Homological S-Duality in 4d N=2 QFTs

Abstract

The $S$-duality group $\mathbb{S}(\mathcal{F})$ of a 4d $\mathcal{N}=2$ supersymmetric theory $\mathcal{F}$ is identified with the group of triangle auto-equivalences of its cluster category $\mathcal{C}(\mathcal{F})$ 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 $\mathcal{N}=2$ 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 $E_r^{(1,1)}(G)$, as well as the asymptotically-free theories $(G,\hat{H})$ (which contain $\mathcal{N}=2$ SQCD as a special case). For the $E_r^{(1,1)}(G)$ 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 much larger group.

arXiv comment102 pages, 12 figure, 6 tables

References in corpus (15)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Four-dimensional wall-crossing via three-dimensional field theory
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
• R-Twisting and 4d/2d Correspondences
• N=2 Quantum Field Theories and Their BPS Quivers
• Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2
• More on the N=2 superconformal systems of type $D_p(G)$
• Infinitely many N=2 SCFT with ADE flavor symmetry
• On Arnold's 14 `exceptional' N=2 superconformal gauge theories
• Categorical Tinkertoys for N=2 Gauge Theories
• Classification of complete N=2 supersymmetric theories in 4 dimensions
• Seiberg-Witten Geometries Revisited
• Tilting theory and cluster algebras
• Higher S-dualities and Shephard-Todd groups

Cited by in corpus (4)

• Lagrangians for generalized Argyres-Douglas theories
• On Irregular Singularity Wave Functions and Superconformal Indices
• Argyres-Douglas matter and N=2 dualities
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I