Scott Collier, Lorenz Eberhardt, Beatrix Muehlmann, Victor A. Rodriguez
SciPost Phys. 16, 057 (2024) ·
published 26 February 2024

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We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as twodimensional quantum gravity on the worldsheet, is equivalent to a doublescaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c$≥$ 25 coupled to timelike Liouville CFT with central charge 26c. The doublescaled matrix integral has as its leading density of states the universal Cardy density of primaries in a twodimensional 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 WeilPetersson 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 threedimensional 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 twodimensional spacetime, and the matching between the leading nonperturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.
SciPost Phys. 15, 151 (2023) ·
published 11 October 2023

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We propose a precise reformulation of 3d quantum gravity with negative cosmological constant in terms of a topological quantum field theory based on the quantization of the Teichmüller space of Riemann surfaces that we refer to as "Virasoro TQFT". This TQFT is similar, but importantly not equivalent, to SL(2, $\mathbb{R}$) ChernSimons theory. This sharpens the folklore that 3d gravity is related to SL(2, $\mathbb{R}$) ChernSimons theory into a precise correspondence, and resolves some wellknown issues with this lore at the quantum level. Our proposal is computationally very useful and provides a powerful tool for the further study of 3d gravity. In particular, we explain how together with standard TQFT surgery techniques this leads to a fully algorithmic procedure for the computation of the gravity partition function on a fixed topology exactly in the central charge. Mathematically, the relation leads to many nontrivial conjectures for hyperbolic 3manifolds, Virasoro conformal blocks and crossing kernels.
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