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

Abstract

The Schur index of a 4 dimensional $$ \mathcal{N} $$ N = 2 superconformal field theory counts (with sign) bosonic and fermionic states that preserve 4 supercharges. We consider the Schur indices of 4d $$ \mathcal{N} $$ N = 4 super Yang-Mills and $$ \mathcal{N} $$ 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” ($$ \mathcal{O} $$ O (N0) as N → ∞) exponential growth, which is much smaller than an $$ \mathcal{O} $$ O (N2) 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.