Decomposition of BPS Moduli Spaces and Asymptotics of Supersymmetric Partition Functions

TL;DR

This paper develops a Wilsonian framework for analyzing the asymptotics of supersymmetric partition functions in non-abelian gauge theories, with a focus on the Cardy-like limit of the 4d superconformal index relevant to black hole microstate counting.

Contribution

It introduces a novel scheme to decompose BPS moduli spaces and localize contributions, providing the most general asymptotic expression for the superconformal index in the Cardy-like limit.

Findings

Derived a scheme-independent asymptotic formula for the superconformal index.

Extended previous results to a more general class of gauge theories.

Applied the framework to black hole microstate counting in AdS$_5$/CFT$_4$.

Abstract

We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by $i)$ decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off $\Lambda$) they support, $ii)$ localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and $iii)$ summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as $\Lambda\to\infty$). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS$_5$/CFT$_4$. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.