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2d Sinh-Gordon model on the infinite cylinder

Colin Guillarmou, Trishen S. Gunaratnam, Vincent Vargas

TL;DR

The paper provides a rigorous probabilistic construction of the massless Sinh-Gordon model on the infinite cylinder by marrying Gaussian Free Field dynamics with Gaussian multiplicative chaos. It defines a self-adjoint Sinh-Gordon Hamiltonian ${\\bf H}$ with discrete spectrum and a positive ground state, and reduces general $R$-dependent constructions to the $R=1$ case via scaling. The main results include a rigorous path integral measure on $C(\mathbb{R}, H^{-s}(\mathbb{T}_R))$, the existence and scaling of vertex correlations, and exponential decay of two-point vertex functions governed by the mass gap $(\\lambda_1-\\lambda_0)/R$. The work advances constructive QFT in 2d by providing explicit PMC-driven potentials and spectral data, with detailed open problems about the full spectrum and the $R\to\infty$ limit, and links to known CFT/Lukyanov–Zamolodchikov results in the massless regime.

Abstract

For $R>0$, we give a rigorous probabilistic construction on the cylinder $\mathbb{R} \times (\mathbb{R}/(2πR\mathbb{Z}))$ 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.

2d Sinh-Gordon model on the infinite cylinder

TL;DR

The paper provides a rigorous probabilistic construction of the massless Sinh-Gordon model on the infinite cylinder by marrying Gaussian Free Field dynamics with Gaussian multiplicative chaos. It defines a self-adjoint Sinh-Gordon Hamiltonian with discrete spectrum and a positive ground state, and reduces general -dependent constructions to the case via scaling. The main results include a rigorous path integral measure on , the existence and scaling of vertex correlations, and exponential decay of two-point vertex functions governed by the mass gap . The work advances constructive QFT in 2d by providing explicit PMC-driven potentials and spectral data, with detailed open problems about the full spectrum and the limit, and links to known CFT/Lukyanov–Zamolodchikov results in the massless regime.

Abstract

For , we give a rigorous probabilistic construction on the cylinder of the (massless) Sinh-Gordon model. In particular we define the -point correlation functions of the model and show that these exhibit a scaling relation with respect to . 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.
Paper Structure (26 sections, 18 theorems, 221 equations)

This paper contains 26 sections, 18 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. The Sinh-Gordon model
  3. Main result on the Sinh-Gordon Hamiltonian
  4. Main results on the Sinh-Gordon path integral
  5. Perspectives and open problems
  6. On the spectrum of the theory
  7. The infinite volume limit $R \to \infty$
  8. Gaussian free field and Gaussian multiplicative chaos
  9. Path integral and the GFF on $\mathcal{C}_R$
  10. Markov process associated to the GFF
  11. Gaussian multiplicative chaos on $\mathcal{C}_R$
  12. Connection with the GFF and GMC on $\mathbb{C}$
  13. The Sinh-Gordon Hamiltonian on the infinite cylinder
  14. Hilbert space and Free Hamiltonian
  15. The Sinh-Gordon Hamiltonian
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\mu > 0$ and $\gamma\in(0,2)$. The Hamiltonian ${\bf H}$ generating the semigroup ${\bf T}_t$ is a self-adjoint, positive operator acting on a dense domain $\mathcal{D}(\mathbf{H}) \subset \mathcal{H}$. Furthermore, the following properties hold:

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1: Free Hamiltonian
  • Lemma 3.2
  • proof
  • ...and 29 more