Potentials for Moduli Spaces of A_m-local Systems on Surfaces

TL;DR

This paper investigates the structure of potentials on quivers from cluster coordinates on moduli spaces of local systems, describing their classification and proposing geometric realizations in higher rank cases.

Contribution

It characterizes the space of primitive and generic potentials on quivers associated with moduli spaces, extending known results from rank 1 to higher ranks, and proposes candidate Calabi-Yau 3-folds for geometric realization.

Findings

Describes the space of primitive potentials on quivers.

Provides a finite-dimensional description of generic potentials for m=2.

Proposes candidate Calabi-Yau 3-folds for higher rank cases.

Abstract

We study properties of potentials on quivers $Q_{\mathcal{T},m}$ arising from cluster coordinates on moduli spaces of $PGL_{m+1}$-local systems on a topological surface with punctures. To every quiver with potential one can associate a $3d$ Calabi-Yau $A_\infty$-category in such a way that a natural notion of equivalence for quivers with potentials (called "right-equivalence") translates to $A_\infty$-equivalence of associated categories. For any quiver one can define a notion of a "primitive" potential. Our first result is the description of the space of equivalence classes of primitive potentials on quivers $Q_{\mathcal{T}, m}$. Then we provide a full description of the space of equivalence classes of all \emph{generic} potentials for the case $m = 2$ (corresponds to $PGL_3$-local systems). In particular, we show that it is finite-dimensional. This claim extends results of Gei\ss, Labardini-Fragoso and Schr\"oer who have proved analogous statement in $m=1$ case. In many cases $3d$ Calabi-Yau $A_\infty$-categories constructed from quivers with potentials are expected to be realized geometrically as Fukaya categories of certain Calabi-Yau $3$-folds. Bridgeland and Smith gave an explicit construction of Fukaya categories for quivers $Q_{\mathcal{T},m=1}$. We propose a candidate for Calabi-Yau $3$-folds that would play analogous role in higher rank cases, $m > 1$. We study their (co)homology and describe a construction of collections of $3$-dimensional spheres that should play a role of generating collections of Lagrangian spheres in corresponding Fukaya categories.