Residues, modularity, and the Cardy limit of the 4d $\mathcal{N}=4$ superconformal index

TL;DR

This paper computes the superconformal index of 4d $ ext{SU}(N)$ $ ext{N}=4$ SYM using residue calculus, revealing universal contributions in the Cardy limit that match black hole entropy and exploring modularity beyond this limit.

Contribution

It introduces a residue-based formula for the superconformal index that explicitly includes all poles without special parameter choices, enabling detailed analysis of the Cardy limit and modular properties.

Findings

All residues contribute at leading order in the Cardy limit.

Universal residues match black hole entropy functions.

The formula facilitates studying modularity effects beyond the Cardy limit.

Abstract

We compute the superconformal index of the $\mathcal{N}=4$ $SU(N)$ Yang-Mills theory through a residue calculation. The method is similar in spirit to the Bethe Ansatz formalism, except that all poles are explicitly known, and we do not require specialization of any of the chemical potentials. Our expression for the index allows us to revisit the Cardy limit using modular properties of four-dimensional supersymmetric partition functions. We find that all residues contribute at leading order in the Cardy limit. In a specific region of flavour chemical potential space, close to the two unrefined points, in fact all residues contribute universally. These universal residues precisely agree with the entropy functions of the asymptotically AdS$_5$ black hole and its "twin saddle" respectively. Finally, we discuss how our formula is suited to study the implications of four-dimensional modularity for the index beyond the Cardy limit.