The Superconformal Index of Theories of Class $$\mathcal {S}$$

Abstract

We review different aspects of the superconformal index of N = 2 superconformal theories of class S. In particular we discuss the relation of the index of class S theories to topological QFTs and integrable models, and review how this relation can be harnessed to completely determine the index. This is part of a combined review on 2d-4d relations, edited by J. Teschner.

E‐String Theory on Riemann Surfaces

“…The complete index can be formed in E 7 characters where it reads 7 7 In case the reader is not familiar with suprsymmetric index [45,46] nomenclature we recommend [47] for beautiful exposition and [48] for a review, and we will use the notations of the latter.…”

Spectral curves of N $$ \mathcal{N} $$ = 1 theories of class S k $$ {\mathcal{S}}_k $$

“…Equations (4.10) show that v k + (t→1) diverges and therefore t = 1 is a branching point. This is connected to the point t = t b 14) where the solution v k − vanishes while v k + remains finite…”

“…The superconformal index of the 4D theory associated with the Riemann surface can be reinterpreted as a correlation function in a 2D topological QFT living on the Riemann surface. See [14] for a recent review. Invariance of the index under generalized S-duality transformations translates into associativity of the operator algebra of the 2D TQFT.…”

“…The a(u) in the IIA/M-theory picture correspond to the positions of the D4/M5 branes. After orbifolding, the gauge invariant operators whose vevs parameterize the Coulomb branch of the theory are 3 tr 14) where the index c = 1, . .…”

“…In the strong coupling limit q → 1 the discriminant (3.30) factorizes as 14) which means that two of the four branch points approach the position t = 1. From the convention (A.5), we identify these two points to be t 2 and t 3 .…”