2d Sinh-Gordon model on the infinite cylinder

TL;DR

This paper rigorously constructs the massless Sinh-Gordon model on an infinite cylinder using probabilistic methods, spectral analysis, and Gaussian multiplicative chaos, revealing key properties of correlation functions and the model's spectrum.

Contribution

It provides the first rigorous probabilistic construction of the massless Sinh-Gordon model on the cylinder, including correlation functions and spectral properties.

Findings

Correlation functions exhibit a scaling relation with respect to the cylinder radius R.

The associated quantum operator has a discrete spectrum.

The operator possesses a strictly positive ground state.

Abstract

For $R>0$, we give a rigorous probabilistic construction on the cylinder $\mathbb{R} \times (\mathbb{R}/(2\pi R\mathbb{Z}))$ of the (massless) Sinh-Gordon model. In particular we define the $n$-point correlation functions of the model and show that these exhibit a scaling relation with respect to $R$. The construction, which relies on the massless Gaussian Free Field, is based on the spectral analysis of a quantum operator associated to the model. Using the theory of Gaussian multiplicative chaos, we prove that this operator has discrete spectrum and a strictly positive ground state.