Stochastic quantization of Liouville conformal field theory

Abstract

We study a nonlinear stochastic heat equation forced by a space-time white noise on closed surfaces, with nonlinearity $e^{\beta u}$. This equation corresponds to the stochastic quantization of the Liouville quantum gravity (LQG) measure. (i) We first revisit the construction of the LQG measure in Liouville conformal field theory (LCFT) in the $L^2$ regime $0<\beta <\sqrt{2}$. This uniformizes in this regime the approaches of David-Kupiainen-Rhodes-Vargas (2016), David-Rhodes-Vargas (2016) and Guillarmou-Rhodes-Vargas (2019) which treated the case of a closed surface with genus 0, 1 and $>1$ respectively. Moreover, our argument shows that this measure is independent of the approximation procedure for a large class of smooth approximations. (ii) We prove almost sure global well-posedness of the parabolic stochastic dynamics, and invariance of the measure under this stochastic flow. In particular, our results improve previous results obtained by Garban (2020) in the cases of the sphere and the torus with their canonical metric, and are new in the case of closed surfaces with higher genus.

arXiv comment77 pages, minor corrections

References in corpus (8)

• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
• Liouville Quantum Gravity on the complex tori
• Critical Liouville measure as a limit of subcritical measures
• Stochastic quantization associated with the $\exp (\Phi {)}_2$-quantum field model driven by space-time white noise on the torus
• Random Morse functions and spectral geometry
• Stochastic Ricci Flow on Compact Surfaces
• On the parabolic and hyperbolic Liouville equations

Cited by in corpus (2)

• Strichartz inequalities with white noise potential on compact surfaces
• Stochastic quantization associated with the $\exp (\Phi {)}_2$-quantum field model driven by space-time white noise on the torus in the full $L^1$-regime