The grin of Cheshire cat resurgence from supersymmetric localization

First we compute the \mbox{S}^2 S 2 partition function of the supersymmetric \mathbb{CP}^{N-1} ℂ ℙ 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 ℂ ℙ 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. The perturbative expansion of the deformed model becomes asymptotic and we analyse it within the framework of resurgence theory. Although the perturbative series truncates when we send the deformation parameter to zero we can still reconstruct non-perturbative physics out of the perturbative data in a nice example of Cheshire cat resurgence in quantum field theory. We also show that the same type of resurgence takes place when we consider an analytic continuation in the number of chiral fields from N N to r\in\mathbb{R} r ∈ ℝ . Although for generic real r r supersymmetry is still formally preserved, we find that the perturbative expansion of the supersymmetric partition function becomes asymptotic so that we can use resurgent analysis and only at the end take the limit of integer r r to recover the undeformed model.