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# Advances in Theoretical and Mathematical Physics

## Volume 23 (2019)

### Number 5

### $\mathrm{S}$ duality and framed BPS states via BPS graphs

Pages: 1361 – 1410

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n5.a4

#### Authors

#### Abstract

We study a realization of $\mathrm{S}$ dualities of four-dimensional $\mathcal{N} = 2$ class $\mathcal{S}$ theories based on BPS graphs. $\mathrm{S}$ 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 $\mathrm{S}$ 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.

Published 12 February 2020