Cardy Formula for 4d SUSY Theories and Localization

TL;DR

This work develops a general effective-theory framework for the small-$\beta$ (Cardy-like) limit of 4d $\mathcal{N}=1$ partition functions on $S^1\times \mathcal{M}_3$, deriving a holonomy potential $V^{\rm eff}_{\mathcal{M}_3}(a)$ from KK fermion loops and their SUSY completion. By matching to localization results on $S^3_b$, lens spaces, and torus fibrations, the authors show that the leading behavior is governed by $\mathrm{Tr}(R)$ when $V^{\rm eff}_{\mathcal{M}_3}$ is minimized at the origin, while nontrivial minima introduce corrections via $V^{\rm eff}_{\mathcal{M}_3}(a_{\min})$ and possible exponential growth of the 3d integrand. They provide explicit expressions for $V^{\rm eff}_{\mathcal{M}_3}(a)$ in several geometries and demonstrate how convergence and large-$a$ behavior of the localization integrals reflect the minima structure. Across a wide class of theories, a consistent link emerges between the sign of $\mathrm{Tr}(R)$ and the presence of a local minimum at the origin, with adjoint and fundamental matter illustrating the pattern and higher representations yielding nonzero minima when $\mathrm{Tr}(R)>0$. The results deepen the understanding of Cardy-like limits in 4d SUSY theories and connect holonomy dynamics, localization, and R-symmetry anomalies in a unified framework.

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 $\mathrm{Tr}(R) \leq 0$.