Extended actions, dynamics of edge modes, and entanglement entropy

TL;DR

The paper develops a unified, covariant framework for incorporating edge modes in gauge theories on manifolds with boundaries by introducing an extended phase space and a boundary action that encodes the edge dynamics. By tying the boundary Lagrangian to the extended symplectic structure, it shows how integrating out bulk fields induces boundary dynamics and contributes to entanglement entropy, demonstrated through explicit tests in Chern–Simons, Maxwell, Maxwell–Chern–Simons, and BF theories. The results reveal consistent boundary algebras (notably Kač–Moody) and clarifying how gluing/splitting of regions yields both topological and contact terms in entanglement, with a clear route toward generalization to non-Abelian theories and gravity. The work thus advances a coherent picture in which edge modes are essential for factorization and boundary dynamics, with potential implications for holography and quantum gravity.

Abstract

In this work we propose a simple and systematic framework for including edge modes in gauge theories on manifolds with boundaries. We argue that this is necessary in order to achieve the factorizability of the path integral, the Hilbert space and the phase space, and that it explains how edge modes acquire a boundary dynamics and can contribute to observables such as the entanglement entropy. Our construction starts with a boundary action containing edge modes. In the case of Maxwell theory for example this is equivalent to coupling the gauge field to boundary sources in order to be able to factorize the theory between subregions. We then introduce a new variational principle which produces a systematic boundary contribution to the symplectic structure, and thereby provides a covariant realization of the extended phase space constructions which have appeared previously in the literature. When considering the path integral for the extended bulk + boundary action, integrating out the bulk degrees of freedom with chosen boundary conditions produces a residual boundary dynamics for the edge modes, in agreement with recent observations concerning the contribution of edge modes to the entanglement entropy. We put our proposal to the test with the familiar examples of Chern-Simons and BF theory, and show that it leads to consistent results. This therefore leads us to conjecture that this mechanism is generically true for any gauge theory, which can therefore all be expected to posses a boundary dynamics. We expect to be able to eventually apply this formalism to gravitational theories.

Figures (1)

• Figure 1: On the top, we have a same gauge theory defined on two neighboring manifolds with boundaries. Each factor contains bulk fields and edge modes. Integrating out the edge modes leads to the theory defined over $M\cup\bar{M}$ and, conversely, the splitting of $M\cup\bar{M}$ into subregions requires to introduce edge modes on which Wilson lines can end. Once the theory is split into factors associated with the regions $M$ and $\bar{M}$, one bulk region can be integrated out, thereby leading to an effective boundary dynamics for the edge modes. This latter will in turn be seen by and contribute to the entanglement entropy.