Polysymplectic reduction and the moduli space of flat connections

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, we then apply this framework to show that the moduli space of flat connections on a principal bundle over a compact manifold M is a polysymplectic reduction of the space of all connections by the action of the gauge group with respect to a natural polysymplectic structure with values in an infinite dimensional Banach space. As a consequence, the moduli space inherits a canonical H-2(M)-valued presymplectic structure. Along the way, we establish various properties of polysymplectic manifolds. For example, a Darboux-type theorem asserts that every V-symplectic manifold locally symplectically embeds in a standard polysymplectic manifold 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.