SIGMA 22 (2026), 112, 45 pages      arXiv:2410.22620      https://doi.org/10.3842/SIGMA.2026.112

Geometric Leaves of Symplectic Groupoids

Erik Brodsky ^a, Leonid Chekhov ^b, Pranjal Dangwal ^c, Skyler Hamlin ^d, Xu Lian ^e, Michael Shapiro ^b, Samuel Sottile ^d and Jenny Zhan ^f
a) Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA
^b) Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA
^c) 360 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA
^d) Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA
^e) School of Mathematical Sciences, Nankai University, No 94. Weijin Road, Tianjin, 300071, P.R. China
^f) Department of Mathematics, University of Washington, Box 354350, C-138 Padelford, Seattle, WA 98195-4350, USA

Received March 05, 2025, in final form October 26, 2026; Published online November 11, 2026

Abstract
A symplectic groupoid of triangular bilinear forms consists of pairs of matrices $(B,A)$ with $A\in \mathcal A$—the set of real unipotent upper-triangular matrices—and $B\in {\rm SL}_n({\mathbb R})$ such that $\widetilde{A}=BAB^{\mathsf T}\in\mathcal A$. The symplectic groupoid possesses a nondegenerate symplectic structure which induces a reflection-equation Poisson algebra on $\mathcal A$. Fock and Chekhov defined a Poisson map from the Teichmüller space ${\mathcal T}_{g,s}$ of genus $g$ surfaces with $s$ (one or two) holes to the space of unipotent upper-triangular $n\times n$ matrices; the image of this map forms a geometric locus. A matrix $A$ in the geometric locus satisfies the rank condition: $\operatorname{rank}\bigl(A+A^{\mathsf T}\bigr)\le 4$ and an additional positivity condition. We describe the coisotropic reduction of the Poisson cluster variety of $\mathcal A$ by the rank condition for $n=5$ and $6$ and show that these coisotropic reductions yield the corresponding geometric loci. Moreover, we analyze in both cases the cluster structures obtained by coisotropic reductions and recover the celebrated cluster structure of ${\mathcal T}_{2,1}$ for $n=5$ and ${\mathcal T}_{2,2}$ for $n=6$. We also discuss the positivity of the geometric cluster variables for $n=3,4,5,6$.

Key words: symplectic groupoid; Poisson structure; Goldman bracket; coisotropic reduction.

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