Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity

TL;DR

The authors develop a systematic, gauge-invariant extended phase space for gauge theories with boundaries by introducing boundary degrees of freedom, following the covariant Hamiltonian framework. They show how finite, large gauge transformations leak information through boundaries and how extending the pre-symplectic potential with new boundary fields yields boundary observables that satisfy affine Kac–Moody algebras. Applying the construction to Abelian Chern–Simons theory and to first-order 3D gravity, they demonstrate that the new observables are dressed versions of known boundary currents, and that the gauge generators can be made to vanish on-shell while preserving a nontrivial boundary symmetry. This formalism clarifies the separation between gauge transformations and boundary symmetries, offering a coherent route to understanding and quantizing boundary degrees of freedom in finite regions of spacetime. It also points to future work on extending the approach to four-dimensional gravity and to diffeomorphism-based boundary algebras, with potential implications for black hole entropy and holography.

Abstract

Boundaries in gauge field theories are known to be the locus of a wealth of interesting phenomena, as illustrated for example by the holographic principle or by the AdS/CFT and bulk-boundary correspondences. In particular, it has been acknowledged for quite some time that boundaries can break gauge invariance, and thereby turn gauge degrees of freedom into physical ones. There is however no known systematic way of identifying these degrees of freedom and possible associated boundary observables. Following recent work by Donnelly and Freidel, we show that this can be achieved by extending the covariant Hamiltonian formalism so as to make it gauge-invariant under arbitrary large gauge transformations. This can be done at the expense of extending the phase space by introducing new boundary fields, which in turn determine new boundary symmetries and observables. We present the general framework behind this construction, and find the conditions under which it can be applied to an arbitrary Lagrangian. By studying the examples of Abelian Chern-Simons theory and first order three-dimensional gravity, we then show that the new boundary observables satisfy the known corresponding Kac-Moody affine algebras. This shows that this new extended phase space formulation does indeed properly describe the dynamical boundary degrees of freedom, and gives credit to the results which have been previously derived in the case of diffeomorphism symmetry. We expect that this systematic understanding of the boundary symmetries will play a major role for the quantization of gravity in finite regions.