S duality and Framed BPS States via BPS Graphs

Authors: D. Gang, P. Longhi and M. Yamazaki

Preprint number: UUITP-42/17

We study a realization of S dualities of four-dimensional N=2 class 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 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.