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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(T Q, 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.

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We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in an essential way the action of the space of coefficients. We define quantization with respect to a polarization and to a spin c structure. In the presence of a complex polarization, it is shown that the polysymplectic Guillemin-Sternberg conjecture is false. We conclude with potential extensions and applications.

We introduce a novel symplectic reduction scheme for C ∞ (M, ω) that extends in a straightforward manner to the multisymplectic setting. Specifically, we exhibit a reduction of the L ∞ -algebra of observables on a premultisymplectic manifold (M, ω) in the presence of a compatible Lie algebra action g M and subset N ⊆ M . In the symplectic setting, this reproduces the Dirac, Śniatycki-Weinstein, and Arms-Cushman-Gotay reduced Poisson algebra whenever the Marsden-Weinstein quotient exists. We examine our construction in the context of various examples, including multicotangent bundles and multiphase spaces, and conclude with a discussion of applications to classical field theories and quantization.