# Wick squares of the Gaussian Free Field and Riemannian rigidity

February 19, 2019

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.

open access link
article file

@article{Dang:2019wgg,
author = "Dang, Nguyen Viet",
title = "{Wick squares of the Gaussian Free Field and Riemannian
rigidity}",
year = "2019",
eprint = "1902.07315",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1902.07315;%%"
}

Keywords:

*none*