High-temperature asymptotics of supersymmetric partition functions

Abstract

We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean $S^3\times S_{\beta }^1$, with $S^3$ the unit-radius squashed three-sphere, and $\beta$ the circumference of the circle. For superconformal theories, this partition function coincides (up to a Casimir energy factor) with the 4d superconformal index. The partition function can be computed exactly using supersymmetric localization of the gauge theory path-integral. It takes the form of an elliptic hypergeometric integral, which may be viewed as a matrix-integral over the moduli space of the holonomies of the gauge fields around $S_{\beta }^1$. At high temperatures ($\beta \to 0$, corresponding to the hyperbolic limit of the elliptic hypergeometric integral) we obtain from the matrix-integral a quantum effective potential for the holonomies. The effective potential is proportional to the temperature. Therefore the high-temperature limit further localizes the matrix-integral to the locus of the minima of the potential. If the effective potential is positive semi-definite, the leading high-temperature asymptotics of the partition function is given by the formula of Di Pietro and Komargodski, and the subleading asymptotics is connected to the Coulomb branch dynamics on $R^3\times S^1$. In theories where the effective potential is not positive semi-definite, the Di Pietro-Komargodski formula needs to be modified. In particular, this modification occurs in the SU(2) theory of Intriligator-Seiberg-Shenker, and the SO(N) theory of Brodie-Cho-Intriligator, both believed to exhibit "misleading" anomaly matchings, and both believed to yield interacting superconformal field theories with $c<a$. Two new simple tests for dualities between 4d supersymmetric gauge theories emerge as byproducts of our analysis.

54+10 pages; 9 figures. v8: more typos corrected; analysis of the BCI model corrected thanks to input from Jonas Jagminas. Supersedes published version

References in corpus (13)

• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Cardy Formulae for SUSY Theories in d=4 and d=6
• Localization on Hopf surfaces
• Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT
• Supersymmetric states of N=4 Yang-Mills from giant gravitons
• (In)validity of large N orientifold equivalence
• On the Superconformal Index of N=1 IR Fixed Points: A Holographic Check
• High-temperature asymptotics of supersymmetric partition functions
• Chiral gauge dynamics and dynamical supersymmetry breaking
• On the ISS model of dynamical SUSY breaking
• $c-a$ from the $N=1$ superconformal index
• Central charges from the $\mathcal{N}=1$ superconformal index

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