IAS Quantum Aspects of Black Holes Group Meeting

Abstract: Is it possible to read off the quantum gravity dual of a CFT directly from its operator algebra? In this talk, I will present a step-by-step recipe for building up 3D quantum spacetimes dual to 2D CFTs, synthesizing results and techniques from conformal bootstrap, topological symmetries, real-space tensor network renormalization, and the asymptotics of quantum 6j symbols. Quantum Liouville theory serves as a simple and explicit example, illustrating how the exact CFT/gravitational path integral can be built up from triangulation and local pieces of BCFT correlation functions, which we call the ``BCFT Legos''. This is essentially a tensor network that admits an interpretation of a state-sum of a 3D topological theory constructed with quantum 6j symbols of Uq(SL(2,R)) with non-trivial boundary conditions, and it reduces to a sum over 3D geometries weighted by the Einstein-Hilbert action to leading order in large c. The triangulation coincides with producing a network of geodesics in the AdS bulk, which can be changed making use of the pentagon identity and orthogonality condition satisfied by the 6j symbols, and arranged into a precise holographic tensor network. The constructive map between gravity and CFT naturally and explicitly bridges local geometrical data, algebraic structures, and quantum entanglement. I will also briefly discuss how this approach can be used to study black hole physics, specifically by considering 3D BTZ black holes as Maldacena-Maoz wormholes to identify the Hartle-Hawking state using the Liouville ZZ boundary state, and elucidate the connection to 2D JT gravity.