Wick squares of the Gaussian Free Field and Riemannian rigidity

TL;DR

This paper demonstrates that the renormalized partition function of a Gaussian Free Field on negatively curved compact Riemannian manifolds uniquely determines the length spectrum and constrains the geometric structure, especially in dimensions up to four.

Contribution

It establishes a link between the Gaussian Free Field's partition function and the geometric and spectral properties of negatively curved manifolds, including finiteness of isometry classes.

Findings

Partition function determines the length spectrum.

Finiteness of isometry classes with given partition function.

Results extend to certain cases in dimensions less than four.

Abstract

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.