The Moduli Space of Flat Connections over Higher Dimensional Manifolds
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 vector-valued 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 vector-valued symplectic geometry. In addition to its applications to the moduli space of flat connections, we show that the vector-valued symplectic formalism has a rich structure that does not always reflect its real-valued counterpart. We prove two symplectic reduction theorems and investigate the vector-valued 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 $G$.