The Virasoro minimal string

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c ≥ 25 coupled to timelike Liouville CFT with central charge 26 − c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter c and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical — and in special cases, analytic — integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.