Wick squares of the Gaussian Free Field and Riemannian rigidity

Nguyen Viet Dang (1)

1 PSPM - Probabilités, statistique, physique mathématique

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 Zg(λ) of a massive Gaussian Free Field or the fluctuations of the integral of the Wick square M : φ 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 Zg(λ) or such that the random variable M : φ 2 : dv has given law.

Domaines: Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]; Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Nguyen Viet Dang. Wick squares of the Gaussian Free Field and Riemannian rigidity. 2019. ⟨hal-02068497⟩