Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions

TL;DR

This paper investigates sub-leading structures in the superconformal index of ${ m N}=4$ SYM, revealing new saddle points, logarithmic corrections, and their universality, which enhances understanding of AdS$_5$ black holes.

Contribution

It introduces a systematic analysis of sub-leading saddle points and logarithmic corrections in superconformal indices, extending methods to a broad class of ${ m N}=1$ theories.

Findings

Identifies subdominant saddles related to SU(N) Chern-Simons theory.

Determines the logarithmic correction as $ ext{log} N$ with exact agreement between approaches.

Shows universality of the $ ext{log} N$ correction across different superconformal theories.

Abstract

We systematically study various sub-leading structures in the superconformal index of ${\cal N}=4$ supersymmetric Yang-Mills theory with SU($N$) gauge group. We concentrate in the superconformal index description as a matrix model of elliptic gamma functions and in the Bethe-Ansatz presentation. Our saddle-point approximation goes beyond the Cardy-like limit and we uncover various saddles governed by a matrix model corresponding to SU($N$) Chern-Simons theory. The dominant saddle, however, leads to perfect agreement with the Bethe-Ansatz approach. We also determine the logarithmic correction to the superconformal index to be $\log N$, finding precise agreement between the saddle-point and Bethe-Ansatz approaches in their respective approximations. We generalize the two approaches to cover a large class of 4d ${\cal N}=1$ superconformal theories. We find that also in this case both approximations agree all the way down to a universal contribution of the form $\log N$. The universality of this last result constitutes a robust signature of this ultraviolet description of asymptotically AdS$_5$ black holes and could be tested by low-energy IIB supergravity.