%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage{amsmath, amssymb, amsthm, amscd} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \numberwithin{equation}{section} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{remark}{Remark} \theoremstyle{definition} \newtheorem{definition}{Definition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \parindent 4mm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\C{{\mathbb C}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\N{{\mathbb N}} \def\F{{\cal F}} \def\B{{\cal B}} \def\K{{\cal K}} \def\wt{\widetilde} \def\ti{\tilde} \def\ov{\overline} \def\t{\theta} \def\d{\delta} \def\o{\omega} \def\O{\Omega} \def\bi{\begin{itemize}} \def\ei{\end{itemize}} \def\la{\label} \def\nn{\nonumber} \def\be{\begin{equation}} \def\ee{\end{equation}} \newcommand{\ba}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \DeclareMathOperator{\ad}{ad} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Perturbative analysis of Chern Simons invariants for lens spaces} \author{Carlos Florentino$^\dagger$, Marcos Mari\~no$^\ddagger$, Jos\'e Mour\~ao$^{\dagger}$ \\ and Jo\~ao P. Nunes$^\dagger$} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle \begin{abstract} We study perturbatively the structure of Chern Simons invariants on lens spaces for flat reducible connections. \end{abstract} \newpage \tableofcontents \newpage \section{Introduction}\label{s1} \section{The space of connections} \subsection{Twisted differentials on manifolds with finite fundamental group} Let $M$ be a $n$--dimensional manifold with finite fundamental group $\Gamma$ and $A_{(0)}$ be a flat connection on a principal $G$--bundle $P(M,G)$ on $M$. In this section we will recall mostly from \cite{Ra} that the covariant derivative defined by $A_{(0)}$ of sections of associated vector bundles is equivalent to the restriction of the exterior differential $d$ on the universal covering $\wt M$ of $M$ to appropriate subspaces of (equivariant vector--valued) functions on $\wt M$. The connection $A_{(0)}$ defines (and is defined by) a (equivalence class of) homomorphism $H$, \be \la{hom} H \ : \ \Gamma \rightarrow G . \ee The principal bundle $P(M,G)$ can then be reduced to the holonomy group $H(\Gamma)$ with total space $Q$ being a finite covering of $M$. Let $\rho$ be a representation of $G$ on the vector space $V$ and $E_\rho = P \times V/G$ be the corresponding vector bundle on $M$ associated to $P$ and $\Gamma(M, E_\rho)$ be the space of sections of $E_\rho$. This space is naturally isomorphic with the space of $V$--valued $\tilde \rho$--equivariant functions on $Q$, where $\tilde \rho$ denotes the representation of $\Gamma$ on $V$ given by $\tilde \rho = \rho \circ H$. By choosing a covering map from $\wt M$ to $Q$ and taking the pull back of functions we obtain that $\Gamma(M, E_\rho)$ is naturally isomorphic with the space $C_{\tilde \rho}^\infty(\wt M, V)$, of $V$-- value $\tilde \rho$--equivariant functions on $\wt M$: $$ C_{\tilde \rho}^\infty(\wt M, V) = \left\{f \in C^\infty(\wt M, V) \ : \ f(p \gamma) = \tilde \rho(\gamma)^{-1} f(p) \ , \ p\in \wt M, \gamma \in \Gamma \right\}. $$ Analogously, the space of differential forms on $M$ with values on $E_\rho$, i.e. sections of $\wedge^* T^*M \otimes E_\rho$, is naturally isomorphic with the following subspace of forms on $\wt M$ with values in $V$, $$ \Omega_{\tilde \rho}^*(\wt M, V) = \left\{\omega \in \Omega^*(\wt M, V) \ : \ \gamma^* \omega = \tilde \rho(\gamma)^{-1} \omega \ , \gamma \in \Gamma \right\}. $$ An advantage of this description consists in the fact that, under this isomorphism, the covariant derivative associated with $A_{(0)}$ $$ \nabla^{A_{(0)}} \ : \ \Gamma(M,\wedge^pT^*M \otimes E_\rho) \ \rightarrow \ \Gamma(M,\wedge^{p+1}T^*M\otimes E_\rho) . $$ is mapped to the usual (untwisted) operator $d$ of exterior diferentiation of (vector--)valued forms on $\wt M$ $$ d \ : \ \Omega_{\tilde \rho}^p(\wt M,V) \ \rightarrow \ \Omega_{\tilde \rho}^{p+1}(\wt M,V) , $$ restricted to the subspace of $\tilde \rho$--equivariant forms. A natural projection operator $P_{\tilde \rho} \ : \ \Omega^*(\wt M,V) \rightarrow \Omega_{\tilde \rho}^*(\wt M,V)$, which commutes with $d$, is given by twisted averaging over $\Gamma$: $$ P_{\tilde \rho} (\omega ) = \frac 1{|\Gamma|} \sum_{\gamma \in \Gamma} \tilde \rho(\gamma)^{-1} (\gamma^{-1})^* \omega . $$ We note that for $V=\C$, $\Omega^*(\wt M,\C)$ is just the space of complex valued forms, which we will denote by $\Omega^*(\wt M)$. The set of one dimensional representations of $\Gamma$ coincides with the set $Ch(\Gamma)$ of characters of $\Gamma$ and the following decomposition is valid \cite{Ra} $$ \Omega^*(\wt M) = \oplus_{\chi \in Ch(\Gamma)} \ \Omega_\chi^*(\wt M) . $$ \subsection{Reducible $SU(N)$ connections on lens spaces} In the following we will restrict ourselves to the case of interest in the present paper. Let $M = L_p = S^3/\Z_p$ and $G=SU(N)$ (the case of arbitrary compact semisimple Lie group is similar). The universal covering $\wt M$ is therefore $S^3 \cong SU(2)$. The geometry of $SU(2)$ as a Lie group will be usefull in finding the spectrum and eigenvalues of the gauge fixed kinetic operator of Chern--Simons (CS) theory on $L_p$. We will be interested in the perturbative expansion of the CS invariant, $\log (Z(L_p))[A_{(0)}]$, around a reducible connection $A_{(0)}$ on a fixed principal bundle $P=P(S^3,SU(N))$. Let $H_{(0)}$ denote (a choice of) the homomorphism (\ref{hom}). It can be brought to the following form: \be \la{hom2} C = H_{(0)}(\overline 1)= \left( \begin{array}{ccccccccc} \o_1 & \dots & 0 & 0 & \dots & 0 & 0 & \dots & 0\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ 0 & \dots & \omega_1 & 0 & \dots & 0 & 0 & \dots & 0\\ 0 & \dots & 0 & & \dots & & 0 & \dots & 0\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ 0 & \dots & 0 & & \dots & & 0 & \dots & 0 \\ 0 & \dots & 0 & 0 & \dots & 0 & \omega_f & \dots & 0\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ 0 & \dots & 0 & 0 & \dots & 0 & 0 & \dots & \o_f \end{array} \right) \ , \ee where $\o_1, \dots , \o_f$ are $f$ different $p$--roots of unity with degeneracies $N_1, \dots , N_f$, respectively, satisfying the constraint $\o_1^{N_1} \cdots \o_f^{N_f}=1$. The space of connections on $P$ is an affine space given by \ba {\cal A}_P&=&\left\{ A = A_{(0)} + B , \ B\in \Gamma(L_p,\wedge^1T^*L_p \otimes Ad \ P) \right\} = \\ &=& A_{(0)} + \wt {\cal F} \ea According to the previous subsection the space $\cal F$ is naturally isomorphic to \be {\cal F} = \O^1_{\ti \rho}(S^3, su(N)) = \left\{\o \in \O^1(S^3,su(N)) \ : \ \gamma_{\overline r}^* \o = C^{-r}\o C^r\right\} , \ee where $su(N)$ denotes the Lie algebra of $SU(N)$ realized as the space of anti--Hermitian traceless $N \times N$ matrices. The space $\F$ decomposes into a sum of eigenspaces of $\gamma_{\ov r}^*$ as follows \be \la{dec1} \F = \O^1_1(S^3) \otimes \B \oplus \left(\oplus_{r,s=1}^f \O^1_{\chi_{r,s}}(S^3) \otimes \B_{rs}\right), \ee where $\chi_{rs}$ is the character of $\Z_p$ defined by $\chi(\ov 1)= \frac {\o_s}{\o_r}$, the trivial charachter $\chi(\ov 1)$ is denoted by $1$ and $\B_{rs}$ is the subspace of $su(N)$ of matrices with all entries equal to zero outside the $rs$ block corresponding to the structure of the matrix $C$ in (\ref{hom2}). If $r=s$ the matrices in $\B_{rr}$ are assumed to be traceless so that the spaces $\B_{rr}$ are isomorphic to $su(N_r), 1\leq r \leq f$. For simplicity we will denote them by $su(N_r)$. We can rewrite (\ref{dec1}) in the form \be \la{dec2} \F = \O^1_1(S^3) \otimes \K \oplus \left(\oplus_{r \neq s}^f \O^1_{\chi_{r,s}}(S^3) \otimes \B_{rs} \right), \ee where $\K = su(N_1) \oplus \dots \oplus su(N_f) \oplus u(1)$ is the Lie algebra of the residual symmetry group in the background of $A_{(0)}$. \newpage \begin{thebibliography}{MMMM} \bibitem[DGKV]{DGKV} R.~Dijkgraaf, S.~Gukov, V.~A.~Kazakov and C.~Vafa, ``Perturbative analysis of gauged matrix models,'' hep-th/0210238, Phys.\ Rev.\ D {\bf 68} (2003) 045007. \bibitem[Ma]{Ma} M.~Mari\~no, ``Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants,'' hep-th/0207096, Commun.\ Math.\ Phys.\ {\bf 253} (2004) 25--49. \bibitem[NC]{NC} C. Nash, D. O' Connor, ``Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function'', hep-th/9212022, J. Math. Phys. {\bf 36} (1995) 1462--1505 \bibitem[Ra]{Ra} D. B. Ray, ``Reidemeister torsion and the Laplacian on lens spaces.'', Adv. Math. {\bf 4} (1970) 109--126. \end{thebibliography} \end{document}