of a polytope $\cP$ can be regarded as a group of {\it simplicial symmetries} of $\partial P$, i.e., automorphisms of $\partial P$ equipped with its simplicial complex structure. This renders plausible the relevance of the  result below.
\end{remark}

The proof of the following result requires, for a prime $p$ and an integer $n\ge -1$, the notion of a {\it cohomology $n$-sphere over $\bZ/p$}. Cohomology $n$-manifolds are introduced in \cite[Definition I.3.3]{borel-seminar} and are the central subject of study in said reference. They coincide with the objects introduced by Smith in \cite{smth-per2} to provide the appropriate context for his study of finite transformation groups.

For brevity, we refer to a cohomology $n$-manifold with the same homology as the sphere in $\bZ/p$ as a {\it cohomology $n$-sphere over $\bZ/p$}.

We do not recall the full definition of cohomology manifolds here, pausing only to note that
\begin{itemize}
\item simplicial $n$-spheres are cohomology $n$-spheres (over every $\bZ/p$);
\item cohomology $0$-spheres are just plain $0$-spheres, i.e. disjoint unions of two points (this follows for instance from \cite[5.3]{smth-per2}).
\end{itemize}

We will also be referring to the {\it link} $\mathrm{lk}_K(\Delta)$ of a simplex $\Delta$ in a simplicial complex $K$. This is usually defined via the {\it star} $\mathrm{st}_K(\Delta)$:
\begin{equation*}
  \mathrm{st}_K(\Delta) = \text{ subcomplex of }K\text{ generated by the simplices }\Delta'\text{ such that }\Delta\subseteq \Delta'\subseteq K.
\end{equation*}
Then, the subcomplex of $\mathrm{st}_K(\Delta)$ consisting only of simplices disjoint from $\Delta$ is the link $\mathrm{lk}_K(\Delta)$. See e.g. \cite[discussion preceding Proposition 3.2.12]{fp-cell} or \cite[Definition 2.4.2]{maun-at}, though in the latter reference our `stars' are `nerves'. The introductory material in \cite[\S 2]{bry-pl} is another reference for these notions.

It follows for instance from \cite[Theorem X.6.9]{wild-man} that in a simplicial complex that is a cohomology $n$-manifold, the link on a $d$-simplex is a cohomology $(n-d-1)$-sphere.

The following result is essentially \cite[second corollary, p.107]{smth}, but it was not obvious to us how it follows from the discussion preceding it, so we include a proof for completeness. See also \Cref{re:p2} for further discussion.

\begin{theorem}\label{pr.simpl-sph}
  Let $p$ be a prime, and let $\Gamma=(\bZ/p)^r$ be an elementary abelian $p$-group acting simplicially and effectively on a simplicial $n$-sphere. Then, we have
  \begin{itemize}
  \item $r\le n+1$ if $p=2$;
  \item $2r\le n+1$ otherwise.
  \end{itemize}
\end{theorem}
\begin{proof}
It will be convenient, throughout the proof, to denote
  \begin{equation*}
    e=
    \begin{cases}
      1 &\text{ if }p=2\\
      \frac 12 &\text{ otherwise}.
    \end{cases}
  \end{equation*}
  We prove the claim that $r\le e(n+1)$ by induction on $n$ for an action on a simplicial cohomology $n$-sphere $\cP$.

  \vspace{.5cm}

  {\bf Base case $n=0$.} We have observed above that cohomology $0$-spheres are just plain $0$-spheres. so the result is obvious: $\bZ/2$ can act effectively by permuting the two points that constitute the $0$-sphere, while $\bZ/p$, $p>2$ cannot.

  \vspace{.5cm}

  {\bf Induction step.} We can assume $r\geq 2$. Since according to \cite[first corollary, p.107]{smth} a non-cyclic abelian group cannot act freely on a cohomology sphere, there must be a cyclic subgroup
  \begin{equation*}
    \bZ/p\cong \langle \sigma\rangle = G\le \Gamma
  \end{equation*}
  with non-empty fixed-point set $\cF$. It is a non-trivial result that the simplicial subcomplex $\cF\subset\cP$ must again be a cohomology $m$-sphere for some $0\le m\le n$:
  \begin{itemize}
  \item It is shown in \cite[\S IV.4.3]{borel-seminar} that $\cF$ has the same $(\bZ/p)$-homology as an $m$-sphere;
  \item \cite[Theorem V.2.2]{borel-seminar} proves that $\cF$ is an orientable cohomology manifold;
  \item \cite[\S I.3.4]{borel-seminar} then shows that its dimension as a complex (and as a cohomology manifold) must be $m$.
  \end{itemize}
  We choose $\sigma$ such that the dimension of $\cF$ is maximal, and consider the pointwise isotropy group $\Gamma_{\cF}$ of $\cF$. Because $\Gamma$ is elementary abelian we have a decomposition
  \begin{equation*}
    \Gamma=\Gamma_{\cF}\oplus \widetilde{\Gamma},
  \end{equation*}
  and $\widetilde{\Gamma}$ acts on $\cF$ (because the latter is the fixed-point set of $\Gamma_{\cF}$ and $\widetilde{\Gamma}$ commutes with the latter). We first claim that this action of $\widetilde{\Gamma}$ on the ($m$-dimensional) cohomology sphere $\cF$ is faithful.

  Indeed, were it not, there would be an element $\widetilde{\sigma}\in \widetilde{\Gamma}$ that fixes a set containing $\cF$. This could not be proper containment  (for that fixed-point set would then be a strictly higher-dimensional cohomology sphere thus contradicting the maximality of $\dim \cF$), and hence
  \begin{equation*}
    \cP^{\widetilde{\sigma}} = \cP^{\sigma} = \cF.
  \end{equation*}
  But then \cite[Theorem]{smth} would imply that $\widetilde{\sigma}\in \Gamma=\langle\sigma\rangle$, i.e. a contradiction.

  But by induction, the faithfulness of the action of $\widetilde{\Gamma}$ on $\cF$ then gives
  \begin{equation}\label{eq:2}
    \rank~\widetilde{\Gamma}\le e(m+1).
  \end{equation}
  On the other hand, the fact that $\Gamma_{\cF}$ itself acts effectively on all of $\cP$ but trivially on $\cF$ implies that the resulting action of $\Gamma_{\cF}$ on the link of some $m$-simplex in $\cF\subset \cP$ is effective. We observed in the above discussion that said link is a simplicial cohomology sphere of dimension $m'$ satisfying
  \begin{equation}\label{eq:3}
    (m+1)+(m'+1)=n+1.
  \end{equation}
  The inductive step applied to {\it this} action gives us
  \begin{equation}\label{eq:4}
    \rank~\Gamma_{\cF}\le e(m'+1),
  \end{equation}
  and adding together \Cref{eq:2,eq:4} produces, via \Cref{eq:3}, precisely the desired inequality
  \begin{equation*}
    r=
    \rank~\Gamma_{\cF}+\rank~\widetilde{\Gamma}
    \le
    e(n+1).
  \end{equation*}
  This finishes the proof.
\end{proof}

\begin{remark}\label{re:p2}
  As mentioned above, \Cref{pr.simpl-sph} is essentially \cite[second corollary, p.107]{smth}. The latter, though, states only the $2r\le n+1$ inequality, which cannot be correct in full generality: $\bZ/2$, of rank $r=1$, acts on the zero-sphere by the obvious permutation; in that case the inequality would read $2\le 1$.

  There are hints in \cite{smth} that perhaps only odd primes are considered (see e.g. footnote 6 therein), but this does not seem to be stated explicitly.
\end{remark}

\begin{remark}
  The inductive argument in the proof of \Cref{pr.simpl-sph} is very much in the spirit of \cite[Lemmas 2.2 and 2.3]{mz}.
\end{remark}

As a consequence we have the following corollary.

\begin{corollary}\label{cor.comb}
For each $n$ there is a finite group that cannot be realized as $\Gamma(\cP)$ for any polytope with $\dim \cP<n$.
\end{corollary}
\begin{proof}
  Immediate from \Cref{pr.simpl-sph} and \Cref{re:act-on-sphr} above, to the effect that $\Gamma(\cP)$ acts effectively on the simplicial sphere $\partial \cP$.
\end{proof}

\Cref{cor.comb} gives an affirmative answer to \cite[Open Question 1]{ssw}. On the other hand, since finite groups of isometries can be regarded as acting effectively and simplicially on simplicial spheres, \Cref{pr.simpl-sph} also proves an affirmative answer to \cite[Open Question 2]{ssw}:

\begin{corollary}\label{cor.geom}
  For each $n$ there is a finite group that cannot be realized as the geometric symmetry group $G(\cP)$ for any polytope with $\dim \cP<n$.  \mbox{}\hfill\ensuremath{\blacksquare}
\end{corollary}

In fact, the last corollary and the geometric version of \Cref{pr.simpl-sph} follow more swiftly from the observation that an abelian group with minimal number $r$ of generators can not be embedded into $\GL(d,\bC)$ for $d < r$ (see \Cref{th.bd} below, proof of lower bound).

