The Dynamical Sine-Gordon Model

Abstract

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter ${\beta}$, which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when ${\beta^{2} \in (0, \frac{16\pi}{3})}$ the Wick renormalised equation is well-posed. In the regime ${\beta^{2} \in (0, 4\pi)}$, the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for ${\beta^{2} \in [4\pi, \frac{16\pi}{3})}$, the solution theory is provided via the theory of regularity structures [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of ${2 + 1}$ -dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al. [Commun Math Phys 337(2):569–632, 2015].