Liouville Quantum Gravity on the complex tori

TL;DR

This paper constructs Liouville Quantum Field Theory on complex tori, explores its modular properties, and connects it to random planar maps of genus one, extending previous work on simpler geometries.

Contribution

It extends the construction of Liouville Quantum Field Theory to complex tori, analyzing modular properties and relating it to random planar maps of genus one.

Findings

Derived the law of the random Liouville modulus.

Extended formulae for LQFT on tori beyond previous physicists' results.

Made conjectures linking LQFT on tori to conformal field theories and random maps.

Abstract

In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov. Our approach follows the construction carried out by the authors together with A. Kupiainen in the case of the Riemann sphere. The difference is here that the moduli space for complex tori is non trivial. Modular properties of LQFT are thus investigated. This allows us to sum up the LQFT on complex tori over the moduli space, to compute the law of the random Liouville modulus, therefore recovering (and extending) formulae obtained by physicists, and make conjectures about the relationship with random planar maps of genus one, eventually weighted by a conformal field theory and conformally embedded onto the torus.