Root of unity asymptotics for Schur indices of 4d Lagrangian theories

TL;DR

This paper analyzes the asymptotic behavior of Schur indices for 4d superconformal theories at roots of unity, revealing small exponential growth patterns that do not correspond to black hole states in holography.

Contribution

It provides the first detailed calculation of root of unity asymptotics for Schur indices in 4d $ ext{SCFTs}$, highlighting their limited growth and implications for holographic duals.

Findings

Some indices show small exponential growth, not matching black hole entropy.

The dominant asymptotic behavior depends on the parity of the gauge group rank.

Indices do not exhibit the large $ ext{O}(N^2)$ growth expected for black hole microstates.

Abstract

The Schur index of a $4$ dimensional $\mathcal{N}=2$ superconformal field theory counts (with sign) bosonic and fermionic states that preserve $4$ supercharges. We consider the Schur indices of $4$d $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=2$ circular quiver gauge theories with gauge groups $U(N)$ or $SU(N)$. We calculate the exponentially dominant part of their asymptotic expansions as the index parameter $q$ approaches any root of unity. We find that some of the indices exhibit ``small" ($\mathcal{O}(N^0)$ as $N \rightarrow \infty$) exponential growth, which is much smaller than an $\mathcal{O}(N^2)$ exponential growth of states that is indicative of a black hole. This implies that the indices do not capture a growth of states that would correspond to a supersymmetric black hole that preserves 4 supercharges in the holographic dual AdS theory. Interestingly, the exponentially dominant part in the Schur asymptotics we consider, depends on the parity of the rank $N$.