The grin of Cheshire cat resurgence from supersymmetric localization

Understand

First we compute the $\mbox{S}^2$ partition function of the supersymmetric $\mathbb{CP}^{N-1}$ model via localization and as a check we show that the chiral ring structure can be correctly reproduced.

• For the $\mathbb{CP}^1$ case we provide a concrete realisation of this ring in terms of Bessel functions.
• We consider a weak coupling expansion in each topological sector and write it as a finite number of perturbative corrections plus an infinite series of instanton-anti-instanton contributions.
• To be able to apply resurgent analysis we then consider a non-supersymmetric deformation of the localized model by introducing a small unbalance between the number of bosons and fermions.