\section{The geometric convex polytope dimension}\label{subse.dim}

\begin{definition}
  Let $\Gamma$ be a group.  Let $\gcpd(\Gamma)$ (the \textit{geometric convex polytope dimension of $\Gamma$}) be the minimum integer $d$ such that there is a convex polytope $\mathcal{P}$ in $\mathbb{R}^d$ such that $G (\mathcal{P}) = \Gamma$.
\end{definition}

\begin{proposition}\label{pr.cpd-leq-gcpd}
  When $\gcpd(\Gamma) \geq 3$, then $\cpd(\Gamma) \leq \gcpd(\Gamma)$.
\end{proposition}
\begin{proof}
  This follows from \Cref{theorem-symmetrybreaks}, Part~(d).
\end{proof}

\Cref{fig:c4polygones} indicates two constructions of polygons with geometric symmetry group the cyclic group of order $4$.  The same constructions apply to all cyclic groups $\bZ/n$ of order $n \geq3$, so $\gcpd(\bZ/n)=2$.  On the other hand, the combinatorial automorphism group of a polygon is a dihedral group, so $\cpd(\bZ/n)\geq 3$.  By \Cref{theorem-symmetrybreaks} or otherwise, we have $\cpd(\bZ/n) = 3$. Similarly, we have $\gcpd((\bZ/2)^2) = 2$ (take a rectangle), but $\cpd( (\bZ/2)^2) = 3$. It follows that the conclusion of \Cref{pr.cpd-leq-gcpd} is wrong only for cyclic groups of order $\geq 3$, and the Klein four-group $(\bZ/2)^2$.

\begin{figure}
\centering
  \begin{tikzpicture}[scale=0.7]
   \tikzset{pnts/.style={circle, draw, fill=black,
                          inner sep=0pt, minimum width=4pt}}      \newcommand{15}{15}
      \newcommand{2.7}{2.7}
      \newcommand{2.3}{2.3}

      \foreach \i in {0,1,2,3}{
         \coordinate (A\i) at (90*\i + 15:2.7);
         \coordinate (B\i) at (90*\i - 15:2.3);
      }

      \filldraw[fill=red!30]
        (A0)  node[pnts] {}
        \foreach \i in {1,2,3}{
          -- (B\i)
              node[pnts] {}
          -- (A\i)
              node[pnts] {}
        }
        -- (B0)
           node[pnts]{}
        -- cycle;
      \node [draw, circle through=(B0)] at (0,0) {};

      \begin{scope}[xshift=7cm]
         \newcommand{15}{15}
         \newcommand{40}{40}
         \newcommand{2.3}{2.3}

         \foreach \i in {0,1,2,3}{
             \coordinate (A\i) at (90*\i:2.3);
             \coordinate (B\i) at (90*\i + 15:2.3);
             \coordinate (C\i) at (90*\i + 40:2.3);
         }

         \filldraw[fill=red!30]
              (A0) node[pnts] {}
               \foreach \i in {0,1,2,3}{
                   -- (A\i)  node[pnts] {}
                   -- (B\i)  node[pnts] {}
                   -- (C\i)  node[pnts] {}
                   }
                   -- cycle;
         \node [draw, circle through=(B0)] at (0,0) {};
      \end{scope}
    \end{tikzpicture}
    \caption{Two polygons with $\bZ/4$ as geometric symmetry group}
    \label{fig:c4polygones}
\end{figure}

For geometric symmetries, we have the following alternative version of \Cref{theorem-symmetrybreaks}, which also works in dimension~$d\leq 2$.  The gist of this result is that we need to add at most one orbit of vertices to break undesired symmetries.  The proof is adapted from an argument by Isaacs~\cite{Isaacs77}.

\begin{proposition}\label{pr.geosymbreak}
  Let $\cQ \subset \bR^d$ be a finite, convex $d$-polytope and $\Gamma$ a subgroup of $G(\cQ)$.  Then there exists an orbit $X$ of $\Gamma$ on $\bR^d$ such that for the polytope $\cP = \conv(\cQ \cup X)$, we have $\Gamma=G(\cP)$.
\end{proposition}
\begin{proof}
  Let $t$ be the translation that sends the barycenter of $\cQ$ to the origin.  Then $G(t(\cQ))=tG(\cQ)t^{-1}$ and $t\Gamma t^{-1}$ is a subgroup of $G(t(\cQ))$.  So after replacing $\cQ$ by a translated copy, we may assume without loss of generality that the barycenter of $\cQ$ is the origin.  In this case, we have $G(\cQ) \subset \orth_d(\bR)$.  Let $Y$ be the vertex set of $\cQ$.  For every $1 \neq g\in G(\cQ)$, we have that $\ker(g-1)$ (the eigenspace of $g$ associated with the eigenvalue~$1$) is a proper subspace of $\bR^d$.  As $\bR^d$ is not the union of finitely many proper subspaces, we can find a vector $x$ such that no non-identity element of $G(\cQ)$ fixes $x$ and such that $x$ is not contained in a proper subspace which is spanned by some subset of $Y$.  Let $X = \Gamma x$ be the $\Gamma$-orbit of $x$.  For $\lambda>0$, set $\cP_{\lambda}= \conv(Y\cup \lambda X)$.  We will show that there is some $\lambda >0$ such that $G(\cP_{\lambda}) = \Gamma$.

  Consider the ray $\{\lambda x \colon \lambda >0\}$ spanned by $x$.  As $x$ is not contained in a proper subspace spanned by a subset of $Y$, this ray meets the boundary of $\cQ$ in the interior of a facet.  Thus there is a nonempty interval $I$ of positive real numbers such that the vertex set of $\cP_{\lambda}= \conv(Y\cup \lambda X)$ is exactly $Y\cup \lambda X$ for all $\lambda\in I$.

  Set $s = (1/\lvert G \rvert) \sum_{g\in \Gamma} gx$.  The barycenter of $\cP_{\lambda}$ is $\lambda s$.  For any $y\in Y$, there are at most two $\lambda$'s that solve the quadratic equation $\abs{ y-\lambda s}^2 = \abs{ \lambda x - \lambda s}^2$.  As $Y$ is finite, there are infinitely many $\lambda \in I$ such that $\abs{y-\lambda s} \neq \abs{ \lambda x - \lambda s}$ for all $y\in Y$.  We claim that $G(\cP_{\lambda}) = \Gamma$ for these $\lambda$.  By construction, $\Gamma \leq G(\cP_{\lambda})$ for any $\lambda$.  As $Y\cup \lambda X$ is the vertex set of $\cP_{\lambda}$, the isometry group $G(\cP_{\lambda})$ is the set of isometries mapping $Y \cup \lambda X$ to itself.  By the choice of $\lambda$, we have that $G(\cP_{\lambda})$ stabilizes $\lambda X$, and thus also $Y$.  Therefore, $G(\cP_{\lambda}) \subseteq G(\cQ)$.  By the choice of $x$, no non-identity element of $G(\cQ)$ fixes $x$.  Thus $\abs{G(\cP_{\lambda})} = \abs{G(\cP_{\lambda})x} \leq \abs{X} \leq \abs{\Gamma} $.  Since $\Gamma\subseteq G(\cP_{\lambda})$ we have $G(\cP_{\lambda}) = \Gamma$.
\end{proof}

Isaacs also showed that when some finite group $\Gamma \subset \orth_d(\bR)$ acts \emph{absolutely irreducibly} on $\bR^d$, then $\Gamma$ is the geometric symmetry group of a vertex-transitive polytope.  This can be generalized to some groups not necessarily acting absolutely irreducibly \cite[Corollary~5.8]{FrieseLadisch18}, but not to arbitrary subgroups of $\orth_d(\bR)$.

\begin{corollary}\label{cor.gcpdlin}
  Let $\Gamma$ be a finite group.  Then $\gcpd(\Gamma)$ equals the smallest $d$ such that $\Gamma$ embeds into $\GL(d,\bR)$.
\end{corollary}
\begin{proof}
  When $\Gamma\cong G(\cP)$ for some $d$-polytope~$\cP$, then $\Gamma$ embeds into $\orth_d(\bR) \subset \GL(d,\bR)$.

