Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 5

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

Pages: 1361 – 1410



Dongmin Gang (Center for Theoretical Physics, Seoul National University, Seoul, South Korea)

Pietro Longhi (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden)

Masahito Yamazaki (Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Chiba, Japan)


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