Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

TL;DR

This paper explores the local Poisson structure at corners in four-dimensional gravity using the coframe formalism, revealing an Atiyah algebroid that links internal symmetries with diffeomorphisms, and relates to BF theory.

Contribution

It introduces a novel Poisson structure framework for corners in 4D gravity within the coframe formalism, utilizing the BFV approach and connecting to Atiyah algebroids and BF theory.

Findings

Identifies a local Poisson structure associated with corners in 4D gravity.

Establishes a connection between the corner structure and Atiyah algebroids.

Describes the relation between the corner structure and BF theory.

Abstract

This note describes a local Poisson structure (up to homotopy) associated to corners in four-dimensional gravity in the coframe (Palatini--Cartan) formalism. This is achieved through the use of the BFV formalism. The corner structure contains in particular an Atiyah algebroid that couples the internal symmetries to diffeomorphisms. The relation with $BF$ theory is also described.