High-temperature asymptotics of supersymmetric partition functions

Abstract

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.

Cardy formula for 4d SUSY theories and localization

“…The possible divergences manifest in the limit β → 0 through a modification of the asymptotic formula (1.1). This was first observed in [1].…”

“…If the matter content is symmetric under ρ f ↔ −ρ f , the potential reduces to the single term proportional to the density L M 3 . In this case, the analysis of [1] shows that the limit β → 0 is dominated by the minima of V eff M 3 (a). If V eff M 3 (a) has a local minimum in the origin, where it vanishes, then the only contribution in (1.2) comes from the prefactor, and (1.1) is valid.…”

“…, r = rank(G). V eff S 3 (a) was computed in [1] starting from the integral representation of the partition function on S 1 × S 3 . We will show an alternative way to compute the effective potential, that does not rely on the knowledge of the matrix model, and readily generalizes to an arbitrary supersymmetry-preserving M 3 .…”

“…1 The derivation in [28] used the existence of a certain mixed Chern-Simons term of order 1 β in the effective action on M3 in the limit β → 0. The existence of this term has been recently related to global anomalies in [29].…”

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

“…As noticed first in [24], in this case the indices reduce to partition functions of three-dimensional field theories up to a diverging exponential, which was shown in [40] to be related to anomaly coefficients in the combination a − c. The results of [21] were applied also in the context of formal holomorphic block factorisations of 4d superconformal indices [41] with the corresponding β → 0 3d-reduction. A more detailed consideration of this limit for several different theories was given in [42]. Analytically, the limit p, q → 1 reduces the elliptic hypergeometric integrals to the hyperbolic integrals.…”