Genus Two Modular Bootstrap

We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.Comment: 23 pages, 6 figures. v2: updated references, typos corrected. v3-v5: minor typos correcte

Recursive Representations of Arbitrary Virasoro Conformal Blocks

We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.Comment: 39 pages, 8 figures, v2: comments on references added, reference added, typos corrected, v3: comments on the relation between the plumbing and the Schottky parameters added, v4: typos correcte

Solving 3d Gravity with Virasoro TQFT

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 Teichm\"uller space of Riemann surfaces that we refer to as ``Virasoro TQFT.'' This TQFT is similar, but importantly not equivalent, to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory. This sharpens the folklore that 3d gravity is related to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory into a precise correspondence, and resolves some well-known 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 3-manifolds, Virasoro conformal blocks and crossing kernels.Comment: 72 page