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The grin of Cheshire cat resurgence from supersymmetric localization
by Daniele Dorigoni, Philip Glass
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Submission summary
Authors (as registered SciPost users):  Daniele Dorigoni 
Submission information  

Preprint Link:  http://arxiv.org/abs/1711.04802v3 (pdf) 
Date accepted:  20180118 
Date submitted:  20180111 01:00 
Submitted by:  Dorigoni, Daniele 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
First we compute the $\mbox{S}^2$ partition function of the supersymmetric $\mathbb{CP}^{N1}$ 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 instantonantiinstanton contributions. To be able to apply resurgent analysis we then consider a nonsupersymmetric 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 nonperturbative 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 $N$ to $r\in\mathbb{R}$. Although for generic real $r$ 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 $r$ to recover the undeformed model.
Published as SciPost Phys. 4, 012 (2018)
Author comments upon resubmission
List of changes
We have addressed all the minor typos pointed out by the referee.
On page 10, equation (2.16) we added the factor $e^{4 \pi \xi} $ multiplying the second term
Second line on page 26 we added the missing "."
In the second paragraph of page 26, we corrected the reference from eq. (6.8) to eq. (5.1)