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# Advances in Theoretical and Mathematical Physics

## Volume 22 (2018)

### Number 7

### Homological $S$-duality in 4d $\mathcal{N}=2$ QFTs

Pages: 1593 – 1711

DOI: https://dx.doi.org/10.4310/ATMP.2018.v22.n7.a1

#### Authors

#### Abstract

The $S$-duality group $\mathbb{S} (\mathcal{F})$ of a 4d $N = 2$ supersymmetric theory $\mathcal{F}$ is identified with the group of triangle equivalences of its cluster category $\mathscr{C} (\mathcal{F})$ modulo the subgroup acting trivially on the physical quantities. $\mathbb{S} (\mathcal{F})$ is a discrete group commensurable to a subgroup of the Siegel modular group $Sp (2g, \mathbb{Z})$ ($g$ being the dimension of the Coulomb branch). This identification reduces the determination of the $S$-duality group of a given $N = 2$ theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of $\mathcal{N} = 2$ QFTs. The group $\mathbb{S} (\mathcal{F})$ is naturally presented as a generalized braid group.

The $S$-duality groups are often larger than expected. In some models the enhancement of $S$-duality is quite spectacular. For instance, a QFT with a huge $S$-duality group is the Lagrangian SCFT with gauge group $SO(8) \times SO(5)^3 \times SO(3)^6$ and half-hypermultiplets in the bi- and tri-spinor representations.

We focus on four families of examples: the $\mathcal{N} = 2$ SCFTs of the form $(G, G^{\prime})$, $D_p (G)$, and $E^{(1,1)}_r (G)$, as well as the asymptotically-free theories $(G, \hat{H})$ (which contain $\mathcal{N} = 2$ SQCD as a special case). For the $E^{(1,1)}_r (G)$ models we confirm the presence of the $PSL (2, \mathbb{Z})$ $S$-duality group predicted by Del Zotto, Vafa and Xie, but for most models in this class $S$-duality gets enhanced to a larger group.

Published 18 June 2019