Lorentz-diffeomorphism edge modes in 3d gravity

TL;DR

This paper constructs an extended phase space for 3D gravity incorporating edge modes that restore gauge invariance at boundaries and explores the boundary symmetry algebra in first order variables.

Contribution

It introduces Lorentz-diffeomorphism edge modes in 3D gravity's phase space and analyzes their boundary symmetries using first order connection and triad variables.

Findings

Boundary symmetry algebra matches metric formulation.

Extended phase space includes boundary coordinate and Lorentz frames.

Gauge invariance of the symplectic structure is established.

Abstract

The proper definition of subsystems in gauge theory and gravity requires an extension of the local phase space by including edge mode fields. Their role is on the one hand to restore gauge invariance with respect to gauge transformations supported on the boundary, and on the other hand to parametrize the largest set of boundary symmetries which can arise if both the gauge parameters and the dynamical fields are unconstrained at the boundary. In this work we construct the extended phase space for three-dimensional gravity in first order connection and triad variables. There, the edge mode fields consist of a choice of coordinate frame on the boundary and a choice of Lorentz frame on the bundle, which together constitute the Lorentz-diffeomorphism edge modes. After constructing the extended symplectic structure and proving its gauge invariance, we study the boundary symmetries and the integrability of their generators. We find that the infinite-dimensional algebra of boundary symmetries with first order variables is the same as that with metric variables, and explain how this can be traced back to the expressions for the diffeomorphism Noether charge in both formulations. This concludes the study of extended phase spaces and edge modes in three-dimensional gravity, which was done previously by the author in the BF and Chern-Simons formulations.