The exact Schur index in closed form

TL;DR

This paper derives a closed-form, finite-sum expression for the Schur index of 4D N=2 superconformal theories, revealing new analytical formulas and conjectures for various classes of theories.

Contribution

It introduces an elementary elliptic function approach to evaluate the Schur index exactly, including formulas for class S theories and conjectures for N=4 super Yang-Mills.

Findings

Finite sum formulas for class S theories of type A1

Conjecture for unflavored Schur indices of N=4 SYM with SU(N)

Closed-form expressions for low-rank gauge theories

Abstract

The Schur limit of the superconformal index of a four-dimensional N = 2 superconformal field theory encodes rich physical information about the protected spectrum of the theory. For a Lagrangian model, this limit of the index can be computed by a contour integral of a multivariate elliptic function. However, surprisingly, so far it has eluded exact evaluation in closed, analytical form. In this paper we propose an elementary approach to bring to heel a large class of these integrals by exploiting the ellipticity of their integrand. Our results take the form of a finite sum of (products of) the well-studied flavored Eisenstein series. In particular, we derive a compact formula for the fully flavored Schur index of all theories of class S of type a1, we put forward a conjecture for the unflavored Schur indices of all N=4 super Yang-Mills theories with gauge group SU(N), and we present closed-form expressions for the index of various other gauge theories of low ranks. We also discuss applications to non-Lagrangian theories, modular properties, and defect indices.