  Conversely, suppose $\Gamma$ embeds into $\GL(d,\bR)$.  By \Cref{pr.inv_innprod}, it follows that $\Gamma$ is isomorphic to a subgroup~$\widetilde{\Gamma}$ of $\orth_d(\bR)$.  As in the proof of \Cref{cor.geosym-givengroup}, we can find a polytope $\cQ$ such that $G(\cQ)$ contains the given subgroup $\widetilde{\Gamma}$ isomorphic to $\Gamma$.  Then by \Cref{pr.geosymbreak} (or \Cref{cor.geosym-givengroup} when $d\geq 3$), $\widetilde{\Gamma}$ is the isometry group of some $d$-polytope.
\end{proof}

\begin{remark}
  When $d$ is minimal such that the given group $\Gamma$ embeds into $\GL(d,\bR)$, then one can actually show that $\bR^d$ can be generated by a single $\Gamma$-orbit on $\bR^d$.  It follows that there is a $d$-polytope~$\cP$ with $\Gamma\cong G(\cP)$ and such that $\Gamma$ has at most two orbits on the set of vertices of $\cP$.  On the other hand, not every group is isomorphic to the geometric automorphism group of a vertex-transitive polytope~\cite{Babai77}.
\end{remark}

As an example, we can compute $\gcpd$ for abelian groups.

\begin{theorem}\label{th.bd}
  Let
  \begin{equation}\label{eq:1}
    \Gamma=\bZ/n_1\times\cdots\times \bZ/n_r
  \end{equation}
  be an abelian group, where
  \begin{equation*}
    n_1 \mid n_2 \mid \cdots \mid n_r
  \end{equation*}
  are the invariant factors of $\Gamma$. Let $s$ be the number of factors $n_i$ such that $n_i=2$ and let $t$ the number of factors $n_i$ such that $n_i>2$ (so $r=s+t$).  Then $\gcpd(\Gamma) =s+2t$, that is, the minimal dimension of a polytope $\cP$ such that $G(\cP)\cong \Gamma$ is $s+2t$.
\end{theorem}
\begin{proof}
  By \Cref{cor.gcpdlin}, we need to show that $d=s+2t$ is the minimum dimension of a faithful linear representation $\rho\colon \Gamma \to \GL(d,\bR)$.  The fact that this is a sharp bound entails two inequalities, which we prove separately.

  \vspace{.3cm}

  {\bf Lower bound.}  Suppose the linear representation $\rho\colon \Gamma\to \GL(d,\bR)$ is faithful.  So when decomposing the complexification of $\rho$ (denoted by the same symbol, for brevity) as a sum of irreducible (hence one-dimensional) characters, the summands $\chi_i$ must generate the Pontryagin dual group
  \begin{equation*}
    \widehat{\Gamma}:=\Hom(\Gamma,\bS^1)\cong \Gamma.
  \end{equation*}
  It follows from this that there is, among the $\chi_i$,
  a minimal set of generators $\chi_1$ up to $\chi_h$,
  so that $r \leq h \leq d$.
  Minimality implies that no two $\chi_i$ and $\chi_j$,
  $1\le i\ne j\le h$ can be mutually conjugate (since conjugation means taking the inverse in $\widehat{\Gamma}$).

All self-conjugate $\chi_i=\overline{\chi_i}$ are trivial on the $t$-factor abelian group $2\Gamma$, so among the $h$ $\chi_i$ we must have at least $t$ non-self-conjugate characters. But because $\rho$ is a real representation, the $t$ conjugates $\overline{\chi_i}$ must be among the summands of $\rho$ as well, meaning that
\begin{equation*}
  \dim \rho\ge h+t \ge r+t=s+2t,
\end{equation*}
as claimed.

  \vspace{.3cm}

  {\bf Upper bound.} Fix an isomorphism
  \begin{equation*}
    \widehat{\Gamma}\cong \Gamma\cong \bZ/n_1\times\cdots\times \bZ/n_r
  \end{equation*}
  and select a set $\chi_i$, $1\le i\le r$ of generators for the $r$ factors. The $\chi_i$ of order two ($s$ of them, in the notation of the statement) are realizable over the real numbers, while the rest can be regarded as representations on $\bR^2\cong \bC$. Summing the characters we thus obtain an orthogonal representation of $\Gamma$ on $\bR^{s+2t}$, which is faithful because its simple constituents generate the dual group.
\end{proof}

We have similar sharp bounds in the centrally symmetric case:

\begin{theorem}\label{th.bd-sym}
  Let $\Gamma$ be a finite abelian group as in \Cref{eq:1}, $\sigma\in \Gamma$ a non-trivial involution, and $s$ and $t$ as in the statement of \Cref{th.bd}. When $\sigma$ is not a square in $\Gamma$, then the conclusion of \Cref{th.bd} holds for polytopes equipped with $\Gamma$-actions with $\sigma\mapsto -1$.  When $\sigma $ is a square, then the minimal dimension of a polytope $\cP$ with geometric symmetry group $\Gamma$ such that $\sigma$ acts as $-1$ is $2r=2(s+t)$.
\end{theorem}
\begin{proof}
  In any case, the fact that $\dim\cP\ge s+2t$ follows from \Cref{th.bd}, since the actions with $\sigma\mapsto -1$ form a subclass of isometric actions.

  When $\sigma$ is not a square in $\Gamma$, then we can write $\Gamma = \langle \sigma \rangle \times \Gamma_0$.  Thus there is a real character $\chi$ taking value $-1$ on $\sigma$.  Now choose generators $\chi_1=\chi$, $\chi_2$, $\dotsc$, $\chi_r$ as before. By replacing $\chi_i $ by $\chi\chi_i$ if necessary, we can ensure that $\chi_i(\sigma)=-1$ for all $i$.  Thus again the lower bound is achieved.

  Now suppose that $\sigma$ is a square in $\Gamma$. Then for every irreducible character $\chi$ taking value $-1$ on $\sigma$ we necessarily have $\chi\neq \overline{\chi}$. Thus in this case a faithful representation $\rho\colon \Gamma\to \GL(d,\bR)$ with $\rho(\sigma)=-I$ has at least $2r=2(s+t)$ summands.  Conversely, we can find a generating set $\chi_1$, $\dotsc$, $\chi_r$ with $\chi_i(\sigma)=-1$ for all $i$ (as above), so the bound is achieved.
\end{proof}

Going back to $\cpd$, we now know enough to determine it precisely for elementary abelian $p$-groups.

\begin{theorem}\label{th:cpd-sharp}
  Let $p$ be a prime. For $\Gamma=(\bZ/p)^r$ we have $\cpd(\Gamma)=\gcpd(\Gamma)$ except in the following cases:
  \begin{itemize}
  \item $r=1$ and $p>2$, when $\cpd(\Gamma)=3$ and $\gcpd(\Gamma)=2$;
  \item $r=2$ and $p=2$, when again $\cpd(\Gamma)=3$ and $\gcpd(\Gamma)=2$.
  \end{itemize}
\end{theorem}
\begin{proof}
  By \Cref{pr.simpl-sph}, the dimension $n+1$ of a polytope $\cP$ on whose boundary spherical simplicial complex $\partial\cP$ the group $\Gamma$ can act effectively is at least the value of $\gcpd$ as computed in \Cref{th.bd} (namely $r$ when $p=2$ and $2r$ otherwise). In short, for elementary abelian $p$-groups $\Gamma$ we always have
  \begin{equation}\label{eq:5}
    \cpd(\Gamma)\ge \gcpd(\Gamma).
  \end{equation}
  The two exceptions in the statement were noted and analyzed in the discussion following \Cref{pr.cpd-leq-gcpd}. As for the other cases, \Cref{th.bd} says that we then have $\gcpd(\Gamma)\ge 3$ and hence $\cpd(\Gamma)\le \gcpd(\Gamma)$ by \Cref{pr.cpd-leq-gcpd}. Combining this with \Cref{eq:5} gives the conclusion.
\end{proof}

