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 , 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.

Quantization of polysymplectic manifolds

“…Let M be a smooth manifold of dimension at least 3 and let P be a G-principle bundle on M . In an earlier work [12], we show that the regular part of the moduli space of flat connections M(P ) possesses a natural H 2 (M )valued presymplectic form ω M(P ) , obtained as the reduced 2-form of a canonical Ω 2 (M )/B 2 (M )-symplectic structure ω A(P ) on the space of connections A(P ). This generalizes the situation in which M is a surface and ω M(P ) is a symplectic structure.…”

“…Let us briefly review the V -Hamiltonian formalism. The following conventions, terminology, and notation are broadly consistent with the more comprehensive presentation in our earlier work [12]. Definition 3.1.…”

“…The polysymplectic reduction theorem, which guarantees the existence and uniqueness of the reduced 2-form ω 0 , was first proved in [51] and constitutes the polysymplectic analogue of the celebrated symplectic reduction theorem of Marsden and Weinstein [52]. We refer to [12] for a statement and proof in the V -symplectic setting.…”

Cotangent bundle reduction and Routh reduction for polysymplectic manifolds

“…the references in the recent monograph [14] on this topic). Recent contributions, both on the side of the mathematical fundamentals [29,3,6] and on the side of the physics applications [28,4,2], testify that polysymplectic geometry is today an active field of research.…”

A global version of Gunther's polysymplectic formalism using vertical projections

“…Later expositions and continuations of Günther's work like that of [23] also forgo its original geometric foundation in favor of other, more modern approaches (like k-symplectic structures). Though [3] provides a much needed update to the original at a much greater level of mathematical sophistication, to this author's knowledge nowhere is Günther's original polysymplectic structure in its original form used to produce the full range of phenomena of covariant Hamiltonian field theory in the general (global) case. The first goal of this paper is fill in this missing link.…”