Poisson structures on certain moduli spaces for bundles on a surface

Abstract

Let Σ be a closed surface, G a compact Lie group, with Lie algebra g, and ξ: P → Σ a principal G-bundle. In earlier work we have shown that the moduli space N(ξ) of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from N(ξ) onto a certain representation space Rep ξ (Γ, G), in fact a diffeomorphism, with reference to suitable smooth structures C ∞ (N(ξ)) and C ∞ Rep ξ (Γ, G) , where Γ denotes the universal central extension of the fundamental group of Σ. Given a coadjoint action invariant symmetric bilinear form on g * , we construct here Poisson structures on C ∞ (N(ξ)) and C ∞ Rep ξ (Γ, G) in such a way that the mentioned diffeomorphism identifies them. When the form on g * is non-degenerate the Poisson structures are compatible with the stratifications where Rep ξ (Γ, G) is endowed with the corresponding stratification and, furthermore, yield structures of a stratified symplectic space, preserved by the induced action of the mapping class group of Σ.

Poisson geometry of flat connections for SU(2)-bundles on surfaces

“…The space H φ together with the Poisson algebra (C ∞ (H φ ), {·, ·} φ ) is our local model for the moduli space N near the point represented by φ, as a stratified symplectic space. See [10] and [11] for details.…”

“…While the space N is known to have a structure of a normal projective variety [19], it is unclear whether and how the stratification (0.1) relates to the corresponding complex analytic one; I am indebted to M. S. Narasimhan for this comment. In another earlier paper [11], we proved that the space N inherits a structure of a stratified symplectic space in the sense of Sjamaar-Lerman [21]; in particular, we constructed a Poisson algebra (C ∞ N, {·, ·}) of continuous functions on N which, on each stratum, restricts to a symplectic Poisson algebra of smooth functions in the ordinary sense. In the present paper we describe the strata locally explicitly in terms of certain related classical constrained systems; in particular, for the special case of genus two, we describe the Poisson algebra and resulting Poisson geometry of the space N explicitly.…”

“…Our method is to study the local models obtained in our earlier papers [7] - [11] by means of invariant theory and to exploit a result of Kempf-Ness [14] and Kirwan [15], thereby playing off against each other invariant theory over the reals and over the complex numbers. To get our hands on the local models we calculate various group cohomology spaces as suitable unitary representations explicitly in Section 2 below.…”

Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve

“…Thus, every semistable rank 2 bundle on Γ that is not stable is of the form ξ ⊕ ξ −1 , where ξ is a line bundle of degree zero on Γ. The case g = 2 is exceptional because the moduli space M(Γ) is P 3 , so it is non-singular (in a certain sense, however, it is singular along Kum(Γ), see [7]). …”

Poisson geometry of certain moduli spaces for bundles on a surface

“…In this note, we present an extension of structures of this kind to the whole space, its singularities included, phrased in terms of the symplectic or, more generally, Poisson geometry of certain related classical constrained systems [5][6][7][8][9][10], which are formally similar to what are called "collective Hamiltonian systems" in the mathematical physics literature. It is known that the spaces N(n, k) have structure of a normal projective variety [15,16].…”