Rémi Rhodes: Polyakov’s formulation of 2d string 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 (also called 2d bosonic string theory). More specifically, we show the convergence of Polyakov's partition function over the moduli space of Riemann surfaces with fixed genus in the case of D boson with D less than 1. 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 the discrete Gaussian Free Field.