Talk 71 - Topological Toy Models for the Emergence of Spacetime.

Abstract

We study entanglement entropy in (2+1)-dimensional gravity as a window into larger open questions regarding entanglement entropy in gravity. (2+1)-dimensional gravity can be rewritten as a topological field theory, which makes it a more tractable model to study. In these topological theories, there remain key questions which we seek to answer in this work, such as the questions 1) What is the entropy of the physical algebra of observables in a subregion, 2) How do we define a factorization map such that the entropy of the resulting factors agrees with this algebraic entropy, and 3) Can we use these insights to build a tensor network that exhibits non-commuting areas? We investigate non-Abelian toric codes / Levin-Wen models as a toy model for black hole entropy in Chern Simons theory. These differ from the usual model in that the stabilizers are implemented as constraints. By enforcing constraints for both Gauss' Law and the flatness of the gauge field, we obtain a choice of algebra that contains only topological operators. The desirable properties of this model are twofold: first, we produce the finiteness of black hole entropy described in previous literature while providing a natural algebraic motivation for this result. Secondly, we obtain non-commuting area operators on a toy model with the topology of a torus.