Exceptional Indices

TL;DR

This paper explores the superconformal index in class S theories, providing criteria for free components, enhanced symmetries, and classifying theories as good, bad, or ugly, with explicit computations for E_n flavor symmetries.

Contribution

It introduces new criteria for analyzing class S theories and computes indices for complex theories with E_n flavor symmetry.

Findings

Criteria for decoupled free components and enhanced flavor symmetry.

A classification scheme for theories as good, bad, or ugly.

Explicit superconformal index expressions for E_n flavor symmetry theories.

Abstract

Recently a prescription to compute the superconformal index for all theories of class S was proposed. In this paper we discuss some of the physical information which can be extracted from this index. We derive a simple criterion for the given theory of class S to have a decoupled free component and for it to have enhanced flavor symmetry. Furthermore, we establish a criterion for the "good", the "bad", and the "ugly" trichotomy of the theories. After interpreting the prescription to compute the index with non-maximal flavor symmetry as a residue calculus we address the computation of the index of the bad theories. In particular we suggest explicit expressions for the superconformal index of higher rank theories with E_n flavor symmetry, i.e. for the Hilbert series of the multi-instanton moduli space of E_n.