# Wick squares of the Gaussian Free Field and Riemannian rigidity

February 19, 2019

In the present paper,
we show that on a compact Riemannian manifold $(M,g)$ of dimension $d\leqslant 4$ whose metric has negative curvature,
the renormalized partition function
$Z_g(\lambda)$ of a massive Gaussian Free Field
determines the length 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
bounded from below and above and whose isometry group is trivial,
there is only a \textbf{finite number of isometry classes} of metrics with given partition function $Z_g(\lambda)$.
When $d<4$, the same result holds true if the random variable
$\int_M:\phi^2:dv$
has given probability distribution and without the lower bound on the sectional curvatures.

Keywords:

*none*