Electronic Journal of Probability

Renormalizability of Liouville quantum field theory at the Seiberg bound

François David, Antti Kupiainen, Rémi Rhodes, and Vincent Vargas

Full-text: Open access

Abstract

Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics $e^{\phi (z)}|dz|^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $\phi $ in LQFT depends on weights $\alpha \in \mathbb{R} $ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: $\alpha <Q$ where $Q$ parametrizes the random surface model in question. These correspond to studying uniformized surfaces with conical singularities in the classical geometrical setup. An interesting limiting case in classical geometry are the cusp singularities. In the random setup this corresponds to the case when the Seiberg bound is saturated. In this paper, we construct LQFT in the case when the Seiberg bound is saturated which can be seen as the probabilistic version of Riemann surfaces with cusp singularities. The construction involves methods from Gaussian Multiplicative Chaos theory at criticality.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 93, 26 pp.

Dates
Received: 28 November 2016
Accepted: 25 September 2017
First available in Project Euclid: 1 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1509501716

Digital Object Identifier
doi:10.1214/17-EJP113

Mathematical Reviews number (MathSciNet)
MR3724561

Zentralblatt MATH identifier
06827070

Subjects
Primary: 81T40: Two-dimensional field theories, conformal field theories, etc. 81T20: Quantum field theory on curved space backgrounds 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Liouville quantum field theory Gaussian multiplicative chaos KPZ formula Polyakov formula punctures cusp singularity uniformization theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent. Renormalizability of Liouville quantum field theory at the Seiberg bound. Electron. J. Probab. 22 (2017), paper no. 93, 26 pp. doi:10.1214/17-EJP113. https://projecteuclid.org/euclid.ejp/1509501716


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