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Morita equivalences, moduli spaces and flag varieties

Daniel Álvarez

TL;DR

The paper develops a global Poisson-geometric picture for generalized double Bruhat cells by realizing configuration Poisson groupoids of flags as decorated moduli spaces of flat $G$-bundles on a disc. It explicitly integrates these Poisson groupoids to complex algebraic symplectic double groupoids $\mathcal{G}_{2n}$, with dual side groupoids $\widetilde{F}_{2n}^*\rightrightarrows F_n$, and proves a Morita equivalence between $\mathcal{G}_{2n}$ and $\mathcal{G}_{2m}$ for all $n,m$, while also restricting this equivalence to symplectic subgroupoids $\mathcal{H}_{2n}$ governing total configuration groupoids $\Gamma_{2n}$. The framework identifies $\Gamma_{2n}$ as a Lie-Dirac submanifold of the ambient Poisson groupoid $\widetilde{F}_{2n}$ and provides a concrete integration via symplectic double subgroupoids, with toric actions and quasi-Hamiltonian interpretations lifting to the double groupoid level. Connections to Boalch’s meromorphic connections and fission-space constructions, together with potential links to cluster doubles and higher-geometric structures, suggest rich interactions with wild character varieties and higher Teichmüller theory.

Abstract

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.

Morita equivalences, moduli spaces and flag varieties

TL;DR

The paper develops a global Poisson-geometric picture for generalized double Bruhat cells by realizing configuration Poisson groupoids of flags as decorated moduli spaces of flat -bundles on a disc. It explicitly integrates these Poisson groupoids to complex algebraic symplectic double groupoids , with dual side groupoids , and proves a Morita equivalence between and for all , while also restricting this equivalence to symplectic subgroupoids governing total configuration groupoids . The framework identifies as a Lie-Dirac submanifold of the ambient Poisson groupoid and provides a concrete integration via symplectic double subgroupoids, with toric actions and quasi-Hamiltonian interpretations lifting to the double groupoid level. Connections to Boalch’s meromorphic connections and fission-space constructions, together with potential links to cluster doubles and higher-geometric structures, suggest rich interactions with wild character varieties and higher Teichmüller theory.

Abstract

Double Bruhat cells in a connected complex semisimple Lie group 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 , where is the flag variety of . The spaces are Poisson groupoids over 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 as decorated moduli spaces of flat -bundles over a disc. This perspective yields the following results: (1) We explicitly integrate the Poisson groupoids to symplectic double groupoids, which are complex algebraic varieties. Furthermore, we show that these integrations are symplectically Morita equivalent for all . (2) Using this construction, we integrate the Poisson subgroupoids of 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 to the double groupoid level and show that they correspond to the quasi-Hamiltonian actions on the fission spaces of irregular singularities.

Paper Structure

This paper contains 23 sections, 22 theorems, 86 equations, 10 figures.

Table of Contents

  1. Preliminaries
  2. Quasi-Poisson structures on spaces of representations and boundary decorations
  3. Moment maps and boundary decorations
  4. Poisson groupoids from symmetric decorations
  5. Configuration Poisson groupoids of flags as decorated moduli spaces
  6. The standard Poisson-Lie structure on complex semi-simple Lie groups
  7. Poisson groupoids over products of flag varieties
  8. Decorated discs and configuration Poisson groupoids of flags
  9. Decorated moduli spaces and Morita equivalences over flag varieties
  10. Morita equivalences of symplectic double groupoids
  11. A symplectic Morita equivalence between $\mathcal{G}_{2n}$ and $\mathcal{G}_{2m}$
  12. Restriction of the Morita equivalence to symplectic subgroupoids of $\widetilde{F}_{2n}$
  13. Symplectic groupoids over configuration Poisson groupoids of flags
  14. Integration of the total configuration Poisson groupoids of flags
  15. Lie-Dirac subgroupoids of Poisson groupoids and slim symplectic double groupoids
  16. ...and 8 more sections

Key Result

Theorem 1

The Poisson groupoid $\widetilde{F}_{2n} \rightrightarrows F_n$ is integrable by an explicit complex algebraic symplectic double groupoid $\mathcal{G}_{2n}$ with dual Poisson groupoid $\widetilde{F}_{2n}^* \rightrightarrows F_n$. Moreover, there is a distinguished symplectic Morita equivalence $\mat

Figures (10)

  • Figure 1: A skeleton of the disc $(\Sigma_4,V_4)$
  • Figure 2: Left: The double surface $(\widehat{\Sigma_2},\widehat{V_2})$ corresponding to the disc with three marked points $(\Sigma_2,V_2)$ determines an integration of $\widetilde{F}_2$. Right: A boundary component decorated with $(a,b^{-1})\in L_v$
  • Figure 3: The double surface of a disc with three marked points; by cutting it along the dashed arc we obtain the marked surface $\widehat{\Sigma_{0,1}}$ corresponding to $\widetilde{F}^*_2$
  • Figure 4: The values that specify $\rho'\in\hom(\Pi_1(\widehat{\Sigma_{0,m}},\widehat{V_{0,m}}),G)$
  • Figure 5: Restriction of a horizontal Morita equivalence to an action of vertical orbits
  • ...and 5 more figures

Theorems & Definitions (56)

  • Theorem
  • Theorem
  • Theorem
  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['pro:conflaqpoi']}
  • Remark 2.3
  • Proposition 3.1
  • ...and 46 more