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

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.