Polyakov's formulation of $2d$ bosonic string theory

TL;DR

This paper rigorously defines and analyzes Polyakov's 2D bosonic string theory using probabilistic methods, establishing convergence results and connecting to Liouville quantum gravity and random planar maps.

Contribution

It provides the first mathematical proof of convergence for string theory partition functions and introduces a non-perturbative probabilistic framework for 2D bosonic strings.

Findings

Proves convergence of Polyakov's partition function for genus ≥ 2 surfaces with D ≤ 1 bosons.

Defines Liouville quantum field theory on Riemann surfaces probabilistically.

Suggests a connection between string theory scaling limits and random planar maps.

Abstract

Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $\mathbf{g}\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic string theory \cite{Pol} (also called $2d$ bosonic string theory) and to Liouville quantum gravity. Then we show the convergence of Polyakov's partition function over the moduli space of Riemann surfaces in genus $\mathbf{g}\geq 2$ in the case of $D\leq 1$ boson. This is done by performing a careful analysis of the behaviour of the partition function at the boundary of moduli space. An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by Conformal Field Theories: we make precise conjectures about this statement at the end of the paper.