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

Abstract

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗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 in the tensor square of the Lie algebra g 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, in terms of explicit algebraic expressions, 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 arising from the data including the symmetric 2-tensor in g⊗g. 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 Kähler 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. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.