  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.
