High-temperature expansion of the Schur index and modularity

TL;DR

This paper extends the high-temperature analysis of 4d superconformal indices to include exponentially suppressed terms using RG-inspired methods, revealing patterns of logarithms and confirming modularity conjectures.

Contribution

It introduces a novel RG-inspired approach to analyze exponentially suppressed terms in Schur indices, explaining observed logarithmic patterns and proving rationality of VOA character dimensions.

Findings

Extended high-temperature expansion including exponential corrections.

Confirmed compatibility with modular linear differential equations.

Proved rationality of conformal dimensions of VOA characters.

Abstract

High-temperature ($q\to1$) asymptotics of 4d superconformal indices of Lagrangian theories have been recently analyzed up to exponentially suppressed corrections. Here we use RG-inspired tools to extend the analysis to the exponentially suppressed terms in the context of Schur indices of $N=2$ SCFTs. In particular, our approach explains the curious patterns of logarithms (polynomials in $1/\log q$) found by Dedushenko and Fluder in their numerical study of the high-temperature expansion of rank-$1$ theories. We also demonstrate compatibility of our results with the conjecture of Beem and Rastelli that Schur indices satisfy finite-order, possibly twisted, modular linear differential equations (MLDEs), and discuss the interplay between our approach and the MLDE approach to the high-temperature expansion. The expansions for $q$ near roots of unity are also treated. A byproduct of our analysis is a proof (for Lagrangian theories) of rationality of the conformal dimensions of all characters of the associated VOA, that mix with the Schur index under modular transformations.