Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

TL;DR

This paper develops a generalized theory of quasi Poisson and quasi Hamiltonian structures, establishing a correspondence between them and applying this framework to various moduli spaces, including those of vector bundles and Higgs bundles.

Contribution

It introduces a novel approach to quasi Poisson and quasi Hamiltonian structures, including a new momentum map concept, and applies it to moduli spaces, extending their Poisson and symplectic geometry.

Findings

Established a bijective correspondence between non-degenerate quasi Poisson and quasi Hamiltonian structures.

Applied the theory to moduli spaces of flat connections, vector bundles, and Higgs bundles.

Derived Poisson structures on moduli spaces via reduction, recovering known symplectic structures.

Abstract

Let G be a Lie group and g its Lie algebra. We develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in the tensor square of g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a K\"ahler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. A side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.