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 4d \( \mathcal{N} \) = 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” (\( \mathcal{O} \)(N0) as N → ∞) exponential growth, which is much smaller than an \( \mathcal{O} \)(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.
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Eleftheriou, G. Root of unity asymptotics for Schur indices of 4d Lagrangian theories. J. High Energ. Phys. 2023, 81 (2023). https://doi.org/10.1007/JHEP01(2023)081
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DOI: https://doi.org/10.1007/JHEP01(2023)081