Geometric leaf of symplectic groupoid

TL;DR

The paper analyzes the geometric leaf of the symplectic groupoid of unipotent upper-triangular matrices by applying a rank condition to the image of the Fock–Chekhov-type Poisson map from Teichmüller spaces. It develops log-canonical coordinates on the Bondal Poisson bracket via Fock–Goncharov parameters, and constructs a Hamiltonian reduction for $n=5$ and $n=6$ that inherits cluster structure. The reductions recover the classical cluster algebras associated with Teichmüller spaces $\mathcal{T}_{2,1}$ and $\mathcal{T}_{2,2}$, with seeds labeled by ideal triangulations of hyperbolic surfaces of genus $\lfloor n/2\rfloor$ and $par(n)$ holes. The work combines transport-matrix representations, skein relations for geodesic functions, and mutation sequences to relate the geometric locus to Teichmüller-type cluster algebras, while also highlighting Weyl-group–type actions on Casimirs in the higher-$n$ cases.

Abstract

We consider the symplectic groupoid of pairs $(B, A)$ with $A$ real unipotent upper-triangular matrix and $B\in GL_n$ being such that $\tilde A=BAB^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of Teichmüller space ${\mathcal T_{g,s}$ of genus $g$ surfaces with $s$ holes into the space of unipotent upper-triangular $n\times n$ matrices whose image forms the \emph{geometric locus}. The elements of geometric locus satisfy \emph{rank condition}. We describe the Hamiltonian reduction of the Poisson cluster variety of symplectic groupoid by the rank condition for $n=5$ and $6$. In both cases, we analyze the induced cluster structures on the results of Hamiltonian reduction and recover celebrated cluster structure on ${\mathcal T}_{2,1}$ for $n=5$ and ${\mathcal T}_{2,2}$ for $n=6$.

Figures (26)

• Figure 1: Factorization of deck transformation operator. The deck transformation along path $\gamma$ is expressed as a product of the matrix sequence $M_\gamma = \ldots\cdot L\cdot X(x_t)\cdot R\cdot\ldots$
• Figure 2: The Fock-Goncharov parameters organized in the triangular lattice. The quiver with vertices labeled by the Fock-Goncharov parameters $Z_{\alpha,\beta,\gamma}$ encodes the Poisson bracket between $Z$s: $\{Z_{\alpha,\beta,\gamma},Z_{\alpha',\beta',\gamma'}\}=\varkappa\cdot Z_{\alpha,\beta,\gamma}\cdot Z_{\alpha',\beta',\gamma'}$, here $\varkappa$ denotes the algebraic sum of weights of arrows between $(\alpha,\beta,\gamma)$ and $(\alpha',\beta',\gamma')$. The dashed arrow contributes to $\varkappa$ weight $\frac{1}{2}$ in the arrow's direction and $-\frac{1}{2}$ in the opposite direction. The solid arrow contributes weight $\pm 1$.
• Figure 3: The plabic graph $G$ dual to the quiver of Fock-Goncharov parameters Face weights $Z_{600}, Z_{060}, Z_{006}$ are added.
• Figure 4: The quiver shows the collection of amalgamated Fock--Goncharov parameters. The amalgamated pairs of variables are marked in the same color and connected by an additional thick curve of the same color. This amalgamated quiver is the one depicted in Fig. \ref{['fig:clusterA6']} where parameters $a_t,b_t$ and $c_t$ coincide either with $Z_{ijk}$ or with amalgamated $Z_{k,0,n-k} Z_{n-k,k,0}$ of this Figure.
• Figure 5: Ribbon Graph $\Gamma_5$ describes the surface of genus 2 with one hole. Gray curve represents the loop $\gamma_{35}$. The matrix of the monodromy operator along the loop $\gamma_{35}$ is $M_{\gamma_{35}}=X(x_5) R X(x_8) L X(x_3) R X(x_7) L$.

Theorems & Definitions (71)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Definition 2.9
• Example 2.10