Wick squares of the Gaussian Free Field and Riemannian rigidity

In this short note, we show that on a compact Riemannian manifold $(M,g)$ of dimension $(d=2,3)$ whose metric has negative curvature, the partition function $Z_g(\lambda)$ of a massive Gaussian Free Field or the fluctuations of the integral of the Wick square $\int_M:\phi^2:dv$ determine the lenght spectrum of $(M,g)$ and imposes some strong geometric constraints on the Riemannian structure of $(M,g)$. In any finite dimensional family of Riemannian metrics of negative sectional curvature, there is only a finite number of isometry classes of metrics with given partition function $Z_g(\lambda)$ or such that the random variable $\int_M:\phi^2:dv$ has given law.