Local subsystems in gauge theory and gravity

TL;DR

The paper advances a gauge-invariant framework for defining localized subsystems in gauge theory and gravity by introducing boundary degrees of freedom that yield an extended phase space and a nontrivial boundary symmetry group. It develops explicit Yang–Mills and gravitational constructions, identifying boundary observables, generators, and a fusion/entangling product that enables consistent gluing of regions and a meaningful notion of entanglement entropy in the presence of gauge symmetries. A key result is the emergence of a boundary group G_S = Diff(S) ⋉ SL(2,$\mathbb{R}$)^S, whose representation theory governs the entanglement structure and may illuminate the origin of black hole entropy via surface degrees of freedom and gluing degeneracies. The work provides a concrete classical and quantum blueprint for quantifying local subsystems in quantum gravity and suggests deep connections to holography, extended TQFT ideas, and the microstate counting underlying geometric entropy, while outlining substantial open questions about dynamics, translations, and higher-curvature generalizations.

Abstract

We consider the problem of defining localized subsystems in gauge theory and gravity. Such systems are associated to spacelike hypersurfaces with boundaries and provide the natural setting for studying entanglement entropy of regions of space. We present a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary by introducing new degrees of freedom on the boundary. In Yang-Mills theory the new degrees of freedom are a choice of gauge on the boundary, transformations of which are generated by the normal component of the nonabelian electric field. In general relativity the new degrees of freedom are the location of a codimension-2 surface and a choice of conformal normal frame. These degrees of freedom transform under a group of surface symmetries, consisting of diffeomorphisms of the codimension-2 boundary, and position-dependent linear deformations of its normal plane. We find the observables which generate these symmetries, consisting of the conformal normal metric and curvature of the normal connection. We discuss the implications for the problem of defining entanglement entropy in quantum gravity. Our work suggests that the Bekenstein-Hawking entropy may arise from the different ways of gluing together two partial Cauchy surfaces at a cross-section of the horizon.