Abstract
We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean S 3 × S 1 β , with S 3 the unit-radius squashed three-sphere, and β 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 the 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 1 β . At high temperatures (β → 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 × 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.
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Ardehali, A.A. High-temperature asymptotics of supersymmetric partition functions. J. High Energ. Phys. 2016, 25 (2016). https://doi.org/10.1007/JHEP07(2016)025
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DOI: https://doi.org/10.1007/JHEP07(2016)025