We introduce the dynamical sine-Gordon equation in two space dimensions
with parameter β, which is the natural dynamic associated to the usual quantum sine-
Gordon model. It is shown that when β2 ∈ (0, 16π ) the Wick renormalised equation 3
is well-posed. In the regime β2 ∈ (0,4π), the Da Prato–Debussche method [J Funct
Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for
β2 ∈ [4π, 16π ), the solution theory is provided via the theory of regularity structures 3
[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.