In principle, \Cref{cor.gcpdlin} allows us to compute $\gcpd(\Gamma)$ from the character table of $\Gamma$.  A few more (elementary) examples which show that $\gcpd(\Gamma)$ can be arbitrarily large:
\begin{theorem}\label{th.bd_sn_etal}
  \begin{enumerate}
  \item $\gcpd(S_n) = n-1$, where $S_n$ is the symmetric group on $n$ letters.
  \item Let $\Gamma$ be an extraspecial $p$-group of order $p^{2k+1}$, $p$ odd.  Then $\gcpd(\Gamma) = 2 p^k$.
  \item Let $\Gamma = \Aff(\bF_q)$ be the affine group of degree~$1$ over the field $\bF_q$ with $q$ elements, that is, the set of all maps $\bF_q\to\bF_q$ of the form $x\mapsto ax+b$, $a$, $b\in \bF_q$, $a\neq 0$.  Then $\gcpd(\Gamma)= q-1$.
  \end{enumerate}
\end{theorem}
\begin{proof}
  \begin{enumerate}
  \item $S_n$ is the geometric (and combinatorial) automorphism group of a regular $(n-1)$-simplex.  The one-dimensional characters of $S_n$ have the alternating group in the kernel.  For $n\neq 4$, the symmetric group $S_n$ has no irreducible representations of degree between $1$ and $n-1$ \cite[pp.~466--468]{BurnsideTGFO}, and for $n=4$, the irreducible representation of degree~$2$ is not faithful.  Thus $\gcpd(S_n)=n-1$.
  \item The irreducible character degrees of an extraspecial $p$-group are $1$ and $p^k$ \cite[7.6(b)]{huppCT}.  The one-dimensional characters have the commutator subgroup in the kernel, so are not faithful.  Every character $\chi$ of degree $p^k$ is faithful, but $\chi\neq \overline{\chi}$ as $p$ is odd.  The representation $\Gamma\to \GL(p^k,\bC)$ affording $\chi$ can be viewed as a representation $\Gamma \to \GL(2p^k,\bR)$, so $\gcpd(\Gamma)=2p^k$.
  \item The affine group is not abelian and has only one nonlinear character, of degree~$q-1$ \cite[7.9(c)]{huppCT}.  This character is afforded by a representation over $\bR$ (in fact over $\bQ$), so $\gcpd(\Gamma)=q-1$.
  \end{enumerate}
\end{proof}

\section{Some open questions}
\label{sec.openqu}

By the results in the last section, we can (in principle) compute the geometric convex polytope dimension $\gcpd(\Gamma)$ of a group using representation theory.  For the convex polytope dimension, the situation is less clear. We have sharp bounds for elementary abelian $p$-groups by \Cref{th:cpd-sharp}, but the analogous question arises naturally for other families:

\begin{question}\label{qu.exact-cpd}
  What is the exact value of $\cpd(S_n)$ or $\cpd(\Aff(\bF_q))$ ?
\end{question}

Notice that for symmetric groups we do at least know that $\cpd(S_n)\to \infty$ for $n\to \infty$, as $S_n$ contains a subgroup $(\bZ/2)^{\lfloor n/2 \rfloor}$. For $\cpd(\Aff(\bF_p))$, $p$ prime, we do not known even this, since abelian subgroups of $\Aff(\bF_p)$ are cyclic.

By \Cref{pr.cpd-leq-gcpd}, we have $\cpd(\Gamma) \leq \gcpd(\Gamma)$ for all groups $\Gamma$ except cyclic groups of order $\geq 3$ and the Klein four-group.  This motivates the following question:

\begin{question}\label{qu.cpd-gcpd}
  Is there a group $\Gamma$ such that $\cpd(\Gamma) < \gcpd(\Gamma)$ ?
\end{question}

When such a group $\Gamma$ exists, then there must be a polytope~$\cP$ of dimension $d=\cpd(\Gamma)$, such that $\Gamma\cong \Gamma(\cP)$ is not isomorphic to a subgroup of $\GL(d,\bR)$.  Conversely, when $\cP$ is a convex $d$-polytope~$\cP$ such that $\Gamma=\Gamma(\cP)$ is not isomorphic to a subgroup of $\GL(d,\bR)$, then $\cpd(\Gamma) \leq d < \gcpd(\Gamma)$, as $\GL(k,\bR) \subset \GL(d,\bR)$ for $k < d$.

Of course, a negative answer to \Cref{qu.cpd-gcpd} would also settle \Cref{qu.exact-cpd}.

Bokowski, Ewald and Kleinschmidt~\cite{BokowskiEK84} found the first example of a polytope~$\cP$ (in dimension~$4$) which has a combinatorial automorphism~$\varphi$ that can not be realized geometrically in the sense that the polytope has no geometric realization that admits $\varphi$ as geometric automorphism. Mani~\cite{Mani71} showed that in dimension~$3$, for every convex polytope there is a combinatorially equivalent polytope, such that all its combinatorial automorphisms come from geometric automorphisms.  It follows that when $\gcpd(\Gamma) > 3$, then also $\cpd(\Gamma) >3$.  Not much else seems to be known.

If additional geometric structure is required on the realization of $G$ as a group of combinatorial symmetries of a polytope $\cP$, it can imply restrictions on the dimension of $\cP$.  Interestingly, such conditions are more diverse than just lower bounds on the dimension.  For example, if $\sigma$ is a central involution and there exists an element $\gamma$ such that $\gamma^2 = \sigma$, then for $\sigma$ to act as a central symmetry we need the ambient dimension to be even.  This is implied by a simple degree argument \cite[Proof of Theorem~4.2]{ssw}. Therefore, if we impose the way that a subgroup $A \subset G$ has to act on our polytope $\cP$, it can heavily restrict on which dimensions admit such a polytope.

\addcontentsline{toc}{section}{References}
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\section{Introduction}

Quantum Chromodynamics (QCD), the fundamental theory which describes the strong interaction among
quarks and gluons, can be solved non-perturbatively by numerical simulation of the theory defined on a 4-dimensional Euclidean lattice.  Parton distribution functions (PDFs) encode important information on the internal structure of hadrons such as how the spin and momentum are distributed among the quarks and gluons.
However, these key quantities are presently only determined from global fits
to experimental data mostly for the proton.  Lattice QCD is thus the natural candidate for a reliable
computation of PDFs. However, since they are defined on the light-cone it was thought until recently to be impossible to compute PDFs directly on the lattice.
A method to extract PDFs from lattice QCD  was proposed by Ji few years ago and is based on the large-momentum effective theory
(LaMET) \cite{Ji:2013dva,Ji:2014gla}. In this approach, instead of calculating the light-cone quark
correlations in a hadron, one computes matrix elements with quarks separated by a space-like distance in
a boosted hadron, which is feasible on the lattice. Fourier transforming these spatial matrix elements
yields --after a suitable non-perturbative renormalization--
the so-called quasi-PDFs that  have the same infrared physics as  the light-cone PDFs. Because of this property,
one can relate quasi-PDFs and PDFs using perturbation theory, through a procedure called  matching.
If the hadron is boosted with high enough momentum, the matching equations should correctly
bring the quasi-PDFs to the PDFs, up to $O(M^2/P^2_z, \Lambda^2_{QCD}/P^2_z)$ corrections, where $M$ and $P_z$ are the
mass and momentum of the boosted hadron. The hadron mass corrections are known to all orders in $M^2/P_z^2$, while
the higher twist corrections are still not  calculated. Besides quasi-PDFs, there are other approaches to accessthe $x$-dependence PDFs from lattice QCD\cite{Detmold:2005gg,
	Braun:2007wv,Ma:2014jla,Ma:2017pxb,Radyushkin:2017cyf}. We refer the interested Reader to the recent
review of Ref.~\cite{Cichy:2018mum}.

