Expanding 3d N $$ \mathcal{N} $$ = 2 theories around the round sphere

Abstract We study a perturbative expansion of the squashed 3-sphere s b 3 $$ \left({s}_b^3\right) $$ partition function of 3d N $$ \mathcal{N} $$ = 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 → 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.