The Coulomb and Higgs branches of $$ \mathcal{N} $$ = 1 theories of Class $$ {\mathcal{S}}_k $$

Abstract

Even though for generic $$ \mathcal{N} $$ N = 1 theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $$ \mathcal{N} $$ N = 1 SCFTs, these of Class $$ {\mathcal{S}}_k $$ S k for which it is possible to define Coulomb and Higgs branches precisely as for the $$ \mathcal{N} $$ N = 2 theories of Class $$ \mathcal{S} $$ S from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional (1, 1) Little String theory can be deconstructed from a toroidal quiver in Class $$ {\mathcal{S}}_k $$ S k .

𝒩 = 1 Curves on Generalized Coulomb Branches of Supersymmetric Gauge Theories

“…They can also be engineered using M-theory on R 3,1 × (CY 2 × C)/Z k × R with M5 branes lying along R 3,1 × C with the Riemann surface C ⊂ CY 2 /Z k . It is an active field of research to examine N = 1 theories of class S k and the broader theories of class S Γ where the Z k singularity is generalized to any ADE orbifold singularity Γ [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

“…For generic N = 1 theories it is not possible to separate distinct branches of supersymmetric vacua, thus the study of their moduli space is in general complicated. Nonetheless, it has been recently understood [15][16][17]19,20] that there is a special but very broad and interesing class of N = 1 superconformal theories (SCFTs), the so called N = 1 theories of class S k [12], for which it is possible to distinguish between generalized Coulomb and Higgs branches precisely as for N = 2 gauge theories. This is due to the fact that in addition to the R-symmetry they enjoy, by construction, extra global symmetries.…”

“…This is due to the fact that in addition to the R-symmetry they enjoy, by construction, extra global symmetries. Consequently, a generalized Coulomb branch can be isolated and has been successfully studied [15][16][17]19,20].…”

Expanding on the Cardy-like limit of the SCI of 4d $$ \mathcal{N} $$ = 1 ABCD SCFTs

“…The SCI for the C 3 /Z k × Z l orbifold theory has been computed in certain limits (albeit different from the Cardy-like one) in[55].…”

Four-dimensional N = 1 theories, S-fold constraints on T-branes, and behaviors in IR and UV

“…Studies of D3-branes probing F-theory singularities can be found, for example, in[47][48][49][50][51] 5. For recent studies on 4d N = 1 theories, see, for example,[16,[52][53][54][55][56].…”