Quantum edge modes in 3d gravity and 2+1d topological phases of matter

TL;DR

This paper investigates the quantum structure of edge modes in 3D gravity, revealing dual statistical models and their integrability, with implications for topological phases of matter and quantum geometry.

Contribution

It introduces a dual pair of integrable statistical models describing edge modes in 3D quantum gravity and interprets them in terms of topological phases of matter.

Findings

Edge modes are encoded in dual vertex and face models.

Duality reflects a fundamental structure in classical and quantum theories.

Quantum analogue of Carlip's diffeomorphisms identified.

Abstract

We analyze the edge mode structure of Euclidean three dimensional gravity from within the quantum theory as embodied by a Ponzano-Regge-Turaev-Viro discrete state sum with Gibbons-=-Hawking-York boundary conditions. This structure is encoded in a pair of dual statistical models of the vertex and face kind, which for specific choices of boundary conditions turn out to be integrable. The duality is just the manifestation of a pervasive dual structure which manifests at different levels of the classical and quantum theories. Emphasis will be put on the geometrical interpretation of the edge modes which leads in particular to the identification of the quantum analogue of Carlip's would-be normal diffeomorphisms. We also provide a reinterpretation of our construction in terms of a non-Abelian 2+1 topological phase with electric boundary conditions.