Resurgence in $\eta$-deformed Principal Chiral Models

TL;DR

This paper investigates the effects of an integrable $ ext{eta}$-deformation on the $SU(2)$ Principal Chiral Model, revealing modified fracton events, novel saddles, and connections to supersymmetric gauge theories through resurgent analysis.

Contribution

It introduces a resurgent analysis of the $ ext{eta}$-deformed PCM, identifying new fracton behaviors, $SL(2, ext{C})$ saddles, and links to complexified path integrals and supersymmetric gauge theories.

Findings

Modified fractons due to $ ext{eta}$-deformation

Discovery of $SL(2, ext{C})$ saddle points

Connection to massive $ ext{N}=2$ $SU(2)$ gauge theory with $N_f=2$

Abstract

We study the $SU(2)$ Principal Chiral Model (PCM) in the presence of an integrable $\eta$-deformation. We put the theory on $\mathbb{R}\times S^1$ with twisted boundary conditions and then reduce the circle to obtain an effective quantum mechanics associated with the Whittaker-Hill equation. Using resurgent analysis we study the large order behaviour of perturbation theory and recover the fracton events responsible for IR renormalons. The fractons are modified from the standard PCM due to the presence of this $\eta$-deformation but they are still the constituents of uniton-like solutions in the deformed quantum field theory. We also find novel $SL(2,\mathbb{C})$ saddles, thus strengthening the conjecture that the semi-classical expansion of the path integral gives rise to a resurgent transseries once written as a sum over Lefschetz thimbles living in a complexification of the field space. We conclude by connecting our quantum mechanics to a massive deformation of the $\mathcal{N}=2~$ $4$-d gauge theory with gauge group $SU(2)$ and $N_f=2$.