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

Abstract

We review different aspects of the superconformal index of $\mathcal{N}=2$ superconformal theories of class $\mathcal{S}$. In particular we discuss the relation of the index of class $\mathcal{S}$ 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.

A citation of the form [V:x] refers to article number x in this volume.

Appendices

Appendix 1: Plethystics

In this appendix we collect the definitions of some special functions and combinatorial objects used in the bulk of the review. The Pochammer symbol is defined as

(8.1)

The theta-function is given by

(8.2)

The plethystic exponential is given by

(8.3)

In particular

$$\begin{aligned} \mathrm{PE}[x]=\frac{1}{1-x}\,,\qquad \mathrm{PE}[-x]=1-x. \end{aligned}$$

(8.4)

The inverse of the plethystic exponential is the logarithm, given by

(8.5)

where $\mu (\ell )$ is the Mobius mu-function. Finally the elliptic Gamma function is defined as

(8.6)

Appendix 2: $\mathcal{N}=2$ Superconformal Representation Theory

In this appendix (adapted from [5]) we review the classification of short representations of the four-dimensional $\mathcal {N}=2$ superconformal algebra [2, 39, 86].

Short representations occur when the norm of a superconformal descendant state in what would otherwise be a long representation is rendered null by a conspiracy of quantum numbers. The unitarity bounds for a superconformal primary operator are given by

$$\begin{aligned} E&\geqslant E_i,\qquad&j_i\ne 0~,\nonumber \\ E&= E_i-2~~ \text{ or } ~~E\geqslant E_i~, \qquad&j_i=0, \end{aligned}$$

(8.7)

where we have defined

$$\begin{aligned} E_1=2+2j_1+2R+ r~, \qquad E_2=2+2j_2+2R- r~, \end{aligned}$$

(8.8)

and short representations occur when one or more of these bounds are saturated. The different ways in which this can happen correspond to different combinations of Poincaré supercharges that will annihilate the superconformal primary state in the representation. There are two types of shortening conditions, each of which has four incarnations corresponding to an $\textit{SU}(2)_R$ doublet’s worth of conditions for each supercharge chirality:

(8.9)

(8.10)

(8.11)

(8.12)

The different admissible combinations of shortening conditions that can be simultaneously realized by a single unitary representation are summarized in Table 4, where the reader can also find the precise relations that must be satisfied by the quantum numbers $(E,j_1,j_2,r,R)$ of the superconformal primary operator, as well as the notations used to designate the different representations in [39] (DO) and [2] (KMMR).³⁰

Table 4 Unitary irreducible representations of the $\mathcal {N}=2$ superconformal algebra

At the level of group theory, it is possible for a collection of short representations to recombine into a generic long representation whose dimension is equal to one of the unitarity bounds of (8.7). In the DO notation, the generic recombinations are as follows:

(8.13)

(8.14)

(8.15)

There are special cases when the quantum numbers of the long multiplet at threshold are such that some Lorentz quantum numbers in (8.13) would be negative and unphysical:

(8.16)

(8.17)

(8.18)

(8.19)

(8.20)

The last three recombinations involve multiplets that make an appearance in the associated chiral algebra described in this work. Note that the $\mathcal {E}$, $\bar{\mathcal {E}}$, $\hat{\mathcal{B}}_{\frac{1}{2}}$, $\hat{\mathcal{B}}_{1}$, $\hat{\mathcal{B}}_{\frac{3}{2}}$, $\mathcal {D}_0$, $\bar{\mathcal {D}}_0$, $\mathcal {D}_{\frac{1}{2}}$ and $\bar{\mathcal {D}}_{\frac{1}{2}}$ multiplets can never recombine, along with $\mathcal {B}_{\frac{1}{2},r(0,j_2)}$ and $\bar{\mathcal {B}}_{\frac{1}{2},r(j_1,0)}$.