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Poisson groupoids and moduli spaces of flat bundles over surfaces

Daniel Álvarez

Abstract

Let $Σ$ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $Σ$ which are trivialized at a finite subset of $\partialΣ$ carries a natural quasi-Hamiltonian structure which was introduced by Li-Bland and Severa. By a suitable restriction of the holonomy over $\partial Σ$ and of the gauge action, which is called a decoration of $\partial Σ$, it is possible to obtain a number of interesting Poisson structures as subquotients of this family of quasi-Hamiltonian structures. In this work we use this quasi-Hamiltonian structure to construct Poisson and symplectic groupoids in a systematic fashion by means of two observations: (1) gluing two copies of the same decorated surface along suitable subspaces of their boundaries determines a groupoid structure on the moduli space associated to the new surface, this procedure can be iterated by gluing four copies of the same surface, thereby inducing a double Poisson groupoid structure; (2) on the other hand, we can suppose that $G$ is a Lie 2-group, then the groupoid structure on $G$ descends to a groupoid structure on the moduli space of flat $G$-bundles over $Σ$. These two observations can be combined to produce up to three distinct and compatible groupoid structures on the associated moduli spaces. We illustrate these methods by considering symplectic groupoids over Bruhat cells, twisted moduli spaces and Poisson 2-groups besides the classical examples.

Poisson groupoids and moduli spaces of flat bundles over surfaces

Abstract

Let be a compact connected and oriented surface with nonempty boundary and let be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal -bundles over which are trivialized at a finite subset of carries a natural quasi-Hamiltonian structure which was introduced by Li-Bland and Severa. By a suitable restriction of the holonomy over and of the gauge action, which is called a decoration of , it is possible to obtain a number of interesting Poisson structures as subquotients of this family of quasi-Hamiltonian structures. In this work we use this quasi-Hamiltonian structure to construct Poisson and symplectic groupoids in a systematic fashion by means of two observations: (1) gluing two copies of the same decorated surface along suitable subspaces of their boundaries determines a groupoid structure on the moduli space associated to the new surface, this procedure can be iterated by gluing four copies of the same surface, thereby inducing a double Poisson groupoid structure; (2) on the other hand, we can suppose that is a Lie 2-group, then the groupoid structure on descends to a groupoid structure on the moduli space of flat -bundles over . These two observations can be combined to produce up to three distinct and compatible groupoid structures on the associated moduli spaces. We illustrate these methods by considering symplectic groupoids over Bruhat cells, twisted moduli spaces and Poisson 2-groups besides the classical examples.

Paper Structure

This paper contains 35 sections, 17 theorems, 114 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Poisson structures and Poisson groupoids
  4. Poisson groupoids and Hamiltonian spaces for Manin pairs
  5. Manin pairs and moment maps
  6. Multiplicative Manin pairs and Poisson groupoids
  7. Poisson groupoids and flat bundles over decorated surfaces
  8. The $\Gamma$-twisted Cartan-Dirac structure
  9. Boundary decorations
  10. Poisson and symplectic groupoids via gluing and decorating surfaces
  11. Direct description of the Lie groupoid structure on $\hom(\Pi_1(\widehat{\Sigma},\widehat{V}),G)$
  12. Direct construction of the groupoid structure on $\mathfrak{M}_G(\widehat{\Sigma},\widehat{V})_{\widehat{H},\widehat{\mathcal{A}}}$
  13. The symplectic leaves of $\mathfrak{M}_G(\widehat{\Sigma},\widehat{V})_{\widehat{H},\widehat{\mathcal{A} }}$
  14. Quasi-Poisson structures on spaces of representations
  15. Quasi-Poisson manifolds and quasi-Poisson groupoids
  16. ...and 20 more sections

Key Result

Proposition 2.7

Let $(\mathcal{E},\mathcal{L} )$ be an exact multiplicative Manin pair and let $\mathcal{A}\subset \mathcal{E}$ be a multiplicative Dirac structure such that $\mathcal{L}\cap \mathcal{A}$ is of constant rank. Let $\mathcal{G}\rightrightarrows M$ be a multiplicative quasi-Hamiltonian space for $(\ma

Figures (7)

  • Figure 1: A decorated disk which induces a Poisson group structure on $\mathfrak{M}_G(\widehat{\Sigma},\widehat{V})_{\widehat{H},\widehat{\mathcal{A} }}$
  • Figure 2: If $S$ is one of the boundary components of the annulus $\Sigma$ with one marked point on each boundary component, then $\widehat{\Sigma}$ is identified with $\Sigma$ itself; the base of the Poisson groupoid structure corresponding to a symmetric decoration is determined by $\Sigma$ with only one marked point
  • Figure 3: The CA-groupoid structure on $\mathcal{E}_{\widehat{\Gamma }}$ corresponding to a disk with four marked points $(\Sigma,V)$ and $S$ consisting of one edge
  • Figure 4: The gluing of $\widehat{\Sigma}=\Sigma\cup_S \Sigma$
  • Figure 5: The decorated surface $(\widehat{\Sigma},\widehat{V} )$.
  • ...and 2 more figures

Theorems & Definitions (77)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • Definition 3.1: quisur2
  • ...and 67 more