Decomposition of BPS Moduli Spaces and Asymptotics of Supersymmetric Partition Functions

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.

arXiv comment56 pages, 2 figures. A short presentation of Section 2 is available at http://scgp.stonybrook.edu/video_portal/video.php?id=4953 v5: minor (sign) changes

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• Supersymmetric phases of AdS${}_4$/CFT${}_3$
• Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• 3d SUSY enhancement and non-semisimple TQFTs from four dimensions
• Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• S-duality in the Cardy-like limit of the superconformal index
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Large $N$ limits of supersymmetric quantum field theories: A pedagogical overview