Since Ji's proposal, considerable progress has been made. In particular progress in  the non-perturbative renormalization of
quasi-PDFs \cite{Alexandrou:2017huk,Green:2017xeu,Ishikawa:2017faj} and in the calculation of the one-loop matching coefficient~\cite{Izubuchi:2018srq,Alexandrou:2018pbm} has enabled the extraction of PDFs from quasi-PDFs for the proton and pion.
 Results on the proton unpolarized PDFs have been obtained using also simulations  with physical pion mass~\cite{Chen:2018xof,Alexandrou:2019lfo,Alexandrou:2018pbm}.
 In this work we present first results on  the $\Delta (1232)$ baryon, the lightest baryon resonance of a
nucleon-pion system. Being a strongly decaying resonance its structure is not directly accessible experimentally. Thus lattice QCD can provide important information on its structure by evaluating the  $\Delta$ electromagnetic and axial form factors and the $\Delta$-nucleon transition form factors.
 In this work we present first results on the PDFs of the $\Delta(1232)$ baryon. Such a study
 is important in order to test the conjecture made in Ref.~\cite{Ethier:2018efr} that spontaneous breaking of chiral symmetry should lead to a significant difference between
the proton and the $\Delta^+$ flavor asymmetry for the isovector distribution $\overline{d}(x) - \overline{u}(x)$. Due to the short lifetime of the $\Delta$, lattice QCD offers the exciting possibility to study the $\Delta$ unpolarized PDF in order to understand the mechanism behind the breaking of the flavor symmetry between
light quarks in the nucleon sea.

\section{Theoretical setup}
The unpolarized PDF, denoted by $q(x)$, is defined on the light cone as~\cite{Collins:2011zzd}
\begin{equation}
q(x)=\int_{-\infty}^{+\infty} \frac{d \xi^{-}}{4 \pi} e^{-i x P^{+} \xi^{-}}\left\langle h\left|\overline{\psi}\left(0\right) \gamma^{+} W\left(0,\xi^{-}\right) \psi(\xi^{-})\right| h\right\rangle
\end{equation}
where the light-cone vectors are taken as $\xi^{ \pm}=\left(\xi^{0} \pm \xi^{3}\right) / \sqrt{2}$. $W\left(0,\xi^{-}\right)=e^{-i g \int_{0}^{\xi^{-}} d \eta^{-} A^{+}\left(\eta^{-}\right)}$ is the Wilson line required for gauge invariance and $x$ is the momentum fraction
carried by the quarks in the hadron. The plus component of the  momentum $P^{+}$ is $\left(P^{0}+P^{3}\right) / \sqrt{2}$ and  $|h\rangle$ the hadron state of interest. Deep inelastic scattering is light-cone
dominated, meaning that $\xi^{2}=t^{2}-\vec{r}^{2}\approxeq 0$. However, in Euclidean space time, this reduces to a
single point, which makes it impossible to compute PDFs in lattice QCD directly. In the large momentum effective theory, the PDF can be extracted from the quasi-PDF defined by
\begin{equation}
\tilde{q}\left(x, {P}_{z}, \mu\right)=\int_{-\infty}^{+\infty} \frac{d z}{4 \pi} e^{-i x P_{z} z}\langle h(P_z)|\overline{\psi}(0) \Gamma {W}(0,{z}) \psi(z)| h(P_z)\rangle,
\end{equation}
where $|h(P_z)\rangle$ is the boosted hadron state with finite momentum $P=\left(E, 0,0, P_{z}\right)$, and $W(0,z)$ is the Wilson line along the boosted direction. $\mu$ is the renormalization scale.The Dirac structure $\Gamma$ defines the type of PDF. For the unpolarized PDF, $\Gamma$ is chosen to be $\gamma^0$ to avoid operator mixing~\cite{Constantinou:2017sej}.
\par
Since the infrared physics is the same for quasi-PDFs and light-cone PDFs and the difference is only in
the perturbative region \cite{Ji:2014gla},  the quasi-PDFs are related to light-cone PDFs through the matching equation
\begin{equation}
\tilde{q}\left(x, P_{z}, \mu\right)=\int_{-1}^{1} \frac{d y}{|y|} C\left(\frac{x}{y}, \frac{\mu}{P_{z}}\right) q(y, \mu)+O\left(\frac{M^{2}}{P_{z}^{2}}, \frac{\Lambda_{Q C D}^{2}}{P_{z}^{2}}\right),
\end{equation}where $q(y,\mu)$ is the light-cone PDF at the scale $\mu$, C is the matching kernel,
which can be calculated perturbatively and has been evaluated to one-loop level.
How large the momentum $P_{z}$ needs to be for the  validity of  LaMET
has to be tested. Exploratory studies have been performed showing a great potential for this approach, as e.g. in Ref.~ \cite{Alexandrou:2016jqi,Alexandrou:2019lfo,Chen:2016utp,Chen:2018xof}.

\section{Lattice details}
\par
In this study, we evaluate the isovector unpolarized quasi-PDFs, $\overline{u}(x) - \overline{d}(x)$ of $\Delta^+$. The matrix elements of interest are given by
\begin{equation}\label{equ:square}
h(P_z, z)=\left\langle h\left|\overline{\psi}(0) \gamma^{0} W(0,z) \psi(z)\right| h\right\rangle
\end{equation}
for a straight Wilson line $W$, with varying length from $z=0$ up to half of the spatial extension $L/2$.
In order to extract the matrix element of Eq. (\ref{equ:square}) we construct ratios of suitable three-point functions and two-point functions, averaged over the gauge field ensemble.
The two-point and three-point functions are given by
\begin{align}
C_{\sigma\rho}^{2 \mathrm{pt}}(\mathcal{P},\mathbf{P}; t_s, 0) &=\mathcal{P}_{\alpha \beta} \sum_{\mathbf{x}} e^{-i \mathbf{P} \cdot \mathbf{x}}\left\langle 0\left|{\cal J}_{\sigma \alpha}(\mathbf{x}, t_s) \overline{{\cal J}}_{\rho\beta}(\mathbf{0}, 0)\right| 0\right\rangle \\ C^{3 \mathrm{pt}}_{\sigma0\rho}\left(\tilde{\mathcal{P}},\mathbf{P} ; t_{s}, \tau, 0\right) &=\tilde{\mathcal{P}}_{\alpha \beta} \sum_{\mathbf{x}, \mathbf{y}} e^{-i \mathbf{P} \cdot \mathbf{x}}\left\langle 0\left|{\cal J}_{\sigma\alpha}\left(\mathbf{x}, t_{s}\right) \mathcal{O}(\mathbf{y}, \tau ; z) \overline{{\cal J}}_{\rho\beta}(\mathbf{0}, 0)\right| 0\right\rangle
\end{align}
where $t_s$ is the time separation of the sink relative to the source, $\tau$ is the insertion time of the $\mathcal{O}$ operator, and ${\cal J}_{\sigma\alpha}(x)$ is the $\Delta^+$ interpolating operator,
\begin{equation}
{\cal J}_{\sigma\alpha}(x)=\frac{1}{\sqrt{3}} \epsilon^{a b c}\left[2\left(u^{a^{T}}(x) C \gamma_{\sigma} d^{b}(x)\right)u_{\alpha}^{c}(x)+\left(u^{a^{T}}(x) C \gamma_{\sigma} u^{b}(x)\right) d_{\alpha}^{c}(x) \right],
\end{equation}
with $C=\gamma^0\gamma^2$ being the charge conjugation matrix.

We use $\mathcal{P}=\tilde{\mathcal{P}}=\frac{1+\gamma^{0}}{4}$, and sum over the space components of
$\sigma$,$\rho$ when computing the ratio of the three to the two point functions:

\begin{equation}
h\left(\mathbf{P}, z\right)\stackrel{ \tau, t_s>>1}{=}\frac{\sum_{\sigma=1}^{3} C_{\sigma 0 \sigma}^{3pt}\left(\tilde{\mathcal{P}},\mathbf{P}; t_s,\tau,0\right)}{ \sum_{\sigma=1}^{3} C^{2pt}_{\sigma \sigma}(\mathcal{P},\mathbf{P};t_s,0)}\; .
\label{eq:matrixelement}
\end{equation}
In principle,
the interpolating field also overlaps with the heavier spin-1/2 excitations.
However, it is known form previous studies that the overlap with spin-1/2 state is small, and can be neglected.

