High-temperature asymptotics of supersymmetric partition functions

TL;DR

The paper develops a comprehensive framework for the high-temperature (β→0) asymptotics of four-dimensional supersymmetric partition functions Z^{SUSY}(b,β) for theories with U(1)_R symmetry, using SUSY localization to rewrite Z^{SUSY} as an elliptic-hypergeometric matrix integral. A temperature-proportional quantum effective potential V^{eff} emerges for the holonomies, localizing the integral to the minima of the Rains function L_h and to the quantum Coulomb branch h_{qu} when flat directions exist; the leading DK formula is recovered when L_h is minimized at the origin and L_h_min≥0, while negative minima trigger modified asymptotics and subleading Coulomb-branch contributions. The work provides two practical duality tests—comparing L_h_min and the dimension of h_{qu} across dual theories—applied to a variety of Lagrangian and non-Lagrangian models, including A_k and SO(2N+1) theories, orbifolds, N=4 SYM, and the E_6 SCFT. It also relates high-T behavior to 3d reductions, Schur limits, and holographic contexts, offering a unified picture of when the 4d index encodes universal Cardy-like data and when new phenomena arise from unlifted Coulomb branches.

Abstract

We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean $S^3\times S_β^1$, 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 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 $S_β^1$. At high temperatures ($β\to 0$, 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.

Figures (9)

• Figure 1: The even function $\vartheta(x)$ (on the left) and the odd function $\kappa(x)$ (on the right). Both are continuous and periodic, and their fundamental domain can be taken to be $[-1/2,1/2]$.
• Figure 2: The Rains function of the $A_1$ SU($3$) theory---also known as SU($3$) SQCD---for $N_f>3$. Note that the minimum lies at $x_1=x_2=0$.
• Figure 3: The Rains function of the Sp($4$) SQCD theory for $N_f>3$. Note that the minimum lies at $x_1=x_2=0$.
• Figure 4: The Rains function of the SU($3$) magnetic Pouliot theory with $N_f=7$. Note that the minimum lies at $x_1=x_2=0$.
• Figure 5: The $Q_h$ function of the SU($3$) magnetic Pouliot theory with $N_f=7$. Note the stationarity at $x_1=x_2=0$.