S duality and Framed BPS States via BPS Graphs

TL;DR

This paper explores how S dualities in 4D $ ext{N}=2$ class $ ext{S}$ theories can be understood through BPS graphs, linking topological graph transitions to algebraic cluster transformations and analyzing their effects on line operators.

Contribution

It provides a general framework for realizing S dualities via BPS graphs for all class $ ext{S}$ theories, including those with non-maximal flavor symmetry, extending previous triangulation-based methods.

Findings

S duality acts as topological transitions on BPS graphs.

S duality transformations correspond to cluster algebra mutations.

Matching of S duality with the mapping class group on UV line operators.

Abstract

We study a realization of S dualities of four-dimensional $\mathcal{N}=2$ class $\mathcal{S}$ theories based on BPS graphs. S duality transformations of the UV curve are explicitly expressed as a sequence of topological transitions of the graph, and translated into cluster transformations of the algebra associated to the dual BPS quiver. Our construction applies to generic class $\mathcal{S}$ theories, including those with non-maximal flavor symmetry, generalizing previous results based on higher triangulations. We study the the action of S duality on UV line operators, and show that it matches precisely with the mapping class group, by a careful analysis of framed wall-crossing. We comment on the implications of our results for the computation of three-manifold invariants via cluster partition functions.