Testing our understanding of SCFTs: a catalogue of rank-2 $$ \mathcal{N} $$ = 2 theories in four dimensions

Abstract

In this paper we begin mapping out the space of rank-2 $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) in four dimensions. This represents an ideal set of theories which can be potentially classified using purely quantum field-theoretic tools, thus providing a precious case study to probe the completeness of the current understanding of SCFTs, primarily derived from string theory constructions. Here, we collect and systematize a large amount of field theoretic data characterizing each theory. We also provide a detailed description of each case and determine the theories’ Coulomb, Higgs and Mixed branch stratification. The theories naturally organize themselves into series connected by RG flows but which have gaps suggesting that our current understanding is not complete.

Probing 7-branes on orbifolds

“…The CB operators have dimension For N = 1 we recover the rank-two SU(8) × SU(2) theory which is known to flow to the E 7 Minahan-Nemeschansky theory upon turning on a nilpotent vev for the SU(2) moment map [54], in agreement with our general expectations. By higgsing the SU(8) global symmetry, we find a family of theories with flavor symmetry containing E 6 × U(1), with magnetic quiver…”

“…By comparing with the partial classification results at rank two, in particular [54,58], we notice that in the present work all the models in the e 8 − so (20) appear. The other theories, at least most of them, are expected to arise by including S-fold quotients in our F-theory setup.…”

“…The allowed holonomies for n = 2 break E 8 to E 8 , E 7 × SU(2) and SO (16). For N = 1 and SO( 16)-preserving holonomy we can immediately identify the worldvolume theory on the D3 brane with the rank two SO(16) × SU(2) theory, which has precisely the expected global symmetry and is known to flow to the rank-one E 8 Minahan-Nemeschansky theory by turning on a nilpotent vev for the SU(2) moment map [54]. Said differently, the theory flows to the worldvolume theory of a single D3-brane probing a 7-brane of type E 8 in flat space with transverse slice C 2 /Z 2 .…”

“…(3.36) 18 For N = 2 the non-abelian global symmetry enhances to SU (10): this theory has two CB operators of dimensions 4 and 5. See also [54,58] for a detailed discussion about this theory. 19 There are two extra U(1)'s which enhance to SO(4) for the minimal number of probe branes, N = 2.…”

Exceptional moduli spaces for exceptional $$ \mathcal{N} $$ = 3 theories

“…Given the central charge, one may then apply the formula of [49,50], which relates the central charge of the theory at the origin of the moduli space to the data of the rank-1 theories living on the codimension-1 singular strata, 8…”