Quantum Sine(h)-Gordon Model and Classical Integrable Equations

TL;DR

This paper explores the relationship between classical solutions of a modified sinh-Gordon equation and quantum integrable models, revealing that certain connection coefficients match the quantum $Q$-function of sine-Gordon and sinh-Gordon models.

Contribution

It establishes a novel link between classical integrable equations and quantum $Q$-functions in sine-Gordon and sinh-Gordon models.

Findings

Connection coefficients match quantum $Q$-functions.

Classical solutions relate to quantum integrable structures.

Provides insights into classical-quantum correspondence in integrable systems.

Abstract

We study a family of classical solutions of modified sinh-Gordon equation, $\partial_z\partial_{{\bar z}} \eta-\re^{2\eta}+p(z)\,p({\bar z})\ \re^{-2\eta}=0$ with $p(z)=z^{2\alpha}-s^{2\alpha}$. We show that certain connection coefficients for solutions of the associated linear problem coincide with the $Q$-function of the quantum sine-Gordon $(\alpha>0)$ or sinh-Gordon $(\alpha<-1)$ models.