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

The Schur index of a 4 dimensional N = 2 superconformal field theory counts (with sign) bosonic and fermionic states that preserve 4 supercharges. We consider the Schur indices of 4d N = 4 super Yang-Mills and 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 (O(N-0) as N -> infinity) exponential growth, which is much smaller than an 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.

Automated summary

The Schur index is calculated for a 4-dimensional N = 2 superconformal field theory, counting the bosonic and fermionic states that preserve 4 supercharges. The Schur indices of 4d N = 4 super Yang-Mills and N = 2 circular quiver gauge theories are examined, with the gauge groups U(N) or SU(N). The exponentially dominant part of their asymptotic expansions is calculated as the index parameter q approaches any root of unity. The results show that the indices do not capture the growth of states corresponding to a supersymmetric black hole that preserves 4 supercharges.