# Conformal Bootstrap in dS/CFT and Topological Quantum Gravity

## TL;DR

This paper extends the holographic correspondence to de Sitter space, revealing new symmetries, instanton solutions, and connections to black hole entropy and topological quantum gravity, enriching the understanding of quantum gravity in dS/CFT.

## Contribution

It introduces Kac-Moodions as new instanton solutions in dS3, links R-matrices to crossing kernels and 6j-symbols, and establishes novel correspondences between instantons, black hole entropy, and topological quantum gravity.

## Key findings

- Identification of Kac-Moodions as new instantons in Euclidean dS3
- Connection of R-matrix to SU(1,1) 6j-symbols and crossing kernels
- Correlation between instantons and black hole horizon punctures

## Abstract

We show that the correspondence among $AdS_{3}/CFT_{2}$, the 1D Schwarzian Model, Sachdev-Ye-Kitaev model and 2+1D Topological Quantum Gravity can be extended to the case of $dS_{3}/CFT_{2}$. The $R$-matrix, related to the gravitational scattering amplitude near the horizon of $dS_{3}$ black hole, corresponds (on the side of the holographic projection) to a crossing kernel in the Schwarzian Model. The $R$-matrix is related to the 6j-symbol of SU$(1,1)$. We also find that in the Euclidean $dS_{3}$ a new Kac-Moody symmetry of instantons emerges out. We dub these new solutions {\it Kac-Moodions}. A one-to-one correspondence of Kac-Moodion levels and SU$(2)$ spin representations is established. Every instanton then corresponds to spin representations deployed in Topological Quantum Gravity. The instantons are directly connected to the Black Hole entropy, as punctures on its horizon. This strongly supports the recent proposal, in arXiv:1707.00347, that a Kac-Moody symmetry of gravitational instantons is related to the black hole information processing. We also comment on a further correspondence that can be established between the Schwarzian Model and non-commutative spacetimes in 2+1D, passing through the equivalence with Topological Quantum Gravity with cosmological constant, in the limit when the latter vanishes.