Categorified canonical bases and framed BPS states

TL;DR

This paper constructs a canonical basis for the algebra of functions on a cluster variety from a triangulated surface, linking it to measured laminations and framed BPS states in certain quantum field theories.

Contribution

It introduces a categorification of canonical bases in cluster algebras associated with surfaces, connecting geometric laminations to BPS state counts.

Findings

Canonical basis parametrized by measured laminations

Graded vector spaces encode basis expansions

Connection established with framed BPS states

Abstract

We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space basis parametrized by certain measured laminations on the surface. To each lamination, we associate a graded vector space, and we prove that the graded dimension of this vector space gives the expansion in cluster coordinates of the corresponding basis element. We discuss the relation to framed BPS states in $\mathcal{N}=2$ field theories of class $\mathcal{S}$.