\section{Results at $m_\pi=330$ MeV}
As a first benchmark computation, we
use a $24^3 \times 48$ twisted mass ensemble with $a=0.096$ fm generated by the Extended Twisted Mass (ETM) collaboration \cite{Alexandrou:2018egz}. The twisted mass parameter is $\mu=0.0053$, which corresponds to a pion mass of $m_\pi =330 $ MeV. Since this is a first computation of the $\Delta$ PDF
we are interested in exploring the feasibility of the calculation and, in particular, the momentum
smearing technique to reduce the statistical errors due to a boosted $\Delta$.

It is well known that the overlap with the ground state  is improved by smearing the quark fields of the interpolating function. We use Gaussian smearing on the quark fields accompanied with APE smearing on the gauge fields that enter the Gaussian smearing function. For stationary hadron states this provides  an efficient approach to improve ground state dominance. However, in the computation of matrix elements of quasi-PDFs, we need the overlap of a  boosted particle state with the
interpolating field. Thus, we use momentum smearing~\cite{Bali:2016lva} which has proven to improve significantly the overlap of a  boosted nucleon state with the interpolating field. The momentum smearing modifies the standard Gaussian smearing  by a complex phase, given by
\setlength\abovedisplayskip{1pt}
\setlength\belowdisplayskip{1pt}
\begin{equation}
S_{mom}=\frac{1}{1+6\alpha} \Big( \psi(x)+\alpha \sum_j U_j(x) e^{-i\xi P \cdot j} \psi(x+\hat{j}) \Big),
\end{equation}
where $U_j$ is the gauge link in the spatial j-direction. The momentum smearing parameter $\xi$ needs to be tuned in order to optimize the overlap with the boosted particle state.

The necessity to boost the particle with relatively large momentum makes the excited state contamination more severe, since the spectrum gets denser. We use the source-sink time separations of {9a, 10a, 11a, 12a}
corresponding to {0.86, 0.96, 1.05, 1.15} fm, in order to study the
excited state contamination. In this preliminary study we employ
momentum $P_z$ = $2\pi/L \approx$  0.54 GeV. In Fig.
\ref{differnet time separation}  we show the comparison of results among four different separations. The results for both the real part of the matrix element, for the four source-sink time separations are all consistent within our current precision. For the imaginary part there is a decreasing trend becoming consistent for source-sink time separation $t_s=10 a$. Therefore, we fix the $t_s$ to $10 a \approx 0.96$ fm throughout this work.
\begin{figure}[H]
	\centering
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/24_moment001_real_different_sep.png}}
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/24_moment001_ima_different_sep.png}}
	\caption{The real (left) and imaginary (right) part of the ratio in
eq.~(\ref{eq:matrixelement}) yielding the isovector unpolarized quasi-PDFs of the $\Delta$ with boosted momentum $P_z=2\pi/L$ for different values of the source-sink time separation as a function of $z/a$.}
	\label{differnet time separation}
\end{figure}

\section{Results at $m_\pi = 250$ MeV}
In this section we present results using  a $32^3 \times 64$ lattice with lattice spacing  $a=0.096$ fm and pion mass  $m_\pi = 250$~MeV. At this value of the pion mass  the ($\bar{d}-\bar{u}$) asymmetry is
expected to be larger, as pointed out in Ref.~\cite{Ethier:2018efr}. All the results presented here
are computed with a source-sink time separation of $10a$. We extract matrix elements for the first two momenta $\{2\pi/L,4\pi/L\}$, which correspond to $\{0.41,0.82\}$ GeV with $\{906,3600\}$ measurements.

In the computation of the matrix elements, we apply three-dimensional stout smearing to the gauge links that enter  in the Wilson line of the operator. This reduces the power divergence in the matrix elements and brings the necessary renormalization factors closer to tree-level value.
In Fig.~\ref{matrix element}, we show the bare matrix elements for momentum $4\pi/L$ with different stout smearing steps. As can be seen, the stout smearing increases the value of matrix elements. The results for different stout smearing converge as the smearing steps increase.

\begin{figure}[h]
	\centering
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/32_moment002_real.png}}
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/32_moment002_ima.png}}
	\caption{The real (left) and imaginary (right) part of the ratio yielding the isovector unpolarized  quasi-PDF of the $\Delta$  for different smearing steps, for  momentum  $P_z=4\pi/L$.}
	\label{matrix element}
\end{figure}

For obtaining physical results, the operator must be renormalized to eliminate divergences. We adopt a non-perturbative RI-MOM renormalization scheme~\cite{Martinelli:1994ty,Alexandrou:2017huk}. The renormalization factors $Z$ are converted to the $M\overline{MS}$ scheme~\cite{Alexandrou:2019lfo} and evolved to $\mu=2$ GeV. The final values of the  $Z$-factors are extracted by chirally extrapolating to zero pion mass.  The $Z$-factors are shown in Fig.\ref{renormalization factor} for different stout smearing steps.
\begin{figure}[h]
	\centering
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/renormalation_factor_real.png}}
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/renormalation_factor_ima.png}}
	\caption{Real part (left) and imaginary (right) parts of the renormalization factor for different stout smearing steps.}
	\label{renormalization factor}
\end{figure}
The renormalized matrix elements are shown in Fig.\ref{renormalization matrix} for momentum $P_z={4\pi}/{L} \approx 0.82$ GeV. As expected, the  renormalized matrix elements for different stout smearing steps are consistent, with stout smearing  clearly
reducing the errors in the renormalization matrix elements. The agreement between different smearing steps verifies the effectiveness of our renormalization procedure.
\begin{figure}[h]
	\centering
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/renormalation_matrix_mom002_real.png}}
	\subfigure{\includegraphics[width=0.45\textwidth]{Fig/renormalation_matrix_mom002_ima.png}}
	\caption{Renormalized matrix of quasi-PDF with boosted momentum $4\pi/L$}
	\label{renormalization matrix}
\end{figure}
\section{Summary and outlook}
In this study, we perform a first and exploratory study of the renormalized matrix
elements for the unpolarized isovector PDF of  the $\Delta^+$.
The momentum boosts used are 0.41 GeV and 0.82 GeV for the
$m_\pi=250$ MeV ensemble, and 0.54 GeV for the $m_\pi=330$ MeV
ensemble. These values are rather small and the next step is
therefore to boost the $ \Delta^+$ to higher momentum. In particular, we aim at reaching
values of the momentum similar to those used in our studies of the nucleons quasi-PDFs.
This will enable us to  compute the ratio $(\overline{d}(x) - \overline{u}(x))^{\Delta^+}/(\overline{d}(x) - \overline{u}(x))^{p^+}$ in order to check the asymmetry enhancement prediction.
 If the light quark asymmetry in the nucleon sea has its origin in the spontaneous breaking of chiral symmetry, this ratio should be as large as 2 for $x\approxeq 0.1$. This would provide  a striking
explanation of the physical mechanism responsible for $\overline{d}(x) > \overline{u}(x)$ in the nucleon.

\noindent
\textbf{Acknowledgments:} We would like to thank all members of Extended Twisted Mass Collaboration
for their constant and pleasant collaboration. Y.C., X.F., C.L., Y.L. and S.X. were supported in part by NSFC of China under Grant No. 11775002 and No. 2015CB856702. It is also supported in part by DFG grant no. TRR~110 and NSFC grant No. 11621131001. This work has received funding from the European Union's Horizon 2020
research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement
No 642069 (HPC-LEAP).
K.C.\ was supported by National Science Centre (Poland) grant SONATA
BIS no.\ 2016/22/E/ST2/00013. F.S.\ was funded by DFG project number 392578569.
M.C. acknowledges financial support by the U.S. Department of Energy, Office of Nuclear Physics, within
the framework of the TMD Topical Collaboration, as well as, by the National Science Foundation
under Grant No.\ PHY-1714407. This research used resources of TianHe-3 (prototype) at
Chinese National Supercomputer Center in Tianjin, and
JUWELS at  the Juelich Supercomputing Centre,  under project id ECY00.


