Poisson groupoids and moduli spaces of flat bundles over surfaces

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 ∂Σ carries a natural quasi-Hamiltonian structure which was introduced by Li-Bland and Ševera. By a suitable restriction of the holonomy over ∂Σ and of the gauge action, which is called a decoration of ∂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: 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; 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.