Laminations of punctured surfaces as $\tau$-regular irreducible components

TL;DR

This paper establishes a natural isomorphism between laminations on punctured surfaces and certain irreducible components of decorated representation varieties of associated Jacobian algebras, linking geometric and algebraic structures.

Contribution

It constructs a canonical isomorphism between lamination sets and irreducible components of Jacobian algebra representations, intertwining dual shear coordinates and $g$-vectors.

Findings

Isomorphism between lamination space and irreducible components

Intertwining of dual shear coordinates with $g$-vectors

Identification of monoid structures in geometric and algebraic contexts

Abstract

Let $\boldsymbol{\Sigma}:=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset\partial\Sigma\neq\varnothing$ on the boundary, and punctures $\mathbb{P}\subset\Sigma\setminus\partial\Sigma$, and $T$ an arbitrary tagged triangulation of $\boldsymbol{\Sigma}$ in the sense of Fomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$ corresponding to the non-degenerate potential $W(T)$ defined by Cerulli Irelli and the second author is tame, as shown by Schr\"{o}er and the first two authors. In this paper, we show that there is a natural isomorphism $\pi_T:\operatorname{Lam}(\boldsymbol{\Sigma})\rightarrow\operatorname{DecIrr}^\tau(A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\operatorname{Lam}(\boldsymbol{\Sigma})$ is the set of laminations of $\boldsymbol{\Sigma}$ considered by Musiker-Schiffler-Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, $\operatorname{DecIrr}^\tau(A(T))$ denotes the set of generically $\tau$-regular irreducible components of the decorated representation varieties of $A(T)$, with the direct sum of generically $E$-orthogonal irreducible components as partial monoid operation, where $E$ is the symmetrized $E$-invariant of Derksen-Weyman-Zelevinsky, $E(-,\bullet)=\dim\operatorname{Hom}_{A(T)}(-,\tau(\bullet))+\dim\operatorname{Hom}_{A(T)}(\bullet,\tau(-))$.