Polysymplectic Reduction and the Moduli Space of Flat Connections

TL;DR

This paper introduces a polysymplectic framework for analyzing the moduli space of flat connections, extending classical symplectic reduction techniques to higher-dimensional and vector-valued forms.

Contribution

It develops the theory of polysymplectic reduction, applies it to moduli spaces of flat connections, and establishes foundational properties of polysymplectic manifolds.

Findings

Moduli space of flat connections is a polysymplectic reduction of the space of all connections.

A Darboux-type theorem for polysymplectic manifolds is proved.

Classical properties like the Arnold conjecture and convexity of the moment map do not hold in the polysymplectic setting.

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.