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Gaussian free fields and Riemannian rigidity

Nguyen Viet Dang

Vol. 3 (2022), No. 1, 1–34

On a compact Riemannian manifold (M,g) of dimension d 4, we present a rigorous construction of the renormalized partition function Zg(λ) of a massive Gaussian free field where we explicitly determine the local counterterms using microlocal methods. Then we show that Zg(λ) 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).

renormalization, Gaussian free field, quantum field theory, geodesic flows, Wick squares
Mathematical Subject Classification
Primary: 11M36, 58J50, 60G15, 81T16, 81T20
Secondary: 58J40
Received: 6 April 2020
Revised: 29 June 2021
Accepted: 5 October 2021
Published: 11 May 2022
Nguyen Viet Dang
Institut de Mathématiques de Jussieu, UMR 7586 du CNRS
Sorbonne Université