Liouville Quantum Gravity and Polyakov's formulation of $2d$ string theory in genus $g\geq 2$

We define the partition function of Liouville quantum field theory (LQFT) on surfaces of genus ${\bf g}\geq 2$ by using the theory of Gaussian multiplicative chaos. Such a construction produces a conformal field theory with central charge ${\bf c}\geq 25$, which depends on the conformal class of the background metric put on the surface. This construction is used to define the Liouville quantum gravity (LQG) partition function by coupling LQFT with a matter field theory and by integrating over the space of conformal structures, i.e moduli space. We analyze the behaviour of the LQFT partition function at the boundary of the compactification of moduli space and we show integrability of the LQG partition function over moduli space when ${\bf c}>25$. For ${\bf c}=25$ (the critical value of the theory), the analysis becomes particularly interesting as it gives a mathematical sense to Polyakov's partition function for (bosonic) $2d$ string theory on hyperbolic surfaces. In this case, we prove convergence of the integral over moduli space for genus ${\bf g}=2$