Cardy Formula for 4d SUSY Theories and Localization

Abstract

We study 4d $\mathcal{N}=1$ supersymmetric theories on a compact Euclidean manifold of the form $S^1\times {\mathcal{M}}_3$. Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold ${\mathcal{M}}_3$. Taking the limit of shrinking $S^1$, we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of arXiv:1512.03376. The limit is given in terms of an effective potential for the holonomies around the $S^1$, whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardy-like behavior fixed by $\mathrm{Tr}(R)$, while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if $\mathrm{Tr}(R)\le 0$.

43 pages, v2: references added, changed slightly discussion in section 5

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• Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae
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• Explorations of Universality in the Entropy and Hawking Radiation of Non-Extremal Kerr AdS${}_4$ Black Holes