Macdonald Identities and Exact Formulas for Superconformal Indices in Super Yang-Mills Theories

TL;DR

This paper derives exact formulas for superconformal indices in 4d N=1 and N=2 Super Yang-Mills theories using Macdonald identities, revealing new algebraic structures and connections to various physical and mathematical frameworks.

Contribution

It introduces a novel application of Macdonald identities to obtain uniform closed-form expressions for superconformal indices across all simple gauge groups.

Findings

Exact formulas for superconformal indices as q-series and eta-quotients.

Discovery of a bilinear structure in the full Schur index.

Connections established between indices, line operators, and Coulomb branch K-theory.

Abstract

We present exact evaluations of superconformal indices for 4d N =1 and N =2 pure Super Yang-Mills theories with arbitrary simple gauge group G. Our approach applies the Macdonald identities for untwisted affine Lie algebras to the integral formulas of the indices, yielding uniform closed formulas valid for all G, expressed both as q-series and as eta-quotients, related through specialized Macdonald identities. Using similar techniques, we also derive exact expressions for half Schur indices with Neumann boundary conditions and uncover a bilinear structure of the full Schur index. Within the framework of holomorphic-topological twists, we further explore connections to the category of line operators, the K-theoretical Coulomb branch, Schur quantization, IR formulas for the BPS spectrum, and class S constructions.