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The Moduli Space of Flat Connections over Higher Dimensional Manifolds
 Author(s): Blacker, Casey Alexander
 Advisor(s): Dai, Xianzhe
 et al.
Abstract
Let $M$ be a smooth manifold of dimension at least $3$, let $G$ be a compact Lie group, and let $P$ be a $G$principal bundle on $M$. This work is motivated by two aims:
1. Exhibit the moduli space $\M(P)$ of flat connections on $P$ as a generalized symplectic reduction of the space $\A(P)$ of connection on $P$ by the action of the gauge group $\G$.
2. Compute the symplectic volume of the moduli space $\M_G(M)$ of all flat $G$connections on $M$.
We show that the appropriate adaptation of the Hamiltonian formalism in this context is to consider a natural $\Omega^2(M)/B^2(M)$valued symplectic form $\omega$ on $\A(P)$. With The action of the gauge group $\G$ on the space of connections $(\A,\omega)$ admits a natural moment map $\mu$, and the reduction of the vectorvalued Hamiltonian system $(\A,\omega,\G,\mu)$ is the moduli space of flat connections $\M$. The reduced form $\omega_0$, which may be nondegenerate, takes values in the second cohomology $H^2(M)$ of the underlying manifold $M$.
Several chapters are devoted to the theory of vectorvalued symplectic geometry. In addition to its applications to the moduli space of flat connections, we show that the vectorvalued symplectic formalism has a rich structure that does not always reflect its realvalued counterpart. We prove two symplectic reduction theorems and investigate the vectorvalued analogues of Hamiltonian and Lagrangian mechanics.
We also compute the volume of the moduli space $\M(M)$ of all $G$connections on a symplectic manifold $(M,\omega)$ in two special cases. First, we assume that the structure group $G$ is abelian; second, that $G$ is semisimple and the fundamental group $\pi_1(M)$ is free abelian. The expression of the volume is a function of the covolume of the lattice $H^1(M,\Z)$ in $H^1(M,\R)$, the rank of the homology $H^1(M,\R)$, and the structure group $G$.
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