Abstract

On a compact Riemannian manifold $(M,g)$ of dimension $d\le 4$, we present a rigorous construction of the renormalized partition function $Z_g(\lambda )$ of a massive Gaussian free field where we explicitly determine the local counterterms using microlocal methods. Then we show that $Z_g(\lambda )$ determines the Laplace spectrum of $(M,g)$ and hence imposes some strong geometric constraints on the Riemannian structure of $(M,g)$. From this observation, using classical results in Riemannian geometry, we illustrate how the partition function allows us to probe the Riemannian structure of the underlying manifold $(M,g)$.

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