Finite cut-off JT and Liouville quantum gravities on the disk at one loop

Within the path integral formalism, we compute the disk partition functions of two dimensional Liouville and JT quantum gravity theories coupled to a matter CFT of central charge $c$, with cosmological constant $\Lambda$, in the limit $c\rightarrow -\infty$, $|\Lambda|\rightarrow\infty$, for fixed $\Lambda/c$ and fixed and finite disk boundary length $\ell$, to leading and first subleading order in the $1/|c|$ expansion. In the case of Liouville theory, we find perfect agreement with the asymptotic expansion of the known exact FZZT partition function. In the case of JT gravity, we obtain the first explicit results for the partition functions at finite cut-off, in the three versions (negative, zero and positive curvature) of the model. Our findings are in agreement with predictions from the recent proposal for a microscopic definition of JT gravity, including the $c\rightarrow -\infty$ expansion of the Hausdorff dimension of the boundary. In the negative curvature case, we also provide evidence for the emergence of an effective Schwarzian description at length scales much greater than the curvature length scale.