Polysymplectic Reduction and the Moduli Space of Flat Connections

Abstract

A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space $V$, then apply this framework to show that the moduli space $\mathcal{M}(P)$ of flat connections on a principal bundle $P$ over a compact manifold $M$ is a polysymplectic reduction of the space $\mathcal{A}(P)$ of all connections on $P$ by the action of the gauge group $\mathcal{G}$ with respect to a natural $\Omega^2(M)/B^2(M)$-valued symplectic structure on $\mathcal{A}(P)$. This extends to the setting of higher-dimensional base spaces $M$ the process by which Atiyah and Bott identify the moduli space of flat connection on a principal bundle over a closed surface $\Sigma$ as the symplectic reduction of the space of all connections. Along the way, we establish various properties of polysymplectic manifolds. For example, a Darboux-type theorem asserts that every $V$-symplectic manifold $(M,\omega)$ locally symplectically embeds in a standard polysymplectic manifold $\mathrm{Hom}(TQ,V)$. We also show that both the Arnold conjecture and the well-known convexity properties of the classical moment map fail to hold in the polysymplectic setting.