# Local phase space and edge modes for diffeomorphism-invariant theories

## TL;DR

This paper develops a gauge-invariant framework using extended phase space and edge modes to characterize local degrees of freedom and symmetries at boundaries in diffeomorphism-invariant theories, crucial for understanding gravitational observables.

## Contribution

It introduces a general construction for edge mode symplectic structure and demonstrates the universal surface symmetry algebra in diffeomorphism-invariant theories.

## Key findings

- Edge mode fields satisfy a surface symmetry algebra.
- The algebra includes diffeomorphisms, $SL(2, ext{R})$ transformations, and shearing.
- Boundary conditions can lead to a central extension of the algebra.

## Abstract

We discuss an approach to characterizing local degrees of freedom of a subregion in diffeomorphism-invariant theories using the extended phase space of Donnelly and Freidel, [JHEP 2016 (2016) 102]. Such a characterization is important for defining local observables and entanglement entropy in gravitational theories. Traditional phase space constructions for subregions are not invariant with respect to diffeomorphisms that act at the boundary. The extended phase space remedies this problem by introducing edge mode fields at the boundary whose transformations under diffeomorphisms render the extended symplectic structure fully gauge invariant. In this work, we present a general construction for the edge mode symplectic structure. We show that the new fields satisfy a surface symmetry algebra generated by the Noether charges associated with the edge mode fields. For surface-preserving symmetries, the algebra is universal for all diffeomorphism-invariant theories, comprised of diffeomorphisms of the boundary, $SL(2,\mathbb{R})$ transformations of the normal plane, and, in some cases, normal shearing transformations. We also show that if boundary conditions are chosen such that surface translations are symmetries, the algebra acquires a central extension.