Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere
Dongmin Gang, Masahito Yamazaki
Abstract
A bstract We study a perturbative expansion of the squashed 3-sphere $$ \left({s}_b^3\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:msubsup> <mml:mi>s</mml:mi> <mml:mi>b</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mfenced> </mml:math> partition function of 3d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.