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

Abstract

The S-duality group S(F ) of a 4d N = 2 supersymmetric theory F is identified with the group of triangle equivalences of its cluster category C (F ) modulo the subgroup acting trivially on the physical quantities. S(F ) is a discrete group commensurable to a subgroup of the Siegel modular group Sp(2g, 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 N = 2 QFTs. The group S(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) × SO(5) 3 × SO(3) 6 and half-hypermultiplets in the bi-and tri-spinor representations.

Homological classification of 4d $$ \mathcal{N} $$ = 2 QFT. Rank-1 revisited

“…For SU (2) N = 2 * , Aut(C )/(phy.triv.) is a Z 2 extension of the modular group P SL(2, Z) [32,33]. As in the previous examples, a finite subgroup of Aut(C )/(phy.triv.)…”

“…To a given N = 2 QFT there are attached several triangle categories describing its BPS sector [32,33]. These categories are fairly well understood when the QFT has the so-called BPS-quiver property [23,24,30].…”

“…Pragmatically, this amounts to requiring the 4d/2d correspondence to be satisfied by the covering cluster categoryC of our UV 2-CY category C . To keep the statement as simple as possible, we shall not state the criterion in its most general form (the interested reader may look at [32,33,53]) but only its simplified version which applies to rank-1 SCFT. In our tables we shall insert back the rank-1 asymptotically-free categories.…”

“…As an auto-equivalence of nil C(4-cycle) its satisfiesτ 4 = Id. 53 For details see §.6.2.1 of [32]. 54 S 1 is the simple module of CD 4 with support on node 1 of the Dynkin quiver (8.1).…”

“…This is not a problem according the discussion in §.7.1, since in this case h = 1. For h ≥ 1 the only condition is that the discrete gauge group G should be an unobstructed subgroup of the homological S-duality group [32,33] equal to the group of auto-equivalences Aut(C ) of the UV category C of the parent theory (modulo its subgroup acting trivially on physical observables).…”