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.
