Expanding 3d $\mathcal{N}=2$ Theories around the Round Sphere

TL;DR

This paper introduces a perturbative expansion method for the 3d $ ext{N}=2$ gauge theories' partition function on squashed spheres, enabling efficient computation of conformal data without complex integrals.

Contribution

It proposes a novel saddle-point based expansion around the round sphere, simplifying the calculation of IR superconformal field theory data.

Findings

Provides a finite-sum saddle-point expansion for the partition function.

Enables practical computation of CFT data like $F$, $C_T$, and $C_{JJ}$.

Offers an efficient alternative to traditional localization integrals.

Abstract

We study a perturbative expansion of the squashed 3-sphere ($S^3_b$) partition function of 3d $\mathcal{N}=2$ gauge theories around the squashing parameter $b= 1$. Our proposal gives the coefficients of the perturbative expansion as a finite sum over the saddle points of the supersymmetric-localization integral in the limit $b \rightarrow 0$ (the so-called Bethe vacua), and the contribution from each Bethe vacua can be systematically computed using saddle-point methods. Our expansion provides an efficient and practical method for computing basic CFT data ($F,C_T,C_{JJ}$ and higher-point correlation functions of the stress-energy tensor) of the IR superconformal field theory without performing the localization integrals.