Morita equivalences, moduli spaces and flag varieties

Double Bruhat cells in a connected complex semisimple Lie group $G$ emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. These cells are special instances of a broader class of cluster varieties known as generalized double Bruhat cells, which can be studied collectively as Poisson subvarieties of $\widetilde{F}_{2n} = \mathcal{B}^{2n-1} \times G$, where $\mathcal{B}$ is the flag variety of $G$. The spaces $\widetilde{F}_{2n}$ are Poisson groupoids over $\mathcal{B}^n$ and were introduced by J.-H. Lu, V. Mouquin, and S. Yu in the study of configuration Poisson groupoids of flags. In this work, we describe the spaces $\widetilde{F}_{2n}$ as decorated moduli spaces of flat $G$-bundles over a disc. This perspective yields the following results: (1) We explicitly integrate the Poisson groupoids $\widetilde{F}_{2n}$ to symplectic double groupoids, which are complex algebraic varieties. Furthermore, we show that these integrations are symplectically Morita equivalent for all $n$. (2) Using this construction, we integrate the Poisson subgroupoids of $\widetilde{F}_{2n}$ formed by unions of generalized double Bruhat cells to explicit symplectic double groupoids. As a corollary, we obtain integrations for the top-dimensional generalized double Bruhat cells contained therein. (3) Finally, we relate our integration to the work of P. Boalch on meromorphic connections. We lift the torus actions on $\widetilde{F}_{2n}$ to the double groupoid level and show that they correspond to the quasi-Hamiltonian actions on the fission spaces of irregular singularities.