\section{Introduction}

Many of the $\sim$200 known extrasolar planets are
``Hot Jupiters'' (i.e., giant planets with star-planet
distances less than about 0.1 AU) that have been identified
from perturbations in their host star's radial velocities.
Other ways in which a close-in extrasolar giant planet
(CEGP) can influence its star include tidal distortions and
magnetic interactions.
These effects can produce enhanced stellar flare activity
(Rubenstein \& Schaefer 2000),
chromospheric and coronal emission (Cuntz et al.\  2000), and
magnetospheric radio emission (Zarka et al.\  2001).
Tidal and magnetic interactions have been proposed to explain
similar phenomena observed in RS~CVn close binary systems (e.g.,
Simon et al.\  1980; Ferreira \& Mendoza-Brice\~{n}o 2005),
stars with possibly unseen companions exhibiting
``superflares'' (Schaefer et al.\  2000),
and even in the aurora of Jupiter where there are magnetic
connections to the inner Galilean moons (Clarke et al.\  2002).
We refer to Saar et al.\  (2004) for a recent review of
observations (or the lack thereof) of various kinds of
CEGP-star interactions and possible theoretical
explanations for these effects.

In this presentation we focus on the chromospheric
(Ca H \& K line) enhancements on a star due to magnetic
interactions between the stellar field and the CEGP's
magnetosphere.
We used the empirically derived solar magnetic field, as a function
of the 11-year solar cycle, as a proxy for the field geometry and
field strength of CEGP host stars.
We simulated light curves that represent the sum of the
rotational modulation of the star's own H\&K line emission and
the added emission from the small set of flux tubes connected
to the planet's magnetosphere.

\section{Observed Ca H \& K Enhancements}

Several early attempts to identify planet-induced
chromospheric emission (reviewed by Saar et al.\  2004)
were unsuccessful mainly because the observations were not
designed specifically for this purpose.
Shkolnik et al.\  (2003, 2005) used high-resolution
($\lambda / \Delta\lambda \sim 10^5$) Ca~II H \& K
spectroscopy to identify two cases of chromospheric enhancement that
are phased with the planet's orbit.
HD 179949 (F8 V, $P_{\rm rot} \approx 9$ days) has a CEGP
with $P_{\rm orb} = 3.092$ days, $M \sin i = 0.98 \,
M_{\rm Jup}$, and semimajor axis $d = 0.045$ AU.
The Ca K enhancement for
HD 179949 occurs at a phase shift $\Delta \phi$ of almost 0.2
{\em ahead} of the sub-planet point reaching stellar disk-center.
This raised suspicions that the star's magnetic field may be
swept back into a ``Parker spiral'' at the orbit of the CEGP,
but for a solar-type magnetic field and wind this is unlikely.
On the other hand, HD 179949 is more active than the Sun
(e.g., $F_{\rm X}/F_{\odot} \sim 20$) and of earlier spectral
type ($T_{\rm eff} \approx 6170$ K; Valenti \& Fischer 2005)
suggesting both the stellar magnetic structure and wind could
be quite different from the solar case.
Shkolnik et al.'s data from 2003 do not show the strong orbital
modulation of the 2001--2002 data, but there is still a
weak maximum at $\phi \sim 0.8$ in agreement with the earlier
epochs.
Similar data for $\upsilon$~And (F7 V,
$P_{\rm rot} \approx 14$ days,
$P_{\rm orb} = 4.6$ days) reveals a slightly weaker
H \& K amplitude, but a maximum with phase shift
$\Delta \phi \approx 0.5$, i.e., when the planet is behind the star
(see also Harrington et al.\  2006).

\section{Theoretical Explanations?}

The periodicity and strength of the HD 179949 and $\upsilon$~And
variations indicate that {\em magnetic interaction}
is the most likely mechanism.
Saar et al.\  (2004) found that other mechanisms such as tidal
distortions or Io-like inductance would have produced even stronger
modulations in other star-planet systems for which no planet-phased
variations have been found (e.g., $\tau$~Boo).
In addition, tidal distortions would have caused two peaks in the
light curve per orbit, rather than just one as is observed.
The energy released in the Ca H\&K enhancement ($\sim$10$^{27}$ erg/s)
is similar to that released in typical solar flares.
Ip et al.\  (2004) modeled CEGP magnetospheres and found that
this level of power generation can be expected from magnetic
reconnection in the most likely geometries.

McIvor et al.\  (2006) predicted the shapes of Ca H\&K light curves
consistent with magnetic field lines connecting a planet and an
inclined, stretched dipolar stellar field.
Phase shifts similar to that of HD 179949 can be reproduced by
misalignment between magnetic and rotation axes.
Also, Preusse et al.\  (2006) were able to model similar phase
shifts by assuming the CEGP to be inside the star's Alfv\'{e}n
radius (i.e., where the Alfv\'{e}n speed exceeds the outflow
speed; $r \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}} 10 \, R_{\odot}$ for the solar wind) and that
the perturbations travel from the planet to the star along the
{\em longitudinally varying} inward Alfv\'{e}n characteristics.

Magnetic interaction models depend on the product of the relative
velocity between the two fields (known from the orbital dynamics),
the stellar magnetic field strength (which can be deduced from
Zeeman splitting or estimated from known activity scalings), and
the planetary magnetic field strength, which is not yet observed,
though modeling progresses (e.g.,
S\'anchez-Lavega 2004; Stevens 2005).

\section{Exploratory Models}

Our goal has been to produce more realistic simulations of
Ca~K light curves for a rotating solar-type star undergoing a CEGP
magnetic interaction.
We use modeled three-dimensional solar magnetic field configurations
(as a function of the 11-year solar cycle) to simulate both the
stellar and planetary light curve components.
Observed maps of the photospheric magnetic flux were extrapolated
into the extended corona using the Potential Field Source Surface
(PFSS) method.
The corona is assumed to remain current-free out to a spherical
``source surface''
at $R_{\rm ss} = 2.5 \, R_{\odot}$ where the nonradial field
components are set to zero, simulating the magnetohydrodynamic
(MHD) expansion of the solar wind (e.g.,
Schatten et al.\  1969; Altschuler \& Newkirk 1969;
Hoeksema \& Scherrer 1986; Wang \& Sheeley 1990).

The highest multipole components fall off the most rapidly with
increasing height, resulting in the photospheric field being much
more complex than the field at $r = R_{\rm ss}$.
Data and reconstruction coefficients were obtained for 11 solar
rotations (one per year over a solar cycle)
from Wilcox Solar Observatory.\footnote{%
http://sun.stanford.edu/$\sim$wso/}
Somewhat arbitrarily, we used the solar magnetic field from the
month of August in each of the years of solar cycle 22 (1986--1996);
i.e., the specific Carrington Rotations 1778, 1792, 1805, 1819,
1832, 1845, 1859, 1872, 1886, 1899, and 1913.

The following subsections describe how the stellar ({\S}~4.1)
and planetary ({\S}~4.2) light curve components were computed.

\subsection{Stellar Light Curve}

\begin{figure}[!ht]
\plotone{cranmer_cs14_poster_f1.eps}
\caption{{\em{Top:}} monthly raw sunspot number over cycle 22.
{\em{Bottom:}} Ca~II K-line ratio of core to wing intensity
(see text).  Red points in both panels define the 11 specific
solar rotations that we chose to model in detail.}
\end{figure}

Figure 1 shows the variation of sunspot number\footnote{%
http://sidc.oma.be/sunspot-data/}
and Ca K-line emission over solar cycle 22.
The larger number of sunspots at solar maximum corresponds
directly to the larger number of chromospheric plage and active
regions which enhance the disk-integrated K-line emission.
The Ca K-line data came from daily observations at the
Sacramento Peak Observatory of the U.S. Air Force Phillips
Laboratory.\footnote{%
http://www.ngdc.noaa.gov/stp/SOLAR/ftpcalcium.html}
The Sac Peak $K_3$ index was divided by a fiducial
wing-to-continuum ratio of 0.40 in order to obtain the
same kind of core-to-wing intensity ratio used by
Shkolnik et al.\  (2003, 2005).
These values were then calibrated against the
hemisphere-averaged magnetic flux densities
$\langle B \rangle$ (measured at the solar surface) over
the modeled epochs.
We performed a similar fit between these two quantities as
did Schrijver et al.\  (1989), and found
\begin{equation}
  (I_{\rm core} / I_{\rm wing} ) \, = \,
  0.13 \, + 0.024 \langle B \rangle^{0.63}
\end{equation}
where $\langle B \rangle$, measured in Gauss, was computed
from the PFSS models over the 11 modeled rotations at times
corresponding to the K-line observations.
This relation was then used to compute the theoretical
stellar-rotation light curves in Figures 3 and 4
from the ``exactly known'' photospheric magnetic fields.

