Local phase space and edge modes for diffeomorphism-invariant theories

TL;DR

This work extends the Donnelly–Freidel edge-mode framework to arbitrary diffeomorphism-invariant theories to define gauge-invariant local subregion data and entanglement. By introducing boundary edge modes via Diff$(M)$-valued maps $X$ and pulling back the Lagrangian, the authors construct a gauge-invariant extended phase space with a boundary-rich symplectic form and a multivalued symplectic potential, revealing nontrivial topology of the reduced phase space. They derive the universal surface symmetry algebra $\text{Diff}(\partial\Sigma)\ltimes\bigl(\text{SL}(2,\mathbb{R})\ltimes \mathbb{R}^{2\cdot(d-2)}\bigr)^{\partial\Sigma}$ (with possible reductions) and show how ambiguities (JKM) in $L$, $\theta$, and $Q_\xi$ shape the edge-mode structure and central extensions, particularly when surface translations are allowed. The framework offers a pathway to compute edge-mode contributions to entanglement entropy in gravity, includes higher-curvature corrections, and connects to holographic entropy concepts, while leaving open questions about boundary conditions, cohomology, and vielbein formulations for future exploration.

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.