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

Understand

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.