A quadratic BSDE approach to normalization for the finite volume 2D sine-Gordon model in the finite ultraviolet regime
Shanjian Tang, Rundong Xu
TL;DR
The paper develops a novel probabilistic construction of the two-dimensional sine-Gordon measure on bounded domains in the finite ultraviolet regime by solving a family of quadratic BSDEs driven by a cylindrical Wiener process with generators of purely quadratic growth in $Z$. The terminal data are Wick-ordered cosines of a Gaussian free field and converge, in the UV limit, to the real part of imaginary multiplicative chaos tested against a test function, enabling a BSDE-based description of the sine-Gordon measure as absolutely continuous with respect to the Gaussian free field via a Radon-Nikodym derivative $\Gamma(ρ)$. A key result is that the UV limit of the approximate measures yields a limiting sine-Gordon measure $\mu_{SG}^{ρ}$ with $\mathbb{E}_{GFF}[\Gamma(ρ)]=1$, and the limit is governed by a limit BSDE with terminal $\langle \cos(βW_1), ρ \rangle$. The framework yields a probabilistic link between the sine-Gordon construction and the scaling limits of the critical planar XOR-Ising model and of two-dimensional log-gases, through renormalized partition functions and the associated forward–backward SDE structure. The results extend the toolkit for constructive quantum field theory in low dimensions by providing a flexible UV-regularization approach that can accommodate bounded domains and connects to canonical objects like complex multiplicative chaos and GFF-based measures. The paper also outlines potential extensions to larger coupling regimes and highlights the role of Wick renormalization as the natural normalization in the subcritical UV regime $β^2\in[0,2)$.
Abstract
This paper is devoted to a new construction of the two-dimensional sine-Gordon model on bounded domains by a novel normalization technique in the finite ultraviolet regime. Our methodology involves a family of backward stochastic differential equations (BSDEs for short) driven by a cylindrical Wiener process, whose generators are purely quadratic functions of the second unknown variable. The terminal conditions of the quadratic BSDEs are uniformly bounded and converge in probability to the real part of imaginary multiplicative chaos tested against an arbitrarily given test function, which helps us describe our sine-Gordon measure through some delicate estimates concerning bounded mean oscillation martingales. As the ultraviolet cutoffs are vanishing, the quadratic BSDEs converge to a quadratic BSDE that completely characterizes the absolute continuity of our sine-Gordon measure with respect to the law of Gaussian free fields. Our approach can also be used effectively to establish the connection between our sine-Gordon measure and the scaling limit of correlation functions of the critical planar XOR-Ising model and to prove the weak convergence of the normalized charge distributions of two-dimensional log-gases.