\subsection{Planetary Light Curve}

To model the enhancement due to planet-star coupling, we assumed
the power released in magnetic interaction scales as
\begin{equation}
  P \, \sim \, \frac{B_{\ast} B_{\rm P}}{8\pi}
  \, ( \pi r_{\rm mag}^{2} )
  \, V_{\rm rel}
\end{equation}
where the stellar magnetic field ($B_{\ast}$) and planetary field
($B_{\rm P}$) are measured at the planet's magnetosphere, which has
a projected area $\pi r_{\rm mag}^{2}$ intercepted
by the stellar field (e.g., Saar et al.\  2004).
Later, we use $r_{\rm mag} \approx 5 \, R_{\rm Jup}$, which was
also assumed by Ip et al.\  (2004).
The two magnetic fields move through one another at
relative velocity $V_{\rm rel}$ ($\sim$100 km/s for HD 179949).
We assumed that $B_{\rm P}$, $r_{\rm mag}$, and $V_{\rm rel}$ remain
fixed over multiple rotations and the activity cycle, and thus
that {\em time variations} in $P$ are sensitive only to $B_{\ast}$.

\begin{figure}[!ht]
\plotone{cranmer_cs14_poster_f2.eps}
\caption{Cartoon illustrating the coarse array of magnetic flux
tubes traced down from the CEGP magnetosphere to the stellar
surface.  Not to scale.}
\end{figure}

For each of the 11 epochs, we created a fine grid of planetary
longitudes (300 points around the circular orbit) defined relative
to the fixed Carrington longitude coordinates of the modeled
magnetic field.
For each planetary position, we traced an envelope of 5
flux tubes from the planet's orbit ($d > R_{\rm ss}$) down to
the stellar surface.
Figure 2 shows the cross-like configuration of flux tubes that
we used to resolve the magnetosphere; note that the pattern of
flux tube ``footpoints'' on the stellar surface is likely to
have a more complicated shape.
We used more than one flux tube so that ``bifurcation'' events
(when the finite-sized magnetosphere connects down to two widely
separated areas on the stellar surface) could be at least
approximately resolved.
Future work should of course use a more finely sampled envelope
of flux tubes so that the detailed shapes of the resulting
star spots can be computed.

The mapped flux tubes were then input into a time-resolved model
of both the planet's orbit and the star's rotation.
The instantaneous angular positions of the star, planet, and
observer were tracked, and the angle $\theta$ (measured between
the stellar surface normal at the flux-tube footpoint and the
observer's line of sight) was computed as a function of time.
The relative contribution from each flux tube was thus
assumed to be proportional to the stellar field strength
at the magnetosphere ($B_{\ast}$) modulated by the spot's
projection factor; i.e.,
\begin{equation}
  (I_{\rm core} / I_{\rm wing}) \, \propto
  \, \sum \, \left( B_{\ast} \, \cos\theta \right)
  \,\,\,\,\,
  \mbox{(summed over 5 footpoints)}
\end{equation}
The above procedure contains the assumption that the magnetic
``communication'' between star and planet is essentially
instantaneous.
This would be a valid assumption if the energy were carried
by, e.g., suprathermal particles as in strong solar flares,
but not if the energy had to propagate along more slowly as
an MHD wave (see Preusse et al.\  2006).

We adopted dynamical parameters roughly analogous to HD 179949
(e.g., $P_{\rm rot} = 2.9 P_{\rm orb}$, not an exact multiple),
and we set the overall
normalization for $I_{\rm core} / I_{\rm wing}$ to resemble
the observed amplitude (Shkolnik et al.\  2005).
This arbitrary normalization is of course a weak point of
the present models, but without a specific predictive model of
the magnetic interaction not much more was possible.

\section{Results}

\begin{figure}
\plotone{cranmer_cs14_poster_f3.eps}
\caption{For years 1986--1991, the top panel shows
the star's magnetic field out to $R_{\rm ss}$, with positive
[negative] polarity in blue [red].
The middle panel shows a representative light curve over 5
planetary orbits, and the bottom panel shows a coarser random
sampling over 200 orbits all phased together.
In both, the full star$+$planet light curve is in
black (solid curves, filled circles), and the stellar
rotational component only is
in green (dotted curves, X's).}
\end{figure}

\begin{figure}
\plotone{cranmer_cs14_poster_f4.eps}
\caption{For years 1992--1996, the top panel shows
the star's magnetic field out to $R_{\rm ss}$, with positive
[negative] polarity in blue [red].
The middle panel shows a representative light curve over 5
planetary orbits, and the bottom panel shows a coarser random
sampling over 200 orbits all phased together.
In both, the full star$+$planet light curve is in
black (solid curves, filled circles), and the stellar
rotational component only is
in green (dotted curves, X's).}
\end{figure}

Figures 3 and 4 show the magnetic field configurations and
light curves for the 11 modeled epochs.
The overall polarity ``flip'' is evident over the 11-year
solar cycle.
Both the complete light curve (star plus planet) and just the
stellar rotational component are shown in order to clearly
see the planetary enhancement.
The light curve is shown in detail over five planetary orbits.
The planetary enhancements generally occur when the planet is
in front of the star ($\phi \sim 0$) but occasional planetary
enhancements occur at other phases, or are absent at phase zero.
Rapid changes sometimes occur over less than 0.05 in phase
(i.e., over just several hours in time).
We also show a much coarser kind of ``light curve'' that was
randomly sampled with 200 points spread over a time-span of
200 planetary orbits.
In these plots, the stellar rotation light curve is not resolved
except as a fuzzy ``band'' which the planetary enhancement tends
to exceed around phase zero.

\begin{figure}[!ht]
\plotone{cranmer_cs14_poster_f5.eps}
\caption{Correlations between statistical properties of the
stellar (abscissa) and planetary (ordinate) light curves over
the 11 modeled epochs.}
\end{figure}

Because the amplitudes of {\em both} the stellar-rotation light
curve and the planetary enhancements are larger at solar maximum
than at solar minimum, there is no systematic trend for the
planetary enhancement to ``poke its head above the fuzz''
more at either maximum or minimum.
Thus, there does not seem to be any preferred cycle-phase
for the planetary enhancement to be more or less distinguishable
from the stellar rotational modulation.

Figure 5 shows some statistical properties of the light curves,
with one point in each panel characterizing the variations over
each of the 11 epochs.
The minimum, maximum, and average intensity ratios
($I_{\rm core} / I_{\rm wing}$) were determined for the separate
stellar and planetary light curves and correlated with one another.
The tendency for the amplitudes of the stellar and planetary
light curves to rise and fall together was discussed above
and is evident in the direct correlations.
Note specifically the nearly linear relationship between the
maximum planet enhancement and the total (max $-$ min) amplitude of
the stellar light curve.

\section{Conclusions}

Some of the more interesting features of the modeled light
curves are summarized here:
\begin{enumerate}
\item
Because of the complex nature of the multipole fields,
the modeled light curves do not repeat exactly
from orbit to orbit, and sometimes the planetary enhancement
seems to disappear altogether.
This may be a possible explanation for the 2003 disappearance of
the strong orbital modulation seen in 2001--2002 for HD 179949
(Shkolnik et al.\  2005).
\item
The planetary enhancements are often
non-monotonic in shape (i.e., multiply peaked).
This occurs because the ``spots'' on the stellar surface that
connect to the planet's magnetosphere may be bifurcated or
substantially swept around in longitude and/or latitude.
\item
For sparsely sampled data, the inferred phase shift
between light-curve maximum and planetary meridian-passage
may range between $-0.2$ and $+0.2$.
\end{enumerate}
Better observational phase coverage (even using multiple
observatories to obtain round-the-clock time series)
over consecutive orbits and stellar rotations
would yield much better tests of the magnetic interaction
paradigm.
More work needs to be done, of course, to model how the stellar
magnetic fields differ from the fiducial solar case modeled here.
Additional kinds of observations, such as X-rays that may be
phased with the planet's orbit (Saar et al., these proceedings),
will help constrain the dynamics and energetics of the
interaction.

\acknowledgements This work was supported by the National
Aeronautics and Space Administration (NASA) under grants
{NNG\-04\-GE77G} and {NNG\-04\-GL54G}
to the Smithsonian Astrophysical Observatory.