Double-Scaling Limit in Principal Chiral Model: a New Non-Critical String?

TL;DR

This paper explores a non-perturbative large-N expansion in the 2D SU(N) Principal Chiral Model, revealing a double-scaling limit that suggests a duality with a non-critical string theory in higher dimensions.

Contribution

It introduces a systematic method to analyze the large-N expansion in PCM and uncovers a novel double-scaling limit akin to non-critical string theories.

Findings

Double-scaling limit resembles $c=1$ non-critical string theory

Emergence of an extra dimension from the SU(N) Dynkin diagram

Potential duality between PCM and a non-critical string in 3D

Abstract

We initiate a systematic, non-perturbative study of the large-$N$ expansion in the two-dimensional $\text{SU}(N)\times \text{SU}(N)$ Principal Chiral Model (PCM). Starting with the known infinite-$N$ solution for the ground state at fixed chemical potential \cite{Fateev:1994dp,Fateev:1994ai}, we devise an iterative procedure to solve the Bethe ansatz equations order by order in $1/N$. The first few orders, which we explicitly compute, reveal a systematic enhancement pattern at strong coupling calling for the near-threshold resummation of the large-$N$ expansion. The resulting double-scaling limit bears striking similarities to the $c=1$ non-critical string theory and suggests that the double-scaled PCM is dual to a non-critical string with a $(2+1)$-dimensional target space where an additional dimension emerges dynamically from the $\text{SU}(N)$ Dynkin diagram.