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

TL;DR

The paper tackles the problem of obtaining exact, uniform formulas for the four-dimensional superconformal index of 4d $\mathcal{N}=1$ and $\mathcal{N}=2$ pure SYM theories for arbitrary simple gauge groups $G$. It achieves this by applying Macdonald identities for untwisted affine Lie algebras to the index integrals, producing dual representations as $q$-series and eta-quotients and revealing a deep link to affine-Grassmannian combinatorics and $N$-core-like structures. A key result is the exact evaluation of both the full index and the half-Schur index with Neumann boundary conditions, plus a bilinear decomposition of the full Schur index into half-indices with Wilson lines, with rich interpretations in holomorphic twists, class-$\mathcal{S}$ TQFT, BPS spectra, and Schur quantization. The work also clarifies the connections between UV gauge theory data and IR BPS data, offering a unifying framework that ties representation theory, modular forms, and geometric structures like the affine Grassmannian to protected QFT observables, and suggests several avenues for extension to twisted algebras, integrable systems, and holography.

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.